LMFDB - Embedded newform 4028.2.c.a.3497.12

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			3497.12
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.71

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46723i q^{3} -4.35102i q^{5} +1.12403 q^{7} -3.08725 q^{9} +O(q^{10})$	$q-2.46723i q^{3} -4.35102i q^{5} +1.12403 q^{7} -3.08725 q^{9} -0.799448 q^{11} +7.10995 q^{13} -10.7350 q^{15} +5.99464 q^{17} -1.00000i q^{19} -2.77325i q^{21} +0.333262i q^{23} -13.9314 q^{25} +0.215258i q^{27} -6.55636 q^{29} -3.36396i q^{31} +1.97243i q^{33} -4.89068i q^{35} -10.2091 q^{37} -17.5419i q^{39} +4.96823i q^{41} +6.65351 q^{43} +13.4327i q^{45} -5.65151 q^{47} -5.73655 q^{49} -14.7902i q^{51} +(-3.17877 + 6.54946i) q^{53} +3.47842i q^{55} -2.46723 q^{57} +8.81155 q^{59} -12.2919i q^{61} -3.47016 q^{63} -30.9356i q^{65} -10.5470i q^{67} +0.822236 q^{69} +8.10535i q^{71} +9.90391i q^{73} +34.3720i q^{75} -0.898605 q^{77} -3.69444i q^{79} -8.73065 q^{81} -7.54897i q^{83} -26.0828i q^{85} +16.1761i q^{87} -3.95256 q^{89} +7.99181 q^{91} -8.29967 q^{93} -4.35102 q^{95} -3.60480 q^{97} +2.46809 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.46723i		−	1.42446i	−0.701947	−	0.712229i	$-0.747686\pi$
$3$		−	2.46723i		−	1.42446i	0.701947	−	0.712229i	$-0.252314\pi$
$4$		0			0
$5$		−	4.35102i		−	1.94584i	−0.231151	−	0.972918i	$-0.574249\pi$
$5$		−	4.35102i		−	1.94584i	0.231151	−	0.972918i	$-0.425751\pi$
$6$		0			0
$7$	1.12403			0.424844			0.212422	−	0.977178i	$-0.431865\pi$
$7$	1.12403			0.424844			0.212422	+	0.977178i	$0.431865\pi$
$8$		0			0
$9$	−3.08725			−1.02908
$10$		0			0
$11$	−0.799448			−0.241043			−0.120521	−	0.992711i	$-0.538457\pi$
$11$	−0.799448			−0.241043			−0.120521	+	0.992711i	$0.538457\pi$
$12$		0			0
$13$	7.10995			1.97195			0.985973	−	0.166906i	$-0.0533776\pi$
$13$	7.10995			1.97195			0.985973	+	0.166906i	$0.0533776\pi$
$14$		0			0
$15$	−10.7350			−2.77176
$16$		0			0
$17$	5.99464			1.45391			0.726957	−	0.686683i	$-0.240934\pi$
$17$	5.99464			1.45391			0.726957	+	0.686683i	$0.240934\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$		−	2.77325i		−	0.605173i
$22$		0			0
$23$			0.333262i			0.0694900i	0.999396	+	0.0347450i	$0.0110619\pi$
$23$			0.333262i			0.0694900i	−0.999396	+	0.0347450i	$0.988938\pi$
$24$		0			0
$25$	−13.9314			−2.78628
$26$		0			0
$27$			0.215258i			0.0414264i
$28$		0			0
$29$	−6.55636			−1.21748			−0.608742	−	0.793368i	$-0.708326\pi$
$29$	−6.55636			−1.21748			−0.608742	+	0.793368i	$0.708326\pi$
$30$		0			0
$31$		−	3.36396i		−	0.604185i	−0.953279	−	0.302092i	$-0.902315\pi$
$31$		−	3.36396i		−	0.604185i	0.953279	−	0.302092i	$-0.0976851\pi$
$32$		0			0
$33$			1.97243i			0.343355i
$34$		0			0
$35$		−	4.89068i		−	0.826677i
$36$		0			0
$37$	−10.2091			−1.67837			−0.839186	−	0.543845i	$-0.816968\pi$
$37$	−10.2091			−1.67837			−0.839186	+	0.543845i	$0.816968\pi$
$38$		0			0
$39$		−	17.5419i		−	2.80895i
$40$		0			0
$41$			4.96823i			0.775908i	0.921679	+	0.387954i	$0.126818\pi$
$41$			4.96823i			0.775908i	−0.921679	+	0.387954i	$0.873182\pi$
$42$		0			0
$43$	6.65351			1.01465			0.507325	−	0.861755i	$-0.330634\pi$
$43$	6.65351			1.01465			0.507325	+	0.861755i	$0.330634\pi$
$44$		0			0
$45$			13.4327i			2.00243i
$46$		0			0
$47$	−5.65151			−0.824358			−0.412179	−	0.911103i	$-0.635232\pi$
$47$	−5.65151			−0.824358			−0.412179	+	0.911103i	$0.635232\pi$
$48$		0			0
$49$	−5.73655			−0.819508
$50$		0			0
$51$		−	14.7902i		−	2.07104i
$52$		0			0
$53$	−3.17877	+	6.54946i	−0.436638	+	0.899637i
$53$	−3.17877	+	6.54946i	−0.436638	+	0.899637i
$54$		0			0
$55$			3.47842i			0.469030i
$56$		0			0
$57$	−2.46723			−0.326793
$58$		0			0
$59$	8.81155			1.14717			0.573583	−	0.819147i	$-0.305553\pi$
$59$	8.81155			1.14717			0.573583	+	0.819147i	$0.305553\pi$
$60$		0			0
$61$		−	12.2919i		−	1.57382i	−0.617068	−	0.786910i	$-0.711680\pi$
$61$		−	12.2919i		−	1.57382i	0.617068	−	0.786910i	$-0.288320\pi$
$62$		0			0
$63$	−3.47016			−0.437199
$64$		0			0
$65$		−	30.9356i		−	3.83708i
$66$		0			0
$67$		−	10.5470i		−	1.28852i	−0.764807	−	0.644259i	$-0.777166\pi$
$67$		−	10.5470i		−	1.28852i	0.764807	−	0.644259i	$-0.222834\pi$
$68$		0			0
$69$	0.822236			0.0989856
$70$		0			0
$71$			8.10535i			0.961928i	0.876740	+	0.480964i	$0.159713\pi$
$71$			8.10535i			0.961928i	−0.876740	+	0.480964i	$0.840287\pi$
$72$		0			0
$73$			9.90391i			1.15917i	0.814913	+	0.579583i	$0.196784\pi$
$73$			9.90391i			1.15917i	−0.814913	+	0.579583i	$0.803216\pi$
$74$		0			0
$75$			34.3720i			3.96894i
$76$		0			0
$77$	−0.898605			−0.102406
$78$		0			0
$79$		−	3.69444i		−	0.415657i	−0.978165	−	0.207828i	$-0.933360\pi$
$79$		−	3.69444i		−	0.415657i	0.978165	−	0.207828i	$-0.0666396\pi$
$80$		0			0
$81$	−8.73065			−0.970072
$82$		0			0
$83$		−	7.54897i		−	0.828608i	−0.910139	−	0.414304i	$-0.864025\pi$
$83$		−	7.54897i		−	0.828608i	0.910139	−	0.414304i	$-0.135975\pi$
$84$		0			0
$85$		−	26.0828i		−	2.82908i
$86$		0			0
$87$			16.1761i			1.73426i
$88$		0			0
$89$	−3.95256			−0.418970			−0.209485	−	0.977812i	$-0.567179\pi$
