LMFDB - Embedded newform 4805.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4805,2,Mod(1,4805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4805, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4805.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4805 = 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4805.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:	$38.3681181712$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 29 x^{18} + 29 x^{17} + 348 x^{16} - 341 x^{15} - 2245 x^{14} + 2101 x^{13} + \cdots - 29 $ x^20 - x^19 - 29x^18 + 29x^17 + 348x^16 - 341x^15 - 2245x^14 + 2101x^13 + 8472x^12 - 7300x^11 - 19098x^10 + 14255x^9 + 25337x^8 - 14501x^7 - 19080x^6 + 6101x^5 + 7773x^4 - 89x^3 - 1349x^2 - 364x - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.51863$ of defining polynomial
Character	$\chi$	$=$	4805.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51863 q^{2} +1.94255 q^{3} +4.34352 q^{4} +1.00000 q^{5} -4.89257 q^{6} -0.835075 q^{7} -5.90247 q^{8} +0.773495 q^{9} +O(q^{10})$	\(q-2.51863 q^{2} +1.94255 q^{3} +4.34352 q^{4} +1.00000 q^{5} -4.89257 q^{6} -0.835075 q^{7} -5.90247 q^{8} +0.773495 q^{9} -2.51863 q^{10} -4.53630 q^{11} +8.43750 q^{12} -6.29628 q^{13} +2.10325 q^{14} +1.94255 q^{15} +6.17912 q^{16} +1.29398 q^{17} -1.94815 q^{18} +0.576198 q^{19} +4.34352 q^{20} -1.62217 q^{21} +11.4253 q^{22} +2.36152 q^{23} -11.4658 q^{24} +1.00000 q^{25} +15.8580 q^{26} -4.32509 q^{27} -3.62716 q^{28} +0.791240 q^{29} -4.89257 q^{30} -3.75801 q^{32} -8.81198 q^{33} -3.25905 q^{34} -0.835075 q^{35} +3.35969 q^{36} +11.2885 q^{37} -1.45123 q^{38} -12.2308 q^{39} -5.90247 q^{40} +11.2526 q^{41} +4.08566 q^{42} +11.1735 q^{43} -19.7035 q^{44} +0.773495 q^{45} -5.94780 q^{46} -8.32997 q^{47} +12.0032 q^{48} -6.30265 q^{49} -2.51863 q^{50} +2.51361 q^{51} -27.3480 q^{52} -5.33827 q^{53} +10.8933 q^{54} -4.53630 q^{55} +4.92900 q^{56} +1.11929 q^{57} -1.99284 q^{58} -5.33667 q^{59} +8.43750 q^{60} -4.18717 q^{61} -0.645926 q^{63} -2.89318 q^{64} -6.29628 q^{65} +22.1942 q^{66} +7.04476 q^{67} +5.62041 q^{68} +4.58736 q^{69} +2.10325 q^{70} -3.01510 q^{71} -4.56553 q^{72} +10.3532 q^{73} -28.4317 q^{74} +1.94255 q^{75} +2.50273 q^{76} +3.78815 q^{77} +30.8050 q^{78} +1.27967 q^{79} +6.17912 q^{80} -10.7222 q^{81} -28.3412 q^{82} +13.1165 q^{83} -7.04594 q^{84} +1.29398 q^{85} -28.1419 q^{86} +1.53702 q^{87} +26.7754 q^{88} +3.69780 q^{89} -1.94815 q^{90} +5.25787 q^{91} +10.2573 q^{92} +20.9802 q^{94} +0.576198 q^{95} -7.30012 q^{96} -5.42095 q^{97} +15.8741 q^{98} -3.50881 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9}+O(q^{10}) $ 20 * q + q^2 + 12 * q^3 + 19 * q^4 + 20 * q^5 + 12 * q^6 - 3 * q^7 - 3 * q^8 + 24 * q^9	\( 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9} + q^{10} + 19 q^{11} - 9 q^{12} + 23 q^{13} - 10 q^{14} + 12 q^{15} + 17 q^{16} + 6 q^{17} - 12 q^{18} + 9 q^{19} + 19 q^{20} - 6 q^{21} + 42 q^{22} + 33 q^{23} + 40 q^{24} + 20 q^{25} + 4 q^{26} + 63 q^{27} - 8 q^{28} + 4 q^{29} + 12 q^{30} - 30 q^{32} - 8 q^{33} - 19 q^{34} - 3 q^{35} + q^{36} + 38 q^{37} - 13 q^{38} + 15 q^{39} - 3 q^{40} + 9 q^{41} + 48 q^{42} + 64 q^{43} + 27 q^{44} + 24 q^{45} - 20 q^{46} - 8 q^{47} + 5 q^{48} - 3 q^{49} + q^{50} - 18 q^{51} - 5 q^{52} + 7 q^{53} + 16 q^{54} + 19 q^{55} - 5 q^{56} + 36 q^{57} + 49 q^{58} - 26 q^{59} - 9 q^{60} + 34 q^{61} - 5 q^{63} + q^{64} + 23 q^{65} + 35 q^{66} + 2 q^{67} + 28 q^{68} + 18 q^{69} - 10 q^{70} + 55 q^{71} - 14 q^{72} + 50 q^{73} - 30 q^{74} + 12 q^{75} - 89 q^{76} + 8 q^{77} + 69 q^{78} + 41 q^{79} + 17 q^{80} + 40 q^{81} - 24 q^{82} + 63 q^{83} - 7 q^{84} + 6 q^{85} + 7 q^{86} + 36 q^{87} + 124 q^{88} + 32 q^{89} - 12 q^{90} + 24 q^{91} + 55 q^{92} - 47 q^{94} + 9 q^{95} + 18 q^{96} + 13 q^{97} + 13 q^{98} - 17 q^{99}+O(q^{100}) \) 20 * q + q^2 + 12 * q^3 + 19 * q^4 + 20 * q^5 + 12 * q^6 - 3 * q^7 - 3 * q^8 + 24 * q^9 + q^10 + 19 * q^11 - 9 * q^12 + 23 * q^13 - 10 * q^14 + 12 * q^15 + 17 * q^16 + 6 * q^17 - 12 * q^18 + 9 * q^19 + 19 * q^20 - 6 * q^21 + 42 * q^22 + 33 * q^23 + 40 * q^24 + 20 * q^25 + 4 * q^26 + 63 * q^27 - 8 * q^28 + 4 * q^29 + 12 * q^30 - 30 * q^32 - 8 * q^33 - 19 * q^34 - 3 * q^35 + q^36 + 38 * q^37 - 13 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 48 * q^42 + 64 * q^43 + 27 * q^44 + 24 * q^45 - 20 * q^46 - 8 * q^47 + 5 * q^48 - 3 * q^49 + q^50 - 18 * q^51 - 5 * q^52 + 7 * q^53 + 16 * q^54 + 19 * q^55 - 5 * q^56 + 36 * q^57 + 49 * q^58 - 26 * q^59 - 9 * q^60 + 34 * q^61 - 5 * q^63 + q^64 + 23 * q^65 + 35 * q^66 + 2 * q^67 + 28 * q^68 + 18 * q^69 - 10 * q^70 + 55 * q^71 - 14 * q^72 + 50 * q^73 - 30 * q^74 + 12 * q^75 - 89 * q^76 + 8 * q^77 + 69 * q^78 + 41 * q^79 + 17 * q^80 + 40 * q^81 - 24 * q^82 + 63 * q^83 - 7 * q^84 + 6 * q^85 + 7 * q^86 + 36 * q^87 + 124 * q^88 + 32 * q^89 - 12 * q^90 + 24 * q^91 + 55 * q^92 - 47 * q^94 + 9 * q^95 + 18 * q^96 + 13 * q^97 + 13 * q^98 - 17 * 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 $

$2$	−2.51863	−1.78094	−0.890472	−	0.455039i	$-0.849625\pi$
$2$	−2.51863	−1.78094	−0.890472	+	0.455039i	$0.849625\pi$
$3$	1.94255	1.12153	0.560765	−	0.827975i	$-0.310507\pi$
$3$	1.94255	1.12153	0.560765	+	0.827975i	$0.310507\pi$
$4$	4.34352	2.17176
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−4.89257	−1.99738
$7$	−0.835075	−0.315629	−0.157814	−	0.987469i	$-0.550445\pi$
$7$	−0.835075	−0.315629	−0.157814	+	0.987469i	$0.550445\pi$
$8$	−5.90247	−2.08684
$9$	0.773495	0.257832
$10$	−2.51863	−0.796462
$11$	−4.53630	−1.36775	−0.683873	−	0.729601i	$-0.739706\pi$
$11$	−4.53630	−1.36775	−0.683873	+	0.729601i	$0.739706\pi$
$12$	8.43750	2.43570
$13$	−6.29628	−1.74627	−0.873137	−	0.487474i	$-0.837918\pi$
$13$	−6.29628	−1.74627	−0.873137	+	0.487474i	$0.837918\pi$
$14$	2.10325	0.562117
$15$	1.94255	0.501564
$16$	6.17912	1.54478
$17$	1.29398	0.313835	0.156918	−	0.987612i	$-0.449844\pi$
$17$	1.29398	0.313835	0.156918	+	0.987612i	$0.449844\pi$
$18$	−1.94815	−0.459184
$19$	0.576198	0.132189	0.0660944	−	0.997813i	$-0.478946\pi$
$19$	0.576198	0.132189	0.0660944	+	0.997813i	$0.478946\pi$
$20$	4.34352	0.971240
$21$	−1.62217	−0.353987
$22$	11.4253	2.43588
$23$	2.36152	0.492410	0.246205	−	0.969218i	$-0.420816\pi$
