LMFDB - Embedded newform 72.22.a.d.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [72,22,Mod(1,72)] 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("72.1"); S:= CuspForms(chi, 22); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	72.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-2080026] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$201.223687887$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2295485x - 828958533 $ x^3 - x^2 - 2295485*x - 828958533
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-386.305$ of defining polynomial
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)

$f(q)$	$=$	$q+3.08294e7 q^{5} -1.10597e9 q^{7} -1.54673e11 q^{11} +5.78668e11 q^{13} -1.59754e12 q^{17} +7.29390e12 q^{19} +1.56234e14 q^{23} +4.73615e14 q^{25} +1.80995e15 q^{29} +5.98096e15 q^{31} -3.40965e16 q^{35} +1.92628e16 q^{37} +1.01900e17 q^{41} -2.34697e17 q^{43} -1.31288e17 q^{47} +6.64628e17 q^{49} -2.51169e18 q^{53} -4.76847e18 q^{55} -4.37160e18 q^{59} -6.88287e18 q^{61} +1.78400e19 q^{65} +1.11554e19 q^{67} +4.02859e19 q^{71} +3.07636e19 q^{73} +1.71064e20 q^{77} -1.48672e20 q^{79} -9.43003e19 q^{83} -4.92511e19 q^{85} +2.18162e20 q^{89} -6.39991e20 q^{91} +2.24866e20 q^{95} +1.66723e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2080026 q^{5} - 1205282064 q^{7} - 13839247500 q^{11} + 718855551690 q^{13} - 2135189843046 q^{17} - 40122324686988 q^{19} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25} - 442708167991794 q^{29}+ \cdots - 15\!\cdots\!22 q^{97}+O(q^{100}) $ 3 * q - 2080026 * q^5 - 1205282064 * q^7 - 13839247500 * q^11 + 718855551690 * q^13 - 2135189843046 * q^17 - 40122324686988 * q^19 - 278424417682632 * q^23 + 1348043260553901 * q^25 - 442708167991794 * q^29 + 8016070162990152 * q^31 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 125648125186340562 * q^41 - 229052541499074612 * q^43 + 448613782068047712 * q^47 + 365221903446092427 * q^49 - 1406206217208267066 * q^53 - 3437829264920292504 * q^55 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 19537666262756991444 * q^65 - 33491023693155020652 * q^67 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 255759806110876888128 * q^77 - 197919973704661098024 * q^79 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 731008175840243483058 * q^89 - 546304658547739915488 * q^91 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97

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
$5$	3.08294e7	1.41182	0.705911	−	0.708300i	$-0.250538\pi$
$5$	3.08294e7	1.41182	0.705911	+	0.708300i	$0.250538\pi$
$6$	0	0
$7$	−1.10597e9	−1.47984	−0.739920	−	0.672695i	$-0.765137\pi$
$7$	−1.10597e9	−1.47984	−0.739920	+	0.672695i	$0.765137\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.54673e11	−1.79801	−0.899003	−	0.437943i	$-0.855707\pi$
$11$	−1.54673e11	−1.79801	−0.899003	+	0.437943i	$0.855707\pi$
$12$	0	0
$13$	5.78668e11	1.16419	0.582095	−	0.813120i	$-0.302233\pi$
$13$	5.78668e11	1.16419	0.582095	+	0.813120i	$0.302233\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.59754e12	−0.192193	−0.0960964	−	0.995372i	$-0.530636\pi$
$17$	−1.59754e12	−0.192193	−0.0960964	+	0.995372i	$0.530636\pi$
$18$	0	0
$19$	7.29390e12	0.272927	0.136464	−	0.990645i	$-0.456426\pi$
$19$	7.29390e12	0.272927	0.136464	+	0.990645i	$0.456426\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.56234e14	0.786381	0.393190	−	0.919457i	$-0.371371\pi$
$23$	1.56234e14	0.786381	0.393190	+	0.919457i	$0.371371\pi$
$24$	0	0
$25$	4.73615e14	0.993242
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.80995e15	0.798890	0.399445	−	0.916757i	$-0.369203\pi$
$29$	1.80995e15	0.798890	0.399445	+	0.916757i	$0.369203\pi$
$30$	0	0
$31$	5.98096e15	1.31061	0.655304	−	0.755365i	$-0.272541\pi$
$31$	5.98096e15	1.31061	0.655304	+	0.755365i	$0.272541\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.40965e16	−2.08927
$36$	0	0
$37$	1.92628e16	0.658568	0.329284	−	0.944231i	$-0.393193\pi$
$37$	1.92628e16	0.658568	0.329284	+	0.944231i	$0.393193\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.01900e17	1.18561	0.592805	−	0.805346i	$-0.298020\pi$
$41$	1.01900e17	1.18561	0.592805	+	0.805346i	$0.298020\pi$
$42$	0	0
$43$	−2.34697e17	−1.65611	−0.828053	−	0.560650i	$-0.810551\pi$
$43$	−2.34697e17	−1.65611	−0.828053	+	0.560650i	$0.810551\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.31288e17	−0.364080	−0.182040	−	0.983291i	$-0.558270\pi$
$47$	−1.31288e17	−0.364080	−0.182040	+	0.983291i	$0.558270\pi$
$48$	0	0
$49$	6.64628e17	1.18993
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.51169e18	−1.97274	−0.986368	−	0.164554i	$-0.947382\pi$
$53$	−2.51169e18	−1.97274	−0.986368	+	0.164554i	$0.947382\pi$
$54$	0	0
$55$	−4.76847e18	−2.53846
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.37160e18	−1.11351	−0.556755	−	0.830677i	$-0.687954\pi$
$59$	−4.37160e18	−1.11351	−0.556755	+	0.830677i	$0.687954\pi$
