LMFDB - Embedded newform 4030.2.a.j.1.2

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 20x^{4} + 9x^{3} - 37x^{2} - 3x + 20 $ x^7 - 3x^6 - 6x^5 + 20x^4 + 9x^3 - 37x^2 - 3x + 20
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64004$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.33934 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.33934 q^{6} -3.11075 q^{7} +1.00000 q^{8} +2.47253 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.33934 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.33934 q^{6} -3.11075 q^{7} +1.00000 q^{8} +2.47253 q^{9} -1.00000 q^{10} +2.74955 q^{11} -2.33934 q^{12} -1.00000 q^{13} -3.11075 q^{14} +2.33934 q^{15} +1.00000 q^{16} +3.35086 q^{17} +2.47253 q^{18} -3.63296 q^{19} -1.00000 q^{20} +7.27711 q^{21} +2.74955 q^{22} +5.74331 q^{23} -2.33934 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.23394 q^{27} -3.11075 q^{28} -9.75425 q^{29} +2.33934 q^{30} +1.00000 q^{31} +1.00000 q^{32} -6.43214 q^{33} +3.35086 q^{34} +3.11075 q^{35} +2.47253 q^{36} +1.88752 q^{37} -3.63296 q^{38} +2.33934 q^{39} -1.00000 q^{40} +11.5507 q^{41} +7.27711 q^{42} -0.185051 q^{43} +2.74955 q^{44} -2.47253 q^{45} +5.74331 q^{46} +3.99290 q^{47} -2.33934 q^{48} +2.67676 q^{49} +1.00000 q^{50} -7.83882 q^{51} -1.00000 q^{52} +3.72564 q^{53} +1.23394 q^{54} -2.74955 q^{55} -3.11075 q^{56} +8.49873 q^{57} -9.75425 q^{58} -7.90093 q^{59} +2.33934 q^{60} -15.4252 q^{61} +1.00000 q^{62} -7.69141 q^{63} +1.00000 q^{64} +1.00000 q^{65} -6.43214 q^{66} -13.0916 q^{67} +3.35086 q^{68} -13.4356 q^{69} +3.11075 q^{70} -1.92637 q^{71} +2.47253 q^{72} +14.2020 q^{73} +1.88752 q^{74} -2.33934 q^{75} -3.63296 q^{76} -8.55316 q^{77} +2.33934 q^{78} -16.0527 q^{79} -1.00000 q^{80} -10.3042 q^{81} +11.5507 q^{82} +11.9663 q^{83} +7.27711 q^{84} -3.35086 q^{85} -0.185051 q^{86} +22.8185 q^{87} +2.74955 q^{88} +10.5879 q^{89} -2.47253 q^{90} +3.11075 q^{91} +5.74331 q^{92} -2.33934 q^{93} +3.99290 q^{94} +3.63296 q^{95} -2.33934 q^{96} +4.01604 q^{97} +2.67676 q^{98} +6.79834 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9}+O(q^{10}) $ 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^7 + 7 * q^8 + 4 * q^9	\( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9} - 7 q^{10} - 10 q^{11} - 3 q^{12} - 7 q^{13} + 4 q^{14} + 3 q^{15} + 7 q^{16} - 6 q^{17} + 4 q^{18} - 5 q^{19} - 7 q^{20} - 15 q^{21} - 10 q^{22} - 11 q^{23} - 3 q^{24} + 7 q^{25} - 7 q^{26} - 21 q^{27} + 4 q^{28} - 18 q^{29} + 3 q^{30} + 7 q^{31} + 7 q^{32} + 4 q^{33} - 6 q^{34} - 4 q^{35} + 4 q^{36} - 8 q^{37} - 5 q^{38} + 3 q^{39} - 7 q^{40} - 12 q^{41} - 15 q^{42} - 5 q^{43} - 10 q^{44} - 4 q^{45} - 11 q^{46} - 10 q^{47} - 3 q^{48} + 7 q^{49} + 7 q^{50} - 29 q^{51} - 7 q^{52} - 18 q^{53} - 21 q^{54} + 10 q^{55} + 4 q^{56} + 13 q^{57} - 18 q^{58} - 11 q^{59} + 3 q^{60} - 25 q^{61} + 7 q^{62} + 17 q^{63} + 7 q^{64} + 7 q^{65} + 4 q^{66} - 22 q^{67} - 6 q^{68} - 18 q^{69} - 4 q^{70} - 22 q^{71} + 4 q^{72} + 19 q^{73} - 8 q^{74} - 3 q^{75} - 5 q^{76} - 47 q^{77} + 3 q^{78} - 20 q^{79} - 7 q^{80} + 7 q^{81} - 12 q^{82} - 8 q^{83} - 15 q^{84} + 6 q^{85} - 5 q^{86} + 29 q^{87} - 10 q^{88} - 4 q^{89} - 4 q^{90} - 4 q^{91} - 11 q^{92} - 3 q^{93} - 10 q^{94} + 5 q^{95} - 3 q^{96} + 13 q^{97} + 7 q^{98} - 27 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^7 + 7 * q^8 + 4 * q^9 - 7 * q^10 - 10 * q^11 - 3 * q^12 - 7 * q^13 + 4 * q^14 + 3 * q^15 + 7 * q^16 - 6 * q^17 + 4 * q^18 - 5 * q^19 - 7 * q^20 - 15 * q^21 - 10 * q^22 - 11 * q^23 - 3 * q^24 + 7 * q^25 - 7 * q^26 - 21 * q^27 + 4 * q^28 - 18 * q^29 + 3 * q^30 + 7 * q^31 + 7 * q^32 + 4 * q^33 - 6 * q^34 - 4 * q^35 + 4 * q^36 - 8 * q^37 - 5 * q^38 + 3 * q^39 - 7 * q^40 - 12 * q^41 - 15 * q^42 - 5 * q^43 - 10 * q^44 - 4 * q^45 - 11 * q^46 - 10 * q^47 - 3 * q^48 + 7 * q^49 + 7 * q^50 - 29 * q^51 - 7 * q^52 - 18 * q^53 - 21 * q^54 + 10 * q^55 + 4 * q^56 + 13 * q^57 - 18 * q^58 - 11 * q^59 + 3 * q^60 - 25 * q^61 + 7 * q^62 + 17 * q^63 + 7 * q^64 + 7 * q^65 + 4 * q^66 - 22 * q^67 - 6 * q^68 - 18 * q^69 - 4 * q^70 - 22 * q^71 + 4 * q^72 + 19 * q^73 - 8 * q^74 - 3 * q^75 - 5 * q^76 - 47 * q^77 + 3 * q^78 - 20 * q^79 - 7 * q^80 + 7 * q^81 - 12 * q^82 - 8 * q^83 - 15 * q^84 + 6 * q^85 - 5 * q^86 + 29 * q^87 - 10 * q^88 - 4 * q^89 - 4 * q^90 - 4 * q^91 - 11 * q^92 - 3 * q^93 - 10 * q^94 + 5 * q^95 - 3 * q^96 + 13 * q^97 + 7 * q^98 - 27 * 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
$3$	−2.33934	−1.35062	−0.675310	−	0.737534i	$-0.735990\pi$
$3$	−2.33934	−1.35062	−0.675310	+	0.737534i	$0.735990\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.33934	−0.955033
$7$	−3.11075	−1.17575	−0.587876	−	0.808951i	$-0.700036\pi$
$7$	−3.11075	−1.17575	−0.587876	+	0.808951i	$0.700036\pi$
$8$	1.00000	0.353553
$9$	2.47253	0.824175
$10$	−1.00000	−0.316228
$11$	2.74955	0.829021	0.414511	−	0.910045i	$-0.363953\pi$
$11$	2.74955	0.829021	0.414511	+	0.910045i	$0.363953\pi$
$12$	−2.33934	−0.675310
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.11075	−0.831383
$15$	2.33934	0.604016
$16$	1.00000	0.250000
$17$	3.35086	0.812704	0.406352	−	0.913717i	$-0.366801\pi$
$17$	3.35086	0.812704	0.406352	+	0.913717i	$0.366801\pi$
$18$	2.47253	0.582780
$19$	−3.63296	−0.833457	−0.416729	−	0.909031i	$-0.636823\pi$
$19$	−3.63296	−0.833457	−0.416729	+	0.909031i	$0.636823\pi$
$20$	−1.00000	−0.223607
$21$	7.27711	1.58800
$22$	2.74955	0.586206
$23$	5.74331	1.19756	0.598781	−	0.800913i	$-0.295652\pi$
$23$	5.74331	1.19756	0.598781	+	0.800913i	$0.295652\pi$
$24$	−2.33934	−0.477516
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	1.23394	0.237473
$28$	−3.11075	−0.587876
$29$	−9.75425	−1.81132	−0.905659	−	0.424006i	$-0.860623\pi$
$29$	−9.75425	−1.81132	−0.905659	+	0.424006i	$0.860623\pi$
$30$	2.33934	0.427104
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	−6.43214	−1.11969
$34$	3.35086	0.574668
$35$	3.11075	0.525812
$36$	2.47253	0.412088
$37$	1.88752	0.310306	0.155153	−	0.987890i	$-0.450413\pi$
$37$	1.88752	0.310306	0.155153	+	0.987890i	$0.450413\pi$
$38$	−3.63296	−0.589343
$39$	2.33934	0.374595
$40$	−1.00000	−0.158114
