LMFDB - Embedded newform 6240.2.a.ce.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.32973$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +4.06562 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +4.06562 q^{7} +1.00000 q^{9} -1.71694 q^{11} +1.00000 q^{13} -1.00000 q^{15} +6.06562 q^{17} -4.46365 q^{19} -4.06562 q^{21} +4.06562 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.46365 q^{29} +1.71694 q^{33} +4.06562 q^{35} +0.283061 q^{37} -1.00000 q^{39} +3.71694 q^{41} +6.81233 q^{43} +1.00000 q^{45} +9.52927 q^{49} -6.06562 q^{51} +7.09539 q^{53} -1.71694 q^{55} +4.46365 q^{57} -8.69736 q^{59} -7.84818 q^{61} +4.06562 q^{63} +1.00000 q^{65} +6.24621 q^{67} -4.06562 q^{69} -3.03585 q^{71} +8.34868 q^{73} -1.00000 q^{75} -6.98042 q^{77} +3.60197 q^{79} +1.00000 q^{81} +10.2462 q^{83} +6.06562 q^{85} +2.46365 q^{87} +5.60197 q^{89} +4.06562 q^{91} -4.46365 q^{95} -2.86168 q^{97} -1.71694 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.06562	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$7$	4.06562	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.71694	−0.517677	−0.258838	−	0.965921i	$-0.583340\pi$
$11$	−1.71694	−0.517677	−0.258838	+	0.965921i	$0.583340\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.06562	1.47113	0.735565	−	0.677455i	$-0.236917\pi$
$17$	6.06562	1.47113	0.735565	+	0.677455i	$0.236917\pi$
$18$	0	0
$19$	−4.46365	−1.02403	−0.512016	−	0.858976i	$-0.671101\pi$
$19$	−4.46365	−1.02403	−0.512016	+	0.858976i	$0.671101\pi$
$20$	0	0
$21$	−4.06562	−0.887191
$22$	0	0
$23$	4.06562	0.847741	0.423870	−	0.905723i	$-0.360671\pi$
$23$	4.06562	0.847741	0.423870	+	0.905723i	$0.360671\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.46365	−0.457489	−0.228744	−	0.973487i	$-0.573462\pi$
$29$	−2.46365	−0.457489	−0.228744	+	0.973487i	$0.573462\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.71694	0.298881
$34$	0	0
$35$	4.06562	0.687215
$36$	0	0
$37$	0.283061	0.0465349	0.0232675	−	0.999729i	$-0.492593\pi$
$37$	0.283061	0.0465349	0.0232675	+	0.999729i	$0.492593\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	3.71694	0.580488	0.290244	−	0.956953i	$-0.406263\pi$
$41$	3.71694	0.580488	0.290244	+	0.956953i	$0.406263\pi$
$42$	0	0
$43$	6.81233	1.03887	0.519436	−	0.854510i	$-0.326142\pi$
$43$	6.81233	1.03887	0.519436	+	0.854510i	$0.326142\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	9.52927	1.36132
$50$	0	0
$51$	−6.06562	−0.849357
$52$	0	0
$53$	7.09539	0.974627	0.487314	−	0.873227i	$-0.337977\pi$
$53$	7.09539	0.974627	0.487314	+	0.873227i	$0.337977\pi$
$54$	0	0
$55$	−1.71694	−0.231512
$56$	0	0
$57$	4.46365	0.591225
$58$	0	0
$59$	−8.69736	−1.13230	−0.566150	−	0.824302i	$-0.691568\pi$
$59$	−8.69736	−1.13230	−0.566150	+	0.824302i	$0.691568\pi$
$60$	0	0
$61$	−7.84818	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$61$	−7.84818	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$62$	0	0
$63$	4.06562	0.512220
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	6.24621	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$67$	6.24621	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$68$	0	0
$69$	−4.06562	−0.489443
$70$	0	0
$71$	−3.03585	−0.360289	−0.180144	−	0.983640i	$-0.557657\pi$
$71$	−3.03585	−0.360289	−0.180144	+	0.983640i	$0.557657\pi$
$72$	0	0
$73$	8.34868	0.977139	0.488570	−	0.872525i	$-0.337519\pi$
$73$	8.34868	0.977139	0.488570	+	0.872525i	$0.337519\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−6.98042	−0.795493
$78$	0	0
$79$	3.60197	0.405253	0.202627	−	0.979256i	$-0.435052\pi$
$79$	3.60197	0.405253	0.202627	+	0.979256i	$0.435052\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.2462	1.12467	0.562334	−	0.826910i	$-0.309904\pi$
$83$	10.2462	1.12467	0.562334	+	0.826910i	$0.309904\pi$
$84$	0	0
$85$	6.06562	0.657909
$86$	0	0
$87$	2.46365	0.264131
$88$	0	0
$89$	5.60197	0.593808	0.296904	−	0.954907i	$-0.404046\pi$
$89$	5.60197	0.593808	0.296904	+	0.954907i	$0.404046\pi$
$90$	0	0
$91$	4.06562	0.426193
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.46365	−0.457961
$96$	0	0
$97$	−2.86168	−0.290560	−0.145280	−	0.989391i	$-0.546408\pi$
$97$	−2.86168	−0.290560	−0.145280	+	0.989391i	$0.546408\pi$
$98$	0	0
$99$	−1.71694	−0.172559
$100$	0	0
$101$	9.10147	0.905630	0.452815	−	0.891604i	$-0.350420\pi$
$101$	9.10147	0.905630	0.452815	+	0.891604i	$0.350420\pi$
$102$	0	0
$103$	−3.37845	−0.332889	−0.166444	−	0.986051i	$-0.553229\pi$
$103$	−3.37845	−0.332889	−0.166444	+	0.986051i	$0.553229\pi$
