LMFDB - Embedded newform 1232.2.j.b.111.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(111,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1232.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.j (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:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.17
Character	$\chi$	$=$	1232.111
Dual form			1232.2.j.b.111.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03324 q^{3} +2.56834i q^{5} +(0.330305 + 2.62505i) q^{7} -1.93241 q^{9} -1.00000i q^{11} +5.65534i q^{13} +2.65372i q^{15} -6.89874i q^{17} +1.66593 q^{19} +(0.341285 + 2.71232i) q^{21} +4.31770i q^{23} -1.59638 q^{25} -5.09638 q^{27} -6.82475 q^{29} -0.123697 q^{31} -1.03324i q^{33} +(-6.74203 + 0.848336i) q^{35} -4.80039 q^{37} +5.84334i q^{39} +1.10144i q^{41} +10.6896i q^{43} -4.96309i q^{45} -0.692255 q^{47} +(-6.78180 + 1.73413i) q^{49} -7.12808i q^{51} +6.59218 q^{53} +2.56834 q^{55} +1.72131 q^{57} +2.59067 q^{59} -13.1465i q^{61} +(-0.638284 - 5.07267i) q^{63} -14.5249 q^{65} +10.1610i q^{67} +4.46123i q^{69} -1.82448i q^{71} +0.300337i q^{73} -1.64945 q^{75} +(2.62505 - 0.330305i) q^{77} +10.8148i q^{79} +0.531427 q^{81} +11.8395 q^{83} +17.7183 q^{85} -7.05162 q^{87} -1.88568i q^{89} +(-14.8456 + 1.86799i) q^{91} -0.127809 q^{93} +4.27869i q^{95} -6.52449i q^{97} +1.93241i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 16 q^{9} + 20 q^{21} - 56 q^{29} + 40 q^{37} - 8 q^{49} + 16 q^{53} + 8 q^{57} - 96 q^{65} - 4 q^{77} + 40 q^{81} - 32 q^{85} + 8 q^{93}+O(q^{100}) $ 24 * q + 16 * q^9 + 20 * q^21 - 56 * q^29 + 40 * q^37 - 8 * q^49 + 16 * q^53 + 8 * q^57 - 96 * q^65 - 4 * q^77 + 40 * q^81 - 32 * q^85 + 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times$.

$n$	$309$	$353$	$463$	$673$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	1.03324			0.596543			0.298272	−	0.954481i	$-0.403590\pi$
$3$	1.03324			0.596543			0.298272	+	0.954481i	$0.403590\pi$
$4$		0			0
$5$			2.56834i			1.14860i	0.818646	+	0.574299i	$0.194725\pi$
$5$			2.56834i			1.14860i	−0.818646	+	0.574299i	$0.805275\pi$
$6$		0			0
$7$	0.330305	+	2.62505i	0.124843	+	0.992176i
$7$	0.330305	+	2.62505i	0.124843	+	0.992176i
$8$		0			0
$9$	−1.93241			−0.644136
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$			5.65534i			1.56851i	0.620439	+	0.784255i	$0.286954\pi$
$13$			5.65534i			1.56851i	−0.620439	+	0.784255i	$0.713046\pi$
$14$		0			0
$15$			2.65372i			0.685188i
$16$		0			0
$17$		−	6.89874i		−	1.67319i	−0.547822	−	0.836595i	$-0.684543\pi$
$17$		−	6.89874i		−	1.67319i	0.547822	−	0.836595i	$-0.315457\pi$
$18$		0			0
$19$	1.66593			0.382191			0.191096	−	0.981571i	$-0.438796\pi$
$19$	1.66593			0.382191			0.191096	+	0.981571i	$0.438796\pi$
$20$		0			0
$21$	0.341285	+	2.71232i	0.0744745	+	0.591876i
$22$		0			0
$23$			4.31770i			0.900302i	0.892952	+	0.450151i	$0.148630\pi$
$23$			4.31770i			0.900302i	−0.892952	+	0.450151i	$0.851370\pi$
$24$		0			0
$25$	−1.59638			−0.319276
$26$		0			0
$27$	−5.09638			−0.980798
$28$		0			0
$29$	−6.82475			−1.26732			−0.633662	−	0.773610i	$-0.718449\pi$
$29$	−6.82475			−1.26732			−0.633662	+	0.773610i	$0.718449\pi$
$30$		0			0
$31$	−0.123697			−0.0222166			−0.0111083	−	0.999938i	$-0.503536\pi$
$31$	−0.123697			−0.0222166			−0.0111083	+	0.999938i	$0.503536\pi$
$32$		0			0
$33$		−	1.03324i		−	0.179865i
$34$		0			0
$35$	−6.74203	+	0.848336i	−1.13961	+	0.143395i
$36$		0			0
$37$	−4.80039			−0.789180			−0.394590	−	0.918857i	$-0.629113\pi$
$37$	−4.80039			−0.789180			−0.394590	+	0.918857i	$0.629113\pi$
$38$		0			0
$39$			5.84334i			0.935684i
$40$		0			0
$41$			1.10144i			0.172017i	0.996294	+	0.0860084i	$0.0274112\pi$
$41$			1.10144i			0.172017i	−0.996294	+	0.0860084i	$0.972589\pi$
$42$		0			0
$43$			10.6896i			1.63014i	0.579360	+	0.815072i	$0.303302\pi$
$43$			10.6896i			1.63014i	−0.579360	+	0.815072i	$0.696698\pi$
$44$		0			0
$45$		−	4.96309i		−	0.739853i
$46$		0			0
$47$	−0.692255			−0.100976			−0.0504879	−	0.998725i	$-0.516078\pi$
$47$	−0.692255			−0.100976			−0.0504879	+	0.998725i	$0.516078\pi$
$48$		0			0
$49$	−6.78180	+	1.73413i	−0.968828	+	0.247733i
$50$		0			0
$51$		−	7.12808i		−	0.998130i
$52$		0			0
$53$	6.59218			0.905505			0.452753	−	0.891636i	$-0.350442\pi$
$53$	6.59218			0.905505			0.452753	+	0.891636i	$0.350442\pi$
$54$		0			0
$55$	2.56834			0.346315
$56$		0			0
$57$	1.72131			0.227994
$58$		0			0
$59$	2.59067			0.337276			0.168638	−	0.985678i	$-0.446063\pi$
$59$	2.59067			0.337276			0.168638	+	0.985678i	$0.446063\pi$
$60$		0			0
$61$		−	13.1465i		−	1.68323i	−0.540076	−	0.841616i	$-0.681605\pi$
$61$		−	13.1465i		−	1.68323i	0.540076	−	0.841616i	$-0.318395\pi$
$62$		0			0
$63$	−0.638284	−	5.07267i	−0.0804162	−	0.639097i
$64$		0			0
$65$	−14.5249			−1.80159
$66$		0			0
$67$			10.1610i			1.24137i	0.784061	+	0.620684i	$0.213145\pi$
$67$			10.1610i			1.24137i	−0.784061	+	0.620684i	$0.786855\pi$
$68$		0			0
$69$			4.46123i			0.537069i
$70$		0			0
$71$		−	1.82448i		−	0.216526i	−0.994122	−	0.108263i	$-0.965471\pi$
$71$		−	1.82448i		−	0.216526i	0.994122	−	0.108263i	$-0.0345289\pi$
$72$		0			0
$73$			0.300337i			0.0351518i	0.999846	+	0.0175759i	$0.00559488\pi$
$73$			0.300337i			0.0351518i	−0.999846	+	0.0175759i	$0.994405\pi$
$74$		0			0
$75$	−1.64945			−0.190462
$76$		0			0
$77$	2.62505	−	0.330305i	0.299152	−	0.0376417i
$78$		0			0