$89$	−3.95256			−0.418970			−0.209485	+	0.977812i	$0.567179\pi$
$90$		0			0
$91$	7.99181			0.837769
$92$		0			0
$93$	−8.29967			−0.860636
$94$		0			0
$95$	−4.35102			−0.446405
$96$		0			0
$97$	−3.60480			−0.366012			−0.183006	−	0.983112i	$-0.558583\pi$
$97$	−3.60480			−0.366012			−0.183006	+	0.983112i	$0.558583\pi$
$98$		0			0
$99$	2.46809			0.248053
$100$		0			0
$101$		−	6.00383i		−	0.597403i	−0.954347	−	0.298702i	$-0.903446\pi$
$101$		−	6.00383i		−	0.597403i	0.954347	−	0.298702i	$-0.0965535\pi$
$102$		0			0
$103$		−	9.32269i		−	0.918591i	−0.888283	−	0.459296i	$-0.848102\pi$
$103$		−	9.32269i		−	0.918591i	0.888283	−	0.459296i	$-0.151898\pi$
$104$		0			0
$105$	−12.0665			−1.17757
$106$		0			0
$107$	15.7521			1.52281			0.761406	−	0.648276i	$-0.224510\pi$
$107$	15.7521			1.52281			0.761406	+	0.648276i	$0.224510\pi$
$108$		0			0
$109$		−	3.31508i		−	0.317527i	−0.987317	−	0.158764i	$-0.949249\pi$
$109$		−	3.31508i		−	0.317527i	0.987317	−	0.158764i	$-0.0507508\pi$
$110$		0			0
$111$			25.1883i			2.39077i
$112$		0			0
$113$	−4.01932			−0.378106			−0.189053	−	0.981967i	$-0.560542\pi$
$113$	−4.01932			−0.378106			−0.189053	+	0.981967i	$0.560542\pi$
$114$		0			0
$115$	1.45003			0.135216
$116$		0			0
$117$	−21.9502			−2.02929
$118$		0			0
$119$	6.73817			0.617687
$120$		0			0
$121$	−10.3609			−0.941898
$122$		0			0
$123$	12.2578			1.10525
$124$		0			0
$125$			38.8607i			3.47580i
$126$		0			0
$127$			11.8113i			1.04808i	0.851693	+	0.524042i	$0.175576\pi$
$127$			11.8113i			1.04808i	−0.851693	+	0.524042i	$0.824424\pi$
$128$		0			0
$129$		−	16.4158i		−	1.44533i
$130$		0			0
$131$	−22.8521			−1.99660			−0.998298	−	0.0583252i	$-0.981424\pi$
$131$	−22.8521			−1.99660			−0.998298	+	0.0583252i	$0.981424\pi$
$132$		0			0
$133$		−	1.12403i		−	0.0974659i
$134$		0			0
$135$	0.936593			0.0806091
$136$		0			0
$137$		−	3.67465i		−	0.313946i	−0.987603	−	0.156973i	$-0.949826\pi$
$137$		−	3.67465i		−	0.313946i	0.987603	−	0.156973i	$-0.0501736\pi$
$138$		0			0
$139$		−	14.6803i		−	1.24517i	−0.782554	−	0.622583i	$-0.786083\pi$
$139$		−	14.6803i		−	1.24517i	0.782554	−	0.622583i	$-0.213917\pi$
$140$		0			0
$141$			13.9436i			1.17426i
$142$		0			0
$143$	−5.68404			−0.475323
$144$		0			0
$145$			28.5268i			2.36903i
$146$		0			0
$147$			14.1534i			1.16735i
$148$		0			0
$149$	17.5594			1.43852			0.719259	−	0.694742i	$-0.244481\pi$
$149$	17.5594			1.43852			0.719259	+	0.694742i	$0.244481\pi$
$150$		0			0
$151$		−	16.5791i		−	1.34918i	−0.738191	−	0.674592i	$-0.764320\pi$
$151$		−	16.5791i		−	1.34918i	0.738191	−	0.674592i	$-0.235680\pi$
$152$		0			0
$153$	−18.5069			−1.49620
$154$		0			0
$155$	−14.6367			−1.17564
$156$		0			0
$157$			14.0601i			1.12212i	0.827775	+	0.561060i	$0.189606\pi$
$157$			14.0601i			1.12212i	−0.827775	+	0.561060i	$0.810394\pi$
$158$		0			0
$159$	16.1591	+	7.84278i	1.28150	+	0.621973i
$160$		0			0
$161$			0.374597i			0.0295224i
$162$		0			0
$163$	12.5711			0.984644			0.492322	−	0.870413i	$-0.336148\pi$
$163$	12.5711			0.984644			0.492322	+	0.870413i	$0.336148\pi$
$164$		0			0
$165$	8.58207			0.668113
$166$		0			0
$167$			14.5056i			1.12248i	0.827654	+	0.561239i	$0.189675\pi$
$167$			14.5056i			1.12248i	−0.827654	+	0.561239i	$0.810325\pi$
$168$		0			0
$169$	37.5514			2.88857
$170$		0			0
$171$			3.08725i			0.236088i
$172$		0			0
$173$			10.9586i			0.833164i	0.909098	+	0.416582i	$0.136772\pi$
$173$			10.9586i			0.833164i	−0.909098	+	0.416582i	$0.863228\pi$
$174$		0			0
$175$	−15.6593			−1.18373
$176$		0			0
$177$		−	21.7402i		−	1.63409i
$178$		0			0
$179$		−	5.07052i		−	0.378988i	−0.981882	−	0.189494i	$-0.939315\pi$
$179$		−	5.07052i		−	0.378988i	0.981882	−	0.189494i	$-0.0606848\pi$
$180$		0			0
$181$			0.195193i			0.0145086i	0.999974	+	0.00725430i	$0.00230914\pi$
$181$			0.195193i			0.0145086i	−0.999974	+	0.00725430i	$0.997691\pi$
$182$		0			0
$183$	−30.3271			−2.24184
$184$		0			0
$185$			44.4202i			3.26584i
$186$		0			0
$187$	−4.79241			−0.350455
$188$		0			0
$189$			0.241957i			0.0175998i
$190$		0			0
$191$		−	21.2536i		−	1.53786i	−0.639334	−	0.768929i	$-0.720790\pi$
$191$		−	21.2536i		−	1.53786i	0.639334	−	0.768929i	$-0.279210\pi$
$192$		0			0
$193$			1.31833i			0.0948951i	0.998874	+	0.0474476i	$0.0151087\pi$
$193$			1.31833i			0.0948951i	−0.998874	+	0.0474476i	$0.984891\pi$
$194$		0			0
$195$	−76.3253			−5.46577
$196$		0			0
$197$	16.8466			1.20027			0.600134	−	0.799900i	$-0.295114\pi$
$197$	16.8466			1.20027			0.600134	+	0.799900i	$0.295114\pi$
$198$		0			0
$199$	16.4864			1.16869			0.584347	−	0.811504i	$-0.301351\pi$
$199$	16.4864			1.16869			0.584347	+	0.811504i	$0.301351\pi$
$200$		0			0
$201$	−26.0219			−1.83544
$202$		0			0
$203$	−7.36955			−0.517241
$204$		0			0
$205$	21.6169			1.50979
$206$		0			0
$207$		−	1.02886i		−	0.0715109i
$208$		0			0
$209$			0.799448i			0.0552990i
$210$		0			0
$211$	6.04700			0.416292			0.208146	−	0.978098i	$-0.433257\pi$
$211$	6.04700			0.416292			0.208146	+	0.978098i	$0.433257\pi$
$212$		0			0
$213$	19.9978			1.37023
$214$		0			0
