LMFDB - Embedded newform 315.10.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 648x^{2} + 6926x - 8308 $ x^4 - x^3 - 648x^2 + 6926x - 8308
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-29.3917$ of defining polynomial
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)

$f(q)$	$=$	\(q-15.4436 q^{2} -273.496 q^{4} +625.000 q^{5} -2401.00 q^{7} +12130.9 q^{8} -9652.23 q^{10} +8060.89 q^{11} -137129. q^{13} +37080.0 q^{14} -47313.7 q^{16} +23676.1 q^{17} +572389. q^{19} -170935. q^{20} -124489. q^{22} +997700. q^{23} +390625. q^{25} +2.11777e6 q^{26} +656664. q^{28} -1.97927e6 q^{29} -8.47740e6 q^{31} -5.48031e6 q^{32} -365643. q^{34} -1.50062e6 q^{35} -4.33062e6 q^{37} -8.83972e6 q^{38} +7.58179e6 q^{40} +1.48554e7 q^{41} +3.14842e7 q^{43} -2.20462e6 q^{44} -1.54081e7 q^{46} -2.31045e7 q^{47} +5.76480e6 q^{49} -6.03264e6 q^{50} +3.75044e7 q^{52} +6.79444e6 q^{53} +5.03805e6 q^{55} -2.91262e7 q^{56} +3.05671e7 q^{58} -8.85117e7 q^{59} +1.24823e8 q^{61} +1.30921e8 q^{62} +1.08860e8 q^{64} -8.57059e7 q^{65} +9.58712e7 q^{67} -6.47531e6 q^{68} +2.31750e7 q^{70} +2.16795e8 q^{71} -1.50701e8 q^{73} +6.68803e7 q^{74} -1.56546e8 q^{76} -1.93542e7 q^{77} -3.89487e8 q^{79} -2.95711e7 q^{80} -2.29420e8 q^{82} +7.43467e8 q^{83} +1.47975e7 q^{85} -4.86228e8 q^{86} +9.77855e7 q^{88} -2.64429e8 q^{89} +3.29248e8 q^{91} -2.72867e8 q^{92} +3.56816e8 q^{94} +3.57743e8 q^{95} +1.39343e9 q^{97} -8.90291e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 19 q^{2} + 1729 q^{4} + 2500 q^{5} - 9604 q^{7} + 30495 q^{8} + 11875 q^{10} - 82438 q^{11} - 72962 q^{13} - 45619 q^{14} + 64257 q^{16} + 357542 q^{17} + 300732 q^{19} + 1080625 q^{20} - 5595068 q^{22}+ \cdots + 109531219 q^{98}+O(q^{100}) $ 4 * q + 19 * q^2 + 1729 * q^4 + 2500 * q^5 - 9604 * q^7 + 30495 * q^8 + 11875 * q^10 - 82438 * q^11 - 72962 * q^13 - 45619 * q^14 + 64257 * q^16 + 357542 * q^17 + 300732 * q^19 + 1080625 * q^20 - 5595068 * q^22 + 2340836 * q^23 + 1562500 * q^25 - 1305666 * q^26 - 4151329 * q^28 + 1372022 * q^29 - 4590888 * q^31 + 10529439 * q^32 - 7199478 * q^34 - 6002500 * q^35 - 38868456 * q^37 + 15477280 * q^38 + 19059375 * q^40 - 57287084 * q^41 - 43403452 * q^43 - 87639348 * q^44 + 28892416 * q^46 - 91822222 * q^47 + 23059204 * q^49 + 7421875 * q^50 + 1953386 * q^52 + 61086884 * q^53 - 51523750 * q^55 - 73218495 * q^56 + 29615390 * q^58 - 72569680 * q^59 - 225681036 * q^61 - 190286472 * q^62 + 141443393 * q^64 - 45601250 * q^65 - 4429720 * q^67 - 15339698 * q^68 - 28511875 * q^70 + 470468984 * q^71 - 168326464 * q^73 - 927809502 * q^74 - 892265344 * q^76 + 197933638 * q^77 - 598805646 * q^79 + 40160625 * q^80 - 1821993318 * q^82 + 1159074304 * q^83 + 223463750 * q^85 - 706791156 * q^86 - 1992103900 * q^88 + 1380153012 * q^89 + 175181762 * q^91 - 710319264 * q^92 + 2466744152 * q^94 + 187957500 * q^95 + 754874082 * q^97 + 109531219 * q^98

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 $

$2$	−15.4436	−0.682516	−0.341258	−	0.939970i	$-0.610853\pi$
$2$	−15.4436	−0.682516	−0.341258	+	0.939970i	$0.610853\pi$
$3$	0	0
$3$	0	0
$4$	−273.496	−0.534172
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	12130.9	1.04710
$9$	0	0
$10$	−9652.23	−0.305230
$11$	8060.89	0.166003	0.0830015	−	0.996549i	$-0.473549\pi$
$11$	8060.89	0.166003	0.0830015	+	0.996549i	$0.473549\pi$
$12$	0	0
$13$	−137129.	−1.33164	−0.665818	−	0.746114i	$-0.731917\pi$
$13$	−137129.	−1.33164	−0.665818	+	0.746114i	$0.731917\pi$
$14$	37080.0	0.257967
$15$	0	0
$16$	−47313.7	−0.180488
$17$	23676.1	0.0687526	0.0343763	−	0.999409i	$-0.489056\pi$
$17$	23676.1	0.0687526	0.0343763	+	0.999409i	$0.489056\pi$
$18$	0	0
$19$	572389.	1.00763	0.503814	−	0.863812i	$-0.331930\pi$
$19$	572389.	1.00763	0.503814	+	0.863812i	$0.331930\pi$
$20$	−170935.	−0.238889
$21$	0	0
$22$	−124489.	−0.113300
$23$	997700.	0.743404	0.371702	−	0.928352i	$-0.378774\pi$
$23$	997700.	0.743404	0.371702	+	0.928352i	$0.378774\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	2.11777e6	0.908862
$27$	0	0
$28$	656664.	0.201898
$29$	−1.97927e6	−0.519655	−0.259827	−	0.965655i	$-0.583666\pi$
$29$	−1.97927e6	−0.519655	−0.259827	+	0.965655i	$0.583666\pi$
$30$	0	0
$31$	−8.47740e6	−1.64867	−0.824337	−	0.566099i	$-0.808452\pi$
$31$	−8.47740e6	−1.64867	−0.824337	+	0.566099i	$0.808452\pi$
$32$	−5.48031e6	−0.923911
$33$	0	0
$34$	−365643.	−0.0469247
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−4.33062e6	−0.379877	−0.189938	−	0.981796i	$-0.560829\pi$
$37$	−4.33062e6	−0.379877	−0.189938	+	0.981796i	$0.560829\pi$
$38$	−8.83972e6	−0.687721
$39$	0	0
$40$	7.58179e6	0.468276
$41$	1.48554e7	0.821024	0.410512	−	0.911855i	$-0.365350\pi$
$41$	1.48554e7	0.821024	0.410512	+	0.911855i	$0.365350\pi$
$42$	0	0
$43$	3.14842e7	1.40438	0.702189	−	0.711991i	$-0.252206\pi$
$43$	3.14842e7	1.40438	0.702189	+	0.711991i	$0.252206\pi$
$44$	−2.20462e6	−0.0886742
$45$	0	0
$46$	−1.54081e7	−0.507385
$47$	−2.31045e7	−0.690647	−0.345324	−	0.938484i	$-0.612231\pi$
$47$	−2.31045e7	−0.690647	−0.345324	+	0.938484i	$0.612231\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	−6.03264e6	−0.136503
$51$	0	0
$52$	3.75044e7	0.711323
$53$	6.79444e6	0.118280	0.0591401	−	0.998250i	$-0.481164\pi$
$53$	6.79444e6	0.118280	0.0591401	+	0.998250i	$0.481164\pi$
$54$	0	0
$55$	5.03805e6	0.0742388
$56$	−2.91262e7	−0.395765
$57$	0	0
$58$	3.05671e7	0.354673
$59$	−8.85117e7	−0.950969	−0.475485	−	0.879724i	$-0.657727\pi$
$59$	−8.85117e7	−0.950969	−0.475485	+	0.879724i	$0.657727\pi$
$60$	0	0