$79$			10.8148i			1.21676i	0.793648	+	0.608378i	$0.208179\pi$
$79$			10.8148i			1.21676i	−0.793648	+	0.608378i	$0.791821\pi$
$80$		0			0
$81$	0.531427			0.0590474
$82$		0			0
$83$	11.8395			1.29955			0.649777	−	0.760125i	$-0.274862\pi$
$83$	11.8395			1.29955			0.649777	+	0.760125i	$0.274862\pi$
$84$		0			0
$85$	17.7183			1.92182
$86$		0			0
$87$	−7.05162			−0.756013
$88$		0			0
$89$		−	1.88568i		−	0.199882i	−0.994993	−	0.0999409i	$-0.968135\pi$
$89$		−	1.88568i		−	0.199882i	0.994993	−	0.0999409i	$-0.0318654\pi$
$90$		0			0
$91$	−14.8456	+	1.86799i	−1.55624	+	0.195818i
$92$		0			0
$93$	−0.127809			−0.0132532
$94$		0			0
$95$			4.27869i			0.438984i
$96$		0			0
$97$		−	6.52449i		−	0.662462i	−0.943550	−	0.331231i	$-0.892536\pi$
$97$		−	6.52449i		−	0.662462i	0.943550	−	0.331231i	$-0.107464\pi$
$98$		0			0
$99$			1.93241i			0.194214i
$100$		0			0
$101$			2.45583i			0.244365i	0.992508	+	0.122182i	$0.0389892\pi$
$101$			2.45583i			0.244365i	−0.992508	+	0.122182i	$0.961011\pi$
$102$		0			0
$103$	7.80667			0.769214			0.384607	−	0.923080i	$-0.374337\pi$
$103$	7.80667			0.769214			0.384607	+	0.923080i	$0.374337\pi$
$104$		0			0
$105$	−6.96616	+	0.876537i	−0.679828	+	0.0855413i
$106$		0			0
$107$			7.56360i			0.731200i	0.930772	+	0.365600i	$0.119136\pi$
$107$			7.56360i			0.731200i	−0.930772	+	0.365600i	$0.880864\pi$
$108$		0			0
$109$	3.20689			0.307164			0.153582	−	0.988136i	$-0.450919\pi$
$109$	3.20689			0.307164			0.153582	+	0.988136i	$0.450919\pi$
$110$		0			0
$111$	−4.95998			−0.470780
$112$		0			0
$113$	−9.34645			−0.879240			−0.439620	−	0.898184i	$-0.644887\pi$
$113$	−9.34645			−0.879240			−0.439620	+	0.898184i	$0.644887\pi$
$114$		0			0
$115$	−11.0893			−1.03408
$116$		0			0
$117$		−	10.9284i		−	1.01033i
$118$		0			0
$119$	18.1095	−	2.27869i	1.66010	−	0.208887i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$			1.13806i			0.102615i
$124$		0			0
$125$			8.74166i			0.781877i
$126$		0			0
$127$		−	0.504172i		−	0.0447380i	−0.999750	−	0.0223690i	$-0.992879\pi$
$127$		−	0.504172i		−	0.0447380i	0.999750	−	0.0223690i	$-0.00712087\pi$
$128$		0			0
$129$			11.0449i			0.972451i
$130$		0			0
$131$	20.2061			1.76541			0.882706	−	0.469925i	$-0.155719\pi$
$131$	20.2061			1.76541			0.882706	+	0.469925i	$0.155719\pi$
$132$		0			0
$133$	0.550266	+	4.37316i	0.0477141	+	0.379201i
$134$		0			0
$135$		−	13.0892i		−	1.12654i
$136$		0			0
$137$	10.2860			0.878790			0.439395	−	0.898294i	$-0.355193\pi$
$137$	10.2860			0.878790			0.439395	+	0.898294i	$0.355193\pi$
$138$		0			0
$139$	11.0425			0.936611			0.468306	−	0.883566i	$-0.344865\pi$
$139$	11.0425			0.936611			0.468306	+	0.883566i	$0.344865\pi$
$140$		0			0
$141$	−0.715268			−0.0602364
$142$		0			0
$143$	5.65534			0.472923
$144$		0			0
$145$		−	17.5283i		−	1.45564i
$146$		0			0
$147$	−7.00725	+	1.79178i	−0.577948	+	0.147784i
$148$		0			0
$149$	1.59665			0.130803			0.0654013	−	0.997859i	$-0.479167\pi$
$149$	1.59665			0.130803			0.0654013	+	0.997859i	$0.479167\pi$
$150$		0			0
$151$			15.2616i			1.24197i	0.783822	+	0.620985i	$0.213267\pi$
$151$			15.2616i			1.24197i	−0.783822	+	0.620985i	$0.786733\pi$
$152$		0			0
$153$			13.3312i			1.07776i
$154$		0			0
$155$		−	0.317696i		−	0.0255180i
$156$		0			0
$157$			14.5610i			1.16209i	0.813870	+	0.581047i	$0.197357\pi$
$157$			14.5610i			1.16209i	−0.813870	+	0.581047i	$0.802643\pi$
$158$		0			0
$159$	6.81132			0.540173
$160$		0			0
$161$	−11.3342	+	1.42616i	−0.893258	+	0.112397i
$162$		0			0
$163$		−	16.0995i		−	1.26101i	−0.776185	−	0.630505i	$-0.782848\pi$
$163$		−	16.0995i		−	1.26101i	0.776185	−	0.630505i	$-0.217152\pi$
$164$		0			0
$165$	2.65372			0.206592
$166$		0			0
$167$	13.6205			1.05398			0.526992	−	0.849870i	$-0.323320\pi$
$167$	13.6205			1.05398			0.526992	+	0.849870i	$0.323320\pi$
$168$		0			0
$169$	−18.9829			−1.46022
$170$		0			0
$171$	−3.21926			−0.246183
$172$		0			0
$173$			21.8928i			1.66448i	0.554417	+	0.832239i	$0.312941\pi$
$173$			21.8928i			1.66448i	−0.554417	+	0.832239i	$0.687059\pi$
$174$		0			0
$175$	−0.527293	−	4.19059i	−0.0398596	−	0.316779i
$176$		0			0
$177$	2.67679			0.201200
$178$		0			0
$179$		−	3.19661i		−	0.238926i	−0.992839	−	0.119463i	$-0.961883\pi$
$179$		−	3.19661i		−	0.238926i	0.992839	−	0.119463i	$-0.0381173\pi$
$180$		0			0
$181$		−	12.7053i		−	0.944374i	−0.881498	−	0.472187i	$-0.843465\pi$
$181$		−	12.7053i		−	0.944374i	0.881498	−	0.472187i	$-0.156535\pi$
$182$		0			0
$183$		−	13.5835i		−	1.00412i
$184$		0			0
$185$		−	12.3291i		−	0.906450i
$186$		0			0
$187$	−6.89874			−0.504486
$188$		0			0
$189$	−1.68336	−	13.3783i	−0.122446	−	0.973125i
$190$		0			0
$191$		−	25.7678i		−	1.86450i	−0.361820	−	0.932248i	$-0.617844\pi$
$191$		−	25.7678i		−	1.86450i	0.361820	−	0.932248i	$-0.382156\pi$
$192$		0			0
$193$	−4.86479			−0.350175			−0.175088	−	0.984553i	$-0.556021\pi$
$193$	−4.86479			−0.350175			−0.175088	+	0.984553i	$0.556021\pi$
$194$		0			0
$195$	−15.0077			−1.07472
$196$		0			0
$197$	20.6424			1.47071			0.735355	−	0.677683i	$-0.237016\pi$
$197$	20.6424			1.47071			0.735355	+	0.677683i	$0.237016\pi$
$198$		0			0
$199$	−26.5679			−1.88335			−0.941675	−	0.336524i	$-0.890749\pi$
$199$	−26.5679			−1.88335			−0.941675	+	0.336524i	$0.890749\pi$
$200$		0			0
$201$			10.4988i			0.740530i
$202$		0			0
$203$	−2.25425	−	17.9153i	−0.158217	−	1.25741i
$204$		0			0
