LMFDB - Embedded newform 1472.2.h.c.735.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,2,Mod(735,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.735"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			735.5
Character	$\chi$	$=$	1472.735
Dual form			1472.2.h.c.735.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11141 q^{3} -2.66434 q^{5} +1.46622 q^{7} +1.45804 q^{9} +1.58217i q^{11} +3.11316i q^{13} +5.62551 q^{15} +1.72966i q^{17} -3.35877i q^{19} -3.09579 q^{21} +(4.35133 + 2.01641i) q^{23} +2.09870 q^{25} +3.25570 q^{27} +5.01617i q^{29} -3.47445i q^{31} -3.34060i q^{33} -3.90651 q^{35} -6.45456 q^{37} -6.57315i q^{39} -1.73936 q^{41} -2.59580i q^{43} -3.88472 q^{45} +0.722956i q^{47} -4.85019 q^{49} -3.65201i q^{51} +5.13424 q^{53} -4.21543i q^{55} +7.09173i q^{57} -0.758714 q^{59} -7.39852 q^{61} +2.13781 q^{63} -8.29451i q^{65} -16.0297i q^{67} +(-9.18744 - 4.25745i) q^{69} +5.65707i q^{71} -11.5896 q^{73} -4.43122 q^{75} +2.31981i q^{77} -7.61860 q^{79} -11.2482 q^{81} -14.3913i q^{83} -4.60840i q^{85} -10.5912i q^{87} -7.61860i q^{89} +4.56458i q^{91} +7.33598i q^{93} +8.94890i q^{95} -14.5916i q^{97} +2.30687i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 40 q^{9} + 32 q^{25} + 8 q^{41} + 48 q^{49} - 104 q^{73} + 112 q^{81}+O(q^{100}) $ 32 * q + 40 * q^9 + 32 * q^25 + 8 * q^41 + 48 * q^49 - 104 * q^73 + 112 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$.

$n$	$645$	$833$	$1151$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−2.11141			−1.21902			−0.609511	−	0.792778i	$-0.708634\pi$
$3$	−2.11141			−1.21902			−0.609511	+	0.792778i	$0.708634\pi$
$4$		0			0
$5$	−2.66434			−1.19153			−0.595764	−	0.803159i	$-0.703151\pi$
$5$	−2.66434			−1.19153			−0.595764	+	0.803159i	$0.703151\pi$
$6$		0			0
$7$	1.46622			0.554180			0.277090	−	0.960844i	$-0.410630\pi$
$7$	1.46622			0.554180			0.277090	+	0.960844i	$0.410630\pi$
$8$		0			0
$9$	1.45804			0.486014
$10$		0			0
$11$			1.58217i			0.477041i	0.971137	+	0.238521i	$0.0766625\pi$
$11$			1.58217i			0.477041i	−0.971137	+	0.238521i	$0.923338\pi$
$12$		0			0
$13$			3.11316i			0.863435i	0.902009	+	0.431718i	$0.142092\pi$
$13$			3.11316i			0.863435i	−0.902009	+	0.431718i	$0.857908\pi$
$14$		0			0
$15$	5.62551			1.45250
$16$		0			0
$17$			1.72966i			0.419504i	0.977755	+	0.209752i	$0.0672656\pi$
$17$			1.72966i			0.419504i	−0.977755	+	0.209752i	$0.932734\pi$
$18$		0			0
$19$		−	3.35877i		−	0.770554i	−0.922801	−	0.385277i	$-0.874106\pi$
$19$		−	3.35877i		−	0.770554i	0.922801	−	0.385277i	$-0.125894\pi$
$20$		0			0
$21$	−3.09579			−0.675557
$22$		0			0
$23$	4.35133	+	2.01641i	0.907316	+	0.420450i
$23$	4.35133	+	2.01641i	0.907316	+	0.420450i
$24$		0			0
$25$	2.09870			0.419740
$26$		0			0
$27$	3.25570			0.626560
$28$		0			0
$29$			5.01617i			0.931479i	0.884922	+	0.465739i	$0.154212\pi$
$29$			5.01617i			0.931479i	−0.884922	+	0.465739i	$0.845788\pi$
$30$		0			0
$31$		−	3.47445i		−	0.624029i	−0.950077	−	0.312015i	$-0.898996\pi$
$31$		−	3.47445i		−	0.624029i	0.950077	−	0.312015i	$-0.101004\pi$
$32$		0			0
$33$		−	3.34060i		−	0.581524i
$34$		0			0
$35$	−3.90651			−0.660321
$36$		0			0
$37$	−6.45456			−1.06112			−0.530562	−	0.847646i	$-0.678019\pi$
$37$	−6.45456			−1.06112			−0.530562	+	0.847646i	$0.678019\pi$
$38$		0			0
$39$		−	6.57315i		−	1.05255i
$40$		0			0
$41$	−1.73936			−0.271643			−0.135821	−	0.990733i	$-0.543367\pi$
$41$	−1.73936			−0.271643			−0.135821	+	0.990733i	$0.543367\pi$
$42$		0			0
$43$		−	2.59580i		−	0.395855i	−0.980217	−	0.197928i	$-0.936579\pi$
$43$		−	2.59580i		−	0.395855i	0.980217	−	0.197928i	$-0.0634211\pi$
$44$		0			0
$45$	−3.88472			−0.579100
$46$		0			0
$47$			0.722956i			0.105454i	0.998609	+	0.0527270i	$0.0167913\pi$
$47$			0.722956i			0.105454i	−0.998609	+	0.0527270i	$0.983209\pi$
$48$		0			0
$49$	−4.85019			−0.692885
$50$		0			0
$51$		−	3.65201i		−	0.511384i
$52$		0			0
$53$	5.13424			0.705243			0.352621	−	0.935766i	$-0.385290\pi$
$53$	5.13424			0.705243			0.352621	+	0.935766i	$0.385290\pi$
$54$		0			0
$55$		−	4.21543i		−	0.568408i
$56$		0			0
$57$			7.09173i			0.939322i
$58$		0			0
$59$	−0.758714			−0.0987761			−0.0493881	−	0.998780i	$-0.515727\pi$
$59$	−0.758714			−0.0987761			−0.0493881	+	0.998780i	$0.515727\pi$
$60$		0			0
$61$	−7.39852			−0.947284			−0.473642	−	0.880717i	$-0.657061\pi$
$61$	−7.39852			−0.947284			−0.473642	+	0.880717i	$0.657061\pi$
$62$		0			0
$63$	2.13781			0.269339
$64$		0			0
$65$		−	8.29451i		−	1.02881i
$66$		0			0
$67$		−	16.0297i		−	1.95834i	−0.203030	−	0.979172i	$-0.565079\pi$
$67$		−	16.0297i		−	1.95834i	0.203030	−	0.979172i	$-0.434921\pi$
$68$		0			0
$69$	−9.18744	−	4.25745i	−1.10604	−	0.512537i
$70$		0			0
$71$			5.65707i			0.671370i	0.941974	+	0.335685i	$0.108968\pi$
$71$			5.65707i			0.671370i	−0.941974	+	0.335685i	$0.891032\pi$
$72$		0			0
$73$	−11.5896			−1.35645			−0.678227	−	0.734852i	$-0.737252\pi$
$73$	−11.5896			−1.35645			−0.678227	+	0.734852i	$0.737252\pi$
$74$		0			0
$75$	−4.43122			−0.511673
$76$		0			0
$77$			2.31981i			0.264367i
$78$		0			0
$79$	−7.61860			−0.857159			−0.428580	−	0.903504i	$-0.640986\pi$
$79$	−7.61860			−0.857159			−0.428580	+	0.903504i	$0.640986\pi$
$80$		0			0
$81$	−11.2482			−1.24980
