LMFDB - Embedded newform 6240.2.a.bw.1.2

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:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.11491$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} -3.58774 q^{11} +1.00000 q^{13} +1.00000 q^{15} -7.81756 q^{17} -4.94567 q^{19} +1.35793 q^{21} -0.0737791 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.51396 q^{29} -4.00000 q^{31} +3.58774 q^{33} +1.35793 q^{35} -1.58774 q^{37} -1.00000 q^{39} -5.58774 q^{41} -7.17548 q^{43} -1.00000 q^{45} -2.56829 q^{47} -5.15604 q^{49} +7.81756 q^{51} +12.7632 q^{53} +3.58774 q^{55} +4.94567 q^{57} +8.45963 q^{59} -4.30359 q^{61} -1.35793 q^{63} -1.00000 q^{65} +0.0737791 q^{69} +0.871889 q^{71} +6.79811 q^{73} -1.00000 q^{75} +4.87189 q^{77} -12.8719 q^{79} +1.00000 q^{81} +5.43171 q^{83} +7.81756 q^{85} -5.51396 q^{87} -6.87189 q^{89} -1.35793 q^{91} +4.00000 q^{93} +4.94567 q^{95} +7.35793 q^{97} -3.58774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 12 * q^31 - q^33 + 5 * q^35 + 7 * q^37 - 3 * q^39 - 5 * q^41 + 2 * q^43 - 3 * q^45 - 4 * q^47 - q^51 + 3 * q^53 - q^55 + 4 * q^57 - 3 * q^61 - 5 * q^63 - 3 * q^65 + 3 * q^69 - 11 * q^71 + 4 * q^73 - 3 * q^75 + q^77 - 25 * q^79 + 3 * q^81 + 20 * q^83 - q^85 - 2 * q^87 - 7 * q^89 - 5 * q^91 + 12 * q^93 + 4 * q^95 + 23 * q^97 + 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
$7$	−1.35793	−0.513248	−0.256624	−	0.966511i	$-0.582610\pi$
$7$	−1.35793	−0.513248	−0.256624	+	0.966511i	$0.582610\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.58774	−1.08174	−0.540872	−	0.841105i	$-0.681906\pi$
$11$	−3.58774	−1.08174	−0.540872	+	0.841105i	$0.681906\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−7.81756	−1.89604	−0.948018	−	0.318217i	$-0.896916\pi$
$17$	−7.81756	−1.89604	−0.948018	+	0.318217i	$0.896916\pi$
$18$	0	0
$19$	−4.94567	−1.13461	−0.567307	−	0.823506i	$-0.692015\pi$
$19$	−4.94567	−1.13461	−0.567307	+	0.823506i	$0.692015\pi$
$20$	0	0
$21$	1.35793	0.296324
$22$	0	0
$23$	−0.0737791	−0.0153840	−0.00769200	−	0.999970i	$-0.502448\pi$
$23$	−0.0737791	−0.0153840	−0.00769200	+	0.999970i	$0.502448\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.51396	1.02392	0.511959	−	0.859010i	$-0.328920\pi$
$29$	5.51396	1.02392	0.511959	+	0.859010i	$0.328920\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	3.58774	0.624546
$34$	0	0
$35$	1.35793	0.229531
$36$	0	0
$37$	−1.58774	−0.261023	−0.130512	−	0.991447i	$-0.541662\pi$
$37$	−1.58774	−0.261023	−0.130512	+	0.991447i	$0.541662\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−5.58774	−0.872659	−0.436329	−	0.899787i	$-0.643722\pi$
$41$	−5.58774	−0.872659	−0.436329	+	0.899787i	$0.643722\pi$
$42$	0	0
$43$	−7.17548	−1.09425	−0.547125	−	0.837051i	$-0.684278\pi$
$43$	−7.17548	−1.09425	−0.547125	+	0.837051i	$0.684278\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.56829	−0.374624	−0.187312	−	0.982300i	$-0.559978\pi$
$47$	−2.56829	−0.374624	−0.187312	+	0.982300i	$0.559978\pi$
$48$	0	0
$49$	−5.15604	−0.736577
$50$	0	0
$51$	7.81756	1.09468
$52$	0	0
$53$	12.7632	1.75316	0.876582	−	0.481253i	$-0.159818\pi$
$53$	12.7632	1.75316	0.876582	+	0.481253i	$0.159818\pi$
$54$	0	0
$55$	3.58774	0.483771
$56$	0	0
$57$	4.94567	0.655070
$58$	0	0
$59$	8.45963	1.10135	0.550675	−	0.834720i	$-0.314370\pi$
$59$	8.45963	1.10135	0.550675	+	0.834720i	$0.314370\pi$
$60$	0	0
$61$	−4.30359	−0.551019	−0.275509	−	0.961298i	$-0.588847\pi$
$61$	−4.30359	−0.551019	−0.275509	+	0.961298i	$0.588847\pi$
$62$	0	0
$63$	−1.35793	−0.171083
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0.0737791	0.00888196
$70$	0	0
$71$	0.871889	0.103474	0.0517371	−	0.998661i	$-0.483524\pi$
$71$	0.871889	0.103474	0.0517371	+	0.998661i	$0.483524\pi$
$72$	0	0
$73$	6.79811	0.795659	0.397829	−	0.917459i	$-0.369764\pi$
$73$	6.79811	0.795659	0.397829	+	0.917459i	$0.369764\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.87189	0.555203
$78$	0	0
$79$	−12.8719	−1.44820	−0.724100	−	0.689695i	$-0.757745\pi$
$79$	−12.8719	−1.44820	−0.724100	+	0.689695i	$0.757745\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.43171	0.596207	0.298104	−	0.954534i	$-0.403646\pi$
$83$	5.43171	0.596207	0.298104	+	0.954534i	$0.403646\pi$
$84$	0	0
$85$	7.81756	0.847933
$86$	0	0
$87$	−5.51396	−0.591159
$88$	0	0
$89$	−6.87189	−0.728419	−0.364209	−	0.931317i	$-0.618661\pi$
$89$	−6.87189	−0.728419	−0.364209	+	0.931317i	$0.618661\pi$
$90$	0	0
$91$	−1.35793	−0.142349
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	4.94567	0.507415
