LMFDB - Embedded newform 8004.2.a.i.1.15

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} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$4.02229$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} -3.42405 q^{11} +4.92009 q^{13} -4.02229 q^{15} +4.60807 q^{17} +7.63146 q^{19} -4.04331 q^{21} +1.00000 q^{23} +11.1788 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.21724 q^{31} +3.42405 q^{33} +16.2634 q^{35} +3.25376 q^{37} -4.92009 q^{39} -3.81231 q^{41} -4.87928 q^{43} +4.02229 q^{45} +4.28191 q^{47} +9.34836 q^{49} -4.60807 q^{51} +11.5936 q^{53} -13.7725 q^{55} -7.63146 q^{57} +4.70683 q^{59} -10.5681 q^{61} +4.04331 q^{63} +19.7900 q^{65} -15.2771 q^{67} -1.00000 q^{69} -7.01655 q^{71} -11.3905 q^{73} -11.1788 q^{75} -13.8445 q^{77} -5.09508 q^{79} +1.00000 q^{81} -0.476321 q^{83} +18.5350 q^{85} -1.00000 q^{87} +8.93310 q^{89} +19.8935 q^{91} +4.21724 q^{93} +30.6959 q^{95} +14.9323 q^{97} -3.42405 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * 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
$5$	4.02229	1.79882	0.899411	−	0.437104i	$-0.143996\pi$
$5$	4.02229	1.79882	0.899411	+	0.437104i	$0.143996\pi$
$6$	0	0
$7$	4.04331	1.52823	0.764114	−	0.645081i	$-0.223177\pi$
$7$	4.04331	1.52823	0.764114	+	0.645081i	$0.223177\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.42405	−1.03239	−0.516195	−	0.856471i	$-0.672652\pi$
$11$	−3.42405	−1.03239	−0.516195	+	0.856471i	$0.672652\pi$
$12$	0	0
$13$	4.92009	1.36459	0.682294	−	0.731078i	$-0.260982\pi$
$13$	4.92009	1.36459	0.682294	+	0.731078i	$0.260982\pi$
$14$	0	0
$15$	−4.02229	−1.03855
$16$	0	0
$17$	4.60807	1.11762	0.558810	−	0.829295i	$-0.311258\pi$
$17$	4.60807	1.11762	0.558810	+	0.829295i	$0.311258\pi$
$18$	0	0
$19$	7.63146	1.75078	0.875388	−	0.483421i	$-0.160606\pi$
$19$	7.63146	1.75078	0.875388	+	0.483421i	$0.160606\pi$
$20$	0	0
$21$	−4.04331	−0.882323
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	11.1788	2.23576
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.21724	−0.757438	−0.378719	−	0.925512i	$-0.623635\pi$
$31$	−4.21724	−0.757438	−0.378719	+	0.925512i	$0.623635\pi$
$32$	0	0
$33$	3.42405	0.596050
$34$	0	0
$35$	16.2634	2.74901
$36$	0	0
$37$	3.25376	0.534915	0.267458	−	0.963570i	$-0.413816\pi$
$37$	3.25376	0.534915	0.267458	+	0.963570i	$0.413816\pi$
$38$	0	0
$39$	−4.92009	−0.787845
$40$	0	0
$41$	−3.81231	−0.595383	−0.297692	−	0.954662i	$-0.596217\pi$
$41$	−3.81231	−0.595383	−0.297692	+	0.954662i	$0.596217\pi$
$42$	0	0
$43$	−4.87928	−0.744083	−0.372041	−	0.928216i	$-0.621342\pi$
$43$	−4.87928	−0.744083	−0.372041	+	0.928216i	$0.621342\pi$
$44$	0	0
$45$	4.02229	0.599607
$46$	0	0
$47$	4.28191	0.624581	0.312290	−	0.949987i	$-0.398904\pi$
$47$	4.28191	0.624581	0.312290	+	0.949987i	$0.398904\pi$
$48$	0	0
$49$	9.34836	1.33548
$50$	0	0
$51$	−4.60807	−0.645259
$52$	0	0
$53$	11.5936	1.59251	0.796253	−	0.604964i	$-0.206812\pi$
$53$	11.5936	1.59251	0.796253	+	0.604964i	$0.206812\pi$
$54$	0	0
$55$	−13.7725	−1.85709
$56$	0	0
$57$	−7.63146	−1.01081
$58$	0	0
$59$	4.70683	0.612777	0.306389	−	0.951907i	$-0.400879\pi$
$59$	4.70683	0.612777	0.306389	+	0.951907i	$0.400879\pi$
$60$	0	0
$61$	−10.5681	−1.35311	−0.676554	−	0.736393i	$-0.736527\pi$
$61$	−10.5681	−1.35311	−0.676554	+	0.736393i	$0.736527\pi$
$62$	0	0
$63$	4.04331	0.509409
$64$	0	0
$65$	19.7900	2.45465
$66$	0	0
$67$	−15.2771	−1.86639	−0.933197	−	0.359365i	$-0.882993\pi$
$67$	−15.2771	−1.86639	−0.933197	+	0.359365i	$0.882993\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−7.01655	−0.832711	−0.416356	−	0.909202i	$-0.636693\pi$
$71$	−7.01655	−0.832711	−0.416356	+	0.909202i	$0.636693\pi$
$72$	0	0
$73$	−11.3905	−1.33316	−0.666580	−	0.745434i	$-0.732242\pi$
$73$	−11.3905	−1.33316	−0.666580	+	0.745434i	$0.732242\pi$
$74$	0	0
$75$	−11.1788	−1.29082
$76$	0	0
$77$	−13.8445	−1.57773
$78$	0	0
$79$	−5.09508	−0.573241	−0.286621	−	0.958044i	$-0.592532\pi$
$79$	−5.09508	−0.573241	−0.286621	+	0.958044i	$0.592532\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.476321	−0.0522830	−0.0261415	−	0.999658i	$-0.508322\pi$
$83$	−0.476321	−0.0522830	−0.0261415	+	0.999658i	$0.508322\pi$
$84$	0	0
$85$	18.5350	2.01040
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	8.93310	0.946907	0.473453	−	0.880819i	$-0.343007\pi$
$89$	8.93310	0.946907	0.473453	+	0.880819i	$0.343007\pi$
$90$	0	0
$91$	19.8935	2.08540