$205$	−2.82889			−0.197578
$206$		0			0
$207$		−	8.34355i		−	0.579917i
$208$		0			0
$209$		−	1.66593i		−	0.115235i
$210$		0			0
$211$			5.53329i			0.380927i	0.981694	+	0.190464i	$0.0609991\pi$
$211$			5.53329i			0.380927i	−0.981694	+	0.190464i	$0.939001\pi$
$212$		0			0
$213$		−	1.88513i		−	0.129167i
$214$		0			0
$215$	−27.4545			−1.87238
$216$		0			0
$217$	−0.0408577	−	0.324711i	−0.00277360	−	0.0220428i
$218$		0			0
$219$			0.310322i			0.0209696i
$220$		0			0
$221$	39.0147			2.62441
$222$		0			0
$223$	−17.9287			−1.20060			−0.600299	−	0.799776i	$-0.704952\pi$
$223$	−17.9287			−1.20060			−0.600299	+	0.799776i	$0.704952\pi$
$224$		0			0
$225$	3.08486			0.205657
$226$		0			0
$227$	−5.50686			−0.365503			−0.182752	−	0.983159i	$-0.558500\pi$
$227$	−5.50686			−0.365503			−0.182752	+	0.983159i	$0.558500\pi$
$228$		0			0
$229$		−	10.1077i		−	0.667933i	−0.942585	−	0.333966i	$-0.891613\pi$
$229$		−	10.1077i		−	0.667933i	0.942585	−	0.333966i	$-0.108387\pi$
$230$		0			0
$231$	2.71232	−	0.341285i	0.178457	−	0.0224549i
$232$		0			0
$233$	6.25458			0.409751			0.204875	−	0.978788i	$-0.434321\pi$
$233$	6.25458			0.409751			0.204875	+	0.978788i	$0.434321\pi$
$234$		0			0
$235$		−	1.77795i		−	0.115981i
$236$		0			0
$237$			11.1743i			0.725847i
$238$		0			0
$239$		−	14.6194i		−	0.945649i	−0.881157	−	0.472825i	$-0.843234\pi$
$239$		−	14.6194i		−	0.945649i	0.881157	−	0.472825i	$-0.156766\pi$
$240$		0			0
$241$		−	10.8953i		−	0.701829i	−0.936408	−	0.350914i	$-0.885871\pi$
$241$		−	10.8953i		−	0.701829i	0.936408	−	0.350914i	$-0.114129\pi$
$242$		0			0
$243$	15.8382			1.01602
$244$		0			0
$245$	−4.45385	−	17.4180i	−0.284546	−	1.11279i
$246$		0			0
$247$			9.42142i			0.599471i
$248$		0			0
$249$	12.2331			0.775240
$250$		0			0
$251$	4.37307			0.276026			0.138013	−	0.990430i	$-0.455928\pi$
$251$	4.37307			0.276026			0.138013	+	0.990430i	$0.455928\pi$
$252$		0			0
$253$	4.31770			0.271451
$254$		0			0
$255$	18.3073			1.14645
$256$		0			0
$257$			27.8008i			1.73417i	0.498164	+	0.867083i	$0.334008\pi$
$257$			27.8008i			1.73417i	−0.498164	+	0.867083i	$0.665992\pi$
$258$		0			0
$259$	−1.58559	−	12.6013i	−0.0985240	−	0.783006i
$260$		0			0
$261$	13.1882			0.816329
$262$		0			0
$263$			5.55290i			0.342407i	0.985236	+	0.171203i	$0.0547655\pi$
$263$			5.55290i			0.342407i	−0.985236	+	0.171203i	$0.945235\pi$
$264$		0			0
$265$			16.9310i			1.04006i
$266$		0			0
$267$		−	1.94837i		−	0.119238i
$268$		0			0
$269$		−	0.221248i		−	0.0134897i	−0.999977	−	0.00674486i	$-0.997853\pi$
$269$		−	0.221248i		−	0.0134897i	0.999977	−	0.00674486i	$-0.00214697\pi$
$270$		0			0
$271$	−4.23577			−0.257305			−0.128652	−	0.991690i	$-0.541065\pi$
$271$	−4.23577			−0.257305			−0.128652	+	0.991690i	$0.541065\pi$
$272$		0			0
$273$	−15.3391	+	1.93008i	−0.928363	+	0.116814i
$274$		0			0
$275$			1.59638i			0.0962655i
$276$		0			0
$277$	28.8794			1.73519			0.867596	−	0.497269i	$-0.165664\pi$
$277$	28.8794			1.73519			0.867596	+	0.497269i	$0.165664\pi$
$278$		0			0
$279$	0.239033			0.0143105
$280$		0			0
$281$	21.1251			1.26022			0.630108	−	0.776507i	$-0.283010\pi$
$281$	21.1251			1.26022			0.630108	+	0.776507i	$0.283010\pi$
$282$		0			0
$283$	20.8403			1.23883			0.619415	−	0.785064i	$-0.287370\pi$
$283$	20.8403			1.23883			0.619415	+	0.785064i	$0.287370\pi$
$284$		0			0
$285$			4.42092i			0.261873i
$286$		0			0
$287$	−2.89135	+	0.363812i	−0.170671	+	0.0214752i
$288$		0			0
$289$	−30.5926			−1.79956
$290$		0			0
$291$		−	6.74139i		−	0.395187i
$292$		0			0
$293$		−	6.02385i		−	0.351917i	−0.984398	−	0.175959i	$-0.943697\pi$
$293$		−	6.02385i		−	0.351917i	0.984398	−	0.175959i	$-0.0563025\pi$
$294$		0			0
$295$			6.65372i			0.387395i
$296$		0			0
$297$			5.09638i			0.295722i
$298$		0			0
$299$	−24.4180			−1.41213
$300$		0			0
$301$	−28.0607	+	3.53081i	−1.61739	+	0.203513i
$302$		0			0
$303$			2.53747i			0.145774i
$304$		0			0
$305$	33.7646			1.93336
$306$		0			0
$307$	32.6711			1.86464			0.932320	−	0.361634i	$-0.117781\pi$
$307$	32.6711			1.86464			0.932320	+	0.361634i	$0.117781\pi$
$308$		0			0
$309$	8.06619			0.458870
$310$		0			0
$311$	29.5383			1.67496			0.837482	−	0.546465i	$-0.184027\pi$
$311$	29.5383			1.67496			0.837482	+	0.546465i	$0.184027\pi$
$312$		0			0
$313$			17.5767i			0.993496i	0.867895	+	0.496748i	$0.165473\pi$
$313$			17.5767i			0.993496i	−0.867895	+	0.496748i	$0.834527\pi$
$314$		0			0
$315$	13.0284	−	1.63933i	0.734065	−	0.0923658i
$316$		0			0
$317$	−26.0280			−1.46188			−0.730938	−	0.682444i	$-0.760917\pi$
$317$	−26.0280			−1.46188			−0.730938	+	0.682444i	$0.760917\pi$
$318$		0			0
$319$			6.82475i			0.382112i
$320$		0			0
$321$			7.81503i			0.436193i
$322$		0			0
$323$		−	11.4928i		−	0.639478i
$324$		0			0
$325$		−	9.02808i		−	0.500788i
$326$		0			0
$327$	3.31350			0.183237
$328$		0			0
$329$	−0.228655	−	1.81721i	−0.0126062	−	0.100186i
$330$		0			0
$331$		−	23.3821i		−	1.28519i	−0.766204	−	0.642597i	$-0.777857\pi$
$331$		−	23.3821i		−	1.28519i	0.766204	−	0.642597i	$-0.222143\pi$
$332$		0			0
$333$	9.27632			0.508339
$334$		0			0
$335$	−26.0970			−1.42583
$336$		0			0
$337$	−10.3551			−0.564080			−0.282040	−	0.959403i	$-0.591011\pi$
$337$	−10.3551			−0.564080			−0.282040	+	0.959403i	$0.591011\pi$
$338$		0			0
$339$	−9.65716			−0.524505
$340$		0			0
$341$			0.123697i			0.00669856i
$342$		0			0
$343$	−6.79225	−	17.2298i	−0.366747	−	0.930321i
