LMFDB - Embedded newform 45.8.a.h.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,8,Mod(1,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	45.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-20] 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:	$14.0573261468$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	45.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.28220 q^{2} -126.356 q^{4} +125.000 q^{5} -538.197 q^{7} +326.136 q^{8} -160.275 q^{10} +1215.12 q^{11} +7070.42 q^{13} +690.077 q^{14} +15755.4 q^{16} +3348.13 q^{17} +22169.7 q^{19} -15794.5 q^{20} -1558.03 q^{22} +58513.1 q^{23} +15625.0 q^{25} -9065.71 q^{26} +68004.4 q^{28} +206301. q^{29} +177822. q^{31} -61947.0 q^{32} -4292.98 q^{34} -67274.6 q^{35} -284128. q^{37} -28426.0 q^{38} +40767.0 q^{40} -627353. q^{41} -164889. q^{43} -153538. q^{44} -75025.7 q^{46} +449355. q^{47} -533887. q^{49} -20034.4 q^{50} -893390. q^{52} +730190. q^{53} +151890. q^{55} -175525. q^{56} -264520. q^{58} -1.42202e6 q^{59} -266326. q^{61} -228004. q^{62} -1.93726e6 q^{64} +883803. q^{65} +2.95028e6 q^{67} -423056. q^{68} +86259.6 q^{70} -921138. q^{71} +4.25657e6 q^{73} +364309. q^{74} -2.80127e6 q^{76} -653973. q^{77} +6.28551e6 q^{79} +1.96942e6 q^{80} +804393. q^{82} +9.17165e6 q^{83} +418516. q^{85} +211421. q^{86} +396294. q^{88} -242643. q^{89} -3.80528e6 q^{91} -7.39348e6 q^{92} -576164. q^{94} +2.77121e6 q^{95} -2.59198e6 q^{97} +684551. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{2} + 96 q^{4} + 250 q^{5} - 100 q^{7} - 1440 q^{8} - 2500 q^{10} - 4544 q^{11} + 3540 q^{13} - 7512 q^{14} + 20352 q^{16} + 27340 q^{17} + 38760 q^{19} + 12000 q^{20} + 106240 q^{22} + 124140 q^{23}+ \cdots + 12505340 q^{98}+O(q^{100}) $ 2 * q - 20 * q^2 + 96 * q^4 + 250 * q^5 - 100 * q^7 - 1440 * q^8 - 2500 * q^10 - 4544 * q^11 + 3540 * q^13 - 7512 * q^14 + 20352 * q^16 + 27340 * q^17 + 38760 * q^19 + 12000 * q^20 + 106240 * q^22 + 124140 * q^23 + 31250 * q^25 + 57016 * q^26 + 165440 * q^28 + 72260 * q^29 + 306824 * q^31 + 78080 * q^32 - 453368 * q^34 - 12500 * q^35 - 123020 * q^37 - 338960 * q^38 - 180000 * q^40 - 264364 * q^41 + 423300 * q^43 - 1434112 * q^44 - 1303416 * q^46 + 105460 * q^47 - 1165414 * q^49 - 312500 * q^50 - 1678400 * q^52 + 2391580 * q^53 - 568000 * q^55 - 949440 * q^56 + 2244440 * q^58 + 1120120 * q^59 + 2257044 * q^61 - 2642640 * q^62 - 5146624 * q^64 + 442500 * q^65 + 4516460 * q^67 + 4911680 * q^68 - 939000 * q^70 - 621784 * q^71 + 4569060 * q^73 - 2651272 * q^74 + 887680 * q^76 - 3177600 * q^77 + 4333040 * q^79 + 2544000 * q^80 - 5989960 * q^82 + 9793020 * q^83 + 3417500 * q^85 - 10798184 * q^86 + 10567680 * q^88 - 6025620 * q^89 - 5352296 * q^91 + 7199040 * q^92 + 5860792 * q^94 + 4845000 * q^95 + 4609540 * q^97 + 12505340 * 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 $

$2$	−1.28220	−0.113332	−0.0566659	−	0.998393i	$-0.518047\pi$
$2$	−1.28220	−0.113332	−0.0566659	+	0.998393i	$0.518047\pi$
$3$	0	0
$3$	0	0
$4$	−126.356	−0.987156
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	−538.197	−0.593059	−0.296529	−	0.955024i	$-0.595829\pi$
$7$	−538.197	−0.593059	−0.296529	+	0.955024i	$0.595829\pi$
$8$	326.136	0.225208
$9$	0	0
$10$	−160.275	−0.0506835
$11$	1215.12	0.275261	0.137630	−	0.990484i	$-0.456051\pi$
$11$	1215.12	0.275261	0.137630	+	0.990484i	$0.456051\pi$
$12$	0	0
$13$	7070.42	0.892573	0.446286	−	0.894890i	$-0.352746\pi$
$13$	7070.42	0.892573	0.446286	+	0.894890i	$0.352746\pi$
$14$	690.077	0.0672124
$15$	0	0
$16$	15755.4	0.961633
$17$	3348.13	0.165284	0.0826420	−	0.996579i	$-0.473664\pi$
$17$	3348.13	0.165284	0.0826420	+	0.996579i	$0.473664\pi$
$18$	0	0
$19$	22169.7	0.741519	0.370759	−	0.928729i	$-0.379097\pi$
$19$	22169.7	0.741519	0.370759	+	0.928729i	$0.379097\pi$
$20$	−15794.5	−0.441470
$21$	0	0
$22$	−1558.03	−0.0311958
$23$	58513.1	1.00278	0.501390	−	0.865221i	$-0.332822\pi$
$23$	58513.1	1.00278	0.501390	+	0.865221i	$0.332822\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	−9065.71	−0.101157
$27$	0	0
$28$	68004.4	0.585442
$29$	206301.	1.57076	0.785379	−	0.619015i	$-0.212468\pi$
$29$	206301.	1.57076	0.785379	+	0.619015i	$0.212468\pi$
$30$	0	0
$31$	177822.	1.07206	0.536030	−	0.844199i	$-0.319923\pi$
$31$	177822.	1.07206	0.536030	+	0.844199i	$0.319923\pi$
$32$	−61947.0	−0.334191
$33$	0	0
$34$	−4292.98	−0.0187319
$35$	−67274.6	−0.265224
$36$	0	0
$37$	−284128.	−0.922163	−0.461081	−	0.887358i	$-0.652538\pi$
$37$	−284128.	−0.922163	−0.461081	+	0.887358i	$0.652538\pi$
$38$	−28426.0	−0.0840376
$39$	0	0
$40$	40767.0	0.100716
$41$	−627353.	−1.42157	−0.710785	−	0.703409i	$-0.751660\pi$
$41$	−627353.	−1.42157	−0.710785	+	0.703409i	$0.751660\pi$
$42$	0	0
$43$	−164889.	−0.316266	−0.158133	−	0.987418i	$-0.550547\pi$
$43$	−164889.	−0.316266	−0.158133	+	0.987418i	$0.550547\pi$
$44$	−153538.	−0.271725
$45$	0	0
$46$	−75025.7	−0.113647
$47$	449355.	0.631316	0.315658	−	0.948873i	$-0.397775\pi$
$47$	449355.	0.631316	0.315658	+	0.948873i	$0.397775\pi$
$48$	0	0
$49$	−533887.	−0.648281
$50$	−20034.4	−0.0226663
$51$	0	0
$52$	−893390.	−0.881108
$53$	730190.	0.673706	0.336853	−	0.941557i	$-0.390638\pi$
$53$	730190.	0.673706	0.336853	+	0.941557i	$0.390638\pi$
$54$	0	0
$55$	151890.	0.123100
$56$	−175525.	−0.133562
$57$	0	0
$58$	−264520.	−0.178017
$59$	−1.42202e6	−0.901412	−0.450706	−	0.892673i	$-0.648828\pi$
$59$	−1.42202e6	−0.901412	−0.450706	+	0.892673i	$0.648828\pi$
$60$	0	0
$61$	−266326.	−0.150231	−0.0751153	−	0.997175i	$-0.523932\pi$