$92$	0	0
$93$	4.21724	0.437307
$94$	0	0
$95$	30.6959	3.14934
$96$	0	0
$97$	14.9323	1.51615	0.758075	−	0.652168i	$-0.226140\pi$
$97$	14.9323	1.51615	0.758075	+	0.652168i	$0.226140\pi$
$98$	0	0
$99$	−3.42405	−0.344130
$100$	0	0
$101$	−10.2354	−1.01847	−0.509233	−	0.860629i	$-0.670071\pi$
$101$	−10.2354	−1.01847	−0.509233	+	0.860629i	$0.670071\pi$
$102$	0	0
$103$	−11.1878	−1.10237	−0.551183	−	0.834384i	$-0.685823\pi$
$103$	−11.1878	−1.10237	−0.551183	+	0.834384i	$0.685823\pi$
$104$	0	0
$105$	−16.2634	−1.58714
$106$	0	0
$107$	15.6748	1.51534	0.757671	−	0.652637i	$-0.226337\pi$
$107$	15.6748	1.51534	0.757671	+	0.652637i	$0.226337\pi$
$108$	0	0
$109$	−16.3071	−1.56194	−0.780968	−	0.624572i	$-0.785274\pi$
$109$	−16.3071	−1.56194	−0.780968	+	0.624572i	$0.785274\pi$
$110$	0	0
$111$	−3.25376	−0.308833
$112$	0	0
$113$	15.1546	1.42562	0.712812	−	0.701355i	$-0.247421\pi$
$113$	15.1546	1.42562	0.712812	+	0.701355i	$0.247421\pi$
$114$	0	0
$115$	4.02229	0.375080
$116$	0	0
$117$	4.92009	0.454863
$118$	0	0
$119$	18.6319	1.70798
$120$	0	0
$121$	0.724108	0.0658280
$122$	0	0
$123$	3.81231	0.343745
$124$	0	0
$125$	24.8529	2.22291
$126$	0	0
$127$	−19.7464	−1.75221	−0.876104	−	0.482123i	$-0.839866\pi$
$127$	−19.7464	−1.75221	−0.876104	+	0.482123i	$0.839866\pi$
$128$	0	0
$129$	4.87928	0.429597
$130$	0	0
$131$	1.99863	0.174621	0.0873107	−	0.996181i	$-0.472173\pi$
$131$	1.99863	0.174621	0.0873107	+	0.996181i	$0.472173\pi$
$132$	0	0
$133$	30.8564	2.67559
$134$	0	0
$135$	−4.02229	−0.346183
$136$	0	0
$137$	−3.85067	−0.328985	−0.164492	−	0.986378i	$-0.552599\pi$
$137$	−3.85067	−0.328985	−0.164492	+	0.986378i	$0.552599\pi$
$138$	0	0
$139$	5.64898	0.479140	0.239570	−	0.970879i	$-0.422993\pi$
$139$	5.64898	0.479140	0.239570	+	0.970879i	$0.422993\pi$
$140$	0	0
$141$	−4.28191	−0.360602
$142$	0	0
$143$	−16.8466	−1.40879
$144$	0	0
$145$	4.02229	0.334033
$146$	0	0
$147$	−9.34836	−0.771040
$148$	0	0
$149$	−14.9170	−1.22205	−0.611023	−	0.791613i	$-0.709242\pi$
$149$	−14.9170	−1.22205	−0.611023	+	0.791613i	$0.709242\pi$
$150$	0	0
$151$	−5.91177	−0.481093	−0.240547	−	0.970638i	$-0.577327\pi$
$151$	−5.91177	−0.481093	−0.240547	+	0.970638i	$0.577327\pi$
$152$	0	0
$153$	4.60807	0.372540
$154$	0	0
$155$	−16.9629	−1.36250
$156$	0	0
$157$	−16.9191	−1.35029	−0.675146	−	0.737684i	$-0.735920\pi$
$157$	−16.9191	−1.35029	−0.675146	+	0.737684i	$0.735920\pi$
$158$	0	0
$159$	−11.5936	−0.919434
$160$	0	0
$161$	4.04331	0.318658
$162$	0	0
$163$	12.0575	0.944412	0.472206	−	0.881488i	$-0.343458\pi$
$163$	12.0575	0.944412	0.472206	+	0.881488i	$0.343458\pi$
$164$	0	0
$165$	13.7725	1.07219
$166$	0	0
$167$	−8.41234	−0.650966	−0.325483	−	0.945548i	$-0.605527\pi$
$167$	−8.41234	−0.650966	−0.325483	+	0.945548i	$0.605527\pi$
$168$	0	0
$169$	11.2073	0.862101
$170$	0	0
$171$	7.63146	0.583592
$172$	0	0
$173$	−20.2440	−1.53913	−0.769563	−	0.638571i	$-0.779526\pi$
$173$	−20.2440	−1.53913	−0.769563	+	0.638571i	$0.779526\pi$
$174$	0	0
$175$	45.1994	3.41675
$176$	0	0
$177$	−4.70683	−0.353787
$178$	0	0
$179$	7.01898	0.524623	0.262312	−	0.964983i	$-0.415515\pi$
$179$	7.01898	0.524623	0.262312	+	0.964983i	$0.415515\pi$
$180$	0	0
$181$	−13.5088	−1.00410	−0.502050	−	0.864838i	$-0.667421\pi$
$181$	−13.5088	−1.00410	−0.502050	+	0.864838i	$0.667421\pi$
$182$	0	0
$183$	10.5681	0.781217
$184$	0	0
$185$	13.0876	0.962217
$186$	0	0
$187$	−15.7783	−1.15382
$188$	0	0
$189$	−4.04331	−0.294108
$190$	0	0
$191$	−14.6319	−1.05872	−0.529362	−	0.848396i	$-0.677569\pi$
$191$	−14.6319	−1.05872	−0.529362	+	0.848396i	$0.677569\pi$
$192$	0	0
$193$	−5.90206	−0.424840	−0.212420	−	0.977178i	$-0.568134\pi$
$193$	−5.90206	−0.424840	−0.212420	+	0.977178i	$0.568134\pi$
$194$	0	0
$195$	−19.7900	−1.41719
$196$	0	0
$197$	3.58043	0.255095	0.127548	−	0.991832i	$-0.459289\pi$
$197$	3.58043	0.255095	0.127548	+	0.991832i	$0.459289\pi$
$198$	0	0
$199$	−0.206157	−0.0146141	−0.00730703	−	0.999973i	$-0.502326\pi$
$199$	−0.206157	−0.0146141	−0.00730703	+	0.999973i	$0.502326\pi$
$200$	0	0
$201$	15.2771	1.07756
$202$	0	0
$203$	4.04331	0.283785
$204$	0	0
$205$	−15.3342	−1.07099
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−26.1305	−1.80748
$210$	0	0
$211$	−26.7071	−1.83860	−0.919298	−	0.393563i	$-0.871242\pi$
$211$	−26.7071	−1.83860	−0.919298	+	0.393563i	$0.871242\pi$
$212$	0	0
$213$	7.01655	0.480766
$214$	0	0
$215$	−19.6259	−1.33847
