LMFDB - Embedded newform 1932.2.k.a.1609.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1609,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1932.1609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.k (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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.6
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +3.63843 q^{5} +(0.122243 - 2.64293i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +3.63843 q^{5} +(0.122243 - 2.64293i) q^{7} -1.00000 q^{9} -1.30191i q^{11} -0.939615i q^{13} -3.63843i q^{15} +5.95096 q^{17} -7.57909 q^{19} +(-2.64293 - 0.122243i) q^{21} +(0.194694 - 4.79188i) q^{23} +8.23814 q^{25} +1.00000i q^{27} -1.09093 q^{29} -4.06591i q^{31} -1.30191 q^{33} +(0.444771 - 9.61609i) q^{35} +7.18677i q^{37} -0.939615 q^{39} -7.59867i q^{41} -0.163763i q^{43} -3.63843 q^{45} -3.48683i q^{47} +(-6.97011 - 0.646157i) q^{49} -5.95096i q^{51} +4.81985i q^{53} -4.73690i q^{55} +7.57909i q^{57} +9.59814i q^{59} +5.67350 q^{61} +(-0.122243 + 2.64293i) q^{63} -3.41872i q^{65} +9.71338i q^{67} +(-4.79188 - 0.194694i) q^{69} +9.72675 q^{71} -11.9713i q^{73} -8.23814i q^{75} +(-3.44085 - 0.159149i) q^{77} +0.0817388i q^{79} +1.00000 q^{81} -11.5225 q^{83} +21.6521 q^{85} +1.09093i q^{87} +9.86670 q^{89} +(-2.48333 - 0.114861i) q^{91} -4.06591 q^{93} -27.5760 q^{95} -12.6867 q^{97} +1.30191i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 32 q^{9}+O(q^{10}) $ 32 * q - 32 * q^9	$ 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) $ 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	3.63843			1.62715			0.813577	−	0.581458i	$-0.197517\pi$
$5$	3.63843			1.62715			0.813577	+	0.581458i	$0.197517\pi$
$6$		0			0
$7$	0.122243	−	2.64293i	0.0462034	−	0.998932i
$7$	0.122243	−	2.64293i	0.0462034	−	0.998932i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	1.30191i		−	0.392541i	−0.980550	−	0.196270i	$-0.937117\pi$
$11$		−	1.30191i		−	0.392541i	0.980550	−	0.196270i	$-0.0628830\pi$
$12$		0			0
$13$		−	0.939615i		−	0.260602i	−0.991474	−	0.130301i	$-0.958406\pi$
$13$		−	0.939615i		−	0.260602i	0.991474	−	0.130301i	$-0.0415944\pi$
$14$		0			0
$15$		−	3.63843i		−	0.939437i
$16$		0			0
$17$	5.95096			1.44332			0.721659	−	0.692248i	$-0.243380\pi$
$17$	5.95096			1.44332			0.721659	+	0.692248i	$0.243380\pi$
$18$		0			0
$19$	−7.57909			−1.73876			−0.869381	−	0.494142i	$-0.835482\pi$
$19$	−7.57909			−1.73876			−0.869381	+	0.494142i	$0.835482\pi$
$20$		0			0
$21$	−2.64293	−	0.122243i	−0.576734	−	0.0266755i
$22$		0			0
$23$	0.194694	−	4.79188i	0.0405966	−	0.999176i
$23$	0.194694	−	4.79188i	0.0405966	−	0.999176i
$24$		0			0
$25$	8.23814			1.64763
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−1.09093			−0.202580			−0.101290	−	0.994857i	$-0.532297\pi$
$29$	−1.09093			−0.202580			−0.101290	+	0.994857i	$0.532297\pi$
$30$		0			0
$31$		−	4.06591i		−	0.730258i	−0.930957	−	0.365129i	$-0.881025\pi$
$31$		−	4.06591i		−	0.730258i	0.930957	−	0.365129i	$-0.118975\pi$
$32$		0			0
$33$	−1.30191			−0.226633
$34$		0			0
$35$	0.444771	−	9.61609i	0.0751800	−	1.62542i
$36$		0			0
$37$			7.18677i			1.18150i	0.806856	+	0.590749i	$0.201168\pi$
$37$			7.18677i			1.18150i	−0.806856	+	0.590749i	$0.798832\pi$
$38$		0			0
$39$	−0.939615			−0.150459
$40$		0			0
$41$		−	7.59867i		−	1.18671i	−0.804940	−	0.593357i	$-0.797802\pi$
$41$		−	7.59867i		−	1.18671i	0.804940	−	0.593357i	$-0.202198\pi$
$42$		0			0
$43$		−	0.163763i		−	0.0249736i	−0.999922	−	0.0124868i	$-0.996025\pi$
$43$		−	0.163763i		−	0.0249736i	0.999922	−	0.0124868i	$-0.00397478\pi$
$44$		0			0
$45$	−3.63843			−0.542384
$46$		0			0
$47$		−	3.48683i		−	0.508606i	−0.967125	−	0.254303i	$-0.918154\pi$
$47$		−	3.48683i		−	0.508606i	0.967125	−	0.254303i	$-0.0818460\pi$
$48$		0			0
$49$	−6.97011	−	0.646157i	−0.995730	−	0.0923081i
$50$		0			0
$51$		−	5.95096i		−	0.833300i
$52$		0			0
$53$			4.81985i			0.662057i	0.943621	+	0.331029i	$0.107396\pi$
$53$			4.81985i			0.662057i	−0.943621	+	0.331029i	$0.892604\pi$
$54$		0			0
$55$		−	4.73690i		−	0.638724i
$56$		0			0
$57$			7.57909i			1.00387i
$58$		0			0
$59$			9.59814i			1.24957i	0.780796	+	0.624786i	$0.214814\pi$
$59$			9.59814i			1.24957i	−0.780796	+	0.624786i	$0.785186\pi$
$60$		0			0
$61$	5.67350			0.726417			0.363208	−	0.931708i	$-0.381681\pi$
$61$	5.67350			0.726417			0.363208	+	0.931708i	$0.381681\pi$
$62$		0			0
$63$	−0.122243	+	2.64293i	−0.0154011	+	0.332977i
$64$		0			0
$65$		−	3.41872i		−	0.424040i
$66$		0			0
$67$			9.71338i			1.18668i	0.804953	+	0.593339i	$0.202191\pi$
$67$			9.71338i			1.18668i	−0.804953	+	0.593339i	$0.797809\pi$
$68$		0			0
$69$	−4.79188	−	0.194694i	−0.576874	−	0.0234385i
$70$		0			0
$71$	9.72675			1.15435			0.577176	−	0.816620i	$-0.304155\pi$
$71$	9.72675			1.15435			0.577176	+	0.816620i	$0.304155\pi$
$72$		0			0
$73$		−	11.9713i		−	1.40113i	−0.713586	−	0.700567i	$-0.752930\pi$
$73$		−	11.9713i		−	1.40113i	0.713586	−	0.700567i	$-0.247070\pi$
$74$		0			0
$75$		−	8.23814i		−	0.951258i
$76$		0			0
$77$	−3.44085	−	0.159149i	−0.392121	−	0.0181367i
$78$		0			0
$79$			0.0817388i			0.00919634i	0.999989	+	0.00459817i	$0.00146365\pi$
$79$			0.0817388i			0.00919634i	−0.999989	+	0.00459817i	$0.998536\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−11.5225			−1.26476			−0.632378	−	0.774660i	$-0.717921\pi$
$83$	−11.5225			−1.26476			−0.632378	+	0.774660i	$0.717921\pi$
$84$		0			0
