LMFDB - Embedded newform 315.10.a.q.1.7

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 = [10,7] 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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 3667 x^{8} + 7323 x^{7} + 4338847 x^{6} - 6510663 x^{5} - 1903644413 x^{4} + \cdots - 2925217654100 $ x^10 - 3x^9 - 3667x^8 + 7323x^7 + 4338847x^6 - 6510663x^5 - 1903644413x^4 + 3604897833x^3 + 273645512708x^2 - 984581099130*x - 2925217654100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{12}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-19.3422$ 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)

$f(q)$	$=$	\(q+20.3422 q^{2} -98.1948 q^{4} +625.000 q^{5} -2401.00 q^{7} -12412.7 q^{8} +12713.9 q^{10} +94596.5 q^{11} -5461.83 q^{13} -48841.6 q^{14} -202226. q^{16} -138708. q^{17} +134259. q^{19} -61371.8 q^{20} +1.92430e6 q^{22} -1.31904e6 q^{23} +390625. q^{25} -111106. q^{26} +235766. q^{28} +3.88336e6 q^{29} -2.86754e6 q^{31} +2.24158e6 q^{32} -2.82164e6 q^{34} -1.50062e6 q^{35} -5.69905e6 q^{37} +2.73113e6 q^{38} -7.75794e6 q^{40} -1.01484e7 q^{41} +1.64962e7 q^{43} -9.28889e6 q^{44} -2.68322e7 q^{46} -5.05341e7 q^{47} +5.76480e6 q^{49} +7.94617e6 q^{50} +536323. q^{52} -1.67715e7 q^{53} +5.91228e7 q^{55} +2.98029e7 q^{56} +7.89961e7 q^{58} +9.62623e7 q^{59} -4.76406e7 q^{61} -5.83321e7 q^{62} +1.49138e8 q^{64} -3.41364e6 q^{65} +1.75074e8 q^{67} +1.36204e7 q^{68} -3.05260e7 q^{70} +3.56669e8 q^{71} +4.49220e8 q^{73} -1.15931e8 q^{74} -1.31836e7 q^{76} -2.27126e8 q^{77} +2.53417e8 q^{79} -1.26391e8 q^{80} -2.06441e8 q^{82} +5.49724e7 q^{83} -8.66928e7 q^{85} +3.35569e8 q^{86} -1.17420e9 q^{88} -7.53460e8 q^{89} +1.31139e7 q^{91} +1.29523e8 q^{92} -1.02797e9 q^{94} +8.39121e7 q^{95} +8.46560e8 q^{97} +1.17269e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 7 q^{2} + 2227 q^{4} + 6250 q^{5} - 24010 q^{7} + 3801 q^{8} + 4375 q^{10} + 66164 q^{11} + 11494 q^{13} - 16807 q^{14} + 918867 q^{16} + 193034 q^{17} + 673344 q^{19} + 1391875 q^{20} - 1386368 q^{22}+ \cdots + 40353607 q^{98}+O(q^{100}) $ 10 * q + 7 * q^2 + 2227 * q^4 + 6250 * q^5 - 24010 * q^7 + 3801 * q^8 + 4375 * q^10 + 66164 * q^11 + 11494 * q^13 - 16807 * q^14 + 918867 * q^16 + 193034 * q^17 + 673344 * q^19 + 1391875 * q^20 - 1386368 * q^22 - 747586 * q^23 + 3906250 * q^25 - 10565862 * q^26 - 5347027 * q^28 + 1684046 * q^29 + 9842238 * q^31 - 4156131 * q^32 + 23016000 * q^34 - 15006250 * q^35 + 9741882 * q^37 - 12152762 * q^38 + 2375625 * q^40 + 14958856 * q^41 + 39625550 * q^43 + 74115048 * q^44 + 111188116 * q^46 + 69877502 * q^47 + 57648010 * q^49 + 2734375 * q^50 - 89168254 * q^52 + 151032896 * q^53 + 41352500 * q^55 - 9126201 * q^56 - 37012702 * q^58 + 81055712 * q^59 + 133907802 * q^61 + 541710636 * q^62 + 360737543 * q^64 + 7183750 * q^65 + 389884430 * q^67 + 770623072 * q^68 - 10504375 * q^70 + 318130274 * q^71 - 97314154 * q^73 + 569155188 * q^74 + 1120827470 * q^76 - 158859764 * q^77 - 1247170680 * q^79 + 574291875 * q^80 - 1310498994 * q^82 - 207205436 * q^83 + 120646250 * q^85 + 3002223966 * q^86 - 2070483952 * q^88 + 660264792 * q^89 - 27597094 * q^91 - 1308446244 * q^92 - 1420882918 * q^94 + 420840000 * q^95 + 253464078 * q^97 + 40353607 * 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 $

$2$	20.3422	0.899007	0.449503	−	0.893279i	$-0.351601\pi$
$2$	20.3422	0.899007	0.449503	+	0.893279i	$0.351601\pi$
$3$	0	0
$3$	0	0
$4$	−98.1948	−0.191787
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−12412.7	−1.07142
$9$	0	0
$10$	12713.9	0.402048
$11$	94596.5	1.94809	0.974043	−	0.226363i	$-0.0726834\pi$
$11$	94596.5	1.94809	0.974043	+	0.226363i	$0.0726834\pi$
$12$	0	0
$13$	−5461.83	−0.0530387	−0.0265194	−	0.999648i	$-0.508442\pi$
$13$	−5461.83	−0.0530387	−0.0265194	+	0.999648i	$0.508442\pi$
$14$	−48841.6	−0.339793
$15$	0	0
$16$	−202226.	−0.771431
$17$	−138708.	−0.402794	−0.201397	−	0.979510i	$-0.564548\pi$
$17$	−138708.	−0.402794	−0.201397	+	0.979510i	$0.564548\pi$
$18$	0	0
$19$	134259.	0.236349	0.118174	−	0.992993i	$-0.462296\pi$
$19$	134259.	0.236349	0.118174	+	0.992993i	$0.462296\pi$
$20$	−61371.8	−0.0857696
$21$	0	0
$22$	1.92430e6	1.75134
$23$	−1.31904e6	−0.982840	−0.491420	−	0.870923i	$-0.663522\pi$
$23$	−1.31904e6	−0.982840	−0.491420	+	0.870923i	$0.663522\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−111106.	−0.0476822
$27$	0	0
$28$	235766.	0.0724886
$29$	3.88336e6	1.01957	0.509784	−	0.860302i	$-0.329725\pi$
$29$	3.88336e6	1.01957	0.509784	+	0.860302i	$0.329725\pi$
$30$	0	0
$31$	−2.86754e6	−0.557676	−0.278838	−	0.960338i	$-0.589949\pi$
$31$	−2.86754e6	−0.557676	−0.278838	+	0.960338i	$0.589949\pi$
$32$	2.24158e6	0.377903
$33$	0	0
$34$	−2.82164e6	−0.362114
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−5.69905e6	−0.499914	−0.249957	−	0.968257i	$-0.580416\pi$
$37$	−5.69905e6	−0.499914	−0.249957	+	0.968257i	$0.580416\pi$
$38$	2.73113e6	0.212479
$39$	0	0
$40$	−7.75794e6	−0.479156
$41$	−1.01484e7	−0.560882	−0.280441	−	0.959871i	$-0.590481\pi$
$41$	−1.01484e7	−0.560882	−0.280441	+	0.959871i	$0.590481\pi$
$42$	0	0
$43$	1.64962e7	0.735828	0.367914	−	0.929860i	$-0.380072\pi$
$43$	1.64962e7	0.735828	0.367914	+	0.929860i	$0.380072\pi$
$44$	−9.28889e6	−0.373617
$45$	0	0
$46$	−2.68322e7	−0.883580
$47$	−5.05341e7	−1.51058	−0.755290	−	0.655390i	$-0.772504\pi$
$47$	−5.05341e7	−1.51058	−0.755290	+	0.655390i	$0.772504\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	7.94617e6	0.179801
$51$	0	0
$52$	536323.	0.0101721
