LMFDB - Embedded newform 224.8.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [224,8,Mod(1,224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(224, 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("224.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 224 = 2^{5} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	224.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:	$69.9742457084$
Analytic rank:	$1$
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} - 1809x^{3} + 6482x^{2} + 488753x + 1733184 $ x^5 - 2x^4 - 1809x^3 + 6482x^2 + 488753x + 1733184
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$35.7694$ of defining polynomial
Character	$\chi$	$=$	224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.5387 q^{3} +494.197 q^{5} +343.000 q^{7} +4461.57 q^{9} -8583.04 q^{11} +5316.37 q^{13} -40296.2 q^{15} -32820.0 q^{17} +1218.80 q^{19} -27967.8 q^{21} +68309.2 q^{23} +166105. q^{25} -185465. q^{27} +104109. q^{29} -32952.8 q^{31} +699850. q^{33} +169509. q^{35} -442903. q^{37} -433490. q^{39} -67469.0 q^{41} +98408.3 q^{43} +2.20489e6 q^{45} +297845. q^{47} +117649. q^{49} +2.67610e6 q^{51} -986996. q^{53} -4.24171e6 q^{55} -99379.4 q^{57} +653762. q^{59} -1.85164e6 q^{61} +1.53032e6 q^{63} +2.62733e6 q^{65} +1.68434e6 q^{67} -5.56984e6 q^{69} +2.80714e6 q^{71} +1.59400e6 q^{73} -1.35440e7 q^{75} -2.94398e6 q^{77} -4.37861e6 q^{79} +5.36516e6 q^{81} -7.35285e6 q^{83} -1.62196e7 q^{85} -8.48894e6 q^{87} +4.31510e6 q^{89} +1.82351e6 q^{91} +2.68693e6 q^{93} +602327. q^{95} -1.71223e7 q^{97} -3.82938e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 54 q^{3} + 84 q^{5} + 1715 q^{7} + 4133 q^{9} - 5324 q^{11} - 9880 q^{13} - 30128 q^{15} - 44322 q^{17} - 22898 q^{19} - 18522 q^{21} + 141016 q^{23} + 150679 q^{25} - 135972 q^{27} - 18998 q^{29}+ \cdots - 58451180 q^{99}+O(q^{100}) $ 5 * q - 54 * q^3 + 84 * q^5 + 1715 * q^7 + 4133 * q^9 - 5324 * q^11 - 9880 * q^13 - 30128 * q^15 - 44322 * q^17 - 22898 * q^19 - 18522 * q^21 + 141016 * q^23 + 150679 * q^25 - 135972 * q^27 - 18998 * q^29 + 167668 * q^31 + 830696 * q^33 + 28812 * q^35 - 434206 * q^37 - 1100728 * q^39 + 44278 * q^41 - 465076 * q^43 + 1823220 * q^45 - 7508 * q^47 + 588245 * q^49 - 65012 * q^51 - 1901986 * q^53 - 4619216 * q^55 - 3116556 * q^57 - 2342994 * q^59 - 553300 * q^61 + 1417619 * q^63 + 1749208 * q^65 - 1908152 * q^67 - 554944 * q^69 - 1721104 * q^71 + 1237634 * q^73 - 19901682 * q^75 - 1826132 * q^77 - 9787808 * q^79 + 4138481 * q^81 - 17717898 * q^83 - 13248184 * q^85 - 25003324 * q^87 + 7408138 * q^89 - 3388840 * q^91 + 13173560 * q^93 - 16659216 * q^95 - 17027554 * q^97 - 58451180 * 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
$3$	−81.5387	−1.74357	−0.871785	−	0.489888i	$-0.837038\pi$
$3$	−81.5387	−1.74357	−0.871785	+	0.489888i	$0.837038\pi$
$4$	0	0
$5$	494.197	1.76809	0.884046	−	0.467400i	$-0.154809\pi$
$5$	494.197	1.76809	0.884046	+	0.467400i	$0.154809\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	4461.57	2.04004
$10$	0	0
$11$	−8583.04	−1.94431	−0.972157	−	0.234329i	$-0.924711\pi$
$11$	−8583.04	−1.94431	−0.972157	+	0.234329i	$0.924711\pi$
$12$	0	0
$13$	5316.37	0.671140	0.335570	−	0.942015i	$-0.391071\pi$
$13$	5316.37	0.671140	0.335570	+	0.942015i	$0.391071\pi$
$14$	0	0
$15$	−40296.2	−3.08279
$16$	0	0
$17$	−32820.0	−1.62020	−0.810099	−	0.586294i	$-0.800586\pi$
$17$	−32820.0	−1.62020	−0.810099	+	0.586294i	$0.800586\pi$
$18$	0	0
$19$	1218.80	0.0407657	0.0203828	−	0.999792i	$-0.493511\pi$
$19$	1218.80	0.0407657	0.0203828	+	0.999792i	$0.493511\pi$
$20$	0	0
$21$	−27967.8	−0.659008
$22$	0	0
$23$	68309.2	1.17066	0.585331	−	0.810794i	$-0.300965\pi$
$23$	68309.2	1.17066	0.585331	+	0.810794i	$0.300965\pi$
$24$	0	0
$25$	166105.	2.12615
$26$	0	0
$27$	−185465.	−1.81338
$28$	0	0
$29$	104109.	0.792677	0.396339	−	0.918104i	$-0.370281\pi$
$29$	104109.	0.792677	0.396339	+	0.918104i	$0.370281\pi$
$30$	0	0
$31$	−32952.8	−0.198668	−0.0993338	−	0.995054i	$-0.531671\pi$
$31$	−32952.8	−0.198668	−0.0993338	+	0.995054i	$0.531671\pi$
$32$	0	0
$33$	699850.	3.39005
$34$	0	0
$35$	169509.	0.668276
$36$	0	0
$37$	−442903.	−1.43748	−0.718742	−	0.695277i	$-0.755282\pi$
$37$	−442903.	−1.43748	−0.718742	+	0.695277i	$0.755282\pi$
$38$	0	0
$39$	−433490.	−1.17018
$40$	0	0
$41$	−67469.0	−0.152884	−0.0764418	−	0.997074i	$-0.524356\pi$
$41$	−67469.0	−0.152884	−0.0764418	+	0.997074i	$0.524356\pi$
$42$	0	0
$43$	98408.3	0.188752	0.0943761	−	0.995537i	$-0.469914\pi$
$43$	98408.3	0.188752	0.0943761	+	0.995537i	$0.469914\pi$
$44$	0	0
$45$	2.20489e6	3.60698
$46$	0	0
$47$	297845.	0.418455	0.209227	−	0.977867i	$-0.432905\pi$
$47$	297845.	0.418455	0.209227	+	0.977867i	$0.432905\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	2.67610e6	2.82493
$52$	0	0
$53$	−986996.	−0.910647	−0.455323	−	0.890326i	$-0.650476\pi$
$53$	−986996.	−0.910647	−0.455323	+	0.890326i	$0.650476\pi$
$54$	0	0
$55$	−4.24171e6	−3.43773
$56$	0	0
