LMFDB - Embedded newform 8004.2.a.j.1.14

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + \cdots + 3888 $ x^16 - 3x^15 - 49x^14 + 130x^13 + 932x^12 - 2028x^11 - 8965x^10 + 14400x^9 + 46229x^8 - 47547x^7 - 122604x^6 + 65278x^5 + 151028x^4 - 17988x^3 - 67608x^2 - 8424*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$3.50323$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.50323 q^{5} +0.271751 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.50323 q^{5} +0.271751 q^{7} +1.00000 q^{9} -3.15022 q^{11} +5.54253 q^{13} +3.50323 q^{15} +7.90123 q^{17} -0.447567 q^{19} +0.271751 q^{21} +1.00000 q^{23} +7.27263 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.65859 q^{31} -3.15022 q^{33} +0.952005 q^{35} +5.73526 q^{37} +5.54253 q^{39} +7.82330 q^{41} -3.05826 q^{43} +3.50323 q^{45} -6.56523 q^{47} -6.92615 q^{49} +7.90123 q^{51} -3.63270 q^{53} -11.0359 q^{55} -0.447567 q^{57} -9.18341 q^{59} +0.878199 q^{61} +0.271751 q^{63} +19.4168 q^{65} -0.508596 q^{67} +1.00000 q^{69} +7.95084 q^{71} -10.3911 q^{73} +7.27263 q^{75} -0.856073 q^{77} -15.0267 q^{79} +1.00000 q^{81} -8.08550 q^{83} +27.6798 q^{85} -1.00000 q^{87} -13.2657 q^{89} +1.50619 q^{91} +6.65859 q^{93} -1.56793 q^{95} +2.98895 q^{97} -3.15022 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	$ 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * 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
$5$	3.50323	1.56669	0.783346	−	0.621585i	$-0.213511\pi$
$5$	3.50323	1.56669	0.783346	+	0.621585i	$0.213511\pi$
$6$	0	0
$7$	0.271751	0.102712	0.0513560	−	0.998680i	$-0.483646\pi$
$7$	0.271751	0.102712	0.0513560	+	0.998680i	$0.483646\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.15022	−0.949826	−0.474913	−	0.880033i	$-0.657520\pi$
$11$	−3.15022	−0.949826	−0.474913	+	0.880033i	$0.657520\pi$
$12$	0	0
$13$	5.54253	1.53722	0.768611	−	0.639717i	$-0.220948\pi$
$13$	5.54253	1.53722	0.768611	+	0.639717i	$0.220948\pi$
$14$	0	0
$15$	3.50323	0.904530
$16$	0	0
$17$	7.90123	1.91633	0.958165	−	0.286218i	$-0.0923981\pi$
$17$	7.90123	1.91633	0.958165	+	0.286218i	$0.0923981\pi$
$18$	0	0
$19$	−0.447567	−0.102679	−0.0513394	−	0.998681i	$-0.516349\pi$
$19$	−0.447567	−0.102679	−0.0513394	+	0.998681i	$0.516349\pi$
$20$	0	0
$21$	0.271751	0.0593008
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	7.27263	1.45453
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.65859	1.19592	0.597959	−	0.801527i	$-0.295979\pi$
$31$	6.65859	1.19592	0.597959	+	0.801527i	$0.295979\pi$
$32$	0	0
$33$	−3.15022	−0.548382
$34$	0	0
$35$	0.952005	0.160918
$36$	0	0
$37$	5.73526	0.942871	0.471435	−	0.881901i	$-0.343736\pi$
$37$	5.73526	0.942871	0.471435	+	0.881901i	$0.343736\pi$
$38$	0	0
$39$	5.54253	0.887515
$40$	0	0
$41$	7.82330	1.22179	0.610897	−	0.791710i	$-0.290809\pi$
$41$	7.82330	1.22179	0.610897	+	0.791710i	$0.290809\pi$
$42$	0	0
$43$	−3.05826	−0.466380	−0.233190	−	0.972431i	$-0.574916\pi$
$43$	−3.05826	−0.466380	−0.233190	+	0.972431i	$0.574916\pi$
$44$	0	0
$45$	3.50323	0.522231
$46$	0	0
$47$	−6.56523	−0.957638	−0.478819	−	0.877914i	$-0.658935\pi$
$47$	−6.56523	−0.957638	−0.478819	+	0.877914i	$0.658935\pi$
$48$	0	0
$49$	−6.92615	−0.989450
$50$	0	0
$51$	7.90123	1.10639
$52$	0	0
$53$	−3.63270	−0.498990	−0.249495	−	0.968376i	$-0.580265\pi$
$53$	−3.63270	−0.498990	−0.249495	+	0.968376i	$0.580265\pi$
$54$	0	0
$55$	−11.0359	−1.48808
$56$	0	0
$57$	−0.447567	−0.0592817
$58$	0	0
$59$	−9.18341	−1.19558	−0.597789	−	0.801654i	$-0.703954\pi$
$59$	−9.18341	−1.19558	−0.597789	+	0.801654i	$0.703954\pi$
$60$	0	0
$61$	0.878199	0.112442	0.0562209	−	0.998418i	$-0.482095\pi$
$61$	0.878199	0.112442	0.0562209	+	0.998418i	$0.482095\pi$
$62$	0	0
$63$	0.271751	0.0342374
$64$	0	0
$65$	19.4168	2.40835
$66$	0	0
$67$	−0.508596	−0.0621349	−0.0310675	−	0.999517i	$-0.509891\pi$
$67$	−0.508596	−0.0621349	−0.0310675	+	0.999517i	$0.509891\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	7.95084	0.943591	0.471795	−	0.881708i	$-0.343606\pi$
$71$	7.95084	0.943591	0.471795	+	0.881708i	$0.343606\pi$
$72$	0	0
$73$	−10.3911	−1.21619	−0.608095	−	0.793864i	$-0.708066\pi$
$73$	−10.3911	−1.21619	−0.608095	+	0.793864i	$0.708066\pi$
$74$	0	0
$75$	7.27263	0.839771
$76$	0	0
$77$	−0.856073	−0.0975586
$78$	0	0
$79$	−15.0267	−1.69064	−0.845321	−	0.534259i	$-0.820591\pi$
$79$	−15.0267	−1.69064	−0.845321	+	0.534259i	$0.820591\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.08550	−0.887499	−0.443749	−	0.896151i	$-0.646352\pi$
$83$	−8.08550	−0.887499	−0.443749	+	0.896151i	$0.646352\pi$
$84$	0	0
$85$	27.6798	3.00230
