LMFDB - Embedded newform 8008.2.a.z.1.15

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-3.26824$ 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)

$f(q)$	$=$	$q+3.26824 q^{3} +2.75302 q^{5} -1.00000 q^{7} +7.68141 q^{9} +O(q^{10})$	$q+3.26824 q^{3} +2.75302 q^{5} -1.00000 q^{7} +7.68141 q^{9} -1.00000 q^{11} +1.00000 q^{13} +8.99753 q^{15} +6.66165 q^{17} +2.23263 q^{19} -3.26824 q^{21} +5.35883 q^{23} +2.57911 q^{25} +15.3000 q^{27} +2.68188 q^{29} -6.52216 q^{31} -3.26824 q^{33} -2.75302 q^{35} -5.88468 q^{37} +3.26824 q^{39} -9.57061 q^{41} -7.83455 q^{43} +21.1471 q^{45} +12.5653 q^{47} +1.00000 q^{49} +21.7719 q^{51} -3.27646 q^{53} -2.75302 q^{55} +7.29676 q^{57} -7.10908 q^{59} +10.8960 q^{61} -7.68141 q^{63} +2.75302 q^{65} +11.4496 q^{67} +17.5140 q^{69} -10.9659 q^{71} -11.0779 q^{73} +8.42916 q^{75} +1.00000 q^{77} -10.6125 q^{79} +26.9599 q^{81} -10.4733 q^{83} +18.3397 q^{85} +8.76505 q^{87} +2.43201 q^{89} -1.00000 q^{91} -21.3160 q^{93} +6.14646 q^{95} -5.98878 q^{97} -7.68141 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	3.26824	1.88692	0.943461	−	0.331485i	$-0.107550\pi$
$3$	3.26824	1.88692	0.943461	+	0.331485i	$0.107550\pi$
$4$	0	0
$5$	2.75302	1.23119	0.615594	−	0.788064i	$-0.288916\pi$
$5$	2.75302	1.23119	0.615594	+	0.788064i	$0.288916\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	7.68141	2.56047
$10$	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
$15$	8.99753	2.32315
$16$	0	0
$17$	6.66165	1.61569	0.807844	−	0.589396i	$-0.200634\pi$
$17$	6.66165	1.61569	0.807844	+	0.589396i	$0.200634\pi$
$18$	0	0
$19$	2.23263	0.512199	0.256100	−	0.966650i	$-0.417562\pi$
$19$	2.23263	0.512199	0.256100	+	0.966650i	$0.417562\pi$
$20$	0	0
$21$	−3.26824	−0.713189
$22$	0	0
$23$	5.35883	1.11739	0.558697	−	0.829372i	$-0.311301\pi$
$23$	5.35883	1.11739	0.558697	+	0.829372i	$0.311301\pi$
$24$	0	0
$25$	2.57911	0.515822
$26$	0	0
$27$	15.3000	2.94449
$28$	0	0
$29$	2.68188	0.498013	0.249007	−	0.968502i	$-0.419896\pi$
$29$	2.68188	0.498013	0.249007	+	0.968502i	$0.419896\pi$
$30$	0	0
$31$	−6.52216	−1.17141	−0.585707	−	0.810523i	$-0.699183\pi$
$31$	−6.52216	−1.17141	−0.585707	+	0.810523i	$0.699183\pi$
$32$	0	0
$33$	−3.26824	−0.568928
$34$	0	0
$35$	−2.75302	−0.465345
$36$	0	0
$37$	−5.88468	−0.967435	−0.483717	−	0.875224i	$-0.660714\pi$
$37$	−5.88468	−0.967435	−0.483717	+	0.875224i	$0.660714\pi$
$38$	0	0
$39$	3.26824	0.523338
$40$	0	0
$41$	−9.57061	−1.49468	−0.747339	−	0.664443i	$-0.768669\pi$
$41$	−9.57061	−1.49468	−0.747339	+	0.664443i	$0.768669\pi$
$42$	0	0
$43$	−7.83455	−1.19476	−0.597379	−	0.801959i	$-0.703791\pi$
$43$	−7.83455	−1.19476	−0.597379	+	0.801959i	$0.703791\pi$
$44$	0	0
$45$	21.1471	3.15242
$46$	0	0
$47$	12.5653	1.83284	0.916422	−	0.400213i	$-0.131064\pi$
$47$	12.5653	1.83284	0.916422	+	0.400213i	$0.131064\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	21.7719	3.04868
$52$	0	0
$53$	−3.27646	−0.450057	−0.225028	−	0.974352i	$-0.572248\pi$
$53$	−3.27646	−0.450057	−0.225028	+	0.974352i	$0.572248\pi$
$54$	0	0
$55$	−2.75302	−0.371217
$56$	0	0
$57$	7.29676	0.966480
$58$	0	0
$59$	−7.10908	−0.925523	−0.462762	−	0.886483i	$-0.653141\pi$
$59$	−7.10908	−0.925523	−0.462762	+	0.886483i	$0.653141\pi$
$60$	0	0
$61$	10.8960	1.39509	0.697543	−	0.716543i	$-0.254277\pi$
$61$	10.8960	1.39509	0.697543	+	0.716543i	$0.254277\pi$
$62$	0	0
$63$	−7.68141	−0.967767
$64$	0	0
$65$	2.75302	0.341470
$66$	0	0
$67$	11.4496	1.39880	0.699399	−	0.714732i	$-0.253451\pi$
$67$	11.4496	1.39880	0.699399	+	0.714732i	$0.253451\pi$
$68$	0	0
$69$	17.5140	2.10843
$70$	0	0
$71$	−10.9659	−1.30141	−0.650705	−	0.759331i	$-0.725527\pi$
$71$	−10.9659	−1.30141	−0.650705	+	0.759331i	$0.725527\pi$
$72$	0	0
$73$	−11.0779	−1.29657	−0.648287	−	0.761396i	$-0.724514\pi$
$73$	−11.0779	−1.29657	−0.648287	+	0.761396i	$0.724514\pi$
$74$	0	0
$75$	8.42916	0.973316
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−10.6125	−1.19400	−0.596999	−	0.802242i	$-0.703640\pi$
$79$	−10.6125	−1.19400	−0.596999	+	0.802242i	$0.703640\pi$
$80$	0	0
$81$	26.9599	2.99554
$82$	0	0
$83$	−10.4733	−1.14959	−0.574796	−	0.818297i	$-0.694919\pi$
$83$	−10.4733	−1.14959	−0.574796	+	0.818297i	$0.694919\pi$
$84$	0	0
$85$	18.3397	1.98922
$86$	0	0
$87$	8.76505	0.939712
$88$	0	0
$89$	2.43201	0.257793	0.128896	−	0.991658i	$-0.458857\pi$
$89$	2.43201	0.257793	0.128896	+	0.991658i	$0.458857\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−21.3160	−2.21037
