LMFDB - Embedded newform 6045.2.a.x.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6045,2,Mod(1,6045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6045 = 3 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6045.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:	$48.2695680219$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 7 x^{10} + 43 x^{9} - 5 x^{8} - 141 x^{7} + 90 x^{6} + 165 x^{5} - 141 x^{4} + \cdots - 2 $ x^12 - 4x^11 - 7x^10 + 43x^9 - 5x^8 - 141x^7 + 90x^6 + 165x^5 - 141x^4 - 56x^3 + 64x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.697642$ of defining polynomial
Character	$\chi$	$=$	6045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.697642 q^{2} +1.00000 q^{3} -1.51330 q^{4} -1.00000 q^{5} -0.697642 q^{6} -2.51812 q^{7} +2.45102 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.697642 q^{2} +1.00000 q^{3} -1.51330 q^{4} -1.00000 q^{5} -0.697642 q^{6} -2.51812 q^{7} +2.45102 q^{8} +1.00000 q^{9} +0.697642 q^{10} -1.15996 q^{11} -1.51330 q^{12} -1.00000 q^{13} +1.75674 q^{14} -1.00000 q^{15} +1.31666 q^{16} +2.05634 q^{17} -0.697642 q^{18} +0.508654 q^{19} +1.51330 q^{20} -2.51812 q^{21} +0.809234 q^{22} -5.57157 q^{23} +2.45102 q^{24} +1.00000 q^{25} +0.697642 q^{26} +1.00000 q^{27} +3.81066 q^{28} +4.70198 q^{29} +0.697642 q^{30} +1.00000 q^{31} -5.82060 q^{32} -1.15996 q^{33} -1.43459 q^{34} +2.51812 q^{35} -1.51330 q^{36} +5.80038 q^{37} -0.354858 q^{38} -1.00000 q^{39} -2.45102 q^{40} +6.04331 q^{41} +1.75674 q^{42} -5.12449 q^{43} +1.75536 q^{44} -1.00000 q^{45} +3.88696 q^{46} +5.16799 q^{47} +1.31666 q^{48} -0.659082 q^{49} -0.697642 q^{50} +2.05634 q^{51} +1.51330 q^{52} +12.1584 q^{53} -0.697642 q^{54} +1.15996 q^{55} -6.17196 q^{56} +0.508654 q^{57} -3.28030 q^{58} -10.1341 q^{59} +1.51330 q^{60} -2.55578 q^{61} -0.697642 q^{62} -2.51812 q^{63} +1.42738 q^{64} +1.00000 q^{65} +0.809234 q^{66} +2.65609 q^{67} -3.11185 q^{68} -5.57157 q^{69} -1.75674 q^{70} -6.48905 q^{71} +2.45102 q^{72} +8.03301 q^{73} -4.04659 q^{74} +1.00000 q^{75} -0.769743 q^{76} +2.92091 q^{77} +0.697642 q^{78} -11.8765 q^{79} -1.31666 q^{80} +1.00000 q^{81} -4.21606 q^{82} +2.25615 q^{83} +3.81066 q^{84} -2.05634 q^{85} +3.57506 q^{86} +4.70198 q^{87} -2.84308 q^{88} -3.13534 q^{89} +0.697642 q^{90} +2.51812 q^{91} +8.43144 q^{92} +1.00000 q^{93} -3.60541 q^{94} -0.508654 q^{95} -5.82060 q^{96} +5.44141 q^{97} +0.459803 q^{98} -1.15996 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 12 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) $ 12 * q - 4 * q^2 + 12 * q^3 + 6 * q^4 - 12 * q^5 - 4 * q^6 + 3 * q^7 - 3 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} + 12 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 12 q^{9} + 4 q^{10} - 10 q^{11} + 6 q^{12} - 12 q^{13} - 19 q^{14} - 12 q^{15} + 2 q^{16} - 7 q^{17} - 4 q^{18} + 4 q^{19} - 6 q^{20} + 3 q^{21} - 11 q^{22} - 3 q^{23} - 3 q^{24} + 12 q^{25} + 4 q^{26} + 12 q^{27} + 8 q^{28} - 23 q^{29} + 4 q^{30} + 12 q^{31} - 10 q^{33} + 23 q^{34} - 3 q^{35} + 6 q^{36} + 9 q^{37} + 30 q^{38} - 12 q^{39} + 3 q^{40} - 9 q^{41} - 19 q^{42} - 20 q^{43} - 7 q^{44} - 12 q^{45} - 2 q^{46} - 9 q^{47} + 2 q^{48} + 9 q^{49} - 4 q^{50} - 7 q^{51} - 6 q^{52} - 20 q^{53} - 4 q^{54} + 10 q^{55} - 53 q^{56} + 4 q^{57} - 2 q^{58} - 40 q^{59} - 6 q^{60} - 2 q^{61} - 4 q^{62} + 3 q^{63} - 5 q^{64} + 12 q^{65} - 11 q^{66} - 19 q^{68} - 3 q^{69} + 19 q^{70} - 17 q^{71} - 3 q^{72} + 3 q^{73} + 4 q^{74} + 12 q^{75} - 12 q^{76} - 32 q^{77} + 4 q^{78} - 31 q^{79} - 2 q^{80} + 12 q^{81} - 39 q^{82} + 3 q^{83} + 8 q^{84} + 7 q^{85} - 60 q^{86} - 23 q^{87} - 23 q^{88} - 39 q^{89} + 4 q^{90} - 3 q^{91} - 53 q^{92} + 12 q^{93} - 24 q^{94} - 4 q^{95} - 23 q^{97} - 32 q^{98} - 10 q^{99}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^3 + 6 * q^4 - 12 * q^5 - 4 * q^6 + 3 * q^7 - 3 * q^8 + 12 * q^9 + 4 * q^10 - 10 * q^11 + 6 * q^12 - 12 * q^13 - 19 * q^14 - 12 * q^15 + 2 * q^16 - 7 * q^17 - 4 * q^18 + 4 * q^19 - 6 * q^20 + 3 * q^21 - 11 * q^22 - 3 * q^23 - 3 * q^24 + 12 * q^25 + 4 * q^26 + 12 * q^27 + 8 * q^28 - 23 * q^29 + 4 * q^30 + 12 * q^31 - 10 * q^33 + 23 * q^34 - 3 * q^35 + 6 * q^36 + 9 * q^37 + 30 * q^38 - 12 * q^39 + 3 * q^40 - 9 * q^41 - 19 * q^42 - 20 * q^43 - 7 * q^44 - 12 * q^45 - 2 * q^46 - 9 * q^47 + 2 * q^48 + 9 * q^49 - 4 * q^50 - 7 * q^51 - 6 * q^52 - 20 * q^53 - 4 * q^54 + 10 * q^55 - 53 * q^56 + 4 * q^57 - 2 * q^58 - 40 * q^59 - 6 * q^60 - 2 * q^61 - 4 * q^62 + 3 * q^63 - 5 * q^64 + 12 * q^65 - 11 * q^66 - 19 * q^68 - 3 * q^69 + 19 * q^70 - 17 * q^71 - 3 * q^72 + 3 * q^73 + 4 * q^74 + 12 * q^75 - 12 * q^76 - 32 * q^77 + 4 * q^78 - 31 * q^79 - 2 * q^80 + 12 * q^81 - 39 * q^82 + 3 * q^83 + 8 * q^84 + 7 * q^85 - 60 * q^86 - 23 * q^87 - 23 * q^88 - 39 * q^89 + 4 * q^90 - 3 * q^91 - 53 * q^92 + 12 * q^93 - 24 * q^94 - 4 * q^95 - 23 * q^97 - 32 * q^98 - 10 * 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 $

$2$	−0.697642	−0.493307	−0.246654	−	0.969104i	$-0.579331\pi$
$2$	−0.697642	−0.493307	−0.246654	+	0.969104i	$0.579331\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.51330	−0.756648
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.697642	−0.284811
$7$	−2.51812	−0.951759	−0.475880	−	0.879510i	$-0.657870\pi$
$7$	−2.51812	−0.951759	−0.475880	+	0.879510i	$0.657870\pi$
$8$	2.45102	0.866567
$9$	1.00000	0.333333
$10$	0.697642	0.220614
$11$	−1.15996	−0.349740	−0.174870	−	0.984592i	$-0.555951\pi$
$11$	−1.15996	−0.349740	−0.174870	+	0.984592i	$0.555951\pi$
$12$	−1.51330	−0.436851
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.75674	0.469510
$15$	−1.00000	−0.258199
$16$	1.31666	0.329164
$17$	2.05634	0.498736	0.249368	−	0.968409i	$-0.419777\pi$
$17$	2.05634	0.498736	0.249368	+	0.968409i	$0.419777\pi$
$18$	−0.697642	−0.164436
$19$	0.508654	0.116693	0.0583466	−	0.998296i	$-0.481417\pi$
$19$	0.508654	0.116693	0.0583466	+	0.998296i	$0.481417\pi$
$20$	1.51330	0.338383
$21$	−2.51812	−0.549498
$22$	0.809234	0.172529
$23$	−5.57157	−1.16175	−0.580877	−	0.813992i	$-0.697290\pi$
$23$	−5.57157	−1.16175	−0.580877	+	0.813992i	$0.697290\pi$
$24$	2.45102	0.500313
$25$	1.00000	0.200000
$26$	0.697642	0.136819
$27$	1.00000	0.192450
$28$	3.81066	0.720147