$344$		0			0
$345$	−11.4580			−0.616876
$346$		0			0
$347$		−	10.4742i		−	0.562287i	−0.959666	−	0.281143i	$-0.909286\pi$
$347$		−	10.4742i		−	0.562287i	0.959666	−	0.281143i	$-0.0907136\pi$
$348$		0			0
$349$		−	25.6330i		−	1.37210i	−0.727554	−	0.686050i	$-0.759343\pi$
$349$		−	25.6330i		−	1.37210i	0.727554	−	0.686050i	$-0.240657\pi$
$350$		0			0
$351$		−	28.8218i		−	1.53839i
$352$		0			0
$353$		−	33.9304i		−	1.80593i	−0.429709	−	0.902967i	$-0.641384\pi$
$353$		−	33.9304i		−	1.80593i	0.429709	−	0.902967i	$-0.358616\pi$
$354$		0			0
$355$	4.68589			0.248701
$356$		0			0
$357$	18.7116	−	2.35444i	0.990321	−	0.124610i
$358$		0			0
$359$		−	12.8824i		−	0.679905i	−0.940443	−	0.339952i	$-0.889589\pi$
$359$		−	12.8824i		−	0.679905i	0.940443	−	0.339952i	$-0.110411\pi$
$360$		0			0
$361$	−16.2247			−0.853930
$362$		0			0
$363$	−1.03324			−0.0542312
$364$		0			0
$365$	−0.771369			−0.0403753
$366$		0			0
$367$	19.2390			1.00427			0.502133	−	0.864790i	$-0.332549\pi$
$367$	19.2390			1.00427			0.502133	+	0.864790i	$0.332549\pi$
$368$		0			0
$369$		−	2.12844i		−	0.110802i
$370$		0			0
$371$	2.17743	+	17.3048i	0.113046	+	0.898421i
$372$		0			0
$373$	11.1961			0.579713			0.289856	−	0.957070i	$-0.406392\pi$
$373$	11.1961			0.579713			0.289856	+	0.957070i	$0.406392\pi$
$374$		0			0
$375$			9.03226i			0.466424i
$376$		0			0
$377$		−	38.5963i		−	1.98781i
$378$		0			0
$379$			33.0992i			1.70019i	0.526630	+	0.850095i	$0.323455\pi$
$379$			33.0992i			1.70019i	−0.526630	+	0.850095i	$0.676545\pi$
$380$		0			0
$381$		−	0.520932i		−	0.0266882i
$382$		0			0
$383$	−31.7510			−1.62240			−0.811200	−	0.584769i	$-0.801185\pi$
$383$	−31.7510			−1.62240			−0.811200	+	0.584769i	$0.801185\pi$
$384$		0			0
$385$	0.848336	+	6.74203i	0.0432352	+	0.343606i
$386$		0			0
$387$		−	20.6566i		−	1.05003i
$388$		0			0
$389$	−13.9441			−0.706996			−0.353498	−	0.935435i	$-0.615008\pi$
$389$	−13.9441			−0.706996			−0.353498	+	0.935435i	$0.615008\pi$
$390$		0			0
$391$	29.7867			1.50638
$392$		0			0
$393$	20.8778			1.05314
$394$		0			0
$395$	−27.7760			−1.39756
$396$		0			0
$397$			29.3923i			1.47516i	0.675262	+	0.737578i	$0.264030\pi$
$397$			29.3923i			1.47516i	−0.675262	+	0.737578i	$0.735970\pi$
$398$		0			0
$399$	0.568558	+	4.51854i	0.0284635	+	0.226210i
$400$		0			0
$401$	−18.6982			−0.933744			−0.466872	−	0.884325i	$-0.654619\pi$
$401$	−18.6982			−0.933744			−0.466872	+	0.884325i	$0.654619\pi$
$402$		0			0
$403$		−	0.699548i		−	0.0348470i
$404$		0			0
$405$			1.36489i			0.0678218i
$406$		0			0
$407$			4.80039i			0.237947i
$408$		0			0
$409$			29.4308i			1.45526i	0.685970	+	0.727630i	$0.259378\pi$
$409$			29.4308i			1.45526i	−0.685970	+	0.727630i	$0.740622\pi$
$410$		0			0
$411$	10.6279			0.524236
$412$		0			0
$413$	0.855710	+	6.80064i	0.0421067	+	0.334638i
$414$		0			0
$415$			30.4079i			1.49266i
$416$		0			0
$417$	11.4096			0.558729
$418$		0			0
$419$	27.6027			1.34848			0.674240	−	0.738513i	$-0.264472\pi$
$419$	27.6027			1.34848			0.674240	+	0.738513i	$0.264472\pi$
$420$		0			0
$421$	14.3711			0.700404			0.350202	−	0.936674i	$-0.386113\pi$
$421$	14.3711			0.700404			0.350202	+	0.936674i	$0.386113\pi$
$422$		0			0
$423$	1.33772			0.0650421
$424$		0			0
$425$			11.0130i			0.534210i
$426$		0			0
$427$	34.5101	−	4.34234i	1.67006	−	0.210141i
$428$		0			0
$429$	5.84334			0.282119
$430$		0			0
$431$			3.74252i			0.180271i	0.995930	+	0.0901355i	$0.0287300\pi$
$431$			3.74252i			0.180271i	−0.995930	+	0.0901355i	$0.971270\pi$
$432$		0			0
$433$		−	34.0829i		−	1.63792i	−0.573850	−	0.818961i	$-0.694551\pi$
$433$		−	34.0829i		−	1.63792i	0.573850	−	0.818961i	$-0.305449\pi$
$434$		0			0
$435$		−	18.1110i		−	0.868355i
$436$		0			0
$437$			7.19299i			0.344087i
$438$		0			0
$439$	11.7494			0.560766			0.280383	−	0.959888i	$-0.409538\pi$
$439$	11.7494			0.560766			0.280383	+	0.959888i	$0.409538\pi$
$440$		0			0
$441$	13.1052	−	3.35106i	0.624057	−	0.159574i
$442$		0			0
$443$		−	37.4757i		−	1.78053i	−0.455447	−	0.890263i	$-0.650520\pi$
$443$		−	37.4757i		−	1.78053i	0.455447	−	0.890263i	$-0.349480\pi$
$444$		0			0
$445$	4.84308			0.229584
$446$		0			0
$447$	1.64973			0.0780294
$448$		0			0
$449$	−11.2634			−0.531553			−0.265777	−	0.964035i	$-0.585628\pi$
$449$	−11.2634			−0.531553			−0.265777	+	0.964035i	$0.585628\pi$
$450$		0			0
$451$	1.10144			0.0518650
$452$		0			0
$453$			15.7689i			0.740889i
$454$		0			0
$455$	−4.79763	−	38.1285i	−0.224916	−	1.78749i
$456$		0			0
$457$	−36.3818			−1.70187			−0.850934	−	0.525272i	$-0.823963\pi$
$457$	−36.3818			−1.70187			−0.850934	+	0.525272i	$0.823963\pi$
$458$		0			0
$459$			35.1586i			1.64106i
$460$		0			0
$461$			1.36268i			0.0634664i	0.999496	+	0.0317332i	$0.0101027\pi$
$461$			1.36268i			0.0634664i	−0.999496	+	0.0317332i	$0.989897\pi$
$462$		0			0
$463$			32.9746i			1.53246i	0.642565	+	0.766231i	$0.277870\pi$
$463$			32.9746i			1.53246i	−0.642565	+	0.766231i	$0.722130\pi$
$464$		0			0
$465$		−	0.328257i		−	0.0152226i
$466$		0			0
$467$	11.8571			0.548681			0.274341	−	0.961633i	$-0.411540\pi$
$467$	11.8571			0.548681			0.274341	+	0.961633i	$0.411540\pi$
$468$		0			0
$469$	−26.6733	+	3.35624i	−1.23166	+	0.154977i
$470$		0			0
$471$			15.0451i			0.693240i
$472$		0			0
$473$	10.6896			0.491507
$474$		0			0
$475$	−2.65947			−0.122025
$476$		0			0
$477$	−12.7388			−0.583269
