LMFDB - Embedded newform 4026.2.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4026,2,Mod(1,4026)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4026, 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("4026.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4026 = 2 \cdot 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4026.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:	$32.1477718538$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.36234$ of defining polynomial
Character	$\chi$	$=$	4026.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.21831 q^{5} +1.00000 q^{6} -1.29741 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.21831 q^{5} +1.00000 q^{6} -1.29741 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.21831 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.33090 q^{13} -1.29741 q^{14} -2.21831 q^{15} +1.00000 q^{16} +4.16130 q^{17} +1.00000 q^{18} +3.24040 q^{19} -2.21831 q^{20} -1.29741 q^{21} -1.00000 q^{22} -8.08702 q^{23} +1.00000 q^{24} -0.0791023 q^{25} -1.33090 q^{26} +1.00000 q^{27} -1.29741 q^{28} -5.42727 q^{29} -2.21831 q^{30} -3.95438 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.16130 q^{34} +2.87806 q^{35} +1.00000 q^{36} -4.70055 q^{37} +3.24040 q^{38} -1.33090 q^{39} -2.21831 q^{40} -2.74821 q^{41} -1.29741 q^{42} -8.72264 q^{43} -1.00000 q^{44} -2.21831 q^{45} -8.08702 q^{46} +6.82105 q^{47} +1.00000 q^{48} -5.31672 q^{49} -0.0791023 q^{50} +4.16130 q^{51} -1.33090 q^{52} +4.40796 q^{53} +1.00000 q^{54} +2.21831 q^{55} -1.29741 q^{56} +3.24040 q^{57} -5.42727 q^{58} +2.69777 q^{59} -2.21831 q^{60} -1.00000 q^{61} -3.95438 q^{62} -1.29741 q^{63} +1.00000 q^{64} +2.95234 q^{65} -1.00000 q^{66} -6.30533 q^{67} +4.16130 q^{68} -8.08702 q^{69} +2.87806 q^{70} +1.68388 q^{71} +1.00000 q^{72} +11.7232 q^{73} -4.70055 q^{74} -0.0791023 q^{75} +3.24040 q^{76} +1.29741 q^{77} -1.33090 q^{78} -3.47946 q^{79} -2.21831 q^{80} +1.00000 q^{81} -2.74821 q^{82} +6.20692 q^{83} -1.29741 q^{84} -9.23105 q^{85} -8.72264 q^{86} -5.42727 q^{87} -1.00000 q^{88} -7.66767 q^{89} -2.21831 q^{90} +1.72672 q^{91} -8.08702 q^{92} -3.95438 q^{93} +6.82105 q^{94} -7.18821 q^{95} +1.00000 q^{96} -14.3319 q^{97} -5.31672 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 6 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{12} - 5 q^{13} - 6 q^{14} - 4 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 8 q^{19} - 4 q^{20} - 6 q^{21} - 4 q^{22} - 7 q^{23} + 4 q^{24} - 6 q^{25} - 5 q^{26} + 4 q^{27} - 6 q^{28} - 4 q^{29} - 4 q^{30} - 9 q^{31} + 4 q^{32} - 4 q^{33} - 10 q^{34} - q^{35} + 4 q^{36} - 11 q^{37} - 8 q^{38} - 5 q^{39} - 4 q^{40} - 17 q^{41} - 6 q^{42} - 11 q^{43} - 4 q^{44} - 4 q^{45} - 7 q^{46} - 7 q^{47} + 4 q^{48} - 6 q^{49} - 6 q^{50} - 10 q^{51} - 5 q^{52} + 16 q^{53} + 4 q^{54} + 4 q^{55} - 6 q^{56} - 8 q^{57} - 4 q^{58} + 3 q^{59} - 4 q^{60} - 4 q^{61} - 9 q^{62} - 6 q^{63} + 4 q^{64} - 2 q^{65} - 4 q^{66} + 5 q^{67} - 10 q^{68} - 7 q^{69} - q^{70} - 10 q^{71} + 4 q^{72} - 9 q^{73} - 11 q^{74} - 6 q^{75} - 8 q^{76} + 6 q^{77} - 5 q^{78} - 11 q^{79} - 4 q^{80} + 4 q^{81} - 17 q^{82} + 5 q^{83} - 6 q^{84} - 8 q^{85} - 11 q^{86} - 4 q^{87} - 4 q^{88} + 8 q^{89} - 4 q^{90} - 3 q^{91} - 7 q^{92} - 9 q^{93} - 7 q^{94} + 7 q^{95} + 4 q^{96} - 12 q^{97} - 6 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 6 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^11 + 4 * q^12 - 5 * q^13 - 6 * q^14 - 4 * q^15 + 4 * q^16 - 10 * q^17 + 4 * q^18 - 8 * q^19 - 4 * q^20 - 6 * q^21 - 4 * q^22 - 7 * q^23 + 4 * q^24 - 6 * q^25 - 5 * q^26 + 4 * q^27 - 6 * q^28 - 4 * q^29 - 4 * q^30 - 9 * q^31 + 4 * q^32 - 4 * q^33 - 10 * q^34 - q^35 + 4 * q^36 - 11 * q^37 - 8 * q^38 - 5 * q^39 - 4 * q^40 - 17 * q^41 - 6 * q^42 - 11 * q^43 - 4 * q^44 - 4 * q^45 - 7 * q^46 - 7 * q^47 + 4 * q^48 - 6 * q^49 - 6 * q^50 - 10 * q^51 - 5 * q^52 + 16 * q^53 + 4 * q^54 + 4 * q^55 - 6 * q^56 - 8 * q^57 - 4 * q^58 + 3 * q^59 - 4 * q^60 - 4 * q^61 - 9 * q^62 - 6 * q^63 + 4 * q^64 - 2 * q^65 - 4 * q^66 + 5 * q^67 - 10 * q^68 - 7 * q^69 - q^70 - 10 * q^71 + 4 * q^72 - 9 * q^73 - 11 * q^74 - 6 * q^75 - 8 * q^76 + 6 * q^77 - 5 * q^78 - 11 * q^79 - 4 * q^80 + 4 * q^81 - 17 * q^82 + 5 * q^83 - 6 * q^84 - 8 * q^85 - 11 * q^86 - 4 * q^87 - 4 * q^88 + 8 * q^89 - 4 * q^90 - 3 * q^91 - 7 * q^92 - 9 * q^93 - 7 * q^94 + 7 * q^95 + 4 * q^96 - 12 * q^97 - 6 * q^98 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.21831	−0.992058	−0.496029	−	0.868306i	$-0.665209\pi$
$5$	−2.21831	−0.992058	−0.496029	+	0.868306i	$0.665209\pi$
$6$	1.00000	0.408248
$7$	−1.29741	−0.490376	−0.245188	−	0.969476i	$-0.578850\pi$
$7$	−1.29741	−0.490376	−0.245188	+	0.969476i	$0.578850\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.21831	−0.701491
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−1.33090	−0.369124	−0.184562	−	0.982821i	$-0.559087\pi$
$13$	−1.33090	−0.369124	−0.184562	+	0.982821i	$0.559087\pi$
$14$	−1.29741	−0.346748
$15$	−2.21831	−0.572765
$16$	1.00000	0.250000
$17$	4.16130	1.00926	0.504632	−	0.863335i	$-0.331628\pi$
$17$	4.16130	1.00926	0.504632	+	0.863335i	$0.331628\pi$
$18$	1.00000	0.235702
$19$	3.24040	0.743399	0.371700	−	0.928353i	$-0.378775\pi$
$19$	3.24040	0.743399	0.371700	+	0.928353i	$0.378775\pi$
$20$	−2.21831	−0.496029
$21$	−1.29741	−0.283118
$22$	−1.00000	−0.213201
$23$	−8.08702	−1.68626	−0.843130	−	0.537710i	$-0.819290\pi$
$23$	−8.08702	−1.68626	−0.843130	+	0.537710i	$0.819290\pi$
$24$	1.00000	0.204124
$25$	−0.0791023	−0.0158205
$26$	−1.33090	−0.261010
$27$	1.00000	0.192450
$28$	−1.29741	−0.245188
$29$	−5.42727	−1.00782	−0.503909	−	0.863757i	$-0.668105\pi$
$29$	−5.42727	−1.00782	−0.503909	+	0.863757i	$0.668105\pi$
$30$	−2.21831	−0.405006
$31$	−3.95438	−0.710228	−0.355114	−	0.934823i	$-0.615558\pi$
$31$	−3.95438	−0.710228	−0.355114	+	0.934823i	$0.615558\pi$
$32$	1.00000	0.176777
$33$	−1.00000	−0.174078
$34$	4.16130	0.713657
$35$	2.87806	0.486481
$36$	1.00000	0.166667
$37$	−4.70055	−0.772765	−0.386383	−	0.922339i	$-0.626276\pi$
$37$	−4.70055	−0.772765	−0.386383	+	0.922339i	$0.626276\pi$
$38$	3.24040	0.525662
