LMFDB - Embedded newform 4028.2.c.a.3497.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(3497,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4028.3497");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.c (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:	$32.1637419342$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3497.15
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.68

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31112i q^{3} +0.310021i q^{5} +2.57819 q^{7} -2.34128 q^{9} +O(q^{10})$	$q-2.31112i q^{3} +0.310021i q^{5} +2.57819 q^{7} -2.34128 q^{9} -2.35966 q^{11} -1.30741 q^{13} +0.716495 q^{15} -2.17348 q^{17} -1.00000i q^{19} -5.95850i q^{21} -5.23636i q^{23} +4.90389 q^{25} -1.52238i q^{27} +2.38324 q^{29} +1.15503i q^{31} +5.45346i q^{33} +0.799292i q^{35} -2.21604 q^{37} +3.02157i q^{39} -4.26678i q^{41} +2.21276 q^{43} -0.725846i q^{45} -10.4435 q^{47} -0.352949 q^{49} +5.02318i q^{51} +(0.994105 + 7.21192i) q^{53} -0.731543i q^{55} -2.31112 q^{57} -12.1492 q^{59} -3.76328i q^{61} -6.03626 q^{63} -0.405323i q^{65} -6.84758i q^{67} -12.1019 q^{69} -9.96963i q^{71} -7.56564i q^{73} -11.3335i q^{75} -6.08364 q^{77} -1.26974i q^{79} -10.5422 q^{81} -5.17336i q^{83} -0.673825i q^{85} -5.50796i q^{87} +12.3751 q^{89} -3.37074 q^{91} +2.66942 q^{93} +0.310021 q^{95} +8.73657 q^{97} +5.52462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) $ 82 * q - 8 * q^7 - 82 * q^9	$ 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) $ 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times$.

$n$	$2015$	$2281$	$2757$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.31112i		−	1.33433i	−0.744912	−	0.667163i	$-0.767508\pi$
$3$		−	2.31112i		−	1.33433i	0.744912	−	0.667163i	$-0.232492\pi$
$4$		0			0
$5$			0.310021i			0.138645i	0.997594	+	0.0693227i	$0.0220838\pi$
$5$			0.310021i			0.138645i	−0.997594	+	0.0693227i	$0.977916\pi$
$6$		0			0
$7$	2.57819			0.974463			0.487232	−	0.873273i	$-0.338007\pi$
$7$	2.57819			0.974463			0.487232	+	0.873273i	$0.338007\pi$
$8$		0			0
$9$	−2.34128			−0.780427
$10$		0			0
$11$	−2.35966			−0.711464			−0.355732	−	0.934588i	$-0.615768\pi$
$11$	−2.35966			−0.711464			−0.355732	+	0.934588i	$0.615768\pi$
$12$		0			0
$13$	−1.30741			−0.362609			−0.181305	−	0.983427i	$-0.558032\pi$
$13$	−1.30741			−0.362609			−0.181305	+	0.983427i	$0.558032\pi$
$14$		0			0
$15$	0.716495			0.184998
$16$		0			0
$17$	−2.17348			−0.527147			−0.263574	−	0.964639i	$-0.584901\pi$
$17$	−2.17348			−0.527147			−0.263574	+	0.964639i	$0.584901\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$		−	5.95850i		−	1.30025i
$22$		0			0
$23$		−	5.23636i		−	1.09186i	−0.837832	−	0.545928i	$-0.816177\pi$
$23$		−	5.23636i		−	1.09186i	0.837832	−	0.545928i	$-0.183823\pi$
$24$		0			0
$25$	4.90389			0.980777
$26$		0			0
$27$		−	1.52238i		−	0.292982i
$28$		0			0
$29$	2.38324			0.442557			0.221279	−	0.975211i	$-0.428977\pi$
$29$	2.38324			0.442557			0.221279	+	0.975211i	$0.428977\pi$
$30$		0			0
$31$			1.15503i			0.207450i	0.994606	+	0.103725i	$0.0330762\pi$
$31$			1.15503i			0.207450i	−0.994606	+	0.103725i	$0.966924\pi$
$32$		0			0
$33$			5.45346i			0.949325i
$34$		0			0
$35$			0.799292i			0.135105i
$36$		0			0
$37$	−2.21604			−0.364315			−0.182157	−	0.983269i	$-0.558308\pi$
$37$	−2.21604			−0.364315			−0.182157	+	0.983269i	$0.558308\pi$
$38$		0			0
$39$			3.02157i			0.483839i
$40$		0			0
$41$		−	4.26678i		−	0.666359i	−0.942863	−	0.333179i	$-0.891879\pi$
$41$		−	4.26678i		−	0.666359i	0.942863	−	0.333179i	$-0.108121\pi$
$42$		0			0
$43$	2.21276			0.337443			0.168722	−	0.985664i	$-0.446036\pi$
$43$	2.21276			0.337443			0.168722	+	0.985664i	$0.446036\pi$
$44$		0			0
$45$		−	0.725846i		−	0.108203i
$46$		0			0
$47$	−10.4435			−1.52335			−0.761673	−	0.647962i	$-0.775621\pi$
$47$	−10.4435			−1.52335			−0.761673	+	0.647962i	$0.775621\pi$
$48$		0			0
$49$	−0.352949			−0.0504212
$50$		0			0
$51$			5.02318i			0.703386i
$52$		0			0
$53$	0.994105	+	7.21192i	0.136551	+	0.990633i
$53$	0.994105	+	7.21192i	0.136551	+	0.990633i
$54$		0			0
$55$		−	0.731543i		−	0.0986413i
$56$		0			0
$57$	−2.31112			−0.306115
$58$		0			0
$59$	−12.1492			−1.58169			−0.790843	−	0.612019i	$-0.790357\pi$
$59$	−12.1492			−1.58169			−0.790843	+	0.612019i	$0.790357\pi$
$60$		0			0
$61$		−	3.76328i		−	0.481838i	−0.970545	−	0.240919i	$-0.922551\pi$
$61$		−	3.76328i		−	0.481838i	0.970545	−	0.240919i	$-0.0774489\pi$
$62$		0			0
$63$	−6.03626			−0.760497
$64$		0			0
$65$		−	0.405323i		−	0.0502741i
$66$		0			0
$67$		−	6.84758i		−	0.836565i	−0.908317	−	0.418282i	$-0.862632\pi$
$67$		−	6.84758i		−	0.836565i	0.908317	−	0.418282i	$-0.137368\pi$
$68$		0			0
$69$	−12.1019			−1.45689
$70$		0			0
$71$		−	9.96963i		−	1.18318i	−0.806240	−	0.591589i	$-0.798501\pi$
$71$		−	9.96963i		−	1.18318i	0.806240	−	0.591589i	$-0.201499\pi$
$72$		0			0
$73$		−	7.56564i		−	0.885491i	−0.896647	−	0.442745i	$-0.854005\pi$
$73$		−	7.56564i		−	0.885491i	0.896647	−	0.442745i	$-0.145995\pi$
$74$		0			0
$75$		−	11.3335i		−	1.30868i
$76$		0			0
$77$	−6.08364			−0.693296
$78$		0			0
$79$		−	1.26974i		−	0.142856i	−0.997446	−	0.0714282i	$-0.977244\pi$