$478$		0			0
$479$	9.84814			0.449973			0.224987	−	0.974362i	$-0.427766\pi$
$479$	9.84814			0.449973			0.224987	+	0.974362i	$0.427766\pi$
$480$		0			0
$481$		−	27.1479i		−	1.23784i
$482$		0			0
$483$	−11.7110	+	1.47357i	−0.532867	+	0.0670496i
$484$		0			0
$485$	16.7571			0.760902
$486$		0			0
$487$			6.86298i			0.310991i	0.987837	+	0.155496i	$0.0496974\pi$
$487$			6.86298i			0.310991i	−0.987837	+	0.155496i	$0.950303\pi$
$488$		0			0
$489$		−	16.6347i		−	0.752247i
$490$		0			0
$491$			12.3303i			0.556458i	0.960515	+	0.278229i	$0.0897475\pi$
$491$			12.3303i			0.556458i	−0.960515	+	0.278229i	$0.910253\pi$
$492$		0			0
$493$			47.0821i			2.12047i
$494$		0			0
$495$	−4.96309			−0.223074
$496$		0			0
$497$	4.78936	−	0.602634i	0.214832	−	0.0270318i
$498$		0			0
$499$			22.2649i			0.996714i	0.866972	+	0.498357i	$0.166063\pi$
$499$			22.2649i			0.996714i	−0.866972	+	0.498357i	$0.833937\pi$
$500$		0			0
$501$	14.0733			0.628747
$502$		0			0
$503$	−29.8936			−1.33289			−0.666444	−	0.745555i	$-0.732184\pi$
$503$	−29.8936			−1.33289			−0.666444	+	0.745555i	$0.732184\pi$
$504$		0			0
$505$	−6.30742			−0.280677
$506$		0			0
$507$	−19.6139			−0.871085
$508$		0			0
$509$		−	24.8759i		−	1.10261i	−0.834305	−	0.551303i	$-0.814131\pi$
$509$		−	24.8759i		−	1.10261i	0.834305	−	0.551303i	$-0.185869\pi$
$510$		0			0
$511$	−0.788401	+	0.0992029i	−0.0348768	+	0.00438848i
$512$		0			0
$513$	−8.49022			−0.374853
$514$		0			0
$515$			20.0502i			0.883518i
$516$		0			0
$517$			0.692255i			0.0304453i
$518$		0			0
$519$			22.6206i			0.992933i
$520$		0			0
$521$		−	30.3413i		−	1.32927i	−0.747166	−	0.664637i	$-0.768586\pi$
$521$		−	30.3413i		−	1.32927i	0.747166	−	0.664637i	$-0.231414\pi$
$522$		0			0
$523$	−36.2616			−1.58561			−0.792804	−	0.609476i	$-0.791380\pi$
$523$	−36.2616			−1.58561			−0.792804	+	0.609476i	$0.791380\pi$
$524$		0			0
$525$	−0.544821	−	4.32989i	−0.0237780	−	0.188972i
$526$		0			0
$527$			0.853353i			0.0371726i
$528$		0			0
$529$	4.35750			0.189457
$530$		0			0
$531$	−5.00623			−0.217252
$532$		0			0
$533$	−6.22904			−0.269810
$534$		0			0
$535$	−19.4259			−0.839855
$536$		0			0
$537$		−	3.30288i		−	0.142530i
$538$		0			0
$539$	1.73413	+	6.78180i	0.0746945	+	0.292113i
$540$		0			0
$541$	−9.46437			−0.406905			−0.203452	−	0.979085i	$-0.565216\pi$
$541$	−9.46437			−0.406905			−0.203452	+	0.979085i	$0.565216\pi$
$542$		0			0
$543$		−	13.1276i		−	0.563360i
$544$		0			0
$545$			8.23639i			0.352808i
$546$		0			0
$547$			34.3136i			1.46714i	0.679612	+	0.733572i	$0.262148\pi$
$547$			34.3136i			1.46714i	−0.679612	+	0.733572i	$0.737852\pi$
$548$		0			0
$549$			25.4043i			1.08423i
$550$		0			0
$551$	−11.3696			−0.484360
$552$		0			0
$553$	−28.3893	+	3.57217i	−1.20724	+	0.151904i
$554$		0			0
$555$		−	12.7389i		−	0.540737i
$556$		0			0
$557$	−11.9210			−0.505111			−0.252555	−	0.967582i	$-0.581271\pi$
$557$	−11.9210			−0.505111			−0.252555	+	0.967582i	$0.581271\pi$
$558$		0			0
$559$	−60.4531			−2.55689
$560$		0			0
$561$	−7.12808			−0.300948
$562$		0			0
$563$	−32.3576			−1.36371			−0.681855	−	0.731487i	$-0.738826\pi$
$563$	−32.3576			−1.36371			−0.681855	+	0.731487i	$0.738826\pi$
$564$		0			0
$565$		−	24.0049i		−	1.00989i
$566$		0			0
$567$	0.175533	+	1.39502i	0.00737169	+	0.0585855i
$568$		0			0
$569$	23.0580			0.966643			0.483321	−	0.875443i	$-0.339430\pi$
$569$	23.0580			0.966643			0.483321	+	0.875443i	$0.339430\pi$
$570$		0			0
$571$			14.0722i			0.588904i	0.955666	+	0.294452i	$0.0951371\pi$
$571$			14.0722i			0.588904i	−0.955666	+	0.294452i	$0.904863\pi$
$572$		0			0
$573$		−	26.6245i		−	1.11225i
$574$		0			0
$575$		−	6.89269i		−	0.287445i
$576$		0			0
$577$		−	3.28465i		−	0.136742i	−0.997660	−	0.0683710i	$-0.978220\pi$
$577$		−	3.28465i		−	0.136742i	0.997660	−	0.0683710i	$-0.0217802\pi$
$578$		0			0
$579$	−5.02651			−0.208895
$580$		0			0
$581$	3.91064	+	31.0793i	0.162241	+	1.28939i
$582$		0			0
$583$		−	6.59218i		−	0.273020i
$584$		0			0
$585$	28.0679			1.16047
$586$		0			0
$587$	4.30261			0.177588			0.0887940	−	0.996050i	$-0.471699\pi$
$587$	4.30261			0.177588			0.0887940	+	0.996050i	$0.471699\pi$
$588$		0			0
$589$	−0.206071			−0.00849100
$590$		0			0
$591$	21.3286			0.877342
$592$		0			0
$593$			19.8929i			0.816905i	0.912780	+	0.408452i	$0.133931\pi$
$593$			19.8929i			0.816905i	−0.912780	+	0.408452i	$0.866069\pi$
$594$		0			0
$595$	5.85245	+	46.5115i	0.239927	+	1.90679i
$596$		0			0
$597$	−27.4511			−1.12350
$598$		0			0
$599$			25.1206i			1.02640i	0.858269	+	0.513200i	$0.171540\pi$
$599$			25.1206i			1.02640i	−0.858269	+	0.513200i	$0.828460\pi$
$600$		0			0
$601$			19.2710i			0.786079i	0.919522	+	0.393039i	$0.128576\pi$
$601$			19.2710i			0.786079i	−0.919522	+	0.393039i	$0.871424\pi$
$602$		0			0
$603$		−	19.6353i		−	0.799610i
$604$		0			0
$605$		−	2.56834i		−	0.104418i
$606$		0			0
$607$	24.8640			1.00920			0.504599	−	0.863354i	$-0.331640\pi$
$607$	24.8640			1.00920			0.504599	+	0.863354i	$0.331640\pi$
$608$		0			0
$609$	−2.32918	−	18.5109i	−0.0943833	−	0.750099i
$610$		0			0
$611$		−	3.91494i		−	0.158381i
$612$		0			0
$613$	25.5282			1.03107			0.515537	−	0.856868i	$-0.327593\pi$
$613$	25.5282			1.03107			0.515537	+	0.856868i	$0.327593\pi$
$614$		0			0
$615$	−2.92293			−0.117864
$616$		0			0
$617$	28.6758			1.15444			0.577222	−	0.816587i	$-0.304137\pi$