$82$		0			0
$83$		−	14.3913i		−	1.57966i	−0.613328	−	0.789828i	$-0.710170\pi$
$83$		−	14.3913i		−	1.57966i	0.613328	−	0.789828i	$-0.289830\pi$
$84$		0			0
$85$		−	4.60840i		−	0.499851i
$86$		0			0
$87$		−	10.5912i		−	1.13549i
$88$		0			0
$89$		−	7.61860i		−	0.807570i	−0.914854	−	0.403785i	$-0.867694\pi$
$89$		−	7.61860i		−	0.807570i	0.914854	−	0.403785i	$-0.132306\pi$
$90$		0			0
$91$			4.56458i			0.478498i
$92$		0			0
$93$			7.33598i			0.760705i
$94$		0			0
$95$			8.94890i			0.918137i
$96$		0			0
$97$		−	14.5916i		−	1.48155i	−0.671751	−	0.740777i	$-0.734458\pi$
$97$		−	14.5916i		−	1.48155i	0.671751	−	0.740777i	$-0.265542\pi$
$98$		0			0
$99$			2.30687i			0.231849i
$100$		0			0
$101$		−	0.316303i		−	0.0314733i	−0.999876	−	0.0157367i	$-0.994991\pi$
$101$		−	0.316303i		−	0.0314733i	0.999876	−	0.0157367i	$-0.00500934\pi$
$102$		0			0
$103$	−6.94701			−0.684509			−0.342255	−	0.939607i	$-0.611191\pi$
$103$	−6.94701			−0.684509			−0.342255	+	0.939607i	$0.611191\pi$
$104$		0			0
$105$	8.24824			0.804946
$106$		0			0
$107$		−	0.680989i		−	0.0658337i	−0.999458	−	0.0329169i	$-0.989520\pi$
$107$		−	0.680989i		−	0.0658337i	0.999458	−	0.0329169i	$-0.0104797\pi$
$108$		0			0
$109$	2.48334			0.237861			0.118931	−	0.992903i	$-0.462053\pi$
$109$	2.48334			0.237861			0.118931	+	0.992903i	$0.462053\pi$
$110$		0			0
$111$	13.6282			1.29353
$112$		0			0
$113$		−	18.7509i		−	1.76394i	−0.471310	−	0.881968i	$-0.656219\pi$
$113$		−	18.7509i		−	1.76394i	0.471310	−	0.881968i	$-0.343781\pi$
$114$		0			0
$115$	−11.5934	−	5.37239i	−1.08109	−	0.500978i
$116$		0			0
$117$			4.53912i			0.419642i
$118$		0			0
$119$			2.53606i			0.232480i
$120$		0			0
$121$	8.49675			0.772432
$122$		0			0
$123$	3.67250			0.331138
$124$		0			0
$125$	7.73004			0.691396
$126$		0			0
$127$		−	4.03709i		−	0.358234i	−0.983828	−	0.179117i	$-0.942676\pi$
$127$		−	4.03709i		−	0.358234i	0.983828	−	0.179117i	$-0.0573240\pi$
$128$		0			0
$129$			5.48079i			0.482556i
$130$		0			0
$131$	−8.72331			−0.762159			−0.381079	−	0.924542i	$-0.624448\pi$
$131$	−8.72331			−0.762159			−0.381079	+	0.924542i	$0.624448\pi$
$132$		0			0
$133$		−	4.92470i		−	0.427026i
$134$		0			0
$135$	−8.67429			−0.746564
$136$		0			0
$137$			12.0673i			1.03098i	0.856895	+	0.515490i	$0.172390\pi$
$137$			12.0673i			1.03098i	−0.856895	+	0.515490i	$0.827610\pi$
$138$		0			0
$139$	−4.01442			−0.340498			−0.170249	−	0.985401i	$-0.554457\pi$
$139$	−4.01442			−0.340498			−0.170249	+	0.985401i	$0.554457\pi$
$140$		0			0
$141$		−	1.52646i		−	0.128551i
$142$		0			0
$143$	−4.92554			−0.411894
$144$		0			0
$145$		−	13.3648i		−	1.10988i
$146$		0			0
$147$	10.2407			0.844642
$148$		0			0
$149$	10.3133			0.844900			0.422450	−	0.906386i	$-0.361170\pi$
$149$	10.3133			0.844900			0.422450	+	0.906386i	$0.361170\pi$
$150$		0			0
$151$			10.2197i			0.831668i	0.909441	+	0.415834i	$0.136510\pi$
$151$			10.2197i			0.831668i	−0.909441	+	0.415834i	$0.863490\pi$
$152$		0			0
$153$			2.52191i			0.203885i
$154$		0			0
$155$			9.25711i			0.743549i
$156$		0			0
$157$	−13.7666			−1.09869			−0.549347	−	0.835595i	$-0.685123\pi$
$157$	−13.7666			−1.09869			−0.549347	+	0.835595i	$0.685123\pi$
$158$		0			0
$159$	−10.8405			−0.859706
$160$		0			0
$161$	6.38002	+	2.95650i	0.502816	+	0.233005i
$162$		0			0
$163$	−14.1472			−1.10810			−0.554049	−	0.832484i	$-0.686918\pi$
$163$	−14.1472			−1.10810			−0.554049	+	0.832484i	$0.686918\pi$
$164$		0			0
$165$			8.90049i			0.692902i
$166$		0			0
$167$			15.5443i			1.20286i	0.798926	+	0.601429i	$0.205402\pi$
$167$			15.5443i			1.20286i	−0.798926	+	0.601429i	$0.794598\pi$
$168$		0			0
$169$	3.30824			0.254480
$170$		0			0
$171$		−	4.89722i		−	0.374500i
$172$		0			0
$173$		−	13.9388i		−	1.05975i	−0.848075	−	0.529875i	$-0.822239\pi$
$173$		−	13.9388i		−	1.05975i	0.848075	−	0.529875i	$-0.177761\pi$
$174$		0			0
$175$	3.07716			0.232612
$176$		0			0
$177$	1.60195			0.120410
$178$		0			0
$179$	7.53535			0.563218			0.281609	−	0.959529i	$-0.409132\pi$
$179$	7.53535			0.563218			0.281609	+	0.959529i	$0.409132\pi$
$180$		0			0
$181$	5.82867			0.433242			0.216621	−	0.976256i	$-0.430496\pi$
$181$	5.82867			0.433242			0.216621	+	0.976256i	$0.430496\pi$
$182$		0			0
$183$	15.6213			1.15476
$184$		0			0
$185$	17.1971			1.26436
$186$		0			0
$187$	−2.73661			−0.200121
$188$		0			0
$189$	4.77358			0.347227
$190$		0			0
$191$	−9.37426			−0.678298			−0.339149	−	0.940733i	$-0.610139\pi$
$191$	−9.37426			−0.678298			−0.339149	+	0.940733i	$0.610139\pi$
$192$		0			0
$193$	−26.2326			−1.88826			−0.944132	−	0.329567i	$-0.893097\pi$
$193$	−26.2326			−1.88826			−0.944132	+	0.329567i	$0.893097\pi$
$194$		0			0
$195$			17.5131i			1.25414i
$196$		0			0
$197$			25.3147i			1.80360i	0.432156	+	0.901799i	$0.357753\pi$
$197$			25.3147i			1.80360i	−0.432156	+	0.901799i	$0.642247\pi$
$198$		0			0
$199$	15.3819			1.09040			0.545198	−	0.838308i	$-0.316455\pi$
$199$	15.3819			1.09040			0.545198	+	0.838308i	$0.316455\pi$
$200$		0			0
$201$			33.8453i			2.38726i
$202$		0			0
$203$			7.35481i			0.516207i
$204$		0			0
$205$	4.63425			0.323670
$206$		0			0
$207$	6.34443	+	2.94000i	0.440968	+	0.204344i
$208$		0			0
$209$	5.31413			0.367586
$210$		0			0
