LMFDB - Embedded newform 72.22.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,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, 22, names="a")

//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);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 22 $
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:	$201.223687887$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{358549}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 89637 $ x^2 - x - 89637
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$299.895$ 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)

$f(q)$	$=$	$q-3.35342e6 q^{5} +7.07779e8 q^{7} +O(q^{10})$	$q-3.35342e6 q^{5} +7.07779e8 q^{7} -8.75119e10 q^{11} -7.41212e11 q^{13} +6.82853e12 q^{17} +5.17730e13 q^{19} +3.13428e14 q^{23} -4.65592e14 q^{25} -1.46400e15 q^{29} -6.42150e15 q^{31} -2.37348e15 q^{35} -6.93987e15 q^{37} -3.25339e16 q^{41} +4.37536e16 q^{43} +5.34426e16 q^{47} -5.75946e16 q^{49} +1.19249e18 q^{53} +2.93464e17 q^{55} +4.48047e17 q^{59} -9.05611e17 q^{61} +2.48560e18 q^{65} -6.06820e17 q^{67} +2.18611e19 q^{71} -6.65250e19 q^{73} -6.19391e19 q^{77} -1.81938e18 q^{79} +2.58951e20 q^{83} -2.28989e19 q^{85} +1.80968e20 q^{89} -5.24614e20 q^{91} -1.73617e20 q^{95} +4.09632e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2108140 q^{5} + 444771792 q^{7}+O(q^{10}) $ 2 * q - 2108140 * q^5 + 444771792 * q^7	$ 2 q - 2108140 q^{5} + 444771792 q^{7} - 53806403320 q^{11} - 490366676932 q^{13} + 6593864672092 q^{17} + 19302397925320 q^{19} + 409737865776272 q^{23} - 940878149007650 q^{25} + 24\!\cdots\!60 q^{29}+ \cdots + 78\!\cdots\!88 q^{97}+O(q^{100}) $ 2 * q - 2108140 * q^5 + 444771792 * q^7 - 53806403320 * q^11 - 490366676932 * q^13 + 6593864672092 * q^17 + 19302397925320 * q^19 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2404787522145060 * q^29 - 8689907170559168 * q^31 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 68178038573558676 * q^41 + 264529652266004024 * q^43 - 426494411558622432 * q^47 - 546967577640131534 * q^49 + 3055980275589518132 * q^53 + 335437371694825040 * q^55 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 + 2797969869756700760 * q^65 + 16674123174011538088 * q^67 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 70803926825553273024 * q^77 - 85280702218715897824 * q^79 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 59742932430695979660 * q^89 - 590588534777956016544 * q^91 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−3.35342e6	−0.153569	−0.0767844	−	0.997048i	$-0.524465\pi$
$5$	−3.35342e6	−0.153569	−0.0767844	+	0.997048i	$0.524465\pi$
$6$	0	0
$7$	7.07779e8	0.947040	0.473520	−	0.880783i	$-0.342983\pi$
$7$	7.07779e8	0.947040	0.473520	+	0.880783i	$0.342983\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.75119e10	−1.01729	−0.508644	−	0.860977i	$-0.669853\pi$
$11$	−8.75119e10	−1.01729	−0.508644	+	0.860977i	$0.669853\pi$
$12$	0	0
$13$	−7.41212e11	−1.49120	−0.745602	−	0.666391i	$-0.767838\pi$
$13$	−7.41212e11	−1.49120	−0.745602	+	0.666391i	$0.767838\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.82853e12	0.821511	0.410756	−	0.911746i	$-0.365265\pi$
$17$	6.82853e12	0.821511	0.410756	+	0.911746i	$0.365265\pi$
$18$	0	0
$19$	5.17730e13	1.93727	0.968635	−	0.248487i	$-0.0799334\pi$
$19$	5.17730e13	1.93727	0.968635	+	0.248487i	$0.0799334\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.13428e14	1.57760	0.788798	−	0.614653i	$-0.210704\pi$
$23$	3.13428e14	1.57760	0.788798	+	0.614653i	$0.210704\pi$
$24$	0	0
$25$	−4.65592e14	−0.976417
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.46400e15	−0.646195	−0.323097	−	0.946366i	$-0.604724\pi$
$29$	−1.46400e15	−0.646195	−0.323097	+	0.946366i	$0.604724\pi$
$30$	0	0
$31$	−6.42150e15	−1.40714	−0.703572	−	0.710624i	$-0.748413\pi$
$31$	−6.42150e15	−1.40714	−0.703572	+	0.710624i	$0.748413\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.37348e15	−0.145436
$36$	0	0
$37$	−6.93987e15	−0.237265	−0.118632	−	0.992938i	$-0.537851\pi$
$37$	−6.93987e15	−0.237265	−0.118632	+	0.992938i	$0.537851\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.25339e16	−0.378534	−0.189267	−	0.981926i	$-0.560611\pi$
$41$	−3.25339e16	−0.378534	−0.189267	+	0.981926i	$0.560611\pi$
$42$	0	0
$43$	4.37536e16	0.308741	0.154371	−	0.988013i	$-0.450665\pi$
$43$	4.37536e16	0.308741	0.154371	+	0.988013i	$0.450665\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.34426e16	0.148204	0.0741020	−	0.997251i	$-0.476391\pi$
$47$	5.34426e16	0.148204	0.0741020	+	0.997251i	$0.476391\pi$
$48$	0	0
$49$	−5.75946e16	−0.103115
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.19249e18	0.936611	0.468306	−	0.883567i	$-0.344865\pi$
$53$	1.19249e18	0.936611	0.468306	+	0.883567i	$0.344865\pi$
$54$	0	0
$55$	2.93464e17	0.156224
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.48047e17	0.114124	0.0570620	−	0.998371i	$-0.481827\pi$
$59$	4.48047e17	0.114124	0.0570620	+	0.998371i	$0.481827\pi$
