LMFDB - Embedded newform 225.7.c.e.26.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,7,Mod(26,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	225.c (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:	$51.7621688145$
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} + 654x^{10} + 151557x^{8} + 15450132x^{6} + 718595460x^{4} + 14140615200x^{2} + 82024960000 $ x^12 + 654x^10 + 151557x^8 + 15450132x^6 + 718595460x^4 + 14140615200*x^2 + 82024960000
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{20}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.8
Root			$5.92942i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.7.c.e.26.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.51521i q^{2} +43.6129 q^{4} +130.042 q^{7} +485.894i q^{8} +O(q^{10})$	$q+4.51521i q^{2} +43.6129 q^{4} +130.042 q^{7} +485.894i q^{8} -1400.37i q^{11} -3702.66 q^{13} +587.168i q^{14} +597.314 q^{16} -4140.41i q^{17} -8484.57 q^{19} +6322.94 q^{22} +8055.65i q^{23} -16718.3i q^{26} +5671.53 q^{28} -14441.5i q^{29} -14636.6 q^{31} +33794.2i q^{32} +18694.8 q^{34} -78295.6 q^{37} -38309.6i q^{38} +31231.6i q^{41} -19606.0 q^{43} -61074.1i q^{44} -36372.9 q^{46} +72029.7i q^{47} -100738. q^{49} -161484. q^{52} +126654. i q^{53} +63186.9i q^{56} +65206.4 q^{58} -203990. i q^{59} +145782. q^{61} -66087.3i q^{62} -114360. q^{64} +451975. q^{67} -180575. i q^{68} -641853. i q^{71} -531893. q^{73} -353521. i q^{74} -370037. q^{76} -182107. i q^{77} -950795. q^{79} -141017. q^{82} +512362. i q^{83} -88525.1i q^{86} +680430. q^{88} -1.20447e6i q^{89} -481503. q^{91} +351330. i q^{92} -325229. q^{94} -32575.6 q^{97} -454853. i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 516 q^{4}+O(q^{10}) $ 12 * q - 516 * q^4	$ 12 q - 516 q^{4} + 36372 q^{16} - 4320 q^{19} + 60192 q^{31} - 106296 q^{34} - 1078968 q^{46} - 711516 q^{49} - 449784 q^{61} - 3964572 q^{64} - 584400 q^{76} - 4324608 q^{79} + 631152 q^{91} - 5793408 q^{94}+O(q^{100}) $ 12 * q - 516 * q^4 + 36372 * q^16 - 4320 * q^19 + 60192 * q^31 - 106296 * q^34 - 1078968 * q^46 - 711516 * q^49 - 449784 * q^61 - 3964572 * q^64 - 584400 * q^76 - 4324608 * q^79 + 631152 * q^91 - 5793408 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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 $

$2$	4.51521i	0.564401i	0.959355	+	0.282200i	$0.0910643\pi$
$2$	4.51521i	0.564401i	−0.959355	+	0.282200i	$0.908936\pi$
$3$	0	0
$3$	0	0
$4$	43.6129	0.681452
$5$	0	0
$5$	0	0
$6$	0	0
$7$	130.042	0.379132	0.189566	−	0.981868i	$-0.439292\pi$
$7$	130.042	0.379132	0.189566	+	0.981868i	$0.439292\pi$
$8$	485.894i	0.949013i
$9$	0	0
$10$	0	0
$11$	− 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
$13$	−3702.66	−1.68533	−0.842663	−	0.538441i	$-0.819014\pi$
$13$	−3702.66	−1.68533	−0.842663	+	0.538441i	$0.819014\pi$
$14$	587.168i	0.213983i
$15$	0	0
$16$	597.314	0.145829
$17$	− 4140.41i	− 0.842746i	−0.906887	−	0.421373i	$-0.861548\pi$
$17$	− 4140.41i	− 0.842746i	0.906887	−	0.421373i	$-0.138452\pi$
$18$	0	0
$19$	−8484.57	−1.23700	−0.618499	−	0.785786i	$-0.712259\pi$
$19$	−8484.57	−1.23700	−0.618499	+	0.785786i	$0.712259\pi$
$20$	0	0
$21$	0	0
$22$	6322.94	0.593815
$23$	8055.65i	0.662090i	0.943615	+	0.331045i	$0.107401\pi$
$23$	8055.65i	0.662090i	−0.943615	+	0.331045i	$0.892599\pi$
$24$	0	0
$25$	0	0
$26$	− 16718.3i	− 0.951199i
$27$	0	0
$28$	5671.53	0.258361
$29$	− 14441.5i	− 0.592133i	−0.955167	−	0.296066i	$-0.904325\pi$
$29$	− 14441.5i	− 0.592133i	0.955167	−	0.296066i	$-0.0956749\pi$
$30$	0	0
$31$	−14636.6	−0.491310	−0.245655	−	0.969357i	$-0.579003\pi$
$31$	−14636.6	−0.491310	−0.245655	+	0.969357i	$0.579003\pi$
$32$	33794.2i	1.03132i
$33$	0	0
$34$	18694.8	0.475646
$35$	0	0
$36$	0	0
$37$	−78295.6	−1.54572	−0.772862	−	0.634574i	$-0.781176\pi$
$37$	−78295.6	−1.54572	−0.772862	+	0.634574i	$0.781176\pi$
$38$	− 38309.6i	− 0.698162i
$39$	0	0
$40$	0	0
$41$	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
$43$	−19606.0	−0.246595	−0.123297	−	0.992370i	$-0.539347\pi$
$43$	−19606.0	−0.246595	−0.123297	+	0.992370i	$0.539347\pi$
$44$	− 61074.1i	− 0.716966i
$45$	0	0
$46$	−36372.9	−0.373684
$47$	72029.7i	0.693774i	0.937907	+	0.346887i	$0.112761\pi$
$47$	72029.7i	0.693774i	−0.937907	+	0.346887i	$0.887239\pi$
$48$	0	0
$49$	−100738.	−0.856259
$50$	0	0
$51$	0	0
$52$	−161484.	−1.14847
$53$	126654.i	0.850727i	0.905023	+	0.425363i	$0.139854\pi$
$53$	126654.i	0.850727i	−0.905023	+	0.425363i	$0.860146\pi$
$54$	0	0
$55$	0	0
$56$	63186.9i	0.359801i
$57$	0	0
$58$	65206.4	0.334200
$59$	− 203990.i	− 0.993238i	−0.867969	−	0.496619i	$-0.834575\pi$
$59$	− 203990.i	− 0.993238i	0.867969	−	0.496619i	$-0.165425\pi$
$60$	0	0
$61$	145782.	0.642266	0.321133	−	0.947034i	$-0.395936\pi$