$216$	0	0
$217$	−17.0516	−1.15754
$218$	0	0
$219$	11.3905	0.769700
$220$	0	0
$221$	22.6721	1.52509
$222$	0	0
$223$	6.90704	0.462529	0.231265	−	0.972891i	$-0.425714\pi$
$223$	6.90704	0.462529	0.231265	+	0.972891i	$0.425714\pi$
$224$	0	0
$225$	11.1788	0.745254
$226$	0	0
$227$	−0.0517430	−0.00343430	−0.00171715	−	0.999999i	$-0.500547\pi$
$227$	−0.0517430	−0.00343430	−0.00171715	+	0.999999i	$0.500547\pi$
$228$	0	0
$229$	3.67060	0.242560	0.121280	−	0.992618i	$-0.461300\pi$
$229$	3.67060	0.242560	0.121280	+	0.992618i	$0.461300\pi$
$230$	0	0
$231$	13.8445	0.910901
$232$	0	0
$233$	18.6446	1.22145	0.610723	−	0.791844i	$-0.290879\pi$
$233$	18.6446	1.22145	0.610723	+	0.791844i	$0.290879\pi$
$234$	0	0
$235$	17.2231	1.12351
$236$	0	0
$237$	5.09508	0.330961
$238$	0	0
$239$	−17.1647	−1.11029	−0.555147	−	0.831752i	$-0.687338\pi$
$239$	−17.1647	−1.11029	−0.555147	+	0.831752i	$0.687338\pi$
$240$	0	0
$241$	−15.5050	−0.998762	−0.499381	−	0.866383i	$-0.666439\pi$
$241$	−15.5050	−0.998762	−0.499381	+	0.866383i	$0.666439\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	37.6018	2.40229
$246$	0	0
$247$	37.5475	2.38909
$248$	0	0
$249$	0.476321	0.0301856
$250$	0	0
$251$	−2.04662	−0.129181	−0.0645907	−	0.997912i	$-0.520574\pi$
$251$	−2.04662	−0.129181	−0.0645907	+	0.997912i	$0.520574\pi$
$252$	0	0
$253$	−3.42405	−0.215268
$254$	0	0
$255$	−18.5350	−1.16071
$256$	0	0
$257$	25.5277	1.59238	0.796188	−	0.605049i	$-0.206847\pi$
$257$	25.5277	1.59238	0.796188	+	0.605049i	$0.206847\pi$
$258$	0	0
$259$	13.1560	0.817472
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−16.2098	−0.999539	−0.499770	−	0.866158i	$-0.666582\pi$
$263$	−16.2098	−0.999539	−0.499770	+	0.866158i	$0.666582\pi$
$264$	0	0
$265$	46.6329	2.86464
$266$	0	0
$267$	−8.93310	−0.546697
$268$	0	0
$269$	−16.0182	−0.976646	−0.488323	−	0.872663i	$-0.662391\pi$
$269$	−16.0182	−0.976646	−0.488323	+	0.872663i	$0.662391\pi$
$270$	0	0
$271$	−3.64736	−0.221561	−0.110781	−	0.993845i	$-0.535335\pi$
$271$	−3.64736	−0.221561	−0.110781	+	0.993845i	$0.535335\pi$
$272$	0	0
$273$	−19.8935	−1.20401
$274$	0	0
$275$	−38.2768	−2.30818
$276$	0	0
$277$	−8.19668	−0.492491	−0.246245	−	0.969208i	$-0.579197\pi$
$277$	−8.19668	−0.492491	−0.246245	+	0.969208i	$0.579197\pi$
$278$	0	0
$279$	−4.21724	−0.252479
$280$	0	0
$281$	−30.6032	−1.82563	−0.912817	−	0.408368i	$-0.866098\pi$
$281$	−30.6032	−1.82563	−0.912817	+	0.408368i	$0.866098\pi$
$282$	0	0
$283$	−18.1893	−1.08124	−0.540622	−	0.841266i	$-0.681811\pi$
$283$	−18.1893	−1.08124	−0.540622	+	0.841266i	$0.681811\pi$
$284$	0	0
$285$	−30.6959	−1.81827
$286$	0	0
$287$	−15.4144	−0.909881
$288$	0	0
$289$	4.23430	0.249077
$290$	0	0
$291$	−14.9323	−0.875349
$292$	0	0
$293$	25.5008	1.48977	0.744887	−	0.667191i	$-0.232503\pi$
$293$	25.5008	1.48977	0.744887	+	0.667191i	$0.232503\pi$
$294$	0	0
$295$	18.9322	1.10228
$296$	0	0
$297$	3.42405	0.198683
$298$	0	0
$299$	4.92009	0.284536
$300$	0	0
$301$	−19.7284	−1.13713
$302$	0	0
$303$	10.2354	0.588011
$304$	0	0
$305$	−42.5080	−2.43400
$306$	0	0
$307$	6.49238	0.370540	0.185270	−	0.982688i	$-0.440684\pi$
$307$	6.49238	0.370540	0.185270	+	0.982688i	$0.440684\pi$
$308$	0	0
$309$	11.1878	0.636451
$310$	0	0
$311$	−31.0601	−1.76126	−0.880630	−	0.473805i	$-0.842880\pi$
$311$	−31.0601	−1.76126	−0.880630	+	0.473805i	$0.842880\pi$
$312$	0	0
$313$	21.1142	1.19344	0.596722	−	0.802448i	$-0.296470\pi$
$313$	21.1142	1.19344	0.596722	+	0.802448i	$0.296470\pi$
$314$	0	0
$315$	16.2634	0.916337
$316$	0	0
$317$	14.5266	0.815897	0.407948	−	0.913005i	$-0.366244\pi$
$317$	14.5266	0.815897	0.407948	+	0.913005i	$0.366244\pi$
$318$	0	0
$319$	−3.42405	−0.191710
$320$	0	0
$321$	−15.6748	−0.874883
$322$	0	0
$323$	35.1663	1.95670
$324$	0	0
$325$	55.0008	3.05089
$326$	0	0
$327$	16.3071	0.901784
$328$	0	0
$329$	17.3131	0.954502
$330$	0	0
$331$	−17.6417	−0.969678	−0.484839	−	0.874603i	$-0.661122\pi$
$331$	−17.6417	−0.969678	−0.484839	+	0.874603i	$0.661122\pi$
$332$	0	0
$333$	3.25376	0.178305
$334$	0	0
$335$	−61.4489	−3.35731
$336$	0	0
$337$	−18.7050	−1.01893	−0.509464	−	0.860492i	$-0.670156\pi$
$337$	−18.7050	−1.01893	−0.509464	+	0.860492i	$0.670156\pi$
$338$	0	0
$339$	−15.1546	−0.823085
$340$	0	0
$341$	14.4400	0.781971
$342$	0	0
$343$	9.49516	0.512690
$344$	0	0
$345$	−4.02229	−0.216553
$346$	0	0
$347$	−23.6707	−1.27071	−0.635355	−	0.772220i	$-0.719146\pi$
