LMFDB - Embedded newform 8008.2.a.s.1.1

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			1.1
Root			$3.44977$ 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)

$f(q)$	$=$	$q-3.44977 q^{3} +0.464152 q^{5} +1.00000 q^{7} +8.90091 q^{9} +O(q^{10})$	$q-3.44977 q^{3} +0.464152 q^{5} +1.00000 q^{7} +8.90091 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.60122 q^{15} -4.01824 q^{17} +6.85112 q^{19} -3.44977 q^{21} +3.62707 q^{23} -4.78456 q^{25} -20.3568 q^{27} -1.88820 q^{29} +5.04704 q^{31} -3.44977 q^{33} +0.464152 q^{35} +5.77398 q^{37} +3.44977 q^{39} -8.56773 q^{41} +8.51421 q^{43} +4.13138 q^{45} -10.2560 q^{47} +1.00000 q^{49} +13.8620 q^{51} -13.0695 q^{53} +0.464152 q^{55} -23.6348 q^{57} -14.3325 q^{59} +7.95444 q^{61} +8.90091 q^{63} -0.464152 q^{65} -1.54325 q^{67} -12.5126 q^{69} -7.47512 q^{71} -15.9923 q^{73} +16.5056 q^{75} +1.00000 q^{77} +6.34164 q^{79} +43.5235 q^{81} -0.125744 q^{83} -1.86508 q^{85} +6.51385 q^{87} -15.4457 q^{89} -1.00000 q^{91} -17.4111 q^{93} +3.17996 q^{95} +3.98079 q^{97} +8.90091 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.44977	−1.99173	−0.995863	−	0.0908701i	$-0.971035\pi$
$3$	−3.44977	−1.99173	−0.995863	+	0.0908701i	$0.971035\pi$
$4$	0	0
$5$	0.464152	0.207575	0.103788	−	0.994599i	$-0.466904\pi$
$5$	0.464152	0.207575	0.103788	+	0.994599i	$0.466904\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.90091	2.96697
$10$	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
$15$	−1.60122	−0.413433
$16$	0	0
$17$	−4.01824	−0.974567	−0.487284	−	0.873244i	$-0.662012\pi$
$17$	−4.01824	−0.974567	−0.487284	+	0.873244i	$0.662012\pi$
$18$	0	0
$19$	6.85112	1.57176	0.785878	−	0.618382i	$-0.212211\pi$
$19$	6.85112	1.57176	0.785878	+	0.618382i	$0.212211\pi$
$20$	0	0
$21$	−3.44977	−0.752801
$22$	0	0
$23$	3.62707	0.756297	0.378148	−	0.925745i	$-0.376561\pi$
$23$	3.62707	0.756297	0.378148	+	0.925745i	$0.376561\pi$
$24$	0	0
$25$	−4.78456	−0.956913
$26$	0	0
$27$	−20.3568	−3.91767
$28$	0	0
$29$	−1.88820	−0.350630	−0.175315	−	0.984512i	$-0.556094\pi$
$29$	−1.88820	−0.350630	−0.175315	+	0.984512i	$0.556094\pi$
$30$	0	0
$31$	5.04704	0.906475	0.453238	−	0.891390i	$-0.350269\pi$
$31$	5.04704	0.906475	0.453238	+	0.891390i	$0.350269\pi$
$32$	0	0
$33$	−3.44977	−0.600528
$34$	0	0
$35$	0.464152	0.0784560
$36$	0	0
$37$	5.77398	0.949237	0.474618	−	0.880192i	$-0.342586\pi$
$37$	5.77398	0.949237	0.474618	+	0.880192i	$0.342586\pi$
$38$	0	0
$39$	3.44977	0.552405
$40$	0	0
$41$	−8.56773	−1.33806	−0.669028	−	0.743238i	$-0.733289\pi$
$41$	−8.56773	−1.33806	−0.669028	+	0.743238i	$0.733289\pi$
$42$	0	0
$43$	8.51421	1.29840	0.649202	−	0.760616i	$-0.275103\pi$
$43$	8.51421	1.29840	0.649202	+	0.760616i	$0.275103\pi$
$44$	0	0
$45$	4.13138	0.615869
$46$	0	0
$47$	−10.2560	−1.49599	−0.747993	−	0.663707i	$-0.768982\pi$
$47$	−10.2560	−1.49599	−0.747993	+	0.663707i	$0.768982\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.8620	1.94107
$52$	0	0
$53$	−13.0695	−1.79524	−0.897619	−	0.440772i	$-0.854705\pi$
$53$	−13.0695	−1.79524	−0.897619	+	0.440772i	$0.854705\pi$
$54$	0	0
$55$	0.464152	0.0625862
$56$	0	0
$57$	−23.6348	−3.13051
$58$	0	0
$59$	−14.3325	−1.86593	−0.932967	−	0.359962i	$-0.882790\pi$
$59$	−14.3325	−1.86593	−0.932967	+	0.359962i	$0.882790\pi$
$60$	0	0
$61$	7.95444	1.01846	0.509231	−	0.860630i	$-0.329930\pi$
$61$	7.95444	1.01846	0.509231	+	0.860630i	$0.329930\pi$
$62$	0	0
$63$	8.90091	1.12141
$64$	0	0
$65$	−0.464152	−0.0575710
$66$	0	0
$67$	−1.54325	−0.188538	−0.0942691	−	0.995547i	$-0.530051\pi$
$67$	−1.54325	−0.188538	−0.0942691	+	0.995547i	$0.530051\pi$
$68$	0	0
$69$	−12.5126	−1.50634
$70$	0	0
$71$	−7.47512	−0.887133	−0.443567	−	0.896241i	$-0.646287\pi$
$71$	−7.47512	−0.887133	−0.443567	+	0.896241i	$0.646287\pi$
$72$	0	0
$73$	−15.9923	−1.87175	−0.935877	−	0.352328i	$-0.885390\pi$
$73$	−15.9923	−1.87175	−0.935877	+	0.352328i	$0.885390\pi$
$74$	0	0
$75$	16.5056	1.90591
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	6.34164	0.713490	0.356745	−	0.934202i	$-0.383886\pi$
$79$	6.34164	0.713490	0.356745	+	0.934202i	$0.383886\pi$
$80$	0	0
$81$	43.5235	4.83594
$82$	0	0
$83$	−0.125744	−0.0138022	−0.00690108	−	0.999976i	$-0.502197\pi$
$83$	−0.125744	−0.0138022	−0.00690108	+	0.999976i	$0.502197\pi$
$84$	0	0
$85$	−1.86508	−0.202296
$86$	0	0
$87$	6.51385	0.698358
$88$	0	0
$89$	−15.4457	−1.63724	−0.818622	−	0.574333i	$-0.805261\pi$
