LMFDB - Embedded newform 72.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,6,Mod(1,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	72.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$11.5476350265$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	72.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−38.0000	−0.679765	−0.339882	−	0.940468i	$-0.610387\pi$
$5$	−38.0000	−0.679765	−0.339882	+	0.940468i	$0.610387\pi$
$6$	0	0
$7$	120.000	0.925627	0.462814	−	0.886456i	$-0.346840\pi$
$7$	120.000	0.925627	0.462814	+	0.886456i	$0.346840\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−524.000	−1.30572	−0.652859	−	0.757479i	$-0.726431\pi$
$11$	−524.000	−1.30572	−0.652859	+	0.757479i	$0.726431\pi$
$12$	0	0
$13$	−962.000	−1.57876	−0.789381	−	0.613904i	$-0.789598\pi$
$13$	−962.000	−1.57876	−0.789381	+	0.613904i	$0.789598\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1358.00	1.13967	0.569833	−	0.821761i	$-0.307008\pi$
$17$	1358.00	1.13967	0.569833	+	0.821761i	$0.307008\pi$
$18$	0	0
$19$	−2284.00	−1.45148	−0.725742	−	0.687967i	$-0.758503\pi$
$19$	−2284.00	−1.45148	−0.725742	+	0.687967i	$0.758503\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2552.00	−1.00591	−0.502957	−	0.864311i	$-0.667755\pi$
$23$	−2552.00	−1.00591	−0.502957	+	0.864311i	$0.667755\pi$
$24$	0	0
$25$	−1681.00	−0.537920
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3966.00	−0.875705	−0.437852	−	0.899047i	$-0.644261\pi$
$29$	−3966.00	−0.875705	−0.437852	+	0.899047i	$0.644261\pi$
$30$	0	0
$31$	−2992.00	−0.559187	−0.279594	−	0.960118i	$-0.590200\pi$
$31$	−2992.00	−0.559187	−0.279594	+	0.960118i	$0.590200\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4560.00	−0.629209
$36$	0	0
$37$	13206.0	1.58587	0.792934	−	0.609308i	$-0.208553\pi$
$37$	13206.0	1.58587	0.792934	+	0.609308i	$0.208553\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	15126.0	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$41$	15126.0	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$42$	0	0
$43$	−7316.00	−0.603396	−0.301698	−	0.953404i	$-0.597553\pi$
$43$	−7316.00	−0.603396	−0.301698	+	0.953404i	$0.597553\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6960.00	0.459584	0.229792	−	0.973240i	$-0.426195\pi$
$47$	6960.00	0.459584	0.229792	+	0.973240i	$0.426195\pi$
$48$	0	0
$49$	−2407.00	−0.143214
$50$	0	0
$51$	0	0
$52$	0	0
$53$	17482.0	0.854873	0.427436	−	0.904045i	$-0.359417\pi$
$53$	17482.0	0.854873	0.427436	+	0.904045i	$0.359417\pi$
$54$	0	0
$55$	19912.0	0.887581
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−33884.0	−1.26726	−0.633628	−	0.773638i	$-0.718435\pi$
$59$	−33884.0	−1.26726	−0.633628	+	0.773638i	$0.718435\pi$
$60$	0	0
$61$	39118.0	1.34602	0.673011	−	0.739633i	$-0.265001\pi$
$61$	39118.0	1.34602	0.673011	+	0.739633i	$0.265001\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	36556.0	1.07319
$66$	0	0
$67$	32996.0	0.897996	0.448998	−	0.893533i	$-0.351781\pi$
$67$	32996.0	0.897996	0.448998	+	0.893533i	$0.351781\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14248.0	−0.335435	−0.167717	−	0.985835i	$-0.553640\pi$
$71$	−14248.0	−0.335435	−0.167717	+	0.985835i	$0.553640\pi$
$72$	0	0
$73$	−35990.0	−0.790451	−0.395225	−	0.918584i	$-0.629333\pi$
$73$	−35990.0	−0.790451	−0.395225	+	0.918584i	$0.629333\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−62880.0	−1.20861
$78$	0	0
$79$	−29888.0	−0.538802	−0.269401	−	0.963028i	$-0.586826\pi$
$79$	−29888.0	−0.538802	−0.269401	+	0.963028i	$0.586826\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	51884.0	0.826682	0.413341	−	0.910576i	$-0.364362\pi$
$83$	51884.0	0.826682	0.413341	+	0.910576i	$0.364362\pi$
$84$	0	0
$85$	−51604.0	−0.774704
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−30714.0	−0.411018	−0.205509	−	0.978655i	$-0.565885\pi$
$89$	−30714.0	−0.411018	−0.205509	+	0.978655i	$0.565885\pi$
$90$	0	0
$91$	−115440.	−1.46135
$92$	0	0
$93$	0	0
$94$	0	0
$95$	86792.0	0.986667
$96$	0	0
$97$	−48478.0	−0.523137	−0.261568	−	0.965185i	$-0.584240\pi$
$97$	−48478.0	−0.523137	−0.261568	+	0.965185i	$0.584240\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−93222.0	−0.909316	−0.454658	−	0.890666i	$-0.650239\pi$
$101$	−93222.0	−0.909316	−0.454658	+	0.890666i	$0.650239\pi$
$102$	0	0
$103$	2296.00	0.0213245	0.0106622	−	0.999943i	$-0.496606\pi$