$23$	2.36152	0.492410	0.246205	+	0.969218i	$0.420816\pi$
$24$	−11.4658	−2.34045
$25$	1.00000	0.200000
$26$	15.8580	3.11002
$27$	−4.32509	−0.832365
$28$	−3.62716	−0.685469
$29$	0.791240	0.146930	0.0734648	−	0.997298i	$-0.476594\pi$
$29$	0.791240	0.146930	0.0734648	+	0.997298i	$0.476594\pi$
$30$	−4.89257	−0.893257
$31$	0	0
$31$	0	0
$32$	−3.75801	−0.664329
$33$	−8.81198	−1.53397
$34$	−3.25905	−0.558923
$35$	−0.835075	−0.141153
$36$	3.35969	0.559948
$37$	11.2885	1.85582	0.927912	−	0.372799i	$-0.121602\pi$
$37$	11.2885	1.85582	0.927912	+	0.372799i	$0.121602\pi$
$38$	−1.45123	−0.235421
$39$	−12.2308	−1.95850
$40$	−5.90247	−0.933262
$41$	11.2526	1.75736	0.878682	−	0.477407i	$-0.158423\pi$
$41$	11.2526	1.75736	0.878682	+	0.477407i	$0.158423\pi$
$42$	4.08566	0.630431
$43$	11.1735	1.70394	0.851969	−	0.523591i	$-0.175408\pi$
$43$	11.1735	1.70394	0.851969	+	0.523591i	$0.175408\pi$
$44$	−19.7035	−2.97042
$45$	0.773495	0.115306
$46$	−5.94780	−0.876955
$47$	−8.32997	−1.21505	−0.607526	−	0.794300i	$-0.707838\pi$
$47$	−8.32997	−1.21505	−0.607526	+	0.794300i	$0.707838\pi$
$48$	12.0032	1.73252
$49$	−6.30265	−0.900379
$50$	−2.51863	−0.356189
$51$	2.51361	0.351976
$52$	−27.3480	−3.79249
$53$	−5.33827	−0.733268	−0.366634	−	0.930365i	$-0.619490\pi$
$53$	−5.33827	−0.733268	−0.366634	+	0.930365i	$0.619490\pi$
$54$	10.8933	1.48239
$55$	−4.53630	−0.611675
$56$	4.92900	0.658666
$57$	1.11929	0.148254
$58$	−1.99284	−0.261673
$59$	−5.33667	−0.694775	−0.347388	−	0.937722i	$-0.612931\pi$
$59$	−5.33667	−0.694775	−0.347388	+	0.937722i	$0.612931\pi$
$60$	8.43750	1.08928
$61$	−4.18717	−0.536112	−0.268056	−	0.963403i	$-0.586381\pi$
$61$	−4.18717	−0.536112	−0.268056	+	0.963403i	$0.586381\pi$
$62$	0	0
$63$	−0.645926	−0.0813790
$64$	−2.89318	−0.361648
$65$	−6.29628	−0.780958
$66$	22.1942	2.73191
$67$	7.04476	0.860654	0.430327	−	0.902673i	$-0.358398\pi$
$67$	7.04476	0.860654	0.430327	+	0.902673i	$0.358398\pi$
$68$	5.62041	0.681575
$69$	4.58736	0.552253
$70$	2.10325	0.251386
$71$	−3.01510	−0.357826	−0.178913	−	0.983865i	$-0.557258\pi$
$71$	−3.01510	−0.357826	−0.178913	+	0.983865i	$0.557258\pi$
$72$	−4.56553	−0.538053
$73$	10.3532	1.21175	0.605874	−	0.795561i	$-0.292824\pi$
$73$	10.3532	1.21175	0.605874	+	0.795561i	$0.292824\pi$
$74$	−28.4317	−3.30512
$75$	1.94255	0.224306
$76$	2.50273	0.287082
$77$	3.78815	0.431700
$78$	30.8050	3.48798
$79$	1.27967	0.143974	0.0719871	−	0.997406i	$-0.477066\pi$
$79$	1.27967	0.143974	0.0719871	+	0.997406i	$0.477066\pi$
$80$	6.17912	0.690847
$81$	−10.7222	−1.19135
$82$	−28.3412	−3.12977
$83$	13.1165	1.43972	0.719860	−	0.694119i	$-0.244206\pi$
$83$	13.1165	1.43972	0.719860	+	0.694119i	$0.244206\pi$
$84$	−7.04594	−0.768775
$85$	1.29398	0.140351
$86$	−28.1419	−3.03462
$87$	1.53702	0.164786
$88$	26.7754	2.85426
$89$	3.69780	0.391966	0.195983	−	0.980607i	$-0.437210\pi$
$89$	3.69780	0.391966	0.195983	+	0.980607i	$0.437210\pi$
$90$	−1.94815	−0.205353
$91$	5.25787	0.551174
$92$	10.2573	1.06940
$93$	0	0
$94$	20.9802	2.16394
$95$	0.576198	0.0591167
$96$	−7.30012	−0.745066
$97$	−5.42095	−0.550414	−0.275207	−	0.961385i	$-0.588746\pi$
$97$	−5.42095	−0.550414	−0.275207	+	0.961385i	$0.588746\pi$
$98$	15.8741	1.60352
$99$	−3.50881	−0.352648
$100$	4.34352	0.434352
$101$	2.28428	0.227295	0.113647	−	0.993521i	$-0.463747\pi$
$101$	2.28428	0.227295	0.113647	+	0.993521i	$0.463747\pi$
$102$	−6.33087	−0.626849
$103$	−6.55110	−0.645499	−0.322750	−	0.946484i	$-0.604607\pi$
$103$	−6.55110	−0.645499	−0.322750	+	0.946484i	$0.604607\pi$
$104$	37.1636	3.64419
$105$	−1.62217	−0.158308
$106$	13.4451	1.30591
$107$	−4.63517	−0.448099	−0.224049	−	0.974578i	$-0.571928\pi$
$107$	−4.63517	−0.448099	−0.224049	+	0.974578i	$0.571928\pi$
$108$	−18.7861	−1.80770
$109$	−0.593703	−0.0568664	−0.0284332	−	0.999596i	$-0.509052\pi$
$109$	−0.593703	−0.0568664	−0.0284332	+	0.999596i	$0.509052\pi$
$110$	11.4253	1.08936
$111$	21.9285	2.08136
$112$	−5.16003	−0.487577
$113$	−9.65595	−0.908355	−0.454178	−	0.890911i	$-0.650067\pi$
$113$	−9.65595	−0.908355	−0.454178	+	0.890911i	$0.650067\pi$
$114$	−2.81909	−0.264032
$115$	2.36152	0.220213
$116$	3.43677	0.319096
$117$	−4.87014	−0.450245
$118$	13.4411	1.23736
$119$	−1.08057	−0.0990553
$120$	−11.4658	−1.04668
$121$	9.57803	0.870730
$122$	10.5460	0.954786
$123$	21.8588	1.97094
$124$	0	0
$125$	1.00000	0.0894427
$126$	1.62685	0.144931
$127$	12.2738	1.08913	0.544563	−	0.838720i	$-0.316696\pi$
$127$	12.2738	1.08913	0.544563	+	0.838720i	$0.316696\pi$
$128$	14.8029	1.30840
$129$	21.7050	1.91102
$130$	15.8580	1.39084
$131$	16.6130	1.45148	0.725742	−	0.687967i	$-0.241496\pi$
$131$	16.6130	1.45148	0.725742	+	0.687967i	$0.241496\pi$
$132$	−38.2750	−3.33141
$133$	−0.481168	−0.0417226
$134$	−17.7432	−1.53278
$135$	−4.32509	−0.372245
$136$	−7.63765	−0.654923
$137$	2.92972	0.250303	0.125151	−	0.992138i	$-0.460058\pi$
$137$	2.92972	0.250303	0.125151	+	0.992138i	$0.460058\pi$
$138$	−11.5539	−0.983532
$139$	21.4363	1.81820	0.909102	−	0.416573i	$-0.136769\pi$
$139$	21.4363	1.81820	0.909102	+	0.416573i	$0.136769\pi$
$140$	−3.62716	−0.306551
$141$	−16.1814	−1.36272
$142$	7.59392	0.637268
$143$	28.5618	2.38846
$144$	4.77952	0.398293
$145$	0.791240	0.0657089
$146$	−26.0759	−2.15805
$147$	−12.2432	−1.00980
$148$	49.0320	4.03040
$149$	−7.89936	−0.647141	−0.323571	−	0.946204i	$-0.604883\pi$
$149$	−7.89936	−0.647141	−0.323571	+	0.946204i	$0.604883\pi$
$150$	−4.89257	−0.399477
$151$	−2.22010	−0.180669	−0.0903344	−	0.995911i	$-0.528794\pi$
$151$	−2.22010	−0.180669	−0.0903344	+	0.995911i	$0.528794\pi$
$152$	−3.40099	−0.275857
$153$	1.00088	0.0809166
$154$	−9.54097	−0.768833
$155$	0	0
$156$	−53.1249	−4.25339
$157$	5.22940	0.417351	0.208676	−	0.977985i	$-0.433085\pi$
$157$	5.22940	0.417351	0.208676	+	0.977985i	$0.433085\pi$
$158$	−3.22302	−0.256410
$159$	−10.3698	−0.822382
$160$	−3.75801	−0.297097
$161$	−1.97204	−0.155419
$162$	27.0053	2.12174
$163$	13.2780	1.04002	0.520008	−	0.854162i	$-0.325929\pi$
$163$	13.2780	1.04002	0.520008	+	0.854162i	$0.325929\pi$
$164$	48.8760	3.81657
$165$	−8.81198	−0.686012
$166$	−33.0356	−2.56406
$167$	6.35272	0.491588	0.245794	−	0.969322i	$-0.420951\pi$
$167$	6.35272	0.491588	0.245794	+	0.969322i	$0.420951\pi$
$168$	9.57483	0.738714
