LMFDB - Embedded newform 8004.2.a.d.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.a (trivial)

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:	yes
Analytic conductor:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 $ x^8 - 3x^7 - 8x^6 + 19x^5 + 19x^4 - 35x^3 - 10x^2 + 18*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.411238$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.71910 q^{5} +0.210007 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.71910 q^{5} +0.210007 q^{7} +1.00000 q^{9} -2.37289 q^{11} -3.22411 q^{13} +1.71910 q^{15} -1.22874 q^{17} -7.29204 q^{19} +0.210007 q^{21} +1.00000 q^{23} -2.04468 q^{25} +1.00000 q^{27} +1.00000 q^{29} +3.68744 q^{31} -2.37289 q^{33} +0.361024 q^{35} +7.65306 q^{37} -3.22411 q^{39} +2.82840 q^{41} +9.02791 q^{43} +1.71910 q^{45} -5.43500 q^{47} -6.95590 q^{49} -1.22874 q^{51} -2.49073 q^{53} -4.07924 q^{55} -7.29204 q^{57} -3.55638 q^{59} -7.27294 q^{61} +0.210007 q^{63} -5.54258 q^{65} +9.41000 q^{67} +1.00000 q^{69} -7.53494 q^{71} +3.75783 q^{73} -2.04468 q^{75} -0.498323 q^{77} +0.807221 q^{79} +1.00000 q^{81} +5.58455 q^{83} -2.11233 q^{85} +1.00000 q^{87} -11.2438 q^{89} -0.677086 q^{91} +3.68744 q^{93} -12.5358 q^{95} -11.5915 q^{97} -2.37289 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.71910	0.768807	0.384403	−	0.923165i	$-0.374407\pi$
$5$	1.71910	0.768807	0.384403	+	0.923165i	$0.374407\pi$
$6$	0	0
$7$	0.210007	0.0793752	0.0396876	−	0.999212i	$-0.487364\pi$
$7$	0.210007	0.0793752	0.0396876	+	0.999212i	$0.487364\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.37289	−0.715453	−0.357726	−	0.933826i	$-0.616448\pi$
$11$	−2.37289	−0.715453	−0.357726	+	0.933826i	$0.616448\pi$
$12$	0	0
$13$	−3.22411	−0.894207	−0.447104	−	0.894482i	$-0.647545\pi$
$13$	−3.22411	−0.894207	−0.447104	+	0.894482i	$0.647545\pi$
$14$	0	0
$15$	1.71910	0.443871
$16$	0	0
$17$	−1.22874	−0.298013	−0.149007	−	0.988836i	$-0.547608\pi$
$17$	−1.22874	−0.298013	−0.149007	+	0.988836i	$0.547608\pi$
$18$	0	0
$19$	−7.29204	−1.67291	−0.836454	−	0.548037i	$-0.815375\pi$
$19$	−7.29204	−1.67291	−0.836454	+	0.548037i	$0.815375\pi$
$20$	0	0
$21$	0.210007	0.0458273
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.04468	−0.408937
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	3.68744	0.662283	0.331142	−	0.943581i	$-0.392566\pi$
$31$	3.68744	0.662283	0.331142	+	0.943581i	$0.392566\pi$
$32$	0	0
$33$	−2.37289	−0.413067
$34$	0	0
$35$	0.361024	0.0610241
$36$	0	0
$37$	7.65306	1.25816	0.629078	−	0.777342i	$-0.283433\pi$
$37$	7.65306	1.25816	0.629078	+	0.777342i	$0.283433\pi$
$38$	0	0
$39$	−3.22411	−0.516271
$40$	0	0
$41$	2.82840	0.441722	0.220861	−	0.975305i	$-0.429113\pi$
$41$	2.82840	0.441722	0.220861	+	0.975305i	$0.429113\pi$
$42$	0	0
$43$	9.02791	1.37674	0.688372	−	0.725358i	$-0.258326\pi$
$43$	9.02791	1.37674	0.688372	+	0.725358i	$0.258326\pi$
$44$	0	0
$45$	1.71910	0.256269
$46$	0	0
$47$	−5.43500	−0.792776	−0.396388	−	0.918083i	$-0.629736\pi$
$47$	−5.43500	−0.792776	−0.396388	+	0.918083i	$0.629736\pi$
$48$	0	0
$49$	−6.95590	−0.993700
$50$	0	0
$51$	−1.22874	−0.172058
$52$	0	0
$53$	−2.49073	−0.342128	−0.171064	−	0.985260i	$-0.554720\pi$
$53$	−2.49073	−0.342128	−0.171064	+	0.985260i	$0.554720\pi$
$54$	0	0
$55$	−4.07924	−0.550045
$56$	0	0
$57$	−7.29204	−0.965854
$58$	0	0
$59$	−3.55638	−0.463002	−0.231501	−	0.972835i	$-0.574364\pi$
$59$	−3.55638	−0.463002	−0.231501	+	0.972835i	$0.574364\pi$
$60$	0	0
$61$	−7.27294	−0.931205	−0.465602	−	0.884994i	$-0.654162\pi$
$61$	−7.27294	−0.931205	−0.465602	+	0.884994i	$0.654162\pi$
$62$	0	0
$63$	0.210007	0.0264584
$64$	0	0
$65$	−5.54258	−0.687472
$66$	0	0
$67$	9.41000	1.14961	0.574807	−	0.818289i	$-0.305077\pi$
$67$	9.41000	1.14961	0.574807	+	0.818289i	$0.305077\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.53494	−0.894233	−0.447117	−	0.894476i	$-0.647549\pi$
$71$	−7.53494	−0.894233	−0.447117	+	0.894476i	$0.647549\pi$
$72$	0	0
$73$	3.75783	0.439820	0.219910	−	0.975520i	$-0.429424\pi$
$73$	3.75783	0.439820	0.219910	+	0.975520i	$0.429424\pi$
$74$	0	0
$75$	−2.04468	−0.236100
$76$	0	0
$77$	−0.498323	−0.0567892
$78$	0	0
$79$	0.807221	0.0908194	0.0454097	−	0.998968i	$-0.485541\pi$
$79$	0.807221	0.0908194	0.0454097	+	0.998968i	$0.485541\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.58455	0.612984	0.306492	−	0.951873i	$-0.400845\pi$
$83$	5.58455	0.612984	0.306492	+	0.951873i	$0.400845\pi$
$84$	0	0
$85$	−2.11233	−0.229114
