LMFDB - Embedded newform 4.36.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,36,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, 36, names="a")

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

chi := DirichletCharacter("4.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 4 = 2^{2} $
Weight:	$ k $	$=$	$ 36 $
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:	$31.0380522535$
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} - 1597028177x + 23572260890640 $ x^3 - x^2 - 1597028177*x + 23572260890640
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{4}\cdot 5\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$26763.9$ 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)

$f(q)$	$=$	$q+9.85025e7 q^{3} -2.11237e12 q^{5} +1.18095e15 q^{7} -4.03288e16 q^{9} +O(q^{10})$	$q+9.85025e7 q^{3} -2.11237e12 q^{5} +1.18095e15 q^{7} -4.03288e16 q^{9} -7.76322e17 q^{11} -2.10523e18 q^{13} -2.08074e20 q^{15} +4.70050e21 q^{17} -3.36016e22 q^{19} +1.16326e23 q^{21} -5.13239e23 q^{23} +1.55172e24 q^{25} -8.90072e24 q^{27} -2.68188e25 q^{29} -1.14473e26 q^{31} -7.64696e25 q^{33} -2.49459e27 q^{35} -3.17148e27 q^{37} -2.07371e26 q^{39} -2.95081e28 q^{41} +1.40702e28 q^{43} +8.51893e28 q^{45} -1.59169e29 q^{47} +1.01581e30 q^{49} +4.63011e29 q^{51} -7.01884e29 q^{53} +1.63988e30 q^{55} -3.30984e30 q^{57} -4.07975e30 q^{59} -1.37474e31 q^{61} -4.76261e31 q^{63} +4.44703e30 q^{65} +1.04711e32 q^{67} -5.05553e31 q^{69} +1.43111e32 q^{71} +6.55256e32 q^{73} +1.52848e32 q^{75} -9.16793e32 q^{77} +2.85812e32 q^{79} +1.14097e33 q^{81} -5.01380e33 q^{83} -9.92920e33 q^{85} -2.64172e33 q^{87} +2.12280e34 q^{89} -2.48616e33 q^{91} -1.12759e34 q^{93} +7.09789e34 q^{95} +2.50370e32 q^{97} +3.13081e34 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	$ 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * 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
$3$	9.85025e7	0.440378	0.220189	−	0.975457i	$-0.429333\pi$
$3$	9.85025e7	0.440378	0.220189	+	0.975457i	$0.429333\pi$
$4$	0	0
$5$	−2.11237e12	−1.23821	−0.619106	−	0.785308i	$-0.712505\pi$
$5$	−2.11237e12	−1.23821	−0.619106	+	0.785308i	$0.712505\pi$
$6$	0	0
$7$	1.18095e15	1.91873	0.959365	−	0.282167i	$-0.0910532\pi$
$7$	1.18095e15	1.91873	0.959365	+	0.282167i	$0.0910532\pi$
$8$	0	0
$9$	−4.03288e16	−0.806068
$10$	0	0
$11$	−7.76322e17	−0.463095	−0.231547	−	0.972824i	$-0.574379\pi$
$11$	−7.76322e17	−0.463095	−0.231547	+	0.972824i	$0.574379\pi$
$12$	0	0
$13$	−2.10523e18	−0.0674981	−0.0337490	−	0.999430i	$-0.510745\pi$
$13$	−2.10523e18	−0.0674981	−0.0337490	+	0.999430i	$0.510745\pi$
$14$	0	0
$15$	−2.08074e20	−0.545280
$16$	0	0
$17$	4.70050e21	1.37812	0.689062	−	0.724702i	$-0.258023\pi$
$17$	4.70050e21	1.37812	0.689062	+	0.724702i	$0.258023\pi$
$18$	0	0
$19$	−3.36016e22	−1.40660	−0.703301	−	0.710892i	$-0.748291\pi$
$19$	−3.36016e22	−1.40660	−0.703301	+	0.710892i	$0.748291\pi$
$20$	0	0
$21$	1.16326e23	0.844966
$22$	0	0
$23$	−5.13239e23	−0.758721	−0.379361	−	0.925249i	$-0.623856\pi$
$23$	−5.13239e23	−0.758721	−0.379361	+	0.925249i	$0.623856\pi$
$24$	0	0
$25$	1.55172e24	0.533167
$26$	0	0
$27$	−8.90072e24	−0.795352
$28$	0	0
$29$	−2.68188e25	−0.686238	−0.343119	−	0.939292i	$-0.611483\pi$
$29$	−2.68188e25	−0.686238	−0.343119	+	0.939292i	$0.611483\pi$
$30$	0	0
$31$	−1.14473e26	−0.911748	−0.455874	−	0.890044i	$-0.650673\pi$
$31$	−1.14473e26	−0.911748	−0.455874	+	0.890044i	$0.650673\pi$
$32$	0	0
$33$	−7.64696e25	−0.203936
$34$	0	0
$35$	−2.49459e27	−2.37579
$36$	0	0
$37$	−3.17148e27	−1.14217	−0.571086	−	0.820891i	$-0.693478\pi$
$37$	−3.17148e27	−1.14217	−0.571086	+	0.820891i	$0.693478\pi$
$38$	0	0
$39$	−2.07371e26	−0.0297246
$40$	0	0
$41$	−2.95081e28	−1.76288	−0.881442	−	0.472293i	$-0.843427\pi$
$41$	−2.95081e28	−1.76288	−0.881442	+	0.472293i	$0.843427\pi$
$42$	0	0
$43$	1.40702e28	0.365260	0.182630	−	0.983182i	$-0.441539\pi$
$43$	1.40702e28	0.365260	0.182630	+	0.983182i	$0.441539\pi$
$44$	0	0
$45$	8.51893e28	0.998082
$46$	0	0
$47$	−1.59169e29	−0.871256	−0.435628	−	0.900127i	$-0.643474\pi$
$47$	−1.59169e29	−0.871256	−0.435628	+	0.900127i	$0.643474\pi$
$48$	0	0
$49$	1.01581e30	2.68153
$50$	0	0
$51$	4.63011e29	0.606895
$52$	0	0
$53$	−7.01884e29	−0.469286	−0.234643	−	0.972082i	$-0.575392\pi$
$53$	−7.01884e29	−0.469286	−0.234643	+	0.972082i	$0.575392\pi$
$54$	0	0
$55$	1.63988e30	0.573409
$56$	0	0
$57$	−3.30984e30	−0.619436
$58$	0	0
$59$	−4.07975e30	−0.417565	−0.208782	−	0.977962i	$-0.566950\pi$
$59$	−4.07975e30	−0.417565	−0.208782	+	0.977962i	$0.566950\pi$
$60$	0	0
$61$	−1.37474e31	−0.785140	−0.392570	−	0.919722i	$-0.628414\pi$
$61$	−1.37474e31	−0.785140	−0.392570	+	0.919722i	$0.628414\pi$
$62$	0	0
$63$	−4.76261e31	−1.54663
$64$	0	0
$65$	4.44703e30	0.0835769
$66$	0	0
$67$	1.04711e32	1.15793	0.578964	−	0.815353i	$-0.303457\pi$
$67$	1.04711e32	1.15793	0.578964	+	0.815353i	$0.303457\pi$
$68$	0	0
$69$	−5.05553e31	−0.334124
$70$	0	0
$71$	1.43111e32	0.573655	0.286827	−	0.957982i	$-0.407399\pi$
$71$	1.43111e32	0.573655	0.286827	+	0.957982i	$0.407399\pi$
$72$	0	0
$73$	6.55256e32	1.61533	0.807664	−	0.589643i	$-0.200732\pi$
