LMFDB - Embedded newform 315.10.a.d.1.4

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,-5] 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} - 1495x^{2} + 193x + 99862 $ x^4 - x^3 - 1495x^2 + 193x + 99862
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-37.2973$ 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)

$f(q)$	$=$	\(q+36.2973 q^{2} +805.496 q^{4} +625.000 q^{5} -2401.00 q^{7} +10653.1 q^{8} +22685.8 q^{10} +19579.7 q^{11} +2870.11 q^{13} -87149.9 q^{14} -25734.4 q^{16} +7239.36 q^{17} -947859. q^{19} +503435. q^{20} +710691. q^{22} -1.45312e6 q^{23} +390625. q^{25} +104177. q^{26} -1.93400e6 q^{28} -5.00626e6 q^{29} -370292. q^{31} -6.38848e6 q^{32} +262769. q^{34} -1.50062e6 q^{35} +1.37062e7 q^{37} -3.44047e7 q^{38} +6.65820e6 q^{40} -3.39969e6 q^{41} +2.47379e7 q^{43} +1.57714e7 q^{44} -5.27442e7 q^{46} -4.78294e6 q^{47} +5.76480e6 q^{49} +1.41786e7 q^{50} +2.31186e6 q^{52} -5.98102e7 q^{53} +1.22373e7 q^{55} -2.55781e7 q^{56} -1.81714e8 q^{58} +7.58688e7 q^{59} -1.25815e7 q^{61} -1.34406e7 q^{62} -2.18709e8 q^{64} +1.79382e6 q^{65} -2.06500e8 q^{67} +5.83127e6 q^{68} -5.44687e7 q^{70} -6.77606e7 q^{71} -1.23201e8 q^{73} +4.97499e8 q^{74} -7.63496e8 q^{76} -4.70109e7 q^{77} -3.22710e8 q^{79} -1.60840e7 q^{80} -1.23400e8 q^{82} -6.83702e8 q^{83} +4.52460e6 q^{85} +8.97920e8 q^{86} +2.08585e8 q^{88} -4.20580e8 q^{89} -6.89114e6 q^{91} -1.17048e9 q^{92} -1.73608e8 q^{94} -5.92412e8 q^{95} -8.46873e8 q^{97} +2.09247e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) $ 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * 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 $

$2$	36.2973	1.60413	0.802065	−	0.597237i	$-0.203735\pi$
$2$	36.2973	1.60413	0.802065	+	0.597237i	$0.203735\pi$
$3$	0	0
$3$	0	0
$4$	805.496	1.57323
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	10653.1	0.919542
$9$	0	0
$10$	22685.8	0.717389
$11$	19579.7	0.403217	0.201609	−	0.979466i	$-0.435383\pi$
$11$	19579.7	0.403217	0.201609	+	0.979466i	$0.435383\pi$
$12$	0	0
$13$	2870.11	0.0278711	0.0139355	−	0.999903i	$-0.495564\pi$
$13$	2870.11	0.0278711	0.0139355	+	0.999903i	$0.495564\pi$
$14$	−87149.9	−0.606304
$15$	0	0
$16$	−25734.4	−0.0981688
$17$	7239.36	0.0210223	0.0105111	−	0.999945i	$-0.496654\pi$
$17$	7239.36	0.0210223	0.0105111	+	0.999945i	$0.496654\pi$
$18$	0	0
$19$	−947859.	−1.66860	−0.834300	−	0.551310i	$-0.814128\pi$
$19$	−947859.	−1.66860	−0.834300	+	0.551310i	$0.814128\pi$
$20$	503435.	0.703572
$21$	0	0
$22$	710691.	0.646813
$23$	−1.45312e6	−1.08274	−0.541371	−	0.840784i	$-0.682094\pi$
$23$	−1.45312e6	−1.08274	−0.541371	+	0.840784i	$0.682094\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	104177.	0.0447088
$27$	0	0
$28$	−1.93400e6	−0.594627
$29$	−5.00626e6	−1.31438	−0.657192	−	0.753723i	$-0.728256\pi$
$29$	−5.00626e6	−1.31438	−0.657192	+	0.753723i	$0.728256\pi$
$30$	0	0
$31$	−370292.	−0.0720140	−0.0360070	−	0.999352i	$-0.511464\pi$
$31$	−370292.	−0.0720140	−0.0360070	+	0.999352i	$0.511464\pi$
$32$	−6.38848e6	−1.07702
$33$	0	0
$34$	262769.	0.0337225
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	1.37062e7	1.20229	0.601145	−	0.799140i	$-0.294711\pi$
$37$	1.37062e7	1.20229	0.601145	+	0.799140i	$0.294711\pi$
$38$	−3.44047e7	−2.67665
$39$	0	0
$40$	6.65820e6	0.411232
$41$	−3.39969e6	−0.187894	−0.0939468	−	0.995577i	$-0.529948\pi$
$41$	−3.39969e6	−0.187894	−0.0939468	+	0.995577i	$0.529948\pi$
$42$	0	0
$43$	2.47379e7	1.10346	0.551728	−	0.834024i	$-0.313969\pi$
$43$	2.47379e7	1.10346	0.551728	+	0.834024i	$0.313969\pi$
$44$	1.57714e7	0.634355
$45$	0	0
$46$	−5.27442e7	−1.73686
$47$	−4.78294e6	−0.142973	−0.0714866	−	0.997442i	$-0.522774\pi$
$47$	−4.78294e6	−0.142973	−0.0714866	+	0.997442i	$0.522774\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	1.41786e7	0.320826
$51$	0	0
$52$	2.31186e6	0.0438477
$53$	−5.98102e7	−1.04120	−0.520600	−	0.853801i	$-0.674292\pi$
$53$	−5.98102e7	−1.04120	−0.520600	+	0.853801i	$0.674292\pi$
$54$	0	0
$55$	1.22373e7	0.180324
$56$	−2.55781e7	−0.347554
$57$	0	0
$58$	−1.81714e8	−2.10844
$59$	7.58688e7	0.815134	0.407567	−	0.913175i	$-0.366377\pi$
$59$	7.58688e7	0.815134	0.407567	+	0.913175i	$0.366377\pi$
$60$	0	0
$61$	−1.25815e7	−0.116345	−0.0581726	−	0.998307i	$-0.518527\pi$
$61$	−1.25815e7	−0.116345	−0.0581726	+	0.998307i	$0.518527\pi$
$62$	−1.34406e7	−0.115520
$63$	0	0
$64$	−2.18709e8	−1.62951
$65$	1.79382e6	0.0124643
$66$	0	0
$67$	−2.06500e8	−1.25194	−0.625971	−	0.779846i	$-0.715297\pi$
$67$	−2.06500e8	−1.25194	−0.625971	+	0.779846i	$0.715297\pi$
$68$	5.83127e6	0.0330730
$69$	0	0
$70$	−5.44687e7	−0.271148
$71$	−6.77606e7	−0.316457	−0.158228	−	0.987403i	$-0.550578\pi$
$71$	−6.77606e7	−0.316457	−0.158228	+	0.987403i	$0.550578\pi$
$72$	0	0
$73$	−1.23201e8	−0.507764	−0.253882	−	0.967235i	$-0.581707\pi$
$73$	−1.23201e8	−0.507764	−0.253882	+	0.967235i	$0.581707\pi$
$74$	4.97499e8	1.92863
$75$	0	0
$76$	−7.63496e8	−2.62510
$77$	−4.70109e7	−0.152402
$78$	0	0
$79$	−3.22710e8	−0.932161	−0.466081	−	0.884742i	$-0.654334\pi$
$79$	−3.22710e8	−0.932161	−0.466081	+	0.884742i	$0.654334\pi$
$80$	−1.60840e7	−0.0439024
$81$	0	0
$82$	−1.23400e8	−0.301406
$83$	−6.83702e8	−1.58130	−0.790652	−	0.612266i	$-0.790258\pi$
$83$	−6.83702e8	−1.58130	−0.790652	+	0.612266i	$0.790258\pi$
$84$	0	0
$85$	4.52460e6	0.00940145
