LMFDB - Embedded newform 4368.2.h.e.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not 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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 273)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.e.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +(2.00000 + 3.00000i) q^{13} +3.00000i q^{15} -2.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +5.00000i q^{31} -3.00000 q^{35} +8.00000i q^{37} +(-2.00000 - 3.00000i) q^{39} +10.0000i q^{41} -9.00000 q^{43} -3.00000i q^{45} +7.00000i q^{47} -1.00000 q^{49} +2.00000 q^{51} +9.00000 q^{53} -1.00000i q^{57} -4.00000i q^{59} -8.00000 q^{61} -1.00000i q^{63} +(9.00000 - 6.00000i) q^{65} +2.00000i q^{67} -1.00000 q^{69} -9.00000i q^{73} +4.00000 q^{75} -15.0000 q^{79} +1.00000 q^{81} +9.00000i q^{83} +6.00000i q^{85} -5.00000 q^{87} +9.00000i q^{89} +(3.00000 - 2.00000i) q^{91} -5.00000i q^{93} +3.00000 q^{95} +13.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} - 6 q^{35} - 4 q^{39} - 18 q^{43} - 2 q^{49} + 4 q^{51} + 18 q^{53} - 16 q^{61} + 18 q^{65} - 2 q^{69} + 8 q^{75} - 30 q^{79} + 2 q^{81} - 10 q^{87} + 6 q^{91} + 6 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 - 6 * q^35 - 4 * q^39 - 18 * q^43 - 2 * q^49 + 4 * q^51 + 18 * q^53 - 16 * q^61 + 18 * q^65 - 2 * q^69 + 8 * q^75 - 30 * q^79 + 2 * q^81 - 10 * q^87 + 6 * q^91 + 6 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$		0			0
$15$			3.00000i			0.774597i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	1.00000			0.208514			0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000			0.208514			0.104257	+	0.994550i	$0.466753\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	5.00000			0.928477			0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000			0.928477			0.464238	+	0.885710i	$0.346328\pi$
$30$		0			0
$31$			5.00000i			0.898027i	0.893525	+	0.449013i	$0.148224\pi$
$31$			5.00000i			0.898027i	−0.893525	+	0.449013i	$0.851776\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000			−0.507093
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$	−2.00000	−	3.00000i	−0.320256	−	0.480384i
$40$		0			0
$41$			10.0000i			1.56174i	0.624695	+	0.780869i	$0.285223\pi$
$41$			10.0000i			1.56174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	−9.00000			−1.37249			−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000			−1.37249			−0.686244	+	0.727372i	$0.740742\pi$
$44$		0			0
$45$		−	3.00000i		−	0.447214i
$46$		0			0
$47$			7.00000i			1.02105i	0.859861	+	0.510527i	$0.170550\pi$
$47$			7.00000i			1.02105i	−0.859861	+	0.510527i	$0.829450\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	2.00000			0.280056
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$		−	4.00000i		−	0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$		−	4.00000i		−	0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		−	1.00000i		−	0.125988i
$64$		0			0
$65$	9.00000	−	6.00000i	1.11631	−	0.744208i
$66$		0			0
$67$			2.00000i			0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$			2.00000i			0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$		0			0
$69$	−1.00000			−0.120386
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	9.00000i		−	1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$		−	9.00000i		−	1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−15.0000			−1.68763			−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000			−1.68763			−0.843816	+	0.536633i	$0.819696\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			9.00000i			0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$			9.00000i			0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$		0			0
$85$			6.00000i			0.650791i
$86$		0			0
$87$	−5.00000			−0.536056
$88$		0			0
$89$			9.00000i			0.953998i	0.878904	+	0.476999i	$0.158275\pi$
$89$			9.00000i			0.953998i	−0.878904	+	0.476999i	$0.841725\pi$
$90$		0			0
$91$	3.00000	−	2.00000i	0.314485	−	0.209657i
$92$		0			0
$93$		−	5.00000i		−	0.518476i
$94$		0			0
$95$	3.00000			0.307794
$96$		0			0
$97$			13.0000i			1.31995i	0.751288	+	0.659975i	$0.229433\pi$
$97$			13.0000i			1.31995i	−0.751288	+	0.659975i	$0.770567\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000			−0.796030			−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000			−0.796030			−0.398015	+	0.917379i	$0.630301\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		0			0
$105$	3.00000			0.292770
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$		−	8.00000i		−	0.759326i
$112$		0			0
$113$	−21.0000			−1.97551			−0.987757	−	0.156001i	$-0.950140\pi$
