LMFDB - Embedded newform 8008.2.a.q.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$3.21682$ 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)

$f(q)$	$=$	$q+3.21682 q^{3} +1.89991 q^{5} +1.00000 q^{7} +7.34791 q^{9} +O(q^{10})$	$q+3.21682 q^{3} +1.89991 q^{5} +1.00000 q^{7} +7.34791 q^{9} +1.00000 q^{11} -1.00000 q^{13} +6.11165 q^{15} +4.99575 q^{17} +5.59210 q^{19} +3.21682 q^{21} -0.631734 q^{23} -1.39035 q^{25} +13.9864 q^{27} -0.143567 q^{29} -4.47853 q^{31} +3.21682 q^{33} +1.89991 q^{35} -1.35440 q^{37} -3.21682 q^{39} +0.221065 q^{41} -7.36809 q^{43} +13.9603 q^{45} +0.631734 q^{47} +1.00000 q^{49} +16.0704 q^{51} -3.13932 q^{53} +1.89991 q^{55} +17.9888 q^{57} +0.632241 q^{59} +2.16739 q^{61} +7.34791 q^{63} -1.89991 q^{65} +3.15860 q^{67} -2.03217 q^{69} -8.21559 q^{71} -4.68200 q^{73} -4.47252 q^{75} +1.00000 q^{77} +8.15876 q^{79} +22.9481 q^{81} -0.585434 q^{83} +9.49146 q^{85} -0.461830 q^{87} -3.86148 q^{89} -1.00000 q^{91} -14.4066 q^{93} +10.6245 q^{95} +13.3061 q^{97} +7.34791 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.21682	1.85723	0.928615	−	0.371045i	$-0.121000\pi$
$3$	3.21682	1.85723	0.928615	+	0.371045i	$0.121000\pi$
$4$	0	0
$5$	1.89991	0.849664	0.424832	−	0.905272i	$-0.360333\pi$
$5$	1.89991	0.849664	0.424832	+	0.905272i	$0.360333\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	7.34791	2.44930
$10$	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
$15$	6.11165	1.57802
$16$	0	0
$17$	4.99575	1.21165	0.605824	−	0.795599i	$-0.292844\pi$
$17$	4.99575	1.21165	0.605824	+	0.795599i	$0.292844\pi$
$18$	0	0
$19$	5.59210	1.28292	0.641458	−	0.767158i	$-0.278330\pi$
$19$	5.59210	1.28292	0.641458	+	0.767158i	$0.278330\pi$
$20$	0	0
$21$	3.21682	0.701967
$22$	0	0
$23$	−0.631734	−0.131726	−0.0658628	−	0.997829i	$-0.520980\pi$
$23$	−0.631734	−0.131726	−0.0658628	+	0.997829i	$0.520980\pi$
$24$	0	0
$25$	−1.39035	−0.278071
$26$	0	0
$27$	13.9864	2.69169
$28$	0	0
$29$	−0.143567	−0.0266598	−0.0133299	−	0.999911i	$-0.504243\pi$
$29$	−0.143567	−0.0266598	−0.0133299	+	0.999911i	$0.504243\pi$
$30$	0	0
$31$	−4.47853	−0.804368	−0.402184	−	0.915559i	$-0.631749\pi$
$31$	−4.47853	−0.804368	−0.402184	+	0.915559i	$0.631749\pi$
$32$	0	0
$33$	3.21682	0.559976
$34$	0	0
$35$	1.89991	0.321143
$36$	0	0
$37$	−1.35440	−0.222663	−0.111331	−	0.993783i	$-0.535511\pi$
$37$	−1.35440	−0.222663	−0.111331	+	0.993783i	$0.535511\pi$
$38$	0	0
$39$	−3.21682	−0.515103
$40$	0	0
$41$	0.221065	0.0345246	0.0172623	−	0.999851i	$-0.494505\pi$
$41$	0.221065	0.0345246	0.0172623	+	0.999851i	$0.494505\pi$
$42$	0	0
$43$	−7.36809	−1.12362	−0.561811	−	0.827265i	$-0.689895\pi$
$43$	−7.36809	−1.12362	−0.561811	+	0.827265i	$0.689895\pi$
$44$	0	0
$45$	13.9603	2.08109
$46$	0	0
$47$	0.631734	0.0921479	0.0460740	−	0.998938i	$-0.485329\pi$
$47$	0.631734	0.0921479	0.0460740	+	0.998938i	$0.485329\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	16.0704	2.25031
$52$	0	0
$53$	−3.13932	−0.431219	−0.215609	−	0.976480i	$-0.569174\pi$
$53$	−3.13932	−0.431219	−0.215609	+	0.976480i	$0.569174\pi$
$54$	0	0
$55$	1.89991	0.256183
$56$	0	0
$57$	17.9888	2.38267
$58$	0	0
$59$	0.632241	0.0823108	0.0411554	−	0.999153i	$-0.486896\pi$
$59$	0.632241	0.0823108	0.0411554	+	0.999153i	$0.486896\pi$
$60$	0	0
$61$	2.16739	0.277506	0.138753	−	0.990327i	$-0.455691\pi$
$61$	2.16739	0.277506	0.138753	+	0.990327i	$0.455691\pi$
$62$	0	0
$63$	7.34791	0.925750
$64$	0	0
$65$	−1.89991	−0.235654
$66$	0	0
$67$	3.15860	0.385885	0.192942	−	0.981210i	$-0.438197\pi$
$67$	3.15860	0.385885	0.192942	+	0.981210i	$0.438197\pi$
$68$	0	0
$69$	−2.03217	−0.244645
$70$	0	0
$71$	−8.21559	−0.975011	−0.487505	−	0.873120i	$-0.662093\pi$
$71$	−8.21559	−0.975011	−0.487505	+	0.873120i	$0.662093\pi$
$72$	0	0
$73$	−4.68200	−0.547986	−0.273993	−	0.961732i	$-0.588345\pi$
$73$	−4.68200	−0.547986	−0.273993	+	0.961732i	$0.588345\pi$
$74$	0	0
$75$	−4.47252	−0.516442
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	8.15876	0.917932	0.458966	−	0.888454i	$-0.348220\pi$
$79$	8.15876	0.917932	0.458966	+	0.888454i	$0.348220\pi$
$80$	0	0
$81$	22.9481	2.54978
$82$	0	0
$83$	−0.585434	−0.0642598	−0.0321299	−	0.999484i	$-0.510229\pi$
$83$	−0.585434	−0.0642598	−0.0321299	+	0.999484i	$0.510229\pi$
$84$	0	0
$85$	9.49146	1.02949
$86$	0	0
$87$	−0.461830	−0.0495134
$88$	0	0
$89$	−3.86148	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$89$	−3.86148	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−14.4066	−1.49390