$86$	8.97920e8	1.77009
$87$	0	0
$88$	2.08585e8	0.370775
$89$	−4.20580e8	−0.710548	−0.355274	−	0.934762i	$-0.615612\pi$
$89$	−4.20580e8	−0.710548	−0.355274	+	0.934762i	$0.615612\pi$
$90$	0	0
$91$	−6.89114e6	−0.0105343
$92$	−1.17048e9	−1.70341
$93$	0	0
$94$	−1.73608e8	−0.229348
$95$	−5.92412e8	−0.746221
$96$	0	0
$97$	−8.46873e8	−0.971283	−0.485641	−	0.874158i	$-0.661414\pi$
$97$	−8.46873e8	−0.971283	−0.485641	+	0.874158i	$0.661414\pi$
$98$	2.09247e8	0.229161
$99$	0	0
$100$	3.14647e8	0.314647
$101$	1.02952e9	0.984441	0.492220	−	0.870471i	$-0.336185\pi$
$101$	1.02952e9	0.984441	0.492220	+	0.870471i	$0.336185\pi$
$102$	0	0
$103$	2.62280e8	0.229613	0.114807	−	0.993388i	$-0.463375\pi$
$103$	2.62280e8	0.229613	0.114807	+	0.993388i	$0.463375\pi$
$104$	3.05756e7	0.0256286
$105$	0	0
$106$	−2.17095e9	−1.67022
$107$	1.85516e9	1.36821	0.684106	−	0.729383i	$-0.260193\pi$
$107$	1.85516e9	1.36821	0.684106	+	0.729383i	$0.260193\pi$
$108$	0	0
$109$	1.72285e9	1.16904	0.584520	−	0.811379i	$-0.301283\pi$
$109$	1.72285e9	1.16904	0.584520	+	0.811379i	$0.301283\pi$
$110$	4.44182e8	0.289264
$111$	0	0
$112$	6.17882e7	0.0371043
$113$	−3.78289e7	−0.0218258	−0.0109129	−	0.999940i	$-0.503474\pi$
$113$	−3.78289e7	−0.0218258	−0.0109129	+	0.999940i	$0.503474\pi$
$114$	0	0
$115$	−9.08197e8	−0.484217
$116$	−4.03252e9	−2.06783
$117$	0	0
$118$	2.75383e9	1.30758
$119$	−1.73817e7	−0.00794568
$120$	0	0
$121$	−1.97458e9	−0.837416
$122$	−4.56675e8	−0.186633
$123$	0	0
$124$	−2.98269e8	−0.113295
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	2.24240e8	0.0764885	0.0382443	−	0.999268i	$-0.487823\pi$
$127$	2.24240e8	0.0764885	0.0382443	+	0.999268i	$0.487823\pi$
$128$	−4.66764e9	−1.53693
$129$	0	0
$130$	6.51109e7	0.0199944
$131$	3.94134e9	1.16929	0.584646	−	0.811288i	$-0.301233\pi$
$131$	3.94134e9	1.16929	0.584646	+	0.811288i	$0.301233\pi$
$132$	0	0
$133$	2.27581e9	0.630672
$134$	−7.49542e9	−2.00828
$135$	0	0
$136$	7.71217e7	0.0193309
$137$	−7.20426e9	−1.74722	−0.873608	−	0.486629i	$-0.838226\pi$
$137$	−7.20426e9	−1.74722	−0.873608	+	0.486629i	$0.838226\pi$
$138$	0	0
$139$	1.50981e9	0.343049	0.171524	−	0.985180i	$-0.445131\pi$
$139$	1.50981e9	0.343049	0.171524	+	0.985180i	$0.445131\pi$
$140$	−1.20875e9	−0.265925
$141$	0	0
$142$	−2.45953e9	−0.507638
$143$	5.61960e7	0.0112381
$144$	0	0
$145$	−3.12891e9	−0.587811
$146$	−4.47187e9	−0.814519
$147$	0	0
$148$	1.10403e10	1.89148
$149$	−3.19847e9	−0.531624	−0.265812	−	0.964025i	$-0.585640\pi$
$149$	−3.19847e9	−0.531624	−0.265812	+	0.964025i	$0.585640\pi$
$150$	0	0
$151$	6.11037e9	0.956469	0.478235	−	0.878232i	$-0.341277\pi$
$151$	6.11037e9	0.956469	0.478235	+	0.878232i	$0.341277\pi$
$152$	−1.00976e10	−1.53435
$153$	0	0
$154$	−1.70637e9	−0.244472
$155$	−2.31433e8	−0.0322056
$156$	0	0
$157$	−6.29783e9	−0.827261	−0.413630	−	0.910445i	$-0.635739\pi$
$157$	−6.29783e9	−0.827261	−0.413630	+	0.910445i	$0.635739\pi$
$158$	−1.17135e10	−1.49531
$159$	0	0
$160$	−3.99280e9	−0.481657
$161$	3.48893e9	0.409238
$162$	0	0
$163$	−5.91633e9	−0.656460	−0.328230	−	0.944598i	$-0.606452\pi$
$163$	−5.91633e9	−0.656460	−0.328230	+	0.944598i	$0.606452\pi$
$164$	−2.73844e9	−0.295601
$165$	0	0
$166$	−2.48165e10	−2.53662
$167$	−7.99428e9	−0.795344	−0.397672	−	0.917528i	$-0.630182\pi$
$167$	−7.99428e9	−0.795344	−0.397672	+	0.917528i	$0.630182\pi$
$168$	0	0
$169$	−1.05963e10	−0.999223
$170$	1.64231e8	0.0150812
$171$	0	0
$172$	1.99263e10	1.73599
$173$	1.46122e10	1.24025	0.620124	−	0.784504i	$-0.287082\pi$
$173$	1.46122e10	1.24025	0.620124	+	0.784504i	$0.287082\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−5.03871e8	−0.0395834
$177$	0	0
$178$	−1.52659e10	−1.13981
$179$	1.18780e10	0.864778	0.432389	−	0.901687i	$-0.357671\pi$
$179$	1.18780e10	0.864778	0.432389	+	0.901687i	$0.357671\pi$
$180$	0	0
$181$	−5.54593e9	−0.384079	−0.192040	−	0.981387i	$-0.561510\pi$
$181$	−5.54593e9	−0.384079	−0.192040	+	0.981387i	$0.561510\pi$
$182$	−2.50130e8	−0.0168984
$183$	0	0
$184$	−1.54802e10	−0.995626
$185$	8.56638e9	0.537681
$186$	0	0
$187$	1.41745e8	0.00847655
$188$	−3.85264e9	−0.224930
$189$	0	0
$190$	−2.15030e10	−1.19704
$191$	1.50997e10	0.820954	0.410477	−	0.911871i	$-0.365362\pi$
$191$	1.50997e10	0.820954	0.410477	+	0.911871i	$0.365362\pi$
$192$	0	0
$193$	−3.02320e10	−1.56841	−0.784203	−	0.620505i	$-0.786928\pi$
$193$	−3.02320e10	−1.56841	−0.784203	+	0.620505i	$0.786928\pi$
$194$	−3.07392e10	−1.55806
$195$	0	0
$196$	4.64352e9	0.224748
$197$	−3.22125e10	−1.52379	−0.761897	−	0.647698i	$-0.775732\pi$
$197$	−3.22125e10	−1.52379	−0.761897	+	0.647698i	$0.775732\pi$
$198$	0	0
$199$	−2.88753e10	−1.30523	−0.652616	−	0.757689i	$-0.726328\pi$
$199$	−2.88753e10	−1.30523	−0.652616	+	0.757689i	$0.726328\pi$
$200$	4.16137e9	0.183908
$201$	0	0
$202$	3.73689e10	1.57917
$203$	1.20200e10	0.496791
$204$	0	0
$205$	−2.12481e9	−0.0840286
$206$	9.52004e9	0.368329
$207$	0	0
$208$	−7.38605e7	−0.00273607
$209$	−1.85588e10	−0.672809
$210$	0	0
$211$	−2.27282e10	−0.789396	−0.394698	−	0.918811i	$-0.629151\pi$
$211$	−2.27282e10	−0.789396	−0.394698	+	0.918811i	$0.629151\pi$
$212$	−4.81769e10	−1.63805
$213$	0	0
$214$	6.73372e10	2.19479
$215$	1.54612e10	0.493480
$216$	0	0
$217$	8.89071e8	0.0272187
$218$	6.25350e10	1.87529
$219$	0	0
$220$	9.85711e9	0.283692
$221$	2.07778e7	0.000585914 0
$222$	0	0
$223$	2.65531e10	0.719023	0.359512	−	0.933141i	$-0.382943\pi$