$60$	0	0
$61$	−6.88287e18	−1.23540	−0.617698	−	0.786415i	$-0.711935\pi$
$61$	−6.88287e18	−1.23540	−0.617698	+	0.786415i	$0.711935\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.78400e19	1.64363
$66$	0	0
$67$	1.11554e19	0.747650	0.373825	−	0.927499i	$-0.378046\pi$
$67$	1.11554e19	0.747650	0.373825	+	0.927499i	$0.378046\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.02859e19	1.46872	0.734362	−	0.678758i	$-0.237481\pi$
$71$	4.02859e19	1.46872	0.734362	+	0.678758i	$0.237481\pi$
$72$	0	0
$73$	3.07636e19	0.837812	0.418906	−	0.908030i	$-0.362414\pi$
$73$	3.07636e19	0.837812	0.418906	+	0.908030i	$0.362414\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.71064e20	2.66076
$78$	0	0
$79$	−1.48672e20	−1.76662	−0.883312	−	0.468785i	$-0.844692\pi$
$79$	−1.48672e20	−1.76662	−0.883312	+	0.468785i	$0.844692\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.43003e19	−0.667104	−0.333552	−	0.942732i	$-0.608247\pi$
$83$	−9.43003e19	−0.667104	−0.333552	+	0.942732i	$0.608247\pi$
$84$	0	0
$85$	−4.92511e19	−0.271342
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.18162e20	0.741626	0.370813	−	0.928708i	$-0.379079\pi$
$89$	2.18162e20	0.741626	0.370813	+	0.928708i	$0.379079\pi$
$90$	0	0
$91$	−6.39991e20	−1.72282
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.24866e20	0.385325
$96$	0	0
$97$	1.66723e20	0.229557	0.114779	−	0.993391i	$-0.463384\pi$
$97$	1.66723e20	0.229557	0.114779	+	0.993391i	$0.463384\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.30649e20	0.117688	0.0588440	−	0.998267i	$-0.481259\pi$
$101$	1.30649e20	0.117688	0.0588440	+	0.998267i	$0.481259\pi$
$102$	0	0
$103$	−1.79611e21	−1.31687	−0.658434	−	0.752639i	$-0.728781\pi$
$103$	−1.79611e21	−1.31687	−0.658434	+	0.752639i	$0.728781\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.52295e19	−0.00748437	−0.00374218	−	0.999993i	$-0.501191\pi$
$107$	−1.52295e19	−0.00748437	−0.00374218	+	0.999993i	$0.501191\pi$
$108$	0	0
$109$	−1.75681e20	−0.0710799	−0.0355400	−	0.999368i	$-0.511315\pi$
$109$	−1.75681e20	−0.0710799	−0.0355400	+	0.999368i	$0.511315\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.54442e21	−1.53650	−0.768250	−	0.640150i	$-0.778872\pi$
$113$	−5.54442e21	−1.53650	−0.768250	+	0.640150i	$0.778872\pi$
$114$	0	0
$115$	4.81660e21	1.11023
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.76683e21	0.284415
$120$	0	0
$121$	1.65235e22	2.23282
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.93400e19	−0.00954045
$126$	0	0
$127$	−7.59121e21	−0.617123	−0.308562	−	0.951204i	$-0.599848\pi$
$127$	−7.59121e21	−0.617123	−0.308562	+	0.951204i	$0.599848\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.26195e22	0.740789	0.370395	−	0.928874i	$-0.379222\pi$
$131$	1.26195e22	0.740789	0.370395	+	0.928874i	$0.379222\pi$
$132$	0	0
$133$	−8.06684e21	−0.403888
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.04751e22	−0.751036	−0.375518	−	0.926815i	$-0.622535\pi$
$137$	−2.04751e22	−0.751036	−0.375518	+	0.926815i	$0.622535\pi$
$138$	0	0
$139$	−8.05555e21	−0.253770	−0.126885	−	0.991917i	$-0.540498\pi$
$139$	−8.05555e21	−0.253770	−0.126885	+	0.991917i	$0.540498\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.95042e22	−2.09322
$144$	0	0
$145$	5.57996e22	1.12789
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.91681e21	0.0291155	0.0145577	−	0.999894i	$-0.495366\pi$
$149$	1.91681e21	0.0291155	0.0145577	+	0.999894i	$0.495366\pi$
$150$	0	0
$151$	4.49230e22	0.593213	0.296606	−	0.955000i	$-0.404145\pi$
$151$	4.49230e22	0.593213	0.296606	+	0.955000i	$0.404145\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.84389e23	1.85035
$156$	0	0
$157$	−1.35200e23	−1.18585	−0.592926	−	0.805257i	$-0.702027\pi$
$157$	−1.35200e23	−1.18585	−0.592926	+	0.805257i	$0.702027\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.72790e23	−1.16372
$162$	0	0
$163$	−2.95535e23	−1.74839	−0.874197	−	0.485572i	$-0.838611\pi$
$163$	−2.95535e23	−1.74839	−0.874197	+	0.485572i	$0.838611\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.71414e23	1.24483	0.622413	−	0.782689i	$-0.286152\pi$
$167$	2.71414e23	1.24483	0.622413	+	0.782689i	$0.286152\pi$
$168$	0	0
$169$	8.77920e22	0.355340
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.84952e23	−1.21877	−0.609386	−	0.792873i	$-0.708584\pi$
$173$	−3.84952e23	−1.21877	−0.609386	+	0.792873i	$0.708584\pi$
$174$	0	0
$175$	−5.23805e23	−1.46984
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.60520e23	1.24061	0.620304	−	0.784362i	$-0.287009\pi$
$179$	5.60520e23	1.24061	0.620304	+	0.784362i	$0.287009\pi$
$180$	0	0
$181$	−7.65962e23	−1.50863	−0.754314	−	0.656514i	$-0.772030\pi$
$181$	−7.65962e23	−1.50863	−0.754314	+	0.656514i	$0.772030\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.93860e23	0.929781
$186$	0	0
$187$	2.47096e23	0.345564
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.28534e23	0.479883	0.239941	−	0.970787i	$-0.422872\pi$
