LMFDB - Embedded newform 5.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5,6,Mod(1,5)] 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("5.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	5.a (trivial)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	yes
Analytic conductor:	$0.801919099065$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} -4.00000 q^{3} -28.0000 q^{4} +25.0000 q^{5} -8.00000 q^{6} +192.000 q^{7} -120.000 q^{8} -227.000 q^{9} +50.0000 q^{10} -148.000 q^{11} +112.000 q^{12} +286.000 q^{13} +384.000 q^{14} -100.000 q^{15} +656.000 q^{16} -1678.00 q^{17} -454.000 q^{18} +1060.00 q^{19} -700.000 q^{20} -768.000 q^{21} -296.000 q^{22} +2976.00 q^{23} +480.000 q^{24} +625.000 q^{25} +572.000 q^{26} +1880.00 q^{27} -5376.00 q^{28} -3410.00 q^{29} -200.000 q^{30} -2448.00 q^{31} +5152.00 q^{32} +592.000 q^{33} -3356.00 q^{34} +4800.00 q^{35} +6356.00 q^{36} +182.000 q^{37} +2120.00 q^{38} -1144.00 q^{39} -3000.00 q^{40} -9398.00 q^{41} -1536.00 q^{42} -1244.00 q^{43} +4144.00 q^{44} -5675.00 q^{45} +5952.00 q^{46} -12088.0 q^{47} -2624.00 q^{48} +20057.0 q^{49} +1250.00 q^{50} +6712.00 q^{51} -8008.00 q^{52} +23846.0 q^{53} +3760.00 q^{54} -3700.00 q^{55} -23040.0 q^{56} -4240.00 q^{57} -6820.00 q^{58} -20020.0 q^{59} +2800.00 q^{60} +32302.0 q^{61} -4896.00 q^{62} -43584.0 q^{63} -10688.0 q^{64} +7150.00 q^{65} +1184.00 q^{66} +60972.0 q^{67} +46984.0 q^{68} -11904.0 q^{69} +9600.00 q^{70} -32648.0 q^{71} +27240.0 q^{72} -38774.0 q^{73} +364.000 q^{74} -2500.00 q^{75} -29680.0 q^{76} -28416.0 q^{77} -2288.00 q^{78} -33360.0 q^{79} +16400.0 q^{80} +47641.0 q^{81} -18796.0 q^{82} +16716.0 q^{83} +21504.0 q^{84} -41950.0 q^{85} -2488.00 q^{86} +13640.0 q^{87} +17760.0 q^{88} +101370. q^{89} -11350.0 q^{90} +54912.0 q^{91} -83328.0 q^{92} +9792.00 q^{93} -24176.0 q^{94} +26500.0 q^{95} -20608.0 q^{96} -119038. q^{97} +40114.0 q^{98} +33596.0 q^{99} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	−4.00000	−0.256600	−0.128300	−	0.991735i	$-0.540952\pi$
$3$	−4.00000	−0.256600	−0.128300	+	0.991735i	$0.540952\pi$
$4$	−28.0000	−0.875000
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	−8.00000	−0.0907218
$7$	192.000	1.48100	0.740502	−	0.672054i	$-0.234588\pi$
$7$	192.000	1.48100	0.740502	+	0.672054i	$0.234588\pi$
$8$	−120.000	−0.662913
$9$	−227.000	−0.934156
$10$	50.0000	0.158114
$11$	−148.000	−0.368791	−0.184395	−	0.982852i	$-0.559033\pi$
$11$	−148.000	−0.368791	−0.184395	+	0.982852i	$0.559033\pi$
$12$	112.000	0.224525
$13$	286.000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	286.000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	384.000	0.523614
$15$	−100.000	−0.114755
$16$	656.000	0.640625
$17$	−1678.00	−1.40822	−0.704109	−	0.710092i	$-0.748653\pi$
$17$	−1678.00	−1.40822	−0.704109	+	0.710092i	$0.748653\pi$
$18$	−454.000	−0.330274
$19$	1060.00	0.673631	0.336815	−	0.941571i	$-0.390650\pi$
$19$	1060.00	0.673631	0.336815	+	0.941571i	$0.390650\pi$
$20$	−700.000	−0.391312
$21$	−768.000	−0.380026
$22$	−296.000	−0.130387
$23$	2976.00	1.17304	0.586521	−	0.809934i	$-0.300497\pi$
$23$	2976.00	1.17304	0.586521	+	0.809934i	$0.300497\pi$
$24$	480.000	0.170103
$25$	625.000	0.200000
$26$	572.000	0.165944
$27$	1880.00	0.496305
$28$	−5376.00	−1.29588
$29$	−3410.00	−0.752938	−0.376469	−	0.926429i	$-0.622862\pi$
$29$	−3410.00	−0.752938	−0.376469	+	0.926429i	$0.622862\pi$
$30$	−200.000	−0.0405720
$31$	−2448.00	−0.457517	−0.228758	−	0.973483i	$-0.573467\pi$
$31$	−2448.00	−0.457517	−0.228758	+	0.973483i	$0.573467\pi$
$32$	5152.00	0.889408
$33$	592.000	0.0946317
$34$	−3356.00	−0.497880
$35$	4800.00	0.662325
$36$	6356.00	0.817387
$37$	182.000	0.0218558	0.0109279	−	0.999940i	$-0.496521\pi$
$37$	182.000	0.0218558	0.0109279	+	0.999940i	$0.496521\pi$
$38$	2120.00	0.238164
$39$	−1144.00	−0.120438
$40$	−3000.00	−0.296464
$41$	−9398.00	−0.873124	−0.436562	−	0.899674i	$-0.643804\pi$
$41$	−9398.00	−0.873124	−0.436562	+	0.899674i	$0.643804\pi$
$42$	−1536.00	−0.134359
$43$	−1244.00	−0.102600	−0.0513002	−	0.998683i	$-0.516337\pi$
$43$	−1244.00	−0.102600	−0.0513002	+	0.998683i	$0.516337\pi$
$44$	4144.00	0.322692
$45$	−5675.00	−0.417767
$46$	5952.00	0.414733
$47$	−12088.0	−0.798196	−0.399098	−	0.916908i	$-0.630677\pi$
$47$	−12088.0	−0.798196	−0.399098	+	0.916908i	$0.630677\pi$
$48$	−2624.00	−0.164384
$49$	20057.0	1.19337
$50$	1250.00	0.0707107
$51$	6712.00	0.361349
$52$	−8008.00	−0.410691
$53$	23846.0	1.16607	0.583037	−	0.812446i	$-0.301864\pi$
$53$	23846.0	1.16607	0.583037	+	0.812446i	$0.301864\pi$
$54$	3760.00	0.175470
$55$	−3700.00	−0.164928
$56$	−23040.0	−0.981776
$57$	−4240.00	−0.172854
$58$	−6820.00	−0.266204
$59$	−20020.0	−0.748745	−0.374373	−	0.927278i	$-0.622142\pi$
$59$	−20020.0	−0.748745	−0.374373	+	0.927278i	$0.622142\pi$
$60$	2800.00	0.100411
$61$	32302.0	1.11149	0.555744	−	0.831353i	$-0.312433\pi$
$61$	32302.0	1.11149	0.555744	+	0.831353i	$0.312433\pi$
$62$	−4896.00	−0.161757
$63$	−43584.0	−1.38349
$64$	−10688.0	−0.326172
$65$	7150.00	0.209905
$66$	1184.00	0.0334574
$67$	60972.0	1.65937	0.829685	−	0.558231i	$-0.188520\pi$
$67$	60972.0	1.65937	0.829685	+	0.558231i	$0.188520\pi$
$68$	46984.0	1.23219
$69$	−11904.0	−0.301003
$70$	9600.00	0.234167
$71$	−32648.0	−0.768618	−0.384309	−	0.923204i	$-0.625560\pi$
$71$	−32648.0	−0.768618	−0.384309	+	0.923204i	$0.625560\pi$
$72$	27240.0	0.619264
$73$	−38774.0	−0.851596	−0.425798	−	0.904818i	$-0.640007\pi$
$73$	−38774.0	−0.851596	−0.425798	+	0.904818i	$0.640007\pi$
$74$	364.000	0.00772720
$75$	−2500.00	−0.0513200
$76$	−29680.0	−0.589427
$77$	−28416.0	−0.546180
$78$	−2288.00	−0.0425814