$169$	26.6432	2.04947
$170$	−3.25905	−0.249958
$171$	0.445686	0.0340825
$172$	48.5322	3.70055
$173$	2.52131	0.191691	0.0958456	−	0.995396i	$-0.469444\pi$
$173$	2.52131	0.191691	0.0958456	+	0.995396i	$0.469444\pi$
$174$	−3.87120	−0.293475
$175$	−0.835075	−0.0631257
$176$	−28.0304	−2.11287
$177$	−10.3667	−0.779212
$178$	−9.31340	−0.698069
$179$	−11.9518	−0.893317	−0.446658	−	0.894705i	$-0.647386\pi$
$179$	−11.9518	−0.893317	−0.446658	+	0.894705i	$0.647386\pi$
$180$	3.35969	0.250417
$181$	14.0948	1.04766	0.523830	−	0.851823i	$-0.324503\pi$
$181$	14.0948	1.04766	0.523830	+	0.851823i	$0.324503\pi$
$182$	−13.2426	−0.981610
$183$	−8.13378	−0.601267
$184$	−13.9388	−1.02758
$185$	11.2885	0.829950
$186$	0	0
$187$	−5.86986	−0.429247
$188$	−36.1814	−2.63880
$189$	3.61178	0.262718
$190$	−1.45123	−0.105283
$191$	0.281408	0.0203620	0.0101810	−	0.999948i	$-0.496759\pi$
$191$	0.281408	0.0203620	0.0101810	+	0.999948i	$0.496759\pi$
$192$	−5.62015	−0.405599
$193$	24.1896	1.74121	0.870603	−	0.491986i	$-0.163729\pi$
$193$	24.1896	1.74121	0.870603	+	0.491986i	$0.163729\pi$
$194$	13.6534	0.980256
$195$	−12.2308	−0.875868
$196$	−27.3757	−1.95541
$197$	−12.7405	−0.907724	−0.453862	−	0.891072i	$-0.649954\pi$
$197$	−12.7405	−0.907724	−0.453862	+	0.891072i	$0.649954\pi$
$198$	8.83740	0.628047
$199$	10.8047	0.765923	0.382962	−	0.923764i	$-0.374904\pi$
$199$	10.8047	0.765923	0.382962	+	0.923764i	$0.374904\pi$
$200$	−5.90247	−0.417368
$201$	13.6848	0.965250
$202$	−5.75327	−0.404799
$203$	−0.660744	−0.0463752
$204$	10.9179	0.764407
$205$	11.2526	0.785917
$206$	16.4998	1.14960
$207$	1.82662	0.126959
$208$	−38.9055	−2.69761
$209$	−2.61381	−0.180801
$210$	4.08566	0.281937
$211$	−4.39815	−0.302781	−0.151391	−	0.988474i	$-0.548375\pi$
$211$	−4.39815	−0.302781	−0.151391	+	0.988474i	$0.548375\pi$
$212$	−23.1869	−1.59248
$213$	−5.85697	−0.401313
$214$	11.6743	0.798038
$215$	11.1735	0.762025
$216$	25.5287	1.73701
$217$	0	0
$218$	1.49532	0.101276
$219$	20.1116	1.35901
$220$	−19.7035	−1.32841
$221$	−8.14724	−0.548042
$222$	−55.2300	−3.70679
$223$	17.5976	1.17842	0.589210	−	0.807980i	$-0.299439\pi$
$223$	17.5976	1.17842	0.589210	+	0.807980i	$0.299439\pi$
$224$	3.13822	0.209681
$225$	0.773495	0.0515663
$226$	24.3198	1.61773
$227$	22.3421	1.48290	0.741449	−	0.671009i	$-0.234139\pi$
$227$	22.3421	1.48290	0.741449	+	0.671009i	$0.234139\pi$
$228$	4.86167	0.321972
$229$	0.146243	0.00966399	0.00483199	−	0.999988i	$-0.498462\pi$
$229$	0.146243	0.00966399	0.00483199	+	0.999988i	$0.498462\pi$
$230$	−5.94780	−0.392186
$231$	7.35867	0.484165
$232$	−4.67027	−0.306618
$233$	8.68948	0.569267	0.284633	−	0.958636i	$-0.408128\pi$
$233$	8.68948	0.569267	0.284633	+	0.958636i	$0.408128\pi$
$234$	12.2661	0.801861
$235$	−8.32997	−0.543387
$236$	−23.1799	−1.50889
$237$	2.48582	0.161471
$238$	2.72155	0.176412
$239$	−10.4622	−0.676742	−0.338371	−	0.941013i	$-0.609876\pi$
$239$	−10.4622	−0.676742	−0.338371	+	0.941013i	$0.609876\pi$
$240$	12.0032	0.774806
$241$	5.44068	0.350465	0.175232	−	0.984527i	$-0.443932\pi$
$241$	5.44068	0.350465	0.175232	+	0.984527i	$0.443932\pi$
$242$	−24.1235	−1.55072
$243$	−7.85309	−0.503776
$244$	−18.1871	−1.16431
$245$	−6.30265	−0.402662
$246$	−55.0542	−3.51013
$247$	−3.62790	−0.230838
$248$	0	0
$249$	25.4794	1.61469
$250$	−2.51863	−0.159292
$251$	−11.5588	−0.729587	−0.364793	−	0.931088i	$-0.618860\pi$
$251$	−11.5588	−0.729587	−0.364793	+	0.931088i	$0.618860\pi$
$252$	−2.80559	−0.176736
$253$	−10.7125	−0.673492
$254$	−30.9133	−1.93967
$255$	2.51361	0.157408
$256$	−31.4967	−1.96855
$257$	18.9720	1.18344	0.591721	−	0.806143i	$-0.298449\pi$
$257$	18.9720	1.18344	0.591721	+	0.806143i	$0.298449\pi$
$258$	−54.6670	−3.40342
$259$	−9.42677	−0.585751
$260$	−27.3480	−1.69605
$261$	0.612020	0.0378831
$262$	−41.8421	−2.58501
$263$	5.21477	0.321556	0.160778	−	0.986991i	$-0.448600\pi$
$263$	5.21477	0.321556	0.160778	+	0.986991i	$0.448600\pi$
$264$	52.0125	3.20115
$265$	−5.33827	−0.327927
$266$	1.21189	0.0743056
$267$	7.18315	0.439602
$268$	30.5990	1.86913
$269$	1.14977	0.0701028	0.0350514	−	0.999386i	$-0.488841\pi$
$269$	1.14977	0.0701028	0.0350514	+	0.999386i	$0.488841\pi$
$270$	10.8933	0.662947
$271$	12.5753	0.763897	0.381949	−	0.924184i	$-0.375253\pi$
$271$	12.5753	0.763897	0.381949	+	0.924184i	$0.375253\pi$
$272$	7.99563	0.484806
$273$	10.2137	0.618159
$274$	−7.37889	−0.445775
$275$	−4.53630	−0.273549
$276$	19.9253	1.19936
$277$	−21.8170	−1.31086	−0.655429	−	0.755256i	$-0.727512\pi$
$277$	−21.8170	−1.31086	−0.655429	+	0.755256i	$0.727512\pi$
$278$	−53.9902	−3.23812
$279$	0	0
$280$	4.92900	0.294564
$281$	−31.1183	−1.85636	−0.928182	−	0.372126i	$-0.878629\pi$
$281$	−31.1183	−1.85636	−0.928182	+	0.372126i	$0.878629\pi$
$282$	40.7550	2.42692
$283$	18.1298	1.07771	0.538854	−	0.842399i	$-0.318858\pi$
$283$	18.1298	1.07771	0.538854	+	0.842399i	$0.318858\pi$
$284$	−13.0961	−0.777112
$285$	1.11929	0.0663012
$286$	−71.9368	−4.25371
$287$	−9.39678	−0.554674
$288$	−2.90680	−0.171285
$289$	−15.3256	−0.901507
$290$	−1.99284	−0.117024
$291$	−10.5305	−0.617306
$292$	44.9692	2.63163
$293$	−11.7062	−0.683886	−0.341943	−	0.939721i	$-0.611085\pi$
$293$	−11.7062	−0.683886	−0.341943	+	0.939721i	$0.611085\pi$
$294$	30.8362	1.79840
$295$	−5.33667	−0.310713
$296$	−66.6302	−3.87280
$297$	19.6199	1.13846
$298$	19.8956	1.15252
$299$	−14.8688	−0.859883
$300$	8.43750	0.487139
$301$	−9.33069	−0.537812
$302$	5.59161	0.321761
$303$	4.43733	0.254918
$304$	3.56040	0.204203
$305$	−4.18717	−0.239757
$306$	−2.52086	−0.144108
$307$	−11.7435	−0.670239	−0.335120	−	0.942176i	$-0.608777\pi$
$307$	−11.7435	−0.670239	−0.335120	+	0.942176i	$0.608777\pi$
$308$	16.4539	0.937548
$309$	−12.7258	−0.723947
$310$	0	0
$311$	−0.833220	−0.0472476	−0.0236238	−	0.999721i	$-0.507520\pi$
$311$	−0.833220	−0.0472476	−0.0236238	+	0.999721i	$0.507520\pi$
$312$	72.1921	4.08707
$313$	10.5958	0.598908	0.299454	−	0.954111i	$-0.403196\pi$
$313$	10.5958	0.598908	0.299454	+	0.954111i	$0.403196\pi$
$314$	−13.1709	−0.743279
$315$	−0.645926	−0.0363938
$316$	5.55827	0.312677
$317$	31.1065	1.74712	0.873558	−	0.486720i	$-0.161807\pi$
$317$	31.1065	1.74712	0.873558	+	0.486720i	$0.161807\pi$
$318$	26.1179	1.46462
$319$	−3.58930	−0.200962
