LMFDB - Embedded newform 4.22.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,22,Mod(1,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	4.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$11.1790937715$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2161}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 540 $ x^2 - x - 540
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$23.7433$ of defining polynomial
Character	$\chi$	$=$	4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-92135.9 q^{3} -1.86463e7 q^{5} +6.05236e8 q^{7} -1.97134e9 q^{9} +O(q^{10})$	$q-92135.9 q^{3} -1.86463e7 q^{5} +6.05236e8 q^{7} -1.97134e9 q^{9} +1.12010e11 q^{11} +7.55030e11 q^{13} +1.71800e12 q^{15} +1.15238e13 q^{17} -4.15598e13 q^{19} -5.57640e13 q^{21} -8.14133e13 q^{23} -1.29151e14 q^{25} +1.14540e15 q^{27} +2.46131e15 q^{29} +6.30301e14 q^{31} -1.03201e16 q^{33} -1.12854e16 q^{35} +1.40093e16 q^{37} -6.95654e16 q^{39} +1.20777e17 q^{41} +1.67373e17 q^{43} +3.67582e16 q^{45} -2.73232e17 q^{47} -1.92235e17 q^{49} -1.06176e18 q^{51} +1.24102e18 q^{53} -2.08857e18 q^{55} +3.82915e18 q^{57} -3.96154e17 q^{59} +8.53713e18 q^{61} -1.19312e18 q^{63} -1.40785e19 q^{65} -1.76971e19 q^{67} +7.50108e18 q^{69} +1.03284e19 q^{71} -1.55389e19 q^{73} +1.18995e19 q^{75} +6.77923e19 q^{77} +7.30491e19 q^{79} -8.49120e19 q^{81} +8.17274e19 q^{83} -2.14877e20 q^{85} -2.26775e20 q^{87} -2.23635e20 q^{89} +4.56972e20 q^{91} -5.80733e19 q^{93} +7.74938e20 q^{95} -2.78850e20 q^{97} -2.20809e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) $ 2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9	$ 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + 145435963320 q^{11} + 1428900417340 q^{13} + 6819782714352 q^{15} + 1840620576420 q^{17} - 16780743928568 q^{19} - 192357511002048 q^{21} - 319691925426960 q^{23} + 439606295919326 q^{25} + 17\!\cdots\!40 q^{27}+ \cdots + 26\!\cdots\!20 q^{99}+O(q^{100}) $ 2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9 + 145435963320 * q^11 + 1428900417340 * q^13 + 6819782714352 * q^15 + 1840620576420 * q^17 - 16780743928568 * q^19 - 192357511002048 * q^21 - 319691925426960 * q^23 + 439606295919326 * q^25 + 1772171769223440 * q^27 + 3742111775766492 * q^29 + 112042353462592 * q^31 - 5046260260851360 * q^33 - 39279847462424352 * q^35 - 33362705637547220 * q^37 + 36755051864070192 * q^39 + 175129744323133332 * q^41 + 15346613416528120 * q^43 + 503454557153115324 * q^45 - 684848819288455200 * q^47 - 1267753878311886 * q^49 - 2589533339352286896 * q^51 + 675305394244421580 * q^53 - 1007711578141009200 * q^55 + 7738689471355209120 * q^57 + 1042445250435434904 * q^59 + 9065997829736468764 * q^61 - 13688298514444201200 * q^63 + 7711482582413403048 * q^65 - 30464301046802775320 * q^67 - 30093532090729291584 * q^69 - 8199093502830518064 * q^71 + 25415086659374793940 * q^73 + 101635704862879905048 * q^75 + 38853770260581178560 * q^77 + 121204353225060164896 * q^79 - 136996120065469586382 * q^81 + 108742936757033809800 * q^83 - 527989845644919754344 * q^85 - 24695689277789874000 * q^87 - 184207999274965368972 * q^89 - 126427419539014666912 * q^91 - 139841964206001442560 * q^93 + 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 + 261627767970577922520 * q^99

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
$3$	−92135.9	−0.900856	−0.450428	−	0.892813i	$-0.648729\pi$
$3$	−92135.9	−0.900856	−0.450428	+	0.892813i	$0.648729\pi$
$4$	0	0
$5$	−1.86463e7	−0.853903	−0.426951	−	0.904275i	$-0.640412\pi$
$5$	−1.86463e7	−0.853903	−0.426951	+	0.904275i	$0.640412\pi$
$6$	0	0
$7$	6.05236e8	0.809833	0.404916	−	0.914354i	$-0.367301\pi$
$7$	6.05236e8	0.809833	0.404916	+	0.914354i	$0.367301\pi$
$8$	0	0
$9$	−1.97134e9	−0.188458
$10$	0	0
$11$	1.12010e11	1.30206	0.651032	−	0.759050i	$-0.274336\pi$
$11$	1.12010e11	1.30206	0.651032	+	0.759050i	$0.274336\pi$
$12$	0	0
$13$	7.55030e11	1.51901	0.759503	−	0.650504i	$-0.225442\pi$
$13$	7.55030e11	1.51901	0.759503	+	0.650504i	$0.225442\pi$
$14$	0	0
$15$	1.71800e12	0.769244
$16$	0	0
$17$	1.15238e13	1.38638	0.693191	−	0.720754i	$-0.256204\pi$
$17$	1.15238e13	1.38638	0.693191	+	0.720754i	$0.256204\pi$
$18$	0	0
$19$	−4.15598e13	−1.55511	−0.777554	−	0.628816i	$-0.783540\pi$
$19$	−4.15598e13	−1.55511	−0.777554	+	0.628816i	$0.783540\pi$
$20$	0	0
$21$	−5.57640e13	−0.729543
$22$	0	0
$23$	−8.14133e13	−0.409782	−0.204891	−	0.978785i	$-0.565684\pi$
$23$	−8.14133e13	−0.409782	−0.204891	+	0.978785i	$0.565684\pi$
$24$	0	0
$25$	−1.29151e14	−0.270850
$26$	0	0
$27$	1.14540e15	1.07063
$28$	0	0
$29$	2.46131e15	1.08639	0.543197	−	0.839605i	$-0.317214\pi$
$29$	2.46131e15	1.08639	0.543197	+	0.839605i	$0.317214\pi$
$30$	0	0
$31$	6.30301e14	0.138118	0.0690590	−	0.997613i	$-0.478000\pi$
$31$	6.30301e14	0.138118	0.0690590	+	0.997613i	$0.478000\pi$
$32$	0	0
$33$	−1.03201e16	−1.17297
$34$	0	0
$35$	−1.12854e16	−0.691519
$36$	0	0
$37$	1.40093e16	0.478959	0.239480	−	0.970901i	$-0.423023\pi$
$37$	1.40093e16	0.478959	0.239480	+	0.970901i	$0.423023\pi$
$38$	0	0
$39$	−6.95654e16	−1.36841
$40$	0	0
$41$	1.20777e17	1.40526	0.702628	−	0.711557i	$-0.252010\pi$
$41$	1.20777e17	1.40526	0.702628	+	0.711557i	$0.252010\pi$
$42$	0	0
$43$	1.67373e17	1.18104	0.590522	−	0.807022i	$-0.298922\pi$
