LMFDB - Embedded newform 1932.2.k.a.1609.20

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			1609.20
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.12739 q^{5} +(-2.43376 + 1.03769i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +2.12739 q^{5} +(-2.43376 + 1.03769i) q^{7} -1.00000 q^{9} -1.02219i q^{11} -1.76109i q^{13} +2.12739i q^{15} -0.210591 q^{17} +4.64139 q^{19} +(-1.03769 - 2.43376i) q^{21} +(3.56324 + 3.20988i) q^{23} -0.474227 q^{25} -1.00000i q^{27} +10.2285 q^{29} +1.19079i q^{31} +1.02219 q^{33} +(-5.17755 + 2.20758i) q^{35} +10.7727i q^{37} +1.76109 q^{39} +1.14009i q^{41} +9.64945i q^{43} -2.12739 q^{45} +5.56619i q^{47} +(4.84638 - 5.05100i) q^{49} -0.210591i q^{51} -1.32211i q^{53} -2.17459i q^{55} +4.64139i q^{57} +9.90536i q^{59} -2.05584 q^{61} +(2.43376 - 1.03769i) q^{63} -3.74651i q^{65} +2.73382i q^{67} +(-3.20988 + 3.56324i) q^{69} +6.57204 q^{71} -12.6369i q^{73} -0.474227i q^{75} +(1.06072 + 2.48776i) q^{77} -1.45891i q^{79} +1.00000 q^{81} -6.66137 q^{83} -0.448008 q^{85} +10.2285i q^{87} -6.85521 q^{89} +(1.82747 + 4.28606i) q^{91} -1.19079 q^{93} +9.87404 q^{95} -1.67955 q^{97} +1.02219i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.12739			0.951396			0.475698	−	0.879609i	$-0.342196\pi$
$5$	2.12739			0.951396			0.475698	+	0.879609i	$0.342196\pi$
$6$		0			0
$7$	−2.43376	+	1.03769i	−0.919875	+	0.392212i
$7$	−2.43376	+	1.03769i	−0.919875	+	0.392212i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	1.02219i		−	0.308202i	−0.988055	−	0.154101i	$-0.950752\pi$
$11$		−	1.02219i		−	0.308202i	0.988055	−	0.154101i	$-0.0492481\pi$
$12$		0			0
$13$		−	1.76109i		−	0.488437i	−0.969720	−	0.244219i	$-0.921469\pi$
$13$		−	1.76109i		−	0.488437i	0.969720	−	0.244219i	$-0.0785315\pi$
$14$		0			0
$15$			2.12739i			0.549289i
$16$		0			0
$17$	−0.210591			−0.0510758			−0.0255379	−	0.999674i	$-0.508130\pi$
$17$	−0.210591			−0.0510758			−0.0255379	+	0.999674i	$0.508130\pi$
$18$		0			0
$19$	4.64139			1.06481			0.532404	−	0.846490i	$-0.321289\pi$
$19$	4.64139			1.06481			0.532404	+	0.846490i	$0.321289\pi$
$20$		0			0
$21$	−1.03769	−	2.43376i	−0.226444	−	0.531090i
$22$		0			0
$23$	3.56324	+	3.20988i	0.742986	+	0.669307i
$23$	3.56324	+	3.20988i	0.742986	+	0.669307i
$24$		0			0
$25$	−0.474227			−0.0948455
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	10.2285			1.89939			0.949697	−	0.313170i	$-0.101391\pi$
$29$	10.2285			1.89939			0.949697	+	0.313170i	$0.101391\pi$
$30$		0			0
$31$			1.19079i			0.213873i	0.994266	+	0.106936i	$0.0341041\pi$
$31$			1.19079i			0.213873i	−0.994266	+	0.106936i	$0.965896\pi$
$32$		0			0
$33$	1.02219			0.177940
$34$		0			0
$35$	−5.17755	+	2.20758i	−0.875165	+	0.373149i
$36$		0			0
$37$			10.7727i			1.77102i	0.464618	+	0.885511i	$0.346191\pi$
$37$			10.7727i			1.77102i	−0.464618	+	0.885511i	$0.653809\pi$
$38$		0			0
$39$	1.76109			0.281999
$40$		0			0
$41$			1.14009i			0.178051i	0.996029	+	0.0890257i	$0.0283753\pi$
$41$			1.14009i			0.178051i	−0.996029	+	0.0890257i	$0.971625\pi$
$42$		0			0
$43$			9.64945i			1.47153i	0.677238	+	0.735764i	$0.263177\pi$
$43$			9.64945i			1.47153i	−0.677238	+	0.735764i	$0.736823\pi$
$44$		0			0
$45$	−2.12739			−0.317132
$46$		0			0
$47$			5.56619i			0.811913i	0.913892	+	0.405956i	$0.133061\pi$
$47$			5.56619i			0.811913i	−0.913892	+	0.405956i	$0.866939\pi$
$48$		0			0
$49$	4.84638	−	5.05100i	0.692340	−	0.721572i
$50$		0			0
$51$		−	0.210591i		−	0.0294886i
$52$		0			0
$53$		−	1.32211i		−	0.181605i	−0.995869	−	0.0908026i	$-0.971057\pi$
$53$		−	1.32211i		−	0.181605i	0.995869	−	0.0908026i	$-0.0289432\pi$
$54$		0			0
$55$		−	2.17459i		−	0.293222i
$56$		0			0
$57$			4.64139i			0.614768i
$58$		0			0
$59$			9.90536i			1.28957i	0.764365	+	0.644784i	$0.223053\pi$
$59$			9.90536i			1.28957i	−0.764365	+	0.644784i	$0.776947\pi$
$60$		0			0
$61$	−2.05584			−0.263224			−0.131612	−	0.991301i	$-0.542015\pi$
$61$	−2.05584			−0.263224			−0.131612	+	0.991301i	$0.542015\pi$
$62$		0			0
$63$	2.43376	−	1.03769i	0.306625	−	0.130737i
$64$		0			0
$65$		−	3.74651i		−	0.464697i
$66$		0			0
$67$			2.73382i			0.333989i	0.985958	+	0.166994i	$0.0534062\pi$
$67$			2.73382i			0.333989i	−0.985958	+	0.166994i	$0.946594\pi$
$68$		0			0
$69$	−3.20988	+	3.56324i	−0.386425	+	0.428963i
$70$		0			0
$71$	6.57204			0.779958			0.389979	−	0.920824i	$-0.372482\pi$
$71$	6.57204			0.779958			0.389979	+	0.920824i	$0.372482\pi$
$72$		0			0
$73$		−	12.6369i		−	1.47904i	−0.673133	−	0.739522i	$-0.735052\pi$
$73$		−	12.6369i		−	1.47904i	0.673133	−	0.739522i	$-0.264948\pi$
$74$		0			0
$75$		−	0.474227i		−	0.0547591i
$76$		0			0
$77$	1.06072	+	2.48776i	0.120880	+	0.283507i
$78$		0			0
$79$		−	1.45891i		−	0.164140i	−0.996627	−	0.0820699i	$-0.973847\pi$
$79$		−	1.45891i		−	0.164140i	0.996627	−	0.0820699i	$-0.0261531\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−6.66137			−0.731180			−0.365590	−	0.930776i	$-0.619133\pi$
$83$	−6.66137			−0.731180			−0.365590	+	0.930776i	$0.619133\pi$
$84$		0			0
$85$	−0.448008			−0.0485933
$86$		0			0
$87$			10.2285i			1.09662i
$88$		0			0
$89$	−6.85521			−0.726650			−0.363325	−	0.931662i	$-0.618359\pi$
$89$	−6.85521			−0.726650			−0.363325	+	0.931662i	$0.618359\pi$
$90$		0			0
$91$	1.82747	+	4.28606i	0.191571	+	0.449301i
$92$		0			0
$93$	−1.19079			−0.123480
$94$		0			0
