LMFDB - Embedded newform 2415.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,2,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, 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("2415.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2415.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:	$19.2838720881$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.15751800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 6x^{3} + 16x^{2} - 7x - 1 $ x^6 - x^5 - 8x^4 + 6x^3 + 16x^2 - 7x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.527344$ of defining polynomial
Character	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96273 q^{2} -1.00000 q^{3} +1.85229 q^{4} -1.00000 q^{5} +1.96273 q^{6} +1.00000 q^{7} +0.289915 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.96273 q^{2} -1.00000 q^{3} +1.85229 q^{4} -1.00000 q^{5} +1.96273 q^{6} +1.00000 q^{7} +0.289915 q^{8} +1.00000 q^{9} +1.96273 q^{10} -1.88144 q^{11} -1.85229 q^{12} -6.73932 q^{13} -1.96273 q^{14} +1.00000 q^{15} -4.27360 q^{16} +7.00958 q^{17} -1.96273 q^{18} +2.74271 q^{19} -1.85229 q^{20} -1.00000 q^{21} +3.69276 q^{22} -1.00000 q^{23} -0.289915 q^{24} +1.00000 q^{25} +13.2274 q^{26} -1.00000 q^{27} +1.85229 q^{28} -4.49850 q^{29} -1.96273 q^{30} -8.78672 q^{31} +7.80808 q^{32} +1.88144 q^{33} -13.7579 q^{34} -1.00000 q^{35} +1.85229 q^{36} +5.04650 q^{37} -5.38318 q^{38} +6.73932 q^{39} -0.289915 q^{40} +4.15170 q^{41} +1.96273 q^{42} +4.10457 q^{43} -3.48498 q^{44} -1.00000 q^{45} +1.96273 q^{46} +8.03482 q^{47} +4.27360 q^{48} +1.00000 q^{49} -1.96273 q^{50} -7.00958 q^{51} -12.4832 q^{52} +1.23693 q^{53} +1.96273 q^{54} +1.88144 q^{55} +0.289915 q^{56} -2.74271 q^{57} +8.82933 q^{58} -11.0298 q^{59} +1.85229 q^{60} -11.8215 q^{61} +17.2459 q^{62} +1.00000 q^{63} -6.77790 q^{64} +6.73932 q^{65} -3.69276 q^{66} -13.4453 q^{67} +12.9838 q^{68} +1.00000 q^{69} +1.96273 q^{70} +5.58843 q^{71} +0.289915 q^{72} -3.39981 q^{73} -9.90488 q^{74} -1.00000 q^{75} +5.08029 q^{76} -1.88144 q^{77} -13.2274 q^{78} -1.34290 q^{79} +4.27360 q^{80} +1.00000 q^{81} -8.14865 q^{82} +3.79511 q^{83} -1.85229 q^{84} -7.00958 q^{85} -8.05615 q^{86} +4.49850 q^{87} -0.545459 q^{88} +16.8310 q^{89} +1.96273 q^{90} -6.73932 q^{91} -1.85229 q^{92} +8.78672 q^{93} -15.7702 q^{94} -2.74271 q^{95} -7.80808 q^{96} -8.03091 q^{97} -1.96273 q^{98} -1.88144 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) $ 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9} - 3 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} + 11 q^{16} + 8 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} - 6 q^{21} + 18 q^{22} - 6 q^{23} - 9 q^{24} + 6 q^{25} + 13 q^{26} - 6 q^{27} + 5 q^{28} + 8 q^{29} + 3 q^{30} - 16 q^{31} + 26 q^{32} - 6 q^{33} - 13 q^{34} - 6 q^{35} + 5 q^{36} + 8 q^{37} - 14 q^{38} + 6 q^{39} - 9 q^{40} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 15 q^{44} - 6 q^{45} - 3 q^{46} + 10 q^{47} - 11 q^{48} + 6 q^{49} + 3 q^{50} - 8 q^{51} + 28 q^{53} - 3 q^{54} - 6 q^{55} + 9 q^{56} + 8 q^{57} + 14 q^{58} - 6 q^{59} + 5 q^{60} - 12 q^{61} + 18 q^{62} + 6 q^{63} + 15 q^{64} + 6 q^{65} - 18 q^{66} + 18 q^{67} + 18 q^{68} + 6 q^{69} - 3 q^{70} + 10 q^{71} + 9 q^{72} - 2 q^{73} + q^{74} - 6 q^{75} + 25 q^{76} + 6 q^{77} - 13 q^{78} - 2 q^{79} - 11 q^{80} + 6 q^{81} - 12 q^{82} + 26 q^{83} - 5 q^{84} - 8 q^{85} + 16 q^{86} - 8 q^{87} + 21 q^{88} + 8 q^{89} - 3 q^{90} - 6 q^{91} - 5 q^{92} + 16 q^{93} - 8 q^{94} + 8 q^{95} - 26 q^{96} + 20 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9 - 3 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 3 * q^14 + 6 * q^15 + 11 * q^16 + 8 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 - 6 * q^21 + 18 * q^22 - 6 * q^23 - 9 * q^24 + 6 * q^25 + 13 * q^26 - 6 * q^27 + 5 * q^28 + 8 * q^29 + 3 * q^30 - 16 * q^31 + 26 * q^32 - 6 * q^33 - 13 * q^34 - 6 * q^35 + 5 * q^36 + 8 * q^37 - 14 * q^38 + 6 * q^39 - 9 * q^40 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 15 * q^44 - 6 * q^45 - 3 * q^46 + 10 * q^47 - 11 * q^48 + 6 * q^49 + 3 * q^50 - 8 * q^51 + 28 * q^53 - 3 * q^54 - 6 * q^55 + 9 * q^56 + 8 * q^57 + 14 * q^58 - 6 * q^59 + 5 * q^60 - 12 * q^61 + 18 * q^62 + 6 * q^63 + 15 * q^64 + 6 * q^65 - 18 * q^66 + 18 * q^67 + 18 * q^68 + 6 * q^69 - 3 * q^70 + 10 * q^71 + 9 * q^72 - 2 * q^73 + q^74 - 6 * q^75 + 25 * q^76 + 6 * q^77 - 13 * q^78 - 2 * q^79 - 11 * q^80 + 6 * q^81 - 12 * q^82 + 26 * q^83 - 5 * q^84 - 8 * q^85 + 16 * q^86 - 8 * q^87 + 21 * q^88 + 8 * q^89 - 3 * q^90 - 6 * q^91 - 5 * q^92 + 16 * q^93 - 8 * q^94 + 8 * q^95 - 26 * q^96 + 20 * q^97 + 3 * q^98 + 6 * 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 $

$2$	−1.96273	−1.38786	−0.693928	−	0.720044i	$-0.744121\pi$
$2$	−1.96273	−1.38786	−0.693928	+	0.720044i	$0.744121\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.85229	0.926145
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.96273	0.801279
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.289915	0.102500
$9$	1.00000	0.333333
$10$	1.96273	0.620668
$11$	−1.88144	−0.567276	−0.283638	−	0.958931i	$-0.591541\pi$
$11$	−1.88144	−0.567276	−0.283638	+	0.958931i	$0.591541\pi$
$12$	−1.85229	−0.534710
$13$	−6.73932	−1.86915	−0.934576	−	0.355764i	$-0.884221\pi$
$13$	−6.73932	−1.86915	−0.934576	+	0.355764i	$0.884221\pi$
$14$	−1.96273	−0.524560
$15$	1.00000	0.258199
$16$	−4.27360	−1.06840
$17$	7.00958	1.70007	0.850036	−	0.526724i	$-0.176580\pi$
$17$	7.00958	1.70007	0.850036	+	0.526724i	$0.176580\pi$
$18$	−1.96273	−0.462619
$19$	2.74271	0.629221	0.314610	−	0.949221i	$-0.398126\pi$
$19$	2.74271	0.629221	0.314610	+	0.949221i	$0.398126\pi$
$20$	−1.85229	−0.414185
$21$	−1.00000	−0.218218
$22$	3.69276	0.787298
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−0.289915	−0.0591787
$25$	1.00000	0.200000
$26$	13.2274	2.59411
$27$	−1.00000	−0.192450
$28$	1.85229	0.350050
$29$	−4.49850	−0.835351	−0.417676	−	0.908596i	$-0.637155\pi$
$29$	−4.49850	−0.835351	−0.417676	+	0.908596i	$0.637155\pi$
$30$	−1.96273	−0.358343
$31$	−8.78672	−1.57814	−0.789070	−	0.614303i	$-0.789437\pi$
$31$	−8.78672	−1.57814	−0.789070	+	0.614303i	$0.789437\pi$
