LMFDB - Embedded newform 72.22.a.c.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:	$3$
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} - 12529199x - 17012391021 $ x^3 - x^2 - 12529199*x - 17012391021
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1949.23$ 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)

$f(q)$	$=$	$q-2.53316e7 q^{5} +1.44995e9 q^{7} +O(q^{10})$	$q-2.53316e7 q^{5} +1.44995e9 q^{7} -1.01435e11 q^{11} +7.80720e11 q^{13} -3.44095e12 q^{17} -2.35478e13 q^{19} -2.44645e13 q^{23} +1.64855e14 q^{25} +7.98755e14 q^{29} -2.09891e15 q^{31} -3.67297e16 q^{35} -4.00244e16 q^{37} +1.03288e17 q^{41} -2.02220e17 q^{43} +6.52667e17 q^{47} +1.54382e18 q^{49} -7.19244e17 q^{53} +2.56952e18 q^{55} +5.11836e18 q^{59} -2.91385e18 q^{61} -1.97769e19 q^{65} +2.16038e19 q^{67} +2.65322e18 q^{71} -3.94411e19 q^{73} -1.47076e20 q^{77} +2.10057e19 q^{79} -2.17942e20 q^{83} +8.71649e19 q^{85} -1.76121e20 q^{89} +1.13201e21 q^{91} +5.96505e20 q^{95} +5.80167e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5280498 q^{5} + 852542376 q^{7}+O(q^{10}) $ 3 * q - 5280498 * q^5 + 852542376 * q^7	$ 3 q - 5280498 q^{5} + 852542376 q^{7} - 62490757668 q^{11} + 203765207802 q^{13} - 695827819926 q^{17} + 4955504123196 q^{19} - 150867407938152 q^{23} + 678194854969869 q^{25} - 32\!\cdots\!46 q^{29}+ \cdots + 72\!\cdots\!02 q^{97}+O(q^{100}) $ 3 * q - 5280498 * q^5 + 852542376 * q^7 - 62490757668 * q^11 + 203765207802 * q^13 - 695827819926 * q^17 + 4955504123196 * q^19 - 150867407938152 * q^23 + 678194854969869 * q^25 - 3242224905226746 * q^29 + 652508601550896 * q^31 - 15443719285839600 * q^35 - 20693271836916222 * q^37 + 47608600312552002 * q^41 - 277110211230415548 * q^43 + 535942507326221328 * q^47 + 1192822454236615755 * q^49 - 2849297892348987426 * q^53 + 2255425539904757592 * q^55 - 881046377434726548 * q^59 + 10620890455615553322 * q^61 - 24923333196516863004 * q^65 + 14840080728867715116 * q^67 - 14608944838311941496 * q^71 + 13575086263949291454 * q^73 - 173887987291330337760 * q^77 + 278096577517755548160 * q^79 - 585716918283456830460 * q^83 + 442542794823708598596 * q^85 - 477818701792662526830 * q^89 + 1318091663263676539056 * q^91 - 698400977547585407208 * q^95 + 722411740013721590502 * 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
$5$	−2.53316e7	−1.16005	−0.580027	−	0.814597i	$-0.696958\pi$
$5$	−2.53316e7	−1.16005	−0.580027	+	0.814597i	$0.696958\pi$
$6$	0	0
$7$	1.44995e9	1.94010	0.970052	−	0.242898i	$-0.0780979\pi$
$7$	1.44995e9	1.94010	0.970052	+	0.242898i	$0.0780979\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.01435e11	−1.17914	−0.589569	−	0.807718i	$-0.700702\pi$
$11$	−1.01435e11	−1.17914	−0.589569	+	0.807718i	$0.700702\pi$
$12$	0	0
$13$	7.80720e11	1.57069	0.785344	−	0.619059i	$-0.212486\pi$
$13$	7.80720e11	1.57069	0.785344	+	0.619059i	$0.212486\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.44095e12	−0.413966	−0.206983	−	0.978345i	$-0.566364\pi$
$17$	−3.44095e12	−0.413966	−0.206983	+	0.978345i	$0.566364\pi$
$18$	0	0
$19$	−2.35478e13	−0.881126	−0.440563	−	0.897722i	$-0.645221\pi$
$19$	−2.35478e13	−0.881126	−0.440563	+	0.897722i	$0.645221\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.44645e13	−0.123138	−0.0615692	−	0.998103i	$-0.519610\pi$
$23$	−2.44645e13	−0.123138	−0.0615692	+	0.998103i	$0.519610\pi$
$24$	0	0
$25$	1.64855e14	0.345725
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.98755e14	0.352561	0.176281	−	0.984340i	$-0.443593\pi$
$29$	7.98755e14	0.352561	0.176281	+	0.984340i	$0.443593\pi$
$30$	0	0
$31$	−2.09891e15	−0.459935	−0.229967	−	0.973198i	$-0.573862\pi$
$31$	−2.09891e15	−0.459935	−0.229967	+	0.973198i	$0.573862\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.67297e16	−2.25063
$36$	0	0
$37$	−4.00244e16	−1.36838	−0.684191	−	0.729303i	$-0.739844\pi$
$37$	−4.00244e16	−1.36838	−0.684191	+	0.729303i	$0.739844\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.03288e17	1.20176	0.600881	−	0.799338i	$-0.294816\pi$
$41$	1.03288e17	1.20176	0.600881	+	0.799338i	$0.294816\pi$
$42$	0	0
$43$	−2.02220e17	−1.42694	−0.713468	−	0.700688i	$-0.752877\pi$
$43$	−2.02220e17	−1.42694	−0.713468	+	0.700688i	$0.752877\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.52667e17	1.80994	0.904970	−	0.425476i	$-0.139893\pi$
$47$	6.52667e17	1.80994	0.904970	+	0.425476i	$0.139893\pi$
$48$	0	0
$49$	1.54382e18	2.76400
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−7.19244e17	−0.564910	−0.282455	−	0.959280i	$-0.591149\pi$
$53$	−7.19244e17	−0.564910	−0.282455	+	0.959280i	$0.591149\pi$
$54$	0	0
$55$	2.56952e18	1.36786
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.11836e18	1.30372	0.651861	−	0.758338i	$-0.273989\pi$
$59$	5.11836e18	1.30372	0.651861	+	0.758338i	$0.273989\pi$
$60$	0	0
$61$	−2.91385e18	−0.523003	−0.261501	−	0.965203i	$-0.584218\pi$