$94$	0	0
$95$	6.14646	0.630613
$96$	0	0
$97$	−5.98878	−0.608069	−0.304034	−	0.952661i	$-0.598334\pi$
$97$	−5.98878	−0.608069	−0.304034	+	0.952661i	$0.598334\pi$
$98$	0	0
$99$	−7.68141	−0.772011
$100$	0	0
$101$	−1.48787	−0.148049	−0.0740243	−	0.997256i	$-0.523584\pi$
$101$	−1.48787	−0.148049	−0.0740243	+	0.997256i	$0.523584\pi$
$102$	0	0
$103$	11.4559	1.12878	0.564389	−	0.825509i	$-0.309112\pi$
$103$	11.4559	1.12878	0.564389	+	0.825509i	$0.309112\pi$
$104$	0	0
$105$	−8.99753	−0.878069
$106$	0	0
$107$	−18.3026	−1.76938	−0.884688	−	0.466183i	$-0.845629\pi$
$107$	−18.3026	−1.76938	−0.884688	+	0.466183i	$0.845629\pi$
$108$	0	0
$109$	12.3467	1.18259	0.591297	−	0.806454i	$-0.298616\pi$
$109$	12.3467	1.18259	0.591297	+	0.806454i	$0.298616\pi$
$110$	0	0
$111$	−19.2326	−1.82547
$112$	0	0
$113$	−14.0471	−1.32144	−0.660722	−	0.750631i	$-0.729750\pi$
$113$	−14.0471	−1.32144	−0.660722	+	0.750631i	$0.729750\pi$
$114$	0	0
$115$	14.7530	1.37572
$116$	0	0
$117$	7.68141	0.710147
$118$	0	0
$119$	−6.66165	−0.610673
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−31.2791	−2.82034
$124$	0	0
$125$	−6.66475	−0.596114
$126$	0	0
$127$	6.67710	0.592497	0.296248	−	0.955111i	$-0.404264\pi$
$127$	6.67710	0.592497	0.296248	+	0.955111i	$0.404264\pi$
$128$	0	0
$129$	−25.6052	−2.25441
$130$	0	0
$131$	−12.3028	−1.07490	−0.537449	−	0.843296i	$-0.680612\pi$
$131$	−12.3028	−1.07490	−0.537449	+	0.843296i	$0.680612\pi$
$132$	0	0
$133$	−2.23263	−0.193593
$134$	0	0
$135$	42.1212	3.62521
$136$	0	0
$137$	16.8790	1.44207	0.721035	−	0.692899i	$-0.243667\pi$
$137$	16.8790	1.44207	0.721035	+	0.692899i	$0.243667\pi$
$138$	0	0
$139$	−20.4847	−1.73749	−0.868743	−	0.495262i	$-0.835072\pi$
$139$	−20.4847	−1.73749	−0.868743	+	0.495262i	$0.835072\pi$
$140$	0	0
$141$	41.0666	3.45843
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	7.38328	0.613148
$146$	0	0
$147$	3.26824	0.269560
$148$	0	0
$149$	−10.6243	−0.870379	−0.435189	−	0.900339i	$-0.643319\pi$
$149$	−10.6243	−0.870379	−0.435189	+	0.900339i	$0.643319\pi$
$150$	0	0
$151$	2.00328	0.163025	0.0815125	−	0.996672i	$-0.474025\pi$
$151$	2.00328	0.163025	0.0815125	+	0.996672i	$0.474025\pi$
$152$	0	0
$153$	51.1709	4.13692
$154$	0	0
$155$	−17.9556	−1.44223
$156$	0	0
$157$	22.9858	1.83446	0.917232	−	0.398352i	$-0.130418\pi$
$157$	22.9858	1.83446	0.917232	+	0.398352i	$0.130418\pi$
$158$	0	0
$159$	−10.7083	−0.849222
$160$	0	0
$161$	−5.35883	−0.422335
$162$	0	0
$163$	−14.6692	−1.14898	−0.574490	−	0.818512i	$-0.694800\pi$
$163$	−14.6692	−1.14898	−0.574490	+	0.818512i	$0.694800\pi$
$164$	0	0
$165$	−8.99753	−0.700457
$166$	0	0
$167$	0.313623	0.0242689	0.0121344	−	0.999926i	$-0.496137\pi$
$167$	0.313623	0.0242689	0.0121344	+	0.999926i	$0.496137\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	17.1497	1.31147
$172$	0	0
$173$	11.8283	0.899292	0.449646	−	0.893207i	$-0.351550\pi$
$173$	11.8283	0.899292	0.449646	+	0.893207i	$0.351550\pi$
$174$	0	0
$175$	−2.57911	−0.194962
$176$	0	0
$177$	−23.2342	−1.74639
$178$	0	0
$179$	14.3557	1.07300	0.536499	−	0.843901i	$-0.319747\pi$
$179$	14.3557	1.07300	0.536499	+	0.843901i	$0.319747\pi$
$180$	0	0
$181$	19.4625	1.44663	0.723317	−	0.690516i	$-0.242616\pi$
$181$	19.4625	1.44663	0.723317	+	0.690516i	$0.242616\pi$
$182$	0	0
$183$	35.6107	2.63242
$184$	0	0
$185$	−16.2006	−1.19109
$186$	0	0
$187$	−6.66165	−0.487148
$188$	0	0
$189$	−15.3000	−1.11291
$190$	0	0
$191$	−3.74449	−0.270942	−0.135471	−	0.990781i	$-0.543255\pi$
$191$	−3.74449	−0.270942	−0.135471	+	0.990781i	$0.543255\pi$
$192$	0	0
$193$	12.1736	0.876278	0.438139	−	0.898907i	$-0.355638\pi$
$193$	12.1736	0.876278	0.438139	+	0.898907i	$0.355638\pi$
$194$	0	0
$195$	8.99753	0.644327
$196$	0	0
$197$	−13.4888	−0.961038	−0.480519	−	0.876984i	$-0.659552\pi$
$197$	−13.4888	−0.961038	−0.480519	+	0.876984i	$0.659552\pi$
$198$	0	0
$199$	23.0425	1.63344	0.816720	−	0.577034i	$-0.195790\pi$
$199$	23.0425	1.63344	0.816720	+	0.577034i	$0.195790\pi$
$200$	0	0
$201$	37.4202	2.63942
$202$	0	0
$203$	−2.68188	−0.188231
$204$	0	0
$205$	−26.3481	−1.84023
$206$	0	0
$207$	41.1634	2.86106
$208$	0	0
$209$	−2.23263	−0.154434
$210$	0	0
$211$	−9.05003	−0.623030	−0.311515	−	0.950241i	$-0.600836\pi$
$211$	−9.05003	−0.623030	−0.311515	+	0.950241i	$0.600836\pi$
$212$	0	0
$213$	−35.8391	−2.45566
$214$	0	0
$215$	−21.5687	−1.47097
$216$	0	0