$57$	−99379.4	−0.0710779
$58$	0	0
$59$	653762.	0.414417	0.207209	−	0.978297i	$-0.433562\pi$
$59$	653762.	0.414417	0.207209	+	0.978297i	$0.433562\pi$
$60$	0	0
$61$	−1.85164e6	−1.04448	−0.522242	−	0.852797i	$-0.674904\pi$
$61$	−1.85164e6	−1.04448	−0.522242	+	0.852797i	$0.674904\pi$
$62$	0	0
$63$	1.53032e6	0.771062
$64$	0	0
$65$	2.62733e6	1.18664
$66$	0	0
$67$	1.68434e6	0.684175	0.342087	−	0.939668i	$-0.388866\pi$
$67$	1.68434e6	0.684175	0.342087	+	0.939668i	$0.388866\pi$
$68$	0	0
$69$	−5.56984e6	−2.04113
$70$	0	0
$71$	2.80714e6	0.930806	0.465403	−	0.885099i	$-0.345909\pi$
$71$	2.80714e6	0.930806	0.465403	+	0.885099i	$0.345909\pi$
$72$	0	0
$73$	1.59400e6	0.479577	0.239788	−	0.970825i	$-0.422922\pi$
$73$	1.59400e6	0.479577	0.239788	+	0.970825i	$0.422922\pi$
$74$	0	0
$75$	−1.35440e7	−3.70709
$76$	0	0
$77$	−2.94398e6	−0.734882
$78$	0	0
$79$	−4.37861e6	−0.999174	−0.499587	−	0.866264i	$-0.666515\pi$
$79$	−4.37861e6	−0.999174	−0.499587	+	0.866264i	$0.666515\pi$
$80$	0	0
$81$	5.36516e6	1.12172
$82$	0	0
$83$	−7.35285e6	−1.41151	−0.705753	−	0.708458i	$-0.749391\pi$
$83$	−7.35285e6	−1.41151	−0.705753	+	0.708458i	$0.749391\pi$
$84$	0	0
$85$	−1.62196e7	−2.86466
$86$	0	0
$87$	−8.48894e6	−1.38209
$88$	0	0
$89$	4.31510e6	0.648822	0.324411	−	0.945916i	$-0.394834\pi$
$89$	4.31510e6	0.648822	0.324411	+	0.945916i	$0.394834\pi$
$90$	0	0
$91$	1.82351e6	0.253667
$92$	0	0
$93$	2.68693e6	0.346391
$94$	0	0
$95$	602327.	0.0720775
$96$	0	0
$97$	−1.71223e7	−1.90485	−0.952426	−	0.304769i	$-0.901421\pi$
$97$	−1.71223e7	−1.90485	−0.952426	+	0.304769i	$0.901421\pi$
$98$	0	0
$99$	−3.82938e7	−3.96648
$100$	0	0
$101$	5.06209e6	0.488883	0.244441	−	0.969664i	$-0.421395\pi$
$101$	5.06209e6	0.488883	0.244441	+	0.969664i	$0.421395\pi$
$102$	0	0
$103$	6.78295e6	0.611629	0.305815	−	0.952091i	$-0.401071\pi$
$103$	6.78295e6	0.611629	0.305815	+	0.952091i	$0.401071\pi$
$104$	0	0
$105$	−1.38216e7	−1.16519
$106$	0	0
$107$	−9.85591e6	−0.777774	−0.388887	−	0.921285i	$-0.627140\pi$
$107$	−9.85591e6	−0.777774	−0.388887	+	0.921285i	$0.627140\pi$
$108$	0	0
$109$	−1.47575e7	−1.09149	−0.545746	−	0.837951i	$-0.683754\pi$
$109$	−1.47575e7	−1.09149	−0.545746	+	0.837951i	$0.683754\pi$
$110$	0	0
$111$	3.61138e7	2.50635
$112$	0	0
$113$	1.44387e6	0.0941357	0.0470679	−	0.998892i	$-0.485012\pi$
$113$	1.44387e6	0.0941357	0.0470679	+	0.998892i	$0.485012\pi$
$114$	0	0
$115$	3.37582e7	2.06984
$116$	0	0
$117$	2.37193e7	1.36915
$118$	0	0
$119$	−1.12573e7	−0.612377
$120$	0	0
$121$	5.41813e7	2.78036
$122$	0	0
$123$	5.50134e6	0.266563
$124$	0	0
$125$	4.34796e7	1.99113
$126$	0	0
$127$	6.92406e6	0.299949	0.149975	−	0.988690i	$-0.452081\pi$
$127$	6.92406e6	0.299949	0.149975	+	0.988690i	$0.452081\pi$
$128$	0	0
$129$	−8.02408e6	−0.329103
$130$	0	0
$131$	−2.13864e7	−0.831167	−0.415583	−	0.909555i	$-0.636423\pi$
$131$	−2.13864e7	−0.831167	−0.415583	+	0.909555i	$0.636423\pi$
$132$	0	0
$133$	418048.	0.0154080
$134$	0	0
$135$	−9.16563e7	−3.20623
$136$	0	0
$137$	3.68253e7	1.22356	0.611779	−	0.791029i	$-0.290454\pi$
$137$	3.68253e7	1.22356	0.611779	+	0.791029i	$0.290454\pi$
$138$	0	0
$139$	−4.90069e7	−1.54777	−0.773883	−	0.633328i	$-0.781688\pi$
$139$	−4.90069e7	−1.54777	−0.773883	+	0.633328i	$0.781688\pi$
$140$	0	0
$141$	−2.42859e7	−0.729605
$142$	0	0
$143$	−4.56306e7	−1.30491
$144$	0	0
$145$	5.14504e7	1.40153
$146$	0	0
$147$	−9.59295e6	−0.249082
$148$	0	0
$149$	−4.38098e7	−1.08497	−0.542487	−	0.840064i	$-0.682517\pi$
$149$	−4.38098e7	−1.08497	−0.542487	+	0.840064i	$0.682517\pi$
$150$	0	0
$151$	−2.52691e7	−0.597270	−0.298635	−	0.954367i	$-0.596531\pi$
$151$	−2.52691e7	−0.597270	−0.298635	+	0.954367i	$0.596531\pi$
$152$	0	0
$153$	−1.46429e8	−3.30527
$154$	0	0
$155$	−1.62852e7	−0.351262
$156$	0	0
$157$	−4.99725e7	−1.03058	−0.515291	−	0.857015i	$-0.672316\pi$
$157$	−4.99725e7	−1.03058	−0.515291	+	0.857015i	$0.672316\pi$
$158$	0	0
$159$	8.04784e7	1.58778
$160$	0	0
$161$	2.34300e7	0.442469
$162$	0	0
$163$	−7.22419e7	−1.30657	−0.653285	−	0.757112i	$-0.726609\pi$
$163$	−7.22419e7	−1.30657	−0.653285	+	0.757112i	$0.726609\pi$
$164$	0	0
$165$	3.45863e8	5.99392
$166$	0	0
$167$	−6.45130e7	−1.07186	−0.535932	−	0.844261i	$-0.680040\pi$
$167$	−6.45130e7	−1.07186	−0.535932	+	0.844261i	$0.680040\pi$
$168$	0	0
$169$	−3.44847e7	−0.549571
$170$	0	0
$171$	5.43776e6	0.0831636
$172$	0	0
$173$	1.56056e7	0.229150	0.114575	−	0.993415i	$-0.463449\pi$
$173$	1.56056e7	0.229150	0.114575	+	0.993415i	$0.463449\pi$
$174$	0	0
$175$	5.69741e7	0.803608
$176$	0	0
$177$	−5.33070e7	−0.722566
$178$	0	0
$179$	−1.31729e8	−1.71671	−0.858354	−	0.513057i	$-0.828513\pi$
$179$	−1.31729e8	−1.71671	−0.858354	+	0.513057i	$0.828513\pi$
$180$	0	0
$181$	−1.58581e7	−0.198781	−0.0993906	−	0.995048i	$-0.531689\pi$
$181$	−1.58581e7	−0.198781	−0.0993906	+	0.995048i	$0.531689\pi$
$182$	0	0
$183$	1.50980e8	1.82113
$184$	0	0
$185$	−2.18881e8	−2.54160
$186$	0	0
$187$	2.81696e8	3.15017
$188$	0	0
$189$	−6.36146e7	−0.685394
$190$	0	0