$61$	−266326.	−0.150231	−0.0751153	+	0.997175i	$0.523932\pi$
$62$	−228004.	−0.121498
$63$	0	0
$64$	−1.93726e6	−0.923758
$65$	883803.	0.399171
$66$	0	0
$67$	2.95028e6	1.19840	0.599200	−	0.800599i	$-0.295485\pi$
$67$	2.95028e6	1.19840	0.599200	+	0.800599i	$0.295485\pi$
$68$	−423056.	−0.163161
$69$	0	0
$70$	86259.6	0.0300583
$71$	−921138.	−0.305436	−0.152718	−	0.988270i	$-0.548803\pi$
$71$	−921138.	−0.305436	−0.152718	+	0.988270i	$0.548803\pi$
$72$	0	0
$73$	4.25657e6	1.28065	0.640323	−	0.768105i	$-0.278800\pi$
$73$	4.25657e6	1.28065	0.640323	+	0.768105i	$0.278800\pi$
$74$	364309.	0.104510
$75$	0	0
$76$	−2.80127e6	−0.731995
$77$	−653973.	−0.163246
$78$	0	0
$79$	6.28551e6	1.43432	0.717159	−	0.696910i	$-0.245442\pi$
$79$	6.28551e6	1.43432	0.717159	+	0.696910i	$0.245442\pi$
$80$	1.96942e6	0.430055
$81$	0	0
$82$	804393.	0.161109
$83$	9.17165e6	1.76065	0.880327	−	0.474367i	$-0.157323\pi$
$83$	9.17165e6	1.76065	0.880327	+	0.474367i	$0.157323\pi$
$84$	0	0
$85$	418516.	0.0739172
$86$	211421.	0.0358430
$87$	0	0
$88$	396294.	0.0619909
$89$	−242643.	−0.0364840	−0.0182420	−	0.999834i	$-0.505807\pi$
$89$	−242643.	−0.0364840	−0.0182420	+	0.999834i	$0.505807\pi$
$90$	0	0
$91$	−3.80528e6	−0.529348
$92$	−7.39348e6	−0.989901
$93$	0	0
$94$	−576164.	−0.0715482
$95$	2.77121e6	0.331617
$96$	0	0
$97$	−2.59198e6	−0.288357	−0.144179	−	0.989552i	$-0.546054\pi$
$97$	−2.59198e6	−0.288357	−0.144179	+	0.989552i	$0.546054\pi$
$98$	684551.	0.0734708
$99$	0	0
$100$	−1.97431e6	−0.197431
$101$	−3.69169e6	−0.356534	−0.178267	−	0.983982i	$-0.557049\pi$
$101$	−3.69169e6	−0.356534	−0.178267	+	0.983982i	$0.557049\pi$
$102$	0	0
$103$	8.68203e6	0.782873	0.391436	−	0.920205i	$-0.371978\pi$
$103$	8.68203e6	0.782873	0.391436	+	0.920205i	$0.371978\pi$
$104$	2.30592e6	0.201014
$105$	0	0
$106$	−936251.	−0.0763522
$107$	−1.52118e7	−1.20043	−0.600216	−	0.799838i	$-0.704919\pi$
$107$	−1.52118e7	−1.20043	−0.600216	+	0.799838i	$0.704919\pi$
$108$	0	0
$109$	−1.60843e6	−0.118963	−0.0594813	−	0.998229i	$-0.518945\pi$
$109$	−1.60843e6	−0.118963	−0.0594813	+	0.998229i	$0.518945\pi$
$110$	−194754.	−0.0139512
$111$	0	0
$112$	−8.47950e6	−0.570305
$113$	2.62766e7	1.71315	0.856574	−	0.516024i	$-0.172589\pi$
$113$	2.62766e7	1.71315	0.856574	+	0.516024i	$0.172589\pi$
$114$	0	0
$115$	7.31414e6	0.448457
$116$	−2.60674e7	−1.55058
$117$	0	0
$118$	1.82332e6	0.102159
$119$	−1.80195e6	−0.0980231
$120$	0	0
$121$	−1.80107e7	−0.924231
$122$	341483.	0.0170259
$123$	0	0
$124$	−2.24688e7	−1.05829
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−1.91864e7	−0.831152	−0.415576	−	0.909559i	$-0.636420\pi$
$127$	−1.91864e7	−0.831152	−0.415576	+	0.909559i	$0.636420\pi$
$128$	1.04132e7	0.438882
$129$	0	0
$130$	−1.13321e6	−0.0452387
$131$	−3.07018e7	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$131$	−3.07018e7	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$132$	0	0
$133$	−1.19317e7	−0.439764
$134$	−3.78286e6	−0.135817
$135$	0	0
$136$	1.09194e6	0.0372232
$137$	3.16847e7	1.05276	0.526378	−	0.850251i	$-0.323550\pi$
$137$	3.16847e7	1.05276	0.526378	+	0.850251i	$0.323550\pi$
$138$	0	0
$139$	2.07581e7	0.655596	0.327798	−	0.944748i	$-0.393693\pi$
$139$	2.07581e7	0.655596	0.327798	+	0.944748i	$0.393693\pi$
$140$	8.50054e6	0.261817
$141$	0	0
$142$	1.18108e6	0.0346156
$143$	8.59140e6	0.245690
$144$	0	0
$145$	2.57877e7	0.702464
$146$	−5.45778e6	−0.145138
$147$	0	0
$148$	3.59012e7	0.910318
$149$	−2.94607e7	−0.729611	−0.364806	−	0.931084i	$-0.618865\pi$
$149$	−2.94607e7	−0.729611	−0.364806	+	0.931084i	$0.618865\pi$
$150$	0	0
$151$	4.60002e7	1.08728	0.543638	−	0.839320i	$-0.317046\pi$
$151$	4.60002e7	1.08728	0.543638	+	0.839320i	$0.317046\pi$
$152$	7.23033e6	0.166996
$153$	0	0
$154$	838526.	0.0185009
$155$	2.22277e7	0.479440
$156$	0	0
$157$	1.92998e7	0.398019	0.199010	−	0.979998i	$-0.436227\pi$
$157$	1.92998e7	0.398019	0.199010	+	0.979998i	$0.436227\pi$
$158$	−8.05929e6	−0.162554
$159$	0	0
$160$	−7.74337e6	−0.149455
$161$	−3.14916e7	−0.594708
$162$	0	0
$163$	2.35624e7	0.426151	0.213076	−	0.977036i	$-0.431652\pi$
$163$	2.35624e7	0.426151	0.213076	+	0.977036i	$0.431652\pi$
$164$	7.92698e7	1.40331
$165$	0	0
$166$	−1.17599e7	−0.199538
$167$	−2.18824e7	−0.363570	−0.181785	−	0.983338i	$-0.558187\pi$
$167$	−2.18824e7	−0.363570	−0.181785	+	0.983338i	$0.558187\pi$
$168$	0	0
$169$	−1.27577e7	−0.203314
$170$	−536622.	−0.00837717
$171$	0	0
$172$	2.08347e7	0.312204
$173$	9.62312e7	1.41304	0.706520	−	0.707693i	$-0.250264\pi$
$173$	9.62312e7	1.41304	0.706520	+	0.707693i	$0.250264\pi$
$174$	0	0
$175$	−8.40932e6	−0.118612
$176$	1.91447e7	0.264700
$177$	0	0
$178$	311117.	0.00413479
$179$	−8.46776e7	−1.10353	−0.551763	−	0.834001i	$-0.686045\pi$
$179$	−8.46776e7	−1.10353	−0.551763	+	0.834001i	$0.686045\pi$
$180$	0	0
$181$	−9.65249e7	−1.20994	−0.604971	−	0.796248i	$-0.706815\pi$
$181$	−9.65249e7	−1.20994	−0.604971	+	0.796248i	$0.706815\pi$
$182$	4.87913e6	0.0599919
$183$	0	0
$184$	1.90832e7	0.225834
$185$	−3.55160e7	−0.412404
$186$	0	0
$187$	4.06837e6	0.0454962
$188$	−5.67787e7	−0.623208
$189$	0	0
$190$	−3.55325e6	−0.0375828
$191$	1.46467e8	1.52098	0.760491	−	0.649348i	$-0.224958\pi$
$191$	1.46467e8	1.52098	0.760491	+	0.649348i	$0.224958\pi$
$192$	0	0
$193$	−8.53275e7	−0.854355	−0.427177	−	0.904168i	$-0.640492\pi$
$193$	−8.53275e7	−0.854355	−0.427177	+	0.904168i	$0.640492\pi$
$194$	3.32345e6	0.0326800
$195$	0	0
$196$	6.74598e7	0.639954
$197$	−1.44802e8	−1.34941	−0.674705	−	0.738087i	$-0.735729\pi$