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−13.2657	−1.40616	−0.703081	−	0.711110i	$-0.748193\pi$
$89$	−13.2657	−1.40616	−0.703081	+	0.711110i	$0.748193\pi$
$90$	0	0
$91$	1.50619	0.157891
$92$	0	0
$93$	6.65859	0.690464
$94$	0	0
$95$	−1.56793	−0.160866
$96$	0	0
$97$	2.98895	0.303482	0.151741	−	0.988420i	$-0.451512\pi$
$97$	2.98895	0.303482	0.151741	+	0.988420i	$0.451512\pi$
$98$	0	0
$99$	−3.15022	−0.316609
$100$	0	0
$101$	−14.3045	−1.42335	−0.711673	−	0.702511i	$-0.752062\pi$
$101$	−14.3045	−1.42335	−0.711673	+	0.702511i	$0.752062\pi$
$102$	0	0
$103$	6.81537	0.671538	0.335769	−	0.941944i	$-0.391004\pi$
$103$	6.81537	0.671538	0.335769	+	0.941944i	$0.391004\pi$
$104$	0	0
$105$	0.952005	0.0929062
$106$	0	0
$107$	−13.9056	−1.34430	−0.672151	−	0.740415i	$-0.734629\pi$
$107$	−13.9056	−1.34430	−0.672151	+	0.740415i	$0.734629\pi$
$108$	0	0
$109$	13.3468	1.27839	0.639197	−	0.769043i	$-0.279267\pi$
$109$	13.3468	1.27839	0.639197	+	0.769043i	$0.279267\pi$
$110$	0	0
$111$	5.73526	0.544367
$112$	0	0
$113$	−0.661719	−0.0622493	−0.0311246	−	0.999516i	$-0.509909\pi$
$113$	−0.661719	−0.0622493	−0.0311246	+	0.999516i	$0.509909\pi$
$114$	0	0
$115$	3.50323	0.326678
$116$	0	0
$117$	5.54253	0.512407
$118$	0	0
$119$	2.14716	0.196830
$120$	0	0
$121$	−1.07614	−0.0978312
$122$	0	0
$123$	7.82330	0.705403
$124$	0	0
$125$	7.96155	0.712102
$126$	0	0
$127$	19.8990	1.76575	0.882874	−	0.469610i	$-0.155605\pi$
$127$	19.8990	1.76575	0.882874	+	0.469610i	$0.155605\pi$
$128$	0	0
$129$	−3.05826	−0.269265
$130$	0	0
$131$	15.8984	1.38905	0.694525	−	0.719468i	$-0.255614\pi$
$131$	15.8984	1.38905	0.694525	+	0.719468i	$0.255614\pi$
$132$	0	0
$133$	−0.121627	−0.0105464
$134$	0	0
$135$	3.50323	0.301510
$136$	0	0
$137$	16.8613	1.44056	0.720279	−	0.693685i	$-0.244014\pi$
$137$	16.8613	1.44056	0.720279	+	0.693685i	$0.244014\pi$
$138$	0	0
$139$	−10.8288	−0.918483	−0.459241	−	0.888312i	$-0.651879\pi$
$139$	−10.8288	−0.918483	−0.459241	+	0.888312i	$0.651879\pi$
$140$	0	0
$141$	−6.56523	−0.552892
$142$	0	0
$143$	−17.4602	−1.46009
$144$	0	0
$145$	−3.50323	−0.290928
$146$	0	0
$147$	−6.92615	−0.571259
$148$	0	0
$149$	−10.2153	−0.836870	−0.418435	−	0.908247i	$-0.637421\pi$
$149$	−10.2153	−0.836870	−0.418435	+	0.908247i	$0.637421\pi$
$150$	0	0
$151$	−16.7767	−1.36527	−0.682636	−	0.730759i	$-0.739167\pi$
$151$	−16.7767	−1.36527	−0.682636	+	0.730759i	$0.739167\pi$
$152$	0	0
$153$	7.90123	0.638776
$154$	0	0
$155$	23.3266	1.87364
$156$	0	0
$157$	21.9816	1.75432	0.877162	−	0.480195i	$-0.159434\pi$
$157$	21.9816	1.75432	0.877162	+	0.480195i	$0.159434\pi$
$158$	0	0
$159$	−3.63270	−0.288092
$160$	0	0
$161$	0.271751	0.0214169
$162$	0	0
$163$	−13.1982	−1.03376	−0.516881	−	0.856057i	$-0.672907\pi$
$163$	−13.1982	−1.03376	−0.516881	+	0.856057i	$0.672907\pi$
$164$	0	0
$165$	−11.0359	−0.859146
$166$	0	0
$167$	10.0383	0.776790	0.388395	−	0.921493i	$-0.373030\pi$
$167$	10.0383	0.776790	0.388395	+	0.921493i	$0.373030\pi$
$168$	0	0
$169$	17.7196	1.36305
$170$	0	0
$171$	−0.447567	−0.0342263
$172$	0	0
$173$	−2.62068	−0.199247	−0.0996234	−	0.995025i	$-0.531764\pi$
$173$	−2.62068	−0.199247	−0.0996234	+	0.995025i	$0.531764\pi$
$174$	0	0
$175$	1.97634	0.149397
$176$	0	0
$177$	−9.18341	−0.690267
$178$	0	0
$179$	19.2949	1.44217	0.721085	−	0.692846i	$-0.243644\pi$
$179$	19.2949	1.44217	0.721085	+	0.692846i	$0.243644\pi$
$180$	0	0
$181$	1.68199	0.125021	0.0625105	−	0.998044i	$-0.480089\pi$
$181$	1.68199	0.125021	0.0625105	+	0.998044i	$0.480089\pi$
$182$	0	0
$183$	0.878199	0.0649183
$184$	0	0
$185$	20.0919	1.47719
$186$	0	0
$187$	−24.8906	−1.82018
$188$	0	0
$189$	0.271751	0.0197669
$190$	0	0
$191$	−5.91755	−0.428179	−0.214089	−	0.976814i	$-0.568678\pi$
$191$	−5.91755	−0.428179	−0.214089	+	0.976814i	$0.568678\pi$
$192$	0	0
$193$	−9.40360	−0.676886	−0.338443	−	0.940987i	$-0.609900\pi$
$193$	−9.40360	−0.676886	−0.338443	+	0.940987i	$0.609900\pi$
$194$	0	0
$195$	19.4168	1.39046
$196$	0	0
$197$	16.1496	1.15061	0.575305	−	0.817939i	$-0.304883\pi$
$197$	16.1496	1.15061	0.575305	+	0.817939i	$0.304883\pi$
$198$	0	0
$199$	6.67857	0.473431	0.236716	−	0.971579i	$-0.423929\pi$
$199$	6.67857	0.473431	0.236716	+	0.971579i	$0.423929\pi$
$200$	0	0
$201$	−0.508596	−0.0358736
$202$	0	0
$203$	−0.271751	−0.0190732
$204$	0	0
$205$	27.4068	1.91418
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.40993	0.0975270
$210$	0	0
$211$	6.03725	0.415621	0.207811	−	0.978169i	$-0.433366\pi$
$211$	6.03725	0.415621	0.207811	+	0.978169i	$0.433366\pi$
$212$	0	0
$213$	7.95084	0.544782
$214$	0	0
$215$	−10.7138	−0.730674
$216$	0	0
$217$	1.80948	0.122835
$218$	0	0
$219$	−10.3911	−0.702168