$215$		−	28.9495i		−	1.97434i
$216$		0			0
$217$		−	3.78119i		−	0.256684i
$218$		0			0
$219$	24.4353			1.65118
$220$		0			0
$221$	42.6216			2.86704
$222$		0			0
$223$	13.9707			0.935546			0.467773	−	0.883849i	$-0.345056\pi$
$223$	13.9707			0.935546			0.467773	+	0.883849i	$0.345056\pi$
$224$		0			0
$225$	43.0096			2.86731
$226$		0			0
$227$	8.90088			0.590772			0.295386	−	0.955378i	$-0.404552\pi$
$227$	8.90088			0.590772			0.295386	+	0.955378i	$0.404552\pi$
$228$		0			0
$229$	1.25597			0.0829965			0.0414983	−	0.999139i	$-0.486787\pi$
$229$	1.25597			0.0829965			0.0414983	+	0.999139i	$0.486787\pi$
$230$		0			0
$231$			2.21707i			0.145872i
$232$		0			0
$233$			19.8034i			1.29736i	0.761059	+	0.648682i	$0.224680\pi$
$233$			19.8034i			1.29736i	−0.761059	+	0.648682i	$0.775320\pi$
$234$		0			0
$235$			24.5898i			1.60406i
$236$		0			0
$237$	−9.11505			−0.592086
$238$		0			0
$239$			16.6074i			1.07424i	0.843505	+	0.537121i	$0.180488\pi$
$239$			16.6074i			1.07424i	−0.843505	+	0.537121i	$0.819512\pi$
$240$		0			0
$241$	−17.2728			−1.11264			−0.556319	−	0.830969i	$-0.687787\pi$
$241$	−17.2728			−1.11264			−0.556319	+	0.830969i	$0.687787\pi$
$242$		0			0
$243$			22.1863i			1.42325i
$244$		0			0
$245$			24.9599i			1.59463i
$246$		0			0
$247$		−	7.10995i		−	0.452395i
$248$		0			0
$249$	−18.6251			−1.18032
$250$		0			0
$251$			21.1738i			1.33648i	0.743946	+	0.668240i	$0.232952\pi$
$251$			21.1738i			1.33648i	−0.743946	+	0.668240i	$0.767048\pi$
$252$		0			0
$253$		−	0.266426i		−	0.0167501i
$254$		0			0
$255$	−64.3524			−4.02991
$256$		0			0
$257$			4.43990i			0.276954i	0.990366	+	0.138477i	$0.0442206\pi$
$257$			4.43990i			0.276954i	−0.990366	+	0.138477i	$0.955779\pi$
$258$		0			0
$259$	−11.4754			−0.713046
$260$		0			0
$261$	20.2411			1.25289
$262$		0			0
$263$			26.5056i			1.63441i	0.576349	+	0.817204i	$0.304477\pi$
$263$			26.5056i			1.63441i	−0.576349	+	0.817204i	$0.695523\pi$
$264$		0			0
$265$	28.4968	+	13.8309i	1.75055	+	0.849626i
$266$		0			0
$267$			9.75189i			0.596806i
$268$		0			0
$269$	3.32516			0.202738			0.101369	−	0.994849i	$-0.467678\pi$
$269$	3.32516			0.202738			0.101369	+	0.994849i	$0.467678\pi$
$270$		0			0
$271$	23.7057			1.44002			0.720009	−	0.693965i	$-0.244138\pi$
$271$	23.7057			1.44002			0.720009	+	0.693965i	$0.244138\pi$
$272$		0			0
$273$		−	19.7177i		−	1.19337i
$274$		0			0
$275$	11.1374			0.671612
$276$		0			0
$277$		−	5.21164i		−	0.313137i	−0.987667	−	0.156568i	$-0.949957\pi$
$277$		−	5.21164i		−	0.313137i	0.987667	−	0.156568i	$-0.0500432\pi$
$278$		0			0
$279$			10.3854i			0.621756i
$280$		0			0
$281$	9.63957			0.575049			0.287524	−	0.957773i	$-0.407168\pi$
$281$	9.63957			0.575049			0.287524	+	0.957773i	$0.407168\pi$
$282$		0			0
$283$		−	27.0307i		−	1.60681i	−0.595433	−	0.803405i	$-0.703019\pi$
$283$		−	27.0307i		−	1.60681i	0.595433	−	0.803405i	$-0.296981\pi$
$284$		0			0
$285$			10.7350i			0.635886i
$286$		0			0
$287$			5.58445i			0.329640i
$288$		0			0
$289$	18.9357			1.11387
$290$		0			0
$291$			8.89390i			0.521369i
$292$		0			0
$293$	−14.1003			−0.823746			−0.411873	−	0.911241i	$-0.635125\pi$
$293$	−14.1003			−0.823746			−0.411873	+	0.911241i	$0.635125\pi$
$294$		0			0
$295$		−	38.3393i		−	2.23220i
$296$		0			0
$297$		−	0.172088i		−	0.00998554i
$298$		0			0
$299$			2.36948i			0.137031i
$300$		0			0
$301$	7.47875			0.431068
$302$		0			0
$303$	−14.8129			−0.850976
$304$		0			0
$305$	−53.4824			−3.06239
$306$		0			0
$307$	23.5676			1.34507			0.672536	−	0.740064i	$-0.265205\pi$
$307$	23.5676			1.34507			0.672536	+	0.740064i	$0.265205\pi$
$308$		0			0
$309$	−23.0013			−1.30850
$310$		0			0
$311$	0.0856000			0.00485393			0.00242696	−	0.999997i	$-0.499227\pi$
$311$	0.0856000			0.00485393			0.00242696	+	0.999997i	$0.499227\pi$
$312$		0			0
$313$		−	28.0803i		−	1.58719i	−0.608444	−	0.793597i	$-0.708206\pi$
$313$		−	28.0803i		−	1.58719i	0.608444	−	0.793597i	$-0.291794\pi$
$314$		0			0
$315$			15.0987i			0.850718i
$316$		0			0
$317$	−8.85550			−0.497374			−0.248687	−	0.968584i	$-0.579999\pi$
$317$	−8.85550			−0.497374			−0.248687	+	0.968584i	$0.579999\pi$
$318$		0			0
$319$	5.24147			0.293466
$320$		0			0
$321$		−	38.8641i		−	2.16918i
$322$		0			0
$323$		−	5.99464i		−	0.333551i
$324$		0			0
$325$	−99.0515			−5.49439
$326$		0			0
$327$	−8.17908			−0.452304
$328$		0			0
$329$	−6.35248			−0.350223
$330$		0			0
$331$	−7.89230			−0.433800			−0.216900	−	0.976194i	$-0.569595\pi$
$331$	−7.89230			−0.433800			−0.216900	+	0.976194i	$0.569595\pi$
$332$		0			0
$333$	31.5181			1.72718
$334$		0			0
$335$	−45.8901			−2.50725
$336$		0			0
$337$		−	4.33311i		−	0.236040i	−0.993011	−	0.118020i	$-0.962345\pi$
$337$		−	4.33311i		−	0.236040i	0.993011	−	0.118020i	$-0.0376546\pi$
$338$		0			0
$339$			9.91660i			0.538596i
$340$		0			0
$341$			2.68931i			0.145634i
$342$		0			0
$343$	−14.3163			−0.773007
$344$		0			0
$345$		−	3.57757i		−	0.192610i
$346$		0			0
$347$	15.3427			0.823638			0.411819	−	0.911266i	$-0.364894\pi$
$347$	15.3427			0.823638			0.411819	+	0.911266i	$0.364894\pi$
$348$		0			0