$191$	4.28534e23	0.479883	0.239941	+	0.970787i	$0.422872\pi$
$192$	0	0
$193$	1.58535e24	1.59138	0.795689	−	0.605705i	$-0.207109\pi$
$193$	1.58535e24	1.59138	0.795689	+	0.605705i	$0.207109\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.58432e24	−1.28217	−0.641086	−	0.767469i	$-0.721516\pi$
$197$	−1.58432e24	−1.28217	−0.641086	+	0.767469i	$0.721516\pi$
$198$	0	0
$199$	−1.61590e24	−1.17614	−0.588068	−	0.808811i	$-0.700111\pi$
$199$	−1.61590e24	−1.17614	−0.588068	+	0.808811i	$0.700111\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.00175e24	−1.18223
$204$	0	0
$205$	3.14150e24	1.67387
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.12817e24	−0.490724
$210$	0	0
$211$	−4.40467e24	−1.73359	−0.866797	−	0.498661i	$-0.833825\pi$
$211$	−4.40467e24	−1.73359	−0.866797	+	0.498661i	$0.833825\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.23556e24	−2.33813
$216$	0	0
$217$	−6.61477e24	−1.93949
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9.24444e23	−0.223749
$222$	0	0
$223$	2.38725e24	0.525650	0.262825	−	0.964843i	$-0.415346\pi$
$223$	2.38725e24	0.525650	0.262825	+	0.964843i	$0.415346\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.41267e24	−0.258089	−0.129045	−	0.991639i	$-0.541191\pi$
$227$	−1.41267e24	−0.258089	−0.129045	+	0.991639i	$0.541191\pi$
$228$	0	0
$229$	−6.13078e24	−1.02151	−0.510755	−	0.859726i	$-0.670634\pi$
$229$	−6.13078e24	−1.02151	−0.510755	+	0.859726i	$0.670634\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.43514e24	−0.477208	−0.238604	−	0.971117i	$-0.576690\pi$
$233$	−3.43514e24	−0.477208	−0.238604	+	0.971117i	$0.576690\pi$
$234$	0	0
$235$	−4.04753e24	−0.514016
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.65401e23	0.0282309	0.0141154	−	0.999900i	$-0.495507\pi$
$239$	2.65401e23	0.0282309	0.0141154	+	0.999900i	$0.495507\pi$
$240$	0	0
$241$	−4.64377e24	−0.452577	−0.226288	−	0.974060i	$-0.572659\pi$
$241$	−4.64377e24	−0.452577	−0.226288	+	0.974060i	$0.572659\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.04901e25	1.67996
$246$	0	0
$247$	4.22074e24	0.317739
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.13351e24	0.580850	0.290425	−	0.956898i	$-0.406203\pi$
$251$	9.13351e24	0.580850	0.290425	+	0.956898i	$0.406203\pi$
$252$	0	0
$253$	−2.41651e25	−1.41392
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.42762e25	−1.20471	−0.602355	−	0.798228i	$-0.705771\pi$
$257$	−2.42762e25	−1.20471	−0.602355	+	0.798228i	$0.705771\pi$
$258$	0	0
$259$	−2.13041e25	−0.974575
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.72262e24	0.222874	0.111437	−	0.993772i	$-0.464455\pi$
$263$	5.72262e24	0.222874	0.111437	+	0.993772i	$0.464455\pi$
$264$	0	0
$265$	−7.74338e25	−2.78515
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.61502e25	0.803668	0.401834	−	0.915712i	$-0.368373\pi$
$269$	2.61502e25	0.803668	0.401834	+	0.915712i	$0.368373\pi$
$270$	0	0
$271$	−1.46407e25	−0.416279	−0.208139	−	0.978099i	$-0.566741\pi$
$271$	−1.46407e25	−0.416279	−0.208139	+	0.978099i	$0.566741\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−7.32554e25	−1.78586
$276$	0	0
$277$	−4.49242e25	−1.01495	−0.507473	−	0.861668i	$-0.669420\pi$
$277$	−4.49242e25	−1.01495	−0.507473	+	0.861668i	$0.669420\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.52906e25	−1.07457	−0.537286	−	0.843400i	$-0.680550\pi$
$281$	−5.52906e25	−1.07457	−0.537286	+	0.843400i	$0.680550\pi$
$282$	0	0
$283$	4.21386e25	0.760192	0.380096	−	0.924947i	$-0.375891\pi$
$283$	4.21386e25	0.760192	0.380096	+	0.924947i	$0.375891\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.12698e26	−1.75451
$288$	0	0
$289$	−6.65398e25	−0.963062
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.37530e25	0.798713	0.399356	−	0.916796i	$-0.369234\pi$
$293$	6.37530e25	0.798713	0.399356	+	0.916796i	$0.369234\pi$
$294$	0	0
$295$	−1.34774e26	−1.57208
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.04075e25	0.915497
$300$	0	0
$301$	2.59568e26	2.45077
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.12195e26	−1.74416
$306$	0	0
$307$	7.93138e25	0.608689	0.304345	−	0.952562i	$-0.401563\pi$
$307$	7.93138e25	0.608689	0.304345	+	0.952562i	$0.401563\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.28045e26	−0.857783	−0.428892	−	0.903356i	$-0.641096\pi$
$311$	−1.28045e26	−0.857783	−0.428892	+	0.903356i	$0.641096\pi$
$312$	0	0
$313$	1.71687e26	1.07528	0.537640	−	0.843174i	$-0.319316\pi$
$313$	1.71687e26	1.07528	0.537640	+	0.843174i	$0.319316\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.54215e26	1.94153	0.970766	−	0.240028i	$-0.0771565\pi$
$317$	3.54215e26	1.94153	0.970766	+	0.240028i	$0.0771565\pi$
$318$	0	0
$319$	−2.79950e26	−1.43641
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.16523e25	−0.0524547
$324$	0	0
$325$	2.74066e26	1.15632
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.45201e26	0.538780