$320$	−2.89318	−0.161734
$321$	−9.00404	−0.502556
$322$	4.96685	0.276792
$323$	0.745586	0.0414855
$324$	−46.5720	−2.58734
$325$	−6.29628	−0.349255
$326$	−33.4425	−1.85221
$327$	−1.15330	−0.0637775
$328$	−66.4183	−3.66733
$329$	6.95615	0.383505
$330$	22.1942	1.22175
$331$	−8.93557	−0.491143	−0.245572	−	0.969378i	$-0.578976\pi$
$331$	−8.93557	−0.491143	−0.245572	+	0.969378i	$0.578976\pi$
$332$	56.9716	3.12673
$333$	8.73163	0.478490
$334$	−16.0002	−0.875491
$335$	7.04476	0.384896
$336$	−10.0236	−0.546833
$337$	23.3018	1.26933	0.634664	−	0.772788i	$-0.281139\pi$
$337$	23.3018	1.26933	0.634664	+	0.772788i	$0.281139\pi$
$338$	−67.1044	−3.65000
$339$	−18.7571	−1.01875
$340$	5.62041	0.304809
$341$	0	0
$342$	−1.12252	−0.0606990
$343$	11.1087	0.599814
$344$	−65.9511	−3.55584
$345$	4.58736	0.246975
$346$	−6.35025	−0.341391
$347$	−19.2651	−1.03421	−0.517103	−	0.855923i	$-0.672989\pi$
$347$	−19.2651	−1.03421	−0.517103	+	0.855923i	$0.672989\pi$
$348$	6.67608	0.357876
$349$	32.6225	1.74624	0.873122	−	0.487501i	$-0.162092\pi$
$349$	32.6225	1.74624	0.873122	+	0.487501i	$0.162092\pi$
$350$	2.10325	0.112423
$351$	27.2320	1.45354
$352$	17.0475	0.908634
$353$	−12.7241	−0.677236	−0.338618	−	0.940924i	$-0.609959\pi$
$353$	−12.7241	−0.677236	−0.338618	+	0.940924i	$0.609959\pi$
$354$	26.1100	1.38773
$355$	−3.01510	−0.160025
$356$	16.0615	0.851255
$357$	−2.09905	−0.111094
$358$	30.1021	1.59095
$359$	−6.41197	−0.338411	−0.169205	−	0.985581i	$-0.554120\pi$
$359$	−6.41197	−0.338411	−0.169205	+	0.985581i	$0.554120\pi$
$360$	−4.56553	−0.240625
$361$	−18.6680	−0.982526
$362$	−35.4997	−1.86582
$363$	18.6058	0.976550
$364$	22.8376	1.19702
$365$	10.3532	0.541910
$366$	20.4860	1.07082
$367$	21.8942	1.14287	0.571435	−	0.820647i	$-0.306387\pi$
$367$	21.8942	1.14287	0.571435	+	0.820647i	$0.306387\pi$
$368$	14.5921	0.760666
$369$	8.70385	0.453104
$370$	−28.4317	−1.47809
$371$	4.45785	0.231440
$372$	0	0
$373$	4.62775	0.239616	0.119808	−	0.992797i	$-0.461772\pi$
$373$	4.62775	0.239616	0.119808	+	0.992797i	$0.461772\pi$
$374$	14.7840	0.764464
$375$	1.94255	0.100313
$376$	49.1674	2.53562
$377$	−4.98187	−0.256579
$378$	−9.09675	−0.467886
$379$	−14.4500	−0.742245	−0.371122	−	0.928584i	$-0.621027\pi$
$379$	−14.4500	−0.742245	−0.371122	+	0.928584i	$0.621027\pi$
$380$	2.50273	0.128387
$381$	23.8425	1.22149
$382$	−0.708764	−0.0362635
$383$	20.4019	1.04249	0.521243	−	0.853408i	$-0.325468\pi$
$383$	20.4019	1.04249	0.521243	+	0.853408i	$0.325468\pi$
$384$	28.7553	1.46742
$385$	3.78815	0.193062
$386$	−60.9248	−3.10099
$387$	8.64263	0.439329
$388$	−23.5460	−1.19537
$389$	6.24665	0.316718	0.158359	−	0.987382i	$-0.449380\pi$
$389$	6.24665	0.316718	0.158359	+	0.987382i	$0.449380\pi$
$390$	30.8050	1.55987
$391$	3.05574	0.154536
$392$	37.2012	1.87894
$393$	32.2716	1.62789
$394$	32.0887	1.61661
$395$	1.27967	0.0643872
$396$	−15.2406	−0.765867
$397$	−31.5910	−1.58551	−0.792753	−	0.609543i	$-0.791353\pi$
$397$	−31.5910	−1.58551	−0.792753	+	0.609543i	$0.791353\pi$
$398$	−27.2130	−1.36407
$399$	−0.934693	−0.0467932
$400$	6.17912	0.308956
$401$	2.07164	0.103453	0.0517264	−	0.998661i	$-0.483528\pi$
$401$	2.07164	0.103453	0.0517264	+	0.998661i	$0.483528\pi$
$402$	−34.4670	−1.71906
$403$	0	0
$404$	9.92182	0.493629
$405$	−10.7222	−0.532790
$406$	1.66417	0.0825916
$407$	−51.2082	−2.53830
$408$	−14.8365	−0.734517
$409$	23.9119	1.18237	0.591184	−	0.806537i	$-0.298661\pi$
$409$	23.9119	1.18237	0.591184	+	0.806537i	$0.298661\pi$
$410$	−28.3412	−1.39967
$411$	5.69112	0.280722
$412$	−28.4548	−1.40187
$413$	4.45652	0.219291
$414$	−4.60059	−0.226107
$415$	13.1165	0.643862
$416$	23.6615	1.16010
$417$	41.6411	2.03917
$418$	6.58322	0.321996
$419$	−30.5648	−1.49319	−0.746594	−	0.665280i	$-0.768312\pi$
$419$	−30.5648	−1.49319	−0.746594	+	0.665280i	$0.768312\pi$
$420$	−7.04594	−0.343807
$421$	22.4509	1.09419	0.547095	−	0.837071i	$-0.315734\pi$
$421$	22.4509	1.09419	0.547095	+	0.837071i	$0.315734\pi$
$422$	11.0773	0.539236
$423$	−6.44319	−0.313279
$424$	31.5090	1.53021
$425$	1.29398	0.0627670
$426$	14.7516	0.714716
$427$	3.49660	0.169212
$428$	−20.1329	−0.973162
$429$	55.4827	2.67873
$430$	−28.1419	−1.35712
$431$	−16.4789	−0.793759	−0.396880	−	0.917871i	$-0.629907\pi$
$431$	−16.4789	−0.793759	−0.396880	+	0.917871i	$0.629907\pi$
$432$	−26.7253	−1.28582
$433$	30.3372	1.45791	0.728957	−	0.684560i	$-0.240006\pi$
$433$	30.3372	1.45791	0.728957	+	0.684560i	$0.240006\pi$
$434$	0	0
$435$	1.53702	0.0736946
$436$	−2.57876	−0.123500
$437$	1.36070	0.0650912
$438$	−50.6536	−2.42032
$439$	−16.5039	−0.787690	−0.393845	−	0.919177i	$-0.628855\pi$
$439$	−16.5039	−0.787690	−0.393845	+	0.919177i	$0.628855\pi$
$440$	26.7754	1.27647
$441$	−4.87507	−0.232146
$442$	20.5199	0.976032
$443$	−25.2453	−1.19944	−0.599719	−	0.800210i	$-0.704721\pi$
$443$	−25.2453	−1.19944	−0.599719	+	0.800210i	$0.704721\pi$
$444$	95.2470	4.52022
$445$	3.69780	0.175292
$446$	−44.3218	−2.09870
$447$	−15.3449	−0.725789
$448$	2.41602	0.114146
$449$	2.61039	0.123192	0.0615960	−	0.998101i	$-0.480381\pi$
$449$	2.61039	0.123192	0.0615960	+	0.998101i	$0.480381\pi$
$450$	−1.94815	−0.0918367
$451$	−51.0453	−2.40363
$452$	−41.9408	−1.97273
$453$	−4.31264	−0.202626
$454$	−56.2716	−2.64096
$455$	5.25787	0.246493
$456$	−6.60659	−0.309382
$457$	−9.36213	−0.437942	−0.218971	−	0.975731i	$-0.570270\pi$
$457$	−9.36213	−0.437942	−0.218971	+	0.975731i	$0.570270\pi$
$458$	−0.368332	−0.0172110
$459$	−5.59657	−0.261225
$460$	10.2573	0.478249
$461$	−17.6338	−0.821286	−0.410643	−	0.911796i	$-0.634696\pi$
$461$	−17.6338	−0.821286	−0.410643	+	0.911796i	$0.634696\pi$
$462$	−18.5338	−0.862270
$463$	5.81279	0.270143	0.135072	−	0.990836i	$-0.456874\pi$
$463$	5.81279	0.270143	0.135072	+	0.990836i	$0.456874\pi$
$464$	4.88917	0.226974
$465$	0	0
$466$	−21.8856	−1.01383
$467$	8.08921	0.374324	0.187162	−	0.982329i	$-0.440071\pi$
$467$	8.08921	0.374324	0.187162	+	0.982329i	$0.440071\pi$
$468$	−21.1536	−0.977824
$469$	−5.88290	−0.271647
$470$	20.9802	0.967742
$471$	10.1584	0.468072
$472$	31.4995	1.44988
$473$	−50.6862	−2.33056
$474$	−6.26088	−0.287572
$475$	0.576198	0.0264378
$476$	−4.69346	−0.215124
$477$	−4.12912	−0.189060
$478$	26.3504	1.20524
$479$	9.10527	0.416031	0.208015	−	0.978126i	$-0.433300\pi$