$60$	0	0
$61$	−9.05611e17	−0.162547	−0.0812734	−	0.996692i	$-0.525899\pi$
$61$	−9.05611e17	−0.162547	−0.0812734	+	0.996692i	$0.525899\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.48560e18	0.229003
$66$	0	0
$67$	−6.06820e17	−0.0406700	−0.0203350	−	0.999793i	$-0.506473\pi$
$67$	−6.06820e17	−0.0406700	−0.0203350	+	0.999793i	$0.506473\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.18611e19	0.797003	0.398501	−	0.917168i	$-0.369530\pi$
$71$	2.18611e19	0.797003	0.398501	+	0.917168i	$0.369530\pi$
$72$	0	0
$73$	−6.65250e19	−1.81174	−0.905868	−	0.423559i	$-0.860781\pi$
$73$	−6.65250e19	−1.81174	−0.905868	+	0.423559i	$0.860781\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.19391e19	−0.963413
$78$	0	0
$79$	−1.81938e18	−0.0216191	−0.0108096	−	0.999942i	$-0.503441\pi$
$79$	−1.81938e18	−0.0216191	−0.0108096	+	0.999942i	$0.503441\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.58951e20	1.83188	0.915940	−	0.401316i	$-0.131447\pi$
$83$	2.58951e20	1.83188	0.915940	+	0.401316i	$0.131447\pi$
$84$	0	0
$85$	−2.28989e19	−0.126159
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.80968e20	0.615187	0.307594	−	0.951518i	$-0.400476\pi$
$89$	1.80968e20	0.615187	0.307594	+	0.951518i	$0.400476\pi$
$90$	0	0
$91$	−5.24614e20	−1.41223
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.73617e20	−0.297504
$96$	0	0
$97$	4.09632e20	0.564016	0.282008	−	0.959412i	$-0.409000\pi$
$97$	4.09632e20	0.564016	0.282008	+	0.959412i	$0.409000\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.49956e21	1.35080	0.675398	−	0.737454i	$-0.263972\pi$
$101$	1.49956e21	1.35080	0.675398	+	0.737454i	$0.263972\pi$
$102$	0	0
$103$	5.14179e20	0.376984	0.188492	−	0.982075i	$-0.439640\pi$
$103$	5.14179e20	0.376984	0.188492	+	0.982075i	$0.439640\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.11062e21	−0.545801	−0.272900	−	0.962042i	$-0.587983\pi$
$107$	−1.11062e21	−0.545801	−0.272900	+	0.962042i	$0.587983\pi$
$108$	0	0
$109$	−1.75592e21	−0.710437	−0.355219	−	0.934783i	$-0.615594\pi$
$109$	−1.75592e21	−0.710437	−0.355219	+	0.934783i	$0.615594\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.38798e21	−1.49315	−0.746573	−	0.665303i	$-0.768302\pi$
$113$	−5.38798e21	−1.49315	−0.746573	+	0.665303i	$0.768302\pi$
$114$	0	0
$115$	−1.05106e21	−0.242270
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.83309e21	0.778004
$120$	0	0
$121$	2.58090e20	0.0348758
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.16036e21	0.303516
$126$	0	0
$127$	1.11104e22	0.903212	0.451606	−	0.892217i	$-0.350851\pi$
$127$	1.11104e22	0.903212	0.451606	+	0.892217i	$0.350851\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.53513e22	−1.48817	−0.744084	−	0.668086i	$-0.767114\pi$
$131$	−2.53513e22	−1.48817	−0.744084	+	0.668086i	$0.767114\pi$
$132$	0	0
$133$	3.66438e22	1.83467
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.99626e22	−0.732235	−0.366117	−	0.930569i	$-0.619313\pi$
$137$	−1.99626e22	−0.732235	−0.366117	+	0.930569i	$0.619313\pi$
$138$	0	0
$139$	−2.94117e22	−0.926540	−0.463270	−	0.886217i	$-0.653324\pi$
$139$	−2.94117e22	−0.926540	−0.463270	+	0.886217i	$0.653324\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.48649e22	1.51699
$144$	0	0
$145$	4.90942e21	0.0992354
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.21478e23	−1.84520	−0.922598	−	0.385762i	$-0.873939\pi$
$149$	−1.21478e23	−1.84520	−0.922598	+	0.385762i	$0.873939\pi$
$150$	0	0
$151$	−5.95169e22	−0.785928	−0.392964	−	0.919554i	$-0.628550\pi$
$151$	−5.95169e22	−0.785928	−0.392964	+	0.919554i	$0.628550\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.15340e22	0.216093
$156$	0	0
$157$	−1.28748e23	−1.12926	−0.564630	−	0.825344i	$-0.690981\pi$
$157$	−1.28748e23	−1.12926	−0.564630	+	0.825344i	$0.690981\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.21838e23	1.49405
$162$	0	0
$163$	−5.66875e22	−0.335365	−0.167682	−	0.985841i	$-0.553628\pi$
$163$	−5.66875e22	−0.335365	−0.167682	+	0.985841i	$0.553628\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.99516e23	0.915071	0.457535	−	0.889191i	$-0.348732\pi$
$167$	1.99516e23	0.915071	0.457535	+	0.889191i	$0.348732\pi$
$168$	0	0
$169$	3.02331e23	1.22369
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.56107e23	−1.12745	−0.563723	−	0.825964i	$-0.690632\pi$
$173$	−3.56107e23	−1.12745	−0.563723	+	0.825964i	$0.690632\pi$
$174$	0	0
$175$	−3.29536e23	−0.924706
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1.56436e23	−0.346243	−0.173122	−	0.984900i	$-0.555385\pi$
$179$	−1.56436e23	−0.346243	−0.173122	+	0.984900i	$0.555385\pi$
$180$	0	0
$181$	−7.11852e23	−1.40205	−0.701027	−	0.713135i	$-0.747275\pi$
$181$	−7.11852e23	−1.40205	−0.701027	+	0.713135i	$0.747275\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.32723e22	0.0364365
$186$	0	0
$187$	−5.97578e23	−0.835714
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.15222e23	0.129029	0.0645143	−	0.997917i	$-0.479450\pi$