$191$	−2.94776e7	−0.306108	−0.153054	−	0.988218i	$-0.548911\pi$
$191$	−2.94776e7	−0.306108	−0.153054	+	0.988218i	$0.548911\pi$
$192$	0	0
$193$	−5.78934e7	−0.579667	−0.289834	−	0.957077i	$-0.593600\pi$
$193$	−5.78934e7	−0.579667	−0.289834	+	0.957077i	$0.593600\pi$
$194$	0	0
$195$	−2.14229e8	−2.06899
$196$	0	0
$197$	−1.41579e8	−1.31937	−0.659685	−	0.751542i	$-0.729310\pi$
$197$	−1.41579e8	−1.31937	−0.659685	+	0.751542i	$0.729310\pi$
$198$	0	0
$199$	1.08966e8	0.980176	0.490088	−	0.871673i	$-0.336965\pi$
$199$	1.08966e8	0.980176	0.490088	+	0.871673i	$0.336965\pi$
$200$	0	0
$201$	−1.37339e8	−1.19291
$202$	0	0
$203$	3.57095e7	0.299604
$204$	0	0
$205$	−3.33430e7	−0.270312
$206$	0	0
$207$	3.04766e8	2.38820
$208$	0	0
$209$	−1.04610e7	−0.0792613
$210$	0	0
$211$	−3.89813e7	−0.285672	−0.142836	−	0.989746i	$-0.545622\pi$
$211$	−3.89813e7	−0.285672	−0.142836	+	0.989746i	$0.545622\pi$
$212$	0	0
$213$	−2.28890e8	−1.62293
$214$	0	0
$215$	4.86330e7	0.333731
$216$	0	0
$217$	−1.13028e7	−0.0750893
$218$	0	0
$219$	−1.29973e8	−0.836176
$220$	0	0
$221$	−1.74483e8	−1.08738
$222$	0	0
$223$	4.99584e7	0.301677	0.150838	−	0.988558i	$-0.451803\pi$
$223$	4.99584e7	0.301677	0.150838	+	0.988558i	$0.451803\pi$
$224$	0	0
$225$	7.41090e8	4.33743
$226$	0	0
$227$	1.18812e8	0.674172	0.337086	−	0.941474i	$-0.390559\pi$
$227$	1.18812e8	0.674172	0.337086	+	0.941474i	$0.390559\pi$
$228$	0	0
$229$	1.46428e8	0.805751	0.402875	−	0.915255i	$-0.368011\pi$
$229$	1.46428e8	0.805751	0.402875	+	0.915255i	$0.368011\pi$
$230$	0	0
$231$	2.40049e8	1.28132
$232$	0	0
$233$	−1.65394e8	−0.856592	−0.428296	−	0.903638i	$-0.640886\pi$
$233$	−1.65394e8	−0.856592	−0.428296	+	0.903638i	$0.640886\pi$
$234$	0	0
$235$	1.47194e8	0.739866
$236$	0	0
$237$	3.57026e8	1.74213
$238$	0	0
$239$	−3.54140e8	−1.67796	−0.838981	−	0.544161i	$-0.816848\pi$
$239$	−3.54140e8	−1.67796	−0.838981	+	0.544161i	$0.816848\pi$
$240$	0	0
$241$	1.11585e8	0.513509	0.256754	−	0.966477i	$-0.417347\pi$
$241$	1.11585e8	0.513509	0.256754	+	0.966477i	$0.417347\pi$
$242$	0	0
$243$	−3.18557e7	−0.142418
$244$	0	0
$245$	5.81417e7	0.252585
$246$	0	0
$247$	6.47959e6	0.0273595
$248$	0	0
$249$	5.99543e8	2.46106
$250$	0	0
$251$	5.76216e7	0.230000	0.115000	−	0.993366i	$-0.463313\pi$
$251$	5.76216e7	0.230000	0.115000	+	0.993366i	$0.463313\pi$
$252$	0	0
$253$	−5.86300e8	−2.27614
$254$	0	0
$255$	1.32252e9	4.99473
$256$	0	0
$257$	1.29067e7	0.0474294	0.0237147	−	0.999719i	$-0.492451\pi$
$257$	1.29067e7	0.0474294	0.0237147	+	0.999719i	$0.492451\pi$
$258$	0	0
$259$	−1.51916e8	−0.543318
$260$	0	0
$261$	4.64490e8	1.61709
$262$	0	0
$263$	4.80008e8	1.62706	0.813530	−	0.581524i	$-0.197543\pi$
$263$	4.80008e8	1.62706	0.813530	+	0.581524i	$0.197543\pi$
$264$	0	0
$265$	−4.87770e8	−1.61011
$266$	0	0
$267$	−3.51848e8	−1.13127
$268$	0	0
$269$	−8.00072e7	−0.250609	−0.125304	−	0.992118i	$-0.539991\pi$
$269$	−8.00072e7	−0.250609	−0.125304	+	0.992118i	$0.539991\pi$
$270$	0	0
$271$	−2.54744e8	−0.777520	−0.388760	−	0.921339i	$-0.627096\pi$
$271$	−2.54744e8	−0.777520	−0.388760	+	0.921339i	$0.627096\pi$
$272$	0	0
$273$	−1.48687e8	−0.442287
$274$	0	0
$275$	−1.42569e9	−4.13390
$276$	0	0
$277$	−2.02665e8	−0.572929	−0.286464	−	0.958091i	$-0.592480\pi$
$277$	−2.02665e8	−0.572929	−0.286464	+	0.958091i	$0.592480\pi$
$278$	0	0
$279$	−1.47021e8	−0.405290
$280$	0	0
$281$	−4.19708e7	−0.112843	−0.0564216	−	0.998407i	$-0.517969\pi$
$281$	−4.19708e7	−0.112843	−0.0564216	+	0.998407i	$0.517969\pi$
$282$	0	0
$283$	−9.14935e7	−0.239959	−0.119980	−	0.992776i	$-0.538283\pi$
$283$	−9.14935e7	−0.239959	−0.119980	+	0.992776i	$0.538283\pi$
$284$	0	0
$285$	−4.91130e7	−0.125672
$286$	0	0
$287$	−2.31419e7	−0.0577846
$288$	0	0
$289$	6.66816e8	1.62504
$290$	0	0
$291$	1.39613e9	3.32125
$292$	0	0
$293$	5.14253e8	1.19437	0.597187	−	0.802102i	$-0.296285\pi$
$293$	5.14253e8	1.19437	0.597187	+	0.802102i	$0.296285\pi$
$294$	0	0
$295$	3.23087e8	0.732728
$296$	0	0
$297$	1.59185e9	3.52579
$298$	0	0
$299$	3.63157e8	0.785679
$300$	0	0
$301$	3.37540e7	0.0713416
$302$	0	0
$303$	−4.12756e8	−0.852402
$304$	0	0
$305$	−9.15074e8	−1.84674
$306$	0	0
$307$	1.99321e8	0.393160	0.196580	−	0.980488i	$-0.437017\pi$
$307$	1.99321e8	0.393160	0.196580	+	0.980488i	$0.437017\pi$
$308$	0	0
$309$	−5.53073e8	−1.06642
$310$	0	0
$311$	2.30346e8	0.434229	0.217115	−	0.976146i	$-0.430335\pi$
$311$	2.30346e8	0.434229	0.217115	+	0.976146i	$0.430335\pi$
$312$	0	0
$313$	−6.42371e8	−1.18408	−0.592040	−	0.805909i	$-0.701677\pi$
$313$	−6.42371e8	−1.18408	−0.592040	+	0.805909i	$0.701677\pi$
$314$	0	0
$315$	7.56278e8	1.36331
$316$	0	0
$317$	−6.82230e8	−1.20288	−0.601441	−	0.798917i	$-0.705407\pi$
$317$	−6.82230e8	−1.20288	−0.601441	+	0.798917i	$0.705407\pi$
$318$	0	0
$319$	−8.93574e8	−1.54121
$320$	0	0
$321$	8.03639e8	1.35610
$322$	0	0
$323$	−4.00010e7	−0.0660484
$324$	0	0
$325$	8.83077e8	1.42694
$326$	0	0
$327$	1.20331e9	1.90309
$328$	0	0
$329$	1.02161e8	0.158161
$330$	0	0
$331$	1.30159e8	0.197276	0.0986382	−	0.995123i	$-0.468551\pi$