$61$	1.24823e8	1.15428	0.577138	−	0.816647i	$-0.304169\pi$
$61$	1.24823e8	1.15428	0.577138	+	0.816647i	$0.304169\pi$
$62$	1.30921e8	1.12525
$63$	0	0
$64$	1.08860e8	0.811071
$65$	−8.57059e7	−0.595526
$66$	0	0
$67$	9.58712e7	0.581235	0.290617	−	0.956839i	$-0.406139\pi$
$67$	9.58712e7	0.581235	0.290617	+	0.956839i	$0.406139\pi$
$68$	−6.47531e6	−0.0367257
$69$	0	0
$70$	2.31750e7	0.115366
$71$	2.16795e8	1.01248	0.506241	−	0.862392i	$-0.331035\pi$
$71$	2.16795e8	1.01248	0.506241	+	0.862392i	$0.331035\pi$
$72$	0	0
$73$	−1.50701e8	−0.621101	−0.310551	−	0.950557i	$-0.600513\pi$
$73$	−1.50701e8	−0.621101	−0.310551	+	0.950557i	$0.600513\pi$
$74$	6.68803e7	0.259272
$75$	0	0
$76$	−1.56546e8	−0.538247
$77$	−1.93542e7	−0.0627432
$78$	0	0
$79$	−3.89487e8	−1.12505	−0.562523	−	0.826781i	$-0.690169\pi$
$79$	−3.89487e8	−1.12505	−0.562523	+	0.826781i	$0.690169\pi$
$80$	−2.95711e7	−0.0807165
$81$	0	0
$82$	−2.29420e8	−0.560362
$83$	7.43467e8	1.71953	0.859766	−	0.510688i	$-0.170609\pi$
$83$	7.43467e8	1.71953	0.859766	+	0.510688i	$0.170609\pi$
$84$	0	0
$85$	1.47975e7	0.0307471
$86$	−4.86228e8	−0.958510
$87$	0	0
$88$	9.77855e7	0.173821
$89$	−2.64429e8	−0.446739	−0.223369	−	0.974734i	$-0.571706\pi$
$89$	−2.64429e8	−0.446739	−0.223369	+	0.974734i	$0.571706\pi$
$90$	0	0
$91$	3.29248e8	0.503311
$92$	−2.72867e8	−0.397106
$93$	0	0
$94$	3.56816e8	0.471378
$95$	3.57743e8	0.450625
$96$	0	0
$97$	1.39343e9	1.59813	0.799063	−	0.601248i	$-0.205329\pi$
$97$	1.39343e9	1.59813	0.799063	+	0.601248i	$0.205329\pi$
$98$	−8.90291e7	−0.0975022
$99$	0	0
$100$	−1.06834e8	−0.106834
$101$	6.43863e8	0.615669	0.307834	−	0.951440i	$-0.400396\pi$
$101$	6.43863e8	0.615669	0.307834	+	0.951440i	$0.400396\pi$
$102$	0	0
$103$	−9.64216e8	−0.844125	−0.422063	−	0.906567i	$-0.638694\pi$
$103$	−9.64216e8	−0.844125	−0.422063	+	0.906567i	$0.638694\pi$
$104$	−1.66350e9	−1.39435
$105$	0	0
$106$	−1.04930e8	−0.0807281
$107$	1.97477e9	1.45643	0.728215	−	0.685348i	$-0.240350\pi$
$107$	1.97477e9	1.45643	0.728215	+	0.685348i	$0.240350\pi$
$108$	0	0
$109$	1.03957e9	0.705401	0.352701	−	0.935736i	$-0.385263\pi$
$109$	1.03957e9	0.705401	0.352701	+	0.935736i	$0.385263\pi$
$110$	−7.78055e7	−0.0506691
$111$	0	0
$112$	1.13600e8	0.0682179
$113$	−5.48265e8	−0.316328	−0.158164	−	0.987413i	$-0.550557\pi$
$113$	−5.48265e8	−0.316328	−0.158164	+	0.987413i	$0.550557\pi$
$114$	0	0
$115$	6.23563e8	0.332460
$116$	5.41324e8	0.277585
$117$	0	0
$118$	1.36694e9	0.649052
$119$	−5.68462e7	−0.0259860
$120$	0	0
$121$	−2.29297e9	−0.972443
$122$	−1.92771e9	−0.787811
$123$	0	0
$124$	2.31854e9	0.880676
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−5.40565e9	−1.84387	−0.921936	−	0.387341i	$-0.873394\pi$
$127$	−5.40565e9	−1.84387	−0.921936	+	0.387341i	$0.873394\pi$
$128$	1.12473e9	0.370342
$129$	0	0
$130$	1.32360e9	0.406456
$131$	1.81287e9	0.537832	0.268916	−	0.963164i	$-0.413335\pi$
$131$	1.81287e9	0.537832	0.268916	+	0.963164i	$0.413335\pi$
$132$	0	0
$133$	−1.37431e9	−0.380847
$134$	−1.48059e9	−0.396702
$135$	0	0
$136$	2.87211e8	0.0719906
$137$	7.50191e8	0.181940	0.0909702	−	0.995854i	$-0.471003\pi$
$137$	7.50191e8	0.181940	0.0909702	+	0.995854i	$0.471003\pi$
$138$	0	0
$139$	6.59090e9	1.49754	0.748769	−	0.662831i	$-0.230645\pi$
$139$	6.59090e9	1.49754	0.748769	+	0.662831i	$0.230645\pi$
$140$	4.10415e8	0.0902916
$141$	0	0
$142$	−3.34809e9	−0.691035
$143$	−1.10538e9	−0.221055
$144$	0	0
$145$	−1.23705e9	−0.232397
$146$	2.32736e9	0.423911
$147$	0	0
$148$	1.18441e9	0.202920
$149$	3.29437e9	0.547564	0.273782	−	0.961792i	$-0.411725\pi$
$149$	3.29437e9	0.547564	0.273782	+	0.961792i	$0.411725\pi$
$150$	0	0
$151$	−1.04401e10	−1.63421	−0.817103	−	0.576492i	$-0.804421\pi$
$151$	−1.04401e10	−1.63421	−0.817103	+	0.576492i	$0.804421\pi$
$152$	6.94357e9	1.05508
$153$	0	0
$154$	2.98898e8	0.0428232
$155$	−5.29837e9	−0.737310
$156$	0	0
$157$	−4.88247e9	−0.641344	−0.320672	−	0.947190i	$-0.603909\pi$
$157$	−4.88247e9	−0.641344	−0.320672	+	0.947190i	$0.603909\pi$
$158$	6.01506e9	0.767862
$159$	0	0
$160$	−3.42519e9	−0.413186
$161$	−2.39548e9	−0.280980
$162$	0	0
$163$	−1.00685e10	−1.11717	−0.558584	−	0.829448i	$-0.688655\pi$
$163$	−1.00685e10	−1.11717	−0.558584	+	0.829448i	$0.688655\pi$
$164$	−4.06289e9	−0.438568
$165$	0	0
$166$	−1.14818e10	−1.17361
$167$	−1.11826e10	−1.11255	−0.556273	−	0.831000i	$-0.687769\pi$
$167$	−1.11826e10	−1.11255	−0.556273	+	0.831000i	$0.687769\pi$
$168$	0	0
$169$	8.19997e9	0.773254
$170$	−2.28527e8	−0.0209854
$171$	0	0
$172$	−8.61080e9	−0.750180
$173$	1.55595e10	1.32065	0.660325	−	0.750980i	$-0.270418\pi$
$173$	1.55595e10	1.32065	0.660325	+	0.750980i	$0.270418\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−3.81391e8	−0.0299615
$177$	0	0
$178$	4.08372e9	0.304906
$179$	1.14811e10	0.835884	0.417942	−	0.908474i	$-0.362752\pi$
$179$	1.14811e10	0.835884	0.417942	+	0.908474i	$0.362752\pi$
$180$	0	0
$181$	1.64859e10	1.14172	0.570860	−	0.821047i	$-0.306610\pi$
$181$	1.64859e10	1.14172	0.570860	+	0.821047i	$0.306610\pi$
$182$	−5.08476e9	−0.343518
$183$	0	0
$184$	1.21030e10	0.778416
$185$	−2.70664e9	−0.169886
$186$	0	0
$187$	1.90850e8	0.0114131
$188$	6.31900e9	0.368925
$189$	0	0
$190$	−5.52483e9	−0.307558
$191$	−1.75186e10	−0.952466	−0.476233	−	0.879319i	$-0.657998\pi$
$191$	−1.75186e10	−0.952466	−0.476233	+	0.879319i	$0.657998\pi$
$192$	0	0
$193$	−2.51278e10	−1.30361	−0.651805	−	0.758387i	$-0.725988\pi$
$193$	−2.51278e10	−1.30361	−0.651805	+	0.758387i	$0.725988\pi$
$194$	−2.15195e10	−1.09075
$195$	0	0
$196$	−1.57665e9	−0.0763103