$197$	−1.44802e8	−1.34941	−0.674705	+	0.738087i	$0.735729\pi$
$198$	0	0
$199$	1.26870e7	0.114123	0.0570617	−	0.998371i	$-0.481827\pi$
$199$	1.26870e7	0.114123	0.0570617	+	0.998371i	$0.481827\pi$
$200$	5.09587e6	0.0450416
$201$	0	0
$202$	4.73349e6	0.0404066
$203$	−1.11031e8	−0.931552
$204$	0	0
$205$	−7.84191e7	−0.635746
$206$	−1.11321e7	−0.0887243
$207$	0	0
$208$	1.11397e8	0.858327
$209$	2.69388e7	0.204111
$210$	0	0
$211$	−9.47331e7	−0.694246	−0.347123	−	0.937820i	$-0.612841\pi$
$211$	−9.47331e7	−0.694246	−0.347123	+	0.937820i	$0.612841\pi$
$212$	−9.22638e7	−0.665053
$213$	0	0
$214$	1.95046e7	0.136047
$215$	−2.06111e7	−0.141438
$216$	0	0
$217$	−9.57031e7	−0.635795
$218$	2.06234e6	0.0134822
$219$	0	0
$220$	−1.91922e7	−0.121519
$221$	2.36727e7	0.147528
$222$	0	0
$223$	−2.28554e8	−1.38014	−0.690069	−	0.723744i	$-0.742420\pi$
$223$	−2.28554e8	−1.38014	−0.690069	+	0.723744i	$0.742420\pi$
$224$	3.33397e7	0.198195
$225$	0	0
$226$	−3.36919e7	−0.194154
$227$	3.00587e8	1.70561	0.852806	−	0.522229i	$-0.174899\pi$
$227$	3.00587e8	1.70561	0.852806	+	0.522229i	$0.174899\pi$
$228$	0	0
$229$	−1.05343e8	−0.579669	−0.289835	−	0.957077i	$-0.593600\pi$
$229$	−1.05343e8	−0.579669	−0.289835	+	0.957077i	$0.593600\pi$
$230$	−9.37821e6	−0.0508244
$231$	0	0
$232$	6.72823e7	0.353747
$233$	−3.27196e8	−1.69458	−0.847291	−	0.531130i	$-0.821768\pi$
$233$	−3.27196e8	−1.69458	−0.847291	+	0.531130i	$0.821768\pi$
$234$	0	0
$235$	5.61694e7	0.282333
$236$	1.79681e8	0.889834
$237$	0	0
$238$	2.31047e6	0.0111091
$239$	5.72888e7	0.271442	0.135721	−	0.990747i	$-0.456665\pi$
$239$	5.72888e7	0.271442	0.135721	+	0.990747i	$0.456665\pi$
$240$	0	0
$241$	2.84287e8	1.30827	0.654134	−	0.756379i	$-0.273033\pi$
$241$	2.84287e8	1.30827	0.654134	+	0.756379i	$0.273033\pi$
$242$	2.30933e7	0.104745
$243$	0	0
$244$	3.36518e7	0.148301
$245$	−6.67359e7	−0.289920
$246$	0	0
$247$	1.56749e8	0.661859
$248$	5.79941e7	0.241436
$249$	0	0
$250$	−2.50430e6	−0.0101367
$251$	2.69178e7	0.107444	0.0537220	−	0.998556i	$-0.482892\pi$
$251$	2.69178e7	0.107444	0.0537220	+	0.998556i	$0.482892\pi$
$252$	0	0
$253$	7.11004e7	0.276026
$254$	2.46008e7	0.0941959
$255$	0	0
$256$	2.34618e8	0.874019
$257$	3.28749e8	1.20809	0.604044	−	0.796951i	$-0.293555\pi$
$257$	3.28749e8	1.20809	0.604044	+	0.796951i	$0.293555\pi$
$258$	0	0
$259$	1.52917e8	0.546897
$260$	−1.11674e8	−0.394044
$261$	0	0
$262$	3.93660e7	0.135228
$263$	−1.85590e8	−0.629084	−0.314542	−	0.949244i	$-0.601851\pi$
$263$	−1.85590e8	−0.629084	−0.314542	+	0.949244i	$0.601851\pi$
$264$	0	0
$265$	9.12737e7	0.301290
$266$	1.52988e7	0.0498393
$267$	0	0
$268$	−3.72786e8	−1.18301
$269$	−2.70876e8	−0.848471	−0.424236	−	0.905552i	$-0.639457\pi$
$269$	−2.70876e8	−0.848471	−0.424236	+	0.905552i	$0.639457\pi$
$270$	0	0
$271$	2.30743e8	0.704263	0.352132	−	0.935950i	$-0.385457\pi$
$271$	2.30743e8	0.704263	0.352132	+	0.935950i	$0.385457\pi$
$272$	5.27511e7	0.158942
$273$	0	0
$274$	−4.06262e7	−0.119311
$275$	1.89862e7	0.0550522
$276$	0	0
$277$	1.42059e8	0.401595	0.200798	−	0.979633i	$-0.435647\pi$
$277$	1.42059e8	0.401595	0.200798	+	0.979633i	$0.435647\pi$
$278$	−2.66161e7	−0.0742998
$279$	0	0
$280$	−2.19406e7	−0.0597305
$281$	1.15362e8	0.310164	0.155082	−	0.987902i	$-0.450436\pi$
$281$	1.15362e8	0.310164	0.155082	+	0.987902i	$0.450436\pi$
$282$	0	0
$283$	2.18080e8	0.571958	0.285979	−	0.958236i	$-0.407681\pi$
$283$	2.18080e8	0.571958	0.285979	+	0.958236i	$0.407681\pi$
$284$	1.16391e8	0.301513
$285$	0	0
$286$	−1.10159e7	−0.0278445
$287$	3.37639e8	0.843075
$288$	0	0
$289$	−3.99129e8	−0.972681
$290$	−3.30650e7	−0.0796115
$291$	0	0
$292$	−5.37842e8	−1.26420
$293$	−4.12384e8	−0.957778	−0.478889	−	0.877875i	$-0.658960\pi$
$293$	−4.12384e8	−0.957778	−0.478889	+	0.877875i	$0.658960\pi$
$294$	0	0
$295$	−1.77752e8	−0.403124
$296$	−9.26642e7	−0.207678
$297$	0	0
$298$	3.77746e7	0.0826881
$299$	4.13713e8	0.895055
$300$	0	0
$301$	8.87428e7	0.187564
$302$	−5.89815e7	−0.123223
$303$	0	0
$304$	3.49292e8	0.713069
$305$	−3.32907e7	−0.0671852
$306$	0	0
$307$	1.57983e8	0.311621	0.155810	−	0.987787i	$-0.450201\pi$
$307$	1.57983e8	0.311621	0.155810	+	0.987787i	$0.450201\pi$
$308$	8.26334e7	0.161149
$309$	0	0
$310$	−2.85004e7	−0.0543357
$311$	1.69169e8	0.318904	0.159452	−	0.987206i	$-0.449027\pi$
$311$	1.69169e8	0.318904	0.159452	+	0.987206i	$0.449027\pi$
$312$	0	0
$313$	−5.22117e8	−0.962415	−0.481208	−	0.876607i	$-0.659802\pi$
$313$	−5.22117e8	−0.962415	−0.481208	+	0.876607i	$0.659802\pi$
$314$	−2.47463e7	−0.0451082
$315$	0	0
$316$	−7.94211e8	−1.41590
$317$	−4.73280e8	−0.834470	−0.417235	−	0.908799i	$-0.637001\pi$
$317$	−4.73280e8	−0.834470	−0.417235	+	0.908799i	$0.637001\pi$
$318$	0	0
$319$	2.50681e8	0.432368
$320$	−2.42158e8	−0.413117
$321$	0	0
$322$	4.03786e7	0.0673993
$323$	7.42270e7	0.122561
$324$	0	0
$325$	1.10475e8	0.178515
$326$	−3.02118e7	−0.0482965
$327$	0	0
$328$	−2.04602e8	−0.320149
$329$	−2.41841e8	−0.374408
$330$	0	0
$331$	−8.75892e7	−0.132756	−0.0663778	−	0.997795i	$-0.521144\pi$
$331$	−8.75892e7	−0.132756	−0.0663778	+	0.997795i	$0.521144\pi$
$332$	−1.15889e9	−1.73804
$333$	0	0
$334$	2.80577e7	0.0412040
$335$	3.68785e8	0.535941
$336$	0	0
$337$	−3.04119e8	−0.432851	−0.216426	−	0.976299i	$-0.569440\pi$
$337$	−3.04119e8	−0.432851	−0.216426	+	0.976299i	$0.569440\pi$
$338$	1.63579e7	0.0230419
$339$	0	0
$340$	−5.28820e7	−0.0729678
$341$	2.16075e8	0.295096
$342$	0	0
$343$	7.30565e8	0.977528
$344$	−5.37762e7	−0.0712256
$345$	0	0
$346$	−1.23388e8	−0.160142
$347$	−3.15347e8	−0.405169	−0.202584	−	0.979265i	$-0.564934\pi$