$223$	2.65531e10	0.719023	0.359512	+	0.933141i	$0.382943\pi$
$224$	1.53387e10	0.407074
$225$	0	0
$226$	−1.37309e9	−0.0350114
$227$	−2.90626e9	−0.0726470	−0.0363235	−	0.999340i	$-0.511565\pi$
$227$	−2.90626e9	−0.0726470	−0.0363235	+	0.999340i	$0.511565\pi$
$228$	0	0
$229$	3.00071e10	0.721047	0.360524	−	0.932750i	$-0.382598\pi$
$229$	3.00071e10	0.721047	0.360524	+	0.932750i	$0.382598\pi$
$230$	−3.29651e10	−0.776747
$231$	0	0
$232$	−5.33323e10	−1.20863
$233$	8.11462e10	1.80371	0.901855	−	0.432039i	$-0.142206\pi$
$233$	8.11462e10	1.80371	0.901855	+	0.432039i	$0.142206\pi$
$234$	0	0
$235$	−2.98934e9	−0.0639396
$236$	6.11120e10	1.28240
$237$	0	0
$238$	−6.30909e8	−0.0127459
$239$	7.77239e9	0.154086	0.0770431	−	0.997028i	$-0.475452\pi$
$239$	7.77239e9	0.154086	0.0770431	+	0.997028i	$0.475452\pi$
$240$	0	0
$241$	−2.02011e10	−0.385744	−0.192872	−	0.981224i	$-0.561780\pi$
$241$	−2.02011e10	−0.385744	−0.192872	+	0.981224i	$0.561780\pi$
$242$	−7.16721e10	−1.34332
$243$	0	0
$244$	−1.01344e10	−0.183038
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−2.72046e9	−0.0465057
$248$	−3.94476e9	−0.0662199
$249$	0	0
$250$	8.86165e9	0.143478
$251$	4.65390e10	0.740092	0.370046	−	0.929013i	$-0.379342\pi$
$251$	4.65390e10	0.740092	0.370046	+	0.929013i	$0.379342\pi$
$252$	0	0
$253$	−2.84516e10	−0.436580
$254$	8.13931e9	0.122698
$255$	0	0
$256$	−5.74440e10	−0.835920
$257$	1.19658e11	1.71097	0.855483	−	0.517831i	$-0.173261\pi$
$257$	1.19658e11	1.71097	0.855483	+	0.517831i	$0.173261\pi$
$258$	0	0
$259$	−3.29086e10	−0.454423
$260$	1.44491e9	0.0196093
$261$	0	0
$262$	1.43060e11	1.87570
$263$	3.24071e10	0.417676	0.208838	−	0.977950i	$-0.433032\pi$
$263$	3.24071e10	0.417676	0.208838	+	0.977950i	$0.433032\pi$
$264$	0	0
$265$	−3.73814e10	−0.465639
$266$	8.26058e10	1.01168
$267$	0	0
$268$	−1.66335e11	−1.96960
$269$	−1.65575e11	−1.92801	−0.964006	−	0.265882i	$-0.914337\pi$
$269$	−1.65575e11	−1.92801	−0.964006	+	0.265882i	$0.914337\pi$
$270$	0	0
$271$	1.70467e10	0.191990	0.0959952	−	0.995382i	$-0.469397\pi$
$271$	1.70467e10	0.191990	0.0959952	+	0.995382i	$0.469397\pi$
$272$	−1.86300e8	−0.00206373
$273$	0	0
$274$	−2.61495e11	−2.80276
$275$	7.64833e9	0.0806435
$276$	0	0
$277$	1.29473e11	1.32136	0.660679	−	0.750669i	$-0.270269\pi$
$277$	1.29473e11	1.32136	0.660679	+	0.750669i	$0.270269\pi$
$278$	5.48021e10	0.550295
$279$	0	0
$280$	−1.59863e10	−0.155431
$281$	1.18853e11	1.13719	0.568594	−	0.822618i	$-0.307487\pi$
$281$	1.18853e11	1.13719	0.568594	+	0.822618i	$0.307487\pi$
$282$	0	0
$283$	2.01507e11	1.86746	0.933732	−	0.357973i	$-0.116532\pi$
$283$	2.01507e11	1.86746	0.933732	+	0.357973i	$0.116532\pi$
$284$	−5.45809e10	−0.497861
$285$	0	0
$286$	2.03976e9	0.0180274
$287$	8.16266e9	0.0710171
$288$	0	0
$289$	−1.18535e11	−0.999558
$290$	−1.13571e11	−0.942925
$291$	0	0
$292$	−9.92380e10	−0.798831
$293$	−1.17343e10	−0.0930153	−0.0465077	−	0.998918i	$-0.514809\pi$
$293$	−1.17343e10	−0.0930153	−0.0465077	+	0.998918i	$0.514809\pi$
$294$	0	0
$295$	4.74180e10	0.364539
$296$	1.46014e11	1.10556
$297$	0	0
$298$	−1.16096e11	−0.852794
$299$	−4.17060e9	−0.0301772
$300$	0	0
$301$	−5.93957e10	−0.417067
$302$	2.21790e11	1.53430
$303$	0	0
$304$	2.43925e10	0.163805
$305$	−7.86344e9	−0.0520312
$306$	0	0
$307$	1.11013e11	0.713264	0.356632	−	0.934245i	$-0.383925\pi$
$307$	1.11013e11	0.713264	0.356632	+	0.934245i	$0.383925\pi$
$308$	−3.78671e10	−0.239764
$309$	0	0
$310$	−8.40038e9	−0.0516620
$311$	1.06444e11	0.645205	0.322603	−	0.946534i	$-0.395442\pi$
$311$	1.06444e11	0.645205	0.322603	+	0.946534i	$0.395442\pi$
$312$	0	0
$313$	−4.73991e10	−0.279139	−0.139570	−	0.990212i	$-0.544572\pi$
$313$	−4.73991e10	−0.279139	−0.139570	+	0.990212i	$0.544572\pi$
$314$	−2.28594e11	−1.32703
$315$	0	0
$316$	−2.59942e11	−1.46651
$317$	4.51985e10	0.251395	0.125698	−	0.992069i	$-0.459883\pi$
$317$	4.51985e10	0.251395	0.125698	+	0.992069i	$0.459883\pi$
$318$	0	0
$319$	−9.80212e10	−0.529983
$320$	−1.36693e11	−0.728738
$321$	0	0
$322$	1.26639e11	0.656471
$323$	−6.86189e9	−0.0350778
$324$	0	0
$325$	1.12114e9	0.00557422
$326$	−2.14747e11	−1.05305
$327$	0	0
$328$	−3.62173e10	−0.172776
$329$	1.14838e10	0.0540388
$330$	0	0
$331$	−1.46701e11	−0.671749	−0.335875	−	0.941907i	$-0.609032\pi$
$331$	−1.46701e11	−0.671749	−0.335875	+	0.941907i	$0.609032\pi$
$332$	−5.50719e11	−2.48776
$333$	0	0
$334$	−2.90171e11	−1.27584
$335$	−1.29063e11	−0.559886
$336$	0	0
$337$	−2.10070e10	−0.0887218	−0.0443609	−	0.999016i	$-0.514125\pi$
$337$	−2.10070e10	−0.0887218	−0.0443609	+	0.999016i	$0.514125\pi$
$338$	−3.84616e11	−1.60288
$339$	0	0
$340$	3.64455e9	0.0147907
$341$	−7.25021e9	−0.0290373
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	2.63536e11	1.01467
$345$	0	0
$346$	5.30385e11	1.98952
$347$	−1.66328e11	−0.615861	−0.307930	−	0.951409i	$-0.599636\pi$
$347$	−1.66328e11	−0.615861	−0.307930	+	0.951409i	$0.599636\pi$
$348$	0	0
$349$	6.09896e10	0.220060	0.110030	−	0.993928i	$-0.464905\pi$
$349$	6.09896e10	0.220060	0.110030	+	0.993928i	$0.464905\pi$
$350$	−3.40429e10	−0.121261
$351$	0	0
$352$	−1.25085e11	−0.434272
$353$	1.27837e11	0.438199	0.219099	−	0.975703i	$-0.429688\pi$
$353$	1.27837e11	0.438199	0.219099	+	0.975703i	$0.429688\pi$
$354$	0	0
$355$	−4.23504e10	−0.141524
$356$	−3.38775e11	−1.11786
$357$	0	0
$358$	4.31140e11	1.38722
