LMFDB - Embedded newform 8008.2.a.t.1.3

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			1.3
Root			$-1.29966$ 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)

$f(q)$	$=$	$q-1.29966 q^{3} +0.367835 q^{5} +1.00000 q^{7} -1.31089 q^{9} +O(q^{10})$	$q-1.29966 q^{3} +0.367835 q^{5} +1.00000 q^{7} -1.31089 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.478060 q^{15} -3.16609 q^{17} -2.94919 q^{19} -1.29966 q^{21} -2.33452 q^{23} -4.86470 q^{25} +5.60268 q^{27} -5.14289 q^{29} -5.54381 q^{31} +1.29966 q^{33} +0.367835 q^{35} +2.80577 q^{37} -1.29966 q^{39} +4.11321 q^{41} +3.28334 q^{43} -0.482189 q^{45} -1.06091 q^{47} +1.00000 q^{49} +4.11484 q^{51} +1.85799 q^{53} -0.367835 q^{55} +3.83295 q^{57} +3.95270 q^{59} -3.98803 q^{61} -1.31089 q^{63} +0.367835 q^{65} -3.64220 q^{67} +3.03408 q^{69} +10.5516 q^{71} +12.3988 q^{73} +6.32245 q^{75} -1.00000 q^{77} +6.64211 q^{79} -3.34892 q^{81} -13.3597 q^{83} -1.16460 q^{85} +6.68401 q^{87} +9.60576 q^{89} +1.00000 q^{91} +7.20507 q^{93} -1.08482 q^{95} -16.6689 q^{97} +1.31089 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.29966	−0.750359	−0.375179	−	0.926952i	$-0.622419\pi$
$3$	−1.29966	−0.750359	−0.375179	+	0.926952i	$0.622419\pi$
$4$	0	0
$5$	0.367835	0.164501	0.0822503	−	0.996612i	$-0.473789\pi$
$5$	0.367835	0.164501	0.0822503	+	0.996612i	$0.473789\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.31089	−0.436962
$10$	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
$15$	−0.478060	−0.123434
$16$	0	0
$17$	−3.16609	−0.767891	−0.383945	−	0.923356i	$-0.625435\pi$
$17$	−3.16609	−0.767891	−0.383945	+	0.923356i	$0.625435\pi$
$18$	0	0
$19$	−2.94919	−0.676591	−0.338296	−	0.941040i	$-0.609850\pi$
$19$	−2.94919	−0.676591	−0.338296	+	0.941040i	$0.609850\pi$
$20$	0	0
$21$	−1.29966	−0.283609
$22$	0	0
$23$	−2.33452	−0.486782	−0.243391	−	0.969928i	$-0.578260\pi$
$23$	−2.33452	−0.486782	−0.243391	+	0.969928i	$0.578260\pi$
$24$	0	0
$25$	−4.86470	−0.972940
$26$	0	0
$27$	5.60268	1.07824
$28$	0	0
$29$	−5.14289	−0.955011	−0.477506	−	0.878629i	$-0.658459\pi$
$29$	−5.14289	−0.955011	−0.477506	+	0.878629i	$0.658459\pi$
$30$	0	0
$31$	−5.54381	−0.995698	−0.497849	−	0.867264i	$-0.665877\pi$
$31$	−5.54381	−0.995698	−0.497849	+	0.867264i	$0.665877\pi$
$32$	0	0
$33$	1.29966	0.226242
$34$	0	0
$35$	0.367835	0.0621754
$36$	0	0
$37$	2.80577	0.461266	0.230633	−	0.973041i	$-0.425920\pi$
$37$	2.80577	0.461266	0.230633	+	0.973041i	$0.425920\pi$
$38$	0	0
$39$	−1.29966	−0.208112
$40$	0	0
$41$	4.11321	0.642375	0.321188	−	0.947016i	$-0.395918\pi$
$41$	4.11321	0.642375	0.321188	+	0.947016i	$0.395918\pi$
$42$	0	0
$43$	3.28334	0.500704	0.250352	−	0.968155i	$-0.419454\pi$
$43$	3.28334	0.500704	0.250352	+	0.968155i	$0.419454\pi$
$44$	0	0
$45$	−0.482189	−0.0718805
$46$	0	0
$47$	−1.06091	−0.154749	−0.0773746	−	0.997002i	$-0.524654\pi$
$47$	−1.06091	−0.154749	−0.0773746	+	0.997002i	$0.524654\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.11484	0.576193
$52$	0	0
$53$	1.85799	0.255214	0.127607	−	0.991825i	$-0.459270\pi$
$53$	1.85799	0.255214	0.127607	+	0.991825i	$0.459270\pi$
$54$	0	0
$55$	−0.367835	−0.0495988
$56$	0	0
$57$	3.83295	0.507686
$58$	0	0
$59$	3.95270	0.514597	0.257299	−	0.966332i	$-0.417168\pi$
$59$	3.95270	0.514597	0.257299	+	0.966332i	$0.417168\pi$
$60$	0	0
$61$	−3.98803	−0.510614	−0.255307	−	0.966860i	$-0.582177\pi$
$61$	−3.98803	−0.510614	−0.255307	+	0.966860i	$0.582177\pi$
$62$	0	0
$63$	−1.31089	−0.165156
$64$	0	0
$65$	0.367835	0.0456243
$66$	0	0
$67$	−3.64220	−0.444965	−0.222482	−	0.974937i	$-0.571416\pi$
$67$	−3.64220	−0.444965	−0.222482	+	0.974937i	$0.571416\pi$
$68$	0	0
$69$	3.03408	0.365261
$70$	0	0
$71$	10.5516	1.25224	0.626120	−	0.779727i	$-0.284642\pi$
$71$	10.5516	1.25224	0.626120	+	0.779727i	$0.284642\pi$
$72$	0	0
$73$	12.3988	1.45117	0.725583	−	0.688135i	$-0.241570\pi$
$73$	12.3988	1.45117	0.725583	+	0.688135i	$0.241570\pi$
$74$	0	0
$75$	6.32245	0.730054
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	6.64211	0.747296	0.373648	−	0.927571i	$-0.378107\pi$
$79$	6.64211	0.747296	0.373648	+	0.927571i	$0.378107\pi$
$80$	0	0
$81$	−3.34892	−0.372102
$82$	0	0
$83$	−13.3597	−1.46642	−0.733211	−	0.680001i	$-0.761979\pi$
$83$	−13.3597	−1.46642	−0.733211	+	0.680001i	$0.761979\pi$
$84$	0	0
$85$	−1.16460	−0.126318
$86$	0	0
$87$	6.68401	0.716601
$88$	0	0
$89$	9.60576	1.01821	0.509104	−	0.860705i	$-0.329977\pi$
