LMFDB - Embedded newform 4030.2.a.o.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 12x^{6} + 40x^{5} + 24x^{4} - 118x^{3} + 25x^{2} + 30x + 4 $ x^8 - 3x^7 - 12x^6 + 40x^5 + 24x^4 - 118x^3 + 25x^2 + 30*x + 4
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			1.8
Root			$2.79411$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.79411 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79411 q^{6} +2.66175 q^{7} +1.00000 q^{8} +4.80704 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.79411 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79411 q^{6} +2.66175 q^{7} +1.00000 q^{8} +4.80704 q^{9} +1.00000 q^{10} +2.35851 q^{11} +2.79411 q^{12} -1.00000 q^{13} +2.66175 q^{14} +2.79411 q^{15} +1.00000 q^{16} +4.88536 q^{17} +4.80704 q^{18} -7.14690 q^{19} +1.00000 q^{20} +7.43721 q^{21} +2.35851 q^{22} -5.71081 q^{23} +2.79411 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.04906 q^{27} +2.66175 q^{28} -7.88697 q^{29} +2.79411 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.58992 q^{33} +4.88536 q^{34} +2.66175 q^{35} +4.80704 q^{36} +2.16249 q^{37} -7.14690 q^{38} -2.79411 q^{39} +1.00000 q^{40} +2.59883 q^{41} +7.43721 q^{42} +3.49049 q^{43} +2.35851 q^{44} +4.80704 q^{45} -5.71081 q^{46} +1.34606 q^{47} +2.79411 q^{48} +0.0849075 q^{49} +1.00000 q^{50} +13.6502 q^{51} -1.00000 q^{52} -2.66750 q^{53} +5.04906 q^{54} +2.35851 q^{55} +2.66175 q^{56} -19.9692 q^{57} -7.88697 q^{58} +5.78840 q^{59} +2.79411 q^{60} +8.65847 q^{61} +1.00000 q^{62} +12.7951 q^{63} +1.00000 q^{64} -1.00000 q^{65} +6.58992 q^{66} -11.2345 q^{67} +4.88536 q^{68} -15.9566 q^{69} +2.66175 q^{70} +1.34315 q^{71} +4.80704 q^{72} -12.1761 q^{73} +2.16249 q^{74} +2.79411 q^{75} -7.14690 q^{76} +6.27775 q^{77} -2.79411 q^{78} +0.0613807 q^{79} +1.00000 q^{80} -0.313490 q^{81} +2.59883 q^{82} -9.28658 q^{83} +7.43721 q^{84} +4.88536 q^{85} +3.49049 q^{86} -22.0370 q^{87} +2.35851 q^{88} -5.37378 q^{89} +4.80704 q^{90} -2.66175 q^{91} -5.71081 q^{92} +2.79411 q^{93} +1.34606 q^{94} -7.14690 q^{95} +2.79411 q^{96} +3.41086 q^{97} +0.0849075 q^{98} +11.3374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	2.79411	1.61318	0.806589	−	0.591112i	$-0.201311\pi$
$3$	2.79411	1.61318	0.806589	+	0.591112i	$0.201311\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	2.79411	1.14069
$7$	2.66175	1.00605	0.503023	−	0.864273i	$-0.332221\pi$
$7$	2.66175	1.00605	0.503023	+	0.864273i	$0.332221\pi$
$8$	1.00000	0.353553
$9$	4.80704	1.60235
$10$	1.00000	0.316228
$11$	2.35851	0.711116	0.355558	−	0.934654i	$-0.384291\pi$
$11$	2.35851	0.711116	0.355558	+	0.934654i	$0.384291\pi$
$12$	2.79411	0.806589
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.66175	0.711382
$15$	2.79411	0.721436
$16$	1.00000	0.250000
$17$	4.88536	1.18487	0.592436	−	0.805617i	$-0.298166\pi$
$17$	4.88536	1.18487	0.592436	+	0.805617i	$0.298166\pi$
$18$	4.80704	1.13303
$19$	−7.14690	−1.63961	−0.819806	−	0.572641i	$-0.805919\pi$
$19$	−7.14690	−1.63961	−0.819806	+	0.572641i	$0.805919\pi$
$20$	1.00000	0.223607
$21$	7.43721	1.62293
$22$	2.35851	0.502835
$23$	−5.71081	−1.19079	−0.595393	−	0.803434i	$-0.703004\pi$
$23$	−5.71081	−1.19079	−0.595393	+	0.803434i	$0.703004\pi$
$24$	2.79411	0.570345
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	5.04906	0.971693
$28$	2.66175	0.503023
$29$	−7.88697	−1.46457	−0.732287	−	0.680997i	$-0.761547\pi$
$29$	−7.88697	−1.46457	−0.732287	+	0.680997i	$0.761547\pi$
$30$	2.79411	0.510132
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	6.58992	1.14716
$34$	4.88536	0.837832
$35$	2.66175	0.449918
$36$	4.80704	0.801173
$37$	2.16249	0.355511	0.177756	−	0.984075i	$-0.443116\pi$
$37$	2.16249	0.355511	0.177756	+	0.984075i	$0.443116\pi$
$38$	−7.14690	−1.15938
$39$	−2.79411	−0.447415
$40$	1.00000	0.158114
$41$	2.59883	0.405868	0.202934	−	0.979192i	$-0.434952\pi$
$41$	2.59883	0.405868	0.202934	+	0.979192i	$0.434952\pi$
$42$	7.43721	1.14759
$43$	3.49049	0.532295	0.266148	−	0.963932i	$-0.414249\pi$
$43$	3.49049	0.532295	0.266148	+	0.963932i	$0.414249\pi$
$44$	2.35851	0.355558
$45$	4.80704	0.716591
$46$	−5.71081	−0.842013
$47$	1.34606	0.196343	0.0981714	−	0.995170i	$-0.468701\pi$
$47$	1.34606	0.196343	0.0981714	+	0.995170i	$0.468701\pi$
$48$	2.79411	0.403295
$49$	0.0849075	0.0121296
$50$	1.00000	0.141421
$51$	13.6502	1.91141
$52$	−1.00000	−0.138675
$53$	−2.66750	−0.366410	−0.183205	−	0.983075i	$-0.558647\pi$
$53$	−2.66750	−0.366410	−0.183205	+	0.983075i	$0.558647\pi$
$54$	5.04906	0.687090
$55$	2.35851	0.318021
$56$	2.66175	0.355691
$57$	−19.9692	−2.64499
$58$	−7.88697	−1.03561
$59$	5.78840	0.753585	0.376793	−	0.926298i	$-0.377027\pi$
$59$	5.78840	0.753585	0.376793	+	0.926298i	$0.377027\pi$
$60$	2.79411	0.360718
$61$	8.65847	1.10860	0.554302	−	0.832316i	$-0.312985\pi$
$61$	8.65847	1.10860	0.554302	+	0.832316i	$0.312985\pi$
$62$	1.00000	0.127000
$63$	12.7951	1.61204
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	6.58992	0.811163
$67$	−11.2345	−1.37251	−0.686256	−	0.727360i	$-0.740747\pi$
$67$	−11.2345	−1.37251	−0.686256	+	0.727360i	$0.740747\pi$
$68$	4.88536	0.592436
$69$	−15.9566	−1.92095
$70$	2.66175	0.318140
$71$	1.34315	0.159403	0.0797013	−	0.996819i	$-0.474603\pi$
$71$	1.34315	0.159403	0.0797013	+	0.996819i	$0.474603\pi$
$72$	4.80704	0.566515
$73$	−12.1761	−1.42511	−0.712553	−	0.701619i	$-0.752461\pi$
$73$	−12.1761	−1.42511	−0.712553	+	0.701619i	$0.752461\pi$
$74$	2.16249	0.251385
$75$	2.79411	0.322636
$76$	−7.14690	−0.819806
$77$	6.27775	0.715416
$78$	−2.79411	−0.316370
$79$	0.0613807	0.00690587	0.00345293	−	0.999994i	$-0.498901\pi$
$79$	0.0613807	0.00690587	0.00345293	+	0.999994i	$0.498901\pi$
$80$	1.00000	0.111803
$81$	−0.313490	−0.0348322
$82$	2.59883	0.286992
$83$	−9.28658	−1.01934	−0.509668	−	0.860371i	$-0.670232\pi$
