LMFDB - Embedded newform 8008.2.a.m.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 11x^{7} + 46x^{6} + 37x^{5} - 169x^{4} - 18x^{3} + 195x^{2} - 72x + 7 $ x^9 - 4x^8 - 11x^7 + 46x^6 + 37x^5 - 169x^4 - 18x^3 + 195x^2 - 72x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.82684$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.82684 q^{3} -0.265205 q^{5} -1.00000 q^{7} +4.99105 q^{9} +O(q^{10})$	$q+2.82684 q^{3} -0.265205 q^{5} -1.00000 q^{7} +4.99105 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.749692 q^{15} -0.785132 q^{17} -3.08068 q^{19} -2.82684 q^{21} -9.36931 q^{23} -4.92967 q^{25} +5.62839 q^{27} +0.297572 q^{29} +10.2590 q^{31} -2.82684 q^{33} +0.265205 q^{35} -5.76202 q^{37} +2.82684 q^{39} -6.19755 q^{41} -12.6754 q^{43} -1.32365 q^{45} -13.3898 q^{47} +1.00000 q^{49} -2.21944 q^{51} -2.48756 q^{53} +0.265205 q^{55} -8.70861 q^{57} -11.4753 q^{59} +12.9236 q^{61} -4.99105 q^{63} -0.265205 q^{65} +10.7680 q^{67} -26.4856 q^{69} +5.10840 q^{71} +12.8388 q^{73} -13.9354 q^{75} +1.00000 q^{77} +1.88025 q^{79} +0.937431 q^{81} -4.99097 q^{83} +0.208220 q^{85} +0.841189 q^{87} -0.731996 q^{89} -1.00000 q^{91} +29.0005 q^{93} +0.817011 q^{95} -0.817154 q^{97} -4.99105 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) $ 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * 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
$3$	2.82684	1.63208	0.816040	−	0.577996i	$-0.196165\pi$
$3$	2.82684	1.63208	0.816040	+	0.577996i	$0.196165\pi$
$4$	0	0
$5$	−0.265205	−0.118603	−0.0593015	−	0.998240i	$-0.518887\pi$
$5$	−0.265205	−0.118603	−0.0593015	+	0.998240i	$0.518887\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.99105	1.66368
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−0.749692	−0.193570
$16$	0	0
$17$	−0.785132	−0.190422	−0.0952112	−	0.995457i	$-0.530353\pi$
$17$	−0.785132	−0.190422	−0.0952112	+	0.995457i	$0.530353\pi$
$18$	0	0
$19$	−3.08068	−0.706757	−0.353379	−	0.935480i	$-0.614967\pi$
$19$	−3.08068	−0.706757	−0.353379	+	0.935480i	$0.614967\pi$
$20$	0	0
$21$	−2.82684	−0.616868
$22$	0	0
$23$	−9.36931	−1.95364	−0.976818	−	0.214073i	$-0.931327\pi$
$23$	−9.36931	−1.95364	−0.976818	+	0.214073i	$0.931327\pi$
$24$	0	0
$25$	−4.92967	−0.985933
$26$	0	0
$27$	5.62839	1.08318
$28$	0	0
$29$	0.297572	0.0552577	0.0276289	−	0.999618i	$-0.491204\pi$
$29$	0.297572	0.0552577	0.0276289	+	0.999618i	$0.491204\pi$
$30$	0	0
$31$	10.2590	1.84256	0.921282	−	0.388894i	$-0.127143\pi$
$31$	10.2590	1.84256	0.921282	+	0.388894i	$0.127143\pi$
$32$	0	0
$33$	−2.82684	−0.492090
$34$	0	0
$35$	0.265205	0.0448278
$36$	0	0
$37$	−5.76202	−0.947271	−0.473635	−	0.880721i	$-0.657059\pi$
$37$	−5.76202	−0.947271	−0.473635	+	0.880721i	$0.657059\pi$
$38$	0	0
$39$	2.82684	0.452657
$40$	0	0
$41$	−6.19755	−0.967894	−0.483947	−	0.875097i	$-0.660797\pi$
$41$	−6.19755	−0.967894	−0.483947	+	0.875097i	$0.660797\pi$
$42$	0	0
$43$	−12.6754	−1.93298	−0.966488	−	0.256713i	$-0.917360\pi$
$43$	−12.6754	−1.93298	−0.966488	+	0.256713i	$0.917360\pi$
$44$	0	0
$45$	−1.32365	−0.197318
$46$	0	0
$47$	−13.3898	−1.95310	−0.976549	−	0.215296i	$-0.930928\pi$
$47$	−13.3898	−1.95310	−0.976549	+	0.215296i	$0.930928\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.21944	−0.310784
$52$	0	0
$53$	−2.48756	−0.341693	−0.170846	−	0.985298i	$-0.554650\pi$
$53$	−2.48756	−0.341693	−0.170846	+	0.985298i	$0.554650\pi$
$54$	0	0
$55$	0.265205	0.0357602
$56$	0	0
$57$	−8.70861	−1.15348
$58$	0	0
$59$	−11.4753	−1.49395	−0.746976	−	0.664851i	$-0.768495\pi$
$59$	−11.4753	−1.49395	−0.746976	+	0.664851i	$0.768495\pi$
$60$	0	0
$61$	12.9236	1.65470	0.827350	−	0.561687i	$-0.189848\pi$
$61$	12.9236	1.65470	0.827350	+	0.561687i	$0.189848\pi$
$62$	0	0
$63$	−4.99105	−0.628813
$64$	0	0
$65$	−0.265205	−0.0328946
$66$	0	0
$67$	10.7680	1.31552	0.657762	−	0.753226i	$-0.271503\pi$
$67$	10.7680	1.31552	0.657762	+	0.753226i	$0.271503\pi$
$68$	0	0
$69$	−26.4856	−3.18849
$70$	0	0
$71$	5.10840	0.606255	0.303128	−	0.952950i	$-0.401969\pi$
$71$	5.10840	0.606255	0.303128	+	0.952950i	$0.401969\pi$
$72$	0	0
$73$	12.8388	1.50266	0.751331	−	0.659926i	$-0.229412\pi$
$73$	12.8388	1.50266	0.751331	+	0.659926i	$0.229412\pi$
$74$	0	0
$75$	−13.9354	−1.60912
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	1.88025	0.211545	0.105773	−	0.994390i	$-0.466268\pi$
$79$	1.88025	0.211545	0.105773	+	0.994390i	$0.466268\pi$
$80$	0	0
$81$	0.937431	0.104159
$82$	0	0
$83$	−4.99097	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$83$	−4.99097	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$84$	0	0
$85$	0.208220	0.0225847
$86$	0	0