$32$	7.80808	1.38029
$33$	1.88144	0.327517
$34$	−13.7579	−2.35946
$35$	−1.00000	−0.169031
$36$	1.85229	0.308715
$37$	5.04650	0.829639	0.414819	−	0.909904i	$-0.363845\pi$
$37$	5.04650	0.829639	0.414819	+	0.909904i	$0.363845\pi$
$38$	−5.38318	−0.873268
$39$	6.73932	1.07916
$40$	−0.289915	−0.0458396
$41$	4.15170	0.648387	0.324193	−	0.945991i	$-0.394907\pi$
$41$	4.15170	0.648387	0.324193	+	0.945991i	$0.394907\pi$
$42$	1.96273	0.302855
$43$	4.10457	0.625941	0.312971	−	0.949763i	$-0.398676\pi$
$43$	4.10457	0.625941	0.312971	+	0.949763i	$0.398676\pi$
$44$	−3.48498	−0.525380
$45$	−1.00000	−0.149071
$46$	1.96273	0.289388
$47$	8.03482	1.17200	0.586000	−	0.810311i	$-0.300702\pi$
$47$	8.03482	1.17200	0.586000	+	0.810311i	$0.300702\pi$
$48$	4.27360	0.616841
$49$	1.00000	0.142857
$50$	−1.96273	−0.277571
$51$	−7.00958	−0.981537
$52$	−12.4832	−1.73110
$53$	1.23693	0.169906	0.0849528	−	0.996385i	$-0.472926\pi$
$53$	1.23693	0.169906	0.0849528	+	0.996385i	$0.472926\pi$
$54$	1.96273	0.267093
$55$	1.88144	0.253694
$56$	0.289915	0.0387415
$57$	−2.74271	−0.363281
$58$	8.82933	1.15935
$59$	−11.0298	−1.43596	−0.717981	−	0.696062i	$-0.754934\pi$
$59$	−11.0298	−1.43596	−0.717981	+	0.696062i	$0.754934\pi$
$60$	1.85229	0.239130
$61$	−11.8215	−1.51358	−0.756791	−	0.653657i	$-0.773234\pi$
$61$	−11.8215	−1.51358	−0.756791	+	0.653657i	$0.773234\pi$
$62$	17.2459	2.19023
$63$	1.00000	0.125988
$64$	−6.77790	−0.847238
$65$	6.73932	0.835910
$66$	−3.69276	−0.454547
$67$	−13.4453	−1.64260	−0.821302	−	0.570494i	$-0.806752\pi$
$67$	−13.4453	−1.64260	−0.821302	+	0.570494i	$0.806752\pi$
$68$	12.9838	1.57451
$69$	1.00000	0.120386
$70$	1.96273	0.234591
$71$	5.58843	0.663224	0.331612	−	0.943416i	$-0.392407\pi$
$71$	5.58843	0.663224	0.331612	+	0.943416i	$0.392407\pi$
$72$	0.289915	0.0341668
$73$	−3.39981	−0.397918	−0.198959	−	0.980008i	$-0.563756\pi$
$73$	−3.39981	−0.397918	−0.198959	+	0.980008i	$0.563756\pi$
$74$	−9.90488	−1.15142
$75$	−1.00000	−0.115470
$76$	5.08029	0.582749
$77$	−1.88144	−0.214410
$78$	−13.2274	−1.49771
$79$	−1.34290	−0.151088	−0.0755440	−	0.997142i	$-0.524069\pi$
$79$	−1.34290	−0.151088	−0.0755440	+	0.997142i	$0.524069\pi$
$80$	4.27360	0.477803
$81$	1.00000	0.111111
$82$	−8.14865	−0.899868
$83$	3.79511	0.416567	0.208284	−	0.978068i	$-0.433212\pi$
$83$	3.79511	0.416567	0.208284	+	0.978068i	$0.433212\pi$
$84$	−1.85229	−0.202101
$85$	−7.00958	−0.760296
$86$	−8.05615	−0.868716
$87$	4.49850	0.482290
$88$	−0.545459	−0.0581461
$89$	16.8310	1.78409	0.892043	−	0.451950i	$-0.149272\pi$
$89$	16.8310	1.78409	0.892043	+	0.451950i	$0.149272\pi$
$90$	1.96273	0.206889
$91$	−6.73932	−0.706473
$92$	−1.85229	−0.193115
$93$	8.78672	0.911140
$94$	−15.7702	−1.62657
$95$	−2.74271	−0.281396
$96$	−7.80808	−0.796908
$97$	−8.03091	−0.815416	−0.407708	−	0.913112i	$-0.633672\pi$
$97$	−8.03091	−0.815416	−0.407708	+	0.913112i	$0.633672\pi$
$98$	−1.96273	−0.198265
$99$	−1.88144	−0.189092
$100$	1.85229	0.185229
$101$	−6.31556	−0.628422	−0.314211	−	0.949353i	$-0.601740\pi$
$101$	−6.31556	−0.628422	−0.314211	+	0.949353i	$0.601740\pi$
$102$	13.7579	1.36223
$103$	4.73791	0.466840	0.233420	−	0.972376i	$-0.425008\pi$
$103$	4.73791	0.466840	0.233420	+	0.972376i	$0.425008\pi$
$104$	−1.95383	−0.191589
$105$	1.00000	0.0975900
$106$	−2.42776	−0.235805
$107$	−7.31608	−0.707272	−0.353636	−	0.935383i	$-0.615055\pi$
$107$	−7.31608	−0.707272	−0.353636	+	0.935383i	$0.615055\pi$
$108$	−1.85229	−0.178237
$109$	−12.2210	−1.17056	−0.585281	−	0.810830i	$-0.699016\pi$
$109$	−12.2210	−1.17056	−0.585281	+	0.810830i	$0.699016\pi$
$110$	−3.69276	−0.352090
$111$	−5.04650	−0.478992
$112$	−4.27360	−0.403817
$113$	18.5848	1.74831	0.874157	−	0.485643i	$-0.161414\pi$
$113$	18.5848	1.74831	0.874157	+	0.485643i	$0.161414\pi$
$114$	5.38318	0.504181
$115$	1.00000	0.0932505
$116$	−8.33253	−0.773656
$117$	−6.73932	−0.623050
$118$	21.6485	1.99291
$119$	7.00958	0.642567
$120$	0.289915	0.0264655
$121$	−7.46017	−0.678197
$122$	23.2023	2.10063
$123$	−4.15170	−0.374346
$124$	−16.2755	−1.46159
$125$	−1.00000	−0.0894427
$126$	−1.96273	−0.174853
$127$	5.57098	0.494344	0.247172	−	0.968972i	$-0.420499\pi$
$127$	5.57098	0.494344	0.247172	+	0.968972i	$0.420499\pi$
$128$	−2.31300	−0.204442
$129$	−4.10457	−0.361387
$130$	−13.2274	−1.16012
$131$	−5.85289	−0.511370	−0.255685	−	0.966760i	$-0.582301\pi$
$131$	−5.85289	−0.511370	−0.255685	+	0.966760i	$0.582301\pi$
$132$	3.48498	0.303328
$133$	2.74271	0.237823
$134$	26.3894	2.27970
$135$	1.00000	0.0860663
$136$	2.03218	0.174258
$137$	−15.7098	−1.34217	−0.671087	−	0.741378i	$-0.734172\pi$
$137$	−15.7098	−1.34217	−0.671087	+	0.741378i	$0.734172\pi$
$138$	−1.96273	−0.167078
$139$	12.1323	1.02905	0.514526	−	0.857475i	$-0.327968\pi$
$139$	12.1323	1.02905	0.514526	+	0.857475i	$0.327968\pi$
$140$	−1.85229	−0.156547
$141$	−8.03482	−0.676654
$142$	−10.9685	−0.920460
$143$	12.6796	1.06033
$144$	−4.27360	−0.356134
$145$	4.49850	0.373580
$146$	6.67289	0.552253
$147$	−1.00000	−0.0824786
$148$	9.34757	0.768366
$149$	9.08584	0.744341	0.372170	−	0.928164i	$-0.378614\pi$
$149$	9.08584	0.744341	0.372170	+	0.928164i	$0.378614\pi$
$150$	1.96273	0.160256
$151$	20.4871	1.66722	0.833608	−	0.552357i	$-0.186271\pi$
$151$	20.4871	1.66722	0.833608	+	0.552357i	$0.186271\pi$
$152$	0.795153	0.0644954
$153$	7.00958	0.566691
$154$	3.69276	0.297571
$155$	8.78672	0.705766
$156$	12.4832	0.999454
$157$	21.6367	1.72680	0.863399	−	0.504522i	$-0.168331\pi$
$157$	21.6367	1.72680	0.863399	+	0.504522i	$0.168331\pi$
$158$	2.63574	0.209688
$159$	−1.23693	−0.0980951
$160$	−7.80808	−0.617283
$161$	−1.00000	−0.0788110
$162$	−1.96273	−0.154206
$163$	25.0185	1.95960	0.979798	−	0.199988i	$-0.0640904\pi$
$163$	25.0185	1.95960	0.979798	+	0.199988i	$0.0640904\pi$
$164$	7.69015	0.600500
$165$	−1.88144	−0.146470
$166$	−7.44876	−0.578136
$167$	0.516869	0.0399965	0.0199983	−	0.999800i	$-0.493634\pi$
$167$	0.516869	0.0399965	0.0199983	+	0.999800i	$0.493634\pi$
$168$	−0.289915	−0.0223674
$169$	32.4184	2.49373
$170$	13.7579	1.05518
$171$	2.74271	0.209740
$172$	7.60285	0.579712
$173$	−1.28351	−0.0975835	−0.0487918	−	0.998809i	$-0.515537\pi$
$173$	−1.28351	−0.0975835	−0.0487918	+	0.998809i	$0.515537\pi$