$103$	2296.00	0.0213245	0.0106622	+	0.999943i	$0.496606\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−38988.0	−0.329209	−0.164604	−	0.986360i	$-0.552635\pi$
$107$	−38988.0	−0.329209	−0.164604	+	0.986360i	$0.552635\pi$
$108$	0	0
$109$	6238.00	0.0502897	0.0251449	−	0.999684i	$-0.491995\pi$
$109$	6238.00	0.0502897	0.0251449	+	0.999684i	$0.491995\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−213618.	−1.57377	−0.786886	−	0.617099i	$-0.788308\pi$
$113$	−213618.	−1.57377	−0.786886	+	0.617099i	$0.788308\pi$
$114$	0	0
$115$	96976.0	0.683785
$116$	0	0
$117$	0	0
$118$	0	0
$119$	162960.	1.05491
$120$	0	0
$121$	113525.	0.704901
$122$	0	0
$123$	0	0
$124$	0	0
$125$	182628.	1.04542
$126$	0	0
$127$	205072.	1.12823	0.564114	−	0.825697i	$-0.309218\pi$
$127$	205072.	1.12823	0.564114	+	0.825697i	$0.309218\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−350116.	−1.78252	−0.891259	−	0.453495i	$-0.850177\pi$
$131$	−350116.	−1.78252	−0.891259	+	0.453495i	$0.850177\pi$
$132$	0	0
$133$	−274080.	−1.34353
$134$	0	0
$135$	0	0
$136$	0	0
$137$	234486.	1.06737	0.533686	−	0.845683i	$-0.320807\pi$
$137$	234486.	1.06737	0.533686	+	0.845683i	$0.320807\pi$
$138$	0	0
$139$	16428.0	0.0721187	0.0360593	−	0.999350i	$-0.488519\pi$
$139$	16428.0	0.0721187	0.0360593	+	0.999350i	$0.488519\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	504088.	2.06142
$144$	0	0
$145$	150708.	0.595273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	92426.0	0.341058	0.170529	−	0.985353i	$-0.445452\pi$
$149$	92426.0	0.341058	0.170529	+	0.985353i	$0.445452\pi$
$150$	0	0
$151$	350984.	1.25269	0.626347	−	0.779544i	$-0.284549\pi$
$151$	350984.	1.25269	0.626347	+	0.779544i	$0.284549\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	113696.	0.380116
$156$	0	0
$157$	46318.0	0.149969	0.0749844	−	0.997185i	$-0.476109\pi$
$157$	46318.0	0.149969	0.0749844	+	0.997185i	$0.476109\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−306240.	−0.931102
$162$	0	0
$163$	−394908.	−1.16420	−0.582099	−	0.813118i	$-0.697768\pi$
$163$	−394908.	−1.16420	−0.582099	+	0.813118i	$0.697768\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−514344.	−1.42713	−0.713563	−	0.700591i	$-0.752920\pi$
$167$	−514344.	−1.42713	−0.713563	+	0.700591i	$0.752920\pi$
$168$	0	0
$169$	554151.	1.49249
$170$	0	0
$171$	0	0
$172$	0	0
$173$	497874.	1.26475	0.632374	−	0.774663i	$-0.282080\pi$
$173$	497874.	1.26475	0.632374	+	0.774663i	$0.282080\pi$
$174$	0	0
$175$	−201720.	−0.497913
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−711252.	−1.65917	−0.829585	−	0.558380i	$-0.811423\pi$
$179$	−711252.	−1.65917	−0.829585	+	0.558380i	$0.811423\pi$
$180$	0	0
$181$	471366.	1.06945	0.534727	−	0.845025i	$-0.320415\pi$
$181$	471366.	1.06945	0.534727	+	0.845025i	$0.320415\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−501828.	−1.07802
$186$	0	0
$187$	−711592.	−1.48808
$188$	0	0
$189$	0	0
$190$	0	0
$191$	646080.	1.28145	0.640727	−	0.767769i	$-0.278633\pi$
$191$	646080.	1.28145	0.640727	+	0.767769i	$0.278633\pi$
$192$	0	0
$193$	−826558.	−1.59728	−0.798638	−	0.601811i	$-0.794446\pi$
$193$	−826558.	−1.59728	−0.798638	+	0.601811i	$0.794446\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	126138.	0.231569	0.115784	−	0.993274i	$-0.463062\pi$
$197$	126138.	0.231569	0.115784	+	0.993274i	$0.463062\pi$
$198$	0	0
$199$	−119144.	−0.213275	−0.106637	−	0.994298i	$-0.534008\pi$
$199$	−119144.	−0.213275	−0.106637	+	0.994298i	$0.534008\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−475920.	−0.810576
$204$	0	0
$205$	−574788.	−0.955263
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.19682e6	1.89523
$210$	0	0
$211$	341620.	0.528247	0.264124	−	0.964489i	$-0.414917\pi$
$211$	341620.	0.528247	0.264124	+	0.964489i	$0.414917\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	278008.	0.410167
$216$	0	0
$217$	−359040.	−0.517599
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.30640e6	−1.79926
$222$	0	0
$223$	−523088.	−0.704389	−0.352195	−	0.935927i	$-0.614564\pi$
$223$	−523088.	−0.704389	−0.352195	+	0.935927i	$0.614564\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.03261e6	−1.33006	−0.665032	−	0.746815i	$-0.731582\pi$
$227$	−1.03261e6	−1.33006	−0.665032	+	0.746815i	$0.731582\pi$
$228$	0	0
$229$	50422.0	0.0635377	0.0317688	−	0.999495i	$-0.489886\pi$
$229$	50422.0	0.0635377	0.0317688	+	0.999495i	$0.489886\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	991030.	1.19591	0.597953	−	0.801531i	$-0.295981\pi$