$29$	4.70198	0.873136	0.436568	−	0.899671i	$-0.356194\pi$
$29$	4.70198	0.873136	0.436568	+	0.899671i	$0.356194\pi$
$30$	0.697642	0.127371
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−5.82060	−1.02895
$33$	−1.15996	−0.201923
$34$	−1.43459	−0.246030
$35$	2.51812	0.425640
$36$	−1.51330	−0.252216
$37$	5.80038	0.953576	0.476788	−	0.879018i	$-0.341801\pi$
$37$	5.80038	0.953576	0.476788	+	0.879018i	$0.341801\pi$
$38$	−0.354858	−0.0575656
$39$	−1.00000	−0.160128
$40$	−2.45102	−0.387541
$41$	6.04331	0.943806	0.471903	−	0.881651i	$-0.343567\pi$
$41$	6.04331	0.943806	0.471903	+	0.881651i	$0.343567\pi$
$42$	1.75674	0.271072
$43$	−5.12449	−0.781477	−0.390738	−	0.920502i	$-0.627780\pi$
$43$	−5.12449	−0.781477	−0.390738	+	0.920502i	$0.627780\pi$
$44$	1.75536	0.264630
$45$	−1.00000	−0.149071
$46$	3.88696	0.573101
$47$	5.16799	0.753829	0.376915	−	0.926248i	$-0.376985\pi$
$47$	5.16799	0.753829	0.376915	+	0.926248i	$0.376985\pi$
$48$	1.31666	0.190043
$49$	−0.659082	−0.0941546
$50$	−0.697642	−0.0986614
$51$	2.05634	0.287945
$52$	1.51330	0.209856
$53$	12.1584	1.67008	0.835040	−	0.550190i	$-0.185445\pi$
$53$	12.1584	1.67008	0.835040	+	0.550190i	$0.185445\pi$
$54$	−0.697642	−0.0949370
$55$	1.15996	0.156409
$56$	−6.17196	−0.824763
$57$	0.508654	0.0673728
$58$	−3.28030	−0.430725
$59$	−10.1341	−1.31935	−0.659674	−	0.751552i	$-0.729306\pi$
$59$	−10.1341	−1.31935	−0.659674	+	0.751552i	$0.729306\pi$
$60$	1.51330	0.195366
$61$	−2.55578	−0.327234	−0.163617	−	0.986524i	$-0.552316\pi$
$61$	−2.55578	−0.327234	−0.163617	+	0.986524i	$0.552316\pi$
$62$	−0.697642	−0.0886006
$63$	−2.51812	−0.317253
$64$	1.42738	0.178422
$65$	1.00000	0.124035
$66$	0.809234	0.0996099
$67$	2.65609	0.324494	0.162247	−	0.986750i	$-0.448126\pi$
$67$	2.65609	0.324494	0.162247	+	0.986750i	$0.448126\pi$
$68$	−3.11185	−0.377367
$69$	−5.57157	−0.670739
$70$	−1.75674	−0.209971
$71$	−6.48905	−0.770109	−0.385054	−	0.922894i	$-0.625817\pi$
$71$	−6.48905	−0.770109	−0.385054	+	0.922894i	$0.625817\pi$
$72$	2.45102	0.288856
$73$	8.03301	0.940192	0.470096	−	0.882615i	$-0.344219\pi$
$73$	8.03301	0.940192	0.470096	+	0.882615i	$0.344219\pi$
$74$	−4.04659	−0.470406
$75$	1.00000	0.115470
$76$	−0.769743	−0.0882956
$77$	2.92091	0.332868
$78$	0.697642	0.0789924
$79$	−11.8765	−1.33621	−0.668105	−	0.744067i	$-0.732894\pi$
$79$	−11.8765	−1.33621	−0.668105	+	0.744067i	$0.732894\pi$
$80$	−1.31666	−0.147207
$81$	1.00000	0.111111
$82$	−4.21606	−0.465586
$83$	2.25615	0.247645	0.123822	−	0.992304i	$-0.460485\pi$
$83$	2.25615	0.247645	0.123822	+	0.992304i	$0.460485\pi$
$84$	3.81066	0.415777
$85$	−2.05634	−0.223041
$86$	3.57506	0.385508
$87$	4.70198	0.504106
$88$	−2.84308	−0.303073
$89$	−3.13534	−0.332345	−0.166173	−	0.986097i	$-0.553141\pi$
$89$	−3.13534	−0.332345	−0.166173	+	0.986097i	$0.553141\pi$
$90$	0.697642	0.0735379
$91$	2.51812	0.263970
$92$	8.43144	0.879038
$93$	1.00000	0.103695
$94$	−3.60541	−0.371870
$95$	−0.508654	−0.0521868
$96$	−5.82060	−0.594062
$97$	5.44141	0.552491	0.276246	−	0.961087i	$-0.410910\pi$
$97$	5.44141	0.552491	0.276246	+	0.961087i	$0.410910\pi$
$98$	0.459803	0.0464471
$99$	−1.15996	−0.116580
$100$	−1.51330	−0.151330
$101$	19.3950	1.92987	0.964937	−	0.262480i	$-0.0845404\pi$
$101$	19.3950	1.92987	0.964937	+	0.262480i	$0.0845404\pi$
$102$	−1.43459	−0.142045
$103$	−3.14084	−0.309476	−0.154738	−	0.987956i	$-0.549453\pi$
$103$	−3.14084	−0.309476	−0.154738	+	0.987956i	$0.549453\pi$
$104$	−2.45102	−0.240342
$105$	2.51812	0.245743
$106$	−8.48218	−0.823862
$107$	−17.0084	−1.64427	−0.822133	−	0.569295i	$-0.807216\pi$
$107$	−17.0084	−1.64427	−0.822133	+	0.569295i	$0.807216\pi$
$108$	−1.51330	−0.145617
$109$	5.55510	0.532082	0.266041	−	0.963962i	$-0.414284\pi$
$109$	5.55510	0.532082	0.266041	+	0.963962i	$0.414284\pi$
$110$	−0.809234	−0.0771575
$111$	5.80038	0.550548
$112$	−3.31550	−0.313285
$113$	−9.65590	−0.908351	−0.454175	−	0.890912i	$-0.650066\pi$
$113$	−9.65590	−0.908351	−0.454175	+	0.890912i	$0.650066\pi$
$114$	−0.354858	−0.0332355
$115$	5.57157	0.519552
$116$	−7.11549	−0.660657
$117$	−1.00000	−0.0924500
$118$	7.06997	0.650844
$119$	−5.17810	−0.474676
$120$	−2.45102	−0.223747
$121$	−9.65450	−0.877682
$122$	1.78302	0.161427
$123$	6.04331	0.544906
$124$	−1.51330	−0.135898
$125$	−1.00000	−0.0894427
$126$	1.75674	0.156503
$127$	−18.1972	−1.61474	−0.807372	−	0.590043i	$-0.799111\pi$
$127$	−18.1972	−1.61474	−0.807372	+	0.590043i	$0.799111\pi$
$128$	10.6454	0.940929
$129$	−5.12449	−0.451186
$130$	−0.697642	−0.0611872
$131$	13.4133	1.17192	0.585961	−	0.810339i	$-0.300717\pi$
$131$	13.4133	1.17192	0.585961	+	0.810339i	$0.300717\pi$
$132$	1.75536	0.152784
$133$	−1.28085	−0.111064
$134$	−1.85300	−0.160075
$135$	−1.00000	−0.0860663
$136$	5.04013	0.432188
$137$	2.24002	0.191378	0.0956888	−	0.995411i	$-0.469495\pi$
$137$	2.24002	0.191378	0.0956888	+	0.995411i	$0.469495\pi$
$138$	3.88696	0.330880
$139$	8.93151	0.757561	0.378781	−	0.925487i	$-0.376344\pi$
$139$	8.93151	0.757561	0.378781	+	0.925487i	$0.376344\pi$
$140$	−3.81066	−0.322059
$141$	5.16799	0.435224
$142$	4.52703	0.379900
$143$	1.15996	0.0970005
$144$	1.31666	0.109721
$145$	−4.70198	−0.390478
$146$	−5.60416	−0.463804
$147$	−0.659082	−0.0543602
$148$	−8.77769	−0.721522
$149$	−19.4968	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$149$	−19.4968	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$150$	−0.697642	−0.0569622
$151$	−0.411591	−0.0334948	−0.0167474	−	0.999860i	$-0.505331\pi$
$151$	−0.411591	−0.0334948	−0.0167474	+	0.999860i	$0.505331\pi$
$152$	1.24672	0.101122
$153$	2.05634	0.166245
$154$	−2.03775	−0.164206
$155$	−1.00000	−0.0803219
$156$	1.51330	0.121161
$157$	8.17071	0.652094	0.326047	−	0.945354i	$-0.394283\pi$
$157$	8.17071	0.652094	0.326047	+	0.945354i	$0.394283\pi$
$158$	8.28553	0.659162
$159$	12.1584	0.964221
$160$	5.82060	0.460159
$161$	14.0299	1.10571
$162$	−0.697642	−0.0548119
$163$	1.92142	0.150497	0.0752486	−	0.997165i	$-0.476025\pi$
$163$	1.92142	0.150497	0.0752486	+	0.997165i	$0.476025\pi$
$164$	−9.14531	−0.714129
$165$	1.15996	0.0903025
$166$	−1.57399	−0.122165
$167$	−12.8540	−0.994670	−0.497335	−	0.867559i	$-0.665688\pi$
$167$	−12.8540	−0.994670	−0.497335	+	0.867559i	$0.665688\pi$
$168$	−6.17196	−0.476177
$169$	1.00000	0.0769231
$170$	1.43459	0.110028