$85$	21.6521			2.34850
$86$		0			0
$87$			1.09093i			0.116960i
$88$		0			0
$89$	9.86670			1.04587			0.522934	−	0.852373i	$-0.324837\pi$
$89$	9.86670			1.04587			0.522934	+	0.852373i	$0.324837\pi$
$90$		0			0
$91$	−2.48333	−	0.114861i	−0.260324	−	0.0120407i
$92$		0			0
$93$	−4.06591			−0.421615
$94$		0			0
$95$	−27.5760			−2.82923
$96$		0			0
$97$	−12.6867			−1.28814			−0.644070	−	0.764967i	$-0.722755\pi$
$97$	−12.6867			−1.28814			−0.644070	+	0.764967i	$0.722755\pi$
$98$		0			0
$99$			1.30191i			0.130847i
$100$		0			0
$101$		−	0.229157i		−	0.0228020i	−0.999935	−	0.0114010i	$-0.996371\pi$
$101$		−	0.229157i		−	0.0228020i	0.999935	−	0.0114010i	$-0.00362912\pi$
$102$		0			0
$103$	14.3191			1.41091			0.705454	−	0.708756i	$-0.250743\pi$
$103$	14.3191			1.41091			0.705454	+	0.708756i	$0.250743\pi$
$104$		0			0
$105$	−9.61609	−	0.444771i	−0.938434	−	0.0434052i
$106$		0			0
$107$			14.2476i			1.37736i	0.725063	+	0.688682i	$0.241811\pi$
$107$			14.2476i			1.37736i	−0.725063	+	0.688682i	$0.758189\pi$
$108$		0			0
$109$		−	5.70114i		−	0.546070i	−0.962004	−	0.273035i	$-0.911972\pi$
$109$		−	5.70114i		−	0.546070i	0.962004	−	0.273035i	$-0.0880275\pi$
$110$		0			0
$111$	7.18677			0.682138
$112$		0			0
$113$			5.90336i			0.555342i	0.960676	+	0.277671i	$0.0895625\pi$
$113$			5.90336i			0.555342i	−0.960676	+	0.277671i	$0.910438\pi$
$114$		0			0
$115$	0.708381	−	17.4349i	0.0660569	−	1.62581i
$116$		0			0
$117$			0.939615i			0.0868675i
$118$		0			0
$119$	0.727461	−	15.7279i	0.0666862	−	1.44178i
$120$		0			0
$121$	9.30503			0.845912
$122$		0			0
$123$	−7.59867			−0.685149
$124$		0			0
$125$	11.7817			1.05379
$126$		0			0
$127$	3.56946			0.316738			0.158369	−	0.987380i	$-0.449376\pi$
$127$	3.56946			0.316738			0.158369	+	0.987380i	$0.449376\pi$
$128$		0			0
$129$	−0.163763			−0.0144185
$130$		0			0
$131$		−	9.14622i		−	0.799109i	−0.916709	−	0.399555i	$-0.869165\pi$
$131$		−	9.14622i		−	0.799109i	0.916709	−	0.399555i	$-0.130835\pi$
$132$		0			0
$133$	−0.926488	+	20.0310i	−0.0803367	+	1.73691i
$134$		0			0
$135$			3.63843i			0.313146i
$136$		0			0
$137$			10.6412i			0.909137i	0.890712	+	0.454569i	$0.150207\pi$
$137$			10.6412i			0.909137i	−0.890712	+	0.454569i	$0.849793\pi$
$138$		0			0
$139$		−	5.45797i		−	0.462939i	−0.972842	−	0.231470i	$-0.925647\pi$
$139$		−	5.45797i		−	0.462939i	0.972842	−	0.231470i	$-0.0743534\pi$
$140$		0			0
$141$	−3.48683			−0.293644
$142$		0			0
$143$	−1.22329			−0.102297
$144$		0			0
$145$	−3.96926			−0.329629
$146$		0			0
$147$	−0.646157	+	6.97011i	−0.0532941	+	0.574885i
$148$		0			0
$149$		−	9.69409i		−	0.794171i	−0.917782	−	0.397085i	$-0.870022\pi$
$149$		−	9.69409i		−	0.794171i	0.917782	−	0.397085i	$-0.129978\pi$
$150$		0			0
$151$	−17.6246			−1.43427			−0.717134	−	0.696935i	$-0.754546\pi$
$151$	−17.6246			−1.43427			−0.717134	+	0.696935i	$0.754546\pi$
$152$		0			0
$153$	−5.95096			−0.481106
$154$		0			0
$155$		−	14.7935i		−	1.18824i
$156$		0			0
$157$	−6.79781			−0.542525			−0.271262	−	0.962505i	$-0.587441\pi$
$157$	−6.79781			−0.542525			−0.271262	+	0.962505i	$0.587441\pi$
$158$		0			0
$159$	4.81985			0.382239
$160$		0			0
$161$	−12.6408	−	1.10033i	−0.996233	−	0.0867185i
$162$		0			0
$163$	18.4836			1.44775			0.723873	−	0.689933i	$-0.242360\pi$
$163$	18.4836			1.44775			0.723873	+	0.689933i	$0.242360\pi$
$164$		0			0
$165$	−4.73690			−0.368767
$166$		0			0
$167$		−	6.48821i		−	0.502073i	−0.967978	−	0.251037i	$-0.919229\pi$
$167$		−	6.48821i		−	0.502073i	0.967978	−	0.251037i	$-0.0807714\pi$
$168$		0			0
$169$	12.1171			0.932086
$170$		0			0
$171$	7.57909			0.579587
$172$		0			0
$173$		−	9.64653i		−	0.733412i	−0.930337	−	0.366706i	$-0.880486\pi$
$173$		−	9.64653i		−	0.733412i	0.930337	−	0.366706i	$-0.119514\pi$
$174$		0			0
$175$	1.00705	−	21.7728i	0.0761260	−	1.64587i
$176$		0			0
$177$	9.59814			0.721440
$178$		0			0
$179$	−12.3782			−0.925193			−0.462597	−	0.886569i	$-0.653082\pi$
$179$	−12.3782			−0.925193			−0.462597	+	0.886569i	$0.653082\pi$
$180$		0			0
$181$	−7.39251			−0.549481			−0.274740	−	0.961518i	$-0.588592\pi$
$181$	−7.39251			−0.549481			−0.274740	+	0.961518i	$0.588592\pi$
$182$		0			0
$183$		−	5.67350i		−	0.419397i
$184$		0			0
$185$			26.1485i			1.92248i
$186$		0			0
$187$		−	7.74761i		−	0.566561i
$188$		0			0
$189$	2.64293	+	0.122243i	0.192245	+	0.00889185i
$190$		0			0
$191$		−	15.1768i		−	1.09816i	−0.835771	−	0.549078i	$-0.814979\pi$
$191$		−	15.1768i		−	1.09816i	0.835771	−	0.549078i	$-0.185021\pi$
$192$		0			0
$193$	−6.41857			−0.462019			−0.231009	−	0.972952i	$-0.574203\pi$
$193$	−6.41857			−0.462019			−0.231009	+	0.972952i	$0.574203\pi$
$194$		0			0
$195$	−3.41872			−0.244820
$196$		0			0
$197$	7.85892			0.559925			0.279963	−	0.960011i	$-0.409678\pi$
$197$	7.85892			0.559925			0.279963	+	0.960011i	$0.409678\pi$
$198$		0			0
$199$	6.12725			0.434349			0.217174	−	0.976133i	$-0.430316\pi$
$199$	6.12725			0.434349			0.217174	+	0.976133i	$0.430316\pi$
$200$		0			0
$201$	9.71338			0.685129
$202$		0			0
$203$	−0.133358	+	2.88324i	−0.00935989	+	0.202364i
$204$		0			0
$205$		−	27.6472i		−	1.93096i
$206$		0			0
$207$	−0.194694	+	4.79188i	−0.0135322	+	0.333059i
$208$		0			0
$209$			9.86729i			0.682535i
$210$		0			0
$211$	7.23950			0.498388			0.249194	−	0.968454i	$-0.419834\pi$
$211$	7.23950			0.498388			0.249194	+	0.968454i	$0.419834\pi$
$212$		0			0
$213$		−	9.72675i		−	0.666466i
$214$		0			0
$215$		−	0.595840i		−	0.0406359i
