LMFDB - Embedded newform 2352.4.a.cr.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.145408.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 24x^{2} + 142 $ x^4 - 24*x^2 + 142
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.25358$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -11.4452 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -11.4452 q^{5} +9.00000 q^{9} +52.5886 q^{11} +5.48395 q^{13} -34.3355 q^{15} -85.3636 q^{17} -110.620 q^{19} +209.332 q^{23} +5.99160 q^{25} +27.0000 q^{27} -132.843 q^{29} -49.3139 q^{31} +157.766 q^{33} +160.351 q^{37} +16.4518 q^{39} -138.120 q^{41} +365.073 q^{43} -103.006 q^{45} +131.370 q^{47} -256.091 q^{51} -561.227 q^{53} -601.885 q^{55} -331.859 q^{57} -436.354 q^{59} +291.961 q^{61} -62.7646 q^{65} +593.683 q^{67} +627.995 q^{69} -775.073 q^{71} -330.907 q^{73} +17.9748 q^{75} -243.473 q^{79} +81.0000 q^{81} +332.477 q^{83} +976.999 q^{85} -398.529 q^{87} +979.115 q^{89} -147.942 q^{93} +1266.06 q^{95} -466.790 q^{97} +473.298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	$ 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} - 152 q^{17} - 224 q^{19} + 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} + 120 q^{33} - 304 q^{37} + 144 q^{39} - 152 q^{41} - 160 q^{43} + 72 q^{45} - 544 q^{47} - 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} + 896 q^{61} - 648 q^{65} + 416 q^{67} + 24 q^{69} - 248 q^{71} - 752 q^{73} - 84 q^{75} - 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} + 2936 q^{89} - 1200 q^{93} + 80 q^{95} + 144 q^{97} + 360 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 - 152 * q^17 - 224 * q^19 + 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 + 120 * q^33 - 304 * q^37 + 144 * q^39 - 152 * q^41 - 160 * q^43 + 72 * q^45 - 544 * q^47 - 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 + 896 * q^61 - 648 * q^65 + 416 * q^67 + 24 * q^69 - 248 * q^71 - 752 * q^73 - 84 * q^75 - 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 + 2936 * q^89 - 1200 * q^93 + 80 * q^95 + 144 * q^97 + 360 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−11.4452	−1.02369	−0.511843	−	0.859079i	$-0.671037\pi$
$5$	−11.4452	−1.02369	−0.511843	+	0.859079i	$0.671037\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	52.5886	1.44146	0.720730	−	0.693216i	$-0.243807\pi$
$11$	52.5886	1.44146	0.720730	+	0.693216i	$0.243807\pi$
$12$	0	0
$13$	5.48395	0.116998	0.0584990	−	0.998287i	$-0.481369\pi$
$13$	5.48395	0.116998	0.0584990	+	0.998287i	$0.481369\pi$
$14$	0	0
$15$	−34.3355	−0.591025
$16$	0	0
$17$	−85.3636	−1.21786	−0.608932	−	0.793222i	$-0.708402\pi$
$17$	−85.3636	−1.21786	−0.608932	+	0.793222i	$0.708402\pi$
$18$	0	0
$19$	−110.620	−1.33568	−0.667839	−	0.744306i	$-0.732780\pi$
$19$	−110.620	−1.33568	−0.667839	+	0.744306i	$0.732780\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	209.332	1.89777	0.948884	−	0.315624i	$-0.102214\pi$
$23$	209.332	1.89777	0.948884	+	0.315624i	$0.102214\pi$
$24$	0	0
$25$	5.99160	0.0479328
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−132.843	−0.850632	−0.425316	−	0.905045i	$-0.639837\pi$
$29$	−132.843	−0.850632	−0.425316	+	0.905045i	$0.639837\pi$
$30$	0	0
$31$	−49.3139	−0.285711	−0.142856	−	0.989744i	$-0.545628\pi$
$31$	−49.3139	−0.285711	−0.142856	+	0.989744i	$0.545628\pi$
$32$	0	0
$33$	157.766	0.832228
$34$	0	0
$35$	0	0
$36$	0	0
$37$	160.351	0.712474	0.356237	−	0.934396i	$-0.384060\pi$
$37$	160.351	0.712474	0.356237	+	0.934396i	$0.384060\pi$
$38$	0	0
$39$	16.4518	0.0675488
$40$	0	0
$41$	−138.120	−0.526116	−0.263058	−	0.964780i	$-0.584731\pi$
$41$	−138.120	−0.526116	−0.263058	+	0.964780i	$0.584731\pi$
$42$	0	0
$43$	365.073	1.29472	0.647362	−	0.762183i	$-0.275872\pi$
$43$	365.073	1.29472	0.647362	+	0.762183i	$0.275872\pi$
$44$	0	0
$45$	−103.006	−0.341229
$46$	0	0
$47$	131.370	0.407708	0.203854	−	0.979001i	$-0.434653\pi$
$47$	131.370	0.407708	0.203854	+	0.979001i	$0.434653\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−256.091	−0.703135
$52$	0	0
$53$	−561.227	−1.45454	−0.727268	−	0.686353i	$-0.759210\pi$
$53$	−561.227	−1.45454	−0.727268	+	0.686353i	$0.759210\pi$
$54$	0	0
$55$	−601.885	−1.47560
$56$	0	0
$57$	−331.859	−0.771154
$58$	0	0
$59$	−436.354	−0.962854	−0.481427	−	0.876486i	$-0.659881\pi$
$59$	−436.354	−0.962854	−0.481427	+	0.876486i	$0.659881\pi$
$60$	0	0
$61$	291.961	0.612816	0.306408	−	0.951900i	$-0.400873\pi$
$61$	291.961	0.612816	0.306408	+	0.951900i	$0.400873\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−62.7646	−0.119769
$66$	0	0
$67$	593.683	1.08254	0.541268	−	0.840850i	$-0.317944\pi$
$67$	593.683	1.08254	0.541268	+	0.840850i	$0.317944\pi$
$68$	0	0
$69$	627.995	1.09568