$87$	0.841189	0.0901850
$88$	0	0
$89$	−0.731996	−0.0775915	−0.0387957	−	0.999247i	$-0.512352\pi$
$89$	−0.731996	−0.0775915	−0.0387957	+	0.999247i	$0.512352\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	29.0005	3.00721
$94$	0	0
$95$	0.817011	0.0838236
$96$	0	0
$97$	−0.817154	−0.0829694	−0.0414847	−	0.999139i	$-0.513209\pi$
$97$	−0.817154	−0.0829694	−0.0414847	+	0.999139i	$0.513209\pi$
$98$	0	0
$99$	−4.99105	−0.501619
$100$	0	0
$101$	−3.80467	−0.378579	−0.189290	−	0.981921i	$-0.560618\pi$
$101$	−3.80467	−0.378579	−0.189290	+	0.981921i	$0.560618\pi$
$102$	0	0
$103$	−16.9162	−1.66680	−0.833401	−	0.552669i	$-0.813609\pi$
$103$	−16.9162	−1.66680	−0.833401	+	0.552669i	$0.813609\pi$
$104$	0	0
$105$	0.749692	0.0731625
$106$	0	0
$107$	3.26529	0.315667	0.157834	−	0.987466i	$-0.449549\pi$
$107$	3.26529	0.315667	0.157834	+	0.987466i	$0.449549\pi$
$108$	0	0
$109$	−0.523088	−0.0501028	−0.0250514	−	0.999686i	$-0.507975\pi$
$109$	−0.523088	−0.0501028	−0.0250514	+	0.999686i	$0.507975\pi$
$110$	0	0
$111$	−16.2883	−1.54602
$112$	0	0
$113$	1.26187	0.118707	0.0593534	−	0.998237i	$-0.481096\pi$
$113$	1.26187	0.118707	0.0593534	+	0.998237i	$0.481096\pi$
$114$	0	0
$115$	2.48478	0.231707
$116$	0	0
$117$	4.99105	0.461423
$118$	0	0
$119$	0.785132	0.0719729
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−17.5195	−1.57968
$124$	0	0
$125$	2.63339	0.235538
$126$	0	0
$127$	7.95537	0.705925	0.352963	−	0.935637i	$-0.385174\pi$
$127$	7.95537	0.705925	0.352963	+	0.935637i	$0.385174\pi$
$128$	0	0
$129$	−35.8313	−3.15477
$130$	0	0
$131$	−16.4454	−1.43684	−0.718419	−	0.695611i	$-0.755134\pi$
$131$	−16.4454	−1.43684	−0.718419	+	0.695611i	$0.755134\pi$
$132$	0	0
$133$	3.08068	0.267129
$134$	0	0
$135$	−1.49267	−0.128469
$136$	0	0
$137$	−4.22395	−0.360876	−0.180438	−	0.983586i	$-0.557752\pi$
$137$	−4.22395	−0.360876	−0.180438	+	0.983586i	$0.557752\pi$
$138$	0	0
$139$	−6.51993	−0.553014	−0.276507	−	0.961012i	$-0.589177\pi$
$139$	−6.51993	−0.553014	−0.276507	+	0.961012i	$0.589177\pi$
$140$	0	0
$141$	−37.8508	−3.18761
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−0.0789174	−0.00655374
$146$	0	0
$147$	2.82684	0.233154
$148$	0	0
$149$	7.58253	0.621185	0.310592	−	0.950543i	$-0.399473\pi$
$149$	7.58253	0.621185	0.310592	+	0.950543i	$0.399473\pi$
$150$	0	0
$151$	5.45056	0.443560	0.221780	−	0.975097i	$-0.428813\pi$
$151$	5.45056	0.443560	0.221780	+	0.975097i	$0.428813\pi$
$152$	0	0
$153$	−3.91863	−0.316803
$154$	0	0
$155$	−2.72073	−0.218534
$156$	0	0
$157$	−4.98784	−0.398073	−0.199037	−	0.979992i	$-0.563781\pi$
$157$	−4.98784	−0.398073	−0.199037	+	0.979992i	$0.563781\pi$
$158$	0	0
$159$	−7.03194	−0.557669
$160$	0	0
$161$	9.36931	0.738405
$162$	0	0
$163$	15.5417	1.21732	0.608660	−	0.793431i	$-0.291707\pi$
$163$	15.5417	1.21732	0.608660	+	0.793431i	$0.291707\pi$
$164$	0	0
$165$	0.749692	0.0583635
$166$	0	0
$167$	4.19858	0.324896	0.162448	−	0.986717i	$-0.448061\pi$
$167$	4.19858	0.324896	0.162448	+	0.986717i	$0.448061\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.3758	−1.17582
$172$	0	0
$173$	10.3920	0.790091	0.395046	−	0.918662i	$-0.370729\pi$
$173$	10.3920	0.790091	0.395046	+	0.918662i	$0.370729\pi$
$174$	0	0
$175$	4.92967	0.372648
$176$	0	0
$177$	−32.4388	−2.43825
$178$	0	0
$179$	21.3803	1.59804	0.799020	−	0.601305i	$-0.205352\pi$
$179$	21.3803	1.59804	0.799020	+	0.601305i	$0.205352\pi$
$180$	0	0
$181$	4.09450	0.304341	0.152171	−	0.988354i	$-0.451374\pi$
$181$	4.09450	0.304341	0.152171	+	0.988354i	$0.451374\pi$
$182$	0	0
$183$	36.5331	2.70060
$184$	0	0
$185$	1.52812	0.112349
$186$	0	0
$187$	0.785132	0.0574145
$188$	0	0
$189$	−5.62839	−0.409405
$190$	0	0
$191$	4.11743	0.297927	0.148963	−	0.988843i	$-0.452406\pi$
$191$	4.11743	0.297927	0.148963	+	0.988843i	$0.452406\pi$
$192$	0	0
$193$	−12.2314	−0.880438	−0.440219	−	0.897890i	$-0.645099\pi$
$193$	−12.2314	−0.880438	−0.440219	+	0.897890i	$0.645099\pi$
$194$	0	0
$195$	−0.749692	−0.0536866
$196$	0	0
$197$	17.4917	1.24623	0.623115	−	0.782130i	$-0.285867\pi$
$197$	17.4917	1.24623	0.623115	+	0.782130i	$0.285867\pi$
$198$	0	0
$199$	8.73777	0.619404	0.309702	−	0.950834i	$-0.399771\pi$
$199$	8.73777	0.619404	0.309702	+	0.950834i	$0.399771\pi$
$200$	0	0
$201$	30.4396	2.14704
$202$	0	0
$203$	−0.297572	−0.0208855
$204$	0	0
$205$	1.64362	0.114795
$206$	0	0
$207$	−46.7627	−3.25023
$208$	0	0
$209$	3.08068	0.213095
$210$	0	0
$211$	−20.3994	−1.40436	−0.702178	−	0.712002i	$-0.747789\pi$
$211$	−20.3994	−1.40436	−0.702178	+	0.712002i	$0.747789\pi$
$212$	0	0
$213$	14.4407	0.989457
$214$	0	0
$215$	3.36157	0.229257
$216$	0	0
$217$	−10.2590	−0.696424
$218$	0	0