$216$		0			0
$217$	−10.7459	−	0.497027i	−0.729478	−	0.0337404i
$218$		0			0
$219$	−11.9713			−0.808946
$220$		0			0
$221$		−	5.59161i		−	0.376132i
$222$		0			0
$223$		−	14.1188i		−	0.945462i	−0.881207	−	0.472731i	$-0.843268\pi$
$223$		−	14.1188i		−	0.945462i	0.881207	−	0.472731i	$-0.156732\pi$
$224$		0			0
$225$	−8.23814			−0.549209
$226$		0			0
$227$	−8.89366			−0.590293			−0.295147	−	0.955452i	$-0.595368\pi$
$227$	−8.89366			−0.590293			−0.295147	+	0.955452i	$0.595368\pi$
$228$		0			0
$229$	24.4390			1.61498			0.807488	−	0.589884i	$-0.200826\pi$
$229$	24.4390			1.61498			0.807488	+	0.589884i	$0.200826\pi$
$230$		0			0
$231$	−0.159149	+	3.44085i	−0.0104712	+	0.226391i
$232$		0			0
$233$	−13.8624			−0.908157			−0.454079	−	0.890962i	$-0.650031\pi$
$233$	−13.8624			−0.908157			−0.454079	+	0.890962i	$0.650031\pi$
$234$		0			0
$235$		−	12.6866i		−	0.827579i
$236$		0			0
$237$	0.0817388			0.00530951
$238$		0			0
$239$	16.7231			1.08173			0.540863	−	0.841111i	$-0.318098\pi$
$239$	16.7231			1.08173			0.540863	+	0.841111i	$0.318098\pi$
$240$		0			0
$241$	0.874251			0.0563155			0.0281577	−	0.999603i	$-0.491036\pi$
$241$	0.874251			0.0563155			0.0281577	+	0.999603i	$0.491036\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−25.3602	−	2.35099i	−1.62021	−	0.150199i
$246$		0			0
$247$			7.12143i			0.453126i
$248$		0			0
$249$			11.5225i			0.730207i
$250$		0			0
$251$	−22.3262			−1.40922			−0.704610	−	0.709595i	$-0.748878\pi$
$251$	−22.3262			−1.40922			−0.704610	+	0.709595i	$0.748878\pi$
$252$		0			0
$253$	−6.23859	−	0.253475i	−0.392217	−	0.0159358i
$254$		0			0
$255$		−	21.6521i		−	1.35591i
$256$		0			0
$257$		−	6.66094i		−	0.415498i	−0.978182	−	0.207749i	$-0.933386\pi$
$257$		−	6.66094i		−	0.415498i	0.978182	−	0.207749i	$-0.0666137\pi$
$258$		0			0
$259$	18.9941	+	0.878530i	1.18024	+	0.0545892i
$260$		0			0
$261$	1.09093			0.0675267
$262$		0			0
$263$			20.6827i			1.27535i	0.770305	+	0.637676i	$0.220104\pi$
$263$			20.6827i			1.27535i	−0.770305	+	0.637676i	$0.779896\pi$
$264$		0			0
$265$			17.5367i			1.07727i
$266$		0			0
$267$		−	9.86670i		−	0.603832i
$268$		0			0
$269$			22.7936i			1.38975i	0.719129	+	0.694876i	$0.244541\pi$
$269$			22.7936i			1.38975i	−0.719129	+	0.694876i	$0.755459\pi$
$270$		0			0
$271$			23.8831i			1.45080i	0.688330	+	0.725398i	$0.258344\pi$
$271$			23.8831i			1.45080i	−0.688330	+	0.725398i	$0.741656\pi$
$272$		0			0
$273$	−0.114861	+	2.48333i	−0.00695171	+	0.150298i
$274$		0			0
$275$		−	10.7253i		−	0.646761i
$276$		0			0
$277$	20.6653			1.24166			0.620828	−	0.783947i	$-0.286796\pi$
$277$	20.6653			1.24166			0.620828	+	0.783947i	$0.286796\pi$
$278$		0			0
$279$			4.06591i			0.243419i
$280$		0			0
$281$			29.1238i			1.73738i	0.495358	+	0.868689i	$0.335037\pi$
$281$			29.1238i			1.73738i	−0.495358	+	0.868689i	$0.664963\pi$
$282$		0			0
$283$	33.2160			1.97449			0.987243	−	0.159222i	$-0.0508986\pi$
$283$	33.2160			1.97449			0.987243	+	0.159222i	$0.0508986\pi$
$284$		0			0
$285$			27.5760i			1.63346i
$286$		0			0
$287$	−20.0827	−	0.928882i	−1.18545	−	0.0548302i
$288$		0			0
$289$	18.4139			1.08317
$290$		0			0
$291$			12.6867i			0.743708i
$292$		0			0
$293$	−9.20987			−0.538047			−0.269023	−	0.963134i	$-0.586701\pi$
$293$	−9.20987			−0.538047			−0.269023	+	0.963134i	$0.586701\pi$
$294$		0			0
$295$			34.9221i			2.03324i
$296$		0			0
$297$	1.30191			0.0755445
$298$		0			0
$299$	−4.50252	−	0.182938i	−0.260388	−	0.0105796i
$300$		0			0
$301$	−0.432814	−	0.0200188i	−0.0249470	−	0.00115387i
$302$		0			0
$303$	−0.229157			−0.0131647
$304$		0			0
$305$	20.6426			1.18199
$306$		0			0
$307$			17.0684i			0.974144i	0.873362	+	0.487072i	$0.161935\pi$
$307$			17.0684i			0.974144i	−0.873362	+	0.487072i	$0.838065\pi$
$308$		0			0
$309$		−	14.3191i		−	0.814588i
$310$		0			0
$311$		−	14.0489i		−	0.796638i	−0.917247	−	0.398319i	$-0.869594\pi$
$311$		−	14.0489i		−	0.796638i	0.917247	−	0.398319i	$-0.130406\pi$
$312$		0			0
$313$	−5.96783			−0.337321			−0.168661	−	0.985674i	$-0.553944\pi$
$313$	−5.96783			−0.337321			−0.168661	+	0.985674i	$0.553944\pi$
$314$		0			0
$315$	−0.444771	+	9.61609i	−0.0250600	+	0.541805i
$316$		0			0
$317$	−23.5015			−1.31997			−0.659987	−	0.751277i	$-0.729438\pi$
$317$	−23.5015			−1.31997			−0.659987	+	0.751277i	$0.729438\pi$
$318$		0			0
$319$			1.42029i			0.0795209i
$320$		0			0
$321$	14.2476			0.795222
$322$		0			0
$323$	−45.1028			−2.50959
$324$		0			0
$325$		−	7.74068i		−	0.429376i
$326$		0			0
$327$	−5.70114			−0.315274
$328$		0			0
$329$	−9.21542	−	0.426239i	−0.508063	−	0.0234993i
$330$		0			0
$331$	27.4401			1.50824			0.754121	−	0.656735i	$-0.228063\pi$
$331$	27.4401			1.50824			0.754121	+	0.656735i	$0.228063\pi$
$332$		0			0
$333$		−	7.18677i		−	0.393833i
$334$		0			0
$335$			35.3414i			1.93091i
$336$		0			0
$337$			0.420104i			0.0228845i	0.999935	+	0.0114423i	$0.00364226\pi$
$337$			0.420104i			0.0228845i	−0.999935	+	0.0114423i	$0.996358\pi$
$338$		0			0
$339$	5.90336			0.320627
$340$		0			0
$341$	−5.29344			−0.286656
$342$		0			0
$343$	−2.55979	+	18.3425i	−0.138216	+	0.990402i
$344$		0			0
$345$	−17.4349	−	0.708381i	−0.938663	−	0.0381380i
$346$		0			0
$347$	18.4798			0.992046			0.496023	−	0.868309i	$-0.334793\pi$
$347$	18.4798			0.992046			0.496023	+	0.868309i	$0.334793\pi$
$348$		0			0
$349$			24.8945i			1.33257i	0.745696	+	0.666286i	$0.232117\pi$
$349$			24.8945i			1.33257i	−0.745696	+	0.666286i	$0.767883\pi$