$83$	−9.28658	−1.01934	−0.509668	+	0.860371i	$0.670232\pi$
$84$	7.43721	0.811467
$85$	4.88536	0.529891
$86$	3.49049	0.376389
$87$	−22.0370	−2.36262
$88$	2.35851	0.251418
$89$	−5.37378	−0.569620	−0.284810	−	0.958584i	$-0.591931\pi$
$89$	−5.37378	−0.569620	−0.284810	+	0.958584i	$0.591931\pi$
$90$	4.80704	0.506706
$91$	−2.66175	−0.279027
$92$	−5.71081	−0.595393
$93$	2.79411	0.289736
$94$	1.34606	0.138835
$95$	−7.14690	−0.733257
$96$	2.79411	0.285172
$97$	3.41086	0.346320	0.173160	−	0.984894i	$-0.444602\pi$
$97$	3.41086	0.346320	0.173160	+	0.984894i	$0.444602\pi$
$98$	0.0849075	0.00857696
$99$	11.3374	1.13945
$100$	1.00000	0.100000
$101$	−19.0241	−1.89297	−0.946485	−	0.322749i	$-0.895393\pi$
$101$	−19.0241	−1.89297	−0.946485	+	0.322749i	$0.895393\pi$
$102$	13.6502	1.35157
$103$	11.9266	1.17516	0.587581	−	0.809166i	$-0.300080\pi$
$103$	11.9266	1.17516	0.587581	+	0.809166i	$0.300080\pi$
$104$	−1.00000	−0.0980581
$105$	7.43721	0.725798
$106$	−2.66750	−0.259091
$107$	3.50725	0.339059	0.169529	−	0.985525i	$-0.445775\pi$
$107$	3.50725	0.339059	0.169529	+	0.985525i	$0.445775\pi$
$108$	5.04906	0.485846
$109$	−6.66079	−0.637988	−0.318994	−	0.947757i	$-0.603345\pi$
$109$	−6.66079	−0.637988	−0.318994	+	0.947757i	$0.603345\pi$
$110$	2.35851	0.224875
$111$	6.04224	0.573504
$112$	2.66175	0.251512
$113$	−3.44949	−0.324501	−0.162250	−	0.986750i	$-0.551875\pi$
$113$	−3.44949	−0.324501	−0.162250	+	0.986750i	$0.551875\pi$
$114$	−19.9692	−1.87029
$115$	−5.71081	−0.532536
$116$	−7.88697	−0.732287
$117$	−4.80704	−0.444411
$118$	5.78840	0.532865
$119$	13.0036	1.19204
$120$	2.79411	0.255066
$121$	−5.43745	−0.494314
$122$	8.65847	0.783901
$123$	7.26140	0.654738
$124$	1.00000	0.0898027
$125$	1.00000	0.0894427
$126$	12.7951	1.13988
$127$	6.20177	0.550318	0.275159	−	0.961399i	$-0.411269\pi$
$127$	6.20177	0.550318	0.275159	+	0.961399i	$0.411269\pi$
$128$	1.00000	0.0883883
$129$	9.75281	0.858687
$130$	−1.00000	−0.0877058
$131$	−2.18518	−0.190920	−0.0954601	−	0.995433i	$-0.530432\pi$
$131$	−2.18518	−0.190920	−0.0954601	+	0.995433i	$0.530432\pi$
$132$	6.58992	0.573579
$133$	−19.0233	−1.64953
$134$	−11.2345	−0.970512
$135$	5.04906	0.434554
$136$	4.88536	0.418916
$137$	−10.7136	−0.915328	−0.457664	−	0.889125i	$-0.651314\pi$
$137$	−10.7136	−0.915328	−0.457664	+	0.889125i	$0.651314\pi$
$138$	−15.9566	−1.35832
$139$	−3.70871	−0.314569	−0.157284	−	0.987553i	$-0.550274\pi$
$139$	−3.70871	−0.314569	−0.157284	+	0.987553i	$0.550274\pi$
$140$	2.66175	0.224959
$141$	3.76103	0.316736
$142$	1.34315	0.112715
$143$	−2.35851	−0.197228
$144$	4.80704	0.400587
$145$	−7.88697	−0.654977
$146$	−12.1761	−1.00770
$147$	0.237241	0.0195673
$148$	2.16249	0.177756
$149$	21.7746	1.78384	0.891921	−	0.452192i	$-0.149358\pi$
$149$	21.7746	1.78384	0.891921	+	0.452192i	$0.149358\pi$
$150$	2.79411	0.228138
$151$	4.81472	0.391817	0.195908	−	0.980622i	$-0.437235\pi$
$151$	4.81472	0.391817	0.195908	+	0.980622i	$0.437235\pi$
$152$	−7.14690	−0.579690
$153$	23.4841	1.89858
$154$	6.27775	0.505876
$155$	1.00000	0.0803219
$156$	−2.79411	−0.223708
$157$	−0.278598	−0.0222345	−0.0111173	−	0.999938i	$-0.503539\pi$
$157$	−0.278598	−0.0222345	−0.0111173	+	0.999938i	$0.503539\pi$
$158$	0.0613807	0.00488319
$159$	−7.45329	−0.591085
$160$	1.00000	0.0790569
$161$	−15.2007	−1.19799
$162$	−0.313490	−0.0246301
$163$	−8.47146	−0.663536	−0.331768	−	0.943361i	$-0.607645\pi$
$163$	−8.47146	−0.663536	−0.331768	+	0.943361i	$0.607645\pi$
$164$	2.59883	0.202934
$165$	6.58992	0.513025
$166$	−9.28658	−0.720779
$167$	17.6391	1.36495	0.682477	−	0.730907i	$-0.260903\pi$
$167$	17.6391	1.36495	0.682477	+	0.730907i	$0.260903\pi$
$168$	7.43721	0.573794
$169$	1.00000	0.0769231
$170$	4.88536	0.374690
$171$	−34.3554	−2.62723
$172$	3.49049	0.266148
$173$	9.72731	0.739553	0.369777	−	0.929121i	$-0.379434\pi$
$173$	9.72731	0.739553	0.369777	+	0.929121i	$0.379434\pi$
$174$	−22.0370	−1.67062
$175$	2.66175	0.201209
$176$	2.35851	0.177779
$177$	16.1734	1.21567
$178$	−5.37378	−0.402782
$179$	−5.02018	−0.375226	−0.187613	−	0.982243i	$-0.560075\pi$
$179$	−5.02018	−0.375226	−0.187613	+	0.982243i	$0.560075\pi$
$180$	4.80704	0.358296
$181$	15.6455	1.16292	0.581459	−	0.813575i	$-0.302482\pi$
$181$	15.6455	1.16292	0.581459	+	0.813575i	$0.302482\pi$
$182$	−2.66175	−0.197302
$183$	24.1927	1.78838
$184$	−5.71081	−0.421007
$185$	2.16249	0.158990
$186$	2.79411	0.204874
$187$	11.5221	0.842582
$188$	1.34606	0.0981714
$189$	13.4393	0.977568
$190$	−7.14690	−0.518491
$191$	−7.23146	−0.523250	−0.261625	−	0.965170i	$-0.584258\pi$
$191$	−7.23146	−0.523250	−0.261625	+	0.965170i	$0.584258\pi$
$192$	2.79411	0.201647
$193$	−17.8081	−1.28186	−0.640928	−	0.767601i	$-0.721450\pi$
$193$	−17.8081	−1.28186	−0.640928	+	0.767601i	$0.721450\pi$
$194$	3.41086	0.244885
$195$	−2.79411	−0.200090
$196$	0.0849075	0.00606482
$197$	19.6361	1.39901	0.699507	−	0.714626i	$-0.253403\pi$
$197$	19.6361	1.39901	0.699507	+	0.714626i	$0.253403\pi$
$198$	11.3374	0.805716
$199$	21.1576	1.49982	0.749911	−	0.661539i	$-0.230096\pi$
$199$	21.1576	1.49982	0.749911	+	0.661539i	$0.230096\pi$
$200$	1.00000	0.0707107
$201$	−31.3904	−2.21411
$202$	−19.0241	−1.33853
$203$	−20.9931	−1.47343
$204$	13.6502	0.955706
$205$	2.59883	0.181510
$206$	11.9266	0.830965
$207$	−27.4521	−1.90805
$208$	−1.00000	−0.0693375
$209$	−16.8560	−1.16595
$210$	7.43721	0.513217
$211$	−4.03472	−0.277762	−0.138881	−	0.990309i	$-0.544351\pi$
$211$	−4.03472	−0.277762	−0.138881	+	0.990309i	$0.544351\pi$
$212$	−2.66750	−0.183205
$213$	3.75290	0.257145
$214$	3.50725	0.239751
$215$	3.49049	0.238050
$216$	5.04906	0.343545
$217$	2.66175	0.180691
$218$	−6.66079	−0.451126
$219$	−34.0214	−2.29895
$220$	2.35851	0.159010
$221$	−4.88536	−0.328625
$222$	6.04224	0.405528
$223$	3.69526	0.247453	0.123727	−	0.992316i	$-0.460515\pi$
$223$	3.69526	0.247453	0.123727	+	0.992316i	$0.460515\pi$
