LMFDB - Embedded newform 1932.2.k.a.1609.10

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			1609.10
Character	$\chi$	$=$	1932.1609
Dual form			1932.2.k.a.1609.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.52823 q^{5} +(-2.25542 + 1.38314i) q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.52823 q^{5} +(-2.25542 + 1.38314i) q^{7} -1.00000 q^{9} -0.480134i q^{11} -0.157873i q^{13} -2.52823i q^{15} -5.77931 q^{17} -6.77661 q^{19} +(1.38314 + 2.25542i) q^{21} +(-4.15621 + 2.39289i) q^{23} +1.39192 q^{25} +1.00000i q^{27} -0.536894 q^{29} +8.94066i q^{31} -0.480134 q^{33} +(-5.70222 + 3.49688i) q^{35} -1.22741i q^{37} -0.157873 q^{39} -11.2504i q^{41} +6.31616i q^{43} -2.52823 q^{45} -1.75990i q^{47} +(3.17387 - 6.23911i) q^{49} +5.77931i q^{51} +1.82973i q^{53} -1.21389i q^{55} +6.77661i q^{57} -8.60869i q^{59} -14.1394 q^{61} +(2.25542 - 1.38314i) q^{63} -0.399138i q^{65} -8.94056i q^{67} +(2.39289 + 4.15621i) q^{69} -8.38636 q^{71} -12.2061i q^{73} -1.39192i q^{75} +(0.664091 + 1.08291i) q^{77} +16.5866i q^{79} +1.00000 q^{81} +0.812475 q^{83} -14.6114 q^{85} +0.536894i q^{87} +14.7848 q^{89} +(0.218359 + 0.356070i) q^{91} +8.94066 q^{93} -17.1328 q^{95} +3.39930 q^{97} +0.480134i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	2.52823			1.13066			0.565328	−	0.824866i	$-0.308749\pi$
$5$	2.52823			1.13066			0.565328	+	0.824866i	$0.308749\pi$
$6$		0			0
$7$	−2.25542	+	1.38314i	−0.852470	+	0.522776i
$7$	−2.25542	+	1.38314i	−0.852470	+	0.522776i
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$		−	0.480134i		−	0.144766i	−0.997377	−	0.0723829i	$-0.976940\pi$
$11$		−	0.480134i		−	0.144766i	0.997377	−	0.0723829i	$-0.0230604\pi$
$12$		0			0
$13$		−	0.157873i		−	0.0437860i	−0.999760	−	0.0218930i	$-0.993031\pi$
$13$		−	0.157873i		−	0.0437860i	0.999760	−	0.0218930i	$-0.00696932\pi$
$14$		0			0
$15$		−	2.52823i		−	0.652785i
$16$		0			0
$17$	−5.77931			−1.40169			−0.700845	−	0.713314i	$-0.747193\pi$
$17$	−5.77931			−1.40169			−0.700845	+	0.713314i	$0.747193\pi$
$18$		0			0
$19$	−6.77661			−1.55466			−0.777330	−	0.629093i	$-0.783427\pi$
$19$	−6.77661			−1.55466			−0.777330	+	0.629093i	$0.783427\pi$
$20$		0			0
$21$	1.38314	+	2.25542i	0.301825	+	0.492174i
$22$		0			0
$23$	−4.15621	+	2.39289i	−0.866629	+	0.498953i
$23$	−4.15621	+	2.39289i	−0.866629	+	0.498953i
$24$		0			0
$25$	1.39192			0.278385
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	−0.536894			−0.0996988			−0.0498494	−	0.998757i	$-0.515874\pi$
$29$	−0.536894			−0.0996988			−0.0498494	+	0.998757i	$0.515874\pi$
$30$		0			0
$31$			8.94066i			1.60579i	0.596121	+	0.802895i	$0.296708\pi$
$31$			8.94066i			1.60579i	−0.596121	+	0.802895i	$0.703292\pi$
$32$		0			0
$33$	−0.480134			−0.0835806
$34$		0			0
$35$	−5.70222	+	3.49688i	−0.963851	+	0.591080i
$36$		0			0
$37$		−	1.22741i		−	0.201786i	−0.994897	−	0.100893i	$-0.967830\pi$
$37$		−	1.22741i		−	0.201786i	0.994897	−	0.100893i	$-0.0321699\pi$
$38$		0			0
$39$	−0.157873			−0.0252799
$40$		0			0
$41$		−	11.2504i		−	1.75701i	−0.477733	−	0.878505i	$-0.658542\pi$
$41$		−	11.2504i		−	1.75701i	0.477733	−	0.878505i	$-0.341458\pi$
$42$		0			0
$43$			6.31616i			0.963206i	0.876390	+	0.481603i	$0.159945\pi$
$43$			6.31616i			0.963206i	−0.876390	+	0.481603i	$0.840055\pi$
$44$		0			0
$45$	−2.52823			−0.376886
$46$		0			0
$47$		−	1.75990i		−	0.256708i	−0.991728	−	0.128354i	$-0.959031\pi$
$47$		−	1.75990i		−	0.256708i	0.991728	−	0.128354i	$-0.0409693\pi$
$48$		0			0
$49$	3.17387	−	6.23911i	0.453410	−	0.891302i
$50$		0			0
$51$			5.77931i			0.809266i
$52$		0			0
$53$			1.82973i			0.251333i	0.992073	+	0.125667i	$0.0401069\pi$
$53$			1.82973i			0.251333i	−0.992073	+	0.125667i	$0.959893\pi$
$54$		0			0
$55$		−	1.21389i		−	0.163681i
$56$		0			0
$57$			6.77661i			0.897584i
$58$		0			0
$59$		−	8.60869i		−	1.12076i	−0.828237	−	0.560378i	$-0.810656\pi$
$59$		−	8.60869i		−	1.12076i	0.828237	−	0.560378i	$-0.189344\pi$
$60$		0			0
$61$	−14.1394			−1.81037			−0.905183	−	0.425023i	$-0.860266\pi$
$61$	−14.1394			−1.81037			−0.905183	+	0.425023i	$0.860266\pi$
$62$		0			0
$63$	2.25542	−	1.38314i	0.284157	−	0.174259i
$64$		0			0
$65$		−	0.399138i		−	0.0495070i
$66$		0			0
$67$		−	8.94056i		−	1.09226i	−0.837699	−	0.546132i	$-0.816100\pi$
$67$		−	8.94056i		−	1.09226i	0.837699	−	0.546132i	$-0.183900\pi$
$68$		0			0
$69$	2.39289	+	4.15621i	0.288070	+	0.500349i
$70$		0			0
$71$	−8.38636			−0.995278			−0.497639	−	0.867384i	$-0.665799\pi$
$71$	−8.38636			−0.995278			−0.497639	+	0.867384i	$0.665799\pi$
$72$		0			0
$73$		−	12.2061i		−	1.42861i	−0.699834	−	0.714306i	$-0.746743\pi$
$73$		−	12.2061i		−	1.42861i	0.699834	−	0.714306i	$-0.253257\pi$
$74$		0			0
$75$		−	1.39192i		−	0.160726i
$76$		0			0
$77$	0.664091	+	1.08291i	0.0756801	+	0.123409i
$78$		0			0
$79$			16.5866i			1.86614i	0.359692	+	0.933071i	$0.382882\pi$
$79$			16.5866i			1.86614i	−0.359692	+	0.933071i	$0.617118\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	0.812475			0.0891807			0.0445904	−	0.999005i	$-0.485802\pi$
$83$	0.812475			0.0891807			0.0445904	+	0.999005i	$0.485802\pi$
$84$		0			0
$85$	−14.6114			−1.58483
$86$		0			0
$87$			0.536894i			0.0575611i
$88$		0			0
$89$	14.7848			1.56718			0.783592	−	0.621275i	$-0.213385\pi$
$89$	14.7848			1.56718			0.783592	+	0.621275i	$0.213385\pi$
$90$		0			0
