LMFDB - Embedded newform 72.22.a.c.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,-5280498] 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} - 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			1.3
Root			$-2135.93$ 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)

$f(q)$	$=$	$q+3.51850e7 q^{5} +2.43346e8 q^{7} +5.47083e9 q^{11} -2.75801e11 q^{13} +7.88815e12 q^{17} -1.71614e13 q^{19} -1.02245e14 q^{23} +7.61145e14 q^{25} -3.25807e15 q^{29} -5.25673e15 q^{31} +8.56212e15 q^{35} -1.32886e15 q^{37} -1.25989e17 q^{41} -8.18452e16 q^{43} -3.29399e17 q^{47} -4.99329e17 q^{49} -1.26060e18 q^{53} +1.92491e17 q^{55} -5.01887e18 q^{59} +7.90880e18 q^{61} -9.70403e18 q^{65} -9.13054e18 q^{67} +3.27073e19 q^{71} +2.66934e19 q^{73} +1.33131e18 q^{77} +1.51094e20 q^{79} -2.78125e20 q^{83} +2.77544e20 q^{85} +1.69366e20 q^{89} -6.71149e19 q^{91} -6.03824e20 q^{95} +5.36145e20 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 652508601550896 q^{31}+ \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

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
$5$	3.51850e7	1.61128	0.805642	−	0.592403i	$-0.201821\pi$
$5$	3.51850e7	1.61128	0.805642	+	0.592403i	$0.201821\pi$
$6$	0	0
$7$	2.43346e8	0.325608	0.162804	−	0.986658i	$-0.447946\pi$
$7$	2.43346e8	0.325608	0.162804	+	0.986658i	$0.447946\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.47083e9	0.0635961	0.0317980	−	0.999494i	$-0.489877\pi$
$11$	5.47083e9	0.0635961	0.0317980	+	0.999494i	$0.489877\pi$
$12$	0	0
$13$	−2.75801e11	−0.554868	−0.277434	−	0.960745i	$-0.589484\pi$
$13$	−2.75801e11	−0.554868	−0.277434	+	0.960745i	$0.589484\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.88815e12	0.948990	0.474495	−	0.880258i	$-0.342631\pi$
$17$	7.88815e12	0.948990	0.474495	+	0.880258i	$0.342631\pi$
$18$	0	0
$19$	−1.71614e13	−0.642156	−0.321078	−	0.947053i	$-0.604045\pi$
$19$	−1.71614e13	−0.642156	−0.321078	+	0.947053i	$0.604045\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.02245e14	−0.514635	−0.257318	−	0.966327i	$-0.582839\pi$
$23$	−1.02245e14	−0.514635	−0.257318	+	0.966327i	$0.582839\pi$
$24$	0	0
$25$	7.61145e14	1.59624
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.25807e15	−1.43807	−0.719037	−	0.694972i	$-0.755417\pi$
$29$	−3.25807e15	−1.43807	−0.719037	+	0.694972i	$0.755417\pi$
$30$	0	0
$31$	−5.25673e15	−1.15191	−0.575954	−	0.817482i	$-0.695369\pi$
$31$	−5.25673e15	−1.15191	−0.575954	+	0.817482i	$0.695369\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.56212e15	0.524647
$36$	0	0
$37$	−1.32886e15	−0.0454319	−0.0227160	−	0.999742i	$-0.507231\pi$
$37$	−1.32886e15	−0.0454319	−0.0227160	+	0.999742i	$0.507231\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.25989e17	−1.46589	−0.732944	−	0.680289i	$-0.761854\pi$
$41$	−1.25989e17	−1.46589	−0.732944	+	0.680289i	$0.761854\pi$
$42$	0	0
$43$	−8.18452e16	−0.577530	−0.288765	−	0.957400i	$-0.593245\pi$
$43$	−8.18452e16	−0.577530	−0.288765	+	0.957400i	$0.593245\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.29399e17	−0.913471	−0.456735	−	0.889603i	$-0.650981\pi$
$47$	−3.29399e17	−0.913471	−0.456735	+	0.889603i	$0.650981\pi$
$48$	0	0
$49$	−4.99329e17	−0.893980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.26060e18	−0.990103	−0.495052	−	0.868864i	$-0.664851\pi$
$53$	−1.26060e18	−0.990103	−0.495052	+	0.868864i	$0.664851\pi$
$54$	0	0
$55$	1.92491e17	0.102471
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.01887e18	−1.27838	−0.639190	−	0.769049i	$-0.720730\pi$
$59$	−5.01887e18	−1.27838	−0.639190	+	0.769049i	$0.720730\pi$
$60$	0	0
$61$	7.90880e18	1.41954	0.709769	−	0.704434i	$-0.248799\pi$
$61$	7.90880e18	1.41954	0.709769	+	0.704434i	$0.248799\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−9.70403e18	−0.894050
$66$	0	0
$67$	−9.13054e18	−0.611943	−0.305972	−	0.952041i	$-0.598981\pi$
$67$	−9.13054e18	−0.611943	−0.305972	+	0.952041i	$0.598981\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.27073e19	1.19243	0.596215	−	0.802825i	$-0.296671\pi$
$71$	3.27073e19	1.19243	0.596215	+	0.802825i	$0.296671\pi$
$72$	0	0
$73$	2.66934e19	0.726966	0.363483	−	0.931601i	$-0.381587\pi$
$73$	2.66934e19	0.726966	0.363483	+	0.931601i	$0.381587\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.33131e18	0.0207074
$78$	0	0
$79$	1.51094e20	1.79541	0.897706	−	0.440595i	$-0.145232\pi$
$79$	1.51094e20	1.79541	0.897706	+	0.440595i	$0.145232\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.78125e20	−1.96752	−0.983761	−	0.179482i	$-0.942558\pi$
$83$	−2.78125e20	−1.96752	−0.983761	+	0.179482i	$0.942558\pi$
$84$	0	0
$85$	2.77544e20	1.52909