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−11.2438	−1.19184	−0.595919	−	0.803044i	$-0.703212\pi$
$89$	−11.2438	−1.19184	−0.595919	+	0.803044i	$0.703212\pi$
$90$	0	0
$91$	−0.677086	−0.0709779
$92$	0	0
$93$	3.68744	0.382370
$94$	0	0
$95$	−12.5358	−1.28614
$96$	0	0
$97$	−11.5915	−1.17694	−0.588471	−	0.808519i	$-0.700270\pi$
$97$	−11.5915	−1.17694	−0.588471	+	0.808519i	$0.700270\pi$
$98$	0	0
$99$	−2.37289	−0.238484
$100$	0	0
$101$	−1.54449	−0.153683	−0.0768413	−	0.997043i	$-0.524483\pi$
$101$	−1.54449	−0.153683	−0.0768413	+	0.997043i	$0.524483\pi$
$102$	0	0
$103$	−14.7680	−1.45513	−0.727567	−	0.686037i	$-0.759349\pi$
$103$	−14.7680	−1.45513	−0.727567	+	0.686037i	$0.759349\pi$
$104$	0	0
$105$	0.361024	0.0352323
$106$	0	0
$107$	−12.0255	−1.16255	−0.581273	−	0.813708i	$-0.697445\pi$
$107$	−12.0255	−1.16255	−0.581273	+	0.813708i	$0.697445\pi$
$108$	0	0
$109$	−3.79224	−0.363230	−0.181615	−	0.983370i	$-0.558133\pi$
$109$	−3.79224	−0.363230	−0.181615	+	0.983370i	$0.558133\pi$
$110$	0	0
$111$	7.65306	0.726396
$112$	0	0
$113$	−15.1915	−1.42909	−0.714546	−	0.699588i	$-0.753367\pi$
$113$	−15.1915	−1.42909	−0.714546	+	0.699588i	$0.753367\pi$
$114$	0	0
$115$	1.71910	0.160307
$116$	0	0
$117$	−3.22411	−0.298069
$118$	0	0
$119$	−0.258044	−0.0236548
$120$	0	0
$121$	−5.36940	−0.488128
$122$	0	0
$123$	2.82840	0.255029
$124$	0	0
$125$	−12.1105	−1.08320
$126$	0	0
$127$	−13.6613	−1.21225	−0.606123	−	0.795371i	$-0.707276\pi$
$127$	−13.6613	−1.21225	−0.606123	+	0.795371i	$0.707276\pi$
$128$	0	0
$129$	9.02791	0.794863
$130$	0	0
$131$	0.920826	0.0804530	0.0402265	−	0.999191i	$-0.487192\pi$
$131$	0.920826	0.0804530	0.0402265	+	0.999191i	$0.487192\pi$
$132$	0	0
$133$	−1.53138	−0.132787
$134$	0	0
$135$	1.71910	0.147957
$136$	0	0
$137$	−1.57814	−0.134830	−0.0674148	−	0.997725i	$-0.521475\pi$
$137$	−1.57814	−0.134830	−0.0674148	+	0.997725i	$0.521475\pi$
$138$	0	0
$139$	7.64986	0.648852	0.324426	−	0.945911i	$-0.394829\pi$
$139$	7.64986	0.648852	0.324426	+	0.945911i	$0.394829\pi$
$140$	0	0
$141$	−5.43500	−0.457709
$142$	0	0
$143$	7.65045	0.639763
$144$	0	0
$145$	1.71910	0.142764
$146$	0	0
$147$	−6.95590	−0.573713
$148$	0	0
$149$	0.562591	0.0460893	0.0230446	−	0.999734i	$-0.492664\pi$
$149$	0.562591	0.0460893	0.0230446	+	0.999734i	$0.492664\pi$
$150$	0	0
$151$	−4.59101	−0.373611	−0.186806	−	0.982397i	$-0.559813\pi$
$151$	−4.59101	−0.373611	−0.186806	+	0.982397i	$0.559813\pi$
$152$	0	0
$153$	−1.22874	−0.0993377
$154$	0	0
$155$	6.33909	0.509168
$156$	0	0
$157$	−17.6359	−1.40750	−0.703750	−	0.710448i	$-0.748492\pi$
$157$	−17.6359	−1.40750	−0.703750	+	0.710448i	$0.748492\pi$
$158$	0	0
$159$	−2.49073	−0.197528
$160$	0	0
$161$	0.210007	0.0165509
$162$	0	0
$163$	−9.54677	−0.747761	−0.373880	−	0.927477i	$-0.621973\pi$
$163$	−9.54677	−0.747761	−0.373880	+	0.927477i	$0.621973\pi$
$164$	0	0
$165$	−4.07924	−0.317568
$166$	0	0
$167$	−18.8167	−1.45608	−0.728042	−	0.685533i	$-0.759569\pi$
$167$	−18.8167	−1.45608	−0.728042	+	0.685533i	$0.759569\pi$
$168$	0	0
$169$	−2.60511	−0.200393
$170$	0	0
$171$	−7.29204	−0.557636
$172$	0	0
$173$	0.0827733	0.00629314	0.00314657	−	0.999995i	$-0.498998\pi$
$173$	0.0827733	0.00629314	0.00314657	+	0.999995i	$0.498998\pi$
$174$	0	0
$175$	−0.429398	−0.0324594
$176$	0	0
$177$	−3.55638	−0.267314
$178$	0	0
$179$	9.20992	0.688382	0.344191	−	0.938900i	$-0.388153\pi$
$179$	9.20992	0.688382	0.344191	+	0.938900i	$0.388153\pi$
$180$	0	0
$181$	−9.17306	−0.681828	−0.340914	−	0.940094i	$-0.610736\pi$
$181$	−9.17306	−0.681828	−0.340914	+	0.940094i	$0.610736\pi$
$182$	0	0
$183$	−7.27294	−0.537631
$184$	0	0
$185$	13.1564	0.967278
$186$	0	0
$187$	2.91566	0.213214
$188$	0	0
$189$	0.210007	0.0152758
$190$	0	0
$191$	−1.71415	−0.124031	−0.0620156	−	0.998075i	$-0.519753\pi$
$191$	−1.71415	−0.124031	−0.0620156	+	0.998075i	$0.519753\pi$
$192$	0	0
$193$	3.98986	0.287196	0.143598	−	0.989636i	$-0.454133\pi$
$193$	3.98986	0.287196	0.143598	+	0.989636i	$0.454133\pi$
$194$	0	0
$195$	−5.54258	−0.396912
$196$	0	0
$197$	−3.34272	−0.238159	−0.119080	−	0.992885i	$-0.537994\pi$
$197$	−3.34272	−0.238159	−0.119080	+	0.992885i	$0.537994\pi$
$198$	0	0
$199$	18.4507	1.30793	0.653967	−	0.756523i	$-0.273104\pi$
$199$	18.4507	1.30793	0.653967	+	0.756523i	$0.273104\pi$
$200$	0	0
$201$	9.41000	0.663730
$202$	0	0
$203$	0.210007	0.0147396
$204$	0	0
$205$	4.86232	0.339599
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	17.3032	1.19689
$210$	0	0