$617$	28.6758			1.15444			0.577222	+	0.816587i	$0.304137\pi$
$618$		0			0
$619$	−4.75382			−0.191072			−0.0955361	−	0.995426i	$-0.530457\pi$
$619$	−4.75382			−0.191072			−0.0955361	+	0.995426i	$0.530457\pi$
$620$		0			0
$621$		−	22.0046i		−	0.883015i
$622$		0			0
$623$	4.95001	−	0.622850i	0.198318	−	0.0249539i
$624$		0			0
$625$	−30.4335			−1.21734
$626$		0			0
$627$		−	1.72131i		−	0.0687427i
$628$		0			0
$629$			33.1167i			1.32045i
$630$		0			0
$631$		−	3.70192i		−	0.147371i	−0.997282	−	0.0736856i	$-0.976524\pi$
$631$		−	3.70192i		−	0.147371i	0.997282	−	0.0736856i	$-0.0234761\pi$
$632$		0			0
$633$			5.71723i			0.227240i
$634$		0			0
$635$	1.29489			0.0513860
$636$		0			0
$637$	−9.80712	−	38.3534i	−0.388572	−	1.51962i
$638$		0			0
$639$			3.52564i			0.139472i
$640$		0			0
$641$	−37.8206			−1.49382			−0.746912	−	0.664922i	$-0.768465\pi$
$641$	−37.8206			−1.49382			−0.746912	+	0.664922i	$0.768465\pi$
$642$		0			0
$643$	3.36636			0.132756			0.0663782	−	0.997795i	$-0.478856\pi$
$643$	3.36636			0.132756			0.0663782	+	0.997795i	$0.478856\pi$
$644$		0			0
$645$	−28.3671			−1.11695
$646$		0			0
$647$	20.3678			0.800742			0.400371	−	0.916353i	$-0.368881\pi$
$647$	20.3678			0.800742			0.400371	+	0.916353i	$0.368881\pi$
$648$		0			0
$649$		−	2.59067i		−	0.101693i
$650$		0			0
$651$	−0.0422159	−	0.335505i	−0.00165457	−	0.0131495i
$652$		0			0
$653$	−32.7229			−1.28055			−0.640274	−	0.768147i	$-0.721179\pi$
$653$	−32.7229			−1.28055			−0.640274	+	0.768147i	$0.721179\pi$
$654$		0			0
$655$			51.8961i			2.02775i
$656$		0			0
$657$		−	0.580375i		−	0.0226426i
$658$		0			0
$659$			25.5161i			0.993965i	0.867760	+	0.496983i	$0.165559\pi$
$659$			25.5161i			0.993965i	−0.867760	+	0.496983i	$0.834441\pi$
$660$		0			0
$661$			28.9252i			1.12506i	0.826776	+	0.562531i	$0.190172\pi$
$661$			28.9252i			1.12506i	−0.826776	+	0.562531i	$0.809828\pi$
$662$		0			0
$663$	40.3117			1.56558
$664$		0			0
$665$	−11.2318	+	1.41327i	−0.435550	+	0.0548043i
$666$		0			0
$667$		−	29.4672i		−	1.14097i
$668$		0			0
$669$	−18.5248			−0.716208
$670$		0			0
$671$	−13.1465			−0.507514
$672$		0			0
$673$	15.1886			0.585477			0.292738	−	0.956193i	$-0.405433\pi$
$673$	15.1886			0.585477			0.292738	+	0.956193i	$0.405433\pi$
$674$		0			0
$675$	8.13577			0.313146
$676$		0			0
$677$			26.1796i			1.00616i	0.864239	+	0.503082i	$0.167801\pi$
$677$			26.1796i			1.00616i	−0.864239	+	0.503082i	$0.832199\pi$
$678$		0			0
$679$	17.1271	−	2.15507i	0.657279	−	0.0827040i
$680$		0			0
$681$	−5.68993			−0.218038
$682$		0			0
$683$		−	5.73500i		−	0.219444i	−0.993962	−	0.109722i	$-0.965004\pi$
$683$		−	5.73500i		−	0.219444i	0.993962	−	0.109722i	$-0.0349960\pi$
$684$		0			0
$685$			26.4179i			1.00938i
$686$		0			0
$687$		−	10.4437i		−	0.398451i
$688$		0			0
$689$			37.2810i			1.42029i
$690$		0			0
$691$	3.04444			0.115816			0.0579080	−	0.998322i	$-0.481557\pi$
$691$	3.04444			0.115816			0.0579080	+	0.998322i	$0.481557\pi$
$692$		0			0
$693$	−5.07267	+	0.638284i	−0.192695	+	0.0242464i
$694$		0			0
$695$			28.3609i			1.07579i
$696$		0			0
$697$	7.59858			0.287817
$698$		0			0
$699$	6.46250			0.244434
$700$		0			0
$701$	8.52274			0.321900			0.160950	−	0.986963i	$-0.448544\pi$
$701$	8.52274			0.321900			0.160950	+	0.986963i	$0.448544\pi$
$702$		0			0
$703$	−7.99713			−0.301618
$704$		0			0
$705$		−	1.83705i		−	0.0691874i
$706$		0			0
$707$	−6.44669	+	0.811173i	−0.242453	+	0.0305073i
$708$		0			0
$709$	−22.6340			−0.850039			−0.425020	−	0.905184i	$-0.639733\pi$
$709$	−22.6340			−0.850039			−0.425020	+	0.905184i	$0.639733\pi$
$710$		0			0
$711$		−	20.8985i		−	0.783756i
$712$		0			0
$713$		−	0.534086i		−	0.0200017i
$714$		0			0
$715$			14.5249i			0.543199i
$716$		0			0
$717$		−	15.1054i		−	0.564121i
$718$		0			0
$719$	−4.00245			−0.149266			−0.0746332	−	0.997211i	$-0.523779\pi$
$719$	−4.00245			−0.149266			−0.0746332	+	0.997211i	$0.523779\pi$
$720$		0			0
$721$	2.57858	+	20.4929i	0.0960314	+	0.763196i
$722$		0			0
$723$		−	11.2575i		−	0.418671i
$724$		0			0
$725$	10.8949			0.404627
$726$		0			0
$727$	−2.40403			−0.0891604			−0.0445802	−	0.999006i	$-0.514195\pi$
$727$	−2.40403			−0.0891604			−0.0445802	+	0.999006i	$0.514195\pi$
$728$		0			0
$729$	14.7705			0.547054
$730$		0			0
$731$	73.7445			2.72754
$732$		0			0
$733$		−	27.7267i		−	1.02411i	−0.858953	−	0.512054i	$-0.828885\pi$
$733$		−	27.7267i		−	1.02411i	0.858953	−	0.512054i	$-0.171115\pi$
$734$		0			0
$735$	−4.60191	−	17.9970i	−0.169744	−	0.663830i
$736$		0			0
$737$	10.1610			0.374287
$738$		0			0
$739$			22.2649i			0.819028i	0.912304	+	0.409514i	$0.134302\pi$
$739$			22.2649i			0.819028i	−0.912304	+	0.409514i	$0.865698\pi$
$740$		0			0
$741$			9.73462i			0.357610i
$742$		0			0
$743$		−	37.6151i		−	1.37996i	−0.723827	−	0.689982i	$-0.757618\pi$
$743$		−	37.6151i		−	1.37996i	0.723827	−	0.689982i	$-0.242382\pi$
$744$		0			0
$745$			4.10074i			0.150239i
$746$		0			0
$747$	−22.8787			−0.837090
$748$		0			0
$749$	−19.8548	+	2.49829i	−0.725480	+	0.0912856i
$750$		0			0
$751$			26.6026i			0.970742i	0.874308	+	0.485371i	$0.161315\pi$
$751$			26.6026i			0.970742i	−0.874308	+	0.485371i	$0.838685\pi$
$752$		0			0
$753$	4.51845			0.164661
$754$		0			0
$755$	−39.1970			−1.42652
$756$		0			0
$757$	−10.1671			−0.369530			−0.184765	−	0.982783i	$-0.559152\pi$
$757$	−10.1671			−0.369530			−0.184765	+	0.982783i	$0.559152\pi$
