LMFDB - Embedded newform 45.7.d.a.44.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,7,Mod(44,45)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 7, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("45.44");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	45.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.3524337629$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 630x^{10} + 143853x^{8} - 14514820x^{6} + 700911828x^{4} - 15238290240x^{2} + 141093389376 $ x^12 - 630x^10 + 143853x^8 - 14514820x^6 + 700911828x^4 - 15238290240*x^2 + 141093389376
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{20}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			44.7
Root			$4.51521 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	45.44
Dual form			45.7.d.a.44.8

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.51521 q^{2} -43.6129 q^{4} +(106.922 - 64.7503i) q^{5} -130.042i q^{7} -485.894 q^{8} +O(q^{10})$	\(q+4.51521 q^{2} -43.6129 q^{4} +(106.922 - 64.7503i) q^{5} -130.042i q^{7} -485.894 q^{8} +(482.777 - 292.361i) q^{10} -1400.37i q^{11} -3702.66i q^{13} -587.168i q^{14} +597.314 q^{16} -4140.41 q^{17} +8484.57 q^{19} +(-4663.20 + 2823.95i) q^{20} -6322.94i q^{22} -8055.65 q^{23} +(7239.81 - 13846.5i) q^{25} -16718.3i q^{26} +5671.53i q^{28} +14441.5i q^{29} -14636.6 q^{31} +33794.2 q^{32} -18694.8 q^{34} +(-8420.28 - 13904.5i) q^{35} +78295.6i q^{37} +38309.6 q^{38} +(-51953.0 + 31461.8i) q^{40} +31231.6i q^{41} -19606.0i q^{43} +61074.1i q^{44} -36372.9 q^{46} +72029.7 q^{47} +100738. q^{49} +(32689.2 - 62519.8i) q^{50} +161484. i q^{52} -126654. q^{53} +(-90674.1 - 149731. i) q^{55} +63186.9i q^{56} +65206.4i q^{58} +203990. i q^{59} +145782. q^{61} -66087.3 q^{62} +114360. q^{64} +(-239748. - 395898. i) q^{65} -451975. i q^{67} +180575. q^{68} +(-38019.3 - 62781.5i) q^{70} -641853. i q^{71} -531893. i q^{73} +353521. i q^{74} -370037. q^{76} -182107. q^{77} +950795. q^{79} +(63866.2 - 38676.2i) q^{80} +141017. i q^{82} -512362. q^{83} +(-442703. + 268093. i) q^{85} -88525.1i q^{86} +680430. i q^{88} +1.20447e6i q^{89} -481503. q^{91} +351330. q^{92} +325229. q^{94} +(907190. - 549378. i) q^{95} +32575.6i q^{97} +454853. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 516 q^{4}+O(q^{10}) $ 12 * q + 516 * q^4	$ 12 q + 516 q^{4} - 1368 q^{10} + 36372 q^{16} + 4320 q^{19} - 63384 q^{25} + 60192 q^{31} + 106296 q^{34} - 221772 q^{40} - 1078968 q^{46} + 711516 q^{49} - 104112 q^{55} - 449784 q^{61} + 3964572 q^{64} - 3326616 q^{70} - 584400 q^{76} + 4324608 q^{79} - 3305772 q^{85} + 631152 q^{91} + 5793408 q^{94}+O(q^{100}) $ 12 * q + 516 * q^4 - 1368 * q^10 + 36372 * q^16 + 4320 * q^19 - 63384 * q^25 + 60192 * q^31 + 106296 * q^34 - 221772 * q^40 - 1078968 * q^46 + 711516 * q^49 - 104112 * q^55 - 449784 * q^61 + 3964572 * q^64 - 3326616 * q^70 - 584400 * q^76 + 4324608 * q^79 - 3305772 * q^85 + 631152 * q^91 + 5793408 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	4.51521			0.564401			0.282200	−	0.959355i	$-0.408936\pi$
$2$	4.51521			0.564401			0.282200	+	0.959355i	$0.408936\pi$
$3$		0			0
$3$		0			0
$4$	−43.6129			−0.681452
$5$	106.922	−	64.7503i	0.855379	−	0.518002i
$5$	106.922	−	64.7503i	0.855379	−	0.518002i
$6$		0			0
$7$		−	130.042i		−	0.379132i	−0.981868	−	0.189566i	$-0.939292\pi$
$7$		−	130.042i		−	0.379132i	0.981868	−	0.189566i	$-0.0607082\pi$
$8$	−485.894			−0.949013
$9$		0			0
$10$	482.777	−	292.361i	0.482777	−	0.292361i
$11$		−	1400.37i		−	1.05212i	−0.850449	−	0.526058i	$-0.823669\pi$
$11$		−	1400.37i		−	1.05212i	0.850449	−	0.526058i	$-0.176331\pi$
$12$		0			0
$13$		−	3702.66i		−	1.68533i	−0.538441	−	0.842663i	$-0.680986\pi$
$13$		−	3702.66i		−	1.68533i	0.538441	−	0.842663i	$-0.319014\pi$
$14$		−	587.168i		−	0.213983i
$15$		0			0
$16$	597.314			0.145829
$17$	−4140.41			−0.842746			−0.421373	−	0.906887i	$-0.638452\pi$
$17$	−4140.41			−0.842746			−0.421373	+	0.906887i	$0.638452\pi$
$18$		0			0
$19$	8484.57			1.23700			0.618499	−	0.785786i	$-0.287741\pi$
$19$	8484.57			1.23700			0.618499	+	0.785786i	$0.287741\pi$
$20$	−4663.20	+	2823.95i	−0.582900	+	0.352993i
$21$		0			0
$22$		−	6322.94i		−	0.593815i
$23$	−8055.65			−0.662090			−0.331045	−	0.943615i	$-0.607401\pi$
$23$	−8055.65			−0.662090			−0.331045	+	0.943615i	$0.607401\pi$
$24$		0			0
$25$	7239.81	−	13846.5i	0.463348	−	0.886177i
$26$		−	16718.3i		−	0.951199i
$27$		0			0
$28$			5671.53i			0.258361i
$29$			14441.5i			0.592133i	0.955167	+	0.296066i	$0.0956749\pi$
$29$			14441.5i			0.592133i	−0.955167	+	0.296066i	$0.904325\pi$
$30$		0			0
$31$	−14636.6			−0.491310			−0.245655	−	0.969357i	$-0.579003\pi$
$31$	−14636.6			−0.491310			−0.245655	+	0.969357i	$0.579003\pi$
$32$	33794.2			1.03132
$33$		0			0
$34$	−18694.8			−0.475646
$35$	−8420.28	−	13904.5i	−0.196391	−	0.324302i
$36$		0			0
$37$			78295.6i			1.54572i	0.634574	+	0.772862i	$0.281176\pi$
$37$			78295.6i			1.54572i	−0.634574	+	0.772862i	$0.718824\pi$
$38$	38309.6			0.698162
$39$		0			0
$40$	−51953.0	+	31461.8i	−0.811766	+	0.491590i
$41$			31231.6i			0.453151i	0.973994	+	0.226575i	$0.0727530\pi$
$41$			31231.6i			0.453151i	−0.973994	+	0.226575i	$0.927247\pi$
$42$		0			0
$43$		−	19606.0i		−	0.246595i	−0.992370	−	0.123297i	$-0.960653\pi$
$43$		−	19606.0i		−	0.246595i	0.992370	−	0.123297i	$-0.0393469\pi$
$44$			61074.1i			0.716966i
$45$		0			0
$46$	−36372.9			−0.373684
$47$	72029.7			0.693774			0.346887	−	0.937907i	$-0.387239\pi$
$47$	72029.7			0.693774			0.346887	+	0.937907i	$0.387239\pi$
$48$		0			0
$49$	100738.			0.856259
$50$	32689.2	−	62519.8i	0.261514	−	0.500159i
$51$		0			0
$52$			161484.i			1.14847i
$53$	−126654.			−0.850727			−0.425363	−	0.905023i	$-0.639854\pi$
$53$	−126654.			−0.850727			−0.425363	+	0.905023i	$0.639854\pi$
$54$		0			0
$55$	−90674.1	−	149731.i	−0.544998	−	0.899958i
$56$			63186.9i			0.359801i
$57$		0			0
$58$			65206.4i			0.334200i
$59$			203990.i			0.993238i	0.867969	+	0.496619i	$0.165425\pi$
$59$			203990.i			0.993238i	−0.867969	+	0.496619i	$0.834575\pi$
$60$		0			0
$61$	145782.			0.642266			0.321133	−	0.947034i	$-0.395936\pi$
$61$	145782.			0.642266			0.321133	+	0.947034i	$0.395936\pi$
$62$	−66087.3			−0.277296
$63$		0			0
$64$	114360.			0.436248
$65$	−239748.	−	395898.i	−0.873003	−	1.44159i
$66$		0			0
$67$		−	451975.i		−	1.50276i	−0.659870	−	0.751380i	$-0.729388\pi$
