LMFDB - Embedded newform 315.8.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,8,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	315.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:	$98.4012830275$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -124.000 q^{4} +125.000 q^{5} +343.000 q^{7} -504.000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} -124.000 q^{4} +125.000 q^{5} +343.000 q^{7} -504.000 q^{8} +250.000 q^{10} -2724.00 q^{11} +2874.00 q^{13} +686.000 q^{14} +14864.0 q^{16} -9278.00 q^{17} -4304.00 q^{19} -15500.0 q^{20} -5448.00 q^{22} +41500.0 q^{23} +15625.0 q^{25} +5748.00 q^{26} -42532.0 q^{28} +35498.0 q^{29} -52940.0 q^{31} +94240.0 q^{32} -18556.0 q^{34} +42875.0 q^{35} -84098.0 q^{37} -8608.00 q^{38} -63000.0 q^{40} -180342. q^{41} -33452.0 q^{43} +337776. q^{44} +83000.0 q^{46} +136120. q^{47} +117649. q^{49} +31250.0 q^{50} -356376. q^{52} +1.27062e6 q^{53} -340500. q^{55} -172872. q^{56} +70996.0 q^{58} +1.55325e6 q^{59} +213598. q^{61} -105880. q^{62} -1.71411e6 q^{64} +359250. q^{65} +487228. q^{67} +1.15047e6 q^{68} +85750.0 q^{70} -1.08600e6 q^{71} -5.92198e6 q^{73} -168196. q^{74} +533696. q^{76} -934332. q^{77} -5.42982e6 q^{79} +1.85800e6 q^{80} -360684. q^{82} -6.93340e6 q^{83} -1.15975e6 q^{85} -66904.0 q^{86} +1.37290e6 q^{88} -262614. q^{89} +985782. q^{91} -5.14600e6 q^{92} +272240. q^{94} -538000. q^{95} -522234. q^{97} +235298. q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.176777	0.0883883	−	0.996086i	$-0.471828\pi$
$2$	2.00000	0.176777	0.0883883	+	0.996086i	$0.471828\pi$
$3$	0	0
$3$	0	0
$4$	−124.000	−0.968750
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	−504.000	−0.348029
$9$	0	0
$10$	250.000	0.0790569
$11$	−2724.00	−0.617068	−0.308534	−	0.951213i	$-0.599838\pi$
$11$	−2724.00	−0.617068	−0.308534	+	0.951213i	$0.599838\pi$
$12$	0	0
$13$	2874.00	0.362815	0.181407	−	0.983408i	$-0.441935\pi$
$13$	2874.00	0.362815	0.181407	+	0.983408i	$0.441935\pi$
$14$	686.000	0.0668153
$15$	0	0
$16$	14864.0	0.907227
$17$	−9278.00	−0.458019	−0.229009	−	0.973424i	$-0.573549\pi$
$17$	−9278.00	−0.458019	−0.229009	+	0.973424i	$0.573549\pi$
$18$	0	0
$19$	−4304.00	−0.143958	−0.0719788	−	0.997406i	$-0.522931\pi$
$19$	−4304.00	−0.143958	−0.0719788	+	0.997406i	$0.522931\pi$
$20$	−15500.0	−0.433238
$21$	0	0
$22$	−5448.00	−0.109083
$23$	41500.0	0.711215	0.355607	−	0.934635i	$-0.384274\pi$
$23$	41500.0	0.711215	0.355607	+	0.934635i	$0.384274\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	5748.00	0.0641372
$27$	0	0
$28$	−42532.0	−0.366153
$29$	35498.0	0.270278	0.135139	−	0.990827i	$-0.456852\pi$
$29$	35498.0	0.270278	0.135139	+	0.990827i	$0.456852\pi$
$30$	0	0
$31$	−52940.0	−0.319167	−0.159584	−	0.987184i	$-0.551015\pi$
$31$	−52940.0	−0.319167	−0.159584	+	0.987184i	$0.551015\pi$
$32$	94240.0	0.508406
$33$	0	0
$34$	−18556.0	−0.0809670
$35$	42875.0	0.169031
$36$	0	0
$37$	−84098.0	−0.272948	−0.136474	−	0.990644i	$-0.543577\pi$
$37$	−84098.0	−0.272948	−0.136474	+	0.990644i	$0.543577\pi$
$38$	−8608.00	−0.0254484
$39$	0	0
$40$	−63000.0	−0.155643
$41$	−180342.	−0.408652	−0.204326	−	0.978903i	$-0.565500\pi$
$41$	−180342.	−0.408652	−0.204326	+	0.978903i	$0.565500\pi$
$42$	0	0
$43$	−33452.0	−0.0641627	−0.0320813	−	0.999485i	$-0.510214\pi$
$43$	−33452.0	−0.0641627	−0.0320813	+	0.999485i	$0.510214\pi$
$44$	337776.	0.597784
$45$	0	0
$46$	83000.0	0.125726
$47$	136120.	0.191240	0.0956202	−	0.995418i	$-0.469517\pi$
$47$	136120.	0.191240	0.0956202	+	0.995418i	$0.469517\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	31250.0	0.0353553
$51$	0	0
$52$	−356376.	−0.351477
$53$	1.27062e6	1.17233	0.586166	−	0.810191i	$-0.300637\pi$
$53$	1.27062e6	1.17233	0.586166	+	0.810191i	$0.300637\pi$
$54$	0	0
$55$	−340500.	−0.275961
$56$	−172872.	−0.131543
$57$	0	0
$58$	70996.0	0.0477789
$59$	1.55325e6	0.984600	0.492300	−	0.870426i	$-0.336156\pi$
$59$	1.55325e6	0.984600	0.492300	+	0.870426i	$0.336156\pi$
$60$	0	0
$61$	213598.	0.120488	0.0602439	−	0.998184i	$-0.480812\pi$
$61$	213598.	0.120488	0.0602439	+	0.998184i	$0.480812\pi$
$62$	−105880.	−0.0564213
$63$	0	0
$64$	−1.71411e6	−0.817352
$65$	359250.	0.162256
$66$	0	0
$67$	487228.	0.197911	0.0989556	−	0.995092i	$-0.468450\pi$
$67$	487228.	0.197911	0.0989556	+	0.995092i	$0.468450\pi$
$68$	1.15047e6	0.443706
$69$	0	0
$70$	85750.0	0.0298807
$71$	−1.08600e6	−0.360102	−0.180051	−	0.983657i	$-0.557626\pi$
$71$	−1.08600e6	−0.360102	−0.180051	+	0.983657i	$0.557626\pi$
$72$	0	0
$73$	−5.92198e6	−1.78171	−0.890855	−	0.454289i	$-0.849893\pi$
$73$	−5.92198e6	−1.78171	−0.890855	+	0.454289i	$0.849893\pi$
$74$	−168196.	−0.0482508
$75$	0	0
$76$	533696.	0.139459
$77$	−934332.	−0.233230
$78$	0	0
$79$	−5.42982e6	−1.23906	−0.619528	−	0.784975i	$-0.712676\pi$
$79$	−5.42982e6	−1.23906	−0.619528	+	0.784975i	$0.712676\pi$
$80$	1.85800e6	0.405724
$81$	0	0
$82$	−360684.	−0.0722401