$89$	−15.4457	−1.63724	−0.818622	+	0.574333i	$0.805261\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−17.4111	−1.80545
$94$	0	0
$95$	3.17996	0.326257
$96$	0	0
$97$	3.98079	0.404188	0.202094	−	0.979366i	$-0.435225\pi$
$97$	3.98079	0.404188	0.202094	+	0.979366i	$0.435225\pi$
$98$	0	0
$99$	8.90091	0.894575
$100$	0	0
$101$	7.28577	0.724961	0.362481	−	0.931991i	$-0.381930\pi$
$101$	7.28577	0.724961	0.362481	+	0.931991i	$0.381930\pi$
$102$	0	0
$103$	15.8439	1.56114	0.780571	−	0.625067i	$-0.214928\pi$
$103$	15.8439	1.56114	0.780571	+	0.625067i	$0.214928\pi$
$104$	0	0
$105$	−1.60122	−0.156263
$106$	0	0
$107$	−1.86181	−0.179988	−0.0899939	−	0.995942i	$-0.528685\pi$
$107$	−1.86181	−0.179988	−0.0899939	+	0.995942i	$0.528685\pi$
$108$	0	0
$109$	13.0226	1.24734	0.623670	−	0.781688i	$-0.285641\pi$
$109$	13.0226	1.24734	0.623670	+	0.781688i	$0.285641\pi$
$110$	0	0
$111$	−19.9189	−1.89062
$112$	0	0
$113$	−15.4808	−1.45632	−0.728158	−	0.685410i	$-0.759623\pi$
$113$	−15.4808	−1.45632	−0.728158	+	0.685410i	$0.759623\pi$
$114$	0	0
$115$	1.68351	0.156988
$116$	0	0
$117$	−8.90091	−0.822890
$118$	0	0
$119$	−4.01824	−0.368352
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	29.5567	2.66504
$124$	0	0
$125$	−4.54152	−0.406206
$126$	0	0
$127$	9.89830	0.878332	0.439166	−	0.898406i	$-0.355274\pi$
$127$	9.89830	0.878332	0.439166	+	0.898406i	$0.355274\pi$
$128$	0	0
$129$	−29.3721	−2.58607
$130$	0	0
$131$	16.6398	1.45383	0.726913	−	0.686729i	$-0.240954\pi$
$131$	16.6398	1.45383	0.726913	+	0.686729i	$0.240954\pi$
$132$	0	0
$133$	6.85112	0.594068
$134$	0	0
$135$	−9.44864	−0.813210
$136$	0	0
$137$	−7.96239	−0.680273	−0.340137	−	0.940376i	$-0.610473\pi$
$137$	−7.96239	−0.680273	−0.340137	+	0.940376i	$0.610473\pi$
$138$	0	0
$139$	−9.21128	−0.781290	−0.390645	−	0.920541i	$-0.627748\pi$
$139$	−9.21128	−0.781290	−0.390645	+	0.920541i	$0.627748\pi$
$140$	0	0
$141$	35.3807	2.97959
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−0.876411	−0.0727820
$146$	0	0
$147$	−3.44977	−0.284532
$148$	0	0
$149$	8.63918	0.707749	0.353875	−	0.935293i	$-0.384864\pi$
$149$	8.63918	0.707749	0.353875	+	0.935293i	$0.384864\pi$
$150$	0	0
$151$	−12.2770	−0.999085	−0.499542	−	0.866289i	$-0.666499\pi$
$151$	−12.2770	−0.999085	−0.499542	+	0.866289i	$0.666499\pi$
$152$	0	0
$153$	−35.7660	−2.89151
$154$	0	0
$155$	2.34259	0.188162
$156$	0	0
$157$	9.05219	0.722443	0.361222	−	0.932480i	$-0.382360\pi$
$157$	9.05219	0.722443	0.361222	+	0.932480i	$0.382360\pi$
$158$	0	0
$159$	45.0869	3.57562
$160$	0	0
$161$	3.62707	0.285853
$162$	0	0
$163$	−18.8656	−1.47767	−0.738835	−	0.673886i	$-0.764624\pi$
$163$	−18.8656	−1.47767	−0.738835	+	0.673886i	$0.764624\pi$
$164$	0	0
$165$	−1.60122	−0.124655
$166$	0	0
$167$	18.6142	1.44041	0.720203	−	0.693763i	$-0.244048\pi$
$167$	18.6142	1.44041	0.720203	+	0.693763i	$0.244048\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	60.9812	4.66335
$172$	0	0
$173$	5.95191	0.452516	0.226258	−	0.974067i	$-0.427351\pi$
$173$	5.95191	0.452516	0.226258	+	0.974067i	$0.427351\pi$
$174$	0	0
$175$	−4.78456	−0.361679
$176$	0	0
$177$	49.4439	3.71643
$178$	0	0
$179$	11.6682	0.872125	0.436063	−	0.899916i	$-0.356373\pi$
$179$	11.6682	0.872125	0.436063	+	0.899916i	$0.356373\pi$
$180$	0	0
$181$	−11.7379	−0.872468	−0.436234	−	0.899833i	$-0.643688\pi$
$181$	−11.7379	−0.872468	−0.436234	+	0.899833i	$0.643688\pi$
$182$	0	0
$183$	−27.4410	−2.02850
$184$	0	0
$185$	2.68000	0.197038
$186$	0	0
$187$	−4.01824	−0.293843
$188$	0	0
$189$	−20.3568	−1.48074
$190$	0	0
$191$	6.00661	0.434623	0.217311	−	0.976102i	$-0.430271\pi$
$191$	6.00661	0.434623	0.217311	+	0.976102i	$0.430271\pi$
$192$	0	0
$193$	−0.121344	−0.00873452	−0.00436726	−	0.999990i	$-0.501390\pi$
$193$	−0.121344	−0.00873452	−0.00436726	+	0.999990i	$0.501390\pi$
$194$	0	0
$195$	1.60122	0.114666
$196$	0	0
$197$	4.49732	0.320421	0.160210	−	0.987083i	$-0.448783\pi$
$197$	4.49732	0.320421	0.160210	+	0.987083i	$0.448783\pi$
$198$	0	0
$199$	−1.53053	−0.108497	−0.0542483	−	0.998527i	$-0.517276\pi$
$199$	−1.53053	−0.108497	−0.0542483	+	0.998527i	$0.517276\pi$
$200$	0	0
$201$	5.32386	0.375516
$202$	0	0
$203$	−1.88820	−0.132526
$204$	0	0
$205$	−3.97673	−0.277747
$206$	0	0
$207$	32.2842	2.24391
$208$	0	0
$209$	6.85112	0.473902
$210$	0	0
$211$	10.2398	0.704937	0.352468	−	0.935824i	$-0.385342\pi$
$211$	10.2398	0.704937	0.352468	+	0.935824i	$0.385342\pi$
$212$	0	0
$213$	25.7874	1.76693
$214$	0	0