$70$	0	0
$71$	−775.073	−1.29555	−0.647776	−	0.761831i	$-0.724301\pi$
$71$	−775.073	−1.29555	−0.647776	+	0.761831i	$0.724301\pi$
$72$	0	0
$73$	−330.907	−0.530544	−0.265272	−	0.964174i	$-0.585462\pi$
$73$	−330.907	−0.530544	−0.265272	+	0.964174i	$0.585462\pi$
$74$	0	0
$75$	17.9748	0.0276740
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−243.473	−0.346745	−0.173373	−	0.984856i	$-0.555467\pi$
$79$	−243.473	−0.346745	−0.173373	+	0.984856i	$0.555467\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	332.477	0.439688	0.219844	−	0.975535i	$-0.429445\pi$
$83$	332.477	0.439688	0.219844	+	0.975535i	$0.429445\pi$
$84$	0	0
$85$	976.999	1.24671
$86$	0	0
$87$	−398.529	−0.491113
$88$	0	0
$89$	979.115	1.16614	0.583068	−	0.812424i	$-0.301852\pi$
$89$	979.115	1.16614	0.583068	+	0.812424i	$0.301852\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−147.942	−0.164955
$94$	0	0
$95$	1266.06	1.36731
$96$	0	0
$97$	−466.790	−0.488612	−0.244306	−	0.969698i	$-0.578560\pi$
$97$	−466.790	−0.488612	−0.244306	+	0.969698i	$0.578560\pi$
$98$	0	0
$99$	473.298	0.480487
$100$	0	0
$101$	−866.429	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$101$	−866.429	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$102$	0	0
$103$	926.755	0.886562	0.443281	−	0.896383i	$-0.353814\pi$
$103$	926.755	0.886562	0.443281	+	0.896383i	$0.353814\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1046.38	−0.945395	−0.472698	−	0.881225i	$-0.656720\pi$
$107$	−1046.38	−0.945395	−0.472698	+	0.881225i	$0.656720\pi$
$108$	0	0
$109$	−486.397	−0.427416	−0.213708	−	0.976898i	$-0.568554\pi$
$109$	−486.397	−0.427416	−0.213708	+	0.976898i	$0.568554\pi$
$110$	0	0
$111$	481.053	0.411347
$112$	0	0
$113$	1955.03	1.62755	0.813777	−	0.581178i	$-0.197408\pi$
$113$	1955.03	1.62755	0.813777	+	0.581178i	$0.197408\pi$
$114$	0	0
$115$	−2395.83	−1.94272
$116$	0	0
$117$	49.3555	0.0389993
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1434.56	1.07781
$122$	0	0
$123$	−414.360	−0.303753
$124$	0	0
$125$	1362.07	0.974618
$126$	0	0
$127$	−1255.53	−0.877246	−0.438623	−	0.898671i	$-0.644534\pi$
$127$	−1255.53	−0.877246	−0.438623	+	0.898671i	$0.644534\pi$
$128$	0	0
$129$	1095.22	0.747509
$130$	0	0
$131$	−1029.41	−0.686561	−0.343281	−	0.939233i	$-0.611538\pi$
$131$	−1029.41	−0.686561	−0.343281	+	0.939233i	$0.611538\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−309.019	−0.197008
$136$	0	0
$137$	−1931.36	−1.20443	−0.602217	−	0.798332i	$-0.705716\pi$
$137$	−1931.36	−1.20443	−0.602217	+	0.798332i	$0.705716\pi$
$138$	0	0
$139$	−2926.04	−1.78549	−0.892747	−	0.450559i	$-0.851225\pi$
$139$	−2926.04	−1.78549	−0.892747	+	0.450559i	$0.851225\pi$
$140$	0	0
$141$	394.110	0.235390
$142$	0	0
$143$	288.393	0.168648
$144$	0	0
$145$	1520.41	0.870780
$146$	0	0
$147$	0	0
$148$	0	0
$149$	131.809	0.0724712	0.0362356	−	0.999343i	$-0.488463\pi$
$149$	131.809	0.0724712	0.0362356	+	0.999343i	$0.488463\pi$
$150$	0	0
$151$	−177.956	−0.0959066	−0.0479533	−	0.998850i	$-0.515270\pi$
$151$	−177.956	−0.0959066	−0.0479533	+	0.998850i	$0.515270\pi$
$152$	0	0
$153$	−768.272	−0.405955
$154$	0	0
$155$	564.406	0.292479
$156$	0	0
$157$	3607.47	1.83381	0.916903	−	0.399111i	$-0.130681\pi$
$157$	3607.47	1.83381	0.916903	+	0.399111i	$0.130681\pi$
$158$	0	0
$159$	−1683.68	−0.839777
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1540.88	−0.740438	−0.370219	−	0.928945i	$-0.620717\pi$
$163$	−1540.88	−0.740438	−0.370219	+	0.928945i	$0.620717\pi$
$164$	0	0
$165$	−1805.66	−0.851940
$166$	0	0
$167$	761.798	0.352992	0.176496	−	0.984301i	$-0.443524\pi$
$167$	761.798	0.352992	0.176496	+	0.984301i	$0.443524\pi$
$168$	0	0
$169$	−2166.93	−0.986311
$170$	0	0
$171$	−995.576	−0.445226
$172$	0	0
$173$	2691.32	1.18276	0.591379	−	0.806394i	$-0.298584\pi$
$173$	2691.32	1.18276	0.591379	+	0.806394i	$0.298584\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1309.06	−0.555904
$178$	0	0
$179$	−1123.25	−0.469027	−0.234513	−	0.972113i	$-0.575350\pi$
$179$	−1123.25	−0.469027	−0.234513	+	0.972113i	$0.575350\pi$
$180$	0	0
$181$	−4221.85	−1.73374	−0.866872	−	0.498531i	$-0.833873\pi$
$181$	−4221.85	−1.73374	−0.866872	+	0.498531i	$0.833873\pi$
$182$	0	0
$183$	875.882	0.353809
$184$	0	0
$185$	−1835.24	−0.729350
$186$	0	0
$187$	−4489.15	−1.75550
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3293.81	−1.24781	−0.623904	−	0.781501i	$-0.714455\pi$
$191$	−3293.81	−1.24781	−0.623904	+	0.781501i	$0.714455\pi$
$192$	0	0
$193$	−4623.91	−1.72454	−0.862270	−	0.506448i	$-0.830958\pi$
$193$	−4623.91	−1.72454	−0.862270	+	0.506448i	$0.830958\pi$
$194$	0	0
$195$	−188.294	−0.0691487
$196$	0	0
$197$	2644.18	0.956293	0.478147	−	0.878280i	$-0.341309\pi$