$113$	−21.0000			−1.97551			−0.987757	+	0.156001i	$0.950140\pi$
$114$		0			0
$115$		−	3.00000i		−	0.279751i
$116$		0			0
$117$	2.00000	+	3.00000i	0.184900	+	0.277350i
$118$		0			0
$119$			2.00000i			0.183340i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		−	10.0000i		−	0.901670i
$124$		0			0
$125$		−	3.00000i		−	0.268328i
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$		0			0
$129$	9.00000			0.792406
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$	1.00000			0.0867110
$134$		0			0
$135$			3.00000i			0.258199i
$136$		0			0
$137$		−	12.0000i		−	1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$		−	12.0000i		−	1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$		0			0
$141$		−	7.00000i		−	0.589506i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		−	15.0000i		−	1.24568i
$146$		0			0
$147$	1.00000			0.0824786
$148$		0			0
$149$			4.00000i			0.327693i	0.986486	+	0.163846i	$0.0523901\pi$
$149$			4.00000i			0.327693i	−0.986486	+	0.163846i	$0.947610\pi$
$150$		0			0
$151$		−	20.0000i		−	1.62758i	−0.581161	−	0.813788i	$-0.697401\pi$
$151$		−	20.0000i		−	1.62758i	0.581161	−	0.813788i	$-0.302599\pi$
$152$		0			0
$153$	−2.00000			−0.161690
$154$		0			0
$155$	15.0000			1.20483
$156$		0			0
$157$	−2.00000			−0.159617			−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000			−0.159617			−0.0798087	+	0.996810i	$0.525431\pi$
$158$		0			0
$159$	−9.00000			−0.713746
$160$		0			0
$161$		−	1.00000i		−	0.0788110i
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	3.00000i		−	0.232147i	−0.993241	−	0.116073i	$-0.962969\pi$
$167$		−	3.00000i		−	0.232147i	0.993241	−	0.116073i	$-0.0370308\pi$
$168$		0			0
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$		0			0
$171$			1.00000i			0.0764719i
$172$		0			0
$173$	24.0000			1.82469			0.912343	−	0.409426i	$-0.134271\pi$
$173$	24.0000			1.82469			0.912343	+	0.409426i	$0.134271\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$			4.00000i			0.300658i
$178$		0			0
$179$	5.00000			0.373718			0.186859	−	0.982387i	$-0.440169\pi$
$179$	5.00000			0.373718			0.186859	+	0.982387i	$0.440169\pi$
$180$		0			0
$181$	−8.00000			−0.594635			−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000			−0.594635			−0.297318	+	0.954779i	$0.596092\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$	24.0000			1.76452
$186$		0			0
$187$		0			0
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	8.00000			0.578860			0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000			0.578860			0.289430	+	0.957199i	$0.406534\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$		0			0
$195$	−9.00000	+	6.00000i	−0.644503	+	0.429669i
$196$		0			0
$197$		−	12.0000i		−	0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$		−	12.0000i		−	0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		0			0
$201$		−	2.00000i		−	0.141069i
$202$		0			0
$203$		−	5.00000i		−	0.350931i
$204$		0			0
$205$	30.0000			2.09529
$206$		0			0
$207$	1.00000			0.0695048
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0000			0.894957			0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000			0.894957			0.447478	+	0.894295i	$0.352322\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			27.0000i			1.84138i
$216$		0			0
$217$	5.00000			0.339422
$218$		0			0
$219$			9.00000i			0.608164i
$220$		0			0
$221$	−4.00000	−	6.00000i	−0.269069	−	0.403604i
$222$		0			0
$223$			19.0000i			1.27233i	0.771551	+	0.636167i	$0.219481\pi$
$223$			19.0000i			1.27233i	−0.771551	+	0.636167i	$0.780519\pi$
$224$		0			0
$225$	−4.00000			−0.266667
$226$		0			0
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.00000			−0.0655122			−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000			−0.0655122			−0.0327561	+	0.999463i	$0.510428\pi$
$234$		0			0
$235$	21.0000			1.36989
$236$		0			0
$237$	15.0000			0.974355
$238$		0			0
$239$			6.00000i			0.388108i	0.980991	+	0.194054i	$0.0621637\pi$
$239$			6.00000i			0.388108i	−0.980991	+	0.194054i	$0.937836\pi$
$240$		0			0
$241$			15.0000i			0.966235i	0.875556	+	0.483117i	$0.160496\pi$
$241$			15.0000i			0.966235i	−0.875556	+	0.483117i	$0.839504\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$			3.00000i			0.191663i
$246$		0			0
$247$	−3.00000	+	2.00000i	−0.190885	+	0.127257i
$248$		0			0
$249$		−	9.00000i		−	0.570352i
$250$		0			0
$251$	28.0000			1.76734			0.883672	−	0.468106i	$-0.155064\pi$
$251$	28.0000			1.76734			0.883672	+	0.468106i	$0.155064\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		−	6.00000i		−	0.375735i
$256$		0			0