$39$	−1.33090	−0.213114
$40$	−2.21831	−0.350746
$41$	−2.74821	−0.429198	−0.214599	−	0.976702i	$-0.568844\pi$
$41$	−2.74821	−0.429198	−0.214599	+	0.976702i	$0.568844\pi$
$42$	−1.29741	−0.200195
$43$	−8.72264	−1.33019	−0.665095	−	0.746759i	$-0.731609\pi$
$43$	−8.72264	−1.33019	−0.665095	+	0.746759i	$0.731609\pi$
$44$	−1.00000	−0.150756
$45$	−2.21831	−0.330686
$46$	−8.08702	−1.19237
$47$	6.82105	0.994953	0.497476	−	0.867478i	$-0.334260\pi$
$47$	6.82105	0.994953	0.497476	+	0.867478i	$0.334260\pi$
$48$	1.00000	0.144338
$49$	−5.31672	−0.759532
$50$	−0.0791023	−0.0111868
$51$	4.16130	0.582698
$52$	−1.33090	−0.184562
$53$	4.40796	0.605480	0.302740	−	0.953073i	$-0.402099\pi$
$53$	4.40796	0.605480	0.302740	+	0.953073i	$0.402099\pi$
$54$	1.00000	0.136083
$55$	2.21831	0.299117
$56$	−1.29741	−0.173374
$57$	3.24040	0.429202
$58$	−5.42727	−0.712635
$59$	2.69777	0.351219	0.175610	−	0.984460i	$-0.443810\pi$
$59$	2.69777	0.351219	0.175610	+	0.984460i	$0.443810\pi$
$60$	−2.21831	−0.286383
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−3.95438	−0.502207
$63$	−1.29741	−0.163459
$64$	1.00000	0.125000
$65$	2.95234	0.366193
$66$	−1.00000	−0.123091
$67$	−6.30533	−0.770319	−0.385159	−	0.922850i	$-0.625853\pi$
$67$	−6.30533	−0.770319	−0.385159	+	0.922850i	$0.625853\pi$
$68$	4.16130	0.504632
$69$	−8.08702	−0.973563
$70$	2.87806	0.343994
$71$	1.68388	0.199840	0.0999202	−	0.994995i	$-0.468141\pi$
$71$	1.68388	0.199840	0.0999202	+	0.994995i	$0.468141\pi$
$72$	1.00000	0.117851
$73$	11.7232	1.37210	0.686051	−	0.727553i	$-0.259343\pi$
$73$	11.7232	1.37210	0.686051	+	0.727553i	$0.259343\pi$
$74$	−4.70055	−0.546427
$75$	−0.0791023	−0.00913394
$76$	3.24040	0.371700
$77$	1.29741	0.147854
$78$	−1.33090	−0.150694
$79$	−3.47946	−0.391469	−0.195735	−	0.980657i	$-0.562709\pi$
$79$	−3.47946	−0.391469	−0.195735	+	0.980657i	$0.562709\pi$
$80$	−2.21831	−0.248015
$81$	1.00000	0.111111
$82$	−2.74821	−0.303489
$83$	6.20692	0.681298	0.340649	−	0.940191i	$-0.389353\pi$
$83$	6.20692	0.681298	0.340649	+	0.940191i	$0.389353\pi$
$84$	−1.29741	−0.141559
$85$	−9.23105	−1.00125
$86$	−8.72264	−0.940586
$87$	−5.42727	−0.581864
$88$	−1.00000	−0.106600
$89$	−7.66767	−0.812771	−0.406386	−	0.913702i	$-0.633211\pi$
$89$	−7.66767	−0.812771	−0.406386	+	0.913702i	$0.633211\pi$
$90$	−2.21831	−0.233830
$91$	1.72672	0.181010
$92$	−8.08702	−0.843130
$93$	−3.95438	−0.410050
$94$	6.82105	0.703538
$95$	−7.18821	−0.737495
$96$	1.00000	0.102062
$97$	−14.3319	−1.45519	−0.727594	−	0.686007i	$-0.759362\pi$
$97$	−14.3319	−1.45519	−0.727594	+	0.686007i	$0.759362\pi$
$98$	−5.31672	−0.537070
$99$	−1.00000	−0.100504
$100$	−0.0791023	−0.00791023
$101$	−4.25458	−0.423346	−0.211673	−	0.977341i	$-0.567891\pi$
$101$	−4.25458	−0.423346	−0.211673	+	0.977341i	$0.567891\pi$
$102$	4.16130	0.412030
$103$	−8.87602	−0.874580	−0.437290	−	0.899320i	$-0.644062\pi$
$103$	−8.87602	−0.874580	−0.437290	+	0.899320i	$0.644062\pi$
$104$	−1.33090	−0.130505
$105$	2.87806	0.280870
$106$	4.40796	0.428139
$107$	−3.97104	−0.383895	−0.191948	−	0.981405i	$-0.561480\pi$
$107$	−3.97104	−0.383895	−0.191948	+	0.981405i	$0.561480\pi$
$108$	1.00000	0.0962250
$109$	−5.84911	−0.560243	−0.280121	−	0.959965i	$-0.590375\pi$
$109$	−5.84911	−0.560243	−0.280121	+	0.959965i	$0.590375\pi$
$110$	2.21831	0.211508
$111$	−4.70055	−0.446156
$112$	−1.29741	−0.122594
$113$	4.16816	0.392108	0.196054	−	0.980593i	$-0.437187\pi$
$113$	4.16816	0.392108	0.196054	+	0.980593i	$0.437187\pi$
$114$	3.24040	0.303491
$115$	17.9395	1.67287
$116$	−5.42727	−0.503909
$117$	−1.33090	−0.123041
$118$	2.69777	0.248349
$119$	−5.39892	−0.494918
$120$	−2.21831	−0.202503
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	−2.74821	−0.247797
$124$	−3.95438	−0.355114
$125$	11.2670	1.00775
$126$	−1.29741	−0.115583
$127$	1.45532	0.129139	0.0645695	−	0.997913i	$-0.479433\pi$
$127$	1.45532	0.129139	0.0645695	+	0.997913i	$0.479433\pi$
$128$	1.00000	0.0883883
$129$	−8.72264	−0.767985
$130$	2.95234	0.258937
$131$	−14.1948	−1.24020	−0.620102	−	0.784521i	$-0.712909\pi$
$131$	−14.1948	−1.24020	−0.620102	+	0.784521i	$0.712909\pi$
$132$	−1.00000	−0.0870388
$133$	−4.20414	−0.364545
$134$	−6.30533	−0.544697
$135$	−2.21831	−0.190922
$136$	4.16130	0.356828
$137$	−17.9066	−1.52987	−0.764934	−	0.644109i	$-0.777228\pi$
$137$	−17.9066	−1.52987	−0.764934	+	0.644109i	$0.777228\pi$
$138$	−8.08702	−0.688413
$139$	14.3033	1.21319	0.606595	−	0.795011i	$-0.292535\pi$
$139$	14.3033	1.21319	0.606595	+	0.795011i	$0.292535\pi$
$140$	2.87806	0.243241
$141$	6.82105	0.574436
$142$	1.68388	0.141308
$143$	1.33090	0.111295
$144$	1.00000	0.0833333
$145$	12.0394	0.999814
$146$	11.7232	0.970223
$147$	−5.31672	−0.438516
$148$	−4.70055	−0.386383
$149$	−13.8867	−1.13764	−0.568822	−	0.822461i	$-0.692601\pi$
$149$	−13.8867	−1.13764	−0.568822	+	0.822461i	$0.692601\pi$
$150$	−0.0791023	−0.00645867
$151$	−8.32260	−0.677283	−0.338642	−	0.940915i	$-0.609967\pi$
$151$	−8.32260	−0.677283	−0.338642	+	0.940915i	$0.609967\pi$
$152$	3.24040	0.262831
$153$	4.16130	0.336421
$154$	1.29741	0.104548
$155$	8.77204	0.704588
$156$	−1.33090	−0.106557
$157$	−21.5288	−1.71818	−0.859092	−	0.511822i	$-0.828971\pi$
$157$	−21.5288	−1.71818	−0.859092	+	0.511822i	$0.828971\pi$
$158$	−3.47946	−0.276811
$159$	4.40796	0.349574
$160$	−2.21831	−0.175373
$161$	10.4922	0.826901
$162$	1.00000	0.0785674
$163$	−19.9750	−1.56457	−0.782283	−	0.622923i	$-0.785945\pi$
$163$	−19.9750	−1.56457	−0.782283	+	0.622923i	$0.785945\pi$
$164$	−2.74821	−0.214599
$165$	2.21831	0.172695
$166$	6.20692	0.481750
$167$	−9.55055	−0.739044	−0.369522	−	0.929222i	$-0.620479\pi$
$167$	−9.55055	−0.739044	−0.369522	+	0.929222i	$0.620479\pi$
$168$	−1.29741	−0.100098
$169$	−11.2287	−0.863747
$170$	−9.23105	−0.707989
$171$	3.24040	0.247800
$172$	−8.72264	−0.665095
$173$	−5.10467	−0.388101	−0.194050	−	0.980992i	$-0.562163\pi$
$173$	−5.10467	−0.388101	−0.194050	+	0.980992i	$0.562163\pi$
$174$	−5.42727	−0.411440
$175$	0.102628	0.00775797
$176$	−1.00000	−0.0753778
$177$	2.69777	0.202776
$178$	−7.66767	−0.574716
$179$	26.0759	1.94901	0.974504	−	0.224372i	$-0.0720330\pi$
$179$	26.0759	1.94901	0.974504	+	0.224372i	$0.0720330\pi$