$197$	2644.18	0.956293	0.478147	+	0.878280i	$0.341309\pi$
$198$	0	0
$199$	−1382.33	−0.492417	−0.246209	−	0.969217i	$-0.579185\pi$
$199$	−1382.33	−0.492417	−0.246209	+	0.969217i	$0.579185\pi$
$200$	0	0
$201$	1781.05	0.625002
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1580.81	0.538577
$206$	0	0
$207$	1883.99	0.632590
$208$	0	0
$209$	−5817.33	−1.92533
$210$	0	0
$211$	−4533.28	−1.47907	−0.739535	−	0.673118i	$-0.764955\pi$
$211$	−4533.28	−1.47907	−0.739535	+	0.673118i	$0.764955\pi$
$212$	0	0
$213$	−2325.22	−0.747987
$214$	0	0
$215$	−4178.32	−1.32539
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−992.721	−0.306310
$220$	0	0
$221$	−468.129	−0.142488
$222$	0	0
$223$	−3052.46	−0.916628	−0.458314	−	0.888790i	$-0.651547\pi$
$223$	−3052.46	−0.916628	−0.458314	+	0.888790i	$0.651547\pi$
$224$	0	0
$225$	53.9244	0.0159776
$226$	0	0
$227$	−5194.86	−1.51892	−0.759461	−	0.650553i	$-0.774537\pi$
$227$	−5194.86	−1.51892	−0.759461	+	0.650553i	$0.774537\pi$
$228$	0	0
$229$	−5497.05	−1.58627	−0.793133	−	0.609048i	$-0.791552\pi$
$229$	−5497.05	−1.58627	−0.793133	+	0.609048i	$0.791552\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5457.17	−1.53438	−0.767191	−	0.641419i	$-0.778346\pi$
$233$	−5457.17	−1.53438	−0.767191	+	0.641419i	$0.778346\pi$
$234$	0	0
$235$	−1503.55	−0.417365
$236$	0	0
$237$	−730.420	−0.200194
$238$	0	0
$239$	5622.17	1.52162	0.760812	−	0.648972i	$-0.224801\pi$
$239$	5622.17	1.52162	0.760812	+	0.648972i	$0.224801\pi$
$240$	0	0
$241$	−4083.02	−1.09133	−0.545665	−	0.838003i	$-0.683723\pi$
$241$	−4083.02	−1.09133	−0.545665	+	0.838003i	$0.683723\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−606.632	−0.156271
$248$	0	0
$249$	997.430	0.253854
$250$	0	0
$251$	−1751.99	−0.440576	−0.220288	−	0.975435i	$-0.570700\pi$
$251$	−1751.99	−0.440576	−0.220288	+	0.975435i	$0.570700\pi$
$252$	0	0
$253$	11008.5	2.73556
$254$	0	0
$255$	2931.00	0.719789
$256$	0	0
$257$	3825.26	0.928457	0.464228	−	0.885716i	$-0.346332\pi$
$257$	3825.26	0.928457	0.464228	+	0.885716i	$0.346332\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1195.59	−0.283544
$262$	0	0
$263$	5272.57	1.23620	0.618099	−	0.786100i	$-0.287903\pi$
$263$	5272.57	1.23620	0.618099	+	0.786100i	$0.287903\pi$
$264$	0	0
$265$	6423.33	1.48899
$266$	0	0
$267$	2937.35	0.673268
$268$	0	0
$269$	4535.04	1.02790	0.513952	−	0.857819i	$-0.328181\pi$
$269$	4535.04	1.02790	0.513952	+	0.857819i	$0.328181\pi$
$270$	0	0
$271$	−1439.10	−0.322580	−0.161290	−	0.986907i	$-0.551565\pi$
$271$	−1439.10	−0.322580	−0.161290	+	0.986907i	$0.551565\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	315.090	0.0690932
$276$	0	0
$277$	−5984.85	−1.29818	−0.649088	−	0.760713i	$-0.724849\pi$
$277$	−5984.85	−1.29818	−0.649088	+	0.760713i	$0.724849\pi$
$278$	0	0
$279$	−443.826	−0.0952371
$280$	0	0
$281$	−2647.87	−0.562131	−0.281066	−	0.959689i	$-0.590688\pi$
$281$	−2647.87	−0.562131	−0.281066	+	0.959689i	$0.590688\pi$
$282$	0	0
$283$	−400.822	−0.0841922	−0.0420961	−	0.999114i	$-0.513404\pi$
$283$	−400.822	−0.0841922	−0.0420961	+	0.999114i	$0.513404\pi$
$284$	0	0
$285$	3798.17	0.789419
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2373.94	0.483195
$290$	0	0
$291$	−1400.37	−0.282100
$292$	0	0
$293$	5307.12	1.05817	0.529087	−	0.848567i	$-0.322534\pi$
$293$	5307.12	1.05817	0.529087	+	0.848567i	$0.322534\pi$
$294$	0	0
$295$	4994.14	0.985660
$296$	0	0
$297$	1419.89	0.277409
$298$	0	0
$299$	1147.96	0.222035
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2599.29	−0.492822
$304$	0	0
$305$	−3341.54	−0.627331
$306$	0	0
$307$	4130.45	0.767873	0.383936	−	0.923360i	$-0.374568\pi$
$307$	4130.45	0.767873	0.383936	+	0.923360i	$0.374568\pi$
$308$	0	0
$309$	2780.26	0.511857
$310$	0	0
$311$	−7653.87	−1.39553	−0.697767	−	0.716325i	$-0.745823\pi$
$311$	−7653.87	−1.39553	−0.697767	+	0.716325i	$0.745823\pi$
$312$	0	0
$313$	−8546.27	−1.54333	−0.771667	−	0.636026i	$-0.780577\pi$
$313$	−8546.27	−1.54333	−0.771667	+	0.636026i	$0.780577\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7643.65	1.35429	0.677145	−	0.735850i	$-0.263217\pi$
$317$	7643.65	1.35429	0.677145	+	0.735850i	$0.263217\pi$
$318$	0	0
$319$	−6986.03	−1.22615
$320$	0	0
$321$	−3139.14	−0.545824
$322$	0	0
$323$	9442.88	1.62667
$324$	0	0
$325$	32.8576	0.00560804
$326$	0	0
$327$	−1459.19	−0.246769
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7550.18	−1.25376	−0.626881	−	0.779115i	$-0.715669\pi$
$331$	−7550.18	−1.25376	−0.626881	+	0.779115i	$0.715669\pi$
$332$	0	0
$333$	1443.16	0.237491
$334$	0	0
$335$	−6794.79	−1.10818
$336$	0	0
$337$	9391.92	1.51813	0.759066	−	0.651014i	$-0.225656\pi$
$337$	9391.92	1.51813	0.759066	+	0.651014i	$0.225656\pi$
$338$	0	0
$339$	5865.08	0.939668