$79$	−33360.0	−0.601393	−0.300696	−	0.953720i	$-0.597219\pi$
$79$	−33360.0	−0.601393	−0.300696	+	0.953720i	$0.597219\pi$
$80$	16400.0	0.286496
$81$	47641.0	0.806805
$82$	−18796.0	−0.308696
$83$	16716.0	0.266340	0.133170	−	0.991093i	$-0.457484\pi$
$83$	16716.0	0.266340	0.133170	+	0.991093i	$0.457484\pi$
$84$	21504.0	0.332522
$85$	−41950.0	−0.629774
$86$	−2488.00	−0.0362747
$87$	13640.0	0.193204
$88$	17760.0	0.244476
$89$	101370.	1.35655	0.678273	−	0.734810i	$-0.262729\pi$
$89$	101370.	1.35655	0.678273	+	0.734810i	$0.262729\pi$
$90$	−11350.0	−0.147703
$91$	54912.0	0.695126
$92$	−83328.0	−1.02641
$93$	9792.00	0.117399
$94$	−24176.0	−0.282205
$95$	26500.0	0.301257
$96$	−20608.0	−0.228222
$97$	−119038.	−1.28457	−0.642283	−	0.766468i	$-0.722013\pi$
$97$	−119038.	−1.28457	−0.642283	+	0.766468i	$0.722013\pi$
$98$	40114.0	0.421921
$99$	33596.0	0.344508
$100$	−17500.0	−0.175000
$101$	−89898.0	−0.876893	−0.438446	−	0.898757i	$-0.644471\pi$
$101$	−89898.0	−0.876893	−0.438446	+	0.898757i	$0.644471\pi$
$102$	13424.0	0.127756
$103$	−19504.0	−0.181147	−0.0905734	−	0.995890i	$-0.528870\pi$
$103$	−19504.0	−0.181147	−0.0905734	+	0.995890i	$0.528870\pi$
$104$	−34320.0	−0.311146
$105$	−19200.0	−0.169953
$106$	47692.0	0.412269
$107$	158292.	1.33659	0.668297	−	0.743895i	$-0.267024\pi$
$107$	158292.	1.33659	0.668297	+	0.743895i	$0.267024\pi$
$108$	−52640.0	−0.434267
$109$	36830.0	0.296917	0.148459	−	0.988919i	$-0.452569\pi$
$109$	36830.0	0.296917	0.148459	+	0.988919i	$0.452569\pi$
$110$	−7400.00	−0.0583109
$111$	−728.000	−0.00560821
$112$	125952.	0.948768
$113$	11186.0	0.0824098	0.0412049	−	0.999151i	$-0.486880\pi$
$113$	11186.0	0.0824098	0.0412049	+	0.999151i	$0.486880\pi$
$114$	−8480.00	−0.0611130
$115$	74400.0	0.524600
$116$	95480.0	0.658821
$117$	−64922.0	−0.438457
$118$	−40040.0	−0.264721
$119$	−322176.	−2.08557
$120$	12000.0	0.0760726
$121$	−139147.	−0.863993
$122$	64604.0	0.392970
$123$	37592.0	0.224044
$124$	68544.0	0.400327
$125$	15625.0	0.0894427
$126$	−87168.0	−0.489137
$127$	70552.0	0.388150	0.194075	−	0.980987i	$-0.437829\pi$
$127$	70552.0	0.388150	0.194075	+	0.980987i	$0.437829\pi$
$128$	−186240.	−1.00473
$129$	4976.00	0.0263273
$130$	14300.0	0.0742126
$131$	76452.0	0.389234	0.194617	−	0.980879i	$-0.437654\pi$
$131$	76452.0	0.389234	0.194617	+	0.980879i	$0.437654\pi$
$132$	−16576.0	−0.0828028
$133$	203520.	0.997650
$134$	121944.	0.586676
$135$	47000.0	0.221954
$136$	201360.	0.933525
$137$	−144918.	−0.659661	−0.329831	−	0.944040i	$-0.606992\pi$
$137$	−144918.	−0.659661	−0.329831	+	0.944040i	$0.606992\pi$
$138$	−23808.0	−0.106420
$139$	112220.	0.492644	0.246322	−	0.969188i	$-0.420778\pi$
$139$	112220.	0.492644	0.246322	+	0.969188i	$0.420778\pi$
$140$	−134400.	−0.579534
$141$	48352.0	0.204817
$142$	−65296.0	−0.271748
$143$	−42328.0	−0.173096
$144$	−148912.	−0.598444
$145$	−85250.0	−0.336724
$146$	−77548.0	−0.301085
$147$	−80228.0	−0.306219
$148$	−5096.00	−0.0191238
$149$	403750.	1.48986	0.744932	−	0.667140i	$-0.232482\pi$
$149$	403750.	1.48986	0.744932	+	0.667140i	$0.232482\pi$
$150$	−5000.00	−0.0181444
$151$	−446648.	−1.59413	−0.797064	−	0.603895i	$-0.793615\pi$
$151$	−446648.	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	−127200.	−0.446558
$153$	380906.	1.31550
$154$	−56832.0	−0.193104
$155$	−61200.0	−0.204608
$156$	32032.0	0.105383
$157$	−262258.	−0.849141	−0.424570	−	0.905395i	$-0.639575\pi$
$157$	−262258.	−0.849141	−0.424570	+	0.905395i	$0.639575\pi$
$158$	−66720.0	−0.212625
$159$	−95384.0	−0.299215
$160$	128800.	0.397755
$161$	571392.	1.73728
$162$	95282.0	0.285248
$163$	−154564.	−0.455658	−0.227829	−	0.973701i	$-0.573163\pi$
$163$	−154564.	−0.455658	−0.227829	+	0.973701i	$0.573163\pi$
$164$	263144.	0.763983
$165$	14800.0	0.0423206
$166$	33432.0	0.0941656
$167$	396672.	1.10063	0.550314	−	0.834958i	$-0.314508\pi$
$167$	396672.	1.10063	0.550314	+	0.834958i	$0.314508\pi$
$168$	92160.0	0.251924
$169$	−289497.	−0.779700
$170$	−83900.0	−0.222659
$171$	−240620.	−0.629276
$172$	34832.0	0.0897754
$173$	−573474.	−1.45680	−0.728398	−	0.685155i	$-0.759735\pi$
$173$	−573474.	−1.45680	−0.728398	+	0.685155i	$0.759735\pi$
$174$	27280.0	0.0683079
$175$	120000.	0.296201
$176$	−97088.0	−0.236257
$177$	80080.0	0.192128
$178$	202740.	0.479611
$179$	−594460.	−1.38672	−0.693362	−	0.720589i	$-0.743871\pi$
$179$	−594460.	−1.38672	−0.693362	+	0.720589i	$0.743871\pi$
$180$	158900.	0.365547
$181$	−107098.	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$181$	−107098.	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$182$	109824.	0.245764
$183$	−129208.	−0.285208
$184$	−357120.	−0.777624
$185$	4550.00	0.00977422
$186$	19584.0	0.0415068
$187$	248344.	0.519337
$188$	338464.	0.698422
$189$	360960.	0.735029
$190$	53000.0	0.106510
$191$	469552.	0.931323	0.465661	−	0.884963i	$-0.345816\pi$
$191$	469552.	0.931323	0.465661	+	0.884963i	$0.345816\pi$
$192$	42752.0	0.0836957
$193$	52706.0	0.101851	0.0509257	−	0.998702i	$-0.483783\pi$
$193$	52706.0	0.101851	0.0509257	+	0.998702i	$0.483783\pi$
$194$	−238076.	−0.454163
$195$	−28600.0	−0.0538616
$196$	−561596.	−1.04420
$197$	455862.	0.836889	0.418444	−	0.908242i	$-0.362575\pi$
$197$	455862.	0.836889	0.418444	+	0.908242i	$0.362575\pi$
$198$	67192.0	0.121802
$199$	865000.	1.54840	0.774200	−	0.632940i	$-0.218152\pi$
$199$	865000.	1.54840	0.774200	+	0.632940i	$0.218152\pi$
$200$	−75000.0	−0.132583
$201$	−243888.	−0.425795
$202$	−179796.	−0.310028
$203$	−654720.	−1.11510
$204$	−187936.	−0.316180
$205$	−234950.	−0.390473
$206$	−39008.0	−0.0640451
$207$	−675552.	−1.09580
$208$	187616.	0.300685
$209$	−156880.	−0.248429
$210$	−38400.0	−0.0600873