$191$	1.15222e23	0.129029	0.0645143	+	0.997917i	$0.479450\pi$
$192$	0	0
$193$	9.86701e23	0.990453	0.495227	−	0.868764i	$-0.335085\pi$
$193$	9.86701e23	0.990453	0.495227	+	0.868764i	$0.335085\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4.06615e23	0.329070	0.164535	−	0.986371i	$-0.447388\pi$
$197$	4.06615e23	0.329070	0.164535	+	0.986371i	$0.447388\pi$
$198$	0	0
$199$	−1.80332e24	−1.31255	−0.656276	−	0.754521i	$-0.727869\pi$
$199$	−1.80332e24	−1.31255	−0.656276	+	0.754521i	$0.727869\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.03619e24	−0.611972
$204$	0	0
$205$	1.09100e23	0.0581311
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.53075e24	−1.97076
$210$	0	0
$211$	3.74550e23	0.147416	0.0737080	−	0.997280i	$-0.476517\pi$
$211$	3.74550e23	0.147416	0.0737080	+	0.997280i	$0.476517\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.46724e23	−0.0474131
$216$	0	0
$217$	−4.54500e24	−1.33262
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.06139e24	−1.22504
$222$	0	0
$223$	3.09034e24	0.680465	0.340232	−	0.940341i	$-0.389494\pi$
$223$	3.09034e24	0.680465	0.340232	+	0.940341i	$0.389494\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.67101e23	0.176685	0.0883426	−	0.996090i	$-0.471843\pi$
$227$	9.67101e23	0.176685	0.0883426	+	0.996090i	$0.471843\pi$
$228$	0	0
$229$	−3.37848e24	−0.562922	−0.281461	−	0.959573i	$-0.590819\pi$
$229$	−3.37848e24	−0.562922	−0.281461	+	0.959573i	$0.590819\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.86566e24	1.23161	0.615806	−	0.787898i	$-0.288831\pi$
$233$	8.86566e24	1.23161	0.615806	+	0.787898i	$0.288831\pi$
$234$	0	0
$235$	−1.79215e23	−0.0227595
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−9.61276e24	−1.02252	−0.511258	−	0.859427i	$-0.670820\pi$
$239$	−9.61276e24	−1.02252	−0.511258	+	0.859427i	$0.670820\pi$
$240$	0	0
$241$	−1.04750e25	−1.02088	−0.510442	−	0.859912i	$-0.670518\pi$
$241$	−1.04750e25	−1.02088	−0.510442	+	0.859912i	$0.670518\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.93139e23	0.0158353
$246$	0	0
$247$	−3.83747e25	−2.88887
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.60451e25	−1.65635	−0.828174	−	0.560471i	$-0.810620\pi$
$251$	−2.60451e25	−1.65635	−0.828174	+	0.560471i	$0.810620\pi$
$252$	0	0
$253$	−2.74287e25	−1.60487
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.83013e25	0.908208	0.454104	−	0.890949i	$-0.349959\pi$
$257$	1.83013e25	0.908208	0.454104	+	0.890949i	$0.349959\pi$
$258$	0	0
$259$	−4.91189e24	−0.224699
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.36593e24	−0.286874	−0.143437	−	0.989659i	$-0.545816\pi$
$263$	−7.36593e24	−0.286874	−0.143437	+	0.989659i	$0.545816\pi$
$264$	0	0
$265$	−3.99893e24	−0.143834
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2.31544e25	−0.711599	−0.355800	−	0.934562i	$-0.615791\pi$
$269$	−2.31544e25	−0.711599	−0.355800	+	0.934562i	$0.615791\pi$
$270$	0	0
$271$	1.17274e25	0.333445	0.166722	−	0.986004i	$-0.446682\pi$
$271$	1.17274e25	0.333445	0.166722	+	0.986004i	$0.446682\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.07448e25	0.993297
$276$	0	0
$277$	−5.46772e25	−1.23529	−0.617644	−	0.786458i	$-0.711913\pi$
$277$	−5.46772e25	−1.23529	−0.617644	+	0.786458i	$0.711913\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.73811e24	−0.130955	−0.0654775	−	0.997854i	$-0.520857\pi$
$281$	−6.73811e24	−0.130955	−0.0654775	+	0.997854i	$0.520857\pi$
$282$	0	0
$283$	−3.72254e25	−0.671555	−0.335778	−	0.941941i	$-0.608999\pi$
$283$	−3.72254e25	−0.671555	−0.335778	+	0.941941i	$0.608999\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.30268e25	−0.358487
$288$	0	0
$289$	−2.24631e25	−0.325119
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6.42074e24	−0.0804406	−0.0402203	−	0.999191i	$-0.512806\pi$
$293$	−6.42074e24	−0.0804406	−0.0402203	+	0.999191i	$0.512806\pi$
$294$	0	0
$295$	−1.50249e24	−0.0175259
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.32317e26	−2.35252
$300$	0	0
$301$	3.09679e25	0.292390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.03690e24	0.0249621
$306$	0	0
$307$	−7.52469e25	−0.577478	−0.288739	−	0.957408i	$-0.593236\pi$
$307$	−7.52469e25	−0.577478	−0.288739	+	0.957408i	$0.593236\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.48646e25	0.166570	0.0832850	−	0.996526i	$-0.473459\pi$
$311$	2.48646e25	0.166570	0.0832850	+	0.996526i	$0.473459\pi$
$312$	0	0
$313$	−1.07554e26	−0.673615	−0.336808	−	0.941574i	$-0.609347\pi$
$313$	−1.07554e26	−0.673615	−0.336808	+	0.941574i	$0.609347\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.82324e26	−1.54748	−0.773742	−	0.633501i	$-0.781617\pi$
$317$	−2.82324e26	−1.54748	−0.773742	+	0.633501i	$0.781617\pi$
$318$	0	0
$319$	1.28118e26	0.657366
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.53533e26	1.59149
$324$	0	0
$325$	3.45102e26	1.45604
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.78255e25	0.140355
$330$	0	0
