LMFDB - Embedded newform 304.8.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5014x^{3} + 113222x^{2} - 625803x + 567036 $ x^5 - 2x^4 - 5014x^3 + 113222x^2 - 625803x + 567036
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	no (minimal twist has level 76)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.12286$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-73.9019 q^{3} -358.964 q^{5} +109.800 q^{7} +3274.48 q^{9} +O(q^{10})$	$q-73.9019 q^{3} -358.964 q^{5} +109.800 q^{7} +3274.48 q^{9} +7153.73 q^{11} -3123.17 q^{13} +26528.1 q^{15} -23898.6 q^{17} +6859.00 q^{19} -8114.45 q^{21} -50622.1 q^{23} +50730.0 q^{25} -80367.1 q^{27} -134788. q^{29} -199472. q^{31} -528674. q^{33} -39414.3 q^{35} -95421.5 q^{37} +230808. q^{39} -488804. q^{41} -162905. q^{43} -1.17542e6 q^{45} +832599. q^{47} -811487. q^{49} +1.76615e6 q^{51} +953158. q^{53} -2.56793e6 q^{55} -506893. q^{57} -2.46032e6 q^{59} +1.15629e6 q^{61} +359539. q^{63} +1.12110e6 q^{65} +1.59366e6 q^{67} +3.74107e6 q^{69} +5.90057e6 q^{71} -1.17056e6 q^{73} -3.74904e6 q^{75} +785482. q^{77} -7.58482e6 q^{79} -1.22202e6 q^{81} -7.26944e6 q^{83} +8.57871e6 q^{85} +9.96106e6 q^{87} -7.80137e6 q^{89} -342925. q^{91} +1.47414e7 q^{93} -2.46213e6 q^{95} -6.56007e6 q^{97} +2.34248e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9}+O(q^{10}) $ 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9	$ 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9} + 2662 q^{11} - 602 q^{13} + 20800 q^{15} - 27366 q^{17} + 34295 q^{19} - 59964 q^{21} + 67096 q^{23} - 109115 q^{25} + 178778 q^{27} - 372398 q^{29} + 271372 q^{31} - 792700 q^{33} + 608250 q^{35} - 562630 q^{37} + 963904 q^{39} - 956714 q^{41} + 827362 q^{43} - 1165100 q^{45} + 1812982 q^{47} - 862031 q^{49} + 2458254 q^{51} + 486998 q^{53} - 467930 q^{55} + 96026 q^{57} + 367182 q^{59} + 1879732 q^{61} + 1007274 q^{63} + 1790920 q^{65} + 1046394 q^{67} + 7261712 q^{69} + 4664572 q^{71} + 4224942 q^{73} - 8194850 q^{75} + 8611110 q^{77} - 9574024 q^{79} + 11351813 q^{81} - 11754804 q^{83} + 18711750 q^{85} - 3801472 q^{87} + 2782542 q^{89} - 7385214 q^{91} + 29535004 q^{93} - 1920520 q^{95} + 1291574 q^{97} + 9760310 q^{99}+O(q^{100}) $ 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9 + 2662 * q^11 - 602 * q^13 + 20800 * q^15 - 27366 * q^17 + 34295 * q^19 - 59964 * q^21 + 67096 * q^23 - 109115 * q^25 + 178778 * q^27 - 372398 * q^29 + 271372 * q^31 - 792700 * q^33 + 608250 * q^35 - 562630 * q^37 + 963904 * q^39 - 956714 * q^41 + 827362 * q^43 - 1165100 * q^45 + 1812982 * q^47 - 862031 * q^49 + 2458254 * q^51 + 486998 * q^53 - 467930 * q^55 + 96026 * q^57 + 367182 * q^59 + 1879732 * q^61 + 1007274 * q^63 + 1790920 * q^65 + 1046394 * q^67 + 7261712 * q^69 + 4664572 * q^71 + 4224942 * q^73 - 8194850 * q^75 + 8611110 * q^77 - 9574024 * q^79 + 11351813 * q^81 - 11754804 * q^83 + 18711750 * q^85 - 3801472 * q^87 + 2782542 * q^89 - 7385214 * q^91 + 29535004 * q^93 - 1920520 * q^95 + 1291574 * q^97 + 9760310 * 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
$3$	−73.9019	−1.58027	−0.790134	−	0.612934i	$-0.789989\pi$
$3$	−73.9019	−1.58027	−0.790134	+	0.612934i	$0.789989\pi$
$4$	0	0
$5$	−358.964	−1.28427	−0.642134	−	0.766592i	$-0.721951\pi$
$5$	−358.964	−1.28427	−0.642134	+	0.766592i	$0.721951\pi$
$6$	0	0
$7$	109.800	0.120993	0.0604965	−	0.998168i	$-0.480732\pi$
$7$	109.800	0.120993	0.0604965	+	0.998168i	$0.480732\pi$
$8$	0	0
$9$	3274.48	1.49725
$10$	0	0
$11$	7153.73	1.62053	0.810267	−	0.586061i	$-0.199322\pi$
$11$	7153.73	1.62053	0.810267	+	0.586061i	$0.199322\pi$
$12$	0	0
$13$	−3123.17	−0.394270	−0.197135	−	0.980376i	$-0.563164\pi$
$13$	−3123.17	−0.394270	−0.197135	+	0.980376i	$0.563164\pi$
$14$	0	0
$15$	26528.1	2.02949
$16$	0	0
$17$	−23898.6	−1.17978	−0.589889	−	0.807484i	$-0.700829\pi$
$17$	−23898.6	−1.17978	−0.589889	+	0.807484i	$0.700829\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	−8114.45	−0.191202
$22$	0	0
$23$	−50622.1	−0.867547	−0.433773	−	0.901022i	$-0.642818\pi$
$23$	−50622.1	−0.867547	−0.433773	+	0.901022i	$0.642818\pi$
$24$	0	0
$25$	50730.0	0.649344
$26$	0	0
$27$	−80367.1	−0.785788
$28$	0	0
$29$	−134788.	−1.02626	−0.513130	−	0.858311i	$-0.671514\pi$
$29$	−134788.	−1.02626	−0.513130	+	0.858311i	$0.671514\pi$
$30$	0	0
$31$	−199472.	−1.20259	−0.601293	−	0.799029i	$-0.705348\pi$
$31$	−199472.	−1.20259	−0.601293	+	0.799029i	$0.705348\pi$
$32$	0	0
$33$	−528674.	−2.56088
$34$	0	0
$35$	−39414.3	−0.155388
$36$	0	0
$37$	−95421.5	−0.309699	−0.154850	−	0.987938i	$-0.549489\pi$
$37$	−95421.5	−0.309699	−0.154850	+	0.987938i	$0.549489\pi$
$38$	0	0
$39$	230808.	0.623052
$40$	0	0
$41$	−488804.	−1.10762	−0.553811	−	0.832642i	$-0.686827\pi$
$41$	−488804.	−1.10762	−0.553811	+	0.832642i	$0.686827\pi$
$42$	0	0
$43$	−162905.	−0.312460	−0.156230	−	0.987721i	$-0.549934\pi$
$43$	−162905.	−0.312460	−0.156230	+	0.987721i	$0.549934\pi$
$44$	0	0
$45$	−1.17542e6	−1.92287
$46$	0	0
$47$	832599.	1.16975	0.584875	−	0.811123i	$-0.301143\pi$
$47$	832599.	1.16975	0.584875	+	0.811123i	$0.301143\pi$