$197$	−3.28481e10	−1.55386	−0.776930	−	0.629587i	$-0.783224\pi$
$197$	−3.28481e10	−1.55386	−0.776930	+	0.629587i	$0.783224\pi$
$198$	0	0
$199$	−3.56924e9	−0.161338	−0.0806691	−	0.996741i	$-0.525706\pi$
$199$	−3.56924e9	−0.161338	−0.0806691	+	0.996741i	$0.525706\pi$
$200$	4.73862e9	0.209419
$201$	0	0
$202$	−9.94354e9	−0.420204
$203$	4.75224e9	0.196411
$204$	0	0
$205$	9.28460e9	0.367173
$206$	1.48909e10	0.576129
$207$	0	0
$208$	6.48810e9	0.240344
$209$	4.61396e9	0.167269
$210$	0	0
$211$	−8.88255e9	−0.308508	−0.154254	−	0.988031i	$-0.549297\pi$
$211$	−8.88255e9	−0.308508	−0.154254	+	0.988031i	$0.549297\pi$
$212$	−1.85825e9	−0.0631820
$213$	0	0
$214$	−3.04975e10	−0.994037
$215$	1.96776e10	0.628057
$216$	0	0
$217$	2.03542e10	0.623140
$218$	−1.60547e10	−0.481447
$219$	0	0
$220$	−1.37789e9	−0.0396563
$221$	−3.24668e9	−0.0915534
$222$	0	0
$223$	1.85138e10	0.501330	0.250665	−	0.968074i	$-0.419351\pi$
$223$	1.85138e10	0.501330	0.250665	+	0.968074i	$0.419351\pi$
$224$	1.31582e10	0.349206
$225$	0	0
$226$	8.46716e9	0.215899
$227$	−6.91501e10	−1.72853	−0.864265	−	0.503037i	$-0.832216\pi$
$227$	−6.91501e10	−1.72853	−0.864265	+	0.503037i	$0.832216\pi$
$228$	0	0
$229$	−4.14990e10	−0.997190	−0.498595	−	0.866835i	$-0.666150\pi$
$229$	−4.14990e10	−0.997190	−0.498595	+	0.866835i	$0.666150\pi$
$230$	−9.63003e9	−0.226909
$231$	0	0
$232$	−2.40103e10	−0.544129
$233$	−1.06684e10	−0.237136	−0.118568	−	0.992946i	$-0.537830\pi$
$233$	−1.06684e10	−0.237136	−0.118568	+	0.992946i	$0.537830\pi$
$234$	0	0
$235$	−1.44403e10	−0.308867
$236$	2.42076e10	0.507982
$237$	0	0
$238$	8.77908e8	0.0177359
$239$	−6.89311e9	−0.136655	−0.0683274	−	0.997663i	$-0.521766\pi$
$239$	−6.89311e9	−0.136655	−0.0683274	+	0.997663i	$0.521766\pi$
$240$	0	0
$241$	−2.72593e10	−0.520520	−0.260260	−	0.965539i	$-0.583808\pi$
$241$	−2.72593e10	−0.520520	−0.260260	+	0.965539i	$0.583808\pi$
$242$	3.54116e10	0.663708
$243$	0	0
$244$	−3.41386e10	−0.616582
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−7.84913e10	−1.34179
$248$	−1.02838e11	−1.72632
$249$	0	0
$250$	−3.77040e9	−0.0610461
$251$	−1.11185e11	−1.76812	−0.884062	−	0.467370i	$-0.845202\pi$
$251$	−1.11185e11	−1.76812	−0.884062	+	0.467370i	$0.845202\pi$
$252$	0	0
$253$	8.04235e9	0.123407
$254$	8.34825e10	1.25847
$255$	0	0
$256$	−7.31062e10	−1.06384
$257$	3.84555e10	0.549869	0.274935	−	0.961463i	$-0.411344\pi$
$257$	3.84555e10	0.549869	0.274935	+	0.961463i	$0.411344\pi$
$258$	0	0
$259$	1.03978e10	0.143580
$260$	2.34402e10	0.318113
$261$	0	0
$262$	−2.79972e10	−0.367079
$263$	−1.50225e11	−1.93616	−0.968082	−	0.250634i	$-0.919361\pi$
$263$	−1.50225e11	−1.93616	−0.968082	+	0.250634i	$0.919361\pi$
$264$	0	0
$265$	4.24653e9	0.0528965
$266$	2.12242e10	0.259934
$267$	0	0
$268$	−2.62204e10	−0.310479
$269$	−9.22583e10	−1.07429	−0.537143	−	0.843491i	$-0.680497\pi$
$269$	−9.22583e10	−1.07429	−0.537143	+	0.843491i	$0.680497\pi$
$270$	0	0
$271$	−1.10392e11	−1.24330	−0.621649	−	0.783296i	$-0.713537\pi$
$271$	−1.10392e11	−1.24330	−0.621649	+	0.783296i	$0.713537\pi$
$272$	−1.12020e9	−0.0124090
$273$	0	0
$274$	−1.15856e10	−0.124177
$275$	3.14878e9	0.0332006
$276$	0	0
$277$	1.06151e11	1.08334	0.541672	−	0.840590i	$-0.317792\pi$
$277$	1.06151e11	1.08334	0.541672	+	0.840590i	$0.317792\pi$
$278$	−1.01787e11	−1.02209
$279$	0	0
$280$	−1.82039e10	−0.176992
$281$	−1.69943e11	−1.62602	−0.813008	−	0.582252i	$-0.802172\pi$
$281$	−1.69943e11	−1.62602	−0.813008	+	0.582252i	$0.802172\pi$
$282$	0	0
$283$	−6.63477e10	−0.614875	−0.307438	−	0.951568i	$-0.599472\pi$
$283$	−6.63477e10	−0.614875	−0.307438	+	0.951568i	$0.599472\pi$
$284$	−5.92927e10	−0.540840
$285$	0	0
$286$	1.70711e10	0.150874
$287$	−3.56677e10	−0.310318
$288$	0	0
$289$	−1.18027e11	−0.995273
$290$	1.91044e10	0.158614
$291$	0	0
$292$	4.12161e10	0.331775
$293$	−7.54377e9	−0.0597976	−0.0298988	−	0.999553i	$-0.509519\pi$
$293$	−7.54377e9	−0.0597976	−0.0298988	+	0.999553i	$0.509519\pi$
$294$	0	0
$295$	−5.53198e10	−0.425286
$296$	−5.25342e10	−0.397768
$297$	0	0
$298$	−5.08769e10	−0.373721
$299$	−1.36814e11	−0.989943
$300$	0	0
$301$	−7.55935e10	−0.530805
$302$	1.61232e11	1.11537
$303$	0	0
$304$	−2.70818e10	−0.181864
$305$	7.80142e10	0.516208
$306$	0	0
$307$	−9.90848e10	−0.636626	−0.318313	−	0.947986i	$-0.603116\pi$
$307$	−9.90848e10	−0.636626	−0.318313	+	0.947986i	$0.603116\pi$
$308$	5.29330e9	0.0335157
$309$	0	0
$310$	8.18258e10	0.503225
$311$	2.78000e10	0.168509	0.0842545	−	0.996444i	$-0.473149\pi$
$311$	2.78000e10	0.168509	0.0842545	+	0.996444i	$0.473149\pi$
$312$	0	0
$313$	7.09223e10	0.417670	0.208835	−	0.977951i	$-0.433033\pi$
$313$	7.09223e10	0.417670	0.208835	+	0.977951i	$0.433033\pi$
$314$	7.54027e10	0.437727
$315$	0	0
$316$	1.06523e11	0.600969
$317$	3.11422e11	1.73214	0.866068	−	0.499926i	$-0.166640\pi$
$317$	3.11422e11	1.73214	0.866068	+	0.499926i	$0.166640\pi$
$318$	0	0
$319$	−1.59547e10	−0.0862643
$320$	6.80376e10	0.362722
$321$	0	0
$322$	3.69947e10	0.191773
$323$	1.35519e10	0.0692770
$324$	0	0
$325$	−5.35662e10	−0.266327
$326$	1.55493e11	0.762485
$327$	0	0
$328$	1.80208e11	0.859692
$329$	5.54739e10	0.261040
$330$	0	0
$331$	2.94782e11	1.34982	0.674908	−	0.737902i	$-0.264183\pi$
$331$	2.94782e11	1.34982	0.674908	+	0.737902i	$0.264183\pi$
$332$	−2.03336e11	−0.918527
$333$	0	0
$334$	1.72699e11	0.759330
$335$	5.99195e10	0.259936
$336$	0	0
$337$	−2.02609e11	−0.855705	−0.427852	−	0.903849i	$-0.640730\pi$
$337$	−2.02609e11	−0.855705	−0.427852	+	0.903849i	$0.640730\pi$
$338$	−1.26637e11	−0.527758
$339$	0	0