$211$	1.10565e6	1.70967	0.854835	−	0.518900i	$-0.173658\pi$
$211$	1.10565e6	1.70967	0.854835	+	0.518900i	$0.173658\pi$
$212$	−667688.	−1.02031
$213$	130592.	0.197228
$214$	316584.	0.472557
$215$	−31100.0	−0.0458843
$216$	−225600.	−0.329007
$217$	−470016.	−0.677584
$218$	73660.0	0.104976
$219$	155096.	0.218520
$220$	103600.	0.144312
$221$	−479908.	−0.660963
$222$	−1456.00	−0.00198280
$223$	1.12158e6	1.51031	0.755156	−	0.655545i	$-0.227561\pi$
$223$	1.12158e6	1.51031	0.755156	+	0.655545i	$0.227561\pi$
$224$	989184.	1.31722
$225$	−141875.	−0.186831
$226$	22372.0	0.0291363
$227$	−23348.0	−0.0300736	−0.0150368	−	0.999887i	$-0.504787\pi$
$227$	−23348.0	−0.0300736	−0.0150368	+	0.999887i	$0.504787\pi$
$228$	118720.	0.151247
$229$	−596010.	−0.751043	−0.375522	−	0.926814i	$-0.622536\pi$
$229$	−596010.	−0.751043	−0.375522	+	0.926814i	$0.622536\pi$
$230$	148800.	0.185474
$231$	113664.	0.140150
$232$	409200.	0.499132
$233$	−485334.	−0.585667	−0.292834	−	0.956163i	$-0.594598\pi$
$233$	−485334.	−0.585667	−0.292834	+	0.956163i	$0.594598\pi$
$234$	−129844.	−0.155018
$235$	−302200.	−0.356964
$236$	560560.	0.655152
$237$	133440.	0.154317
$238$	−644352.	−0.737362
$239$	−48880.0	−0.0553524	−0.0276762	−	0.999617i	$-0.508811\pi$
$239$	−48880.0	−0.0553524	−0.0276762	+	0.999617i	$0.508811\pi$
$240$	−65600.0	−0.0735150
$241$	−110798.	−0.122882	−0.0614411	−	0.998111i	$-0.519570\pi$
$241$	−110798.	−0.122882	−0.0614411	+	0.998111i	$0.519570\pi$
$242$	−278294.	−0.305468
$243$	−647404.	−0.703331
$244$	−904456.	−0.972552
$245$	501425.	0.533692
$246$	75184.0	0.0792114
$247$	303160.	0.316176
$248$	293760.	0.303294
$249$	−66864.0	−0.0683430
$250$	31250.0	0.0316228
$251$	−1.64375e6	−1.64684	−0.823419	−	0.567434i	$-0.807936\pi$
$251$	−1.64375e6	−1.64684	−0.823419	+	0.567434i	$0.807936\pi$
$252$	1.22035e6	1.21055
$253$	−440448.	−0.432607
$254$	141104.	0.137232
$255$	167800.	0.161600
$256$	−30464.0	−0.0290527
$257$	1.30624e6	1.23365	0.616823	−	0.787102i	$-0.288419\pi$
$257$	1.30624e6	1.23365	0.616823	+	0.787102i	$0.288419\pi$
$258$	9952.00	0.00930810
$259$	34944.0	0.0323685
$260$	−200200.	−0.183667
$261$	774070.	0.703362
$262$	152904.	0.137615
$263$	2.12834e6	1.89736	0.948682	−	0.316231i	$-0.102417\pi$
$263$	2.12834e6	1.89736	0.948682	+	0.316231i	$0.102417\pi$
$264$	−71040.0	−0.0627326
$265$	596150.	0.521484
$266$	407040.	0.352722
$267$	−405480.	−0.348090
$268$	−1.70722e6	−1.45195
$269$	−1.44109e6	−1.21426	−0.607128	−	0.794604i	$-0.707679\pi$
$269$	−1.44109e6	−1.21426	−0.607128	+	0.794604i	$0.707679\pi$
$270$	94000.0	0.0784727
$271$	−93248.0	−0.0771288	−0.0385644	−	0.999256i	$-0.512278\pi$
$271$	−93248.0	−0.0771288	−0.0385644	+	0.999256i	$0.512278\pi$
$272$	−1.10077e6	−0.902139
$273$	−219648.	−0.178370
$274$	−289836.	−0.233225
$275$	−92500.0	−0.0737581
$276$	333312.	0.263377
$277$	−110298.	−0.0863711	−0.0431855	−	0.999067i	$-0.513751\pi$
$277$	−110298.	−0.0863711	−0.0431855	+	0.999067i	$0.513751\pi$
$278$	224440.	0.174176
$279$	555696.	0.427392
$280$	−576000.	−0.439064
$281$	−192198.	−0.145205	−0.0726027	−	0.997361i	$-0.523131\pi$
$281$	−192198.	−0.145205	−0.0726027	+	0.997361i	$0.523131\pi$
$282$	96704.0	0.0724139
$283$	−331884.	−0.246332	−0.123166	−	0.992386i	$-0.539305\pi$
$283$	−331884.	−0.246332	−0.123166	+	0.992386i	$0.539305\pi$
$284$	914144.	0.672541
$285$	−106000.	−0.0773025
$286$	−84656.0	−0.0611988
$287$	−1.80442e6	−1.29310
$288$	−1.16950e6	−0.830846
$289$	1.39583e6	0.983076
$290$	−170500.	−0.119050
$291$	476152.	0.329620
$292$	1.08567e6	0.745146
$293$	2.19481e6	1.49358	0.746788	−	0.665063i	$-0.231595\pi$
$293$	2.19481e6	1.49358	0.746788	+	0.665063i	$0.231595\pi$
$294$	−160456.	−0.108265
$295$	−500500.	−0.334849
$296$	−21840.0	−0.0144885
$297$	−278240.	−0.183033
$298$	807500.	0.526747
$299$	851136.	0.550581
$300$	70000.0	0.0449050
$301$	−238848.	−0.151952
$302$	−893296.	−0.563609
$303$	359592.	0.225011
$304$	695360.	0.431545
$305$	807550.	0.497073
$306$	761812.	0.465098
$307$	−2.37751e6	−1.43971	−0.719857	−	0.694123i	$-0.755793\pi$
$307$	−2.37751e6	−1.43971	−0.719857	+	0.694123i	$0.755793\pi$
$308$	795648.	0.477908
$309$	78016.0	0.0464823
$310$	−122400.	−0.0723398
$311$	−2.37305e6	−1.39125	−0.695626	−	0.718405i	$-0.744873\pi$
$311$	−2.37305e6	−1.39125	−0.695626	+	0.718405i	$0.744873\pi$
$312$	137280.	0.0798400
$313$	−1.42941e6	−0.824702	−0.412351	−	0.911025i	$-0.635292\pi$
$313$	−1.42941e6	−0.824702	−0.412351	+	0.911025i	$0.635292\pi$
$314$	−524516.	−0.300217
$315$	−1.08960e6	−0.618715
$316$	934080.	0.526219
$317$	2.12462e6	1.18750	0.593750	−	0.804650i	$-0.297647\pi$
$317$	2.12462e6	1.18750	0.593750	+	0.804650i	$0.297647\pi$
$318$	−190768.	−0.105788
$319$	504680.	0.277677
$320$	−267200.	−0.145868
$321$	−633168.	−0.342970
$322$	1.14278e6	0.614221
$323$	−1.77868e6	−0.948618
$324$	−1.33395e6	−0.705954
$325$	178750.	0.0938723
$326$	−309128.	−0.161100
$327$	−147320.	−0.0761890
$328$	1.12776e6	0.578805
$329$	−2.32090e6	−1.18213
$330$	29600.0	0.0149626
$331$	3.09985e6	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$331$	3.09985e6	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$332$	−468048.	−0.233048
$333$	−41314.0	−0.0204168
$334$	793344.	0.389131
$335$	1.52430e6	0.742093
$336$	−503808.	−0.243454
$337$	2.40008e6	1.15120	0.575601	−	0.817731i	$-0.304768\pi$
$337$	2.40008e6	1.15120	0.575601	+	0.817731i	$0.304768\pi$
$338$	−578994.	−0.275665
$339$	−44744.0	−0.0211464
$340$	1.17460e6	0.551052
$341$	362304.	0.168728
$342$	−481240.	−0.222483
$343$	624000.	0.286384
$344$	149280.	0.0680151
$345$	−297600.	−0.134612
$346$	−1.14695e6	−0.515055
$347$	1.77741e6	0.792436	0.396218	−	0.918156i	$-0.370322\pi$