$104$	0	0
$105$	−4.06562	−0.396764
$106$	0	0
$107$	−7.21036	−0.697052	−0.348526	−	0.937299i	$-0.613318\pi$
$107$	−7.21036	−0.697052	−0.348526	+	0.937299i	$0.613318\pi$
$108$	0	0
$109$	−3.78256	−0.362304	−0.181152	−	0.983455i	$-0.557983\pi$
$109$	−3.78256	−0.362304	−0.181152	+	0.983455i	$0.557983\pi$
$110$	0	0
$111$	−0.283061	−0.0268669
$112$	0	0
$113$	10.9865	1.03352	0.516761	−	0.856129i	$-0.327137\pi$
$113$	10.9865	1.03352	0.516761	+	0.856129i	$0.327137\pi$
$114$	0	0
$115$	4.06562	0.379121
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	24.6605	2.26063
$120$	0	0
$121$	−8.05212	−0.732011
$122$	0	0
$123$	−3.71694	−0.335145
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.0163	−1.24374	−0.621871	−	0.783119i	$-0.713627\pi$
$127$	−14.0163	−1.24374	−0.621871	+	0.783119i	$0.713627\pi$
$128$	0	0
$129$	−6.81233	−0.599792
$130$	0	0
$131$	−2.34868	−0.205205	−0.102603	−	0.994722i	$-0.532717\pi$
$131$	−2.34868	−0.205205	−0.102603	+	0.994722i	$0.532717\pi$
$132$	0	0
$133$	−18.1475	−1.57359
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	0.681091	0.0581896	0.0290948	−	0.999577i	$-0.490738\pi$
$137$	0.681091	0.0581896	0.0290948	+	0.999577i	$0.490738\pi$
$138$	0	0
$139$	−8.92088	−0.756659	−0.378330	−	0.925671i	$-0.623501\pi$
$139$	−8.92088	−0.756659	−0.378330	+	0.925671i	$0.623501\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.71694	−0.143578
$144$	0	0
$145$	−2.46365	−0.204595
$146$	0	0
$147$	−9.52927	−0.785961
$148$	0	0
$149$	13.9077	1.13937	0.569683	−	0.821865i	$-0.307066\pi$
$149$	13.9077	1.13937	0.569683	+	0.821865i	$0.307066\pi$
$150$	0	0
$151$	−16.2625	−1.32342	−0.661711	−	0.749759i	$-0.730169\pi$
$151$	−16.2625	−1.32342	−0.661711	+	0.749759i	$0.730169\pi$
$152$	0	0
$153$	6.06562	0.490376
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−12.9436	−1.03301	−0.516505	−	0.856284i	$-0.672767\pi$
$157$	−12.9436	−1.03301	−0.516505	+	0.856284i	$0.672767\pi$
$158$	0	0
$159$	−7.09539	−0.562701
$160$	0	0
$161$	16.5293	1.30269
$162$	0	0
$163$	7.73321	0.605712	0.302856	−	0.953036i	$-0.402060\pi$
$163$	7.73321	0.605712	0.302856	+	0.953036i	$0.402060\pi$
$164$	0	0
$165$	1.71694	0.133664
$166$	0	0
$167$	−18.3220	−1.41780	−0.708901	−	0.705308i	$-0.750809\pi$
$167$	−18.3220	−1.41780	−0.708901	+	0.705308i	$0.750809\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.46365	−0.341344
$172$	0	0
$173$	12.9436	0.984082	0.492041	−	0.870572i	$-0.336251\pi$
$173$	12.9436	0.984082	0.492041	+	0.870572i	$0.336251\pi$
$174$	0	0
$175$	4.06562	0.307332
$176$	0	0
$177$	8.69736	0.653734
$178$	0	0
$179$	9.55262	0.713996	0.356998	−	0.934105i	$-0.383800\pi$
$179$	9.55262	0.713996	0.356998	+	0.934105i	$0.383800\pi$
$180$	0	0
$181$	21.1117	1.56922	0.784609	−	0.619991i	$-0.212864\pi$
$181$	21.1117	1.56922	0.784609	+	0.619991i	$0.212864\pi$
$182$	0	0
$183$	7.84818	0.580154
$184$	0	0
$185$	0.283061	0.0208110
$186$	0	0
$187$	−10.4143	−0.761569
$188$	0	0
$189$	−4.06562	−0.295730
$190$	0	0
$191$	−19.6964	−1.42518	−0.712589	−	0.701581i	$-0.752478\pi$
$191$	−19.6964	−1.42518	−0.712589	+	0.701581i	$0.752478\pi$
$192$	0	0
$193$	9.49950	0.683789	0.341894	−	0.939738i	$-0.388931\pi$
$193$	9.49950	0.683789	0.341894	+	0.939738i	$0.388931\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	16.9436	1.20718	0.603590	−	0.797295i	$-0.293736\pi$
$197$	16.9436	1.20718	0.603590	+	0.797295i	$0.293736\pi$
$198$	0	0
$199$	−18.5087	−1.31205	−0.656023	−	0.754741i	$-0.727763\pi$
$199$	−18.5087	−1.31205	−0.656023	+	0.754741i	$0.727763\pi$
$200$	0	0
$201$	−6.24621	−0.440574
$202$	0	0
$203$	−10.0163	−0.703004
$204$	0	0
$205$	3.71694	0.259602
$206$	0	0
$207$	4.06562	0.282580
$208$	0	0
$209$	7.66382	0.530117
$210$	0	0
$211$	−10.8678	−0.748167	−0.374084	−	0.927395i	$-0.622043\pi$
$211$	−10.8678	−0.748167	−0.374084	+	0.927395i	$0.622043\pi$
$212$	0	0
$213$	3.03585	0.208013
$214$	0	0
$215$	6.81233	0.464597
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.34868	−0.564152
$220$	0	0
$221$	6.06562	0.408018
$222$	0	0
$223$	19.9733	1.33751	0.668757	−	0.743481i	$-0.266827\pi$
$223$	19.9733	1.33751	0.668757	+	0.743481i	$0.266827\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−3.20394	−0.212653	−0.106326	−	0.994331i	$-0.533909\pi$
$227$	−3.20394	−0.212653	−0.106326	+	0.994331i	$0.533909\pi$
$228$	0	0
$229$	27.3477	1.80719	0.903593	−	0.428392i	$-0.140920\pi$
$229$	27.3477	1.80719	0.903593	+	0.428392i	$0.140920\pi$
$230$	0	0
$231$	6.98042	0.459278
$232$	0	0