$340$	0	0
$341$	−2593.35	−0.411841
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7187.50	−1.12163
$346$	0	0
$347$	−1005.66	−0.155580	−0.0777902	−	0.996970i	$-0.524786\pi$
$347$	−1005.66	−0.155580	−0.0777902	+	0.996970i	$0.524786\pi$
$348$	0	0
$349$	3109.75	0.476966	0.238483	−	0.971147i	$-0.423350\pi$
$349$	3109.75	0.476966	0.238483	+	0.971147i	$0.423350\pi$
$350$	0	0
$351$	148.067	0.0225163
$352$	0	0
$353$	−7449.10	−1.12316	−0.561580	−	0.827423i	$-0.689806\pi$
$353$	−7449.10	−1.12316	−0.561580	+	0.827423i	$0.689806\pi$
$354$	0	0
$355$	8870.83	1.32624
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6427.61	0.944948	0.472474	−	0.881344i	$-0.343361\pi$
$359$	6427.61	0.944948	0.472474	+	0.881344i	$0.343361\pi$
$360$	0	0
$361$	5377.69	0.784034
$362$	0	0
$363$	4303.69	0.622273
$364$	0	0
$365$	3787.28	0.543111
$366$	0	0
$367$	−10767.3	−1.53147	−0.765734	−	0.643157i	$-0.777624\pi$
$367$	−10767.3	−1.53147	−0.765734	+	0.643157i	$0.777624\pi$
$368$	0	0
$369$	−1243.08	−0.175372
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3601.03	−0.499877	−0.249938	−	0.968262i	$-0.580410\pi$
$373$	−3601.03	−0.499877	−0.249938	+	0.968262i	$0.580410\pi$
$374$	0	0
$375$	4086.21	0.562696
$376$	0	0
$377$	−728.504	−0.0995222
$378$	0	0
$379$	−6832.02	−0.925956	−0.462978	−	0.886370i	$-0.653219\pi$
$379$	−6832.02	−0.925956	−0.462978	+	0.886370i	$0.653219\pi$
$380$	0	0
$381$	−3766.59	−0.506478
$382$	0	0
$383$	3339.87	0.445586	0.222793	−	0.974866i	$-0.428483\pi$
$383$	3339.87	0.445586	0.222793	+	0.974866i	$0.428483\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3285.66	0.431575
$388$	0	0
$389$	4701.31	0.612765	0.306383	−	0.951908i	$-0.400881\pi$
$389$	4701.31	0.612765	0.306383	+	0.951908i	$0.400881\pi$
$390$	0	0
$391$	−17869.3	−2.31123
$392$	0	0
$393$	−3088.22	−0.396386
$394$	0	0
$395$	2786.59	0.354958
$396$	0	0
$397$	−11715.8	−1.48110	−0.740552	−	0.671999i	$-0.765436\pi$
$397$	−11715.8	−1.48110	−0.740552	+	0.671999i	$0.765436\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1643.72	−0.204697	−0.102349	−	0.994749i	$-0.532636\pi$
$401$	−1643.72	−0.204697	−0.102349	+	0.994749i	$0.532636\pi$
$402$	0	0
$403$	−270.435	−0.0334276
$404$	0	0
$405$	−927.058	−0.113743
$406$	0	0
$407$	8432.64	1.02700
$408$	0	0
$409$	−12206.6	−1.47574	−0.737869	−	0.674944i	$-0.764168\pi$
$409$	−12206.6	−1.47574	−0.737869	+	0.674944i	$0.764168\pi$
$410$	0	0
$411$	−5794.09	−0.695380
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3805.25	−0.450102
$416$	0	0
$417$	−8778.12	−1.03086
$418$	0	0
$419$	11189.5	1.30464	0.652319	−	0.757945i	$-0.273796\pi$
$419$	11189.5	1.30464	0.652319	+	0.757945i	$0.273796\pi$
$420$	0	0
$421$	2718.12	0.314663	0.157331	−	0.987546i	$-0.449711\pi$
$421$	2718.12	0.314663	0.157331	+	0.987546i	$0.449711\pi$
$422$	0	0
$423$	1182.33	0.135903
$424$	0	0
$425$	−511.464	−0.0583757
$426$	0	0
$427$	0	0
$428$	0	0
$429$	865.179	0.0973689
$430$	0	0
$431$	−11085.2	−1.23888	−0.619439	−	0.785045i	$-0.712640\pi$
$431$	−11085.2	−1.23888	−0.619439	+	0.785045i	$0.712640\pi$
$432$	0	0
$433$	−10880.6	−1.20759	−0.603795	−	0.797140i	$-0.706345\pi$
$433$	−10880.6	−1.20759	−0.603795	+	0.797140i	$0.706345\pi$
$434$	0	0
$435$	4561.23	0.502745
$436$	0	0
$437$	−23156.2	−2.53481
$438$	0	0
$439$	14902.3	1.62015	0.810075	−	0.586326i	$-0.199426\pi$
$439$	14902.3	1.62015	0.810075	+	0.586326i	$0.199426\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12504.3	1.34108	0.670538	−	0.741875i	$-0.266063\pi$
$443$	12504.3	1.34108	0.670538	+	0.741875i	$0.266063\pi$
$444$	0	0
$445$	−11206.1	−1.19376
$446$	0	0
$447$	395.427	0.0418413
$448$	0	0
$449$	−1612.21	−0.169454	−0.0847268	−	0.996404i	$-0.527002\pi$
$449$	−1612.21	−0.169454	−0.0847268	+	0.996404i	$0.527002\pi$
$450$	0	0
$451$	−7263.55	−0.758375
$452$	0	0
$453$	−533.869	−0.0553717
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1275.51	0.130560	0.0652800	−	0.997867i	$-0.479206\pi$
$457$	1275.51	0.130560	0.0652800	+	0.997867i	$0.479206\pi$
$458$	0	0
$459$	−2304.82	−0.234378
$460$	0	0
$461$	−9267.92	−0.936335	−0.468167	−	0.883640i	$-0.655086\pi$
$461$	−9267.92	−0.936335	−0.468167	+	0.883640i	$0.655086\pi$
$462$	0	0
$463$	12615.3	1.26627	0.633136	−	0.774041i	$-0.281767\pi$
$463$	12615.3	1.26627	0.633136	+	0.774041i	$0.281767\pi$
$464$	0	0
$465$	1693.22	0.168863
$466$	0	0
$467$	−14407.1	−1.42758	−0.713790	−	0.700359i	$-0.753023\pi$
$467$	−14407.1	−1.42758	−0.713790	+	0.700359i	$0.753023\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	10822.4	1.05875
$472$	0	0
$473$	19198.7	1.86629
$474$	0	0
$475$	−662.788	−0.0640228
$476$	0	0
$477$	−5051.04	−0.484845
$478$	0	0
$479$	−284.296	−0.0271186	−0.0135593	−	0.999908i	$-0.504316\pi$