$349$		−	3.05386i		−	0.163469i	−0.996654	−	0.0817347i	$-0.973954\pi$
$349$		−	3.05386i		−	0.163469i	0.996654	−	0.0817347i	$-0.0260460\pi$
$350$		0			0
$351$			1.53047i			0.0816907i
$352$		0			0
$353$		−	11.0742i		−	0.589421i	−0.955587	−	0.294711i	$-0.904777\pi$
$353$		−	11.0742i		−	0.589421i	0.955587	−	0.294711i	$-0.0952232\pi$
$354$		0			0
$355$	35.2666			1.87175
$356$		0			0
$357$		−	16.6246i		−	0.879869i
$358$		0			0
$359$		−	5.90570i		−	0.311691i	−0.987781	−	0.155845i	$-0.950190\pi$
$359$		−	5.90570i		−	0.311691i	0.987781	−	0.155845i	$-0.0498102\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			25.5627i			1.34170i
$364$		0			0
$365$	43.0921			2.25555
$366$		0			0
$367$	−24.2944			−1.26816			−0.634080	−	0.773268i	$-0.718621\pi$
$367$	−24.2944			−1.26816			−0.634080	+	0.773268i	$0.718621\pi$
$368$		0			0
$369$		−	15.3382i		−	0.798473i
$370$		0			0
$371$	−3.57304	+	7.36180i	−0.185503	+	0.382205i
$372$		0			0
$373$			18.9477i			0.981076i	0.871420	+	0.490538i	$0.163200\pi$
$373$			18.9477i			0.981076i	−0.871420	+	0.490538i	$0.836800\pi$
$374$		0			0
$375$	95.8784			4.95114
$376$		0			0
$377$	−46.6154			−2.40081
$378$		0			0
$379$		−	14.5267i		−	0.746187i	−0.927794	−	0.373094i	$-0.878297\pi$
$379$		−	14.5267i		−	0.746187i	0.927794	−	0.373094i	$-0.121703\pi$
$380$		0			0
$381$	29.1413			1.49295
$382$		0			0
$383$			17.1599i			0.876828i	0.898773	+	0.438414i	$0.144460\pi$
$383$			17.1599i			0.876828i	−0.898773	+	0.438414i	$0.855540\pi$
$384$		0			0
$385$			3.90985i			0.199264i
$386$		0			0
$387$	−20.5410			−1.04416
$388$		0			0
$389$			11.2500i			0.570398i	0.958468	+	0.285199i	$0.0920597\pi$
$389$			11.2500i			0.570398i	−0.958468	+	0.285199i	$0.907940\pi$
$390$		0			0
$391$			1.99779i			0.101033i
$392$		0			0
$393$			56.3814i			2.84407i
$394$		0			0
$395$	−16.0746			−0.808800
$396$		0			0
$397$		−	11.1600i		−	0.560102i	−0.959985	−	0.280051i	$-0.909649\pi$
$397$		−	11.1600i		−	0.560102i	0.959985	−	0.280051i	$-0.0903514\pi$
$398$		0			0
$399$	−2.77325			−0.138836
$400$		0			0
$401$		−	19.9036i		−	0.993940i	−0.867768	−	0.496970i	$-0.834446\pi$
$401$		−	19.9036i		−	0.993940i	0.867768	−	0.496970i	$-0.165554\pi$
$402$		0			0
$403$		−	23.9176i		−	1.19142i
$404$		0			0
$405$			37.9872i			1.88760i
$406$		0			0
$407$	8.16168			0.404559
$408$		0			0
$409$	23.9676			1.18512			0.592560	−	0.805526i	$-0.298117\pi$
$409$	23.9676			1.18512			0.592560	+	0.805526i	$0.298117\pi$
$410$		0			0
$411$	−9.06622			−0.447204
$412$		0			0
$413$	9.90446			0.487367
$414$		0			0
$415$	−32.8457			−1.61233
$416$		0			0
$417$	−36.2197			−1.77369
$418$		0			0
$419$		−	8.48249i		−	0.414397i	−0.978299	−	0.207198i	$-0.933565\pi$
$419$		−	8.48249i		−	0.414397i	0.978299	−	0.207198i	$-0.0664345\pi$
$420$		0			0
$421$			0.117834i			0.00574287i	0.999996	+	0.00287143i	$0.000914007\pi$
$421$			0.117834i			0.00574287i	−0.999996	+	0.00287143i	$0.999086\pi$
$422$		0			0
$423$	17.4476			0.848332
$424$		0			0
$425$	−83.5137			−4.05101
$426$		0			0
$427$		−	13.8165i		−	0.668628i
$428$		0			0
$429$			14.0239i			0.677078i
$430$		0			0
$431$	−2.88258			−0.138849			−0.0694246	−	0.997587i	$-0.522116\pi$
$431$	−2.88258			−0.138849			−0.0694246	+	0.997587i	$0.522116\pi$
$432$		0			0
$433$	21.4341			1.03006			0.515030	−	0.857172i	$-0.327781\pi$
$433$	21.4341			1.03006			0.515030	+	0.857172i	$0.327781\pi$
$434$		0			0
$435$	70.3824			3.37458
$436$		0			0
$437$	0.333262			0.0159421
$438$		0			0
$439$	31.5508			1.50584			0.752918	−	0.658114i	$-0.228646\pi$
$439$	31.5508			1.50584			0.752918	+	0.658114i	$0.228646\pi$
$440$		0			0
$441$	17.7102			0.843341
$442$		0			0
$443$			14.5530i			0.691436i	0.938339	+	0.345718i	$0.112365\pi$
$443$			14.5530i			0.691436i	−0.938339	+	0.345718i	$0.887635\pi$
$444$		0			0
$445$			17.1977i			0.815248i
$446$		0			0
$447$		−	43.3230i		−	2.04911i
$448$		0			0
$449$	6.98908			0.329835			0.164917	−	0.986307i	$-0.447264\pi$
$449$	6.98908			0.329835			0.164917	+	0.986307i	$0.447264\pi$
$450$		0			0
$451$		−	3.97185i		−	0.187027i
$452$		0			0
$453$	−40.9044			−1.92186
$454$		0			0
$455$		−	34.7725i		−	1.63016i
$456$		0			0
$457$		−	37.5696i		−	1.75743i	−0.477343	−	0.878717i	$-0.658400\pi$
$457$		−	37.5696i		−	1.75743i	0.477343	−	0.878717i	$-0.341600\pi$
$458$		0			0
$459$			1.29040i			0.0602305i
$460$		0			0
$461$	18.1987			0.847599			0.423799	−	0.905756i	$-0.360696\pi$
$461$	18.1987			0.847599			0.423799	+	0.905756i	$0.360696\pi$
$462$		0			0
$463$		−	23.9973i		−	1.11525i	−0.830094	−	0.557624i	$-0.811713\pi$
$463$		−	23.9973i		−	1.11525i	0.830094	−	0.557624i	$-0.188287\pi$
$464$		0			0
$465$			36.1121i			1.67466i
$466$		0			0
$467$	21.9992			1.01800			0.509001	−	0.860766i	$-0.330015\pi$
$467$	21.9992			1.01800			0.509001	+	0.860766i	$0.330015\pi$
$468$		0			0
$469$		−	11.8551i		−	0.547419i
$470$		0			0
$471$	34.6896			1.59841
$472$		0			0
$473$	−5.31913			−0.244574
$474$		0			0
$475$			13.9314i			0.639216i
$476$		0			0
$477$	9.81365	−	20.2198i	0.449336	−	0.925801i
$478$		0			0