$73$	6.55256e32	1.61533	0.807664	+	0.589643i	$0.200732\pi$
$74$	0	0
$75$	1.52848e32	0.234795
$76$	0	0
$77$	−9.16793e32	−0.888554
$78$	0	0
$79$	2.85812e32	0.176850	0.0884252	−	0.996083i	$-0.471817\pi$
$79$	2.85812e32	0.176850	0.0884252	+	0.996083i	$0.471817\pi$
$80$	0	0
$81$	1.14097e33	0.455813
$82$	0	0
$83$	−5.01380e33	−1.30707	−0.653537	−	0.756894i	$-0.726716\pi$
$83$	−5.01380e33	−1.30707	−0.653537	+	0.756894i	$0.726716\pi$
$84$	0	0
$85$	−9.92920e33	−1.70641
$86$	0	0
$87$	−2.64172e33	−0.302204
$88$	0	0
$89$	2.12280e34	1.63149	0.815744	−	0.578413i	$-0.196328\pi$
$89$	2.12280e34	1.63149	0.815744	+	0.578413i	$0.196328\pi$
$90$	0	0
$91$	−2.48616e33	−0.129511
$92$	0	0
$93$	−1.12759e34	−0.401513
$94$	0	0
$95$	7.09789e34	1.74167
$96$	0	0
$97$	2.50370e32	0.00426656	0.00213328	−	0.999998i	$-0.499321\pi$
$97$	2.50370e32	0.00426656	0.00213328	+	0.999998i	$0.499321\pi$
$98$	0	0
$99$	3.13081e34	0.373286
$100$	0	0
$101$	−1.37029e34	−0.115130	−0.0575652	−	0.998342i	$-0.518334\pi$
$101$	−1.37029e34	−0.115130	−0.0575652	+	0.998342i	$0.518334\pi$
$102$	0	0
$103$	7.10144e34	0.423345	0.211673	−	0.977341i	$-0.432109\pi$
$103$	7.10144e34	0.423345	0.211673	+	0.977341i	$0.432109\pi$
$104$	0	0
$105$	−2.45723e35	−1.04625
$106$	0	0
$107$	−5.90581e35	−1.80744	−0.903719	−	0.428126i	$-0.859174\pi$
$107$	−5.90581e35	−1.80744	−0.903719	+	0.428126i	$0.859174\pi$
$108$	0	0
$109$	4.29257e35	0.950067	0.475033	−	0.879968i	$-0.342436\pi$
$109$	4.29257e35	0.950067	0.475033	+	0.879968i	$0.342436\pi$
$110$	0	0
$111$	−3.12398e35	−0.502987
$112$	0	0
$113$	−8.60072e35	−1.01312	−0.506561	−	0.862204i	$-0.669084\pi$
$113$	−8.60072e35	−1.01312	−0.506561	+	0.862204i	$0.669084\pi$
$114$	0	0
$115$	1.08415e36	0.939457
$116$	0	0
$117$	8.49015e34	0.0544080
$118$	0	0
$119$	5.55104e36	2.64425
$120$	0	0
$121$	−2.20757e36	−0.785543
$122$	0	0
$123$	−2.90662e36	−0.776334
$124$	0	0
$125$	2.87000e36	0.578038
$126$	0	0
$127$	4.23158e36	0.645561	0.322781	−	0.946474i	$-0.395382\pi$
$127$	4.23158e36	0.645561	0.322781	+	0.946474i	$0.395382\pi$
$128$	0	0
$129$	1.38595e36	0.160852
$130$	0	0
$131$	−8.04308e36	−0.713140	−0.356570	−	0.934269i	$-0.616054\pi$
$131$	−8.04308e36	−0.713140	−0.356570	+	0.934269i	$0.616054\pi$
$132$	0	0
$133$	−3.96816e37	−2.69889
$134$	0	0
$135$	1.88016e37	0.984813
$136$	0	0
$137$	−2.30532e37	−0.933514	−0.466757	−	0.884386i	$-0.654578\pi$
$137$	−2.30532e37	−0.933514	−0.466757	+	0.884386i	$0.654578\pi$
$138$	0	0
$139$	4.20073e37	1.31997	0.659987	−	0.751277i	$-0.270562\pi$
$139$	4.20073e37	1.31997	0.659987	+	0.751277i	$0.270562\pi$
$140$	0	0
$141$	−1.56785e37	−0.383682
$142$	0	0
$143$	1.63434e36	0.0312580
$144$	0	0
$145$	5.66512e37	0.849707
$146$	0	0
$147$	1.00060e38	1.18088
$148$	0	0
$149$	2.48012e34	0.000231053 0	0.000115527	−	1.00000i	$-0.499963\pi$
$149$	2.48012e34	0.000231053 0	0.000115527		1.00000i	$0.499963\pi$
$150$	0	0
$151$	3.39920e37	0.250773	0.125386	−	0.992108i	$-0.459983\pi$
$151$	3.39920e37	0.250773	0.125386	+	0.992108i	$0.459983\pi$
$152$	0	0
$153$	−1.89566e38	−1.11086
$154$	0	0
$155$	2.41810e38	1.12894
$156$	0	0
$157$	−2.81299e38	−1.04936	−0.524680	−	0.851300i	$-0.675815\pi$
$157$	−2.81299e38	−1.04936	−0.524680	+	0.851300i	$0.675815\pi$
$158$	0	0
$159$	−6.91373e37	−0.206663
$160$	0	0
$161$	−6.06107e38	−1.45578
$162$	0	0
$163$	1.76753e38	0.342045	0.171023	−	0.985267i	$-0.445293\pi$
$163$	1.76753e38	0.342045	0.171023	+	0.985267i	$0.445293\pi$
$164$	0	0
$165$	1.61532e38	0.252516
$166$	0	0
$167$	−4.42192e38	−0.559851	−0.279925	−	0.960022i	$-0.590310\pi$
$167$	−4.42192e38	−0.559851	−0.279925	+	0.960022i	$0.590310\pi$
$168$	0	0
$169$	−9.68354e38	−0.995444
$170$	0	0
$171$	1.35511e39	1.13382
$172$	0	0
$173$	2.81076e39	1.91875	0.959374	−	0.282139i	$-0.0910440\pi$
$173$	2.81076e39	1.91875	0.959374	+	0.282139i	$0.0910440\pi$
$174$	0	0
$175$	1.83249e39	1.02300
$176$	0	0
$177$	−4.01866e38	−0.183886
$178$	0	0
$179$	−3.59826e39	−1.35259	−0.676293	−	0.736633i	$-0.736415\pi$
$179$	−3.59826e39	−1.35259	−0.676293	+	0.736633i	$0.736415\pi$
$180$	0	0
$181$	9.21352e38	0.285135	0.142568	−	0.989785i	$-0.454464\pi$
$181$	9.21352e38	0.285135	0.142568	+	0.989785i	$0.454464\pi$
$182$	0	0
$183$	−1.35415e39	−0.345758
$184$	0	0
$185$	6.69933e39	1.41425
$186$	0	0
$187$	−3.64910e39	−0.638202
$188$	0	0
$189$	−1.05113e40	−1.52607
$190$	0	0
$191$	−7.26257e39	−0.877013	−0.438507	−	0.898728i	$-0.644492\pi$
$191$	−7.26257e39	−0.877013	−0.438507	+	0.898728i	$0.644492\pi$
$192$	0	0
$193$	−1.51495e39	−0.152456	−0.0762281	−	0.997090i	$-0.524288\pi$
$193$	−1.51495e39	−0.152456	−0.0762281	+	0.997090i	$0.524288\pi$
$194$	0	0
$195$	4.38043e38	0.0368054
$196$	0	0
$197$	8.20333e39	0.576543	0.288271	−	0.957549i	$-0.406920\pi$
$197$	8.20333e39	0.576543	0.288271	+	0.957549i	$0.406920\pi$
$198$	0	0
$199$	5.35938e39	0.315635	0.157818	−	0.987468i	$-0.449554\pi$
$199$	5.35938e39	0.315635	0.157818	+	0.987468i	$0.449554\pi$
$200$	0	0
$201$	1.03143e40	0.509926
$202$	0	0
$203$	−3.16715e40	−1.31671
$204$	0	0
$205$	6.23320e40	2.18282
$206$	0	0
$207$	2.06983e40	0.611581
$208$	0	0
$209$	2.60856e40	0.651390
$210$	0	0