$180$	−2.21831	−0.165343
$181$	15.3039	1.13753	0.568765	−	0.822500i	$-0.307421\pi$
$181$	15.3039	1.13753	0.568765	+	0.822500i	$0.307421\pi$
$182$	1.72672	0.127993
$183$	−1.00000	−0.0739221
$184$	−8.08702	−0.596183
$185$	10.4273	0.766628
$186$	−3.95438	−0.289949
$187$	−4.16130	−0.304304
$188$	6.82105	0.497476
$189$	−1.29741	−0.0943728
$190$	−7.18821	−0.521488
$191$	2.34364	0.169580	0.0847898	−	0.996399i	$-0.472978\pi$
$191$	2.34364	0.169580	0.0847898	+	0.996399i	$0.472978\pi$
$192$	1.00000	0.0721688
$193$	4.76065	0.342679	0.171340	−	0.985212i	$-0.445190\pi$
$193$	4.76065	0.342679	0.171340	+	0.985212i	$0.445190\pi$
$194$	−14.3319	−1.02897
$195$	2.95234	0.211422
$196$	−5.31672	−0.379766
$197$	−3.20618	−0.228431	−0.114215	−	0.993456i	$-0.536435\pi$
$197$	−3.20618	−0.228431	−0.114215	+	0.993456i	$0.536435\pi$
$198$	−1.00000	−0.0710669
$199$	0.688414	0.0488004	0.0244002	−	0.999702i	$-0.492232\pi$
$199$	0.688414	0.0488004	0.0244002	+	0.999702i	$0.492232\pi$
$200$	−0.0791023	−0.00559338
$201$	−6.30533	−0.444744
$202$	−4.25458	−0.299351
$203$	7.04140	0.494210
$204$	4.16130	0.291349
$205$	6.09637	0.425789
$206$	−8.87602	−0.618422
$207$	−8.08702	−0.562087
$208$	−1.33090	−0.0922811
$209$	−3.24040	−0.224143
$210$	2.87806	0.198605
$211$	18.3668	1.26442	0.632211	−	0.774797i	$-0.282148\pi$
$211$	18.3668	1.26442	0.632211	+	0.774797i	$0.282148\pi$
$212$	4.40796	0.302740
$213$	1.68388	0.115378
$214$	−3.97104	−0.271455
$215$	19.3495	1.31963
$216$	1.00000	0.0680414
$217$	5.13046	0.348279
$218$	−5.84911	−0.396151
$219$	11.7232	0.792184
$220$	2.21831	0.149558
$221$	−5.53826	−0.372544
$222$	−4.70055	−0.315480
$223$	10.6103	0.710521	0.355261	−	0.934767i	$-0.384392\pi$
$223$	10.6103	0.710521	0.355261	+	0.934767i	$0.384392\pi$
$224$	−1.29741	−0.0866870
$225$	−0.0791023	−0.00527349
$226$	4.16816	0.277262
$227$	2.93332	0.194692	0.0973458	−	0.995251i	$-0.468965\pi$
$227$	2.93332	0.194692	0.0973458	+	0.995251i	$0.468965\pi$
$228$	3.24040	0.214601
$229$	0.923992	0.0610591	0.0305296	−	0.999534i	$-0.490281\pi$
$229$	0.923992	0.0610591	0.0305296	+	0.999534i	$0.490281\pi$
$230$	17.9395	1.18290
$231$	1.29741	0.0853634
$232$	−5.42727	−0.356318
$233$	26.1457	1.71286	0.856431	−	0.516262i	$-0.172677\pi$
$233$	26.1457	1.71286	0.856431	+	0.516262i	$0.172677\pi$
$234$	−1.33090	−0.0870034
$235$	−15.1312	−0.987051
$236$	2.69777	0.175610
$237$	−3.47946	−0.226015
$238$	−5.39892	−0.349960
$239$	−16.2303	−1.04985	−0.524926	−	0.851148i	$-0.675907\pi$
$239$	−16.2303	−1.04985	−0.524926	+	0.851148i	$0.675907\pi$
$240$	−2.21831	−0.143191
$241$	−14.5962	−0.940222	−0.470111	−	0.882607i	$-0.655786\pi$
$241$	−14.5962	−0.940222	−0.470111	+	0.882607i	$0.655786\pi$
$242$	1.00000	0.0642824
$243$	1.00000	0.0641500
$244$	−1.00000	−0.0640184
$245$	11.7941	0.753500
$246$	−2.74821	−0.175219
$247$	−4.31264	−0.274407
$248$	−3.95438	−0.251104
$249$	6.20692	0.393347
$250$	11.2670	0.712589
$251$	3.95921	0.249903	0.124951	−	0.992163i	$-0.460123\pi$
$251$	3.95921	0.249903	0.124951	+	0.992163i	$0.460123\pi$
$252$	−1.29741	−0.0817293
$253$	8.08702	0.508427
$254$	1.45532	0.0913151
$255$	−9.23105	−0.578071
$256$	1.00000	0.0625000
$257$	13.6532	0.851662	0.425831	−	0.904803i	$-0.359982\pi$
$257$	13.6532	0.851662	0.425831	+	0.904803i	$0.359982\pi$
$258$	−8.72264	−0.543048
$259$	6.09855	0.378945
$260$	2.95234	0.183096
$261$	−5.42727	−0.335939
$262$	−14.1948	−0.876957
$263$	28.9741	1.78662	0.893308	−	0.449444i	$-0.148378\pi$
$263$	28.9741	1.78662	0.893308	+	0.449444i	$0.148378\pi$
$264$	−1.00000	−0.0615457
$265$	−9.77821	−0.600671
$266$	−4.20414	−0.257772
$267$	−7.66767	−0.469254
$268$	−6.30533	−0.385159
$269$	2.95409	0.180114	0.0900570	−	0.995937i	$-0.471295\pi$
$269$	2.95409	0.180114	0.0900570	+	0.995937i	$0.471295\pi$
$270$	−2.21831	−0.135002
$271$	−27.1972	−1.65211	−0.826056	−	0.563589i	$-0.809420\pi$
$271$	−27.1972	−1.65211	−0.826056	+	0.563589i	$0.809420\pi$
$272$	4.16130	0.252316
$273$	1.72672	0.104506
$274$	−17.9066	−1.08178
$275$	0.0791023	0.00477005
$276$	−8.08702	−0.486781
$277$	6.27258	0.376883	0.188442	−	0.982084i	$-0.439656\pi$
$277$	6.27258	0.376883	0.188442	+	0.982084i	$0.439656\pi$
$278$	14.3033	0.857854
$279$	−3.95438	−0.236743
$280$	2.87806	0.171997
$281$	26.1569	1.56039	0.780196	−	0.625536i	$-0.215120\pi$
$281$	26.1569	1.56039	0.780196	+	0.625536i	$0.215120\pi$
$282$	6.82105	0.406188
$283$	−1.01592	−0.0603903	−0.0301951	−	0.999544i	$-0.509613\pi$
$283$	−1.01592	−0.0603903	−0.0301951	+	0.999544i	$0.509613\pi$
$284$	1.68388	0.0999202
$285$	−7.18821	−0.425793
$286$	1.33090	0.0786976
$287$	3.56555	0.210468
$288$	1.00000	0.0589256
$289$	0.316409	0.0186123
$290$	12.0394	0.706976
$291$	−14.3319	−0.840154
$292$	11.7232	0.686051
$293$	14.8428	0.867128	0.433564	−	0.901123i	$-0.357256\pi$
$293$	14.8428	0.867128	0.433564	+	0.901123i	$0.357256\pi$
$294$	−5.31672	−0.310078
$295$	−5.98448	−0.348430
$296$	−4.70055	−0.273214
$297$	−1.00000	−0.0580259
$298$	−13.8867	−0.804436
$299$	10.7630	0.622440
$300$	−0.0791023	−0.00456697
$301$	11.3169	0.652293
$302$	−8.32260	−0.478912
$303$	−4.25458	−0.244419
$304$	3.24040	0.185850
$305$	2.21831	0.127020
$306$	4.16130	0.237886
$307$	17.1047	0.976219	0.488109	−	0.872782i	$-0.337687\pi$
$307$	17.1047	0.976219	0.488109	+	0.872782i	$0.337687\pi$
$308$	1.29741	0.0739269
$309$	−8.87602	−0.504939
$310$	8.77204	0.498219
$311$	−5.21205	−0.295548	−0.147774	−	0.989021i	$-0.547211\pi$
$311$	−5.21205	−0.295548	−0.147774	+	0.989021i	$0.547211\pi$
$312$	−1.33090	−0.0753472
$313$	−25.5057	−1.44167	−0.720833	−	0.693109i	$-0.756241\pi$
$313$	−25.5057	−1.44167	−0.720833	+	0.693109i	$0.756241\pi$
$314$	−21.5288	−1.21494
$315$	2.87806	0.162160
$316$	−3.47946	−0.195735
$317$	3.66571	0.205887	0.102944	−	0.994687i	$-0.467174\pi$
$317$	3.66571	0.205887	0.102944	+	0.994687i	$0.467174\pi$
$318$	4.40796	0.247186
$319$	5.42727	0.303869
$320$	−2.21831	−0.124007
$321$	−3.97104	−0.221642
$322$	10.4922	0.584707
$323$	13.4843	0.750285
$324$	1.00000	0.0555556
$325$	0.105277	0.00583972
$326$	−19.9750	−1.10632
$327$	−5.84911	−0.323456
$328$	−2.74821	−0.151744
$329$	−8.84971	−0.487900
$330$	2.21831	0.122114