$79$		−	1.26974i		−	0.142856i	0.997446	−	0.0714282i	$-0.0227557\pi$
$80$		0			0
$81$	−10.5422			−1.17136
$82$		0			0
$83$		−	5.17336i		−	0.567850i	−0.958846	−	0.283925i	$-0.908363\pi$
$83$		−	5.17336i		−	0.567850i	0.958846	−	0.283925i	$-0.0916367\pi$
$84$		0			0
$85$		−	0.673825i		−	0.0730866i
$86$		0			0
$87$		−	5.50796i		−	0.590515i
$88$		0			0
$89$	12.3751			1.31176			0.655878	−	0.754867i	$-0.272299\pi$
$89$	12.3751			1.31176			0.655878	+	0.754867i	$0.272299\pi$
$90$		0			0
$91$	−3.37074			−0.353349
$92$		0			0
$93$	2.66942			0.276806
$94$		0			0
$95$	0.310021			0.0318075
$96$		0			0
$97$	8.73657			0.887064			0.443532	−	0.896259i	$-0.353725\pi$
$97$	8.73657			0.887064			0.443532	+	0.896259i	$0.353725\pi$
$98$		0			0
$99$	5.52462			0.555246
$100$		0			0
$101$		−	2.24177i		−	0.223064i	−0.993761	−	0.111532i	$-0.964424\pi$
$101$		−	2.24177i		−	0.223064i	0.993761	−	0.111532i	$-0.0355757\pi$
$102$		0			0
$103$		−	16.1258i		−	1.58892i	−0.607316	−	0.794461i	$-0.707754\pi$
$103$		−	16.1258i		−	1.58892i	0.607316	−	0.794461i	$-0.292246\pi$
$104$		0			0
$105$	1.84726			0.180274
$106$		0			0
$107$	−5.39956			−0.521995			−0.260998	−	0.965339i	$-0.584051\pi$
$107$	−5.39956			−0.521995			−0.260998	+	0.965339i	$0.584051\pi$
$108$		0			0
$109$			10.2236i			0.979247i	0.871934	+	0.489623i	$0.162866\pi$
$109$			10.2236i			0.979247i	−0.871934	+	0.489623i	$0.837134\pi$
$110$		0			0
$111$			5.12154i			0.486115i
$112$		0			0
$113$	−11.2920			−1.06226			−0.531130	−	0.847290i	$-0.678232\pi$
$113$	−11.2920			−1.06226			−0.531130	+	0.847290i	$0.678232\pi$
$114$		0			0
$115$	1.62338			0.151381
$116$		0			0
$117$	3.06100			0.282990
$118$		0			0
$119$	−5.60365			−0.513686
$120$		0			0
$121$	−5.43201			−0.493819
$122$		0			0
$123$	−9.86104			−0.889140
$124$		0			0
$125$			3.07041i			0.274626i
$126$		0			0
$127$			0.902103i			0.0800487i	0.999199	+	0.0400243i	$0.0127435\pi$
$127$			0.902103i			0.0800487i	−0.999199	+	0.0400243i	$0.987256\pi$
$128$		0			0
$129$		−	5.11396i		−	0.450259i
$130$		0			0
$131$	−3.87187			−0.338287			−0.169144	−	0.985591i	$-0.554100\pi$
$131$	−3.87187			−0.338287			−0.169144	+	0.985591i	$0.554100\pi$
$132$		0			0
$133$		−	2.57819i		−	0.223557i
$134$		0			0
$135$	0.471969			0.0406206
$136$		0			0
$137$		−	13.3363i		−	1.13940i	−0.821853	−	0.569699i	$-0.807060\pi$
$137$		−	13.3363i		−	1.13940i	0.821853	−	0.569699i	$-0.192940\pi$
$138$		0			0
$139$			6.30093i			0.534438i	0.963636	+	0.267219i	$0.0861046\pi$
$139$			6.30093i			0.534438i	−0.963636	+	0.267219i	$0.913895\pi$
$140$		0			0
$141$			24.1363i			2.03264i
$142$		0			0
$143$	3.08503			0.257983
$144$		0			0
$145$			0.738855i			0.0613585i
$146$		0			0
$147$			0.815707i			0.0672784i
$148$		0			0
$149$	8.45598			0.692741			0.346371	−	0.938098i	$-0.387414\pi$
$149$	8.45598			0.692741			0.346371	+	0.938098i	$0.387414\pi$
$150$		0			0
$151$		−	13.9210i		−	1.13287i	−0.824105	−	0.566437i	$-0.808321\pi$
$151$		−	13.9210i		−	1.13287i	0.824105	−	0.566437i	$-0.191679\pi$
$152$		0			0
$153$	5.08874			0.411400
$154$		0			0
$155$	−0.358084			−0.0287620
$156$		0			0
$157$			8.23917i			0.657557i	0.944407	+	0.328779i	$0.106637\pi$
$157$			8.23917i			0.657557i	−0.944407	+	0.328779i	$0.893363\pi$
$158$		0			0
$159$	16.6676	−	2.29750i	1.32183	−	0.182203i
$160$		0			0
$161$		−	13.5003i		−	1.06397i
$162$		0			0
$163$	−2.32678			−0.182247			−0.0911237	−	0.995840i	$-0.529046\pi$
$163$	−2.32678			−0.182247			−0.0911237	+	0.995840i	$0.529046\pi$
$164$		0			0
$165$	−1.69068			−0.131620
$166$		0			0
$167$			14.0509i			1.08729i	0.839315	+	0.543646i	$0.182957\pi$
$167$			14.0509i			1.08729i	−0.839315	+	0.543646i	$0.817043\pi$
$168$		0			0
$169$	−11.2907			−0.868515
$170$		0			0
$171$			2.34128i			0.179042i
$172$		0			0
$173$			6.67837i			0.507747i	0.967237	+	0.253874i	$0.0817047\pi$
$173$			6.67837i			0.507747i	−0.967237	+	0.253874i	$0.918295\pi$
$174$		0			0
$175$	12.6431			0.955732
$176$		0			0
$177$			28.0782i			2.11049i
$178$		0			0
$179$			6.76580i			0.505700i	0.967505	+	0.252850i	$0.0813679\pi$
$179$			6.76580i			0.505700i	−0.967505	+	0.252850i	$0.918632\pi$
$180$		0			0
$181$		−	9.38702i		−	0.697731i	−0.937173	−	0.348866i	$-0.886567\pi$
$181$		−	9.38702i		−	0.697731i	0.937173	−	0.348866i	$-0.113433\pi$
$182$		0			0
$183$	−8.69739			−0.642930
$184$		0			0
$185$		−	0.687018i		−	0.0505106i
$186$		0			0
$187$	5.12868			0.375046
$188$		0			0
$189$		−	3.92498i		−	0.285500i
$190$		0			0
$191$			1.24049i			0.0897585i	0.998992	+	0.0448792i	$0.0142903\pi$
$191$			1.24049i			0.0897585i	−0.998992	+	0.0448792i	$0.985710\pi$
$192$		0			0
$193$			8.27409i			0.595582i	0.954631	+	0.297791i	$0.0962498\pi$
$193$			8.27409i			0.595582i	−0.954631	+	0.297791i	$0.903750\pi$
$194$		0			0
$195$	−0.936750			−0.0670821
$196$		0			0
$197$	−22.8609			−1.62877			−0.814385	−	0.580325i	$-0.802925\pi$
$197$	−22.8609			−1.62877			−0.814385	+	0.580325i	$0.802925\pi$
$198$		0			0
$199$	5.65631			0.400965			0.200483	−	0.979697i	$-0.435749\pi$
$199$	5.65631			0.400965			0.200483	+	0.979697i	$0.435749\pi$
$200$		0			0
$201$	−15.8256			−1.11625
$202$		0			0
$203$	6.14445			0.431256
$204$		0			0
$205$	1.32279			0.0923876