$211$	−4.17762e40	−0.883052	−0.441526	−	0.897249i	$-0.645563\pi$
$211$	−4.17762e40	−0.883052	−0.441526	+	0.897249i	$0.645563\pi$
$212$	0	0
$213$	1.40967e40	0.252625
$214$	0	0
$215$	−2.97214e40	−0.452268
$216$	0	0
$217$	−1.35187e41	−1.74940
$218$	0	0
$219$	6.45443e40	0.711354
$220$	0	0
$221$	−9.89565e39	−0.0930207
$222$	0	0
$223$	−1.07086e41	−0.859796	−0.429898	−	0.902877i	$-0.641451\pi$
$223$	−1.07086e41	−0.859796	−0.429898	+	0.902877i	$0.641451\pi$
$224$	0	0
$225$	−6.25790e40	−0.429768
$226$	0	0
$227$	6.99277e40	0.411336	0.205668	−	0.978622i	$-0.434063\pi$
$227$	6.99277e40	0.411336	0.205668	+	0.978622i	$0.434063\pi$
$228$	0	0
$229$	1.01309e41	0.511123	0.255562	−	0.966793i	$-0.417740\pi$
$229$	1.01309e41	0.511123	0.255562	+	0.966793i	$0.417740\pi$
$230$	0	0
$231$	−9.03064e40	−0.391299
$232$	0	0
$233$	−9.80943e40	−0.365523	−0.182762	−	0.983157i	$-0.558504\pi$
$233$	−9.80943e40	−0.365523	−0.182762	+	0.983157i	$0.558504\pi$
$234$	0	0
$235$	3.36223e41	1.07880
$236$	0	0
$237$	2.81532e40	0.0778809
$238$	0	0
$239$	4.72945e41	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$239$	4.72945e41	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$240$	0	0
$241$	6.97733e41	1.44009	0.720047	−	0.693925i	$-0.244120\pi$
$241$	6.97733e41	1.44009	0.720047	+	0.693925i	$0.244120\pi$
$242$	0	0
$243$	5.57705e41	0.996081
$244$	0	0
$245$	−2.14577e42	−3.32030
$246$	0	0
$247$	7.07391e40	0.0949429
$248$	0	0
$249$	−4.93872e41	−0.575606
$250$	0	0
$251$	5.30558e41	0.537580	0.268790	−	0.963199i	$-0.413376\pi$
$251$	5.30558e41	0.537580	0.268790	+	0.963199i	$0.413376\pi$
$252$	0	0
$253$	3.98438e41	0.351360
$254$	0	0
$255$	−9.78051e41	−0.751464
$256$	0	0
$257$	−5.50615e41	−0.368993	−0.184497	−	0.982833i	$-0.559065\pi$
$257$	−5.50615e41	−0.368993	−0.184497	+	0.982833i	$0.559065\pi$
$258$	0	0
$259$	−3.74534e42	−2.19152
$260$	0	0
$261$	1.08157e42	0.553154
$262$	0	0
$263$	3.30715e42	1.47989	0.739943	−	0.672670i	$-0.234852\pi$
$263$	3.30715e42	1.47989	0.739943	+	0.672670i	$0.234852\pi$
$264$	0	0
$265$	1.48264e42	0.581075
$266$	0	0
$267$	2.09101e42	0.718471
$268$	0	0
$269$	−2.48975e42	−0.750742	−0.375371	−	0.926875i	$-0.622485\pi$
$269$	−2.48975e42	−0.750742	−0.375371	+	0.926875i	$0.622485\pi$
$270$	0	0
$271$	2.69421e42	0.713622	0.356811	−	0.934177i	$-0.383864\pi$
$271$	2.69421e42	0.713622	0.356811	+	0.934177i	$0.383864\pi$
$272$	0	0
$273$	−2.44893e41	−0.0570336
$274$	0	0
$275$	−1.20463e42	−0.246907
$276$	0	0
$277$	6.93182e42	1.25156	0.625780	−	0.780000i	$-0.284781\pi$
$277$	6.93182e42	1.25156	0.625780	+	0.780000i	$0.284781\pi$
$278$	0	0
$279$	4.61657e42	0.734930
$280$	0	0
$281$	−3.42638e42	−0.481366	−0.240683	−	0.970604i	$-0.577371\pi$
$281$	−3.42638e42	−0.481366	−0.240683	+	0.970604i	$0.577371\pi$
$282$	0	0
$283$	−6.74711e42	−0.837250	−0.418625	−	0.908159i	$-0.637488\pi$
$283$	−6.74711e42	−0.837250	−0.418625	+	0.908159i	$0.637488\pi$
$284$	0	0
$285$	6.99159e42	0.766993
$286$	0	0
$287$	−3.48474e43	−3.38250
$288$	0	0
$289$	1.04612e43	0.899226
$290$	0	0
$291$	2.46621e40	0.00187890
$292$	0	0
$293$	−1.63253e43	−1.10326	−0.551629	−	0.834090i	$-0.685993\pi$
$293$	−1.63253e43	−1.10326	−0.551629	+	0.834090i	$0.685993\pi$
$294$	0	0
$295$	8.61794e42	0.517033
$296$	0	0
$297$	6.90982e42	0.368323
$298$	0	0
$299$	1.08049e42	0.0512122
$300$	0	0
$301$	1.66161e43	0.700835
$302$	0	0
$303$	−1.34977e42	−0.0507008
$304$	0	0
$305$	2.90395e43	0.972169
$306$	0	0
$307$	−5.40714e42	−0.161453	−0.0807264	−	0.996736i	$-0.525724\pi$
$307$	−5.40714e42	−0.161453	−0.0807264	+	0.996736i	$0.525724\pi$
$308$	0	0
$309$	6.99509e42	0.186432
$310$	0	0
$311$	−6.67143e42	−0.158823	−0.0794113	−	0.996842i	$-0.525304\pi$
$311$	−6.67143e42	−0.158823	−0.0794113	+	0.996842i	$0.525304\pi$
$312$	0	0
$313$	3.71666e42	0.0790910	0.0395455	−	0.999218i	$-0.487409\pi$
$313$	3.71666e42	0.0790910	0.0395455	+	0.999218i	$0.487409\pi$
$314$	0	0
$315$	1.00604e44	1.91505
$316$	0	0
$317$	−5.72779e43	−0.976000	−0.488000	−	0.872844i	$-0.662273\pi$
$317$	−5.72779e43	−0.976000	−0.488000	+	0.872844i	$0.662273\pi$
$318$	0	0
$319$	2.08200e43	0.317793
$320$	0	0
$321$	−5.81737e43	−0.795955
$322$	0	0
$323$	−1.57944e44	−1.93847
$324$	0	0
$325$	−3.26673e42	−0.0359877
$326$	0	0
$327$	4.22829e43	0.418388
$328$	0	0
$329$	−1.87970e44	−1.67171
$330$	0	0
$331$	1.84564e44	1.47624	0.738120	−	0.674669i	$-0.235714\pi$
$331$	1.84564e44	1.47624	0.738120	+	0.674669i	$0.235714\pi$
$332$	0	0
$333$	1.27902e44	0.920667
$334$	0	0
$335$	−2.21187e44	−1.43376
$336$	0	0
$337$	−9.54914e43	−0.557752	−0.278876	−	0.960327i	$-0.589962\pi$
$337$	−9.54914e43	−0.557752	−0.278876	+	0.960327i	$0.589962\pi$
$338$	0	0
$339$	−8.47192e43	−0.446157
$340$	0	0
$341$	8.88681e43	0.422225
$342$	0	0
$343$	7.52255e44	3.22640
$344$	0	0
$345$	1.06791e44	0.413716
$346$	0	0
$347$	1.39215e44	0.487437	0.243719	−	0.969846i	$-0.421633\pi$
$347$	1.39215e44	0.487437	0.243719	+	0.969846i	$0.421633\pi$
$348$	0	0
$349$	−2.64283e44	−0.836805	−0.418403	−	0.908262i	$-0.637410\pi$
$349$	−2.64283e44	−0.836805	−0.418403	+	0.908262i	$0.637410\pi$
$350$	0	0
$351$	1.87381e43	0.0536847
$352$	0	0