$211$	20.7086	1.42564	0.712818	−	0.701349i	$-0.247418\pi$
$211$	20.7086	1.42564	0.712818	+	0.701349i	$0.247418\pi$
$212$	0	0
$213$	−7.53494	−0.516286
$214$	0	0
$215$	15.5199	1.05845
$216$	0	0
$217$	0.774388	0.0525689
$218$	0	0
$219$	3.75783	0.253930
$220$	0	0
$221$	3.96159	0.266486
$222$	0	0
$223$	−12.1547	−0.813936	−0.406968	−	0.913442i	$-0.633414\pi$
$223$	−12.1547	−0.813936	−0.406968	+	0.913442i	$0.633414\pi$
$224$	0	0
$225$	−2.04468	−0.136312
$226$	0	0
$227$	12.9879	0.862035	0.431017	−	0.902344i	$-0.358155\pi$
$227$	12.9879	0.862035	0.431017	+	0.902344i	$0.358155\pi$
$228$	0	0
$229$	9.91961	0.655506	0.327753	−	0.944763i	$-0.393709\pi$
$229$	9.91961	0.655506	0.327753	+	0.944763i	$0.393709\pi$
$230$	0	0
$231$	−0.498323	−0.0327872
$232$	0	0
$233$	13.0401	0.854285	0.427143	−	0.904184i	$-0.359520\pi$
$233$	13.0401	0.854285	0.427143	+	0.904184i	$0.359520\pi$
$234$	0	0
$235$	−9.34332	−0.609491
$236$	0	0
$237$	0.807221	0.0524346
$238$	0	0
$239$	−4.25253	−0.275073	−0.137537	−	0.990497i	$-0.543918\pi$
$239$	−4.25253	−0.275073	−0.137537	+	0.990497i	$0.543918\pi$
$240$	0	0
$241$	−1.22996	−0.0792289	−0.0396144	−	0.999215i	$-0.512613\pi$
$241$	−1.22996	−0.0792289	−0.0396144	+	0.999215i	$0.512613\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−11.9579	−0.763963
$246$	0	0
$247$	23.5103	1.49593
$248$	0	0
$249$	5.58455	0.353907
$250$	0	0
$251$	12.4225	0.784100	0.392050	−	0.919944i	$-0.371766\pi$
$251$	12.4225	0.784100	0.392050	+	0.919944i	$0.371766\pi$
$252$	0	0
$253$	−2.37289	−0.149182
$254$	0	0
$255$	−2.11233	−0.132279
$256$	0	0
$257$	3.16912	0.197684	0.0988422	−	0.995103i	$-0.468486\pi$
$257$	3.16912	0.197684	0.0988422	+	0.995103i	$0.468486\pi$
$258$	0	0
$259$	1.60720	0.0998663
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−11.4038	−0.703190	−0.351595	−	0.936152i	$-0.614361\pi$
$263$	−11.4038	−0.703190	−0.351595	+	0.936152i	$0.614361\pi$
$264$	0	0
$265$	−4.28182	−0.263030
$266$	0	0
$267$	−11.2438	−0.688108
$268$	0	0
$269$	3.50827	0.213903	0.106952	−	0.994264i	$-0.465891\pi$
$269$	3.50827	0.213903	0.106952	+	0.994264i	$0.465891\pi$
$270$	0	0
$271$	2.74651	0.166839	0.0834193	−	0.996515i	$-0.473416\pi$
$271$	2.74651	0.166839	0.0834193	+	0.996515i	$0.473416\pi$
$272$	0	0
$273$	−0.677086	−0.0409791
$274$	0	0
$275$	4.85180	0.292575
$276$	0	0
$277$	−26.2225	−1.57556	−0.787780	−	0.615957i	$-0.788769\pi$
$277$	−26.2225	−1.57556	−0.787780	+	0.615957i	$0.788769\pi$
$278$	0	0
$279$	3.68744	0.220761
$280$	0	0
$281$	13.9533	0.832382	0.416191	−	0.909277i	$-0.363365\pi$
$281$	13.9533	0.832382	0.416191	+	0.909277i	$0.363365\pi$
$282$	0	0
$283$	0.0948723	0.00563957	0.00281979	−	0.999996i	$-0.499102\pi$
$283$	0.0948723	0.00563957	0.00281979	+	0.999996i	$0.499102\pi$
$284$	0	0
$285$	−12.5358	−0.742555
$286$	0	0
$287$	0.593984	0.0350618
$288$	0	0
$289$	−15.4902	−0.911188
$290$	0	0
$291$	−11.5915	−0.679507
$292$	0	0
$293$	−19.3283	−1.12917	−0.564586	−	0.825374i	$-0.690964\pi$
$293$	−19.3283	−1.12917	−0.564586	+	0.825374i	$0.690964\pi$
$294$	0	0
$295$	−6.11379	−0.355959
$296$	0	0
$297$	−2.37289	−0.137689
$298$	0	0
$299$	−3.22411	−0.186455
$300$	0	0
$301$	1.89592	0.109279
$302$	0	0
$303$	−1.54449	−0.0887287
$304$	0	0
$305$	−12.5029	−0.715916
$306$	0	0
$307$	−27.8115	−1.58729	−0.793644	−	0.608382i	$-0.791819\pi$
$307$	−27.8115	−1.58729	−0.793644	+	0.608382i	$0.791819\pi$
$308$	0	0
$309$	−14.7680	−0.840122
$310$	0	0
$311$	4.91082	0.278467	0.139234	−	0.990260i	$-0.455536\pi$
$311$	4.91082	0.278467	0.139234	+	0.990260i	$0.455536\pi$
$312$	0	0
$313$	3.32210	0.187776	0.0938881	−	0.995583i	$-0.470070\pi$
$313$	3.32210	0.187776	0.0938881	+	0.995583i	$0.470070\pi$
$314$	0	0
$315$	0.361024	0.0203414
$316$	0	0
$317$	11.0295	0.619477	0.309738	−	0.950822i	$-0.399759\pi$
$317$	11.0295	0.619477	0.309738	+	0.950822i	$0.399759\pi$
$318$	0	0
$319$	−2.37289	−0.132856
$320$	0	0
$321$	−12.0255	−0.671197
$322$	0	0
$323$	8.96002	0.498549
$324$	0	0
$325$	6.59228	0.365674
$326$	0	0
$327$	−3.79224	−0.209711
$328$	0	0
$329$	−1.14139	−0.0629267
$330$	0	0
$331$	−14.1235	−0.776295	−0.388148	−	0.921597i	$-0.626885\pi$
$331$	−14.1235	−0.776295	−0.388148	+	0.921597i	$0.626885\pi$
$332$	0	0
$333$	7.65306	0.419385
$334$	0	0
$335$	16.1768	0.883831
$336$	0	0
$337$	−27.3246	−1.48846	−0.744232	−	0.667921i	$-0.767184\pi$
$337$	−27.3246	−1.48846	−0.744232	+	0.667921i	$0.767184\pi$
$338$	0	0
$339$	−15.1915	−0.825087
$340$	0	0
$341$	−8.74988	−0.473832
$342$	0	0