$48$	0	0
$49$	−811487.	−0.985361
$50$	0	0
$51$	1.76615e6	1.86437
$52$	0	0
$53$	953158.	0.879426	0.439713	−	0.898138i	$-0.355080\pi$
$53$	953158.	0.879426	0.439713	+	0.898138i	$0.355080\pi$
$54$	0	0
$55$	−2.56793e6	−2.08120
$56$	0	0
$57$	−506893.	−0.362539
$58$	0	0
$59$	−2.46032e6	−1.55958	−0.779792	−	0.626039i	$-0.784675\pi$
$59$	−2.46032e6	−1.55958	−0.779792	+	0.626039i	$0.784675\pi$
$60$	0	0
$61$	1.15629e6	0.652245	0.326122	−	0.945328i	$-0.394258\pi$
$61$	1.15629e6	0.652245	0.326122	+	0.945328i	$0.394258\pi$
$62$	0	0
$63$	359539.	0.181157
$64$	0	0
$65$	1.12110e6	0.506348
$66$	0	0
$67$	1.59366e6	0.647344	0.323672	−	0.946169i	$-0.395083\pi$
$67$	1.59366e6	0.647344	0.323672	+	0.946169i	$0.395083\pi$
$68$	0	0
$69$	3.74107e6	1.37096
$70$	0	0
$71$	5.90057e6	1.95654	0.978272	−	0.207326i	$-0.0664759\pi$
$71$	5.90057e6	1.95654	0.978272	+	0.207326i	$0.0664759\pi$
$72$	0	0
$73$	−1.17056e6	−0.352178	−0.176089	−	0.984374i	$-0.556345\pi$
$73$	−1.17056e6	−0.352178	−0.176089	+	0.984374i	$0.556345\pi$
$74$	0	0
$75$	−3.74904e6	−1.02614
$76$	0	0
$77$	785482.	0.196073
$78$	0	0
$79$	−7.58482e6	−1.73082	−0.865408	−	0.501069i	$-0.832940\pi$
$79$	−7.58482e6	−1.73082	−0.865408	+	0.501069i	$0.832940\pi$
$80$	0	0
$81$	−1.22202e6	−0.255494
$82$	0	0
$83$	−7.26944e6	−1.39549	−0.697747	−	0.716345i	$-0.745814\pi$
$83$	−7.26944e6	−1.39549	−0.697747	+	0.716345i	$0.745814\pi$
$84$	0	0
$85$	8.57871e6	1.51515
$86$	0	0
$87$	9.96106e6	1.62177
$88$	0	0
$89$	−7.80137e6	−1.17302	−0.586511	−	0.809941i	$-0.699499\pi$
$89$	−7.80137e6	−1.17302	−0.586511	+	0.809941i	$0.699499\pi$
$90$	0	0
$91$	−342925.	−0.0477039
$92$	0	0
$93$	1.47414e7	1.90041
$94$	0	0
$95$	−2.46213e6	−0.294631
$96$	0	0
$97$	−6.56007e6	−0.729806	−0.364903	−	0.931046i	$-0.618898\pi$
$97$	−6.56007e6	−0.729806	−0.364903	+	0.931046i	$0.618898\pi$
$98$	0	0
$99$	2.34248e7	2.42634
$100$	0	0
$101$	−149684.	−0.0144561	−0.00722805	−	0.999974i	$-0.502301\pi$
$101$	−149684.	−0.0144561	−0.00722805	+	0.999974i	$0.502301\pi$
$102$	0	0
$103$	−9.12098e6	−0.822453	−0.411226	−	0.911533i	$-0.634899\pi$
$103$	−9.12098e6	−0.822453	−0.411226	+	0.911533i	$0.634899\pi$
$104$	0	0
$105$	2.91279e6	0.245554
$106$	0	0
$107$	−8.96589e6	−0.707538	−0.353769	−	0.935333i	$-0.615100\pi$
$107$	−8.96589e6	−0.707538	−0.353769	+	0.935333i	$0.615100\pi$
$108$	0	0
$109$	1.55911e7	1.15314	0.576572	−	0.817047i	$-0.304390\pi$
$109$	1.55911e7	1.15314	0.576572	+	0.817047i	$0.304390\pi$
$110$	0	0
$111$	7.05182e6	0.489408
$112$	0	0
$113$	1.68951e7	1.10150	0.550752	−	0.834669i	$-0.314341\pi$
$113$	1.68951e7	1.10150	0.550752	+	0.834669i	$0.314341\pi$
$114$	0	0
$115$	1.81715e7	1.11416
$116$	0	0
$117$	−1.02268e7	−0.590320
$118$	0	0
$119$	−2.62407e6	−0.142745
$120$	0	0
$121$	3.16887e7	1.62613
$122$	0	0
$123$	3.61235e7	1.75034
$124$	0	0
$125$	9.83381e6	0.450336
$126$	0	0
$127$	−1.19961e7	−0.519668	−0.259834	−	0.965653i	$-0.583668\pi$
$127$	−1.19961e7	−0.519668	−0.259834	+	0.965653i	$0.583668\pi$
$128$	0	0
$129$	1.20390e7	0.493771
$130$	0	0
$131$	1.52350e7	0.592096	0.296048	−	0.955173i	$-0.404331\pi$
$131$	1.52350e7	0.592096	0.296048	+	0.955173i	$0.404331\pi$
$132$	0	0
$133$	753120.	0.0277577
$134$	0	0
$135$	2.88489e7	1.00916
$136$	0	0
$137$	−4.60519e7	−1.53012	−0.765060	−	0.643959i	$-0.777291\pi$
$137$	−4.60519e7	−1.53012	−0.765060	+	0.643959i	$0.777291\pi$
$138$	0	0
$139$	2.26208e7	0.714424	0.357212	−	0.934023i	$-0.383727\pi$
$139$	2.26208e7	0.714424	0.357212	+	0.934023i	$0.383727\pi$
$140$	0	0
$141$	−6.15306e7	−1.84852
$142$	0	0
$143$	−2.23423e7	−0.638927
$144$	0	0
$145$	4.83839e7	1.31799
$146$	0	0
$147$	5.99704e7	1.55713
$148$	0	0
$149$	4.97987e7	1.23329	0.616646	−	0.787240i	$-0.288491\pi$
$149$	4.97987e7	1.23329	0.616646	+	0.787240i	$0.288491\pi$
$150$	0	0
$151$	3.51439e7	0.830675	0.415337	−	0.909667i	$-0.363663\pi$
$151$	3.51439e7	0.830675	0.415337	+	0.909667i	$0.363663\pi$
$152$	0	0
$153$	−7.82554e7	−1.76642
$154$	0	0
$155$	7.16032e7	1.54444
$156$	0	0
$157$	−7.17788e7	−1.48029	−0.740146	−	0.672447i	$-0.765243\pi$
$157$	−7.17788e7	−1.48029	−0.740146	+	0.672447i	$0.765243\pi$
$158$	0	0
$159$	−7.04401e7	−1.38973
$160$	0	0
$161$	−5.55833e6	−0.104967
$162$	0	0
$163$	9.56289e7	1.72955	0.864773	−	0.502162i	$-0.167462\pi$
$163$	9.56289e7	1.72955	0.864773	+	0.502162i	$0.167462\pi$
$164$	0	0
$165$	1.89775e8	3.28885
$166$	0	0
$167$	−7.29897e7	−1.21270	−0.606351	−	0.795197i	$-0.707367\pi$
$167$	−7.29897e7	−1.21270	−0.606351	+	0.795197i	$0.707367\pi$
$168$	0	0
$169$	−5.29944e7	−0.844551
$170$	0	0
$171$	2.24597e7	0.343493
$172$	0	0
$173$	5.12303e7	0.752257	0.376128	−	0.926568i	$-0.377255\pi$
$173$	5.12303e7	0.752257	0.376128	+	0.926568i	$0.377255\pi$
$174$	0	0
$175$	5.57017e6	0.0785661
$176$	0	0
$177$	1.81822e8	2.46456
$178$	0	0
$179$	−4.58207e7	−0.597140	−0.298570	−	0.954388i	$-0.596510\pi$
$179$	−4.58207e7	−0.597140	−0.298570	+	0.954388i	$0.596510\pi$
$180$	0	0
$181$	7.39469e7	0.926925	0.463463	−	0.886116i	$-0.346607\pi$
$181$	7.39469e7	0.926925	0.463463	+	0.886116i	$0.346607\pi$
$182$	0	0
$183$	−8.54516e7	−1.03072