$233$	991030.	1.19591	0.597953	+	0.801531i	$0.295981\pi$
$234$	0	0
$235$	−264480.	−0.312409
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−514864.	−0.583039	−0.291520	−	0.956565i	$-0.594161\pi$
$239$	−514864.	−0.583039	−0.291520	+	0.956565i	$0.594161\pi$
$240$	0	0
$241$	480498.	0.532904	0.266452	−	0.963848i	$-0.414149\pi$
$241$	480498.	0.532904	0.266452	+	0.963848i	$0.414149\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	91466.0	0.0973519
$246$	0	0
$247$	2.19721e6	2.29155
$248$	0	0
$249$	0	0
$250$	0	0
$251$	768260.	0.769704	0.384852	−	0.922978i	$-0.374252\pi$
$251$	768260.	0.769704	0.384852	+	0.922978i	$0.374252\pi$
$252$	0	0
$253$	1.33725e6	1.31344
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−316162.	−0.298591	−0.149296	−	0.988793i	$-0.547701\pi$
$257$	−316162.	−0.298591	−0.149296	+	0.988793i	$0.547701\pi$
$258$	0	0
$259$	1.58472e6	1.46792
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.33017e6	1.18582	0.592908	−	0.805270i	$-0.297980\pi$
$263$	1.33017e6	1.18582	0.592908	+	0.805270i	$0.297980\pi$
$264$	0	0
$265$	−664316.	−0.581112
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−812590.	−0.684685	−0.342342	−	0.939575i	$-0.611220\pi$
$269$	−812590.	−0.684685	−0.342342	+	0.939575i	$0.611220\pi$
$270$	0	0
$271$	−1.99235e6	−1.64795	−0.823973	−	0.566629i	$-0.808247\pi$
$271$	−1.99235e6	−1.64795	−0.823973	+	0.566629i	$0.808247\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	880844.	0.702372
$276$	0	0
$277$	356134.	0.278878	0.139439	−	0.990231i	$-0.455470\pi$
$277$	356134.	0.278878	0.139439	+	0.990231i	$0.455470\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−644986.	−0.487287	−0.243643	−	0.969865i	$-0.578343\pi$
$281$	−644986.	−0.487287	−0.243643	+	0.969865i	$0.578343\pi$
$282$	0	0
$283$	−677188.	−0.502624	−0.251312	−	0.967906i	$-0.580862\pi$
$283$	−677188.	−0.502624	−0.251312	+	0.967906i	$0.580862\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.81512e6	1.30077
$288$	0	0
$289$	424307.	0.298838
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−414822.	−0.282288	−0.141144	−	0.989989i	$-0.545078\pi$
$293$	−414822.	−0.282288	−0.141144	+	0.989989i	$0.545078\pi$
$294$	0	0
$295$	1.28759e6	0.861436
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.45502e6	1.58810
$300$	0	0
$301$	−877920.	−0.558520
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.48648e6	−0.914978
$306$	0	0
$307$	−362028.	−0.219228	−0.109614	−	0.993974i	$-0.534961\pi$
$307$	−362028.	−0.219228	−0.109614	+	0.993974i	$0.534961\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.54342e6	−0.904861	−0.452431	−	0.891800i	$-0.649443\pi$
$311$	−1.54342e6	−0.904861	−0.452431	+	0.891800i	$0.649443\pi$
$312$	0	0
$313$	−1.54225e6	−0.889801	−0.444900	−	0.895580i	$-0.646761\pi$
$313$	−1.54225e6	−0.889801	−0.444900	+	0.895580i	$0.646761\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−33246.0	−0.0185819	−0.00929097	−	0.999957i	$-0.502957\pi$
$317$	−33246.0	−0.0185819	−0.00929097	+	0.999957i	$0.502957\pi$
$318$	0	0
$319$	2.07818e6	1.14342
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.10167e6	−1.65421
$324$	0	0
$325$	1.61712e6	0.849248
$326$	0	0
$327$	0	0
$328$	0	0
$329$	835200.	0.425403
$330$	0	0
$331$	−1.60738e6	−0.806396	−0.403198	−	0.915113i	$-0.632101\pi$
$331$	−1.60738e6	−0.806396	−0.403198	+	0.915113i	$0.632101\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.25385e6	−0.610426
$336$	0	0
$337$	−1.44958e6	−0.695293	−0.347647	−	0.937626i	$-0.613019\pi$
$337$	−1.44958e6	−0.695293	−0.347647	+	0.937626i	$0.613019\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.56781e6	0.730141
$342$	0	0
$343$	−2.30568e6	−1.05819
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−474588.	−0.211589	−0.105794	−	0.994388i	$-0.533739\pi$
$347$	−474588.	−0.211589	−0.105794	+	0.994388i	$0.533739\pi$
$348$	0	0
$349$	−2.98869e6	−1.31346	−0.656731	−	0.754125i	$-0.728061\pi$
$349$	−2.98869e6	−1.31346	−0.656731	+	0.754125i	$0.728061\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.26480e6	−1.39451	−0.697253	−	0.716826i	$-0.745594\pi$
$353$	−3.26480e6	−1.39451	−0.697253	+	0.716826i	$0.745594\pi$
$354$	0	0
$355$	541424.	0.228017
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.92430e6	−0.788017	−0.394009	−	0.919107i	$-0.628912\pi$
$359$	−1.92430e6	−0.788017	−0.394009	+	0.919107i	$0.628912\pi$