$94$	0	0
$95$	10.6245	1.09005
$96$	0	0
$97$	13.3061	1.35103	0.675517	−	0.737345i	$-0.263921\pi$
$97$	13.3061	1.35103	0.675517	+	0.737345i	$0.263921\pi$
$98$	0	0
$99$	7.34791	0.738493
$100$	0	0
$101$	−10.8699	−1.08159	−0.540797	−	0.841153i	$-0.681877\pi$
$101$	−10.8699	−1.08159	−0.540797	+	0.841153i	$0.681877\pi$
$102$	0	0
$103$	−18.3575	−1.80882	−0.904408	−	0.426668i	$-0.859687\pi$
$103$	−18.3575	−1.80882	−0.904408	+	0.426668i	$0.859687\pi$
$104$	0	0
$105$	6.11165	0.596436
$106$	0	0
$107$	5.62207	0.543506	0.271753	−	0.962367i	$-0.412397\pi$
$107$	5.62207	0.543506	0.271753	+	0.962367i	$0.412397\pi$
$108$	0	0
$109$	−13.5045	−1.29349	−0.646746	−	0.762705i	$-0.723871\pi$
$109$	−13.5045	−1.29349	−0.646746	+	0.762705i	$0.723871\pi$
$110$	0	0
$111$	−4.35687	−0.413536
$112$	0	0
$113$	−9.50499	−0.894154	−0.447077	−	0.894495i	$-0.647535\pi$
$113$	−9.50499	−0.894154	−0.447077	+	0.894495i	$0.647535\pi$
$114$	0	0
$115$	−1.20024	−0.111923
$116$	0	0
$117$	−7.34791	−0.679315
$118$	0	0
$119$	4.99575	0.457960
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.711126	0.0641201
$124$	0	0
$125$	−12.1411	−1.08593
$126$	0	0
$127$	9.10888	0.808282	0.404141	−	0.914697i	$-0.367570\pi$
$127$	9.10888	0.808282	0.404141	+	0.914697i	$0.367570\pi$
$128$	0	0
$129$	−23.7018	−2.08683
$130$	0	0
$131$	−13.9605	−1.21973	−0.609866	−	0.792504i	$-0.708777\pi$
$131$	−13.9605	−1.21973	−0.609866	+	0.792504i	$0.708777\pi$
$132$	0	0
$133$	5.59210	0.484897
$134$	0	0
$135$	26.5729	2.28703
$136$	0	0
$137$	2.24698	0.191972	0.0959862	−	0.995383i	$-0.469400\pi$
$137$	2.24698	0.191972	0.0959862	+	0.995383i	$0.469400\pi$
$138$	0	0
$139$	15.1281	1.28315	0.641576	−	0.767059i	$-0.278281\pi$
$139$	15.1281	1.28315	0.641576	+	0.767059i	$0.278281\pi$
$140$	0	0
$141$	2.03217	0.171140
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−0.272765	−0.0226519
$146$	0	0
$147$	3.21682	0.265319
$148$	0	0
$149$	5.81483	0.476369	0.238185	−	0.971220i	$-0.423448\pi$
$149$	5.81483	0.476369	0.238185	+	0.971220i	$0.423448\pi$
$150$	0	0
$151$	7.62755	0.620721	0.310361	−	0.950619i	$-0.399550\pi$
$151$	7.62755	0.620721	0.310361	+	0.950619i	$0.399550\pi$
$152$	0	0
$153$	36.7083	2.96769
$154$	0	0
$155$	−8.50879	−0.683442
$156$	0	0
$157$	−14.2821	−1.13984	−0.569918	−	0.821702i	$-0.693025\pi$
$157$	−14.2821	−1.13984	−0.569918	+	0.821702i	$0.693025\pi$
$158$	0	0
$159$	−10.0986	−0.800872
$160$	0	0
$161$	−0.631734	−0.0497876
$162$	0	0
$163$	4.02468	0.315238	0.157619	−	0.987500i	$-0.449618\pi$
$163$	4.02468	0.315238	0.157619	+	0.987500i	$0.449618\pi$
$164$	0	0
$165$	6.11165	0.475791
$166$	0	0
$167$	−11.7863	−0.912050	−0.456025	−	0.889967i	$-0.650727\pi$
$167$	−11.7863	−0.912050	−0.456025	+	0.889967i	$0.650727\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	41.0903	3.14225
$172$	0	0
$173$	−1.99993	−0.152052	−0.0760259	−	0.997106i	$-0.524223\pi$
$173$	−1.99993	−0.152052	−0.0760259	+	0.997106i	$0.524223\pi$
$174$	0	0
$175$	−1.39035	−0.105101
$176$	0	0
$177$	2.03380	0.152870
$178$	0	0
$179$	−21.9901	−1.64362	−0.821808	−	0.569765i	$-0.807034\pi$
$179$	−21.9901	−1.64362	−0.821808	+	0.569765i	$0.807034\pi$
$180$	0	0
$181$	3.85084	0.286230	0.143115	−	0.989706i	$-0.454288\pi$
$181$	3.85084	0.286230	0.143115	+	0.989706i	$0.454288\pi$
$182$	0	0
$183$	6.97211	0.515393
$184$	0	0
$185$	−2.57324	−0.189189
$186$	0	0
$187$	4.99575	0.365326
$188$	0	0
$189$	13.9864	1.01736
$190$	0	0
$191$	4.85623	0.351385	0.175692	−	0.984445i	$-0.443784\pi$
$191$	4.85623	0.351385	0.175692	+	0.984445i	$0.443784\pi$
$192$	0	0
$193$	−11.3337	−0.815821	−0.407910	−	0.913022i	$-0.633742\pi$
$193$	−11.3337	−0.815821	−0.407910	+	0.913022i	$0.633742\pi$
$194$	0	0
$195$	−6.11165	−0.437664
$196$	0	0
$197$	11.8958	0.847540	0.423770	−	0.905770i	$-0.360706\pi$
$197$	11.8958	0.847540	0.423770	+	0.905770i	$0.360706\pi$
$198$	0	0
$199$	−0.573404	−0.0406475	−0.0203238	−	0.999793i	$-0.506470\pi$
$199$	−0.573404	−0.0406475	−0.0203238	+	0.999793i	$0.506470\pi$
$200$	0	0
$201$	10.1606	0.716677
$202$	0	0
$203$	−0.143567	−0.0100765
$204$	0	0
$205$	0.420003	0.0293343
$206$	0	0
$207$	−4.64193	−0.322636
$208$	0	0
$209$	5.59210	0.386814
$210$	0	0
$211$	−0.0874973	−0.00602356	−0.00301178	−	0.999995i	$-0.500959\pi$
$211$	−0.0874973	−0.00602356	−0.00301178	+	0.999995i	$0.500959\pi$
$212$	0	0
$213$	−26.4280	−1.81082
$214$	0	0