$61$	145782.	0.642266	0.321133	+	0.947034i	$0.395936\pi$
$62$	− 66087.3i	− 0.277296i
$63$	0	0
$64$	−114360.	−0.436248
$65$	0	0
$66$	0	0
$67$	451975.	1.50276	0.751380	−	0.659870i	$-0.229388\pi$
$67$	451975.	1.50276	0.751380	+	0.659870i	$0.229388\pi$
$68$	− 180575.i	− 0.574291i
$69$	0	0
$70$	0	0
$71$	− 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.	−1.36728	−0.683638	−	0.729822i	$-0.739603\pi$
$73$	−531893.	−1.36728	−0.683638	+	0.729822i	$0.739603\pi$
$74$	− 353521.i	− 0.872408i
$75$	0	0
$76$	−370037.	−0.842954
$77$	− 182107.i	− 0.398891i
$78$	0	0
$79$	−950795.	−1.92844	−0.964219	−	0.265106i	$-0.914593\pi$
$79$	−950795.	−1.92844	−0.964219	+	0.265106i	$0.914593\pi$
$80$	0	0
$81$	0	0
$82$	−141017.	−0.255759
$83$	512362.i	0.896072i	0.894016	+	0.448036i	$0.147876\pi$
$83$	512362.i	0.896072i	−0.894016	+	0.448036i	$0.852124\pi$
$84$	0	0
$85$	0	0
$86$	− 88525.1i	− 0.139178i
$87$	0	0
$88$	680430.	0.998471
$89$	− 1.20447e6i	− 1.70854i	−0.519830	−	0.854269i	$-0.674005\pi$
$89$	− 1.20447e6i	− 1.70854i	0.519830	−	0.854269i	$-0.325995\pi$
$90$	0	0
$91$	−481503.	−0.638962
$92$	351330.i	0.451182i
$93$	0	0
$94$	−325229.	−0.391567
$95$	0	0
$96$	0	0
$97$	−32575.6	−0.0356926	−0.0178463	−	0.999841i	$-0.505681\pi$
$97$	−32575.6	−0.0356926	−0.0178463	+	0.999841i	$0.505681\pi$
$98$	− 454853.i	− 0.483273i
$99$	0	0
$100$	0	0
$101$	− 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.	0.404188	0.202094	−	0.979366i	$-0.435225\pi$
$103$	441667.	0.404188	0.202094	+	0.979366i	$0.435225\pi$
$104$	− 1.79910e6i	− 1.59940i
$105$	0	0
$106$	−571867.	−0.480151
$107$	− 1.09771e6i	− 0.896060i	−0.894018	−	0.448030i	$-0.852126\pi$
$107$	− 1.09771e6i	− 0.896060i	0.894018	−	0.448030i	$-0.147874\pi$
$108$	0	0
$109$	−81620.4	−0.0630260	−0.0315130	−	0.999503i	$-0.510033\pi$
$109$	−81620.4	−0.0630260	−0.0315130	+	0.999503i	$0.510033\pi$
$110$	0	0
$111$	0	0
$112$	77676.1	0.0552883
$113$	− 2.21958e6i	− 1.53828i	−0.639082	−	0.769139i	$-0.720685\pi$
$113$	− 2.21958e6i	− 1.53828i	0.639082	−	0.769139i	$-0.279315\pi$
$114$	0	0
$115$	0	0
$116$	− 629837.i	− 0.403510i
$117$	0	0
$118$	921058.	0.560584
$119$	− 538429.i	− 0.319512i
$120$	0	0
$121$	−189465.	−0.106948
$122$	658236.i	0.362495i
$123$	0	0
$124$	−638345.	−0.334804
$125$	0	0
$126$	0	0
$127$	1.21685e6	0.594053	0.297027	−	0.954869i	$-0.404005\pi$
$127$	1.21685e6	0.594053	0.297027	+	0.954869i	$0.404005\pi$
$128$	1.64647e6i	0.785100i
$129$	0	0
$130$	0	0
$131$	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
$133$	−1.10335e6	−0.468986
$134$	2.04076e6i	0.848159i
$135$	0	0
$136$	2.01180e6	0.799776
$137$	179426.i	0.0697787i	0.999391	+	0.0348894i	$0.0111079\pi$
$137$	179426.i	0.0697787i	−0.999391	+	0.0348894i	$0.988892\pi$
$138$	0	0
$139$	−2.77728e6	−1.03413	−0.517064	−	0.855947i	$-0.672975\pi$
$139$	−2.77728e6	−1.03413	−0.517064	+	0.855947i	$0.672975\pi$
$140$	0	0
$141$	0	0
$142$	2.89810e6	1.01216
$143$	5.18508e6i	1.77316i
$144$	0	0
$145$	0	0
$146$	− 2.40161e6i	− 0.771691i
$147$	0	0
$148$	−3.41470e6	−1.05334
$149$	− 1.75084e6i	− 0.529282i	−0.964347	−	0.264641i	$-0.914747\pi$
$149$	− 1.75084e6i	− 0.529282i	0.964347	−	0.264641i	$-0.0852535\pi$
$150$	0	0
$151$	1.35692e6	0.394117	0.197058	−	0.980392i	$-0.436861\pi$
$151$	1.35692e6	0.394117	0.197058	+	0.980392i	$0.436861\pi$
$152$	− 4.12260e6i	− 1.17393i
$153$	0	0
$154$	822251.	0.225135
$155$	0	0
$156$	0	0
$157$	−6.20484e6	−1.60336	−0.801681	−	0.597752i	$-0.796061\pi$
$157$	−6.20484e6	−1.60336	−0.801681	+	0.597752i	$0.796061\pi$
$158$	− 4.29304e6i	− 1.08841i
$159$	0	0
$160$	0	0
$161$	1.04758e6i	0.251020i
$162$	0	0
$163$	−1.46152e6	−0.337475	−0.168738	−	0.985661i	$-0.553969\pi$
$163$	−1.46152e6	−0.337475	−0.168738	+	0.985661i	$0.553969\pi$
$164$	1.36210e6i	0.308801i
$165$	0	0
$166$	−2.31342e6	−0.505744
$167$	8.40236e6i	1.80406i	0.431670	+	0.902031i	$0.357924\pi$
$167$	8.40236e6i	1.80406i	−0.431670	+	0.902031i	$0.642076\pi$
$168$	0	0
$169$	8.88290e6	1.84033
$170$	0	0
$171$	0	0
$172$	−855075.	−0.168042
$173$	− 3.07497e6i	− 0.593886i	−0.954895	−	0.296943i	$-0.904033\pi$
$173$	− 3.07497e6i	− 0.593886i	0.954895	−	0.296943i	$-0.0959671\pi$
$174$	0	0
$175$	0	0
$176$	− 836458.i	− 0.153429i
$177$	0	0
$178$	5.43842e6	0.964301
$179$	8.35195e6i	1.45623i	0.685457	+	0.728113i	$0.259602\pi$
$179$	8.35195e6i	1.45623i	−0.685457	+	0.728113i	$0.740398\pi$
$180$	0	0
$181$	−1.63307e6	−0.275404	−0.137702	−	0.990474i	$-0.543972\pi$
$181$	−1.63307e6	−0.275404	−0.137702	+	0.990474i	$0.543972\pi$
$182$	− 2.17409e6i	− 0.360631i
$183$	0	0
$184$	−3.91419e6	−0.628332
$185$	0	0
$186$	0	0
$187$	−5.79809e6	−0.886666
$188$	3.14143e6i	0.472774i
$189$	0	0
$190$	0	0
$191$	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
$193$	7.03200e6	0.978154	0.489077	−	0.872241i	$-0.337334\pi$