$41$	11.5507	1.80391	0.901957	−	0.431827i	$-0.142131\pi$
$41$	11.5507	1.80391	0.901957	+	0.431827i	$0.142131\pi$
$42$	7.27711	1.12288
$43$	−0.185051	−0.0282200	−0.0141100	−	0.999900i	$-0.504491\pi$
$43$	−0.185051	−0.0282200	−0.0141100	+	0.999900i	$0.504491\pi$
$44$	2.74955	0.414511
$45$	−2.47253	−0.368582
$46$	5.74331	0.846804
$47$	3.99290	0.582425	0.291213	−	0.956658i	$-0.405941\pi$
$47$	3.99290	0.582425	0.291213	+	0.956658i	$0.405941\pi$
$48$	−2.33934	−0.337655
$49$	2.67676	0.382394
$50$	1.00000	0.141421
$51$	−7.83882	−1.09765
$52$	−1.00000	−0.138675
$53$	3.72564	0.511756	0.255878	−	0.966709i	$-0.417635\pi$
$53$	3.72564	0.511756	0.255878	+	0.966709i	$0.417635\pi$
$54$	1.23394	0.167918
$55$	−2.74955	−0.370749
$56$	−3.11075	−0.415691
$57$	8.49873	1.12568
$58$	−9.75425	−1.28080
$59$	−7.90093	−1.02861	−0.514307	−	0.857606i	$-0.671951\pi$
$59$	−7.90093	−1.02861	−0.514307	+	0.857606i	$0.671951\pi$
$60$	2.33934	0.302008
$61$	−15.4252	−1.97500	−0.987499	−	0.157624i	$-0.949617\pi$
$61$	−15.4252	−1.97500	−0.987499	+	0.157624i	$0.949617\pi$
$62$	1.00000	0.127000
$63$	−7.69141	−0.969026
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−6.43214	−0.791742
$67$	−13.0916	−1.59939	−0.799695	−	0.600406i	$-0.795006\pi$
$67$	−13.0916	−1.59939	−0.799695	+	0.600406i	$0.795006\pi$
$68$	3.35086	0.406352
$69$	−13.4356	−1.61745
$70$	3.11075	0.371806
$71$	−1.92637	−0.228618	−0.114309	−	0.993445i	$-0.536465\pi$
$71$	−1.92637	−0.228618	−0.114309	+	0.993445i	$0.536465\pi$
$72$	2.47253	0.291390
$73$	14.2020	1.66222	0.831109	−	0.556110i	$-0.187707\pi$
$73$	14.2020	1.66222	0.831109	+	0.556110i	$0.187707\pi$
$74$	1.88752	0.219420
$75$	−2.33934	−0.270124
$76$	−3.63296	−0.416729
$77$	−8.55316	−0.974723
$78$	2.33934	0.264878
$79$	−16.0527	−1.80607	−0.903033	−	0.429572i	$-0.858665\pi$
$79$	−16.0527	−1.80607	−0.903033	+	0.429572i	$0.858665\pi$
$80$	−1.00000	−0.111803
$81$	−10.3042	−1.14491
$82$	11.5507	1.27556
$83$	11.9663	1.31347	0.656734	−	0.754123i	$-0.271938\pi$
$83$	11.9663	1.31347	0.656734	+	0.754123i	$0.271938\pi$
$84$	7.27711	0.793998
$85$	−3.35086	−0.363452
$86$	−0.185051	−0.0199545
$87$	22.8185	2.44640
$88$	2.74955	0.293103
$89$	10.5879	1.12232	0.561159	−	0.827708i	$-0.310356\pi$
$89$	10.5879	1.12232	0.561159	+	0.827708i	$0.310356\pi$
$90$	−2.47253	−0.260627
$91$	3.11075	0.326095
$92$	5.74331	0.598781
$93$	−2.33934	−0.242579
$94$	3.99290	0.411837
$95$	3.63296	0.372733
$96$	−2.33934	−0.238758
$97$	4.01604	0.407767	0.203883	−	0.978995i	$-0.434644\pi$
$97$	4.01604	0.407767	0.203883	+	0.978995i	$0.434644\pi$
$98$	2.67676	0.270393
$99$	6.79834	0.683259
$100$	1.00000	0.100000
$101$	0.463235	0.0460936	0.0230468	−	0.999734i	$-0.492663\pi$
$101$	0.463235	0.0460936	0.0230468	+	0.999734i	$0.492663\pi$
$102$	−7.83882	−0.776159
$103$	−2.11141	−0.208044	−0.104022	−	0.994575i	$-0.533171\pi$
$103$	−2.11141	−0.208044	−0.104022	+	0.994575i	$0.533171\pi$
$104$	−1.00000	−0.0980581
$105$	−7.27711	−0.710173
$106$	3.72564	0.361866
$107$	−19.9664	−1.93023	−0.965114	−	0.261830i	$-0.915674\pi$
$107$	−19.9664	−1.93023	−0.965114	+	0.261830i	$0.915674\pi$
$108$	1.23394	0.118736
$109$	−6.54433	−0.626833	−0.313417	−	0.949616i	$-0.601474\pi$
$109$	−6.54433	−0.626833	−0.313417	+	0.949616i	$0.601474\pi$
$110$	−2.74955	−0.262159
$111$	−4.41556	−0.419106
$112$	−3.11075	−0.293938
$113$	−1.42842	−0.134374	−0.0671871	−	0.997740i	$-0.521402\pi$
$113$	−1.42842	−0.134374	−0.0671871	+	0.997740i	$0.521402\pi$
$114$	8.49873	0.795979
$115$	−5.74331	−0.535566
$116$	−9.75425	−0.905659
$117$	−2.47253	−0.228585
$118$	−7.90093	−0.727339
$119$	−10.4237	−0.955538
$120$	2.33934	0.213552
$121$	−3.43997	−0.312724
$122$	−15.4252	−1.39653
$123$	−27.0210	−2.43640
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	−7.69141	−0.685205
$127$	1.53678	0.136367	0.0681837	−	0.997673i	$-0.478280\pi$
$127$	1.53678	0.136367	0.0681837	+	0.997673i	$0.478280\pi$
$128$	1.00000	0.0883883
$129$	0.432897	0.0381145
$130$	1.00000	0.0877058
$131$	3.09049	0.270017	0.135009	−	0.990844i	$-0.456894\pi$
$131$	3.09049	0.270017	0.135009	+	0.990844i	$0.456894\pi$
$132$	−6.43214	−0.559846
$133$	11.3012	0.979939
$134$	−13.0916	−1.13094
$135$	−1.23394	−0.106201
$136$	3.35086	0.287334
$137$	−17.1136	−1.46212	−0.731058	−	0.682315i	$-0.760973\pi$
$137$	−17.1136	−1.46212	−0.731058	+	0.682315i	$0.760973\pi$
$138$	−13.4356	−1.14371
$139$	−6.59843	−0.559672	−0.279836	−	0.960048i	$-0.590280\pi$
$139$	−6.59843	−0.559672	−0.279836	+	0.960048i	$0.590280\pi$
$140$	3.11075	0.262906
$141$	−9.34077	−0.786635
$142$	−1.92637	−0.161657
$143$	−2.74955	−0.229929
$144$	2.47253	0.206044
$145$	9.75425	0.810046
$146$	14.2020	1.17537
$147$	−6.26185	−0.516469
$148$	1.88752	0.155153
$149$	2.61443	0.214182	0.107091	−	0.994249i	$-0.465846\pi$
$149$	2.61443	0.214182	0.107091	+	0.994249i	$0.465846\pi$
$150$	−2.33934	−0.191007
$151$	−24.2929	−1.97693	−0.988465	−	0.151452i	$-0.951605\pi$
$151$	−24.2929	−1.97693	−0.988465	+	0.151452i	$0.951605\pi$
$152$	−3.63296	−0.294672
$153$	8.28509	0.669810
$154$	−8.55316	−0.689234
$155$	−1.00000	−0.0803219
$156$	2.33934	0.187297
$157$	−12.8465	−1.02526	−0.512631	−	0.858609i	$-0.671329\pi$
$157$	−12.8465	−1.02526	−0.512631	+	0.858609i	$0.671329\pi$
$158$	−16.0527	−1.27708
$159$	−8.71555	−0.691188
$160$	−1.00000	−0.0790569
$161$	−17.8660	−1.40804
$162$	−10.3042	−0.809574
$163$	−2.54345	−0.199218	−0.0996090	−	0.995027i	$-0.531759\pi$
$163$	−2.54345	−0.199218	−0.0996090	+	0.995027i	$0.531759\pi$
$164$	11.5507	0.901957
$165$	6.43214	0.500742
$166$	11.9663	0.928761
$167$	−12.5265	−0.969331	−0.484665	−	0.874700i	$-0.661059\pi$
$167$	−12.5265	−0.969331	−0.484665	+	0.874700i	$0.661059\pi$
$168$	7.27711	0.561441
$169$	1.00000	0.0769231
$170$	−3.35086	−0.256999
$171$	−8.98258	−0.686915
$172$	−0.185051	−0.0141100
$173$	9.60505	0.730258	0.365129	−	0.930957i	$-0.381025\pi$
$173$	9.60505	0.730258	0.365129	+	0.930957i	$0.381025\pi$
$174$	22.8185	1.72987
$175$	−3.11075	−0.235150
$176$	2.74955	0.207255
$177$	18.4830	1.38927
$178$	10.5879	0.793599
$179$	18.3788	1.37369	0.686846	−	0.726803i	$-0.258995\pi$