$350$		0			0
$351$	0.939615			0.0501530
$352$		0			0
$353$			4.64405i			0.247178i	0.992333	+	0.123589i	$0.0394404\pi$
$353$			4.64405i			0.247178i	−0.992333	+	0.123589i	$0.960560\pi$
$354$		0			0
$355$	35.3900			1.87831
$356$		0			0
$357$	−15.7279	−	0.727461i	−0.832411	−	0.0385013i
$358$		0			0
$359$			32.5475i			1.71779i	0.512150	+	0.858896i	$0.328849\pi$
$359$			32.5475i			1.71779i	−0.512150	+	0.858896i	$0.671151\pi$
$360$		0			0
$361$	38.4426			2.02329
$362$		0			0
$363$		−	9.30503i		−	0.488387i
$364$		0			0
$365$		−	43.5567i		−	2.27986i
$366$		0			0
$367$	−8.09205			−0.422402			−0.211201	−	0.977443i	$-0.567737\pi$
$367$	−8.09205			−0.422402			−0.211201	+	0.977443i	$0.567737\pi$
$368$		0			0
$369$			7.59867i			0.395571i
$370$		0			0
$371$	12.7385	+	0.589191i	0.661350	+	0.0305893i
$372$		0			0
$373$		−	20.5258i		−	1.06278i	−0.847126	−	0.531392i	$-0.821669\pi$
$373$		−	20.5258i		−	1.06278i	0.847126	−	0.531392i	$-0.178331\pi$
$374$		0			0
$375$		−	11.7817i		−	0.608405i
$376$		0			0
$377$			1.02505i			0.0527929i
$378$		0			0
$379$			32.6693i			1.67811i	0.544049	+	0.839053i	$0.316891\pi$
$379$			32.6693i			1.67811i	−0.544049	+	0.839053i	$0.683109\pi$
$380$		0			0
$381$		−	3.56946i		−	0.182869i
$382$		0			0
$383$	12.7205			0.649987			0.324993	−	0.945716i	$-0.394638\pi$
$383$	12.7205			0.649987			0.324993	+	0.945716i	$0.394638\pi$
$384$		0			0
$385$	−12.5193	−	0.579051i	−0.638042	−	0.0295112i
$386$		0			0
$387$			0.163763i			0.00832455i
$388$		0			0
$389$		−	7.95915i		−	0.403545i	−0.979432	−	0.201772i	$-0.935330\pi$
$389$		−	7.95915i		−	0.403545i	0.979432	−	0.201772i	$-0.0646701\pi$
$390$		0			0
$391$	1.15862	−	28.5163i	0.0585938	−	1.44213i
$392$		0			0
$393$	−9.14622			−0.461366
$394$		0			0
$395$			0.297401i			0.0149639i
$396$		0			0
$397$			10.0549i			0.504639i	0.967644	+	0.252319i	$0.0811934\pi$
$397$			10.0549i			0.504639i	−0.967644	+	0.252319i	$0.918807\pi$
$398$		0			0
$399$	20.0310	+	0.926488i	1.00280	+	0.0463824i
$400$		0			0
$401$			24.5670i			1.22682i	0.789766	+	0.613408i	$0.210202\pi$
$401$			24.5670i			1.22682i	−0.789766	+	0.613408i	$0.789798\pi$
$402$		0			0
$403$	−3.82039			−0.190307
$404$		0			0
$405$	3.63843			0.180795
$406$		0			0
$407$	9.35653			0.463786
$408$		0			0
$409$			21.0942i			1.04304i	0.853239	+	0.521520i	$0.174635\pi$
$409$			21.0942i			1.04304i	−0.853239	+	0.521520i	$0.825365\pi$
$410$		0			0
$411$	10.6412			0.524891
$412$		0			0
$413$	25.3672	+	1.17330i	1.24824	+	0.0577344i
$414$		0			0
$415$	−41.9236			−2.05795
$416$		0			0
$417$	−5.45797			−0.267278
$418$		0			0
$419$	−15.8106			−0.772399			−0.386199	−	0.922415i	$-0.626212\pi$
$419$	−15.8106			−0.772399			−0.386199	+	0.922415i	$0.626212\pi$
$420$		0			0
$421$		−	2.96189i		−	0.144354i	−0.997392	−	0.0721769i	$-0.977005\pi$
$421$		−	2.96189i		−	0.144354i	0.997392	−	0.0721769i	$-0.0229946\pi$
$422$		0			0
$423$			3.48683i			0.169535i
$424$		0			0
$425$	49.0248			2.37805
$426$		0			0
$427$	0.693543	−	14.9946i	0.0335629	−	0.725641i
$428$		0			0
$429$			1.22329i			0.0590612i
$430$		0			0
$431$			15.1634i			0.730395i	0.930930	+	0.365198i	$0.118999\pi$
$431$			15.1634i			0.730395i	−0.930930	+	0.365198i	$0.881001\pi$
$432$		0			0
$433$	12.6898			0.609831			0.304915	−	0.952379i	$-0.401372\pi$
$433$	12.6898			0.609831			0.304915	+	0.952379i	$0.401372\pi$
$434$		0			0
$435$			3.96926i			0.190311i
$436$		0			0
$437$	−1.47561	+	36.3181i	−0.0705878	+	1.73733i
$438$		0			0
$439$			13.8025i			0.658756i	0.944198	+	0.329378i	$0.106839\pi$
$439$			13.8025i			0.658756i	−0.944198	+	0.329378i	$0.893161\pi$
$440$		0			0
$441$	6.97011	+	0.646157i	0.331910	+	0.0307694i
$442$		0			0
$443$	39.1564			1.86037			0.930187	−	0.367085i	$-0.119644\pi$
$443$	39.1564			1.86037			0.930187	+	0.367085i	$0.119644\pi$
$444$		0			0
$445$	35.8993			1.70179
$446$		0			0
$447$	−9.69409			−0.458515
$448$		0			0
$449$	−24.3544			−1.14935			−0.574677	−	0.818381i	$-0.694872\pi$
$449$	−24.3544			−1.14935			−0.574677	+	0.818381i	$0.694872\pi$
$450$		0			0
$451$	−9.89279			−0.465833
$452$		0			0
$453$			17.6246i			0.828075i
$454$		0			0
$455$	−9.03542	−	0.417913i	−0.423587	−	0.0195921i
$456$		0			0
$457$			29.1727i			1.36464i	0.731052	+	0.682322i	$0.239030\pi$
$457$			29.1727i			1.36464i	−0.731052	+	0.682322i	$0.760970\pi$
$458$		0			0
$459$			5.95096i			0.277767i
$460$		0			0
$461$			28.9743i			1.34947i	0.738061	+	0.674734i	$0.235742\pi$
$461$			28.9743i			1.34947i	−0.738061	+	0.674734i	$0.764258\pi$
$462$		0			0
$463$	−5.34242			−0.248283			−0.124142	−	0.992264i	$-0.539618\pi$
$463$	−5.34242			−0.248283			−0.124142	+	0.992264i	$0.539618\pi$
$464$		0			0
$465$	−14.7935			−0.686032
$466$		0			0
$467$	26.4783			1.22527			0.612634	−	0.790367i	$-0.290110\pi$
$467$	26.4783			1.22527			0.612634	+	0.790367i	$0.290110\pi$
$468$		0			0
$469$	25.6717	+	1.18739i	1.18541	+	0.0548286i
$470$		0			0
$471$			6.79781i			0.313227i
$472$		0			0
$473$	−0.213205			−0.00980317
$474$		0			0
$475$	−62.4376			−2.86483
$476$		0			0
$477$		−	4.81985i		−	0.220686i
$478$		0			0
$479$	37.1934			1.69941			0.849706	−	0.527257i	$-0.176779\pi$
$479$	37.1934			1.69941			0.849706	+	0.527257i	$0.176779\pi$
$480$		0			0
$481$	6.75280			0.307901
$482$		0			0
$483$	−1.10033	+	12.6408i	−0.0500670	+	0.575175i
$484$		0			0
$485$	−46.1596			−2.09600
$486$		0			0
$487$	2.93800			0.133134			0.0665668	−	0.997782i	$-0.478795\pi$