$257$	−12.0000			−0.748539			−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000			−0.748539			−0.374270	+	0.927320i	$0.622107\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	5.00000			0.309492
$262$		0			0
$263$	−19.0000			−1.17159			−0.585795	−	0.810459i	$-0.699218\pi$
$263$	−19.0000			−1.17159			−0.585795	+	0.810459i	$0.699218\pi$
$264$		0			0
$265$		−	27.0000i		−	1.65860i
$266$		0			0
$267$		−	9.00000i		−	0.550791i
$268$		0			0
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$			20.0000i			1.21491i	0.794353	+	0.607457i	$0.207810\pi$
$271$			20.0000i			1.21491i	−0.794353	+	0.607457i	$0.792190\pi$
$272$		0			0
$273$	−3.00000	+	2.00000i	−0.181568	+	0.121046i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.00000			0.180253			0.0901263	−	0.995930i	$-0.471273\pi$
$277$	3.00000			0.180253			0.0901263	+	0.995930i	$0.471273\pi$
$278$		0			0
$279$			5.00000i			0.299342i
$280$		0			0
$281$		−	10.0000i		−	0.596550i	−0.954480	−	0.298275i	$-0.903589\pi$
$281$		−	10.0000i		−	0.596550i	0.954480	−	0.298275i	$-0.0964112\pi$
$282$		0			0
$283$	−24.0000			−1.42665			−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000			−1.42665			−0.713326	+	0.700832i	$0.752812\pi$
$284$		0			0
$285$	−3.00000			−0.177705
$286$		0			0
$287$	10.0000			0.590281
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	13.0000i		−	0.762073i
$292$		0			0
$293$			31.0000i			1.81104i	0.424304	+	0.905520i	$0.360519\pi$
$293$			31.0000i			1.81104i	−0.424304	+	0.905520i	$0.639481\pi$
$294$		0			0
$295$	−12.0000			−0.698667
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.00000	+	3.00000i	0.115663	+	0.173494i
$300$		0			0
$301$			9.00000i			0.518751i
$302$		0			0
$303$	8.00000			0.459588
$304$		0			0
$305$			24.0000i			1.37424i
$306$		0			0
$307$			17.0000i			0.970241i	0.874447	+	0.485121i	$0.161224\pi$
$307$			17.0000i			0.970241i	−0.874447	+	0.485121i	$0.838776\pi$
$308$		0			0
$309$	−16.0000			−0.910208
$310$		0			0
$311$	−2.00000			−0.113410			−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000			−0.113410			−0.0567048	+	0.998391i	$0.518059\pi$
$312$		0			0
$313$	4.00000			0.226093			0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000			0.226093			0.113047	+	0.993590i	$0.463939\pi$
$314$		0			0
$315$	−3.00000			−0.169031
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.0000			−0.669775
$322$		0			0
$323$		−	2.00000i		−	0.111283i
$324$		0			0
$325$	−8.00000	−	12.0000i	−0.443760	−	0.665640i
$326$		0			0
$327$		−	14.0000i		−	0.774202i
$328$		0			0
$329$	7.00000			0.385922
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		0			0
$333$			8.00000i			0.438397i
$334$		0			0
$335$	6.00000			0.327815
$336$		0			0
$337$	23.0000			1.25289			0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000			1.25289			0.626445	+	0.779466i	$0.284509\pi$
$338$		0			0
$339$	21.0000			1.14056
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			3.00000i			0.161515i
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$		−	1.00000i		−	0.0535288i	−0.999642	−	0.0267644i	$-0.991480\pi$
$349$		−	1.00000i		−	0.0535288i	0.999642	−	0.0267644i	$-0.00852039\pi$
$350$		0			0
$351$	−2.00000	−	3.00000i	−0.106752	−	0.160128i
$352$		0			0
$353$			6.00000i			0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$			6.00000i			0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	2.00000i		−	0.105851i
$358$		0			0
$359$		−	14.0000i		−	0.738892i	−0.929252	−	0.369446i	$-0.879548\pi$
$359$		−	14.0000i		−	0.738892i	0.929252	−	0.369446i	$-0.120452\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	−27.0000			−1.41324
$366$		0			0
$367$	−18.0000			−0.939592			−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000			−0.939592			−0.469796	+	0.882775i	$0.655673\pi$
$368$		0			0
$369$			10.0000i			0.520579i
$370$		0			0
$371$		−	9.00000i		−	0.467257i
$372$		0			0
$373$	−26.0000			−1.34623			−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000			−1.34623			−0.673114	+	0.739538i	$0.735044\pi$
$374$		0			0
$375$			3.00000i			0.154919i
$376$		0			0
$377$	10.0000	+	15.0000i	0.515026	+	0.772539i
$378$		0			0
$379$		−	4.00000i		−	0.205466i	−0.994709	−	0.102733i	$-0.967241\pi$
$379$		−	4.00000i		−	0.205466i	0.994709	−	0.102733i	$-0.0327588\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$		−	16.0000i		−	0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$		−	16.0000i		−	0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−9.00000			−0.457496
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	−2.00000			−0.101144
$392$		0			0
$393$	−18.0000			−0.907980
$394$		0			0
$395$			45.0000i			2.26420i
$396$		0			0
$397$		−	17.0000i		−	0.853206i	−0.904439	−	0.426603i	$-0.859710\pi$