$211$	16.6186			1.14407			0.572037	−	0.820228i	$-0.306153\pi$
$211$	16.6186			1.14407			0.572037	+	0.820228i	$0.306153\pi$
$212$		0			0
$213$		−	11.9444i		−	0.818415i
$214$		0			0
$215$			6.91608i			0.471673i
$216$		0			0
$217$		−	5.09431i		−	0.345824i
$218$		0			0
$219$	24.4703			1.65355
$220$		0			0
$221$	−5.38470			−0.362214
$222$		0			0
$223$		−	3.06913i		−	0.205524i	−0.994706	−	0.102762i	$-0.967232\pi$
$223$		−	3.06913i		−	0.205524i	0.994706	−	0.102762i	$-0.0327680\pi$
$224$		0			0
$225$	3.06000			0.204000
$226$		0			0
$227$			9.68856i			0.643052i	0.946901	+	0.321526i	$0.104196\pi$
$227$			9.68856i			0.643052i	−0.946901	+	0.321526i	$0.895804\pi$
$228$		0			0
$229$	−7.81202			−0.516233			−0.258116	−	0.966114i	$-0.583102\pi$
$229$	−7.81202			−0.516233			−0.258116	+	0.966114i	$0.583102\pi$
$230$		0			0
$231$		−	4.89806i		−	0.322269i
$232$		0			0
$233$	10.7063			0.701392			0.350696	−	0.936489i	$-0.385945\pi$
$233$	10.7063			0.701392			0.350696	+	0.936489i	$0.385945\pi$
$234$		0			0
$235$		−	1.92620i		−	0.125651i
$236$		0			0
$237$	16.0860			1.04490
$238$		0			0
$239$		−	27.2014i		−	1.75951i	−0.475425	−	0.879756i	$-0.657706\pi$
$239$		−	27.2014i		−	1.75951i	0.475425	−	0.879756i	$-0.342294\pi$
$240$		0			0
$241$			3.05116i			0.196543i	0.995160	+	0.0982713i	$0.0313313\pi$
$241$			3.05116i			0.196543i	−0.995160	+	0.0982713i	$0.968669\pi$
$242$		0			0
$243$	13.9825			0.896979
$244$		0			0
$245$	12.9226			0.825592
$246$		0			0
$247$	10.4564			0.665324
$248$		0			0
$249$			30.3860i			1.92563i
$250$		0			0
$251$			2.20805i			0.139371i	0.997569	+	0.0696856i	$0.0221996\pi$
$251$			2.20805i			0.139371i	−0.997569	+	0.0696856i	$0.977800\pi$
$252$		0			0
$253$	−3.19029	+	6.88454i	−0.200572	+	0.432827i
$254$		0			0
$255$			9.73020i			0.609329i
$256$		0			0
$257$	−15.7902			−0.984965			−0.492483	−	0.870322i	$-0.663911\pi$
$257$	−15.7902			−0.984965			−0.492483	+	0.870322i	$0.663911\pi$
$258$		0			0
$259$	−9.46382			−0.588053
$260$		0			0
$261$			7.31378i			0.452712i
$262$		0			0
$263$	−22.0915			−1.36222			−0.681110	−	0.732181i	$-0.738502\pi$
$263$	−22.0915			−1.36222			−0.681110	+	0.732181i	$0.738502\pi$
$264$		0			0
$265$	−13.6794			−0.840317
$266$		0			0
$267$			16.0860i			0.984445i
$268$		0			0
$269$		−	24.7133i		−	1.50680i	−0.657564	−	0.753399i	$-0.728413\pi$
$269$		−	24.7133i		−	1.50680i	0.657564	−	0.753399i	$-0.271587\pi$
$270$		0			0
$271$		−	27.7746i		−	1.68718i	−0.536984	−	0.843592i	$-0.680437\pi$
$271$		−	27.7746i		−	1.68718i	0.536984	−	0.843592i	$-0.319563\pi$
$272$		0			0
$273$		−	9.63770i		−	0.583300i
$274$		0			0
$275$			3.32050i			0.200233i
$276$		0			0
$277$		−	5.01617i		−	0.301392i	−0.988580	−	0.150696i	$-0.951848\pi$
$277$		−	5.01617i		−	0.301392i	0.988580	−	0.150696i	$-0.0481515\pi$
$278$		0			0
$279$		−	5.06589i		−	0.303287i
$280$		0			0
$281$			31.1806i			1.86008i	0.367457	+	0.930040i	$0.380228\pi$
$281$			31.1806i			1.86008i	−0.367457	+	0.930040i	$0.619772\pi$
$282$		0			0
$283$		−	11.5640i		−	0.687407i	−0.939078	−	0.343703i	$-0.888319\pi$
$283$		−	11.5640i		−	0.687407i	0.939078	−	0.343703i	$-0.111681\pi$
$284$		0			0
$285$		−	18.8948i		−	1.11923i
$286$		0			0
$287$	−2.55029			−0.150539
$288$		0			0
$289$	14.0083			0.824017
$290$		0			0
$291$			30.8088i			1.80605i
$292$		0			0
$293$	−12.2384			−0.714975			−0.357487	−	0.933918i	$-0.616367\pi$
$293$	−12.2384			−0.714975			−0.357487	+	0.933918i	$0.616367\pi$
$294$		0			0
$295$	2.02147			0.117695
$296$		0			0
$297$			5.15106i			0.298895i
$298$		0			0
$299$	−6.27739	+	13.5464i	−0.363031	+	0.783408i
$300$		0			0
$301$		−	3.80602i		−	0.219375i
$302$		0			0
$303$			0.667845i			0.0383667i
$304$		0			0
$305$	19.7122			1.12872
$306$		0			0
$307$	12.6116			0.719783			0.359891	−	0.932994i	$-0.382814\pi$
$307$	12.6116			0.719783			0.359891	+	0.932994i	$0.382814\pi$
$308$		0			0
$309$	14.6680			0.834431
$310$		0			0
$311$		−	23.9349i		−	1.35722i	−0.734498	−	0.678611i	$-0.762582\pi$
$311$		−	23.9349i		−	1.35722i	0.734498	−	0.678611i	$-0.237418\pi$
$312$		0			0
$313$		−	18.4374i		−	1.04214i	−0.853513	−	0.521072i	$-0.825532\pi$
$313$		−	18.4374i		−	1.04214i	0.853513	−	0.521072i	$-0.174468\pi$
$314$		0			0
$315$	−5.69586			−0.320925
$316$		0			0
$317$		−	28.1428i		−	1.58066i	−0.612684	−	0.790328i	$-0.709910\pi$
$317$		−	28.1428i		−	1.58066i	0.612684	−	0.790328i	$-0.290090\pi$
$318$		0			0
$319$	−7.93641			−0.444354
$320$		0			0
$321$			1.43785i			0.0802527i
$322$		0			0
$323$	5.80952			0.323250
$324$		0			0
$325$			6.53359i			0.362419i
$326$		0			0
$327$	−5.24335			−0.289958
$328$		0			0
$329$			1.06001i			0.0584405i
$330$		0			0
$331$	6.23373			0.342637			0.171318	−	0.985216i	$-0.445197\pi$
$331$	6.23373			0.342637			0.171318	+	0.985216i	$0.445197\pi$
$332$		0			0
$333$	−9.41102			−0.515721
$334$		0			0
$335$			42.7087i			2.33342i
$336$		0			0
$337$			28.0992i			1.53066i	0.643639	+	0.765329i	$0.277424\pi$
$337$			28.0992i			1.53066i	−0.643639	+	0.765329i	$0.722576\pi$
$338$		0			0
$339$			39.5908i			2.15028i
$340$		0			0
$341$	5.49716			0.297688
$342$		0			0
$343$	−17.3750			−0.938163
$344$		0			0
$345$	24.4785	+	11.3433i	1.31788	+	0.610703i
$346$		0			0
$347$	9.79092			0.525604			0.262802	−	0.964850i	$-0.415353\pi$