$219$	36.2932	2.45246
$220$	0	0
$221$	−0.785132	−0.0528137
$222$	0	0
$223$	−14.8087	−0.991664	−0.495832	−	0.868418i	$-0.665137\pi$
$223$	−14.8087	−0.991664	−0.495832	+	0.868418i	$0.665137\pi$
$224$	0	0
$225$	−24.6042	−1.64028
$226$	0	0
$227$	−15.9610	−1.05937	−0.529683	−	0.848196i	$-0.677689\pi$
$227$	−15.9610	−1.05937	−0.529683	+	0.848196i	$0.677689\pi$
$228$	0	0
$229$	−18.5798	−1.22779	−0.613893	−	0.789389i	$-0.710397\pi$
$229$	−18.5798	−1.22779	−0.613893	+	0.789389i	$0.710397\pi$
$230$	0	0
$231$	2.82684	0.185993
$232$	0	0
$233$	1.79238	0.117423	0.0587115	−	0.998275i	$-0.481301\pi$
$233$	1.79238	0.117423	0.0587115	+	0.998275i	$0.481301\pi$
$234$	0	0
$235$	3.55103	0.231643
$236$	0	0
$237$	5.31519	0.345258
$238$	0	0
$239$	11.0205	0.712857	0.356429	−	0.934323i	$-0.383994\pi$
$239$	11.0205	0.712857	0.356429	+	0.934323i	$0.383994\pi$
$240$	0	0
$241$	27.7693	1.78878	0.894388	−	0.447291i	$-0.147611\pi$
$241$	27.7693	1.78878	0.894388	+	0.447291i	$0.147611\pi$
$242$	0	0
$243$	−14.2352	−0.913188
$244$	0	0
$245$	−0.265205	−0.0169433
$246$	0	0
$247$	−3.08068	−0.196019
$248$	0	0
$249$	−14.1087	−0.894102
$250$	0	0
$251$	17.3406	1.09453	0.547264	−	0.836960i	$-0.315669\pi$
$251$	17.3406	1.09453	0.547264	+	0.836960i	$0.315669\pi$
$252$	0	0
$253$	9.36931	0.589043
$254$	0	0
$255$	0.588607	0.0368600
$256$	0	0
$257$	−26.6080	−1.65976	−0.829881	−	0.557941i	$-0.811592\pi$
$257$	−26.6080	−1.65976	−0.829881	+	0.557941i	$0.811592\pi$
$258$	0	0
$259$	5.76202	0.358035
$260$	0	0
$261$	1.48520	0.0919313
$262$	0	0
$263$	−0.692617	−0.0427086	−0.0213543	−	0.999772i	$-0.506798\pi$
$263$	−0.692617	−0.0427086	−0.0213543	+	0.999772i	$0.506798\pi$
$264$	0	0
$265$	0.659712	0.0405258
$266$	0	0
$267$	−2.06924	−0.126635
$268$	0	0
$269$	7.98912	0.487105	0.243553	−	0.969888i	$-0.421687\pi$
$269$	7.98912	0.487105	0.243553	+	0.969888i	$0.421687\pi$
$270$	0	0
$271$	23.8637	1.44961	0.724807	−	0.688952i	$-0.241929\pi$
$271$	23.8637	1.44961	0.724807	+	0.688952i	$0.241929\pi$
$272$	0	0
$273$	−2.82684	−0.171088
$274$	0	0
$275$	4.92967	0.297270
$276$	0	0
$277$	−17.1496	−1.03042	−0.515211	−	0.857064i	$-0.672286\pi$
$277$	−17.1496	−1.03042	−0.515211	+	0.857064i	$0.672286\pi$
$278$	0	0
$279$	51.2030	3.06544
$280$	0	0
$281$	−2.67201	−0.159399	−0.0796993	−	0.996819i	$-0.525396\pi$
$281$	−2.67201	−0.159399	−0.0796993	+	0.996819i	$0.525396\pi$
$282$	0	0
$283$	17.9001	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$283$	17.9001	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$284$	0	0
$285$	2.30956	0.136807
$286$	0	0
$287$	6.19755	0.365830
$288$	0	0
$289$	−16.3836	−0.963739
$290$	0	0
$291$	−2.30997	−0.135413
$292$	0	0
$293$	−16.2504	−0.949356	−0.474678	−	0.880160i	$-0.657435\pi$
$293$	−16.2504	−0.949356	−0.474678	+	0.880160i	$0.657435\pi$
$294$	0	0
$295$	3.04329	0.177187
$296$	0	0
$297$	−5.62839	−0.326592
$298$	0	0
$299$	−9.36931	−0.541841
$300$	0	0
$301$	12.6754	0.730596
$302$	0	0
$303$	−10.7552	−0.617871
$304$	0	0
$305$	−3.42740	−0.196252
$306$	0	0
$307$	−12.6558	−0.722307	−0.361153	−	0.932506i	$-0.617617\pi$
$307$	−12.6558	−0.722307	−0.361153	+	0.932506i	$0.617617\pi$
$308$	0	0
$309$	−47.8194	−2.72035
$310$	0	0
$311$	−4.45674	−0.252719	−0.126359	−	0.991985i	$-0.540329\pi$
$311$	−4.45674	−0.252719	−0.126359	+	0.991985i	$0.540329\pi$
$312$	0	0
$313$	−32.7096	−1.84886	−0.924428	−	0.381356i	$-0.875457\pi$
$313$	−32.7096	−1.84886	−0.924428	+	0.381356i	$0.875457\pi$
$314$	0	0
$315$	1.32365	0.0745792
$316$	0	0
$317$	9.56706	0.537340	0.268670	−	0.963232i	$-0.413416\pi$
$317$	9.56706	0.537340	0.268670	+	0.963232i	$0.413416\pi$
$318$	0	0
$319$	−0.297572	−0.0166608
$320$	0	0
$321$	9.23046	0.515194
$322$	0	0
$323$	2.41874	0.134582
$324$	0	0
$325$	−4.92967	−0.273449
$326$	0	0
$327$	−1.47869	−0.0817717
$328$	0	0
$329$	13.3898	0.738202
$330$	0	0
$331$	−23.4703	−1.29005	−0.645023	−	0.764163i	$-0.723152\pi$
$331$	−23.4703	−1.29005	−0.645023	+	0.764163i	$0.723152\pi$
$332$	0	0
$333$	−28.7585	−1.57596
$334$	0	0
$335$	−2.85573	−0.156025
$336$	0	0
$337$	−11.1936	−0.609757	−0.304878	−	0.952391i	$-0.598616\pi$
$337$	−11.1936	−0.609757	−0.304878	+	0.952391i	$0.598616\pi$
$338$	0	0
$339$	3.56711	0.193739
$340$	0	0
$341$	−10.2590	−0.555554
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.02409	0.378165
$346$	0	0
$347$	−12.9843	−0.697034	−0.348517	−	0.937302i	$-0.613315\pi$
$347$	−12.9843	−0.697034	−0.348517	+	0.937302i	$0.613315\pi$
$348$	0	0
$349$	26.7567	1.43225	0.716127	−	0.697970i	$-0.245913\pi$
$349$	26.7567	1.43225	0.716127	+	0.697970i	$0.245913\pi$
$350$	0	0
$351$	5.62839	0.300421
$352$	0	0
