LMFDB - Embedded newform 304.8.a.g.1.5

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			1.5
Root			$57.9656$ 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)

$f(q)$	$=$	$q+84.8970 q^{3} +72.4872 q^{5} +210.006 q^{7} +5020.49 q^{9} +O(q^{10})$	$q+84.8970 q^{3} +72.4872 q^{5} +210.006 q^{7} +5020.49 q^{9} -2350.13 q^{11} +2219.57 q^{13} +6153.94 q^{15} +22406.1 q^{17} +6859.00 q^{19} +17828.8 q^{21} +33836.7 q^{23} -72870.6 q^{25} +240555. q^{27} -93654.9 q^{29} +222646. q^{31} -199519. q^{33} +15222.7 q^{35} +228916. q^{37} +188435. q^{39} +67353.8 q^{41} -162691. q^{43} +363922. q^{45} -81632.8 q^{47} -779441. q^{49} +1.90221e6 q^{51} +905874. q^{53} -170354. q^{55} +582308. q^{57} +1.42600e6 q^{59} +2.18670e6 q^{61} +1.05433e6 q^{63} +160891. q^{65} +4.73974e6 q^{67} +2.87263e6 q^{69} +3.58523e6 q^{71} -3.22283e6 q^{73} -6.18649e6 q^{75} -493541. q^{77} -334481. q^{79} +9.44257e6 q^{81} -9.68747e6 q^{83} +1.62416e6 q^{85} -7.95102e6 q^{87} -6.74280e6 q^{89} +466123. q^{91} +1.89020e7 q^{93} +497190. q^{95} -483068. q^{97} -1.17988e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	84.8970	1.81538	0.907690	−	0.419640i	$-0.137844\pi$
$3$	84.8970	1.81538	0.907690	+	0.419640i	$0.137844\pi$
$4$	0	0
$5$	72.4872	0.259338	0.129669	−	0.991557i	$-0.458608\pi$
$5$	72.4872	0.259338	0.129669	+	0.991557i	$0.458608\pi$
$6$	0	0
$7$	210.006	0.231413	0.115707	−	0.993283i	$-0.463087\pi$
$7$	210.006	0.231413	0.115707	+	0.993283i	$0.463087\pi$
$8$	0	0
$9$	5020.49	2.29561
$10$	0	0
$11$	−2350.13	−0.532375	−0.266188	−	0.963921i	$-0.585764\pi$
$11$	−2350.13	−0.532375	−0.266188	+	0.963921i	$0.585764\pi$
$12$	0	0
$13$	2219.57	0.280200	0.140100	−	0.990137i	$-0.455258\pi$
$13$	2219.57	0.280200	0.140100	+	0.990137i	$0.455258\pi$
$14$	0	0
$15$	6153.94	0.470798
$16$	0	0
$17$	22406.1	1.10610	0.553051	−	0.833147i	$-0.313464\pi$
$17$	22406.1	1.10610	0.553051	+	0.833147i	$0.313464\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	17828.8	0.420103
$22$	0	0
$23$	33836.7	0.579883	0.289941	−	0.957044i	$-0.406364\pi$
$23$	33836.7	0.579883	0.289941	+	0.957044i	$0.406364\pi$
$24$	0	0
$25$	−72870.6	−0.932744
$26$	0	0
$27$	240555.	2.35202
$28$	0	0
$29$	−93654.9	−0.713079	−0.356539	−	0.934280i	$-0.616043\pi$
$29$	−93654.9	−0.713079	−0.356539	+	0.934280i	$0.616043\pi$
$30$	0	0
$31$	222646.	1.34230	0.671150	−	0.741322i	$-0.265801\pi$
$31$	222646.	1.34230	0.671150	+	0.741322i	$0.265801\pi$
$32$	0	0
$33$	−199519.	−0.966464
$34$	0	0
$35$	15222.7	0.0600142
$36$	0	0
$37$	228916.	0.742968	0.371484	−	0.928439i	$-0.378849\pi$
$37$	228916.	0.742968	0.371484	+	0.928439i	$0.378849\pi$
$38$	0	0
$39$	188435.	0.508669
$40$	0	0
$41$	67353.8	0.152623	0.0763113	−	0.997084i	$-0.475686\pi$
$41$	67353.8	0.152623	0.0763113	+	0.997084i	$0.475686\pi$
$42$	0	0
$43$	−162691.	−0.312049	−0.156025	−	0.987753i	$-0.549868\pi$
$43$	−162691.	−0.312049	−0.156025	+	0.987753i	$0.549868\pi$
$44$	0	0
$45$	363922.	0.595339
$46$	0	0
$47$	−81632.8	−0.114689	−0.0573445	−	0.998354i	$-0.518263\pi$
$47$	−81632.8	−0.114689	−0.0573445	+	0.998354i	$0.518263\pi$
$48$	0	0
$49$	−779441.	−0.946448
$50$	0	0
$51$	1.90221e6	2.00800
$52$	0	0
$53$	905874.	0.835800	0.417900	−	0.908493i	$-0.362766\pi$
$53$	905874.	0.835800	0.417900	+	0.908493i	$0.362766\pi$
$54$	0	0
$55$	−170354.	−0.138065
$56$	0	0
$57$	582308.	0.416477
$58$	0	0
$59$	1.42600e6	0.903934	0.451967	−	0.892035i	$-0.350723\pi$
$59$	1.42600e6	0.903934	0.451967	+	0.892035i	$0.350723\pi$
$60$	0	0
$61$	2.18670e6	1.23349	0.616745	−	0.787163i	$-0.288451\pi$
$61$	2.18670e6	1.23349	0.616745	+	0.787163i	$0.288451\pi$
$62$	0	0
$63$	1.05433e6	0.531234
$64$	0	0
$65$	160891.	0.0726665
$66$	0	0
$67$	4.73974e6	1.92527	0.962636	−	0.270797i	$-0.0872872\pi$
$67$	4.73974e6	1.92527	0.962636	+	0.270797i	$0.0872872\pi$
$68$	0	0
$69$	2.87263e6	1.05271
$70$	0	0
$71$	3.58523e6	1.18881	0.594406	−	0.804165i	$-0.297387\pi$
$71$	3.58523e6	1.18881	0.594406	+	0.804165i	$0.297387\pi$
$72$	0	0
$73$	−3.22283e6	−0.969632	−0.484816	−	0.874616i	$-0.661113\pi$
$73$	−3.22283e6	−0.969632	−0.484816	+	0.874616i	$0.661113\pi$
$74$	0	0
$75$	−6.18649e6	−1.69329
$76$	0	0
$77$	−493541.	−0.123199
$78$	0	0
$79$	−334481.	−0.0763268	−0.0381634	−	0.999272i	$-0.512151\pi$
$79$	−334481.	−0.0763268	−0.0381634	+	0.999272i	$0.512151\pi$
$80$	0	0
$81$	9.44257e6	1.97421
$82$	0	0
$83$	−9.68747e6	−1.85967	−0.929837	−	0.367972i	$-0.880052\pi$
$83$	−9.68747e6	−1.85967	−0.929837	+	0.367972i	$0.880052\pi$
$84$	0	0
$85$	1.62416e6	0.286854
$86$	0	0
$87$	−7.95102e6	−1.29451
$88$	0	0
$89$	−6.74280e6	−1.01385	−0.506927	−	0.861989i	$-0.669219\pi$