$220$	0	0
$221$	43.7928	2.94582
$222$	0	0
$223$	−25.6466	−1.71742	−0.858712	−	0.512458i	$-0.828735\pi$
$223$	−25.6466	−1.71742	−0.858712	+	0.512458i	$0.828735\pi$
$224$	0	0
$225$	7.27263	0.484842
$226$	0	0
$227$	0.782140	0.0519125	0.0259562	−	0.999663i	$-0.491737\pi$
$227$	0.782140	0.0519125	0.0259562	+	0.999663i	$0.491737\pi$
$228$	0	0
$229$	−17.9486	−1.18608	−0.593039	−	0.805173i	$-0.702072\pi$
$229$	−17.9486	−1.18608	−0.593039	+	0.805173i	$0.702072\pi$
$230$	0	0
$231$	−0.856073	−0.0563255
$232$	0	0
$233$	16.2525	1.06473	0.532367	−	0.846514i	$-0.321303\pi$
$233$	16.2525	1.06473	0.532367	+	0.846514i	$0.321303\pi$
$234$	0	0
$235$	−22.9995	−1.50032
$236$	0	0
$237$	−15.0267	−0.976092
$238$	0	0
$239$	−20.0022	−1.29384	−0.646918	−	0.762560i	$-0.723942\pi$
$239$	−20.0022	−1.29384	−0.646918	+	0.762560i	$0.723942\pi$
$240$	0	0
$241$	−18.5415	−1.19437	−0.597183	−	0.802105i	$-0.703713\pi$
$241$	−18.5415	−1.19437	−0.597183	+	0.802105i	$0.703713\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−24.2639	−1.55016
$246$	0	0
$247$	−2.48065	−0.157840
$248$	0	0
$249$	−8.08550	−0.512398
$250$	0	0
$251$	2.34485	0.148006	0.0740028	−	0.997258i	$-0.476423\pi$
$251$	2.34485	0.148006	0.0740028	+	0.997258i	$0.476423\pi$
$252$	0	0
$253$	−3.15022	−0.198052
$254$	0	0
$255$	27.6798	1.73338
$256$	0	0
$257$	5.81081	0.362468	0.181234	−	0.983440i	$-0.441991\pi$
$257$	5.81081	0.362468	0.181234	+	0.983440i	$0.441991\pi$
$258$	0	0
$259$	1.55856	0.0968442
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	18.5997	1.14691	0.573453	−	0.819239i	$-0.305604\pi$
$263$	18.5997	1.14691	0.573453	+	0.819239i	$0.305604\pi$
$264$	0	0
$265$	−12.7262	−0.781763
$266$	0	0
$267$	−13.2657	−0.811848
$268$	0	0
$269$	−9.99523	−0.609420	−0.304710	−	0.952445i	$-0.598560\pi$
$269$	−9.99523	−0.609420	−0.304710	+	0.952445i	$0.598560\pi$
$270$	0	0
$271$	13.4259	0.815568	0.407784	−	0.913079i	$-0.366302\pi$
$271$	13.4259	0.815568	0.407784	+	0.913079i	$0.366302\pi$
$272$	0	0
$273$	1.50619	0.0911585
$274$	0	0
$275$	−22.9103	−1.38155
$276$	0	0
$277$	26.3946	1.58590	0.792948	−	0.609289i	$-0.208545\pi$
$277$	26.3946	1.58590	0.792948	+	0.609289i	$0.208545\pi$
$278$	0	0
$279$	6.65859	0.398639
$280$	0	0
$281$	4.24115	0.253006	0.126503	−	0.991966i	$-0.459625\pi$
$281$	4.24115	0.253006	0.126503	+	0.991966i	$0.459625\pi$
$282$	0	0
$283$	29.1562	1.73315	0.866577	−	0.499043i	$-0.166315\pi$
$283$	29.1562	1.73315	0.866577	+	0.499043i	$0.166315\pi$
$284$	0	0
$285$	−1.56793	−0.0928761
$286$	0	0
$287$	2.12599	0.125493
$288$	0	0
$289$	45.4294	2.67232
$290$	0	0
$291$	2.98895	0.175215
$292$	0	0
$293$	7.66800	0.447969	0.223985	−	0.974593i	$-0.428093\pi$
$293$	7.66800	0.447969	0.223985	+	0.974593i	$0.428093\pi$
$294$	0	0
$295$	−32.1716	−1.87310
$296$	0	0
$297$	−3.15022	−0.182794
$298$	0	0
$299$	5.54253	0.320533
$300$	0	0
$301$	−0.831084	−0.0479029
$302$	0	0
$303$	−14.3045	−0.821770
$304$	0	0
$305$	3.07653	0.176162
$306$	0	0
$307$	11.5950	0.661759	0.330879	−	0.943673i	$-0.392655\pi$
$307$	11.5950	0.661759	0.330879	+	0.943673i	$0.392655\pi$
$308$	0	0
$309$	6.81537	0.387713
$310$	0	0
$311$	3.22439	0.182838	0.0914192	−	0.995812i	$-0.470860\pi$
$311$	3.22439	0.182838	0.0914192	+	0.995812i	$0.470860\pi$
$312$	0	0
$313$	−18.5121	−1.04637	−0.523184	−	0.852220i	$-0.675256\pi$
$313$	−18.5121	−1.04637	−0.523184	+	0.852220i	$0.675256\pi$
$314$	0	0
$315$	0.952005	0.0536394
$316$	0	0
$317$	34.8481	1.95726	0.978632	−	0.205618i	$-0.0659204\pi$
$317$	34.8481	1.95726	0.978632	+	0.205618i	$0.0659204\pi$
$318$	0	0
$319$	3.15022	0.176378
$320$	0	0
$321$	−13.9056	−0.776133
$322$	0	0
$323$	−3.53633	−0.196766
$324$	0	0
$325$	40.3088	2.23593
$326$	0	0
$327$	13.3468	0.738081
$328$	0	0
$329$	−1.78411	−0.0983610
$330$	0	0
$331$	13.7037	0.753224	0.376612	−	0.926371i	$-0.377089\pi$
$331$	13.7037	0.753224	0.376612	+	0.926371i	$0.377089\pi$
$332$	0	0
$333$	5.73526	0.314290
$334$	0	0
$335$	−1.78173	−0.0973463
$336$	0	0
$337$	−2.68235	−0.146117	−0.0730586	−	0.997328i	$-0.523276\pi$
$337$	−2.68235	−0.146117	−0.0730586	+	0.997328i	$0.523276\pi$
$338$	0	0
$339$	−0.661719	−0.0359396
$340$	0	0
$341$	−20.9760	−1.13591
$342$	0	0
$343$	−3.78444	−0.204341
$344$	0	0
$345$	3.50323	0.188608
$346$	0	0
$347$	13.3778	0.718156	0.359078	−	0.933308i	$-0.383091\pi$
$347$	13.3778	0.718156	0.359078	+	0.933308i	$0.383091\pi$
$348$	0	0
$349$	8.56716	0.458590	0.229295	−	0.973357i	$-0.426358\pi$
$349$	8.56716	0.458590	0.229295	+	0.973357i	$0.426358\pi$
$350$	0	0
$351$	5.54253	0.295838
$352$	0	0
$353$	−35.1739	−1.87212	−0.936060	−	0.351841i	$-0.885556\pi$
