LMFDB - Embedded newform 8008.2.a.t.1.8

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 $ x^10 - x^9 - 17x^8 + 13x^7 + 96x^6 - 49x^5 - 207x^4 + 58x^3 + 169x^2 - 12x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.38780$ 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)

$f(q)$	$=$	$q+1.38780 q^{3} +2.74127 q^{5} +1.00000 q^{7} -1.07400 q^{9} +O(q^{10})$	$q+1.38780 q^{3} +2.74127 q^{5} +1.00000 q^{7} -1.07400 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.80434 q^{15} +6.95391 q^{17} +2.55250 q^{19} +1.38780 q^{21} -6.24233 q^{23} +2.51454 q^{25} -5.65391 q^{27} -10.2678 q^{29} +7.77194 q^{31} -1.38780 q^{33} +2.74127 q^{35} -4.57503 q^{37} +1.38780 q^{39} +12.1297 q^{41} +3.77923 q^{43} -2.94411 q^{45} +6.53139 q^{47} +1.00000 q^{49} +9.65066 q^{51} +10.2578 q^{53} -2.74127 q^{55} +3.54237 q^{57} +4.29618 q^{59} -6.39090 q^{61} -1.07400 q^{63} +2.74127 q^{65} +14.0977 q^{67} -8.66313 q^{69} -5.53764 q^{71} -2.81316 q^{73} +3.48968 q^{75} -1.00000 q^{77} +13.8361 q^{79} -4.62453 q^{81} +16.8269 q^{83} +19.0625 q^{85} -14.2497 q^{87} -8.60422 q^{89} +1.00000 q^{91} +10.7859 q^{93} +6.99707 q^{95} -3.28756 q^{97} +1.07400 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * 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
$3$	1.38780	0.801249	0.400625	−	0.916242i	$-0.368793\pi$
$3$	1.38780	0.801249	0.400625	+	0.916242i	$0.368793\pi$
$4$	0	0
$5$	2.74127	1.22593	0.612966	−	0.790110i	$-0.289976\pi$
$5$	2.74127	1.22593	0.612966	+	0.790110i	$0.289976\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.07400	−0.357999
$10$	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
$15$	3.80434	0.982277
$16$	0	0
$17$	6.95391	1.68657	0.843285	−	0.537467i	$-0.180619\pi$
$17$	6.95391	1.68657	0.843285	+	0.537467i	$0.180619\pi$
$18$	0	0
$19$	2.55250	0.585583	0.292792	−	0.956176i	$-0.405416\pi$
$19$	2.55250	0.585583	0.292792	+	0.956176i	$0.405416\pi$
$20$	0	0
$21$	1.38780	0.302844
$22$	0	0
$23$	−6.24233	−1.30162	−0.650808	−	0.759243i	$-0.725570\pi$
$23$	−6.24233	−1.30162	−0.650808	+	0.759243i	$0.725570\pi$
$24$	0	0
$25$	2.51454	0.502907
$26$	0	0
$27$	−5.65391	−1.08810
$28$	0	0
$29$	−10.2678	−1.90668	−0.953339	−	0.301901i	$-0.902379\pi$
$29$	−10.2678	−1.90668	−0.953339	+	0.301901i	$0.902379\pi$
$30$	0	0
$31$	7.77194	1.39588	0.697941	−	0.716155i	$-0.254100\pi$
$31$	7.77194	1.39588	0.697941	+	0.716155i	$0.254100\pi$
$32$	0	0
$33$	−1.38780	−0.241586
$34$	0	0
$35$	2.74127	0.463358
$36$	0	0
$37$	−4.57503	−0.752130	−0.376065	−	0.926593i	$-0.622723\pi$
$37$	−4.57503	−0.752130	−0.376065	+	0.926593i	$0.622723\pi$
$38$	0	0
$39$	1.38780	0.222227
$40$	0	0
$41$	12.1297	1.89434	0.947171	−	0.320729i	$-0.103928\pi$
$41$	12.1297	1.89434	0.947171	+	0.320729i	$0.103928\pi$
$42$	0	0
$43$	3.77923	0.576327	0.288163	−	0.957581i	$-0.406955\pi$
$43$	3.77923	0.576327	0.288163	+	0.957581i	$0.406955\pi$
$44$	0	0
$45$	−2.94411	−0.438883
$46$	0	0
$47$	6.53139	0.952701	0.476350	−	0.879256i	$-0.341959\pi$
$47$	6.53139	0.952701	0.476350	+	0.879256i	$0.341959\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.65066	1.35136
$52$	0	0
$53$	10.2578	1.40901	0.704505	−	0.709699i	$-0.251169\pi$
$53$	10.2578	1.40901	0.704505	+	0.709699i	$0.251169\pi$
$54$	0	0
$55$	−2.74127	−0.369632
$56$	0	0
$57$	3.54237	0.469198
$58$	0	0
$59$	4.29618	0.559315	0.279657	−	0.960100i	$-0.409779\pi$
$59$	4.29618	0.559315	0.279657	+	0.960100i	$0.409779\pi$
$60$	0	0
$61$	−6.39090	−0.818271	−0.409136	−	0.912474i	$-0.634170\pi$
$61$	−6.39090	−0.818271	−0.409136	+	0.912474i	$0.634170\pi$
$62$	0	0
$63$	−1.07400	−0.135311
$64$	0	0
$65$	2.74127	0.340012
$66$	0	0
$67$	14.0977	1.72231	0.861157	−	0.508340i	$-0.169740\pi$
$67$	14.0977	1.72231	0.861157	+	0.508340i	$0.169740\pi$
$68$	0	0
$69$	−8.66313	−1.04292
$70$	0	0
$71$	−5.53764	−0.657197	−0.328599	−	0.944470i	$-0.606576\pi$
$71$	−5.53764	−0.657197	−0.328599	+	0.944470i	$0.606576\pi$
$72$	0	0
$73$	−2.81316	−0.329255	−0.164627	−	0.986356i	$-0.552642\pi$
$73$	−2.81316	−0.329255	−0.164627	+	0.986356i	$0.552642\pi$
$74$	0	0
$75$	3.48968	0.402954
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.8361	1.55668	0.778339	−	0.627844i	$-0.216062\pi$
$79$	13.8361	1.55668	0.778339	+	0.627844i	$0.216062\pi$
$80$	0	0
$81$	−4.62453	−0.513837
$82$	0	0
$83$	16.8269	1.84699	0.923495	−	0.383612i	$-0.125320\pi$
$83$	16.8269	1.84699	0.923495	+	0.383612i	$0.125320\pi$
$84$	0	0
$85$	19.0625	2.06762