$174$	−8.82933	−0.669350
$175$	1.00000	0.0755929
$176$	8.04054	0.606078
$177$	11.0298	0.829053
$178$	−33.0347	−2.47606
$179$	13.5380	1.01188	0.505940	−	0.862569i	$-0.331146\pi$
$179$	13.5380	1.01188	0.505940	+	0.862569i	$0.331146\pi$
$180$	−1.85229	−0.138062
$181$	4.64451	0.345224	0.172612	−	0.984990i	$-0.444779\pi$
$181$	4.64451	0.345224	0.172612	+	0.984990i	$0.444779\pi$
$182$	13.2274	0.980483
$183$	11.8215	0.873867
$184$	−0.289915	−0.0213728
$185$	−5.04650	−0.371026
$186$	−17.2459	−1.26453
$187$	−13.1881	−0.964411
$188$	14.8828	1.08544
$189$	−1.00000	−0.0727393
$190$	5.38318	0.390537
$191$	−17.8427	−1.29105	−0.645527	−	0.763738i	$-0.723362\pi$
$191$	−17.8427	−1.29105	−0.645527	+	0.763738i	$0.723362\pi$
$192$	6.77790	0.489153
$193$	−4.12911	−0.297220	−0.148610	−	0.988896i	$-0.547480\pi$
$193$	−4.12911	−0.297220	−0.148610	+	0.988896i	$0.547480\pi$
$194$	15.7625	1.13168
$195$	−6.73932	−0.482613
$196$	1.85229	0.132306
$197$	25.6140	1.82492	0.912459	−	0.409168i	$-0.134181\pi$
$197$	25.6140	1.82492	0.912459	+	0.409168i	$0.134181\pi$
$198$	3.69276	0.262433
$199$	−16.8191	−1.19228	−0.596139	−	0.802881i	$-0.703299\pi$
$199$	−16.8191	−1.19228	−0.596139	+	0.802881i	$0.703299\pi$
$200$	0.289915	0.0205001
$201$	13.4453	0.948358
$202$	12.3957	0.872159
$203$	−4.49850	−0.315733
$204$	−12.9838	−0.909046
$205$	−4.15170	−0.289967
$206$	−9.29921	−0.647906
$207$	−1.00000	−0.0695048
$208$	28.8012	1.99700
$209$	−5.16025	−0.356942
$210$	−1.96273	−0.135441
$211$	18.4015	1.26681	0.633406	−	0.773820i	$-0.281656\pi$
$211$	18.4015	1.26681	0.633406	+	0.773820i	$0.281656\pi$
$212$	2.29116	0.157357
$213$	−5.58843	−0.382913
$214$	14.3594	0.981592
$215$	−4.10457	−0.279929
$216$	−0.289915	−0.0197262
$217$	−8.78672	−0.596481
$218$	23.9865	1.62457
$219$	3.39981	0.229738
$220$	3.48498	0.234957
$221$	−47.2398	−3.17769
$222$	9.90488	0.664772
$223$	−16.9884	−1.13763	−0.568813	−	0.822467i	$-0.692597\pi$
$223$	−16.9884	−1.13763	−0.568813	+	0.822467i	$0.692597\pi$
$224$	7.80808	0.521699
$225$	1.00000	0.0666667
$226$	−36.4769	−2.42641
$227$	8.01149	0.531741	0.265871	−	0.964009i	$-0.414341\pi$
$227$	8.01149	0.531741	0.265871	+	0.964009i	$0.414341\pi$
$228$	−5.08029	−0.336450
$229$	−11.2656	−0.744454	−0.372227	−	0.928142i	$-0.621406\pi$
$229$	−11.2656	−0.744454	−0.372227	+	0.928142i	$0.621406\pi$
$230$	−1.96273	−0.129418
$231$	1.88144	0.123790
$232$	−1.30418	−0.0856239
$233$	24.2473	1.58849	0.794246	−	0.607596i	$-0.207866\pi$
$233$	24.2473	1.58849	0.794246	+	0.607596i	$0.207866\pi$
$234$	13.2274	0.864704
$235$	−8.03482	−0.524134
$236$	−20.4305	−1.32991
$237$	1.34290	0.0872306
$238$	−13.7579	−0.891791
$239$	17.4225	1.12697	0.563483	−	0.826128i	$-0.309461\pi$
$239$	17.4225	1.12697	0.563483	+	0.826128i	$0.309461\pi$
$240$	−4.27360	−0.275860
$241$	−6.02267	−0.387954	−0.193977	−	0.981006i	$-0.562139\pi$
$241$	−6.02267	−0.387954	−0.193977	+	0.981006i	$0.562139\pi$
$242$	14.6423	0.941241
$243$	−1.00000	−0.0641500
$244$	−21.8968	−1.40180
$245$	−1.00000	−0.0638877
$246$	8.14865	0.519539
$247$	−18.4840	−1.17611
$248$	−2.54740	−0.161760
$249$	−3.79511	−0.240505
$250$	1.96273	0.124134
$251$	−17.9480	−1.13287	−0.566434	−	0.824107i	$-0.691678\pi$
$251$	−17.9480	−1.13287	−0.566434	+	0.824107i	$0.691678\pi$
$252$	1.85229	0.116683
$253$	1.88144	0.118285
$254$	−10.9343	−0.686079
$255$	7.00958	0.438957
$256$	18.0956	1.13097
$257$	26.6198	1.66050	0.830250	−	0.557392i	$-0.188198\pi$
$257$	26.6198	1.66050	0.830250	+	0.557392i	$0.188198\pi$
$258$	8.05615	0.501554
$259$	5.04650	0.313574
$260$	12.4832	0.774174
$261$	−4.49850	−0.278450
$262$	11.4876	0.709707
$263$	29.7779	1.83619	0.918093	−	0.396366i	$-0.129729\pi$
$263$	29.7779	1.83619	0.918093	+	0.396366i	$0.129729\pi$
$264$	0.545459	0.0335707
$265$	−1.23693	−0.0759841
$266$	−5.38318	−0.330064
$267$	−16.8310	−1.03004
$268$	−24.9046	−1.52129
$269$	−12.5465	−0.764972	−0.382486	−	0.923961i	$-0.624932\pi$
$269$	−12.5465	−0.764972	−0.382486	+	0.923961i	$0.624932\pi$
$270$	−1.96273	−0.119448
$271$	28.8498	1.75250	0.876249	−	0.481858i	$-0.160038\pi$
$271$	28.8498	1.75250	0.876249	+	0.481858i	$0.160038\pi$
$272$	−29.9562	−1.81636
$273$	6.73932	0.407882
$274$	30.8339	1.86275
$275$	−1.88144	−0.113455
$276$	1.85229	0.111495
$277$	5.27842	0.317150	0.158575	−	0.987347i	$-0.449310\pi$
$277$	5.27842	0.317150	0.158575	+	0.987347i	$0.449310\pi$
$278$	−23.8124	−1.42818
$279$	−8.78672	−0.526047
$280$	−0.289915	−0.0173257
$281$	23.3196	1.39113	0.695566	−	0.718462i	$-0.255154\pi$
$281$	23.3196	1.39113	0.695566	+	0.718462i	$0.255154\pi$
$282$	15.7702	0.939099
$283$	20.2709	1.20498	0.602491	−	0.798125i	$-0.294175\pi$
$283$	20.2709	1.20498	0.602491	+	0.798125i	$0.294175\pi$
$284$	10.3514	0.614242
$285$	2.74271	0.162464
$286$	−24.8867	−1.47158
$287$	4.15170	0.245067
$288$	7.80808	0.460095
$289$	32.1342	1.89025
$290$	−8.82933	−0.518476
$291$	8.03091	0.470780
$292$	−6.29743	−0.368529
$293$	7.09432	0.414454	0.207227	−	0.978293i	$-0.433556\pi$
$293$	7.09432	0.414454	0.207227	+	0.978293i	$0.433556\pi$
$294$	1.96273	0.114468
$295$	11.0298	0.642182
$296$	1.46306	0.0850384
$297$	1.88144	0.109172
$298$	−17.8330	−1.03304
$299$	6.73932	0.389745
$300$	−1.85229	−0.106942
$301$	4.10457	0.236584
$302$	−40.2105	−2.31386
$303$	6.31556	0.362820
$304$	−11.7212	−0.672260
$305$	11.8215	0.676895
$306$	−13.7579	−0.786485
$307$	−11.6887	−0.667111	−0.333556	−	0.942730i	$-0.608249\pi$
$307$	−11.6887	−0.667111	−0.333556	+	0.942730i	$0.608249\pi$
$308$	−3.48498	−0.198575
$309$	−4.73791	−0.269530
$310$	−17.2459	−0.979502
$311$	−11.3272	−0.642308	−0.321154	−	0.947027i	$-0.604071\pi$
$311$	−11.3272	−0.642308	−0.321154	+	0.947027i	$0.604071\pi$
$312$	1.95383	0.110614
$313$	24.4382	1.38133	0.690663	−	0.723176i	$-0.257319\pi$
$313$	24.4382	1.38133	0.690663	+	0.723176i	$0.257319\pi$
$314$	−42.4669	−2.39655
$315$	−1.00000	−0.0563436
$316$	−2.48744	−0.139929
$317$	16.2044	0.910129	0.455065	−	0.890458i	$-0.349616\pi$
$317$	16.2044	0.910129	0.455065	+	0.890458i	$0.349616\pi$
$318$	2.42776	0.136142
$319$	8.46368	0.473875
$320$	6.77790	0.378896
$321$	7.31608	0.408344
$322$	1.96273	0.109378
$323$	19.2252	1.06972