$347$	−3.15347e8	−0.405169	−0.202584	+	0.979265i	$0.564934\pi$
$348$	0	0
$349$	−5.23418e8	−0.659112	−0.329556	−	0.944136i	$-0.606899\pi$
$349$	−5.23418e8	−0.659112	−0.329556	+	0.944136i	$0.606899\pi$
$350$	1.07825e7	0.0134425
$351$	0	0
$352$	−7.52730e7	−0.0919898
$353$	4.13964e8	0.500900	0.250450	−	0.968130i	$-0.419421\pi$
$353$	4.13964e8	0.500900	0.250450	+	0.968130i	$0.419421\pi$
$354$	0	0
$355$	−1.15142e8	−0.136595
$356$	3.06593e7	0.0360154
$357$	0	0
$358$	1.08574e8	0.125065
$359$	2.48768e8	0.283768	0.141884	−	0.989883i	$-0.454684\pi$
$359$	2.48768e8	0.283768	0.141884	+	0.989883i	$0.454684\pi$
$360$	0	0
$361$	−4.02376e8	−0.450150
$362$	1.23764e8	0.137125
$363$	0	0
$364$	4.80819e8	0.522549
$365$	5.32071e8	0.572723
$366$	0	0
$367$	1.45884e9	1.54056	0.770278	−	0.637708i	$-0.220117\pi$
$367$	1.45884e9	1.54056	0.770278	+	0.637708i	$0.220117\pi$
$368$	9.21897e8	0.964307
$369$	0	0
$370$	4.55386e7	0.0467384
$371$	−3.92986e8	−0.399547
$372$	0	0
$373$	6.29125e8	0.627706	0.313853	−	0.949472i	$-0.398380\pi$
$373$	6.29125e8	0.627706	0.313853	+	0.949472i	$0.398380\pi$
$374$	−5.21648e6	−0.00515616
$375$	0	0
$376$	1.46551e8	0.142177
$377$	1.45864e9	1.40202
$378$	0	0
$379$	8.54658e7	0.0806409	0.0403204	−	0.999187i	$-0.487162\pi$
$379$	8.54658e7	0.0806409	0.0403204	+	0.999187i	$0.487162\pi$
$380$	−3.50159e8	−0.327358
$381$	0	0
$382$	−1.87801e8	−0.172376
$383$	7.56340e8	0.687893	0.343947	−	0.938989i	$-0.388236\pi$
$383$	7.56340e8	0.687893	0.343947	+	0.938989i	$0.388236\pi$
$384$	0	0
$385$	−8.17466e7	−0.0730058
$386$	1.09407e8	0.0968255
$387$	0	0
$388$	3.27512e8	0.284654
$389$	−1.42837e9	−1.23032	−0.615160	−	0.788402i	$-0.710908\pi$
$389$	−1.42837e9	−1.23032	−0.615160	+	0.788402i	$0.710908\pi$
$390$	0	0
$391$	1.95909e8	0.165744
$392$	−1.74120e8	−0.145998
$393$	0	0
$394$	1.85666e8	0.152931
$395$	7.85688e8	0.641446
$396$	0	0
$397$	−1.67152e8	−0.134074	−0.0670372	−	0.997750i	$-0.521355\pi$
$397$	−1.67152e8	−0.134074	−0.0670372	+	0.997750i	$0.521355\pi$
$398$	−1.62673e7	−0.0129338
$399$	0	0
$400$	2.46178e8	0.192327
$401$	−2.32051e9	−1.79712	−0.898561	−	0.438848i	$-0.855387\pi$
$401$	−2.32051e9	−1.79712	−0.898561	+	0.438848i	$0.855387\pi$
$402$	0	0
$403$	1.25728e9	0.956892
$404$	4.66467e8	0.351954
$405$	0	0
$406$	1.42364e8	0.105574
$407$	−3.45249e8	−0.253835
$408$	0	0
$409$	−8.24735e8	−0.596050	−0.298025	−	0.954558i	$-0.596328\pi$
$409$	−8.24735e8	−0.596050	−0.298025	+	0.954558i	$0.596328\pi$
$410$	1.00549e8	0.0720502
$411$	0	0
$412$	−1.09703e9	−0.772817
$413$	7.65326e8	0.534590
$414$	0	0
$415$	1.14646e9	0.787389
$416$	−4.37991e8	−0.298290
$417$	0	0
$418$	−3.45410e7	−0.0231323
$419$	−8.23968e8	−0.547219	−0.273610	−	0.961841i	$-0.588218\pi$
$419$	−8.23968e8	−0.547219	−0.273610	+	0.961841i	$0.588218\pi$
$420$	0	0
$421$	−2.12046e8	−0.138498	−0.0692488	−	0.997599i	$-0.522060\pi$
$421$	−2.12046e8	−0.138498	−0.0692488	+	0.997599i	$0.522060\pi$
$422$	1.21467e8	0.0786801
$423$	0	0
$424$	2.38141e8	0.151724
$425$	5.23145e7	0.0330568
$426$	0	0
$427$	1.43336e8	0.0890956
$428$	1.92210e9	1.18501
$429$	0	0
$430$	2.64276e7	0.0160295
$431$	−7.44023e8	−0.447627	−0.223813	−	0.974632i	$-0.571851\pi$
$431$	−7.44023e8	−0.447627	−0.223813	+	0.974632i	$0.571851\pi$
$432$	0	0
$433$	−1.93573e9	−1.14588	−0.572938	−	0.819598i	$-0.694197\pi$
$433$	−1.93573e9	−1.14588	−0.572938	+	0.819598i	$0.694197\pi$
$434$	1.22711e8	0.0720557
$435$	0	0
$436$	2.03235e8	0.117435
$437$	1.29722e9	0.743581
$438$	0	0
$439$	−1.35485e9	−0.764300	−0.382150	−	0.924100i	$-0.624816\pi$
$439$	−1.35485e9	−0.764300	−0.382150	+	0.924100i	$0.624816\pi$
$440$	4.95367e7	0.0277232
$441$	0	0
$442$	−3.03531e7	−0.0167196
$443$	−1.05985e9	−0.579203	−0.289601	−	0.957147i	$-0.593523\pi$
$443$	−1.05985e9	−0.579203	−0.289601	+	0.957147i	$0.593523\pi$
$444$	0	0
$445$	−3.03303e7	−0.0163161
$446$	2.93053e8	0.156413
$447$	0	0
$448$	1.04263e9	0.547843
$449$	1.70272e9	0.887730	0.443865	−	0.896094i	$-0.353607\pi$
$449$	1.70272e9	0.887730	0.443865	+	0.896094i	$0.353607\pi$
$450$	0	0
$451$	−7.62309e8	−0.391303
$452$	−3.32021e9	−1.69114
$453$	0	0
$454$	−3.85414e8	−0.193300
$455$	−4.75660e8	−0.236732
$456$	0	0
$457$	−3.76431e9	−1.84493	−0.922463	−	0.386085i	$-0.873827\pi$
$457$	−3.76431e9	−1.84493	−0.922463	+	0.386085i	$0.873827\pi$
$458$	1.35071e8	0.0656949
$459$	0	0
$460$	−9.24185e8	−0.442697
$461$	−3.59084e9	−1.70704	−0.853519	−	0.521062i	$-0.825536\pi$
$461$	−3.59084e9	−1.70704	−0.853519	+	0.521062i	$0.825536\pi$
$462$	0	0
$463$	−2.45649e8	−0.115022	−0.0575111	−	0.998345i	$-0.518316\pi$
$463$	−2.45649e8	−0.115022	−0.0575111	+	0.998345i	$0.518316\pi$
$464$	3.25036e9	1.51049
$465$	0	0
$466$	4.19532e8	0.192050
$467$	4.04985e8	0.184005	0.0920025	−	0.995759i	$-0.470673\pi$
$467$	4.04985e8	0.184005	0.0920025	+	0.995759i	$0.470673\pi$
$468$	0	0
$469$	−1.58783e9	−0.710722
$470$	−7.20205e7	−0.0319973
$471$	0	0
$472$	−4.63771e8	−0.203005
$473$	−2.00360e8	−0.0870556
$474$	0	0
$475$	3.46401e8	0.148304
$476$	2.27687e8	0.0967641
$477$	0	0
$478$	−7.34559e7	−0.0307630
$479$	−4.18334e9	−1.73920	−0.869598	−	0.493760i	$-0.835622\pi$
$479$	−4.18334e9	−1.73920	−0.869598	+	0.493760i	$0.835622\pi$
$480$	0	0
$481$	−2.00890e9	−0.823097
$482$	−3.64513e8	−0.148268
$483$	0	0
$484$	2.27575e9	0.912361
$485$	−3.23998e8	−0.128957
$486$	0	0
$487$	3.25089e9	1.27541	0.637706	−	0.770280i	$-0.279883\pi$
$487$	3.25089e9	1.27541	0.637706	+	0.770280i	$0.279883\pi$
$488$	−8.68583e7	−0.0338331
$489$	0	0
$490$	8.55689e7	0.0328571