$67$		−	451975.i		−	1.50276i	0.659870	−	0.751380i	$-0.270612\pi$
$68$	180575.			0.574291
$69$		0			0
$70$	−38019.3	−	62781.5i	−0.110843	−	0.183036i
$71$		−	641853.i		−	1.79333i	−0.442708	−	0.896666i	$-0.645982\pi$
$71$		−	641853.i		−	1.79333i	0.442708	−	0.896666i	$-0.354018\pi$
$72$		0			0
$73$		−	531893.i		−	1.36728i	−0.729822	−	0.683638i	$-0.760397\pi$
$73$		−	531893.i		−	1.36728i	0.729822	−	0.683638i	$-0.239603\pi$
$74$			353521.i			0.872408i
$75$		0			0
$76$	−370037.			−0.842954
$77$	−182107.			−0.398891
$78$		0			0
$79$	950795.			1.92844			0.964219	−	0.265106i	$-0.0854071\pi$
$79$	950795.			1.92844			0.964219	+	0.265106i	$0.0854071\pi$
$80$	63866.2	−	38676.2i	0.124739	−	0.0755395i
$81$		0			0
$82$			141017.i			0.255759i
$83$	−512362.			−0.896072			−0.448036	−	0.894016i	$-0.647876\pi$
$83$	−512362.			−0.896072			−0.448036	+	0.894016i	$0.647876\pi$
$84$		0			0
$85$	−442703.	+	268093.i	−0.720867	+	0.436544i
$86$		−	88525.1i		−	0.139178i
$87$		0			0
$88$			680430.i			0.998471i
$89$			1.20447e6i			1.70854i	0.519830	+	0.854269i	$0.325995\pi$
$89$			1.20447e6i			1.70854i	−0.519830	+	0.854269i	$0.674005\pi$
$90$		0			0
$91$	−481503.			−0.638962
$92$	351330.			0.451182
$93$		0			0
$94$	325229.			0.391567
$95$	907190.	−	549378.i	1.05810	−	0.640767i
$96$		0			0
$97$			32575.6i			0.0356926i	0.999841	+	0.0178463i	$0.00568095\pi$
$97$			32575.6i			0.0356926i	−0.999841	+	0.0178463i	$0.994319\pi$
$98$	454853.			0.483273
$99$		0			0
$100$	−315749.	+	603887.i	−0.315749	+	0.603887i
$101$		−	432461.i		−	0.419742i	−0.977729	−	0.209871i	$-0.932696\pi$
$101$		−	432461.i		−	0.419742i	0.977729	−	0.209871i	$-0.0673044\pi$
$102$		0			0
$103$			441667.i			0.404188i	0.979366	+	0.202094i	$0.0647747\pi$
$103$			441667.i			0.404188i	−0.979366	+	0.202094i	$0.935225\pi$
$104$			1.79910e6i			1.59940i
$105$		0			0
$106$	−571867.			−0.480151
$107$	−1.09771e6			−0.896060			−0.448030	−	0.894018i	$-0.647874\pi$
$107$	−1.09771e6			−0.896060			−0.448030	+	0.894018i	$0.647874\pi$
$108$		0			0
$109$	81620.4			0.0630260			0.0315130	−	0.999503i	$-0.489967\pi$
$109$	81620.4			0.0630260			0.0315130	+	0.999503i	$0.489967\pi$
$110$	−409412.	−	676064.i	−0.307597	−	0.507937i
$111$		0			0
$112$		−	77676.1i		−	0.0552883i
$113$	2.21958e6			1.53828			0.769139	−	0.639082i	$-0.220685\pi$
$113$	2.21958e6			1.53828			0.769139	+	0.639082i	$0.220685\pi$
$114$		0			0
$115$	−861329.	+	521605.i	−0.566338	+	0.342964i
$116$		−	629837.i		−	0.403510i
$117$		0			0
$118$			921058.i			0.560584i
$119$			538429.i			0.319512i
$120$		0			0
$121$	−189465.			−0.106948
$122$	658236.			0.362495
$123$		0			0
$124$	638345.			0.334804
$125$	−122467.	−	1.94928e6i	−0.0627033	−	0.998032i
$126$		0			0
$127$		−	1.21685e6i		−	0.594053i	−0.954869	−	0.297027i	$-0.904005\pi$
$127$		−	1.21685e6i		−	0.594053i	0.954869	−	0.297027i	$-0.0959950\pi$
$128$	−1.64647e6			−0.785100
$129$		0			0
$130$	−1.08251e6	−	1.78756e6i	−0.492723	−	0.813636i
$131$			2.16432e6i			0.962737i	0.876518	+	0.481368i	$0.159860\pi$
$131$			2.16432e6i			0.962737i	−0.876518	+	0.481368i	$0.840140\pi$
$132$		0			0
$133$		−	1.10335e6i		−	0.468986i
$134$		−	2.04076e6i		−	0.848159i
$135$		0			0
$136$	2.01180e6			0.799776
$137$	179426.			0.0697787			0.0348894	−	0.999391i	$-0.488892\pi$
$137$	179426.			0.0697787			0.0348894	+	0.999391i	$0.488892\pi$
$138$		0			0
$139$	2.77728e6			1.03413			0.517064	−	0.855947i	$-0.327025\pi$
$139$	2.77728e6			1.03413			0.517064	+	0.855947i	$0.327025\pi$
$140$	367233.	+	606414.i	0.133831	+	0.220996i
$141$		0			0
$142$		−	2.89810e6i		−	1.01216i
$143$	−5.18508e6			−1.77316
$144$		0			0
$145$	935092.	+	1.54412e6i	0.306726	+	0.506498i
$146$		−	2.40161e6i		−	0.771691i
$147$		0			0
$148$		−	3.41470e6i		−	1.05334i
$149$			1.75084e6i			0.529282i	0.964347	+	0.264641i	$0.0852535\pi$
$149$			1.75084e6i			0.529282i	−0.964347	+	0.264641i	$0.914747\pi$
$150$		0			0
$151$	1.35692e6			0.394117			0.197058	−	0.980392i	$-0.436861\pi$
$151$	1.35692e6			0.394117			0.197058	+	0.980392i	$0.436861\pi$
$152$	−4.12260e6			−1.17393
$153$		0			0
$154$	−822251.			−0.225135
$155$	−1.56498e6	+	947724.i	−0.420256	+	0.254499i
$156$		0			0
$157$			6.20484e6i			1.60336i	0.597752	+	0.801681i	$0.296061\pi$
$157$			6.20484e6i			1.60336i	−0.597752	+	0.801681i	$0.703939\pi$
$158$	4.29304e6			1.08841
$159$		0			0
$160$	3.61336e6	−	2.18819e6i	0.882168	−	0.534225i
$161$			1.04758e6i			0.251020i
$162$		0			0
$163$		−	1.46152e6i		−	0.337475i	−0.985661	−	0.168738i	$-0.946031\pi$
$163$		−	1.46152e6i		−	0.337475i	0.985661	−	0.168738i	$-0.0539691\pi$
$164$		−	1.36210e6i		−	0.308801i
$165$		0			0
$166$	−2.31342e6			−0.505744
$167$	8.40236e6			1.80406			0.902031	−	0.431670i	$-0.142076\pi$
$167$	8.40236e6			1.80406			0.902031	+	0.431670i	$0.142076\pi$
$168$		0			0
$169$	−8.88290e6			−1.84033
$170$	−1.99889e6	+	1.21049e6i	−0.406858	+	0.246386i
$171$		0			0
$172$			855075.i			0.168042i
$173$	3.07497e6			0.593886			0.296943	−	0.954895i	$-0.404033\pi$
$173$	3.07497e6			0.593886			0.296943	+	0.954895i	$0.404033\pi$
$174$		0			0
$175$	−1.80063e6	−	941482.i	−0.335978	−	0.175670i
$176$		−	836458.i		−	0.153429i
$177$		0			0
$178$			5.43842e6i			0.964301i
$179$		−	8.35195e6i		−	1.45623i	−0.685457	−	0.728113i	$-0.740398\pi$
$179$		−	8.35195e6i		−	1.45623i	0.685457	−	0.728113i	$-0.259602\pi$
$180$		0			0
$181$	−1.63307e6			−0.275404			−0.137702	−	0.990474i	$-0.543972\pi$
$181$	−1.63307e6			−0.275404			−0.137702	+	0.990474i	$0.543972\pi$
$182$	−2.17409e6			−0.360631
$183$		0			0
$184$	3.91419e6			0.628332
$185$	5.06966e6	+	8.37155e6i	0.800688	+	1.32218i
$186$		0			0
$187$			5.79809e6i			0.886666i
$188$	−3.14143e6			−0.472774
$189$		0			0
$190$	4.09615e6	−	2.48055e6i	0.597194	−	0.361649i
$191$			3.80673e6i			0.546327i	0.961968	+	0.273163i	$0.0880699\pi$
$191$			3.80673e6i			0.546327i	−0.961968	+	0.273163i	$0.911930\pi$
$192$		0			0
$193$			7.03200e6i			0.978154i	0.872241	+	0.489077i	$0.162666\pi$
$193$			7.03200e6i			0.978154i	−0.872241	+	0.489077i	$0.837334\pi$
$194$			147086.i			0.0201449i
$195$		0			0
$196$	−4.39348e6			−0.583499
$197$	4.80780e6			0.628850			0.314425	−	0.949282i	$-0.398188\pi$
$197$	4.80780e6			0.628850			0.314425	+	0.949282i	$0.398188\pi$
$198$		0			0
$199$	−1.05811e7			−1.34268			−0.671339	−	0.741150i	$-0.734281\pi$
