LMFDB - Embedded newform 28.8.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [28,8,Mod(1,28)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(28, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("28.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 28 = 2^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	28.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:	$8.74678071356$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{3529}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 882 $ x^2 - x - 882
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-29.2027$ of defining polynomial
Character	$\chi$	$=$	28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+52.4054 q^{3} +199.216 q^{5} +343.000 q^{7} +559.325 q^{9} +1218.97 q^{11} +6449.65 q^{13} +10440.0 q^{15} +26786.9 q^{17} +14910.9 q^{19} +17975.0 q^{21} -73500.9 q^{23} -38437.9 q^{25} -85299.0 q^{27} -106453. q^{29} +66816.7 q^{31} +63880.8 q^{33} +68331.1 q^{35} -479502. q^{37} +337996. q^{39} -644539. q^{41} +145165. q^{43} +111426. q^{45} +1.01085e6 q^{47} +117649. q^{49} +1.40378e6 q^{51} +34208.5 q^{53} +242839. q^{55} +781414. q^{57} -443317. q^{59} -1.14043e6 q^{61} +191848. q^{63} +1.28487e6 q^{65} -4.31928e6 q^{67} -3.85185e6 q^{69} +2.54867e6 q^{71} -3.67723e6 q^{73} -2.01435e6 q^{75} +418108. q^{77} +8.55564e6 q^{79} -5.69337e6 q^{81} -1.79721e6 q^{83} +5.33639e6 q^{85} -5.57873e6 q^{87} +5.56317e6 q^{89} +2.21223e6 q^{91} +3.50155e6 q^{93} +2.97050e6 q^{95} -1.72057e7 q^{97} +681802. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} + 42 q^{5} + 686 q^{7} + 2782 q^{9} + 7428 q^{11} + 11830 q^{13} + 20880 q^{15} + 15792 q^{17} + 26614 q^{19} - 4802 q^{21} + 32640 q^{23} - 91846 q^{25} - 87668 q^{27} - 158016 q^{29} - 180740 q^{31}+ \cdots + 14482452 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 + 42 * q^5 + 686 * q^7 + 2782 * q^9 + 7428 * q^11 + 11830 * q^13 + 20880 * q^15 + 15792 * q^17 + 26614 * q^19 - 4802 * q^21 + 32640 * q^23 - 91846 * q^25 - 87668 * q^27 - 158016 * q^29 - 180740 * q^31 - 348432 * q^33 + 14406 * q^35 - 45824 * q^37 - 19288 * q^39 - 321720 * q^41 + 1023868 * q^43 - 238014 * q^45 + 1665972 * q^47 + 235298 * q^49 + 2133900 * q^51 - 410628 * q^53 - 733320 * q^55 + 4268 * q^57 + 1702134 * q^59 - 547526 * q^61 + 954226 * q^63 + 438996 * q^65 - 2590616 * q^67 - 10900176 * q^69 + 4129272 * q^71 - 8008868 * q^73 + 1532230 * q^75 + 2547804 * q^77 + 2470456 * q^79 - 10397042 * q^81 - 9900786 * q^83 + 7064964 * q^85 - 2154684 * q^87 + 15423492 * q^89 + 4057690 * q^91 + 19940648 * q^93 + 1130592 * q^95 - 17377472 * q^97 + 14482452 * q^99

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
$3$	52.4054	1.12060	0.560301	−	0.828289i	$-0.310685\pi$
$3$	52.4054	1.12060	0.560301	+	0.828289i	$0.310685\pi$
$4$	0	0
$5$	199.216	0.712737	0.356369	−	0.934345i	$-0.384015\pi$
$5$	199.216	0.712737	0.356369	+	0.934345i	$0.384015\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	559.325	0.255750
$10$	0	0
$11$	1218.97	0.276134	0.138067	−	0.990423i	$-0.455911\pi$
$11$	1218.97	0.276134	0.138067	+	0.990423i	$0.455911\pi$
$12$	0	0
$13$	6449.65	0.814206	0.407103	−	0.913382i	$-0.366539\pi$
$13$	6449.65	0.814206	0.407103	+	0.913382i	$0.366539\pi$
$14$	0	0
$15$	10440.0	0.798695
$16$	0	0
$17$	26786.9	1.32237	0.661183	−	0.750225i	$-0.270055\pi$
$17$	26786.9	1.32237	0.661183	+	0.750225i	$0.270055\pi$
$18$	0	0
$19$	14910.9	0.498732	0.249366	−	0.968409i	$-0.419778\pi$
$19$	14910.9	0.498732	0.249366	+	0.968409i	$0.419778\pi$
$20$	0	0
$21$	17975.0	0.423548
$22$	0	0
$23$	−73500.9	−1.25964	−0.629819	−	0.776742i	$-0.716871\pi$
$23$	−73500.9	−1.25964	−0.629819	+	0.776742i	$0.716871\pi$
$24$	0	0
$25$	−38437.9	−0.492005
$26$	0	0
$27$	−85299.0	−0.834009
$28$	0	0
$29$	−106453.	−0.810524	−0.405262	−	0.914200i	$-0.632820\pi$
$29$	−106453.	−0.810524	−0.405262	+	0.914200i	$0.632820\pi$
$30$	0	0
$31$	66816.7	0.402827	0.201414	−	0.979506i	$-0.435446\pi$
$31$	66816.7	0.402827	0.201414	+	0.979506i	$0.435446\pi$
$32$	0	0
$33$	63880.8	0.309436
$34$	0	0
$35$	68331.1	0.269389
$36$	0	0
$37$	−479502.	−1.55627	−0.778134	−	0.628099i	$-0.783833\pi$
$37$	−479502.	−1.55627	−0.778134	+	0.628099i	$0.783833\pi$
$38$	0	0
$39$	337996.	0.912401
$40$	0	0
$41$	−644539.	−1.46051	−0.730257	−	0.683173i	$-0.760599\pi$
$41$	−644539.	−1.46051	−0.730257	+	0.683173i	$0.760599\pi$
$42$	0	0
$43$	145165.	0.278434	0.139217	−	0.990262i	$-0.455541\pi$
$43$	145165.	0.278434	0.139217	+	0.990262i	$0.455541\pi$
$44$	0	0
$45$	111426.	0.182282
$46$	0	0
$47$	1.01085e6	1.42018	0.710088	−	0.704113i	$-0.248655\pi$
$47$	1.01085e6	1.42018	0.710088	+	0.704113i	$0.248655\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	1.40378e6	1.48185
$52$	0	0
$53$	34208.5	0.0315623	0.0157812	−	0.999875i	$-0.494976\pi$
$53$	34208.5	0.0315623	0.0157812	+	0.999875i	$0.494976\pi$
$54$	0	0
$55$	242839.	0.196811
$56$	0	0
$57$	781414.	0.558881
$58$	0	0
$59$	−443317.	−0.281017	−0.140508	−	0.990079i	$-0.544874\pi$