$43$	1.67373e17	1.18104	0.590522	+	0.807022i	$0.298922\pi$
$44$	0	0
$45$	3.67582e16	0.160925
$46$	0	0
$47$	−2.73232e17	−0.757712	−0.378856	−	0.925456i	$-0.623682\pi$
$47$	−2.73232e17	−0.757712	−0.378856	+	0.925456i	$0.623682\pi$
$48$	0	0
$49$	−1.92235e17	−0.344171
$50$	0	0
$51$	−1.06176e18	−1.24893
$52$	0	0
$53$	1.24102e18	0.974726	0.487363	−	0.873199i	$-0.337959\pi$
$53$	1.24102e18	0.974726	0.487363	+	0.873199i	$0.337959\pi$
$54$	0	0
$55$	−2.08857e18	−1.11184
$56$	0	0
$57$	3.82915e18	1.40093
$58$	0	0
$59$	−3.96154e17	−0.100906	−0.0504531	−	0.998726i	$-0.516067\pi$
$59$	−3.96154e17	−0.100906	−0.0504531	+	0.998726i	$0.516067\pi$
$60$	0	0
$61$	8.53713e18	1.53232	0.766158	−	0.642652i	$-0.222166\pi$
$61$	8.53713e18	1.53232	0.766158	+	0.642652i	$0.222166\pi$
$62$	0	0
$63$	−1.19312e18	−0.152619
$64$	0	0
$65$	−1.40785e19	−1.29708
$66$	0	0
$67$	−1.76971e19	−1.18609	−0.593043	−	0.805171i	$-0.702074\pi$
$67$	−1.76971e19	−1.18609	−0.593043	+	0.805171i	$0.702074\pi$
$68$	0	0
$69$	7.50108e18	0.369155
$70$	0	0
$71$	1.03284e19	0.376548	0.188274	−	0.982117i	$-0.439711\pi$
$71$	1.03284e19	0.376548	0.188274	+	0.982117i	$0.439711\pi$
$72$	0	0
$73$	−1.55389e19	−0.423185	−0.211593	−	0.977358i	$-0.567865\pi$
$73$	−1.55389e19	−0.423185	−0.211593	+	0.977358i	$0.567865\pi$
$74$	0	0
$75$	1.18995e19	0.243997
$76$	0	0
$77$	6.77923e19	1.05445
$78$	0	0
$79$	7.30491e19	0.868022	0.434011	−	0.900908i	$-0.357098\pi$
$79$	7.30491e19	0.868022	0.434011	+	0.900908i	$0.357098\pi$
$80$	0	0
$81$	−8.49120e19	−0.776026
$82$	0	0
$83$	8.17274e19	0.578160	0.289080	−	0.957305i	$-0.406651\pi$
$83$	8.17274e19	0.578160	0.289080	+	0.957305i	$0.406651\pi$
$84$	0	0
$85$	−2.14877e20	−1.18384
$86$	0	0
$87$	−2.26775e20	−0.978685
$88$	0	0
$89$	−2.23635e20	−0.760229	−0.380114	−	0.924939i	$-0.624115\pi$
$89$	−2.23635e20	−0.760229	−0.380114	+	0.924939i	$0.624115\pi$
$90$	0	0
$91$	4.56972e20	1.23014
$92$	0	0
$93$	−5.80733e19	−0.124424
$94$	0	0
$95$	7.74938e20	1.32791
$96$	0	0
$97$	−2.78850e20	−0.383943	−0.191972	−	0.981400i	$-0.561488\pi$
$97$	−2.78850e20	−0.383943	−0.191972	+	0.981400i	$0.561488\pi$
$98$	0	0
$99$	−2.20809e20	−0.245384
$100$	0	0
$101$	1.66280e21	1.49784	0.748920	−	0.662660i	$-0.230573\pi$
$101$	1.66280e21	1.49784	0.748920	+	0.662660i	$0.230573\pi$
$102$	0	0
$103$	−1.51135e21	−1.10809	−0.554046	−	0.832486i	$-0.686917\pi$
$103$	−1.51135e21	−1.10809	−0.554046	+	0.832486i	$0.686917\pi$
$104$	0	0
$105$	1.03979e21	0.622959
$106$	0	0
$107$	−2.26434e21	−1.11279	−0.556393	−	0.830920i	$-0.687815\pi$
$107$	−2.26434e21	−1.11279	−0.556393	+	0.830920i	$0.687815\pi$
$108$	0	0
$109$	−3.92748e21	−1.58904	−0.794522	−	0.607235i	$-0.792279\pi$
$109$	−3.92748e21	−1.58904	−0.794522	+	0.607235i	$0.792279\pi$
$110$	0	0
$111$	−1.29076e21	−0.431474
$112$	0	0
$113$	6.00234e21	1.66340	0.831701	−	0.555224i	$-0.187367\pi$
$113$	6.00234e21	1.66340	0.831701	+	0.555224i	$0.187367\pi$
$114$	0	0
$115$	1.51806e21	0.349914
$116$	0	0
$117$	−1.48842e21	−0.286268
$118$	0	0
$119$	6.97464e21	1.12274
$120$	0	0
$121$	5.14593e21	0.695372
$122$	0	0
$123$	−1.11279e22	−1.26593
$124$	0	0
$125$	1.12995e22	1.08518
$126$	0	0
$127$	−1.32453e22	−1.07677	−0.538385	−	0.842699i	$-0.680965\pi$
$127$	−1.32453e22	−1.07677	−0.538385	+	0.842699i	$0.680965\pi$
$128$	0	0
$129$	−1.54210e22	−1.06395
$130$	0	0
$131$	2.38495e22	1.40001	0.700004	−	0.714139i	$-0.253182\pi$
$131$	2.38495e22	1.40001	0.700004	+	0.714139i	$0.253182\pi$
$132$	0	0
$133$	−2.51535e22	−1.25938
$134$	0	0
$135$	−2.13576e22	−0.914214
$136$	0	0
$137$	−2.16852e21	−0.0795423	−0.0397712	−	0.999209i	$-0.512663\pi$
$137$	−2.16852e21	−0.0795423	−0.0397712	+	0.999209i	$0.512663\pi$
$138$	0	0
$139$	2.92783e22	0.922337	0.461169	−	0.887313i	$-0.347430\pi$
$139$	2.92783e22	0.922337	0.461169	+	0.887313i	$0.347430\pi$
$140$	0	0
$141$	2.51745e22	0.682590
$142$	0	0
$143$	8.45707e22	1.97784
$144$	0	0
$145$	−4.58944e22	−0.927675
$146$	0	0
$147$	1.77117e22	0.310048
$148$	0	0
$149$	−2.50216e22	−0.380066	−0.190033	−	0.981778i	$-0.560860\pi$
$149$	−2.50216e22	−0.380066	−0.190033	+	0.981778i	$0.560860\pi$
$150$	0	0
$151$	−7.09599e22	−0.937033	−0.468517	−	0.883455i	$-0.655211\pi$
$151$	−7.09599e22	−0.937033	−0.468517	+	0.883455i	$0.655211\pi$
$152$	0	0
$153$	−2.27173e22	−0.261275
$154$	0	0
$155$	−1.17528e22	−0.117939
$156$	0	0
$157$	4.23299e22	0.371280	0.185640	−	0.982618i	$-0.440564\pi$
$157$	4.23299e22	0.371280	0.185640	+	0.982618i	$0.440564\pi$
$158$	0	0
$159$	−1.14343e23	−0.878088
$160$	0	0
$161$	−4.92743e22	−0.331855
$162$	0	0
$163$	−2.03046e22	−0.120123	−0.0600614	−	0.998195i	$-0.519130\pi$
$163$	−2.03046e22	−0.120123	−0.0600614	+	0.998195i	$0.519130\pi$
$164$	0	0
$165$	1.92432e23	1.00161
$166$	0	0
$167$	−1.47752e19	−6.77660e−5 0	−3.38830e−5	−	1.00000i	$-0.500011\pi$
$167$	−1.47752e19	−6.77660e−5 0	−3.38830e−5		1.00000i	$0.500011\pi$
$168$	0	0
$169$	3.23006e23	1.30738
$170$	0	0
$171$	8.19283e22	0.293072
$172$	0	0
$173$	1.87075e22	0.0592287	0.0296143	−	0.999561i	$-0.490572\pi$
$173$	1.87075e22	0.0592287	0.0296143	+	0.999561i	$0.490572\pi$
$174$	0	0
$175$	−7.81671e22	−0.219343
$176$	0	0
$177$	3.65000e22	0.0909020
$178$	0	0
$179$	−2.14720e23	−0.475244	−0.237622	−	0.971358i	$-0.576368\pi$
$179$	−2.14720e23	−0.475244	−0.237622	+	0.971358i	$0.576368\pi$