$199$	−1.05811e7			−1.34268			−0.671339	+	0.741150i	$0.734281\pi$
$200$	−3.51778e6	+	6.72794e6i	−0.439723	+	0.840993i
$201$		0			0
$202$		−	1.95265e6i		−	0.236903i
$203$	1.87801e6			0.224497
$204$		0			0
$205$	2.02226e6	+	3.33936e6i	0.234733	+	0.387616i
$206$			1.99422e6i			0.228124i
$207$		0			0
$208$		−	2.21165e6i		−	0.245769i
$209$		−	1.18815e7i		−	1.30146i
$210$		0			0
$211$	1.20825e7			1.28620			0.643102	−	0.765780i	$-0.277647\pi$
$211$	1.20825e7			1.28620			0.643102	+	0.765780i	$0.277647\pi$
$212$	5.52374e6			0.579729
$213$		0			0
$214$	−4.95640e6			−0.505737
$215$	−1.26949e6	−	2.09632e6i	−0.127737	−	0.210932i
$216$		0			0
$217$			1.90338e6i			0.186271i
$218$	368533.			0.0355719
$219$		0			0
$220$	3.95456e6	+	6.53019e6i	0.371390	+	0.613278i
$221$			1.53305e7i			1.42030i
$222$		0			0
$223$		−	2.06849e7i		−	1.86526i	−0.360831	−	0.932631i	$-0.617507\pi$
$223$		−	2.06849e7i		−	1.86526i	0.360831	−	0.932631i	$-0.382493\pi$
$224$		−	4.39468e6i		−	0.391006i
$225$		0			0
$226$	1.00218e7			0.868205
$227$	−1.97084e7			−1.68490			−0.842448	−	0.538777i	$-0.818887\pi$
$227$	−1.97084e7			−1.68490			−0.842448	+	0.538777i	$0.818887\pi$
$228$		0			0
$229$	−3.65151e6			−0.304064			−0.152032	−	0.988376i	$-0.548582\pi$
$229$	−3.65151e6			−0.304064			−0.152032	+	0.988376i	$0.548582\pi$
$230$	−3.88908e6	+	2.35515e6i	−0.319642	+	0.193569i
$231$		0			0
$232$		−	7.01705e6i		−	0.561941i
$233$	−1.33460e7			−1.05507			−0.527536	−	0.849533i	$-0.676884\pi$
$233$	−1.33460e7			−1.05507			−0.527536	+	0.849533i	$0.676884\pi$
$234$		0			0
$235$	7.70159e6	−	4.66394e6i	0.593440	−	0.359376i
$236$		−	8.89661e6i		−	0.676844i
$237$		0			0
$238$			2.43112e6i			0.180333i
$239$		−	2.02954e7i		−	1.48663i	−0.668940	−	0.743316i	$-0.733252\pi$
$239$		−	2.02954e7i		−	1.48663i	0.668940	−	0.743316i	$-0.266748\pi$
$240$		0			0
$241$	6.49603e6			0.464084			0.232042	−	0.972706i	$-0.425459\pi$
$241$	6.49603e6			0.464084			0.232042	+	0.972706i	$0.425459\pi$
$242$	−855473.			−0.0603615
$243$		0			0
$244$	−6.35798e6			−0.437673
$245$	1.07711e7	−	6.52281e6i	0.732426	−	0.443544i
$246$		0			0
$247$		−	3.14155e7i		−	2.08474i
$248$	7.11185e6			0.466259
$249$		0			0
$250$	−552965.	−	8.80141e6i	−0.0353898	−	0.563290i
$251$			2.99930e6i			0.189670i	0.995493	+	0.0948350i	$0.0302323\pi$
$251$			2.99930e6i			0.189670i	−0.995493	+	0.0948350i	$0.969768\pi$
$252$		0			0
$253$			1.12809e7i			0.696595i
$254$		−	5.49432e6i		−	0.335284i
$255$		0			0
$256$	−1.47532e7			−0.879359
$257$	−4.64810e6			−0.273827			−0.136914	−	0.990583i	$-0.543718\pi$
$257$	−4.64810e6			−0.273827			−0.136914	+	0.990583i	$0.543718\pi$
$258$		0			0
$259$	1.01817e7			0.586034
$260$	1.04561e7	+	1.72663e7i	0.594909	+	0.982377i
$261$		0			0
$262$			9.77235e6i			0.543369i
$263$	5.31944e6			0.292414			0.146207	−	0.989254i	$-0.453293\pi$
$263$	5.31944e6			0.292414			0.146207	+	0.989254i	$0.453293\pi$
$264$		0			0
$265$	−1.35421e7	+	8.20086e6i	−0.727694	+	0.440678i
$266$		−	4.98187e6i		−	0.264696i
$267$		0			0
$268$			1.97119e7i			1.02406i
$269$			4.28696e6i			0.220238i	0.993918	+	0.110119i	$0.0351232\pi$
$269$			4.28696e6i			0.220238i	−0.993918	+	0.110119i	$0.964877\pi$
$270$		0			0
$271$	1.26290e7			0.634541			0.317271	−	0.948335i	$-0.397234\pi$
$271$	1.26290e7			0.634541			0.317271	+	0.948335i	$0.397234\pi$
$272$	−2.47312e6			−0.122896
$273$		0			0
$274$	810144.			0.0393832
$275$	−1.93902e7	−	1.01384e7i	−0.932361	−	0.487495i
$276$		0			0
$277$			7.68330e6i			0.361500i	0.983529	+	0.180750i	$0.0578525\pi$
$277$			7.68330e6i			0.361500i	−0.983529	+	0.180750i	$0.942148\pi$
$278$	1.25400e7			0.583663
$279$		0			0
$280$	4.09137e6	+	6.75610e6i	0.186378	+	0.307767i
$281$			2.37951e7i			1.07243i	0.844082	+	0.536214i	$0.180146\pi$
$281$			2.37951e7i			1.07243i	−0.844082	+	0.536214i	$0.819854\pi$
$282$		0			0
$283$		−	1.20245e7i		−	0.530526i	−0.964176	−	0.265263i	$-0.914541\pi$
$283$		−	1.20245e7i		−	0.530526i	0.964176	−	0.265263i	$-0.0854588\pi$
$284$			2.79931e7i			1.22207i
$285$		0			0
$286$	−2.34117e7			−1.00077
$287$	4.06144e6			0.171804
$288$		0			0
$289$	−6.99458e6			−0.289780
$290$	4.22213e6	+	6.97203e6i	0.173116	+	0.285868i
$291$		0			0
$292$			2.31974e7i			0.931732i
$293$	2.95220e6			0.117366			0.0586831	−	0.998277i	$-0.481310\pi$
$293$	2.95220e6			0.117366			0.0586831	+	0.998277i	$0.481310\pi$
$294$		0			0
$295$	1.32084e7	+	2.18111e7i	0.514499	+	0.849595i
$296$		−	3.80434e7i		−	1.46691i
$297$		0			0
$298$			7.90540e6i			0.298727i
$299$			2.98273e7i			1.11584i
$300$		0			0
$301$	−2.54961e6			−0.0934921
$302$	6.12679e6			0.222440
$303$		0			0
$304$	5.06795e6			0.180390
$305$	1.55874e7	−	9.43943e6i	0.549381	−	0.332695i
$306$		0			0
$307$		−	1.64727e7i		−	0.569310i	−0.958630	−	0.284655i	$-0.908121\pi$
$307$		−	1.64727e7i		−	0.569310i	0.958630	−	0.284655i	$-0.0918791\pi$
$308$	7.94222e6			0.271825
$309$		0			0
$310$	−7.06621e6	+	4.27917e6i	−0.237193	+	0.143640i
$311$		−	1.64394e7i		−	0.546519i	−0.961940	−	0.273259i	$-0.911898\pi$
$311$		−	1.64394e7i		−	0.546519i	0.961940	−	0.273259i	$-0.0881017\pi$
$312$		0			0
$313$			1.01004e7i			0.329388i	0.986345	+	0.164694i	$0.0526636\pi$
$313$			1.01004e7i			0.329388i	−0.986345	+	0.164694i	$0.947336\pi$
$314$			2.80161e7i			0.904939i
$315$		0			0
$316$	−4.14670e7			−1.31414
$317$	2.96743e7			0.931543			0.465772	−	0.884905i	$-0.345777\pi$
$317$	2.96743e7			0.931543			0.465772	+	0.884905i	$0.345777\pi$
$318$		0			0
$319$	2.02234e7			0.622992
$320$	1.22276e7	−	7.40483e6i	0.373158	−	0.225977i
$321$		0			0
$322$			4.73002e6i			0.141676i
$323$	−3.51296e7			−1.04247
$324$		0			0
$325$	−5.12690e7	−	2.68066e7i	−1.49350	−	0.780892i
$326$		−	6.59907e6i		−	0.190471i
$327$		0			0
$328$		−	1.51753e7i		−	0.430046i
$329$		−	9.36692e6i		−	0.263032i
$330$		0			0
$331$	2.42693e7			0.669227			0.334614	−	0.942355i	$-0.391394\pi$
$331$	2.42693e7			0.669227			0.334614	+	0.942355i	$0.391394\pi$
$332$	2.23456e7			0.610630
$333$		0			0
$334$	3.79384e7			1.01821
$335$	−2.92655e7	−	4.83262e7i	−0.778433	−	1.28543i
$336$		0			0
$337$			3.44964e7i			0.901331i	0.892693	+	0.450665i	$0.148813\pi$
$337$			3.44964e7i			0.901331i	−0.892693	+	0.450665i	$0.851187\pi$
$338$	−4.01081e7			−1.03868
$339$		0			0
$340$	1.93076e7	−	1.16923e7i	0.491236	−	0.297484i
$341$			2.04966e7i			0.516915i
$342$		0			0
$343$		−	2.83996e7i		−	0.703768i