$331$	−8.70208e25	−0.302991	−0.151495	−	0.988458i	$-0.548409\pi$
$331$	−8.70208e25	−0.302991	−0.151495	+	0.988458i	$0.548409\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.03492e24	0.00624565
$336$	0	0
$337$	1.22441e26	0.353030	0.176515	−	0.984298i	$-0.443518\pi$
$337$	1.22441e26	0.353030	0.176515	+	0.984298i	$0.443518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.61958e26	1.43147
$342$	0	0
$343$	−4.36091e26	−1.04469
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.10647e26	−0.658882	−0.329441	−	0.944176i	$-0.606860\pi$
$347$	−3.10647e26	−0.658882	−0.329441	+	0.944176i	$0.606860\pi$
$348$	0	0
$349$	−3.62929e26	−0.724695	−0.362347	−	0.932043i	$-0.618025\pi$
$349$	−3.62929e26	−0.724695	−0.362347	+	0.932043i	$0.618025\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.02974e25	0.159971	0.0799854	−	0.996796i	$-0.474513\pi$
$353$	9.02974e25	0.159971	0.0799854	+	0.996796i	$0.474513\pi$
$354$	0	0
$355$	−7.33096e25	−0.122395
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.84199e26	0.867100	0.433550	−	0.901130i	$-0.357261\pi$
$359$	5.84199e26	0.867100	0.433550	+	0.901130i	$0.357261\pi$
$360$	0	0
$361$	1.96623e27	2.75302
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.23087e26	0.278226
$366$	0	0
$367$	7.68446e26	0.904940	0.452470	−	0.891780i	$-0.350543\pi$
$367$	7.68446e26	0.904940	0.452470	+	0.891780i	$0.350543\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.44022e26	0.887008
$372$	0	0
$373$	−6.12745e26	−0.608607	−0.304304	−	0.952575i	$-0.598424\pi$
$373$	−6.12745e26	−0.608607	−0.304304	+	0.952575i	$0.598424\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.08514e27	0.963609
$378$	0	0
$379$	−1.02134e27	−0.857945	−0.428972	−	0.903318i	$-0.641124\pi$
$379$	−1.02134e27	−0.857945	−0.428972	+	0.903318i	$0.641124\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.10822e27	−1.58609	−0.793045	−	0.609163i	$-0.791506\pi$
$383$	−2.10822e27	−1.58609	−0.793045	+	0.609163i	$0.791506\pi$
$384$	0	0
$385$	2.07708e26	0.147950
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.16494e26	−0.330061	−0.165030	−	0.986288i	$-0.552772\pi$
$389$	−5.16494e26	−0.330061	−0.165030	+	0.986288i	$0.552772\pi$
$390$	0	0
$391$	2.14025e27	1.29601
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.10113e24	0.00332002
$396$	0	0
$397$	2.07736e27	1.07204	0.536019	−	0.844206i	$-0.319927\pi$
$397$	2.07736e27	1.07204	0.536019	+	0.844206i	$0.319927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.03566e27	1.41006	0.705029	−	0.709179i	$-0.250934\pi$
$401$	3.03566e27	1.41006	0.705029	+	0.709179i	$0.250934\pi$
$402$	0	0
$403$	4.75969e27	2.09834
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.07321e26	0.241367
$408$	0	0
$409$	1.38404e27	0.522462	0.261231	−	0.965276i	$-0.415872\pi$
$409$	1.38404e27	0.522462	0.261231	+	0.965276i	$0.415872\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.17118e26	0.108080
$414$	0	0
$415$	−8.68370e26	−0.281320
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.05547e27	−0.602093	−0.301046	−	0.953610i	$-0.597336\pi$
$419$	−2.05547e27	−0.602093	−0.301046	+	0.953610i	$0.597336\pi$
$420$	0	0
$421$	4.61061e27	1.28468	0.642341	−	0.766419i	$-0.277963\pi$
$421$	4.61061e27	1.28468	0.642341	+	0.766419i	$0.277963\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.17931e27	−0.802137
$426$	0	0
$427$	−6.40972e26	−0.153938
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.49501e27	−1.19662	−0.598312	−	0.801264i	$-0.704161\pi$
$431$	−5.49501e27	−1.19662	−0.598312	+	0.801264i	$0.704161\pi$
$432$	0	0
$433$	−5.65719e27	−1.17349	−0.586743	−	0.809773i	$-0.699590\pi$
$433$	−5.65719e27	−1.17349	−0.586743	+	0.809773i	$0.699590\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.62271e28	3.05623
$438$	0	0
$439$	−2.13975e27	−0.384136	−0.192068	−	0.981382i	$-0.561519\pi$
$439$	−2.13975e27	−0.384136	−0.192068	+	0.981382i	$0.561519\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.22577e27	−0.852926	−0.426463	−	0.904505i	$-0.640241\pi$
$443$	−5.22577e27	−0.852926	−0.426463	+	0.904505i	$0.640241\pi$
$444$	0	0
$445$	−6.06863e26	−0.0944736
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.58499e27	−0.508044	−0.254022	−	0.967198i	$-0.581754\pi$
$449$	−3.58499e27	−0.508044	−0.254022	+	0.967198i	$0.581754\pi$
$450$	0	0
$451$	2.84710e27	0.385079
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.75925e27	0.216875
$456$	0	0
$457$	1.08470e27	0.127700	0.0638498	−	0.997960i	$-0.479662\pi$
$457$	1.08470e27	0.127700	0.0638498	+	0.997960i	$0.479662\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.61325e28	1.73317	0.866585	−	0.499029i	$-0.166310\pi$
$461$	1.61325e28	1.73317	0.866585	+	0.499029i	$0.166310\pi$
$462$	0	0
$463$	−1.26030e28	−1.29382	−0.646909	−	0.762567i	$-0.723939\pi$
$463$	−1.26030e28	−1.29382	−0.646909	+	0.762567i	$0.723939\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.01065e28	−0.947923	−0.473961	−	0.880546i	$-0.657176\pi$
$467$	−1.01065e28	−0.947923	−0.473961	+	0.880546i	$0.657176\pi$