$347$	−23.6707	−1.27071	−0.635355	+	0.772220i	$0.719146\pi$
$348$	0	0
$349$	−20.5123	−1.09800	−0.548999	−	0.835823i	$-0.684991\pi$
$349$	−20.5123	−1.09800	−0.548999	+	0.835823i	$0.684991\pi$
$350$	0	0
$351$	−4.92009	−0.262615
$352$	0	0
$353$	33.8808	1.80329	0.901646	−	0.432475i	$-0.142360\pi$
$353$	33.8808	1.80329	0.901646	+	0.432475i	$0.142360\pi$
$354$	0	0
$355$	−28.2226	−1.49790
$356$	0	0
$357$	−18.6319	−0.986102
$358$	0	0
$359$	5.37177	0.283511	0.141755	−	0.989902i	$-0.454725\pi$
$359$	5.37177	0.283511	0.141755	+	0.989902i	$0.454725\pi$
$360$	0	0
$361$	39.2391	2.06522
$362$	0	0
$363$	−0.724108	−0.0380058
$364$	0	0
$365$	−45.8160	−2.39812
$366$	0	0
$367$	−23.2226	−1.21221	−0.606106	−	0.795384i	$-0.707269\pi$
$367$	−23.2226	−1.21221	−0.606106	+	0.795384i	$0.707269\pi$
$368$	0	0
$369$	−3.81231	−0.198461
$370$	0	0
$371$	46.8766	2.43371
$372$	0	0
$373$	37.2210	1.92723	0.963615	−	0.267293i	$-0.0861290\pi$
$373$	37.2210	1.92723	0.963615	+	0.267293i	$0.0861290\pi$
$374$	0	0
$375$	−24.8529	−1.28340
$376$	0	0
$377$	4.92009	0.253398
$378$	0	0
$379$	30.5601	1.56977	0.784884	−	0.619643i	$-0.212723\pi$
$379$	30.5601	1.56977	0.784884	+	0.619643i	$0.212723\pi$
$380$	0	0
$381$	19.7464	1.01164
$382$	0	0
$383$	25.5302	1.30453	0.652266	−	0.757990i	$-0.273818\pi$
$383$	25.5302	1.30453	0.652266	+	0.757990i	$0.273818\pi$
$384$	0	0
$385$	−55.6865	−2.83805
$386$	0	0
$387$	−4.87928	−0.248028
$388$	0	0
$389$	−30.3878	−1.54072	−0.770362	−	0.637606i	$-0.779925\pi$
$389$	−30.3878	−1.54072	−0.770362	+	0.637606i	$0.779925\pi$
$390$	0	0
$391$	4.60807	0.233040
$392$	0	0
$393$	−1.99863	−0.100818
$394$	0	0
$395$	−20.4939	−1.03116
$396$	0	0
$397$	3.14774	0.157981	0.0789903	−	0.996875i	$-0.474830\pi$
$397$	3.14774	0.157981	0.0789903	+	0.996875i	$0.474830\pi$
$398$	0	0
$399$	−30.8564	−1.54475
$400$	0	0
$401$	−7.83957	−0.391489	−0.195745	−	0.980655i	$-0.562712\pi$
$401$	−7.83957	−0.391489	−0.195745	+	0.980655i	$0.562712\pi$
$402$	0	0
$403$	−20.7492	−1.03359
$404$	0	0
$405$	4.02229	0.199869
$406$	0	0
$407$	−11.1410	−0.552241
$408$	0	0
$409$	−4.94580	−0.244554	−0.122277	−	0.992496i	$-0.539020\pi$
$409$	−4.94580	−0.244554	−0.122277	+	0.992496i	$0.539020\pi$
$410$	0	0
$411$	3.85067	0.189939
$412$	0	0
$413$	19.0312	0.936463
$414$	0	0
$415$	−1.91590	−0.0940478
$416$	0	0
$417$	−5.64898	−0.276632
$418$	0	0
$419$	21.0781	1.02973	0.514867	−	0.857270i	$-0.327841\pi$
$419$	21.0781	1.02973	0.514867	+	0.857270i	$0.327841\pi$
$420$	0	0
$421$	−18.5556	−0.904343	−0.452172	−	0.891931i	$-0.649351\pi$
$421$	−18.5556	−0.904343	−0.452172	+	0.891931i	$0.649351\pi$
$422$	0	0
$423$	4.28191	0.208194
$424$	0	0
$425$	51.5127	2.49873
$426$	0	0
$427$	−42.7301	−2.06786
$428$	0	0
$429$	16.8466	0.813363
$430$	0	0
$431$	−0.248683	−0.0119786	−0.00598931	−	0.999982i	$-0.501906\pi$
$431$	−0.248683	−0.0119786	−0.00598931	+	0.999982i	$0.501906\pi$
$432$	0	0
$433$	−31.8676	−1.53146	−0.765731	−	0.643161i	$-0.777622\pi$
$433$	−31.8676	−1.53146	−0.765731	+	0.643161i	$0.777622\pi$
$434$	0	0
$435$	−4.02229	−0.192854
$436$	0	0
$437$	7.63146	0.365062
$438$	0	0
$439$	17.1853	0.820208	0.410104	−	0.912039i	$-0.365492\pi$
$439$	17.1853	0.820208	0.410104	+	0.912039i	$0.365492\pi$
$440$	0	0
$441$	9.34836	0.445160
$442$	0	0
$443$	26.7251	1.26975	0.634875	−	0.772615i	$-0.281052\pi$
$443$	26.7251	1.26975	0.634875	+	0.772615i	$0.281052\pi$
$444$	0	0
$445$	35.9315	1.70332
$446$	0	0
$447$	14.9170	0.705548
$448$	0	0
$449$	25.0993	1.18451	0.592254	−	0.805751i	$-0.298238\pi$
$449$	25.0993	1.18451	0.592254	+	0.805751i	$0.298238\pi$
$450$	0	0
$451$	13.0535	0.614667
$452$	0	0
$453$	5.91177	0.277759
$454$	0	0
$455$	80.0173	3.75127
$456$	0	0
$457$	4.68062	0.218950	0.109475	−	0.993990i	$-0.465083\pi$
$457$	4.68062	0.218950	0.109475	+	0.993990i	$0.465083\pi$
$458$	0	0
$459$	−4.60807	−0.215086
$460$	0	0
$461$	−8.78731	−0.409266	−0.204633	−	0.978839i	$-0.565600\pi$
$461$	−8.78731	−0.409266	−0.204633	+	0.978839i	$0.565600\pi$
$462$	0	0
$463$	31.6797	1.47228	0.736139	−	0.676830i	$-0.236647\pi$
$463$	31.6797	1.47228	0.736139	+	0.676830i	$0.236647\pi$
$464$	0	0
$465$	16.9629	0.786637
$466$	0	0
$467$	17.0114	0.787196	0.393598	−	0.919283i	$-0.371230\pi$
$467$	17.0114	0.787196	0.393598	+	0.919283i	$0.371230\pi$
$468$	0	0
$469$	−61.7701	−2.85228
$470$	0	0
$471$	16.9191	0.779592
$472$	0	0
$473$	16.7069	0.768183
$474$	0	0
$475$	85.3106	3.91432
$476$	0	0