$479$		−	6.19442i		−	0.283030i	−0.989936	−	0.141515i	$-0.954803\pi$
$479$		−	6.19442i		−	0.283030i	0.989936	−	0.141515i	$-0.0451974\pi$
$480$		0			0
$481$	−72.5865			−3.30966
$482$		0			0
$483$	0.924219			0.0420534
$484$		0			0
$485$			15.6846i			0.712200i
$486$		0			0
$487$	−28.5011			−1.29151			−0.645755	−	0.763545i	$-0.723457\pi$
$487$	−28.5011			−1.29151			−0.645755	+	0.763545i	$0.723457\pi$
$488$		0			0
$489$		−	31.0158i		−	1.40258i
$490$		0			0
$491$		−	26.3098i		−	1.18734i	−0.804708	−	0.593671i	$-0.797678\pi$
$491$		−	26.3098i		−	1.18734i	0.804708	−	0.593671i	$-0.202322\pi$
$492$		0			0
$493$	−39.3030			−1.77012
$494$		0			0
$495$		−	10.7387i		−	0.482670i
$496$		0			0
$497$			9.11067i			0.408669i
$498$		0			0
$499$			42.5145i			1.90321i	0.307321	+	0.951606i	$0.400568\pi$
$499$			42.5145i			1.90321i	−0.307321	+	0.951606i	$0.599432\pi$
$500$		0			0
$501$	35.7888			1.59892
$502$		0			0
$503$		−	13.8489i		−	0.617493i	−0.951144	−	0.308746i	$-0.900091\pi$
$503$		−	13.8489i		−	0.617493i	0.951144	−	0.308746i	$-0.0999094\pi$
$504$		0			0
$505$	−26.1228			−1.16245
$506$		0			0
$507$		−	92.6481i		−	4.11465i
$508$		0			0
$509$		−	17.7897i		−	0.788515i	−0.919000	−	0.394257i	$-0.871002\pi$
$509$		−	17.7897i		−	0.788515i	0.919000	−	0.394257i	$-0.128998\pi$
$510$		0			0
$511$			11.1323i			0.492464i
$512$		0			0
$513$	0.215258			0.00950388
$514$		0			0
$515$	−40.5632			−1.78743
$516$		0			0
$517$	4.51809			0.198705
$518$		0			0
$519$	27.0373			1.18681
$520$		0			0
$521$	39.3978			1.72605			0.863024	−	0.505163i	$-0.168568\pi$
$521$	39.3978			1.72605			0.863024	+	0.505163i	$0.168568\pi$
$522$		0			0
$523$	−30.8615			−1.34948			−0.674739	−	0.738057i	$-0.735744\pi$
$523$	−30.8615			−1.34948			−0.674739	+	0.738057i	$0.735744\pi$
$524$		0			0
$525$			38.6352i			1.68618i
$526$		0			0
$527$		−	20.1657i		−	0.878433i
$528$		0			0
$529$	22.8889			0.995171
$530$		0			0
$531$	−27.2034			−1.18053
$532$		0			0
$533$			35.3239i			1.53005i
$534$		0			0
$535$		−	68.5377i		−	2.96314i
$536$		0			0
$537$	−12.5102			−0.539853
$538$		0			0
$539$	4.58608			0.197536
$540$		0			0
$541$	−18.4291			−0.792331			−0.396165	−	0.918179i	$-0.629659\pi$
$541$	−18.4291			−0.792331			−0.396165	+	0.918179i	$0.629659\pi$
$542$		0			0
$543$	0.481588			0.0206669
$544$		0			0
$545$	−14.4240			−0.617856
$546$		0			0
$547$	−4.08161			−0.174517			−0.0872585	−	0.996186i	$-0.527811\pi$
$547$	−4.08161			−0.174517			−0.0872585	+	0.996186i	$0.527811\pi$
$548$		0			0
$549$			37.9482i			1.61959i
$550$		0			0
$551$			6.55636i			0.279310i
$552$		0			0
$553$		−	4.15267i		−	0.176589i
$554$		0			0
$555$	109.595			4.65205
$556$		0			0
$557$			23.5063i			0.995995i	0.867178	+	0.497998i	$0.165931\pi$
$557$			23.5063i			0.995995i	−0.867178	+	0.497998i	$0.834069\pi$
$558$		0			0
$559$	47.3061			2.00084
$560$		0			0
$561$			11.8240i			0.499209i
$562$		0			0
$563$		−	34.0375i		−	1.43451i	−0.696811	−	0.717255i	$-0.745398\pi$
$563$		−	34.0375i		−	1.43451i	0.696811	−	0.717255i	$-0.254602\pi$
$564$		0			0
$565$			17.4881i			0.735732i
$566$		0			0
$567$	−9.81352			−0.412129
$568$		0			0
$569$		−	11.8506i		−	0.496803i	−0.968657	−	0.248402i	$-0.920095\pi$
$569$		−	11.8506i		−	0.496803i	0.968657	−	0.248402i	$-0.0799053\pi$
$570$		0			0
$571$		−	11.7548i		−	0.491921i	−0.969280	−	0.245961i	$-0.920897\pi$
$571$		−	11.7548i		−	0.491921i	0.969280	−	0.245961i	$-0.0791034\pi$
$572$		0			0
$573$	−52.4377			−2.19061
$574$		0			0
$575$		−	4.64281i		−	0.193618i
$576$		0			0
$577$	3.13371			0.130458			0.0652291	−	0.997870i	$-0.479222\pi$
$577$	3.13371			0.130458			0.0652291	+	0.997870i	$0.479222\pi$
$578$		0			0
$579$	3.25262			0.135174
$580$		0			0
$581$		−	8.48528i		−	0.352029i
$582$		0			0
$583$	2.54126	−	5.23595i	0.105248	−	0.216851i
$584$		0			0
$585$			95.5057i			3.94867i
$586$		0			0
$587$	19.5255			0.805902			0.402951	−	0.915222i	$-0.367985\pi$
$587$	19.5255			0.805902			0.402951	+	0.915222i	$0.367985\pi$
$588$		0			0
$589$	−3.36396			−0.138609
$590$		0			0
$591$		−	41.5644i		−	1.70973i
$592$		0			0
$593$	−21.3680			−0.877478			−0.438739	−	0.898614i	$-0.644575\pi$
$593$	−21.3680			−0.877478			−0.438739	+	0.898614i	$0.644575\pi$
$594$		0			0
$595$		−	29.3179i		−	1.20192i
$596$		0			0
$597$		−	40.6759i		−	1.66475i
$598$		0			0
$599$	23.6978			0.968266			0.484133	−	0.874995i	$-0.339135\pi$
$599$	23.6978			0.968266			0.484133	+	0.874995i	$0.339135\pi$
$600$		0			0
$601$			41.7233i			1.70193i	0.525224	+	0.850964i	$0.323981\pi$
$601$			41.7233i			1.70193i	−0.525224	+	0.850964i	$0.676019\pi$
$602$		0			0
$603$			32.5611i			1.32599i
$604$		0			0
$605$			45.0804i			1.83278i
$606$		0			0
$607$	2.78387			0.112994			0.0564969	−	0.998403i	$-0.482007\pi$
$607$	2.78387			0.112994			0.0564969	+	0.998403i	$0.482007\pi$
$608$		0			0
$609$			18.1824i			0.736788i
$610$		0			0
$611$	−40.1820			−1.62559
$612$		0			0
$613$		−	41.8172i		−	1.68898i	−0.535571	−	0.844490i	$-0.679904\pi$
$613$		−	41.8172i		−	1.68898i	0.535571	−	0.844490i	$-0.320096\pi$
$614$		0			0
$615$		−	53.3339i		−	2.15063i
$616$		0			0