$217$	6.52216	0.442753
$218$	0	0
$219$	−36.2054	−2.44653
$220$	0	0
$221$	6.66165	0.448111
$222$	0	0
$223$	−4.85048	−0.324812	−0.162406	−	0.986724i	$-0.551925\pi$
$223$	−4.85048	−0.324812	−0.162406	+	0.986724i	$0.551925\pi$
$224$	0	0
$225$	19.8112	1.32075
$226$	0	0
$227$	2.51538	0.166952	0.0834759	−	0.996510i	$-0.473398\pi$
$227$	2.51538	0.166952	0.0834759	+	0.996510i	$0.473398\pi$
$228$	0	0
$229$	−0.185154	−0.0122353	−0.00611767	−	0.999981i	$-0.501947\pi$
$229$	−0.185154	−0.0122353	−0.00611767	+	0.999981i	$0.501947\pi$
$230$	0	0
$231$	3.26824	0.215035
$232$	0	0
$233$	24.4063	1.59891	0.799454	−	0.600728i	$-0.205122\pi$
$233$	24.4063	1.59891	0.799454	+	0.600728i	$0.205122\pi$
$234$	0	0
$235$	34.5926	2.25657
$236$	0	0
$237$	−34.6842	−2.25298
$238$	0	0
$239$	21.5522	1.39409	0.697047	−	0.717025i	$-0.254497\pi$
$239$	21.5522	1.39409	0.697047	+	0.717025i	$0.254497\pi$
$240$	0	0
$241$	3.25952	0.209964	0.104982	−	0.994474i	$-0.466522\pi$
$241$	3.25952	0.209964	0.104982	+	0.994474i	$0.466522\pi$
$242$	0	0
$243$	42.2114	2.70786
$244$	0	0
$245$	2.75302	0.175884
$246$	0	0
$247$	2.23263	0.142059
$248$	0	0
$249$	−34.2292	−2.16919
$250$	0	0
$251$	19.1611	1.20944	0.604719	−	0.796439i	$-0.293285\pi$
$251$	19.1611	1.20944	0.604719	+	0.796439i	$0.293285\pi$
$252$	0	0
$253$	−5.35883	−0.336907
$254$	0	0
$255$	59.9385	3.75349
$256$	0	0
$257$	24.0401	1.49958	0.749791	−	0.661675i	$-0.230154\pi$
$257$	24.0401	1.49958	0.749791	+	0.661675i	$0.230154\pi$
$258$	0	0
$259$	5.88468	0.365656
$260$	0	0
$261$	20.6007	1.27515
$262$	0	0
$263$	−8.32266	−0.513197	−0.256599	−	0.966518i	$-0.582602\pi$
$263$	−8.32266	−0.513197	−0.256599	+	0.966518i	$0.582602\pi$
$264$	0	0
$265$	−9.02016	−0.554104
$266$	0	0
$267$	7.94840	0.486434
$268$	0	0
$269$	−5.34437	−0.325852	−0.162926	−	0.986638i	$-0.552093\pi$
$269$	−5.34437	−0.325852	−0.162926	+	0.986638i	$0.552093\pi$
$270$	0	0
$271$	−23.9724	−1.45622	−0.728111	−	0.685459i	$-0.759602\pi$
$271$	−23.9724	−1.45622	−0.728111	+	0.685459i	$0.759602\pi$
$272$	0	0
$273$	−3.26824	−0.197803
$274$	0	0
$275$	−2.57911	−0.155526
$276$	0	0
$277$	−2.90993	−0.174841	−0.0874204	−	0.996172i	$-0.527862\pi$
$277$	−2.90993	−0.174841	−0.0874204	+	0.996172i	$0.527862\pi$
$278$	0	0
$279$	−50.0994	−2.99937
$280$	0	0
$281$	2.83622	0.169195	0.0845975	−	0.996415i	$-0.473040\pi$
$281$	2.83622	0.169195	0.0845975	+	0.996415i	$0.473040\pi$
$282$	0	0
$283$	3.74101	0.222380	0.111190	−	0.993799i	$-0.464534\pi$
$283$	3.74101	0.222380	0.111190	+	0.993799i	$0.464534\pi$
$284$	0	0
$285$	20.0881	1.18992
$286$	0	0
$287$	9.57061	0.564935
$288$	0	0
$289$	27.3776	1.61045
$290$	0	0
$291$	−19.5728	−1.14738
$292$	0	0
$293$	−21.0992	−1.23263	−0.616314	−	0.787501i	$-0.711375\pi$
$293$	−21.0992	−1.23263	−0.616314	+	0.787501i	$0.711375\pi$
$294$	0	0
$295$	−19.5714	−1.13949
$296$	0	0
$297$	−15.3000	−0.887796
$298$	0	0
$299$	5.35883	0.309909
$300$	0	0
$301$	7.83455	0.451576
$302$	0	0
$303$	−4.86272	−0.279356
$304$	0	0
$305$	29.9968	1.71761
$306$	0	0
$307$	−22.3221	−1.27399	−0.636996	−	0.770867i	$-0.719823\pi$
$307$	−22.3221	−1.27399	−0.636996	+	0.770867i	$0.719823\pi$
$308$	0	0
$309$	37.4405	2.12992
$310$	0	0
$311$	−14.0889	−0.798909	−0.399454	−	0.916753i	$-0.630800\pi$
$311$	−14.0889	−0.798909	−0.399454	+	0.916753i	$0.630800\pi$
$312$	0	0
$313$	−22.7823	−1.28773	−0.643865	−	0.765139i	$-0.722670\pi$
$313$	−22.7823	−1.28773	−0.643865	+	0.765139i	$0.722670\pi$
$314$	0	0
$315$	−21.1471	−1.19150
$316$	0	0
$317$	0.715916	0.0402098	0.0201049	−	0.999798i	$-0.493600\pi$
$317$	0.715916	0.0402098	0.0201049	+	0.999798i	$0.493600\pi$
$318$	0	0
$319$	−2.68188	−0.150157
$320$	0	0
$321$	−59.8173	−3.33867
$322$	0	0
$323$	14.8730	0.827555
$324$	0	0
$325$	2.57911	0.143063
$326$	0	0
$327$	40.3519	2.23146
$328$	0	0
$329$	−12.5653	−0.692750
$330$	0	0
$331$	33.3112	1.83095	0.915474	−	0.402377i	$-0.131816\pi$
$331$	33.3112	1.83095	0.915474	+	0.402377i	$0.131816\pi$
$332$	0	0
$333$	−45.2026	−2.47709
$334$	0	0
$335$	31.5211	1.72218
$336$	0	0
$337$	−1.00583	−0.0547912	−0.0273956	−	0.999625i	$-0.508721\pi$
$337$	−1.00583	−0.0547912	−0.0273956	+	0.999625i	$0.508721\pi$
$338$	0	0
$339$	−45.9095	−2.49346
$340$	0	0
$341$	6.52216	0.353195
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	48.2163	2.59588
$346$	0	0
$347$	−10.3214	−0.554081	−0.277040	−	0.960858i	$-0.589354\pi$
$347$	−10.3214	−0.554081	−0.277040	+	0.960858i	$0.589354\pi$