$344$			9.52645e6i			0.234021i
$345$		0			0
$346$	1.38841e7			0.335189
$347$	−7.48839e7			−1.79226			−0.896128	−	0.443795i	$-0.853632\pi$
$347$	−7.48839e7			−1.79226			−0.896128	+	0.443795i	$0.853632\pi$
$348$		0			0
$349$	4.24411e7			0.998414			0.499207	−	0.866483i	$-0.333625\pi$
$349$	4.24411e7			0.998414			0.499207	+	0.866483i	$0.333625\pi$
$350$	−8.13023e6	−	4.25099e6i	−0.189626	−	0.0991483i
$351$		0			0
$352$		−	4.73243e7i		−	1.08507i
$353$	7.17043e7			1.63013			0.815063	−	0.579373i	$-0.196703\pi$
$353$	7.17043e7			1.63013			0.815063	+	0.579373i	$0.196703\pi$
$354$		0			0
$355$	−4.15602e7	−	6.86285e7i	−0.928950	−	1.53398i
$356$		−	5.25303e7i		−	1.16429i
$357$		0			0
$358$		−	3.77108e7i		−	0.821895i
$359$			4.25715e7i			0.920101i	0.887893	+	0.460050i	$0.152169\pi$
$359$			4.25715e7i			0.920101i	−0.887893	+	0.460050i	$0.847831\pi$
$360$		0			0
$361$	2.49420e7			0.530163
$362$	−7.37367e6			−0.155438
$363$		0			0
$364$	2.09998e7			0.435422
$365$	−3.44402e7	−	5.68713e7i	−0.708251	−	1.16954i
$366$		0			0
$367$			1.04626e7i			0.211662i	0.994384	+	0.105831i	$0.0337503\pi$
$367$			1.04626e7i			0.211662i	−0.994384	+	0.105831i	$0.966250\pi$
$368$	−4.81175e6			−0.0965516
$369$		0			0
$370$	2.28906e7	+	3.77993e7i	0.451909	+	0.746240i
$371$			1.64703e7i			0.322538i
$372$		0			0
$373$			2.55798e7i			0.492913i	0.969154	+	0.246457i	$0.0792663\pi$
$373$			2.55798e7i			0.492913i	−0.969154	+	0.246457i	$0.920734\pi$
$374$			2.61796e7i			0.500435i
$375$		0			0
$376$	−3.49988e7			−0.658400
$377$	5.34721e7			0.997937
$378$		0			0
$379$	−4.43077e6			−0.0813882			−0.0406941	−	0.999172i	$-0.512957\pi$
$379$	−4.43077e6			−0.0813882			−0.0406941	+	0.999172i	$0.512957\pi$
$380$	−3.95652e7	+	2.39600e7i	−0.721046	+	0.436652i
$381$		0			0
$382$			1.71882e7i			0.308347i
$383$	−3.83146e7			−0.681974			−0.340987	−	0.940068i	$-0.610761\pi$
$383$	−3.83146e7			−0.681974			−0.340987	+	0.940068i	$0.610761\pi$
$384$		0			0
$385$	−1.94713e7	+	1.17915e7i	−0.341203	+	0.206627i
$386$			3.17509e7i			0.552071i
$387$		0			0
$388$		−	1.42072e6i		−	0.0243228i
$389$			5.22674e7i			0.887937i	0.896042	+	0.443968i	$0.146430\pi$
$389$			5.22674e7i			0.887937i	−0.896042	+	0.443968i	$0.853570\pi$
$390$		0			0
$391$	3.33537e7			0.557973
$392$	−4.89480e7			−0.812600
$393$		0			0
$394$	2.17082e7			0.354924
$395$	1.01661e8	−	6.15642e7i	1.64955	−	0.998935i
$396$		0			0
$397$			5.17848e7i			0.827619i	0.910364	+	0.413809i	$0.135802\pi$
$397$			5.17848e7i			0.827619i	−0.910364	+	0.413809i	$0.864198\pi$
$398$	−4.77759e7			−0.757809
$399$		0			0
$400$	4.32444e6	−	8.27071e6i	0.0675693	−	0.129230i
$401$			2.43160e7i			0.377101i	0.982063	+	0.188551i	$0.0603790\pi$
$401$			2.43160e7i			0.377101i	−0.982063	+	0.188551i	$0.939621\pi$
$402$		0			0
$403$			5.41944e7i			0.828017i
$404$			1.88609e7i			0.286034i
$405$		0			0
$406$	8.47960e6			0.126706
$407$	1.09642e8			1.62628
$408$		0			0
$409$	−8.89691e7			−1.30038			−0.650189	−	0.759773i	$-0.725310\pi$
$409$	−8.89691e7			−1.30038			−0.650189	+	0.759773i	$0.725310\pi$
$410$	9.13090e6	+	1.50779e7i	0.132484	+	0.218771i
$411$		0			0
$412$		−	1.92624e7i		−	0.275435i
$413$	2.65274e7			0.376569
$414$		0			0
$415$	−5.47830e7	+	3.31756e7i	−0.766481	+	0.464167i
$416$		−	1.25129e8i		−	1.73811i
$417$		0			0
$418$		−	5.36474e7i		−	0.734548i
$419$		−	6.91160e7i		−	0.939586i	−0.882777	−	0.469793i	$-0.844329\pi$
$419$		−	6.91160e7i		−	0.939586i	0.882777	−	0.469793i	$-0.155671\pi$
$420$		0			0
$421$	−1.43936e8			−1.92896			−0.964478	−	0.264162i	$-0.914905\pi$
$421$	−1.43936e8			−1.92896			−0.964478	+	0.264162i	$0.914905\pi$
$422$	5.45550e7			0.725935
$423$		0			0
$424$	6.15403e7			0.807350
$425$	−2.99758e7	+	5.73302e7i	−0.390484	+	0.746822i
$426$		0			0
$427$		−	1.89579e7i		−	0.243504i
$428$	4.78744e7			0.610622
$429$		0			0
$430$	−5.73203e6	−	9.46532e6i	−0.0720946	−	0.119050i
$431$			9.94058e7i			1.24159i	0.783971	+	0.620797i	$0.213191\pi$
$431$			9.94058e7i			1.24159i	−0.783971	+	0.620797i	$0.786809\pi$
$432$		0			0
$433$			6.91542e7i			0.851834i	0.904762	+	0.425917i	$0.140048\pi$
$433$			6.91542e7i			0.851834i	−0.904762	+	0.425917i	$0.859952\pi$
$434$			8.59415e6i			0.105132i
$435$		0			0
$436$	−3.55971e6			−0.0429492
$437$	−6.83487e7			−0.819003
$438$		0			0
$439$	1.98760e7			0.234928			0.117464	−	0.993077i	$-0.462524\pi$
$439$	1.98760e7			0.234928			0.117464	+	0.993077i	$0.462524\pi$
$440$	4.40580e7	+	7.27532e7i	0.517210	+	0.854072i
$441$		0			0
$442$			6.92205e7i			0.801619i
$443$	1.05447e8			1.21290			0.606449	−	0.795122i	$-0.292593\pi$
$443$	1.05447e8			1.21290			0.606449	+	0.795122i	$0.292593\pi$
$444$		0			0
$445$	7.79896e7	+	1.28785e8i	0.885027	+	1.46145i
$446$		−	9.33968e7i		−	1.05276i
$447$		0			0
$448$		−	1.48716e7i		−	0.165396i
$449$		−	1.96471e7i		−	0.217050i	−0.994094	−	0.108525i	$-0.965387\pi$
$449$		−	1.96471e7i		−	0.217050i	0.994094	−	0.108525i	$-0.0346127\pi$
$450$		0			0
$451$	4.37357e7			0.476767
$452$	−9.68022e7			−1.04826
$453$		0			0
$454$	−8.89874e7			−0.950957
$455$	−5.14835e7	+	3.11775e7i	−0.546555	+	0.330984i
$456$		0			0
$457$		−	8.58264e7i		−	0.899234i	−0.893222	−	0.449617i	$-0.851561\pi$
$457$		−	8.58264e7i		−	0.899234i	0.893222	−	0.449617i	$-0.148439\pi$
$458$	−1.64873e7			−0.171614
$459$		0			0
$460$	3.75651e7	−	2.27487e7i	0.385932	−	0.233713i
$461$			1.00409e8i			1.02488i	0.858724	+	0.512438i	$0.171258\pi$
$461$			1.00409e8i			1.02488i	−0.858724	+	0.512438i	$0.828742\pi$
$462$		0			0
$463$			1.10413e8i			1.11244i	0.831035	+	0.556220i	$0.187749\pi$
$463$			1.10413e8i			1.11244i	−0.831035	+	0.556220i	$0.812251\pi$
$464$			8.62612e6i			0.0863498i
$465$		0			0
$466$	−6.02597e7			−0.595483
$467$	−8.18293e7			−0.803449			−0.401724	−	0.915761i	$-0.631589\pi$
$467$	−8.18293e7			−0.803449			−0.401724	+	0.915761i	$0.631589\pi$
$468$		0			0
$469$	−5.87759e7			−0.569745
$470$	3.47743e7	−	2.10587e7i	0.334938	−	0.202832i
$471$		0			0
$472$		−	9.91177e7i		−	0.942595i
$473$	−2.74556e7			−0.259446
$474$		0			0
$475$	6.14266e7	−	1.17482e8i	0.573160	−	1.09620i
$476$		−	2.34825e7i		−	0.217732i
$477$		0			0
$478$		−	9.16378e7i		−	0.839056i
$479$		−	9.96028e7i		−	0.906286i	−0.891438	−	0.453143i	$-0.850303\pi$
$479$		−	9.96028e7i		−	0.906286i	0.891438	−	0.453143i	$-0.149697\pi$
$480$		0			0
$481$	2.89902e8			2.60505
$482$	2.93309e7			0.261929
$483$		0			0