$758$		0			0
$759$	4.46123			0.161932
$760$		0			0
$761$		−	29.4369i		−	1.06709i	−0.845773	−	0.533543i	$-0.820860\pi$
$761$		−	29.4369i		−	1.06709i	0.845773	−	0.533543i	$-0.179140\pi$
$762$		0			0
$763$	1.05925	+	8.41825i	0.0383474	+	0.304761i
$764$		0			0
$765$	−34.2390			−1.23791
$766$		0			0
$767$			14.6511i			0.529021i
$768$		0			0
$769$			12.9246i			0.466073i	0.972468	+	0.233036i	$0.0748661\pi$
$769$			12.9246i			0.466073i	−0.972468	+	0.233036i	$0.925134\pi$
$770$		0			0
$771$			28.7250i			1.03450i
$772$		0			0
$773$		−	25.2123i		−	0.906823i	−0.891301	−	0.453411i	$-0.850207\pi$
$773$		−	25.2123i		−	0.906823i	0.891301	−	0.453411i	$-0.149793\pi$
$774$		0			0
$775$	0.197467			0.00709324
$776$		0			0
$777$	−1.63830	−	13.0202i	−0.0587738	−	0.467097i
$778$		0			0
$779$			1.83493i			0.0657433i
$780$		0			0
$781$	−1.82448			−0.0652850
$782$		0			0
$783$	34.7815			1.24299
$784$		0			0
$785$	−37.3976			−1.33478
$786$		0			0
$787$	−34.7440			−1.23849			−0.619245	−	0.785197i	$-0.712561\pi$
$787$	−34.7440			−1.23849			−0.619245	+	0.785197i	$0.712561\pi$
$788$		0			0
$789$			5.73750i			0.204260i
$790$		0			0
$791$	−3.08718	−	24.5349i	−0.109767	−	0.872362i
$792$		0			0
$793$	74.3477			2.64017
$794$		0			0
$795$			17.4938i			0.620441i
$796$		0			0
$797$		−	23.0793i		−	0.817511i	−0.912644	−	0.408755i	$-0.865963\pi$
$797$		−	23.0793i		−	0.817511i	0.912644	−	0.408755i	$-0.134037\pi$
$798$		0			0
$799$			4.77569i			0.168952i
$800$		0			0
$801$			3.64391i			0.128751i
$802$		0			0
$803$	0.300337			0.0105987
$804$		0			0
$805$	−3.66286	−	29.1100i	−0.129099	−	1.02599i
$806$		0			0
$807$		−	0.228603i		−	0.00804721i
$808$		0			0
$809$	10.6783			0.375427			0.187714	−	0.982224i	$-0.439892\pi$
$809$	10.6783			0.375427			0.187714	+	0.982224i	$0.439892\pi$
$810$		0			0
$811$	−47.1085			−1.65420			−0.827102	−	0.562051i	$-0.810012\pi$
$811$	−47.1085			−1.65420			−0.827102	+	0.562051i	$0.810012\pi$
$812$		0			0
$813$	−4.37659			−0.153494
$814$		0			0
$815$	41.3490			1.44839
$816$		0			0
$817$			17.8081i			0.623026i
$818$		0			0
$819$	28.6877	−	3.60971i	1.00243	−	0.126134i
$820$		0			0
$821$	−9.75162			−0.340334			−0.170167	−	0.985415i	$-0.554431\pi$
$821$	−9.75162			−0.340334			−0.170167	+	0.985415i	$0.554431\pi$
$822$		0			0
$823$		−	8.64641i		−	0.301395i	−0.988580	−	0.150698i	$-0.951848\pi$
$823$		−	8.64641i		−	0.301395i	0.988580	−	0.150698i	$-0.0481519\pi$
$824$		0			0
$825$			1.64945i			0.0574265i
$826$		0			0
$827$			31.4994i			1.09534i	0.836694	+	0.547671i	$0.184486\pi$
$827$			31.4994i			1.09534i	−0.836694	+	0.547671i	$0.815514\pi$
$828$		0			0
$829$		−	39.1594i		−	1.36006i	−0.733184	−	0.680031i	$-0.761966\pi$
$829$		−	39.1594i		−	1.36006i	0.733184	−	0.680031i	$-0.238034\pi$
$830$		0			0
$831$	29.8394			1.03512
$832$		0			0
$833$	11.9633	+	46.7858i	0.414505	+	1.62103i
$834$		0			0
$835$			34.9820i			1.21060i
$836$		0			0
$837$	0.630406			0.0217900
$838$		0			0
$839$	13.2453			0.457280			0.228640	−	0.973511i	$-0.426572\pi$
$839$	13.2453			0.457280			0.228640	+	0.973511i	$0.426572\pi$
$840$		0			0
$841$	17.5772			0.606109
$842$		0			0
$843$	21.8273			0.751774
$844$		0			0
$845$		−	48.7545i		−	1.67721i
$846$		0			0
$847$	−0.330305	−	2.62505i	−0.0113494	−	0.0901979i
$848$		0			0
$849$	21.5331			0.739016
$850$		0			0
$851$		−	20.7266i		−	0.710500i
$852$		0			0
$853$			37.2833i			1.27656i	0.769806	+	0.638278i	$0.220353\pi$
$853$			37.2833i			1.27656i	−0.769806	+	0.638278i	$0.779647\pi$
$854$		0			0
$855$		−	8.26817i		−	0.282765i
$856$		0			0
$857$		−	38.3501i		−	1.31001i	−0.755623	−	0.655007i	$-0.772666\pi$
$857$		−	38.3501i		−	1.31001i	0.755623	−	0.655007i	$-0.227334\pi$
$858$		0			0
$859$	−12.0408			−0.410828			−0.205414	−	0.978675i	$-0.565854\pi$
$859$	−12.0408			−0.410828			−0.205414	+	0.978675i	$0.565854\pi$
$860$		0			0
$861$	−2.98747	+	0.375907i	−0.101813	+	0.0128109i
$862$		0			0
$863$			50.8073i			1.72950i	0.502203	+	0.864750i	$0.332523\pi$
$863$			50.8073i			1.72950i	−0.502203	+	0.864750i	$0.667477\pi$
$864$		0			0
$865$	−56.2282			−1.91181
$866$		0			0
$867$	−31.6096			−1.07352
$868$		0			0
$869$	10.8148			0.366866
$870$		0			0
$871$	−57.4641			−1.94710
$872$		0			0
$873$			12.6080i			0.426716i
$874$		0			0
$875$	−22.9473	+	2.88741i	−0.775760	+	0.0976123i
$876$		0			0
$877$	22.5865			0.762694			0.381347	−	0.924432i	$-0.375460\pi$
$877$	22.5865			0.762694			0.381347	+	0.924432i	$0.375460\pi$
$878$		0			0
$879$		−	6.22411i		−	0.209934i
$880$		0			0
$881$			48.7337i			1.64188i	0.571015	+	0.820940i	$0.306550\pi$
$881$			48.7337i			1.64188i	−0.571015	+	0.820940i	$0.693450\pi$
$882$		0			0
$883$		−	30.9872i		−	1.04280i	−0.853312	−	0.521401i	$-0.825410\pi$
$883$		−	30.9872i		−	1.04280i	0.853312	−	0.521401i	$-0.174590\pi$
$884$		0			0
$885$			6.87491i			0.231098i
$886$		0			0
$887$	6.53593			0.219455			0.109728	−	0.993962i	$-0.465002\pi$
$887$	6.53593			0.219455			0.109728	+	0.993962i	$0.465002\pi$
$888$		0			0
$889$	1.32348	−	0.166530i	0.0443880	−	0.00558525i
$890$		0			0
$891$		−	0.531427i		−	0.0178035i
$892$		0			0
$893$	−1.15325			−0.0385921
$894$		0			0
$895$	8.21000			0.274430
$896$		0			0
$897$	−25.2298			−0.842398
$898$		0			0
$899$	0.844200			0.0281556
$900$		0			0
$901$		−	45.4777i		−	1.51508i
$902$		0			0
$903$	−28.9935	+	3.64819i	−0.964843	+	0.121404i
$904$		0			0
$905$	32.6315			1.08471