$340$	−4.04707e9	−0.0164243
$341$	−6.83353e10	−0.273685
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	3.81930e11	1.47052
$345$	0	0
$346$	−2.40294e11	−0.901365
$347$	1.52000e11	0.562810	0.281405	−	0.959589i	$-0.409200\pi$
$347$	1.52000e11	0.562810	0.281405	+	0.959589i	$0.409200\pi$
$348$	0	0
$349$	−2.82049e11	−1.01768	−0.508840	−	0.860861i	$-0.669925\pi$
$349$	−2.82049e11	−1.01768	−0.508840	+	0.860861i	$0.669925\pi$
$350$	1.44844e10	0.0515933
$351$	0	0
$352$	−4.41762e10	−0.153372
$353$	4.32937e11	1.48402	0.742008	−	0.670391i	$-0.233874\pi$
$353$	4.32937e11	1.48402	0.742008	+	0.670391i	$0.233874\pi$
$354$	0	0
$355$	1.35497e11	0.452796
$356$	7.23203e10	0.238636
$357$	0	0
$358$	−1.77310e11	−0.570504
$359$	−1.35500e10	−0.0430542	−0.0215271	−	0.999768i	$-0.506853\pi$
$359$	−1.35500e10	−0.0430542	−0.0215271	+	0.999768i	$0.506853\pi$
$360$	0	0
$361$	4.94112e9	0.0153124
$362$	−2.54601e11	−0.779242
$363$	0	0
$364$	−9.00480e10	−0.268855
$365$	−9.41879e10	−0.277765
$366$	0	0
$367$	−1.26149e11	−0.362984	−0.181492	−	0.983392i	$-0.558093\pi$
$367$	−1.26149e11	−0.362984	−0.181492	+	0.983392i	$0.558093\pi$
$368$	−4.72049e10	−0.134175
$369$	0	0
$370$	4.18002e10	0.115950
$371$	−1.63135e10	−0.0447057
$372$	0	0
$373$	−2.45212e11	−0.655923	−0.327961	−	0.944691i	$-0.606362\pi$
$373$	−2.45212e11	−0.655923	−0.327961	+	0.944691i	$0.606362\pi$
$374$	−2.94741e9	−0.00778964
$375$	0	0
$376$	−2.80278e11	−0.723175
$377$	2.71417e11	0.691991
$378$	0	0
$379$	3.62591e11	0.902694	0.451347	−	0.892348i	$-0.350944\pi$
$379$	3.62591e11	0.902694	0.451347	+	0.892348i	$0.350944\pi$
$380$	−9.78413e10	−0.240711
$381$	0	0
$382$	2.70550e11	0.650073
$383$	−7.23039e10	−0.171699	−0.0858494	−	0.996308i	$-0.527360\pi$
$383$	−7.23039e10	−0.171699	−0.0858494	+	0.996308i	$0.527360\pi$
$384$	0	0
$385$	−1.20964e10	−0.0280596
$386$	3.88064e11	0.889734
$387$	0	0
$388$	−3.81097e11	−0.853674
$389$	−3.00609e11	−0.665624	−0.332812	−	0.942993i	$-0.607998\pi$
$389$	−3.00609e11	−0.665624	−0.332812	+	0.942993i	$0.607998\pi$
$390$	0	0
$391$	2.36216e10	0.0511110
$392$	6.99320e10	0.149585
$393$	0	0
$394$	5.07291e11	1.06053
$395$	−2.43429e11	−0.503136
$396$	0	0
$397$	−2.86335e11	−0.578518	−0.289259	−	0.957251i	$-0.593409\pi$
$397$	−2.86335e11	−0.578518	−0.289259	+	0.957251i	$0.593409\pi$
$398$	5.51219e10	0.110116
$399$	0	0
$400$	−1.84819e10	−0.0360975
$401$	−8.19964e11	−1.58360	−0.791800	−	0.610781i	$-0.790856\pi$
$401$	−8.19964e11	−1.58360	−0.791800	+	0.610781i	$0.790856\pi$
$402$	0	0
$403$	1.16250e12	2.19543
$404$	−1.76094e11	−0.328873
$405$	0	0
$406$	−7.33915e10	−0.134054
$407$	−3.49087e10	−0.0630607
$408$	0	0
$409$	9.67528e11	1.70966	0.854828	−	0.518911i	$-0.173663\pi$
$409$	9.67528e11	1.70966	0.854828	+	0.518911i	$0.173663\pi$
$410$	−1.43387e11	−0.250601
$411$	0	0
$412$	2.63710e11	0.450908
$413$	2.12517e11	0.359433
$414$	0	0
$415$	4.64667e11	0.768998
$416$	7.51511e11	1.23031
$417$	0	0
$418$	−7.12560e10	−0.114164
$419$	8.51357e10	0.134942	0.0674712	−	0.997721i	$-0.478507\pi$
$419$	8.51357e10	0.134942	0.0674712	+	0.997721i	$0.478507\pi$
$420$	0	0
$421$	5.61043e10	0.0870415	0.0435207	−	0.999053i	$-0.486143\pi$
$421$	5.61043e10	0.0870415	0.0435207	+	0.999053i	$0.486143\pi$
$422$	1.37178e11	0.210562
$423$	0	0
$424$	8.24224e10	0.123851
$425$	9.24846e9	0.0137505
$426$	0	0
$427$	−2.99700e11	−0.436275
$428$	−5.40093e11	−0.777985
$429$	0	0
$430$	−3.03892e11	−0.428659
$431$	1.36440e11	0.190456	0.0952278	−	0.995456i	$-0.469642\pi$
$431$	1.36440e11	0.190456	0.0952278	+	0.995456i	$0.469642\pi$
$432$	0	0
$433$	3.25710e11	0.445282	0.222641	−	0.974900i	$-0.428532\pi$
$433$	3.25710e11	0.445282	0.222641	+	0.974900i	$0.428532\pi$
$434$	−3.14342e11	−0.425303
$435$	0	0
$436$	−2.84320e11	−0.376806
$437$	5.71072e11	0.749074
$438$	0	0
$439$	1.44336e11	0.185474	0.0927372	−	0.995691i	$-0.470438\pi$
$439$	1.44336e11	0.185474	0.0927372	+	0.995691i	$0.470438\pi$
$440$	6.11160e10	0.0777352
$441$	0	0
$442$	5.01404e10	0.0624867
$443$	9.59823e11	1.18406	0.592031	−	0.805915i	$-0.298326\pi$
$443$	9.59823e11	1.18406	0.592031	+	0.805915i	$0.298326\pi$
$444$	0	0
$445$	−1.65268e11	−0.199788
$446$	−2.85919e11	−0.342166
$447$	0	0
$448$	−2.61373e11	−0.306556
$449$	1.02004e12	1.18443	0.592215	−	0.805780i	$-0.298253\pi$
$449$	1.02004e12	1.18443	0.592215	+	0.805780i	$0.298253\pi$
$450$	0	0
$451$	1.19747e11	0.136292
$452$	1.49948e11	0.168974
$453$	0	0
$454$	1.06792e12	1.17975
$455$	2.05780e11	0.225088
$456$	0	0
$457$	−6.45199e11	−0.691944	−0.345972	−	0.938245i	$-0.612451\pi$
$457$	−6.45199e11	−0.691944	−0.345972	+	0.938245i	$0.612451\pi$
$458$	6.40892e11	0.680598
$459$	0	0
$460$	−1.70542e11	−0.177591
$461$	−1.04575e12	−1.07838	−0.539192	−	0.842183i	$-0.681270\pi$
$461$	−1.04575e12	−1.07838	−0.539192	+	0.842183i	$0.681270\pi$
$462$	0	0
$463$	6.01465e11	0.608269	0.304134	−	0.952629i	$-0.401633\pi$
$463$	6.01465e11	0.608269	0.304134	+	0.952629i	$0.401633\pi$
$464$	9.36469e10	0.0937913
$465$	0	0
$466$	1.64758e11	0.161849
$467$	−9.53000e11	−0.927186	−0.463593	−	0.886048i	$-0.653440\pi$
$467$	−9.53000e11	−0.927186	−0.463593	+	0.886048i	$0.653440\pi$
$468$	0	0
$469$	−2.30187e11	−0.219686
$470$	2.23010e11	0.210806
$471$	0	0
$472$	−1.07372e12	−0.995757
$473$	2.53790e11	0.233131
$474$	0	0
$475$	2.23589e11	0.201525
$476$	1.55472e10	0.0138810
$477$	0	0
$478$	1.06454e11	0.0932690
$479$	−1.99157e12	−1.72857	−0.864283	−	0.503005i	$-0.832228\pi$
$479$	−1.99157e12	−1.72857	−0.864283	+	0.503005i	$0.832228\pi$
$480$	0	0
$481$	5.93856e11	0.505857