$484$	8.26311e6			0.0728799
$485$	2.10928e6	+	3.48307e6i	0.0184888	+	0.0305307i
$486$		0			0
$487$		−	7.74497e7i		−	0.670553i	−0.942120	−	0.335276i	$-0.891170\pi$
$487$		−	7.74497e7i		−	0.670553i	0.942120	−	0.335276i	$-0.108830\pi$
$488$	−7.08347e7			−0.609518
$489$		0			0
$490$	4.86339e7	−	2.94518e7i	0.413382	−	0.250336i
$491$			3.65838e7i			0.309061i	0.987988	+	0.154530i	$0.0493864\pi$
$491$			3.65838e7i			0.309061i	−0.987988	+	0.154530i	$0.950614\pi$
$492$		0			0
$493$		−	5.97938e7i		−	0.499017i
$494$		−	1.41847e8i		−	1.17663i
$495$		0			0
$496$	−8.74265e6			−0.0716470
$497$	−8.34682e7			−0.679910
$498$		0			0
$499$	−5.15087e7			−0.414552			−0.207276	−	0.978282i	$-0.566460\pi$
$499$	−5.15087e7			−0.414552			−0.207276	+	0.978282i	$0.566460\pi$
$500$	5.34116e6	+	8.50139e7i	0.0427293	+	0.680111i
$501$		0			0
$502$			1.35425e7i			0.107050i
$503$	−1.43061e8			−1.12413			−0.562065	−	0.827093i	$-0.689993\pi$
$503$	−1.43061e8			−1.12413			−0.562065	+	0.827093i	$0.689993\pi$
$504$		0			0
$505$	−2.80019e7	−	4.62397e7i	−0.217427	−	0.359039i
$506$			5.09354e7i			0.393159i
$507$		0			0
$508$			5.30703e7i			0.404819i
$509$		−	1.68915e8i		−	1.28090i	−0.768001	−	0.640449i	$-0.778748\pi$
$509$		−	1.68915e8i		−	1.28090i	0.768001	−	0.640449i	$-0.221252\pi$
$510$		0			0
$511$	−6.91687e7			−0.518378
$512$	3.87606e7			0.288789
$513$		0			0
$514$	−2.09871e7			−0.154548
$515$	2.85981e7	+	4.72241e7i	0.209370	+	0.345734i
$516$		0			0
$517$		−	1.00868e8i		−	0.729931i
$518$	4.59727e7			0.330758
$519$		0			0
$520$	1.16492e8	+	1.92364e8i	0.828491	+	1.36809i
$521$		−	5.52389e7i		−	0.390600i	−0.980744	−	0.195300i	$-0.937432\pi$
$521$		−	5.52389e7i		−	0.390600i	0.980744	−	0.195300i	$-0.0625680\pi$
$522$		0			0
$523$		−	1.58150e8i		−	1.10551i	−0.833343	−	0.552756i	$-0.813576\pi$
$523$		−	1.58150e8i		−	1.10551i	0.833343	−	0.552756i	$-0.186424\pi$
$524$		−	9.43923e7i		−	0.656059i
$525$		0			0
$526$	2.40183e7			0.165039
$527$	6.06016e7			0.414049
$528$		0			0
$529$	−8.31424e7			−0.561637
$530$	−6.11454e7	+	3.70286e7i	−0.410711	+	0.248719i
$531$		0			0
$532$			4.81205e7i			0.319591i
$533$	1.15640e8			0.763707
$534$		0			0
$535$	−1.17370e8	+	7.10772e7i	−0.766471	+	0.464161i
$536$			2.19612e8i			1.42614i
$537$		0			0
$538$			1.93565e7i			0.124303i
$539$		−	1.41070e8i		−	0.900883i
$540$		0			0
$541$	1.50872e8			0.952833			0.476417	−	0.879220i	$-0.341935\pi$
$541$	1.50872e8			0.952833			0.476417	+	0.879220i	$0.341935\pi$
$542$	5.70224e7			0.358136
$543$		0			0
$544$	−1.39922e8			−0.869139
$545$	8.72705e6	−	5.28494e6i	0.0539111	−	0.0326476i
$546$		0			0
$547$			2.02215e8i			1.23552i	0.786366	+	0.617761i	$0.211960\pi$
$547$			2.02215e8i			1.23552i	−0.786366	+	0.617761i	$0.788040\pi$
$548$	−7.82528e6			−0.0475509
$549$		0			0
$550$	−8.75507e7	−	4.57769e7i	−0.526225	−	0.275143i
$551$			1.22530e8i			0.732466i
$552$		0			0
$553$		−	1.23644e8i		−	0.731134i
$554$			3.46917e7i			0.204031i
$555$		0			0
$556$	−1.21125e8			−0.704709
$557$	−1.43873e7			−0.0832559			−0.0416279	−	0.999133i	$-0.513254\pi$
$557$	−1.43873e7			−0.0832559			−0.0416279	+	0.999133i	$0.513254\pi$
$558$		0			0
$559$	−7.25944e7			−0.415593
$560$	−5.02955e6	−	8.30532e6i	−0.0286395	−	0.0472925i
$561$		0			0
$562$			1.07440e8i			0.605279i
$563$	1.59667e8			0.894727			0.447363	−	0.894352i	$-0.352363\pi$
$563$	1.59667e8			0.894727			0.447363	+	0.894352i	$0.352363\pi$
$564$		0			0
$565$	2.37322e8	−	1.43718e8i	1.31581	−	0.796831i
$566$		−	5.42930e7i		−	0.299429i
$567$		0			0
$568$			3.11873e8i			1.70189i
$569$		−	1.10665e8i		−	0.600724i	−0.953825	−	0.300362i	$-0.902893\pi$
$569$		−	1.10665e8i		−	0.600724i	0.953825	−	0.300362i	$-0.0971075\pi$
$570$		0			0
$571$	2.79498e7			0.150131			0.0750655	−	0.997179i	$-0.476083\pi$
$571$	2.79498e7			0.150131			0.0750655	+	0.997179i	$0.476083\pi$
$572$	2.26137e8			1.20832
$573$		0			0
$574$	1.83382e7			0.0969664
$575$	−5.83213e7	+	1.11543e8i	−0.306778	+	0.586728i
$576$		0			0
$577$			7.01646e7i			0.365250i	0.983183	+	0.182625i	$0.0584595\pi$
$577$			7.01646e7i			0.365250i	−0.983183	+	0.182625i	$0.941541\pi$
$578$	−3.15820e7			−0.163552
$579$		0			0
$580$	−4.07821e7	−	6.73437e7i	−0.209019	−	0.345154i
$581$			6.66288e7i			0.339730i
$582$		0			0
$583$			1.77362e8i			0.895063i
$584$			2.58444e8i			1.29756i
$585$		0			0
$586$	1.33298e7			0.0662416
$587$	−5.87531e7			−0.290480			−0.145240	−	0.989396i	$-0.546395\pi$
$587$	−5.87531e7			−0.290480			−0.145240	+	0.989396i	$0.546395\pi$
$588$		0			0
$589$	−1.24185e8			−0.607749
$590$	5.96387e7	+	9.84817e7i	0.290384	+	0.479512i
$591$		0			0
$592$			4.67670e7i			0.225411i
$593$	−3.95366e8			−1.89598			−0.947992	−	0.318294i	$-0.896890\pi$
$593$	−3.95366e8			−1.89598			−0.947992	+	0.318294i	$0.896890\pi$
$594$		0			0
$595$	3.48634e7	+	5.75701e7i	0.165508	+	0.273304i
$596$		−	7.63592e7i		−	0.360680i
$597$		0			0
$598$			1.34677e8i			0.629780i
$599$			2.61309e8i			1.21583i	0.794001	+	0.607916i	$0.207994\pi$
$599$			2.61309e8i			1.21583i	−0.794001	+	0.607916i	$0.792006\pi$
$600$		0			0
$601$	−2.00171e8			−0.922099			−0.461050	−	0.887374i	$-0.652527\pi$
$601$	−2.00171e8			−0.922099			−0.461050	+	0.887374i	$0.652527\pi$
$602$	−1.15120e7			−0.0527670
$603$		0			0
$604$	−5.91794e7			−0.268572
$605$	−2.02580e7	+	1.22679e7i	−0.0914811	+	0.0553993i
$606$		0			0
$607$			2.31807e8i			1.03648i	0.855235	+	0.518240i	$0.173412\pi$
$607$			2.31807e8i			1.03648i	−0.855235	+	0.518240i	$0.826588\pi$
$608$	2.86729e8			1.27574
$609$		0			0
$610$	7.03802e7	−	4.26210e7i	0.310071	−	0.187773i
$611$		−	2.66702e8i		−	1.16924i
$612$		0			0
$613$		−	1.74366e8i		−	0.756974i	−0.925607	−	0.378487i	$-0.876445\pi$
$613$		−	1.74366e8i		−	0.756974i	0.925607	−	0.378487i	$-0.123555\pi$
$614$		−	7.43775e7i		−	0.321319i
$615$		0			0
$616$	8.84848e7			0.378553
$617$	2.49715e8			1.06314			0.531568	−	0.847015i	$-0.321603\pi$
$617$	2.49715e8			1.06314			0.531568	+	0.847015i	$0.321603\pi$
$618$		0			0
$619$	2.80511e8			1.18271			0.591355	−	0.806411i	$-0.298593\pi$
$619$	2.80511e8			1.18271			0.591355	+	0.806411i	$0.298593\pi$
$620$	6.82534e7	−	4.13330e7i	0.286384	−	0.173429i
$621$		0			0
$622$		−	7.42273e7i		−	0.308455i
$623$	1.56632e8			0.647763
$624$		0			0
$625$	−1.39311e8	−	2.00492e8i	−0.570618	−	0.821216i
$626$			4.56056e7i			0.185907i
$627$		0			0
$628$		−	2.70611e8i		−	1.09261i