$487$	2.93800			0.133134			0.0665668	+	0.997782i	$0.478795\pi$
$488$		0			0
$489$		−	18.4836i		−	0.835856i
$490$		0			0
$491$	−17.5214			−0.790729			−0.395364	−	0.918524i	$-0.629382\pi$
$491$	−17.5214			−0.790729			−0.395364	+	0.918524i	$0.629382\pi$
$492$		0			0
$493$	−6.49206			−0.292388
$494$		0			0
$495$			4.73690i			0.212908i
$496$		0			0
$497$	1.18902	−	25.7071i	0.0533350	−	1.15312i
$498$		0			0
$499$	−3.22468			−0.144357			−0.0721783	−	0.997392i	$-0.522995\pi$
$499$	−3.22468			−0.144357			−0.0721783	+	0.997392i	$0.522995\pi$
$500$		0			0
$501$	−6.48821			−0.289872
$502$		0			0
$503$	0.297856			0.0132807			0.00664037	−	0.999978i	$-0.497886\pi$
$503$	0.297856			0.0132807			0.00664037	+	0.999978i	$0.497886\pi$
$504$		0			0
$505$		−	0.833770i		−	0.0371023i
$506$		0			0
$507$		−	12.1171i		−	0.538140i
$508$		0			0
$509$		−	2.46804i		−	0.109394i	−0.998503	−	0.0546970i	$-0.982581\pi$
$509$		−	2.46804i		−	0.109394i	0.998503	−	0.0546970i	$-0.0174193\pi$
$510$		0			0
$511$	−31.6393	−	1.46340i	−1.39964	−	0.0647372i
$512$		0			0
$513$		−	7.57909i		−	0.334625i
$514$		0			0
$515$	52.0992			2.29576
$516$		0			0
$517$	−4.53953			−0.199648
$518$		0			0
$519$	−9.64653			−0.423435
$520$		0			0
$521$	−19.0579			−0.834941			−0.417470	−	0.908690i	$-0.637083\pi$
$521$	−19.0579			−0.834941			−0.417470	+	0.908690i	$0.637083\pi$
$522$		0			0
$523$	16.6181			0.726658			0.363329	−	0.931661i	$-0.381640\pi$
$523$	16.6181			0.726658			0.363329	+	0.931661i	$0.381640\pi$
$524$		0			0
$525$	−21.7728	−	1.00705i	−0.950242	−	0.0439513i
$526$		0			0
$527$		−	24.1960i		−	1.05400i
$528$		0			0
$529$	−22.9242	−	1.86590i	−0.996704	−	0.0811263i
$530$		0			0
$531$		−	9.59814i		−	0.416524i
$532$		0			0
$533$	−7.13983			−0.309260
$534$		0			0
$535$			51.8387i			2.24118i
$536$		0			0
$537$			12.3782i			0.534161i
$538$		0			0
$539$	−0.841238	+	9.07446i	−0.0362347	+	0.390865i
$540$		0			0
$541$	−20.9403			−0.900295			−0.450147	−	0.892954i	$-0.648629\pi$
$541$	−20.9403			−0.900295			−0.450147	+	0.892954i	$0.648629\pi$
$542$		0			0
$543$			7.39251i			0.317243i
$544$		0			0
$545$		−	20.7432i		−	0.888540i
$546$		0			0
$547$	−35.6100			−1.52257			−0.761286	−	0.648416i	$-0.775432\pi$
$547$	−35.6100			−1.52257			−0.761286	+	0.648416i	$0.775432\pi$
$548$		0			0
$549$	−5.67350			−0.242139
$550$		0			0
$551$	8.26824			0.352239
$552$		0			0
$553$	0.216030	+	0.00999197i	0.00918652	+	0.000424902i
$554$		0			0
$555$	26.1485			1.10994
$556$		0			0
$557$		−	35.3291i		−	1.49694i	−0.663169	−	0.748470i	$-0.730789\pi$
$557$		−	35.3291i		−	1.49694i	0.663169	−	0.748470i	$-0.269211\pi$
$558$		0			0
$559$	−0.153874			−0.00650819
$560$		0			0
$561$	−7.74761			−0.327104
$562$		0			0
$563$	3.09012			0.130233			0.0651165	−	0.997878i	$-0.479258\pi$
$563$	3.09012			0.130233			0.0651165	+	0.997878i	$0.479258\pi$
$564$		0			0
$565$			21.4790i			0.903626i
$566$		0			0
$567$	0.122243	−	2.64293i	0.00513371	−	0.110992i
$568$		0			0
$569$		−	38.9189i		−	1.63156i	−0.578360	−	0.815782i	$-0.696307\pi$
$569$		−	38.9189i		−	1.63156i	0.578360	−	0.815782i	$-0.303693\pi$
$570$		0			0
$571$		−	27.7218i		−	1.16012i	−0.814574	−	0.580060i	$-0.803029\pi$
$571$		−	27.7218i		−	1.16012i	0.814574	−	0.580060i	$-0.196971\pi$
$572$		0			0
$573$	−15.1768			−0.634021
$574$		0			0
$575$	1.60392	−	39.4761i	0.0668881	−	1.64627i
$576$		0			0
$577$			25.5830i			1.06503i	0.846419	+	0.532517i	$0.178754\pi$
$577$			25.5830i			1.06503i	−0.846419	+	0.532517i	$0.821246\pi$
$578$		0			0
$579$			6.41857i			0.266747i
$580$		0			0
$581$	−1.40854	+	30.4530i	−0.0584360	+	1.26340i
$582$		0			0
$583$	6.27501			0.259884
$584$		0			0
$585$			3.41872i			0.141347i
$586$		0			0
$587$		−	17.7845i		−	0.734043i	−0.930212	−	0.367022i	$-0.880378\pi$
$587$		−	17.7845i		−	0.734043i	0.930212	−	0.367022i	$-0.119622\pi$
$588$		0			0
$589$			30.8159i			1.26975i
$590$		0			0
$591$		−	7.85892i		−	0.323273i
$592$		0			0
$593$		−	35.1899i		−	1.44508i	−0.691332	−	0.722538i	$-0.742976\pi$
$593$		−	35.1899i		−	1.44508i	0.691332	−	0.722538i	$-0.257024\pi$
$594$		0			0
$595$	2.64681	−	57.2249i	0.108509	−	2.34599i
$596$		0			0
$597$		−	6.12725i		−	0.250772i
$598$		0			0
$599$	−24.3500			−0.994912			−0.497456	−	0.867489i	$-0.665732\pi$
$599$	−24.3500			−0.994912			−0.497456	+	0.867489i	$0.665732\pi$
$600$		0			0
$601$		−	45.8988i		−	1.87225i	−0.351666	−	0.936126i	$-0.614385\pi$
$601$		−	45.8988i		−	1.87225i	0.351666	−	0.936126i	$-0.385615\pi$
$602$		0			0
$603$		−	9.71338i		−	0.395559i
$604$		0			0
$605$	33.8557			1.37643
$606$		0			0
$607$		−	40.2085i		−	1.63201i	−0.578042	−	0.816007i	$-0.696183\pi$
$607$		−	40.2085i		−	1.63201i	0.578042	−	0.816007i	$-0.303817\pi$
$608$		0			0
$609$	2.88324	+	0.133358i	0.116835	+	0.00540393i
$610$		0			0
$611$	−3.27627			−0.132544
$612$		0			0
$613$			22.1857i			0.896071i	0.894016	+	0.448036i	$0.147876\pi$
$613$			22.1857i			0.896071i	−0.894016	+	0.448036i	$0.852124\pi$
$614$		0			0
$615$	−27.6472			−1.11484
$616$		0			0
$617$		−	15.1120i		−	0.608387i	−0.952610	−	0.304193i	$-0.901613\pi$
$617$		−	15.1120i		−	0.608387i	0.952610	−	0.304193i	$-0.0983869\pi$
$618$		0			0
$619$	−39.3761			−1.58266			−0.791329	−	0.611390i	$-0.790611\pi$
$619$	−39.3761			−1.58266			−0.791329	+	0.611390i	$0.790611\pi$
$620$		0			0
$621$	4.79188	+	0.194694i	0.192291	+	0.00781282i
$622$		0			0
$623$	1.20613	−	26.0770i	0.0483227	−	1.04475i
$624$		0			0
$625$	1.67622			0.0670488