$53$	−1.67715e7	−0.291965	−0.145983	−	0.989287i	$-0.546634\pi$
$53$	−1.67715e7	−0.291965	−0.145983	+	0.989287i	$0.546634\pi$
$54$	0	0
$55$	5.91228e7	0.871211
$56$	2.98029e7	0.404960
$57$	0	0
$58$	7.89961e7	0.916599
$59$	9.62623e7	1.03424	0.517121	−	0.855912i	$-0.327004\pi$
$59$	9.62623e7	1.03424	0.517121	+	0.855912i	$0.327004\pi$
$60$	0	0
$61$	−4.76406e7	−0.440547	−0.220274	−	0.975438i	$-0.570695\pi$
$61$	−4.76406e7	−0.440547	−0.220274	+	0.975438i	$0.570695\pi$
$62$	−5.83321e7	−0.501354
$63$	0	0
$64$	1.49138e8	1.11117
$65$	−3.41364e6	−0.0237196
$66$	0	0
$67$	1.75074e8	1.06142	0.530708	−	0.847555i	$-0.321926\pi$
$67$	1.75074e8	1.06142	0.530708	+	0.847555i	$0.321926\pi$
$68$	1.36204e7	0.0772505
$69$	0	0
$70$	−3.05260e7	−0.151960
$71$	3.56669e8	1.66572	0.832862	−	0.553481i	$-0.186701\pi$
$71$	3.56669e8	1.66572	0.832862	+	0.553481i	$0.186701\pi$
$72$	0	0
$73$	4.49220e8	1.85142	0.925712	−	0.378229i	$-0.123467\pi$
$73$	4.49220e8	1.85142	0.925712	+	0.378229i	$0.123467\pi$
$74$	−1.15931e8	−0.449426
$75$	0	0
$76$	−1.31836e7	−0.0453286
$77$	−2.27126e8	−0.736307
$78$	0	0
$79$	2.53417e8	0.732005	0.366003	−	0.930614i	$-0.380726\pi$
$79$	2.53417e8	0.732005	0.366003	+	0.930614i	$0.380726\pi$
$80$	−1.26391e8	−0.344994
$81$	0	0
$82$	−2.06441e8	−0.504237
$83$	5.49724e7	0.127143	0.0635717	−	0.997977i	$-0.479751\pi$
$83$	5.49724e7	0.127143	0.0635717	+	0.997977i	$0.479751\pi$
$84$	0	0
$85$	−8.66928e7	−0.180135
$86$	3.35569e8	0.661514
$87$	0	0
$88$	−1.17420e9	−2.08723
$89$	−7.53460e8	−1.27293	−0.636466	−	0.771304i	$-0.719605\pi$
$89$	−7.53460e8	−1.27293	−0.636466	+	0.771304i	$0.719605\pi$
$90$	0	0
$91$	1.31139e7	0.0200468
$92$	1.29523e8	0.188496
$93$	0	0
$94$	−1.02797e9	−1.35802
$95$	8.39121e7	0.105698
$96$	0	0
$97$	8.46560e8	0.970923	0.485462	−	0.874258i	$-0.338652\pi$
$97$	8.46560e8	0.970923	0.485462	+	0.874258i	$0.338652\pi$
$98$	1.17269e8	0.128430
$99$	0	0
$100$	−3.83573e7	−0.0383573
$101$	1.74311e9	1.66678	0.833391	−	0.552684i	$-0.186396\pi$
$101$	1.74311e9	1.66678	0.833391	+	0.552684i	$0.186396\pi$
$102$	0	0
$103$	−1.43939e8	−0.126011	−0.0630057	−	0.998013i	$-0.520069\pi$
$103$	−1.43939e8	−0.126011	−0.0630057	+	0.998013i	$0.520069\pi$
$104$	6.77961e7	0.0568270
$105$	0	0
$106$	−3.41170e8	−0.262479
$107$	−2.40851e9	−1.77632	−0.888160	−	0.459535i	$-0.848016\pi$
$107$	−2.40851e9	−1.77632	−0.888160	+	0.459535i	$0.848016\pi$
$108$	0	0
$109$	2.18680e9	1.48385	0.741926	−	0.670482i	$-0.233912\pi$
$109$	2.18680e9	1.48385	0.741926	+	0.670482i	$0.233912\pi$
$110$	1.20269e9	0.783224
$111$	0	0
$112$	4.85545e8	0.291574
$113$	1.86324e8	0.107502	0.0537510	−	0.998554i	$-0.482882\pi$
$113$	1.86324e8	0.107502	0.0537510	+	0.998554i	$0.482882\pi$
$114$	0	0
$115$	−8.24400e8	−0.439539
$116$	−3.81326e8	−0.195540
$117$	0	0
$118$	1.95819e9	0.929791
$119$	3.33039e8	0.152242
$120$	0	0
$121$	6.59056e9	2.79504
$122$	−9.69114e8	−0.396055
$123$	0	0
$124$	2.81577e8	0.106955
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	1.15165e9	0.392829	0.196414	−	0.980521i	$-0.437070\pi$
$127$	1.15165e9	0.392829	0.196414	+	0.980521i	$0.437070\pi$
$128$	1.88611e9	0.621045
$129$	0	0
$130$	−6.94410e7	−0.0213241
$131$	3.26859e9	0.969706	0.484853	−	0.874596i	$-0.338873\pi$
$131$	3.26859e9	0.969706	0.484853	+	0.874596i	$0.338873\pi$
$132$	0	0
$133$	−3.22357e8	−0.0893314
$134$	3.56139e9	0.954219
$135$	0	0
$136$	1.72175e9	0.431563
$137$	−5.55375e9	−1.34693	−0.673463	−	0.739221i	$-0.735194\pi$
$137$	−5.55375e9	−1.34693	−0.673463	+	0.739221i	$0.735194\pi$
$138$	0	0
$139$	4.54790e9	1.03334	0.516671	−	0.856184i	$-0.327171\pi$
$139$	4.54790e9	1.03334	0.516671	+	0.856184i	$0.327171\pi$
$140$	1.47354e8	0.0324179
$141$	0	0
$142$	7.25543e9	1.49750
$143$	−5.16670e8	−0.103324
$144$	0	0
$145$	2.42710e9	0.455965
$146$	9.13812e9	1.66444
$147$	0	0
$148$	5.59617e8	0.0958768
$149$	5.93288e9	0.986115	0.493058	−	0.869997i	$-0.335879\pi$
$149$	5.93288e9	0.986115	0.493058	+	0.869997i	$0.335879\pi$
$150$	0	0
$151$	−4.50028e9	−0.704439	−0.352219	−	0.935917i	$-0.614573\pi$
$151$	−4.50028e9	−0.704439	−0.352219	+	0.935917i	$0.614573\pi$
$152$	−1.66652e9	−0.253230
$153$	0	0
$154$	−4.62025e9	−0.661945
$155$	−1.79221e9	−0.249400
$156$	0	0
$157$	−1.07661e10	−1.41419	−0.707096	−	0.707118i	$-0.749995\pi$
$157$	−1.07661e10	−1.41419	−0.707096	+	0.707118i	$0.749995\pi$
$158$	5.15507e9	0.658078
$159$	0	0
$160$	1.40099e9	0.169003
$161$	3.16702e9	0.371479
$162$	0	0
$163$	1.09421e10	1.21411	0.607054	−	0.794661i	$-0.292351\pi$
$163$	1.09421e10	1.21411	0.607054	+	0.794661i	$0.292351\pi$
$164$	9.96523e8	0.107570
$165$	0	0
$166$	1.11826e9	0.114303
$167$	−3.53222e9	−0.351417	−0.175709	−	0.984442i	$-0.556222\pi$
$167$	−3.53222e9	−0.351417	−0.175709	+	0.984442i	$0.556222\pi$
$168$	0	0
$169$	−1.05747e10	−0.997187
$170$	−1.76352e9	−0.161942
$171$	0	0
$172$	−1.61984e9	−0.141122
$173$	1.39082e10	1.18049	0.590245	−	0.807224i	$-0.299031\pi$
$173$	1.39082e10	1.18049	0.590245	+	0.807224i	$0.299031\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−1.91299e10	−1.50281
$177$	0	0
$178$	−1.53270e10	−1.14438
$179$	1.51920e10	1.10605	0.553026	−	0.833164i	$-0.313473\pi$
$179$	1.51920e10	1.10605	0.553026	+	0.833164i	$0.313473\pi$
$180$	0	0
$181$	7.57219e9	0.524407	0.262203	−	0.965013i	$-0.415551\pi$
$181$	7.57219e9	0.524407	0.262203	+	0.965013i	$0.415551\pi$
$182$	2.66765e8	0.0180222
$183$	0	0
$184$	1.63729e10	1.05304
$185$	−3.56191e9	−0.223568
$186$	0	0
$187$	−1.31213e10	−0.784677
$188$	4.96218e9	0.289709
$189$	0	0
$190$	1.70696e9	0.0950236
$191$	7.70084e9	0.418685	0.209343	−	0.977842i	$-0.432868\pi$