$89$	9.60576	1.01821	0.509104	+	0.860705i	$0.329977\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	7.20507	0.747131
$94$	0	0
$95$	−1.08482	−0.111300
$96$	0	0
$97$	−16.6689	−1.69247	−0.846233	−	0.532813i	$-0.821135\pi$
$97$	−16.6689	−1.69247	−0.846233	+	0.532813i	$0.821135\pi$
$98$	0	0
$99$	1.31089	0.131749
$100$	0	0
$101$	−15.5728	−1.54955	−0.774776	−	0.632236i	$-0.782137\pi$
$101$	−15.5728	−1.54955	−0.774776	+	0.632236i	$0.782137\pi$
$102$	0	0
$103$	17.1091	1.68581	0.842906	−	0.538061i	$-0.180843\pi$
$103$	17.1091	1.68581	0.842906	+	0.538061i	$0.180843\pi$
$104$	0	0
$105$	−0.478060	−0.0466538
$106$	0	0
$107$	6.20743	0.600095	0.300047	−	0.953924i	$-0.402998\pi$
$107$	6.20743	0.600095	0.300047	+	0.953924i	$0.402998\pi$
$108$	0	0
$109$	3.57081	0.342021	0.171011	−	0.985269i	$-0.445297\pi$
$109$	3.57081	0.342021	0.171011	+	0.985269i	$0.445297\pi$
$110$	0	0
$111$	−3.64654	−0.346115
$112$	0	0
$113$	11.0151	1.03621	0.518106	−	0.855316i	$-0.326637\pi$
$113$	11.0151	1.03621	0.518106	+	0.855316i	$0.326637\pi$
$114$	0	0
$115$	−0.858718	−0.0800759
$116$	0	0
$117$	−1.31089	−0.121191
$118$	0	0
$119$	−3.16609	−0.290235
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.34577	−0.482012
$124$	0	0
$125$	−3.62858	−0.324550
$126$	0	0
$127$	−10.8010	−0.958436	−0.479218	−	0.877696i	$-0.659080\pi$
$127$	−10.8010	−0.958436	−0.479218	+	0.877696i	$0.659080\pi$
$128$	0	0
$129$	−4.26722	−0.375708
$130$	0	0
$131$	22.1122	1.93195	0.965976	−	0.258631i	$-0.0832713\pi$
$131$	22.1122	1.93195	0.965976	+	0.258631i	$0.0832713\pi$
$132$	0	0
$133$	−2.94919	−0.255728
$134$	0	0
$135$	2.06086	0.177371
$136$	0	0
$137$	−5.87131	−0.501620	−0.250810	−	0.968036i	$-0.580697\pi$
$137$	−5.87131	−0.501620	−0.250810	+	0.968036i	$0.580697\pi$
$138$	0	0
$139$	−10.9189	−0.926129	−0.463065	−	0.886324i	$-0.653250\pi$
$139$	−10.9189	−0.926129	−0.463065	+	0.886324i	$0.653250\pi$
$140$	0	0
$141$	1.37882	0.116117
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−1.89173	−0.157100
$146$	0	0
$147$	−1.29966	−0.107194
$148$	0	0
$149$	16.4376	1.34662	0.673311	−	0.739359i	$-0.264871\pi$
$149$	16.4376	1.34662	0.673311	+	0.739359i	$0.264871\pi$
$150$	0	0
$151$	12.9191	1.05134	0.525669	−	0.850689i	$-0.323815\pi$
$151$	12.9191	1.05134	0.525669	+	0.850689i	$0.323815\pi$
$152$	0	0
$153$	4.15039	0.335539
$154$	0	0
$155$	−2.03921	−0.163793
$156$	0	0
$157$	−15.4780	−1.23528	−0.617641	−	0.786460i	$-0.711911\pi$
$157$	−15.4780	−1.23528	−0.617641	+	0.786460i	$0.711911\pi$
$158$	0	0
$159$	−2.41475	−0.191502
$160$	0	0
$161$	−2.33452	−0.183986
$162$	0	0
$163$	0.0360998	0.00282756	0.00141378	−	0.999999i	$-0.499550\pi$
$163$	0.0360998	0.00282756	0.00141378	+	0.999999i	$0.499550\pi$
$164$	0	0
$165$	0.478060	0.0372169
$166$	0	0
$167$	19.4309	1.50361	0.751804	−	0.659386i	$-0.229184\pi$
$167$	19.4309	1.50361	0.751804	+	0.659386i	$0.229184\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.86606	0.295645
$172$	0	0
$173$	−8.39797	−0.638486	−0.319243	−	0.947673i	$-0.603429\pi$
$173$	−8.39797	−0.638486	−0.319243	+	0.947673i	$0.603429\pi$
$174$	0	0
$175$	−4.86470	−0.367737
$176$	0	0
$177$	−5.13716	−0.386133
$178$	0	0
$179$	3.08054	0.230250	0.115125	−	0.993351i	$-0.463273\pi$
$179$	3.08054	0.230250	0.115125	+	0.993351i	$0.463273\pi$
$180$	0	0
$181$	−18.1763	−1.35103	−0.675517	−	0.737344i	$-0.736080\pi$
$181$	−18.1763	−1.35103	−0.675517	+	0.737344i	$0.736080\pi$
$182$	0	0
$183$	5.18308	0.383144
$184$	0	0
$185$	1.03206	0.0758785
$186$	0	0
$187$	3.16609	0.231528
$188$	0	0
$189$	5.60268	0.407535
$190$	0	0
$191$	−8.59910	−0.622209	−0.311104	−	0.950376i	$-0.600699\pi$
$191$	−8.59910	−0.622209	−0.311104	+	0.950376i	$0.600699\pi$
$192$	0	0
$193$	−26.4916	−1.90691	−0.953455	−	0.301534i	$-0.902501\pi$
$193$	−26.4916	−1.90691	−0.953455	+	0.301534i	$0.902501\pi$
$194$	0	0
$195$	−0.478060	−0.0342346
$196$	0	0
$197$	11.1829	0.796745	0.398373	−	0.917224i	$-0.369575\pi$
$197$	11.1829	0.796745	0.398373	+	0.917224i	$0.369575\pi$
$198$	0	0
$199$	12.0016	0.850771	0.425386	−	0.905012i	$-0.360138\pi$
$199$	12.0016	0.850771	0.425386	+	0.905012i	$0.360138\pi$
$200$	0	0
$201$	4.73361	0.333883
$202$	0	0
$203$	−5.14289	−0.360960
$204$	0	0
$205$	1.51298	0.105671
$206$	0	0
$207$	3.06029	0.212705
$208$	0	0
$209$	2.94919	0.204000
$210$	0	0
$211$	14.7747	1.01713	0.508567	−	0.861023i	$-0.330176\pi$
$211$	14.7747	1.01713	0.508567	+	0.861023i	$0.330176\pi$
$212$	0	0
$213$	−13.7134	−0.939629