$626$		0			0
$627$	9.86729			0.394062
$628$		0			0
$629$			42.7681i			1.70528i
$630$		0			0
$631$		−	19.5806i		−	0.779490i	−0.920923	−	0.389745i	$-0.872563\pi$
$631$		−	19.5806i		−	0.779490i	0.920923	−	0.389745i	$-0.127437\pi$
$632$		0			0
$633$		−	7.23950i		−	0.287744i
$634$		0			0
$635$	12.9872			0.515381
$636$		0			0
$637$	−0.607139	+	6.54923i	−0.0240557	+	0.259490i
$638$		0			0
$639$	−9.72675			−0.384784
$640$		0			0
$641$			41.1131i			1.62387i	0.583748	+	0.811935i	$0.301586\pi$
$641$			41.1131i			1.62387i	−0.583748	+	0.811935i	$0.698414\pi$
$642$		0			0
$643$	17.4583			0.688487			0.344243	−	0.938880i	$-0.388135\pi$
$643$	17.4583			0.688487			0.344243	+	0.938880i	$0.388135\pi$
$644$		0			0
$645$	−0.595840			−0.0234612
$646$		0			0
$647$			28.1146i			1.10530i	0.833414	+	0.552650i	$0.186383\pi$
$647$			28.1146i			1.10530i	−0.833414	+	0.552650i	$0.813617\pi$
$648$		0			0
$649$	12.4959			0.490507
$650$		0			0
$651$	−0.497027	+	10.7459i	−0.0194800	+	0.421165i
$652$		0			0
$653$	11.1125			0.434864			0.217432	−	0.976075i	$-0.430232\pi$
$653$	11.1125			0.434864			0.217432	+	0.976075i	$0.430232\pi$
$654$		0			0
$655$		−	33.2778i		−	1.30027i
$656$		0			0
$657$			11.9713i			0.467045i
$658$		0			0
$659$		−	8.73700i		−	0.340345i	−0.985414	−	0.170173i	$-0.945567\pi$
$659$		−	8.73700i		−	0.340345i	0.985414	−	0.170173i	$-0.0544325\pi$
$660$		0			0
$661$	27.7390			1.07892			0.539461	−	0.842010i	$-0.318628\pi$
$661$	27.7390			1.07892			0.539461	+	0.842010i	$0.318628\pi$
$662$		0			0
$663$	−5.59161			−0.217160
$664$		0			0
$665$	−3.37096	+	72.8812i	−0.130720	+	2.82621i
$666$		0			0
$667$	−0.212398	+	5.22759i	−0.00822407	+	0.202413i
$668$		0			0
$669$	−14.1188			−0.545863
$670$		0			0
$671$		−	7.38638i		−	0.285148i
$672$		0			0
$673$	21.8006			0.840353			0.420177	−	0.907442i	$-0.361968\pi$
$673$	21.8006			0.840353			0.420177	+	0.907442i	$0.361968\pi$
$674$		0			0
$675$			8.23814i			0.317086i
$676$		0			0
$677$	−5.69171			−0.218750			−0.109375	−	0.994001i	$-0.534885\pi$
$677$	−5.69171			−0.218750			−0.109375	+	0.994001i	$0.534885\pi$
$678$		0			0
$679$	−1.55086	+	33.5300i	−0.0595164	+	1.28676i
$680$		0			0
$681$			8.89366i			0.340806i
$682$		0			0
$683$	−37.7523			−1.44455			−0.722275	−	0.691606i	$-0.756904\pi$
$683$	−37.7523			−1.44455			−0.722275	+	0.691606i	$0.756904\pi$
$684$		0			0
$685$			38.7171i			1.47931i
$686$		0			0
$687$		−	24.4390i		−	0.932407i
$688$		0			0
$689$	4.52880			0.172534
$690$		0			0
$691$			43.8174i			1.66689i	0.552602	+	0.833445i	$0.313635\pi$
$691$			43.8174i			1.66689i	−0.552602	+	0.833445i	$0.686365\pi$
$692$		0			0
$693$	3.44085	+	0.159149i	0.130707	+	0.00604557i
$694$		0			0
$695$		−	19.8584i		−	0.753273i
$696$		0			0
$697$		−	45.2194i		−	1.71281i
$698$		0			0
$699$			13.8624i			0.524325i
$700$		0			0
$701$			38.3110i			1.44699i	0.690331	+	0.723494i	$0.257465\pi$
$701$			38.3110i			1.44699i	−0.690331	+	0.723494i	$0.742535\pi$
$702$		0			0
$703$		−	54.4692i		−	2.05434i
$704$		0			0
$705$	−12.6866			−0.477803
$706$		0			0
$707$	−0.605645	−	0.0280127i	−0.0227776	−	0.00105353i
$708$		0			0
$709$			36.4971i			1.37068i	0.728224	+	0.685339i	$0.240346\pi$
$709$			36.4971i			1.37068i	−0.728224	+	0.685339i	$0.759654\pi$
$710$		0			0
$711$		−	0.0817388i		−	0.00306545i
$712$		0			0
$713$	−19.4833	−	0.791609i	−0.729656	−	0.0296460i
$714$		0			0
$715$	−4.45087			−0.166453
$716$		0			0
$717$		−	16.7231i		−	0.624534i
$718$		0			0
$719$		−	14.6988i		−	0.548171i	−0.961705	−	0.274086i	$-0.911625\pi$
$719$		−	14.6988i		−	0.548171i	0.961705	−	0.274086i	$-0.0883752\pi$
$720$		0			0
$721$	1.75041	−	37.8444i	0.0651887	−	1.40940i
$722$		0			0
$723$		−	0.874251i		−	0.0325138i
$724$		0			0
$725$	−8.98721			−0.333777
$726$		0			0
$727$	−19.6778			−0.729811			−0.364905	−	0.931045i	$-0.618899\pi$
$727$	−19.6778			−0.729811			−0.364905	+	0.931045i	$0.618899\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	0.974547i		−	0.0360449i
$732$		0			0
$733$	15.6429			0.577785			0.288893	−	0.957362i	$-0.406713\pi$
$733$	15.6429			0.577785			0.288893	+	0.957362i	$0.406713\pi$
$734$		0			0
$735$	−2.35099	+	25.3602i	−0.0867177	+	0.935426i
$736$		0			0
$737$	12.6459			0.465819
$738$		0			0
$739$	2.40027			0.0882954			0.0441477	−	0.999025i	$-0.485943\pi$
$739$	2.40027			0.0882954			0.0441477	+	0.999025i	$0.485943\pi$
$740$		0			0
$741$	7.12143			0.261612
$742$		0			0
$743$			23.8234i			0.873994i	0.899463	+	0.436997i	$0.143958\pi$
$743$			23.8234i			0.873994i	−0.899463	+	0.436997i	$0.856042\pi$
$744$		0			0
$745$		−	35.2712i		−	1.29224i
$746$		0			0
$747$	11.5225			0.421585
$748$		0			0
$749$	37.6553	+	1.74166i	1.37589	+	0.0636389i
$750$		0			0
$751$			26.9836i			0.984644i	0.870413	+	0.492322i	$0.163852\pi$
$751$			26.9836i			0.984644i	−0.870413	+	0.492322i	$0.836148\pi$
$752$		0			0
$753$			22.3262i			0.813614i
$754$		0			0
$755$	−64.1257			−2.33377
$756$		0			0
$757$			47.7940i			1.73710i	0.495600	+	0.868551i	$0.334948\pi$
$757$			47.7940i			1.73710i	−0.495600	+	0.868551i	$0.665052\pi$
$758$		0			0
$759$	−0.253475	+	6.23859i	−0.00920055	+	0.226447i
$760$		0			0
$761$		−	39.6828i		−	1.43850i	−0.694750	−	0.719251i	$-0.744485\pi$
$761$		−	39.6828i		−	1.43850i	0.694750	−	0.719251i	$-0.255515\pi$
$762$		0			0
$763$	−15.0677	−	0.696923i	−0.545487	−	0.0252303i
$764$		0			0
$765$	−21.6521			−0.782834
$766$		0			0
$767$	9.01856			0.325641
$768$		0			0