$353$	−4.43123e44	−1.14939	−0.574694	−	0.818369i	$-0.694879\pi$
$353$	−4.43123e44	−1.14939	−0.574694	+	0.818369i	$0.694879\pi$
$354$	0	0
$355$	−3.02302e44	−0.710305
$356$	0	0
$357$	5.46791e44	1.16447
$358$	0	0
$359$	−1.31325e44	−0.253627	−0.126813	−	0.991927i	$-0.540475\pi$
$359$	−1.31325e44	−0.253627	−0.126813	+	0.991927i	$0.540475\pi$
$360$	0	0
$361$	5.58406e44	0.978530
$362$	0	0
$363$	−2.17451e44	−0.345936
$364$	0	0
$365$	−1.38414e45	−2.00012
$366$	0	0
$367$	−9.20275e44	−1.20854	−0.604271	−	0.796779i	$-0.706535\pi$
$367$	−9.20275e44	−1.20854	−0.604271	+	0.796779i	$0.706535\pi$
$368$	0	0
$369$	1.19003e45	1.42100
$370$	0	0
$371$	−8.28887e44	−0.900434
$372$	0	0
$373$	−4.53964e44	−0.448866	−0.224433	−	0.974489i	$-0.572053\pi$
$373$	−4.53964e44	−0.448866	−0.224433	+	0.974489i	$0.572053\pi$
$374$	0	0
$375$	2.82702e44	0.254555
$376$	0	0
$377$	5.64598e43	0.0463197
$378$	0	0
$379$	1.70247e45	1.27319	0.636595	−	0.771198i	$-0.280342\pi$
$379$	1.70247e45	1.27319	0.636595	+	0.771198i	$0.280342\pi$
$380$	0	0
$381$	4.16821e44	0.284291
$382$	0	0
$383$	7.11531e43	0.0442808	0.0221404	−	0.999755i	$-0.492952\pi$
$383$	7.11531e43	0.0442808	0.0221404	+	0.999755i	$0.492952\pi$
$384$	0	0
$385$	1.93661e45	1.10022
$386$	0	0
$387$	−5.67433e44	−0.294424
$388$	0	0
$389$	3.46080e45	1.64081	0.820405	−	0.571783i	$-0.193748\pi$
$389$	3.46080e45	1.64081	0.820405	+	0.571783i	$0.193748\pi$
$390$	0	0
$391$	−2.41248e45	−1.04561
$392$	0	0
$393$	−7.92263e44	−0.314051
$394$	0	0
$395$	−6.03740e44	−0.218978
$396$	0	0
$397$	2.48041e45	0.823551	0.411775	−	0.911285i	$-0.364909\pi$
$397$	2.48041e45	0.823551	0.411775	+	0.911285i	$0.364909\pi$
$398$	0	0
$399$	−3.90873e45	−1.18853
$400$	0	0
$401$	−1.15693e45	−0.322315	−0.161158	−	0.986929i	$-0.551523\pi$
$401$	−1.15693e45	−0.322315	−0.161158	+	0.986929i	$0.551523\pi$
$402$	0	0
$403$	2.40993e44	0.0615412
$404$	0	0
$405$	−2.41015e45	−0.564392
$406$	0	0
$407$	2.46209e45	0.528933
$408$	0	0
$409$	−8.50380e45	−1.67670	−0.838350	−	0.545133i	$-0.816479\pi$
$409$	−8.50380e45	−1.67670	−0.838350	+	0.545133i	$0.816479\pi$
$410$	0	0
$411$	−2.27079e45	−0.411098
$412$	0	0
$413$	−4.81797e45	−0.801194
$414$	0	0
$415$	1.05910e46	1.61843
$416$	0	0
$417$	4.13782e45	0.581287
$418$	0	0
$419$	3.54279e45	0.457720	0.228860	−	0.973459i	$-0.426500\pi$
$419$	3.54279e45	0.457720	0.228860	+	0.973459i	$0.426500\pi$
$420$	0	0
$421$	−5.92954e45	−0.704830	−0.352415	−	0.935844i	$-0.614639\pi$
$421$	−5.92954e45	−0.704830	−0.352415	+	0.935844i	$0.614639\pi$
$422$	0	0
$423$	6.41909e45	0.702291
$424$	0	0
$425$	7.29386e45	0.734770
$426$	0	0
$427$	−1.62349e46	−1.50647
$428$	0	0
$429$	1.60986e44	0.0137653
$430$	0	0
$431$	2.37879e46	1.87501	0.937506	−	0.347968i	$-0.113128\pi$
$431$	2.37879e46	1.87501	0.937506	+	0.347968i	$0.113128\pi$
$432$	0	0
$433$	1.29475e46	0.941129	0.470565	−	0.882366i	$-0.344050\pi$
$433$	1.29475e46	0.941129	0.470565	+	0.882366i	$0.344050\pi$
$434$	0	0
$435$	5.58028e45	0.374192
$436$	0	0
$437$	1.72456e46	1.06722
$438$	0	0
$439$	−1.43870e46	−0.821945	−0.410972	−	0.911648i	$-0.634811\pi$
$439$	−1.43870e46	−0.821945	−0.410972	+	0.911648i	$0.634811\pi$
$440$	0	0
$441$	−4.09665e46	−2.16149
$442$	0	0
$443$	−1.49528e45	−0.0728880	−0.0364440	−	0.999336i	$-0.511603\pi$
$443$	−1.49528e45	−0.0728880	−0.0364440	+	0.999336i	$0.511603\pi$
$444$	0	0
$445$	−4.48414e46	−2.02013
$446$	0	0
$447$	2.44298e42	0.000101751 0
$448$	0	0
$449$	8.31879e45	0.320442	0.160221	−	0.987081i	$-0.448779\pi$
$449$	8.31879e45	0.320442	0.160221	+	0.987081i	$0.448779\pi$
$450$	0	0
$451$	2.29078e46	0.816382
$452$	0	0
$453$	3.34830e45	0.110435
$454$	0	0
$455$	5.25170e45	0.160362
$456$	0	0
$457$	−2.23166e46	−0.631093	−0.315547	−	0.948910i	$-0.602188\pi$
$457$	−2.23166e46	−0.631093	−0.315547	+	0.948910i	$0.602188\pi$
$458$	0	0
$459$	−4.18379e46	−1.09609
$460$	0	0
$461$	2.04747e46	0.497108	0.248554	−	0.968618i	$-0.420045\pi$
$461$	2.04747e46	0.497108	0.248554	+	0.968618i	$0.420045\pi$
$462$	0	0
$463$	−7.55079e45	−0.169952	−0.0849758	−	0.996383i	$-0.527081\pi$
$463$	−7.55079e45	−0.169952	−0.0849758	+	0.996383i	$0.527081\pi$
$464$	0	0
$465$	2.38189e46	0.497158
$466$	0	0
$467$	2.70181e46	0.523131	0.261565	−	0.965186i	$-0.415761\pi$
$467$	2.70181e46	0.523131	0.261565	+	0.965186i	$0.415761\pi$
$468$	0	0
$469$	1.23657e47	2.22175
$470$	0	0
$471$	−2.77086e46	−0.462114
$472$	0	0
$473$	−1.09230e46	−0.169150
$474$	0	0
$475$	−5.21402e46	−0.749953
$476$	0	0
$477$	2.83062e46	0.378276
$478$	0	0
$479$	1.17438e47	1.45861	0.729303	−	0.684191i	$-0.239845\pi$
$479$	1.17438e47	1.45861	0.729303	+	0.684191i	$0.239845\pi$
$480$	0	0
$481$	6.67669e45	0.0770944
$482$	0	0
$483$	−5.97030e46	−0.641094
$484$	0	0
$485$	−5.28875e44	−0.00528290
$486$	0	0
$487$	−1.21915e47	−1.13318	−0.566592	−	0.823999i	$-0.691738\pi$
$487$	−1.21915e47	−1.13318	−0.566592	+	0.823999i	$0.691738\pi$
$488$	0	0
$489$	1.74106e46	0.150629
$490$	0	0
$491$	−5.21185e46	−0.419822	−0.209911	−	0.977720i	$-0.567317\pi$
$491$	−5.21185e46	−0.419822	−0.209911	+	0.977720i	$0.567317\pi$
$492$	0	0
$493$	−1.26062e47	−0.945720
$494$	0	0
$495$	−6.61343e46	−0.462206
$496$	0	0
$497$	1.69006e47	1.10069