$215$	3.95189	0.269516
$216$	0	0
$217$	5.04704	0.342615
$218$	0	0
$219$	55.1696	3.72802
$220$	0	0
$221$	4.01824	0.270296
$222$	0	0
$223$	−10.6982	−0.716403	−0.358202	−	0.933644i	$-0.616610\pi$
$223$	−10.6982	−0.716403	−0.358202	+	0.933644i	$0.616610\pi$
$224$	0	0
$225$	−42.5870	−2.83913
$226$	0	0
$227$	7.61811	0.505632	0.252816	−	0.967514i	$-0.418643\pi$
$227$	7.61811	0.505632	0.252816	+	0.967514i	$0.418643\pi$
$228$	0	0
$229$	−4.17177	−0.275678	−0.137839	−	0.990455i	$-0.544016\pi$
$229$	−4.17177	−0.275678	−0.137839	+	0.990455i	$0.544016\pi$
$230$	0	0
$231$	−3.44977	−0.226978
$232$	0	0
$233$	12.3332	0.807973	0.403986	−	0.914765i	$-0.367624\pi$
$233$	12.3332	0.807973	0.403986	+	0.914765i	$0.367624\pi$
$234$	0	0
$235$	−4.76033	−0.310529
$236$	0	0
$237$	−21.8772	−1.42108
$238$	0	0
$239$	24.5462	1.58776	0.793882	−	0.608072i	$-0.208057\pi$
$239$	24.5462	1.58776	0.793882	+	0.608072i	$0.208057\pi$
$240$	0	0
$241$	−12.4143	−0.799673	−0.399836	−	0.916587i	$-0.630933\pi$
$241$	−12.4143	−0.799673	−0.399836	+	0.916587i	$0.630933\pi$
$242$	0	0
$243$	−89.0757	−5.71421
$244$	0	0
$245$	0.464152	0.0296536
$246$	0	0
$247$	−6.85112	−0.435927
$248$	0	0
$249$	0.433787	0.0274901
$250$	0	0
$251$	15.0779	0.951711	0.475855	−	0.879523i	$-0.342139\pi$
$251$	15.0779	0.951711	0.475855	+	0.879523i	$0.342139\pi$
$252$	0	0
$253$	3.62707	0.228032
$254$	0	0
$255$	6.43408	0.402918
$256$	0	0
$257$	−7.55359	−0.471180	−0.235590	−	0.971853i	$-0.575702\pi$
$257$	−7.55359	−0.471180	−0.235590	+	0.971853i	$0.575702\pi$
$258$	0	0
$259$	5.77398	0.358778
$260$	0	0
$261$	−16.8067	−1.04031
$262$	0	0
$263$	−14.8281	−0.914339	−0.457170	−	0.889380i	$-0.651137\pi$
$263$	−14.8281	−0.914339	−0.457170	+	0.889380i	$0.651137\pi$
$264$	0	0
$265$	−6.06625	−0.372647
$266$	0	0
$267$	53.2842	3.26094
$268$	0	0
$269$	−28.5276	−1.73936	−0.869680	−	0.493616i	$-0.835675\pi$
$269$	−28.5276	−1.73936	−0.869680	+	0.493616i	$0.835675\pi$
$270$	0	0
$271$	−23.9477	−1.45472	−0.727360	−	0.686256i	$-0.759253\pi$
$271$	−23.9477	−1.45472	−0.727360	+	0.686256i	$0.759253\pi$
$272$	0	0
$273$	3.44977	0.208790
$274$	0	0
$275$	−4.78456	−0.288520
$276$	0	0
$277$	−12.8081	−0.769563	−0.384782	−	0.923008i	$-0.625723\pi$
$277$	−12.8081	−0.769563	−0.384782	+	0.923008i	$0.625723\pi$
$278$	0	0
$279$	44.9233	2.68948
$280$	0	0
$281$	−19.5408	−1.16571	−0.582854	−	0.812577i	$-0.698064\pi$
$281$	−19.5408	−1.16571	−0.582854	+	0.812577i	$0.698064\pi$
$282$	0	0
$283$	2.09133	0.124317	0.0621584	−	0.998066i	$-0.480202\pi$
$283$	2.09133	0.124317	0.0621584	+	0.998066i	$0.480202\pi$
$284$	0	0
$285$	−10.9701	−0.649815
$286$	0	0
$287$	−8.56773	−0.505737
$288$	0	0
$289$	−0.853724	−0.0502191
$290$	0	0
$291$	−13.7328	−0.805031
$292$	0	0
$293$	7.71077	0.450468	0.225234	−	0.974305i	$-0.427685\pi$
$293$	7.71077	0.450468	0.225234	+	0.974305i	$0.427685\pi$
$294$	0	0
$295$	−6.65246	−0.387321
$296$	0	0
$297$	−20.3568	−1.18122
$298$	0	0
$299$	−3.62707	−0.209759
$300$	0	0
$301$	8.51421	0.490751
$302$	0	0
$303$	−25.1342	−1.44392
$304$	0	0
$305$	3.69207	0.211407
$306$	0	0
$307$	−20.1550	−1.15031	−0.575154	−	0.818045i	$-0.695058\pi$
$307$	−20.1550	−1.15031	−0.575154	+	0.818045i	$0.695058\pi$
$308$	0	0
$309$	−54.6577	−3.10937
$310$	0	0
$311$	−21.7429	−1.23293	−0.616463	−	0.787384i	$-0.711435\pi$
$311$	−21.7429	−1.23293	−0.616463	+	0.787384i	$0.711435\pi$
$312$	0	0
$313$	−12.4448	−0.703424	−0.351712	−	0.936108i	$-0.614400\pi$
$313$	−12.4448	−0.703424	−0.351712	+	0.936108i	$0.614400\pi$
$314$	0	0
$315$	4.13138	0.232777
$316$	0	0
$317$	−11.2603	−0.632442	−0.316221	−	0.948685i	$-0.602414\pi$
$317$	−11.2603	−0.632442	−0.316221	+	0.948685i	$0.602414\pi$
$318$	0	0
$319$	−1.88820	−0.105719
$320$	0	0
$321$	6.42281	0.358486
$322$	0	0
$323$	−27.5295	−1.53178
$324$	0	0
$325$	4.78456	0.265400
$326$	0	0
$327$	−44.9250	−2.48436
$328$	0	0
$329$	−10.2560	−0.565430
$330$	0	0
$331$	26.9595	1.48183	0.740914	−	0.671600i	$-0.234393\pi$
$331$	26.9595	1.48183	0.740914	+	0.671600i	$0.234393\pi$
$332$	0	0
$333$	51.3937	2.81636
$334$	0	0
$335$	−0.716303	−0.0391358
$336$	0	0
$337$	−23.8892	−1.30133	−0.650663	−	0.759367i	$-0.725509\pi$
$337$	−23.8892	−1.30133	−0.650663	+	0.759367i	$0.725509\pi$
$338$	0	0
$339$	53.4054	2.90058
$340$	0	0
$341$	5.04704	0.273313
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−5.80773	−0.312678
$346$	0	0
$347$	−22.4485	−1.20510	−0.602550	−	0.798081i	$-0.705849\pi$
$347$	−22.4485	−1.20510	−0.602550	+	0.798081i	$0.705849\pi$