$769$	15.4471			0.557037			0.278519	−	0.960431i	$-0.410157\pi$
$769$	15.4471			0.557037			0.278519	+	0.960431i	$0.410157\pi$
$770$		0			0
$771$	−6.66094			−0.239888
$772$		0			0
$773$	−21.6351			−0.778160			−0.389080	−	0.921204i	$-0.627207\pi$
$773$	−21.6351			−0.778160			−0.389080	+	0.921204i	$0.627207\pi$
$774$		0			0
$775$		−	33.4955i		−	1.20319i
$776$		0			0
$777$	0.878530	−	18.9941i	0.0315171	−	0.681409i
$778$		0			0
$779$			57.5910i			2.06341i
$780$		0			0
$781$		−	12.6633i		−	0.453130i
$782$		0			0
$783$		−	1.09093i		−	0.0389866i
$784$		0			0
$785$	−24.7333			−0.882771
$786$		0			0
$787$	35.0043			1.24777			0.623885	−	0.781516i	$-0.285553\pi$
$787$	35.0043			1.24777			0.623885	+	0.781516i	$0.285553\pi$
$788$		0			0
$789$	20.6827			0.736325
$790$		0			0
$791$	15.6022	+	0.721643i	0.554749	+	0.0256587i
$792$		0			0
$793$		−	5.33090i		−	0.189306i
$794$		0			0
$795$	17.5367			0.621961
$796$		0			0
$797$	20.0786			0.711220			0.355610	−	0.934635i	$-0.384273\pi$
$797$	20.0786			0.711220			0.355610	+	0.934635i	$0.384273\pi$
$798$		0			0
$799$		−	20.7499i		−	0.734080i
$800$		0			0
$801$	−9.86670			−0.348623
$802$		0			0
$803$	−15.5856			−0.550002
$804$		0			0
$805$	−45.9925	−	4.00349i	−1.62102	−	0.141104i
$806$		0			0
$807$	22.7936			0.802374
$808$		0			0
$809$	−19.4535			−0.683948			−0.341974	−	0.939709i	$-0.611096\pi$
$809$	−19.4535			−0.683948			−0.341974	+	0.939709i	$0.611096\pi$
$810$		0			0
$811$			9.50360i			0.333717i	0.985981	+	0.166858i	$0.0533623\pi$
$811$			9.50360i			0.333717i	−0.985981	+	0.166858i	$0.946638\pi$
$812$		0			0
$813$	23.8831			0.837617
$814$		0			0
$815$	67.2511			2.35570
$816$		0			0
$817$			1.24118i			0.0434232i
$818$		0			0
$819$	2.48333	+	0.114861i	0.0867747	+	0.00401357i
$820$		0			0
$821$	33.4896			1.16880			0.584398	−	0.811468i	$-0.301331\pi$
$821$	33.4896			1.16880			0.584398	+	0.811468i	$0.301331\pi$
$822$		0			0
$823$	−32.5552			−1.13480			−0.567402	−	0.823441i	$-0.692051\pi$
$823$	−32.5552			−1.13480			−0.567402	+	0.823441i	$0.692051\pi$
$824$		0			0
$825$	−10.7253			−0.373407
$826$		0			0
$827$		−	0.917615i		−	0.0319086i	−0.999873	−	0.0159543i	$-0.994921\pi$
$827$		−	0.917615i		−	0.0319086i	0.999873	−	0.0159543i	$-0.00507863\pi$
$828$		0			0
$829$		−	20.4626i		−	0.710694i	−0.934735	−	0.355347i	$-0.884363\pi$
$829$		−	20.4626i		−	0.710694i	0.934735	−	0.355347i	$-0.115637\pi$
$830$		0			0
$831$		−	20.6653i		−	0.716871i
$832$		0			0
$833$	−41.4788	−	3.84525i	−1.43716	−	0.133230i
$834$		0			0
$835$		−	23.6069i		−	0.816950i
$836$		0			0
$837$	4.06591			0.140538
$838$		0			0
$839$	51.4869			1.77753			0.888763	−	0.458366i	$-0.151565\pi$
$839$	51.4869			1.77753			0.888763	+	0.458366i	$0.151565\pi$
$840$		0			0
$841$	−27.8099			−0.958961
$842$		0			0
$843$	29.1238			1.00308
$844$		0			0
$845$	44.0872			1.51665
$846$		0			0
$847$	1.13747	−	24.5925i	0.0390840	−	0.845008i
$848$		0			0
$849$		−	33.2160i		−	1.13997i
$850$		0			0
$851$	34.4381	+	1.39922i	1.18052	+	0.0479648i
$852$		0			0
$853$			14.0958i			0.482632i	0.970447	+	0.241316i	$0.0775790\pi$
$853$			14.0958i			0.482632i	−0.970447	+	0.241316i	$0.922421\pi$
$854$		0			0
$855$	27.5760			0.943078
$856$		0			0
$857$		−	49.0859i		−	1.67674i	−0.545099	−	0.838371i	$-0.683508\pi$
$857$		−	49.0859i		−	1.67674i	0.545099	−	0.838371i	$-0.316492\pi$
$858$		0			0
$859$		−	46.7930i		−	1.59656i	−0.602288	−	0.798279i	$-0.705744\pi$
$859$		−	46.7930i		−	1.59656i	0.602288	−	0.798279i	$-0.294256\pi$
$860$		0			0
$861$	−0.928882	+	20.0827i	−0.0316562	+	0.684417i
$862$		0			0
$863$	37.0112			1.25987			0.629937	−	0.776646i	$-0.283081\pi$
$863$	37.0112			1.25987			0.629937	+	0.776646i	$0.283081\pi$
$864$		0			0
$865$		−	35.0982i		−	1.19337i
$866$		0			0
$867$		−	18.4139i		−	0.625368i
$868$		0			0
$869$	0.106417			0.00360994
$870$		0			0
$871$	9.12684			0.309251
$872$		0			0
$873$	12.6867			0.429380
$874$		0			0
$875$	1.44023	−	31.1382i	0.0486886	−	1.05266i
$876$		0			0
$877$	−23.0244			−0.777479			−0.388739	−	0.921348i	$-0.627089\pi$
$877$	−23.0244			−0.777479			−0.388739	+	0.921348i	$0.627089\pi$
$878$		0			0
$879$			9.20987i			0.310641i
$880$		0			0
$881$	14.2414			0.479805			0.239902	−	0.970797i	$-0.422885\pi$
$881$	14.2414			0.479805			0.239902	+	0.970797i	$0.422885\pi$
$882$		0			0
$883$	−48.4134			−1.62924			−0.814620	−	0.579994i	$-0.803055\pi$
$883$	−48.4134			−1.62924			−0.814620	+	0.579994i	$0.803055\pi$
$884$		0			0
$885$	34.9221			1.17389
$886$		0			0
$887$			11.7253i			0.393697i	0.980434	+	0.196849i	$0.0630707\pi$
$887$			11.7253i			0.393697i	−0.980434	+	0.196849i	$0.936929\pi$
$888$		0			0
$889$	0.436340	−	9.43381i	0.0146344	−	0.316400i
$890$		0			0
$891$		−	1.30191i		−	0.0436156i
$892$		0			0
$893$			26.4270i			0.884345i
$894$		0			0
$895$	−45.0373			−1.50543
$896$		0			0
$897$	−0.182938	+	4.50252i	−0.00610812	+	0.150335i
$898$		0			0
$899$			4.43561i			0.147936i
$900$		0			0
$901$			28.6827i			0.955560i
$902$		0			0
$903$	−0.0200188	+	0.432814i	−0.000666185	+	0.0144031i
$904$		0			0
$905$	−26.8971			−0.894089
$906$		0			0
$907$			43.3773i			1.44032i	0.693809	+	0.720159i	$0.255931\pi$
$907$			43.3773i			1.44032i	−0.693809	+	0.720159i	$0.744069\pi$
$908$		0			0
$909$			0.229157i			0.00760065i
$910$		0			0
$911$		−	23.8808i		−	0.791207i	−0.918421	−	0.395604i	$-0.870535\pi$
$911$		−	23.8808i		−	0.791207i	0.918421	−	0.395604i	$-0.129465\pi$
$912$		0			0
$913$			15.0012i			0.496468i