$331$	26.8984	1.47847	0.739236	−	0.673447i	$-0.235187\pi$
$331$	26.8984	1.47847	0.739236	+	0.673447i	$0.235187\pi$
$332$	6.20692	0.340649
$333$	−4.70055	−0.257588
$334$	−9.55055	−0.522583
$335$	13.9872	0.764201
$336$	−1.29741	−0.0707796
$337$	34.4013	1.87396	0.936980	−	0.349384i	$-0.113609\pi$
$337$	34.4013	1.87396	0.936980	+	0.349384i	$0.113609\pi$
$338$	−11.2287	−0.610762
$339$	4.16816	0.226384
$340$	−9.23105	−0.500624
$341$	3.95438	0.214142
$342$	3.24040	0.175221
$343$	15.9799	0.862831
$344$	−8.72264	−0.470293
$345$	17.9395	0.965831
$346$	−5.10467	−0.274429
$347$	−33.0472	−1.77407	−0.887033	−	0.461706i	$-0.847238\pi$
$347$	−33.0472	−1.77407	−0.887033	+	0.461706i	$0.847238\pi$
$348$	−5.42727	−0.290932
$349$	−23.2016	−1.24196	−0.620978	−	0.783828i	$-0.713264\pi$
$349$	−23.2016	−1.24196	−0.620978	+	0.783828i	$0.713264\pi$
$350$	0.102628	0.00548571
$351$	−1.33090	−0.0710380
$352$	−1.00000	−0.0533002
$353$	20.6276	1.09790	0.548949	−	0.835856i	$-0.315028\pi$
$353$	20.6276	1.09790	0.548949	+	0.835856i	$0.315028\pi$
$354$	2.69777	0.143385
$355$	−3.73538	−0.198253
$356$	−7.66767	−0.406386
$357$	−5.39892	−0.285741
$358$	26.0759	1.37816
$359$	−18.8535	−0.995052	−0.497526	−	0.867449i	$-0.665758\pi$
$359$	−18.8535	−0.995052	−0.497526	+	0.867449i	$0.665758\pi$
$360$	−2.21831	−0.116915
$361$	−8.49980	−0.447358
$362$	15.3039	0.804355
$363$	1.00000	0.0524864
$364$	1.72672	0.0905048
$365$	−26.0058	−1.36121
$366$	−1.00000	−0.0522708
$367$	−17.2676	−0.901363	−0.450681	−	0.892685i	$-0.648819\pi$
$367$	−17.2676	−0.901363	−0.450681	+	0.892685i	$0.648819\pi$
$368$	−8.08702	−0.421565
$369$	−2.74821	−0.143066
$370$	10.4273	0.542088
$371$	−5.71894	−0.296912
$372$	−3.95438	−0.205025
$373$	−10.9993	−0.569522	−0.284761	−	0.958598i	$-0.591914\pi$
$373$	−10.9993	−0.569522	−0.284761	+	0.958598i	$0.591914\pi$
$374$	−4.16130	−0.215176
$375$	11.2670	0.581826
$376$	6.82105	0.351769
$377$	7.22313	0.372010
$378$	−1.29741	−0.0667317
$379$	−6.90354	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$379$	−6.90354	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$380$	−7.18821	−0.368748
$381$	1.45532	0.0745584
$382$	2.34364	0.119911
$383$	5.17577	0.264469	0.132235	−	0.991218i	$-0.457785\pi$
$383$	5.17577	0.264469	0.132235	+	0.991218i	$0.457785\pi$
$384$	1.00000	0.0510310
$385$	−2.87806	−0.146680
$386$	4.76065	0.242311
$387$	−8.72264	−0.443397
$388$	−14.3319	−0.727594
$389$	−12.6327	−0.640506	−0.320253	−	0.947332i	$-0.603768\pi$
$389$	−12.6327	−0.640506	−0.320253	+	0.947332i	$0.603768\pi$
$390$	2.95234	0.149498
$391$	−33.6525	−1.70188
$392$	−5.31672	−0.268535
$393$	−14.1948	−0.716032
$394$	−3.20618	−0.161525
$395$	7.71851	0.388360
$396$	−1.00000	−0.0502519
$397$	32.2525	1.61870	0.809352	−	0.587323i	$-0.199818\pi$
$397$	32.2525	1.61870	0.809352	+	0.587323i	$0.199818\pi$
$398$	0.688414	0.0345071
$399$	−4.20414	−0.210470
$400$	−0.0791023	−0.00395511
$401$	10.0038	0.499568	0.249784	−	0.968302i	$-0.419640\pi$
$401$	10.0038	0.499568	0.249784	+	0.968302i	$0.419640\pi$
$402$	−6.30533	−0.314481
$403$	5.26287	0.262162
$404$	−4.25458	−0.211673
$405$	−2.21831	−0.110229
$406$	7.04140	0.349459
$407$	4.70055	0.232997
$408$	4.16130	0.206015
$409$	10.5928	0.523780	0.261890	−	0.965098i	$-0.415654\pi$
$409$	10.5928	0.523780	0.261890	+	0.965098i	$0.415654\pi$
$410$	6.09637	0.301078
$411$	−17.9066	−0.883269
$412$	−8.87602	−0.437290
$413$	−3.50011	−0.172229
$414$	−8.08702	−0.397455
$415$	−13.7689	−0.675887
$416$	−1.33090	−0.0652526
$417$	14.3033	0.700435
$418$	−3.24040	−0.158493
$419$	31.1754	1.52302	0.761509	−	0.648155i	$-0.224459\pi$
$419$	31.1754	1.52302	0.761509	+	0.648155i	$0.224459\pi$
$420$	2.87806	0.140435
$421$	26.7108	1.30181	0.650903	−	0.759161i	$-0.274391\pi$
$421$	26.7108	1.30181	0.650903	+	0.759161i	$0.274391\pi$
$422$	18.3668	0.894081
$423$	6.82105	0.331651
$424$	4.40796	0.214069
$425$	−0.329168	−0.0159670
$426$	1.68388	0.0815845
$427$	1.29741	0.0627862
$428$	−3.97104	−0.191948
$429$	1.33090	0.0642563
$430$	19.3495	0.933116
$431$	36.5264	1.75941	0.879706	−	0.475518i	$-0.157739\pi$
$431$	36.5264	1.75941	0.879706	+	0.475518i	$0.157739\pi$
$432$	1.00000	0.0481125
$433$	31.0106	1.49027	0.745137	−	0.666911i	$-0.232384\pi$
$433$	31.0106	1.49027	0.745137	+	0.666911i	$0.232384\pi$
$434$	5.13046	0.246270
$435$	12.0394	0.577243
$436$	−5.84911	−0.280121
$437$	−26.2052	−1.25356
$438$	11.7232	0.560158
$439$	21.0853	1.00635	0.503173	−	0.864186i	$-0.332166\pi$
$439$	21.0853	1.00635	0.503173	+	0.864186i	$0.332166\pi$
$440$	2.21831	0.105754
$441$	−5.31672	−0.253177
$442$	−5.53826	−0.263428
$443$	−15.7813	−0.749793	−0.374896	−	0.927067i	$-0.622322\pi$
$443$	−15.7813	−0.749793	−0.374896	+	0.927067i	$0.622322\pi$
$444$	−4.70055	−0.223078
$445$	17.0093	0.806316
$446$	10.6103	0.502414
$447$	−13.8867	−0.656819
$448$	−1.29741	−0.0612970
$449$	11.3471	0.535503	0.267752	−	0.963488i	$-0.413719\pi$
$449$	11.3471	0.535503	0.267752	+	0.963488i	$0.413719\pi$
$450$	−0.0791023	−0.00372892
$451$	2.74821	0.129408
$452$	4.16816	0.196054
$453$	−8.32260	−0.391030
$454$	2.93332	0.137668
$455$	−3.83040	−0.179572
$456$	3.24040	0.151746
$457$	29.8176	1.39481	0.697404	−	0.716679i	$-0.254339\pi$
$457$	29.8176	1.39481	0.697404	+	0.716679i	$0.254339\pi$
$458$	0.923992	0.0431753
$459$	4.16130	0.194233
$460$	17.9395	0.836434
$461$	30.5005	1.42055	0.710274	−	0.703925i	$-0.248571\pi$
$461$	30.5005	1.42055	0.710274	+	0.703925i	$0.248571\pi$
$462$	1.29741	0.0603611
$463$	−0.350340	−0.0162817	−0.00814083	−	0.999967i	$-0.502591\pi$
$463$	−0.350340	−0.0162817	−0.00814083	+	0.999967i	$0.502591\pi$
$464$	−5.42727	−0.251955
$465$	8.77204	0.406794
$466$	26.1457	1.21118
$467$	35.1893	1.62837	0.814183	−	0.580609i	$-0.197185\pi$
$467$	35.1893	1.62837	0.814183	+	0.580609i	$0.197185\pi$
$468$	−1.33090	−0.0615207
$469$	8.18061	0.377745
$470$	−15.1312	−0.697950
$471$	−21.5288	−0.991994
$472$	2.69777	0.124175
$473$	8.72264	0.401067
$474$	−3.47946	−0.159817
$475$	−0.256323	−0.0117609
$476$	−5.39892	−0.247459
$477$	4.40796	0.201827
$478$	−16.2303	−0.742357
$479$	6.66767	0.304654	0.152327	−	0.988330i	$-0.451323\pi$