$330$	0	0
$331$	−2.50034e26	−0.870573	−0.435286	−	0.900292i	$-0.643353\pi$
$331$	−2.50034e26	−0.870573	−0.435286	+	0.900292i	$0.643353\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.43913e26	1.05555
$336$	0	0
$337$	−1.04705e26	−0.301893	−0.150947	−	0.988542i	$-0.548232\pi$
$337$	−1.04705e26	−0.301893	−0.150947	+	0.988542i	$0.548232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.25092e26	−2.35648
$342$	0	0
$343$	−1.17324e26	−0.281061
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7.57106e25	−0.160582	−0.0802910	−	0.996771i	$-0.525585\pi$
$347$	−7.57106e25	−0.160582	−0.0802910	+	0.996771i	$0.525585\pi$
$348$	0	0
$349$	−6.14974e26	−1.22798	−0.613988	−	0.789315i	$-0.710436\pi$
$349$	−6.14974e26	−1.22798	−0.613988	+	0.789315i	$0.710436\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.59357e25	0.0636637	0.0318319	−	0.999493i	$-0.489866\pi$
$353$	3.59357e25	0.0636637	0.0318319	+	0.999493i	$0.489866\pi$
$354$	0	0
$355$	1.24199e27	2.07358
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.77822e26	0.412359	0.206180	−	0.978514i	$-0.433897\pi$
$359$	2.77822e26	0.412359	0.206180	+	0.978514i	$0.433897\pi$
$360$	0	0
$361$	−6.61009e26	−0.925511
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.48422e26	1.18284
$366$	0	0
$367$	−2.81108e26	−0.331039	−0.165520	−	0.986206i	$-0.552930\pi$
$367$	−2.81108e26	−0.331039	−0.165520	+	0.986206i	$0.552930\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.77786e27	2.91933
$372$	0	0
$373$	1.46468e27	1.45479	0.727396	−	0.686218i	$-0.240730\pi$
$373$	1.46468e27	1.45479	0.727396	+	0.686218i	$0.240730\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.04736e27	0.930060
$378$	0	0
$379$	4.24665e26	0.356726	0.178363	−	0.983965i	$-0.442920\pi$
$379$	4.24665e26	0.356726	0.178363	+	0.983965i	$0.442920\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−9.06730e26	−0.682167	−0.341083	−	0.940033i	$-0.610794\pi$
$383$	−9.06730e26	−0.682167	−0.341083	+	0.940033i	$0.610794\pi$
$384$	0	0
$385$	5.27380e27	3.75652
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.38481e26	−0.0884951	−0.0442476	−	0.999021i	$-0.514089\pi$
$389$	−1.38481e26	−0.0884951	−0.0442476	+	0.999021i	$0.514089\pi$
$390$	0	0
$391$	−2.49589e26	−0.151137
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.58346e27	−2.49416
$396$	0	0
$397$	−3.07714e27	−1.58799	−0.793993	−	0.607927i	$-0.792001\pi$
$397$	−3.07714e27	−1.58799	−0.793993	+	0.607927i	$0.792001\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.92618e27	1.35920	0.679602	−	0.733581i	$-0.262153\pi$
$401$	2.92618e27	1.35920	0.679602	+	0.733581i	$0.262153\pi$
$402$	0	0
$403$	3.46099e27	1.52580
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.97943e27	−1.18411
$408$	0	0
$409$	3.30155e27	1.24630	0.623151	−	0.782102i	$-0.285852\pi$
$409$	3.30155e27	1.24630	0.623151	+	0.782102i	$0.285852\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.83487e27	1.64782
$414$	0	0
$415$	−2.90722e27	−0.941832
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.60698e27	−1.05656	−0.528282	−	0.849069i	$-0.677164\pi$
$419$	−3.60698e27	−1.05656	−0.528282	+	0.849069i	$0.677164\pi$
$420$	0	0
$421$	1.13735e27	0.316908	0.158454	−	0.987366i	$-0.449349\pi$
$421$	1.13735e27	0.316908	0.158454	+	0.987366i	$0.449349\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.56617e26	−0.190894
$426$	0	0
$427$	7.61226e27	1.82819
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.65432e27	0.360254	0.180127	−	0.983643i	$-0.442349\pi$
$431$	1.65432e27	0.360254	0.180127	+	0.983643i	$0.442349\pi$
$432$	0	0
$433$	−3.17892e27	−0.659412	−0.329706	−	0.944084i	$-0.606950\pi$
$433$	−3.17892e27	−0.659412	−0.329706	+	0.944084i	$0.606950\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.13955e27	0.214625
$438$	0	0
$439$	2.89492e27	0.519707	0.259853	−	0.965648i	$-0.416326\pi$
$439$	2.89492e27	0.519707	0.259853	+	0.965648i	$0.416326\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.00624e27	0.490665	0.245332	−	0.969439i	$-0.421103\pi$
$443$	3.00624e27	0.490665	0.245332	+	0.969439i	$0.421103\pi$
$444$	0	0
$445$	6.72582e27	1.04704
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.37879e27	0.337109	0.168554	−	0.985692i	$-0.446090\pi$
$449$	2.37879e27	0.337109	0.168554	+	0.985692i	$0.446090\pi$
$450$	0	0
$451$	−1.57611e28	−2.13173
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.97305e28	−2.43231
$456$	0	0
$457$	−8.43469e27	−0.992999	−0.496500	−	0.868037i	$-0.665382\pi$
$457$	−8.43469e27	−0.992999	−0.496500	+	0.868037i	$0.665382\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.82405e27	0.195965	0.0979823	−	0.995188i	$-0.468761\pi$
$461$	1.82405e27	0.195965	0.0979823	+	0.995188i	$0.468761\pi$
$462$	0	0
$463$	−1.00432e28	−1.03103	−0.515516	−	0.856880i	$-0.672400\pi$
$463$	−1.00432e28	−1.03103	−0.515516	+	0.856880i	$0.672400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.94652e28	1.82571	0.912856	−	0.408282i	$-0.133872\pi$