$95$	9.87404			1.01305
$96$		0			0
$97$	−1.67955			−0.170533			−0.0852663	−	0.996358i	$-0.527174\pi$
$97$	−1.67955			−0.170533			−0.0852663	+	0.996358i	$0.527174\pi$
$98$		0			0
$99$			1.02219i			0.102734i
$100$		0			0
$101$		−	11.5432i		−	1.14859i	−0.818647	−	0.574297i	$-0.805275\pi$
$101$		−	11.5432i		−	1.14859i	0.818647	−	0.574297i	$-0.194725\pi$
$102$		0			0
$103$	−11.7886			−1.16156			−0.580781	−	0.814060i	$-0.697253\pi$
$103$	−11.7886			−1.16156			−0.580781	+	0.814060i	$0.697253\pi$
$104$		0			0
$105$	−2.20758	−	5.17755i	−0.215438	−	0.505277i
$106$		0			0
$107$			18.1052i			1.75029i	0.483859	+	0.875146i	$0.339235\pi$
$107$			18.1052i			1.75029i	−0.483859	+	0.875146i	$0.660765\pi$
$108$		0			0
$109$		−	2.28162i		−	0.218540i	−0.994012	−	0.109270i	$-0.965149\pi$
$109$		−	2.28162i		−	0.218540i	0.994012	−	0.109270i	$-0.0348513\pi$
$110$		0			0
$111$	−10.7727			−1.02250
$112$		0			0
$113$			1.03963i			0.0977998i	0.998804	+	0.0488999i	$0.0155715\pi$
$113$			1.03963i			0.0977998i	−0.998804	+	0.0488999i	$0.984428\pi$
$114$		0			0
$115$	7.58038	+	6.82866i	0.706874	+	0.636776i
$116$		0			0
$117$			1.76109i			0.162812i
$118$		0			0
$119$	0.512528	−	0.218529i	0.0469834	−	0.0200325i
$120$		0			0
$121$	9.95513			0.905012
$122$		0			0
$123$	−1.14009			−0.102798
$124$		0			0
$125$	−11.6458			−1.04163
$126$		0			0
$127$	15.4167			1.36801			0.684005	−	0.729478i	$-0.260237\pi$
$127$	15.4167			1.36801			0.684005	+	0.729478i	$0.260237\pi$
$128$		0			0
$129$	−9.64945			−0.849587
$130$		0			0
$131$		−	0.484240i		−	0.0423083i	−0.999776	−	0.0211541i	$-0.993266\pi$
$131$		−	0.484240i		−	0.0423083i	0.999776	−	0.0211541i	$-0.00673407\pi$
$132$		0			0
$133$	−11.2960	+	4.81635i	−0.979491	+	0.417631i
$134$		0			0
$135$		−	2.12739i		−	0.183096i
$136$		0			0
$137$		−	10.2651i		−	0.877006i	−0.898730	−	0.438503i	$-0.855509\pi$
$137$		−	10.2651i		−	0.877006i	0.898730	−	0.438503i	$-0.144491\pi$
$138$		0			0
$139$			3.05403i			0.259040i	0.991577	+	0.129520i	$0.0413436\pi$
$139$			3.05403i			0.259040i	−0.991577	+	0.129520i	$0.958656\pi$
$140$		0			0
$141$	−5.56619			−0.468758
$142$		0			0
$143$	−1.80016			−0.150537
$144$		0			0
$145$	21.7601			1.80708
$146$		0			0
$147$	5.05100	+	4.84638i	0.416600	+	0.399723i
$148$		0			0
$149$		−	7.11262i		−	0.582689i	−0.956618	−	0.291344i	$-0.905897\pi$
$149$		−	7.11262i		−	0.582689i	0.956618	−	0.291344i	$-0.0941025\pi$
$150$		0			0
$151$	4.67321			0.380300			0.190150	−	0.981755i	$-0.439103\pi$
$151$	4.67321			0.380300			0.190150	+	0.981755i	$0.439103\pi$
$152$		0			0
$153$	0.210591			0.0170253
$154$		0			0
$155$			2.53328i			0.203478i
$156$		0			0
$157$	11.1758			0.891928			0.445964	−	0.895051i	$-0.352861\pi$
$157$	11.1758			0.891928			0.445964	+	0.895051i	$0.352861\pi$
$158$		0			0
$159$	1.32211			0.104850
$160$		0			0
$161$	−12.0029	−	4.11454i	−0.945964	−	0.324271i
$162$		0			0
$163$	−1.03867			−0.0813550			−0.0406775	−	0.999172i	$-0.512952\pi$
$163$	−1.03867			−0.0813550			−0.0406775	+	0.999172i	$0.512952\pi$
$164$		0			0
$165$	2.17459			0.169292
$166$		0			0
$167$		−	11.2150i		−	0.867844i	−0.900951	−	0.433922i	$-0.857129\pi$
$167$		−	11.2150i		−	0.867844i	0.900951	−	0.433922i	$-0.142871\pi$
$168$		0			0
$169$	9.89858			0.761429
$170$		0			0
$171$	−4.64139			−0.354936
$172$		0			0
$173$			17.1904i			1.30697i	0.756941	+	0.653483i	$0.226693\pi$
$173$			17.1904i			1.30697i	−0.756941	+	0.653483i	$0.773307\pi$
$174$		0			0
$175$	1.15416	−	0.492103i	0.0872460	−	0.0371995i
$176$		0			0
$177$	−9.90536			−0.744532
$178$		0			0
$179$	−8.05571			−0.602112			−0.301056	−	0.953607i	$-0.597339\pi$
$179$	−8.05571			−0.602112			−0.301056	+	0.953607i	$0.597339\pi$
$180$		0			0
$181$	13.9644			1.03797			0.518983	−	0.854784i	$-0.326311\pi$
$181$	13.9644			1.03797			0.518983	+	0.854784i	$0.326311\pi$
$182$		0			0
$183$		−	2.05584i		−	0.151972i
$184$		0			0
$185$			22.9177i			1.68494i
$186$		0			0
$187$			0.215264i			0.0157417i
$188$		0			0
$189$	1.03769	+	2.43376i	0.0754812	+	0.177030i
$190$		0			0
$191$			7.43937i			0.538294i	0.963099	+	0.269147i	$0.0867418\pi$
$191$			7.43937i			0.538294i	−0.963099	+	0.269147i	$0.913258\pi$
$192$		0			0
$193$	17.3939			1.25204			0.626021	−	0.779806i	$-0.284682\pi$
$193$	17.3939			1.25204			0.626021	+	0.779806i	$0.284682\pi$
$194$		0			0
$195$	3.74651			0.268293
$196$		0			0
$197$	−21.2563			−1.51445			−0.757224	−	0.653155i	$-0.773445\pi$
$197$	−21.2563			−1.51445			−0.757224	+	0.653155i	$0.773445\pi$
$198$		0			0
$199$	−14.8203			−1.05058			−0.525292	−	0.850922i	$-0.676044\pi$
$199$	−14.8203			−1.05058			−0.525292	+	0.850922i	$0.676044\pi$
$200$		0			0
$201$	−2.73382			−0.192828
$202$		0			0
$203$	−24.8938	+	10.6141i	−1.74720	+	0.744965i
$204$		0			0
$205$			2.42540i			0.169397i
$206$		0			0
$207$	−3.56324	−	3.20988i	−0.247662	−	0.223102i
$208$		0			0
$209$		−	4.74439i		−	0.328176i
$210$		0			0
$211$	15.0565			1.03654			0.518268	−	0.855218i	$-0.326577\pi$
$211$	15.0565			1.03654			0.518268	+	0.855218i	$0.326577\pi$
$212$		0			0
$213$			6.57204i			0.450309i
$214$		0			0
$215$			20.5281i			1.40001i
$216$		0			0
$217$	−1.23568	−	2.89811i	−0.0838834	−	0.196736i
$218$		0			0
$219$	12.6369			0.853926
$220$		0			0
$221$			0.370869i			0.0249473i
$222$		0			0
$223$		−	19.7776i		−	1.32441i	−0.749324	−	0.662203i	$-0.769622\pi$