$61$	−2.91385e18	−0.523003	−0.261501	+	0.965203i	$0.584218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.97769e19	−1.82208
$66$	0	0
$67$	2.16038e19	1.44792	0.723959	−	0.689843i	$-0.242320\pi$
$67$	2.16038e19	1.44792	0.723959	+	0.689843i	$0.242320\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.65322e18	0.0967298	0.0483649	−	0.998830i	$-0.484599\pi$
$71$	2.65322e18	0.0967298	0.0483649	+	0.998830i	$0.484599\pi$
$72$	0	0
$73$	−3.94411e19	−1.07414	−0.537068	−	0.843539i	$-0.680468\pi$
$73$	−3.94411e19	−1.07414	−0.537068	+	0.843539i	$0.680468\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.47076e20	−2.28765
$78$	0	0
$79$	2.10057e19	0.249605	0.124802	−	0.992182i	$-0.460170\pi$
$79$	2.10057e19	0.249605	0.124802	+	0.992182i	$0.460170\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.17942e20	−1.54177	−0.770886	−	0.636973i	$-0.780186\pi$
$83$	−2.17942e20	−1.54177	−0.770886	+	0.636973i	$0.780186\pi$
$84$	0	0
$85$	8.71649e19	0.480223
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.76121e20	−0.598708	−0.299354	−	0.954142i	$-0.596771\pi$
$89$	−1.76121e20	−0.598708	−0.299354	+	0.954142i	$0.596771\pi$
$90$	0	0
$91$	1.13201e21	3.04730
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.96505e20	1.02215
$96$	0	0
$97$	5.80167e20	0.798821	0.399411	−	0.916772i	$-0.369215\pi$
$97$	5.80167e20	0.798821	0.399411	+	0.916772i	$0.369215\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.59894e20	0.414270	0.207135	−	0.978312i	$-0.433586\pi$
$101$	4.59894e20	0.414270	0.207135	+	0.978312i	$0.433586\pi$
$102$	0	0
$103$	−7.23758e20	−0.530643	−0.265322	−	0.964160i	$-0.585478\pi$
$103$	−7.23758e20	−0.530643	−0.265322	+	0.964160i	$0.585478\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.69849e21	−0.834708	−0.417354	−	0.908744i	$-0.637042\pi$
$107$	−1.69849e21	−0.834708	−0.417354	+	0.908744i	$0.637042\pi$
$108$	0	0
$109$	−4.52952e21	−1.83263	−0.916313	−	0.400464i	$-0.868849\pi$
$109$	−4.52952e21	−1.83263	−0.916313	+	0.400464i	$0.868849\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.74943e21	0.484812	0.242406	−	0.970175i	$-0.422063\pi$
$113$	1.74943e21	0.484812	0.242406	+	0.970175i	$0.422063\pi$
$114$	0	0
$115$	6.19725e20	0.142847
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.98922e21	−0.803137
$120$	0	0
$121$	2.88882e21	0.390368
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.90303e21	0.758994
$126$	0	0
$127$	−4.73201e21	−0.384686	−0.192343	−	0.981328i	$-0.561609\pi$
$127$	−4.73201e21	−0.384686	−0.192343	+	0.981328i	$0.561609\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.10825e22	1.82460	0.912300	−	0.409523i	$-0.134305\pi$
$131$	3.10825e22	1.82460	0.912300	+	0.409523i	$0.134305\pi$
$132$	0	0
$133$	−3.41433e22	−1.70948
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.90411e21	0.179885	0.0899424	−	0.995947i	$-0.471332\pi$
$137$	4.90411e21	0.179885	0.0899424	+	0.995947i	$0.471332\pi$
$138$	0	0
$139$	4.72044e21	0.148705	0.0743527	−	0.997232i	$-0.476311\pi$
$139$	4.72044e21	0.148705	0.0743527	+	0.997232i	$0.476311\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−7.91924e22	−1.85206
$144$	0	0
$145$	−2.02338e22	−0.408990
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.55599e22	0.995824	0.497912	−	0.867227i	$-0.334100\pi$
$149$	6.55599e22	0.995824	0.497912	+	0.867227i	$0.334100\pi$
$150$	0	0
$151$	4.48928e22	0.592815	0.296407	−	0.955062i	$-0.404211\pi$
$151$	4.48928e22	0.592815	0.296407	+	0.955062i	$0.404211\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.31689e22	0.533549
$156$	0	0
$157$	−7.46659e22	−0.654902	−0.327451	−	0.944868i	$-0.606190\pi$
$157$	−7.46659e22	−0.654902	−0.327451	+	0.944868i	$0.606190\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.54724e22	−0.238901
$162$	0	0
$163$	1.75448e22	0.103796	0.0518978	−	0.998652i	$-0.483473\pi$
$163$	1.75448e22	0.103796	0.0518978	+	0.998652i	$0.483473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.45094e23	−1.58276	−0.791378	−	0.611327i	$-0.790636\pi$
$167$	−3.45094e23	−1.58276	−0.791378	+	0.611327i	$0.790636\pi$
$168$	0	0
$169$	3.62459e23	1.46706
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.87876e22	−0.0911425	−0.0455712	−	0.998961i	$-0.514511\pi$
$173$	−2.87876e22	−0.0911425	−0.0455712	+	0.998961i	$0.514511\pi$
$174$	0	0
$175$	2.39032e23	0.670743
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−7.55279e23	−1.67167	−0.835835	−	0.548980i	$-0.815016\pi$
$179$	−7.55279e23	−1.67167	−0.835835	+	0.548980i	$0.815016\pi$
$180$	0	0
$181$	−6.28292e23	−1.23748	−0.618738	−	0.785597i	$-0.712356\pi$
$181$	−6.28292e23	−1.23748	−0.618738	+	0.785597i	$0.712356\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.01388e24	1.58740
$186$	0	0
$187$	3.49033e23	0.488123
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.25757e24	−1.40826	−0.704131	−	0.710070i	$-0.748663\pi$
$191$	−1.25757e24	−1.40826	−0.704131	+	0.710070i	$0.748663\pi$
$192$	0	0