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.69366e20	0.575747	0.287873	−	0.957669i	$-0.407052\pi$
$89$	1.69366e20	0.575747	0.287873	+	0.957669i	$0.407052\pi$
$90$	0	0
$91$	−6.71149e19	−0.180669
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.03824e20	−1.03470
$96$	0	0
$97$	5.36145e20	0.738209	0.369104	−	0.929388i	$-0.379664\pi$
$97$	5.36145e20	0.738209	0.369104	+	0.929388i	$0.379664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.66650e21	−1.50117	−0.750587	−	0.660771i	$-0.770229\pi$
$101$	−1.66650e21	−1.50117	−0.750587	+	0.660771i	$0.770229\pi$
$102$	0	0
$103$	−1.73073e21	−1.26893	−0.634465	−	0.772952i	$-0.718779\pi$
$103$	−1.73073e21	−1.26893	−0.634465	+	0.772952i	$0.718779\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.95507e21	−0.960800	−0.480400	−	0.877050i	$-0.659508\pi$
$107$	−1.95507e21	−0.960800	−0.480400	+	0.877050i	$0.659508\pi$
$108$	0	0
$109$	2.31264e21	0.935687	0.467844	−	0.883811i	$-0.345031\pi$
$109$	2.31264e21	0.935687	0.467844	+	0.883811i	$0.345031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.49465e21	1.52271	0.761354	−	0.648336i	$-0.224535\pi$
$113$	5.49465e21	1.52271	0.761354	+	0.648336i	$0.224535\pi$
$114$	0	0
$115$	−3.59749e21	−0.829224
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.91955e21	0.308998
$120$	0	0
$121$	−7.37032e21	−0.995956
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00033e22	0.960705
$126$	0	0
$127$	8.26429e21	0.671841	0.335921	−	0.941890i	$-0.390953\pi$
$127$	8.26429e21	0.671841	0.335921	+	0.941890i	$0.390953\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.36696e22	0.802433	0.401216	−	0.915983i	$-0.368588\pi$
$131$	1.36696e22	0.802433	0.401216	+	0.915983i	$0.368588\pi$
$132$	0	0
$133$	−4.17616e21	−0.209091
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.55734e22	1.67165	0.835826	−	0.548994i	$-0.184989\pi$
$137$	4.55734e22	1.67165	0.835826	+	0.548994i	$0.184989\pi$
$138$	0	0
$139$	−1.19577e22	−0.376697	−0.188349	−	0.982102i	$-0.560314\pi$
$139$	−1.19577e22	−0.376697	−0.188349	+	0.982102i	$0.560314\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.50886e21	−0.0352874
$144$	0	0
$145$	−1.14635e23	−2.31715
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.81889e22	−1.49144	−0.745721	−	0.666258i	$-0.767895\pi$
$149$	−9.81889e22	−1.49144	−0.745721	+	0.666258i	$0.767895\pi$
$150$	0	0
$151$	1.19351e23	1.57605	0.788025	−	0.615644i	$-0.211104\pi$
$151$	1.19351e23	1.57605	0.788025	+	0.615644i	$0.211104\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.84958e23	−1.85605
$156$	0	0
$157$	−7.47687e22	−0.655803	−0.327901	−	0.944712i	$-0.606341\pi$
$157$	−7.47687e22	−0.655803	−0.327901	+	0.944712i	$0.606341\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.48809e22	−0.167569
$162$	0	0
$163$	2.03663e23	1.20488	0.602438	−	0.798166i	$-0.294196\pi$
$163$	2.03663e23	1.20488	0.602438	+	0.798166i	$0.294196\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.50641e23	−0.690907	−0.345453	−	0.938436i	$-0.612275\pi$
$167$	−1.50641e23	−0.690907	−0.345453	+	0.938436i	$0.612275\pi$
$168$	0	0
$169$	−1.70999e23	−0.692121
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.13849e23	0.677054	0.338527	−	0.940957i	$-0.390071\pi$
$173$	2.13849e23	0.677054	0.338527	+	0.940957i	$0.390071\pi$
$174$	0	0
$175$	1.85221e23	0.519747
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.14152e22	0.0695318	0.0347659	−	0.999395i	$-0.488931\pi$
$179$	3.14152e22	0.0695318	0.0347659	+	0.999395i	$0.488931\pi$
$180$	0	0
$181$	6.03303e22	0.118826	0.0594128	−	0.998233i	$-0.481077\pi$
$181$	6.03303e22	0.118826	0.0594128	+	0.998233i	$0.481077\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.67559e22	−0.0732038
$186$	0	0
$187$	4.31547e22	0.0603520
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.09055e24	−1.22123	−0.610613	−	0.791929i	$-0.709077\pi$
$191$	−1.09055e24	−1.22123	−0.610613	+	0.791929i	$0.709077\pi$
$192$	0	0
$193$	3.10755e23	0.311936	0.155968	−	0.987762i	$-0.450150\pi$
$193$	3.10755e23	0.311936	0.155968	+	0.987762i	$0.450150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.68710e24	1.36535	0.682676	−	0.730721i	$-0.260816\pi$
$197$	1.68710e24	1.36535	0.682676	+	0.730721i	$0.260816\pi$
$198$	0	0
$199$	−1.51134e24	−1.10003	−0.550015	−	0.835155i	$-0.685378\pi$
$199$	−1.51134e24	−1.10003	−0.550015	+	0.835155i	$0.685378\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.92838e23	−0.468248
$204$	0	0
$205$	−4.43291e24	−2.36196
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.38872e22	−0.0408386
$210$	0	0
$211$	−4.04443e24	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$211$	−4.04443e24	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.87972e24	−0.930564