$324$	1.85229	0.102905
$325$	−6.73932	−0.373830
$326$	−49.1043	−2.71964
$327$	12.2210	0.675825
$328$	1.20364	0.0664600
$329$	8.03482	0.442974
$330$	3.69276	0.203279
$331$	7.17385	0.394311	0.197155	−	0.980372i	$-0.436830\pi$
$331$	7.17385	0.394311	0.197155	+	0.980372i	$0.436830\pi$
$332$	7.02964	0.385802
$333$	5.04650	0.276546
$334$	−1.01447	−0.0555095
$335$	13.4453	0.734595
$336$	4.27360	0.233144
$337$	−19.3317	−1.05306	−0.526532	−	0.850155i	$-0.676508\pi$
$337$	−19.3317	−1.05306	−0.526532	+	0.850155i	$0.676508\pi$
$338$	−63.6285	−3.46093
$339$	−18.5848	−1.00939
$340$	−12.9838	−0.704144
$341$	16.5317	0.895242
$342$	−5.38318	−0.291089
$343$	1.00000	0.0539949
$344$	1.18998	0.0641593
$345$	−1.00000	−0.0538382
$346$	2.51918	0.135432
$347$	−6.69208	−0.359250	−0.179625	−	0.983735i	$-0.557488\pi$
$347$	−6.69208	−0.359250	−0.179625	+	0.983735i	$0.557488\pi$
$348$	8.33253	0.446671
$349$	26.8193	1.43560	0.717802	−	0.696247i	$-0.245148\pi$
$349$	26.8193	1.43560	0.717802	+	0.696247i	$0.245148\pi$
$350$	−1.96273	−0.104912
$351$	6.73932	0.359718
$352$	−14.6905	−0.783004
$353$	−0.00680511	−0.000362199 0	−0.000181100	−	1.00000i	$-0.500058\pi$
$353$	−0.00680511	−0.000362199 0	−0.000181100		1.00000i	$0.500058\pi$
$354$	−21.6485	−1.15061
$355$	−5.58843	−0.296603
$356$	31.1760	1.65232
$357$	−7.00958	−0.370986
$358$	−26.5714	−1.40434
$359$	−16.1485	−0.852284	−0.426142	−	0.904656i	$-0.640128\pi$
$359$	−16.1485	−0.852284	−0.426142	+	0.904656i	$0.640128\pi$
$360$	−0.289915	−0.0152799
$361$	−11.4775	−0.604081
$362$	−9.11590	−0.479121
$363$	7.46017	0.391558
$364$	−12.4832	−0.654296
$365$	3.39981	0.177954
$366$	−23.2023	−1.21280
$367$	14.6103	0.762653	0.381327	−	0.924440i	$-0.375467\pi$
$367$	14.6103	0.762653	0.381327	+	0.924440i	$0.375467\pi$
$368$	4.27360	0.222777
$369$	4.15170	0.216129
$370$	9.90488	0.514930
$371$	1.23693	0.0642183
$372$	16.2755	0.843848
$373$	23.6520	1.22465	0.612327	−	0.790605i	$-0.290234\pi$
$373$	23.6520	1.22465	0.612327	+	0.790605i	$0.290234\pi$
$374$	25.8847	1.33846
$375$	1.00000	0.0516398
$376$	2.32942	0.120131
$377$	30.3169	1.56140
$378$	1.96273	0.100952
$379$	−23.7989	−1.22247	−0.611233	−	0.791450i	$-0.709326\pi$
$379$	−23.7989	−1.22247	−0.611233	+	0.791450i	$0.709326\pi$
$380$	−5.08029	−0.260613
$381$	−5.57098	−0.285410
$382$	35.0203	1.79180
$383$	−7.64039	−0.390406	−0.195203	−	0.980763i	$-0.562537\pi$
$383$	−7.64039	−0.390406	−0.195203	+	0.980763i	$0.562537\pi$
$384$	2.31300	0.118035
$385$	1.88144	0.0958872
$386$	8.10431	0.412498
$387$	4.10457	0.208647
$388$	−14.8756	−0.755193
$389$	−30.3840	−1.54053	−0.770265	−	0.637724i	$-0.779876\pi$
$389$	−30.3840	−1.54053	−0.770265	+	0.637724i	$0.779876\pi$
$390$	13.2274	0.669797
$391$	−7.00958	−0.354490
$392$	0.289915	0.0146429
$393$	5.85289	0.295239
$394$	−50.2731	−2.53272
$395$	1.34290	0.0675686
$396$	−3.48498	−0.175127
$397$	−23.6752	−1.18823	−0.594113	−	0.804381i	$-0.702497\pi$
$397$	−23.6752	−1.18823	−0.594113	+	0.804381i	$0.702497\pi$
$398$	33.0114	1.65471
$399$	−2.74271	−0.137307
$400$	−4.27360	−0.213680
$401$	−2.38003	−0.118853	−0.0594264	−	0.998233i	$-0.518927\pi$
$401$	−2.38003	−0.118853	−0.0594264	+	0.998233i	$0.518927\pi$
$402$	−26.3894	−1.31618
$403$	59.2165	2.94978
$404$	−11.6982	−0.582010
$405$	−1.00000	−0.0496904
$406$	8.82933	0.438192
$407$	−9.49470	−0.470635
$408$	−2.03218	−0.100608
$409$	26.4202	1.30640	0.653198	−	0.757187i	$-0.273427\pi$
$409$	26.4202	1.30640	0.653198	+	0.757187i	$0.273427\pi$
$410$	8.14865	0.402433
$411$	15.7098	0.774905
$412$	8.77597	0.432361
$413$	−11.0298	−0.542743
$414$	1.96273	0.0964627
$415$	−3.79511	−0.186295
$416$	−52.6211	−2.57996
$417$	−12.1323	−0.594123
$418$	10.1282	0.495384
$419$	35.4114	1.72996	0.864979	−	0.501808i	$-0.167332\pi$
$419$	35.4114	1.72996	0.864979	+	0.501808i	$0.167332\pi$
$420$	1.85229	0.0903825
$421$	28.1562	1.37225	0.686124	−	0.727485i	$-0.259311\pi$
$421$	28.1562	1.37225	0.686124	+	0.727485i	$0.259311\pi$
$422$	−36.1171	−1.75815
$423$	8.03482	0.390667
$424$	0.358605	0.0174154
$425$	7.00958	0.340015
$426$	10.9685	0.531428
$427$	−11.8215	−0.572080
$428$	−13.5515	−0.655036
$429$	−12.6796	−0.612179
$430$	8.05615	0.388502
$431$	−13.6124	−0.655685	−0.327842	−	0.944732i	$-0.606322\pi$
$431$	−13.6124	−0.655685	−0.327842	+	0.944732i	$0.606322\pi$
$432$	4.27360	0.205614
$433$	−38.0720	−1.82962	−0.914812	−	0.403880i	$-0.867661\pi$
$433$	−38.0720	−1.82962	−0.914812	+	0.403880i	$0.867661\pi$
$434$	17.2459	0.827830
$435$	−4.49850	−0.215687
$436$	−22.6369	−1.08411
$437$	−2.74271	−0.131202
$438$	−6.67289	−0.318843
$439$	−16.0900	−0.767936	−0.383968	−	0.923346i	$-0.625443\pi$
$439$	−16.0900	−0.767936	−0.383968	+	0.923346i	$0.625443\pi$
$440$	0.545459	0.0260037
$441$	1.00000	0.0476190
$442$	92.7187	4.41018
$443$	34.1527	1.62264	0.811321	−	0.584601i	$-0.198749\pi$
$443$	34.1527	1.62264	0.811321	+	0.584601i	$0.198749\pi$
$444$	−9.34757	−0.443616
$445$	−16.8310	−0.797868
$446$	33.3435	1.57886
$447$	−9.08584	−0.429745
$448$	−6.77790	−0.320226
$449$	8.43536	0.398089	0.199045	−	0.979990i	$-0.436216\pi$
$449$	8.43536	0.398089	0.199045	+	0.979990i	$0.436216\pi$
$450$	−1.96273	−0.0925237
$451$	−7.81119	−0.367815
$452$	34.4245	1.61919
$453$	−20.4871	−0.962568
$454$	−15.7244	−0.737980
$455$	6.73932	0.315944
$456$	−0.795153	−0.0372364
$457$	23.8348	1.11495	0.557473	−	0.830195i	$-0.311771\pi$
$457$	23.8348	1.11495	0.557473	+	0.830195i	$0.311771\pi$
$458$	22.1113	1.03320
$459$	−7.00958	−0.327179
$460$	1.85229	0.0863634
$461$	−0.511791	−0.0238365	−0.0119182	−	0.999929i	$-0.503794\pi$
$461$	−0.511791	−0.0238365	−0.0119182	+	0.999929i	$0.503794\pi$
$462$	−3.69276	−0.171803
$463$	−24.9074	−1.15754	−0.578772	−	0.815490i	$-0.696468\pi$
$463$	−24.9074	−1.15754	−0.578772	+	0.815490i	$0.696468\pi$
$464$	19.2248	0.892490
$465$	−8.78672	−0.407474
$466$	−47.5908	−2.20460
$467$	15.4648	0.715625	0.357812	−	0.933793i	$-0.383523\pi$
$467$	15.4648	0.715625	0.357812	+	0.933793i	$0.383523\pi$
$468$	−12.4832	−0.577035
$469$	−13.4453	−0.620846
$470$	15.7702	0.727423
$471$	−21.6367	−0.996967
$472$	−3.19772	−0.147187
$473$	−7.72252	−0.355082
$474$	−2.63574	−0.121064
$475$	2.74271	0.125844
$476$	12.9838	0.595110