$193$	−1.48490e24	−1.49055	−0.745275	−	0.666758i	$-0.767682\pi$
$193$	−1.48490e24	−1.49055	−0.745275	+	0.666758i	$0.767682\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.60010e24	−1.29495	−0.647473	−	0.762089i	$-0.724174\pi$
$197$	−1.60010e24	−1.29495	−0.647473	+	0.762089i	$0.724174\pi$
$198$	0	0
$199$	−1.47292e24	−1.07206	−0.536032	−	0.844198i	$-0.680077\pi$
$199$	−1.47292e24	−1.07206	−0.536032	+	0.844198i	$0.680077\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.15816e24	0.684005
$204$	0	0
$205$	−2.61645e24	−1.39411
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.38858e24	1.03897
$210$	0	0
$211$	−1.59562e24	−0.628004	−0.314002	−	0.949422i	$-0.601670\pi$
$211$	−1.59562e24	−0.628004	−0.314002	+	0.949422i	$0.601670\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.12255e24	1.65532
$216$	0	0
$217$	−3.04333e24	−0.892321
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.68642e24	−0.650212
$222$	0	0
$223$	4.75834e24	1.04774	0.523871	−	0.851798i	$-0.324487\pi$
$223$	4.75834e24	1.04774	0.523871	+	0.851798i	$0.324487\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.58914e23	−0.102111	−0.0510556	−	0.998696i	$-0.516259\pi$
$227$	−5.58914e23	−0.102111	−0.0510556	+	0.998696i	$0.516259\pi$
$228$	0	0
$229$	−6.85427e24	−1.14206	−0.571030	−	0.820929i	$-0.693456\pi$
$229$	−6.85427e24	−1.14206	−0.571030	+	0.820929i	$0.693456\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.88251e23	−0.0956114	−0.0478057	−	0.998857i	$-0.515223\pi$
$233$	−6.88251e23	−0.0956114	−0.0478057	+	0.998857i	$0.515223\pi$
$234$	0	0
$235$	−1.65331e25	−2.09963
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.98899e24	−0.743424	−0.371712	−	0.928348i	$-0.621229\pi$
$239$	−6.98899e24	−0.743424	−0.371712	+	0.928348i	$0.621229\pi$
$240$	0	0
$241$	9.03567e24	0.880606	0.440303	−	0.897849i	$-0.354871\pi$
$241$	9.03567e24	0.880606	0.440303	+	0.897849i	$0.354871\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.91076e25	−3.20639
$246$	0	0
$247$	−1.83843e25	−1.38397
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5.95713e24	−0.378846	−0.189423	−	0.981896i	$-0.560662\pi$
$251$	−5.95713e24	−0.378846	−0.189423	+	0.981896i	$0.560662\pi$
$252$	0	0
$253$	2.48156e24	0.145197
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.93837e23	0.00961918	0.00480959	−	0.999988i	$-0.498469\pi$
$257$	1.93837e23	0.00961918	0.00480959	+	0.999988i	$0.498469\pi$
$258$	0	0
$259$	−5.80336e25	−2.65480
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.90485e25	−1.91025	−0.955126	−	0.296200i	$-0.904280\pi$
$263$	−4.90485e25	−1.91025	−0.955126	+	0.296200i	$0.904280\pi$
$264$	0	0
$265$	1.82196e25	0.655327
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.15236e25	1.58346	0.791731	−	0.610870i	$-0.209180\pi$
$269$	5.15236e25	1.58346	0.791731	+	0.610870i	$0.209180\pi$
$270$	0	0
$271$	1.41661e25	0.402783	0.201392	−	0.979511i	$-0.435454\pi$
$271$	1.41661e25	0.402783	0.201392	+	0.979511i	$0.435454\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.67220e25	−0.407658
$276$	0	0
$277$	1.29345e25	0.292221	0.146111	−	0.989268i	$-0.453324\pi$
$277$	1.29345e25	0.292221	0.146111	+	0.989268i	$0.453324\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.90093e25	−0.563795	−0.281897	−	0.959445i	$-0.590964\pi$
$281$	−2.90093e25	−0.563795	−0.281897	+	0.959445i	$0.590964\pi$
$282$	0	0
$283$	3.35634e25	0.605492	0.302746	−	0.953071i	$-0.402097\pi$
$283$	3.35634e25	0.605492	0.302746	+	0.953071i	$0.402097\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.49763e26	2.33154
$288$	0	0
$289$	−5.72518e25	−0.828632
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.28365e26	−1.60819	−0.804095	−	0.594501i	$-0.797350\pi$
$293$	−1.28365e26	−1.60819	−0.804095	+	0.594501i	$0.797350\pi$
$294$	0	0
$295$	−1.29657e26	−1.51239
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.90999e25	−0.193412
$300$	0	0
$301$	−2.93209e26	−2.76840
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.38126e25	0.606712
$306$	0	0
$307$	−4.20094e24	−0.0322399	−0.0161200	−	0.999870i	$-0.505131\pi$
$307$	−4.20094e24	−0.0322399	−0.0161200	+	0.999870i	$0.505131\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.80869e25	−0.322139	−0.161069	−	0.986943i	$-0.551494\pi$
$311$	−4.80869e25	−0.322139	−0.161069	+	0.986943i	$0.551494\pi$
$312$	0	0
$313$	−1.21754e26	−0.762551	−0.381275	−	0.924462i	$-0.624515\pi$
$313$	−1.21754e26	−0.762551	−0.381275	+	0.924462i	$0.624515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3.40097e26	−1.86415	−0.932076	−	0.362264i	$-0.882004\pi$
$317$	−3.40097e26	−1.86415	−0.932076	+	0.362264i	$0.882004\pi$
$318$	0	0
$319$	−8.10217e25	−0.415718
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.10269e25	0.364756
$324$	0	0
$325$	1.28705e26	0.543027
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.46337e26	3.51147
$330$	0	0
$331$	−3.67522e25	−0.127964	−0.0639822	−	0.997951i	$-0.520380\pi$