$359$	−5.48375e11	−1.74242	−0.871209	−	0.490911i	$-0.836664\pi$
$359$	−5.48375e11	−1.74242	−0.871209	+	0.490911i	$0.836664\pi$
$360$	0	0
$361$	5.75749e11	1.78423
$362$	−2.01302e11	−0.616113
$363$	0	0
$364$	−5.55078e9	−0.0165729
$365$	−7.70007e10	−0.227079
$366$	0	0
$367$	−2.78567e11	−0.801554	−0.400777	−	0.916176i	$-0.631260\pi$
$367$	−2.78567e11	−0.801554	−0.400777	+	0.916176i	$0.631260\pi$
$368$	3.73950e10	0.106291
$369$	0	0
$370$	3.10937e11	0.862510
$371$	1.43604e11	0.393536
$372$	0	0
$373$	−6.73026e11	−1.80029	−0.900144	−	0.435593i	$-0.856539\pi$
$373$	−6.73026e11	−1.80029	−0.900144	+	0.435593i	$0.856539\pi$
$374$	5.14495e9	0.0135975
$375$	0	0
$376$	−5.09532e10	−0.131470
$377$	−1.43685e10	−0.0366333
$378$	0	0
$379$	−2.57537e11	−0.641154	−0.320577	−	0.947222i	$-0.603877\pi$
$379$	−2.57537e11	−0.641154	−0.320577	+	0.947222i	$0.603877\pi$
$380$	−4.77185e11	−1.17398
$381$	0	0
$382$	5.48079e11	1.31692
$383$	5.64360e11	1.34018	0.670088	−	0.742281i	$-0.266256\pi$
$383$	5.64360e11	1.34018	0.670088	+	0.742281i	$0.266256\pi$
$384$	0	0
$385$	−2.93818e10	−0.0681562
$386$	−1.09734e12	−2.51593
$387$	0	0
$388$	−6.82153e11	−1.52806
$389$	−4.53690e10	−0.100458	−0.0502291	−	0.998738i	$-0.515995\pi$
$389$	−4.53690e10	−0.100458	−0.0502291	+	0.998738i	$0.515995\pi$
$390$	0	0
$391$	−1.05196e10	−0.0227617
$392$	6.14131e10	0.131363
$393$	0	0
$394$	−1.16923e12	−2.44437
$395$	−2.01694e11	−0.416875
$396$	0	0
$397$	8.32333e11	1.68167	0.840833	−	0.541294i	$-0.182066\pi$
$397$	8.32333e11	1.68167	0.840833	+	0.541294i	$0.182066\pi$
$398$	−1.04810e12	−2.09376
$399$	0	0
$400$	−1.00525e10	−0.0196338
$401$	2.51693e10	0.0486095	0.0243047	−	0.999705i	$-0.492263\pi$
$401$	2.51693e10	0.0486095	0.0243047	+	0.999705i	$0.492263\pi$
$402$	0	0
$403$	−1.06278e9	−0.00200711
$404$	8.29276e11	1.54876
$405$	0	0
$406$	4.36295e11	0.796917
$407$	2.68364e11	0.484784
$408$	0	0
$409$	4.25724e11	0.752268	0.376134	−	0.926565i	$-0.377253\pi$
$409$	4.25724e11	0.752268	0.376134	+	0.926565i	$0.377253\pi$
$410$	−7.71248e10	−0.134793
$411$	0	0
$412$	2.11265e11	0.361235
$413$	−1.82161e11	−0.308092
$414$	0	0
$415$	−4.27314e11	−0.707180
$416$	−1.83357e10	−0.0300176
$417$	0	0
$418$	−6.73635e11	−1.07927
$419$	9.31828e11	1.47697	0.738487	−	0.674268i	$-0.235541\pi$
$419$	9.31828e11	1.47697	0.738487	+	0.674268i	$0.235541\pi$
$420$	0	0
$421$	1.01576e12	1.57587	0.787934	−	0.615759i	$-0.211151\pi$
$421$	1.01576e12	1.57587	0.787934	+	0.615759i	$0.211151\pi$
$422$	−8.24974e11	−1.26629
$423$	0	0
$424$	−6.37165e11	−0.957427
$425$	2.82788e9	0.00420446
$426$	0	0
$427$	3.02082e10	0.0439744
$428$	1.49432e12	2.15252
$429$	0	0
$430$	5.61200e11	0.791607
$431$	−1.79917e11	−0.251144	−0.125572	−	0.992084i	$-0.540077\pi$
$431$	−1.79917e11	−0.251144	−0.125572	+	0.992084i	$0.540077\pi$
$432$	0	0
$433$	−5.18545e11	−0.708909	−0.354455	−	0.935073i	$-0.615333\pi$
$433$	−5.18545e11	−0.708909	−0.354455	+	0.935073i	$0.615333\pi$
$434$	3.22709e10	0.0436624
$435$	0	0
$436$	1.38775e12	1.83917
$437$	1.37735e12	1.80666
$438$	0	0
$439$	−4.26837e11	−0.548493	−0.274247	−	0.961659i	$-0.588428\pi$
$439$	−4.26837e11	−0.548493	−0.274247	+	0.961659i	$0.588428\pi$
$440$	1.30366e11	0.165816
$441$	0	0
$442$	7.54178e8	0.000939882 0
$443$	9.07073e11	1.11899	0.559494	−	0.828834i	$-0.310995\pi$
$443$	9.07073e11	1.11899	0.559494	+	0.828834i	$0.310995\pi$
$444$	0	0
$445$	−2.62862e11	−0.317767
$446$	9.63806e11	1.15341
$447$	0	0
$448$	5.25120e11	0.615896
$449$	6.02137e11	0.699176	0.349588	−	0.936904i	$-0.386322\pi$
$449$	6.02137e11	0.699176	0.349588	+	0.936904i	$0.386322\pi$
$450$	0	0
$451$	−6.65650e10	−0.0757620
$452$	−3.04710e10	−0.0343371
$453$	0	0
$454$	−1.05489e11	−0.116535
$455$	−4.30696e9	−0.00471107
$456$	0	0
$457$	−7.51523e11	−0.805971	−0.402986	−	0.915206i	$-0.632028\pi$
$457$	−7.51523e11	−0.805971	−0.402986	+	0.915206i	$0.632028\pi$
$458$	1.08918e12	1.15665
$459$	0	0
$460$	−7.31549e11	−0.761786
$461$	1.23967e12	1.27835	0.639176	−	0.769061i	$-0.279276\pi$
$461$	1.23967e12	1.27835	0.639176	+	0.769061i	$0.279276\pi$
$462$	0	0
$463$	−6.44926e11	−0.652222	−0.326111	−	0.945332i	$-0.605738\pi$
$463$	−6.44926e11	−0.652222	−0.326111	+	0.945332i	$0.605738\pi$
$464$	1.28833e11	0.129032
$465$	0	0
$466$	2.94539e12	2.89339
$467$	5.27360e11	0.513076	0.256538	−	0.966534i	$-0.417418\pi$
$467$	5.27360e11	0.513076	0.256538	+	0.966534i	$0.417418\pi$
$468$	0	0
$469$	4.95808e11	0.473190
$470$	−1.08505e11	−0.102567
$471$	0	0
$472$	8.08239e11	0.749550
$473$	4.84361e11	0.444932
$474$	0	0
$475$	−3.70257e11	−0.333720
$476$	−1.40009e10	−0.0125004
$477$	0	0
$478$	2.82117e11	0.247174
$479$	1.82383e12	1.58298	0.791490	−	0.611182i	$-0.209306\pi$
$479$	1.82383e12	1.58298	0.791490	+	0.611182i	$0.209306\pi$
$480$	0	0
$481$	3.93383e10	0.0335091
$482$	−7.33248e11	−0.618784
$483$	0	0
$484$	−1.59052e12	−1.31745
$485$	−5.29296e11	−0.434371
$486$	0	0
$487$	2.28006e11	0.183682	0.0918409	−	0.995774i	$-0.470725\pi$
$487$	2.28006e11	0.183682	0.0918409	+	0.995774i	$0.470725\pi$
$488$	−1.34032e11	−0.106984
$489$	0	0
$490$	1.30779e11	0.102484
$491$	2.55761e10	0.0198595	0.00992975	−	0.999951i	$-0.496839\pi$
$491$	2.55761e10	0.0198595	0.00992975	+	0.999951i	$0.496839\pi$
$492$	0	0
$493$	−3.62421e10	−0.0276314
$494$	−9.87455e10	−0.0746012
$495$	0	0
$496$	9.52923e9	0.00706953
$497$	1.62693e11	0.119609