$215$	−13.9987	−0.954702
$216$	0	0
$217$	−4.47853	−0.304022
$218$	0	0
$219$	−15.0611	−1.01774
$220$	0	0
$221$	−4.99575	−0.336051
$222$	0	0
$223$	−0.727143	−0.0486931	−0.0243465	−	0.999704i	$-0.507751\pi$
$223$	−0.727143	−0.0486931	−0.0243465	+	0.999704i	$0.507751\pi$
$224$	0	0
$225$	−10.2162	−0.681080
$226$	0	0
$227$	25.5356	1.69486	0.847430	−	0.530908i	$-0.178149\pi$
$227$	25.5356	1.69486	0.847430	+	0.530908i	$0.178149\pi$
$228$	0	0
$229$	14.3791	0.950198	0.475099	−	0.879932i	$-0.342412\pi$
$229$	14.3791	0.950198	0.475099	+	0.879932i	$0.342412\pi$
$230$	0	0
$231$	3.21682	0.211651
$232$	0	0
$233$	9.40619	0.616220	0.308110	−	0.951351i	$-0.400303\pi$
$233$	9.40619	0.616220	0.308110	+	0.951351i	$0.400303\pi$
$234$	0	0
$235$	1.20024	0.0782948
$236$	0	0
$237$	26.2452	1.70481
$238$	0	0
$239$	13.5369	0.875631	0.437815	−	0.899065i	$-0.355752\pi$
$239$	13.5369	0.875631	0.437815	+	0.899065i	$0.355752\pi$
$240$	0	0
$241$	21.0340	1.35492	0.677460	−	0.735559i	$-0.263080\pi$
$241$	21.0340	1.35492	0.677460	+	0.735559i	$0.263080\pi$
$242$	0	0
$243$	31.8604	2.04385
$244$	0	0
$245$	1.89991	0.121381
$246$	0	0
$247$	−5.59210	−0.355817
$248$	0	0
$249$	−1.88324	−0.119345
$250$	0	0
$251$	8.18638	0.516720	0.258360	−	0.966049i	$-0.416818\pi$
$251$	8.18638	0.516720	0.258360	+	0.966049i	$0.416818\pi$
$252$	0	0
$253$	−0.631734	−0.0397168
$254$	0	0
$255$	30.5323	1.91201
$256$	0	0
$257$	3.91638	0.244297	0.122149	−	0.992512i	$-0.461022\pi$
$257$	3.91638	0.244297	0.122149	+	0.992512i	$0.461022\pi$
$258$	0	0
$259$	−1.35440	−0.0841586
$260$	0	0
$261$	−1.05492	−0.0652979
$262$	0	0
$263$	0.791286	0.0487928	0.0243964	−	0.999702i	$-0.492234\pi$
$263$	0.791286	0.0487928	0.0243964	+	0.999702i	$0.492234\pi$
$264$	0	0
$265$	−5.96441	−0.366391
$266$	0	0
$267$	−12.4217	−0.760195
$268$	0	0
$269$	15.2719	0.931146	0.465573	−	0.885009i	$-0.345848\pi$
$269$	15.2719	0.931146	0.465573	+	0.885009i	$0.345848\pi$
$270$	0	0
$271$	−13.7120	−0.832942	−0.416471	−	0.909149i	$-0.636733\pi$
$271$	−13.7120	−0.832942	−0.416471	+	0.909149i	$0.636733\pi$
$272$	0	0
$273$	−3.21682	−0.194691
$274$	0	0
$275$	−1.39035	−0.0838415
$276$	0	0
$277$	−10.4040	−0.625117	−0.312559	−	0.949898i	$-0.601186\pi$
$277$	−10.4040	−0.625117	−0.312559	+	0.949898i	$0.601186\pi$
$278$	0	0
$279$	−32.9078	−1.97014
$280$	0	0
$281$	−33.0656	−1.97253	−0.986264	−	0.165175i	$-0.947181\pi$
$281$	−33.0656	−1.97253	−0.986264	+	0.165175i	$0.947181\pi$
$282$	0	0
$283$	−24.2358	−1.44067	−0.720334	−	0.693627i	$-0.756011\pi$
$283$	−24.2358	−1.44067	−0.720334	+	0.693627i	$0.756011\pi$
$284$	0	0
$285$	34.1770	2.02447
$286$	0	0
$287$	0.221065	0.0130491
$288$	0	0
$289$	7.95753	0.468090
$290$	0	0
$291$	42.8034	2.50918
$292$	0	0
$293$	1.78278	0.104151	0.0520755	−	0.998643i	$-0.483416\pi$
$293$	1.78278	0.104151	0.0520755	+	0.998643i	$0.483416\pi$
$294$	0	0
$295$	1.20120	0.0699365
$296$	0	0
$297$	13.9864	0.811575
$298$	0	0
$299$	0.631734	0.0365341
$300$	0	0
$301$	−7.36809	−0.424689
$302$	0	0
$303$	−34.9664	−2.00877
$304$	0	0
$305$	4.11784	0.235787
$306$	0	0
$307$	19.7129	1.12508	0.562538	−	0.826772i	$-0.309825\pi$
$307$	19.7129	1.12508	0.562538	+	0.826772i	$0.309825\pi$
$308$	0	0
$309$	−59.0527	−3.35939
$310$	0	0
$311$	−9.14341	−0.518475	−0.259238	−	0.965814i	$-0.583471\pi$
$311$	−9.14341	−0.518475	−0.259238	+	0.965814i	$0.583471\pi$
$312$	0	0
$313$	−20.6257	−1.16584	−0.582918	−	0.812531i	$-0.698089\pi$
$313$	−20.6257	−1.16584	−0.582918	+	0.812531i	$0.698089\pi$
$314$	0	0
$315$	13.9603	0.786576
$316$	0	0
$317$	1.54663	0.0868674	0.0434337	−	0.999056i	$-0.486170\pi$
$317$	1.54663	0.0868674	0.0434337	+	0.999056i	$0.486170\pi$
$318$	0	0
$319$	−0.143567	−0.00803823
$320$	0	0
$321$	18.0852	1.00942
$322$	0	0
$323$	27.9367	1.55444
$324$	0	0
$325$	1.39035	0.0771230
$326$	0	0
$327$	−43.4414	−2.40231
$328$	0	0
$329$	0.631734	0.0348286
$330$	0	0
$331$	−21.5863	−1.18649	−0.593247	−	0.805021i	$-0.702154\pi$
$331$	−21.5863	−1.18649	−0.593247	+	0.805021i	$0.702154\pi$
$332$	0	0
$333$	−9.95204	−0.545369
$334$	0	0
$335$	6.00105	0.327872
$336$	0	0
$337$	30.2320	1.64684	0.823421	−	0.567431i	$-0.192062\pi$
$337$	30.2320	1.64684	0.823421	+	0.567431i	$0.192062\pi$
$338$	0	0
$339$	−30.5758	−1.66065
$340$	0	0
$341$	−4.47853	−0.242526
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.86094	−0.207866
$346$	0	0
$347$	14.2375	0.764307	0.382153	−	0.924099i	$-0.375183\pi$
$347$	14.2375	0.764307	0.382153	+	0.924099i	$0.375183\pi$