$214$	0	0
$215$	1.20773	0.0823662
$216$	0	0
$217$	−5.54381	−0.376338
$218$	0	0
$219$	−16.1142	−1.08889
$220$	0	0
$221$	−3.16609	−0.212975
$222$	0	0
$223$	26.1032	1.74800	0.874001	−	0.485923i	$-0.161517\pi$
$223$	26.1032	1.74800	0.874001	+	0.485923i	$0.161517\pi$
$224$	0	0
$225$	6.37706	0.425138
$226$	0	0
$227$	−7.04039	−0.467287	−0.233643	−	0.972322i	$-0.575065\pi$
$227$	−7.04039	−0.467287	−0.233643	+	0.972322i	$0.575065\pi$
$228$	0	0
$229$	26.5857	1.75683	0.878417	−	0.477895i	$-0.158600\pi$
$229$	26.5857	1.75683	0.878417	+	0.477895i	$0.158600\pi$
$230$	0	0
$231$	1.29966	0.0855113
$232$	0	0
$233$	−28.3070	−1.85445	−0.927227	−	0.374501i	$-0.877814\pi$
$233$	−28.3070	−1.85445	−0.927227	+	0.374501i	$0.877814\pi$
$234$	0	0
$235$	−0.390238	−0.0254563
$236$	0	0
$237$	−8.63248	−0.560740
$238$	0	0
$239$	−4.79811	−0.310364	−0.155182	−	0.987886i	$-0.549596\pi$
$239$	−4.79811	−0.310364	−0.155182	+	0.987886i	$0.549596\pi$
$240$	0	0
$241$	3.05264	0.196638	0.0983190	−	0.995155i	$-0.468653\pi$
$241$	3.05264	0.196638	0.0983190	+	0.995155i	$0.468653\pi$
$242$	0	0
$243$	−12.4556	−0.799027
$244$	0	0
$245$	0.367835	0.0235001
$246$	0	0
$247$	−2.94919	−0.187653
$248$	0	0
$249$	17.3631	1.10034
$250$	0	0
$251$	10.0894	0.636840	0.318420	−	0.947950i	$-0.396848\pi$
$251$	10.0894	0.636840	0.318420	+	0.947950i	$0.396848\pi$
$252$	0	0
$253$	2.33452	0.146770
$254$	0	0
$255$	1.51358	0.0947842
$256$	0	0
$257$	18.5642	1.15800	0.579000	−	0.815327i	$-0.303443\pi$
$257$	18.5642	1.15800	0.579000	+	0.815327i	$0.303443\pi$
$258$	0	0
$259$	2.80577	0.174342
$260$	0	0
$261$	6.74175	0.417304
$262$	0	0
$263$	−12.4686	−0.768844	−0.384422	−	0.923157i	$-0.625599\pi$
$263$	−12.4686	−0.768844	−0.384422	+	0.923157i	$0.625599\pi$
$264$	0	0
$265$	0.683431	0.0419829
$266$	0	0
$267$	−12.4842	−0.764021
$268$	0	0
$269$	−0.562951	−0.0343237	−0.0171619	−	0.999853i	$-0.505463\pi$
$269$	−0.562951	−0.0343237	−0.0171619	+	0.999853i	$0.505463\pi$
$270$	0	0
$271$	−5.05485	−0.307061	−0.153530	−	0.988144i	$-0.549064\pi$
$271$	−5.05485	−0.307061	−0.153530	+	0.988144i	$0.549064\pi$
$272$	0	0
$273$	−1.29966	−0.0786590
$274$	0	0
$275$	4.86470	0.293352
$276$	0	0
$277$	11.7908	0.708439	0.354220	−	0.935162i	$-0.384747\pi$
$277$	11.7908	0.708439	0.354220	+	0.935162i	$0.384747\pi$
$278$	0	0
$279$	7.26730	0.435082
$280$	0	0
$281$	−7.40423	−0.441699	−0.220850	−	0.975308i	$-0.570883\pi$
$281$	−7.40423	−0.441699	−0.220850	+	0.975308i	$0.570883\pi$
$282$	0	0
$283$	12.6161	0.749952	0.374976	−	0.927035i	$-0.377651\pi$
$283$	12.6161	0.749952	0.374976	+	0.927035i	$0.377651\pi$
$284$	0	0
$285$	1.40989	0.0835147
$286$	0	0
$287$	4.11321	0.242795
$288$	0	0
$289$	−6.97585	−0.410344
$290$	0	0
$291$	21.6638	1.26996
$292$	0	0
$293$	22.1681	1.29507	0.647537	−	0.762034i	$-0.275799\pi$
$293$	22.1681	1.29507	0.647537	+	0.762034i	$0.275799\pi$
$294$	0	0
$295$	1.45394	0.0846516
$296$	0	0
$297$	−5.60268	−0.325101
$298$	0	0
$299$	−2.33452	−0.135009
$300$	0	0
$301$	3.28334	0.189248
$302$	0	0
$303$	20.2393	1.16272
$304$	0	0
$305$	−1.46693	−0.0839964
$306$	0	0
$307$	16.9922	0.969798	0.484899	−	0.874570i	$-0.338856\pi$
$307$	16.9922	0.969798	0.484899	+	0.874570i	$0.338856\pi$
$308$	0	0
$309$	−22.2360	−1.26496
$310$	0	0
$311$	10.4004	0.589751	0.294875	−	0.955536i	$-0.404722\pi$
$311$	10.4004	0.589751	0.294875	+	0.955536i	$0.404722\pi$
$312$	0	0
$313$	−14.4997	−0.819569	−0.409784	−	0.912182i	$-0.634396\pi$
$313$	−14.4997	−0.819569	−0.409784	+	0.912182i	$0.634396\pi$
$314$	0	0
$315$	−0.482189	−0.0271683
$316$	0	0
$317$	−0.629938	−0.0353809	−0.0176904	−	0.999844i	$-0.505631\pi$
$317$	−0.629938	−0.0353809	−0.0176904	+	0.999844i	$0.505631\pi$
$318$	0	0
$319$	5.14289	0.287947
$320$	0	0
$321$	−8.06754	−0.450286
$322$	0	0
$323$	9.33742	0.519548
$324$	0	0
$325$	−4.86470	−0.269845
$326$	0	0
$327$	−4.64083	−0.256639
$328$	0	0
$329$	−1.06091	−0.0584897
$330$	0	0
$331$	−10.0927	−0.554746	−0.277373	−	0.960762i	$-0.589464\pi$
$331$	−10.0927	−0.554746	−0.277373	+	0.960762i	$0.589464\pi$
$332$	0	0
$333$	−3.67804	−0.201556
$334$	0	0
$335$	−1.33973	−0.0731970
$336$	0	0
$337$	15.3535	0.836358	0.418179	−	0.908365i	$-0.362668\pi$
$337$	15.3535	0.836358	0.418179	+	0.908365i	$0.362668\pi$
$338$	0	0
$339$	−14.3159	−0.777531
$340$	0	0
$341$	5.54381	0.300214
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.11604	0.0600856
$346$	0	0
$347$	29.8634	1.60315	0.801575	−	0.597895i	$-0.203996\pi$