$171$	0.508654	0.0388977
$172$	7.75487	0.591303
$173$	18.1180	1.37748	0.688742	−	0.725006i	$-0.258163\pi$
$173$	18.1180	1.37748	0.688742	+	0.725006i	$0.258163\pi$
$174$	−3.28030	−0.248679
$175$	−2.51812	−0.190352
$176$	−1.52726	−0.115122
$177$	−10.1341	−0.761726
$178$	2.18734	0.163948
$179$	−13.8098	−1.03219	−0.516097	−	0.856530i	$-0.672616\pi$
$179$	−13.8098	−1.03219	−0.516097	+	0.856530i	$0.672616\pi$
$180$	1.51330	0.112794
$181$	−15.0475	−1.11847	−0.559235	−	0.829009i	$-0.688905\pi$
$181$	−15.0475	−1.11847	−0.559235	+	0.829009i	$0.688905\pi$
$182$	−1.75674	−0.130219
$183$	−2.55578	−0.188929
$184$	−13.6560	−1.00674
$185$	−5.80038	−0.426452
$186$	−0.697642	−0.0511536
$187$	−2.38526	−0.174428
$188$	−7.82071	−0.570384
$189$	−2.51812	−0.183166
$190$	0.354858	0.0257441
$191$	8.12473	0.587885	0.293942	−	0.955823i	$-0.405033\pi$
$191$	8.12473	0.587885	0.293942	+	0.955823i	$0.405033\pi$
$192$	1.42738	0.103012
$193$	−10.8479	−0.780849	−0.390425	−	0.920635i	$-0.627672\pi$
$193$	−10.8479	−0.780849	−0.390425	+	0.920635i	$0.627672\pi$
$194$	−3.79615	−0.272548
$195$	1.00000	0.0716115
$196$	0.997386	0.0712419
$197$	−9.23074	−0.657663	−0.328832	−	0.944389i	$-0.606655\pi$
$197$	−9.23074	−0.657663	−0.328832	+	0.944389i	$0.606655\pi$
$198$	0.809234	0.0575098
$199$	−16.0489	−1.13768	−0.568839	−	0.822449i	$-0.692607\pi$
$199$	−16.0489	−1.13768	−0.568839	+	0.822449i	$0.692607\pi$
$200$	2.45102	0.173313
$201$	2.65609	0.187346
$202$	−13.5308	−0.952021
$203$	−11.8401	−0.831016
$204$	−3.11185	−0.217873
$205$	−6.04331	−0.422083
$206$	2.19118	0.152667
$207$	−5.57157	−0.387251
$208$	−1.31666	−0.0912937
$209$	−0.590016	−0.0408123
$210$	−1.75674	−0.121227
$211$	−19.4982	−1.34231	−0.671156	−	0.741316i	$-0.734202\pi$
$211$	−19.4982	−1.34231	−0.671156	+	0.741316i	$0.734202\pi$
$212$	−18.3992	−1.26366
$213$	−6.48905	−0.444623
$214$	11.8658	0.811128
$215$	5.12449	0.349487
$216$	2.45102	0.166771
$217$	−2.51812	−0.170941
$218$	−3.87547	−0.262480
$219$	8.03301	0.542820
$220$	−1.75536	−0.118346
$221$	−2.05634	−0.138324
$222$	−4.04659	−0.271589
$223$	6.02868	0.403710	0.201855	−	0.979415i	$-0.435303\pi$
$223$	6.02868	0.403710	0.201855	+	0.979415i	$0.435303\pi$
$224$	14.6570	0.979309
$225$	1.00000	0.0666667
$226$	6.73636	0.448096
$227$	−5.62958	−0.373648	−0.186824	−	0.982393i	$-0.559819\pi$
$227$	−5.62958	−0.373648	−0.186824	+	0.982393i	$0.559819\pi$
$228$	−0.769743	−0.0509775
$229$	24.0282	1.58783	0.793913	−	0.608032i	$-0.208041\pi$
$229$	24.0282	1.58783	0.793913	+	0.608032i	$0.208041\pi$
$230$	−3.88696	−0.256299
$231$	2.92091	0.192182
$232$	11.5247	0.756631
$233$	5.36054	0.351181	0.175590	−	0.984463i	$-0.443817\pi$
$233$	5.36054	0.351181	0.175590	+	0.984463i	$0.443817\pi$
$234$	0.697642	0.0456063
$235$	−5.16799	−0.337123
$236$	15.3359	0.998282
$237$	−11.8765	−0.771461
$238$	3.61246	0.234161
$239$	−9.75975	−0.631305	−0.315653	−	0.948875i	$-0.602223\pi$
$239$	−9.75975	−0.631305	−0.315653	+	0.948875i	$0.602223\pi$
$240$	−1.31666	−0.0849898
$241$	30.3443	1.95465	0.977325	−	0.211743i	$-0.0679138\pi$
$241$	30.3443	1.95465	0.977325	+	0.211743i	$0.0679138\pi$
$242$	6.73538	0.432967
$243$	1.00000	0.0641500
$244$	3.86765	0.247601
$245$	0.659082	0.0421072
$246$	−4.21606	−0.268806
$247$	−0.508654	−0.0323648
$248$	2.45102	0.155640
$249$	2.25615	0.142978
$250$	0.697642	0.0441227
$251$	−20.3095	−1.28192	−0.640962	−	0.767573i	$-0.721464\pi$
$251$	−20.3095	−1.28192	−0.640962	+	0.767573i	$0.721464\pi$
$252$	3.81066	0.240049
$253$	6.46278	0.406312
$254$	12.6952	0.796565
$255$	−2.05634	−0.128773
$256$	−10.2814	−0.642590
$257$	−26.6013	−1.65934	−0.829672	−	0.558251i	$-0.811473\pi$
$257$	−26.6013	−1.65934	−0.829672	+	0.558251i	$0.811473\pi$
$258$	3.57506	0.222573
$259$	−14.6060	−0.907575
$260$	−1.51330	−0.0938506
$261$	4.70198	0.291045
$262$	−9.35765	−0.578117
$263$	−0.631739	−0.0389547	−0.0194773	−	0.999810i	$-0.506200\pi$
$263$	−0.631739	−0.0389547	−0.0194773	+	0.999810i	$0.506200\pi$
$264$	−2.84308	−0.174979
$265$	−12.1584	−0.746882
$266$	0.893574	0.0547885
$267$	−3.13534	−0.191880
$268$	−4.01946	−0.245527
$269$	−9.73289	−0.593425	−0.296712	−	0.954967i	$-0.595890\pi$
$269$	−9.73289	−0.593425	−0.296712	+	0.954967i	$0.595890\pi$
$270$	0.697642	0.0424571
$271$	1.53188	0.0930552	0.0465276	−	0.998917i	$-0.485184\pi$
$271$	1.53188	0.0930552	0.0465276	+	0.998917i	$0.485184\pi$
$272$	2.70749	0.164166
$273$	2.51812	0.152403
$274$	−1.56273	−0.0944080
$275$	−1.15996	−0.0699480
$276$	8.43144	0.507513
$277$	−14.3703	−0.863427	−0.431714	−	0.902011i	$-0.642091\pi$
$277$	−14.3703	−0.863427	−0.431714	+	0.902011i	$0.642091\pi$
$278$	−6.23100	−0.373710
$279$	1.00000	0.0598684
$280$	6.17196	0.368845
$281$	−15.3903	−0.918107	−0.459054	−	0.888409i	$-0.651811\pi$
$281$	−15.3903	−0.918107	−0.459054	+	0.888409i	$0.651811\pi$
$282$	−3.60541	−0.214699
$283$	−12.7761	−0.759461	−0.379730	−	0.925097i	$-0.623983\pi$
$283$	−12.7761	−0.759461	−0.379730	+	0.925097i	$0.623983\pi$
$284$	9.81986	0.582701
$285$	−0.508654	−0.0301300
$286$	−0.809234	−0.0478510
$287$	−15.2178	−0.898276
$288$	−5.82060	−0.342982
$289$	−12.7715	−0.751263
$290$	3.28030	0.192626
$291$	5.44141	0.318981
$292$	−12.1563	−0.711395
$293$	17.6272	1.02979	0.514896	−	0.857253i	$-0.327831\pi$
$293$	17.6272	1.02979	0.514896	+	0.857253i	$0.327831\pi$
$294$	0.459803	0.0268163
$295$	10.1341	0.590030
$296$	14.2169	0.826338
$297$	−1.15996	−0.0673075
$298$	13.6018	0.787931
$299$	5.57157	0.322212
$300$	−1.51330	−0.0873702
$301$	12.9041	0.743778
$302$	0.287143	0.0165232
$303$	19.3950	1.11421
$304$	0.669722	0.0384112
$305$	2.55578	0.146343
$306$	−1.43459	−0.0820099
$307$	−9.36715	−0.534612	−0.267306	−	0.963612i	$-0.586133\pi$
$307$	−9.36715	−0.534612	−0.267306	+	0.963612i	$0.586133\pi$
$308$	−4.42020	−0.251864
$309$	−3.14084	−0.178676
$310$	0.697642	0.0396234
$311$	−6.20566	−0.351890	−0.175945	−	0.984400i	$-0.556298\pi$
$311$	−6.20566	−0.351890	−0.175945	+	0.984400i	$0.556298\pi$
$312$	−2.45102	−0.138762
$313$	−4.36870	−0.246934	−0.123467	−	0.992349i	$-0.539401\pi$
$313$	−4.36870	−0.246934	−0.123467	+	0.992349i	$0.539401\pi$
$314$	−5.70023	−0.321683
$315$	2.51812	0.141880
$316$	17.9726	1.01104
$317$	−2.10287	−0.118109	−0.0590545	−	0.998255i	$-0.518809\pi$
$317$	−2.10287	−0.118109	−0.0590545	+	0.998255i	$0.518809\pi$