$223$		−	19.7776i		−	1.32441i	0.749324	−	0.662203i	$-0.230378\pi$
$224$		0			0
$225$	0.474227			0.0316152
$226$		0			0
$227$	19.3055			1.28135			0.640675	−	0.767812i	$-0.278654\pi$
$227$	19.3055			1.28135			0.640675	+	0.767812i	$0.278654\pi$
$228$		0			0
$229$	27.5303			1.81925			0.909627	−	0.415427i	$-0.136368\pi$
$229$	27.5303			1.81925			0.909627	+	0.415427i	$0.136368\pi$
$230$		0			0
$231$	−2.48776	+	1.06072i	−0.163683	+	0.0697903i
$232$		0			0
$233$	−5.62682			−0.368625			−0.184313	−	0.982868i	$-0.559006\pi$
$233$	−5.62682			−0.368625			−0.184313	+	0.982868i	$0.559006\pi$
$234$		0			0
$235$			11.8414i			0.772450i
$236$		0			0
$237$	1.45891			0.0947662
$238$		0			0
$239$	−19.4843			−1.26033			−0.630166	−	0.776461i	$-0.717013\pi$
$239$	−19.4843			−1.26033			−0.630166	+	0.776461i	$0.717013\pi$
$240$		0			0
$241$	5.53521			0.356554			0.178277	−	0.983980i	$-0.442948\pi$
$241$	5.53521			0.356554			0.178277	+	0.983980i	$0.442948\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	10.3101	−	10.7454i	0.658689	−	0.686500i
$246$		0			0
$247$		−	8.17389i		−	0.520092i
$248$		0			0
$249$		−	6.66137i		−	0.422147i
$250$		0			0
$251$	−14.4857			−0.914326			−0.457163	−	0.889383i	$-0.651134\pi$
$251$	−14.4857			−0.914326			−0.457163	+	0.889383i	$0.651134\pi$
$252$		0			0
$253$	3.28111	−	3.64230i	0.206282	−	0.228990i
$254$		0			0
$255$		−	0.448008i		−	0.0280554i
$256$		0			0
$257$			10.3246i			0.644032i	0.946734	+	0.322016i	$0.104361\pi$
$257$			10.3246i			0.644032i	−0.946734	+	0.322016i	$0.895639\pi$
$258$		0			0
$259$	−11.1788	−	26.2182i	−0.694616	−	1.62912i
$260$		0			0
$261$	−10.2285			−0.633131
$262$		0			0
$263$		−	14.4868i		−	0.893296i	−0.894710	−	0.446648i	$-0.852618\pi$
$263$		−	14.4868i		−	0.893296i	0.894710	−	0.446648i	$-0.147382\pi$
$264$		0			0
$265$		−	2.81263i		−	0.172779i
$266$		0			0
$267$		−	6.85521i		−	0.419532i
$268$		0			0
$269$		−	10.9629i		−	0.668420i	−0.942499	−	0.334210i	$-0.891531\pi$
$269$		−	10.9629i		−	0.668420i	0.942499	−	0.334210i	$-0.108469\pi$
$270$		0			0
$271$			30.5793i			1.85756i	0.370630	+	0.928781i	$0.379142\pi$
$271$			30.5793i			1.85756i	−0.370630	+	0.928781i	$0.620858\pi$
$272$		0			0
$273$	−4.28606	+	1.82747i	−0.259404	+	0.110603i
$274$		0			0
$275$			0.484750i			0.0292315i
$276$		0			0
$277$	−19.4095			−1.16620			−0.583101	−	0.812399i	$-0.698161\pi$
$277$	−19.4095			−1.16620			−0.583101	+	0.812399i	$0.698161\pi$
$278$		0			0
$279$		−	1.19079i		−	0.0712909i
$280$		0			0
$281$			28.8785i			1.72275i	0.507973	+	0.861373i	$0.330395\pi$
$281$			28.8785i			1.72275i	−0.507973	+	0.861373i	$0.669605\pi$
$282$		0			0
$283$	9.39898			0.558711			0.279356	−	0.960188i	$-0.409879\pi$
$283$	9.39898			0.558711			0.279356	+	0.960188i	$0.409879\pi$
$284$		0			0
$285$			9.87404i			0.584888i
$286$		0			0
$287$	−1.18306	−	2.77469i	−0.0698339	−	0.163785i
$288$		0			0
$289$	−16.9557			−0.997391
$290$		0			0
$291$		−	1.67955i		−	0.0984570i
$292$		0			0
$293$	21.7420			1.27018			0.635092	−	0.772437i	$-0.280962\pi$
$293$	21.7420			1.27018			0.635092	+	0.772437i	$0.280962\pi$
$294$		0			0
$295$			21.0725i			1.22689i
$296$		0			0
$297$	−1.02219			−0.0593135
$298$		0			0
$299$	5.65288	−	6.27516i	0.326914	−	0.362902i
$300$		0			0
$301$	−10.0132	−	23.4844i	−0.577150	−	1.35362i
$302$		0			0
$303$	11.5432			0.663141
$304$		0			0
$305$	−4.37357			−0.250430
$306$		0			0
$307$		−	10.5915i		−	0.604490i	−0.953230	−	0.302245i	$-0.902264\pi$
$307$		−	10.5915i		−	0.604490i	0.953230	−	0.302245i	$-0.0977361\pi$
$308$		0			0
$309$		−	11.7886i		−	0.670629i
$310$		0			0
$311$		−	27.2504i		−	1.54523i	−0.634875	−	0.772615i	$-0.718948\pi$
$311$		−	27.2504i		−	1.54523i	0.634875	−	0.772615i	$-0.281052\pi$
$312$		0			0
$313$	25.3449			1.43258			0.716288	−	0.697805i	$-0.245840\pi$
$313$	25.3449			1.43258			0.716288	+	0.697805i	$0.245840\pi$
$314$		0			0
$315$	5.17755	−	2.20758i	0.291722	−	0.124383i
$316$		0			0
$317$	−20.9786			−1.17828			−0.589138	−	0.808032i	$-0.700533\pi$
$317$	−20.9786			−1.17828			−0.589138	+	0.808032i	$0.700533\pi$
$318$		0			0
$319$		−	10.4555i		−	0.585397i
$320$		0			0
$321$	−18.1052			−1.01053
$322$		0			0
$323$	−0.977436			−0.0543860
$324$		0			0
$325$			0.835155i			0.0463261i
$326$		0			0
$327$	2.28162			0.126174
$328$		0			0
$329$	−5.77601	−	13.5468i	−0.318442	−	0.746858i
$330$		0			0
$331$	19.2100			1.05588			0.527938	−	0.849283i	$-0.322965\pi$
$331$	19.2100			1.05588			0.527938	+	0.849283i	$0.322965\pi$
$332$		0			0
$333$		−	10.7727i		−	0.590341i
$334$		0			0
$335$			5.81588i			0.317756i
$336$		0			0
$337$		−	13.9168i		−	0.758096i	−0.925377	−	0.379048i	$-0.876251\pi$
$337$		−	13.9168i		−	0.758096i	0.925377	−	0.379048i	$-0.123749\pi$
$338$		0			0
$339$	−1.03963			−0.0564647
$340$		0			0
$341$	1.21722			0.0659160
$342$		0			0
$343$	−6.55353	+	17.3220i	−0.353857	+	0.935299i
$344$		0			0
$345$	−6.82866	+	7.58038i	−0.367643	+	0.408114i
$346$		0			0
$347$	21.9347			1.17751			0.588757	−	0.808310i	$-0.299618\pi$
$347$	21.9347			1.17751			0.588757	+	0.808310i	$0.299618\pi$
$348$		0			0
$349$		−	16.1113i		−	0.862420i	−0.902252	−	0.431210i	$-0.858087\pi$
$349$		−	16.1113i		−	0.862420i	0.902252	−	0.431210i	$-0.141913\pi$
$350$		0			0
$351$	−1.76109			−0.0939998
$352$		0			0
$353$		−	15.7897i		−	0.840402i	−0.907431	−	0.420201i	$-0.861960\pi$
$353$		−	15.7897i		−	0.840402i	0.907431	−	0.420201i	$-0.138040\pi$
