LMFDB - Embedded newform 4028.2.c.a.3497.17

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			3497.17
Character	$\chi$	$=$	4028.3497
Dual form			4028.2.c.a.3497.66

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19213i q^{3} +3.88947i q^{5} -3.90251 q^{7} -1.80545 q^{9} +O(q^{10})$	$q-2.19213i q^{3} +3.88947i q^{5} -3.90251 q^{7} -1.80545 q^{9} +0.589846 q^{11} -0.305058 q^{13} +8.52624 q^{15} -1.55787 q^{17} -1.00000i q^{19} +8.55483i q^{21} +6.85587i q^{23} -10.1280 q^{25} -2.61862i q^{27} -3.58876 q^{29} -5.43913i q^{31} -1.29302i q^{33} -15.1787i q^{35} +0.681176 q^{37} +0.668727i q^{39} +1.44568i q^{41} +0.617222 q^{43} -7.02224i q^{45} +6.33715 q^{47} +8.22961 q^{49} +3.41505i q^{51} +(6.55673 + 3.16375i) q^{53} +2.29419i q^{55} -2.19213 q^{57} -1.55005 q^{59} +10.8134i q^{61} +7.04578 q^{63} -1.18651i q^{65} -8.26551i q^{67} +15.0290 q^{69} -14.1586i q^{71} -0.943558i q^{73} +22.2019i q^{75} -2.30188 q^{77} -11.3958i q^{79} -11.1567 q^{81} -1.30974i q^{83} -6.05928i q^{85} +7.86704i q^{87} -3.73946 q^{89} +1.19049 q^{91} -11.9233 q^{93} +3.88947 q^{95} -3.22997 q^{97} -1.06494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	2.19213i		−	1.26563i	−0.774304	−	0.632814i	$-0.781900\pi$
$3$		−	2.19213i		−	1.26563i	0.774304	−	0.632814i	$-0.218100\pi$
$4$		0			0
$5$			3.88947i			1.73943i	0.493559	+	0.869713i	$0.335696\pi$
$5$			3.88947i			1.73943i	−0.493559	+	0.869713i	$0.664304\pi$
$6$		0			0
$7$	−3.90251			−1.47501			−0.737506	−	0.675341i	$-0.763996\pi$
$7$	−3.90251			−1.47501			−0.737506	+	0.675341i	$0.763996\pi$
$8$		0			0
$9$	−1.80545			−0.601816
$10$		0			0
$11$	0.589846			0.177845			0.0889227	−	0.996039i	$-0.471658\pi$
$11$	0.589846			0.177845			0.0889227	+	0.996039i	$0.471658\pi$
$12$		0			0
$13$	−0.305058			−0.0846078			−0.0423039	−	0.999105i	$-0.513470\pi$
$13$	−0.305058			−0.0846078			−0.0423039	+	0.999105i	$0.513470\pi$
$14$		0			0
$15$	8.52624			2.20147
$16$		0			0
$17$	−1.55787			−0.377838			−0.188919	−	0.981993i	$-0.560498\pi$
$17$	−1.55787			−0.377838			−0.188919	+	0.981993i	$0.560498\pi$
$18$		0			0
$19$		−	1.00000i		−	0.229416i
$19$		−	1.00000i		−	0.229416i
$20$		0			0
$21$			8.55483i			1.86682i
$22$		0			0
$23$			6.85587i			1.42955i	0.699355	+	0.714774i	$0.253470\pi$
$23$			6.85587i			1.42955i	−0.699355	+	0.714774i	$0.746530\pi$
$24$		0			0
$25$	−10.1280			−2.02560
$26$		0			0
$27$		−	2.61862i		−	0.503953i
$28$		0			0
$29$	−3.58876			−0.666416			−0.333208	−	0.942853i	$-0.608131\pi$
$29$	−3.58876			−0.666416			−0.333208	+	0.942853i	$0.608131\pi$
$30$		0			0
$31$		−	5.43913i		−	0.976897i	−0.872593	−	0.488449i	$-0.837563\pi$
$31$		−	5.43913i		−	0.976897i	0.872593	−	0.488449i	$-0.162437\pi$
$32$		0			0
$33$		−	1.29302i		−	0.225086i
$34$		0			0
$35$		−	15.1787i		−	2.56567i
$36$		0			0
$37$	0.681176			0.111985			0.0559923	−	0.998431i	$-0.482168\pi$
$37$	0.681176			0.111985			0.0559923	+	0.998431i	$0.482168\pi$
$38$		0			0
$39$			0.668727i			0.107082i
$40$		0			0
$41$			1.44568i			0.225777i	0.993608	+	0.112888i	$0.0360103\pi$
$41$			1.44568i			0.225777i	−0.993608	+	0.112888i	$0.963990\pi$
$42$		0			0
$43$	0.617222			0.0941255			0.0470628	−	0.998892i	$-0.485014\pi$
$43$	0.617222			0.0941255			0.0470628	+	0.998892i	$0.485014\pi$
$44$		0			0
$45$		−	7.02224i		−	1.04681i
$46$		0			0
$47$	6.33715			0.924368			0.462184	−	0.886784i	$-0.347066\pi$
$47$	6.33715			0.924368			0.462184	+	0.886784i	$0.347066\pi$
$48$		0			0
$49$	8.22961			1.17566
$50$		0			0
$51$			3.41505i			0.478203i
$52$		0			0
$53$	6.55673	+	3.16375i	0.900636	+	0.434574i
$53$	6.55673	+	3.16375i	0.900636	+	0.434574i
$54$		0			0
$55$			2.29419i			0.309349i
$56$		0			0
$57$	−2.19213			−0.290355
$58$		0			0
$59$	−1.55005			−0.201799			−0.100899	−	0.994897i	$-0.532172\pi$
$59$	−1.55005			−0.201799			−0.100899	+	0.994897i	$0.532172\pi$
$60$		0			0
$61$			10.8134i			1.38452i	0.721650	+	0.692258i	$0.243384\pi$
$61$			10.8134i			1.38452i	−0.721650	+	0.692258i	$0.756616\pi$
$62$		0			0
$63$	7.04578			0.887685
$64$		0			0
$65$		−	1.18651i		−	0.147169i
$66$		0			0
$67$		−	8.26551i		−	1.00979i	−0.863180	−	0.504896i	$-0.831531\pi$
$67$		−	8.26551i		−	1.00979i	0.863180	−	0.504896i	$-0.168469\pi$
$68$		0			0
$69$	15.0290			1.80928
$70$		0			0
$71$		−	14.1586i		−	1.68032i	−0.542340	−	0.840159i	$-0.682461\pi$
$71$		−	14.1586i		−	1.68032i	0.542340	−	0.840159i	$-0.317539\pi$
$72$		0			0
$73$		−	0.943558i		−	0.110435i	−0.998474	−	0.0552176i	$-0.982415\pi$
$73$		−	0.943558i		−	0.110435i	0.998474	−	0.0552176i	$-0.0175852\pi$
$74$		0			0
$75$			22.2019i			2.56366i
$76$		0			0
$77$	−2.30188			−0.262324
$78$		0			0
$79$		−	11.3958i		−	1.28213i	−0.767486	−	0.641066i	$-0.778493\pi$
$79$		−	11.3958i		−	1.28213i	0.767486	−	0.641066i	$-0.221507\pi$
$80$		0			0
$81$	−11.1567			−1.23963
$82$		0			0
$83$		−	1.30974i		−	0.143762i	−0.997413	−	0.0718811i	$-0.977100\pi$
$83$		−	1.30974i		−	0.143762i	0.997413	−	0.0718811i	$-0.0229002\pi$
$84$		0			0
$85$		−	6.05928i		−	0.657222i
$86$		0			0
$87$			7.86704i			0.843435i
$88$		0			0