$491$	3.37360e9	1.28620	0.643100	−	0.765782i	$-0.277648\pi$
$491$	3.37360e9	1.28620	0.643100	+	0.765782i	$0.277648\pi$
$492$	0	0
$493$	6.90723e8	0.259621
$494$	−2.00984e8	−0.0750097
$495$	0	0
$496$	2.80165e9	1.03093
$497$	4.95753e8	0.181142
$498$	0	0
$499$	3.74951e9	1.35090	0.675449	−	0.737406i	$-0.263950\pi$
$499$	3.74951e9	1.35090	0.675449	+	0.737406i	$0.263950\pi$
$500$	−2.46789e8	−0.0882939
$501$	0	0
$502$	−3.45141e7	−0.0121768
$503$	3.09301e9	1.08366	0.541830	−	0.840488i	$-0.317732\pi$
$503$	3.09301e9	1.08366	0.541830	+	0.840488i	$0.317732\pi$
$504$	0	0
$505$	−4.61461e8	−0.159447
$506$	−9.11651e7	−0.0312825
$507$	0	0
$508$	2.42432e9	0.820476
$509$	−1.36862e9	−0.460015	−0.230007	−	0.973189i	$-0.573875\pi$
$509$	−1.36862e9	−0.460015	−0.230007	+	0.973189i	$0.573875\pi$
$510$	0	0
$511$	−2.29087e9	−0.759499
$512$	−1.63371e9	−0.537937
$513$	0	0
$514$	−4.21523e8	−0.136915
$515$	1.08525e9	0.350111
$516$	0	0
$517$	5.46020e8	0.173777
$518$	−1.96070e8	−0.0619808
$519$	0	0
$520$	2.88240e8	0.0898963
$521$	4.47414e9	1.38604	0.693022	−	0.720916i	$-0.256279\pi$
$521$	4.47414e9	1.38604	0.693022	+	0.720916i	$0.256279\pi$
$522$	0	0
$523$	−3.01993e9	−0.923085	−0.461542	−	0.887118i	$-0.652704\pi$
$523$	−3.01993e9	−0.923085	−0.461542	+	0.887118i	$0.652704\pi$
$524$	3.87936e9	1.17788
$525$	0	0
$526$	2.37963e8	0.0712952
$527$	5.95370e8	0.177194
$528$	0	0
$529$	1.89619e7	0.00556913
$530$	−1.17031e8	−0.0341458
$531$	0	0
$532$	1.50764e9	0.434116
$533$	−4.43565e9	−1.26885
$534$	0	0
$535$	−1.90148e9	−0.536850
$536$	9.62193e8	0.269889
$537$	0	0
$538$	3.47318e8	0.0961587
$539$	−6.48737e8	−0.178446
$540$	0	0
$541$	3.23946e9	0.879593	0.439796	−	0.898097i	$-0.355051\pi$
$541$	3.23946e9	0.879593	0.439796	+	0.898097i	$0.355051\pi$
$542$	−2.95859e8	−0.0798154
$543$	0	0
$544$	−2.07406e8	−0.0552365
$545$	−2.01054e8	−0.0532017
$546$	0	0
$547$	4.25742e9	1.11222	0.556111	−	0.831108i	$-0.312293\pi$
$547$	4.25742e9	1.11222	0.556111	+	0.831108i	$0.312293\pi$
$548$	−4.00355e9	−1.03923
$549$	0	0
$550$	−2.43442e7	−0.00623916
$551$	4.57364e9	1.16475
$552$	0	0
$553$	−3.38284e9	−0.850635
$554$	−1.82148e8	−0.0455135
$555$	0	0
$556$	−2.62291e9	−0.647175
$557$	5.65828e9	1.38737	0.693684	−	0.720280i	$-0.255987\pi$
$557$	5.65828e9	1.38737	0.693684	+	0.720280i	$0.255987\pi$
$558$	0	0
$559$	−1.16584e9	−0.282290
$560$	−1.05994e9	−0.255048
$561$	0	0
$562$	−1.47918e8	−0.0351515
$563$	−4.37511e9	−1.03326	−0.516630	−	0.856209i	$-0.672814\pi$
$563$	−4.37511e9	−1.03326	−0.516630	+	0.856209i	$0.672814\pi$
$564$	0	0
$565$	3.28458e9	0.766143
$566$	−2.79623e8	−0.0648210
$567$	0	0
$568$	−3.00416e8	−0.0687866
$569$	−1.67226e9	−0.380550	−0.190275	−	0.981731i	$-0.560938\pi$
$569$	−1.67226e9	−0.380550	−0.190275	+	0.981731i	$0.560938\pi$
$570$	0	0
$571$	6.39802e8	0.143820	0.0719100	−	0.997411i	$-0.477091\pi$
$571$	6.39802e8	0.143820	0.0719100	+	0.997411i	$0.477091\pi$
$572$	−1.08558e9	−0.242535
$573$	0	0
$574$	−4.32922e8	−0.0955472
$575$	9.14268e8	0.200556
$576$	0	0
$577$	4.87101e9	1.05561	0.527805	−	0.849366i	$-0.323015\pi$
$577$	4.87101e9	1.05561	0.527805	+	0.849366i	$0.323015\pi$
$578$	5.11764e8	0.110236
$579$	0	0
$580$	−3.25843e9	−0.693442
$581$	−4.93615e9	−1.04417
$582$	0	0
$583$	8.87268e8	0.185445
$584$	1.38822e9	0.288412
$585$	0	0
$586$	5.28760e8	0.108547
$587$	2.32406e9	0.474258	0.237129	−	0.971478i	$-0.423794\pi$
$587$	2.32406e9	0.474258	0.237129	+	0.971478i	$0.423794\pi$
$588$	0	0
$589$	3.94226e9	0.794953
$590$	2.27914e8	0.0456867
$591$	0	0
$592$	−4.47654e9	−0.886782
$593$	−4.27080e9	−0.841043	−0.420522	−	0.907283i	$-0.638153\pi$
$593$	−4.27080e9	−0.841043	−0.420522	+	0.907283i	$0.638153\pi$
$594$	0	0
$595$	−2.25244e8	−0.0438373
$596$	3.72254e9	0.720240
$597$	0	0
$598$	−5.30463e8	−0.101438
$599$	−3.50335e9	−0.666023	−0.333012	−	0.942923i	$-0.608065\pi$
$599$	−3.50335e9	−0.666023	−0.333012	+	0.942923i	$0.608065\pi$
$600$	0	0
$601$	2.98866e9	0.561585	0.280792	−	0.959769i	$-0.409403\pi$
$601$	2.98866e9	0.561585	0.280792	+	0.959769i	$0.409403\pi$
$602$	−1.13786e8	−0.0212570
$603$	0	0
$604$	−5.81240e9	−1.07331
$605$	−2.25133e9	−0.413329
$606$	0	0
$607$	−7.31901e9	−1.32829	−0.664143	−	0.747605i	$-0.731203\pi$
$607$	−7.31901e9	−1.32829	−0.664143	+	0.747605i	$0.731203\pi$
$608$	−1.37335e9	−0.247809
$609$	0	0
$610$	4.26854e7	0.00761421
$611$	3.17713e9	0.563496
$612$	0	0
$613$	−8.32799e9	−1.46026	−0.730128	−	0.683311i	$-0.760539\pi$
$613$	−8.32799e9	−1.46026	−0.730128	+	0.683311i	$0.760539\pi$
$614$	−2.02566e8	−0.0353165
$615$	0	0
$616$	−2.13284e8	−0.0367643
$617$	2.88456e9	0.494403	0.247201	−	0.968964i	$-0.420489\pi$
$617$	2.88456e9	0.494403	0.247201	+	0.968964i	$0.420489\pi$
$618$	0	0
$619$	−2.27213e9	−0.385049	−0.192525	−	0.981292i	$-0.561668\pi$
$619$	−2.27213e9	−0.385049	−0.192525	+	0.981292i	$0.561668\pi$
$620$	−2.80861e9	−0.473282
$621$	0	0
$622$	−2.16909e8	−0.0361420
$623$	1.30589e8	0.0216371
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	6.69459e8	0.109072
$627$	0	0
$628$	−2.43865e9	−0.392907
$629$	−9.51296e8	−0.152419
$630$	0	0
$631$	−2.90856e7	−0.00460867	−0.00230434	−	0.999997i	$-0.500733\pi$
$631$	−2.90856e7	−0.00460867	−0.00230434	+	0.999997i	$0.500733\pi$
$632$	2.04993e9	0.323020
$633$	0	0
$634$	6.06841e8	0.0945719
$635$	−2.39830e9	−0.371702
$636$	0	0
$637$	−3.77481e9	−0.578638
$638$	−3.21423e8	−0.0490010
$639$	0	0
$640$	1.30165e9	0.196274
$641$	3.12634e9	0.468849	0.234424	−	0.972134i	$-0.424680\pi$
$641$	3.12634e9	0.468849	0.234424	+	0.972134i	$0.424680\pi$
$642$	0	0