$331$	−3.67522e25	−0.127964	−0.0639822	+	0.997951i	$0.520380\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5.47259e26	−1.67966
$336$	0	0
$337$	4.05557e26	1.16933	0.584667	−	0.811274i	$-0.301225\pi$
$337$	4.05557e26	1.16933	0.584667	+	0.811274i	$0.301225\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.12903e26	0.542327
$342$	0	0
$343$	1.42861e27	3.42235
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.16106e26	−1.30676	−0.653380	−	0.757030i	$-0.726650\pi$
$347$	−6.16106e26	−1.30676	−0.653380	+	0.757030i	$0.726650\pi$
$348$	0	0
$349$	3.03715e26	0.606457	0.303228	−	0.952918i	$-0.401936\pi$
$349$	3.03715e26	0.606457	0.303228	+	0.952918i	$0.401936\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.29761e25	−0.164716	−0.0823582	−	0.996603i	$-0.526245\pi$
$353$	−9.29761e25	−0.164716	−0.0823582	+	0.996603i	$0.526245\pi$
$354$	0	0
$355$	−6.72104e25	−0.112212
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.30849e26	0.194213	0.0971067	−	0.995274i	$-0.469041\pi$
$359$	1.30849e26	0.194213	0.0971067	+	0.995274i	$0.469041\pi$
$360$	0	0
$361$	−1.59709e26	−0.223617
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.99109e26	1.24606
$366$	0	0
$367$	−5.49911e26	−0.647588	−0.323794	−	0.946128i	$-0.604958\pi$
$367$	−5.49911e26	−0.647588	−0.323794	+	0.946128i	$0.604958\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.04287e27	−1.09598
$372$	0	0
$373$	3.52147e26	0.349769	0.174885	−	0.984589i	$-0.444045\pi$
$373$	3.52147e26	0.349769	0.174885	+	0.984589i	$0.444045\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.23604e26	0.553764
$378$	0	0
$379$	1.69895e26	0.142714	0.0713572	−	0.997451i	$-0.477267\pi$
$379$	1.69895e26	0.142714	0.0713572	+	0.997451i	$0.477267\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.62080e27	1.21939	0.609696	−	0.792636i	$-0.291292\pi$
$383$	1.62080e27	1.21939	0.609696	+	0.792636i	$0.291292\pi$
$384$	0	0
$385$	3.72568e27	2.65380
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.43976e27	−0.920066	−0.460033	−	0.887902i	$-0.652162\pi$
$389$	−1.43976e27	−0.920066	−0.460033	+	0.887902i	$0.652162\pi$
$390$	0	0
$391$	8.41810e25	0.0509751
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.32109e26	−0.289555
$396$	0	0
$397$	1.07209e27	0.553263	0.276632	−	0.960976i	$-0.410782\pi$
$397$	1.07209e27	0.553263	0.276632	+	0.960976i	$0.410782\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.39135e26	−0.250427	−0.125214	−	0.992130i	$-0.539962\pi$
$401$	−5.39135e26	−0.250427	−0.125214	+	0.992130i	$0.539962\pi$
$402$	0	0
$403$	−1.63866e27	−0.722414
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.05988e27	1.61351
$408$	0	0
$409$	−3.87676e27	−1.46344	−0.731718	−	0.681608i	$-0.761281\pi$
$409$	−3.87676e27	−1.46344	−0.731718	+	0.681608i	$0.761281\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.42140e27	2.52936
$414$	0	0
$415$	5.52082e27	1.78854
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.74399e26	−0.138962	−0.0694810	−	0.997583i	$-0.522134\pi$
$419$	−4.74399e26	−0.138962	−0.0694810	+	0.997583i	$0.522134\pi$
$420$	0	0
$421$	2.22867e27	0.620988	0.310494	−	0.950575i	$-0.399506\pi$
$421$	2.22867e27	0.620988	0.310494	+	0.950575i	$0.399506\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.67257e26	−0.143119
$426$	0	0
$427$	−4.22495e27	−1.01468
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.26909e27	0.711893	0.355947	−	0.934506i	$-0.384158\pi$
$431$	3.26909e27	0.711893	0.355947	+	0.934506i	$0.384158\pi$
$432$	0	0
$433$	3.11860e27	0.646900	0.323450	−	0.946245i	$-0.395157\pi$
$433$	3.11860e27	0.646900	0.323450	+	0.946245i	$0.395157\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.76085e26	0.108501
$438$	0	0
$439$	−6.30672e26	−0.113221	−0.0566104	−	0.998396i	$-0.518029\pi$
$439$	−6.30672e26	−0.113221	−0.0566104	+	0.998396i	$0.518029\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.95450e27	0.319005	0.159502	−	0.987198i	$-0.449011\pi$
$443$	1.95450e27	0.319005	0.159502	+	0.987198i	$0.449011\pi$
$444$	0	0
$445$	4.46142e27	0.694533
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.08453e28	−1.53694	−0.768470	−	0.639886i	$-0.778981\pi$
$449$	−1.08453e28	−1.53694	−0.768470	+	0.639886i	$0.778981\pi$
$450$	0	0
$451$	−1.04770e28	−1.41704
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.86756e28	−3.53503
$456$	0	0
$457$	9.52639e26	0.112152	0.0560762	−	0.998426i	$-0.482141\pi$
$457$	9.52639e26	0.112152	0.0560762	+	0.998426i	$0.482141\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.42034e28	1.52592	0.762960	−	0.646445i	$-0.223745\pi$
$461$	1.42034e28	1.52592	0.762960	+	0.646445i	$0.223745\pi$
$462$	0	0
$463$	2.64083e27	0.271107	0.135553	−	0.990770i	$-0.456719\pi$
$463$	2.64083e27	0.271107	0.135553	+	0.990770i	$0.456719\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.33597e28	−1.25305	−0.626526	−	0.779401i	$-0.715524\pi$