$89$	−6.74280e6	−1.01385	−0.506927	+	0.861989i	$0.669219\pi$
$90$	0	0
$91$	466123.	0.0648419
$92$	0	0
$93$	1.89020e7	2.43679
$94$	0	0
$95$	497190.	0.0594963
$96$	0	0
$97$	−483068.	−0.0537412	−0.0268706	−	0.999639i	$-0.508554\pi$
$97$	−483068.	−0.0537412	−0.0268706	+	0.999639i	$0.508554\pi$
$98$	0	0
$99$	−1.17988e7	−1.22212
$100$	0	0
$101$	−1.87119e7	−1.80714	−0.903571	−	0.428439i	$-0.859064\pi$
$101$	−1.87119e7	−1.80714	−0.903571	+	0.428439i	$0.859064\pi$
$102$	0	0
$103$	−4.79581e6	−0.432446	−0.216223	−	0.976344i	$-0.569374\pi$
$103$	−4.79581e6	−0.432446	−0.216223	+	0.976344i	$0.569374\pi$
$104$	0	0
$105$	1.29236e6	0.108949
$106$	0	0
$107$	9.12089e6	0.719770	0.359885	−	0.932997i	$-0.382816\pi$
$107$	9.12089e6	0.719770	0.359885	+	0.932997i	$0.382816\pi$
$108$	0	0
$109$	−1.07583e7	−0.795706	−0.397853	−	0.917449i	$-0.630245\pi$
$109$	−1.07583e7	−0.795706	−0.397853	+	0.917449i	$0.630245\pi$
$110$	0	0
$111$	1.94343e7	1.34877
$112$	0	0
$113$	1.08959e7	0.710378	0.355189	−	0.934795i	$-0.384417\pi$
$113$	1.08959e7	0.710378	0.355189	+	0.934795i	$0.384417\pi$
$114$	0	0
$115$	2.45273e6	0.150386
$116$	0	0
$117$	1.11434e7	0.643229
$118$	0	0
$119$	4.70541e6	0.255966
$120$	0	0
$121$	−1.39641e7	−0.716577
$122$	0	0
$123$	5.71814e6	0.277068
$124$	0	0
$125$	−1.09453e7	−0.501234
$126$	0	0
$127$	3.16808e7	1.37241	0.686204	−	0.727409i	$-0.259276\pi$
$127$	3.16808e7	1.37241	0.686204	+	0.727409i	$0.259276\pi$
$128$	0	0
$129$	−1.38119e7	−0.566488
$130$	0	0
$131$	4.08202e7	1.58645	0.793223	−	0.608931i	$-0.208402\pi$
$131$	4.08202e7	1.58645	0.793223	+	0.608931i	$0.208402\pi$
$132$	0	0
$133$	1.44043e6	0.0530898
$134$	0	0
$135$	1.74372e7	0.609969
$136$	0	0
$137$	2.17104e7	0.721351	0.360676	−	0.932691i	$-0.382546\pi$
$137$	2.17104e7	0.721351	0.360676	+	0.932691i	$0.382546\pi$
$138$	0	0
$139$	−3.68005e7	−1.16226	−0.581128	−	0.813812i	$-0.697388\pi$
$139$	−3.68005e7	−1.16226	−0.581128	+	0.813812i	$0.697388\pi$
$140$	0	0
$141$	−6.93037e6	−0.208204
$142$	0	0
$143$	−5.21629e6	−0.149171
$144$	0	0
$145$	−6.78878e6	−0.184929
$146$	0	0
$147$	−6.61721e7	−1.71816
$148$	0	0
$149$	−7.59158e6	−0.188010	−0.0940049	−	0.995572i	$-0.529967\pi$
$149$	−7.59158e6	−0.188010	−0.0940049	+	0.995572i	$0.529967\pi$
$150$	0	0
$151$	1.51149e7	0.357260	0.178630	−	0.983916i	$-0.442833\pi$
$151$	1.51149e7	0.357260	0.178630	+	0.983916i	$0.442833\pi$
$152$	0	0
$153$	1.12490e8	2.53918
$154$	0	0
$155$	1.61390e7	0.348110
$156$	0	0
$157$	−3.20106e7	−0.660154	−0.330077	−	0.943954i	$-0.607075\pi$
$157$	−3.20106e7	−0.660154	−0.330077	+	0.943954i	$0.607075\pi$
$158$	0	0
$159$	7.69060e7	1.51730
$160$	0	0
$161$	7.10589e6	0.134192
$162$	0	0
$163$	−2.86614e7	−0.518372	−0.259186	−	0.965827i	$-0.583454\pi$
$163$	−2.86614e7	−0.518372	−0.259186	+	0.965827i	$0.583454\pi$
$164$	0	0
$165$	−1.44626e7	−0.250641
$166$	0	0
$167$	1.00827e8	1.67522	0.837608	−	0.546271i	$-0.183953\pi$
$167$	1.00827e8	1.67522	0.837608	+	0.546271i	$0.183953\pi$
$168$	0	0
$169$	−5.78220e7	−0.921488
$170$	0	0
$171$	3.44356e7	0.526648
$172$	0	0
$173$	−3.62991e6	−0.0533009	−0.0266505	−	0.999645i	$-0.508484\pi$
$173$	−3.62991e6	−0.0533009	−0.0266505	+	0.999645i	$0.508484\pi$
$174$	0	0
$175$	−1.53032e7	−0.215849
$176$	0	0
$177$	1.21063e8	1.64098
$178$	0	0
$179$	6.65481e7	0.867261	0.433630	−	0.901091i	$-0.357232\pi$
$179$	6.65481e7	0.867261	0.433630	+	0.901091i	$0.357232\pi$
$180$	0	0
$181$	1.41404e8	1.77250	0.886250	−	0.463207i	$-0.153301\pi$
$181$	1.41404e8	1.77250	0.886250	+	0.463207i	$0.153301\pi$
$182$	0	0
$183$	1.85645e8	2.23925
$184$	0	0
$185$	1.65935e7	0.192680
$186$	0	0
$187$	−5.26573e7	−0.588861
$188$	0	0
$189$	5.05179e7	0.544288
$190$	0	0
$191$	−1.12407e8	−1.16728	−0.583641	−	0.812012i	$-0.698373\pi$
$191$	−1.12407e8	−1.16728	−0.583641	+	0.812012i	$0.698373\pi$
$192$	0	0
$193$	5.05226e7	0.505866	0.252933	−	0.967484i	$-0.418605\pi$
$193$	5.05226e7	0.505866	0.252933	+	0.967484i	$0.418605\pi$
$194$	0	0
$195$	1.36591e7	0.131917
$196$	0	0
$197$	−8.40182e7	−0.782964	−0.391482	−	0.920186i	$-0.628037\pi$
$197$	−8.40182e7	−0.782964	−0.391482	+	0.920186i	$0.628037\pi$
$198$	0	0
$199$	1.15328e8	1.03740	0.518701	−	0.854956i	$-0.326416\pi$
$199$	1.15328e8	1.03740	0.518701	+	0.854956i	$0.326416\pi$
$200$	0	0
$201$	4.02389e8	3.49510
$202$	0	0
$203$	−1.96681e7	−0.165016
$204$	0	0
$205$	4.88229e6	0.0395809
$206$	0	0
$207$	1.69877e8	1.33118
$208$	0	0
$209$	−1.61196e7	−0.122135
$210$	0	0
$211$	−2.20711e6	−0.0161747	−0.00808733	−	0.999967i	$-0.502574\pi$
$211$	−2.20711e6	−0.0161747	−0.00808733	+	0.999967i	$0.502574\pi$
$212$	0	0
$213$	3.04375e8	2.15815
$214$	0	0
$215$	−1.17930e7	−0.0809262