$96$	0	0
$97$	7.35793	0.747084	0.373542	−	0.927613i	$-0.378143\pi$
$97$	7.35793	0.747084	0.373542	+	0.927613i	$0.378143\pi$
$98$	0	0
$99$	−3.58774	−0.360582
$100$	0	0
$101$	−6.94567	−0.691120	−0.345560	−	0.938397i	$-0.612311\pi$
$101$	−6.94567	−0.691120	−0.345560	+	0.938397i	$0.612311\pi$
$102$	0	0
$103$	−10.5683	−1.04133	−0.520663	−	0.853763i	$-0.674315\pi$
$103$	−10.5683	−1.04133	−0.520663	+	0.853763i	$0.674315\pi$
$104$	0	0
$105$	−1.35793	−0.132520
$106$	0	0
$107$	1.69641	0.163998	0.0819989	−	0.996632i	$-0.473870\pi$
$107$	1.69641	0.163998	0.0819989	+	0.996632i	$0.473870\pi$
$108$	0	0
$109$	−3.40530	−0.326168	−0.163084	−	0.986612i	$-0.552144\pi$
$109$	−3.40530	−0.326168	−0.163084	+	0.986612i	$0.552144\pi$
$110$	0	0
$111$	1.58774	0.150702
$112$	0	0
$113$	16.6894	1.57001	0.785005	−	0.619489i	$-0.212660\pi$
$113$	16.6894	1.57001	0.785005	+	0.619489i	$0.212660\pi$
$114$	0	0
$115$	0.0737791	0.00687994
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.6157	0.973137
$120$	0	0
$121$	1.87189	0.170172
$122$	0	0
$123$	5.58774	0.503830
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.7827	−1.75543	−0.877714	−	0.479185i	$-0.840932\pi$
$127$	−19.7827	−1.75543	−0.877714	+	0.479185i	$0.840932\pi$
$128$	0	0
$129$	7.17548	0.631766
$130$	0	0
$131$	0.798110	0.0697312	0.0348656	−	0.999392i	$-0.488900\pi$
$131$	0.798110	0.0697312	0.0348656	+	0.999392i	$0.488900\pi$
$132$	0	0
$133$	6.71585	0.582338
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	13.0279	1.11305	0.556525	−	0.830831i	$-0.312134\pi$
$137$	13.0279	1.11305	0.556525	+	0.830831i	$0.312134\pi$
$138$	0	0
$139$	8.19493	0.695085	0.347542	−	0.937664i	$-0.387016\pi$
$139$	8.19493	0.695085	0.347542	+	0.937664i	$0.387016\pi$
$140$	0	0
$141$	2.56829	0.216289
$142$	0	0
$143$	−3.58774	−0.300022
$144$	0	0
$145$	−5.51396	−0.457910
$146$	0	0
$147$	5.15604	0.425263
$148$	0	0
$149$	12.1560	0.995861	0.497931	−	0.867217i	$-0.334093\pi$
$149$	12.1560	0.995861	0.497931	+	0.867217i	$0.334093\pi$
$150$	0	0
$151$	12.9193	1.05135	0.525677	−	0.850684i	$-0.323812\pi$
$151$	12.9193	1.05135	0.525677	+	0.850684i	$0.323812\pi$
$152$	0	0
$153$	−7.81756	−0.632012
$154$	0	0
$155$	4.00000	0.321288
$156$	0	0
$157$	9.32304	0.744060	0.372030	−	0.928221i	$-0.378662\pi$
$157$	9.32304	0.744060	0.372030	+	0.928221i	$0.378662\pi$
$158$	0	0
$159$	−12.7632	−1.01219
$160$	0	0
$161$	0.100187	0.00789581
$162$	0	0
$163$	14.7632	1.15634	0.578172	−	0.815915i	$-0.303766\pi$
$163$	14.7632	1.15634	0.578172	+	0.815915i	$0.303766\pi$
$164$	0	0
$165$	−3.58774	−0.279305
$166$	0	0
$167$	−4.45963	−0.345097	−0.172548	−	0.985001i	$-0.555200\pi$
$167$	−4.45963	−0.345097	−0.172548	+	0.985001i	$0.555200\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.94567	−0.378205
$172$	0	0
$173$	−6.45963	−0.491117	−0.245558	−	0.969382i	$-0.578971\pi$
$173$	−6.45963	−0.491117	−0.245558	+	0.969382i	$0.578971\pi$
$174$	0	0
$175$	−1.35793	−0.102650
$176$	0	0
$177$	−8.45963	−0.635865
$178$	0	0
$179$	8.94567	0.668631	0.334315	−	0.942461i	$-0.391495\pi$
$179$	8.94567	0.668631	0.334315	+	0.942461i	$0.391495\pi$
$180$	0	0
$181$	−19.3315	−1.43690	−0.718450	−	0.695578i	$-0.755148\pi$
$181$	−19.3315	−1.43690	−0.718450	+	0.695578i	$0.755148\pi$
$182$	0	0
$183$	4.30359	0.318131
$184$	0	0
$185$	1.58774	0.116733
$186$	0	0
$187$	28.0474	2.05103
$188$	0	0
$189$	1.35793	0.0987746
$190$	0	0
$191$	−9.74378	−0.705035	−0.352517	−	0.935805i	$-0.614674\pi$
$191$	−9.74378	−0.705035	−0.352517	+	0.935805i	$0.614674\pi$
$192$	0	0
$193$	−4.27719	−0.307879	−0.153939	−	0.988080i	$-0.549196\pi$
$193$	−4.27719	−0.307879	−0.153939	+	0.988080i	$0.549196\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	1.54037	0.109747	0.0548734	−	0.998493i	$-0.482524\pi$
$197$	1.54037	0.109747	0.0548734	+	0.998493i	$0.482524\pi$
$198$	0	0
$199$	−16.9193	−1.19937	−0.599687	−	0.800234i	$-0.704708\pi$
$199$	−16.9193	−1.19937	−0.599687	+	0.800234i	$0.704708\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−7.48755	−0.525523
$204$	0	0
$205$	5.58774	0.390265
$206$	0	0
$207$	−0.0737791	−0.00512800
$208$	0	0
$209$	17.7438	1.22736
$210$	0	0
$211$	−6.56829	−0.452180	−0.226090	−	0.974106i	$-0.572594\pi$
$211$	−6.56829	−0.452180	−0.226090	+	0.974106i	$0.572594\pi$
$212$	0	0
$213$	−0.871889	−0.0597408
$214$	0	0
$215$	7.17548	0.489364
$216$	0	0
$217$	5.43171	0.368728
$218$	0	0
$219$	−6.79811	−0.459374
$220$	0	0
$221$	−7.81756	−0.525866
$222$	0	0
$223$	−4.33848	−0.290526	−0.145263	−	0.989393i	$-0.546403\pi$