$477$	1.23693	0.0566352
$478$	−34.1955	−1.56407
$479$	−26.5630	−1.21369	−0.606847	−	0.794819i	$-0.707566\pi$
$479$	−26.5630	−1.21369	−0.606847	+	0.794819i	$0.707566\pi$
$480$	7.80808	0.356388
$481$	−34.0100	−1.55072
$482$	11.8208	0.538424
$483$	1.00000	0.0455016
$484$	−13.8184	−0.628109
$485$	8.03091	0.364665
$486$	1.96273	0.0890310
$487$	−24.3058	−1.10140	−0.550700	−	0.834703i	$-0.685639\pi$
$487$	−24.3058	−1.10140	−0.550700	+	0.834703i	$0.685639\pi$
$488$	−3.42722	−0.155143
$489$	−25.0185	−1.13137
$490$	1.96273	0.0886669
$491$	6.29663	0.284163	0.142081	−	0.989855i	$-0.454621\pi$
$491$	6.29663	0.284163	0.142081	+	0.989855i	$0.454621\pi$
$492$	−7.69015	−0.346699
$493$	−31.5326	−1.42016
$494$	36.2790	1.63227
$495$	1.88144	0.0845646
$496$	37.5509	1.68609
$497$	5.58843	0.250675
$498$	7.44876	0.333787
$499$	20.5102	0.918163	0.459082	−	0.888394i	$-0.348179\pi$
$499$	20.5102	0.918163	0.459082	+	0.888394i	$0.348179\pi$
$500$	−1.85229	−0.0828369
$501$	−0.516869	−0.0230920
$502$	35.2270	1.57226
$503$	−6.15713	−0.274533	−0.137267	−	0.990534i	$-0.543832\pi$
$503$	−6.15713	−0.274533	−0.137267	+	0.990534i	$0.543832\pi$
$504$	0.289915	0.0129138
$505$	6.31556	0.281039
$506$	−3.69276	−0.164163
$507$	−32.4184	−1.43975
$508$	10.3191	0.457835
$509$	31.0409	1.37586	0.687932	−	0.725775i	$-0.258519\pi$
$509$	31.0409	1.37586	0.687932	+	0.725775i	$0.258519\pi$
$510$	−13.7579	−0.609209
$511$	−3.39981	−0.150399
$512$	−30.8907	−1.36519
$513$	−2.74271	−0.121094
$514$	−52.2474	−2.30453
$515$	−4.73791	−0.208777
$516$	−7.60285	−0.334697
$517$	−15.1171	−0.664848
$518$	−9.90488	−0.435196
$519$	1.28351	0.0563399
$520$	1.95383	0.0856812
$521$	9.77360	0.428189	0.214095	−	0.976813i	$-0.431320\pi$
$521$	9.77360	0.428189	0.214095	+	0.976813i	$0.431320\pi$
$522$	8.82933	0.386449
$523$	−29.4299	−1.28688	−0.643440	−	0.765497i	$-0.722493\pi$
$523$	−29.4299	−1.28688	−0.643440	+	0.765497i	$0.722493\pi$
$524$	−10.8413	−0.473602
$525$	−1.00000	−0.0436436
$526$	−58.4459	−2.54836
$527$	−61.5912	−2.68295
$528$	−8.04054	−0.349920
$529$	1.00000	0.0434783
$530$	2.42776	0.105455
$531$	−11.0298	−0.478654
$532$	5.08029	0.220259
$533$	−27.9796	−1.21193
$534$	33.0347	1.42955
$535$	7.31608	0.316302
$536$	−3.89799	−0.168368
$537$	−13.5380	−0.584209
$538$	24.6253	1.06167
$539$	−1.88144	−0.0810395
$540$	1.85229	0.0797098
$541$	−13.4867	−0.579839	−0.289920	−	0.957051i	$-0.593629\pi$
$541$	−13.4867	−0.579839	−0.289920	+	0.957051i	$0.593629\pi$
$542$	−56.6241	−2.43222
$543$	−4.64451	−0.199315
$544$	54.7313	2.34659
$545$	12.2210	0.523492
$546$	−13.2274	−0.566082
$547$	12.0421	0.514881	0.257441	−	0.966294i	$-0.417121\pi$
$547$	12.0421	0.514881	0.257441	+	0.966294i	$0.417121\pi$
$548$	−29.0990	−1.24305
$549$	−11.8215	−0.504528
$550$	3.69276	0.157460
$551$	−12.3381	−0.525620
$552$	0.289915	0.0123396
$553$	−1.34290	−0.0571059
$554$	−10.3601	−0.440158
$555$	5.04650	0.214212
$556$	22.4726	0.953050
$557$	12.0468	0.510441	0.255221	−	0.966883i	$-0.417852\pi$
$557$	12.0468	0.510441	0.255221	+	0.966883i	$0.417852\pi$
$558$	17.2459	0.730077
$559$	−27.6620	−1.16998
$560$	4.27360	0.180593
$561$	13.1881	0.556803
$562$	−45.7700	−1.93069
$563$	−29.4668	−1.24188	−0.620939	−	0.783858i	$-0.713249\pi$
$563$	−29.4668	−1.24188	−0.620939	+	0.783858i	$0.713249\pi$
$564$	−14.8828	−0.626680
$565$	−18.5848	−0.781870
$566$	−39.7863	−1.67234
$567$	1.00000	0.0419961
$568$	1.62017	0.0679808
$569$	36.0979	1.51330	0.756651	−	0.653819i	$-0.226834\pi$
$569$	36.0979	1.51330	0.756651	+	0.653819i	$0.226834\pi$
$570$	−5.38318	−0.225477
$571$	13.6991	0.573289	0.286644	−	0.958037i	$-0.407460\pi$
$571$	13.6991	0.573289	0.286644	+	0.958037i	$0.407460\pi$
$572$	23.4864	0.982015
$573$	17.8427	0.745390
$574$	−8.14865	−0.340118
$575$	−1.00000	−0.0417029
$576$	−6.77790	−0.282413
$577$	−6.27784	−0.261350	−0.130675	−	0.991425i	$-0.541714\pi$
$577$	−6.27784	−0.261350	−0.130675	+	0.991425i	$0.541714\pi$
$578$	−63.0706	−2.62339
$579$	4.12911	0.171600
$580$	8.33253	0.345990
$581$	3.79511	0.157448
$582$	−15.7625	−0.653376
$583$	−2.32722	−0.0963835
$584$	−0.985656	−0.0407868
$585$	6.73932	0.278637
$586$	−13.9242	−0.575203
$587$	−33.1572	−1.36854	−0.684272	−	0.729227i	$-0.739880\pi$
$587$	−33.1572	−1.36854	−0.684272	+	0.729227i	$0.739880\pi$
$588$	−1.85229	−0.0763871
$589$	−24.0994	−0.992999
$590$	−21.6485	−0.891256
$591$	−25.6140	−1.05362
$592$	−21.5667	−0.886387
$593$	−14.0093	−0.575294	−0.287647	−	0.957737i	$-0.592873\pi$
$593$	−14.0093	−0.575294	−0.287647	+	0.957737i	$0.592873\pi$
$594$	−3.69276	−0.151516
$595$	−7.00958	−0.287365
$596$	16.8296	0.689367
$597$	16.8191	0.688362
$598$	−13.2274	−0.540910
$599$	19.9301	0.814320	0.407160	−	0.913357i	$-0.366519\pi$
$599$	19.9301	0.814320	0.407160	+	0.913357i	$0.366519\pi$
$600$	−0.289915	−0.0118357
$601$	−6.66385	−0.271824	−0.135912	−	0.990721i	$-0.543396\pi$
$601$	−6.66385	−0.271824	−0.135912	+	0.990721i	$0.543396\pi$
$602$	−8.05615	−0.328344
$603$	−13.4453	−0.547534
$604$	37.9480	1.54408
$605$	7.46017	0.303299
$606$	−12.3957	−0.503541
$607$	−21.7575	−0.883110	−0.441555	−	0.897234i	$-0.645573\pi$
$607$	−21.7575	−0.883110	−0.441555	+	0.897234i	$0.645573\pi$
$608$	21.4153	0.868504
$609$	4.49850	0.182289
$610$	−23.2023	−0.939433
$611$	−54.1493	−2.19064
$612$	12.9838	0.524838
$613$	25.9815	1.04938	0.524692	−	0.851292i	$-0.324181\pi$
$613$	25.9815	1.04938	0.524692	+	0.851292i	$0.324181\pi$
$614$	22.9418	0.925855
$615$	4.15170	0.167413
$616$	−0.545459	−0.0219772
$617$	6.02246	0.242455	0.121228	−	0.992625i	$-0.461317\pi$
$617$	6.02246	0.242455	0.121228	+	0.992625i	$0.461317\pi$
$618$	9.29921	0.374069
$619$	19.2385	0.773259	0.386629	−	0.922235i	$-0.373639\pi$
$619$	19.2385	0.773259	0.386629	+	0.922235i	$0.373639\pi$
$620$	16.2755	0.653641
$621$	1.00000	0.0401286
$622$	22.2322	0.891432
$623$	16.8310	0.674321
$624$	−28.8012	−1.15297
$625$	1.00000	0.0400000
$626$	−47.9654	−1.91708
$627$	5.16025	0.206081
$628$	40.0775	1.59926
$629$	35.3738	1.41045
$630$	1.96273	0.0781968
$631$	10.1227	0.402978	0.201489	−	0.979491i	$-0.435422\pi$
$631$	10.1227	0.402978	0.201489	+	0.979491i	$0.435422\pi$
$632$	−0.389327	−0.0154866
$633$	−18.4015	−0.731394