$216$	0	0
$217$	−1.27920e24	−0.375070
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.17556e24	−0.526564
$222$	0	0
$223$	−3.23296e24	−0.711868	−0.355934	−	0.934511i	$-0.615837\pi$
$223$	−3.23296e24	−0.711868	−0.355934	+	0.934511i	$0.615837\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.71374e24	1.04388	0.521938	−	0.852983i	$-0.325209\pi$
$227$	5.71374e24	1.04388	0.521938	+	0.852983i	$0.325209\pi$
$228$	0	0
$229$	−1.35658e24	−0.226033	−0.113016	−	0.993593i	$-0.536051\pi$
$229$	−1.35658e24	−0.226033	−0.113016	+	0.993593i	$0.536051\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.19554e24	−0.582842	−0.291421	−	0.956595i	$-0.594128\pi$
$233$	−4.19554e24	−0.582842	−0.291421	+	0.956595i	$0.594128\pi$
$234$	0	0
$235$	−1.15899e25	−1.47186
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.83395e24	0.407820	0.203910	−	0.978990i	$-0.434635\pi$
$239$	3.83395e24	0.407820	0.203910	+	0.978990i	$0.434635\pi$
$240$	0	0
$241$	1.11880e24	0.109037	0.0545186	−	0.998513i	$-0.482638\pi$
$241$	1.11880e24	0.109037	0.0545186	+	0.998513i	$0.482638\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.75689e25	−1.44045
$246$	0	0
$247$	4.73313e24	0.356312
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.18420e24	−0.138905	−0.0694527	−	0.997585i	$-0.522125\pi$
$251$	−2.18420e24	−0.138905	−0.0694527	+	0.997585i	$0.522125\pi$
$252$	0	0
$253$	−5.59365e23	−0.0327288
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.30434e24	−0.412105	−0.206052	−	0.978541i	$-0.566062\pi$
$257$	−8.30434e24	−0.412105	−0.206052	+	0.978541i	$0.566062\pi$
$258$	0	0
$259$	−3.23373e23	−0.0147930
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.63856e25	−1.02762	−0.513809	−	0.857904i	$-0.671766\pi$
$263$	−2.63856e25	−1.02762	−0.513809	+	0.857904i	$0.671766\pi$
$264$	0	0
$265$	−4.43542e25	−1.59534
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.85272e25	−1.18405	−0.592023	−	0.805921i	$-0.701670\pi$
$269$	−3.85272e25	−1.18405	−0.592023	+	0.805921i	$0.701670\pi$
$270$	0	0
$271$	−4.43179e25	−1.26009	−0.630046	−	0.776558i	$-0.716964\pi$
$271$	−4.43179e25	−1.26009	−0.630046	+	0.776558i	$0.716964\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.16410e24	0.101514
$276$	0	0
$277$	5.73390e24	0.129542	0.0647712	−	0.997900i	$-0.479368\pi$
$277$	5.73390e24	0.129542	0.0647712	+	0.997900i	$0.479368\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.65196e25	−0.904107	−0.452054	−	0.891991i	$-0.649308\pi$
$281$	−4.65196e25	−0.904107	−0.452054	+	0.891991i	$0.649308\pi$
$282$	0	0
$283$	−2.23171e25	−0.402605	−0.201303	−	0.979529i	$-0.564517\pi$
$283$	−2.23171e25	−0.402605	−0.201303	+	0.979529i	$0.564517\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.06588e25	−0.477305
$288$	0	0
$289$	−6.86904e24	−0.0994189
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.10928e25	−0.514820	−0.257410	−	0.966302i	$-0.582869\pi$
$293$	−4.10928e25	−0.514820	−0.257410	+	0.966302i	$0.582869\pi$
$294$	0	0
$295$	−1.76589e26	−2.05983
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.81992e25	0.285555
$300$	0	0
$301$	−1.99167e25	−0.188048
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.78271e26	2.28728
$306$	0	0
$307$	−7.68334e25	−0.589654	−0.294827	−	0.955551i	$-0.595262\pi$
$307$	−7.68334e25	−0.589654	−0.294827	+	0.955551i	$0.595262\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.06898e25	0.473557	0.236779	−	0.971564i	$-0.423908\pi$
$311$	7.06898e25	0.473557	0.236779	+	0.971564i	$0.423908\pi$
$312$	0	0
$313$	−2.82026e25	−0.176634	−0.0883168	−	0.996092i	$-0.528149\pi$
$313$	−2.82026e25	−0.176634	−0.0883168	+	0.996092i	$0.528149\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.32837e26	1.27623	0.638115	−	0.769941i	$-0.279714\pi$
$317$	2.32837e26	1.27623	0.638115	+	0.769941i	$0.279714\pi$
$318$	0	0
$319$	−1.78244e25	−0.0914559
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.35372e26	−0.609399
$324$	0	0
$325$	−2.09924e26	−0.885701
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8.01579e25	−0.297433
$330$	0	0
$331$	5.36145e26	1.86676	0.933379	−	0.358892i	$-0.116845\pi$
$331$	5.36145e26	1.86676	0.933379	+	0.358892i	$0.116845\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.21258e26	−0.986014
$336$	0	0
$337$	5.24070e26	1.51104	0.755519	−	0.655126i	$-0.227385\pi$
$337$	5.24070e26	1.51104	0.755519	+	0.655126i	$0.227385\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.87587e25	−0.0732568
$342$	0	0
$343$	−2.57429e26	−0.616694
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.79992e26	−0.805962	−0.402981	−	0.915208i	$-0.632026\pi$