$86$	0	0
$87$	−14.2497	−1.52773
$88$	0	0
$89$	−8.60422	−0.912045	−0.456023	−	0.889968i	$-0.650726\pi$
$89$	−8.60422	−0.912045	−0.456023	+	0.889968i	$0.650726\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	10.7859	1.11845
$94$	0	0
$95$	6.99707	0.717885
$96$	0	0
$97$	−3.28756	−0.333801	−0.166900	−	0.985974i	$-0.553376\pi$
$97$	−3.28756	−0.333801	−0.166900	+	0.985974i	$0.553376\pi$
$98$	0	0
$99$	1.07400	0.107941
$100$	0	0
$101$	−4.47513	−0.445292	−0.222646	−	0.974899i	$-0.571469\pi$
$101$	−4.47513	−0.445292	−0.222646	+	0.974899i	$0.571469\pi$
$102$	0	0
$103$	−9.13598	−0.900195	−0.450098	−	0.892979i	$-0.648611\pi$
$103$	−9.13598	−0.900195	−0.450098	+	0.892979i	$0.648611\pi$
$104$	0	0
$105$	3.80434	0.371266
$106$	0	0
$107$	10.1625	0.982446	0.491223	−	0.871034i	$-0.336550\pi$
$107$	10.1625	0.982446	0.491223	+	0.871034i	$0.336550\pi$
$108$	0	0
$109$	20.6824	1.98101	0.990506	−	0.137470i	$-0.0438970\pi$
$109$	20.6824	1.98101	0.990506	+	0.137470i	$0.0438970\pi$
$110$	0	0
$111$	−6.34925	−0.602644
$112$	0	0
$113$	−14.7579	−1.38831	−0.694155	−	0.719825i	$-0.744222\pi$
$113$	−14.7579	−1.38831	−0.694155	+	0.719825i	$0.744222\pi$
$114$	0	0
$115$	−17.1119	−1.59569
$116$	0	0
$117$	−1.07400	−0.0992912
$118$	0	0
$119$	6.95391	0.637464
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.8337	1.51784
$124$	0	0
$125$	−6.81332	−0.609402
$126$	0	0
$127$	−13.3229	−1.18222	−0.591109	−	0.806591i	$-0.701310\pi$
$127$	−13.3229	−1.18222	−0.591109	+	0.806591i	$0.701310\pi$
$128$	0	0
$129$	5.24483	0.461782
$130$	0	0
$131$	0.837885	0.0732064	0.0366032	−	0.999330i	$-0.488346\pi$
$131$	0.837885	0.0732064	0.0366032	+	0.999330i	$0.488346\pi$
$132$	0	0
$133$	2.55250	0.221330
$134$	0	0
$135$	−15.4989	−1.33393
$136$	0	0
$137$	20.4678	1.74868	0.874342	−	0.485311i	$-0.161294\pi$
$137$	20.4678	1.74868	0.874342	+	0.485311i	$0.161294\pi$
$138$	0	0
$139$	−15.3965	−1.30591	−0.652956	−	0.757395i	$-0.726472\pi$
$139$	−15.3965	−1.30591	−0.652956	+	0.757395i	$0.726472\pi$
$140$	0	0
$141$	9.06429	0.763351
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−28.1467	−2.33746
$146$	0	0
$147$	1.38780	0.114464
$148$	0	0
$149$	17.6970	1.44979	0.724895	−	0.688859i	$-0.241888\pi$
$149$	17.6970	1.44979	0.724895	+	0.688859i	$0.241888\pi$
$150$	0	0
$151$	−10.9316	−0.889604	−0.444802	−	0.895629i	$-0.646726\pi$
$151$	−10.9316	−0.889604	−0.444802	+	0.895629i	$0.646726\pi$
$152$	0	0
$153$	−7.46848	−0.603791
$154$	0	0
$155$	21.3050	1.71126
$156$	0	0
$157$	6.45987	0.515554	0.257777	−	0.966204i	$-0.417010\pi$
$157$	6.45987	0.515554	0.257777	+	0.966204i	$0.417010\pi$
$158$	0	0
$159$	14.2358	1.12897
$160$	0	0
$161$	−6.24233	−0.491964
$162$	0	0
$163$	5.80873	0.454975	0.227487	−	0.973781i	$-0.426949\pi$
$163$	5.80873	0.454975	0.227487	+	0.973781i	$0.426949\pi$
$164$	0	0
$165$	−3.80434	−0.296168
$166$	0	0
$167$	6.74382	0.521852	0.260926	−	0.965359i	$-0.415972\pi$
$167$	6.74382	0.521852	0.260926	+	0.965359i	$0.415972\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.74138	−0.209638
$172$	0	0
$173$	−17.0064	−1.29297	−0.646485	−	0.762927i	$-0.723762\pi$
$173$	−17.0064	−1.29297	−0.646485	+	0.762927i	$0.723762\pi$
$174$	0	0
$175$	2.51454	0.190081
$176$	0	0
$177$	5.96226	0.448151
$178$	0	0
$179$	16.9227	1.26486	0.632432	−	0.774616i	$-0.282057\pi$
$179$	16.9227	1.26486	0.632432	+	0.774616i	$0.282057\pi$
$180$	0	0
$181$	12.8327	0.953845	0.476923	−	0.878945i	$-0.341752\pi$
$181$	12.8327	0.953845	0.476923	+	0.878945i	$0.341752\pi$
$182$	0	0
$183$	−8.86933	−0.655639
$184$	0	0
$185$	−12.5414	−0.922060
$186$	0	0
$187$	−6.95391	−0.508520
$188$	0	0
$189$	−5.65391	−0.411262
$190$	0	0
$191$	−18.3827	−1.33012	−0.665061	−	0.746789i	$-0.731595\pi$
$191$	−18.3827	−1.33012	−0.665061	+	0.746789i	$0.731595\pi$
$192$	0	0
$193$	−1.20939	−0.0870536	−0.0435268	−	0.999052i	$-0.513859\pi$
$193$	−1.20939	−0.0870536	−0.0435268	+	0.999052i	$0.513859\pi$
$194$	0	0
$195$	3.80434	0.272434
$196$	0	0
$197$	−20.7279	−1.47680	−0.738402	−	0.674361i	$-0.764419\pi$
$197$	−20.7279	−1.47680	−0.738402	+	0.674361i	$0.764419\pi$
$198$	0	0
$199$	7.15801	0.507418	0.253709	−	0.967281i	$-0.418349\pi$
$199$	7.15801	0.507418	0.253709	+	0.967281i	$0.418349\pi$
$200$	0	0
$201$	19.5649	1.38000
$202$	0	0
$203$	−10.2678	−0.720657
$204$	0	0
$205$	33.2507	2.32233
$206$	0	0
$207$	6.70425	0.465978
$208$	0	0
$209$	−2.55250	−0.176560
$210$	0	0
$211$	−20.2063	−1.39106	−0.695528	−	0.718499i	$-0.744830\pi$