$193$	7.03200e6	0.978154	0.489077	+	0.872241i	$0.337334\pi$
$194$	− 147086.i	− 0.0201449i
$195$	0	0
$196$	−4.39348e6	−0.583499
$197$	4.80780e6i	0.628850i	0.949282	+	0.314425i	$0.101812\pi$
$197$	4.80780e6i	0.628850i	−0.949282	+	0.314425i	$0.898188\pi$
$198$	0	0
$199$	1.05811e7	1.34268	0.671339	−	0.741150i	$-0.265719\pi$
$199$	1.05811e7	1.34268	0.671339	+	0.741150i	$0.265719\pi$
$200$	0	0
$201$	0	0
$202$	1.95265e6	0.236903
$203$	− 1.87801e6i	− 0.224497i
$204$	0	0
$205$	0	0
$206$	1.99422e6i	0.228124i
$207$	0	0
$208$	−2.21165e6	−0.245769
$209$	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	$0.277647\pi$
$212$	5.52374e6i	0.579729i
$213$	0	0
$214$	4.95640e6	0.505737
$215$	0	0
$216$	0	0
$217$	−1.90338e6	−0.186271
$218$	− 368533.i	− 0.0355719i
$219$	0	0
$220$	0	0
$221$	1.53305e7i	1.42030i
$222$	0	0
$223$	−2.06849e7	−1.86526	−0.932631	−	0.360831i	$-0.882493\pi$
$223$	−2.06849e7	−1.86526	−0.932631	+	0.360831i	$0.882493\pi$
$224$	4.39468e6i	0.391006i
$225$	0	0
$226$	1.00218e7	0.868205
$227$	− 1.97084e7i	− 1.68490i	−0.538777	−	0.842448i	$-0.681113\pi$
$227$	− 1.97084e7i	− 1.68490i	0.538777	−	0.842448i	$-0.318887\pi$
$228$	0	0
$229$	3.65151e6	0.304064	0.152032	−	0.988376i	$-0.451418\pi$
$229$	3.65151e6	0.304064	0.152032	+	0.988376i	$0.451418\pi$
$230$	0	0
$231$	0	0
$232$	7.01705e6	0.561941
$233$	1.33460e7i	1.05507i	0.849533	+	0.527536i	$0.176884\pi$
$233$	1.33460e7i	1.05507i	−0.849533	+	0.527536i	$0.823116\pi$
$234$	0	0
$235$	0	0
$236$	− 8.89661e6i	− 0.676844i
$237$	0	0
$238$	2.43112e6	0.180333
$239$	2.02954e7i	1.48663i	0.668940	+	0.743316i	$0.266748\pi$
$239$	2.02954e7i	1.48663i	−0.668940	+	0.743316i	$0.733252\pi$
$240$	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.i	− 0.0603615i
$243$	0	0
$244$	6.35798e6	0.437673
$245$	0	0
$246$	0	0
$247$	3.14155e7	2.08474
$248$	− 7.11185e6i	− 0.466259i
$249$	0	0
$250$	0	0
$251$	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
$253$	1.12809e7	0.696595
$254$	5.49432e6i	0.335284i
$255$	0	0
$256$	−1.47532e7	−0.879359
$257$	− 4.64810e6i	− 0.273827i	−0.990583	−	0.136914i	$-0.956282\pi$
$257$	− 4.64810e6i	− 0.273827i	0.990583	−	0.136914i	$-0.0437183\pi$
$258$	0	0
$259$	−1.01817e7	−0.586034
$260$	0	0
$261$	0	0
$262$	−9.77235e6	−0.543369
$263$	− 5.31944e6i	− 0.292414i	−0.989254	−	0.146207i	$-0.953293\pi$
$263$	− 5.31944e6i	− 0.292414i	0.989254	−	0.146207i	$-0.0467066\pi$
$264$	0	0
$265$	0	0
$266$	− 4.98187e6i	− 0.264696i
$267$	0	0
$268$	1.97119e7	1.02406
$269$	− 4.28696e6i	− 0.220238i	−0.993918	−	0.110119i	$-0.964877\pi$
$269$	− 4.28696e6i	− 0.220238i	0.993918	−	0.110119i	$-0.0351232\pi$
$270$	0	0
$271$	1.26290e7	0.634541	0.317271	−	0.948335i	$-0.397234\pi$
$271$	1.26290e7	0.634541	0.317271	+	0.948335i	$0.397234\pi$
$272$	− 2.47312e6i	− 0.122896i
$273$	0	0
$274$	−810144.	−0.0393832
$275$	0	0
$276$	0	0
$277$	−7.68330e6	−0.361500	−0.180750	−	0.983529i	$-0.557852\pi$
$277$	−7.68330e6	−0.361500	−0.180750	+	0.983529i	$0.557852\pi$
$278$	− 1.25400e7i	− 0.583663i
$279$	0	0
$280$	0	0
$281$	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
$283$	−1.20245e7	−0.530526	−0.265263	−	0.964176i	$-0.585459\pi$
$283$	−1.20245e7	−0.530526	−0.265263	+	0.964176i	$0.585459\pi$
$284$	− 2.79931e7i	− 1.22207i
$285$	0	0
$286$	−2.34117e7	−1.00077
$287$	4.06144e6i	0.171804i
$288$	0	0
$289$	6.99458e6	0.289780
$290$	0	0
$291$	0	0
$292$	−2.31974e7	−0.931732
$293$	− 2.95220e6i	− 0.117366i	−0.998277	−	0.0586831i	$-0.981310\pi$
$293$	− 2.95220e6i	− 0.117366i	0.998277	−	0.0586831i	$-0.0186902\pi$
$294$	0	0
$295$	0	0
$296$	− 3.80434e7i	− 1.46691i
$297$	0	0
$298$	7.90540e6	0.298727
$299$	− 2.98273e7i	− 1.11584i
$300$	0	0
$301$	−2.54961e6	−0.0934921
$302$	6.12679e6i	0.222440i
$303$	0	0
$304$	−5.06795e6	−0.180390
$305$	0	0
$306$	0	0
$307$	1.64727e7	0.569310	0.284655	−	0.958630i	$-0.408121\pi$
$307$	1.64727e7	0.569310	0.284655	+	0.958630i	$0.408121\pi$
$308$	− 7.94222e6i	− 0.271825i
$309$	0	0
$310$	0	0
$311$	− 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
$313$	1.01004e7	0.329388	0.164694	−	0.986345i	$-0.447336\pi$
$313$	1.01004e7	0.329388	0.164694	+	0.986345i	$0.447336\pi$
$314$	− 2.80161e7i	− 0.904939i
$315$	0	0
$316$	−4.14670e7	−1.31414
$317$	2.96743e7i	0.931543i	0.884905	+	0.465772i	$0.154223\pi$
$317$	2.96743e7i	0.931543i	−0.884905	+	0.465772i	$0.845777\pi$
$318$	0	0
$319$	−2.02234e7	−0.622992
$320$	0	0
$321$	0	0
$322$	−4.73002e6	−0.141676
$323$	3.51296e7i	1.04247i
$324$	0	0
$325$	0	0
$326$	− 6.59907e6i	− 0.190471i
$327$	0	0
$328$	−1.51753e7	−0.430046
$329$	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	$0.391394\pi$
$332$	2.23456e7i	0.610630i
$333$	0	0
$334$	−3.79384e7	−1.01821
$335$	0	0
$336$	0	0
$337$	−3.44964e7	−0.901331	−0.450665	−	0.892693i	$-0.648813\pi$