$331$	1.30159e8	0.197276	0.0986382	+	0.995123i	$0.468551\pi$
$332$	0	0
$333$	−1.97604e9	−2.93252
$334$	0	0
$335$	8.32393e8	1.20968
$336$	0	0
$337$	−9.46752e8	−1.34751	−0.673754	−	0.738955i	$-0.735319\pi$
$337$	−9.46752e8	−1.34751	−0.673754	+	0.738955i	$0.735319\pi$
$338$	0	0
$339$	−1.17732e8	−0.164132
$340$	0	0
$341$	2.82835e8	0.386272
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	−2.75260e9	−3.60891
$346$	0	0
$347$	3.12829e8	0.401933	0.200967	−	0.979598i	$-0.435592\pi$
$347$	3.12829e8	0.401933	0.200967	+	0.979598i	$0.435592\pi$
$348$	0	0
$349$	−2.38409e8	−0.300215	−0.150108	−	0.988670i	$-0.547962\pi$
$349$	−2.38409e8	−0.300215	−0.150108	+	0.988670i	$0.547962\pi$
$350$	0	0
$351$	−9.86002e8	−1.21703
$352$	0	0
$353$	7.89064e8	0.954774	0.477387	−	0.878693i	$-0.341584\pi$
$353$	7.89064e8	0.954774	0.477387	+	0.878693i	$0.341584\pi$
$354$	0	0
$355$	1.38728e9	1.64575
$356$	0	0
$357$	9.17904e8	1.06772
$358$	0	0
$359$	−1.51523e9	−1.72841	−0.864207	−	0.503136i	$-0.832179\pi$
$359$	−1.51523e9	−1.72841	−0.864207	+	0.503136i	$0.832179\pi$
$360$	0	0
$361$	−8.92386e8	−0.998338
$362$	0	0
$363$	−4.41788e9	−4.84775
$364$	0	0
$365$	7.87749e8	0.847936
$366$	0	0
$367$	1.68484e8	0.177921	0.0889607	−	0.996035i	$-0.471645\pi$
$367$	1.68484e8	0.177921	0.0889607	+	0.996035i	$0.471645\pi$
$368$	0	0
$369$	−3.01017e8	−0.311889
$370$	0	0
$371$	−3.38540e8	−0.344192
$372$	0	0
$373$	6.45207e8	0.643752	0.321876	−	0.946782i	$-0.395687\pi$
$373$	6.45207e8	0.643752	0.321876	+	0.946782i	$0.395687\pi$
$374$	0	0
$375$	−3.54527e9	−3.47168
$376$	0	0
$377$	5.53483e8	0.531998
$378$	0	0
$379$	6.17222e8	0.582377	0.291189	−	0.956666i	$-0.405949\pi$
$379$	6.17222e8	0.582377	0.291189	+	0.956666i	$0.405949\pi$
$380$	0	0
$381$	−5.64579e8	−0.522982
$382$	0	0
$383$	1.25018e9	1.13705	0.568524	−	0.822667i	$-0.307515\pi$
$383$	1.25018e9	1.13705	0.568524	+	0.822667i	$0.307515\pi$
$384$	0	0
$385$	−1.45491e9	−1.29934
$386$	0	0
$387$	4.39055e8	0.385062
$388$	0	0
$389$	−1.03870e9	−0.894680	−0.447340	−	0.894364i	$-0.647629\pi$
$389$	−1.03870e9	−0.894680	−0.447340	+	0.894364i	$0.647629\pi$
$390$	0	0
$391$	−2.24191e9	−1.89670
$392$	0	0
$393$	1.74382e9	1.44920
$394$	0	0
$395$	−2.16389e9	−1.76663
$396$	0	0
$397$	5.25716e8	0.421681	0.210840	−	0.977520i	$-0.432380\pi$
$397$	5.25716e8	0.421681	0.210840	+	0.977520i	$0.432380\pi$
$398$	0	0
$399$	−3.40871e7	−0.0268649
$400$	0	0
$401$	4.94457e8	0.382934	0.191467	−	0.981499i	$-0.438676\pi$
$401$	4.94457e8	0.382934	0.191467	+	0.981499i	$0.438676\pi$
$402$	0	0
$403$	−1.75189e8	−0.133334
$404$	0	0
$405$	2.65144e9	1.98331
$406$	0	0
$407$	3.80146e9	2.79492
$408$	0	0
$409$	−1.53968e9	−1.11275	−0.556377	−	0.830930i	$-0.687809\pi$
$409$	−1.53968e9	−1.11275	−0.556377	+	0.830930i	$0.687809\pi$
$410$	0	0
$411$	−3.00269e9	−2.13336
$412$	0	0
$413$	2.24241e8	0.156635
$414$	0	0
$415$	−3.63376e9	−2.49567
$416$	0	0
$417$	3.99596e9	2.69864
$418$	0	0
$419$	2.34483e9	1.55726	0.778631	−	0.627482i	$-0.215914\pi$
$419$	2.34483e9	1.55726	0.778631	+	0.627482i	$0.215914\pi$
$420$	0	0
$421$	−6.13759e7	−0.0400877	−0.0200438	−	0.999799i	$-0.506381\pi$
$421$	−6.13759e7	−0.0400877	−0.0200438	+	0.999799i	$0.506381\pi$
$422$	0	0
$423$	1.32886e9	0.853664
$424$	0	0
$425$	−5.45158e9	−3.44478
$426$	0	0
$427$	−6.35112e8	−0.394778
$428$	0	0
$429$	3.72066e9	2.27520
$430$	0	0
$431$	−2.41932e8	−0.145554	−0.0727769	−	0.997348i	$-0.523186\pi$
$431$	−2.41932e8	−0.145554	−0.0727769	+	0.997348i	$0.523186\pi$
$432$	0	0
$433$	2.29127e9	1.35634	0.678170	−	0.734905i	$-0.262773\pi$
$433$	2.29127e9	1.35634	0.678170	+	0.734905i	$0.262773\pi$
$434$	0	0
$435$	−4.19520e9	−2.44366
$436$	0	0
$437$	8.32552e7	0.0477228
$438$	0	0
$439$	8.22541e8	0.464015	0.232007	−	0.972714i	$-0.425471\pi$
$439$	8.22541e8	0.464015	0.232007	+	0.972714i	$0.425471\pi$
$440$	0	0
$441$	5.24899e8	0.291434
$442$	0	0
$443$	2.35100e9	1.28481	0.642405	−	0.766366i	$-0.277937\pi$
$443$	2.35100e9	1.28481	0.642405	+	0.766366i	$0.277937\pi$
$444$	0	0
$445$	2.13251e9	1.14718
$446$	0	0
$447$	3.57220e9	1.89173
$448$	0	0
$449$	2.51179e9	1.30955	0.654774	−	0.755824i	$-0.272764\pi$
$449$	2.51179e9	1.30955	0.654774	+	0.755824i	$0.272764\pi$
$450$	0	0
$451$	5.79089e8	0.297254
$452$	0	0
$453$	2.06041e9	1.04138
$454$	0	0
$455$	9.01175e8	0.448507
$456$	0	0
$457$	−3.57198e9	−1.75066	−0.875331	−	0.483525i	$-0.839356\pi$
$457$	−3.57198e9	−1.75066	−0.875331	+	0.483525i	$0.839356\pi$
$458$	0	0
$459$	6.08698e9	2.93804
$460$	0	0
$461$	−3.97021e8	−0.188738	−0.0943692	−	0.995537i	$-0.530083\pi$
$461$	−3.97021e8	−0.188738	−0.0943692	+	0.995537i	$0.530083\pi$
$462$	0	0
$463$	−3.36771e9	−1.57689	−0.788445	−	0.615106i	$-0.789113\pi$
$463$	−3.36771e9	−1.57689	−0.788445	+	0.615106i	$0.789113\pi$
$464$	0	0
$465$	1.32787e9	0.612451
$466$	0	0
$467$	1.95270e9	0.887208	0.443604	−	0.896223i	$-0.353700\pi$
$467$	1.95270e9	0.887208	0.443604	+	0.896223i	$0.353700\pi$
$468$	0	0
$469$	5.77727e8	0.258594
$470$	0	0