$89$	−3.73946			−0.396382			−0.198191	−	0.980163i	$-0.563507\pi$
$89$	−3.73946			−0.396382			−0.198191	+	0.980163i	$0.563507\pi$
$90$		0			0
$91$	1.19049			0.124797
$92$		0			0
$93$	−11.9233			−1.23639
$94$		0			0
$95$	3.88947			0.399051
$96$		0			0
$97$	−3.22997			−0.327953			−0.163977	−	0.986464i	$-0.552432\pi$
$97$	−3.22997			−0.327953			−0.163977	+	0.986464i	$0.552432\pi$
$98$		0			0
$99$	−1.06494			−0.107030
$100$		0			0
$101$		−	9.91589i		−	0.986668i	−0.869840	−	0.493334i	$-0.835778\pi$
$101$		−	9.91589i		−	0.986668i	0.869840	−	0.493334i	$-0.164222\pi$
$102$		0			0
$103$		−	15.9915i		−	1.57569i	−0.615874	−	0.787845i	$-0.711197\pi$
$103$		−	15.9915i		−	1.57569i	0.615874	−	0.787845i	$-0.288803\pi$
$104$		0			0
$105$	−33.2738			−3.24719
$106$		0			0
$107$	11.0028			1.06368			0.531839	−	0.846845i	$-0.321501\pi$
$107$	11.0028			1.06368			0.531839	+	0.846845i	$0.321501\pi$
$108$		0			0
$109$		−	10.5483i		−	1.01035i	−0.863017	−	0.505174i	$-0.831428\pi$
$109$		−	10.5483i		−	1.01035i	0.863017	−	0.505174i	$-0.168572\pi$
$110$		0			0
$111$		−	1.49323i		−	0.141731i
$112$		0			0
$113$	8.60386			0.809383			0.404692	−	0.914453i	$-0.367379\pi$
$113$	8.60386			0.809383			0.404692	+	0.914453i	$0.367379\pi$
$114$		0			0
$115$	−26.6657			−2.48659
$116$		0			0
$117$	0.550766			0.0509183
$118$		0			0
$119$	6.07960			0.557316
$120$		0			0
$121$	−10.6521			−0.968371
$122$		0			0
$123$	3.16912			0.285750
$124$		0			0
$125$		−	19.9452i		−	1.78395i
$126$		0			0
$127$		−	12.4313i		−	1.10310i	−0.834142	−	0.551550i	$-0.814037\pi$
$127$		−	12.4313i		−	1.10310i	0.834142	−	0.551550i	$-0.185963\pi$
$128$		0			0
$129$		−	1.35303i		−	0.119128i
$130$		0			0
$131$	−1.46502			−0.127999			−0.0639997	−	0.997950i	$-0.520386\pi$
$131$	−1.46502			−0.127999			−0.0639997	+	0.997950i	$0.520386\pi$
$132$		0			0
$133$			3.90251i			0.338391i
$134$		0			0
$135$	10.1850			0.876589
$136$		0			0
$137$		−	13.2272i		−	1.13007i	−0.825066	−	0.565037i	$-0.808862\pi$
$137$		−	13.2272i		−	1.13007i	0.825066	−	0.565037i	$-0.191138\pi$
$138$		0			0
$139$			4.69461i			0.398192i	0.979980	+	0.199096i	$0.0638005\pi$
$139$			4.69461i			0.398192i	−0.979980	+	0.199096i	$0.936199\pi$
$140$		0			0
$141$		−	13.8919i		−	1.16991i
$142$		0			0
$143$	−0.179937			−0.0150471
$144$		0			0
$145$		−	13.9584i		−	1.15918i
$146$		0			0
$147$		−	18.0404i		−	1.48795i
$148$		0			0
$149$	−3.07616			−0.252009			−0.126005	−	0.992030i	$-0.540215\pi$
$149$	−3.07616			−0.252009			−0.126005	+	0.992030i	$0.540215\pi$
$150$		0			0
$151$			5.86802i			0.477533i	0.971077	+	0.238766i	$0.0767430\pi$
$151$			5.86802i			0.477533i	−0.971077	+	0.238766i	$0.923257\pi$
$152$		0			0
$153$	2.81265			0.227389
$154$		0			0
$155$	21.1554			1.69924
$156$		0			0
$157$		−	17.5472i		−	1.40042i	−0.713939	−	0.700208i	$-0.753091\pi$
$157$		−	17.5472i		−	1.40042i	0.713939	−	0.700208i	$-0.246909\pi$
$158$		0			0
$159$	6.93536	−	14.3732i	0.550010	−	1.13987i
$160$		0			0
$161$		−	26.7551i		−	2.10860i
$162$		0			0
$163$	−17.8867			−1.40100			−0.700499	−	0.713654i	$-0.747039\pi$
$163$	−17.8867			−1.40100			−0.700499	+	0.713654i	$0.747039\pi$
$164$		0			0
$165$	5.02917			0.391521
$166$		0			0
$167$			14.2851i			1.10542i	0.833375	+	0.552708i	$0.186406\pi$
$167$			14.2851i			1.10542i	−0.833375	+	0.552708i	$0.813594\pi$
$168$		0			0
$169$	−12.9069			−0.992842
$170$		0			0
$171$			1.80545i			0.138066i
$172$		0			0
$173$		−	13.0761i		−	0.994161i	−0.867704	−	0.497081i	$-0.834405\pi$
$173$		−	13.0761i		−	0.994161i	0.867704	−	0.497081i	$-0.165595\pi$
$174$		0			0
$175$	39.5246			2.98778
$176$		0			0
$177$			3.39791i			0.255402i
$178$		0			0
$179$			0.933524i			0.0697748i	0.999391	+	0.0348874i	$0.0111073\pi$
$179$			0.933524i			0.0697748i	−0.999391	+	0.0348874i	$0.988893\pi$
$180$		0			0
$181$		−	20.9403i		−	1.55648i	−0.627967	−	0.778240i	$-0.716113\pi$
$181$		−	20.9403i		−	1.55648i	0.627967	−	0.778240i	$-0.283887\pi$
$182$		0			0
$183$	23.7045			1.75228
$184$		0			0
$185$			2.64941i			0.194789i
$186$		0			0
$187$	−0.918903			−0.0671968
$188$		0			0
$189$			10.2192i			0.743337i
$190$		0			0
$191$			16.0073i			1.15824i	0.815241	+	0.579122i	$0.196605\pi$
$191$			16.0073i			1.15824i	−0.815241	+	0.579122i	$0.803395\pi$
$192$		0			0
$193$		−	2.83611i		−	0.204148i	−0.994777	−	0.102074i	$-0.967452\pi$
$193$		−	2.83611i		−	0.204148i	0.994777	−	0.102074i	$-0.0325478\pi$
$194$		0			0
$195$	−2.60100			−0.186261
$196$		0			0
$197$	16.0055			1.14034			0.570172	−	0.821525i	$-0.306876\pi$
$197$	16.0055			1.14034			0.570172	+	0.821525i	$0.306876\pi$
$198$		0			0
$199$	22.1568			1.57065			0.785327	−	0.619081i	$-0.212495\pi$
$199$	22.1568			1.57065			0.785327	+	0.619081i	$0.212495\pi$
$200$		0			0
$201$	−18.1191			−1.27802
$202$		0			0
$203$	14.0052			0.982971
$204$		0			0
$205$	−5.62292			−0.392722
$206$		0			0
$207$		−	12.3779i		−	0.860325i
$208$		0			0
$209$		−	0.589846i		−	0.0408005i
$210$		0			0
$211$	−11.8615			−0.816579			−0.408289	−	0.912853i	$-0.633875\pi$
$211$	−11.8615			−0.816579			−0.408289	+	0.912853i	$0.633875\pi$
$212$		0			0
$213$	−31.0376			−2.12666