$206$		0			0
$207$			12.2598i			0.852114i
$208$		0			0
$209$			2.35966i			0.163221i
$210$		0			0
$211$	−8.26224			−0.568796			−0.284398	−	0.958706i	$-0.591794\pi$
$211$	−8.26224			−0.568796			−0.284398	+	0.958706i	$0.591794\pi$
$212$		0			0
$213$	−23.0410			−1.57874
$214$		0			0
$215$			0.686002i			0.0467850i
$216$		0			0
$217$			2.97789i			0.202153i
$218$		0			0
$219$	−17.4851			−1.18153
$220$		0			0
$221$	2.84163			0.191148
$222$		0			0
$223$	20.2135			1.35360			0.676799	−	0.736168i	$-0.263366\pi$
$223$	20.2135			1.35360			0.676799	+	0.736168i	$0.263366\pi$
$224$		0			0
$225$	−11.4814			−0.765425
$226$		0			0
$227$	−12.0634			−0.800675			−0.400337	−	0.916368i	$-0.631107\pi$
$227$	−12.0634			−0.800675			−0.400337	+	0.916368i	$0.631107\pi$
$228$		0			0
$229$	14.8022			0.978160			0.489080	−	0.872239i	$-0.337333\pi$
$229$	14.8022			0.978160			0.489080	+	0.872239i	$0.337333\pi$
$230$		0			0
$231$			14.0600i			0.925083i
$232$		0			0
$233$			7.85027i			0.514288i	0.966373	+	0.257144i	$0.0827815\pi$
$233$			7.85027i			0.514288i	−0.966373	+	0.257144i	$0.917219\pi$
$234$		0			0
$235$		−	3.23771i		−	0.211205i
$236$		0			0
$237$	−2.93451			−0.190617
$238$		0			0
$239$		−	7.56632i		−	0.489425i	−0.969596	−	0.244712i	$-0.921306\pi$
$239$		−	7.56632i		−	0.489425i	0.969596	−	0.244712i	$-0.0786935\pi$
$240$		0			0
$241$	−1.77793			−0.114526			−0.0572631	−	0.998359i	$-0.518237\pi$
$241$	−1.77793			−0.114526			−0.0572631	+	0.998359i	$0.518237\pi$
$242$		0			0
$243$			19.7973i			1.27000i
$244$		0			0
$245$		−	0.109421i		−	0.00699068i
$246$		0			0
$247$			1.30741i			0.0831883i
$248$		0			0
$249$	−11.9563			−0.757698
$250$		0			0
$251$		−	11.0518i		−	0.697584i	−0.937200	−	0.348792i	$-0.886592\pi$
$251$		−	11.0518i		−	0.697584i	0.937200	−	0.348792i	$-0.113408\pi$
$252$		0			0
$253$			12.3560i			0.776816i
$254$		0			0
$255$	−1.55729			−0.0975213
$256$		0			0
$257$		−	15.6598i		−	0.976834i	−0.872610	−	0.488417i	$-0.837575\pi$
$257$		−	15.6598i		−	0.976834i	0.872610	−	0.488417i	$-0.162425\pi$
$258$		0			0
$259$	−5.71337			−0.355011
$260$		0			0
$261$	−5.57984			−0.345383
$262$		0			0
$263$		−	26.7948i		−	1.65224i	−0.563494	−	0.826120i	$-0.690543\pi$
$263$		−	26.7948i		−	1.65224i	0.563494	−	0.826120i	$-0.309457\pi$
$264$		0			0
$265$	−2.23584	+	0.308193i	−0.137347	+	0.0189322i
$266$		0			0
$267$		−	28.6003i		−	1.75031i
$268$		0			0
$269$	24.0768			1.46799			0.733993	−	0.679157i	$-0.237655\pi$
$269$	24.0768			1.46799			0.733993	+	0.679157i	$0.237655\pi$
$270$		0			0
$271$	4.73104			0.287390			0.143695	−	0.989622i	$-0.454102\pi$
$271$	4.73104			0.287390			0.143695	+	0.989622i	$0.454102\pi$
$272$		0			0
$273$			7.79018i			0.471483i
$274$		0			0
$275$	−11.5715			−0.697788
$276$		0			0
$277$			1.10955i			0.0666662i	0.999444	+	0.0333331i	$0.0106122\pi$
$277$			1.10955i			0.0666662i	−0.999444	+	0.0333331i	$0.989388\pi$
$278$		0			0
$279$		−	2.70426i		−	0.161900i
$280$		0			0
$281$	2.81329			0.167827			0.0839135	−	0.996473i	$-0.473258\pi$
$281$	2.81329			0.167827			0.0839135	+	0.996473i	$0.473258\pi$
$282$		0			0
$283$			9.81979i			0.583726i	0.956460	+	0.291863i	$0.0942752\pi$
$283$			9.81979i			0.583726i	−0.956460	+	0.291863i	$0.905725\pi$
$284$		0			0
$285$		−	0.716495i		−	0.0424415i
$286$		0			0
$287$		−	11.0006i		−	0.649342i
$288$		0			0
$289$	−12.2760			−0.722116
$290$		0			0
$291$		−	20.1913i		−	1.18363i
$292$		0			0
$293$	−29.5578			−1.72678			−0.863392	−	0.504534i	$-0.831664\pi$
$293$	−29.5578			−1.72678			−0.863392	+	0.504534i	$0.831664\pi$
$294$		0			0
$295$		−	3.76649i		−	0.219294i
$296$		0			0
$297$			3.59230i			0.208446i
$298$		0			0
$299$			6.84605i			0.395917i
$300$		0			0
$301$	5.70492			0.328826
$302$		0			0
$303$	−5.18099			−0.297640
$304$		0			0
$305$	1.16669			0.0668047
$306$		0			0
$307$	10.6171			0.605952			0.302976	−	0.952998i	$-0.402020\pi$
$307$	10.6171			0.605952			0.302976	+	0.952998i	$0.402020\pi$
$308$		0			0
$309$	−37.2687			−2.12014
$310$		0			0
$311$	1.99691			0.113234			0.0566171	−	0.998396i	$-0.481969\pi$
$311$	1.99691			0.113234			0.0566171	+	0.998396i	$0.481969\pi$
$312$		0			0
$313$			27.9450i			1.57954i	0.613401	+	0.789771i	$0.289801\pi$
$313$			27.9450i			1.57954i	−0.613401	+	0.789771i	$0.710199\pi$
$314$		0			0
$315$		−	1.87137i		−	0.105440i
$316$		0			0
$317$	0.413178			0.0232064			0.0116032	−	0.999933i	$-0.496307\pi$
$317$	0.413178			0.0232064			0.0116032	+	0.999933i	$0.496307\pi$
$318$		0			0
$319$	−5.62364			−0.314863
$320$		0			0
$321$			12.4790i			0.696512i
$322$		0			0
$323$			2.17348i			0.120936i
$324$		0			0
$325$	−6.41137			−0.355639
$326$		0			0
$327$	23.6281			1.30664
$328$		0			0
$329$	−26.9254			−1.48444
$330$		0			0
$331$	22.8866			1.25796			0.628981	−	0.777420i	$-0.283472\pi$
$331$	22.8866			1.25796			0.628981	+	0.777420i	$0.283472\pi$
$332$		0			0
$333$	5.18837			0.284321
$334$		0			0
$335$	2.12289			0.115986
$336$		0			0
$337$		−	3.46090i		−	0.188527i	−0.995547	−	0.0942637i	$-0.969950\pi$
$337$		−	3.46090i		−	0.188527i	0.995547	−	0.0942637i	$-0.0300497\pi$
$338$		0			0
$339$			26.0971i			1.41740i
$340$		0			0
$341$		−	2.72548i		−	0.147593i
$342$		0			0
$343$	−18.9573			−1.02360