$471$	4.07470e9	1.79689
$472$	0	0
$473$	−8.44642e8	−0.366994
$474$	0	0
$475$	2.02449e8	0.0866739
$476$	0	0
$477$	−4.40355e9	−1.85775
$478$	0	0
$479$	−3.34441e9	−1.39042	−0.695208	−	0.718808i	$-0.744688\pi$
$479$	−3.34441e9	−1.39042	−0.695208	+	0.718808i	$0.744688\pi$
$480$	0	0
$481$	−2.35464e9	−0.964753
$482$	0	0
$483$	−1.91046e9	−0.771476
$484$	0	0
$485$	−8.46179e9	−3.36795
$486$	0	0
$487$	−4.19199e9	−1.64463	−0.822317	−	0.569029i	$-0.807319\pi$
$487$	−4.19199e9	−1.64463	−0.822317	+	0.569029i	$0.807319\pi$
$488$	0	0
$489$	5.89051e9	2.27810
$490$	0	0
$491$	8.96000e8	0.341604	0.170802	−	0.985305i	$-0.445364\pi$
$491$	8.96000e8	0.341604	0.170802	+	0.985305i	$0.445364\pi$
$492$	0	0
$493$	−3.41687e9	−1.28429
$494$	0	0
$495$	−1.89247e10	−7.01310
$496$	0	0
$497$	9.62848e8	0.351812
$498$	0	0
$499$	−1.09257e9	−0.393640	−0.196820	−	0.980440i	$-0.563061\pi$
$499$	−1.09257e9	−0.393640	−0.196820	+	0.980440i	$0.563061\pi$
$500$	0	0
$501$	5.26031e9	1.86887
$502$	0	0
$503$	5.02202e9	1.75950	0.879752	−	0.475432i	$-0.157708\pi$
$503$	5.02202e9	1.75950	0.879752	+	0.475432i	$0.157708\pi$
$504$	0	0
$505$	2.50167e9	0.864389
$506$	0	0
$507$	2.81184e9	0.958215
$508$	0	0
$509$	−1.51540e9	−0.509348	−0.254674	−	0.967027i	$-0.581968\pi$
$509$	−1.51540e9	−0.509348	−0.254674	+	0.967027i	$0.581968\pi$
$510$	0	0
$511$	5.46742e8	0.181263
$512$	0	0
$513$	−2.26045e8	−0.0739238
$514$	0	0
$515$	3.35211e9	1.08142
$516$	0	0
$517$	−2.55642e9	−0.813608
$518$	0	0
$519$	−1.27246e9	−0.399539
$520$	0	0
$521$	−2.68389e9	−0.831444	−0.415722	−	0.909492i	$-0.636471\pi$
$521$	−2.68389e9	−0.831444	−0.415722	+	0.909492i	$0.636471\pi$
$522$	0	0
$523$	6.06556e9	1.85402	0.927012	−	0.375032i	$-0.122368\pi$
$523$	6.06556e9	1.85402	0.927012	+	0.375032i	$0.122368\pi$
$524$	0	0
$525$	−4.64560e9	−1.40115
$526$	0	0
$527$	1.08151e9	0.321881
$528$	0	0
$529$	1.26132e9	0.370450
$530$	0	0
$531$	2.91680e9	0.845428
$532$	0	0
$533$	−3.58690e8	−0.102606
$534$	0	0
$535$	−4.87076e9	−1.37518
$536$	0	0
$537$	1.07410e10	2.99320
$538$	0	0
$539$	−1.00979e9	−0.277759
$540$	0	0
$541$	2.42694e6	0.000658975 0	0.000329488	−	1.00000i	$-0.499895\pi$
$541$	2.42694e6	0.000658975 0	0.000329488		1.00000i	$0.499895\pi$
$542$	0	0
$543$	1.29305e9	0.346589
$544$	0	0
$545$	−7.29311e9	−1.92986
$546$	0	0
$547$	2.65266e8	0.0692988	0.0346494	−	0.999400i	$-0.488969\pi$
$547$	2.65266e8	0.0692988	0.0346494	+	0.999400i	$0.488969\pi$
$548$	0	0
$549$	−8.26121e9	−2.13079
$550$	0	0
$551$	1.26888e8	0.0323140
$552$	0	0
$553$	−1.50186e9	−0.377652
$554$	0	0
$555$	1.78473e10	4.43146
$556$	0	0
$557$	2.51235e9	0.616009	0.308004	−	0.951385i	$-0.400339\pi$
$557$	2.51235e9	0.616009	0.308004	+	0.951385i	$0.400339\pi$
$558$	0	0
$559$	5.23175e8	0.126679
$560$	0	0
$561$	−2.29691e10	−5.49255
$562$	0	0
$563$	−7.12182e9	−1.68195	−0.840973	−	0.541078i	$-0.818016\pi$
$563$	−7.12182e9	−1.68195	−0.840973	+	0.541078i	$0.818016\pi$
$564$	0	0
$565$	7.13557e8	0.166441
$566$	0	0
$567$	1.84025e9	0.423971
$568$	0	0
$569$	6.42906e9	1.46303	0.731517	−	0.681823i	$-0.238813\pi$
$569$	6.42906e9	1.46303	0.731517	+	0.681823i	$0.238813\pi$
$570$	0	0
$571$	−4.65069e9	−1.04542	−0.522710	−	0.852510i	$-0.675079\pi$
$571$	−4.65069e9	−1.04542	−0.522710	+	0.852510i	$0.675079\pi$
$572$	0	0
$573$	2.40356e9	0.533721
$574$	0	0
$575$	1.13465e10	2.48900
$576$	0	0
$577$	3.28509e9	0.711921	0.355961	−	0.934501i	$-0.384154\pi$
$577$	3.28509e9	0.711921	0.355961	+	0.934501i	$0.384154\pi$
$578$	0	0
$579$	4.72056e9	1.01069
$580$	0	0
$581$	−2.52203e9	−0.533499
$582$	0	0
$583$	8.47142e9	1.77058
$584$	0	0
$585$	1.17220e10	2.42079
$586$	0	0
$587$	−8.16294e9	−1.66576	−0.832882	−	0.553451i	$-0.813311\pi$
$587$	−8.16294e9	−1.66576	−0.832882	+	0.553451i	$0.813311\pi$
$588$	0	0
$589$	−4.01629e7	−0.00809882
$590$	0	0
$591$	1.15442e10	2.30042
$592$	0	0
$593$	6.28232e9	1.23717	0.618584	−	0.785719i	$-0.287707\pi$
$593$	6.28232e9	1.23717	0.618584	+	0.785719i	$0.287707\pi$
$594$	0	0
$595$	−5.56331e9	−1.08274
$596$	0	0
$597$	−8.88492e9	−1.70901
$598$	0	0
$599$	−9.23181e9	−1.75506	−0.877532	−	0.479519i	$-0.840811\pi$
$599$	−9.23181e9	−1.75506	−0.877532	+	0.479519i	$0.840811\pi$
$600$	0	0
$601$	−8.46741e8	−0.159107	−0.0795536	−	0.996831i	$-0.525349\pi$
$601$	−8.46741e8	−0.159107	−0.0795536	+	0.996831i	$0.525349\pi$
$602$	0	0
$603$	7.51478e9	1.39574
$604$	0	0
$605$	2.67762e10	4.91593
$606$	0	0
$607$	−5.08997e9	−0.923751	−0.461876	−	0.886945i	$-0.652823\pi$
$607$	−5.08997e9	−0.923751	−0.461876	+	0.886945i	$0.652823\pi$
$608$	0	0
$609$	−2.91171e9	−0.522381
$610$	0	0
$611$	1.58346e9	0.280842
$612$	0	0
$613$	7.13836e9	1.25166	0.625830	−	0.779959i	$-0.284760\pi$
$613$	7.13836e9	1.25166	0.625830	+	0.779959i	$0.284760\pi$
$614$	0	0
$615$	2.71874e9	0.471308
$616$	0	0
$617$	−2.73074e8	−0.0468039	−0.0234019	−	0.999726i	$-0.507450\pi$
$617$	−2.73074e8	−0.0468039	−0.0234019	+	0.999726i	$0.507450\pi$
$618$	0	0
$619$	4.24318e8	0.0719074	0.0359537	−	0.999353i	$-0.488553\pi$
