LMFDB - Embedded newform 8008.2.a.y.1.13

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.46105$ 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)

$f(q)$	$=$	$q+2.46105 q^{3} -1.12857 q^{5} +1.00000 q^{7} +3.05676 q^{9} +O(q^{10})$	$q+2.46105 q^{3} -1.12857 q^{5} +1.00000 q^{7} +3.05676 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.77747 q^{15} +6.31528 q^{17} -1.83477 q^{19} +2.46105 q^{21} -8.30939 q^{23} -3.72632 q^{25} +0.139682 q^{27} -7.64782 q^{29} -5.82327 q^{31} -2.46105 q^{33} -1.12857 q^{35} +4.38927 q^{37} -2.46105 q^{39} -1.61087 q^{41} -9.78285 q^{43} -3.44977 q^{45} +3.50039 q^{47} +1.00000 q^{49} +15.5422 q^{51} +10.6895 q^{53} +1.12857 q^{55} -4.51547 q^{57} -11.5145 q^{59} +7.27639 q^{61} +3.05676 q^{63} +1.12857 q^{65} -10.5388 q^{67} -20.4498 q^{69} +13.2627 q^{71} -14.6993 q^{73} -9.17066 q^{75} -1.00000 q^{77} -8.87583 q^{79} -8.82651 q^{81} -14.7237 q^{83} -7.12725 q^{85} -18.8216 q^{87} +5.76039 q^{89} -1.00000 q^{91} -14.3313 q^{93} +2.07068 q^{95} -7.88319 q^{97} -3.05676 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	2.46105	1.42089	0.710443	−	0.703754i	$-0.248494\pi$
$3$	2.46105	1.42089	0.710443	+	0.703754i	$0.248494\pi$
$4$	0	0
$5$	−1.12857	−0.504713	−0.252357	−	0.967634i	$-0.581206\pi$
$5$	−1.12857	−0.504713	−0.252357	+	0.967634i	$0.581206\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.05676	1.01892
$10$	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
$15$	−2.77747	−0.717140
$16$	0	0
$17$	6.31528	1.53168	0.765840	−	0.643032i	$-0.222324\pi$
$17$	6.31528	1.53168	0.765840	+	0.643032i	$0.222324\pi$
$18$	0	0
$19$	−1.83477	−0.420926	−0.210463	−	0.977602i	$-0.567497\pi$
$19$	−1.83477	−0.420926	−0.210463	+	0.977602i	$0.567497\pi$
$20$	0	0
$21$	2.46105	0.537045
$22$	0	0
$23$	−8.30939	−1.73263	−0.866314	−	0.499500i	$-0.833517\pi$
$23$	−8.30939	−1.73263	−0.866314	+	0.499500i	$0.833517\pi$
$24$	0	0
$25$	−3.72632	−0.745265
$26$	0	0
$27$	0.139682	0.0268819
$28$	0	0
$29$	−7.64782	−1.42016	−0.710082	−	0.704119i	$-0.751342\pi$
$29$	−7.64782	−1.42016	−0.710082	+	0.704119i	$0.751342\pi$
$30$	0	0
$31$	−5.82327	−1.04589	−0.522945	−	0.852367i	$-0.675167\pi$
$31$	−5.82327	−1.04589	−0.522945	+	0.852367i	$0.675167\pi$
$32$	0	0
$33$	−2.46105	−0.428413
$34$	0	0
$35$	−1.12857	−0.190764
$36$	0	0
$37$	4.38927	0.721592	0.360796	−	0.932645i	$-0.382505\pi$
$37$	4.38927	0.721592	0.360796	+	0.932645i	$0.382505\pi$
$38$	0	0
$39$	−2.46105	−0.394083
$40$	0	0
$41$	−1.61087	−0.251576	−0.125788	−	0.992057i	$-0.540146\pi$
$41$	−1.61087	−0.251576	−0.125788	+	0.992057i	$0.540146\pi$
$42$	0	0
$43$	−9.78285	−1.49187	−0.745936	−	0.666018i	$-0.767997\pi$
$43$	−9.78285	−1.49187	−0.745936	+	0.666018i	$0.767997\pi$
$44$	0	0
$45$	−3.44977	−0.514262
$46$	0	0
$47$	3.50039	0.510584	0.255292	−	0.966864i	$-0.417828\pi$
$47$	3.50039	0.510584	0.255292	+	0.966864i	$0.417828\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.5422	2.17634
$52$	0	0
$53$	10.6895	1.46832	0.734160	−	0.678976i	$-0.237576\pi$
$53$	10.6895	1.46832	0.734160	+	0.678976i	$0.237576\pi$
$54$	0	0
$55$	1.12857	0.152177
$56$	0	0
$57$	−4.51547	−0.598088
$58$	0	0
$59$	−11.5145	−1.49906	−0.749532	−	0.661968i	$-0.769722\pi$
$59$	−11.5145	−1.49906	−0.749532	+	0.661968i	$0.769722\pi$
$60$	0	0
$61$	7.27639	0.931646	0.465823	−	0.884878i	$-0.345758\pi$
$61$	7.27639	0.931646	0.465823	+	0.884878i	$0.345758\pi$
$62$	0	0
$63$	3.05676	0.385115
$64$	0	0
$65$	1.12857	0.139982
$66$	0	0
$67$	−10.5388	−1.28752	−0.643762	−	0.765226i	$-0.722627\pi$
$67$	−10.5388	−1.28752	−0.643762	+	0.765226i	$0.722627\pi$
$68$	0	0
$69$	−20.4498	−2.46187
$70$	0	0
$71$	13.2627	1.57399	0.786997	−	0.616957i	$-0.211635\pi$
$71$	13.2627	1.57399	0.786997	+	0.616957i	$0.211635\pi$
$72$	0	0
$73$	−14.6993	−1.72042	−0.860209	−	0.509942i	$-0.829667\pi$
$73$	−14.6993	−1.72042	−0.860209	+	0.509942i	$0.829667\pi$
$74$	0	0
$75$	−9.17066	−1.05894
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−8.87583	−0.998609	−0.499304	−	0.866427i	$-0.666411\pi$
$79$	−8.87583	−0.998609	−0.499304	+	0.866427i	$0.666411\pi$
$80$	0	0
$81$	−8.82651	−0.980723
$82$	0	0
$83$	−14.7237	−1.61614	−0.808068	−	0.589089i	$-0.799487\pi$
$83$	−14.7237	−1.61614	−0.808068	+	0.589089i	$0.799487\pi$
$84$	0	0
$85$	−7.12725	−0.773059
$86$	0	0
$87$	−18.8216	−2.01789
$88$	0	0
$89$	5.76039	0.610600	0.305300	−	0.952256i	$-0.401243\pi$