$184$	0	0
$185$	3.42529e7	0.397737
$186$	0	0
$187$	−1.70964e8	−1.91187
$188$	0	0
$189$	−8.82433e6	−0.0950748
$190$	0	0
$191$	1.01526e8	1.05429	0.527144	−	0.849776i	$-0.323263\pi$
$191$	1.01526e8	1.05429	0.527144	+	0.849776i	$0.323263\pi$
$192$	0	0
$193$	−4.98679e7	−0.499310	−0.249655	−	0.968335i	$-0.580317\pi$
$193$	−4.98679e7	−0.499310	−0.249655	+	0.968335i	$0.580317\pi$
$194$	0	0
$195$	−8.28516e7	−0.800166
$196$	0	0
$197$	−1.39368e8	−1.29876	−0.649382	−	0.760463i	$-0.724972\pi$
$197$	−1.39368e8	−1.29876	−0.649382	+	0.760463i	$0.724972\pi$
$198$	0	0
$199$	1.01709e8	0.914898	0.457449	−	0.889236i	$-0.348763\pi$
$199$	1.01709e8	0.914898	0.457449	+	0.889236i	$0.348763\pi$
$200$	0	0
$201$	−1.17775e8	−1.02298
$202$	0	0
$203$	−1.47997e7	−0.124170
$204$	0	0
$205$	1.75463e8	1.42248
$206$	0	0
$207$	−1.65761e8	−1.29893
$208$	0	0
$209$	4.90674e7	0.371776
$210$	0	0
$211$	−9.48482e7	−0.695089	−0.347545	−	0.937663i	$-0.612985\pi$
$211$	−9.48482e7	−0.695089	−0.347545	+	0.937663i	$0.612985\pi$
$212$	0	0
$213$	−4.36063e8	−3.09187
$214$	0	0
$215$	5.84769e7	0.401282
$216$	0	0
$217$	−2.19021e7	−0.145505
$218$	0	0
$219$	8.65063e7	0.556536
$220$	0	0
$221$	7.46391e7	0.465151
$222$	0	0
$223$	−1.13197e8	−0.683548	−0.341774	−	0.939782i	$-0.611028\pi$
$223$	−1.13197e8	−0.683548	−0.341774	+	0.939782i	$0.611028\pi$
$224$	0	0
$225$	1.66115e8	0.972230
$226$	0	0
$227$	3.23618e8	1.83629	0.918147	−	0.396241i	$-0.129686\pi$
$227$	3.23618e8	1.83629	0.918147	+	0.396241i	$0.129686\pi$
$228$	0	0
$229$	2.63829e7	0.145177	0.0725887	−	0.997362i	$-0.476874\pi$
$229$	2.63829e7	0.145177	0.0725887	+	0.997362i	$0.476874\pi$
$230$	0	0
$231$	−5.80486e7	−0.309849
$232$	0	0
$233$	−2.91009e8	−1.50716	−0.753581	−	0.657355i	$-0.771675\pi$
$233$	−2.91009e8	−1.50716	−0.753581	+	0.657355i	$0.771675\pi$
$234$	0	0
$235$	−2.98873e8	−1.50227
$236$	0	0
$237$	5.60533e8	2.73515
$238$	0	0
$239$	3.00022e8	1.42155	0.710773	−	0.703422i	$-0.248345\pi$
$239$	3.00022e8	1.42155	0.710773	+	0.703422i	$0.248345\pi$
$240$	0	0
$241$	1.30849e8	0.602157	0.301078	−	0.953599i	$-0.402653\pi$
$241$	1.30849e8	0.602157	0.301078	+	0.953599i	$0.402653\pi$
$242$	0	0
$243$	2.66072e8	1.18954
$244$	0	0
$245$	2.91294e8	1.26547
$246$	0	0
$247$	−2.14218e7	−0.0904517
$248$	0	0
$249$	5.37225e8	2.20525
$250$	0	0
$251$	4.29176e8	1.71308	0.856539	−	0.516083i	$-0.172610\pi$
$251$	4.29176e8	1.71308	0.856539	+	0.516083i	$0.172610\pi$
$252$	0	0
$253$	−3.62137e8	−1.40589
$254$	0	0
$255$	−6.33983e8	−2.39435
$256$	0	0
$257$	−3.14319e8	−1.15506	−0.577530	−	0.816370i	$-0.695983\pi$
$257$	−3.14319e8	−1.15506	−0.577530	+	0.816370i	$0.695983\pi$
$258$	0	0
$259$	−1.04773e7	−0.0374715
$260$	0	0
$261$	−4.41360e8	−1.53657
$262$	0	0
$263$	4.31366e8	1.46218	0.731090	−	0.682281i	$-0.239012\pi$
$263$	4.31366e8	1.46218	0.731090	+	0.682281i	$0.239012\pi$
$264$	0	0
$265$	−3.42149e8	−1.12942
$266$	0	0
$267$	5.76536e8	1.85369
$268$	0	0
$269$	1.12548e8	0.352537	0.176269	−	0.984342i	$-0.443597\pi$
$269$	1.12548e8	0.352537	0.176269	+	0.984342i	$0.443597\pi$
$270$	0	0
$271$	−4.25465e8	−1.29859	−0.649294	−	0.760537i	$-0.724936\pi$
$271$	−4.25465e8	−1.29859	−0.649294	+	0.760537i	$0.724936\pi$
$272$	0	0
$273$	2.53428e7	0.0753850
$274$	0	0
$275$	3.62909e8	1.05228
$276$	0	0
$277$	−3.22706e8	−0.912280	−0.456140	−	0.889908i	$-0.650768\pi$
$277$	−3.22706e8	−0.912280	−0.456140	+	0.889908i	$0.650768\pi$
$278$	0	0
$279$	−6.53168e8	−1.80057
$280$	0	0
$281$	3.95029e8	1.06208	0.531039	−	0.847347i	$-0.321802\pi$
$281$	3.95029e8	1.06208	0.531039	+	0.847347i	$0.321802\pi$
$282$	0	0
$283$	−2.22370e7	−0.0583209	−0.0291605	−	0.999575i	$-0.509283\pi$
$283$	−2.22370e7	−0.0583209	−0.0291605	+	0.999575i	$0.509283\pi$
$284$	0	0
$285$	1.81956e8	0.465597
$286$	0	0
$287$	−5.36709e7	−0.134015
$288$	0	0
$289$	1.60802e8	0.391877
$290$	0	0
$291$	4.84801e8	1.15329
$292$	0	0
$293$	−2.90759e8	−0.675299	−0.337649	−	0.941272i	$-0.609632\pi$
$293$	−2.90759e8	−0.675299	−0.337649	+	0.941272i	$0.609632\pi$
$294$	0	0
$295$	8.83164e8	2.00292
$296$	0	0
$297$	−5.74925e8	−1.27340
$298$	0	0
$299$	1.58101e8	0.342047
$300$	0	0
$301$	−1.78870e7	−0.0378055
$302$	0	0
$303$	1.10619e7	0.0228445
$304$	0	0
$305$	−4.15064e8	−0.837657
$306$	0	0
$307$	3.45260e8	0.681023	0.340512	−	0.940240i	$-0.389400\pi$
$307$	3.45260e8	0.681023	0.340512	+	0.940240i	$0.389400\pi$
$308$	0	0
$309$	6.74057e8	1.29970
$310$	0	0
$311$	1.53537e8	0.289435	0.144718	−	0.989473i	$-0.453773\pi$
$311$	1.53537e8	0.289435	0.144718	+	0.989473i	$0.453773\pi$
$312$	0	0
$313$	3.62886e8	0.668906	0.334453	−	0.942412i	$-0.391449\pi$
$313$	3.62886e8	0.668906	0.334453	+	0.942412i	$0.391449\pi$
$314$	0	0
$315$	−1.29062e8	−0.232654
$316$	0	0
$317$	−2.06047e8	−0.363294	−0.181647	−	0.983364i	$-0.558143\pi$
$317$	−2.06047e8	−0.363294	−0.181647	+	0.983364i	$0.558143\pi$
$318$	0	0
$319$	−9.64234e8	−1.66309
$320$	0	0
$321$	6.62596e8	1.11810
$322$	0	0
$323$	−1.63920e8	−0.270660
$324$	0	0
$325$	−1.58438e8	−0.256017
$326$	0	0
$327$	−1.15221e9	−1.82228
$328$	0	0
$329$	9.14196e7	0.141532
$330$	0	0