$337$	−3.44964e7	−0.901331	−0.450665	+	0.892693i	$0.648813\pi$
$338$	4.01081e7i	1.03868i
$339$	0	0
$340$	0	0
$341$	2.04966e7i	0.516915i
$342$	0	0
$343$	−2.83996e7	−0.703768
$344$	− 9.52645e6i	− 0.234021i
$345$	0	0
$346$	1.38841e7	0.335189
$347$	− 7.48839e7i	− 1.79226i	−0.443795	−	0.896128i	$-0.646368\pi$
$347$	− 7.48839e7i	− 1.79226i	0.443795	−	0.896128i	$-0.353632\pi$
$348$	0	0
$349$	−4.24411e7	−0.998414	−0.499207	−	0.866483i	$-0.666375\pi$
$349$	−4.24411e7	−0.998414	−0.499207	+	0.866483i	$0.666375\pi$
$350$	0	0
$351$	0	0
$352$	4.73243e7	1.08507
$353$	− 7.17043e7i	− 1.63013i	−0.579373	−	0.815063i	$-0.696703\pi$
$353$	− 7.17043e7i	− 1.63013i	0.579373	−	0.815063i	$-0.303297\pi$
$354$	0	0
$355$	0	0
$356$	− 5.25303e7i	− 1.16429i
$357$	0	0
$358$	−3.77108e7	−0.821895
$359$	− 4.25715e7i	− 0.920101i	−0.887893	−	0.460050i	$-0.847831\pi$
$359$	− 4.25715e7i	− 0.920101i	0.887893	−	0.460050i	$-0.152169\pi$
$360$	0	0
$361$	2.49420e7	0.530163
$362$	− 7.37367e6i	− 0.155438i
$363$	0	0
$364$	−2.09998e7	−0.435422
$365$	0	0
$366$	0	0
$367$	−1.04626e7	−0.211662	−0.105831	−	0.994384i	$-0.533750\pi$
$367$	−1.04626e7	−0.211662	−0.105831	+	0.994384i	$0.533750\pi$
$368$	4.81175e6i	0.0965516i
$369$	0	0
$370$	0	0
$371$	1.64703e7i	0.322538i
$372$	0	0
$373$	2.55798e7	0.492913	0.246457	−	0.969154i	$-0.420734\pi$
$373$	2.55798e7	0.492913	0.246457	+	0.969154i	$0.420734\pi$
$374$	− 2.61796e7i	− 0.500435i
$375$	0	0
$376$	−3.49988e7	−0.658400
$377$	5.34721e7i	0.997937i
$378$	0	0
$379$	4.43077e6	0.0813882	0.0406941	−	0.999172i	$-0.487043\pi$
$379$	4.43077e6	0.0813882	0.0406941	+	0.999172i	$0.487043\pi$
$380$	0	0
$381$	0	0
$382$	−1.71882e7	−0.308347
$383$	3.83146e7i	0.681974i	0.940068	+	0.340987i	$0.110761\pi$
$383$	3.83146e7i	0.681974i	−0.940068	+	0.340987i	$0.889239\pi$
$384$	0	0
$385$	0	0
$386$	3.17509e7i	0.552071i
$387$	0	0
$388$	−1.42072e6	−0.0243228
$389$	− 5.22674e7i	− 0.887937i	−0.896042	−	0.443968i	$-0.853570\pi$
$389$	− 5.22674e7i	− 0.887937i	0.896042	−	0.443968i	$-0.146430\pi$
$390$	0	0
$391$	3.33537e7	0.557973
$392$	− 4.89480e7i	− 0.812600i
$393$	0	0
$394$	−2.17082e7	−0.354924
$395$	0	0
$396$	0	0
$397$	−5.17848e7	−0.827619	−0.413809	−	0.910364i	$-0.635802\pi$
$397$	−5.17848e7	−0.827619	−0.413809	+	0.910364i	$0.635802\pi$
$398$	4.77759e7i	0.757809i
$399$	0	0
$400$	0	0
$401$	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
$403$	5.41944e7	0.828017
$404$	− 1.88609e7i	− 0.286034i
$405$	0	0
$406$	8.47960e6	0.126706
$407$	1.09642e8i	1.62628i
$408$	0	0
$409$	8.89691e7	1.30038	0.650189	−	0.759773i	$-0.274690\pi$
$409$	8.89691e7	1.30038	0.650189	+	0.759773i	$0.274690\pi$
$410$	0	0
$411$	0	0
$412$	1.92624e7	0.275435
$413$	− 2.65274e7i	− 0.376569i
$414$	0	0
$415$	0	0
$416$	− 1.25129e8i	− 1.73811i
$417$	0	0
$418$	−5.36474e7	−0.734548
$419$	6.91160e7i	0.939586i	0.882777	+	0.469793i	$0.155671\pi$
$419$	6.91160e7i	0.939586i	−0.882777	+	0.469793i	$0.844329\pi$
$420$	0	0
$421$	−1.43936e8	−1.92896	−0.964478	−	0.264162i	$-0.914905\pi$
$421$	−1.43936e8	−1.92896	−0.964478	+	0.264162i	$0.914905\pi$
$422$	5.45550e7i	0.725935i
$423$	0	0
$424$	−6.15403e7	−0.807350
$425$	0	0
$426$	0	0
$427$	1.89579e7	0.243504
$428$	− 4.78744e7i	− 0.610622i
$429$	0	0
$430$	0	0
$431$	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
$433$	6.91542e7	0.851834	0.425917	−	0.904762i	$-0.359952\pi$
$433$	6.91542e7	0.851834	0.425917	+	0.904762i	$0.359952\pi$
$434$	− 8.59415e6i	− 0.105132i
$435$	0	0
$436$	−3.55971e6	−0.0429492
$437$	− 6.83487e7i	− 0.819003i
$438$	0	0
$439$	−1.98760e7	−0.234928	−0.117464	−	0.993077i	$-0.537476\pi$
$439$	−1.98760e7	−0.234928	−0.117464	+	0.993077i	$0.537476\pi$
$440$	0	0
$441$	0	0
$442$	−6.92205e7	−0.801619
$443$	− 1.05447e8i	− 1.21290i	−0.795122	−	0.606449i	$-0.792593\pi$
$443$	− 1.05447e8i	− 1.21290i	0.795122	−	0.606449i	$-0.207407\pi$
$444$	0	0
$445$	0	0
$446$	− 9.33968e7i	− 1.05276i
$447$	0	0
$448$	−1.48716e7	−0.165396
$449$	1.96471e7i	0.217050i	0.994094	+	0.108525i	$0.0346127\pi$
$449$	1.96471e7i	0.217050i	−0.994094	+	0.108525i	$0.965387\pi$
$450$	0	0
$451$	4.37357e7	0.476767
$452$	− 9.68022e7i	− 1.04826i
$453$	0	0
$454$	8.89874e7	0.950957
$455$	0	0
$456$	0	0
$457$	8.58264e7	0.899234	0.449617	−	0.893222i	$-0.351561\pi$
$457$	8.58264e7	0.899234	0.449617	+	0.893222i	$0.351561\pi$
$458$	1.64873e7i	0.171614i
$459$	0	0
$460$	0	0
$461$	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
$463$	1.10413e8	1.11244	0.556220	−	0.831035i	$-0.312251\pi$
$463$	1.10413e8	1.11244	0.556220	+	0.831035i	$0.312251\pi$
$464$	− 8.62612e6i	− 0.0863498i
$465$	0	0
$466$	−6.02597e7	−0.595483
$467$	− 8.18293e7i	− 0.803449i	−0.915761	−	0.401724i	$-0.868411\pi$
$467$	− 8.18293e7i	− 0.803449i	0.915761	−	0.401724i	$-0.131589\pi$
$468$	0	0
$469$	5.87759e7	0.569745
$470$	0	0
$471$	0	0