$360$	0	0
$361$	2.74056e6	1.10680
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.36762e6	0.537320
$366$	0	0
$367$	−1.50013e6	−0.581384	−0.290692	−	0.956817i	$-0.593886\pi$
$367$	−1.50013e6	−0.581384	−0.290692	+	0.956817i	$0.593886\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.09784e6	0.791293
$372$	0	0
$373$	−4.70185e6	−1.74983	−0.874917	−	0.484273i	$-0.839084\pi$
$373$	−4.70185e6	−1.74983	−0.874917	+	0.484273i	$0.839084\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.81529e6	1.38253
$378$	0	0
$379$	1.51526e6	0.541863	0.270931	−	0.962599i	$-0.412668\pi$
$379$	1.51526e6	0.541863	0.270931	+	0.962599i	$0.412668\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−155520.	−0.0541738	−0.0270869	−	0.999633i	$-0.508623\pi$
$383$	−155520.	−0.0541738	−0.0270869	+	0.999633i	$0.508623\pi$
$384$	0	0
$385$	2.38944e6	0.821569
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.05084e6	1.02222	0.511112	−	0.859514i	$-0.329234\pi$
$389$	3.05084e6	1.02222	0.511112	+	0.859514i	$0.329234\pi$
$390$	0	0
$391$	−3.46562e6	−1.14641
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.13574e6	0.366258
$396$	0	0
$397$	196574.	0.0625965	0.0312982	−	0.999510i	$-0.490036\pi$
$397$	196574.	0.0625965	0.0312982	+	0.999510i	$0.490036\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	752910.	0.233820	0.116910	−	0.993142i	$-0.462701\pi$
$401$	752910.	0.233820	0.116910	+	0.993142i	$0.462701\pi$
$402$	0	0
$403$	2.87830e6	0.882824
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.91994e6	−2.07070
$408$	0	0
$409$	5.61695e6	1.66032	0.830162	−	0.557523i	$-0.188248\pi$
$409$	5.61695e6	1.66032	0.830162	+	0.557523i	$0.188248\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.06608e6	−1.17301
$414$	0	0
$415$	−1.97159e6	−0.561949
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6.35267e6	1.76775	0.883876	−	0.467722i	$-0.154925\pi$
$419$	6.35267e6	1.76775	0.883876	+	0.467722i	$0.154925\pi$
$420$	0	0
$421$	5.80991e6	1.59759	0.798793	−	0.601606i	$-0.205472\pi$
$421$	5.80991e6	1.59759	0.798793	+	0.601606i	$0.205472\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.28280e6	−0.613049
$426$	0	0
$427$	4.69416e6	1.24591
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.31099e6	−0.339944	−0.169972	−	0.985449i	$-0.554368\pi$
$431$	−1.31099e6	−0.339944	−0.169972	+	0.985449i	$0.554368\pi$
$432$	0	0
$433$	3.76127e6	0.964083	0.482041	−	0.876148i	$-0.339895\pi$
$433$	3.76127e6	0.964083	0.482041	+	0.876148i	$0.339895\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.82877e6	1.46007
$438$	0	0
$439$	1.00116e6	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$439$	1.00116e6	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.01542e6	−0.972124	−0.486062	−	0.873924i	$-0.661567\pi$
$443$	−4.01542e6	−0.972124	−0.486062	+	0.873924i	$0.661567\pi$
$444$	0	0
$445$	1.16713e6	0.279396
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−322466.	−0.0754863	−0.0377431	−	0.999287i	$-0.512017\pi$
$449$	−322466.	−0.0754863	−0.0377431	+	0.999287i	$0.512017\pi$
$450$	0	0
$451$	−7.92602e6	−1.83491
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.38672e6	0.993371
$456$	0	0
$457$	−6.64994e6	−1.48945	−0.744727	−	0.667369i	$-0.767421\pi$
$457$	−6.64994e6	−1.48945	−0.744727	+	0.667369i	$0.767421\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.17793e6	−0.258148	−0.129074	−	0.991635i	$-0.541200\pi$
$461$	−1.17793e6	−0.258148	−0.129074	+	0.991635i	$0.541200\pi$
$462$	0	0
$463$	5.85949e6	1.27030	0.635151	−	0.772388i	$-0.280938\pi$
$463$	5.85949e6	1.27030	0.635151	+	0.772388i	$0.280938\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.28056e6	0.908255	0.454128	−	0.890937i	$-0.349951\pi$
$467$	4.28056e6	0.908255	0.454128	+	0.890937i	$0.349951\pi$
$468$	0	0
$469$	3.95952e6	0.831209
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.83358e6	0.787866
$474$	0	0
$475$	3.83940e6	0.780782
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.75622e6	−1.14630	−0.573151	−	0.819450i	$-0.694279\pi$
$479$	−5.75622e6	−1.14630	−0.573151	+	0.819450i	$0.694279\pi$
$480$	0	0
$481$	−1.27042e7	−2.50371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.84216e6	0.355610
$486$	0	0
$487$	−5.63127e6	−1.07593	−0.537965	−	0.842967i	$-0.680807\pi$
$487$	−5.63127e6	−1.07593	−0.537965	+	0.842967i	$0.680807\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.46885e6	0.649355	0.324677	−	0.945825i	$-0.394744\pi$