$216$	0	0
$217$	4.67570e7	0.310626
$218$	0	0
$219$	−2.73608e8	−1.76025
$220$	0	0
$221$	4.97320e7	0.309929
$222$	0	0
$223$	−1.35844e8	−0.820300	−0.410150	−	0.912018i	$-0.634524\pi$
$223$	−1.35844e8	−0.820300	−0.410150	+	0.912018i	$0.634524\pi$
$224$	0	0
$225$	−3.65846e8	−2.14121
$226$	0	0
$227$	−2.40869e8	−1.36675	−0.683377	−	0.730066i	$-0.739489\pi$
$227$	−2.40869e8	−1.36675	−0.683377	+	0.730066i	$0.739489\pi$
$228$	0	0
$229$	2.98261e8	1.64124	0.820621	−	0.571472i	$-0.193628\pi$
$229$	2.98261e8	1.64124	0.820621	+	0.571472i	$0.193628\pi$
$230$	0	0
$231$	−4.19001e7	−0.223652
$232$	0	0
$233$	−1.54139e8	−0.798302	−0.399151	−	0.916885i	$-0.630695\pi$
$233$	−1.54139e8	−0.798302	−0.399151	+	0.916885i	$0.630695\pi$
$234$	0	0
$235$	−5.91733e6	−0.0297433
$236$	0	0
$237$	−2.83965e7	−0.138562
$238$	0	0
$239$	1.87849e8	0.890054	0.445027	−	0.895517i	$-0.353194\pi$
$239$	1.87849e8	0.890054	0.445027	+	0.895517i	$0.353194\pi$
$240$	0	0
$241$	−3.93989e8	−1.81311	−0.906555	−	0.422089i	$-0.861297\pi$
$241$	−3.93989e8	−1.81311	−0.906555	+	0.422089i	$0.861297\pi$
$242$	0	0
$243$	2.75552e8	1.23192
$244$	0	0
$245$	−5.64995e7	−0.245450
$246$	0	0
$247$	1.52240e7	0.0642822
$248$	0	0
$249$	−8.22436e8	−3.37602
$250$	0	0
$251$	−3.64401e8	−1.45453	−0.727263	−	0.686359i	$-0.759208\pi$
$251$	−3.64401e8	−1.45453	−0.727263	+	0.686359i	$0.759208\pi$
$252$	0	0
$253$	−7.95206e7	−0.308715
$254$	0	0
$255$	1.37886e8	0.520750
$256$	0	0
$257$	−1.31627e8	−0.483702	−0.241851	−	0.970313i	$-0.577755\pi$
$257$	−1.31627e8	−0.483702	−0.241851	+	0.970313i	$0.577755\pi$
$258$	0	0
$259$	4.80737e7	0.171933
$260$	0	0
$261$	−4.70194e8	−1.63695
$262$	0	0
$263$	5.06319e8	1.71625	0.858123	−	0.513445i	$-0.171631\pi$
$263$	5.06319e8	1.71625	0.858123	+	0.513445i	$0.171631\pi$
$264$	0	0
$265$	6.56643e7	0.216755
$266$	0	0
$267$	−5.72443e8	−1.84053
$268$	0	0
$269$	3.36604e7	0.105435	0.0527176	−	0.998609i	$-0.483212\pi$
$269$	3.36604e7	0.105435	0.0527176	+	0.998609i	$0.483212\pi$
$270$	0	0
$271$	−5.12274e8	−1.56354	−0.781772	−	0.623564i	$-0.785684\pi$
$271$	−5.12274e8	−1.56354	−0.781772	+	0.623564i	$0.785684\pi$
$272$	0	0
$273$	3.95724e7	0.117713
$274$	0	0
$275$	1.71255e8	0.496570
$276$	0	0
$277$	6.54393e8	1.84995	0.924974	−	0.380030i	$-0.124086\pi$
$277$	6.54393e8	1.84995	0.924974	+	0.380030i	$0.124086\pi$
$278$	0	0
$279$	1.11779e9	3.08139
$280$	0	0
$281$	−1.27528e8	−0.342874	−0.171437	−	0.985195i	$-0.554841\pi$
$281$	−1.27528e8	−0.342874	−0.171437	+	0.985195i	$0.554841\pi$
$282$	0	0
$283$	−4.21241e8	−1.10479	−0.552393	−	0.833584i	$-0.686285\pi$
$283$	−4.21241e8	−1.10479	−0.552393	+	0.833584i	$0.686285\pi$
$284$	0	0
$285$	4.22099e7	0.108008
$286$	0	0
$287$	1.41447e7	0.0353189
$288$	0	0
$289$	9.16947e7	0.223461
$290$	0	0
$291$	−4.10110e7	−0.0975608
$292$	0	0
$293$	−3.75311e8	−0.871675	−0.435837	−	0.900025i	$-0.643548\pi$
$293$	−3.75311e8	−0.871675	−0.435837	+	0.900025i	$0.643548\pi$
$294$	0	0
$295$	1.03367e8	0.234425
$296$	0	0
$297$	−5.65336e8	−1.25216
$298$	0	0
$299$	7.51030e7	0.162483
$300$	0	0
$301$	−3.41659e7	−0.0722122
$302$	0	0
$303$	−1.58858e9	−3.28065
$304$	0	0
$305$	1.58508e8	0.319891
$306$	0	0
$307$	−2.15576e7	−0.0425223	−0.0212611	−	0.999774i	$-0.506768\pi$
$307$	−2.15576e7	−0.0425223	−0.0212611	+	0.999774i	$0.506768\pi$
$308$	0	0
$309$	−4.07150e8	−0.785054
$310$	0	0
$311$	−5.96289e8	−1.12408	−0.562038	−	0.827111i	$-0.689983\pi$
$311$	−5.96289e8	−1.12408	−0.562038	+	0.827111i	$0.689983\pi$
$312$	0	0
$313$	5.04393e8	0.929746	0.464873	−	0.885377i	$-0.346100\pi$
$313$	5.04393e8	0.929746	0.464873	+	0.885377i	$0.346100\pi$
$314$	0	0
$315$	7.64256e7	0.137769
$316$	0	0
$317$	−7.71147e8	−1.35966	−0.679829	−	0.733371i	$-0.737946\pi$
$317$	−7.71147e8	−1.35966	−0.679829	+	0.733371i	$0.737946\pi$
$318$	0	0
$319$	2.20101e8	0.379625
$320$	0	0
$321$	7.74336e8	1.30666
$322$	0	0
$323$	1.53683e8	0.253757
$324$	0	0
$325$	−1.61742e8	−0.261354
$326$	0	0
$327$	−9.13351e8	−1.44451
$328$	0	0
$329$	−1.71433e7	−0.0265405
$330$	0	0
$331$	−3.04843e8	−0.462039	−0.231020	−	0.972949i	$-0.574206\pi$
$331$	−3.04843e8	−0.462039	−0.231020	+	0.972949i	$0.574206\pi$
$332$	0	0
$333$	1.14927e9	1.70556
$334$	0	0
$335$	3.43570e8	0.499297
$336$	0	0
$337$	−7.56300e8	−1.07644	−0.538219	−	0.842805i	$-0.680903\pi$
$337$	−7.56300e8	−1.07644	−0.538219	+	0.842805i	$0.680903\pi$
$338$	0	0
$339$	9.25031e8	1.28961
$340$	0	0
$341$	−5.23248e8	−0.714607
$342$	0	0
$343$	−3.36636e8	−0.450433
$344$	0	0
$345$	2.08229e8	0.273007
$346$	0	0
$347$	−1.93595e8	−0.248737	−0.124368	−	0.992236i	$-0.539690\pi$
$347$	−1.93595e8	−0.248737	−0.124368	+	0.992236i	$0.539690\pi$
$348$	0	0