$180$	0	0
$181$	−5.51937e23	−1.08709	−0.543544	−	0.839381i	$-0.682918\pi$
$181$	−5.51937e23	−1.08709	−0.543544	+	0.839381i	$0.682918\pi$
$182$	0	0
$183$	−7.86576e23	−1.38040
$184$	0	0
$185$	−2.61222e23	−0.408985
$186$	0	0
$187$	1.29078e24	1.80516
$188$	0	0
$189$	6.93240e23	0.867031
$190$	0	0
$191$	−1.45180e24	−1.62577	−0.812883	−	0.582428i	$-0.802103\pi$
$191$	−1.45180e24	−1.62577	−0.812883	+	0.582428i	$0.802103\pi$
$192$	0	0
$193$	5.88032e23	0.590268	0.295134	−	0.955456i	$-0.404636\pi$
$193$	5.88032e23	0.590268	0.295134	+	0.955456i	$0.404636\pi$
$194$	0	0
$195$	1.29714e24	1.16849
$196$	0	0
$197$	−1.28721e24	−1.04172	−0.520862	−	0.853641i	$-0.674390\pi$
$197$	−1.28721e24	−1.04172	−0.520862	+	0.853641i	$0.674390\pi$
$198$	0	0
$199$	1.02349e24	0.744948	0.372474	−	0.928043i	$-0.378510\pi$
$199$	1.02349e24	0.744948	0.372474	+	0.928043i	$0.378510\pi$
$200$	0	0
$201$	1.63054e24	1.06849
$202$	0	0
$203$	1.48967e24	0.879798
$204$	0	0
$205$	−2.25206e24	−1.19995
$206$	0	0
$207$	1.60493e23	0.0772267
$208$	0	0
$209$	−4.65510e24	−2.02485
$210$	0	0
$211$	9.95941e23	0.391984	0.195992	−	0.980606i	$-0.437207\pi$
$211$	9.95941e23	0.391984	0.195992	+	0.980606i	$0.437207\pi$
$212$	0	0
$213$	−9.51616e23	−0.339216
$214$	0	0
$215$	−3.12089e24	−1.00850
$216$	0	0
$217$	3.81481e23	0.111852
$218$	0	0
$219$	1.43169e24	0.381229
$220$	0	0
$221$	8.70084e24	2.10592
$222$	0	0
$223$	−4.42854e24	−0.975122	−0.487561	−	0.873089i	$-0.662113\pi$
$223$	−4.42854e24	−0.975122	−0.487561	+	0.873089i	$0.662113\pi$
$224$	0	0
$225$	2.54601e23	0.0510438
$226$	0	0
$227$	4.23549e24	0.773806	0.386903	−	0.922120i	$-0.373545\pi$
$227$	4.23549e24	0.773806	0.386903	+	0.922120i	$0.373545\pi$
$228$	0	0
$229$	6.57364e24	1.09530	0.547650	−	0.836708i	$-0.315523\pi$
$229$	6.57364e24	1.09530	0.547650	+	0.836708i	$0.315523\pi$
$230$	0	0
$231$	−6.24611e24	−0.949912
$232$	0	0
$233$	−1.03875e24	−0.144302	−0.0721509	−	0.997394i	$-0.522986\pi$
$233$	−1.03875e24	−0.144302	−0.0721509	+	0.997394i	$0.522986\pi$
$234$	0	0
$235$	5.09478e24	0.647012
$236$	0	0
$237$	−6.73044e24	−0.781963
$238$	0	0
$239$	−5.83947e24	−0.621149	−0.310574	−	0.950549i	$-0.600521\pi$
$239$	−5.83947e24	−0.621149	−0.310574	+	0.950549i	$0.600521\pi$
$240$	0	0
$241$	3.84920e24	0.375138	0.187569	−	0.982251i	$-0.439939\pi$
$241$	3.84920e24	0.375138	0.187569	+	0.982251i	$0.439939\pi$
$242$	0	0
$243$	−4.15790e24	−0.371542
$244$	0	0
$245$	3.58448e24	0.293888
$246$	0	0
$247$	−3.13789e25	−2.36222
$248$	0	0
$249$	−7.53002e24	−0.520839
$250$	0	0
$251$	2.86677e25	1.82313	0.911567	−	0.411151i	$-0.134873\pi$
$251$	2.86677e25	1.82313	0.911567	+	0.411151i	$0.134873\pi$
$252$	0	0
$253$	−9.11908e24	−0.533563
$254$	0	0
$255$	1.97979e25	1.06647
$256$	0	0
$257$	3.67250e25	1.82249	0.911243	−	0.411868i	$-0.135124\pi$
$257$	3.67250e25	1.82249	0.911243	+	0.411868i	$0.135124\pi$
$258$	0	0
$259$	8.47894e24	0.387877
$260$	0	0
$261$	−4.85207e24	−0.204740
$262$	0	0
$263$	−1.79474e25	−0.698983	−0.349492	−	0.936939i	$-0.613646\pi$
$263$	−1.79474e25	−0.698983	−0.349492	+	0.936939i	$0.613646\pi$
$264$	0	0
$265$	−2.31405e25	−0.832321
$266$	0	0
$267$	2.06048e25	0.684857
$268$	0	0
$269$	−6.85890e24	−0.210793	−0.105396	−	0.994430i	$-0.533611\pi$
$269$	−6.85890e24	−0.210793	−0.105396	+	0.994430i	$0.533611\pi$
$270$	0	0
$271$	9.95982e24	0.283187	0.141594	−	0.989925i	$-0.454777\pi$
$271$	9.95982e24	0.283187	0.141594	+	0.989925i	$0.454777\pi$
$272$	0	0
$273$	−4.21035e25	−1.10818
$274$	0	0
$275$	−1.44662e25	−0.352664
$276$	0	0
$277$	−1.13250e25	−0.255858	−0.127929	−	0.991783i	$-0.540833\pi$
$277$	−1.13250e25	−0.255858	−0.127929	+	0.991783i	$0.540833\pi$
$278$	0	0
$279$	−1.24253e24	−0.0260294
$280$	0	0
$281$	−1.74755e23	−0.00339636	−0.00169818	−	0.999999i	$-0.500541\pi$
$281$	−1.74755e23	−0.00339636	−0.00169818	+	0.999999i	$0.500541\pi$
$282$	0	0
$283$	−2.85617e23	−0.00515260	−0.00257630	−	0.999997i	$-0.500820\pi$
$283$	−2.85617e23	−0.00515260	−0.00257630	+	0.999997i	$0.500820\pi$
$284$	0	0
$285$	−7.13996e25	−1.19626
$286$	0	0
$287$	7.30989e25	1.13802
$288$	0	0
$289$	6.37066e25	0.922055
$290$	0	0
$291$	2.56921e25	0.345878
$292$	0	0
$293$	2.21385e25	0.277357	0.138678	−	0.990337i	$-0.455715\pi$
$293$	2.21385e25	0.277357	0.138678	+	0.990337i	$0.455715\pi$
$294$	0	0
$295$	7.38682e24	0.0861641
$296$	0	0
$297$	1.28296e26	1.39403
$298$	0	0
$299$	−6.14695e25	−0.622461
$300$	0	0
$301$	1.01300e26	0.956448
$302$	0	0
$303$	−1.53204e26	−1.34934
$304$	0	0
$305$	−1.59186e26	−1.30845
$306$	0	0
$307$	−1.00481e26	−0.771137	−0.385569	−	0.922679i	$-0.625995\pi$
$307$	−1.00481e26	−0.771137	−0.385569	+	0.922679i	$0.625995\pi$
$308$	0	0
$309$	1.39250e26	0.998231
$310$	0	0
$311$	−1.29128e26	−0.865043	−0.432521	−	0.901624i	$-0.642376\pi$
$311$	−1.29128e26	−0.865043	−0.432521	+	0.901624i	$0.642376\pi$
$312$	0	0
$313$	−1.59492e26	−0.998906	−0.499453	−	0.866341i	$-0.666466\pi$
$313$	−1.59492e26	−0.998906	−0.499453	+	0.866341i	$0.666466\pi$
$314$	0	0
$315$	2.22474e25	0.130322
$316$	0	0
$317$	8.73329e25	0.478692	0.239346	−	0.970934i	$-0.423067\pi$
$317$	8.73329e25	0.478692	0.239346	+	0.970934i	$0.423067\pi$
$318$	0	0
$319$	2.75691e26	1.41456
$320$	0	0
$321$	2.08627e26	1.00246
$322$	0	0
$323$	−4.78928e26	−2.15598
$324$	0	0
$325$	−9.75132e25	−0.411423