$318$	−8.48218	−0.475657
$319$	−5.45410	−0.305371
$320$	−1.42738	−0.0797929
$321$	−17.0084	−0.949318
$322$	−9.78783	−0.545454
$323$	1.04596	0.0581990
$324$	−1.51330	−0.0840720
$325$	−1.00000	−0.0554700
$326$	−1.34046	−0.0742414
$327$	5.55510	0.307198
$328$	14.8123	0.817871
$329$	−13.0136	−0.717464
$330$	−0.809234	−0.0445469
$331$	14.7456	0.810489	0.405245	−	0.914208i	$-0.367186\pi$
$331$	14.7456	0.810489	0.405245	+	0.914208i	$0.367186\pi$
$332$	−3.41423	−0.187380
$333$	5.80038	0.317859
$334$	8.96746	0.490678
$335$	−2.65609	−0.145118
$336$	−3.31550	−0.180875
$337$	−25.1882	−1.37209	−0.686046	−	0.727558i	$-0.740655\pi$
$337$	−25.1882	−1.37209	−0.686046	+	0.727558i	$0.740655\pi$
$338$	−0.697642	−0.0379467
$339$	−9.65590	−0.524437
$340$	3.11185	0.168764
$341$	−1.15996	−0.0628152
$342$	−0.354858	−0.0191885
$343$	19.2865	1.04137
$344$	−12.5602	−0.677202
$345$	5.57157	0.299963
$346$	−12.6399	−0.679523
$347$	−26.4666	−1.42080	−0.710400	−	0.703798i	$-0.751486\pi$
$347$	−26.4666	−1.42080	−0.710400	+	0.703798i	$0.751486\pi$
$348$	−7.11549	−0.381430
$349$	−27.2632	−1.45936	−0.729682	−	0.683787i	$-0.760332\pi$
$349$	−27.2632	−1.45936	−0.729682	+	0.683787i	$0.760332\pi$
$350$	1.75674	0.0939019
$351$	−1.00000	−0.0533761
$352$	6.75164	0.359864
$353$	−34.5584	−1.83936	−0.919680	−	0.392670i	$-0.871552\pi$
$353$	−34.5584	−1.83936	−0.919680	+	0.392670i	$0.871552\pi$
$354$	7.06997	0.375765
$355$	6.48905	0.344403
$356$	4.74469	0.251468
$357$	−5.17810	−0.274054
$358$	9.63430	0.509188
$359$	−26.1302	−1.37910	−0.689550	−	0.724238i	$-0.742191\pi$
$359$	−26.1302	−1.37910	−0.689550	+	0.724238i	$0.742191\pi$
$360$	−2.45102	−0.129180
$361$	−18.7413	−0.986383
$362$	10.4977	0.551749
$363$	−9.65450	−0.506730
$364$	−3.81066	−0.199733
$365$	−8.03301	−0.420467
$366$	1.78302	0.0931998
$367$	28.0594	1.46469	0.732344	−	0.680935i	$-0.238426\pi$
$367$	28.0594	1.46469	0.732344	+	0.680935i	$0.238426\pi$
$368$	−7.33585	−0.382408
$369$	6.04331	0.314602
$370$	4.04659	0.210372
$371$	−30.6162	−1.58951
$372$	−1.51330	−0.0784607
$373$	19.2421	0.996319	0.498160	−	0.867085i	$-0.334009\pi$
$373$	19.2421	0.996319	0.498160	+	0.867085i	$0.334009\pi$
$374$	1.66406	0.0860465
$375$	−1.00000	−0.0516398
$376$	12.6669	0.653244
$377$	−4.70198	−0.242164
$378$	1.75674	0.0903572
$379$	29.0798	1.49373	0.746865	−	0.664975i	$-0.231558\pi$
$379$	29.0798	1.49373	0.746865	+	0.664975i	$0.231558\pi$
$380$	0.769743	0.0394870
$381$	−18.1972	−0.932273
$382$	−5.66815	−0.290008
$383$	−7.58325	−0.387486	−0.193743	−	0.981052i	$-0.562063\pi$
$383$	−7.58325	−0.387486	−0.193743	+	0.981052i	$0.562063\pi$
$384$	10.6454	0.543246
$385$	−2.92091	−0.148863
$386$	7.56795	0.385199
$387$	−5.12449	−0.260492
$388$	−8.23446	−0.418041
$389$	−14.0372	−0.711713	−0.355857	−	0.934541i	$-0.615811\pi$
$389$	−14.0372	−0.711713	−0.355857	+	0.934541i	$0.615811\pi$
$390$	−0.697642	−0.0353265
$391$	−11.4570	−0.579408
$392$	−1.61542	−0.0815913
$393$	13.4133	0.676609
$394$	6.43975	0.324430
$395$	11.8765	0.597571
$396$	1.75536	0.0882101
$397$	9.09892	0.456662	0.228331	−	0.973584i	$-0.426673\pi$
$397$	9.09892	0.456662	0.228331	+	0.973584i	$0.426673\pi$
$398$	11.1964	0.561224
$399$	−1.28085	−0.0641227
$400$	1.31666	0.0658328
$401$	−19.6138	−0.979467	−0.489733	−	0.871872i	$-0.662906\pi$
$401$	−19.6138	−0.979467	−0.489733	+	0.871872i	$0.662906\pi$
$402$	−1.85300	−0.0924194
$403$	−1.00000	−0.0498135
$404$	−29.3504	−1.46024
$405$	−1.00000	−0.0496904
$406$	8.26018	0.409946
$407$	−6.72819	−0.333504
$408$	5.04013	0.249524
$409$	−17.7495	−0.877654	−0.438827	−	0.898572i	$-0.644606\pi$
$409$	−17.7495	−0.877654	−0.438827	+	0.898572i	$0.644606\pi$
$410$	4.21606	0.208216
$411$	2.24002	0.110492
$412$	4.75301	0.234164
$413$	25.5189	1.25570
$414$	3.88696	0.191034
$415$	−2.25615	−0.110750
$416$	5.82060	0.285378
$417$	8.93151	0.437378
$418$	0.411620	0.0201330
$419$	−30.1607	−1.47345	−0.736723	−	0.676195i	$-0.763628\pi$
$419$	−30.1607	−1.47345	−0.736723	+	0.676195i	$0.763628\pi$
$420$	−3.81066	−0.185941
$421$	24.3287	1.18571	0.592854	−	0.805310i	$-0.298001\pi$
$421$	24.3287	1.18571	0.592854	+	0.805310i	$0.298001\pi$
$422$	13.6028	0.662172
$423$	5.16799	0.251276
$424$	29.8004	1.44724
$425$	2.05634	0.0997471
$426$	4.52703	0.219336
$427$	6.43575	0.311448
$428$	25.7388	1.24413
$429$	1.15996	0.0560032
$430$	−3.57506	−0.172405
$431$	10.7862	0.519552	0.259776	−	0.965669i	$-0.416351\pi$
$431$	10.7862	0.519552	0.259776	+	0.965669i	$0.416351\pi$
$432$	1.31666	0.0633477
$433$	19.5055	0.937375	0.468688	−	0.883364i	$-0.344727\pi$
$433$	19.5055	0.937375	0.468688	+	0.883364i	$0.344727\pi$
$434$	1.75674	0.0843264
$435$	−4.70198	−0.225443
$436$	−8.40651	−0.402599
$437$	−2.83400	−0.135569
$438$	−5.60416	−0.267777
$439$	5.90233	0.281703	0.140851	−	0.990031i	$-0.455016\pi$
$439$	5.90233	0.281703	0.140851	+	0.990031i	$0.455016\pi$
$440$	2.84308	0.135539
$441$	−0.659082	−0.0313849
$442$	1.43459	0.0682364
$443$	−17.9045	−0.850668	−0.425334	−	0.905036i	$-0.639843\pi$
$443$	−17.9045	−0.850668	−0.425334	+	0.905036i	$0.639843\pi$
$444$	−8.77769	−0.416571
$445$	3.13534	0.148629
$446$	−4.20586	−0.199153
$447$	−19.4968	−0.922168
$448$	−3.59431	−0.169815
$449$	−9.58722	−0.452449	−0.226224	−	0.974075i	$-0.572638\pi$
$449$	−9.58722	−0.452449	−0.226224	+	0.974075i	$0.572638\pi$
$450$	−0.697642	−0.0328871
$451$	−7.00997	−0.330087
$452$	14.6122	0.687302
$453$	−0.411591	−0.0193382
$454$	3.92743	0.184323
$455$	−2.51812	−0.118051
$456$	1.24672	0.0583831
$457$	−0.854352	−0.0399649	−0.0199824	−	0.999800i	$-0.506361\pi$
$457$	−0.854352	−0.0399649	−0.0199824	+	0.999800i	$0.506361\pi$
$458$	−16.7630	−0.783286
$459$	2.05634	0.0959817
$460$	−8.43144	−0.393118
$461$	−36.0672	−1.67982	−0.839908	−	0.542728i	$-0.817391\pi$
$461$	−36.0672	−1.67982	−0.839908	+	0.542728i	$0.817391\pi$
$462$	−2.03775	−0.0948046
$463$	−2.18339	−0.101471	−0.0507353	−	0.998712i	$-0.516156\pi$
$463$	−2.18339	−0.101471	−0.0507353	+	0.998712i	$0.516156\pi$
$464$	6.19090	0.287405
$465$	−1.00000	−0.0463739
$466$	−3.73974	−0.173240
$467$	−2.69215	−0.124578	−0.0622890	−	0.998058i	$-0.519840\pi$
$467$	−2.69215	−0.124578	−0.0622890	+	0.998058i	$0.519840\pi$
$468$	1.51330	0.0699521
$469$	−6.68836	−0.308840
$470$	3.60541	0.166305
