LMFDB - Embedded newform 8008.2.a.s.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$10$
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} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 $ x^10 - 3x^9 - 15x^8 + 43x^7 + 66x^6 - 173x^5 - 127x^4 + 246x^3 + 99x^2 - 82*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.99972$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99972 q^{3} -1.77646 q^{5} +1.00000 q^{7} +5.99829 q^{9} +O(q^{10})$	$q+2.99972 q^{3} -1.77646 q^{5} +1.00000 q^{7} +5.99829 q^{9} +1.00000 q^{11} -1.00000 q^{13} -5.32888 q^{15} -2.47445 q^{17} -4.42130 q^{19} +2.99972 q^{21} -3.48707 q^{23} -1.84418 q^{25} +8.99403 q^{27} -6.55906 q^{29} -4.76106 q^{31} +2.99972 q^{33} -1.77646 q^{35} -2.65246 q^{37} -2.99972 q^{39} -6.12788 q^{41} -0.885031 q^{43} -10.6557 q^{45} -5.64454 q^{47} +1.00000 q^{49} -7.42264 q^{51} -5.48205 q^{53} -1.77646 q^{55} -13.2626 q^{57} +1.10886 q^{59} +3.44379 q^{61} +5.99829 q^{63} +1.77646 q^{65} +8.10527 q^{67} -10.4602 q^{69} -11.1486 q^{71} -6.25405 q^{73} -5.53202 q^{75} +1.00000 q^{77} +13.8226 q^{79} +8.98465 q^{81} -2.86946 q^{83} +4.39576 q^{85} -19.6753 q^{87} +7.26143 q^{89} -1.00000 q^{91} -14.2818 q^{93} +7.85427 q^{95} +10.5658 q^{97} +5.99829 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	$ 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.99972	1.73189	0.865943	−	0.500142i	$-0.166719\pi$
$3$	2.99972	1.73189	0.865943	+	0.500142i	$0.166719\pi$
$4$	0	0
$5$	−1.77646	−0.794458	−0.397229	−	0.917719i	$-0.630028\pi$
$5$	−1.77646	−0.794458	−0.397229	+	0.917719i	$0.630028\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.99829	1.99943
$10$	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
$15$	−5.32888	−1.37591
$16$	0	0
$17$	−2.47445	−0.600142	−0.300071	−	0.953917i	$-0.597010\pi$
$17$	−2.47445	−0.600142	−0.300071	+	0.953917i	$0.597010\pi$
$18$	0	0
$19$	−4.42130	−1.01432	−0.507158	−	0.861853i	$-0.669304\pi$
$19$	−4.42130	−1.01432	−0.507158	+	0.861853i	$0.669304\pi$
$20$	0	0
$21$	2.99972	0.654592
$22$	0	0
$23$	−3.48707	−0.727103	−0.363552	−	0.931574i	$-0.618436\pi$
$23$	−3.48707	−0.727103	−0.363552	+	0.931574i	$0.618436\pi$
$24$	0	0
$25$	−1.84418	−0.368836
$26$	0	0
$27$	8.99403	1.73090
$28$	0	0
$29$	−6.55906	−1.21799	−0.608993	−	0.793175i	$-0.708426\pi$
$29$	−6.55906	−1.21799	−0.608993	+	0.793175i	$0.708426\pi$
$30$	0	0
$31$	−4.76106	−0.855111	−0.427556	−	0.903989i	$-0.640625\pi$
$31$	−4.76106	−0.855111	−0.427556	+	0.903989i	$0.640625\pi$
$32$	0	0
$33$	2.99972	0.522183
$34$	0	0
$35$	−1.77646	−0.300277
$36$	0	0
$37$	−2.65246	−0.436061	−0.218031	−	0.975942i	$-0.569963\pi$
$37$	−2.65246	−0.436061	−0.218031	+	0.975942i	$0.569963\pi$
$38$	0	0
$39$	−2.99972	−0.480339
$40$	0	0
$41$	−6.12788	−0.957015	−0.478507	−	0.878084i	$-0.658822\pi$
$41$	−6.12788	−0.957015	−0.478507	+	0.878084i	$0.658822\pi$
$42$	0	0
$43$	−0.885031	−0.134966	−0.0674830	−	0.997720i	$-0.521497\pi$
$43$	−0.885031	−0.134966	−0.0674830	+	0.997720i	$0.521497\pi$
$44$	0	0
$45$	−10.6557	−1.58846
$46$	0	0
$47$	−5.64454	−0.823341	−0.411671	−	0.911333i	$-0.635055\pi$
$47$	−5.64454	−0.823341	−0.411671	+	0.911333i	$0.635055\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.42264	−1.03938
$52$	0	0
$53$	−5.48205	−0.753017	−0.376509	−	0.926413i	$-0.622875\pi$
$53$	−5.48205	−0.753017	−0.376509	+	0.926413i	$0.622875\pi$
$54$	0	0
$55$	−1.77646	−0.239538
$56$	0	0
$57$	−13.2626	−1.75668
$58$	0	0
$59$	1.10886	0.144361	0.0721803	−	0.997392i	$-0.477004\pi$
$59$	1.10886	0.144361	0.0721803	+	0.997392i	$0.477004\pi$
$60$	0	0
$61$	3.44379	0.440932	0.220466	−	0.975395i	$-0.429242\pi$
$61$	3.44379	0.440932	0.220466	+	0.975395i	$0.429242\pi$
$62$	0	0
$63$	5.99829	0.755714
$64$	0	0
$65$	1.77646	0.220343
$66$	0	0
$67$	8.10527	0.990216	0.495108	−	0.868831i	$-0.335128\pi$
$67$	8.10527	0.990216	0.495108	+	0.868831i	$0.335128\pi$
$68$	0	0
$69$	−10.4602	−1.25926
$70$	0	0
$71$	−11.1486	−1.32309	−0.661545	−	0.749906i	$-0.730099\pi$
$71$	−11.1486	−1.32309	−0.661545	+	0.749906i	$0.730099\pi$
$72$	0	0
$73$	−6.25405	−0.731982	−0.365991	−	0.930618i	$-0.619270\pi$
$73$	−6.25405	−0.731982	−0.365991	+	0.930618i	$0.619270\pi$
$74$	0	0
$75$	−5.53202	−0.638782
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	13.8226	1.55516	0.777580	−	0.628784i	$-0.216447\pi$
$79$	13.8226	1.55516	0.777580	+	0.628784i	$0.216447\pi$
$80$	0	0
$81$	8.98465	0.998294
$82$	0	0
$83$	−2.86946	−0.314964	−0.157482	−	0.987522i	$-0.550338\pi$
$83$	−2.86946	−0.314964	−0.157482	+	0.987522i	$0.550338\pi$