$223$	−4.33848	−0.290526	−0.145263	+	0.989393i	$0.546403\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.0947	−1.33373	−0.666867	−	0.745176i	$-0.732365\pi$
$227$	−20.0947	−1.33373	−0.666867	+	0.745176i	$0.732365\pi$
$228$	0	0
$229$	−0.837003	−0.0553107	−0.0276554	−	0.999618i	$-0.508804\pi$
$229$	−0.837003	−0.0553107	−0.0276554	+	0.999618i	$0.508804\pi$
$230$	0	0
$231$	−4.87189	−0.320547
$232$	0	0
$233$	−3.21037	−0.210318	−0.105159	−	0.994455i	$-0.533535\pi$
$233$	−3.21037	−0.210318	−0.105159	+	0.994455i	$0.533535\pi$
$234$	0	0
$235$	2.56829	0.167537
$236$	0	0
$237$	12.8719	0.836119
$238$	0	0
$239$	10.3036	0.666484	0.333242	−	0.942841i	$-0.391857\pi$
$239$	10.3036	0.666484	0.333242	+	0.942841i	$0.391857\pi$
$240$	0	0
$241$	23.5264	1.51547	0.757736	−	0.652561i	$-0.226306\pi$
$241$	23.5264	1.51547	0.757736	+	0.652561i	$0.226306\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.15604	0.329407
$246$	0	0
$247$	−4.94567	−0.314685
$248$	0	0
$249$	−5.43171	−0.344220
$250$	0	0
$251$	17.7702	1.12164	0.560822	−	0.827936i	$-0.310485\pi$
$251$	17.7702	1.12164	0.560822	+	0.827936i	$0.310485\pi$
$252$	0	0
$253$	0.264700	0.0166416
$254$	0	0
$255$	−7.81756	−0.489554
$256$	0	0
$257$	−17.4442	−1.08814	−0.544069	−	0.839040i	$-0.683117\pi$
$257$	−17.4442	−1.08814	−0.544069	+	0.839040i	$0.683117\pi$
$258$	0	0
$259$	2.15604	0.133970
$260$	0	0
$261$	5.51396	0.341306
$262$	0	0
$263$	−11.1491	−0.687481	−0.343741	−	0.939065i	$-0.611694\pi$
$263$	−11.1491	−0.687481	−0.343741	+	0.939065i	$0.611694\pi$
$264$	0	0
$265$	−12.7632	−0.784039
$266$	0	0
$267$	6.87189	0.420553
$268$	0	0
$269$	−13.2966	−0.810710	−0.405355	−	0.914159i	$-0.632852\pi$
$269$	−13.2966	−0.810710	−0.405355	+	0.914159i	$0.632852\pi$
$270$	0	0
$271$	12.0947	0.734703	0.367352	−	0.930082i	$-0.380265\pi$
$271$	12.0947	0.734703	0.367352	+	0.930082i	$0.380265\pi$
$272$	0	0
$273$	1.35793	0.0821854
$274$	0	0
$275$	−3.58774	−0.216349
$276$	0	0
$277$	27.0668	1.62629	0.813144	−	0.582063i	$-0.197754\pi$
$277$	27.0668	1.62629	0.813144	+	0.582063i	$0.197754\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	17.7827	1.06083	0.530413	−	0.847740i	$-0.322037\pi$
$281$	17.7827	1.06083	0.530413	+	0.847740i	$0.322037\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	−4.94567	−0.292956
$286$	0	0
$287$	7.58774	0.447890
$288$	0	0
$289$	44.1142	2.59495
$290$	0	0
$291$	−7.35793	−0.431329
$292$	0	0
$293$	16.7159	0.976551	0.488275	−	0.872690i	$-0.337626\pi$
$293$	16.7159	0.976551	0.488275	+	0.872690i	$0.337626\pi$
$294$	0	0
$295$	−8.45963	−0.492539
$296$	0	0
$297$	3.58774	0.208182
$298$	0	0
$299$	−0.0737791	−0.00426676
$300$	0	0
$301$	9.74378	0.561622
$302$	0	0
$303$	6.94567	0.399018
$304$	0	0
$305$	4.30359	0.246423
$306$	0	0
$307$	22.0863	1.26053	0.630265	−	0.776380i	$-0.282946\pi$
$307$	22.0863	1.26053	0.630265	+	0.776380i	$0.282946\pi$
$308$	0	0
$309$	10.5683	0.601209
$310$	0	0
$311$	4.60719	0.261250	0.130625	−	0.991432i	$-0.458302\pi$
$311$	4.60719	0.261250	0.130625	+	0.991432i	$0.458302\pi$
$312$	0	0
$313$	−16.8106	−0.950191	−0.475096	−	0.879934i	$-0.657587\pi$
$313$	−16.8106	−0.950191	−0.475096	+	0.879934i	$0.657587\pi$
$314$	0	0
$315$	1.35793	0.0765105
$316$	0	0
$317$	32.3510	1.81701	0.908506	−	0.417873i	$-0.137224\pi$
$317$	32.3510	1.81701	0.908506	+	0.417873i	$0.137224\pi$
$318$	0	0
$319$	−19.7827	−1.10762
$320$	0	0
$321$	−1.69641	−0.0946841
$322$	0	0
$323$	38.6630	2.15127
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	3.40530	0.188313
$328$	0	0
$329$	3.48755	0.192275
$330$	0	0
$331$	−28.2159	−1.55089	−0.775443	−	0.631418i	$-0.782473\pi$
$331$	−28.2159	−1.55089	−0.775443	+	0.631418i	$0.782473\pi$
$332$	0	0
$333$	−1.58774	−0.0870077
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.5962	1.06747	0.533737	−	0.845651i	$-0.320787\pi$
$337$	19.5962	1.06747	0.533737	+	0.845651i	$0.320787\pi$
$338$	0	0
$339$	−16.6894	−0.906446
$340$	0	0
$341$	14.3510	0.777148
$342$	0	0
$343$	16.5070	0.891294
$344$	0	0
$345$	−0.0737791	−0.00397213
$346$	0	0
$347$	33.6436	1.80608	0.903041	−	0.429554i	$-0.141329\pi$
$347$	33.6436	1.80608	0.903041	+	0.429554i	$0.141329\pi$
$348$	0	0
$349$	23.5529	1.26076	0.630378	−	0.776289i	$-0.282900\pi$
$349$	23.5529	1.26076	0.630378	+	0.776289i	$0.282900\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	20.2034	1.07532	0.537659	−	0.843162i	$-0.319309\pi$
$353$	20.2034	1.07532	0.537659	+	0.843162i	$0.319309\pi$
$354$	0	0
$355$	−0.871889	−0.0462750
$356$	0	0
$357$	−10.6157	−0.561841
$358$	0	0
$359$	1.43171	0.0755625	0.0377813	−	0.999286i	$-0.487971\pi$