$214$		0			0
$215$			2.40067i			0.163724i
$216$		0			0
$217$			21.2263i			1.44093i
$218$		0			0
$219$	−2.06841			−0.139770
$220$		0			0
$221$	0.475239			0.0319681
$222$		0			0
$223$	−1.58699			−0.106273			−0.0531363	−	0.998587i	$-0.516922\pi$
$223$	−1.58699			−0.106273			−0.0531363	+	0.998587i	$0.516922\pi$
$224$		0			0
$225$	18.2856			1.21904
$226$		0			0
$227$	−6.14089			−0.407585			−0.203793	−	0.979014i	$-0.565327\pi$
$227$	−6.14089			−0.407585			−0.203793	+	0.979014i	$0.565327\pi$
$228$		0			0
$229$	4.87084			0.321874			0.160937	−	0.986965i	$-0.448548\pi$
$229$	4.87084			0.321874			0.160937	+	0.986965i	$0.448548\pi$
$230$		0			0
$231$			5.04603i			0.332005i
$232$		0			0
$233$			12.4439i			0.815228i	0.913154	+	0.407614i	$0.133639\pi$
$233$			12.4439i			0.815228i	−0.913154	+	0.407614i	$0.866361\pi$
$234$		0			0
$235$			24.6482i			1.60787i
$236$		0			0
$237$	−24.9812			−1.62270
$238$		0			0
$239$			14.9919i			0.969743i	0.874585	+	0.484871i	$0.161134\pi$
$239$			14.9919i			0.969743i	−0.874585	+	0.484871i	$0.838866\pi$
$240$		0			0
$241$	−26.3395			−1.69668			−0.848339	−	0.529453i	$-0.822397\pi$
$241$	−26.3395			−1.69668			−0.848339	+	0.529453i	$0.822397\pi$
$242$		0			0
$243$			16.6011i			1.06496i
$244$		0			0
$245$			32.0088i			2.04497i
$246$		0			0
$247$			0.305058i			0.0194104i
$248$		0			0
$249$	−2.87112			−0.181950
$250$		0			0
$251$		−	24.6145i		−	1.55365i	−0.629716	−	0.776826i	$-0.716829\pi$
$251$		−	24.6145i		−	1.55365i	0.629716	−	0.776826i	$-0.283171\pi$
$252$		0			0
$253$			4.04391i			0.254239i
$254$		0			0
$255$	−13.2828			−0.831799
$256$		0			0
$257$		−	1.42854i		−	0.0891102i	−0.999007	−	0.0445551i	$-0.985813\pi$
$257$		−	1.42854i		−	0.0891102i	0.999007	−	0.0445551i	$-0.0141870\pi$
$258$		0			0
$259$	−2.65830			−0.165179
$260$		0			0
$261$	6.47932			0.401060
$262$		0			0
$263$		−	27.1245i		−	1.67257i	−0.548294	−	0.836286i	$-0.684723\pi$
$263$		−	27.1245i		−	1.67257i	0.548294	−	0.836286i	$-0.315277\pi$
$264$		0			0
$265$	−12.3053	+	25.5022i	−0.755910	+	1.56659i
$266$		0			0
$267$			8.19740i			0.501673i
$268$		0			0
$269$	8.01789			0.488859			0.244430	−	0.969667i	$-0.421399\pi$
$269$	8.01789			0.488859			0.244430	+	0.969667i	$0.421399\pi$
$270$		0			0
$271$	−2.02016			−0.122716			−0.0613581	−	0.998116i	$-0.519543\pi$
$271$	−2.02016			−0.122716			−0.0613581	+	0.998116i	$0.519543\pi$
$272$		0			0
$273$		−	2.60972i		−	0.157947i
$274$		0			0
$275$	−5.97396			−0.360244
$276$		0			0
$277$			20.1595i			1.21127i	0.795744	+	0.605633i	$0.207080\pi$
$277$			20.1595i			1.21127i	−0.795744	+	0.605633i	$0.792920\pi$
$278$		0			0
$279$			9.82007i			0.587912i
$280$		0			0
$281$	9.40390			0.560990			0.280495	−	0.959856i	$-0.409501\pi$
$281$	9.40390			0.560990			0.280495	+	0.959856i	$0.409501\pi$
$282$		0			0
$283$			17.8287i			1.05981i	0.848058	+	0.529904i	$0.177772\pi$
$283$			17.8287i			1.05981i	−0.848058	+	0.529904i	$0.822228\pi$
$284$		0			0
$285$		−	8.52624i		−	0.505051i
$286$		0			0
$287$		−	5.64178i		−	0.333024i
$288$		0			0
$289$	−14.5730			−0.857238
$290$		0			0
$291$			7.08052i			0.415067i
$292$		0			0
$293$	22.6467			1.32304			0.661518	−	0.749929i	$-0.269912\pi$
$293$	22.6467			1.32304			0.661518	+	0.749929i	$0.269912\pi$
$294$		0			0
$295$		−	6.02887i		−	0.351014i
$296$		0			0
$297$		−	1.54458i		−	0.0896258i
$298$		0			0
$299$		−	2.09144i		−	0.120951i
$300$		0			0
$301$	−2.40872			−0.138836
$302$		0			0
$303$	−21.7370			−1.24876
$304$		0			0
$305$	−42.0585			−2.40826
$306$		0			0
$307$	33.1610			1.89260			0.946300	−	0.323289i	$-0.104789\pi$
$307$	33.1610			1.89260			0.946300	+	0.323289i	$0.104789\pi$
$308$		0			0
$309$	−35.0555			−1.99424
$310$		0			0
$311$	−15.5106			−0.879524			−0.439762	−	0.898114i	$-0.644937\pi$
$311$	−15.5106			−0.879524			−0.439762	+	0.898114i	$0.644937\pi$
$312$		0			0
$313$		−	9.96113i		−	0.563036i	−0.959556	−	0.281518i	$-0.909162\pi$
$313$		−	9.96113i		−	0.563036i	0.959556	−	0.281518i	$-0.0908380\pi$
$314$		0			0
$315$			27.4044i			1.54406i
$316$		0			0
$317$	4.11660			0.231211			0.115606	−	0.993295i	$-0.463119\pi$
$317$	4.11660			0.231211			0.115606	+	0.993295i	$0.463119\pi$
$318$		0			0
$319$	−2.11682			−0.118519
$320$		0			0
$321$		−	24.1196i		−	1.34622i
$322$		0			0
$323$			1.55787i			0.0866821i
$324$		0			0
$325$	3.08962			0.171381
$326$		0			0
$327$	−23.1234			−1.27873
$328$		0			0
$329$	−24.7308			−1.36345
$330$		0			0
$331$	13.1412			0.722305			0.361152	−	0.932507i	$-0.382383\pi$
$331$	13.1412			0.722305			0.361152	+	0.932507i	$0.382383\pi$
$332$		0			0
$333$	−1.22983			−0.0673941
$334$		0			0
$335$	32.1485			1.75646
$336$		0			0
$337$		−	8.53780i		−	0.465083i	−0.972586	−	0.232542i	$-0.925296\pi$
$337$		−	8.53780i		−	0.465083i	0.972586	−	0.232542i	$-0.0747042\pi$
$338$		0			0
$339$		−	18.8608i		−	1.02438i
$340$		0			0
$341$		−	3.20825i		−	0.173737i
$342$		0			0
$343$	−4.79855			−0.259098
$344$		0			0
$345$			58.4548i			3.14710i
$346$		0			0
$347$	−7.64074			−0.410177			−0.205088	−	0.978743i	$-0.565748\pi$
$347$	−7.64074			−0.410177			−0.205088	+	0.978743i	$0.565748\pi$