$347$	−3.79992e26	−0.805962	−0.402981	+	0.915208i	$0.632026\pi$
$348$	0	0
$349$	−2.61472e26	−0.522106	−0.261053	−	0.965324i	$-0.584070\pi$
$349$	−2.61472e26	−0.522106	−0.261053	+	0.965324i	$0.584070\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.68732e26	−0.476086	−0.238043	−	0.971255i	$-0.576506\pi$
$353$	−2.68732e26	−0.476086	−0.238043	+	0.971255i	$0.576506\pi$
$354$	0	0
$355$	1.15081e27	1.92134
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.16873e26	−0.618745	−0.309373	−	0.950941i	$-0.600119\pi$
$359$	−4.16873e26	−0.618745	−0.309373	+	0.950941i	$0.600119\pi$
$360$	0	0
$361$	−4.19695e26	−0.587636
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.39207e26	1.17135
$366$	0	0
$367$	8.62599e26	1.01582	0.507908	−	0.861411i	$-0.330419\pi$
$367$	8.62599e26	1.01582	0.507908	+	0.861411i	$0.330419\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.06762e26	−0.322385
$372$	0	0
$373$	1.44945e27	1.43966	0.719830	−	0.694151i	$-0.244220\pi$
$373$	1.44945e27	1.43966	0.719830	+	0.694151i	$0.244220\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.98577e26	0.797942
$378$	0	0
$379$	1.00826e27	0.846952	0.423476	−	0.905907i	$-0.360810\pi$
$379$	1.00826e27	0.846952	0.423476	+	0.905907i	$0.360810\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.55682e26	0.117125	0.0585627	−	0.998284i	$-0.481348\pi$
$383$	1.55682e26	0.117125	0.0585627	+	0.998284i	$0.481348\pi$
$384$	0	0
$385$	4.68419e25	0.0333655
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.90693e26	0.185765	0.0928825	−	0.995677i	$-0.470392\pi$
$389$	2.90693e26	0.185765	0.0928825	+	0.995677i	$0.470392\pi$
$390$	0	0
$391$	−8.06524e26	−0.488384
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.31625e27	2.89292
$396$	0	0
$397$	1.52566e27	0.787331	0.393665	−	0.919254i	$-0.371207\pi$
$397$	1.52566e27	0.787331	0.393665	+	0.919254i	$0.371207\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.96782e27	0.914048	0.457024	−	0.889454i	$-0.348915\pi$
$401$	1.96782e27	0.914048	0.457024	+	0.889454i	$0.348915\pi$
$402$	0	0
$403$	1.44981e27	0.639157
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.26998e24	−0.00288929
$408$	0	0
$409$	3.91042e27	1.47615	0.738073	−	0.674721i	$-0.235736\pi$
$409$	3.91042e27	1.47615	0.738073	+	0.674721i	$0.235736\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.22132e27	−0.416250
$414$	0	0
$415$	−9.78581e27	−3.17024
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−2.38402e27	−0.698334	−0.349167	−	0.937060i	$-0.613535\pi$
$419$	−2.38402e27	−0.698334	−0.349167	+	0.937060i	$0.613535\pi$
$420$	0	0
$421$	5.54346e27	1.54461	0.772305	−	0.635252i	$-0.219104\pi$
$421$	5.54346e27	1.54461	0.772305	+	0.635252i	$0.219104\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.00402e27	1.51481
$426$	0	0
$427$	1.92457e27	0.462213
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.83540e27	−1.92404	−0.962022	−	0.272971i	$-0.911994\pi$
$431$	−8.83540e27	−1.92404	−0.962022	+	0.272971i	$0.911994\pi$
$432$	0	0
$433$	−6.01615e27	−1.24794	−0.623972	−	0.781446i	$-0.714482\pi$
$433$	−6.01615e27	−1.24794	−0.623972	+	0.781446i	$0.714482\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.75467e27	0.330476
$438$	0	0
$439$	−5.36549e27	−0.963234	−0.481617	−	0.876382i	$-0.659950\pi$
$439$	−5.36549e27	−0.963234	−0.481617	+	0.876382i	$0.659950\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.01374e25	−0.00328673	−0.00164337	−	0.999999i	$-0.500523\pi$
$443$	−2.01374e25	−0.00328673	−0.00164337	+	0.999999i	$0.500523\pi$
$444$	0	0
$445$	5.95914e27	0.927691
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.26698e27	−0.604691	−0.302346	−	0.953198i	$-0.597770\pi$
$449$	−4.26698e27	−0.604691	−0.302346	+	0.953198i	$0.597770\pi$
$450$	0	0
$451$	−6.89263e26	−0.0932248
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.36144e27	−0.291110
$456$	0	0
$457$	1.37046e28	1.61341	0.806707	−	0.590952i	$-0.201248\pi$
$457$	1.37046e28	1.61341	0.806707	+	0.590952i	$0.201248\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.95760e26	−0.0425179	−0.0212590	−	0.999774i	$-0.506767\pi$
$461$	−3.95760e26	−0.0425179	−0.0212590	+	0.999774i	$0.506767\pi$
$462$	0	0
$463$	1.64489e28	1.68864	0.844320	−	0.535839i	$-0.180005\pi$
$463$	1.64489e28	1.68864	0.844320	+	0.535839i	$0.180005\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.81167e27	−0.451303	−0.225651	−	0.974208i	$-0.572451\pi$
$467$	−4.81167e27	−0.451303	−0.225651	+	0.974208i	$0.572451\pi$
$468$	0	0
$469$	−2.22188e27	−0.199253
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−4.47761e26	−0.0367286
$474$	0	0
$475$	−1.30623e28	−1.02503