$354$		0			0
$355$	13.9813			0.742049
$356$		0			0
$357$	0.218529	+	0.512528i	0.0115658	+	0.0271259i
$358$		0			0
$359$		−	5.95677i		−	0.314386i	−0.987568	−	0.157193i	$-0.949755\pi$
$359$		−	5.95677i		−	0.314386i	0.987568	−	0.157193i	$-0.0502445\pi$
$360$		0			0
$361$	2.54254			0.133818
$362$		0			0
$363$			9.95513i			0.522509i
$364$		0			0
$365$		−	26.8837i		−	1.40716i
$366$		0			0
$367$	−2.73646			−0.142842			−0.0714212	−	0.997446i	$-0.522753\pi$
$367$	−2.73646			−0.142842			−0.0714212	+	0.997446i	$0.522753\pi$
$368$		0			0
$369$		−	1.14009i		−	0.0593505i
$370$		0			0
$371$	1.37194	+	3.21769i	0.0712277	+	0.167054i
$372$		0			0
$373$		−	5.74849i		−	0.297646i	−0.988864	−	0.148823i	$-0.952452\pi$
$373$		−	5.74849i		−	0.297646i	0.988864	−	0.148823i	$-0.0475484\pi$
$374$		0			0
$375$		−	11.6458i		−	0.601386i
$376$		0			0
$377$		−	18.0134i		−	0.927735i
$378$		0			0
$379$		−	35.0596i		−	1.80089i	−0.434972	−	0.900444i	$-0.643242\pi$
$379$		−	35.0596i		−	1.80089i	0.434972	−	0.900444i	$-0.356758\pi$
$380$		0			0
$381$			15.4167i			0.789820i
$382$		0			0
$383$	−16.9904			−0.868171			−0.434086	−	0.900872i	$-0.642928\pi$
$383$	−16.9904			−0.868171			−0.434086	+	0.900872i	$0.642928\pi$
$384$		0			0
$385$	2.25656	+	5.29244i	0.115005	+	0.269728i
$386$		0			0
$387$		−	9.64945i		−	0.490509i
$388$		0			0
$389$		−	0.130944i		−	0.00663913i	−0.999994	−	0.00331957i	$-0.998943\pi$
$389$		−	0.130944i		−	0.00663913i	0.999994	−	0.00331957i	$-0.00105665\pi$
$390$		0			0
$391$	−0.750385	−	0.675972i	−0.0379486	−	0.0341854i
$392$		0			0
$393$	0.484240			0.0244267
$394$		0			0
$395$		−	3.10366i		−	0.156162i
$396$		0			0
$397$		−	28.9414i		−	1.45253i	−0.687416	−	0.726264i	$-0.741255\pi$
$397$		−	28.9414i		−	1.45253i	0.687416	−	0.726264i	$-0.258745\pi$
$398$		0			0
$399$	−4.81635	−	11.2960i	−0.241119	−	0.565509i
$400$		0			0
$401$		−	16.2810i		−	0.813037i	−0.913643	−	0.406518i	$-0.866743\pi$
$401$		−	16.2810i		−	0.813037i	0.913643	−	0.406518i	$-0.133257\pi$
$402$		0			0
$403$	2.09709			0.104463
$404$		0			0
$405$	2.12739			0.105711
$406$		0			0
$407$	11.0117			0.545832
$408$		0			0
$409$		−	18.4959i		−	0.914562i	−0.889322	−	0.457281i	$-0.848823\pi$
$409$		−	18.4959i		−	0.914562i	0.889322	−	0.457281i	$-0.151177\pi$
$410$		0			0
$411$	10.2651			0.506340
$412$		0			0
$413$	−10.2787	−	24.1073i	−0.505784	−	1.18624i
$414$		0			0
$415$	−14.1713			−0.695642
$416$		0			0
$417$	−3.05403			−0.149557
$418$		0			0
$419$	−2.34468			−0.114545			−0.0572727	−	0.998359i	$-0.518240\pi$
$419$	−2.34468			−0.114545			−0.0572727	+	0.998359i	$0.518240\pi$
$420$		0			0
$421$		−	17.1711i		−	0.836869i	−0.908247	−	0.418435i	$-0.862579\pi$
$421$		−	17.1711i		−	0.836869i	0.908247	−	0.418435i	$-0.137421\pi$
$422$		0			0
$423$		−	5.56619i		−	0.270638i
$424$		0			0
$425$	0.0998680			0.00484431
$426$		0			0
$427$	5.00343	−	2.13334i	0.242133	−	0.103239i
$428$		0			0
$429$		−	1.80016i		−	0.0869127i
$430$		0			0
$431$		−	0.00452204i		−	0.000217819i	−1.00000		0.000108910i	$-0.999965\pi$
$431$		−	0.00452204i		−	0.000217819i	1.00000		0.000108910i	$-3.46670e-5\pi$
$432$		0			0
$433$	−24.7806			−1.19088			−0.595439	−	0.803400i	$-0.703022\pi$
$433$	−24.7806			−1.19088			−0.595439	+	0.803400i	$0.703022\pi$
$434$		0			0
$435$			21.7601i			1.04332i
$436$		0			0
$437$	16.5384	+	14.8983i	0.791138	+	0.712684i
$438$		0			0
$439$		−	25.1567i		−	1.20066i	−0.799751	−	0.600332i	$-0.795035\pi$
$439$		−	25.1567i		−	1.20066i	0.799751	−	0.600332i	$-0.204965\pi$
$440$		0			0
$441$	−4.84638	+	5.05100i	−0.230780	+	0.240524i
$442$		0			0
$443$	−0.978145			−0.0464731			−0.0232365	−	0.999730i	$-0.507397\pi$
$443$	−0.978145			−0.0464731			−0.0232365	+	0.999730i	$0.507397\pi$
$444$		0			0
$445$	−14.5837			−0.691332
$446$		0			0
$447$	7.11262			0.336415
$448$		0			0
$449$	−20.5888			−0.971646			−0.485823	−	0.874057i	$-0.661480\pi$
$449$	−20.5888			−0.971646			−0.485823	+	0.874057i	$0.661480\pi$
$450$		0			0
$451$	1.16538			0.0548758
$452$		0			0
$453$			4.67321i			0.219567i
$454$		0			0
$455$	3.88773	+	9.11811i	0.182260	+	0.427463i
$456$		0			0
$457$			1.92751i			0.0901651i	0.998983	+	0.0450825i	$0.0143551\pi$
$457$			1.92751i			0.0901651i	−0.998983	+	0.0450825i	$0.985645\pi$
$458$		0			0
$459$			0.210591i			0.00982954i
$460$		0			0
$461$			11.8675i			0.552722i	0.961054	+	0.276361i	$0.0891286\pi$
$461$			11.8675i			0.552722i	−0.961054	+	0.276361i	$0.910871\pi$
$462$		0			0
$463$	9.31784			0.433037			0.216518	−	0.976279i	$-0.430530\pi$
$463$	9.31784			0.433037			0.216518	+	0.976279i	$0.430530\pi$
$464$		0			0
$465$	−2.53328			−0.117478
$466$		0			0
$467$	6.57001			0.304024			0.152012	−	0.988379i	$-0.451425\pi$
$467$	6.57001			0.304024			0.152012	+	0.988379i	$0.451425\pi$
$468$		0			0
$469$	−2.83687	−	6.65345i	−0.130994	−	0.307228i
$470$		0			0
$471$			11.1758i			0.514955i
$472$		0			0
$473$	9.86357			0.453527
$474$		0			0
$475$	−2.20108			−0.100992
$476$		0			0
$477$			1.32211i			0.0605351i
$478$		0			0
$479$	2.48711			0.113639			0.0568194	−	0.998384i	$-0.481904\pi$
$479$	2.48711			0.113639			0.0568194	+	0.998384i	$0.481904\pi$
$480$		0			0
$481$	18.9717			0.865033
$482$		0			0
$483$	4.11454	−	12.0029i	0.187218	−	0.546153i
$484$		0			0
$485$	−3.57305			−0.162244
$486$		0			0
$487$	−28.9885			−1.31359			−0.656796	−	0.754068i	$-0.728089\pi$