$83$	−6.93340e6	−1.33099	−0.665493	−	0.746405i	$-0.731778\pi$
$83$	−6.93340e6	−1.33099	−0.665493	+	0.746405i	$0.731778\pi$
$84$	0	0
$85$	−1.15975e6	−0.204832
$86$	−66904.0	−0.0113425
$87$	0	0
$88$	1.37290e6	0.214757
$89$	−262614.	−0.0394869	−0.0197434	−	0.999805i	$-0.506285\pi$
$89$	−262614.	−0.0394869	−0.0197434	+	0.999805i	$0.506285\pi$
$90$	0	0
$91$	985782.	0.137131
$92$	−5.14600e6	−0.688989
$93$	0	0
$94$	272240.	0.0338068
$95$	−538000.	−0.0643798
$96$	0	0
$97$	−522234.	−0.0580984	−0.0290492	−	0.999578i	$-0.509248\pi$
$97$	−522234.	−0.0580984	−0.0290492	+	0.999578i	$0.509248\pi$
$98$	235298.	0.0252538
$99$	0	0
$100$	−1.93750e6	−0.193750
$101$	1.63961e6	0.158349	0.0791744	−	0.996861i	$-0.474772\pi$
$101$	1.63961e6	0.158349	0.0791744	+	0.996861i	$0.474772\pi$
$102$	0	0
$103$	−6.57506e6	−0.592883	−0.296442	−	0.955051i	$-0.595800\pi$
$103$	−6.57506e6	−0.592883	−0.296442	+	0.955051i	$0.595800\pi$
$104$	−1.44850e6	−0.126270
$105$	0	0
$106$	2.54124e6	0.207241
$107$	−1.21480e7	−0.958654	−0.479327	−	0.877636i	$-0.659119\pi$
$107$	−1.21480e7	−0.958654	−0.479327	+	0.877636i	$0.659119\pi$
$108$	0	0
$109$	−9.68452e6	−0.716284	−0.358142	−	0.933667i	$-0.616590\pi$
$109$	−9.68452e6	−0.716284	−0.358142	+	0.933667i	$0.616590\pi$
$110$	−681000.	−0.0487835
$111$	0	0
$112$	5.09835e6	0.342899
$113$	−2.50296e7	−1.63184	−0.815922	−	0.578162i	$-0.803770\pi$
$113$	−2.50296e7	−1.63184	−0.815922	+	0.578162i	$0.803770\pi$
$114$	0	0
$115$	5.18750e6	0.318065
$116$	−4.40175e6	−0.261832
$117$	0	0
$118$	3.10650e6	0.174054
$119$	−3.18235e6	−0.173115
$120$	0	0
$121$	−1.20670e7	−0.619228
$122$	427196.	0.0212994
$123$	0	0
$124$	6.56456e6	0.309193
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	3.05960e6	0.132541	0.0662707	−	0.997802i	$-0.478890\pi$
$127$	3.05960e6	0.132541	0.0662707	+	0.997802i	$0.478890\pi$
$128$	−1.54909e7	−0.652894
$129$	0	0
$130$	718500.	0.0286830
$131$	3.56912e6	0.138711	0.0693555	−	0.997592i	$-0.477906\pi$
$131$	3.56912e6	0.138711	0.0693555	+	0.997592i	$0.477906\pi$
$132$	0	0
$133$	−1.47627e6	−0.0544109
$134$	974456.	0.0349861
$135$	0	0
$136$	4.67611e6	0.159404
$137$	−4.83736e6	−0.160726	−0.0803630	−	0.996766i	$-0.525608\pi$
$137$	−4.83736e6	−0.160726	−0.0803630	+	0.996766i	$0.525608\pi$
$138$	0	0
$139$	2.48573e6	0.0785058	0.0392529	−	0.999229i	$-0.487502\pi$
$139$	2.48573e6	0.0785058	0.0392529	+	0.999229i	$0.487502\pi$
$140$	−5.31650e6	−0.163749
$141$	0	0
$142$	−2.17200e6	−0.0636577
$143$	−7.82878e6	−0.223881
$144$	0	0
$145$	4.43725e6	0.120872
$146$	−1.18440e7	−0.314965
$147$	0	0
$148$	1.04282e7	0.264418
$149$	3.99895e7	0.990362	0.495181	−	0.868790i	$-0.335102\pi$
$149$	3.99895e7	0.990362	0.495181	+	0.868790i	$0.335102\pi$
$150$	0	0
$151$	−5.37919e7	−1.27145	−0.635723	−	0.771918i	$-0.719298\pi$
$151$	−5.37919e7	−1.27145	−0.635723	+	0.771918i	$0.719298\pi$
$152$	2.16922e6	0.0501014
$153$	0	0
$154$	−1.86866e6	−0.0412296
$155$	−6.61750e6	−0.142736
$156$	0	0
$157$	9.64019e6	0.198809	0.0994046	−	0.995047i	$-0.468306\pi$
$157$	9.64019e6	0.198809	0.0994046	+	0.995047i	$0.468306\pi$
$158$	−1.08596e7	−0.219036
$159$	0	0
$160$	1.17800e7	0.227366
$161$	1.42345e7	0.268814
$162$	0	0
$163$	−7.67343e7	−1.38782	−0.693910	−	0.720062i	$-0.744113\pi$
$163$	−7.67343e7	−1.38782	−0.693910	+	0.720062i	$0.744113\pi$
$164$	2.23624e7	0.395881
$165$	0	0
$166$	−1.38668e7	−0.235287
$167$	−1.03856e7	−0.172554	−0.0862770	−	0.996271i	$-0.527497\pi$
$167$	−1.03856e7	−0.172554	−0.0862770	+	0.996271i	$0.527497\pi$
$168$	0	0
$169$	−5.44886e7	−0.868365
$170$	−2.31950e6	−0.0362096
$171$	0	0
$172$	4.14805e6	0.0621576
$173$	5.64927e6	0.0829528	0.0414764	−	0.999139i	$-0.486794\pi$
$173$	5.64927e6	0.0829528	0.0414764	+	0.999139i	$0.486794\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	−4.04895e7	−0.559820
$177$	0	0
$178$	−525228.	−0.00698036
$179$	1.04373e8	1.36020	0.680098	−	0.733121i	$-0.261937\pi$
$179$	1.04373e8	1.36020	0.680098	+	0.733121i	$0.261937\pi$
$180$	0	0
$181$	−1.42262e8	−1.78326	−0.891628	−	0.452769i	$-0.850436\pi$
$181$	−1.42262e8	−1.78326	−0.891628	+	0.452769i	$0.850436\pi$
$182$	1.97156e6	0.0242416
$183$	0	0
$184$	−2.09160e7	−0.247523
$185$	−1.05122e7	−0.122066
$186$	0	0
$187$	2.52733e7	0.282628
$188$	−1.68789e7	−0.185264
$189$	0	0
$190$	−1.07600e6	−0.0113808
$191$	3.35533e7	0.348433	0.174216	−	0.984707i	$-0.444261\pi$
$191$	3.35533e7	0.348433	0.174216	+	0.984707i	$0.444261\pi$
$192$	0	0
$193$	−1.70599e8	−1.70815	−0.854073	−	0.520153i	$-0.825875\pi$
$193$	−1.70599e8	−1.70815	−0.854073	+	0.520153i	$0.825875\pi$
$194$	−1.04447e6	−0.0102704
$195$	0	0
$196$	−1.45885e7	−0.138393
$197$	−6.92904e7	−0.645716	−0.322858	−	0.946447i	$-0.604644\pi$
$197$	−6.92904e7	−0.645716	−0.322858	+	0.946447i	$0.604644\pi$
$198$	0	0
$199$	−6.71910e7	−0.604401	−0.302201	−	0.953244i	$-0.597721\pi$
$199$	−6.71910e7	−0.604401	−0.302201	+	0.953244i	$0.597721\pi$
$200$	−7.87500e6	−0.0696058
$201$	0	0
$202$	3.27921e6	0.0279924