$89$	5.76039	0.610600	0.305300	+	0.952256i	$0.401243\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−14.3313	−1.48609
$94$	0	0
$95$	2.07068	0.212447
$96$	0	0
$97$	−7.88319	−0.800416	−0.400208	−	0.916424i	$-0.631062\pi$
$97$	−7.88319	−0.800416	−0.400208	+	0.916424i	$0.631062\pi$
$98$	0	0
$99$	−3.05676	−0.307216
$100$	0	0
$101$	11.6940	1.16359	0.581797	−	0.813334i	$-0.302350\pi$
$101$	11.6940	1.16359	0.581797	+	0.813334i	$0.302350\pi$
$102$	0	0
$103$	14.2920	1.40824	0.704118	−	0.710083i	$-0.251343\pi$
$103$	14.2920	1.40824	0.704118	+	0.710083i	$0.251343\pi$
$104$	0	0
$105$	−2.77747	−0.271053
$106$	0	0
$107$	−17.6760	−1.70881	−0.854404	−	0.519610i	$-0.826077\pi$
$107$	−17.6760	−1.70881	−0.854404	+	0.519610i	$0.826077\pi$
$108$	0	0
$109$	13.7778	1.31967	0.659837	−	0.751408i	$-0.270625\pi$
$109$	13.7778	1.31967	0.659837	+	0.751408i	$0.270625\pi$
$110$	0	0
$111$	10.8022	1.02530
$112$	0	0
$113$	4.96334	0.466911	0.233456	−	0.972367i	$-0.424997\pi$
$113$	4.96334	0.466911	0.233456	+	0.972367i	$0.424997\pi$
$114$	0	0
$115$	9.37775	0.874480
$116$	0	0
$117$	−3.05676	−0.282597
$118$	0	0
$119$	6.31528	0.578920
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.96444	−0.357461
$124$	0	0
$125$	9.84829	0.880858
$126$	0	0
$127$	−14.9712	−1.32848	−0.664240	−	0.747519i	$-0.731245\pi$
$127$	−14.9712	−1.32848	−0.664240	+	0.747519i	$0.731245\pi$
$128$	0	0
$129$	−24.0761	−2.11978
$130$	0	0
$131$	15.3791	1.34368	0.671838	−	0.740698i	$-0.265505\pi$
$131$	15.3791	1.34368	0.671838	+	0.740698i	$0.265505\pi$
$132$	0	0
$133$	−1.83477	−0.159095
$134$	0	0
$135$	−0.157642	−0.0135676
$136$	0	0
$137$	−12.0238	−1.02726	−0.513632	−	0.858010i	$-0.671700\pi$
$137$	−12.0238	−1.02726	−0.513632	+	0.858010i	$0.671700\pi$
$138$	0	0
$139$	20.9661	1.77832	0.889162	−	0.457593i	$-0.151288\pi$
$139$	20.9661	1.77832	0.889162	+	0.457593i	$0.151288\pi$
$140$	0	0
$141$	8.61462	0.725482
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	8.63112	0.716775
$146$	0	0
$147$	2.46105	0.202984
$148$	0	0
$149$	−14.6526	−1.20039	−0.600195	−	0.799854i	$-0.704910\pi$
$149$	−14.6526	−1.20039	−0.600195	+	0.799854i	$0.704910\pi$
$150$	0	0
$151$	13.0826	1.06465	0.532323	−	0.846542i	$-0.321319\pi$
$151$	13.0826	1.06465	0.532323	+	0.846542i	$0.321319\pi$
$152$	0	0
$153$	19.3043	1.56066
$154$	0	0
$155$	6.57198	0.527874
$156$	0	0
$157$	−1.44812	−0.115572	−0.0577862	−	0.998329i	$-0.518404\pi$
$157$	−1.44812	−0.115572	−0.0577862	+	0.998329i	$0.518404\pi$
$158$	0	0
$159$	26.3075	2.08632
$160$	0	0
$161$	−8.30939	−0.654872
$162$	0	0
$163$	6.97261	0.546137	0.273069	−	0.961995i	$-0.411961\pi$
$163$	6.97261	0.546137	0.273069	+	0.961995i	$0.411961\pi$
$164$	0	0
$165$	2.77747	0.216226
$166$	0	0
$167$	5.02380	0.388754	0.194377	−	0.980927i	$-0.437732\pi$
$167$	5.02380	0.388754	0.194377	+	0.980927i	$0.437732\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.60846	−0.428890
$172$	0	0
$173$	−12.1426	−0.923184	−0.461592	−	0.887092i	$-0.652722\pi$
$173$	−12.1426	−0.923184	−0.461592	+	0.887092i	$0.652722\pi$
$174$	0	0
$175$	−3.72632	−0.281684
$176$	0	0
$177$	−28.3378	−2.13000
$178$	0	0
$179$	−6.10770	−0.456511	−0.228255	−	0.973601i	$-0.573302\pi$
$179$	−6.10770	−0.456511	−0.228255	+	0.973601i	$0.573302\pi$
$180$	0	0
$181$	9.69721	0.720788	0.360394	−	0.932800i	$-0.382642\pi$
$181$	9.69721	0.720788	0.360394	+	0.932800i	$0.382642\pi$
$182$	0	0
$183$	17.9075	1.32376
$184$	0	0
$185$	−4.95361	−0.364197
$186$	0	0
$187$	−6.31528	−0.461819
$188$	0	0
$189$	0.139682	0.0101604
$190$	0	0
$191$	−9.52272	−0.689040	−0.344520	−	0.938779i	$-0.611958\pi$
$191$	−9.52272	−0.689040	−0.344520	+	0.938779i	$0.611958\pi$
$192$	0	0
$193$	−10.1215	−0.728564	−0.364282	−	0.931289i	$-0.618686\pi$
$193$	−10.1215	−0.728564	−0.364282	+	0.931289i	$0.618686\pi$
$194$	0	0
$195$	2.77747	0.198899
$196$	0	0
$197$	−7.40852	−0.527835	−0.263917	−	0.964545i	$-0.585015\pi$
$197$	−7.40852	−0.527835	−0.263917	+	0.964545i	$0.585015\pi$
$198$	0	0
$199$	−3.83670	−0.271977	−0.135988	−	0.990710i	$-0.543421\pi$
$199$	−3.83670	−0.271977	−0.135988	+	0.990710i	$0.543421\pi$
$200$	0	0
$201$	−25.9366	−1.82943
$202$	0	0
$203$	−7.64782	−0.536771
$204$	0	0
$205$	1.81799	0.126974
$206$	0	0
$207$	−25.3998	−1.76541
$208$	0	0
$209$	1.83477	0.126914
$210$	0	0
$211$	−5.48435	−0.377558	−0.188779	−	0.982020i	$-0.560453\pi$
$211$	−5.48435	−0.377558	−0.188779	+	0.982020i	$0.560453\pi$
$212$	0	0
$213$	32.6402	2.23647
$214$	0	0
$215$	11.0407	0.752967