$348$	0	0
$349$	3.84238	0.205678	0.102839	−	0.994698i	$-0.467207\pi$
$349$	3.84238	0.205678	0.102839	+	0.994698i	$0.467207\pi$
$350$	0	0
$351$	15.3000	0.816654
$352$	0	0
$353$	−3.58462	−0.190790	−0.0953951	−	0.995439i	$-0.530411\pi$
$353$	−3.58462	−0.190790	−0.0953951	+	0.995439i	$0.530411\pi$
$354$	0	0
$355$	−30.1893	−1.60228
$356$	0	0
$357$	−21.7719	−1.15229
$358$	0	0
$359$	0.00330067	0.000174202 0	8.71012e−5	−	1.00000i	$-0.499972\pi$
$359$	0.00330067	0.000174202 0	8.71012e−5		1.00000i	$0.499972\pi$
$360$	0	0
$361$	−14.0154	−0.737652
$362$	0	0
$363$	3.26824	0.171538
$364$	0	0
$365$	−30.4977	−1.59632
$366$	0	0
$367$	6.31764	0.329778	0.164889	−	0.986312i	$-0.447273\pi$
$367$	6.31764	0.329778	0.164889	+	0.986312i	$0.447273\pi$
$368$	0	0
$369$	−73.5158	−3.82708
$370$	0	0
$371$	3.27646	0.170106
$372$	0	0
$373$	0.519831	0.0269158	0.0134579	−	0.999909i	$-0.495716\pi$
$373$	0.519831	0.0269158	0.0134579	+	0.999909i	$0.495716\pi$
$374$	0	0
$375$	−21.7820	−1.12482
$376$	0	0
$377$	2.68188	0.138124
$378$	0	0
$379$	1.85989	0.0955361	0.0477681	−	0.998858i	$-0.484789\pi$
$379$	1.85989	0.0955361	0.0477681	+	0.998858i	$0.484789\pi$
$380$	0	0
$381$	21.8224	1.11799
$382$	0	0
$383$	−3.38623	−0.173028	−0.0865140	−	0.996251i	$-0.527573\pi$
$383$	−3.38623	−0.173028	−0.0865140	+	0.996251i	$0.527573\pi$
$384$	0	0
$385$	2.75302	0.140307
$386$	0	0
$387$	−60.1804	−3.05914
$388$	0	0
$389$	−35.2284	−1.78615	−0.893076	−	0.449906i	$-0.851457\pi$
$389$	−35.2284	−1.78615	−0.893076	+	0.449906i	$0.851457\pi$
$390$	0	0
$391$	35.6987	1.80536
$392$	0	0
$393$	−40.2085	−2.02825
$394$	0	0
$395$	−29.2164	−1.47003
$396$	0	0
$397$	28.3876	1.42473	0.712367	−	0.701807i	$-0.247623\pi$
$397$	28.3876	1.42473	0.712367	+	0.701807i	$0.247623\pi$
$398$	0	0
$399$	−7.29676	−0.365295
$400$	0	0
$401$	16.5887	0.828399	0.414200	−	0.910186i	$-0.364062\pi$
$401$	16.5887	0.828399	0.414200	+	0.910186i	$0.364062\pi$
$402$	0	0
$403$	−6.52216	−0.324892
$404$	0	0
$405$	74.2210	3.68807
$406$	0	0
$407$	5.88468	0.291693
$408$	0	0
$409$	30.2121	1.49389	0.746946	−	0.664884i	$-0.231519\pi$
$409$	30.2121	1.49389	0.746946	+	0.664884i	$0.231519\pi$
$410$	0	0
$411$	55.1647	2.72107
$412$	0	0
$413$	7.10908	0.349815
$414$	0	0
$415$	−28.8331	−1.41536
$416$	0	0
$417$	−66.9489	−3.27850
$418$	0	0
$419$	21.8194	1.06595	0.532973	−	0.846132i	$-0.321075\pi$
$419$	21.8194	1.06595	0.532973	+	0.846132i	$0.321075\pi$
$420$	0	0
$421$	−15.3538	−0.748300	−0.374150	−	0.927368i	$-0.622065\pi$
$421$	−15.3538	−0.748300	−0.374150	+	0.927368i	$0.622065\pi$
$422$	0	0
$423$	96.5196	4.69295
$424$	0	0
$425$	17.1811	0.833408
$426$	0	0
$427$	−10.8960	−0.527293
$428$	0	0
$429$	−3.26824	−0.157792
$430$	0	0
$431$	13.6247	0.656280	0.328140	−	0.944629i	$-0.393578\pi$
$431$	13.6247	0.656280	0.328140	+	0.944629i	$0.393578\pi$
$432$	0	0
$433$	−32.3230	−1.55334	−0.776672	−	0.629905i	$-0.783094\pi$
$433$	−32.3230	−1.55334	−0.776672	+	0.629905i	$0.783094\pi$
$434$	0	0
$435$	24.1303	1.15696
$436$	0	0
$437$	11.9643	0.572329
$438$	0	0
$439$	26.4324	1.26155	0.630774	−	0.775967i	$-0.282738\pi$
$439$	26.4324	1.26155	0.630774	+	0.775967i	$0.282738\pi$
$440$	0	0
$441$	7.68141	0.365782
$442$	0	0
$443$	−3.54119	−0.168247	−0.0841234	−	0.996455i	$-0.526809\pi$
$443$	−3.54119	−0.168247	−0.0841234	+	0.996455i	$0.526809\pi$
$444$	0	0
$445$	6.69537	0.317391
$446$	0	0
$447$	−34.7229	−1.64234
$448$	0	0
$449$	32.2280	1.52093	0.760466	−	0.649377i	$-0.224970\pi$
$449$	32.2280	1.52093	0.760466	+	0.649377i	$0.224970\pi$
$450$	0	0
$451$	9.57061	0.450662
$452$	0	0
$453$	6.54722	0.307615
$454$	0	0
$455$	−2.75302	−0.129063
$456$	0	0
$457$	−19.0416	−0.890727	−0.445363	−	0.895350i	$-0.646926\pi$
$457$	−19.0416	−0.890727	−0.445363	+	0.895350i	$0.646926\pi$
$458$	0	0
$459$	101.923	4.75737
$460$	0	0
$461$	39.3280	1.83169	0.915843	−	0.401537i	$-0.131524\pi$
$461$	39.3280	1.83169	0.915843	+	0.401537i	$0.131524\pi$
$462$	0	0
$463$	−8.90218	−0.413719	−0.206860	−	0.978371i	$-0.566324\pi$
$463$	−8.90218	−0.413719	−0.206860	+	0.978371i	$0.566324\pi$
$464$	0	0
$465$	−58.6834	−2.72138
$466$	0	0
$467$	−24.5145	−1.13439	−0.567197	−	0.823582i	$-0.691972\pi$
$467$	−24.5145	−1.13439	−0.567197	+	0.823582i	$0.691972\pi$
$468$	0	0
$469$	−11.4496	−0.528696
$470$	0	0
$471$	75.1231	3.46149
$472$	0	0
$473$	7.83455	0.360233
$474$	0	0
$475$	5.75819	0.264204
$476$	0	0
$477$	−25.1679	−1.15236
$478$	0	0