$472$	9.91177e7	0.942595
$473$	2.74556e7i	0.259446i
$474$	0	0
$475$	0	0
$476$	− 2.34825e7i	− 0.217732i
$477$	0	0
$478$	−9.16378e7	−0.839056
$479$	9.96028e7i	0.906286i	0.891438	+	0.453143i	$0.149697\pi$
$479$	9.96028e7i	0.906286i	−0.891438	+	0.453143i	$0.850303\pi$
$480$	0	0
$481$	2.89902e8	2.60505
$482$	2.93309e7i	0.261929i
$483$	0	0
$484$	−8.26311e6	−0.0728799
$485$	0	0
$486$	0	0
$487$	7.74497e7	0.670553	0.335276	−	0.942120i	$-0.391170\pi$
$487$	7.74497e7	0.670553	0.335276	+	0.942120i	$0.391170\pi$
$488$	7.08347e7i	0.609518i
$489$	0	0
$490$	0	0
$491$	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
$493$	−5.97938e7	−0.499017
$494$	1.41847e8i	1.17663i
$495$	0	0
$496$	−8.74265e6	−0.0716470
$497$	− 8.34682e7i	− 0.679910i
$498$	0	0
$499$	5.15087e7	0.414552	0.207276	−	0.978282i	$-0.433540\pi$
$499$	5.15087e7	0.414552	0.207276	+	0.978282i	$0.433540\pi$
$500$	0	0
$501$	0	0
$502$	−1.35425e7	−0.107050
$503$	1.43061e8i	1.12413i	0.827093	+	0.562065i	$0.189993\pi$
$503$	1.43061e8i	1.12413i	−0.827093	+	0.562065i	$0.810007\pi$
$504$	0	0
$505$	0	0
$506$	5.09354e7i	0.393159i
$507$	0	0
$508$	5.30703e7	0.404819
$509$	1.68915e8i	1.28090i	0.768001	+	0.640449i	$0.221252\pi$
$509$	1.68915e8i	1.28090i	−0.768001	+	0.640449i	$0.778748\pi$
$510$	0	0
$511$	−6.91687e7	−0.518378
$512$	3.87606e7i	0.288789i
$513$	0	0
$514$	2.09871e7	0.154548
$515$	0	0
$516$	0	0
$517$	1.00868e8	0.729931
$518$	− 4.59727e7i	− 0.330758i
$519$	0	0
$520$	0	0
$521$	− 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
$523$	−1.58150e8	−1.10551	−0.552756	−	0.833343i	$-0.686424\pi$
$523$	−1.58150e8	−1.10551	−0.552756	+	0.833343i	$0.686424\pi$
$524$	9.43923e7i	0.656059i
$525$	0	0
$526$	2.40183e7	0.165039
$527$	6.06016e7i	0.414049i
$528$	0	0
$529$	8.31424e7	0.561637
$530$	0	0
$531$	0	0
$532$	−4.81205e7	−0.319591
$533$	− 1.15640e8i	− 0.763707i
$534$	0	0
$535$	0	0
$536$	2.19612e8i	1.42614i
$537$	0	0
$538$	1.93565e7	0.124303
$539$	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	$0.341935\pi$
$542$	5.70224e7i	0.358136i
$543$	0	0
$544$	1.39922e8	0.869139
$545$	0	0
$546$	0	0
$547$	−2.02215e8	−1.23552	−0.617761	−	0.786366i	$-0.711960\pi$
$547$	−2.02215e8	−1.23552	−0.617761	+	0.786366i	$0.711960\pi$
$548$	7.82528e6i	0.0475509i
$549$	0	0
$550$	0	0
$551$	1.22530e8i	0.732466i
$552$	0	0
$553$	−1.23644e8	−0.731134
$554$	− 3.46917e7i	− 0.204031i
$555$	0	0
$556$	−1.21125e8	−0.704709
$557$	− 1.43873e7i	− 0.0832559i	−0.999133	−	0.0416279i	$-0.986746\pi$
$557$	− 1.43873e7i	− 0.0832559i	0.999133	−	0.0416279i	$-0.0132544\pi$
$558$	0	0
$559$	7.25944e7	0.415593
$560$	0	0
$561$	0	0
$562$	−1.07440e8	−0.605279
$563$	− 1.59667e8i	− 0.894727i	−0.894352	−	0.447363i	$-0.852363\pi$
$563$	− 1.59667e8i	− 0.894727i	0.894352	−	0.447363i	$-0.147637\pi$
$564$	0	0
$565$	0	0
$566$	− 5.42930e7i	− 0.299429i
$567$	0	0
$568$	3.11873e8	1.70189
$569$	1.10665e8i	0.600724i	0.953825	+	0.300362i	$0.0971075\pi$
$569$	1.10665e8i	0.600724i	−0.953825	+	0.300362i	$0.902893\pi$
$570$	0	0
$571$	2.79498e7	0.150131	0.0750655	−	0.997179i	$-0.476083\pi$
$571$	2.79498e7	0.150131	0.0750655	+	0.997179i	$0.476083\pi$
$572$	2.26137e8i	1.20832i
$573$	0	0
$574$	−1.83382e7	−0.0969664
$575$	0	0
$576$	0	0
$577$	−7.01646e7	−0.365250	−0.182625	−	0.983183i	$-0.558459\pi$
$577$	−7.01646e7	−0.365250	−0.182625	+	0.983183i	$0.558459\pi$
$578$	3.15820e7i	0.163552i
$579$	0	0
$580$	0	0
$581$	6.66288e7i	0.339730i
$582$	0	0
$583$	1.77362e8	0.895063
$584$	− 2.58444e8i	− 1.29756i
$585$	0	0
$586$	1.33298e7	0.0662416
$587$	− 5.87531e7i	− 0.290480i	−0.989396	−	0.145240i	$-0.953605\pi$
$587$	− 5.87531e7i	− 0.290480i	0.989396	−	0.145240i	$-0.0463955\pi$
$588$	0	0
$589$	1.24185e8	0.607749
$590$	0	0
$591$	0	0
$592$	−4.67670e7	−0.225411
$593$	3.95366e8i	1.89598i	0.318294	+	0.947992i	$0.396890\pi$
$593$	3.95366e8i	1.89598i	−0.318294	+	0.947992i	$0.603110\pi$
$594$	0	0
$595$	0	0
$596$	− 7.63592e7i	− 0.360680i
$597$	0	0
$598$	1.34677e8	0.629780
$599$	− 2.61309e8i	− 1.21583i	−0.794001	−	0.607916i	$-0.792006\pi$
$599$	− 2.61309e8i	− 1.21583i	0.794001	−	0.607916i	$-0.207994\pi$
$600$	0	0
$601$	−2.00171e8	−0.922099	−0.461050	−	0.887374i	$-0.652527\pi$
$601$	−2.00171e8	−0.922099	−0.461050	+	0.887374i	$0.652527\pi$
$602$	− 1.15120e7i	− 0.0527670i
$603$	0	0
$604$	5.91794e7	0.268572
$605$	0	0
$606$	0	0
$607$	−2.31807e8	−1.03648	−0.518240	−	0.855235i	$-0.673412\pi$
$607$	−2.31807e8	−1.03648	−0.518240	+	0.855235i	$0.673412\pi$
$608$	− 2.86729e8i	− 1.27574i
$609$	0	0
$610$	0	0
$611$	− 2.66702e8i	− 1.16924i
$612$	0	0
$613$	−1.74366e8	−0.756974	−0.378487	−	0.925607i	$-0.623555\pi$
$613$	−1.74366e8	−0.756974	−0.378487	+	0.925607i	$0.623555\pi$
$614$	7.43775e7i	0.321319i
$615$	0	0
$616$	8.84848e7	0.378553
$617$	2.49715e8i	1.06314i	0.847015	+	0.531568i	$0.178397\pi$