$491$	3.46885e6	0.649355	0.324677	+	0.945825i	$0.394744\pi$
$492$	0	0
$493$	−5.38583e6	−0.998011
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.70976e6	−0.310488
$498$	0	0
$499$	5.98837e6	1.07661	0.538304	−	0.842751i	$-0.319065\pi$
$499$	5.98837e6	1.07661	0.538304	+	0.842751i	$0.319065\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.13058e6	−0.551703	−0.275852	−	0.961200i	$-0.588960\pi$
$503$	−3.13058e6	−0.551703	−0.275852	+	0.961200i	$0.588960\pi$
$504$	0	0
$505$	3.54244e6	0.618121
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.11965e6	−0.533717	−0.266858	−	0.963736i	$-0.585986\pi$
$509$	−3.11965e6	−0.533717	−0.266858	+	0.963736i	$0.585986\pi$
$510$	0	0
$511$	−4.31880e6	−0.731663
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−87248.0	−0.0144956
$516$	0	0
$517$	−3.64704e6	−0.600087
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−4.34939e6	−0.701994	−0.350997	−	0.936377i	$-0.614157\pi$
$521$	−4.34939e6	−0.701994	−0.350997	+	0.936377i	$0.614157\pi$
$522$	0	0
$523$	−2.08524e6	−0.333350	−0.166675	−	0.986012i	$-0.553303\pi$
$523$	−2.08524e6	−0.333350	−0.166675	+	0.986012i	$0.553303\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.06314e6	−0.637287
$528$	0	0
$529$	76361.0	0.0118640
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.45512e7	−2.21861
$534$	0	0
$535$	1.48154e6	0.223785
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.26127e6	0.186997
$540$	0	0
$541$	−2.16722e6	−0.318353	−0.159177	−	0.987250i	$-0.550884\pi$
$541$	−2.16722e6	−0.318353	−0.159177	+	0.987250i	$0.550884\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−237044.	−0.0341852
$546$	0	0
$547$	−1.02512e6	−0.146489	−0.0732444	−	0.997314i	$-0.523335\pi$
$547$	−1.02512e6	−0.146489	−0.0732444	+	0.997314i	$0.523335\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.05834e6	1.27107
$552$	0	0
$553$	−3.58656e6	−0.498730
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.08401e6	−1.24062	−0.620311	−	0.784356i	$-0.712994\pi$
$557$	−9.08401e6	−1.24062	−0.620311	+	0.784356i	$0.712994\pi$
$558$	0	0
$559$	7.03799e6	0.952619
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.45921e6	1.12476	0.562379	−	0.826880i	$-0.309886\pi$
$563$	8.45921e6	1.12476	0.562379	+	0.826880i	$0.309886\pi$
$564$	0	0
$565$	8.11748e6	1.06979
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.16334e7	1.50634	0.753172	−	0.657824i	$-0.228523\pi$
$569$	1.16334e7	1.50634	0.753172	+	0.657824i	$0.228523\pi$
$570$	0	0
$571$	−4.01840e6	−0.515779	−0.257889	−	0.966174i	$-0.583027\pi$
$571$	−4.01840e6	−0.515779	−0.257889	+	0.966174i	$0.583027\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.28991e6	0.541102
$576$	0	0
$577$	−3.55296e6	−0.444274	−0.222137	−	0.975016i	$-0.571303\pi$
$577$	−3.55296e6	−0.444274	−0.222137	+	0.975016i	$0.571303\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.22608e6	0.765199
$582$	0	0
$583$	−9.16057e6	−1.11622
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.29496e6	−0.754045	−0.377023	−	0.926204i	$-0.623052\pi$
$587$	−6.29496e6	−0.754045	−0.377023	+	0.926204i	$0.623052\pi$
$588$	0	0
$589$	6.83373e6	0.811651
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.01935e6	−1.05327	−0.526633	−	0.850093i	$-0.676546\pi$
$593$	−9.01935e6	−1.05327	−0.526633	+	0.850093i	$0.676546\pi$
$594$	0	0
$595$	−6.19248e6	−0.717088
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.24315e7	1.41565	0.707826	−	0.706387i	$-0.249676\pi$
$599$	1.24315e7	1.41565	0.707826	+	0.706387i	$0.249676\pi$
$600$	0	0
$601$	4.74476e6	0.535832	0.267916	−	0.963442i	$-0.413665\pi$
$601$	4.74476e6	0.535832	0.267916	+	0.963442i	$0.413665\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.31395e6	−0.479167
$606$	0	0
$607$	3.49784e6	0.385326	0.192663	−	0.981265i	$-0.438288\pi$
$607$	3.49784e6	0.385326	0.192663	+	0.981265i	$0.438288\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.69552e6	−0.725573
$612$	0	0
$613$	−358762.	−0.0385616	−0.0192808	−	0.999814i	$-0.506138\pi$
$613$	−358762.	−0.0385616	−0.0192808	+	0.999814i	$0.506138\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.26388e7	1.33658	0.668289	−	0.743902i	$-0.267027\pi$
$617$	1.26388e7	1.33658	0.668289	+	0.743902i	$0.267027\pi$
$618$	0	0
$619$	1.06705e7	1.11933	0.559664	−	0.828720i	$-0.310930\pi$
$619$	1.06705e7	1.11933	0.559664	+	0.828720i	$0.310930\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.68568e6	−0.380450