$359$	1.43171	0.0755625	0.0377813	+	0.999286i	$0.487971\pi$
$360$	0	0
$361$	5.45963	0.287349
$362$	0	0
$363$	−1.87189	−0.0982487
$364$	0	0
$365$	−6.79811	−0.355829
$366$	0	0
$367$	9.74378	0.508621	0.254311	−	0.967123i	$-0.418151\pi$
$367$	9.74378	0.508621	0.254311	+	0.967123i	$0.418151\pi$
$368$	0	0
$369$	−5.58774	−0.290886
$370$	0	0
$371$	−17.3315	−0.899808
$372$	0	0
$373$	15.6740	0.811569	0.405785	−	0.913969i	$-0.366998\pi$
$373$	15.6740	0.811569	0.405785	+	0.913969i	$0.366998\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	5.51396	0.283984
$378$	0	0
$379$	−27.1491	−1.39455	−0.697277	−	0.716802i	$-0.745605\pi$
$379$	−27.1491	−1.39455	−0.697277	+	0.716802i	$0.745605\pi$
$380$	0	0
$381$	19.7827	1.01350
$382$	0	0
$383$	9.06682	0.463293	0.231646	−	0.972800i	$-0.425589\pi$
$383$	9.06682	0.463293	0.231646	+	0.972800i	$0.425589\pi$
$384$	0	0
$385$	−4.87189	−0.248294
$386$	0	0
$387$	−7.17548	−0.364750
$388$	0	0
$389$	15.7004	0.796043	0.398021	−	0.917376i	$-0.369697\pi$
$389$	15.7004	0.796043	0.398021	+	0.917376i	$0.369697\pi$
$390$	0	0
$391$	0.576772	0.0291686
$392$	0	0
$393$	−0.798110	−0.0402593
$394$	0	0
$395$	12.8719	0.647655
$396$	0	0
$397$	11.5488	0.579620	0.289810	−	0.957084i	$-0.406408\pi$
$397$	11.5488	0.579620	0.289810	+	0.957084i	$0.406408\pi$
$398$	0	0
$399$	−6.71585	−0.336213
$400$	0	0
$401$	−16.8106	−0.839481	−0.419741	−	0.907644i	$-0.637879\pi$
$401$	−16.8106	−0.839481	−0.419741	+	0.907644i	$0.637879\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	5.69641	0.282360
$408$	0	0
$409$	16.2034	0.801207	0.400603	−	0.916252i	$-0.368801\pi$
$409$	16.2034	0.801207	0.400603	+	0.916252i	$0.368801\pi$
$410$	0	0
$411$	−13.0279	−0.642620
$412$	0	0
$413$	−11.4876	−0.565266
$414$	0	0
$415$	−5.43171	−0.266632
$416$	0	0
$417$	−8.19493	−0.401307
$418$	0	0
$419$	−34.1600	−1.66883	−0.834414	−	0.551139i	$-0.814194\pi$
$419$	−34.1600	−1.66883	−0.834414	+	0.551139i	$0.814194\pi$
$420$	0	0
$421$	−1.80908	−0.0881691	−0.0440846	−	0.999028i	$-0.514037\pi$
$421$	−1.80908	−0.0881691	−0.0440846	+	0.999028i	$0.514037\pi$
$422$	0	0
$423$	−2.56829	−0.124875
$424$	0	0
$425$	−7.81756	−0.379207
$426$	0	0
$427$	5.84396	0.282809
$428$	0	0
$429$	3.58774	0.173218
$430$	0	0
$431$	13.1366	0.632767	0.316384	−	0.948631i	$-0.397531\pi$
$431$	13.1366	0.632767	0.316384	+	0.948631i	$0.397531\pi$
$432$	0	0
$433$	−18.7547	−0.901296	−0.450648	−	0.892702i	$-0.648807\pi$
$433$	−18.7547	−0.901296	−0.450648	+	0.892702i	$0.648807\pi$
$434$	0	0
$435$	5.51396	0.264374
$436$	0	0
$437$	0.364887	0.0174549
$438$	0	0
$439$	15.2229	0.726547	0.363274	−	0.931683i	$-0.381659\pi$
$439$	15.2229	0.726547	0.363274	+	0.931683i	$0.381659\pi$
$440$	0	0
$441$	−5.15604	−0.245526
$442$	0	0
$443$	−17.1142	−0.813120	−0.406560	−	0.913624i	$-0.633272\pi$
$443$	−17.1142	−0.813120	−0.406560	+	0.913624i	$0.633272\pi$
$444$	0	0
$445$	6.87189	0.325759
$446$	0	0
$447$	−12.1560	−0.574961
$448$	0	0
$449$	−22.9666	−1.08386	−0.541931	−	0.840423i	$-0.682307\pi$
$449$	−22.9666	−1.08386	−0.541931	+	0.840423i	$0.682307\pi$
$450$	0	0
$451$	20.0474	0.943994
$452$	0	0
$453$	−12.9193	−0.607000
$454$	0	0
$455$	1.35793	0.0636606
$456$	0	0
$457$	−11.2104	−0.524399	−0.262199	−	0.965014i	$-0.584448\pi$
$457$	−11.2104	−0.524399	−0.262199	+	0.965014i	$0.584448\pi$
$458$	0	0
$459$	7.81756	0.364892
$460$	0	0
$461$	−20.6157	−0.960167	−0.480084	−	0.877223i	$-0.659394\pi$
$461$	−20.6157	−0.960167	−0.480084	+	0.877223i	$0.659394\pi$
$462$	0	0
$463$	−5.81756	−0.270365	−0.135182	−	0.990821i	$-0.543162\pi$
$463$	−5.81756	−0.270365	−0.135182	+	0.990821i	$0.543162\pi$
$464$	0	0
$465$	−4.00000	−0.185496
$466$	0	0
$467$	12.2647	0.567543	0.283771	−	0.958892i	$-0.408414\pi$
$467$	12.2647	0.567543	0.283771	+	0.958892i	$0.408414\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−9.32304	−0.429583
$472$	0	0
$473$	25.7438	1.18370
$474$	0	0
$475$	−4.94567	−0.226923
$476$	0	0
$477$	12.7632	0.584388
$478$	0	0
$479$	14.9108	0.681291	0.340646	−	0.940192i	$-0.389354\pi$
$479$	14.9108	0.681291	0.340646	+	0.940192i	$0.389354\pi$
$480$	0	0
$481$	−1.58774	−0.0723948
$482$	0	0
$483$	−0.100187	−0.00455865
$484$	0	0
$485$	−7.35793	−0.334106
$486$	0	0
$487$	−24.5334	−1.11171	−0.555857	−	0.831278i	$-0.687610\pi$
$487$	−24.5334	−1.11171	−0.555857	+	0.831278i	$0.687610\pi$
$488$	0	0
$489$	−14.7632	−0.667616
$490$	0	0
$491$	30.3246	1.36853	0.684264	−	0.729234i	$-0.260124\pi$
$491$	30.3246	1.36853	0.684264	+	0.729234i	$0.260124\pi$
$492$	0	0
$493$	−43.1057	−1.94138
$494$	0	0