$216$	0	0
$217$	−5.82327	−0.395309
$218$	0	0
$219$	−36.1756	−2.44452
$220$	0	0
$221$	−6.31528	−0.424811
$222$	0	0
$223$	10.6506	0.713219	0.356610	−	0.934254i	$-0.383933\pi$
$223$	10.6506	0.713219	0.356610	+	0.934254i	$0.383933\pi$
$224$	0	0
$225$	−11.3905	−0.759364
$226$	0	0
$227$	−18.0320	−1.19682	−0.598412	−	0.801189i	$-0.704201\pi$
$227$	−18.0320	−1.19682	−0.598412	+	0.801189i	$0.704201\pi$
$228$	0	0
$229$	5.70503	0.376999	0.188499	−	0.982073i	$-0.439638\pi$
$229$	5.70503	0.376999	0.188499	+	0.982073i	$0.439638\pi$
$230$	0	0
$231$	−2.46105	−0.161925
$232$	0	0
$233$	2.49616	0.163529	0.0817645	−	0.996652i	$-0.473944\pi$
$233$	2.49616	0.163529	0.0817645	+	0.996652i	$0.473944\pi$
$234$	0	0
$235$	−3.95044	−0.257698
$236$	0	0
$237$	−21.8438	−1.41891
$238$	0	0
$239$	−7.30917	−0.472791	−0.236395	−	0.971657i	$-0.575966\pi$
$239$	−7.30917	−0.472791	−0.236395	+	0.971657i	$0.575966\pi$
$240$	0	0
$241$	−2.84554	−0.183297	−0.0916486	−	0.995791i	$-0.529214\pi$
$241$	−2.84554	−0.183297	−0.0916486	+	0.995791i	$0.529214\pi$
$242$	0	0
$243$	−22.1415	−1.42038
$244$	0	0
$245$	−1.12857	−0.0721019
$246$	0	0
$247$	1.83477	0.116744
$248$	0	0
$249$	−36.2357	−2.29635
$250$	0	0
$251$	−4.21343	−0.265949	−0.132975	−	0.991119i	$-0.542453\pi$
$251$	−4.21343	−0.265949	−0.132975	+	0.991119i	$0.542453\pi$
$252$	0	0
$253$	8.30939	0.522407
$254$	0	0
$255$	−17.5405	−1.09843
$256$	0	0
$257$	7.22890	0.450926	0.225463	−	0.974252i	$-0.427610\pi$
$257$	7.22890	0.450926	0.225463	+	0.974252i	$0.427610\pi$
$258$	0	0
$259$	4.38927	0.272736
$260$	0	0
$261$	−23.3775	−1.44703
$262$	0	0
$263$	31.2574	1.92742	0.963708	−	0.266957i	$-0.0860183\pi$
$263$	31.2574	1.92742	0.963708	+	0.266957i	$0.0860183\pi$
$264$	0	0
$265$	−12.0639	−0.741080
$266$	0	0
$267$	14.1766	0.867593
$268$	0	0
$269$	−6.79410	−0.414244	−0.207122	−	0.978315i	$-0.566410\pi$
$269$	−6.79410	−0.414244	−0.207122	+	0.978315i	$0.566410\pi$
$270$	0	0
$271$	3.25998	0.198029	0.0990147	−	0.995086i	$-0.468431\pi$
$271$	3.25998	0.198029	0.0990147	+	0.995086i	$0.468431\pi$
$272$	0	0
$273$	−2.46105	−0.148949
$274$	0	0
$275$	3.72632	0.224706
$276$	0	0
$277$	−2.94766	−0.177108	−0.0885539	−	0.996071i	$-0.528225\pi$
$277$	−2.94766	−0.177108	−0.0885539	+	0.996071i	$0.528225\pi$
$278$	0	0
$279$	−17.8003	−1.06568
$280$	0	0
$281$	10.4355	0.622528	0.311264	−	0.950323i	$-0.399248\pi$
$281$	10.4355	0.622528	0.311264	+	0.950323i	$0.399248\pi$
$282$	0	0
$283$	21.3817	1.27101	0.635505	−	0.772097i	$-0.280792\pi$
$283$	21.3817	1.27101	0.635505	+	0.772097i	$0.280792\pi$
$284$	0	0
$285$	5.09603	0.301863
$286$	0	0
$287$	−1.61087	−0.0950868
$288$	0	0
$289$	22.8827	1.34604
$290$	0	0
$291$	−19.4009	−1.13730
$292$	0	0
$293$	1.12807	0.0659025	0.0329512	−	0.999457i	$-0.489509\pi$
$293$	1.12807	0.0659025	0.0329512	+	0.999457i	$0.489509\pi$
$294$	0	0
$295$	12.9950	0.756597
$296$	0	0
$297$	−0.139682	−0.00810520
$298$	0	0
$299$	8.30939	0.480545
$300$	0	0
$301$	−9.78285	−0.563874
$302$	0	0
$303$	28.7794	1.65333
$304$	0	0
$305$	−8.21193	−0.470214
$306$	0	0
$307$	17.8790	1.02041	0.510206	−	0.860052i	$-0.329569\pi$
$307$	17.8790	1.02041	0.510206	+	0.860052i	$0.329569\pi$
$308$	0	0
$309$	35.1734	2.00094
$310$	0	0
$311$	−15.2644	−0.865563	−0.432782	−	0.901499i	$-0.642468\pi$
$311$	−15.2644	−0.865563	−0.432782	+	0.901499i	$0.642468\pi$
$312$	0	0
$313$	11.7555	0.664459	0.332230	−	0.943199i	$-0.392199\pi$
$313$	11.7555	0.664459	0.332230	+	0.943199i	$0.392199\pi$
$314$	0	0
$315$	−3.44977	−0.194373
$316$	0	0
$317$	−33.0135	−1.85422	−0.927110	−	0.374789i	$-0.877715\pi$
$317$	−33.0135	−1.85422	−0.927110	+	0.374789i	$0.877715\pi$
$318$	0	0
$319$	7.64782	0.428195
$320$	0	0
$321$	−43.5016	−2.42802
$322$	0	0
$323$	−11.5871	−0.644724
$324$	0	0
$325$	3.72632	0.206699
$326$	0	0
$327$	33.9079	1.87511
$328$	0	0
$329$	3.50039	0.192982
$330$	0	0
$331$	−22.1492	−1.21743	−0.608714	−	0.793390i	$-0.708314\pi$
$331$	−22.1492	−1.21743	−0.608714	+	0.793390i	$0.708314\pi$
$332$	0	0
$333$	13.4169	0.735244
$334$	0	0
$335$	11.8938	0.649830
$336$	0	0
$337$	25.1041	1.36751	0.683755	−	0.729712i	$-0.260346\pi$
$337$	25.1041	1.36751	0.683755	+	0.729712i	$0.260346\pi$
$338$	0	0
$339$	12.2150	0.663428
$340$	0	0
$341$	5.82327	0.315348
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	23.0791	1.24254
$346$	0	0
$347$	−24.0730	−1.29230	−0.646152	−	0.763209i	$-0.723623\pi$
$347$	−24.0730	−1.29230	−0.646152	+	0.763209i	$0.723623\pi$