$467$	−1.33597e28	−1.25305	−0.626526	+	0.779401i	$0.715524\pi$
$468$	0	0
$469$	3.13245e28	2.80911
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.05122e28	1.68255
$474$	0	0
$475$	−3.88197e27	−0.304628
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.67850e27	−0.192473	−0.0962364	−	0.995359i	$-0.530680\pi$
$479$	−2.67850e27	−0.192473	−0.0962364	+	0.995359i	$0.530680\pi$
$480$	0	0
$481$	−3.12479e28	−2.14930
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.46966e28	−0.926676
$486$	0	0
$487$	−5.56727e26	−0.0336193	−0.0168096	−	0.999859i	$-0.505351\pi$
$487$	−5.56727e26	−0.0336193	−0.0168096	+	0.999859i	$0.505351\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.86683e27	−0.380540	−0.190270	−	0.981732i	$-0.560936\pi$
$491$	−6.86683e27	−0.380540	−0.190270	+	0.981732i	$0.560936\pi$
$492$	0	0
$493$	−2.74847e27	−0.145948
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.84705e27	0.187666
$498$	0	0
$499$	−2.97473e28	−1.39121	−0.695604	−	0.718426i	$-0.744863\pi$
$499$	−2.97473e28	−1.39121	−0.695604	+	0.718426i	$0.744863\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.56614e28	−1.10361	−0.551806	−	0.833972i	$-0.686061\pi$
$503$	−2.56614e28	−1.10361	−0.551806	+	0.833972i	$0.686061\pi$
$504$	0	0
$505$	−1.16499e28	−0.480575
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.13487e27	−0.119037	−0.0595186	−	0.998227i	$-0.518957\pi$
$509$	−3.13487e27	−0.119037	−0.0595186	+	0.998227i	$0.518957\pi$
$510$	0	0
$511$	−5.71879e28	−2.08394
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.83340e28	0.615575
$516$	0	0
$517$	−6.62033e28	−2.13417
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.96111e28	1.17766	0.588831	−	0.808256i	$-0.299588\pi$
$521$	3.96111e28	1.17766	0.588831	+	0.808256i	$0.299588\pi$
$522$	0	0
$523$	3.75430e28	1.07217	0.536083	−	0.844166i	$-0.319904\pi$
$523$	3.75430e28	1.07217	0.536083	+	0.844166i	$0.319904\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.22225e27	0.190397
$528$	0	0
$529$	−3.88731e28	−0.984837
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.06389e28	1.88760
$534$	0	0
$535$	4.30257e28	0.968306
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.56598e29	−3.25914
$540$	0	0
$541$	6.39269e28	1.27971	0.639856	−	0.768495i	$-0.278994\pi$
$541$	6.39269e28	1.27971	0.639856	+	0.768495i	$0.278994\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.14740e29	2.12594
$546$	0	0
$547$	−2.12874e28	−0.379539	−0.189769	−	0.981829i	$-0.560774\pi$
$547$	−2.12874e28	−0.379539	−0.189769	+	0.981829i	$0.560774\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.88089e28	−0.310651
$552$	0	0
$553$	3.04573e28	0.484259
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.46737e28	−1.10075	−0.550374	−	0.834918i	$-0.685515\pi$
$557$	−7.46737e28	−1.10075	−0.550374	+	0.834918i	$0.685515\pi$
$558$	0	0
$559$	−1.57877e29	−2.24127
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.57294e28	−0.602361	−0.301181	−	0.953567i	$-0.597381\pi$
$563$	−4.57294e28	−0.602361	−0.301181	+	0.953567i	$0.597381\pi$
$564$	0	0
$565$	−4.43159e28	−0.562408
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.17440e28	0.138400	0.0692000	−	0.997603i	$-0.477955\pi$
$569$	1.17440e28	0.138400	0.0692000	+	0.997603i	$0.477955\pi$
$570$	0	0
$571$	−2.02098e28	−0.229553	−0.114776	−	0.993391i	$-0.536615\pi$
$571$	−2.02098e28	−0.229553	−0.114776	+	0.993391i	$0.536615\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.03309e27	−0.0425721
$576$	0	0
$577$	1.10351e29	1.12313	0.561564	−	0.827433i	$-0.310200\pi$
$577$	1.10351e29	1.12313	0.561564	+	0.827433i	$0.310200\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.16006e29	−2.99120
$582$	0	0
$583$	7.29565e28	0.666108
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.79271e28	−0.322292	−0.161146	−	0.986931i	$-0.551519\pi$
$587$	−3.79271e28	−0.322292	−0.161146	+	0.986931i	$0.551519\pi$
$588$	0	0
$589$	4.94248e28	0.405261
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.79354e29	1.36973	0.684867	−	0.728668i	$-0.259860\pi$
$593$	1.79354e29	1.36973	0.684867	+	0.728668i	$0.259860\pi$
$594$	0	0
$595$	1.26385e29	0.931682
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.40104e28	0.233685	0.116842	−	0.993150i	$-0.462723\pi$
$599$	3.40104e28	0.233685	0.116842	+	0.993150i	$0.462723\pi$
$600$	0	0
$601$	2.72570e29	1.80841	0.904203	−	0.427103i	$-0.140466\pi$
$601$	2.72570e29	1.80841	0.904203	+	0.427103i	$0.140466\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.31785e28	−0.452848
$606$	0	0
$607$	1.46897e29	0.878073	0.439036	−	0.898469i	$-0.355320\pi$
$607$	1.46897e29	0.878073	0.439036	+	0.898469i	$0.355320\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.09550e29	2.84285
$612$	0	0
$613$	2.43388e28	0.131209	0.0656046	−	0.997846i	$-0.479102\pi$
$613$	2.43388e28	0.131209	0.0656046	+	0.997846i	$0.479102\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.53731e29	1.78106	0.890530	−	0.454925i	$-0.150334\pi$