$191$	7.70084e9	0.418685	0.209343	+	0.977842i	$0.432868\pi$
$192$	0	0
$193$	2.80464e9	0.145502	0.0727512	−	0.997350i	$-0.476822\pi$
$193$	2.80464e9	0.145502	0.0727512	+	0.997350i	$0.476822\pi$
$194$	1.72209e10	0.872867
$195$	0	0
$196$	−5.66074e8	−0.0273981
$197$	2.20023e10	1.04081	0.520403	−	0.853921i	$-0.325782\pi$
$197$	2.20023e10	1.04081	0.520403	+	0.853921i	$0.325782\pi$
$198$	0	0
$199$	7.10947e9	0.321365	0.160682	−	0.987006i	$-0.448631\pi$
$199$	7.10947e9	0.321365	0.160682	+	0.987006i	$0.448631\pi$
$200$	−4.84871e9	−0.214285
$201$	0	0
$202$	3.54587e10	1.49845
$203$	−9.32395e9	−0.385361
$204$	0	0
$205$	−6.34277e9	−0.250834
$206$	−2.92803e9	−0.113285
$207$	0	0
$208$	1.10452e9	0.0409157
$209$	1.27005e10	0.460428
$210$	0	0
$211$	7.61592e8	0.0264516	0.0132258	−	0.999913i	$-0.495790\pi$
$211$	7.61592e8	0.0264516	0.0132258	+	0.999913i	$0.495790\pi$
$212$	1.64688e9	0.0559950
$213$	0	0
$214$	−4.89944e10	−1.59692
$215$	1.03101e10	0.329072
$216$	0	0
$217$	6.88496e9	0.210782
$218$	4.44844e10	1.33399
$219$	0	0
$220$	−5.80556e9	−0.167087
$221$	7.57602e8	0.0213637
$222$	0	0
$223$	−3.07068e10	−0.831500	−0.415750	−	0.909479i	$-0.636481\pi$
$223$	−3.07068e10	−0.831500	−0.415750	+	0.909479i	$0.636481\pi$
$224$	−5.38204e9	−0.142834
$225$	0	0
$226$	3.79025e9	0.0966450
$227$	5.14276e10	1.28552	0.642762	−	0.766066i	$-0.277788\pi$
$227$	5.14276e10	1.28552	0.642762	+	0.766066i	$0.277788\pi$
$228$	0	0
$229$	1.93108e10	0.464024	0.232012	−	0.972713i	$-0.425469\pi$
$229$	1.93108e10	0.464024	0.232012	+	0.972713i	$0.425469\pi$
$230$	−1.67701e10	−0.395149
$231$	0	0
$232$	−4.82030e10	−1.09239
$233$	−6.20033e9	−0.137820	−0.0689101	−	0.997623i	$-0.521952\pi$
$233$	−6.20033e9	−0.137820	−0.0689101	+	0.997623i	$0.521952\pi$
$234$	0	0
$235$	−3.15838e10	−0.675552
$236$	−9.45246e9	−0.198354
$237$	0	0
$238$	6.77475e9	0.136866
$239$	7.10156e10	1.40787	0.703936	−	0.710263i	$-0.251424\pi$
$239$	7.10156e10	1.40787	0.703936	+	0.710263i	$0.251424\pi$
$240$	0	0
$241$	−7.98625e10	−1.52499	−0.762494	−	0.646995i	$-0.776025\pi$
$241$	−7.98625e10	−1.52499	−0.762494	+	0.646995i	$0.776025\pi$
$242$	1.34066e11	2.51276
$243$	0	0
$244$	4.67806e9	0.0844911
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−7.33302e8	−0.0125356
$248$	3.55939e10	0.597507
$249$	0	0
$250$	4.96636e9	0.0804096
$251$	−5.56254e8	−0.00884589	−0.00442294	−	0.999990i	$-0.501408\pi$
$251$	−5.56254e8	−0.00884589	−0.00442294	+	0.999990i	$0.501408\pi$
$252$	0	0
$253$	−1.24777e11	−1.91466
$254$	2.34271e10	0.353156
$255$	0	0
$256$	−3.79912e10	−0.552844
$257$	7.50599e9	0.107327	0.0536635	−	0.998559i	$-0.482910\pi$
$257$	7.50599e9	0.107327	0.0536635	+	0.998559i	$0.482910\pi$
$258$	0	0
$259$	1.36834e10	0.188950
$260$	3.35202e8	0.00454911
$261$	0	0
$262$	6.64904e10	0.871772
$263$	−3.84009e10	−0.494927	−0.247463	−	0.968897i	$-0.579597\pi$
$263$	−3.84009e10	−0.494927	−0.247463	+	0.968897i	$0.579597\pi$
$264$	0	0
$265$	−1.04822e10	−0.130571
$266$	−6.55745e9	−0.0803096
$267$	0	0
$268$	−1.71914e10	−0.203565
$269$	−1.47581e10	−0.171848	−0.0859238	−	0.996302i	$-0.527384\pi$
$269$	−1.47581e10	−0.171848	−0.0859238	+	0.996302i	$0.527384\pi$
$270$	0	0
$271$	1.66552e11	1.87580	0.937901	−	0.346903i	$-0.112767\pi$
$271$	1.66552e11	1.87580	0.937901	+	0.346903i	$0.112767\pi$
$272$	2.80505e10	0.310728
$273$	0	0
$274$	−1.12976e11	−1.21090
$275$	3.69518e10	0.389617
$276$	0	0
$277$	−8.01981e9	−0.0818475	−0.0409237	−	0.999162i	$-0.513030\pi$
$277$	−8.01981e9	−0.0818475	−0.0409237	+	0.999162i	$0.513030\pi$
$278$	9.25143e10	0.928982
$279$	0	0
$280$	1.86268e10	0.181104
$281$	1.05528e10	0.100969	0.0504847	−	0.998725i	$-0.483923\pi$
$281$	1.05528e10	0.100969	0.0504847	+	0.998725i	$0.483923\pi$
$282$	0	0
$283$	1.77172e11	1.64194	0.820969	−	0.570973i	$-0.193434\pi$
$283$	1.77172e11	1.64194	0.820969	+	0.570973i	$0.193434\pi$
$284$	−3.50231e10	−0.319464
$285$	0	0
$286$	−1.05102e10	−0.0928890
$287$	2.43664e10	0.211993
$288$	0	0
$289$	−9.93478e10	−0.837757
$290$	4.93726e10	0.409916
$291$	0	0
$292$	−4.41110e10	−0.355079
$293$	2.35923e11	1.87010	0.935052	−	0.354510i	$-0.115352\pi$
$293$	2.35923e11	1.87010	0.935052	+	0.354510i	$0.115352\pi$
$294$	0	0
$295$	6.01639e10	0.462527
$296$	7.07407e10	0.535620
$297$	0	0
$298$	1.20688e11	0.886524
$299$	7.20437e9	0.0521286
$300$	0	0
$301$	−3.96074e10	−0.278117
$302$	−9.15456e10	−0.633295
$303$	0	0
$304$	−2.71507e10	−0.182327
$305$	−2.97754e10	−0.197019
$306$	0	0
$307$	−2.60119e11	−1.67128	−0.835639	−	0.549279i	$-0.814902\pi$
$307$	−2.60119e11	−1.67128	−0.835639	+	0.549279i	$0.814902\pi$
$308$	2.23026e10	0.141214
$309$	0	0
$310$	−3.64575e10	−0.224212
$311$	2.08486e11	1.26373	0.631867	−	0.775077i	$-0.282289\pi$
$311$	2.08486e11	1.26373	0.631867	+	0.775077i	$0.282289\pi$
$312$	0	0
$313$	1.02185e10	0.0601782	0.0300891	−	0.999547i	$-0.490421\pi$
$313$	1.02185e10	0.0601782	0.0300891	+	0.999547i	$0.490421\pi$
$314$	−2.19005e11	−1.27137
$315$	0	0
$316$	−2.48843e10	−0.140389
$317$	9.48999e10	0.527836	0.263918	−	0.964545i	$-0.414985\pi$
$317$	9.48999e10	0.527836	0.263918	+	0.964545i	$0.414985\pi$
$318$	0	0
$319$	3.67352e11	1.98621
$320$	9.32115e10	0.496929
$321$	0	0
$322$	6.44241e10	0.333962
$323$	−1.86229e10	−0.0951998
$324$	0	0
$325$	−2.13353e9	−0.0106077
$326$	2.22587e11	1.09149
$327$	0	0
$328$	1.25969e11	0.600942
$329$	1.21332e11	0.570946
$330$	0	0
$331$	−2.17983e11	−0.998151	−0.499076	−	0.866559i	$-0.666327\pi$
$331$	−2.17983e11	−0.998151	−0.499076	+	0.866559i	$0.666327\pi$
$332$	−5.39801e9	−0.0243844
$333$	0	0