$348$	0	0
$349$	11.9430	0.639292	0.319646	−	0.947537i	$-0.396436\pi$
$349$	11.9430	0.639292	0.319646	+	0.947537i	$0.396436\pi$
$350$	0	0
$351$	20.3568	1.08656
$352$	0	0
$353$	13.7292	0.730731	0.365366	−	0.930864i	$-0.380944\pi$
$353$	13.7292	0.730731	0.365366	+	0.930864i	$0.380944\pi$
$354$	0	0
$355$	−3.46959	−0.184147
$356$	0	0
$357$	13.8620	0.733656
$358$	0	0
$359$	−12.0107	−0.633899	−0.316950	−	0.948442i	$-0.602659\pi$
$359$	−12.0107	−0.633899	−0.316950	+	0.948442i	$0.602659\pi$
$360$	0	0
$361$	27.9379	1.47042
$362$	0	0
$363$	−3.44977	−0.181066
$364$	0	0
$365$	−7.42284	−0.388529
$366$	0	0
$367$	−14.2149	−0.742014	−0.371007	−	0.928630i	$-0.620987\pi$
$367$	−14.2149	−0.742014	−0.371007	+	0.928630i	$0.620987\pi$
$368$	0	0
$369$	−76.2606	−3.96997
$370$	0	0
$371$	−13.0695	−0.678536
$372$	0	0
$373$	−10.7390	−0.556046	−0.278023	−	0.960574i	$-0.589679\pi$
$373$	−10.7390	−0.556046	−0.278023	+	0.960574i	$0.589679\pi$
$374$	0	0
$375$	15.6672	0.809051
$376$	0	0
$377$	1.88820	0.0972472
$378$	0	0
$379$	−5.67763	−0.291640	−0.145820	−	0.989311i	$-0.546582\pi$
$379$	−5.67763	−0.291640	−0.145820	+	0.989311i	$0.546582\pi$
$380$	0	0
$381$	−34.1469	−1.74940
$382$	0	0
$383$	−36.9832	−1.88975	−0.944876	−	0.327430i	$-0.893818\pi$
$383$	−36.9832	−1.88975	−0.944876	+	0.327430i	$0.893818\pi$
$384$	0	0
$385$	0.464152	0.0236554
$386$	0	0
$387$	75.7842	3.85233
$388$	0	0
$389$	29.3873	1.49000	0.744998	−	0.667067i	$-0.232450\pi$
$389$	29.3873	1.49000	0.744998	+	0.667067i	$0.232450\pi$
$390$	0	0
$391$	−14.5745	−0.737062
$392$	0	0
$393$	−57.4035	−2.89562
$394$	0	0
$395$	2.94348	0.148103
$396$	0	0
$397$	35.1993	1.76660	0.883302	−	0.468805i	$-0.155315\pi$
$397$	35.1993	1.76660	0.883302	+	0.468805i	$0.155315\pi$
$398$	0	0
$399$	−23.6348	−1.18322
$400$	0	0
$401$	31.3236	1.56423	0.782114	−	0.623136i	$-0.214142\pi$
$401$	31.3236	1.56423	0.782114	+	0.623136i	$0.214142\pi$
$402$	0	0
$403$	−5.04704	−0.251411
$404$	0	0
$405$	20.2015	1.00382
$406$	0	0
$407$	5.77398	0.286206
$408$	0	0
$409$	5.73048	0.283354	0.141677	−	0.989913i	$-0.454751\pi$
$409$	5.73048	0.283354	0.141677	+	0.989913i	$0.454751\pi$
$410$	0	0
$411$	27.4684	1.35492
$412$	0	0
$413$	−14.3325	−0.705257
$414$	0	0
$415$	−0.0583642	−0.00286499
$416$	0	0
$417$	31.7768	1.55612
$418$	0	0
$419$	6.75381	0.329945	0.164973	−	0.986298i	$-0.447246\pi$
$419$	6.75381	0.329945	0.164973	+	0.986298i	$0.447246\pi$
$420$	0	0
$421$	−34.5756	−1.68511	−0.842557	−	0.538607i	$-0.818951\pi$
$421$	−34.5756	−1.68511	−0.842557	+	0.538607i	$0.818951\pi$
$422$	0	0
$423$	−91.2874	−4.43855
$424$	0	0
$425$	19.2255	0.932575
$426$	0	0
$427$	7.95444	0.384942
$428$	0	0
$429$	3.44977	0.166556
$430$	0	0
$431$	−36.8349	−1.77427	−0.887137	−	0.461506i	$-0.847309\pi$
$431$	−36.8349	−1.77427	−0.887137	+	0.461506i	$0.847309\pi$
$432$	0	0
$433$	−13.4150	−0.644686	−0.322343	−	0.946623i	$-0.604470\pi$
$433$	−13.4150	−0.644686	−0.322343	+	0.946623i	$0.604470\pi$
$434$	0	0
$435$	3.02342	0.144962
$436$	0	0
$437$	24.8495	1.18871
$438$	0	0
$439$	−4.31362	−0.205878	−0.102939	−	0.994688i	$-0.532825\pi$
$439$	−4.31362	−0.205878	−0.102939	+	0.994688i	$0.532825\pi$
$440$	0	0
$441$	8.90091	0.423853
$442$	0	0
$443$	−14.2885	−0.678867	−0.339433	−	0.940630i	$-0.610235\pi$
$443$	−14.2885	−0.678867	−0.339433	+	0.940630i	$0.610235\pi$
$444$	0	0
$445$	−7.16916	−0.339851
$446$	0	0
$447$	−29.8032	−1.40964
$448$	0	0
$449$	22.4707	1.06046	0.530229	−	0.847855i	$-0.322106\pi$
$449$	22.4707	1.06046	0.530229	+	0.847855i	$0.322106\pi$
$450$	0	0
$451$	−8.56773	−0.403439
$452$	0	0
$453$	42.3527	1.98990
$454$	0	0
$455$	−0.464152	−0.0217598
$456$	0	0
$457$	7.32637	0.342713	0.171357	−	0.985209i	$-0.445185\pi$
$457$	7.32637	0.342713	0.171357	+	0.985209i	$0.445185\pi$
$458$	0	0
$459$	81.7985	3.81803
$460$	0	0
$461$	0.347289	0.0161749	0.00808743	−	0.999967i	$-0.497426\pi$
$461$	0.347289	0.0161749	0.00808743	+	0.999967i	$0.497426\pi$
$462$	0	0
$463$	−1.43110	−0.0665088	−0.0332544	−	0.999447i	$-0.510587\pi$
$463$	−1.43110	−0.0665088	−0.0332544	+	0.999447i	$0.510587\pi$
$464$	0	0
$465$	−8.08141	−0.374766
$466$	0	0
$467$	−18.9205	−0.875536	−0.437768	−	0.899088i	$-0.644231\pi$
$467$	−18.9205	−0.875536	−0.437768	+	0.899088i	$0.644231\pi$
$468$	0	0
$469$	−1.54325	−0.0712607
$470$	0	0
$471$	−31.2280	−1.43891
$472$	0	0
$473$	8.51421	0.391484
$474$	0	0
$475$	−32.7796	−1.50403
$476$	0	0
$477$	−116.331	−5.32642
$478$	0	0