$479$	9.10527	0.416031	0.208015	+	0.978126i	$0.433300\pi$
$480$	−7.30012	−0.333204
$481$	−71.0758	−3.24078
$482$	−13.7031	−0.624158
$483$	−3.83079	−0.174307
$484$	41.6023	1.89102
$485$	−5.42095	−0.246153
$486$	19.7791	0.897197
$487$	41.1471	1.86455	0.932275	−	0.361749i	$-0.117820\pi$
$487$	41.1471	1.86455	0.932275	+	0.361749i	$0.117820\pi$
$488$	24.7146	1.11878
$489$	25.7932	1.16641
$490$	15.8741	0.717117
$491$	18.9228	0.853973	0.426986	−	0.904258i	$-0.359575\pi$
$491$	18.9228	0.853973	0.426986	+	0.904258i	$0.359575\pi$
$492$	94.9440	4.28041
$493$	1.02385	0.0461117
$494$	9.13737	0.411110
$495$	−3.50881	−0.157709
$496$	0	0
$497$	2.51783	0.112940
$498$	−64.1732	−2.87567
$499$	7.95800	0.356249	0.178124	−	0.984008i	$-0.442997\pi$
$499$	7.95800	0.356249	0.178124	+	0.984008i	$0.442997\pi$
$500$	4.34352	0.194248
$501$	12.3405	0.551331
$502$	29.1125	1.29935
$503$	−7.52349	−0.335456	−0.167728	−	0.985833i	$-0.553643\pi$
$503$	−7.52349	−0.335456	−0.167728	+	0.985833i	$0.553643\pi$
$504$	3.81256	0.169825
$505$	2.28428	0.101649
$506$	26.9810	1.19945
$507$	51.7557	2.29855
$508$	53.3116	2.36532
$509$	13.4310	0.595320	0.297660	−	0.954672i	$-0.403794\pi$
$509$	13.4310	0.595320	0.297660	+	0.954672i	$0.403794\pi$
$510$	−6.33087	−0.280335
$511$	−8.64568	−0.382462
$512$	49.7229	2.19746
$513$	−2.49211	−0.110029
$514$	−47.7836	−2.10764
$515$	−6.55110	−0.288676
$516$	94.2762	4.15028
$517$	37.7873	1.66188
$518$	23.7426	1.04319
$519$	4.89776	0.214988
$520$	37.1636	1.62973
$521$	−19.4911	−0.853922	−0.426961	−	0.904270i	$-0.640416\pi$
$521$	−19.4911	−0.853922	−0.426961	+	0.904270i	$0.640416\pi$
$522$	−1.54145	−0.0674676
$523$	−6.26080	−0.273766	−0.136883	−	0.990587i	$-0.543708\pi$
$523$	−6.26080	−0.273766	−0.136883	+	0.990587i	$0.543708\pi$
$524$	72.1589	3.15228
$525$	−1.62217	−0.0707974
$526$	−13.1341	−0.572674
$527$	0	0
$528$	−54.4503	−2.36965
$529$	−17.4232	−0.757532
$530$	13.4451	0.584020
$531$	−4.12789	−0.179135
$532$	−2.08996	−0.0906114
$533$	−70.8497	−3.06884
$534$	−18.0917	−0.782906
$535$	−4.63517	−0.200396
$536$	−41.5815	−1.79605
$537$	−23.2169	−1.00188
$538$	−2.89585	−0.124849
$539$	28.5907	1.23149
$540$	−18.7861	−0.808426
$541$	36.5452	1.57120	0.785601	−	0.618734i	$-0.212354\pi$
$541$	36.5452	1.57120	0.785601	+	0.618734i	$0.212354\pi$
$542$	−31.6727	−1.36046
$543$	27.3799	1.17498
$544$	−4.86278	−0.208490
$545$	−0.593703	−0.0254314
$546$	−25.7245	−1.10091
$547$	11.3749	0.486357	0.243178	−	0.969982i	$-0.421810\pi$
$547$	11.3749	0.486357	0.243178	+	0.969982i	$0.421810\pi$
$548$	12.7253	0.543597
$549$	−3.23876	−0.138227
$550$	11.4253	0.487176
$551$	0.455911	0.0194225
$552$	−27.0768	−1.15246
$553$	−1.06862	−0.0454424
$554$	54.9491	2.33457
$555$	21.9285	0.930814
$556$	93.1091	3.94870
$557$	17.9263	0.759560	0.379780	−	0.925077i	$-0.376000\pi$
$557$	17.9263	0.759560	0.379780	+	0.925077i	$0.376000\pi$
$558$	0	0
$559$	−70.3514	−2.97555
$560$	−5.16003	−0.218051
$561$	−11.4025	−0.481414
$562$	78.3757	3.30608
$563$	33.1937	1.39895	0.699474	−	0.714658i	$-0.253418\pi$
$563$	33.1937	1.39895	0.699474	+	0.714658i	$0.253418\pi$
$564$	−70.2841	−2.95950
$565$	−9.65595	−0.406229
$566$	−45.6625	−1.91934
$567$	8.95383	0.376026
$568$	17.7965	0.746725
$569$	27.2963	1.14432	0.572160	−	0.820142i	$-0.306106\pi$
$569$	27.2963	1.14432	0.572160	+	0.820142i	$0.306106\pi$
$570$	−2.81909	−0.118079
$571$	29.6640	1.24140	0.620700	−	0.784048i	$-0.286848\pi$
$571$	29.6640	1.24140	0.620700	+	0.784048i	$0.286848\pi$
$572$	124.059	5.18716
$573$	0.546649	0.0228366
$574$	23.6671	0.987844
$575$	2.36152	0.0984820
$576$	−2.23786	−0.0932443
$577$	−2.01933	−0.0840658	−0.0420329	−	0.999116i	$-0.513383\pi$
$577$	−2.01933	−0.0840658	−0.0420329	+	0.999116i	$0.513383\pi$
$578$	38.5997	1.60553
$579$	46.9895	1.95282
$580$	3.43677	0.142704
$581$	−10.9532	−0.454417
$582$	26.5224	1.09939
$583$	24.2160	1.00292
$584$	−61.1093	−2.52872
$585$	−4.87014	−0.201356
$586$	29.4838	1.21796
$587$	3.41140	0.140804	0.0704018	−	0.997519i	$-0.477572\pi$
$587$	3.41140	0.140804	0.0704018	+	0.997519i	$0.477572\pi$
$588$	−53.1786	−2.19305
$589$	0	0
$590$	13.4411	0.553362
$591$	−24.7491	−1.01804
$592$	69.7533	2.86684
$593$	−14.5117	−0.595924	−0.297962	−	0.954578i	$-0.596307\pi$
$593$	−14.5117	−0.595924	−0.297962	+	0.954578i	$0.596307\pi$
$594$	−49.4154	−2.02754
$595$	−1.08057	−0.0442989
$596$	−34.3110	−1.40544
$597$	20.9886	0.859006
$598$	37.4490	1.53140
$599$	0.564285	0.0230561	0.0115280	−	0.999934i	$-0.496330\pi$
$599$	0.564285	0.0230561	0.0115280	+	0.999934i	$0.496330\pi$
$600$	−11.4658	−0.468091
$601$	−21.7838	−0.888581	−0.444291	−	0.895883i	$-0.646544\pi$
$601$	−21.7838	−0.888581	−0.444291	+	0.895883i	$0.646544\pi$
$602$	23.5006	0.957813
$603$	5.44908	0.221904
$604$	−9.64303	−0.392369
$605$	9.57803	0.389402
$606$	−11.1760	−0.453994
$607$	−10.4251	−0.423141	−0.211571	−	0.977363i	$-0.567858\pi$
$607$	−10.4251	−0.423141	−0.211571	+	0.977363i	$0.567858\pi$
$608$	−2.16536	−0.0878169
$609$	−1.28353	−0.0520112
$610$	10.5460	0.426993
$611$	52.4479	2.12181
$612$	4.34736	0.175731
$613$	−16.5840	−0.669821	−0.334911	−	0.942250i	$-0.608706\pi$
$613$	−16.5840	−0.669821	−0.334911	+	0.942250i	$0.608706\pi$
$614$	29.5777	1.19366
$615$	21.8588	0.881431
$616$	−22.3594	−0.900888
$617$	15.4160	0.620624	0.310312	−	0.950635i	$-0.399567\pi$
$617$	15.4160	0.620624	0.310312	+	0.950635i	$0.399567\pi$
$618$	32.0517	1.28931
$619$	40.5205	1.62866	0.814328	−	0.580406i	$-0.197106\pi$
$619$	40.5205	1.62866	0.814328	+	0.580406i	$0.197106\pi$
$620$	0	0
$621$	−10.2138	−0.409865
$622$	2.09858	0.0841453
$623$	−3.08794	−0.123716
$624$	−75.5758	−3.02545
$625$	1.00000	0.0400000
$626$	−26.6868	−1.06662
$627$	−5.07745	−0.202774
$628$	22.7140	0.906387
$629$	14.6071	0.582423
$630$	1.62685	0.0648153
$631$	−7.93456	−0.315870	−0.157935	−	0.987450i	$-0.550484\pi$
$631$	−7.93456	−0.315870	−0.157935	+	0.987450i	$0.550484\pi$
$632$	−7.55322	−0.300451
$633$	−8.54362	−0.339578
$634$	−78.3459	−3.11151
$635$	12.2738	0.487072
$636$	−45.0416	−1.78602
$637$	39.6833	1.57231
$638$	9.04014	0.357903
$639$	−2.33216	−0.0922589
$640$	14.8029	0.585136
$641$	−32.4666	−1.28235	−0.641177	−	0.767393i	$-0.721554\pi$
$641$	−32.4666	−1.28235	−0.641177	+	0.767393i	$0.721554\pi$