$347$	9.79092			0.525604			0.262802	+	0.964850i	$0.415353\pi$
$348$		0			0
$349$			6.93723i			0.371341i	0.982612	+	0.185671i	$0.0594458\pi$
$349$			6.93723i			0.371341i	−0.982612	+	0.185671i	$0.940554\pi$
$350$		0			0
$351$			10.1355i			0.540994i
$352$		0			0
$353$	−19.6224			−1.04439			−0.522197	−	0.852825i	$-0.674887\pi$
$353$	−19.6224			−1.04439			−0.522197	+	0.852825i	$0.674887\pi$
$354$		0			0
$355$		−	15.0723i		−	0.799957i
$356$		0			0
$357$		−	5.35466i		−	0.283399i
$358$		0			0
$359$	−9.92951			−0.524059			−0.262030	−	0.965060i	$-0.584392\pi$
$359$	−9.92951			−0.524059			−0.262030	+	0.965060i	$0.584392\pi$
$360$		0			0
$361$	7.71868			0.406246
$362$		0			0
$363$	−17.9401			−0.941611
$364$		0			0
$365$	30.8785			1.61625
$366$		0			0
$367$	23.0308			1.20220			0.601100	−	0.799174i	$-0.294729\pi$
$367$	23.0308			1.20220			0.601100	+	0.799174i	$0.294729\pi$
$368$		0			0
$369$	−2.53606			−0.132022
$370$		0			0
$371$	7.52794			0.390831
$372$		0			0
$373$	19.7874			1.02456			0.512278	−	0.858820i	$-0.328802\pi$
$373$	19.7874			1.02456			0.512278	+	0.858820i	$0.328802\pi$
$374$		0			0
$375$	−16.3213			−0.842827
$376$		0			0
$377$	−15.6161			−0.804272
$378$		0			0
$379$		−	14.4027i		−	0.739819i	−0.929068	−	0.369910i	$-0.879389\pi$
$379$		−	14.4027i		−	0.739819i	0.929068	−	0.369910i	$-0.120611\pi$
$380$		0			0
$381$			8.52394i			0.436694i
$382$		0			0
$383$	−20.0483			−1.02442			−0.512211	−	0.858860i	$-0.671174\pi$
$383$	−20.0483			−1.02442			−0.512211	+	0.858860i	$0.671174\pi$
$384$		0			0
$385$		−	6.18075i		−	0.315000i
$386$		0			0
$387$		−	3.78478i		−	0.192391i
$388$		0			0
$389$	25.3993			1.28780			0.643898	−	0.765111i	$-0.277316\pi$
$389$	25.3993			1.28780			0.643898	+	0.765111i	$0.277316\pi$
$390$		0			0
$391$	−3.48769	+	7.52632i	−0.176380	+	0.380622i
$392$		0			0
$393$	18.4185			0.929088
$394$		0			0
$395$	20.2985			1.02133
$396$		0			0
$397$			21.1050i			1.05923i	0.848238	+	0.529616i	$0.177664\pi$
$397$			21.1050i			1.05923i	−0.848238	+	0.529616i	$0.822336\pi$
$398$		0			0
$399$			10.3980i			0.520553i
$400$		0			0
$401$		−	25.5268i		−	1.27475i	−0.770556	−	0.637373i	$-0.780021\pi$
$401$		−	25.5268i		−	1.27475i	0.770556	−	0.637373i	$-0.219979\pi$
$402$		0			0
$403$	10.8165			0.538809
$404$		0			0
$405$	29.9691			1.48918
$406$		0			0
$407$		−	10.2122i		−	0.506199i
$408$		0			0
$409$	−14.0204			−0.693265			−0.346632	−	0.938001i	$-0.612675\pi$
$409$	−14.0204			−0.693265			−0.346632	+	0.938001i	$0.612675\pi$
$410$		0			0
$411$		−	25.4790i		−	1.25679i
$412$		0			0
$413$	−1.11244			−0.0547397
$414$		0			0
$415$			38.3434i			1.88221i
$416$		0			0
$417$	8.47607			0.415075
$418$		0			0
$419$			28.0377i			1.36973i	0.728670	+	0.684865i	$0.240139\pi$
$419$			28.0377i			1.36973i	−0.728670	+	0.684865i	$0.759861\pi$
$420$		0			0
$421$	−33.8484			−1.64967			−0.824835	−	0.565373i	$-0.808732\pi$
$421$	−33.8484			−1.64967			−0.824835	+	0.565373i	$0.808732\pi$
$422$		0			0
$423$			1.05410i			0.0512521i
$424$		0			0
$425$			3.63004i			0.176083i
$426$		0			0
$427$	−10.8479			−0.524966
$428$		0			0
$429$	10.3998			0.502108
$430$		0			0
$431$	22.0915			1.06411			0.532055	−	0.846710i	$-0.321420\pi$
$431$	22.0915			1.06411			0.532055	+	0.846710i	$0.321420\pi$
$432$		0			0
$433$		−	21.7780i		−	1.04658i	−0.852153	−	0.523292i	$-0.824704\pi$
$433$		−	21.7780i		−	1.04658i	0.852153	−	0.523292i	$-0.175296\pi$
$434$		0			0
$435$			28.2185i			1.35297i
$436$		0			0
$437$	6.77264	−	14.6151i	0.323979	−	0.699136i
$438$		0			0
$439$			27.7194i			1.32298i	0.749955	+	0.661489i	$0.230075\pi$
$439$			27.7194i			1.32298i	−0.749955	+	0.661489i	$0.769925\pi$
$440$		0			0
$441$	−7.07179			−0.336752
$442$		0			0
$443$	−39.1547			−1.86029			−0.930147	−	0.367187i	$-0.880321\pi$
$443$	−39.1547			−1.86029			−0.930147	+	0.367187i	$0.880321\pi$
$444$		0			0
$445$			20.2985i			0.962243i
$446$		0			0
$447$	−21.7756			−1.02995
$448$		0			0
$449$	−20.0742			−0.947359			−0.473679	−	0.880697i	$-0.657074\pi$
$449$	−20.0742			−0.947359			−0.473679	+	0.880697i	$0.657074\pi$
$450$		0			0
$451$		−	2.75196i		−	0.129585i
$452$		0			0
$453$		−	21.5780i		−	1.01382i
$454$		0			0
$455$		−	12.1616i		−	0.570144i
$456$		0			0
$457$		−	19.0187i		−	0.889655i	−0.895616	−	0.444828i	$-0.853265\pi$
$457$		−	19.0187i		−	0.889655i	0.895616	−	0.444828i	$-0.146735\pi$
$458$		0			0
$459$			5.63125i			0.262844i
$460$		0			0
$461$			11.7746i			0.548398i	0.961673	+	0.274199i	$0.0884126\pi$
$461$			11.7746i			0.548398i	−0.961673	+	0.274199i	$0.911587\pi$
$462$		0			0
$463$		−	5.32036i		−	0.247258i	−0.992328	−	0.123629i	$-0.960547\pi$
$463$		−	5.32036i		−	0.247258i	0.992328	−	0.123629i	$-0.0394533\pi$
$464$		0			0
$465$		−	19.5455i		−	0.906402i
$466$		0			0
$467$		−	16.3781i		−	0.757888i	−0.925420	−	0.378944i	$-0.876287\pi$
$467$		−	16.3781i		−	0.757888i	0.925420	−	0.378944i	$-0.123713\pi$
$468$		0			0
$469$		−	23.5032i		−	1.08528i
$470$		0			0
$471$	29.0669			1.33933
$472$		0			0
$473$	4.10698			0.188839
$474$		0			0
$475$		−	7.04905i		−	0.323433i
$476$		0			0
$477$	7.48594			0.342758
$478$		0			0
$479$	−14.5959			−0.666905			−0.333453	−	0.942767i	$-0.608214\pi$
$479$	−14.5959			−0.666905			−0.333453	+	0.942767i	$0.608214\pi$
$480$		0			0