$91$	0.218359	+	0.356070i	0.0228903	+	0.0373263i
$92$		0			0
$93$	8.94066			0.927103
$94$		0			0
$95$	−17.1328			−1.75779
$96$		0			0
$97$	3.39930			0.345146			0.172573	−	0.984997i	$-0.444792\pi$
$97$	3.39930			0.345146			0.172573	+	0.984997i	$0.444792\pi$
$98$		0			0
$99$			0.480134i			0.0482553i
$100$		0			0
$101$			7.49957i			0.746235i	0.927784	+	0.373118i	$0.121711\pi$
$101$			7.49957i			0.746235i	−0.927784	+	0.373118i	$0.878289\pi$
$102$		0			0
$103$	−7.70522			−0.759218			−0.379609	−	0.925147i	$-0.623941\pi$
$103$	−7.70522			−0.759218			−0.379609	+	0.925147i	$0.623941\pi$
$104$		0			0
$105$	3.49688	+	5.70222i	0.341260	+	0.556480i
$106$		0			0
$107$			6.29579i			0.608637i	0.952570	+	0.304318i	$0.0984286\pi$
$107$			6.29579i			0.608637i	−0.952570	+	0.304318i	$0.901571\pi$
$108$		0			0
$109$		−	0.812061i		−	0.0777813i	−0.999243	−	0.0388906i	$-0.987618\pi$
$109$		−	0.812061i		−	0.0777813i	0.999243	−	0.0388906i	$-0.0123824\pi$
$110$		0			0
$111$	−1.22741			−0.116501
$112$		0			0
$113$			18.5773i			1.74761i	0.486279	+	0.873804i	$0.338354\pi$
$113$			18.5773i			1.74761i	−0.486279	+	0.873804i	$0.661646\pi$
$114$		0			0
$115$	−10.5078	+	6.04977i	−0.979860	+	0.564144i
$116$		0			0
$117$			0.157873i			0.0145953i
$118$		0			0
$119$	13.0348	−	7.99357i	1.19490	−	0.732770i
$120$		0			0
$121$	10.7695			0.979043
$122$		0			0
$123$	−11.2504			−1.01441
$124$		0			0
$125$	−9.12203			−0.815899
$126$		0			0
$127$	−0.441763			−0.0392001			−0.0196001	−	0.999808i	$-0.506239\pi$
$127$	−0.441763			−0.0392001			−0.0196001	+	0.999808i	$0.506239\pi$
$128$		0			0
$129$	6.31616			0.556107
$130$		0			0
$131$			18.4632i			1.61314i	0.591142	+	0.806568i	$0.298677\pi$
$131$			18.4632i			1.61314i	−0.591142	+	0.806568i	$0.701323\pi$
$132$		0			0
$133$	15.2841	−	9.37297i	1.32530	−	0.812740i
$134$		0			0
$135$			2.52823i			0.217595i
$136$		0			0
$137$			0.205196i			0.0175311i	0.999962	+	0.00876555i	$0.00279020\pi$
$137$			0.205196i			0.0175311i	−0.999962	+	0.00876555i	$0.997210\pi$
$138$		0			0
$139$		−	22.9307i		−	1.94496i	−0.232986	−	0.972480i	$-0.574850\pi$
$139$		−	22.9307i		−	1.94496i	0.232986	−	0.972480i	$-0.425150\pi$
$140$		0			0
$141$	−1.75990			−0.148210
$142$		0			0
$143$	−0.0758001			−0.00633872
$144$		0			0
$145$	−1.35739			−0.112725
$146$		0			0
$147$	−6.23911	−	3.17387i	−0.514593	−	0.261777i
$148$		0			0
$149$		−	8.90986i		−	0.729924i	−0.931022	−	0.364962i	$-0.881082\pi$
$149$		−	8.90986i		−	0.729924i	0.931022	−	0.364962i	$-0.118918\pi$
$150$		0			0
$151$	8.43791			0.686667			0.343334	−	0.939213i	$-0.388444\pi$
$151$	8.43791			0.686667			0.343334	+	0.939213i	$0.388444\pi$
$152$		0			0
$153$	5.77931			0.467230
$154$		0			0
$155$			22.6040i			1.81560i
$156$		0			0
$157$	8.25082			0.658487			0.329244	−	0.944245i	$-0.393206\pi$
$157$	8.25082			0.658487			0.329244	+	0.944245i	$0.393206\pi$
$158$		0			0
$159$	1.82973			0.145107
$160$		0			0
$161$	6.06431	−	11.1456i	0.477935	−	0.878395i
$162$		0			0
$163$	−18.5371			−1.45194			−0.725970	−	0.687727i	$-0.758609\pi$
$163$	−18.5371			−1.45194			−0.725970	+	0.687727i	$0.758609\pi$
$164$		0			0
$165$	−1.21389			−0.0945010
$166$		0			0
$167$			2.15409i			0.166688i	0.996521	+	0.0833441i	$0.0265601\pi$
$167$			2.15409i			0.166688i	−0.996521	+	0.0833441i	$0.973440\pi$
$168$		0			0
$169$	12.9751			0.998083
$170$		0			0
$171$	6.77661			0.518220
$172$		0			0
$173$			14.9120i			1.13374i	0.823808	+	0.566869i	$0.191845\pi$
$173$			14.9120i			1.13374i	−0.823808	+	0.566869i	$0.808155\pi$
$174$		0			0
$175$	−3.13938	+	1.92522i	−0.237315	+	0.145533i
$176$		0			0
$177$	−8.60869			−0.647068
$178$		0			0
$179$	22.4921			1.68114			0.840571	−	0.541701i	$-0.182220\pi$
$179$	22.4921			1.68114			0.840571	+	0.541701i	$0.182220\pi$
$180$		0			0
$181$	−9.66975			−0.718747			−0.359373	−	0.933194i	$-0.617010\pi$
$181$	−9.66975			−0.718747			−0.359373	+	0.933194i	$0.617010\pi$
$182$		0			0
$183$			14.1394i			1.04521i
$184$		0			0
$185$		−	3.10318i		−	0.228150i
$186$		0			0
$187$			2.77484i			0.202917i
$188$		0			0
$189$	−1.38314	−	2.25542i	−0.100608	−	0.164058i
$190$		0			0
$191$			17.1153i			1.23842i	0.785226	+	0.619209i	$0.212547\pi$
$191$			17.1153i			1.23842i	−0.785226	+	0.619209i	$0.787453\pi$
$192$		0			0
$193$	−21.3868			−1.53946			−0.769728	−	0.638371i	$-0.779608\pi$
$193$	−21.3868			−1.53946			−0.769728	+	0.638371i	$0.779608\pi$
$194$		0			0
$195$	−0.399138			−0.0285829
$196$		0			0
$197$	−19.1434			−1.36391			−0.681956	−	0.731393i	$-0.738871\pi$
$197$	−19.1434			−1.36391			−0.681956	+	0.731393i	$0.738871\pi$
$198$		0			0
$199$	13.5204			0.958432			0.479216	−	0.877697i	$-0.340921\pi$
$199$	13.5204			0.958432			0.479216	+	0.877697i	$0.340921\pi$
$200$		0			0
$201$	−8.94056			−0.630619
$202$		0			0
$203$	1.21092	−	0.742598i	0.0849902	−	0.0521201i
$204$		0			0
$205$		−	28.4434i		−	1.98657i
$206$		0			0
$207$	4.15621	−	2.39289i	0.288876	−	0.166318i
$208$		0			0
$209$			3.25368i			0.225062i
$210$		0			0
$211$	−24.6125			−1.69440			−0.847199	−	0.531276i	$-0.821713\pi$
$211$	−24.6125			−1.69440			−0.847199	+	0.531276i	$0.821713\pi$
$212$		0			0
$213$			8.38636i			0.574624i
$214$		0			0
$215$			15.9687i			1.08905i
$216$		0			0
$217$	−12.3661	−	20.1650i	−0.839468	−	1.36889i
$218$		0			0
$219$	−12.2061			−0.824809
$220$		0			0
$221$			0.912396i			0.0613744i
$222$		0			0