$642$	22.6779	0.895025
$643$	−35.7326	−1.40916	−0.704578	−	0.709627i	$-0.748863\pi$
$643$	−35.7326	−1.40916	−0.704578	+	0.709627i	$0.748863\pi$
$644$	−8.56561	−0.337532
$645$	21.7050	0.854634
$646$	−1.87786	−0.0738834
$647$	−4.69934	−0.184750	−0.0923750	−	0.995724i	$-0.529446\pi$
$647$	−4.69934	−0.184750	−0.0923750	+	0.995724i	$0.529446\pi$
$648$	63.2874	2.48616
$649$	24.2087	0.950276
$650$	15.8580	0.622003
$651$	0	0
$652$	57.6734	2.25866
$653$	44.2313	1.73090	0.865451	−	0.500993i	$-0.167032\pi$
$653$	44.2313	1.73090	0.865451	+	0.500993i	$0.167032\pi$
$654$	2.90473	0.113584
$655$	16.6130	0.649124
$656$	69.5313	2.71474
$657$	8.00813	0.312427
$658$	−17.5200	−0.683001
$659$	−14.3094	−0.557416	−0.278708	−	0.960376i	$-0.589906\pi$
$659$	−14.3094	−0.557416	−0.278708	+	0.960376i	$0.589906\pi$
$660$	−38.2750	−1.48985
$661$	−19.4905	−0.758094	−0.379047	−	0.925377i	$-0.623748\pi$
$661$	−19.4905	−0.758094	−0.379047	+	0.925377i	$0.623748\pi$
$662$	22.5054	0.874699
$663$	−15.8264	−0.614646
$664$	−77.4196	−3.00446
$665$	−0.481168	−0.0186589
$666$	−21.9918	−0.852164
$667$	1.86853	0.0723496
$668$	27.5932	1.06761
$669$	34.1841	1.32163
$670$	−17.7432	−0.685478
$671$	18.9943	0.733266
$672$	6.09615	0.235164
$673$	−10.4725	−0.403684	−0.201842	−	0.979418i	$-0.564693\pi$
$673$	−10.4725	−0.403684	−0.201842	+	0.979418i	$0.564693\pi$
$674$	−58.6886	−2.26060
$675$	−4.32509	−0.166473
$676$	115.725	4.45097
$677$	−28.2809	−1.08692	−0.543462	−	0.839434i	$-0.682887\pi$
$677$	−28.2809	−1.08692	−0.543462	+	0.839434i	$0.682887\pi$
$678$	47.2424	1.81433
$679$	4.52690	0.173726
$680$	−7.63765	−0.292891
$681$	43.4006	1.66312
$682$	0	0
$683$	2.92111	0.111773	0.0558866	−	0.998437i	$-0.482201\pi$
$683$	2.92111	0.111773	0.0558866	+	0.998437i	$0.482201\pi$
$684$	1.93585	0.0740189
$685$	2.92972	0.111939
$686$	−27.9788	−1.06823
$687$	0.284083	0.0108385
$688$	69.0423	2.63221
$689$	33.6112	1.28049
$690$	−11.5539	−0.439849
$691$	−22.5973	−0.859640	−0.429820	−	0.902915i	$-0.641423\pi$
$691$	−22.5973	−0.859640	−0.429820	+	0.902915i	$0.641423\pi$
$692$	10.9513	0.416307
$693$	2.93012	0.111306
$694$	48.5218	1.84186
$695$	21.4363	0.813126
$696$	−9.07222	−0.343882
$697$	14.5606	0.551523
$698$	−82.1643	−3.10996
$699$	16.8797	0.638450
$700$	−3.62716	−0.137094
$701$	−28.2991	−1.06884	−0.534421	−	0.845218i	$-0.679470\pi$
$701$	−28.2991	−1.06884	−0.534421	+	0.845218i	$0.679470\pi$
$702$	−68.5875	−2.58867
$703$	6.50443	0.245319
$704$	13.1243	0.494642
$705$	−16.1814	−0.609426
$706$	32.0474	1.20612
$707$	−1.90755	−0.0717407
$708$	−45.0281	−1.69226
$709$	−16.0694	−0.603499	−0.301750	−	0.953387i	$-0.597571\pi$
$709$	−16.0694	−0.603499	−0.301750	+	0.953387i	$0.597571\pi$
$710$	7.59392	0.284995
$711$	0.989819	0.0371211
$712$	−21.8261	−0.817969
$713$	0	0
$714$	5.28675	0.197851
$715$	28.5618	1.06815
$716$	−51.9127	−1.94007
$717$	−20.3233	−0.758987
$718$	16.1494	0.602690
$719$	28.5219	1.06369	0.531843	−	0.846843i	$-0.321500\pi$
$719$	28.5219	1.06369	0.531843	+	0.846843i	$0.321500\pi$
$720$	4.77952	0.178122
$721$	5.47066	0.203738
$722$	47.0179	1.74982
$723$	10.5688	0.393057
$724$	61.2212	2.27527
$725$	0.791240	0.0293859
$726$	−46.8612	−1.73918
$727$	5.49484	0.203792	0.101896	−	0.994795i	$-0.467509\pi$
$727$	5.49484	0.203792	0.101896	+	0.994795i	$0.467509\pi$
$728$	−31.0344	−1.15021
$729$	16.9116	0.626354
$730$	−26.0759	−0.965111
$731$	14.4582	0.534756
$732$	−35.3292	−1.30581
$733$	41.0460	1.51607	0.758033	−	0.652216i	$-0.226160\pi$
$733$	41.0460	1.51607	0.758033	+	0.652216i	$0.226160\pi$
$734$	−55.1436	−2.03539
$735$	−12.2432	−0.451597
$736$	−8.87461	−0.327123
$737$	−31.9571	−1.17716
$738$	−21.9218	−0.806953
$739$	−12.8579	−0.472984	−0.236492	−	0.971633i	$-0.575998\pi$
$739$	−12.8579	−0.472984	−0.236492	+	0.971633i	$0.575998\pi$
$740$	49.0320	1.80245
$741$	−7.04738	−0.258892
$742$	−11.2277	−0.412182
$743$	5.08104	0.186405	0.0932026	−	0.995647i	$-0.470290\pi$
$743$	5.08104	0.186405	0.0932026	+	0.995647i	$0.470290\pi$
$744$	0	0
$745$	−7.89936	−0.289410
$746$	−11.6556	−0.426742
$747$	10.1455	0.371205
$748$	−25.4959	−0.932221
$749$	3.87071	0.141433
$750$	−4.89257	−0.178651
$751$	−6.97125	−0.254384	−0.127192	−	0.991878i	$-0.540597\pi$
$751$	−6.97125	−0.254384	−0.127192	+	0.991878i	$0.540597\pi$
$752$	−51.4719	−1.87699
$753$	−22.4536	−0.818254
$754$	12.5475	0.456953
$755$	−2.22010	−0.0807976
$756$	15.6878	0.570561
$757$	−14.6551	−0.532648	−0.266324	−	0.963884i	$-0.585809\pi$
$757$	−14.6551	−0.532648	−0.266324	+	0.963884i	$0.585809\pi$
$758$	36.3942	1.32190
$759$	−20.8096	−0.755342
$760$	−3.40099	−0.123367
$761$	31.5193	1.14257	0.571286	−	0.820751i	$-0.306445\pi$
$761$	31.5193	1.14257	0.571286	+	0.820751i	$0.306445\pi$
$762$	−60.0506	−2.17540
$763$	0.495786	0.0179487
$764$	1.22230	0.0442213
$765$	1.00088	0.0361870
$766$	−51.3848	−1.85661
$767$	33.6012	1.21327
$768$	−61.1839	−2.20778
$769$	43.7791	1.57872	0.789358	−	0.613933i	$-0.210413\pi$
$769$	43.7791	1.57872	0.789358	+	0.613933i	$0.210413\pi$
$770$	−9.54097	−0.343833
$771$	36.8541	1.32727
$772$	105.068	3.78148
$773$	24.2956	0.873851	0.436926	−	0.899498i	$-0.356067\pi$
$773$	24.2956	0.873851	0.436926	+	0.899498i	$0.356067\pi$
$774$	−21.7676	−0.782421
$775$	0	0
$776$	31.9970	1.14862
$777$	−18.3120	−0.656938
$778$	−15.7330	−0.564057
$779$	6.48374	0.232304
$780$	−53.1249	−1.90218
$781$	13.6774	0.489415
$782$	−7.69630	−0.275219
$783$	−3.42219	−0.122299
$784$	−38.9449	−1.39089
$785$	5.22940	0.186645
$786$	−81.2803	−2.89917
$787$	14.5193	0.517556	0.258778	−	0.965937i	$-0.416680\pi$
$787$	14.5193	0.517556	0.258778	+	0.965937i	$0.416680\pi$
$788$	−55.3387	−1.97136
$789$	10.1299	0.360635
$790$	−3.22302	−0.114670
$791$	8.06344	0.286703
$792$	20.7106	0.735920
$793$	26.3636	0.936199
$794$	79.5661	2.82370
$795$	−10.3698	−0.367781
$796$	46.9303	1.66340
$797$	−21.8434	−0.773732	−0.386866	−	0.922136i	$-0.626442\pi$
$797$	−21.8434	−0.773732	−0.386866	+	0.922136i	$0.626442\pi$
$798$	2.35415	0.0833360
$799$	−10.7788	−0.381326
$800$	−3.75801	−0.132866
$801$	2.86023	0.101061
$802$	−5.21770	−0.184243
$803$	−46.9651	−1.65736
$804$	59.4401	2.09629
$805$	−1.97204	−0.0695054
$806$	0	0
$807$	2.23349	0.0786224
$808$	−13.4829	−0.474327
$809$	19.6058	0.689305	0.344652	−	0.938730i	$-0.387997\pi$