$224$	2.66175	0.177846
$225$	4.80704	0.320469
$226$	−3.44949	−0.229457
$227$	−0.292760	−0.0194312	−0.00971558	−	0.999953i	$-0.503093\pi$
$227$	−0.292760	−0.0194312	−0.00971558	+	0.999953i	$0.503093\pi$
$228$	−19.9692	−1.32249
$229$	16.2872	1.07629	0.538145	−	0.842852i	$-0.319125\pi$
$229$	16.2872	1.07629	0.538145	+	0.842852i	$0.319125\pi$
$230$	−5.71081	−0.376560
$231$	17.5407	1.15409
$232$	−7.88697	−0.517805
$233$	20.6623	1.35363	0.676816	−	0.736152i	$-0.263359\pi$
$233$	20.6623	1.35363	0.676816	+	0.736152i	$0.263359\pi$
$234$	−4.80704	−0.314246
$235$	1.34606	0.0878072
$236$	5.78840	0.376793
$237$	0.171504	0.0111404
$238$	13.0036	0.842898
$239$	3.65003	0.236101	0.118051	−	0.993008i	$-0.462336\pi$
$239$	3.65003	0.236101	0.118051	+	0.993008i	$0.462336\pi$
$240$	2.79411	0.180359
$241$	23.2818	1.49971	0.749856	−	0.661601i	$-0.230123\pi$
$241$	23.2818	1.49971	0.749856	+	0.661601i	$0.230123\pi$
$242$	−5.43745	−0.349533
$243$	−16.0231	−1.02788
$244$	8.65847	0.554302
$245$	0.0849075	0.00542454
$246$	7.26140	0.462970
$247$	7.14690	0.454747
$248$	1.00000	0.0635001
$249$	−25.9477	−1.64437
$250$	1.00000	0.0632456
$251$	7.71137	0.486737	0.243369	−	0.969934i	$-0.421748\pi$
$251$	7.71137	0.486737	0.243369	+	0.969934i	$0.421748\pi$
$252$	12.7951	0.806018
$253$	−13.4690	−0.846788
$254$	6.20177	0.389134
$255$	13.6502	0.854809
$256$	1.00000	0.0625000
$257$	−13.1094	−0.817743	−0.408872	−	0.912592i	$-0.634078\pi$
$257$	−13.1094	−0.817743	−0.408872	+	0.912592i	$0.634078\pi$
$258$	9.75281	0.607184
$259$	5.75601	0.357661
$260$	−1.00000	−0.0620174
$261$	−37.9130	−2.34675
$262$	−2.18518	−0.135001
$263$	−31.2369	−1.92615	−0.963074	−	0.269236i	$-0.913229\pi$
$263$	−31.2369	−1.92615	−0.963074	+	0.269236i	$0.913229\pi$
$264$	6.58992	0.405582
$265$	−2.66750	−0.163863
$266$	−19.0233	−1.16639
$267$	−15.0149	−0.918899
$268$	−11.2345	−0.686256
$269$	2.92778	0.178510	0.0892550	−	0.996009i	$-0.471551\pi$
$269$	2.92778	0.178510	0.0892550	+	0.996009i	$0.471551\pi$
$270$	5.04906	0.307276
$271$	−10.4456	−0.634525	−0.317263	−	0.948338i	$-0.602764\pi$
$271$	−10.4456	−0.634525	−0.317263	+	0.948338i	$0.602764\pi$
$272$	4.88536	0.296218
$273$	−7.43721	−0.450121
$274$	−10.7136	−0.647235
$275$	2.35851	0.142223
$276$	−15.9566	−0.960476
$277$	17.3403	1.04188	0.520940	−	0.853593i	$-0.325581\pi$
$277$	17.3403	1.04188	0.520940	+	0.853593i	$0.325581\pi$
$278$	−3.70871	−0.222434
$279$	4.80704	0.287790
$280$	2.66175	0.159070
$281$	14.6944	0.876593	0.438296	−	0.898830i	$-0.355582\pi$
$281$	14.6944	0.876593	0.438296	+	0.898830i	$0.355582\pi$
$282$	3.76103	0.223966
$283$	23.3052	1.38535	0.692675	−	0.721250i	$-0.256432\pi$
$283$	23.3052	1.38535	0.692675	+	0.721250i	$0.256432\pi$
$284$	1.34315	0.0797013
$285$	−19.9692	−1.18287
$286$	−2.35851	−0.139461
$287$	6.91742	0.408322
$288$	4.80704	0.283258
$289$	6.86670	0.403923
$290$	−7.88697	−0.463139
$291$	9.53030	0.558676
$292$	−12.1761	−0.712553
$293$	17.1061	0.999349	0.499675	−	0.866213i	$-0.333453\pi$
$293$	17.1061	0.999349	0.499675	+	0.866213i	$0.333453\pi$
$294$	0.237241	0.0138362
$295$	5.78840	0.337013
$296$	2.16249	0.125692
$297$	11.9082	0.690986
$298$	21.7746	1.26137
$299$	5.71081	0.330265
$300$	2.79411	0.161318
$301$	9.29081	0.535514
$302$	4.81472	0.277056
$303$	−53.1554	−3.05370
$304$	−7.14690	−0.409903
$305$	8.65847	0.495783
$306$	23.4841	1.34250
$307$	20.9194	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$307$	20.9194	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$308$	6.27775	0.357708
$309$	33.3242	1.89575
$310$	1.00000	0.0567962
$311$	−14.5430	−0.824658	−0.412329	−	0.911035i	$-0.635285\pi$
$311$	−14.5430	−0.824658	−0.412329	+	0.911035i	$0.635285\pi$
$312$	−2.79411	−0.158185
$313$	12.8599	0.726886	0.363443	−	0.931616i	$-0.381601\pi$
$313$	12.8599	0.726886	0.363443	+	0.931616i	$0.381601\pi$
$314$	−0.278598	−0.0157222
$315$	12.7951	0.720924
$316$	0.0613807	0.00345293
$317$	−19.0811	−1.07170	−0.535850	−	0.844314i	$-0.680009\pi$
$317$	−19.0811	−1.07170	−0.535850	+	0.844314i	$0.680009\pi$
$318$	−7.45329	−0.417960
$319$	−18.6015	−1.04148
$320$	1.00000	0.0559017
$321$	9.79964	0.546963
$322$	−15.2007	−0.847105
$323$	−34.9152	−1.94273
$324$	−0.313490	−0.0174161
$325$	−1.00000	−0.0554700
$326$	−8.47146	−0.469191
$327$	−18.6110	−1.02919
$328$	2.59883	0.143496
$329$	3.58287	0.197530
$330$	6.58992	0.362763
$331$	18.7359	1.02982	0.514910	−	0.857244i	$-0.327825\pi$
$331$	18.7359	1.02982	0.514910	+	0.857244i	$0.327825\pi$
$332$	−9.28658	−0.509668
$333$	10.3952	0.569653
$334$	17.6391	0.965168
$335$	−11.2345	−0.613806
$336$	7.43721	0.405733
$337$	−35.1199	−1.91310	−0.956551	−	0.291567i	$-0.905824\pi$
$337$	−35.1199	−1.91310	−0.956551	+	0.291567i	$0.905824\pi$
$338$	1.00000	0.0543928
$339$	−9.63825	−0.523478
$340$	4.88536	0.264946
$341$	2.35851	0.127720
$342$	−34.3554	−1.85773
$343$	−18.4062	−0.993844
$344$	3.49049	0.188195
$345$	−15.9566	−0.859076
$346$	9.72731	0.522943
$347$	0.460709	0.0247321	0.0123661	−	0.999924i	$-0.496064\pi$
$347$	0.460709	0.0247321	0.0123661	+	0.999924i	$0.496064\pi$
$348$	−22.0370	−1.18131
$349$	−12.2597	−0.656246	−0.328123	−	0.944635i	$-0.606416\pi$
$349$	−12.2597	−0.656246	−0.328123	+	0.944635i	$0.606416\pi$
$350$	2.66175	0.142276
$351$	−5.04906	−0.269499
$352$	2.35851	0.125709
$353$	15.5481	0.827541	0.413770	−	0.910381i	$-0.364212\pi$
$353$	15.5481	0.827541	0.413770	+	0.910381i	$0.364212\pi$
$354$	16.1734	0.859607
$355$	1.34315	0.0712870
$356$	−5.37378	−0.284810
$357$	36.3334	1.92297
$358$	−5.02018	−0.265325
$359$	18.2708	0.964295	0.482148	−	0.876090i	$-0.339857\pi$
$359$	18.2708	0.964295	0.482148	+	0.876090i	$0.339857\pi$
$360$	4.80704	0.253353
$361$	32.0782	1.68833
$362$	15.6455	0.822308
$363$	−15.1928	−0.797416
$364$	−2.66175	−0.139514
$365$	−12.1761	−0.637327
$366$	24.1927	1.26457