$481$		−	20.0941i		−	0.916211i
$482$		0			0
$483$	−13.4708	−	6.24237i	−0.612944	−	0.284038i
$484$		0			0
$485$			38.8770i			1.76531i
$486$		0			0
$487$		−	12.6716i		−	0.574204i	−0.957900	−	0.287102i	$-0.907308\pi$
$487$		−	12.6716i		−	0.574204i	0.957900	−	0.287102i	$-0.0926919\pi$
$488$		0			0
$489$	29.8706			1.35080
$490$		0			0
$491$	19.8743			0.896916			0.448458	−	0.893804i	$-0.351973\pi$
$491$	19.8743			0.896916			0.448458	+	0.893804i	$0.351973\pi$
$492$		0			0
$493$	−8.67626			−0.390759
$494$		0			0
$495$		−	6.14627i		−	0.276254i
$496$		0			0
$497$			8.29451i			0.372060i
$498$		0			0
$499$	−18.7044			−0.837325			−0.418662	−	0.908142i	$-0.637501\pi$
$499$	−18.7044			−0.837325			−0.418662	+	0.908142i	$0.637501\pi$
$500$		0			0
$501$		−	32.8204i		−	1.46631i
$502$		0			0
$503$	−27.2828			−1.21648			−0.608241	−	0.793752i	$-0.708125\pi$
$503$	−27.2828			−1.21648			−0.608241	+	0.793752i	$0.708125\pi$
$504$		0			0
$505$			0.842739i			0.0375014i
$506$		0			0
$507$	−6.98503			−0.310216
$508$		0			0
$509$		−	24.1960i		−	1.07247i	−0.844069	−	0.536235i	$-0.819846\pi$
$509$		−	24.1960i		−	1.07247i	0.844069	−	0.536235i	$-0.180154\pi$
$510$		0			0
$511$	−16.9929			−0.751720
$512$		0			0
$513$		−	10.9351i		−	0.482798i
$514$		0			0
$515$	18.5092			0.815612
$516$		0			0
$517$	−1.14384			−0.0503059
$518$		0			0
$519$			29.4306i			1.29186i
$520$		0			0
$521$			0.794631i			0.0348134i	0.999848	+	0.0174067i	$0.00554100\pi$
$521$			0.794631i			0.0348134i	−0.999848	+	0.0174067i	$0.994459\pi$
$522$		0			0
$523$		−	37.9703i		−	1.66033i	−0.557520	−	0.830164i	$-0.688247\pi$
$523$		−	37.9703i		−	1.66033i	0.557520	−	0.830164i	$-0.311753\pi$
$524$		0			0
$525$	−6.49714			−0.283559
$526$		0			0
$527$	6.00961			0.261783
$528$		0			0
$529$	14.8682	+	17.5481i	0.646444	+	0.762961i
$530$		0			0
$531$	−1.10624			−0.0480066
$532$		0			0
$533$		−	5.41491i		−	0.234546i
$534$		0			0
$535$			1.81439i			0.0784428i
$536$		0			0
$537$	−15.9102			−0.686575
$538$		0			0
$539$		−	7.67381i		−	0.330535i
$540$		0			0
$541$			36.8184i			1.58295i	0.611203	+	0.791474i	$0.290686\pi$
$541$			36.8184i			1.58295i	−0.611203	+	0.791474i	$0.709314\pi$
$542$		0			0
$543$	−12.3067			−0.528131
$544$		0			0
$545$	−6.61647			−0.283418
$546$		0			0
$547$	−16.0683			−0.687031			−0.343516	−	0.939147i	$-0.611618\pi$
$547$	−16.0683			−0.687031			−0.343516	+	0.939147i	$0.611618\pi$
$548$		0			0
$549$	−10.7874			−0.460393
$550$		0			0
$551$	16.8481			0.717755
$552$		0			0
$553$	−11.1706			−0.475020
$554$		0			0
$555$	−36.3102			−1.54128
$556$		0			0
$557$	6.24669			0.264681			0.132340	−	0.991204i	$-0.457751\pi$
$557$	6.24669			0.264681			0.132340	+	0.991204i	$0.457751\pi$
$558$		0			0
$559$	8.08113			0.341795
$560$		0			0
$561$	5.77809			0.243951
$562$		0			0
$563$		−	9.92670i		−	0.418361i	−0.977877	−	0.209180i	$-0.932920\pi$
$563$		−	9.92670i		−	0.418361i	0.977877	−	0.209180i	$-0.0670796\pi$
$564$		0			0
$565$			49.9587i			2.10178i
$566$		0			0
$567$	−16.4924			−0.692616
$568$		0			0
$569$		−	1.82432i		−	0.0764795i	−0.999269	−	0.0382398i	$-0.987825\pi$
$569$		−	1.82432i		−	0.0764795i	0.999269	−	0.0382398i	$-0.0121751\pi$
$570$		0			0
$571$			12.2767i			0.513763i	0.966443	+	0.256881i	$0.0826949\pi$
$571$			12.2767i			0.513763i	−0.966443	+	0.256881i	$0.917305\pi$
$572$		0			0
$573$	19.7929			0.826860
$574$		0			0
$575$	9.13215	+	4.23183i	0.380837	+	0.176480i
$576$		0			0
$577$	14.8857			0.619698			0.309849	−	0.950786i	$-0.399721\pi$
$577$	14.8857			0.619698			0.309849	+	0.950786i	$0.399721\pi$
$578$		0			0
$579$	55.3877			2.30183
$580$		0			0
$581$		−	21.1009i		−	0.875413i
$582$		0			0
$583$			8.12323i			0.336430i
$584$		0			0
$585$		−	12.0937i		−	0.500015i
$586$		0			0
$587$	30.4240			1.25573			0.627866	−	0.778322i	$-0.283929\pi$
$587$	30.4240			1.25573			0.627866	+	0.778322i	$0.283929\pi$
$588$		0			0
$589$	−11.6699			−0.480848
$590$		0			0
$591$		−	53.4497i		−	2.19862i
$592$		0			0
$593$	−39.8878			−1.63799			−0.818997	−	0.573797i	$-0.805470\pi$
$593$	−39.8878			−1.63799			−0.818997	+	0.573797i	$0.805470\pi$
$594$		0			0
$595$		−	6.75693i		−	0.277007i
$596$		0			0
$597$	−32.4775			−1.32922
$598$		0			0
$599$			5.15809i			0.210754i	0.994432	+	0.105377i	$0.0336049\pi$
$599$			5.15809i			0.210754i	−0.994432	+	0.105377i	$0.966395\pi$
$600$		0			0
$601$	16.6407			0.678787			0.339393	−	0.940645i	$-0.389778\pi$
$601$	16.6407			0.678787			0.339393	+	0.940645i	$0.389778\pi$
$602$		0			0
$603$		−	23.3720i		−	0.951783i
$604$		0			0
$605$	−22.6382			−0.920374
$606$		0			0
$607$		−	33.0231i		−	1.34037i	−0.742196	−	0.670183i	$-0.766216\pi$
$607$		−	33.0231i		−	1.34037i	0.742196	−	0.670183i	$-0.233784\pi$
$608$		0			0
$609$		−	15.5290i		−	0.629267i
$610$		0			0
$611$	−2.25068			−0.0910527
$612$		0			0
$613$	−26.6072			−1.07465			−0.537327	−	0.843374i	$-0.680566\pi$
$613$	−26.6072			−1.07465			−0.537327	+	0.843374i	$0.680566\pi$
$614$		0			0
$615$	−9.78479			−0.394561
$616$		0			0
$617$			36.9911i			1.48921i	0.667507	+	0.744604i	$0.267362\pi$
$617$			36.9911i			1.48921i	−0.667507	+	0.744604i	$0.732638\pi$
$618$		0			0
$619$		−	15.2679i		−	0.613669i	−0.951763	−	0.306834i	$-0.900730\pi$
$619$		−	15.2679i		−	0.613669i	0.951763	−	0.306834i	$-0.0992697\pi$