$809$	19.6058	0.689305	0.344652	+	0.938730i	$0.387997\pi$
$810$	27.0053	0.948869
$811$	0.894826	0.0314216	0.0157108	−	0.999877i	$-0.494999\pi$
$811$	0.894826	0.0314216	0.0157108	+	0.999877i	$0.494999\pi$
$812$	−2.86996	−0.100716
$813$	24.4282	0.856734
$814$	128.975	4.52056
$815$	13.2780	0.465109
$816$	15.5319	0.543725
$817$	6.43813	0.225242
$818$	−60.2253	−2.10573
$819$	4.06693	0.142110
$820$	48.8760	1.70682
$821$	−37.4329	−1.30642	−0.653208	−	0.757179i	$-0.726577\pi$
$821$	−37.4329	−1.30642	−0.653208	+	0.757179i	$0.726577\pi$
$822$	−14.3338	−0.499950
$823$	−30.4983	−1.06311	−0.531553	−	0.847025i	$-0.678391\pi$
$823$	−30.4983	−1.06311	−0.531553	+	0.847025i	$0.678391\pi$
$824$	38.6677	1.34705
$825$	−8.81198	−0.306794
$826$	−11.2243	−0.390545
$827$	26.5029	0.921595	0.460798	−	0.887505i	$-0.347563\pi$
$827$	26.5029	0.921595	0.460798	+	0.887505i	$0.347563\pi$
$828$	7.93396	0.275724
$829$	−36.6503	−1.27292	−0.636459	−	0.771310i	$-0.719602\pi$
$829$	−36.6503	−1.27292	−0.636459	+	0.771310i	$0.719602\pi$
$830$	−33.0356	−1.14668
$831$	−42.3807	−1.47017
$832$	18.2163	0.631536
$833$	−8.15547	−0.282570
$834$	−104.879	−3.63165
$835$	6.35272	0.219845
$836$	−11.3531	−0.392656
$837$	0	0
$838$	76.9815	2.65928
$839$	2.26253	0.0781112	0.0390556	−	0.999237i	$-0.487565\pi$
$839$	2.26253	0.0781112	0.0390556	+	0.999237i	$0.487565\pi$
$840$	9.57483	0.330363
$841$	−28.3739	−0.978412
$842$	−56.5456	−1.94869
$843$	−60.4489	−2.08197
$844$	−19.1034	−0.657568
$845$	26.6432	0.916553
$846$	16.2280	0.557932
$847$	−7.99837	−0.274827
$848$	−32.9858	−1.13274
$849$	35.2181	1.20868
$850$	−3.25905	−0.111785
$851$	26.6581	0.913827
$852$	−25.4399	−0.871555
$853$	−6.61780	−0.226589	−0.113295	−	0.993561i	$-0.536140\pi$
$853$	−6.61780	−0.226589	−0.113295	+	0.993561i	$0.536140\pi$
$854$	−8.80666	−0.301358
$855$	0.445686	0.0152421
$856$	27.3589	0.935109
$857$	−37.6699	−1.28678	−0.643389	−	0.765539i	$-0.722472\pi$
$857$	−37.6699	−1.28678	−0.643389	+	0.765539i	$0.722472\pi$
$858$	−139.741	−4.77067
$859$	16.4279	0.560513	0.280256	−	0.959925i	$-0.409581\pi$
$859$	16.4279	0.560513	0.280256	+	0.959925i	$0.409581\pi$
$860$	48.5322	1.65493
$861$	−18.2537	−0.622085
$862$	41.5042	1.41364
$863$	25.7105	0.875195	0.437598	−	0.899171i	$-0.355829\pi$
$863$	25.7105	0.875195	0.437598	+	0.899171i	$0.355829\pi$
$864$	16.2538	0.552964
$865$	2.52131	0.0857269
$866$	−76.4083	−2.59646
$867$	−29.7708	−1.01107
$868$	0	0
$869$	−5.80497	−0.196920
$870$	−3.87120	−0.131246
$871$	−44.3558	−1.50294
$872$	3.50431	0.118671
$873$	−4.19308	−0.141914
$874$	−3.42711	−0.115924
$875$	−0.835075	−0.0282307
$876$	87.3549	2.95145
$877$	−17.5065	−0.591152	−0.295576	−	0.955319i	$-0.595512\pi$
$877$	−17.5065	−0.591152	−0.295576	+	0.955319i	$0.595512\pi$
$878$	41.5674	1.40283
$879$	−22.7400	−0.767000
$880$	−28.0304	−0.944903
$881$	−9.48275	−0.319482	−0.159741	−	0.987159i	$-0.551066\pi$
$881$	−9.48275	−0.319482	−0.159741	+	0.987159i	$0.551066\pi$
$882$	12.2785	0.413439
$883$	22.1412	0.745109	0.372555	−	0.928010i	$-0.378482\pi$
$883$	22.1412	0.745109	0.372555	+	0.928010i	$0.378482\pi$
$884$	−35.3877	−1.19022
$885$	−10.3667	−0.348474
$886$	63.5836	2.13613
$887$	−31.9850	−1.07395	−0.536976	−	0.843597i	$-0.680434\pi$
$887$	−31.9850	−1.07395	−0.536976	+	0.843597i	$0.680434\pi$
$888$	−129.432	−4.34347
$889$	−10.2496	−0.343759
$890$	−9.31340	−0.312186
$891$	48.6391	1.62947
$892$	76.4354	2.55925
$893$	−4.79971	−0.160616
$894$	38.6482	1.29259
$895$	−11.9518	−0.399503
$896$	−12.3615	−0.412970
$897$	−28.8833	−0.964386
$898$	−6.57462	−0.219398
$899$	0	0
$900$	3.35969	0.111990
$901$	−6.90759	−0.230125
$902$	128.564	4.28073
$903$	−18.1253	−0.603173
$904$	56.9939	1.89559
$905$	14.0948	0.468528
$906$	10.8620	0.360865
$907$	−33.4520	−1.11076	−0.555379	−	0.831598i	$-0.687427\pi$
$907$	−33.4520	−1.11076	−0.555379	+	0.831598i	$0.687427\pi$
$908$	97.0434	3.22050
$909$	1.76688	0.0586037
$910$	−13.2426	−0.438989
$911$	−44.7899	−1.48396	−0.741978	−	0.670424i	$-0.766112\pi$
$911$	−44.7899	−1.48396	−0.741978	+	0.670424i	$0.766112\pi$
$912$	6.91625	0.229020
$913$	−59.5003	−1.96917
$914$	23.5798	0.779950
$915$	−8.13378	−0.268895
$916$	0.635208	0.0209879
$917$	−13.8731	−0.458130
$918$	14.0957	0.465228
$919$	−28.5264	−0.941001	−0.470500	−	0.882400i	$-0.655927\pi$
$919$	−28.5264	−0.941001	−0.470500	+	0.882400i	$0.655927\pi$
$920$	−13.9388	−0.459548
$921$	−22.8124	−0.751694
$922$	44.4130	1.46266
$923$	18.9839	0.624862
$924$	31.9625	1.05149
$925$	11.2885	0.371165
$926$	−14.6403	−0.481110
$927$	−5.06724	−0.166430
$928$	−2.97349	−0.0976096
$929$	−47.0642	−1.54413	−0.772064	−	0.635545i	$-0.780775\pi$
$929$	−47.0642	−1.54413	−0.772064	+	0.635545i	$0.780775\pi$
$930$	0	0
$931$	−3.63157	−0.119020
$932$	37.7429	1.23631
$933$	−1.61857	−0.0529896
$934$	−20.3738	−0.666650
$935$	−5.86986	−0.191965
$936$	28.7459	0.939588
$937$	−30.8032	−1.00630	−0.503148	−	0.864200i	$-0.667825\pi$
$937$	−30.8032	−1.00630	−0.503148	+	0.864200i	$0.667825\pi$
$938$	14.8169	0.483788
$939$	20.5828	0.671693
$940$	−36.1814	−1.18011
$941$	2.84339	0.0926919	0.0463459	−	0.998925i	$-0.485242\pi$
$941$	2.84339	0.0926919	0.0463459	+	0.998925i	$0.485242\pi$
$942$	−25.5852	−0.833610
$943$	26.5733	0.865344
$944$	−32.9759	−1.07328
$945$	3.61178	0.117491
$946$	127.660	4.15059
$947$	10.4261	0.338803	0.169402	−	0.985547i	$-0.445816\pi$
$947$	10.4261	0.338803	0.169402	+	0.985547i	$0.445816\pi$
$948$	10.7972	0.350677
$949$	−65.1865	−2.11604
$950$	−1.45123	−0.0470842
$951$	60.4259	1.95944
$952$	6.37801	0.206712
$953$	−47.3840	−1.53492	−0.767460	−	0.641097i	$-0.778479\pi$
$953$	−47.3840	−1.53492	−0.767460	+	0.641097i	$0.778479\pi$
$954$	10.3998	0.336704
$955$	0.281408	0.00910616
$956$	−45.4427	−1.46972
$957$	−6.97239	−0.225385
$958$	−22.9329	−0.740927
$959$	−2.44653	−0.0790027
$960$	−5.62015	−0.181390
$961$	0	0
$962$	179.014	5.77164
$963$	−3.58528	−0.115534
$964$	23.6317	0.761125
$965$	24.1896	0.778691
$966$	9.64836	0.310431
$967$	−22.7052	−0.730150	−0.365075	−	0.930978i	$-0.618957\pi$
$967$	−22.7052	−0.730150	−0.365075	+	0.930978i	$0.618957\pi$
$968$	−56.5340	−1.81707
$969$	1.44834	0.0465273
$970$	13.6534	0.438384
$971$	−55.0522	−1.76671	−0.883354	−	0.468707i	$-0.844720\pi$