$367$	−25.7391	−1.34357	−0.671786	−	0.740745i	$-0.734473\pi$
$367$	−25.7391	−1.34357	−0.671786	+	0.740745i	$0.734473\pi$
$368$	−5.71081	−0.297697
$369$	12.4927	0.650342
$370$	2.16249	0.112423
$371$	−7.10023	−0.368625
$372$	2.79411	0.144868
$373$	−19.6430	−1.01708	−0.508539	−	0.861039i	$-0.669814\pi$
$373$	−19.6430	−1.01708	−0.508539	+	0.861039i	$0.669814\pi$
$374$	11.5221	0.595796
$375$	2.79411	0.144287
$376$	1.34606	0.0694177
$377$	7.88697	0.406199
$378$	13.4393	0.691245
$379$	22.7378	1.16796	0.583981	−	0.811768i	$-0.301494\pi$
$379$	22.7378	1.16796	0.583981	+	0.811768i	$0.301494\pi$
$380$	−7.14690	−0.366628
$381$	17.3284	0.887761
$382$	−7.23146	−0.369994
$383$	−27.1222	−1.38588	−0.692940	−	0.720996i	$-0.743685\pi$
$383$	−27.1222	−1.38588	−0.692940	+	0.720996i	$0.743685\pi$
$384$	2.79411	0.142586
$385$	6.27775	0.319944
$386$	−17.8081	−0.906408
$387$	16.7789	0.852921
$388$	3.41086	0.173160
$389$	0.168917	0.00856444	0.00428222	−	0.999991i	$-0.498637\pi$
$389$	0.168917	0.00856444	0.00428222	+	0.999991i	$0.498637\pi$
$390$	−2.79411	−0.141485
$391$	−27.8993	−1.41093
$392$	0.0849075	0.00428848
$393$	−6.10564	−0.307989
$394$	19.6361	0.989252
$395$	0.0613807	0.00308840
$396$	11.3374	0.569727
$397$	32.1776	1.61495	0.807474	−	0.589903i	$-0.200834\pi$
$397$	32.1776	1.61495	0.807474	+	0.589903i	$0.200834\pi$
$398$	21.1576	1.06053
$399$	−53.1530	−2.66098
$400$	1.00000	0.0500000
$401$	3.96755	0.198130	0.0990651	−	0.995081i	$-0.468415\pi$
$401$	3.96755	0.198130	0.0990651	+	0.995081i	$0.468415\pi$
$402$	−31.3904	−1.56561
$403$	−1.00000	−0.0498135
$404$	−19.0241	−0.946485
$405$	−0.313490	−0.0155774
$406$	−20.9931	−1.04187
$407$	5.10025	0.252810
$408$	13.6502	0.675786
$409$	−28.9700	−1.43248	−0.716238	−	0.697856i	$-0.754138\pi$
$409$	−28.9700	−1.43248	−0.716238	+	0.697856i	$0.754138\pi$
$410$	2.59883	0.128347
$411$	−29.9351	−1.47659
$412$	11.9266	0.587581
$413$	15.4073	0.758142
$414$	−27.4521	−1.34920
$415$	−9.28658	−0.455861
$416$	−1.00000	−0.0490290
$417$	−10.3625	−0.507456
$418$	−16.8560	−0.824454
$419$	31.0560	1.51719	0.758593	−	0.651565i	$-0.225887\pi$
$419$	31.0560	1.51719	0.758593	+	0.651565i	$0.225887\pi$
$420$	7.43721	0.362899
$421$	−3.04267	−0.148291	−0.0741454	−	0.997247i	$-0.523623\pi$
$421$	−3.04267	−0.148291	−0.0741454	+	0.997247i	$0.523623\pi$
$422$	−4.03472	−0.196407
$423$	6.47056	0.314609
$424$	−2.66750	−0.129545
$425$	4.88536	0.236975
$426$	3.75290	0.181829
$427$	23.0467	1.11531
$428$	3.50725	0.169529
$429$	−6.58992	−0.318164
$430$	3.49049	0.168326
$431$	25.9373	1.24936	0.624678	−	0.780882i	$-0.285230\pi$
$431$	25.9373	1.24936	0.624678	+	0.780882i	$0.285230\pi$
$432$	5.04906	0.242923
$433$	−27.2208	−1.30815	−0.654075	−	0.756430i	$-0.726942\pi$
$433$	−27.2208	−1.30815	−0.654075	+	0.756430i	$0.726942\pi$
$434$	2.66175	0.127768
$435$	−22.0370	−1.05660
$436$	−6.66079	−0.318994
$437$	40.8146	1.95243
$438$	−34.0214	−1.62560
$439$	−27.3067	−1.30328	−0.651640	−	0.758529i	$-0.725919\pi$
$439$	−27.3067	−1.30328	−0.651640	+	0.758529i	$0.725919\pi$
$440$	2.35851	0.112437
$441$	0.408154	0.0194359
$442$	−4.88536	−0.232373
$443$	−12.5763	−0.597519	−0.298760	−	0.954328i	$-0.596573\pi$
$443$	−12.5763	−0.597519	−0.298760	+	0.954328i	$0.596573\pi$
$444$	6.04224	0.286752
$445$	−5.37378	−0.254742
$446$	3.69526	0.174976
$447$	60.8405	2.87766
$448$	2.66175	0.125756
$449$	−32.8601	−1.55076	−0.775382	−	0.631492i	$-0.782443\pi$
$449$	−32.8601	−1.55076	−0.775382	+	0.631492i	$0.782443\pi$
$450$	4.80704	0.226606
$451$	6.12935	0.288620
$452$	−3.44949	−0.162250
$453$	13.4529	0.632070
$454$	−0.292760	−0.0137399
$455$	−2.66175	−0.124785
$456$	−19.9692	−0.935144
$457$	23.4358	1.09628	0.548141	−	0.836386i	$-0.315336\pi$
$457$	23.4358	1.09628	0.548141	+	0.836386i	$0.315336\pi$
$458$	16.2872	0.761052
$459$	24.6665	1.15133
$460$	−5.71081	−0.266268
$461$	−20.0953	−0.935934	−0.467967	−	0.883746i	$-0.655013\pi$
$461$	−20.0953	−0.935934	−0.467967	+	0.883746i	$0.655013\pi$
$462$	17.5407	0.816068
$463$	−17.2142	−0.800013	−0.400007	−	0.916512i	$-0.630992\pi$
$463$	−17.2142	−0.800013	−0.400007	+	0.916512i	$0.630992\pi$
$464$	−7.88697	−0.366143
$465$	2.79411	0.129574
$466$	20.6623	0.957162
$467$	−32.6392	−1.51036	−0.755180	−	0.655517i	$-0.772451\pi$
$467$	−32.6392	−1.51036	−0.755180	+	0.655517i	$0.772451\pi$
$468$	−4.80704	−0.222205
$469$	−29.9034	−1.38081
$470$	1.34606	0.0620891
$471$	−0.778432	−0.0358682
$472$	5.78840	0.266433
$473$	8.23235	0.378524
$474$	0.171504	0.00787745
$475$	−7.14690	−0.327922
$476$	13.0036	0.596019
$477$	−12.8228	−0.587116
$478$	3.65003	0.166949
$479$	27.2146	1.24347	0.621734	−	0.783228i	$-0.286428\pi$
$479$	27.2146	1.24347	0.621734	+	0.783228i	$0.286428\pi$
$480$	2.79411	0.127533
$481$	−2.16249	−0.0986011
$482$	23.2818	1.06046
$483$	−42.4725	−1.93257
$484$	−5.43745	−0.247157
$485$	3.41086	0.154879
$486$	−16.0231	−0.726823
$487$	−14.2623	−0.646287	−0.323144	−	0.946350i	$-0.604740\pi$
$487$	−14.2623	−0.646287	−0.323144	+	0.946350i	$0.604740\pi$
$488$	8.65847	0.391951
$489$	−23.6702	−1.07040
$490$	0.0849075	0.00383573
$491$	18.6806	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$491$	18.6806	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$492$	7.26140	0.327369
$493$	−38.5306	−1.73533
$494$	7.14690	0.321554
$495$	11.3374	0.509580
$496$	1.00000	0.0449013
$497$	3.57513	0.160366
$498$	−25.9477	−1.16275
$499$	7.21159	0.322835	0.161417	−	0.986886i	$-0.448393\pi$
$499$	7.21159	0.322835	0.161417	+	0.986886i	$0.448393\pi$
$500$	1.00000	0.0447214
$501$	49.2855	2.20191
$502$	7.71137	0.344175
$503$	−14.8185	−0.660725	−0.330362	−	0.943854i	$-0.607171\pi$
$503$	−14.8185	−0.660725	−0.330362	+	0.943854i	$0.607171\pi$
$504$	12.7951	0.569940
$505$	−19.0241	−0.846562
$506$	−13.4690	−0.598769
$507$	2.79411	0.124091
$508$	6.20177	0.275159
$509$	−16.7978	−0.744551	−0.372276	−	0.928122i	$-0.621422\pi$