$203$	1.21758e7	0.102156
$204$	0	0
$205$	−2.25428e7	−0.182755
$206$	−1.31501e7	−0.104808
$207$	0	0
$208$	4.27191e7	0.329155
$209$	1.17241e7	0.0888316
$210$	0	0
$211$	−5.14714e7	−0.377205	−0.188602	−	0.982054i	$-0.560396\pi$
$211$	−5.14714e7	−0.377205	−0.188602	+	0.982054i	$0.560396\pi$
$212$	−1.57557e8	−1.13570
$213$	0	0
$214$	−2.42960e7	−0.169468
$215$	−4.18150e6	−0.0286944
$216$	0	0
$217$	−1.81584e7	−0.120634
$218$	−1.93690e7	−0.126622
$219$	0	0
$220$	4.22220e7	0.267337
$221$	−2.66650e7	−0.166176
$222$	0	0
$223$	−2.49372e8	−1.50585	−0.752923	−	0.658109i	$-0.771357\pi$
$223$	−2.49372e8	−1.50585	−0.752923	+	0.658109i	$0.771357\pi$
$224$	3.23243e7	0.192159
$225$	0	0
$226$	−5.00591e7	−0.288472
$227$	−3.93928e7	−0.223525	−0.111762	−	0.993735i	$-0.535650\pi$
$227$	−3.93928e7	−0.223525	−0.111762	+	0.993735i	$0.535650\pi$
$228$	0	0
$229$	1.25945e8	0.693035	0.346517	−	0.938044i	$-0.387364\pi$
$229$	1.25945e8	0.693035	0.346517	+	0.938044i	$0.387364\pi$
$230$	1.03750e7	0.0562265
$231$	0	0
$232$	−1.78910e7	−0.0940647
$233$	1.55290e8	0.804265	0.402132	−	0.915582i	$-0.368269\pi$
$233$	1.55290e8	0.804265	0.402132	+	0.915582i	$0.368269\pi$
$234$	0	0
$235$	1.70150e7	0.0855253
$236$	−1.92603e8	−0.953831
$237$	0	0
$238$	−6.36471e6	−0.0306027
$239$	1.53561e8	0.727593	0.363797	−	0.931478i	$-0.381480\pi$
$239$	1.53561e8	0.727593	0.363797	+	0.931478i	$0.381480\pi$
$240$	0	0
$241$	1.67184e8	0.769369	0.384685	−	0.923048i	$-0.374310\pi$
$241$	1.67184e8	0.769369	0.384685	+	0.923048i	$0.374310\pi$
$242$	−2.41340e7	−0.109465
$243$	0	0
$244$	−2.64862e7	−0.116722
$245$	1.47061e7	0.0638877
$246$	0	0
$247$	−1.23697e7	−0.0522300
$248$	2.66818e7	0.111079
$249$	0	0
$250$	3.90625e6	0.0158114
$251$	4.29028e7	0.171249	0.0856245	−	0.996327i	$-0.472711\pi$
$251$	4.29028e7	0.171249	0.0856245	+	0.996327i	$0.472711\pi$
$252$	0	0
$253$	−1.13046e8	−0.438867
$254$	6.11920e6	0.0234302
$255$	0	0
$256$	1.88424e8	0.701936
$257$	−4.70837e8	−1.73023	−0.865116	−	0.501571i	$-0.832755\pi$
$257$	−4.70837e8	−1.73023	−0.865116	+	0.501571i	$0.832755\pi$
$258$	0	0
$259$	−2.88456e7	−0.103165
$260$	−4.45470e7	−0.157185
$261$	0	0
$262$	7.13823e6	0.0245209
$263$	1.02607e8	0.347802	0.173901	−	0.984763i	$-0.444363\pi$
$263$	1.02607e8	0.347802	0.173901	+	0.984763i	$0.444363\pi$
$264$	0	0
$265$	1.58828e8	0.524283
$266$	−2.95254e6	−0.00961857
$267$	0	0
$268$	−6.04163e7	−0.191727
$269$	5.49478e8	1.72114	0.860571	−	0.509330i	$-0.170107\pi$
$269$	5.49478e8	1.72114	0.860571	+	0.509330i	$0.170107\pi$
$270$	0	0
$271$	3.38205e8	1.03225	0.516127	−	0.856512i	$-0.327373\pi$
$271$	3.38205e8	1.03225	0.516127	+	0.856512i	$0.327373\pi$
$272$	−1.37908e8	−0.415527
$273$	0	0
$274$	−9.67472e6	−0.0284126
$275$	−4.25625e7	−0.123414
$276$	0	0
$277$	3.85029e8	1.08846	0.544232	−	0.838934i	$-0.316821\pi$
$277$	3.85029e8	1.08846	0.544232	+	0.838934i	$0.316821\pi$
$278$	4.97146e6	0.0138780
$279$	0	0
$280$	−2.16090e7	−0.0588277
$281$	−3.58731e8	−0.964488	−0.482244	−	0.876037i	$-0.660178\pi$
$281$	−3.58731e8	−0.964488	−0.482244	+	0.876037i	$0.660178\pi$
$282$	0	0
$283$	5.81811e7	0.152591	0.0762955	−	0.997085i	$-0.475691\pi$
$283$	5.81811e7	0.152591	0.0762955	+	0.997085i	$0.475691\pi$
$284$	1.34664e8	0.348849
$285$	0	0
$286$	−1.56576e7	−0.0395770
$287$	−6.18573e7	−0.154456
$288$	0	0
$289$	−3.24257e8	−0.790219
$290$	8.87450e6	0.0213674
$291$	0	0
$292$	7.34325e8	1.72603
$293$	−5.57054e8	−1.29378	−0.646890	−	0.762583i	$-0.723931\pi$
$293$	−5.57054e8	−1.29378	−0.646890	+	0.762583i	$0.723931\pi$
$294$	0	0
$295$	1.94156e8	0.440327
$296$	4.23854e7	0.0949938
$297$	0	0
$298$	7.99790e7	0.175073
$299$	1.19271e8	0.258039
$300$	0	0
$301$	−1.14740e7	−0.0242512
$302$	−1.07584e8	−0.224762
$303$	0	0
$304$	−6.39747e7	−0.130602
$305$	2.66998e7	0.0538837
$306$	0	0
$307$	5.19798e8	1.02530	0.512649	−	0.858598i	$-0.328664\pi$
$307$	5.19798e8	1.02530	0.512649	+	0.858598i	$0.328664\pi$
$308$	1.15857e8	0.225941
$309$	0	0
$310$	−1.32350e7	−0.0252324
$311$	3.93391e8	0.741589	0.370794	−	0.928715i	$-0.379085\pi$
$311$	3.93391e8	0.741589	0.370794	+	0.928715i	$0.379085\pi$
$312$	0	0
$313$	7.46643e8	1.37628	0.688142	−	0.725576i	$-0.258427\pi$
$313$	7.46643e8	1.37628	0.688142	+	0.725576i	$0.258427\pi$
$314$	1.92804e7	0.0351448
$315$	0	0
$316$	6.73298e8	1.20034
$317$	−1.90220e8	−0.335389	−0.167694	−	0.985839i	$-0.553632\pi$
$317$	−1.90220e8	−0.335389	−0.167694	+	0.985839i	$0.553632\pi$
$318$	0	0
$319$	−9.66966e7	−0.166780
$320$	−2.14264e8	−0.365531
$321$	0	0
$322$	2.84690e7	0.0475200
$323$	3.99325e7	0.0659353
$324$	0	0
$325$	4.49063e7	0.0725630
$326$	−1.53469e8	−0.245334
$327$	0	0
$328$	9.08924e7	0.142223
$329$	4.66892e7	0.0722820
$330$	0	0
$331$	−1.09993e9	−1.66712	−0.833559	−	0.552430i	$-0.813701\pi$
$331$	−1.09993e9	−1.66712	−0.833559	+	0.552430i	$0.813701\pi$
$332$	8.59742e8	1.28939
$333$	0	0
$334$	−2.07712e7	−0.0305035
$335$	6.09035e7	0.0885086
$336$	0	0