$353$	−35.1739	−1.87212	−0.936060	+	0.351841i	$0.885556\pi$
$354$	0	0
$355$	27.8536	1.47832
$356$	0	0
$357$	2.14716	0.113640
$358$	0	0
$359$	−6.45448	−0.340654	−0.170327	−	0.985388i	$-0.554482\pi$
$359$	−6.45448	−0.340654	−0.170327	+	0.985388i	$0.554482\pi$
$360$	0	0
$361$	−18.7997	−0.989457
$362$	0	0
$363$	−1.07614	−0.0564829
$364$	0	0
$365$	−36.4025	−1.90540
$366$	0	0
$367$	−19.1539	−0.999826	−0.499913	−	0.866076i	$-0.666635\pi$
$367$	−19.1539	−0.999826	−0.499913	+	0.866076i	$0.666635\pi$
$368$	0	0
$369$	7.82330	0.407265
$370$	0	0
$371$	−0.987188	−0.0512523
$372$	0	0
$373$	−4.99898	−0.258837	−0.129419	−	0.991590i	$-0.541311\pi$
$373$	−4.99898	−0.258837	−0.129419	+	0.991590i	$0.541311\pi$
$374$	0	0
$375$	7.96155	0.411133
$376$	0	0
$377$	−5.54253	−0.285455
$378$	0	0
$379$	11.5365	0.592588	0.296294	−	0.955097i	$-0.404249\pi$
$379$	11.5365	0.592588	0.296294	+	0.955097i	$0.404249\pi$
$380$	0	0
$381$	19.8990	1.01946
$382$	0	0
$383$	18.8685	0.964135	0.482068	−	0.876134i	$-0.339886\pi$
$383$	18.8685	0.964135	0.482068	+	0.876134i	$0.339886\pi$
$384$	0	0
$385$	−2.99902	−0.152844
$386$	0	0
$387$	−3.05826	−0.155460
$388$	0	0
$389$	−18.3970	−0.932765	−0.466383	−	0.884583i	$-0.654443\pi$
$389$	−18.3970	−0.932765	−0.466383	+	0.884583i	$0.654443\pi$
$390$	0	0
$391$	7.90123	0.399582
$392$	0	0
$393$	15.8984	0.801969
$394$	0	0
$395$	−52.6422	−2.64872
$396$	0	0
$397$	−20.1948	−1.01355	−0.506774	−	0.862079i	$-0.669162\pi$
$397$	−20.1948	−1.01355	−0.506774	+	0.862079i	$0.669162\pi$
$398$	0	0
$399$	−0.121627	−0.00608894
$400$	0	0
$401$	−26.4958	−1.32314	−0.661570	−	0.749884i	$-0.730109\pi$
$401$	−26.4958	−1.32314	−0.661570	+	0.749884i	$0.730109\pi$
$402$	0	0
$403$	36.9054	1.83839
$404$	0	0
$405$	3.50323	0.174077
$406$	0	0
$407$	−18.0673	−0.895563
$408$	0	0
$409$	−23.0830	−1.14138	−0.570690	−	0.821166i	$-0.693324\pi$
$409$	−23.0830	−1.14138	−0.570690	+	0.821166i	$0.693324\pi$
$410$	0	0
$411$	16.8613	0.831707
$412$	0	0
$413$	−2.49560	−0.122800
$414$	0	0
$415$	−28.3254	−1.39044
$416$	0	0
$417$	−10.8288	−0.530286
$418$	0	0
$419$	14.8361	0.724792	0.362396	−	0.932024i	$-0.381959\pi$
$419$	14.8361	0.724792	0.362396	+	0.932024i	$0.381959\pi$
$420$	0	0
$421$	31.1076	1.51609	0.758046	−	0.652201i	$-0.226154\pi$
$421$	31.1076	1.51609	0.758046	+	0.652201i	$0.226154\pi$
$422$	0	0
$423$	−6.56523	−0.319213
$424$	0	0
$425$	57.4627	2.78735
$426$	0	0
$427$	0.238651	0.0115491
$428$	0	0
$429$	−17.4602	−0.842985
$430$	0	0
$431$	−38.9602	−1.87664	−0.938322	−	0.345762i	$-0.887621\pi$
$431$	−38.9602	−1.87664	−0.938322	+	0.345762i	$0.887621\pi$
$432$	0	0
$433$	−9.73144	−0.467663	−0.233832	−	0.972277i	$-0.575126\pi$
$433$	−9.73144	−0.467663	−0.233832	+	0.972277i	$0.575126\pi$
$434$	0	0
$435$	−3.50323	−0.167967
$436$	0	0
$437$	−0.447567	−0.0214100
$438$	0	0
$439$	2.02051	0.0964336	0.0482168	−	0.998837i	$-0.484646\pi$
$439$	2.02051	0.0964336	0.0482168	+	0.998837i	$0.484646\pi$
$440$	0	0
$441$	−6.92615	−0.329817
$442$	0	0
$443$	−8.39353	−0.398789	−0.199394	−	0.979919i	$-0.563897\pi$
$443$	−8.39353	−0.398789	−0.199394	+	0.979919i	$0.563897\pi$
$444$	0	0
$445$	−46.4728	−2.20302
$446$	0	0
$447$	−10.2153	−0.483167
$448$	0	0
$449$	−25.6101	−1.20861	−0.604307	−	0.796752i	$-0.706550\pi$
$449$	−25.6101	−1.20861	−0.604307	+	0.796752i	$0.706550\pi$
$450$	0	0
$451$	−24.6451	−1.16049
$452$	0	0
$453$	−16.7767	−0.788240
$454$	0	0
$455$	5.27652	0.247367
$456$	0	0
$457$	−8.26415	−0.386580	−0.193290	−	0.981142i	$-0.561916\pi$
$457$	−8.26415	−0.386580	−0.193290	+	0.981142i	$0.561916\pi$
$458$	0	0
$459$	7.90123	0.368798
$460$	0	0
$461$	−23.7842	−1.10774	−0.553870	−	0.832603i	$-0.686850\pi$
$461$	−23.7842	−1.10774	−0.553870	+	0.832603i	$0.686850\pi$
$462$	0	0
$463$	28.4554	1.32243	0.661217	−	0.750195i	$-0.270040\pi$
$463$	28.4554	1.32243	0.661217	+	0.750195i	$0.270040\pi$
$464$	0	0
$465$	23.3266	1.08174
$466$	0	0
$467$	−35.5643	−1.64572	−0.822859	−	0.568245i	$-0.807622\pi$
$467$	−35.5643	−1.64572	−0.822859	+	0.568245i	$0.807622\pi$
$468$	0	0
$469$	−0.138211	−0.00638201
$470$	0	0
$471$	21.9816	1.01286
$472$	0	0
$473$	9.63418	0.442980
$474$	0	0
$475$	−3.25499	−0.149349
$476$	0	0
$477$	−3.63270	−0.166330
$478$	0	0
$479$	42.1906	1.92774	0.963868	−	0.266380i	$-0.0858276\pi$
$479$	42.1906	1.92774	0.963868	+	0.266380i	$0.0858276\pi$
$480$	0	0
$481$	31.7878	1.44940
$482$	0	0
$483$	0.271751	0.0123651
$484$	0	0
$485$	10.4710	0.475463
$486$	0	0
$487$	−14.6201	−0.662499	−0.331249	−	0.943543i	$-0.607470\pi$
$487$	−14.6201	−0.662499	−0.331249	+	0.943543i	$0.607470\pi$
$488$	0	0
$489$	−13.1982	−0.596843
$490$	0	0
$491$	−9.22278	−0.416218	−0.208109	−	0.978106i	$-0.566731\pi$