$509$	−16.7978	−0.744551	−0.372276	+	0.928122i	$0.621422\pi$
$510$	13.6502	0.604441
$511$	−32.4097	−1.43372
$512$	1.00000	0.0441942
$513$	−36.0852	−1.59320
$514$	−13.1094	−0.578232
$515$	11.9266	0.525548
$516$	9.75281	0.429344
$517$	3.17469	0.139623
$518$	5.75601	0.252905
$519$	27.1791	1.19303
$520$	−1.00000	−0.0438529
$521$	−29.9060	−1.31021	−0.655104	−	0.755539i	$-0.727375\pi$
$521$	−29.9060	−1.31021	−0.655104	+	0.755539i	$0.727375\pi$
$522$	−37.9130	−1.65941
$523$	−33.6782	−1.47265	−0.736323	−	0.676631i	$-0.763439\pi$
$523$	−33.6782	−1.47265	−0.736323	+	0.676631i	$0.763439\pi$
$524$	−2.18518	−0.0954601
$525$	7.43721	0.324587
$526$	−31.2369	−1.36199
$527$	4.88536	0.212809
$528$	6.58992	0.286789
$529$	9.61338	0.417973
$530$	−2.66750	−0.115869
$531$	27.8251	1.20750
$532$	−19.0233	−0.824763
$533$	−2.59883	−0.112568
$534$	−15.0149	−0.649760
$535$	3.50725	0.151632
$536$	−11.2345	−0.485256
$537$	−14.0269	−0.605306
$538$	2.92778	0.126226
$539$	0.200255	0.00862559
$540$	5.04906	0.217277
$541$	−34.4820	−1.48250	−0.741249	−	0.671230i	$-0.765766\pi$
$541$	−34.4820	−1.48250	−0.741249	+	0.671230i	$0.765766\pi$
$542$	−10.4456	−0.448677
$543$	43.7151	1.87600
$544$	4.88536	0.209458
$545$	−6.66079	−0.285317
$546$	−7.43721	−0.318283
$547$	−18.3566	−0.784873	−0.392436	−	0.919779i	$-0.628368\pi$
$547$	−18.3566	−0.784873	−0.392436	+	0.919779i	$0.628368\pi$
$548$	−10.7136	−0.457664
$549$	41.6216	1.77637
$550$	2.35851	0.100567
$551$	56.3674	2.40133
$552$	−15.9566	−0.679159
$553$	0.163380	0.00694762
$554$	17.3403	0.736721
$555$	6.04224	0.256479
$556$	−3.70871	−0.157284
$557$	19.1318	0.810641	0.405321	−	0.914175i	$-0.367160\pi$
$557$	19.1318	0.810641	0.405321	+	0.914175i	$0.367160\pi$
$558$	4.80704	0.203498
$559$	−3.49049	−0.147632
$560$	2.66175	0.112479
$561$	32.1941	1.35924
$562$	14.6944	0.619845
$563$	25.1886	1.06157	0.530787	−	0.847505i	$-0.321897\pi$
$563$	25.1886	1.06157	0.530787	+	0.847505i	$0.321897\pi$
$564$	3.76103	0.158368
$565$	−3.44949	−0.145121
$566$	23.3052	0.979591
$567$	−0.834432	−0.0350429
$568$	1.34315	0.0563573
$569$	−1.02419	−0.0429364	−0.0214682	−	0.999770i	$-0.506834\pi$
$569$	−1.02419	−0.0429364	−0.0214682	+	0.999770i	$0.506834\pi$
$570$	−19.9692	−0.836418
$571$	−8.93041	−0.373726	−0.186863	−	0.982386i	$-0.559832\pi$
$571$	−8.93041	−0.373726	−0.186863	+	0.982386i	$0.559832\pi$
$572$	−2.35851	−0.0986141
$573$	−20.2055	−0.844096
$574$	6.91742	0.288728
$575$	−5.71081	−0.238157
$576$	4.80704	0.200293
$577$	−34.2689	−1.42663	−0.713316	−	0.700842i	$-0.752808\pi$
$577$	−34.2689	−1.42663	−0.713316	+	0.700842i	$0.752808\pi$
$578$	6.86670	0.285617
$579$	−49.7578	−2.06786
$580$	−7.88697	−0.327488
$581$	−24.7186	−1.02550
$582$	9.53030	0.395044
$583$	−6.29132	−0.260560
$584$	−12.1761	−0.503851
$585$	−4.80704	−0.198747
$586$	17.1061	0.706647
$587$	18.9280	0.781244	0.390622	−	0.920551i	$-0.372260\pi$
$587$	18.9280	0.781244	0.390622	+	0.920551i	$0.372260\pi$
$588$	0.237241	0.00978365
$589$	−7.14690	−0.294483
$590$	5.78840	0.238305
$591$	54.8654	2.25686
$592$	2.16249	0.0888779
$593$	−21.2614	−0.873102	−0.436551	−	0.899679i	$-0.643800\pi$
$593$	−21.2614	−0.873102	−0.436551	+	0.899679i	$0.643800\pi$
$594$	11.9082	0.488601
$595$	13.0036	0.533095
$596$	21.7746	0.891921
$597$	59.1166	2.41948
$598$	5.71081	0.233532
$599$	37.6873	1.53986	0.769930	−	0.638128i	$-0.220291\pi$
$599$	37.6873	1.53986	0.769930	+	0.638128i	$0.220291\pi$
$600$	2.79411	0.114069
$601$	−18.7533	−0.764964	−0.382482	−	0.923963i	$-0.624931\pi$
$601$	−18.7533	−0.764964	−0.382482	+	0.923963i	$0.624931\pi$
$602$	9.29081	0.378665
$603$	−54.0046	−2.19924
$604$	4.81472	0.195908
$605$	−5.43745	−0.221064
$606$	−53.1554	−2.15929
$607$	−21.0855	−0.855833	−0.427916	−	0.903818i	$-0.640752\pi$
$607$	−21.0855	−0.855833	−0.427916	+	0.903818i	$0.640752\pi$
$608$	−7.14690	−0.289845
$609$	−58.6571	−2.37690
$610$	8.65847	0.350571
$611$	−1.34606	−0.0544557
$612$	23.4841	0.949288
$613$	42.3406	1.71012	0.855060	−	0.518529i	$-0.173520\pi$
$613$	42.3406	1.71012	0.855060	+	0.518529i	$0.173520\pi$
$614$	20.9194	0.844240
$615$	7.26140	0.292808
$616$	6.27775	0.252938
$617$	5.22254	0.210251	0.105126	−	0.994459i	$-0.466475\pi$
$617$	5.22254	0.210251	0.105126	+	0.994459i	$0.466475\pi$
$618$	33.3242	1.34049
$619$	47.8371	1.92273	0.961367	−	0.275270i	$-0.0887672\pi$
$619$	47.8371	1.92273	0.961367	+	0.275270i	$0.0887672\pi$
$620$	1.00000	0.0401610
$621$	−28.8343	−1.15708
$622$	−14.5430	−0.583121
$623$	−14.3037	−0.573064
$624$	−2.79411	−0.111854
$625$	1.00000	0.0400000
$626$	12.8599	0.513986
$627$	−47.0975	−1.88089
$628$	−0.278598	−0.0111173
$629$	10.5645	0.421236
$630$	12.7951	0.509770
$631$	−33.1936	−1.32142	−0.660708	−	0.750643i	$-0.729744\pi$
$631$	−33.1936	−1.32142	−0.660708	+	0.750643i	$0.729744\pi$
$632$	0.0613807	0.00244159
$633$	−11.2735	−0.448079
$634$	−19.0811	−0.757806
$635$	6.20177	0.246110
$636$	−7.45329	−0.295542
$637$	−0.0849075	−0.00336416
$638$	−18.6015	−0.736439
$639$	6.45657	0.255418
$640$	1.00000	0.0395285
$641$	30.5188	1.20542	0.602710	−	0.797960i	$-0.294087\pi$
$641$	30.5188	1.20542	0.602710	+	0.797960i	$0.294087\pi$
$642$	9.79964	0.386761
$643$	−0.735376	−0.0290004	−0.0145002	−	0.999895i	$-0.504616\pi$
$643$	−0.735376	−0.0290004	−0.0145002	+	0.999895i	$0.504616\pi$
$644$	−15.2007	−0.598993
$645$	9.75281	0.384017
$646$	−34.9152	−1.37372
$647$	32.4752	1.27673	0.638366	−	0.769733i	$-0.279610\pi$
$647$	32.4752	1.27673	0.638366	+	0.769733i	$0.279610\pi$
$648$	−0.313490	−0.0123151
$649$	13.6520	0.535887
$650$	−1.00000	−0.0392232
$651$	7.43721	0.291487
$652$	−8.47146	−0.331768
$653$	10.8171	0.423307	0.211653	−	0.977345i	$-0.432115\pi$
$653$	10.8171	0.423307	0.211653	+	0.977345i	$0.432115\pi$
$654$	−18.6110	−0.727747
$655$	−2.18518	−0.0853821
$656$	2.59883	0.101467
$657$	−58.5310	−2.28351