$84$	0	0
$85$	4.39576	0.476788
$86$	0	0
$87$	−19.6753	−2.10941
$88$	0	0
$89$	7.26143	0.769710	0.384855	−	0.922977i	$-0.374251\pi$
$89$	7.26143	0.769710	0.384855	+	0.922977i	$0.374251\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−14.2818	−1.48096
$94$	0	0
$95$	7.85427	0.805831
$96$	0	0
$97$	10.5658	1.07279	0.536396	−	0.843966i	$-0.319785\pi$
$97$	10.5658	1.07279	0.536396	+	0.843966i	$0.319785\pi$
$98$	0	0
$99$	5.99829	0.602851
$100$	0	0
$101$	−1.66936	−0.166108	−0.0830539	−	0.996545i	$-0.526467\pi$
$101$	−1.66936	−0.166108	−0.0830539	+	0.996545i	$0.526467\pi$
$102$	0	0
$103$	3.66769	0.361388	0.180694	−	0.983539i	$-0.442166\pi$
$103$	3.66769	0.361388	0.180694	+	0.983539i	$0.442166\pi$
$104$	0	0
$105$	−5.32888	−0.520046
$106$	0	0
$107$	0.647908	0.0626356	0.0313178	−	0.999509i	$-0.490030\pi$
$107$	0.647908	0.0626356	0.0313178	+	0.999509i	$0.490030\pi$
$108$	0	0
$109$	−12.6692	−1.21349	−0.606744	−	0.794897i	$-0.707525\pi$
$109$	−12.6692	−1.21349	−0.606744	+	0.794897i	$0.707525\pi$
$110$	0	0
$111$	−7.95662	−0.755209
$112$	0	0
$113$	8.05325	0.757586	0.378793	−	0.925481i	$-0.376339\pi$
$113$	8.05325	0.757586	0.378793	+	0.925481i	$0.376339\pi$
$114$	0	0
$115$	6.19464	0.577653
$116$	0	0
$117$	−5.99829	−0.554542
$118$	0	0
$119$	−2.47445	−0.226832
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−18.3819	−1.65744
$124$	0	0
$125$	12.1584	1.08748
$126$	0	0
$127$	−21.0959	−1.87195	−0.935977	−	0.352060i	$-0.885481\pi$
$127$	−21.0959	−1.87195	−0.935977	+	0.352060i	$0.885481\pi$
$128$	0	0
$129$	−2.65484	−0.233746
$130$	0	0
$131$	−13.9140	−1.21567	−0.607837	−	0.794062i	$-0.707962\pi$
$131$	−13.9140	−1.21567	−0.607837	+	0.794062i	$0.707962\pi$
$132$	0	0
$133$	−4.42130	−0.383375
$134$	0	0
$135$	−15.9776	−1.37513
$136$	0	0
$137$	−0.0929259	−0.00793919	−0.00396960	−	0.999992i	$-0.501264\pi$
$137$	−0.0929259	−0.00793919	−0.00396960	+	0.999992i	$0.501264\pi$
$138$	0	0
$139$	2.04542	0.173490	0.0867449	−	0.996231i	$-0.472353\pi$
$139$	2.04542	0.173490	0.0867449	+	0.996231i	$0.472353\pi$
$140$	0	0
$141$	−16.9320	−1.42593
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	11.6519	0.967639
$146$	0	0
$147$	2.99972	0.247412
$148$	0	0
$149$	−10.2240	−0.837579	−0.418789	−	0.908083i	$-0.637545\pi$
$149$	−10.2240	−0.837579	−0.418789	+	0.908083i	$0.637545\pi$
$150$	0	0
$151$	13.9424	1.13462	0.567308	−	0.823505i	$-0.307985\pi$
$151$	13.9424	1.13462	0.567308	+	0.823505i	$0.307985\pi$
$152$	0	0
$153$	−14.8425	−1.19994
$154$	0	0
$155$	8.45784	0.679350
$156$	0	0
$157$	14.7884	1.18024	0.590121	−	0.807315i	$-0.299080\pi$
$157$	14.7884	1.18024	0.590121	+	0.807315i	$0.299080\pi$
$158$	0	0
$159$	−16.4446	−1.30414
$160$	0	0
$161$	−3.48707	−0.274819
$162$	0	0
$163$	−12.8809	−1.00891	−0.504456	−	0.863437i	$-0.668307\pi$
$163$	−12.8809	−1.00891	−0.504456	+	0.863437i	$0.668307\pi$
$164$	0	0
$165$	−5.32888	−0.414853
$166$	0	0
$167$	4.28545	0.331618	0.165809	−	0.986158i	$-0.446976\pi$
$167$	4.28545	0.331618	0.165809	+	0.986158i	$0.446976\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−26.5202	−2.02805
$172$	0	0
$173$	16.7195	1.27116	0.635581	−	0.772034i	$-0.280760\pi$
$173$	16.7195	1.27116	0.635581	+	0.772034i	$0.280760\pi$
$174$	0	0
$175$	−1.84418	−0.139407
$176$	0	0
$177$	3.32625	0.250016
$178$	0	0
$179$	20.6431	1.54294	0.771468	−	0.636269i	$-0.219523\pi$
$179$	20.6431	1.54294	0.771468	+	0.636269i	$0.219523\pi$
$180$	0	0
$181$	−2.04156	−0.151748	−0.0758739	−	0.997117i	$-0.524175\pi$
$181$	−2.04156	−0.151748	−0.0758739	+	0.997117i	$0.524175\pi$
$182$	0	0
$183$	10.3304	0.763643
$184$	0	0
$185$	4.71199	0.346433
$186$	0	0
$187$	−2.47445	−0.180950
$188$	0	0
$189$	8.99403	0.654219
$190$	0	0
$191$	−8.49716	−0.614833	−0.307416	−	0.951575i	$-0.599464\pi$
$191$	−8.49716	−0.614833	−0.307416	+	0.951575i	$0.599464\pi$
$192$	0	0
$193$	15.8876	1.14361	0.571807	−	0.820388i	$-0.306243\pi$
$193$	15.8876	1.14361	0.571807	+	0.820388i	$0.306243\pi$
$194$	0	0
$195$	5.32888	0.381609
$196$	0	0
$197$	−5.62959	−0.401092	−0.200546	−	0.979684i	$-0.564272\pi$
$197$	−5.62959	−0.401092	−0.200546	+	0.979684i	$0.564272\pi$
$198$	0	0
$199$	−5.39454	−0.382408	−0.191204	−	0.981550i	$-0.561239\pi$
$199$	−5.39454	−0.382408	−0.191204	+	0.981550i	$0.561239\pi$
$200$	0	0
$201$	24.3135	1.71494
$202$	0	0
$203$	−6.55906	−0.460356
$204$	0	0
$205$	10.8860	0.760308
$206$	0	0
$207$	−20.9164	−1.45379
$208$	0	0
$209$	−4.42130	−0.305828
$210$	0	0
$211$	26.8140	1.84595	0.922975	−	0.384860i	$-0.125750\pi$
$211$	26.8140	1.84595	0.922975	+	0.384860i	$0.125750\pi$