$334$	−7.18531e10	−0.315927
$335$	1.09421e11	0.474679
$336$	0	0
$337$	−9.16330e10	−0.387006	−0.193503	−	0.981100i	$-0.561985\pi$
$337$	−9.16330e10	−0.387006	−0.193503	+	0.981100i	$0.561985\pi$
$338$	−2.15112e11	−0.896478
$339$	0	0
$340$	8.51278e9	0.0345475
$341$	−2.71259e11	−1.08640
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	−2.04763e11	−0.788384
$345$	0	0
$346$	2.82923e11	1.06127
$347$	−3.32280e11	−1.23033	−0.615165	−	0.788398i	$-0.710911\pi$
$347$	−3.32280e11	−1.23033	−0.615165	+	0.788398i	$0.710911\pi$
$348$	0	0
$349$	−3.21762e11	−1.16097	−0.580484	−	0.814271i	$-0.697137\pi$
$349$	−3.21762e11	−1.16097	−0.580484	+	0.814271i	$0.697137\pi$
$350$	−1.90788e10	−0.0679585
$351$	0	0
$352$	2.12046e11	0.736187
$353$	−8.38477e10	−0.287412	−0.143706	−	0.989620i	$-0.545902\pi$
$353$	−8.38477e10	−0.287412	−0.143706	+	0.989620i	$0.545902\pi$
$354$	0	0
$355$	2.22918e11	0.744934
$356$	7.39859e10	0.244132
$357$	0	0
$358$	3.09038e11	0.994349
$359$	1.75368e11	0.557219	0.278610	−	0.960404i	$-0.410126\pi$
$359$	1.75368e11	0.557219	0.278610	+	0.960404i	$0.410126\pi$
$360$	0	0
$361$	−3.04662e11	−0.944139
$362$	1.54035e11	0.471445
$363$	0	0
$364$	−1.28771e9	−0.00384470
$365$	2.80762e11	0.827982
$366$	0	0
$367$	4.73587e11	1.36271	0.681354	−	0.731954i	$-0.261392\pi$
$367$	4.73587e11	1.36271	0.681354	+	0.731954i	$0.261392\pi$
$368$	2.66744e11	0.758193
$369$	0	0
$370$	−7.24571e10	−0.200989
$371$	4.02684e10	0.110352
$372$	0	0
$373$	−2.66752e11	−0.713540	−0.356770	−	0.934192i	$-0.616122\pi$
$373$	−2.66752e11	−0.713540	−0.356770	+	0.934192i	$0.616122\pi$
$374$	−2.66917e11	−0.705430
$375$	0	0
$376$	6.27265e11	1.61847
$377$	−2.12102e10	−0.0540766
$378$	0	0
$379$	−4.45189e10	−0.110833	−0.0554163	−	0.998463i	$-0.517649\pi$
$379$	−4.45189e10	−0.110833	−0.0554163	+	0.998463i	$0.517649\pi$
$380$	−8.23973e9	−0.0202715
$381$	0	0
$382$	1.56652e11	0.376401
$383$	4.63051e10	0.109960	0.0549799	−	0.998487i	$-0.482491\pi$
$383$	4.63051e10	0.109960	0.0549799	+	0.998487i	$0.482491\pi$
$384$	0	0
$385$	−1.41954e11	−0.329287
$386$	5.70527e10	0.130808
$387$	0	0
$388$	−8.31278e10	−0.186210
$389$	2.27563e11	0.503881	0.251940	−	0.967743i	$-0.418931\pi$
$389$	2.27563e11	0.503881	0.251940	+	0.967743i	$0.418931\pi$
$390$	0	0
$391$	1.82962e11	0.395882
$392$	−7.15568e10	−0.153061
$393$	0	0
$394$	4.47575e11	0.935691
$395$	1.58386e11	0.327363
$396$	0	0
$397$	−6.33956e11	−1.28086	−0.640431	−	0.768016i	$-0.721244\pi$
$397$	−6.33956e11	−1.28086	−0.640431	+	0.768016i	$0.721244\pi$
$398$	1.44622e11	0.288909
$399$	0	0
$400$	−7.89945e10	−0.154286
$401$	−5.00873e11	−0.967337	−0.483669	−	0.875251i	$-0.660696\pi$
$401$	−5.00873e11	−0.967337	−0.483669	+	0.875251i	$0.660696\pi$
$402$	0	0
$403$	1.56620e10	0.0295784
$404$	−1.71164e11	−0.319667
$405$	0	0
$406$	−1.89670e11	−0.346442
$407$	−5.39111e11	−0.973875
$408$	0	0
$409$	4.11851e11	0.727754	0.363877	−	0.931447i	$-0.381453\pi$
$409$	4.11851e11	0.727754	0.363877	+	0.931447i	$0.381453\pi$
$410$	−1.29026e11	−0.225501
$411$	0	0
$412$	1.41340e10	0.0241673
$413$	−2.31126e11	−0.390907
$414$	0	0
$415$	3.43578e10	0.0568602
$416$	−1.22431e10	−0.0200435
$417$	0	0
$418$	2.58356e11	0.413928
$419$	−2.98490e11	−0.473115	−0.236557	−	0.971618i	$-0.576019\pi$
$419$	−2.98490e11	−0.473115	−0.236557	+	0.971618i	$0.576019\pi$
$420$	0	0
$421$	7.84245e11	1.21670	0.608349	−	0.793670i	$-0.291832\pi$
$421$	7.84245e11	1.21670	0.608349	+	0.793670i	$0.291832\pi$
$422$	1.54925e10	0.0237801
$423$	0	0
$424$	2.08180e11	0.312819
$425$	−5.41830e10	−0.0805587
$426$	0	0
$427$	1.14385e11	0.166511
$428$	2.36503e11	0.340675
$429$	0	0
$430$	2.09731e11	0.295838
$431$	−9.38806e11	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$431$	−9.38806e11	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$432$	0	0
$433$	−4.26650e11	−0.583279	−0.291640	−	0.956528i	$-0.594201\pi$
$433$	−4.26650e11	−0.583279	−0.291640	+	0.956528i	$0.594201\pi$
$434$	1.40055e11	0.189494
$435$	0	0
$436$	−2.14733e11	−0.284583
$437$	−1.77093e11	−0.232293
$438$	0	0
$439$	−1.24959e12	−1.60574	−0.802871	−	0.596153i	$-0.796695\pi$
$439$	−1.24959e12	−1.60574	−0.802871	+	0.596153i	$0.796695\pi$
$440$	−7.33874e11	−0.933436
$441$	0	0
$442$	1.54113e10	0.0192061
$443$	−2.49206e11	−0.307426	−0.153713	−	0.988116i	$-0.549123\pi$
$443$	−2.49206e11	−0.307426	−0.153713	+	0.988116i	$0.549123\pi$
$444$	0	0
$445$	−4.70913e11	−0.569273
$446$	−6.24644e11	−0.747524
$447$	0	0
$448$	−3.58081e11	−0.419982
$449$	1.15882e12	1.34557	0.672787	−	0.739836i	$-0.265097\pi$
$449$	1.15882e12	1.34557	0.672787	+	0.739836i	$0.265097\pi$
$450$	0	0
$451$	−9.60006e11	−1.09265
$452$	−1.82961e10	−0.0206175
$453$	0	0
$454$	1.04615e12	1.15570
$455$	8.19616e9	0.00896518
$456$	0	0
$457$	1.55301e12	1.66552	0.832762	−	0.553631i	$-0.186758\pi$
$457$	1.55301e12	1.66552	0.832762	+	0.553631i	$0.186758\pi$
$458$	3.92824e11	0.417161
$459$	0	0
$460$	8.09518e10	0.0842978
$461$	−1.16000e12	−1.19619	−0.598097	−	0.801423i	$-0.704076\pi$
$461$	−1.16000e12	−1.19619	−0.598097	+	0.801423i	$0.704076\pi$
$462$	0	0
$463$	7.80681e11	0.789513	0.394757	−	0.918786i	$-0.370829\pi$
$463$	7.80681e11	0.789513	0.394757	+	0.918786i	$0.370829\pi$
$464$	−7.85316e11	−0.786527
$465$	0	0
$466$	−1.26128e11	−0.123901
$467$	−4.85747e11	−0.472590	−0.236295	−	0.971681i	$-0.575933\pi$
$467$	−4.85747e11	−0.472590	−0.236295	+	0.971681i	$0.575933\pi$
$468$	0	0
$469$	−4.20353e11	−0.401177
$470$	−6.42484e11	−0.607326
$471$	0	0
$472$	−1.19488e12	−1.10811
$473$	1.56049e12	1.43346
$474$	0	0
$475$	5.24451e10	0.0472698