$348$	0	0
$349$	11.5217	0.616742	0.308371	−	0.951266i	$-0.400216\pi$
$349$	11.5217	0.616742	0.308371	+	0.951266i	$0.400216\pi$
$350$	0	0
$351$	−13.9864	−0.746540
$352$	0	0
$353$	−15.2627	−0.812353	−0.406176	−	0.913795i	$-0.633138\pi$
$353$	−15.2627	−0.812353	−0.406176	+	0.913795i	$0.633138\pi$
$354$	0	0
$355$	−15.6088	−0.828432
$356$	0	0
$357$	16.0704	0.850537
$358$	0	0
$359$	4.17458	0.220326	0.110163	−	0.993914i	$-0.464863\pi$
$359$	4.17458	0.220326	0.110163	+	0.993914i	$0.464863\pi$
$360$	0	0
$361$	12.2716	0.645873
$362$	0	0
$363$	3.21682	0.168839
$364$	0	0
$365$	−8.89535	−0.465604
$366$	0	0
$367$	6.14618	0.320828	0.160414	−	0.987050i	$-0.448717\pi$
$367$	6.14618	0.320828	0.160414	+	0.987050i	$0.448717\pi$
$368$	0	0
$369$	1.62437	0.0845612
$370$	0	0
$371$	−3.13932	−0.162985
$372$	0	0
$373$	26.4364	1.36883	0.684413	−	0.729095i	$-0.260059\pi$
$373$	26.4364	1.36883	0.684413	+	0.729095i	$0.260059\pi$
$374$	0	0
$375$	−39.0556	−2.01682
$376$	0	0
$377$	0.143567	0.00739409
$378$	0	0
$379$	−16.9117	−0.868695	−0.434348	−	0.900745i	$-0.643021\pi$
$379$	−16.9117	−0.868695	−0.434348	+	0.900745i	$0.643021\pi$
$380$	0	0
$381$	29.3016	1.50117
$382$	0	0
$383$	−18.4963	−0.945118	−0.472559	−	0.881299i	$-0.656670\pi$
$383$	−18.4963	−0.945118	−0.472559	+	0.881299i	$0.656670\pi$
$384$	0	0
$385$	1.89991	0.0968282
$386$	0	0
$387$	−54.1400	−2.75209
$388$	0	0
$389$	−16.6645	−0.844925	−0.422463	−	0.906380i	$-0.638834\pi$
$389$	−16.6645	−0.844925	−0.422463	+	0.906380i	$0.638834\pi$
$390$	0	0
$391$	−3.15599	−0.159605
$392$	0	0
$393$	−44.9083	−2.26532
$394$	0	0
$395$	15.5009	0.779934
$396$	0	0
$397$	−16.4918	−0.827702	−0.413851	−	0.910345i	$-0.635816\pi$
$397$	−16.4918	−0.827702	−0.413851	+	0.910345i	$0.635816\pi$
$398$	0	0
$399$	17.9888	0.900565
$400$	0	0
$401$	5.58597	0.278950	0.139475	−	0.990226i	$-0.455459\pi$
$401$	5.58597	0.278950	0.139475	+	0.990226i	$0.455459\pi$
$402$	0	0
$403$	4.47853	0.223091
$404$	0	0
$405$	43.5992	2.16646
$406$	0	0
$407$	−1.35440	−0.0671354
$408$	0	0
$409$	18.8964	0.934366	0.467183	−	0.884161i	$-0.345269\pi$
$409$	18.8964	0.934366	0.467183	+	0.884161i	$0.345269\pi$
$410$	0	0
$411$	7.22812	0.356537
$412$	0	0
$413$	0.632241	0.0311105
$414$	0	0
$415$	−1.11227	−0.0545992
$416$	0	0
$417$	48.6645	2.38311
$418$	0	0
$419$	−2.38323	−0.116429	−0.0582143	−	0.998304i	$-0.518541\pi$
$419$	−2.38323	−0.116429	−0.0582143	+	0.998304i	$0.518541\pi$
$420$	0	0
$421$	24.5964	1.19875	0.599377	−	0.800467i	$-0.295415\pi$
$421$	24.5964	1.19875	0.599377	+	0.800467i	$0.295415\pi$
$422$	0	0
$423$	4.64193	0.225698
$424$	0	0
$425$	−6.94587	−0.336924
$426$	0	0
$427$	2.16739	0.104887
$428$	0	0
$429$	−3.21682	−0.155309
$430$	0	0
$431$	21.9705	1.05828	0.529142	−	0.848534i	$-0.322514\pi$
$431$	21.9705	1.05828	0.529142	+	0.848534i	$0.322514\pi$
$432$	0	0
$433$	−9.00252	−0.432634	−0.216317	−	0.976323i	$-0.569404\pi$
$433$	−9.00252	−0.432634	−0.216317	+	0.976323i	$0.569404\pi$
$434$	0	0
$435$	−0.877434	−0.0420697
$436$	0	0
$437$	−3.53272	−0.168993
$438$	0	0
$439$	1.69665	0.0809767	0.0404883	−	0.999180i	$-0.487109\pi$
$439$	1.69665	0.0809767	0.0404883	+	0.999180i	$0.487109\pi$
$440$	0	0
$441$	7.34791	0.349900
$442$	0	0
$443$	38.8388	1.84529	0.922643	−	0.385656i	$-0.126025\pi$
$443$	38.8388	1.84529	0.922643	+	0.385656i	$0.126025\pi$
$444$	0	0
$445$	−7.33646	−0.347781
$446$	0	0
$447$	18.7052	0.884727
$448$	0	0
$449$	−17.8939	−0.844464	−0.422232	−	0.906488i	$-0.638753\pi$
$449$	−17.8939	−0.844464	−0.422232	+	0.906488i	$0.638753\pi$
$450$	0	0
$451$	0.221065	0.0104096
$452$	0	0
$453$	24.5364	1.15282
$454$	0	0
$455$	−1.89991	−0.0890690
$456$	0	0
$457$	27.2464	1.27453	0.637266	−	0.770644i	$-0.280065\pi$
$457$	27.2464	1.27453	0.637266	+	0.770644i	$0.280065\pi$
$458$	0	0
$459$	69.8727	3.26138
$460$	0	0
$461$	10.0838	0.469649	0.234825	−	0.972038i	$-0.424548\pi$
$461$	10.0838	0.469649	0.234825	+	0.972038i	$0.424548\pi$
$462$	0	0
$463$	−1.39089	−0.0646403	−0.0323202	−	0.999478i	$-0.510290\pi$
$463$	−1.39089	−0.0646403	−0.0323202	+	0.999478i	$0.510290\pi$
$464$	0	0
$465$	−27.3712	−1.26931
$466$	0	0
$467$	−16.6725	−0.771511	−0.385755	−	0.922601i	$-0.626059\pi$
$467$	−16.6725	−0.771511	−0.385755	+	0.922601i	$0.626059\pi$
$468$	0	0
$469$	3.15860	0.145851
$470$	0	0
$471$	−45.9429	−2.11694
$472$	0	0
$473$	−7.36809	−0.338785
$474$	0	0
$475$	−7.77500	−0.356742
$476$	0	0
$477$	−23.0674	−1.05619