$326$	0	0
$327$	3.61862e26	1.43150
$328$	0	0
$329$	−1.65370e26	−0.613620
$330$	0	0
$331$	5.50377e26	1.91631	0.958157	−	0.286245i	$-0.0924070\pi$
$331$	5.50377e26	1.91631	0.958157	+	0.286245i	$0.0924070\pi$
$332$	0	0
$333$	−2.76170e25	−0.0902636
$334$	0	0
$335$	3.29986e26	1.01280
$336$	0	0
$337$	5.51815e25	0.159103	0.0795517	−	0.996831i	$-0.474651\pi$
$337$	5.51815e25	0.159103	0.0795517	+	0.996831i	$0.474651\pi$
$338$	0	0
$339$	−5.53031e26	−1.49849
$340$	0	0
$341$	7.05998e25	0.179838
$342$	0	0
$343$	−4.54400e26	−1.08855
$344$	0	0
$345$	−1.39868e26	−0.315222
$346$	0	0
$347$	−4.14487e26	−0.879126	−0.439563	−	0.898212i	$-0.644867\pi$
$347$	−4.14487e26	−0.879126	−0.439563	+	0.898212i	$0.644867\pi$
$348$	0	0
$349$	−4.30244e26	−0.859108	−0.429554	−	0.903041i	$-0.641329\pi$
$349$	−4.30244e26	−0.859108	−0.429554	+	0.903041i	$0.641329\pi$
$350$	0	0
$351$	8.64815e26	1.62629
$352$	0	0
$353$	−4.04943e26	−0.717398	−0.358699	−	0.933453i	$-0.616780\pi$
$353$	−4.04943e26	−0.717398	−0.358699	+	0.933453i	$0.616780\pi$
$354$	0	0
$355$	−1.92587e26	−0.321535
$356$	0	0
$357$	−6.42614e26	−1.01143
$358$	0	0
$359$	1.27535e27	1.89294	0.946469	−	0.322794i	$-0.104622\pi$
$359$	1.27535e27	1.89294	0.946469	+	0.322794i	$0.104622\pi$
$360$	0	0
$361$	1.01301e27	1.41836
$362$	0	0
$363$	−4.74125e26	−0.626431
$364$	0	0
$365$	2.89744e26	0.361359
$366$	0	0
$367$	4.71084e26	0.554759	0.277380	−	0.960760i	$-0.410534\pi$
$367$	4.71084e26	0.554759	0.277380	+	0.960760i	$0.410534\pi$
$368$	0	0
$369$	−2.38093e26	−0.264832
$370$	0	0
$371$	7.51111e26	0.789365
$372$	0	0
$373$	−9.52311e26	−0.945880	−0.472940	−	0.881095i	$-0.656807\pi$
$373$	−9.52311e26	−0.945880	−0.472940	+	0.881095i	$0.656807\pi$
$374$	0	0
$375$	−1.04109e27	−0.977593
$376$	0	0
$377$	1.85836e27	1.65024
$378$	0	0
$379$	−6.39167e26	−0.536912	−0.268456	−	0.963292i	$-0.586513\pi$
$379$	−6.39167e26	−0.536912	−0.268456	+	0.963292i	$0.586513\pi$
$380$	0	0
$381$	1.22037e27	0.970016
$382$	0	0
$383$	−2.07948e27	−1.56447	−0.782237	−	0.622981i	$-0.785922\pi$
$383$	−2.07948e27	−1.56447	−0.782237	+	0.622981i	$0.785922\pi$
$384$	0	0
$385$	−1.26408e27	−0.900402
$386$	0	0
$387$	−3.29948e26	−0.222577
$388$	0	0
$389$	9.59448e26	0.613127	0.306564	−	0.951850i	$-0.400821\pi$
$389$	9.59448e26	0.613127	0.306564	+	0.951850i	$0.400821\pi$
$390$	0	0
$391$	−9.38193e26	−0.568115
$392$	0	0
$393$	−2.19739e27	−1.26121
$394$	0	0
$395$	−1.36210e27	−0.741206
$396$	0	0
$397$	1.07265e27	0.553551	0.276776	−	0.960935i	$-0.410734\pi$
$397$	1.07265e27	0.553551	0.276776	+	0.960935i	$0.410734\pi$
$398$	0	0
$399$	2.31754e27	1.13452
$400$	0	0
$401$	3.20976e27	1.49093	0.745464	−	0.666546i	$-0.232228\pi$
$401$	3.20976e27	1.49093	0.745464	+	0.666546i	$0.232228\pi$
$402$	0	0
$403$	4.75896e26	0.209802
$404$	0	0
$405$	1.58330e27	0.662651
$406$	0	0
$407$	1.56918e27	0.623636
$408$	0	0
$409$	−1.21280e27	−0.457820	−0.228910	−	0.973448i	$-0.573516\pi$
$409$	−1.21280e27	−0.457820	−0.228910	+	0.973448i	$0.573516\pi$
$410$	0	0
$411$	1.99799e26	0.0716562
$412$	0	0
$413$	−2.39767e26	−0.0817172
$414$	0	0
$415$	−1.52392e27	−0.493692
$416$	0	0
$417$	−2.69758e27	−0.830893
$418$	0	0
$419$	1.87689e27	0.549784	0.274892	−	0.961475i	$-0.411358\pi$
$419$	1.87689e27	0.549784	0.274892	+	0.961475i	$0.411358\pi$
$420$	0	0
$421$	−1.61930e27	−0.451195	−0.225598	−	0.974221i	$-0.572433\pi$
$421$	−1.61930e27	−0.451195	−0.225598	+	0.974221i	$0.572433\pi$
$422$	0	0
$423$	5.38632e26	0.142797
$424$	0	0
$425$	−1.48832e27	−0.375502
$426$	0	0
$427$	5.16698e27	1.24092
$428$	0	0
$429$	−7.79200e27	−1.78175
$430$	0	0
$431$	−7.51808e27	−1.63718	−0.818589	−	0.574380i	$-0.805243\pi$
$431$	−7.51808e27	−1.63718	−0.818589	+	0.574380i	$0.805243\pi$
$432$	0	0
$433$	−5.53633e25	−0.0114842	−0.00574208	−	0.999984i	$-0.501828\pi$
$433$	−5.53633e25	−0.0114842	−0.00574208	+	0.999984i	$0.501828\pi$
$434$	0	0
$435$	4.22852e27	0.835702
$436$	0	0
$437$	3.38352e27	0.637256
$438$	0	0
$439$	2.79922e27	0.502527	0.251264	−	0.967919i	$-0.419154\pi$
$439$	2.79922e27	0.502527	0.251264	+	0.967919i	$0.419154\pi$
$440$	0	0
$441$	3.78960e26	0.0648616
$442$	0	0
$443$	−6.75198e27	−1.10203	−0.551014	−	0.834496i	$-0.685759\pi$
$443$	−6.75198e27	−1.10203	−0.551014	+	0.834496i	$0.685759\pi$
$444$	0	0
$445$	4.16997e27	0.649162
$446$	0	0
$447$	2.30539e27	0.342385
$448$	0	0
$449$	3.45343e27	0.489400	0.244700	−	0.969599i	$-0.421311\pi$
$449$	3.45343e27	0.489400	0.244700	+	0.969599i	$0.421311\pi$
$450$	0	0
$451$	1.35283e28	1.82973
$452$	0	0
$453$	6.53795e27	0.844132
$454$	0	0
$455$	−8.52085e27	−1.05042
$456$	0	0
$457$	−5.82771e27	−0.686085	−0.343042	−	0.939320i	$-0.611457\pi$
$457$	−5.82771e27	−0.686085	−0.343042	+	0.939320i	$0.611457\pi$
$458$	0	0
$459$	1.31994e28	1.48430
$460$	0	0
$461$	−9.26600e27	−0.995480	−0.497740	−	0.867326i	$-0.665837\pi$
$461$	−9.26600e27	−0.995480	−0.497740	+	0.867326i	$0.665837\pi$
$462$	0	0
$463$	−1.43306e28	−1.47117	−0.735586	−	0.677431i	$-0.763093\pi$
$463$	−1.43306e28	−1.47117	−0.735586	+	0.677431i	$0.763093\pi$
$464$	0	0
$465$	1.08285e27	0.106246
$466$	0	0
$467$	−1.24220e28	−1.16510	−0.582552	−	0.812794i	$-0.697946\pi$
$467$	−1.24220e28	−1.16510	−0.582552	+	0.812794i	$0.697946\pi$
$468$	0	0
$469$	−1.07109e28	−0.960532
$470$	0	0
$471$	−3.90010e27	−0.334470