$343$	−2.93084	−0.158250
$344$	0	0
$345$	1.71910	0.0925534
$346$	0	0
$347$	−11.1666	−0.599456	−0.299728	−	0.954025i	$-0.596896\pi$
$347$	−11.1666	−0.599456	−0.299728	+	0.954025i	$0.596896\pi$
$348$	0	0
$349$	16.7330	0.895697	0.447849	−	0.894109i	$-0.352190\pi$
$349$	16.7330	0.895697	0.447849	+	0.894109i	$0.352190\pi$
$350$	0	0
$351$	−3.22411	−0.172090
$352$	0	0
$353$	28.3617	1.50954	0.754771	−	0.655988i	$-0.227748\pi$
$353$	28.3617	1.50954	0.754771	+	0.655988i	$0.227748\pi$
$354$	0	0
$355$	−12.9533	−0.687492
$356$	0	0
$357$	−0.258044	−0.0136571
$358$	0	0
$359$	26.6653	1.40734	0.703670	−	0.710527i	$-0.251543\pi$
$359$	26.6653	1.40734	0.703670	+	0.710527i	$0.251543\pi$
$360$	0	0
$361$	34.1738	1.79862
$362$	0	0
$363$	−5.36940	−0.281821
$364$	0	0
$365$	6.46009	0.338137
$366$	0	0
$367$	23.0394	1.20265	0.601323	−	0.799006i	$-0.294641\pi$
$367$	23.0394	1.20265	0.601323	+	0.799006i	$0.294641\pi$
$368$	0	0
$369$	2.82840	0.147241
$370$	0	0
$371$	−0.523070	−0.0271565
$372$	0	0
$373$	19.9479	1.03286	0.516432	−	0.856328i	$-0.327260\pi$
$373$	19.9479	1.03286	0.516432	+	0.856328i	$0.327260\pi$
$374$	0	0
$375$	−12.1105	−0.625386
$376$	0	0
$377$	−3.22411	−0.166050
$378$	0	0
$379$	−5.94273	−0.305257	−0.152629	−	0.988284i	$-0.548774\pi$
$379$	−5.94273	−0.305257	−0.152629	+	0.988284i	$0.548774\pi$
$380$	0	0
$381$	−13.6613	−0.699891
$382$	0	0
$383$	17.1706	0.877377	0.438688	−	0.898639i	$-0.355443\pi$
$383$	17.1706	0.877377	0.438688	+	0.898639i	$0.355443\pi$
$384$	0	0
$385$	−0.856669	−0.0436599
$386$	0	0
$387$	9.02791	0.458914
$388$	0	0
$389$	−9.89922	−0.501910	−0.250955	−	0.967999i	$-0.580745\pi$
$389$	−9.89922	−0.501910	−0.250955	+	0.967999i	$0.580745\pi$
$390$	0	0
$391$	−1.22874	−0.0621400
$392$	0	0
$393$	0.920826	0.0464496
$394$	0	0
$395$	1.38770	0.0698226
$396$	0	0
$397$	−6.53349	−0.327906	−0.163953	−	0.986468i	$-0.552425\pi$
$397$	−6.53349	−0.327906	−0.163953	+	0.986468i	$0.552425\pi$
$398$	0	0
$399$	−1.53138	−0.0766648
$400$	0	0
$401$	−28.3783	−1.41715	−0.708573	−	0.705638i	$-0.750661\pi$
$401$	−28.3783	−1.41715	−0.708573	+	0.705638i	$0.750661\pi$
$402$	0	0
$403$	−11.8887	−0.592219
$404$	0	0
$405$	1.71910	0.0854229
$406$	0	0
$407$	−18.1599	−0.900151
$408$	0	0
$409$	−11.9632	−0.591543	−0.295771	−	0.955259i	$-0.595577\pi$
$409$	−11.9632	−0.591543	−0.295771	+	0.955259i	$0.595577\pi$
$410$	0	0
$411$	−1.57814	−0.0778439
$412$	0	0
$413$	−0.746865	−0.0367508
$414$	0	0
$415$	9.60042	0.471266
$416$	0	0
$417$	7.64986	0.374615
$418$	0	0
$419$	25.4630	1.24395	0.621974	−	0.783038i	$-0.286331\pi$
$419$	25.4630	1.24395	0.621974	+	0.783038i	$0.286331\pi$
$420$	0	0
$421$	−26.3023	−1.28190	−0.640949	−	0.767584i	$-0.721459\pi$
$421$	−26.3023	−1.28190	−0.640949	+	0.767584i	$0.721459\pi$
$422$	0	0
$423$	−5.43500	−0.264259
$424$	0	0
$425$	2.51238	0.121868
$426$	0	0
$427$	−1.52737	−0.0739145
$428$	0	0
$429$	7.65045	0.369367
$430$	0	0
$431$	20.9906	1.01108	0.505540	−	0.862803i	$-0.331293\pi$
$431$	20.9906	1.01108	0.505540	+	0.862803i	$0.331293\pi$
$432$	0	0
$433$	12.3754	0.594722	0.297361	−	0.954765i	$-0.403894\pi$
$433$	12.3754	0.594722	0.297361	+	0.954765i	$0.403894\pi$
$434$	0	0
$435$	1.71910	0.0824247
$436$	0	0
$437$	−7.29204	−0.348825
$438$	0	0
$439$	24.0738	1.14898	0.574490	−	0.818512i	$-0.305200\pi$
$439$	24.0738	1.14898	0.574490	+	0.818512i	$0.305200\pi$
$440$	0	0
$441$	−6.95590	−0.331233
$442$	0	0
$443$	5.14360	0.244380	0.122190	−	0.992507i	$-0.461008\pi$
$443$	5.14360	0.244380	0.122190	+	0.992507i	$0.461008\pi$
$444$	0	0
$445$	−19.3292	−0.916293
$446$	0	0
$447$	0.562591	0.0266097
$448$	0	0
$449$	18.0027	0.849601	0.424800	−	0.905287i	$-0.360344\pi$
$449$	18.0027	0.849601	0.424800	+	0.905287i	$0.360344\pi$
$450$	0	0
$451$	−6.71148	−0.316031
$452$	0	0
$453$	−4.59101	−0.215705
$454$	0	0
$455$	−1.16398	−0.0545682
$456$	0	0
$457$	34.1329	1.59667	0.798334	−	0.602215i	$-0.205715\pi$
$457$	34.1329	1.59667	0.798334	+	0.602215i	$0.205715\pi$
$458$	0	0
$459$	−1.22874	−0.0573527
$460$	0	0
$461$	10.0677	0.468899	0.234449	−	0.972128i	$-0.424671\pi$
$461$	10.0677	0.468899	0.234449	+	0.972128i	$0.424671\pi$
$462$	0	0
$463$	−11.9720	−0.556387	−0.278194	−	0.960525i	$-0.589736\pi$
$463$	−11.9720	−0.556387	−0.278194	+	0.960525i	$0.589736\pi$
$464$	0	0
$465$	6.33909	0.293968
$466$	0	0
$467$	−16.8685	−0.780582	−0.390291	−	0.920692i	$-0.627626\pi$
$467$	−16.8685	−0.780582	−0.390291	+	0.920692i	$0.627626\pi$
$468$	0	0
$469$	1.97617	0.0912508
$470$	0	0