$59$	−443317.	−0.281017	−0.140508	+	0.990079i	$0.544874\pi$
$60$	0	0
$61$	−1.14043e6	−0.643300	−0.321650	−	0.946859i	$-0.604237\pi$
$61$	−1.14043e6	−0.643300	−0.321650	+	0.946859i	$0.604237\pi$
$62$	0	0
$63$	191848.	0.0966643
$64$	0	0
$65$	1.28487e6	0.580315
$66$	0	0
$67$	−4.31928e6	−1.75448	−0.877242	−	0.480048i	$-0.840619\pi$
$67$	−4.31928e6	−1.75448	−0.877242	+	0.480048i	$0.840619\pi$
$68$	0	0
$69$	−3.85185e6	−1.41155
$70$	0	0
$71$	2.54867e6	0.845103	0.422551	−	0.906339i	$-0.361135\pi$
$71$	2.54867e6	0.845103	0.422551	+	0.906339i	$0.361135\pi$
$72$	0	0
$73$	−3.67723e6	−1.10635	−0.553173	−	0.833067i	$-0.686583\pi$
$73$	−3.67723e6	−1.10635	−0.553173	+	0.833067i	$0.686583\pi$
$74$	0	0
$75$	−2.01435e6	−0.551342
$76$	0	0
$77$	418108.	0.104369
$78$	0	0
$79$	8.55564e6	1.95235	0.976174	−	0.216987i	$-0.0696230\pi$
$79$	8.55564e6	1.95235	0.976174	+	0.216987i	$0.0696230\pi$
$80$	0	0
$81$	−5.69337e6	−1.19034
$82$	0	0
$83$	−1.79721e6	−0.345006	−0.172503	−	0.985009i	$-0.555185\pi$
$83$	−1.79721e6	−0.345006	−0.172503	+	0.985009i	$0.555185\pi$
$84$	0	0
$85$	5.33639e6	0.942499
$86$	0	0
$87$	−5.57873e6	−0.908276
$88$	0	0
$89$	5.56317e6	0.836484	0.418242	−	0.908336i	$-0.362646\pi$
$89$	5.56317e6	0.836484	0.418242	+	0.908336i	$0.362646\pi$
$90$	0	0
$91$	2.21223e6	0.307741
$92$	0	0
$93$	3.50155e6	0.451409
$94$	0	0
$95$	2.97050e6	0.355465
$96$	0	0
$97$	−1.72057e7	−1.91413	−0.957064	−	0.289877i	$-0.906386\pi$
$97$	−1.72057e7	−1.91413	−0.957064	+	0.289877i	$0.906386\pi$
$98$	0	0
$99$	681802.	0.0706212
$100$	0	0
$101$	1.18725e7	1.14661	0.573306	−	0.819341i	$-0.305661\pi$
$101$	1.18725e7	1.14661	0.573306	+	0.819341i	$0.305661\pi$
$102$	0	0
$103$	1.60523e7	1.44746	0.723731	−	0.690082i	$-0.242426\pi$
$103$	1.60523e7	1.44746	0.723731	+	0.690082i	$0.242426\pi$
$104$	0	0
$105$	3.58092e6	0.301878
$106$	0	0
$107$	1.04274e7	0.822870	0.411435	−	0.911439i	$-0.365028\pi$
$107$	1.04274e7	0.822870	0.411435	+	0.911439i	$0.365028\pi$
$108$	0	0
$109$	7.51810e6	0.556052	0.278026	−	0.960574i	$-0.410320\pi$
$109$	7.51810e6	0.556052	0.278026	+	0.960574i	$0.410320\pi$
$110$	0	0
$111$	−2.51285e7	−1.74396
$112$	0	0
$113$	2.53184e7	1.65067	0.825337	−	0.564640i	$-0.190985\pi$
$113$	2.53184e7	1.65067	0.825337	+	0.564640i	$0.190985\pi$
$114$	0	0
$115$	−1.46426e7	−0.897791
$116$	0	0
$117$	3.60745e6	0.208233
$118$	0	0
$119$	9.18791e6	0.499807
$120$	0	0
$121$	−1.80013e7	−0.923750
$122$	0	0
$123$	−3.37773e7	−1.63665
$124$	0	0
$125$	−2.32212e7	−1.06341
$126$	0	0
$127$	−3.98433e7	−1.72600	−0.863002	−	0.505201i	$-0.831419\pi$
$127$	−3.98433e7	−1.72600	−0.863002	+	0.505201i	$0.831419\pi$
$128$	0	0
$129$	7.60744e6	0.312014
$130$	0	0
$131$	1.30645e7	0.507741	0.253870	−	0.967238i	$-0.418296\pi$
$131$	1.30645e7	0.507741	0.253870	+	0.967238i	$0.418296\pi$
$132$	0	0
$133$	5.11445e6	0.188503
$134$	0	0
$135$	−1.69929e7	−0.594429
$136$	0	0
$137$	3.35662e6	0.111527	0.0557636	−	0.998444i	$-0.482241\pi$
$137$	3.35662e6	0.111527	0.0557636	+	0.998444i	$0.482241\pi$
$138$	0	0
$139$	−9.65184e6	−0.304830	−0.152415	−	0.988317i	$-0.548705\pi$
$139$	−9.65184e6	−0.304830	−0.152415	+	0.988317i	$0.548705\pi$
$140$	0	0
$141$	5.29738e7	1.59145
$142$	0	0
$143$	7.86195e6	0.224830
$144$	0	0
$145$	−2.12072e7	−0.577691
$146$	0	0
$147$	6.16544e6	0.160086
$148$	0	0
$149$	5.61846e7	1.39144	0.695722	−	0.718311i	$-0.255085\pi$
$149$	5.61846e7	1.39144	0.695722	+	0.718311i	$0.255085\pi$
$150$	0	0
$151$	−2.33811e7	−0.552644	−0.276322	−	0.961065i	$-0.589116\pi$
$151$	−2.33811e7	−0.552644	−0.276322	+	0.961065i	$0.589116\pi$
$152$	0	0
$153$	1.49826e7	0.338195
$154$	0	0
$155$	1.33110e7	0.287110
$156$	0	0
$157$	−2.10986e7	−0.435116	−0.217558	−	0.976047i	$-0.569809\pi$
$157$	−2.10986e7	−0.435116	−0.217558	+	0.976047i	$0.569809\pi$
$158$	0	0
$159$	1.79271e6	0.0353688
$160$	0	0
$161$	−2.52108e7	−0.476098
$162$	0	0
$163$	6.03996e7	1.09239	0.546195	−	0.837658i	$-0.316076\pi$
$163$	6.03996e7	1.09239	0.546195	+	0.837658i	$0.316076\pi$
$164$	0	0
$165$	1.27261e7	0.220547
$166$	0	0
$167$	6.88696e7	1.14425	0.572124	−	0.820167i	$-0.306120\pi$
$167$	6.88696e7	1.14425	0.572124	+	0.820167i	$0.306120\pi$
$168$	0	0
$169$	−2.11506e7	−0.337069
$170$	0	0
$171$	8.34006e6	0.127551
$172$	0	0
$173$	−2.94427e7	−0.432331	−0.216166	−	0.976357i	$-0.569355\pi$
$173$	−2.94427e7	−0.432331	−0.216166	+	0.976357i	$0.569355\pi$
$174$	0	0
$175$	−1.31842e7	−0.185961
$176$	0	0
$177$	−2.32322e7	−0.314908
$178$	0	0
$179$	8.98500e7	1.17093	0.585467	−	0.810696i	$-0.300911\pi$
$179$	8.98500e7	1.17093	0.585467	+	0.810696i	$0.300911\pi$
$180$	0	0
$181$	4.60929e7	0.577775	0.288888	−	0.957363i	$-0.406715\pi$
$181$	4.60929e7	0.577775	0.288888	+	0.957363i	$0.406715\pi$
$182$	0	0
$183$	−5.97646e7	−0.720883
$184$	0	0
$185$	−9.55245e7	−1.10921
$186$	0	0
$187$	3.26525e7	0.365150
$188$	0	0
$189$	−2.92575e7	−0.315226
$190$	0	0
$191$	−3.90211e7	−0.405213	−0.202606	−	0.979260i	$-0.564941\pi$
$191$	−3.90211e7	−0.405213	−0.202606	+	0.979260i	$0.564941\pi$
$192$	0	0
$193$	−1.65472e7	−0.165681	−0.0828406	−	0.996563i	$-0.526399\pi$