$223$			1.20632i			0.0807809i	0.999184	+	0.0403905i	$0.0128602\pi$
$223$			1.20632i			0.0807809i	−0.999184	+	0.0403905i	$0.987140\pi$
$224$		0			0
$225$	−1.39192			−0.0927949
$226$		0			0
$227$	−23.1968			−1.53962			−0.769812	−	0.638270i	$-0.779650\pi$
$227$	−23.1968			−1.53962			−0.769812	+	0.638270i	$0.779650\pi$
$228$		0			0
$229$	−14.5997			−0.964775			−0.482388	−	0.875958i	$-0.660230\pi$
$229$	−14.5997			−0.964775			−0.482388	+	0.875958i	$0.660230\pi$
$230$		0			0
$231$	1.08291	−	0.664091i	0.0712500	−	0.0436939i
$232$		0			0
$233$	14.4763			0.948372			0.474186	−	0.880425i	$-0.342742\pi$
$233$	14.4763			0.948372			0.474186	+	0.880425i	$0.342742\pi$
$234$		0			0
$235$		−	4.44942i		−	0.290248i
$236$		0			0
$237$	16.5866			1.07742
$238$		0			0
$239$	11.2973			0.730759			0.365379	−	0.930859i	$-0.380939\pi$
$239$	11.2973			0.730759			0.365379	+	0.930859i	$0.380939\pi$
$240$		0			0
$241$	−15.4230			−0.993484			−0.496742	−	0.867898i	$-0.665470\pi$
$241$	−15.4230			−0.993484			−0.496742	+	0.867898i	$0.665470\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	8.02426	−	15.7739i	0.512651	−	1.00776i
$246$		0			0
$247$			1.06984i			0.0680724i
$248$		0			0
$249$		−	0.812475i		−	0.0514885i
$250$		0			0
$251$	−8.10888			−0.511828			−0.255914	−	0.966700i	$-0.582376\pi$
$251$	−8.10888			−0.511828			−0.255914	+	0.966700i	$0.582376\pi$
$252$		0			0
$253$	1.14891	+	1.99554i	0.0722313	+	0.125458i
$254$		0			0
$255$			14.6114i			0.915002i
$256$		0			0
$257$			19.4938i			1.21599i	0.793940	+	0.607996i	$0.208026\pi$
$257$			19.4938i			1.21599i	−0.793940	+	0.607996i	$0.791974\pi$
$258$		0			0
$259$	1.69768	+	2.76834i	0.105489	+	0.172016i
$260$		0			0
$261$	0.536894			0.0332329
$262$		0			0
$263$		−	8.54961i		−	0.527191i	−0.964633	−	0.263596i	$-0.915092\pi$
$263$		−	8.54961i		−	0.527191i	0.964633	−	0.263596i	$-0.0849085\pi$
$264$		0			0
$265$			4.62597i			0.284171i
$266$		0			0
$267$		−	14.7848i		−	0.904815i
$268$		0			0
$269$			19.3468i			1.17960i	0.807550	+	0.589799i	$0.200793\pi$
$269$			19.3468i			1.17960i	−0.807550	+	0.589799i	$0.799207\pi$
$270$		0			0
$271$		−	27.5658i		−	1.67450i	−0.546818	−	0.837251i	$-0.684161\pi$
$271$		−	27.5658i		−	1.67450i	0.546818	−	0.837251i	$-0.315839\pi$
$272$		0			0
$273$	0.356070	−	0.218359i	0.0215503	−	0.0132157i
$274$		0			0
$275$		−	0.668310i		−	0.0403006i
$276$		0			0
$277$	7.85710			0.472087			0.236044	−	0.971742i	$-0.424149\pi$
$277$	7.85710			0.472087			0.236044	+	0.971742i	$0.424149\pi$
$278$		0			0
$279$		−	8.94066i		−	0.535263i
$280$		0			0
$281$		−	17.5640i		−	1.04778i	−0.851785	−	0.523891i	$-0.824480\pi$
$281$		−	17.5640i		−	1.04778i	0.851785	−	0.523891i	$-0.175520\pi$
$282$		0			0
$283$	2.14307			0.127393			0.0636963	−	0.997969i	$-0.479711\pi$
$283$	2.14307			0.127393			0.0636963	+	0.997969i	$0.479711\pi$
$284$		0			0
$285$			17.1328i			1.01486i
$286$		0			0
$287$	15.5608	+	25.3743i	0.918523	+	1.49780i
$288$		0			0
$289$	16.4005			0.964733
$290$		0			0
$291$		−	3.39930i		−	0.199270i
$292$		0			0
$293$	25.1287			1.46804			0.734018	−	0.679130i	$-0.237643\pi$
$293$	25.1287			1.46804			0.734018	+	0.679130i	$0.237643\pi$
$294$		0			0
$295$		−	21.7647i		−	1.26719i
$296$		0			0
$297$	0.480134			0.0278602
$298$		0			0
$299$	0.377772	+	0.656152i	0.0218471	+	0.0379462i
$300$		0			0
$301$	−8.73611	−	14.2456i	−0.503541	−	0.821104i
$302$		0			0
$303$	7.49957			0.430839
$304$		0			0
$305$	−35.7476			−2.04690
$306$		0			0
$307$		−	2.58458i		−	0.147510i	−0.997276	−	0.0737550i	$-0.976502\pi$
$307$		−	2.58458i		−	0.147510i	0.997276	−	0.0737550i	$-0.0234983\pi$
$308$		0			0
$309$			7.70522i			0.438335i
$310$		0			0
$311$		−	28.9877i		−	1.64374i	−0.569675	−	0.821870i	$-0.692931\pi$
$311$		−	28.9877i		−	1.64374i	0.569675	−	0.821870i	$-0.307069\pi$
$312$		0			0
$313$	8.17704			0.462194			0.231097	−	0.972931i	$-0.425769\pi$
$313$	8.17704			0.462194			0.231097	+	0.972931i	$0.425769\pi$
$314$		0			0
$315$	5.70222	−	3.49688i	0.321284	−	0.197027i
$316$		0			0
$317$	16.4518			0.924026			0.462013	−	0.886873i	$-0.347127\pi$
$317$	16.4518			0.924026			0.462013	+	0.886873i	$0.347127\pi$
$318$		0			0
$319$			0.257781i			0.0144330i
$320$		0			0
$321$	6.29579			0.351397
$322$		0			0
$323$	39.1641			2.17915
$324$		0			0
$325$		−	0.219747i		−	0.0121894i
$326$		0			0
$327$	−0.812061			−0.0449071
$328$		0			0
$329$	2.43418	+	3.96932i	0.134201	+	0.218836i
$330$		0			0
$331$	20.4830			1.12585			0.562924	−	0.826509i	$-0.309677\pi$
$331$	20.4830			1.12585			0.562924	+	0.826509i	$0.309677\pi$
$332$		0			0
$333$			1.22741i			0.0672619i
$334$		0			0
$335$		−	22.6038i		−	1.23498i
$336$		0			0
$337$		−	27.9979i		−	1.52514i	−0.646905	−	0.762571i	$-0.723937\pi$
$337$		−	27.9979i		−	1.52514i	0.646905	−	0.762571i	$-0.276063\pi$
$338$		0			0
$339$	18.5773			1.00898
$340$		0			0
$341$	4.29271			0.232463
$342$		0			0
$343$	1.47112	+	18.4617i	0.0794328	+	0.996840i
$344$		0			0
$345$	6.04977	+	10.5078i	0.325709	+	0.565723i
$346$		0			0
$347$	3.45100			0.185260			0.0926298	−	0.995701i	$-0.470473\pi$
$347$	3.45100			0.185260			0.0926298	+	0.995701i	$0.470473\pi$
$348$		0			0
$349$		−	6.96070i		−	0.372598i	−0.982493	−	0.186299i	$-0.940351\pi$
$349$		−	6.96070i		−	0.372598i	0.982493	−	0.186299i	$-0.0596492\pi$
$350$		0			0
$351$	0.157873			0.00842662
$352$		0			0
$353$		−	13.2681i		−	0.706187i	−0.935588	−	0.353094i	$-0.885130\pi$