$397$		−	17.0000i		−	0.853206i	0.904439	−	0.426603i	$-0.140290\pi$
$398$		0			0
$399$	−1.00000			−0.0500626
$400$		0			0
$401$			10.0000i			0.499376i	0.968326	+	0.249688i	$0.0803281\pi$
$401$			10.0000i			0.499376i	−0.968326	+	0.249688i	$0.919672\pi$
$402$		0			0
$403$	−15.0000	+	10.0000i	−0.747203	+	0.498135i
$404$		0			0
$405$		−	3.00000i		−	0.149071i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	31.0000i		−	1.53285i	−0.642333	−	0.766426i	$-0.722033\pi$
$409$		−	31.0000i		−	1.53285i	0.642333	−	0.766426i	$-0.277967\pi$
$410$		0			0
$411$			12.0000i			0.591916i
$412$		0			0
$413$	−4.00000			−0.196827
$414$		0			0
$415$	27.0000			1.32538
$416$		0			0
$417$	−20.0000			−0.979404
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$		−	10.0000i		−	0.487370i	−0.969854	−	0.243685i	$-0.921644\pi$
$421$		−	10.0000i		−	0.487370i	0.969854	−	0.243685i	$-0.0783563\pi$
$422$		0			0
$423$			7.00000i			0.340352i
$424$		0			0
$425$	8.00000			0.388057
$426$		0			0
$427$			8.00000i			0.387147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			20.0000i			0.963366i	0.876346	+	0.481683i	$0.159974\pi$
$431$			20.0000i			0.963366i	−0.876346	+	0.481683i	$0.840026\pi$
$432$		0			0
$433$	−6.00000			−0.288342			−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000			−0.288342			−0.144171	+	0.989553i	$0.546051\pi$
$434$		0			0
$435$			15.0000i			0.719195i
$436$		0			0
$437$			1.00000i			0.0478365i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	31.0000			1.47285			0.736427	−	0.676517i	$-0.236511\pi$
$443$	31.0000			1.47285			0.736427	+	0.676517i	$0.236511\pi$
$444$		0			0
$445$	27.0000			1.27992
$446$		0			0
$447$		−	4.00000i		−	0.189194i
$448$		0			0
$449$			34.0000i			1.60456i	0.596948	+	0.802280i	$0.296380\pi$
$449$			34.0000i			1.60456i	−0.596948	+	0.802280i	$0.703620\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			20.0000i			0.939682i
$454$		0			0
$455$	−6.00000	−	9.00000i	−0.281284	−	0.421927i
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$		0			0
$459$	2.00000			0.0933520
$460$		0			0
$461$			10.0000i			0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$			10.0000i			0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$		0			0
$463$		−	26.0000i		−	1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$		−	26.0000i		−	1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$		0			0
$465$	−15.0000			−0.695608
$466$		0			0
$467$	−18.0000			−0.832941			−0.416470	−	0.909149i	$-0.636733\pi$
$467$	−18.0000			−0.832941			−0.416470	+	0.909149i	$0.636733\pi$
$468$		0			0
$469$	2.00000			0.0923514
$470$		0			0
$471$	2.00000			0.0921551
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	4.00000i		−	0.183533i
$476$		0			0
$477$	9.00000			0.412082
$478$		0			0
$479$			21.0000i			0.959514i	0.877401	+	0.479757i	$0.159275\pi$
$479$			21.0000i			0.959514i	−0.877401	+	0.479757i	$0.840725\pi$
$480$		0			0
$481$	−24.0000	+	16.0000i	−1.09431	+	0.729537i
$482$		0			0
$483$			1.00000i			0.0455016i
$484$		0			0
$485$	39.0000			1.77090
$486$		0			0
$487$		−	8.00000i		−	0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$		−	8.00000i		−	0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$	−32.0000			−1.44414			−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000			−1.44414			−0.722070	+	0.691820i	$0.756809\pi$
$492$		0			0
$493$	−10.0000			−0.450377
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	14.0000i		−	0.626726i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$		−	14.0000i		−	0.626726i	0.949633	−	0.313363i	$-0.101456\pi$
$500$		0			0
$501$			3.00000i			0.134030i
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$			24.0000i			1.06799i
$506$		0			0
$507$	5.00000	−	12.0000i	0.222058	−	0.532939i
$508$		0			0
$509$			9.00000i			0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$			9.00000i			0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$		0			0
$511$	−9.00000			−0.398137
$512$		0			0
$513$		−	1.00000i		−	0.0441511i
$514$		0			0
$515$		−	48.0000i		−	2.11513i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−24.0000			−1.05348
$520$		0			0
$521$	12.0000			0.525730			0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000			0.525730			0.262865	+	0.964833i	$0.415333\pi$
$522$		0			0
$523$	−4.00000			−0.174908			−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000			−0.174908			−0.0874539	+	0.996169i	$0.527873\pi$
$524$		0			0
$525$		−	4.00000i		−	0.174574i
$526$		0			0
$527$		−	10.0000i		−	0.435607i
$528$		0			0
$529$	−22.0000			−0.956522
$530$		0			0
$531$		−	4.00000i		−	0.173585i
$532$		0			0