$347$	29.8634	1.60315	0.801575	+	0.597895i	$0.203996\pi$
$348$	0	0
$349$	33.2270	1.77860	0.889301	−	0.457322i	$-0.151191\pi$
$349$	33.2270	1.77860	0.889301	+	0.457322i	$0.151191\pi$
$350$	0	0
$351$	5.60268	0.299049
$352$	0	0
$353$	−16.4874	−0.877533	−0.438767	−	0.898601i	$-0.644585\pi$
$353$	−16.4874	−0.877533	−0.438767	+	0.898601i	$0.644585\pi$
$354$	0	0
$355$	3.88123	0.205994
$356$	0	0
$357$	4.11484	0.217781
$358$	0	0
$359$	26.7000	1.40917	0.704585	−	0.709619i	$-0.251133\pi$
$359$	26.7000	1.40917	0.704585	+	0.709619i	$0.251133\pi$
$360$	0	0
$361$	−10.3023	−0.542224
$362$	0	0
$363$	−1.29966	−0.0682144
$364$	0	0
$365$	4.56069	0.238718
$366$	0	0
$367$	−9.55771	−0.498909	−0.249454	−	0.968387i	$-0.580251\pi$
$367$	−9.55771	−0.498909	−0.249454	+	0.968387i	$0.580251\pi$
$368$	0	0
$369$	−5.39195	−0.280694
$370$	0	0
$371$	1.85799	0.0964618
$372$	0	0
$373$	−5.80818	−0.300736	−0.150368	−	0.988630i	$-0.548046\pi$
$373$	−5.80818	−0.300736	−0.150368	+	0.988630i	$0.548046\pi$
$374$	0	0
$375$	4.71591	0.243529
$376$	0	0
$377$	−5.14289	−0.264872
$378$	0	0
$379$	−17.0635	−0.876495	−0.438248	−	0.898854i	$-0.644401\pi$
$379$	−17.0635	−0.876495	−0.438248	+	0.898854i	$0.644401\pi$
$380$	0	0
$381$	14.0377	0.719171
$382$	0	0
$383$	2.04893	0.104695	0.0523477	−	0.998629i	$-0.483330\pi$
$383$	2.04893	0.104695	0.0523477	+	0.998629i	$0.483330\pi$
$384$	0	0
$385$	−0.367835	−0.0187466
$386$	0	0
$387$	−4.30408	−0.218789
$388$	0	0
$389$	33.8917	1.71838	0.859189	−	0.511658i	$-0.170968\pi$
$389$	33.8917	1.71838	0.859189	+	0.511658i	$0.170968\pi$
$390$	0	0
$391$	7.39132	0.373795
$392$	0	0
$393$	−28.7383	−1.44966
$394$	0	0
$395$	2.44320	0.122931
$396$	0	0
$397$	16.3217	0.819165	0.409583	−	0.912273i	$-0.365674\pi$
$397$	16.3217	0.819165	0.409583	+	0.912273i	$0.365674\pi$
$398$	0	0
$399$	3.83295	0.191887
$400$	0	0
$401$	12.1105	0.604771	0.302386	−	0.953186i	$-0.402217\pi$
$401$	12.1105	0.604771	0.302386	+	0.953186i	$0.402217\pi$
$402$	0	0
$403$	−5.54381	−0.276157
$404$	0	0
$405$	−1.23185	−0.0612111
$406$	0	0
$407$	−2.80577	−0.139077
$408$	0	0
$409$	−11.0318	−0.545490	−0.272745	−	0.962086i	$-0.587931\pi$
$409$	−11.0318	−0.545490	−0.272745	+	0.962086i	$0.587931\pi$
$410$	0	0
$411$	7.63070	0.376395
$412$	0	0
$413$	3.95270	0.194500
$414$	0	0
$415$	−4.91418	−0.241227
$416$	0	0
$417$	14.1909	0.694929
$418$	0	0
$419$	36.9120	1.80327	0.901633	−	0.432501i	$-0.142369\pi$
$419$	36.9120	1.80327	0.901633	+	0.432501i	$0.142369\pi$
$420$	0	0
$421$	−21.1033	−1.02851	−0.514257	−	0.857636i	$-0.671932\pi$
$421$	−21.1033	−1.02851	−0.514257	+	0.857636i	$0.671932\pi$
$422$	0	0
$423$	1.39073	0.0676195
$424$	0	0
$425$	15.4021	0.747111
$426$	0	0
$427$	−3.98803	−0.192994
$428$	0	0
$429$	1.29966	0.0627481
$430$	0	0
$431$	−13.3621	−0.643631	−0.321816	−	0.946802i	$-0.604293\pi$
$431$	−13.3621	−0.643631	−0.321816	+	0.946802i	$0.604293\pi$
$432$	0	0
$433$	−33.0154	−1.58662	−0.793310	−	0.608818i	$-0.791644\pi$
$433$	−33.0154	−1.58662	−0.793310	+	0.608818i	$0.791644\pi$
$434$	0	0
$435$	2.45861	0.117881
$436$	0	0
$437$	6.88496	0.329352
$438$	0	0
$439$	−13.4206	−0.640528	−0.320264	−	0.947328i	$-0.603772\pi$
$439$	−13.4206	−0.640528	−0.320264	+	0.947328i	$0.603772\pi$
$440$	0	0
$441$	−1.31089	−0.0624231
$442$	0	0
$443$	11.7878	0.560055	0.280028	−	0.959992i	$-0.409656\pi$
$443$	11.7878	0.560055	0.280028	+	0.959992i	$0.409656\pi$
$444$	0	0
$445$	3.53333	0.167496
$446$	0	0
$447$	−21.3633	−1.01045
$448$	0	0
$449$	−35.3033	−1.66607	−0.833033	−	0.553223i	$-0.813398\pi$
$449$	−35.3033	−1.66607	−0.833033	+	0.553223i	$0.813398\pi$
$450$	0	0
$451$	−4.11321	−0.193683
$452$	0	0
$453$	−16.7904	−0.788881
$454$	0	0
$455$	0.367835	0.0172443
$456$	0	0
$457$	1.25748	0.0588225	0.0294113	−	0.999567i	$-0.490637\pi$
$457$	1.25748	0.0588225	0.0294113	+	0.999567i	$0.490637\pi$
$458$	0	0
$459$	−17.7386	−0.827968
$460$	0	0
$461$	31.8863	1.48509	0.742547	−	0.669794i	$-0.233618\pi$
$461$	31.8863	1.48509	0.742547	+	0.669794i	$0.233618\pi$
$462$	0	0
$463$	8.47985	0.394092	0.197046	−	0.980394i	$-0.436865\pi$
$463$	8.47985	0.394092	0.197046	+	0.980394i	$0.436865\pi$
$464$	0	0
$465$	2.65027	0.122903
$466$	0	0
$467$	33.1522	1.53410	0.767050	−	0.641587i	$-0.221724\pi$
$467$	33.1522	1.53410	0.767050	+	0.641587i	$0.221724\pi$
$468$	0	0
$469$	−3.64220	−0.168181
$470$	0	0
$471$	20.1162	0.926904
$472$	0	0
$473$	−3.28334	−0.150968
$474$	0	0
$475$	14.3469	0.658283
$476$	0	0
$477$	−2.43561	−0.111519