$211$	−20.2063	−1.39106	−0.695528	+	0.718499i	$0.744830\pi$
$212$	0	0
$213$	−7.68517	−0.526579
$214$	0	0
$215$	10.3599	0.706537
$216$	0	0
$217$	7.77194	0.527594
$218$	0	0
$219$	−3.90411	−0.263815
$220$	0	0
$221$	6.95391	0.467770
$222$	0	0
$223$	28.6490	1.91848	0.959238	−	0.282599i	$-0.0911964\pi$
$223$	28.6490	1.91848	0.959238	+	0.282599i	$0.0911964\pi$
$224$	0	0
$225$	−2.70061	−0.180040
$226$	0	0
$227$	8.06233	0.535115	0.267558	−	0.963542i	$-0.413783\pi$
$227$	8.06233	0.535115	0.267558	+	0.963542i	$0.413783\pi$
$228$	0	0
$229$	−13.3907	−0.884885	−0.442443	−	0.896797i	$-0.645888\pi$
$229$	−13.3907	−0.884885	−0.442443	+	0.896797i	$0.645888\pi$
$230$	0	0
$231$	−1.38780	−0.0913108
$232$	0	0
$233$	−8.17712	−0.535701	−0.267850	−	0.963461i	$-0.586313\pi$
$233$	−8.17712	−0.535701	−0.267850	+	0.963461i	$0.586313\pi$
$234$	0	0
$235$	17.9043	1.16795
$236$	0	0
$237$	19.2017	1.24729
$238$	0	0
$239$	25.0837	1.62253	0.811264	−	0.584680i	$-0.198780\pi$
$239$	25.0837	1.62253	0.811264	+	0.584680i	$0.198780\pi$
$240$	0	0
$241$	−12.2224	−0.787313	−0.393657	−	0.919258i	$-0.628790\pi$
$241$	−12.2224	−0.787313	−0.393657	+	0.919258i	$0.628790\pi$
$242$	0	0
$243$	10.5438	0.676385
$244$	0	0
$245$	2.74127	0.175133
$246$	0	0
$247$	2.55250	0.162412
$248$	0	0
$249$	23.3524	1.47990
$250$	0	0
$251$	3.07149	0.193870	0.0969352	−	0.995291i	$-0.469096\pi$
$251$	3.07149	0.193870	0.0969352	+	0.995291i	$0.469096\pi$
$252$	0	0
$253$	6.24233	0.392452
$254$	0	0
$255$	26.4550	1.65668
$256$	0	0
$257$	23.2004	1.44720	0.723600	−	0.690220i	$-0.242486\pi$
$257$	23.2004	1.44720	0.723600	+	0.690220i	$0.242486\pi$
$258$	0	0
$259$	−4.57503	−0.284279
$260$	0	0
$261$	11.0276	0.682590
$262$	0	0
$263$	3.43069	0.211545	0.105773	−	0.994390i	$-0.466268\pi$
$263$	3.43069	0.211545	0.105773	+	0.994390i	$0.466268\pi$
$264$	0	0
$265$	28.1192	1.72735
$266$	0	0
$267$	−11.9410	−0.730776
$268$	0	0
$269$	18.2760	1.11431	0.557153	−	0.830410i	$-0.311894\pi$
$269$	18.2760	1.11431	0.557153	+	0.830410i	$0.311894\pi$
$270$	0	0
$271$	7.31409	0.444299	0.222150	−	0.975013i	$-0.428693\pi$
$271$	7.31409	0.444299	0.222150	+	0.975013i	$0.428693\pi$
$272$	0	0
$273$	1.38780	0.0839938
$274$	0	0
$275$	−2.51454	−0.151632
$276$	0	0
$277$	−33.0033	−1.98298	−0.991489	−	0.130194i	$-0.958440\pi$
$277$	−33.0033	−1.98298	−0.991489	+	0.130194i	$0.958440\pi$
$278$	0	0
$279$	−8.34705	−0.499725
$280$	0	0
$281$	−7.25744	−0.432943	−0.216471	−	0.976289i	$-0.569455\pi$
$281$	−7.25744	−0.432943	−0.216471	+	0.976289i	$0.569455\pi$
$282$	0	0
$283$	−23.2776	−1.38371	−0.691856	−	0.722036i	$-0.743207\pi$
$283$	−23.2776	−1.38371	−0.691856	+	0.722036i	$0.743207\pi$
$284$	0	0
$285$	9.71057	0.575205
$286$	0	0
$287$	12.1297	0.715994
$288$	0	0
$289$	31.3568	1.84452
$290$	0	0
$291$	−4.56248	−0.267458
$292$	0	0
$293$	−22.0273	−1.28685	−0.643424	−	0.765510i	$-0.722487\pi$
$293$	−22.0273	−1.28685	−0.643424	+	0.765510i	$0.722487\pi$
$294$	0	0
$295$	11.7770	0.685681
$296$	0	0
$297$	5.65391	0.328073
$298$	0	0
$299$	−6.24233	−0.361003
$300$	0	0
$301$	3.77923	0.217831
$302$	0	0
$303$	−6.21061	−0.356790
$304$	0	0
$305$	−17.5192	−1.00314
$306$	0	0
$307$	−10.6519	−0.607936	−0.303968	−	0.952682i	$-0.598312\pi$
$307$	−10.6519	−0.607936	−0.303968	+	0.952682i	$0.598312\pi$
$308$	0	0
$309$	−12.6790	−0.721281
$310$	0	0
$311$	−9.04447	−0.512865	−0.256432	−	0.966562i	$-0.582547\pi$
$311$	−9.04447	−0.512865	−0.256432	+	0.966562i	$0.582547\pi$
$312$	0	0
$313$	−13.4038	−0.757626	−0.378813	−	0.925473i	$-0.623668\pi$
$313$	−13.4038	−0.757626	−0.378813	+	0.925473i	$0.623668\pi$
$314$	0	0
$315$	−2.94411	−0.165882
$316$	0	0
$317$	0.433334	0.0243385	0.0121692	−	0.999926i	$-0.496126\pi$
$317$	0.433334	0.0243385	0.0121692	+	0.999926i	$0.496126\pi$
$318$	0	0
$319$	10.2678	0.574885
$320$	0	0
$321$	14.1036	0.787184
$322$	0	0
$323$	17.7498	0.987627
$324$	0	0
$325$	2.51454	0.139481
$326$	0	0
$327$	28.7031	1.58728
$328$	0	0
$329$	6.53139	0.360087
$330$	0	0
$331$	−16.2816	−0.894916	−0.447458	−	0.894305i	$-0.647671\pi$
$331$	−16.2816	−0.894916	−0.447458	+	0.894305i	$0.647671\pi$
$332$	0	0
$333$	4.91357	0.269262
$334$	0	0
$335$	38.6457	2.11144
$336$	0	0
$337$	9.05662	0.493346	0.246673	−	0.969099i	$-0.420663\pi$
$337$	9.05662	0.493346	0.246673	+	0.969099i	$0.420663\pi$
$338$	0	0
$339$	−20.4811	−1.11238
$340$	0	0
$341$	−7.77194	−0.420874
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−23.7479	−1.27855