$487$	−28.9885			−1.31359			−0.656796	+	0.754068i	$0.728089\pi$
$488$		0			0
$489$		−	1.03867i		−	0.0469703i
$490$		0			0
$491$	−17.3523			−0.783100			−0.391550	−	0.920157i	$-0.628061\pi$
$491$	−17.3523			−0.783100			−0.391550	+	0.920157i	$0.628061\pi$
$492$		0			0
$493$	−2.15404			−0.0970131
$494$		0			0
$495$			2.17459i			0.0977407i
$496$		0			0
$497$	−15.9948	+	6.81977i	−0.717464	+	0.305909i
$498$		0			0
$499$	14.8025			0.662652			0.331326	−	0.943516i	$-0.392504\pi$
$499$	14.8025			0.662652			0.331326	+	0.943516i	$0.392504\pi$
$500$		0			0
$501$	11.2150			0.501050
$502$		0			0
$503$	−27.9943			−1.24820			−0.624101	−	0.781344i	$-0.714535\pi$
$503$	−27.9943			−1.24820			−0.624101	+	0.781344i	$0.714535\pi$
$504$		0			0
$505$		−	24.5569i		−	1.09277i
$506$		0			0
$507$			9.89858i			0.439611i
$508$		0			0
$509$			39.9223i			1.76952i	0.466042	+	0.884762i	$0.345679\pi$
$509$			39.9223i			1.76952i	−0.466042	+	0.884762i	$0.654321\pi$
$510$		0			0
$511$	13.1133	+	30.7553i	0.580098	+	1.36053i
$512$		0			0
$513$		−	4.64139i		−	0.204923i
$514$		0			0
$515$	−25.0789			−1.10511
$516$		0			0
$517$	5.68970			0.250233
$518$		0			0
$519$	−17.1904			−0.754577
$520$		0			0
$521$	−1.14012			−0.0499494			−0.0249747	−	0.999688i	$-0.507951\pi$
$521$	−1.14012			−0.0499494			−0.0249747	+	0.999688i	$0.507951\pi$
$522$		0			0
$523$	5.12708			0.224191			0.112096	−	0.993697i	$-0.464244\pi$
$523$	5.12708			0.224191			0.112096	+	0.993697i	$0.464244\pi$
$524$		0			0
$525$	0.492103	+	1.15416i	0.0214771	+	0.0503715i
$526$		0			0
$527$		−	0.250770i		−	0.0109237i
$528$		0			0
$529$	2.39330	+	22.8751i	0.104057	+	0.994571i
$530$		0			0
$531$		−	9.90536i		−	0.429856i
$532$		0			0
$533$	2.00779			0.0869670
$534$		0			0
$535$			38.5167i			1.66522i
$536$		0			0
$537$		−	8.05571i		−	0.347629i
$538$		0			0
$539$	−5.16308	−	4.95392i	−0.222390	−	0.213380i
$540$		0			0
$541$	−16.7103			−0.718431			−0.359215	−	0.933255i	$-0.616956\pi$
$541$	−16.7103			−0.718431			−0.359215	+	0.933255i	$0.616956\pi$
$542$		0			0
$543$			13.9644i			0.599270i
$544$		0			0
$545$		−	4.85389i		−	0.207918i
$546$		0			0
$547$	−36.2468			−1.54980			−0.774901	−	0.632083i	$-0.782200\pi$
$547$	−36.2468			−1.54980			−0.774901	+	0.632083i	$0.782200\pi$
$548$		0			0
$549$	2.05584			0.0877412
$550$		0			0
$551$	47.4747			2.02249
$552$		0			0
$553$	1.51390	+	3.55063i	0.0643776	+	0.150988i
$554$		0			0
$555$	−22.9177			−0.972802
$556$		0			0
$557$		−	15.4759i		−	0.655734i	−0.944724	−	0.327867i	$-0.893670\pi$
$557$		−	15.4759i		−	0.655734i	0.944724	−	0.327867i	$-0.106330\pi$
$558$		0			0
$559$	16.9935			0.718749
$560$		0			0
$561$	−0.215264			−0.00908845
$562$		0			0
$563$	19.3351			0.814878			0.407439	−	0.913232i	$-0.366422\pi$
$563$	19.3351			0.814878			0.407439	+	0.913232i	$0.366422\pi$
$564$		0			0
$565$			2.21169i			0.0930464i
$566$		0			0
$567$	−2.43376	+	1.03769i	−0.102208	+	0.0435791i
$568$		0			0
$569$		−	32.6250i		−	1.36771i	−0.729618	−	0.683855i	$-0.760302\pi$
$569$		−	32.6250i		−	1.36771i	0.729618	−	0.683855i	$-0.239698\pi$
$570$		0			0
$571$		−	16.7266i		−	0.699985i	−0.936752	−	0.349993i	$-0.886184\pi$
$571$		−	16.7266i		−	0.699985i	0.936752	−	0.349993i	$-0.113816\pi$
$572$		0			0
$573$	−7.43937			−0.310784
$574$		0			0
$575$	−1.68978	−	1.52221i	−0.0704689	−	0.0634807i
$576$		0			0
$577$			15.2582i			0.635206i	0.948224	+	0.317603i	$0.102878\pi$
$577$			15.2582i			0.635206i	−0.948224	+	0.317603i	$0.897122\pi$
$578$		0			0
$579$			17.3939i			0.722867i
$580$		0			0
$581$	16.2122	−	6.91247i	0.672594	−	0.286777i
$582$		0			0
$583$	−1.35144			−0.0559711
$584$		0			0
$585$			3.74651i			0.154899i
$586$		0			0
$587$		−	45.1841i		−	1.86495i	−0.361237	−	0.932474i	$-0.617646\pi$
$587$		−	45.1841i		−	1.86495i	0.361237	−	0.932474i	$-0.382354\pi$
$588$		0			0
$589$			5.52694i			0.227734i
$590$		0			0
$591$		−	21.2563i		−	0.874367i
$592$		0			0
$593$			44.8289i			1.84090i	0.390859	+	0.920450i	$0.372178\pi$
$593$			44.8289i			1.84090i	−0.390859	+	0.920450i	$0.627822\pi$
$594$		0			0
$595$	1.09034	−	0.464896i	0.0446998	−	0.0190589i
$596$		0			0
$597$		−	14.8203i		−	0.606555i
$598$		0			0
$599$	7.15981			0.292542			0.146271	−	0.989245i	$-0.453273\pi$
$599$	7.15981			0.292542			0.146271	+	0.989245i	$0.453273\pi$
$600$		0			0
$601$		−	5.79246i		−	0.236279i	−0.992997	−	0.118140i	$-0.962307\pi$
$601$		−	5.79246i		−	0.236279i	0.992997	−	0.118140i	$-0.0376930\pi$
$602$		0			0
$603$		−	2.73382i		−	0.111330i
$604$		0			0
$605$	21.1784			0.861025
$606$		0			0
$607$			25.2265i			1.02391i	0.859012	+	0.511956i	$0.171079\pi$
$607$			25.2265i			1.02391i	−0.859012	+	0.511956i	$0.828921\pi$
$608$		0			0
$609$	−10.6141	−	24.8938i	−0.430106	−	1.00875i
$610$		0			0
$611$	9.80254			0.396568
$612$		0			0
$613$			46.3625i			1.87257i	0.351247	+	0.936283i	$0.385758\pi$
$613$			46.3625i			1.87257i	−0.351247	+	0.936283i	$0.614242\pi$
$614$		0			0
$615$	−2.42540			−0.0978017
$616$		0			0
$617$		−	5.72843i		−	0.230618i	−0.993330	−	0.115309i	$-0.963214\pi$
$617$		−	5.72843i		−	0.230618i	0.993330	−	0.115309i	$-0.0367858\pi$
$618$		0			0
$619$	−10.7333			−0.431408			−0.215704	−	0.976459i	$-0.569205\pi$
$619$	−10.7333			−0.431408			−0.215704	+	0.976459i	$0.569205\pi$
$620$		0			0
$621$	3.20988	−	3.56324i	0.128808	−	0.142988i
$622$		0			0