$348$	0	0
$349$	−19.3655	−1.03661	−0.518307	−	0.855195i	$-0.673437\pi$
$349$	−19.3655	−1.03661	−0.518307	+	0.855195i	$0.673437\pi$
$350$	0	0
$351$	−0.139682	−0.00745570
$352$	0	0
$353$	−5.02927	−0.267681	−0.133840	−	0.991003i	$-0.542731\pi$
$353$	−5.02927	−0.267681	−0.133840	+	0.991003i	$0.542731\pi$
$354$	0	0
$355$	−14.9679	−0.794415
$356$	0	0
$357$	15.5422	0.822580
$358$	0	0
$359$	−6.30526	−0.332779	−0.166389	−	0.986060i	$-0.553211\pi$
$359$	−6.30526	−0.332779	−0.166389	+	0.986060i	$0.553211\pi$
$360$	0	0
$361$	−15.6336	−0.822821
$362$	0	0
$363$	2.46105	0.129172
$364$	0	0
$365$	16.5892	0.868317
$366$	0	0
$367$	−2.60896	−0.136187	−0.0680933	−	0.997679i	$-0.521692\pi$
$367$	−2.60896	−0.136187	−0.0680933	+	0.997679i	$0.521692\pi$
$368$	0	0
$369$	−4.92405	−0.256336
$370$	0	0
$371$	10.6895	0.554973
$372$	0	0
$373$	10.2717	0.531848	0.265924	−	0.963994i	$-0.414323\pi$
$373$	10.2717	0.531848	0.265924	+	0.963994i	$0.414323\pi$
$374$	0	0
$375$	24.2371	1.25160
$376$	0	0
$377$	7.64782	0.393883
$378$	0	0
$379$	−25.4041	−1.30492	−0.652460	−	0.757823i	$-0.726263\pi$
$379$	−25.4041	−1.30492	−0.652460	+	0.757823i	$0.726263\pi$
$380$	0	0
$381$	−36.8449	−1.88762
$382$	0	0
$383$	−18.2432	−0.932183	−0.466092	−	0.884736i	$-0.654338\pi$
$383$	−18.2432	−0.932183	−0.466092	+	0.884736i	$0.654338\pi$
$384$	0	0
$385$	1.12857	0.0575174
$386$	0	0
$387$	−29.9038	−1.52010
$388$	0	0
$389$	−34.2961	−1.73888	−0.869441	−	0.494038i	$-0.835520\pi$
$389$	−34.2961	−1.73888	−0.869441	+	0.494038i	$0.835520\pi$
$390$	0	0
$391$	−52.4761	−2.65383
$392$	0	0
$393$	37.8487	1.90921
$394$	0	0
$395$	10.0170	0.504011
$396$	0	0
$397$	−4.69756	−0.235764	−0.117882	−	0.993028i	$-0.537610\pi$
$397$	−4.69756	−0.235764	−0.117882	+	0.993028i	$0.537610\pi$
$398$	0	0
$399$	−4.51547	−0.226056
$400$	0	0
$401$	−16.7387	−0.835889	−0.417944	−	0.908473i	$-0.637249\pi$
$401$	−16.7387	−0.835889	−0.417944	+	0.908473i	$0.637249\pi$
$402$	0	0
$403$	5.82327	0.290078
$404$	0	0
$405$	9.96135	0.494984
$406$	0	0
$407$	−4.38927	−0.217568
$408$	0	0
$409$	−34.0983	−1.68605	−0.843027	−	0.537871i	$-0.819229\pi$
$409$	−34.0983	−1.68605	−0.843027	+	0.537871i	$0.819229\pi$
$410$	0	0
$411$	−29.5912	−1.45963
$412$	0	0
$413$	−11.5145	−0.566593
$414$	0	0
$415$	16.6168	0.815685
$416$	0	0
$417$	51.5986	2.52680
$418$	0	0
$419$	−18.3810	−0.897971	−0.448985	−	0.893539i	$-0.648214\pi$
$419$	−18.3810	−0.897971	−0.448985	+	0.893539i	$0.648214\pi$
$420$	0	0
$421$	6.61356	0.322325	0.161163	−	0.986928i	$-0.448476\pi$
$421$	6.61356	0.322325	0.161163	+	0.986928i	$0.448476\pi$
$422$	0	0
$423$	10.6998	0.520243
$424$	0	0
$425$	−23.5328	−1.14151
$426$	0	0
$427$	7.27639	0.352129
$428$	0	0
$429$	2.46105	0.118821
$430$	0	0
$431$	35.9318	1.73077	0.865386	−	0.501105i	$-0.167073\pi$
$431$	35.9318	1.73077	0.865386	+	0.501105i	$0.167073\pi$
$432$	0	0
$433$	−3.63283	−0.174583	−0.0872913	−	0.996183i	$-0.527821\pi$
$433$	−3.63283	−0.174583	−0.0872913	+	0.996183i	$0.527821\pi$
$434$	0	0
$435$	21.2416	1.01846
$436$	0	0
$437$	15.2459	0.729308
$438$	0	0
$439$	39.0982	1.86606	0.933028	−	0.359804i	$-0.117156\pi$
$439$	39.0982	1.86606	0.933028	+	0.359804i	$0.117156\pi$
$440$	0	0
$441$	3.05676	0.145560
$442$	0	0
$443$	−9.86352	−0.468630	−0.234315	−	0.972161i	$-0.575285\pi$
$443$	−9.86352	−0.468630	−0.234315	+	0.972161i	$0.575285\pi$
$444$	0	0
$445$	−6.50102	−0.308178
$446$	0	0
$447$	−36.0608	−1.70562
$448$	0	0
$449$	29.8132	1.40697	0.703487	−	0.710708i	$-0.251625\pi$
$449$	29.8132	1.40697	0.703487	+	0.710708i	$0.251625\pi$
$450$	0	0
$451$	1.61087	0.0758531
$452$	0	0
$453$	32.1969	1.51274
$454$	0	0
$455$	1.12857	0.0529083
$456$	0	0
$457$	15.6736	0.733178	0.366589	−	0.930383i	$-0.380526\pi$
$457$	15.6736	0.733178	0.366589	+	0.930383i	$0.380526\pi$
$458$	0	0
$459$	0.882133	0.0411745
$460$	0	0
$461$	1.65607	0.0771310	0.0385655	−	0.999256i	$-0.487721\pi$
$461$	1.65607	0.0771310	0.0385655	+	0.999256i	$0.487721\pi$
$462$	0	0
$463$	−35.6461	−1.65661	−0.828306	−	0.560276i	$-0.810695\pi$
$463$	−35.6461	−1.65661	−0.828306	+	0.560276i	$0.810695\pi$
$464$	0	0
$465$	16.1740	0.750049
$466$	0	0
$467$	22.2596	1.03005	0.515025	−	0.857175i	$-0.327783\pi$
$467$	22.2596	1.03005	0.515025	+	0.857175i	$0.327783\pi$
$468$	0	0
$469$	−10.5388	−0.486638
$470$	0	0
$471$	−3.56388	−0.164215
$472$	0	0
$473$	9.78285	0.449816
$474$	0	0
$475$	6.83696	0.313701
$476$	0	0
$477$	32.6753	1.49610
$478$	0	0