$634$	−31.8048	−1.26313
$635$	−5.57098	−0.221078
$636$	−2.29116	−0.0908502
$637$	−6.73932	−0.267022
$638$	−16.6119	−0.657670
$639$	5.58843	0.221075
$640$	2.31300	0.0914292
$641$	15.9066	0.628273	0.314136	−	0.949378i	$-0.398285\pi$
$641$	15.9066	0.628273	0.314136	+	0.949378i	$0.398285\pi$
$642$	−14.3594	−0.566722
$643$	18.5716	0.732394	0.366197	−	0.930537i	$-0.380660\pi$
$643$	18.5716	0.732394	0.366197	+	0.930537i	$0.380660\pi$
$644$	−1.85229	−0.0729904
$645$	4.10457	0.161617
$646$	−37.7339	−1.48462
$647$	−31.8456	−1.25198	−0.625990	−	0.779831i	$-0.715305\pi$
$647$	−31.8456	−1.25198	−0.625990	+	0.779831i	$0.715305\pi$
$648$	0.289915	0.0113889
$649$	20.7520	0.814588
$650$	13.2274	0.518823
$651$	8.78672	0.344379
$652$	46.3414	1.81487
$653$	−8.11417	−0.317532	−0.158766	−	0.987316i	$-0.550752\pi$
$653$	−8.11417	−0.317532	−0.158766	+	0.987316i	$0.550752\pi$
$654$	−23.9865	−0.937948
$655$	5.85289	0.228691
$656$	−17.7427	−0.692737
$657$	−3.39981	−0.132639
$658$	−15.7702	−0.614785
$659$	−38.1449	−1.48591	−0.742956	−	0.669340i	$-0.766577\pi$
$659$	−38.1449	−1.48591	−0.742956	+	0.669340i	$0.766577\pi$
$660$	−3.48498	−0.135653
$661$	−22.6727	−0.881867	−0.440933	−	0.897540i	$-0.645352\pi$
$661$	−22.6727	−0.881867	−0.440933	+	0.897540i	$0.645352\pi$
$662$	−14.0803	−0.547246
$663$	47.2398	1.83464
$664$	1.10026	0.0426984
$665$	−2.74271	−0.106358
$666$	−9.90488	−0.383806
$667$	4.49850	0.174183
$668$	0.957391	0.0370426
$669$	16.9884	0.656809
$670$	−26.3894	−1.01951
$671$	22.2414	0.858620
$672$	−7.80808	−0.301203
$673$	22.4442	0.865160	0.432580	−	0.901596i	$-0.357603\pi$
$673$	22.4442	0.865160	0.432580	+	0.901596i	$0.357603\pi$
$674$	37.9428	1.46150
$675$	−1.00000	−0.0384900
$676$	60.0483	2.30955
$677$	19.9877	0.768188	0.384094	−	0.923294i	$-0.374514\pi$
$677$	19.9877	0.768188	0.384094	+	0.923294i	$0.374514\pi$
$678$	36.4769	1.40089
$679$	−8.03091	−0.308198
$680$	−2.03218	−0.0779307
$681$	−8.01149	−0.307001
$682$	−32.4472	−1.24247
$683$	−43.8454	−1.67770	−0.838849	−	0.544364i	$-0.816771\pi$
$683$	−43.8454	−1.67770	−0.838849	+	0.544364i	$0.816771\pi$
$684$	5.08029	0.194250
$685$	15.7098	0.600239
$686$	−1.96273	−0.0749372
$687$	11.2656	0.429811
$688$	−17.5413	−0.668756
$689$	−8.33608	−0.317579
$690$	1.96273	0.0747197
$691$	1.15506	0.0439404	0.0219702	−	0.999759i	$-0.493006\pi$
$691$	1.15506	0.0439404	0.0219702	+	0.999759i	$0.493006\pi$
$692$	−2.37743	−0.0903765
$693$	−1.88144	−0.0714701
$694$	13.1347	0.498587
$695$	−12.1323	−0.460206
$696$	1.30418	0.0494350
$697$	29.1017	1.10230
$698$	−52.6389	−1.99241
$699$	−24.2473	−0.917117
$700$	1.85229	0.0700100
$701$	−18.3620	−0.693524	−0.346762	−	0.937953i	$-0.612719\pi$
$701$	−18.3620	−0.693524	−0.346762	+	0.937953i	$0.612719\pi$
$702$	−13.2274	−0.499237
$703$	13.8411	0.522026
$704$	12.7522	0.480618
$705$	8.03482	0.302609
$706$	0.0133566	0.000502681 0
$707$	−6.31556	−0.237521
$708$	20.4305	0.767824
$709$	20.7381	0.778837	0.389418	−	0.921061i	$-0.372676\pi$
$709$	20.7381	0.778837	0.389418	+	0.921061i	$0.372676\pi$
$710$	10.9685	0.411642
$711$	−1.34290	−0.0503626
$712$	4.87957	0.182870
$713$	8.78672	0.329065
$714$	13.7579	0.514876
$715$	−12.6796	−0.474192
$716$	25.0764	0.937147
$717$	−17.4225	−0.650654
$718$	31.6950	1.18285
$719$	7.65837	0.285609	0.142805	−	0.989751i	$-0.454388\pi$
$719$	7.65837	0.285609	0.142805	+	0.989751i	$0.454388\pi$
$720$	4.27360	0.159268
$721$	4.73791	0.176449
$722$	22.5273	0.838378
$723$	6.02267	0.223985
$724$	8.60298	0.319727
$725$	−4.49850	−0.167070
$726$	−14.6423	−0.543425
$727$	5.38950	0.199885	0.0999427	−	0.994993i	$-0.468134\pi$
$727$	5.38950	0.199885	0.0999427	+	0.994993i	$0.468134\pi$
$728$	−1.95383	−0.0724138
$729$	1.00000	0.0370370
$730$	−6.67289	−0.246975
$731$	28.7713	1.06415
$732$	21.8968	0.809328
$733$	28.0140	1.03472	0.517361	−	0.855767i	$-0.326914\pi$
$733$	28.0140	1.03472	0.517361	+	0.855767i	$0.326914\pi$
$734$	−28.6761	−1.05845
$735$	1.00000	0.0368856
$736$	−7.80808	−0.287810
$737$	25.2965	0.931810
$738$	−8.14865	−0.299956
$739$	35.6871	1.31277	0.656386	−	0.754425i	$-0.272084\pi$
$739$	35.6871	1.31277	0.656386	+	0.754425i	$0.272084\pi$
$740$	−9.34757	−0.343624
$741$	18.4840	0.679027
$742$	−2.42776	−0.0891258
$743$	−19.4748	−0.714462	−0.357231	−	0.934016i	$-0.616279\pi$
$743$	−19.4748	−0.714462	−0.357231	+	0.934016i	$0.616279\pi$
$744$	2.54740	0.0933923
$745$	−9.08584	−0.332879
$746$	−46.4224	−1.69964
$747$	3.79511	0.138856
$748$	−24.4282	−0.893184
$749$	−7.31608	−0.267324
$750$	−1.96273	−0.0716686
$751$	10.1154	0.369115	0.184557	−	0.982822i	$-0.440915\pi$
$751$	10.1154	0.369115	0.184557	+	0.982822i	$0.440915\pi$
$752$	−34.3376	−1.25217
$753$	17.9480	0.654062
$754$	−59.5037	−2.16700
$755$	−20.4871	−0.745602
$756$	−1.85229	−0.0673671
$757$	29.5859	1.07532	0.537660	−	0.843162i	$-0.319309\pi$
$757$	29.5859	1.07532	0.537660	+	0.843162i	$0.319309\pi$
$758$	46.7107	1.69661
$759$	−1.88144	−0.0682921
$760$	−0.795153	−0.0288432
$761$	2.64150	0.0957542	0.0478771	−	0.998853i	$-0.484754\pi$
$761$	2.64150	0.0957542	0.0478771	+	0.998853i	$0.484754\pi$
$762$	10.9343	0.396108
$763$	−12.2210	−0.442431
$764$	−33.0499	−1.19570
$765$	−7.00958	−0.253432
$766$	14.9960	0.541827
$767$	74.3336	2.68403
$768$	−18.0956	−0.652968
$769$	3.76293	0.135695	0.0678474	−	0.997696i	$-0.478387\pi$
$769$	3.76293	0.135695	0.0678474	+	0.997696i	$0.478387\pi$
$770$	−3.69276	−0.133078
$771$	−26.6198	−0.958690
$772$	−7.64831	−0.275269
$773$	−5.88779	−0.211769	−0.105884	−	0.994378i	$-0.533767\pi$
$773$	−5.88779	−0.211769	−0.105884	+	0.994378i	$0.533767\pi$
$774$	−8.05615	−0.289572
$775$	−8.78672	−0.315628
$776$	−2.32828	−0.0835805
$777$	−5.04650	−0.181042
$778$	59.6354	2.13803
$779$	11.3869	0.407978
$780$	−12.4832	−0.446969
$781$	−10.5143	−0.376232
$782$	13.7579	0.491981
$783$	4.49850	0.160763
$784$	−4.27360	−0.152629
$785$	−21.6367	−0.772248
$786$	−11.4876	−0.409750
$787$	−14.4101	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$787$	−14.4101	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$788$	47.4445	1.69014
$789$	−29.7779	−1.06012
$790$	−2.63574	−0.0937754
$791$	18.5848	0.660801
$792$	−0.545459	−0.0193820
$793$	79.6686	2.82911