$346$	0	0
$347$	−14.1565	−0.759959	−0.379980	−	0.924995i	$-0.624069\pi$
$347$	−14.1565	−0.759959	−0.379980	+	0.924995i	$0.624069\pi$
$348$	0	0
$349$	18.6251	0.996981	0.498491	−	0.866895i	$-0.333888\pi$
$349$	18.6251	0.996981	0.498491	+	0.866895i	$0.333888\pi$
$350$	0	0
$351$	−5.65391	−0.301784
$352$	0	0
$353$	−30.8819	−1.64368	−0.821840	−	0.569719i	$-0.807052\pi$
$353$	−30.8819	−1.64368	−0.821840	+	0.569719i	$0.807052\pi$
$354$	0	0
$355$	−15.1802	−0.805679
$356$	0	0
$357$	9.65066	0.510767
$358$	0	0
$359$	−10.9506	−0.577953	−0.288976	−	0.957336i	$-0.593315\pi$
$359$	−10.9506	−0.577953	−0.288976	+	0.957336i	$0.593315\pi$
$360$	0	0
$361$	−12.4848	−0.657092
$362$	0	0
$363$	1.38780	0.0728409
$364$	0	0
$365$	−7.71161	−0.403644
$366$	0	0
$367$	−4.49768	−0.234777	−0.117389	−	0.993086i	$-0.537452\pi$
$367$	−4.49768	−0.234777	−0.117389	+	0.993086i	$0.537452\pi$
$368$	0	0
$369$	−13.0273	−0.678173
$370$	0	0
$371$	10.2578	0.532556
$372$	0	0
$373$	14.8396	0.768367	0.384184	−	0.923257i	$-0.374483\pi$
$373$	14.8396	0.768367	0.384184	+	0.923257i	$0.374483\pi$
$374$	0	0
$375$	−9.45555	−0.488283
$376$	0	0
$377$	−10.2678	−0.528818
$378$	0	0
$379$	4.59098	0.235823	0.117911	−	0.993024i	$-0.462380\pi$
$379$	4.59098	0.235823	0.117911	+	0.993024i	$0.462380\pi$
$380$	0	0
$381$	−18.4896	−0.947252
$382$	0	0
$383$	−20.7519	−1.06037	−0.530186	−	0.847882i	$-0.677878\pi$
$383$	−20.7519	−1.06037	−0.530186	+	0.847882i	$0.677878\pi$
$384$	0	0
$385$	−2.74127	−0.139708
$386$	0	0
$387$	−4.05888	−0.206325
$388$	0	0
$389$	−17.6014	−0.892425	−0.446212	−	0.894927i	$-0.647227\pi$
$389$	−17.6014	−0.892425	−0.446212	+	0.894927i	$0.647227\pi$
$390$	0	0
$391$	−43.4086	−2.19527
$392$	0	0
$393$	1.16282	0.0586566
$394$	0	0
$395$	37.9283	1.90838
$396$	0	0
$397$	−9.17484	−0.460472	−0.230236	−	0.973135i	$-0.573950\pi$
$397$	−9.17484	−0.460472	−0.230236	+	0.973135i	$0.573950\pi$
$398$	0	0
$399$	3.54237	0.177340
$400$	0	0
$401$	−37.9082	−1.89304	−0.946521	−	0.322641i	$-0.895429\pi$
$401$	−37.9082	−1.89304	−0.946521	+	0.322641i	$0.895429\pi$
$402$	0	0
$403$	7.77194	0.387148
$404$	0	0
$405$	−12.6771	−0.629929
$406$	0	0
$407$	4.57503	0.226776
$408$	0	0
$409$	−24.9169	−1.23206	−0.616030	−	0.787723i	$-0.711260\pi$
$409$	−24.9169	−1.23206	−0.616030	+	0.787723i	$0.711260\pi$
$410$	0	0
$411$	28.4053	1.40113
$412$	0	0
$413$	4.29618	0.211401
$414$	0	0
$415$	46.1269	2.26428
$416$	0	0
$417$	−21.3673	−1.04636
$418$	0	0
$419$	7.98588	0.390136	0.195068	−	0.980790i	$-0.437507\pi$
$419$	7.98588	0.390136	0.195068	+	0.980790i	$0.437507\pi$
$420$	0	0
$421$	−22.5967	−1.10130	−0.550648	−	0.834738i	$-0.685619\pi$
$421$	−22.5967	−1.10130	−0.550648	+	0.834738i	$0.685619\pi$
$422$	0	0
$423$	−7.01470	−0.341066
$424$	0	0
$425$	17.4858	0.848188
$426$	0	0
$427$	−6.39090	−0.309278
$428$	0	0
$429$	−1.38780	−0.0670038
$430$	0	0
$431$	−0.904118	−0.0435498	−0.0217749	−	0.999763i	$-0.506932\pi$
$431$	−0.904118	−0.0435498	−0.0217749	+	0.999763i	$0.506932\pi$
$432$	0	0
$433$	3.32773	0.159921	0.0799603	−	0.996798i	$-0.474521\pi$
$433$	3.32773	0.159921	0.0799603	+	0.996798i	$0.474521\pi$
$434$	0	0
$435$	−39.0621	−1.87289
$436$	0	0
$437$	−15.9335	−0.762204
$438$	0	0
$439$	−11.3152	−0.540043	−0.270022	−	0.962854i	$-0.587031\pi$
$439$	−11.3152	−0.540043	−0.270022	+	0.962854i	$0.587031\pi$
$440$	0	0
$441$	−1.07400	−0.0511428
$442$	0	0
$443$	−15.6329	−0.742740	−0.371370	−	0.928485i	$-0.621112\pi$
$443$	−15.6329	−0.742740	−0.371370	+	0.928485i	$0.621112\pi$
$444$	0	0
$445$	−23.5864	−1.11810
$446$	0	0
$447$	24.5599	1.16164
$448$	0	0
$449$	27.9062	1.31697	0.658486	−	0.752593i	$-0.271197\pi$
$449$	27.9062	1.31697	0.658486	+	0.752593i	$0.271197\pi$
$450$	0	0
$451$	−12.1297	−0.571166
$452$	0	0
$453$	−15.1710	−0.712794
$454$	0	0
$455$	2.74127	0.128512
$456$	0	0
$457$	16.1061	0.753411	0.376705	−	0.926333i	$-0.377057\pi$
$457$	16.1061	0.753411	0.376705	+	0.926333i	$0.377057\pi$
$458$	0	0
$459$	−39.3168	−1.83515
$460$	0	0
$461$	−8.31022	−0.387046	−0.193523	−	0.981096i	$-0.561991\pi$
$461$	−8.31022	−0.387046	−0.193523	+	0.981096i	$0.561991\pi$
$462$	0	0
$463$	14.3623	0.667473	0.333736	−	0.942666i	$-0.391690\pi$
$463$	14.3623	0.667473	0.333736	+	0.942666i	$0.391690\pi$
$464$	0	0
$465$	29.5671	1.37114
$466$	0	0
$467$	18.4835	0.855312	0.427656	−	0.903941i	$-0.359339\pi$
$467$	18.4835	0.855312	0.427656	+	0.903941i	$0.359339\pi$
$468$	0	0
$469$	14.0977	0.650973
$470$	0	0
$471$	8.96504	0.413087