$479$	6.66767	0.304654	0.152327	+	0.988330i	$0.451323\pi$
$480$	−2.21831	−0.101252
$481$	6.25594	0.285246
$482$	−14.5962	−0.664837
$483$	10.4922	0.477411
$484$	1.00000	0.0454545
$485$	31.7927	1.44363
$486$	1.00000	0.0453609
$487$	−6.32886	−0.286788	−0.143394	−	0.989666i	$-0.545802\pi$
$487$	−6.32886	−0.286788	−0.143394	+	0.989666i	$0.545802\pi$
$488$	−1.00000	−0.0452679
$489$	−19.9750	−0.903303
$490$	11.7941	0.532805
$491$	7.91751	0.357312	0.178656	−	0.983912i	$-0.442825\pi$
$491$	7.91751	0.357312	0.178656	+	0.983912i	$0.442825\pi$
$492$	−2.74821	−0.123899
$493$	−22.5845	−1.01715
$494$	−4.31264	−0.194035
$495$	2.21831	0.0997056
$496$	−3.95438	−0.177557
$497$	−2.18469	−0.0979968
$498$	6.20692	0.278139
$499$	13.1675	0.589457	0.294728	−	0.955581i	$-0.404771\pi$
$499$	13.1675	0.589457	0.294728	+	0.955581i	$0.404771\pi$
$500$	11.2670	0.503877
$501$	−9.55055	−0.426687
$502$	3.95921	0.176708
$503$	−22.1954	−0.989644	−0.494822	−	0.868994i	$-0.664767\pi$
$503$	−22.1954	−0.989644	−0.494822	+	0.868994i	$0.664767\pi$
$504$	−1.29741	−0.0577913
$505$	9.43797	0.419984
$506$	8.08702	0.359512
$507$	−11.2287	−0.498685
$508$	1.45532	0.0645695
$509$	9.73991	0.431714	0.215857	−	0.976425i	$-0.430746\pi$
$509$	9.73991	0.431714	0.215857	+	0.976425i	$0.430746\pi$
$510$	−9.23105	−0.408758
$511$	−15.2099	−0.672845
$512$	1.00000	0.0441942
$513$	3.24040	0.143067
$514$	13.6532	0.602216
$515$	19.6898	0.867635
$516$	−8.72264	−0.383993
$517$	−6.82105	−0.299989
$518$	6.09855	0.267955
$519$	−5.10467	−0.224070
$520$	2.95234	0.129469
$521$	−7.33195	−0.321219	−0.160609	−	0.987018i	$-0.551346\pi$
$521$	−7.33195	−0.321219	−0.160609	+	0.987018i	$0.551346\pi$
$522$	−5.42727	−0.237545
$523$	−6.65975	−0.291211	−0.145605	−	0.989343i	$-0.546513\pi$
$523$	−6.65975	−0.291211	−0.145605	+	0.989343i	$0.546513\pi$
$524$	−14.1948	−0.620102
$525$	0.102628	0.00447906
$526$	28.9741	1.26333
$527$	−16.4554	−0.716807
$528$	−1.00000	−0.0435194
$529$	42.3999	1.84347
$530$	−9.77821	−0.424738
$531$	2.69777	0.117073
$532$	−4.20414	−0.182272
$533$	3.65758	0.158427
$534$	−7.66767	−0.331812
$535$	8.80901	0.380847
$536$	−6.30533	−0.272349
$537$	26.0759	1.12526
$538$	2.95409	0.127360
$539$	5.31672	0.229007
$540$	−2.21831	−0.0954608
$541$	−37.4790	−1.61135	−0.805674	−	0.592360i	$-0.798197\pi$
$541$	−37.4790	−1.61135	−0.805674	+	0.592360i	$0.798197\pi$
$542$	−27.1972	−1.16822
$543$	15.3039	0.656753
$544$	4.16130	0.178414
$545$	12.9751	0.555793
$546$	1.72672	0.0738968
$547$	−14.7534	−0.630809	−0.315405	−	0.948957i	$-0.602140\pi$
$547$	−14.7534	−0.630809	−0.315405	+	0.948957i	$0.602140\pi$
$548$	−17.9066	−0.764934
$549$	−1.00000	−0.0426790
$550$	0.0791023	0.00337293
$551$	−17.5865	−0.749211
$552$	−8.08702	−0.344206
$553$	4.51429	0.191967
$554$	6.27258	0.266497
$555$	10.4273	0.442613
$556$	14.3033	0.606595
$557$	14.0427	0.595010	0.297505	−	0.954720i	$-0.403845\pi$
$557$	14.0427	0.595010	0.297505	+	0.954720i	$0.403845\pi$
$558$	−3.95438	−0.167402
$559$	11.6089	0.491005
$560$	2.87806	0.121620
$561$	−4.16130	−0.175690
$562$	26.1569	1.10336
$563$	15.2586	0.643072	0.321536	−	0.946897i	$-0.395801\pi$
$563$	15.2586	0.643072	0.321536	+	0.946897i	$0.395801\pi$
$564$	6.82105	0.287218
$565$	−9.24628	−0.388994
$566$	−1.01592	−0.0427024
$567$	−1.29741	−0.0544862
$568$	1.68388	0.0706542
$569$	−19.9164	−0.834938	−0.417469	−	0.908691i	$-0.637083\pi$
$569$	−19.9164	−0.834938	−0.417469	+	0.908691i	$0.637083\pi$
$570$	−7.18821	−0.301081
$571$	−25.4747	−1.06608	−0.533041	−	0.846089i	$-0.678951\pi$
$571$	−25.4747	−1.06608	−0.533041	+	0.846089i	$0.678951\pi$
$572$	1.33090	0.0556476
$573$	2.34364	0.0979068
$574$	3.56555	0.148823
$575$	0.639702	0.0266774
$576$	1.00000	0.0416667
$577$	−25.1933	−1.04881	−0.524406	−	0.851468i	$-0.675713\pi$
$577$	−25.1933	−1.04881	−0.524406	+	0.851468i	$0.675713\pi$
$578$	0.316409	0.0131609
$579$	4.76065	0.197846
$580$	12.0394	0.499907
$581$	−8.05293	−0.334092
$582$	−14.3319	−0.594078
$583$	−4.40796	−0.182559
$584$	11.7232	0.485111
$585$	2.95234	0.122064
$586$	14.8428	0.613152
$587$	1.57509	0.0650108	0.0325054	−	0.999472i	$-0.489651\pi$
$587$	1.57509	0.0650108	0.0325054	+	0.999472i	$0.489651\pi$
$588$	−5.31672	−0.219258
$589$	−12.8138	−0.527983
$590$	−5.98448	−0.246377
$591$	−3.20618	−0.131884
$592$	−4.70055	−0.193191
$593$	−41.1003	−1.68779	−0.843893	−	0.536511i	$-0.819742\pi$
$593$	−41.1003	−1.68779	−0.843893	+	0.536511i	$0.819742\pi$
$594$	−1.00000	−0.0410305
$595$	11.9765	0.490988
$596$	−13.8867	−0.568822
$597$	0.688414	0.0281749
$598$	10.7630	0.440131
$599$	4.74896	0.194037	0.0970187	−	0.995283i	$-0.469069\pi$
$599$	4.74896	0.194037	0.0970187	+	0.995283i	$0.469069\pi$
$600$	−0.0791023	−0.00322934
$601$	−26.9635	−1.09986	−0.549932	−	0.835210i	$-0.685346\pi$
$601$	−26.9635	−1.09986	−0.549932	+	0.835210i	$0.685346\pi$
$602$	11.3169	0.461241
$603$	−6.30533	−0.256773
$604$	−8.32260	−0.338642
$605$	−2.21831	−0.0901871
$606$	−4.25458	−0.172830
$607$	−25.9087	−1.05160	−0.525800	−	0.850608i	$-0.676234\pi$
$607$	−25.9087	−1.05160	−0.525800	+	0.850608i	$0.676234\pi$
$608$	3.24040	0.131416
$609$	7.04140	0.285332
$610$	2.21831	0.0898167
$611$	−9.07811	−0.367261
$612$	4.16130	0.168211
$613$	−43.3770	−1.75198	−0.875991	−	0.482328i	$-0.839792\pi$
$613$	−43.3770	−1.75198	−0.875991	+	0.482328i	$0.839792\pi$
$614$	17.1047	0.690291
$615$	6.09637	0.245829
$616$	1.29741	0.0522742
$617$	37.8718	1.52466	0.762331	−	0.647188i	$-0.224055\pi$
$617$	37.8718	1.52466	0.762331	+	0.647188i	$0.224055\pi$
$618$	−8.87602	−0.357046
$619$	−6.45696	−0.259527	−0.129764	−	0.991545i	$-0.541422\pi$
$619$	−6.45696	−0.259527	−0.129764	+	0.991545i	$0.541422\pi$
$620$	8.77204	0.352294
$621$	−8.08702	−0.324521
$622$	−5.21205	−0.208984
$623$	9.94813	0.398563
$624$	−1.33090	−0.0532785
$625$	−24.5982	−0.983929
$626$	−25.5057	−1.01941
$627$	−3.24040	−0.129409
$628$	−21.5288	−0.859092
$629$	−19.5604	−0.779923
$630$	2.87806	0.114665
$631$	29.5883	1.17789	0.588946	−	0.808172i	$-0.299543\pi$
$631$	29.5883	1.17789	0.588946	+	0.808172i	$0.299543\pi$
$632$	−3.47946	−0.138405
$633$	18.3668	0.730014
$634$	3.66571	0.145584
$635$	−3.22836	−0.128113
$636$	4.40796	0.174787