$467$	1.94652e28	1.82571	0.912856	+	0.408282i	$0.133872\pi$
$468$	0	0
$469$	−1.23375e28	−1.10640
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.63012e28	2.97769
$474$	0	0
$475$	3.45450e27	0.271083
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.75628e28	−1.26204	−0.631018	−	0.775768i	$-0.717362\pi$
$479$	−1.75628e28	−1.26204	−0.631018	+	0.775768i	$0.717362\pi$
$480$	0	0
$481$	1.11467e28	0.766699
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.13996e27	0.324094
$486$	0	0
$487$	−1.86000e28	−1.12320	−0.561602	−	0.827407i	$-0.689815\pi$
$487$	−1.86000e28	−1.12320	−0.561602	+	0.827407i	$0.689815\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.81388e27	−0.211354	−0.105677	−	0.994400i	$-0.533701\pi$
$491$	−3.81388e27	−0.211354	−0.105677	+	0.994400i	$0.533701\pi$
$492$	0	0
$493$	−2.89146e27	−0.153541
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.45551e28	−2.17348
$498$	0	0
$499$	−9.64495e27	−0.451070	−0.225535	−	0.974235i	$-0.572413\pi$
$499$	−9.64495e27	−0.451070	−0.225535	+	0.974235i	$0.572413\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−1.64738e27	−0.0708485	−0.0354243	−	0.999372i	$-0.511278\pi$
$503$	−1.64738e27	−0.0708485	−0.0354243	+	0.999372i	$0.511278\pi$
$504$	0	0
$505$	4.02784e27	0.166155
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.39315e28	−0.908725	−0.454363	−	0.890817i	$-0.650133\pi$
$509$	−2.39315e28	−0.908725	−0.454363	+	0.890817i	$0.650133\pi$
$510$	0	0
$511$	−3.40236e28	−1.23983
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.53730e28	−1.85918
$516$	0	0
$517$	2.03067e28	0.654618
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.46945e28	−0.436876	−0.218438	−	0.975851i	$-0.570096\pi$
$521$	−1.46945e28	−0.436876	−0.218438	+	0.975851i	$0.570096\pi$
$522$	0	0
$523$	2.10247e27	0.0600430	0.0300215	−	0.999549i	$-0.490442\pi$
$523$	2.10247e27	0.0600430	0.0300215	+	0.999549i	$0.490442\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.55480e27	−0.251890
$528$	0	0
$529$	−1.50626e28	−0.381605
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.89660e28	1.38028
$534$	0	0
$535$	−4.69515e26	−0.0105666
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.02800e29	−2.13949
$540$	0	0
$541$	−7.08289e27	−0.141788	−0.0708939	−	0.997484i	$-0.522585\pi$
$541$	−7.08289e27	−0.141788	−0.0708939	+	0.997484i	$0.522585\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.41614e27	−0.100352
$546$	0	0
$547$	−6.08725e28	−1.08531	−0.542655	−	0.839955i	$-0.682581\pi$
$547$	−6.08725e28	−1.08531	−0.542655	+	0.839955i	$0.682581\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.32016e28	0.218039
$552$	0	0
$553$	1.64427e29	2.61432
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.08630e29	1.60129	0.800643	−	0.599142i	$-0.204492\pi$
$557$	1.08630e29	1.60129	0.800643	+	0.599142i	$0.204492\pi$
$558$	0	0
$559$	−1.35811e29	−1.92802
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.40541e28	0.975463	0.487732	−	0.872994i	$-0.337824\pi$
$563$	7.40541e28	0.975463	0.487732	+	0.872994i	$0.337824\pi$
$564$	0	0
$565$	−1.70931e29	−2.16927
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.96450e28	−0.349359	−0.174680	−	0.984625i	$-0.555889\pi$
$569$	−2.96450e28	−0.349359	−0.174680	+	0.984625i	$0.555889\pi$
$570$	0	0
$571$	−1.55764e29	−1.76924	−0.884622	−	0.466309i	$-0.845584\pi$
$571$	−1.55764e29	−1.76924	−0.884622	+	0.466309i	$0.845584\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.39947e28	0.781067
$576$	0	0
$577$	1.06756e29	1.08654	0.543270	−	0.839558i	$-0.317186\pi$
$577$	1.06756e29	1.08654	0.543270	+	0.839558i	$0.317186\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.04294e29	0.987207
$582$	0	0
$583$	3.88490e29	3.54699
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.34711e29	−1.14473	−0.572365	−	0.819999i	$-0.693974\pi$
$587$	−1.34711e29	−1.14473	−0.572365	+	0.819999i	$0.693974\pi$
$588$	0	0
$589$	4.36245e28	0.357700
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.79961e28	−0.137437	−0.0687185	−	0.997636i	$-0.521891\pi$
$593$	−1.79961e28	−0.137437	−0.0687185	+	0.997636i	$0.521891\pi$
$594$	0	0
$595$	5.44704e28	0.401543
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.20344e28	−0.288817	−0.144409	−	0.989518i	$-0.546128\pi$
$599$	−4.20344e28	−0.288817	−0.144409	+	0.989518i	$0.546128\pi$
$600$	0	0
$601$	−2.69124e28	−0.178554	−0.0892772	−	0.996007i	$-0.528456\pi$
$601$	−2.69124e28	−0.178554	−0.0892772	+	0.996007i	$0.528456\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.09408e29	3.15235
$606$	0	0
$607$	−1.40422e29	−0.839373	−0.419687	−	0.907669i	$-0.637860\pi$
$607$	−1.40422e29	−0.839373	−0.419687	+	0.907669i	$0.637860\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.59720e28	−0.423859
$612$	0	0
$613$	−2.71738e29	−1.46493	−0.732464	−	0.680806i	$-0.761630\pi$