$233$	−13.6307	−0.892980	−0.446490	−	0.894789i	$-0.647326\pi$
$233$	−13.6307	−0.892980	−0.446490	+	0.894789i	$0.647326\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.60197	−0.233973
$238$	0	0
$239$	19.2104	1.24262	0.621308	−	0.783567i	$-0.286602\pi$
$239$	19.2104	1.24262	0.621308	+	0.783567i	$0.286602\pi$
$240$	0	0
$241$	−7.05854	−0.454681	−0.227340	−	0.973815i	$-0.573003\pi$
$241$	−7.05854	−0.454681	−0.227340	+	0.973815i	$0.573003\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	9.52927	0.608803
$246$	0	0
$247$	−4.46365	−0.284015
$248$	0	0
$249$	−10.2462	−0.649327
$250$	0	0
$251$	−13.9138	−0.878231	−0.439116	−	0.898431i	$-0.644708\pi$
$251$	−13.9138	−0.878231	−0.439116	+	0.898431i	$0.644708\pi$
$252$	0	0
$253$	−6.98042	−0.438855
$254$	0	0
$255$	−6.06562	−0.379844
$256$	0	0
$257$	21.9733	1.37066	0.685330	−	0.728233i	$-0.259658\pi$
$257$	21.9733	1.37066	0.685330	+	0.728233i	$0.259658\pi$
$258$	0	0
$259$	1.15082	0.0715083
$260$	0	0
$261$	−2.46365	−0.152496
$262$	0	0
$263$	30.6112	1.88757	0.943783	−	0.330567i	$-0.107240\pi$
$263$	30.6112	1.88757	0.943783	+	0.330567i	$0.107240\pi$
$264$	0	0
$265$	7.09539	0.435867
$266$	0	0
$267$	−5.60197	−0.342835
$268$	0	0
$269$	−31.4235	−1.91592	−0.957962	−	0.286894i	$-0.907377\pi$
$269$	−31.4235	−1.91592	−0.957962	+	0.286894i	$0.907377\pi$
$270$	0	0
$271$	−12.8286	−0.779282	−0.389641	−	0.920967i	$-0.627401\pi$
$271$	−12.8286	−0.779282	−0.389641	+	0.920967i	$0.627401\pi$
$272$	0	0
$273$	−4.06562	−0.246063
$274$	0	0
$275$	−1.71694	−0.103535
$276$	0	0
$277$	−13.7396	−0.825535	−0.412767	−	0.910836i	$-0.635438\pi$
$277$	−13.7396	−0.825535	−0.412767	+	0.910836i	$0.635438\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.0163	1.90993	0.954965	−	0.296717i	$-0.0958919\pi$
$281$	32.0163	1.90993	0.954965	+	0.296717i	$0.0958919\pi$
$282$	0	0
$283$	16.6541	0.989983	0.494991	−	0.868898i	$-0.335171\pi$
$283$	16.6541	0.989983	0.494991	+	0.868898i	$0.335171\pi$
$284$	0	0
$285$	4.46365	0.264404
$286$	0	0
$287$	15.1117	0.892013
$288$	0	0
$289$	19.7918	1.16422
$290$	0	0
$291$	2.86168	0.167755
$292$	0	0
$293$	−13.7396	−0.802678	−0.401339	−	0.915930i	$-0.631455\pi$
$293$	−13.7396	−0.802678	−0.401339	+	0.915930i	$0.631455\pi$
$294$	0	0
$295$	−8.69736	−0.506380
$296$	0	0
$297$	1.71694	0.0996269
$298$	0	0
$299$	4.06562	0.235121
$300$	0	0
$301$	27.6964	1.59639
$302$	0	0
$303$	−9.10147	−0.522866
$304$	0	0
$305$	−7.84818	−0.449386
$306$	0	0
$307$	−28.0944	−1.60343	−0.801716	−	0.597705i	$-0.796079\pi$
$307$	−28.0944	−1.60343	−0.801716	+	0.597705i	$0.796079\pi$
$308$	0	0
$309$	3.37845	0.192194
$310$	0	0
$311$	15.2039	0.862136	0.431068	−	0.902319i	$-0.358137\pi$
$311$	15.2039	0.862136	0.431068	+	0.902319i	$0.358137\pi$
$312$	0	0
$313$	−15.1898	−0.858577	−0.429289	−	0.903167i	$-0.641236\pi$
$313$	−15.1898	−0.858577	−0.429289	+	0.903167i	$0.641236\pi$
$314$	0	0
$315$	4.06562	0.229072
$316$	0	0
$317$	−15.6247	−0.877569	−0.438784	−	0.898592i	$-0.644591\pi$
$317$	−15.6247	−0.877569	−0.438784	+	0.898592i	$0.644591\pi$
$318$	0	0
$319$	4.22994	0.236831
$320$	0	0
$321$	7.21036	0.402443
$322$	0	0
$323$	−27.0748	−1.50648
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	3.78256	0.209176
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.7261	−1.57893	−0.789466	−	0.613795i	$-0.789642\pi$
$331$	−28.7261	−1.57893	−0.789466	+	0.613795i	$0.789642\pi$
$332$	0	0
$333$	0.283061	0.0155116
$334$	0	0
$335$	6.24621	0.341267
$336$	0	0
$337$	−11.9405	−0.650438	−0.325219	−	0.945639i	$-0.605438\pi$
$337$	−11.9405	−0.650438	−0.325219	+	0.945639i	$0.605438\pi$
$338$	0	0
$339$	−10.9865	−0.596705
$340$	0	0
$341$	0	0
$342$	0	0
$343$	10.2831	0.555233
$344$	0	0
$345$	−4.06562	−0.218886
$346$	0	0
$347$	16.4738	0.884362	0.442181	−	0.896926i	$-0.354205\pi$
$347$	16.4738	0.884362	0.442181	+	0.896926i	$0.354205\pi$
$348$	0	0
$349$	17.0460	0.912454	0.456227	−	0.889863i	$-0.349201\pi$
$349$	17.0460	0.912454	0.456227	+	0.889863i	$0.349201\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	17.7396	0.944186	0.472093	−	0.881549i	$-0.343499\pi$
$353$	17.7396	0.944186	0.472093	+	0.881549i	$0.343499\pi$
$354$	0	0
$355$	−3.03585	−0.161126
$356$	0	0
$357$	−24.6605	−1.30517
$358$	0	0
$359$	30.2462	1.59633	0.798167	−	0.602436i	$-0.205803\pi$
$359$	30.2462	1.59633	0.798167	+	0.602436i	$0.205803\pi$
$360$	0	0
$361$	0.924183	0.0486412
$362$	0	0
$363$	8.05212	0.422627
$364$	0	0
$365$	8.34868	0.436990
$366$	0	0
$367$	2.58239	0.134800	0.0673999	−	0.997726i	$-0.478530\pi$