$637$	7.07601	0.280362
$638$	5.42727	0.214868
$639$	1.68388	0.0666135
$640$	−2.21831	−0.0876864
$641$	−23.6570	−0.934394	−0.467197	−	0.884153i	$-0.654736\pi$
$641$	−23.6570	−0.934394	−0.467197	+	0.884153i	$0.654736\pi$
$642$	−3.97104	−0.156725
$643$	−3.28775	−0.129656	−0.0648280	−	0.997896i	$-0.520650\pi$
$643$	−3.28775	−0.129656	−0.0648280	+	0.997896i	$0.520650\pi$
$644$	10.4922	0.413450
$645$	19.3495	0.761886
$646$	13.4843	0.530532
$647$	−0.430495	−0.0169245	−0.00846225	−	0.999964i	$-0.502694\pi$
$647$	−0.430495	−0.0169245	−0.00846225	+	0.999964i	$0.502694\pi$
$648$	1.00000	0.0392837
$649$	−2.69777	−0.105897
$650$	0.105277	0.00412930
$651$	5.13046	0.201079
$652$	−19.9750	−0.782283
$653$	16.7827	0.656759	0.328380	−	0.944546i	$-0.393497\pi$
$653$	16.7827	0.656759	0.328380	+	0.944546i	$0.393497\pi$
$654$	−5.84911	−0.228718
$655$	31.4884	1.23035
$656$	−2.74821	−0.107299
$657$	11.7232	0.457367
$658$	−8.84971	−0.344998
$659$	−15.0581	−0.586579	−0.293289	−	0.956024i	$-0.594750\pi$
$659$	−15.0581	−0.586579	−0.293289	+	0.956024i	$0.594750\pi$
$660$	2.21831	0.0863476
$661$	−29.2914	−1.13930	−0.569651	−	0.821887i	$-0.692922\pi$
$661$	−29.2914	−1.13930	−0.569651	+	0.821887i	$0.692922\pi$
$662$	26.8984	1.04544
$663$	−5.53826	−0.215088
$664$	6.20692	0.240875
$665$	9.32607	0.361650
$666$	−4.70055	−0.182142
$667$	43.8904	1.69944
$668$	−9.55055	−0.369522
$669$	10.6103	0.410220
$670$	13.9872	0.540372
$671$	1.00000	0.0386046
$672$	−1.29741	−0.0500488
$673$	−6.58969	−0.254014	−0.127007	−	0.991902i	$-0.540537\pi$
$673$	−6.58969	−0.254014	−0.127007	+	0.991902i	$0.540537\pi$
$674$	34.4013	1.32509
$675$	−0.0791023	−0.00304465
$676$	−11.2287	−0.431874
$677$	17.0945	0.656995	0.328497	−	0.944505i	$-0.393458\pi$
$677$	17.0945	0.656995	0.328497	+	0.944505i	$0.393458\pi$
$678$	4.16816	0.160077
$679$	18.5944	0.713589
$680$	−9.23105	−0.353995
$681$	2.93332	0.112405
$682$	3.95438	0.151421
$683$	15.5499	0.595000	0.297500	−	0.954722i	$-0.403847\pi$
$683$	15.5499	0.595000	0.297500	+	0.954722i	$0.403847\pi$
$684$	3.24040	0.123900
$685$	39.7225	1.51772
$686$	15.9799	0.610114
$687$	0.923992	0.0352525
$688$	−8.72264	−0.332547
$689$	−5.86654	−0.223497
$690$	17.9395	0.682946
$691$	4.83318	0.183863	0.0919315	−	0.995765i	$-0.470696\pi$
$691$	4.83318	0.183863	0.0919315	+	0.995765i	$0.470696\pi$
$692$	−5.10467	−0.194050
$693$	1.29741	0.0492846
$694$	−33.0472	−1.25445
$695$	−31.7291	−1.20355
$696$	−5.42727	−0.205720
$697$	−11.4361	−0.433173
$698$	−23.2016	−0.878195
$699$	26.1457	0.988921
$700$	0.102628	0.00387898
$701$	−2.27530	−0.0859369	−0.0429685	−	0.999076i	$-0.513682\pi$
$701$	−2.27530	−0.0859369	−0.0429685	+	0.999076i	$0.513682\pi$
$702$	−1.33090	−0.0502315
$703$	−15.2317	−0.574473
$704$	−1.00000	−0.0376889
$705$	−15.1312	−0.569874
$706$	20.6276	0.776330
$707$	5.51994	0.207599
$708$	2.69777	0.101388
$709$	−40.4466	−1.51901	−0.759503	−	0.650504i	$-0.774558\pi$
$709$	−40.4466	−1.51901	−0.759503	+	0.650504i	$0.774558\pi$
$710$	−3.73538	−0.140186
$711$	−3.47946	−0.130490
$712$	−7.66767	−0.287358
$713$	31.9792	1.19763
$714$	−5.39892	−0.202049
$715$	−2.95234	−0.110411
$716$	26.0759	0.974504
$717$	−16.2303	−0.606132
$718$	−18.8535	−0.703608
$719$	−20.4866	−0.764022	−0.382011	−	0.924158i	$-0.624768\pi$
$719$	−20.4866	−0.764022	−0.382011	+	0.924158i	$0.624768\pi$
$720$	−2.21831	−0.0826715
$721$	11.5159	0.428873
$722$	−8.49980	−0.316330
$723$	−14.5962	−0.542837
$724$	15.3039	0.568765
$725$	0.429309	0.0159441
$726$	1.00000	0.0371135
$727$	8.50215	0.315327	0.157664	−	0.987493i	$-0.449604\pi$
$727$	8.50215	0.315327	0.157664	+	0.987493i	$0.449604\pi$
$728$	1.72672	0.0639965
$729$	1.00000	0.0370370
$730$	−26.0058	−0.962517
$731$	−36.2975	−1.34251
$732$	−1.00000	−0.0369611
$733$	−37.3877	−1.38095	−0.690474	−	0.723357i	$-0.742598\pi$
$733$	−37.3877	−1.38095	−0.690474	+	0.723357i	$0.742598\pi$
$734$	−17.2676	−0.637360
$735$	11.7941	0.435033
$736$	−8.08702	−0.298091
$737$	6.30533	0.232260
$738$	−2.74821	−0.101163
$739$	−12.1167	−0.445721	−0.222861	−	0.974850i	$-0.571539\pi$
$739$	−12.1167	−0.445721	−0.222861	+	0.974850i	$0.571539\pi$
$740$	10.4273	0.383314
$741$	−4.31264	−0.158429
$742$	−5.71894	−0.209949
$743$	−30.8873	−1.13315	−0.566573	−	0.824011i	$-0.691731\pi$
$743$	−30.8873	−1.13315	−0.566573	+	0.824011i	$0.691731\pi$
$744$	−3.95438	−0.144975
$745$	30.8050	1.12861
$746$	−10.9993	−0.402713
$747$	6.20692	0.227099
$748$	−4.16130	−0.152152
$749$	5.15208	0.188253
$750$	11.2670	0.411413
$751$	6.66668	0.243271	0.121635	−	0.992575i	$-0.461186\pi$
$751$	6.66668	0.243271	0.121635	+	0.992575i	$0.461186\pi$
$752$	6.82105	0.248738
$753$	3.95921	0.144281
$754$	7.22313	0.263051
$755$	18.4621	0.671905
$756$	−1.29741	−0.0471864
$757$	0.402601	0.0146328	0.00731640	−	0.999973i	$-0.497671\pi$
$757$	0.402601	0.0146328	0.00731640	+	0.999973i	$0.497671\pi$
$758$	−6.90354	−0.250748
$759$	8.08702	0.293540
$760$	−7.18821	−0.260744
$761$	−42.7285	−1.54891	−0.774454	−	0.632630i	$-0.781975\pi$
$761$	−42.7285	−1.54891	−0.774454	+	0.632630i	$0.781975\pi$
$762$	1.45532	0.0527208
$763$	7.58870	0.274729
$764$	2.34364	0.0847898
$765$	−9.23105	−0.333749
$766$	5.17577	0.187008
$767$	−3.59045	−0.129644
$768$	1.00000	0.0360844
$769$	−11.2977	−0.407406	−0.203703	−	0.979033i	$-0.565298\pi$
$769$	−11.2977	−0.407406	−0.203703	+	0.979033i	$0.565298\pi$
$770$	−2.87806	−0.103718
$771$	13.6532	0.491707
$772$	4.76065	0.171340
$773$	31.9466	1.14904	0.574520	−	0.818491i	$-0.305189\pi$
$773$	31.9466	1.14904	0.574520	+	0.818491i	$0.305189\pi$
$774$	−8.72264	−0.313529
$775$	0.312801	0.0112361
$776$	−14.3319	−0.514487
$777$	6.09855	0.218784
$778$	−12.6327	−0.452906
$779$	−8.90529	−0.319065
$780$	2.95234	0.105711
$781$	−1.68388	−0.0602541
$782$	−33.6525	−1.20341
$783$	−5.42727	−0.193955
$784$	−5.31672	−0.189883
$785$	47.7575	1.70454
$786$	−14.1948	−0.506311
$787$	54.6951	1.94967	0.974835	−	0.222928i	$-0.0715615\pi$
$787$	54.6951	1.94967	0.974835	+	0.222928i	$0.0715615\pi$
$788$	−3.20618	−0.114215
$789$	28.9741	1.03150
$790$	7.71851	0.274612
$791$	−5.40782	−0.192280
$792$	−1.00000	−0.0355335
$793$	1.33090	0.0472615
$794$	32.2525	1.14460