$349$	−7.55673e8	−0.951579	−0.475790	−	0.879559i	$-0.657838\pi$
$349$	−7.55673e8	−0.951579	−0.475790	+	0.879559i	$0.657838\pi$
$350$	0	0
$351$	5.33929e8	0.659036
$352$	0	0
$353$	−3.97258e7	−0.0480686	−0.0240343	−	0.999711i	$-0.507651\pi$
$353$	−3.97258e7	−0.0480686	−0.0240343	+	0.999711i	$0.507651\pi$
$354$	0	0
$355$	2.59884e8	0.308304
$356$	0	0
$357$	3.99475e8	0.464676
$358$	0	0
$359$	1.08901e9	1.24223	0.621113	−	0.783721i	$-0.286681\pi$
$359$	1.08901e9	1.24223	0.621113	+	0.783721i	$0.286681\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−1.18551e9	−1.30086
$364$	0	0
$365$	−2.33614e8	−0.251463
$366$	0	0
$367$	−1.47931e9	−1.56216	−0.781082	−	0.624428i	$-0.785332\pi$
$367$	−1.47931e9	−1.56216	−0.781082	+	0.624428i	$0.785332\pi$
$368$	0	0
$369$	3.38149e8	0.350362
$370$	0	0
$371$	1.90239e8	0.193415
$372$	0	0
$373$	1.36968e9	1.36659	0.683294	−	0.730144i	$-0.260547\pi$
$373$	1.36968e9	1.36659	0.683294	+	0.730144i	$0.260547\pi$
$374$	0	0
$375$	−9.29219e8	−0.909931
$376$	0	0
$377$	−2.07874e8	−0.199804
$378$	0	0
$379$	3.11167e7	0.0293600	0.0146800	−	0.999892i	$-0.495327\pi$
$379$	3.11167e7	0.0293600	0.0146800	+	0.999892i	$0.495327\pi$
$380$	0	0
$381$	2.68961e9	2.49144
$382$	0	0
$383$	8.84422e8	0.804384	0.402192	−	0.915555i	$-0.368248\pi$
$383$	8.84422e8	0.804384	0.402192	+	0.915555i	$0.368248\pi$
$384$	0	0
$385$	−3.57754e7	−0.0319501
$386$	0	0
$387$	−8.16787e8	−0.716342
$388$	0	0
$389$	−1.48902e9	−1.28256	−0.641280	−	0.767307i	$-0.721596\pi$
$389$	−1.48902e9	−1.28256	−0.641280	+	0.767307i	$0.721596\pi$
$390$	0	0
$391$	7.58148e8	0.641409
$392$	0	0
$393$	3.46551e9	2.88000
$394$	0	0
$395$	−2.42456e7	−0.0197945
$396$	0	0
$397$	−1.12415e9	−0.901692	−0.450846	−	0.892602i	$-0.648878\pi$
$397$	−1.12415e9	−0.901692	−0.450846	+	0.892602i	$0.648878\pi$
$398$	0	0
$399$	1.22288e8	0.0963782
$400$	0	0
$401$	−1.49136e9	−1.15499	−0.577494	−	0.816395i	$-0.695969\pi$
$401$	−1.49136e9	−1.15499	−0.577494	+	0.816395i	$0.695969\pi$
$402$	0	0
$403$	4.94180e8	0.376112
$404$	0	0
$405$	6.84466e8	0.511987
$406$	0	0
$407$	−5.37983e8	−0.395538
$408$	0	0
$409$	1.49909e9	1.08342	0.541710	−	0.840566i	$-0.317777\pi$
$409$	1.49909e9	1.08342	0.541710	+	0.840566i	$0.317777\pi$
$410$	0	0
$411$	1.84315e9	1.30953
$412$	0	0
$413$	2.99467e8	0.209182
$414$	0	0
$415$	−7.02217e8	−0.482284
$416$	0	0
$417$	−3.12425e9	−2.10994
$418$	0	0
$419$	2.26383e9	1.50347	0.751735	−	0.659465i	$-0.229217\pi$
$419$	2.26383e9	1.50347	0.751735	+	0.659465i	$0.229217\pi$
$420$	0	0
$421$	−7.52298e8	−0.491363	−0.245682	−	0.969351i	$-0.579012\pi$
$421$	−7.52298e8	−0.491363	−0.245682	+	0.969351i	$0.579012\pi$
$422$	0	0
$423$	−4.09837e8	−0.263281
$424$	0	0
$425$	−1.63275e9	−1.03171
$426$	0	0
$427$	4.59220e8	0.285446
$428$	0	0
$429$	−4.42847e8	−0.270803
$430$	0	0
$431$	−2.07720e9	−1.24970	−0.624852	−	0.780743i	$-0.714841\pi$
$431$	−2.07720e9	−1.24970	−0.624852	+	0.780743i	$0.714841\pi$
$432$	0	0
$433$	−2.52531e9	−1.49488	−0.747440	−	0.664329i	$-0.768717\pi$
$433$	−2.52531e9	−1.49488	−0.747440	+	0.664329i	$0.768717\pi$
$434$	0	0
$435$	−5.76347e8	−0.335716
$436$	0	0
$437$	2.32086e8	0.133034
$438$	0	0
$439$	−1.91118e9	−1.07814	−0.539071	−	0.842260i	$-0.681225\pi$
$439$	−1.91118e9	−1.07814	−0.539071	+	0.842260i	$0.681225\pi$
$440$	0	0
$441$	−3.91318e9	−2.17267
$442$	0	0
$443$	−3.04554e7	−0.0166438	−0.00832188	−	0.999965i	$-0.502649\pi$
$443$	−3.04554e7	−0.0166438	−0.00832188	+	0.999965i	$0.502649\pi$
$444$	0	0
$445$	−4.88767e8	−0.262931
$446$	0	0
$447$	−6.44502e8	−0.341309
$448$	0	0
$449$	−3.70483e9	−1.93155	−0.965775	−	0.259381i	$-0.916481\pi$
$449$	−3.70483e9	−1.93155	−0.965775	+	0.259381i	$0.916481\pi$
$450$	0	0
$451$	−1.58290e8	−0.0812525
$452$	0	0
$453$	1.28321e9	0.648564
$454$	0	0
$455$	3.37879e7	0.0168160
$456$	0	0
$457$	−1.34064e9	−0.657059	−0.328529	−	0.944494i	$-0.606553\pi$
$457$	−1.34064e9	−0.657059	−0.328529	+	0.944494i	$0.606553\pi$
$458$	0	0
$459$	5.38990e9	2.60157
$460$	0	0
$461$	−2.91246e9	−1.38454	−0.692272	−	0.721637i	$-0.743390\pi$
$461$	−2.91246e9	−1.38454	−0.692272	+	0.721637i	$0.743390\pi$
$462$	0	0
$463$	−7.56134e8	−0.354051	−0.177025	−	0.984206i	$-0.556647\pi$
$463$	−7.56134e8	−0.354051	−0.177025	+	0.984206i	$0.556647\pi$
$464$	0	0
$465$	1.37015e9	0.631951
$466$	0	0
$467$	2.01789e9	0.916831	0.458415	−	0.888738i	$-0.348417\pi$
$467$	2.01789e9	0.916831	0.458415	+	0.888738i	$0.348417\pi$
$468$	0	0
$469$	9.95371e8	0.445533
$470$	0	0
$471$	−2.71761e9	−1.19843
$472$	0	0
$473$	3.82344e8	0.166127
$474$	0	0
$475$	−4.99819e8	−0.213986
$476$	0	0
$477$	4.54794e9	1.91867
$478$	0	0
$479$	1.39778e9	0.581118	0.290559	−	0.956857i	$-0.406159\pi$