$472$	0	0
$473$	1.87474e28	1.53780
$474$	0	0
$475$	5.36751e27	0.421201
$476$	0	0
$477$	−2.44647e27	−0.183695
$478$	0	0
$479$	−1.55141e28	−1.11482	−0.557410	−	0.830237i	$-0.688205\pi$
$479$	−1.55141e28	−1.11482	−0.557410	+	0.830237i	$0.688205\pi$
$480$	0	0
$481$	1.05775e28	0.727542
$482$	0	0
$483$	4.53993e27	0.298954
$484$	0	0
$485$	5.19953e27	0.327850
$486$	0	0
$487$	−1.10212e26	−0.00665542	−0.00332771	−	0.999994i	$-0.501059\pi$
$487$	−1.10212e26	−0.00665542	−0.00332771	+	0.999994i	$0.501059\pi$
$488$	0	0
$489$	1.87079e27	0.108213
$490$	0	0
$491$	−1.45763e28	−0.807779	−0.403890	−	0.914808i	$-0.632342\pi$
$491$	−1.45763e28	−0.807779	−0.403890	+	0.914808i	$0.632342\pi$
$492$	0	0
$493$	2.83637e28	1.50616
$494$	0	0
$495$	4.11727e27	0.209534
$496$	0	0
$497$	6.25112e27	0.304941
$498$	0	0
$499$	1.13652e28	0.531522	0.265761	−	0.964039i	$-0.414377\pi$
$499$	1.13652e28	0.531522	0.265761	+	0.964039i	$0.414377\pi$
$500$	0	0
$501$	1.36133e24	6.10474e−5 0
$502$	0	0
$503$	3.78149e28	1.62630	0.813148	−	0.582057i	$-0.197752\pi$
$503$	3.78149e28	1.62630	0.813148	+	0.582057i	$0.197752\pi$
$504$	0	0
$505$	−3.10051e28	−1.27901
$506$	0	0
$507$	−2.97605e28	−1.17776
$508$	0	0
$509$	−9.31841e27	−0.353839	−0.176919	−	0.984225i	$-0.556613\pi$
$509$	−9.31841e27	−0.353839	−0.176919	+	0.984225i	$0.556613\pi$
$510$	0	0
$511$	−9.40471e27	−0.342710
$512$	0	0
$513$	−4.76028e28	−1.66495
$514$	0	0
$515$	2.81812e28	0.946202
$516$	0	0
$517$	−3.06046e28	−0.986590
$518$	0	0
$519$	−1.72363e27	−0.0533565
$520$	0	0
$521$	−2.08795e25	−0.000620762 0	−0.000310381	−	1.00000i	$-0.500099\pi$
$521$	−2.08795e25	−0.000620762 0	−0.000310381		1.00000i	$0.500099\pi$
$522$	0	0
$523$	1.53000e28	0.436941	0.218470	−	0.975844i	$-0.429893\pi$
$523$	1.53000e28	0.436941	0.218470	+	0.975844i	$0.429893\pi$
$524$	0	0
$525$	7.20199e27	0.197597
$526$	0	0
$527$	7.26347e27	0.191484
$528$	0	0
$529$	−3.28435e28	−0.832079
$530$	0	0
$531$	7.80952e26	0.0190166
$532$	0	0
$533$	9.11907e28	2.13459
$534$	0	0
$535$	4.22216e28	0.950210
$536$	0	0
$537$	1.97834e28	0.428126
$538$	0	0
$539$	−2.15322e28	−0.448132
$540$	0	0
$541$	−7.80431e28	−1.56230	−0.781148	−	0.624346i	$-0.785365\pi$
$541$	−7.80431e28	−1.56230	−0.781148	+	0.624346i	$0.785365\pi$
$542$	0	0
$543$	5.08532e28	0.979310
$544$	0	0
$545$	7.32332e28	1.35689
$546$	0	0
$547$	−4.32938e28	−0.771897	−0.385948	−	0.922520i	$-0.626126\pi$
$547$	−4.32938e28	−0.771897	−0.385948	+	0.922520i	$0.626126\pi$
$548$	0	0
$549$	−1.68295e28	−0.288777
$550$	0	0
$551$	−1.02292e29	−1.68946
$552$	0	0
$553$	4.42120e28	0.702953
$554$	0	0
$555$	2.40679e28	0.368436
$556$	0	0
$557$	1.94436e28	0.286614	0.143307	−	0.989678i	$-0.454226\pi$
$557$	1.94436e28	0.286614	0.143307	+	0.989678i	$0.454226\pi$
$558$	0	0
$559$	1.26372e29	1.79401
$560$	0	0
$561$	−1.18927e29	−1.62619
$562$	0	0
$563$	−1.12608e29	−1.48331	−0.741655	−	0.670781i	$-0.765959\pi$
$563$	−1.12608e29	−1.48331	−0.741655	+	0.670781i	$0.765959\pi$
$564$	0	0
$565$	−1.11922e29	−1.42038
$566$	0	0
$567$	−5.13918e28	−0.628451
$568$	0	0
$569$	6.23100e28	0.734309	0.367154	−	0.930160i	$-0.380332\pi$
$569$	6.23100e28	0.734309	0.367154	+	0.930160i	$0.380332\pi$
$570$	0	0
$571$	−1.12398e29	−1.27667	−0.638334	−	0.769760i	$-0.720376\pi$
$571$	−1.12398e29	−1.27667	−0.638334	+	0.769760i	$0.720376\pi$
$572$	0	0
$573$	1.33763e29	1.46458
$574$	0	0
$575$	1.05146e28	0.110990
$576$	0	0
$577$	1.21433e29	1.23592	0.617960	−	0.786210i	$-0.287960\pi$
$577$	1.21433e29	1.23592	0.617960	+	0.786210i	$0.287960\pi$
$578$	0	0
$579$	−5.41788e28	−0.531747
$580$	0	0
$581$	4.94644e28	0.468213
$582$	0	0
$583$	1.39006e29	1.26916
$584$	0	0
$585$	2.77535e28	0.244445
$586$	0	0
$587$	6.84042e28	0.581276	0.290638	−	0.956833i	$-0.406132\pi$
$587$	6.84042e28	0.581276	0.290638	+	0.956833i	$0.406132\pi$
$588$	0	0
$589$	−2.61952e28	−0.214788
$590$	0	0
$591$	1.18598e29	0.938444
$592$	0	0
$593$	−5.52013e28	−0.421575	−0.210787	−	0.977532i	$-0.567603\pi$
$593$	−5.52013e28	−0.421575	−0.210787	+	0.977532i	$0.567603\pi$
$594$	0	0
$595$	−1.30051e29	−0.958709
$596$	0	0
$597$	−9.43001e28	−0.671091
$598$	0	0
$599$	−1.03055e29	−0.708091	−0.354046	−	0.935228i	$-0.615194\pi$
$599$	−1.03055e29	−0.708091	−0.354046	+	0.935228i	$0.615194\pi$
$600$	0	0
$601$	1.81265e29	1.20263	0.601314	−	0.799013i	$-0.294644\pi$
$601$	1.81265e29	1.20263	0.601314	+	0.799013i	$0.294644\pi$
$602$	0	0
$603$	3.48869e28	0.223527
$604$	0	0
$605$	−9.59527e28	−0.593780
$606$	0	0
$607$	−3.69189e28	−0.220682	−0.110341	−	0.993894i	$-0.535194\pi$
$607$	−3.69189e28	−0.220682	−0.110341	+	0.993894i	$0.535194\pi$
$608$	0	0
$609$	−1.37252e29	−0.792572
$610$	0	0
$611$	−2.06298e29	−1.15097
$612$	0	0
$613$	1.42961e29	0.770696	0.385348	−	0.922771i	$-0.374081\pi$
$613$	1.42961e29	0.770696	0.385348	+	0.922771i	$0.374081\pi$
$614$	0	0
$615$	2.07495e29	1.08098
$616$	0	0
$617$	−3.41658e29	−1.72027	−0.860137	−	0.510063i	$-0.829622\pi$
$617$	−3.41658e29	−1.72027	−0.860137	+	0.510063i	$0.829622\pi$
$618$	0	0
$619$	1.26714e29	0.616700	0.308350	−	0.951273i	$-0.400223\pi$
$619$	1.26714e29	0.616700	0.308350	+	0.951273i	$0.400223\pi$
$620$	0	0
$621$	−9.32511e28	−0.438725
$622$	0	0
$623$	−1.35352e29	−0.615658
$624$	0	0
$625$	−1.49109e29	−0.655790
$626$	0	0
$627$	4.28902e29	1.82410
$628$	0	0