$906$		0			0
$907$		−	41.4719i		−	1.37705i	−0.725212	−	0.688526i	$-0.758258\pi$
$907$		−	41.4719i		−	1.37705i	0.725212	−	0.688526i	$-0.241742\pi$
$908$		0			0
$909$		−	4.74567i		−	0.157404i
$910$		0			0
$911$		−	31.5421i		−	1.04504i	−0.852628	−	0.522519i	$-0.824993\pi$
$911$		−	31.5421i		−	1.04504i	0.852628	−	0.522519i	$-0.175007\pi$
$912$		0			0
$913$		−	11.8395i		−	0.391830i
$914$		0			0
$915$	34.8871			1.15333
$916$		0			0
$917$	6.67416	+	53.0420i	0.220400	+	1.75160i
$918$		0			0
$919$			50.6805i			1.67179i	0.548886	+	0.835897i	$0.315052\pi$
$919$			50.6805i			1.67179i	−0.548886	+	0.835897i	$0.684948\pi$
$920$		0			0
$921$	33.7572			1.11234
$922$		0			0
$923$	10.3181			0.339623
$924$		0			0
$925$	7.66326			0.251967
$926$		0			0
$927$	−15.0857			−0.495479
$928$		0			0
$929$		−	15.4072i		−	0.505494i	−0.967532	−	0.252747i	$-0.918666\pi$
$929$		−	15.4072i		−	0.505494i	0.967532	−	0.252747i	$-0.0813340\pi$
$930$		0			0
$931$	−11.2980	+	2.88895i	−0.370278	+	0.0946816i
$932$		0			0
$933$	30.5203			0.999189
$934$		0			0
$935$		−	17.7183i		−	0.579451i
$936$		0			0
$937$		−	35.7306i		−	1.16727i	−0.812017	−	0.583633i	$-0.801631\pi$
$937$		−	35.7306i		−	1.16727i	0.812017	−	0.583633i	$-0.198369\pi$
$938$		0			0
$939$			18.1610i			0.592663i
$940$		0			0
$941$		−	37.9057i		−	1.23569i	−0.786300	−	0.617845i	$-0.788006\pi$
$941$		−	37.9057i		−	1.23569i	0.786300	−	0.617845i	$-0.211994\pi$
$942$		0			0
$943$	−4.75570			−0.154867
$944$		0			0
$945$	34.3599	−	4.32344i	1.11773	−	0.140641i
$946$		0			0
$947$		−	54.7739i		−	1.77991i	−0.456047	−	0.889956i	$-0.650735\pi$
$947$		−	54.7739i		−	1.77991i	0.456047	−	0.889956i	$-0.349265\pi$
$948$		0			0
$949$	−1.69851			−0.0551360
$950$		0			0
$951$	−26.8932			−0.872072
$952$		0			0
$953$	8.28906			0.268509			0.134255	−	0.990947i	$-0.457136\pi$
$953$	8.28906			0.268509			0.134255	+	0.990947i	$0.457136\pi$
$954$		0			0
$955$	66.1807			2.14156
$956$		0			0
$957$			7.05162i			0.227947i
$958$		0			0
$959$	3.39750	+	27.0012i	0.109711	+	0.871914i
$960$		0			0
$961$	−30.9847			−0.999506
$962$		0			0
$963$		−	14.6160i		−	0.470993i
$964$		0			0
$965$		−	12.4945i		−	0.402211i
$966$		0			0
$967$		−	31.8644i		−	1.02469i	−0.858780	−	0.512345i	$-0.828777\pi$
$967$		−	31.8644i		−	1.02469i	0.858780	−	0.512345i	$-0.171223\pi$
$968$		0			0
$969$		−	11.8749i		−	0.381477i
$970$		0			0
$971$	−6.49583			−0.208461			−0.104231	−	0.994553i	$-0.533238\pi$
$971$	−6.49583			−0.208461			−0.104231	+	0.994553i	$0.533238\pi$
$972$		0			0
$973$	3.64739	+	28.9871i	0.116930	+	0.929284i
$974$		0			0
$975$		−	9.32821i		−	0.298742i
$976$		0			0
$977$	59.0021			1.88764			0.943822	−	0.330455i	$-0.107202\pi$
$977$	59.0021			1.88764			0.943822	+	0.330455i	$0.107202\pi$
$978$		0			0
$979$	−1.88568			−0.0602667
$980$		0			0
$981$	−6.19702			−0.197856
$982$		0			0
$983$	−33.3603			−1.06403			−0.532014	−	0.846735i	$-0.678565\pi$
$983$	−33.3603			−1.06403			−0.532014	+	0.846735i	$0.678565\pi$
$984$		0			0
$985$			53.0167i			1.68925i
$986$		0			0
$987$	−0.236256	−	1.87762i	−0.00752012	−	0.0597652i
$988$		0			0
$989$	−46.1543			−1.46762
$990$		0			0
$991$			6.80745i			0.216246i	0.994138	+	0.108123i	$0.0344840\pi$
$991$			6.80745i			0.216246i	−0.994138	+	0.108123i	$0.965516\pi$
$992$		0			0
$993$		−	24.1593i		−	0.766674i
$994$		0			0
$995$		−	68.2355i		−	2.16321i
$996$		0			0
$997$		−	32.2250i		−	1.02058i	−0.860003	−	0.510288i	$-0.829539\pi$
$997$		−	32.2250i		−	1.02058i	0.860003	−	0.510288i	$-0.170461\pi$
$998$		0			0
$999$	24.4646			0.774026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.j.b.111.17	yes	24
4.3	odd	2	inner	1232.2.j.b.111.7	✓	24
7.6	odd	2	inner	1232.2.j.b.111.8	yes	24
28.27	even	2	inner	1232.2.j.b.111.18	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1232.2.j.b.111.7	✓	24	4.3	odd	2	inner
1232.2.j.b.111.8	yes	24	7.6	odd	2	inner
1232.2.j.b.111.17	yes	24	1.1	even	1	trivial
1232.2.j.b.111.18	yes	24	28.27	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.03324 q^{3} +2.56834i q^{5} +(0.330305 + 2.62505i) q^{7} -1.93241 q^{9} -1.00000i q^{11} +5.65534i q^{13} +2.65372i q^{15} -6.89874i q^{17} +1.66593 q^{19} +(0.341285 + 2.71232i) q^{21} +4.31770i q^{23} -1.59638 q^{25} -5.09638 q^{27} -6.82475 q^{29} -0.123697 q^{31} -1.03324i q^{33} +(-6.74203 + 0.848336i) q^{35} -4.80039 q^{37} +5.84334i q^{39} +1.10144i q^{41} +10.6896i q^{43} -4.96309i q^{45} -0.692255 q^{47} +(-6.78180 + 1.73413i) q^{49} -7.12808i q^{51} +6.59218 q^{53} +2.56834 q^{55} +1.72131 q^{57} +2.59067 q^{59} -13.1465i q^{61} +(-0.638284 - 5.07267i) q^{63} -14.5249 q^{65} +10.1610i q^{67} +4.46123i q^{69} -1.82448i q^{71} +0.300337i q^{73} -1.64945 q^{75} +(2.62505 - 0.330305i) q^{77} +10.8148i q^{79} +0.531427 q^{81} +11.8395 q^{83} +17.7183 q^{85} -7.05162 q^{87} -1.88568i q^{89} +(-14.8456 + 1.86799i) q^{91} -0.127809 q^{93} +4.27869i q^{95} -6.52449i q^{97} +1.93241i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{9} + 20 q^{21} - 56 q^{29} + 40 q^{37} - 8 q^{49} + 16 q^{53} + 8 q^{57} - 96 q^{65} - 4 q^{77} + 40 q^{81} - 32 q^{85} + 8 q^{93}+O(q^{100}) \) 24 * q + 16 * q^9 + 20 * q^21 - 56 * q^29 + 40 * q^37 - 8 * q^49 + 16 * q^53 + 8 * q^57 - 96 * q^65 - 4 * q^77 + 40 * q^81 - 32 * q^85 + 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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