$794$	46.4680	1.64909
$795$	1.23693	0.0438694
$796$	−31.1539	−1.10422
$797$	−11.0306	−0.390724	−0.195362	−	0.980731i	$-0.562588\pi$
$797$	−11.0306	−0.390724	−0.195362	+	0.980731i	$0.562588\pi$
$798$	5.38318	0.190563
$799$	56.3207	1.99248
$800$	7.80808	0.276057
$801$	16.8310	0.594695
$802$	4.67134	0.164951
$803$	6.39655	0.225729
$804$	24.9046	0.878316
$805$	1.00000	0.0352454
$806$	−116.226	−4.09388
$807$	12.5465	0.441657
$808$	−1.83098	−0.0644135
$809$	14.7087	0.517131	0.258566	−	0.965994i	$-0.416750\pi$
$809$	14.7087	0.517131	0.258566	+	0.965994i	$0.416750\pi$
$810$	1.96273	0.0689631
$811$	−49.7502	−1.74697	−0.873483	−	0.486854i	$-0.838144\pi$
$811$	−49.7502	−1.74697	−0.873483	+	0.486854i	$0.838144\pi$
$812$	−8.33253	−0.292415
$813$	−28.8498	−1.01181
$814$	18.6355	0.653173
$815$	−25.0185	−0.876358
$816$	29.9562	1.04868
$817$	11.2576	0.393855
$818$	−51.8556	−1.81309
$819$	−6.73932	−0.235491
$820$	−7.69015	−0.268552
$821$	−49.9002	−1.74153	−0.870764	−	0.491700i	$-0.836376\pi$
$821$	−49.9002	−1.74153	−0.870764	+	0.491700i	$0.836376\pi$
$822$	−30.8339	−1.07546
$823$	−45.1092	−1.57241	−0.786204	−	0.617967i	$-0.787956\pi$
$823$	−45.1092	−1.57241	−0.786204	+	0.617967i	$0.787956\pi$
$824$	1.37359	0.0478513
$825$	1.88144	0.0655034
$826$	21.6485	0.753249
$827$	10.1711	0.353685	0.176842	−	0.984239i	$-0.443412\pi$
$827$	10.1711	0.353685	0.176842	+	0.984239i	$0.443412\pi$
$828$	−1.85229	−0.0643715
$829$	42.1450	1.46376	0.731879	−	0.681435i	$-0.238644\pi$
$829$	42.1450	1.46376	0.731879	+	0.681435i	$0.238644\pi$
$830$	7.44876	0.258550
$831$	−5.27842	−0.183106
$832$	45.6785	1.58362
$833$	7.00958	0.242868
$834$	23.8124	0.824557
$835$	−0.516869	−0.0178870
$836$	−9.55828	−0.330580
$837$	8.78672	0.303713
$838$	−69.5028	−2.40093
$839$	−3.54653	−0.122440	−0.0612199	−	0.998124i	$-0.519499\pi$
$839$	−3.54653	−0.122440	−0.0612199	+	0.998124i	$0.519499\pi$
$840$	0.289915	0.0100030
$841$	−8.76346	−0.302188
$842$	−55.2628	−1.90448
$843$	−23.3196	−0.803170
$844$	34.0849	1.17325
$845$	−32.4184	−1.11523
$846$	−15.7702	−0.542189
$847$	−7.46017	−0.256335
$848$	−5.28615	−0.181527
$849$	−20.2709	−0.695697
$850$	−13.7579	−0.471891
$851$	−5.04650	−0.172992
$852$	−10.3514	−0.354633
$853$	−39.8554	−1.36462	−0.682311	−	0.731062i	$-0.739025\pi$
$853$	−39.8554	−1.36462	−0.682311	+	0.731062i	$0.739025\pi$
$854$	23.2023	0.793965
$855$	−2.74271	−0.0937987
$856$	−2.12104	−0.0724957
$857$	20.4533	0.698671	0.349335	−	0.936998i	$-0.386407\pi$
$857$	20.4533	0.698671	0.349335	+	0.936998i	$0.386407\pi$
$858$	24.8867	0.849617
$859$	18.2150	0.621489	0.310744	−	0.950493i	$-0.399422\pi$
$859$	18.2150	0.621489	0.310744	+	0.950493i	$0.399422\pi$
$860$	−7.60285	−0.259255
$861$	−4.15170	−0.141490
$862$	26.7173	0.909996
$863$	−38.7361	−1.31859	−0.659295	−	0.751884i	$-0.729145\pi$
$863$	−38.7361	−1.31859	−0.659295	+	0.751884i	$0.729145\pi$
$864$	−7.80808	−0.265636
$865$	1.28351	0.0436407
$866$	74.7249	2.53926
$867$	−32.1342	−1.09133
$868$	−16.2755	−0.552428
$869$	2.52659	0.0857086
$870$	8.82933	0.299342
$871$	90.6121	3.07027
$872$	−3.54306	−0.119983
$873$	−8.03091	−0.271805
$874$	5.38318	0.182089
$875$	−1.00000	−0.0338062
$876$	6.29743	0.212771
$877$	3.26479	0.110244	0.0551220	−	0.998480i	$-0.482445\pi$
$877$	3.26479	0.110244	0.0551220	+	0.998480i	$0.482445\pi$
$878$	31.5803	1.06578
$879$	−7.09432	−0.239285
$880$	−8.04054	−0.271047
$881$	40.8856	1.37747	0.688736	−	0.725012i	$-0.258166\pi$
$881$	40.8856	1.37747	0.688736	+	0.725012i	$0.258166\pi$
$882$	−1.96273	−0.0660884
$883$	48.6229	1.63629	0.818146	−	0.575010i	$-0.195002\pi$
$883$	48.6229	1.63629	0.818146	+	0.575010i	$0.195002\pi$
$884$	−87.5018	−2.94300
$885$	−11.0298	−0.370764
$886$	−67.0323	−2.25199
$887$	48.4837	1.62792	0.813962	−	0.580918i	$-0.197306\pi$
$887$	48.4837	1.62792	0.813962	+	0.580918i	$0.197306\pi$
$888$	−1.46306	−0.0490969
$889$	5.57098	0.186845
$890$	33.0347	1.10733
$891$	−1.88144	−0.0630307
$892$	−31.4674	−1.05361
$893$	22.0372	0.737446
$894$	17.8330	0.596425
$895$	−13.5380	−0.452526
$896$	−2.31300	−0.0772718
$897$	−6.73932	−0.225019
$898$	−16.5563	−0.552490
$899$	39.5271	1.31830
$900$	1.85229	0.0617430
$901$	8.67037	0.288852
$902$	15.3312	0.510474
$903$	−4.10457	−0.136592
$904$	5.38803	0.179203
$905$	−4.64451	−0.154389
$906$	40.2105	1.33591
$907$	−2.28527	−0.0758811	−0.0379405	−	0.999280i	$-0.512080\pi$
$907$	−2.28527	−0.0758811	−0.0379405	+	0.999280i	$0.512080\pi$
$908$	14.8396	0.492469
$909$	−6.31556	−0.209474
$910$	−13.2274	−0.438485
$911$	2.64407	0.0876020	0.0438010	−	0.999040i	$-0.486053\pi$
$911$	2.64407	0.0876020	0.0438010	+	0.999040i	$0.486053\pi$
$912$	11.7212	0.388129
$913$	−7.14028	−0.236309
$914$	−46.7812	−1.54739
$915$	−11.8215	−0.390805
$916$	−20.8672	−0.689472
$917$	−5.85289	−0.193280
$918$	13.7579	0.454078
$919$	−7.35469	−0.242609	−0.121304	−	0.992615i	$-0.538708\pi$
$919$	−7.35469	−0.242609	−0.121304	+	0.992615i	$0.538708\pi$
$920$	0.289915	0.00955822
$921$	11.6887	0.385157
$922$	1.00450	0.0330816
$923$	−37.6622	−1.23967
$924$	3.48498	0.114647
$925$	5.04650	0.165928
$926$	48.8863	1.60650
$927$	4.73791	0.155613
$928$	−35.1247	−1.15302
$929$	19.9623	0.654942	0.327471	−	0.944861i	$-0.393804\pi$
$929$	19.9623	0.654942	0.327471	+	0.944861i	$0.393804\pi$
$930$	17.2459	0.565516
$931$	2.74271	0.0898887
$932$	44.9130	1.47117
$933$	11.3272	0.370837
$934$	−30.3531	−0.993184
$935$	13.1881	0.431298
$936$	−1.95383	−0.0638630
$937$	−0.325598	−0.0106368	−0.00531841	−	0.999986i	$-0.501693\pi$
$937$	−0.325598	−0.0106368	−0.00531841	+	0.999986i	$0.501693\pi$
$938$	26.3894	0.861645
$939$	−24.4382	−0.797509
$940$	−14.8828	−0.485424
$941$	18.5685	0.605317	0.302658	−	0.953099i	$-0.402126\pi$
$941$	18.5685	0.605317	0.302658	+	0.953099i	$0.402126\pi$
$942$	42.4669	1.38365
$943$	−4.15170	−0.135198
$944$	47.1372	1.53418
$945$	1.00000	0.0325300
$946$	15.1572	0.492802
$947$	1.85177	0.0601743	0.0300871	−	0.999547i	$-0.490422\pi$
$947$	1.85177	0.0601743	0.0300871	+	0.999547i	$0.490422\pi$
$948$	2.48744	0.0807882
$949$	22.9124	0.743768
$950$	−5.38318	−0.174654
$951$	−16.2044	−0.525463
$952$	2.03218	0.0658634
$953$	2.75696	0.0893068	0.0446534	−	0.999003i	$-0.485782\pi$