$617$	2.49715e8i	1.06314i	−0.847015	+	0.531568i	$0.821603\pi$
$618$	0	0
$619$	−2.80511e8	−1.18271	−0.591355	−	0.806411i	$-0.701407\pi$
$619$	−2.80511e8	−1.18271	−0.591355	+	0.806411i	$0.701407\pi$
$620$	0	0
$621$	0	0
$622$	7.42273e7	0.308455
$623$	− 1.56632e8i	− 0.647763i
$624$	0	0
$625$	0	0
$626$	4.56056e7i	0.185907i
$627$	0	0
$628$	−2.70611e8	−1.09261
$629$	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	$0.573603\pi$
$632$	− 4.61986e8i	− 1.83011i
$633$	0	0
$634$	−1.33986e8	−0.525764
$635$	0	0
$636$	0	0
$637$	3.72999e8	1.44308
$638$	− 9.13129e7i	− 0.351617i
$639$	0	0
$640$	0	0
$641$	− 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
$643$	−4.01619e8	−1.51071	−0.755356	−	0.655315i	$-0.772536\pi$
$643$	−4.01619e8	−1.51071	−0.755356	+	0.655315i	$0.772536\pi$
$644$	4.56878e7i	0.171058i
$645$	0	0
$646$	−1.58617e8	−0.588373
$647$	5.99705e7i	0.221424i	0.993853	+	0.110712i	$0.0353131\pi$
$647$	5.99705e7i	0.221424i	−0.993853	+	0.110712i	$0.964687\pi$
$648$	0	0
$649$	−2.85661e8	−1.04500
$650$	0	0
$651$	0	0
$652$	−6.37412e7	−0.229973
$653$	3.83377e8i	1.37685i	0.725307	+	0.688425i	$0.241698\pi$
$653$	3.83377e8i	1.37685i	−0.725307	+	0.688425i	$0.758302\pi$
$654$	0	0
$655$	0	0
$656$	1.86551e7i	0.0660823i
$657$	0	0
$658$	−4.22936e7	−0.148456
$659$	− 4.96645e8i	− 1.73536i	−0.497124	−	0.867680i	$-0.665610\pi$
$659$	− 4.96645e8i	− 1.73536i	0.497124	−	0.867680i	$-0.334390\pi$
$660$	0	0
$661$	1.58072e8	0.547332	0.273666	−	0.961825i	$-0.411764\pi$
$661$	1.58072e8	0.547332	0.273666	+	0.961825i	$0.411764\pi$
$662$	1.09581e8i	0.377712i
$663$	0	0
$664$	−2.48954e8	−0.850383
$665$	0	0
$666$	0	0
$667$	1.16336e8	0.392045
$668$	3.66451e8i	1.22938i
$669$	0	0
$670$	0	0
$671$	− 2.04148e8i	− 0.675738i
$672$	0	0
$673$	2.06670e7	0.0678004	0.0339002	−	0.999425i	$-0.489207\pi$
$673$	2.06670e7	0.0678004	0.0339002	+	0.999425i	$0.489207\pi$
$674$	− 1.55758e8i	− 0.508712i
$675$	0	0
$676$	3.87409e8	1.25409
$677$	2.70842e7i	0.0872870i	0.999047	+	0.0436435i	$0.0138966\pi$
$677$	2.70842e7i	0.0872870i	−0.999047	+	0.0436435i	$0.986103\pi$
$678$	0	0
$679$	−4.23622e6	−0.0135322
$680$	0	0
$681$	0	0
$682$	−9.25464e7	−0.291747
$683$	− 1.53065e7i	− 0.0480413i	−0.999711	−	0.0240207i	$-0.992353\pi$
$683$	− 1.53065e7i	− 0.0480413i	0.999711	−	0.0240207i	$-0.00764675\pi$
$684$	0	0
$685$	0	0
$686$	− 1.28230e8i	− 0.397207i
$687$	0	0
$688$	−1.17109e7	−0.0359605
$689$	− 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	$0.817109\pi$
$692$	− 1.34109e8i	− 0.404704i
$693$	0	0
$694$	3.38116e8	1.01155
$695$	0	0
$696$	0	0
$697$	1.29312e8	0.381891
$698$	− 1.91630e8i	− 0.563505i
$699$	0	0
$700$	0	0
$701$	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
$703$	6.64304e8	1.91206
$704$	1.60146e8i	0.458984i
$705$	0	0
$706$	3.23760e8	0.920044
$707$	− 5.62382e7i	− 0.159138i
$708$	0	0
$709$	−4.85341e8	−1.36178	−0.680892	−	0.732384i	$-0.738408\pi$
$709$	−4.85341e8	−1.36178	−0.680892	+	0.732384i	$0.738408\pi$
$710$	0	0
$711$	0	0
$712$	5.85244e8	1.62142
$713$	− 1.17907e8i	− 0.325291i
$714$	0	0
$715$	0	0
$716$	3.64253e8i	0.992348i
$717$	0	0
$718$	1.92219e8	0.519305
$719$	− 5.52306e8i	− 1.48591i	−0.669340	−	0.742956i	$-0.733423\pi$
$719$	− 5.52306e8i	− 1.48591i	0.669340	−	0.742956i	$-0.266577\pi$
$720$	0	0
$721$	5.74355e7	0.153241
$722$	1.12618e8i	0.299224i
$723$	0	0
$724$	−7.12231e7	−0.187675
$725$	0	0
$726$	0	0
$727$	−3.69905e8	−0.962690	−0.481345	−	0.876531i	$-0.659852\pi$
$727$	−3.69905e8	−0.962690	−0.481345	+	0.876531i	$0.659852\pi$
$728$	− 2.33960e8i	− 0.606383i
$729$	0	0
$730$	0	0
$731$	8.11769e7i	0.207817i
$732$	0	0
$733$	1.99062e8	0.505447	0.252724	−	0.967539i	$-0.418674\pi$
$733$	1.99062e8	0.505447	0.252724	+	0.967539i	$0.418674\pi$
$734$	− 4.72410e7i	− 0.119462i
$735$	0	0
$736$	−2.72234e8	−0.682825
$737$	− 6.32930e8i	− 1.58108i
$738$	0	0
$739$	1.22941e8	0.304625	0.152312	−	0.988332i	$-0.451328\pi$
$739$	1.22941e8	0.304625	0.152312	+	0.988332i	$0.451328\pi$
$740$	0	0
$741$	0	0
$742$	−7.43670e7	−0.182041
$743$	− 5.00243e8i	− 1.21959i	−0.792558	−	0.609796i	$-0.791251\pi$
$743$	− 5.00243e8i	− 1.21959i	0.792558	−	0.609796i	$-0.208749\pi$
$744$	0	0
$745$	0	0
$746$	1.15498e8i	0.278201i
$747$	0	0
$748$	−2.52872e8	−0.604220
$749$	− 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	$0.533709\pi$
$752$	4.30243e7i	0.101172i
$753$	0	0
$754$	−2.41437e8	−0.563236
$755$	0	0
$756$	0	0
$757$	−6.33166e8	−1.45959	−0.729793	−	0.683668i	$-0.760384\pi$
$757$	−6.33166e8	−1.45959	−0.729793	+	0.683668i	$0.760384\pi$
$758$	2.00058e7i	0.0459356i
$759$	0	0
$760$	0	0
$761$	− 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
$763$	−1.06141e7	−0.0238952
$764$	1.66023e8i	0.372295i
$765$	0	0
$766$	−1.72998e8	−0.384907
$767$	7.55307e8i	1.67393i