$479$	−30.7375	−1.40443	−0.702216	−	0.711964i	$-0.747806\pi$
$479$	−30.7375	−1.40443	−0.702216	+	0.711964i	$0.747806\pi$
$480$	0	0
$481$	−5.88468	−0.268318
$482$	0	0
$483$	−17.5140	−0.796913
$484$	0	0
$485$	−16.4872	−0.748647
$486$	0	0
$487$	−17.7603	−0.804794	−0.402397	−	0.915465i	$-0.631823\pi$
$487$	−17.7603	−0.804794	−0.402397	+	0.915465i	$0.631823\pi$
$488$	0	0
$489$	−47.9425	−2.16803
$490$	0	0
$491$	−17.8988	−0.807763	−0.403882	−	0.914811i	$-0.632339\pi$
$491$	−17.8988	−0.807763	−0.403882	+	0.914811i	$0.632339\pi$
$492$	0	0
$493$	17.8658	0.804634
$494$	0	0
$495$	−21.1471	−0.950490
$496$	0	0
$497$	10.9659	0.491887
$498$	0	0
$499$	5.67091	0.253865	0.126932	−	0.991911i	$-0.459487\pi$
$499$	5.67091	0.253865	0.126932	+	0.991911i	$0.459487\pi$
$500$	0	0
$501$	1.02500	0.0457935
$502$	0	0
$503$	−5.11003	−0.227845	−0.113922	−	0.993490i	$-0.536342\pi$
$503$	−5.11003	−0.227845	−0.113922	+	0.993490i	$0.536342\pi$
$504$	0	0
$505$	−4.09614	−0.182276
$506$	0	0
$507$	3.26824	0.145148
$508$	0	0
$509$	−1.79758	−0.0796763	−0.0398381	−	0.999206i	$-0.512684\pi$
$509$	−1.79758	−0.0796763	−0.0398381	+	0.999206i	$0.512684\pi$
$510$	0	0
$511$	11.0779	0.490059
$512$	0	0
$513$	34.1592	1.50816
$514$	0	0
$515$	31.5382	1.38974
$516$	0	0
$517$	−12.5653	−0.552623
$518$	0	0
$519$	38.6579	1.69689
$520$	0	0
$521$	−23.4890	−1.02907	−0.514535	−	0.857469i	$-0.672035\pi$
$521$	−23.4890	−1.02907	−0.514535	+	0.857469i	$0.672035\pi$
$522$	0	0
$523$	39.4965	1.72706	0.863530	−	0.504297i	$-0.168248\pi$
$523$	39.4965	1.72706	0.863530	+	0.504297i	$0.168248\pi$
$524$	0	0
$525$	−8.42916	−0.367879
$526$	0	0
$527$	−43.4484	−1.89264
$528$	0	0
$529$	5.71710	0.248570
$530$	0	0
$531$	−54.6078	−2.36978
$532$	0	0
$533$	−9.57061	−0.414549
$534$	0	0
$535$	−50.3873	−2.17843
$536$	0	0
$537$	46.9180	2.02466
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	2.03823	0.0876304	0.0438152	−	0.999040i	$-0.486049\pi$
$541$	2.03823	0.0876304	0.0438152	+	0.999040i	$0.486049\pi$
$542$	0	0
$543$	63.6081	2.72969
$544$	0	0
$545$	33.9906	1.45600
$546$	0	0
$547$	−24.5519	−1.04976	−0.524882	−	0.851175i	$-0.675891\pi$
$547$	−24.5519	−1.04976	−0.524882	+	0.851175i	$0.675891\pi$
$548$	0	0
$549$	83.6964	3.57208
$550$	0	0
$551$	5.98764	0.255082
$552$	0	0
$553$	10.6125	0.451289
$554$	0	0
$555$	−52.9476	−2.24750
$556$	0	0
$557$	−27.3699	−1.15970	−0.579850	−	0.814723i	$-0.696889\pi$
$557$	−27.3699	−1.15970	−0.579850	+	0.814723i	$0.696889\pi$
$558$	0	0
$559$	−7.83455	−0.331366
$560$	0	0
$561$	−21.7719	−0.919211
$562$	0	0
$563$	−25.4770	−1.07373	−0.536863	−	0.843670i	$-0.680391\pi$
$563$	−25.4770	−1.07373	−0.536863	+	0.843670i	$0.680391\pi$
$564$	0	0
$565$	−38.6720	−1.62694
$566$	0	0
$567$	−26.9599	−1.13221
$568$	0	0
$569$	3.25396	0.136413	0.0682066	−	0.997671i	$-0.478272\pi$
$569$	3.25396	0.136413	0.0682066	+	0.997671i	$0.478272\pi$
$570$	0	0
$571$	−34.5283	−1.44496	−0.722481	−	0.691390i	$-0.756999\pi$
$571$	−34.5283	−1.44496	−0.722481	+	0.691390i	$0.756999\pi$
$572$	0	0
$573$	−12.2379	−0.511246
$574$	0	0
$575$	13.8210	0.576377
$576$	0	0
$577$	−4.31488	−0.179631	−0.0898153	−	0.995958i	$-0.528628\pi$
$577$	−4.31488	−0.179631	−0.0898153	+	0.995958i	$0.528628\pi$
$578$	0	0
$579$	39.7864	1.65347
$580$	0	0
$581$	10.4733	0.434505
$582$	0	0
$583$	3.27646	0.135697
$584$	0	0
$585$	21.1471	0.874324
$586$	0	0
$587$	−18.9078	−0.780409	−0.390205	−	0.920728i	$-0.627596\pi$
$587$	−18.9078	−0.780409	−0.390205	+	0.920728i	$0.627596\pi$
$588$	0	0
$589$	−14.5615	−0.599998
$590$	0	0
$591$	−44.0847	−1.81340
$592$	0	0
$593$	33.8443	1.38982	0.694909	−	0.719097i	$-0.255444\pi$
$593$	33.8443	1.38982	0.694909	+	0.719097i	$0.255444\pi$
$594$	0	0
$595$	−18.3397	−0.751853
$596$	0	0
$597$	75.3085	3.08217
$598$	0	0
$599$	10.9222	0.446267	0.223134	−	0.974788i	$-0.428371\pi$
$599$	10.9222	0.446267	0.223134	+	0.974788i	$0.428371\pi$
$600$	0	0
$601$	−8.65265	−0.352949	−0.176474	−	0.984305i	$-0.556469\pi$
$601$	−8.65265	−0.352949	−0.176474	+	0.984305i	$0.556469\pi$
$602$	0	0
$603$	87.9495	3.58158
$604$	0	0
$605$	2.75302	0.111926
$606$	0	0
$607$	11.0619	0.448988	0.224494	−	0.974475i	$-0.427927\pi$
$607$	11.0619	0.448988	0.224494	+	0.974475i	$0.427927\pi$
$608$	0	0
$609$	−8.76505	−0.355178
$610$	0	0
$611$	12.5653	0.508340
$612$	0	0
$613$	−28.9290	−1.16843	−0.584216	−	0.811598i	$-0.698598\pi$
$613$	−28.9290	−1.16843	−0.584216	+	0.811598i	$0.698598\pi$
$614$	0	0
$615$	−86.1119	−3.47237