$619$	4.24318e8	0.0719074	0.0359537	+	0.999353i	$0.488553\pi$
$620$	0	0
$621$	−1.26690e10	−2.12286
$622$	0	0
$623$	1.48008e9	0.245232
$624$	0	0
$625$	8.51048e9	1.39436
$626$	0	0
$627$	8.52977e8	0.138198
$628$	0	0
$629$	1.45361e10	2.32901
$630$	0	0
$631$	9.29957e9	1.47353	0.736767	−	0.676147i	$-0.236351\pi$
$631$	9.29957e9	1.47353	0.736767	+	0.676147i	$0.236351\pi$
$632$	0	0
$633$	3.17849e9	0.498090
$634$	0	0
$635$	3.42185e9	0.530337
$636$	0	0
$637$	6.25465e8	0.0958772
$638$	0	0
$639$	1.25242e10	1.89888
$640$	0	0
$641$	−5.82292e9	−0.873249	−0.436624	−	0.899644i	$-0.643826\pi$
$641$	−5.82292e9	−0.873249	−0.436624	+	0.899644i	$0.643826\pi$
$642$	0	0
$643$	−1.70599e9	−0.253068	−0.126534	−	0.991962i	$-0.540385\pi$
$643$	−1.70599e9	−0.253068	−0.126534	+	0.991962i	$0.540385\pi$
$644$	0	0
$645$	−3.96548e9	−0.581884
$646$	0	0
$647$	−9.72632e9	−1.41183	−0.705917	−	0.708295i	$-0.749465\pi$
$647$	−9.72632e9	−1.41183	−0.705917	+	0.708295i	$0.749465\pi$
$648$	0	0
$649$	−5.61127e9	−0.805758
$650$	0	0
$651$	9.21618e8	0.130923
$652$	0	0
$653$	7.16287e9	1.00668	0.503339	−	0.864089i	$-0.332105\pi$
$653$	7.16287e9	1.00668	0.503339	+	0.864089i	$0.332105\pi$
$654$	0	0
$655$	−1.05691e10	−1.46958
$656$	0	0
$657$	7.11173e9	0.978356
$658$	0	0
$659$	5.64729e9	0.768672	0.384336	−	0.923193i	$-0.374430\pi$
$659$	5.64729e9	0.768672	0.384336	+	0.923193i	$0.374430\pi$
$660$	0	0
$661$	−1.33551e10	−1.79862	−0.899312	−	0.437307i	$-0.855932\pi$
$661$	−1.33551e10	−1.79862	−0.899312	+	0.437307i	$0.855932\pi$
$662$	0	0
$663$	1.42272e10	1.89592
$664$	0	0
$665$	2.06598e8	0.0272427
$666$	0	0
$667$	7.11162e9	0.927957
$668$	0	0
$669$	−4.07355e9	−0.525995
$670$	0	0
$671$	1.58927e10	2.03081
$672$	0	0
$673$	−5.36937e8	−0.0679001	−0.0339500	−	0.999424i	$-0.510809\pi$
$673$	−5.36937e8	−0.0679001	−0.0339500	+	0.999424i	$0.510809\pi$
$674$	0	0
$675$	−3.08068e10	−3.85552
$676$	0	0
$677$	−2.65173e9	−0.328450	−0.164225	−	0.986423i	$-0.552512\pi$
$677$	−2.65173e9	−0.328450	−0.164225	+	0.986423i	$0.552512\pi$
$678$	0	0
$679$	−5.87295e9	−0.719967
$680$	0	0
$681$	−9.68780e9	−1.17547
$682$	0	0
$683$	1.51906e10	1.82432	0.912162	−	0.409829i	$-0.134412\pi$
$683$	1.51906e10	1.82432	0.912162	+	0.409829i	$0.134412\pi$
$684$	0	0
$685$	1.81989e10	2.16336
$686$	0	0
$687$	−1.19396e10	−1.40488
$688$	0	0
$689$	−5.24723e9	−0.611172
$690$	0	0
$691$	−3.82626e9	−0.441165	−0.220582	−	0.975368i	$-0.570796\pi$
$691$	−3.82626e9	−0.441165	−0.220582	+	0.975368i	$0.570796\pi$
$692$	0	0
$693$	−1.31348e10	−1.49919
$694$	0	0
$695$	−2.42190e10	−2.73659
$696$	0	0
$697$	2.21434e9	0.247702
$698$	0	0
$699$	1.34860e10	1.49353
$700$	0	0
$701$	2.30422e9	0.252645	0.126322	−	0.991989i	$-0.459683\pi$
$701$	2.30422e9	0.252645	0.126322	+	0.991989i	$0.459683\pi$
$702$	0	0
$703$	−5.39810e8	−0.0586000
$704$	0	0
$705$	−1.20020e10	−1.29001
$706$	0	0
$707$	1.73630e9	0.184780
$708$	0	0
$709$	4.24453e9	0.447268	0.223634	−	0.974673i	$-0.428208\pi$
$709$	4.24453e9	0.447268	0.223634	+	0.974673i	$0.428208\pi$
$710$	0	0
$711$	−1.95354e10	−2.03835
$712$	0	0
$713$	−2.25098e9	−0.232573
$714$	0	0
$715$	−2.25505e10	−2.30720
$716$	0	0
$717$	2.88761e10	2.92564
$718$	0	0
$719$	−5.72123e8	−0.0574035	−0.0287018	−	0.999588i	$-0.509137\pi$
$719$	−5.72123e8	−0.0574035	−0.0287018	+	0.999588i	$0.509137\pi$
$720$	0	0
$721$	2.32655e9	0.231174
$722$	0	0
$723$	−9.09853e9	−0.895338
$724$	0	0
$725$	1.72931e10	1.68535
$726$	0	0
$727$	1.21447e10	1.17224	0.586118	−	0.810226i	$-0.300656\pi$
$727$	1.21447e10	1.17224	0.586118	+	0.810226i	$0.300656\pi$
$728$	0	0
$729$	−9.13613e9	−0.873405
$730$	0	0
$731$	−3.22976e9	−0.305816
$732$	0	0
$733$	1.52761e10	1.43268	0.716338	−	0.697754i	$-0.245817\pi$
$733$	1.52761e10	1.43268	0.716338	+	0.697754i	$0.245817\pi$
$734$	0	0
$735$	−4.74080e9	−0.440399
$736$	0	0
$737$	−1.44567e10	−1.33025
$738$	0	0
$739$	1.48055e9	0.134949	0.0674743	−	0.997721i	$-0.478506\pi$
$739$	1.48055e9	0.134949	0.0674743	+	0.997721i	$0.478506\pi$
$740$	0	0
$741$	−5.28337e8	−0.0477032
$742$	0	0
$743$	7.03372e9	0.629106	0.314553	−	0.949240i	$-0.398145\pi$
$743$	7.03372e9	0.629106	0.314553	+	0.949240i	$0.398145\pi$
$744$	0	0
$745$	−2.16507e10	−1.91833
$746$	0	0
$747$	−3.28052e10	−2.87953
$748$	0	0
$749$	−3.38058e9	−0.293971
$750$	0	0
$751$	−4.20345e9	−0.362131	−0.181066	−	0.983471i	$-0.557955\pi$
$751$	−4.20345e9	−0.362131	−0.181066	+	0.983471i	$0.557955\pi$
$752$	0	0
$753$	−4.69839e9	−0.401021
$754$	0	0
$755$	−1.24879e10	−1.05603
$756$	0	0
$757$	9.75496e9	0.817316	0.408658	−	0.912688i	$-0.365997\pi$
$757$	9.75496e9	0.817316	0.408658	+	0.912688i	$0.365997\pi$
$758$	0	0
$759$	4.78062e10	3.96860
$760$	0	0
$761$	1.16501e9	0.0958259	0.0479129	−	0.998852i	$-0.484743\pi$
$761$	1.16501e9	0.0958259	0.0479129	+	0.998852i	$0.484743\pi$
$762$	0	0
$763$	−5.06183e9	−0.412545
$764$	0	0
$765$	−7.23646e10	−5.84401
$766$	0	0
$767$	3.47564e9	0.278132
$768$	0	0
$769$	1.33869e10	1.06154	0.530770	−	0.847516i	$-0.321903\pi$
$769$	1.33869e10	1.06154	0.530770	+	0.847516i	$0.321903\pi$