$495$	3.58774	0.161257
$496$	0	0
$497$	−1.18396	−0.0531079
$498$	0	0
$499$	10.0823	0.451344	0.225672	−	0.974203i	$-0.427542\pi$
$499$	10.0823	0.451344	0.225672	+	0.974203i	$0.427542\pi$
$500$	0	0
$501$	4.45963	0.199242
$502$	0	0
$503$	−38.9317	−1.73588	−0.867940	−	0.496668i	$-0.834557\pi$
$503$	−38.9317	−1.73588	−0.867940	+	0.496668i	$0.834557\pi$
$504$	0	0
$505$	6.94567	0.309078
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−14.3594	−0.636471	−0.318236	−	0.948012i	$-0.603090\pi$
$509$	−14.3594	−0.636471	−0.318236	+	0.948012i	$0.603090\pi$
$510$	0	0
$511$	−9.23133	−0.408370
$512$	0	0
$513$	4.94567	0.218357
$514$	0	0
$515$	10.5683	0.465695
$516$	0	0
$517$	9.21438	0.405248
$518$	0	0
$519$	6.45963	0.283546
$520$	0	0
$521$	3.59622	0.157553	0.0787766	−	0.996892i	$-0.474899\pi$
$521$	3.59622	0.157553	0.0787766	+	0.996892i	$0.474899\pi$
$522$	0	0
$523$	−7.68793	−0.336170	−0.168085	−	0.985773i	$-0.553758\pi$
$523$	−7.68793	−0.336170	−0.168085	+	0.985773i	$0.553758\pi$
$524$	0	0
$525$	1.35793	0.0592648
$526$	0	0
$527$	31.2702	1.36215
$528$	0	0
$529$	−22.9946	−0.999763
$530$	0	0
$531$	8.45963	0.367117
$532$	0	0
$533$	−5.58774	−0.242032
$534$	0	0
$535$	−1.69641	−0.0733420
$536$	0	0
$537$	−8.94567	−0.386034
$538$	0	0
$539$	18.4985	0.796788
$540$	0	0
$541$	−1.41922	−0.0610170	−0.0305085	−	0.999535i	$-0.509713\pi$
$541$	−1.41922	−0.0610170	−0.0305085	+	0.999535i	$0.509713\pi$
$542$	0	0
$543$	19.3315	0.829595
$544$	0	0
$545$	3.40530	0.145867
$546$	0	0
$547$	6.86341	0.293458	0.146729	−	0.989177i	$-0.453125\pi$
$547$	6.86341	0.293458	0.146729	+	0.989177i	$0.453125\pi$
$548$	0	0
$549$	−4.30359	−0.183673
$550$	0	0
$551$	−27.2702	−1.16175
$552$	0	0
$553$	17.4791	0.743286
$554$	0	0
$555$	−1.58774	−0.0673959
$556$	0	0
$557$	35.7438	1.51451	0.757256	−	0.653118i	$-0.226539\pi$
$557$	35.7438	1.51451	0.757256	+	0.653118i	$0.226539\pi$
$558$	0	0
$559$	−7.17548	−0.303491
$560$	0	0
$561$	−28.0474	−1.18416
$562$	0	0
$563$	36.8021	1.55102	0.775512	−	0.631333i	$-0.217492\pi$
$563$	36.8021	1.55102	0.775512	+	0.631333i	$0.217492\pi$
$564$	0	0
$565$	−16.6894	−0.702130
$566$	0	0
$567$	−1.35793	−0.0570275
$568$	0	0
$569$	6.45963	0.270802	0.135401	−	0.990791i	$-0.456768\pi$
$569$	6.45963	0.270802	0.135401	+	0.990791i	$0.456768\pi$
$570$	0	0
$571$	31.3704	1.31281	0.656405	−	0.754408i	$-0.272076\pi$
$571$	31.3704	1.31281	0.656405	+	0.754408i	$0.272076\pi$
$572$	0	0
$573$	9.74378	0.407052
$574$	0	0
$575$	−0.0737791	−0.00307680
$576$	0	0
$577$	40.9541	1.70494	0.852472	−	0.522773i	$-0.175103\pi$
$577$	40.9541	1.70494	0.852472	+	0.522773i	$0.175103\pi$
$578$	0	0
$579$	4.27719	0.177754
$580$	0	0
$581$	−7.37586	−0.306002
$582$	0	0
$583$	−45.7911	−1.89648
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	36.3899	1.50197	0.750985	−	0.660319i	$-0.229579\pi$
$587$	36.3899	1.50197	0.750985	+	0.660319i	$0.229579\pi$
$588$	0	0
$589$	19.7827	0.815131
$590$	0	0
$591$	−1.54037	−0.0633623
$592$	0	0
$593$	8.03889	0.330118	0.165059	−	0.986284i	$-0.447219\pi$
$593$	8.03889	0.330118	0.165059	+	0.986284i	$0.447219\pi$
$594$	0	0
$595$	−10.6157	−0.435200
$596$	0	0
$597$	16.9193	0.692459
$598$	0	0
$599$	44.7019	1.82647	0.913236	−	0.407432i	$-0.133576\pi$
$599$	44.7019	1.82647	0.913236	+	0.407432i	$0.133576\pi$
$600$	0	0
$601$	−23.9387	−0.976480	−0.488240	−	0.872709i	$-0.662361\pi$
$601$	−23.9387	−0.976480	−0.488240	+	0.872709i	$0.662361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.87189	−0.0761031
$606$	0	0
$607$	−6.25622	−0.253932	−0.126966	−	0.991907i	$-0.540524\pi$
$607$	−6.25622	−0.253932	−0.126966	+	0.991907i	$0.540524\pi$
$608$	0	0
$609$	7.48755	0.303411
$610$	0	0
$611$	−2.56829	−0.103902
$612$	0	0
$613$	−18.4123	−0.743664	−0.371832	−	0.928300i	$-0.621270\pi$
$613$	−18.4123	−0.743664	−0.371832	+	0.928300i	$0.621270\pi$
$614$	0	0
$615$	−5.58774	−0.225319
$616$	0	0
$617$	−11.5962	−0.466846	−0.233423	−	0.972375i	$-0.574993\pi$
$617$	−11.5962	−0.466846	−0.233423	+	0.972375i	$0.574993\pi$
$618$	0	0
$619$	19.8260	0.796876	0.398438	−	0.917195i	$-0.369552\pi$
$619$	19.8260	0.796876	0.398438	+	0.917195i	$0.369552\pi$
$620$	0	0
$621$	0.0737791	0.00296065
$622$	0	0
$623$	9.33152	0.373859
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.7438	−0.708618
$628$	0	0
$629$	12.4123	0.494909
$630$	0	0
$631$	−33.0838	−1.31704	−0.658522	−	0.752561i	$-0.728818\pi$
$631$	−33.0838	−1.31704	−0.658522	+	0.752561i	$0.728818\pi$
$632$	0	0
$633$	6.56829	0.261066
$634$	0	0
$635$	19.7827	0.785051
$636$	0	0
$637$	−5.15604	−0.204290