$472$	0	0
$473$	−3.77923	−0.173769
$474$	0	0
$475$	6.41835	0.294494
$476$	0	0
$477$	−11.0168	−0.504425
$478$	0	0
$479$	−0.671272	−0.0306712	−0.0153356	−	0.999882i	$-0.504882\pi$
$479$	−0.671272	−0.0306712	−0.0153356	+	0.999882i	$0.504882\pi$
$480$	0	0
$481$	−4.57503	−0.208603
$482$	0	0
$483$	−8.66313	−0.394186
$484$	0	0
$485$	−9.01206	−0.409217
$486$	0	0
$487$	28.1025	1.27345	0.636724	−	0.771092i	$-0.280289\pi$
$487$	28.1025	1.27345	0.636724	+	0.771092i	$0.280289\pi$
$488$	0	0
$489$	8.06138	0.364548
$490$	0	0
$491$	2.07701	0.0937341	0.0468671	−	0.998901i	$-0.485076\pi$
$491$	2.07701	0.0937341	0.0468671	+	0.998901i	$0.485076\pi$
$492$	0	0
$493$	−71.4012	−3.21575
$494$	0	0
$495$	2.94411	0.132328
$496$	0	0
$497$	−5.53764	−0.248397
$498$	0	0
$499$	−7.80131	−0.349234	−0.174617	−	0.984636i	$-0.555869\pi$
$499$	−7.80131	−0.349234	−0.174617	+	0.984636i	$0.555869\pi$
$500$	0	0
$501$	9.35910	0.418134
$502$	0	0
$503$	0.615390	0.0274389	0.0137195	−	0.999906i	$-0.495633\pi$
$503$	0.615390	0.0274389	0.0137195	+	0.999906i	$0.495633\pi$
$504$	0	0
$505$	−12.2675	−0.545897
$506$	0	0
$507$	1.38780	0.0616346
$508$	0	0
$509$	14.7185	0.652386	0.326193	−	0.945303i	$-0.394234\pi$
$509$	14.7185	0.652386	0.326193	+	0.945303i	$0.394234\pi$
$510$	0	0
$511$	−2.81316	−0.124447
$512$	0	0
$513$	−14.4316	−0.637171
$514$	0	0
$515$	−25.0442	−1.10358
$516$	0	0
$517$	−6.53139	−0.287250
$518$	0	0
$519$	−23.6015	−1.03599
$520$	0	0
$521$	−7.62950	−0.334254	−0.167127	−	0.985935i	$-0.553449\pi$
$521$	−7.62950	−0.334254	−0.167127	+	0.985935i	$0.553449\pi$
$522$	0	0
$523$	−1.90130	−0.0831379	−0.0415690	−	0.999136i	$-0.513236\pi$
$523$	−1.90130	−0.0831379	−0.0415690	+	0.999136i	$0.513236\pi$
$524$	0	0
$525$	3.48968	0.152302
$526$	0	0
$527$	54.0454	2.35425
$528$	0	0
$529$	15.9667	0.694203
$530$	0	0
$531$	−4.61409	−0.200234
$532$	0	0
$533$	12.1297	0.525396
$534$	0	0
$535$	27.8581	1.20441
$536$	0	0
$537$	23.4854	1.01347
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	17.6174	0.757432	0.378716	−	0.925513i	$-0.376366\pi$
$541$	17.6174	0.757432	0.378716	+	0.925513i	$0.376366\pi$
$542$	0	0
$543$	17.8092	0.764268
$544$	0	0
$545$	56.6959	2.42858
$546$	0	0
$547$	−21.3048	−0.910926	−0.455463	−	0.890255i	$-0.650526\pi$
$547$	−21.3048	−0.910926	−0.455463	+	0.890255i	$0.650526\pi$
$548$	0	0
$549$	6.86382	0.292941
$550$	0	0
$551$	−26.2085	−1.11652
$552$	0	0
$553$	13.8361	0.588369
$554$	0	0
$555$	−17.4050	−0.738800
$556$	0	0
$557$	−12.5232	−0.530625	−0.265313	−	0.964162i	$-0.585475\pi$
$557$	−12.5232	−0.530625	−0.265313	+	0.964162i	$0.585475\pi$
$558$	0	0
$559$	3.77923	0.159844
$560$	0	0
$561$	−9.65066	−0.407451
$562$	0	0
$563$	23.5617	0.993007	0.496503	−	0.868035i	$-0.334617\pi$
$563$	23.5617	0.993007	0.496503	+	0.868035i	$0.334617\pi$
$564$	0	0
$565$	−40.4554	−1.70197
$566$	0	0
$567$	−4.62453	−0.194212
$568$	0	0
$569$	−37.3787	−1.56700	−0.783498	−	0.621394i	$-0.786567\pi$
$569$	−37.3787	−1.56700	−0.783498	+	0.621394i	$0.786567\pi$
$570$	0	0
$571$	42.4063	1.77465	0.887325	−	0.461145i	$-0.152561\pi$
$571$	42.4063	1.77465	0.887325	+	0.461145i	$0.152561\pi$
$572$	0	0
$573$	−25.5115	−1.06576
$574$	0	0
$575$	−15.6966	−0.654592
$576$	0	0
$577$	−0.0389215	−0.00162032	−0.000810162	−	1.00000i	$-0.500258\pi$
$577$	−0.0389215	−0.00162032	−0.000810162		1.00000i	$0.500258\pi$
$578$	0	0
$579$	−1.67839	−0.0697516
$580$	0	0
$581$	16.8269	0.698096
$582$	0	0
$583$	−10.2578	−0.424833
$584$	0	0
$585$	−2.94411	−0.121724
$586$	0	0
$587$	20.3189	0.838650	0.419325	−	0.907836i	$-0.362267\pi$
$587$	20.3189	0.838650	0.419325	+	0.907836i	$0.362267\pi$
$588$	0	0
$589$	19.8379	0.817405
$590$	0	0
$591$	−28.7663	−1.18329
$592$	0	0
$593$	35.4093	1.45409	0.727043	−	0.686592i	$-0.240894\pi$
$593$	35.4093	1.45409	0.727043	+	0.686592i	$0.240894\pi$
$594$	0	0
$595$	19.0625	0.781486
$596$	0	0
$597$	9.93393	0.406569
$598$	0	0
$599$	−5.71534	−0.233523	−0.116761	−	0.993160i	$-0.537251\pi$
$599$	−5.71534	−0.233523	−0.116761	+	0.993160i	$0.537251\pi$
$600$	0	0
$601$	−40.0034	−1.63177	−0.815886	−	0.578213i	$-0.803750\pi$
$601$	−40.0034	−1.63177	−0.815886	+	0.578213i	$0.803750\pi$
$602$	0	0
$603$	−15.1410	−0.616587
$604$	0	0
$605$	2.74127	0.111448
$606$	0	0
$607$	−28.7615	−1.16739	−0.583697	−	0.811971i	$-0.698395\pi$
$607$	−28.7615	−1.16739	−0.583697	+	0.811971i	$0.698395\pi$
$608$	0	0
$609$	−14.2497	−0.577426
$610$	0	0
$611$	6.53139	0.264232
$612$	0	0