$353$	−17.6616	−0.940030	−0.470015	−	0.882659i	$-0.655751\pi$
$353$	−17.6616	−0.940030	−0.470015	+	0.882659i	$0.655751\pi$
$354$	0	0
$355$	−1.35477	−0.0719038
$356$	0	0
$357$	2.21944	0.117465
$358$	0	0
$359$	21.3485	1.12673	0.563367	−	0.826207i	$-0.309506\pi$
$359$	21.3485	1.12673	0.563367	+	0.826207i	$0.309506\pi$
$360$	0	0
$361$	−9.50939	−0.500494
$362$	0	0
$363$	2.82684	0.148371
$364$	0	0
$365$	−3.40490	−0.178220
$366$	0	0
$367$	−3.12107	−0.162919	−0.0814593	−	0.996677i	$-0.525958\pi$
$367$	−3.12107	−0.162919	−0.0814593	+	0.996677i	$0.525958\pi$
$368$	0	0
$369$	−30.9323	−1.61027
$370$	0	0
$371$	2.48756	0.129148
$372$	0	0
$373$	−10.5661	−0.547091	−0.273546	−	0.961859i	$-0.588196\pi$
$373$	−10.5661	−0.547091	−0.273546	+	0.961859i	$0.588196\pi$
$374$	0	0
$375$	7.44419	0.384416
$376$	0	0
$377$	0.297572	0.0153257
$378$	0	0
$379$	−8.94369	−0.459406	−0.229703	−	0.973261i	$-0.573776\pi$
$379$	−8.94369	−0.459406	−0.229703	+	0.973261i	$0.573776\pi$
$380$	0	0
$381$	22.4886	1.15213
$382$	0	0
$383$	−33.5323	−1.71342	−0.856711	−	0.515796i	$-0.827496\pi$
$383$	−33.5323	−1.71342	−0.856711	+	0.515796i	$0.827496\pi$
$384$	0	0
$385$	−0.265205	−0.0135161
$386$	0	0
$387$	−63.2634	−3.21586
$388$	0	0
$389$	−1.22393	−0.0620555	−0.0310278	−	0.999519i	$-0.509878\pi$
$389$	−1.22393	−0.0620555	−0.0310278	+	0.999519i	$0.509878\pi$
$390$	0	0
$391$	7.35614	0.372016
$392$	0	0
$393$	−46.4885	−2.34503
$394$	0	0
$395$	−0.498652	−0.0250899
$396$	0	0
$397$	−16.2798	−0.817059	−0.408529	−	0.912745i	$-0.633958\pi$
$397$	−16.2798	−0.817059	−0.408529	+	0.912745i	$0.633958\pi$
$398$	0	0
$399$	8.70861	0.435976
$400$	0	0
$401$	6.03846	0.301546	0.150773	−	0.988568i	$-0.451824\pi$
$401$	6.03846	0.301546	0.150773	+	0.988568i	$0.451824\pi$
$402$	0	0
$403$	10.2590	0.511036
$404$	0	0
$405$	−0.248611	−0.0123536
$406$	0	0
$407$	5.76202	0.285613
$408$	0	0
$409$	26.3627	1.30355	0.651776	−	0.758412i	$-0.274024\pi$
$409$	26.3627	1.30355	0.651776	+	0.758412i	$0.274024\pi$
$410$	0	0
$411$	−11.9405	−0.588979
$412$	0	0
$413$	11.4753	0.564661
$414$	0	0
$415$	1.32363	0.0649743
$416$	0	0
$417$	−18.4308	−0.902562
$418$	0	0
$419$	−1.97301	−0.0963881	−0.0481940	−	0.998838i	$-0.515347\pi$
$419$	−1.97301	−0.0963881	−0.0481940	+	0.998838i	$0.515347\pi$
$420$	0	0
$421$	11.1395	0.542905	0.271453	−	0.962452i	$-0.412496\pi$
$421$	11.1395	0.542905	0.271453	+	0.962452i	$0.412496\pi$
$422$	0	0
$423$	−66.8290	−3.24934
$424$	0	0
$425$	3.87044	0.187744
$426$	0	0
$427$	−12.9236	−0.625418
$428$	0	0
$429$	−2.82684	−0.136481
$430$	0	0
$431$	−3.48318	−0.167779	−0.0838894	−	0.996475i	$-0.526734\pi$
$431$	−3.48318	−0.167779	−0.0838894	+	0.996475i	$0.526734\pi$
$432$	0	0
$433$	−16.7879	−0.806774	−0.403387	−	0.915029i	$-0.632167\pi$
$433$	−16.7879	−0.806774	−0.403387	+	0.915029i	$0.632167\pi$
$434$	0	0
$435$	−0.223087	−0.0106962
$436$	0	0
$437$	28.8639	1.38075
$438$	0	0
$439$	13.4866	0.643682	0.321841	−	0.946794i	$-0.395698\pi$
$439$	13.4866	0.643682	0.321841	+	0.946794i	$0.395698\pi$
$440$	0	0
$441$	4.99105	0.237669
$442$	0	0
$443$	−18.2572	−0.867424	−0.433712	−	0.901052i	$-0.642796\pi$
$443$	−18.2572	−0.867424	−0.433712	+	0.901052i	$0.642796\pi$
$444$	0	0
$445$	0.194129	0.00920259
$446$	0	0
$447$	21.4346	1.01382
$448$	0	0
$449$	−35.6341	−1.68167	−0.840837	−	0.541288i	$-0.817937\pi$
$449$	−35.6341	−1.68167	−0.840837	+	0.541288i	$0.817937\pi$
$450$	0	0
$451$	6.19755	0.291831
$452$	0	0
$453$	15.4079	0.723925
$454$	0	0
$455$	0.265205	0.0124330
$456$	0	0
$457$	34.9442	1.63462	0.817311	−	0.576197i	$-0.195464\pi$
$457$	34.9442	1.63462	0.817311	+	0.576197i	$0.195464\pi$
$458$	0	0
$459$	−4.41903	−0.206262
$460$	0	0
$461$	35.9168	1.67281	0.836406	−	0.548110i	$-0.184652\pi$
$461$	35.9168	1.67281	0.836406	+	0.548110i	$0.184652\pi$
$462$	0	0
$463$	21.1254	0.981780	0.490890	−	0.871222i	$-0.336672\pi$
$463$	21.1254	0.981780	0.490890	+	0.871222i	$0.336672\pi$
$464$	0	0
$465$	−7.69107	−0.356665
$466$	0	0
$467$	−6.51161	−0.301321	−0.150661	−	0.988586i	$-0.548140\pi$
$467$	−6.51161	−0.301321	−0.150661	+	0.988586i	$0.548140\pi$
$468$	0	0
$469$	−10.7680	−0.497222
$470$	0	0
$471$	−14.0999	−0.649687
$472$	0	0
$473$	12.6754	0.582814
$474$	0	0
$475$	15.1867	0.696816
$476$	0	0
$477$	−12.4155	−0.568468
$478$	0	0
$479$	−2.97917	−0.136122	−0.0680610	−	0.997681i	$-0.521681\pi$
$479$	−2.97917	−0.136122	−0.0680610	+	0.997681i	$0.521681\pi$
$480$	0	0
$481$	−5.76202	−0.262726
$482$	0	0
$483$	26.4856	1.20514
$484$	0	0
$485$	0.216713	0.00984043
$486$	0	0
$487$	15.8323	0.717428	0.358714	−	0.933448i	$-0.383215\pi$
$487$	15.8323	0.717428	0.358714	+	0.933448i	$0.383215\pi$
$488$	0	0
$489$	43.9340	1.98676