$348$		0			0
$349$			26.3012i			1.40787i	0.710265	+	0.703934i	$0.248575\pi$
$349$			26.3012i			1.40787i	−0.710265	+	0.703934i	$0.751425\pi$
$350$		0			0
$351$			0.798829i			0.0426384i
$352$		0			0
$353$		−	25.1203i		−	1.33702i	−0.743703	−	0.668510i	$-0.766932\pi$
$353$		−	25.1203i		−	1.33702i	0.743703	−	0.668510i	$-0.233068\pi$
$354$		0			0
$355$	55.0695			2.92279
$356$		0			0
$357$		−	13.3273i		−	0.705355i
$358$		0			0
$359$		−	0.0877235i		−	0.00462987i	−0.999997	−	0.00231494i	$-0.999263\pi$
$359$		−	0.0877235i		−	0.00462987i	0.999997	−	0.00231494i	$-0.000736867\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$		0			0
$363$			23.3508i			1.22560i
$364$		0			0
$365$	3.66994			0.192094
$366$		0			0
$367$	−21.3396			−1.11392			−0.556960	−	0.830539i	$-0.688032\pi$
$367$	−21.3396			−1.11392			−0.556960	+	0.830539i	$0.688032\pi$
$368$		0			0
$369$		−	2.61010i		−	0.135876i
$370$		0			0
$371$	−25.5877	−	12.3466i	−1.32845	−	0.641002i
$372$		0			0
$373$		−	11.9089i		−	0.616621i	−0.951286	−	0.308310i	$-0.900236\pi$
$373$		−	11.9089i		−	0.616621i	0.951286	−	0.308310i	$-0.0997635\pi$
$374$		0			0
$375$	−43.7226			−2.25782
$376$		0			0
$377$	1.09478			0.0563839
$378$		0			0
$379$		−	31.0982i		−	1.59741i	−0.601723	−	0.798705i	$-0.705519\pi$
$379$		−	31.0982i		−	1.59741i	0.601723	−	0.798705i	$-0.294481\pi$
$380$		0			0
$381$	−27.2511			−1.39612
$382$		0			0
$383$			4.11291i			0.210160i	0.994464	+	0.105080i	$0.0335099\pi$
$383$			4.11291i			0.210160i	−0.994464	+	0.105080i	$0.966490\pi$
$384$		0			0
$385$		−	8.95311i		−	0.456293i
$386$		0			0
$387$	−1.11436			−0.0566462
$388$		0			0
$389$			6.47949i			0.328523i	0.986417	+	0.164262i	$0.0525241\pi$
$389$			6.47949i			0.328523i	−0.986417	+	0.164262i	$0.947476\pi$
$390$		0			0
$391$		−	10.6805i		−	0.540138i
$392$		0			0
$393$			3.21152i			0.162000i
$394$		0			0
$395$	44.3238			2.23017
$396$		0			0
$397$		−	4.25013i		−	0.213308i	−0.994296	−	0.106654i	$-0.965986\pi$
$397$		−	4.25013i		−	0.213308i	0.994296	−	0.106654i	$-0.0340137\pi$
$398$		0			0
$399$	8.55483			0.428277
$400$		0			0
$401$		−	7.79715i		−	0.389371i	−0.980866	−	0.194685i	$-0.937631\pi$
$401$		−	7.79715i		−	0.389371i	0.980866	−	0.194685i	$-0.0623686\pi$
$402$		0			0
$403$			1.65925i			0.0826531i
$404$		0			0
$405$		−	43.3937i		−	2.15625i
$406$		0			0
$407$	0.401789			0.0199159
$408$		0			0
$409$	0.444104			0.0219595			0.0109798	−	0.999940i	$-0.496505\pi$
$409$	0.444104			0.0219595			0.0109798	+	0.999940i	$0.496505\pi$
$410$		0			0
$411$	−28.9957			−1.43025
$412$		0			0
$413$	6.04908			0.297656
$414$		0			0
$415$	5.09418			0.250064
$416$		0			0
$417$	10.2912			0.503963
$418$		0			0
$419$			13.9760i			0.682772i	0.939923	+	0.341386i	$0.110896\pi$
$419$			13.9760i			0.682772i	−0.939923	+	0.341386i	$0.889104\pi$
$420$		0			0
$421$			11.5841i			0.564575i	0.959330	+	0.282287i	$0.0910931\pi$
$421$			11.5841i			0.564575i	−0.959330	+	0.282287i	$0.908907\pi$
$422$		0			0
$423$	−11.4414			−0.556299
$424$		0			0
$425$	15.7781			0.765350
$426$		0			0
$427$		−	42.1995i		−	2.04218i
$428$		0			0
$429$			0.394446i			0.0190440i
$430$		0			0
$431$	3.59227			0.173034			0.0865169	−	0.996250i	$-0.472426\pi$
$431$	3.59227			0.173034			0.0865169	+	0.996250i	$0.472426\pi$
$432$		0			0
$433$	−3.21288			−0.154401			−0.0772005	−	0.997016i	$-0.524598\pi$
$433$	−3.21288			−0.154401			−0.0772005	+	0.997016i	$0.524598\pi$
$434$		0			0
$435$	−30.5986			−1.46709
$436$		0			0
$437$	6.85587			0.327961
$438$		0			0
$439$	32.0108			1.52779			0.763896	−	0.645339i	$-0.223284\pi$
$439$	32.0108			1.52779			0.763896	+	0.645339i	$0.223284\pi$
$440$		0			0
$441$	−14.8581			−0.707530
$442$		0			0
$443$			17.4677i			0.829913i	0.909841	+	0.414957i	$0.136203\pi$
$443$			17.4677i			0.829913i	−0.909841	+	0.414957i	$0.863797\pi$
$444$		0			0
$445$		−	14.5445i		−	0.689477i
$446$		0			0
$447$			6.74336i			0.318950i
$448$		0			0
$449$	−23.7883			−1.12264			−0.561320	−	0.827599i	$-0.689706\pi$
$449$	−23.7883			−1.12264			−0.561320	+	0.827599i	$0.689706\pi$
$450$		0			0
$451$			0.852728i			0.0401534i
$452$		0			0
$453$	12.8635			0.604379
$454$		0			0
$455$			4.63038i			0.217076i
$456$		0			0
$457$			23.3763i			1.09350i	0.837297	+	0.546749i	$0.184135\pi$
$457$			23.3763i			1.09350i	−0.837297	+	0.546749i	$0.815865\pi$
$458$		0			0
$459$			4.07946i			0.190413i
$460$		0			0
$461$	−11.3681			−0.529466			−0.264733	−	0.964322i	$-0.585284\pi$
$461$	−11.3681			−0.529466			−0.264733	+	0.964322i	$0.585284\pi$
$462$		0			0
$463$		−	13.7873i		−	0.640752i	−0.947290	−	0.320376i	$-0.896191\pi$
$463$		−	13.7873i		−	0.640752i	0.947290	−	0.320376i	$-0.103809\pi$
$464$		0			0
$465$		−	46.3754i		−	2.15061i
$466$		0			0
$467$	0.764250			0.0353653			0.0176826	−	0.999844i	$-0.494371\pi$
$467$	0.764250			0.0353653			0.0176826	+	0.999844i	$0.494371\pi$
$468$		0			0
$469$			32.2563i			1.48946i
$470$		0			0
$471$	−38.4657			−1.77241
$472$		0			0
$473$	0.364066			0.0167398
$474$		0			0
$475$			10.1280i			0.464704i
$476$		0			0
$477$	−11.8378	−	5.71198i	−0.542017	−	0.261534i