$331$	−5.59579e8	−0.848131	−0.424066	−	0.905631i	$-0.639397\pi$
$331$	−5.59579e8	−0.848131	−0.424066	+	0.905631i	$0.639397\pi$
$332$	0	0
$333$	−3.12456e8	−0.463697
$334$	0	0
$335$	−5.72068e8	−0.831363
$336$	0	0
$337$	−4.56120e8	−0.649195	−0.324597	−	0.945852i	$-0.605229\pi$
$337$	−4.56120e8	−0.649195	−0.324597	+	0.945852i	$0.605229\pi$
$338$	0	0
$339$	−1.24858e9	−1.74067
$340$	0	0
$341$	−1.42697e9	−1.94883
$342$	0	0
$343$	−1.79527e8	−0.240215
$344$	0	0
$345$	−1.34291e9	−1.76068
$346$	0	0
$347$	−9.84759e8	−1.26525	−0.632626	−	0.774458i	$-0.718023\pi$
$347$	−9.84759e8	−1.26525	−0.632626	+	0.774458i	$0.718023\pi$
$348$	0	0
$349$	6.02914e8	0.759218	0.379609	−	0.925147i	$-0.376059\pi$
$349$	6.02914e8	0.759218	0.379609	+	0.925147i	$0.376059\pi$
$350$	0	0
$351$	2.51000e8	0.309812
$352$	0	0
$353$	2.23097e7	0.0269949	0.0134975	−	0.999909i	$-0.495703\pi$
$353$	2.23097e7	0.0269949	0.0134975	+	0.999909i	$0.495703\pi$
$354$	0	0
$355$	−2.11809e9	−2.51273
$356$	0	0
$357$	1.93924e8	0.225575
$358$	0	0
$359$	9.61552e8	1.09684	0.548418	−	0.836204i	$-0.315230\pi$
$359$	9.61552e8	1.09684	0.548418	+	0.836204i	$0.315230\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−2.34185e9	−2.56972
$364$	0	0
$365$	4.20188e8	0.452291
$366$	0	0
$367$	−6.58001e8	−0.694856	−0.347428	−	0.937707i	$-0.612945\pi$
$367$	−6.58001e8	−0.694856	−0.347428	+	0.937707i	$0.612945\pi$
$368$	0	0
$369$	−1.60058e9	−1.65839
$370$	0	0
$371$	1.04657e8	0.106404
$372$	0	0
$373$	−1.20948e9	−1.20675	−0.603373	−	0.797459i	$-0.706177\pi$
$373$	−1.20948e9	−1.20675	−0.603373	+	0.797459i	$0.706177\pi$
$374$	0	0
$375$	−7.26737e8	−0.711652
$376$	0	0
$377$	4.20964e8	0.404623
$378$	0	0
$379$	−1.22771e9	−1.15840	−0.579198	−	0.815187i	$-0.696634\pi$
$379$	−1.22771e9	−1.15840	−0.579198	+	0.815187i	$0.696634\pi$
$380$	0	0
$381$	8.86531e8	0.821214
$382$	0	0
$383$	1.21666e9	1.10655	0.553276	−	0.832998i	$-0.313377\pi$
$383$	1.21666e9	1.10655	0.553276	+	0.832998i	$0.313377\pi$
$384$	0	0
$385$	−2.81960e8	−0.251811
$386$	0	0
$387$	−5.33429e8	−0.467830
$388$	0	0
$389$	1.34163e9	1.15561	0.577803	−	0.816176i	$-0.303910\pi$
$389$	1.34163e9	1.15561	0.577803	+	0.816176i	$0.303910\pi$
$390$	0	0
$391$	1.20980e9	1.02351
$392$	0	0
$393$	−1.12589e9	−0.935670
$394$	0	0
$395$	2.72268e9	2.22283
$396$	0	0
$397$	−1.91153e9	−1.53325	−0.766626	−	0.642094i	$-0.778066\pi$
$397$	−1.91153e9	−1.53325	−0.766626	+	0.642094i	$0.778066\pi$
$398$	0	0
$399$	−5.56570e7	−0.0438647
$400$	0	0
$401$	−1.14237e9	−0.884708	−0.442354	−	0.896841i	$-0.645856\pi$
$401$	−1.14237e9	−0.884708	−0.442354	+	0.896841i	$0.645856\pi$
$402$	0	0
$403$	6.22984e8	0.474143
$404$	0	0
$405$	4.38661e8	0.328123
$406$	0	0
$407$	−6.82619e8	−0.501878
$408$	0	0
$409$	2.00435e9	1.44858	0.724290	−	0.689496i	$-0.242168\pi$
$409$	2.00435e9	1.44858	0.724290	+	0.689496i	$0.242168\pi$
$410$	0	0
$411$	3.40332e9	2.41800
$412$	0	0
$413$	−2.70143e8	−0.188699
$414$	0	0
$415$	2.60947e9	1.79219
$416$	0	0
$417$	−1.67172e9	−1.12898
$418$	0	0
$419$	−1.02610e9	−0.681461	−0.340730	−	0.940161i	$-0.610674\pi$
$419$	−1.02610e9	−0.681461	−0.340730	+	0.940161i	$0.610674\pi$
$420$	0	0
$421$	2.29081e9	1.49624	0.748122	−	0.663562i	$-0.230956\pi$
$421$	2.29081e9	1.49624	0.748122	+	0.663562i	$0.230956\pi$
$422$	0	0
$423$	2.72633e9	1.75141
$424$	0	0
$425$	−1.21237e9	−0.766082
$426$	0	0
$427$	1.26960e8	0.0789171
$428$	0	0
$429$	1.65114e9	1.00968
$430$	0	0
$431$	4.13824e8	0.248969	0.124484	−	0.992222i	$-0.460272\pi$
$431$	4.13824e8	0.248969	0.124484	+	0.992222i	$0.460272\pi$
$432$	0	0
$433$	2.83448e9	1.67790	0.838950	−	0.544209i	$-0.183170\pi$
$433$	2.83448e9	1.67790	0.838950	+	0.544209i	$0.183170\pi$
$434$	0	0
$435$	−3.57566e9	−2.08278
$436$	0	0
$437$	−3.47217e8	−0.199029
$438$	0	0
$439$	2.83393e9	1.59869	0.799344	−	0.600874i	$-0.205181\pi$
$439$	2.83393e9	1.59869	0.799344	+	0.600874i	$0.205181\pi$
$440$	0	0
$441$	−2.65720e9	−1.47533
$442$	0	0
$443$	−2.13580e9	−1.16720	−0.583602	−	0.812040i	$-0.698357\pi$
$443$	−2.13580e9	−1.16720	−0.583602	+	0.812040i	$0.698357\pi$
$444$	0	0
$445$	2.80041e9	1.50647
$446$	0	0
$447$	−3.68022e9	−1.94893
$448$	0	0
$449$	8.43253e8	0.439638	0.219819	−	0.975541i	$-0.429453\pi$
$449$	8.43253e8	0.439638	0.219819	+	0.975541i	$0.429453\pi$
$450$	0	0
$451$	−3.49677e9	−1.79494
$452$	0	0
$453$	−2.59720e9	−1.31269
$454$	0	0
$455$	1.23098e8	0.0612646
$456$	0	0
$457$	2.59270e9	1.27071	0.635354	−	0.772221i	$-0.280854\pi$
$457$	2.59270e9	1.27071	0.635354	+	0.772221i	$0.280854\pi$
$458$	0	0
$459$	1.92066e9	0.927055
$460$	0	0
$461$	1.25804e9	0.598053	0.299027	−	0.954245i	$-0.403338\pi$
$461$	1.25804e9	0.598053	0.299027	+	0.954245i	$0.403338\pi$
$462$	0	0
$463$	2.70904e9	1.26847	0.634236	−	0.773139i	$-0.281315\pi$
$463$	2.70904e9	1.26847	0.634236	+	0.773139i	$0.281315\pi$
$464$	0	0
$465$	−5.29161e9	−2.44063
$466$	0	0
$467$	1.94781e9	0.884987	0.442494	−	0.896772i	$-0.354094\pi$
$467$	1.94781e9	0.884987	0.442494	+	0.896772i	$0.354094\pi$
$468$	0	0
$469$	1.74985e8	0.0783241
$470$	0	0
$471$	5.30458e9	2.33926
$472$	0	0
$473$	−1.16538e9	−0.506352
$474$	0	0
$475$	3.47957e8	0.148970