$914$		0			0
$915$		−	20.6426i		−	0.682423i
$916$		0			0
$917$	−24.1728	−	1.11806i	−0.798256	−	0.0369215i
$918$		0			0
$919$			4.77818i			0.157618i	0.996890	+	0.0788088i	$0.0251117\pi$
$919$			4.77818i			0.157618i	−0.996890	+	0.0788088i	$0.974888\pi$
$920$		0			0
$921$	17.0684			0.562422
$922$		0			0
$923$		−	9.13940i		−	0.300827i
$924$		0			0
$925$			59.2056i			1.94667i
$926$		0			0
$927$	−14.3191			−0.470303
$928$		0			0
$929$			26.6681i			0.874953i	0.899230	+	0.437476i	$0.144128\pi$
$929$			26.6681i			0.874953i	−0.899230	+	0.437476i	$0.855872\pi$
$930$		0			0
$931$	52.8271	+	4.89728i	1.73134	+	0.160502i
$932$		0			0
$933$	−14.0489			−0.459939
$934$		0			0
$935$		−	28.1891i		−	0.921882i
$936$		0			0
$937$	11.3961			0.372294			0.186147	−	0.982522i	$-0.440400\pi$
$937$	11.3961			0.372294			0.186147	+	0.982522i	$0.440400\pi$
$938$		0			0
$939$			5.96783i			0.194753i
$940$		0			0
$941$	−27.8062			−0.906456			−0.453228	−	0.891395i	$-0.649728\pi$
$941$	−27.8062			−0.906456			−0.453228	+	0.891395i	$0.649728\pi$
$942$		0			0
$943$	−36.4119	−	1.47942i	−1.18573	−	0.0481765i
$944$		0			0
$945$	9.61609	+	0.444771i	0.312811	+	0.0144684i
$946$		0			0
$947$	16.5153			0.536676			0.268338	−	0.963325i	$-0.413526\pi$
$947$	16.5153			0.536676			0.268338	+	0.963325i	$0.413526\pi$
$948$		0			0
$949$	−11.2484			−0.365139
$950$		0			0
$951$			23.5015i			0.762088i
$952$		0			0
$953$		−	16.1459i		−	0.523019i	−0.965201	−	0.261509i	$-0.915780\pi$
$953$		−	16.1459i		−	0.523019i	0.965201	−	0.261509i	$-0.0842202\pi$
$954$		0			0
$955$		−	55.2197i		−	1.78687i
$956$		0			0
$957$	1.42029			0.0459114
$958$		0			0
$959$	28.1239	+	1.30081i	0.908166	+	0.0420052i
$960$		0			0
$961$	14.4684			0.466723
$962$		0			0
$963$		−	14.2476i		−	0.459122i
$964$		0			0
$965$	−23.3535			−0.751775
$966$		0			0
$967$	6.00388			0.193072			0.0965359	−	0.995330i	$-0.469224\pi$
$967$	6.00388			0.193072			0.0965359	+	0.995330i	$0.469224\pi$
$968$		0			0
$969$			45.1028i			1.44891i
$970$		0			0
$971$	2.27132			0.0728902			0.0364451	−	0.999336i	$-0.488397\pi$
$971$	2.27132			0.0728902			0.0364451	+	0.999336i	$0.488397\pi$
$972$		0			0
$973$	−14.4250	−	0.667197i	−0.462445	−	0.0213894i
$974$		0			0
$975$	−7.74068			−0.247900
$976$		0			0
$977$			14.2965i			0.457386i	0.973499	+	0.228693i	$0.0734452\pi$
$977$			14.2965i			0.457386i	−0.973499	+	0.228693i	$0.926555\pi$
$978$		0			0
$979$		−	12.8456i		−	0.410546i
$980$		0			0
$981$			5.70114i			0.182023i
$982$		0			0
$983$	−20.2331			−0.645337			−0.322668	−	0.946512i	$-0.604580\pi$
$983$	−20.2331			−0.645337			−0.322668	+	0.946512i	$0.604580\pi$
$984$		0			0
$985$	28.5941			0.911084
$986$		0			0
$987$	−0.426239	+	9.21542i	−0.0135673	+	0.293330i
$988$		0			0
$989$	−0.784733	−	0.0318838i	−0.0249531	−	0.00101385i
$990$		0			0
$991$	−12.5716			−0.399351			−0.199675	−	0.979862i	$-0.563989\pi$
$991$	−12.5716			−0.399351			−0.199675	+	0.979862i	$0.563989\pi$
$992$		0			0
$993$		−	27.4401i		−	0.870784i
$994$		0			0
$995$	22.2935			0.706752
$996$		0			0
$997$		−	28.7007i		−	0.908960i	−0.890757	−	0.454480i	$-0.849825\pi$
$997$		−	28.7007i		−	0.908960i	0.890757	−	0.454480i	$-0.150175\pi$
$998$		0			0
$999$	−7.18677			−0.227379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.6	yes	32
3.2	odd	2		5796.2.k.d.5473.3		32
7.6	odd	2	inner	1932.2.k.a.1609.7	yes	32
21.20	even	2		5796.2.k.d.5473.29		32
23.22	odd	2	inner	1932.2.k.a.1609.5	✓	32
69.68	even	2		5796.2.k.d.5473.30		32
161.160	even	2	inner	1932.2.k.a.1609.8	yes	32
483.482	odd	2		5796.2.k.d.5473.4		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.5	✓	32	23.22	odd	2	inner
1932.2.k.a.1609.6	yes	32	1.1	even	1	trivial
1932.2.k.a.1609.7	yes	32	7.6	odd	2	inner
1932.2.k.a.1609.8	yes	32	161.160	even	2	inner
5796.2.k.d.5473.3		32	3.2	odd	2
5796.2.k.d.5473.4		32	483.482	odd	2
5796.2.k.d.5473.29		32	21.20	even	2
5796.2.k.d.5473.30		32	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +3.63843 q^{5} +(0.122243 - 2.64293i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +3.63843 q^{5} +(0.122243 - 2.64293i) q^{7} -1.00000 q^{9} -1.30191i q^{11} -0.939615i q^{13} -3.63843i q^{15} +5.95096 q^{17} -7.57909 q^{19} +(-2.64293 - 0.122243i) q^{21} +(0.194694 - 4.79188i) q^{23} +8.23814 q^{25} +1.00000i q^{27} -1.09093 q^{29} -4.06591i q^{31} -1.30191 q^{33} +(0.444771 - 9.61609i) q^{35} +7.18677i q^{37} -0.939615 q^{39} -7.59867i q^{41} -0.163763i q^{43} -3.63843 q^{45} -3.48683i q^{47} +(-6.97011 - 0.646157i) q^{49} -5.95096i q^{51} +4.81985i q^{53} -4.73690i q^{55} +7.57909i q^{57} +9.59814i q^{59} +5.67350 q^{61} +(-0.122243 + 2.64293i) q^{63} -3.41872i q^{65} +9.71338i q^{67} +(-4.79188 - 0.194694i) q^{69} +9.72675 q^{71} -11.9713i q^{73} -8.23814i q^{75} +(-3.44085 - 0.159149i) q^{77} +0.0817388i q^{79} +1.00000 q^{81} -11.5225 q^{83} +21.6521 q^{85} +1.09093i q^{87} +9.86670 q^{89} +(-2.48333 - 0.114861i) q^{91} -4.06591 q^{93} -27.5760 q^{95} -12.6867 q^{97} +1.30191i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 32 * q - 32 * q^9	\( 32 q - 32 q^{9} + 12 q^{23} + 40 q^{25} - 16 q^{29} + 8 q^{35} + 32 q^{71} + 24 q^{77} + 32 q^{81} + 8 q^{85} + 8 q^{93} - 64 q^{95}+O(q^{100}) \) 32 * q - 32 * q^9 + 12 * q^23 + 40 * q^25 - 16 * q^29 + 8 * q^35 + 32 * q^71 + 24 * q^77 + 32 * q^81 + 8 * q^85 + 8 * q^93 - 64 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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