$471$	−17.6359	−0.812620
$472$	0	0
$473$	−21.4222	−0.984995
$474$	0	0
$475$	14.9099	0.684113
$476$	0	0
$477$	−2.49073	−0.114043
$478$	0	0
$479$	38.7594	1.77096	0.885480	−	0.464677i	$-0.153830\pi$
$479$	38.7594	1.77096	0.885480	+	0.464677i	$0.153830\pi$
$480$	0	0
$481$	−24.6743	−1.12505
$482$	0	0
$483$	0.210007	0.00955565
$484$	0	0
$485$	−19.9270	−0.904840
$486$	0	0
$487$	−3.17969	−0.144085	−0.0720427	−	0.997402i	$-0.522952\pi$
$487$	−3.17969	−0.144085	−0.0720427	+	0.997402i	$0.522952\pi$
$488$	0	0
$489$	−9.54677	−0.431720
$490$	0	0
$491$	1.43125	0.0645914	0.0322957	−	0.999478i	$-0.489718\pi$
$491$	1.43125	0.0645914	0.0322957	+	0.999478i	$0.489718\pi$
$492$	0	0
$493$	−1.22874	−0.0553397
$494$	0	0
$495$	−4.07924	−0.183348
$496$	0	0
$497$	−1.58239	−0.0709799
$498$	0	0
$499$	28.9140	1.29437	0.647184	−	0.762334i	$-0.275946\pi$
$499$	28.9140	1.29437	0.647184	+	0.762334i	$0.275946\pi$
$500$	0	0
$501$	−18.8167	−0.840670
$502$	0	0
$503$	6.83555	0.304782	0.152391	−	0.988320i	$-0.451303\pi$
$503$	6.83555	0.304782	0.152391	+	0.988320i	$0.451303\pi$
$504$	0	0
$505$	−2.65514	−0.118152
$506$	0	0
$507$	−2.60511	−0.115697
$508$	0	0
$509$	−16.3988	−0.726862	−0.363431	−	0.931621i	$-0.618395\pi$
$509$	−16.3988	−0.726862	−0.363431	+	0.931621i	$0.618395\pi$
$510$	0	0
$511$	0.789170	0.0349108
$512$	0	0
$513$	−7.29204	−0.321951
$514$	0	0
$515$	−25.3877	−1.11872
$516$	0	0
$517$	12.8966	0.567193
$518$	0	0
$519$	0.0827733	0.00363334
$520$	0	0
$521$	45.1845	1.97957	0.989784	−	0.142575i	$-0.0455382\pi$
$521$	45.1845	1.97957	0.989784	+	0.142575i	$0.0455382\pi$
$522$	0	0
$523$	−8.82991	−0.386105	−0.193052	−	0.981188i	$-0.561839\pi$
$523$	−8.82991	−0.386105	−0.193052	+	0.981188i	$0.561839\pi$
$524$	0	0
$525$	−0.429398	−0.0187404
$526$	0	0
$527$	−4.53090	−0.197369
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.55638	−0.154334
$532$	0	0
$533$	−9.11909	−0.394991
$534$	0	0
$535$	−20.6730	−0.893774
$536$	0	0
$537$	9.20992	0.397437
$538$	0	0
$539$	16.5056	0.710945
$540$	0	0
$541$	−15.9832	−0.687173	−0.343586	−	0.939121i	$-0.611642\pi$
$541$	−15.9832	−0.687173	−0.343586	+	0.939121i	$0.611642\pi$
$542$	0	0
$543$	−9.17306	−0.393654
$544$	0	0
$545$	−6.51925	−0.279254
$546$	0	0
$547$	26.8308	1.14720	0.573601	−	0.819135i	$-0.305546\pi$
$547$	26.8308	1.14720	0.573601	+	0.819135i	$0.305546\pi$
$548$	0	0
$549$	−7.27294	−0.310402
$550$	0	0
$551$	−7.29204	−0.310651
$552$	0	0
$553$	0.169522	0.00720881
$554$	0	0
$555$	13.1564	0.558458
$556$	0	0
$557$	−3.96427	−0.167971	−0.0839857	−	0.996467i	$-0.526765\pi$
$557$	−3.96427	−0.167971	−0.0839857	+	0.996467i	$0.526765\pi$
$558$	0	0
$559$	−29.1070	−1.23109
$560$	0	0
$561$	2.91566	0.123099
$562$	0	0
$563$	−1.13810	−0.0479653	−0.0239827	−	0.999712i	$-0.507635\pi$
$563$	−1.13810	−0.0479653	−0.0239827	+	0.999712i	$0.507635\pi$
$564$	0	0
$565$	−26.1157	−1.09870
$566$	0	0
$567$	0.210007	0.00881946
$568$	0	0
$569$	−19.6531	−0.823900	−0.411950	−	0.911206i	$-0.635152\pi$
$569$	−19.6531	−0.823900	−0.411950	+	0.911206i	$0.635152\pi$
$570$	0	0
$571$	−10.0484	−0.420511	−0.210256	−	0.977646i	$-0.567430\pi$
$571$	−10.0484	−0.420511	−0.210256	+	0.977646i	$0.567430\pi$
$572$	0	0
$573$	−1.71415	−0.0716095
$574$	0	0
$575$	−2.04468	−0.0852692
$576$	0	0
$577$	23.9270	0.996093	0.498046	−	0.867150i	$-0.334051\pi$
$577$	23.9270	0.996093	0.498046	+	0.867150i	$0.334051\pi$
$578$	0	0
$579$	3.98986	0.165813
$580$	0	0
$581$	1.17279	0.0486557
$582$	0	0
$583$	5.91022	0.244776
$584$	0	0
$585$	−5.54258	−0.229157
$586$	0	0
$587$	−45.4351	−1.87531	−0.937654	−	0.347571i	$-0.887007\pi$
$587$	−45.4351	−1.87531	−0.937654	+	0.347571i	$0.887007\pi$
$588$	0	0
$589$	−26.8889	−1.10794
$590$	0	0
$591$	−3.34272	−0.137501
$592$	0	0
$593$	33.5765	1.37882	0.689411	−	0.724370i	$-0.257869\pi$
$593$	33.5765	1.37882	0.689411	+	0.724370i	$0.257869\pi$
$594$	0	0
$595$	−0.443604	−0.0181860
$596$	0	0
$597$	18.4507	0.755136
$598$	0	0
$599$	−10.5064	−0.429278	−0.214639	−	0.976693i	$-0.568858\pi$
$599$	−10.5064	−0.429278	−0.214639	+	0.976693i	$0.568858\pi$
$600$	0	0
$601$	−9.26956	−0.378113	−0.189057	−	0.981966i	$-0.560543\pi$
$601$	−9.26956	−0.378113	−0.189057	+	0.981966i	$0.560543\pi$
$602$	0	0
$603$	9.41000	0.383205
$604$	0	0
$605$	−9.23056	−0.375276
$606$	0	0
$607$	41.7857	1.69603	0.848014	−	0.529973i	$-0.177798\pi$
$607$	41.7857	1.69603	0.848014	+	0.529973i	$0.177798\pi$
$608$	0	0
$609$	0.210007	0.00850991
$610$	0	0
$611$	17.5230	0.708906
$612$	0	0