$482$	4.20981e11	0.355263
$483$	0	0
$484$	6.27119e11	0.519452
$485$	8.70891e11	0.714703
$486$	0	0
$487$	−2.40847e12	−1.94027	−0.970134	−	0.242570i	$-0.922010\pi$
$487$	−2.40847e12	−1.94027	−0.970134	+	0.242570i	$0.922010\pi$
$488$	1.51421e12	1.20864
$489$	0	0
$490$	−5.56432e10	−0.0436043
$491$	−2.85374e11	−0.221589	−0.110794	−	0.993843i	$-0.535339\pi$
$491$	−2.85374e11	−0.221589	−0.110794	+	0.993843i	$0.535339\pi$
$492$	0	0
$493$	−4.68614e10	−0.0357276
$494$	1.21219e12	0.915794
$495$	0	0
$496$	4.01097e11	0.297565
$497$	−5.20526e11	−0.382682
$498$	0	0
$499$	2.11594e12	1.52775	0.763873	−	0.645367i	$-0.223296\pi$
$499$	2.11594e12	1.52775	0.763873	+	0.645367i	$0.223296\pi$
$500$	−6.67715e10	−0.0477778
$501$	0	0
$502$	1.71709e12	1.20677
$503$	−7.41865e11	−0.516736	−0.258368	−	0.966047i	$-0.583185\pi$
$503$	−7.41865e11	−0.516736	−0.258368	+	0.966047i	$0.583185\pi$
$504$	0	0
$505$	4.02414e11	0.275335
$506$	−1.24203e11	−0.0842274
$507$	0	0
$508$	1.47842e12	0.984946
$509$	−1.62377e12	−1.07225	−0.536124	−	0.844140i	$-0.680112\pi$
$509$	−1.62377e12	−1.07225	−0.536124	+	0.844140i	$0.680112\pi$
$510$	0	0
$511$	3.61832e11	0.234754
$512$	5.53160e11	0.355742
$513$	0	0
$514$	−5.93890e11	−0.375294
$515$	−6.02635e11	−0.377504
$516$	0	0
$517$	−1.86243e11	−0.114649
$518$	−1.60580e11	−0.0979955
$519$	0	0
$520$	−1.03969e12	−0.623573
$521$	−1.48817e12	−0.884877	−0.442439	−	0.896799i	$-0.645887\pi$
$521$	−1.48817e12	−0.884877	−0.442439	+	0.896799i	$0.645887\pi$
$522$	0	0
$523$	2.41360e12	1.41061	0.705306	−	0.708903i	$-0.250810\pi$
$523$	2.41360e12	1.41061	0.705306	+	0.708903i	$0.250810\pi$
$524$	−4.95814e11	−0.287295
$525$	0	0
$526$	2.32001e12	1.32146
$527$	−2.00711e11	−0.113351
$528$	0	0
$529$	−8.05747e11	−0.447351
$530$	−6.55815e10	−0.0361027
$531$	0	0
$532$	3.75867e11	0.203438
$533$	−2.03711e12	−1.09331
$534$	0	0
$535$	1.23423e12	0.651336
$536$	1.16300e12	0.608609
$537$	0	0
$538$	1.42480e12	0.733218
$539$	4.64694e10	0.0237147
$540$	0	0
$541$	−2.14334e12	−1.07573	−0.537864	−	0.843032i	$-0.680769\pi$
$541$	−2.14334e12	−1.07573	−0.537864	+	0.843032i	$0.680769\pi$
$542$	1.70484e12	0.848570
$543$	0	0
$544$	−1.29752e11	−0.0635213
$545$	6.49734e11	0.315465
$546$	0	0
$547$	1.02462e12	0.489349	0.244675	−	0.969605i	$-0.421319\pi$
$547$	1.02462e12	0.489349	0.244675	+	0.969605i	$0.421319\pi$
$548$	−2.05174e11	−0.0971875
$549$	0	0
$550$	−4.86285e10	−0.0226599
$551$	−1.13291e12	−0.523618
$552$	0	0
$553$	9.35157e11	0.425228
$554$	−1.63935e12	−0.739399
$555$	0	0
$556$	−1.80259e12	−0.799944
$557$	3.18977e12	1.40414	0.702070	−	0.712108i	$-0.252259\pi$
$557$	3.18977e12	1.40414	0.702070	+	0.712108i	$0.252259\pi$
$558$	0	0
$559$	−4.31740e12	−1.87012
$560$	7.10002e10	0.0305080
$561$	0	0
$562$	2.62453e12	1.10978
$563$	−1.08826e12	−0.456503	−0.228252	−	0.973602i	$-0.573301\pi$
$563$	−1.08826e12	−0.456503	−0.228252	+	0.973602i	$0.573301\pi$
$564$	0	0
$565$	−3.42666e11	−0.141466
$566$	1.02465e12	0.419662
$567$	0	0
$568$	2.62992e12	1.06017
$569$	3.28834e12	1.31514	0.657569	−	0.753394i	$-0.271585\pi$
$569$	3.28834e12	1.31514	0.657569	+	0.753394i	$0.271585\pi$
$570$	0	0
$571$	−1.88806e12	−0.743283	−0.371641	−	0.928376i	$-0.621205\pi$
$571$	−1.88806e12	−0.743283	−0.371641	+	0.928376i	$0.621205\pi$
$572$	3.02319e11	0.118082
$573$	0	0
$574$	5.50837e11	0.211797
$575$	3.89727e11	0.148681
$576$	0	0
$577$	1.12881e12	0.423963	0.211982	−	0.977274i	$-0.432008\pi$
$577$	1.12881e12	0.423963	0.211982	+	0.977274i	$0.432008\pi$
$578$	1.82276e12	0.679289
$579$	0	0
$580$	3.38328e11	0.124140
$581$	−1.78506e12	−0.649922
$582$	0	0
$583$	5.47692e10	0.0196349
$584$	−1.82813e12	−0.650353
$585$	0	0
$586$	1.16503e11	0.0408128
$587$	−6.31776e11	−0.219630	−0.109815	−	0.993952i	$-0.535026\pi$
$587$	−6.31776e11	−0.219630	−0.109815	+	0.993952i	$0.535026\pi$
$588$	0	0
$589$	−4.85237e12	−1.66125
$590$	8.54335e11	0.290265
$591$	0	0
$592$	2.04898e11	0.0685630
$593$	4.97504e12	1.65215	0.826077	−	0.563557i	$-0.190568\pi$
$593$	4.97504e12	1.65215	0.826077	+	0.563557i	$0.190568\pi$
$594$	0	0
$595$	−3.55289e10	−0.0116213
$596$	−9.00999e11	−0.292493
$597$	0	0
$598$	2.11290e12	0.675652
$599$	4.55686e12	1.44626	0.723128	−	0.690714i	$-0.242704\pi$
$599$	4.55686e12	1.44626	0.723128	+	0.690714i	$0.242704\pi$
$600$	0	0
$601$	−5.78188e12	−1.80773	−0.903866	−	0.427815i	$-0.859283\pi$
$601$	−5.78188e12	−1.80773	−0.903866	+	0.427815i	$0.859283\pi$
$602$	1.16743e12	0.362283
$603$	0	0
$604$	2.85532e12	0.872948
$605$	−1.43311e12	−0.434890
$606$	0	0
$607$	−7.37332e11	−0.220452	−0.110226	−	0.993907i	$-0.535157\pi$
$607$	−7.37332e11	−0.220452	−0.110226	+	0.993907i	$0.535157\pi$
$608$	−3.13687e12	−0.930958
$609$	0	0
$610$	−1.20482e12	−0.352320
$611$	3.16831e12	0.919691
$612$	0	0
$613$	−3.50587e12	−1.00282	−0.501410	−	0.865210i	$-0.667185\pi$
$613$	−3.50587e12	−1.00282	−0.501410	+	0.865210i	$0.667185\pi$
$614$	1.53022e12	0.434507
$615$	0	0
$616$	−2.34783e11	−0.0656982
$617$	−2.47222e12	−0.686759	−0.343380	−	0.939197i	$-0.611572\pi$
$617$	−2.47222e12	−0.686759	−0.343380	+	0.939197i	$0.611572\pi$
$618$	0	0
$619$	−1.57589e12	−0.431436	−0.215718	−	0.976456i	$-0.569209\pi$
$619$	−1.57589e12	−0.431436	−0.215718	+	0.976456i	$0.569209\pi$
$620$	1.44908e12	0.393850
$621$	0	0
$622$	−4.29331e11	−0.115010
$623$	6.34894e11	0.168851
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.09529e12	−0.285066
$627$	0	0
$628$	1.33534e12	0.342588
$629$	−1.02532e11	−0.0261175
$630$	0	0
$631$	4.95895e11	0.124525	0.0622627	−	0.998060i	$-0.480168\pi$