$479$	1.39778e9	0.581118	0.290559	+	0.956857i	$0.406159\pi$
$480$	0	0
$481$	5.08096e8	0.208179
$482$	0	0
$483$	6.03269e8	0.243610
$484$	0	0
$485$	−3.50163e7	−0.0139371
$486$	0	0
$487$	−1.34378e9	−0.527203	−0.263602	−	0.964632i	$-0.584910\pi$
$487$	−1.34378e9	−0.527203	−0.263602	+	0.964632i	$0.584910\pi$
$488$	0	0
$489$	−2.43327e9	−0.941042
$490$	0	0
$491$	−9.27643e8	−0.353668	−0.176834	−	0.984241i	$-0.556586\pi$
$491$	−9.27643e8	−0.353668	−0.176834	+	0.984241i	$0.556586\pi$
$492$	0	0
$493$	−2.09844e9	−0.788738
$494$	0	0
$495$	−8.55264e8	−0.316943
$496$	0	0
$497$	7.52919e8	0.275107
$498$	0	0
$499$	−2.84992e9	−1.02679	−0.513394	−	0.858153i	$-0.671612\pi$
$499$	−2.84992e9	−1.02679	−0.513394	+	0.858153i	$0.671612\pi$
$500$	0	0
$501$	8.55994e9	3.04116
$502$	0	0
$503$	−2.38596e9	−0.835940	−0.417970	−	0.908461i	$-0.637258\pi$
$503$	−2.38596e9	−0.835940	−0.417970	+	0.908461i	$0.637258\pi$
$504$	0	0
$505$	−1.35637e9	−0.468661
$506$	0	0
$507$	−4.90891e9	−1.67285
$508$	0	0
$509$	5.57715e9	1.87456	0.937282	−	0.348573i	$-0.113333\pi$
$509$	5.57715e9	1.87456	0.937282	+	0.348573i	$0.113333\pi$
$510$	0	0
$511$	−6.76812e8	−0.224385
$512$	0	0
$513$	1.64997e9	0.539591
$514$	0	0
$515$	−3.47635e8	−0.112150
$516$	0	0
$517$	1.91848e8	0.0610576
$518$	0	0
$519$	−3.08168e8	−0.0967615
$520$	0	0
$521$	3.55947e9	1.10269	0.551344	−	0.834278i	$-0.314115\pi$
$521$	3.55947e9	1.10269	0.551344	+	0.834278i	$0.314115\pi$
$522$	0	0
$523$	3.38191e9	1.03373	0.516864	−	0.856067i	$-0.327099\pi$
$523$	3.38191e9	1.03373	0.516864	+	0.856067i	$0.327099\pi$
$524$	0	0
$525$	−1.29920e9	−0.391848
$526$	0	0
$527$	4.98864e9	1.48472
$528$	0	0
$529$	−2.25990e9	−0.663736
$530$	0	0
$531$	7.15921e9	2.07508
$532$	0	0
$533$	1.49497e8	0.0427648
$534$	0	0
$535$	6.61148e8	0.186664
$536$	0	0
$537$	5.64973e9	1.57441
$538$	0	0
$539$	1.83179e9	0.503865
$540$	0	0
$541$	2.48160e9	0.673817	0.336908	−	0.941537i	$-0.390619\pi$
$541$	2.48160e9	0.673817	0.336908	+	0.941537i	$0.390619\pi$
$542$	0	0
$543$	1.20048e10	3.21776
$544$	0	0
$545$	−7.79843e8	−0.206357
$546$	0	0
$547$	−4.17148e9	−1.08977	−0.544884	−	0.838511i	$-0.683427\pi$
$547$	−4.17148e9	−1.08977	−0.544884	+	0.838511i	$0.683427\pi$
$548$	0	0
$549$	1.09783e10	2.83161
$550$	0	0
$551$	−6.42379e8	−0.163591
$552$	0	0
$553$	−7.02430e7	−0.0176630
$554$	0	0
$555$	1.40874e9	0.349788
$556$	0	0
$557$	−1.77249e9	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$557$	−1.77249e9	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$558$	0	0
$559$	−3.61104e8	−0.0874360
$560$	0	0
$561$	−4.47044e9	−1.06901
$562$	0	0
$563$	−5.96074e9	−1.40774	−0.703868	−	0.710331i	$-0.748545\pi$
$563$	−5.96074e9	−1.40774	−0.703868	+	0.710331i	$0.748545\pi$
$564$	0	0
$565$	7.89815e8	0.184228
$566$	0	0
$567$	1.98299e9	0.456857
$568$	0	0
$569$	2.59626e9	0.590819	0.295410	−	0.955371i	$-0.404544\pi$
$569$	2.59626e9	0.590819	0.295410	+	0.955371i	$0.404544\pi$
$570$	0	0
$571$	2.53949e9	0.570848	0.285424	−	0.958401i	$-0.407866\pi$
$571$	2.53949e9	0.570848	0.285424	+	0.958401i	$0.407866\pi$
$572$	0	0
$573$	−9.54300e9	−2.11906
$574$	0	0
$575$	−2.46570e9	−0.540882
$576$	0	0
$577$	5.51763e9	1.19574	0.597871	−	0.801592i	$-0.296014\pi$
$577$	5.51763e9	1.19574	0.597871	+	0.801592i	$0.296014\pi$
$578$	0	0
$579$	4.28922e9	0.918339
$580$	0	0
$581$	−2.03442e9	−0.430353
$582$	0	0
$583$	−2.12892e9	−0.444959
$584$	0	0
$585$	8.07751e8	0.166814
$586$	0	0
$587$	−6.29754e9	−1.28510	−0.642550	−	0.766243i	$-0.722124\pi$
$587$	−6.29754e9	−1.28510	−0.642550	+	0.766243i	$0.722124\pi$
$588$	0	0
$589$	1.52713e9	0.307945
$590$	0	0
$591$	−7.13289e9	−1.42138
$592$	0	0
$593$	−4.02283e9	−0.792209	−0.396105	−	0.918205i	$-0.629638\pi$
$593$	−4.02283e9	−0.792209	−0.396105	+	0.918205i	$0.629638\pi$
$594$	0	0
$595$	3.41082e8	0.0663819
$596$	0	0
$597$	9.79096e9	1.88328
$598$	0	0
$599$	−1.25980e9	−0.239502	−0.119751	−	0.992804i	$-0.538210\pi$
$599$	−1.25980e9	−0.239502	−0.119751	+	0.992804i	$0.538210\pi$
$600$	0	0
$601$	−2.33022e9	−0.437860	−0.218930	−	0.975741i	$-0.570257\pi$
$601$	−2.33022e9	−0.437860	−0.218930	+	0.975741i	$0.570257\pi$
$602$	0	0
$603$	2.37958e10	4.41967
$604$	0	0
$605$	−1.01222e9	−0.185836
$606$	0	0
$607$	7.72032e8	0.140112	0.0700559	−	0.997543i	$-0.477682\pi$
$607$	7.72032e8	0.140112	0.0700559	+	0.997543i	$0.477682\pi$
$608$	0	0
$609$	−1.66976e9	−0.299566
$610$	0	0
$611$	−1.81190e8	−0.0321358
$612$	0	0
$613$	1.24050e9	0.217514	0.108757	−	0.994068i	$-0.465313\pi$
$613$	1.24050e9	0.217514	0.108757	+	0.994068i	$0.465313\pi$
$614$	0	0
$615$	4.14492e8	0.0718543
$616$	0	0
$617$	−3.30230e8	−0.0566003	−0.0283002	−	0.999599i	$-0.509009\pi$