$344$		0			0
$345$		−	3.75183i		−	0.201992i
$346$		0			0
$347$	−16.7819			−0.900899			−0.450450	−	0.892802i	$-0.648736\pi$
$347$	−16.7819			−0.900899			−0.450450	+	0.892802i	$0.648736\pi$
$348$		0			0
$349$			1.45232i			0.0777409i	0.999244	+	0.0388705i	$0.0123760\pi$
$349$			1.45232i			0.0777409i	−0.999244	+	0.0388705i	$0.987624\pi$
$350$		0			0
$351$			1.99037i			0.106238i
$352$		0			0
$353$			1.70539i			0.0907688i	0.998970	+	0.0453844i	$0.0144513\pi$
$353$			1.70539i			0.0907688i	−0.998970	+	0.0453844i	$0.985549\pi$
$354$		0			0
$355$	3.09079			0.164042
$356$		0			0
$357$			12.9507i			0.685424i
$358$		0			0
$359$			25.1893i			1.32944i	0.747093	+	0.664719i	$0.231449\pi$
$359$			25.1893i			1.32944i	−0.747093	+	0.664719i	$0.768551\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			12.5540i			0.658916i
$364$		0			0
$365$	2.34550			0.122769
$366$		0			0
$367$	0.120978			0.00631498			0.00315749	−	0.999995i	$-0.498995\pi$
$367$	0.120978			0.00631498			0.00315749	+	0.999995i	$0.498995\pi$
$368$		0			0
$369$			9.98972i			0.520044i
$370$		0			0
$371$	2.56299	+	18.5937i	0.133064	+	0.965336i
$372$		0			0
$373$			17.6534i			0.914058i	0.889452	+	0.457029i	$0.151086\pi$
$373$			17.6534i			0.914058i	−0.889452	+	0.457029i	$0.848914\pi$
$374$		0			0
$375$	7.09609			0.366440
$376$		0			0
$377$	−3.11587			−0.160475
$378$		0			0
$379$		−	0.161504i		−	0.00829590i	−0.999991	−	0.00414795i	$-0.998680\pi$
$379$		−	0.161504i		−	0.00829590i	0.999991	−	0.00414795i	$-0.00132034\pi$
$380$		0			0
$381$	2.08487			0.106811
$382$		0			0
$383$		−	21.8949i		−	1.11878i	−0.828906	−	0.559389i	$-0.811036\pi$
$383$		−	21.8949i		−	1.11878i	0.828906	−	0.559389i	$-0.188964\pi$
$384$		0			0
$385$		−	1.88606i		−	0.0961223i
$386$		0			0
$387$	−5.18070			−0.263350
$388$		0			0
$389$			2.92655i			0.148382i	0.997244	+	0.0741911i	$0.0236375\pi$
$389$			2.92655i			0.148382i	−0.997244	+	0.0741911i	$0.976363\pi$
$390$		0			0
$391$			11.3811i			0.575569i
$392$		0			0
$393$			8.94837i			0.451386i
$394$		0			0
$395$	0.393644			0.0198064
$396$		0			0
$397$		−	0.728626i		−	0.0365687i	−0.999833	−	0.0182844i	$-0.994180\pi$
$397$		−	0.728626i		−	0.0365687i	0.999833	−	0.0182844i	$-0.00582041\pi$
$398$		0			0
$399$	−5.95850			−0.298298
$400$		0			0
$401$			13.8457i			0.691423i	0.938341	+	0.345712i	$0.112362\pi$
$401$			13.8457i			0.691423i	−0.938341	+	0.345712i	$0.887638\pi$
$402$		0			0
$403$		−	1.51010i		−	0.0752233i
$404$		0			0
$405$		−	3.26831i		−	0.162404i
$406$		0			0
$407$	5.22910			0.259197
$408$		0			0
$409$	−22.6420			−1.11957			−0.559787	−	0.828637i	$-0.689117\pi$
$409$	−22.6420			−1.11957			−0.559787	+	0.828637i	$0.689117\pi$
$410$		0			0
$411$	−30.8218			−1.52033
$412$		0			0
$413$	−31.3228			−1.54129
$414$		0			0
$415$	1.60385			0.0787299
$416$		0			0
$417$	14.5622			0.713114
$418$		0			0
$419$		−	1.13103i		−	0.0552545i	−0.999618	−	0.0276272i	$-0.991205\pi$
$419$		−	1.13103i		−	0.0552545i	0.999618	−	0.0276272i	$-0.00879515\pi$
$420$		0			0
$421$		−	4.04926i		−	0.197349i	−0.995120	−	0.0986745i	$-0.968540\pi$
$421$		−	4.04926i		−	0.197349i	0.995120	−	0.0986745i	$-0.0314603\pi$
$422$		0			0
$423$	24.4512			1.18886
$424$		0			0
$425$	−10.6585			−0.517014
$426$		0			0
$427$		−	9.70244i		−	0.469534i
$428$		0			0
$429$		−	7.12988i		−	0.344234i
$430$		0			0
$431$	27.0550			1.30319			0.651596	−	0.758567i	$-0.274100\pi$
$431$	27.0550			1.30319			0.651596	+	0.758567i	$0.274100\pi$
$432$		0			0
$433$	−3.20278			−0.153916			−0.0769580	−	0.997034i	$-0.524521\pi$
$433$	−3.20278			−0.153916			−0.0769580	+	0.997034i	$0.524521\pi$
$434$		0			0
$435$	1.70758			0.0818723
$436$		0			0
$437$	−5.23636			−0.250489
$438$		0			0
$439$	37.2496			1.77783			0.888914	−	0.458075i	$-0.151461\pi$
$439$	37.2496			1.77783			0.888914	+	0.458075i	$0.151461\pi$
$440$		0			0
$441$	0.826352			0.0393501
$442$		0			0
$443$		−	4.23398i		−	0.201162i	−0.994929	−	0.100581i	$-0.967930\pi$
$443$		−	4.23398i		−	0.201162i	0.994929	−	0.100581i	$-0.0320702\pi$
$444$		0			0
$445$			3.83653i			0.181869i
$446$		0			0
$447$		−	19.5428i		−	0.924343i
$448$		0			0
$449$	20.3969			0.962590			0.481295	−	0.876559i	$-0.340167\pi$
$449$	20.3969			0.962590			0.481295	+	0.876559i	$0.340167\pi$
$450$		0			0
$451$			10.0681i			0.474090i
$452$		0			0
$453$	−32.1731			−1.51162
$454$		0			0
$455$		−	1.04500i		−	0.0489903i
$456$		0			0
$457$		−	31.8077i		−	1.48790i	−0.668235	−	0.743950i	$-0.732950\pi$
$457$		−	31.8077i		−	1.48790i	0.668235	−	0.743950i	$-0.267050\pi$
$458$		0			0
$459$			3.30887i			0.154445i
$460$		0			0
$461$	−27.9341			−1.30102			−0.650509	−	0.759498i	$-0.725445\pi$
$461$	−27.9341			−1.30102			−0.650509	+	0.759498i	$0.725445\pi$
$462$		0			0
$463$			4.52859i			0.210461i	0.994448	+	0.105231i	$0.0335581\pi$
$463$			4.52859i			0.210461i	−0.994448	+	0.105231i	$0.966442\pi$
$464$		0			0
$465$			0.827576i			0.0383779i
$466$		0			0
$467$	17.0570			0.789302			0.394651	−	0.918831i	$-0.370866\pi$
$467$	17.0570			0.789302			0.394651	+	0.918831i	$0.370866\pi$
$468$		0			0
$469$		−	17.6543i		−	0.815202i
$470$		0			0
$471$	19.0417			0.877396
$472$		0			0
$473$	−5.22137			−0.240079
$474$		0			0