$468$	0	0
$469$	−4.29495e26	−0.0385162
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.82896e27	−0.314079
$474$	0	0
$475$	−2.41051e28	−1.89158
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.85064e27	0.564134	0.282067	−	0.959395i	$-0.408980\pi$
$479$	7.85064e27	0.564134	0.282067	+	0.959395i	$0.408980\pi$
$480$	0	0
$481$	5.14391e27	0.353810
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.37367e27	−0.0866152
$486$	0	0
$487$	1.35819e28	0.820172	0.410086	−	0.912047i	$-0.365499\pi$
$487$	1.35819e28	0.820172	0.410086	+	0.912047i	$0.365499\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.02246e28	−1.67496	−0.837482	−	0.546465i	$-0.815973\pi$
$491$	−3.02246e28	−1.67496	−0.837482	+	0.546465i	$0.815973\pi$
$492$	0	0
$493$	−9.99700e27	−0.530856
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.54728e28	0.754793
$498$	0	0
$499$	−1.52172e28	−0.711669	−0.355835	−	0.934549i	$-0.615803\pi$
$499$	−1.52172e28	−0.711669	−0.355835	+	0.934549i	$0.615803\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.52959e27	−0.409836	−0.204918	−	0.978779i	$-0.565693\pi$
$503$	−9.52959e27	−0.409836	−0.204918	+	0.978779i	$0.565693\pi$
$504$	0	0
$505$	−5.02866e27	−0.207440
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.87204e28	−1.47029	−0.735146	−	0.677909i	$-0.762886\pi$
$509$	−3.87204e28	−1.47029	−0.735146	+	0.677909i	$0.762886\pi$
$510$	0	0
$511$	−4.70850e28	−1.71579
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.72426e27	−0.0578931
$516$	0	0
$517$	−4.67686e27	−0.150766
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.03085e26	−0.0238762	−0.0119381	−	0.999929i	$-0.503800\pi$
$521$	−8.03085e26	−0.0238762	−0.0119381	+	0.999929i	$0.503800\pi$
$522$	0	0
$523$	−1.91554e28	−0.547047	−0.273523	−	0.961865i	$-0.588189\pi$
$523$	−1.91554e28	−0.547047	−0.273523	+	0.961865i	$0.588189\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.38494e28	−1.15598
$528$	0	0
$529$	5.87656e28	1.48881
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.41145e28	0.564472
$534$	0	0
$535$	3.72436e27	0.0838180
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.04021e27	0.104898
$540$	0	0
$541$	−3.68313e27	−0.0737303	−0.0368651	−	0.999320i	$-0.511737\pi$
$541$	−3.68313e27	−0.0737303	−0.0368651	+	0.999320i	$0.511737\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.88833e27	0.109101
$546$	0	0
$547$	−4.83617e28	−0.862253	−0.431127	−	0.902291i	$-0.641884\pi$
$547$	−4.83617e28	−0.862253	−0.431127	+	0.902291i	$0.641884\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−7.57959e28	−1.25185
$552$	0	0
$553$	−1.28772e27	−0.0204742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.17549e28	0.910316	0.455158	−	0.890411i	$-0.349583\pi$
$557$	6.17549e28	0.910316	0.455158	+	0.890411i	$0.349583\pi$
$558$	0	0
$559$	−3.24307e28	−0.460397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.51350e28	0.462809	0.231405	−	0.972858i	$-0.425668\pi$
$563$	3.51350e28	0.462809	0.231405	+	0.972858i	$0.425668\pi$
$564$	0	0
$565$	1.80682e28	0.229301
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−1.45054e28	−0.170943	−0.0854716	−	0.996341i	$-0.527240\pi$
$569$	−1.45054e28	−0.170943	−0.0854716	+	0.996341i	$0.527240\pi$
$570$	0	0
$571$	6.07413e28	0.689930	0.344965	−	0.938616i	$-0.387891\pi$
$571$	6.07413e28	0.689930	0.344965	+	0.938616i	$0.387891\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.45930e29	−1.54039
$576$	0	0
$577$	2.71266e28	0.276089	0.138044	−	0.990426i	$-0.455918\pi$
$577$	2.71266e28	0.276089	0.138044	+	0.990426i	$0.455918\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.83280e29	1.73486
$582$	0	0
$583$	−1.04357e29	−0.952804
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−2.51262e28	−0.213514	−0.106757	−	0.994285i	$-0.534047\pi$
$587$	−2.51262e28	−0.213514	−0.106757	+	0.994285i	$0.534047\pi$
$588$	0	0
$589$	−3.32460e29	−2.72602
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.95530e29	1.49328	0.746638	−	0.665231i	$-0.231667\pi$
$593$	1.95530e29	1.49328	0.746638	+	0.665231i	$0.231667\pi$
$594$	0	0
$595$	−1.62074e28	−0.119477
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.11818e29	0.768299	0.384149	−	0.923271i	$-0.374495\pi$
$599$	1.11818e29	0.768299	0.384149	+	0.923271i	$0.374495\pi$
$600$	0	0
$601$	2.32726e29	1.54405	0.772027	−	0.635590i	$-0.219243\pi$
$601$	2.32726e29	1.54405	0.772027	+	0.635590i	$0.219243\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.65483e26	−0.00535583
$606$	0	0
$607$	2.39690e29	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$607$	2.39690e29	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.96123e28	−0.221002
$612$	0	0
$613$	1.89238e29	1.02017	0.510086	−	0.860123i	$-0.329613\pi$
$613$	1.89238e29	1.02017	0.510086	+	0.860123i	$0.329613\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.13109e29	−0.569510	−0.284755	−	0.958600i	$-0.591912\pi$
$617$	−1.13109e29	−0.569510	−0.284755	+	0.958600i	$0.591912\pi$
$618$	0	0