$768$	0	0
$769$	−4.02615e7	−0.0885341	−0.0442671	−	0.999020i	$-0.514095\pi$
$769$	−4.02615e7	−0.0885341	−0.0442671	+	0.999020i	$0.514095\pi$
$770$	0	0
$771$	0	0
$772$	3.06686e8	0.666565
$773$	− 3.12803e8i	− 0.677225i	−0.940926	−	0.338612i	$-0.890042\pi$
$773$	− 3.12803e8i	− 0.677225i	0.940926	−	0.338612i	$-0.109958\pi$
$774$	0	0
$775$	0	0
$776$	− 1.58283e7i	− 0.0338727i
$777$	0	0
$778$	2.35998e8	0.501152
$779$	− 2.64987e8i	− 0.560547i
$780$	0	0
$781$	−8.98830e8	−1.88679
$782$	1.50599e8i	0.314921i
$783$	0	0
$784$	−6.01722e7	−0.124867
$785$	0	0
$786$	0	0
$787$	3.05972e8	0.627708	0.313854	−	0.949471i	$-0.398380\pi$
$787$	3.05972e8	0.627708	0.313854	+	0.949471i	$0.398380\pi$
$788$	2.09682e8i	0.428531i
$789$	0	0
$790$	0	0
$791$	− 2.88639e8i	− 0.583211i
$792$	0	0
$793$	−5.39782e8	−1.08243
$794$	− 2.33819e8i	− 0.467109i
$795$	0	0
$796$	4.61473e8	0.914971
$797$	− 5.28096e8i	− 1.04313i	−0.853212	−	0.521564i	$-0.825349\pi$
$797$	− 5.28096e8i	− 1.04313i	0.853212	−	0.521564i	$-0.174651\pi$
$798$	0	0
$799$	2.98232e8	0.584675
$800$	0	0
$801$	0	0
$802$	−1.09792e8	−0.212836
$803$	7.44845e8i	1.43853i
$804$	0	0
$805$	0	0
$806$	2.44699e8i	0.467334i
$807$	0	0
$808$	2.10130e8	0.398340
$809$	2.33257e8i	0.440543i	0.975439	+	0.220272i	$0.0706944\pi$
$809$	2.33257e8i	0.440543i	−0.975439	+	0.220272i	$0.929306\pi$
$810$	0	0
$811$	5.54237e8	1.03904	0.519521	−	0.854458i	$-0.326110\pi$
$811$	5.54237e8	1.03904	0.519521	+	0.854458i	$0.326110\pi$
$812$	− 8.19055e7i	− 0.152984i
$813$	0	0
$814$	−4.95058e8	−0.917874
$815$	0	0
$816$	0	0
$817$	1.66348e8	0.305037
$818$	4.01714e8i	0.733934i
$819$	0	0
$820$	0	0
$821$	− 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
$823$	3.22124e7	0.0577861	0.0288930	−	0.999583i	$-0.490802\pi$
$823$	3.22124e7	0.0577861	0.0288930	+	0.999583i	$0.490802\pi$
$824$	2.14604e8i	0.383580i
$825$	0	0
$826$	1.19777e8	0.212536
$827$	2.62206e8i	0.463582i	0.972766	+	0.231791i	$0.0744585\pi$
$827$	2.62206e8i	0.463582i	−0.972766	+	0.231791i	$0.925542\pi$
$828$	0	0
$829$	1.95536e8	0.343213	0.171606	−	0.985166i	$-0.445104\pi$
$829$	1.95536e8	0.343213	0.171606	+	0.985166i	$0.445104\pi$
$830$	0	0
$831$	0	0
$832$	4.23436e8	0.735221
$833$	4.17096e8i	0.721608i
$834$	0	0
$835$	0	0
$836$	5.18187e8i	0.886886i
$837$	0	0
$838$	−3.12073e8	−0.530303
$839$	− 3.22370e8i	− 0.545844i	−0.962036	−	0.272922i	$-0.912010\pi$
$839$	− 3.22370e8i	− 0.545844i	0.962036	−	0.272922i	$-0.0879902\pi$
$840$	0	0
$841$	3.86266e8	0.649379
$842$	− 6.49900e8i	− 1.08870i
$843$	0	0
$844$	5.26954e8	0.876486
$845$	0	0
$846$	0	0
$847$	−2.46385e7	−0.0405474
$848$	7.56520e7i	0.124060i
$849$	0	0
$850$	0	0
$851$	− 6.30721e8i	− 1.02341i
$852$	0	0
$853$	−7.53062e8	−1.21334	−0.606672	−	0.794952i	$-0.707496\pi$
$853$	−7.53062e8	−1.21334	−0.606672	+	0.794952i	$0.707496\pi$
$854$	8.55986e7i	0.137434i
$855$	0	0
$856$	5.33372e8	0.850373
$857$	6.84693e8i	1.08781i	0.839147	+	0.543905i	$0.183055\pi$
$857$	6.84693e8i	1.08781i	−0.839147	+	0.543905i	$0.816945\pi$
$858$	0	0
$859$	−7.60604e8	−1.19999	−0.599997	−	0.800002i	$-0.704832\pi$
$859$	−7.60604e8	−1.19999	−0.599997	+	0.800002i	$0.704832\pi$
$860$	0	0
$861$	0	0
$862$	−4.48838e8	−0.700757
$863$	− 8.36400e8i	− 1.30131i	−0.759372	−	0.650657i	$-0.774494\pi$
$863$	− 8.36400e8i	− 1.30131i	0.759372	−	0.650657i	$-0.225506\pi$
$864$	0	0
$865$	0	0
$866$	3.12246e8i	0.480776i
$867$	0	0
$868$	−8.30120e7	−0.126935
$869$	1.33146e9i	2.02894i
$870$	0	0
$871$	−1.67351e9	−2.53264
$872$	− 3.96589e7i	− 0.0598124i
$873$	0	0
$874$	3.08608e8	0.462246
$875$	0	0
$876$	0	0
$877$	1.77565e8	0.263245	0.131622	−	0.991300i	$-0.457981\pi$
$877$	1.77565e8	0.263245	0.131622	+	0.991300i	$0.457981\pi$
$878$	− 8.97441e7i	− 0.132593i
$879$	0	0
$880$	0	0
$881$	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
$883$	3.92924e8	0.570724	0.285362	−	0.958420i	$-0.407886\pi$
$883$	3.92924e8	0.570724	0.285362	+	0.958420i	$0.407886\pi$
$884$	6.68610e8i	0.967867i
$885$	0	0
$886$	4.76117e8	0.684561
$887$	− 2.03664e8i	− 0.291839i	−0.989296	−	0.145919i	$-0.953386\pi$
$887$	− 2.03664e8i	− 0.291839i	0.989296	−	0.145919i	$-0.0466140\pi$
$888$	0	0
$889$	1.58242e8	0.225225
$890$	0	0
$891$	0	0
$892$	−9.02131e8	−1.27109
$893$	− 6.11141e8i	− 0.858197i
$894$	0	0
$895$	0	0
$896$	2.14111e8i	0.297657i
$897$	0	0
$898$	−8.87107e7	−0.122503
$899$	2.11375e8i	0.290920i
$900$	0	0
$901$	5.24398e8	0.716946
$902$	1.97476e8i	0.269088i
$903$	0	0
$904$	1.07848e9	1.45984
$905$	0	0
$906$	0	0
$907$	7.07620e7	0.0948370	0.0474185	−	0.998875i	$-0.484901\pi$
$907$	7.07620e7	0.0948370	0.0474185	+	0.998875i	$0.484901\pi$
$908$	− 8.59540e8i	− 1.14818i
$909$	0	0
$910$	0	0
$911$	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
$913$	7.17495e8	0.942772
$914$	3.87524e8i	0.507528i
$915$	0	0