$475$		−	4.90389i		−	0.225006i
$476$		0			0
$477$	−2.32748	−	16.8851i	−0.106568	−	0.773117i
$478$		0			0
$479$			2.98380i			0.136333i	0.997674	+	0.0681666i	$0.0217150\pi$
$479$			2.98380i			0.136333i	−0.997674	+	0.0681666i	$0.978285\pi$
$480$		0			0
$481$	2.89726			0.132104
$482$		0			0
$483$	−31.2009			−1.41969
$484$		0			0
$485$			2.70852i			0.122987i
$486$		0			0
$487$	22.2004			1.00600			0.502998	−	0.864288i	$-0.332230\pi$
$487$	22.2004			1.00600			0.502998	+	0.864288i	$0.332230\pi$
$488$		0			0
$489$			5.37747i			0.243178i
$490$		0			0
$491$			19.2352i			0.868073i	0.900895	+	0.434037i	$0.142911\pi$
$491$			19.2352i			0.868073i	−0.900895	+	0.434037i	$0.857089\pi$
$492$		0			0
$493$	−5.17994			−0.233293
$494$		0			0
$495$			1.71275i			0.0769823i
$496$		0			0
$497$		−	25.7036i		−	1.15296i
$498$		0			0
$499$			25.3322i			1.13403i	0.823708	+	0.567014i	$0.191901\pi$
$499$			25.3322i			1.13403i	−0.823708	+	0.567014i	$0.808099\pi$
$500$		0			0
$501$	32.4733			1.45080
$502$		0			0
$503$		−	34.1054i		−	1.52069i	−0.649522	−	0.760343i	$-0.725031\pi$
$503$		−	34.1054i		−	1.52069i	0.649522	−	0.760343i	$-0.274969\pi$
$504$		0			0
$505$	0.694994			0.0309268
$506$		0			0
$507$			26.0942i			1.15888i
$508$		0			0
$509$			25.5998i			1.13469i	0.823479	+	0.567346i	$0.192030\pi$
$509$			25.5998i			1.13469i	−0.823479	+	0.567346i	$0.807970\pi$
$510$		0			0
$511$		−	19.5056i		−	0.862878i
$512$		0			0
$513$	−1.52238			−0.0672147
$514$		0			0
$515$	4.99933			0.220297
$516$		0			0
$517$	24.6432			1.08381
$518$		0			0
$519$	15.4345			0.677501
$520$		0			0
$521$	6.82255			0.298901			0.149451	−	0.988769i	$-0.452249\pi$
$521$	6.82255			0.298901			0.149451	+	0.988769i	$0.452249\pi$
$522$		0			0
$523$	21.0239			0.919311			0.459656	−	0.888097i	$-0.347973\pi$
$523$	21.0239			0.919311			0.459656	+	0.888097i	$0.347973\pi$
$524$		0			0
$525$		−	29.2198i		−	1.27526i
$526$		0			0
$527$		−	2.51045i		−	0.109357i
$528$		0			0
$529$	−4.41944			−0.192150
$530$		0			0
$531$	28.4446			1.23439
$532$		0			0
$533$			5.57841i			0.241628i
$534$		0			0
$535$		−	1.67398i		−	0.0723723i
$536$		0			0
$537$	15.6366			0.674769
$538$		0			0
$539$	0.832838			0.0358729
$540$		0			0
$541$	4.18948			0.180120			0.0900600	−	0.995936i	$-0.471294\pi$
$541$	4.18948			0.180120			0.0900600	+	0.995936i	$0.471294\pi$
$542$		0			0
$543$	−21.6945			−0.931001
$544$		0			0
$545$	−3.16954			−0.135768
$546$		0			0
$547$	33.1063			1.41552			0.707761	−	0.706452i	$-0.249705\pi$
$547$	33.1063			1.41552			0.707761	+	0.706452i	$0.249705\pi$
$548$		0			0
$549$			8.81089i			0.376040i
$550$		0			0
$551$		−	2.38324i		−	0.101530i
$552$		0			0
$553$		−	3.27362i		−	0.139208i
$554$		0			0
$555$	−1.58778			−0.0673976
$556$		0			0
$557$		−	23.7837i		−	1.00775i	−0.863777	−	0.503874i	$-0.831908\pi$
$557$		−	23.7837i		−	1.00775i	0.863777	−	0.503874i	$-0.168092\pi$
$558$		0			0
$559$	−2.89298			−0.122360
$560$		0			0
$561$		−	11.8530i		−	0.500434i
$562$		0			0
$563$		−	32.8995i		−	1.38655i	−0.720674	−	0.693274i	$-0.756168\pi$
$563$		−	32.8995i		−	1.38655i	0.720674	−	0.693274i	$-0.243832\pi$
$564$		0			0
$565$		−	3.50075i		−	0.147278i
$566$		0			0
$567$	−27.1799			−1.14145
$568$		0			0
$569$			14.3413i			0.601219i	0.953747	+	0.300610i	$0.0971901\pi$
$569$			14.3413i			0.601219i	−0.953747	+	0.300610i	$0.902810\pi$
$570$		0			0
$571$			6.54333i			0.273830i	0.990583	+	0.136915i	$0.0437187\pi$
$571$			6.54333i			0.273830i	−0.990583	+	0.136915i	$0.956281\pi$
$572$		0			0
$573$	2.86692			0.119767
$574$		0			0
$575$		−	25.6785i		−	1.07087i
$576$		0			0
$577$	1.61500			0.0672332			0.0336166	−	0.999435i	$-0.489297\pi$
$577$	1.61500			0.0672332			0.0336166	+	0.999435i	$0.489297\pi$
$578$		0			0
$579$	19.1224			0.794701
$580$		0			0
$581$		−	13.3379i		−	0.553349i
$582$		0			0
$583$	−2.34575	−	17.0177i	−0.0971510	−	0.704800i
$584$		0			0
$585$			0.948975i			0.0392353i
$586$		0			0
$587$	24.8954			1.02754			0.513772	−	0.857927i	$-0.328248\pi$
$587$	24.8954			1.02754			0.513772	+	0.857927i	$0.328248\pi$
$588$		0			0
$589$	1.15503			0.0475923
$590$		0			0
$591$			52.8342i			2.17331i
$592$		0			0
$593$	29.4991			1.21138			0.605691	−	0.795700i	$-0.292897\pi$
$593$	29.4991			1.21138			0.605691	+	0.795700i	$0.292897\pi$
$594$		0			0
$595$		−	1.73725i		−	0.0712202i
$596$		0			0
$597$		−	13.0724i		−	0.535018i
$598$		0			0
$599$	5.08450			0.207747			0.103873	−	0.994591i	$-0.466876\pi$
$599$	5.08450			0.207747			0.103873	+	0.994591i	$0.466876\pi$
$600$		0			0
$601$		−	15.6904i		−	0.640023i	−0.947414	−	0.320011i	$-0.896313\pi$
$601$		−	15.6904i		−	0.640023i	0.947414	−	0.320011i	$-0.103687\pi$
$602$		0			0
$603$			16.0321i			0.652878i
$604$		0			0
$605$		−	1.68404i		−	0.0684658i
$606$		0			0
$607$	13.0272			0.528758			0.264379	−	0.964419i	$-0.414833\pi$
$607$	13.0272			0.528758			0.264379	+	0.964419i	$0.414833\pi$
$608$		0			0
$609$		−	14.2006i		−	0.575436i
$610$		0			0
$611$	13.6539			0.552379
$612$		0			0
$613$		−	9.95998i		−	0.402280i	−0.979563	−	0.201140i	$-0.935535\pi$
$613$		−	9.95998i		−	0.402280i	0.979563	−	0.201140i	$-0.0644646\pi$
$614$		0			0
$615$		−	3.05713i		−	0.123275i
$616$		0			0