$617$	3.53731e29	1.78106	0.890530	+	0.454925i	$0.150334\pi$
$618$	0	0
$619$	−1.27384e29	−0.619958	−0.309979	−	0.950743i	$-0.600322\pi$
$619$	−1.27384e29	−0.619958	−0.309979	+	0.950743i	$0.600322\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−2.55367e29	−1.16156
$624$	0	0
$625$	−2.78805e29	−1.22620
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.37722e29	0.566463
$630$	0	0
$631$	−3.40698e29	−1.35538	−0.677691	−	0.735347i	$-0.737019\pi$
$631$	−3.40698e29	−1.35538	−0.677691	+	0.735347i	$0.737019\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.19870e29	0.446257
$636$	0	0
$637$	1.20529e30	4.34139
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.95178e28	−0.335654	−0.167827	−	0.985816i	$-0.553675\pi$
$641$	−9.95178e28	−0.335654	−0.167827	+	0.985816i	$0.553675\pi$
$642$	0	0
$643$	5.85921e29	1.91260	0.956300	−	0.292388i	$-0.0944500\pi$
$643$	5.85921e29	1.91260	0.956300	+	0.292388i	$0.0944500\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.18098e28	0.127874	0.0639371	−	0.997954i	$-0.479634\pi$
$647$	4.18098e28	0.127874	0.0639371	+	0.997954i	$0.479634\pi$
$648$	0	0
$649$	−5.19181e29	−1.53727
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.65567e28	−0.268037	−0.134018	−	0.990979i	$-0.542788\pi$
$653$	−9.65567e28	−0.268037	−0.134018	+	0.990979i	$0.542788\pi$
$654$	0	0
$655$	−7.87371e29	−2.11663
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.52577e29	−0.384763	−0.192382	−	0.981320i	$-0.561621\pi$
$659$	−1.52577e29	−0.384763	−0.192382	+	0.981320i	$0.561621\pi$
$660$	0	0
$661$	3.72545e25	9.10047e−5 0	4.55023e−5	−	1.00000i	$-0.499986\pi$
$661$	3.72545e25	9.10047e−5 0	4.55023e−5		1.00000i	$0.499986\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.64905e29	1.98308
$666$	0	0
$667$	−1.95411e28	−0.0434138
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.95567e29	0.616693
$672$	0	0
$673$	−5.65681e29	−1.14397	−0.571984	−	0.820265i	$-0.693826\pi$
$673$	−5.65681e29	−1.14397	−0.571984	+	0.820265i	$0.693826\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.36606e29	−1.39976	−0.699881	−	0.714259i	$-0.746764\pi$
$677$	−7.36606e29	−1.39976	−0.699881	+	0.714259i	$0.746764\pi$
$678$	0	0
$679$	8.41215e29	1.54980
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.48146e29	−0.256609	−0.128304	−	0.991735i	$-0.540953\pi$
$683$	−1.48146e29	−0.256609	−0.128304	+	0.991735i	$0.540953\pi$
$684$	0	0
$685$	−1.24229e29	−0.208676
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.61528e29	−0.887298
$690$	0	0
$691$	1.84866e29	0.283359	0.141680	−	0.989913i	$-0.454750\pi$
$691$	1.84866e29	0.283359	0.141680	+	0.989913i	$0.454750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.19576e29	−0.172506
$696$	0	0
$697$	−3.55408e29	−0.497489
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.60796e29	1.00283	0.501417	−	0.865206i	$-0.332812\pi$
$701$	7.60796e29	1.00283	0.501417	+	0.865206i	$0.332812\pi$
$702$	0	0
$703$	9.42489e29	1.20572
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.66825e29	0.803727
$708$	0	0
$709$	1.21798e30	1.42513	0.712566	−	0.701605i	$-0.247533\pi$
$709$	1.21798e30	1.42513	0.712566	+	0.701605i	$0.247533\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.13488e28	0.0566357
$714$	0	0
$715$	2.00607e30	2.14849
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.14372e30	−1.15522	−0.577611	−	0.816312i	$-0.696015\pi$
$719$	−1.14372e30	−1.15522	−0.577611	+	0.816312i	$0.696015\pi$
$720$	0	0
$721$	−1.04942e30	−1.02950
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.31678e29	0.121889
$726$	0	0
$727$	1.16930e30	1.05151	0.525755	−	0.850636i	$-0.323783\pi$
$727$	1.16930e30	1.05151	0.525755	+	0.850636i	$0.323783\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.95827e29	0.590703
$732$	0	0
$733$	−1.78437e30	−1.47195	−0.735974	−	0.677010i	$-0.763276\pi$
$733$	−1.78437e30	−1.47195	−0.735974	+	0.677010i	$0.763276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.19138e30	−1.70730
$738$	0	0
$739$	−1.95595e30	−1.48112	−0.740561	−	0.671989i	$-0.765440\pi$
$739$	−1.95595e30	−1.48112	−0.740561	+	0.671989i	$0.765440\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.86810e29	0.205217	0.102608	−	0.994722i	$-0.467281\pi$
$743$	2.86810e29	0.205217	0.102608	+	0.994722i	$0.467281\pi$
$744$	0	0
$745$	−1.66074e30	−1.15521
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.46274e30	−1.61942
$750$	0	0
$751$	−8.76672e28	−0.0560555	−0.0280277	−	0.999607i	$-0.508923\pi$
$751$	−8.76672e28	−0.0560555	−0.0280277	+	0.999607i	$0.508923\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.13721e30	−0.687697
$756$	0	0
$757$	1.33126e29	0.0782990	0.0391495	−	0.999233i	$-0.487535\pi$
$757$	1.33126e29	0.0782990	0.0391495	+	0.999233i	$0.487535\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.46231e29	−0.415274	−0.207637	−	0.978206i	$-0.566577\pi$
$761$	−7.46231e29	−0.415274	−0.207637	+	0.978206i	$0.566577\pi$