$619$	1.89565e29	0.922585	0.461293	−	0.887248i	$-0.347386\pi$
$619$	1.89565e29	0.922585	0.461293	+	0.887248i	$0.347386\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.28086e29	0.582607
$624$	0	0
$625$	2.11413e29	0.929806
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.73891e28	−0.194916
$630$	0	0
$631$	1.39864e29	0.556415	0.278208	−	0.960521i	$-0.410260\pi$
$631$	1.39864e29	0.556415	0.278208	+	0.960521i	$0.410260\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.72578e28	−0.138705
$636$	0	0
$637$	4.26898e28	0.153766
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.73898e28	0.261020	0.130510	−	0.991447i	$-0.458338\pi$
$641$	7.73898e28	0.261020	0.130510	+	0.991447i	$0.458338\pi$
$642$	0	0
$643$	2.02401e29	0.660691	0.330345	−	0.943860i	$-0.392835\pi$
$643$	2.02401e29	0.660691	0.330345	+	0.943860i	$0.392835\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.36949e28	0.194809	0.0974046	−	0.995245i	$-0.468946\pi$
$647$	6.36949e28	0.194809	0.0974046	+	0.995245i	$0.468946\pi$
$648$	0	0
$649$	−3.92094e28	−0.116097
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.78655e29	0.495938	0.247969	−	0.968768i	$-0.420237\pi$
$653$	1.78655e29	0.495938	0.247969	+	0.968768i	$0.420237\pi$
$654$	0	0
$655$	8.50137e28	0.228536
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.55707e29	−1.40136	−0.700678	−	0.713477i	$-0.747119\pi$
$659$	−5.55707e29	−1.40136	−0.700678	+	0.713477i	$0.747119\pi$
$660$	0	0
$661$	2.72969e29	0.666804	0.333402	−	0.942785i	$-0.391803\pi$
$661$	2.72969e29	0.666804	0.333402	+	0.942785i	$0.391803\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.22882e29	−0.281749
$666$	0	0
$667$	−4.58860e29	−1.01943
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.92518e28	0.165357
$672$	0	0
$673$	3.87928e29	0.784499	0.392250	−	0.919859i	$-0.371697\pi$
$673$	3.87928e29	0.784499	0.392250	+	0.919859i	$0.371697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.66988e29	1.45750	0.728748	−	0.684782i	$-0.240102\pi$
$677$	7.66988e29	1.45750	0.728748	+	0.684782i	$0.240102\pi$
$678$	0	0
$679$	2.89929e29	0.534145
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.57644e29	0.965917	0.482959	−	0.875643i	$-0.339562\pi$
$683$	5.57644e29	0.965917	0.482959	+	0.875643i	$0.339562\pi$
$684$	0	0
$685$	6.69429e28	0.112448
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.83890e29	−1.39668
$690$	0	0
$691$	6.52122e29	0.999561	0.499781	−	0.866152i	$-0.333414\pi$
$691$	6.52122e29	0.999561	0.499781	+	0.866152i	$0.333414\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.86297e28	0.142288
$696$	0	0
$697$	−2.22158e29	−0.310970
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.49391e29	−0.196917	−0.0984586	−	0.995141i	$-0.531391\pi$
$701$	−1.49391e29	−0.196917	−0.0984586	+	0.995141i	$0.531391\pi$
$702$	0	0
$703$	−3.59297e29	−0.459646
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.06136e30	1.27926
$708$	0	0
$709$	−1.42826e30	−1.67117	−0.835585	−	0.549361i	$-0.814871\pi$
$709$	−1.42826e30	−1.67117	−0.835585	+	0.549361i	$0.814871\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.01268e30	−2.21990
$714$	0	0
$715$	−2.17519e29	−0.232962
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.67477e30	1.69162	0.845810	−	0.533484i	$-0.179118\pi$
$719$	1.67477e30	1.69162	0.845810	+	0.533484i	$0.179118\pi$
$720$	0	0
$721$	3.63925e29	0.357019
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.81628e29	0.630955
$726$	0	0
$727$	−1.28121e30	−1.15215	−0.576073	−	0.817398i	$-0.695416\pi$
$727$	−1.28121e30	−1.15215	−0.576073	+	0.817398i	$0.695416\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.98773e29	0.253634
$732$	0	0
$733$	−3.55290e29	−0.293083	−0.146542	−	0.989205i	$-0.546814\pi$
$733$	−3.55290e29	−0.293083	−0.146542	+	0.989205i	$0.546814\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.31040e28	0.0413732
$738$	0	0
$739$	3.02616e29	0.229153	0.114576	−	0.993414i	$-0.463449\pi$
$739$	3.02616e29	0.229153	0.114576	+	0.993414i	$0.463449\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.53318e29	−0.252803	−0.126402	−	0.991979i	$-0.540343\pi$
$743$	−3.53318e29	−0.252803	−0.126402	+	0.991979i	$0.540343\pi$
$744$	0	0
$745$	4.07368e29	0.283365
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7.86071e29	−0.516895
$750$	0	0
$751$	−1.66286e30	−1.06325	−0.531625	−	0.846980i	$-0.678419\pi$
$751$	−1.66286e30	−1.06325	−0.531625	+	0.846980i	$0.678419\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.99585e29	0.120694
$756$	0	0
$757$	−1.26493e30	−0.743979	−0.371989	−	0.928237i	$-0.621324\pi$
$757$	−1.26493e30	−0.743979	−0.371989	+	0.928237i	$0.621324\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.19527e30	−1.77815	−0.889075	−	0.457762i	$-0.848651\pi$
$761$	−3.19527e30	−1.77815	−0.889075	+	0.457762i	$0.848651\pi$
$762$	0	0
$763$	−1.24280e30	−0.672812
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.32098e29	−0.170182
$768$	0	0
$769$	−2.81246e30	−1.40236	−0.701182	−	0.712983i	$-0.747344\pi$