$478$	0	0
$479$	4.06267	0.185628	0.0928140	−	0.995683i	$-0.470414\pi$
$479$	4.06267	0.185628	0.0928140	+	0.995683i	$0.470414\pi$
$480$	0	0
$481$	2.80577	0.127932
$482$	0	0
$483$	3.03408	0.138056
$484$	0	0
$485$	−6.13138	−0.278412
$486$	0	0
$487$	−0.952812	−0.0431760	−0.0215880	−	0.999767i	$-0.506872\pi$
$487$	−0.952812	−0.0431760	−0.0215880	+	0.999767i	$0.506872\pi$
$488$	0	0
$489$	−0.0469175	−0.00212168
$490$	0	0
$491$	−22.3621	−1.00919	−0.504594	−	0.863357i	$-0.668358\pi$
$491$	−22.3621	−1.00919	−0.504594	+	0.863357i	$0.668358\pi$
$492$	0	0
$493$	16.2829	0.733344
$494$	0	0
$495$	0.482189	0.0216728
$496$	0	0
$497$	10.5516	0.473302
$498$	0	0
$499$	34.0574	1.52462	0.762309	−	0.647213i	$-0.224066\pi$
$499$	34.0574	1.52462	0.762309	+	0.647213i	$0.224066\pi$
$500$	0	0
$501$	−25.2536	−1.12825
$502$	0	0
$503$	−4.96353	−0.221313	−0.110657	−	0.993859i	$-0.535295\pi$
$503$	−4.96353	−0.221313	−0.110657	+	0.993859i	$0.535295\pi$
$504$	0	0
$505$	−5.72822	−0.254902
$506$	0	0
$507$	−1.29966	−0.0577199
$508$	0	0
$509$	29.2355	1.29584	0.647921	−	0.761708i	$-0.275639\pi$
$509$	29.2355	1.29584	0.647921	+	0.761708i	$0.275639\pi$
$510$	0	0
$511$	12.3988	0.548489
$512$	0	0
$513$	−16.5234	−0.729526
$514$	0	0
$515$	6.29333	0.277317
$516$	0	0
$517$	1.06091	0.0466586
$518$	0	0
$519$	10.9145	0.479093
$520$	0	0
$521$	31.6037	1.38459	0.692293	−	0.721617i	$-0.256601\pi$
$521$	31.6037	1.38459	0.692293	+	0.721617i	$0.256601\pi$
$522$	0	0
$523$	−4.33942	−0.189750	−0.0948748	−	0.995489i	$-0.530245\pi$
$523$	−4.33942	−0.189750	−0.0948748	+	0.995489i	$0.530245\pi$
$524$	0	0
$525$	6.32245	0.275934
$526$	0	0
$527$	17.5522	0.764587
$528$	0	0
$529$	−17.5500	−0.763044
$530$	0	0
$531$	−5.18154	−0.224860
$532$	0	0
$533$	4.11321	0.178163
$534$	0	0
$535$	2.28331	0.0987159
$536$	0	0
$537$	−4.00365	−0.172770
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	15.3221	0.658749	0.329374	−	0.944199i	$-0.393162\pi$
$541$	15.3221	0.658749	0.329374	+	0.944199i	$0.393162\pi$
$542$	0	0
$543$	23.6230	1.01376
$544$	0	0
$545$	1.31347	0.0562627
$546$	0	0
$547$	2.47200	0.105695	0.0528476	−	0.998603i	$-0.483170\pi$
$547$	2.47200	0.105695	0.0528476	+	0.998603i	$0.483170\pi$
$548$	0	0
$549$	5.22785	0.223119
$550$	0	0
$551$	15.1674	0.646153
$552$	0	0
$553$	6.64211	0.282451
$554$	0	0
$555$	−1.34132	−0.0569361
$556$	0	0
$557$	11.5478	0.489296	0.244648	−	0.969612i	$-0.421328\pi$
$557$	11.5478	0.489296	0.244648	+	0.969612i	$0.421328\pi$
$558$	0	0
$559$	3.28334	0.138870
$560$	0	0
$561$	−4.11484	−0.173729
$562$	0	0
$563$	8.26277	0.348234	0.174117	−	0.984725i	$-0.444293\pi$
$563$	8.26277	0.348234	0.174117	+	0.984725i	$0.444293\pi$
$564$	0	0
$565$	4.05173	0.170458
$566$	0	0
$567$	−3.34892	−0.140641
$568$	0	0
$569$	−40.3764	−1.69267	−0.846334	−	0.532653i	$-0.821195\pi$
$569$	−40.3764	−1.69267	−0.846334	+	0.532653i	$0.821195\pi$
$570$	0	0
$571$	5.30577	0.222040	0.111020	−	0.993818i	$-0.464588\pi$
$571$	5.30577	0.222040	0.111020	+	0.993818i	$0.464588\pi$
$572$	0	0
$573$	11.1759	0.466880
$574$	0	0
$575$	11.3567	0.473609
$576$	0	0
$577$	17.7045	0.737048	0.368524	−	0.929618i	$-0.379863\pi$
$577$	17.7045	0.737048	0.368524	+	0.929618i	$0.379863\pi$
$578$	0	0
$579$	34.4301	1.43087
$580$	0	0
$581$	−13.3597	−0.554256
$582$	0	0
$583$	−1.85799	−0.0769499
$584$	0	0
$585$	−0.482189	−0.0199361
$586$	0	0
$587$	16.4500	0.678964	0.339482	−	0.940613i	$-0.389748\pi$
$587$	16.4500	0.678964	0.339482	+	0.940613i	$0.389748\pi$
$588$	0	0
$589$	16.3498	0.673681
$590$	0	0
$591$	−14.5339	−0.597845
$592$	0	0
$593$	−41.9935	−1.72447	−0.862234	−	0.506511i	$-0.830935\pi$
$593$	−41.9935	−1.72447	−0.862234	+	0.506511i	$0.830935\pi$
$594$	0	0
$595$	−1.16460	−0.0477439
$596$	0	0
$597$	−15.5980	−0.638384
$598$	0	0
$599$	−34.8809	−1.42519	−0.712597	−	0.701574i	$-0.752481\pi$
$599$	−34.8809	−1.42519	−0.712597	+	0.701574i	$0.752481\pi$
$600$	0	0
$601$	−33.1340	−1.35156	−0.675782	−	0.737101i	$-0.736194\pi$
$601$	−33.1340	−1.35156	−0.675782	+	0.737101i	$0.736194\pi$
$602$	0	0
$603$	4.77450	0.194433
$604$	0	0
$605$	0.367835	0.0149546
$606$	0	0
$607$	28.8484	1.17092	0.585460	−	0.810701i	$-0.300914\pi$
$607$	28.8484	1.17092	0.585460	+	0.810701i	$0.300914\pi$
$608$	0	0
$609$	6.68401	0.270850
$610$	0	0
$611$	−1.06091	−0.0429197
$612$	0	0
$613$	−22.1713	−0.895490	−0.447745	−	0.894161i	$-0.647773\pi$
$613$	−22.1713	−0.895490	−0.447745	+	0.894161i	$0.647773\pi$
$614$	0	0