$179$	18.3788	1.37369	0.686846	+	0.726803i	$0.258995\pi$
$180$	−2.47253	−0.184291
$181$	−4.98907	−0.370834	−0.185417	−	0.982660i	$-0.559364\pi$
$181$	−4.98907	−0.370834	−0.185417	+	0.982660i	$0.559364\pi$
$182$	3.11075	0.230584
$183$	36.0849	2.66747
$184$	5.74331	0.423402
$185$	−1.88752	−0.138773
$186$	−2.33934	−0.171529
$187$	9.21337	0.673748
$188$	3.99290	0.291213
$189$	−3.83849	−0.279209
$190$	3.63296	0.263562
$191$	−16.3834	−1.18546	−0.592731	−	0.805401i	$-0.701950\pi$
$191$	−16.3834	−1.18546	−0.592731	+	0.805401i	$0.701950\pi$
$192$	−2.33934	−0.168828
$193$	8.68164	0.624918	0.312459	−	0.949931i	$-0.398847\pi$
$193$	8.68164	0.624918	0.312459	+	0.949931i	$0.398847\pi$
$194$	4.01604	0.288335
$195$	−2.33934	−0.167524
$196$	2.67676	0.191197
$197$	−8.78856	−0.626159	−0.313079	−	0.949727i	$-0.601361\pi$
$197$	−8.78856	−0.626159	−0.313079	+	0.949727i	$0.601361\pi$
$198$	6.79834	0.483137
$199$	−11.9538	−0.847381	−0.423690	−	0.905807i	$-0.639266\pi$
$199$	−11.9538	−0.847381	−0.423690	+	0.905807i	$0.639266\pi$
$200$	1.00000	0.0707107
$201$	30.6257	2.16017
$202$	0.463235	0.0325931
$203$	30.3430	2.12966
$204$	−7.83882	−0.548827
$205$	−11.5507	−0.806734
$206$	−2.11141	−0.147109
$207$	14.2005	0.987001
$208$	−1.00000	−0.0693375
$209$	−9.98900	−0.690954
$210$	−7.27711	−0.502168
$211$	−27.5547	−1.89694	−0.948471	−	0.316865i	$-0.897370\pi$
$211$	−27.5547	−1.89694	−0.948471	+	0.316865i	$0.897370\pi$
$212$	3.72564	0.255878
$213$	4.50643	0.308776
$214$	−19.9664	−1.36488
$215$	0.185051	0.0126204
$216$	1.23394	0.0839592
$217$	−3.11075	−0.211171
$218$	−6.54433	−0.443238
$219$	−33.2233	−2.24502
$220$	−2.74955	−0.185375
$221$	−3.35086	−0.225403
$222$	−4.41556	−0.296353
$223$	−2.73737	−0.183308	−0.0916541	−	0.995791i	$-0.529215\pi$
$223$	−2.73737	−0.183308	−0.0916541	+	0.995791i	$0.529215\pi$
$224$	−3.11075	−0.207846
$225$	2.47253	0.164835
$226$	−1.42842	−0.0950169
$227$	−21.3711	−1.41845	−0.709225	−	0.704983i	$-0.750955\pi$
$227$	−21.3711	−1.41845	−0.709225	+	0.704983i	$0.750955\pi$
$228$	8.49873	0.562842
$229$	−15.5317	−1.02636	−0.513180	−	0.858281i	$-0.671533\pi$
$229$	−15.5317	−1.02636	−0.513180	+	0.858281i	$0.671533\pi$
$230$	−5.74331	−0.378702
$231$	20.0088	1.31648
$232$	−9.75425	−0.640398
$233$	−7.61532	−0.498896	−0.249448	−	0.968388i	$-0.580249\pi$
$233$	−7.61532	−0.498896	−0.249448	+	0.968388i	$0.580249\pi$
$234$	−2.47253	−0.161634
$235$	−3.99290	−0.260468
$236$	−7.90093	−0.514307
$237$	37.5527	2.43931
$238$	−10.4237	−0.675668
$239$	−18.5072	−1.19713	−0.598565	−	0.801074i	$-0.704262\pi$
$239$	−18.5072	−1.19713	−0.598565	+	0.801074i	$0.704262\pi$
$240$	2.33934	0.151004
$241$	12.6736	0.816379	0.408189	−	0.912897i	$-0.366160\pi$
$241$	12.6736	0.816379	0.408189	+	0.912897i	$0.366160\pi$
$242$	−3.43997	−0.221129
$243$	20.4032	1.30887
$244$	−15.4252	−0.987499
$245$	−2.67676	−0.171012
$246$	−27.0210	−1.72280
$247$	3.63296	0.231159
$248$	1.00000	0.0635001
$249$	−27.9932	−1.77400
$250$	−1.00000	−0.0632456
$251$	29.3675	1.85366	0.926830	−	0.375482i	$-0.122523\pi$
$251$	29.3675	1.85366	0.926830	+	0.375482i	$0.122523\pi$
$252$	−7.69141	−0.484513
$253$	15.7915	0.992804
$254$	1.53678	0.0964264
$255$	7.83882	0.490886
$256$	1.00000	0.0625000
$257$	23.7925	1.48414	0.742068	−	0.670324i	$-0.233845\pi$
$257$	23.7925	1.48414	0.742068	+	0.670324i	$0.233845\pi$
$258$	0.432897	0.0269510
$259$	−5.87160	−0.364844
$260$	1.00000	0.0620174
$261$	−24.1176	−1.49284
$262$	3.09049	0.190931
$263$	19.8314	1.22285	0.611427	−	0.791301i	$-0.290596\pi$
$263$	19.8314	1.22285	0.611427	+	0.791301i	$0.290596\pi$
$264$	−6.43214	−0.395871
$265$	−3.72564	−0.228864
$266$	11.3012	0.692922
$267$	−24.7688	−1.51583
$268$	−13.0916	−0.799695
$269$	21.2657	1.29659	0.648296	−	0.761388i	$-0.275482\pi$
$269$	21.2657	1.29659	0.648296	+	0.761388i	$0.275482\pi$
$270$	−1.23394	−0.0750954
$271$	−3.34630	−0.203273	−0.101637	−	0.994822i	$-0.532408\pi$
$271$	−3.34630	−0.203273	−0.101637	+	0.994822i	$0.532408\pi$
$272$	3.35086	0.203176
$273$	−7.27711	−0.440431
$274$	−17.1136	−1.03387
$275$	2.74955	0.165804
$276$	−13.4356	−0.808726
$277$	24.7214	1.48536	0.742681	−	0.669645i	$-0.233554\pi$
$277$	24.7214	1.48536	0.742681	+	0.669645i	$0.233554\pi$
$278$	−6.59843	−0.395748
$279$	2.47253	0.148026
$280$	3.11075	0.185903
$281$	19.9003	1.18715	0.593576	−	0.804778i	$-0.297715\pi$
$281$	19.9003	1.18715	0.593576	+	0.804778i	$0.297715\pi$
$282$	−9.34077	−0.556235
$283$	−9.45092	−0.561799	−0.280899	−	0.959737i	$-0.590633\pi$
$283$	−9.45092	−0.561799	−0.280899	+	0.959737i	$0.590633\pi$
$284$	−1.92637	−0.114309
$285$	−8.49873	−0.503421
$286$	−2.74955	−0.162584
$287$	−35.9313	−2.12096
$288$	2.47253	0.145695
$289$	−5.77172	−0.339513
$290$	9.75425	0.572789
$291$	−9.39489	−0.550738
$292$	14.2020	0.831109
$293$	−7.14610	−0.417480	−0.208740	−	0.977971i	$-0.566936\pi$
$293$	−7.14610	−0.417480	−0.208740	+	0.977971i	$0.566936\pi$
$294$	−6.26185	−0.365199
$295$	7.90093	0.460010
$296$	1.88752	0.109710
$297$	3.39279	0.196870
$298$	2.61443	0.151450
$299$	−5.74331	−0.332144
$300$	−2.33934	−0.135062
$301$	0.575647	0.0331797
$302$	−24.2929	−1.39790
$303$	−1.08367	−0.0622549
$304$	−3.63296	−0.208364
$305$	15.4252	0.883246
$306$	8.28509	0.473627
$307$	−23.3970	−1.33534	−0.667669	−	0.744458i	$-0.732708\pi$
$307$	−23.3970	−1.33534	−0.667669	+	0.744458i	$0.732708\pi$
$308$	−8.55316	−0.487362
$309$	4.93932	0.280988
$310$	−1.00000	−0.0567962
$311$	−28.5918	−1.62129	−0.810646	−	0.585536i	$-0.800884\pi$
$311$	−28.5918	−1.62129	−0.810646	+	0.585536i	$0.800884\pi$
$312$	2.33934	0.132439
$313$	7.68092	0.434151	0.217076	−	0.976155i	$-0.430348\pi$
$313$	7.68092	0.434151	0.217076	+	0.976155i	$0.430348\pi$
$314$	−12.8465	−0.724970
$315$	7.69141	0.433362
$316$	−16.0527	−0.903033
$317$	19.8470	1.11472	0.557360	−	0.830271i	$-0.311814\pi$
$317$	19.8470	1.11472	0.557360	+	0.830271i	$0.311814\pi$
$318$	−8.71555	−0.488744
$319$	−26.8198	−1.50162
$320$	−1.00000	−0.0559017
$321$	46.7083	2.60701
$322$	−17.8660	−0.995632
$323$	−12.1735	−0.677354
$324$	−10.3042	−0.572455
$325$	−1.00000	−0.0554700
$326$	−2.54345	−0.140868
$327$	15.3094	0.846614
$328$	11.5507	0.637780