$477$	11.5936	0.530835
$478$	0	0
$479$	28.3153	1.29376	0.646880	−	0.762592i	$-0.276073\pi$
$479$	28.3153	1.29376	0.646880	+	0.762592i	$0.276073\pi$
$480$	0	0
$481$	16.0088	0.729939
$482$	0	0
$483$	−4.04331	−0.183977
$484$	0	0
$485$	60.0622	2.72728
$486$	0	0
$487$	19.5510	0.885942	0.442971	−	0.896536i	$-0.353925\pi$
$487$	19.5510	0.885942	0.442971	+	0.896536i	$0.353925\pi$
$488$	0	0
$489$	−12.0575	−0.545257
$490$	0	0
$491$	−30.8233	−1.39103	−0.695517	−	0.718510i	$-0.744825\pi$
$491$	−30.8233	−1.39103	−0.695517	+	0.718510i	$0.744825\pi$
$492$	0	0
$493$	4.60807	0.207537
$494$	0	0
$495$	−13.7725	−0.619028
$496$	0	0
$497$	−28.3701	−1.27257
$498$	0	0
$499$	−32.3081	−1.44631	−0.723154	−	0.690687i	$-0.757308\pi$
$499$	−32.3081	−1.44631	−0.723154	+	0.690687i	$0.757308\pi$
$500$	0	0
$501$	8.41234	0.375836
$502$	0	0
$503$	−21.6976	−0.967449	−0.483725	−	0.875220i	$-0.660716\pi$
$503$	−21.6976	−0.967449	−0.483725	+	0.875220i	$0.660716\pi$
$504$	0	0
$505$	−41.1699	−1.83204
$506$	0	0
$507$	−11.2073	−0.497734
$508$	0	0
$509$	−6.16165	−0.273110	−0.136555	−	0.990632i	$-0.543603\pi$
$509$	−6.16165	−0.273110	−0.136555	+	0.990632i	$0.543603\pi$
$510$	0	0
$511$	−46.0554	−2.03737
$512$	0	0
$513$	−7.63146	−0.336937
$514$	0	0
$515$	−45.0005	−1.98296
$516$	0	0
$517$	−14.6615	−0.644810
$518$	0	0
$519$	20.2440	0.888615
$520$	0	0
$521$	−8.32842	−0.364875	−0.182437	−	0.983217i	$-0.558399\pi$
$521$	−8.32842	−0.364875	−0.182437	+	0.983217i	$0.558399\pi$
$522$	0	0
$523$	13.4234	0.586965	0.293482	−	0.955964i	$-0.405186\pi$
$523$	13.4234	0.586965	0.293482	+	0.955964i	$0.405186\pi$
$524$	0	0
$525$	−45.1994	−1.97266
$526$	0	0
$527$	−19.4333	−0.846528
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.70683	0.204259
$532$	0	0
$533$	−18.7569	−0.812453
$534$	0	0
$535$	63.0486	2.72583
$536$	0	0
$537$	−7.01898	−0.302892
$538$	0	0
$539$	−32.0092	−1.37874
$540$	0	0
$541$	−20.0066	−0.860153	−0.430076	−	0.902792i	$-0.641513\pi$
$541$	−20.0066	−0.860153	−0.430076	+	0.902792i	$0.641513\pi$
$542$	0	0
$543$	13.5088	0.579718
$544$	0	0
$545$	−65.5918	−2.80964
$546$	0	0
$547$	29.1367	1.24580	0.622898	−	0.782303i	$-0.285955\pi$
$547$	29.1367	1.24580	0.622898	+	0.782303i	$0.285955\pi$
$548$	0	0
$549$	−10.5681	−0.451036
$550$	0	0
$551$	7.63146	0.325111
$552$	0	0
$553$	−20.6010	−0.876043
$554$	0	0
$555$	−13.0876	−0.555536
$556$	0	0
$557$	36.2619	1.53646	0.768232	−	0.640171i	$-0.221137\pi$
$557$	36.2619	1.53646	0.768232	+	0.640171i	$0.221137\pi$
$558$	0	0
$559$	−24.0065	−1.01537
$560$	0	0
$561$	15.7783	0.666158
$562$	0	0
$563$	23.3050	0.982190	0.491095	−	0.871106i	$-0.336597\pi$
$563$	23.3050	0.982190	0.491095	+	0.871106i	$0.336597\pi$
$564$	0	0
$565$	60.9562	2.56444
$566$	0	0
$567$	4.04331	0.169803
$568$	0	0
$569$	−16.6105	−0.696350	−0.348175	−	0.937430i	$-0.613198\pi$
$569$	−16.6105	−0.696350	−0.348175	+	0.937430i	$0.613198\pi$
$570$	0	0
$571$	18.2191	0.762444	0.381222	−	0.924483i	$-0.375503\pi$
$571$	18.2191	0.762444	0.381222	+	0.924483i	$0.375503\pi$
$572$	0	0
$573$	14.6319	0.611254
$574$	0	0
$575$	11.1788	0.466188
$576$	0	0
$577$	26.5487	1.10524	0.552619	−	0.833434i	$-0.313628\pi$
$577$	26.5487	1.10524	0.552619	+	0.833434i	$0.313628\pi$
$578$	0	0
$579$	5.90206	0.245281
$580$	0	0
$581$	−1.92591	−0.0799003
$582$	0	0
$583$	−39.6971	−1.64409
$584$	0	0
$585$	19.7900	0.818217
$586$	0	0
$587$	−6.48638	−0.267722	−0.133861	−	0.991000i	$-0.542738\pi$
$587$	−6.48638	−0.267722	−0.133861	+	0.991000i	$0.542738\pi$
$588$	0	0
$589$	−32.1837	−1.32610
$590$	0	0
$591$	−3.58043	−0.147279
$592$	0	0
$593$	37.2027	1.52773	0.763866	−	0.645375i	$-0.223299\pi$
$593$	37.2027	1.52773	0.763866	+	0.645375i	$0.223299\pi$
$594$	0	0
$595$	74.9427	3.07235
$596$	0	0
$597$	0.206157	0.00843744
$598$	0	0
$599$	−47.8509	−1.95514	−0.977568	−	0.210618i	$-0.932452\pi$
$599$	−47.8509	−1.95514	−0.977568	+	0.210618i	$0.932452\pi$
$600$	0	0
$601$	18.4176	0.751270	0.375635	−	0.926768i	$-0.377425\pi$
$601$	18.4176	0.751270	0.375635	+	0.926768i	$0.377425\pi$
$602$	0	0
$603$	−15.2771	−0.622131
$604$	0	0
$605$	2.91257	0.118413
$606$	0	0
$607$	16.4814	0.668959	0.334480	−	0.942403i	$-0.391439\pi$
$607$	16.4814	0.668959	0.334480	+	0.942403i	$0.391439\pi$
$608$	0	0
$609$	−4.04331	−0.163843
$610$	0	0
$611$	21.0674	0.852295
$612$	0	0
$613$	−19.2142	−0.776056	−0.388028	−	0.921648i	$-0.626844\pi$
$613$	−19.2142	−0.776056	−0.388028	+	0.921648i	$0.626844\pi$