$623$	16.6839	−	7.11361i	0.668428	−	0.285001i
$624$		0			0
$625$	−22.4040			−0.896159
$626$		0			0
$627$	4.74439			0.189472
$628$		0			0
$629$		−	2.26863i		−	0.0904564i
$630$		0			0
$631$		−	43.5034i		−	1.73184i	−0.500178	−	0.865922i	$-0.666732\pi$
$631$		−	43.5034i		−	1.73184i	0.500178	−	0.865922i	$-0.333268\pi$
$632$		0			0
$633$			15.0565i			0.598444i
$634$		0			0
$635$	32.7972			1.30152
$636$		0			0
$637$	−8.89525	−	8.53489i	−0.352442	−	0.338165i
$638$		0			0
$639$	−6.57204			−0.259986
$640$		0			0
$641$			22.3815i			0.884016i	0.897011	+	0.442008i	$0.145734\pi$
$641$			22.3815i			0.884016i	−0.897011	+	0.442008i	$0.854266\pi$
$642$		0			0
$643$	−37.5558			−1.48106			−0.740528	−	0.672026i	$-0.765424\pi$
$643$	−37.5558			−1.48106			−0.740528	+	0.672026i	$0.765424\pi$
$644$		0			0
$645$	−20.5281			−0.808293
$646$		0			0
$647$			20.6665i			0.812482i	0.913766	+	0.406241i	$0.133161\pi$
$647$			20.6665i			0.812482i	−0.913766	+	0.406241i	$0.866839\pi$
$648$		0			0
$649$	10.1252			0.397447
$650$		0			0
$651$	2.89811	−	1.23568i	0.113586	−	0.0484301i
$652$		0			0
$653$	−18.3578			−0.718395			−0.359198	−	0.933261i	$-0.616950\pi$
$653$	−18.3578			−0.718395			−0.359198	+	0.933261i	$0.616950\pi$
$654$		0			0
$655$		−	1.03017i		−	0.0402519i
$656$		0			0
$657$			12.6369i			0.493014i
$658$		0			0
$659$			10.8207i			0.421513i	0.977539	+	0.210757i	$0.0675927\pi$
$659$			10.8207i			0.421513i	−0.977539	+	0.210757i	$0.932407\pi$
$660$		0			0
$661$	−4.38781			−0.170666			−0.0853330	−	0.996352i	$-0.527195\pi$
$661$	−4.38781			−0.170666			−0.0853330	+	0.996352i	$0.527195\pi$
$662$		0			0
$663$	−0.370869			−0.0144033
$664$		0			0
$665$	−24.0310	+	10.2462i	−0.931884	+	0.397332i
$666$		0			0
$667$	36.4467	+	32.8324i	1.41122	+	1.27128i
$668$		0			0
$669$	19.7776			0.764646
$670$		0			0
$671$			2.10146i			0.0811260i
$672$		0			0
$673$	−6.59561			−0.254242			−0.127121	−	0.991887i	$-0.540574\pi$
$673$	−6.59561			−0.254242			−0.127121	+	0.991887i	$0.540574\pi$
$674$		0			0
$675$			0.474227i			0.0182530i
$676$		0			0
$677$	−23.4456			−0.901089			−0.450545	−	0.892754i	$-0.648770\pi$
$677$	−23.4456			−0.901089			−0.450545	+	0.892754i	$0.648770\pi$
$678$		0			0
$679$	4.08763	−	1.74286i	0.156869	−	0.0668849i
$680$		0			0
$681$			19.3055i			0.739788i
$682$		0			0
$683$	22.5410			0.862509			0.431254	−	0.902230i	$-0.358071\pi$
$683$	22.5410			0.862509			0.431254	+	0.902230i	$0.358071\pi$
$684$		0			0
$685$		−	21.8378i		−	0.834381i
$686$		0			0
$687$			27.5303i			1.05035i
$688$		0			0
$689$	−2.32834			−0.0887028
$690$		0			0
$691$			30.4737i			1.15928i	0.814874	+	0.579638i	$0.196806\pi$
$691$			30.4737i			1.15928i	−0.814874	+	0.579638i	$0.803194\pi$
$692$		0			0
$693$	−1.06072	−	2.48776i	−0.0402935	−	0.0945024i
$694$		0			0
$695$			6.49711i			0.246449i
$696$		0			0
$697$		−	0.240092i		−	0.00909412i
$698$		0			0
$699$		−	5.62682i		−	0.212826i
$700$		0			0
$701$		−	34.4228i		−	1.30013i	−0.759877	−	0.650066i	$-0.774741\pi$
$701$		−	34.4228i		−	1.30013i	0.759877	−	0.650066i	$-0.225259\pi$
$702$		0			0
$703$			50.0004i			1.88580i
$704$		0			0
$705$	−11.8414			−0.445974
$706$		0			0
$707$	11.9783	+	28.0934i	0.450492	+	1.05656i
$708$		0			0
$709$		−	42.5227i		−	1.59697i	−0.602012	−	0.798487i	$-0.705634\pi$
$709$		−	42.5227i		−	1.59697i	0.602012	−	0.798487i	$-0.294366\pi$
$710$		0			0
$711$			1.45891i			0.0547133i
$712$		0			0
$713$	−3.82231	+	4.24308i	−0.143147	+	0.158905i
$714$		0			0
$715$	−3.82964			−0.143221
$716$		0			0
$717$		−	19.4843i		−	0.727653i
$718$		0			0
$719$			12.1537i			0.453257i	0.973981	+	0.226629i	$0.0727704\pi$
$719$			12.1537i			0.453257i	−0.973981	+	0.226629i	$0.927230\pi$
$720$		0			0
$721$	28.6906	−	12.2329i	1.06849	−	0.455579i
$722$		0			0
$723$			5.53521i			0.205857i
$724$		0			0
$725$	−4.85066			−0.180149
$726$		0			0
$727$	−52.1295			−1.93338			−0.966688	−	0.255956i	$-0.917610\pi$
$727$	−52.1295			−1.93338			−0.966688	+	0.255956i	$0.917610\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	2.03209i		−	0.0751594i
$732$		0			0
$733$	4.37140			0.161461			0.0807307	−	0.996736i	$-0.474275\pi$
$733$	4.37140			0.161461			0.0807307	+	0.996736i	$0.474275\pi$
$734$		0			0
$735$	10.7454	+	10.3101i	0.396351	+	0.380295i
$736$		0			0
$737$	2.79448			0.102936
$738$		0			0
$739$	−8.40711			−0.309261			−0.154630	−	0.987972i	$-0.549419\pi$
$739$	−8.40711			−0.309261			−0.154630	+	0.987972i	$0.549419\pi$
$740$		0			0
$741$	8.17389			0.300275
$742$		0			0
$743$		−	29.1935i		−	1.07100i	−0.844534	−	0.535502i	$-0.820122\pi$
$743$		−	29.1935i		−	1.07100i	0.844534	−	0.535502i	$-0.179878\pi$
$744$		0			0
$745$		−	15.1313i		−	0.554368i
$746$		0			0
$747$	6.66137			0.243727
$748$		0			0
$749$	−18.7876	−	44.0636i	−0.686485	−	1.61005i
$750$		0			0
$751$			18.3047i			0.667948i	0.942582	+	0.333974i	$0.108390\pi$
$751$			18.3047i			0.667948i	−0.942582	+	0.333974i	$0.891610\pi$
$752$		0			0
$753$		−	14.4857i		−	0.527887i
$754$		0			0
$755$	9.94172			0.361816
$756$		0			0
$757$			8.82459i			0.320735i	0.987057	+	0.160368i	$0.0512680\pi$
$757$			8.82459i			0.320735i	−0.987057	+	0.160368i	$0.948732\pi$
$758$		0			0
$759$	3.64230	+	3.28111i	0.132207	+	0.119097i
$760$		0			0
$761$			8.87026i			0.321547i	0.986991	+	0.160773i	$0.0513988\pi$
$761$			8.87026i			0.321547i	−0.986991	+	0.160773i	$0.948601\pi$
$762$		0			0