$353$		−	13.2681i		−	0.706187i	0.935588	−	0.353094i	$-0.114870\pi$
$354$		0			0
$355$	−21.2026			−1.12532
$356$		0			0
$357$	−7.99357	−	13.0348i	−0.423065	−	0.689875i
$358$		0			0
$359$			4.17550i			0.220375i	0.993911	+	0.110187i	$0.0351451\pi$
$359$			4.17550i			0.220375i	−0.993911	+	0.110187i	$0.964855\pi$
$360$		0			0
$361$	26.9224			1.41697
$362$		0			0
$363$		−	10.7695i		−	0.565251i
$364$		0			0
$365$		−	30.8597i		−	1.61527i
$366$		0			0
$367$	−15.1124			−0.788859			−0.394429	−	0.918926i	$-0.629058\pi$
$367$	−15.1124			−0.788859			−0.394429	+	0.918926i	$0.629058\pi$
$368$		0			0
$369$			11.2504i			0.585670i
$370$		0			0
$371$	−2.53077	−	4.12682i	−0.131391	−	0.214254i
$372$		0			0
$373$		−	27.3736i		−	1.41735i	−0.705535	−	0.708675i	$-0.749293\pi$
$373$		−	27.3736i		−	1.41735i	0.705535	−	0.708675i	$-0.250707\pi$
$374$		0			0
$375$			9.12203i			0.471060i
$376$		0			0
$377$			0.0847610i			0.00436541i
$378$		0			0
$379$		−	16.2060i		−	0.832444i	−0.909263	−	0.416222i	$-0.863354\pi$
$379$		−	16.2060i		−	0.832444i	0.909263	−	0.416222i	$-0.136646\pi$
$380$		0			0
$381$			0.441763i			0.0226322i
$382$		0			0
$383$	1.30497			0.0666809			0.0333405	−	0.999444i	$-0.489385\pi$
$383$	1.30497			0.0666809			0.0333405	+	0.999444i	$0.489385\pi$
$384$		0			0
$385$	1.67897	+	2.73783i	0.0855683	+	0.139533i
$386$		0			0
$387$		−	6.31616i		−	0.321069i
$388$		0			0
$389$		−	25.8967i		−	1.31301i	−0.754320	−	0.656507i	$-0.772033\pi$
$389$		−	25.8967i		−	1.31301i	0.754320	−	0.656507i	$-0.227967\pi$
$390$		0			0
$391$	24.0200	−	13.8293i	1.21475	−	0.699377i
$392$		0			0
$393$	18.4632			0.931344
$394$		0			0
$395$			41.9348i			2.10997i
$396$		0			0
$397$			36.9780i			1.85587i	0.372741	+	0.927935i	$0.378418\pi$
$397$			36.9780i			1.85587i	−0.372741	+	0.927935i	$0.621582\pi$
$398$		0			0
$399$	−9.37297	−	15.2841i	−0.469235	−	0.765163i
$400$		0			0
$401$			7.86431i			0.392725i	0.980531	+	0.196363i	$0.0629130\pi$
$401$			7.86431i			0.392725i	−0.980531	+	0.196363i	$0.937087\pi$
$402$		0			0
$403$	1.41149			0.0703111
$404$		0			0
$405$	2.52823			0.125629
$406$		0			0
$407$	−0.589323			−0.0292117
$408$		0			0
$409$		−	0.877853i		−	0.0434070i	−0.999764	−	0.0217035i	$-0.993091\pi$
$409$		−	0.877853i		−	0.0434070i	0.999764	−	0.0217035i	$-0.00690899\pi$
$410$		0			0
$411$	0.205196			0.0101216
$412$		0			0
$413$	11.9070	+	19.4162i	0.585904	+	0.955410i
$414$		0			0
$415$	2.05412			0.100833
$416$		0			0
$417$	−22.9307			−1.12292
$418$		0			0
$419$	33.5414			1.63860			0.819302	−	0.573362i	$-0.194361\pi$
$419$	33.5414			1.63860			0.819302	+	0.573362i	$0.194361\pi$
$420$		0			0
$421$		−	9.64976i		−	0.470300i	−0.971959	−	0.235150i	$-0.924442\pi$
$421$		−	9.64976i		−	0.470300i	0.971959	−	0.235150i	$-0.0755582\pi$
$422$		0			0
$423$			1.75990i			0.0855692i
$424$		0			0
$425$	−8.04436			−0.390209
$426$		0			0
$427$	31.8904	−	19.5567i	1.54328	−	0.946416i
$428$		0			0
$429$			0.0758001i			0.00365966i
$430$		0			0
$431$		−	12.9319i		−	0.622905i	−0.950262	−	0.311453i	$-0.899184\pi$
$431$		−	12.9319i		−	0.622905i	0.950262	−	0.311453i	$-0.100816\pi$
$432$		0			0
$433$	−19.8408			−0.953487			−0.476743	−	0.879043i	$-0.658183\pi$
$433$	−19.8408			−0.953487			−0.476743	+	0.879043i	$0.658183\pi$
$434$		0			0
$435$			1.35739i			0.0650819i
$436$		0			0
$437$	28.1650	−	16.2157i	1.34731	−	0.775702i
$438$		0			0
$439$			30.2031i			1.44152i	0.693187	+	0.720758i	$0.256206\pi$
$439$			30.2031i			1.44152i	−0.693187	+	0.720758i	$0.743794\pi$
$440$		0			0
$441$	−3.17387	+	6.23911i	−0.151137	+	0.297101i
$442$		0			0
$443$	−21.5419			−1.02349			−0.511743	−	0.859139i	$-0.671000\pi$
$443$	−21.5419			−1.02349			−0.511743	+	0.859139i	$0.671000\pi$
$444$		0			0
$445$	37.3793			1.77195
$446$		0			0
$447$	−8.90986			−0.421422
$448$		0			0
$449$	20.6163			0.972942			0.486471	−	0.873697i	$-0.338284\pi$
$449$	20.6163			0.972942			0.486471	+	0.873697i	$0.338284\pi$
$450$		0			0
$451$	−5.40168			−0.254355
$452$		0			0
$453$		−	8.43791i		−	0.396448i
$454$		0			0
$455$	0.552062	+	0.900225i	0.0258811	+	0.0422032i
$456$		0			0
$457$			17.9285i			0.838662i	0.907833	+	0.419331i	$0.137735\pi$
$457$			17.9285i			0.838662i	−0.907833	+	0.419331i	$0.862265\pi$
$458$		0			0
$459$		−	5.77931i		−	0.269755i
$460$		0			0
$461$			8.09754i			0.377140i	0.982060	+	0.188570i	$0.0603853\pi$
$461$			8.09754i			0.377140i	−0.982060	+	0.188570i	$0.939615\pi$
$462$		0			0
$463$	17.3159			0.804738			0.402369	−	0.915478i	$-0.368187\pi$
$463$	17.3159			0.804738			0.402369	+	0.915478i	$0.368187\pi$
$464$		0			0
$465$	22.6040			1.04824
$466$		0			0
$467$	−25.3173			−1.17155			−0.585774	−	0.810475i	$-0.699209\pi$
$467$	−25.3173			−1.17155			−0.585774	+	0.810475i	$0.699209\pi$
$468$		0			0
$469$	12.3660	+	20.1648i	0.571009	+	0.931122i
$470$		0			0
$471$		−	8.25082i		−	0.380178i
$472$		0			0
$473$	3.03260			0.139439
$474$		0			0
$475$	−9.43252			−0.432794
$476$		0			0
$477$		−	1.82973i		−	0.0837777i
$478$		0			0
$479$	−18.2662			−0.834606			−0.417303	−	0.908767i	$-0.637025\pi$
$479$	−18.2662			−0.834606			−0.417303	+	0.908767i	$0.637025\pi$
$480$		0			0
$481$	−0.193775			−0.00883539
$482$		0			0
$483$	−11.1456	−	6.06431i	−0.507142	−	0.275936i
$484$		0			0
$485$	8.59418			0.390242
$486$		0			0
$487$	−19.9712			−0.904983			−0.452491	−	0.891769i	$-0.649465\pi$