$613$	−2.71738e29	−1.46493	−0.732464	+	0.680806i	$0.761630\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.15284e29	−1.08397	−0.541984	−	0.840389i	$-0.682327\pi$
$617$	−2.15284e29	−1.08397	−0.541984	+	0.840389i	$0.682327\pi$
$618$	0	0
$619$	1.12642e29	0.548212	0.274106	−	0.961699i	$-0.411618\pi$
$619$	1.12642e29	0.548212	0.274106	+	0.961699i	$0.411618\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.41282e29	−1.09749
$624$	0	0
$625$	−2.28900e29	−1.00671
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.07730e28	−0.126572
$630$	0	0
$631$	4.46959e29	1.77811	0.889056	−	0.457798i	$-0.151362\pi$
$631$	4.46959e29	1.77811	0.889056	+	0.457798i	$0.151362\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.34033e29	−0.871269
$636$	0	0
$637$	3.84599e29	1.38530
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.05893e29	0.694435	0.347218	−	0.937785i	$-0.387127\pi$
$641$	2.05893e29	0.694435	0.347218	+	0.937785i	$0.387127\pi$
$642$	0	0
$643$	1.28958e29	0.420951	0.210476	−	0.977599i	$-0.432499\pi$
$643$	1.28958e29	0.420951	0.210476	+	0.977599i	$0.432499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−2.06845e28	−0.0632629	−0.0316315	−	0.999500i	$-0.510070\pi$
$647$	−2.06845e28	−0.0632629	−0.0316315	+	0.999500i	$0.510070\pi$
$648$	0	0
$649$	6.76168e29	2.00210
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.59343e28	0.0442328	0.0221164	−	0.999755i	$-0.492960\pi$
$653$	1.59343e28	0.0442328	0.0221164	+	0.999755i	$0.492960\pi$
$654$	0	0
$655$	3.89053e29	1.04586
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.85081e29	1.47543	0.737716	−	0.675111i	$-0.235904\pi$
$659$	5.85081e29	1.47543	0.737716	+	0.675111i	$0.235904\pi$
$660$	0	0
$661$	6.08260e29	1.48585	0.742923	−	0.669377i	$-0.233439\pi$
$661$	6.08260e29	1.48585	0.742923	+	0.669377i	$0.233439\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.48696e29	−0.570219
$666$	0	0
$667$	2.82775e29	0.628231
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.06459e30	2.22125
$672$	0	0
$673$	1.17199e29	0.237009	0.118505	−	0.992953i	$-0.462190\pi$
$673$	1.17199e29	0.237009	0.118505	+	0.992953i	$0.462190\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.66070e29	0.315581	0.157790	−	0.987473i	$-0.449563\pi$
$677$	1.66070e29	0.315581	0.157790	+	0.987473i	$0.449563\pi$
$678$	0	0
$679$	−1.84391e29	−0.339708
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3.97118e29	0.687863	0.343931	−	0.938995i	$-0.388241\pi$
$683$	3.97118e29	0.687863	0.343931	+	0.938995i	$0.388241\pi$
$684$	0	0
$685$	−6.31236e29	−1.06033
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.45343e30	−2.29664
$690$	0	0
$691$	−5.43975e29	−0.833795	−0.416897	−	0.908954i	$-0.636883\pi$
$691$	−5.43975e29	−0.833795	−0.416897	+	0.908954i	$0.636883\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.48348e29	−0.358278
$696$	0	0
$697$	−1.62788e29	−0.227866
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.32602e29	−0.702043	−0.351022	−	0.936367i	$-0.614166\pi$
$701$	−5.32602e29	−0.702043	−0.351022	+	0.936367i	$0.614166\pi$
$702$	0	0
$703$	1.40501e29	0.179741
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.44494e29	−0.174159
$708$	0	0
$709$	−1.45514e30	−1.70262	−0.851312	−	0.524660i	$-0.824193\pi$
$709$	−1.45514e30	−1.70262	−0.851312	+	0.524660i	$0.824193\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.34428e29	1.03064
$714$	0	0
$715$	−2.75936e30	−2.95526
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.61730e29	−0.567381	−0.283690	−	0.958916i	$-0.591559\pi$
$719$	−5.61730e29	−0.567381	−0.283690	+	0.958916i	$0.591559\pi$
$720$	0	0
$721$	1.98645e30	1.94875
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.57218e29	0.793491
$726$	0	0
$727$	−9.24645e29	−0.831503	−0.415751	−	0.909478i	$-0.636481\pi$
$727$	−9.24645e29	−0.831503	−0.415751	+	0.909478i	$0.636481\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.74937e29	0.318292
$732$	0	0
$733$	−1.10815e30	−0.914124	−0.457062	−	0.889435i	$-0.651098\pi$
$733$	−1.10815e30	−0.914124	−0.457062	+	0.889435i	$0.651098\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.72543e30	−1.34428
$738$	0	0
$739$	2.39445e30	1.81317	0.906585	−	0.422023i	$-0.138680\pi$
$739$	2.39445e30	1.81317	0.906585	+	0.422023i	$0.138680\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.95615e30	1.39965	0.699825	−	0.714314i	$-0.253261\pi$
$743$	1.95615e30	1.39965	0.699825	+	0.714314i	$0.253261\pi$
$744$	0	0
$745$	5.90942e28	0.0411059
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.68434e28	0.0110757
$750$	0	0
$751$	−1.69999e30	−1.08699	−0.543497	−	0.839411i	$-0.682900\pi$
$751$	−1.69999e30	−1.08699	−0.543497	+	0.839411i	$0.682900\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.38495e30	0.837511
$756$	0	0
$757$	−1.95617e30	−1.15054	−0.575269	−	0.817964i	$-0.695103\pi$
$757$	−1.95617e30	−1.15054	−0.575269	+	0.817964i	$0.695103\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.47879e29	−0.360541	−0.180271	−	0.983617i	$-0.557697\pi$