$643$	2.48120e9	0.368064	0.184032	−	0.982920i	$-0.441085\pi$
$643$	2.48120e9	0.368064	0.184032	+	0.982920i	$0.441085\pi$
$644$	3.97915e9	0.587070
$645$	0	0
$646$	−9.51740e7	−0.0138901
$647$	−3.10080e9	−0.450100	−0.225050	−	0.974347i	$-0.572254\pi$
$647$	−3.10080e9	−0.450100	−0.225050	+	0.974347i	$0.572254\pi$
$648$	0	0
$649$	−1.72792e9	−0.248123
$650$	−1.41652e8	−0.0202314
$651$	0	0
$652$	−2.97726e9	−0.420678
$653$	−9.64911e8	−0.135610	−0.0678049	−	0.997699i	$-0.521600\pi$
$653$	−9.64911e8	−0.135610	−0.0678049	+	0.997699i	$0.521600\pi$
$654$	0	0
$655$	−3.83773e9	−0.533617
$656$	−9.88419e9	−1.36703
$657$	0	0
$658$	3.10089e8	0.0424323
$659$	7.70039e9	1.04813	0.524063	−	0.851679i	$-0.324416\pi$
$659$	7.70039e9	1.04813	0.524063	+	0.851679i	$0.324416\pi$
$660$	0	0
$661$	1.30650e10	1.75956	0.879779	−	0.475382i	$-0.157690\pi$
$661$	1.30650e10	1.75956	0.879779	+	0.475382i	$0.157690\pi$
$662$	1.12307e8	0.0150454
$663$	0	0
$664$	2.99120e9	0.396513
$665$	−1.49146e9	−0.196669
$666$	0	0
$667$	1.20713e10	1.57513
$668$	2.76498e9	0.358900
$669$	0	0
$670$	−4.72857e8	−0.0607391
$671$	−3.23617e8	−0.0413526
$672$	0	0
$673$	1.31771e10	1.66636	0.833179	−	0.553003i	$-0.186518\pi$
$673$	1.31771e10	1.66636	0.833179	+	0.553003i	$0.186518\pi$
$674$	3.89942e8	0.0490558
$675$	0	0
$676$	1.61201e9	0.200703
$677$	3.18151e9	0.394069	0.197035	−	0.980397i	$-0.436869\pi$
$677$	3.18151e9	0.394069	0.197035	+	0.980397i	$0.436869\pi$
$678$	0	0
$679$	1.39500e9	0.171013
$680$	1.36493e8	0.0166467
$681$	0	0
$682$	−2.77051e8	−0.0334438
$683$	3.33139e8	0.0400086	0.0200043	−	0.999800i	$-0.493632\pi$
$683$	3.33139e8	0.0400086	0.0200043	+	0.999800i	$0.493632\pi$
$684$	0	0
$685$	3.96059e9	0.470807
$686$	−9.36731e8	−0.110785
$687$	0	0
$688$	−2.59789e9	−0.304132
$689$	5.16275e9	0.601331
$690$	0	0
$691$	−4.45248e9	−0.513368	−0.256684	−	0.966495i	$-0.582630\pi$
$691$	−4.45248e9	−0.513368	−0.256684	+	0.966495i	$0.582630\pi$
$692$	−1.21594e10	−1.39489
$693$	0	0
$694$	4.04339e8	0.0459185
$695$	2.59476e9	0.293191
$696$	0	0
$697$	−2.10046e9	−0.234963
$698$	6.71127e8	0.0746983
$699$	0	0
$700$	1.06257e9	0.117088
$701$	1.91380e9	0.209838	0.104919	−	0.994481i	$-0.466542\pi$
$701$	1.91380e9	0.209838	0.104919	+	0.994481i	$0.466542\pi$
$702$	0	0
$703$	−6.29902e9	−0.683801
$704$	−2.35400e9	−0.254274
$705$	0	0
$706$	−5.30785e8	−0.0567679
$707$	1.98686e9	0.211445
$708$	0	0
$709$	1.38704e9	0.146160	0.0730799	−	0.997326i	$-0.476717\pi$
$709$	1.38704e9	0.146160	0.0730799	+	0.997326i	$0.476717\pi$
$710$	1.47636e8	0.0154806
$711$	0	0
$712$	−7.91344e7	−0.00821647
$713$	1.04049e10	1.07504
$714$	0	0
$715$	1.07393e9	0.109876
$716$	1.06995e10	1.08935
$717$	0	0
$718$	−3.18970e8	−0.0321599
$719$	−9.42958e9	−0.946109	−0.473055	−	0.881033i	$-0.656849\pi$
$719$	−9.42958e9	−0.946109	−0.473055	+	0.881033i	$0.656849\pi$
$720$	0	0
$721$	−4.67264e9	−0.464290
$722$	5.15928e8	0.0510163
$723$	0	0
$724$	1.21965e10	1.19440
$725$	3.22346e9	0.314152
$726$	0	0
$727$	−2.99387e9	−0.288976	−0.144488	−	0.989507i	$-0.546154\pi$
$727$	−2.99387e9	−0.288976	−0.144488	+	0.989507i	$0.546154\pi$
$728$	−1.24104e9	−0.119213
$729$	0	0
$730$	−6.82222e8	−0.0649076
$731$	−5.52070e8	−0.0522737
$732$	0	0
$733$	1.11711e10	1.04768	0.523842	−	0.851815i	$-0.324498\pi$
$733$	1.11711e10	1.04768	0.523842	+	0.851815i	$0.324498\pi$
$734$	−1.87053e9	−0.174594
$735$	0	0
$736$	−3.62471e9	−0.335121
$737$	3.58495e9	0.329873
$738$	0	0
$739$	−6.96449e9	−0.634796	−0.317398	−	0.948292i	$-0.602809\pi$
$739$	−6.96449e9	−0.634796	−0.317398	+	0.948292i	$0.602809\pi$
$740$	4.48765e9	0.407107
$741$	0	0
$742$	5.03887e8	0.0452814
$743$	−1.27161e9	−0.113734	−0.0568672	−	0.998382i	$-0.518111\pi$
$743$	−1.27161e9	−0.113734	−0.0568672	+	0.998382i	$0.518111\pi$
$744$	0	0
$745$	−3.68259e9	−0.326292
$746$	−8.06666e8	−0.0711390
$747$	0	0
$748$	−5.14063e8	−0.0449119
$749$	8.18695e9	0.711927
$750$	0	0
$751$	−1.70771e10	−1.47121	−0.735603	−	0.677413i	$-0.763101\pi$
$751$	−1.70771e10	−1.47121	−0.735603	+	0.677413i	$0.763101\pi$
$752$	7.07976e9	0.607094
$753$	0	0
$754$	−1.87027e9	−0.158893
$755$	5.75002e9	0.486245
$756$	0	0
$757$	1.37933e10	1.15567	0.577833	−	0.816155i	$-0.303899\pi$
$757$	1.37933e10	1.15567	0.577833	+	0.816155i	$0.303899\pi$
$758$	−1.09584e8	−0.00913917
$759$	0	0
$760$	9.03791e8	0.0746828
$761$	−1.71364e10	−1.40953	−0.704765	−	0.709441i	$-0.748947\pi$
$761$	−1.71364e10	−1.40953	−0.704765	+	0.709441i	$0.748947\pi$
$762$	0	0
$763$	8.65654e8	0.0705518
$764$	−1.85070e10	−1.50145
$765$	0	0
$766$	−9.69780e8	−0.0779602
$767$	−1.00543e10	−0.804575
$768$	0	0
$769$	−4.41529e9	−0.350120	−0.175060	−	0.984558i	$-0.556012\pi$
$769$	−4.41529e9	−0.350120	−0.175060	+	0.984558i	$0.556012\pi$
$770$	1.04816e8	0.00827387
$771$	0	0
$772$	1.07816e10	0.843381
$773$	5.87300e8	0.0457332	0.0228666	−	0.999739i	$-0.492721\pi$
$773$	5.87300e8	0.0457332	0.0228666	+	0.999739i	$0.492721\pi$
$774$	0	0
$775$	2.77847e9	0.214412
$776$	−8.45338e8	−0.0649403
$777$	0	0
$778$	1.83146e9	0.139434
$779$	−1.39082e10	−1.05412
$780$	0	0
$781$	−1.11929e9	−0.0840746
$782$	−2.51195e8	−0.0187840
$783$	0	0
$784$	−8.41160e9	−0.623408
$785$	2.41248e9	0.178000
$786$	0	0
$787$	−2.30069e10	−1.68247	−0.841235	−	0.540670i	$-0.818171\pi$
$787$	−2.30069e10	−1.68247	−0.841235	+	0.540670i	$0.818171\pi$
$788$	1.82967e10	1.33208
$789$	0	0
$790$	−1.00741e9	−0.0726962
$791$	−1.41420e10	−1.01600
$792$	0	0
$793$	−1.88303e9	−0.134092
$794$	2.14323e8	0.0151949
$795$	0	0
$796$	−1.60308e9	−0.112658