$193$	−1.65472e7	−0.165681	−0.0828406	+	0.996563i	$0.526399\pi$
$194$	0	0
$195$	6.73343e7	0.650302
$196$	0	0
$197$	−2.12903e7	−0.198403	−0.0992017	−	0.995067i	$-0.531629\pi$
$197$	−2.12903e7	−0.198403	−0.0992017	+	0.995067i	$0.531629\pi$
$198$	0	0
$199$	2.28518e7	0.205558	0.102779	−	0.994704i	$-0.467227\pi$
$199$	2.28518e7	0.205558	0.102779	+	0.994704i	$0.467227\pi$
$200$	0	0
$201$	−2.26354e8	−1.96608
$202$	0	0
$203$	−3.65135e7	−0.306349
$204$	0	0
$205$	−1.28403e8	−1.04096
$206$	0	0
$207$	−4.11109e7	−0.322152
$208$	0	0
$209$	1.81761e7	0.137717
$210$	0	0
$211$	−1.49824e8	−1.09797	−0.548987	−	0.835831i	$-0.684986\pi$
$211$	−1.49824e8	−1.09797	−0.548987	+	0.835831i	$0.684986\pi$
$212$	0	0
$213$	1.33564e8	0.947024
$214$	0	0
$215$	2.89192e7	0.198451
$216$	0	0
$217$	2.29181e7	0.152254
$218$	0	0
$219$	−1.92707e8	−1.23977
$220$	0	0
$221$	1.72766e8	1.07668
$222$	0	0
$223$	1.85577e8	1.12062	0.560308	−	0.828284i	$-0.310683\pi$
$223$	1.85577e8	1.12062	0.560308	+	0.828284i	$0.310683\pi$
$224$	0	0
$225$	−2.14993e7	−0.125830
$226$	0	0
$227$	3.17593e8	1.80211	0.901053	−	0.433708i	$-0.142795\pi$
$227$	3.17593e8	1.80211	0.901053	+	0.433708i	$0.142795\pi$
$228$	0	0
$229$	2.35342e8	1.29502	0.647509	−	0.762057i	$-0.275811\pi$
$229$	2.35342e8	1.29502	0.647509	+	0.762057i	$0.275811\pi$
$230$	0	0
$231$	2.19111e7	0.116956
$232$	0	0
$233$	−3.16797e8	−1.64072	−0.820362	−	0.571844i	$-0.806228\pi$
$233$	−3.16797e8	−1.64072	−0.820362	+	0.571844i	$0.806228\pi$
$234$	0	0
$235$	2.01377e8	1.01221
$236$	0	0
$237$	4.48361e8	2.18781
$238$	0	0
$239$	−8.35562e7	−0.395900	−0.197950	−	0.980212i	$-0.563428\pi$
$239$	−8.35562e7	−0.395900	−0.197950	+	0.980212i	$0.563428\pi$
$240$	0	0
$241$	−1.04444e8	−0.480644	−0.240322	−	0.970693i	$-0.577253\pi$
$241$	−1.04444e8	−0.480644	−0.240322	+	0.970693i	$0.577253\pi$
$242$	0	0
$243$	−1.11814e8	−0.499891
$244$	0	0
$245$	2.34376e7	0.101820
$246$	0	0
$247$	9.61704e7	0.406071
$248$	0	0
$249$	−9.41837e7	−0.386614
$250$	0	0
$251$	−2.74656e8	−1.09630	−0.548152	−	0.836379i	$-0.684669\pi$
$251$	−2.74656e8	−1.09630	−0.548152	+	0.836379i	$0.684669\pi$
$252$	0	0
$253$	−8.95957e7	−0.347829
$254$	0	0
$255$	2.79655e8	1.05617
$256$	0	0
$257$	−4.74786e7	−0.174474	−0.0872372	−	0.996188i	$-0.527804\pi$
$257$	−4.74786e7	−0.174474	−0.0872372	+	0.996188i	$0.527804\pi$
$258$	0	0
$259$	−1.64469e8	−0.588214
$260$	0	0
$261$	−5.95419e7	−0.207291
$262$	0	0
$263$	−1.32034e7	−0.0447548	−0.0223774	−	0.999750i	$-0.507124\pi$
$263$	−1.32034e7	−0.0447548	−0.0223774	+	0.999750i	$0.507124\pi$
$264$	0	0
$265$	6.81489e6	0.0224956
$266$	0	0
$267$	2.91540e8	0.937366
$268$	0	0
$269$	−4.66144e8	−1.46012	−0.730058	−	0.683386i	$-0.760507\pi$
$269$	−4.66144e8	−1.46012	−0.730058	+	0.683386i	$0.760507\pi$
$270$	0	0
$271$	−1.64684e8	−0.502642	−0.251321	−	0.967904i	$-0.580865\pi$
$271$	−1.64684e8	−0.502642	−0.251321	+	0.967904i	$0.580865\pi$
$272$	0	0
$273$	1.15933e8	0.344855
$274$	0	0
$275$	−4.68548e7	−0.135859
$276$	0	0
$277$	6.26721e8	1.77172	0.885861	−	0.463952i	$-0.153569\pi$
$277$	6.26721e8	1.77172	0.885861	+	0.463952i	$0.153569\pi$
$278$	0	0
$279$	3.73722e7	0.103023
$280$	0	0
$281$	−5.30935e8	−1.42748	−0.713739	−	0.700412i	$-0.753000\pi$
$281$	−5.30935e8	−1.42748	−0.713739	+	0.700412i	$0.753000\pi$
$282$	0	0
$283$	−1.73440e8	−0.454880	−0.227440	−	0.973792i	$-0.573036\pi$
$283$	−1.73440e8	−0.454880	−0.227440	+	0.973792i	$0.573036\pi$
$284$	0	0
$285$	1.55670e8	0.398335
$286$	0	0
$287$	−2.21077e8	−0.552022
$288$	0	0
$289$	3.07200e8	0.748650
$290$	0	0
$291$	−9.01671e8	−2.14498
$292$	0	0
$293$	−1.19128e8	−0.276680	−0.138340	−	0.990385i	$-0.544177\pi$
$293$	−1.19128e8	−0.276680	−0.138340	+	0.990385i	$0.544177\pi$
$294$	0	0
$295$	−8.83159e7	−0.200291
$296$	0	0
$297$	−1.03977e8	−0.230298
$298$	0	0
$299$	−4.74055e8	−1.02560
$300$	0	0
$301$	4.97916e7	0.105238
$302$	0	0
$303$	6.22182e8	1.28490
$304$	0	0
$305$	−2.27192e8	−0.458504
$306$	0	0
$307$	−3.98641e8	−0.786318	−0.393159	−	0.919471i	$-0.628618\pi$
$307$	−3.98641e8	−0.786318	−0.393159	+	0.919471i	$0.628618\pi$
$308$	0	0
$309$	8.41227e8	1.62203
$310$	0	0
$311$	−3.21786e8	−0.606606	−0.303303	−	0.952894i	$-0.598089\pi$
$311$	−3.21786e8	−0.606606	−0.303303	+	0.952894i	$0.598089\pi$
$312$	0	0
$313$	2.80257e8	0.516596	0.258298	−	0.966065i	$-0.416838\pi$
$313$	2.80257e8	0.516596	0.258298	+	0.966065i	$0.416838\pi$
$314$	0	0
$315$	3.82193e7	0.0688963
$316$	0	0
$317$	8.05173e8	1.41965	0.709826	−	0.704377i	$-0.248774\pi$
$317$	8.05173e8	1.41965	0.709826	+	0.704377i	$0.248774\pi$
$318$	0	0
$319$	−1.29764e8	−0.223813
$320$	0	0
$321$	5.46450e8	0.922110
$322$	0	0
$323$	3.99418e8	0.659506
$324$	0	0
$325$	−2.47911e8	−0.400594
$326$	0	0
$327$	3.93989e8	0.623113
$328$	0	0
$329$	3.46720e8	0.536776
$330$	0	0
$331$	2.31397e8	0.350719	0.175359	−	0.984504i	$-0.443891\pi$
$331$	2.31397e8	0.350719	0.175359	+	0.984504i	$0.443891\pi$
$332$	0	0
$333$	−2.68197e8	−0.398015
$334$	0	0
$335$	−8.60470e8	−1.25049
$336$	0	0
$337$	1.12688e9	1.60388	0.801940	−	0.597405i	$-0.203801\pi$