$329$	−12.4209	−0.684788
$330$	6.43214	0.354078
$331$	−5.95269	−0.327190	−0.163595	−	0.986528i	$-0.552309\pi$
$331$	−5.95269	−0.327190	−0.163595	+	0.986528i	$0.552309\pi$
$332$	11.9663	0.656734
$333$	4.66694	0.255747
$334$	−12.5265	−0.685420
$335$	13.0916	0.715269
$336$	7.27711	0.396999
$337$	8.11752	0.442190	0.221095	−	0.975252i	$-0.429037\pi$
$337$	8.11752	0.442190	0.221095	+	0.975252i	$0.429037\pi$
$338$	1.00000	0.0543928
$339$	3.34156	0.181488
$340$	−3.35086	−0.181726
$341$	2.74955	0.148897
$342$	−8.98258	−0.485722
$343$	13.4485	0.726152
$344$	−0.185051	−0.00997727
$345$	13.4356	0.723346
$346$	9.60505	0.516370
$347$	−21.3908	−1.14832	−0.574160	−	0.818743i	$-0.694671\pi$
$347$	−21.3908	−1.14832	−0.574160	+	0.818743i	$0.694671\pi$
$348$	22.8185	1.22320
$349$	8.40541	0.449931	0.224966	−	0.974367i	$-0.427773\pi$
$349$	8.40541	0.449931	0.224966	+	0.974367i	$0.427773\pi$
$350$	−3.11075	−0.166277
$351$	−1.23394	−0.0658630
$352$	2.74955	0.146552
$353$	−13.3579	−0.710968	−0.355484	−	0.934682i	$-0.615684\pi$
$353$	−13.3579	−0.710968	−0.355484	+	0.934682i	$0.615684\pi$
$354$	18.4830	0.982359
$355$	1.92637	0.102241
$356$	10.5879	0.561159
$357$	24.3846	1.29057
$358$	18.3788	0.971347
$359$	15.5841	0.822498	0.411249	−	0.911523i	$-0.365093\pi$
$359$	15.5841	0.822498	0.411249	+	0.911523i	$0.365093\pi$
$360$	−2.47253	−0.130314
$361$	−5.80163	−0.305349
$362$	−4.98907	−0.262220
$363$	8.04726	0.422372
$364$	3.11075	0.163048
$365$	−14.2020	−0.743366
$366$	36.0849	1.88619
$367$	−23.1792	−1.20995	−0.604973	−	0.796246i	$-0.706816\pi$
$367$	−23.1792	−1.20995	−0.604973	+	0.796246i	$0.706816\pi$
$368$	5.74331	0.299391
$369$	28.5594	1.48674
$370$	−1.88752	−0.0981275
$371$	−11.5895	−0.601698
$372$	−2.33934	−0.121289
$373$	−24.1268	−1.24924	−0.624618	−	0.780931i	$-0.714745\pi$
$373$	−24.1268	−1.24924	−0.624618	+	0.780931i	$0.714745\pi$
$374$	9.21337	0.476412
$375$	2.33934	0.120803
$376$	3.99290	0.205918
$377$	9.75425	0.502369
$378$	−3.83849	−0.197431
$379$	4.08395	0.209779	0.104889	−	0.994484i	$-0.466551\pi$
$379$	4.08395	0.209779	0.104889	+	0.994484i	$0.466551\pi$
$380$	3.63296	0.186367
$381$	−3.59506	−0.184181
$382$	−16.3834	−0.838248
$383$	−11.4751	−0.586349	−0.293175	−	0.956059i	$-0.594712\pi$
$383$	−11.4751	−0.586349	−0.293175	+	0.956059i	$0.594712\pi$
$384$	−2.33934	−0.119379
$385$	8.55316	0.435910
$386$	8.68164	0.441884
$387$	−0.457543	−0.0232582
$388$	4.01604	0.203883
$389$	12.4424	0.630853	0.315427	−	0.948950i	$-0.397852\pi$
$389$	12.4424	0.630853	0.315427	+	0.948950i	$0.397852\pi$
$390$	−2.33934	−0.118457
$391$	19.2450	0.973263
$392$	2.67676	0.135197
$393$	−7.22971	−0.364691
$394$	−8.78856	−0.442761
$395$	16.0527	0.807697
$396$	6.79834	0.341629
$397$	27.2087	1.36556	0.682782	−	0.730622i	$-0.260770\pi$
$397$	27.2087	1.36556	0.682782	+	0.730622i	$0.260770\pi$
$398$	−11.9538	−0.599189
$399$	−26.4374	−1.32353
$400$	1.00000	0.0500000
$401$	−29.0213	−1.44926	−0.724628	−	0.689140i	$-0.757988\pi$
$401$	−29.0213	−1.44926	−0.724628	+	0.689140i	$0.757988\pi$
$402$	30.6257	1.52747
$403$	−1.00000	−0.0498135
$404$	0.463235	0.0230468
$405$	10.3042	0.512020
$406$	30.3430	1.50590
$407$	5.18983	0.257251
$408$	−7.83882	−0.388079
$409$	−27.5582	−1.36266	−0.681332	−	0.731974i	$-0.738599\pi$
$409$	−27.5582	−1.36266	−0.681332	+	0.731974i	$0.738599\pi$
$410$	−11.5507	−0.570447
$411$	40.0346	1.97476
$412$	−2.11141	−0.104022
$413$	24.5778	1.20939
$414$	14.2005	0.697915
$415$	−11.9663	−0.587400
$416$	−1.00000	−0.0490290
$417$	15.4360	0.755904
$418$	−9.98900	−0.488578
$419$	26.1508	1.27755	0.638775	−	0.769393i	$-0.279441\pi$
$419$	26.1508	1.27755	0.638775	+	0.769393i	$0.279441\pi$
$420$	−7.27711	−0.355086
$421$	12.6211	0.615113	0.307556	−	0.951530i	$-0.400489\pi$
$421$	12.6211	0.615113	0.307556	+	0.951530i	$0.400489\pi$
$422$	−27.5547	−1.34134
$423$	9.87256	0.480020
$424$	3.72564	0.180933
$425$	3.35086	0.162541
$426$	4.50643	0.218337
$427$	47.9840	2.32211
$428$	−19.9664	−0.965114
$429$	6.43214	0.310547
$430$	0.185051	0.00892394
$431$	37.5643	1.80941	0.904704	−	0.426041i	$-0.140092\pi$
$431$	37.5643	1.80941	0.904704	+	0.426041i	$0.140092\pi$
$432$	1.23394	0.0593681
$433$	−8.84893	−0.425252	−0.212626	−	0.977134i	$-0.568202\pi$
$433$	−8.84893	−0.425252	−0.212626	+	0.977134i	$0.568202\pi$
$434$	−3.11075	−0.149321
$435$	−22.8185	−1.09406
$436$	−6.54433	−0.313417
$437$	−20.8652	−0.998117
$438$	−33.2233	−1.58747
$439$	27.1661	1.29657	0.648283	−	0.761399i	$-0.275487\pi$
$439$	27.1661	1.29657	0.648283	+	0.761399i	$0.275487\pi$
$440$	−2.74955	−0.131080
$441$	6.61835	0.315159
$442$	−3.35086	−0.159384
$443$	−18.7448	−0.890593	−0.445297	−	0.895383i	$-0.646902\pi$
$443$	−18.7448	−0.890593	−0.445297	+	0.895383i	$0.646902\pi$
$444$	−4.41556	−0.209553
$445$	−10.5879	−0.501916
$446$	−2.73737	−0.129618
$447$	−6.11604	−0.289279
$448$	−3.11075	−0.146969
$449$	−32.0617	−1.51309	−0.756543	−	0.653944i	$-0.773113\pi$
$449$	−32.0617	−1.51309	−0.756543	+	0.653944i	$0.773113\pi$
$450$	2.47253	0.116556
$451$	31.7592	1.49548
$452$	−1.42842	−0.0671871
$453$	56.8295	2.67008
$454$	−21.3711	−1.00300
$455$	−3.11075	−0.145834
$456$	8.49873	0.397989
$457$	4.54839	0.212765	0.106382	−	0.994325i	$-0.466073\pi$
$457$	4.54839	0.212765	0.106382	+	0.994325i	$0.466073\pi$
$458$	−15.5317	−0.725747
$459$	4.13477	0.192995
$460$	−5.74331	−0.267783
$461$	4.90157	0.228289	0.114144	−	0.993464i	$-0.463587\pi$
$461$	4.90157	0.228289	0.114144	+	0.993464i	$0.463587\pi$
$462$	20.0088	0.930893
$463$	−28.3363	−1.31690	−0.658451	−	0.752624i	$-0.728788\pi$
$463$	−28.3363	−1.31690	−0.658451	+	0.752624i	$0.728788\pi$
$464$	−9.75425	−0.452830
$465$	2.33934	0.108484
$466$	−7.61532	−0.352773
$467$	29.9987	1.38818	0.694088	−	0.719891i	$-0.255808\pi$
$467$	29.9987	1.38818	0.694088	+	0.719891i	$0.255808\pi$
$468$	−2.47253	−0.114293
$469$	40.7246	1.88049
$470$	−3.99290	−0.184179
$471$	30.0524	1.38474
$472$	−7.90093	−0.363670
$473$	−0.508807	−0.0233950
$474$	37.5527	1.72485
$475$	−3.63296	−0.166691
$476$	−10.4237	−0.477769
$477$	9.21174	0.421777
$478$	−18.5072	−0.846499
$479$	−35.3442	−1.61492	−0.807460	−	0.589923i	$-0.799158\pi$