$624$	0	0
$625$	−1.68674e6	−0.172722
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.79337e7	1.80736
$630$	0	0
$631$	1.32621e7	1.32598	0.662991	−	0.748628i	$-0.269287\pi$
$631$	1.32621e7	1.32598	0.662991	+	0.748628i	$0.269287\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.79274e6	−0.766930
$636$	0	0
$637$	2.31553e6	0.226101
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.29879e6	−0.701626	−0.350813	−	0.936446i	$-0.614095\pi$
$641$	−7.29879e6	−0.701626	−0.350813	+	0.936446i	$0.614095\pi$
$642$	0	0
$643$	−1.97747e6	−0.188618	−0.0943088	−	0.995543i	$-0.530064\pi$
$643$	−1.97747e6	−0.188618	−0.0943088	+	0.995543i	$0.530064\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	272792.	0.0256195	0.0128098	−	0.999918i	$-0.495922\pi$
$647$	272792.	0.0256195	0.0128098	+	0.999918i	$0.495922\pi$
$648$	0	0
$649$	1.77552e7	1.65468
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.13845e6	−0.379799	−0.189900	−	0.981803i	$-0.560816\pi$
$653$	−4.13845e6	−0.379799	−0.189900	+	0.981803i	$0.560816\pi$
$654$	0	0
$655$	1.33044e7	1.21169
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−445812.	−0.0399888	−0.0199944	−	0.999800i	$-0.506365\pi$
$659$	−445812.	−0.0399888	−0.0199944	+	0.999800i	$0.506365\pi$
$660$	0	0
$661$	−3.65881e6	−0.325714	−0.162857	−	0.986650i	$-0.552071\pi$
$661$	−3.65881e6	−0.325714	−0.162857	+	0.986650i	$0.552071\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.04150e7	0.913286
$666$	0	0
$667$	1.01212e7	0.880884
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.04978e7	−1.75753
$672$	0	0
$673$	−1.42444e7	−1.21229	−0.606147	−	0.795353i	$-0.707286\pi$
$673$	−1.42444e7	−1.21229	−0.606147	+	0.795353i	$0.707286\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.33128e7	1.11634	0.558170	−	0.829727i	$-0.311504\pi$
$677$	1.33128e7	1.11634	0.558170	+	0.829727i	$0.311504\pi$
$678$	0	0
$679$	−5.81736e6	−0.484230
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.49468e6	0.122601	0.0613007	−	0.998119i	$-0.480475\pi$
$683$	1.49468e6	0.122601	0.0613007	+	0.998119i	$0.480475\pi$
$684$	0	0
$685$	−8.91047e6	−0.725561
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.68177e7	−1.34964
$690$	0	0
$691$	−1.11320e7	−0.886910	−0.443455	−	0.896297i	$-0.646247\pi$
$691$	−1.11320e7	−0.886910	−0.443455	+	0.896297i	$0.646247\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−624264.	−0.0490237
$696$	0	0
$697$	2.05411e7	1.60156
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.52327e7	1.17080	0.585398	−	0.810746i	$-0.300938\pi$
$701$	1.52327e7	1.17080	0.585398	+	0.810746i	$0.300938\pi$
$702$	0	0
$703$	−3.01625e7	−2.30186
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.11866e7	−0.841688
$708$	0	0
$709$	−8.59921e6	−0.642455	−0.321228	−	0.947002i	$-0.604095\pi$
$709$	−8.59921e6	−0.642455	−0.321228	+	0.947002i	$0.604095\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.63558e6	0.562495
$714$	0	0
$715$	−1.91553e7	−1.40128
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.84891e6	0.205521	0.102761	−	0.994706i	$-0.467232\pi$
$719$	2.84891e6	0.205521	0.102761	+	0.994706i	$0.467232\pi$
$720$	0	0
$721$	275520.	0.0197385
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.66685e6	0.471059
$726$	0	0
$727$	−8.11615e6	−0.569527	−0.284763	−	0.958598i	$-0.591915\pi$
$727$	−8.11615e6	−0.569527	−0.284763	+	0.958598i	$0.591915\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−9.93513e6	−0.687670
$732$	0	0
$733$	−1.14038e7	−0.783954	−0.391977	−	0.919975i	$-0.628209\pi$
$733$	−1.14038e7	−0.783954	−0.391977	+	0.919975i	$0.628209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.72899e7	−1.17253
$738$	0	0
$739$	−2.28780e6	−0.154102	−0.0770509	−	0.997027i	$-0.524550\pi$
$739$	−2.28780e6	−0.154102	−0.0770509	+	0.997027i	$0.524550\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.92359e7	1.94287	0.971437	−	0.237296i	$-0.0762610\pi$
$743$	2.92359e7	1.94287	0.971437	+	0.237296i	$0.0762610\pi$
$744$	0	0
$745$	−3.51219e6	−0.231839
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.67856e6	−0.304725
$750$	0	0
$751$	1.71311e7	1.10837	0.554186	−	0.832393i	$-0.313030\pi$
$751$	1.71311e7	1.10837	0.554186	+	0.832393i	$0.313030\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.33374e7	−0.851537
$756$	0	0
$757$	3.31732e6	0.210401	0.105200	−	0.994451i	$-0.466452\pi$