$498$	0	0
$499$	−2.52314e47	−1.53173	−0.765865	−	0.643001i	$-0.777689\pi$
$499$	−2.52314e47	−1.53173	−0.765865	+	0.643001i	$0.777689\pi$
$500$	0	0
$501$	−4.35570e46	−0.246546
$502$	0	0
$503$	1.27081e47	0.670875	0.335438	−	0.942062i	$-0.391116\pi$
$503$	1.27081e47	0.670875	0.335438	+	0.942062i	$0.391116\pi$
$504$	0	0
$505$	2.89457e46	0.142556
$506$	0	0
$507$	−9.53853e46	−0.438371
$508$	0	0
$509$	−3.25034e47	−1.39434	−0.697168	−	0.716908i	$-0.745557\pi$
$509$	−3.25034e47	−1.39434	−0.697168	+	0.716908i	$0.745557\pi$
$510$	0	0
$511$	7.73821e47	3.09938
$512$	0	0
$513$	2.99078e47	1.11874
$514$	0	0
$515$	−1.50009e47	−0.524191
$516$	0	0
$517$	1.23566e47	0.403474
$518$	0	0
$519$	2.76867e47	0.844973
$520$	0	0
$521$	−3.11762e47	−0.889536	−0.444768	−	0.895646i	$-0.646714\pi$
$521$	−3.11762e47	−0.889536	−0.444768	+	0.895646i	$0.646714\pi$
$522$	0	0
$523$	5.68420e46	0.151666	0.0758332	−	0.997121i	$-0.475838\pi$
$523$	5.68420e46	0.151666	0.0758332	+	0.997121i	$0.475838\pi$
$524$	0	0
$525$	1.80505e47	0.450508
$526$	0	0
$527$	−5.38082e47	−1.25650
$528$	0	0
$529$	−1.94174e47	−0.424342
$530$	0	0
$531$	1.64532e47	0.336585
$532$	0	0
$533$	6.21214e46	0.118991
$534$	0	0
$535$	1.24753e48	2.23799
$536$	0	0
$537$	−3.54438e47	−0.595648
$538$	0	0
$539$	−7.88598e47	−1.24180
$540$	0	0
$541$	−1.28815e48	−1.90115	−0.950574	−	0.310498i	$-0.899504\pi$
$541$	−1.28815e48	−1.90115	−0.950574	+	0.310498i	$0.899504\pi$
$542$	0	0
$543$	9.07555e46	0.125567
$544$	0	0
$545$	−9.06750e47	−1.17638
$546$	0	0
$547$	−9.33750e47	−1.13619	−0.568097	−	0.822962i	$-0.692320\pi$
$547$	−9.33750e47	−1.13619	−0.568097	+	0.822962i	$0.692320\pi$
$548$	0	0
$549$	5.54414e47	0.632876
$550$	0	0
$551$	9.01153e47	0.965263
$552$	0	0
$553$	3.37528e47	0.339328
$554$	0	0
$555$	6.59900e47	0.622803
$556$	0	0
$557$	1.50170e48	1.33081	0.665405	−	0.746483i	$-0.268259\pi$
$557$	1.50170e48	1.33081	0.665405	+	0.746483i	$0.268259\pi$
$558$	0	0
$559$	−2.96210e46	−0.0246543
$560$	0	0
$561$	−3.59446e47	−0.281050
$562$	0	0
$563$	−2.35805e48	−1.73243	−0.866215	−	0.499671i	$-0.833454\pi$
$563$	−2.35805e48	−1.73243	−0.866215	+	0.499671i	$0.833454\pi$
$564$	0	0
$565$	1.81679e48	1.25446
$566$	0	0
$567$	1.34742e48	0.874582
$568$	0	0
$569$	9.82515e47	0.599618	0.299809	−	0.953999i	$-0.403077\pi$
$569$	9.82515e47	0.599618	0.299809	+	0.953999i	$0.403077\pi$
$570$	0	0
$571$	1.91558e48	1.09943	0.549715	−	0.835352i	$-0.314736\pi$
$571$	1.91558e48	1.09943	0.549715	+	0.835352i	$0.314736\pi$
$572$	0	0
$573$	−7.15381e47	−0.386217
$574$	0	0
$575$	−7.96402e47	−0.404525
$576$	0	0
$577$	1.31929e48	0.630613	0.315306	−	0.948990i	$-0.397893\pi$
$577$	1.31929e48	0.630613	0.315306	+	0.948990i	$0.397893\pi$
$578$	0	0
$579$	−1.49226e47	−0.0671383
$580$	0	0
$581$	−5.92102e48	−2.50792
$582$	0	0
$583$	5.44888e47	0.217324
$584$	0	0
$585$	−1.79343e47	−0.0673686
$586$	0	0
$587$	−3.80204e48	−1.34540	−0.672698	−	0.739917i	$-0.734865\pi$
$587$	−3.80204e48	−1.34540	−0.672698	+	0.739917i	$0.734865\pi$
$588$	0	0
$589$	3.84648e48	1.28247
$590$	0	0
$591$	8.08048e47	0.253896
$592$	0	0
$593$	3.11869e48	0.923665	0.461832	−	0.886967i	$-0.347192\pi$
$593$	3.11869e48	0.923665	0.461832	+	0.886967i	$0.347192\pi$
$594$	0	0
$595$	−1.17258e49	−3.27414
$596$	0	0
$597$	5.27912e47	0.138999
$598$	0	0
$599$	3.47614e48	0.863233	0.431617	−	0.902057i	$-0.357943\pi$
$599$	3.47614e48	0.863233	0.431617	+	0.902057i	$0.357943\pi$
$600$	0	0
$601$	9.89423e47	0.231781	0.115891	−	0.993262i	$-0.463028\pi$
$601$	9.89423e47	0.231781	0.115891	+	0.993262i	$0.463028\pi$
$602$	0	0
$603$	−4.22285e48	−0.933368
$604$	0	0
$605$	4.66320e48	0.972668
$606$	0	0
$607$	4.69546e48	0.924434	0.462217	−	0.886767i	$-0.347054\pi$
$607$	4.69546e48	0.924434	0.462217	+	0.886767i	$0.347054\pi$
$608$	0	0
$609$	−3.11972e48	−0.579847
$610$	0	0
$611$	3.35088e47	0.0588081
$612$	0	0
$613$	2.81285e48	0.466217	0.233109	−	0.972451i	$-0.425110\pi$
$613$	2.81285e48	0.466217	0.233109	+	0.972451i	$0.425110\pi$
$614$	0	0
$615$	6.13985e48	0.961266
$616$	0	0
$617$	−7.26687e47	−0.107487	−0.0537437	−	0.998555i	$-0.517115\pi$
$617$	−7.26687e47	−0.107487	−0.0537437	+	0.998555i	$0.517115\pi$
$618$	0	0
$619$	−2.60272e48	−0.363782	−0.181891	−	0.983319i	$-0.558222\pi$
$619$	−2.60272e48	−0.363782	−0.181891	+	0.983319i	$0.558222\pi$
$620$	0	0
$621$	4.56819e48	0.603450
$622$	0	0
$623$	2.50691e49	3.13039
$624$	0	0
$625$	−1.05786e49	−1.24890
$626$	0	0
$627$	2.56950e48	0.286858
$628$	0	0
$629$	−1.49075e49	−1.57405
$630$	0	0
$631$	−1.09091e49	−1.08962	−0.544811	−	0.838559i	$-0.683399\pi$
$631$	−1.09091e49	−1.08962	−0.544811	+	0.838559i	$0.683399\pi$
$632$	0	0
$633$	−4.11506e48	−0.388876
$634$	0	0
$635$	−8.93866e48	−0.799341
$636$	0	0
$637$	−2.13852e48	−0.180998
$638$	0	0
$639$	−5.77148e48	−0.462404
$640$	0	0
$641$	9.10478e48	0.690643	0.345322	−	0.938484i	$-0.387770\pi$
$641$	9.10478e48	0.690643	0.345322	+	0.938484i	$0.387770\pi$
$642$	0	0
$643$	1.48922e48	0.106971	0.0534856	−	0.998569i	$-0.482967\pi$
$643$	1.48922e48	0.106971	0.0534856	+	0.998569i	$0.482967\pi$
$644$	0	0
$645$	−2.92763e48	−0.199169
$646$	0	0
$647$	4.08990e48	0.263565	0.131782	−	0.991279i	$-0.457930\pi$