$479$	−35.3442	−1.61492	−0.807460	+	0.589923i	$0.799158\pi$
$480$	2.33934	0.106776
$481$	−1.88752	−0.0860635
$482$	12.6736	0.577267
$483$	41.7947	1.90172
$484$	−3.43997	−0.156362
$485$	−4.01604	−0.182359
$486$	20.4032	0.925509
$487$	−7.22669	−0.327472	−0.163736	−	0.986504i	$-0.552355\pi$
$487$	−7.22669	−0.327472	−0.163736	+	0.986504i	$0.552355\pi$
$488$	−15.4252	−0.698267
$489$	5.94999	0.269068
$490$	−2.67676	−0.120924
$491$	−27.9404	−1.26093	−0.630467	−	0.776216i	$-0.717136\pi$
$491$	−27.9404	−1.26093	−0.630467	+	0.776216i	$0.717136\pi$
$492$	−27.0210	−1.21820
$493$	−32.6851	−1.47206
$494$	3.63296	0.163454
$495$	−6.79834	−0.305563
$496$	1.00000	0.0449013
$497$	5.99244	0.268798
$498$	−27.9932	−1.25440
$499$	−13.6185	−0.609647	−0.304823	−	0.952409i	$-0.598597\pi$
$499$	−13.6185	−0.609647	−0.304823	+	0.952409i	$0.598597\pi$
$500$	−1.00000	−0.0447214
$501$	29.3038	1.30920
$502$	29.3675	1.31074
$503$	−4.36171	−0.194479	−0.0972394	−	0.995261i	$-0.531001\pi$
$503$	−4.36171	−0.194479	−0.0972394	+	0.995261i	$0.531001\pi$
$504$	−7.69141	−0.342602
$505$	−0.463235	−0.0206137
$506$	15.7915	0.702019
$507$	−2.33934	−0.103894
$508$	1.53678	0.0681837
$509$	−18.9278	−0.838959	−0.419479	−	0.907765i	$-0.637787\pi$
$509$	−18.9278	−0.838959	−0.419479	+	0.907765i	$0.637787\pi$
$510$	7.83882	0.347109
$511$	−44.1788	−1.95436
$512$	1.00000	0.0441942
$513$	−4.48286	−0.197923
$514$	23.7925	1.04944
$515$	2.11141	0.0930399
$516$	0.432897	0.0190572
$517$	10.9787	0.482843
$518$	−5.87160	−0.257983
$519$	−22.4695	−0.986301
$520$	1.00000	0.0438529
$521$	30.0758	1.31765	0.658823	−	0.752298i	$-0.271055\pi$
$521$	30.0758	1.31765	0.658823	+	0.752298i	$0.271055\pi$
$522$	−24.1176	−1.05560
$523$	7.61035	0.332777	0.166389	−	0.986060i	$-0.446789\pi$
$523$	7.61035	0.332777	0.166389	+	0.986060i	$0.446789\pi$
$524$	3.09049	0.135009
$525$	7.27711	0.317599
$526$	19.8314	0.864689
$527$	3.35086	0.145966
$528$	−6.43214	−0.279923
$529$	9.98557	0.434155
$530$	−3.72564	−0.161831
$531$	−19.5352	−0.847757
$532$	11.3012	0.489970
$533$	−11.5507	−0.500315
$534$	−24.7688	−1.07185
$535$	19.9664	0.863224
$536$	−13.0916	−0.565470
$537$	−42.9942	−1.85534
$538$	21.2657	0.916829
$539$	7.35988	0.317012
$540$	−1.23394	−0.0531005
$541$	−19.5677	−0.841282	−0.420641	−	0.907227i	$-0.638195\pi$
$541$	−19.5677	−0.841282	−0.420641	+	0.907227i	$0.638195\pi$
$542$	−3.34630	−0.143736
$543$	11.6711	0.500857
$544$	3.35086	0.143667
$545$	6.54433	0.280328
$546$	−7.27711	−0.311431
$547$	20.0339	0.856587	0.428293	−	0.903640i	$-0.359115\pi$
$547$	20.0339	0.856587	0.428293	+	0.903640i	$0.359115\pi$
$548$	−17.1136	−0.731058
$549$	−38.1393	−1.62774
$550$	2.74955	0.117241
$551$	35.4368	1.50966
$552$	−13.4356	−0.571856
$553$	49.9358	2.12349
$554$	24.7214	1.05031
$555$	4.41556	0.187430
$556$	−6.59843	−0.279836
$557$	−35.5103	−1.50462	−0.752310	−	0.658810i	$-0.771060\pi$
$557$	−35.5103	−1.50462	−0.752310	+	0.658810i	$0.771060\pi$
$558$	2.47253	0.104670
$559$	0.185051	0.00782682
$560$	3.11075	0.131453
$561$	−21.5532	−0.909978
$562$	19.9003	0.839444
$563$	40.2658	1.69700	0.848501	−	0.529193i	$-0.177505\pi$
$563$	40.2658	1.69700	0.848501	+	0.529193i	$0.177505\pi$
$564$	−9.34077	−0.393318
$565$	1.42842	0.0600939
$566$	−9.45092	−0.397252
$567$	32.0538	1.34613
$568$	−1.92637	−0.0808286
$569$	5.35081	0.224318	0.112159	−	0.993690i	$-0.464223\pi$
$569$	5.35081	0.224318	0.112159	+	0.993690i	$0.464223\pi$
$570$	−8.49873	−0.355973
$571$	22.3056	0.933459	0.466729	−	0.884400i	$-0.345432\pi$
$571$	22.3056	0.933459	0.466729	+	0.884400i	$0.345432\pi$
$572$	−2.74955	−0.114965
$573$	38.3264	1.60111
$574$	−35.9313	−1.49974
$575$	5.74331	0.239512
$576$	2.47253	0.103022
$577$	−22.3046	−0.928552	−0.464276	−	0.885691i	$-0.653685\pi$
$577$	−22.3046	−0.928552	−0.464276	+	0.885691i	$0.653685\pi$
$578$	−5.77172	−0.240072
$579$	−20.3093	−0.844027
$580$	9.75425	0.405023
$581$	−37.2240	−1.54431
$582$	−9.39489	−0.389431
$583$	10.2438	0.424257
$584$	14.2020	0.587683
$585$	2.47253	0.102226
$586$	−7.14610	−0.295203
$587$	−39.2271	−1.61908	−0.809538	−	0.587068i	$-0.800282\pi$
$587$	−39.2271	−1.61908	−0.809538	+	0.587068i	$0.800282\pi$
$588$	−6.26185	−0.258234
$589$	−3.63296	−0.149693
$590$	7.90093	0.325276
$591$	20.5595	0.845703
$592$	1.88752	0.0775766
$593$	−33.5553	−1.37795	−0.688976	−	0.724784i	$-0.741939\pi$
$593$	−33.5553	−1.37795	−0.688976	+	0.724784i	$0.741939\pi$
$594$	3.39279	0.139208
$595$	10.4237	0.427330
$596$	2.61443	0.107091
$597$	27.9640	1.14449
$598$	−5.74331	−0.234861
$599$	36.9194	1.50849	0.754243	−	0.656595i	$-0.228004\pi$
$599$	36.9194	1.50849	0.754243	+	0.656595i	$0.228004\pi$
$600$	−2.33934	−0.0955033
$601$	18.8116	0.767340	0.383670	−	0.923470i	$-0.374660\pi$
$601$	18.8116	0.767340	0.383670	+	0.923470i	$0.374660\pi$
$602$	0.575647	0.0234616
$603$	−32.3693	−1.31818
$604$	−24.2929	−0.988465
$605$	3.43997	0.139855
$606$	−1.08367	−0.0440209
$607$	16.3410	0.663259	0.331630	−	0.943410i	$-0.392402\pi$
$607$	16.3410	0.663259	0.331630	+	0.943410i	$0.392402\pi$
$608$	−3.63296	−0.147336
$609$	−70.9827	−2.87636
$610$	15.4252	0.624549
$611$	−3.99290	−0.161536
$612$	8.28509	0.334905
$613$	−10.0663	−0.406574	−0.203287	−	0.979119i	$-0.565162\pi$
$613$	−10.0663	−0.406574	−0.203287	+	0.979119i	$0.565162\pi$
$614$	−23.3970	−0.944226
$615$	27.0210	1.08959
$616$	−8.55316	−0.344617
$617$	−9.31059	−0.374830	−0.187415	−	0.982281i	$-0.560011\pi$
$617$	−9.31059	−0.374830	−0.187415	+	0.982281i	$0.560011\pi$
$618$	4.93932	0.198688
$619$	−3.09653	−0.124460	−0.0622301	−	0.998062i	$-0.519821\pi$
$619$	−3.09653	−0.124460	−0.0622301	+	0.998062i	$0.519821\pi$
$620$	−1.00000	−0.0401610
$621$	7.08692	0.284388
$622$	−28.5918	−1.14643
$623$	−32.9364	−1.31957
$624$	2.33934	0.0936487
$625$	1.00000	0.0400000
$626$	7.68092	0.306991
$627$	23.3677	0.933216
$628$	−12.8465	−0.512631
$629$	6.32482	0.252187
$630$	7.69141	0.306433
$631$	13.9463	0.555193	0.277597	−	0.960698i	$-0.410462\pi$
$631$	13.9463	0.555193	0.277597	+	0.960698i	$0.410462\pi$
$632$	−16.0527	−0.638541
$633$	64.4598	2.56205
$634$	19.8470	0.788227
$635$	−1.53678	−0.0609854
$636$	−8.71555	−0.345594