$614$	0	0
$615$	15.3342	0.618336
$616$	0	0
$617$	−5.92586	−0.238566	−0.119283	−	0.992860i	$-0.538060\pi$
$617$	−5.92586	−0.238566	−0.119283	+	0.992860i	$0.538060\pi$
$618$	0	0
$619$	43.0096	1.72870	0.864351	−	0.502889i	$-0.167729\pi$
$619$	43.0096	1.72870	0.864351	+	0.502889i	$0.167729\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	36.1193	1.44709
$624$	0	0
$625$	44.0717	1.76287
$626$	0	0
$627$	26.1305	1.04355
$628$	0	0
$629$	14.9936	0.597832
$630$	0	0
$631$	−29.8343	−1.18769	−0.593843	−	0.804581i	$-0.702390\pi$
$631$	−29.8343	−1.18769	−0.593843	+	0.804581i	$0.702390\pi$
$632$	0	0
$633$	26.7071	1.06151
$634$	0	0
$635$	−79.4256	−3.15191
$636$	0	0
$637$	45.9948	1.82238
$638$	0	0
$639$	−7.01655	−0.277570
$640$	0	0
$641$	34.1137	1.34741	0.673705	−	0.739000i	$-0.264702\pi$
$641$	34.1137	1.34741	0.673705	+	0.739000i	$0.264702\pi$
$642$	0	0
$643$	26.9368	1.06229	0.531143	−	0.847282i	$-0.321763\pi$
$643$	26.9368	1.06229	0.531143	+	0.847282i	$0.321763\pi$
$644$	0	0
$645$	19.6259	0.772768
$646$	0	0
$647$	3.38305	0.133001	0.0665006	−	0.997786i	$-0.478817\pi$
$647$	3.38305	0.133001	0.0665006	+	0.997786i	$0.478817\pi$
$648$	0	0
$649$	−16.1164	−0.632625
$650$	0	0
$651$	17.0516	0.668305
$652$	0	0
$653$	20.1857	0.789929	0.394964	−	0.918696i	$-0.370757\pi$
$653$	20.1857	0.789929	0.394964	+	0.918696i	$0.370757\pi$
$654$	0	0
$655$	8.03908	0.314113
$656$	0	0
$657$	−11.3905	−0.444386
$658$	0	0
$659$	18.3517	0.714881	0.357441	−	0.933936i	$-0.383649\pi$
$659$	18.3517	0.714881	0.357441	+	0.933936i	$0.383649\pi$
$660$	0	0
$661$	29.7589	1.15749	0.578744	−	0.815509i	$-0.303543\pi$
$661$	29.7589	1.15749	0.578744	+	0.815509i	$0.303543\pi$
$662$	0	0
$663$	−22.6721	−0.880513
$664$	0	0
$665$	124.113	4.81290
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−6.90704	−0.267041
$670$	0	0
$671$	36.1857	1.39693
$672$	0	0
$673$	−5.25389	−0.202523	−0.101261	−	0.994860i	$-0.532288\pi$
$673$	−5.25389	−0.202523	−0.101261	+	0.994860i	$0.532288\pi$
$674$	0	0
$675$	−11.1788	−0.430272
$676$	0	0
$677$	34.9045	1.34149	0.670745	−	0.741688i	$-0.265975\pi$
$677$	34.9045	1.34149	0.670745	+	0.741688i	$0.265975\pi$
$678$	0	0
$679$	60.3761	2.31702
$680$	0	0
$681$	0.0517430	0.00198280
$682$	0	0
$683$	−9.35781	−0.358067	−0.179033	−	0.983843i	$-0.557297\pi$
$683$	−9.35781	−0.358067	−0.179033	+	0.983843i	$0.557297\pi$
$684$	0	0
$685$	−15.4885	−0.591785
$686$	0	0
$687$	−3.67060	−0.140042
$688$	0	0
$689$	57.0417	2.17312
$690$	0	0
$691$	4.96336	0.188815	0.0944076	−	0.995534i	$-0.469904\pi$
$691$	4.96336	0.188815	0.0944076	+	0.995534i	$0.469904\pi$
$692$	0	0
$693$	−13.8445	−0.525909
$694$	0	0
$695$	22.7218	0.861888
$696$	0	0
$697$	−17.5674	−0.665413
$698$	0	0
$699$	−18.6446	−0.705202
$700$	0	0
$701$	18.6860	0.705762	0.352881	−	0.935668i	$-0.385202\pi$
$701$	18.6860	0.705762	0.352881	+	0.935668i	$0.385202\pi$
$702$	0	0
$703$	24.8309	0.936517
$704$	0	0
$705$	−17.2231	−0.648659
$706$	0	0
$707$	−41.3851	−1.55645
$708$	0	0
$709$	33.8457	1.27110	0.635551	−	0.772059i	$-0.280773\pi$
$709$	33.8457	1.27110	0.635551	+	0.772059i	$0.280773\pi$
$710$	0	0
$711$	−5.09508	−0.191080
$712$	0	0
$713$	−4.21724	−0.157937
$714$	0	0
$715$	−67.7620	−2.53416
$716$	0	0
$717$	17.1647	0.641029
$718$	0	0
$719$	9.94275	0.370802	0.185401	−	0.982663i	$-0.440642\pi$
$719$	9.94275	0.370802	0.185401	+	0.982663i	$0.440642\pi$
$720$	0	0
$721$	−45.2357	−1.68467
$722$	0	0
$723$	15.5050	0.576635
$724$	0	0
$725$	11.1788	0.415170
$726$	0	0
$727$	−5.49962	−0.203970	−0.101985	−	0.994786i	$-0.532519\pi$
$727$	−5.49962	−0.203970	−0.101985	+	0.994786i	$0.532519\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.4841	−0.831603
$732$	0	0
$733$	35.7108	1.31901	0.659504	−	0.751701i	$-0.270766\pi$
$733$	35.7108	1.31901	0.659504	+	0.751701i	$0.270766\pi$
$734$	0	0
$735$	−37.6018	−1.38696
$736$	0	0
$737$	52.3095	1.92685
$738$	0	0
$739$	40.9990	1.50817	0.754087	−	0.656775i	$-0.228080\pi$
$739$	40.9990	1.50817	0.754087	+	0.656775i	$0.228080\pi$
$740$	0	0
$741$	−37.5475	−1.37934
$742$	0	0
$743$	−3.54115	−0.129912	−0.0649562	−	0.997888i	$-0.520691\pi$
$743$	−3.54115	−0.129912	−0.0649562	+	0.997888i	$0.520691\pi$
$744$	0	0
$745$	−60.0004	−2.19824
$746$	0	0
$747$	−0.476321	−0.0174277
$748$	0	0
$749$	63.3782	2.31579
$750$	0	0
$751$	0.915056	0.0333909	0.0166954	−	0.999861i	$-0.494685\pi$