$770$	0	0
$771$	−1.05239e9	−0.0826965
$772$	0	0
$773$	−1.31246e10	−1.02202	−0.511009	−	0.859575i	$-0.670728\pi$
$773$	−1.31246e10	−1.02202	−0.511009	+	0.859575i	$0.670728\pi$
$774$	0	0
$775$	−5.47364e9	−0.422397
$776$	0	0
$777$	1.23870e10	0.947313
$778$	0	0
$779$	−8.22312e7	−0.00623240
$780$	0	0
$781$	−2.40937e10	−1.80978
$782$	0	0
$783$	−1.93087e10	−1.43743
$784$	0	0
$785$	−2.46963e10	−1.82216
$786$	0	0
$787$	2.03749e10	1.48999	0.744997	−	0.667068i	$-0.232451\pi$
$787$	2.03749e10	1.48999	0.744997	+	0.667068i	$0.232451\pi$
$788$	0	0
$789$	−3.91392e10	−2.83689
$790$	0	0
$791$	4.95248e8	0.0355800
$792$	0	0
$793$	−9.84400e9	−0.700996
$794$	0	0
$795$	3.97722e10	2.80733
$796$	0	0
$797$	4.64240e9	0.324817	0.162408	−	0.986724i	$-0.448074\pi$
$797$	4.64240e9	0.324817	0.162408	+	0.986724i	$0.448074\pi$
$798$	0	0
$799$	−9.77530e9	−0.677979
$800$	0	0
$801$	1.92521e10	1.32362
$802$	0	0
$803$	−1.36814e10	−0.932448
$804$	0	0
$805$	1.15791e10	0.782325
$806$	0	0
$807$	6.52369e9	0.436954
$808$	0	0
$809$	1.05503e10	0.700562	0.350281	−	0.936645i	$-0.386086\pi$
$809$	1.05503e10	0.700562	0.350281	+	0.936645i	$0.386086\pi$
$810$	0	0
$811$	−2.52257e9	−0.166062	−0.0830312	−	0.996547i	$-0.526460\pi$
$811$	−2.52257e9	−0.166062	−0.0830312	+	0.996547i	$0.526460\pi$
$812$	0	0
$813$	2.07715e10	1.35566
$814$	0	0
$815$	−3.57017e10	−2.31013
$816$	0	0
$817$	1.19940e8	0.00769461
$818$	0	0
$819$	8.13573e9	0.517491
$820$	0	0
$821$	−6.45008e9	−0.406784	−0.203392	−	0.979097i	$-0.565197\pi$
$821$	−6.45008e9	−0.406784	−0.203392	+	0.979097i	$0.565197\pi$
$822$	0	0
$823$	1.50941e10	0.943862	0.471931	−	0.881636i	$-0.343557\pi$
$823$	1.50941e10	0.943862	0.471931	+	0.881636i	$0.343557\pi$
$824$	0	0
$825$	1.16249e11	7.20775
$826$	0	0
$827$	−1.46469e10	−0.900487	−0.450244	−	0.892906i	$-0.648663\pi$
$827$	−1.46469e10	−0.900487	−0.450244	+	0.892906i	$0.648663\pi$
$828$	0	0
$829$	1.11678e10	0.680812	0.340406	−	0.940279i	$-0.389435\pi$
$829$	1.11678e10	0.680812	0.340406	+	0.940279i	$0.389435\pi$
$830$	0	0
$831$	1.65251e10	0.998942
$832$	0	0
$833$	−3.86124e9	−0.231457
$834$	0	0
$835$	−3.18821e10	−1.89515
$836$	0	0
$837$	6.11161e9	0.360260
$838$	0	0
$839$	−2.34877e9	−0.137301	−0.0686504	−	0.997641i	$-0.521869\pi$
$839$	−2.34877e9	−0.137301	−0.0686504	+	0.997641i	$0.521869\pi$
$840$	0	0
$841$	−6.41114e9	−0.371663
$842$	0	0
$843$	3.42225e9	0.196750
$844$	0	0
$845$	−1.70422e10	−0.971691
$846$	0	0
$847$	1.85842e10	1.05088
$848$	0	0
$849$	7.46027e9	0.418386
$850$	0	0
$851$	−3.02544e10	−1.68281
$852$	0	0
$853$	1.54112e10	0.850186	0.425093	−	0.905150i	$-0.360241\pi$
$853$	1.54112e10	0.850186	0.425093	+	0.905150i	$0.360241\pi$
$854$	0	0
$855$	2.68732e9	0.147041
$856$	0	0
$857$	1.42297e10	0.772260	0.386130	−	0.922444i	$-0.373812\pi$
$857$	1.42297e10	0.772260	0.386130	+	0.922444i	$0.373812\pi$
$858$	0	0
$859$	5.26078e8	0.0283188	0.0141594	−	0.999900i	$-0.495493\pi$
$859$	5.26078e8	0.0283188	0.0141594	+	0.999900i	$0.495493\pi$
$860$	0	0
$861$	1.88696e9	0.100751
$862$	0	0
$863$	2.63506e10	1.39557	0.697786	−	0.716306i	$-0.254169\pi$
$863$	2.63506e10	1.39557	0.697786	+	0.716306i	$0.254169\pi$
$864$	0	0
$865$	7.71225e9	0.405158
$866$	0	0
$867$	−5.43713e10	−2.83337
$868$	0	0
$869$	3.75817e10	1.94271
$870$	0	0
$871$	8.95455e9	0.459177
$872$	0	0
$873$	−7.63923e10	−3.88597
$874$	0	0
$875$	1.49135e10	0.752577
$876$	0	0
$877$	3.02458e10	1.51414	0.757070	−	0.653333i	$-0.226630\pi$
$877$	3.02458e10	1.51414	0.757070	+	0.653333i	$0.226630\pi$
$878$	0	0
$879$	−4.19315e10	−2.08247
$880$	0	0
$881$	−8.05520e8	−0.0396881	−0.0198441	−	0.999803i	$-0.506317\pi$
$881$	−8.05520e8	−0.0396881	−0.0198441	+	0.999803i	$0.506317\pi$
$882$	0	0
$883$	−2.92968e10	−1.43205	−0.716025	−	0.698074i	$-0.754041\pi$
$883$	−2.92968e10	−1.43205	−0.716025	+	0.698074i	$0.754041\pi$
$884$	0	0
$885$	−2.63441e10	−1.27756
$886$	0	0
$887$	−7.16915e9	−0.344933	−0.172467	−	0.985015i	$-0.555174\pi$
$887$	−7.16915e9	−0.344933	−0.172467	+	0.985015i	$0.555174\pi$
$888$	0	0
$889$	2.37495e9	0.113370
$890$	0	0
$891$	−4.60493e10	−2.18098
$892$	0	0
$893$	3.63014e8	0.0170586
$894$	0	0
$895$	−6.51002e10	−3.03530
$896$	0	0
$897$	−2.96113e10	−1.36989
$898$	0	0
$899$	−3.43070e9	−0.157479
$900$	0	0
$901$	3.23932e10	1.47543
$902$	0	0
$903$	−2.75226e9	−0.124389
$904$	0	0
$905$	−7.83701e9	−0.351463
$906$	0	0
$907$	−3.31249e10	−1.47411	−0.737053	−	0.675835i	$-0.763783\pi$
$907$	−3.31249e10	−1.47411	−0.737053	+	0.675835i	$0.763783\pi$
$908$	0	0
$909$	2.25848e10	0.997340
$910$	0	0
$911$	2.81243e9	0.123245	0.0616223	−	0.998100i	$-0.480373\pi$
$911$	2.81243e9	0.123245	0.0616223	+	0.998100i	$0.480373\pi$
$912$	0	0
$913$	6.31098e10	2.74441
$914$	0	0
$915$	7.46140e10	3.21993
$916$	0	0
$917$	−7.33554e9	−0.314152
$918$	0	0
$919$	−1.67812e10	−0.713212	−0.356606	−	0.934255i	$-0.616066\pi$
$919$	−1.67812e10	−0.713212	−0.356606	+	0.934255i	$0.616066\pi$
$920$	0	0
$921$	−1.62524e10	−0.685502
$922$	0	0
$923$	1.49238e10	0.624702