$491$	−9.22278	−0.416218	−0.208109	+	0.978106i	$0.566731\pi$
$492$	0	0
$493$	−7.90123	−0.355853
$494$	0	0
$495$	−11.0359	−0.496028
$496$	0	0
$497$	2.16064	0.0969182
$498$	0	0
$499$	−28.1745	−1.26126	−0.630632	−	0.776082i	$-0.717204\pi$
$499$	−28.1745	−1.26126	−0.630632	+	0.776082i	$0.717204\pi$
$500$	0	0
$501$	10.0383	0.448480
$502$	0	0
$503$	−15.8419	−0.706354	−0.353177	−	0.935557i	$-0.614899\pi$
$503$	−15.8419	−0.706354	−0.353177	+	0.935557i	$0.614899\pi$
$504$	0	0
$505$	−50.1118	−2.22995
$506$	0	0
$507$	17.7196	0.786957
$508$	0	0
$509$	−3.22033	−0.142739	−0.0713693	−	0.997450i	$-0.522737\pi$
$509$	−3.22033	−0.142739	−0.0713693	+	0.997450i	$0.522737\pi$
$510$	0	0
$511$	−2.82380	−0.124917
$512$	0	0
$513$	−0.447567	−0.0197606
$514$	0	0
$515$	23.8758	1.05209
$516$	0	0
$517$	20.6819	0.909589
$518$	0	0
$519$	−2.62068	−0.115035
$520$	0	0
$521$	−31.0912	−1.36213	−0.681064	−	0.732224i	$-0.738483\pi$
$521$	−31.0912	−1.36213	−0.681064	+	0.732224i	$0.738483\pi$
$522$	0	0
$523$	4.78853	0.209388	0.104694	−	0.994505i	$-0.466614\pi$
$523$	4.78853	0.209388	0.104694	+	0.994505i	$0.466614\pi$
$524$	0	0
$525$	1.97634	0.0862546
$526$	0	0
$527$	52.6110	2.29177
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.18341	−0.398526
$532$	0	0
$533$	43.3609	1.87817
$534$	0	0
$535$	−48.7144	−2.10611
$536$	0	0
$537$	19.2949	0.832637
$538$	0	0
$539$	21.8189	0.939805
$540$	0	0
$541$	2.14598	0.0922628	0.0461314	−	0.998935i	$-0.485311\pi$
$541$	2.14598	0.0922628	0.0461314	+	0.998935i	$0.485311\pi$
$542$	0	0
$543$	1.68199	0.0721809
$544$	0	0
$545$	46.7570	2.00285
$546$	0	0
$547$	11.1544	0.476926	0.238463	−	0.971152i	$-0.423356\pi$
$547$	11.1544	0.476926	0.238463	+	0.971152i	$0.423356\pi$
$548$	0	0
$549$	0.878199	0.0374806
$550$	0	0
$551$	0.447567	0.0190670
$552$	0	0
$553$	−4.08353	−0.173649
$554$	0	0
$555$	20.0919	0.852855
$556$	0	0
$557$	−16.7212	−0.708498	−0.354249	−	0.935151i	$-0.615264\pi$
$557$	−16.7212	−0.708498	−0.354249	+	0.935151i	$0.615264\pi$
$558$	0	0
$559$	−16.9505	−0.716930
$560$	0	0
$561$	−24.8906	−1.05088
$562$	0	0
$563$	−14.8989	−0.627915	−0.313957	−	0.949437i	$-0.601655\pi$
$563$	−14.8989	−0.627915	−0.313957	+	0.949437i	$0.601655\pi$
$564$	0	0
$565$	−2.31815	−0.0975255
$566$	0	0
$567$	0.271751	0.0114125
$568$	0	0
$569$	−28.7000	−1.20317	−0.601584	−	0.798809i	$-0.705464\pi$
$569$	−28.7000	−1.20317	−0.601584	+	0.798809i	$0.705464\pi$
$570$	0	0
$571$	−3.99280	−0.167093	−0.0835467	−	0.996504i	$-0.526625\pi$
$571$	−3.99280	−0.167093	−0.0835467	+	0.996504i	$0.526625\pi$
$572$	0	0
$573$	−5.91755	−0.247209
$574$	0	0
$575$	7.27263	0.303290
$576$	0	0
$577$	34.4039	1.43226	0.716128	−	0.697969i	$-0.245913\pi$
$577$	34.4039	1.43226	0.716128	+	0.697969i	$0.245913\pi$
$578$	0	0
$579$	−9.40360	−0.390800
$580$	0	0
$581$	−2.19724	−0.0911569
$582$	0	0
$583$	11.4438	0.473953
$584$	0	0
$585$	19.4168	0.802785
$586$	0	0
$587$	−31.8196	−1.31334	−0.656668	−	0.754180i	$-0.728035\pi$
$587$	−31.8196	−1.31334	−0.656668	+	0.754180i	$0.728035\pi$
$588$	0	0
$589$	−2.98016	−0.122795
$590$	0	0
$591$	16.1496	0.664305
$592$	0	0
$593$	−16.8271	−0.691004	−0.345502	−	0.938418i	$-0.612291\pi$
$593$	−16.8271	−0.691004	−0.345502	+	0.938418i	$0.612291\pi$
$594$	0	0
$595$	7.52201	0.308372
$596$	0	0
$597$	6.67857	0.273336
$598$	0	0
$599$	9.44446	0.385890	0.192945	−	0.981210i	$-0.438196\pi$
$599$	9.44446	0.385890	0.192945	+	0.981210i	$0.438196\pi$
$600$	0	0
$601$	−33.7434	−1.37642	−0.688210	−	0.725511i	$-0.741603\pi$
$601$	−33.7434	−1.37642	−0.688210	+	0.725511i	$0.741603\pi$
$602$	0	0
$603$	−0.508596	−0.0207116
$604$	0	0
$605$	−3.76998	−0.153271
$606$	0	0
$607$	−23.8922	−0.969756	−0.484878	−	0.874582i	$-0.661136\pi$
$607$	−23.8922	−0.969756	−0.484878	+	0.874582i	$0.661136\pi$
$608$	0	0
$609$	−0.271751	−0.0110119
$610$	0	0
$611$	−36.3880	−1.47210
$612$	0	0
$613$	44.2826	1.78856	0.894278	−	0.447512i	$-0.147690\pi$
$613$	44.2826	1.78856	0.894278	+	0.447512i	$0.147690\pi$
$614$	0	0
$615$	27.4068	1.10515
$616$	0	0
$617$	−28.6113	−1.15185	−0.575925	−	0.817503i	$-0.695358\pi$
$617$	−28.6113	−1.15185	−0.575925	+	0.817503i	$0.695358\pi$
$618$	0	0
$619$	−18.7046	−0.751800	−0.375900	−	0.926660i	$-0.622666\pi$
$619$	−18.7046	−0.751800	−0.375900	+	0.926660i	$0.622666\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−3.60496	−0.144430
$624$	0	0
$625$	−8.47201	−0.338880
$626$	0	0
$627$	1.40993	0.0563072
$628$	0	0
$629$	45.3156	1.80685
$630$	0	0
$631$	36.0637	1.43567	0.717837	−	0.696211i	$-0.245132\pi$
$631$	36.0637	1.43567	0.717837	+	0.696211i	$0.245132\pi$
$632$	0	0
$633$	6.03725	0.239959