$337$	1.12688e9	1.60388	0.801940	+	0.597405i	$0.203801\pi$
$338$	0	0
$339$	1.32682e9	1.84975
$340$	0	0
$341$	8.14477e7	0.111234
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−7.67350e8	−1.00607
$346$	0	0
$347$	2.75688e8	0.354213	0.177107	−	0.984192i	$-0.443326\pi$
$347$	2.75688e8	0.354213	0.177107	+	0.984192i	$0.443326\pi$
$348$	0	0
$349$	−8.85168e8	−1.11465	−0.557323	−	0.830296i	$-0.688171\pi$
$349$	−8.85168e8	−1.11465	−0.557323	+	0.830296i	$0.688171\pi$
$350$	0	0
$351$	−5.50148e8	−0.679055
$352$	0	0
$353$	−9.12174e8	−1.10374	−0.551869	−	0.833930i	$-0.686085\pi$
$353$	−9.12174e8	−1.10374	−0.551869	+	0.833930i	$0.686085\pi$
$354$	0	0
$355$	5.07736e8	0.602336
$356$	0	0
$357$	4.81496e8	0.560085
$358$	0	0
$359$	5.87418e8	0.670065	0.335032	−	0.942207i	$-0.391253\pi$
$359$	5.87418e8	0.670065	0.335032	+	0.942207i	$0.391253\pi$
$360$	0	0
$361$	−6.71535e8	−0.751266
$362$	0	0
$363$	−9.43364e8	−1.03516
$364$	0	0
$365$	−7.32563e8	−0.788534
$366$	0	0
$367$	7.02558e8	0.741910	0.370955	−	0.928651i	$-0.379030\pi$
$367$	7.02558e8	0.741910	0.370955	+	0.928651i	$0.379030\pi$
$368$	0	0
$369$	−3.60506e8	−0.373526
$370$	0	0
$371$	1.17335e7	0.0119294
$372$	0	0
$373$	−6.39508e8	−0.638065	−0.319033	−	0.947744i	$-0.603358\pi$
$373$	−6.39508e8	−0.638065	−0.319033	+	0.947744i	$0.603358\pi$
$374$	0	0
$375$	−1.21692e9	−1.19166
$376$	0	0
$377$	−6.86586e8	−0.659934
$378$	0	0
$379$	3.55496e8	0.335426	0.167713	−	0.985836i	$-0.446362\pi$
$379$	3.55496e8	0.335426	0.167713	+	0.985836i	$0.446362\pi$
$380$	0	0
$381$	−2.08800e9	−1.93416
$382$	0	0
$383$	1.73345e9	1.57658	0.788290	−	0.615304i	$-0.210967\pi$
$383$	1.73345e9	1.57658	0.788290	+	0.615304i	$0.210967\pi$
$384$	0	0
$385$	8.32939e7	0.0743876
$386$	0	0
$387$	8.11944e7	0.0712095
$388$	0	0
$389$	1.70118e9	1.46530	0.732650	−	0.680605i	$-0.238283\pi$
$389$	1.70118e9	1.46530	0.732650	+	0.680605i	$0.238283\pi$
$390$	0	0
$391$	−1.96886e9	−1.66570
$392$	0	0
$393$	6.84648e8	0.568975
$394$	0	0
$395$	1.70442e9	1.39151
$396$	0	0
$397$	1.44068e8	0.115558	0.0577792	−	0.998329i	$-0.481598\pi$
$397$	1.44068e8	0.115558	0.0577792	+	0.998329i	$0.481598\pi$
$398$	0	0
$399$	2.68025e8	0.211237
$400$	0	0
$401$	−2.79743e8	−0.216648	−0.108324	−	0.994116i	$-0.534548\pi$
$401$	−2.79743e8	−0.216648	−0.108324	+	0.994116i	$0.534548\pi$
$402$	0	0
$403$	4.30944e8	0.327984
$404$	0	0
$405$	−1.13421e9	−0.848401
$406$	0	0
$407$	−5.84500e8	−0.429738
$408$	0	0
$409$	−1.00801e8	−0.0728507	−0.0364254	−	0.999336i	$-0.511597\pi$
$409$	−1.00801e8	−0.0728507	−0.0364254	+	0.999336i	$0.511597\pi$
$410$	0	0
$411$	1.75905e8	0.124978
$412$	0	0
$413$	−1.52058e8	−0.106214
$414$	0	0
$415$	−3.58034e8	−0.245899
$416$	0	0
$417$	−5.05808e8	−0.341594
$418$	0	0
$419$	1.27396e9	0.846071	0.423036	−	0.906113i	$-0.360965\pi$
$419$	1.27396e9	0.846071	0.423036	+	0.906113i	$0.360965\pi$
$420$	0	0
$421$	7.46709e8	0.487713	0.243856	−	0.969811i	$-0.421587\pi$
$421$	7.46709e8	0.487713	0.243856	+	0.969811i	$0.421587\pi$
$422$	0	0
$423$	5.65391e8	0.363210
$424$	0	0
$425$	−1.02963e9	−0.650611
$426$	0	0
$427$	−3.91167e8	−0.243145
$428$	0	0
$429$	4.12009e8	0.251945
$430$	0	0
$431$	−3.22414e7	−0.0193974	−0.00969871	−	0.999953i	$-0.503087\pi$
$431$	−3.22414e7	−0.0193974	−0.00969871	+	0.999953i	$0.503087\pi$
$432$	0	0
$433$	2.64567e9	1.56613	0.783066	−	0.621938i	$-0.213655\pi$
$433$	2.64567e9	1.56613	0.783066	+	0.621938i	$0.213655\pi$
$434$	0	0
$435$	−1.11137e9	−0.647362
$436$	0	0
$437$	−1.09597e9	−0.628222
$438$	0	0
$439$	1.43248e8	0.0808095	0.0404047	−	0.999183i	$-0.487135\pi$
$439$	1.43248e8	0.0808095	0.0404047	+	0.999183i	$0.487135\pi$
$440$	0	0
$441$	6.58040e7	0.0365357
$442$	0	0
$443$	3.37159e9	1.84256	0.921279	−	0.388903i	$-0.127146\pi$
$443$	3.37159e9	1.84256	0.921279	+	0.388903i	$0.127146\pi$
$444$	0	0
$445$	1.10827e9	0.596193
$446$	0	0
$447$	2.94438e9	1.55926
$448$	0	0
$449$	3.50835e9	1.82911	0.914557	−	0.404456i	$-0.132539\pi$
$449$	3.50835e9	1.82911	0.914557	+	0.404456i	$0.132539\pi$
$450$	0	0
$451$	−7.85676e8	−0.403297
$452$	0	0
$453$	−1.22529e9	−0.619294
$454$	0	0
$455$	4.40712e8	0.219338
$456$	0	0
$457$	2.16582e8	0.106149	0.0530745	−	0.998591i	$-0.483098\pi$
$457$	2.16582e8	0.106149	0.0530745	+	0.998591i	$0.483098\pi$
$458$	0	0
$459$	−2.28490e9	−1.10286
$460$	0	0
$461$	2.47502e8	0.117659	0.0588296	−	0.998268i	$-0.481263\pi$
$461$	2.47502e8	0.117659	0.0588296	+	0.998268i	$0.481263\pi$
$462$	0	0
$463$	−3.15378e9	−1.47672	−0.738359	−	0.674408i	$-0.764399\pi$
$463$	−3.15378e9	−1.47672	−0.738359	+	0.674408i	$0.764399\pi$
$464$	0	0
$465$	6.97566e8	0.321736
$466$	0	0
$467$	−8.44807e8	−0.383838	−0.191919	−	0.981411i	$-0.561471\pi$
$467$	−8.44807e8	−0.383838	−0.191919	+	0.981411i	$0.561471\pi$
$468$	0	0
$469$	−1.48151e9	−0.663133
$470$	0	0
$471$	−1.10568e9	−0.487593
$472$	0	0
$473$	1.76952e8	0.0768852
$474$	0	0
$475$	−5.73146e8	−0.245379
$476$	0	0
$477$	1.91337e7	0.00807205
$478$	0	0
$479$	1.97235e9	0.819992	0.409996	−	0.912087i	$-0.365530\pi$