$476$	0	0
$477$	3.12110e9	1.31672
$478$	0	0
$479$	1.46097e9	0.607390	0.303695	−	0.952769i	$-0.401780\pi$
$479$	1.46097e9	0.607390	0.303695	+	0.952769i	$0.401780\pi$
$480$	0	0
$481$	2.98017e8	0.122105
$482$	0	0
$483$	4.10771e8	0.165876
$484$	0	0
$485$	2.35483e9	0.937266
$486$	0	0
$487$	−1.26613e9	−0.496736	−0.248368	−	0.968666i	$-0.579894\pi$
$487$	−1.26613e9	−0.496736	−0.248368	+	0.968666i	$0.579894\pi$
$488$	0	0
$489$	−7.06715e9	−2.73315
$490$	0	0
$491$	3.83771e9	1.46314	0.731572	−	0.681765i	$-0.238787\pi$
$491$	3.83771e9	1.46314	0.731572	+	0.681765i	$0.238787\pi$
$492$	0	0
$493$	3.22123e9	1.21076
$494$	0	0
$495$	−8.40865e9	−3.11607
$496$	0	0
$497$	6.47884e8	0.236728
$498$	0	0
$499$	3.49904e9	1.26066	0.630328	−	0.776329i	$-0.282920\pi$
$499$	3.49904e9	1.26066	0.630328	+	0.776329i	$0.282920\pi$
$500$	0	0
$501$	5.39408e9	1.91640
$502$	0	0
$503$	2.11713e9	0.741754	0.370877	−	0.928682i	$-0.379057\pi$
$503$	2.11713e9	0.741754	0.370877	+	0.928682i	$0.379057\pi$
$504$	0	0
$505$	5.37312e7	0.0185655
$506$	0	0
$507$	3.91638e9	1.33462
$508$	0	0
$509$	−1.01744e9	−0.341977	−0.170988	−	0.985273i	$-0.554696\pi$
$509$	−1.01744e9	−0.341977	−0.170988	+	0.985273i	$0.554696\pi$
$510$	0	0
$511$	−1.28528e8	−0.0426111
$512$	0	0
$513$	−5.51238e8	−0.180272
$514$	0	0
$515$	3.27410e9	1.05625
$516$	0	0
$517$	5.95619e9	1.89562
$518$	0	0
$519$	−3.78602e9	−1.18877
$520$	0	0
$521$	−2.02605e9	−0.627650	−0.313825	−	0.949481i	$-0.601611\pi$
$521$	−2.02605e9	−0.627650	−0.313825	+	0.949481i	$0.601611\pi$
$522$	0	0
$523$	1.04115e8	0.0318241	0.0159120	−	0.999873i	$-0.494935\pi$
$523$	1.04115e8	0.0318241	0.0159120	+	0.999873i	$0.494935\pi$
$524$	0	0
$525$	−4.11646e8	−0.124156
$526$	0	0
$527$	4.76709e9	1.41878
$528$	0	0
$529$	−8.42227e8	−0.247363
$530$	0	0
$531$	−8.05626e9	−2.33509
$532$	0	0
$533$	1.52662e9	0.436702
$534$	0	0
$535$	3.21843e9	0.908669
$536$	0	0
$537$	3.38623e9	0.943641
$538$	0	0
$539$	−5.80516e9	−1.59681
$540$	0	0
$541$	2.86709e9	0.778487	0.389243	−	0.921135i	$-0.372737\pi$
$541$	2.86709e9	0.778487	0.389243	+	0.921135i	$0.372737\pi$
$542$	0	0
$543$	−5.46481e9	−1.46479
$544$	0	0
$545$	−5.59663e9	−1.48095
$546$	0	0
$547$	−4.20539e9	−1.09863	−0.549313	−	0.835616i	$-0.685111\pi$
$547$	−4.20539e9	−1.09863	−0.549313	+	0.835616i	$0.685111\pi$
$548$	0	0
$549$	3.78624e9	0.976573
$550$	0	0
$551$	−9.24508e8	−0.235440
$552$	0	0
$553$	−8.32816e8	−0.209417
$554$	0	0
$555$	−2.53135e9	−0.628531
$556$	0	0
$557$	6.90397e9	1.69280	0.846400	−	0.532548i	$-0.178765\pi$
$557$	6.90397e9	1.69280	0.846400	+	0.532548i	$0.178765\pi$
$558$	0	0
$559$	5.08779e8	0.123193
$560$	0	0
$561$	1.26345e10	3.02127
$562$	0	0
$563$	2.07530e9	0.490119	0.245060	−	0.969508i	$-0.421192\pi$
$563$	2.07530e9	0.490119	0.245060	+	0.969508i	$0.421192\pi$
$564$	0	0
$565$	−6.06473e9	−1.41463
$566$	0	0
$567$	−1.34178e8	−0.0309130
$568$	0	0
$569$	3.50410e9	0.797414	0.398707	−	0.917078i	$-0.369459\pi$
$569$	3.50410e9	0.797414	0.398707	+	0.917078i	$0.369459\pi$
$570$	0	0
$571$	−5.15559e9	−1.15892	−0.579458	−	0.815002i	$-0.696736\pi$
$571$	−5.15559e9	−1.15892	−0.579458	+	0.815002i	$0.696736\pi$
$572$	0	0
$573$	−7.50294e9	−1.66606
$574$	0	0
$575$	−2.56806e9	−0.563336
$576$	0	0
$577$	−6.08613e9	−1.31894	−0.659471	−	0.751730i	$-0.729220\pi$
$577$	−6.08613e9	−1.31894	−0.659471	+	0.751730i	$0.729220\pi$
$578$	0	0
$579$	3.68533e9	0.789045
$580$	0	0
$581$	−7.98187e8	−0.168845
$582$	0	0
$583$	6.81863e9	1.42514
$584$	0	0
$585$	3.67104e9	0.758129
$586$	0	0
$587$	−3.10318e9	−0.633247	−0.316623	−	0.948551i	$-0.602549\pi$
$587$	−3.10318e9	−0.633247	−0.316623	+	0.948551i	$0.602549\pi$
$588$	0	0
$589$	−1.36818e9	−0.275892
$590$	0	0
$591$	1.02995e10	2.05240
$592$	0	0
$593$	8.15698e9	1.60634	0.803170	−	0.595749i	$-0.203145\pi$
$593$	8.15698e9	1.60634	0.803170	+	0.595749i	$0.203145\pi$
$594$	0	0
$595$	9.41946e8	0.183323
$596$	0	0
$597$	−7.51647e9	−1.44578
$598$	0	0
$599$	3.83681e9	0.729418	0.364709	−	0.931122i	$-0.381169\pi$
$599$	3.83681e9	0.729418	0.364709	+	0.931122i	$0.381169\pi$
$600$	0	0
$601$	−5.74255e9	−1.07906	−0.539529	−	0.841967i	$-0.681398\pi$
$601$	−5.74255e9	−1.07906	−0.539529	+	0.841967i	$0.681398\pi$
$602$	0	0
$603$	5.21843e9	0.969235
$604$	0	0
$605$	−1.13751e10	−2.08839
$606$	0	0
$607$	−2.66716e9	−0.484048	−0.242024	−	0.970270i	$-0.577811\pi$
$607$	−2.66716e9	−0.484048	−0.242024	+	0.970270i	$0.577811\pi$
$608$	0	0
$609$	1.09373e9	0.196222
$610$	0	0
$611$	−2.60034e9	−0.461197
$612$	0	0
$613$	2.97013e9	0.520791	0.260396	−	0.965502i	$-0.416147\pi$
$613$	2.97013e9	0.520791	0.260396	+	0.965502i	$0.416147\pi$
$614$	0	0
$615$	−1.29670e10	−2.24791
$616$	0	0
$617$	−6.26837e9	−1.07438	−0.537189	−	0.843462i	$-0.680514\pi$
$617$	−6.26837e9	−1.07438	−0.537189	+	0.843462i	$0.680514\pi$
$618$	0	0
$619$	6.46559e9	1.09570	0.547849	−	0.836577i	$-0.315447\pi$
$619$	6.46559e9	1.09570	0.547849	+	0.836577i	$0.315447\pi$
$620$	0	0
$621$	4.06835e9	0.681707
$622$	0	0
$623$	−8.56594e8	−0.141928
$624$	0	0
$625$	−7.49326e9	−1.22770
$626$	0	0
$627$	−3.62617e9	−0.587506
$628$	0	0
$629$	2.28043e9	0.365376