$634$	0	0
$635$	69.7107	2.76638
$636$	0	0
$637$	−38.3884	−1.52100
$638$	0	0
$639$	7.95084	0.314530
$640$	0	0
$641$	0.607972	0.0240135	0.0120067	−	0.999928i	$-0.496178\pi$
$641$	0.607972	0.0240135	0.0120067	+	0.999928i	$0.496178\pi$
$642$	0	0
$643$	6.14478	0.242326	0.121163	−	0.992633i	$-0.461338\pi$
$643$	6.14478	0.242326	0.121163	+	0.992633i	$0.461338\pi$
$644$	0	0
$645$	−10.7138	−0.421855
$646$	0	0
$647$	42.7096	1.67909	0.839544	−	0.543292i	$-0.182822\pi$
$647$	42.7096	1.67909	0.839544	+	0.543292i	$0.182822\pi$
$648$	0	0
$649$	28.9297	1.13559
$650$	0	0
$651$	1.80948	0.0709189
$652$	0	0
$653$	30.3806	1.18888	0.594442	−	0.804138i	$-0.297373\pi$
$653$	30.3806	1.18888	0.594442	+	0.804138i	$0.297373\pi$
$654$	0	0
$655$	55.6958	2.17622
$656$	0	0
$657$	−10.3911	−0.405397
$658$	0	0
$659$	−28.7657	−1.12055	−0.560276	−	0.828306i	$-0.689305\pi$
$659$	−28.7657	−1.12055	−0.560276	+	0.828306i	$0.689305\pi$
$660$	0	0
$661$	33.0608	1.28592	0.642958	−	0.765901i	$-0.277707\pi$
$661$	33.0608	1.28592	0.642958	+	0.765901i	$0.277707\pi$
$662$	0	0
$663$	43.7928	1.70077
$664$	0	0
$665$	−0.426086	−0.0165229
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	−25.6466	−0.991556
$670$	0	0
$671$	−2.76652	−0.106800
$672$	0	0
$673$	12.6691	0.488359	0.244180	−	0.969730i	$-0.421481\pi$
$673$	12.6691	0.488359	0.244180	+	0.969730i	$0.421481\pi$
$674$	0	0
$675$	7.27263	0.279924
$676$	0	0
$677$	−4.77767	−0.183621	−0.0918104	−	0.995777i	$-0.529265\pi$
$677$	−4.77767	−0.183621	−0.0918104	+	0.995777i	$0.529265\pi$
$678$	0	0
$679$	0.812249	0.0311712
$680$	0	0
$681$	0.782140	0.0299717
$682$	0	0
$683$	7.86732	0.301035	0.150517	−	0.988607i	$-0.451906\pi$
$683$	7.86732	0.301035	0.150517	+	0.988607i	$0.451906\pi$
$684$	0	0
$685$	59.0690	2.25691
$686$	0	0
$687$	−17.9486	−0.684783
$688$	0	0
$689$	−20.1344	−0.767058
$690$	0	0
$691$	25.4938	0.969830	0.484915	−	0.874561i	$-0.338851\pi$
$691$	25.4938	0.969830	0.484915	+	0.874561i	$0.338851\pi$
$692$	0	0
$693$	−0.856073	−0.0325195
$694$	0	0
$695$	−37.9356	−1.43898
$696$	0	0
$697$	61.8137	2.34136
$698$	0	0
$699$	16.2525	0.614724
$700$	0	0
$701$	−35.3843	−1.33645	−0.668223	−	0.743961i	$-0.732945\pi$
$701$	−35.3843	−1.33645	−0.668223	+	0.743961i	$0.732945\pi$
$702$	0	0
$703$	−2.56691	−0.0968129
$704$	0	0
$705$	−22.9995	−0.866212
$706$	0	0
$707$	−3.88724	−0.146195
$708$	0	0
$709$	−1.31981	−0.0495665	−0.0247832	−	0.999693i	$-0.507890\pi$
$709$	−1.31981	−0.0495665	−0.0247832	+	0.999693i	$0.507890\pi$
$710$	0	0
$711$	−15.0267	−0.563547
$712$	0	0
$713$	6.65859	0.249366
$714$	0	0
$715$	−61.1670	−2.28752
$716$	0	0
$717$	−20.0022	−0.746996
$718$	0	0
$719$	43.7563	1.63184	0.815918	−	0.578168i	$-0.196232\pi$
$719$	43.7563	1.63184	0.815918	+	0.578168i	$0.196232\pi$
$720$	0	0
$721$	1.85208	0.0689751
$722$	0	0
$723$	−18.5415	−0.689567
$724$	0	0
$725$	−7.27263	−0.270099
$726$	0	0
$727$	36.4333	1.35124	0.675618	−	0.737252i	$-0.263877\pi$
$727$	36.4333	1.35124	0.675618	+	0.737252i	$0.263877\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−24.1640	−0.893738
$732$	0	0
$733$	34.8828	1.28842	0.644212	−	0.764847i	$-0.277185\pi$
$733$	34.8828	1.28842	0.644212	+	0.764847i	$0.277185\pi$
$734$	0	0
$735$	−24.2639	−0.894988
$736$	0	0
$737$	1.60219	0.0590173
$738$	0	0
$739$	−8.65983	−0.318557	−0.159278	−	0.987234i	$-0.550917\pi$
$739$	−8.65983	−0.318557	−0.159278	+	0.987234i	$0.550917\pi$
$740$	0	0
$741$	−2.48065	−0.0911290
$742$	0	0
$743$	40.4395	1.48358	0.741792	−	0.670631i	$-0.233976\pi$
$743$	40.4395	1.48358	0.741792	+	0.670631i	$0.233976\pi$
$744$	0	0
$745$	−35.7866	−1.31112
$746$	0	0
$747$	−8.08550	−0.295833
$748$	0	0
$749$	−3.77884	−0.138076
$750$	0	0
$751$	−36.1760	−1.32008	−0.660041	−	0.751230i	$-0.729461\pi$
$751$	−36.1760	−1.32008	−0.660041	+	0.751230i	$0.729461\pi$
$752$	0	0
$753$	2.34485	0.0854511
$754$	0	0
$755$	−58.7728	−2.13896
$756$	0	0
$757$	−33.6143	−1.22173	−0.610866	−	0.791734i	$-0.709178\pi$
$757$	−33.6143	−1.22173	−0.610866	+	0.791734i	$0.709178\pi$
$758$	0	0
$759$	−3.15022	−0.114346
$760$	0	0
$761$	8.72408	0.316248	0.158124	−	0.987419i	$-0.449455\pi$
$761$	8.72408	0.316248	0.158124	+	0.987419i	$0.449455\pi$
$762$	0	0
$763$	3.62701	0.131306
$764$	0	0
$765$	27.6798	1.00077
$766$	0	0
$767$	−50.8993	−1.83787
$768$	0	0
$769$	37.1907	1.34113	0.670566	−	0.741850i	$-0.266051\pi$
$769$	37.1907	1.34113	0.670566	+	0.741850i	$0.266051\pi$
$770$	0	0
$771$	5.81081	0.209271
$772$	0	0
$773$	−16.8445	−0.605855	−0.302928	−	0.953014i	$-0.597964\pi$
$773$	−16.8445	−0.605855	−0.302928	+	0.953014i	$0.597964\pi$
$774$	0	0
$775$	48.4255	1.73949
$776$	0	0
$777$	1.55856	0.0559130