$498$	0	0
$499$	2.46191e12	1.77754	0.888769	−	0.458355i	$-0.151561\pi$
$499$	2.46191e12	1.77754	0.888769	+	0.458355i	$0.151561\pi$
$500$	1.96654e11	0.140714
$501$	0	0
$502$	1.68924e12	1.18720
$503$	−6.97292e11	−0.485689	−0.242845	−	0.970065i	$-0.578081\pi$
$503$	−6.97292e11	−0.485689	−0.242845	+	0.970065i	$0.578081\pi$
$504$	0	0
$505$	6.43452e11	0.440255
$506$	−1.03272e12	−0.700332
$507$	0	0
$508$	1.80624e11	0.120334
$509$	−3.21991e11	−0.212625	−0.106312	−	0.994333i	$-0.533904\pi$
$509$	−3.21991e11	−0.212625	−0.106312	+	0.994333i	$0.533904\pi$
$510$	0	0
$511$	2.95806e11	0.191917
$512$	3.04769e11	0.196000
$513$	0	0
$514$	4.34325e12	2.74461
$515$	1.63925e11	0.102686
$516$	0	0
$517$	−9.36487e10	−0.0576493
$518$	−1.19449e12	−0.728954
$519$	0	0
$520$	1.91098e10	0.0114615
$521$	−2.33202e11	−0.138664	−0.0693318	−	0.997594i	$-0.522087\pi$
$521$	−2.33202e11	−0.138664	−0.0693318	+	0.997594i	$0.522087\pi$
$522$	0	0
$523$	−7.96910e11	−0.465749	−0.232874	−	0.972507i	$-0.574813\pi$
$523$	−7.96910e11	−0.465749	−0.232874	+	0.972507i	$0.574813\pi$
$524$	3.17473e12	1.83957
$525$	0	0
$526$	1.17629e12	0.670007
$527$	−2.68068e9	−0.00151390
$528$	0	0
$529$	3.10391e11	0.172329
$530$	−1.35684e12	−0.746945
$531$	0	0
$532$	1.83315e12	0.992194
$533$	−9.75749e9	−0.00523680
$534$	0	0
$535$	1.15947e12	0.611883
$536$	−2.19987e12	−1.15121
$537$	0	0
$538$	−6.00993e12	−3.09278
$539$	1.12873e11	0.0576025
$540$	0	0
$541$	−3.42715e12	−1.72007	−0.860034	−	0.510237i	$-0.829558\pi$
$541$	−3.42715e12	−1.72007	−0.860034	+	0.510237i	$0.829558\pi$
$542$	6.18751e11	0.307978
$543$	0	0
$544$	−4.62485e10	−0.0226414
$545$	1.07678e12	0.522810
$546$	0	0
$547$	−6.69117e11	−0.319565	−0.159782	−	0.987152i	$-0.551079\pi$
$547$	−6.69117e11	−0.319565	−0.159782	+	0.987152i	$0.551079\pi$
$548$	−5.80300e12	−2.74878
$549$	0	0
$550$	2.77614e11	0.129363
$551$	4.74523e12	2.19318
$552$	0	0
$553$	7.74828e11	0.352324
$554$	4.69952e12	2.11963
$555$	0	0
$556$	1.21615e12	0.539696
$557$	−2.12962e12	−0.937460	−0.468730	−	0.883341i	$-0.655288\pi$
$557$	−2.12962e12	−0.937460	−0.468730	+	0.883341i	$0.655288\pi$
$558$	0	0
$559$	7.10006e10	0.0307545
$560$	3.86176e10	0.0165936
$561$	0	0
$562$	4.31405e12	1.82420
$563$	2.31439e12	0.970841	0.485421	−	0.874281i	$-0.338667\pi$
$563$	2.31439e12	0.970841	0.485421	+	0.874281i	$0.338667\pi$
$564$	0	0
$565$	−2.36430e10	−0.00976080
$566$	7.31418e12	2.99566
$567$	0	0
$568$	−7.21861e11	−0.290995
$569$	−3.36477e12	−1.34571	−0.672853	−	0.739776i	$-0.734931\pi$
$569$	−3.36477e12	−1.34571	−0.672853	+	0.739776i	$0.734931\pi$
$570$	0	0
$571$	−1.60789e12	−0.632985	−0.316492	−	0.948595i	$-0.602505\pi$
$571$	−1.60789e12	−0.632985	−0.316492	+	0.948595i	$0.602505\pi$
$572$	4.52656e10	0.0176802
$573$	0	0
$574$	2.96283e11	0.113921
$575$	−5.67623e11	−0.216548
$576$	0	0
$577$	3.56168e12	1.33772	0.668858	−	0.743390i	$-0.266783\pi$
$577$	3.56168e12	1.33772	0.668858	+	0.743390i	$0.266783\pi$
$578$	−4.30252e12	−1.60342
$579$	0	0
$580$	−2.52033e12	−0.924764
$581$	1.64157e12	0.597677
$582$	0	0
$583$	−1.17107e12	−0.419830
$584$	−1.31248e12	−0.466910
$585$	0	0
$586$	−4.25925e11	−0.149209
$587$	2.13937e12	0.743729	0.371864	−	0.928287i	$-0.378719\pi$
$587$	2.13937e12	0.743729	0.371864	+	0.928287i	$0.378719\pi$
$588$	0	0
$589$	3.50985e11	0.120163
$590$	1.72115e12	0.584768
$591$	0	0
$592$	−3.52720e11	−0.118027
$593$	−5.70730e12	−1.89533	−0.947664	−	0.319269i	$-0.896563\pi$
$593$	−5.70730e12	−1.89533	−0.947664	+	0.319269i	$0.896563\pi$
$594$	0	0
$595$	−1.08636e10	−0.00355342
$596$	−2.57636e12	−0.836368
$597$	0	0
$598$	−1.51382e11	−0.0484081
$599$	1.75098e12	0.555725	0.277863	−	0.960621i	$-0.410374\pi$
$599$	1.75098e12	0.555725	0.277863	+	0.960621i	$0.410374\pi$
$600$	0	0
$601$	−1.55382e12	−0.485810	−0.242905	−	0.970050i	$-0.578100\pi$
$601$	−1.55382e12	−0.485810	−0.242905	+	0.970050i	$0.578100\pi$
$602$	−2.15591e12	−0.669030
$603$	0	0
$604$	4.92188e12	1.50475
$605$	−1.23411e12	−0.374504
$606$	0	0
$607$	2.89741e12	0.866284	0.433142	−	0.901326i	$-0.357405\pi$
$607$	2.89741e12	0.866284	0.433142	+	0.901326i	$0.357405\pi$
$608$	6.05538e12	1.79711
$609$	0	0
$610$	−2.85422e11	−0.0834648
$611$	−1.37276e10	−0.00398482
$612$	0	0
$613$	−8.72329e9	−0.00249522	−0.00124761	−	0.999999i	$-0.500397\pi$
$613$	−8.72329e9	−0.00249522	−0.00124761	+	0.999999i	$0.500397\pi$
$614$	4.02947e12	1.14417
$615$	0	0
$616$	−5.00812e11	−0.140140
$617$	7.53217e11	0.209236	0.104618	−	0.994512i	$-0.466638\pi$
$617$	7.53217e11	0.209236	0.104618	+	0.994512i	$0.466638\pi$
$618$	0	0
$619$	−1.08273e12	−0.296424	−0.148212	−	0.988956i	$-0.547352\pi$
$619$	−1.08273e12	−0.296424	−0.148212	+	0.988956i	$0.547352\pi$
$620$	−1.86418e11	−0.0506670
$621$	0	0
$622$	3.86362e12	1.03499
$623$	1.00981e12	0.268562
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.72046e12	−0.447776
$627$	0	0
$628$	−5.07287e12	−1.30147
$629$	9.92241e10	0.0252749
$630$	0	0
$631$	3.89280e12	0.977529	0.488765	−	0.872416i	$-0.337448\pi$
$631$	3.89280e12	0.977529	0.488765	+	0.872416i	$0.337448\pi$
$632$	−3.43787e12	−0.857161
$633$	0	0
$634$	1.64058e12	0.403271
$635$	1.40150e11	0.0342067
$636$	0	0
$637$	1.65456e10	0.00398158
$638$	−3.55791e12	−0.850162
$639$	0	0
$640$	−2.91728e12	−0.687334
$641$	−1.08503e12	−0.253853	−0.126926	−	0.991912i	$-0.540511\pi$
$641$	−1.08503e12	−0.253853	−0.126926	+	0.991912i	$0.540511\pi$