$615$	−1.96636	−0.0792912
$616$	0	0
$617$	15.9337	0.641467	0.320734	−	0.947169i	$-0.396071\pi$
$617$	15.9337	0.641467	0.320734	+	0.947169i	$0.396071\pi$
$618$	0	0
$619$	25.4103	1.02133	0.510664	−	0.859781i	$-0.329400\pi$
$619$	25.4103	1.02133	0.510664	+	0.859781i	$0.329400\pi$
$620$	0	0
$621$	−13.0796	−0.524866
$622$	0	0
$623$	9.60576	0.384847
$624$	0	0
$625$	22.9888	0.919551
$626$	0	0
$627$	−3.83295	−0.153073
$628$	0	0
$629$	−8.88333	−0.354201
$630$	0	0
$631$	5.56181	0.221412	0.110706	−	0.993853i	$-0.464689\pi$
$631$	5.56181	0.221412	0.110706	+	0.993853i	$0.464689\pi$
$632$	0	0
$633$	−19.2021	−0.763215
$634$	0	0
$635$	−3.97299	−0.157663
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−13.8319	−0.547181
$640$	0	0
$641$	36.8545	1.45566	0.727832	−	0.685755i	$-0.240528\pi$
$641$	36.8545	1.45566	0.727832	+	0.685755i	$0.240528\pi$
$642$	0	0
$643$	20.8823	0.823518	0.411759	−	0.911293i	$-0.364915\pi$
$643$	20.8823	0.823518	0.411759	+	0.911293i	$0.364915\pi$
$644$	0	0
$645$	−1.56963	−0.0618042
$646$	0	0
$647$	3.55948	0.139938	0.0699688	−	0.997549i	$-0.477710\pi$
$647$	3.55948	0.139938	0.0699688	+	0.997549i	$0.477710\pi$
$648$	0	0
$649$	−3.95270	−0.155157
$650$	0	0
$651$	7.20507	0.282389
$652$	0	0
$653$	−13.3651	−0.523018	−0.261509	−	0.965201i	$-0.584220\pi$
$653$	−13.3651	−0.523018	−0.261509	+	0.965201i	$0.584220\pi$
$654$	0	0
$655$	8.13363	0.317807
$656$	0	0
$657$	−16.2534	−0.634104
$658$	0	0
$659$	24.6684	0.960945	0.480473	−	0.877010i	$-0.340465\pi$
$659$	24.6684	0.960945	0.480473	+	0.877010i	$0.340465\pi$
$660$	0	0
$661$	−29.3381	−1.14112	−0.570559	−	0.821256i	$-0.693274\pi$
$661$	−29.3381	−1.14112	−0.570559	+	0.821256i	$0.693274\pi$
$662$	0	0
$663$	4.11484	0.159807
$664$	0	0
$665$	−1.08482	−0.0420673
$666$	0	0
$667$	12.0062	0.464882
$668$	0	0
$669$	−33.9253	−1.31163
$670$	0	0
$671$	3.98803	0.153956
$672$	0	0
$673$	29.3732	1.13225	0.566126	−	0.824319i	$-0.308441\pi$
$673$	29.3732	1.13225	0.566126	+	0.824319i	$0.308441\pi$
$674$	0	0
$675$	−27.2554	−1.04906
$676$	0	0
$677$	25.6509	0.985844	0.492922	−	0.870074i	$-0.335929\pi$
$677$	25.6509	0.985844	0.492922	+	0.870074i	$0.335929\pi$
$678$	0	0
$679$	−16.6689	−0.639692
$680$	0	0
$681$	9.15011	0.350633
$682$	0	0
$683$	14.0604	0.538005	0.269003	−	0.963139i	$-0.413306\pi$
$683$	14.0604	0.538005	0.269003	+	0.963139i	$0.413306\pi$
$684$	0	0
$685$	−2.15967	−0.0825168
$686$	0	0
$687$	−34.5524	−1.31826
$688$	0	0
$689$	1.85799	0.0707836
$690$	0	0
$691$	2.69056	0.102354	0.0511769	−	0.998690i	$-0.483703\pi$
$691$	2.69056	0.102354	0.0511769	+	0.998690i	$0.483703\pi$
$692$	0	0
$693$	1.31089	0.0497964
$694$	0	0
$695$	−4.01635	−0.152349
$696$	0	0
$697$	−13.0228	−0.493274
$698$	0	0
$699$	36.7895	1.39150
$700$	0	0
$701$	−32.4478	−1.22554	−0.612769	−	0.790262i	$-0.709944\pi$
$701$	−32.4478	−1.22554	−0.612769	+	0.790262i	$0.709944\pi$
$702$	0	0
$703$	−8.27476	−0.312088
$704$	0	0
$705$	0.507177	0.0191014
$706$	0	0
$707$	−15.5728	−0.585676
$708$	0	0
$709$	−50.0649	−1.88023	−0.940113	−	0.340862i	$-0.889281\pi$
$709$	−50.0649	−1.88023	−0.940113	+	0.340862i	$0.889281\pi$
$710$	0	0
$711$	−8.70705	−0.326540
$712$	0	0
$713$	12.9422	0.484687
$714$	0	0
$715$	−0.367835	−0.0137562
$716$	0	0
$717$	6.23591	0.232884
$718$	0	0
$719$	24.3937	0.909731	0.454865	−	0.890560i	$-0.349687\pi$
$719$	24.3937	0.909731	0.454865	+	0.890560i	$0.349687\pi$
$720$	0	0
$721$	17.1091	0.637177
$722$	0	0
$723$	−3.96739	−0.147549
$724$	0	0
$725$	25.0186	0.929168
$726$	0	0
$727$	10.0503	0.372746	0.186373	−	0.982479i	$-0.440327\pi$
$727$	10.0503	0.372746	0.186373	+	0.982479i	$0.440327\pi$
$728$	0	0
$729$	26.2348	0.971659
$730$	0	0
$731$	−10.3954	−0.384486
$732$	0	0
$733$	29.3299	1.08332	0.541662	−	0.840596i	$-0.317795\pi$
$733$	29.3299	1.08332	0.541662	+	0.840596i	$0.317795\pi$
$734$	0	0
$735$	−0.478060	−0.0176335
$736$	0	0
$737$	3.64220	0.134162
$738$	0	0
$739$	31.4469	1.15679	0.578396	−	0.815756i	$-0.303679\pi$
$739$	31.4469	1.15679	0.578396	+	0.815756i	$0.303679\pi$
$740$	0	0
$741$	3.83295	0.140807
$742$	0	0
$743$	−28.1700	−1.03346	−0.516729	−	0.856149i	$-0.672851\pi$
$743$	−28.1700	−1.03346	−0.516729	+	0.856149i	$0.672851\pi$
$744$	0	0
$745$	6.04632	0.221520
$746$	0	0
$747$	17.5131	0.640771
$748$	0	0
$749$	6.20743	0.226814
$750$	0	0
$751$	−1.20795	−0.0440788	−0.0220394	−	0.999757i	$-0.507016\pi$
$751$	−1.20795	−0.0440788	−0.0220394	+	0.999757i	$0.507016\pi$
$752$	0	0
$753$	−13.1128	−0.477859