$479$	−17.2604	−0.788647	−0.394323	−	0.918972i	$-0.629021\pi$
$479$	−17.2604	−0.788647	−0.394323	+	0.918972i	$0.629021\pi$
$480$	0	0
$481$	−4.38927	−0.200134
$482$	0	0
$483$	−20.4498	−0.930499
$484$	0	0
$485$	8.89675	0.403981
$486$	0	0
$487$	−21.1130	−0.956722	−0.478361	−	0.878163i	$-0.658769\pi$
$487$	−21.1130	−0.956722	−0.478361	+	0.878163i	$0.658769\pi$
$488$	0	0
$489$	17.1599	0.775999
$490$	0	0
$491$	10.9835	0.495678	0.247839	−	0.968801i	$-0.420280\pi$
$491$	10.9835	0.495678	0.247839	+	0.968801i	$0.420280\pi$
$492$	0	0
$493$	−48.2981	−2.17524
$494$	0	0
$495$	3.44977	0.155056
$496$	0	0
$497$	13.2627	0.594914
$498$	0	0
$499$	19.7724	0.885132	0.442566	−	0.896736i	$-0.354068\pi$
$499$	19.7724	0.885132	0.442566	+	0.896736i	$0.354068\pi$
$500$	0	0
$501$	12.3638	0.552375
$502$	0	0
$503$	17.2151	0.767582	0.383791	−	0.923420i	$-0.374618\pi$
$503$	17.2151	0.767582	0.383791	+	0.923420i	$0.374618\pi$
$504$	0	0
$505$	−13.1975	−0.587281
$506$	0	0
$507$	2.46105	0.109299
$508$	0	0
$509$	−32.2352	−1.42880	−0.714400	−	0.699738i	$-0.753300\pi$
$509$	−32.2352	−1.42880	−0.714400	+	0.699738i	$0.753300\pi$
$510$	0	0
$511$	−14.6993	−0.650257
$512$	0	0
$513$	−0.256286	−0.0113153
$514$	0	0
$515$	−16.1296	−0.710755
$516$	0	0
$517$	−3.50039	−0.153947
$518$	0	0
$519$	−29.8835	−1.31174
$520$	0	0
$521$	12.4244	0.544325	0.272162	−	0.962251i	$-0.412261\pi$
$521$	12.4244	0.544325	0.272162	+	0.962251i	$0.412261\pi$
$522$	0	0
$523$	−14.8783	−0.650582	−0.325291	−	0.945614i	$-0.605462\pi$
$523$	−14.8783	−0.650582	−0.325291	+	0.945614i	$0.605462\pi$
$524$	0	0
$525$	−9.17066	−0.400240
$526$	0	0
$527$	−36.7755	−1.60197
$528$	0	0
$529$	46.0460	2.00200
$530$	0	0
$531$	−35.1971	−1.52743
$532$	0	0
$533$	1.61087	0.0697747
$534$	0	0
$535$	19.9487	0.862457
$536$	0	0
$537$	−15.0313	−0.648650
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	11.0299	0.474212	0.237106	−	0.971484i	$-0.423801\pi$
$541$	11.0299	0.474212	0.237106	+	0.971484i	$0.423801\pi$
$542$	0	0
$543$	23.8653	1.02416
$544$	0	0
$545$	−15.5493	−0.666057
$546$	0	0
$547$	−18.1982	−0.778101	−0.389050	−	0.921217i	$-0.627197\pi$
$547$	−18.1982	−0.778101	−0.389050	+	0.921217i	$0.627197\pi$
$548$	0	0
$549$	22.2422	0.949272
$550$	0	0
$551$	14.0320	0.597784
$552$	0	0
$553$	−8.87583	−0.377439
$554$	0	0
$555$	−12.1911	−0.517482
$556$	0	0
$557$	20.7801	0.880483	0.440241	−	0.897879i	$-0.354893\pi$
$557$	20.7801	0.880483	0.440241	+	0.897879i	$0.354893\pi$
$558$	0	0
$559$	9.78285	0.413771
$560$	0	0
$561$	−15.5422	−0.656192
$562$	0	0
$563$	−6.41182	−0.270226	−0.135113	−	0.990830i	$-0.543140\pi$
$563$	−6.41182	−0.270226	−0.135113	+	0.990830i	$0.543140\pi$
$564$	0	0
$565$	−5.60149	−0.235656
$566$	0	0
$567$	−8.82651	−0.370678
$568$	0	0
$569$	42.0250	1.76178	0.880889	−	0.473323i	$-0.156946\pi$
$569$	42.0250	1.76178	0.880889	+	0.473323i	$0.156946\pi$
$570$	0	0
$571$	21.9071	0.916785	0.458393	−	0.888750i	$-0.348425\pi$
$571$	21.9071	0.916785	0.458393	+	0.888750i	$0.348425\pi$
$572$	0	0
$573$	−23.4359	−0.979048
$574$	0	0
$575$	30.9635	1.29127
$576$	0	0
$577$	23.2586	0.968269	0.484134	−	0.874994i	$-0.339135\pi$
$577$	23.2586	0.968269	0.484134	+	0.874994i	$0.339135\pi$
$578$	0	0
$579$	−24.9096	−1.03521
$580$	0	0
$581$	−14.7237	−0.610842
$582$	0	0
$583$	−10.6895	−0.442715
$584$	0	0
$585$	3.44977	0.142631
$586$	0	0
$587$	−30.4207	−1.25560	−0.627799	−	0.778375i	$-0.716044\pi$
$587$	−30.4207	−1.25560	−0.627799	+	0.778375i	$0.716044\pi$
$588$	0	0
$589$	10.6844	0.440242
$590$	0	0
$591$	−18.2327	−0.749994
$592$	0	0
$593$	20.5517	0.843957	0.421979	−	0.906606i	$-0.361336\pi$
$593$	20.5517	0.843957	0.421979	+	0.906606i	$0.361336\pi$
$594$	0	0
$595$	−7.12725	−0.292189
$596$	0	0
$597$	−9.44231	−0.386448
$598$	0	0
$599$	−3.38583	−0.138341	−0.0691707	−	0.997605i	$-0.522035\pi$
$599$	−3.38583	−0.138341	−0.0691707	+	0.997605i	$0.522035\pi$
$600$	0	0
$601$	33.7434	1.37642	0.688211	−	0.725511i	$-0.258396\pi$
$601$	33.7434	1.37642	0.688211	+	0.725511i	$0.258396\pi$
$602$	0	0
$603$	−32.2147	−1.31188
$604$	0	0
$605$	−1.12857	−0.0458830
$606$	0	0
$607$	26.2209	1.06427	0.532137	−	0.846658i	$-0.321389\pi$
$607$	26.2209	1.06427	0.532137	+	0.846658i	$0.321389\pi$
$608$	0	0
$609$	−18.8216	−0.762691
$610$	0	0
$611$	−3.50039	−0.141610
$612$	0	0
$613$	17.1368	0.692149	0.346075	−	0.938207i	$-0.387514\pi$
$613$	17.1368	0.692149	0.346075	+	0.938207i	$0.387514\pi$
$614$	0	0
$615$	4.47415	0.180415
$616$	0	0