$638$	0	0
$639$	0.871889	0.0344914
$640$	0	0
$641$	−11.4317	−0.451525	−0.225763	−	0.974182i	$-0.572487\pi$
$641$	−11.4317	−0.451525	−0.225763	+	0.974182i	$0.572487\pi$
$642$	0	0
$643$	−8.26470	−0.325928	−0.162964	−	0.986632i	$-0.552105\pi$
$643$	−8.26470	−0.325928	−0.162964	+	0.986632i	$0.552105\pi$
$644$	0	0
$645$	−7.17548	−0.282534
$646$	0	0
$647$	−19.7089	−0.774837	−0.387418	−	0.921904i	$-0.626633\pi$
$647$	−19.7089	−0.774837	−0.387418	+	0.921904i	$0.626633\pi$
$648$	0	0
$649$	−30.3510	−1.19138
$650$	0	0
$651$	−5.43171	−0.212885
$652$	0	0
$653$	−22.0778	−0.863971	−0.431985	−	0.901881i	$-0.642187\pi$
$653$	−22.0778	−0.863971	−0.431985	+	0.901881i	$0.642187\pi$
$654$	0	0
$655$	−0.798110	−0.0311847
$656$	0	0
$657$	6.79811	0.265220
$658$	0	0
$659$	−35.6087	−1.38712	−0.693559	−	0.720400i	$-0.743958\pi$
$659$	−35.6087	−1.38712	−0.693559	+	0.720400i	$0.743958\pi$
$660$	0	0
$661$	−51.2577	−1.99370	−0.996848	−	0.0793414i	$-0.974718\pi$
$661$	−51.2577	−1.99370	−0.996848	+	0.0793414i	$0.974718\pi$
$662$	0	0
$663$	7.81756	0.303609
$664$	0	0
$665$	−6.71585	−0.260430
$666$	0	0
$667$	−0.406815	−0.0157519
$668$	0	0
$669$	4.33848	0.167735
$670$	0	0
$671$	15.4402	0.596062
$672$	0	0
$673$	25.1755	0.970444	0.485222	−	0.874391i	$-0.338739\pi$
$673$	25.1755	0.970444	0.485222	+	0.874391i	$0.338739\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	13.7353	0.527890	0.263945	−	0.964538i	$-0.414976\pi$
$677$	13.7353	0.527890	0.263945	+	0.964538i	$0.414976\pi$
$678$	0	0
$679$	−9.99152	−0.383439
$680$	0	0
$681$	20.0947	0.770032
$682$	0	0
$683$	40.3899	1.54548	0.772738	−	0.634726i	$-0.218887\pi$
$683$	40.3899	1.54548	0.772738	+	0.634726i	$0.218887\pi$
$684$	0	0
$685$	−13.0279	−0.497771
$686$	0	0
$687$	0.837003	0.0319337
$688$	0	0
$689$	12.7632	0.486240
$690$	0	0
$691$	−41.9427	−1.59558	−0.797788	−	0.602938i	$-0.793997\pi$
$691$	−41.9427	−1.59558	−0.797788	+	0.602938i	$0.793997\pi$
$692$	0	0
$693$	4.87189	0.185068
$694$	0	0
$695$	−8.19493	−0.310851
$696$	0	0
$697$	43.6825	1.65459
$698$	0	0
$699$	3.21037	0.121427
$700$	0	0
$701$	−42.7283	−1.61383	−0.806914	−	0.590670i	$-0.798864\pi$
$701$	−42.7283	−1.61383	−0.806914	+	0.590670i	$0.798864\pi$
$702$	0	0
$703$	7.85244	0.296160
$704$	0	0
$705$	−2.56829	−0.0967276
$706$	0	0
$707$	9.43171	0.354716
$708$	0	0
$709$	−11.1102	−0.417252	−0.208626	−	0.977996i	$-0.566899\pi$
$709$	−11.1102	−0.417252	−0.208626	+	0.977996i	$0.566899\pi$
$710$	0	0
$711$	−12.8719	−0.482734
$712$	0	0
$713$	0.295116	0.0110522
$714$	0	0
$715$	3.58774	0.134174
$716$	0	0
$717$	−10.3036	−0.384795
$718$	0	0
$719$	8.67696	0.323596	0.161798	−	0.986824i	$-0.448271\pi$
$719$	8.67696	0.323596	0.161798	+	0.986824i	$0.448271\pi$
$720$	0	0
$721$	14.3510	0.534458
$722$	0	0
$723$	−23.5264	−0.874958
$724$	0	0
$725$	5.51396	0.204783
$726$	0	0
$727$	14.4985	0.537720	0.268860	−	0.963179i	$-0.413353\pi$
$727$	14.4985	0.537720	0.268860	+	0.963179i	$0.413353\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	56.0947	2.07474
$732$	0	0
$733$	−0.668481	−0.0246909	−0.0123455	−	0.999924i	$-0.503930\pi$
$733$	−0.668481	−0.0246909	−0.0123455	+	0.999924i	$0.503930\pi$
$734$	0	0
$735$	−5.15604	−0.190183
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−17.1102	−0.629408	−0.314704	−	0.949190i	$-0.601905\pi$
$739$	−17.1102	−0.629408	−0.314704	+	0.949190i	$0.601905\pi$
$740$	0	0
$741$	4.94567	0.181684
$742$	0	0
$743$	−5.57926	−0.204683	−0.102342	−	0.994749i	$-0.532634\pi$
$743$	−5.57926	−0.204683	−0.102342	+	0.994749i	$0.532634\pi$
$744$	0	0
$745$	−12.1560	−0.445363
$746$	0	0
$747$	5.43171	0.198736
$748$	0	0
$749$	−2.30359	−0.0841715
$750$	0	0
$751$	−5.47908	−0.199934	−0.0999672	−	0.994991i	$-0.531874\pi$
$751$	−5.47908	−0.199934	−0.0999672	+	0.994991i	$0.531874\pi$
$752$	0	0
$753$	−17.7702	−0.647582
$754$	0	0
$755$	−12.9193	−0.470180
$756$	0	0
$757$	−48.7408	−1.77152	−0.885758	−	0.464148i	$-0.846361\pi$
$757$	−48.7408	−1.77152	−0.885758	+	0.464148i	$0.846361\pi$
$758$	0	0
$759$	−0.264700	−0.00960801
$760$	0	0
$761$	−29.9472	−1.08558	−0.542792	−	0.839867i	$-0.682633\pi$
$761$	−29.9472	−1.08558	−0.542792	+	0.839867i	$0.682633\pi$
$762$	0	0
$763$	4.62414	0.167405
$764$	0	0
$765$	7.81756	0.282644
$766$	0	0
$767$	8.45963	0.305460
$768$	0	0
$769$	−24.1336	−0.870281	−0.435141	−	0.900363i	$-0.643301\pi$
$769$	−24.1336	−0.870281	−0.435141	+	0.900363i	$0.643301\pi$
$770$	0	0
$771$	17.4442	0.628237
$772$	0	0
$773$	42.5544	1.53057	0.765287	−	0.643689i	$-0.222597\pi$