$479$	3.17521	0.145079	0.0725395	−	0.997366i	$-0.476890\pi$
$479$	3.17521	0.145079	0.0725395	+	0.997366i	$0.476890\pi$
$480$	0	0
$481$	−5.77398	−0.263271
$482$	0	0
$483$	−12.5126	−0.569341
$484$	0	0
$485$	1.84769	0.0838993
$486$	0	0
$487$	−20.5400	−0.930756	−0.465378	−	0.885112i	$-0.654082\pi$
$487$	−20.5400	−0.930756	−0.465378	+	0.885112i	$0.654082\pi$
$488$	0	0
$489$	65.0821	2.94311
$490$	0	0
$491$	−14.5189	−0.655229	−0.327615	−	0.944811i	$-0.606245\pi$
$491$	−14.5189	−0.655229	−0.327615	+	0.944811i	$0.606245\pi$
$492$	0	0
$493$	7.58724	0.341712
$494$	0	0
$495$	4.13138	0.185692
$496$	0	0
$497$	−7.47512	−0.335305
$498$	0	0
$499$	36.6505	1.64070	0.820350	−	0.571861i	$-0.193778\pi$
$499$	36.6505	1.64070	0.820350	+	0.571861i	$0.193778\pi$
$500$	0	0
$501$	−64.2146	−2.86889
$502$	0	0
$503$	14.4041	0.642245	0.321123	−	0.947038i	$-0.395940\pi$
$503$	14.4041	0.642245	0.321123	+	0.947038i	$0.395940\pi$
$504$	0	0
$505$	3.38171	0.150484
$506$	0	0
$507$	−3.44977	−0.153210
$508$	0	0
$509$	−24.9517	−1.10596	−0.552982	−	0.833193i	$-0.686510\pi$
$509$	−24.9517	−1.10596	−0.552982	+	0.833193i	$0.686510\pi$
$510$	0	0
$511$	−15.9923	−0.707456
$512$	0	0
$513$	−139.467	−6.15761
$514$	0	0
$515$	7.35396	0.324054
$516$	0	0
$517$	−10.2560	−0.451057
$518$	0	0
$519$	−20.5327	−0.901287
$520$	0	0
$521$	−5.94259	−0.260350	−0.130175	−	0.991491i	$-0.541554\pi$
$521$	−5.94259	−0.260350	−0.130175	+	0.991491i	$0.541554\pi$
$522$	0	0
$523$	−32.0009	−1.39930	−0.699650	−	0.714486i	$-0.746661\pi$
$523$	−32.0009	−1.39930	−0.699650	+	0.714486i	$0.746661\pi$
$524$	0	0
$525$	16.5056	0.720365
$526$	0	0
$527$	−20.2802	−0.883421
$528$	0	0
$529$	−9.84435	−0.428015
$530$	0	0
$531$	−127.572	−5.53617
$532$	0	0
$533$	8.56773	0.371110
$534$	0	0
$535$	−0.864162	−0.0373610
$536$	0	0
$537$	−40.2527	−1.73703
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−13.9300	−0.598898	−0.299449	−	0.954112i	$-0.596803\pi$
$541$	−13.9300	−0.598898	−0.299449	+	0.954112i	$0.596803\pi$
$542$	0	0
$543$	40.4929	1.73772
$544$	0	0
$545$	6.04447	0.258917
$546$	0	0
$547$	1.35443	0.0579111	0.0289555	−	0.999581i	$-0.490782\pi$
$547$	1.35443	0.0579111	0.0289555	+	0.999581i	$0.490782\pi$
$548$	0	0
$549$	70.8017	3.02174
$550$	0	0
$551$	−12.9363	−0.551104
$552$	0	0
$553$	6.34164	0.269674
$554$	0	0
$555$	−9.24540	−0.392445
$556$	0	0
$557$	−3.25317	−0.137841	−0.0689207	−	0.997622i	$-0.521956\pi$
$557$	−3.25317	−0.137841	−0.0689207	+	0.997622i	$0.521956\pi$
$558$	0	0
$559$	−8.51421	−0.360113
$560$	0	0
$561$	13.8620	0.585255
$562$	0	0
$563$	41.0660	1.73073	0.865363	−	0.501145i	$-0.167088\pi$
$563$	41.0660	1.73073	0.865363	+	0.501145i	$0.167088\pi$
$564$	0	0
$565$	−7.18547	−0.302295
$566$	0	0
$567$	43.5235	1.82781
$568$	0	0
$569$	−8.25733	−0.346165	−0.173083	−	0.984907i	$-0.555373\pi$
$569$	−8.25733	−0.346165	−0.173083	+	0.984907i	$0.555373\pi$
$570$	0	0
$571$	−18.2587	−0.764103	−0.382051	−	0.924141i	$-0.624782\pi$
$571$	−18.2587	−0.764103	−0.382051	+	0.924141i	$0.624782\pi$
$572$	0	0
$573$	−20.7214	−0.865649
$574$	0	0
$575$	−17.3540	−0.723710
$576$	0	0
$577$	−14.0453	−0.584715	−0.292357	−	0.956309i	$-0.594440\pi$
$577$	−14.0453	−0.584715	−0.292357	+	0.956309i	$0.594440\pi$
$578$	0	0
$579$	0.418608	0.0173968
$580$	0	0
$581$	−0.125744	−0.00521673
$582$	0	0
$583$	−13.0695	−0.541285
$584$	0	0
$585$	−4.13138	−0.170811
$586$	0	0
$587$	9.98054	0.411941	0.205971	−	0.978558i	$-0.433965\pi$
$587$	9.98054	0.411941	0.205971	+	0.978558i	$0.433965\pi$
$588$	0	0
$589$	34.5779	1.42476
$590$	0	0
$591$	−15.5147	−0.638191
$592$	0	0
$593$	−22.4635	−0.922465	−0.461233	−	0.887279i	$-0.652593\pi$
$593$	−22.4635	−0.922465	−0.461233	+	0.887279i	$0.652593\pi$
$594$	0	0
$595$	−1.86508	−0.0764606
$596$	0	0
$597$	5.27999	0.216095
$598$	0	0
$599$	24.9432	1.01915	0.509576	−	0.860425i	$-0.329802\pi$
$599$	24.9432	1.01915	0.509576	+	0.860425i	$0.329802\pi$
$600$	0	0
$601$	31.5852	1.28839	0.644193	−	0.764863i	$-0.277193\pi$
$601$	31.5852	1.28839	0.644193	+	0.764863i	$0.277193\pi$
$602$	0	0
$603$	−13.7363	−0.559387
$604$	0	0
$605$	0.464152	0.0188705
$606$	0	0
$607$	−6.87940	−0.279226	−0.139613	−	0.990206i	$-0.544586\pi$
$607$	−6.87940	−0.279226	−0.139613	+	0.990206i	$0.544586\pi$
$608$	0	0
$609$	6.51385	0.263955
$610$	0	0
$611$	10.2560	0.414912
$612$	0	0
$613$	39.5379	1.59692	0.798460	−	0.602048i	$-0.205648\pi$
$613$	39.5379	1.59692	0.798460	+	0.602048i	$0.205648\pi$
$614$	0	0