$367$	2.58239	0.134800	0.0673999	+	0.997726i	$0.478530\pi$
$368$	0	0
$369$	3.71694	0.193496
$370$	0	0
$371$	28.8472	1.49767
$372$	0	0
$373$	22.3058	1.15495	0.577474	−	0.816409i	$-0.304038\pi$
$373$	22.3058	1.15495	0.577474	+	0.816409i	$0.304038\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−2.46365	−0.126884
$378$	0	0
$379$	29.7847	1.52994	0.764968	−	0.644068i	$-0.222755\pi$
$379$	29.7847	1.52994	0.764968	+	0.644068i	$0.222755\pi$
$380$	0	0
$381$	14.0163	0.718075
$382$	0	0
$383$	−1.72336	−0.0880598	−0.0440299	−	0.999030i	$-0.514020\pi$
$383$	−1.72336	−0.0880598	−0.0440299	+	0.999030i	$0.514020\pi$
$384$	0	0
$385$	−6.98042	−0.355755
$386$	0	0
$387$	6.81233	0.346290
$388$	0	0
$389$	−7.16101	−0.363078	−0.181539	−	0.983384i	$-0.558108\pi$
$389$	−7.16101	−0.363078	−0.181539	+	0.983384i	$0.558108\pi$
$390$	0	0
$391$	24.6605	1.24714
$392$	0	0
$393$	2.34868	0.118475
$394$	0	0
$395$	3.60197	0.181235
$396$	0	0
$397$	33.2679	1.66967	0.834834	−	0.550502i	$-0.185563\pi$
$397$	33.2679	1.66967	0.834834	+	0.550502i	$0.185563\pi$
$398$	0	0
$399$	18.1475	0.908512
$400$	0	0
$401$	−14.9598	−0.747059	−0.373530	−	0.927618i	$-0.621853\pi$
$401$	−14.9598	−0.747059	−0.373530	+	0.927618i	$0.621853\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−0.485998	−0.0240900
$408$	0	0
$409$	6.36118	0.314540	0.157270	−	0.987556i	$-0.449731\pi$
$409$	6.36118	0.314540	0.157270	+	0.987556i	$0.449731\pi$
$410$	0	0
$411$	−0.681091	−0.0335958
$412$	0	0
$413$	−35.3602	−1.73996
$414$	0	0
$415$	10.2462	0.502967
$416$	0	0
$417$	8.92088	0.436857
$418$	0	0
$419$	−0.420377	−0.0205368	−0.0102684	−	0.999947i	$-0.503269\pi$
$419$	−0.420377	−0.0205368	−0.0102684	+	0.999947i	$0.503269\pi$
$420$	0	0
$421$	23.7826	1.15909	0.579546	−	0.814940i	$-0.303230\pi$
$421$	23.7826	1.15909	0.579546	+	0.814940i	$0.303230\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.06562	0.294226
$426$	0	0
$427$	−31.9077	−1.54412
$428$	0	0
$429$	1.71694	0.0828946
$430$	0	0
$431$	1.24309	0.0598776	0.0299388	−	0.999552i	$-0.490469\pi$
$431$	1.24309	0.0598776	0.0299388	+	0.999552i	$0.490469\pi$
$432$	0	0
$433$	−10.6974	−0.514082	−0.257041	−	0.966400i	$-0.582748\pi$
$433$	−10.6974	−0.514082	−0.257041	+	0.966400i	$0.582748\pi$
$434$	0	0
$435$	2.46365	0.118123
$436$	0	0
$437$	−18.1475	−0.868113
$438$	0	0
$439$	18.1844	0.867892	0.433946	−	0.900939i	$-0.357121\pi$
$439$	18.1844	0.867892	0.433946	+	0.900939i	$0.357121\pi$
$440$	0	0
$441$	9.52927	0.453775
$442$	0	0
$443$	19.1117	0.908023	0.454011	−	0.890996i	$-0.349993\pi$
$443$	19.1117	0.908023	0.454011	+	0.890996i	$0.349993\pi$
$444$	0	0
$445$	5.60197	0.265559
$446$	0	0
$447$	−13.9077	−0.657813
$448$	0	0
$449$	−34.5868	−1.63225	−0.816126	−	0.577874i	$-0.803883\pi$
$449$	−34.5868	−1.63225	−0.816126	+	0.577874i	$0.803883\pi$
$450$	0	0
$451$	−6.38176	−0.300505
$452$	0	0
$453$	16.2625	0.764078
$454$	0	0
$455$	4.06562	0.190599
$456$	0	0
$457$	−27.1242	−1.26881	−0.634407	−	0.772999i	$-0.718756\pi$
$457$	−27.1242	−1.26881	−0.634407	+	0.772999i	$0.718756\pi$
$458$	0	0
$459$	−6.06562	−0.283119
$460$	0	0
$461$	35.7745	1.66618	0.833092	−	0.553135i	$-0.186569\pi$
$461$	35.7745	1.66618	0.833092	+	0.553135i	$0.186569\pi$
$462$	0	0
$463$	7.36826	0.342432	0.171216	−	0.985234i	$-0.445230\pi$
$463$	7.36826	0.342432	0.171216	+	0.985234i	$0.445230\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9.51200	0.440163	0.220081	−	0.975482i	$-0.429368\pi$
$467$	9.51200	0.440163	0.220081	+	0.975482i	$0.429368\pi$
$468$	0	0
$469$	25.3947	1.17262
$470$	0	0
$471$	12.9436	0.596408
$472$	0	0
$473$	−11.6964	−0.537799
$474$	0	0
$475$	−4.46365	−0.204806
$476$	0	0
$477$	7.09539	0.324876
$478$	0	0
$479$	26.4143	1.20690	0.603450	−	0.797401i	$-0.293792\pi$
$479$	26.4143	1.20690	0.603450	+	0.797401i	$0.293792\pi$
$480$	0	0
$481$	0.283061	0.0129065
$482$	0	0
$483$	−16.5293	−0.752108
$484$	0	0
$485$	−2.86168	−0.129942
$486$	0	0
$487$	14.2564	0.646020	0.323010	−	0.946396i	$-0.395305\pi$
$487$	14.2564	0.646020	0.323010	+	0.946396i	$0.395305\pi$
$488$	0	0
$489$	−7.73321	−0.349708
$490$	0	0
$491$	14.7099	0.663847	0.331923	−	0.943306i	$-0.392302\pi$
$491$	14.7099	0.663847	0.331923	+	0.943306i	$0.392302\pi$
$492$	0	0
$493$	−14.9436	−0.673025
$494$	0	0
$495$	−1.71694	−0.0771707
$496$	0	0
$497$	−12.3426	−0.553642
$498$	0	0
$499$	2.87153	0.128547	0.0642736	−	0.997932i	$-0.479527\pi$
$499$	2.87153	0.128547	0.0642736	+	0.997932i	$0.479527\pi$
$500$	0	0
$501$	18.3220	0.818568
$502$	0	0