$613$	−13.8443	−0.559168	−0.279584	−	0.960121i	$-0.590197\pi$
$613$	−13.8443	−0.559168	−0.279584	+	0.960121i	$0.590197\pi$
$614$	0	0
$615$	4.86232	0.196068
$616$	0	0
$617$	−31.6630	−1.27470	−0.637352	−	0.770573i	$-0.719970\pi$
$617$	−31.6630	−1.27470	−0.637352	+	0.770573i	$0.719970\pi$
$618$	0	0
$619$	−44.1516	−1.77460	−0.887301	−	0.461191i	$-0.847422\pi$
$619$	−44.1516	−1.77460	−0.887301	+	0.461191i	$0.847422\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−2.36127	−0.0946024
$624$	0	0
$625$	−10.5959	−0.423834
$626$	0	0
$627$	17.3032	0.691023
$628$	0	0
$629$	−9.40362	−0.374947
$630$	0	0
$631$	0.420214	0.0167284	0.00836422	−	0.999965i	$-0.497338\pi$
$631$	0.420214	0.0167284	0.00836422	+	0.999965i	$0.497338\pi$
$632$	0	0
$633$	20.7086	0.823092
$634$	0	0
$635$	−23.4852	−0.931983
$636$	0	0
$637$	22.4266	0.888574
$638$	0	0
$639$	−7.53494	−0.298078
$640$	0	0
$641$	−9.31380	−0.367873	−0.183937	−	0.982938i	$-0.558884\pi$
$641$	−9.31380	−0.367873	−0.183937	+	0.982938i	$0.558884\pi$
$642$	0	0
$643$	43.1468	1.70154	0.850772	−	0.525535i	$-0.176135\pi$
$643$	43.1468	1.70154	0.850772	+	0.525535i	$0.176135\pi$
$644$	0	0
$645$	15.5199	0.611096
$646$	0	0
$647$	13.0052	0.511287	0.255644	−	0.966771i	$-0.417713\pi$
$647$	13.0052	0.511287	0.255644	+	0.966771i	$0.417713\pi$
$648$	0	0
$649$	8.43890	0.331256
$650$	0	0
$651$	0.774388	0.0303506
$652$	0	0
$653$	24.4920	0.958444	0.479222	−	0.877694i	$-0.340919\pi$
$653$	24.4920	0.958444	0.479222	+	0.877694i	$0.340919\pi$
$654$	0	0
$655$	1.58300	0.0618528
$656$	0	0
$657$	3.75783	0.146607
$658$	0	0
$659$	5.15236	0.200707	0.100354	−	0.994952i	$-0.468003\pi$
$659$	5.15236	0.200707	0.100354	+	0.994952i	$0.468003\pi$
$660$	0	0
$661$	−12.5643	−0.488693	−0.244347	−	0.969688i	$-0.578574\pi$
$661$	−12.5643	−0.488693	−0.244347	+	0.969688i	$0.578574\pi$
$662$	0	0
$663$	3.96159	0.153856
$664$	0	0
$665$	−2.63260	−0.102088
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−12.1547	−0.469926
$670$	0	0
$671$	17.2579	0.666233
$672$	0	0
$673$	−19.9920	−0.770633	−0.385317	−	0.922784i	$-0.625908\pi$
$673$	−19.9920	−0.770633	−0.385317	+	0.922784i	$0.625908\pi$
$674$	0	0
$675$	−2.04468	−0.0786999
$676$	0	0
$677$	36.2945	1.39491	0.697456	−	0.716627i	$-0.254315\pi$
$677$	36.2945	1.39491	0.697456	+	0.716627i	$0.254315\pi$
$678$	0	0
$679$	−2.43430	−0.0934199
$680$	0	0
$681$	12.9879	0.497696
$682$	0	0
$683$	−26.0186	−0.995573	−0.497787	−	0.867300i	$-0.665854\pi$
$683$	−26.0186	−0.995573	−0.497787	+	0.867300i	$0.665854\pi$
$684$	0	0
$685$	−2.71299	−0.103658
$686$	0	0
$687$	9.91961	0.378457
$688$	0	0
$689$	8.03038	0.305933
$690$	0	0
$691$	−12.8459	−0.488679	−0.244340	−	0.969690i	$-0.578571\pi$
$691$	−12.8459	−0.488679	−0.244340	+	0.969690i	$0.578571\pi$
$692$	0	0
$693$	−0.498323	−0.0189297
$694$	0	0
$695$	13.1509	0.498842
$696$	0	0
$697$	−3.47537	−0.131639
$698$	0	0
$699$	13.0401	0.493222
$700$	0	0
$701$	−23.7174	−0.895792	−0.447896	−	0.894086i	$-0.647827\pi$
$701$	−23.7174	−0.895792	−0.447896	+	0.894086i	$0.647827\pi$
$702$	0	0
$703$	−55.8064	−2.10478
$704$	0	0
$705$	−9.34332	−0.351890
$706$	0	0
$707$	−0.324354	−0.0121986
$708$	0	0
$709$	−14.4256	−0.541765	−0.270883	−	0.962612i	$-0.587316\pi$
$709$	−14.4256	−0.541765	−0.270883	+	0.962612i	$0.587316\pi$
$710$	0	0
$711$	0.807221	0.0302731
$712$	0	0
$713$	3.68744	0.138096
$714$	0	0
$715$	13.1519	0.491854
$716$	0	0
$717$	−4.25253	−0.158814
$718$	0	0
$719$	−17.6876	−0.659636	−0.329818	−	0.944044i	$-0.606987\pi$
$719$	−17.6876	−0.659636	−0.329818	+	0.944044i	$0.606987\pi$
$720$	0	0
$721$	−3.10138	−0.115501
$722$	0	0
$723$	−1.22996	−0.0457428
$724$	0	0
$725$	−2.04468	−0.0759376
$726$	0	0
$727$	35.0547	1.30011	0.650053	−	0.759889i	$-0.274747\pi$
$727$	35.0547	1.30011	0.650053	+	0.759889i	$0.274747\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.0930	−0.410288
$732$	0	0
$733$	−1.31412	−0.0485383	−0.0242691	−	0.999705i	$-0.507726\pi$
$733$	−1.31412	−0.0485383	−0.0242691	+	0.999705i	$0.507726\pi$
$734$	0	0
$735$	−11.9579	−0.441074
$736$	0	0
$737$	−22.3289	−0.822495
$738$	0	0
$739$	−9.56395	−0.351816	−0.175908	−	0.984407i	$-0.556286\pi$
$739$	−9.56395	−0.351816	−0.175908	+	0.984407i	$0.556286\pi$
$740$	0	0
$741$	23.5103	0.863674
$742$	0	0
$743$	−15.5554	−0.570673	−0.285336	−	0.958427i	$-0.592105\pi$
$743$	−15.5554	−0.570673	−0.285336	+	0.958427i	$0.592105\pi$
$744$	0	0
$745$	0.967153	0.0354337
$746$	0	0
$747$	5.58455	0.204328
$748$	0	0
$749$	−2.52543	−0.0922773
$750$	0	0