$478$		0			0
$479$		−	2.13846i		−	0.0977088i	−0.998806	−	0.0488544i	$-0.984443\pi$
$479$		−	2.13846i		−	0.0977088i	0.998806	−	0.0488544i	$-0.0155570\pi$
$480$		0			0
$481$	−0.207798			−0.00947477
$482$		0			0
$483$	−58.6508			−2.66870
$484$		0			0
$485$		−	12.5629i		−	0.570450i
$486$		0			0
$487$	−14.2737			−0.646802			−0.323401	−	0.946262i	$-0.604826\pi$
$487$	−14.2737			−0.646802			−0.323401	+	0.946262i	$0.604826\pi$
$488$		0			0
$489$			39.2101i			1.77314i
$490$		0			0
$491$		−	33.4003i		−	1.50734i	−0.657255	−	0.753668i	$-0.728283\pi$
$491$		−	33.4003i		−	1.50734i	0.657255	−	0.753668i	$-0.271717\pi$
$492$		0			0
$493$	5.59081			0.251797
$494$		0			0
$495$		−	4.14204i		−	0.186171i
$496$		0			0
$497$			55.2542i			2.47849i
$498$		0			0
$499$			15.9555i			0.714268i	0.934053	+	0.357134i	$0.116246\pi$
$499$			15.9555i			0.714268i	−0.934053	+	0.357134i	$0.883754\pi$
$500$		0			0
$501$	31.3149			1.39905
$502$		0			0
$503$			18.2042i			0.811685i	0.913943	+	0.405843i	$0.133022\pi$
$503$			18.2042i			0.811685i	−0.913943	+	0.405843i	$0.866978\pi$
$504$		0			0
$505$	38.5676			1.71624
$506$		0			0
$507$			28.2937i			1.25657i
$508$		0			0
$509$			36.6831i			1.62595i	0.582297	+	0.812976i	$0.302154\pi$
$509$			36.6831i			1.62595i	−0.582297	+	0.812976i	$0.697846\pi$
$510$		0			0
$511$			3.68225i			0.162893i
$512$		0			0
$513$	−2.61862			−0.115615
$514$		0			0
$515$	62.1985			2.74079
$516$		0			0
$517$	3.73795			0.164395
$518$		0			0
$519$	−28.6647			−1.25824
$520$		0			0
$521$	−7.89431			−0.345856			−0.172928	−	0.984934i	$-0.555323\pi$
$521$	−7.89431			−0.345856			−0.172928	+	0.984934i	$0.555323\pi$
$522$		0			0
$523$	−34.1228			−1.49209			−0.746043	−	0.665897i	$-0.768049\pi$
$523$	−34.1228			−1.49209			−0.746043	+	0.665897i	$0.768049\pi$
$524$		0			0
$525$		−	86.6433i		−	3.78142i
$526$		0			0
$527$			8.47345i			0.369109i
$528$		0			0
$529$	−24.0030			−1.04361
$530$		0			0
$531$	2.79853			0.121446
$532$		0			0
$533$		−	0.441015i		−	0.0191025i
$534$		0			0
$535$			42.7950i			1.85019i
$536$		0			0
$537$	2.04641			0.0883090
$538$		0			0
$539$	4.85420			0.209085
$540$		0			0
$541$	10.1125			0.434769			0.217384	−	0.976086i	$-0.430248\pi$
$541$	10.1125			0.434769			0.217384	+	0.976086i	$0.430248\pi$
$542$		0			0
$543$	−45.9039			−1.96992
$544$		0			0
$545$	41.0275			1.75742
$546$		0			0
$547$	−32.5585			−1.39210			−0.696050	−	0.717993i	$-0.745061\pi$
$547$	−32.5585			−1.39210			−0.696050	+	0.717993i	$0.745061\pi$
$548$		0			0
$549$		−	19.5231i		−	0.833224i
$550$		0			0
$551$			3.58876i			0.152886i
$552$		0			0
$553$			44.4724i			1.89116i
$554$		0			0
$555$	5.80787			0.246530
$556$		0			0
$557$			10.1052i			0.428173i	0.976815	+	0.214087i	$0.0686775\pi$
$557$			10.1052i			0.428173i	−0.976815	+	0.214087i	$0.931323\pi$
$558$		0			0
$559$	−0.188288			−0.00796375
$560$		0			0
$561$			2.01436i			0.0850462i
$562$		0			0
$563$			16.4964i			0.695242i	0.937635	+	0.347621i	$0.113010\pi$
$563$			16.4964i			0.695242i	−0.937635	+	0.347621i	$0.886990\pi$
$564$		0			0
$565$			33.4645i			1.40786i
$566$		0			0
$567$	43.5392			1.82847
$568$		0			0
$569$		−	33.0419i		−	1.38519i	−0.721327	−	0.692595i	$-0.756467\pi$
$569$		−	33.0419i		−	1.38519i	0.721327	−	0.692595i	$-0.243533\pi$
$570$		0			0
$571$			7.29697i			0.305369i	0.988275	+	0.152684i	$0.0487918\pi$
$571$			7.29697i			0.305369i	−0.988275	+	0.152684i	$0.951208\pi$
$572$		0			0
$573$	35.0900			1.46591
$574$		0			0
$575$		−	69.4363i		−	2.89569i
$576$		0			0
$577$	22.7640			0.947679			0.473840	−	0.880611i	$-0.342868\pi$
$577$	22.7640			0.947679			0.473840	+	0.880611i	$0.342868\pi$
$578$		0			0
$579$	−6.21713			−0.258375
$580$		0			0
$581$			5.11126i			0.212051i
$582$		0			0
$583$	3.86746	+	1.86613i	0.160174	+	0.0772871i
$584$		0			0
$585$			2.14219i			0.0885686i
$586$		0			0
$587$	−2.26127			−0.0933324			−0.0466662	−	0.998911i	$-0.514860\pi$
$587$	−2.26127			−0.0933324			−0.0466662	+	0.998911i	$0.514860\pi$
$588$		0			0
$589$	−5.43913			−0.224116
$590$		0			0
$591$		−	35.0862i		−	1.44325i
$592$		0			0
$593$	−28.4207			−1.16710			−0.583549	−	0.812078i	$-0.698336\pi$
$593$	−28.4207			−1.16710			−0.583549	+	0.812078i	$0.698336\pi$
$594$		0			0
$595$			23.6464i			0.969409i
$596$		0			0
$597$		−	48.5707i		−	1.98787i
$598$		0			0
$599$	−23.6805			−0.967561			−0.483780	−	0.875190i	$-0.660737\pi$
$599$	−23.6805			−0.967561			−0.483780	+	0.875190i	$0.660737\pi$
$600$		0			0
$601$		−	20.8985i		−	0.852467i	−0.904613	−	0.426234i	$-0.859840\pi$
$601$		−	20.8985i		−	0.852467i	0.904613	−	0.426234i	$-0.140160\pi$
$602$		0			0
$603$			14.9229i			0.607709i
$604$		0			0
$605$		−	41.4310i		−	1.68441i
$606$		0			0
$607$	48.8968			1.98466			0.992330	−	0.123614i	$-0.0394486\pi$
$607$	48.8968			1.98466			0.992330	+	0.123614i	$0.0394486\pi$
$608$		0			0
$609$		−	30.7012i		−	1.24408i
$610$		0			0
$611$	−1.93320			−0.0782087
$612$		0			0
$613$			30.3114i			1.22426i	0.790755	+	0.612132i	$0.209688\pi$
$613$			30.3114i			1.22426i	−0.790755	+	0.612132i	$0.790312\pi$
$614$		0			0
$615$			12.3262i			0.497040i