$503$	13.2164	0.589292	0.294646	−	0.955606i	$-0.404798\pi$
$503$	13.2164	0.589292	0.294646	+	0.955606i	$0.404798\pi$
$504$	0	0
$505$	9.10147	0.405010
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−19.2821	−0.854662	−0.427331	−	0.904095i	$-0.640546\pi$
$509$	−19.2821	−0.854662	−0.427331	+	0.904095i	$0.640546\pi$
$510$	0	0
$511$	33.9426	1.50153
$512$	0	0
$513$	4.46365	0.197075
$514$	0	0
$515$	−3.37845	−0.148872
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−12.9436	−0.568160
$520$	0	0
$521$	34.9273	1.53019	0.765096	−	0.643916i	$-0.222691\pi$
$521$	34.9273	1.53019	0.765096	+	0.643916i	$0.222691\pi$
$522$	0	0
$523$	11.9263	0.521501	0.260750	−	0.965406i	$-0.416030\pi$
$523$	11.9263	0.521501	0.260750	+	0.965406i	$0.416030\pi$
$524$	0	0
$525$	−4.06562	−0.177438
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−6.47073	−0.281336
$530$	0	0
$531$	−8.69736	−0.377433
$532$	0	0
$533$	3.71694	0.160999
$534$	0	0
$535$	−7.21036	−0.311731
$536$	0	0
$537$	−9.55262	−0.412226
$538$	0	0
$539$	−16.3612	−0.704726
$540$	0	0
$541$	27.2164	1.17013	0.585063	−	0.810988i	$-0.301070\pi$
$541$	27.2164	1.17013	0.585063	+	0.810988i	$0.301070\pi$
$542$	0	0
$543$	−21.1117	−0.905988
$544$	0	0
$545$	−3.78256	−0.162027
$546$	0	0
$547$	22.9003	0.979146	0.489573	−	0.871962i	$-0.337153\pi$
$547$	22.9003	0.979146	0.489573	+	0.871962i	$0.337153\pi$
$548$	0	0
$549$	−7.84818	−0.334952
$550$	0	0
$551$	10.9969	0.468483
$552$	0	0
$553$	14.6442	0.622737
$554$	0	0
$555$	−0.283061	−0.0120153
$556$	0	0
$557$	−16.0717	−0.680980	−0.340490	−	0.940248i	$-0.610593\pi$
$557$	−16.0717	−0.680980	−0.340490	+	0.940248i	$0.610593\pi$
$558$	0	0
$559$	6.81233	0.288131
$560$	0	0
$561$	10.4143	0.439692
$562$	0	0
$563$	−27.0380	−1.13951	−0.569757	−	0.821813i	$-0.692963\pi$
$563$	−27.0380	−1.13951	−0.569757	+	0.821813i	$0.692963\pi$
$564$	0	0
$565$	10.9865	0.462205
$566$	0	0
$567$	4.06562	0.170740
$568$	0	0
$569$	−4.28949	−0.179825	−0.0899123	−	0.995950i	$-0.528659\pi$
$569$	−4.28949	−0.179825	−0.0899123	+	0.995950i	$0.528659\pi$
$570$	0	0
$571$	8.13767	0.340551	0.170275	−	0.985397i	$-0.445534\pi$
$571$	8.13767	0.340551	0.170275	+	0.985397i	$0.445534\pi$
$572$	0	0
$573$	19.6964	0.822827
$574$	0	0
$575$	4.06562	0.169548
$576$	0	0
$577$	14.3281	0.596487	0.298243	−	0.954490i	$-0.403599\pi$
$577$	14.3281	0.596487	0.298243	+	0.954490i	$0.403599\pi$
$578$	0	0
$579$	−9.49950	−0.394786
$580$	0	0
$581$	41.6572	1.72823
$582$	0	0
$583$	−12.1824	−0.504542
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	6.97400	0.287848	0.143924	−	0.989589i	$-0.454028\pi$
$587$	6.97400	0.287848	0.143924	+	0.989589i	$0.454028\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−16.9436	−0.696966
$592$	0	0
$593$	16.8123	0.690400	0.345200	−	0.938529i	$-0.387811\pi$
$593$	16.8123	0.690400	0.345200	+	0.938529i	$0.387811\pi$
$594$	0	0
$595$	24.6605	1.01098
$596$	0	0
$597$	18.5087	0.757510
$598$	0	0
$599$	22.8678	0.934351	0.467176	−	0.884165i	$-0.345272\pi$
$599$	22.8678	0.934351	0.467176	+	0.884165i	$0.345272\pi$
$600$	0	0
$601$	−12.6010	−0.514004	−0.257002	−	0.966411i	$-0.582735\pi$
$601$	−12.6010	−0.514004	−0.257002	+	0.966411i	$0.582735\pi$
$602$	0	0
$603$	6.24621	0.254365
$604$	0	0
$605$	−8.05212	−0.327365
$606$	0	0
$607$	−41.4502	−1.68241	−0.841205	−	0.540717i	$-0.818153\pi$
$607$	−41.4502	−1.68241	−0.841205	+	0.540717i	$0.818153\pi$
$608$	0	0
$609$	10.0163	0.405880
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.28306	−0.172991	−0.0864956	−	0.996252i	$-0.527567\pi$
$613$	−4.28306	−0.172991	−0.0864956	+	0.996252i	$0.527567\pi$
$614$	0	0
$615$	−3.71694	−0.149881
$616$	0	0
$617$	−12.6811	−0.510522	−0.255261	−	0.966872i	$-0.582161\pi$
$617$	−12.6811	−0.510522	−0.255261	+	0.966872i	$0.582161\pi$
$618$	0	0
$619$	−44.8574	−1.80297	−0.901485	−	0.432810i	$-0.857522\pi$
$619$	−44.8574	−1.80297	−0.901485	+	0.432810i	$0.857522\pi$
$620$	0	0
$621$	−4.06562	−0.163148
$622$	0	0
$623$	22.7755	0.912480
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.66382	−0.306063
$628$	0	0
$629$	1.71694	0.0684589
$630$	0	0
$631$	14.2029	0.565410	0.282705	−	0.959207i	$-0.408768\pi$
$631$	14.2029	0.565410	0.282705	+	0.959207i	$0.408768\pi$
$632$	0	0
$633$	10.8678	0.431955
$634$	0	0
$635$	−14.0163	−0.556219
$636$	0	0
$637$	9.52927	0.377564
$638$	0	0
$639$	−3.03585	−0.120096
$640$	0	0
$641$	−0.986848	−0.0389782	−0.0194891	−	0.999810i	$-0.506204\pi$
$641$	−0.986848	−0.0389782	−0.0194891	+	0.999810i	$0.506204\pi$
$642$	0	0