$615$	13.7188	0.553196
$616$	0	0
$617$	0.341357	0.0137425	0.00687126	−	0.999976i	$-0.497813\pi$
$617$	0.341357	0.0137425	0.00687126	+	0.999976i	$0.497813\pi$
$618$	0	0
$619$	−7.12382	−0.286330	−0.143165	−	0.989699i	$-0.545728\pi$
$619$	−7.12382	−0.286330	−0.143165	+	0.989699i	$0.545728\pi$
$620$	0	0
$621$	−73.8355	−2.96292
$622$	0	0
$623$	−15.4457	−0.618820
$624$	0	0
$625$	21.8149	0.872594
$626$	0	0
$627$	−23.6348	−0.943883
$628$	0	0
$629$	−23.2013	−0.925095
$630$	0	0
$631$	46.0122	1.83172	0.915858	−	0.401503i	$-0.131512\pi$
$631$	46.0122	1.83172	0.915858	+	0.401503i	$0.131512\pi$
$632$	0	0
$633$	−35.3249	−1.40404
$634$	0	0
$635$	4.59432	0.182320
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−66.5353	−2.63210
$640$	0	0
$641$	−2.54896	−0.100678	−0.0503389	−	0.998732i	$-0.516030\pi$
$641$	−2.54896	−0.100678	−0.0503389	+	0.998732i	$0.516030\pi$
$642$	0	0
$643$	−9.86489	−0.389033	−0.194517	−	0.980899i	$-0.562314\pi$
$643$	−9.86489	−0.389033	−0.194517	+	0.980899i	$0.562314\pi$
$644$	0	0
$645$	−13.6331	−0.536803
$646$	0	0
$647$	−25.2269	−0.991773	−0.495887	−	0.868387i	$-0.665157\pi$
$647$	−25.2269	−0.991773	−0.495887	+	0.868387i	$0.665157\pi$
$648$	0	0
$649$	−14.3325	−0.562600
$650$	0	0
$651$	−17.4111	−0.682396
$652$	0	0
$653$	−9.92732	−0.388486	−0.194243	−	0.980953i	$-0.562225\pi$
$653$	−9.92732	−0.388486	−0.194243	+	0.980953i	$0.562225\pi$
$654$	0	0
$655$	7.72340	0.301778
$656$	0	0
$657$	−142.346	−5.55344
$658$	0	0
$659$	−34.0329	−1.32573	−0.662867	−	0.748737i	$-0.730661\pi$
$659$	−34.0329	−1.32573	−0.662867	+	0.748737i	$0.730661\pi$
$660$	0	0
$661$	−27.6691	−1.07620	−0.538101	−	0.842880i	$-0.680858\pi$
$661$	−27.6691	−1.07620	−0.538101	+	0.842880i	$0.680858\pi$
$662$	0	0
$663$	−13.8620	−0.538356
$664$	0	0
$665$	3.17996	0.123314
$666$	0	0
$667$	−6.84863	−0.265180
$668$	0	0
$669$	36.9063	1.42688
$670$	0	0
$671$	7.95444	0.307078
$672$	0	0
$673$	−35.2291	−1.35798	−0.678991	−	0.734147i	$-0.737582\pi$
$673$	−35.2291	−1.35798	−0.678991	+	0.734147i	$0.737582\pi$
$674$	0	0
$675$	97.3983	3.74886
$676$	0	0
$677$	−30.2013	−1.16073	−0.580364	−	0.814357i	$-0.697090\pi$
$677$	−30.2013	−1.16073	−0.580364	+	0.814357i	$0.697090\pi$
$678$	0	0
$679$	3.98079	0.152769
$680$	0	0
$681$	−26.2807	−1.00708
$682$	0	0
$683$	11.4441	0.437898	0.218949	−	0.975736i	$-0.429737\pi$
$683$	11.4441	0.437898	0.218949	+	0.975736i	$0.429737\pi$
$684$	0	0
$685$	−3.69576	−0.141208
$686$	0	0
$687$	14.3917	0.549076
$688$	0	0
$689$	13.0695	0.497909
$690$	0	0
$691$	−27.3611	−1.04087	−0.520433	−	0.853902i	$-0.674230\pi$
$691$	−27.3611	−1.04087	−0.520433	+	0.853902i	$0.674230\pi$
$692$	0	0
$693$	8.90091	0.338118
$694$	0	0
$695$	−4.27543	−0.162176
$696$	0	0
$697$	34.4272	1.30402
$698$	0	0
$699$	−42.5466	−1.60926
$700$	0	0
$701$	48.2865	1.82376	0.911878	−	0.410461i	$-0.134632\pi$
$701$	48.2865	1.82376	0.911878	+	0.410461i	$0.134632\pi$
$702$	0	0
$703$	39.5583	1.49197
$704$	0	0
$705$	16.4220	0.618489
$706$	0	0
$707$	7.28577	0.274010
$708$	0	0
$709$	0.298891	0.0112251	0.00561254	−	0.999984i	$-0.498213\pi$
$709$	0.298891	0.0112251	0.00561254	+	0.999984i	$0.498213\pi$
$710$	0	0
$711$	56.4464	2.11690
$712$	0	0
$713$	18.3060	0.685564
$714$	0	0
$715$	−0.464152	−0.0173583
$716$	0	0
$717$	−84.6789	−3.16239
$718$	0	0
$719$	−9.88406	−0.368613	−0.184307	−	0.982869i	$-0.559004\pi$
$719$	−9.88406	−0.368613	−0.184307	+	0.982869i	$0.559004\pi$
$720$	0	0
$721$	15.8439	0.590056
$722$	0	0
$723$	42.8263	1.59273
$724$	0	0
$725$	9.03420	0.335522
$726$	0	0
$727$	45.8135	1.69913	0.849564	−	0.527486i	$-0.176865\pi$
$727$	45.8135	1.69913	0.849564	+	0.527486i	$0.176865\pi$
$728$	0	0
$729$	176.720	6.54519
$730$	0	0
$731$	−34.2122	−1.26538
$732$	0	0
$733$	−47.0323	−1.73718	−0.868589	−	0.495534i	$-0.834973\pi$
$733$	−47.0323	−1.73718	−0.868589	+	0.495534i	$0.834973\pi$
$734$	0	0
$735$	−1.60122	−0.0590618
$736$	0	0
$737$	−1.54325	−0.0568464
$738$	0	0
$739$	22.4554	0.826035	0.413017	−	0.910723i	$-0.364475\pi$
$739$	22.4554	0.826035	0.413017	+	0.910723i	$0.364475\pi$
$740$	0	0
$741$	23.6348	0.868246
$742$	0	0
$743$	−13.7741	−0.505323	−0.252661	−	0.967555i	$-0.581306\pi$
$743$	−13.7741	−0.505323	−0.252661	+	0.967555i	$0.581306\pi$
$744$	0	0
$745$	4.00989	0.146911
$746$	0	0
$747$	−1.11923	−0.0409506
$748$	0	0
$749$	−1.86181	−0.0680290
$750$	0	0
$751$	32.2987	1.17860	0.589298	−	0.807915i	$-0.299404\pi$
$751$	32.2987	1.17860	0.589298	+	0.807915i	$0.299404\pi$
$752$	0	0