$642$	0	0
$643$	2.65784e12	0.613168	0.306584	−	0.951844i	$-0.400814\pi$
$643$	2.65784e12	0.613168	0.306584	+	0.951844i	$0.400814\pi$
$644$	2.81032e12	0.643827
$645$	0	0
$646$	−2.49068e11	−0.0562694
$647$	1.48568e12	0.333316	0.166658	−	0.986015i	$-0.446702\pi$
$647$	1.48568e12	0.333316	0.166658	+	0.986015i	$0.446702\pi$
$648$	0	0
$649$	1.48549e12	0.328676
$650$	4.06943e10	0.00894177
$651$	0	0
$652$	−4.76558e12	−1.03277
$653$	−6.47488e12	−1.39355	−0.696775	−	0.717290i	$-0.745383\pi$
$653$	−6.47488e12	−1.39355	−0.696775	+	0.717290i	$0.745383\pi$
$654$	0	0
$655$	2.46334e12	0.522924
$656$	8.74889e10	0.0184453
$657$	0	0
$658$	4.16833e11	0.0866853
$659$	−5.69212e12	−1.17568	−0.587841	−	0.808976i	$-0.700022\pi$
$659$	−5.69212e12	−1.17568	−0.587841	+	0.808976i	$0.700022\pi$
$660$	0	0
$661$	3.81989e12	0.778296	0.389148	−	0.921175i	$-0.372770\pi$
$661$	3.81989e12	0.778296	0.389148	+	0.921175i	$0.372770\pi$
$662$	−5.32485e12	−1.07757
$663$	0	0
$664$	−7.28355e12	−1.45407
$665$	1.42238e12	0.282045
$666$	0	0
$667$	7.27467e12	1.42314
$668$	−6.43936e12	−1.25126
$669$	0	0
$670$	−4.68463e12	−0.898130
$671$	−2.46342e11	−0.0469124
$672$	0	0
$673$	−5.72322e12	−1.07541	−0.537703	−	0.843134i	$-0.680708\pi$
$673$	−5.72322e12	−1.07541	−0.537703	+	0.843134i	$0.680708\pi$
$674$	−7.62499e11	−0.142321
$675$	0	0
$676$	−8.53524e12	−1.57201
$677$	−2.31892e12	−0.424265	−0.212132	−	0.977241i	$-0.568041\pi$
$677$	−2.31892e12	−0.424265	−0.212132	+	0.977241i	$0.568041\pi$
$678$	0	0
$679$	2.03334e12	0.367110
$680$	4.82011e10	0.00864503
$681$	0	0
$682$	−2.63163e11	−0.0465796
$683$	−8.75141e12	−1.53881	−0.769405	−	0.638761i	$-0.779447\pi$
$683$	−8.75141e12	−1.53881	−0.769405	+	0.638761i	$0.779447\pi$
$684$	0	0
$685$	−4.50266e12	−0.781379
$686$	−5.02402e11	−0.0866149
$687$	0	0
$688$	−6.36614e11	−0.108325
$689$	−1.71662e11	−0.0290194
$690$	0	0
$691$	−3.06847e12	−0.512000	−0.256000	−	0.966677i	$-0.582405\pi$
$691$	−3.06847e12	−0.512000	−0.256000	+	0.966677i	$0.582405\pi$
$692$	1.17701e13	1.95120
$693$	0	0
$694$	−6.03726e12	−0.987921
$695$	9.43631e11	0.153416
$696$	0	0
$697$	−2.46116e10	−0.00394995
$698$	2.21376e12	0.353005
$699$	0	0
$700$	−7.55467e11	−0.118925
$701$	−6.79179e12	−1.06231	−0.531157	−	0.847273i	$-0.678243\pi$
$701$	−6.79179e12	−1.06231	−0.531157	+	0.847273i	$0.678243\pi$
$702$	0	0
$703$	−1.29915e13	−2.00614
$704$	−4.28226e12	−0.657046
$705$	0	0
$706$	4.64015e12	0.702928
$707$	−2.47188e12	−0.372084
$708$	0	0
$709$	9.87334e12	1.46743	0.733713	−	0.679460i	$-0.237786\pi$
$709$	9.87334e12	1.46743	0.733713	+	0.679460i	$0.237786\pi$
$710$	−1.53721e12	−0.227023
$711$	0	0
$712$	−4.48048e12	−0.653379
$713$	5.38077e11	0.0779725
$714$	0	0
$715$	3.51225e10	0.00502583
$716$	9.56768e12	1.36050
$717$	0	0
$718$	−1.99045e13	−2.79507
$719$	−9.96873e12	−1.39110	−0.695552	−	0.718475i	$-0.744840\pi$
$719$	−9.96873e12	−1.39110	−0.695552	+	0.718475i	$0.744840\pi$
$720$	0	0
$721$	−6.29733e11	−0.0867856
$722$	2.08981e13	2.86214
$723$	0	0
$724$	−4.46722e12	−0.604247
$725$	−1.95557e12	−0.262877
$726$	0	0
$727$	1.18582e13	1.57439	0.787197	−	0.616701i	$-0.211531\pi$
$727$	1.18582e13	1.57439	0.787197	+	0.616701i	$0.211531\pi$
$728$	−7.34121e10	−0.00968671
$729$	0	0
$730$	−2.79492e12	−0.364264
$731$	1.79087e11	0.0231972
$732$	0	0
$733$	−1.34166e13	−1.71662	−0.858308	−	0.513135i	$-0.828484\pi$
$733$	−1.34166e13	−1.71662	−0.858308	+	0.513135i	$0.828484\pi$
$734$	−1.01112e13	−1.28580
$735$	0	0
$736$	9.28320e12	1.16613
$737$	−4.04322e12	−0.504805
$738$	0	0
$739$	7.69859e11	0.0949535	0.0474768	−	0.998872i	$-0.484882\pi$
$739$	7.69859e11	0.0949535	0.0474768	+	0.998872i	$0.484882\pi$
$740$	6.90018e12	0.845897
$741$	0	0
$742$	5.21245e12	0.631284
$743$	−3.57491e12	−0.430344	−0.215172	−	0.976576i	$-0.569031\pi$
$743$	−3.57491e12	−0.430344	−0.215172	+	0.976576i	$0.569031\pi$
$744$	0	0
$745$	−1.99904e12	−0.237749
$746$	−2.44290e13	−2.88790
$747$	0	0
$748$	1.14175e11	0.0133356
$749$	−4.45423e12	−0.517136
$750$	0	0
$751$	−3.35759e12	−0.385166	−0.192583	−	0.981281i	$-0.561686\pi$
$751$	−3.35759e12	−0.385166	−0.192583	+	0.981281i	$0.561686\pi$
$752$	1.23086e11	0.0140355
$753$	0	0
$754$	−5.21539e11	−0.0587646
$755$	3.81898e12	0.427746
$756$	0	0
$757$	−8.30063e12	−0.918713	−0.459356	−	0.888252i	$-0.651920\pi$
$757$	−8.30063e12	−0.918713	−0.459356	+	0.888252i	$0.651920\pi$
$758$	−9.34789e12	−1.02849
$759$	0	0
$760$	−6.31103e12	−0.686182
$761$	1.52318e13	1.64634	0.823170	−	0.567795i	$-0.192204\pi$
$761$	1.52318e13	1.64634	0.823170	+	0.567795i	$0.192204\pi$
$762$	0	0
$763$	−4.13657e12	−0.441855
$764$	1.21628e13	1.29155
$765$	0	0
$766$	2.04848e13	2.14982
$767$	2.17752e11	0.0227187
$768$	0	0
$769$	1.39127e13	1.43464	0.717321	−	0.696743i	$-0.245368\pi$
$769$	1.39127e13	1.43464	0.717321	+	0.696743i	$0.245368\pi$
$770$	−1.06648e12	−0.109331
$771$	0	0
$772$	−2.43517e13	−2.46747
$773$	1.39207e13	1.40234	0.701172	−	0.712992i	$-0.252660\pi$
$773$	1.39207e13	1.40234	0.701172	+	0.712992i	$0.252660\pi$
$774$	0	0
$775$	−1.44645e11	−0.0144028
$776$	−9.02184e12	−0.893135
$777$	0	0
$778$	−1.64677e12	−0.161148
$779$	3.22243e12	0.313520
$780$	0	0
$781$	−1.32673e12	−0.127601
$782$	−3.81834e11	−0.0365127
$783$	0	0
$784$	−1.48353e11	−0.0140241
$785$	−3.93614e12	−0.369962
$786$	0	0
$787$	−1.97103e13	−1.83150	−0.915751	−	0.401746i	$-0.868404\pi$