$471$	8.17071	0.376486
$472$	−24.8389	−1.14330
$473$	5.94418	0.273314
$474$	8.28553	0.380567
$475$	0.508654	0.0233386
$476$	7.83601	0.359163
$477$	12.1584	0.556693
$478$	6.80881	0.311428
$479$	−33.0150	−1.50849	−0.754246	−	0.656592i	$-0.771997\pi$
$479$	−33.0150	−1.50849	−0.754246	+	0.656592i	$0.771997\pi$
$480$	5.82060	0.265673
$481$	−5.80038	−0.264475
$482$	−21.1695	−0.964243
$483$	14.0299	0.638382
$484$	14.6101	0.664096
$485$	−5.44141	−0.247082
$486$	−0.697642	−0.0316457
$487$	0.0287749	0.00130392	0.000651958	−	1.00000i	$-0.499792\pi$
$487$	0.0287749	0.00130392	0.000651958		1.00000i	$0.499792\pi$
$488$	−6.26427	−0.283570
$489$	1.92142	0.0868896
$490$	−0.459803	−0.0207718
$491$	36.2483	1.63586	0.817931	−	0.575317i	$-0.195121\pi$
$491$	36.2483	1.63586	0.817931	+	0.575317i	$0.195121\pi$
$492$	−9.14531	−0.412302
$493$	9.66887	0.435464
$494$	0.354858	0.0159658
$495$	1.15996	0.0521362
$496$	1.31666	0.0591196
$497$	16.3402	0.732958
$498$	−1.57399	−0.0705320
$499$	17.3840	0.778216	0.389108	−	0.921192i	$-0.372783\pi$
$499$	17.3840	0.778216	0.389108	+	0.921192i	$0.372783\pi$
$500$	1.51330	0.0676767
$501$	−12.8540	−0.574273
$502$	14.1688	0.632382
$503$	26.2081	1.16856	0.584280	−	0.811552i	$-0.301377\pi$
$503$	26.2081	1.16856	0.584280	+	0.811552i	$0.301377\pi$
$504$	−6.17196	−0.274921
$505$	−19.3950	−0.863066
$506$	−4.50871	−0.200437
$507$	1.00000	0.0444116
$508$	27.5378	1.22179
$509$	32.0889	1.42231	0.711157	−	0.703033i	$-0.248171\pi$
$509$	32.0889	1.42231	0.711157	+	0.703033i	$0.248171\pi$
$510$	1.43459	0.0635246
$511$	−20.2281	−0.894836
$512$	−14.1180	−0.623935
$513$	0.508654	0.0224576
$514$	18.5582	0.818567
$515$	3.14084	0.138402
$516$	7.75487	0.341389
$517$	−5.99465	−0.263644
$518$	10.1898	0.447713
$519$	18.1180	0.795291
$520$	2.45102	0.107484
$521$	−20.3949	−0.893517	−0.446758	−	0.894655i	$-0.647422\pi$
$521$	−20.3949	−0.893517	−0.446758	+	0.894655i	$0.647422\pi$
$522$	−3.28030	−0.143575
$523$	11.5355	0.504411	0.252205	−	0.967674i	$-0.418844\pi$
$523$	11.5355	0.504411	0.252205	+	0.967674i	$0.418844\pi$
$524$	−20.2982	−0.886732
$525$	−2.51812	−0.109900
$526$	0.440727	0.0192166
$527$	2.05634	0.0895755
$528$	−1.52726	−0.0664657
$529$	8.04242	0.349671
$530$	8.48218	0.368442
$531$	−10.1341	−0.439783
$532$	1.93830	0.0840362
$533$	−6.04331	−0.261765
$534$	2.18734	0.0946556
$535$	17.0084	0.735338
$536$	6.51015	0.281196
$537$	−13.8098	−0.595937
$538$	6.79007	0.292741
$539$	0.764507	0.0329296
$540$	1.51330	0.0651219
$541$	−13.3505	−0.573982	−0.286991	−	0.957933i	$-0.592655\pi$
$541$	−13.3505	−0.573982	−0.286991	+	0.957933i	$0.592655\pi$
$542$	−1.06871	−0.0459048
$543$	−15.0475	−0.645749
$544$	−11.9691	−0.513172
$545$	−5.55510	−0.237954
$546$	−1.75674	−0.0751817
$547$	3.37212	0.144181	0.0720907	−	0.997398i	$-0.477033\pi$
$547$	3.37212	0.144181	0.0720907	+	0.997398i	$0.477033\pi$
$548$	−3.38981	−0.144806
$549$	−2.55578	−0.109078
$550$	0.809234	0.0345059
$551$	2.39168	0.101889
$552$	−13.6560	−0.581240
$553$	29.9064	1.27175
$554$	10.0253	0.425935
$555$	−5.80038	−0.246212
$556$	−13.5160	−0.573207
$557$	−22.2052	−0.940866	−0.470433	−	0.882436i	$-0.655902\pi$
$557$	−22.2052	−0.940866	−0.470433	+	0.882436i	$0.655902\pi$
$558$	−0.697642	−0.0295335
$559$	5.12449	0.216743
$560$	3.31550	0.140105
$561$	−2.38526	−0.100706
$562$	10.7369	0.452909
$563$	−36.5220	−1.53922	−0.769609	−	0.638516i	$-0.779549\pi$
$563$	−36.5220	−1.53922	−0.769609	+	0.638516i	$0.779549\pi$
$564$	−7.82071	−0.329311
$565$	9.65590	0.406227
$566$	8.91315	0.374648
$567$	−2.51812	−0.105751
$568$	−15.9048	−0.667351
$569$	12.6198	0.529050	0.264525	−	0.964379i	$-0.414785\pi$
$569$	12.6198	0.529050	0.264525	+	0.964379i	$0.414785\pi$
$570$	0.354858	0.0148634
$571$	23.0104	0.962954	0.481477	−	0.876459i	$-0.340100\pi$
$571$	23.0104	0.962954	0.481477	+	0.876459i	$0.340100\pi$
$572$	−1.75536	−0.0733952
$573$	8.12473	0.339416
$574$	10.6165	0.443126
$575$	−5.57157	−0.232351
$576$	1.42738	0.0594741
$577$	29.3566	1.22213	0.611066	−	0.791579i	$-0.290741\pi$
$577$	29.3566	1.22213	0.611066	+	0.791579i	$0.290741\pi$
$578$	8.90991	0.370603
$579$	−10.8479	−0.450823
$580$	7.11549	0.295455
$581$	−5.68126	−0.235698
$582$	−3.79615	−0.157356
$583$	−14.1032	−0.584094
$584$	19.6891	0.814740
$585$	1.00000	0.0413449
$586$	−12.2975	−0.508004
$587$	26.4688	1.09248	0.546242	−	0.837627i	$-0.316058\pi$
$587$	26.4688	1.09248	0.546242	+	0.837627i	$0.316058\pi$
$588$	0.997386	0.0411315
$589$	0.508654	0.0209587
$590$	−7.06997	−0.291066
$591$	−9.23074	−0.379702
$592$	7.63711	0.313883
$593$	28.2521	1.16018	0.580088	−	0.814554i	$-0.303018\pi$
$593$	28.2521	1.16018	0.580088	+	0.814554i	$0.303018\pi$
$594$	0.809234	0.0332033
$595$	5.17810	0.212282
$596$	29.5045	1.20855
$597$	−16.0489	−0.656838
$598$	−3.88696	−0.158950
$599$	−33.5380	−1.37032	−0.685162	−	0.728391i	$-0.740269\pi$
$599$	−33.5380	−1.37032	−0.685162	+	0.728391i	$0.740269\pi$
$600$	2.45102	0.100063
$601$	32.7830	1.33725	0.668623	−	0.743601i	$-0.266884\pi$
$601$	32.7830	1.33725	0.668623	+	0.743601i	$0.266884\pi$
$602$	−9.00241	−0.366911
$603$	2.65609	0.108165
$604$	0.622859	0.0253438
$605$	9.65450	0.392511
$606$	−13.5308	−0.549650
$607$	−45.5264	−1.84786	−0.923929	−	0.382563i	$-0.875042\pi$
$607$	−45.5264	−1.84786	−0.923929	+	0.382563i	$0.875042\pi$
$608$	−2.96067	−0.120071
$609$	−11.8401	−0.479787
$610$	−1.78302	−0.0721923
$611$	−5.16799	−0.209075
$612$	−3.11185	−0.125789
$613$	−35.9401	−1.45161	−0.725804	−	0.687901i	$-0.758532\pi$
$613$	−35.9401	−1.45161	−0.725804	+	0.687901i	$0.758532\pi$
$614$	6.53492	0.263728
$615$	−6.04331	−0.243690
$616$	7.15921	0.288453
$617$	14.7393	0.593384	0.296692	−	0.954973i	$-0.404117\pi$
$617$	14.7393	0.593384	0.296692	+	0.954973i	$0.404117\pi$
$618$	2.19118	0.0881421
$619$	19.3993	0.779726	0.389863	−	0.920873i	$-0.372522\pi$
$619$	19.3993	0.779726	0.389863	+	0.920873i	$0.372522\pi$
$620$	1.51330	0.0607754
$621$	−5.57157	−0.223580
$622$	4.32932	0.173590
$623$	7.89515	0.316312
$624$	−1.31666	−0.0527084
$625$	1.00000	0.0400000
$626$	3.04779	0.121814
$627$	−0.590016	−0.0235630
$628$	−12.3647	−0.493405
$629$	11.9275	0.475582
$630$	−1.75674	−0.0699904
$631$	−1.73913	−0.0692335	−0.0346167	−	0.999401i	$-0.511021\pi$
$631$	−1.73913	−0.0692335	−0.0346167	+	0.999401i	$0.511021\pi$