$751$	27.3525	0.998106	0.499053	−	0.866572i	$-0.333681\pi$
$751$	27.3525	0.998106	0.499053	+	0.866572i	$0.333681\pi$
$752$	0	0
$753$	12.4225	0.452700
$754$	0	0
$755$	−7.89243	−0.287235
$756$	0	0
$757$	17.2917	0.628479	0.314240	−	0.949344i	$-0.398250\pi$
$757$	17.2917	0.628479	0.314240	+	0.949344i	$0.398250\pi$
$758$	0	0
$759$	−2.37289	−0.0861304
$760$	0	0
$761$	32.3174	1.17150	0.585752	−	0.810490i	$-0.300799\pi$
$761$	32.3174	1.17150	0.585752	+	0.810490i	$0.300799\pi$
$762$	0	0
$763$	−0.796396	−0.0288315
$764$	0	0
$765$	−2.11233	−0.0763715
$766$	0	0
$767$	11.4662	0.414020
$768$	0	0
$769$	−5.88519	−0.212225	−0.106113	−	0.994354i	$-0.533840\pi$
$769$	−5.88519	−0.212225	−0.106113	+	0.994354i	$0.533840\pi$
$770$	0	0
$771$	3.16912	0.114133
$772$	0	0
$773$	19.0255	0.684298	0.342149	−	0.939646i	$-0.388845\pi$
$773$	19.0255	0.684298	0.342149	+	0.939646i	$0.388845\pi$
$774$	0	0
$775$	−7.53964	−0.270832
$776$	0	0
$777$	1.60720	0.0576578
$778$	0	0
$779$	−20.6248	−0.738961
$780$	0	0
$781$	17.8796	0.639781
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−30.3180	−1.08209
$786$	0	0
$787$	18.6197	0.663720	0.331860	−	0.943329i	$-0.392324\pi$
$787$	18.6197	0.663720	0.331860	+	0.943329i	$0.392324\pi$
$788$	0	0
$789$	−11.4038	−0.405987
$790$	0	0
$791$	−3.19031	−0.113434
$792$	0	0
$793$	23.4488	0.832690
$794$	0	0
$795$	−4.28182	−0.151860
$796$	0	0
$797$	20.6132	0.730158	0.365079	−	0.930977i	$-0.381042\pi$
$797$	20.6132	0.730158	0.365079	+	0.930977i	$0.381042\pi$
$798$	0	0
$799$	6.67819	0.236258
$800$	0	0
$801$	−11.2438	−0.397280
$802$	0	0
$803$	−8.91690	−0.314671
$804$	0	0
$805$	0.361024	0.0127244
$806$	0	0
$807$	3.50827	0.123497
$808$	0	0
$809$	−22.9018	−0.805184	−0.402592	−	0.915379i	$-0.631891\pi$
$809$	−22.9018	−0.805184	−0.402592	+	0.915379i	$0.631891\pi$
$810$	0	0
$811$	−21.5774	−0.757685	−0.378843	−	0.925461i	$-0.623678\pi$
$811$	−21.5774	−0.757685	−0.378843	+	0.925461i	$0.623678\pi$
$812$	0	0
$813$	2.74651	0.0963243
$814$	0	0
$815$	−16.4119	−0.574883
$816$	0	0
$817$	−65.8319	−2.30317
$818$	0	0
$819$	−0.677086	−0.0236593
$820$	0	0
$821$	11.3416	0.395824	0.197912	−	0.980220i	$-0.436584\pi$
$821$	11.3416	0.395824	0.197912	+	0.980220i	$0.436584\pi$
$822$	0	0
$823$	4.53006	0.157908	0.0789540	−	0.996878i	$-0.474842\pi$
$823$	4.53006	0.157908	0.0789540	+	0.996878i	$0.474842\pi$
$824$	0	0
$825$	4.85180	0.168918
$826$	0	0
$827$	2.90044	0.100858	0.0504290	−	0.998728i	$-0.483941\pi$
$827$	2.90044	0.100858	0.0504290	+	0.998728i	$0.483941\pi$
$828$	0	0
$829$	−5.31154	−0.184477	−0.0922387	−	0.995737i	$-0.529402\pi$
$829$	−5.31154	−0.184477	−0.0922387	+	0.995737i	$0.529402\pi$
$830$	0	0
$831$	−26.2225	−0.909650
$832$	0	0
$833$	8.54699	0.296136
$834$	0	0
$835$	−32.3479	−1.11945
$836$	0	0
$837$	3.68744	0.127457
$838$	0	0
$839$	52.2724	1.80464	0.902322	−	0.431063i	$-0.141861\pi$
$839$	52.2724	1.80464	0.902322	+	0.431063i	$0.141861\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	13.9533	0.480576
$844$	0	0
$845$	−4.47845	−0.154063
$846$	0	0
$847$	−1.12761	−0.0387452
$848$	0	0
$849$	0.0948723	0.00325601
$850$	0	0
$851$	7.65306	0.262344
$852$	0	0
$853$	14.2670	0.488492	0.244246	−	0.969713i	$-0.421460\pi$
$853$	14.2670	0.488492	0.244246	+	0.969713i	$0.421460\pi$
$854$	0	0
$855$	−12.5358	−0.428714
$856$	0	0
$857$	9.13021	0.311882	0.155941	−	0.987766i	$-0.450159\pi$
$857$	9.13021	0.311882	0.155941	+	0.987766i	$0.450159\pi$
$858$	0	0
$859$	16.4652	0.561785	0.280892	−	0.959739i	$-0.409370\pi$
$859$	16.4652	0.561785	0.280892	+	0.959739i	$0.409370\pi$
$860$	0	0
$861$	0.593984	0.0202429
$862$	0	0
$863$	15.6882	0.534032	0.267016	−	0.963692i	$-0.413962\pi$
$863$	15.6882	0.534032	0.267016	+	0.963692i	$0.413962\pi$
$864$	0	0
$865$	0.142296	0.00483820
$866$	0	0
$867$	−15.4902	−0.526075
$868$	0	0
$869$	−1.91544	−0.0649770
$870$	0	0
$871$	−30.3389	−1.02799
$872$	0	0
$873$	−11.5915	−0.392314
$874$	0	0
$875$	−2.54330	−0.0859791
$876$	0	0
$877$	−45.6806	−1.54252	−0.771262	−	0.636518i	$-0.780374\pi$
$877$	−45.6806	−1.54252	−0.771262	+	0.636518i	$0.780374\pi$
$878$	0	0
$879$	−19.3283	−0.651928
$880$	0	0
$881$	58.8054	1.98120	0.990602	−	0.136777i	$-0.0436744\pi$
$881$	58.8054	1.98120	0.990602	+	0.136777i	$0.0436744\pi$
$882$	0	0
$883$	−20.5660	−0.692099	−0.346050	−	0.938216i	$-0.612477\pi$
$883$	−20.5660	−0.692099	−0.346050	+	0.938216i	$0.612477\pi$
$884$	0	0
$885$	−6.11379	−0.205513
$886$	0	0
$887$	12.0199	0.403587	0.201794	−	0.979428i	$-0.435323\pi$