$617$	−28.7678	−1.15815	−0.579074	−	0.815275i	$-0.696586\pi$
$617$	−28.7678	−1.15815	−0.579074	+	0.815275i	$0.696586\pi$
$618$	0	0
$619$	−4.70624	−0.189160	−0.0945800	−	0.995517i	$-0.530151\pi$
$619$	−4.70624	−0.189160	−0.0945800	+	0.995517i	$0.530151\pi$
$620$	0	0
$621$	−1.16068	−0.0465763
$622$	0	0
$623$	5.76039	0.230785
$624$	0	0
$625$	7.51711	0.300684
$626$	0	0
$627$	4.51547	0.180330
$628$	0	0
$629$	27.7195	1.10525
$630$	0	0
$631$	−9.62586	−0.383199	−0.191600	−	0.981473i	$-0.561368\pi$
$631$	−9.62586	−0.383199	−0.191600	+	0.981473i	$0.561368\pi$
$632$	0	0
$633$	−13.4972	−0.536467
$634$	0	0
$635$	16.8961	0.670501
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	40.5409	1.60377
$640$	0	0
$641$	−3.47126	−0.137107	−0.0685534	−	0.997647i	$-0.521838\pi$
$641$	−3.47126	−0.137107	−0.0685534	+	0.997647i	$0.521838\pi$
$642$	0	0
$643$	12.9442	0.510470	0.255235	−	0.966879i	$-0.417847\pi$
$643$	12.9442	0.510470	0.255235	+	0.966879i	$0.417847\pi$
$644$	0	0
$645$	27.1716	1.06988
$646$	0	0
$647$	−23.2745	−0.915016	−0.457508	−	0.889206i	$-0.651258\pi$
$647$	−23.2745	−0.915016	−0.457508	+	0.889206i	$0.651258\pi$
$648$	0	0
$649$	11.5145	0.451985
$650$	0	0
$651$	−14.3313	−0.561689
$652$	0	0
$653$	17.5041	0.684987	0.342493	−	0.939520i	$-0.388729\pi$
$653$	17.5041	0.684987	0.342493	+	0.939520i	$0.388729\pi$
$654$	0	0
$655$	−17.3564	−0.678171
$656$	0	0
$657$	−44.9320	−1.75297
$658$	0	0
$659$	−9.94490	−0.387398	−0.193699	−	0.981061i	$-0.562049\pi$
$659$	−9.94490	−0.387398	−0.193699	+	0.981061i	$0.562049\pi$
$660$	0	0
$661$	13.0326	0.506908	0.253454	−	0.967347i	$-0.418433\pi$
$661$	13.0326	0.506908	0.253454	+	0.967347i	$0.418433\pi$
$662$	0	0
$663$	−15.5422	−0.603609
$664$	0	0
$665$	2.07068	0.0802974
$666$	0	0
$667$	63.5487	2.46062
$668$	0	0
$669$	26.2117	1.01340
$670$	0	0
$671$	−7.27639	−0.280902
$672$	0	0
$673$	−34.6008	−1.33376	−0.666881	−	0.745165i	$-0.732371\pi$
$673$	−34.6008	−1.33376	−0.666881	+	0.745165i	$0.732371\pi$
$674$	0	0
$675$	−0.520502	−0.0200341
$676$	0	0
$677$	−31.3813	−1.20608	−0.603040	−	0.797711i	$-0.706044\pi$
$677$	−31.3813	−1.20608	−0.603040	+	0.797711i	$0.706044\pi$
$678$	0	0
$679$	−7.88319	−0.302529
$680$	0	0
$681$	−44.3775	−1.70055
$682$	0	0
$683$	8.50776	0.325540	0.162770	−	0.986664i	$-0.447957\pi$
$683$	8.50776	0.325540	0.162770	+	0.986664i	$0.447957\pi$
$684$	0	0
$685$	13.5698	0.518474
$686$	0	0
$687$	14.0403	0.535673
$688$	0	0
$689$	−10.6895	−0.407239
$690$	0	0
$691$	33.2712	1.26570	0.632848	−	0.774276i	$-0.281886\pi$
$691$	33.2712	1.26570	0.632848	+	0.774276i	$0.281886\pi$
$692$	0	0
$693$	−3.05676	−0.116117
$694$	0	0
$695$	−23.6618	−0.897543
$696$	0	0
$697$	−10.1731	−0.385334
$698$	0	0
$699$	6.14317	0.232356
$700$	0	0
$701$	49.2054	1.85846	0.929231	−	0.369500i	$-0.120471\pi$
$701$	49.2054	1.85846	0.929231	+	0.369500i	$0.120471\pi$
$702$	0	0
$703$	−8.05332	−0.303737
$704$	0	0
$705$	−9.72222	−0.366160
$706$	0	0
$707$	11.6940	0.439797
$708$	0	0
$709$	−25.3934	−0.953671	−0.476835	−	0.878993i	$-0.658216\pi$
$709$	−25.3934	−0.953671	−0.476835	+	0.878993i	$0.658216\pi$
$710$	0	0
$711$	−27.1313	−1.01750
$712$	0	0
$713$	48.3878	1.81214
$714$	0	0
$715$	−1.12857	−0.0422062
$716$	0	0
$717$	−17.9882	−0.671782
$718$	0	0
$719$	−4.03613	−0.150522	−0.0752611	−	0.997164i	$-0.523979\pi$
$719$	−4.03613	−0.150522	−0.0752611	+	0.997164i	$0.523979\pi$
$720$	0	0
$721$	14.2920	0.532263
$722$	0	0
$723$	−7.00300	−0.260444
$724$	0	0
$725$	28.4982	1.05840
$726$	0	0
$727$	36.8804	1.36782	0.683909	−	0.729567i	$-0.260279\pi$
$727$	36.8804	1.36782	0.683909	+	0.729567i	$0.260279\pi$
$728$	0	0
$729$	−28.0118	−1.03747
$730$	0	0
$731$	−61.7814	−2.28507
$732$	0	0
$733$	46.0408	1.70056	0.850278	−	0.526334i	$-0.176434\pi$
$733$	46.0408	1.70056	0.850278	+	0.526334i	$0.176434\pi$
$734$	0	0
$735$	−2.77747	−0.102449
$736$	0	0
$737$	10.5388	0.388203
$738$	0	0
$739$	−33.7700	−1.24225	−0.621124	−	0.783712i	$-0.713324\pi$
$739$	−33.7700	−1.24225	−0.621124	+	0.783712i	$0.713324\pi$
$740$	0	0
$741$	4.51547	0.165880
$742$	0	0
$743$	−32.8431	−1.20490	−0.602448	−	0.798158i	$-0.705808\pi$
$743$	−32.8431	−1.20490	−0.602448	+	0.798158i	$0.705808\pi$
$744$	0	0
$745$	16.5366	0.605853
$746$	0	0
$747$	−45.0068	−1.64671
$748$	0	0
$749$	−17.6760	−0.645868
$750$	0	0
$751$	−44.4424	−1.62172	−0.810862	−	0.585237i	$-0.801001\pi$
$751$	−44.4424	−1.62172	−0.810862	+	0.585237i	$0.801001\pi$
$752$	0	0
$753$	−10.3694	−0.377883