$613$	−36.3957	−1.47001	−0.735004	−	0.678063i	$-0.762820\pi$
$613$	−36.3957	−1.47001	−0.735004	+	0.678063i	$0.762820\pi$
$614$	0	0
$615$	46.1455	1.86077
$616$	0	0
$617$	11.0656	0.445484	0.222742	−	0.974877i	$-0.428499\pi$
$617$	11.0656	0.445484	0.222742	+	0.974877i	$0.428499\pi$
$618$	0	0
$619$	27.2470	1.09515	0.547576	−	0.836756i	$-0.315551\pi$
$619$	27.2470	1.09515	0.547576	+	0.836756i	$0.315551\pi$
$620$	0	0
$621$	35.2936	1.41628
$622$	0	0
$623$	−8.60422	−0.344721
$624$	0	0
$625$	−31.2498	−1.24999
$626$	0	0
$627$	−3.54237	−0.141469
$628$	0	0
$629$	−31.8143	−1.26852
$630$	0	0
$631$	36.7169	1.46168	0.730838	−	0.682551i	$-0.239129\pi$
$631$	36.7169	1.46168	0.730838	+	0.682551i	$0.239129\pi$
$632$	0	0
$633$	−28.0424	−1.11458
$634$	0	0
$635$	−36.5217	−1.44932
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	5.94742	0.235276
$640$	0	0
$641$	36.4999	1.44166	0.720829	−	0.693113i	$-0.243761\pi$
$641$	36.4999	1.44166	0.720829	+	0.693113i	$0.243761\pi$
$642$	0	0
$643$	47.9769	1.89202	0.946012	−	0.324132i	$-0.105072\pi$
$643$	47.9769	1.89202	0.946012	+	0.324132i	$0.105072\pi$
$644$	0	0
$645$	14.3775	0.566112
$646$	0	0
$647$	−20.4471	−0.803857	−0.401929	−	0.915671i	$-0.631660\pi$
$647$	−20.4471	−0.803857	−0.401929	+	0.915671i	$0.631660\pi$
$648$	0	0
$649$	−4.29618	−0.168640
$650$	0	0
$651$	10.7859	0.422734
$652$	0	0
$653$	−8.67800	−0.339596	−0.169798	−	0.985479i	$-0.554312\pi$
$653$	−8.67800	−0.339596	−0.169798	+	0.985479i	$0.554312\pi$
$654$	0	0
$655$	2.29687	0.0897460
$656$	0	0
$657$	3.02132	0.117873
$658$	0	0
$659$	−22.0533	−0.859074	−0.429537	−	0.903049i	$-0.641323\pi$
$659$	−22.0533	−0.859074	−0.429537	+	0.903049i	$0.641323\pi$
$660$	0	0
$661$	23.8034	0.925843	0.462922	−	0.886399i	$-0.346801\pi$
$661$	23.8034	0.925843	0.462922	+	0.886399i	$0.346801\pi$
$662$	0	0
$663$	9.65066	0.374801
$664$	0	0
$665$	6.99707	0.271335
$666$	0	0
$667$	64.0949	2.48176
$668$	0	0
$669$	39.7592	1.53718
$670$	0	0
$671$	6.39090	0.246718
$672$	0	0
$673$	45.9809	1.77243	0.886216	−	0.463272i	$-0.153325\pi$
$673$	45.9809	1.77243	0.886216	+	0.463272i	$0.153325\pi$
$674$	0	0
$675$	−14.2170	−0.547211
$676$	0	0
$677$	33.8283	1.30013	0.650063	−	0.759880i	$-0.274742\pi$
$677$	33.8283	1.30013	0.650063	+	0.759880i	$0.274742\pi$
$678$	0	0
$679$	−3.28756	−0.126165
$680$	0	0
$681$	11.1889	0.428761
$682$	0	0
$683$	−12.4917	−0.477981	−0.238991	−	0.971022i	$-0.576817\pi$
$683$	−12.4917	−0.477981	−0.238991	+	0.971022i	$0.576817\pi$
$684$	0	0
$685$	56.1077	2.14377
$686$	0	0
$687$	−18.5837	−0.709014
$688$	0	0
$689$	10.2578	0.390789
$690$	0	0
$691$	2.19589	0.0835356	0.0417678	−	0.999127i	$-0.486701\pi$
$691$	2.19589	0.0835356	0.0417678	+	0.999127i	$0.486701\pi$
$692$	0	0
$693$	1.07400	0.0407978
$694$	0	0
$695$	−42.2059	−1.60096
$696$	0	0
$697$	84.3488	3.19494
$698$	0	0
$699$	−11.3482	−0.429230
$700$	0	0
$701$	−16.2127	−0.612345	−0.306172	−	0.951976i	$-0.599048\pi$
$701$	−16.2127	−0.612345	−0.306172	+	0.951976i	$0.599048\pi$
$702$	0	0
$703$	−11.6778	−0.440435
$704$	0	0
$705$	24.8476	0.935816
$706$	0	0
$707$	−4.47513	−0.168305
$708$	0	0
$709$	32.3894	1.21641	0.608205	−	0.793780i	$-0.291890\pi$
$709$	32.3894	1.21641	0.608205	+	0.793780i	$0.291890\pi$
$710$	0	0
$711$	−14.8599	−0.557290
$712$	0	0
$713$	−48.5150	−1.81690
$714$	0	0
$715$	−2.74127	−0.102518
$716$	0	0
$717$	34.8112	1.30005
$718$	0	0
$719$	−12.2523	−0.456932	−0.228466	−	0.973552i	$-0.573371\pi$
$719$	−12.2523	−0.456932	−0.228466	+	0.973552i	$0.573371\pi$
$720$	0	0
$721$	−9.13598	−0.340242
$722$	0	0
$723$	−16.9623	−0.630834
$724$	0	0
$725$	−25.8187	−0.958882
$726$	0	0
$727$	−28.9408	−1.07335	−0.536677	−	0.843788i	$-0.680321\pi$
$727$	−28.9408	−1.07335	−0.536677	+	0.843788i	$0.680321\pi$
$728$	0	0
$729$	28.5063	1.05579
$730$	0	0
$731$	26.2804	0.972016
$732$	0	0
$733$	38.1957	1.41079	0.705395	−	0.708814i	$-0.250769\pi$
$733$	38.1957	1.41079	0.705395	+	0.708814i	$0.250769\pi$
$734$	0	0
$735$	3.80434	0.140325
$736$	0	0
$737$	−14.0977	−0.519297
$738$	0	0
$739$	29.5144	1.08570	0.542852	−	0.839829i	$-0.317345\pi$
$739$	29.5144	1.08570	0.542852	+	0.839829i	$0.317345\pi$
$740$	0	0
$741$	3.54237	0.130132
$742$	0	0
$743$	20.9017	0.766808	0.383404	−	0.923581i	$-0.374752\pi$
$743$	20.9017	0.766808	0.383404	+	0.923581i	$0.374752\pi$
$744$	0	0
$745$	48.5120	1.77734
$746$	0	0
$747$	−18.0720	−0.661221
$748$	0	0
$749$	10.1625	0.371330
$750$	0	0
$751$	−24.8579	−0.907079	−0.453539	−	0.891236i	$-0.649839\pi$