$632$	−29.1095	−1.15792
$633$	−19.4982	−0.774984
$634$	1.46705	0.0582640
$635$	18.1972	0.722135
$636$	−18.3992	−0.729576
$637$	0.659082	0.0261138
$638$	3.80501	0.150642
$639$	−6.48905	−0.256703
$640$	−10.6454	−0.420796
$641$	−22.3185	−0.881528	−0.440764	−	0.897623i	$-0.645292\pi$
$641$	−22.3185	−0.881528	−0.440764	+	0.897623i	$0.645292\pi$
$642$	11.8658	0.468305
$643$	−45.1577	−1.78085	−0.890424	−	0.455133i	$-0.849592\pi$
$643$	−45.1577	−1.78085	−0.890424	+	0.455133i	$0.849592\pi$
$644$	−21.2314	−0.836633
$645$	5.12449	0.201776
$646$	−0.729708	−0.0287100
$647$	36.9007	1.45072	0.725358	−	0.688372i	$-0.241674\pi$
$647$	36.9007	1.45072	0.725358	+	0.688372i	$0.241674\pi$
$648$	2.45102	0.0962852
$649$	11.7551	0.461429
$650$	0.697642	0.0273638
$651$	−2.51812	−0.0986928
$652$	−2.90768	−0.113873
$653$	−1.31591	−0.0514957	−0.0257478	−	0.999668i	$-0.508197\pi$
$653$	−1.31591	−0.0514957	−0.0257478	+	0.999668i	$0.508197\pi$
$654$	−3.87547	−0.151543
$655$	−13.4133	−0.524099
$656$	7.95696	0.310667
$657$	8.03301	0.313397
$658$	9.07884	0.353930
$659$	−24.8918	−0.969646	−0.484823	−	0.874612i	$-0.661116\pi$
$659$	−24.8918	−0.969646	−0.484823	+	0.874612i	$0.661116\pi$
$660$	−1.75536	−0.0683272
$661$	31.0004	1.20578	0.602889	−	0.797825i	$-0.294016\pi$
$661$	31.0004	1.20578	0.602889	+	0.797825i	$0.294016\pi$
$662$	−10.2871	−0.399820
$663$	−2.05634	−0.0798616
$664$	5.52988	0.214601
$665$	1.28085	0.0496692
$666$	−4.04659	−0.156802
$667$	−26.1974	−1.01437
$668$	19.4519	0.752615
$669$	6.02868	0.233082
$670$	1.85300	0.0715877
$671$	2.96459	0.114447
$672$	14.6570	0.565404
$673$	14.3099	0.551606	0.275803	−	0.961214i	$-0.411056\pi$
$673$	14.3099	0.551606	0.275803	+	0.961214i	$0.411056\pi$
$674$	17.5724	0.676863
$675$	1.00000	0.0384900
$676$	−1.51330	−0.0582037
$677$	50.4067	1.93729	0.968643	−	0.248458i	$-0.0799240\pi$
$677$	50.4067	1.93729	0.968643	+	0.248458i	$0.0799240\pi$
$678$	6.73636	0.258708
$679$	−13.7021	−0.525838
$680$	−5.04013	−0.193280
$681$	−5.62958	−0.215726
$682$	0.809234	0.0309872
$683$	38.8892	1.48805	0.744026	−	0.668150i	$-0.232914\pi$
$683$	38.8892	1.48805	0.744026	+	0.668150i	$0.232914\pi$
$684$	−0.769743	−0.0294319
$685$	−2.24002	−0.0855867
$686$	−13.4550	−0.513716
$687$	24.0282	0.916731
$688$	−6.74719	−0.257234
$689$	−12.1584	−0.463197
$690$	−3.88696	−0.147974
$691$	−48.8475	−1.85825	−0.929124	−	0.369768i	$-0.879437\pi$
$691$	−48.8475	−1.85825	−0.929124	+	0.369768i	$0.879437\pi$
$692$	−27.4179	−1.04227
$693$	2.92091	0.110956
$694$	18.4642	0.700891
$695$	−8.93151	−0.338792
$696$	11.5247	0.436841
$697$	12.4271	0.470709
$698$	19.0199	0.719914
$699$	5.36054	0.202754
$700$	3.81066	0.144029
$701$	−43.1997	−1.63163	−0.815815	−	0.578313i	$-0.803711\pi$
$701$	−43.1997	−1.63163	−0.815815	+	0.578313i	$0.803711\pi$
$702$	0.697642	0.0263308
$703$	2.95038	0.111276
$704$	−1.65570	−0.0624015
$705$	−5.16799	−0.194638
$706$	24.1094	0.907369
$707$	−48.8389	−1.83678
$708$	15.3359	0.576358
$709$	−24.3569	−0.914744	−0.457372	−	0.889276i	$-0.651209\pi$
$709$	−24.3569	−0.914744	−0.457372	+	0.889276i	$0.651209\pi$
$710$	−4.52703	−0.169897
$711$	−11.8765	−0.445403
$712$	−7.68478	−0.287999
$713$	−5.57157	−0.208657
$714$	3.61246	0.135193
$715$	−1.15996	−0.0433799
$716$	20.8983	0.781007
$717$	−9.75975	−0.364484
$718$	18.2295	0.680320
$719$	0.184045	0.00686371	0.00343185	−	0.999994i	$-0.498908\pi$
$719$	0.184045	0.00686371	0.00343185	+	0.999994i	$0.498908\pi$
$720$	−1.31666	−0.0490689
$721$	7.90900	0.294546
$722$	13.0747	0.486590
$723$	30.3443	1.12852
$724$	22.7713	0.846288
$725$	4.70198	0.174627
$726$	6.73538	0.249973
$727$	1.78117	0.0660599	0.0330300	−	0.999454i	$-0.489484\pi$
$727$	1.78117	0.0660599	0.0330300	+	0.999454i	$0.489484\pi$
$728$	6.17196	0.228748
$729$	1.00000	0.0370370
$730$	5.60416	0.207419
$731$	−10.5377	−0.389750
$732$	3.86765	0.142952
$733$	10.7813	0.398216	0.199108	−	0.979978i	$-0.436195\pi$
$733$	10.7813	0.398216	0.199108	+	0.979978i	$0.436195\pi$
$734$	−19.5754	−0.722541
$735$	0.659082	0.0243106
$736$	32.4299	1.19538
$737$	−3.08096	−0.113488
$738$	−4.21606	−0.155195
$739$	−18.9251	−0.696172	−0.348086	−	0.937463i	$-0.613168\pi$
$739$	−18.9251	−0.696172	−0.348086	+	0.937463i	$0.613168\pi$
$740$	8.77769	0.322674
$741$	−0.508654	−0.0186859
$742$	21.3591	0.784119
$743$	38.3850	1.40821	0.704104	−	0.710097i	$-0.251349\pi$
$743$	38.3850	1.40821	0.704104	+	0.710097i	$0.251349\pi$
$744$	2.45102	0.0898588
$745$	19.4968	0.714308
$746$	−13.4241	−0.491491
$747$	2.25615	0.0825483
$748$	3.60961	0.131980
$749$	42.8292	1.56495
$750$	0.697642	0.0254743
$751$	10.7844	0.393530	0.196765	−	0.980451i	$-0.436956\pi$
$751$	10.7844	0.393530	0.196765	+	0.980451i	$0.436956\pi$
$752$	6.80447	0.248134
$753$	−20.3095	−0.740119
$754$	3.28030	0.119461
$755$	0.411591	0.0149793
$756$	3.81066	0.138592
$757$	18.6288	0.677076	0.338538	−	0.940953i	$-0.390068\pi$
$757$	18.6288	0.677076	0.338538	+	0.940953i	$0.390068\pi$
$758$	−20.2873	−0.736868
$759$	6.46278	0.234584
$760$	−1.24672	−0.0452233
$761$	18.5855	0.673723	0.336862	−	0.941554i	$-0.390635\pi$
$761$	18.5855	0.673723	0.336862	+	0.941554i	$0.390635\pi$
$762$	12.6952	0.459897
$763$	−13.9884	−0.506414
$764$	−12.2951	−0.444822
$765$	−2.05634	−0.0743471
$766$	5.29039	0.191150
$767$	10.1341	0.365921
$768$	−10.2814	−0.370999
$769$	41.5367	1.49785	0.748926	−	0.662653i	$-0.230570\pi$
$769$	41.5367	1.49785	0.748926	+	0.662653i	$0.230570\pi$
$770$	2.03775	0.0734353
$771$	−26.6013	−0.958023
$772$	16.4161	0.590828
$773$	−33.7113	−1.21251	−0.606256	−	0.795269i	$-0.707330\pi$
$773$	−33.7113	−1.21251	−0.606256	+	0.795269i	$0.707330\pi$
$774$	3.57506	0.128503
$775$	1.00000	0.0359211
$776$	13.3370	0.478771
$777$	−14.6060	−0.523989
$778$	9.79292	0.351093
$779$	3.07395	0.110136
$780$	−1.51330	−0.0541847
$781$	7.52702	0.269338
$782$	7.99291	0.285826
$783$	4.70198	0.168035
$784$	−0.867785	−0.0309923
$785$	−8.17071	−0.291625
$786$	−9.35765	−0.333776
$787$	12.3444	0.440030	0.220015	−	0.975496i	$-0.429389\pi$
$787$	12.3444	0.440030	0.220015	+	0.975496i	$0.429389\pi$
$788$	13.9688	0.497620
$789$	−0.631739	−0.0224905
$790$	−8.28553	−0.294786
$791$	24.3147	0.864531
$792$	−2.84308	−0.101024
$793$	2.55578	0.0907583
$794$	−6.34779	−0.225275
$795$	−12.1584	−0.431213