$617$			11.9244i			0.480058i	0.970766	+	0.240029i	$0.0771569\pi$
$617$			11.9244i			0.480058i	−0.970766	+	0.240029i	$0.922843\pi$
$618$		0			0
$619$	−12.6939			−0.510211			−0.255105	−	0.966913i	$-0.582110\pi$
$619$	−12.6939			−0.510211			−0.255105	+	0.966913i	$0.582110\pi$
$620$		0			0
$621$	−7.97173			−0.319894
$622$		0			0
$623$	31.9053			1.27826
$624$		0			0
$625$	23.5675			0.942702
$626$		0			0
$627$	5.45346			0.217790
$628$		0			0
$629$	4.81653			0.192047
$630$		0			0
$631$		−	10.4503i		−	0.416019i	−0.978127	−	0.208010i	$-0.933301\pi$
$631$		−	10.4503i		−	0.416019i	0.978127	−	0.208010i	$-0.0666986\pi$
$632$		0			0
$633$			19.0950i			0.758960i
$634$		0			0
$635$	−0.279671			−0.0110984
$636$		0			0
$637$	0.461447			0.0182832
$638$		0			0
$639$			23.3417i			0.923384i
$640$		0			0
$641$			27.3111i			1.07872i	0.842074	+	0.539362i	$0.181335\pi$
$641$			27.3111i			1.07872i	−0.842074	+	0.539362i	$0.818665\pi$
$642$		0			0
$643$	−44.9963			−1.77448			−0.887240	−	0.461309i	$-0.847380\pi$
$643$	−44.9963			−1.77448			−0.887240	+	0.461309i	$0.847380\pi$
$644$		0			0
$645$	1.58543			0.0624264
$646$		0			0
$647$	7.54502			0.296625			0.148313	−	0.988941i	$-0.452616\pi$
$647$	7.54502			0.296625			0.148313	+	0.988941i	$0.452616\pi$
$648$		0			0
$649$	28.6679			1.12531
$650$		0			0
$651$	6.88227			0.269737
$652$		0			0
$653$	27.7678			1.08664			0.543319	−	0.839527i	$-0.317168\pi$
$653$	27.7678			1.08664			0.543319	+	0.839527i	$0.317168\pi$
$654$		0			0
$655$		−	1.20036i		−	0.0469020i
$656$		0			0
$657$			17.7133i			0.691061i
$658$		0			0
$659$			32.8525i			1.27975i	0.768479	+	0.639875i	$0.221014\pi$
$659$			32.8525i			1.27975i	−0.768479	+	0.639875i	$0.778986\pi$
$660$		0			0
$661$	39.0817			1.52010			0.760050	−	0.649865i	$-0.225174\pi$
$661$	39.0817			1.52010			0.760050	+	0.649865i	$0.225174\pi$
$662$		0			0
$663$		−	6.56734i		−	0.255054i
$664$		0			0
$665$	0.799292			0.0309952
$666$		0			0
$667$		−	12.4795i		−	0.483209i
$668$		0			0
$669$		−	46.7159i		−	1.80614i
$670$		0			0
$671$			8.88005i			0.342811i
$672$		0			0
$673$	−29.4237			−1.13420			−0.567100	−	0.823649i	$-0.691935\pi$
$673$	−29.4237			−1.13420			−0.567100	+	0.823649i	$0.691935\pi$
$674$		0			0
$675$		−	7.46558i		−	0.287350i
$676$		0			0
$677$		−	32.6696i		−	1.25559i	−0.778377	−	0.627797i	$-0.783957\pi$
$677$		−	32.6696i		−	1.25559i	0.778377	−	0.627797i	$-0.216043\pi$
$678$		0			0
$679$	22.5245			0.864411
$680$		0			0
$681$			27.8799i			1.06836i
$682$		0			0
$683$	18.5290			0.708994			0.354497	−	0.935057i	$-0.384652\pi$
$683$	18.5290			0.708994			0.354497	+	0.935057i	$0.384652\pi$
$684$		0			0
$685$	4.13453			0.157972
$686$		0			0
$687$		−	34.2098i		−	1.30518i
$688$		0			0
$689$	−1.29970	−	9.42890i	−0.0495146	−	0.359213i
$690$		0			0
$691$		−	42.9523i		−	1.63398i	−0.576651	−	0.816991i	$-0.695641\pi$
$691$		−	42.9523i		−	1.63398i	0.576651	−	0.816991i	$-0.304359\pi$
$692$		0			0
$693$	14.2435			0.541067
$694$		0			0
$695$	−1.95342			−0.0740974
$696$		0			0
$697$			9.27377i			0.351269i
$698$		0			0
$699$	18.1429			0.686228
$700$		0			0
$701$			22.0760i			0.833801i	0.908952	+	0.416900i	$0.136884\pi$
$701$			22.0760i			0.833801i	−0.908952	+	0.416900i	$0.863116\pi$
$702$		0			0
$703$			2.21604i			0.0835795i
$704$		0			0
$705$	−7.48274			−0.281816
$706$		0			0
$707$		−	5.77969i		−	0.217368i
$708$		0			0
$709$		−	45.3842i		−	1.70444i	−0.523184	−	0.852219i	$-0.675256\pi$
$709$		−	45.3842i		−	1.70444i	0.523184	−	0.852219i	$-0.324744\pi$
$710$		0			0
$711$			2.97281i			0.111489i
$712$		0			0
$713$	6.04817			0.226506
$714$		0			0
$715$			0.956424i			0.0357682i
$716$		0			0
$717$	−17.4867			−0.653052
$718$		0			0
$719$		−	31.7037i		−	1.18235i	−0.806543	−	0.591175i	$-0.798664\pi$
$719$		−	31.7037i		−	1.18235i	0.806543	−	0.591175i	$-0.201336\pi$
$720$		0			0
$721$		−	41.5753i		−	1.54835i
$722$		0			0
$723$			4.10900i			0.152815i
$724$		0			0
$725$	11.6872			0.434050
$726$		0			0
$727$	−25.2558			−0.936687			−0.468343	−	0.883547i	$-0.655149\pi$
$727$	−25.2558			−0.936687			−0.468343	+	0.883547i	$0.655149\pi$
$728$		0			0
$729$	14.1271			0.523228
$730$		0			0
$731$	−4.80940			−0.177882
$732$		0			0
$733$	−13.2716			−0.490197			−0.245098	−	0.969498i	$-0.578820\pi$
$733$	−13.2716			−0.490197			−0.245098	+	0.969498i	$0.578820\pi$
$734$		0			0
$735$	−0.252886			−0.00932784
$736$		0			0
$737$			16.1580i			0.595186i
$738$		0			0
$739$			35.5158i			1.30647i	0.757155	+	0.653235i	$0.226589\pi$
$739$			35.5158i			1.30647i	−0.757155	+	0.653235i	$0.773411\pi$
$740$		0			0
$741$	3.02157			0.111000
$742$		0			0
$743$	0.0144279			0.000529308			0.000264654	−	1.00000i	$-0.499916\pi$
$743$	0.0144279			0.000529308			0.000264654		1.00000i	$0.499916\pi$
$744$		0			0
$745$			2.62153i			0.0960454i
$746$		0			0
$747$			12.1123i			0.443166i
$748$		0			0
$749$	−13.9211			−0.508665
$750$		0			0
$751$	16.3413			0.596302			0.298151	−	0.954519i	$-0.403630\pi$
$751$	16.3413			0.596302			0.298151	+	0.954519i	$0.403630\pi$
$752$		0			0
$753$	−25.5421			−0.930805
$754$		0			0
$755$	4.31579			0.157068
$756$		0			0
$757$	5.34006			0.194088			0.0970438	−	0.995280i	$-0.469061\pi$
$757$	5.34006			0.194088			0.0970438	+	0.995280i	$0.469061\pi$