$631$	4.95895e11	0.124525	0.0622627	+	0.998060i	$0.480168\pi$
$632$	−4.72481e12	−1.17803
$633$	0	0
$634$	−4.80946e12	−1.18221
$635$	−3.37853e12	−0.824605
$636$	0	0
$637$	−7.90524e11	−0.190234
$638$	2.46398e11	0.0588767
$639$	0	0
$640$	7.02956e11	0.165622
$641$	−1.72054e12	−0.402535	−0.201268	−	0.979536i	$-0.564506\pi$
$641$	−1.72054e12	−0.402535	−0.201268	+	0.979536i	$0.564506\pi$
$642$	0	0
$643$	8.19457e12	1.89050	0.945250	−	0.326346i	$-0.105818\pi$
$643$	8.19457e12	1.89050	0.945250	+	0.326346i	$0.105818\pi$
$644$	6.55154e11	0.150092
$645$	0	0
$646$	−2.09290e11	−0.0472826
$647$	2.56336e12	0.575097	0.287549	−	0.957766i	$-0.407160\pi$
$647$	2.56336e12	0.575097	0.287549	+	0.957766i	$0.407160\pi$
$648$	0	0
$649$	−7.13483e11	−0.157864
$650$	8.27253e11	0.181772
$651$	0	0
$652$	2.75368e12	0.596760
$653$	2.23256e12	0.480499	0.240250	−	0.970711i	$-0.422771\pi$
$653$	2.23256e12	0.480499	0.240250	+	0.970711i	$0.422771\pi$
$654$	0	0
$655$	1.13305e12	0.240526
$656$	−7.02863e11	−0.148185
$657$	0	0
$658$	−8.56715e11	−0.178164
$659$	−6.47077e12	−1.33651	−0.668254	−	0.743933i	$-0.732958\pi$
$659$	−6.47077e12	−1.33651	−0.668254	+	0.743933i	$0.732958\pi$
$660$	0	0
$661$	−2.35514e12	−0.479856	−0.239928	−	0.970791i	$-0.577124\pi$
$661$	−2.35514e12	−0.479856	−0.239928	+	0.970791i	$0.577124\pi$
$662$	−4.55248e12	−0.921270
$663$	0	0
$664$	9.01890e12	1.80052
$665$	−8.58941e11	−0.170320
$666$	0	0
$667$	−1.97472e12	−0.386314
$668$	3.05839e12	0.594291
$669$	0	0
$670$	−9.25371e11	−0.177410
$671$	1.00618e12	0.191613
$672$	0	0
$673$	−1.00068e13	−1.88030	−0.940149	−	0.340764i	$-0.889314\pi$
$673$	−1.00068e13	−1.88030	−0.940149	+	0.340764i	$0.889314\pi$
$674$	3.12900e12	0.584032
$675$	0	0
$676$	−2.24266e12	−0.413051
$677$	4.75925e12	0.870742	0.435371	−	0.900251i	$-0.356617\pi$
$677$	4.75925e12	0.870742	0.435371	+	0.900251i	$0.356617\pi$
$678$	0	0
$679$	−3.34561e12	−0.604035
$680$	1.79507e11	0.0321952
$681$	0	0
$682$	1.05534e12	0.186794
$683$	−5.44016e11	−0.0956575	−0.0478287	−	0.998856i	$-0.515230\pi$
$683$	−5.44016e11	−0.0956575	−0.0478287	+	0.998856i	$0.515230\pi$
$684$	0	0
$685$	4.68869e11	0.0813662
$686$	2.13759e11	0.0368524
$687$	0	0
$688$	−1.48963e12	−0.253473
$689$	−9.31717e11	−0.157506
$690$	0	0
$691$	8.57339e12	1.43054	0.715272	−	0.698846i	$-0.246303\pi$
$691$	8.57339e12	1.43054	0.715272	+	0.698846i	$0.246303\pi$
$692$	−4.25546e12	−0.705455
$693$	0	0
$694$	−2.34743e12	−0.384127
$695$	4.11931e12	0.669720
$696$	0	0
$697$	3.51716e11	0.0564476
$698$	4.35585e12	0.694582
$699$	0	0
$700$	2.56510e11	0.0403796
$701$	−1.25396e13	−1.96133	−0.980666	−	0.195687i	$-0.937306\pi$
$701$	−1.25396e13	−1.96133	−0.980666	+	0.195687i	$0.937306\pi$
$702$	0	0
$703$	−2.47880e12	−0.382774
$704$	8.77510e11	0.134640
$705$	0	0
$706$	−6.68609e12	−1.01286
$707$	−1.54591e12	−0.232701
$708$	0	0
$709$	4.37869e12	0.650782	0.325391	−	0.945580i	$-0.394504\pi$
$709$	4.37869e12	0.650782	0.325391	+	0.945580i	$0.394504\pi$
$710$	−2.09256e12	−0.309040
$711$	0	0
$712$	−3.20775e12	−0.467779
$713$	−8.45790e12	−1.22563
$714$	0	0
$715$	−6.90865e11	−0.0988590
$716$	−3.14004e12	−0.446506
$717$	0	0
$718$	2.09261e11	0.0293852
$719$	−7.76448e12	−1.08351	−0.541754	−	0.840537i	$-0.682240\pi$
$719$	−7.76448e12	−1.08351	−0.541754	+	0.840537i	$0.682240\pi$
$720$	0	0
$721$	2.31508e12	0.319049
$722$	−7.63086e10	−0.0104510
$723$	0	0
$724$	−4.50884e12	−0.609875
$725$	−7.73154e11	−0.103931
$726$	0	0
$727$	7.08318e11	0.0940423	0.0470211	−	0.998894i	$-0.485027\pi$
$727$	7.08318e11	0.0940423	0.0470211	+	0.998894i	$0.485027\pi$
$728$	3.99406e12	0.527015
$729$	0	0
$730$	1.45460e12	0.189579
$731$	7.45421e11	0.0965547
$732$	0	0
$733$	6.81304e12	0.871712	0.435856	−	0.900017i	$-0.356446\pi$
$733$	6.81304e12	0.871712	0.435856	+	0.900017i	$0.356446\pi$
$734$	1.94820e12	0.247743
$735$	0	0
$736$	−5.46771e12	−0.686839
$737$	7.72807e11	0.0964866
$738$	0	0
$739$	9.20369e12	1.13517	0.567586	−	0.823314i	$-0.307877\pi$
$739$	9.20369e12	1.13517	0.567586	+	0.823314i	$0.307877\pi$
$740$	7.40256e11	0.0907484
$741$	0	0
$742$	2.51938e11	0.0305124
$743$	8.72196e12	1.04994	0.524970	−	0.851121i	$-0.324077\pi$
$743$	8.72196e12	1.04994	0.524970	+	0.851121i	$0.324077\pi$
$744$	0	0
$745$	2.05898e12	0.244878
$746$	3.78695e12	0.447678
$747$	0	0
$748$	−5.21968e10	−0.00609658
$749$	−4.74143e12	−0.550479
$750$	0	0
$751$	−1.63201e13	−1.87216	−0.936078	−	0.351793i	$-0.885572\pi$
$751$	−1.63201e13	−1.87216	−0.936078	+	0.351793i	$0.885572\pi$
$752$	1.09316e12	0.124653
$753$	0	0
$754$	−4.19164e12	−0.472295
$755$	−6.52504e12	−0.730839
$756$	0	0
$757$	−3.23287e12	−0.357813	−0.178907	−	0.983866i	$-0.557256\pi$
$757$	−3.23287e12	−0.357813	−0.178907	+	0.983866i	$0.557256\pi$
$758$	−5.59970e12	−0.616103
$759$	0	0
$760$	4.33973e12	0.471847
$761$	6.46766e11	0.0699063	0.0349531	−	0.999389i	$-0.488872\pi$
$761$	6.46766e11	0.0699063	0.0349531	+	0.999389i	$0.488872\pi$
$762$	0	0
$763$	−2.49602e12	−0.266617
$764$	4.79127e12	0.508781
$765$	0	0
$766$	1.11663e12	0.117187
$767$	1.21376e13	1.26634
$768$	0	0
$769$	−2.42101e12	−0.249648	−0.124824	−	0.992179i	$-0.539837\pi$
$769$	−2.42101e12	−0.249648	−0.124824	+	0.992179i	$0.539837\pi$
$770$	1.86811e11	0.0191511
$771$	0	0
$772$	6.87237e12	0.696352
$773$	4.43910e12	0.447185	0.223592	−	0.974683i	$-0.428222\pi$
$773$	4.43910e12	0.447185	0.223592	+	0.974683i	$0.428222\pi$
$774$	0	0
$775$	−3.31148e12	−0.329735
$776$	1.69035e13	1.67339
$777$	0	0
$778$	4.64248e12	0.454299
$779$	8.50304e12	0.827286
$780$	0	0
$781$	1.74756e12	0.168075