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.27968e27	−0.594964	−0.297482	−	0.954727i	$-0.596147\pi$
$479$	−8.27968e27	−0.594964	−0.297482	+	0.954727i	$0.596147\pi$
$480$	0	0
$481$	3.66500e26	0.0252087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.88642e28	1.18946
$486$	0	0
$487$	−2.94208e28	−1.77664	−0.888321	−	0.459223i	$-0.848128\pi$
$487$	−2.94208e28	−1.77664	−0.888321	+	0.459223i	$0.848128\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.03749e26	0.0279164	0.0139582	−	0.999903i	$-0.495557\pi$
$491$	5.03749e26	0.0279164	0.0139582	+	0.999903i	$0.495557\pi$
$492$	0	0
$493$	−2.57001e28	−1.36472
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.95920e27	0.388264
$498$	0	0
$499$	2.85361e28	1.33456	0.667282	−	0.744805i	$-0.267458\pi$
$499$	2.85361e28	1.33456	0.667282	+	0.744805i	$0.267458\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.16677e28	−1.36192	−0.680962	−	0.732319i	$-0.738438\pi$
$503$	−3.16677e28	−1.36192	−0.680962	+	0.732319i	$0.738438\pi$
$504$	0	0
$505$	−5.86358e28	−2.41882
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−5.83426e27	−0.221538	−0.110769	−	0.993846i	$-0.535331\pi$
$509$	−5.83426e27	−0.221538	−0.110769	+	0.993846i	$0.535331\pi$
$510$	0	0
$511$	6.49574e27	0.236706
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.08955e28	−2.04461
$516$	0	0
$517$	−1.80209e27	−0.0580932
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.34708e28	1.29241	0.646207	−	0.763162i	$-0.276354\pi$
$521$	4.34708e28	1.29241	0.646207	+	0.763162i	$0.276354\pi$
$522$	0	0
$523$	−6.35350e28	−1.81445	−0.907226	−	0.420644i	$-0.861804\pi$
$523$	−6.35350e28	−1.81445	−0.907226	+	0.420644i	$0.861804\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.14659e28	−1.09315
$528$	0	0
$529$	−2.90175e28	−0.735150
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.47477e28	0.813375
$534$	0	0
$535$	−6.87891e28	−1.54812
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.73174e27	−0.0568536
$540$	0	0
$541$	1.40153e28	0.280563	0.140281	−	0.990112i	$-0.455199\pi$
$541$	1.40153e28	0.280563	0.140281	+	0.990112i	$0.455199\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.13703e28	1.50766
$546$	0	0
$547$	−1.55956e28	−0.278058	−0.139029	−	0.990288i	$-0.544398\pi$
$547$	−1.55956e28	−0.278058	−0.139029	+	0.990288i	$0.544398\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.59131e28	0.923468
$552$	0	0
$553$	3.67682e28	0.584600
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.80978e28	−0.856406	−0.428203	−	0.903682i	$-0.640853\pi$
$557$	−5.80978e28	−0.856406	−0.428203	+	0.903682i	$0.640853\pi$
$558$	0	0
$559$	2.25729e28	0.320453
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.80401e28	−1.15969	−0.579846	−	0.814726i	$-0.696887\pi$
$563$	−8.80401e28	−1.15969	−0.579846	+	0.814726i	$0.696887\pi$
$564$	0	0
$565$	1.93329e29	2.45352
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.11084e29	1.30910	0.654551	−	0.756018i	$-0.272858\pi$
$569$	1.11084e29	1.30910	0.654551	+	0.756018i	$0.272858\pi$
$570$	0	0
$571$	−5.41935e28	−0.615557	−0.307778	−	0.951458i	$-0.599586\pi$
$571$	−5.41935e28	−0.615557	−0.307778	+	0.951458i	$0.599586\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.78232e28	−0.821480
$576$	0	0
$577$	4.58822e27	0.0466980	0.0233490	−	0.999727i	$-0.492567\pi$
$577$	4.58822e27	0.0466980	0.0233490	+	0.999727i	$0.492567\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−6.76805e28	−0.640641
$582$	0	0
$583$	−6.89653e27	−0.0629667
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.93390e28	−0.844150	−0.422075	−	0.906561i	$-0.638698\pi$
$587$	−9.93390e28	−0.844150	−0.422075	+	0.906561i	$0.638698\pi$
$588$	0	0
$589$	9.02129e28	0.739704
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.21696e28	0.474793	0.237396	−	0.971413i	$-0.423706\pi$
$593$	6.21696e28	0.474793	0.237396	+	0.971413i	$0.423706\pi$
$594$	0	0
$595$	6.75393e28	0.497884
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.21046e28	0.632849	0.316424	−	0.948618i	$-0.397518\pi$
$599$	9.21046e28	0.632849	0.316424	+	0.948618i	$0.397518\pi$
$600$	0	0
$601$	−5.86893e28	−0.389383	−0.194691	−	0.980865i	$-0.562371\pi$
$601$	−5.86893e28	−0.389383	−0.194691	+	0.980865i	$0.562371\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.59324e29	−1.60477
$606$	0	0
$607$	1.82607e29	1.09153	0.545766	−	0.837938i	$-0.316239\pi$
$607$	1.82607e29	1.09153	0.545766	+	0.837938i	$0.316239\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.08484e28	0.506856
$612$	0	0
$613$	1.99434e29	1.07514	0.537570	−	0.843219i	$-0.319342\pi$