$629$		−	3.24176e8i		−	1.30265i
$630$		0			0
$631$	−1.15156e8			−0.458350			−0.229175	−	0.973385i	$-0.573603\pi$
$631$	−1.15156e8			−0.458350			−0.229175	+	0.973385i	$0.573603\pi$
$632$	−4.61986e8			−1.83011
$633$		0			0
$634$	1.33986e8			0.525764
$635$	−7.87912e7	−	1.30108e8i	−0.307721	−	0.508141i
$636$		0			0
$637$		−	3.72999e8i		−	1.44308i
$638$	9.13129e7			0.351617
$639$		0			0
$640$	−1.76045e8	+	1.06610e8i	−0.671558	+	0.406683i
$641$		−	1.88367e8i		−	0.715204i	−0.933874	−	0.357602i	$-0.883594\pi$
$641$		−	1.88367e8i		−	0.715204i	0.933874	−	0.357602i	$-0.116406\pi$
$642$		0			0
$643$		−	4.01619e8i		−	1.51071i	−0.655315	−	0.755356i	$-0.727464\pi$
$643$		−	4.01619e8i		−	1.51071i	0.655315	−	0.755356i	$-0.272536\pi$
$644$		−	4.56878e7i		−	0.171058i
$645$		0			0
$646$	−1.58617e8			−0.588373
$647$	5.99705e7			0.221424			0.110712	−	0.993853i	$-0.464687\pi$
$647$	5.99705e7			0.221424			0.110712	+	0.993853i	$0.464687\pi$
$648$		0			0
$649$	2.85661e8			1.04500
$650$	−2.31490e8	−	1.21037e8i	−0.842931	−	0.440736i
$651$		0			0
$652$			6.37412e7i			0.229973i
$653$	−3.83377e8			−1.37685			−0.688425	−	0.725307i	$-0.741698\pi$
$653$	−3.83377e8			−1.37685			−0.688425	+	0.725307i	$0.741698\pi$
$654$		0			0
$655$	1.40140e8	+	2.31414e8i	0.498700	+	0.823505i
$656$			1.86551e7i			0.0660823i
$657$		0			0
$658$		−	4.22936e7i		−	0.148456i
$659$			4.96645e8i			1.73536i	0.497124	+	0.867680i	$0.334390\pi$
$659$			4.96645e8i			1.73536i	−0.497124	+	0.867680i	$0.665610\pi$
$660$		0			0
$661$	1.58072e8			0.547332			0.273666	−	0.961825i	$-0.411764\pi$
$661$	1.58072e8			0.547332			0.273666	+	0.961825i	$0.411764\pi$
$662$	1.09581e8			0.377712
$663$		0			0
$664$	2.48954e8			0.850383
$665$	−7.14424e7	−	1.17973e8i	−0.242936	−	0.401161i
$666$		0			0
$667$		−	1.16336e8i		−	0.392045i
$668$	−3.66451e8			−1.22938
$669$		0			0
$670$	−1.32140e8	−	2.18203e8i	−0.439348	−	0.725498i
$671$		−	2.04148e8i		−	0.675738i
$672$		0			0
$673$			2.06670e7i			0.0678004i	0.999425	+	0.0339002i	$0.0107928\pi$
$673$			2.06670e7i			0.0678004i	−0.999425	+	0.0339002i	$0.989207\pi$
$674$			1.55758e8i			0.508712i
$675$		0			0
$676$	3.87409e8			1.25409
$677$	2.70842e7			0.0872870			0.0436435	−	0.999047i	$-0.486103\pi$
$677$	2.70842e7			0.0872870			0.0436435	+	0.999047i	$0.486103\pi$
$678$		0			0
$679$	4.23622e6			0.0135322
$680$	2.15107e8	−	1.30265e8i	0.684112	−	0.414286i
$681$		0			0
$682$			9.25464e7i			0.291747i
$683$	1.53065e7			0.0480413			0.0240207	−	0.999711i	$-0.492353\pi$
$683$	1.53065e7			0.0480413			0.0240207	+	0.999711i	$0.492353\pi$
$684$		0			0
$685$	1.91846e7	−	1.16179e7i	0.0596873	−	0.0361455i
$686$		−	1.28230e8i		−	0.397207i
$687$		0			0
$688$		−	1.17109e7i		−	0.0359605i
$689$			4.68956e8i			1.43375i
$690$		0			0
$691$	−5.53920e8			−1.67885			−0.839427	−	0.543472i	$-0.817109\pi$
$691$	−5.53920e8			−1.67885			−0.839427	+	0.543472i	$0.817109\pi$
$692$	−1.34109e8			−0.404704
$693$		0			0
$694$	−3.38116e8			−1.01155
$695$	2.96953e8	−	1.79829e8i	0.884572	−	0.535681i
$696$		0			0
$697$		−	1.29312e8i		−	0.381891i
$698$	1.91630e8			0.563505
$699$		0			0
$700$	7.85309e7	+	4.10608e7i	0.228953	+	0.119711i
$701$			1.13092e8i			0.328304i	0.986435	+	0.164152i	$0.0524888\pi$
$701$			1.13092e8i			0.328304i	−0.986435	+	0.164152i	$0.947511\pi$
$702$		0			0
$703$			6.64304e8i			1.91206i
$704$		−	1.60146e8i		−	0.458984i
$705$		0			0
$706$	3.23760e8			0.920044
$707$	−5.62382e7			−0.159138
$708$		0			0
$709$	4.85341e8			1.36178			0.680892	−	0.732384i	$-0.261592\pi$
$709$	4.85341e8			1.36178			0.680892	+	0.732384i	$0.261592\pi$
$710$	−1.87653e8	−	3.09872e8i	−0.524300	−	0.865779i
$711$		0			0
$712$		−	5.85244e8i		−	1.62142i
$713$	1.17907e8			0.325291
$714$		0			0
$715$	−5.54402e8	+	3.35736e8i	−1.51672	+	0.918500i
$716$			3.64253e8i			0.992348i
$717$		0			0
$718$			1.92219e8i			0.519305i
$719$			5.52306e8i			1.48591i	0.669340	+	0.742956i	$0.266577\pi$
$719$			5.52306e8i			1.48591i	−0.669340	+	0.742956i	$0.733423\pi$
$720$		0			0
$721$	5.74355e7			0.153241
$722$	1.12618e8			0.299224
$723$		0			0
$724$	7.12231e7			0.187675
$725$	1.99965e8	+	1.04554e8i	0.524734	+	0.274363i
$726$		0			0
$727$			3.69905e8i			0.962690i	0.876531	+	0.481345i	$0.159852\pi$
$727$			3.69905e8i			0.962690i	−0.876531	+	0.481345i	$0.840148\pi$
$728$	2.33960e8			0.606383
$729$		0			0
$730$	−1.55505e8	−	2.56786e8i	−0.399738	−	0.660089i
$731$			8.11769e7i			0.207817i
$732$		0			0
$733$			1.99062e8i			0.505447i	0.967539	+	0.252724i	$0.0813263\pi$
$733$			1.99062e8i			0.505447i	−0.967539	+	0.252724i	$0.918674\pi$
$734$			4.72410e7i			0.119462i
$735$		0			0
$736$	−2.72234e8			−0.682825
$737$	−6.32930e8			−1.58108
$738$		0			0
$739$	−1.22941e8			−0.304625			−0.152312	−	0.988332i	$-0.548672\pi$
$739$	−1.22941e8			−0.304625			−0.152312	+	0.988332i	$0.548672\pi$
$740$	−2.21103e8	−	3.65108e8i	−0.545631	−	0.901002i
$741$		0			0
$742$			7.43670e7i			0.182041i
$743$	5.00243e8			1.21959			0.609796	−	0.792558i	$-0.291251\pi$
$743$	5.00243e8			1.21959			0.609796	+	0.792558i	$0.291251\pi$
$744$		0			0
$745$	1.13367e8	+	1.87204e8i	0.274169	+	0.452737i
$746$			1.15498e8i			0.278201i
$747$		0			0
$748$		−	2.52872e8i		−	0.604220i
$749$			1.42749e8i			0.339726i
$750$		0			0
$751$	−8.95444e7			−0.211407			−0.105703	−	0.994398i	$-0.533709\pi$
$751$	−8.95444e7			−0.211407			−0.105703	+	0.994398i	$0.533709\pi$
$752$	4.30243e7			0.101172
$753$		0			0
$754$	2.41437e8			0.563236
$755$	1.45086e8	−	8.78612e7i	0.337119	−	0.204153i
$756$		0			0
$757$			6.33166e8i			1.45959i	0.683668	+	0.729793i	$0.260384\pi$
$757$			6.33166e8i			1.45959i	−0.683668	+	0.729793i	$0.739616\pi$
$758$	−2.00058e7			−0.0459356
$759$		0			0
$760$	−4.40799e8	+	2.66940e8i	−1.00415	+	0.608096i
$761$		−	5.68758e8i		−	1.29055i	−0.763952	−	0.645273i	$-0.776744\pi$
$761$		−	5.68758e8i		−	1.29055i	0.763952	−	0.645273i	$-0.223256\pi$
$762$		0			0
$763$		−	1.06141e7i		−	0.0238952i
$764$		−	1.66023e8i		−	0.372295i
$765$		0			0
$766$	−1.72998e8			−0.384907
$767$	7.55307e8			1.67393
$768$		0			0
$769$	4.02615e7			0.0885341			0.0442671	−	0.999020i	$-0.485905\pi$
$769$	4.02615e7			0.0885341			0.0442671	+	0.999020i	$0.485905\pi$
$770$	−8.79170e7	+	5.32409e7i	−0.192575	+	0.116620i
$771$		0			0
$772$		−	3.06686e8i		−	0.666565i
$773$	3.12803e8			0.677225			0.338612	−	0.940926i	$-0.390042\pi$