$643$	−6.46973	−0.255141	−0.127571	−	0.991829i	$-0.540718\pi$
$643$	−6.46973	−0.255141	−0.127571	+	0.991829i	$0.540718\pi$
$644$	0	0
$645$	−6.81233	−0.268235
$646$	0	0
$647$	27.0646	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$647$	27.0646	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$648$	0	0
$649$	14.9328	0.586165
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.62155	−0.259121	−0.129561	−	0.991572i	$-0.541357\pi$
$653$	−6.62155	−0.259121	−0.129561	+	0.991572i	$0.541357\pi$
$654$	0	0
$655$	−2.34868	−0.0917706
$656$	0	0
$657$	8.34868	0.325713
$658$	0	0
$659$	0.289136	0.0112631	0.00563157	−	0.999984i	$-0.498207\pi$
$659$	0.289136	0.0112631	0.00563157	+	0.999984i	$0.498207\pi$
$660$	0	0
$661$	−20.1763	−0.784767	−0.392383	−	0.919802i	$-0.628349\pi$
$661$	−20.1763	−0.784767	−0.392383	+	0.919802i	$0.628349\pi$
$662$	0	0
$663$	−6.06562	−0.235569
$664$	0	0
$665$	−18.1475	−0.703730
$666$	0	0
$667$	−10.0163	−0.387832
$668$	0	0
$669$	−19.9733	−0.772214
$670$	0	0
$671$	13.4748	0.520191
$672$	0	0
$673$	−6.33618	−0.244242	−0.122121	−	0.992515i	$-0.538970\pi$
$673$	−6.33618	−0.244242	−0.122121	+	0.992515i	$0.538970\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−3.22664	−0.124010	−0.0620048	−	0.998076i	$-0.519749\pi$
$677$	−3.22664	−0.124010	−0.0620048	+	0.998076i	$0.519749\pi$
$678$	0	0
$679$	−11.6345	−0.446492
$680$	0	0
$681$	3.20394	0.122775
$682$	0	0
$683$	−11.8255	−0.452490	−0.226245	−	0.974070i	$-0.572645\pi$
$683$	−11.8255	−0.452490	−0.226245	+	0.974070i	$0.572645\pi$
$684$	0	0
$685$	0.681091	0.0260232
$686$	0	0
$687$	−27.3477	−1.04338
$688$	0	0
$689$	7.09539	0.270313
$690$	0	0
$691$	−16.2337	−0.617559	−0.308780	−	0.951134i	$-0.599921\pi$
$691$	−16.2337	−0.617559	−0.308780	+	0.951134i	$0.599921\pi$
$692$	0	0
$693$	−6.98042	−0.265164
$694$	0	0
$695$	−8.92088	−0.338388
$696$	0	0
$697$	22.5455	0.853973
$698$	0	0
$699$	13.6307	0.515562
$700$	0	0
$701$	−34.0613	−1.28648	−0.643239	−	0.765665i	$-0.722410\pi$
$701$	−34.0613	−1.28648	−0.643239	+	0.765665i	$0.722410\pi$
$702$	0	0
$703$	−1.26348	−0.0476532
$704$	0	0
$705$	0	0
$706$	0	0
$707$	37.0031	1.39165
$708$	0	0
$709$	27.3477	1.02706	0.513532	−	0.858071i	$-0.328337\pi$
$709$	27.3477	1.02706	0.513532	+	0.858071i	$0.328337\pi$
$710$	0	0
$711$	3.60197	0.135084
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.71694	−0.0642099
$716$	0	0
$717$	−19.2104	−0.717424
$718$	0	0
$719$	−3.82760	−0.142746	−0.0713728	−	0.997450i	$-0.522738\pi$
$719$	−3.82760	−0.142746	−0.0713728	+	0.997450i	$0.522738\pi$
$720$	0	0
$721$	−13.7355	−0.511537
$722$	0	0
$723$	7.05854	0.262510
$724$	0	0
$725$	−2.46365	−0.0914977
$726$	0	0
$727$	−12.7528	−0.472975	−0.236487	−	0.971635i	$-0.575996\pi$
$727$	−12.7528	−0.472975	−0.236487	+	0.971635i	$0.575996\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	41.3210	1.52831
$732$	0	0
$733$	5.18536	0.191526	0.0957629	−	0.995404i	$-0.469471\pi$
$733$	5.18536	0.191526	0.0957629	+	0.995404i	$0.469471\pi$
$734$	0	0
$735$	−9.52927	−0.351492
$736$	0	0
$737$	−10.7244	−0.395037
$738$	0	0
$739$	25.2186	0.927680	0.463840	−	0.885919i	$-0.346471\pi$
$739$	25.2186	0.927680	0.463840	+	0.885919i	$0.346471\pi$
$740$	0	0
$741$	4.46365	0.163976
$742$	0	0
$743$	21.4934	0.788517	0.394259	−	0.919000i	$-0.371001\pi$
$743$	21.4934	0.788517	0.394259	+	0.919000i	$0.371001\pi$
$744$	0	0
$745$	13.9077	0.509540
$746$	0	0
$747$	10.2462	0.374889
$748$	0	0
$749$	−29.3146	−1.07113
$750$	0	0
$751$	14.6768	0.535563	0.267782	−	0.963480i	$-0.413709\pi$
$751$	14.6768	0.535563	0.267782	+	0.963480i	$0.413709\pi$
$752$	0	0
$753$	13.9138	0.507047
$754$	0	0
$755$	−16.2625	−0.591852
$756$	0	0
$757$	−39.5510	−1.43750	−0.718752	−	0.695266i	$-0.755286\pi$
$757$	−39.5510	−1.43750	−0.718752	+	0.695266i	$0.755286\pi$
$758$	0	0
$759$	6.98042	0.253373
$760$	0	0
$761$	1.04015	0.0377056	0.0188528	−	0.999822i	$-0.493999\pi$
$761$	1.04015	0.0377056	0.0188528	+	0.999822i	$0.493999\pi$
$762$	0	0
$763$	−15.3785	−0.556737
$764$	0	0
$765$	6.06562	0.219303
$766$	0	0
$767$	−8.69736	−0.314044
$768$	0	0
$769$	−33.7701	−1.21778	−0.608890	−	0.793255i	$-0.708385\pi$
$769$	−33.7701	−1.21778	−0.608890	+	0.793255i	$0.708385\pi$
$770$	0	0
$771$	−21.9733	−0.791351
$772$	0	0
$773$	32.8699	1.18225	0.591124	−	0.806581i	$-0.298685\pi$
$773$	32.8699	1.18225	0.591124	+	0.806581i	$0.298685\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.15082	−0.0412854
$778$	0	0
$779$	−16.5911	−0.594439
$780$	0	0
$781$	5.21237	0.186513