$487$	−19.9712			−0.904983			−0.452491	+	0.891769i	$0.649465\pi$
$488$		0			0
$489$			18.5371i			0.838277i
$490$		0			0
$491$	−7.53997			−0.340274			−0.170137	−	0.985420i	$-0.554421\pi$
$491$	−7.53997			−0.340274			−0.170137	+	0.985420i	$0.554421\pi$
$492$		0			0
$493$	3.10288			0.139747
$494$		0			0
$495$			1.21389i			0.0545602i
$496$		0			0
$497$	18.9148	−	11.5995i	0.848444	−	0.520307i
$498$		0			0
$499$	−32.6848			−1.46317			−0.731586	−	0.681749i	$-0.761219\pi$
$499$	−32.6848			−1.46317			−0.731586	+	0.681749i	$0.761219\pi$
$500$		0			0
$501$	2.15409			0.0962375
$502$		0			0
$503$	27.2358			1.21438			0.607192	−	0.794555i	$-0.292296\pi$
$503$	27.2358			1.21438			0.607192	+	0.794555i	$0.292296\pi$
$504$		0			0
$505$			18.9606i			0.843736i
$506$		0			0
$507$		−	12.9751i		−	0.576243i
$508$		0			0
$509$		−	4.99793i		−	0.221529i	−0.993847	−	0.110765i	$-0.964670\pi$
$509$		−	4.99793i		−	0.221529i	0.993847	−	0.110765i	$-0.0353300\pi$
$510$		0			0
$511$	16.8826	+	27.5298i	0.746844	+	1.21785i
$512$		0			0
$513$		−	6.77661i		−	0.299195i
$514$		0			0
$515$	−19.4805			−0.858415
$516$		0			0
$517$	−0.844987			−0.0371625
$518$		0			0
$519$	14.9120			0.654563
$520$		0			0
$521$	−8.36778			−0.366599			−0.183300	−	0.983057i	$-0.558678\pi$
$521$	−8.36778			−0.366599			−0.183300	+	0.983057i	$0.558678\pi$
$522$		0			0
$523$	−5.28145			−0.230941			−0.115471	−	0.993311i	$-0.536838\pi$
$523$	−5.28145			−0.230941			−0.115471	+	0.993311i	$0.536838\pi$
$524$		0			0
$525$	1.92522	+	3.13938i	0.0840235	+	0.137014i
$526$		0			0
$527$		−	51.6709i		−	2.25082i
$528$		0			0
$529$	11.5481	−	19.8907i	0.502093	−	0.864814i
$530$		0			0
$531$			8.60869i			0.373585i
$532$		0			0
$533$	−1.77612			−0.0769325
$534$		0			0
$535$			15.9172i			0.688159i
$536$		0			0
$537$		−	22.4921i		−	0.970608i
$538$		0			0
$539$	−2.99561	−	1.52388i	−0.129030	−	0.0656383i
$540$		0			0
$541$	17.0128			0.731438			0.365719	−	0.930725i	$-0.380823\pi$
$541$	17.0128			0.731438			0.365719	+	0.930725i	$0.380823\pi$
$542$		0			0
$543$			9.66975i			0.414969i
$544$		0			0
$545$		−	2.05307i		−	0.0879440i
$546$		0			0
$547$	−18.7534			−0.801838			−0.400919	−	0.916114i	$-0.631309\pi$
$547$	−18.7534			−0.801838			−0.400919	+	0.916114i	$0.631309\pi$
$548$		0			0
$549$	14.1394			0.603455
$550$		0			0
$551$	3.63832			0.154998
$552$		0			0
$553$	−22.9416	−	37.4099i	−0.975575	−	1.59083i
$554$		0			0
$555$	−3.10318			−0.131723
$556$		0			0
$557$			22.0215i			0.933081i	0.884500	+	0.466540i	$0.154500\pi$
$557$			22.0215i			0.933081i	−0.884500	+	0.466540i	$0.845500\pi$
$558$		0			0
$559$	0.997150			0.0421749
$560$		0			0
$561$	2.77484			0.117154
$562$		0			0
$563$	−34.8464			−1.46860			−0.734300	−	0.678825i	$-0.762489\pi$
$563$	−34.8464			−1.46860			−0.734300	+	0.678825i	$0.762489\pi$
$564$		0			0
$565$			46.9677i			1.97594i
$566$		0			0
$567$	−2.25542	+	1.38314i	−0.0947189	+	0.0580862i
$568$		0			0
$569$			25.7487i			1.07944i	0.841844	+	0.539720i	$0.181470\pi$
$569$			25.7487i			1.07944i	−0.841844	+	0.539720i	$0.818530\pi$
$570$		0			0
$571$		−	0.323750i		−	0.0135485i	−0.999977	−	0.00677427i	$-0.997844\pi$
$571$		−	0.323750i		−	0.0135485i	0.999977	−	0.00677427i	$-0.00215633\pi$
$572$		0			0
$573$	17.1153			0.715001
$574$		0			0
$575$	−5.78513	+	3.33072i	−0.241256	+	0.138901i
$576$		0			0
$577$		−	1.53509i		−	0.0639067i	−0.999489	−	0.0319534i	$-0.989827\pi$
$577$		−	1.53509i		−	0.0639067i	0.999489	−	0.0319534i	$-0.0101728\pi$
$578$		0			0
$579$			21.3868i			0.888806i
$580$		0			0
$581$	−1.83248	+	1.12376i	−0.0760239	+	0.0466216i
$582$		0			0
$583$	0.878517			0.0363844
$584$		0			0
$585$			0.399138i			0.0165023i
$586$		0			0
$587$		−	34.4664i		−	1.42258i	−0.702899	−	0.711290i	$-0.748111\pi$
$587$		−	34.4664i		−	1.42258i	0.702899	−	0.711290i	$-0.251889\pi$
$588$		0			0
$589$		−	60.5873i		−	2.49646i
$590$		0			0
$591$			19.1434i			0.787455i
$592$		0			0
$593$			36.0781i			1.48155i	0.671752	+	0.740776i	$0.265542\pi$
$593$			36.0781i			1.48155i	−0.671752	+	0.740776i	$0.734458\pi$
$594$		0			0
$595$	32.9549	−	20.2096i	1.35102	−	0.828511i
$596$		0			0
$597$		−	13.5204i		−	0.553351i
$598$		0			0
$599$	−8.91272			−0.364164			−0.182082	−	0.983283i	$-0.558284\pi$
$599$	−8.91272			−0.364164			−0.182082	+	0.983283i	$0.558284\pi$
$600$		0			0
$601$		−	17.9433i		−	0.731924i	−0.930630	−	0.365962i	$-0.880740\pi$
$601$		−	17.9433i		−	0.731924i	0.930630	−	0.365962i	$-0.119260\pi$
$602$		0			0
$603$			8.94056i			0.364088i
$604$		0			0
$605$	27.2277			1.10696
$606$		0			0
$607$			32.7477i			1.32919i	0.747204	+	0.664595i	$0.231396\pi$
$607$			32.7477i			1.32919i	−0.747204	+	0.664595i	$0.768604\pi$
$608$		0			0
$609$	−0.742598	−	1.21092i	−0.0300916	−	0.0490691i
$610$		0			0
$611$	−0.277840			−0.0112402
$612$		0			0
$613$		−	18.0455i		−	0.728851i	−0.931233	−	0.364426i	$-0.881265\pi$
$613$		−	18.0455i		−	0.728851i	0.931233	−	0.364426i	$-0.118735\pi$
$614$		0			0
$615$	−28.4434			−1.14695
$616$		0			0
$617$		−	16.5047i		−	0.664456i	−0.943199	−	0.332228i	$-0.892200\pi$
$617$		−	16.5047i		−	0.664456i	0.943199	−	0.332228i	$-0.107800\pi$
$618$		0			0
$619$	−5.91872			−0.237893			−0.118947	−	0.992901i	$-0.537952\pi$
$619$	−5.91872			−0.237893			−0.118947	+	0.992901i	$0.537952\pi$
$620$		0			0
$621$	−2.39289	−	4.15621i	−0.0960235	−	0.166783i
$622$		0			0
$623$	−33.3460	+	20.4494i	−1.33598	+	0.819287i