$478$	0	0
$479$	18.1617	0.829831	0.414915	−	0.909860i	$-0.363811\pi$
$479$	18.1617	0.829831	0.414915	+	0.909860i	$0.363811\pi$
$480$	0	0
$481$	1.35440	0.0617555
$482$	0	0
$483$	−2.03217	−0.0924671
$484$	0	0
$485$	25.2804	1.14792
$486$	0	0
$487$	−26.7875	−1.21386	−0.606930	−	0.794756i	$-0.707599\pi$
$487$	−26.7875	−1.21386	−0.606930	+	0.794756i	$0.707599\pi$
$488$	0	0
$489$	12.9467	0.585469
$490$	0	0
$491$	8.94524	0.403693	0.201847	−	0.979417i	$-0.435306\pi$
$491$	8.94524	0.403693	0.201847	+	0.979417i	$0.435306\pi$
$492$	0	0
$493$	−0.717227	−0.0323023
$494$	0	0
$495$	13.9603	0.627471
$496$	0	0
$497$	−8.21559	−0.368519
$498$	0	0
$499$	20.7292	0.927965	0.463982	−	0.885844i	$-0.346420\pi$
$499$	20.7292	0.927965	0.463982	+	0.885844i	$0.346420\pi$
$500$	0	0
$501$	−37.9143	−1.69389
$502$	0	0
$503$	11.7933	0.525836	0.262918	−	0.964818i	$-0.415315\pi$
$503$	11.7933	0.525836	0.262918	+	0.964818i	$0.415315\pi$
$504$	0	0
$505$	−20.6518	−0.918992
$506$	0	0
$507$	3.21682	0.142864
$508$	0	0
$509$	−7.15469	−0.317126	−0.158563	−	0.987349i	$-0.550686\pi$
$509$	−7.15469	−0.317126	−0.158563	+	0.987349i	$0.550686\pi$
$510$	0	0
$511$	−4.68200	−0.207119
$512$	0	0
$513$	78.2135	3.45321
$514$	0	0
$515$	−34.8775	−1.53689
$516$	0	0
$517$	0.631734	0.0277836
$518$	0	0
$519$	−6.43341	−0.282395
$520$	0	0
$521$	13.7964	0.604432	0.302216	−	0.953239i	$-0.402274\pi$
$521$	13.7964	0.604432	0.302216	+	0.953239i	$0.402274\pi$
$522$	0	0
$523$	−37.3832	−1.63465	−0.817327	−	0.576174i	$-0.804545\pi$
$523$	−37.3832	−1.63465	−0.817327	+	0.576174i	$0.804545\pi$
$524$	0	0
$525$	−4.47252	−0.195197
$526$	0	0
$527$	−22.3736	−0.974610
$528$	0	0
$529$	−22.6009	−0.982648
$530$	0	0
$531$	4.64565	0.201604
$532$	0	0
$533$	−0.221065	−0.00957540
$534$	0	0
$535$	10.6814	0.461798
$536$	0	0
$537$	−70.7381	−3.05257
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−28.1042	−1.20829	−0.604147	−	0.796873i	$-0.706486\pi$
$541$	−28.1042	−1.20829	−0.604147	+	0.796873i	$0.706486\pi$
$542$	0	0
$543$	12.3874	0.531596
$544$	0	0
$545$	−25.6572	−1.09903
$546$	0	0
$547$	16.5213	0.706399	0.353200	−	0.935548i	$-0.385094\pi$
$547$	16.5213	0.706399	0.353200	+	0.935548i	$0.385094\pi$
$548$	0	0
$549$	15.9258	0.679697
$550$	0	0
$551$	−0.802843	−0.0342023
$552$	0	0
$553$	8.15876	0.346946
$554$	0	0
$555$	−8.27765	−0.351367
$556$	0	0
$557$	−23.4337	−0.992920	−0.496460	−	0.868060i	$-0.665367\pi$
$557$	−23.4337	−0.992920	−0.496460	+	0.868060i	$0.665367\pi$
$558$	0	0
$559$	7.36809	0.311637
$560$	0	0
$561$	16.0704	0.678494
$562$	0	0
$563$	17.2828	0.728385	0.364192	−	0.931324i	$-0.381345\pi$
$563$	17.2828	0.728385	0.364192	+	0.931324i	$0.381345\pi$
$564$	0	0
$565$	−18.0586	−0.759731
$566$	0	0
$567$	22.9481	0.963728
$568$	0	0
$569$	−31.4709	−1.31933	−0.659664	−	0.751561i	$-0.729301\pi$
$569$	−31.4709	−1.31933	−0.659664	+	0.751561i	$0.729301\pi$
$570$	0	0
$571$	−13.7413	−0.575056	−0.287528	−	0.957772i	$-0.592833\pi$
$571$	−13.7413	−0.575056	−0.287528	+	0.957772i	$0.592833\pi$
$572$	0	0
$573$	15.6216	0.652602
$574$	0	0
$575$	0.878334	0.0366291
$576$	0	0
$577$	−19.0270	−0.792103	−0.396051	−	0.918228i	$-0.629620\pi$
$577$	−19.0270	−0.792103	−0.396051	+	0.918228i	$0.629620\pi$
$578$	0	0
$579$	−36.4586	−1.51517
$580$	0	0
$581$	−0.585434	−0.0242879
$582$	0	0
$583$	−3.13932	−0.130017
$584$	0	0
$585$	−13.9603	−0.577189
$586$	0	0
$587$	31.0286	1.28069	0.640345	−	0.768088i	$-0.278792\pi$
$587$	31.0286	1.28069	0.640345	+	0.768088i	$0.278792\pi$
$588$	0	0
$589$	−25.0444	−1.03194
$590$	0	0
$591$	38.2666	1.57408
$592$	0	0
$593$	24.7849	1.01779	0.508897	−	0.860827i	$-0.330053\pi$
$593$	24.7849	1.01779	0.508897	+	0.860827i	$0.330053\pi$
$594$	0	0
$595$	9.49146	0.389112
$596$	0	0
$597$	−1.84453	−0.0754918
$598$	0	0
$599$	29.2205	1.19392	0.596958	−	0.802272i	$-0.296376\pi$
$599$	29.2205	1.19392	0.596958	+	0.802272i	$0.296376\pi$
$600$	0	0
$601$	−5.84414	−0.238387	−0.119194	−	0.992871i	$-0.538031\pi$
$601$	−5.84414	−0.238387	−0.119194	+	0.992871i	$0.538031\pi$
$602$	0	0
$603$	23.2091	0.945149
$604$	0	0
$605$	1.89991	0.0772422
$606$	0	0
$607$	−31.4989	−1.27850	−0.639250	−	0.768999i	$-0.720755\pi$
$607$	−31.4989	−1.27850	−0.639250	+	0.768999i	$0.720755\pi$
$608$	0	0
$609$	−0.461830	−0.0187143
$610$	0	0
$611$	−0.631734	−0.0255572
$612$	0	0
$613$	47.3100	1.91083	0.955416	−	0.295263i	$-0.0954073\pi$
$613$	47.3100	1.91083	0.955416	+	0.295263i	$0.0954073\pi$
$614$	0	0