$613$	1.99434e29	1.07514	0.537570	+	0.843219i	$0.319342\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.61110e28	−0.282523	−0.141261	−	0.989972i	$-0.545116\pi$
$617$	−5.61110e28	−0.282523	−0.141261	+	0.989972i	$0.545116\pi$
$618$	0	0
$619$	−2.67515e29	−1.30196	−0.650978	−	0.759096i	$-0.725641\pi$
$619$	−2.67515e29	−1.30196	−0.650978	+	0.759096i	$0.725641\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.12146e28	0.187468
$624$	0	0
$625$	−1.09746e28	−0.0482667
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.04823e28	−0.0431144
$630$	0	0
$631$	2.36523e29	0.940948	0.470474	−	0.882414i	$-0.344083\pi$
$631$	2.36523e29	0.940948	0.470474	+	0.882414i	$0.344083\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.90779e29	1.08253
$636$	0	0
$637$	1.37715e29	0.496041
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.06042e29	1.70678	0.853389	−	0.521274i	$-0.174543\pi$
$641$	5.06042e29	1.70678	0.853389	+	0.521274i	$0.174543\pi$
$642$	0	0
$643$	−1.16614e29	−0.380657	−0.190329	−	0.981720i	$-0.560955\pi$
$643$	−1.16614e29	−0.380657	−0.190329	+	0.981720i	$0.560955\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.16701e29	1.88616	0.943082	−	0.332561i	$-0.107913\pi$
$647$	6.16701e29	1.88616	0.943082	+	0.332561i	$0.107913\pi$
$648$	0	0
$649$	−2.74574e28	−0.0813000
$650$	0	0
$651$	0	0
$652$	0	0
$653$	3.77314e29	1.04741	0.523703	−	0.851901i	$-0.324550\pi$
$653$	3.77314e29	1.04741	0.523703	+	0.851901i	$0.324550\pi$
$654$	0	0
$655$	4.80966e29	1.29295
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−2.99810e29	−0.756048	−0.378024	−	0.925796i	$-0.623396\pi$
$659$	−2.99810e29	−0.756048	−0.378024	+	0.925796i	$0.623396\pi$
$660$	0	0
$661$	−4.87781e29	−1.19154	−0.595771	−	0.803154i	$-0.703154\pi$
$661$	−4.87781e29	−1.19154	−0.595771	+	0.803154i	$0.703154\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.46938e29	−0.336905
$666$	0	0
$667$	3.33121e29	0.740084
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.32677e28	0.0902771
$672$	0	0
$673$	9.88058e28	0.199813	0.0999066	−	0.994997i	$-0.468146\pi$
$673$	9.88058e28	0.199813	0.0999066	+	0.994997i	$0.468146\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.03899e29	0.197438	0.0987189	−	0.995115i	$-0.468526\pi$
$677$	1.03899e29	0.197438	0.0987189	+	0.995115i	$0.468526\pi$
$678$	0	0
$679$	1.30469e29	0.240366
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.77427e28	−0.0307328	−0.0153664	−	0.999882i	$-0.504891\pi$
$683$	−1.77427e28	−0.0307328	−0.0153664	+	0.999882i	$0.504891\pi$
$684$	0	0
$685$	1.60350e30	2.69351
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.47674e29	0.549377
$690$	0	0
$691$	−8.85136e29	−1.35672	−0.678361	−	0.734729i	$-0.737309\pi$
$691$	−8.85136e29	−1.35672	−0.678361	+	0.734729i	$0.737309\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.20732e29	−0.606966
$696$	0	0
$697$	−9.93817e29	−1.39111
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.01323e30	−1.33558	−0.667791	−	0.744349i	$-0.732760\pi$
$701$	−1.01323e30	−1.33558	−0.667791	+	0.744349i	$0.732760\pi$
$702$	0	0
$703$	2.28051e28	0.0291744
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.05536e29	−0.488794
$708$	0	0
$709$	−3.34197e27	−0.00391036	−0.00195518	−	0.999998i	$-0.500622\pi$
$709$	−3.34197e27	−0.00391036	−0.00195518	+	0.999998i	$0.500622\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.37474e29	0.592813
$714$	0	0
$715$	−5.30891e28	−0.0568581
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.99416e29	−0.605445	−0.302723	−	0.953079i	$-0.597896\pi$
$719$	−5.99416e29	−0.605445	−0.302723	+	0.953079i	$0.597896\pi$
$720$	0	0
$721$	−4.21165e29	−0.413173
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.47986e30	−2.29551
$726$	0	0
$727$	1.65266e30	1.48618	0.743092	−	0.669189i	$-0.233358\pi$
$727$	1.65266e30	1.48618	0.743092	+	0.669189i	$0.233358\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6.45607e29	−0.548070
$732$	0	0
$733$	−4.23430e29	−0.349293	−0.174646	−	0.984631i	$-0.555878\pi$
$733$	−4.23430e29	−0.349293	−0.174646	+	0.984631i	$0.555878\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.99517e28	−0.0389172
$738$	0	0
$739$	2.50879e30	1.89975	0.949877	−	0.312624i	$-0.101208\pi$
$739$	2.50879e30	1.89975	0.949877	+	0.312624i	$0.101208\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.01564e30	−0.726706	−0.363353	−	0.931652i	$-0.618368\pi$
$743$	−1.01564e30	−0.726706	−0.363353	+	0.931652i	$0.618368\pi$
$744$	0	0
$745$	−3.45477e30	−2.40314
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.75759e29	−0.312844
$750$	0	0
$751$	−2.34721e30	−1.50083	−0.750417	−	0.660964i	$-0.770148\pi$
$751$	−2.34721e30	−1.50083	−0.750417	+	0.660964i	$0.770148\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.19938e30	2.53946