$337$	4.94657e8	0.704043	0.352022	−	0.935992i	$-0.385494\pi$
$337$	4.94657e8	0.704043	0.352022	+	0.935992i	$0.385494\pi$
$338$	−1.08977e8	−0.153507
$339$	0	0
$340$	1.43809e8	0.198431
$341$	1.44209e8	0.196948
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	1.68598e7	0.0223305
$345$	0	0
$346$	1.12985e7	0.0146641
$347$	9.31091e8	1.19630	0.598148	−	0.801385i	$-0.295903\pi$
$347$	9.31091e8	1.19630	0.598148	+	0.801385i	$0.295903\pi$
$348$	0	0
$349$	−4.56646e8	−0.575031	−0.287515	−	0.957776i	$-0.592829\pi$
$349$	−4.56646e8	−0.575031	−0.287515	+	0.957776i	$0.592829\pi$
$350$	1.07188e7	0.0133631
$351$	0	0
$352$	−2.56710e8	−0.313721
$353$	5.02682e8	0.608250	0.304125	−	0.952632i	$-0.401636\pi$
$353$	5.02682e8	0.608250	0.304125	+	0.952632i	$0.401636\pi$
$354$	0	0
$355$	−1.35750e8	−0.161043
$356$	3.25641e7	0.0382529
$357$	0	0
$358$	2.08746e8	0.240451
$359$	1.39665e9	1.59316	0.796578	−	0.604536i	$-0.206641\pi$
$359$	1.39665e9	1.59316	0.796578	+	0.604536i	$0.206641\pi$
$360$	0	0
$361$	−8.75347e8	−0.979276
$362$	−2.84524e8	−0.315238
$363$	0	0
$364$	−1.22237e8	−0.132846
$365$	−7.40247e8	−0.796804
$366$	0	0
$367$	−2.70500e8	−0.285651	−0.142826	−	0.989748i	$-0.545619\pi$
$367$	−2.70500e8	−0.285651	−0.142826	+	0.989748i	$0.545619\pi$
$368$	6.16856e8	0.645233
$369$	0	0
$370$	−2.10245e7	−0.0215784
$371$	4.35823e8	0.443100
$372$	0	0
$373$	−1.06422e9	−1.06182	−0.530912	−	0.847427i	$-0.678150\pi$
$373$	−1.06422e9	−1.06182	−0.530912	+	0.847427i	$0.678150\pi$
$374$	5.05465e7	0.0499621
$375$	0	0
$376$	−6.86045e7	−0.0665572
$377$	1.02021e8	0.0980609
$378$	0	0
$379$	−3.95456e8	−0.373131	−0.186565	−	0.982443i	$-0.559736\pi$
$379$	−3.95456e8	−0.373131	−0.186565	+	0.982443i	$0.559736\pi$
$380$	6.67120e7	0.0623679
$381$	0	0
$382$	6.71067e7	0.0615948
$383$	−1.64793e8	−0.149880	−0.0749400	−	0.997188i	$-0.523877\pi$
$383$	−1.64793e8	−0.149880	−0.0749400	+	0.997188i	$0.523877\pi$
$384$	0	0
$385$	−1.16792e8	−0.104303
$386$	−3.41197e8	−0.301960
$387$	0	0
$388$	6.47570e7	0.0562828
$389$	1.71125e8	0.147398	0.0736989	−	0.997281i	$-0.476520\pi$
$389$	1.71125e8	0.147398	0.0736989	+	0.997281i	$0.476520\pi$
$390$	0	0
$391$	−3.85037e8	−0.325750
$392$	−5.92951e7	−0.0497184
$393$	0	0
$394$	−1.38581e8	−0.114147
$395$	−6.78728e8	−0.554123
$396$	0	0
$397$	1.73552e9	1.39207	0.696037	−	0.718006i	$-0.254945\pi$
$397$	1.73552e9	1.39207	0.696037	+	0.718006i	$0.254945\pi$
$398$	−1.34382e8	−0.106844
$399$	0	0
$400$	2.32250e8	0.181445
$401$	−2.26343e8	−0.175292	−0.0876458	−	0.996152i	$-0.527934\pi$
$401$	−2.26343e8	−0.175292	−0.0876458	+	0.996152i	$0.527934\pi$
$402$	0	0
$403$	−1.52150e8	−0.115799
$404$	−2.03311e8	−0.153400
$405$	0	0
$406$	2.43516e7	0.0180587
$407$	2.29083e8	0.168427
$408$	0	0
$409$	1.14869e9	0.830176	0.415088	−	0.909781i	$-0.363751\pi$
$409$	1.14869e9	0.830176	0.415088	+	0.909781i	$0.363751\pi$
$410$	−4.50855e7	−0.0323068
$411$	0	0
$412$	8.15307e8	0.574356
$413$	5.32765e8	0.372144
$414$	0	0
$415$	−8.66676e8	−0.595235
$416$	2.70846e8	0.184457
$417$	0	0
$418$	2.34482e7	0.0157034
$419$	−1.98365e9	−1.31740	−0.658699	−	0.752407i	$-0.728893\pi$
$419$	−1.98365e9	−1.31740	−0.658699	+	0.752407i	$0.728893\pi$
$420$	0	0
$421$	2.66045e9	1.73767	0.868835	−	0.495103i	$-0.164870\pi$
$421$	2.66045e9	1.73767	0.868835	+	0.495103i	$0.164870\pi$
$422$	−1.02943e8	−0.0666810
$423$	0	0
$424$	−6.40393e8	−0.408006
$425$	−1.44969e8	−0.0916037
$426$	0	0
$427$	7.32641e7	0.0455401
$428$	1.50635e9	0.928696
$429$	0	0
$430$	−8.36300e6	−0.00507251
$431$	1.50478e9	0.905322	0.452661	−	0.891683i	$-0.350475\pi$
$431$	1.50478e9	0.905322	0.452661	+	0.891683i	$0.350475\pi$
$432$	0	0
$433$	1.37803e9	0.815738	0.407869	−	0.913040i	$-0.366272\pi$
$433$	1.37803e9	0.815738	0.407869	+	0.913040i	$0.366272\pi$
$434$	−3.63168e7	−0.0213252
$435$	0	0
$436$	1.20088e9	0.693900
$437$	−1.78616e8	−0.102385
$438$	0	0
$439$	2.15016e9	1.21296	0.606478	−	0.795100i	$-0.292582\pi$
$439$	2.15016e9	1.21296	0.606478	+	0.795100i	$0.292582\pi$
$440$	1.71612e8	0.0960425
$441$	0	0
$442$	−5.33299e7	−0.0293760
$443$	1.01094e9	0.552474	0.276237	−	0.961090i	$-0.410913\pi$
$443$	1.01094e9	0.552474	0.276237	+	0.961090i	$0.410913\pi$
$444$	0	0
$445$	−3.28268e7	−0.0176591
$446$	−4.98744e8	−0.266198
$447$	0	0
$448$	−5.87940e8	−0.308930
$449$	−1.82648e8	−0.0952256	−0.0476128	−	0.998866i	$-0.515161\pi$
$449$	−1.82648e8	−0.0952256	−0.0476128	+	0.998866i	$0.515161\pi$
$450$	0	0
$451$	4.91252e8	0.252166
$452$	3.10367e9	1.58085
$453$	0	0
$454$	−7.87855e7	−0.0395140
$455$	1.23223e8	0.0613269
$456$	0	0
$457$	−1.51611e6	−0.000743060 0	−0.000371530	−	1.00000i	$-0.500118\pi$
$457$	−1.51611e6	−0.000743060 0	−0.000371530		1.00000i	$0.500118\pi$
$458$	2.51889e8	0.122512
$459$	0	0
$460$	−6.43250e8	−0.308125
$461$	4.79940e8	0.228157	0.114079	−	0.993472i	$-0.463608\pi$
$461$	4.79940e8	0.228157	0.114079	+	0.993472i	$0.463608\pi$
$462$	0	0
$463$	−6.86812e8	−0.321591	−0.160796	−	0.986988i	$-0.551406\pi$