$212$	0	0
$213$	−33.4425	−2.29144
$214$	0	0
$215$	1.57223	0.107225
$216$	0	0
$217$	−4.76106	−0.323202
$218$	0	0
$219$	−18.7604	−1.26771
$220$	0	0
$221$	2.47445	0.166449
$222$	0	0
$223$	−12.5937	−0.843340	−0.421670	−	0.906749i	$-0.638556\pi$
$223$	−12.5937	−0.843340	−0.421670	+	0.906749i	$0.638556\pi$
$224$	0	0
$225$	−11.0619	−0.737462
$226$	0	0
$227$	−20.7207	−1.37528	−0.687641	−	0.726051i	$-0.741353\pi$
$227$	−20.7207	−1.37528	−0.687641	+	0.726051i	$0.741353\pi$
$228$	0	0
$229$	2.52256	0.166695	0.0833477	−	0.996521i	$-0.473439\pi$
$229$	2.52256	0.166695	0.0833477	+	0.996521i	$0.473439\pi$
$230$	0	0
$231$	2.99972	0.197367
$232$	0	0
$233$	25.4210	1.66539	0.832693	−	0.553734i	$-0.186798\pi$
$233$	25.4210	1.66539	0.832693	+	0.553734i	$0.186798\pi$
$234$	0	0
$235$	10.0273	0.654110
$236$	0	0
$237$	41.4638	2.69336
$238$	0	0
$239$	1.35059	0.0873627	0.0436813	−	0.999046i	$-0.486091\pi$
$239$	1.35059	0.0873627	0.0436813	+	0.999046i	$0.486091\pi$
$240$	0	0
$241$	−21.9518	−1.41404	−0.707020	−	0.707193i	$-0.749961\pi$
$241$	−21.9518	−1.41404	−0.707020	+	0.707193i	$0.749961\pi$
$242$	0	0
$243$	−0.0306974	−0.00196924
$244$	0	0
$245$	−1.77646	−0.113494
$246$	0	0
$247$	4.42130	0.281320
$248$	0	0
$249$	−8.60756	−0.545482
$250$	0	0
$251$	13.6839	0.863723	0.431861	−	0.901940i	$-0.357857\pi$
$251$	13.6839	0.863723	0.431861	+	0.901940i	$0.357857\pi$
$252$	0	0
$253$	−3.48707	−0.219230
$254$	0	0
$255$	13.1860	0.825742
$256$	0	0
$257$	16.6349	1.03766	0.518828	−	0.854878i	$-0.326368\pi$
$257$	16.6349	1.03766	0.518828	+	0.854878i	$0.326368\pi$
$258$	0	0
$259$	−2.65246	−0.164816
$260$	0	0
$261$	−39.3432	−2.43528
$262$	0	0
$263$	−1.05203	−0.0648711	−0.0324355	−	0.999474i	$-0.510326\pi$
$263$	−1.05203	−0.0648711	−0.0324355	+	0.999474i	$0.510326\pi$
$264$	0	0
$265$	9.73866	0.598241
$266$	0	0
$267$	21.7822	1.33305
$268$	0	0
$269$	11.5759	0.705792	0.352896	−	0.935663i	$-0.385197\pi$
$269$	11.5759	0.705792	0.352896	+	0.935663i	$0.385197\pi$
$270$	0	0
$271$	7.04345	0.427859	0.213930	−	0.976849i	$-0.431374\pi$
$271$	7.04345	0.427859	0.213930	+	0.976849i	$0.431374\pi$
$272$	0	0
$273$	−2.99972	−0.181551
$274$	0	0
$275$	−1.84418	−0.111208
$276$	0	0
$277$	2.42668	0.145805	0.0729026	−	0.997339i	$-0.476774\pi$
$277$	2.42668	0.145805	0.0729026	+	0.997339i	$0.476774\pi$
$278$	0	0
$279$	−28.5582	−1.70974
$280$	0	0
$281$	25.3722	1.51358	0.756789	−	0.653659i	$-0.226767\pi$
$281$	25.3722	1.51358	0.756789	+	0.653659i	$0.226767\pi$
$282$	0	0
$283$	5.32335	0.316440	0.158220	−	0.987404i	$-0.449424\pi$
$283$	5.32335	0.316440	0.158220	+	0.987404i	$0.449424\pi$
$284$	0	0
$285$	23.5606	1.39561
$286$	0	0
$287$	−6.12788	−0.361718
$288$	0	0
$289$	−10.8771	−0.639830
$290$	0	0
$291$	31.6943	1.85795
$292$	0	0
$293$	−21.6274	−1.26349	−0.631744	−	0.775177i	$-0.717660\pi$
$293$	−21.6274	−1.26349	−0.631744	+	0.775177i	$0.717660\pi$
$294$	0	0
$295$	−1.96984	−0.114689
$296$	0	0
$297$	8.99403	0.521887
$298$	0	0
$299$	3.48707	0.201662
$300$	0	0
$301$	−0.885031	−0.0510124
$302$	0	0
$303$	−5.00762	−0.287680
$304$	0	0
$305$	−6.11776	−0.350302
$306$	0	0
$307$	−26.9059	−1.53560	−0.767799	−	0.640690i	$-0.778648\pi$
$307$	−26.9059	−1.53560	−0.767799	+	0.640690i	$0.778648\pi$
$308$	0	0
$309$	11.0020	0.625883
$310$	0	0
$311$	−30.0992	−1.70677	−0.853384	−	0.521282i	$-0.825454\pi$
$311$	−30.0992	−1.70677	−0.853384	+	0.521282i	$0.825454\pi$
$312$	0	0
$313$	24.3646	1.37717	0.688585	−	0.725156i	$-0.258232\pi$
$313$	24.3646	1.37717	0.688585	+	0.725156i	$0.258232\pi$
$314$	0	0
$315$	−10.6557	−0.600383
$316$	0	0
$317$	8.62754	0.484571	0.242285	−	0.970205i	$-0.422103\pi$
$317$	8.62754	0.484571	0.242285	+	0.970205i	$0.422103\pi$
$318$	0	0
$319$	−6.55906	−0.367237
$320$	0	0
$321$	1.94354	0.108478
$322$	0	0
$323$	10.9403	0.608733
$324$	0	0
$325$	1.84418	0.102297
$326$	0	0
$327$	−38.0040	−2.10162
$328$	0	0
$329$	−5.64454	−0.311194
$330$	0	0
$331$	−8.36471	−0.459766	−0.229883	−	0.973218i	$-0.573834\pi$
$331$	−8.36471	−0.459766	−0.229883	+	0.973218i	$0.573834\pi$
$332$	0	0
$333$	−15.9102	−0.871875
$334$	0	0
$335$	−14.3987	−0.786685
$336$	0	0
$337$	−8.63754	−0.470517	−0.235258	−	0.971933i	$-0.575594\pi$
$337$	−8.63754	−0.470517	−0.235258	+	0.971933i	$0.575594\pi$
$338$	0	0
$339$	24.1575	1.31205
$340$	0	0
$341$	−4.76106	−0.257826
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	18.5822	1.00043
$346$	0	0
$347$	−10.5425	−0.565950	−0.282975	−	0.959127i	$-0.591321\pi$
$347$	−10.5425	−0.565950	−0.282975	+	0.959127i	$0.591321\pi$
$348$	0	0