$756$	0	0
$757$	3.29629e29	0.193874	0.0969369	−	0.995291i	$-0.469095\pi$
$757$	3.29629e29	0.193874	0.0969369	+	0.995291i	$0.469095\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.22786e30	1.23979	0.619896	−	0.784684i	$-0.287175\pi$
$761$	2.22786e30	1.23979	0.619896	+	0.784684i	$0.287175\pi$
$762$	0	0
$763$	5.62773e29	0.304667
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.38421e30	0.709333
$768$	0	0
$769$	−2.41780e30	−1.20557	−0.602786	−	0.797903i	$-0.705943\pi$
$769$	−2.41780e30	−1.20557	−0.602786	+	0.797903i	$0.705943\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.22170e30	−0.576871	−0.288435	−	0.957499i	$-0.593135\pi$
$773$	−1.22170e30	−0.576871	−0.288435	+	0.957499i	$0.593135\pi$
$774$	0	0
$775$	−4.00113e30	−1.83872
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.16214e30	0.941329
$780$	0	0
$781$	1.78936e29	0.0758338
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.63073e30	−1.05668
$786$	0	0
$787$	−3.41643e29	−0.133609	−0.0668047	−	0.997766i	$-0.521280\pi$
$787$	−3.41643e29	−0.133609	−0.0668047	+	0.997766i	$0.521280\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.33710e30	0.495806
$792$	0	0
$793$	−2.18125e30	−0.787657
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.47484e30	−1.87525	−0.937625	−	0.347648i	$-0.886980\pi$
$797$	−5.47484e30	−1.87525	−0.937625	+	0.347648i	$0.886980\pi$
$798$	0	0
$799$	−2.59835e30	−0.866874
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.46035e29	0.0462322
$804$	0	0
$805$	−8.75434e29	−0.270002
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.91242e30	−1.73103	−0.865517	−	0.500880i	$-0.833010\pi$
$809$	−5.91242e30	−1.73103	−0.865517	+	0.500880i	$0.833010\pi$
$810$	0	0
$811$	2.38545e30	0.680537	0.340269	−	0.940328i	$-0.389482\pi$
$811$	2.38545e30	0.680537	0.340269	+	0.940328i	$0.389482\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.16588e30	1.94140
$816$	0	0
$817$	1.40458e30	0.370864
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.05222e30	1.26730	0.633650	−	0.773620i	$-0.281556\pi$
$821$	5.05222e30	1.26730	0.633650	+	0.773620i	$0.281556\pi$
$822$	0	0
$823$	8.68241e29	0.212296	0.106148	−	0.994350i	$-0.466148\pi$
$823$	8.68241e29	0.212296	0.106148	+	0.994350i	$0.466148\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.90101e30	−0.906503	−0.453251	−	0.891383i	$-0.649736\pi$
$827$	−3.90101e30	−0.906503	−0.453251	+	0.891383i	$0.649736\pi$
$828$	0	0
$829$	−5.27365e30	−1.19478	−0.597391	−	0.801950i	$-0.703796\pi$
$829$	−5.27365e30	−1.19478	−0.597391	+	0.801950i	$0.703796\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.93878e30	−0.848377
$834$	0	0
$835$	−5.30029e30	−1.11325
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.94809e30	−1.78743	−0.893716	−	0.448632i	$-0.851911\pi$
$839$	−8.94809e30	−1.78743	−0.893716	+	0.448632i	$0.851911\pi$
$840$	0	0
$841$	5.48217e30	1.06806
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.01658e30	−1.11520
$846$	0	0
$847$	−1.79354e30	−0.324291
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.35869e29	0.0233809
$852$	0	0
$853$	5.79287e30	0.972588	0.486294	−	0.873795i	$-0.338348\pi$
$853$	5.79287e30	0.972588	0.486294	+	0.873795i	$0.338348\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.06465e31	1.70181	0.850903	−	0.525323i	$-0.176056\pi$
$857$	1.06465e31	1.70181	0.850903	+	0.525323i	$0.176056\pi$
$858$	0	0
$859$	1.06376e31	1.65927	0.829633	−	0.558309i	$-0.188549\pi$
$859$	1.06376e31	1.65927	0.829633	+	0.558309i	$0.188549\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	7.39788e30	1.09899	0.549495	−	0.835497i	$-0.314820\pi$
$863$	7.39788e30	1.09899	0.549495	+	0.835497i	$0.314820\pi$
$864$	0	0
$865$	7.52427e30	1.09093
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.26613e29	0.114181
$870$	0	0
$871$	2.51821e30	0.339548
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.43427e30	0.312813
$876$	0	0
$877$	1.08447e31	1.36057	0.680284	−	0.732949i	$-0.261857\pi$
$877$	1.08447e31	1.36057	0.680284	+	0.732949i	$0.261857\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−3.94287e30	−0.471591	−0.235796	−	0.971803i	$-0.575770\pi$
$881$	−3.94287e30	−0.471591	−0.235796	+	0.971803i	$0.575770\pi$
$882$	0	0
$883$	1.16137e30	0.135638	0.0678191	−	0.997698i	$-0.478396\pi$
$883$	1.16137e30	0.135638	0.0678191	+	0.997698i	$0.478396\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.66393e29	0.0853599	0.0426799	−	0.999089i	$-0.486410\pi$
$887$	7.66393e29	0.0853599	0.0426799	+	0.999089i	$0.486410\pi$
$888$	0	0
$889$	2.01108e30	0.218757
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.65295e30	0.586590