$658$	3.58287	0.139675
$659$	−3.36659	−0.131144	−0.0655719	−	0.997848i	$-0.520887\pi$
$659$	−3.36659	−0.131144	−0.0655719	+	0.997848i	$0.520887\pi$
$660$	6.58992	0.256512
$661$	−22.5084	−0.875475	−0.437737	−	0.899103i	$-0.644220\pi$
$661$	−22.5084	−0.875475	−0.437737	+	0.899103i	$0.644220\pi$
$662$	18.7359	0.728193
$663$	−13.6502	−0.530130
$664$	−9.28658	−0.360389
$665$	−19.0233	−0.737690
$666$	10.3952	0.402805
$667$	45.0410	1.74399
$668$	17.6391	0.682477
$669$	10.3250	0.399186
$670$	−11.2345	−0.434026
$671$	20.4211	0.788346
$672$	7.43721	0.286897
$673$	31.7382	1.22342	0.611710	−	0.791082i	$-0.290482\pi$
$673$	31.7382	1.22342	0.611710	+	0.791082i	$0.290482\pi$
$674$	−35.1199	−1.35277
$675$	5.04906	0.194339
$676$	1.00000	0.0384615
$677$	2.03883	0.0783585	0.0391792	−	0.999232i	$-0.487526\pi$
$677$	2.03883	0.0783585	0.0391792	+	0.999232i	$0.487526\pi$
$678$	−9.63825	−0.370155
$679$	9.07884	0.348414
$680$	4.88536	0.187345
$681$	−0.818003	−0.0313459
$682$	2.35851	0.0903119
$683$	−6.00740	−0.229867	−0.114933	−	0.993373i	$-0.536665\pi$
$683$	−6.00740	−0.229867	−0.114933	+	0.993373i	$0.536665\pi$
$684$	−34.3554	−1.31361
$685$	−10.7136	−0.409347
$686$	−18.4062	−0.702754
$687$	45.5083	1.73625
$688$	3.49049	0.133074
$689$	2.66750	0.101624
$690$	−15.9566	−0.607458
$691$	13.8615	0.527316	0.263658	−	0.964616i	$-0.415071\pi$
$691$	13.8615	0.527316	0.263658	+	0.964616i	$0.415071\pi$
$692$	9.72731	0.369777
$693$	30.1774	1.14634
$694$	0.460709	0.0174883
$695$	−3.70871	−0.140679
$696$	−22.0370	−0.835312
$697$	12.6962	0.480902
$698$	−12.2597	−0.464036
$699$	57.7327	2.18365
$700$	2.66175	0.100605
$701$	40.7930	1.54073	0.770365	−	0.637603i	$-0.220074\pi$
$701$	40.7930	1.54073	0.770365	+	0.637603i	$0.220074\pi$
$702$	−5.04906	−0.190565
$703$	−15.4551	−0.582901
$704$	2.35851	0.0888895
$705$	3.76103	0.141649
$706$	15.5481	0.585160
$707$	−50.6374	−1.90442
$708$	16.1734	0.607834
$709$	8.05080	0.302354	0.151177	−	0.988507i	$-0.451694\pi$
$709$	8.05080	0.302354	0.151177	+	0.988507i	$0.451694\pi$
$710$	1.34315	0.0504075
$711$	0.295059	0.0110656
$712$	−5.37378	−0.201391
$713$	−5.71081	−0.213872
$714$	36.3334	1.35974
$715$	−2.35851	−0.0882031
$716$	−5.02018	−0.187613
$717$	10.1986	0.380873
$718$	18.2708	0.681860
$719$	−35.0814	−1.30832	−0.654158	−	0.756358i	$-0.726977\pi$
$719$	−35.0814	−1.30832	−0.654158	+	0.756358i	$0.726977\pi$
$720$	4.80704	0.179148
$721$	31.7456	1.18227
$722$	32.0782	1.19383
$723$	65.0519	2.41931
$724$	15.6455	0.581459
$725$	−7.88697	−0.292915
$726$	−15.1928	−0.563859
$727$	−44.7947	−1.66134	−0.830672	−	0.556762i	$-0.812044\pi$
$727$	−44.7947	−1.66134	−0.830672	+	0.556762i	$0.812044\pi$
$728$	−2.66175	−0.0986510
$729$	−43.8298	−1.62333
$730$	−12.1761	−0.450658
$731$	17.0523	0.630702
$732$	24.1927	0.894188
$733$	−28.1660	−1.04033	−0.520167	−	0.854065i	$-0.674130\pi$
$733$	−28.1660	−1.04033	−0.520167	+	0.854065i	$0.674130\pi$
$734$	−25.7391	−0.950049
$735$	0.237241	0.00875076
$736$	−5.71081	−0.210503
$737$	−26.4966	−0.976015
$738$	12.4927	0.459861
$739$	−6.62827	−0.243825	−0.121912	−	0.992541i	$-0.538903\pi$
$739$	−6.62827	−0.243825	−0.121912	+	0.992541i	$0.538903\pi$
$740$	2.16249	0.0794948
$741$	19.9692	0.733588
$742$	−7.10023	−0.260658
$743$	24.8399	0.911287	0.455643	−	0.890162i	$-0.349409\pi$
$743$	24.8399	0.911287	0.455643	+	0.890162i	$0.349409\pi$
$744$	2.79411	0.102437
$745$	21.7746	0.797758
$746$	−19.6430	−0.719182
$747$	−44.6410	−1.63333
$748$	11.5221	0.421291
$749$	9.33543	0.341109
$750$	2.79411	0.102026
$751$	−10.7351	−0.391728	−0.195864	−	0.980631i	$-0.562751\pi$
$751$	−10.7351	−0.391728	−0.195864	+	0.980631i	$0.562751\pi$
$752$	1.34606	0.0490857
$753$	21.5464	0.785194
$754$	7.88697	0.287226
$755$	4.81472	0.175226
$756$	13.4393	0.488784
$757$	−8.83396	−0.321076	−0.160538	−	0.987030i	$-0.551323\pi$
$757$	−8.83396	−0.321076	−0.160538	+	0.987030i	$0.551323\pi$
$758$	22.7378	0.825873
$759$	−37.6338	−1.36602
$760$	−7.14690	−0.259245
$761$	22.6737	0.821922	0.410961	−	0.911653i	$-0.365193\pi$
$761$	22.6737	0.821922	0.410961	+	0.911653i	$0.365193\pi$
$762$	17.3284	0.627742
$763$	−17.7294	−0.641846
$764$	−7.23146	−0.261625
$765$	23.4841	0.849069
$766$	−27.1222	−0.979965
$767$	−5.78840	−0.209007
$768$	2.79411	0.100824
$769$	−54.9675	−1.98218	−0.991090	−	0.133197i	$-0.957476\pi$
$769$	−54.9675	−1.98218	−0.991090	+	0.133197i	$0.957476\pi$
$770$	6.27775	0.226234
$771$	−36.6291	−1.31917
$772$	−17.8081	−0.640928
$773$	22.7623	0.818704	0.409352	−	0.912377i	$-0.365755\pi$
$773$	22.7623	0.818704	0.409352	+	0.912377i	$0.365755\pi$
$774$	16.7789	0.603106
$775$	1.00000	0.0359211
$776$	3.41086	0.122443
$777$	16.0829	0.576971
$778$	0.168917	0.00605597
$779$	−18.5736	−0.665467
$780$	−2.79411	−0.100045
$781$	3.16783	0.113354
$782$	−27.8993	−0.997679
$783$	−39.8218	−1.42311
$784$	0.0849075	0.00303241
$785$	−0.278598	−0.00994357
$786$	−6.10564	−0.217781
$787$	3.97220	0.141594	0.0707968	−	0.997491i	$-0.477446\pi$
$787$	3.97220	0.141594	0.0707968	+	0.997491i	$0.477446\pi$
$788$	19.6361	0.699507
$789$	−87.2792	−3.10722
$790$	0.0613807	0.00218383
$791$	−9.18168	−0.326463
$792$	11.3374	0.402858
$793$	−8.65847	−0.307471
$794$	32.1776	1.14194
$795$	−7.45329	−0.264341
$796$	21.1576	0.749911
$797$	1.00204	0.0354942	0.0177471	−	0.999843i	$-0.494351\pi$
$797$	1.00204	0.0354942	0.0177471	+	0.999843i	$0.494351\pi$
$798$	−53.1530	−1.88160
$799$	6.57597	0.232641
$800$	1.00000	0.0353553
$801$	−25.8320	−0.912729
$802$	3.96755	0.140099
$803$	−28.7174	−1.01342
$804$	−31.3904	−1.10705
$805$	−15.2007	−0.535756
$806$	−1.00000	−0.0352235
$807$	8.18054	0.287969
$808$	−19.0241	−0.669266
$809$	−12.0699	−0.424355	−0.212178	−	0.977231i	$-0.568056\pi$
$809$	−12.0699	−0.424355	−0.212178	+	0.977231i	$0.568056\pi$
$810$	−0.313490	−0.0110149
$811$	40.8742	1.43529	0.717644	−	0.696410i	$-0.245220\pi$