$782$	−3.64802e11	−0.0348840
$783$	0	0
$784$	−2.72754e11	−0.0257839
$785$	−3.05154e12	−0.286818
$786$	0	0
$787$	−1.72989e13	−1.60744	−0.803718	−	0.595011i	$-0.797148\pi$
$787$	−1.72989e13	−1.60744	−0.803718	+	0.595011i	$0.797148\pi$
$788$	8.98382e12	0.830029
$789$	0	0
$790$	3.75941e12	0.343398
$791$	1.31638e12	0.119561
$792$	0	0
$793$	−1.71169e13	−1.53708
$794$	4.42203e12	0.394848
$795$	0	0
$796$	9.76175e11	0.0861825
$797$	1.37776e13	1.20951	0.604756	−	0.796410i	$-0.293270\pi$
$797$	1.37776e13	1.20951	0.604756	+	0.796410i	$0.293270\pi$
$798$	0	0
$799$	−5.47024e11	−0.0474838
$800$	−2.14075e12	−0.184782
$801$	0	0
$802$	1.26632e13	1.08083
$803$	−1.21478e12	−0.103105
$804$	0	0
$805$	−1.49717e12	−0.125658
$806$	−1.79531e13	−1.49842
$807$	0	0
$808$	7.81061e12	0.644665
$809$	−1.38369e13	−1.13572	−0.567860	−	0.823125i	$-0.692228\pi$
$809$	−1.38369e13	−1.13572	−0.567860	+	0.823125i	$0.692228\pi$
$810$	0	0
$811$	−1.96078e13	−1.59161	−0.795804	−	0.605555i	$-0.792951\pi$
$811$	−1.96078e13	−1.59161	−0.795804	+	0.605555i	$0.792951\pi$
$812$	−1.29972e12	−0.104917
$813$	0	0
$814$	5.39114e11	0.0430399
$815$	−6.29278e12	−0.499613
$816$	0	0
$817$	1.80212e13	1.41509
$818$	−1.49421e13	−1.16687
$819$	0	0
$820$	−2.53930e12	−0.196134
$821$	−2.41017e13	−1.85142	−0.925708	−	0.378239i	$-0.876530\pi$
$821$	−2.41017e13	−1.85142	−0.925708	+	0.378239i	$0.876530\pi$
$822$	0	0
$823$	8.20762e12	0.623617	0.311809	−	0.950145i	$-0.399065\pi$
$823$	8.20762e12	0.623617	0.311809	+	0.950145i	$0.399065\pi$
$824$	−1.16968e13	−0.883881
$825$	0	0
$826$	−3.28201e12	−0.245318
$827$	−2.84612e12	−0.211582	−0.105791	−	0.994388i	$-0.533737\pi$
$827$	−2.84612e12	−0.211582	−0.105791	+	0.994388i	$0.533737\pi$
$828$	0	0
$829$	−5.88562e12	−0.432809	−0.216405	−	0.976304i	$-0.569433\pi$
$829$	−5.88562e12	−0.432809	−0.216405	+	0.976304i	$0.569433\pi$
$830$	−7.17612e12	−0.524853
$831$	0	0
$832$	−1.49279e13	−1.08005
$833$	1.36488e11	0.00982180
$834$	0	0
$835$	−6.98911e12	−0.497546
$836$	−1.26190e12	−0.0893505
$837$	0	0
$838$	−1.31480e12	−0.0921003
$839$	8.44799e12	0.588606	0.294303	−	0.955712i	$-0.404913\pi$
$839$	8.44799e12	0.588606	0.294303	+	0.955712i	$0.404913\pi$
$840$	0	0
$841$	−1.05896e13	−0.729959
$842$	−8.66450e11	−0.0594072
$843$	0	0
$844$	2.42934e12	0.164796
$845$	5.12498e12	0.345810
$846$	0	0
$847$	5.50542e12	0.367549
$848$	−3.21470e11	−0.0213481
$849$	0	0
$850$	−1.42829e11	−0.00938495
$851$	−4.32066e12	−0.282402
$852$	0	0
$853$	−7.17311e12	−0.463913	−0.231957	−	0.972726i	$-0.574513\pi$
$853$	−7.17311e12	−0.463913	−0.231957	+	0.972726i	$0.574513\pi$
$854$	4.62843e12	0.297765
$855$	0	0
$856$	2.39557e13	1.52502
$857$	−2.17636e13	−1.37821	−0.689107	−	0.724660i	$-0.741997\pi$
$857$	−2.17636e13	−1.37821	−0.689107	+	0.724660i	$0.741997\pi$
$858$	0	0
$859$	−9.02945e12	−0.565838	−0.282919	−	0.959144i	$-0.591303\pi$
$859$	−9.02945e12	−0.565838	−0.282919	+	0.959144i	$0.591303\pi$
$860$	−5.38175e12	−0.335491
$861$	0	0
$862$	−2.10712e12	−0.129989
$863$	1.84428e13	1.13182	0.565910	−	0.824467i	$-0.308525\pi$
$863$	1.84428e13	1.13182	0.565910	+	0.824467i	$0.308525\pi$
$864$	0	0
$865$	9.72468e12	0.590613
$866$	−5.03012e12	−0.303912
$867$	0	0
$868$	−5.56680e12	−0.332864
$869$	−3.13961e12	−0.186761
$870$	0	0
$871$	−1.31468e13	−0.773993
$872$	1.26109e13	0.738623
$873$	0	0
$874$	−8.81940e12	−0.511255
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−1.12140e13	−0.640120	−0.320060	−	0.947397i	$-0.603703\pi$
$877$	−1.12140e13	−0.640120	−0.320060	+	0.947397i	$0.603703\pi$
$878$	−2.22906e12	−0.126589
$879$	0	0
$880$	−2.38369e11	−0.0133992
$881$	−3.00664e13	−1.68147	−0.840735	−	0.541446i	$-0.817877\pi$
$881$	−3.00664e13	−1.68147	−0.840735	+	0.541446i	$0.817877\pi$
$882$	0	0
$883$	−1.58968e13	−0.880007	−0.440004	−	0.897996i	$-0.645023\pi$
$883$	−1.58968e13	−0.880007	−0.440004	+	0.897996i	$0.645023\pi$
$884$	8.87956e11	0.0489053
$885$	0	0
$886$	−1.48231e13	−0.808141
$887$	−1.46131e13	−0.792658	−0.396329	−	0.918109i	$-0.629716\pi$
$887$	−1.46131e13	−0.792658	−0.396329	+	0.918109i	$0.629716\pi$
$888$	0	0
$889$	1.29790e13	0.696918
$890$	2.55233e12	0.136358
$891$	0	0
$892$	−5.06346e12	−0.267797
$893$	−1.32248e13	−0.695915
$894$	0	0
$895$	7.17570e12	0.373819
$896$	−2.70047e12	−0.139976
$897$	0	0
$898$	−1.57531e13	−0.808393
$899$	1.67791e13	0.856742
$900$	0	0
$901$	1.60866e11	0.00813208
$902$	−1.84933e12	−0.0930217
$903$	0	0
$904$	−6.65093e12	−0.331226
$905$	1.03037e13	0.510593
$906$	0	0
$907$	1.41504e12	0.0694281	0.0347141	−	0.999397i	$-0.488948\pi$
$907$	1.41504e12	0.0694281	0.0347141	+	0.999397i	$0.488948\pi$
$908$	1.89123e13	0.923333
$909$	0	0
$910$	−3.17797e12	−0.153626
$911$	1.61901e13	0.778784	0.389392	−	0.921072i	$-0.372685\pi$
$911$	1.61901e13	0.778784	0.389392	+	0.921072i	$0.372685\pi$
$912$	0	0
$913$	5.99301e12	0.285447
$914$	9.96417e12	0.472262
$915$	0	0
$916$	1.13498e13	0.532671
$917$	−4.35271e12	−0.203281
$918$	0	0
$919$	−1.60413e13	−0.741858	−0.370929	−	0.928661i	$-0.620961\pi$
$919$	−1.60413e13	−0.741858	−0.370929	+	0.928661i	$0.620961\pi$
$920$	7.56435e12	0.348118
$921$	0	0
$922$	1.61501e13	0.736014
$923$	−2.97290e13	−1.34826
$924$	0	0
$925$	−1.69165e12	−0.0759753
$926$	−9.28876e12	−0.415153
$927$	0	0
$928$	1.08470e13	0.480115
$929$	3.98759e13	1.75647	0.878233	−	0.478233i	$-0.158723\pi$
$929$	3.98759e13	1.75647	0.878233	+	0.478233i	$0.158723\pi$
$930$	0	0
$931$	3.29971e12	0.143947
$932$	2.91777e12	0.126672
$933$	0	0
$934$	1.47177e13	0.632819
$935$	1.19281e11	0.00510411
$936$	0	0