$971$	−55.0522	−1.76671	−0.883354	+	0.468707i	$0.844720\pi$
$972$	−34.1101	−1.09408
$973$	−17.9009	−0.573877
$974$	−103.634	−3.32066
$975$	−12.2308	−0.391700
$976$	−25.8730	−0.828176
$977$	13.6862	0.437862	0.218931	−	0.975740i	$-0.429743\pi$
$977$	13.6862	0.437862	0.218931	+	0.975740i	$0.429743\pi$
$978$	−64.9637	−2.07731
$979$	−16.7743	−0.536110
$980$	−27.3757	−0.874484
$981$	−0.459226	−0.0146620
$982$	−47.6595	−1.52088
$983$	41.2269	1.31494	0.657468	−	0.753483i	$-0.271628\pi$
$983$	41.2269	1.31494	0.657468	+	0.753483i	$0.271628\pi$
$984$	−129.021	−4.11303
$985$	−12.7405	−0.405947
$986$	−2.57869	−0.0821223
$987$	13.5127	0.430113
$988$	−15.7579	−0.501325
$989$	26.3863	0.839037
$990$	8.83740	0.280871
$991$	−34.9979	−1.11174	−0.555872	−	0.831268i	$-0.687616\pi$
$991$	−34.9979	−1.11174	−0.555872	+	0.831268i	$0.687616\pi$
$992$	0	0
$993$	−17.3578	−0.550833
$994$	−6.34149	−0.201140
$995$	10.8047	0.342531
$996$	110.670	3.50672
$997$	−2.06452	−0.0653840	−0.0326920	−	0.999465i	$-0.510408\pi$
$997$	−2.06452	−0.0653840	−0.0326920	+	0.999465i	$0.510408\pi$
$998$	−20.0433	−0.634459
$999$	−48.8240	−1.54472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4805.2.a.ba.1.2		20
31.14	even	15		155.2.q.b.41.5	✓	40
31.20	even	15		155.2.q.b.121.5	yes	40
31.30	odd	2		4805.2.a.z.1.2		20
155.14	even	30		775.2.bl.b.351.1		40
155.82	odd	60		775.2.ck.b.524.10		80
155.107	odd	60		775.2.ck.b.599.1		80
155.113	odd	60		775.2.ck.b.524.1		80
155.138	odd	60		775.2.ck.b.599.10		80
155.144	even	30		775.2.bl.b.276.1		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.q.b.41.5	✓	40	31.14	even	15
155.2.q.b.121.5	yes	40	31.20	even	15
775.2.bl.b.276.1		40	155.144	even	30
775.2.bl.b.351.1		40	155.14	even	30
775.2.ck.b.524.1		80	155.113	odd	60
775.2.ck.b.524.10		80	155.82	odd	60
775.2.ck.b.599.1		80	155.107	odd	60
775.2.ck.b.599.10		80	155.138	odd	60
4805.2.a.z.1.2		20	31.30	odd	2
4805.2.a.ba.1.2		20	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:	\( N \)	\(=\)	\( 4805 = 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4805.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51863 q^{2} +1.94255 q^{3} +4.34352 q^{4} +1.00000 q^{5} -4.89257 q^{6} -0.835075 q^{7} -5.90247 q^{8} +0.773495 q^{9} +O(q^{10})\)	\(q-2.51863 q^{2} +1.94255 q^{3} +4.34352 q^{4} +1.00000 q^{5} -4.89257 q^{6} -0.835075 q^{7} -5.90247 q^{8} +0.773495 q^{9} -2.51863 q^{10} -4.53630 q^{11} +8.43750 q^{12} -6.29628 q^{13} +2.10325 q^{14} +1.94255 q^{15} +6.17912 q^{16} +1.29398 q^{17} -1.94815 q^{18} +0.576198 q^{19} +4.34352 q^{20} -1.62217 q^{21} +11.4253 q^{22} +2.36152 q^{23} -11.4658 q^{24} +1.00000 q^{25} +15.8580 q^{26} -4.32509 q^{27} -3.62716 q^{28} +0.791240 q^{29} -4.89257 q^{30} -3.75801 q^{32} -8.81198 q^{33} -3.25905 q^{34} -0.835075 q^{35} +3.35969 q^{36} +11.2885 q^{37} -1.45123 q^{38} -12.2308 q^{39} -5.90247 q^{40} +11.2526 q^{41} +4.08566 q^{42} +11.1735 q^{43} -19.7035 q^{44} +0.773495 q^{45} -5.94780 q^{46} -8.32997 q^{47} +12.0032 q^{48} -6.30265 q^{49} -2.51863 q^{50} +2.51361 q^{51} -27.3480 q^{52} -5.33827 q^{53} +10.8933 q^{54} -4.53630 q^{55} +4.92900 q^{56} +1.11929 q^{57} -1.99284 q^{58} -5.33667 q^{59} +8.43750 q^{60} -4.18717 q^{61} -0.645926 q^{63} -2.89318 q^{64} -6.29628 q^{65} +22.1942 q^{66} +7.04476 q^{67} +5.62041 q^{68} +4.58736 q^{69} +2.10325 q^{70} -3.01510 q^{71} -4.56553 q^{72} +10.3532 q^{73} -28.4317 q^{74} +1.94255 q^{75} +2.50273 q^{76} +3.78815 q^{77} +30.8050 q^{78} +1.27967 q^{79} +6.17912 q^{80} -10.7222 q^{81} -28.3412 q^{82} +13.1165 q^{83} -7.04594 q^{84} +1.29398 q^{85} -28.1419 q^{86} +1.53702 q^{87} +26.7754 q^{88} +3.69780 q^{89} -1.94815 q^{90} +5.25787 q^{91} +10.2573 q^{92} +20.9802 q^{94} +0.576198 q^{95} -7.30012 q^{96} -5.42095 q^{97} +15.8741 q^{98} -3.50881 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9}+O(q^{10}) \) 20 * q + q^2 + 12 * q^3 + 19 * q^4 + 20 * q^5 + 12 * q^6 - 3 * q^7 - 3 * q^8 + 24 * q^9	\( 20 q + q^{2} + 12 q^{3} + 19 q^{4} + 20 q^{5} + 12 q^{6} - 3 q^{7} - 3 q^{8} + 24 q^{9} + q^{10} + 19 q^{11} - 9 q^{12} + 23 q^{13} - 10 q^{14} + 12 q^{15} + 17 q^{16} + 6 q^{17} - 12 q^{18} + 9 q^{19} + 19 q^{20} - 6 q^{21} + 42 q^{22} + 33 q^{23} + 40 q^{24} + 20 q^{25} + 4 q^{26} + 63 q^{27} - 8 q^{28} + 4 q^{29} + 12 q^{30} - 30 q^{32} - 8 q^{33} - 19 q^{34} - 3 q^{35} + q^{36} + 38 q^{37} - 13 q^{38} + 15 q^{39} - 3 q^{40} + 9 q^{41} + 48 q^{42} + 64 q^{43} + 27 q^{44} + 24 q^{45} - 20 q^{46} - 8 q^{47} + 5 q^{48} - 3 q^{49} + q^{50} - 18 q^{51} - 5 q^{52} + 7 q^{53} + 16 q^{54} + 19 q^{55} - 5 q^{56} + 36 q^{57} + 49 q^{58} - 26 q^{59} - 9 q^{60} + 34 q^{61} - 5 q^{63} + q^{64} + 23 q^{65} + 35 q^{66} + 2 q^{67} + 28 q^{68} + 18 q^{69} - 10 q^{70} + 55 q^{71} - 14 q^{72} + 50 q^{73} - 30 q^{74} + 12 q^{75} - 89 q^{76} + 8 q^{77} + 69 q^{78} + 41 q^{79} + 17 q^{80} + 40 q^{81} - 24 q^{82} + 63 q^{83} - 7 q^{84} + 6 q^{85} + 7 q^{86} + 36 q^{87} + 124 q^{88} + 32 q^{89} - 12 q^{90} + 24 q^{91} + 55 q^{92} - 47 q^{94} + 9 q^{95} + 18 q^{96} + 13 q^{97} + 13 q^{98} - 17 q^{99}+O(q^{100}) \) 20 * q + q^2 + 12 * q^3 + 19 * q^4 + 20 * q^5 + 12 * q^6 - 3 * q^7 - 3 * q^8 + 24 * q^9 + q^10 + 19 * q^11 - 9 * q^12 + 23 * q^13 - 10 * q^14 + 12 * q^15 + 17 * q^16 + 6 * q^17 - 12 * q^18 + 9 * q^19 + 19 * q^20 - 6 * q^21 + 42 * q^22 + 33 * q^23 + 40 * q^24 + 20 * q^25 + 4 * q^26 + 63 * q^27 - 8 * q^28 + 4 * q^29 + 12 * q^30 - 30 * q^32 - 8 * q^33 - 19 * q^34 - 3 * q^35 + q^36 + 38 * q^37 - 13 * q^38 + 15 * q^39 - 3 * q^40 + 9 * q^41 + 48 * q^42 + 64 * q^43 + 27 * q^44 + 24 * q^45 - 20 * q^46 - 8 * q^47 + 5 * q^48 - 3 * q^49 + q^50 - 18 * q^51 - 5 * q^52 + 7 * q^53 + 16 * q^54 + 19 * q^55 - 5 * q^56 + 36 * q^57 + 49 * q^58 - 26 * q^59 - 9 * q^60 + 34 * q^61 - 5 * q^63 + q^64 + 23 * q^65 + 35 * q^66 + 2 * q^67 + 28 * q^68 + 18 * q^69 - 10 * q^70 + 55 * q^71 - 14 * q^72 + 50 * q^73 - 30 * q^74 + 12 * q^75 - 89 * q^76 + 8 * q^77 + 69 * q^78 + 41 * q^79 + 17 * q^80 + 40 * q^81 - 24 * q^82 + 63 * q^83 - 7 * q^84 + 6 * q^85 + 7 * q^86 + 36 * q^87 + 124 * q^88 + 32 * q^89 - 12 * q^90 + 24 * q^91 + 55 * q^92 - 47 * q^94 + 9 * q^95 + 18 * q^96 + 13 * q^97 + 13 * q^98 - 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more