$476$	−3.27027e10	−0.0291979
$477$	0	0
$478$	1.44461e12	1.26569
$479$	−4.22167e11	−0.366416	−0.183208	−	0.983074i	$-0.558648\pi$
$479$	−4.22167e11	−0.366416	−0.183208	+	0.983074i	$0.558648\pi$
$480$	0	0
$481$	3.11273e10	0.0265148
$482$	−1.62458e12	−1.37097
$483$	0	0
$484$	−6.47159e11	−0.536052
$485$	5.29100e11	0.434210
$486$	0	0
$487$	4.04896e11	0.326184	0.163092	−	0.986611i	$-0.447853\pi$
$487$	4.04896e11	0.326184	0.163092	+	0.986611i	$0.447853\pi$
$488$	5.91348e11	0.472013
$489$	0	0
$490$	7.32930e10	0.0574354
$491$	−1.02346e12	−0.794698	−0.397349	−	0.917668i	$-0.630070\pi$
$491$	−1.02346e12	−0.794698	−0.397349	+	0.917668i	$0.630070\pi$
$492$	0	0
$493$	−5.38655e11	−0.410676
$494$	−1.49170e10	−0.0112696
$495$	0	0
$496$	5.79891e11	0.430208
$497$	−8.56362e11	−0.629584
$498$	0	0
$499$	−1.00743e12	−0.727383	−0.363691	−	0.931520i	$-0.618484\pi$
$499$	−1.00743e12	−0.727383	−0.363691	+	0.931520i	$0.618484\pi$
$500$	−2.39733e10	−0.0171539
$501$	0	0
$502$	−1.13154e10	−0.00795251
$503$	9.72840e11	0.677619	0.338810	−	0.940855i	$-0.389976\pi$
$503$	9.72840e11	0.677619	0.338810	+	0.940855i	$0.389976\pi$
$504$	0	0
$505$	1.08944e12	0.745408
$506$	−2.53823e12	−1.72129
$507$	0	0
$508$	−1.13086e11	−0.0753394
$509$	3.61105e11	0.238453	0.119227	−	0.992867i	$-0.461959\pi$
$509$	3.61105e11	0.238453	0.119227	+	0.992867i	$0.461959\pi$
$510$	0	0
$511$	−1.07858e12	−0.699772
$512$	−1.73851e12	−1.11806
$513$	0	0
$514$	1.52688e11	0.0964877
$515$	−8.99616e10	−0.0563540
$516$	0	0
$517$	−4.78035e12	−2.94274
$518$	2.78351e11	0.169867
$519$	0	0
$520$	4.23726e10	0.0254138
$521$	−3.12045e12	−1.85544	−0.927722	−	0.373273i	$-0.878236\pi$
$521$	−3.12045e12	−1.85544	−0.927722	+	0.373273i	$0.878236\pi$
$522$	0	0
$523$	−2.52016e12	−1.47289	−0.736446	−	0.676496i	$-0.763497\pi$
$523$	−2.52016e12	−1.47289	−0.736446	+	0.676496i	$0.763497\pi$
$524$	−3.20959e11	−0.185977
$525$	0	0
$526$	−7.81160e11	−0.444943
$527$	3.97752e11	0.224628
$528$	0	0
$529$	−6.12859e10	−0.0340259
$530$	−2.13231e11	−0.117384
$531$	0	0
$532$	3.16538e10	0.0171326
$533$	5.54290e10	0.0297485
$534$	0	0
$535$	−1.50532e12	−0.794394
$536$	−2.17314e12	−1.13723
$537$	0	0
$538$	−3.00211e11	−0.154492
$539$	5.45330e11	0.278298
$540$	0	0
$541$	5.30604e11	0.266307	0.133154	−	0.991095i	$-0.457490\pi$
$541$	5.30604e11	0.266307	0.133154	+	0.991095i	$0.457490\pi$
$542$	3.38803e12	1.68636
$543$	0	0
$544$	−3.10926e11	−0.152217
$545$	1.36675e12	0.663599
$546$	0	0
$547$	−2.33227e11	−0.111388	−0.0556938	−	0.998448i	$-0.517737\pi$
$547$	−2.33227e11	−0.111388	−0.0556938	+	0.998448i	$0.517737\pi$
$548$	5.45349e11	0.258323
$549$	0	0
$550$	7.51680e11	0.350269
$551$	5.21377e11	0.240974
$552$	0	0
$553$	−6.08455e11	−0.276672
$554$	−1.63141e11	−0.0735814
$555$	0	0
$556$	−4.46580e11	−0.198181
$557$	−2.38965e12	−1.05193	−0.525965	−	0.850506i	$-0.676296\pi$
$557$	−2.38965e12	−1.05193	−0.525965	+	0.850506i	$0.676296\pi$
$558$	0	0
$559$	−9.00995e10	−0.0390274
$560$	3.03465e11	0.130396
$561$	0	0
$562$	2.14668e11	0.0907723
$563$	−8.19534e11	−0.343779	−0.171889	−	0.985116i	$-0.554987\pi$
$563$	−8.19534e11	−0.343779	−0.171889	+	0.985116i	$0.554987\pi$
$564$	0	0
$565$	1.16453e11	0.0480764
$566$	3.60407e12	1.47611
$567$	0	0
$568$	−4.42723e12	−1.78470
$569$	−1.22742e12	−0.490896	−0.245448	−	0.969410i	$-0.578935\pi$
$569$	−1.22742e12	−0.490896	−0.245448	+	0.969410i	$0.578935\pi$
$570$	0	0
$571$	2.77719e12	1.09331	0.546654	−	0.837359i	$-0.315901\pi$
$571$	2.77719e12	1.09331	0.546654	+	0.837359i	$0.315901\pi$
$572$	5.07343e10	0.0198162
$573$	0	0
$574$	4.95666e11	0.190584
$575$	−5.15250e11	−0.196568
$576$	0	0
$577$	−1.44846e12	−0.544020	−0.272010	−	0.962294i	$-0.587688\pi$
$577$	−1.44846e12	−0.544020	−0.272010	+	0.962294i	$0.587688\pi$
$578$	−2.02095e12	−0.753149
$579$	0	0
$580$	−2.38329e11	−0.0874481
$581$	−1.31989e11	−0.0480557
$582$	0	0
$583$	−1.58653e12	−0.568773
$584$	−5.57603e12	−1.98366
$585$	0	0
$586$	4.79919e12	1.68124
$587$	−3.31169e12	−1.15127	−0.575636	−	0.817706i	$-0.695246\pi$
$587$	−3.31169e12	−1.15127	−0.575636	+	0.817706i	$0.695246\pi$
$588$	0	0
$589$	−3.84994e11	−0.131806
$590$	1.22387e12	0.415815
$591$	0	0
$592$	1.15250e12	0.385649
$593$	1.32686e12	0.440634	0.220317	−	0.975428i	$-0.429291\pi$
$593$	1.32686e12	0.440634	0.220317	+	0.975428i	$0.429291\pi$
$594$	0	0
$595$	2.08149e11	0.0680846
$596$	−5.82578e11	−0.189124
$597$	0	0
$598$	1.46553e11	0.0468639
$599$	2.19919e12	0.697979	0.348989	−	0.937127i	$-0.386525\pi$
$599$	2.19919e12	0.697979	0.348989	+	0.937127i	$0.386525\pi$
$600$	0	0
$601$	−1.34347e11	−0.0420042	−0.0210021	−	0.999779i	$-0.506686\pi$
$601$	−1.34347e11	−0.0420042	−0.0210021	+	0.999779i	$0.506686\pi$
$602$	−8.05702e11	−0.250029
$603$	0	0
$604$	4.41904e11	0.135102
$605$	4.11910e12	1.24998
$606$	0	0
$607$	1.85820e12	0.555576	0.277788	−	0.960642i	$-0.410399\pi$
$607$	1.85820e12	0.555576	0.277788	+	0.960642i	$0.410399\pi$
$608$	3.00953e11	0.0893168
$609$	0	0
$610$	−6.05696e11	−0.177121
$611$	2.76008e11	0.0801193
$612$	0	0
$613$	3.99243e12	1.14200	0.570999	−	0.820951i	$-0.306556\pi$
$613$	3.99243e12	1.14200	0.570999	+	0.820951i	$0.306556\pi$
$614$	−5.29138e12	−1.50249
$615$	0	0
$616$	2.81925e12	0.788898
$617$	−1.10517e12	−0.307005	−0.153503	−	0.988148i	$-0.549055\pi$
$617$	−1.10517e12	−0.307005	−0.153503	+	0.988148i	$0.549055\pi$
$618$	0	0
$619$	−2.34847e12	−0.642950	−0.321475	−	0.946918i	$-0.604179\pi$
$619$	−2.34847e12	−0.642950	−0.321475	+	0.946918i	$0.604179\pi$
$620$	1.75986e11	0.0478316
$621$	0	0
$622$	4.24107e12	1.13610