$533$	−30.0000	+	20.0000i	−1.29944	+	0.866296i
$534$		0			0
$535$		−	36.0000i		−	1.55642i
$536$		0			0
$537$	−5.00000			−0.215766
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	20.0000i		−	0.859867i	−0.902861	−	0.429934i	$-0.858537\pi$
$541$		−	20.0000i		−	0.859867i	0.902861	−	0.429934i	$-0.141463\pi$
$542$		0			0
$543$	8.00000			0.343313
$544$		0			0
$545$	42.0000			1.79908
$546$		0			0
$547$	−13.0000			−0.555840			−0.277920	−	0.960604i	$-0.589645\pi$
$547$	−13.0000			−0.555840			−0.277920	+	0.960604i	$0.589645\pi$
$548$		0			0
$549$	−8.00000			−0.341432
$550$		0			0
$551$			5.00000i			0.213007i
$552$		0			0
$553$			15.0000i			0.637865i
$554$		0			0
$555$	−24.0000			−1.01874
$556$		0			0
$557$		−	32.0000i		−	1.35588i	−0.735116	−	0.677942i	$-0.762872\pi$
$557$		−	32.0000i		−	1.35588i	0.735116	−	0.677942i	$-0.237128\pi$
$558$		0			0
$559$	−18.0000	−	27.0000i	−0.761319	−	1.14198i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	26.0000			1.09577			0.547885	−	0.836554i	$-0.315433\pi$
$563$	26.0000			1.09577			0.547885	+	0.836554i	$0.315433\pi$
$564$		0			0
$565$			63.0000i			2.65043i
$566$		0			0
$567$		−	1.00000i		−	0.0419961i
$568$		0			0
$569$	−5.00000			−0.209611			−0.104805	−	0.994493i	$-0.533422\pi$
$569$	−5.00000			−0.209611			−0.104805	+	0.994493i	$0.533422\pi$
$570$		0			0
$571$	23.0000			0.962520			0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000			0.962520			0.481260	+	0.876578i	$0.340179\pi$
$572$		0			0
$573$	−8.00000			−0.334205
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$		−	22.0000i		−	0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$		−	22.0000i		−	0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$		0			0
$579$			14.0000i			0.581820i
$580$		0			0
$581$	9.00000			0.373383
$582$		0			0
$583$		0			0
$584$		0			0
$585$	9.00000	−	6.00000i	0.372104	−	0.248069i
$586$		0			0
$587$		−	3.00000i		−	0.123823i	−0.998082	−	0.0619116i	$-0.980280\pi$
$587$		−	3.00000i		−	0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$		0			0
$589$	−5.00000			−0.206021
$590$		0			0
$591$			12.0000i			0.493614i
$592$		0			0
$593$		−	9.00000i		−	0.369586i	−0.982777	−	0.184793i	$-0.940839\pi$
$593$		−	9.00000i		−	0.369586i	0.982777	−	0.184793i	$-0.0591614\pi$
$594$		0			0
$595$	6.00000			0.245976
$596$		0			0
$597$	10.0000			0.409273
$598$		0			0
$599$	35.0000			1.43006			0.715031	−	0.699093i	$-0.246413\pi$
$599$	35.0000			1.43006			0.715031	+	0.699093i	$0.246413\pi$
$600$		0			0
$601$	2.00000			0.0815817			0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000			0.0815817			0.0407909	+	0.999168i	$0.487012\pi$
$602$		0			0
$603$			2.00000i			0.0814463i
$604$		0			0
$605$		−	33.0000i		−	1.34164i
$606$		0			0
$607$	−28.0000			−1.13648			−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000			−1.13648			−0.568242	+	0.822861i	$0.692376\pi$
$608$		0			0
$609$			5.00000i			0.202610i
$610$		0			0
$611$	−21.0000	+	14.0000i	−0.849569	+	0.566379i
$612$		0			0
$613$			26.0000i			1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$			26.0000i			1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$		0			0
$615$	−30.0000			−1.20972
$616$		0			0
$617$			48.0000i			1.93241i	0.257780	+	0.966204i	$0.417009\pi$
$617$			48.0000i			1.93241i	−0.257780	+	0.966204i	$0.582991\pi$
$618$		0			0
$619$		−	44.0000i		−	1.76851i	−0.467005	−	0.884255i	$-0.654667\pi$
$619$		−	44.0000i		−	1.76851i	0.467005	−	0.884255i	$-0.345333\pi$
$620$		0			0
$621$	−1.00000			−0.0401286
$622$		0			0
$623$	9.00000			0.360577
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16.0000i		−	0.637962i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−13.0000			−0.516704
$634$		0			0
$635$			24.0000i			0.952411i
$636$		0			0
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	17.0000			0.671460			0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000			0.671460			0.335730	+	0.941958i	$0.391017\pi$
$642$		0			0
$643$			44.0000i			1.73519i	0.497271	+	0.867595i	$0.334335\pi$
$643$			44.0000i			1.73519i	−0.497271	+	0.867595i	$0.665665\pi$
$644$		0			0
$645$		−	27.0000i		−	1.06312i
$646$		0			0
$647$	2.00000			0.0786281			0.0393141	−	0.999227i	$-0.487483\pi$
$647$	2.00000			0.0786281			0.0393141	+	0.999227i	$0.487483\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−5.00000			−0.195965
$652$		0			0
$653$	14.0000			0.547862			0.273931	−	0.961749i	$-0.411676\pi$
$653$	14.0000			0.547862			0.273931	+	0.961749i	$0.411676\pi$
$654$		0			0
$655$		−	54.0000i		−	2.10995i
$656$		0			0
$657$		−	9.00000i		−	0.351123i
$658$		0			0
$659$	−15.0000			−0.584317			−0.292159	−	0.956370i	$-0.594373\pi$