$479$	1.97235e9	0.819992	0.409996	+	0.912087i	$0.365530\pi$
$480$	0	0
$481$	−3.09262e9	−1.26712
$482$	0	0
$483$	−1.32118e9	−0.533517
$484$	0	0
$485$	−3.42765e9	−1.36427
$486$	0	0
$487$	−2.22754e9	−0.873927	−0.436963	−	0.899479i	$-0.643946\pi$
$487$	−2.22754e9	−0.873927	−0.436963	+	0.899479i	$0.643946\pi$
$488$	0	0
$489$	3.16526e9	1.22413
$490$	0	0
$491$	−4.30359e9	−1.64076	−0.820381	−	0.571817i	$-0.806239\pi$
$491$	−4.30359e9	−1.64076	−0.820381	+	0.571817i	$0.806239\pi$
$492$	0	0
$493$	−2.85155e9	−1.07181
$494$	0	0
$495$	1.35826e8	0.0503344
$496$	0	0
$497$	8.74194e8	0.319419
$498$	0	0
$499$	−1.57110e9	−0.566045	−0.283022	−	0.959113i	$-0.591337\pi$
$499$	−1.57110e9	−0.566045	−0.283022	+	0.959113i	$0.591337\pi$
$500$	0	0
$501$	3.60914e9	1.28225
$502$	0	0
$503$	2.77956e9	0.973842	0.486921	−	0.873446i	$-0.338120\pi$
$503$	2.77956e9	0.973842	0.486921	+	0.873446i	$0.338120\pi$
$504$	0	0
$505$	2.36519e9	0.817234
$506$	0	0
$507$	−1.10840e9	−0.377720
$508$	0	0
$509$	−2.74393e9	−0.922277	−0.461138	−	0.887328i	$-0.652559\pi$
$509$	−2.74393e9	−0.922277	−0.461138	+	0.887328i	$0.652559\pi$
$510$	0	0
$511$	−1.26129e9	−0.418159
$512$	0	0
$513$	−1.27189e9	−0.415947
$514$	0	0
$515$	3.19788e9	1.03166
$516$	0	0
$517$	1.23219e9	0.392159
$518$	0	0
$519$	−1.54296e9	−0.484471
$520$	0	0
$521$	3.32895e9	1.03128	0.515639	−	0.856806i	$-0.327555\pi$
$521$	3.32895e9	1.03128	0.515639	+	0.856806i	$0.327555\pi$
$522$	0	0
$523$	1.70843e9	0.522205	0.261102	−	0.965311i	$-0.415914\pi$
$523$	1.70843e9	0.522205	0.261102	+	0.965311i	$0.415914\pi$
$524$	0	0
$525$	−6.90923e8	−0.208388
$526$	0	0
$527$	1.78981e9	0.532685
$528$	0	0
$529$	1.99756e9	0.586686
$530$	0	0
$531$	−2.47958e8	−0.0718700
$532$	0	0
$533$	−4.15705e9	−1.18916
$534$	0	0
$535$	2.07730e9	0.586490
$536$	0	0
$537$	4.70863e9	1.31215
$538$	0	0
$539$	1.43411e8	0.0394477
$540$	0	0
$541$	−3.29318e9	−0.894181	−0.447090	−	0.894489i	$-0.647540\pi$
$541$	−3.29318e9	−0.894181	−0.447090	+	0.894489i	$0.647540\pi$
$542$	0	0
$543$	2.41552e9	0.647456
$544$	0	0
$545$	1.49773e9	0.396319
$546$	0	0
$547$	−2.11523e9	−0.552589	−0.276294	−	0.961073i	$-0.589107\pi$
$547$	−2.11523e9	−0.552589	−0.276294	+	0.961073i	$0.589107\pi$
$548$	0	0
$549$	−6.37870e8	−0.164524
$550$	0	0
$551$	−1.58732e9	−0.404235
$552$	0	0
$553$	2.93458e9	0.737919
$554$	0	0
$555$	−5.00600e9	−1.24298
$556$	0	0
$557$	5.07633e9	1.24468	0.622339	−	0.782748i	$-0.286183\pi$
$557$	5.07633e9	1.24468	0.622339	+	0.782748i	$0.286183\pi$
$558$	0	0
$559$	9.36264e8	0.226703
$560$	0	0
$561$	1.71117e9	0.409188
$562$	0	0
$563$	3.94053e9	0.930626	0.465313	−	0.885146i	$-0.345942\pi$
$563$	3.94053e9	0.930626	0.465313	+	0.885146i	$0.345942\pi$
$564$	0	0
$565$	5.04383e9	1.17650
$566$	0	0
$567$	−1.95283e9	−0.449907
$568$	0	0
$569$	−4.13724e9	−0.941495	−0.470748	−	0.882268i	$-0.656016\pi$
$569$	−4.13724e9	−0.941495	−0.470748	+	0.882268i	$0.656016\pi$
$570$	0	0
$571$	−3.29082e9	−0.739737	−0.369868	−	0.929084i	$-0.620597\pi$
$571$	−3.29082e9	−0.739737	−0.369868	+	0.929084i	$0.620597\pi$
$572$	0	0
$573$	−2.04492e9	−0.454082
$574$	0	0
$575$	2.82522e9	0.619748
$576$	0	0
$577$	4.39404e9	0.952245	0.476122	−	0.879379i	$-0.342042\pi$
$577$	4.39404e9	0.952245	0.476122	+	0.879379i	$0.342042\pi$
$578$	0	0
$579$	−8.67161e8	−0.185663
$580$	0	0
$581$	−6.16445e8	−0.130400
$582$	0	0
$583$	4.16993e7	0.00871543
$584$	0	0
$585$	7.18662e8	0.148415
$586$	0	0
$587$	5.42998e8	0.110806	0.0554032	−	0.998464i	$-0.482356\pi$
$587$	5.42998e8	0.110806	0.0554032	+	0.998464i	$0.482356\pi$
$588$	0	0
$589$	9.96299e8	0.200903
$590$	0	0
$591$	−1.11572e9	−0.222331
$592$	0	0
$593$	−8.24702e9	−1.62407	−0.812037	−	0.583606i	$-0.801641\pi$
$593$	−8.24702e9	−1.62407	−0.812037	+	0.583606i	$0.801641\pi$
$594$	0	0
$595$	1.83038e9	0.356231
$596$	0	0
$597$	1.19756e9	0.230349
$598$	0	0
$599$	−3.49160e9	−0.663789	−0.331895	−	0.943316i	$-0.607688\pi$
$599$	−3.49160e9	−0.663789	−0.331895	+	0.943316i	$0.607688\pi$
$600$	0	0
$601$	7.27920e9	1.36780	0.683900	−	0.729575i	$-0.260282\pi$
$601$	7.27920e9	1.36780	0.683900	+	0.729575i	$0.260282\pi$
$602$	0	0
$603$	−2.41588e9	−0.448709
$604$	0	0
$605$	−3.58614e9	−0.658391
$606$	0	0
$607$	8.02348e7	0.0145614	0.00728068	−	0.999973i	$-0.497682\pi$
$607$	8.02348e7	0.0145614	0.00728068	+	0.999973i	$0.497682\pi$
$608$	0	0
$609$	−1.91350e9	−0.343296
$610$	0	0
$611$	6.51960e9	1.15632
$612$	0	0
$613$	−7.90903e9	−1.38679	−0.693397	−	0.720556i	$-0.743887\pi$
$613$	−7.90903e9	−1.38679	−0.693397	+	0.720556i	$0.743887\pi$
$614$	0	0
$615$	−6.72898e9	−1.16650
$616$	0	0
$617$	−5.23704e9	−0.897611	−0.448805	−	0.893630i	$-0.648150\pi$
$617$	−5.23704e9	−0.897611	−0.448805	+	0.893630i	$0.648150\pi$
$618$	0	0
$619$	−1.14826e9	−0.194591	−0.0972953	−	0.995256i	$-0.531019\pi$
$619$	−1.14826e9	−0.194591	−0.0972953	+	0.995256i	$0.531019\pi$
$620$	0	0
$621$	6.26955e9	1.05055
$622$	0	0
$623$	1.90817e9	0.316161
$624$	0	0
$625$	−1.62308e9	−0.265925
$626$	0	0
$627$	9.52523e8	0.154326
$628$	0	0
$629$	−1.28444e10	−2.05795
$630$	0	0