$623$	1.80906e12	0.481123
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	2.07867e11	0.0541006
$627$	0	0
$628$	1.05717e12	0.271223
$629$	7.90507e11	0.201362
$630$	0	0
$631$	−1.22889e12	−0.308590	−0.154295	−	0.988025i	$-0.549311\pi$
$631$	−1.22889e12	−0.308590	−0.154295	+	0.988025i	$0.549311\pi$
$632$	−3.14559e12	−0.784288
$633$	0	0
$634$	1.93047e12	0.474528
$635$	7.19781e11	0.175678
$636$	0	0
$637$	−3.14864e10	−0.00757696
$638$	7.47276e12	1.78561
$639$	0	0
$640$	1.17882e12	0.277740
$641$	4.53912e9	0.00106197	0.000530983	−	1.00000i	$-0.499831\pi$
$641$	4.53912e9	0.00106197	0.000530983		1.00000i	$0.499831\pi$
$642$	0	0
$643$	1.46686e12	0.338406	0.169203	−	0.985581i	$-0.445881\pi$
$643$	1.46686e12	0.338406	0.169203	+	0.985581i	$0.445881\pi$
$644$	−3.10984e11	−0.0712447
$645$	0	0
$646$	−3.78831e11	−0.0855853
$647$	5.98026e12	1.34169	0.670843	−	0.741600i	$-0.265933\pi$
$647$	5.98026e12	1.34169	0.670843	+	0.741600i	$0.265933\pi$
$648$	0	0
$649$	9.10608e12	2.01479
$650$	−4.34006e10	−0.00953643
$651$	0	0
$652$	−1.07446e12	−0.232850
$653$	2.38614e12	0.513554	0.256777	−	0.966471i	$-0.417339\pi$
$653$	2.38614e12	0.513554	0.256777	+	0.966471i	$0.417339\pi$
$654$	0	0
$655$	2.04287e12	0.433666
$656$	2.05228e12	0.432682
$657$	0	0
$658$	2.46817e12	0.513284
$659$	2.22402e12	0.459362	0.229681	−	0.973266i	$-0.426232\pi$
$659$	2.22402e12	0.459362	0.229681	+	0.973266i	$0.426232\pi$
$660$	0	0
$661$	2.19776e12	0.447790	0.223895	−	0.974613i	$-0.428123\pi$
$661$	2.19776e12	0.447790	0.223895	+	0.974613i	$0.428123\pi$
$662$	−4.43425e12	−0.897345
$663$	0	0
$664$	−6.82357e11	−0.136224
$665$	−2.01473e11	−0.0399502
$666$	0	0
$667$	−5.12231e12	−1.00207
$668$	3.46845e11	0.0673972
$669$	0	0
$670$	2.22587e12	0.426740
$671$	−4.50663e12	−0.858224
$672$	0	0
$673$	4.38627e11	0.0824191	0.0412095	−	0.999151i	$-0.486879\pi$
$673$	4.38627e11	0.0824191	0.0412095	+	0.999151i	$0.486879\pi$
$674$	−1.86402e12	−0.347921
$675$	0	0
$676$	1.03838e12	0.191247
$677$	8.30205e12	1.51893	0.759463	−	0.650551i	$-0.225462\pi$
$677$	8.30205e12	1.51893	0.759463	+	0.650551i	$0.225462\pi$
$678$	0	0
$679$	−2.03259e12	−0.366975
$680$	1.07609e12	0.193001
$681$	0	0
$682$	−5.51801e12	−0.976681
$683$	9.55861e12	1.68074	0.840372	−	0.542010i	$-0.182337\pi$
$683$	9.55861e12	1.68074	0.840372	+	0.542010i	$0.182337\pi$
$684$	0	0
$685$	−3.47109e12	−0.602364
$686$	−2.81562e11	−0.0485418
$687$	0	0
$688$	−3.33596e12	−0.567641
$689$	9.16032e10	0.0154855
$690$	0	0
$691$	−7.94645e12	−1.32593	−0.662967	−	0.748649i	$-0.730703\pi$
$691$	−7.94645e12	−1.32593	−0.662967	+	0.748649i	$0.730703\pi$
$692$	−1.36571e12	−0.226402
$693$	0	0
$694$	−6.75931e12	−1.10608
$695$	2.84244e12	0.462125
$696$	0	0
$697$	1.40767e12	0.225920
$698$	−6.54535e12	−1.04372
$699$	0	0
$700$	9.20960e10	0.0144977
$701$	7.30842e12	1.14312	0.571561	−	0.820559i	$-0.306338\pi$
$701$	7.30842e12	1.14312	0.571561	+	0.820559i	$0.306338\pi$
$702$	0	0
$703$	−7.65151e11	−0.118154
$704$	1.41080e13	2.16465
$705$	0	0
$706$	−1.70565e12	−0.258385
$707$	−4.18521e12	−0.629984
$708$	0	0
$709$	−4.32977e12	−0.643512	−0.321756	−	0.946823i	$-0.604273\pi$
$709$	−4.32977e12	−0.643512	−0.321756	+	0.946823i	$0.604273\pi$
$710$	4.53465e12	0.669701
$711$	0	0
$712$	9.35248e12	1.36385
$713$	3.78240e12	0.548106
$714$	0	0
$715$	−3.22919e11	−0.0462079
$716$	−1.49177e12	−0.212126
$717$	0	0
$718$	3.56738e12	0.500944
$719$	−1.24222e13	−1.73348	−0.866738	−	0.498763i	$-0.833788\pi$
$719$	−1.24222e13	−1.73348	−0.866738	+	0.498763i	$0.833788\pi$
$720$	0	0
$721$	3.45596e11	0.0476278
$722$	−6.19750e12	−0.848788
$723$	0	0
$724$	−7.43550e11	−0.100574
$725$	1.51694e12	0.203914
$726$	0	0
$727$	−1.09384e13	−1.45227	−0.726136	−	0.687551i	$-0.758686\pi$
$727$	−1.09384e13	−1.45227	−0.726136	+	0.687551i	$0.758686\pi$
$728$	−1.62778e11	−0.0214786
$729$	0	0
$730$	5.71132e12	0.744361
$731$	−2.28816e12	−0.296387
$732$	0	0
$733$	1.28000e13	1.63773	0.818865	−	0.573986i	$-0.194604\pi$
$733$	1.28000e13	1.63773	0.818865	+	0.573986i	$0.194604\pi$
$734$	9.63381e12	1.22508
$735$	0	0
$736$	−2.95674e12	−0.371418
$737$	1.65614e13	2.06773
$738$	0	0
$739$	1.02293e13	1.26167	0.630834	−	0.775917i	$-0.282713\pi$
$739$	1.02293e13	1.26167	0.630834	+	0.775917i	$0.282713\pi$
$740$	3.49761e11	0.0428774
$741$	0	0
$742$	8.19148e11	0.0992076
$743$	1.41180e13	1.69951	0.849757	−	0.527175i	$-0.176749\pi$
$743$	1.41180e13	1.69951	0.849757	+	0.527175i	$0.176749\pi$
$744$	0	0
$745$	3.70805e12	0.441004
$746$	−5.42633e12	−0.641478
$747$	0	0
$748$	1.28845e12	0.150491
$749$	5.78283e12	0.671386
$750$	0	0
$751$	1.24714e13	1.43065	0.715327	−	0.698790i	$-0.246278\pi$
$751$	1.24714e13	1.43065	0.715327	+	0.698790i	$0.246278\pi$
$752$	1.02193e13	1.16531
$753$	0	0
$754$	−4.31463e11	−0.0486153
$755$	−2.81268e12	−0.315035
$756$	0	0
$757$	2.04130e12	0.225931	0.112965	−	0.993599i	$-0.463965\pi$
$757$	2.04130e12	0.225931	0.112965	+	0.993599i	$0.463965\pi$
$758$	−9.05612e11	−0.0996393
$759$	0	0
$760$	−1.04158e12	−0.113248
$761$	−7.40974e12	−0.800888	−0.400444	−	0.916321i	$-0.631144\pi$
$761$	−7.40974e12	−0.800888	−0.400444	+	0.916321i	$0.631144\pi$
$762$	0	0
$763$	−5.25052e12	−0.560843
$764$	−7.56182e11	−0.0802983
$765$	0	0
$766$	9.41947e11	0.0988546
$767$	−5.25768e11	−0.0548549
$768$	0	0
$769$	−6.08626e11	−0.0627598	−0.0313799	−	0.999508i	$-0.509990\pi$
$769$	−6.08626e11	−0.0627598	−0.0313799	+	0.999508i	$0.509990\pi$
$770$	−2.88766e12	−0.296031
$771$	0	0
$772$	−2.75402e11	−0.0279054
$773$	−3.34982e12	−0.337453	−0.168727	−	0.985663i	$-0.553966\pi$