$630$	0	0
$631$	9.86287e9	1.56279	0.781395	−	0.624037i	$-0.214509\pi$
$631$	9.86287e9	1.56279	0.781395	+	0.624037i	$0.214509\pi$
$632$	0	0
$633$	7.00946e9	1.09843
$634$	0	0
$635$	4.30615e9	0.667392
$636$	0	0
$637$	2.53441e9	0.388498
$638$	0	0
$639$	1.93213e10	2.92943
$640$	0	0
$641$	1.04849e10	1.57240	0.786198	−	0.617974i	$-0.212046\pi$
$641$	1.04849e10	1.57240	0.786198	+	0.617974i	$0.212046\pi$
$642$	0	0
$643$	9.27539e8	0.137592	0.0687961	−	0.997631i	$-0.478084\pi$
$643$	9.27539e8	0.137592	0.0687961	+	0.997631i	$0.478084\pi$
$644$	0	0
$645$	−4.32155e9	−0.634134
$646$	0	0
$647$	−1.07688e10	−1.56316	−0.781578	−	0.623808i	$-0.785585\pi$
$647$	−1.07688e10	−1.56316	−0.781578	+	0.623808i	$0.785585\pi$
$648$	0	0
$649$	−1.76004e10	−2.52736
$650$	0	0
$651$	1.61861e9	0.229936
$652$	0	0
$653$	7.05077e9	0.990925	0.495462	−	0.868629i	$-0.334999\pi$
$653$	7.05077e9	0.990925	0.495462	+	0.868629i	$0.334999\pi$
$654$	0	0
$655$	−5.46880e9	−0.760409
$656$	0	0
$657$	−3.83297e9	−0.527299
$658$	0	0
$659$	−9.71506e8	−0.132235	−0.0661175	−	0.997812i	$-0.521061\pi$
$659$	−9.71506e8	−0.132235	−0.0661175	+	0.997812i	$0.521061\pi$
$660$	0	0
$661$	−7.03635e9	−0.947637	−0.473819	−	0.880622i	$-0.657125\pi$
$661$	−7.03635e9	−0.947637	−0.473819	+	0.880622i	$0.657125\pi$
$662$	0	0
$663$	−5.51597e9	−0.735063
$664$	0	0
$665$	−2.70343e8	−0.0356483
$666$	0	0
$667$	6.82324e9	0.890328
$668$	0	0
$669$	8.36548e9	1.08019
$670$	0	0
$671$	8.27175e9	1.05698
$672$	0	0
$673$	9.35466e9	1.18297	0.591487	−	0.806314i	$-0.298541\pi$
$673$	9.35466e9	1.18297	0.591487	+	0.806314i	$0.298541\pi$
$674$	0	0
$675$	−4.07702e9	−0.510246
$676$	0	0
$677$	5.64449e9	0.699140	0.349570	−	0.936910i	$-0.386328\pi$
$677$	5.64449e9	0.699140	0.349570	+	0.936910i	$0.386328\pi$
$678$	0	0
$679$	−7.20298e8	−0.0883015
$680$	0	0
$681$	−2.39160e10	−2.90184
$682$	0	0
$683$	1.50975e10	1.81314	0.906571	−	0.422054i	$-0.138691\pi$
$683$	1.50975e10	1.81314	0.906571	+	0.422054i	$0.138691\pi$
$684$	0	0
$685$	1.65310e10	1.96508
$686$	0	0
$687$	−1.94975e9	−0.229419
$688$	0	0
$689$	−2.97687e9	−0.346731
$690$	0	0
$691$	1.25008e9	0.144133	0.0720666	−	0.997400i	$-0.477041\pi$
$691$	1.25008e9	0.144133	0.0720666	+	0.997400i	$0.477041\pi$
$692$	0	0
$693$	2.57205e9	0.293571
$694$	0	0
$695$	−8.12004e9	−0.917511
$696$	0	0
$697$	1.16817e10	1.30675
$698$	0	0
$699$	2.15061e10	2.38172
$700$	0	0
$701$	4.65921e9	0.510857	0.255428	−	0.966828i	$-0.417784\pi$
$701$	4.65921e9	0.510857	0.255428	+	0.966828i	$0.417784\pi$
$702$	0	0
$703$	−6.54496e8	−0.0710499
$704$	0	0
$705$	2.20873e10	2.37399
$706$	0	0
$707$	−1.64354e7	−0.00174909
$708$	0	0
$709$	−5.70761e9	−0.601441	−0.300720	−	0.953712i	$-0.597227\pi$
$709$	−5.70761e9	−0.601441	−0.300720	+	0.953712i	$0.597227\pi$
$710$	0	0
$711$	−2.48364e10	−2.59146
$712$	0	0
$713$	1.00977e10	1.04330
$714$	0	0
$715$	8.02007e9	0.820554
$716$	0	0
$717$	−2.21722e10	−2.24642
$718$	0	0
$719$	−1.08664e10	−1.09027	−0.545136	−	0.838347i	$-0.683522\pi$
$719$	−1.08664e10	−1.09027	−0.545136	+	0.838347i	$0.683522\pi$
$720$	0	0
$721$	−1.00149e9	−0.0995111
$722$	0	0
$723$	−9.66996e9	−0.951569
$724$	0	0
$725$	−6.83778e9	−0.666395
$726$	0	0
$727$	3.49074e9	0.336936	0.168468	−	0.985707i	$-0.446118\pi$
$727$	3.49074e9	0.336936	0.168468	+	0.985707i	$0.446118\pi$
$728$	0	0
$729$	−1.69907e10	−1.62429
$730$	0	0
$731$	3.89319e9	0.368633
$732$	0	0
$733$	−1.94782e9	−0.182677	−0.0913386	−	0.995820i	$-0.529115\pi$
$733$	−1.94782e9	−0.182677	−0.0913386	+	0.995820i	$0.529115\pi$
$734$	0	0
$735$	−2.15272e10	−1.99978
$736$	0	0
$737$	1.14006e10	1.04904
$738$	0	0
$739$	5.89076e9	0.536928	0.268464	−	0.963290i	$-0.413484\pi$
$739$	5.89076e9	0.536928	0.268464	+	0.963290i	$0.413484\pi$
$740$	0	0
$741$	1.58311e9	0.142938
$742$	0	0
$743$	−1.25115e10	−1.11905	−0.559525	−	0.828813i	$-0.689016\pi$
$743$	−1.25115e10	−1.11905	−0.559525	+	0.828813i	$0.689016\pi$
$744$	0	0
$745$	−1.78759e10	−1.58388
$746$	0	0
$747$	−2.38037e10	−2.08940
$748$	0	0
$749$	−9.84457e8	−0.0856072
$750$	0	0
$751$	−1.24647e10	−1.07385	−0.536925	−	0.843630i	$-0.680414\pi$
$751$	−1.24647e10	−1.07385	−0.536925	+	0.843630i	$0.680414\pi$
$752$	0	0
$753$	−3.17169e10	−2.70712
$754$	0	0
$755$	−1.26154e10	−1.06681
$756$	0	0
$757$	1.27300e10	1.06658	0.533290	−	0.845933i	$-0.320956\pi$
$757$	1.27300e10	1.06658	0.533290	+	0.845933i	$0.320956\pi$
$758$	0	0
$759$	2.67626e10	2.22168
$760$	0	0
$761$	−2.03284e10	−1.67208	−0.836041	−	0.548666i	$-0.815136\pi$
$761$	−2.03284e10	−1.67208	−0.836041	+	0.548666i	$0.815136\pi$
$762$	0	0
$763$	1.71191e9	0.139522
$764$	0	0
$765$	2.80909e10	2.26856
$766$	0	0
$767$	7.68397e9	0.614897
$768$	0	0
$769$	−9.40790e9	−0.746020	−0.373010	−	0.927827i	$-0.621674\pi$
$769$	−9.40790e9	−0.746020	−0.373010	+	0.927827i	$0.621674\pi$
$770$	0	0
$771$	2.32287e10	1.82530
$772$	0	0
$773$	3.86805e9	0.301206	0.150603	−	0.988594i	$-0.451878\pi$
$773$	3.86805e9	0.301206	0.150603	+	0.988594i	$0.451878\pi$
$774$	0	0
$775$	−1.01192e10	−0.780892
$776$	0	0
$777$	7.74293e8	0.0592150
$778$	0	0
$779$	−3.35271e9	−0.254106
$780$	0	0