$761$	−6.47879e29	−0.360541	−0.180271	+	0.983617i	$0.557697\pi$
$762$	0	0
$763$	1.94298e29	0.105187
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.52970e30	−1.29634
$768$	0	0
$769$	3.06878e30	1.53017	0.765083	−	0.643931i	$-0.222698\pi$
$769$	3.06878e30	1.53017	0.765083	+	0.643931i	$0.222698\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3.32643e30	−1.57070	−0.785351	−	0.619050i	$-0.787518\pi$
$773$	−3.32643e30	−1.57070	−0.785351	+	0.619050i	$0.787518\pi$
$774$	0	0
$775$	2.83267e30	1.30175
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.43245e29	0.323585
$780$	0	0
$781$	−6.23113e30	−2.64077
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.16814e30	−1.67421
$786$	0	0
$787$	3.66843e30	1.43465	0.717324	−	0.696740i	$-0.245367\pi$
$787$	3.66843e30	1.43465	0.717324	+	0.696740i	$0.245367\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.13198e30	2.27378
$792$	0	0
$793$	−3.98290e30	−1.43824
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.14635e29	−0.313282	−0.156641	−	0.987656i	$-0.550067\pi$
$797$	−9.14635e29	−0.313282	−0.156641	+	0.987656i	$0.550067\pi$
$798$	0	0
$799$	2.09737e29	0.0699736
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.75829e30	−1.50639
$804$	0	0
$805$	−5.32702e30	−1.64296
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.04306e30	0.305385	0.152693	−	0.988274i	$-0.451206\pi$
$809$	1.04306e30	0.305385	0.152693	+	0.988274i	$0.451206\pi$
$810$	0	0
$811$	4.59742e28	0.0131158	0.00655791	−	0.999978i	$-0.497913\pi$
$811$	4.59742e28	0.0131158	0.00655791	+	0.999978i	$0.497913\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.11117e30	−2.46842
$816$	0	0
$817$	−1.71185e30	−0.451996
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.75888e30	−0.692037	−0.346018	−	0.938228i	$-0.612467\pi$
$821$	−2.75888e30	−0.692037	−0.346018	+	0.938228i	$0.612467\pi$
$822$	0	0
$823$	−2.37654e30	−0.581094	−0.290547	−	0.956861i	$-0.593837\pi$
$823$	−2.37654e30	−0.581094	−0.290547	+	0.956861i	$0.593837\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.99308e30	0.927897	0.463949	−	0.885862i	$-0.346432\pi$
$827$	3.99308e30	0.927897	0.463949	+	0.885862i	$0.346432\pi$
$828$	0	0
$829$	7.15523e30	1.62107	0.810534	−	0.585691i	$-0.199177\pi$
$829$	7.15523e30	1.62107	0.810534	+	0.585691i	$0.199177\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.06177e30	−0.228695
$834$	0	0
$835$	8.36752e30	1.75747
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.19244e30	1.03722	0.518610	−	0.855011i	$-0.326450\pi$
$839$	5.19244e30	1.03722	0.518610	+	0.855011i	$0.326450\pi$
$840$	0	0
$841$	−1.85694e30	−0.361775
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.70657e30	0.501677
$846$	0	0
$847$	−1.82745e31	−3.30422
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.00950e30	0.517885
$852$	0	0
$853$	9.61096e30	1.61362	0.806811	−	0.590809i	$-0.201191\pi$
$853$	9.61096e30	1.61362	0.806811	+	0.590809i	$0.201191\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.96150e30	0.793075	0.396537	−	0.918019i	$-0.370212\pi$
$857$	4.96150e30	0.793075	0.396537	+	0.918019i	$0.370212\pi$
$858$	0	0
$859$	4.43332e30	0.691513	0.345757	−	0.938324i	$-0.387622\pi$
$859$	4.43332e30	0.691513	0.345757	+	0.938324i	$0.387622\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.32513e30	0.493965	0.246982	−	0.969020i	$-0.420561\pi$
$863$	3.32513e30	0.493965	0.246982	+	0.969020i	$0.420561\pi$
$864$	0	0
$865$	−1.18678e31	−1.72069
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.29955e31	3.17640
$870$	0	0
$871$	6.45525e30	0.870408
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.09867e29	0.0141183
$876$	0	0
$877$	4.83563e30	0.606677	0.303339	−	0.952883i	$-0.401899\pi$
$877$	4.83563e30	0.606677	0.303339	+	0.952883i	$0.401899\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.40504e30	−0.168051	−0.0840257	−	0.996464i	$-0.526778\pi$
$881$	−1.40504e30	−0.168051	−0.0840257	+	0.996464i	$0.526778\pi$
$882$	0	0
$883$	−3.49210e30	−0.407849	−0.203924	−	0.978987i	$-0.565370\pi$
$883$	−3.49210e30	−0.407849	−0.203924	+	0.978987i	$0.565370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.00845e30	−0.669213	−0.334607	−	0.942358i	$-0.608603\pi$
$887$	−6.00845e30	−0.669213	−0.334607	+	0.942358i	$0.608603\pi$
$888$	0	0
$889$	8.39567e30	0.913244
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.57600e29	−0.0993673
$894$	0	0
$895$	1.72805e31	1.75152
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.08252e31	1.04703
$900$	0	0
$901$	4.01251e30	0.379146
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.36141e31	−2.12991
$906$	0	0
$907$	−1.46441e31	−1.29058	−0.645290	−	0.763938i	$-0.723263\pi$
$907$	−1.46441e31	−1.29058	−0.645290	+	0.763938i	$0.723263\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.68894e30	−0.731179	−0.365590	−	0.930776i	$-0.619133\pi$
$911$	−8.68894e30	−0.731179	−0.365590	+	0.930776i	$0.619133\pi$