$479$	−284.296	−0.0271186	−0.0135593	+	0.999908i	$0.504316\pi$
$480$	0	0
$481$	879.356	0.0833580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5342.49	0.500185
$486$	0	0
$487$	−10535.7	−0.980323	−0.490161	−	0.871632i	$-0.663062\pi$
$487$	−10535.7	−0.980323	−0.490161	+	0.871632i	$0.663062\pi$
$488$	0	0
$489$	−4622.65	−0.427492
$490$	0	0
$491$	14230.4	1.30796	0.653982	−	0.756510i	$-0.273097\pi$
$491$	14230.4	1.30796	0.653982	+	0.756510i	$0.273097\pi$
$492$	0	0
$493$	11340.0	1.03596
$494$	0	0
$495$	−5416.97	−0.491868
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9796.09	0.878824	0.439412	−	0.898286i	$-0.355187\pi$
$499$	9796.09	0.878824	0.439412	+	0.898286i	$0.355187\pi$
$500$	0	0
$501$	2285.39	0.203800
$502$	0	0
$503$	7285.16	0.645784	0.322892	−	0.946436i	$-0.395345\pi$
$503$	7285.16	0.645784	0.322892	+	0.946436i	$0.395345\pi$
$504$	0	0
$505$	9916.41	0.873811
$506$	0	0
$507$	−6500.78	−0.569447
$508$	0	0
$509$	13269.6	1.15553	0.577764	−	0.816204i	$-0.303925\pi$
$509$	13269.6	1.15553	0.577764	+	0.816204i	$0.303925\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2986.73	−0.257051
$514$	0	0
$515$	−10606.9	−0.907561
$516$	0	0
$517$	6908.56	0.587695
$518$	0	0
$519$	8073.95	0.682865
$520$	0	0
$521$	−14711.8	−1.23711	−0.618556	−	0.785740i	$-0.712282\pi$
$521$	−14711.8	−1.23711	−0.618556	+	0.785740i	$0.712282\pi$
$522$	0	0
$523$	11004.9	0.920093	0.460046	−	0.887895i	$-0.347833\pi$
$523$	11004.9	0.920093	0.460046	+	0.887895i	$0.347833\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4209.61	0.347958
$528$	0	0
$529$	31652.8	2.60153
$530$	0	0
$531$	−3927.18	−0.320951
$532$	0	0
$533$	−757.443	−0.0615544
$534$	0	0
$535$	11976.0	0.967788
$536$	0	0
$537$	−3369.75	−0.270793
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8884.91	−0.706085	−0.353042	−	0.935607i	$-0.614853\pi$
$541$	−8884.91	−0.706085	−0.353042	+	0.935607i	$0.614853\pi$
$542$	0	0
$543$	−12665.5	−1.00098
$544$	0	0
$545$	5566.89	0.437540
$546$	0	0
$547$	−13986.6	−1.09328	−0.546639	−	0.837368i	$-0.684093\pi$
$547$	−13986.6	−1.09328	−0.546639	+	0.837368i	$0.684093\pi$
$548$	0	0
$549$	2627.65	0.204272
$550$	0	0
$551$	14695.0	1.13617
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5505.73	−0.421090
$556$	0	0
$557$	−11252.8	−0.856006	−0.428003	−	0.903777i	$-0.640783\pi$
$557$	−11252.8	−0.856006	−0.428003	+	0.903777i	$0.640783\pi$
$558$	0	0
$559$	2002.04	0.151480
$560$	0	0
$561$	−13467.5	−1.01354
$562$	0	0
$563$	−13559.4	−1.01503	−0.507514	−	0.861643i	$-0.669435\pi$
$563$	−13559.4	−1.01503	−0.507514	+	0.861643i	$0.669435\pi$
$564$	0	0
$565$	−22375.6	−1.66610
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17208.6	−1.26788	−0.633940	−	0.773382i	$-0.718563\pi$
$569$	−17208.6	−1.26788	−0.633940	+	0.773382i	$0.718563\pi$
$570$	0	0
$571$	−12910.3	−0.946196	−0.473098	−	0.881010i	$-0.656864\pi$
$571$	−12910.3	−0.946196	−0.473098	+	0.881010i	$0.656864\pi$
$572$	0	0
$573$	−9881.42	−0.720423
$574$	0	0
$575$	1254.23	0.0909654
$576$	0	0
$577$	16154.8	1.16557	0.582783	−	0.812628i	$-0.301964\pi$
$577$	16154.8	1.16557	0.582783	+	0.812628i	$0.301964\pi$
$578$	0	0
$579$	−13871.7	−0.995664
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−29514.1	−2.09666
$584$	0	0
$585$	−564.881	−0.0399230
$586$	0	0
$587$	3403.75	0.239332	0.119666	−	0.992814i	$-0.461818\pi$
$587$	3403.75	0.239332	0.119666	+	0.992814i	$0.461818\pi$
$588$	0	0
$589$	5455.09	0.381618
$590$	0	0
$591$	7932.53	0.552116
$592$	0	0
$593$	−12884.9	−0.892272	−0.446136	−	0.894965i	$-0.647200\pi$
$593$	−12884.9	−0.892272	−0.446136	+	0.894965i	$0.647200\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4147.00	−0.284297
$598$	0	0
$599$	19002.4	1.29619	0.648095	−	0.761559i	$-0.275566\pi$
$599$	19002.4	1.29619	0.648095	+	0.761559i	$0.275566\pi$
$600$	0	0
$601$	−8159.94	−0.553829	−0.276914	−	0.960895i	$-0.589312\pi$
$601$	−8159.94	−0.553829	−0.276914	+	0.960895i	$0.589312\pi$
$602$	0	0
$603$	5343.15	0.360845
$604$	0	0
$605$	−16418.8	−1.10334
$606$	0	0
$607$	5304.02	0.354668	0.177334	−	0.984151i	$-0.443253\pi$
$607$	5304.02	0.354668	0.177334	+	0.984151i	$0.443253\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	720.425	0.0477009
$612$	0	0
$613$	14871.9	0.979886	0.489943	−	0.871754i	$-0.337018\pi$
$613$	14871.9	0.979886	0.489943	+	0.871754i	$0.337018\pi$
$614$	0	0
$615$	4742.42	0.310948
$616$	0	0
$617$	−6214.11	−0.405463	−0.202732	−	0.979234i	$-0.564982\pi$
$617$	−6214.11	−0.405463	−0.202732	+	0.979234i	$0.564982\pi$
$618$	0	0
$619$	26716.2	1.73475	0.867377	−	0.497652i	$-0.165804\pi$
$619$	26716.2	1.73475	0.867377	+	0.497652i	$0.165804\pi$
$620$	0	0
$621$	5651.96	0.365226
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−16338.1	−1.04564