$624$		0			0
$625$	−30.0222			−1.20089
$626$		0			0
$627$	3.25368			0.129939
$628$		0			0
$629$			7.09361i			0.282841i
$630$		0			0
$631$			8.43557i			0.335815i	0.985803	+	0.167907i	$0.0537009\pi$
$631$			8.43557i			0.335815i	−0.985803	+	0.167907i	$0.946299\pi$
$632$		0			0
$633$			24.6125i			0.978261i
$634$		0			0
$635$	−1.11688			−0.0443219
$636$		0			0
$637$	−0.984986	−	0.501068i	−0.0390266	−	0.0198530i
$638$		0			0
$639$	8.38636			0.331759
$640$		0			0
$641$			23.4495i			0.926201i	0.886306	+	0.463100i	$0.153263\pi$
$641$			23.4495i			0.926201i	−0.886306	+	0.463100i	$0.846737\pi$
$642$		0			0
$643$	1.57467			0.0620989			0.0310494	−	0.999518i	$-0.490115\pi$
$643$	1.57467			0.0620989			0.0310494	+	0.999518i	$0.490115\pi$
$644$		0			0
$645$	15.9687			0.628766
$646$		0			0
$647$			31.3096i			1.23091i	0.788173	+	0.615454i	$0.211027\pi$
$647$			31.3096i			1.23091i	−0.788173	+	0.615454i	$0.788973\pi$
$648$		0			0
$649$	−4.13332			−0.162247
$650$		0			0
$651$	−20.1650	+	12.3661i	−0.790327	+	0.484667i
$652$		0			0
$653$	−3.01516			−0.117992			−0.0589962	−	0.998258i	$-0.518790\pi$
$653$	−3.01516			−0.117992			−0.0589962	+	0.998258i	$0.518790\pi$
$654$		0			0
$655$			46.6791i			1.82390i
$656$		0			0
$657$			12.2061i			0.476204i
$658$		0			0
$659$		−	33.4374i		−	1.30254i	−0.758848	−	0.651268i	$-0.774237\pi$
$659$		−	33.4374i		−	1.30254i	0.758848	−	0.651268i	$-0.225763\pi$
$660$		0			0
$661$	−8.93963			−0.347711			−0.173856	−	0.984771i	$-0.555623\pi$
$661$	−8.93963			−0.347711			−0.173856	+	0.984771i	$0.555623\pi$
$662$		0			0
$663$	0.912396			0.0354345
$664$		0			0
$665$	38.6417	−	23.6970i	1.49846	−	0.918929i
$666$		0			0
$667$	2.23144	−	1.28473i	0.0864019	−	0.0497450i
$668$		0			0
$669$	1.20632			0.0466389
$670$		0			0
$671$			6.78881i			0.262079i
$672$		0			0
$673$	−32.3681			−1.24770			−0.623848	−	0.781545i	$-0.714432\pi$
$673$	−32.3681			−1.24770			−0.623848	+	0.781545i	$0.714432\pi$
$674$		0			0
$675$			1.39192i			0.0535752i
$676$		0			0
$677$	−31.4970			−1.21053			−0.605264	−	0.796025i	$-0.706932\pi$
$677$	−31.4970			−1.21053			−0.605264	+	0.796025i	$0.706932\pi$
$678$		0			0
$679$	−7.66685	+	4.70169i	−0.294227	+	0.180434i
$680$		0			0
$681$			23.1968i			0.888903i
$682$		0			0
$683$	1.09478			0.0418905			0.0209453	−	0.999781i	$-0.493332\pi$
$683$	1.09478			0.0418905			0.0209453	+	0.999781i	$0.493332\pi$
$684$		0			0
$685$			0.518782i			0.0198216i
$686$		0			0
$687$			14.5997i			0.557013i
$688$		0			0
$689$	0.288865			0.0110049
$690$		0			0
$691$		−	3.85121i		−	0.146507i	−0.997313	−	0.0732534i	$-0.976662\pi$
$691$		−	3.85121i		−	0.146507i	0.997313	−	0.0732534i	$-0.0233382\pi$
$692$		0			0
$693$	−0.664091	−	1.08291i	−0.0252267	−	0.0411362i
$694$		0			0
$695$		−	57.9741i		−	2.19908i
$696$		0			0
$697$			65.0193i			2.46278i
$698$		0			0
$699$		−	14.4763i		−	0.547543i
$700$		0			0
$701$			18.7806i			0.709335i	0.934993	+	0.354667i	$0.115406\pi$
$701$			18.7806i			0.709335i	−0.934993	+	0.354667i	$0.884594\pi$
$702$		0			0
$703$			8.31771i			0.313708i
$704$		0			0
$705$	−4.44942			−0.167575
$706$		0			0
$707$	−10.3729	−	16.9147i	−0.390114	−	0.636143i
$708$		0			0
$709$		−	15.1173i		−	0.567743i	−0.958862	−	0.283872i	$-0.908381\pi$
$709$		−	15.1173i		−	0.567743i	0.958862	−	0.283872i	$-0.0916190\pi$
$710$		0			0
$711$		−	16.5866i		−	0.622047i
$712$		0			0
$713$	−21.3940	−	37.1592i	−0.801213	−	1.39162i
$714$		0			0
$715$	−0.191640			−0.00716692
$716$		0			0
$717$		−	11.2973i		−	0.421904i
$718$		0			0
$719$		−	7.32797i		−	0.273287i	−0.990620	−	0.136644i	$-0.956368\pi$
$719$		−	7.32797i		−	0.273287i	0.990620	−	0.136644i	$-0.0436315\pi$
$720$		0			0
$721$	17.3785	−	10.6574i	0.647211	−	0.396901i
$722$		0			0
$723$			15.4230i			0.573588i
$724$		0			0
$725$	−0.747316			−0.0277546
$726$		0			0
$727$	12.8198			0.475461			0.237731	−	0.971331i	$-0.423596\pi$
$727$	12.8198			0.475461			0.237731	+	0.971331i	$0.423596\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	36.5031i		−	1.35012i
$732$		0			0
$733$	1.72956			0.0638829			0.0319414	−	0.999490i	$-0.489831\pi$
$733$	1.72956			0.0638829			0.0319414	+	0.999490i	$0.489831\pi$
$734$		0			0
$735$	−15.7739	−	8.02426i	−0.581829	−	0.295979i
$736$		0			0
$737$	−4.29267			−0.158122
$738$		0			0
$739$	18.5569			0.682628			0.341314	−	0.939949i	$-0.389128\pi$
$739$	18.5569			0.682628			0.341314	+	0.939949i	$0.389128\pi$
$740$		0			0
$741$	1.06984			0.0393016
$742$		0			0
$743$			40.9157i			1.50105i	0.660841	+	0.750526i	$0.270200\pi$
$743$			40.9157i			1.50105i	−0.660841	+	0.750526i	$0.729800\pi$
$744$		0			0
$745$		−	22.5261i		−	0.825293i
$746$		0			0
$747$	−0.812475			−0.0297269
$748$		0			0
$749$	−8.70793	−	14.1997i	−0.318181	−	0.518844i
$750$		0			0
$751$		−	25.1503i		−	0.917748i	−0.888501	−	0.458874i	$-0.848253\pi$
$751$		−	25.1503i		−	0.917748i	0.888501	−	0.458874i	$-0.151747\pi$
$752$		0			0
$753$			8.10888i			0.295504i
$754$		0			0
$755$	21.3329			0.776385
$756$		0			0
$757$			28.0539i			1.01964i	0.860282	+	0.509818i	$0.170287\pi$
$757$			28.0539i			1.01964i	−0.860282	+	0.509818i	$0.829713\pi$
$758$		0			0
$759$	1.99554	−	1.14891i	0.0724334	−	0.0417028i
$760$		0			0
$761$			30.2742i			1.09744i	0.836006	+	0.548720i	$0.184885\pi$
$761$			30.2742i			1.09744i	−0.836006	+	0.548720i	$0.815115\pi$
$762$		0			0