$937$	−5.95427e12	−0.252348	−0.126174	−	0.992008i	$-0.540270\pi$
$937$	−5.95427e12	−0.252348	−0.126174	+	0.992008i	$0.540270\pi$
$938$	3.55490e12	0.149939
$939$	0	0
$940$	3.94937e12	0.164988
$941$	−9.86055e12	−0.409966	−0.204983	−	0.978766i	$-0.565714\pi$
$941$	−9.86055e12	−0.409966	−0.204983	+	0.978766i	$0.565714\pi$
$942$	0	0
$943$	1.48212e13	0.610353
$944$	4.18782e12	0.171638
$945$	0	0
$946$	−3.91943e12	−0.159115
$947$	2.10061e13	0.848732	0.424366	−	0.905491i	$-0.360497\pi$
$947$	2.10061e13	0.848732	0.424366	+	0.905491i	$0.360497\pi$
$948$	0	0
$949$	2.06655e13	0.827080
$950$	−3.45302e12	−0.137544
$951$	0	0
$952$	−6.89594e11	−0.0272099
$953$	4.25218e13	1.66991	0.834955	−	0.550318i	$-0.185494\pi$
$953$	4.25218e13	1.66991	0.834955	+	0.550318i	$0.185494\pi$
$954$	0	0
$955$	−1.09491e13	−0.425956
$956$	1.88524e12	0.0729972
$957$	0	0
$958$	3.07570e13	1.17977
$959$	−1.80121e12	−0.0687670
$960$	0	0
$961$	4.54266e13	1.71813
$962$	−9.17125e12	−0.345256
$963$	0	0
$964$	7.45531e12	0.278048
$965$	−1.57049e13	−0.582992
$966$	0	0
$967$	−1.21906e13	−0.448338	−0.224169	−	0.974550i	$-0.571967\pi$
$967$	−1.21906e13	−0.448338	−0.224169	+	0.974550i	$0.571967\pi$
$968$	−2.78157e13	−1.01824
$969$	0	0
$970$	−1.34497e13	−0.487796
$971$	−5.38639e12	−0.194452	−0.0972258	−	0.995262i	$-0.530997\pi$
$971$	−5.38639e12	−0.194452	−0.0972258	+	0.995262i	$0.530997\pi$
$972$	0	0
$973$	−1.58247e13	−0.566016
$974$	3.71954e13	1.32426
$975$	0	0
$976$	−5.90583e12	−0.208332
$977$	1.89284e13	0.664643	0.332321	−	0.943166i	$-0.392168\pi$
$977$	1.89284e13	0.664643	0.332321	+	0.943166i	$0.392168\pi$
$978$	0	0
$979$	−2.13153e12	−0.0741600
$980$	−9.85407e11	−0.0341270
$981$	0	0
$982$	4.40719e12	0.151238
$983$	−3.98513e13	−1.36129	−0.680647	−	0.732612i	$-0.738301\pi$
$983$	−3.98513e13	−1.36129	−0.680647	+	0.732612i	$0.738301\pi$
$984$	0	0
$985$	−2.05300e13	−0.694907
$986$	7.23708e11	0.0243847
$987$	0	0
$988$	2.14671e13	0.716748
$989$	3.14118e13	1.04402
$990$	0	0
$991$	−3.44424e13	−1.13439	−0.567194	−	0.823584i	$-0.691971\pi$
$991$	−3.44424e13	−1.13439	−0.567194	+	0.823584i	$0.691971\pi$
$992$	4.64588e13	1.52323
$993$	0	0
$994$	8.03877e12	0.261187
$995$	−2.23078e12	−0.0721527
$996$	0	0
$997$	4.12406e13	1.32189	0.660947	−	0.750433i	$-0.270155\pi$
$997$	4.12406e13	1.32189	0.660947	+	0.750433i	$0.270155\pi$
$998$	−3.26777e13	−1.04271
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.g.1.2		4
3.2	odd	2		35.10.a.c.1.3	✓	4
15.2	even	4		175.10.b.e.99.5		8
15.8	even	4		175.10.b.e.99.4		8
15.14	odd	2		175.10.a.e.1.2		4
21.20	even	2		245.10.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.c.1.3	✓	4	3.2	odd	2
175.10.a.e.1.2		4	15.14	odd	2
175.10.b.e.99.4		8	15.8	even	4
175.10.b.e.99.5		8	15.2	even	4
245.10.a.e.1.3		4	21.20	even	2
315.10.a.g.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-15.4436 q^{2} -273.496 q^{4} +625.000 q^{5} -2401.00 q^{7} +12130.9 q^{8} -9652.23 q^{10} +8060.89 q^{11} -137129. q^{13} +37080.0 q^{14} -47313.7 q^{16} +23676.1 q^{17} +572389. q^{19} -170935. q^{20} -124489. q^{22} +997700. q^{23} +390625. q^{25} +2.11777e6 q^{26} +656664. q^{28} -1.97927e6 q^{29} -8.47740e6 q^{31} -5.48031e6 q^{32} -365643. q^{34} -1.50062e6 q^{35} -4.33062e6 q^{37} -8.83972e6 q^{38} +7.58179e6 q^{40} +1.48554e7 q^{41} +3.14842e7 q^{43} -2.20462e6 q^{44} -1.54081e7 q^{46} -2.31045e7 q^{47} +5.76480e6 q^{49} -6.03264e6 q^{50} +3.75044e7 q^{52} +6.79444e6 q^{53} +5.03805e6 q^{55} -2.91262e7 q^{56} +3.05671e7 q^{58} -8.85117e7 q^{59} +1.24823e8 q^{61} +1.30921e8 q^{62} +1.08860e8 q^{64} -8.57059e7 q^{65} +9.58712e7 q^{67} -6.47531e6 q^{68} +2.31750e7 q^{70} +2.16795e8 q^{71} -1.50701e8 q^{73} +6.68803e7 q^{74} -1.56546e8 q^{76} -1.93542e7 q^{77} -3.89487e8 q^{79} -2.95711e7 q^{80} -2.29420e8 q^{82} +7.43467e8 q^{83} +1.47975e7 q^{85} -4.86228e8 q^{86} +9.77855e7 q^{88} -2.64429e8 q^{89} +3.29248e8 q^{91} -2.72867e8 q^{92} +3.56816e8 q^{94} +3.57743e8 q^{95} +1.39343e9 q^{97} -8.90291e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 19 q^{2} + 1729 q^{4} + 2500 q^{5} - 9604 q^{7} + 30495 q^{8} + 11875 q^{10} - 82438 q^{11} - 72962 q^{13} - 45619 q^{14} + 64257 q^{16} + 357542 q^{17} + 300732 q^{19} + 1080625 q^{20} - 5595068 q^{22}+ \cdots + 109531219 q^{98}+O(q^{100}) \) 4 * q + 19 * q^2 + 1729 * q^4 + 2500 * q^5 - 9604 * q^7 + 30495 * q^8 + 11875 * q^10 - 82438 * q^11 - 72962 * q^13 - 45619 * q^14 + 64257 * q^16 + 357542 * q^17 + 300732 * q^19 + 1080625 * q^20 - 5595068 * q^22 + 2340836 * q^23 + 1562500 * q^25 - 1305666 * q^26 - 4151329 * q^28 + 1372022 * q^29 - 4590888 * q^31 + 10529439 * q^32 - 7199478 * q^34 - 6002500 * q^35 - 38868456 * q^37 + 15477280 * q^38 + 19059375 * q^40 - 57287084 * q^41 - 43403452 * q^43 - 87639348 * q^44 + 28892416 * q^46 - 91822222 * q^47 + 23059204 * q^49 + 7421875 * q^50 + 1953386 * q^52 + 61086884 * q^53 - 51523750 * q^55 - 73218495 * q^56 + 29615390 * q^58 - 72569680 * q^59 - 225681036 * q^61 - 190286472 * q^62 + 141443393 * q^64 - 45601250 * q^65 - 4429720 * q^67 - 15339698 * q^68 - 28511875 * q^70 + 470468984 * q^71 - 168326464 * q^73 - 927809502 * q^74 - 892265344 * q^76 + 197933638 * q^77 - 598805646 * q^79 + 40160625 * q^80 - 1821993318 * q^82 + 1159074304 * q^83 + 223463750 * q^85 - 706791156 * q^86 - 1992103900 * q^88 + 1380153012 * q^89 + 175181762 * q^91 - 710319264 * q^92 + 2466744152 * q^94 + 187957500 * q^95 + 754874082 * q^97 + 109531219 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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