$795$	−9.77821	−0.346798
$796$	0.688414	0.0244002
$797$	−25.5610	−0.905419	−0.452709	−	0.891658i	$-0.649542\pi$
$797$	−25.5610	−0.905419	−0.452709	+	0.891658i	$0.649542\pi$
$798$	−4.20414	−0.148825
$799$	28.3844	1.00417
$800$	−0.0791023	−0.00279669
$801$	−7.66767	−0.270924
$802$	10.0038	0.353248
$803$	−11.7232	−0.413704
$804$	−6.30533	−0.222372
$805$	−23.2749	−0.820334
$806$	5.26287	0.185377
$807$	2.95409	0.103989
$808$	−4.25458	−0.149675
$809$	−15.9392	−0.560393	−0.280197	−	0.959943i	$-0.590400\pi$
$809$	−15.9392	−0.560393	−0.280197	+	0.959943i	$0.590400\pi$
$810$	−2.21831	−0.0779435
$811$	−12.6207	−0.443174	−0.221587	−	0.975141i	$-0.571124\pi$
$811$	−12.6207	−0.443174	−0.221587	+	0.975141i	$0.571124\pi$
$812$	7.04140	0.247105
$813$	−27.1972	−0.953847
$814$	4.70055	0.164754
$815$	44.3108	1.55214
$816$	4.16130	0.145675
$817$	−28.2648	−0.988862
$818$	10.5928	0.370369
$819$	1.72672	0.0603365
$820$	6.09637	0.212895
$821$	8.78683	0.306662	0.153331	−	0.988175i	$-0.451000\pi$
$821$	8.78683	0.306662	0.153331	+	0.988175i	$0.451000\pi$
$822$	−17.9066	−0.624566
$823$	−53.1782	−1.85368	−0.926838	−	0.375463i	$-0.877484\pi$
$823$	−53.1782	−1.85368	−0.926838	+	0.375463i	$0.877484\pi$
$824$	−8.87602	−0.309211
$825$	0.0791023	0.00275399
$826$	−3.50011	−0.121785
$827$	−36.6218	−1.27346	−0.636732	−	0.771085i	$-0.719714\pi$
$827$	−36.6218	−1.27346	−0.636732	+	0.771085i	$0.719714\pi$
$828$	−8.08702	−0.281043
$829$	−30.9148	−1.07371	−0.536857	−	0.843673i	$-0.680389\pi$
$829$	−30.9148	−1.07371	−0.536857	+	0.843673i	$0.680389\pi$
$830$	−13.7689	−0.477924
$831$	6.27258	0.217594
$832$	−1.33090	−0.0461405
$833$	−22.1245	−0.766567
$834$	14.3033	0.495282
$835$	21.1861	0.733175
$836$	−3.24040	−0.112072
$837$	−3.95438	−0.136683
$838$	31.1754	1.07694
$839$	−53.5196	−1.84770	−0.923852	−	0.382751i	$-0.874977\pi$
$839$	−53.5196	−1.84770	−0.923852	+	0.382751i	$0.874977\pi$
$840$	2.87806	0.0993026
$841$	0.455235	0.0156978
$842$	26.7108	0.920516
$843$	26.1569	0.900892
$844$	18.3668	0.632211
$845$	24.9088	0.856888
$846$	6.82105	0.234513
$847$	−1.29741	−0.0445796
$848$	4.40796	0.151370
$849$	−1.01592	−0.0348663
$850$	−0.329168	−0.0112904
$851$	38.0134	1.30308
$852$	1.68388	0.0576889
$853$	−33.2896	−1.13981	−0.569906	−	0.821710i	$-0.693021\pi$
$853$	−33.2896	−1.13981	−0.569906	+	0.821710i	$0.693021\pi$
$854$	1.29741	0.0443965
$855$	−7.18821	−0.245832
$856$	−3.97104	−0.135727
$857$	31.3557	1.07109	0.535545	−	0.844507i	$-0.320106\pi$
$857$	31.3557	1.07109	0.535545	+	0.844507i	$0.320106\pi$
$858$	1.33090	0.0454361
$859$	31.1175	1.06171	0.530857	−	0.847461i	$-0.321870\pi$
$859$	31.1175	1.06171	0.530857	+	0.847461i	$0.321870\pi$
$860$	19.3495	0.659813
$861$	3.56555	0.121514
$862$	36.5264	1.24409
$863$	20.4570	0.696363	0.348182	−	0.937427i	$-0.386799\pi$
$863$	20.4570	0.696363	0.348182	+	0.937427i	$0.386799\pi$
$864$	1.00000	0.0340207
$865$	11.3237	0.385019
$866$	31.0106	1.05378
$867$	0.316409	0.0107458
$868$	5.13046	0.174139
$869$	3.47946	0.118032
$870$	12.0394	0.408173
$871$	8.39174	0.284343
$872$	−5.84911	−0.198076
$873$	−14.3319	−0.485063
$874$	−26.2052	−0.886404
$875$	−14.6180	−0.494178
$876$	11.7232	0.396092
$877$	−28.1693	−0.951209	−0.475605	−	0.879659i	$-0.657771\pi$
$877$	−28.1693	−0.951209	−0.475605	+	0.879659i	$0.657771\pi$
$878$	21.0853	0.711594
$879$	14.8428	0.500637
$880$	2.21831	0.0747792
$881$	−43.1712	−1.45448	−0.727238	−	0.686385i	$-0.759196\pi$
$881$	−43.1712	−1.45448	−0.727238	+	0.686385i	$0.759196\pi$
$882$	−5.31672	−0.179023
$883$	4.58939	0.154445	0.0772227	−	0.997014i	$-0.475395\pi$
$883$	4.58939	0.154445	0.0772227	+	0.997014i	$0.475395\pi$
$884$	−5.53826	−0.186272
$885$	−5.98448	−0.201166
$886$	−15.7813	−0.530183
$887$	−0.206755	−0.00694216	−0.00347108	−	0.999994i	$-0.501105\pi$
$887$	−0.206755	−0.00694216	−0.00347108	+	0.999994i	$0.501105\pi$
$888$	−4.70055	−0.157740
$889$	−1.88815	−0.0633266
$890$	17.0093	0.570152
$891$	−1.00000	−0.0335013
$892$	10.6103	0.355261
$893$	22.1029	0.739647
$894$	−13.8867	−0.464441
$895$	−57.8445	−1.93353
$896$	−1.29741	−0.0433435
$897$	10.7630	0.359366
$898$	11.3471	0.378658
$899$	21.4615	0.715781
$900$	−0.0791023	−0.00263674
$901$	18.3428	0.611088
$902$	2.74821	0.0915052
$903$	11.3169	0.376601
$904$	4.16816	0.138631
$905$	−33.9488	−1.12850
$906$	−8.32260	−0.276500
$907$	30.3434	1.00754	0.503768	−	0.863839i	$-0.331947\pi$
$907$	30.3434	1.00754	0.503768	+	0.863839i	$0.331947\pi$
$908$	2.93332	0.0973458
$909$	−4.25458	−0.141115
$910$	−3.83040	−0.126977
$911$	22.0825	0.731626	0.365813	−	0.930688i	$-0.380791\pi$
$911$	22.0825	0.731626	0.365813	+	0.930688i	$0.380791\pi$
$912$	3.24040	0.107300
$913$	−6.20692	−0.205419
$914$	29.8176	0.986278
$915$	2.21831	0.0733351
$916$	0.923992	0.0305296
$917$	18.4165	0.608166
$918$	4.16130	0.137343
$919$	14.8860	0.491045	0.245522	−	0.969391i	$-0.421041\pi$
$919$	14.8860	0.491045	0.245522	+	0.969391i	$0.421041\pi$
$920$	17.9395	0.591448
$921$	17.1047	0.563620
$922$	30.5005	1.00448
$923$	−2.24108	−0.0737659
$924$	1.29741	0.0426817
$925$	0.371824	0.0122255
$926$	−0.350340	−0.0115129
$927$	−8.87602	−0.291527
$928$	−5.42727	−0.178159
$929$	−41.1239	−1.34923	−0.674615	−	0.738170i	$-0.735690\pi$
$929$	−41.1239	−1.34923	−0.674615	+	0.738170i	$0.735690\pi$
$930$	8.77204	0.287647
$931$	−17.2283	−0.564635
$932$	26.1457	0.856431
$933$	−5.21205	−0.170635
$934$	35.1893	1.15143
$935$	9.23105	0.301888
$936$	−1.33090	−0.0435017
$937$	−19.2815	−0.629899	−0.314949	−	0.949108i	$-0.601988\pi$
$937$	−19.2815	−0.629899	−0.314949	+	0.949108i	$0.601988\pi$
$938$	8.18061	0.267106
$939$	−25.5057	−0.832346
$940$	−15.1312	−0.493525
$941$	7.04245	0.229577	0.114789	−	0.993390i	$-0.463381\pi$
$941$	7.04245	0.229577	0.114789	+	0.993390i	$0.463381\pi$
$942$	−21.5288	−0.701445
$943$	22.2248	0.723739
$944$	2.69777	0.0878048
$945$	2.87806	0.0936233
$946$	8.72264	0.283597
$947$	−1.52983	−0.0497127	−0.0248564	−	0.999691i	$-0.507913\pi$
$947$	−1.52983	−0.0497127	−0.0248564	+	0.999691i	$0.507913\pi$
$948$	−3.47946	−0.113007
$949$	−15.6024	−0.506476
$950$	−0.256323	−0.00831622
$951$	3.66571	0.118869
$952$	−5.39892	−0.174980