$617$		−	28.8234i		−	1.16038i	−0.814479	−	0.580192i	$-0.802977\pi$
$617$		−	28.8234i		−	1.16038i	0.814479	−	0.580192i	$-0.197023\pi$
$618$		0			0
$619$	4.62025			0.185704			0.0928518	−	0.995680i	$-0.470402\pi$
$619$	4.62025			0.185704			0.0928518	+	0.995680i	$0.470402\pi$
$620$		0			0
$621$	−0.0717374			−0.00287872
$622$		0			0
$623$	−4.44280			−0.177997
$624$		0			0
$625$	99.4266			3.97707
$626$		0			0
$627$	1.97243			0.0787711
$628$		0			0
$629$	−61.2001			−2.44021
$630$		0			0
$631$		−	18.2552i		−	0.726729i	−0.931647	−	0.363365i	$-0.881628\pi$
$631$		−	18.2552i		−	0.726729i	0.931647	−	0.363365i	$-0.118372\pi$
$632$		0			0
$633$		−	14.9194i		−	0.592991i
$634$		0			0
$635$	51.3912			2.03940
$636$		0			0
$637$	−40.7866			−1.61602
$638$		0			0
$639$		−	25.0232i		−	0.989903i
$640$		0			0
$641$			16.1655i			0.638500i	0.947670	+	0.319250i	$0.103431\pi$
$641$			16.1655i			0.638500i	−0.947670	+	0.319250i	$0.896569\pi$
$642$		0			0
$643$	24.9130			0.982474			0.491237	−	0.871026i	$-0.336545\pi$
$643$	24.9130			0.982474			0.491237	+	0.871026i	$0.336545\pi$
$644$		0			0
$645$	−71.4253			−2.81237
$646$		0			0
$647$	36.8797			1.44989			0.724945	−	0.688807i	$-0.241865\pi$
$647$	36.8797			1.44989			0.724945	+	0.688807i	$0.241865\pi$
$648$		0			0
$649$	−7.04438			−0.276516
$650$		0			0
$651$	−9.32909			−0.365636
$652$		0			0
$653$	−17.2267			−0.674132			−0.337066	−	0.941481i	$-0.609435\pi$
$653$	−17.2267			−0.674132			−0.337066	+	0.941481i	$0.609435\pi$
$654$		0			0
$655$			99.4299i			3.88505i
$656$		0			0
$657$		−	30.5758i		−	1.19288i
$658$		0			0
$659$		−	5.64099i		−	0.219742i	−0.993946	−	0.109871i	$-0.964956\pi$
$659$		−	5.64099i		−	0.219742i	0.993946	−	0.109871i	$-0.0350438\pi$
$660$		0			0
$661$	28.1239			1.09389			0.546947	−	0.837167i	$-0.315790\pi$
$661$	28.1239			1.09389			0.546947	+	0.837167i	$0.315790\pi$
$662$		0			0
$663$		−	105.158i		−	4.08398i
$664$		0			0
$665$	−4.89068			−0.189653
$666$		0			0
$667$		−	2.18499i		−	0.0846030i
$668$		0			0
$669$		−	34.4690i		−	1.33265i
$670$		0			0
$671$			9.82675i			0.379358i
$672$		0			0
$673$	−11.2664			−0.434287			−0.217143	−	0.976140i	$-0.569674\pi$
$673$	−11.2664			−0.434287			−0.217143	+	0.976140i	$0.569674\pi$
$674$		0			0
$675$		−	2.99884i		−	0.115426i
$676$		0			0
$677$		−	2.72673i		−	0.104797i	−0.998626	−	0.0523983i	$-0.983313\pi$
$677$		−	2.72673i		−	0.104797i	0.998626	−	0.0523983i	$-0.0166865\pi$
$678$		0			0
$679$	−4.05191			−0.155498
$680$		0			0
$681$		−	21.9606i		−	0.841530i
$682$		0			0
$683$	4.20070			0.160735			0.0803676	−	0.996765i	$-0.474391\pi$
$683$	4.20070			0.160735			0.0803676	+	0.996765i	$0.474391\pi$
$684$		0			0
$685$	−15.9885			−0.610888
$686$		0			0
$687$		−	3.09876i		−	0.118225i
$688$		0			0
$689$	−22.6009	+	46.5663i	−0.861026	+	1.77404i
$690$		0			0
$691$			18.3582i			0.698380i	0.937052	+	0.349190i	$0.113543\pi$
$691$			18.3582i			0.698380i	−0.937052	+	0.349190i	$0.886457\pi$
$692$		0			0
$693$	2.77421			0.105384
$694$		0			0
$695$	−63.8743			−2.42289
$696$		0			0
$697$			29.7828i			1.12810i
$698$		0			0
$699$	48.8597			1.84804
$700$		0			0
$701$		−	22.0729i		−	0.833681i	−0.908980	−	0.416841i	$-0.863137\pi$
$701$		−	22.0729i		−	0.833681i	0.908980	−	0.416841i	$-0.136863\pi$
$702$		0			0
$703$			10.2091i			0.385045i
$704$		0			0
$705$	60.6689			2.28492
$706$		0			0
$707$		−	6.74849i		−	0.253803i
$708$		0			0
$709$		−	10.9825i		−	0.412456i	−0.978504	−	0.206228i	$-0.933881\pi$
$709$		−	10.9825i		−	0.412456i	0.978504	−	0.206228i	$-0.0661189\pi$
$710$		0			0
$711$			11.4056i			0.427745i
$712$		0			0
$713$	1.12108			0.0419848
$714$		0			0
$715$			24.7314i			0.924901i
$716$		0			0
$717$	40.9743			1.53021
$718$		0			0
$719$			8.53125i			0.318162i	0.987266	+	0.159081i	$0.0508531\pi$
$719$			8.53125i			0.318162i	−0.987266	+	0.159081i	$0.949147\pi$
$720$		0			0
$721$		−	10.4790i		−	0.390258i
$722$		0			0
$723$			42.6160i			1.58491i
$724$		0			0
$725$	91.3392			3.39225
$726$		0			0
$727$	13.9667			0.517997			0.258999	−	0.965878i	$-0.416607\pi$
$727$	13.9667			0.517997			0.258999	+	0.965878i	$0.416607\pi$
$728$		0			0
$729$	28.5469			1.05729
$730$		0			0
$731$	39.8854			1.47521
$732$		0			0
$733$	−44.3695			−1.63883			−0.819413	−	0.573204i	$-0.805700\pi$
$733$	−44.3695			−1.63883			−0.819413	+	0.573204i	$0.805700\pi$
$734$		0			0
$735$	61.5819			2.27148
$736$		0			0
$737$			8.43176i			0.310588i
$738$		0			0
$739$			7.29746i			0.268441i	0.990951	+	0.134221i	$0.0428531\pi$
$739$			7.29746i			0.268441i	−0.990951	+	0.134221i	$0.957147\pi$
$740$		0			0
$741$	−17.5419			−0.644418
$742$		0			0
$743$	42.4881			1.55874			0.779369	−	0.626565i	$-0.215540\pi$
$743$	42.4881			1.55874			0.779369	+	0.626565i	$0.215540\pi$
$744$		0			0
$745$		−	76.4011i		−	2.79912i
$746$		0			0
$747$			23.3055i			0.852705i
$748$		0			0
$749$	17.7058			0.646957
$750$		0			0
$751$	−34.3091			−1.25196			−0.625979	−	0.779840i	$-0.715300\pi$
$751$	−34.3091			−1.25196			−0.625979	+	0.779840i	$0.715300\pi$
$752$		0			0
$753$	52.2408			1.90376
$754$		0			0