$631$	6.33937e9	1.00448	0.502242	−	0.864727i	$-0.332509\pi$
$631$	6.33937e9	1.00448	0.502242	+	0.864727i	$0.332509\pi$
$632$	0	0
$633$	−7.85157e9	−1.23039
$634$	0	0
$635$	−7.93742e9	−1.23019
$636$	0	0
$637$	7.58795e8	0.116315
$638$	0	0
$639$	1.42553e9	0.216135
$640$	0	0
$641$	−1.04479e10	−1.56684	−0.783422	−	0.621491i	$-0.786527\pi$
$641$	−1.04479e10	−1.56684	−0.783422	+	0.621491i	$0.786527\pi$
$642$	0	0
$643$	5.52038e9	0.818899	0.409450	−	0.912333i	$-0.365721\pi$
$643$	5.52038e9	0.818899	0.409450	+	0.912333i	$0.365721\pi$
$644$	0	0
$645$	1.51552e9	0.222384
$646$	0	0
$647$	−1.08730e9	−0.157829	−0.0789143	−	0.996881i	$-0.525145\pi$
$647$	−1.08730e9	−0.157829	−0.0789143	+	0.996881i	$0.525145\pi$
$648$	0	0
$649$	−5.40392e8	−0.0775983
$650$	0	0
$651$	1.20103e9	0.170617
$652$	0	0
$653$	1.07697e10	1.51359	0.756795	−	0.653652i	$-0.226764\pi$
$653$	1.07697e10	1.51359	0.756795	+	0.653652i	$0.226764\pi$
$654$	0	0
$655$	2.60265e9	0.361886
$656$	0	0
$657$	−2.05676e9	−0.282947
$658$	0	0
$659$	1.23205e10	1.67699	0.838495	−	0.544910i	$-0.183436\pi$
$659$	1.23205e10	1.67699	0.838495	+	0.544910i	$0.183436\pi$
$660$	0	0
$661$	−4.43102e9	−0.596759	−0.298379	−	0.954447i	$-0.596446\pi$
$661$	−4.43102e9	−0.596759	−0.298379	+	0.954447i	$0.596446\pi$
$662$	0	0
$663$	9.05388e9	1.20653
$664$	0	0
$665$	1.01888e9	0.134353
$666$	0	0
$667$	7.82442e9	1.02097
$668$	0	0
$669$	9.72523e9	1.25576
$670$	0	0
$671$	−1.39015e9	−0.177637
$672$	0	0
$673$	−2.00633e9	−0.253717	−0.126858	−	0.991921i	$-0.540489\pi$
$673$	−2.00633e9	−0.253717	−0.126858	+	0.991921i	$0.540489\pi$
$674$	0	0
$675$	3.27871e9	0.410337
$676$	0	0
$677$	−7.63334e9	−0.945485	−0.472742	−	0.881201i	$-0.656736\pi$
$677$	−7.63334e9	−0.945485	−0.472742	+	0.881201i	$0.656736\pi$
$678$	0	0
$679$	−5.90155e9	−0.723472
$680$	0	0
$681$	1.66436e10	2.01945
$682$	0	0
$683$	1.54524e9	0.185577	0.0927885	−	0.995686i	$-0.470422\pi$
$683$	1.54524e9	0.185577	0.0927885	+	0.995686i	$0.470422\pi$
$684$	0	0
$685$	6.68693e8	0.0794895
$686$	0	0
$687$	1.23332e10	1.45120
$688$	0	0
$689$	2.20633e8	0.0256982
$690$	0	0
$691$	1.22484e10	1.41223	0.706113	−	0.708099i	$-0.250447\pi$
$691$	1.22484e10	1.41223	0.706113	+	0.708099i	$0.250447\pi$
$692$	0	0
$693$	2.33858e8	0.0266923
$694$	0	0
$695$	−1.92280e9	−0.217264
$696$	0	0
$697$	−1.72652e10	−1.93133
$698$	0	0
$699$	−1.66019e10	−1.83860
$700$	0	0
$701$	−7.41592e9	−0.813115	−0.406558	−	0.913625i	$-0.633271\pi$
$701$	−7.41592e9	−0.813115	−0.406558	+	0.913625i	$0.633271\pi$
$702$	0	0
$703$	−7.14983e9	−0.776161
$704$	0	0
$705$	1.05532e10	1.13429
$706$	0	0
$707$	4.07226e9	0.433379
$708$	0	0
$709$	3.86406e9	0.407176	0.203588	−	0.979057i	$-0.434740\pi$
$709$	3.86406e9	0.407176	0.203588	+	0.979057i	$0.434740\pi$
$710$	0	0
$711$	4.78538e9	0.499313
$712$	0	0
$713$	−4.91109e9	−0.507416
$714$	0	0
$715$	1.56623e9	0.160245
$716$	0	0
$717$	−4.37879e9	−0.443647
$718$	0	0
$719$	−1.63270e10	−1.63816	−0.819079	−	0.573680i	$-0.805515\pi$
$719$	−1.63270e10	−1.63816	−0.819079	+	0.573680i	$0.805515\pi$
$720$	0	0
$721$	5.50594e9	0.547089
$722$	0	0
$723$	−5.47342e9	−0.538611
$724$	0	0
$725$	4.09184e9	0.398782
$726$	0	0
$727$	−1.45389e10	−1.40333	−0.701667	−	0.712505i	$-0.747561\pi$
$727$	−1.45389e10	−1.40333	−0.701667	+	0.712505i	$0.747561\pi$
$728$	0	0
$729$	6.59172e9	0.630163
$730$	0	0
$731$	3.88853e9	0.368192
$732$	0	0
$733$	1.84625e10	1.73152	0.865759	−	0.500460i	$-0.166836\pi$
$733$	1.84625e10	1.73152	0.865759	+	0.500460i	$0.166836\pi$
$734$	0	0
$735$	1.22826e9	0.114099
$736$	0	0
$737$	−5.26509e9	−0.484473
$738$	0	0
$739$	−1.54075e9	−0.140436	−0.0702178	−	0.997532i	$-0.522369\pi$
$739$	−1.54075e9	−0.140436	−0.0702178	+	0.997532i	$0.522369\pi$
$740$	0	0
$741$	5.03984e9	0.455044
$742$	0	0
$743$	−7.42658e9	−0.664245	−0.332122	−	0.943236i	$-0.607765\pi$
$743$	−7.42658e9	−0.664245	−0.332122	+	0.943236i	$0.607765\pi$
$744$	0	0
$745$	1.11929e10	0.991734
$746$	0	0
$747$	−1.00523e9	−0.0882352
$748$	0	0
$749$	3.57659e9	0.311016
$750$	0	0
$751$	−1.23061e10	−1.06018	−0.530090	−	0.847941i	$-0.677842\pi$
$751$	−1.23061e10	−1.06018	−0.530090	+	0.847941i	$0.677842\pi$
$752$	0	0
$753$	−1.43935e10	−1.22852
$754$	0	0
$755$	−4.65789e9	−0.393890
$756$	0	0
$757$	−2.23022e9	−0.186859	−0.0934293	−	0.995626i	$-0.529783\pi$
$757$	−2.23022e9	−0.186859	−0.0934293	+	0.995626i	$0.529783\pi$
$758$	0	0
$759$	−4.69530e9	−0.389778
$760$	0	0
$761$	−5.38368e9	−0.442826	−0.221413	−	0.975180i	$-0.571067\pi$
$761$	−5.38368e9	−0.442826	−0.221413	+	0.975180i	$0.571067\pi$
$762$	0	0
$763$	2.57871e9	0.210168
$764$	0	0
$765$	2.98477e9	0.241044
$766$	0	0
$767$	−2.85924e9	−0.228806
$768$	0	0
$769$	−1.17224e10	−0.929550	−0.464775	−	0.885429i	$-0.653865\pi$
$769$	−1.17224e10	−0.929550	−0.464775	+	0.885429i	$0.653865\pi$
$770$	0	0
$771$	−2.48813e9	−0.195516
$772$	0	0
$773$	1.49447e10	1.16375	0.581875	−	0.813278i	$-0.302319\pi$
$773$	1.49447e10	1.16375	0.581875	+	0.813278i	$0.302319\pi$
$774$	0	0
$775$	−2.56829e9	−0.198193
$776$	0	0
$777$	−8.61907e9	−0.659154
$778$	0	0
$779$	−9.61068e9	−0.728405