$616$	0	0
$617$	−36.6999	−1.47748	−0.738741	−	0.673989i	$-0.764579\pi$
$617$	−36.6999	−1.47748	−0.738741	+	0.673989i	$0.764579\pi$
$618$	0	0
$619$	−27.6326	−1.11065	−0.555323	−	0.831635i	$-0.687405\pi$
$619$	−27.6326	−1.11065	−0.555323	+	0.831635i	$0.687405\pi$
$620$	0	0
$621$	81.9902	3.29015
$622$	0	0
$623$	−2.43201	−0.0974365
$624$	0	0
$625$	−31.2437	−1.24975
$626$	0	0
$627$	−7.29676	−0.291405
$628$	0	0
$629$	−39.2017	−1.56307
$630$	0	0
$631$	27.6036	1.09888	0.549440	−	0.835533i	$-0.314841\pi$
$631$	27.6036	1.09888	0.549440	+	0.835533i	$0.314841\pi$
$632$	0	0
$633$	−29.5777	−1.17561
$634$	0	0
$635$	18.3822	0.729474
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−84.2334	−3.33222
$640$	0	0
$641$	−33.7084	−1.33140	−0.665701	−	0.746218i	$-0.731867\pi$
$641$	−33.7084	−1.33140	−0.665701	+	0.746218i	$0.731867\pi$
$642$	0	0
$643$	−14.6561	−0.577982	−0.288991	−	0.957332i	$-0.593320\pi$
$643$	−14.6561	−0.577982	−0.288991	+	0.957332i	$0.593320\pi$
$644$	0	0
$645$	−70.4916	−2.77561
$646$	0	0
$647$	−7.51969	−0.295629	−0.147815	−	0.989015i	$-0.547224\pi$
$647$	−7.51969	−0.295629	−0.147815	+	0.989015i	$0.547224\pi$
$648$	0	0
$649$	7.10908	0.279056
$650$	0	0
$651$	21.3160	0.835440
$652$	0	0
$653$	−34.5195	−1.35085	−0.675426	−	0.737427i	$-0.736040\pi$
$653$	−34.5195	−1.35085	−0.675426	+	0.737427i	$0.736040\pi$
$654$	0	0
$655$	−33.8698	−1.32340
$656$	0	0
$657$	−85.0941	−3.31984
$658$	0	0
$659$	−41.7416	−1.62602	−0.813011	−	0.582248i	$-0.802173\pi$
$659$	−41.7416	−1.62602	−0.813011	+	0.582248i	$0.802173\pi$
$660$	0	0
$661$	27.3663	1.06442	0.532212	−	0.846611i	$-0.321361\pi$
$661$	27.3663	1.06442	0.532212	+	0.846611i	$0.321361\pi$
$662$	0	0
$663$	21.7719	0.845551
$664$	0	0
$665$	−6.14646	−0.238349
$666$	0	0
$667$	14.3718	0.556477
$668$	0	0
$669$	−15.8526	−0.612895
$670$	0	0
$671$	−10.8960	−0.420634
$672$	0	0
$673$	−27.5400	−1.06159	−0.530795	−	0.847500i	$-0.678107\pi$
$673$	−27.5400	−1.06159	−0.530795	+	0.847500i	$0.678107\pi$
$674$	0	0
$675$	39.4604	1.51883
$676$	0	0
$677$	2.17216	0.0834829	0.0417415	−	0.999128i	$-0.486709\pi$
$677$	2.17216	0.0834829	0.0417415	+	0.999128i	$0.486709\pi$
$678$	0	0
$679$	5.98878	0.229828
$680$	0	0
$681$	8.22088	0.315025
$682$	0	0
$683$	29.3156	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$683$	29.3156	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$684$	0	0
$685$	46.4682	1.77546
$686$	0	0
$687$	−0.605129	−0.0230871
$688$	0	0
$689$	−3.27646	−0.124823
$690$	0	0
$691$	−36.0554	−1.37161	−0.685806	−	0.727785i	$-0.740550\pi$
$691$	−36.0554	−1.37161	−0.685806	+	0.727785i	$0.740550\pi$
$692$	0	0
$693$	7.68141	0.291793
$694$	0	0
$695$	−56.3947	−2.13917
$696$	0	0
$697$	−63.7561	−2.41493
$698$	0	0
$699$	79.7656	3.01701
$700$	0	0
$701$	−6.43537	−0.243061	−0.121530	−	0.992588i	$-0.538780\pi$
$701$	−6.43537	−0.243061	−0.121530	+	0.992588i	$0.538780\pi$
$702$	0	0
$703$	−13.1383	−0.495520
$704$	0	0
$705$	113.057	4.25798
$706$	0	0
$707$	1.48787	0.0559571
$708$	0	0
$709$	11.6784	0.438593	0.219296	−	0.975658i	$-0.429624\pi$
$709$	11.6784	0.438593	0.219296	+	0.975658i	$0.429624\pi$
$710$	0	0
$711$	−81.5189	−3.05720
$712$	0	0
$713$	−34.9512	−1.30893
$714$	0	0
$715$	−2.75302	−0.102957
$716$	0	0
$717$	70.4378	2.63055
$718$	0	0
$719$	−4.58160	−0.170865	−0.0854325	−	0.996344i	$-0.527227\pi$
$719$	−4.58160	−0.170865	−0.0854325	+	0.996344i	$0.527227\pi$
$720$	0	0
$721$	−11.4559	−0.426638
$722$	0	0
$723$	10.6529	0.396185
$724$	0	0
$725$	6.91687	0.256886
$726$	0	0
$727$	16.6131	0.616147	0.308074	−	0.951362i	$-0.400316\pi$
$727$	16.6131	0.616147	0.308074	+	0.951362i	$0.400316\pi$
$728$	0	0
$729$	57.0776	2.11399
$730$	0	0
$731$	−52.1911	−1.93036
$732$	0	0
$733$	26.2503	0.969575	0.484788	−	0.874632i	$-0.338897\pi$
$733$	26.2503	0.969575	0.484788	+	0.874632i	$0.338897\pi$
$734$	0	0
$735$	8.99753	0.331879
$736$	0	0
$737$	−11.4496	−0.421753
$738$	0	0
$739$	28.8360	1.06075	0.530374	−	0.847764i	$-0.322051\pi$
$739$	28.8360	1.06075	0.530374	+	0.847764i	$0.322051\pi$
$740$	0	0
$741$	7.29676	0.268053
$742$	0	0
$743$	−13.4231	−0.492447	−0.246223	−	0.969213i	$-0.579190\pi$
$743$	−13.4231	−0.492447	−0.246223	+	0.969213i	$0.579190\pi$
$744$	0	0
$745$	−29.2490	−1.07160
$746$	0	0
$747$	−80.4496	−2.94350
$748$	0	0
$749$	18.3026	0.668762
$750$	0	0
$751$	−50.9163	−1.85796	−0.928981	−	0.370127i	$-0.879314\pi$
$751$	−50.9163	−1.85796	−0.928981	+	0.370127i	$0.879314\pi$