$811$	40.8742	1.43529	0.717644	+	0.696410i	$0.245220\pi$
$812$	−20.9931	−0.736714
$813$	−29.1861	−1.02360
$814$	5.10025	0.178764
$815$	−8.47146	−0.296742
$816$	13.6502	0.477853
$817$	−24.9462	−0.872757
$818$	−28.9700	−1.01291
$819$	−12.7951	−0.447098
$820$	2.59883	0.0907549
$821$	34.6974	1.21095	0.605474	−	0.795865i	$-0.292984\pi$
$821$	34.6974	1.21095	0.605474	+	0.795865i	$0.292984\pi$
$822$	−29.9351	−1.04411
$823$	41.3736	1.44219	0.721096	−	0.692835i	$-0.243639\pi$
$823$	41.3736	1.44219	0.721096	+	0.692835i	$0.243639\pi$
$824$	11.9266	0.415482
$825$	6.58992	0.229432
$826$	15.4073	0.536087
$827$	29.8460	1.03785	0.518923	−	0.854821i	$-0.326333\pi$
$827$	29.8460	1.03785	0.518923	+	0.854821i	$0.326333\pi$
$828$	−27.4521	−0.954026
$829$	38.7116	1.34451	0.672255	−	0.740319i	$-0.265326\pi$
$829$	38.7116	1.34451	0.672255	+	0.740319i	$0.265326\pi$
$830$	−9.28658	−0.322342
$831$	48.4508	1.68074
$832$	−1.00000	−0.0346688
$833$	0.414804	0.0143721
$834$	−10.3625	−0.358826
$835$	17.6391	0.610426
$836$	−16.8560	−0.582977
$837$	5.04906	0.174521
$838$	31.0560	1.07281
$839$	−28.5740	−0.986484	−0.493242	−	0.869892i	$-0.664188\pi$
$839$	−28.5740	−0.986484	−0.493242	+	0.869892i	$0.664188\pi$
$840$	7.43721	0.256608
$841$	33.2043	1.14497
$842$	−3.04267	−0.104857
$843$	41.0577	1.41410
$844$	−4.03472	−0.138881
$845$	1.00000	0.0344010
$846$	6.47056	0.222462
$847$	−14.4731	−0.497303
$848$	−2.66750	−0.0916025
$849$	65.1172	2.23482
$850$	4.88536	0.167566
$851$	−12.3496	−0.423338
$852$	3.75290	0.128572
$853$	25.4030	0.869782	0.434891	−	0.900483i	$-0.356787\pi$
$853$	25.4030	0.869782	0.434891	+	0.900483i	$0.356787\pi$
$854$	23.0467	0.788641
$855$	−34.3554	−1.17493
$856$	3.50725	0.119875
$857$	29.7801	1.01727	0.508635	−	0.860982i	$-0.330150\pi$
$857$	29.7801	1.01727	0.508635	+	0.860982i	$0.330150\pi$
$858$	−6.58992	−0.224976
$859$	−13.5409	−0.462009	−0.231005	−	0.972953i	$-0.574201\pi$
$859$	−13.5409	−0.462009	−0.231005	+	0.972953i	$0.574201\pi$
$860$	3.49049	0.119025
$861$	19.3280	0.658697
$862$	25.9373	0.883429
$863$	−15.1795	−0.516717	−0.258358	−	0.966049i	$-0.583182\pi$
$863$	−15.1795	−0.516717	−0.258358	+	0.966049i	$0.583182\pi$
$864$	5.04906	0.171773
$865$	9.72731	0.330738
$866$	−27.2208	−0.925002
$867$	19.1863	0.651601
$868$	2.66175	0.0903456
$869$	0.144767	0.00491087
$870$	−22.0370	−0.747126
$871$	11.2345	0.380666
$872$	−6.66079	−0.225563
$873$	16.3961	0.554925
$874$	40.8146	1.38058
$875$	2.66175	0.0899835
$876$	−34.0214	−1.14948
$877$	40.2120	1.35786	0.678931	−	0.734202i	$-0.262443\pi$
$877$	40.2120	1.35786	0.678931	+	0.734202i	$0.262443\pi$
$878$	−27.3067	−0.921558
$879$	47.7963	1.61213
$880$	2.35851	0.0795052
$881$	23.2395	0.782959	0.391479	−	0.920187i	$-0.371963\pi$
$881$	23.2395	0.782959	0.391479	+	0.920187i	$0.371963\pi$
$882$	0.408154	0.0137433
$883$	11.6851	0.393235	0.196618	−	0.980480i	$-0.437004\pi$
$883$	11.6851	0.393235	0.196618	+	0.980480i	$0.437004\pi$
$884$	−4.88536	−0.164312
$885$	16.1734	0.543663
$886$	−12.5763	−0.422510
$887$	−3.60871	−0.121169	−0.0605843	−	0.998163i	$-0.519296\pi$
$887$	−3.60871	−0.121169	−0.0605843	+	0.998163i	$0.519296\pi$
$888$	6.04224	0.202764
$889$	16.5076	0.553646
$890$	−5.37378	−0.180130
$891$	−0.739368	−0.0247698
$892$	3.69526	0.123727
$893$	−9.62015	−0.321926
$894$	60.8405	2.03481
$895$	−5.02018	−0.167806
$896$	2.66175	0.0889228
$897$	15.9566	0.532776
$898$	−32.8601	−1.09656
$899$	−7.88697	−0.263045
$900$	4.80704	0.160235
$901$	−13.0317	−0.434149
$902$	6.12935	0.204085
$903$	25.9595	0.863879
$904$	−3.44949	−0.114728
$905$	15.6455	0.520073
$906$	13.4529	0.446941
$907$	−48.6644	−1.61587	−0.807937	−	0.589268i	$-0.799416\pi$
$907$	−48.6644	−1.61587	−0.807937	+	0.589268i	$0.799416\pi$
$908$	−0.292760	−0.00971558
$909$	−91.4496	−3.03319
$910$	−2.66175	−0.0882361
$911$	−6.71872	−0.222601	−0.111301	−	0.993787i	$-0.535502\pi$
$911$	−6.71872	−0.222601	−0.111301	+	0.993787i	$0.535502\pi$
$912$	−19.9692	−0.661247
$913$	−21.9025	−0.724866
$914$	23.4358	0.775188
$915$	24.1927	0.799786
$916$	16.2872	0.538145
$917$	−5.81641	−0.192075
$918$	24.6665	0.814115
$919$	−23.2048	−0.765455	−0.382727	−	0.923861i	$-0.625015\pi$
$919$	−23.2048	−0.765455	−0.382727	+	0.923861i	$0.625015\pi$
$920$	−5.71081	−0.188280
$921$	58.4512	1.92603
$922$	−20.0953	−0.661805
$923$	−1.34315	−0.0442103
$924$	17.5407	0.577047
$925$	2.16249	0.0711023
$926$	−17.2142	−0.565695
$927$	57.3316	1.88302
$928$	−7.88697	−0.258902
$929$	9.98429	0.327574	0.163787	−	0.986496i	$-0.447629\pi$
$929$	9.98429	0.327574	0.163787	+	0.986496i	$0.447629\pi$
$930$	2.79411	0.0916224
$931$	−0.606826	−0.0198879
$932$	20.6623	0.676816
$933$	−40.6347	−1.33032
$934$	−32.6392	−1.06799
$935$	11.5221	0.376814
$936$	−4.80704	−0.157123
$937$	−11.8698	−0.387768	−0.193884	−	0.981024i	$-0.562109\pi$
$937$	−11.8698	−0.387768	−0.193884	+	0.981024i	$0.562109\pi$
$938$	−29.9034	−0.976380
$939$	35.9320	1.17260
$940$	1.34606	0.0439036
$941$	−22.6949	−0.739833	−0.369916	−	0.929065i	$-0.620614\pi$
$941$	−22.6949	−0.739833	−0.369916	+	0.929065i	$0.620614\pi$
$942$	−0.778432	−0.0253627
$943$	−14.8414	−0.483303
$944$	5.78840	0.188396
$945$	13.4393	0.437182
$946$	8.23235	0.267657
$947$	−17.2278	−0.559827	−0.279914	−	0.960025i	$-0.590306\pi$
$947$	−17.2278	−0.559827	−0.279914	+	0.960025i	$0.590306\pi$
$948$	0.171504	0.00557020
$949$	12.1761	0.395253
$950$	−7.14690	−0.231876
$951$	−53.3145	−1.72884
$952$	13.0036	0.421449
$953$	−48.5135	−1.57151	−0.785753	−	0.618541i	$-0.787724\pi$
$953$	−48.5135	−1.57151	−0.785753	+	0.618541i	$0.787724\pi$
$954$	−12.8228	−0.415153
$955$	−7.23146	−0.234005
$956$	3.65003	0.118051
$957$	−51.9745	−1.68010
$958$	27.2146	0.879265
$959$	−28.5170	−0.920863
$960$	2.79411	0.0901794
$961$	1.00000	0.0322581
$962$	−2.16249	−0.0697215
$963$	16.8595	0.543290
$964$	23.2818	0.749856