$615$	1.35107	0.0544805
$616$	0	0
$617$	38.2065	1.53814	0.769068	−	0.639167i	$-0.220721\pi$
$617$	38.2065	1.53814	0.769068	+	0.639167i	$0.220721\pi$
$618$	0	0
$619$	8.85216	0.355798	0.177899	−	0.984049i	$-0.443070\pi$
$619$	8.85216	0.355798	0.177899	+	0.984049i	$0.443070\pi$
$620$	0	0
$621$	−8.83571	−0.354565
$622$	0	0
$623$	−3.86148	−0.154707
$624$	0	0
$625$	−16.1151	−0.644606
$626$	0	0
$627$	17.9888	0.718402
$628$	0	0
$629$	−6.76627	−0.269789
$630$	0	0
$631$	33.6767	1.34065	0.670323	−	0.742069i	$-0.266155\pi$
$631$	33.6767	1.34065	0.670323	+	0.742069i	$0.266155\pi$
$632$	0	0
$633$	−0.281463	−0.0111871
$634$	0	0
$635$	17.3060	0.686769
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−60.3674	−2.38810
$640$	0	0
$641$	−2.10802	−0.0832618	−0.0416309	−	0.999133i	$-0.513255\pi$
$641$	−2.10802	−0.0832618	−0.0416309	+	0.999133i	$0.513255\pi$
$642$	0	0
$643$	−3.51647	−0.138676	−0.0693379	−	0.997593i	$-0.522089\pi$
$643$	−3.51647	−0.138676	−0.0693379	+	0.997593i	$0.522089\pi$
$644$	0	0
$645$	−45.0312	−1.77310
$646$	0	0
$647$	−34.3184	−1.34920	−0.674598	−	0.738185i	$-0.735683\pi$
$647$	−34.3184	−1.34920	−0.674598	+	0.738185i	$0.735683\pi$
$648$	0	0
$649$	0.632241	0.0248176
$650$	0	0
$651$	−14.4066	−0.564640
$652$	0	0
$653$	4.64965	0.181955	0.0909774	−	0.995853i	$-0.471001\pi$
$653$	4.64965	0.181955	0.0909774	+	0.995853i	$0.471001\pi$
$654$	0	0
$655$	−26.5236	−1.03636
$656$	0	0
$657$	−34.4029	−1.34218
$658$	0	0
$659$	−15.7376	−0.613049	−0.306524	−	0.951863i	$-0.599166\pi$
$659$	−15.7376	−0.613049	−0.306524	+	0.951863i	$0.599166\pi$
$660$	0	0
$661$	−23.8279	−0.926797	−0.463399	−	0.886150i	$-0.653370\pi$
$661$	−23.8279	−0.926797	−0.463399	+	0.886150i	$0.653370\pi$
$662$	0	0
$663$	−16.0704	−0.624123
$664$	0	0
$665$	10.6245	0.411999
$666$	0	0
$667$	0.0906964	0.00351178
$668$	0	0
$669$	−2.33909	−0.0904343
$670$	0	0
$671$	2.16739	0.0836713
$672$	0	0
$673$	−23.3860	−0.901465	−0.450732	−	0.892659i	$-0.648837\pi$
$673$	−23.3860	−0.901465	−0.450732	+	0.892659i	$0.648837\pi$
$674$	0	0
$675$	−19.4461	−0.748481
$676$	0	0
$677$	−31.2152	−1.19970	−0.599849	−	0.800113i	$-0.704773\pi$
$677$	−31.2152	−1.19970	−0.599849	+	0.800113i	$0.704773\pi$
$678$	0	0
$679$	13.3061	0.510642
$680$	0	0
$681$	82.1434	3.14774
$682$	0	0
$683$	−18.4235	−0.704958	−0.352479	−	0.935820i	$-0.614661\pi$
$683$	−18.4235	−0.704958	−0.352479	+	0.935820i	$0.614661\pi$
$684$	0	0
$685$	4.26905	0.163112
$686$	0	0
$687$	46.2549	1.76474
$688$	0	0
$689$	3.13932	0.119599
$690$	0	0
$691$	−4.44398	−0.169057	−0.0845285	−	0.996421i	$-0.526938\pi$
$691$	−4.44398	−0.169057	−0.0845285	+	0.996421i	$0.526938\pi$
$692$	0	0
$693$	7.34791	0.279124
$694$	0	0
$695$	28.7421	1.09025
$696$	0	0
$697$	1.10439	0.0418316
$698$	0	0
$699$	30.2580	1.14446
$700$	0	0
$701$	−46.6181	−1.76074	−0.880370	−	0.474287i	$-0.842706\pi$
$701$	−46.6181	−1.76074	−0.880370	+	0.474287i	$0.842706\pi$
$702$	0	0
$703$	−7.57397	−0.285658
$704$	0	0
$705$	3.86094	0.145411
$706$	0	0
$707$	−10.8699	−0.408804
$708$	0	0
$709$	−18.6249	−0.699472	−0.349736	−	0.936848i	$-0.613729\pi$
$709$	−18.6249	−0.699472	−0.349736	+	0.936848i	$0.613729\pi$
$710$	0	0
$711$	59.9498	2.24829
$712$	0	0
$713$	2.82924	0.105956
$714$	0	0
$715$	−1.89991	−0.0710525
$716$	0	0
$717$	43.5458	1.62625
$718$	0	0
$719$	3.70812	0.138289	0.0691447	−	0.997607i	$-0.477973\pi$
$719$	3.70812	0.138289	0.0691447	+	0.997607i	$0.477973\pi$
$720$	0	0
$721$	−18.3575	−0.683668
$722$	0	0
$723$	67.6626	2.51640
$724$	0	0
$725$	0.199609	0.00741331
$726$	0	0
$727$	−26.6460	−0.988244	−0.494122	−	0.869393i	$-0.664510\pi$
$727$	−26.6460	−0.988244	−0.494122	+	0.869393i	$0.664510\pi$
$728$	0	0
$729$	33.6449	1.24611
$730$	0	0
$731$	−36.8091	−1.36143
$732$	0	0
$733$	−18.1502	−0.670394	−0.335197	−	0.942148i	$-0.608803\pi$
$733$	−18.1502	−0.670394	−0.335197	+	0.942148i	$0.608803\pi$
$734$	0	0
$735$	6.11165	0.225432
$736$	0	0
$737$	3.15860	0.116349
$738$	0	0
$739$	7.00921	0.257838	0.128919	−	0.991655i	$-0.458849\pi$
$739$	7.00921	0.257838	0.128919	+	0.991655i	$0.458849\pi$
$740$	0	0
$741$	−17.9888	−0.660834
$742$	0	0
$743$	13.1825	0.483619	0.241809	−	0.970324i	$-0.422259\pi$
$743$	13.1825	0.483619	0.241809	+	0.970324i	$0.422259\pi$
$744$	0	0
$745$	11.0476	0.404754
$746$	0	0
$747$	−4.30172	−0.157392
$748$	0	0
$749$	5.62207	0.205426
$750$	0	0
$751$	23.9212	0.872896	0.436448	−	0.899729i	$-0.356236\pi$
$751$	23.9212	0.872896	0.436448	+	0.899729i	$0.356236\pi$