$778$	0	0
$779$	−3.50145	−0.125452
$780$	0	0
$781$	−25.0468	−0.896247
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	77.0067	2.74849
$786$	0	0
$787$	6.07688	0.216617	0.108309	−	0.994117i	$-0.465456\pi$
$787$	6.07688	0.216617	0.108309	+	0.994117i	$0.465456\pi$
$788$	0	0
$789$	18.5997	0.662166
$790$	0	0
$791$	−0.179822	−0.00639375
$792$	0	0
$793$	4.86745	0.172848
$794$	0	0
$795$	−12.7262	−0.451351
$796$	0	0
$797$	−3.06586	−0.108598	−0.0542992	−	0.998525i	$-0.517292\pi$
$797$	−3.06586	−0.108598	−0.0542992	+	0.998525i	$0.517292\pi$
$798$	0	0
$799$	−51.8734	−1.83515
$800$	0	0
$801$	−13.2657	−0.468721
$802$	0	0
$803$	32.7343	1.15517
$804$	0	0
$805$	0.952005	0.0335538
$806$	0	0
$807$	−9.99523	−0.351849
$808$	0	0
$809$	5.74699	0.202053	0.101027	−	0.994884i	$-0.467787\pi$
$809$	5.74699	0.202053	0.101027	+	0.994884i	$0.467787\pi$
$810$	0	0
$811$	−10.6321	−0.373345	−0.186672	−	0.982422i	$-0.559770\pi$
$811$	−10.6321	−0.373345	−0.186672	+	0.982422i	$0.559770\pi$
$812$	0	0
$813$	13.4259	0.470868
$814$	0	0
$815$	−46.2364	−1.61959
$816$	0	0
$817$	1.36877	0.0478874
$818$	0	0
$819$	1.50619	0.0526304
$820$	0	0
$821$	−24.2293	−0.845607	−0.422803	−	0.906221i	$-0.638954\pi$
$821$	−24.2293	−0.845607	−0.422803	+	0.906221i	$0.638954\pi$
$822$	0	0
$823$	51.5784	1.79791	0.898955	−	0.438041i	$-0.144327\pi$
$823$	51.5784	1.79791	0.898955	+	0.438041i	$0.144327\pi$
$824$	0	0
$825$	−22.9103	−0.797636
$826$	0	0
$827$	4.15760	0.144574	0.0722869	−	0.997384i	$-0.476970\pi$
$827$	4.15760	0.144574	0.0722869	+	0.997384i	$0.476970\pi$
$828$	0	0
$829$	11.6005	0.402902	0.201451	−	0.979499i	$-0.435434\pi$
$829$	11.6005	0.402902	0.201451	+	0.979499i	$0.435434\pi$
$830$	0	0
$831$	26.3946	0.915618
$832$	0	0
$833$	−54.7251	−1.89611
$834$	0	0
$835$	35.1666	1.21699
$836$	0	0
$837$	6.65859	0.230155
$838$	0	0
$839$	4.00532	0.138279	0.0691395	−	0.997607i	$-0.477975\pi$
$839$	4.00532	0.138279	0.0691395	+	0.997607i	$0.477975\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	4.24115	0.146073
$844$	0	0
$845$	62.0760	2.13548
$846$	0	0
$847$	−0.292443	−0.0100484
$848$	0	0
$849$	29.1562	1.00064
$850$	0	0
$851$	5.73526	0.196602
$852$	0	0
$853$	31.7370	1.08665	0.543327	−	0.839521i	$-0.317165\pi$
$853$	31.7370	1.08665	0.543327	+	0.839521i	$0.317165\pi$
$854$	0	0
$855$	−1.56793	−0.0536221
$856$	0	0
$857$	6.72548	0.229738	0.114869	−	0.993381i	$-0.463355\pi$
$857$	6.72548	0.229738	0.114869	+	0.993381i	$0.463355\pi$
$858$	0	0
$859$	1.42040	0.0484633	0.0242316	−	0.999706i	$-0.492286\pi$
$859$	1.42040	0.0484633	0.0242316	+	0.999706i	$0.492286\pi$
$860$	0	0
$861$	2.12599	0.0724534
$862$	0	0
$863$	17.3070	0.589139	0.294569	−	0.955630i	$-0.404824\pi$
$863$	17.3070	0.589139	0.294569	+	0.955630i	$0.404824\pi$
$864$	0	0
$865$	−9.18085	−0.312158
$866$	0	0
$867$	45.4294	1.54286
$868$	0	0
$869$	47.3375	1.60581
$870$	0	0
$871$	−2.81891	−0.0955151
$872$	0	0
$873$	2.98895	0.101161
$874$	0	0
$875$	2.16356	0.0731415
$876$	0	0
$877$	23.1992	0.783383	0.391691	−	0.920097i	$-0.371890\pi$
$877$	23.1992	0.783383	0.391691	+	0.920097i	$0.371890\pi$
$878$	0	0
$879$	7.66800	0.258635
$880$	0	0
$881$	−32.7212	−1.10241	−0.551203	−	0.834371i	$-0.685831\pi$
$881$	−32.7212	−1.10241	−0.551203	+	0.834371i	$0.685831\pi$
$882$	0	0
$883$	14.2784	0.480506	0.240253	−	0.970710i	$-0.422770\pi$
$883$	14.2784	0.480506	0.240253	+	0.970710i	$0.422770\pi$
$884$	0	0
$885$	−32.1716	−1.08144
$886$	0	0
$887$	−0.290019	−0.00973788	−0.00486894	−	0.999988i	$-0.501550\pi$
$887$	−0.290019	−0.00973788	−0.00486894	+	0.999988i	$0.501550\pi$
$888$	0	0
$889$	5.40756	0.181364
$890$	0	0
$891$	−3.15022	−0.105536
$892$	0	0
$893$	2.93838	0.0983291
$894$	0	0
$895$	67.5946	2.25944
$896$	0	0
$897$	5.54253	0.185060
$898$	0	0
$899$	−6.65859	−0.222076
$900$	0	0
$901$	−28.7028	−0.956229
$902$	0	0
$903$	−0.831084	−0.0276567
$904$	0	0
$905$	5.89238	0.195869
$906$	0	0
$907$	10.2597	0.340666	0.170333	−	0.985387i	$-0.445516\pi$
$907$	10.2597	0.340666	0.170333	+	0.985387i	$0.445516\pi$
$908$	0	0
$909$	−14.3045	−0.474449
$910$	0	0
$911$	29.7117	0.984392	0.492196	−	0.870484i	$-0.336194\pi$
$911$	29.7117	0.984392	0.492196	+	0.870484i	$0.336194\pi$
$912$	0	0
$913$	25.4711	0.842969
$914$	0	0
$915$	3.07653	0.101707
$916$	0	0
$917$	4.32040	0.142672
$918$	0	0
$919$	−5.18904	−0.171171	−0.0855853	−	0.996331i	$-0.527276\pi$
$919$	−5.18904	−0.171171	−0.0855853	+	0.996331i	$0.527276\pi$
$920$	0	0
$921$	11.5950	0.382067
$922$	0	0
$923$	44.0678	1.45051
$924$	0	0
$925$	41.7104	1.37143
$926$	0	0
$927$	6.81537	0.223846
$928$	0	0
$929$	−18.0854	−0.593362	−0.296681	−	0.954977i	$-0.595880\pi$