$463$	−6.86812e8	−0.321591	−0.160796	+	0.986988i	$0.551406\pi$
$464$	5.27642e8	0.245203
$465$	0	0
$466$	3.10581e8	0.142175
$467$	−1.69004e9	−0.767869	−0.383934	−	0.923360i	$-0.625431\pi$
$467$	−1.69004e9	−0.767869	−0.383934	+	0.923360i	$0.625431\pi$
$468$	0	0
$469$	1.67119e8	0.0748034
$470$	3.40300e7	0.0151189
$471$	0	0
$472$	−7.82839e8	−0.342670
$473$	9.11232e7	0.0395927
$474$	0	0
$475$	−6.72500e7	−0.0287915
$476$	3.94612e8	0.167705
$477$	0	0
$478$	3.07122e8	0.128622
$479$	−2.39843e9	−0.997132	−0.498566	−	0.866852i	$-0.666140\pi$
$479$	−2.39843e9	−0.997132	−0.498566	+	0.866852i	$0.666140\pi$
$480$	0	0
$481$	−2.41698e8	−0.0990295
$482$	3.34368e8	0.136007
$483$	0	0
$484$	1.49631e9	0.599877
$485$	−6.52792e7	−0.0259824
$486$	0	0
$487$	−6.76814e8	−0.265533	−0.132766	−	0.991147i	$-0.542386\pi$
$487$	−6.76814e8	−0.265533	−0.132766	+	0.991147i	$0.542386\pi$
$488$	−1.07653e8	−0.0419332
$489$	0	0
$490$	2.94122e7	0.0112938
$491$	−3.97038e9	−1.51372	−0.756862	−	0.653574i	$-0.773269\pi$
$491$	−3.97038e9	−1.51372	−0.756862	+	0.653574i	$0.773269\pi$
$492$	0	0
$493$	−3.29350e8	−0.123792
$494$	−2.47394e7	−0.00923304
$495$	0	0
$496$	−7.86900e8	−0.289557
$497$	−3.72498e8	−0.136106
$498$	0	0
$499$	1.59757e9	0.575584	0.287792	−	0.957693i	$-0.407079\pi$
$499$	1.59757e9	0.575584	0.287792	+	0.957693i	$0.407079\pi$
$500$	−2.42188e8	−0.0866476
$501$	0	0
$502$	8.58056e7	0.0302728
$503$	−1.72479e9	−0.604295	−0.302148	−	0.953261i	$-0.597704\pi$
$503$	−1.72479e9	−0.604295	−0.302148	+	0.953261i	$0.597704\pi$
$504$	0	0
$505$	2.04951e8	0.0708157
$506$	−2.26092e8	−0.0775815
$507$	0	0
$508$	−3.79390e8	−0.128400
$509$	−5.83626e9	−1.96165	−0.980827	−	0.194880i	$-0.937568\pi$
$509$	−5.83626e9	−1.96165	−0.980827	+	0.194880i	$0.937568\pi$
$510$	0	0
$511$	−2.03124e9	−0.673423
$512$	2.35969e9	0.776980
$513$	0	0
$514$	−9.41674e8	−0.305865
$515$	−8.21882e8	−0.265145
$516$	0	0
$517$	−3.70791e8	−0.118008
$518$	−5.76912e7	−0.0182371
$519$	0	0
$520$	−1.81062e8	−0.0564697
$521$	−4.99457e9	−1.54727	−0.773634	−	0.633632i	$-0.781563\pi$
$521$	−4.99457e9	−1.54727	−0.773634	+	0.633632i	$0.781563\pi$
$522$	0	0
$523$	1.03723e9	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$523$	1.03723e9	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$524$	−4.42570e8	−0.134376
$525$	0	0
$526$	2.05214e8	0.0614834
$527$	4.91177e8	0.146184
$528$	0	0
$529$	−1.68258e9	−0.494174
$530$	3.17655e8	0.0926810
$531$	0	0
$532$	1.83058e8	0.0527105
$533$	−5.18303e8	−0.148265
$534$	0	0
$535$	−1.51850e9	−0.428723
$536$	−2.45563e8	−0.0688789
$537$	0	0
$538$	1.09896e9	0.304258
$539$	−3.20476e8	−0.0881525
$540$	0	0
$541$	−2.48336e9	−0.674295	−0.337147	−	0.941452i	$-0.609462\pi$
$541$	−2.48336e9	−0.674295	−0.337147	+	0.941452i	$0.609462\pi$
$542$	6.76409e8	0.182479
$543$	0	0
$544$	−8.74359e8	−0.232859
$545$	−1.21057e9	−0.320332
$546$	0	0
$547$	−3.74874e9	−0.979331	−0.489665	−	0.871910i	$-0.662881\pi$
$547$	−3.74874e9	−0.979331	−0.489665	+	0.871910i	$0.662881\pi$
$548$	5.99832e8	0.155703
$549$	0	0
$550$	−8.51250e7	−0.0218166
$551$	−1.52783e8	−0.0389086
$552$	0	0
$553$	−1.86243e9	−0.468319
$554$	7.70058e8	0.192415
$555$	0	0
$556$	−3.08230e8	−0.0760525
$557$	2.18032e9	0.534596	0.267298	−	0.963614i	$-0.413869\pi$
$557$	2.18032e9	0.534596	0.267298	+	0.963614i	$0.413869\pi$
$558$	0	0
$559$	−9.61410e7	−0.0232792
$560$	6.37294e8	0.153349
$561$	0	0
$562$	−7.17462e8	−0.170499
$563$	−2.56548e9	−0.605883	−0.302941	−	0.953009i	$-0.597969\pi$
$563$	−2.56548e9	−0.605883	−0.302941	+	0.953009i	$0.597969\pi$
$564$	0	0
$565$	−3.12869e9	−0.729783
$566$	1.16362e8	0.0269745
$567$	0	0
$568$	5.47344e8	0.125326
$569$	−1.65916e8	−0.0377568	−0.0188784	−	0.999822i	$-0.506010\pi$
$569$	−1.65916e8	−0.0377568	−0.0188784	+	0.999822i	$0.506010\pi$
$570$	0	0
$571$	−5.11928e9	−1.15075	−0.575377	−	0.817888i	$-0.695145\pi$
$571$	−5.11928e9	−1.15075	−0.575377	+	0.817888i	$0.695145\pi$
$572$	9.70768e8	0.216885
$573$	0	0
$574$	−1.23715e8	−0.0273042
$575$	6.48438e8	0.142243
$576$	0	0
$577$	−6.90669e9	−1.49677	−0.748384	−	0.663265i	$-0.769170\pi$
$577$	−6.90669e9	−1.49677	−0.748384	+	0.663265i	$0.769170\pi$
$578$	−6.48515e8	−0.139692
$579$	0	0
$580$	−5.50219e8	−0.117095
$581$	−2.37816e9	−0.503065
$582$	0	0
$583$	−3.46117e9	−0.723408
$584$	2.98468e9	0.620087
$585$	0	0
$586$	−1.11411e9	−0.228710
$587$	4.92686e9	1.00540	0.502698	−	0.864462i	$-0.332341\pi$
$587$	4.92686e9	1.00540	0.502698	+	0.864462i	$0.332341\pi$
$588$	0	0
$589$	2.27854e8	0.0459465
$590$	3.88313e8	0.0778395
$591$	0	0
$592$	−1.25003e9	−0.247626
$593$	−2.48990e9	−0.490332	−0.245166	−	0.969481i	$-0.578842\pi$
$593$	−2.48990e9	−0.490332	−0.245166	+	0.969481i	$0.578842\pi$
$594$	0	0
$595$	−3.97794e8	−0.0774193
$596$	−4.95870e9	−0.959413
$597$	0	0
$598$	2.38542e8	0.0456153
$599$	1.79975e9	0.342151	0.171076	−	0.985258i	$-0.445276\pi$
$599$	1.79975e9	0.342151	0.171076	+	0.985258i	$0.445276\pi$
$600$	0	0