$349$	−26.6692	−1.42757	−0.713785	−	0.700365i	$-0.753020\pi$
$349$	−26.6692	−1.42757	−0.713785	+	0.700365i	$0.753020\pi$
$350$	0	0
$351$	−8.99403	−0.480066
$352$	0	0
$353$	0.907925	0.0483240	0.0241620	−	0.999708i	$-0.492308\pi$
$353$	0.907925	0.0483240	0.0241620	+	0.999708i	$0.492308\pi$
$354$	0	0
$355$	19.8050	1.05114
$356$	0	0
$357$	−7.42264	−0.392848
$358$	0	0
$359$	7.69256	0.405998	0.202999	−	0.979179i	$-0.434931\pi$
$359$	7.69256	0.405998	0.202999	+	0.979179i	$0.434931\pi$
$360$	0	0
$361$	0.547868	0.0288351
$362$	0	0
$363$	2.99972	0.157444
$364$	0	0
$365$	11.1101	0.581529
$366$	0	0
$367$	−4.21077	−0.219800	−0.109900	−	0.993943i	$-0.535053\pi$
$367$	−4.21077	−0.219800	−0.109900	+	0.993943i	$0.535053\pi$
$368$	0	0
$369$	−36.7568	−1.91349
$370$	0	0
$371$	−5.48205	−0.284614
$372$	0	0
$373$	10.4230	0.539684	0.269842	−	0.962905i	$-0.413028\pi$
$373$	10.4230	0.539684	0.269842	+	0.962905i	$0.413028\pi$
$374$	0	0
$375$	36.4718	1.88340
$376$	0	0
$377$	6.55906	0.337809
$378$	0	0
$379$	−5.66507	−0.290995	−0.145498	−	0.989359i	$-0.546478\pi$
$379$	−5.66507	−0.290995	−0.145498	+	0.989359i	$0.546478\pi$
$380$	0	0
$381$	−63.2816	−3.24201
$382$	0	0
$383$	−4.07048	−0.207992	−0.103996	−	0.994578i	$-0.533163\pi$
$383$	−4.07048	−0.207992	−0.103996	+	0.994578i	$0.533163\pi$
$384$	0	0
$385$	−1.77646	−0.0905369
$386$	0	0
$387$	−5.30868	−0.269855
$388$	0	0
$389$	17.3892	0.881668	0.440834	−	0.897589i	$-0.354683\pi$
$389$	17.3892	0.881668	0.440834	+	0.897589i	$0.354683\pi$
$390$	0	0
$391$	8.62856	0.436365
$392$	0	0
$393$	−41.7381	−2.10541
$394$	0	0
$395$	−24.5553	−1.23551
$396$	0	0
$397$	−2.39739	−0.120322	−0.0601608	−	0.998189i	$-0.519161\pi$
$397$	−2.39739	−0.120322	−0.0601608	+	0.998189i	$0.519161\pi$
$398$	0	0
$399$	−13.2626	−0.663962
$400$	0	0
$401$	1.57672	0.0787374	0.0393687	−	0.999225i	$-0.487465\pi$
$401$	1.57672	0.0787374	0.0393687	+	0.999225i	$0.487465\pi$
$402$	0	0
$403$	4.76106	0.237165
$404$	0	0
$405$	−15.9609	−0.793103
$406$	0	0
$407$	−2.65246	−0.131477
$408$	0	0
$409$	−26.2069	−1.29585	−0.647925	−	0.761705i	$-0.724363\pi$
$409$	−26.2069	−1.29585	−0.647925	+	0.761705i	$0.724363\pi$
$410$	0	0
$411$	−0.278751	−0.0137498
$412$	0	0
$413$	1.10886	0.0545632
$414$	0	0
$415$	5.09749	0.250226
$416$	0	0
$417$	6.13566	0.300465
$418$	0	0
$419$	−6.22271	−0.303999	−0.152000	−	0.988381i	$-0.548571\pi$
$419$	−6.22271	−0.303999	−0.152000	+	0.988381i	$0.548571\pi$
$420$	0	0
$421$	−0.301856	−0.0147116	−0.00735578	−	0.999973i	$-0.502341\pi$
$421$	−0.301856	−0.0147116	−0.00735578	+	0.999973i	$0.502341\pi$
$422$	0	0
$423$	−33.8576	−1.64621
$424$	0	0
$425$	4.56333	0.221354
$426$	0	0
$427$	3.44379	0.166656
$428$	0	0
$429$	−2.99972	−0.144828
$430$	0	0
$431$	−29.5793	−1.42478	−0.712392	−	0.701782i	$-0.752388\pi$
$431$	−29.5793	−1.42478	−0.712392	+	0.701782i	$0.752388\pi$
$432$	0	0
$433$	2.70338	0.129916	0.0649581	−	0.997888i	$-0.479309\pi$
$433$	2.70338	0.129916	0.0649581	+	0.997888i	$0.479309\pi$
$434$	0	0
$435$	34.9524	1.67584
$436$	0	0
$437$	15.4174	0.737512
$438$	0	0
$439$	−20.4462	−0.975843	−0.487921	−	0.872888i	$-0.662245\pi$
$439$	−20.4462	−0.975843	−0.487921	+	0.872888i	$0.662245\pi$
$440$	0	0
$441$	5.99829	0.285633
$442$	0	0
$443$	−9.67690	−0.459763	−0.229882	−	0.973219i	$-0.573834\pi$
$443$	−9.67690	−0.459763	−0.229882	+	0.973219i	$0.573834\pi$
$444$	0	0
$445$	−12.8997	−0.611503
$446$	0	0
$447$	−30.6690	−1.45059
$448$	0	0
$449$	−2.35385	−0.111085	−0.0555425	−	0.998456i	$-0.517689\pi$
$449$	−2.35385	−0.111085	−0.0555425	+	0.998456i	$0.517689\pi$
$450$	0	0
$451$	−6.12788	−0.288551
$452$	0	0
$453$	41.8232	1.96503
$454$	0	0
$455$	1.77646	0.0832819
$456$	0	0
$457$	−8.67968	−0.406018	−0.203009	−	0.979177i	$-0.565072\pi$
$457$	−8.67968	−0.406018	−0.203009	+	0.979177i	$0.565072\pi$
$458$	0	0
$459$	−22.2553	−1.03879
$460$	0	0
$461$	−10.4737	−0.487810	−0.243905	−	0.969799i	$-0.578429\pi$
$461$	−10.4737	−0.487810	−0.243905	+	0.969799i	$0.578429\pi$
$462$	0	0
$463$	24.7010	1.14795	0.573976	−	0.818872i	$-0.305400\pi$
$463$	24.7010	1.14795	0.573976	+	0.818872i	$0.305400\pi$
$464$	0	0
$465$	25.3711	1.17656
$466$	0	0
$467$	−18.7469	−0.867504	−0.433752	−	0.901032i	$-0.642811\pi$
$467$	−18.7469	−0.867504	−0.433752	+	0.901032i	$0.642811\pi$
$468$	0	0
$469$	8.10527	0.374267
$470$	0	0
$471$	44.3610	2.04404
$472$	0	0
$473$	−0.885031	−0.0406938
$474$	0	0
$475$	8.15367	0.374116
$476$	0	0
$477$	−32.8829	−1.50561
$478$	0	0
$479$	−25.2148	−1.15209	−0.576046	−	0.817417i	$-0.695405\pi$
$479$	−25.2148	−1.15209	−0.576046	+	0.817417i	$0.695405\pi$