$490$	0	0
$491$	−16.5495	−0.746867	−0.373434	−	0.927657i	$-0.621820\pi$
$491$	−16.5495	−0.746867	−0.373434	+	0.927657i	$0.621820\pi$
$492$	0	0
$493$	−0.233633	−0.0105223
$494$	0	0
$495$	1.32365	0.0594936
$496$	0	0
$497$	−5.10840	−0.229143
$498$	0	0
$499$	−28.6272	−1.28153	−0.640764	−	0.767738i	$-0.721382\pi$
$499$	−28.6272	−1.28153	−0.640764	+	0.767738i	$0.721382\pi$
$500$	0	0
$501$	11.8687	0.530256
$502$	0	0
$503$	−26.2943	−1.17241	−0.586203	−	0.810164i	$-0.699378\pi$
$503$	−26.2943	−1.17241	−0.586203	+	0.810164i	$0.699378\pi$
$504$	0	0
$505$	1.00902	0.0449007
$506$	0	0
$507$	2.82684	0.125545
$508$	0	0
$509$	12.9786	0.575267	0.287634	−	0.957741i	$-0.407131\pi$
$509$	12.9786	0.575267	0.287634	+	0.957741i	$0.407131\pi$
$510$	0	0
$511$	−12.8388	−0.567953
$512$	0	0
$513$	−17.3393	−0.765548
$514$	0	0
$515$	4.48625	0.197688
$516$	0	0
$517$	13.3898	0.588881
$518$	0	0
$519$	29.3766	1.28949
$520$	0	0
$521$	6.28438	0.275324	0.137662	−	0.990479i	$-0.456041\pi$
$521$	6.28438	0.275324	0.137662	+	0.990479i	$0.456041\pi$
$522$	0	0
$523$	2.56260	0.112055	0.0560273	−	0.998429i	$-0.482157\pi$
$523$	2.56260	0.112055	0.0560273	+	0.998429i	$0.482157\pi$
$524$	0	0
$525$	13.9354	0.608191
$526$	0	0
$527$	−8.05464	−0.350866
$528$	0	0
$529$	64.7839	2.81669
$530$	0	0
$531$	−57.2736	−2.48546
$532$	0	0
$533$	−6.19755	−0.268445
$534$	0	0
$535$	−0.865969	−0.0374391
$536$	0	0
$537$	60.4388	2.60813
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.4334	0.448569	0.224284	−	0.974524i	$-0.427996\pi$
$541$	10.4334	0.448569	0.224284	+	0.974524i	$0.427996\pi$
$542$	0	0
$543$	11.5745	0.496709
$544$	0	0
$545$	0.138725	0.00594235
$546$	0	0
$547$	15.5589	0.665250	0.332625	−	0.943059i	$-0.392066\pi$
$547$	15.5589	0.665250	0.332625	+	0.943059i	$0.392066\pi$
$548$	0	0
$549$	64.5024	2.75290
$550$	0	0
$551$	−0.916725	−0.0390538
$552$	0	0
$553$	−1.88025	−0.0799566
$554$	0	0
$555$	4.31974	0.183363
$556$	0	0
$557$	−17.7248	−0.751025	−0.375513	−	0.926817i	$-0.622533\pi$
$557$	−17.7248	−0.751025	−0.375513	+	0.926817i	$0.622533\pi$
$558$	0	0
$559$	−12.6754	−0.536111
$560$	0	0
$561$	2.21944	0.0937050
$562$	0	0
$563$	−19.8922	−0.838355	−0.419178	−	0.907904i	$-0.637682\pi$
$563$	−19.8922	−0.838355	−0.419178	+	0.907904i	$0.637682\pi$
$564$	0	0
$565$	−0.334654	−0.0140790
$566$	0	0
$567$	−0.937431	−0.0393684
$568$	0	0
$569$	1.26311	0.0529524	0.0264762	−	0.999649i	$-0.491571\pi$
$569$	1.26311	0.0529524	0.0264762	+	0.999649i	$0.491571\pi$
$570$	0	0
$571$	−1.41774	−0.0593306	−0.0296653	−	0.999560i	$-0.509444\pi$
$571$	−1.41774	−0.0593306	−0.0296653	+	0.999560i	$0.509444\pi$
$572$	0	0
$573$	11.6393	0.486240
$574$	0	0
$575$	46.1876	1.92615
$576$	0	0
$577$	−1.37579	−0.0572749	−0.0286374	−	0.999590i	$-0.509117\pi$
$577$	−1.37579	−0.0572749	−0.0286374	+	0.999590i	$0.509117\pi$
$578$	0	0
$579$	−34.5764	−1.43694
$580$	0	0
$581$	4.99097	0.207060
$582$	0	0
$583$	2.48756	0.103024
$584$	0	0
$585$	−1.32365	−0.0547262
$586$	0	0
$587$	35.8824	1.48102	0.740512	−	0.672043i	$-0.234583\pi$
$587$	35.8824	1.48102	0.740512	+	0.672043i	$0.234583\pi$
$588$	0	0
$589$	−31.6046	−1.30225
$590$	0	0
$591$	49.4462	2.03395
$592$	0	0
$593$	22.3695	0.918603	0.459302	−	0.888280i	$-0.348100\pi$
$593$	22.3695	0.918603	0.459302	+	0.888280i	$0.348100\pi$
$594$	0	0
$595$	−0.208220	−0.00853621
$596$	0	0
$597$	24.7003	1.01092
$598$	0	0
$599$	40.3750	1.64968	0.824838	−	0.565369i	$-0.191266\pi$
$599$	40.3750	1.64968	0.824838	+	0.565369i	$0.191266\pi$
$600$	0	0
$601$	−37.7285	−1.53898	−0.769489	−	0.638660i	$-0.779489\pi$
$601$	−37.7285	−1.53898	−0.769489	+	0.638660i	$0.779489\pi$
$602$	0	0
$603$	53.7438	2.18862
$604$	0	0
$605$	−0.265205	−0.0107821
$606$	0	0
$607$	10.6433	0.431998	0.215999	−	0.976394i	$-0.430699\pi$
$607$	10.6433	0.431998	0.215999	+	0.976394i	$0.430699\pi$
$608$	0	0
$609$	−0.841189	−0.0340867
$610$	0	0
$611$	−13.3898	−0.541692
$612$	0	0
$613$	−11.8622	−0.479110	−0.239555	−	0.970883i	$-0.577001\pi$
$613$	−11.8622	−0.479110	−0.239555	+	0.970883i	$0.577001\pi$
$614$	0	0
$615$	4.64625	0.187355
$616$	0	0
$617$	11.3708	0.457771	0.228885	−	0.973453i	$-0.426492\pi$
$617$	11.3708	0.457771	0.228885	+	0.973453i	$0.426492\pi$
$618$	0	0
$619$	−2.10600	−0.0846474	−0.0423237	−	0.999104i	$-0.513476\pi$
$619$	−2.10600	−0.0846474	−0.0423237	+	0.999104i	$0.513476\pi$
$620$	0	0
$621$	−52.7341	−2.11615
$622$	0	0
$623$	0.731996	0.0293268
$624$	0	0
$625$	23.9499	0.957998
$626$	0	0
$627$	8.70861	0.347789
$628$	0	0
$629$	4.52395	0.180382
$630$	0	0
$631$	−24.4152	−0.971954	−0.485977	−	0.873972i	$-0.661536\pi$
$631$	−24.4152	−0.971954	−0.485977	+	0.873972i	$0.661536\pi$