$601$	−9.64250e9	−1.81188	−0.905939	−	0.423409i	$-0.860834\pi$
$601$	−9.64250e9	−1.81188	−0.905939	+	0.423409i	$0.860834\pi$
$602$	−2.29481e7	−0.00428705
$603$	0	0
$604$	6.67020e9	1.23171
$605$	−1.50837e9	−0.276927
$606$	0	0
$607$	−7.59240e9	−1.37790	−0.688951	−	0.724808i	$-0.741929\pi$
$607$	−7.59240e9	−1.37790	−0.688951	+	0.724808i	$0.741929\pi$
$608$	−4.05609e8	−0.0731889
$609$	0	0
$610$	5.33995e7	0.00952539
$611$	3.91209e8	0.0693848
$612$	0	0
$613$	6.21966e9	1.09057	0.545287	−	0.838249i	$-0.316421\pi$
$613$	6.21966e9	1.09057	0.545287	+	0.838249i	$0.316421\pi$
$614$	1.03960e9	0.181249
$615$	0	0
$616$	4.70903e8	0.0811707
$617$	1.01931e10	1.74706	0.873529	−	0.486772i	$-0.161826\pi$
$617$	1.01931e10	1.74706	0.873529	+	0.486772i	$0.161826\pi$
$618$	0	0
$619$	−1.86525e9	−0.316097	−0.158048	−	0.987431i	$-0.550520\pi$
$619$	−1.86525e9	−0.316097	−0.158048	+	0.987431i	$0.550520\pi$
$620$	8.20570e8	0.138275
$621$	0	0
$622$	7.86782e8	0.131096
$623$	−9.00766e7	−0.0149246
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	1.49329e9	0.243295
$627$	0	0
$628$	−1.19538e9	−0.192596
$629$	7.80261e8	0.125015
$630$	0	0
$631$	3.03048e9	0.480185	0.240092	−	0.970750i	$-0.422822\pi$
$631$	3.03048e9	0.480185	0.240092	+	0.970750i	$0.422822\pi$
$632$	2.73663e9	0.431228
$633$	0	0
$634$	−3.80440e8	−0.0592889
$635$	3.82450e8	0.0592743
$636$	0	0
$637$	3.38123e8	0.0518307
$638$	−1.93393e8	−0.0294828
$639$	0	0
$640$	−1.93637e9	−0.291983
$641$	−1.08550e10	−1.62789	−0.813945	−	0.580942i	$-0.802684\pi$
$641$	−1.08550e10	−1.62789	−0.813945	+	0.580942i	$0.802684\pi$
$642$	0	0
$643$	9.35593e8	0.138787	0.0693934	−	0.997589i	$-0.477894\pi$
$643$	9.35593e8	0.138787	0.0693934	+	0.997589i	$0.477894\pi$
$644$	−1.76508e9	−0.260413
$645$	0	0
$646$	7.98650e7	0.0116558
$647$	8.82519e9	1.28103	0.640514	−	0.767946i	$-0.278721\pi$
$647$	8.82519e9	1.28103	0.640514	+	0.767946i	$0.278721\pi$
$648$	0	0
$649$	−4.23106e9	−0.607565
$650$	8.98125e7	0.0128274
$651$	0	0
$652$	9.51506e9	1.34445
$653$	−2.99850e9	−0.421413	−0.210707	−	0.977549i	$-0.567576\pi$
$653$	−2.99850e9	−0.421413	−0.210707	+	0.977549i	$0.567576\pi$
$654$	0	0
$655$	4.46140e8	0.0620335
$656$	−2.68060e9	−0.370740
$657$	0	0
$658$	9.33783e7	0.0127778
$659$	3.93595e9	0.535736	0.267868	−	0.963456i	$-0.413681\pi$
$659$	3.93595e9	0.535736	0.267868	+	0.963456i	$0.413681\pi$
$660$	0	0
$661$	−6.26839e8	−0.0844211	−0.0422106	−	0.999109i	$-0.513440\pi$
$661$	−6.26839e8	−0.0844211	−0.0422106	+	0.999109i	$0.513440\pi$
$662$	−2.19986e9	−0.294708
$663$	0	0
$664$	3.49444e9	0.463222
$665$	−1.84534e8	−0.0243333
$666$	0	0
$667$	1.47317e9	0.192226
$668$	1.28782e9	0.167162
$669$	0	0
$670$	1.21807e8	0.0156463
$671$	−5.81841e8	−0.0743491
$672$	0	0
$673$	4.74113e9	0.599555	0.299777	−	0.954009i	$-0.403088\pi$
$673$	4.74113e9	0.599555	0.299777	+	0.954009i	$0.403088\pi$
$674$	9.89313e8	0.124458
$675$	0	0
$676$	6.75659e9	0.841229
$677$	8.34660e9	1.03383	0.516915	−	0.856037i	$-0.327080\pi$
$677$	8.34660e9	1.03383	0.516915	+	0.856037i	$0.327080\pi$
$678$	0	0
$679$	−1.79126e8	−0.0219591
$680$	5.84514e8	0.0712876
$681$	0	0
$682$	2.88417e8	0.0348157
$683$	−1.00158e10	−1.20286	−0.601428	−	0.798927i	$-0.705401\pi$
$683$	−1.00158e10	−1.20286	−0.601428	+	0.798927i	$0.705401\pi$
$684$	0	0
$685$	−6.04670e8	−0.0718789
$686$	8.07072e7	0.00954504
$687$	0	0
$688$	−4.97231e8	−0.0582101
$689$	3.65177e9	0.425340
$690$	0	0
$691$	−1.16744e10	−1.34605	−0.673024	−	0.739621i	$-0.735005\pi$
$691$	−1.16744e10	−1.34605	−0.673024	+	0.739621i	$0.735005\pi$
$692$	−7.00509e8	−0.0803605
$693$	0	0
$694$	1.86218e9	0.211477
$695$	3.10716e8	0.0351089
$696$	0	0
$697$	1.67321e9	0.187170
$698$	−9.13293e8	−0.101652
$699$	0	0
$700$	−6.64562e8	−0.0732306
$701$	−7.05584e9	−0.773634	−0.386817	−	0.922157i	$-0.626425\pi$
$701$	−7.05584e9	−0.773634	−0.386817	+	0.922157i	$0.626425\pi$
$702$	0	0
$703$	3.61958e8	0.0392929
$704$	4.66924e9	0.504362
$705$	0	0
$706$	1.00536e9	0.107524
$707$	5.62385e8	0.0598502
$708$	0	0
$709$	1.96477e8	0.0207038	0.0103519	−	0.999946i	$-0.496705\pi$
$709$	1.96477e8	0.0207038	0.0103519	+	0.999946i	$0.496705\pi$
$710$	−2.71500e8	−0.0284686
$711$	0	0
$712$	1.32357e8	0.0137426
$713$	−2.19701e9	−0.226996
$714$	0	0
$715$	−9.78597e8	−0.100123
$716$	−1.29422e10	−1.31769
$717$	0	0
$718$	2.79331e9	0.281633
$719$	8.96572e9	0.899569	0.449784	−	0.893137i	$-0.351501\pi$
$719$	8.96572e9	0.899569	0.449784	+	0.893137i	$0.351501\pi$
$720$	0	0
$721$	−2.25524e9	−0.224089
$722$	−1.75069e9	−0.173113
$723$	0	0
$724$	1.76405e10	1.72753
$725$	5.54656e8	0.0540556
$726$	0	0
$727$	−4.44082e9	−0.428640	−0.214320	−	0.976763i	$-0.568754\pi$
$727$	−4.44082e9	−0.428640	−0.214320	+	0.976763i	$0.568754\pi$
$728$	−4.96834e8	−0.0477256
$729$	0	0
$730$	−1.48049e9	−0.140856
$731$	3.10368e8	0.0293877
$732$	0	0
$733$	−8.97113e9	−0.841363	−0.420681	−	0.907208i	$-0.638209\pi$
$733$	−8.97113e9	−0.841363	−0.420681	+	0.907208i	$0.638209\pi$