$647$	4.08990e48	0.263565	0.131782	+	0.991279i	$0.457930\pi$
$648$	0	0
$649$	3.16720e48	0.193372
$650$	0	0
$651$	−1.33162e49	−0.770396
$652$	0	0
$653$	−1.49404e49	−0.819182	−0.409591	−	0.912269i	$-0.634328\pi$
$653$	−1.49404e49	−0.819182	−0.409591	+	0.912269i	$0.634328\pi$
$654$	0	0
$655$	1.69900e49	0.883018
$656$	0	0
$657$	−2.64257e49	−1.30206
$658$	0	0
$659$	3.02435e49	1.41298	0.706492	−	0.707721i	$-0.250277\pi$
$659$	3.02435e49	1.41298	0.706492	+	0.707721i	$0.250277\pi$
$660$	0	0
$661$	−1.45285e49	−0.643719	−0.321859	−	0.946787i	$-0.604308\pi$
$661$	−1.45285e49	−0.643719	−0.321859	+	0.946787i	$0.604308\pi$
$662$	0	0
$663$	−9.74746e47	−0.0409642
$664$	0	0
$665$	8.38221e49	3.34180
$666$	0	0
$667$	1.37644e49	0.520663
$668$	0	0
$669$	−1.05482e49	−0.378635
$670$	0	0
$671$	1.06724e49	0.363594
$672$	0	0
$673$	−2.94583e49	−0.952674	−0.476337	−	0.879263i	$-0.658036\pi$
$673$	−2.94583e49	−0.952674	−0.476337	+	0.879263i	$0.658036\pi$
$674$	0	0
$675$	−1.38114e49	−0.424055
$676$	0	0
$677$	4.88819e48	0.142510	0.0712551	−	0.997458i	$-0.477300\pi$
$677$	4.88819e48	0.142510	0.0712551	+	0.997458i	$0.477300\pi$
$678$	0	0
$679$	2.95674e47	0.00818637
$680$	0	0
$681$	6.88805e48	0.181143
$682$	0	0
$683$	−3.44377e49	−0.860343	−0.430172	−	0.902747i	$-0.641547\pi$
$683$	−3.44377e49	−0.860343	−0.430172	+	0.902747i	$0.641547\pi$
$684$	0	0
$685$	4.86968e49	1.15589
$686$	0	0
$687$	9.97915e48	0.225087
$688$	0	0
$689$	1.47763e48	0.0316759
$690$	0	0
$691$	−7.98436e49	−1.62695	−0.813475	−	0.581600i	$-0.802427\pi$
$691$	−7.98436e49	−1.62695	−0.813475	+	0.581600i	$0.802427\pi$
$692$	0	0
$693$	3.69732e49	0.716235
$694$	0	0
$695$	−8.87348e49	−1.63441
$696$	0	0
$697$	−1.38703e50	−2.42947
$698$	0	0
$699$	−9.66253e48	−0.160968
$700$	0	0
$701$	2.43681e49	0.386151	0.193076	−	0.981184i	$-0.438154\pi$
$701$	2.43681e49	0.386151	0.193076	+	0.981184i	$0.438154\pi$
$702$	0	0
$703$	1.06566e50	1.60658
$704$	0	0
$705$	3.31188e49	0.475079
$706$	0	0
$707$	−1.61824e49	−0.220904
$708$	0	0
$709$	4.35586e49	0.565933	0.282967	−	0.959130i	$-0.408681\pi$
$709$	4.35586e49	0.565933	0.282967	+	0.959130i	$0.408681\pi$
$710$	0	0
$711$	−1.15264e49	−0.142553
$712$	0	0
$713$	5.87521e49	0.691762
$714$	0	0
$715$	−3.45232e48	−0.0387040
$716$	0	0
$717$	4.65863e49	0.497362
$718$	0	0
$719$	−4.73504e49	−0.481469	−0.240734	−	0.970591i	$-0.577388\pi$
$719$	−4.73504e49	−0.481469	−0.240734	+	0.970591i	$0.577388\pi$
$720$	0	0
$721$	8.38641e49	0.812286
$722$	0	0
$723$	6.87284e49	0.634185
$724$	0	0
$725$	−4.16152e49	−0.365879
$726$	0	0
$727$	2.47756e49	0.207574	0.103787	−	0.994600i	$-0.466904\pi$
$727$	2.47756e49	0.207574	0.103787	+	0.994600i	$0.466904\pi$
$728$	0	0
$729$	−2.14916e48	−0.0171608
$730$	0	0
$731$	6.61369e49	0.503373
$732$	0	0
$733$	7.58534e49	0.550371	0.275185	−	0.961391i	$-0.411261\pi$
$733$	7.58534e49	0.550371	0.275185	+	0.961391i	$0.411261\pi$
$734$	0	0
$735$	−2.11364e50	−1.46218
$736$	0	0
$737$	−8.12891e49	−0.536230
$738$	0	0
$739$	8.68078e49	0.546111	0.273056	−	0.961998i	$-0.411966\pi$
$739$	8.68078e49	0.546111	0.273056	+	0.961998i	$0.411966\pi$
$740$	0	0
$741$	6.96797e48	0.0418107
$742$	0	0
$743$	7.20034e49	0.412144	0.206072	−	0.978537i	$-0.433932\pi$
$743$	7.20034e49	0.412144	0.206072	+	0.978537i	$0.433932\pi$
$744$	0	0
$745$	−5.23892e46	−0.000286093 0
$746$	0	0
$747$	2.02201e50	1.05359
$748$	0	0
$749$	−6.97444e50	−3.46799
$750$	0	0
$751$	−2.42999e50	−1.15320	−0.576601	−	0.817026i	$-0.695621\pi$
$751$	−2.42999e50	−1.15320	−0.576601	+	0.817026i	$0.695621\pi$
$752$	0	0
$753$	5.22613e49	0.236738
$754$	0	0
$755$	−7.18036e49	−0.310509
$756$	0	0
$757$	4.27887e48	0.0176665	0.00883326	−	0.999961i	$-0.497188\pi$
$757$	4.27887e48	0.0176665	0.00883326	+	0.999961i	$0.497188\pi$
$758$	0	0
$759$	3.92472e49	0.154731
$760$	0	0
$761$	−2.57272e50	−0.968637	−0.484318	−	0.874892i	$-0.660932\pi$
$761$	−2.57272e50	−0.968637	−0.484318	+	0.874892i	$0.660932\pi$
$762$	0	0
$763$	5.06929e50	1.82292
$764$	0	0
$765$	4.00433e50	1.37548
$766$	0	0
$767$	8.58883e48	0.0281848
$768$	0	0
$769$	2.26037e50	0.708710	0.354355	−	0.935111i	$-0.384700\pi$
$769$	2.26037e50	0.708710	0.354355	+	0.935111i	$0.384700\pi$
$770$	0	0
$771$	−5.42369e49	−0.162496
$772$	0	0
$773$	−1.69135e50	−0.484274	−0.242137	−	0.970242i	$-0.577848\pi$
$773$	−1.69135e50	−0.484274	−0.242137	+	0.970242i	$0.577848\pi$
$774$	0	0
$775$	−1.77630e50	−0.486113
$776$	0	0
$777$	−3.68925e50	−0.965096
$778$	0	0
$779$	9.91518e50	2.47968
$780$	0	0
$781$	−1.11100e50	−0.265656
$782$	0	0
$783$	2.38707e50	0.545800
$784$	0	0
$785$	5.94207e50	1.29933
$786$	0	0
$787$	−1.04783e50	−0.219145	−0.109572	−	0.993979i	$-0.534948\pi$
$787$	−1.04783e50	−0.219145	−0.109572	+	0.993979i	$0.534948\pi$
$788$	0	0
$789$	3.25762e50	0.651709
$790$	0	0
$791$	−1.01570e51	−1.94391
$792$	0	0
$793$	2.89414e49	0.0529954
$794$	0	0
$795$	1.46044e50	0.255893
$796$	0	0
$797$	−3.06687e50	−0.514251	−0.257125	−	0.966378i	$-0.582775\pi$
$797$	−3.06687e50	−0.514251	−0.257125	+	0.966378i	$0.582775\pi$
$798$	0	0
$799$	−7.48174e50	−1.20070
$800$	0	0
$801$	−8.56100e50	−1.31509
$802$	0	0
$803$	−5.08689e50	−0.748050