$616$		0			0
$617$		−	20.5655i		−	0.827935i	−0.910292	−	0.413968i	$-0.864143\pi$
$617$		−	20.5655i		−	0.827935i	0.910292	−	0.413968i	$-0.135857\pi$
$618$		0			0
$619$	0.140772			0.00565810			0.00282905	−	0.999996i	$-0.499099\pi$
$619$	0.140772			0.00565810			0.00282905	+	0.999996i	$0.499099\pi$
$620$		0			0
$621$	17.9529			0.720425
$622$		0			0
$623$	14.5933			0.584668
$624$		0			0
$625$	26.9364			1.07745
$626$		0			0
$627$	−1.29302			−0.0516383
$628$		0			0
$629$	−1.06118			−0.0423121
$630$		0			0
$631$		−	20.8186i		−	0.828775i	−0.910101	−	0.414387i	$-0.863996\pi$
$631$		−	20.8186i		−	0.828775i	0.910101	−	0.414387i	$-0.136004\pi$
$632$		0			0
$633$			26.0020i			1.03349i
$634$		0			0
$635$	48.3512			1.91876
$636$		0			0
$637$	−2.51050			−0.0994698
$638$		0			0
$639$			25.5626i			1.01124i
$640$		0			0
$641$			11.5031i			0.454347i	0.973854	+	0.227173i	$0.0729484\pi$
$641$			11.5031i			0.454347i	−0.973854	+	0.227173i	$0.927052\pi$
$642$		0			0
$643$	−8.17717			−0.322476			−0.161238	−	0.986916i	$-0.551549\pi$
$643$	−8.17717			−0.322476			−0.161238	+	0.986916i	$0.551549\pi$
$644$		0			0
$645$	5.26259			0.207214
$646$		0			0
$647$	36.2458			1.42497			0.712485	−	0.701688i	$-0.247570\pi$
$647$	36.2458			1.42497			0.712485	+	0.701688i	$0.247570\pi$
$648$		0			0
$649$	−0.914290			−0.0358890
$650$		0			0
$651$	46.5309			1.82369
$652$		0			0
$653$	25.3797			0.993184			0.496592	−	0.867984i	$-0.334585\pi$
$653$	25.3797			0.993184			0.496592	+	0.867984i	$0.334585\pi$
$654$		0			0
$655$		−	5.69816i		−	0.222645i
$656$		0			0
$657$			1.70354i			0.0664616i
$658$		0			0
$659$			7.18003i			0.279694i	0.990173	+	0.139847i	$0.0446611\pi$
$659$			7.18003i			0.279694i	−0.990173	+	0.139847i	$0.955339\pi$
$660$		0			0
$661$	−26.3424			−1.02460			−0.512300	−	0.858807i	$-0.671206\pi$
$661$	−26.3424			−1.02460			−0.512300	+	0.858807i	$0.671206\pi$
$662$		0			0
$663$		−	1.04179i		−	0.0404597i
$664$		0			0
$665$	−15.1787			−0.588605
$666$		0			0
$667$		−	24.6041i		−	0.952673i
$668$		0			0
$669$			3.47889i			0.134502i
$670$		0			0
$671$			6.37826i			0.246230i
$672$		0			0
$673$	30.4050			1.17203			0.586014	−	0.810301i	$-0.300696\pi$
$673$	30.4050			1.17203			0.586014	+	0.810301i	$0.300696\pi$
$674$		0			0
$675$			26.5214i			1.02081i
$676$		0			0
$677$		−	5.53529i		−	0.212738i	−0.994327	−	0.106369i	$-0.966077\pi$
$677$		−	5.53529i		−	0.212738i	0.994327	−	0.106369i	$-0.0339225\pi$
$678$		0			0
$679$	12.6050			0.483735
$680$		0			0
$681$			13.4617i			0.515852i
$682$		0			0
$683$	−19.9242			−0.762380			−0.381190	−	0.924497i	$-0.624486\pi$
$683$	−19.9242			−0.762380			−0.381190	+	0.924497i	$0.624486\pi$
$684$		0			0
$685$	51.4467			1.96568
$686$		0			0
$687$		−	10.6775i		−	0.407373i
$688$		0			0
$689$	−2.00018	−	0.965126i	−0.0762008	−	0.0367684i
$690$		0			0
$691$		−	22.3157i		−	0.848929i	−0.905445	−	0.424465i	$-0.860462\pi$
$691$		−	22.3157i		−	0.848929i	0.905445	−	0.424465i	$-0.139538\pi$
$692$		0			0
$693$	4.15593			0.157871
$694$		0			0
$695$	−18.2596			−0.692625
$696$		0			0
$697$		−	2.25218i		−	0.0853072i
$698$		0			0
$699$	27.2787			1.03178
$700$		0			0
$701$		−	35.6102i		−	1.34498i	−0.740107	−	0.672489i	$-0.765225\pi$
$701$		−	35.6102i		−	1.34498i	0.740107	−	0.672489i	$-0.234775\pi$
$702$		0			0
$703$		−	0.681176i		−	0.0256910i
$704$		0			0
$705$	54.0321			2.03497
$706$		0			0
$707$			38.6969i			1.45535i
$708$		0			0
$709$			2.92195i			0.109736i	0.998494	+	0.0548681i	$0.0174738\pi$
$709$			2.92195i			0.109736i	−0.998494	+	0.0548681i	$0.982526\pi$
$710$		0			0
$711$			20.5746i			0.771607i
$712$		0			0
$713$	37.2900			1.39652
$714$		0			0
$715$		−	0.699861i		−	0.0261733i
$716$		0			0
$717$	32.8642			1.22733
$718$		0			0
$719$		−	16.0143i		−	0.597231i	−0.954373	−	0.298616i	$-0.903475\pi$
$719$		−	16.0143i		−	0.597231i	0.954373	−	0.298616i	$-0.0965249\pi$
$720$		0			0
$721$			62.4070i			2.32416i
$722$		0			0
$723$			57.7398i			2.14737i
$724$		0			0
$725$	36.3469			1.34989
$726$		0			0
$727$	1.22014			0.0452526			0.0226263	−	0.999744i	$-0.492797\pi$
$727$	1.22014			0.0452526			0.0226263	+	0.999744i	$0.492797\pi$
$728$		0			0
$729$	2.92176			0.108214
$730$		0			0
$731$	−0.961551			−0.0355642
$732$		0			0
$733$	3.37143			0.124527			0.0622633	−	0.998060i	$-0.480168\pi$
$733$	3.37143			0.124527			0.0622633	+	0.998060i	$0.480168\pi$
$734$		0			0
$735$	70.1676			2.58817
$736$		0			0
$737$		−	4.87538i		−	0.179587i
$738$		0			0
$739$		−	32.7320i		−	1.20406i	−0.798472	−	0.602032i	$-0.794358\pi$
$739$		−	32.7320i		−	1.20406i	0.798472	−	0.602032i	$-0.205642\pi$
$740$		0			0
$741$	0.668727			0.0245663
$742$		0			0
$743$	21.7457			0.797771			0.398886	−	0.917001i	$-0.369397\pi$
$743$	21.7457			0.797771			0.398886	+	0.917001i	$0.369397\pi$
$744$		0			0
$745$		−	11.9647i		−	0.438351i
$746$		0			0
$747$			2.36466i			0.0865184i
$748$		0			0
$749$	−42.9385			−1.56894
$750$		0			0
$751$	15.7685			0.575401			0.287700	−	0.957720i	$-0.407109\pi$
$751$	15.7685			0.575401			0.287700	+	0.957720i	$0.407109\pi$
$752$		0			0
$753$	−53.9582			−1.96635
$754$		0			0