$763$	2.36763	+	5.55293i	0.0857139	+	0.201029i
$764$		0			0
$765$	0.448008			0.0161978
$766$		0			0
$767$	17.4442			0.629873
$768$		0			0
$769$	1.25850			0.0453826			0.0226913	−	0.999743i	$-0.492777\pi$
$769$	1.25850			0.0453826			0.0226913	+	0.999743i	$0.492777\pi$
$770$		0			0
$771$	−10.3246			−0.371832
$772$		0			0
$773$	−21.9871			−0.790822			−0.395411	−	0.918504i	$-0.629398\pi$
$773$	−21.9871			−0.790822			−0.395411	+	0.918504i	$0.629398\pi$
$774$		0			0
$775$		−	0.564707i		−	0.0202849i
$776$		0			0
$777$	26.2182	−	11.1788i	0.940572	−	0.401036i
$778$		0			0
$779$			5.29159i			0.189591i
$780$		0			0
$781$		−	6.71787i		−	0.240384i
$782$		0			0
$783$		−	10.2285i		−	0.365539i
$784$		0			0
$785$	23.7753			0.848577
$786$		0			0
$787$	16.5097			0.588508			0.294254	−	0.955727i	$-0.404929\pi$
$787$	16.5097			0.588508			0.294254	+	0.955727i	$0.404929\pi$
$788$		0			0
$789$	14.4868			0.515745
$790$		0			0
$791$	−1.07881	−	2.53020i	−0.0383582	−	0.0899636i
$792$		0			0
$793$			3.62051i			0.128568i
$794$		0			0
$795$	2.81263			0.0997538
$796$		0			0
$797$	−38.5879			−1.36685			−0.683427	−	0.730019i	$-0.739511\pi$
$797$	−38.5879			−1.36685			−0.683427	+	0.730019i	$0.739511\pi$
$798$		0			0
$799$		−	1.17219i		−	0.0414691i
$800$		0			0
$801$	6.85521			0.242217
$802$		0			0
$803$	−12.9174			−0.455844
$804$		0			0
$805$	−25.5349	−	8.75321i	−0.899987	−	0.308510i
$806$		0			0
$807$	10.9629			0.385912
$808$		0			0
$809$	18.0833			0.635775			0.317887	−	0.948128i	$-0.397027\pi$
$809$	18.0833			0.635775			0.317887	+	0.948128i	$0.397027\pi$
$810$		0			0
$811$		−	26.5607i		−	0.932674i	−0.884607	−	0.466337i	$-0.845573\pi$
$811$		−	26.5607i		−	0.932674i	0.884607	−	0.466337i	$-0.154427\pi$
$812$		0			0
$813$	−30.5793			−1.07246
$814$		0			0
$815$	−2.20965			−0.0774008
$816$		0			0
$817$			44.7869i			1.56690i
$818$		0			0
$819$	−1.82747	−	4.28606i	−0.0638570	−	0.149767i
$820$		0			0
$821$	−6.57505			−0.229471			−0.114735	−	0.993396i	$-0.536602\pi$
$821$	−6.57505			−0.229471			−0.114735	+	0.993396i	$0.536602\pi$
$822$		0			0
$823$	2.36858			0.0825633			0.0412817	−	0.999148i	$-0.486856\pi$
$823$	2.36858			0.0825633			0.0412817	+	0.999148i	$0.486856\pi$
$824$		0			0
$825$	−0.484750			−0.0168768
$826$		0			0
$827$			15.3850i			0.534990i	0.963559	+	0.267495i	$0.0861959\pi$
$827$			15.3850i			0.534990i	−0.963559	+	0.267495i	$0.913804\pi$
$828$		0			0
$829$		−	45.9758i		−	1.59680i	−0.602125	−	0.798402i	$-0.705679\pi$
$829$		−	45.9758i		−	1.59680i	0.602125	−	0.798402i	$-0.294321\pi$
$830$		0			0
$831$		−	19.4095i		−	0.673307i
$832$		0			0
$833$	−1.02060	+	1.06369i	−0.0353618	+	0.0368548i
$834$		0			0
$835$		−	23.8587i		−	0.825663i
$836$		0			0
$837$	1.19079			0.0411598
$838$		0			0
$839$	−14.3431			−0.495178			−0.247589	−	0.968865i	$-0.579638\pi$
$839$	−14.3431			−0.495178			−0.247589	+	0.968865i	$0.579638\pi$
$840$		0			0
$841$	75.6232			2.60770
$842$		0			0
$843$	−28.8785			−0.994628
$844$		0			0
$845$	21.0581			0.724421
$846$		0			0
$847$	−24.2284	+	10.3304i	−0.832498	+	0.354956i
$848$		0			0
$849$			9.39898i			0.322572i
$850$		0			0
$851$	−34.5791	+	38.3857i	−1.18536	+	1.31584i
$852$		0			0
$853$			6.03560i			0.206655i	0.994647	+	0.103328i	$0.0329490\pi$
$853$			6.03560i			0.206655i	−0.994647	+	0.103328i	$0.967051\pi$
$854$		0			0
$855$	−9.87404			−0.337685
$856$		0			0
$857$		−	17.8398i		−	0.609395i	−0.952449	−	0.304698i	$-0.901445\pi$
$857$		−	17.8398i		−	0.609395i	0.952449	−	0.304698i	$-0.0985554\pi$
$858$		0			0
$859$			31.7261i			1.08248i	0.840868	+	0.541241i	$0.182045\pi$
$859$			31.7261i			1.08248i	−0.840868	+	0.541241i	$0.817955\pi$
$860$		0			0
$861$	2.77469	−	1.18306i	0.0945613	−	0.0403186i
$862$		0			0
$863$	9.51364			0.323848			0.161924	−	0.986803i	$-0.448230\pi$
$863$	9.51364			0.323848			0.161924	+	0.986803i	$0.448230\pi$
$864$		0			0
$865$			36.5707i			1.24344i
$866$		0			0
$867$		−	16.9557i		−	0.575844i
$868$		0			0
$869$	−1.49128			−0.0505882
$870$		0			0
$871$	4.81448			0.163133
$872$		0			0
$873$	1.67955			0.0568442
$874$		0			0
$875$	28.3431	−	12.0848i	0.958171	−	0.408540i
$876$		0			0
$877$	46.5160			1.57073			0.785367	−	0.619031i	$-0.212474\pi$
$877$	46.5160			1.57073			0.785367	+	0.619031i	$0.212474\pi$
$878$		0			0
$879$			21.7420i			0.733341i
$880$		0			0
$881$	8.56083			0.288422			0.144211	−	0.989547i	$-0.453936\pi$
$881$	8.56083			0.288422			0.144211	+	0.989547i	$0.453936\pi$
$882$		0			0
$883$	48.1229			1.61947			0.809733	−	0.586799i	$-0.199612\pi$
$883$	48.1229			1.61947			0.809733	+	0.586799i	$0.199612\pi$
$884$		0			0
$885$	−21.0725			−0.708345
$886$		0			0
$887$		−	28.3677i		−	0.952496i	−0.879311	−	0.476248i	$-0.841997\pi$
$887$		−	28.3677i		−	0.952496i	0.879311	−	0.476248i	$-0.158003\pi$
$888$		0			0
$889$	−37.5205	+	15.9978i	−1.25840	+	0.536549i
$890$		0			0
$891$		−	1.02219i		−	0.0342446i
$892$		0			0
$893$			25.8349i			0.864532i
$894$		0			0
$895$	−17.1376			−0.572847
$896$		0			0
$897$	6.27516	+	5.65288i	0.209522	+	0.188744i
$898$		0			0
$899$			12.1801i			0.406229i
$900$		0			0
$901$			0.278424i			0.00927564i
$902$		0			0
$903$	23.4844	−	10.0132i	0.781514	−	0.333218i
$904$		0			0
$905$	29.7077			0.987517
$906$		0			0
$907$			7.67296i			0.254777i	0.991853	+	0.127388i	$0.0406594\pi$
$907$			7.67296i			0.254777i	−0.991853	+	0.127388i	$0.959341\pi$