$787$	−1.97103e13	−1.83150	−0.915751	+	0.401746i	$0.868404\pi$
$788$	−2.59470e13	−2.39729
$789$	0	0
$790$	−7.32095e12	−0.668722
$791$	9.08271e10	0.00824938
$792$	0	0
$793$	−3.61103e10	−0.00324267
$794$	3.02115e13	2.69761
$795$	0	0
$796$	−2.32589e13	−2.05343
$797$	−7.53957e12	−0.661888	−0.330944	−	0.943650i	$-0.607367\pi$
$797$	−7.53957e12	−0.661888	−0.330944	+	0.943650i	$0.607367\pi$
$798$	0	0
$799$	−3.46255e10	−0.00300563
$800$	−2.49550e12	−0.215404
$801$	0	0
$802$	9.13577e11	0.0779759
$803$	−2.41224e12	−0.204739
$804$	0	0
$805$	2.18058e12	0.183017
$806$	−3.85761e10	−0.00321966
$807$	0	0
$808$	1.09676e13	0.905235
$809$	−1.23482e13	−1.01353	−0.506764	−	0.862085i	$-0.669158\pi$
$809$	−1.23482e13	−1.01353	−0.506764	+	0.862085i	$0.669158\pi$
$810$	0	0
$811$	1.03224e13	0.837888	0.418944	−	0.908012i	$-0.362400\pi$
$811$	1.03224e13	0.837888	0.418944	+	0.908012i	$0.362400\pi$
$812$	9.68209e12	0.781568
$813$	0	0
$814$	9.74088e12	0.777657
$815$	−3.69771e12	−0.293578
$816$	0	0
$817$	−2.34480e13	−1.84123
$818$	1.54526e13	1.20674
$819$	0	0
$820$	−1.71152e12	−0.132197
$821$	9.22885e11	0.0708930	0.0354465	−	0.999372i	$-0.488715\pi$
$821$	9.22885e11	0.0708930	0.0354465	+	0.999372i	$0.488715\pi$
$822$	0	0
$823$	−1.56068e13	−1.18581	−0.592905	−	0.805272i	$-0.702019\pi$
$823$	−1.56068e13	−1.18581	−0.592905	+	0.805272i	$0.702019\pi$
$824$	2.79409e12	0.211139
$825$	0	0
$826$	−6.61196e12	−0.494219
$827$	1.58271e13	1.17659	0.588297	−	0.808645i	$-0.299798\pi$
$827$	1.58271e13	1.17659	0.588297	+	0.808645i	$0.299798\pi$
$828$	0	0
$829$	−7.71794e12	−0.567553	−0.283776	−	0.958891i	$-0.591587\pi$
$829$	−7.71794e12	−0.567553	−0.283776	+	0.958891i	$0.591587\pi$
$830$	−1.55103e13	−1.13441
$831$	0	0
$832$	−6.27719e11	−0.0454161
$833$	4.17335e10	0.00300318
$834$	0	0
$835$	−4.99642e12	−0.355689
$836$	−1.49490e13	−1.05849
$837$	0	0
$838$	3.38229e13	2.36926
$839$	−9.87242e11	−0.0687852	−0.0343926	−	0.999408i	$-0.510950\pi$
$839$	−9.87242e11	−0.0687852	−0.0343926	+	0.999408i	$0.510950\pi$
$840$	0	0
$841$	1.05555e13	0.727607
$842$	3.68692e13	2.52790
$843$	0	0
$844$	−1.83075e13	−1.24190
$845$	−6.62266e12	−0.446866
$846$	0	0
$847$	4.74097e12	0.316513
$848$	1.53918e12	0.102213
$849$	0	0
$850$	1.02644e11	0.00674450
$851$	−1.99167e13	−1.30177
$852$	0	0
$853$	1.23982e13	0.801842	0.400921	−	0.916113i	$-0.368690\pi$
$853$	1.23982e13	0.801842	0.400921	+	0.916113i	$0.368690\pi$
$854$	1.09648e12	0.0705406
$855$	0	0
$856$	1.97632e13	1.25813
$857$	2.30727e13	1.46112	0.730559	−	0.682850i	$-0.239260\pi$
$857$	2.30727e13	1.46112	0.730559	+	0.682850i	$0.239260\pi$
$858$	0	0
$859$	−2.78440e13	−1.74487	−0.872433	−	0.488734i	$-0.837459\pi$
$859$	−2.78440e13	−1.74487	−0.872433	+	0.488734i	$0.837459\pi$
$860$	1.24539e13	0.776360
$861$	0	0
$862$	−6.53049e12	−0.402868
$863$	2.50973e13	1.54020	0.770102	−	0.637921i	$-0.220205\pi$
$863$	2.50973e13	1.54020	0.770102	+	0.637921i	$0.220205\pi$
$864$	0	0
$865$	9.13264e12	0.554656
$866$	−1.88218e13	−1.13718
$867$	0	0
$868$	7.16143e11	0.0428214
$869$	−6.31858e12	−0.375864
$870$	0	0
$871$	−5.92680e11	−0.0348930
$872$	1.83538e13	1.07498
$873$	0	0
$874$	4.99941e13	2.89812
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	−6.92376e12	−0.395224	−0.197612	−	0.980280i	$-0.563319\pi$
$877$	−6.92376e12	−0.395224	−0.197612	+	0.980280i	$0.563319\pi$
$878$	−1.54930e13	−0.879855
$879$	0	0
$880$	−3.14920e11	−0.0177022
$881$	−1.15424e12	−0.0645513	−0.0322756	−	0.999479i	$-0.510275\pi$
$881$	−1.15424e12	−0.0645513	−0.0322756	+	0.999479i	$0.510275\pi$
$882$	0	0
$883$	5.32403e12	0.294725	0.147363	−	0.989083i	$-0.452922\pi$
$883$	5.32403e12	0.294725	0.147363	+	0.989083i	$0.452922\pi$
$884$	1.67364e10	0.000921780 0
$885$	0	0
$886$	3.29243e13	1.79500
$887$	−2.63521e13	−1.42942	−0.714709	−	0.699422i	$-0.753441\pi$
$887$	−2.63521e13	−1.42942	−0.714709	+	0.699422i	$0.753441\pi$
$888$	0	0
$889$	−5.38400e11	−0.0289100
$890$	−9.54120e12	−0.509739
$891$	0	0
$892$	2.13884e13	1.13119
$893$	4.53356e12	0.238565
$894$	0	0
$895$	7.42375e12	0.386741
$896$	1.12070e13	0.580903
$897$	0	0
$898$	2.18559e13	1.12157
$899$	1.85378e12	0.0946541
$900$	0	0
$901$	−4.32988e11	−0.0218884
$902$	−2.41613e12	−0.121532
$903$	0	0
$904$	−4.02995e11	−0.0200697
$905$	−3.46621e12	−0.171765
$906$	0	0
$907$	−1.42186e13	−0.697629	−0.348815	−	0.937192i	$-0.613416\pi$
$907$	−1.42186e13	−0.697629	−0.348815	+	0.937192i	$0.613416\pi$
$908$	−2.34098e12	−0.114291
$909$	0	0
$910$	−1.56331e11	−0.00755717
$911$	9.88799e12	0.475637	0.237818	−	0.971310i	$-0.423568\pi$
$911$	9.88799e12	0.475637	0.237818	+	0.971310i	$0.423568\pi$
$912$	0	0
$913$	−1.33867e13	−0.637609
$914$	−2.72783e13	−1.29288
$915$	0	0
$916$	2.41706e13	1.13438
$917$	−9.46316e12	−0.441951
$918$	0	0
$919$	1.15139e13	0.532480	0.266240	−	0.963907i	$-0.414219\pi$
$919$	1.15139e13	0.532480	0.266240	+	0.963907i	$0.414219\pi$
$920$	−9.67512e12	−0.445258
$921$	0	0
$922$	4.49965e13	2.05064
$923$	−1.94481e11	−0.00882000
$924$	0	0
$925$	5.35399e12	0.240458
$926$	−2.34091e13	−1.04625
$927$	0	0
$928$	3.19824e13	1.41562
$929$	−3.05831e13	−1.34713	−0.673566	−	0.739127i	$-0.735238\pi$
$929$	−3.05831e13	−1.34713	−0.673566	+	0.739127i	$0.735238\pi$
$930$	0	0
$931$	−5.46422e12	−0.238372
$932$	6.53629e13	2.83766
$933$	0	0
$934$	1.91418e13	0.823041
$935$	8.85904e10	0.00379083