$480$	0	0
$481$	2.65246	0.120942
$482$	0	0
$483$	−10.4602	−0.475956
$484$	0	0
$485$	−18.7697	−0.852289
$486$	0	0
$487$	−3.85207	−0.174554	−0.0872769	−	0.996184i	$-0.527816\pi$
$487$	−3.85207	−0.174554	−0.0872769	+	0.996184i	$0.527816\pi$
$488$	0	0
$489$	−38.6391	−1.74732
$490$	0	0
$491$	41.0917	1.85444	0.927220	−	0.374516i	$-0.122191\pi$
$491$	41.0917	1.85444	0.927220	+	0.374516i	$0.122191\pi$
$492$	0	0
$493$	16.2300	0.730964
$494$	0	0
$495$	−10.6557	−0.478940
$496$	0	0
$497$	−11.1486	−0.500081
$498$	0	0
$499$	27.2157	1.21834	0.609170	−	0.793039i	$-0.291503\pi$
$499$	27.2157	1.21834	0.609170	+	0.793039i	$0.291503\pi$
$500$	0	0
$501$	12.8551	0.574325
$502$	0	0
$503$	−16.4419	−0.733110	−0.366555	−	0.930396i	$-0.619463\pi$
$503$	−16.4419	−0.733110	−0.366555	+	0.930396i	$0.619463\pi$
$504$	0	0
$505$	2.96556	0.131966
$506$	0	0
$507$	2.99972	0.133222
$508$	0	0
$509$	−4.30745	−0.190924	−0.0954622	−	0.995433i	$-0.530433\pi$
$509$	−4.30745	−0.190924	−0.0954622	+	0.995433i	$0.530433\pi$
$510$	0	0
$511$	−6.25405	−0.276663
$512$	0	0
$513$	−39.7653	−1.75568
$514$	0	0
$515$	−6.51551	−0.287108
$516$	0	0
$517$	−5.64454	−0.248247
$518$	0	0
$519$	50.1539	2.20151
$520$	0	0
$521$	−36.9901	−1.62057	−0.810284	−	0.586038i	$-0.800687\pi$
$521$	−36.9901	−1.62057	−0.810284	+	0.586038i	$0.800687\pi$
$522$	0	0
$523$	38.9692	1.70400	0.852002	−	0.523539i	$-0.175388\pi$
$523$	38.9692	1.70400	0.852002	+	0.523539i	$0.175388\pi$
$524$	0	0
$525$	−5.53202	−0.241437
$526$	0	0
$527$	11.7810	0.513188
$528$	0	0
$529$	−10.8404	−0.471321
$530$	0	0
$531$	6.65124	0.288639
$532$	0	0
$533$	6.12788	0.265428
$534$	0	0
$535$	−1.15098	−0.0497614
$536$	0	0
$537$	61.9233	2.67219
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−14.3813	−0.618300	−0.309150	−	0.951013i	$-0.600044\pi$
$541$	−14.3813	−0.618300	−0.309150	+	0.951013i	$0.600044\pi$
$542$	0	0
$543$	−6.12409	−0.262810
$544$	0	0
$545$	22.5063	0.964066
$546$	0	0
$547$	−10.2535	−0.438409	−0.219205	−	0.975679i	$-0.570346\pi$
$547$	−10.2535	−0.438409	−0.219205	+	0.975679i	$0.570346\pi$
$548$	0	0
$549$	20.6568	0.881612
$550$	0	0
$551$	28.9995	1.23542
$552$	0	0
$553$	13.8226	0.587795
$554$	0	0
$555$	14.1346	0.599982
$556$	0	0
$557$	−31.5338	−1.33613	−0.668065	−	0.744103i	$-0.732877\pi$
$557$	−31.5338	−1.33613	−0.668065	+	0.744103i	$0.732877\pi$
$558$	0	0
$559$	0.885031	0.0374328
$560$	0	0
$561$	−7.42264	−0.313384
$562$	0	0
$563$	−45.4595	−1.91589	−0.957945	−	0.286952i	$-0.907358\pi$
$563$	−45.4595	−1.91589	−0.957945	+	0.286952i	$0.907358\pi$
$564$	0	0
$565$	−14.3063	−0.601871
$566$	0	0
$567$	8.98465	0.377320
$568$	0	0
$569$	−30.9265	−1.29651	−0.648254	−	0.761424i	$-0.724501\pi$
$569$	−30.9265	−1.29651	−0.648254	+	0.761424i	$0.724501\pi$
$570$	0	0
$571$	−26.6150	−1.11380	−0.556901	−	0.830579i	$-0.688010\pi$
$571$	−26.6150	−1.11380	−0.556901	+	0.830579i	$0.688010\pi$
$572$	0	0
$573$	−25.4891	−1.06482
$574$	0	0
$575$	6.43078	0.268182
$576$	0	0
$577$	−27.3660	−1.13926	−0.569631	−	0.821900i	$-0.692914\pi$
$577$	−27.3660	−1.13926	−0.569631	+	0.821900i	$0.692914\pi$
$578$	0	0
$579$	47.6583	1.98061
$580$	0	0
$581$	−2.86946	−0.119045
$582$	0	0
$583$	−5.48205	−0.227043
$584$	0	0
$585$	10.6557	0.440561
$586$	0	0
$587$	3.34466	0.138049	0.0690244	−	0.997615i	$-0.478011\pi$
$587$	3.34466	0.138049	0.0690244	+	0.997615i	$0.478011\pi$
$588$	0	0
$589$	21.0501	0.867352
$590$	0	0
$591$	−16.8872	−0.694645
$592$	0	0
$593$	−39.6660	−1.62889	−0.814444	−	0.580243i	$-0.802958\pi$
$593$	−39.6660	−1.62889	−0.814444	+	0.580243i	$0.802958\pi$
$594$	0	0
$595$	4.39576	0.180209
$596$	0	0
$597$	−16.1821	−0.662288
$598$	0	0
$599$	−44.3887	−1.81367	−0.906837	−	0.421482i	$-0.861510\pi$
$599$	−44.3887	−1.81367	−0.906837	+	0.421482i	$0.861510\pi$
$600$	0	0
$601$	−4.69558	−0.191537	−0.0957683	−	0.995404i	$-0.530531\pi$
$601$	−4.69558	−0.191537	−0.0957683	+	0.995404i	$0.530531\pi$
$602$	0	0
$603$	48.6178	1.97987
$604$	0	0
$605$	−1.77646	−0.0722235
$606$	0	0
$607$	29.9868	1.21713	0.608564	−	0.793505i	$-0.291746\pi$
$607$	29.9868	1.21713	0.608564	+	0.793505i	$0.291746\pi$
$608$	0	0
$609$	−19.6753	−0.797284
$610$	0	0
$611$	5.64454	0.228354
$612$	0	0
$613$	1.32213	0.0534001	0.0267001	−	0.999643i	$-0.491500\pi$
$613$	1.32213	0.0534001	0.0267001	+	0.999643i	$0.491500\pi$
$614$	0	0
$615$	32.6548	1.31677
$616$	0	0
$617$	11.9084	0.479413	0.239706	−	0.970845i	$-0.422949\pi$
$617$	11.9084	0.479413	0.239706	+	0.970845i	$0.422949\pi$
$618$	0	0
$619$	2.59522	0.104311	0.0521553	−	0.998639i	$-0.483391\pi$