$953$	2.75696	0.0893068	0.0446534	+	0.999003i	$0.485782\pi$
$954$	−2.42776	−0.0786015
$955$	17.8427	0.577377
$956$	32.2714	1.04373
$957$	−8.46368	−0.273592
$958$	52.1359	1.68443
$959$	−15.7098	−0.507294
$960$	−6.77790	−0.218756
$961$	46.2064	1.49053
$962$	66.7522	2.15218
$963$	−7.31608	−0.235757
$964$	−11.1557	−0.359302
$965$	4.12911	0.132921
$966$	−1.96273	−0.0631496
$967$	−16.5472	−0.532122	−0.266061	−	0.963956i	$-0.585722\pi$
$967$	−16.5472	−0.532122	−0.266061	+	0.963956i	$0.585722\pi$
$968$	−2.16282	−0.0695156
$969$	−19.2252	−0.617604
$970$	−15.7625	−0.506103
$971$	−13.9435	−0.447468	−0.223734	−	0.974650i	$-0.571825\pi$
$971$	−13.9435	−0.447468	−0.223734	+	0.974650i	$0.571825\pi$
$972$	−1.85229	−0.0594122
$973$	12.1323	0.388945
$974$	47.7055	1.52858
$975$	6.73932	0.215831
$976$	50.5202	1.61711
$977$	39.0767	1.25018	0.625088	−	0.780554i	$-0.285063\pi$
$977$	39.0767	1.25018	0.625088	+	0.780554i	$0.285063\pi$
$978$	49.1043	1.57018
$979$	−31.6666	−1.01207
$980$	−1.85229	−0.0591692
$981$	−12.2210	−0.390188
$982$	−12.3585	−0.394377
$983$	−23.5152	−0.750018	−0.375009	−	0.927021i	$-0.622360\pi$
$983$	−23.5152	−0.750018	−0.375009	+	0.927021i	$0.622360\pi$
$984$	−1.20364	−0.0383707
$985$	−25.6140	−0.816128
$986$	61.8899	1.97098
$987$	−8.03482	−0.255751
$988$	−34.2377	−1.08925
$989$	−4.10457	−0.130518
$990$	−3.69276	−0.117363
$991$	−0.815147	−0.0258940	−0.0129470	−	0.999916i	$-0.504121\pi$
$991$	−0.815147	−0.0258940	−0.0129470	+	0.999916i	$0.504121\pi$
$992$	−68.6074	−2.17829
$993$	−7.17385	−0.227655
$994$	−10.9685	−0.347901
$995$	16.8191	0.533203
$996$	−7.02964	−0.222743
$997$	−47.3282	−1.49890	−0.749449	−	0.662062i	$-0.769682\pi$
$997$	−47.3282	−1.49890	−0.749449	+	0.662062i	$0.769682\pi$
$998$	−40.2559	−1.27428
$999$	−5.04650	−0.159664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.2.a.q.1.1	✓	6
3.2	odd	2		7245.2.a.bj.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.q.1.1	✓	6	1.1	even	1	trivial
7245.2.a.bj.1.6		6	3.2	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:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96273 q^{2} -1.00000 q^{3} +1.85229 q^{4} -1.00000 q^{5} +1.96273 q^{6} +1.00000 q^{7} +0.289915 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.96273 q^{2} -1.00000 q^{3} +1.85229 q^{4} -1.00000 q^{5} +1.96273 q^{6} +1.00000 q^{7} +0.289915 q^{8} +1.00000 q^{9} +1.96273 q^{10} -1.88144 q^{11} -1.85229 q^{12} -6.73932 q^{13} -1.96273 q^{14} +1.00000 q^{15} -4.27360 q^{16} +7.00958 q^{17} -1.96273 q^{18} +2.74271 q^{19} -1.85229 q^{20} -1.00000 q^{21} +3.69276 q^{22} -1.00000 q^{23} -0.289915 q^{24} +1.00000 q^{25} +13.2274 q^{26} -1.00000 q^{27} +1.85229 q^{28} -4.49850 q^{29} -1.96273 q^{30} -8.78672 q^{31} +7.80808 q^{32} +1.88144 q^{33} -13.7579 q^{34} -1.00000 q^{35} +1.85229 q^{36} +5.04650 q^{37} -5.38318 q^{38} +6.73932 q^{39} -0.289915 q^{40} +4.15170 q^{41} +1.96273 q^{42} +4.10457 q^{43} -3.48498 q^{44} -1.00000 q^{45} +1.96273 q^{46} +8.03482 q^{47} +4.27360 q^{48} +1.00000 q^{49} -1.96273 q^{50} -7.00958 q^{51} -12.4832 q^{52} +1.23693 q^{53} +1.96273 q^{54} +1.88144 q^{55} +0.289915 q^{56} -2.74271 q^{57} +8.82933 q^{58} -11.0298 q^{59} +1.85229 q^{60} -11.8215 q^{61} +17.2459 q^{62} +1.00000 q^{63} -6.77790 q^{64} +6.73932 q^{65} -3.69276 q^{66} -13.4453 q^{67} +12.9838 q^{68} +1.00000 q^{69} +1.96273 q^{70} +5.58843 q^{71} +0.289915 q^{72} -3.39981 q^{73} -9.90488 q^{74} -1.00000 q^{75} +5.08029 q^{76} -1.88144 q^{77} -13.2274 q^{78} -1.34290 q^{79} +4.27360 q^{80} +1.00000 q^{81} -8.14865 q^{82} +3.79511 q^{83} -1.85229 q^{84} -7.00958 q^{85} -8.05615 q^{86} +4.49850 q^{87} -0.545459 q^{88} +16.8310 q^{89} +1.96273 q^{90} -6.73932 q^{91} -1.85229 q^{92} +8.78672 q^{93} -15.7702 q^{94} -2.74271 q^{95} -7.80808 q^{96} -8.03091 q^{97} -1.96273 q^{98} -1.88144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9	\( 6 q + 3 q^{2} - 6 q^{3} + 5 q^{4} - 6 q^{5} - 3 q^{6} + 6 q^{7} + 9 q^{8} + 6 q^{9} - 3 q^{10} + 6 q^{11} - 5 q^{12} - 6 q^{13} + 3 q^{14} + 6 q^{15} + 11 q^{16} + 8 q^{17} + 3 q^{18} - 8 q^{19} - 5 q^{20} - 6 q^{21} + 18 q^{22} - 6 q^{23} - 9 q^{24} + 6 q^{25} + 13 q^{26} - 6 q^{27} + 5 q^{28} + 8 q^{29} + 3 q^{30} - 16 q^{31} + 26 q^{32} - 6 q^{33} - 13 q^{34} - 6 q^{35} + 5 q^{36} + 8 q^{37} - 14 q^{38} + 6 q^{39} - 9 q^{40} + 8 q^{41} - 3 q^{42} + 8 q^{43} + 15 q^{44} - 6 q^{45} - 3 q^{46} + 10 q^{47} - 11 q^{48} + 6 q^{49} + 3 q^{50} - 8 q^{51} + 28 q^{53} - 3 q^{54} - 6 q^{55} + 9 q^{56} + 8 q^{57} + 14 q^{58} - 6 q^{59} + 5 q^{60} - 12 q^{61} + 18 q^{62} + 6 q^{63} + 15 q^{64} + 6 q^{65} - 18 q^{66} + 18 q^{67} + 18 q^{68} + 6 q^{69} - 3 q^{70} + 10 q^{71} + 9 q^{72} - 2 q^{73} + q^{74} - 6 q^{75} + 25 q^{76} + 6 q^{77} - 13 q^{78} - 2 q^{79} - 11 q^{80} + 6 q^{81} - 12 q^{82} + 26 q^{83} - 5 q^{84} - 8 q^{85} + 16 q^{86} - 8 q^{87} + 21 q^{88} + 8 q^{89} - 3 q^{90} - 6 q^{91} - 5 q^{92} + 16 q^{93} - 8 q^{94} + 8 q^{95} - 26 q^{96} + 20 q^{97} + 3 q^{98} + 6 q^{99}+O(q^{100}) \) 6 * q + 3 * q^2 - 6 * q^3 + 5 * q^4 - 6 * q^5 - 3 * q^6 + 6 * q^7 + 9 * q^8 + 6 * q^9 - 3 * q^10 + 6 * q^11 - 5 * q^12 - 6 * q^13 + 3 * q^14 + 6 * q^15 + 11 * q^16 + 8 * q^17 + 3 * q^18 - 8 * q^19 - 5 * q^20 - 6 * q^21 + 18 * q^22 - 6 * q^23 - 9 * q^24 + 6 * q^25 + 13 * q^26 - 6 * q^27 + 5 * q^28 + 8 * q^29 + 3 * q^30 - 16 * q^31 + 26 * q^32 - 6 * q^33 - 13 * q^34 - 6 * q^35 + 5 * q^36 + 8 * q^37 - 14 * q^38 + 6 * q^39 - 9 * q^40 + 8 * q^41 - 3 * q^42 + 8 * q^43 + 15 * q^44 - 6 * q^45 - 3 * q^46 + 10 * q^47 - 11 * q^48 + 6 * q^49 + 3 * q^50 - 8 * q^51 + 28 * q^53 - 3 * q^54 - 6 * q^55 + 9 * q^56 + 8 * q^57 + 14 * q^58 - 6 * q^59 + 5 * q^60 - 12 * q^61 + 18 * q^62 + 6 * q^63 + 15 * q^64 + 6 * q^65 - 18 * q^66 + 18 * q^67 + 18 * q^68 + 6 * q^69 - 3 * q^70 + 10 * q^71 + 9 * q^72 - 2 * q^73 + q^74 - 6 * q^75 + 25 * q^76 + 6 * q^77 - 13 * q^78 - 2 * q^79 - 11 * q^80 + 6 * q^81 - 12 * q^82 + 26 * q^83 - 5 * q^84 - 8 * q^85 + 16 * q^86 - 8 * q^87 + 21 * q^88 + 8 * q^89 - 3 * q^90 - 6 * q^91 - 5 * q^92 + 16 * q^93 - 8 * q^94 + 8 * q^95 - 26 * q^96 + 20 * q^97 + 3 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more