$629$	1.61441e29	0.664021
$630$	0	0
$631$	8.54527e28	0.339952	0.169976	−	0.985448i	$-0.445631\pi$
$631$	8.54527e28	0.339952	0.169976	+	0.985448i	$0.445631\pi$
$632$	0	0
$633$	−9.17619e28	−0.353121
$634$	0	0
$635$	2.46977e29	0.919457
$636$	0	0
$637$	−1.45143e29	−0.522797
$638$	0	0
$639$	−2.03607e28	−0.0709634
$640$	0	0
$641$	3.29678e29	1.11194	0.555969	−	0.831203i	$-0.312347\pi$
$641$	3.29678e29	1.11194	0.555969	+	0.831203i	$0.312347\pi$
$642$	0	0
$643$	2.26473e29	0.739268	0.369634	−	0.929177i	$-0.379483\pi$
$643$	2.26473e29	0.739268	0.369634	+	0.929177i	$0.379483\pi$
$644$	0	0
$645$	2.87546e29	0.908511
$646$	0	0
$647$	−1.40922e29	−0.431005	−0.215503	−	0.976503i	$-0.569139\pi$
$647$	−1.40922e29	−0.431005	−0.215503	+	0.976503i	$0.569139\pi$
$648$	0	0
$649$	−4.43731e28	−0.131386
$650$	0	0
$651$	−3.51481e28	−0.100763
$652$	0	0
$653$	−6.00388e29	−1.66665	−0.833323	−	0.552786i	$-0.813565\pi$
$653$	−6.00388e29	−1.66665	−0.833323	+	0.552786i	$0.813565\pi$
$654$	0	0
$655$	−4.44705e29	−1.19547
$656$	0	0
$657$	3.06324e28	0.0797526
$658$	0	0
$659$	−5.22410e29	−1.31739	−0.658695	−	0.752410i	$-0.728891\pi$
$659$	−5.22410e29	−1.31739	−0.658695	+	0.752410i	$0.728891\pi$
$660$	0	0
$661$	4.46063e29	1.08963	0.544817	−	0.838555i	$-0.316599\pi$
$661$	4.46063e29	1.08963	0.544817	+	0.838555i	$0.316599\pi$
$662$	0	0
$663$	−8.01659e29	−1.89713
$664$	0	0
$665$	4.69021e29	1.07539
$666$	0	0
$667$	−2.00383e29	−0.445185
$668$	0	0
$669$	4.08027e29	0.878445
$670$	0	0
$671$	9.56241e29	1.99518
$672$	0	0
$673$	−7.24333e29	−1.46481	−0.732403	−	0.680871i	$-0.761601\pi$
$673$	−7.24333e29	−1.46481	−0.732403	+	0.680871i	$0.761601\pi$
$674$	0	0
$675$	−1.47931e29	−0.289980
$676$	0	0
$677$	2.20060e29	0.418176	0.209088	−	0.977897i	$-0.432950\pi$
$677$	2.20060e29	0.418176	0.209088	+	0.977897i	$0.432950\pi$
$678$	0	0
$679$	−1.68770e29	−0.310930
$680$	0	0
$681$	−3.90240e29	−0.697088
$682$	0	0
$683$	7.92621e29	1.37293	0.686465	−	0.727163i	$-0.259162\pi$
$683$	7.92621e29	1.37293	0.686465	+	0.727163i	$0.259162\pi$
$684$	0	0
$685$	4.04350e28	0.0679214
$686$	0	0
$687$	−6.05668e29	−0.986708
$688$	0	0
$689$	9.37009e29	1.48061
$690$	0	0
$691$	8.05813e29	1.23514	0.617568	−	0.786517i	$-0.288118\pi$
$691$	8.05813e29	1.23514	0.617568	+	0.786517i	$0.288118\pi$
$692$	0	0
$693$	−1.33641e29	−0.198720
$694$	0	0
$695$	−5.45932e29	−0.787586
$696$	0	0
$697$	1.39182e30	1.94822
$698$	0	0
$699$	9.57057e28	0.129995
$700$	0	0
$701$	−8.00523e29	−1.05520	−0.527600	−	0.849493i	$-0.676908\pi$
$701$	−8.00523e29	−1.05520	−0.527600	+	0.849493i	$0.676908\pi$
$702$	0	0
$703$	−5.82224e29	−0.744834
$704$	0	0
$705$	−4.69412e29	−0.582865
$706$	0	0
$707$	1.00639e30	1.21300
$708$	0	0
$709$	−1.20024e30	−1.40437	−0.702185	−	0.711994i	$-0.747792\pi$
$709$	−1.20024e30	−1.40437	−0.702185	+	0.711994i	$0.747792\pi$
$710$	0	0
$711$	−1.44004e29	−0.163585
$712$	0	0
$713$	−5.13149e28	−0.0565983
$714$	0	0
$715$	−1.57693e30	−1.68889
$716$	0	0
$717$	5.38025e29	0.559566
$718$	0	0
$719$	−3.17808e29	−0.321004	−0.160502	−	0.987035i	$-0.551311\pi$
$719$	−3.17808e29	−0.321004	−0.160502	+	0.987035i	$0.551311\pi$
$720$	0	0
$721$	−9.14727e29	−0.897369
$722$	0	0
$723$	−3.54649e29	−0.337946
$724$	0	0
$725$	−3.17882e29	−0.294250
$726$	0	0
$727$	1.42314e30	1.27978	0.639892	−	0.768465i	$-0.278979\pi$
$727$	1.42314e30	1.27978	0.639892	+	0.768465i	$0.278979\pi$
$728$	0	0
$729$	1.27130e30	1.11073
$730$	0	0
$731$	1.92877e30	1.63738
$732$	0	0
$733$	−1.17834e30	−0.972031	−0.486016	−	0.873950i	$-0.661550\pi$
$733$	−1.17834e30	−0.972031	−0.486016	+	0.873950i	$0.661550\pi$
$734$	0	0
$735$	−3.30259e29	−0.264751
$736$	0	0
$737$	−1.98225e30	−1.54436
$738$	0	0
$739$	1.73577e30	1.31440	0.657198	−	0.753718i	$-0.271742\pi$
$739$	1.73577e30	1.31440	0.657198	+	0.753718i	$0.271742\pi$
$740$	0	0
$741$	2.89112e30	2.12802
$742$	0	0
$743$	4.69683e29	0.336064	0.168032	−	0.985782i	$-0.446259\pi$
$743$	4.69683e29	0.336064	0.168032	+	0.985782i	$0.446259\pi$
$744$	0	0
$745$	4.66561e29	0.324540
$746$	0	0
$747$	−1.61112e29	−0.108959
$748$	0	0
$749$	−1.37046e30	−0.901170
$750$	0	0
$751$	−7.47353e29	−0.477866	−0.238933	−	0.971036i	$-0.576798\pi$
$751$	−7.47353e29	−0.477866	−0.238933	+	0.971036i	$0.576798\pi$
$752$	0	0
$753$	−2.64132e30	−1.64238
$754$	0	0
$755$	1.32314e30	0.800135
$756$	0	0
$757$	2.10588e30	1.23859	0.619293	−	0.785160i	$-0.287419\pi$
$757$	2.10588e30	1.23859	0.619293	+	0.785160i	$0.287419\pi$
$758$	0	0
$759$	8.40194e29	0.480664
$760$	0	0
$761$	−9.10629e29	−0.506760	−0.253380	−	0.967367i	$-0.581542\pi$
$761$	−9.10629e29	−0.506760	−0.253380	+	0.967367i	$0.581542\pi$
$762$	0	0
$763$	−2.37706e30	−1.28686
$764$	0	0
$765$	4.23595e29	0.223103
$766$	0	0
$767$	−2.99108e29	−0.153277
$768$	0	0
$769$	2.59434e30	1.29360	0.646801	−	0.762659i	$-0.276106\pi$
$769$	2.59434e30	1.29360	0.646801	+	0.762659i	$0.276106\pi$
$770$	0	0
$771$	−3.38369e30	−1.64180
$772$	0	0
$773$	−1.32692e30	−0.626559	−0.313279	−	0.949661i	$-0.601428\pi$
$773$	−1.32692e30	−0.626559	−0.313279	+	0.949661i	$0.601428\pi$
$774$	0	0
$775$	−8.14042e28	−0.0374092
$776$	0	0
$777$	−7.81215e29	−0.349422
$778$	0	0
$779$	−5.01949e30	−2.18533
$780$	0	0
$781$	1.15688e30	0.490290
$782$	0	0
$783$	2.81920e30	1.16313
$784$	0	0
$785$	−7.89298e29	−0.317037
$786$	0	0