$347$	1.77741e6	0.792436	0.396218	+	0.918156i	$0.370322\pi$
$348$	−381920.	−0.169054
$349$	−2.14805e6	−0.944019	−0.472010	−	0.881593i	$-0.656471\pi$
$349$	−2.14805e6	−0.944019	−0.472010	+	0.881593i	$0.656471\pi$
$350$	240000.	0.104723
$351$	537680.	0.232946
$352$	−762496.	−0.328005
$353$	−661854.	−0.282700	−0.141350	−	0.989960i	$-0.545144\pi$
$353$	−661854.	−0.282700	−0.141350	+	0.989960i	$0.545144\pi$
$354$	160160.	0.0679275
$355$	−816200.	−0.343737
$356$	−2.83836e6	−1.18698
$357$	1.28870e6	0.535159
$358$	−1.18892e6	−0.490281
$359$	−259320.	−0.106194	−0.0530970	−	0.998589i	$-0.516909\pi$
$359$	−259320.	−0.106194	−0.0530970	+	0.998589i	$0.516909\pi$
$360$	681000.	0.276943
$361$	−1.35250e6	−0.546222
$362$	−214196.	−0.0859093
$363$	556588.	0.221701
$364$	−1.53754e6	−0.608236
$365$	−969350.	−0.380845
$366$	−258416.	−0.100836
$367$	−1.49993e6	−0.581307	−0.290653	−	0.956828i	$-0.593873\pi$
$367$	−1.49993e6	−0.581307	−0.290653	+	0.956828i	$0.593873\pi$
$368$	1.95226e6	0.751480
$369$	2.13335e6	0.815634
$370$	9100.00	0.00345571
$371$	4.57843e6	1.72696
$372$	−274176.	−0.102724
$373$	−2.23807e6	−0.832918	−0.416459	−	0.909154i	$-0.636729\pi$
$373$	−2.23807e6	−0.832918	−0.416459	+	0.909154i	$0.636729\pi$
$374$	496688.	0.183614
$375$	−62500.0	−0.0229510
$376$	1.45056e6	0.529135
$377$	−975260.	−0.353400
$378$	721920.	0.259872
$379$	3.15934e6	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$379$	3.15934e6	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$380$	−742000.	−0.263600
$381$	−282208.	−0.0995994
$382$	939104.	0.329272
$383$	342216.	0.119207	0.0596037	−	0.998222i	$-0.481016\pi$
$383$	342216.	0.119207	0.0596037	+	0.998222i	$0.481016\pi$
$384$	744960.	0.257813
$385$	−710400.	−0.244259
$386$	105412.	0.0360099
$387$	282388.	0.0958449
$388$	3.33306e6	1.12399
$389$	88470.0	0.0296430	0.0148215	−	0.999890i	$-0.495282\pi$
$389$	88470.0	0.0296430	0.0148215	+	0.999890i	$0.495282\pi$
$390$	−57200.0	−0.0190430
$391$	−4.99373e6	−1.65190
$392$	−2.40684e6	−0.791101
$393$	−305808.	−0.0998775
$394$	911724.	0.295885
$395$	−834000.	−0.268951
$396$	−940688.	−0.301445
$397$	−5.45674e6	−1.73763	−0.868814	−	0.495138i	$-0.835117\pi$
$397$	−5.45674e6	−1.73763	−0.868814	+	0.495138i	$0.835117\pi$
$398$	1.73000e6	0.547442
$399$	−814080.	−0.255997
$400$	410000.	0.128125
$401$	4.04680e6	1.25676	0.628378	−	0.777908i	$-0.283719\pi$
$401$	4.04680e6	1.25676	0.628378	+	0.777908i	$0.283719\pi$
$402$	−487776.	−0.150541
$403$	−700128.	−0.214741
$404$	2.51714e6	0.767281
$405$	1.19102e6	0.360814
$406$	−1.30944e6	−0.394249
$407$	−26936.0	−0.00806022
$408$	−805440.	−0.239543
$409$	−2.71207e6	−0.801664	−0.400832	−	0.916151i	$-0.631279\pi$
$409$	−2.71207e6	−0.801664	−0.400832	+	0.916151i	$0.631279\pi$
$410$	−469900.	−0.138053
$411$	579672.	0.169269
$412$	546112.	0.158503
$413$	−3.84384e6	−1.10889
$414$	−1.35110e6	−0.387425
$415$	417900.	0.119111
$416$	1.47347e6	0.417454
$417$	−448880.	−0.126413
$418$	−313760.	−0.0878328
$419$	3.71746e6	1.03445	0.517227	−	0.855848i	$-0.326964\pi$
$419$	3.71746e6	1.03445	0.517227	+	0.855848i	$0.326964\pi$
$420$	537600.	0.148709
$421$	3.55250e6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$421$	3.55250e6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$422$	2.21130e6	0.604460
$423$	2.74398e6	0.745640
$424$	−2.86152e6	−0.773005
$425$	−1.04875e6	−0.281643
$426$	261184.	0.0697305
$427$	6.20198e6	1.64612
$428$	−4.43218e6	−1.16952
$429$	169312.	0.0444165
$430$	−62200.0	−0.0162226
$431$	−4.06205e6	−1.05330	−0.526650	−	0.850082i	$-0.676552\pi$
$431$	−4.06205e6	−1.05330	−0.526650	+	0.850082i	$0.676552\pi$
$432$	1.23328e6	0.317945
$433$	7.26287e6	1.86161	0.930804	−	0.365518i	$-0.119108\pi$
$433$	7.26287e6	1.86161	0.930804	+	0.365518i	$0.119108\pi$
$434$	−940032.	−0.239562
$435$	341000.	0.0864035
$436$	−1.03124e6	−0.259803
$437$	3.15456e6	0.790197
$438$	310192.	0.0772583
$439$	−5.41028e6	−1.33986	−0.669928	−	0.742426i	$-0.733675\pi$
$439$	−5.41028e6	−1.33986	−0.669928	+	0.742426i	$0.733675\pi$
$440$	444000.	0.109333
$441$	−4.55294e6	−1.11480
$442$	−959816.	−0.233686
$443$	−6.51524e6	−1.57733	−0.788663	−	0.614826i	$-0.789226\pi$
$443$	−6.51524e6	−1.57733	−0.788663	+	0.614826i	$0.789226\pi$
$444$	20384.0	0.00490718
$445$	2.53425e6	0.606666
$446$	2.24315e6	0.533976
$447$	−1.61500e6	−0.382299
$448$	−2.05210e6	−0.483062
$449$	−509950.	−0.119375	−0.0596873	−	0.998217i	$-0.519010\pi$
$449$	−509950.	−0.119375	−0.0596873	+	0.998217i	$0.519010\pi$
$450$	−283750.	−0.0660548
$451$	1.39090e6	0.322000
$452$	−313208.	−0.0721085
$453$	1.78659e6	0.409053
$454$	−46696.0	−0.0106326
$455$	1.37280e6	0.310870
$456$	508800.	0.114587
$457$	1.22084e6	0.273444	0.136722	−	0.990609i	$-0.456343\pi$
$457$	1.22084e6	0.273444	0.136722	+	0.990609i	$0.456343\pi$
$458$	−1.19202e6	−0.265534
$459$	−3.15464e6	−0.698905
$460$	−2.08320e6	−0.459025
$461$	−4.07210e6	−0.892413	−0.446207	−	0.894930i	$-0.647225\pi$
$461$	−4.07210e6	−0.892413	−0.446207	+	0.894930i	$0.647225\pi$
$462$	227328.	0.0495505
$463$	2.02294e6	0.438561	0.219280	−	0.975662i	$-0.429629\pi$
$463$	2.02294e6	0.438561	0.219280	+	0.975662i	$0.429629\pi$
$464$	−2.23696e6	−0.482351
$465$	244800.	0.0525024
$466$	−970668.	−0.207065
$467$	3.25097e6	0.689797	0.344898	−	0.938640i	$-0.387913\pi$
$467$	3.25097e6	0.689797	0.344898	+	0.938640i	$0.387913\pi$
$468$	1.81782e6	0.383650
$469$	1.17066e7	2.45753
$470$	−604400.	−0.126206
$471$	1.04903e6	0.217890
$472$	2.40240e6	0.496353
$473$	184112.	0.0378381
$474$	266880.	0.0545595
$475$	662500.	0.134726
$476$	9.02093e6	1.82488
$477$	−5.41304e6	−1.08929
$478$	−97760.0	−0.0195700
$479$	−3.27936e6	−0.653056	−0.326528	−	0.945188i	$-0.605879\pi$