$757$	3.31732e6	0.210401	0.105200	+	0.994451i	$0.466452\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.28948e7	−0.807146	−0.403573	−	0.914947i	$-0.632232\pi$
$761$	−1.28948e7	−0.807146	−0.403573	+	0.914947i	$0.632232\pi$
$762$	0	0
$763$	748560.	0.0465495
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.25964e7	2.00070
$768$	0	0
$769$	1.87622e7	1.14411	0.572056	−	0.820214i	$-0.306146\pi$
$769$	1.87622e7	1.14411	0.572056	+	0.820214i	$0.306146\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6.84442e6	−0.411991	−0.205996	−	0.978553i	$-0.566043\pi$
$773$	−6.84442e6	−0.411991	−0.205996	+	0.978553i	$0.566043\pi$
$774$	0	0
$775$	5.02955e6	0.300798
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.45478e7	−2.03975
$780$	0	0
$781$	7.46595e6	0.437983
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1.76008e6	−0.101943
$786$	0	0
$787$	−2.91272e7	−1.67634	−0.838170	−	0.545409i	$-0.816374\pi$
$787$	−2.91272e7	−1.67634	−0.838170	+	0.545409i	$0.816374\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.56342e7	−1.45673
$792$	0	0
$793$	−3.76315e7	−2.12505
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.12485e7	−1.74254	−0.871272	−	0.490800i	$-0.836705\pi$
$797$	−3.12485e7	−1.74254	−0.871272	+	0.490800i	$0.836705\pi$
$798$	0	0
$799$	9.45168e6	0.523772
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.88588e7	1.03211
$804$	0	0
$805$	1.16371e7	0.632930
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.27324e7	0.683972	0.341986	−	0.939705i	$-0.388901\pi$
$809$	1.27324e7	0.683972	0.341986	+	0.939705i	$0.388901\pi$
$810$	0	0
$811$	2.65194e7	1.41583	0.707916	−	0.706297i	$-0.249636\pi$
$811$	2.65194e7	1.41583	0.707916	+	0.706297i	$0.249636\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.50065e7	0.791381
$816$	0	0
$817$	1.67097e7	0.875820
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.84134e7	1.47118	0.735590	−	0.677427i	$-0.236905\pi$
$821$	2.84134e7	1.47118	0.735590	+	0.677427i	$0.236905\pi$
$822$	0	0
$823$	3.21639e7	1.65527	0.827637	−	0.561264i	$-0.189685\pi$
$823$	3.21639e7	1.65527	0.827637	+	0.561264i	$0.189685\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.01436e7	1.53261	0.766304	−	0.642478i	$-0.222094\pi$
$827$	3.01436e7	1.53261	0.766304	+	0.642478i	$0.222094\pi$
$828$	0	0
$829$	−2.01164e7	−1.01663	−0.508315	−	0.861171i	$-0.669732\pi$
$829$	−2.01164e7	−1.01663	−0.508315	+	0.861171i	$0.669732\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.26871e6	−0.163216
$834$	0	0
$835$	1.95451e7	0.970110
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.20685e7	1.08235	0.541175	−	0.840910i	$-0.317980\pi$
$839$	2.20685e7	1.08235	0.541175	+	0.840910i	$0.317980\pi$
$840$	0	0
$841$	−4.78199e6	−0.233141
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.10577e7	−1.01454
$846$	0	0
$847$	1.36230e7	0.652476
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.37017e7	−1.59525
$852$	0	0
$853$	1.12740e7	0.530524	0.265262	−	0.964176i	$-0.414542\pi$
$853$	1.12740e7	0.530524	0.265262	+	0.964176i	$0.414542\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.67412e7	0.778634	0.389317	−	0.921104i	$-0.372711\pi$
$857$	1.67412e7	0.778634	0.389317	+	0.921104i	$0.372711\pi$
$858$	0	0
$859$	−3.26435e7	−1.50943	−0.754716	−	0.656051i	$-0.772225\pi$
$859$	−3.26435e7	−1.50943	−0.754716	+	0.656051i	$0.772225\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.54029e7	1.16106	0.580532	−	0.814237i	$-0.302844\pi$
$863$	2.54029e7	1.16106	0.580532	+	0.814237i	$0.302844\pi$
$864$	0	0
$865$	−1.89192e7	−0.859731
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.56613e7	0.703524
$870$	0	0
$871$	−3.17422e7	−1.41772
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.19154e7	0.967673
$876$	0	0
$877$	3.55846e6	0.156230	0.0781148	−	0.996944i	$-0.475110\pi$
$877$	3.55846e6	0.156230	0.0781148	+	0.996944i	$0.475110\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.10156e7	1.78037	0.890184	−	0.455602i	$-0.150576\pi$
$881$	4.10156e7	1.78037	0.890184	+	0.455602i	$0.150576\pi$
$882$	0	0
$883$	3.51736e7	1.51815	0.759075	−	0.651004i	$-0.225652\pi$
$883$	3.51736e7	1.51815	0.759075	+	0.651004i	$0.225652\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.33071e7	−1.84821	−0.924103	−	0.382144i	$-0.875186\pi$
$887$	−4.33071e7	−1.84821	−0.924103	+	0.382144i	$0.875186\pi$
$888$	0	0
$889$	2.46086e7	1.04432
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.58966e7	−0.667078