$804$	0	0
$805$	1.28032e51	1.80256
$806$	0	0
$807$	−2.45246e50	−0.330610
$808$	0	0
$809$	3.87817e50	0.500643	0.250321	−	0.968163i	$-0.419464\pi$
$809$	3.87817e50	0.500643	0.250321	+	0.968163i	$0.419464\pi$
$810$	0	0
$811$	−2.24232e50	−0.277226	−0.138613	−	0.990347i	$-0.544264\pi$
$811$	−2.24232e50	−0.277226	−0.138613	+	0.990347i	$0.544264\pi$
$812$	0	0
$813$	2.65386e50	0.314263
$814$	0	0
$815$	−3.73367e50	−0.423524
$816$	0	0
$817$	−4.72780e50	−0.513775
$818$	0	0
$819$	1.00264e50	0.104394
$820$	0	0
$821$	−1.54029e51	−1.53673	−0.768363	−	0.640014i	$-0.778929\pi$
$821$	−1.54029e51	−1.53673	−0.768363	+	0.640014i	$0.778929\pi$
$822$	0	0
$823$	1.00824e51	0.963975	0.481987	−	0.876178i	$-0.339915\pi$
$823$	1.00824e51	0.963975	0.481987	+	0.876178i	$0.339915\pi$
$824$	0	0
$825$	−1.18659e50	−0.108732
$826$	0	0
$827$	2.58040e50	0.226642	0.113321	−	0.993558i	$-0.463851\pi$
$827$	2.58040e50	0.226642	0.113321	+	0.993558i	$0.463851\pi$
$828$	0	0
$829$	−9.80356e50	−0.825428	−0.412714	−	0.910861i	$-0.635419\pi$
$829$	−9.80356e50	−0.825428	−0.412714	+	0.910861i	$0.635419\pi$
$830$	0	0
$831$	6.82802e50	0.551159
$832$	0	0
$833$	4.77483e51	3.69548
$834$	0	0
$835$	9.34072e50	0.693213
$836$	0	0
$837$	1.01889e51	0.725160
$838$	0	0
$839$	−1.82331e51	−1.24459	−0.622295	−	0.782783i	$-0.713799\pi$
$839$	−1.82331e51	−1.24459	−0.622295	+	0.782783i	$0.713799\pi$
$840$	0	0
$841$	−8.08071e50	−0.529078
$842$	0	0
$843$	−3.37507e50	−0.211983
$844$	0	0
$845$	2.04552e51	1.23257
$846$	0	0
$847$	−2.60702e51	−1.50725
$848$	0	0
$849$	−6.64607e50	−0.368706
$850$	0	0
$851$	1.62772e51	0.866589
$852$	0	0
$853$	8.23841e50	0.420955	0.210477	−	0.977599i	$-0.432498\pi$
$853$	8.23841e50	0.420955	0.210477	+	0.977599i	$0.432498\pi$
$854$	0	0
$855$	−2.86249e51	−1.40390
$856$	0	0
$857$	3.07964e49	0.0144989	0.00724946	−	0.999974i	$-0.497692\pi$
$857$	3.07964e49	0.0144989	0.00724946	+	0.999974i	$0.497692\pi$
$858$	0	0
$859$	−7.47141e50	−0.337693	−0.168847	−	0.985642i	$-0.554004\pi$
$859$	−7.47141e50	−0.337693	−0.168847	+	0.985642i	$0.554004\pi$
$860$	0	0
$861$	−3.43256e51	−1.48958
$862$	0	0
$863$	4.33141e51	1.80485	0.902424	−	0.430848i	$-0.141786\pi$
$863$	4.33141e51	1.80485	0.902424	+	0.430848i	$0.141786\pi$
$864$	0	0
$865$	−5.93737e51	−2.37581
$866$	0	0
$867$	1.03045e51	0.395999
$868$	0	0
$869$	−2.21882e50	−0.0818984
$870$	0	0
$871$	−2.20440e50	−0.0781579
$872$	0	0
$873$	−1.00971e49	−0.00343913
$874$	0	0
$875$	3.38931e51	1.10910
$876$	0	0
$877$	5.86982e50	0.184557	0.0922787	−	0.995733i	$-0.470585\pi$
$877$	5.86982e50	0.184557	0.0922787	+	0.995733i	$0.470585\pi$
$878$	0	0
$879$	−1.60808e51	−0.485850
$880$	0	0
$881$	−5.49134e51	−1.59441	−0.797205	−	0.603709i	$-0.793689\pi$
$881$	−5.49134e51	−1.59441	−0.797205	+	0.603709i	$0.793689\pi$
$882$	0	0
$883$	4.12828e51	1.15201	0.576005	−	0.817446i	$-0.304611\pi$
$883$	4.12828e51	1.15201	0.576005	+	0.817446i	$0.304611\pi$
$884$	0	0
$885$	8.48889e50	0.227690
$886$	0	0
$887$	−3.72893e51	−0.961437	−0.480719	−	0.876875i	$-0.659624\pi$
$887$	−3.72893e51	−0.961437	−0.480719	+	0.876875i	$0.659624\pi$
$888$	0	0
$889$	4.99726e51	1.23866
$890$	0	0
$891$	−8.85760e50	−0.211084
$892$	0	0
$893$	5.34832e51	1.22551
$894$	0	0
$895$	7.60086e51	1.67479
$896$	0	0
$897$	1.06431e50	0.0225527
$898$	0	0
$899$	3.07004e51	0.625675
$900$	0	0
$901$	−3.29921e51	−0.646735
$902$	0	0
$903$	1.63673e51	0.308632
$904$	0	0
$905$	−1.94624e51	−0.353057
$906$	0	0
$907$	−1.06248e52	−1.85435	−0.927177	−	0.374625i	$-0.877772\pi$
$907$	−1.06248e52	−1.85435	−0.927177	+	0.374625i	$0.877772\pi$
$908$	0	0
$909$	5.52623e50	0.0928028
$910$	0	0
$911$	4.69154e51	0.758130	0.379065	−	0.925370i	$-0.376246\pi$
$911$	4.69154e51	0.758130	0.379065	+	0.925370i	$0.376246\pi$
$912$	0	0
$913$	3.89232e51	0.605299
$914$	0	0
$915$	2.86046e51	0.428121
$916$	0	0
$917$	−9.49844e51	−1.36832
$918$	0	0
$919$	2.94881e51	0.408908	0.204454	−	0.978876i	$-0.434458\pi$
$919$	2.94881e51	0.408908	0.204454	+	0.978876i	$0.434458\pi$
$920$	0	0
$921$	−5.32616e50	−0.0711002
$922$	0	0
$923$	−3.01281e50	−0.0387206
$924$	0	0
$925$	−4.92124e51	−0.608967
$926$	0	0
$927$	−2.86392e51	−0.341245
$928$	0	0
$929$	−4.09887e51	−0.470315	−0.235158	−	0.971957i	$-0.575561\pi$
$929$	−4.09887e51	−0.470315	−0.235158	+	0.971957i	$0.575561\pi$
$930$	0	0
$931$	−3.41329e52	−3.77184
$932$	0	0
$933$	−6.57153e50	−0.0699419
$934$	0	0
$935$	7.70825e51	0.790229
$936$	0	0
$937$	1.52781e52	1.50879	0.754393	−	0.656424i	$-0.227932\pi$
$937$	1.52781e52	1.50879	0.754393	+	0.656424i	$0.227932\pi$
$938$	0	0
$939$	3.66100e50	0.0348299
$940$	0	0
$941$	5.67298e51	0.519988	0.259994	−	0.965610i	$-0.416279\pi$
$941$	5.67298e51	0.519988	0.259994	+	0.965610i	$0.416279\pi$
$942$	0	0
$943$	1.51447e52	1.33754
$944$	0	0
$945$	2.22037e52	1.88959
$946$	0	0
$947$	−1.67140e52	−1.37074	−0.685370	−	0.728195i	$-0.740360\pi$
$947$	−1.67140e52	−1.37074	−0.685370	+	0.728195i	$0.740360\pi$
$948$	0	0
$949$	−1.37947e51	−0.109032
$950$	0	0
$951$	−5.64201e51	−0.429809
$952$	0	0
$953$	3.81538e51	0.280164	0.140082	−	0.990140i	$-0.455263\pi$
$953$	3.81538e51	0.280164	0.140082	+	0.990140i	$0.455263\pi$