$479$	−3.27936e6	−0.653056	−0.326528	+	0.945188i	$0.605879\pi$
$480$	−515200.	−0.102064
$481$	52052.0	0.0102583
$482$	−221596.	−0.0434455
$483$	−2.28557e6	−0.445786
$484$	3.89612e6	0.755994
$485$	−2.97595e6	−0.574475
$486$	−1.29481e6	−0.248665
$487$	−8.53197e6	−1.63015	−0.815074	−	0.579357i	$-0.803304\pi$
$487$	−8.53197e6	−1.63015	−0.815074	+	0.579357i	$0.803304\pi$
$488$	−3.87624e6	−0.736819
$489$	618256.	0.116922
$490$	1.00285e6	0.188689
$491$	1.51265e6	0.283162	0.141581	−	0.989927i	$-0.454781\pi$
$491$	1.51265e6	0.283162	0.141581	+	0.989927i	$0.454781\pi$
$492$	−1.05258e6	−0.196038
$493$	5.72198e6	1.06030
$494$	606320.	0.111785
$495$	839900.	0.154069
$496$	−1.60589e6	−0.293097
$497$	−6.26842e6	−1.13833
$498$	−133728.	−0.0241629
$499$	−6.49190e6	−1.16713	−0.583567	−	0.812065i	$-0.698343\pi$
$499$	−6.49190e6	−1.16713	−0.583567	+	0.812065i	$0.698343\pi$
$500$	−437500.	−0.0782624
$501$	−1.58669e6	−0.282421
$502$	−3.28750e6	−0.582245
$503$	8.61770e6	1.51870	0.759349	−	0.650684i	$-0.225518\pi$
$503$	8.61770e6	1.51870	0.759349	+	0.650684i	$0.225518\pi$
$504$	5.23008e6	0.917132
$505$	−2.24745e6	−0.392158
$506$	−880896.	−0.152950
$507$	1.15799e6	0.200071
$508$	−1.97546e6	−0.339632
$509$	2.67323e6	0.457343	0.228671	−	0.973504i	$-0.426562\pi$
$509$	2.67323e6	0.457343	0.228671	+	0.973504i	$0.426562\pi$
$510$	335600.	0.0571342
$511$	−7.44461e6	−1.26122
$512$	5.89875e6	0.994455
$513$	1.99280e6	0.334326
$514$	2.61248e6	0.436160
$515$	−487600.	−0.0810113
$516$	−139328.	−0.0230364
$517$	1.78902e6	0.294367
$518$	69888.0	0.0114440
$519$	2.29390e6	0.373814
$520$	−858000.	−0.139149
$521$	6.18500e6	0.998264	0.499132	−	0.866526i	$-0.333652\pi$
$521$	6.18500e6	0.998264	0.499132	+	0.866526i	$0.333652\pi$
$522$	1.54814e6	0.248676
$523$	−6.89452e6	−1.10217	−0.551087	−	0.834448i	$-0.685787\pi$
$523$	−6.89452e6	−1.10217	−0.551087	+	0.834448i	$0.685787\pi$
$524$	−2.14066e6	−0.340580
$525$	−480000.	−0.0760051
$526$	4.25667e6	0.670820
$527$	4.10774e6	0.644283
$528$	388352.	0.0606235
$529$	2.42023e6	0.376026
$530$	1.19230e6	0.184372
$531$	4.54454e6	0.699445
$532$	−5.69856e6	−0.872943
$533$	−2.68783e6	−0.409811
$534$	−810960.	−0.123068
$535$	3.95730e6	0.597743
$536$	−7.31664e6	−1.10002
$537$	2.37784e6	0.355834
$538$	−2.88218e6	−0.429304
$539$	−2.96844e6	−0.440104
$540$	−1.31600e6	−0.194210
$541$	155502.	0.0228425	0.0114212	−	0.999935i	$-0.496364\pi$
$541$	155502.	0.0228425	0.0114212	+	0.999935i	$0.496364\pi$
$542$	−186496.	−0.0272691
$543$	428392.	0.0623508
$544$	−8.64506e6	−1.25248
$545$	920750.	0.132785
$546$	−439296.	−0.0630631
$547$	1.26544e7	1.80831	0.904157	−	0.427201i	$-0.140500\pi$
$547$	1.26544e7	1.80831	0.904157	+	0.427201i	$0.140500\pi$
$548$	4.05770e6	0.577204
$549$	−7.33255e6	−1.03830
$550$	−185000.	−0.0260774
$551$	−3.61460e6	−0.507202
$552$	1.42848e6	0.199538
$553$	−6.40512e6	−0.890665
$554$	−220596.	−0.0305368
$555$	−18200.0	−0.00250807
$556$	−3.14216e6	−0.431064
$557$	−7.07786e6	−0.966638	−0.483319	−	0.875444i	$-0.660569\pi$
$557$	−7.07786e6	−0.966638	−0.483319	+	0.875444i	$0.660569\pi$
$558$	1.11139e6	0.151106
$559$	−355784.	−0.0481567
$560$	3.14880e6	0.424302
$561$	−993376.	−0.133262
$562$	−384396.	−0.0513379
$563$	846636.	0.112571	0.0562854	−	0.998415i	$-0.482074\pi$
$563$	846636.	0.112571	0.0562854	+	0.998415i	$0.482074\pi$
$564$	−1.35386e6	−0.179215
$565$	279650.	0.0368548
$566$	−663768.	−0.0870914
$567$	9.14707e6	1.19488
$568$	3.91776e6	0.509527
$569$	4.96041e6	0.642299	0.321149	−	0.947029i	$-0.395931\pi$
$569$	4.96041e6	0.642299	0.321149	+	0.947029i	$0.395931\pi$
$570$	−212000.	−0.0273306
$571$	8.96505e6	1.15070	0.575351	−	0.817907i	$-0.304866\pi$
$571$	8.96505e6	1.15070	0.575351	+	0.817907i	$0.304866\pi$
$572$	1.18518e6	0.151459
$573$	−1.87821e6	−0.238978
$574$	−3.60883e6	−0.457180
$575$	1.86000e6	0.234608
$576$	2.42618e6	0.304696
$577$	−2.86080e6	−0.357724	−0.178862	−	0.983874i	$-0.557242\pi$
$577$	−2.86080e6	−0.357724	−0.178862	+	0.983874i	$0.557242\pi$
$578$	2.79165e6	0.347570
$579$	−210824.	−0.0261351
$580$	2.38700e6	0.294634
$581$	3.20947e6	0.394451
$582$	952304.	0.116538
$583$	−3.52921e6	−0.430037
$584$	4.65288e6	0.564534
$585$	−1.62305e6	−0.196084
$586$	4.38961e6	0.528059
$587$	−6.74027e6	−0.807387	−0.403694	−	0.914894i	$-0.632274\pi$
$587$	−6.74027e6	−0.807387	−0.403694	+	0.914894i	$0.632274\pi$
$588$	2.24638e6	0.267942
$589$	−2.59488e6	−0.308197
$590$	−1.00100e6	−0.118387
$591$	−1.82345e6	−0.214746
$592$	119392.	0.0140014
$593$	−1.78609e6	−0.208578	−0.104289	−	0.994547i	$-0.533257\pi$
$593$	−1.78609e6	−0.208578	−0.104289	+	0.994547i	$0.533257\pi$
$594$	−556480.	−0.0647118
$595$	−8.05440e6	−0.932697
$596$	−1.13050e7	−1.30363
$597$	−3.46000e6	−0.397320
$598$	1.70227e6	0.194660
$599$	4.94620e6	0.563254	0.281627	−	0.959524i	$-0.409126\pi$
$599$	4.94620e6	0.563254	0.281627	+	0.959524i	$0.409126\pi$
$600$	300000.	0.0340207
$601$	−4.58100e6	−0.517337	−0.258669	−	0.965966i	$-0.583284\pi$
$601$	−4.58100e6	−0.517337	−0.258669	+	0.965966i	$0.583284\pi$
$602$	−477696.	−0.0537230
$603$	−1.38406e7	−1.55011
$604$	1.25061e7	1.39486
$605$	−3.47868e6	−0.386390
$606$	719184.	0.0795533
$607$	7.07999e6	0.779940	0.389970	−	0.920828i	$-0.372485\pi$
$607$	7.07999e6	0.779940	0.389970	+	0.920828i	$0.372485\pi$
$608$	5.46112e6	0.599132
$609$	2.61888e6	0.286136
$610$	1.61510e6	0.175742
$611$	−3.45717e6	−0.374643
$612$	−1.06654e7	−1.15106
$613$	5.09609e6	0.547754	0.273877	−	0.961765i	$-0.411694\pi$
$613$	5.09609e6	0.547754	0.273877	+	0.961765i	$0.411694\pi$
$614$	−4.75502e6	−0.509016
$615$	939800.	0.100195
$616$	3.40992e6	0.362070
$617$	−1.30003e7	−1.37480	−0.687400	−	0.726279i	$-0.741248\pi$