$887$	12.0199	0.403587	0.201794	+	0.979428i	$0.435323\pi$
$888$	0	0
$889$	−2.86897	−0.0962222
$890$	0	0
$891$	−2.37289	−0.0794947
$892$	0	0
$893$	39.6322	1.32624
$894$	0	0
$895$	15.8328	0.529232
$896$	0	0
$897$	−3.22411	−0.107650
$898$	0	0
$899$	3.68744	0.122983
$900$	0	0
$901$	3.06046	0.101959
$902$	0	0
$903$	1.89592	0.0630924
$904$	0	0
$905$	−15.7694	−0.524194
$906$	0	0
$907$	0.278333	0.00924189	0.00462094	−	0.999989i	$-0.498529\pi$
$907$	0.278333	0.00924189	0.00462094	+	0.999989i	$0.498529\pi$
$908$	0	0
$909$	−1.54449	−0.0512275
$910$	0	0
$911$	40.1668	1.33079	0.665393	−	0.746493i	$-0.268264\pi$
$911$	40.1668	1.33079	0.665393	+	0.746493i	$0.268264\pi$
$912$	0	0
$913$	−13.2515	−0.438561
$914$	0	0
$915$	−12.5029	−0.413334
$916$	0	0
$917$	0.193380	0.00638597
$918$	0	0
$919$	11.1451	0.367644	0.183822	−	0.982960i	$-0.441153\pi$
$919$	11.1451	0.367644	0.183822	+	0.982960i	$0.441153\pi$
$920$	0	0
$921$	−27.8115	−0.916421
$922$	0	0
$923$	24.2935	0.799630
$924$	0	0
$925$	−15.6481	−0.514506
$926$	0	0
$927$	−14.7680	−0.485045
$928$	0	0
$929$	22.1350	0.726227	0.363113	−	0.931745i	$-0.381714\pi$
$929$	22.1350	0.726227	0.363113	+	0.931745i	$0.381714\pi$
$930$	0	0
$931$	50.7227	1.66237
$932$	0	0
$933$	4.91082	0.160773
$934$	0	0
$935$	5.01232	0.163921
$936$	0	0
$937$	−28.8017	−0.940909	−0.470454	−	0.882424i	$-0.655910\pi$
$937$	−28.8017	−0.940909	−0.470454	+	0.882424i	$0.655910\pi$
$938$	0	0
$939$	3.32210	0.108413
$940$	0	0
$941$	−11.8130	−0.385092	−0.192546	−	0.981288i	$-0.561675\pi$
$941$	−11.8130	−0.385092	−0.192546	+	0.981288i	$0.561675\pi$
$942$	0	0
$943$	2.82840	0.0921055
$944$	0	0
$945$	0.361024	0.0117441
$946$	0	0
$947$	−25.7370	−0.836340	−0.418170	−	0.908369i	$-0.637328\pi$
$947$	−25.7370	−0.836340	−0.418170	+	0.908369i	$0.637328\pi$
$948$	0	0
$949$	−12.1157	−0.393291
$950$	0	0
$951$	11.0295	0.357655
$952$	0	0
$953$	−43.0227	−1.39364	−0.696821	−	0.717245i	$-0.745403\pi$
$953$	−43.0227	−1.39364	−0.696821	+	0.717245i	$0.745403\pi$
$954$	0	0
$955$	−2.94679	−0.0953560
$956$	0	0
$957$	−2.37289	−0.0767046
$958$	0	0
$959$	−0.331420	−0.0107021
$960$	0	0
$961$	−17.4028	−0.561381
$962$	0	0
$963$	−12.0255	−0.387516
$964$	0	0
$965$	6.85898	0.220798
$966$	0	0
$967$	−45.1451	−1.45177	−0.725884	−	0.687817i	$-0.758569\pi$
$967$	−45.1451	−1.45177	−0.725884	+	0.687817i	$0.758569\pi$
$968$	0	0
$969$	8.96002	0.287837
$970$	0	0
$971$	−13.2567	−0.425426	−0.212713	−	0.977115i	$-0.568230\pi$
$971$	−13.2567	−0.425426	−0.212713	+	0.977115i	$0.568230\pi$
$972$	0	0
$973$	1.60652	0.0515028
$974$	0	0
$975$	6.59228	0.211122
$976$	0	0
$977$	25.0117	0.800197	0.400098	−	0.916472i	$-0.368976\pi$
$977$	25.0117	0.800197	0.400098	+	0.916472i	$0.368976\pi$
$978$	0	0
$979$	26.6802	0.852704
$980$	0	0
$981$	−3.79224	−0.121077
$982$	0	0
$983$	35.4337	1.13016	0.565080	−	0.825036i	$-0.308845\pi$
$983$	35.4337	1.13016	0.565080	+	0.825036i	$0.308845\pi$
$984$	0	0
$985$	−5.74649	−0.183098
$986$	0	0
$987$	−1.14139	−0.0363307
$988$	0	0
$989$	9.02791	0.287071
$990$	0	0
$991$	−46.0130	−1.46165	−0.730825	−	0.682565i	$-0.760865\pi$
$991$	−46.0130	−1.46165	−0.730825	+	0.682565i	$0.760865\pi$
$992$	0	0
$993$	−14.1235	−0.448194
$994$	0	0
$995$	31.7186	1.00555
$996$	0	0
$997$	−34.7346	−1.10006	−0.550028	−	0.835146i	$-0.685383\pi$
$997$	−34.7346	−1.10006	−0.550028	+	0.835146i	$0.685383\pi$
$998$	0	0
$999$	7.65306	0.242132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.d.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.7	✓	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.71910 q^{5} +0.210007 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.71910 q^{5} +0.210007 q^{7} +1.00000 q^{9} -2.37289 q^{11} -3.22411 q^{13} +1.71910 q^{15} -1.22874 q^{17} -7.29204 q^{19} +0.210007 q^{21} +1.00000 q^{23} -2.04468 q^{25} +1.00000 q^{27} +1.00000 q^{29} +3.68744 q^{31} -2.37289 q^{33} +0.361024 q^{35} +7.65306 q^{37} -3.22411 q^{39} +2.82840 q^{41} +9.02791 q^{43} +1.71910 q^{45} -5.43500 q^{47} -6.95590 q^{49} -1.22874 q^{51} -2.49073 q^{53} -4.07924 q^{55} -7.29204 q^{57} -3.55638 q^{59} -7.27294 q^{61} +0.210007 q^{63} -5.54258 q^{65} +9.41000 q^{67} +1.00000 q^{69} -7.53494 q^{71} +3.75783 q^{73} -2.04468 q^{75} -0.498323 q^{77} +0.807221 q^{79} +1.00000 q^{81} +5.58455 q^{83} -2.11233 q^{85} +1.00000 q^{87} -11.2438 q^{89} -0.677086 q^{91} +3.68744 q^{93} -12.5358 q^{95} -11.5915 q^{97} -2.37289 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more