$924$	0	0
$925$	−7.35686e10	−3.05630
$926$	0	0
$927$	3.02626e10	1.24775
$928$	0	0
$929$	3.90341e10	1.59731	0.798655	−	0.601790i	$-0.205545\pi$
$929$	3.90341e10	1.59731	0.798655	+	0.601790i	$0.205545\pi$
$930$	0	0
$931$	1.43391e8	0.00582367
$932$	0	0
$933$	−1.87821e10	−0.757110
$934$	0	0
$935$	1.39213e11	5.56979
$936$	0	0
$937$	1.41640e10	0.562466	0.281233	−	0.959639i	$-0.409257\pi$
$937$	1.41640e10	0.562466	0.281233	+	0.959639i	$0.409257\pi$
$938$	0	0
$939$	5.23781e10	2.06453
$940$	0	0
$941$	−2.63419e10	−1.03058	−0.515292	−	0.857015i	$-0.672316\pi$
$941$	−2.63419e10	−1.03058	−0.515292	+	0.857015i	$0.672316\pi$
$942$	0	0
$943$	−4.60875e9	−0.178975
$944$	0	0
$945$	−3.14381e10	−1.21184
$946$	0	0
$947$	2.36334e10	0.904278	0.452139	−	0.891948i	$-0.350661\pi$
$947$	2.36334e10	0.904278	0.452139	+	0.891948i	$0.350661\pi$
$948$	0	0
$949$	8.47429e9	0.321863
$950$	0	0
$951$	5.56281e10	2.09731
$952$	0	0
$953$	4.67146e10	1.74835	0.874174	−	0.485613i	$-0.161403\pi$
$953$	4.67146e10	1.74835	0.874174	+	0.485613i	$0.161403\pi$
$954$	0	0
$955$	−1.45677e10	−0.541227
$956$	0	0
$957$	7.28609e10	2.68722
$958$	0	0
$959$	1.26311e10	0.462461
$960$	0	0
$961$	−2.64267e10	−0.960531
$962$	0	0
$963$	−4.39728e10	−1.58669
$964$	0	0
$965$	−2.86107e10	−1.02490
$966$	0	0
$967$	−8.12566e9	−0.288979	−0.144489	−	0.989506i	$-0.546154\pi$
$967$	−8.12566e9	−0.288979	−0.144489	+	0.989506i	$0.546154\pi$
$968$	0	0
$969$	3.26164e9	0.115160
$970$	0	0
$971$	4.21974e10	1.47917	0.739586	−	0.673062i	$-0.235021\pi$
$971$	4.21974e10	1.47917	0.739586	+	0.673062i	$0.235021\pi$
$972$	0	0
$973$	−1.68094e10	−0.585001
$974$	0	0
$975$	−7.20050e10	−2.48798
$976$	0	0
$977$	5.64581e10	1.93685	0.968423	−	0.249312i	$-0.0802045\pi$
$977$	5.64581e10	1.93685	0.968423	+	0.249312i	$0.0802045\pi$
$978$	0	0
$979$	−3.70367e10	−1.26151
$980$	0	0
$981$	−6.58416e10	−2.22669
$982$	0	0
$983$	3.55771e10	1.19463	0.597314	−	0.802007i	$-0.296234\pi$
$983$	3.55771e10	1.19463	0.597314	+	0.802007i	$0.296234\pi$
$984$	0	0
$985$	−6.99678e10	−2.33277
$986$	0	0
$987$	−8.33008e9	−0.275765
$988$	0	0
$989$	6.72219e9	0.220965
$990$	0	0
$991$	3.06595e10	1.00071	0.500354	−	0.865821i	$-0.333203\pi$
$991$	3.06595e10	1.00071	0.500354	+	0.865821i	$0.333203\pi$
$992$	0	0
$993$	−1.06130e10	−0.343965
$994$	0	0
$995$	5.38505e10	1.73304
$996$	0	0
$997$	3.75012e10	1.19843	0.599214	−	0.800589i	$-0.295480\pi$
$997$	3.75012e10	1.19843	0.599214	+	0.800589i	$0.295480\pi$
$998$	0	0
$999$	8.21432e10	2.60671

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	224.8.a.c.1.1	✓	5
4.3	odd	2		224.8.a.d.1.5	yes	5
8.3	odd	2		448.8.a.ba.1.1		5
8.5	even	2		448.8.a.bb.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.8.a.c.1.1	✓	5	1.1	even	1	trivial
224.8.a.d.1.5	yes	5	4.3	odd	2
448.8.a.ba.1.1		5	8.3	odd	2
448.8.a.bb.1.5		5	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:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)

\(f(q)\)	\(=\)	\(q-81.5387 q^{3} +494.197 q^{5} +343.000 q^{7} +4461.57 q^{9} -8583.04 q^{11} +5316.37 q^{13} -40296.2 q^{15} -32820.0 q^{17} +1218.80 q^{19} -27967.8 q^{21} +68309.2 q^{23} +166105. q^{25} -185465. q^{27} +104109. q^{29} -32952.8 q^{31} +699850. q^{33} +169509. q^{35} -442903. q^{37} -433490. q^{39} -67469.0 q^{41} +98408.3 q^{43} +2.20489e6 q^{45} +297845. q^{47} +117649. q^{49} +2.67610e6 q^{51} -986996. q^{53} -4.24171e6 q^{55} -99379.4 q^{57} +653762. q^{59} -1.85164e6 q^{61} +1.53032e6 q^{63} +2.62733e6 q^{65} +1.68434e6 q^{67} -5.56984e6 q^{69} +2.80714e6 q^{71} +1.59400e6 q^{73} -1.35440e7 q^{75} -2.94398e6 q^{77} -4.37861e6 q^{79} +5.36516e6 q^{81} -7.35285e6 q^{83} -1.62196e7 q^{85} -8.48894e6 q^{87} +4.31510e6 q^{89} +1.82351e6 q^{91} +2.68693e6 q^{93} +602327. q^{95} -1.71223e7 q^{97} -3.82938e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 54 q^{3} + 84 q^{5} + 1715 q^{7} + 4133 q^{9} - 5324 q^{11} - 9880 q^{13} - 30128 q^{15} - 44322 q^{17} - 22898 q^{19} - 18522 q^{21} + 141016 q^{23} + 150679 q^{25} - 135972 q^{27} - 18998 q^{29}+ \cdots - 58451180 q^{99}+O(q^{100}) \) 5 * q - 54 * q^3 + 84 * q^5 + 1715 * q^7 + 4133 * q^9 - 5324 * q^11 - 9880 * q^13 - 30128 * q^15 - 44322 * q^17 - 22898 * q^19 - 18522 * q^21 + 141016 * q^23 + 150679 * q^25 - 135972 * q^27 - 18998 * q^29 + 167668 * q^31 + 830696 * q^33 + 28812 * q^35 - 434206 * q^37 - 1100728 * q^39 + 44278 * q^41 - 465076 * q^43 + 1823220 * q^45 - 7508 * q^47 + 588245 * q^49 - 65012 * q^51 - 1901986 * q^53 - 4619216 * q^55 - 3116556 * q^57 - 2342994 * q^59 - 553300 * q^61 + 1417619 * q^63 + 1749208 * q^65 - 1908152 * q^67 - 554944 * q^69 - 1721104 * q^71 + 1237634 * q^73 - 19901682 * q^75 - 1826132 * q^77 - 9787808 * q^79 + 4138481 * q^81 - 17717898 * q^83 - 13248184 * q^85 - 25003324 * q^87 + 7408138 * q^89 - 3388840 * q^91 + 13173560 * q^93 - 16659216 * q^95 - 17027554 * q^97 - 58451180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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