$620$		0			0
$621$	14.1666	+	6.56481i	0.568488	+	0.263437i
$622$		0			0
$623$		−	11.1706i		−	0.447539i
$624$		0			0
$625$	−31.0890			−1.24356
$626$		0			0
$627$	−11.2203			−0.448095
$628$		0			0
$629$		−	11.1642i		−	0.445145i
$630$		0			0
$631$	35.9855			1.43256			0.716280	−	0.697813i	$-0.245843\pi$
$631$	35.9855			1.43256			0.716280	+	0.697813i	$0.245843\pi$
$632$		0			0
$633$	−35.0887			−1.39465
$634$		0			0
$635$			10.7562i			0.426845i
$636$		0			0
$637$		−	15.0994i		−	0.598261i
$638$		0			0
$639$			8.24824i			0.326295i
$640$		0			0
$641$			23.9182i			0.944712i	0.881408	+	0.472356i	$0.156596\pi$
$641$			23.9182i			0.944712i	−0.881408	+	0.472356i	$0.843404\pi$
$642$		0			0
$643$		−	35.9185i		−	1.41649i	−0.705969	−	0.708243i	$-0.749488\pi$
$643$		−	35.9185i		−	1.41649i	0.705969	−	0.708243i	$-0.250512\pi$
$644$		0			0
$645$		−	14.6027i		−	0.574980i
$646$		0			0
$647$		−	21.8459i		−	0.858852i	−0.903102	−	0.429426i	$-0.858716\pi$
$647$		−	21.8459i		−	0.858852i	0.903102	−	0.429426i	$-0.141284\pi$
$648$		0			0
$649$		−	1.20041i		−	0.0471203i
$650$		0			0
$651$			10.7562i			0.421567i
$652$		0			0
$653$		−	2.12784i		−	0.0832689i	−0.999133	−	0.0416344i	$-0.986744\pi$
$653$		−	2.12784i		−	0.0832689i	0.999133	−	0.0416344i	$-0.0132565\pi$
$654$		0			0
$655$	23.2418			0.908134
$656$		0			0
$657$	−16.8981			−0.659256
$658$		0			0
$659$			26.4501i			1.03035i	0.857085	+	0.515174i	$0.172273\pi$
$659$			26.4501i			1.03035i	−0.857085	+	0.515174i	$0.827727\pi$
$660$		0			0
$661$	−29.2769			−1.13874			−0.569371	−	0.822081i	$-0.692813\pi$
$661$	−29.2769			−1.13874			−0.569371	+	0.822081i	$0.692813\pi$
$662$		0			0
$663$	11.3693			0.441547
$664$		0			0
$665$			13.1211i			0.508813i
$666$		0			0
$667$	−10.1146	+	21.8270i	−0.391640	+	0.845146i
$668$		0			0
$669$			6.48018i			0.250538i
$670$		0			0
$671$		−	11.7057i		−	0.451894i
$672$		0			0
$673$	37.4959			1.44536			0.722680	−	0.691182i	$-0.242910\pi$
$673$	37.4959			1.44536			0.722680	+	0.691182i	$0.242910\pi$
$674$		0			0
$675$	6.83275			0.262993
$676$		0			0
$677$	−13.1407			−0.505038			−0.252519	−	0.967592i	$-0.581259\pi$
$677$	−13.1407			−0.505038			−0.252519	+	0.967592i	$0.581259\pi$
$678$		0			0
$679$		−	21.3945i		−	0.821047i
$680$		0			0
$681$		−	20.4565i		−	0.783895i
$682$		0			0
$683$	−5.96670			−0.228309			−0.114155	−	0.993463i	$-0.536416\pi$
$683$	−5.96670			−0.228309			−0.114155	+	0.993463i	$0.536416\pi$
$684$		0			0
$685$		−	32.1514i		−	1.22844i
$686$		0			0
$687$	16.4944			0.629299
$688$		0			0
$689$			15.9837i			0.608931i
$690$		0			0
$691$	−41.7217			−1.58717			−0.793583	−	0.608462i	$-0.791787\pi$
$691$	−41.7217			−1.58717			−0.793583	+	0.608462i	$0.791787\pi$
$692$		0			0
$693$			3.38238i			0.128486i
$694$		0			0
$695$	10.6958			0.405713
$696$		0			0
$697$		−	3.00850i		−	0.113955i
$698$		0			0
$699$	−22.6053			−0.855012
$700$		0			0
$701$	−23.9934			−0.906219			−0.453109	−	0.891455i	$-0.649685\pi$
$701$	−23.9934			−0.906219			−0.453109	+	0.891455i	$0.649685\pi$
$702$		0			0
$703$			21.6794i			0.817653i
$704$		0			0
$705$			4.06699i			0.153172i
$706$		0			0
$707$		−	0.463771i		−	0.0174419i
$708$		0			0
$709$	2.88884			0.108493			0.0542464	−	0.998528i	$-0.482724\pi$
$709$	2.88884			0.108493			0.0542464	+	0.998528i	$0.482724\pi$
$710$		0			0
$711$	−11.1082			−0.416592
$712$		0			0
$713$	7.00590	−	15.1185i	0.262373	−	0.566192i
$714$		0			0
$715$	13.1233			0.490784
$716$		0			0
$717$			57.4333i			2.14488i
$718$		0			0
$719$		−	17.2400i		−	0.642942i	−0.946919	−	0.321471i	$-0.895823\pi$
$719$		−	17.2400i		−	0.642942i	0.946919	−	0.321471i	$-0.104177\pi$
$720$		0			0
$721$	−10.1859			−0.379341
$722$		0			0
$723$		−	6.44224i		−	0.239590i
$724$		0			0
$725$			10.5274i			0.390979i
$726$		0			0
$727$	50.2190			1.86252			0.931260	−	0.364355i	$-0.118710\pi$
$727$	50.2190			1.86252			0.931260	+	0.364355i	$0.118710\pi$
$728$		0			0
$729$	4.22193			0.156368
$730$		0			0
$731$	4.48984			0.166063
$732$		0			0
$733$	−29.2975			−1.08213			−0.541063	−	0.840982i	$-0.681978\pi$
$733$	−29.2975			−1.08213			−0.541063	+	0.840982i	$0.681978\pi$
$734$		0			0
$735$	−27.2848			−1.00641
$736$		0			0
$737$	25.3617			0.934211
$738$		0			0
$739$	−17.5545			−0.645753			−0.322877	−	0.946441i	$-0.604650\pi$
$739$	−17.5545			−0.645753			−0.322877	+	0.946441i	$0.604650\pi$
$740$		0			0
$741$	−22.0777			−0.811044
$742$		0			0
$743$	13.9441			0.511558			0.255779	−	0.966735i	$-0.417668\pi$
$743$	13.9441			0.511558			0.255779	+	0.966735i	$0.417668\pi$
$744$		0			0
$745$	−27.4782			−1.00672
$746$		0			0
$747$		−	20.9832i		−	0.767735i
$748$		0			0
$749$		−	0.998481i		−	0.0364837i
$750$		0			0
$751$	45.7458			1.66929			0.834644	−	0.550789i	$-0.185673\pi$
$751$	45.7458			1.66929			0.834644	+	0.550789i	$0.185673\pi$
$752$		0			0
$753$		−	4.66210i		−	0.169896i
$754$		0			0
$755$		−	27.2288i		−	0.990956i
$756$		0			0
$757$	−36.3497			−1.32115			−0.660576	−	0.750759i	$-0.729688\pi$
$757$	−36.3497			−1.32115			−0.660576	+	0.750759i	$0.729688\pi$
$758$		0			0
$759$	6.73600	−	14.5361i	0.244501	−	0.527626i
$760$		0			0
$761$	4.32626			0.156827			0.0784134	−	0.996921i	$-0.475015\pi$
$761$	4.32626			0.156827			0.0784134	+	0.996921i	$0.475015\pi$