$754$	0	0
$755$	4.75208	0.172946
$756$	0	0
$757$	44.7402	1.62611	0.813055	−	0.582187i	$-0.197803\pi$
$757$	44.7402	1.62611	0.813055	+	0.582187i	$0.197803\pi$
$758$	0	0
$759$	−3.03408	−0.110130
$760$	0	0
$761$	15.0279	0.544763	0.272381	−	0.962189i	$-0.412189\pi$
$761$	15.0279	0.544763	0.272381	+	0.962189i	$0.412189\pi$
$762$	0	0
$763$	3.57081	0.129272
$764$	0	0
$765$	1.52666	0.0551964
$766$	0	0
$767$	3.95270	0.142724
$768$	0	0
$769$	49.8673	1.79826	0.899130	−	0.437681i	$-0.144200\pi$
$769$	49.8673	1.79826	0.899130	+	0.437681i	$0.144200\pi$
$770$	0	0
$771$	−24.1271	−0.868916
$772$	0	0
$773$	−26.1182	−0.939405	−0.469702	−	0.882825i	$-0.655639\pi$
$773$	−26.1182	−0.939405	−0.469702	+	0.882825i	$0.655639\pi$
$774$	0	0
$775$	26.9690	0.968754
$776$	0	0
$777$	−3.64654	−0.130819
$778$	0	0
$779$	−12.1306	−0.434626
$780$	0	0
$781$	−10.5516	−0.377564
$782$	0	0
$783$	−28.8140	−1.02973
$784$	0	0
$785$	−5.69335	−0.203205
$786$	0	0
$787$	−45.3795	−1.61760	−0.808802	−	0.588082i	$-0.799883\pi$
$787$	−45.3795	−1.61760	−0.808802	+	0.588082i	$0.799883\pi$
$788$	0	0
$789$	16.2049	0.576909
$790$	0	0
$791$	11.0151	0.391652
$792$	0	0
$793$	−3.98803	−0.141619
$794$	0	0
$795$	−0.888228	−0.0315022
$796$	0	0
$797$	−23.6795	−0.838770	−0.419385	−	0.907808i	$-0.637754\pi$
$797$	−23.6795	−0.838770	−0.419385	+	0.907808i	$0.637754\pi$
$798$	0	0
$799$	3.35893	0.118830
$800$	0	0
$801$	−12.5921	−0.444918
$802$	0	0
$803$	−12.3988	−0.437543
$804$	0	0
$805$	−0.858718	−0.0302658
$806$	0	0
$807$	0.731645	0.0257551
$808$	0	0
$809$	24.9237	0.876270	0.438135	−	0.898909i	$-0.355639\pi$
$809$	24.9237	0.876270	0.438135	+	0.898909i	$0.355639\pi$
$810$	0	0
$811$	40.5612	1.42430	0.712148	−	0.702030i	$-0.247723\pi$
$811$	40.5612	1.42430	0.712148	+	0.702030i	$0.247723\pi$
$812$	0	0
$813$	6.56959	0.230405
$814$	0	0
$815$	0.0132788	0.000465135 0
$816$	0	0
$817$	−9.68320	−0.338772
$818$	0	0
$819$	−1.31089	−0.0458061
$820$	0	0
$821$	4.51674	0.157635	0.0788176	−	0.996889i	$-0.474886\pi$
$821$	4.51674	0.157635	0.0788176	+	0.996889i	$0.474886\pi$
$822$	0	0
$823$	12.1067	0.422014	0.211007	−	0.977485i	$-0.432326\pi$
$823$	12.1067	0.422014	0.211007	+	0.977485i	$0.432326\pi$
$824$	0	0
$825$	−6.32245	−0.220119
$826$	0	0
$827$	29.3034	1.01898	0.509490	−	0.860477i	$-0.329834\pi$
$827$	29.3034	1.01898	0.509490	+	0.860477i	$0.329834\pi$
$828$	0	0
$829$	−32.0106	−1.11177	−0.555886	−	0.831258i	$-0.687621\pi$
$829$	−32.0106	−1.11177	−0.555886	+	0.831258i	$0.687621\pi$
$830$	0	0
$831$	−15.3240	−0.531583
$832$	0	0
$833$	−3.16609	−0.109699
$834$	0	0
$835$	7.14736	0.247345
$836$	0	0
$837$	−31.0602	−1.07360
$838$	0	0
$839$	−23.4231	−0.808655	−0.404328	−	0.914614i	$-0.632494\pi$
$839$	−23.4231	−0.808655	−0.404328	+	0.914614i	$0.632494\pi$
$840$	0	0
$841$	−2.55065	−0.0879533
$842$	0	0
$843$	9.62297	0.331433
$844$	0	0
$845$	0.367835	0.0126539
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−16.3967	−0.562733
$850$	0	0
$851$	−6.55013	−0.224536
$852$	0	0
$853$	49.5891	1.69790	0.848949	−	0.528475i	$-0.177236\pi$
$853$	49.5891	1.69790	0.848949	+	0.528475i	$0.177236\pi$
$854$	0	0
$855$	1.42207	0.0486337
$856$	0	0
$857$	7.13125	0.243599	0.121799	−	0.992555i	$-0.461134\pi$
$857$	7.13125	0.243599	0.121799	+	0.992555i	$0.461134\pi$
$858$	0	0
$859$	−8.45820	−0.288590	−0.144295	−	0.989535i	$-0.546091\pi$
$859$	−8.45820	−0.288590	−0.144295	+	0.989535i	$0.546091\pi$
$860$	0	0
$861$	−5.34577	−0.182183
$862$	0	0
$863$	34.4023	1.17107	0.585534	−	0.810648i	$-0.300885\pi$
$863$	34.4023	1.17107	0.585534	+	0.810648i	$0.300885\pi$
$864$	0	0
$865$	−3.08906	−0.105031
$866$	0	0
$867$	9.06623	0.307905
$868$	0	0
$869$	−6.64211	−0.225318
$870$	0	0
$871$	−3.64220	−0.123411
$872$	0	0
$873$	21.8510	0.739543
$874$	0	0
$875$	−3.62858	−0.122668
$876$	0	0
$877$	42.8823	1.44803	0.724017	−	0.689782i	$-0.242294\pi$
$877$	42.8823	1.44803	0.724017	+	0.689782i	$0.242294\pi$
$878$	0	0
$879$	−28.8110	−0.971771
$880$	0	0
$881$	30.9639	1.04320	0.521601	−	0.853190i	$-0.325335\pi$
$881$	30.9639	1.04320	0.521601	+	0.853190i	$0.325335\pi$
$882$	0	0
$883$	−19.7409	−0.664334	−0.332167	−	0.943221i	$-0.607780\pi$
$883$	−19.7409	−0.664334	−0.332167	+	0.943221i	$0.607780\pi$
$884$	0	0
$885$	−1.88963	−0.0635191
$886$	0	0
$887$	10.0173	0.336348	0.168174	−	0.985757i	$-0.446213\pi$
$887$	10.0173	0.336348	0.168174	+	0.985757i	$0.446213\pi$
$888$	0	0
$889$	−10.8010	−0.362255
$890$	0	0