$755$	−72.1358			−2.62529
$756$		0			0
$757$	−22.5103			−0.818150			−0.409075	−	0.912501i	$-0.634149\pi$
$757$	−22.5103			−0.818150			−0.409075	+	0.912501i	$0.634149\pi$
$758$		0			0
$759$	−0.657335			−0.0238598
$760$		0			0
$761$		−	19.3050i		−	0.699806i	−0.936786	−	0.349903i	$-0.886215\pi$
$761$		−	19.3050i		−	0.699806i	0.936786	−	0.349903i	$-0.113785\pi$
$762$		0			0
$763$		−	3.72625i		−	0.134899i
$764$		0			0
$765$			80.5241i			2.91135i
$766$		0			0
$767$	62.6497			2.26215
$768$		0			0
$769$		−	6.48851i		−	0.233982i	−0.993133	−	0.116991i	$-0.962675\pi$
$769$		−	6.48851i		−	0.233982i	0.993133	−	0.116991i	$-0.0373248\pi$
$770$		0			0
$771$	10.9543			0.394509
$772$		0			0
$773$			37.2672i			1.34041i	0.742176	+	0.670205i	$0.233794\pi$
$773$			37.2672i			1.34041i	−0.742176	+	0.670205i	$0.766206\pi$
$774$		0			0
$775$			46.8646i			1.68343i
$776$		0			0
$777$			28.3125i			1.01570i
$778$		0			0
$779$	4.96823			0.178005
$780$		0			0
$781$		−	6.47981i		−	0.231866i
$782$		0			0
$783$		−	1.41131i		−	0.0504361i
$784$		0			0
$785$	61.1759			2.18346
$786$		0			0
$787$			39.8874i			1.42183i	0.703276	+	0.710917i	$0.251720\pi$
$787$			39.8874i			1.42183i	−0.703276	+	0.710917i	$0.748280\pi$
$788$		0			0
$789$	65.3956			2.32815
$790$		0			0
$791$	−4.51784			−0.160636
$792$		0			0
$793$		−	87.3950i		−	3.10349i
$794$		0			0
$795$	34.1241	−	70.3084i	1.21026	−	2.49358i
$796$		0			0
$797$			31.4702i			1.11473i	0.830267	+	0.557365i	$0.188188\pi$
$797$			31.4702i			1.11473i	−0.830267	+	0.557365i	$0.811812\pi$
$798$		0			0
$799$	−33.8788			−1.19855
$800$		0			0
$801$	12.2025			0.431155
$802$		0			0
$803$		−	7.91766i		−	0.279408i
$804$		0			0
$805$	1.62988			0.0574458
$806$		0			0
$807$		−	8.20395i		−	0.288793i
$808$		0			0
$809$			48.1518i			1.69293i	0.532446	+	0.846464i	$0.321273\pi$
$809$			48.1518i			1.69293i	−0.532446	+	0.846464i	$0.678727\pi$
$810$		0			0
$811$	−32.7113			−1.14865			−0.574324	−	0.818628i	$-0.694735\pi$
$811$	−32.7113			−1.14865			−0.574324	+	0.818628i	$0.694735\pi$
$812$		0			0
$813$		−	58.4875i		−	2.05125i
$814$		0			0
$815$		−	54.6971i		−	1.91596i
$816$		0			0
$817$		−	6.65351i		−	0.232777i
$818$		0			0
$819$	−24.6727			−0.862133
$820$		0			0
$821$			11.1657i			0.389684i	0.980835	+	0.194842i	$0.0624195\pi$
$821$			11.1657i			0.389684i	−0.980835	+	0.194842i	$0.937581\pi$
$822$		0			0
$823$	11.0893			0.386547			0.193274	−	0.981145i	$-0.438089\pi$
$823$	11.0893			0.386547			0.193274	+	0.981145i	$0.438089\pi$
$824$		0			0
$825$		−	27.4786i		−	0.956683i
$826$		0			0
$827$		−	20.7674i		−	0.722155i	−0.932536	−	0.361077i	$-0.882409\pi$
$827$		−	20.7674i		−	0.722155i	0.932536	−	0.361077i	$-0.117591\pi$
$828$		0			0
$829$		−	26.2672i		−	0.912299i	−0.889903	−	0.456150i	$-0.849228\pi$
$829$		−	26.2672i		−	0.912299i	0.889903	−	0.456150i	$-0.150772\pi$
$830$		0			0
$831$	−12.8583			−0.446051
$832$		0			0
$833$	−34.3886			−1.19149
$834$		0			0
$835$	63.1143			2.18416
$836$		0			0
$837$	0.724119			0.0250292
$838$		0			0
$839$	1.86134			0.0642606			0.0321303	−	0.999484i	$-0.489771\pi$
$839$	1.86134			0.0642606			0.0321303	+	0.999484i	$0.489771\pi$
$840$		0			0
$841$	13.9858			0.482269
$842$		0			0
$843$		−	23.7831i		−	0.819133i
$844$		0			0
$845$		−	163.387i		−	5.62068i
$846$		0			0
$847$	−11.6460			−0.400160
$848$		0			0
$849$	−66.6911			−2.28883
$850$		0			0
$851$		−	3.40232i		−	0.116630i
$852$		0			0
$853$			56.6400i			1.93932i	0.244466	+	0.969658i	$0.421387\pi$
$853$			56.6400i			1.93932i	−0.244466	+	0.969658i	$0.578613\pi$
$854$		0			0
$855$	13.4327			0.459388
$856$		0			0
$857$	4.63673			0.158388			0.0791939	−	0.996859i	$-0.474765\pi$
$857$	4.63673			0.158388			0.0791939	+	0.996859i	$0.474765\pi$
$858$		0			0
$859$	−21.8934			−0.746993			−0.373497	−	0.927632i	$-0.621841\pi$
$859$	−21.8934			−0.746993			−0.373497	+	0.927632i	$0.621841\pi$
$860$		0			0
$861$	13.7781			0.469558
$862$		0			0
$863$	45.9550			1.56433			0.782164	−	0.623073i	$-0.214116\pi$
$863$	45.9550			1.56433			0.782164	+	0.623073i	$0.214116\pi$
$864$		0			0
$865$	47.6809			1.62120
$866$		0			0
$867$		−	46.7189i		−	1.58666i
$868$		0			0
$869$			2.95351i			0.100191i
$870$		0			0
$871$		−	74.9885i		−	2.54089i
$872$		0			0
$873$	11.1289			0.376657
$874$		0			0
$875$			43.6806i			1.47667i
$876$		0			0
$877$	8.07009			0.272508			0.136254	−	0.990674i	$-0.456494\pi$
$877$	8.07009			0.272508			0.136254	+	0.990674i	$0.456494\pi$
$878$		0			0
$879$			34.7887i			1.17339i
$880$		0			0
$881$			10.3542i			0.348842i	0.984671	+	0.174421i	$0.0558054\pi$
$881$			10.3542i			0.348842i	−0.984671	+	0.174421i	$0.944195\pi$
$882$		0			0
$883$			19.8133i			0.666769i	0.942791	+	0.333385i	$0.108191\pi$
$883$			19.8133i			0.666769i	−0.942791	+	0.333385i	$0.891809\pi$
$884$		0			0
$885$	−94.5920			−3.17967
$886$		0			0
$887$		−	56.7227i		−	1.90456i	−0.305222	−	0.952281i	$-0.598731\pi$
$887$		−	56.7227i		−	1.90456i	0.305222	−	0.952281i	$-0.401269\pi$
$888$		0			0
$889$			13.2763i			0.445272i
$890$		0			0
$891$	6.97970			0.233829
$892$		0			0
$893$			5.65151i			0.189121i