$734$	−5.41000e8	−0.0504965
$735$	0	0
$736$	3.91096e9	0.361586
$737$	−1.32721e9	−0.122125
$738$	0	0
$739$	1.27011e10	1.15767	0.578835	−	0.815445i	$-0.303508\pi$
$739$	1.27011e10	1.15767	0.578835	+	0.815445i	$0.303508\pi$
$740$	1.30352e9	0.118251
$741$	0	0
$742$	8.71647e8	0.0783298
$743$	1.54200e10	1.37919	0.689593	−	0.724197i	$-0.257789\pi$
$743$	1.54200e10	1.37919	0.689593	+	0.724197i	$0.257789\pi$
$744$	0	0
$745$	4.99869e9	0.442903
$746$	−2.12845e9	−0.187706
$747$	0	0
$748$	−3.13389e9	−0.273796
$749$	−4.16677e9	−0.362337
$750$	0	0
$751$	4.99561e9	0.430376	0.215188	−	0.976573i	$-0.430964\pi$
$751$	4.99561e9	0.430376	0.215188	+	0.976573i	$0.430964\pi$
$752$	2.02329e9	0.173498
$753$	0	0
$754$	2.04043e8	0.0173349
$755$	−6.72399e9	−0.568608
$756$	0	0
$757$	1.13781e10	0.953313	0.476657	−	0.879090i	$-0.341848\pi$
$757$	1.13781e10	0.953313	0.476657	+	0.879090i	$0.341848\pi$
$758$	−7.90912e8	−0.0659608
$759$	0	0
$760$	2.71152e8	0.0224060
$761$	−9.31372e9	−0.766084	−0.383042	−	0.923731i	$-0.625124\pi$
$761$	−9.31372e9	−0.766084	−0.383042	+	0.923731i	$0.625124\pi$
$762$	0	0
$763$	−3.32179e9	−0.270730
$764$	−4.16061e9	−0.337544
$765$	0	0
$766$	−3.29586e8	−0.0264953
$767$	4.46405e9	0.357228
$768$	0	0
$769$	−5.75726e9	−0.456535	−0.228267	−	0.973598i	$-0.573306\pi$
$769$	−5.75726e9	−0.456535	−0.228267	+	0.973598i	$0.573306\pi$
$770$	−2.33583e8	−0.0184384
$771$	0	0
$772$	2.11542e10	1.65477
$773$	−1.45658e10	−1.13424	−0.567121	−	0.823634i	$-0.691943\pi$
$773$	−1.45658e10	−1.13424	−0.567121	+	0.823634i	$0.691943\pi$
$774$	0	0
$775$	−8.27188e8	−0.0638334
$776$	2.63206e8	0.0202199
$777$	0	0
$778$	3.42251e8	0.0260565
$779$	7.76192e8	0.0588285
$780$	0	0
$781$	2.95826e9	0.222207
$782$	−7.70074e8	−0.0575849
$783$	0	0
$784$	1.74873e9	0.129604
$785$	1.20502e9	0.0889102
$786$	0	0
$787$	7.81273e9	0.571336	0.285668	−	0.958329i	$-0.407785\pi$
$787$	7.81273e9	0.571336	0.285668	+	0.958329i	$0.407785\pi$
$788$	8.59201e9	0.625537
$789$	0	0
$790$	−1.35746e9	−0.0979560
$791$	−8.58514e9	−0.616779
$792$	0	0
$793$	6.13881e8	0.0437147
$794$	3.47103e9	0.246086
$795$	0	0
$796$	8.33168e9	0.585514
$797$	−5.72212e9	−0.400362	−0.200181	−	0.979759i	$-0.564153\pi$
$797$	−5.72212e9	−0.400362	−0.200181	+	0.979759i	$0.564153\pi$
$798$	0	0
$799$	−1.26292e9	−0.0875916
$800$	1.47250e9	0.101681
$801$	0	0
$802$	−4.52685e8	−0.0309875
$803$	1.61315e10	1.09943
$804$	0	0
$805$	1.77931e9	0.120217
$806$	−3.04299e8	−0.0204705
$807$	0	0
$808$	−8.26361e8	−0.0551100
$809$	9.16070e9	0.608287	0.304144	−	0.952626i	$-0.401630\pi$
$809$	9.16070e9	0.608287	0.304144	+	0.952626i	$0.401630\pi$
$810$	0	0
$811$	−2.04091e10	−1.34354	−0.671771	−	0.740759i	$-0.734466\pi$
$811$	−2.04091e10	−1.34354	−0.671771	+	0.740759i	$0.734466\pi$
$812$	−1.50980e9	−0.0989632
$813$	0	0
$814$	4.58166e8	0.0297740
$815$	−9.59179e9	−0.620652
$816$	0	0
$817$	1.43977e8	0.00923671
$818$	2.29738e9	0.146756
$819$	0	0
$820$	2.79530e9	0.177044
$821$	−6.26680e9	−0.395225	−0.197613	−	0.980280i	$-0.563319\pi$
$821$	−6.26680e9	−0.395225	−0.197613	+	0.980280i	$0.563319\pi$
$822$	0	0
$823$	4.37792e9	0.273759	0.136879	−	0.990588i	$-0.456293\pi$
$823$	4.37792e9	0.273759	0.136879	+	0.990588i	$0.456293\pi$
$824$	3.31383e9	0.206341
$825$	0	0
$826$	1.06553e9	0.0657864
$827$	1.61665e10	0.993912	0.496956	−	0.867776i	$-0.334451\pi$
$827$	1.61665e10	0.993912	0.496956	+	0.867776i	$0.334451\pi$
$828$	0	0
$829$	2.76523e10	1.68574	0.842868	−	0.538120i	$-0.180865\pi$
$829$	2.76523e10	1.68574	0.842868	+	0.538120i	$0.180865\pi$
$830$	−1.73335e9	−0.105224
$831$	0	0
$832$	−4.92636e9	−0.296548
$833$	−1.09155e9	−0.0654312
$834$	0	0
$835$	−1.29820e9	−0.0771685
$836$	−1.45379e9	−0.0860556
$837$	0	0
$838$	−3.96731e9	−0.232885
$839$	1.72137e10	1.00625	0.503126	−	0.864213i	$-0.332183\pi$
$839$	1.72137e10	1.00625	0.503126	+	0.864213i	$0.332183\pi$
$840$	0	0
$841$	−1.59898e10	−0.926950
$842$	5.32089e9	0.307179
$843$	0	0
$844$	6.38245e9	0.365417
$845$	−6.81108e9	−0.388345
$846$	0	0
$847$	−4.13898e9	−0.234046
$848$	1.88865e10	1.06357
$849$	0	0
$850$	−2.89938e8	−0.0161934
$851$	−3.49007e9	−0.194124
$852$	0	0
$853$	6.71439e9	0.370412	0.185206	−	0.982700i	$-0.440705\pi$
$853$	6.71439e9	0.370412	0.185206	+	0.982700i	$0.440705\pi$
$854$	1.46528e8	0.00805042
$855$	0	0
$856$	6.12260e9	0.333639
$857$	9.89153e9	0.536822	0.268411	−	0.963304i	$-0.413501\pi$
$857$	9.89153e9	0.536822	0.268411	+	0.963304i	$0.413501\pi$
$858$	0	0
$859$	−3.35625e10	−1.80667	−0.903335	−	0.428937i	$-0.858888\pi$
$859$	−3.35625e10	−1.80667	−0.903335	+	0.428937i	$0.858888\pi$
$860$	5.18506e8	0.0277977
$861$	0	0
$862$	3.00956e9	0.160040
$863$	−9.99427e9	−0.529314	−0.264657	−	0.964343i	$-0.585259\pi$
$863$	−9.99427e9	−0.529314	−0.264657	+	0.964343i	$0.585259\pi$
$864$	0	0
$865$	7.06159e8	0.0370976
$866$	2.75606e9	0.144203
$867$	0	0
$868$	2.25164e9	0.116864
$869$	1.47908e10	0.764581
$870$	0	0
$871$	1.40029e9	0.0718051