$752$	0	0
$753$	26.3341	0.959668
$754$	0	0
$755$	14.4916	0.527405
$756$	0	0
$757$	−13.9569	−0.507273	−0.253637	−	0.967300i	$-0.581627\pi$
$757$	−13.9569	−0.507273	−0.253637	+	0.967300i	$0.581627\pi$
$758$	0	0
$759$	−2.03217	−0.0737632
$760$	0	0
$761$	15.9000	0.576374	0.288187	−	0.957574i	$-0.406947\pi$
$761$	15.9000	0.576374	0.288187	+	0.957574i	$0.406947\pi$
$762$	0	0
$763$	−13.5045	−0.488894
$764$	0	0
$765$	69.7424	2.52154
$766$	0	0
$767$	−0.632241	−0.0228289
$768$	0	0
$769$	−14.1087	−0.508773	−0.254386	−	0.967103i	$-0.581873\pi$
$769$	−14.1087	−0.508773	−0.254386	+	0.967103i	$0.581873\pi$
$770$	0	0
$771$	12.5983	0.453716
$772$	0	0
$773$	24.7170	0.889009	0.444505	−	0.895777i	$-0.353380\pi$
$773$	24.7170	0.889009	0.444505	+	0.895777i	$0.353380\pi$
$774$	0	0
$775$	6.22674	0.223671
$776$	0	0
$777$	−4.35687	−0.156302
$778$	0	0
$779$	1.23622	0.0442921
$780$	0	0
$781$	−8.21559	−0.293977
$782$	0	0
$783$	−2.00799	−0.0717599
$784$	0	0
$785$	−27.1346	−0.968477
$786$	0	0
$787$	−2.19380	−0.0782006	−0.0391003	−	0.999235i	$-0.512449\pi$
$787$	−2.19380	−0.0782006	−0.0391003	+	0.999235i	$0.512449\pi$
$788$	0	0
$789$	2.54542	0.0906195
$790$	0	0
$791$	−9.50499	−0.337958
$792$	0	0
$793$	−2.16739	−0.0769664
$794$	0	0
$795$	−19.1864	−0.680472
$796$	0	0
$797$	−39.2886	−1.39168	−0.695838	−	0.718199i	$-0.744967\pi$
$797$	−39.2886	−1.39168	−0.695838	+	0.718199i	$0.744967\pi$
$798$	0	0
$799$	3.15599	0.111651
$800$	0	0
$801$	−28.3738	−1.00254
$802$	0	0
$803$	−4.68200	−0.165224
$804$	0	0
$805$	−1.20024	−0.0423028
$806$	0	0
$807$	49.1270	1.72935
$808$	0	0
$809$	−38.0717	−1.33853	−0.669264	−	0.743024i	$-0.733391\pi$
$809$	−38.0717	−1.33853	−0.669264	+	0.743024i	$0.733391\pi$
$810$	0	0
$811$	23.3924	0.821418	0.410709	−	0.911766i	$-0.365281\pi$
$811$	23.3924	0.821418	0.410709	+	0.911766i	$0.365281\pi$
$812$	0	0
$813$	−44.1089	−1.54697
$814$	0	0
$815$	7.64652	0.267846
$816$	0	0
$817$	−41.2031	−1.44151
$818$	0	0
$819$	−7.34791	−0.256757
$820$	0	0
$821$	0.765607	0.0267199	0.0133599	−	0.999911i	$-0.495747\pi$
$821$	0.765607	0.0267199	0.0133599	+	0.999911i	$0.495747\pi$
$822$	0	0
$823$	38.3017	1.33511	0.667556	−	0.744559i	$-0.267340\pi$
$823$	38.3017	1.33511	0.667556	+	0.744559i	$0.267340\pi$
$824$	0	0
$825$	−4.47252	−0.155713
$826$	0	0
$827$	−26.4058	−0.918221	−0.459110	−	0.888379i	$-0.651832\pi$
$827$	−26.4058	−0.918221	−0.459110	+	0.888379i	$0.651832\pi$
$828$	0	0
$829$	16.4420	0.571055	0.285527	−	0.958371i	$-0.407831\pi$
$829$	16.4420	0.571055	0.285527	+	0.958371i	$0.407831\pi$
$830$	0	0
$831$	−33.4679	−1.16099
$832$	0	0
$833$	4.99575	0.173093
$834$	0	0
$835$	−22.3928	−0.774936
$836$	0	0
$837$	−62.6387	−2.16511
$838$	0	0
$839$	−21.8288	−0.753615	−0.376808	−	0.926292i	$-0.622978\pi$
$839$	−21.8288	−0.753615	−0.376808	+	0.926292i	$0.622978\pi$
$840$	0	0
$841$	−28.9794	−0.999289
$842$	0	0
$843$	−106.366	−3.66344
$844$	0	0
$845$	1.89991	0.0653588
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−77.9621	−2.67565
$850$	0	0
$851$	0.855624	0.0293304
$852$	0	0
$853$	34.9932	1.19814	0.599071	−	0.800696i	$-0.295537\pi$
$853$	34.9932	1.19814	0.599071	+	0.800696i	$0.295537\pi$
$854$	0	0
$855$	78.0676	2.66986
$856$	0	0
$857$	50.5463	1.72663	0.863314	−	0.504667i	$-0.168385\pi$
$857$	50.5463	1.72663	0.863314	+	0.504667i	$0.168385\pi$
$858$	0	0
$859$	48.8775	1.66768	0.833840	−	0.552006i	$-0.186137\pi$
$859$	48.8775	1.66768	0.833840	+	0.552006i	$0.186137\pi$
$860$	0	0
$861$	0.711126	0.0242351
$862$	0	0
$863$	−38.0432	−1.29501	−0.647503	−	0.762063i	$-0.724187\pi$
$863$	−38.0432	−1.29501	−0.647503	+	0.762063i	$0.724187\pi$
$864$	0	0
$865$	−3.79968	−0.129193
$866$	0	0
$867$	25.5979	0.869351
$868$	0	0
$869$	8.15876	0.276767
$870$	0	0
$871$	−3.15860	−0.107025
$872$	0	0
$873$	97.7723	3.30909
$874$	0	0
$875$	−12.1411	−0.410443
$876$	0	0
$877$	11.4193	0.385602	0.192801	−	0.981238i	$-0.438243\pi$
$877$	11.4193	0.385602	0.192801	+	0.981238i	$0.438243\pi$
$878$	0	0
$879$	5.73487	0.193432
$880$	0	0
$881$	−40.1016	−1.35106	−0.675529	−	0.737333i	$-0.736085\pi$
$881$	−40.1016	−1.35106	−0.675529	+	0.737333i	$0.736085\pi$
$882$	0	0
$883$	−37.9296	−1.27643	−0.638216	−	0.769857i	$-0.720327\pi$
$883$	−37.9296	−1.27643	−0.638216	+	0.769857i	$0.720327\pi$
$884$	0	0
$885$	3.86404	0.129888
$886$	0	0
$887$	29.3261	0.984676	0.492338	−	0.870404i	$-0.336142\pi$
$887$	29.3261	0.984676	0.492338	+	0.870404i	$0.336142\pi$