$758$		0			0
$759$	28.5563			1.03653
$760$		0			0
$761$		−	35.4334i		−	1.28446i	−0.766512	−	0.642230i	$-0.778010\pi$
$761$		−	35.4334i		−	1.28446i	0.766512	−	0.642230i	$-0.221990\pi$
$762$		0			0
$763$			26.3585i			0.954240i
$764$		0			0
$765$			1.57761i			0.0570387i
$766$		0			0
$767$	15.8839			0.573534
$768$		0			0
$769$			14.2885i			0.515255i	0.966244	+	0.257627i	$0.0829407\pi$
$769$			14.2885i			0.515255i	−0.966244	+	0.257627i	$0.917059\pi$
$770$		0			0
$771$	−36.1918			−1.30341
$772$		0			0
$773$			34.2256i			1.23101i	0.788134	+	0.615504i	$0.211047\pi$
$773$			34.2256i			1.23101i	−0.788134	+	0.615504i	$0.788953\pi$
$774$		0			0
$775$			5.66415i			0.203462i
$776$		0			0
$777$			13.2043i			0.473701i
$778$		0			0
$779$	−4.26678			−0.152873
$780$		0			0
$781$			23.5249i			0.841788i
$782$		0			0
$783$		−	3.62820i		−	0.129661i
$784$		0			0
$785$	−2.55431			−0.0911673
$786$		0			0
$787$		−	46.9626i		−	1.67404i	−0.547174	−	0.837019i	$-0.684296\pi$
$787$		−	46.9626i		−	1.67404i	0.547174	−	0.837019i	$-0.315704\pi$
$788$		0			0
$789$	−61.9261			−2.20463
$790$		0			0
$791$	−29.1128			−1.03513
$792$		0			0
$793$			4.92013i			0.174719i
$794$		0			0
$795$	0.712272	+	5.16731i	0.0252617	+	0.183265i
$796$		0			0
$797$		−	17.9603i		−	0.636185i	−0.948060	−	0.318093i	$-0.896958\pi$
$797$		−	17.9603i		−	0.636185i	0.948060	−	0.318093i	$-0.103042\pi$
$798$		0			0
$799$	22.6988			0.803027
$800$		0			0
$801$	−28.9735			−1.02373
$802$		0			0
$803$			17.8523i			0.629995i
$804$		0			0
$805$	4.18538			0.147515
$806$		0			0
$807$		−	55.6443i		−	1.95877i
$808$		0			0
$809$		−	8.08447i		−	0.284235i	−0.989850	−	0.142117i	$-0.954609\pi$
$809$		−	8.08447i		−	0.284235i	0.989850	−	0.142117i	$-0.0453911\pi$
$810$		0			0
$811$	40.0553			1.40653			0.703265	−	0.710927i	$-0.251725\pi$
$811$	40.0553			1.40653			0.703265	+	0.710927i	$0.251725\pi$
$812$		0			0
$813$		−	10.9340i		−	0.383472i
$814$		0			0
$815$		−	0.721350i		−	0.0252678i
$816$		0			0
$817$		−	2.21276i		−	0.0774148i
$818$		0			0
$819$	7.89184			0.275763
$820$		0			0
$821$		−	56.0192i		−	1.95508i	−0.210744	−	0.977541i	$-0.567589\pi$
$821$		−	56.0192i		−	1.95508i	0.210744	−	0.977541i	$-0.432411\pi$
$822$		0			0
$823$	24.2551			0.845481			0.422740	−	0.906251i	$-0.361068\pi$
$823$	24.2551			0.845481			0.422740	+	0.906251i	$0.361068\pi$
$824$		0			0
$825$			26.7431i			0.931077i
$826$		0			0
$827$			29.5251i			1.02669i	0.858183	+	0.513344i	$0.171594\pi$
$827$			29.5251i			1.02669i	−0.858183	+	0.513344i	$0.828406\pi$
$828$		0			0
$829$		−	7.06136i		−	0.245251i	−0.992453	−	0.122626i	$-0.960869\pi$
$829$		−	7.06136i		−	0.245251i	0.992453	−	0.122626i	$-0.0391314\pi$
$830$		0			0
$831$	2.56430			0.0889545
$832$		0			0
$833$	0.767128			0.0265794
$834$		0			0
$835$	−4.35607			−0.150748
$836$		0			0
$837$	1.75840			0.0607792
$838$		0			0
$839$	32.3786			1.11783			0.558917	−	0.829223i	$-0.311217\pi$
$839$	32.3786			1.11783			0.558917	+	0.829223i	$0.311217\pi$
$840$		0			0
$841$	−23.3202			−0.804143
$842$		0			0
$843$		−	6.50186i		−	0.223936i
$844$		0			0
$845$		−	3.50035i		−	0.120416i
$846$		0			0
$847$	−14.0047			−0.481209
$848$		0			0
$849$	22.6947			0.778881
$850$		0			0
$851$			11.6040i			0.397779i
$852$		0			0
$853$			4.16750i			0.142692i	0.997452	+	0.0713462i	$0.0227295\pi$
$853$			4.16750i			0.142692i	−0.997452	+	0.0713462i	$0.977271\pi$
$854$		0			0
$855$	−0.725846			−0.0248234
$856$		0			0
$857$	−28.0549			−0.958338			−0.479169	−	0.877723i	$-0.659062\pi$
$857$	−28.0549			−0.958338			−0.479169	+	0.877723i	$0.659062\pi$
$858$		0			0
$859$	55.7166			1.90103			0.950513	−	0.310684i	$-0.100558\pi$
$859$	55.7166			1.90103			0.950513	+	0.310684i	$0.100558\pi$
$860$		0			0
$861$	−25.4236			−0.866434
$862$		0			0
$863$	−38.4040			−1.30729			−0.653643	−	0.756803i	$-0.726760\pi$
$863$	−38.4040			−1.30729			−0.653643	+	0.756803i	$0.726760\pi$
$864$		0			0
$865$	−2.07043			−0.0703969
$866$		0			0
$867$			28.3713i			0.963538i
$868$		0			0
$869$			2.99614i			0.101637i
$870$		0			0
$871$			8.95257i			0.303346i
$872$		0			0
$873$	−20.4548			−0.692289
$874$		0			0
$875$			7.91609i			0.267613i
$876$		0			0
$877$	48.2873			1.63055			0.815273	−	0.579077i	$-0.196587\pi$
$877$	48.2873			1.63055			0.815273	+	0.579077i	$0.196587\pi$
$878$		0			0
$879$			68.3116i			2.30409i
$880$		0			0
$881$		−	40.0370i		−	1.34888i	−0.738329	−	0.674441i	$-0.764385\pi$
$881$		−	40.0370i		−	1.34888i	0.738329	−	0.674441i	$-0.235615\pi$
$882$		0			0
$883$		−	45.6512i		−	1.53629i	−0.640278	−	0.768143i	$-0.721181\pi$
$883$		−	45.6512i		−	1.53629i	0.640278	−	0.768143i	$-0.278819\pi$
$884$		0			0
$885$	−8.70482			−0.292609
$886$		0			0
$887$		−	55.6995i		−	1.87020i	−0.354378	−	0.935102i	$-0.615307\pi$
$887$		−	55.6995i		−	1.87020i	0.354378	−	0.935102i	$-0.384693\pi$
$888$		0			0
$889$			2.32579i			0.0780045i
$890$		0			0
$891$	24.8761			0.833381
$892$		0			0
$893$			10.4435i			0.349479i
$894$		0			0
$895$	−2.09754			−0.0701130
$896$		0			0
$897$	15.8220			0.528283
$898$		0			0
$899$			2.75272i			0.0918085i
$900$		0			0
$901$	−2.16067	−	15.6750i	−0.0719824	−	0.522209i
$902$		0			0
$903$		−	13.1848i		−	0.438761i