$954$	0	0
$955$	1.53412e52	1.08593
$956$	0	0
$957$	2.05082e51	0.139949
$958$	0	0
$959$	−2.72245e52	−1.79116
$960$	0	0
$961$	−2.65960e51	−0.168716
$962$	0	0
$963$	2.38174e52	1.45692
$964$	0	0
$965$	3.20013e51	0.188773
$966$	0	0
$967$	2.81440e52	1.60112	0.800558	−	0.599255i	$-0.204537\pi$
$967$	2.81440e52	1.60112	0.800558	+	0.599255i	$0.204537\pi$
$968$	0	0
$969$	−1.55579e52	−0.853660
$970$	0	0
$971$	−1.70192e52	−0.900748	−0.450374	−	0.892840i	$-0.648709\pi$
$971$	−1.70192e52	−0.900748	−0.450374	+	0.892840i	$0.648709\pi$
$972$	0	0
$973$	4.96083e52	2.53267
$974$	0	0
$975$	−3.21781e50	−0.0158482
$976$	0	0
$977$	−4.00155e52	−1.90140	−0.950700	−	0.310114i	$-0.899633\pi$
$977$	−4.00155e52	−1.90140	−0.950700	+	0.310114i	$0.899633\pi$
$978$	0	0
$979$	−1.64798e52	−0.755534
$980$	0	0
$981$	−1.73114e52	−0.765818
$982$	0	0
$983$	1.57112e52	0.670691	0.335345	−	0.942095i	$-0.391147\pi$
$983$	1.57112e52	0.670691	0.335345	+	0.942095i	$0.391147\pi$
$984$	0	0
$985$	−1.73285e52	−0.713881
$986$	0	0
$987$	−1.85155e52	−0.736182
$988$	0	0
$989$	−7.22136e51	−0.277130
$990$	0	0
$991$	−2.36019e52	−0.874296	−0.437148	−	0.899390i	$-0.644011\pi$
$991$	−2.36019e52	−0.874296	−0.437148	+	0.899390i	$0.644011\pi$
$992$	0	0
$993$	1.81800e52	0.650103
$994$	0	0
$995$	−1.13210e52	−0.390823
$996$	0	0
$997$	−7.13164e51	−0.237697	−0.118849	−	0.992912i	$-0.537920\pi$
$997$	−7.13164e51	−0.237697	−0.118849	+	0.992912i	$0.537920\pi$
$998$	0	0
$999$	2.82284e52	0.908428

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4.36.a.a.1.2	✓	3
4.3	odd	2		16.36.a.c.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.36.a.a.1.2	✓	3	1.1	even	1	trivial
16.36.a.c.1.2		3	4.3	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:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	4.a (trivial)

\(f(q)\)	\(=\)	\(q+9.85025e7 q^{3} -2.11237e12 q^{5} +1.18095e15 q^{7} -4.03288e16 q^{9} +O(q^{10})\)	\(q+9.85025e7 q^{3} -2.11237e12 q^{5} +1.18095e15 q^{7} -4.03288e16 q^{9} -7.76322e17 q^{11} -2.10523e18 q^{13} -2.08074e20 q^{15} +4.70050e21 q^{17} -3.36016e22 q^{19} +1.16326e23 q^{21} -5.13239e23 q^{23} +1.55172e24 q^{25} -8.90072e24 q^{27} -2.68188e25 q^{29} -1.14473e26 q^{31} -7.64696e25 q^{33} -2.49459e27 q^{35} -3.17148e27 q^{37} -2.07371e26 q^{39} -2.95081e28 q^{41} +1.40702e28 q^{43} +8.51893e28 q^{45} -1.59169e29 q^{47} +1.01581e30 q^{49} +4.63011e29 q^{51} -7.01884e29 q^{53} +1.63988e30 q^{55} -3.30984e30 q^{57} -4.07975e30 q^{59} -1.37474e31 q^{61} -4.76261e31 q^{63} +4.44703e30 q^{65} +1.04711e32 q^{67} -5.05553e31 q^{69} +1.43111e32 q^{71} +6.55256e32 q^{73} +1.52848e32 q^{75} -9.16793e32 q^{77} +2.85812e32 q^{79} +1.14097e33 q^{81} -5.01380e33 q^{83} -9.92920e33 q^{85} -2.64172e33 q^{87} +2.12280e34 q^{89} -2.48616e33 q^{91} -1.12759e34 q^{93} +7.09789e34 q^{95} +2.50370e32 q^{97} +3.13081e34 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+O(q^{10}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9	\( 3 q + 50908884 q^{3} + 280720890 q^{5} - 5549296289016 q^{7} - 41\!\cdots\!13 q^{9}+ \cdots + 91\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 50908884 * q^3 + 280720890 * q^5 - 5549296289016 * q^7 - 4115694118097313 * q^9 - 420463958869151460 * q^11 - 7370701315414736574 * q^13 + 35737802623493622360 * q^15 + 616802274448822103958 * q^17 - 3523177807465210008348 * q^19 - 35614415089970552016672 * q^21 - 97200569155131383968872 * q^23 - 1397444430311587979767275 * q^25 - 13813263585892101497770872 * q^27 - 72104888547394509330484398 * q^29 - 305306760334252700869662432 * q^31 - 1206118755141042924734246640 * q^33 - 4139993873456897291989313040 * q^35 - 10467805589252853184477978086 * q^37 - 21306312046147482048067771848 * q^39 - 28614809221544804827249675602 * q^41 - 20661331953141752160177135300 * q^43 + 109445534106618058172074201410 * q^45 + 431671100729347757519313060528 * q^47 + 1202271820749265164710734860219 * q^49 + 1812312130969213433267629787496 * q^51 + 2094590682902326557681205190442 * q^53 - 424018895462656971242660883000 * q^55 - 7056738639317962169249569687824 * q^57 - 21898322952557962385462881418676 * q^59 - 46553596324122391351852871731854 * q^61 - 60534256794589553738854966512984 * q^63 - 47301026213190020793582172528260 * q^65 + 75982094975394264303461043768084 * q^67 + 266168516453886472772566159456416 * q^69 + 591751031973420639599491172643144 * q^71 + 840249720714294658198657086588366 * q^73 + 844328642366224046122007927037900 * q^75 + 366867399909875520359123617491360 * q^77 - 1512021839065233136089826558058544 * q^79 - 4715323858342810015899488694567333 * q^81 - 8944825848970974146478286242709212 * q^83 - 11518175153366806194794571426747180 * q^85 - 9434346897547249340232797055364104 * q^87 + 1625382197094823895014829864559198 * q^89 + 28933989451866564710063788057094832 * q^91 + 43351982414907351295090224432607104 * q^93 + 96155354088435090124008638354927160 * q^95 + 83007021412015685382745763811847014 * q^97 + 91112511762343802272300511308928460 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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