$773$	−3.34982e12	−0.337453	−0.168727	+	0.985663i	$0.553966\pi$
$774$	0	0
$775$	−1.12013e12	−0.111535
$776$	−1.05081e13	−1.04027
$777$	0	0
$778$	4.62913e12	0.452992
$779$	−1.36252e12	−0.132564
$780$	0	0
$781$	3.37397e13	3.24497
$782$	3.72185e12	0.355900
$783$	0	0
$784$	−1.16579e12	−0.110204
$785$	−6.72879e12	−0.632446
$786$	0	0
$787$	−1.11762e13	−1.03850	−0.519251	−	0.854622i	$-0.673789\pi$
$787$	−1.11762e13	−1.03850	−0.519251	+	0.854622i	$0.673789\pi$
$788$	−2.16051e12	−0.199613
$789$	0	0
$790$	3.22192e12	0.294301
$791$	−4.47365e11	−0.0406319
$792$	0	0
$793$	2.60205e11	0.0233661
$794$	−1.28961e13	−1.15150
$795$	0	0
$796$	−6.98113e11	−0.0616335
$797$	−2.81409e12	−0.247045	−0.123522	−	0.992342i	$-0.539419\pi$
$797$	−2.81409e12	−0.247045	−0.123522	+	0.992342i	$0.539419\pi$
$798$	0	0
$799$	7.00950e12	0.608452
$800$	8.75618e11	0.0755805
$801$	0	0
$802$	−1.01889e13	−0.869643
$803$	4.24946e13	3.60673
$804$	0	0
$805$	1.97938e12	0.166130
$806$	3.18600e11	0.0265912
$807$	0	0
$808$	−2.16367e13	−1.78583
$809$	1.02388e13	0.840391	0.420196	−	0.907434i	$-0.361961\pi$
$809$	1.02388e13	0.840391	0.420196	+	0.907434i	$0.361961\pi$
$810$	0	0
$811$	−7.13828e12	−0.579428	−0.289714	−	0.957113i	$-0.593560\pi$
$811$	−7.13828e12	−0.579428	−0.289714	+	0.957113i	$0.593560\pi$
$812$	9.15563e11	0.0739071
$813$	0	0
$814$	−1.09667e13	−0.875520
$815$	6.83883e12	0.542966
$816$	0	0
$817$	2.21477e12	0.173912
$818$	8.37795e12	0.654256
$819$	0	0
$820$	6.22827e11	0.0481066
$821$	−1.17624e13	−0.903546	−0.451773	−	0.892133i	$-0.649208\pi$
$821$	−1.17624e13	−0.903546	−0.451773	+	0.892133i	$0.649208\pi$
$822$	0	0
$823$	4.93950e12	0.375305	0.187652	−	0.982236i	$-0.439912\pi$
$823$	4.93950e12	0.375305	0.187652	+	0.982236i	$0.439912\pi$
$824$	1.78667e12	0.135012
$825$	0	0
$826$	−4.70161e12	−0.351428
$827$	−1.42644e13	−1.06043	−0.530213	−	0.847865i	$-0.677888\pi$
$827$	−1.42644e13	−1.06043	−0.530213	+	0.847865i	$0.677888\pi$
$828$	0	0
$829$	−2.12066e13	−1.55947	−0.779734	−	0.626111i	$-0.784646\pi$
$829$	−2.12066e13	−1.55947	−0.779734	+	0.626111i	$0.784646\pi$
$830$	6.98913e11	0.0511177
$831$	0	0
$832$	−8.14569e11	−0.0589349
$833$	−7.99627e11	−0.0575420
$834$	0	0
$835$	−2.20764e12	−0.157159
$836$	−1.24712e12	−0.0883039
$837$	0	0
$838$	−6.07194e12	−0.425333
$839$	1.51532e13	1.05579	0.527894	−	0.849310i	$-0.322982\pi$
$839$	1.51532e13	1.05579	0.527894	+	0.849310i	$0.322982\pi$
$840$	0	0
$841$	5.73335e11	0.0395208
$842$	1.59533e13	1.09382
$843$	0	0
$844$	−7.47844e10	−0.00507306
$845$	−6.60917e12	−0.445956
$846$	0	0
$847$	−1.58239e13	−1.05643
$848$	3.39164e12	0.225231
$849$	0	0
$850$	−1.10220e12	−0.0724229
$851$	7.51728e12	0.491335
$852$	0	0
$853$	−2.33173e13	−1.50802	−0.754009	−	0.656864i	$-0.771883\pi$
$853$	−2.33173e13	−1.50802	−0.754009	+	0.656864i	$0.771883\pi$
$854$	2.32684e12	0.149695
$855$	0	0
$856$	2.98961e13	1.90319
$857$	3.02364e13	1.91477	0.957385	−	0.288814i	$-0.0932610\pi$
$857$	3.02364e13	1.91477	0.957385	+	0.288814i	$0.0932610\pi$
$858$	0	0
$859$	−1.83562e13	−1.15030	−0.575152	−	0.818047i	$-0.695057\pi$
$859$	−1.83562e13	−1.15030	−0.575152	+	0.818047i	$0.695057\pi$
$860$	−1.01240e12	−0.0631117
$861$	0	0
$862$	−1.90974e13	−1.17812
$863$	6.03162e12	0.370156	0.185078	−	0.982724i	$-0.440746\pi$
$863$	6.03162e12	0.370156	0.185078	+	0.982724i	$0.440746\pi$
$864$	0	0
$865$	8.69260e12	0.527931
$866$	−8.67900e12	−0.524372
$867$	0	0
$868$	−6.76067e11	−0.0404251
$869$	2.39724e13	1.42601
$870$	0	0
$871$	−9.56225e11	−0.0562961
$872$	−2.71442e13	−1.58984
$873$	0	0
$874$	−3.60247e12	−0.208833
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−3.73174e12	−0.213016	−0.106508	−	0.994312i	$-0.533967\pi$
$877$	−3.73174e12	−0.213016	−0.106508	+	0.994312i	$0.533967\pi$
$878$	−2.54193e13	−1.44357
$879$	0	0
$880$	−1.19562e13	−0.672079
$881$	8.92664e12	0.499225	0.249613	−	0.968346i	$-0.419697\pi$
$881$	8.92664e12	0.499225	0.249613	+	0.968346i	$0.419697\pi$
$882$	0	0
$883$	−1.02445e13	−0.567108	−0.283554	−	0.958956i	$-0.591513\pi$
$883$	−1.02445e13	−0.567108	−0.283554	+	0.958956i	$0.591513\pi$
$884$	−7.43926e10	−0.00409727
$885$	0	0
$886$	−5.06939e12	−0.276378
$887$	−2.35306e13	−1.27637	−0.638186	−	0.769882i	$-0.720315\pi$
$887$	−2.35306e13	−1.27637	−0.638186	+	0.769882i	$0.720315\pi$
$888$	0	0
$889$	−2.76511e12	−0.148475
$890$	−9.57940e12	−0.511780
$891$	0	0
$892$	3.01525e12	0.159471
$893$	−6.78467e12	−0.357024
$894$	0	0
$895$	9.49499e12	0.494642
$896$	−4.52856e12	−0.234733
$897$	0	0
$898$	2.35730e13	1.20968
$899$	−1.11357e13	−0.568589
$900$	0	0
$901$	2.32635e12	0.117602
$902$	−1.95286e13	−0.982296
$903$	0	0
$904$	−2.31279e12	−0.115180
$905$	4.73262e12	0.234522
$906$	0	0
$907$	−3.14256e12	−0.154188	−0.0770941	−	0.997024i	$-0.524564\pi$
$907$	−3.14256e12	−0.154188	−0.0770941	+	0.997024i	$0.524564\pi$
$908$	−5.04993e12	−0.246546
$909$	0	0
$910$	1.66728e11	0.00805976
$911$	1.66876e13	0.802715	0.401358	−	0.915921i	$-0.368538\pi$
$911$	1.66876e13	0.802715	0.401358	+	0.915921i	$0.368538\pi$
$912$	0	0
$913$	5.20020e12	0.247686
$914$	3.15916e13	1.49732
$915$	0	0
$916$	−1.89622e12	−0.0889937
$917$	−7.84789e12	−0.366514
$918$	0	0
$919$	3.51280e13	1.62455	0.812275	−	0.583274i	$-0.198229\pi$
$919$	3.51280e13	1.62455	0.812275	+	0.583274i	$0.198229\pi$
$920$	1.02330e13	0.470933
$921$	0	0
$922$	−2.35969e13	−1.07539
$923$	−1.94807e12	−0.0883478
$924$	0	0
$925$	−2.22619e12	−0.0999827
$926$	1.58808e13	0.709778
$927$	0	0
$928$	8.70487e12	0.385298
$929$	−3.50454e13	−1.54369	−0.771846	−	0.635809i	$-0.780667\pi$