$781$	4.22111e10	3.17065
$782$	0	0
$783$	1.08325e10	0.806422
$784$	0	0
$785$	2.57660e10	1.90109
$786$	0	0
$787$	−2.08450e10	−1.52437	−0.762185	−	0.647359i	$-0.775873\pi$
$787$	−2.08450e10	−1.52437	−0.762185	+	0.647359i	$0.775873\pi$
$788$	0	0
$789$	−3.18787e10	−2.31064
$790$	0	0
$791$	1.85509e9	0.133274
$792$	0	0
$793$	−3.61127e9	−0.257160
$794$	0	0
$795$	2.52854e10	1.78478
$796$	0	0
$797$	2.08677e9	0.146006	0.0730028	−	0.997332i	$-0.476742\pi$
$797$	2.08677e9	0.146006	0.0730028	+	0.997332i	$0.476742\pi$
$798$	0	0
$799$	−1.98979e10	−1.38005
$800$	0	0
$801$	−2.55455e10	−1.75631
$802$	0	0
$803$	−8.37385e9	−0.570717
$804$	0	0
$805$	1.99524e9	0.134806
$806$	0	0
$807$	−8.31751e9	−0.557104
$808$	0	0
$809$	−1.85609e10	−1.23248	−0.616238	−	0.787560i	$-0.711344\pi$
$809$	−1.85609e10	−1.23248	−0.616238	+	0.787560i	$0.711344\pi$
$810$	0	0
$811$	−1.76811e10	−1.16396	−0.581979	−	0.813204i	$-0.697721\pi$
$811$	−1.76811e10	−1.16396	−0.581979	+	0.813204i	$0.697721\pi$
$812$	0	0
$813$	3.14427e10	2.05212
$814$	0	0
$815$	−3.43273e10	−2.22120
$816$	0	0
$817$	−1.11736e9	−0.0716832
$818$	0	0
$819$	−1.12290e9	−0.0714246
$820$	0	0
$821$	2.32701e10	1.46757	0.733784	−	0.679383i	$-0.237753\pi$
$821$	2.32701e10	1.46757	0.733784	+	0.679383i	$0.237753\pi$
$822$	0	0
$823$	1.10963e10	0.693874	0.346937	−	0.937888i	$-0.387222\pi$
$823$	1.10963e10	0.693874	0.346937	+	0.937888i	$0.387222\pi$
$824$	0	0
$825$	−2.68196e10	−1.66289
$826$	0	0
$827$	−6.85391e9	−0.421375	−0.210688	−	0.977553i	$-0.567570\pi$
$827$	−6.85391e9	−0.421375	−0.210688	+	0.977553i	$0.567570\pi$
$828$	0	0
$829$	−1.93058e9	−0.117692	−0.0588459	−	0.998267i	$-0.518742\pi$
$829$	−1.93058e9	−0.117692	−0.0588459	+	0.998267i	$0.518742\pi$
$830$	0	0
$831$	2.38486e10	1.44165
$832$	0	0
$833$	1.93934e10	1.16251
$834$	0	0
$835$	2.62007e10	1.55743
$836$	0	0
$837$	1.60310e10	0.944977
$838$	0	0
$839$	1.22571e10	0.716508	0.358254	−	0.933624i	$-0.383372\pi$
$839$	1.22571e10	0.716508	0.358254	+	0.933624i	$0.383372\pi$
$840$	0	0
$841$	9.17827e8	0.0532077
$842$	0	0
$843$	−2.91934e10	−1.67837
$844$	0	0
$845$	1.90231e10	1.08463
$846$	0	0
$847$	3.47943e9	0.196751
$848$	0	0
$849$	1.64336e9	0.0921627
$850$	0	0
$851$	4.83044e9	0.268679
$852$	0	0
$853$	−2.04007e10	−1.12545	−0.562723	−	0.826646i	$-0.690246\pi$
$853$	−2.04007e10	−1.12545	−0.562723	+	0.826646i	$0.690246\pi$
$854$	0	0
$855$	−8.06221e9	−0.441136
$856$	0	0
$857$	1.22548e10	0.665080	0.332540	−	0.943089i	$-0.392094\pi$
$857$	1.22548e10	0.665080	0.332540	+	0.943089i	$0.392094\pi$
$858$	0	0
$859$	9.32872e9	0.502164	0.251082	−	0.967966i	$-0.419214\pi$
$859$	9.32872e9	0.502164	0.251082	+	0.967966i	$0.419214\pi$
$860$	0	0
$861$	3.96638e9	0.211779
$862$	0	0
$863$	−5.79054e9	−0.306677	−0.153339	−	0.988174i	$-0.549002\pi$
$863$	−5.79054e9	−0.306677	−0.153339	+	0.988174i	$0.549002\pi$
$864$	0	0
$865$	−1.83898e10	−0.966099
$866$	0	0
$867$	−1.18836e10	−0.619270
$868$	0	0
$869$	−5.42598e10	−2.80484
$870$	0	0
$871$	−4.97728e9	−0.255228
$872$	0	0
$873$	−2.14808e10	−1.09270
$874$	0	0
$875$	1.07976e9	0.0544876
$876$	0	0
$877$	−1.87189e10	−0.937090	−0.468545	−	0.883440i	$-0.655222\pi$
$877$	−1.87189e10	−0.937090	−0.468545	+	0.883440i	$0.655222\pi$
$878$	0	0
$879$	2.14876e10	1.06715
$880$	0	0
$881$	1.71691e10	0.845927	0.422964	−	0.906147i	$-0.360990\pi$
$881$	1.71691e10	0.845927	0.422964	+	0.906147i	$0.360990\pi$
$882$	0	0
$883$	5.07048e9	0.247848	0.123924	−	0.992292i	$-0.460452\pi$
$883$	5.07048e9	0.247848	0.123924	+	0.992292i	$0.460452\pi$
$884$	0	0
$885$	−6.52675e10	−3.16516
$886$	0	0
$887$	−2.31390e10	−1.11330	−0.556650	−	0.830747i	$-0.687914\pi$
$887$	−2.31390e10	−1.11330	−0.556650	+	0.830747i	$0.687914\pi$
$888$	0	0
$889$	−1.31717e9	−0.0628762
$890$	0	0
$891$	−8.74199e9	−0.414036
$892$	0	0
$893$	5.71079e9	0.268359
$894$	0	0
$895$	1.64480e10	0.766887
$896$	0	0
$897$	−1.16840e10	−0.540527
$898$	0	0
$899$	2.68864e10	1.23416
$900$	0	0
$901$	−2.27791e10	−1.03753
$902$	0	0
$903$	1.32188e9	0.0597428
$904$	0	0
$905$	−2.65443e10	−1.19042
$906$	0	0
$907$	−1.75358e10	−0.780369	−0.390184	−	0.920737i	$-0.627589\pi$
$907$	−1.75358e10	−0.780369	−0.390184	+	0.920737i	$0.627589\pi$
$908$	0	0
$909$	−4.90138e8	−0.0216444
$910$	0	0
$911$	−2.89796e10	−1.26992	−0.634962	−	0.772543i	$-0.718984\pi$
$911$	−2.89796e10	−1.26992	−0.634962	+	0.772543i	$0.718984\pi$
$912$	0	0
$913$	−5.20036e10	−2.26144
$914$	0	0
$915$	3.06740e10	1.32372
$916$	0	0
$917$	1.67280e9	0.0716395
$918$	0	0
$919$	−2.54457e10	−1.08146	−0.540729	−	0.841197i	$-0.681852\pi$
$919$	−2.54457e10	−1.08146	−0.540729	+	0.841197i	$0.681852\pi$
$920$	0	0
$921$	−2.55154e10	−1.07620
$922$	0	0
$923$	−1.84285e10	−0.771406
$924$	0	0
$925$	−4.84073e9	−0.201101
$926$	0	0
$927$	−2.98665e10	−1.23142
$928$	0	0
$929$	3.45187e10	1.41254	0.706268	−	0.707945i	$-0.250378\pi$
$929$	3.45187e10	1.41254	0.706268	+	0.707945i	$0.250378\pi$
$930$	0	0
$931$	−5.56599e9	−0.226057
$932$	0	0
$933$	−1.13467e10	−0.457385
$934$	0	0
$935$	6.13698e10	2.45535
$936$	0	0
$937$	1.01764e10	0.404116	0.202058	−	0.979374i	$-0.435237\pi$