$619$	2.59522	0.104311	0.0521553	+	0.998639i	$0.483391\pi$
$620$	0	0
$621$	−31.3628	−1.25854
$622$	0	0
$623$	7.26143	0.290923
$624$	0	0
$625$	−12.3781	−0.495124
$626$	0	0
$627$	−13.2626	−0.529659
$628$	0	0
$629$	6.56337	0.261699
$630$	0	0
$631$	13.3551	0.531658	0.265829	−	0.964020i	$-0.414354\pi$
$631$	13.3551	0.531658	0.265829	+	0.964020i	$0.414354\pi$
$632$	0	0
$633$	80.4343	3.19698
$634$	0	0
$635$	37.4760	1.48719
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−66.8723	−2.64543
$640$	0	0
$641$	26.2654	1.03742	0.518711	−	0.854950i	$-0.326412\pi$
$641$	26.2654	1.03742	0.518711	+	0.854950i	$0.326412\pi$
$642$	0	0
$643$	36.2923	1.43123	0.715614	−	0.698496i	$-0.246147\pi$
$643$	36.2923	1.43123	0.715614	+	0.698496i	$0.246147\pi$
$644$	0	0
$645$	4.71623	0.185701
$646$	0	0
$647$	45.7895	1.80017	0.900085	−	0.435714i	$-0.143504\pi$
$647$	45.7895	1.80017	0.900085	+	0.435714i	$0.143504\pi$
$648$	0	0
$649$	1.10886	0.0435264
$650$	0	0
$651$	−14.2818	−0.559749
$652$	0	0
$653$	−10.5377	−0.412374	−0.206187	−	0.978513i	$-0.566105\pi$
$653$	−10.5377	−0.412374	−0.206187	+	0.978513i	$0.566105\pi$
$654$	0	0
$655$	24.7177	0.965802
$656$	0	0
$657$	−37.5136	−1.46355
$658$	0	0
$659$	31.0801	1.21071	0.605355	−	0.795956i	$-0.293031\pi$
$659$	31.0801	1.21071	0.605355	+	0.795956i	$0.293031\pi$
$660$	0	0
$661$	42.8240	1.66566	0.832831	−	0.553528i	$-0.186719\pi$
$661$	42.8240	1.66566	0.832831	+	0.553528i	$0.186719\pi$
$662$	0	0
$663$	7.42264	0.288271
$664$	0	0
$665$	7.85427	0.304575
$666$	0	0
$667$	22.8719	0.885602
$668$	0	0
$669$	−37.7777	−1.46057
$670$	0	0
$671$	3.44379	0.132946
$672$	0	0
$673$	17.2056	0.663229	0.331614	−	0.943415i	$-0.392407\pi$
$673$	17.2056	0.663229	0.331614	+	0.943415i	$0.392407\pi$
$674$	0	0
$675$	−16.5866	−0.638419
$676$	0	0
$677$	8.13704	0.312732	0.156366	−	0.987699i	$-0.450022\pi$
$677$	8.13704	0.312732	0.156366	+	0.987699i	$0.450022\pi$
$678$	0	0
$679$	10.5658	0.405477
$680$	0	0
$681$	−62.1562	−2.38183
$682$	0	0
$683$	−17.2714	−0.660872	−0.330436	−	0.943828i	$-0.607196\pi$
$683$	−17.2714	−0.660872	−0.330436	+	0.943828i	$0.607196\pi$
$684$	0	0
$685$	0.165079	0.00630736
$686$	0	0
$687$	7.56696	0.288698
$688$	0	0
$689$	5.48205	0.208849
$690$	0	0
$691$	18.3633	0.698574	0.349287	−	0.937016i	$-0.386424\pi$
$691$	18.3633	0.698574	0.349287	+	0.937016i	$0.386424\pi$
$692$	0	0
$693$	5.99829	0.227856
$694$	0	0
$695$	−3.63360	−0.137830
$696$	0	0
$697$	15.1631	0.574345
$698$	0	0
$699$	76.2558	2.88426
$700$	0	0
$701$	−37.2881	−1.40835	−0.704177	−	0.710025i	$-0.748684\pi$
$701$	−37.2881	−1.40835	−0.704177	+	0.710025i	$0.748684\pi$
$702$	0	0
$703$	11.7273	0.442304
$704$	0	0
$705$	30.0791	1.13284
$706$	0	0
$707$	−1.66936	−0.0627829
$708$	0	0
$709$	36.8205	1.38282	0.691411	−	0.722461i	$-0.256989\pi$
$709$	36.8205	1.38282	0.691411	+	0.722461i	$0.256989\pi$
$710$	0	0
$711$	82.9118	3.10943
$712$	0	0
$713$	16.6021	0.621754
$714$	0	0
$715$	1.77646	0.0664359
$716$	0	0
$717$	4.05140	0.151302
$718$	0	0
$719$	−15.8687	−0.591803	−0.295902	−	0.955218i	$-0.595620\pi$
$719$	−15.8687	−0.591803	−0.295902	+	0.955218i	$0.595620\pi$
$720$	0	0
$721$	3.66769	0.136592
$722$	0	0
$723$	−65.8492	−2.44896
$724$	0	0
$725$	12.0961	0.449237
$726$	0	0
$727$	−14.3172	−0.530994	−0.265497	−	0.964112i	$-0.585536\pi$
$727$	−14.3172	−0.530994	−0.265497	+	0.964112i	$0.585536\pi$
$728$	0	0
$729$	−27.0460	−1.00170
$730$	0	0
$731$	2.18996	0.0809988
$732$	0	0
$733$	−40.0784	−1.48033	−0.740166	−	0.672425i	$-0.765253\pi$
$733$	−40.0784	−1.48033	−0.740166	+	0.672425i	$0.765253\pi$
$734$	0	0
$735$	−5.32888	−0.196559
$736$	0	0
$737$	8.10527	0.298561
$738$	0	0
$739$	20.1723	0.742049	0.371024	−	0.928623i	$-0.379007\pi$
$739$	20.1723	0.742049	0.371024	+	0.928623i	$0.379007\pi$
$740$	0	0
$741$	13.2626	0.487215
$742$	0	0
$743$	−28.1864	−1.03406	−0.517029	−	0.855968i	$-0.672962\pi$
$743$	−28.1864	−1.03406	−0.517029	+	0.855968i	$0.672962\pi$
$744$	0	0
$745$	18.1625	0.665422
$746$	0	0
$747$	−17.2119	−0.629749
$748$	0	0
$749$	0.647908	0.0236740
$750$	0	0
$751$	−24.9873	−0.911799	−0.455900	−	0.890031i	$-0.650682\pi$
$751$	−24.9873	−0.911799	−0.455900	+	0.890031i	$0.650682\pi$
$752$	0	0
$753$	41.0479	1.49587
$754$	0	0
$755$	−24.7682	−0.901406
$756$	0	0
$757$	29.8409	1.08459	0.542294	−	0.840189i	$-0.317556\pi$
$757$	29.8409	1.08459	0.542294	+	0.840189i	$0.317556\pi$
$758$	0	0
$759$	−10.4602	−0.379681
$760$	0	0
$761$	7.93547	0.287661	0.143830	−	0.989602i	$-0.454058\pi$
$761$	7.93547	0.287661	0.143830	+	0.989602i	$0.454058\pi$