$904$		0			0
$905$	2.91017			0.0967373
$906$		0			0
$907$	16.8316			0.558883			0.279442	−	0.960163i	$-0.409851\pi$
$907$	16.8316			0.558883			0.279442	+	0.960163i	$0.409851\pi$
$908$		0			0
$909$			5.24860i			0.174085i
$910$		0			0
$911$	7.98811			0.264658			0.132329	−	0.991206i	$-0.457754\pi$
$911$	7.98811			0.264658			0.132329	+	0.991206i	$0.457754\pi$
$912$		0			0
$913$			12.2074i			0.404005i
$914$		0			0
$915$		−	2.69637i		−	0.0891393i
$916$		0			0
$917$	−9.98242			−0.329648
$918$		0			0
$919$		−	0.379604i		−	0.0125220i	−0.999980	−	0.00626099i	$-0.998007\pi$
$919$		−	0.379604i		−	0.0125220i	0.999980	−	0.00626099i	$-0.00199295\pi$
$920$		0			0
$921$		−	24.5375i		−	0.808537i
$922$		0			0
$923$			13.0344i			0.429031i
$924$		0			0
$925$	−10.8672			−0.357312
$926$		0			0
$927$			37.7550i			1.24004i
$928$		0			0
$929$	27.4759			0.901454			0.450727	−	0.892662i	$-0.351165\pi$
$929$	27.4759			0.901454			0.450727	+	0.892662i	$0.351165\pi$
$930$		0			0
$931$			0.352949i			0.0115674i
$932$		0			0
$933$		−	4.61510i		−	0.151091i
$934$		0			0
$935$			1.59000i			0.0519985i
$936$		0			0
$937$	6.02396			0.196794			0.0983971	−	0.995147i	$-0.468628\pi$
$937$	6.02396			0.196794			0.0983971	+	0.995147i	$0.468628\pi$
$938$		0			0
$939$	64.5842			2.10763
$940$		0			0
$941$	−42.3887			−1.38183			−0.690916	−	0.722935i	$-0.742793\pi$
$941$	−42.3887			−1.38183			−0.690916	+	0.722935i	$0.742793\pi$
$942$		0			0
$943$	−22.3424			−0.727568
$944$		0			0
$945$	1.21683			0.0395833
$946$		0			0
$947$	5.24221			0.170349			0.0851744	−	0.996366i	$-0.472855\pi$
$947$	5.24221			0.170349			0.0851744	+	0.996366i	$0.472855\pi$
$948$		0			0
$949$			9.89136i			0.321087i
$950$		0			0
$951$		−	0.954905i		−	0.0309649i
$952$		0			0
$953$	3.86972			0.125353			0.0626763	−	0.998034i	$-0.480036\pi$
$953$	3.86972			0.125353			0.0626763	+	0.998034i	$0.480036\pi$
$954$		0			0
$955$	−0.384577			−0.0124446
$956$		0			0
$957$			12.9969i			0.420130i
$958$		0			0
$959$		−	34.3835i		−	1.11030i
$960$		0			0
$961$	29.6659			0.956964
$962$		0			0
$963$	12.6419			0.407379
$964$		0			0
$965$	−2.56514			−0.0825747
$966$		0			0
$967$	27.7649			0.892858			0.446429	−	0.894819i	$-0.352695\pi$
$967$	27.7649			0.892858			0.446429	+	0.894819i	$0.352695\pi$
$968$		0			0
$969$	5.02318			0.161368
$970$		0			0
$971$	−24.3208			−0.780492			−0.390246	−	0.920711i	$-0.627610\pi$
$971$	−24.3208			−0.780492			−0.390246	+	0.920711i	$0.627610\pi$
$972$		0			0
$973$			16.2450i			0.520790i
$974$		0			0
$975$			14.8175i			0.474538i
$976$		0			0
$977$		−	29.1304i		−	0.931963i	−0.884794	−	0.465982i	$-0.845701\pi$
$977$		−	29.1304i		−	0.931963i	0.884794	−	0.465982i	$-0.154299\pi$
$978$		0			0
$979$	−29.2010			−0.933267
$980$		0			0
$981$		−	23.9364i		−	0.764231i
$982$		0			0
$983$	−40.8903			−1.30420			−0.652100	−	0.758133i	$-0.726112\pi$
$983$	−40.8903			−1.30420			−0.652100	+	0.758133i	$0.726112\pi$
$984$		0			0
$985$		−	7.08734i		−	0.225821i
$986$		0			0
$987$			62.2278i			1.98073i
$988$		0			0
$989$		−	11.5868i		−	0.368439i
$990$		0			0
$991$	5.27099			0.167438			0.0837192	−	0.996489i	$-0.473320\pi$
$991$	5.27099			0.167438			0.0837192	+	0.996489i	$0.473320\pi$
$992$		0			0
$993$		−	52.8938i		−	1.67853i
$994$		0			0
$995$			1.75357i			0.0555920i
$996$		0			0
$997$	3.13045			0.0991423			0.0495711	−	0.998771i	$-0.484215\pi$
$997$	3.13045			0.0991423			0.0495711	+	0.998771i	$0.484215\pi$
$998$		0			0
$999$			3.37365i			0.106738i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.15	✓	82
53.52	even	2	inner	4028.2.c.a.3497.68	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.15	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.68	yes	82	53.52	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.31112i q^{3} +0.310021i q^{5} +2.57819 q^{7} -2.34128 q^{9} +O(q^{10})\)	\(q-2.31112i q^{3} +0.310021i q^{5} +2.57819 q^{7} -2.34128 q^{9} -2.35966 q^{11} -1.30741 q^{13} +0.716495 q^{15} -2.17348 q^{17} -1.00000i q^{19} -5.95850i q^{21} -5.23636i q^{23} +4.90389 q^{25} -1.52238i q^{27} +2.38324 q^{29} +1.15503i q^{31} +5.45346i q^{33} +0.799292i q^{35} -2.21604 q^{37} +3.02157i q^{39} -4.26678i q^{41} +2.21276 q^{43} -0.725846i q^{45} -10.4435 q^{47} -0.352949 q^{49} +5.02318i q^{51} +(0.994105 + 7.21192i) q^{53} -0.731543i q^{55} -2.31112 q^{57} -12.1492 q^{59} -3.76328i q^{61} -6.03626 q^{63} -0.405323i q^{65} -6.84758i q^{67} -12.1019 q^{69} -9.96963i q^{71} -7.56564i q^{73} -11.3335i q^{75} -6.08364 q^{77} -1.26974i q^{79} -10.5422 q^{81} -5.17336i q^{83} -0.673825i q^{85} -5.50796i q^{87} +12.3751 q^{89} -3.37074 q^{91} +2.66942 q^{93} +0.310021 q^{95} +8.73657 q^{97} +5.52462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 82 * q - 8 * q^7 - 82 * q^9	\( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) 82 * q - 8 * q^7 - 82 * q^9 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 58 * q^25 - 16 * q^29 - 12 * q^37 - 32 * q^43 + 8 * q^47 + 98 * q^49 + 6 * q^53 - 4 * q^57 + 4 * q^59 + 8 * q^63 + 28 * q^69 - 8 * q^77 + 154 * q^81 - 20 * q^89 + 48 * q^91 - 56 * q^93 - 44 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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