$762$		0			0
$763$	3.64113			0.131818
$764$		0			0
$765$		−	6.71923i		−	0.242934i
$766$		0			0
$767$		−	2.36200i		−	0.0852868i
$768$		0			0
$769$			12.4755i			0.449877i	0.974373	+	0.224939i	$0.0722182\pi$
$769$			12.4755i			0.449877i	−0.974373	+	0.224939i	$0.927782\pi$
$770$		0			0
$771$	33.3395			1.20069
$772$		0			0
$773$	−14.6923			−0.528445			−0.264222	−	0.964462i	$-0.585115\pi$
$773$	−14.6923			−0.528445			−0.264222	+	0.964462i	$0.585115\pi$
$774$		0			0
$775$		−	7.29183i		−	0.261930i
$776$		0			0
$777$	19.9820			0.716849
$778$		0			0
$779$			5.84211i			0.209315i
$780$		0			0
$781$	−8.95042			−0.320271
$782$		0			0
$783$			16.3311i			0.583627i
$784$		0			0
$785$	36.6789			1.30912
$786$		0			0
$787$		−	10.2156i		−	0.364148i	−0.983285	−	0.182074i	$-0.941719\pi$
$787$		−	10.2156i		−	0.364148i	0.983285	−	0.182074i	$-0.0582811\pi$
$788$		0			0
$789$	46.6441			1.66058
$790$		0			0
$791$		−	27.4930i		−	0.977538i
$792$		0			0
$793$		−	23.0328i		−	0.817918i
$794$		0			0
$795$	28.8827			1.02436
$796$		0			0
$797$	−10.5503			−0.373712			−0.186856	−	0.982387i	$-0.559830\pi$
$797$	−10.5503			−0.373712			−0.186856	+	0.982387i	$0.559830\pi$
$798$		0			0
$799$	−1.25047			−0.0442384
$800$		0			0
$801$		−	11.1082i		−	0.392490i
$802$		0			0
$803$		−	18.3366i		−	0.647085i
$804$		0			0
$805$	−16.9985	−	7.87711i	−0.599120	−	0.277632i
$806$		0			0
$807$			52.1799i			1.83682i
$808$		0			0
$809$	−24.4161			−0.858423			−0.429212	−	0.903204i	$-0.641209\pi$
$809$	−24.4161			−0.858423			−0.429212	+	0.903204i	$0.641209\pi$
$810$		0			0
$811$	−28.3512			−0.995544			−0.497772	−	0.867308i	$-0.665848\pi$
$811$	−28.3512			−0.995544			−0.497772	+	0.867308i	$0.665848\pi$
$812$		0			0
$813$			58.6434i			2.05671i
$814$		0			0
$815$	37.6931			1.32033
$816$		0			0
$817$	−8.71868			−0.305028
$818$		0			0
$819$			6.65535i			0.232557i
$820$		0			0
$821$			12.3341i			0.430463i	0.976563	+	0.215231i	$0.0690505\pi$
$821$			12.3341i			0.430463i	−0.976563	+	0.215231i	$0.930949\pi$
$822$		0			0
$823$		−	53.4349i		−	1.86262i	−0.364225	−	0.931311i	$-0.618666\pi$
$823$		−	53.4349i		−	1.86262i	0.364225	−	0.931311i	$-0.381334\pi$
$824$		0			0
$825$		−	7.01092i		−	0.244089i
$826$		0			0
$827$			42.1604i			1.46606i	0.680196	+	0.733031i	$0.261895\pi$
$827$			42.1604i			1.46606i	−0.680196	+	0.733031i	$0.738105\pi$
$828$		0			0
$829$			50.8436i			1.76587i	0.469496	+	0.882935i	$0.344436\pi$
$829$			50.8436i			1.76587i	−0.469496	+	0.882935i	$0.655564\pi$
$830$		0			0
$831$			10.5912i			0.367404i
$832$		0			0
$833$		−	8.38918i		−	0.290668i
$834$		0			0
$835$		−	41.4154i		−	1.43324i
$836$		0			0
$837$		−	11.3118i		−	0.390992i
$838$		0			0
$839$	−38.0306			−1.31296			−0.656481	−	0.754342i	$-0.727956\pi$
$839$	−38.0306			−1.31296			−0.656481	+	0.754342i	$0.727956\pi$
$840$		0			0
$841$	3.83806			0.132347
$842$		0			0
$843$		−	65.8350i		−	2.26748i
$844$		0			0
$845$	−8.81426			−0.303220
$846$		0			0
$847$	12.4581			0.428066
$848$		0			0
$849$			24.4163i			0.837964i
$850$		0			0
$851$	−28.0859	−	13.0150i	−0.962774	−	0.446149i
$852$		0			0
$853$			52.0135i			1.78091i	0.455074	+	0.890454i	$0.349613\pi$
$853$			52.0135i			1.78091i	−0.455074	+	0.890454i	$0.650387\pi$
$854$		0			0
$855$			13.0479i			0.446228i
$856$		0			0
$857$	24.8381			0.848452			0.424226	−	0.905556i	$-0.360546\pi$
$857$	24.8381			0.848452			0.424226	+	0.905556i	$0.360546\pi$
$858$		0			0
$859$	34.0470			1.16167			0.580835	−	0.814021i	$-0.302726\pi$
$859$	34.0470			1.16167			0.580835	+	0.814021i	$0.302726\pi$
$860$		0			0
$861$	5.38470			0.183510
$862$		0			0
$863$			27.6866i			0.942464i	0.882009	+	0.471232i	$0.156191\pi$
$863$			27.6866i			0.942464i	−0.882009	+	0.471232i	$0.843809\pi$
$864$		0			0
$865$			37.1378i			1.26272i
$866$		0			0
$867$	−29.5772			−1.00449
$868$		0			0
$869$		−	12.0539i		−	0.408900i
$870$		0			0
$871$	49.9032			1.69090
$872$		0			0
$873$		−	21.2752i		−	0.720056i
$874$		0			0
$875$	11.3340			0.383158
$876$		0			0
$877$		−	33.8386i		−	1.14265i	−0.820724	−	0.571325i	$-0.806430\pi$
$877$		−	33.8386i		−	1.14265i	0.820724	−	0.571325i	$-0.193570\pi$
$878$		0			0
$879$	25.8402			0.871570
$880$		0			0
$881$			3.48337i			0.117358i	0.998277	+	0.0586789i	$0.0186888\pi$
$881$			3.48337i			0.117358i	−0.998277	+	0.0586789i	$0.981311\pi$
$882$		0			0
$883$	−23.9456			−0.805833			−0.402916	−	0.915237i	$-0.632003\pi$
$883$	−23.9456			−0.805833			−0.402916	+	0.915237i	$0.632003\pi$
$884$		0			0
$885$	−4.26815			−0.143472
$886$		0			0
$887$			22.2229i			0.746173i	0.927796	+	0.373087i	$0.121701\pi$
$887$			22.2229i			0.746173i	−0.927796	+	0.373087i	$0.878299\pi$
$888$		0			0
$889$		−	5.91927i		−	0.198526i
$890$		0			0
$891$		−	17.7966i		−	0.596208i
$892$		0			0
$893$	2.42824			0.0812580
$894$		0			0
$895$	−20.0767			−0.671091
$896$		0			0
$897$	13.2541	−	28.6020i	0.442543	−	0.954992i
$898$		0			0
$899$	17.4284			0.581270
$900$		0			0
$901$			8.88049i			0.295852i
$902$		0			0
$903$			8.03605i			0.267423i
$904$		0			0
$905$	−15.5296			−0.516220
$906$		0			0
$907$			30.6008i			1.01608i	0.861333	+	0.508041i	$0.169630\pi$
$907$			30.6008i			1.01608i	−0.861333	+	0.508041i	$0.830370\pi$
$908$		0			0
$909$		−	0.461183i		−	0.0152965i
$910$		0			0