$894$		0			0
$895$	−22.0619			−0.737449
$896$		0			0
$897$	5.84606			0.195194
$898$		0			0
$899$			22.0553i			0.735586i
$900$		0			0
$901$	−19.0556	+	39.2617i	−0.634834	+	1.30800i
$902$		0			0
$903$		−	18.4518i		−	0.614038i
$904$		0			0
$905$	0.849290			0.0282314
$906$		0			0
$907$	−3.25269			−0.108004			−0.0540020	−	0.998541i	$-0.517198\pi$
$907$	−3.25269			−0.108004			−0.0540020	+	0.998541i	$0.517198\pi$
$908$		0			0
$909$			18.5353i			0.614777i
$910$		0			0
$911$	−47.6023			−1.57713			−0.788567	−	0.614949i	$-0.789177\pi$
$911$	−47.6023			−1.57713			−0.788567	+	0.614949i	$0.789177\pi$
$912$		0			0
$913$			6.03501i			0.199730i
$914$		0			0
$915$			131.954i			4.36225i
$916$		0			0
$917$	−25.6864			−0.848241
$918$		0			0
$919$		−	35.1078i		−	1.15810i	−0.815293	−	0.579049i	$-0.803424\pi$
$919$		−	35.1078i		−	1.15810i	0.815293	−	0.579049i	$-0.196576\pi$
$920$		0			0
$921$		−	58.1467i		−	1.91600i
$922$		0			0
$923$			57.6287i			1.89687i
$924$		0			0
$925$	142.227			4.67641
$926$		0			0
$927$			28.7814i			0.945306i
$928$		0			0
$929$	2.66566			0.0874575			0.0437288	−	0.999043i	$-0.486076\pi$
$929$	2.66566			0.0874575			0.0437288	+	0.999043i	$0.486076\pi$
$930$		0			0
$931$			5.73655i			0.188008i
$932$		0			0
$933$		−	0.211195i		−	0.00691422i
$934$		0			0
$935$			20.8519i			0.681929i
$936$		0			0
$937$	−14.5138			−0.474145			−0.237073	−	0.971492i	$-0.576188\pi$
$937$	−14.5138			−0.474145			−0.237073	+	0.971492i	$0.576188\pi$
$938$		0			0
$939$	−69.2807			−2.26089
$940$		0			0
$941$	5.71991			0.186464			0.0932319	−	0.995644i	$-0.470280\pi$
$941$	5.71991			0.186464			0.0932319	+	0.995644i	$0.470280\pi$
$942$		0			0
$943$	−1.65573			−0.0539178
$944$		0			0
$945$	1.05276			0.0342463
$946$		0			0
$947$	−19.6649			−0.639024			−0.319512	−	0.947582i	$-0.603519\pi$
$947$	−19.6649			−0.639024			−0.319512	+	0.947582i	$0.603519\pi$
$948$		0			0
$949$			70.4163i			2.28581i
$950$		0			0
$951$			21.8486i			0.708489i
$952$		0			0
$953$	48.5107			1.57141			0.785707	−	0.618598i	$-0.212299\pi$
$953$	48.5107			1.57141			0.785707	+	0.618598i	$0.212299\pi$
$954$		0			0
$955$	−92.4749			−2.99242
$956$		0			0
$957$		−	12.9319i		−	0.418030i
$958$		0			0
$959$		−	4.13042i		−	0.133378i
$960$		0			0
$961$	19.6838			0.634961
$962$		0			0
$963$	−48.6306			−1.56710
$964$		0			0
$965$	5.73606			0.184650
$966$		0			0
$967$	32.8192			1.05540			0.527698	−	0.849432i	$-0.323055\pi$
$967$	32.8192			1.05540			0.527698	+	0.849432i	$0.323055\pi$
$968$		0			0
$969$	−14.7902			−0.475129
$970$		0			0
$971$	−5.21415			−0.167330			−0.0836651	−	0.996494i	$-0.526663\pi$
$971$	−5.21415			−0.167330			−0.0836651	+	0.996494i	$0.526663\pi$
$972$		0			0
$973$		−	16.5011i		−	0.529001i
$974$		0			0
$975$			244.383i			7.82653i
$976$		0			0
$977$			32.2134i			1.03060i	0.857010	+	0.515300i	$0.172319\pi$
$977$			32.2134i			1.03060i	−0.857010	+	0.515300i	$0.827681\pi$
$978$		0			0
$979$	3.15987			0.100990
$980$		0			0
$981$			10.2345i			0.326762i
$982$		0			0
$983$	−11.0382			−0.352063			−0.176032	−	0.984385i	$-0.556326\pi$
$983$	−11.0382			−0.352063			−0.176032	+	0.984385i	$0.556326\pi$
$984$		0			0
$985$		−	73.2997i		−	2.33552i
$986$		0			0
$987$			15.6730i			0.498879i
$988$		0			0
$989$			2.21736i			0.0705080i
$990$		0			0
$991$	−58.3737			−1.85430			−0.927151	−	0.374687i	$-0.877750\pi$
$991$	−58.3737			−1.85430			−0.927151	+	0.374687i	$0.877750\pi$
$992$		0			0
$993$			19.4721i			0.617930i
$994$		0			0
$995$		−	71.7329i		−	2.27409i
$996$		0			0
$997$	21.4790			0.680247			0.340123	−	0.940381i	$-0.389531\pi$
$997$	21.4790			0.680247			0.340123	+	0.940381i	$0.389531\pi$
$998$		0			0
$999$		−	2.19760i		−	0.0695290i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.12	✓	82
53.52	even	2	inner	4028.2.c.a.3497.71	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.12	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.71	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)

\(f(q)\)	\(=\)	\(q-2.46723i q^{3} -4.35102i q^{5} +1.12403 q^{7} -3.08725 q^{9} +O(q^{10})\)	\(q-2.46723i q^{3} -4.35102i q^{5} +1.12403 q^{7} -3.08725 q^{9} -0.799448 q^{11} +7.10995 q^{13} -10.7350 q^{15} +5.99464 q^{17} -1.00000i q^{19} -2.77325i q^{21} +0.333262i q^{23} -13.9314 q^{25} +0.215258i q^{27} -6.55636 q^{29} -3.36396i q^{31} +1.97243i q^{33} -4.89068i q^{35} -10.2091 q^{37} -17.5419i q^{39} +4.96823i q^{41} +6.65351 q^{43} +13.4327i q^{45} -5.65151 q^{47} -5.73655 q^{49} -14.7902i q^{51} +(-3.17877 + 6.54946i) q^{53} +3.47842i q^{55} -2.46723 q^{57} +8.81155 q^{59} -12.2919i q^{61} -3.47016 q^{63} -30.9356i q^{65} -10.5470i q^{67} +0.822236 q^{69} +8.10535i q^{71} +9.90391i q^{73} +34.3720i q^{75} -0.898605 q^{77} -3.69444i q^{79} -8.73065 q^{81} -7.54897i q^{83} -26.0828i q^{85} +16.1761i q^{87} -3.95256 q^{89} +7.99181 q^{91} -8.29967 q^{93} -4.35102 q^{95} -3.60480 q^{97} +2.46809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 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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