$787$	−5.12807e29	−0.200548	−0.100274	−	0.994960i	$-0.531972\pi$
$787$	−5.12807e29	−0.200548	−0.100274	+	0.994960i	$0.531972\pi$
$788$	0	0
$789$	1.65360e30	0.629684
$790$	0	0
$791$	3.63284e30	1.34708
$792$	0	0
$793$	6.44579e30	2.32760
$794$	0	0
$795$	2.13207e30	0.749802
$796$	0	0
$797$	−1.27653e30	−0.437239	−0.218620	−	0.975810i	$-0.570155\pi$
$797$	−1.27653e30	−0.437239	−0.218620	+	0.975810i	$0.570155\pi$
$798$	0	0
$799$	−3.14868e30	−1.05048
$800$	0	0
$801$	4.40859e29	0.143271
$802$	0	0
$803$	−1.74051e30	−0.551015
$804$	0	0
$805$	9.18785e29	0.283372
$806$	0	0
$807$	6.31951e29	0.189894
$808$	0	0
$809$	1.38086e30	0.404287	0.202144	−	0.979356i	$-0.435209\pi$
$809$	1.38086e30	0.404287	0.202144	+	0.979356i	$0.435209\pi$
$810$	0	0
$811$	4.03179e29	0.115021	0.0575107	−	0.998345i	$-0.481684\pi$
$811$	4.03179e29	0.115021	0.0575107	+	0.998345i	$0.481684\pi$
$812$	0	0
$813$	−9.17657e29	−0.255111
$814$	0	0
$815$	3.78607e29	0.102573
$816$	0	0
$817$	−6.95598e30	−1.83665
$818$	0	0
$819$	−9.00845e29	−0.231830
$820$	0	0
$821$	−3.10056e29	−0.0777745	−0.0388873	−	0.999244i	$-0.512381\pi$
$821$	−3.10056e29	−0.0777745	−0.0388873	+	0.999244i	$0.512381\pi$
$822$	0	0
$823$	1.10242e30	0.269555	0.134777	−	0.990876i	$-0.456968\pi$
$823$	1.10242e30	0.269555	0.134777	+	0.990876i	$0.456968\pi$
$824$	0	0
$825$	1.33286e30	0.317700
$826$	0	0
$827$	−9.82951e29	−0.228415	−0.114207	−	0.993457i	$-0.536433\pi$
$827$	−9.82951e29	−0.228415	−0.114207	+	0.993457i	$0.536433\pi$
$828$	0	0
$829$	−5.60776e30	−1.27048	−0.635239	−	0.772316i	$-0.719098\pi$
$829$	−5.60776e30	−1.27048	−0.635239	+	0.772316i	$0.719098\pi$
$830$	0	0
$831$	1.04344e30	0.230492
$832$	0	0
$833$	−2.21528e30	−0.477152
$834$	0	0
$835$	2.75504e26	5.78655e−5 0
$836$	0	0
$837$	7.21949e29	0.147873
$838$	0	0
$839$	3.92515e30	0.784071	0.392036	−	0.919950i	$-0.371771\pi$
$839$	3.92515e30	0.784071	0.392036	+	0.919950i	$0.371771\pi$
$840$	0	0
$841$	9.25212e29	0.180253
$842$	0	0
$843$	1.61012e28	0.00305963
$844$	0	0
$845$	−6.02288e30	−1.11637
$846$	0	0
$847$	3.11450e30	0.563135
$848$	0	0
$849$	2.63155e28	0.00464175
$850$	0	0
$851$	−1.14054e30	−0.196269
$852$	0	0
$853$	9.01715e30	1.51393	0.756963	−	0.653458i	$-0.226683\pi$
$853$	9.01715e30	1.51393	0.756963	+	0.653458i	$0.226683\pi$
$854$	0	0
$855$	−1.52766e30	−0.250255
$856$	0	0
$857$	7.89609e30	1.26216	0.631079	−	0.775719i	$-0.282612\pi$
$857$	7.89609e30	1.26216	0.631079	+	0.775719i	$0.282612\pi$
$858$	0	0
$859$	3.91414e30	0.610531	0.305266	−	0.952267i	$-0.401255\pi$
$859$	3.91414e30	0.610531	0.305266	+	0.952267i	$0.401255\pi$
$860$	0	0
$861$	−6.73503e30	−1.02520
$862$	0	0
$863$	1.32795e31	1.97273	0.986365	−	0.164572i	$-0.0526243\pi$
$863$	1.32795e31	1.97273	0.986365	+	0.164572i	$0.0526243\pi$
$864$	0	0
$865$	−3.48826e29	−0.0505755
$866$	0	0
$867$	−5.86966e30	−0.830639
$868$	0	0
$869$	8.18221e30	1.13022
$870$	0	0
$871$	−1.33618e31	−1.80167
$872$	0	0
$873$	5.49707e29	0.0723571
$874$	0	0
$875$	6.83884e30	0.878816
$876$	0	0
$877$	−1.33474e31	−1.67456	−0.837278	−	0.546778i	$-0.815854\pi$
$877$	−1.33474e31	−1.67456	−0.837278	+	0.546778i	$0.815854\pi$
$878$	0	0
$879$	−2.03975e30	−0.249859
$880$	0	0
$881$	−1.47240e31	−1.76108	−0.880538	−	0.473976i	$-0.842818\pi$
$881$	−1.47240e31	−1.76108	−0.880538	+	0.473976i	$0.842818\pi$
$882$	0	0
$883$	4.20820e29	0.0491484	0.0245742	−	0.999698i	$-0.492177\pi$
$883$	4.20820e29	0.0491484	0.0245742	+	0.999698i	$0.492177\pi$
$884$	0	0
$885$	−6.80591e29	−0.0776215
$886$	0	0
$887$	3.56495e30	0.397060	0.198530	−	0.980095i	$-0.436383\pi$
$887$	3.56495e30	0.397060	0.198530	+	0.980095i	$0.436383\pi$
$888$	0	0
$889$	−8.01654e30	−0.872004
$890$	0	0
$891$	−9.51097e30	−1.01044
$892$	0	0
$893$	1.13555e31	1.17832
$894$	0	0
$895$	4.00375e30	0.405812
$896$	0	0
$897$	5.66355e30	0.560748
$898$	0	0
$899$	1.55137e30	0.150051
$900$	0	0
$901$	1.43013e31	1.35134
$902$	0	0
$903$	−9.33337e30	−0.861623
$904$	0	0
$905$	1.02916e31	0.928267
$906$	0	0
$907$	−8.63550e30	−0.761046	−0.380523	−	0.924771i	$-0.624256\pi$
$907$	−8.63550e30	−0.761046	−0.380523	+	0.924771i	$0.624256\pi$
$908$	0	0
$909$	−3.27794e30	−0.282280
$910$	0	0
$911$	−2.20841e31	−1.85839	−0.929197	−	0.369586i	$-0.879500\pi$
$911$	−2.20841e31	−1.85839	−0.929197	+	0.369586i	$0.879500\pi$
$912$	0	0
$913$	9.15426e30	0.752801
$914$	0	0
$915$	1.46668e31	1.17872
$916$	0	0
$917$	1.44346e31	1.13377
$918$	0	0
$919$	2.56943e30	0.197253	0.0986265	−	0.995125i	$-0.468555\pi$
$919$	2.56943e30	0.197253	0.0986265	+	0.995125i	$0.468555\pi$
$920$	0	0
$921$	9.25792e30	0.694684
$922$	0	0
$923$	7.79825e30	0.571978
$924$	0	0
$925$	−1.80932e30	−0.129726
$926$	0	0
$927$	2.97939e30	0.208828
$928$	0	0
$929$	−1.85518e31	−1.27122	−0.635611	−	0.772010i	$-0.719252\pi$
$929$	−1.85518e31	−1.27122	−0.635611	+	0.772010i	$0.719252\pi$
$930$	0	0
$931$	7.98925e30	0.535223
$932$	0	0
$933$	1.18974e31	0.779279
$934$	0	0
$935$	−2.40683e31	−1.54143
$936$	0	0
$937$	−5.89731e30	−0.369308	−0.184654	−	0.982804i	$-0.559116\pi$
$937$	−5.89731e30	−0.369308	−0.184654	+	0.982804i	$0.559116\pi$
$938$	0	0
$939$	1.46950e31	0.899871
$940$	0	0
$941$	−1.82883e31	−1.09517	−0.547587	−	0.836749i	$-0.684453\pi$
$941$	−1.82883e31	−1.09517	−0.547587	+	0.836749i	$0.684453\pi$
$942$	0	0
$943$	−9.83289e30	−0.575849
$944$	0	0