$965$	−17.8081	−0.573263
$966$	−42.4725	−1.36653
$967$	−30.8270	−0.991328	−0.495664	−	0.868514i	$-0.665075\pi$
$967$	−30.8270	−0.991328	−0.495664	+	0.868514i	$0.665075\pi$
$968$	−5.43745	−0.174766
$969$	−97.5567	−3.13397
$970$	3.41086	0.109516
$971$	9.88965	0.317374	0.158687	−	0.987329i	$-0.449274\pi$
$971$	9.88965	0.317374	0.158687	+	0.987329i	$0.449274\pi$
$972$	−16.0231	−0.513942
$973$	−9.87166	−0.316471
$974$	−14.2623	−0.456994
$975$	−2.79411	−0.0894831
$976$	8.65847	0.277151
$977$	−0.286127	−0.00915401	−0.00457700	−	0.999990i	$-0.501457\pi$
$977$	−0.286127	−0.00915401	−0.00457700	+	0.999990i	$0.501457\pi$
$978$	−23.6702	−0.756888
$979$	−12.6741	−0.405066
$980$	0.0849075	0.00271227
$981$	−32.0187	−1.02228
$982$	18.6806	0.596122
$983$	29.4012	0.937753	0.468877	−	0.883264i	$-0.344659\pi$
$983$	29.4012	0.937753	0.468877	+	0.883264i	$0.344659\pi$
$984$	7.26140	0.231485
$985$	19.6361	0.625658
$986$	−38.5306	−1.22707
$987$	10.0109	0.318651
$988$	7.14690	0.227373
$989$	−19.9335	−0.633850
$990$	11.3374	0.360327
$991$	27.9354	0.887398	0.443699	−	0.896176i	$-0.353666\pi$
$991$	27.9354	0.887398	0.443699	+	0.896176i	$0.353666\pi$
$992$	1.00000	0.0317500
$993$	52.3503	1.66129
$994$	3.57513	0.113396
$995$	21.1576	0.670741
$996$	−25.9477	−0.822185
$997$	17.8941	0.566712	0.283356	−	0.959015i	$-0.408552\pi$
$997$	17.8941	0.566712	0.283356	+	0.959015i	$0.408552\pi$
$998$	7.21159	0.228279
$999$	10.9186	0.345448

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.o.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.o.1.8	✓	8	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:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.79411 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79411 q^{6} +2.66175 q^{7} +1.00000 q^{8} +4.80704 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.79411 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.79411 q^{6} +2.66175 q^{7} +1.00000 q^{8} +4.80704 q^{9} +1.00000 q^{10} +2.35851 q^{11} +2.79411 q^{12} -1.00000 q^{13} +2.66175 q^{14} +2.79411 q^{15} +1.00000 q^{16} +4.88536 q^{17} +4.80704 q^{18} -7.14690 q^{19} +1.00000 q^{20} +7.43721 q^{21} +2.35851 q^{22} -5.71081 q^{23} +2.79411 q^{24} +1.00000 q^{25} -1.00000 q^{26} +5.04906 q^{27} +2.66175 q^{28} -7.88697 q^{29} +2.79411 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.58992 q^{33} +4.88536 q^{34} +2.66175 q^{35} +4.80704 q^{36} +2.16249 q^{37} -7.14690 q^{38} -2.79411 q^{39} +1.00000 q^{40} +2.59883 q^{41} +7.43721 q^{42} +3.49049 q^{43} +2.35851 q^{44} +4.80704 q^{45} -5.71081 q^{46} +1.34606 q^{47} +2.79411 q^{48} +0.0849075 q^{49} +1.00000 q^{50} +13.6502 q^{51} -1.00000 q^{52} -2.66750 q^{53} +5.04906 q^{54} +2.35851 q^{55} +2.66175 q^{56} -19.9692 q^{57} -7.88697 q^{58} +5.78840 q^{59} +2.79411 q^{60} +8.65847 q^{61} +1.00000 q^{62} +12.7951 q^{63} +1.00000 q^{64} -1.00000 q^{65} +6.58992 q^{66} -11.2345 q^{67} +4.88536 q^{68} -15.9566 q^{69} +2.66175 q^{70} +1.34315 q^{71} +4.80704 q^{72} -12.1761 q^{73} +2.16249 q^{74} +2.79411 q^{75} -7.14690 q^{76} +6.27775 q^{77} -2.79411 q^{78} +0.0613807 q^{79} +1.00000 q^{80} -0.313490 q^{81} +2.59883 q^{82} -9.28658 q^{83} +7.43721 q^{84} +4.88536 q^{85} +3.49049 q^{86} -22.0370 q^{87} +2.35851 q^{88} -5.37378 q^{89} +4.80704 q^{90} -2.66175 q^{91} -5.71081 q^{92} +2.79411 q^{93} +1.34606 q^{94} -7.14690 q^{95} +2.79411 q^{96} +3.41086 q^{97} +0.0849075 q^{98} +11.3374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} + 3 q^{3} + 8 q^{4} + 8 q^{5} + 3 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + 8 q^{10} + 10 q^{11} + 3 q^{12} - 8 q^{13} + 7 q^{14} + 3 q^{15} + 8 q^{16} + 11 q^{17} + 9 q^{18} + 2 q^{19} + 8 q^{20} + 5 q^{21} + 10 q^{22} + 12 q^{23} + 3 q^{24} + 8 q^{25} - 8 q^{26} - 3 q^{27} + 7 q^{28} + 9 q^{29} + 3 q^{30} + 8 q^{31} + 8 q^{32} + 6 q^{33} + 11 q^{34} + 7 q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} - 3 q^{39} + 8 q^{40} + 6 q^{41} + 5 q^{42} + 21 q^{43} + 10 q^{44} + 9 q^{45} + 12 q^{46} + q^{47} + 3 q^{48} - 5 q^{49} + 8 q^{50} + 17 q^{51} - 8 q^{52} + 18 q^{53} - 3 q^{54} + 10 q^{55} + 7 q^{56} - 11 q^{57} + 9 q^{58} - 4 q^{59} + 3 q^{60} + 10 q^{61} + 8 q^{62} + 22 q^{63} + 8 q^{64} - 8 q^{65} + 6 q^{66} + 8 q^{67} + 11 q^{68} - 26 q^{69} + 7 q^{70} + 18 q^{71} + 9 q^{72} + 9 q^{73} + 19 q^{74} + 3 q^{75} + 2 q^{76} + 13 q^{77} - 3 q^{78} + 14 q^{79} + 8 q^{80} + 6 q^{82} + 3 q^{83} + 5 q^{84} + 11 q^{85} + 21 q^{86} - 21 q^{87} + 10 q^{88} - 15 q^{89} + 9 q^{90} - 7 q^{91} + 12 q^{92} + 3 q^{93} + q^{94} + 2 q^{95} + 3 q^{96} + 12 q^{97} - 5 q^{98} + 11 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 3 * q^3 + 8 * q^4 + 8 * q^5 + 3 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + 8 * q^10 + 10 * q^11 + 3 * q^12 - 8 * q^13 + 7 * q^14 + 3 * q^15 + 8 * q^16 + 11 * q^17 + 9 * q^18 + 2 * q^19 + 8 * q^20 + 5 * q^21 + 10 * q^22 + 12 * q^23 + 3 * q^24 + 8 * q^25 - 8 * q^26 - 3 * q^27 + 7 * q^28 + 9 * q^29 + 3 * q^30 + 8 * q^31 + 8 * q^32 + 6 * q^33 + 11 * q^34 + 7 * q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 - 3 * q^39 + 8 * q^40 + 6 * q^41 + 5 * q^42 + 21 * q^43 + 10 * q^44 + 9 * q^45 + 12 * q^46 + q^47 + 3 * q^48 - 5 * q^49 + 8 * q^50 + 17 * q^51 - 8 * q^52 + 18 * q^53 - 3 * q^54 + 10 * q^55 + 7 * q^56 - 11 * q^57 + 9 * q^58 - 4 * q^59 + 3 * q^60 + 10 * q^61 + 8 * q^62 + 22 * q^63 + 8 * q^64 - 8 * q^65 + 6 * q^66 + 8 * q^67 + 11 * q^68 - 26 * q^69 + 7 * q^70 + 18 * q^71 + 9 * q^72 + 9 * q^73 + 19 * q^74 + 3 * q^75 + 2 * q^76 + 13 * q^77 - 3 * q^78 + 14 * q^79 + 8 * q^80 + 6 * q^82 + 3 * q^83 + 5 * q^84 + 11 * q^85 + 21 * q^86 - 21 * q^87 + 10 * q^88 - 15 * q^89 + 9 * q^90 - 7 * q^91 + 12 * q^92 + 3 * q^93 + q^94 + 2 * q^95 + 3 * q^96 + 12 * q^97 - 5 * q^98 + 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more