$953$	13.1712	0.426658	0.213329	−	0.976980i	$-0.431569\pi$
$953$	13.1712	0.426658	0.213329	+	0.976980i	$0.431569\pi$
$954$	4.40796	0.142713
$955$	−5.19891	−0.168233
$956$	−16.2303	−0.524926
$957$	5.42727	0.175439
$958$	6.66767	0.215423
$959$	23.2323	0.750210
$960$	−2.21831	−0.0715956
$961$	−15.3629	−0.495576
$962$	6.25594	0.201700
$963$	−3.97104	−0.127965
$964$	−14.5962	−0.470111
$965$	−10.5606	−0.339958
$966$	10.4922	0.337581
$967$	50.8212	1.63430	0.817149	−	0.576426i	$-0.195553\pi$
$967$	50.8212	1.63430	0.817149	+	0.576426i	$0.195553\pi$
$968$	1.00000	0.0321412
$969$	13.4843	0.433177
$970$	31.7927	1.02080
$971$	10.9262	0.350638	0.175319	−	0.984512i	$-0.443904\pi$
$971$	10.9262	0.350638	0.175319	+	0.984512i	$0.443904\pi$
$972$	1.00000	0.0320750
$973$	−18.5573	−0.594918
$974$	−6.32886	−0.202790
$975$	0.105277	0.00337156
$976$	−1.00000	−0.0320092
$977$	35.5735	1.13810	0.569048	−	0.822304i	$-0.307312\pi$
$977$	35.5735	1.13810	0.569048	+	0.822304i	$0.307312\pi$
$978$	−19.9750	−0.638731
$979$	7.66767	0.245060
$980$	11.7941	0.376750
$981$	−5.84911	−0.186748
$982$	7.91751	0.252658
$983$	−42.7035	−1.36203	−0.681015	−	0.732270i	$-0.738461\pi$
$983$	−42.7035	−1.36203	−0.681015	+	0.732270i	$0.738461\pi$
$984$	−2.74821	−0.0876096
$985$	7.11229	0.226616
$986$	−22.5845	−0.719236
$987$	−8.84971	−0.281689
$988$	−4.31264	−0.137203
$989$	70.5401	2.24305
$990$	2.21831	0.0705025
$991$	−29.2031	−0.927669	−0.463834	−	0.885922i	$-0.653527\pi$
$991$	−29.2031	−0.927669	−0.463834	+	0.885922i	$0.653527\pi$
$992$	−3.95438	−0.125552
$993$	26.8984	0.853596
$994$	−2.18469	−0.0692942
$995$	−1.52711	−0.0484128
$996$	6.20692	0.196674
$997$	21.7710	0.689496	0.344748	−	0.938695i	$-0.387964\pi$
$997$	21.7710	0.689496	0.344748	+	0.938695i	$0.387964\pi$
$998$	13.1675	0.416809
$999$	−4.70055	−0.148719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4026.2.a.t.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4026.2.a.t.1.2	✓	4	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:	\( N \)	\(=\)	\( 4026 = 2 \cdot 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4026.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.21831 q^{5} +1.00000 q^{6} -1.29741 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.21831 q^{5} +1.00000 q^{6} -1.29741 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.21831 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.33090 q^{13} -1.29741 q^{14} -2.21831 q^{15} +1.00000 q^{16} +4.16130 q^{17} +1.00000 q^{18} +3.24040 q^{19} -2.21831 q^{20} -1.29741 q^{21} -1.00000 q^{22} -8.08702 q^{23} +1.00000 q^{24} -0.0791023 q^{25} -1.33090 q^{26} +1.00000 q^{27} -1.29741 q^{28} -5.42727 q^{29} -2.21831 q^{30} -3.95438 q^{31} +1.00000 q^{32} -1.00000 q^{33} +4.16130 q^{34} +2.87806 q^{35} +1.00000 q^{36} -4.70055 q^{37} +3.24040 q^{38} -1.33090 q^{39} -2.21831 q^{40} -2.74821 q^{41} -1.29741 q^{42} -8.72264 q^{43} -1.00000 q^{44} -2.21831 q^{45} -8.08702 q^{46} +6.82105 q^{47} +1.00000 q^{48} -5.31672 q^{49} -0.0791023 q^{50} +4.16130 q^{51} -1.33090 q^{52} +4.40796 q^{53} +1.00000 q^{54} +2.21831 q^{55} -1.29741 q^{56} +3.24040 q^{57} -5.42727 q^{58} +2.69777 q^{59} -2.21831 q^{60} -1.00000 q^{61} -3.95438 q^{62} -1.29741 q^{63} +1.00000 q^{64} +2.95234 q^{65} -1.00000 q^{66} -6.30533 q^{67} +4.16130 q^{68} -8.08702 q^{69} +2.87806 q^{70} +1.68388 q^{71} +1.00000 q^{72} +11.7232 q^{73} -4.70055 q^{74} -0.0791023 q^{75} +3.24040 q^{76} +1.29741 q^{77} -1.33090 q^{78} -3.47946 q^{79} -2.21831 q^{80} +1.00000 q^{81} -2.74821 q^{82} +6.20692 q^{83} -1.29741 q^{84} -9.23105 q^{85} -8.72264 q^{86} -5.42727 q^{87} -1.00000 q^{88} -7.66767 q^{89} -2.21831 q^{90} +1.72672 q^{91} -8.08702 q^{92} -3.95438 q^{93} +6.82105 q^{94} -7.18821 q^{95} +1.00000 q^{96} -14.3319 q^{97} -5.31672 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 6 * q^7 + 4 * q^8 + 4 * q^9	\( 4 q + 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} + 4 q^{6} - 6 q^{7} + 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{12} - 5 q^{13} - 6 q^{14} - 4 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 8 q^{19} - 4 q^{20} - 6 q^{21} - 4 q^{22} - 7 q^{23} + 4 q^{24} - 6 q^{25} - 5 q^{26} + 4 q^{27} - 6 q^{28} - 4 q^{29} - 4 q^{30} - 9 q^{31} + 4 q^{32} - 4 q^{33} - 10 q^{34} - q^{35} + 4 q^{36} - 11 q^{37} - 8 q^{38} - 5 q^{39} - 4 q^{40} - 17 q^{41} - 6 q^{42} - 11 q^{43} - 4 q^{44} - 4 q^{45} - 7 q^{46} - 7 q^{47} + 4 q^{48} - 6 q^{49} - 6 q^{50} - 10 q^{51} - 5 q^{52} + 16 q^{53} + 4 q^{54} + 4 q^{55} - 6 q^{56} - 8 q^{57} - 4 q^{58} + 3 q^{59} - 4 q^{60} - 4 q^{61} - 9 q^{62} - 6 q^{63} + 4 q^{64} - 2 q^{65} - 4 q^{66} + 5 q^{67} - 10 q^{68} - 7 q^{69} - q^{70} - 10 q^{71} + 4 q^{72} - 9 q^{73} - 11 q^{74} - 6 q^{75} - 8 q^{76} + 6 q^{77} - 5 q^{78} - 11 q^{79} - 4 q^{80} + 4 q^{81} - 17 q^{82} + 5 q^{83} - 6 q^{84} - 8 q^{85} - 11 q^{86} - 4 q^{87} - 4 q^{88} + 8 q^{89} - 4 q^{90} - 3 q^{91} - 7 q^{92} - 9 q^{93} - 7 q^{94} + 7 q^{95} + 4 q^{96} - 12 q^{97} - 6 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^5 + 4 * q^6 - 6 * q^7 + 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^11 + 4 * q^12 - 5 * q^13 - 6 * q^14 - 4 * q^15 + 4 * q^16 - 10 * q^17 + 4 * q^18 - 8 * q^19 - 4 * q^20 - 6 * q^21 - 4 * q^22 - 7 * q^23 + 4 * q^24 - 6 * q^25 - 5 * q^26 + 4 * q^27 - 6 * q^28 - 4 * q^29 - 4 * q^30 - 9 * q^31 + 4 * q^32 - 4 * q^33 - 10 * q^34 - q^35 + 4 * q^36 - 11 * q^37 - 8 * q^38 - 5 * q^39 - 4 * q^40 - 17 * q^41 - 6 * q^42 - 11 * q^43 - 4 * q^44 - 4 * q^45 - 7 * q^46 - 7 * q^47 + 4 * q^48 - 6 * q^49 - 6 * q^50 - 10 * q^51 - 5 * q^52 + 16 * q^53 + 4 * q^54 + 4 * q^55 - 6 * q^56 - 8 * q^57 - 4 * q^58 + 3 * q^59 - 4 * q^60 - 4 * q^61 - 9 * q^62 - 6 * q^63 + 4 * q^64 - 2 * q^65 - 4 * q^66 + 5 * q^67 - 10 * q^68 - 7 * q^69 - q^70 - 10 * q^71 + 4 * q^72 - 9 * q^73 - 11 * q^74 - 6 * q^75 - 8 * q^76 + 6 * q^77 - 5 * q^78 - 11 * q^79 - 4 * q^80 + 4 * q^81 - 17 * q^82 + 5 * q^83 - 6 * q^84 - 8 * q^85 - 11 * q^86 - 4 * q^87 - 4 * q^88 + 8 * q^89 - 4 * q^90 - 3 * q^91 - 7 * q^92 - 9 * q^93 - 7 * q^94 + 7 * q^95 + 4 * q^96 - 12 * q^97 - 6 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more