$763$	1.12319	+	1.83154i	0.0406622	+	0.0663062i
$764$		0			0
$765$	14.6114			0.528277
$766$		0			0
$767$	−1.35908			−0.0490734
$768$		0			0
$769$	36.6217			1.32061			0.660307	−	0.750996i	$-0.270426\pi$
$769$	36.6217			1.32061			0.660307	+	0.750996i	$0.270426\pi$
$770$		0			0
$771$	19.4938			0.702054
$772$		0			0
$773$	2.95750			0.106374			0.0531869	−	0.998585i	$-0.483062\pi$
$773$	2.95750			0.106374			0.0531869	+	0.998585i	$0.483062\pi$
$774$		0			0
$775$			12.4447i			0.447027i
$776$		0			0
$777$	2.76834	−	1.69768i	0.0993136	−	0.0609039i
$778$		0			0
$779$			76.2392i			2.73155i
$780$		0			0
$781$			4.02658i			0.144082i
$782$		0			0
$783$		−	0.536894i		−	0.0191870i
$784$		0			0
$785$	20.8599			0.744523
$786$		0			0
$787$	27.2278			0.970568			0.485284	−	0.874357i	$-0.338716\pi$
$787$	27.2278			0.970568			0.485284	+	0.874357i	$0.338716\pi$
$788$		0			0
$789$	−8.54961			−0.304374
$790$		0			0
$791$	−25.6950	−	41.8997i	−0.913608	−	1.48978i
$792$		0			0
$793$			2.23223i			0.0792687i
$794$		0			0
$795$	4.62597			0.164066
$796$		0			0
$797$	37.6624			1.33407			0.667036	−	0.745026i	$-0.267563\pi$
$797$	37.6624			1.33407			0.667036	+	0.745026i	$0.267563\pi$
$798$		0			0
$799$			10.1710i			0.359824i
$800$		0			0
$801$	−14.7848			−0.522395
$802$		0			0
$803$	−5.86055			−0.206814
$804$		0			0
$805$	15.3320	−	28.1786i	0.540380	−	0.993164i
$806$		0			0
$807$	19.3468			0.681041
$808$		0			0
$809$	13.0314			0.458160			0.229080	−	0.973408i	$-0.426428\pi$
$809$	13.0314			0.458160			0.229080	+	0.973408i	$0.426428\pi$
$810$		0			0
$811$		−	40.9562i		−	1.43817i	−0.694923	−	0.719084i	$-0.744562\pi$
$811$		−	40.9562i		−	1.43817i	0.694923	−	0.719084i	$-0.255438\pi$
$812$		0			0
$813$	−27.5658			−0.966775
$814$		0			0
$815$	−46.8660			−1.64164
$816$		0			0
$817$		−	42.8022i		−	1.49746i
$818$		0			0
$819$	−0.218359	−	0.356070i	−0.00763010	−	0.0124421i
$820$		0			0
$821$	27.6706			0.965712			0.482856	−	0.875700i	$-0.339599\pi$
$821$	27.6706			0.965712			0.482856	+	0.875700i	$0.339599\pi$
$822$		0			0
$823$	16.1783			0.563941			0.281971	−	0.959423i	$-0.409012\pi$
$823$	16.1783			0.563941			0.281971	+	0.959423i	$0.409012\pi$
$824$		0			0
$825$	−0.668310			−0.0232676
$826$		0			0
$827$			37.2726i			1.29610i	0.761600	+	0.648048i	$0.224414\pi$
$827$			37.2726i			1.29610i	−0.761600	+	0.648048i	$0.775586\pi$
$828$		0			0
$829$			22.5278i			0.782422i	0.920301	+	0.391211i	$0.127944\pi$
$829$			22.5278i			0.782422i	−0.920301	+	0.391211i	$0.872056\pi$
$830$		0			0
$831$		−	7.85710i		−	0.272560i
$832$		0			0
$833$	−18.3428	+	36.0578i	−0.635540	+	1.24933i
$834$		0			0
$835$			5.44602i			0.188467i
$836$		0			0
$837$	−8.94066			−0.309034
$838$		0			0
$839$	−21.6852			−0.748657			−0.374328	−	0.927296i	$-0.622127\pi$
$839$	−21.6852			−0.748657			−0.374328	+	0.927296i	$0.622127\pi$
$840$		0			0
$841$	−28.7117			−0.990060
$842$		0			0
$843$	−17.5640			−0.604937
$844$		0			0
$845$	32.8039			1.12849
$846$		0			0
$847$	−24.2897	+	14.8956i	−0.834605	+	0.511820i
$848$		0			0
$849$		−	2.14307i		−	0.0735501i
$850$		0			0
$851$	2.93707	+	5.10139i	0.100681	+	0.174873i
$852$		0			0
$853$		−	38.1579i		−	1.30650i	−0.757141	−	0.653252i	$-0.773404\pi$
$853$		−	38.1579i		−	1.30650i	0.757141	−	0.653252i	$-0.226596\pi$
$854$		0			0
$855$	17.1328			0.585929
$856$		0			0
$857$			4.57559i			0.156299i	0.996942	+	0.0781496i	$0.0249012\pi$
$857$			4.57559i			0.156299i	−0.996942	+	0.0781496i	$0.975099\pi$
$858$		0			0
$859$		−	5.43478i		−	0.185432i	−0.995693	−	0.0927161i	$-0.970445\pi$
$859$		−	5.43478i		−	0.185432i	0.995693	−	0.0927161i	$-0.0295549\pi$
$860$		0			0
$861$	25.3743	−	15.5608i	0.864754	−	0.530309i
$862$		0			0
$863$	−29.8557			−1.01630			−0.508149	−	0.861269i	$-0.669670\pi$
$863$	−29.8557			−1.01630			−0.508149	+	0.861269i	$0.669670\pi$
$864$		0			0
$865$			37.7009i			1.28187i
$866$		0			0
$867$		−	16.4005i		−	0.556989i
$868$		0			0
$869$	7.96381			0.270154
$870$		0			0
$871$	−1.41147			−0.0478259
$872$		0			0
$873$	−3.39930			−0.115049
$874$		0			0
$875$	20.5740	−	12.6170i	0.695530	−	0.426533i
$876$		0			0
$877$	18.3038			0.618075			0.309037	−	0.951050i	$-0.399993\pi$
$877$	18.3038			0.618075			0.309037	+	0.951050i	$0.399993\pi$
$878$		0			0
$879$		−	25.1287i		−	0.847571i
$880$		0			0
$881$	−32.9337			−1.10957			−0.554783	−	0.831995i	$-0.687199\pi$
$881$	−32.9337			−1.10957			−0.554783	+	0.831995i	$0.687199\pi$
$882$		0			0
$883$	2.72453			0.0916876			0.0458438	−	0.998949i	$-0.485402\pi$
$883$	2.72453			0.0916876			0.0458438	+	0.998949i	$0.485402\pi$
$884$		0			0
$885$	−21.7647			−0.731612
$886$		0			0
$887$		−	30.9050i		−	1.03769i	−0.854868	−	0.518845i	$-0.826362\pi$
$887$		−	30.9050i		−	1.03769i	0.854868	−	0.518845i	$-0.173638\pi$
$888$		0			0
$889$	0.996363	−	0.611018i	0.0334169	−	0.0204929i
$890$		0			0
$891$		−	0.480134i		−	0.0160851i
$892$		0			0
$893$			11.9261i			0.399093i
$894$		0			0
$895$	56.8652			1.90079
$896$		0			0
$897$	0.656152	−	0.377772i	0.0219083	−	0.0126135i
$898$		0			0
$899$		−	4.80019i		−	0.160095i
$900$		0			0
$901$		−	10.5746i		−	0.352291i
$902$		0			0
$903$	−14.2456	+	8.73611i	−0.474065	+	0.290719i
$904$		0			0
$905$	−24.4473			−0.812656
$906$		0			0
$907$			14.6904i			0.487785i	0.969802	+	0.243893i	$0.0784245\pi$
$907$			14.6904i			0.487785i	−0.969802	+	0.243893i	$0.921576\pi$
$908$		0			0
$909$		−	7.49957i		−	0.248745i