$753$	−52.0154	−1.89555
$754$	0	0
$755$	−5.69838	−0.207385
$756$	0	0
$757$	3.78710	0.137644	0.0688222	−	0.997629i	$-0.478076\pi$
$757$	3.78710	0.137644	0.0688222	+	0.997629i	$0.478076\pi$
$758$	0	0
$759$	−12.5126	−0.454177
$760$	0	0
$761$	−8.95730	−0.324702	−0.162351	−	0.986733i	$-0.551908\pi$
$761$	−8.95730	−0.324702	−0.162351	+	0.986733i	$0.551908\pi$
$762$	0	0
$763$	13.0226	0.471450
$764$	0	0
$765$	−16.6009	−0.600206
$766$	0	0
$767$	14.3325	0.517517
$768$	0	0
$769$	−5.45354	−0.196660	−0.0983298	−	0.995154i	$-0.531350\pi$
$769$	−5.45354	−0.196660	−0.0983298	+	0.995154i	$0.531350\pi$
$770$	0	0
$771$	26.0581	0.938461
$772$	0	0
$773$	16.2984	0.586214	0.293107	−	0.956080i	$-0.405311\pi$
$773$	16.2984	0.586214	0.293107	+	0.956080i	$0.405311\pi$
$774$	0	0
$775$	−24.1479	−0.867417
$776$	0	0
$777$	−19.9189	−0.714587
$778$	0	0
$779$	−58.6986	−2.10310
$780$	0	0
$781$	−7.47512	−0.267481
$782$	0	0
$783$	38.4376	1.37365
$784$	0	0
$785$	4.20159	0.149961
$786$	0	0
$787$	−31.9887	−1.14027	−0.570136	−	0.821550i	$-0.693110\pi$
$787$	−31.9887	−1.14027	−0.570136	+	0.821550i	$0.693110\pi$
$788$	0	0
$789$	51.1535	1.82111
$790$	0	0
$791$	−15.4808	−0.550436
$792$	0	0
$793$	−7.95444	−0.282470
$794$	0	0
$795$	20.9272	0.742210
$796$	0	0
$797$	43.7779	1.55069	0.775346	−	0.631537i	$-0.217575\pi$
$797$	43.7779	1.55069	0.775346	+	0.631537i	$0.217575\pi$
$798$	0	0
$799$	41.2110	1.45794
$800$	0	0
$801$	−137.481	−4.85765
$802$	0	0
$803$	−15.9923	−0.564355
$804$	0	0
$805$	1.68351	0.0593360
$806$	0	0
$807$	98.4138	3.46433
$808$	0	0
$809$	−27.5956	−0.970209	−0.485104	−	0.874456i	$-0.661218\pi$
$809$	−27.5956	−0.970209	−0.485104	+	0.874456i	$0.661218\pi$
$810$	0	0
$811$	−34.6769	−1.21767	−0.608835	−	0.793297i	$-0.708363\pi$
$811$	−34.6769	−1.21767	−0.608835	+	0.793297i	$0.708363\pi$
$812$	0	0
$813$	82.6141	2.89740
$814$	0	0
$815$	−8.75652	−0.306728
$816$	0	0
$817$	58.3319	2.04078
$818$	0	0
$819$	−8.90091	−0.311023
$820$	0	0
$821$	−25.9985	−0.907355	−0.453678	−	0.891166i	$-0.649888\pi$
$821$	−25.9985	−0.907355	−0.453678	+	0.891166i	$0.649888\pi$
$822$	0	0
$823$	−16.8341	−0.586800	−0.293400	−	0.955990i	$-0.594787\pi$
$823$	−16.8341	−0.586800	−0.293400	+	0.955990i	$0.594787\pi$
$824$	0	0
$825$	16.5056	0.574653
$826$	0	0
$827$	−29.3044	−1.01902	−0.509508	−	0.860466i	$-0.670172\pi$
$827$	−29.3044	−1.01902	−0.509508	+	0.860466i	$0.670172\pi$
$828$	0	0
$829$	0.130866	0.00454516	0.00227258	−	0.999997i	$-0.499277\pi$
$829$	0.130866	0.00454516	0.00227258	+	0.999997i	$0.499277\pi$
$830$	0	0
$831$	44.1850	1.53276
$832$	0	0
$833$	−4.01824	−0.139224
$834$	0	0
$835$	8.63980	0.298993
$836$	0	0
$837$	−102.742	−3.55127
$838$	0	0
$839$	25.2987	0.873408	0.436704	−	0.899605i	$-0.356146\pi$
$839$	25.2987	0.873408	0.436704	+	0.899605i	$0.356146\pi$
$840$	0	0
$841$	−25.4347	−0.877059
$842$	0	0
$843$	67.4114	2.32177
$844$	0	0
$845$	0.464152	0.0159673
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−7.21462	−0.247605
$850$	0	0
$851$	20.9426	0.717905
$852$	0	0
$853$	50.4402	1.72704	0.863520	−	0.504315i	$-0.168255\pi$
$853$	50.4402	1.72704	0.863520	+	0.504315i	$0.168255\pi$
$854$	0	0
$855$	28.3046	0.967996
$856$	0	0
$857$	−17.4089	−0.594677	−0.297339	−	0.954772i	$-0.596099\pi$
$857$	−17.4089	−0.594677	−0.297339	+	0.954772i	$0.596099\pi$
$858$	0	0
$859$	8.85074	0.301983	0.150992	−	0.988535i	$-0.451753\pi$
$859$	8.85074	0.301983	0.150992	+	0.988535i	$0.451753\pi$
$860$	0	0
$861$	29.5567	1.00729
$862$	0	0
$863$	34.2808	1.16693	0.583465	−	0.812138i	$-0.301696\pi$
$863$	34.2808	1.16693	0.583465	+	0.812138i	$0.301696\pi$
$864$	0	0
$865$	2.76259	0.0939310
$866$	0	0
$867$	2.94515	0.100023
$868$	0	0
$869$	6.34164	0.215125
$870$	0	0
$871$	1.54325	0.0522911
$872$	0	0
$873$	35.4326	1.19921
$874$	0	0
$875$	−4.54152	−0.153532
$876$	0	0
$877$	−34.0575	−1.15004	−0.575019	−	0.818140i	$-0.695005\pi$
$877$	−34.0575	−1.15004	−0.575019	+	0.818140i	$0.695005\pi$
$878$	0	0
$879$	−26.6004	−0.897209
$880$	0	0
$881$	−33.0038	−1.11193	−0.555963	−	0.831207i	$-0.687650\pi$
$881$	−33.0038	−1.11193	−0.555963	+	0.831207i	$0.687650\pi$
$882$	0	0
$883$	33.9109	1.14119	0.570596	−	0.821231i	$-0.306712\pi$
$883$	33.9109	1.14119	0.570596	+	0.821231i	$0.306712\pi$
$884$	0	0
$885$	22.9495	0.771438
$886$	0	0
$887$	16.3537	0.549102	0.274551	−	0.961572i	$-0.411471\pi$
$887$	16.3537	0.549102	0.274551	+	0.961572i	$0.411471\pi$
$888$	0	0
$889$	9.89830	0.331978
$890$	0	0