$912$	0	0
$913$	1.45857e31	1.19946
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.39569e31	−1.09625
$918$	0	0
$919$	1.84101e30	0.141333	0.0706664	−	0.997500i	$-0.477487\pi$
$919$	1.84101e30	0.141333	0.0706664	+	0.997500i	$0.477487\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.33122e31	1.70988
$924$	0	0
$925$	9.12313e30	0.654118
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.19812e31	0.820983	0.410491	−	0.911864i	$-0.365357\pi$
$929$	1.19812e31	0.820983	0.410491	+	0.911864i	$0.365357\pi$
$930$	0	0
$931$	4.84773e30	0.324763
$932$	0	0
$933$	0	0
$934$	0	0
$935$	7.61781e30	0.487875
$936$	0	0
$937$	−3.39787e30	−0.212785	−0.106392	−	0.994324i	$-0.533930\pi$
$937$	−3.39787e30	−0.212785	−0.106392	+	0.994324i	$0.533930\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.36473e31	−1.41609	−0.708045	−	0.706167i	$-0.750423\pi$
$941$	−2.36473e31	−1.41609	−0.708045	+	0.706167i	$0.750423\pi$
$942$	0	0
$943$	1.59202e31	0.932341
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.07342e31	−1.72166	−0.860829	−	0.508894i	$-0.830055\pi$
$947$	−3.07342e31	−1.72166	−0.860829	+	0.508894i	$0.830055\pi$
$948$	0	0
$949$	1.78019e31	0.975373
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.23214e31	−1.17016	−0.585082	−	0.810974i	$-0.698938\pi$
$953$	−2.23214e31	−1.17016	−0.585082	+	0.810974i	$0.698938\pi$
$954$	0	0
$955$	1.32115e31	0.677509
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.26449e31	1.11141
$960$	0	0
$961$	1.49463e31	0.717693
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.88754e31	2.24674
$966$	0	0
$967$	1.97944e31	0.890356	0.445178	−	0.895442i	$-0.353140\pi$
$967$	1.97944e31	0.890356	0.445178	+	0.895442i	$0.353140\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.03370e31	−0.875961	−0.437980	−	0.898985i	$-0.644306\pi$
$971$	−2.03370e31	−0.875961	−0.437980	+	0.898985i	$0.644306\pi$
$972$	0	0
$973$	8.90921e30	0.375538
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.80480e31	1.93991	0.969956	−	0.243280i	$-0.0782233\pi$
$977$	4.80480e31	1.93991	0.969956	+	0.243280i	$0.0782233\pi$
$978$	0	0
$979$	−3.37438e31	−1.33345
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.07092e31	0.405458	0.202729	−	0.979235i	$-0.435019\pi$
$983$	1.07092e31	0.405458	0.202729	+	0.979235i	$0.435019\pi$
$984$	0	0
$985$	−4.88435e31	−1.81020
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.66676e31	−1.30233
$990$	0	0
$991$	−2.08415e31	−0.724694	−0.362347	−	0.932043i	$-0.618024\pi$
$991$	−2.08415e31	−0.724694	−0.362347	+	0.932043i	$0.618024\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.98173e31	−1.66050
$996$	0	0
$997$	−1.63103e31	−0.532306	−0.266153	−	0.963931i	$-0.585753\pi$
$997$	−1.63103e31	−0.532306	−0.266153	+	0.963931i	$0.585753\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.22.a.d.1.3		3
3.2	odd	2		24.22.a.c.1.1	✓	3
12.11	even	2		48.22.a.l.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.c.1.1	✓	3	3.2	odd	2
48.22.a.l.1.1		3	12.11	even	2
72.22.a.d.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	72.a (trivial)

\(f(q)\)	\(=\)	\(q+3.08294e7 q^{5} -1.10597e9 q^{7} -1.54673e11 q^{11} +5.78668e11 q^{13} -1.59754e12 q^{17} +7.29390e12 q^{19} +1.56234e14 q^{23} +4.73615e14 q^{25} +1.80995e15 q^{29} +5.98096e15 q^{31} -3.40965e16 q^{35} +1.92628e16 q^{37} +1.01900e17 q^{41} -2.34697e17 q^{43} -1.31288e17 q^{47} +6.64628e17 q^{49} -2.51169e18 q^{53} -4.76847e18 q^{55} -4.37160e18 q^{59} -6.88287e18 q^{61} +1.78400e19 q^{65} +1.11554e19 q^{67} +4.02859e19 q^{71} +3.07636e19 q^{73} +1.71064e20 q^{77} -1.48672e20 q^{79} -9.43003e19 q^{83} -4.92511e19 q^{85} +2.18162e20 q^{89} -6.39991e20 q^{91} +2.24866e20 q^{95} +1.66723e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2080026 q^{5} - 1205282064 q^{7} - 13839247500 q^{11} + 718855551690 q^{13} - 2135189843046 q^{17} - 40122324686988 q^{19} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25} - 442708167991794 q^{29}+ \cdots - 15\!\cdots\!22 q^{97}+O(q^{100}) \) 3 * q - 2080026 * q^5 - 1205282064 * q^7 - 13839247500 * q^11 + 718855551690 * q^13 - 2135189843046 * q^17 - 40122324686988 * q^19 - 278424417682632 * q^23 + 1348043260553901 * q^25 - 442708167991794 * q^29 + 8016070162990152 * q^31 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 125648125186340562 * q^41 - 229052541499074612 * q^43 + 448613782068047712 * q^47 + 365221903446092427 * q^49 - 1406206217208267066 * q^53 - 3437829264920292504 * q^55 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 19537666262756991444 * q^65 - 33491023693155020652 * q^67 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 255759806110876888128 * q^77 - 197919973704661098024 * q^79 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 731008175840243483058 * q^89 - 546304658547739915488 * q^91 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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