$632$	0	0
$633$	−57.6661	−2.29202
$634$	0	0
$635$	−2.10980	−0.0837249
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	25.4963	1.00862
$640$	0	0
$641$	−1.37574	−0.0543383	−0.0271691	−	0.999631i	$-0.508649\pi$
$641$	−1.37574	−0.0543383	−0.0271691	+	0.999631i	$0.508649\pi$
$642$	0	0
$643$	−27.5500	−1.08647	−0.543234	−	0.839581i	$-0.682800\pi$
$643$	−27.5500	−1.08647	−0.543234	+	0.839581i	$0.682800\pi$
$644$	0	0
$645$	9.50262	0.374165
$646$	0	0
$647$	10.8041	0.424755	0.212377	−	0.977188i	$-0.431879\pi$
$647$	10.8041	0.424755	0.212377	+	0.977188i	$0.431879\pi$
$648$	0	0
$649$	11.4753	0.450443
$650$	0	0
$651$	−29.0005	−1.13662
$652$	0	0
$653$	19.9126	0.779238	0.389619	−	0.920976i	$-0.372607\pi$
$653$	19.9126	0.779238	0.389619	+	0.920976i	$0.372607\pi$
$654$	0	0
$655$	4.36138	0.170413
$656$	0	0
$657$	64.0788	2.49995
$658$	0	0
$659$	3.20418	0.124817	0.0624087	−	0.998051i	$-0.480122\pi$
$659$	3.20418	0.124817	0.0624087	+	0.998051i	$0.480122\pi$
$660$	0	0
$661$	48.4544	1.88466	0.942328	−	0.334690i	$-0.108632\pi$
$661$	48.4544	1.88466	0.942328	+	0.334690i	$0.108632\pi$
$662$	0	0
$663$	−2.21944	−0.0861961
$664$	0	0
$665$	−0.817011	−0.0316823
$666$	0	0
$667$	−2.78804	−0.107953
$668$	0	0
$669$	−41.8619	−1.61848
$670$	0	0
$671$	−12.9236	−0.498911
$672$	0	0
$673$	26.7184	1.02992	0.514960	−	0.857214i	$-0.327807\pi$
$673$	26.7184	1.02992	0.514960	+	0.857214i	$0.327807\pi$
$674$	0	0
$675$	−27.7461	−1.06795
$676$	0	0
$677$	−30.3182	−1.16522	−0.582612	−	0.812750i	$-0.697969\pi$
$677$	−30.3182	−1.16522	−0.582612	+	0.812750i	$0.697969\pi$
$678$	0	0
$679$	0.817154	0.0313595
$680$	0	0
$681$	−45.1191	−1.72897
$682$	0	0
$683$	−27.6491	−1.05796	−0.528982	−	0.848633i	$-0.677426\pi$
$683$	−27.6491	−1.05796	−0.528982	+	0.848633i	$0.677426\pi$
$684$	0	0
$685$	1.12021	0.0428011
$686$	0	0
$687$	−52.5221	−2.00384
$688$	0	0
$689$	−2.48756	−0.0947685
$690$	0	0
$691$	42.0470	1.59954	0.799772	−	0.600304i	$-0.204954\pi$
$691$	42.0470	1.59954	0.799772	+	0.600304i	$0.204954\pi$
$692$	0	0
$693$	4.99105	0.189594
$694$	0	0
$695$	1.72912	0.0655891
$696$	0	0
$697$	4.86589	0.184309
$698$	0	0
$699$	5.06679	0.191644
$700$	0	0
$701$	7.81083	0.295011	0.147506	−	0.989061i	$-0.452876\pi$
$701$	7.81083	0.295011	0.147506	+	0.989061i	$0.452876\pi$
$702$	0	0
$703$	17.7510	0.669491
$704$	0	0
$705$	10.0382	0.378061
$706$	0	0
$707$	3.80467	0.143089
$708$	0	0
$709$	17.7792	0.667712	0.333856	−	0.942624i	$-0.391650\pi$
$709$	17.7792	0.667712	0.333856	+	0.942624i	$0.391650\pi$
$710$	0	0
$711$	9.38444	0.351944
$712$	0	0
$713$	−96.1194	−3.59970
$714$	0	0
$715$	0.265205	0.00991809
$716$	0	0
$717$	31.1532	1.16344
$718$	0	0
$719$	−20.0447	−0.747543	−0.373771	−	0.927521i	$-0.621935\pi$
$719$	−20.0447	−0.747543	−0.373771	+	0.927521i	$0.621935\pi$
$720$	0	0
$721$	16.9162	0.629992
$722$	0	0
$723$	78.4994	2.91943
$724$	0	0
$725$	−1.46693	−0.0544804
$726$	0	0
$727$	0.184313	0.00683578	0.00341789	−	0.999994i	$-0.498912\pi$
$727$	0.184313	0.00683578	0.00341789	+	0.999994i	$0.498912\pi$
$728$	0	0
$729$	−43.0530	−1.59455
$730$	0	0
$731$	9.95183	0.368082
$732$	0	0
$733$	−6.71732	−0.248110	−0.124055	−	0.992275i	$-0.539590\pi$
$733$	−6.71732	−0.248110	−0.124055	+	0.992275i	$0.539590\pi$
$734$	0	0
$735$	−0.749692	−0.0276528
$736$	0	0
$737$	−10.7680	−0.396646
$738$	0	0
$739$	−19.8011	−0.728395	−0.364197	−	0.931322i	$-0.618657\pi$
$739$	−19.8011	−0.728395	−0.364197	+	0.931322i	$0.618657\pi$
$740$	0	0
$741$	−8.70861	−0.319919
$742$	0	0
$743$	−40.1408	−1.47262	−0.736311	−	0.676643i	$-0.763434\pi$
$743$	−40.1408	−1.47262	−0.736311	+	0.676643i	$0.763434\pi$
$744$	0	0
$745$	−2.01092	−0.0736745
$746$	0	0
$747$	−24.9102	−0.911415
$748$	0	0
$749$	−3.26529	−0.119311
$750$	0	0
$751$	8.66278	0.316109	0.158055	−	0.987430i	$-0.449478\pi$
$751$	8.66278	0.316109	0.158055	+	0.987430i	$0.449478\pi$
$752$	0	0
$753$	49.0192	1.78636
$754$	0	0
$755$	−1.44551	−0.0526076
$756$	0	0
$757$	−8.81477	−0.320378	−0.160189	−	0.987086i	$-0.551210\pi$
$757$	−8.81477	−0.320378	−0.160189	+	0.987086i	$0.551210\pi$
$758$	0	0
$759$	26.4856	0.961365
$760$	0	0
$761$	−35.6793	−1.29337	−0.646686	−	0.762756i	$-0.723846\pi$
$761$	−35.6793	−1.29337	−0.646686	+	0.762756i	$0.723846\pi$
$762$	0	0
$763$	0.523088	0.0189371
$764$	0	0
$765$	1.03924	0.0375738
$766$	0	0
$767$	−11.4753	−0.414348
$768$	0	0
$769$	46.2758	1.66875	0.834373	−	0.551199i	$-0.185830\pi$
$769$	46.2758	1.66875	0.834373	+	0.551199i	$0.185830\pi$
$770$	0	0
$771$	−75.2167	−2.70886
$772$	0	0
$773$	−16.5994	−0.597040	−0.298520	−	0.954403i	$-0.596493\pi$
$773$	−16.5994	−0.597040	−0.298520	+	0.954403i	$0.596493\pi$
$774$	0	0