$637$	−2.67676	−0.106057
$638$	−26.8198	−1.06181
$639$	−4.76299	−0.188421
$640$	−1.00000	−0.0395285
$641$	24.2648	0.958402	0.479201	−	0.877705i	$-0.340927\pi$
$641$	24.2648	0.958402	0.479201	+	0.877705i	$0.340927\pi$
$642$	46.7083	1.84343
$643$	−37.8920	−1.49432	−0.747158	−	0.664646i	$-0.768582\pi$
$643$	−37.8920	−1.49432	−0.747158	+	0.664646i	$0.768582\pi$
$644$	−17.8660	−0.704018
$645$	−0.432897	−0.0170453
$646$	−12.1735	−0.478961
$647$	22.9093	0.900659	0.450329	−	0.892863i	$-0.351307\pi$
$647$	22.9093	0.900659	0.450329	+	0.892863i	$0.351307\pi$
$648$	−10.3042	−0.404787
$649$	−21.7240	−0.852742
$650$	−1.00000	−0.0392232
$651$	7.27711	0.285212
$652$	−2.54345	−0.0996090
$653$	−32.3414	−1.26562	−0.632808	−	0.774308i	$-0.718098\pi$
$653$	−32.3414	−1.26562	−0.632808	+	0.774308i	$0.718098\pi$
$654$	15.3094	0.598646
$655$	−3.09049	−0.120755
$656$	11.5507	0.450978
$657$	35.1148	1.36996
$658$	−12.4209	−0.484218
$659$	−35.3404	−1.37666	−0.688332	−	0.725395i	$-0.741657\pi$
$659$	−35.3404	−1.37666	−0.688332	+	0.725395i	$0.741657\pi$
$660$	6.43214	0.250371
$661$	15.8120	0.615014	0.307507	−	0.951546i	$-0.400505\pi$
$661$	15.8120	0.615014	0.307507	+	0.951546i	$0.400505\pi$
$662$	−5.95269	−0.231358
$663$	7.83882	0.304434
$664$	11.9663	0.464381
$665$	−11.3012	−0.438242
$666$	4.66694	0.180840
$667$	−56.0216	−2.16917
$668$	−12.5265	−0.484665
$669$	6.40366	0.247580
$670$	13.0916	0.505772
$671$	−42.4125	−1.63732
$672$	7.27711	0.280721
$673$	45.8835	1.76868	0.884340	−	0.466844i	$-0.154609\pi$
$673$	45.8835	1.76868	0.884340	+	0.466844i	$0.154609\pi$
$674$	8.11752	0.312675
$675$	1.23394	0.0474945
$676$	1.00000	0.0384615
$677$	5.57858	0.214402	0.107201	−	0.994237i	$-0.465811\pi$
$677$	5.57858	0.214402	0.107201	+	0.994237i	$0.465811\pi$
$678$	3.34156	0.128332
$679$	−12.4929	−0.479433
$680$	−3.35086	−0.128500
$681$	49.9943	1.91579
$682$	2.74955	0.105286
$683$	19.8777	0.760600	0.380300	−	0.924863i	$-0.375821\pi$
$683$	19.8777	0.760600	0.380300	+	0.924863i	$0.375821\pi$
$684$	−8.98258	−0.343457
$685$	17.1136	0.653878
$686$	13.4485	0.513467
$687$	36.3339	1.38622
$688$	−0.185051	−0.00705500
$689$	−3.72564	−0.141936
$690$	13.4356	0.511483
$691$	22.1855	0.843976	0.421988	−	0.906601i	$-0.361332\pi$
$691$	22.1855	0.843976	0.421988	+	0.906601i	$0.361332\pi$
$692$	9.60505	0.365129
$693$	−21.1479	−0.803343
$694$	−21.3908	−0.811985
$695$	6.59843	0.250293
$696$	22.8185	0.864934
$697$	38.7047	1.46605
$698$	8.40541	0.318149
$699$	17.8149	0.673820
$700$	−3.11075	−0.117575
$701$	50.7880	1.91824	0.959119	−	0.283004i	$-0.0913311\pi$
$701$	50.7880	1.91824	0.959119	+	0.283004i	$0.0913311\pi$
$702$	−1.23394	−0.0465722
$703$	−6.85728	−0.258627
$704$	2.74955	0.103628
$705$	9.34077	0.351794
$706$	−13.3579	−0.502731
$707$	−1.44101	−0.0541947
$708$	18.4830	0.694633
$709$	−15.0570	−0.565477	−0.282739	−	0.959197i	$-0.591243\pi$
$709$	−15.0570	−0.565477	−0.282739	+	0.959197i	$0.591243\pi$
$710$	1.92637	0.0722953
$711$	−39.6906	−1.48851
$712$	10.5879	0.396799
$713$	5.74331	0.215089
$714$	24.3846	0.912570
$715$	2.74955	0.102827
$716$	18.3788	0.686846
$717$	43.2946	1.61687
$718$	15.5841	0.581594
$719$	−41.6751	−1.55422	−0.777110	−	0.629365i	$-0.783315\pi$
$719$	−41.6751	−1.55422	−0.777110	+	0.629365i	$0.783315\pi$
$720$	−2.47253	−0.0921456
$721$	6.56807	0.244608
$722$	−5.80163	−0.215914
$723$	−29.6479	−1.10262
$724$	−4.98907	−0.185417
$725$	−9.75425	−0.362264
$726$	8.04726	0.298662
$727$	−20.1032	−0.745587	−0.372794	−	0.927914i	$-0.621600\pi$
$727$	−20.1032	−0.745587	−0.372794	+	0.927914i	$0.621600\pi$
$728$	3.11075	0.115292
$729$	−16.8175	−0.622872
$730$	−14.2020	−0.525639
$731$	−0.620080	−0.0229345
$732$	36.0849	1.33374
$733$	−16.3765	−0.604880	−0.302440	−	0.953168i	$-0.597801\pi$
$733$	−16.3765	−0.604880	−0.302440	+	0.953168i	$0.597801\pi$
$734$	−23.1792	−0.855561
$735$	6.26185	0.230972
$736$	5.74331	0.211701
$737$	−35.9960	−1.32593
$738$	28.5594	1.05128
$739$	−24.4530	−0.899519	−0.449759	−	0.893150i	$-0.648490\pi$
$739$	−24.4530	−0.899519	−0.449759	+	0.893150i	$0.648490\pi$
$740$	−1.88752	−0.0693866
$741$	−8.49873	−0.312209
$742$	−11.5895	−0.425465
$743$	0.574740	0.0210852	0.0105426	−	0.999944i	$-0.496644\pi$
$743$	0.574740	0.0210852	0.0105426	+	0.999944i	$0.496644\pi$
$744$	−2.33934	−0.0857645
$745$	−2.61443	−0.0957852
$746$	−24.1268	−0.883343
$747$	29.5869	1.08253
$748$	9.21337	0.336874
$749$	62.1106	2.26947
$750$	2.33934	0.0854207
$751$	40.0773	1.46244	0.731221	−	0.682141i	$-0.238951\pi$
$751$	40.0773	1.46244	0.731221	+	0.682141i	$0.238951\pi$
$752$	3.99290	0.145606
$753$	−68.7006	−2.50359
$754$	9.75425	0.355229
$755$	24.2929	0.884110
$756$	−3.83849	−0.139604
$757$	−18.9859	−0.690054	−0.345027	−	0.938593i	$-0.612130\pi$
$757$	−18.9859	−0.690054	−0.345027	+	0.938593i	$0.612130\pi$
$758$	4.08395	0.148336
$759$	−36.9418	−1.34090
$760$	3.63296	0.131781
$761$	9.21884	0.334183	0.167091	−	0.985941i	$-0.446563\pi$
$761$	9.21884	0.334183	0.167091	+	0.985941i	$0.446563\pi$
$762$	−3.59506	−0.130235
$763$	20.3578	0.737001
$764$	−16.3834	−0.592731
$765$	−8.28509	−0.299548
$766$	−11.4751	−0.414611
$767$	7.90093	0.285286
$768$	−2.33934	−0.0844138
$769$	47.1983	1.70202	0.851008	−	0.525153i	$-0.175992\pi$
$769$	47.1983	1.70202	0.851008	+	0.525153i	$0.175992\pi$
$770$	8.55316	0.308235
$771$	−55.6589	−2.00451
$772$	8.68164	0.312459
$773$	−11.0195	−0.396345	−0.198173	−	0.980167i	$-0.563501\pi$
$773$	−11.0195	−0.396345	−0.198173	+	0.980167i	$0.563501\pi$
$774$	−0.457543	−0.0164460
$775$	1.00000	0.0359211
$776$	4.01604	0.144167
$777$	13.7357	0.492765
$778$	12.4424	0.446081
$779$	−41.9631	−1.50348
$780$	−2.33934	−0.0837619
$781$	−5.29665	−0.189529
$782$	19.2450	0.688201
$783$	−12.0362	−0.430138
$784$	2.67676	0.0955984
$785$	12.8465	0.458511
$786$	−7.22971	−0.257875
$787$	33.6704	1.20022	0.600111	−	0.799917i	$-0.295123\pi$
$787$	33.6704	1.20022	0.600111	+	0.799917i	$0.295123\pi$
$788$	−8.78856	−0.313079
$789$	−46.3924	−1.65161
$790$	16.0527	0.571128
$791$	4.44345	0.157991
$792$	6.79834	0.241568
$793$	15.4252	0.547766
$794$	27.2087	0.965600
$795$	8.71555	0.309109
$796$	−11.9538	−0.423690