$617$	−3.30230e8	−0.0566003	−0.0283002	+	0.999599i	$0.509009\pi$
$618$	0	0
$619$	−5.95644e9	−1.00941	−0.504707	−	0.863291i	$-0.668400\pi$
$619$	−5.95644e9	−1.00941	−0.504707	+	0.863291i	$0.668400\pi$
$620$	0	0
$621$	8.13958e9	1.36390
$622$	0	0
$623$	−1.41603e9	−0.234619
$624$	0	0
$625$	4.89962e9	0.802755
$626$	0	0
$627$	−1.36850e9	−0.221722
$628$	0	0
$629$	5.12912e9	0.821798
$630$	0	0
$631$	2.68160e9	0.424905	0.212452	−	0.977171i	$-0.431855\pi$
$631$	2.68160e9	0.424905	0.212452	+	0.977171i	$0.431855\pi$
$632$	0	0
$633$	−1.87377e8	−0.0293632
$634$	0	0
$635$	2.29646e9	0.355918
$636$	0	0
$637$	−1.73003e9	−0.265194
$638$	0	0
$639$	1.79996e10	2.72905
$640$	0	0
$641$	2.85995e9	0.428899	0.214449	−	0.976735i	$-0.431204\pi$
$641$	2.85995e9	0.428899	0.214449	+	0.976735i	$0.431204\pi$
$642$	0	0
$643$	−1.77154e9	−0.262792	−0.131396	−	0.991330i	$-0.541946\pi$
$643$	−1.77154e9	−0.262792	−0.131396	+	0.991330i	$0.541946\pi$
$644$	0	0
$645$	−1.00119e9	−0.146912
$646$	0	0
$647$	−6.32710e9	−0.918416	−0.459208	−	0.888329i	$-0.651867\pi$
$647$	−6.32710e9	−0.918416	−0.459208	+	0.888329i	$0.651867\pi$
$648$	0	0
$649$	−3.35128e9	−0.481232
$650$	0	0
$651$	3.96953e9	0.563904
$652$	0	0
$653$	1.04854e10	1.47363	0.736817	−	0.676093i	$-0.236328\pi$
$653$	1.04854e10	1.47363	0.736817	+	0.676093i	$0.236328\pi$
$654$	0	0
$655$	2.95894e9	0.411426
$656$	0	0
$657$	−1.61802e10	−2.22589
$658$	0	0
$659$	−5.46117e8	−0.0743339	−0.0371669	−	0.999309i	$-0.511833\pi$
$659$	−5.46117e8	−0.0743339	−0.0371669	+	0.999309i	$0.511833\pi$
$660$	0	0
$661$	−9.98464e9	−1.34471	−0.672353	−	0.740231i	$-0.734716\pi$
$661$	−9.98464e9	−1.34471	−0.672353	+	0.740231i	$0.734716\pi$
$662$	0	0
$663$	4.22209e9	0.562640
$664$	0	0
$665$	1.04413e8	0.0137682
$666$	0	0
$667$	−3.16897e9	−0.413502
$668$	0	0
$669$	−1.15327e10	−1.48916
$670$	0	0
$671$	−5.13904e9	−0.656679
$672$	0	0
$673$	−3.92194e9	−0.495962	−0.247981	−	0.968765i	$-0.579767\pi$
$673$	−3.92194e9	−0.495962	−0.247981	+	0.968765i	$0.579767\pi$
$674$	0	0
$675$	−1.75294e10	−2.19383
$676$	0	0
$677$	1.14925e9	0.142349	0.0711745	−	0.997464i	$-0.477325\pi$
$677$	1.14925e9	0.142349	0.0711745	+	0.997464i	$0.477325\pi$
$678$	0	0
$679$	−1.01447e8	−0.0124364
$680$	0	0
$681$	−2.04490e10	−2.48118
$682$	0	0
$683$	1.45436e10	1.74662	0.873311	−	0.487164i	$-0.161969\pi$
$683$	1.45436e10	1.74662	0.873311	+	0.487164i	$0.161969\pi$
$684$	0	0
$685$	1.57373e9	0.187074
$686$	0	0
$687$	2.53215e10	2.97948
$688$	0	0
$689$	2.01065e9	0.234191
$690$	0	0
$691$	8.07157e8	0.0930647	0.0465323	−	0.998917i	$-0.485183\pi$
$691$	8.07157e8	0.0930647	0.0465323	+	0.998917i	$0.485183\pi$
$692$	0	0
$693$	−2.47782e9	−0.282816
$694$	0	0
$695$	−2.66756e9	−0.301417
$696$	0	0
$697$	1.50914e9	0.168816
$698$	0	0
$699$	−1.30859e10	−1.44922
$700$	0	0
$701$	−1.29139e10	−1.41594	−0.707970	−	0.706242i	$-0.750389\pi$
$701$	−1.29139e10	−1.41594	−0.707970	+	0.706242i	$0.750389\pi$
$702$	0	0
$703$	1.57014e9	0.170449
$704$	0	0
$705$	−5.02364e8	−0.0539953
$706$	0	0
$707$	−3.92960e9	−0.418196
$708$	0	0
$709$	6.99785e9	0.737399	0.368700	−	0.929549i	$-0.379803\pi$
$709$	6.99785e9	0.737399	0.368700	+	0.929549i	$0.379803\pi$
$710$	0	0
$711$	−1.67926e9	−0.175216
$712$	0	0
$713$	7.53361e9	0.778377
$714$	0	0
$715$	−3.78114e8	−0.0386858
$716$	0	0
$717$	1.59478e10	1.61579
$718$	0	0
$719$	−1.02312e10	−1.02654	−0.513271	−	0.858227i	$-0.671566\pi$
$719$	−1.02312e10	−1.02654	−0.513271	+	0.858227i	$0.671566\pi$
$720$	0	0
$721$	−1.00715e9	−0.100074
$722$	0	0
$723$	−3.34484e10	−3.29148
$724$	0	0
$725$	6.82469e9	0.665120
$726$	0	0
$727$	5.55225e9	0.535918	0.267959	−	0.963430i	$-0.413651\pi$
$727$	5.55225e9	0.535918	0.267959	+	0.963430i	$0.413651\pi$
$728$	0	0
$729$	2.74259e9	0.262189
$730$	0	0
$731$	−3.64526e9	−0.345158
$732$	0	0
$733$	−1.22155e10	−1.14564	−0.572818	−	0.819683i	$-0.694150\pi$
$733$	−1.22155e10	−1.14564	−0.572818	+	0.819683i	$0.694150\pi$
$734$	0	0
$735$	−4.79663e9	−0.445585
$736$	0	0
$737$	−1.11390e10	−1.02497
$738$	0	0
$739$	4.14234e9	0.377564	0.188782	−	0.982019i	$-0.439546\pi$
$739$	4.14234e9	0.377564	0.188782	+	0.982019i	$0.439546\pi$
$740$	0	0
$741$	1.29248e9	0.116697
$742$	0	0
$743$	4.49275e9	0.401839	0.200919	−	0.979608i	$-0.435607\pi$
$743$	4.49275e9	0.401839	0.200919	+	0.979608i	$0.435607\pi$
$744$	0	0
$745$	−5.50293e8	−0.0487581
$746$	0	0
$747$	−4.86359e10	−4.26908
$748$	0	0
$749$	1.91544e9	0.166564
$750$	0	0
$751$	−6.17152e9	−0.531683	−0.265841	−	0.964017i	$-0.585650\pi$
$751$	−6.17152e9	−0.531683	−0.265841	+	0.964017i	$0.585650\pi$
$752$	0	0
$753$	−3.09365e10	−2.64052
$754$	0	0
$755$	1.09563e9	0.0926513
$756$	0	0