$754$	0	0
$755$	−14.7646	−0.537340
$756$	0	0
$757$	1.68206	0.0611357	0.0305678	−	0.999533i	$-0.490268\pi$
$757$	1.68206	0.0611357	0.0305678	+	0.999533i	$0.490268\pi$
$758$	0	0
$759$	20.4498	0.742281
$760$	0	0
$761$	−36.3394	−1.31730	−0.658651	−	0.752448i	$-0.728873\pi$
$761$	−36.3394	−1.31730	−0.658651	+	0.752448i	$0.728873\pi$
$762$	0	0
$763$	13.7778	0.498790
$764$	0	0
$765$	−21.7863	−0.787684
$766$	0	0
$767$	11.5145	0.415766
$768$	0	0
$769$	−1.81762	−0.0655449	−0.0327725	−	0.999463i	$-0.510434\pi$
$769$	−1.81762	−0.0655449	−0.0327725	+	0.999463i	$0.510434\pi$
$770$	0	0
$771$	17.7907	0.640715
$772$	0	0
$773$	−24.7208	−0.889146	−0.444573	−	0.895743i	$-0.646645\pi$
$773$	−24.7208	−0.889146	−0.444573	+	0.895743i	$0.646645\pi$
$774$	0	0
$775$	21.6994	0.779465
$776$	0	0
$777$	10.8022	0.387527
$778$	0	0
$779$	2.95559	0.105895
$780$	0	0
$781$	−13.2627	−0.474577
$782$	0	0
$783$	−1.06827	−0.0381767
$784$	0	0
$785$	1.63431	0.0583308
$786$	0	0
$787$	30.5597	1.08933	0.544667	−	0.838652i	$-0.316656\pi$
$787$	30.5597	1.08933	0.544667	+	0.838652i	$0.316656\pi$
$788$	0	0
$789$	76.9261	2.73864
$790$	0	0
$791$	4.96334	0.176476
$792$	0	0
$793$	−7.27639	−0.258392
$794$	0	0
$795$	−29.6899	−1.05299
$796$	0	0
$797$	45.7590	1.62087	0.810433	−	0.585831i	$-0.199232\pi$
$797$	45.7590	1.62087	0.810433	+	0.585831i	$0.199232\pi$
$798$	0	0
$799$	22.1059	0.782050
$800$	0	0
$801$	17.6081	0.622152
$802$	0	0
$803$	14.6993	0.518725
$804$	0	0
$805$	9.37775	0.330522
$806$	0	0
$807$	−16.7206	−0.588593
$808$	0	0
$809$	−16.5329	−0.581264	−0.290632	−	0.956835i	$-0.593866\pi$
$809$	−16.5329	−0.581264	−0.290632	+	0.956835i	$0.593866\pi$
$810$	0	0
$811$	−5.39570	−0.189469	−0.0947343	−	0.995503i	$-0.530200\pi$
$811$	−5.39570	−0.189469	−0.0947343	+	0.995503i	$0.530200\pi$
$812$	0	0
$813$	8.02296	0.281377
$814$	0	0
$815$	−7.86910	−0.275643
$816$	0	0
$817$	17.9493	0.627967
$818$	0	0
$819$	−3.05676	−0.106812
$820$	0	0
$821$	−0.478489	−0.0166994	−0.00834969	−	0.999965i	$-0.502658\pi$
$821$	−0.478489	−0.0166994	−0.00834969	+	0.999965i	$0.502658\pi$
$822$	0	0
$823$	6.60949	0.230392	0.115196	−	0.993343i	$-0.463250\pi$
$823$	6.60949	0.230392	0.115196	+	0.993343i	$0.463250\pi$
$824$	0	0
$825$	9.17066	0.319281
$826$	0	0
$827$	12.3810	0.430530	0.215265	−	0.976556i	$-0.430938\pi$
$827$	12.3810	0.430530	0.215265	+	0.976556i	$0.430938\pi$
$828$	0	0
$829$	−22.0788	−0.766830	−0.383415	−	0.923576i	$-0.625252\pi$
$829$	−22.0788	−0.766830	−0.383415	+	0.923576i	$0.625252\pi$
$830$	0	0
$831$	−7.25434	−0.251650
$832$	0	0
$833$	6.31528	0.218811
$834$	0	0
$835$	−5.66973	−0.196209
$836$	0	0
$837$	−0.813408	−0.0281155
$838$	0	0
$839$	15.8007	0.545500	0.272750	−	0.962085i	$-0.412067\pi$
$839$	15.8007	0.545500	0.272750	+	0.962085i	$0.412067\pi$
$840$	0	0
$841$	29.4891	1.01687
$842$	0	0
$843$	25.6822	0.884542
$844$	0	0
$845$	−1.12857	−0.0388241
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	52.6214	1.80596
$850$	0	0
$851$	−36.4722	−1.25025
$852$	0	0
$853$	−43.5280	−1.49037	−0.745185	−	0.666858i	$-0.767639\pi$
$853$	−43.5280	−1.49037	−0.745185	+	0.666858i	$0.767639\pi$
$854$	0	0
$855$	6.32955	0.216466
$856$	0	0
$857$	−6.61496	−0.225963	−0.112981	−	0.993597i	$-0.536040\pi$
$857$	−6.61496	−0.225963	−0.112981	+	0.993597i	$0.536040\pi$
$858$	0	0
$859$	−5.55129	−0.189408	−0.0947038	−	0.995505i	$-0.530190\pi$
$859$	−5.55129	−0.189408	−0.0947038	+	0.995505i	$0.530190\pi$
$860$	0	0
$861$	−3.96444	−0.135108
$862$	0	0
$863$	−12.8522	−0.437495	−0.218748	−	0.975781i	$-0.570197\pi$
$863$	−12.8522	−0.437495	−0.218748	+	0.975781i	$0.570197\pi$
$864$	0	0
$865$	13.7038	0.465943
$866$	0	0
$867$	56.3155	1.91257
$868$	0	0
$869$	8.87583	0.301092
$870$	0	0
$871$	10.5388	0.357095
$872$	0	0
$873$	−24.0970	−0.815560
$874$	0	0
$875$	9.84829	0.332933
$876$	0	0
$877$	−33.9578	−1.14667	−0.573337	−	0.819320i	$-0.694351\pi$
$877$	−33.9578	−1.14667	−0.573337	+	0.819320i	$0.694351\pi$
$878$	0	0
$879$	2.77623	0.0936399
$880$	0	0
$881$	39.7693	1.33986	0.669931	−	0.742423i	$-0.266324\pi$
$881$	39.7693	1.33986	0.669931	+	0.742423i	$0.266324\pi$
$882$	0	0
$883$	50.8579	1.71150	0.855752	−	0.517387i	$-0.173095\pi$
$883$	50.8579	1.71150	0.855752	+	0.517387i	$0.173095\pi$
$884$	0	0
$885$	31.9813	1.07504
$886$	0	0
$887$	−19.6392	−0.659421	−0.329711	−	0.944082i	$-0.606951\pi$
$887$	−19.6392	−0.659421	−0.329711	+	0.944082i	$0.606951\pi$
$888$	0	0
$889$	−14.9712	−0.502118
$890$	0	0