$797$	−2.49526e10	−1.74587	−0.872935	−	0.487836i	$-0.837787\pi$
$797$	−2.49526e10	−1.74587	−0.872935	+	0.487836i	$0.837787\pi$
$798$	0	0
$799$	1.50450e9	0.104346
$800$	−9.67921e8	−0.0668383
$801$	0	0
$802$	2.97536e9	0.203671
$803$	5.17223e9	0.352512
$804$	0	0
$805$	−3.93645e9	−0.265962
$806$	−1.61208e9	−0.108446
$807$	0	0
$808$	−1.20399e9	−0.0802942
$809$	1.29758e10	0.861616	0.430808	−	0.902443i	$-0.358229\pi$
$809$	1.29758e10	0.861616	0.430808	+	0.902443i	$0.358229\pi$
$810$	0	0
$811$	−2.04887e10	−1.34878	−0.674391	−	0.738374i	$-0.735594\pi$
$811$	−2.04887e10	−1.34878	−0.674391	+	0.738374i	$0.735594\pi$
$812$	1.40294e10	0.919587
$813$	0	0
$814$	4.42679e8	0.0287676
$815$	2.94531e9	0.190581
$816$	0	0
$817$	−3.65554e9	−0.234517
$818$	1.05748e9	0.0675514
$819$	0	0
$820$	9.90872e9	0.627580
$821$	7.97556e9	0.502991	0.251495	−	0.967858i	$-0.419078\pi$
$821$	7.97556e9	0.502991	0.251495	+	0.967858i	$0.419078\pi$
$822$	0	0
$823$	5.64462e9	0.352968	0.176484	−	0.984304i	$-0.443528\pi$
$823$	5.64462e9	0.352968	0.176484	+	0.984304i	$0.443528\pi$
$824$	2.83152e9	0.176309
$825$	0	0
$826$	−9.81302e8	−0.0605860
$827$	−1.22946e10	−0.755866	−0.377933	−	0.925833i	$-0.623365\pi$
$827$	−1.22946e10	−0.755866	−0.377933	+	0.925833i	$0.623365\pi$
$828$	0	0
$829$	−1.72844e10	−1.05369	−0.526845	−	0.849962i	$-0.676625\pi$
$829$	−1.72844e10	−1.05369	−0.526845	+	0.849962i	$0.676625\pi$
$830$	−1.46999e9	−0.0892361
$831$	0	0
$832$	−1.36973e10	−0.824521
$833$	−1.78752e9	−0.107150
$834$	0	0
$835$	−2.73530e9	−0.162593
$836$	−3.40388e9	−0.201489
$837$	0	0
$838$	1.05649e9	0.0620173
$839$	4.87580e9	0.285023	0.142511	−	0.989793i	$-0.454482\pi$
$839$	4.87580e9	0.285023	0.142511	+	0.989793i	$0.454482\pi$
$840$	0	0
$841$	2.53104e10	1.46728
$842$	2.71885e8	0.0156962
$843$	0	0
$844$	1.19701e10	0.685329
$845$	−1.59471e9	−0.0909249
$846$	0	0
$847$	9.69328e9	0.548124
$848$	1.15044e10	0.647857
$849$	0	0
$850$	−6.70777e7	−0.00374638
$851$	−1.66252e10	−0.924727
$852$	0	0
$853$	−1.67542e10	−0.924277	−0.462138	−	0.886808i	$-0.652918\pi$
$853$	−1.67542e10	−0.924277	−0.462138	+	0.886808i	$0.652918\pi$
$854$	−1.83785e8	−0.0100974
$855$	0	0
$856$	−4.96112e9	−0.270347
$857$	1.45198e10	0.788001	0.394001	−	0.919110i	$-0.371091\pi$
$857$	1.45198e10	0.788001	0.394001	+	0.919110i	$0.371091\pi$
$858$	0	0
$859$	2.44108e10	1.31403	0.657016	−	0.753877i	$-0.271819\pi$
$859$	2.44108e10	1.31403	0.657016	+	0.753877i	$0.271819\pi$
$860$	2.60434e9	0.139622
$861$	0	0
$862$	9.53988e8	0.0507303
$863$	2.03914e10	1.07996	0.539981	−	0.841677i	$-0.318431\pi$
$863$	2.03914e10	1.07996	0.539981	+	0.841677i	$0.318431\pi$
$864$	0	0
$865$	1.20289e10	0.631931
$866$	2.48200e9	0.129864
$867$	0	0
$868$	1.20927e10	0.627629
$869$	7.63764e9	0.394812
$870$	0	0
$871$	2.08597e10	1.06966
$872$	−5.24568e8	−0.0267913
$873$	0	0
$874$	−1.66330e9	−0.0842713
$875$	−1.05117e9	−0.0530448
$876$	0	0
$877$	−3.75418e10	−1.87939	−0.939693	−	0.342018i	$-0.888890\pi$
$877$	−3.75418e10	−1.87939	−0.939693	+	0.342018i	$0.888890\pi$
$878$	1.73719e9	0.0866194
$879$	0	0
$880$	2.39308e9	0.118377
$881$	−1.05852e10	−0.521533	−0.260767	−	0.965402i	$-0.583975\pi$
$881$	−1.05852e10	−0.521533	−0.260767	+	0.965402i	$0.583975\pi$
$882$	0	0
$883$	3.45129e10	1.68702	0.843509	−	0.537116i	$-0.180486\pi$
$883$	3.45129e10	1.68702	0.843509	+	0.537116i	$0.180486\pi$
$884$	−2.99118e9	−0.145633
$885$	0	0
$886$	1.35894e9	0.0656420
$887$	−1.68685e10	−0.811602	−0.405801	−	0.913961i	$-0.633007\pi$
$887$	−1.68685e10	−0.811602	−0.405801	+	0.913961i	$0.633007\pi$
$888$	0	0
$889$	1.03261e10	0.492922
$890$	3.88896e7	0.00184913
$891$	0	0
$892$	2.88792e10	1.36241
$893$	9.96206e9	0.468133
$894$	0	0
$895$	−1.05847e10	−0.493512
$896$	−5.60434e9	−0.260283
$897$	0	0
$898$	−2.18323e9	−0.100608
$899$	3.66849e10	1.68395
$900$	0	0
$901$	2.44477e9	0.111353
$902$	9.77434e8	0.0443470
$903$	0	0
$904$	8.56974e9	0.385814
$905$	−1.20656e10	−0.541102
$906$	0	0
$907$	−1.14713e10	−0.510492	−0.255246	−	0.966876i	$-0.582156\pi$
$907$	−1.14713e10	−0.510492	−0.255246	+	0.966876i	$0.582156\pi$
$908$	−3.79810e10	−1.68370
$909$	0	0
$910$	6.09892e8	0.0268292
$911$	−1.62306e10	−0.711248	−0.355624	−	0.934629i	$-0.615732\pi$
$911$	−1.62306e10	−0.711248	−0.355624	+	0.934629i	$0.615732\pi$
$912$	0	0
$913$	1.11446e10	0.484639
$914$	4.82661e9	0.209089
$915$	0	0
$916$	1.33107e10	0.572224
$917$	1.65236e10	0.707641
$918$	0	0
$919$	−4.36259e10	−1.85413	−0.927065	−	0.374900i	$-0.877677\pi$
$919$	−4.36259e10	−1.85413	−0.927065	+	0.374900i	$0.877677\pi$
$920$	2.38540e9	0.100996
$921$	0	0
$922$	4.60419e9	0.193462
$923$	−6.51283e9	−0.272624
$924$	0	0
$925$	−4.43950e9	−0.184433
$926$	3.14972e8	0.0130357
$927$	0	0
$928$	−1.27797e10	−0.524934
$929$	−1.67801e10	−0.686656	−0.343328	−	0.939216i	$-0.611554\pi$
$929$	−1.67801e10	−0.686656	−0.343328	+	0.939216i	$0.611554\pi$
$930$	0	0
$931$	−1.18361e10	−0.480713
$932$	4.13432e10	1.67282
$933$	0	0
$934$	−5.19273e8	−0.0208536
$935$	5.08547e8	0.0203465
$936$	0	0
$937$	1.94934e10	0.774105	0.387053	−	0.922058i	$-0.373493\pi$
$937$	1.94934e10	0.774105	0.387053	+	0.922058i	$0.373493\pi$
$938$	2.03592e9	0.0805474
$939$	0	0
$940$	−7.09733e9	−0.278707
$941$	4.91523e10	1.92300	0.961502	−	0.274797i	$-0.0886108\pi$
$941$	4.91523e10	1.92300	0.961502	+	0.274797i	$0.0886108\pi$
$942$	0	0
$943$	−3.67084e10	−1.42552
$944$	−2.24045e10	−0.866827
$945$	0	0
$946$	2.56902e8	0.00986617
$947$	−3.01570e10	−1.15389	−0.576944	−	0.816784i	$-0.695755\pi$
$947$	−3.01570e10	−1.15389	−0.576944	+	0.816784i	$0.695755\pi$
$948$	0	0