$751$	−24.8579	−0.907079	−0.453539	+	0.891236i	$0.649839\pi$
$752$	0	0
$753$	4.26262	0.155339
$754$	0	0
$755$	−29.9665	−1.09059
$756$	0	0
$757$	−44.3929	−1.61349	−0.806743	−	0.590902i	$-0.798772\pi$
$757$	−44.3929	−1.61349	−0.806743	+	0.590902i	$0.798772\pi$
$758$	0	0
$759$	8.66313	0.314452
$760$	0	0
$761$	9.75959	0.353785	0.176892	−	0.984230i	$-0.443396\pi$
$761$	9.75959	0.353785	0.176892	+	0.984230i	$0.443396\pi$
$762$	0	0
$763$	20.6824	0.748752
$764$	0	0
$765$	−20.4731	−0.740206
$766$	0	0
$767$	4.29618	0.155126
$768$	0	0
$769$	−49.8103	−1.79621	−0.898103	−	0.439785i	$-0.855055\pi$
$769$	−49.8103	−1.79621	−0.898103	+	0.439785i	$0.855055\pi$
$770$	0	0
$771$	32.1976	1.15957
$772$	0	0
$773$	−3.37984	−0.121564	−0.0607821	−	0.998151i	$-0.519359\pi$
$773$	−3.37984	−0.121564	−0.0607821	+	0.998151i	$0.519359\pi$
$774$	0	0
$775$	19.5428	0.701999
$776$	0	0
$777$	−6.34925	−0.227778
$778$	0	0
$779$	30.9610	1.10929
$780$	0	0
$781$	5.53764	0.198152
$782$	0	0
$783$	58.0531	2.07465
$784$	0	0
$785$	17.7082	0.632033
$786$	0	0
$787$	−50.5844	−1.80314	−0.901570	−	0.432633i	$-0.857585\pi$
$787$	−50.5844	−1.80314	−0.901570	+	0.432633i	$0.857585\pi$
$788$	0	0
$789$	4.76112	0.169501
$790$	0	0
$791$	−14.7579	−0.524732
$792$	0	0
$793$	−6.39090	−0.226948
$794$	0	0
$795$	39.0240	1.38404
$796$	0	0
$797$	−37.4478	−1.32647	−0.663235	−	0.748411i	$-0.730817\pi$
$797$	−37.4478	−1.32647	−0.663235	+	0.748411i	$0.730817\pi$
$798$	0	0
$799$	45.4187	1.60680
$800$	0	0
$801$	9.24091	0.326512
$802$	0	0
$803$	2.81316	0.0992741
$804$	0	0
$805$	−17.1119	−0.603114
$806$	0	0
$807$	25.3635	0.892837
$808$	0	0
$809$	−27.9751	−0.983551	−0.491775	−	0.870722i	$-0.663652\pi$
$809$	−27.9751	−0.983551	−0.491775	+	0.870722i	$0.663652\pi$
$810$	0	0
$811$	−15.1442	−0.531786	−0.265893	−	0.964003i	$-0.585667\pi$
$811$	−15.1442	−0.531786	−0.265893	+	0.964003i	$0.585667\pi$
$812$	0	0
$813$	10.1505	0.355995
$814$	0	0
$815$	15.9233	0.557767
$816$	0	0
$817$	9.64647	0.337487
$818$	0	0
$819$	−1.07400	−0.0375285
$820$	0	0
$821$	−17.9640	−0.626946	−0.313473	−	0.949597i	$-0.601493\pi$
$821$	−17.9640	−0.626946	−0.313473	+	0.949597i	$0.601493\pi$
$822$	0	0
$823$	12.7090	0.443009	0.221505	−	0.975159i	$-0.428903\pi$
$823$	12.7090	0.443009	0.221505	+	0.975159i	$0.428903\pi$
$824$	0	0
$825$	−3.48968	−0.121495
$826$	0	0
$827$	−40.7277	−1.41624	−0.708121	−	0.706091i	$-0.750457\pi$
$827$	−40.7277	−1.41624	−0.708121	+	0.706091i	$0.750457\pi$
$828$	0	0
$829$	−9.11331	−0.316518	−0.158259	−	0.987398i	$-0.550588\pi$
$829$	−9.11331	−0.316518	−0.158259	+	0.987398i	$0.550588\pi$
$830$	0	0
$831$	−45.8022	−1.58886
$832$	0	0
$833$	6.95391	0.240939
$834$	0	0
$835$	18.4866	0.639755
$836$	0	0
$837$	−43.9419	−1.51885
$838$	0	0
$839$	−3.86984	−0.133602	−0.0668008	−	0.997766i	$-0.521279\pi$
$839$	−3.86984	−0.133602	−0.0668008	+	0.997766i	$0.521279\pi$
$840$	0	0
$841$	76.4273	2.63542
$842$	0	0
$843$	−10.0719	−0.346895
$844$	0	0
$845$	2.74127	0.0943024
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−32.3048	−1.10870
$850$	0	0
$851$	28.5588	0.978984
$852$	0	0
$853$	−10.6214	−0.363669	−0.181834	−	0.983329i	$-0.558204\pi$
$853$	−10.6214	−0.363669	−0.181834	+	0.983329i	$0.558204\pi$
$854$	0	0
$855$	−7.51484	−0.257002
$856$	0	0
$857$	−17.8575	−0.610002	−0.305001	−	0.952352i	$-0.598657\pi$
$857$	−17.8575	−0.610002	−0.305001	+	0.952352i	$0.598657\pi$
$858$	0	0
$859$	21.8326	0.744918	0.372459	−	0.928049i	$-0.378515\pi$
$859$	21.8326	0.744918	0.372459	+	0.928049i	$0.378515\pi$
$860$	0	0
$861$	16.8337	0.573690
$862$	0	0
$863$	1.73176	0.0589499	0.0294750	−	0.999566i	$-0.490616\pi$
$863$	1.73176	0.0589499	0.0294750	+	0.999566i	$0.490616\pi$
$864$	0	0
$865$	−46.6190	−1.58509
$866$	0	0
$867$	43.5171	1.47792
$868$	0	0
$869$	−13.8361	−0.469356
$870$	0	0
$871$	14.0977	0.477684
$872$	0	0
$873$	3.53083	0.119500
$874$	0	0
$875$	−6.81332	−0.230332
$876$	0	0
$877$	−21.2135	−0.716330	−0.358165	−	0.933658i	$-0.616598\pi$
$877$	−21.2135	−0.716330	−0.358165	+	0.933658i	$0.616598\pi$
$878$	0	0
$879$	−30.5696	−1.03109
$880$	0	0
$881$	−33.2955	−1.12175	−0.560877	−	0.827899i	$-0.689536\pi$
$881$	−33.2955	−1.12175	−0.560877	+	0.827899i	$0.689536\pi$
$882$	0	0
$883$	50.8532	1.71135	0.855674	−	0.517515i	$-0.173143\pi$
$883$	50.8532	1.71135	0.855674	+	0.517515i	$0.173143\pi$
$884$	0	0
$885$	16.3441	0.549402
$886$	0	0
$887$	15.2793	0.513029	0.256514	−	0.966540i	$-0.417426\pi$