$911$	−0.910974			−0.0301819			−0.0150910	−	0.999886i	$-0.504804\pi$
$911$	−0.910974			−0.0301819			−0.0150910	+	0.999886i	$0.504804\pi$
$912$		0			0
$913$	22.7695			0.753561
$914$		0			0
$915$	−41.6204			−1.37593
$916$		0			0
$917$	−12.7903			−0.422373
$918$		0			0
$919$	−53.5999			−1.76810			−0.884049	−	0.467394i	$-0.845193\pi$
$919$	−53.5999			−1.76810			−0.884049	+	0.467394i	$0.845193\pi$
$920$		0			0
$921$	−26.6283			−0.877431
$922$		0			0
$923$	−17.6113			−0.579685
$924$		0			0
$925$	−13.5462			−0.445396
$926$		0			0
$927$	−10.1290			−0.332681
$928$		0			0
$929$	−0.265952			−0.00872559			−0.00436279	−	0.999990i	$-0.501389\pi$
$929$	−0.265952			−0.00872559			−0.00436279	+	0.999990i	$0.501389\pi$
$930$		0			0
$931$			16.2907i			0.533905i
$932$		0			0
$933$			50.5363i			1.65448i
$934$		0			0
$935$	7.29125			0.238449
$936$		0			0
$937$		−	39.7886i		−	1.29984i	−0.760003	−	0.649919i	$-0.774803\pi$
$937$		−	39.7886i		−	1.29984i	0.760003	−	0.649919i	$-0.225197\pi$
$938$		0			0
$939$			38.9289i			1.27040i
$940$		0			0
$941$	−26.0669			−0.849755			−0.424878	−	0.905251i	$-0.639683\pi$
$941$	−26.0669			−0.849755			−0.424878	+	0.905251i	$0.639683\pi$
$942$		0			0
$943$	−7.56854	−	3.50726i	−0.246466	−	0.114212i
$944$		0			0
$945$	−12.7184			−0.413731
$946$		0			0
$947$	−5.78406			−0.187957			−0.0939783	−	0.995574i	$-0.529958\pi$
$947$	−5.78406			−0.187957			−0.0939783	+	0.995574i	$0.529958\pi$
$948$		0			0
$949$		−	36.0801i		−	1.17121i
$950$		0			0
$951$			59.4209i			1.92685i
$952$		0			0
$953$			3.77518i			0.122290i	0.998129	+	0.0611451i	$0.0194752\pi$
$953$			3.77518i			0.122290i	−0.998129	+	0.0611451i	$0.980525\pi$
$954$		0			0
$955$	24.9762			0.808211
$956$		0			0
$957$	16.7570			0.541677
$958$		0			0
$959$			17.6934i			0.571349i
$960$		0			0
$961$	18.9282			0.610588
$962$		0			0
$963$		−	0.992911i		−	0.0319961i
$964$		0			0
$965$	69.8925			2.24992
$966$		0			0
$967$		−	7.04359i		−	0.226507i	−0.993566	−	0.113253i	$-0.963873\pi$
$967$		−	7.04359i		−	0.226507i	0.993566	−	0.113253i	$-0.0361272\pi$
$968$		0			0
$969$	−12.2663			−0.394049
$970$		0			0
$971$		−	10.0392i		−	0.322172i	−0.986940	−	0.161086i	$-0.948500\pi$
$971$		−	10.0392i		−	0.322172i	0.986940	−	0.161086i	$-0.0514996\pi$
$972$		0			0
$973$	−5.88602			−0.188697
$974$		0			0
$975$		−	13.7951i		−	0.441796i
$976$		0			0
$977$			8.19985i			0.262336i	0.991360	+	0.131168i	$0.0418728\pi$
$977$			8.19985i			0.262336i	−0.991360	+	0.131168i	$0.958127\pi$
$978$		0			0
$979$	12.0539			0.385244
$980$		0			0
$981$	3.62082			0.115604
$982$		0			0
$983$	26.2027			0.835735			0.417868	−	0.908508i	$-0.362778\pi$
$983$	26.2027			0.835735			0.417868	+	0.908508i	$0.362778\pi$
$984$		0			0
$985$		−	67.4469i		−	2.14904i
$986$		0			0
$987$		−	2.23812i		−	0.0712402i
$988$		0			0
$989$	5.23418	−	11.2952i	0.166437	−	0.359166i
$990$		0			0
$991$		−	10.3292i		−	0.328117i	−0.986451	−	0.164059i	$-0.947541\pi$
$991$		−	10.3292i		−	0.328117i	0.986451	−	0.164059i	$-0.0524586\pi$
$992$		0			0
$993$	−13.1619			−0.417681
$994$		0			0
$995$	−40.9826			−1.29924
$996$		0			0
$997$			5.29547i			0.167709i	0.996478	+	0.0838546i	$0.0267231\pi$
$997$			5.29547i			0.167709i	−0.996478	+	0.0838546i	$0.973277\pi$
$998$		0			0
$999$	−21.0141			−0.664857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.2.h.c.735.5	✓	32
4.3	odd	2	inner	1472.2.h.c.735.27	yes	32
8.3	odd	2	inner	1472.2.h.c.735.6	yes	32
8.5	even	2	inner	1472.2.h.c.735.28	yes	32
23.22	odd	2	inner	1472.2.h.c.735.7	yes	32
92.91	even	2	inner	1472.2.h.c.735.25	yes	32
184.45	odd	2	inner	1472.2.h.c.735.26	yes	32
184.91	even	2	inner	1472.2.h.c.735.8	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.2.h.c.735.5	✓	32	1.1	even	1	trivial
1472.2.h.c.735.6	yes	32	8.3	odd	2	inner
1472.2.h.c.735.7	yes	32	23.22	odd	2	inner
1472.2.h.c.735.8	yes	32	184.91	even	2	inner
1472.2.h.c.735.25	yes	32	92.91	even	2	inner
1472.2.h.c.735.26	yes	32	184.45	odd	2	inner
1472.2.h.c.735.27	yes	32	4.3	odd	2	inner
1472.2.h.c.735.28	yes	32	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.11141 q^{3} -2.66434 q^{5} +1.46622 q^{7} +1.45804 q^{9} +1.58217i q^{11} +3.11316i q^{13} +5.62551 q^{15} +1.72966i q^{17} -3.35877i q^{19} -3.09579 q^{21} +(4.35133 + 2.01641i) q^{23} +2.09870 q^{25} +3.25570 q^{27} +5.01617i q^{29} -3.47445i q^{31} -3.34060i q^{33} -3.90651 q^{35} -6.45456 q^{37} -6.57315i q^{39} -1.73936 q^{41} -2.59580i q^{43} -3.88472 q^{45} +0.722956i q^{47} -4.85019 q^{49} -3.65201i q^{51} +5.13424 q^{53} -4.21543i q^{55} +7.09173i q^{57} -0.758714 q^{59} -7.39852 q^{61} +2.13781 q^{63} -8.29451i q^{65} -16.0297i q^{67} +(-9.18744 - 4.25745i) q^{69} +5.65707i q^{71} -11.5896 q^{73} -4.43122 q^{75} +2.31981i q^{77} -7.61860 q^{79} -11.2482 q^{81} -14.3913i q^{83} -4.60840i q^{85} -10.5912i q^{87} -7.61860i q^{89} +4.56458i q^{91} +7.33598i q^{93} +8.94890i q^{95} -14.5916i q^{97} +2.30687i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 40 q^{9} + 32 q^{25} + 8 q^{41} + 48 q^{49} - 104 q^{73} + 112 q^{81}+O(q^{100}) \) 32 * q + 40 * q^9 + 32 * q^25 + 8 * q^41 + 48 * q^49 - 104 * q^73 + 112 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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