$762$	0	0
$763$	−12.6692	−0.458655
$764$	0	0
$765$	26.3671	0.953304
$766$	0	0
$767$	−1.10886	−0.0400384
$768$	0	0
$769$	11.5847	0.417756	0.208878	−	0.977942i	$-0.433019\pi$
$769$	11.5847	0.417756	0.208878	+	0.977942i	$0.433019\pi$
$770$	0	0
$771$	49.9000	1.79710
$772$	0	0
$773$	−17.0930	−0.614792	−0.307396	−	0.951582i	$-0.599458\pi$
$773$	−17.0930	−0.614792	−0.307396	+	0.951582i	$0.599458\pi$
$774$	0	0
$775$	8.78025	0.315396
$776$	0	0
$777$	−7.95662	−0.285442
$778$	0	0
$779$	27.0932	0.970714
$780$	0	0
$781$	−11.1486	−0.398927
$782$	0	0
$783$	−58.9923	−2.10821
$784$	0	0
$785$	−26.2710	−0.937653
$786$	0	0
$787$	−1.77350	−0.0632184	−0.0316092	−	0.999500i	$-0.510063\pi$
$787$	−1.77350	−0.0632184	−0.0316092	+	0.999500i	$0.510063\pi$
$788$	0	0
$789$	−3.15580	−0.112349
$790$	0	0
$791$	8.05325	0.286341
$792$	0	0
$793$	−3.44379	−0.122292
$794$	0	0
$795$	29.2132	1.03609
$796$	0	0
$797$	30.4367	1.07812	0.539061	−	0.842266i	$-0.318779\pi$
$797$	30.4367	1.07812	0.539061	+	0.842266i	$0.318779\pi$
$798$	0	0
$799$	13.9671	0.494121
$800$	0	0
$801$	43.5562	1.53898
$802$	0	0
$803$	−6.25405	−0.220701
$804$	0	0
$805$	6.19464	0.218332
$806$	0	0
$807$	34.7243	1.22235
$808$	0	0
$809$	33.0368	1.16151	0.580756	−	0.814077i	$-0.302757\pi$
$809$	33.0368	1.16151	0.580756	+	0.814077i	$0.302757\pi$
$810$	0	0
$811$	37.7359	1.32509	0.662543	−	0.749023i	$-0.269477\pi$
$811$	37.7359	1.32509	0.662543	+	0.749023i	$0.269477\pi$
$812$	0	0
$813$	21.1284	0.741004
$814$	0	0
$815$	22.8825	0.801538
$816$	0	0
$817$	3.91299	0.136898
$818$	0	0
$819$	−5.99829	−0.209597
$820$	0	0
$821$	−33.7092	−1.17646	−0.588230	−	0.808694i	$-0.700175\pi$
$821$	−33.7092	−1.17646	−0.588230	+	0.808694i	$0.700175\pi$
$822$	0	0
$823$	6.59243	0.229798	0.114899	−	0.993377i	$-0.463346\pi$
$823$	6.59243	0.229798	0.114899	+	0.993377i	$0.463346\pi$
$824$	0	0
$825$	−5.53202	−0.192600
$826$	0	0
$827$	48.1980	1.67601	0.838005	−	0.545663i	$-0.183722\pi$
$827$	48.1980	1.67601	0.838005	+	0.545663i	$0.183722\pi$
$828$	0	0
$829$	48.5004	1.68449	0.842244	−	0.539097i	$-0.181234\pi$
$829$	48.5004	1.68449	0.842244	+	0.539097i	$0.181234\pi$
$830$	0	0
$831$	7.27936	0.252518
$832$	0	0
$833$	−2.47445	−0.0857345
$834$	0	0
$835$	−7.61294	−0.263457
$836$	0	0
$837$	−42.8211	−1.48011
$838$	0	0
$839$	9.39621	0.324393	0.162197	−	0.986758i	$-0.448142\pi$
$839$	9.39621	0.324393	0.162197	+	0.986758i	$0.448142\pi$
$840$	0	0
$841$	14.0212	0.483491
$842$	0	0
$843$	76.1094	2.62135
$844$	0	0
$845$	−1.77646	−0.0611122
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	15.9685	0.548039
$850$	0	0
$851$	9.24929	0.317062
$852$	0	0
$853$	13.7796	0.471806	0.235903	−	0.971777i	$-0.424195\pi$
$853$	13.7796	0.471806	0.235903	+	0.971777i	$0.424195\pi$
$854$	0	0
$855$	47.1122	1.61120
$856$	0	0
$857$	−15.1690	−0.518162	−0.259081	−	0.965856i	$-0.583420\pi$
$857$	−15.1690	−0.518162	−0.259081	+	0.965856i	$0.583420\pi$
$858$	0	0
$859$	−22.2357	−0.758671	−0.379335	−	0.925259i	$-0.623847\pi$
$859$	−22.2357	−0.758671	−0.379335	+	0.925259i	$0.623847\pi$
$860$	0	0
$861$	−18.3819	−0.626454
$862$	0	0
$863$	−1.02578	−0.0349180	−0.0174590	−	0.999848i	$-0.505558\pi$
$863$	−1.02578	−0.0349180	−0.0174590	+	0.999848i	$0.505558\pi$
$864$	0	0
$865$	−29.7016	−1.00989
$866$	0	0
$867$	−32.6282	−1.10811
$868$	0	0
$869$	13.8226	0.468898
$870$	0	0
$871$	−8.10527	−0.274637
$872$	0	0
$873$	63.3767	2.14497
$874$	0	0
$875$	12.1584	0.411030
$876$	0	0
$877$	−15.3514	−0.518380	−0.259190	−	0.965826i	$-0.583456\pi$
$877$	−15.3514	−0.518380	−0.259190	+	0.965826i	$0.583456\pi$
$878$	0	0
$879$	−64.8761	−2.18822
$880$	0	0
$881$	12.5377	0.422404	0.211202	−	0.977442i	$-0.432262\pi$
$881$	12.5377	0.422404	0.211202	+	0.977442i	$0.432262\pi$
$882$	0	0
$883$	22.7645	0.766086	0.383043	−	0.923731i	$-0.374876\pi$
$883$	22.7645	0.766086	0.383043	+	0.923731i	$0.374876\pi$
$884$	0	0
$885$	−5.90896	−0.198628
$886$	0	0
$887$	−4.92096	−0.165230	−0.0826148	−	0.996582i	$-0.526327\pi$
$887$	−4.92096	−0.165230	−0.0826148	+	0.996582i	$0.526327\pi$
$888$	0	0
$889$	−21.0959	−0.707532
$890$	0	0
$891$	8.98465	0.300997
$892$	0	0
$893$	24.9562	0.835127
$894$	0	0
$895$	−36.6716	−1.22580
$896$	0	0
$897$	10.4602	0.349256
$898$	0	0
$899$	31.2281	1.04151
$900$	0	0
$901$	13.5650	0.451917
$902$	0	0
$903$	−2.65484	−0.0883476
$904$	0	0
$905$	3.62675	0.120557
$906$	0	0
$907$	41.8327	1.38903	0.694516	−	0.719477i	$-0.255619\pi$
$907$	41.8327	1.38903	0.694516	+	0.719477i	$0.255619\pi$
$908$	0	0
$909$	−10.0133	−0.332121
$910$	0	0