$949$	3.00957e10	1.14307
$950$	−4.44157e8	−0.0168075
$951$	0	0
$952$	−5.87681e8	−0.0220756
$953$	1.86916e10	0.699554	0.349777	−	0.936833i	$-0.386257\pi$
$953$	1.86916e10	0.699554	0.349777	+	0.936833i	$0.386257\pi$
$954$	0	0
$955$	1.83084e10	0.680204
$956$	−7.23879e9	−0.267956
$957$	0	0
$958$	5.36388e9	0.197106
$959$	−1.70526e10	−0.624346
$960$	0	0
$961$	4.10799e9	0.149313
$962$	2.57582e9	0.0932830
$963$	0	0
$964$	−3.59213e10	−1.29146
$965$	−1.06659e10	−0.382079
$966$	0	0
$967$	−7.61328e9	−0.270757	−0.135378	−	0.990794i	$-0.543225\pi$
$967$	−7.61328e9	−0.270757	−0.135378	+	0.990794i	$0.543225\pi$
$968$	−5.87392e9	−0.208144
$969$	0	0
$970$	4.15431e8	0.0146150
$971$	4.83242e10	1.69394	0.846969	−	0.531643i	$-0.178425\pi$
$971$	4.83242e10	1.69394	0.846969	+	0.531643i	$0.178425\pi$
$972$	0	0
$973$	−1.11720e10	−0.388807
$974$	−4.16829e9	−0.144545
$975$	0	0
$976$	−4.19606e9	−0.144467
$977$	−1.42491e9	−0.0488829	−0.0244414	−	0.999701i	$-0.507781\pi$
$977$	−1.42491e9	−0.0488829	−0.0244414	+	0.999701i	$0.507781\pi$
$978$	0	0
$979$	−2.94840e8	−0.0100426
$980$	8.43248e9	0.286196
$981$	0	0
$982$	−4.32564e9	−0.145767
$983$	−4.85675e10	−1.63083	−0.815415	−	0.578877i	$-0.803491\pi$
$983$	−4.85675e10	−1.63083	−0.815415	+	0.578877i	$0.803491\pi$
$984$	0	0
$985$	−1.81003e10	−0.603475
$986$	−8.85647e8	−0.0294233
$987$	0	0
$988$	−1.98062e10	−0.653358
$989$	−9.64818e9	−0.317145
$990$	0	0
$991$	−1.69341e10	−0.552719	−0.276360	−	0.961054i	$-0.589128\pi$
$991$	−1.69341e10	−0.552719	−0.276360	+	0.961054i	$0.589128\pi$
$992$	−1.10155e10	−0.358273
$993$	0	0
$994$	−6.35656e8	−0.0205291
$995$	1.58588e9	0.0510375
$996$	0	0
$997$	2.36849e10	0.756900	0.378450	−	0.925622i	$-0.376457\pi$
$997$	2.36849e10	0.756900	0.378450	+	0.925622i	$0.376457\pi$
$998$	−4.80763e9	−0.153100
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.8.a.h.1.2		2
3.2	odd	2		5.8.a.b.1.1	✓	2
5.2	odd	4		225.8.b.m.199.2		4
5.3	odd	4		225.8.b.m.199.3		4
5.4	even	2		225.8.a.w.1.1		2
12.11	even	2		80.8.a.g.1.1		2
15.2	even	4		25.8.b.c.24.3		4
15.8	even	4		25.8.b.c.24.2		4
15.14	odd	2		25.8.a.b.1.2		2
21.20	even	2		245.8.a.c.1.1		2
24.5	odd	2		320.8.a.l.1.1		2
24.11	even	2		320.8.a.u.1.2		2
33.32	even	2		605.8.a.d.1.2		2
60.23	odd	4		400.8.c.m.49.1		4
60.47	odd	4		400.8.c.m.49.4		4
60.59	even	2		400.8.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.8.a.b.1.1	✓	2	3.2	odd	2
25.8.a.b.1.2		2	15.14	odd	2
25.8.b.c.24.2		4	15.8	even	4
25.8.b.c.24.3		4	15.2	even	4
45.8.a.h.1.2		2	1.1	even	1	trivial
80.8.a.g.1.1		2	12.11	even	2
225.8.a.w.1.1		2	5.4	even	2
225.8.b.m.199.2		4	5.2	odd	4
225.8.b.m.199.3		4	5.3	odd	4
245.8.a.c.1.1		2	21.20	even	2
320.8.a.l.1.1		2	24.5	odd	2
320.8.a.u.1.2		2	24.11	even	2
400.8.a.bb.1.2		2	60.59	even	2
400.8.c.m.49.1		4	60.23	odd	4
400.8.c.m.49.4		4	60.47	odd	4
605.8.a.d.1.2		2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	45.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28220 q^{2} -126.356 q^{4} +125.000 q^{5} -538.197 q^{7} +326.136 q^{8} -160.275 q^{10} +1215.12 q^{11} +7070.42 q^{13} +690.077 q^{14} +15755.4 q^{16} +3348.13 q^{17} +22169.7 q^{19} -15794.5 q^{20} -1558.03 q^{22} +58513.1 q^{23} +15625.0 q^{25} -9065.71 q^{26} +68004.4 q^{28} +206301. q^{29} +177822. q^{31} -61947.0 q^{32} -4292.98 q^{34} -67274.6 q^{35} -284128. q^{37} -28426.0 q^{38} +40767.0 q^{40} -627353. q^{41} -164889. q^{43} -153538. q^{44} -75025.7 q^{46} +449355. q^{47} -533887. q^{49} -20034.4 q^{50} -893390. q^{52} +730190. q^{53} +151890. q^{55} -175525. q^{56} -264520. q^{58} -1.42202e6 q^{59} -266326. q^{61} -228004. q^{62} -1.93726e6 q^{64} +883803. q^{65} +2.95028e6 q^{67} -423056. q^{68} +86259.6 q^{70} -921138. q^{71} +4.25657e6 q^{73} +364309. q^{74} -2.80127e6 q^{76} -653973. q^{77} +6.28551e6 q^{79} +1.96942e6 q^{80} +804393. q^{82} +9.17165e6 q^{83} +418516. q^{85} +211421. q^{86} +396294. q^{88} -242643. q^{89} -3.80528e6 q^{91} -7.39348e6 q^{92} -576164. q^{94} +2.77121e6 q^{95} -2.59198e6 q^{97} +684551. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{2} + 96 q^{4} + 250 q^{5} - 100 q^{7} - 1440 q^{8} - 2500 q^{10} - 4544 q^{11} + 3540 q^{13} - 7512 q^{14} + 20352 q^{16} + 27340 q^{17} + 38760 q^{19} + 12000 q^{20} + 106240 q^{22} + 124140 q^{23}+ \cdots + 12505340 q^{98}+O(q^{100}) \) 2 * q - 20 * q^2 + 96 * q^4 + 250 * q^5 - 100 * q^7 - 1440 * q^8 - 2500 * q^10 - 4544 * q^11 + 3540 * q^13 - 7512 * q^14 + 20352 * q^16 + 27340 * q^17 + 38760 * q^19 + 12000 * q^20 + 106240 * q^22 + 124140 * q^23 + 31250 * q^25 + 57016 * q^26 + 165440 * q^28 + 72260 * q^29 + 306824 * q^31 + 78080 * q^32 - 453368 * q^34 - 12500 * q^35 - 123020 * q^37 - 338960 * q^38 - 180000 * q^40 - 264364 * q^41 + 423300 * q^43 - 1434112 * q^44 - 1303416 * q^46 + 105460 * q^47 - 1165414 * q^49 - 312500 * q^50 - 1678400 * q^52 + 2391580 * q^53 - 568000 * q^55 - 949440 * q^56 + 2244440 * q^58 + 1120120 * q^59 + 2257044 * q^61 - 2642640 * q^62 - 5146624 * q^64 + 442500 * q^65 + 4516460 * q^67 + 4911680 * q^68 - 939000 * q^70 - 621784 * q^71 + 4569060 * q^73 - 2651272 * q^74 + 887680 * q^76 - 3177600 * q^77 + 4333040 * q^79 + 2544000 * q^80 - 5989960 * q^82 + 9793020 * q^83 + 3417500 * q^85 - 10798184 * q^86 + 10567680 * q^88 - 6025620 * q^89 - 5352296 * q^91 + 7199040 * q^92 + 5860792 * q^94 + 4845000 * q^95 + 4609540 * q^97 + 12505340 * q^98

Introduction

L-functions

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Database

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