$659$	−15.0000			−0.584317			−0.292159	+	0.956370i	$0.594373\pi$
$660$		0			0
$661$			15.0000i			0.583432i	0.956505	+	0.291716i	$0.0942263\pi$
$661$			15.0000i			0.583432i	−0.956505	+	0.291716i	$0.905774\pi$
$662$		0			0
$663$	4.00000	+	6.00000i	0.155347	+	0.233021i
$664$		0			0
$665$		−	3.00000i		−	0.116335i
$666$		0			0
$667$	5.00000			0.193601
$668$		0			0
$669$		−	19.0000i		−	0.734582i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$	4.00000			0.153960
$676$		0			0
$677$	18.0000			0.691796			0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000			0.691796			0.345898	+	0.938272i	$0.387574\pi$
$678$		0			0
$679$	13.0000			0.498894
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$			24.0000i			0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$			24.0000i			0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$		0			0
$685$	−36.0000			−1.37549
$686$		0			0
$687$		−	14.0000i		−	0.534133i
$688$		0			0
$689$	18.0000	+	27.0000i	0.685745	+	1.02862i
$690$		0			0
$691$			35.0000i			1.33146i	0.746191	+	0.665731i	$0.231880\pi$
$691$			35.0000i			1.33146i	−0.746191	+	0.665731i	$0.768120\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	60.0000i		−	2.27593i
$696$		0			0
$697$		−	20.0000i		−	0.757554i
$698$		0			0
$699$	1.00000			0.0378235
$700$		0			0
$701$	−23.0000			−0.868698			−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000			−0.868698			−0.434349	+	0.900745i	$0.643022\pi$
$702$		0			0
$703$	−8.00000			−0.301726
$704$		0			0
$705$	−21.0000			−0.790906
$706$		0			0
$707$			8.00000i			0.300871i
$708$		0			0
$709$		−	16.0000i		−	0.600893i	−0.953799	−	0.300446i	$-0.902864\pi$
$709$		−	16.0000i		−	0.600893i	0.953799	−	0.300446i	$-0.0971356\pi$
$710$		0			0
$711$	−15.0000			−0.562544
$712$		0			0
$713$			5.00000i			0.187251i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	6.00000i		−	0.224074i
$718$		0			0
$719$	−50.0000			−1.86469			−0.932343	−	0.361576i	$-0.882239\pi$
$719$	−50.0000			−1.86469			−0.932343	+	0.361576i	$0.882239\pi$
$720$		0			0
$721$		−	16.0000i		−	0.595871i
$722$		0			0
$723$		−	15.0000i		−	0.557856i
$724$		0			0
$725$	−20.0000			−0.742781
$726$		0			0
$727$	2.00000			0.0741759			0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000			0.0741759			0.0370879	+	0.999312i	$0.488192\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	18.0000			0.665754
$732$		0			0
$733$		−	19.0000i		−	0.701781i	−0.936416	−	0.350891i	$-0.885879\pi$
$733$		−	19.0000i		−	0.701781i	0.936416	−	0.350891i	$-0.114121\pi$
$734$		0			0
$735$		−	3.00000i		−	0.110657i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	4.00000i		−	0.147142i	−0.997290	−	0.0735712i	$-0.976560\pi$
$739$		−	4.00000i		−	0.147142i	0.997290	−	0.0735712i	$-0.0234396\pi$
$740$		0			0
$741$	3.00000	−	2.00000i	0.110208	−	0.0734718i
$742$		0			0
$743$		−	26.0000i		−	0.953847i	−0.878945	−	0.476924i	$-0.841752\pi$
$743$		−	26.0000i		−	0.953847i	0.878945	−	0.476924i	$-0.158248\pi$
$744$		0			0
$745$	12.0000			0.439646
$746$		0			0
$747$			9.00000i			0.329293i
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$		0			0
$751$	23.0000			0.839282			0.419641	−	0.907690i	$-0.362156\pi$
$751$	23.0000			0.839282			0.419641	+	0.907690i	$0.362156\pi$
$752$		0			0
$753$	−28.0000			−1.02038
$754$		0			0
$755$	−60.0000			−2.18362
$756$		0			0
$757$	13.0000			0.472493			0.236247	−	0.971693i	$-0.424083\pi$
$757$	13.0000			0.472493			0.236247	+	0.971693i	$0.424083\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	45.0000i		−	1.63125i	−0.578582	−	0.815624i	$-0.696394\pi$
$761$		−	45.0000i		−	1.63125i	0.578582	−	0.815624i	$-0.303606\pi$
$762$		0			0
$763$	14.0000			0.506834
$764$		0			0
$765$			6.00000i			0.216930i
$766$		0			0
$767$	12.0000	−	8.00000i	0.433295	−	0.288863i
$768$		0			0
$769$		−	1.00000i		−	0.0360609i	−0.999837	−	0.0180305i	$-0.994260\pi$
$769$		−	1.00000i		−	0.0360609i	0.999837	−	0.0180305i	$-0.00573959\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$			6.00000i			0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$			6.00000i			0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$		0			0
$775$		−	20.0000i		−	0.718421i
$776$		0			0
$777$	−8.00000			−0.286998
$778$		0			0
$779$	−10.0000			−0.358287
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−5.00000			−0.178685
$784$		0			0
$785$			6.00000i			0.214149i
$786$		0			0
$787$			7.00000i			0.249523i	0.992187	+	0.124762i	$0.0398166\pi$
$787$			7.00000i			0.249523i	−0.992187	+	0.124762i	$0.960183\pi$
$788$		0			0
$789$	19.0000			0.676418
$790$		0			0
$791$			21.0000i			0.746674i