$762$	0	0
$763$	−6.56759e30	−3.55548
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.99601e30	2.04774
$768$	0	0
$769$	−1.39687e30	−0.696511	−0.348255	−	0.937400i	$-0.613226\pi$
$769$	−1.39687e30	−0.696511	−0.348255	+	0.937400i	$0.613226\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.76954e30	−1.30774	−0.653872	−	0.756605i	$-0.726856\pi$
$773$	−2.76954e30	−1.30774	−0.653872	+	0.756605i	$0.726856\pi$
$774$	0	0
$775$	−3.46015e29	−0.159011
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.43220e30	−1.05890
$780$	0	0
$781$	−2.69129e29	−0.114058
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.89141e30	0.759721
$786$	0	0
$787$	3.38800e30	1.32498	0.662489	−	0.749072i	$-0.269500\pi$
$787$	3.38800e30	1.32498	0.662489	+	0.749072i	$0.269500\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.53660e30	0.940585
$792$	0	0
$793$	−2.27490e30	−0.821475
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.64536e30	0.906091	0.453046	−	0.891487i	$-0.350337\pi$
$797$	2.64536e30	0.906091	0.453046	+	0.891487i	$0.350337\pi$
$798$	0	0
$799$	−2.24579e30	−0.749253
$800$	0	0
$801$	0	0
$802$	0	0
$803$	4.00071e30	1.26656
$804$	0	0
$805$	8.98574e29	0.277139
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.70549e30	0.792110	0.396055	−	0.918227i	$-0.370379\pi$
$809$	2.70549e30	0.792110	0.396055	+	0.918227i	$0.370379\pi$
$810$	0	0
$811$	−2.04480e30	−0.583353	−0.291677	−	0.956517i	$-0.594213\pi$
$811$	−2.04480e30	−0.583353	−0.291677	+	0.956517i	$0.594213\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.44439e29	−0.120408
$816$	0	0
$817$	4.76183e30	1.25731
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−5.73168e30	−1.43774	−0.718868	−	0.695147i	$-0.755339\pi$
$821$	−5.73168e30	−1.43774	−0.718868	+	0.695147i	$0.755339\pi$
$822$	0	0
$823$	−6.14845e30	−1.50337	−0.751687	−	0.659520i	$-0.770760\pi$
$823$	−6.14845e30	−1.50337	−0.751687	+	0.659520i	$0.770760\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.86102e30	−0.664833	−0.332416	−	0.943133i	$-0.607864\pi$
$827$	−2.86102e30	−0.664833	−0.332416	+	0.943133i	$0.607864\pi$
$828$	0	0
$829$	3.25927e30	0.738411	0.369205	−	0.929348i	$-0.379630\pi$
$829$	3.25927e30	0.738411	0.369205	+	0.929348i	$0.379630\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.31222e30	−1.14420
$834$	0	0
$835$	8.74178e30	1.83608
$836$	0	0
$837$	0	0
$838$	0	0
$839$	7.23035e30	1.44430	0.722152	−	0.691734i	$-0.243153\pi$
$839$	7.23035e30	1.44430	0.722152	+	0.691734i	$0.243153\pi$
$840$	0	0
$841$	−4.49483e30	−0.875701
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.18169e30	−1.70187
$846$	0	0
$847$	4.18866e30	0.757354
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.79177e29	0.168500
$852$	0	0
$853$	−5.62649e30	−0.944655	−0.472327	−	0.881423i	$-0.656586\pi$
$853$	−5.62649e30	−0.944655	−0.472327	+	0.881423i	$0.656586\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.36845e29	−0.0858124	−0.0429062	−	0.999079i	$-0.513662\pi$
$857$	−5.36845e29	−0.0858124	−0.0429062	+	0.999079i	$0.513662\pi$
$858$	0	0
$859$	−1.05601e31	−1.64718	−0.823590	−	0.567186i	$-0.808032\pi$
$859$	−1.05601e31	−1.64718	−0.823590	+	0.567186i	$0.808032\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.51971e30	−0.522869	−0.261435	−	0.965221i	$-0.584196\pi$
$863$	−3.51971e30	−0.522869	−0.261435	+	0.965221i	$0.584196\pi$
$864$	0	0
$865$	7.29236e29	0.105730
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.13072e30	−0.294319
$870$	0	0
$871$	1.68665e31	2.27423
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.14590e31	1.47253
$876$	0	0
$877$	6.09111e30	0.764189	0.382094	−	0.924123i	$-0.375203\pi$
$877$	6.09111e30	0.764189	0.382094	+	0.924123i	$0.375203\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.37553e30	0.523339	0.261670	−	0.965157i	$-0.415727\pi$
$881$	4.37553e30	0.523339	0.261670	+	0.965157i	$0.415727\pi$
$882$	0	0
$883$	1.34907e31	1.57560	0.787802	−	0.615929i	$-0.211219\pi$
$883$	1.34907e31	1.57560	0.787802	+	0.615929i	$0.211219\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.18552e31	−1.32042	−0.660211	−	0.751080i	$-0.729533\pi$
$887$	−1.18552e31	−1.32042	−0.660211	+	0.751080i	$0.729533\pi$
$888$	0	0
$889$	−6.86120e30	−0.746331
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.53689e31	−1.59478
$894$	0	0
$895$	1.91325e31	1.93923
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.67651e30	−0.162155
$900$	0	0
$901$	2.47488e30	0.233854
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.59157e31	1.43554
$906$	0	0
$907$	3.69200e30	0.325376	0.162688	−	0.986678i	$-0.447984\pi$
$907$	3.69200e30	0.325376	0.162688	+	0.986678i	$0.447984\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.13452e30	0.768675	0.384337	−	0.923193i	$-0.374430\pi$
$911$	9.13452e30	0.768675	0.384337	+	0.923193i	$0.374430\pi$
$912$	0	0
$913$	2.21069e31	1.81796
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.50682e31	3.53991