$929$	−3.50454e13	−1.54369	−0.771846	+	0.635809i	$0.780667\pi$
$930$	0	0
$931$	7.73978e11	0.0337641
$932$	6.08840e11	0.0264321
$933$	0	0
$934$	−9.88117e12	−0.424861
$935$	−8.20084e12	−0.350918
$936$	0	0
$937$	1.09225e12	0.0462909	0.0231455	−	0.999732i	$-0.492632\pi$
$937$	1.09225e12	0.0462909	0.0231455	+	0.999732i	$0.492632\pi$
$938$	−8.55090e12	−0.360661
$939$	0	0
$940$	3.10136e12	0.129562
$941$	1.16562e13	0.484624	0.242312	−	0.970198i	$-0.422094\pi$
$941$	1.16562e13	0.484624	0.242312	+	0.970198i	$0.422094\pi$
$942$	0	0
$943$	1.33862e13	0.551257
$944$	−1.94667e13	−0.797847
$945$	0	0
$946$	3.17437e13	1.28869
$947$	−2.24899e13	−0.908682	−0.454341	−	0.890828i	$-0.650125\pi$
$947$	−2.24899e13	−0.908682	−0.454341	+	0.890828i	$0.650125\pi$
$948$	0	0
$949$	−2.45356e12	−0.0981971
$950$	1.06685e12	0.0424958
$951$	0	0
$952$	−4.13391e12	−0.163116
$953$	4.73683e13	1.86024	0.930121	−	0.367253i	$-0.119702\pi$
$953$	4.73683e13	1.86024	0.930121	+	0.367253i	$0.119702\pi$
$954$	0	0
$955$	4.81302e12	0.187242
$956$	−6.97336e12	−0.270011
$957$	0	0
$958$	−8.58781e12	−0.329411
$959$	1.33346e13	0.509090
$960$	0	0
$961$	−1.82168e13	−0.688998
$962$	6.33197e11	0.0238370
$963$	0	0
$964$	7.84209e12	0.292472
$965$	1.75290e12	0.0650706
$966$	0	0
$967$	2.08057e13	0.765178	0.382589	−	0.923919i	$-0.375033\pi$
$967$	2.08057e13	0.765178	0.382589	+	0.923919i	$0.375033\pi$
$968$	−8.18067e13	−2.99467
$969$	0	0
$970$	1.07631e13	0.390358
$971$	3.04448e13	1.09907	0.549536	−	0.835470i	$-0.314804\pi$
$971$	3.04448e13	1.09907	0.549536	+	0.835470i	$0.314804\pi$
$972$	0	0
$973$	−1.09195e13	−0.390567
$974$	8.23647e12	0.293242
$975$	0	0
$976$	9.63416e12	0.339852
$977$	−2.50664e13	−0.880169	−0.440084	−	0.897956i	$-0.645051\pi$
$977$	−2.50664e13	−0.880169	−0.440084	+	0.897956i	$0.645051\pi$
$978$	0	0
$979$	−7.12748e13	−2.47978
$980$	−3.53796e11	−0.0122528
$981$	0	0
$982$	−2.08193e13	−0.714439
$983$	−3.24268e12	−0.110768	−0.0553838	−	0.998465i	$-0.517638\pi$
$983$	−3.24268e12	−0.110768	−0.0553838	+	0.998465i	$0.517638\pi$
$984$	0	0
$985$	1.37514e13	0.465462
$986$	−1.09574e13	−0.369201
$987$	0	0
$988$	7.20064e10	0.00240417
$989$	−2.17592e13	−0.723201
$990$	0	0
$991$	3.41624e13	1.12517	0.562583	−	0.826741i	$-0.309808\pi$
$991$	3.41624e13	1.12517	0.562583	+	0.826741i	$0.309808\pi$
$992$	−6.42783e12	−0.210747
$993$	0	0
$994$	−1.74203e13	−0.566001
$995$	4.44342e12	0.143719
$996$	0	0
$997$	−2.85092e13	−0.913811	−0.456905	−	0.889515i	$-0.651042\pi$
$997$	−2.85092e13	−0.913811	−0.456905	+	0.889515i	$0.651042\pi$
$998$	−2.04934e13	−0.653922
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.q.1.7	yes	10
3.2	odd	2		315.10.a.p.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.10.a.p.1.4	✓	10	3.2	odd	2
315.10.a.q.1.7	yes	10	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)

\(f(q)\)	\(=\)	\(q+20.3422 q^{2} -98.1948 q^{4} +625.000 q^{5} -2401.00 q^{7} -12412.7 q^{8} +12713.9 q^{10} +94596.5 q^{11} -5461.83 q^{13} -48841.6 q^{14} -202226. q^{16} -138708. q^{17} +134259. q^{19} -61371.8 q^{20} +1.92430e6 q^{22} -1.31904e6 q^{23} +390625. q^{25} -111106. q^{26} +235766. q^{28} +3.88336e6 q^{29} -2.86754e6 q^{31} +2.24158e6 q^{32} -2.82164e6 q^{34} -1.50062e6 q^{35} -5.69905e6 q^{37} +2.73113e6 q^{38} -7.75794e6 q^{40} -1.01484e7 q^{41} +1.64962e7 q^{43} -9.28889e6 q^{44} -2.68322e7 q^{46} -5.05341e7 q^{47} +5.76480e6 q^{49} +7.94617e6 q^{50} +536323. q^{52} -1.67715e7 q^{53} +5.91228e7 q^{55} +2.98029e7 q^{56} +7.89961e7 q^{58} +9.62623e7 q^{59} -4.76406e7 q^{61} -5.83321e7 q^{62} +1.49138e8 q^{64} -3.41364e6 q^{65} +1.75074e8 q^{67} +1.36204e7 q^{68} -3.05260e7 q^{70} +3.56669e8 q^{71} +4.49220e8 q^{73} -1.15931e8 q^{74} -1.31836e7 q^{76} -2.27126e8 q^{77} +2.53417e8 q^{79} -1.26391e8 q^{80} -2.06441e8 q^{82} +5.49724e7 q^{83} -8.66928e7 q^{85} +3.35569e8 q^{86} -1.17420e9 q^{88} -7.53460e8 q^{89} +1.31139e7 q^{91} +1.29523e8 q^{92} -1.02797e9 q^{94} +8.39121e7 q^{95} +8.46560e8 q^{97} +1.17269e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 7 q^{2} + 2227 q^{4} + 6250 q^{5} - 24010 q^{7} + 3801 q^{8} + 4375 q^{10} + 66164 q^{11} + 11494 q^{13} - 16807 q^{14} + 918867 q^{16} + 193034 q^{17} + 673344 q^{19} + 1391875 q^{20} - 1386368 q^{22}+ \cdots + 40353607 q^{98}+O(q^{100}) \) 10 * q + 7 * q^2 + 2227 * q^4 + 6250 * q^5 - 24010 * q^7 + 3801 * q^8 + 4375 * q^10 + 66164 * q^11 + 11494 * q^13 - 16807 * q^14 + 918867 * q^16 + 193034 * q^17 + 673344 * q^19 + 1391875 * q^20 - 1386368 * q^22 - 747586 * q^23 + 3906250 * q^25 - 10565862 * q^26 - 5347027 * q^28 + 1684046 * q^29 + 9842238 * q^31 - 4156131 * q^32 + 23016000 * q^34 - 15006250 * q^35 + 9741882 * q^37 - 12152762 * q^38 + 2375625 * q^40 + 14958856 * q^41 + 39625550 * q^43 + 74115048 * q^44 + 111188116 * q^46 + 69877502 * q^47 + 57648010 * q^49 + 2734375 * q^50 - 89168254 * q^52 + 151032896 * q^53 + 41352500 * q^55 - 9126201 * q^56 - 37012702 * q^58 + 81055712 * q^59 + 133907802 * q^61 + 541710636 * q^62 + 360737543 * q^64 + 7183750 * q^65 + 389884430 * q^67 + 770623072 * q^68 - 10504375 * q^70 + 318130274 * q^71 - 97314154 * q^73 + 569155188 * q^74 + 1120827470 * q^76 - 158859764 * q^77 - 1247170680 * q^79 + 574291875 * q^80 - 1310498994 * q^82 - 207205436 * q^83 + 120646250 * q^85 + 3002223966 * q^86 - 2070483952 * q^88 + 660264792 * q^89 - 27597094 * q^91 - 1308446244 * q^92 - 1420882918 * q^94 + 420840000 * q^95 + 253464078 * q^97 + 40353607 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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