$910$		0			0
$911$		−	50.4298i		−	1.67081i	−0.549633	−	0.835406i	$-0.685232\pi$
$911$		−	50.4298i		−	1.67081i	0.549633	−	0.835406i	$-0.314768\pi$
$912$		0			0
$913$		−	0.390097i		−	0.0129103i
$914$		0			0
$915$			35.7476i			1.18178i
$916$		0			0
$917$	−25.5371	−	41.6423i	−0.843308	−	1.37515i
$918$		0			0
$919$			2.03850i			0.0672440i	0.999435	+	0.0336220i	$0.0107042\pi$
$919$			2.03850i			0.0672440i	−0.999435	+	0.0336220i	$0.989296\pi$
$920$		0			0
$921$	−2.58458			−0.0851649
$922$		0			0
$923$			1.32398i			0.0435792i
$924$		0			0
$925$		−	1.70847i		−	0.0561740i
$926$		0			0
$927$	7.70522			0.253073
$928$		0			0
$929$			11.7046i			0.384014i	0.981394	+	0.192007i	$0.0614997\pi$
$929$			11.7046i			0.384014i	−0.981394	+	0.192007i	$0.938500\pi$
$930$		0			0
$931$	−21.5081	+	42.2800i	−0.704899	+	1.38567i
$932$		0			0
$933$	−28.9877			−0.949014
$934$		0			0
$935$			7.01543i			0.229429i
$936$		0			0
$937$	−30.6036			−0.999775			−0.499887	−	0.866090i	$-0.666625\pi$
$937$	−30.6036			−0.999775			−0.499887	+	0.866090i	$0.666625\pi$
$938$		0			0
$939$		−	8.17704i		−	0.266848i
$940$		0			0
$941$	−15.0811			−0.491628			−0.245814	−	0.969317i	$-0.579055\pi$
$941$	−15.0811			−0.491628			−0.245814	+	0.969317i	$0.579055\pi$
$942$		0			0
$943$	26.9209	+	46.7588i	0.876665	+	1.52268i
$944$		0			0
$945$	−3.49688	−	5.70222i	−0.113753	−	0.185493i
$946$		0			0
$947$	28.5189			0.926741			0.463370	−	0.886165i	$-0.346640\pi$
$947$	28.5189			0.926741			0.463370	+	0.886165i	$0.346640\pi$
$948$		0			0
$949$	−1.92700			−0.0625532
$950$		0			0
$951$		−	16.4518i		−	0.533487i
$952$		0			0
$953$			39.6771i			1.28527i	0.766173	+	0.642634i	$0.222158\pi$
$953$			39.6771i			1.28527i	−0.766173	+	0.642634i	$0.777842\pi$
$954$		0			0
$955$			43.2713i			1.40023i
$956$		0			0
$957$	0.257781			0.00833288
$958$		0			0
$959$	−0.283814	−	0.462804i	−0.00916484	−	0.0149447i
$960$		0			0
$961$	−48.9353			−1.57856
$962$		0			0
$963$		−	6.29579i		−	0.202879i
$964$		0			0
$965$	−54.0707			−1.74060
$966$		0			0
$967$	−51.0059			−1.64024			−0.820120	−	0.572192i	$-0.806093\pi$
$967$	−51.0059			−1.64024			−0.820120	+	0.572192i	$0.806093\pi$
$968$		0			0
$969$		−	39.1641i		−	1.25813i
$970$		0			0
$971$	−9.39979			−0.301654			−0.150827	−	0.988560i	$-0.548194\pi$
$971$	−9.39979			−0.301654			−0.150827	+	0.988560i	$0.548194\pi$
$972$		0			0
$973$	31.7163	+	51.7185i	1.01678	+	1.65802i
$974$		0			0
$975$	−0.219747			−0.00703753
$976$		0			0
$977$			22.9715i			0.734922i	0.930039	+	0.367461i	$0.119773\pi$
$977$			22.9715i			0.734922i	−0.930039	+	0.367461i	$0.880227\pi$
$978$		0			0
$979$		−	7.09868i		−	0.226875i
$980$		0			0
$981$			0.812061i			0.0259271i
$982$		0			0
$983$	0.961599			0.0306703			0.0153351	−	0.999882i	$-0.495118\pi$
$983$	0.961599			0.0306703			0.0153351	+	0.999882i	$0.495118\pi$
$984$		0			0
$985$	−48.3989			−1.54212
$986$		0			0
$987$	3.96932	−	2.43418i	0.126345	−	0.0774807i
$988$		0			0
$989$	−15.1139	−	26.2513i	−0.480594	−	0.834742i
$990$		0			0
$991$	−52.5552			−1.66947			−0.834735	−	0.550652i	$-0.814379\pi$
$991$	−52.5552			−1.66947			−0.834735	+	0.550652i	$0.814379\pi$
$992$		0			0
$993$		−	20.4830i		−	0.650008i
$994$		0			0
$995$	34.1825			1.08366
$996$		0			0
$997$			27.4131i			0.868180i	0.900869	+	0.434090i	$0.142930\pi$
$997$			27.4131i			0.868180i	−0.900869	+	0.434090i	$0.857070\pi$
$998$		0			0
$999$	1.22741			0.0388337

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.k.a.1609.10	yes	32
3.2	odd	2		5796.2.k.d.5473.6		32
7.6	odd	2	inner	1932.2.k.a.1609.11	yes	32
21.20	even	2		5796.2.k.d.5473.28		32
23.22	odd	2	inner	1932.2.k.a.1609.9	✓	32
69.68	even	2		5796.2.k.d.5473.27		32
161.160	even	2	inner	1932.2.k.a.1609.12	yes	32
483.482	odd	2		5796.2.k.d.5473.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.k.a.1609.9	✓	32	23.22	odd	2	inner
1932.2.k.a.1609.10	yes	32	1.1	even	1	trivial
1932.2.k.a.1609.11	yes	32	7.6	odd	2	inner
1932.2.k.a.1609.12	yes	32	161.160	even	2	inner
5796.2.k.d.5473.5		32	483.482	odd	2
5796.2.k.d.5473.6		32	3.2	odd	2
5796.2.k.d.5473.27		32	69.68	even	2
5796.2.k.d.5473.28		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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.52823 q^{5} +(-2.25542 + 1.38314i) q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.52823 q^{5} +(-2.25542 + 1.38314i) q^{7} -1.00000 q^{9} -0.480134i q^{11} -0.157873i q^{13} -2.52823i q^{15} -5.77931 q^{17} -6.77661 q^{19} +(1.38314 + 2.25542i) q^{21} +(-4.15621 + 2.39289i) q^{23} +1.39192 q^{25} +1.00000i q^{27} -0.536894 q^{29} +8.94066i q^{31} -0.480134 q^{33} +(-5.70222 + 3.49688i) q^{35} -1.22741i q^{37} -0.157873 q^{39} -11.2504i q^{41} +6.31616i q^{43} -2.52823 q^{45} -1.75990i q^{47} +(3.17387 - 6.23911i) q^{49} +5.77931i q^{51} +1.82973i q^{53} -1.21389i q^{55} +6.77661i q^{57} -8.60869i q^{59} -14.1394 q^{61} +(2.25542 - 1.38314i) q^{63} -0.399138i q^{65} -8.94056i q^{67} +(2.39289 + 4.15621i) q^{69} -8.38636 q^{71} -12.2061i q^{73} -1.39192i q^{75} +(0.664091 + 1.08291i) q^{77} +16.5866i q^{79} +1.00000 q^{81} +0.812475 q^{83} -14.6114 q^{85} +0.536894i q^{87} +14.7848 q^{89} +(0.218359 + 0.356070i) q^{91} +8.94066 q^{93} -17.1328 q^{95} +3.39930 q^{97} +0.480134i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 32 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more