$617$	−1.30003e7	−1.37480	−0.687400	+	0.726279i	$0.741248\pi$
$618$	156032.	0.0164340
$619$	4.84406e6	0.508139	0.254070	−	0.967186i	$-0.418231\pi$
$619$	4.84406e6	0.508139	0.254070	+	0.967186i	$0.418231\pi$
$620$	1.71360e6	0.179032
$621$	5.59488e6	0.582186
$622$	−4.74610e6	−0.491882
$623$	1.94630e7	2.00905
$624$	−750464.	−0.0771558
$625$	390625.	0.0400000
$626$	−2.85883e6	−0.291576
$627$	627520.	0.0637468
$628$	7.34322e6	0.742998
$629$	−305396.	−0.0307777
$630$	−2.17920e6	−0.218749
$631$	6.22775e6	0.622670	0.311335	−	0.950300i	$-0.399224\pi$
$631$	6.22775e6	0.622670	0.311335	+	0.950300i	$0.399224\pi$
$632$	4.00320e6	0.398671
$633$	−4.42261e6	−0.438702
$634$	4.24924e6	0.419845
$635$	1.76380e6	0.173586
$636$	2.67075e6	0.261813
$637$	5.73630e6	0.560123
$638$	1.00936e6	0.0981735
$639$	7.41110e6	0.718010
$640$	−4.65600e6	−0.449328
$641$	1.53280e6	0.147347	0.0736734	−	0.997282i	$-0.476528\pi$
$641$	1.53280e6	0.147347	0.0736734	+	0.997282i	$0.476528\pi$
$642$	−1.26634e6	−0.121258
$643$	−1.74382e7	−1.66332	−0.831659	−	0.555287i	$-0.812609\pi$
$643$	−1.74382e7	−1.66332	−0.831659	+	0.555287i	$0.812609\pi$
$644$	−1.59990e7	−1.52012
$645$	124400.	0.0117739
$646$	−3.55736e6	−0.335387
$647$	−4.25469e6	−0.399583	−0.199792	−	0.979838i	$-0.564026\pi$
$647$	−4.25469e6	−0.399583	−0.199792	+	0.979838i	$0.564026\pi$
$648$	−5.71692e6	−0.534841
$649$	2.96296e6	0.276130
$650$	357500.	0.0331889
$651$	1.88006e6	0.173868
$652$	4.32779e6	0.398701
$653$	3.01085e6	0.276316	0.138158	−	0.990410i	$-0.455882\pi$
$653$	3.01085e6	0.276316	0.138158	+	0.990410i	$0.455882\pi$
$654$	−294640.	−0.0269369
$655$	1.91130e6	0.174071
$656$	−6.16509e6	−0.559345
$657$	8.80170e6	0.795524
$658$	−4.64179e6	−0.417947
$659$	−8.11462e6	−0.727871	−0.363936	−	0.931424i	$-0.618567\pi$
$659$	−8.11462e6	−0.727871	−0.363936	+	0.931424i	$0.618567\pi$
$660$	−414400.	−0.0370305
$661$	2.47370e6	0.220213	0.110107	−	0.993920i	$-0.464881\pi$
$661$	2.47370e6	0.220213	0.110107	+	0.993920i	$0.464881\pi$
$662$	6.19970e6	0.549827
$663$	1.91963e6	0.169603
$664$	−2.00592e6	−0.176560
$665$	5.08800e6	0.446162
$666$	−82628.0	−0.00721841
$667$	−1.01482e7	−0.883228
$668$	−1.11068e7	−0.963049
$669$	−4.48630e6	−0.387546
$670$	3.04860e6	0.262370
$671$	−4.78070e6	−0.409907
$672$	−3.95674e6	−0.337998
$673$	5.77063e6	0.491117	0.245559	−	0.969382i	$-0.421029\pi$
$673$	5.77063e6	0.491117	0.245559	+	0.969382i	$0.421029\pi$
$674$	4.80016e6	0.407011
$675$	1.17500e6	0.0992610
$676$	8.10592e6	0.682237
$677$	1.67197e7	1.40203	0.701014	−	0.713147i	$-0.252731\pi$
$677$	1.67197e7	1.40203	0.701014	+	0.713147i	$0.252731\pi$
$678$	−89488.0	−0.00747637
$679$	−2.28553e7	−1.90245
$680$	5.03400e6	0.417485
$681$	93392.0	0.00771688
$682$	724608.	0.0596544
$683$	7.14532e6	0.586097	0.293049	−	0.956098i	$-0.405330\pi$
$683$	7.14532e6	0.586097	0.293049	+	0.956098i	$0.405330\pi$
$684$	6.73736e6	0.550617
$685$	−3.62295e6	−0.295009
$686$	1.24800e6	0.101252
$687$	2.38404e6	0.192718
$688$	−816064.	−0.0657284
$689$	6.81996e6	0.547310
$690$	−595200.	−0.0475927
$691$	−8.78395e6	−0.699833	−0.349917	−	0.936781i	$-0.613790\pi$
$691$	−8.78395e6	−0.699833	−0.349917	+	0.936781i	$0.613790\pi$
$692$	1.60573e7	1.27470
$693$	6.45043e6	0.510218
$694$	3.55482e6	0.280169
$695$	2.80550e6	0.220317
$696$	−1.63680e6	−0.128077
$697$	1.57698e7	1.22955
$698$	−4.29610e6	−0.333761
$699$	1.94134e6	0.150282
$700$	−3.36000e6	−0.259176
$701$	−1.60141e7	−1.23086	−0.615428	−	0.788193i	$-0.711017\pi$
$701$	−1.60141e7	−1.23086	−0.615428	+	0.788193i	$0.711017\pi$
$702$	1.07536e6	0.0823590
$703$	192920.	0.0147228
$704$	1.58182e6	0.120289
$705$	1.20880e6	0.0915971
$706$	−1.32371e6	−0.0999495
$707$	−1.72604e7	−1.29868
$708$	−2.24224e6	−0.168112
$709$	−1.91354e7	−1.42962	−0.714811	−	0.699318i	$-0.753487\pi$
$709$	−1.91354e7	−1.42962	−0.714811	+	0.699318i	$0.753487\pi$
$710$	−1.63240e6	−0.121529
$711$	7.57272e6	0.561795
$712$	−1.21644e7	−0.899271
$713$	−7.28525e6	−0.536686
$714$	2.57741e6	0.189207
$715$	−1.05820e6	−0.0774110
$716$	1.66449e7	1.21338
$717$	195520.	0.0142034
$718$	−518640.	−0.0375452
$719$	1.02934e7	0.742566	0.371283	−	0.928520i	$-0.378918\pi$
$719$	1.02934e7	0.742566	0.371283	+	0.928520i	$0.378918\pi$
$720$	−3.72280e6	−0.267632
$721$	−3.74477e6	−0.268279
$722$	−2.70500e6	−0.193119
$723$	443192.	0.0315316
$724$	2.99874e6	0.212615
$725$	−2.13125e6	−0.150588
$726$	1.11318e6	0.0783831
$727$	−1.93264e7	−1.35618	−0.678088	−	0.734981i	$-0.737191\pi$
$727$	−1.93264e7	−1.35618	−0.678088	+	0.734981i	$0.737191\pi$
$728$	−6.58944e6	−0.460808
$729$	−8.98715e6	−0.626330
$730$	−1.93870e6	−0.134649
$731$	2.08743e6	0.144484
$732$	3.61782e6	0.249557
$733$	5.26197e6	0.361733	0.180866	−	0.983508i	$-0.442110\pi$
$733$	5.26197e6	0.361733	0.180866	+	0.983508i	$0.442110\pi$
$734$	−2.99986e6	−0.205523
$735$	−2.00570e6	−0.136945
$736$	1.53324e7	1.04331
$737$	−9.02386e6	−0.611961
$738$	4.26669e6	0.288370
$739$	2.82944e7	1.90585	0.952927	−	0.303199i	$-0.0980548\pi$
$739$	2.82944e7	1.90585	0.952927	+	0.303199i	$0.0980548\pi$
$740$	−127400.	−0.00855244
$741$	−1.21264e6	−0.0811309
$742$	9.15686e6	0.610572
$743$	2.09863e7	1.39464	0.697321	−	0.716759i	$-0.254375\pi$
$743$	2.09863e7	1.39464	0.697321	+	0.716759i	$0.254375\pi$
$744$	−1.17504e6	−0.0778252
$745$	1.00938e7	0.666288
$746$	−4.47615e6	−0.294481
$747$	−3.79453e6	−0.248804
$748$	−6.95363e6	−0.454420
$749$	3.03921e7	1.97950
$750$	−125000.	−0.00811441
$751$	−1.89668e7	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$751$	−1.89668e7	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$752$	−7.92973e6	−0.511345
$753$	6.57499e6	0.422579