$792$		0			0
$793$	−16.0000	−	24.0000i	−0.568177	−	0.852265i
$794$		0			0
$795$			27.0000i			0.957591i
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$		−	14.0000i		−	0.495284i
$800$		0			0
$801$			9.00000i			0.317999i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−3.00000			−0.105736
$806$		0			0
$807$	−10.0000			−0.352017
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$		−	20.0000i		−	0.702295i	−0.936320	−	0.351147i	$-0.885792\pi$
$811$		−	20.0000i		−	0.702295i	0.936320	−	0.351147i	$-0.114208\pi$
$812$		0			0
$813$		−	20.0000i		−	0.701431i
$814$		0			0
$815$	−18.0000			−0.630512
$816$		0			0
$817$		−	9.00000i		−	0.314870i
$818$		0			0
$819$	3.00000	−	2.00000i	0.104828	−	0.0698857i
$820$		0			0
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	16.0000			0.557725			0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000			0.557725			0.278862	+	0.960331i	$0.410043\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			42.0000i			1.46048i	0.683189	+	0.730242i	$0.260592\pi$
$827$			42.0000i			1.46048i	−0.683189	+	0.730242i	$0.739408\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$	−3.00000			−0.104069
$832$		0			0
$833$	2.00000			0.0692959
$834$		0			0
$835$	−9.00000			−0.311458
$836$		0			0
$837$		−	5.00000i		−	0.172825i
$838$		0			0
$839$		−	24.0000i		−	0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$		−	24.0000i		−	0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$		0			0
$841$	−4.00000			−0.137931
$842$		0			0
$843$			10.0000i			0.344418i
$844$		0			0
$845$	36.0000	+	15.0000i	1.23844	+	0.516016i
$846$		0			0
$847$		−	11.0000i		−	0.377964i
$848$		0			0
$849$	24.0000			0.823678
$850$		0			0
$851$			8.00000i			0.274236i
$852$		0			0
$853$			1.00000i			0.0342393i	0.999853	+	0.0171197i	$0.00544963\pi$
$853$			1.00000i			0.0342393i	−0.999853	+	0.0171197i	$0.994550\pi$
$854$		0			0
$855$	3.00000			0.102598
$856$		0			0
$857$	8.00000			0.273275			0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000			0.273275			0.136637	+	0.990621i	$0.456370\pi$
$858$		0			0
$859$	−30.0000			−1.02359			−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000			−1.02359			−0.511793	+	0.859109i	$0.671019\pi$
$860$		0			0
$861$	−10.0000			−0.340799
$862$		0			0
$863$			4.00000i			0.136162i	0.997680	+	0.0680808i	$0.0216876\pi$
$863$			4.00000i			0.136162i	−0.997680	+	0.0680808i	$0.978312\pi$
$864$		0			0
$865$		−	72.0000i		−	2.44807i
$866$		0			0
$867$	13.0000			0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−6.00000	+	4.00000i	−0.203302	+	0.135535i
$872$		0			0
$873$			13.0000i			0.439983i
$874$		0			0
$875$	−3.00000			−0.101419
$876$		0			0
$877$		−	22.0000i		−	0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$		−	22.0000i		−	0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$		0			0
$879$		−	31.0000i		−	1.04560i
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$	16.0000			0.538443			0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000			0.538443			0.269221	+	0.963078i	$0.413234\pi$
$884$		0			0
$885$	12.0000			0.403376
$886$		0			0
$887$	−18.0000			−0.604381			−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000			−0.604381			−0.302190	+	0.953248i	$0.597718\pi$
$888$		0			0
$889$			8.00000i			0.268311i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−7.00000			−0.234246
$894$		0			0
$895$		−	15.0000i		−	0.501395i
$896$		0			0
$897$	−2.00000	−	3.00000i	−0.0667781	−	0.100167i
$898$		0			0
$899$			25.0000i			0.833797i
$900$		0			0
$901$	−18.0000			−0.599667
$902$		0			0
$903$		−	9.00000i		−	0.299501i
$904$		0			0
$905$			24.0000i			0.797787i
$906$		0			0
$907$	27.0000			0.896520			0.448260	−	0.893903i	$-0.352044\pi$
$907$	27.0000			0.896520			0.448260	+	0.893903i	$0.352044\pi$
$908$		0			0
$909$	−8.00000			−0.265343
$910$		0			0
$911$	33.0000			1.09334			0.546669	−	0.837349i	$-0.315895\pi$
$911$	33.0000			1.09334			0.546669	+	0.837349i	$0.315895\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		−	24.0000i		−	0.793416i
$916$		0			0
$917$		−	18.0000i		−	0.594412i
$918$		0			0
$919$	−40.0000			−1.31948			−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000			−1.31948			−0.659739	+	0.751495i	$0.729333\pi$
$920$		0			0
$921$		−	17.0000i		−	0.560169i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		−	32.0000i		−	1.05215i
$926$		0			0
$927$	16.0000			0.525509
$928$		0			0
$929$			29.0000i			0.951459i	0.879592	+	0.475730i	$0.157816\pi$
$929$			29.0000i			0.951459i	−0.879592	+	0.475730i	$0.842184\pi$
$930$		0			0
$931$		−	1.00000i		−	0.0327737i
$932$		0			0
$933$	2.00000			0.0654771
$934$		0			0