$796$	24.2868	0.860821
$797$	−40.6470	−1.43979	−0.719895	−	0.694083i	$-0.755810\pi$
$797$	−40.6470	−1.43979	−0.719895	+	0.694083i	$0.755810\pi$
$798$	0.893574	0.0316322
$799$	10.6272	0.375962
$800$	−5.82060	−0.205789
$801$	−3.13534	−0.110782
$802$	13.6834	0.483178
$803$	−9.31794	−0.328823
$804$	−4.01946	−0.141755
$805$	−14.0299	−0.494488
$806$	0.697642	0.0245734
$807$	−9.73289	−0.342614
$808$	47.5376	1.67237
$809$	0.985090	0.0346339	0.0173170	−	0.999850i	$-0.494488\pi$
$809$	0.985090	0.0346339	0.0173170	+	0.999850i	$0.494488\pi$
$810$	0.697642	0.0245126
$811$	55.6425	1.95387	0.976937	−	0.213530i	$-0.0684960\pi$
$811$	55.6425	1.95387	0.976937	+	0.213530i	$0.0684960\pi$
$812$	17.9177	0.628786
$813$	1.53188	0.0537255
$814$	4.69387	0.164520
$815$	−1.92142	−0.0673044
$816$	2.70749	0.0947812
$817$	−2.60659	−0.0911930
$818$	12.3828	0.432953
$819$	2.51812	0.0879902
$820$	9.14531	0.319368
$821$	−11.9119	−0.415728	−0.207864	−	0.978158i	$-0.566651\pi$
$821$	−11.9119	−0.415728	−0.207864	+	0.978158i	$0.566651\pi$
$822$	−1.56273	−0.0545065
$823$	−21.8964	−0.763260	−0.381630	−	0.924315i	$-0.624637\pi$
$823$	−21.8964	−0.763260	−0.381630	+	0.924315i	$0.624637\pi$
$824$	−7.69826	−0.268182
$825$	−1.15996	−0.0403845
$826$	−17.8030	−0.619446
$827$	−27.6852	−0.962710	−0.481355	−	0.876526i	$-0.659855\pi$
$827$	−27.6852	−0.962710	−0.481355	+	0.876526i	$0.659855\pi$
$828$	8.43144	0.293013
$829$	−18.4451	−0.640626	−0.320313	−	0.947312i	$-0.603788\pi$
$829$	−18.4451	−0.640626	−0.320313	+	0.947312i	$0.603788\pi$
$830$	1.57399	0.0546339
$831$	−14.3703	−0.498500
$832$	−1.42738	−0.0494855
$833$	−1.35530	−0.0469582
$834$	−6.23100	−0.215762
$835$	12.8540	0.444830
$836$	0.892869	0.0308805
$837$	1.00000	0.0345651
$838$	21.0414	0.726862
$839$	−35.8882	−1.23900	−0.619499	−	0.784997i	$-0.712664\pi$
$839$	−35.8882	−1.23900	−0.619499	+	0.784997i	$0.712664\pi$
$840$	6.17196	0.212953
$841$	−6.89135	−0.237633
$842$	−16.9727	−0.584918
$843$	−15.3903	−0.530070
$844$	29.5066	1.01566
$845$	−1.00000	−0.0344010
$846$	−3.60541	−0.123957
$847$	24.3112	0.835342
$848$	16.0084	0.549730
$849$	−12.7761	−0.438475
$850$	−1.43459	−0.0492060
$851$	−32.3172	−1.10782
$852$	9.81986	0.336423
$853$	−51.2593	−1.75509	−0.877543	−	0.479498i	$-0.840819\pi$
$853$	−51.2593	−1.75509	−0.877543	+	0.479498i	$0.840819\pi$
$854$	−4.48985	−0.153639
$855$	−0.508654	−0.0173956
$856$	−41.6880	−1.42487
$857$	−9.74408	−0.332851	−0.166426	−	0.986054i	$-0.553223\pi$
$857$	−9.74408	−0.332851	−0.166426	+	0.986054i	$0.553223\pi$
$858$	−0.809234	−0.0276268
$859$	27.5732	0.940786	0.470393	−	0.882457i	$-0.344112\pi$
$859$	27.5732	0.940786	0.470393	+	0.882457i	$0.344112\pi$
$860$	−7.75487	−0.264439
$861$	−15.2178	−0.518620
$862$	−7.52489	−0.256299
$863$	−3.32242	−0.113096	−0.0565482	−	0.998400i	$-0.518009\pi$
$863$	−3.32242	−0.113096	−0.0565482	+	0.998400i	$0.518009\pi$
$864$	−5.82060	−0.198021
$865$	−18.1180	−0.616030
$866$	−13.6079	−0.462414
$867$	−12.7715	−0.433742
$868$	3.81066	0.129342
$869$	13.7762	0.467326
$870$	3.28030	0.111213
$871$	−2.65609	−0.0899983
$872$	13.6157	0.461085
$873$	5.44141	0.184164
$874$	1.97712	0.0668770
$875$	2.51812	0.0851279
$876$	−12.1563	−0.410724
$877$	31.6592	1.06906	0.534528	−	0.845151i	$-0.320489\pi$
$877$	31.6592	1.06906	0.534528	+	0.845151i	$0.320489\pi$
$878$	−4.11771	−0.138966
$879$	17.6272	0.594550
$880$	1.52726	0.0514841
$881$	−38.2621	−1.28908	−0.644541	−	0.764570i	$-0.722951\pi$
$881$	−38.2621	−1.28908	−0.644541	+	0.764570i	$0.722951\pi$
$882$	0.459803	0.0154824
$883$	24.8341	0.835735	0.417867	−	0.908508i	$-0.362778\pi$
$883$	24.8341	0.835735	0.417867	+	0.908508i	$0.362778\pi$
$884$	3.11185	0.104663
$885$	10.1341	0.340654
$886$	12.4909	0.419641
$887$	39.2591	1.31819	0.659096	−	0.752059i	$-0.270939\pi$
$887$	39.2591	1.31819	0.659096	+	0.752059i	$0.270939\pi$
$888$	14.2169	0.477086
$889$	45.8228	1.53685
$890$	−2.18734	−0.0733199
$891$	−1.15996	−0.0388600
$892$	−9.12318	−0.305467
$893$	2.62872	0.0879667
$894$	13.6018	0.454912
$895$	13.8098	0.461611
$896$	−26.8064	−0.895538
$897$	5.57157	0.186029
$898$	6.68845	0.223196
$899$	4.70198	0.156820
$900$	−1.51330	−0.0504432
$901$	25.0017	0.832928
$902$	4.89045	0.162834
$903$	12.9041	0.429420
$904$	−23.6668	−0.787147
$905$	15.0475	0.500195
$906$	0.287143	0.00953968
$907$	−15.1673	−0.503623	−0.251811	−	0.967776i	$-0.581026\pi$
$907$	−15.1673	−0.503623	−0.251811	+	0.967776i	$0.581026\pi$
$908$	8.51922	0.282720
$909$	19.3950	0.643292
$910$	1.75674	0.0582355
$911$	−13.1710	−0.436375	−0.218188	−	0.975907i	$-0.570014\pi$
$911$	−13.1710	−0.436375	−0.218188	+	0.975907i	$0.570014\pi$
$912$	0.669722	0.0221767
$913$	−2.61704	−0.0866114
$914$	0.596032	0.0197150
$915$	2.55578	0.0844914
$916$	−36.3617	−1.20142
$917$	−33.7762	−1.11539
$918$	−1.43459	−0.0473485
$919$	48.6348	1.60431	0.802157	−	0.597113i	$-0.203685\pi$
$919$	48.6348	1.60431	0.802157	+	0.597113i	$0.203685\pi$
$920$	13.6560	0.450227
$921$	−9.36715	−0.308658
$922$	25.1620	0.828666
$923$	6.48905	0.213590
$924$	−4.42020	−0.145414
$925$	5.80038	0.190715
$926$	1.52322	0.0500562
$927$	−3.14084	−0.103159
$928$	−27.3684	−0.898410
$929$	−41.8219	−1.37213	−0.686066	−	0.727539i	$-0.740664\pi$
$929$	−41.8219	−1.37213	−0.686066	+	0.727539i	$0.740664\pi$
$930$	0.697642	0.0228766
$931$	−0.335244	−0.0109872
$932$	−8.11209	−0.265720
$933$	−6.20566	−0.203164
$934$	1.87816	0.0614553
$935$	2.38526	0.0780065
$936$	−2.45102	−0.0801142
$937$	−42.0380	−1.37332	−0.686661	−	0.726978i	$-0.740924\pi$
$937$	−42.0380	−1.37332	−0.686661	+	0.726978i	$0.740924\pi$
$938$	4.66608	0.152353
$939$	−4.36870	−0.142567
$940$	7.82071	0.255083
$941$	12.5804	0.410111	0.205055	−	0.978750i	$-0.434263\pi$
$941$	12.5804	0.410111	0.205055	+	0.978750i	$0.434263\pi$
$942$	−5.70023	−0.185724
$943$	−33.6707	−1.09647
$944$	−13.3431	−0.434282
$945$	2.51812	0.0819144
$946$	−4.14691	−0.134828
$947$	15.4115	0.500805	0.250403	−	0.968142i	$-0.419437\pi$
$947$	15.4115	0.500805	0.250403	+	0.968142i	$0.419437\pi$
$948$	17.9726	0.583724
$949$	−8.03301	−0.260762
$950$	−0.354858	−0.0115131
$951$	−2.10287	−0.0681902
$952$	−12.6916	−0.411339
$953$	31.9934	1.03637	0.518184	−	0.855269i	$-0.326608\pi$
$953$	31.9934	1.03637	0.518184	+	0.855269i	$0.326608\pi$
$954$	−8.48218	−0.274621