$916$	1.59253e8	0.207205
$917$	2.81453e8i	0.365005i
$918$	0	0
$919$	2.83629e8	0.365430	0.182715	−	0.983166i	$-0.441511\pi$
$919$	2.83629e8	0.365430	0.182715	+	0.983166i	$0.441511\pi$
$920$	0	0
$921$	0	0
$922$	−4.53369e8	−0.578441
$923$	2.37657e9i	3.02235i
$924$	0	0
$925$	0	0
$926$	4.98537e8i	0.627862i
$927$	0	0
$928$	4.88040e8	0.610677
$929$	7.17935e8i	0.895443i	0.894173	+	0.447722i	$0.147764\pi$
$929$	7.17935e8i	0.895443i	−0.894173	+	0.447722i	$0.852236\pi$
$930$	0	0
$931$	8.54718e8	1.05919
$932$	5.82056e8i	0.718981i
$933$	0	0
$934$	3.69476e8	0.453467
$935$	0	0
$936$	0	0
$937$	2.23247e7	0.0271374	0.0135687	−	0.999908i	$-0.495681\pi$
$937$	2.23247e7	0.0271374	0.0135687	+	0.999908i	$0.495681\pi$
$938$	2.65385e8i	0.321565i
$939$	0	0
$940$	0	0
$941$	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
$943$	−2.51591e8	−0.300027
$944$	− 1.21846e8i	− 0.144842i
$945$	0	0
$946$	−1.23968e8	−0.146432
$947$	− 3.99274e8i	− 0.470134i	−0.971979	−	0.235067i	$-0.924469\pi$
$947$	− 3.99274e8i	− 0.470134i	0.971979	−	0.235067i	$-0.0755309\pi$
$948$	0	0
$949$	1.96942e9	2.30431
$950$	0	0
$951$	0	0
$952$	2.61620e8	0.303221
$953$	− 1.24741e9i	− 1.44123i	−0.693338	−	0.720613i	$-0.743861\pi$
$953$	− 1.24741e9i	− 1.44123i	0.693338	−	0.720613i	$-0.256139\pi$
$954$	0	0
$955$	0	0
$956$	8.85141e8i	1.01307i
$957$	0	0
$958$	−4.49727e8	−0.511508
$959$	2.33330e7i	0.0264554i
$960$	0	0
$961$	−6.73273e8	−0.758615
$962$	1.30897e9i	1.47029i
$963$	0	0
$964$	2.83311e8	0.316251
$965$	0	0
$966$	0	0
$967$	−1.37121e9	−1.51643	−0.758216	−	0.652003i	$-0.773929\pi$
$967$	−1.37121e9	−1.51643	−0.758216	+	0.652003i	$0.773929\pi$
$968$	− 9.20599e7i	− 0.101495i
$969$	0	0
$970$	0	0
$971$	− 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
$973$	−3.61164e8	−0.392072
$974$	3.49701e8i	0.378460i
$975$	0	0
$976$	8.70776e7	0.0936607
$977$	− 2.92404e8i	− 0.313545i	−0.987635	−	0.156772i	$-0.949891\pi$
$977$	− 2.92404e8i	− 0.313545i	0.987635	−	0.156772i	$-0.0501089\pi$
$978$	0	0
$979$	−1.68670e9	−1.79758
$980$	0	0
$981$	0	0
$982$	−1.65183e8	−0.174434
$983$	− 3.67838e8i	− 0.387254i	−0.981075	−	0.193627i	$-0.937975\pi$
$983$	− 3.67838e8i	− 0.387254i	0.981075	−	0.193627i	$-0.0620252\pi$
$984$	0	0
$985$	0	0
$986$	− 2.69981e8i	− 0.281646i
$987$	0	0
$988$	1.37012e9	1.42065
$989$	− 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	$0.392391\pi$
$992$	− 4.94633e8i	− 0.506697i
$993$	0	0
$994$	3.76876e8	0.383742
$995$	0	0
$996$	0	0
$997$	2.15401e8	0.217352	0.108676	−	0.994077i	$-0.465339\pi$
$997$	2.15401e8	0.217352	0.108676	+	0.994077i	$0.465339\pi$
$998$	2.32572e8i	0.233973i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.7.d.a.44.5	✓	12	5.2	odd	4
45.7.d.a.44.6	yes	12	15.8	even	4
45.7.d.a.44.7	yes	12	5.3	odd	4
45.7.d.a.44.8	yes	12	15.2	even	4
225.7.c.e.26.5		12	5.4	even	2	inner
225.7.c.e.26.6		12	3.2	odd	2	inner
225.7.c.e.26.7		12	15.14	odd	2	inner
225.7.c.e.26.8		12	1.1	even	1	trivial
720.7.c.a.449.1		12	20.7	even	4
720.7.c.a.449.2		12	60.23	odd	4
720.7.c.a.449.11		12	20.3	even	4
720.7.c.a.449.12		12	60.47	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.51521i q^{2} +43.6129 q^{4} +130.042 q^{7} +485.894i q^{8} +O(q^{10})\)	\(q+4.51521i q^{2} +43.6129 q^{4} +130.042 q^{7} +485.894i q^{8} -1400.37i q^{11} -3702.66 q^{13} +587.168i q^{14} +597.314 q^{16} -4140.41i q^{17} -8484.57 q^{19} +6322.94 q^{22} +8055.65i q^{23} -16718.3i q^{26} +5671.53 q^{28} -14441.5i q^{29} -14636.6 q^{31} +33794.2i q^{32} +18694.8 q^{34} -78295.6 q^{37} -38309.6i q^{38} +31231.6i q^{41} -19606.0 q^{43} -61074.1i q^{44} -36372.9 q^{46} +72029.7i q^{47} -100738. q^{49} -161484. q^{52} +126654. i q^{53} +63186.9i q^{56} +65206.4 q^{58} -203990. i q^{59} +145782. q^{61} -66087.3i q^{62} -114360. q^{64} +451975. q^{67} -180575. i q^{68} -641853. i q^{71} -531893. q^{73} -353521. i q^{74} -370037. q^{76} -182107. i q^{77} -950795. q^{79} -141017. q^{82} +512362. i q^{83} -88525.1i q^{86} +680430. q^{88} -1.20447e6i q^{89} -481503. q^{91} +351330. i q^{92} -325229. q^{94} -32575.6 q^{97} -454853. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 516 q^{4}+O(q^{10}) \) 12 * q - 516 * q^4	\( 12 q - 516 q^{4} + 36372 q^{16} - 4320 q^{19} + 60192 q^{31} - 106296 q^{34} - 1078968 q^{46} - 711516 q^{49} - 449784 q^{61} - 3964572 q^{64} - 584400 q^{76} - 4324608 q^{79} + 631152 q^{91} - 5793408 q^{94}+O(q^{100}) \) 12 * q - 516 * q^4 + 36372 * q^16 - 4320 * q^19 + 60192 * q^31 - 106296 * q^34 - 1078968 * q^46 - 711516 * q^49 - 449784 * q^61 - 3964572 * q^64 - 584400 * q^76 - 4324608 * q^79 + 631152 * q^91 - 5793408 * q^94

Introduction

L-functions

Modular forms

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Database

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