$918$	0	0
$919$	−1.90280e31	−1.46077	−0.730383	−	0.683038i	$-0.760658\pi$
$919$	−1.90280e31	−1.46077	−0.730383	+	0.683038i	$0.760658\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.07142e30	0.151932
$924$	0	0
$925$	−6.59822e30	−0.473084
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.46422e30	0.305901	0.152950	−	0.988234i	$-0.451123\pi$
$929$	4.46422e30	0.305901	0.152950	+	0.988234i	$0.451123\pi$
$930$	0	0
$931$	−3.63537e31	−2.43544
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.84157e30	−0.566249
$936$	0	0
$937$	2.95935e31	1.85324	0.926618	−	0.376004i	$-0.122702\pi$
$937$	2.95935e31	1.85324	0.926618	+	0.376004i	$0.122702\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.13331e31	0.678666	0.339333	−	0.940666i	$-0.389799\pi$
$941$	1.13331e31	0.678666	0.339333	+	0.940666i	$0.389799\pi$
$942$	0	0
$943$	−2.52688e30	−0.147983
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.51385e31	−0.848023	−0.424012	−	0.905657i	$-0.639378\pi$
$947$	−1.51385e31	−0.848023	−0.424012	+	0.905657i	$0.639378\pi$
$948$	0	0
$949$	−3.07925e31	−1.68713
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.90072e30	0.0996421	0.0498211	−	0.998758i	$-0.484135\pi$
$953$	1.90072e30	0.0996421	0.0498211	+	0.998758i	$0.484135\pi$
$954$	0	0
$955$	3.18564e31	1.63366
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.11074e30	0.348995
$960$	0	0
$961$	−1.64201e31	−0.788460
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.76150e31	1.72912
$966$	0	0
$967$	−8.03782e30	−0.361543	−0.180772	−	0.983525i	$-0.557860\pi$
$967$	−8.03782e30	−0.361543	−0.180772	+	0.983525i	$0.557860\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.30207e31	−1.85300	−0.926500	−	0.376296i	$-0.877198\pi$
$971$	−4.30207e31	−1.85300	−0.926500	+	0.376296i	$0.877198\pi$
$972$	0	0
$973$	6.84442e30	0.288504
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.84681e31	0.745640	0.372820	−	0.927904i	$-0.378391\pi$
$977$	1.84681e31	0.745640	0.372820	+	0.927904i	$0.378391\pi$
$978$	0	0
$979$	1.78648e31	0.705959
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.75766e31	−1.04407	−0.522033	−	0.852925i	$-0.674826\pi$
$983$	−2.75766e31	−1.04407	−0.522033	+	0.852925i	$0.674826\pi$
$984$	0	0
$985$	4.05331e31	1.50221
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.94720e30	0.175711
$990$	0	0
$991$	6.85726e30	0.238439	0.119220	−	0.992868i	$-0.461961\pi$
$991$	6.85726e30	0.238439	0.119220	+	0.992868i	$0.461961\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.73114e31	1.24365
$996$	0	0
$997$	1.12397e31	0.366822	0.183411	−	0.983036i	$-0.441286\pi$
$997$	1.12397e31	0.366822	0.183411	+	0.983036i	$0.441286\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.22.a.d.1.3	✓	3	3.2	odd	2
48.22.a.j.1.3		3	12.11	even	2
72.22.a.c.1.1		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.53316e7 q^{5} +1.44995e9 q^{7} +O(q^{10})\)	\(q-2.53316e7 q^{5} +1.44995e9 q^{7} -1.01435e11 q^{11} +7.80720e11 q^{13} -3.44095e12 q^{17} -2.35478e13 q^{19} -2.44645e13 q^{23} +1.64855e14 q^{25} +7.98755e14 q^{29} -2.09891e15 q^{31} -3.67297e16 q^{35} -4.00244e16 q^{37} +1.03288e17 q^{41} -2.02220e17 q^{43} +6.52667e17 q^{47} +1.54382e18 q^{49} -7.19244e17 q^{53} +2.56952e18 q^{55} +5.11836e18 q^{59} -2.91385e18 q^{61} -1.97769e19 q^{65} +2.16038e19 q^{67} +2.65322e18 q^{71} -3.94411e19 q^{73} -1.47076e20 q^{77} +2.10057e19 q^{79} -2.17942e20 q^{83} +8.71649e19 q^{85} -1.76121e20 q^{89} +1.13201e21 q^{91} +5.96505e20 q^{95} +5.80167e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5280498 q^{5} + 852542376 q^{7}+O(q^{10}) \) 3 * q - 5280498 * q^5 + 852542376 * q^7	\( 3 q - 5280498 q^{5} + 852542376 q^{7} - 62490757668 q^{11} + 203765207802 q^{13} - 695827819926 q^{17} + 4955504123196 q^{19} - 150867407938152 q^{23} + 678194854969869 q^{25} - 32\!\cdots\!46 q^{29}+ \cdots + 72\!\cdots\!02 q^{97}+O(q^{100}) \) 3 * q - 5280498 * q^5 + 852542376 * q^7 - 62490757668 * q^11 + 203765207802 * q^13 - 695827819926 * q^17 + 4955504123196 * q^19 - 150867407938152 * q^23 + 678194854969869 * q^25 - 3242224905226746 * q^29 + 652508601550896 * q^31 - 15443719285839600 * q^35 - 20693271836916222 * q^37 + 47608600312552002 * q^41 - 277110211230415548 * q^43 + 535942507326221328 * q^47 + 1192822454236615755 * q^49 - 2849297892348987426 * q^53 + 2255425539904757592 * q^55 - 881046377434726548 * q^59 + 10620890455615553322 * q^61 - 24923333196516863004 * q^65 + 14840080728867715116 * q^67 - 14608944838311941496 * q^71 + 13575086263949291454 * q^73 - 173887987291330337760 * q^77 + 278096577517755548160 * q^79 - 585716918283456830460 * q^83 + 442542794823708598596 * q^85 - 477818701792662526830 * q^89 + 1318091663263676539056 * q^91 - 698400977547585407208 * q^95 + 722411740013721590502 * q^97

Introduction

L-functions

Modular forms

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