$936$	0	0
$937$	2.66802e13	1.13073	0.565367	−	0.824839i	$-0.308735\pi$
$937$	2.66802e13	1.13073	0.565367	+	0.824839i	$0.308735\pi$
$938$	1.79965e13	0.759058
$939$	0	0
$940$	−2.40790e12	−0.100592
$941$	1.87072e13	0.777776	0.388888	−	0.921285i	$-0.372859\pi$
$941$	1.87072e13	0.777776	0.388888	+	0.921285i	$0.372859\pi$
$942$	0	0
$943$	4.94014e12	0.203440
$944$	−1.95243e12	−0.0800207
$945$	0	0
$946$	1.75810e13	0.713730
$947$	4.84514e13	1.95763	0.978817	−	0.204738i	$-0.0656342\pi$
$947$	4.84514e13	1.95763	0.978817	+	0.204738i	$0.0656342\pi$
$948$	0	0
$949$	−3.53601e11	−0.0141519
$950$	−1.34394e13	−0.535331
$951$	0	0
$952$	−1.85169e11	−0.00730639
$953$	2.15103e13	0.844750	0.422375	−	0.906421i	$-0.361197\pi$
$953$	2.15103e13	0.844750	0.422375	+	0.906421i	$0.361197\pi$
$954$	0	0
$955$	9.43732e12	0.367142
$956$	6.26063e12	0.242414
$957$	0	0
$958$	6.62003e13	2.53931
$959$	1.72974e13	0.660386
$960$	0	0
$961$	−2.63025e13	−0.994814
$962$	1.42788e12	0.0537530
$963$	0	0
$964$	−1.62719e13	−0.606866
$965$	−1.88950e13	−0.701412
$966$	0	0
$967$	−1.88700e13	−0.693991	−0.346996	−	0.937867i	$-0.612798\pi$
$967$	−1.88700e13	−0.693991	−0.346996	+	0.937867i	$0.612798\pi$
$968$	−2.10354e13	−0.770039
$969$	0	0
$970$	−1.92120e13	−0.696787
$971$	1.39732e13	0.504440	0.252220	−	0.967670i	$-0.418839\pi$
$971$	1.39732e13	0.504440	0.252220	+	0.967670i	$0.418839\pi$
$972$	0	0
$973$	−3.62505e12	−0.129660
$974$	8.27601e12	0.294649
$975$	0	0
$976$	3.23777e11	0.0114215
$977$	−2.02308e13	−0.710375	−0.355187	−	0.934795i	$-0.615583\pi$
$977$	−2.02308e13	−0.710375	−0.355187	+	0.934795i	$0.615583\pi$
$978$	0	0
$979$	−8.23483e12	−0.286505
$980$	2.90220e12	0.100510
$981$	0	0
$982$	9.28345e11	0.0318572
$983$	−1.66424e13	−0.568492	−0.284246	−	0.958751i	$-0.591743\pi$
$983$	−1.66424e13	−0.568492	−0.284246	+	0.958751i	$0.591743\pi$
$984$	0	0
$985$	−2.01328e13	−0.681462
$986$	−1.31549e12	−0.0443243
$987$	0	0
$988$	−2.19132e12	−0.0731644
$989$	−3.59470e13	−1.19476
$990$	0	0
$991$	−4.62849e13	−1.52443	−0.762215	−	0.647324i	$-0.775888\pi$
$991$	−4.62849e13	−1.52443	−0.762215	+	0.647324i	$0.775888\pi$
$992$	2.36560e12	0.0775603
$993$	0	0
$994$	5.90533e12	0.191869
$995$	−1.80471e13	−0.583717
$996$	0	0
$997$	−3.80995e13	−1.22121	−0.610606	−	0.791934i	$-0.709074\pi$
$997$	−3.80995e13	−1.22121	−0.610606	+	0.791934i	$0.709074\pi$
$998$	8.93606e13	2.85140
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.d.1.4		4
3.2	odd	2		105.10.a.e.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.e.1.1	✓	4	3.2	odd	2
315.10.a.d.1.4		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)

\(f(q)\)	\(=\)	\(q+36.2973 q^{2} +805.496 q^{4} +625.000 q^{5} -2401.00 q^{7} +10653.1 q^{8} +22685.8 q^{10} +19579.7 q^{11} +2870.11 q^{13} -87149.9 q^{14} -25734.4 q^{16} +7239.36 q^{17} -947859. q^{19} +503435. q^{20} +710691. q^{22} -1.45312e6 q^{23} +390625. q^{25} +104177. q^{26} -1.93400e6 q^{28} -5.00626e6 q^{29} -370292. q^{31} -6.38848e6 q^{32} +262769. q^{34} -1.50062e6 q^{35} +1.37062e7 q^{37} -3.44047e7 q^{38} +6.65820e6 q^{40} -3.39969e6 q^{41} +2.47379e7 q^{43} +1.57714e7 q^{44} -5.27442e7 q^{46} -4.78294e6 q^{47} +5.76480e6 q^{49} +1.41786e7 q^{50} +2.31186e6 q^{52} -5.98102e7 q^{53} +1.22373e7 q^{55} -2.55781e7 q^{56} -1.81714e8 q^{58} +7.58688e7 q^{59} -1.25815e7 q^{61} -1.34406e7 q^{62} -2.18709e8 q^{64} +1.79382e6 q^{65} -2.06500e8 q^{67} +5.83127e6 q^{68} -5.44687e7 q^{70} -6.77606e7 q^{71} -1.23201e8 q^{73} +4.97499e8 q^{74} -7.63496e8 q^{76} -4.70109e7 q^{77} -3.22710e8 q^{79} -1.60840e7 q^{80} -1.23400e8 q^{82} -6.83702e8 q^{83} +4.52460e6 q^{85} +8.97920e8 q^{86} +2.08585e8 q^{88} -4.20580e8 q^{89} -6.89114e6 q^{91} -1.17048e9 q^{92} -1.73608e8 q^{94} -5.92412e8 q^{95} -8.46873e8 q^{97} +2.09247e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 5 q^{2} + 949 q^{4} + 2500 q^{5} - 9604 q^{7} - 7767 q^{8} - 3125 q^{10} - 64546 q^{11} - 29390 q^{13} + 12005 q^{14} + 554577 q^{16} - 278788 q^{17} - 929142 q^{19} + 593125 q^{20} + 2767732 q^{22}+ \cdots - 28824005 q^{98}+O(q^{100}) \) 4 * q - 5 * q^2 + 949 * q^4 + 2500 * q^5 - 9604 * q^7 - 7767 * q^8 - 3125 * q^10 - 64546 * q^11 - 29390 * q^13 + 12005 * q^14 + 554577 * q^16 - 278788 * q^17 - 929142 * q^19 + 593125 * q^20 + 2767732 * q^22 + 526064 * q^23 + 1562500 * q^25 + 3920706 * q^26 - 2278549 * q^28 - 365860 * q^29 - 4977954 * q^31 - 7227279 * q^32 - 2976030 * q^34 - 6002500 * q^35 + 22846008 * q^37 - 27001832 * q^38 - 4854375 * q^40 + 28257844 * q^41 + 23603420 * q^43 + 9817860 * q^44 - 136600280 * q^46 + 30058700 * q^47 + 23059204 * q^49 - 1953125 * q^50 - 183429274 * q^52 - 113767294 * q^53 - 40341250 * q^55 + 18648567 * q^56 - 323185726 * q^58 + 151786220 * q^59 - 191130108 * q^61 + 292486944 * q^62 - 423209695 * q^64 - 18368750 * q^65 - 147812356 * q^67 + 266317150 * q^68 + 7503125 * q^70 + 37100486 * q^71 + 163444574 * q^73 + 255454086 * q^74 - 720141040 * q^76 + 154974946 * q^77 - 327433200 * q^79 + 346610625 * q^80 - 686338410 * q^82 - 216775352 * q^83 - 174242500 * q^85 + 739989240 * q^86 - 127433212 * q^88 + 792987912 * q^89 + 70565390 * q^91 + 31184232 * q^92 - 1357334188 * q^94 - 580713750 * q^95 - 640730334 * q^97 - 28824005 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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