$751$	0.915056	0.0333909	0.0166954	+	0.999861i	$0.494685\pi$
$752$	0	0
$753$	2.04662	0.0745829
$754$	0	0
$755$	−23.7789	−0.865401
$756$	0	0
$757$	2.75926	0.100287	0.0501435	−	0.998742i	$-0.484032\pi$
$757$	2.75926	0.100287	0.0501435	+	0.998742i	$0.484032\pi$
$758$	0	0
$759$	3.42405	0.124285
$760$	0	0
$761$	−23.4124	−0.848698	−0.424349	−	0.905499i	$-0.639497\pi$
$761$	−23.4124	−0.848698	−0.424349	+	0.905499i	$0.639497\pi$
$762$	0	0
$763$	−65.9346	−2.38699
$764$	0	0
$765$	18.5350	0.670134
$766$	0	0
$767$	23.1580	0.836189
$768$	0	0
$769$	−11.9546	−0.431095	−0.215547	−	0.976493i	$-0.569154\pi$
$769$	−11.9546	−0.431095	−0.215547	+	0.976493i	$0.569154\pi$
$770$	0	0
$771$	−25.5277	−0.919359
$772$	0	0
$773$	−24.0495	−0.864999	−0.432499	−	0.901634i	$-0.642368\pi$
$773$	−24.0495	−0.864999	−0.432499	+	0.901634i	$0.642368\pi$
$774$	0	0
$775$	−47.1437	−1.69345
$776$	0	0
$777$	−13.1560	−0.471968
$778$	0	0
$779$	−29.0935	−1.04238
$780$	0	0
$781$	24.0250	0.859682
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	−68.0536	−2.42894
$786$	0	0
$787$	−37.7208	−1.34460	−0.672300	−	0.740279i	$-0.734693\pi$
$787$	−37.7208	−1.34460	−0.672300	+	0.740279i	$0.734693\pi$
$788$	0	0
$789$	16.2098	0.577084
$790$	0	0
$791$	61.2747	2.17868
$792$	0	0
$793$	−51.9961	−1.84643
$794$	0	0
$795$	−46.6329	−1.65390
$796$	0	0
$797$	−20.6036	−0.729817	−0.364908	−	0.931043i	$-0.618900\pi$
$797$	−20.6036	−0.729817	−0.364908	+	0.931043i	$0.618900\pi$
$798$	0	0
$799$	19.7313	0.698044
$800$	0	0
$801$	8.93310	0.315636
$802$	0	0
$803$	39.0017	1.37634
$804$	0	0
$805$	16.2634	0.573208
$806$	0	0
$807$	16.0182	0.563867
$808$	0	0
$809$	−45.8191	−1.61091	−0.805457	−	0.592654i	$-0.798080\pi$
$809$	−45.8191	−1.61091	−0.805457	+	0.592654i	$0.798080\pi$
$810$	0	0
$811$	41.2866	1.44977	0.724884	−	0.688871i	$-0.241893\pi$
$811$	41.2866	1.44977	0.724884	+	0.688871i	$0.241893\pi$
$812$	0	0
$813$	3.64736	0.127918
$814$	0	0
$815$	48.4985	1.69883
$816$	0	0
$817$	−37.2360	−1.30272
$818$	0	0
$819$	19.8935	0.695134
$820$	0	0
$821$	50.1840	1.75144	0.875718	−	0.482824i	$-0.160389\pi$
$821$	50.1840	1.75144	0.875718	+	0.482824i	$0.160389\pi$
$822$	0	0
$823$	−47.5287	−1.65675	−0.828374	−	0.560176i	$-0.810734\pi$
$823$	−47.5287	−1.65675	−0.828374	+	0.560176i	$0.810734\pi$
$824$	0	0
$825$	38.2768	1.33263
$826$	0	0
$827$	−14.5045	−0.504370	−0.252185	−	0.967679i	$-0.581149\pi$
$827$	−14.5045	−0.504370	−0.252185	+	0.967679i	$0.581149\pi$
$828$	0	0
$829$	36.4074	1.26448	0.632240	−	0.774772i	$-0.282136\pi$
$829$	36.4074	1.26448	0.632240	+	0.774772i	$0.282136\pi$
$830$	0	0
$831$	8.19668	0.284340
$832$	0	0
$833$	43.0779	1.49256
$834$	0	0
$835$	−33.8369	−1.17097
$836$	0	0
$837$	4.21724	0.145769
$838$	0	0
$839$	5.28119	0.182327	0.0911635	−	0.995836i	$-0.470941\pi$
$839$	5.28119	0.182327	0.0911635	+	0.995836i	$0.470941\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	30.6032	1.05403
$844$	0	0
$845$	45.0790	1.55077
$846$	0	0
$847$	2.92779	0.100600
$848$	0	0
$849$	18.1893	0.624256
$850$	0	0
$851$	3.25376	0.111538
$852$	0	0
$853$	−34.7810	−1.19088	−0.595439	−	0.803401i	$-0.703022\pi$
$853$	−34.7810	−1.19088	−0.595439	+	0.803401i	$0.703022\pi$
$854$	0	0
$855$	30.6959	1.04978
$856$	0	0
$857$	−8.18147	−0.279474	−0.139737	−	0.990189i	$-0.544626\pi$
$857$	−8.18147	−0.279474	−0.139737	+	0.990189i	$0.544626\pi$
$858$	0	0
$859$	3.39184	0.115728	0.0578641	−	0.998324i	$-0.481571\pi$
$859$	3.39184	0.115728	0.0578641	+	0.998324i	$0.481571\pi$
$860$	0	0
$861$	15.4144	0.525320
$862$	0	0
$863$	4.40961	0.150105	0.0750525	−	0.997180i	$-0.476088\pi$
$863$	4.40961	0.150105	0.0750525	+	0.997180i	$0.476088\pi$
$864$	0	0
$865$	−81.4274	−2.76861
$866$	0	0
$867$	−4.23430	−0.143804
$868$	0	0
$869$	17.4458	0.591808
$870$	0	0
$871$	−75.1647	−2.54686
$872$	0	0
$873$	14.9323	0.505383
$874$	0	0
$875$	100.488	3.39712
$876$	0	0
$877$	27.5955	0.931835	0.465917	−	0.884828i	$-0.345724\pi$
$877$	27.5955	0.931835	0.465917	+	0.884828i	$0.345724\pi$
$878$	0	0
$879$	−25.5008	−0.860121
$880$	0	0
$881$	−1.53612	−0.0517533	−0.0258767	−	0.999665i	$-0.508238\pi$
$881$	−1.53612	−0.0517533	−0.0258767	+	0.999665i	$0.508238\pi$
$882$	0	0
$883$	−5.50815	−0.185364	−0.0926820	−	0.995696i	$-0.529544\pi$
$883$	−5.50815	−0.185364	−0.0926820	+	0.995696i	$0.529544\pi$
$884$	0	0
$885$	−18.9322	−0.636400
$886$	0	0
$887$	17.7856	0.597182	0.298591	−	0.954381i	$-0.403483\pi$