$887$	15.2793	0.513029	0.256514	+	0.966540i	$0.417426\pi$
$888$	0	0
$889$	−13.3229	−0.446837
$890$	0	0
$891$	4.62453	0.154928
$892$	0	0
$893$	16.6714	0.557886
$894$	0	0
$895$	46.3897	1.55064
$896$	0	0
$897$	−8.66313	−0.289254
$898$	0	0
$899$	−79.8006	−2.66150
$900$	0	0
$901$	71.3315	2.37640
$902$	0	0
$903$	5.24483	0.174537
$904$	0	0
$905$	35.1778	1.16935
$906$	0	0
$907$	−19.9491	−0.662400	−0.331200	−	0.943561i	$-0.607454\pi$
$907$	−19.9491	−0.662400	−0.331200	+	0.943561i	$0.607454\pi$
$908$	0	0
$909$	4.80628	0.159414
$910$	0	0
$911$	−0.872833	−0.0289182	−0.0144591	−	0.999895i	$-0.504603\pi$
$911$	−0.872833	−0.0289182	−0.0144591	+	0.999895i	$0.504603\pi$
$912$	0	0
$913$	−16.8269	−0.556888
$914$	0	0
$915$	−24.3132	−0.803769
$916$	0	0
$917$	0.837885	0.0276694
$918$	0	0
$919$	29.4831	0.972558	0.486279	−	0.873804i	$-0.338354\pi$
$919$	29.4831	0.972558	0.486279	+	0.873804i	$0.338354\pi$
$920$	0	0
$921$	−14.7827	−0.487108
$922$	0	0
$923$	−5.53764	−0.182274
$924$	0	0
$925$	−11.5041	−0.378252
$926$	0	0
$927$	9.81203	0.322269
$928$	0	0
$929$	−39.1492	−1.28445	−0.642223	−	0.766518i	$-0.721988\pi$
$929$	−39.1492	−1.28445	−0.642223	+	0.766518i	$0.721988\pi$
$930$	0	0
$931$	2.55250	0.0836547
$932$	0	0
$933$	−12.5520	−0.410933
$934$	0	0
$935$	−19.0625	−0.623410
$936$	0	0
$937$	−4.09280	−0.133706	−0.0668530	−	0.997763i	$-0.521296\pi$
$937$	−4.09280	−0.133706	−0.0668530	+	0.997763i	$0.521296\pi$
$938$	0	0
$939$	−18.6018	−0.607048
$940$	0	0
$941$	−13.5138	−0.440537	−0.220269	−	0.975439i	$-0.570693\pi$
$941$	−13.5138	−0.440537	−0.220269	+	0.975439i	$0.570693\pi$
$942$	0	0
$943$	−75.7176	−2.46570
$944$	0	0
$945$	−15.4989	−0.504179
$946$	0	0
$947$	40.9177	1.32965	0.664823	−	0.747001i	$-0.268507\pi$
$947$	40.9177	1.32965	0.664823	+	0.747001i	$0.268507\pi$
$948$	0	0
$949$	−2.81316	−0.0913189
$950$	0	0
$951$	0.601383	0.0195012
$952$	0	0
$953$	−29.5662	−0.957743	−0.478872	−	0.877885i	$-0.658954\pi$
$953$	−29.5662	−0.957743	−0.478872	+	0.877885i	$0.658954\pi$
$954$	0	0
$955$	−50.3918	−1.63064
$956$	0	0
$957$	14.2497	0.460626
$958$	0	0
$959$	20.4678	0.660940
$960$	0	0
$961$	29.4031	0.948488
$962$	0	0
$963$	−10.9145	−0.351715
$964$	0	0
$965$	−3.31525	−0.106722
$966$	0	0
$967$	−26.0329	−0.837161	−0.418580	−	0.908180i	$-0.637472\pi$
$967$	−26.0329	−0.837161	−0.418580	+	0.908180i	$0.637472\pi$
$968$	0	0
$969$	24.6333	0.791336
$970$	0	0
$971$	−6.12348	−0.196512	−0.0982559	−	0.995161i	$-0.531326\pi$
$971$	−6.12348	−0.196512	−0.0982559	+	0.995161i	$0.531326\pi$
$972$	0	0
$973$	−15.3965	−0.493589
$974$	0	0
$975$	3.48968	0.111759
$976$	0	0
$977$	7.44253	0.238108	0.119054	−	0.992888i	$-0.462014\pi$
$977$	7.44253	0.238108	0.119054	+	0.992888i	$0.462014\pi$
$978$	0	0
$979$	8.60422	0.274992
$980$	0	0
$981$	−22.2128	−0.709201
$982$	0	0
$983$	8.20767	0.261784	0.130892	−	0.991397i	$-0.458216\pi$
$983$	8.20767	0.261784	0.130892	+	0.991397i	$0.458216\pi$
$984$	0	0
$985$	−56.8207	−1.81046
$986$	0	0
$987$	9.06429	0.288520
$988$	0	0
$989$	−23.5912	−0.750156
$990$	0	0
$991$	−48.0724	−1.52707	−0.763536	−	0.645766i	$-0.776538\pi$
$991$	−48.0724	−1.52707	−0.763536	+	0.645766i	$0.776538\pi$
$992$	0	0
$993$	−22.5956	−0.717051
$994$	0	0
$995$	19.6220	0.622060
$996$	0	0
$997$	12.6449	0.400468	0.200234	−	0.979748i	$-0.435830\pi$
$997$	12.6449	0.400468	0.200234	+	0.979748i	$0.435830\pi$
$998$	0	0
$999$	25.8668	0.818390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.8	✓	10	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38780 q^{3} +2.74127 q^{5} +1.00000 q^{7} -1.07400 q^{9} +O(q^{10})\)	\(q+1.38780 q^{3} +2.74127 q^{5} +1.00000 q^{7} -1.07400 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.80434 q^{15} +6.95391 q^{17} +2.55250 q^{19} +1.38780 q^{21} -6.24233 q^{23} +2.51454 q^{25} -5.65391 q^{27} -10.2678 q^{29} +7.77194 q^{31} -1.38780 q^{33} +2.74127 q^{35} -4.57503 q^{37} +1.38780 q^{39} +12.1297 q^{41} +3.77923 q^{43} -2.94411 q^{45} +6.53139 q^{47} +1.00000 q^{49} +9.65066 q^{51} +10.2578 q^{53} -2.74127 q^{55} +3.54237 q^{57} +4.29618 q^{59} -6.39090 q^{61} -1.07400 q^{63} +2.74127 q^{65} +14.0977 q^{67} -8.66313 q^{69} -5.53764 q^{71} -2.81316 q^{73} +3.48968 q^{75} -1.00000 q^{77} +13.8361 q^{79} -4.62453 q^{81} +16.8269 q^{83} +19.0625 q^{85} -14.2497 q^{87} -8.60422 q^{89} +1.00000 q^{91} +10.7859 q^{93} +6.99707 q^{95} -3.28756 q^{97} +1.07400 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more