$872$	4.88100e9	0.249288
$873$	0	0
$874$	−3.57232e8	−0.0180992
$875$	6.69922e8	0.0338062
$876$	0	0
$877$	1.24239e8	0.00621953	0.00310977	−	0.999995i	$-0.499010\pi$
$877$	1.24239e8	0.00621953	0.00310977	+	0.999995i	$0.499010\pi$
$878$	4.30032e9	0.214422
$879$	0	0
$880$	−5.06119e9	−0.250359
$881$	−2.45438e10	−1.20928	−0.604640	−	0.796499i	$-0.706683\pi$
$881$	−2.45438e10	−1.20928	−0.604640	+	0.796499i	$0.706683\pi$
$882$	0	0
$883$	9.35492e9	0.457275	0.228638	−	0.973512i	$-0.426573\pi$
$883$	9.35492e9	0.457275	0.228638	+	0.973512i	$0.426573\pi$
$884$	3.30646e9	0.160983
$885$	0	0
$886$	2.02188e9	0.0976646
$887$	3.18320e10	1.53155	0.765776	−	0.643107i	$-0.222355\pi$
$887$	3.18320e10	1.53155	0.765776	+	0.643107i	$0.222355\pi$
$888$	0	0
$889$	1.04944e9	0.0500959
$890$	−6.56535e7	−0.00312171
$891$	0	0
$892$	3.09221e10	1.45879
$893$	−5.85860e8	−0.0275305
$894$	0	0
$895$	1.30466e10	0.608298
$896$	−5.31339e9	−0.246771
$897$	0	0
$898$	−3.65297e8	−0.0168337
$899$	−1.87926e9	−0.0862639
$900$	0	0
$901$	−1.17888e10	−0.536950
$902$	9.82503e8	0.0445770
$903$	0	0
$904$	1.26149e10	0.567929
$905$	−1.77827e10	−0.797496
$906$	0	0
$907$	2.00342e10	0.891552	0.445776	−	0.895144i	$-0.352928\pi$
$907$	2.00342e10	0.891552	0.445776	+	0.895144i	$0.352928\pi$
$908$	4.88470e9	0.216540
$909$	0	0
$910$	2.46446e8	0.0108412
$911$	2.82239e10	1.23681	0.618405	−	0.785859i	$-0.287779\pi$
$911$	2.82239e10	1.23681	0.618405	+	0.785859i	$0.287779\pi$
$912$	0	0
$913$	1.88866e10	0.821308
$914$	−3.03222e6	−0.000131356 0
$915$	0	0
$916$	−1.56171e10	−0.671377
$917$	1.22421e9	0.0524279
$918$	0	0
$919$	−3.40984e10	−1.44921	−0.724603	−	0.689167i	$-0.757977\pi$
$919$	−3.40984e10	−1.44921	−0.724603	+	0.689167i	$0.757977\pi$
$920$	−2.61450e9	−0.110696
$921$	0	0
$922$	9.59881e8	0.0403329
$923$	−3.12116e9	−0.130650
$924$	0	0
$925$	−1.31403e9	−0.0545896
$926$	−1.37362e9	−0.0568499
$927$	0	0
$928$	3.34533e9	0.137411
$929$	2.18397e10	0.893698	0.446849	−	0.894609i	$-0.352546\pi$
$929$	2.18397e10	0.893698	0.446849	+	0.894609i	$0.352546\pi$
$930$	0	0
$931$	−5.06361e8	−0.0205654
$932$	−1.92560e10	−0.779131
$933$	0	0
$934$	−3.38007e9	−0.135741
$935$	3.15916e9	0.126395
$936$	0	0
$937$	−1.29359e10	−0.513699	−0.256849	−	0.966451i	$-0.582684\pi$
$937$	−1.29359e10	−0.513699	−0.256849	+	0.966451i	$0.582684\pi$
$938$	3.34238e8	0.0132235
$939$	0	0
$940$	−2.10986e9	−0.0828526
$941$	1.70996e10	0.668995	0.334497	−	0.942397i	$-0.391433\pi$
$941$	1.70996e10	0.668995	0.334497	+	0.942397i	$0.391433\pi$
$942$	0	0
$943$	−7.48419e9	−0.290639
$944$	2.30875e10	0.893255
$945$	0	0
$946$	1.82246e8	0.00699907
$947$	1.95022e8	0.00746207	0.00373104	−	0.999993i	$-0.498812\pi$
$947$	1.95022e8	0.00746207	0.00373104	+	0.999993i	$0.498812\pi$
$948$	0	0
$949$	−1.70198e10	−0.646430
$950$	−1.34500e8	−0.00508967
$951$	0	0
$952$	1.60391e9	0.0602490
$953$	1.97958e10	0.740878	0.370439	−	0.928857i	$-0.379207\pi$
$953$	1.97958e10	0.740878	0.370439	+	0.928857i	$0.379207\pi$
$954$	0	0
$955$	4.19417e9	0.155824
$956$	−1.90416e10	−0.704856
$957$	0	0
$958$	−4.79686e9	−0.176270
$959$	−1.65921e9	−0.0607487
$960$	0	0
$961$	−2.47100e10	−0.898132
$962$	−4.83395e8	−0.0175061
$963$	0	0
$964$	−2.07308e10	−0.745327
$965$	−2.13248e10	−0.763906
$966$	0	0
$967$	4.18825e10	1.48950	0.744748	−	0.667346i	$-0.232570\pi$
$967$	4.18825e10	1.48950	0.744748	+	0.667346i	$0.232570\pi$
$968$	6.08177e9	0.215509
$969$	0	0
$970$	−1.30559e8	−0.00459308
$971$	1.82766e9	0.0640661	0.0320331	−	0.999487i	$-0.489802\pi$
$971$	1.82766e9	0.0640661	0.0320331	+	0.999487i	$0.489802\pi$
$972$	0	0
$973$	8.52605e8	0.0296724
$974$	−1.35363e9	−0.0469400
$975$	0	0
$976$	3.17492e9	0.109310
$977$	−4.04873e10	−1.38895	−0.694477	−	0.719515i	$-0.744364\pi$
$977$	−4.04873e10	−1.38895	−0.694477	+	0.719515i	$0.744364\pi$
$978$	0	0
$979$	7.15361e8	0.0243661
$980$	−1.82356e9	−0.0618912
$981$	0	0
$982$	−7.94076e9	−0.267591
$983$	−1.79913e9	−0.0604124	−0.0302062	−	0.999544i	$-0.509616\pi$
$983$	−1.79913e9	−0.0604124	−0.0302062	+	0.999544i	$0.509616\pi$
$984$	0	0
$985$	−8.66130e9	−0.288773
$986$	−6.58701e8	−0.0218836
$987$	0	0
$988$	1.53384e9	0.0505978
$989$	−1.38826e9	−0.0456334
$990$	0	0
$991$	−1.80926e10	−0.590532	−0.295266	−	0.955415i	$-0.595408\pi$
$991$	−1.80926e10	−0.590532	−0.295266	+	0.955415i	$0.595408\pi$
$992$	−4.98907e9	−0.162266
$993$	0	0
$994$	−7.44996e8	−0.0240603
$995$	−8.39888e9	−0.270296
$996$	0	0
$997$	4.79210e10	1.53141	0.765707	−	0.643189i	$-0.222389\pi$
$997$	4.79210e10	1.53141	0.765707	+	0.643189i	$0.222389\pi$
$998$	3.19514e9	0.101750
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.b.1.1		1
3.2	odd	2		105.8.a.a.1.1	✓	1
15.14	odd	2		525.8.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.8.a.a.1.1	✓	1	3.2	odd	2
315.8.a.b.1.1		1	1.1	even	1	trivial
525.8.a.c.1.1		1	15.14	odd	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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