$754$	−1.95052e6	−0.124946
$755$	−1.11662e7	−0.712915
$756$	−1.01069e7	−0.643151
$757$	−1.08257e7	−0.686617	−0.343309	−	0.939223i	$-0.611548\pi$
$757$	−1.08257e7	−0.686617	−0.343309	+	0.939223i	$0.611548\pi$
$758$	6.31868e6	0.399442
$759$	1.76179e6	0.111007
$760$	−3.18000e6	−0.199707
$761$	1.90534e7	1.19264	0.596322	−	0.802745i	$-0.296628\pi$
$761$	1.90534e7	1.19264	0.596322	+	0.802745i	$0.296628\pi$
$762$	−564416.	−0.0352137
$763$	7.07136e6	0.439736
$764$	−1.31475e7	−0.814908
$765$	9.52265e6	0.588307
$766$	684432.	0.0421462
$767$	−5.72572e6	−0.351432
$768$	121856.	0.00745494
$769$	−1.57826e7	−0.962415	−0.481208	−	0.876607i	$-0.659802\pi$
$769$	−1.57826e7	−0.962415	−0.481208	+	0.876607i	$0.659802\pi$
$770$	−1.42080e6	−0.0863587
$771$	−5.22497e6	−0.316554
$772$	−1.47577e6	−0.0891199
$773$	−2.44049e7	−1.46902	−0.734510	−	0.678598i	$-0.762588\pi$
$773$	−2.44049e7	−1.46902	−0.734510	+	0.678598i	$0.762588\pi$
$774$	564776.	0.0338863
$775$	−1.53000e6	−0.0915034
$776$	1.42846e7	0.851555
$777$	−139776.	−0.00830577
$778$	176940.	0.0104804
$779$	−9.96188e6	−0.588163
$780$	800800.	0.0471289
$781$	4.83190e6	0.283459
$782$	−9.98746e6	−0.584034
$783$	−6.41080e6	−0.373687
$784$	1.31574e7	0.764504
$785$	−6.55645e6	−0.379747
$786$	−611616.	−0.0353120
$787$	3.37607e7	1.94301	0.971505	−	0.237019i	$-0.0761704\pi$
$787$	3.37607e7	1.94301	0.971505	+	0.237019i	$0.0761704\pi$
$788$	−1.27641e7	−0.732278
$789$	−8.51334e6	−0.486864
$790$	−1.66800e6	−0.0950886
$791$	2.14771e6	0.122049
$792$	−4.03152e6	−0.228379
$793$	9.23837e6	0.521690
$794$	−1.09135e7	−0.614344
$795$	−2.38460e6	−0.133813
$796$	−2.42200e7	−1.35485
$797$	2.19885e7	1.22617	0.613083	−	0.790019i	$-0.289929\pi$
$797$	2.19885e7	1.22617	0.613083	+	0.790019i	$0.289929\pi$
$798$	−1.62816e6	−0.0905086
$799$	2.02837e7	1.12403
$800$	3.22000e6	0.177882
$801$	−2.30110e7	−1.26723
$802$	8.09360e6	0.444330
$803$	5.73855e6	0.314061
$804$	6.82886e6	0.372570
$805$	1.42848e7	0.776935
$806$	−1.40026e6	−0.0759224
$807$	5.76436e6	0.311578
$808$	1.07878e7	0.581303
$809$	−2.93597e7	−1.57717	−0.788587	−	0.614923i	$-0.789187\pi$
$809$	−2.93597e7	−1.57717	−0.788587	+	0.614923i	$0.789187\pi$
$810$	2.38205e6	0.127567
$811$	3.17703e7	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$811$	3.17703e7	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$812$	1.83322e7	0.975716
$813$	372992.	0.0197912
$814$	−53872.0	−0.00284972
$815$	−3.86410e6	−0.203777
$816$	4.40307e6	0.231489
$817$	−1.31864e6	−0.0691148
$818$	−5.42414e6	−0.283431
$819$	−1.24650e7	−0.649357
$820$	6.57860e6	0.341664
$821$	−2.71430e6	−0.140540	−0.0702699	−	0.997528i	$-0.522386\pi$
$821$	−2.71430e6	−0.140540	−0.0702699	+	0.997528i	$0.522386\pi$
$822$	1.15934e6	0.0598457
$823$	−1.25866e7	−0.647753	−0.323877	−	0.946099i	$-0.604986\pi$
$823$	−1.25866e7	−0.647753	−0.323877	+	0.946099i	$0.604986\pi$
$824$	2.34048e6	0.120084
$825$	370000.	0.0189263
$826$	−7.68768e6	−0.392053
$827$	−8.72355e6	−0.443537	−0.221768	−	0.975099i	$-0.571183\pi$
$827$	−8.72355e6	−0.443537	−0.221768	+	0.975099i	$0.571183\pi$
$828$	1.89155e7	0.958829
$829$	−1.06178e7	−0.536597	−0.268299	−	0.963336i	$-0.586461\pi$
$829$	−1.06178e7	−0.536597	−0.268299	+	0.963336i	$0.586461\pi$
$830$	835800.	0.0421121
$831$	441192.	0.0221628
$832$	−3.05677e6	−0.153093
$833$	−3.36556e7	−1.68053
$834$	−897760.	−0.0446936
$835$	9.91680e6	0.492216
$836$	4.39264e6	0.217375
$837$	−4.60224e6	−0.227068
$838$	7.43492e6	0.365735
$839$	1.67765e7	0.822805	0.411403	−	0.911454i	$-0.365039\pi$
$839$	1.67765e7	0.822805	0.411403	+	0.911454i	$0.365039\pi$
$840$	2.30400e6	0.112664
$841$	−8.88305e6	−0.433084
$842$	7.10500e6	0.345370
$843$	768792.	0.0372597
$844$	−3.09583e7	−1.49596
$845$	−7.23742e6	−0.348692
$846$	5.48795e6	0.263624
$847$	−2.67162e7	−1.27958
$848$	1.56430e7	0.747016
$849$	1.32754e6	0.0632087
$850$	−2.09750e6	−0.0995760
$851$	541632.	0.0256378
$852$	−3.65658e6	−0.172574
$853$	−2.20186e7	−1.03613	−0.518067	−	0.855340i	$-0.673348\pi$
$853$	−2.20186e7	−1.03613	−0.518067	+	0.855340i	$0.673348\pi$
$854$	1.24040e7	0.581991
$855$	−6.01550e6	−0.281421
$856$	−1.89950e7	−0.886045
$857$	3.16676e7	1.47287	0.736434	−	0.676510i	$-0.236508\pi$
$857$	3.16676e7	1.47287	0.736434	+	0.676510i	$0.236508\pi$
$858$	338624.	0.0157036
$859$	1.58064e7	0.730886	0.365443	−	0.930834i	$-0.380918\pi$
$859$	1.58064e7	0.730886	0.365443	+	0.930834i	$0.380918\pi$
$860$	870800.	0.0401488
$861$	7.21766e6	0.331809
$862$	−8.12410e6	−0.372398
$863$	−1.44287e7	−0.659476	−0.329738	−	0.944072i	$-0.606960\pi$
$863$	−1.44287e7	−0.659476	−0.329738	+	0.944072i	$0.606960\pi$
$864$	9.68576e6	0.441417
$865$	−1.43368e7	−0.651499
$866$	1.45257e7	0.658178
$867$	−5.58331e6	−0.252257
$868$	1.31604e7	0.592886
$869$	4.93728e6	0.221788
$870$	682000.	0.0305482
$871$	1.74380e7	0.778845
$872$	−4.41960e6	−0.196830
$873$	2.70216e7	1.19999
$874$	6.30912e6	0.279377
$875$	3.00000e6	0.132465
$876$	−4.34269e6	−0.191205
$877$	247902.	0.0108838	0.00544191	−	0.999985i	$-0.498268\pi$
$877$	247902.	0.0108838	0.00544191	+	0.999985i	$0.498268\pi$
$878$	−1.08206e7	−0.473711
$879$	−8.77922e6	−0.383252
$880$	−2.42720e6	−0.105657
$881$	4.10268e7	1.78085	0.890426	−	0.455128i	$-0.150406\pi$
$881$	4.10268e7	1.78085	0.890426	+	0.455128i	$0.150406\pi$
$882$	−9.10588e6	−0.394140
$883$	4.18015e7	1.80422	0.902112	−	0.431503i	$-0.142016\pi$
$883$	4.18015e7	1.80422	0.902112	+	0.431503i	$0.142016\pi$
$884$	1.34374e7	0.578343
$885$	2.00200e6	0.0859223
$886$	−1.30305e7	−0.557669
$887$	−2.10476e7	−0.898241	−0.449120	−	0.893471i	$-0.648263\pi$
$887$	−2.10476e7	−0.898241	−0.449120	+	0.893471i	$0.648263\pi$
$888$	87360.0	0.00371775