$775$	−50.5733	−1.81665
$776$	0	0
$777$	16.2883	0.584341
$778$	0	0
$779$	19.0927	0.684066
$780$	0	0
$781$	−5.10840	−0.182793
$782$	0	0
$783$	1.67485	0.0598543
$784$	0	0
$785$	1.32280	0.0472127
$786$	0	0
$787$	15.2256	0.542735	0.271367	−	0.962476i	$-0.412524\pi$
$787$	15.2256	0.542735	0.271367	+	0.962476i	$0.412524\pi$
$788$	0	0
$789$	−1.95792	−0.0697039
$790$	0	0
$791$	−1.26187	−0.0448669
$792$	0	0
$793$	12.9236	0.458931
$794$	0	0
$795$	1.86490	0.0661413
$796$	0	0
$797$	25.3922	0.899440	0.449720	−	0.893170i	$-0.351524\pi$
$797$	25.3922	0.899440	0.449720	+	0.893170i	$0.351524\pi$
$798$	0	0
$799$	10.5127	0.371914
$800$	0	0
$801$	−3.65343	−0.129088
$802$	0	0
$803$	−12.8388	−0.453070
$804$	0	0
$805$	−2.48478	−0.0875771
$806$	0	0
$807$	22.5840	0.794995
$808$	0	0
$809$	17.7638	0.624542	0.312271	−	0.949993i	$-0.398910\pi$
$809$	17.7638	0.624542	0.312271	+	0.949993i	$0.398910\pi$
$810$	0	0
$811$	12.2833	0.431324	0.215662	−	0.976468i	$-0.430809\pi$
$811$	12.2833	0.431324	0.215662	+	0.976468i	$0.430809\pi$
$812$	0	0
$813$	67.4589	2.36589
$814$	0	0
$815$	−4.12173	−0.144378
$816$	0	0
$817$	39.0488	1.36614
$818$	0	0
$819$	−4.99105	−0.174401
$820$	0	0
$821$	−13.6396	−0.476025	−0.238012	−	0.971262i	$-0.576496\pi$
$821$	−13.6396	−0.476025	−0.238012	+	0.971262i	$0.576496\pi$
$822$	0	0
$823$	−23.4630	−0.817868	−0.408934	−	0.912564i	$-0.634099\pi$
$823$	−23.4630	−0.817868	−0.408934	+	0.912564i	$0.634099\pi$
$824$	0	0
$825$	13.9354	0.485168
$826$	0	0
$827$	22.4736	0.781485	0.390743	−	0.920500i	$-0.372218\pi$
$827$	22.4736	0.781485	0.390743	+	0.920500i	$0.372218\pi$
$828$	0	0
$829$	−25.3067	−0.878939	−0.439470	−	0.898257i	$-0.644834\pi$
$829$	−25.3067	−0.878939	−0.439470	+	0.898257i	$0.644834\pi$
$830$	0	0
$831$	−48.4793	−1.68173
$832$	0	0
$833$	−0.785132	−0.0272032
$834$	0	0
$835$	−1.11348	−0.0385337
$836$	0	0
$837$	57.7415	1.99584
$838$	0	0
$839$	2.99992	0.103569	0.0517844	−	0.998658i	$-0.483509\pi$
$839$	2.99992	0.103569	0.0517844	+	0.998658i	$0.483509\pi$
$840$	0	0
$841$	−28.9115	−0.996947
$842$	0	0
$843$	−7.55335	−0.260151
$844$	0	0
$845$	−0.265205	−0.00912332
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	50.6008	1.73662
$850$	0	0
$851$	53.9862	1.85062
$852$	0	0
$853$	37.0169	1.26743	0.633717	−	0.773565i	$-0.281528\pi$
$853$	37.0169	1.26743	0.633717	+	0.773565i	$0.281528\pi$
$854$	0	0
$855$	4.07775	0.139456
$856$	0	0
$857$	−35.4382	−1.21054	−0.605272	−	0.796019i	$-0.706936\pi$
$857$	−35.4382	−1.21054	−0.605272	+	0.796019i	$0.706936\pi$
$858$	0	0
$859$	−2.74672	−0.0937169	−0.0468584	−	0.998902i	$-0.514921\pi$
$859$	−2.74672	−0.0937169	−0.0468584	+	0.998902i	$0.514921\pi$
$860$	0	0
$861$	17.5195	0.597063
$862$	0	0
$863$	39.0069	1.32781	0.663904	−	0.747817i	$-0.268898\pi$
$863$	39.0069	1.32781	0.663904	+	0.747817i	$0.268898\pi$
$864$	0	0
$865$	−2.75601	−0.0937073
$866$	0	0
$867$	−46.3138	−1.57290
$868$	0	0
$869$	−1.88025	−0.0637833
$870$	0	0
$871$	10.7680	0.364861
$872$	0	0
$873$	−4.07846	−0.138035
$874$	0	0
$875$	−2.63339	−0.0890249
$876$	0	0
$877$	−5.26341	−0.177733	−0.0888663	−	0.996044i	$-0.528324\pi$
$877$	−5.26341	−0.177733	−0.0888663	+	0.996044i	$0.528324\pi$
$878$	0	0
$879$	−45.9372	−1.54942
$880$	0	0
$881$	−40.2436	−1.35584	−0.677921	−	0.735135i	$-0.737119\pi$
$881$	−40.2436	−1.35584	−0.677921	+	0.735135i	$0.737119\pi$
$882$	0	0
$883$	−37.2926	−1.25500	−0.627498	−	0.778618i	$-0.715921\pi$
$883$	−37.2926	−1.25500	−0.627498	+	0.778618i	$0.715921\pi$
$884$	0	0
$885$	8.60291	0.289184
$886$	0	0
$887$	19.1023	0.641393	0.320697	−	0.947182i	$-0.396083\pi$
$887$	19.1023	0.641393	0.320697	+	0.947182i	$0.396083\pi$
$888$	0	0
$889$	−7.95537	−0.266815
$890$	0	0
$891$	−0.937431	−0.0314051
$892$	0	0
$893$	41.2496	1.38037
$894$	0	0
$895$	−5.67016	−0.189532
$896$	0	0
$897$	−26.4856	−0.884327
$898$	0	0
$899$	3.05278	0.101816
$900$	0	0
$901$	1.95306	0.0650659
$902$	0	0
$903$	35.8313	1.19239
$904$	0	0
$905$	−1.08588	−0.0360958
$906$	0	0
$907$	22.5207	0.747787	0.373894	−	0.927472i	$-0.378023\pi$
$907$	22.5207	0.747787	0.373894	+	0.927472i	$0.378023\pi$
$908$	0	0
$909$	−18.9893	−0.629836
$910$	0	0
$911$	−34.0920	−1.12952	−0.564759	−	0.825256i	$-0.691031\pi$
$911$	−34.0920	−1.12952	−0.564759	+	0.825256i	$0.691031\pi$
$912$	0	0
$913$	4.99097	0.165177
$914$	0	0
$915$	−9.68873	−0.320300
$916$	0	0
$917$	16.4454	0.543073
$918$	0	0
$919$	−51.7167	−1.70598	−0.852988	−	0.521931i	$-0.825212\pi$
$919$	−51.7167	−1.70598	−0.852988	+	0.521931i	$0.825212\pi$
$920$	0	0
$921$	−35.7761	−1.17886
$922$	0	0
$923$	5.10840	0.168145
$924$	0	0
$925$	28.4049	0.933946
$926$	0	0
$927$	−84.4295	−2.77303
$928$	0	0