$782$	0	0
$783$	2.46365	0.0880437
$784$	0	0
$785$	−12.9436	−0.461976
$786$	0	0
$787$	−34.9436	−1.24560	−0.622802	−	0.782380i	$-0.714006\pi$
$787$	−34.9436	−1.24560	−0.622802	+	0.782380i	$0.714006\pi$
$788$	0	0
$789$	−30.6112	−1.08979
$790$	0	0
$791$	44.6669	1.58817
$792$	0	0
$793$	−7.84818	−0.278697
$794$	0	0
$795$	−7.09539	−0.251648
$796$	0	0
$797$	2.79175	0.0988890	0.0494445	−	0.998777i	$-0.484255\pi$
$797$	2.79175	0.0988890	0.0494445	+	0.998777i	$0.484255\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5.60197	0.197936
$802$	0	0
$803$	−14.3342	−0.505842
$804$	0	0
$805$	16.5293	0.582580
$806$	0	0
$807$	31.4235	1.10616
$808$	0	0
$809$	18.9848	0.667472	0.333736	−	0.942667i	$-0.391691\pi$
$809$	18.9848	0.667472	0.333736	+	0.942667i	$0.391691\pi$
$810$	0	0
$811$	−50.3508	−1.76806	−0.884028	−	0.467434i	$-0.845178\pi$
$811$	−50.3508	−1.76806	−0.884028	+	0.467434i	$0.845178\pi$
$812$	0	0
$813$	12.8286	0.449919
$814$	0	0
$815$	7.73321	0.270883
$816$	0	0
$817$	−30.4079	−1.06384
$818$	0	0
$819$	4.06562	0.142064
$820$	0	0
$821$	−52.8080	−1.84301	−0.921506	−	0.388363i	$-0.873041\pi$
$821$	−52.8080	−1.84301	−0.921506	+	0.388363i	$0.873041\pi$
$822$	0	0
$823$	−14.7386	−0.513756	−0.256878	−	0.966444i	$-0.582694\pi$
$823$	−14.7386	−0.513756	−0.256878	+	0.966444i	$0.582694\pi$
$824$	0	0
$825$	1.71694	0.0597762
$826$	0	0
$827$	−6.47615	−0.225198	−0.112599	−	0.993641i	$-0.535918\pi$
$827$	−6.47615	−0.225198	−0.112599	+	0.993641i	$0.535918\pi$
$828$	0	0
$829$	−23.6247	−0.820519	−0.410259	−	0.911969i	$-0.634562\pi$
$829$	−23.6247	−0.820519	−0.410259	+	0.911969i	$0.634562\pi$
$830$	0	0
$831$	13.7396	0.476623
$832$	0	0
$833$	57.8009	2.00268
$834$	0	0
$835$	−18.3220	−0.634060
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−36.7939	−1.27027	−0.635133	−	0.772403i	$-0.719055\pi$
$839$	−36.7939	−1.27027	−0.635133	+	0.772403i	$0.719055\pi$
$840$	0	0
$841$	−22.9304	−0.790704
$842$	0	0
$843$	−32.0163	−1.10270
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−32.7369	−1.12485
$848$	0	0
$849$	−16.6541	−0.571567
$850$	0	0
$851$	1.15082	0.0394495
$852$	0	0
$853$	−42.4327	−1.45287	−0.726434	−	0.687236i	$-0.758824\pi$
$853$	−42.4327	−1.45287	−0.726434	+	0.687236i	$0.758824\pi$
$854$	0	0
$855$	−4.46365	−0.152654
$856$	0	0
$857$	−5.57320	−0.190377	−0.0951884	−	0.995459i	$-0.530345\pi$
$857$	−5.57320	−0.190377	−0.0951884	+	0.995459i	$0.530345\pi$
$858$	0	0
$859$	−46.3406	−1.58112	−0.790560	−	0.612384i	$-0.790211\pi$
$859$	−46.3406	−1.58112	−0.790560	+	0.612384i	$0.790211\pi$
$860$	0	0
$861$	−15.1117	−0.515004
$862$	0	0
$863$	−28.8861	−0.983296	−0.491648	−	0.870794i	$-0.663605\pi$
$863$	−28.8861	−0.983296	−0.491648	+	0.870794i	$0.663605\pi$
$864$	0	0
$865$	12.9436	0.440095
$866$	0	0
$867$	−19.7918	−0.672163
$868$	0	0
$869$	−6.18436	−0.209790
$870$	0	0
$871$	6.24621	0.211645
$872$	0	0
$873$	−2.86168	−0.0968533
$874$	0	0
$875$	4.06562	0.137443
$876$	0	0
$877$	−7.01315	−0.236817	−0.118409	−	0.992965i	$-0.537779\pi$
$877$	−7.01315	−0.236817	−0.118409	+	0.992965i	$0.537779\pi$
$878$	0	0
$879$	13.7396	0.463426
$880$	0	0
$881$	−9.03154	−0.304280	−0.152140	−	0.988359i	$-0.548617\pi$
$881$	−9.03154	−0.304280	−0.152140	+	0.988359i	$0.548617\pi$
$882$	0	0
$883$	−14.9003	−0.501435	−0.250718	−	0.968060i	$-0.580667\pi$
$883$	−14.9003	−0.501435	−0.250718	+	0.968060i	$0.580667\pi$
$884$	0	0
$885$	8.69736	0.292359
$886$	0	0
$887$	−4.99292	−0.167646	−0.0838230	−	0.996481i	$-0.526713\pi$
$887$	−4.99292	−0.167646	−0.0838230	+	0.996481i	$0.526713\pi$
$888$	0	0
$889$	−56.9848	−1.91121
$890$	0	0
$891$	−1.71694	−0.0575196
$892$	0	0
$893$	0	0
$894$	0	0
$895$	9.55262	0.319309
$896$	0	0
$897$	−4.06562	−0.135747
$898$	0	0
$899$	0	0
$900$	0	0
$901$	43.0380	1.43380
$902$	0	0
$903$	−27.6964	−0.921677
$904$	0	0
$905$	21.1117	0.701775
$906$	0	0
$907$	54.2916	1.80272	0.901362	−	0.433067i	$-0.142569\pi$
$907$	54.2916	1.80272	0.901362	+	0.433067i	$0.142569\pi$
$908$	0	0
$909$	9.10147	0.301877
$910$	0	0
$911$	−12.3612	−0.409544	−0.204772	−	0.978810i	$-0.565645\pi$
$911$	−12.3612	−0.409544	−0.204772	+	0.978810i	$0.565645\pi$
$912$	0	0
$913$	−17.5921	−0.582214
$914$	0	0
$915$	7.84818	0.259453
$916$	0	0
$917$	−9.54885	−0.315331
$918$	0	0
$919$	−30.0044	−0.989755	−0.494877	−	0.868963i	$-0.664787\pi$
$919$	−30.0044	−0.989755	−0.494877	+	0.868963i	$0.664787\pi$
$920$	0	0
$921$	28.0944	0.925742
$922$	0	0
$923$	−3.03585	−0.0999262
$924$	0	0
$925$	0.283061	0.00930698
$926$	0	0
$927$	−3.37845	−0.110963