$626$	0	0
$627$	−17452.0	−1.11159
$628$	0	0
$629$	−13688.1	−0.867697
$630$	0	0
$631$	1788.18	0.112815	0.0564074	−	0.998408i	$-0.482035\pi$
$631$	1788.18	0.112815	0.0564074	+	0.998408i	$0.482035\pi$
$632$	0	0
$633$	−13599.8	−0.853942
$634$	0	0
$635$	14369.7	0.898025
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6975.65	−0.431851
$640$	0	0
$641$	−4845.85	−0.298595	−0.149298	−	0.988792i	$-0.547701\pi$
$641$	−4845.85	−0.298595	−0.149298	+	0.988792i	$0.547701\pi$
$642$	0	0
$643$	−8154.19	−0.500109	−0.250054	−	0.968232i	$-0.580449\pi$
$643$	−8154.19	−0.500109	−0.250054	+	0.968232i	$0.580449\pi$
$644$	0	0
$645$	−12535.0	−0.765215
$646$	0	0
$647$	14514.1	0.881928	0.440964	−	0.897525i	$-0.354637\pi$
$647$	14514.1	0.881928	0.440964	+	0.897525i	$0.354637\pi$
$648$	0	0
$649$	−22947.2	−1.38792
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−14194.4	−0.850641	−0.425321	−	0.905043i	$-0.639839\pi$
$653$	−14194.4	−0.850641	−0.425321	+	0.905043i	$0.639839\pi$
$654$	0	0
$655$	11781.7	0.702823
$656$	0	0
$657$	−2978.16	−0.176848
$658$	0	0
$659$	−24762.7	−1.46376	−0.731879	−	0.681435i	$-0.761356\pi$
$659$	−24762.7	−1.46376	−0.731879	+	0.681435i	$0.761356\pi$
$660$	0	0
$661$	23482.5	1.38179	0.690897	−	0.722954i	$-0.257216\pi$
$661$	23482.5	1.38179	0.690897	+	0.722954i	$0.257216\pi$
$662$	0	0
$663$	−1404.39	−0.0822653
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−27808.3	−1.61430
$668$	0	0
$669$	−9157.39	−0.529216
$670$	0	0
$671$	15353.8	0.883350
$672$	0	0
$673$	12943.0	0.741333	0.370666	−	0.928766i	$-0.379129\pi$
$673$	12943.0	0.741333	0.370666	+	0.928766i	$0.379129\pi$
$674$	0	0
$675$	161.773	0.00922467
$676$	0	0
$677$	25409.8	1.44251	0.721255	−	0.692669i	$-0.243565\pi$
$677$	25409.8	1.44251	0.721255	+	0.692669i	$0.243565\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15584.6	−0.876950
$682$	0	0
$683$	−5747.89	−0.322016	−0.161008	−	0.986953i	$-0.551474\pi$
$683$	−5747.89	−0.322016	−0.161008	+	0.986953i	$0.551474\pi$
$684$	0	0
$685$	22104.7	1.23296
$686$	0	0
$687$	−16491.1	−0.915831
$688$	0	0
$689$	−3077.74	−0.170178
$690$	0	0
$691$	30578.5	1.68344	0.841722	−	0.539911i	$-0.181542\pi$
$691$	30578.5	1.68344	0.841722	+	0.539911i	$0.181542\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33489.0	1.82778
$696$	0	0
$697$	11790.4	0.640738
$698$	0	0
$699$	−16371.5	−0.885876
$700$	0	0
$701$	11378.9	0.613088	0.306544	−	0.951857i	$-0.400827\pi$
$701$	11378.9	0.613088	0.306544	+	0.951857i	$0.400827\pi$
$702$	0	0
$703$	−17738.0	−0.951636
$704$	0	0
$705$	−4510.65	−0.240966
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16050.7	0.850208	0.425104	−	0.905145i	$-0.360238\pi$
$709$	16050.7	0.850208	0.425104	+	0.905145i	$0.360238\pi$
$710$	0	0
$711$	−2191.26	−0.115582
$712$	0	0
$713$	−10323.0	−0.542214
$714$	0	0
$715$	−3300.70	−0.172642
$716$	0	0
$717$	16866.5	0.878510
$718$	0	0
$719$	10628.6	0.551296	0.275648	−	0.961259i	$-0.411108\pi$
$719$	10628.6	0.551296	0.275648	+	0.961259i	$0.411108\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12249.1	−0.630080
$724$	0	0
$725$	−795.943	−0.0407732
$726$	0	0
$727$	−16912.8	−0.862810	−0.431405	−	0.902158i	$-0.641982\pi$
$727$	−16912.8	−0.862810	−0.431405	+	0.902158i	$0.641982\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−31163.9	−1.57680
$732$	0	0
$733$	32129.7	1.61901	0.809506	−	0.587111i	$-0.199735\pi$
$733$	32129.7	1.61901	0.809506	+	0.587111i	$0.199735\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	31221.0	1.56043
$738$	0	0
$739$	−21286.3	−1.05958	−0.529791	−	0.848128i	$-0.677730\pi$
$739$	−21286.3	−1.05958	−0.529791	+	0.848128i	$0.677730\pi$
$740$	0	0
$741$	−1819.90	−0.0902234
$742$	0	0
$743$	−16441.2	−0.811804	−0.405902	−	0.913917i	$-0.633043\pi$
$743$	−16441.2	−0.811804	−0.405902	+	0.913917i	$0.633043\pi$
$744$	0	0
$745$	−1508.57	−0.0741877
$746$	0	0
$747$	2992.29	0.146563
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−844.181	−0.0410181	−0.0205091	−	0.999790i	$-0.506529\pi$
$751$	−844.181	−0.0410181	−0.0205091	+	0.999790i	$0.506529\pi$
$752$	0	0
$753$	−5255.97	−0.254367
$754$	0	0
$755$	2036.74	0.0981782
$756$	0	0
$757$	−13607.9	−0.653352	−0.326676	−	0.945136i	$-0.605929\pi$
$757$	−13607.9	−0.653352	−0.326676	+	0.945136i	$0.605929\pi$
$758$	0	0
$759$	33025.4	1.57938
$760$	0	0
$761$	−23150.7	−1.10277	−0.551387	−	0.834249i	$-0.685901\pi$
$761$	−23150.7	−1.10277	−0.551387	+	0.834249i	$0.685901\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8792.99	0.415570
$766$	0	0
$767$	−2392.94	−0.112652
$768$	0	0
$769$	39732.0	1.86316	0.931581	−	0.363534i	$-0.118430\pi$
$769$	39732.0	1.86316	0.931581	+	0.363534i	$0.118430\pi$
$770$	0	0