$773$	3.12803e8			0.677225			0.338612	+	0.940926i	$0.390042\pi$
$774$		0			0
$775$	−1.05966e8	+	2.02666e8i	−0.227647	+	0.435387i
$776$		−	1.58283e7i		−	0.0338727i
$777$		0			0
$778$			2.35998e8i			0.501152i
$779$			2.64987e8i			0.560547i
$780$		0			0
$781$	−8.98830e8			−1.88679
$782$	1.50599e8			0.314921
$783$		0			0
$784$	6.01722e7			0.124867
$785$	4.01765e8	+	6.63437e8i	0.830545	+	1.37148i
$786$		0			0
$787$		−	3.05972e8i		−	0.627708i	−0.949471	−	0.313854i	$-0.898380\pi$
$787$		−	3.05972e8i		−	0.627708i	0.949471	−	0.313854i	$-0.101620\pi$
$788$	−2.09682e8			−0.428531
$789$		0			0
$790$	4.59022e8	−	2.77975e8i	0.931005	−	0.563800i
$791$		−	2.88639e8i		−	0.583211i
$792$		0			0
$793$		−	5.39782e8i		−	1.08243i
$794$			2.33819e8i			0.467109i
$795$		0			0
$796$	4.61473e8			0.914971
$797$	−5.28096e8			−1.04313			−0.521564	−	0.853212i	$-0.674651\pi$
$797$	−5.28096e8			−1.04313			−0.521564	+	0.853212i	$0.674651\pi$
$798$		0			0
$799$	−2.98232e8			−0.584675
$800$	2.44664e8	−	4.67932e8i	0.477859	−	0.913930i
$801$		0			0
$802$			1.09792e8i			0.212836i
$803$	−7.44845e8			−1.43853
$804$		0			0
$805$	6.78308e7	+	1.12009e8i	0.130029	+	0.214717i
$806$			2.44699e8i			0.467334i
$807$		0			0
$808$			2.10130e8i			0.398340i
$809$		−	2.33257e8i		−	0.440543i	−0.975439	−	0.220272i	$-0.929306\pi$
$809$		−	2.33257e8i		−	0.440543i	0.975439	−	0.220272i	$-0.0706944\pi$
$810$		0			0
$811$	5.54237e8			1.03904			0.519521	−	0.854458i	$-0.326110\pi$
$811$	5.54237e8			1.03904			0.519521	+	0.854458i	$0.326110\pi$
$812$	−8.19055e7			−0.152984
$813$		0			0
$814$	4.95058e8			0.917874
$815$	−9.46339e7	−	1.56269e8i	−0.174813	−	0.288670i
$816$		0			0
$817$		−	1.66348e8i		−	0.305037i
$818$	−4.01714e8			−0.733934
$819$		0			0
$820$	−8.81965e7	−	1.45639e8i	−0.159959	−	0.264142i
$821$		−	4.90266e8i		−	0.885936i	−0.896537	−	0.442968i	$-0.853925\pi$
$821$		−	4.90266e8i		−	0.885936i	0.896537	−	0.442968i	$-0.146075\pi$
$822$		0			0
$823$			3.22124e7i			0.0577861i	0.999583	+	0.0288930i	$0.00919822\pi$
$823$			3.22124e7i			0.0577861i	−0.999583	+	0.0288930i	$0.990802\pi$
$824$		−	2.14604e8i		−	0.383580i
$825$		0			0
$826$	1.19777e8			0.212536
$827$	2.62206e8			0.463582			0.231791	−	0.972766i	$-0.425542\pi$
$827$	2.62206e8			0.463582			0.231791	+	0.972766i	$0.425542\pi$
$828$		0			0
$829$	−1.95536e8			−0.343213			−0.171606	−	0.985166i	$-0.554896\pi$
$829$	−1.95536e8			−0.343213			−0.171606	+	0.985166i	$0.554896\pi$
$830$	−2.47357e8	+	1.49795e8i	−0.432603	+	0.261976i
$831$		0			0
$832$		−	4.23436e8i		−	0.735221i
$833$	−4.17096e8			−0.721608
$834$		0			0
$835$	8.98400e8	−	5.44055e8i	1.54316	−	0.934508i
$836$			5.18187e8i			0.886886i
$837$		0			0
$838$		−	3.12073e8i		−	0.530303i
$839$			3.22370e8i			0.545844i	0.962036	+	0.272922i	$0.0879902\pi$
$839$			3.22370e8i			0.545844i	−0.962036	+	0.272922i	$0.912010\pi$
$840$		0			0
$841$	3.86266e8			0.649379
$842$	−6.49900e8			−1.08870
$843$		0			0
$844$	−5.26954e8			−0.876486
$845$	−9.49781e8	+	5.75170e8i	−1.57418	+	0.953293i
$846$		0			0
$847$			2.46385e7i			0.0405474i
$848$	−7.56520e7			−0.124060
$849$		0			0
$850$	−1.35347e8	+	2.58858e8i	−0.220390	+	0.421507i
$851$		−	6.30721e8i		−	1.02341i
$852$		0			0
$853$		−	7.53062e8i		−	1.21334i	−0.794952	−	0.606672i	$-0.792504\pi$
$853$		−	7.53062e8i		−	1.21334i	0.794952	−	0.606672i	$-0.207496\pi$
$854$		−	8.55986e7i		−	0.137434i
$855$		0			0
$856$	5.33372e8			0.850373
$857$	6.84693e8			1.08781			0.543905	−	0.839147i	$-0.316945\pi$
$857$	6.84693e8			1.08781			0.543905	+	0.839147i	$0.316945\pi$
$858$		0			0
$859$	7.60604e8			1.19999			0.599997	−	0.800002i	$-0.295168\pi$
$859$	7.60604e8			1.19999			0.599997	+	0.800002i	$0.295168\pi$
$860$	5.53663e7	+	9.14267e7i	0.0870463	+	0.143740i
$861$		0			0
$862$			4.48838e8i			0.700757i
$863$	8.36400e8			1.30131			0.650657	−	0.759372i	$-0.274494\pi$
$863$	8.36400e8			1.30131			0.650657	+	0.759372i	$0.274494\pi$
$864$		0			0
$865$	3.28783e8	−	1.99105e8i	0.507998	−	0.307634i
$866$			3.12246e8i			0.480776i
$867$		0			0
$868$		−	8.30120e7i		−	0.126935i
$869$		−	1.33146e9i		−	2.02894i
$870$		0			0
$871$	−1.67351e9			−2.53264
$872$	−3.96589e7			−0.0598124
$873$		0			0
$874$	−3.08608e8			−0.462246
$875$	−2.53489e8	+	1.59259e7i	−0.378386	+	0.0237728i
$876$		0			0
$877$		−	1.77565e8i		−	0.263245i	−0.991300	−	0.131622i	$-0.957981\pi$
$877$		−	1.77565e8i		−	0.263245i	0.991300	−	0.131622i	$-0.0420186\pi$
$878$	8.97441e7			0.132593
$879$		0			0
$880$	−5.41609e7	−	8.94361e7i	−0.0794763	−	0.131240i
$881$			2.39090e8i			0.349650i	0.984600	+	0.174825i	$0.0559359\pi$
$881$			2.39090e8i			0.349650i	−0.984600	+	0.174825i	$0.944064\pi$
$882$		0			0
$883$			3.92924e8i			0.570724i	0.958420	+	0.285362i	$0.0921138\pi$
$883$			3.92924e8i			0.570724i	−0.958420	+	0.285362i	$0.907886\pi$
$884$		−	6.68610e8i		−	0.967867i
$885$		0			0
$886$	4.76117e8			0.684561
$887$	−2.03664e8			−0.291839			−0.145919	−	0.989296i	$-0.546614\pi$
$887$	−2.03664e8			−0.291839			−0.145919	+	0.989296i	$0.546614\pi$
$888$		0			0
$889$	−1.58242e8			−0.225225
$890$	3.52139e8	+	5.81489e8i	0.499510	+	0.824843i
$891$		0			0
$892$			9.02131e8i			1.27109i
$893$	6.11141e8			0.858197
$894$		0			0
$895$	−5.40791e8	−	8.93011e8i	−0.754328	−	1.24563i
$896$			2.14111e8i			0.297657i
$897$		0			0
$898$		−	8.87107e7i		−	0.122503i
$899$		−	2.11375e8i		−	0.290920i
$900$		0			0
$901$	5.24398e8			0.716946
$902$	1.97476e8			0.269088
$903$		0			0
$904$	−1.07848e9			−1.45984
$905$	−1.74612e8	+	1.05742e8i	−0.235575	+	0.142660i
$906$		0			0
$907$		−	7.07620e7i		−	0.0948370i	−0.998875	−	0.0474185i	$-0.984901\pi$
$907$		−	7.07620e7i		−	0.0948370i	0.998875	−	0.0474185i	$-0.0150994\pi$
$908$	8.59540e8			1.14818
$909$		0			0
$910$	−2.32459e8	+	1.40773e8i	−0.308476	+	0.186807i
$911$			5.01933e8i			0.663882i	0.943300	+	0.331941i	$0.107703\pi$
$911$			5.01933e8i			0.663882i	−0.943300	+	0.331941i	$0.892297\pi$
$912$		0			0
$913$			7.17495e8i			0.942772i
$914$		−	3.87524e8i		−	0.507528i
$915$		0			0
$916$	1.59253e8			0.207205
$917$	2.81453e8			0.365005
$918$		0			0
$919$	−2.83629e8			−0.365430			−0.182715	−	0.983166i	$-0.558489\pi$
$919$	−2.83629e8			−0.365430			−0.182715	+	0.983166i	$0.558489\pi$
$920$	4.18515e8	−	2.53445e8i	0.537462	−	0.325477i
$921$		0			0
$922$			4.53369e8i			0.578441i
$923$	−2.37657e9			−3.02235
$924$		0			0
$925$	1.08412e9	+	5.66845e8i	1.36978	+	0.716208i