$780$	0	0
$781$	3.10676e9	0.233362
$782$	0	0
$783$	9.08035e9	0.675984
$784$	0	0
$785$	−4.20319e9	−0.310124
$786$	0	0
$787$	−1.68671e10	−1.23347	−0.616736	−	0.787170i	$-0.711546\pi$
$787$	−1.68671e10	−1.23347	−0.616736	+	0.787170i	$0.711546\pi$
$788$	0	0
$789$	−6.91927e8	−0.0501523
$790$	0	0
$791$	8.68420e9	0.623896
$792$	0	0
$793$	−7.35536e9	−0.523779
$794$	0	0
$795$	3.57137e8	0.0252087
$796$	0	0
$797$	−1.57820e9	−0.110422	−0.0552112	−	0.998475i	$-0.517583\pi$
$797$	−1.57820e9	−0.110422	−0.0552112	+	0.998475i	$0.517583\pi$
$798$	0	0
$799$	2.70774e10	1.87799
$800$	0	0
$801$	3.11162e9	0.213930
$802$	0	0
$803$	−4.48245e9	−0.305500
$804$	0	0
$805$	−5.02240e9	−0.339333
$806$	0	0
$807$	−2.44285e10	−1.63621
$808$	0	0
$809$	1.98655e10	1.31910	0.659552	−	0.751659i	$-0.270746\pi$
$809$	1.98655e10	1.31910	0.659552	+	0.751659i	$0.270746\pi$
$810$	0	0
$811$	1.87325e10	1.23317	0.616583	−	0.787290i	$-0.288516\pi$
$811$	1.87325e10	1.23317	0.616583	+	0.787290i	$0.288516\pi$
$812$	0	0
$813$	−8.63033e9	−0.563262
$814$	0	0
$815$	1.20326e10	0.778587
$816$	0	0
$817$	2.16455e9	0.138864
$818$	0	0
$819$	1.23735e9	0.0787047
$820$	0	0
$821$	2.71441e10	1.71189	0.855943	−	0.517070i	$-0.172977\pi$
$821$	2.71441e10	1.71189	0.855943	+	0.517070i	$0.172977\pi$
$822$	0	0
$823$	−4.27707e9	−0.267453	−0.133726	−	0.991018i	$-0.542694\pi$
$823$	−4.27707e9	−0.267453	−0.133726	+	0.991018i	$0.542694\pi$
$824$	0	0
$825$	−2.45544e9	−0.152244
$826$	0	0
$827$	−2.17659e10	−1.33816	−0.669078	−	0.743193i	$-0.733311\pi$
$827$	−2.17659e10	−1.33816	−0.669078	+	0.743193i	$0.733311\pi$
$828$	0	0
$829$	2.85448e10	1.74015	0.870073	−	0.492923i	$-0.164072\pi$
$829$	2.85448e10	1.74015	0.870073	+	0.492923i	$0.164072\pi$
$830$	0	0
$831$	3.28436e10	1.98539
$832$	0	0
$833$	3.15145e9	0.188909
$834$	0	0
$835$	1.37199e10	0.815548
$836$	0	0
$837$	−5.69939e9	−0.335961
$838$	0	0
$839$	1.42629e10	0.833761	0.416881	−	0.908961i	$-0.363123\pi$
$839$	1.42629e10	0.833761	0.416881	+	0.908961i	$0.363123\pi$
$840$	0	0
$841$	−5.91757e9	−0.343050
$842$	0	0
$843$	−2.78239e10	−1.59963
$844$	0	0
$845$	−4.21353e9	−0.240241
$846$	0	0
$847$	−6.17444e9	−0.349145
$848$	0	0
$849$	−9.08919e9	−0.509739
$850$	0	0
$851$	3.52438e10	1.96033
$852$	0	0
$853$	1.80907e10	0.998007	0.499004	−	0.866600i	$-0.333699\pi$
$853$	1.80907e10	0.998007	0.499004	+	0.866600i	$0.333699\pi$
$854$	0	0
$855$	1.66147e9	0.0909101
$856$	0	0
$857$	8.64155e9	0.468985	0.234492	−	0.972118i	$-0.424657\pi$
$857$	8.64155e9	0.468985	0.234492	+	0.972118i	$0.424657\pi$
$858$	0	0
$859$	−3.40755e10	−1.83428	−0.917141	−	0.398564i	$-0.869509\pi$
$859$	−3.40755e10	−1.83428	−0.917141	+	0.398564i	$0.869509\pi$
$860$	0	0
$861$	−1.15856e10	−0.618597
$862$	0	0
$863$	2.86099e10	1.51523	0.757614	−	0.652703i	$-0.226365\pi$
$863$	2.86099e10	1.51523	0.757614	+	0.652703i	$0.226365\pi$
$864$	0	0
$865$	−5.86546e9	−0.308138
$866$	0	0
$867$	1.60989e10	0.838939
$868$	0	0
$869$	1.04291e10	0.539110
$870$	0	0
$871$	−2.78578e10	−1.42851
$872$	0	0
$873$	−9.62356e9	−0.489538
$874$	0	0
$875$	−7.96488e9	−0.401930
$876$	0	0
$877$	−2.50723e10	−1.25515	−0.627575	−	0.778556i	$-0.715953\pi$
$877$	−2.50723e10	−1.25515	−0.627575	+	0.778556i	$0.715953\pi$
$878$	0	0
$879$	−6.24295e9	−0.310048
$880$	0	0
$881$	7.00965e9	0.345367	0.172683	−	0.984977i	$-0.444756\pi$
$881$	7.00965e9	0.345367	0.172683	+	0.984977i	$0.444756\pi$
$882$	0	0
$883$	−1.54916e10	−0.757241	−0.378620	−	0.925552i	$-0.623601\pi$
$883$	−1.54916e10	−0.757241	−0.378620	+	0.925552i	$0.623601\pi$
$884$	0	0
$885$	−4.62823e9	−0.224447
$886$	0	0
$887$	−2.99298e10	−1.44003	−0.720015	−	0.693958i	$-0.755865\pi$
$887$	−2.99298e10	−1.44003	−0.720015	+	0.693958i	$0.755865\pi$
$888$	0	0
$889$	−1.36662e10	−0.652368
$890$	0	0
$891$	−6.94007e9	−0.328694
$892$	0	0
$893$	1.50727e10	0.708288
$894$	0	0
$895$	1.78996e10	0.834569
$896$	0	0
$897$	−2.48430e10	−1.14929
$898$	0	0
$899$	−7.11285e9	−0.326501
$900$	0	0
$901$	9.16341e8	0.0417369
$902$	0	0
$903$	2.60935e9	0.117930
$904$	0	0
$905$	9.18245e9	0.411802
$906$	0	0
$907$	2.76012e9	0.122830	0.0614148	−	0.998112i	$-0.480439\pi$
$907$	2.76012e9	0.122830	0.0614148	+	0.998112i	$0.480439\pi$
$908$	0	0
$909$	6.64057e9	0.293246
$910$	0	0
$911$	−8.06506e9	−0.353422	−0.176711	−	0.984263i	$-0.556546\pi$
$911$	−8.06506e9	−0.353422	−0.176711	+	0.984263i	$0.556546\pi$
$912$	0	0
$913$	−2.19076e9	−0.0952679
$914$	0	0
$915$	−1.19061e10	−0.513801
$916$	0	0
$917$	4.48111e9	0.191908
$918$	0	0
$919$	−3.14509e10	−1.33668	−0.668342	−	0.743854i	$-0.732996\pi$
$919$	−3.14509e10	−1.33668	−0.668342	+	0.743854i	$0.732996\pi$
$920$	0	0
$921$	−2.08910e10	−0.881150
$922$	0	0
$923$	1.64380e10	0.688088
$924$	0	0
$925$	1.84311e10	0.765692
$926$	0	0
$927$	8.97845e9	0.370188
$928$	0	0
$929$	−1.61192e10	−0.659610	−0.329805	−	0.944049i	$-0.606983\pi$
$929$	−1.61192e10	−0.659610	−0.329805	+	0.944049i	$0.606983\pi$
$930$	0	0
$931$	1.75426e9	0.0712475
$932$	0	0
$933$	−1.68633e10	−0.679764
$934$	0	0
$935$	6.50491e9	0.260256
$936$	0	0