$771$	11475.8	0.536045
$772$	0	0
$773$	4151.21	0.193155	0.0965773	−	0.995325i	$-0.469210\pi$
$773$	4151.21	0.193155	0.0965773	+	0.995325i	$0.469210\pi$
$774$	0	0
$775$	−295.469	−0.0136949
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15278.8	0.702721
$780$	0	0
$781$	−40760.0	−1.86749
$782$	0	0
$783$	−3586.76	−0.163704
$784$	0	0
$785$	−41288.1	−1.87724
$786$	0	0
$787$	1290.94	0.0584714	0.0292357	−	0.999573i	$-0.490693\pi$
$787$	1290.94	0.0584714	0.0292357	+	0.999573i	$0.490693\pi$
$788$	0	0
$789$	15817.7	0.713720
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1601.10	0.0716981
$794$	0	0
$795$	19270.0	0.859668
$796$	0	0
$797$	−4469.79	−0.198655	−0.0993276	−	0.995055i	$-0.531669\pi$
$797$	−4469.79	−0.198655	−0.0993276	+	0.995055i	$0.531669\pi$
$798$	0	0
$799$	−11214.2	−0.496533
$800$	0	0
$801$	8812.04	0.388712
$802$	0	0
$803$	−17401.9	−0.764759
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13605.1	0.593460
$808$	0	0
$809$	29777.8	1.29411	0.647054	−	0.762445i	$-0.276001\pi$
$809$	29777.8	1.29411	0.647054	+	0.762445i	$0.276001\pi$
$810$	0	0
$811$	4083.89	0.176825	0.0884124	−	0.996084i	$-0.471821\pi$
$811$	4083.89	0.176825	0.0884124	+	0.996084i	$0.471821\pi$
$812$	0	0
$813$	−4317.31	−0.186242
$814$	0	0
$815$	17635.7	0.757975
$816$	0	0
$817$	−40384.2	−1.72933
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16909.2	0.718799	0.359399	−	0.933184i	$-0.382982\pi$
$821$	16909.2	0.718799	0.359399	+	0.933184i	$0.382982\pi$
$822$	0	0
$823$	294.522	0.0124743	0.00623717	−	0.999981i	$-0.498015\pi$
$823$	294.522	0.0124743	0.00623717	+	0.999981i	$0.498015\pi$
$824$	0	0
$825$	945.270	0.0398910
$826$	0	0
$827$	43346.8	1.82263	0.911315	−	0.411711i	$-0.135069\pi$
$827$	43346.8	1.82263	0.911315	+	0.411711i	$0.135069\pi$
$828$	0	0
$829$	3542.74	0.148425	0.0742126	−	0.997242i	$-0.476356\pi$
$829$	3542.74	0.148425	0.0742126	+	0.997242i	$0.476356\pi$
$830$	0	0
$831$	−17954.6	−0.749503
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8718.89	−0.361353
$836$	0	0
$837$	−1331.48	−0.0549851
$838$	0	0
$839$	−26513.0	−1.09098	−0.545489	−	0.838118i	$-0.683656\pi$
$839$	−26513.0	−1.09098	−0.545489	+	0.838118i	$0.683656\pi$
$840$	0	0
$841$	−6741.72	−0.276425
$842$	0	0
$843$	−7943.62	−0.324547
$844$	0	0
$845$	24800.8	1.00967
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1202.47	−0.0486084
$850$	0	0
$851$	33566.6	1.35211
$852$	0	0
$853$	−31158.3	−1.25069	−0.625345	−	0.780348i	$-0.715042\pi$
$853$	−31158.3	−1.25069	−0.625345	+	0.780348i	$0.715042\pi$
$854$	0	0
$855$	11394.5	0.455771
$856$	0	0
$857$	−22611.4	−0.901271	−0.450635	−	0.892708i	$-0.648803\pi$
$857$	−22611.4	−0.901271	−0.450635	+	0.892708i	$0.648803\pi$
$858$	0	0
$859$	−17135.6	−0.680627	−0.340313	−	0.940312i	$-0.610533\pi$
$859$	−17135.6	−0.680627	−0.340313	+	0.940312i	$0.610533\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31801.8	1.25440	0.627199	−	0.778859i	$-0.284201\pi$
$863$	31801.8	1.25440	0.627199	+	0.778859i	$0.284201\pi$
$864$	0	0
$865$	−30802.5	−1.21077
$866$	0	0
$867$	7121.81	0.278973
$868$	0	0
$869$	−12803.9	−0.499820
$870$	0	0
$871$	3255.72	0.126654
$872$	0	0
$873$	−4201.11	−0.162871
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−23693.0	−0.912265	−0.456133	−	0.889912i	$-0.650766\pi$
$877$	−23693.0	−0.912265	−0.456133	+	0.889912i	$0.650766\pi$
$878$	0	0
$879$	15921.4	0.610938
$880$	0	0
$881$	35584.4	1.36080	0.680402	−	0.732839i	$-0.261805\pi$
$881$	35584.4	1.36080	0.680402	+	0.732839i	$0.261805\pi$
$882$	0	0
$883$	−31928.8	−1.21686	−0.608431	−	0.793607i	$-0.708201\pi$
$883$	−31928.8	−1.21686	−0.608431	+	0.793607i	$0.708201\pi$
$884$	0	0
$885$	14982.4	0.569071
$886$	0	0
$887$	−24218.6	−0.916778	−0.458389	−	0.888752i	$-0.651573\pi$
$887$	−24218.6	−0.916778	−0.458389	+	0.888752i	$0.651573\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4259.68	0.160162
$892$	0	0
$893$	−14532.1	−0.544566
$894$	0	0
$895$	12855.8	0.480136
$896$	0	0
$897$	3443.89	0.128192
$898$	0	0
$899$	6551.02	0.243035
$900$	0	0
$901$	47908.3	1.77143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	48319.7	1.77481
$906$	0	0
$907$	37779.4	1.38307	0.691535	−	0.722343i	$-0.256935\pi$
$907$	37779.4	1.38307	0.691535	+	0.722343i	$0.256935\pi$
$908$	0	0
$909$	−7797.86	−0.284531
$910$	0	0
$911$	−33419.0	−1.21539	−0.607695	−	0.794170i	$-0.707906\pi$
$911$	−33419.0	−1.21539	−0.607695	+	0.794170i	$0.707906\pi$
$912$	0	0
$913$	17484.5	0.633792
$914$	0	0
$915$	−10024.6	−0.362190
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−15771.4	−0.566105	−0.283053	−	0.959104i	$-0.591347\pi$
$919$	−15771.4	−0.566105	−0.283053	+	0.959104i	$0.591347\pi$
$920$	0	0
$921$	12391.3	0.443332
$922$	0	0