$929$	−18.0854	−0.593362	−0.296681	+	0.954977i	$0.595880\pi$
$930$	0	0
$931$	3.09991	0.101596
$932$	0	0
$933$	3.22439	0.105562
$934$	0	0
$935$	−87.1974	−2.85166
$936$	0	0
$937$	21.9073	0.715679	0.357840	−	0.933783i	$-0.383513\pi$
$937$	21.9073	0.715679	0.357840	+	0.933783i	$0.383513\pi$
$938$	0	0
$939$	−18.5121	−0.604120
$940$	0	0
$941$	−27.2754	−0.889151	−0.444576	−	0.895741i	$-0.646646\pi$
$941$	−27.2754	−0.889151	−0.444576	+	0.895741i	$0.646646\pi$
$942$	0	0
$943$	7.82330	0.254762
$944$	0	0
$945$	0.952005	0.0309687
$946$	0	0
$947$	−14.4712	−0.470251	−0.235126	−	0.971965i	$-0.575550\pi$
$947$	−14.4712	−0.470251	−0.235126	+	0.971965i	$0.575550\pi$
$948$	0	0
$949$	−57.5932	−1.86955
$950$	0	0
$951$	34.8481	1.13003
$952$	0	0
$953$	13.8054	0.447201	0.223601	−	0.974681i	$-0.428219\pi$
$953$	13.8054	0.447201	0.223601	+	0.974681i	$0.428219\pi$
$954$	0	0
$955$	−20.7305	−0.670825
$956$	0	0
$957$	3.15022	0.101832
$958$	0	0
$959$	4.58207	0.147963
$960$	0	0
$961$	13.3368	0.430220
$962$	0	0
$963$	−13.9056	−0.448100
$964$	0	0
$965$	−32.9430	−1.06047
$966$	0	0
$967$	56.6537	1.82186	0.910930	−	0.412562i	$-0.135366\pi$
$967$	56.6537	1.82186	0.910930	+	0.412562i	$0.135366\pi$
$968$	0	0
$969$	−3.53633	−0.113603
$970$	0	0
$971$	−1.35956	−0.0436302	−0.0218151	−	0.999762i	$-0.506945\pi$
$971$	−1.35956	−0.0436302	−0.0218151	+	0.999762i	$0.506945\pi$
$972$	0	0
$973$	−2.94272	−0.0943393
$974$	0	0
$975$	40.3088	1.29091
$976$	0	0
$977$	−28.3722	−0.907707	−0.453854	−	0.891076i	$-0.649951\pi$
$977$	−28.3722	−0.907707	−0.453854	+	0.891076i	$0.649951\pi$
$978$	0	0
$979$	41.7898	1.33561
$980$	0	0
$981$	13.3468	0.426131
$982$	0	0
$983$	−53.9413	−1.72046	−0.860230	−	0.509906i	$-0.829680\pi$
$983$	−53.9413	−1.72046	−0.860230	+	0.509906i	$0.829680\pi$
$984$	0	0
$985$	56.5757	1.80265
$986$	0	0
$987$	−1.78411	−0.0567887
$988$	0	0
$989$	−3.05826	−0.0972470
$990$	0	0
$991$	−56.5273	−1.79565	−0.897825	−	0.440352i	$-0.854854\pi$
$991$	−56.5273	−1.79565	−0.897825	+	0.440352i	$0.854854\pi$
$992$	0	0
$993$	13.7037	0.434874
$994$	0	0
$995$	23.3966	0.741721
$996$	0	0
$997$	9.47644	0.300122	0.150061	−	0.988677i	$-0.452053\pi$
$997$	9.47644	0.300122	0.150061	+	0.988677i	$0.452053\pi$
$998$	0	0
$999$	5.73526	0.181456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.j.1.14	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.j.1.14	✓	16	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.50323 q^{5} +0.271751 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.50323 q^{5} +0.271751 q^{7} +1.00000 q^{9} -3.15022 q^{11} +5.54253 q^{13} +3.50323 q^{15} +7.90123 q^{17} -0.447567 q^{19} +0.271751 q^{21} +1.00000 q^{23} +7.27263 q^{25} +1.00000 q^{27} -1.00000 q^{29} +6.65859 q^{31} -3.15022 q^{33} +0.952005 q^{35} +5.73526 q^{37} +5.54253 q^{39} +7.82330 q^{41} -3.05826 q^{43} +3.50323 q^{45} -6.56523 q^{47} -6.92615 q^{49} +7.90123 q^{51} -3.63270 q^{53} -11.0359 q^{55} -0.447567 q^{57} -9.18341 q^{59} +0.878199 q^{61} +0.271751 q^{63} +19.4168 q^{65} -0.508596 q^{67} +1.00000 q^{69} +7.95084 q^{71} -10.3911 q^{73} +7.27263 q^{75} -0.856073 q^{77} -15.0267 q^{79} +1.00000 q^{81} -8.08550 q^{83} +27.6798 q^{85} -1.00000 q^{87} -13.2657 q^{89} +1.50619 q^{91} +6.65859 q^{93} -1.56793 q^{95} +2.98895 q^{97} -3.15022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9	\( 16 q + 16 q^{3} + 3 q^{5} + 4 q^{7} + 16 q^{9} + 5 q^{11} + 6 q^{13} + 3 q^{15} + 3 q^{17} + 11 q^{19} + 4 q^{21} + 16 q^{23} + 27 q^{25} + 16 q^{27} - 16 q^{29} + 14 q^{31} + 5 q^{33} + 11 q^{35} + 4 q^{37} + 6 q^{39} + 11 q^{41} + 23 q^{43} + 3 q^{45} - 2 q^{47} + 34 q^{49} + 3 q^{51} + 19 q^{53} + 31 q^{55} + 11 q^{57} + 32 q^{59} + 19 q^{61} + 4 q^{63} + 6 q^{65} + 33 q^{67} + 16 q^{69} - 5 q^{71} + 23 q^{73} + 27 q^{75} + 42 q^{77} + 24 q^{79} + 16 q^{81} + 7 q^{83} - 16 q^{87} - 2 q^{89} + 25 q^{91} + 14 q^{93} + 7 q^{95} + 33 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q + 16 * q^3 + 3 * q^5 + 4 * q^7 + 16 * q^9 + 5 * q^11 + 6 * q^13 + 3 * q^15 + 3 * q^17 + 11 * q^19 + 4 * q^21 + 16 * q^23 + 27 * q^25 + 16 * q^27 - 16 * q^29 + 14 * q^31 + 5 * q^33 + 11 * q^35 + 4 * q^37 + 6 * q^39 + 11 * q^41 + 23 * q^43 + 3 * q^45 - 2 * q^47 + 34 * q^49 + 3 * q^51 + 19 * q^53 + 31 * q^55 + 11 * q^57 + 32 * q^59 + 19 * q^61 + 4 * q^63 + 6 * q^65 + 33 * q^67 + 16 * q^69 - 5 * q^71 + 23 * q^73 + 27 * q^75 + 42 * q^77 + 24 * q^79 + 16 * q^81 + 7 * q^83 - 16 * q^87 - 2 * q^89 + 25 * q^91 + 14 * q^93 + 7 * q^95 + 33 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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