$888$	0	0
$889$	9.10888	0.305502
$890$	0	0
$891$	22.9481	0.768789
$892$	0	0
$893$	3.53272	0.118218
$894$	0	0
$895$	−41.7791	−1.39652
$896$	0	0
$897$	2.03217	0.0678523
$898$	0	0
$899$	0.642971	0.0214443
$900$	0	0
$901$	−15.6833	−0.522485
$902$	0	0
$903$	−23.7018	−0.788746
$904$	0	0
$905$	7.31623	0.243200
$906$	0	0
$907$	41.5890	1.38094	0.690469	−	0.723362i	$-0.257404\pi$
$907$	41.5890	1.38094	0.690469	+	0.723362i	$0.257404\pi$
$908$	0	0
$909$	−79.8710	−2.64915
$910$	0	0
$911$	49.1208	1.62744	0.813722	−	0.581255i	$-0.197438\pi$
$911$	49.1208	1.62744	0.813722	+	0.581255i	$0.197438\pi$
$912$	0	0
$913$	−0.585434	−0.0193750
$914$	0	0
$915$	13.2464	0.437911
$916$	0	0
$917$	−13.9605	−0.461016
$918$	0	0
$919$	−4.94764	−0.163208	−0.0816038	−	0.996665i	$-0.526004\pi$
$919$	−4.94764	−0.163208	−0.0816038	+	0.996665i	$0.526004\pi$
$920$	0	0
$921$	63.4128	2.08952
$922$	0	0
$923$	8.21559	0.270419
$924$	0	0
$925$	1.88310	0.0619160
$926$	0	0
$927$	−134.889	−4.43034
$928$	0	0
$929$	−28.3191	−0.929118	−0.464559	−	0.885542i	$-0.653787\pi$
$929$	−28.3191	−0.929118	−0.464559	+	0.885542i	$0.653787\pi$
$930$	0	0
$931$	5.59210	0.183274
$932$	0	0
$933$	−29.4127	−0.962927
$934$	0	0
$935$	9.49146	0.310404
$936$	0	0
$937$	−40.8751	−1.33533	−0.667666	−	0.744461i	$-0.732707\pi$
$937$	−40.8751	−1.33533	−0.667666	+	0.744461i	$0.732707\pi$
$938$	0	0
$939$	−66.3492	−2.16522
$940$	0	0
$941$	−39.5674	−1.28986	−0.644930	−	0.764242i	$-0.723113\pi$
$941$	−39.5674	−1.28986	−0.644930	+	0.764242i	$0.723113\pi$
$942$	0	0
$943$	−0.139654	−0.00454777
$944$	0	0
$945$	26.5729	0.864417
$946$	0	0
$947$	46.9025	1.52413	0.762063	−	0.647503i	$-0.224187\pi$
$947$	46.9025	1.52413	0.762063	+	0.647503i	$0.224187\pi$
$948$	0	0
$949$	4.68200	0.151984
$950$	0	0
$951$	4.97522	0.161333
$952$	0	0
$953$	29.6776	0.961353	0.480677	−	0.876898i	$-0.340391\pi$
$953$	29.6776	0.961353	0.480677	+	0.876898i	$0.340391\pi$
$954$	0	0
$955$	9.22639	0.298559
$956$	0	0
$957$	−0.461830	−0.0149288
$958$	0	0
$959$	2.24698	0.0725588
$960$	0	0
$961$	−10.9428	−0.352992
$962$	0	0
$963$	41.3105	1.33121
$964$	0	0
$965$	−21.5331	−0.693174
$966$	0	0
$967$	33.4001	1.07408	0.537038	−	0.843558i	$-0.319543\pi$
$967$	33.4001	1.07408	0.537038	+	0.843558i	$0.319543\pi$
$968$	0	0
$969$	89.8674	2.88696
$970$	0	0
$971$	−16.8380	−0.540356	−0.270178	−	0.962810i	$-0.587082\pi$
$971$	−16.8380	−0.540356	−0.270178	+	0.962810i	$0.587082\pi$
$972$	0	0
$973$	15.1281	0.484986
$974$	0	0
$975$	4.47252	0.143235
$976$	0	0
$977$	26.4510	0.846242	0.423121	−	0.906073i	$-0.360935\pi$
$977$	26.4510	0.846242	0.423121	+	0.906073i	$0.360935\pi$
$978$	0	0
$979$	−3.86148	−0.123414
$980$	0	0
$981$	−99.2296	−3.16816
$982$	0	0
$983$	−27.2622	−0.869530	−0.434765	−	0.900544i	$-0.643169\pi$
$983$	−27.2622	−0.869530	−0.434765	+	0.900544i	$0.643169\pi$
$984$	0	0
$985$	22.6009	0.720124
$986$	0	0
$987$	2.03217	0.0646848
$988$	0	0
$989$	4.65467	0.148010
$990$	0	0
$991$	23.6244	0.750453	0.375226	−	0.926933i	$-0.377565\pi$
$991$	23.6244	0.750453	0.375226	+	0.926933i	$0.377565\pi$
$992$	0	0
$993$	−69.4393	−2.20359
$994$	0	0
$995$	−1.08941	−0.0345367
$996$	0	0
$997$	−34.9550	−1.10703	−0.553517	−	0.832838i	$-0.686715\pi$
$997$	−34.9550	−1.10703	−0.553517	+	0.832838i	$0.686715\pi$
$998$	0	0
$999$	−18.9433	−0.599339

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.9	✓	9	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)

\(f(q)\)	\(=\)	\(q+3.21682 q^{3} +1.89991 q^{5} +1.00000 q^{7} +7.34791 q^{9} +O(q^{10})\)	\(q+3.21682 q^{3} +1.89991 q^{5} +1.00000 q^{7} +7.34791 q^{9} +1.00000 q^{11} -1.00000 q^{13} +6.11165 q^{15} +4.99575 q^{17} +5.59210 q^{19} +3.21682 q^{21} -0.631734 q^{23} -1.39035 q^{25} +13.9864 q^{27} -0.143567 q^{29} -4.47853 q^{31} +3.21682 q^{33} +1.89991 q^{35} -1.35440 q^{37} -3.21682 q^{39} +0.221065 q^{41} -7.36809 q^{43} +13.9603 q^{45} +0.631734 q^{47} +1.00000 q^{49} +16.0704 q^{51} -3.13932 q^{53} +1.89991 q^{55} +17.9888 q^{57} +0.632241 q^{59} +2.16739 q^{61} +7.34791 q^{63} -1.89991 q^{65} +3.15860 q^{67} -2.03217 q^{69} -8.21559 q^{71} -4.68200 q^{73} -4.47252 q^{75} +1.00000 q^{77} +8.15876 q^{79} +22.9481 q^{81} -0.585434 q^{83} +9.49146 q^{85} -0.461830 q^{87} -3.86148 q^{89} -1.00000 q^{91} -14.4066 q^{93} +10.6245 q^{95} +13.3061 q^{97} +7.34791 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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