$755$	−22.8235			−0.830632
$756$		0			0
$757$	−31.6342			−1.14977			−0.574883	−	0.818236i	$-0.694952\pi$
$757$	−31.6342			−1.14977			−0.574883	+	0.818236i	$0.694952\pi$
$758$		0			0
$759$	8.86479			0.321772
$760$		0			0
$761$			4.36559i			0.158253i	0.996865	+	0.0791263i	$0.0252130\pi$
$761$			4.36559i			0.158253i	−0.996865	+	0.0791263i	$0.974787\pi$
$762$		0			0
$763$			41.1650i			1.49027i
$764$		0			0
$765$			10.9397i			0.395526i
$766$		0			0
$767$	0.472854			0.0170738
$768$		0			0
$769$			7.62499i			0.274964i	0.990504	+	0.137482i	$0.0439010\pi$
$769$			7.62499i			0.274964i	−0.990504	+	0.137482i	$0.956099\pi$
$770$		0			0
$771$	−3.13156			−0.112780
$772$		0			0
$773$		−	9.03456i		−	0.324950i	−0.986713	−	0.162475i	$-0.948052\pi$
$773$		−	9.03456i		−	0.324950i	0.986713	−	0.162475i	$-0.0519478\pi$
$774$		0			0
$775$			55.0875i			1.97880i
$776$		0			0
$777$			5.82734i			0.209055i
$778$		0			0
$779$	1.44568			0.0517968
$780$		0			0
$781$		−	8.35141i		−	0.298837i
$782$		0			0
$783$			9.39759i			0.335842i
$784$		0			0
$785$	68.2492			2.43592
$786$		0			0
$787$		−	21.0982i		−	0.752070i	−0.926606	−	0.376035i	$-0.877287\pi$
$787$		−	21.0982i		−	0.752070i	0.926606	−	0.376035i	$-0.122713\pi$
$788$		0			0
$789$	−59.4606			−2.11685
$790$		0			0
$791$	−33.5767			−1.19385
$792$		0			0
$793$		−	3.29872i		−	0.117141i
$794$		0			0
$795$	55.9043	+	26.9749i	1.98272	+	0.956701i
$796$		0			0
$797$		−	45.1331i		−	1.59870i	−0.600867	−	0.799349i	$-0.705178\pi$
$797$		−	45.1331i		−	1.59870i	0.600867	−	0.799349i	$-0.294822\pi$
$798$		0			0
$799$	−9.87244			−0.349262
$800$		0			0
$801$	6.75140			0.238549
$802$		0			0
$803$		−	0.556554i		−	0.0196404i
$804$		0			0
$805$	104.063			3.66775
$806$		0			0
$807$		−	17.5763i		−	0.618714i
$808$		0			0
$809$			39.2867i			1.38125i	0.723215	+	0.690623i	$0.242663\pi$
$809$			39.2867i			1.38125i	−0.723215	+	0.690623i	$0.757337\pi$
$810$		0			0
$811$	5.82247			0.204455			0.102227	−	0.994761i	$-0.467403\pi$
$811$	5.82247			0.204455			0.102227	+	0.994761i	$0.467403\pi$
$812$		0			0
$813$			4.42847i			0.155313i
$814$		0			0
$815$		−	69.5700i		−	2.43693i
$816$		0			0
$817$		−	0.617222i		−	0.0215939i
$818$		0			0
$819$	−2.14937			−0.0751051
$820$		0			0
$821$			6.06719i			0.211746i	0.994380	+	0.105873i	$0.0337637\pi$
$821$			6.06719i			0.211746i	−0.994380	+	0.105873i	$0.966236\pi$
$822$		0			0
$823$	13.3562			0.465567			0.232783	−	0.972529i	$-0.425217\pi$
$823$	13.3562			0.465567			0.232783	+	0.972529i	$0.425217\pi$
$824$		0			0
$825$			13.0957i			0.455935i
$826$		0			0
$827$			0.532700i			0.0185238i	0.999957	+	0.00926189i	$0.00294819\pi$
$827$			0.532700i			0.0185238i	−0.999957	+	0.00926189i	$0.997052\pi$
$828$		0			0
$829$		−	26.1766i		−	0.909152i	−0.890708	−	0.454576i	$-0.849791\pi$
$829$		−	26.1766i		−	0.909152i	0.890708	−	0.454576i	$-0.150209\pi$
$830$		0			0
$831$	44.1923			1.53301
$832$		0			0
$833$	−12.8206			−0.444209
$834$		0			0
$835$	−55.5616			−1.92279
$836$		0			0
$837$	−14.2430			−0.492311
$838$		0			0
$839$	−40.1205			−1.38511			−0.692557	−	0.721363i	$-0.743516\pi$
$839$	−40.1205			−1.38511			−0.692557	+	0.721363i	$0.743516\pi$
$840$		0			0
$841$	−16.1208			−0.555890
$842$		0			0
$843$		−	20.6146i		−	0.710005i
$844$		0			0
$845$		−	50.2012i		−	1.72697i
$846$		0			0
$847$	41.5699			1.42836
$848$		0			0
$849$	39.0830			1.34132
$850$		0			0
$851$			4.67005i			0.160087i
$852$		0			0
$853$			39.7457i			1.36087i	0.732810	+	0.680433i	$0.238208\pi$
$853$			39.7457i			1.36087i	−0.732810	+	0.680433i	$0.761792\pi$
$854$		0			0
$855$	−7.02224			−0.240156
$856$		0			0
$857$	22.2110			0.758712			0.379356	−	0.925251i	$-0.376146\pi$
$857$	22.2110			0.758712			0.379356	+	0.925251i	$0.376146\pi$
$858$		0			0
$859$	34.3733			1.17280			0.586401	−	0.810021i	$-0.300544\pi$
$859$	34.3733			1.17280			0.586401	+	0.810021i	$0.300544\pi$
$860$		0			0
$861$	−12.3675			−0.421484
$862$		0			0
$863$	5.28095			0.179766			0.0898829	−	0.995952i	$-0.471351\pi$
$863$	5.28095			0.179766			0.0898829	+	0.995952i	$0.471351\pi$
$864$		0			0
$865$	50.8593			1.72927
$866$		0			0
$867$			31.9461i			1.08495i
$868$		0			0
$869$		−	6.72179i		−	0.228021i
$870$		0			0
$871$			2.52146i			0.0854363i
$872$		0			0
$873$	5.83153			0.197368
$874$		0			0
$875$			77.8365i			2.63135i
$876$		0			0
$877$	−18.9890			−0.641213			−0.320607	−	0.947212i	$-0.603887\pi$
$877$	−18.9890			−0.641213			−0.320607	+	0.947212i	$0.603887\pi$
$878$		0			0
$879$		−	49.6447i		−	1.67447i
$880$		0			0
$881$		−	40.9653i		−	1.38015i	−0.723736	−	0.690077i	$-0.757576\pi$
$881$		−	40.9653i		−	1.38015i	0.723736	−	0.690077i	$-0.242424\pi$
$882$		0			0
$883$		−	46.2617i		−	1.55683i	−0.627750	−	0.778415i	$-0.716024\pi$
$883$		−	46.2617i		−	1.55683i	0.627750	−	0.778415i	$-0.283976\pi$
$884$		0			0
$885$	−13.2161			−0.444254
$886$		0			0
$887$		−	7.18836i		−	0.241361i	−0.992691	−	0.120681i	$-0.961492\pi$
$887$		−	7.18836i		−	0.241361i	0.992691	−	0.120681i	$-0.0385077\pi$
$888$		0			0
$889$			48.5133i			1.62708i
$890$		0			0
$891$	−6.58074			−0.220463
$892$		0			0