$935$		0			0
$936$		0			0
$937$	28.0000			0.914720			0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000			0.914720			0.457360	+	0.889282i	$0.348795\pi$
$938$		0			0
$939$	−4.00000			−0.130535
$940$		0			0
$941$		−	5.00000i		−	0.162995i	−0.996674	−	0.0814977i	$-0.974030\pi$
$941$		−	5.00000i		−	0.162995i	0.996674	−	0.0814977i	$-0.0259703\pi$
$942$		0			0
$943$			10.0000i			0.325645i
$944$		0			0
$945$	3.00000			0.0975900
$946$		0			0
$947$			2.00000i			0.0649913i	0.999472	+	0.0324956i	$0.0103455\pi$
$947$			2.00000i			0.0649913i	−0.999472	+	0.0324956i	$0.989654\pi$
$948$		0			0
$949$	27.0000	−	18.0000i	0.876457	−	0.584305i
$950$		0			0
$951$			2.00000i			0.0648544i
$952$		0			0
$953$	29.0000			0.939402			0.469701	−	0.882826i	$-0.344362\pi$
$953$	29.0000			0.939402			0.469701	+	0.882826i	$0.344362\pi$
$954$		0			0
$955$		−	24.0000i		−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	6.00000			0.193548
$962$		0			0
$963$	12.0000			0.386695
$964$		0			0
$965$	−42.0000			−1.35203
$966$		0			0
$967$			52.0000i			1.67221i	0.548572	+	0.836104i	$0.315172\pi$
$967$			52.0000i			1.67221i	−0.548572	+	0.836104i	$0.684828\pi$
$968$		0			0
$969$			2.00000i			0.0642493i
$970$		0			0
$971$	28.0000			0.898563			0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000			0.898563			0.449281	+	0.893390i	$0.351680\pi$
$972$		0			0
$973$		−	20.0000i		−	0.641171i
$974$		0			0
$975$	8.00000	+	12.0000i	0.256205	+	0.384308i
$976$		0			0
$977$		−	52.0000i		−	1.66363i	−0.555055	−	0.831814i	$-0.687303\pi$
$977$		−	52.0000i		−	1.66363i	0.555055	−	0.831814i	$-0.312697\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			14.0000i			0.446986i
$982$		0			0
$983$		−	31.0000i		−	0.988746i	−0.869250	−	0.494373i	$-0.835398\pi$
$983$		−	31.0000i		−	0.988746i	0.869250	−	0.494373i	$-0.164602\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$	−7.00000			−0.222812
$988$		0			0
$989$	−9.00000			−0.286183
$990$		0			0
$991$	−32.0000			−1.01651			−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000			−1.01651			−0.508257	+	0.861206i	$0.669710\pi$
$992$		0			0
$993$		−	20.0000i		−	0.634681i
$994$		0			0
$995$			30.0000i			0.951064i
$996$		0			0
$997$	−32.0000			−1.01345			−0.506725	−	0.862108i	$-0.669144\pi$
$997$	−32.0000			−1.01345			−0.506725	+	0.862108i	$0.669144\pi$
$998$		0			0
$999$		−	8.00000i		−	0.253109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.e.337.1		2
4.3	odd	2		273.2.c.a.64.1	✓	2
12.11	even	2		819.2.c.a.64.2		2
13.12	even	2	inner	4368.2.h.e.337.2		2
28.27	even	2		1911.2.c.a.883.1		2
52.31	even	4		3549.2.a.a.1.1		1
52.47	even	4		3549.2.a.e.1.1		1
52.51	odd	2		273.2.c.a.64.2	yes	2
156.155	even	2		819.2.c.a.64.1		2
364.363	even	2		1911.2.c.a.883.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.c.a.64.1	✓	2	4.3	odd	2
273.2.c.a.64.2	yes	2	52.51	odd	2
819.2.c.a.64.1		2	156.155	even	2
819.2.c.a.64.2		2	12.11	even	2
1911.2.c.a.883.1		2	28.27	even	2
1911.2.c.a.883.2		2	364.363	even	2
3549.2.a.a.1.1		1	52.31	even	4
3549.2.a.e.1.1		1	52.47	even	4
4368.2.h.e.337.1		2	1.1	even	1	trivial
4368.2.h.e.337.2		2	13.12	even	2	inner

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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.00000i q^{5} -1.00000i q^{7} +1.00000 q^{9} +(2.00000 + 3.00000i) q^{13} +3.00000i q^{15} -2.00000 q^{17} +1.00000i q^{19} +1.00000i q^{21} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} +5.00000 q^{29} +5.00000i q^{31} -3.00000 q^{35} +8.00000i q^{37} +(-2.00000 - 3.00000i) q^{39} +10.0000i q^{41} -9.00000 q^{43} -3.00000i q^{45} +7.00000i q^{47} -1.00000 q^{49} +2.00000 q^{51} +9.00000 q^{53} -1.00000i q^{57} -4.00000i q^{59} -8.00000 q^{61} -1.00000i q^{63} +(9.00000 - 6.00000i) q^{65} +2.00000i q^{67} -1.00000 q^{69} -9.00000i q^{73} +4.00000 q^{75} -15.0000 q^{79} +1.00000 q^{81} +9.00000i q^{83} +6.00000i q^{85} -5.00000 q^{87} +9.00000i q^{89} +(3.00000 - 2.00000i) q^{91} -5.00000i q^{93} +3.00000 q^{95} +13.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 4 q^{17} + 2 q^{23} - 8 q^{25} - 2 q^{27} + 10 q^{29} - 6 q^{35} - 4 q^{39} - 18 q^{43} - 2 q^{49} + 4 q^{51} + 18 q^{53} - 16 q^{61} + 18 q^{65} - 2 q^{69} + 8 q^{75} - 30 q^{79} + 2 q^{81} - 10 q^{87} + 6 q^{91} + 6 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 4 * q^17 + 2 * q^23 - 8 * q^25 - 2 * q^27 + 10 * q^29 - 6 * q^35 - 4 * q^39 - 18 * q^43 - 2 * q^49 + 4 * q^51 + 18 * q^53 - 16 * q^61 + 18 * q^65 - 2 * q^69 + 8 * q^75 - 30 * q^79 + 2 * q^81 - 10 * q^87 + 6 * q^91 + 6 * q^95

Introduction

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