$937$	−1.26173e10	−0.501048	−0.250524	−	0.968110i	$-0.580603\pi$
$937$	−1.26173e10	−0.501048	−0.250524	+	0.968110i	$0.580603\pi$
$938$	0	0
$939$	1.46870e10	0.578898
$940$	0	0
$941$	3.26441e10	1.27715	0.638574	−	0.769560i	$-0.279525\pi$
$941$	3.26441e10	1.27715	0.638574	+	0.769560i	$0.279525\pi$
$942$	0	0
$943$	4.73742e10	1.83972
$944$	0	0
$945$	−5.82858e9	−0.224673
$946$	0	0
$947$	2.95307e10	1.12992	0.564961	−	0.825117i	$-0.308891\pi$
$947$	2.95307e10	1.12992	0.564961	+	0.825117i	$0.308891\pi$
$948$	0	0
$949$	−2.37168e10	−0.900793
$950$	0	0
$951$	4.21954e10	1.59086
$952$	0	0
$953$	−4.15195e10	−1.55391	−0.776957	−	0.629553i	$-0.783238\pi$
$953$	−4.15195e10	−1.55391	−0.776957	+	0.629553i	$0.783238\pi$
$954$	0	0
$955$	−7.77364e9	−0.288810
$956$	0	0
$957$	−6.80032e9	−0.250806
$958$	0	0
$959$	1.15132e9	0.0421533
$960$	0	0
$961$	−2.30481e10	−0.837730
$962$	0	0
$963$	5.83228e9	0.210449
$964$	0	0
$965$	−3.29647e9	−0.118087
$966$	0	0
$967$	−8.25883e9	−0.293715	−0.146857	−	0.989158i	$-0.546916\pi$
$967$	−8.25883e9	−0.293715	−0.146857	+	0.989158i	$0.546916\pi$
$968$	0	0
$969$	2.09317e10	0.739044
$970$	0	0
$971$	−3.17512e10	−1.11299	−0.556496	−	0.830850i	$-0.687855\pi$
$971$	−3.17512e10	−1.11299	−0.556496	+	0.830850i	$0.687855\pi$
$972$	0	0
$973$	−3.31058e9	−0.115215
$974$	0	0
$975$	−1.29919e10	−0.448906
$976$	0	0
$977$	3.97626e10	1.36409	0.682046	−	0.731309i	$-0.261090\pi$
$977$	3.97626e10	1.36409	0.682046	+	0.731309i	$0.261090\pi$
$978$	0	0
$979$	6.78136e9	0.230982
$980$	0	0
$981$	4.20506e9	0.142210
$982$	0	0
$983$	−3.22059e10	−1.08143	−0.540714	−	0.841206i	$-0.681846\pi$
$983$	−3.22059e10	−1.08143	−0.540714	+	0.841206i	$0.681846\pi$
$984$	0	0
$985$	−4.24136e9	−0.141410
$986$	0	0
$987$	1.81700e10	0.601513
$988$	0	0
$989$	−1.06698e10	−0.350726
$990$	0	0
$991$	2.58333e10	0.843183	0.421592	−	0.906786i	$-0.361472\pi$
$991$	2.58333e10	0.843183	0.421592	+	0.906786i	$0.361472\pi$
$992$	0	0
$993$	1.21264e10	0.393016
$994$	0	0
$995$	4.55244e9	0.146509
$996$	0	0
$997$	3.40734e10	1.08889	0.544444	−	0.838797i	$-0.316741\pi$
$997$	3.40734e10	1.08889	0.544444	+	0.838797i	$0.316741\pi$
$998$	0	0
$999$	4.09010e10	1.29794

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	28.8.a.a.1.2	✓	2
3.2	odd	2		252.8.a.e.1.1		2
4.3	odd	2		112.8.a.i.1.1		2
7.2	even	3		196.8.e.d.165.1		4
7.3	odd	6		196.8.e.a.177.2		4
7.4	even	3		196.8.e.d.177.1		4
7.5	odd	6		196.8.e.a.165.2		4
7.6	odd	2		196.8.a.b.1.1		2
8.3	odd	2		448.8.a.n.1.2		2
8.5	even	2		448.8.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.a.a.1.2	✓	2	1.1	even	1	trivial
112.8.a.i.1.1		2	4.3	odd	2
196.8.a.b.1.1		2	7.6	odd	2
196.8.e.a.165.2		4	7.5	odd	6
196.8.e.a.177.2		4	7.3	odd	6
196.8.e.d.165.1		4	7.2	even	3
196.8.e.d.177.1		4	7.4	even	3
252.8.a.e.1.1		2	3.2	odd	2
448.8.a.n.1.2		2	8.3	odd	2
448.8.a.p.1.1		2	8.5	even	2

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:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	28.a (trivial)

\(f(q)\)	\(=\)	\(q+52.4054 q^{3} +199.216 q^{5} +343.000 q^{7} +559.325 q^{9} +1218.97 q^{11} +6449.65 q^{13} +10440.0 q^{15} +26786.9 q^{17} +14910.9 q^{19} +17975.0 q^{21} -73500.9 q^{23} -38437.9 q^{25} -85299.0 q^{27} -106453. q^{29} +66816.7 q^{31} +63880.8 q^{33} +68331.1 q^{35} -479502. q^{37} +337996. q^{39} -644539. q^{41} +145165. q^{43} +111426. q^{45} +1.01085e6 q^{47} +117649. q^{49} +1.40378e6 q^{51} +34208.5 q^{53} +242839. q^{55} +781414. q^{57} -443317. q^{59} -1.14043e6 q^{61} +191848. q^{63} +1.28487e6 q^{65} -4.31928e6 q^{67} -3.85185e6 q^{69} +2.54867e6 q^{71} -3.67723e6 q^{73} -2.01435e6 q^{75} +418108. q^{77} +8.55564e6 q^{79} -5.69337e6 q^{81} -1.79721e6 q^{83} +5.33639e6 q^{85} -5.57873e6 q^{87} +5.56317e6 q^{89} +2.21223e6 q^{91} +3.50155e6 q^{93} +2.97050e6 q^{95} -1.72057e7 q^{97} +681802. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 42 q^{5} + 686 q^{7} + 2782 q^{9} + 7428 q^{11} + 11830 q^{13} + 20880 q^{15} + 15792 q^{17} + 26614 q^{19} - 4802 q^{21} + 32640 q^{23} - 91846 q^{25} - 87668 q^{27} - 158016 q^{29} - 180740 q^{31}+ \cdots + 14482452 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 + 42 * q^5 + 686 * q^7 + 2782 * q^9 + 7428 * q^11 + 11830 * q^13 + 20880 * q^15 + 15792 * q^17 + 26614 * q^19 - 4802 * q^21 + 32640 * q^23 - 91846 * q^25 - 87668 * q^27 - 158016 * q^29 - 180740 * q^31 - 348432 * q^33 + 14406 * q^35 - 45824 * q^37 - 19288 * q^39 - 321720 * q^41 + 1023868 * q^43 - 238014 * q^45 + 1665972 * q^47 + 235298 * q^49 + 2133900 * q^51 - 410628 * q^53 - 733320 * q^55 + 4268 * q^57 + 1702134 * q^59 - 547526 * q^61 + 954226 * q^63 + 438996 * q^65 - 2590616 * q^67 - 10900176 * q^69 + 4129272 * q^71 - 8008868 * q^73 + 1532230 * q^75 + 2547804 * q^77 + 2470456 * q^79 - 10397042 * q^81 - 9900786 * q^83 + 7064964 * q^85 - 2154684 * q^87 + 15423492 * q^89 + 4057690 * q^91 + 19940648 * q^93 + 1130592 * q^95 - 17377472 * q^97 + 14482452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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