$911$	42.6738	1.41385	0.706923	−	0.707290i	$-0.250083\pi$
$911$	42.6738	1.41385	0.706923	+	0.707290i	$0.250083\pi$
$912$	0	0
$913$	−2.86946	−0.0949653
$914$	0	0
$915$	−18.3515	−0.606683
$916$	0	0
$917$	−13.9140	−0.459481
$918$	0	0
$919$	−35.2222	−1.16187	−0.580937	−	0.813949i	$-0.697314\pi$
$919$	−35.2222	−1.16187	−0.580937	+	0.813949i	$0.697314\pi$
$920$	0	0
$921$	−80.7099	−2.65948
$922$	0	0
$923$	11.1486	0.366959
$924$	0	0
$925$	4.89161	0.160835
$926$	0	0
$927$	21.9999	0.722570
$928$	0	0
$929$	−28.7563	−0.943463	−0.471732	−	0.881742i	$-0.656371\pi$
$929$	−28.7563	−0.943463	−0.471732	+	0.881742i	$0.656371\pi$
$930$	0	0
$931$	−4.42130	−0.144902
$932$	0	0
$933$	−90.2890	−2.95593
$934$	0	0
$935$	4.39576	0.143757
$936$	0	0
$937$	−29.4909	−0.963424	−0.481712	−	0.876329i	$-0.659985\pi$
$937$	−29.4909	−0.963424	−0.481712	+	0.876329i	$0.659985\pi$
$938$	0	0
$939$	73.0869	2.38510
$940$	0	0
$941$	−21.2020	−0.691165	−0.345583	−	0.938388i	$-0.612319\pi$
$941$	−21.2020	−0.691165	−0.345583	+	0.938388i	$0.612319\pi$
$942$	0	0
$943$	21.3683	0.695849
$944$	0	0
$945$	−15.9776	−0.519750
$946$	0	0
$947$	27.2940	0.886936	0.443468	−	0.896290i	$-0.353748\pi$
$947$	27.2940	0.886936	0.443468	+	0.896290i	$0.353748\pi$
$948$	0	0
$949$	6.25405	0.203015
$950$	0	0
$951$	25.8802	0.839222
$952$	0	0
$953$	1.06436	0.0344779	0.0172390	−	0.999851i	$-0.494512\pi$
$953$	1.06436	0.0344779	0.0172390	+	0.999851i	$0.494512\pi$
$954$	0	0
$955$	15.0949	0.488459
$956$	0	0
$957$	−19.6753	−0.636012
$958$	0	0
$959$	−0.0929259	−0.00300073
$960$	0	0
$961$	−8.33232	−0.268785
$962$	0	0
$963$	3.88634	0.125236
$964$	0	0
$965$	−28.2237	−0.908553
$966$	0	0
$967$	34.3547	1.10477	0.552386	−	0.833588i	$-0.313717\pi$
$967$	34.3547	1.10477	0.552386	+	0.833588i	$0.313717\pi$
$968$	0	0
$969$	32.8177	1.05426
$970$	0	0
$971$	−18.2292	−0.585004	−0.292502	−	0.956265i	$-0.594488\pi$
$971$	−18.2292	−0.585004	−0.292502	+	0.956265i	$0.594488\pi$
$972$	0	0
$973$	2.04542	0.0655730
$974$	0	0
$975$	5.53202	0.177166
$976$	0	0
$977$	−31.1762	−0.997415	−0.498707	−	0.866770i	$-0.666192\pi$
$977$	−31.1762	−0.997415	−0.498707	+	0.866770i	$0.666192\pi$
$978$	0	0
$979$	7.26143	0.232076
$980$	0	0
$981$	−75.9935	−2.42629
$982$	0	0
$983$	−57.3358	−1.82873	−0.914364	−	0.404894i	$-0.867308\pi$
$983$	−57.3358	−1.82873	−0.914364	+	0.404894i	$0.867308\pi$
$984$	0	0
$985$	10.0008	0.318651
$986$	0	0
$987$	−16.9320	−0.538952
$988$	0	0
$989$	3.08616	0.0981342
$990$	0	0
$991$	16.2523	0.516272	0.258136	−	0.966109i	$-0.416892\pi$
$991$	16.2523	0.516272	0.258136	+	0.966109i	$0.416892\pi$
$992$	0	0
$993$	−25.0918	−0.796263
$994$	0	0
$995$	9.58319	0.303808
$996$	0	0
$997$	6.82077	0.216016	0.108008	−	0.994150i	$-0.465553\pi$
$997$	6.82077	0.216016	0.108008	+	0.994150i	$0.465553\pi$
$998$	0	0
$999$	−23.8563	−0.754780

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.s.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.s.1.10	✓	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)

\(f(q)\)	\(=\)	\(q+2.99972 q^{3} -1.77646 q^{5} +1.00000 q^{7} +5.99829 q^{9} +O(q^{10})\)	\(q+2.99972 q^{3} -1.77646 q^{5} +1.00000 q^{7} +5.99829 q^{9} +1.00000 q^{11} -1.00000 q^{13} -5.32888 q^{15} -2.47445 q^{17} -4.42130 q^{19} +2.99972 q^{21} -3.48707 q^{23} -1.84418 q^{25} +8.99403 q^{27} -6.55906 q^{29} -4.76106 q^{31} +2.99972 q^{33} -1.77646 q^{35} -2.65246 q^{37} -2.99972 q^{39} -6.12788 q^{41} -0.885031 q^{43} -10.6557 q^{45} -5.64454 q^{47} +1.00000 q^{49} -7.42264 q^{51} -5.48205 q^{53} -1.77646 q^{55} -13.2626 q^{57} +1.10886 q^{59} +3.44379 q^{61} +5.99829 q^{63} +1.77646 q^{65} +8.10527 q^{67} -10.4602 q^{69} -11.1486 q^{71} -6.25405 q^{73} -5.53202 q^{75} +1.00000 q^{77} +13.8226 q^{79} +8.98465 q^{81} -2.86946 q^{83} +4.39576 q^{85} -19.6753 q^{87} +7.26143 q^{89} -1.00000 q^{91} -14.2818 q^{93} +7.85427 q^{95} +10.5658 q^{97} +5.99829 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 9 * q^9 + 10 * q^11 - 10 * q^13 - 5 * q^15 - 11 * q^17 + 2 * q^19 - 3 * q^21 - 8 * q^23 + 2 * q^25 - 15 * q^27 - 8 * q^29 - 23 * q^31 - 3 * q^33 - 4 * q^35 + 10 * q^37 + 3 * q^39 - 18 * q^41 + 12 * q^43 - 10 * q^45 - 36 * q^47 + 10 * q^49 + 9 * q^51 - 21 * q^53 - 4 * q^55 - 30 * q^57 - 13 * q^59 - 2 * q^61 + 9 * q^63 + 4 * q^65 - 4 * q^67 - 26 * q^69 - 24 * q^71 - 23 * q^73 - 28 * q^75 + 10 * q^77 + 14 * q^79 + 30 * q^81 - 9 * q^83 - 17 * q^85 + 7 * q^87 - 18 * q^89 - 10 * q^91 + q^93 - 4 * q^95 - 9 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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