$769$	−2.81246e30	−1.40236	−0.701182	+	0.712983i	$0.747344\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.43052e30	−1.14766	−0.573831	−	0.818974i	$-0.694543\pi$
$773$	−2.43052e30	−1.14766	−0.573831	+	0.818974i	$0.694543\pi$
$774$	0	0
$775$	2.98980e30	1.37396
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.68438e30	−0.733323
$780$	0	0
$781$	−1.91311e30	−0.810782
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.31746e29	0.173419
$786$	0	0
$787$	2.79697e30	1.09384	0.546919	−	0.837185i	$-0.315800\pi$
$787$	2.79697e30	1.09384	0.546919	+	0.837185i	$0.315800\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.81350e30	−1.41407
$792$	0	0
$793$	6.71250e29	0.242390
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.37782e30	0.814456	0.407228	−	0.913327i	$-0.366496\pi$
$797$	2.37782e30	0.814456	0.407228	+	0.913327i	$0.366496\pi$
$798$	0	0
$799$	3.64934e29	0.121751
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.82173e30	1.84306
$804$	0	0
$805$	−7.43916e29	−0.229439
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−6.13505e29	−0.179622	−0.0898108	−	0.995959i	$-0.528626\pi$
$809$	−6.13505e29	−0.179622	−0.0898108	+	0.995959i	$0.528626\pi$
$810$	0	0
$811$	−4.24348e30	−1.21061	−0.605303	−	0.795995i	$-0.706948\pi$
$811$	−4.24348e30	−1.21061	−0.605303	+	0.795995i	$0.706948\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.90097e29	0.0515016
$816$	0	0
$817$	2.26525e30	0.598116
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.33090e30	−0.584683	−0.292342	−	0.956314i	$-0.594434\pi$
$821$	−2.33090e30	−0.584683	−0.292342	+	0.956314i	$0.594434\pi$
$822$	0	0
$823$	4.92628e30	1.20454	0.602270	−	0.798292i	$-0.294263\pi$
$823$	4.92628e30	1.20454	0.602270	+	0.798292i	$0.294263\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.48099e30	−0.808899	−0.404450	−	0.914560i	$-0.632537\pi$
$827$	−3.48099e30	−0.808899	−0.404450	+	0.914560i	$0.632537\pi$
$828$	0	0
$829$	−7.46934e30	−1.69223	−0.846116	−	0.532999i	$-0.821065\pi$
$829$	−7.46934e30	−1.69223	−0.846116	+	0.532999i	$0.821065\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.93286e29	−0.0847103
$834$	0	0
$835$	−6.69061e29	−0.140526
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.84188e30	1.56646	0.783231	−	0.621731i	$-0.213570\pi$
$839$	7.84188e30	1.56646	0.783231	+	0.621731i	$0.213570\pi$
$840$	0	0
$841$	−2.98953e30	−0.582432
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.01384e30	−0.187921
$846$	0	0
$847$	1.82670e29	0.0330288
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.17515e30	−0.374308
$852$	0	0
$853$	−1.56244e30	−0.262324	−0.131162	−	0.991361i	$-0.541871\pi$
$853$	−1.56244e30	−0.262324	−0.131162	+	0.991361i	$0.541871\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.39912e29	−0.0543336	−0.0271668	−	0.999631i	$-0.508649\pi$
$857$	−3.39912e29	−0.0543336	−0.0271668	+	0.999631i	$0.508649\pi$
$858$	0	0
$859$	−3.06417e30	−0.477953	−0.238976	−	0.971025i	$-0.576812\pi$
$859$	−3.06417e30	−0.477953	−0.238976	+	0.971025i	$0.576812\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.16441e30	−0.172979	−0.0864896	−	0.996253i	$-0.527565\pi$
$863$	−1.16441e30	−0.172979	−0.0864896	+	0.996253i	$0.527565\pi$
$864$	0	0
$865$	1.19418e30	0.173141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.59217e29	0.0219929
$870$	0	0
$871$	4.49783e29	0.0606474
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.23684e30	0.287442
$876$	0	0
$877$	1.22709e31	1.53950	0.769752	−	0.638343i	$-0.220380\pi$
$877$	1.22709e31	1.53950	0.769752	+	0.638343i	$0.220380\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.15175e30	0.855392	0.427696	−	0.903923i	$-0.359325\pi$
$881$	7.15175e30	0.855392	0.427696	+	0.903923i	$0.359325\pi$
$882$	0	0
$883$	4.43259e30	0.517691	0.258845	−	0.965919i	$-0.416658\pi$
$883$	4.43259e30	0.517691	0.258845	+	0.965919i	$0.416658\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.71592e31	1.91117	0.955586	−	0.294712i	$-0.0952238\pi$
$887$	1.71592e31	1.91117	0.955586	+	0.294712i	$0.0952238\pi$
$888$	0	0
$889$	7.86369e30	0.855378
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.76688e30	0.287111
$894$	0	0
$895$	5.24597e29	0.0531722
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.40110e30	0.909289
$900$	0	0
$901$	8.14298e30	0.769437
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.38714e30	0.215312
$906$	0	0
$907$	1.99778e31	1.76064	0.880322	−	0.474377i	$-0.157327\pi$
$907$	1.99778e31	1.76064	0.880322	+	0.474377i	$0.157327\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.13275e31	−0.953219	−0.476610	−	0.879115i	$-0.658134\pi$
$911$	−1.13275e31	−0.953219	−0.476610	+	0.879115i	$0.658134\pi$
$912$	0	0
$913$	−2.26613e31	−1.86355
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.79431e31	−1.40935
$918$	0	0
$919$	4.22159e30	0.324089	0.162044	−	0.986783i	$-0.448191\pi$
$919$	4.22159e30	0.324089	0.162044	+	0.986783i	$0.448191\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.62037e31	−1.18849