$929$	39.3595	1.29134	0.645672	−	0.763615i	$-0.276577\pi$
$929$	39.3595	1.29134	0.645672	+	0.763615i	$0.276577\pi$
$930$	0	0
$931$	−3.08068	−0.100965
$932$	0	0
$933$	−12.5985	−0.412457
$934$	0	0
$935$	−0.208220	−0.00680954
$936$	0	0
$937$	−3.88634	−0.126961	−0.0634807	−	0.997983i	$-0.520220\pi$
$937$	−3.88634	−0.126961	−0.0634807	+	0.997983i	$0.520220\pi$
$938$	0	0
$939$	−92.4650	−3.01748
$940$	0	0
$941$	−5.41681	−0.176583	−0.0882915	−	0.996095i	$-0.528141\pi$
$941$	−5.41681	−0.176583	−0.0882915	+	0.996095i	$0.528141\pi$
$942$	0	0
$943$	58.0667	1.89091
$944$	0	0
$945$	1.49267	0.0485567
$946$	0	0
$947$	30.6173	0.994927	0.497464	−	0.867485i	$-0.334265\pi$
$947$	30.6173	0.994927	0.497464	+	0.867485i	$0.334265\pi$
$948$	0	0
$949$	12.8388	0.416763
$950$	0	0
$951$	27.0446	0.876981
$952$	0	0
$953$	−31.8538	−1.03185	−0.515924	−	0.856635i	$-0.672551\pi$
$953$	−31.8538	−1.03185	−0.515924	+	0.856635i	$0.672551\pi$
$954$	0	0
$955$	−1.09196	−0.0353351
$956$	0	0
$957$	−0.841189	−0.0271918
$958$	0	0
$959$	4.22395	0.136398
$960$	0	0
$961$	74.2464	2.39505
$962$	0	0
$963$	16.2972	0.525170
$964$	0	0
$965$	3.24383	0.104423
$966$	0	0
$967$	−26.4589	−0.850861	−0.425430	−	0.904991i	$-0.639877\pi$
$967$	−26.4589	−0.850861	−0.425430	+	0.904991i	$0.639877\pi$
$968$	0	0
$969$	6.83741	0.219649
$970$	0	0
$971$	−21.9399	−0.704085	−0.352043	−	0.935984i	$-0.614513\pi$
$971$	−21.9399	−0.704085	−0.352043	+	0.935984i	$0.614513\pi$
$972$	0	0
$973$	6.51993	0.209019
$974$	0	0
$975$	−13.9354	−0.446290
$976$	0	0
$977$	47.6591	1.52475	0.762376	−	0.647135i	$-0.224033\pi$
$977$	47.6591	1.52475	0.762376	+	0.647135i	$0.224033\pi$
$978$	0	0
$979$	0.731996	0.0233947
$980$	0	0
$981$	−2.61076	−0.0833552
$982$	0	0
$983$	48.1635	1.53618	0.768088	−	0.640344i	$-0.221208\pi$
$983$	48.1635	1.53618	0.768088	+	0.640344i	$0.221208\pi$
$984$	0	0
$985$	−4.63887	−0.147807
$986$	0	0
$987$	37.8508	1.20480
$988$	0	0
$989$	118.759	3.77633
$990$	0	0
$991$	−42.1197	−1.33798	−0.668988	−	0.743273i	$-0.733272\pi$
$991$	−42.1197	−1.33798	−0.668988	+	0.743273i	$0.733272\pi$
$992$	0	0
$993$	−66.3469	−2.10546
$994$	0	0
$995$	−2.31730	−0.0734632
$996$	0	0
$997$	43.1360	1.36613	0.683065	−	0.730358i	$-0.260647\pi$
$997$	43.1360	1.36613	0.683065	+	0.730358i	$0.260647\pi$
$998$	0	0
$999$	−32.4309	−1.02607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.m.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.m.1.9	✓	9	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.82684 q^{3} -0.265205 q^{5} -1.00000 q^{7} +4.99105 q^{9} +O(q^{10})\)	\(q+2.82684 q^{3} -0.265205 q^{5} -1.00000 q^{7} +4.99105 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.749692 q^{15} -0.785132 q^{17} -3.08068 q^{19} -2.82684 q^{21} -9.36931 q^{23} -4.92967 q^{25} +5.62839 q^{27} +0.297572 q^{29} +10.2590 q^{31} -2.82684 q^{33} +0.265205 q^{35} -5.76202 q^{37} +2.82684 q^{39} -6.19755 q^{41} -12.6754 q^{43} -1.32365 q^{45} -13.3898 q^{47} +1.00000 q^{49} -2.21944 q^{51} -2.48756 q^{53} +0.265205 q^{55} -8.70861 q^{57} -11.4753 q^{59} +12.9236 q^{61} -4.99105 q^{63} -0.265205 q^{65} +10.7680 q^{67} -26.4856 q^{69} +5.10840 q^{71} +12.8388 q^{73} -13.9354 q^{75} +1.00000 q^{77} +1.88025 q^{79} +0.937431 q^{81} -4.99097 q^{83} +0.208220 q^{85} +0.841189 q^{87} -0.731996 q^{89} -1.00000 q^{91} +29.0005 q^{93} +0.817011 q^{95} -0.817154 q^{97} -4.99105 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 5 q^{3} - 5 q^{5} - 9 q^{7} + 12 q^{9} - 9 q^{11} + 9 q^{13} - q^{15} + 13 q^{17} - 5 q^{19} + 5 q^{21} - 7 q^{23} + 14 q^{25} - 23 q^{27} - 19 q^{29} + 2 q^{31} + 5 q^{33} + 5 q^{35} + 4 q^{37} - 5 q^{39} + 10 q^{41} - 17 q^{43} + 11 q^{45} - 5 q^{47} + 9 q^{49} - 9 q^{51} - 24 q^{53} + 5 q^{55} - 4 q^{57} - 19 q^{59} - 12 q^{63} - 5 q^{65} + 2 q^{67} - 2 q^{69} - 2 q^{71} + 34 q^{73} - 46 q^{75} + 9 q^{77} + 5 q^{79} + 37 q^{81} - 24 q^{83} + 33 q^{85} + 41 q^{87} - 11 q^{89} - 9 q^{91} + 53 q^{93} - 23 q^{95} - 12 q^{99}+O(q^{100}) \) 9 * q - 5 * q^3 - 5 * q^5 - 9 * q^7 + 12 * q^9 - 9 * q^11 + 9 * q^13 - q^15 + 13 * q^17 - 5 * q^19 + 5 * q^21 - 7 * q^23 + 14 * q^25 - 23 * q^27 - 19 * q^29 + 2 * q^31 + 5 * q^33 + 5 * q^35 + 4 * q^37 - 5 * q^39 + 10 * q^41 - 17 * q^43 + 11 * q^45 - 5 * q^47 + 9 * q^49 - 9 * q^51 - 24 * q^53 + 5 * q^55 - 4 * q^57 - 19 * q^59 - 12 * q^63 - 5 * q^65 + 2 * q^67 - 2 * q^69 - 2 * q^71 + 34 * q^73 - 46 * q^75 + 9 * q^77 + 5 * q^79 + 37 * q^81 - 24 * q^83 + 33 * q^85 + 41 * q^87 - 11 * q^89 - 9 * q^91 + 53 * q^93 - 23 * q^95 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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