$797$	−28.9101	−1.02405	−0.512024	−	0.858971i	$-0.671104\pi$
$797$	−28.9101	−1.02405	−0.512024	+	0.858971i	$0.671104\pi$
$798$	−26.4374	−0.935874
$799$	13.3797	0.473339
$800$	1.00000	0.0353553
$801$	26.1789	0.924987
$802$	−29.0213	−1.02478
$803$	39.0491	1.37801
$804$	30.6257	1.08008
$805$	17.8660	0.629693
$806$	−1.00000	−0.0352235
$807$	−49.7478	−1.75120
$808$	0.463235	0.0162965
$809$	11.3869	0.400343	0.200171	−	0.979761i	$-0.435850\pi$
$809$	11.3869	0.400343	0.200171	+	0.979761i	$0.435850\pi$
$810$	10.3042	0.362052
$811$	−22.2379	−0.780877	−0.390439	−	0.920629i	$-0.627677\pi$
$811$	−22.2379	−0.780877	−0.390439	+	0.920629i	$0.627677\pi$
$812$	30.3430	1.06483
$813$	7.82815	0.274545
$814$	5.18983	0.181904
$815$	2.54345	0.0890930
$816$	−7.83882	−0.274413
$817$	0.672281	0.0235202
$818$	−27.5582	−0.963549
$819$	7.69141	0.268759
$820$	−11.5507	−0.403367
$821$	−36.3045	−1.26704	−0.633518	−	0.773728i	$-0.718390\pi$
$821$	−36.3045	−1.26704	−0.633518	+	0.773728i	$0.718390\pi$
$822$	40.0346	1.39637
$823$	−32.8747	−1.14594	−0.572971	−	0.819576i	$-0.694209\pi$
$823$	−32.8747	−1.14594	−0.572971	+	0.819576i	$0.694209\pi$
$824$	−2.11141	−0.0735545
$825$	−6.43214	−0.223939
$826$	24.5778	0.855171
$827$	25.8574	0.899150	0.449575	−	0.893243i	$-0.351576\pi$
$827$	25.8574	0.899150	0.449575	+	0.893243i	$0.351576\pi$
$828$	14.2005	0.493501
$829$	26.7029	0.927431	0.463716	−	0.885984i	$-0.346516\pi$
$829$	26.7029	0.927431	0.463716	+	0.885984i	$0.346516\pi$
$830$	−11.9663	−0.415355
$831$	−57.8317	−2.00616
$832$	−1.00000	−0.0346688
$833$	8.96944	0.310773
$834$	15.4360	0.534505
$835$	12.5265	0.433498
$836$	−9.98900	−0.345477
$837$	1.23394	0.0426513
$838$	26.1508	0.903364
$839$	−11.2860	−0.389638	−0.194819	−	0.980839i	$-0.562412\pi$
$839$	−11.2860	−0.389638	−0.194819	+	0.980839i	$0.562412\pi$
$840$	−7.27711	−0.251084
$841$	66.1454	2.28087
$842$	12.6211	0.434950
$843$	−46.5536	−1.60339
$844$	−27.5547	−0.948471
$845$	−1.00000	−0.0344010
$846$	9.87256	0.339426
$847$	10.7009	0.367686
$848$	3.72564	0.127939
$849$	22.1089	0.758777
$850$	3.35086	0.114934
$851$	10.8406	0.371611
$852$	4.50643	0.154388
$853$	47.6601	1.63185	0.815925	−	0.578157i	$-0.196228\pi$
$853$	47.6601	1.63185	0.815925	+	0.578157i	$0.196228\pi$
$854$	47.9840	1.64198
$855$	8.98258	0.307198
$856$	−19.9664	−0.682439
$857$	25.0693	0.856350	0.428175	−	0.903696i	$-0.359157\pi$
$857$	25.0693	0.856350	0.428175	+	0.903696i	$0.359157\pi$
$858$	6.43214	0.219590
$859$	−24.3226	−0.829876	−0.414938	−	0.909850i	$-0.636197\pi$
$859$	−24.3226	−0.829876	−0.414938	+	0.909850i	$0.636197\pi$
$860$	0.185051	0.00631018
$861$	84.0555	2.86461
$862$	37.5643	1.27944
$863$	−16.6303	−0.566103	−0.283051	−	0.959105i	$-0.591347\pi$
$863$	−16.6303	−0.566103	−0.283051	+	0.959105i	$0.591347\pi$
$864$	1.23394	0.0419796
$865$	−9.60505	−0.326581
$866$	−8.84893	−0.300699
$867$	13.5020	0.458553
$868$	−3.11075	−0.105586
$869$	−44.1376	−1.49727
$870$	−22.8185	−0.773621
$871$	13.0916	0.443591
$872$	−6.54433	−0.221619
$873$	9.92975	0.336071
$874$	−20.8652	−0.705775
$875$	3.11075	0.105162
$876$	−33.2233	−1.12251
$877$	20.0457	0.676897	0.338448	−	0.940985i	$-0.390098\pi$
$877$	20.0457	0.676897	0.338448	+	0.940985i	$0.390098\pi$
$878$	27.1661	0.916811
$879$	16.7172	0.563857
$880$	−2.74955	−0.0926874
$881$	9.12038	0.307274	0.153637	−	0.988127i	$-0.450901\pi$
$881$	9.12038	0.307274	0.153637	+	0.988127i	$0.450901\pi$
$882$	6.61835	0.222851
$883$	19.5315	0.657288	0.328644	−	0.944454i	$-0.393408\pi$
$883$	19.5315	0.657288	0.328644	+	0.944454i	$0.393408\pi$
$884$	−3.35086	−0.112702
$885$	−18.4830	−0.621299
$886$	−18.7448	−0.629744
$887$	−20.4935	−0.688103	−0.344051	−	0.938951i	$-0.611799\pi$
$887$	−20.4935	−0.688103	−0.344051	+	0.938951i	$0.611799\pi$
$888$	−4.41556	−0.148176
$889$	−4.78055	−0.160334
$890$	−10.5879	−0.354908
$891$	−28.3319	−0.949155
$892$	−2.73737	−0.0916541
$893$	−14.5060	−0.485426
$894$	−6.11604	−0.204551
$895$	−18.3788	−0.614334
$896$	−3.11075	−0.103923
$897$	13.4356	0.448600
$898$	−32.0617	−1.06991
$899$	−9.75425	−0.325322
$900$	2.47253	0.0824175
$901$	12.4841	0.415906
$902$	31.7592	1.05747
$903$	−1.34663	−0.0448132
$904$	−1.42842	−0.0475084
$905$	4.98907	0.165842
$906$	56.8295	1.88803
$907$	45.0680	1.49646	0.748230	−	0.663439i	$-0.230904\pi$
$907$	45.0680	1.49646	0.748230	+	0.663439i	$0.230904\pi$
$908$	−21.3711	−0.709225
$909$	1.14536	0.0379892
$910$	−3.11075	−0.103120
$911$	42.4496	1.40642	0.703208	−	0.710984i	$-0.251750\pi$
$911$	42.4496	1.40642	0.703208	+	0.710984i	$0.251750\pi$
$912$	8.49873	0.281421
$913$	32.9018	1.08889
$914$	4.54839	0.150447
$915$	−36.0849	−1.19293
$916$	−15.5317	−0.513180
$917$	−9.61373	−0.317473
$918$	4.13477	0.136468
$919$	25.3220	0.835295	0.417648	−	0.908609i	$-0.362855\pi$
$919$	25.3220	0.835295	0.417648	+	0.908609i	$0.362855\pi$
$920$	−5.74331	−0.189351
$921$	54.7336	1.80353
$922$	4.90157	0.161425
$923$	1.92637	0.0634072
$924$	20.0088	0.658241
$925$	1.88752	0.0620613
$926$	−28.3363	−0.931190
$927$	−5.22052	−0.171464
$928$	−9.75425	−0.320199
$929$	−53.9723	−1.77077	−0.885386	−	0.464856i	$-0.846106\pi$
$929$	−53.9723	−1.77077	−0.885386	+	0.464856i	$0.846106\pi$
$930$	2.33934	0.0767101
$931$	−9.72454	−0.318709
$932$	−7.61532	−0.249448
$933$	66.8861	2.18975
$934$	29.9987	0.981588
$935$	−9.21337	−0.301309
$936$	−2.47253	−0.0808170
$937$	−23.4035	−0.764558	−0.382279	−	0.924047i	$-0.624861\pi$
$937$	−23.4035	−0.764558	−0.382279	+	0.924047i	$0.624861\pi$
$938$	40.7246	1.32971
$939$	−17.9683	−0.586373
$940$	−3.99290	−0.130234
$941$	−52.2512	−1.70334	−0.851671	−	0.524077i	$-0.824410\pi$
$941$	−52.2512	−1.70334	−0.851671	+	0.524077i	$0.824410\pi$
$942$	30.0524	0.979159
$943$	66.3391	2.16030
$944$	−7.90093	−0.257153
$945$	3.83849	0.124866
$946$	−0.508807	−0.0165427
$947$	50.1762	1.63051	0.815253	−	0.579105i	$-0.196598\pi$
$947$	50.1762	1.63051	0.815253	+	0.579105i	$0.196598\pi$
$948$	37.5527	1.21965
$949$	−14.2020	−0.461016
$950$	−3.63296	−0.117869
$951$	−46.4290	−1.50556
$952$	−10.4237	−0.337834
$953$	−30.0118	−0.972179	−0.486089	−	0.873909i	$-0.661577\pi$
$953$	−30.0118	−0.972179	−0.486089	+	0.873909i	$0.661577\pi$