$891$	3.34892	0.112193
$892$	0	0
$893$	3.12882	0.104702
$894$	0	0
$895$	1.13313	0.0378763
$896$	0	0
$897$	3.03408	0.101305
$898$	0	0
$899$	28.5112	0.950903
$900$	0	0
$901$	−5.88256	−0.195976
$902$	0	0
$903$	−4.26722	−0.142004
$904$	0	0
$905$	−6.68587	−0.222246
$906$	0	0
$907$	−47.5856	−1.58005	−0.790027	−	0.613073i	$-0.789933\pi$
$907$	−47.5856	−1.58005	−0.790027	+	0.613073i	$0.789933\pi$
$908$	0	0
$909$	20.4142	0.677095
$910$	0	0
$911$	14.0309	0.464865	0.232432	−	0.972613i	$-0.425332\pi$
$911$	14.0309	0.464865	0.232432	+	0.972613i	$0.425332\pi$
$912$	0	0
$913$	13.3597	0.442143
$914$	0	0
$915$	1.90651	0.0630274
$916$	0	0
$917$	22.1122	0.730209
$918$	0	0
$919$	−15.6168	−0.515149	−0.257575	−	0.966258i	$-0.582923\pi$
$919$	−15.6168	−0.515149	−0.257575	+	0.966258i	$0.582923\pi$
$920$	0	0
$921$	−22.0841	−0.727696
$922$	0	0
$923$	10.5516	0.347309
$924$	0	0
$925$	−13.6492	−0.448783
$926$	0	0
$927$	−22.4281	−0.736636
$928$	0	0
$929$	−22.3978	−0.734848	−0.367424	−	0.930054i	$-0.619760\pi$
$929$	−22.3978	−0.734848	−0.367424	+	0.930054i	$0.619760\pi$
$930$	0	0
$931$	−2.94919	−0.0966559
$932$	0	0
$933$	−13.5169	−0.442525
$934$	0	0
$935$	1.16460	0.0380865
$936$	0	0
$937$	−33.3952	−1.09097	−0.545487	−	0.838119i	$-0.683655\pi$
$937$	−33.3952	−1.09097	−0.545487	+	0.838119i	$0.683655\pi$
$938$	0	0
$939$	18.8446	0.614971
$940$	0	0
$941$	32.2319	1.05073	0.525365	−	0.850877i	$-0.323929\pi$
$941$	32.2319	1.05073	0.525365	+	0.850877i	$0.323929\pi$
$942$	0	0
$943$	−9.60238	−0.312696
$944$	0	0
$945$	2.06086	0.0670398
$946$	0	0
$947$	−15.0257	−0.488270	−0.244135	−	0.969741i	$-0.578504\pi$
$947$	−15.0257	−0.488270	−0.244135	+	0.969741i	$0.578504\pi$
$948$	0	0
$949$	12.3988	0.402481
$950$	0	0
$951$	0.818705	0.0265483
$952$	0	0
$953$	−39.8672	−1.29142	−0.645712	−	0.763581i	$-0.723439\pi$
$953$	−39.8672	−1.29142	−0.645712	+	0.763581i	$0.723439\pi$
$954$	0	0
$955$	−3.16305	−0.102354
$956$	0	0
$957$	−6.68401	−0.216063
$958$	0	0
$959$	−5.87131	−0.189595
$960$	0	0
$961$	−0.266152	−0.00858554
$962$	0	0
$963$	−8.13723	−0.262219
$964$	0	0
$965$	−9.74454	−0.313688
$966$	0	0
$967$	−34.4073	−1.10646	−0.553232	−	0.833027i	$-0.686606\pi$
$967$	−34.4073	−1.10646	−0.553232	+	0.833027i	$0.686606\pi$
$968$	0	0
$969$	−12.1355	−0.389847
$970$	0	0
$971$	3.85885	0.123836	0.0619182	−	0.998081i	$-0.480278\pi$
$971$	3.85885	0.123836	0.0619182	+	0.998081i	$0.480278\pi$
$972$	0	0
$973$	−10.9189	−0.350044
$974$	0	0
$975$	6.32245	0.202480
$976$	0	0
$977$	42.5821	1.36232	0.681162	−	0.732133i	$-0.261475\pi$
$977$	42.5821	1.36232	0.681162	+	0.732133i	$0.261475\pi$
$978$	0	0
$979$	−9.60576	−0.307001
$980$	0	0
$981$	−4.68092	−0.149450
$982$	0	0
$983$	15.3500	0.489588	0.244794	−	0.969575i	$-0.421280\pi$
$983$	15.3500	0.489588	0.244794	+	0.969575i	$0.421280\pi$
$984$	0	0
$985$	4.11344	0.131065
$986$	0	0
$987$	1.37882	0.0438882
$988$	0	0
$989$	−7.66503	−0.243734
$990$	0	0
$991$	−57.8331	−1.83713	−0.918564	−	0.395272i	$-0.870650\pi$
$991$	−57.8331	−1.83713	−0.918564	+	0.395272i	$0.870650\pi$
$992$	0	0
$993$	13.1171	0.416259
$994$	0	0
$995$	4.41461	0.139952
$996$	0	0
$997$	4.25300	0.134694	0.0673469	−	0.997730i	$-0.478547\pi$
$997$	4.25300	0.134694	0.0673469	+	0.997730i	$0.478547\pi$
$998$	0	0
$999$	15.7198	0.497353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.3	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.3	✓	10	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)

\(f(q)\)	\(=\)	\(q-1.29966 q^{3} +0.367835 q^{5} +1.00000 q^{7} -1.31089 q^{9} +O(q^{10})\)	\(q-1.29966 q^{3} +0.367835 q^{5} +1.00000 q^{7} -1.31089 q^{9} -1.00000 q^{11} +1.00000 q^{13} -0.478060 q^{15} -3.16609 q^{17} -2.94919 q^{19} -1.29966 q^{21} -2.33452 q^{23} -4.86470 q^{25} +5.60268 q^{27} -5.14289 q^{29} -5.54381 q^{31} +1.29966 q^{33} +0.367835 q^{35} +2.80577 q^{37} -1.29966 q^{39} +4.11321 q^{41} +3.28334 q^{43} -0.482189 q^{45} -1.06091 q^{47} +1.00000 q^{49} +4.11484 q^{51} +1.85799 q^{53} -0.367835 q^{55} +3.83295 q^{57} +3.95270 q^{59} -3.98803 q^{61} -1.31089 q^{63} +0.367835 q^{65} -3.64220 q^{67} +3.03408 q^{69} +10.5516 q^{71} +12.3988 q^{73} +6.32245 q^{75} -1.00000 q^{77} +6.64211 q^{79} -3.34892 q^{81} -13.3597 q^{83} -1.16460 q^{85} +6.68401 q^{87} +9.60576 q^{89} +1.00000 q^{91} +7.20507 q^{93} -1.08482 q^{95} -16.6689 q^{97} +1.31089 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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