$937$	1.01764e10	0.404116	0.202058	+	0.979374i	$0.435237\pi$
$938$	0	0
$939$	−2.68179e10	−1.05705
$940$	0	0
$941$	5.96144e9	0.233232	0.116616	−	0.993177i	$-0.462795\pi$
$941$	5.96144e9	0.233232	0.116616	+	0.993177i	$0.462795\pi$
$942$	0	0
$943$	2.47443e10	0.960914
$944$	0	0
$945$	3.16762e9	0.122102
$946$	0	0
$947$	4.69268e10	1.79554	0.897772	−	0.440461i	$-0.145185\pi$
$947$	4.69268e10	1.79554	0.897772	+	0.440461i	$0.145185\pi$
$948$	0	0
$949$	3.65584e9	0.138853
$950$	0	0
$951$	1.52273e10	0.574103
$952$	0	0
$953$	−5.55628e9	−0.207950	−0.103975	−	0.994580i	$-0.533156\pi$
$953$	−5.55628e9	−0.207950	−0.103975	+	0.994580i	$0.533156\pi$
$954$	0	0
$955$	−3.64440e10	−1.35399
$956$	0	0
$957$	7.12587e10	2.62813
$958$	0	0
$959$	−5.05652e9	−0.185134
$960$	0	0
$961$	1.22765e10	0.446213
$962$	0	0
$963$	−2.93587e10	−1.05936
$964$	0	0
$965$	1.79008e10	0.641248
$966$	0	0
$967$	−2.14955e10	−0.764461	−0.382231	−	0.924067i	$-0.624844\pi$
$967$	−2.14955e10	−0.764461	−0.382231	+	0.924067i	$0.624844\pi$
$968$	0	0
$969$	1.21140e10	0.427715
$970$	0	0
$971$	1.64141e10	0.575375	0.287688	−	0.957724i	$-0.407114\pi$
$971$	1.64141e10	0.575375	0.287688	+	0.957724i	$0.407114\pi$
$972$	0	0
$973$	2.48377e9	0.0864403
$974$	0	0
$975$	1.17089e10	0.404575
$976$	0	0
$977$	3.51550e10	1.20602	0.603012	−	0.797732i	$-0.293967\pi$
$977$	3.51550e10	1.20602	0.603012	+	0.797732i	$0.293967\pi$
$978$	0	0
$979$	−5.58089e10	−1.90092
$980$	0	0
$981$	5.10527e10	1.72654
$982$	0	0
$983$	1.34333e10	0.451073	0.225536	−	0.974235i	$-0.427587\pi$
$983$	1.34333e10	0.451073	0.225536	+	0.974235i	$0.427587\pi$
$984$	0	0
$985$	5.00279e10	1.66796
$986$	0	0
$987$	−6.75608e9	−0.223658
$988$	0	0
$989$	8.24659e9	0.271074
$990$	0	0
$991$	4.25370e10	1.38838	0.694191	−	0.719791i	$-0.255762\pi$
$991$	4.25370e10	1.38838	0.694191	+	0.719791i	$0.255762\pi$
$992$	0	0
$993$	4.13539e10	1.34028
$994$	0	0
$995$	−3.65098e10	−1.17497
$996$	0	0
$997$	2.44581e10	0.781609	0.390805	−	0.920474i	$-0.372197\pi$
$997$	2.44581e10	0.781609	0.390805	+	0.920474i	$0.372197\pi$
$998$	0	0
$999$	7.66875e9	0.243358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.8.a.g.1.1		5
4.3	odd	2		76.8.a.a.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.a.1.5	✓	5	4.3	odd	2
304.8.a.g.1.1		5	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-73.9019 q^{3} -358.964 q^{5} +109.800 q^{7} +3274.48 q^{9} +O(q^{10})\)	\(q-73.9019 q^{3} -358.964 q^{5} +109.800 q^{7} +3274.48 q^{9} +7153.73 q^{11} -3123.17 q^{13} +26528.1 q^{15} -23898.6 q^{17} +6859.00 q^{19} -8114.45 q^{21} -50622.1 q^{23} +50730.0 q^{25} -80367.1 q^{27} -134788. q^{29} -199472. q^{31} -528674. q^{33} -39414.3 q^{35} -95421.5 q^{37} +230808. q^{39} -488804. q^{41} -162905. q^{43} -1.17542e6 q^{45} +832599. q^{47} -811487. q^{49} +1.76615e6 q^{51} +953158. q^{53} -2.56793e6 q^{55} -506893. q^{57} -2.46032e6 q^{59} +1.15629e6 q^{61} +359539. q^{63} +1.12110e6 q^{65} +1.59366e6 q^{67} +3.74107e6 q^{69} +5.90057e6 q^{71} -1.17056e6 q^{73} -3.74904e6 q^{75} +785482. q^{77} -7.58482e6 q^{79} -1.22202e6 q^{81} -7.26944e6 q^{83} +8.57871e6 q^{85} +9.96106e6 q^{87} -7.80137e6 q^{89} -342925. q^{91} +1.47414e7 q^{93} -2.46213e6 q^{95} -6.56007e6 q^{97} +2.34248e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9}+O(q^{10}) \) 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9	\( 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9} + 2662 q^{11} - 602 q^{13} + 20800 q^{15} - 27366 q^{17} + 34295 q^{19} - 59964 q^{21} + 67096 q^{23} - 109115 q^{25} + 178778 q^{27} - 372398 q^{29} + 271372 q^{31} - 792700 q^{33} + 608250 q^{35} - 562630 q^{37} + 963904 q^{39} - 956714 q^{41} + 827362 q^{43} - 1165100 q^{45} + 1812982 q^{47} - 862031 q^{49} + 2458254 q^{51} + 486998 q^{53} - 467930 q^{55} + 96026 q^{57} + 367182 q^{59} + 1879732 q^{61} + 1007274 q^{63} + 1790920 q^{65} + 1046394 q^{67} + 7261712 q^{69} + 4664572 q^{71} + 4224942 q^{73} - 8194850 q^{75} + 8611110 q^{77} - 9574024 q^{79} + 11351813 q^{81} - 11754804 q^{83} + 18711750 q^{85} - 3801472 q^{87} + 2782542 q^{89} - 7385214 q^{91} + 29535004 q^{93} - 1920520 q^{95} + 1291574 q^{97} + 9760310 q^{99}+O(q^{100}) \) 5 * q + 14 * q^3 - 280 * q^5 - 414 * q^7 + 3779 * q^9 + 2662 * q^11 - 602 * q^13 + 20800 * q^15 - 27366 * q^17 + 34295 * q^19 - 59964 * q^21 + 67096 * q^23 - 109115 * q^25 + 178778 * q^27 - 372398 * q^29 + 271372 * q^31 - 792700 * q^33 + 608250 * q^35 - 562630 * q^37 + 963904 * q^39 - 956714 * q^41 + 827362 * q^43 - 1165100 * q^45 + 1812982 * q^47 - 862031 * q^49 + 2458254 * q^51 + 486998 * q^53 - 467930 * q^55 + 96026 * q^57 + 367182 * q^59 + 1879732 * q^61 + 1007274 * q^63 + 1790920 * q^65 + 1046394 * q^67 + 7261712 * q^69 + 4664572 * q^71 + 4224942 * q^73 - 8194850 * q^75 + 8611110 * q^77 - 9574024 * q^79 + 11351813 * q^81 - 11754804 * q^83 + 18711750 * q^85 - 3801472 * q^87 + 2782542 * q^89 - 7385214 * q^91 + 29535004 * q^93 - 1920520 * q^95 + 1291574 * q^97 + 9760310 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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