$955$	−8.12473	−0.262910
$956$	14.7694	0.477676
$957$	−5.45410	−0.176306
$958$	23.0326	0.744150
$959$	−5.64063	−0.182145
$960$	−1.42738	−0.0460685
$961$	1.00000	0.0322581
$962$	4.04659	0.130467
$963$	−17.0084	−0.548089
$964$	−45.9200	−1.47898
$965$	10.8479	0.349206
$966$	−9.78783	−0.314918
$967$	46.6212	1.49924	0.749618	−	0.661871i	$-0.230237\pi$
$967$	46.6212	1.49924	0.749618	+	0.661871i	$0.230237\pi$
$968$	−23.6634	−0.760570
$969$	1.04596	0.0336012
$970$	3.79615	0.121887
$971$	−19.7839	−0.634897	−0.317448	−	0.948276i	$-0.602826\pi$
$971$	−19.7839	−0.634897	−0.317448	+	0.948276i	$0.602826\pi$
$972$	−1.51330	−0.0485390
$973$	−22.4906	−0.721016
$974$	−0.0200746	−0.000643232 0
$975$	−1.00000	−0.0320256
$976$	−3.36508	−0.107714
$977$	−34.1297	−1.09191	−0.545953	−	0.837816i	$-0.683832\pi$
$977$	−34.1297	−1.09191	−0.545953	+	0.837816i	$0.683832\pi$
$978$	−1.34046	−0.0428633
$979$	3.63686	0.116234
$980$	−0.997386	−0.0318603
$981$	5.55510	0.177361
$982$	−25.2883	−0.806982
$983$	54.8590	1.74973	0.874866	−	0.484365i	$-0.160949\pi$
$983$	54.8590	1.74973	0.874866	+	0.484365i	$0.160949\pi$
$984$	14.8123	0.472198
$985$	9.23074	0.294116
$986$	−6.74541	−0.214818
$987$	−13.0136	−0.414228
$988$	0.769743	0.0244888
$989$	28.5515	0.907883
$990$	−0.809234	−0.0257192
$991$	2.23200	0.0709018	0.0354509	−	0.999371i	$-0.488713\pi$
$991$	2.23200	0.0709018	0.0354509	+	0.999371i	$0.488713\pi$
$992$	−5.82060	−0.184804
$993$	14.7456	0.467936
$994$	−11.3996	−0.361574
$995$	16.0489	0.508785
$996$	−3.41423	−0.108184
$997$	11.8204	0.374355	0.187177	−	0.982326i	$-0.440066\pi$
$997$	11.8204	0.374355	0.187177	+	0.982326i	$0.440066\pi$
$998$	−12.1278	−0.383900
$999$	5.80038	0.183516

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6045.2.a.x.1.6	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6045.2.a.x.1.6	✓	12	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:	\( N \)	\(=\)	\( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6045.a (trivial)

\(f(q)\)	\(=\)	\(q-0.697642 q^{2} +1.00000 q^{3} -1.51330 q^{4} -1.00000 q^{5} -0.697642 q^{6} -2.51812 q^{7} +2.45102 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.697642 q^{2} +1.00000 q^{3} -1.51330 q^{4} -1.00000 q^{5} -0.697642 q^{6} -2.51812 q^{7} +2.45102 q^{8} +1.00000 q^{9} +0.697642 q^{10} -1.15996 q^{11} -1.51330 q^{12} -1.00000 q^{13} +1.75674 q^{14} -1.00000 q^{15} +1.31666 q^{16} +2.05634 q^{17} -0.697642 q^{18} +0.508654 q^{19} +1.51330 q^{20} -2.51812 q^{21} +0.809234 q^{22} -5.57157 q^{23} +2.45102 q^{24} +1.00000 q^{25} +0.697642 q^{26} +1.00000 q^{27} +3.81066 q^{28} +4.70198 q^{29} +0.697642 q^{30} +1.00000 q^{31} -5.82060 q^{32} -1.15996 q^{33} -1.43459 q^{34} +2.51812 q^{35} -1.51330 q^{36} +5.80038 q^{37} -0.354858 q^{38} -1.00000 q^{39} -2.45102 q^{40} +6.04331 q^{41} +1.75674 q^{42} -5.12449 q^{43} +1.75536 q^{44} -1.00000 q^{45} +3.88696 q^{46} +5.16799 q^{47} +1.31666 q^{48} -0.659082 q^{49} -0.697642 q^{50} +2.05634 q^{51} +1.51330 q^{52} +12.1584 q^{53} -0.697642 q^{54} +1.15996 q^{55} -6.17196 q^{56} +0.508654 q^{57} -3.28030 q^{58} -10.1341 q^{59} +1.51330 q^{60} -2.55578 q^{61} -0.697642 q^{62} -2.51812 q^{63} +1.42738 q^{64} +1.00000 q^{65} +0.809234 q^{66} +2.65609 q^{67} -3.11185 q^{68} -5.57157 q^{69} -1.75674 q^{70} -6.48905 q^{71} +2.45102 q^{72} +8.03301 q^{73} -4.04659 q^{74} +1.00000 q^{75} -0.769743 q^{76} +2.92091 q^{77} +0.697642 q^{78} -11.8765 q^{79} -1.31666 q^{80} +1.00000 q^{81} -4.21606 q^{82} +2.25615 q^{83} +3.81066 q^{84} -2.05634 q^{85} +3.57506 q^{86} +4.70198 q^{87} -2.84308 q^{88} -3.13534 q^{89} +0.697642 q^{90} +2.51812 q^{91} +8.43144 q^{92} +1.00000 q^{93} -3.60541 q^{94} -0.508654 q^{95} -5.82060 q^{96} +5.44141 q^{97} +0.459803 q^{98} -1.15996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 12 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) 12 * q - 4 * q^2 + 12 * q^3 + 6 * q^4 - 12 * q^5 - 4 * q^6 + 3 * q^7 - 3 * q^8 + 12 * q^9	\( 12 q - 4 q^{2} + 12 q^{3} + 6 q^{4} - 12 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 12 q^{9} + 4 q^{10} - 10 q^{11} + 6 q^{12} - 12 q^{13} - 19 q^{14} - 12 q^{15} + 2 q^{16} - 7 q^{17} - 4 q^{18} + 4 q^{19} - 6 q^{20} + 3 q^{21} - 11 q^{22} - 3 q^{23} - 3 q^{24} + 12 q^{25} + 4 q^{26} + 12 q^{27} + 8 q^{28} - 23 q^{29} + 4 q^{30} + 12 q^{31} - 10 q^{33} + 23 q^{34} - 3 q^{35} + 6 q^{36} + 9 q^{37} + 30 q^{38} - 12 q^{39} + 3 q^{40} - 9 q^{41} - 19 q^{42} - 20 q^{43} - 7 q^{44} - 12 q^{45} - 2 q^{46} - 9 q^{47} + 2 q^{48} + 9 q^{49} - 4 q^{50} - 7 q^{51} - 6 q^{52} - 20 q^{53} - 4 q^{54} + 10 q^{55} - 53 q^{56} + 4 q^{57} - 2 q^{58} - 40 q^{59} - 6 q^{60} - 2 q^{61} - 4 q^{62} + 3 q^{63} - 5 q^{64} + 12 q^{65} - 11 q^{66} - 19 q^{68} - 3 q^{69} + 19 q^{70} - 17 q^{71} - 3 q^{72} + 3 q^{73} + 4 q^{74} + 12 q^{75} - 12 q^{76} - 32 q^{77} + 4 q^{78} - 31 q^{79} - 2 q^{80} + 12 q^{81} - 39 q^{82} + 3 q^{83} + 8 q^{84} + 7 q^{85} - 60 q^{86} - 23 q^{87} - 23 q^{88} - 39 q^{89} + 4 q^{90} - 3 q^{91} - 53 q^{92} + 12 q^{93} - 24 q^{94} - 4 q^{95} - 23 q^{97} - 32 q^{98} - 10 q^{99}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^3 + 6 * q^4 - 12 * q^5 - 4 * q^6 + 3 * q^7 - 3 * q^8 + 12 * q^9 + 4 * q^10 - 10 * q^11 + 6 * q^12 - 12 * q^13 - 19 * q^14 - 12 * q^15 + 2 * q^16 - 7 * q^17 - 4 * q^18 + 4 * q^19 - 6 * q^20 + 3 * q^21 - 11 * q^22 - 3 * q^23 - 3 * q^24 + 12 * q^25 + 4 * q^26 + 12 * q^27 + 8 * q^28 - 23 * q^29 + 4 * q^30 + 12 * q^31 - 10 * q^33 + 23 * q^34 - 3 * q^35 + 6 * q^36 + 9 * q^37 + 30 * q^38 - 12 * q^39 + 3 * q^40 - 9 * q^41 - 19 * q^42 - 20 * q^43 - 7 * q^44 - 12 * q^45 - 2 * q^46 - 9 * q^47 + 2 * q^48 + 9 * q^49 - 4 * q^50 - 7 * q^51 - 6 * q^52 - 20 * q^53 - 4 * q^54 + 10 * q^55 - 53 * q^56 + 4 * q^57 - 2 * q^58 - 40 * q^59 - 6 * q^60 - 2 * q^61 - 4 * q^62 + 3 * q^63 - 5 * q^64 + 12 * q^65 - 11 * q^66 - 19 * q^68 - 3 * q^69 + 19 * q^70 - 17 * q^71 - 3 * q^72 + 3 * q^73 + 4 * q^74 + 12 * q^75 - 12 * q^76 - 32 * q^77 + 4 * q^78 - 31 * q^79 - 2 * q^80 + 12 * q^81 - 39 * q^82 + 3 * q^83 + 8 * q^84 + 7 * q^85 - 60 * q^86 - 23 * q^87 - 23 * q^88 - 39 * q^89 + 4 * q^90 - 3 * q^91 - 53 * q^92 + 12 * q^93 - 24 * q^94 - 4 * q^95 - 23 * q^97 - 32 * q^98 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more