$752$	0	0
$753$	62.6232	2.28212
$754$	0	0
$755$	5.51508	0.200714
$756$	0	0
$757$	−26.7606	−0.972632	−0.486316	−	0.873783i	$-0.661660\pi$
$757$	−26.7606	−0.972632	−0.486316	+	0.873783i	$0.661660\pi$
$758$	0	0
$759$	−17.5140	−0.635717
$760$	0	0
$761$	29.9302	1.08497	0.542485	−	0.840065i	$-0.317483\pi$
$761$	29.9302	1.08497	0.542485	+	0.840065i	$0.317483\pi$
$762$	0	0
$763$	−12.3467	−0.446979
$764$	0	0
$765$	140.874	5.09333
$766$	0	0
$767$	−7.10908	−0.256694
$768$	0	0
$769$	15.5035	0.559071	0.279535	−	0.960135i	$-0.409820\pi$
$769$	15.5035	0.559071	0.279535	+	0.960135i	$0.409820\pi$
$770$	0	0
$771$	78.5690	2.82959
$772$	0	0
$773$	23.9002	0.859631	0.429816	−	0.902917i	$-0.358579\pi$
$773$	23.9002	0.859631	0.429816	+	0.902917i	$0.358579\pi$
$774$	0	0
$775$	−16.8214	−0.604242
$776$	0	0
$777$	19.2326	0.689964
$778$	0	0
$779$	−21.3676	−0.765573
$780$	0	0
$781$	10.9659	0.392390
$782$	0	0
$783$	41.0328	1.46639
$784$	0	0
$785$	63.2803	2.25857
$786$	0	0
$787$	16.2525	0.579338	0.289669	−	0.957127i	$-0.406455\pi$
$787$	16.2525	0.579338	0.289669	+	0.957127i	$0.406455\pi$
$788$	0	0
$789$	−27.2005	−0.968363
$790$	0	0
$791$	14.0471	0.499459
$792$	0	0
$793$	10.8960	0.386927
$794$	0	0
$795$	−29.4801	−1.04555
$796$	0	0
$797$	12.3933	0.438994	0.219497	−	0.975613i	$-0.429558\pi$
$797$	12.3933	0.438994	0.219497	+	0.975613i	$0.429558\pi$
$798$	0	0
$799$	83.7060	2.96131
$800$	0	0
$801$	18.6813	0.660071
$802$	0	0
$803$	11.0779	0.390932
$804$	0	0
$805$	−14.7530	−0.519974
$806$	0	0
$807$	−17.4667	−0.614857
$808$	0	0
$809$	−6.70826	−0.235850	−0.117925	−	0.993023i	$-0.537624\pi$
$809$	−6.70826	−0.235850	−0.117925	+	0.993023i	$0.537624\pi$
$810$	0	0
$811$	22.7647	0.799376	0.399688	−	0.916651i	$-0.369118\pi$
$811$	22.7647	0.799376	0.399688	+	0.916651i	$0.369118\pi$
$812$	0	0
$813$	−78.3478	−2.74778
$814$	0	0
$815$	−40.3846	−1.41461
$816$	0	0
$817$	−17.4916	−0.611954
$818$	0	0
$819$	−7.68141	−0.268410
$820$	0	0
$821$	28.9984	1.01205	0.506026	−	0.862518i	$-0.331114\pi$
$821$	28.9984	1.01205	0.506026	+	0.862518i	$0.331114\pi$
$822$	0	0
$823$	−22.9275	−0.799202	−0.399601	−	0.916689i	$-0.630851\pi$
$823$	−22.9275	−0.799202	−0.399601	+	0.916689i	$0.630851\pi$
$824$	0	0
$825$	−8.42916	−0.293466
$826$	0	0
$827$	−42.7646	−1.48707	−0.743535	−	0.668697i	$-0.766852\pi$
$827$	−42.7646	−1.48707	−0.743535	+	0.668697i	$0.766852\pi$
$828$	0	0
$829$	12.6059	0.437822	0.218911	−	0.975745i	$-0.429750\pi$
$829$	12.6059	0.437822	0.218911	+	0.975745i	$0.429750\pi$
$830$	0	0
$831$	−9.51036	−0.329911
$832$	0	0
$833$	6.66165	0.230813
$834$	0	0
$835$	0.863410	0.0298795
$836$	0	0
$837$	−99.7891	−3.44921
$838$	0	0
$839$	−10.4653	−0.361303	−0.180651	−	0.983547i	$-0.557821\pi$
$839$	−10.4653	−0.361303	−0.180651	+	0.983547i	$0.557821\pi$
$840$	0	0
$841$	−21.8075	−0.751983
$842$	0	0
$843$	9.26947	0.319258
$844$	0	0
$845$	2.75302	0.0947067
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	12.2265	0.419614
$850$	0	0
$851$	−31.5350	−1.08101
$852$	0	0
$853$	42.4511	1.45350	0.726749	−	0.686903i	$-0.241030\pi$
$853$	42.4511	1.45350	0.726749	+	0.686903i	$0.241030\pi$
$854$	0	0
$855$	47.2135	1.61467
$856$	0	0
$857$	35.3972	1.20915	0.604573	−	0.796550i	$-0.293344\pi$
$857$	35.3972	1.20915	0.604573	+	0.796550i	$0.293344\pi$
$858$	0	0
$859$	−48.3507	−1.64971	−0.824853	−	0.565348i	$-0.808742\pi$
$859$	−48.3507	−1.64971	−0.824853	+	0.565348i	$0.808742\pi$
$860$	0	0
$861$	31.2791	1.06599
$862$	0	0
$863$	28.8195	0.981026	0.490513	−	0.871434i	$-0.336809\pi$
$863$	28.8195	0.981026	0.490513	+	0.871434i	$0.336809\pi$
$864$	0	0
$865$	32.5636	1.10720
$866$	0	0
$867$	89.4768	3.03879
$868$	0	0
$869$	10.6125	0.360004
$870$	0	0
$871$	11.4496	0.387956
$872$	0	0
$873$	−46.0023	−1.55694
$874$	0	0
$875$	6.66475	0.225310
$876$	0	0
$877$	31.3636	1.05907	0.529537	−	0.848287i	$-0.322366\pi$
$877$	31.3636	1.05907	0.529537	+	0.848287i	$0.322366\pi$
$878$	0	0
$879$	−68.9573	−2.32587
$880$	0	0
$881$	−16.4236	−0.553326	−0.276663	−	0.960967i	$-0.589229\pi$
$881$	−16.4236	−0.553326	−0.276663	+	0.960967i	$0.589229\pi$
$882$	0	0
$883$	46.8549	1.57679	0.788396	−	0.615168i	$-0.210912\pi$
$883$	46.8549	1.57679	0.788396	+	0.615168i	$0.210912\pi$
$884$	0	0
$885$	−63.9642	−2.15013
$886$	0	0
$887$	−6.50852	−0.218535	−0.109267	−	0.994012i	$-0.534850\pi$
$887$	−6.50852	−0.218535	−0.109267	+	0.994012i	$0.534850\pi$
$888$	0	0
$889$	−6.67710	−0.223943