$889$	1.35460e7	0.574852
$890$	5.06850e6	0.214489
$891$	−7.05087e6	−0.297542
$892$	−3.14041e7	−1.32152
$893$	−1.28133e7	−0.537690
$894$	−3.23000e6	−0.135163
$895$	−1.48615e7	−0.620162
$896$	−3.57581e7	−1.48800
$897$	−3.40454e6	−0.141279
$898$	−1.01990e6	−0.0422053
$899$	8.34768e6	0.344482
$900$	3.97250e6	0.163477
$901$	−4.00136e7	−1.64208
$902$	2.78181e6	0.113844
$903$	955392.	0.0389908
$904$	−1.34232e6	−0.0546305
$905$	−2.67745e6	−0.108668
$906$	3.57318e6	0.144622
$907$	7.48309e6	0.302039	0.151019	−	0.988531i	$-0.451744\pi$
$907$	7.48309e6	0.302039	0.151019	+	0.988531i	$0.451744\pi$
$908$	653744.	0.0263144
$909$	2.04068e7	0.819155
$910$	2.74560e6	0.109909
$911$	−6.63165e6	−0.264744	−0.132372	−	0.991200i	$-0.542259\pi$
$911$	−6.63165e6	−0.264744	−0.132372	+	0.991200i	$0.542259\pi$
$912$	−2.78144e6	−0.110734
$913$	−2.47397e6	−0.0982239
$914$	2.44168e6	0.0966772
$915$	−3.23020e6	−0.127549
$916$	1.66883e7	0.657163
$917$	1.46788e7	0.576457
$918$	−6.30928e6	−0.247100
$919$	−1.68976e7	−0.659990	−0.329995	−	0.943983i	$-0.607047\pi$
$919$	−1.68976e7	−0.659990	−0.329995	+	0.943983i	$0.607047\pi$
$920$	−8.92800e6	−0.347764
$921$	9.51003e6	0.369431
$922$	−8.14420e6	−0.315516
$923$	−9.33733e6	−0.360760
$924$	−3.18259e6	−0.122631
$925$	113750.	0.00437116
$926$	4.04587e6	0.155055
$927$	4.42741e6	0.169219
$928$	−1.75683e7	−0.669669
$929$	−1.28653e7	−0.489081	−0.244541	−	0.969639i	$-0.578637\pi$
$929$	−1.28653e7	−0.489081	−0.244541	+	0.969639i	$0.578637\pi$
$930$	489600.	0.0185624
$931$	2.12604e7	0.803892
$932$	1.35894e7	0.512459
$933$	9.49219e6	0.356995
$934$	6.50194e6	0.243880
$935$	6.20860e6	0.232255
$936$	7.79064e6	0.290659
$937$	1.06887e7	0.397718	0.198859	−	0.980028i	$-0.436276\pi$
$937$	1.06887e7	0.397718	0.198859	+	0.980028i	$0.436276\pi$
$938$	2.34132e7	0.868870
$939$	5.71766e6	0.211619
$940$	8.46160e6	0.312344
$941$	2.82455e7	1.03986	0.519930	−	0.854209i	$-0.325958\pi$
$941$	2.82455e7	1.03986	0.519930	+	0.854209i	$0.325958\pi$
$942$	2.09806e6	0.0770356
$943$	−2.79684e7	−1.02421
$944$	−1.31331e7	−0.479665
$945$	9.02400e6	0.328715
$946$	368224.	0.0133778
$947$	−1.70892e7	−0.619222	−0.309611	−	0.950863i	$-0.600199\pi$
$947$	−1.70892e7	−0.619222	−0.309611	+	0.950863i	$0.600199\pi$
$948$	−3.73632e6	−0.135028
$949$	−1.10894e7	−0.399706
$950$	1.32500e6	0.0476329
$951$	−8.49849e6	−0.304713
$952$	3.86611e7	1.38255
$953$	2.22259e7	0.792735	0.396367	−	0.918092i	$-0.370271\pi$
$953$	2.22259e7	0.792735	0.396367	+	0.918092i	$0.370271\pi$
$954$	−1.08261e7	−0.385124
$955$	1.17388e7	0.416500
$956$	1.36864e6	0.0484333
$957$	−2.01872e6	−0.0712519
$958$	−6.55872e6	−0.230890
$959$	−2.78243e7	−0.976961
$960$	1.06880e6	0.0374299
$961$	−2.26364e7	−0.790678
$962$	104104.	0.00362685
$963$	−3.59323e7	−1.24859
$964$	3.10234e6	0.107522
$965$	1.31765e6	0.0455493
$966$	−4.57114e6	−0.157609
$967$	2.41551e7	0.830696	0.415348	−	0.909663i	$-0.363660\pi$
$967$	2.41551e7	0.830696	0.415348	+	0.909663i	$0.363660\pi$
$968$	1.66976e7	0.572752
$969$	7.11472e6	0.243416
$970$	−5.95190e6	−0.203108
$971$	−5.48313e7	−1.86630	−0.933149	−	0.359491i	$-0.882950\pi$
$971$	−5.48313e7	−1.86630	−0.933149	+	0.359491i	$0.882950\pi$
$972$	1.81273e7	0.615415
$973$	2.15462e7	0.729608
$974$	−1.70639e7	−0.576344
$975$	−715000.	−0.0240877
$976$	2.11901e7	0.712047
$977$	−1.56612e7	−0.524915	−0.262457	−	0.964944i	$-0.584533\pi$
$977$	−1.56612e7	−0.524915	−0.262457	+	0.964944i	$0.584533\pi$
$978$	1.23651e6	0.0413382
$979$	−1.50028e7	−0.500281
$980$	−1.40399e7	−0.466981
$981$	−8.36041e6	−0.277367
$982$	3.02530e6	0.100113
$983$	−1.63420e7	−0.539412	−0.269706	−	0.962943i	$-0.586927\pi$
$983$	−1.63420e7	−0.539412	−0.269706	+	0.962943i	$0.586927\pi$
$984$	−4.51104e6	−0.148521
$985$	1.13966e7	0.374268
$986$	1.14440e7	0.374873
$987$	9.28358e6	0.303335
$988$	−8.48848e6	−0.276654
$989$	−3.70214e6	−0.120355
$990$	1.67980e6	0.0544715
$991$	1.37576e7	0.444997	0.222498	−	0.974933i	$-0.428579\pi$
$991$	1.37576e7	0.444997	0.222498	+	0.974933i	$0.428579\pi$
$992$	−1.26121e7	−0.406919
$993$	−1.23994e7	−0.399050
$994$	−1.25368e7	−0.402459
$995$	2.16250e7	0.692466
$996$	1.87219e6	0.0598001
$997$	−1.29097e7	−0.411320	−0.205660	−	0.978624i	$-0.565934\pi$
$997$	−1.29097e7	−0.411320	−0.205660	+	0.978624i	$0.565934\pi$
$998$	−1.29838e7	−0.412644
$999$	342160.	0.0108471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.6.a.a.1.1	✓	1
3.2	odd	2		45.6.a.b.1.1		1
4.3	odd	2		80.6.a.e.1.1		1
5.2	odd	4		25.6.b.a.24.2		2
5.3	odd	4		25.6.b.a.24.1		2
5.4	even	2		25.6.a.a.1.1		1
7.6	odd	2		245.6.a.b.1.1		1
8.3	odd	2		320.6.a.g.1.1		1
8.5	even	2		320.6.a.j.1.1		1
11.10	odd	2		605.6.a.a.1.1		1
12.11	even	2		720.6.a.a.1.1		1
13.12	even	2		845.6.a.b.1.1		1
15.2	even	4		225.6.b.e.199.1		2
15.8	even	4		225.6.b.e.199.2		2
15.14	odd	2		225.6.a.f.1.1		1
20.3	even	4		400.6.c.j.49.2		2
20.7	even	4		400.6.c.j.49.1		2
20.19	odd	2		400.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	1.1	even	1	trivial
25.6.a.a.1.1		1	5.4	even	2
25.6.b.a.24.1		2	5.3	odd	4
25.6.b.a.24.2		2	5.2	odd	4
45.6.a.b.1.1		1	3.2	odd	2
80.6.a.e.1.1		1	4.3	odd	2
225.6.a.f.1.1		1	15.14	odd	2
225.6.b.e.199.1		2	15.2	even	4
225.6.b.e.199.2		2	15.8	even	4
245.6.a.b.1.1		1	7.6	odd	2
320.6.a.g.1.1		1	8.3	odd	2
320.6.a.j.1.1		1	8.5	even	2
400.6.a.g.1.1		1	20.19	odd	2
400.6.c.j.49.1		2	20.7	even	4
400.6.c.j.49.2		2	20.3	even	4
605.6.a.a.1.1		1	11.10	odd	2
720.6.a.a.1.1		1	12.11	even	2
845.6.a.b.1.1		1	13.12	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:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	5.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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