$924$	0	0
$925$	3.23114e30	0.231669
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−7.45447e30	−0.510801	−0.255400	−	0.966835i	$-0.582207\pi$
$929$	−7.45447e30	−0.510801	−0.255400	+	0.966835i	$0.582207\pi$
$930$	0	0
$931$	−2.98184e30	−0.199762
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.00393e30	0.128340
$936$	0	0
$937$	2.86894e31	1.79662	0.898308	−	0.439366i	$-0.144797\pi$
$937$	2.86894e31	1.79662	0.898308	+	0.439366i	$0.144797\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.96096e31	1.17430	0.587148	−	0.809480i	$-0.300251\pi$
$941$	1.96096e31	1.17430	0.587148	+	0.809480i	$0.300251\pi$
$942$	0	0
$943$	−1.01970e31	−0.597174
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.20608e30	0.347650	0.173825	−	0.984777i	$-0.444387\pi$
$947$	6.20608e30	0.347650	0.173825	+	0.984777i	$0.444387\pi$
$948$	0	0
$949$	4.93092e31	2.70167
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.74365e31	1.43831	0.719156	−	0.694849i	$-0.244529\pi$
$953$	2.74365e31	1.43831	0.719156	+	0.694849i	$0.244529\pi$
$954$	0	0
$955$	−3.86389e29	−0.0198148
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.41291e31	−0.693456
$960$	0	0
$961$	2.04101e31	0.980054
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−3.30883e30	−0.152103
$966$	0	0
$967$	−2.94115e31	−1.32294	−0.661469	−	0.749973i	$-0.730067\pi$
$967$	−2.94115e31	−1.32294	−0.661469	+	0.749973i	$0.730067\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.73435e30	0.160847	0.0804236	−	0.996761i	$-0.474373\pi$
$971$	3.73435e30	0.160847	0.0804236	+	0.996761i	$0.474373\pi$
$972$	0	0
$973$	−2.08170e31	−0.877470
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.55570e30	−0.183934	−0.0919670	−	0.995762i	$-0.529315\pi$
$977$	−4.55570e30	−0.183934	−0.0919670	+	0.995762i	$0.529315\pi$
$978$	0	0
$979$	−1.58369e31	−0.625823
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.81915e31	−1.82456	−0.912280	−	0.409567i	$-0.865680\pi$
$983$	−4.81915e31	−1.82456	−0.912280	+	0.409567i	$0.865680\pi$
$984$	0	0
$985$	−1.36355e30	−0.0505349
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.37136e31	0.487069
$990$	0	0
$991$	2.54898e31	0.886324	0.443162	−	0.896441i	$-0.353857\pi$
$991$	2.54898e31	0.886324	0.443162	+	0.896441i	$0.353857\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6.04731e30	0.201567
$996$	0	0
$997$	−4.55000e31	−1.48495	−0.742476	−	0.669873i	$-0.766349\pi$
$997$	−4.55000e31	−1.48495	−0.742476	+	0.669873i	$0.766349\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.22.a.b.1.1		2
3.2	odd	2		8.22.a.a.1.1	✓	2
12.11	even	2		16.22.a.e.1.2		2
24.5	odd	2		64.22.a.k.1.2		2
24.11	even	2		64.22.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.22.a.a.1.1	✓	2	3.2	odd	2
16.22.a.e.1.2		2	12.11	even	2
64.22.a.h.1.1		2	24.11	even	2
64.22.a.k.1.2		2	24.5	odd	2
72.22.a.b.1.1		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.35342e6 q^{5} +7.07779e8 q^{7} +O(q^{10})\)	\(q-3.35342e6 q^{5} +7.07779e8 q^{7} -8.75119e10 q^{11} -7.41212e11 q^{13} +6.82853e12 q^{17} +5.17730e13 q^{19} +3.13428e14 q^{23} -4.65592e14 q^{25} -1.46400e15 q^{29} -6.42150e15 q^{31} -2.37348e15 q^{35} -6.93987e15 q^{37} -3.25339e16 q^{41} +4.37536e16 q^{43} +5.34426e16 q^{47} -5.75946e16 q^{49} +1.19249e18 q^{53} +2.93464e17 q^{55} +4.48047e17 q^{59} -9.05611e17 q^{61} +2.48560e18 q^{65} -6.06820e17 q^{67} +2.18611e19 q^{71} -6.65250e19 q^{73} -6.19391e19 q^{77} -1.81938e18 q^{79} +2.58951e20 q^{83} -2.28989e19 q^{85} +1.80968e20 q^{89} -5.24614e20 q^{91} -1.73617e20 q^{95} +4.09632e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2108140 q^{5} + 444771792 q^{7}+O(q^{10}) \) 2 * q - 2108140 * q^5 + 444771792 * q^7	\( 2 q - 2108140 q^{5} + 444771792 q^{7} - 53806403320 q^{11} - 490366676932 q^{13} + 6593864672092 q^{17} + 19302397925320 q^{19} + 409737865776272 q^{23} - 940878149007650 q^{25} + 24\!\cdots\!60 q^{29}+ \cdots + 78\!\cdots\!88 q^{97}+O(q^{100}) \) 2 * q - 2108140 * q^5 + 444771792 * q^7 - 53806403320 * q^11 - 490366676932 * q^13 + 6593864672092 * q^17 + 19302397925320 * q^19 + 409737865776272 * q^23 - 940878149007650 * q^25 + 2404787522145060 * q^29 - 8689907170559168 * q^31 - 2701000503537120 * q^35 - 2186204096251860 * q^37 - 68178038573558676 * q^41 + 264529652266004024 * q^43 - 426494411558622432 * q^47 - 546967577640131534 * q^49 + 3055980275589518132 * q^53 + 335437371694825040 * q^55 - 783424997522814424 * q^59 - 7177279049078597092 * q^61 + 2797969869756700760 * q^65 + 16674123174011538088 * q^67 - 9448263149848716368 * q^71 - 11586140334503007532 * q^73 - 70803926825553273024 * q^77 - 85280702218715897824 * q^79 + 381814622040086245816 * q^83 - 23191160664932407400 * q^85 + 59742932430695979660 * q^89 - 590588534777956016544 * q^91 - 214051632007883873840 * q^95 + 783394660926711950788 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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