$945$	−1.29264e31	−0.740360
$946$	0	0
$947$	−5.73590e30	−0.321312	−0.160656	−	0.987010i	$-0.551361\pi$
$947$	−5.73590e30	−0.321312	−0.160656	+	0.987010i	$0.551361\pi$
$948$	0	0
$949$	−1.17324e31	−0.642821
$950$	0	0
$951$	−8.04649e30	−0.431232
$952$	0	0
$953$	1.27548e31	0.668647	0.334324	−	0.942458i	$-0.391492\pi$
$953$	1.27548e31	0.668647	0.334324	+	0.942458i	$0.391492\pi$
$954$	0	0
$955$	2.70708e31	1.38825
$956$	0	0
$957$	−2.54010e31	−1.27431
$958$	0	0
$959$	−1.31247e30	−0.0644160
$960$	0	0
$961$	−2.04282e31	−0.980923
$962$	0	0
$963$	4.46377e30	0.209713
$964$	0	0
$965$	−1.09646e31	−0.504031
$966$	0	0
$967$	2.48789e31	1.11906	0.559531	−	0.828810i	$-0.310981\pi$
$967$	2.48789e31	1.11906	0.559531	+	0.828810i	$0.310981\pi$
$968$	0	0
$969$	4.41264e31	1.94222
$970$	0	0
$971$	2.93374e31	1.26363	0.631814	−	0.775120i	$-0.282311\pi$
$971$	2.93374e31	1.26363	0.631814	+	0.775120i	$0.282311\pi$
$972$	0	0
$973$	1.77203e31	0.746939
$974$	0	0
$975$	8.98446e30	0.370633
$976$	0	0
$977$	−1.41713e31	−0.572159	−0.286080	−	0.958206i	$-0.592352\pi$
$977$	−1.41713e31	−0.572159	−0.286080	+	0.958206i	$0.592352\pi$
$978$	0	0
$979$	−2.50493e31	−0.989867
$980$	0	0
$981$	7.74239e30	0.299468
$982$	0	0
$983$	−3.05921e31	−1.15824	−0.579119	−	0.815243i	$-0.696603\pi$
$983$	−3.05921e31	−1.15824	−0.579119	+	0.815243i	$0.696603\pi$
$984$	0	0
$985$	2.40017e31	0.889532
$986$	0	0
$987$	1.52365e31	0.552784
$988$	0	0
$989$	−1.36264e31	−0.483971
$990$	0	0
$991$	4.24783e31	1.47704	0.738522	−	0.674229i	$-0.235524\pi$
$991$	4.24783e31	1.47704	0.738522	+	0.674229i	$0.235524\pi$
$992$	0	0
$993$	−5.07095e31	−1.72632
$994$	0	0
$995$	−1.90843e31	−0.636113
$996$	0	0
$997$	−1.55461e31	−0.507369	−0.253684	−	0.967287i	$-0.581642\pi$
$997$	−1.55461e31	−0.507369	−0.253684	+	0.967287i	$0.581642\pi$
$998$	0	0
$999$	1.60463e31	0.512788

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.22.a.a.1.1	✓	2
3.2	odd	2		36.22.a.c.1.2		2
4.3	odd	2		16.22.a.d.1.2		2
8.3	odd	2		64.22.a.j.1.1		2
8.5	even	2		64.22.a.i.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.22.a.a.1.1	✓	2	1.1	even	1	trivial
16.22.a.d.1.2		2	4.3	odd	2
36.22.a.c.1.2		2	3.2	odd	2
64.22.a.i.1.2		2	8.5	even	2
64.22.a.j.1.1		2	8.3	odd	2

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:	\( N \)	\(=\)	\( 4 = 2^{2} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q-92135.9 q^{3} -1.86463e7 q^{5} +6.05236e8 q^{7} -1.97134e9 q^{9} +O(q^{10})\)	\(q-92135.9 q^{3} -1.86463e7 q^{5} +6.05236e8 q^{7} -1.97134e9 q^{9} +1.12010e11 q^{11} +7.55030e11 q^{13} +1.71800e12 q^{15} +1.15238e13 q^{17} -4.15598e13 q^{19} -5.57640e13 q^{21} -8.14133e13 q^{23} -1.29151e14 q^{25} +1.14540e15 q^{27} +2.46131e15 q^{29} +6.30301e14 q^{31} -1.03201e16 q^{33} -1.12854e16 q^{35} +1.40093e16 q^{37} -6.95654e16 q^{39} +1.20777e17 q^{41} +1.67373e17 q^{43} +3.67582e16 q^{45} -2.73232e17 q^{47} -1.92235e17 q^{49} -1.06176e18 q^{51} +1.24102e18 q^{53} -2.08857e18 q^{55} +3.82915e18 q^{57} -3.96154e17 q^{59} +8.53713e18 q^{61} -1.19312e18 q^{63} -1.40785e19 q^{65} -1.76971e19 q^{67} +7.50108e18 q^{69} +1.03284e19 q^{71} -1.55389e19 q^{73} +1.18995e19 q^{75} +6.77923e19 q^{77} +7.30491e19 q^{79} -8.49120e19 q^{81} +8.17274e19 q^{83} -2.14877e20 q^{85} -2.26775e20 q^{87} -2.23635e20 q^{89} +4.56972e20 q^{91} -5.80733e19 q^{93} +7.74938e20 q^{95} -2.78850e20 q^{97} -2.20809e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) \) 2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9	\( 2 q + 65640 q^{3} + 13689324 q^{5} - 260508080 q^{7} + 12461535162 q^{9} + 145435963320 q^{11} + 1428900417340 q^{13} + 6819782714352 q^{15} + 1840620576420 q^{17} - 16780743928568 q^{19} - 192357511002048 q^{21} - 319691925426960 q^{23} + 439606295919326 q^{25} + 17\!\cdots\!40 q^{27}+ \cdots + 26\!\cdots\!20 q^{99}+O(q^{100}) \) 2 * q + 65640 * q^3 + 13689324 * q^5 - 260508080 * q^7 + 12461535162 * q^9 + 145435963320 * q^11 + 1428900417340 * q^13 + 6819782714352 * q^15 + 1840620576420 * q^17 - 16780743928568 * q^19 - 192357511002048 * q^21 - 319691925426960 * q^23 + 439606295919326 * q^25 + 1772171769223440 * q^27 + 3742111775766492 * q^29 + 112042353462592 * q^31 - 5046260260851360 * q^33 - 39279847462424352 * q^35 - 33362705637547220 * q^37 + 36755051864070192 * q^39 + 175129744323133332 * q^41 + 15346613416528120 * q^43 + 503454557153115324 * q^45 - 684848819288455200 * q^47 - 1267753878311886 * q^49 - 2589533339352286896 * q^51 + 675305394244421580 * q^53 - 1007711578141009200 * q^55 + 7738689471355209120 * q^57 + 1042445250435434904 * q^59 + 9065997829736468764 * q^61 - 13688298514444201200 * q^63 + 7711482582413403048 * q^65 - 30464301046802775320 * q^67 - 30093532090729291584 * q^69 - 8199093502830518064 * q^71 + 25415086659374793940 * q^73 + 101635704862879905048 * q^75 + 38853770260581178560 * q^77 + 121204353225060164896 * q^79 - 136996120065469586382 * q^81 + 108742936757033809800 * q^83 - 527989845644919754344 * q^85 - 24695689277789874000 * q^87 - 184207999274965368972 * q^89 - 126427419539014666912 * q^91 - 139841964206001442560 * q^93 + 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 + 261627767970577922520 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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