$890$	0	0
$891$	−26.9599	−0.903190
$892$	0	0
$893$	28.0537	0.938782
$894$	0	0
$895$	39.5216	1.32106
$896$	0	0
$897$	17.5140	0.584775
$898$	0	0
$899$	−17.4917	−0.583380
$900$	0	0
$901$	−21.8267	−0.727152
$902$	0	0
$903$	25.6052	0.852088
$904$	0	0
$905$	53.5806	1.78108
$906$	0	0
$907$	36.6797	1.21793	0.608964	−	0.793198i	$-0.291585\pi$
$907$	36.6797	1.21793	0.608964	+	0.793198i	$0.291585\pi$
$908$	0	0
$909$	−11.4290	−0.379074
$910$	0	0
$911$	32.1842	1.06631	0.533155	−	0.846018i	$-0.321006\pi$
$911$	32.1842	1.06631	0.533155	+	0.846018i	$0.321006\pi$
$912$	0	0
$913$	10.4733	0.346615
$914$	0	0
$915$	98.0368	3.24100
$916$	0	0
$917$	12.3028	0.406274
$918$	0	0
$919$	52.6452	1.73660	0.868302	−	0.496036i	$-0.165212\pi$
$919$	52.6452	1.73660	0.868302	+	0.496036i	$0.165212\pi$
$920$	0	0
$921$	−72.9542	−2.40392
$922$	0	0
$923$	−10.9659	−0.360946
$924$	0	0
$925$	−15.1772	−0.499024
$926$	0	0
$927$	87.9972	2.89021
$928$	0	0
$929$	32.7512	1.07453	0.537265	−	0.843413i	$-0.319457\pi$
$929$	32.7512	1.07453	0.537265	+	0.843413i	$0.319457\pi$
$930$	0	0
$931$	2.23263	0.0731713
$932$	0	0
$933$	−46.0460	−1.50748
$934$	0	0
$935$	−18.3397	−0.599771
$936$	0	0
$937$	−1.43983	−0.0470373	−0.0235186	−	0.999723i	$-0.507487\pi$
$937$	−1.43983	−0.0470373	−0.0235186	+	0.999723i	$0.507487\pi$
$938$	0	0
$939$	−74.4580	−2.42984
$940$	0	0
$941$	−0.0905142	−0.00295068	−0.00147534	−	0.999999i	$-0.500470\pi$
$941$	−0.0905142	−0.00295068	−0.00147534	+	0.999999i	$0.500470\pi$
$942$	0	0
$943$	−51.2873	−1.67014
$944$	0	0
$945$	−42.1212	−1.37020
$946$	0	0
$947$	−35.8930	−1.16637	−0.583183	−	0.812341i	$-0.698193\pi$
$947$	−35.8930	−1.16637	−0.583183	+	0.812341i	$0.698193\pi$
$948$	0	0
$949$	−11.0779	−0.359605
$950$	0	0
$951$	2.33979	0.0758728
$952$	0	0
$953$	26.5767	0.860902	0.430451	−	0.902614i	$-0.358354\pi$
$953$	26.5767	0.860902	0.430451	+	0.902614i	$0.358354\pi$
$954$	0	0
$955$	−10.3087	−0.333580
$956$	0	0
$957$	−8.76505	−0.283334
$958$	0	0
$959$	−16.8790	−0.545051
$960$	0	0
$961$	11.5386	0.372213
$962$	0	0
$963$	−140.590	−4.53044
$964$	0	0
$965$	33.5143	1.07886
$966$	0	0
$967$	−40.1957	−1.29261	−0.646303	−	0.763081i	$-0.723686\pi$
$967$	−40.1957	−1.29261	−0.646303	+	0.763081i	$0.723686\pi$
$968$	0	0
$969$	48.6085	1.56153
$970$	0	0
$971$	26.0946	0.837416	0.418708	−	0.908121i	$-0.362483\pi$
$971$	26.0946	0.837416	0.418708	+	0.908121i	$0.362483\pi$
$972$	0	0
$973$	20.4847	0.656708
$974$	0	0
$975$	8.42916	0.269949
$976$	0	0
$977$	20.2396	0.647522	0.323761	−	0.946139i	$-0.395053\pi$
$977$	20.2396	0.647522	0.323761	+	0.946139i	$0.395053\pi$
$978$	0	0
$979$	−2.43201	−0.0777274
$980$	0	0
$981$	94.8397	3.02800
$982$	0	0
$983$	−35.2267	−1.12356	−0.561778	−	0.827288i	$-0.689883\pi$
$983$	−35.2267	−1.12356	−0.561778	+	0.827288i	$0.689883\pi$
$984$	0	0
$985$	−37.1349	−1.18322
$986$	0	0
$987$	−41.0666	−1.30716
$988$	0	0
$989$	−41.9841	−1.33502
$990$	0	0
$991$	−15.3245	−0.486797	−0.243399	−	0.969926i	$-0.578262\pi$
$991$	−15.3245	−0.486797	−0.243399	+	0.969926i	$0.578262\pi$
$992$	0	0
$993$	108.869	3.45486
$994$	0	0
$995$	63.4364	2.01107
$996$	0	0
$997$	22.4955	0.712441	0.356220	−	0.934402i	$-0.384065\pi$
$997$	22.4955	0.712441	0.356220	+	0.934402i	$0.384065\pi$
$998$	0	0
$999$	−90.0355	−2.84860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.15	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.15	✓	15	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.26824 q^{3} +2.75302 q^{5} -1.00000 q^{7} +7.68141 q^{9} +O(q^{10})\)	\(q+3.26824 q^{3} +2.75302 q^{5} -1.00000 q^{7} +7.68141 q^{9} -1.00000 q^{11} +1.00000 q^{13} +8.99753 q^{15} +6.66165 q^{17} +2.23263 q^{19} -3.26824 q^{21} +5.35883 q^{23} +2.57911 q^{25} +15.3000 q^{27} +2.68188 q^{29} -6.52216 q^{31} -3.26824 q^{33} -2.75302 q^{35} -5.88468 q^{37} +3.26824 q^{39} -9.57061 q^{41} -7.83455 q^{43} +21.1471 q^{45} +12.5653 q^{47} +1.00000 q^{49} +21.7719 q^{51} -3.27646 q^{53} -2.75302 q^{55} +7.29676 q^{57} -7.10908 q^{59} +10.8960 q^{61} -7.68141 q^{63} +2.75302 q^{65} +11.4496 q^{67} +17.5140 q^{69} -10.9659 q^{71} -11.0779 q^{73} +8.42916 q^{75} +1.00000 q^{77} -10.6125 q^{79} +26.9599 q^{81} -10.4733 q^{83} +18.3397 q^{85} +8.76505 q^{87} +2.43201 q^{89} -1.00000 q^{91} -21.3160 q^{93} +6.14646 q^{95} -5.98878 q^{97} -7.68141 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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