$926$			4.98537e8i			0.627862i
$927$		0			0
$928$			4.88040e8i			0.610677i
$929$		−	7.17935e8i		−	0.895443i	−0.894173	−	0.447722i	$-0.852236\pi$
$929$		−	7.17935e8i		−	0.895443i	0.894173	−	0.447722i	$-0.147764\pi$
$930$		0			0
$931$	8.54718e8			1.05919
$932$	5.82056e8			0.718981
$933$		0			0
$934$	−3.69476e8			−0.453467
$935$	3.75428e8	+	6.19946e8i	0.459295	+	0.758436i
$936$		0			0
$937$		−	2.23247e7i		−	0.0271374i	−0.999908	−	0.0135687i	$-0.995681\pi$
$937$		−	2.23247e7i		−	0.0271374i	0.999908	−	0.0135687i	$-0.00431918\pi$
$938$	−2.65385e8			−0.321565
$939$		0			0
$940$	−3.35889e8	+	2.03408e8i	−0.404401	+	0.244898i
$941$			7.70717e8i			0.924966i	0.886628	+	0.462483i	$0.153041\pi$
$941$			7.70717e8i			0.924966i	−0.886628	+	0.462483i	$0.846959\pi$
$942$		0			0
$943$		−	2.51591e8i		−	0.300027i
$944$			1.21846e8i			0.144842i
$945$		0			0
$946$	−1.23968e8			−0.146432
$947$	−3.99274e8			−0.470134			−0.235067	−	0.971979i	$-0.575531\pi$
$947$	−3.99274e8			−0.470134			−0.235067	+	0.971979i	$0.575531\pi$
$948$		0			0
$949$	−1.96942e9			−2.30431
$950$	2.77354e8	−	5.30454e8i	0.323492	−	0.618695i
$951$		0			0
$952$		−	2.61620e8i		−	0.303221i
$953$	1.24741e9			1.44123			0.720613	−	0.693338i	$-0.243861\pi$
$953$	1.24741e9			1.44123			0.720613	+	0.693338i	$0.243861\pi$
$954$		0			0
$955$	2.46487e8	+	4.07025e8i	0.282998	+	0.467317i
$956$			8.85141e8i			1.01307i
$957$		0			0
$958$		−	4.49727e8i		−	0.511508i
$959$		−	2.33330e7i		−	0.0264554i
$960$		0			0
$961$	−6.73273e8			−0.758615
$962$	1.30897e9			1.47029
$963$		0			0
$964$	−2.83311e8			−0.316251
$965$	4.55324e8	+	7.51879e8i	0.506686	+	0.836693i
$966$		0			0
$967$			1.37121e9i			1.51643i	0.652003	+	0.758216i	$0.273929\pi$
$967$			1.37121e9i			1.51643i	−0.652003	+	0.758216i	$0.726071\pi$
$968$	9.20599e7			0.101495
$969$		0			0
$970$	9.52384e6	+	1.57268e7i	0.0104351	+	0.0172315i
$971$		−	8.95883e8i		−	0.978574i	−0.872123	−	0.489287i	$-0.837257\pi$
$971$		−	8.95883e8i		−	0.978574i	0.872123	−	0.489287i	$-0.162743\pi$
$972$		0			0
$973$		−	3.61164e8i		−	0.392072i
$974$		−	3.49701e8i		−	0.378460i
$975$		0			0
$976$	8.70776e7			0.0936607
$977$	−2.92404e8			−0.313545			−0.156772	−	0.987635i	$-0.550109\pi$
$977$	−2.92404e8			−0.313545			−0.156772	+	0.987635i	$0.550109\pi$
$978$		0			0
$979$	1.68670e9			1.79758
$980$	−4.69761e8	+	2.84479e8i	−0.499113	+	0.302254i
$981$		0			0
$982$			1.65183e8i			0.174434i
$983$	3.67838e8			0.387254			0.193627	−	0.981075i	$-0.437975\pi$
$983$	3.67838e8			0.387254			0.193627	+	0.981075i	$0.437975\pi$
$984$		0			0
$985$	5.14061e8	−	3.11306e8i	0.537906	−	0.325746i
$986$		−	2.69981e8i		−	0.281646i
$987$		0			0
$988$			1.37012e9i			1.42065i
$989$			1.57939e8i			0.163268i
$990$		0			0
$991$	6.45576e8			0.663325			0.331662	−	0.943398i	$-0.392391\pi$
$991$	6.45576e8			0.663325			0.331662	+	0.943398i	$0.392391\pi$
$992$	−4.94633e8			−0.506697
$993$		0			0
$994$	−3.76876e8			−0.383742
$995$	−1.13136e9	+	6.85130e8i	−1.14850	+	0.695510i
$996$		0			0
$997$		−	2.15401e8i		−	0.217352i	−0.994077	−	0.108676i	$-0.965339\pi$
$997$		−	2.15401e8i		−	0.217352i	0.994077	−	0.108676i	$-0.0346610\pi$
$998$	−2.32572e8			−0.233973
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.7.d.a.44.7	yes	12
3.2	odd	2	inner	45.7.d.a.44.6	yes	12
4.3	odd	2		720.7.c.a.449.11		12
5.2	odd	4		225.7.c.e.26.8		12
5.3	odd	4		225.7.c.e.26.5		12
5.4	even	2	inner	45.7.d.a.44.5	✓	12
12.11	even	2		720.7.c.a.449.2		12
15.2	even	4		225.7.c.e.26.6		12
15.8	even	4		225.7.c.e.26.7		12
15.14	odd	2	inner	45.7.d.a.44.8	yes	12
20.19	odd	2		720.7.c.a.449.1		12
60.59	even	2		720.7.c.a.449.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.7.d.a.44.5	✓	12	5.4	even	2	inner
45.7.d.a.44.6	yes	12	3.2	odd	2	inner
45.7.d.a.44.7	yes	12	1.1	even	1	trivial
45.7.d.a.44.8	yes	12	15.14	odd	2	inner
225.7.c.e.26.5		12	5.3	odd	4
225.7.c.e.26.6		12	15.2	even	4
225.7.c.e.26.7		12	15.8	even	4
225.7.c.e.26.8		12	5.2	odd	4
720.7.c.a.449.1		12	20.19	odd	2
720.7.c.a.449.2		12	12.11	even	2
720.7.c.a.449.11		12	4.3	odd	2
720.7.c.a.449.12		12	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	45.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.51521 q^{2} -43.6129 q^{4} +(106.922 - 64.7503i) q^{5} -130.042i q^{7} -485.894 q^{8} +O(q^{10})\)	\(q+4.51521 q^{2} -43.6129 q^{4} +(106.922 - 64.7503i) q^{5} -130.042i q^{7} -485.894 q^{8} +(482.777 - 292.361i) q^{10} -1400.37i q^{11} -3702.66i q^{13} -587.168i q^{14} +597.314 q^{16} -4140.41 q^{17} +8484.57 q^{19} +(-4663.20 + 2823.95i) q^{20} -6322.94i q^{22} -8055.65 q^{23} +(7239.81 - 13846.5i) q^{25} -16718.3i q^{26} +5671.53i q^{28} +14441.5i q^{29} -14636.6 q^{31} +33794.2 q^{32} -18694.8 q^{34} +(-8420.28 - 13904.5i) q^{35} +78295.6i q^{37} +38309.6 q^{38} +(-51953.0 + 31461.8i) q^{40} +31231.6i q^{41} -19606.0i q^{43} +61074.1i q^{44} -36372.9 q^{46} +72029.7 q^{47} +100738. q^{49} +(32689.2 - 62519.8i) q^{50} +161484. i q^{52} -126654. q^{53} +(-90674.1 - 149731. i) q^{55} +63186.9i q^{56} +65206.4i q^{58} +203990. i q^{59} +145782. q^{61} -66087.3 q^{62} +114360. q^{64} +(-239748. - 395898. i) q^{65} -451975. i q^{67} +180575. q^{68} +(-38019.3 - 62781.5i) q^{70} -641853. i q^{71} -531893. i q^{73} +353521. i q^{74} -370037. q^{76} -182107. q^{77} +950795. q^{79} +(63866.2 - 38676.2i) q^{80} +141017. i q^{82} -512362. q^{83} +(-442703. + 268093. i) q^{85} -88525.1i q^{86} +680430. i q^{88} +1.20447e6i q^{89} -481503. q^{91} +351330. q^{92} +325229. q^{94} +(907190. - 549378. i) q^{95} +32575.6i q^{97} +454853. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 516 q^{4}+O(q^{10}) \) 12 * q + 516 * q^4	\( 12 q + 516 q^{4} - 1368 q^{10} + 36372 q^{16} + 4320 q^{19} - 63384 q^{25} + 60192 q^{31} + 106296 q^{34} - 221772 q^{40} - 1078968 q^{46} + 711516 q^{49} - 104112 q^{55} - 449784 q^{61} + 3964572 q^{64} - 3326616 q^{70} - 584400 q^{76} + 4324608 q^{79} - 3305772 q^{85} + 631152 q^{91} + 5793408 q^{94}+O(q^{100}) \) 12 * q + 516 * q^4 - 1368 * q^10 + 36372 * q^16 + 4320 * q^19 - 63384 * q^25 + 60192 * q^31 + 106296 * q^34 - 221772 * q^40 - 1078968 * q^46 + 711516 * q^49 - 104112 * q^55 - 449784 * q^61 + 3964572 * q^64 - 3326616 * q^70 - 584400 * q^76 + 4324608 * q^79 - 3305772 * q^85 + 631152 * q^91 + 5793408 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more