$757$	−1.23563e10	−1.03527	−0.517634	−	0.855602i	$-0.673187\pi$
$757$	−1.23563e10	−1.03527	−0.517634	+	0.855602i	$0.673187\pi$
$758$	0	0
$759$	−6.75106e9	−0.560436
$760$	0	0
$761$	1.46973e10	1.20891	0.604453	−	0.796641i	$-0.293392\pi$
$761$	1.46973e10	1.20891	0.604453	+	0.796641i	$0.293392\pi$
$762$	0	0
$763$	−2.25931e9	−0.184137
$764$	0	0
$765$	8.15407e9	0.658505
$766$	0	0
$767$	3.16510e9	0.253282
$768$	0	0
$769$	−1.99912e10	−1.58525	−0.792625	−	0.609710i	$-0.791286\pi$
$769$	−1.99912e10	−1.58525	−0.792625	+	0.609710i	$0.791286\pi$
$770$	0	0
$771$	−1.11747e10	−0.878104
$772$	0	0
$773$	1.87659e10	1.46130	0.730651	−	0.682751i	$-0.239217\pi$
$773$	1.87659e10	1.46130	0.730651	+	0.682751i	$0.239217\pi$
$774$	0	0
$775$	−1.62244e10	−1.25202
$776$	0	0
$777$	4.08131e9	0.312123
$778$	0	0
$779$	4.61980e8	0.0350140
$780$	0	0
$781$	−8.42577e9	−0.632894
$782$	0	0
$783$	−2.25292e10	−1.67718
$784$	0	0
$785$	−2.32036e9	−0.171203
$786$	0	0
$787$	1.33857e10	0.978877	0.489439	−	0.872038i	$-0.337202\pi$
$787$	1.33857e10	0.978877	0.489439	+	0.872038i	$0.337202\pi$
$788$	0	0
$789$	4.29850e10	3.11564
$790$	0	0
$791$	2.28821e9	0.164391
$792$	0	0
$793$	4.85355e9	0.345624
$794$	0	0
$795$	5.57470e9	0.393493
$796$	0	0
$797$	2.57977e10	1.80500	0.902498	−	0.430693i	$-0.141731\pi$
$797$	2.57977e10	1.80500	0.902498	+	0.430693i	$0.141731\pi$
$798$	0	0
$799$	−1.82907e9	−0.126858
$800$	0	0
$801$	−3.38522e10	−2.32741
$802$	0	0
$803$	7.57406e9	0.516208
$804$	0	0
$805$	5.15087e8	0.0348012
$806$	0	0
$807$	2.85766e9	0.191405
$808$	0	0
$809$	1.24018e10	0.823504	0.411752	−	0.911296i	$-0.364917\pi$
$809$	1.24018e10	0.823504	0.411752	+	0.911296i	$0.364917\pi$
$810$	0	0
$811$	1.28908e10	0.848607	0.424303	−	0.905520i	$-0.360519\pi$
$811$	1.28908e10	0.848607	0.424303	+	0.905520i	$0.360519\pi$
$812$	0	0
$813$	−4.34905e10	−2.83843
$814$	0	0
$815$	−2.07759e9	−0.134434
$816$	0	0
$817$	−1.11589e9	−0.0715890
$818$	0	0
$819$	2.34017e9	0.148851
$820$	0	0
$821$	−1.73179e9	−0.109218	−0.0546089	−	0.998508i	$-0.517391\pi$
$821$	−1.73179e9	−0.109218	−0.0546089	+	0.998508i	$0.517391\pi$
$822$	0	0
$823$	1.33148e10	0.832599	0.416299	−	0.909228i	$-0.363327\pi$
$823$	1.33148e10	0.832599	0.416299	+	0.909228i	$0.363327\pi$
$824$	0	0
$825$	1.45391e10	0.901463
$826$	0	0
$827$	−7.23050e9	−0.444528	−0.222264	−	0.974987i	$-0.571345\pi$
$827$	−7.23050e9	−0.444528	−0.222264	+	0.974987i	$0.571345\pi$
$828$	0	0
$829$	−2.85814e10	−1.74238	−0.871189	−	0.490947i	$-0.836651\pi$
$829$	−2.85814e10	−1.74238	−0.871189	+	0.490947i	$0.836651\pi$
$830$	0	0
$831$	5.55560e10	3.35836
$832$	0	0
$833$	−1.74642e10	−1.04687
$834$	0	0
$835$	7.30870e9	0.434448
$836$	0	0
$837$	5.35587e10	3.15712
$838$	0	0
$839$	2.31016e10	1.35044	0.675219	−	0.737618i	$-0.264049\pi$
$839$	2.31016e10	1.35044	0.675219	+	0.737618i	$0.264049\pi$
$840$	0	0
$841$	−8.47864e9	−0.491519
$842$	0	0
$843$	−1.08268e10	−0.622447
$844$	0	0
$845$	−4.19136e9	−0.238977
$846$	0	0
$847$	−2.93253e9	−0.165825
$848$	0	0
$849$	−3.57621e10	−2.00561
$850$	0	0
$851$	7.74576e9	0.430834
$852$	0	0
$853$	3.69001e9	0.203566	0.101783	−	0.994807i	$-0.467545\pi$
$853$	3.69001e9	0.203566	0.101783	+	0.994807i	$0.467545\pi$
$854$	0	0
$855$	2.49614e9	0.136580
$856$	0	0
$857$	8.27437e9	0.449057	0.224529	−	0.974467i	$-0.427916\pi$
$857$	8.27437e9	0.449057	0.224529	+	0.974467i	$0.427916\pi$
$858$	0	0
$859$	−3.37678e10	−1.81772	−0.908859	−	0.417104i	$-0.863045\pi$
$859$	−3.37678e10	−1.81772	−0.908859	+	0.417104i	$0.863045\pi$
$860$	0	0
$861$	1.20084e9	0.0641172
$862$	0	0
$863$	7.31538e9	0.387435	0.193718	−	0.981057i	$-0.437945\pi$
$863$	7.31538e9	0.387435	0.193718	+	0.981057i	$0.437945\pi$
$864$	0	0
$865$	−2.63122e8	−0.0138230
$866$	0	0
$867$	7.78460e9	0.405667
$868$	0	0
$869$	7.86075e8	0.0406345
$870$	0	0
$871$	1.05202e10	0.539461
$872$	0	0
$873$	−2.42524e9	−0.123369
$874$	0	0
$875$	−2.29856e9	−0.115992
$876$	0	0
$877$	2.90785e10	1.45570	0.727852	−	0.685734i	$-0.240519\pi$
$877$	2.90785e10	1.45570	0.727852	+	0.685734i	$0.240519\pi$
$878$	0	0
$879$	−3.18628e10	−1.58242
$880$	0	0
$881$	1.37853e10	0.679205	0.339602	−	0.940569i	$-0.389708\pi$
$881$	1.37853e10	0.679205	0.339602	+	0.940569i	$0.389708\pi$
$882$	0	0
$883$	1.41202e10	0.690208	0.345104	−	0.938564i	$-0.387844\pi$
$883$	1.41202e10	0.690208	0.345104	+	0.938564i	$0.387844\pi$
$884$	0	0
$885$	8.77551e9	0.425570
$886$	0	0
$887$	3.17818e10	1.52914	0.764568	−	0.644543i	$-0.222952\pi$
$887$	3.17818e10	1.52914	0.764568	+	0.644543i	$0.222952\pi$
$888$	0	0
$889$	6.65315e9	0.317593
$890$	0	0
$891$	−2.21913e10	−1.05102
$892$	0	0
$893$	−5.59919e8	−0.0263115
$894$	0	0
$895$	4.82388e9	0.224914