$928$	0	0
$929$	−7.41118	−0.243153	−0.121577	−	0.992582i	$-0.538795\pi$
$929$	−7.41118	−0.243153	−0.121577	+	0.992582i	$0.538795\pi$
$930$	0	0
$931$	−42.5353	−1.39404
$932$	0	0
$933$	−15.2039	−0.497755
$934$	0	0
$935$	−10.4143	−0.340584
$936$	0	0
$937$	16.6832	0.545017	0.272508	−	0.962153i	$-0.412147\pi$
$937$	16.6832	0.545017	0.272508	+	0.962153i	$0.412147\pi$
$938$	0	0
$939$	15.1898	0.495700
$940$	0	0
$941$	34.0511	1.11003	0.555017	−	0.831839i	$-0.312712\pi$
$941$	34.0511	1.11003	0.555017	+	0.831839i	$0.312712\pi$
$942$	0	0
$943$	15.1117	0.492104
$944$	0	0
$945$	−4.06562	−0.132255
$946$	0	0
$947$	4.51070	0.146578	0.0732890	−	0.997311i	$-0.476650\pi$
$947$	4.51070	0.146578	0.0732890	+	0.997311i	$0.476650\pi$
$948$	0	0
$949$	8.34868	0.271010
$950$	0	0
$951$	15.6247	0.506664
$952$	0	0
$953$	−6.16432	−0.199682	−0.0998409	−	0.995003i	$-0.531833\pi$
$953$	−6.16432	−0.199682	−0.0998409	+	0.995003i	$0.531833\pi$
$954$	0	0
$955$	−19.6964	−0.637359
$956$	0	0
$957$	−4.22994	−0.136735
$958$	0	0
$959$	2.76906	0.0894176
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−7.21036	−0.232351
$964$	0	0
$965$	9.49950	0.305800
$966$	0	0
$967$	37.1631	1.19509	0.597543	−	0.801837i	$-0.296144\pi$
$967$	37.1631	1.19509	0.597543	+	0.801837i	$0.296144\pi$
$968$	0	0
$969$	27.0748	0.869768
$970$	0	0
$971$	34.0776	1.09360	0.546801	−	0.837263i	$-0.315845\pi$
$971$	34.0776	1.09360	0.546801	+	0.837263i	$0.315845\pi$
$972$	0	0
$973$	−36.2689	−1.16273
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	−34.8840	−1.11604	−0.558019	−	0.829828i	$-0.688439\pi$
$977$	−34.8840	−1.11604	−0.558019	+	0.829828i	$0.688439\pi$
$978$	0	0
$979$	−9.61824	−0.307400
$980$	0	0
$981$	−3.78256	−0.120768
$982$	0	0
$983$	7.18355	0.229120	0.114560	−	0.993416i	$-0.463454\pi$
$983$	7.18355	0.229120	0.114560	+	0.993416i	$0.463454\pi$
$984$	0	0
$985$	16.9436	0.539867
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.6964	0.880693
$990$	0	0
$991$	7.29833	0.231839	0.115920	−	0.993259i	$-0.463019\pi$
$991$	7.29833	0.231839	0.115920	+	0.993259i	$0.463019\pi$
$992$	0	0
$993$	28.7261	0.911596
$994$	0	0
$995$	−18.5087	−0.586765
$996$	0	0
$997$	−43.2778	−1.37062	−0.685310	−	0.728251i	$-0.740333\pi$
$997$	−43.2778	−1.37062	−0.685310	+	0.728251i	$0.740333\pi$
$998$	0	0
$999$	−0.283061	−0.00895565

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.ce.1.4	✓	4
4.3	odd	2		6240.2.a.cf.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.ce.1.4	✓	4	1.1	even	1	trivial
6240.2.a.cf.1.1	yes	4	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.06562 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +4.06562 q^{7} +1.00000 q^{9} -1.71694 q^{11} +1.00000 q^{13} -1.00000 q^{15} +6.06562 q^{17} -4.46365 q^{19} -4.06562 q^{21} +4.06562 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.46365 q^{29} +1.71694 q^{33} +4.06562 q^{35} +0.283061 q^{37} -1.00000 q^{39} +3.71694 q^{41} +6.81233 q^{43} +1.00000 q^{45} +9.52927 q^{49} -6.06562 q^{51} +7.09539 q^{53} -1.71694 q^{55} +4.46365 q^{57} -8.69736 q^{59} -7.84818 q^{61} +4.06562 q^{63} +1.00000 q^{65} +6.24621 q^{67} -4.06562 q^{69} -3.03585 q^{71} +8.34868 q^{73} -1.00000 q^{75} -6.98042 q^{77} +3.60197 q^{79} +1.00000 q^{81} +10.2462 q^{83} +6.06562 q^{85} +2.46365 q^{87} +5.60197 q^{89} +4.06562 q^{91} -4.46365 q^{95} -2.86168 q^{97} -1.71694 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 4 q^{15} + 5 q^{17} - 8 q^{19} + 3 q^{21} - 3 q^{23} + 4 q^{25} - 4 q^{27} + 3 q^{33} - 3 q^{35} + 5 q^{37} - 4 q^{39} + 11 q^{41} + 2 q^{43} + 4 q^{45} + 9 q^{49} - 5 q^{51} + 7 q^{53} - 3 q^{55} + 8 q^{57} - 4 q^{59} + 11 q^{61} - 3 q^{63} + 4 q^{65} - 8 q^{67} + 3 q^{69} + 5 q^{71} + 18 q^{73} - 4 q^{75} - q^{77} + 5 q^{79} + 4 q^{81} + 8 q^{83} + 5 q^{85} + 13 q^{89} - 3 q^{91} - 8 q^{95} - 11 q^{97} - 3 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 3 * q^11 + 4 * q^13 - 4 * q^15 + 5 * q^17 - 8 * q^19 + 3 * q^21 - 3 * q^23 + 4 * q^25 - 4 * q^27 + 3 * q^33 - 3 * q^35 + 5 * q^37 - 4 * q^39 + 11 * q^41 + 2 * q^43 + 4 * q^45 + 9 * q^49 - 5 * q^51 + 7 * q^53 - 3 * q^55 + 8 * q^57 - 4 * q^59 + 11 * q^61 - 3 * q^63 + 4 * q^65 - 8 * q^67 + 3 * q^69 + 5 * q^71 + 18 * q^73 - 4 * q^75 - q^77 + 5 * q^79 + 4 * q^81 + 8 * q^83 + 5 * q^85 + 13 * q^89 - 3 * q^91 - 8 * q^95 - 11 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

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Database

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