$894$	0	0
$895$	2.70276e7	1.12785
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.18663e7	0.489683
$900$	0	0
$901$	2.37406e7	0.974269
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.79119e7	−0.726977
$906$	0	0
$907$	−1.33192e7	−0.537599	−0.268800	−	0.963196i	$-0.586627\pi$
$907$	−1.33192e7	−0.537599	−0.268800	+	0.963196i	$0.586627\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.92578e6	−0.356328	−0.178164	−	0.984001i	$-0.557016\pi$
$911$	−8.92578e6	−0.356328	−0.178164	+	0.984001i	$0.557016\pi$
$912$	0	0
$913$	−2.71872e7	−1.07941
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.20139e7	−1.64995
$918$	0	0
$919$	4.13982e7	1.61694	0.808468	−	0.588540i	$-0.200297\pi$
$919$	4.13982e7	1.61694	0.808468	+	0.588540i	$0.200297\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.37066e7	0.529572
$924$	0	0
$925$	−2.21993e7	−0.853070
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.42481e7	−1.30196	−0.650980	−	0.759095i	$-0.725642\pi$
$929$	−3.42481e7	−1.30196	−0.650980	+	0.759095i	$0.725642\pi$
$930$	0	0
$931$	5.49759e6	0.207873
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.70405e7	1.01155
$936$	0	0
$937$	5.44468e6	0.202593	0.101296	−	0.994856i	$-0.467701\pi$
$937$	5.44468e6	0.202593	0.101296	+	0.994856i	$0.467701\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.54752e7	−1.30602	−0.653012	−	0.757347i	$-0.726495\pi$
$941$	−3.54752e7	−1.30602	−0.653012	+	0.757347i	$0.726495\pi$
$942$	0	0
$943$	−3.86016e7	−1.41360
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.22505e7	−0.443892	−0.221946	−	0.975059i	$-0.571241\pi$
$947$	−1.22505e7	−0.443892	−0.221946	+	0.975059i	$0.571241\pi$
$948$	0	0
$949$	3.46224e7	1.24793
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.77139e6	−0.170181	−0.0850907	−	0.996373i	$-0.527118\pi$
$953$	−4.77139e6	−0.170181	−0.0850907	+	0.996373i	$0.527118\pi$
$954$	0	0
$955$	−2.45510e7	−0.871087
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.81383e7	0.987988
$960$	0	0
$961$	−1.96771e7	−0.687309
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.14092e7	1.08577
$966$	0	0
$967$	−4.31939e7	−1.48544	−0.742722	−	0.669600i	$-0.766466\pi$
$967$	−4.31939e7	−1.48544	−0.742722	+	0.669600i	$0.766466\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.73630e7	−0.590984	−0.295492	−	0.955345i	$-0.595484\pi$
$971$	−1.73630e7	−0.590984	−0.295492	+	0.955345i	$0.595484\pi$
$972$	0	0
$973$	1.97136e6	0.0667550
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.71680e7	0.575416	0.287708	−	0.957718i	$-0.407107\pi$
$977$	1.71680e7	0.575416	0.287708	+	0.957718i	$0.407107\pi$
$978$	0	0
$979$	1.60941e7	0.536674
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.53993e7	−1.16845	−0.584225	−	0.811592i	$-0.698602\pi$
$983$	−3.53993e7	−1.16845	−0.584225	+	0.811592i	$0.698602\pi$
$984$	0	0
$985$	−4.79324e6	−0.157412
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.86704e7	0.606965
$990$	0	0
$991$	−5.23340e7	−1.69278	−0.846389	−	0.532566i	$-0.821228\pi$
$991$	−5.23340e7	−1.69278	−0.846389	+	0.532566i	$0.821228\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.52747e6	0.144977
$996$	0	0
$997$	−7.21035e6	−0.229730	−0.114865	−	0.993381i	$-0.536644\pi$
$997$	−7.21035e6	−0.229730	−0.114865	+	0.993381i	$0.536644\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.6.a.b.1.1		1
3.2	odd	2		24.6.a.c.1.1	✓	1
4.3	odd	2		144.6.a.d.1.1		1
8.3	odd	2		576.6.a.ba.1.1		1
8.5	even	2		576.6.a.bb.1.1		1
12.11	even	2		48.6.a.b.1.1		1
15.2	even	4		600.6.f.h.49.1		2
15.8	even	4		600.6.f.h.49.2		2
15.14	odd	2		600.6.a.a.1.1		1
24.5	odd	2		192.6.a.b.1.1		1
24.11	even	2		192.6.a.j.1.1		1
48.5	odd	4		768.6.d.f.385.1		2
48.11	even	4		768.6.d.m.385.2		2
48.29	odd	4		768.6.d.f.385.2		2
48.35	even	4		768.6.d.m.385.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.c.1.1	✓	1	3.2	odd	2
48.6.a.b.1.1		1	12.11	even	2
72.6.a.b.1.1		1	1.1	even	1	trivial
144.6.a.d.1.1		1	4.3	odd	2
192.6.a.b.1.1		1	24.5	odd	2
192.6.a.j.1.1		1	24.11	even	2
576.6.a.ba.1.1		1	8.3	odd	2
576.6.a.bb.1.1		1	8.5	even	2
600.6.a.a.1.1		1	15.14	odd	2
600.6.f.h.49.1		2	15.2	even	4
600.6.f.h.49.2		2	15.8	even	4
768.6.d.f.385.1		2	48.5	odd	4
768.6.d.f.385.2		2	48.29	odd	4
768.6.d.m.385.1		2	48.35	even	4
768.6.d.m.385.2		2	48.11	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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