$908$		0			0
$909$			11.5432i			0.382865i
$910$		0			0
$911$			42.3052i			1.40163i	0.713341	+	0.700817i	$0.247181\pi$
$911$			42.3052i			1.40163i	−0.713341	+	0.700817i	$0.752819\pi$
$912$		0			0
$913$			6.80918i			0.225351i
$914$		0			0
$915$		−	4.37357i		−	0.144586i
$916$		0			0
$917$	0.502493	+	1.17852i	0.0165938	+	0.0389183i
$918$		0			0
$919$		−	46.3516i		−	1.52900i	−0.644625	−	0.764499i	$-0.722987\pi$
$919$		−	46.3516i		−	1.52900i	0.644625	−	0.764499i	$-0.277013\pi$
$920$		0			0
$921$	10.5915			0.349003
$922$		0			0
$923$		−	11.5739i		−	0.380961i
$924$		0			0
$925$		−	5.10871i		−	0.167973i
$926$		0			0
$927$	11.7886			0.387188
$928$		0			0
$929$		−	32.1843i		−	1.05593i	−0.849265	−	0.527967i	$-0.822955\pi$
$929$		−	32.1843i		−	1.05593i	0.849265	−	0.527967i	$-0.177045\pi$
$930$		0			0
$931$	22.4940	−	23.4437i	0.737210	−	0.768336i
$932$		0			0
$933$	27.2504			0.892139
$934$		0			0
$935$			0.457949i			0.0149765i
$936$		0			0
$937$	33.7281			1.10185			0.550924	−	0.834555i	$-0.314275\pi$
$937$	33.7281			1.10185			0.550924	+	0.834555i	$0.314275\pi$
$938$		0			0
$939$			25.3449i			0.827098i
$940$		0			0
$941$	−42.9871			−1.40134			−0.700670	−	0.713485i	$-0.747116\pi$
$941$	−42.9871			−1.40134			−0.700670	+	0.713485i	$0.747116\pi$
$942$		0			0
$943$	−3.65954	+	4.06239i	−0.119171	+	0.132290i
$944$		0			0
$945$	2.20758	+	5.17755i	0.0718125	+	0.168426i
$946$		0			0
$947$	44.2188			1.43692			0.718460	−	0.695569i	$-0.244848\pi$
$947$	44.2188			1.43692			0.718460	+	0.695569i	$0.244848\pi$
$948$		0			0
$949$	−22.2548			−0.722420
$950$		0			0
$951$		−	20.9786i		−	0.680278i
$952$		0			0
$953$			49.7390i			1.61120i	0.592457	+	0.805602i	$0.298158\pi$
$953$			49.7390i			1.61120i	−0.592457	+	0.805602i	$0.701842\pi$
$954$		0			0
$955$			15.8264i			0.512131i
$956$		0			0
$957$	10.4555			0.337979
$958$		0			0
$959$	10.6520	+	24.9828i	0.343972	+	0.806736i
$960$		0			0
$961$	29.5820			0.954258
$962$		0			0
$963$		−	18.1052i		−	0.583431i
$964$		0			0
$965$	37.0036			1.19119
$966$		0			0
$967$	−27.2894			−0.877569			−0.438784	−	0.898592i	$-0.644591\pi$
$967$	−27.2894			−0.877569			−0.438784	+	0.898592i	$0.644591\pi$
$968$		0			0
$969$		−	0.977436i		−	0.0313998i
$970$		0			0
$971$	−20.1643			−0.647103			−0.323552	−	0.946211i	$-0.604877\pi$
$971$	−20.1643			−0.647103			−0.323552	+	0.946211i	$0.604877\pi$
$972$		0			0
$973$	−3.16915	−	7.43278i	−0.101598	−	0.238284i
$974$		0			0
$975$	−0.835155			−0.0267464
$976$		0			0
$977$		−	21.7343i		−	0.695341i	−0.937617	−	0.347670i	$-0.886973\pi$
$977$		−	21.7343i		−	0.695341i	0.937617	−	0.347670i	$-0.113027\pi$
$978$		0			0
$979$			7.00732i			0.223955i
$980$		0			0
$981$			2.28162i			0.0728466i
$982$		0			0
$983$	51.3288			1.63713			0.818567	−	0.574412i	$-0.194769\pi$
$983$	51.3288			1.63713			0.818567	+	0.574412i	$0.194769\pi$
$984$		0			0
$985$	−45.2203			−1.44084
$986$		0			0
$987$	13.5468	−	5.77601i	0.431199	−	0.183852i
$988$		0			0
$989$	−30.9736	+	34.3833i	−0.984903	+	1.09332i
$990$		0			0
$991$	−24.9900			−0.793833			−0.396917	−	0.917855i	$-0.629920\pi$
$991$	−24.9900			−0.793833			−0.396917	+	0.917855i	$0.629920\pi$
$992$		0			0
$993$			19.2100i			0.609610i
$994$		0			0
$995$	−31.5285			−0.999521
$996$		0			0
$997$			4.91451i			0.155644i	0.996967	+	0.0778220i	$0.0247966\pi$
$997$			4.91451i			0.155644i	−0.996967	+	0.0778220i	$0.975203\pi$
$998$		0			0
$999$	10.7727			0.340833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.20	yes	32
3.2	odd	2		5796.2.k.d.5473.10		32
7.6	odd	2	inner	1932.2.k.a.1609.17	✓	32
21.20	even	2		5796.2.k.d.5473.24		32
23.22	odd	2	inner	1932.2.k.a.1609.19	yes	32
69.68	even	2		5796.2.k.d.5473.23		32
161.160	even	2	inner	1932.2.k.a.1609.18	yes	32
483.482	odd	2		5796.2.k.d.5473.9		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.17	✓	32	7.6	odd	2	inner
1932.2.k.a.1609.18	yes	32	161.160	even	2	inner
1932.2.k.a.1609.19	yes	32	23.22	odd	2	inner
1932.2.k.a.1609.20	yes	32	1.1	even	1	trivial
5796.2.k.d.5473.9		32	483.482	odd	2
5796.2.k.d.5473.10		32	3.2	odd	2
5796.2.k.d.5473.23		32	69.68	even	2
5796.2.k.d.5473.24		32	21.20	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.12739 q^{5} +(-2.43376 + 1.03769i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +2.12739 q^{5} +(-2.43376 + 1.03769i) q^{7} -1.00000 q^{9} -1.02219i q^{11} -1.76109i q^{13} +2.12739i q^{15} -0.210591 q^{17} +4.64139 q^{19} +(-1.03769 - 2.43376i) q^{21} +(3.56324 + 3.20988i) q^{23} -0.474227 q^{25} -1.00000i q^{27} +10.2285 q^{29} +1.19079i q^{31} +1.02219 q^{33} +(-5.17755 + 2.20758i) q^{35} +10.7727i q^{37} +1.76109 q^{39} +1.14009i q^{41} +9.64945i q^{43} -2.12739 q^{45} +5.56619i q^{47} +(4.84638 - 5.05100i) q^{49} -0.210591i q^{51} -1.32211i q^{53} -2.17459i q^{55} +4.64139i q^{57} +9.90536i q^{59} -2.05584 q^{61} +(2.43376 - 1.03769i) q^{63} -3.74651i q^{65} +2.73382i q^{67} +(-3.20988 + 3.56324i) q^{69} +6.57204 q^{71} -12.6369i q^{73} -0.474227i q^{75} +(1.06072 + 2.48776i) q^{77} -1.45891i q^{79} +1.00000 q^{81} -6.66137 q^{83} -0.448008 q^{85} +10.2285i q^{87} -6.85521 q^{89} +(1.82747 + 4.28606i) q^{91} -1.19079 q^{93} +9.87404 q^{95} -1.67955 q^{97} +1.02219i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 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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