$954$	9.21174	0.298241
$955$	16.3834	0.530155
$956$	−18.5072	−0.598565
$957$	62.7407	2.02812
$958$	−35.3442	−1.14192
$959$	53.2362	1.71909
$960$	2.33934	0.0755020
$961$	1.00000	0.0322581
$962$	−1.88752	−0.0608561
$963$	−49.3675	−1.59085
$964$	12.6736	0.408189
$965$	−8.68164	−0.279472
$966$	41.7947	1.34472
$967$	−34.9499	−1.12391	−0.561957	−	0.827167i	$-0.689951\pi$
$967$	−34.9499	−1.12391	−0.561957	+	0.827167i	$0.689951\pi$
$968$	−3.43997	−0.110565
$969$	28.4781	0.914848
$970$	−4.01604	−0.128947
$971$	35.1115	1.12678	0.563391	−	0.826191i	$-0.309497\pi$
$971$	35.1115	1.12678	0.563391	+	0.826191i	$0.309497\pi$
$972$	20.4032	0.654433
$973$	20.5261	0.658035
$974$	−7.22669	−0.231558
$975$	2.33934	0.0749189
$976$	−15.4252	−0.493750
$977$	42.6697	1.36512	0.682562	−	0.730828i	$-0.260866\pi$
$977$	42.6697	1.36512	0.682562	+	0.730828i	$0.260866\pi$
$978$	5.94999	0.190260
$979$	29.1120	0.930425
$980$	−2.67676	−0.0855058
$981$	−16.1810	−0.516621
$982$	−27.9404	−0.891614
$983$	43.2660	1.37997	0.689986	−	0.723823i	$-0.257617\pi$
$983$	43.2660	1.37997	0.689986	+	0.723823i	$0.257617\pi$
$984$	−27.0210	−0.861398
$985$	8.78856	0.280027
$986$	−32.6851	−1.04091
$987$	29.0568	0.924888
$988$	3.63296	0.115580
$989$	−1.06280	−0.0337952
$990$	−6.79834	−0.216065
$991$	−1.99316	−0.0633148	−0.0316574	−	0.999499i	$-0.510079\pi$
$991$	−1.99316	−0.0633148	−0.0316574	+	0.999499i	$0.510079\pi$
$992$	1.00000	0.0317500
$993$	13.9254	0.441909
$994$	5.99244	0.190069
$995$	11.9538	0.378960
$996$	−27.9932	−0.886998
$997$	−50.6652	−1.60458	−0.802292	−	0.596932i	$-0.796386\pi$
$997$	−50.6652	−1.60458	−0.802292	+	0.596932i	$0.796386\pi$
$998$	−13.6185	−0.431085
$999$	2.32909	0.0736893

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.j.1.2	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.j.1.2	✓	7	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.33934 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.33934 q^{6} -3.11075 q^{7} +1.00000 q^{8} +2.47253 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.33934 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.33934 q^{6} -3.11075 q^{7} +1.00000 q^{8} +2.47253 q^{9} -1.00000 q^{10} +2.74955 q^{11} -2.33934 q^{12} -1.00000 q^{13} -3.11075 q^{14} +2.33934 q^{15} +1.00000 q^{16} +3.35086 q^{17} +2.47253 q^{18} -3.63296 q^{19} -1.00000 q^{20} +7.27711 q^{21} +2.74955 q^{22} +5.74331 q^{23} -2.33934 q^{24} +1.00000 q^{25} -1.00000 q^{26} +1.23394 q^{27} -3.11075 q^{28} -9.75425 q^{29} +2.33934 q^{30} +1.00000 q^{31} +1.00000 q^{32} -6.43214 q^{33} +3.35086 q^{34} +3.11075 q^{35} +2.47253 q^{36} +1.88752 q^{37} -3.63296 q^{38} +2.33934 q^{39} -1.00000 q^{40} +11.5507 q^{41} +7.27711 q^{42} -0.185051 q^{43} +2.74955 q^{44} -2.47253 q^{45} +5.74331 q^{46} +3.99290 q^{47} -2.33934 q^{48} +2.67676 q^{49} +1.00000 q^{50} -7.83882 q^{51} -1.00000 q^{52} +3.72564 q^{53} +1.23394 q^{54} -2.74955 q^{55} -3.11075 q^{56} +8.49873 q^{57} -9.75425 q^{58} -7.90093 q^{59} +2.33934 q^{60} -15.4252 q^{61} +1.00000 q^{62} -7.69141 q^{63} +1.00000 q^{64} +1.00000 q^{65} -6.43214 q^{66} -13.0916 q^{67} +3.35086 q^{68} -13.4356 q^{69} +3.11075 q^{70} -1.92637 q^{71} +2.47253 q^{72} +14.2020 q^{73} +1.88752 q^{74} -2.33934 q^{75} -3.63296 q^{76} -8.55316 q^{77} +2.33934 q^{78} -16.0527 q^{79} -1.00000 q^{80} -10.3042 q^{81} +11.5507 q^{82} +11.9663 q^{83} +7.27711 q^{84} -3.35086 q^{85} -0.185051 q^{86} +22.8185 q^{87} +2.74955 q^{88} +10.5879 q^{89} -2.47253 q^{90} +3.11075 q^{91} +5.74331 q^{92} -2.33934 q^{93} +3.99290 q^{94} +3.63296 q^{95} -2.33934 q^{96} +4.01604 q^{97} +2.67676 q^{98} +6.79834 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9}+O(q^{10}) \) 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^7 + 7 * q^8 + 4 * q^9	\( 7 q + 7 q^{2} - 3 q^{3} + 7 q^{4} - 7 q^{5} - 3 q^{6} + 4 q^{7} + 7 q^{8} + 4 q^{9} - 7 q^{10} - 10 q^{11} - 3 q^{12} - 7 q^{13} + 4 q^{14} + 3 q^{15} + 7 q^{16} - 6 q^{17} + 4 q^{18} - 5 q^{19} - 7 q^{20} - 15 q^{21} - 10 q^{22} - 11 q^{23} - 3 q^{24} + 7 q^{25} - 7 q^{26} - 21 q^{27} + 4 q^{28} - 18 q^{29} + 3 q^{30} + 7 q^{31} + 7 q^{32} + 4 q^{33} - 6 q^{34} - 4 q^{35} + 4 q^{36} - 8 q^{37} - 5 q^{38} + 3 q^{39} - 7 q^{40} - 12 q^{41} - 15 q^{42} - 5 q^{43} - 10 q^{44} - 4 q^{45} - 11 q^{46} - 10 q^{47} - 3 q^{48} + 7 q^{49} + 7 q^{50} - 29 q^{51} - 7 q^{52} - 18 q^{53} - 21 q^{54} + 10 q^{55} + 4 q^{56} + 13 q^{57} - 18 q^{58} - 11 q^{59} + 3 q^{60} - 25 q^{61} + 7 q^{62} + 17 q^{63} + 7 q^{64} + 7 q^{65} + 4 q^{66} - 22 q^{67} - 6 q^{68} - 18 q^{69} - 4 q^{70} - 22 q^{71} + 4 q^{72} + 19 q^{73} - 8 q^{74} - 3 q^{75} - 5 q^{76} - 47 q^{77} + 3 q^{78} - 20 q^{79} - 7 q^{80} + 7 q^{81} - 12 q^{82} - 8 q^{83} - 15 q^{84} + 6 q^{85} - 5 q^{86} + 29 q^{87} - 10 q^{88} - 4 q^{89} - 4 q^{90} - 4 q^{91} - 11 q^{92} - 3 q^{93} - 10 q^{94} + 5 q^{95} - 3 q^{96} + 13 q^{97} + 7 q^{98} - 27 q^{99}+O(q^{100}) \) 7 * q + 7 * q^2 - 3 * q^3 + 7 * q^4 - 7 * q^5 - 3 * q^6 + 4 * q^7 + 7 * q^8 + 4 * q^9 - 7 * q^10 - 10 * q^11 - 3 * q^12 - 7 * q^13 + 4 * q^14 + 3 * q^15 + 7 * q^16 - 6 * q^17 + 4 * q^18 - 5 * q^19 - 7 * q^20 - 15 * q^21 - 10 * q^22 - 11 * q^23 - 3 * q^24 + 7 * q^25 - 7 * q^26 - 21 * q^27 + 4 * q^28 - 18 * q^29 + 3 * q^30 + 7 * q^31 + 7 * q^32 + 4 * q^33 - 6 * q^34 - 4 * q^35 + 4 * q^36 - 8 * q^37 - 5 * q^38 + 3 * q^39 - 7 * q^40 - 12 * q^41 - 15 * q^42 - 5 * q^43 - 10 * q^44 - 4 * q^45 - 11 * q^46 - 10 * q^47 - 3 * q^48 + 7 * q^49 + 7 * q^50 - 29 * q^51 - 7 * q^52 - 18 * q^53 - 21 * q^54 + 10 * q^55 + 4 * q^56 + 13 * q^57 - 18 * q^58 - 11 * q^59 + 3 * q^60 - 25 * q^61 + 7 * q^62 + 17 * q^63 + 7 * q^64 + 7 * q^65 + 4 * q^66 - 22 * q^67 - 6 * q^68 - 18 * q^69 - 4 * q^70 - 22 * q^71 + 4 * q^72 + 19 * q^73 - 8 * q^74 - 3 * q^75 - 5 * q^76 - 47 * q^77 + 3 * q^78 - 20 * q^79 - 7 * q^80 + 7 * q^81 - 12 * q^82 - 8 * q^83 - 15 * q^84 + 6 * q^85 - 5 * q^86 + 29 * q^87 - 10 * q^88 - 4 * q^89 - 4 * q^90 - 4 * q^91 - 11 * q^92 - 3 * q^93 - 10 * q^94 + 5 * q^95 - 3 * q^96 + 13 * q^97 + 7 * q^98 - 27 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more