$887$	17.7856	0.597182	0.298591	+	0.954381i	$0.403483\pi$
$888$	0	0
$889$	−79.8407	−2.67777
$890$	0	0
$891$	−3.42405	−0.114710
$892$	0	0
$893$	32.6772	1.09350
$894$	0	0
$895$	28.2324	0.943704
$896$	0	0
$897$	−4.92009	−0.164277
$898$	0	0
$899$	−4.21724	−0.140653
$900$	0	0
$901$	53.4242	1.77982
$902$	0	0
$903$	19.7284	0.656521
$904$	0	0
$905$	−54.3363	−1.80620
$906$	0	0
$907$	28.5259	0.947186	0.473593	−	0.880744i	$-0.342957\pi$
$907$	28.5259	0.947186	0.473593	+	0.880744i	$0.342957\pi$
$908$	0	0
$909$	−10.2354	−0.339488
$910$	0	0
$911$	−49.1530	−1.62851	−0.814255	−	0.580507i	$-0.802854\pi$
$911$	−49.1530	−1.62851	−0.814255	+	0.580507i	$0.802854\pi$
$912$	0	0
$913$	1.63095	0.0539764
$914$	0	0
$915$	42.5080	1.40527
$916$	0	0
$917$	8.08110	0.266861
$918$	0	0
$919$	43.9059	1.44832	0.724162	−	0.689630i	$-0.242227\pi$
$919$	43.9059	1.44832	0.724162	+	0.689630i	$0.242227\pi$
$920$	0	0
$921$	−6.49238	−0.213931
$922$	0	0
$923$	−34.5221	−1.13631
$924$	0	0
$925$	36.3732	1.19594
$926$	0	0
$927$	−11.1878	−0.367455
$928$	0	0
$929$	13.3119	0.436749	0.218375	−	0.975865i	$-0.429925\pi$
$929$	13.3119	0.436749	0.218375	+	0.975865i	$0.429925\pi$
$930$	0	0
$931$	71.3416	2.33813
$932$	0	0
$933$	31.0601	1.01686
$934$	0	0
$935$	−63.4647	−2.07552
$936$	0	0
$937$	−16.2816	−0.531896	−0.265948	−	0.963987i	$-0.585685\pi$
$937$	−16.2816	−0.531896	−0.265948	+	0.963987i	$0.585685\pi$
$938$	0	0
$939$	−21.1142	−0.689035
$940$	0	0
$941$	−17.7132	−0.577434	−0.288717	−	0.957415i	$-0.593229\pi$
$941$	−17.7132	−0.577434	−0.288717	+	0.957415i	$0.593229\pi$
$942$	0	0
$943$	−3.81231	−0.124146
$944$	0	0
$945$	−16.2634	−0.529047
$946$	0	0
$947$	15.9640	0.518761	0.259381	−	0.965775i	$-0.416482\pi$
$947$	15.9640	0.518761	0.259381	+	0.965775i	$0.416482\pi$
$948$	0	0
$949$	−56.0424	−1.81921
$950$	0	0
$951$	−14.5266	−0.471058
$952$	0	0
$953$	54.4117	1.76257	0.881284	−	0.472587i	$-0.156680\pi$
$953$	54.4117	1.76257	0.881284	+	0.472587i	$0.156680\pi$
$954$	0	0
$955$	−58.8535	−1.90446
$956$	0	0
$957$	3.42405	0.110684
$958$	0	0
$959$	−15.5694	−0.502763
$960$	0	0
$961$	−13.2149	−0.426288
$962$	0	0
$963$	15.6748	0.505114
$964$	0	0
$965$	−23.7398	−0.764211
$966$	0	0
$967$	6.88545	0.221421	0.110711	−	0.993853i	$-0.464687\pi$
$967$	6.88545	0.221421	0.110711	+	0.993853i	$0.464687\pi$
$968$	0	0
$969$	−35.1663	−1.12970
$970$	0	0
$971$	−3.07284	−0.0986123	−0.0493061	−	0.998784i	$-0.515701\pi$
$971$	−3.07284	−0.0986123	−0.0493061	+	0.998784i	$0.515701\pi$
$972$	0	0
$973$	22.8406	0.732236
$974$	0	0
$975$	−55.0008	−1.76143
$976$	0	0
$977$	39.3904	1.26021	0.630106	−	0.776509i	$-0.283012\pi$
$977$	39.3904	1.26021	0.630106	+	0.776509i	$0.283012\pi$
$978$	0	0
$979$	−30.5874	−0.977577
$980$	0	0
$981$	−16.3071	−0.520645
$982$	0	0
$983$	45.5701	1.45346	0.726731	−	0.686922i	$-0.241039\pi$
$983$	45.5701	1.45346	0.726731	+	0.686922i	$0.241039\pi$
$984$	0	0
$985$	14.4015	0.458871
$986$	0	0
$987$	−17.3131	−0.551082
$988$	0	0
$989$	−4.87928	−0.155152
$990$	0	0
$991$	−11.3994	−0.362114	−0.181057	−	0.983473i	$-0.557952\pi$
$991$	−11.3994	−0.362114	−0.181057	+	0.983473i	$0.557952\pi$
$992$	0	0
$993$	17.6417	0.559844
$994$	0	0
$995$	−0.829222	−0.0262881
$996$	0	0
$997$	36.4117	1.15317	0.576585	−	0.817037i	$-0.304385\pi$
$997$	36.4117	1.15317	0.576585	+	0.817037i	$0.304385\pi$
$998$	0	0
$999$	−3.25376	−0.102944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.15	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.15	✓	16	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.02229 q^{5} +4.04331 q^{7} +1.00000 q^{9} -3.42405 q^{11} +4.92009 q^{13} -4.02229 q^{15} +4.60807 q^{17} +7.63146 q^{19} -4.04331 q^{21} +1.00000 q^{23} +11.1788 q^{25} -1.00000 q^{27} +1.00000 q^{29} -4.21724 q^{31} +3.42405 q^{33} +16.2634 q^{35} +3.25376 q^{37} -4.92009 q^{39} -3.81231 q^{41} -4.87928 q^{43} +4.02229 q^{45} +4.28191 q^{47} +9.34836 q^{49} -4.60807 q^{51} +11.5936 q^{53} -13.7725 q^{55} -7.63146 q^{57} +4.70683 q^{59} -10.5681 q^{61} +4.04331 q^{63} +19.7900 q^{65} -15.2771 q^{67} -1.00000 q^{69} -7.01655 q^{71} -11.3905 q^{73} -11.1788 q^{75} -13.8445 q^{77} -5.09508 q^{79} +1.00000 q^{81} -0.476321 q^{83} +18.5350 q^{85} -1.00000 q^{87} +8.93310 q^{89} +19.8935 q^{91} +4.21724 q^{93} +30.6959 q^{95} +14.9323 q^{97} -3.42405 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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