$894$	0	0
$895$	1.10534e30	0.112035
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.71268e31	1.65653
$900$	0	0
$901$	−9.94380e30	−0.939598
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.12272e30	0.191462
$906$	0	0
$907$	1.52281e31	1.34205	0.671027	−	0.741432i	$-0.265853\pi$
$907$	1.52281e31	1.34205	0.671027	+	0.741432i	$0.265853\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.10597e31	−1.77218	−0.886092	−	0.463510i	$-0.846590\pi$
$911$	−2.10597e31	−1.77218	−0.886092	+	0.463510i	$0.846590\pi$
$912$	0	0
$913$	−1.52157e30	−0.125127
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.32645e30	0.261278
$918$	0	0
$919$	9.82307e29	0.0754109	0.0377055	−	0.999289i	$-0.487995\pi$
$919$	9.82307e29	0.0754109	0.0377055	+	0.999289i	$0.487995\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−9.02070e30	−0.661641
$924$	0	0
$925$	−1.01146e30	−0.0725201
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.88415e31	1.29107	0.645535	−	0.763731i	$-0.276635\pi$
$929$	1.88415e31	1.29107	0.645535	+	0.763731i	$0.276635\pi$
$930$	0	0
$931$	8.56919e30	0.574074
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.51840e30	0.0972442
$936$	0	0
$937$	−2.47039e31	−1.54703	−0.773516	−	0.633777i	$-0.781504\pi$
$937$	−2.47039e31	−1.54703	−0.773516	+	0.633777i	$0.781504\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.65576e31	0.991530	0.495765	−	0.868457i	$-0.334888\pi$
$941$	1.65576e31	0.991530	0.495765	+	0.868457i	$0.334888\pi$
$942$	0	0
$943$	1.28817e31	0.754398
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.58984e30	−0.481183	−0.240592	−	0.970626i	$-0.577341\pi$
$947$	−8.58984e30	−0.481183	−0.240592	+	0.970626i	$0.577341\pi$
$948$	0	0
$949$	−7.36206e30	−0.403370
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.50226e30	0.183600	0.0918000	−	0.995777i	$-0.470738\pi$
$953$	3.50226e30	0.183600	0.0918000	+	0.995777i	$0.470738\pi$
$954$	0	0
$955$	−3.83711e31	−1.96774
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.10901e31	0.544303
$960$	0	0
$961$	6.80771e30	0.326893
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.09339e31	0.502618
$966$	0	0
$967$	2.62968e31	1.18284	0.591419	−	0.806364i	$-0.298568\pi$
$967$	2.62968e31	1.18284	0.591419	+	0.806364i	$0.298568\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.10311e30	0.305947	0.152974	−	0.988230i	$-0.451115\pi$
$971$	7.10311e30	0.305947	0.152974	+	0.988230i	$0.451115\pi$
$972$	0	0
$973$	−2.90986e30	−0.122656
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.54851e30	−0.183644	−0.0918218	−	0.995775i	$-0.529269\pi$
$977$	−4.54851e30	−0.183644	−0.0918218	+	0.995775i	$0.529269\pi$
$978$	0	0
$979$	9.26574e29	0.0366152
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.59763e30	−0.325512	−0.162756	−	0.986666i	$-0.552038\pi$
$983$	−8.59763e30	−0.325512	−0.162756	+	0.986666i	$0.552038\pi$
$984$	0	0
$985$	5.93604e31	2.19997
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.36826e30	0.297217
$990$	0	0
$991$	3.32226e31	1.15521	0.577604	−	0.816317i	$-0.303988\pi$
$991$	3.32226e31	1.15521	0.577604	+	0.816317i	$0.303988\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.31764e31	−1.77246
$996$	0	0
$997$	−8.79911e30	−0.287170	−0.143585	−	0.989638i	$-0.545863\pi$
$997$	−8.79911e30	−0.287170	−0.143585	+	0.989638i	$0.545863\pi$
$998$	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.1.3		3
3.2	odd	2		24.22.a.d.1.1	✓	3
12.11	even	2		48.22.a.j.1.1		3

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

\(f(q)\)	\(=\)	\(q+3.51850e7 q^{5} +2.43346e8 q^{7} +5.47083e9 q^{11} -2.75801e11 q^{13} +7.88815e12 q^{17} -1.71614e13 q^{19} -1.02245e14 q^{23} +7.61145e14 q^{25} -3.25807e15 q^{29} -5.25673e15 q^{31} +8.56212e15 q^{35} -1.32886e15 q^{37} -1.25989e17 q^{41} -8.18452e16 q^{43} -3.29399e17 q^{47} -4.99329e17 q^{49} -1.26060e18 q^{53} +1.92491e17 q^{55} -5.01887e18 q^{59} +7.90880e18 q^{61} -9.70403e18 q^{65} -9.13054e18 q^{67} +3.27073e19 q^{71} +2.66934e19 q^{73} +1.33131e18 q^{77} +1.51094e20 q^{79} -2.78125e20 q^{83} +2.77544e20 q^{85} +1.69366e20 q^{89} -6.71149e19 q^{91} -6.03824e20 q^{95} +5.36145e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5280498 q^{5} + 852542376 q^{7} - 62490757668 q^{11} + 203765207802 q^{13} - 695827819926 q^{17} + 4955504123196 q^{19} - 150867407938152 q^{23} + 678194854969869 q^{25} - 32\!\cdots\!46 q^{29} + 652508601550896 q^{31}+ \cdots + 72\!\cdots\!02 q^{97}+O(q^{100}) \) 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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Database

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