$773$	42.5544	1.53057	0.765287	+	0.643689i	$0.222597\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	−2.15604	−0.0773474
$778$	0	0
$779$	27.6351	0.990131
$780$	0	0
$781$	−3.12811	−0.111933
$782$	0	0
$783$	−5.51396	−0.197053
$784$	0	0
$785$	−9.32304	−0.332754
$786$	0	0
$787$	0.676959	0.0241310	0.0120655	−	0.999927i	$-0.496159\pi$
$787$	0.676959	0.0241310	0.0120655	+	0.999927i	$0.496159\pi$
$788$	0	0
$789$	11.1491	0.396918
$790$	0	0
$791$	−22.6630	−0.805805
$792$	0	0
$793$	−4.30359	−0.152825
$794$	0	0
$795$	12.7632	0.452665
$796$	0	0
$797$	−7.84396	−0.277847	−0.138924	−	0.990303i	$-0.544364\pi$
$797$	−7.84396	−0.277847	−0.138924	+	0.990303i	$0.544364\pi$
$798$	0	0
$799$	20.0778	0.710301
$800$	0	0
$801$	−6.87189	−0.242806
$802$	0	0
$803$	−24.3899	−0.860699
$804$	0	0
$805$	−0.100187	−0.00353111
$806$	0	0
$807$	13.2966	0.468064
$808$	0	0
$809$	7.13659	0.250909	0.125455	−	0.992099i	$-0.459961\pi$
$809$	7.13659	0.250909	0.125455	+	0.992099i	$0.459961\pi$
$810$	0	0
$811$	16.4860	0.578903	0.289452	−	0.957193i	$-0.406527\pi$
$811$	16.4860	0.578903	0.289452	+	0.957193i	$0.406527\pi$
$812$	0	0
$813$	−12.0947	−0.424181
$814$	0	0
$815$	−14.7632	−0.517133
$816$	0	0
$817$	35.4876	1.24155
$818$	0	0
$819$	−1.35793	−0.0474498
$820$	0	0
$821$	−9.80507	−0.342199	−0.171100	−	0.985254i	$-0.554732\pi$
$821$	−9.80507	−0.342199	−0.171100	+	0.985254i	$0.554732\pi$
$822$	0	0
$823$	−26.5683	−0.926113	−0.463056	−	0.886329i	$-0.653247\pi$
$823$	−26.5683	−0.926113	−0.463056	+	0.886329i	$0.653247\pi$
$824$	0	0
$825$	3.58774	0.124909
$826$	0	0
$827$	17.6490	0.613717	0.306859	−	0.951755i	$-0.400722\pi$
$827$	17.6490	0.613717	0.306859	+	0.951755i	$0.400722\pi$
$828$	0	0
$829$	54.1895	1.88208	0.941039	−	0.338297i	$-0.109851\pi$
$829$	54.1895	1.88208	0.941039	+	0.338297i	$0.109851\pi$
$830$	0	0
$831$	−27.0668	−0.938938
$832$	0	0
$833$	40.3076	1.39658
$834$	0	0
$835$	4.45963	0.154332
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	−47.8300	−1.65128	−0.825638	−	0.564200i	$-0.809185\pi$
$839$	−47.8300	−1.65128	−0.825638	+	0.564200i	$0.809185\pi$
$840$	0	0
$841$	1.40378	0.0484062
$842$	0	0
$843$	−17.7827	−0.612468
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−2.54189	−0.0873403
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.117142	0.00401558
$852$	0	0
$853$	9.07530	0.310732	0.155366	−	0.987857i	$-0.450344\pi$
$853$	9.07530	0.310732	0.155366	+	0.987857i	$0.450344\pi$
$854$	0	0
$855$	4.94567	0.169138
$856$	0	0
$857$	1.54438	0.0527550	0.0263775	−	0.999652i	$-0.491603\pi$
$857$	1.54438	0.0527550	0.0263775	+	0.999652i	$0.491603\pi$
$858$	0	0
$859$	−9.69641	−0.330837	−0.165419	−	0.986223i	$-0.552897\pi$
$859$	−9.69641	−0.330837	−0.165419	+	0.986223i	$0.552897\pi$
$860$	0	0
$861$	−7.58774	−0.258590
$862$	0	0
$863$	37.9083	1.29041	0.645207	−	0.764008i	$-0.276771\pi$
$863$	37.9083	1.29041	0.645207	+	0.764008i	$0.276771\pi$
$864$	0	0
$865$	6.45963	0.219634
$866$	0	0
$867$	−44.1142	−1.49820
$868$	0	0
$869$	46.1810	1.56658
$870$	0	0
$871$	0	0
$872$	0	0
$873$	7.35793	0.249028
$874$	0	0
$875$	1.35793	0.0459063
$876$	0	0
$877$	−22.4068	−0.756624	−0.378312	−	0.925678i	$-0.623495\pi$
$877$	−22.4068	−0.756624	−0.378312	+	0.925678i	$0.623495\pi$
$878$	0	0
$879$	−16.7159	−0.563812
$880$	0	0
$881$	−32.7159	−1.10223	−0.551113	−	0.834431i	$-0.685797\pi$
$881$	−32.7159	−1.10223	−0.551113	+	0.834431i	$0.685797\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	8.45963	0.284367
$886$	0	0
$887$	49.5305	1.66307	0.831535	−	0.555472i	$-0.187463\pi$
$887$	49.5305	1.66307	0.831535	+	0.555472i	$0.187463\pi$
$888$	0	0
$889$	26.8634	0.900970
$890$	0	0
$891$	−3.58774	−0.120194
$892$	0	0
$893$	12.7019	0.425054
$894$	0	0
$895$	−8.94567	−0.299021
$896$	0	0
$897$	0.0737791	0.00246341
$898$	0	0
$899$	−22.0558	−0.735604
$900$	0	0
$901$	−99.7772	−3.32406
$902$	0	0
$903$	−9.74378	−0.324253
$904$	0	0
$905$	19.3315	0.642601
$906$	0	0
$907$	50.3679	1.67244	0.836220	−	0.548395i	$-0.184761\pi$
$907$	50.3679	1.67244	0.836220	+	0.548395i	$0.184761\pi$
$908$	0	0
$909$	−6.94567	−0.230373
$910$	0	0
$911$	−16.7717	−0.555671	−0.277836	−	0.960629i	$-0.589617\pi$
$911$	−16.7717	−0.555671	−0.277836	+	0.960629i	$0.589617\pi$
$912$	0	0
$913$	−19.4876	−0.644944
$914$	0	0
$915$	−4.30359	−0.142272
$916$	0	0
$917$	−1.08377	−0.0357894
$918$	0	0
$919$	30.1032	0.993014	0.496507	−	0.868033i	$-0.334616\pi$
$919$	30.1032	0.993014	0.496507	+	0.868033i	$0.334616\pi$
$920$	0	0
$921$	−22.0863	−0.727767
$922$	0	0
$923$	0.871889	0.0286986
$924$	0	0
$925$	−1.58774	−0.0522046
$926$	0	0