$923$	−4250.46	−0.151577
$924$	0	0
$925$	960.759	0.0341509
$926$	0	0
$927$	8340.79	0.295521
$928$	0	0
$929$	44216.4	1.56157	0.780783	−	0.624802i	$-0.214820\pi$
$929$	44216.4	1.56157	0.780783	+	0.624802i	$0.214820\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22961.6	−0.805712
$934$	0	0
$935$	51379.1	1.79709
$936$	0	0
$937$	−8691.91	−0.303044	−0.151522	−	0.988454i	$-0.548417\pi$
$937$	−8691.91	−0.303044	−0.151522	+	0.988454i	$0.548417\pi$
$938$	0	0
$939$	−25638.8	−0.891045
$940$	0	0
$941$	11824.1	0.409622	0.204811	−	0.978802i	$-0.434342\pi$
$941$	11824.1	0.409622	0.204811	+	0.978802i	$0.434342\pi$
$942$	0	0
$943$	−28912.9	−0.998446
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4468.88	−0.153346	−0.0766732	−	0.997056i	$-0.524430\pi$
$947$	−4468.88	−0.153346	−0.0766732	+	0.997056i	$0.524430\pi$
$948$	0	0
$949$	−1814.68	−0.0620726
$950$	0	0
$951$	22930.9	0.781900
$952$	0	0
$953$	18172.0	0.617680	0.308840	−	0.951114i	$-0.400059\pi$
$953$	18172.0	0.617680	0.308840	+	0.951114i	$0.400059\pi$
$954$	0	0
$955$	37698.1	1.27736
$956$	0	0
$957$	−20958.1	−0.707920
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27359.1	−0.918369
$962$	0	0
$963$	−9417.41	−0.315132
$964$	0	0
$965$	52921.4	1.76539
$966$	0	0
$967$	−53267.4	−1.77142	−0.885710	−	0.464239i	$-0.846328\pi$
$967$	−53267.4	−1.77142	−0.885710	+	0.464239i	$0.846328\pi$
$968$	0	0
$969$	28328.6	0.939161
$970$	0	0
$971$	14899.8	0.492439	0.246220	−	0.969214i	$-0.420812\pi$
$971$	14899.8	0.492439	0.246220	+	0.969214i	$0.420812\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	98.5728	0.00323780
$976$	0	0
$977$	50912.6	1.66718	0.833592	−	0.552380i	$-0.186280\pi$
$977$	50912.6	1.66718	0.833592	+	0.552380i	$0.186280\pi$
$978$	0	0
$979$	51490.3	1.68094
$980$	0	0
$981$	−4377.57	−0.142472
$982$	0	0
$983$	−49617.9	−1.60993	−0.804967	−	0.593320i	$-0.797817\pi$
$983$	−49617.9	−1.60993	−0.804967	+	0.593320i	$0.797817\pi$
$984$	0	0
$985$	−30263.0	−0.978944
$986$	0	0
$987$	0	0
$988$	0	0
$989$	76421.4	2.45709
$990$	0	0
$991$	16435.7	0.526839	0.263419	−	0.964681i	$-0.415150\pi$
$991$	16435.7	0.526839	0.263419	+	0.964681i	$0.415150\pi$
$992$	0	0
$993$	−22650.5	−0.723860
$994$	0	0
$995$	15821.0	0.504080
$996$	0	0
$997$	−634.476	−0.0201545	−0.0100773	−	0.999949i	$-0.503208\pi$
$997$	−634.476	−0.0201545	−0.0100773	+	0.999949i	$0.503208\pi$
$998$	0	0
$999$	4329.48	0.137116

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.cr.1.1		4
4.3	odd	2		1176.4.a.bb.1.1	✓	4
7.6	odd	2		2352.4.a.ck.1.4		4
28.27	even	2		1176.4.a.bc.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.bb.1.1	✓	4	4.3	odd	2
1176.4.a.bc.1.4	yes	4	28.27	even	2
2352.4.a.ck.1.4		4	7.6	odd	2
2352.4.a.cr.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -11.4452 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -11.4452 q^{5} +9.00000 q^{9} +52.5886 q^{11} +5.48395 q^{13} -34.3355 q^{15} -85.3636 q^{17} -110.620 q^{19} +209.332 q^{23} +5.99160 q^{25} +27.0000 q^{27} -132.843 q^{29} -49.3139 q^{31} +157.766 q^{33} +160.351 q^{37} +16.4518 q^{39} -138.120 q^{41} +365.073 q^{43} -103.006 q^{45} +131.370 q^{47} -256.091 q^{51} -561.227 q^{53} -601.885 q^{55} -331.859 q^{57} -436.354 q^{59} +291.961 q^{61} -62.7646 q^{65} +593.683 q^{67} +627.995 q^{69} -775.073 q^{71} -330.907 q^{73} +17.9748 q^{75} -243.473 q^{79} +81.0000 q^{81} +332.477 q^{83} +976.999 q^{85} -398.529 q^{87} +979.115 q^{89} -147.942 q^{93} +1266.06 q^{95} -466.790 q^{97} +473.298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9	\( 4 q + 12 q^{3} + 8 q^{5} + 36 q^{9} + 40 q^{11} + 48 q^{13} + 24 q^{15} - 152 q^{17} - 224 q^{19} + 8 q^{23} - 28 q^{25} + 108 q^{27} - 144 q^{29} - 400 q^{31} + 120 q^{33} - 304 q^{37} + 144 q^{39} - 152 q^{41} - 160 q^{43} + 72 q^{45} - 544 q^{47} - 456 q^{51} - 1320 q^{53} - 16 q^{55} - 672 q^{57} - 1040 q^{59} + 896 q^{61} - 648 q^{65} + 416 q^{67} + 24 q^{69} - 248 q^{71} - 752 q^{73} - 84 q^{75} - 864 q^{79} + 324 q^{81} - 1456 q^{83} - 1608 q^{85} - 432 q^{87} + 2936 q^{89} - 1200 q^{93} + 80 q^{95} + 144 q^{97} + 360 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 8 * q^5 + 36 * q^9 + 40 * q^11 + 48 * q^13 + 24 * q^15 - 152 * q^17 - 224 * q^19 + 8 * q^23 - 28 * q^25 + 108 * q^27 - 144 * q^29 - 400 * q^31 + 120 * q^33 - 304 * q^37 + 144 * q^39 - 152 * q^41 - 160 * q^43 + 72 * q^45 - 544 * q^47 - 456 * q^51 - 1320 * q^53 - 16 * q^55 - 672 * q^57 - 1040 * q^59 + 896 * q^61 - 648 * q^65 + 416 * q^67 + 24 * q^69 - 248 * q^71 - 752 * q^73 - 84 * q^75 - 864 * q^79 + 324 * q^81 - 1456 * q^83 - 1608 * q^85 - 432 * q^87 + 2936 * q^89 - 1200 * q^93 + 80 * q^95 + 144 * q^97 + 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more