$896$	0	0
$897$	6.37601e9	0.294969
$898$	0	0
$899$	−2.08519e10	−0.957166
$900$	0	0
$901$	2.02971e10	0.924480
$902$	0	0
$903$	−2.90058e9	−0.131093
$904$	0	0
$905$	1.02500e10	0.459677
$906$	0	0
$907$	3.87339e10	1.72372	0.861858	−	0.507149i	$-0.169301\pi$
$907$	3.87339e10	1.72372	0.861858	+	0.507149i	$0.169301\pi$
$908$	0	0
$909$	−9.39428e10	−4.14849
$910$	0	0
$911$	−4.43662e10	−1.94419	−0.972093	−	0.234597i	$-0.924623\pi$
$911$	−4.43662e10	−1.94419	−0.972093	+	0.234597i	$0.924623\pi$
$912$	0	0
$913$	2.27668e10	0.990044
$914$	0	0
$915$	1.34569e10	0.580724
$916$	0	0
$917$	8.57247e9	0.367124
$918$	0	0
$919$	1.49991e10	0.637474	0.318737	−	0.947843i	$-0.396741\pi$
$919$	1.49991e10	0.637474	0.318737	+	0.947843i	$0.396741\pi$
$920$	0	0
$921$	−1.83018e9	−0.0771941
$922$	0	0
$923$	7.95768e9	0.333105
$924$	0	0
$925$	−1.66812e10	−0.692999
$926$	0	0
$927$	−2.40773e10	−0.992726
$928$	0	0
$929$	−2.26273e9	−0.0925931	−0.0462965	−	0.998928i	$-0.514742\pi$
$929$	−2.26273e9	−0.0925931	−0.0462965	+	0.998928i	$0.514742\pi$
$930$	0	0
$931$	−5.34618e9	−0.217130
$932$	0	0
$933$	−5.06232e10	−2.04063
$934$	0	0
$935$	−3.81698e9	−0.152714
$936$	0	0
$937$	−1.38459e10	−0.549834	−0.274917	−	0.961468i	$-0.588650\pi$
$937$	−1.38459e10	−0.549834	−0.274917	+	0.961468i	$0.588650\pi$
$938$	0	0
$939$	4.28215e10	1.68784
$940$	0	0
$941$	−1.00343e10	−0.392574	−0.196287	−	0.980547i	$-0.562888\pi$
$941$	−1.00343e10	−0.392574	−0.196287	+	0.980547i	$0.562888\pi$
$942$	0	0
$943$	2.27903e9	0.0885032
$944$	0	0
$945$	3.66190e9	0.141155
$946$	0	0
$947$	1.99331e10	0.762693	0.381347	−	0.924432i	$-0.375460\pi$
$947$	1.99331e10	0.762693	0.381347	+	0.924432i	$0.375460\pi$
$948$	0	0
$949$	−7.15330e9	−0.271691
$950$	0	0
$951$	−6.54680e10	−2.46830
$952$	0	0
$953$	5.95607e9	0.222913	0.111456	−	0.993769i	$-0.464448\pi$
$953$	5.95607e9	0.222913	0.111456	+	0.993769i	$0.464448\pi$
$954$	0	0
$955$	−8.14806e9	−0.302721
$956$	0	0
$957$	1.86859e10	0.689165
$958$	0	0
$959$	4.55932e9	0.166930
$960$	0	0
$961$	2.20588e10	0.801769
$962$	0	0
$963$	4.57914e10	1.65231
$964$	0	0
$965$	3.66225e9	0.131190
$966$	0	0
$967$	−1.33084e10	−0.473295	−0.236648	−	0.971596i	$-0.576049\pi$
$967$	−1.33084e10	−0.473295	−0.236648	+	0.971596i	$0.576049\pi$
$968$	0	0
$969$	1.30473e10	0.460666
$970$	0	0
$971$	−7.00846e9	−0.245672	−0.122836	−	0.992427i	$-0.539199\pi$
$971$	−7.00846e9	−0.245672	−0.122836	+	0.992427i	$0.539199\pi$
$972$	0	0
$973$	−7.72830e9	−0.268961
$974$	0	0
$975$	−1.37314e10	−0.474458
$976$	0	0
$977$	4.45829e10	1.52946	0.764729	−	0.644352i	$-0.222873\pi$
$977$	4.45829e10	1.52946	0.764729	+	0.644352i	$0.222873\pi$
$978$	0	0
$979$	1.58465e10	0.539750
$980$	0	0
$981$	−5.40122e10	−1.82663
$982$	0	0
$983$	1.18845e10	0.399066	0.199533	−	0.979891i	$-0.436057\pi$
$983$	1.18845e10	0.399066	0.199533	+	0.979891i	$0.436057\pi$
$984$	0	0
$985$	−6.09025e9	−0.203052
$986$	0	0
$987$	−1.45542e9	−0.0481812
$988$	0	0
$989$	−5.50491e9	−0.180952
$990$	0	0
$991$	−8.83738e9	−0.288447	−0.144223	−	0.989545i	$-0.546068\pi$
$991$	−8.83738e9	−0.288447	−0.144223	+	0.989545i	$0.546068\pi$
$992$	0	0
$993$	−2.58803e10	−0.838777
$994$	0	0
$995$	8.35977e9	0.269038
$996$	0	0
$997$	4.97053e10	1.58844	0.794219	−	0.607632i	$-0.207880\pi$
$997$	4.97053e10	1.58844	0.794219	+	0.607632i	$0.207880\pi$
$998$	0	0
$999$	5.50669e10	1.74748

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.a.1.1	✓	5	4.3	odd	2
304.8.a.g.1.5		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)

\(f(q)\)	\(=\)	\(q+84.8970 q^{3} +72.4872 q^{5} +210.006 q^{7} +5020.49 q^{9} +O(q^{10})\)	\(q+84.8970 q^{3} +72.4872 q^{5} +210.006 q^{7} +5020.49 q^{9} -2350.13 q^{11} +2219.57 q^{13} +6153.94 q^{15} +22406.1 q^{17} +6859.00 q^{19} +17828.8 q^{21} +33836.7 q^{23} -72870.6 q^{25} +240555. q^{27} -93654.9 q^{29} +222646. q^{31} -199519. q^{33} +15222.7 q^{35} +228916. q^{37} +188435. q^{39} +67353.8 q^{41} -162691. q^{43} +363922. q^{45} -81632.8 q^{47} -779441. q^{49} +1.90221e6 q^{51} +905874. q^{53} -170354. q^{55} +582308. q^{57} +1.42600e6 q^{59} +2.18670e6 q^{61} +1.05433e6 q^{63} +160891. q^{65} +4.73974e6 q^{67} +2.87263e6 q^{69} +3.58523e6 q^{71} -3.22283e6 q^{73} -6.18649e6 q^{75} -493541. q^{77} -334481. q^{79} +9.44257e6 q^{81} -9.68747e6 q^{83} +1.62416e6 q^{85} -7.95102e6 q^{87} -6.74280e6 q^{89} +466123. q^{91} +1.89020e7 q^{93} +497190. q^{95} -483068. q^{97} -1.17988e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 14 q^{3} - 280 q^{5} - 414 q^{7} + 3779 q^{9}+O(q^{10}) \) 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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