$927$	−10.5683	−0.347108
$928$	0	0
$929$	16.3983	0.538012	0.269006	−	0.963139i	$-0.413305\pi$
$929$	16.3983	0.538012	0.269006	+	0.963139i	$0.413305\pi$
$930$	0	0
$931$	25.5000	0.835730
$932$	0	0
$933$	−4.60719	−0.150833
$934$	0	0
$935$	−28.0474	−0.917247
$936$	0	0
$937$	−23.3091	−0.761476	−0.380738	−	0.924683i	$-0.624330\pi$
$937$	−23.3091	−0.761476	−0.380738	+	0.924683i	$0.624330\pi$
$938$	0	0
$939$	16.8106	0.548593
$940$	0	0
$941$	33.1531	1.08076	0.540380	−	0.841421i	$-0.318281\pi$
$941$	33.1531	1.08076	0.540380	+	0.841421i	$0.318281\pi$
$942$	0	0
$943$	0.412259	0.0134250
$944$	0	0
$945$	−1.35793	−0.0441733
$946$	0	0
$947$	43.7827	1.42275	0.711373	−	0.702815i	$-0.248074\pi$
$947$	43.7827	1.42275	0.711373	+	0.702815i	$0.248074\pi$
$948$	0	0
$949$	6.79811	0.220676
$950$	0	0
$951$	−32.3510	−1.04905
$952$	0	0
$953$	−29.2493	−0.947477	−0.473738	−	0.880666i	$-0.657096\pi$
$953$	−29.2493	−0.947477	−0.473738	+	0.880666i	$0.657096\pi$
$954$	0	0
$955$	9.74378	0.315301
$956$	0	0
$957$	19.7827	0.639483
$958$	0	0
$959$	−17.6910	−0.571271
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	1.69641	0.0546659
$964$	0	0
$965$	4.27719	0.137688
$966$	0	0
$967$	−28.5808	−0.919096	−0.459548	−	0.888153i	$-0.651989\pi$
$967$	−28.5808	−0.919096	−0.459548	+	0.888153i	$0.651989\pi$
$968$	0	0
$969$	−38.6630	−1.24204
$970$	0	0
$971$	−31.6785	−1.01661	−0.508305	−	0.861177i	$-0.669728\pi$
$971$	−31.6785	−1.01661	−0.508305	+	0.861177i	$0.669728\pi$
$972$	0	0
$973$	−11.1281	−0.356751
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	13.2702	0.424552	0.212276	−	0.977210i	$-0.431912\pi$
$977$	13.2702	0.424552	0.212276	+	0.977210i	$0.431912\pi$
$978$	0	0
$979$	24.6546	0.787963
$980$	0	0
$981$	−3.40530	−0.108723
$982$	0	0
$983$	−2.18645	−0.0697370	−0.0348685	−	0.999392i	$-0.511101\pi$
$983$	−2.18645	−0.0697370	−0.0348685	+	0.999392i	$0.511101\pi$
$984$	0	0
$985$	−1.54037	−0.0490803
$986$	0	0
$987$	−3.48755	−0.111010
$988$	0	0
$989$	0.529401	0.0168340
$990$	0	0
$991$	16.9497	0.538424	0.269212	−	0.963081i	$-0.413237\pi$
$991$	16.9497	0.538424	0.269212	+	0.963081i	$0.413237\pi$
$992$	0	0
$993$	28.2159	0.895404
$994$	0	0
$995$	16.9193	0.536377
$996$	0	0
$997$	−20.7936	−0.658541	−0.329271	−	0.944236i	$-0.606803\pi$
$997$	−20.7936	−0.658541	−0.329271	+	0.944236i	$0.606803\pi$
$998$	0	0
$999$	1.58774	0.0502339

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bw.1.2	✓	3
4.3	odd	2		6240.2.a.cb.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bw.1.2	✓	3	1.1	even	1	trivial
6240.2.a.cb.1.2	yes	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.35793 q^{7} +1.00000 q^{9} -3.58774 q^{11} +1.00000 q^{13} +1.00000 q^{15} -7.81756 q^{17} -4.94567 q^{19} +1.35793 q^{21} -0.0737791 q^{23} +1.00000 q^{25} -1.00000 q^{27} +5.51396 q^{29} -4.00000 q^{31} +3.58774 q^{33} +1.35793 q^{35} -1.58774 q^{37} -1.00000 q^{39} -5.58774 q^{41} -7.17548 q^{43} -1.00000 q^{45} -2.56829 q^{47} -5.15604 q^{49} +7.81756 q^{51} +12.7632 q^{53} +3.58774 q^{55} +4.94567 q^{57} +8.45963 q^{59} -4.30359 q^{61} -1.35793 q^{63} -1.00000 q^{65} +0.0737791 q^{69} +0.871889 q^{71} +6.79811 q^{73} -1.00000 q^{75} +4.87189 q^{77} -12.8719 q^{79} +1.00000 q^{81} +5.43171 q^{83} +7.81756 q^{85} -5.51396 q^{87} -6.87189 q^{89} -1.35793 q^{91} +4.00000 q^{93} +4.94567 q^{95} +7.35793 q^{97} -3.58774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} - 5 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} + q^{17} - 4 q^{19} + 5 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} + 2 q^{29} - 12 q^{31} - q^{33} + 5 q^{35} + 7 q^{37} - 3 q^{39} - 5 q^{41} + 2 q^{43} - 3 q^{45} - 4 q^{47} - q^{51} + 3 q^{53} - q^{55} + 4 q^{57} - 3 q^{61} - 5 q^{63} - 3 q^{65} + 3 q^{69} - 11 q^{71} + 4 q^{73} - 3 q^{75} + q^{77} - 25 q^{79} + 3 q^{81} + 20 q^{83} - q^{85} - 2 q^{87} - 7 q^{89} - 5 q^{91} + 12 q^{93} + 4 q^{95} + 23 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 5 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + 3 * q^15 + q^17 - 4 * q^19 + 5 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 + 2 * q^29 - 12 * q^31 - q^33 + 5 * q^35 + 7 * q^37 - 3 * q^39 - 5 * q^41 + 2 * q^43 - 3 * q^45 - 4 * q^47 - q^51 + 3 * q^53 - q^55 + 4 * q^57 - 3 * q^61 - 5 * q^63 - 3 * q^65 + 3 * q^69 - 11 * q^71 + 4 * q^73 - 3 * q^75 + q^77 - 25 * q^79 + 3 * q^81 + 20 * q^83 - q^85 - 2 * q^87 - 7 * q^89 - 5 * q^91 + 12 * q^93 + 4 * q^95 + 23 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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