$891$	8.82651	0.295699
$892$	0	0
$893$	−6.42241	−0.214918
$894$	0	0
$895$	6.89298	0.230407
$896$	0	0
$897$	20.4498	0.682799
$898$	0	0
$899$	44.5353	1.48533
$900$	0	0
$901$	67.5073	2.24900
$902$	0	0
$903$	−24.0761	−0.801202
$904$	0	0
$905$	−10.9440	−0.363791
$906$	0	0
$907$	39.9066	1.32508	0.662539	−	0.749028i	$-0.269479\pi$
$907$	39.9066	1.32508	0.662539	+	0.749028i	$0.269479\pi$
$908$	0	0
$909$	35.7456	1.18561
$910$	0	0
$911$	−5.32300	−0.176359	−0.0881794	−	0.996105i	$-0.528105\pi$
$911$	−5.32300	−0.176359	−0.0881794	+	0.996105i	$0.528105\pi$
$912$	0	0
$913$	14.7237	0.487283
$914$	0	0
$915$	−20.2100	−0.668121
$916$	0	0
$917$	15.3791	0.507862
$918$	0	0
$919$	24.7182	0.815379	0.407690	−	0.913121i	$-0.366335\pi$
$919$	24.7182	0.815379	0.407690	+	0.913121i	$0.366335\pi$
$920$	0	0
$921$	44.0012	1.44989
$922$	0	0
$923$	−13.2627	−0.436547
$924$	0	0
$925$	−16.3558	−0.537777
$926$	0	0
$927$	43.6873	1.43488
$928$	0	0
$929$	−26.0883	−0.855931	−0.427965	−	0.903795i	$-0.640769\pi$
$929$	−26.0883	−0.855931	−0.427965	+	0.903795i	$0.640769\pi$
$930$	0	0
$931$	−1.83477	−0.0601323
$932$	0	0
$933$	−37.5663	−1.22987
$934$	0	0
$935$	7.12725	0.233086
$936$	0	0
$937$	0.787111	0.0257138	0.0128569	−	0.999917i	$-0.495907\pi$
$937$	0.787111	0.0257138	0.0128569	+	0.999917i	$0.495907\pi$
$938$	0	0
$939$	28.9308	0.944122
$940$	0	0
$941$	2.09811	0.0683964	0.0341982	−	0.999415i	$-0.489112\pi$
$941$	2.09811	0.0683964	0.0341982	+	0.999415i	$0.489112\pi$
$942$	0	0
$943$	13.3854	0.435888
$944$	0	0
$945$	−0.157642	−0.00512809
$946$	0	0
$947$	−8.80450	−0.286108	−0.143054	−	0.989715i	$-0.545692\pi$
$947$	−8.80450	−0.286108	−0.143054	+	0.989715i	$0.545692\pi$
$948$	0	0
$949$	14.6993	0.477158
$950$	0	0
$951$	−81.2477	−2.63464
$952$	0	0
$953$	42.1631	1.36580	0.682899	−	0.730513i	$-0.260719\pi$
$953$	42.1631	1.36580	0.682899	+	0.730513i	$0.260719\pi$
$954$	0	0
$955$	10.7471	0.347767
$956$	0	0
$957$	18.8216	0.608417
$958$	0	0
$959$	−12.0238	−0.388270
$960$	0	0
$961$	2.91044	0.0938850
$962$	0	0
$963$	−54.0314	−1.74114
$964$	0	0
$965$	11.4229	0.367716
$966$	0	0
$967$	33.8790	1.08947	0.544737	−	0.838607i	$-0.316629\pi$
$967$	33.8790	1.08947	0.544737	+	0.838607i	$0.316629\pi$
$968$	0	0
$969$	−28.5164	−0.916079
$970$	0	0
$971$	−3.12178	−0.100183	−0.0500913	−	0.998745i	$-0.515951\pi$
$971$	−3.12178	−0.100183	−0.0500913	+	0.998745i	$0.515951\pi$
$972$	0	0
$973$	20.9661	0.672143
$974$	0	0
$975$	9.17066	0.293696
$976$	0	0
$977$	−47.2461	−1.51154	−0.755768	−	0.654840i	$-0.772736\pi$
$977$	−47.2461	−1.51154	−0.755768	+	0.654840i	$0.772736\pi$
$978$	0	0
$979$	−5.76039	−0.184103
$980$	0	0
$981$	42.1154	1.34464
$982$	0	0
$983$	6.17996	0.197110	0.0985551	−	0.995132i	$-0.468578\pi$
$983$	6.17996	0.197110	0.0985551	+	0.995132i	$0.468578\pi$
$984$	0	0
$985$	8.36105	0.266405
$986$	0	0
$987$	8.61462	0.274206
$988$	0	0
$989$	81.2896	2.58486
$990$	0	0
$991$	−34.3570	−1.09139	−0.545693	−	0.837985i	$-0.683733\pi$
$991$	−34.3570	−1.09139	−0.545693	+	0.837985i	$0.683733\pi$
$992$	0	0
$993$	−54.5102	−1.72983
$994$	0	0
$995$	4.33000	0.137270
$996$	0	0
$997$	−17.6230	−0.558126	−0.279063	−	0.960273i	$-0.590024\pi$
$997$	−17.6230	−0.558126	−0.279063	+	0.960273i	$0.590024\pi$
$998$	0	0
$999$	0.613104	0.0193978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.13	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.13	✓	14	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)

\(f(q)\)	\(=\)	\(q+2.46105 q^{3} -1.12857 q^{5} +1.00000 q^{7} +3.05676 q^{9} +O(q^{10})\)	\(q+2.46105 q^{3} -1.12857 q^{5} +1.00000 q^{7} +3.05676 q^{9} -1.00000 q^{11} -1.00000 q^{13} -2.77747 q^{15} +6.31528 q^{17} -1.83477 q^{19} +2.46105 q^{21} -8.30939 q^{23} -3.72632 q^{25} +0.139682 q^{27} -7.64782 q^{29} -5.82327 q^{31} -2.46105 q^{33} -1.12857 q^{35} +4.38927 q^{37} -2.46105 q^{39} -1.61087 q^{41} -9.78285 q^{43} -3.44977 q^{45} +3.50039 q^{47} +1.00000 q^{49} +15.5422 q^{51} +10.6895 q^{53} +1.12857 q^{55} -4.51547 q^{57} -11.5145 q^{59} +7.27639 q^{61} +3.05676 q^{63} +1.12857 q^{65} -10.5388 q^{67} -20.4498 q^{69} +13.2627 q^{71} -14.6993 q^{73} -9.17066 q^{75} -1.00000 q^{77} -8.87583 q^{79} -8.82651 q^{81} -14.7237 q^{83} -7.12725 q^{85} -18.8216 q^{87} +5.76039 q^{89} -1.00000 q^{91} -14.3313 q^{93} +2.07068 q^{95} -7.88319 q^{97} -3.05676 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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