$891$	43.5235	1.45809
$892$	0	0
$893$	−70.2649	−2.35132
$894$	0	0
$895$	5.41584	0.181031
$896$	0	0
$897$	12.5126	0.417782
$898$	0	0
$899$	−9.52981	−0.317837
$900$	0	0
$901$	52.5165	1.74958
$902$	0	0
$903$	−29.3721	−0.977441
$904$	0	0
$905$	−5.44815	−0.181103
$906$	0	0
$907$	16.7771	0.557075	0.278537	−	0.960425i	$-0.410150\pi$
$907$	16.7771	0.557075	0.278537	+	0.960425i	$0.410150\pi$
$908$	0	0
$909$	64.8500	2.15094
$910$	0	0
$911$	8.82693	0.292449	0.146225	−	0.989251i	$-0.453288\pi$
$911$	8.82693	0.292449	0.146225	+	0.989251i	$0.453288\pi$
$912$	0	0
$913$	−0.125744	−0.00416151
$914$	0	0
$915$	−12.7368	−0.421065
$916$	0	0
$917$	16.6398	0.549495
$918$	0	0
$919$	2.00991	0.0663009	0.0331504	−	0.999450i	$-0.489446\pi$
$919$	2.00991	0.0663009	0.0331504	+	0.999450i	$0.489446\pi$
$920$	0	0
$921$	69.5302	2.29110
$922$	0	0
$923$	7.47512	0.246046
$924$	0	0
$925$	−27.6260	−0.908336
$926$	0	0
$927$	141.025	4.63186
$928$	0	0
$929$	−59.0602	−1.93770	−0.968852	−	0.247641i	$-0.920345\pi$
$929$	−59.0602	−1.93770	−0.968852	+	0.247641i	$0.920345\pi$
$930$	0	0
$931$	6.85112	0.224537
$932$	0	0
$933$	75.0080	2.45565
$934$	0	0
$935$	−1.86508	−0.0609945
$936$	0	0
$937$	48.2928	1.57766	0.788828	−	0.614614i	$-0.210688\pi$
$937$	48.2928	1.57766	0.788828	+	0.614614i	$0.210688\pi$
$938$	0	0
$939$	42.9319	1.40103
$940$	0	0
$941$	26.0679	0.849790	0.424895	−	0.905243i	$-0.360311\pi$
$941$	26.0679	0.849790	0.424895	+	0.905243i	$0.360311\pi$
$942$	0	0
$943$	−31.0758	−1.01197
$944$	0	0
$945$	−9.44864	−0.307364
$946$	0	0
$947$	57.2953	1.86185	0.930923	−	0.365215i	$-0.119004\pi$
$947$	57.2953	1.86185	0.930923	+	0.365215i	$0.119004\pi$
$948$	0	0
$949$	15.9923	0.519131
$950$	0	0
$951$	38.8455	1.25965
$952$	0	0
$953$	−47.8246	−1.54919	−0.774595	−	0.632458i	$-0.782046\pi$
$953$	−47.8246	−1.54919	−0.774595	+	0.632458i	$0.782046\pi$
$954$	0	0
$955$	2.78798	0.0902168
$956$	0	0
$957$	6.51385	0.210563
$958$	0	0
$959$	−7.96239	−0.257119
$960$	0	0
$961$	−5.52739	−0.178303
$962$	0	0
$963$	−16.5718	−0.534018
$964$	0	0
$965$	−0.0563220	−0.00181307
$966$	0	0
$967$	−38.6971	−1.24441	−0.622207	−	0.782853i	$-0.713764\pi$
$967$	−38.6971	−1.24441	−0.622207	+	0.782853i	$0.713764\pi$
$968$	0	0
$969$	94.9704	3.05089
$970$	0	0
$971$	−3.21911	−0.103306	−0.0516531	−	0.998665i	$-0.516449\pi$
$971$	−3.21911	−0.103306	−0.0516531	+	0.998665i	$0.516449\pi$
$972$	0	0
$973$	−9.21128	−0.295300
$974$	0	0
$975$	−16.5056	−0.528604
$976$	0	0
$977$	−38.5069	−1.23194	−0.615972	−	0.787768i	$-0.711237\pi$
$977$	−38.5069	−1.23194	−0.615972	+	0.787768i	$0.711237\pi$
$978$	0	0
$979$	−15.4457	−0.493648
$980$	0	0
$981$	115.913	3.70082
$982$	0	0
$983$	15.4477	0.492704	0.246352	−	0.969180i	$-0.420768\pi$
$983$	15.4477	0.492704	0.246352	+	0.969180i	$0.420768\pi$
$984$	0	0
$985$	2.08744	0.0665114
$986$	0	0
$987$	35.3807	1.12618
$988$	0	0
$989$	30.8817	0.981979
$990$	0	0
$991$	21.2743	0.675799	0.337900	−	0.941182i	$-0.390284\pi$
$991$	21.2743	0.675799	0.337900	+	0.941182i	$0.390284\pi$
$992$	0	0
$993$	−93.0041	−2.95139
$994$	0	0
$995$	−0.710400	−0.0225212
$996$	0	0
$997$	−13.0657	−0.413796	−0.206898	−	0.978363i	$-0.566337\pi$
$997$	−13.0657	−0.413796	−0.206898	+	0.978363i	$0.566337\pi$
$998$	0	0
$999$	−117.540	−3.71879

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.s.1.1	✓	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)

\(f(q)\)	\(=\)	\(q-3.44977 q^{3} +0.464152 q^{5} +1.00000 q^{7} +8.90091 q^{9} +O(q^{10})\)	\(q-3.44977 q^{3} +0.464152 q^{5} +1.00000 q^{7} +8.90091 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.60122 q^{15} -4.01824 q^{17} +6.85112 q^{19} -3.44977 q^{21} +3.62707 q^{23} -4.78456 q^{25} -20.3568 q^{27} -1.88820 q^{29} +5.04704 q^{31} -3.44977 q^{33} +0.464152 q^{35} +5.77398 q^{37} +3.44977 q^{39} -8.56773 q^{41} +8.51421 q^{43} +4.13138 q^{45} -10.2560 q^{47} +1.00000 q^{49} +13.8620 q^{51} -13.0695 q^{53} +0.464152 q^{55} -23.6348 q^{57} -14.3325 q^{59} +7.95444 q^{61} +8.90091 q^{63} -0.464152 q^{65} -1.54325 q^{67} -12.5126 q^{69} -7.47512 q^{71} -15.9923 q^{73} +16.5056 q^{75} +1.00000 q^{77} +6.34164 q^{79} +43.5235 q^{81} -0.125744 q^{83} -1.86508 q^{85} +6.51385 q^{87} -15.4457 q^{89} -1.00000 q^{91} -17.4111 q^{93} +3.17996 q^{95} +3.98079 q^{97} +8.90091 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) 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

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