$893$		−	6.33715i		−	0.212065i
$894$		0			0
$895$	−3.63092			−0.121368
$896$		0			0
$897$	−4.58471			−0.153079
$898$		0			0
$899$			19.5197i			0.651020i
$900$		0			0
$901$	−10.2145	−	4.92870i	−0.340295	−	0.164199i
$902$		0			0
$903$			5.28023i			0.175715i
$904$		0			0
$905$	81.4467			2.70738
$906$		0			0
$907$	−47.9780			−1.59308			−0.796542	−	0.604584i	$-0.793339\pi$
$907$	−47.9780			−1.59308			−0.796542	+	0.604584i	$0.793339\pi$
$908$		0			0
$909$			17.9026i			0.593793i
$910$		0			0
$911$	−26.2328			−0.869132			−0.434566	−	0.900640i	$-0.643098\pi$
$911$	−26.2328			−0.869132			−0.434566	+	0.900640i	$0.643098\pi$
$912$		0			0
$913$		−	0.772543i		−	0.0255675i
$914$		0			0
$915$			92.1979i			3.04797i
$916$		0			0
$917$	5.71726			0.188801
$918$		0			0
$919$			36.1021i			1.19090i	0.803393	+	0.595449i	$0.203026\pi$
$919$			36.1021i			1.19090i	−0.803393	+	0.595449i	$0.796974\pi$
$920$		0			0
$921$		−	72.6934i		−	2.39533i
$922$		0			0
$923$			4.31919i			0.142168i
$924$		0			0
$925$	−6.89895			−0.226836
$926$		0			0
$927$			28.8718i			0.948275i
$928$		0			0
$929$	8.41667			0.276142			0.138071	−	0.990422i	$-0.455910\pi$
$929$	8.41667			0.276142			0.138071	+	0.990422i	$0.455910\pi$
$930$		0			0
$931$		−	8.22961i		−	0.269714i
$932$		0			0
$933$			34.0012i			1.11315i
$934$		0			0
$935$		−	3.57405i		−	0.116884i
$936$		0			0
$937$	26.5700			0.868003			0.434001	−	0.900912i	$-0.357101\pi$
$937$	26.5700			0.868003			0.434001	+	0.900912i	$0.357101\pi$
$938$		0			0
$939$	−21.8361			−0.712595
$940$		0			0
$941$	−2.19990			−0.0717146			−0.0358573	−	0.999357i	$-0.511416\pi$
$941$	−2.19990			−0.0717146			−0.0358573	+	0.999357i	$0.511416\pi$
$942$		0			0
$943$	−9.91138			−0.322759
$944$		0			0
$945$	−39.7473			−1.29298
$946$		0			0
$947$	4.10001			0.133232			0.0666162	−	0.997779i	$-0.478780\pi$
$947$	4.10001			0.133232			0.0666162	+	0.997779i	$0.478780\pi$
$948$		0			0
$949$			0.287840i			0.00934367i
$950$		0			0
$951$		−	9.02414i		−	0.292628i
$952$		0			0
$953$	15.9095			0.515359			0.257680	−	0.966230i	$-0.417042\pi$
$953$	15.9095			0.515359			0.257680	+	0.966230i	$0.417042\pi$
$954$		0			0
$955$	−62.2598			−2.01468
$956$		0			0
$957$			4.64034i			0.150001i
$958$		0			0
$959$			51.6192i			1.66687i
$960$		0			0
$961$	1.41582			0.0456716
$962$		0			0
$963$	−19.8649			−0.640139
$964$		0			0
$965$	11.0310			0.355100
$966$		0			0
$967$	46.8814			1.50760			0.753802	−	0.657101i	$-0.228218\pi$
$967$	46.8814			1.50760			0.753802	+	0.657101i	$0.228218\pi$
$968$		0			0
$969$	3.41505			0.109707
$970$		0			0
$971$	−49.4213			−1.58601			−0.793003	−	0.609218i	$-0.791484\pi$
$971$	−49.4213			−1.58601			−0.793003	+	0.609218i	$0.791484\pi$
$972$		0			0
$973$		−	18.3208i		−	0.587337i
$974$		0			0
$975$		−	6.77287i		−	0.216905i
$976$		0			0
$977$			13.9017i			0.444755i	0.974961	+	0.222378i	$0.0713818\pi$
$977$			13.9017i			0.444755i	−0.974961	+	0.222378i	$0.928618\pi$
$978$		0			0
$979$	−2.20571			−0.0704947
$980$		0			0
$981$			19.0445i			0.608044i
$982$		0			0
$983$	−42.7713			−1.36419			−0.682096	−	0.731262i	$-0.738931\pi$
$983$	−42.7713			−1.36419			−0.682096	+	0.731262i	$0.738931\pi$
$984$		0			0
$985$			62.2530i			1.98354i
$986$		0			0
$987$			54.2132i			1.72563i
$988$		0			0
$989$			4.23160i			0.134557i
$990$		0			0
$991$	−42.0586			−1.33604			−0.668018	−	0.744145i	$-0.732857\pi$
$991$	−42.0586			−1.33604			−0.668018	+	0.744145i	$0.732857\pi$
$992$		0			0
$993$		−	28.8072i		−	0.914169i
$994$		0			0
$995$			86.1783i			2.73204i
$996$		0			0
$997$	−16.2751			−0.515436			−0.257718	−	0.966220i	$-0.582971\pi$
$997$	−16.2751			−0.515436			−0.257718	+	0.966220i	$0.582971\pi$
$998$		0			0
$999$		−	1.78374i		−	0.0564350i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.c.a.3497.17	✓	82
53.52	even	2	inner	4028.2.c.a.3497.66	yes	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.c.a.3497.17	✓	82	1.1	even	1	trivial
4028.2.c.a.3497.66	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)

\(f(q)\)	\(=\)	\(q-2.19213i q^{3} +3.88947i q^{5} -3.90251 q^{7} -1.80545 q^{9} +O(q^{10})\)	\(q-2.19213i q^{3} +3.88947i q^{5} -3.90251 q^{7} -1.80545 q^{9} +0.589846 q^{11} -0.305058 q^{13} +8.52624 q^{15} -1.55787 q^{17} -1.00000i q^{19} +8.55483i q^{21} +6.85587i q^{23} -10.1280 q^{25} -2.61862i q^{27} -3.58876 q^{29} -5.43913i q^{31} -1.29302i q^{33} -15.1787i q^{35} +0.681176 q^{37} +0.668727i q^{39} +1.44568i q^{41} +0.617222 q^{43} -7.02224i q^{45} +6.33715 q^{47} +8.22961 q^{49} +3.41505i q^{51} +(6.55673 + 3.16375i) q^{53} +2.29419i q^{55} -2.19213 q^{57} -1.55005 q^{59} +10.8134i q^{61} +7.04578 q^{63} -1.18651i q^{65} -8.26551i q^{67} +15.0290 q^{69} -14.1586i q^{71} -0.943558i q^{73} +22.2019i q^{75} -2.30188 q^{77} -11.3958i q^{79} -11.1567 q^{81} -1.30974i q^{83} -6.05928i q^{85} +7.86704i q^{87} -3.73946 q^{89} +1.19049 q^{91} -11.9233 q^{93} +3.88947 q^{95} -3.22997 q^{97} -1.06494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) 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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