LMFDB - Embedded newform 5054.2.a.bl.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, 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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 24 x^{10} + 68 x^{9} + 213 x^{8} - 555 x^{7} - 814 x^{6} + 1911 x^{5} + 1005 x^{4} + \cdots + 1 $ x^12 - 3x^11 - 24x^10 + 68x^9 + 213x^8 - 555x^7 - 814x^6 + 1911x^5 + 1005x^4 - 2326x^3 + 570x^2 + 105*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.113874$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.113874 q^{3} +1.00000 q^{4} -3.74313 q^{5} -0.113874 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.98703 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.113874 q^{3} +1.00000 q^{4} -3.74313 q^{5} -0.113874 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.98703 q^{9} +3.74313 q^{10} +0.445283 q^{11} +0.113874 q^{12} +4.99602 q^{13} -1.00000 q^{14} -0.426244 q^{15} +1.00000 q^{16} -6.32279 q^{17} +2.98703 q^{18} -3.74313 q^{20} +0.113874 q^{21} -0.445283 q^{22} -1.13076 q^{23} -0.113874 q^{24} +9.01099 q^{25} -4.99602 q^{26} -0.681767 q^{27} +1.00000 q^{28} +4.92240 q^{29} +0.426244 q^{30} -9.61879 q^{31} -1.00000 q^{32} +0.0507061 q^{33} +6.32279 q^{34} -3.74313 q^{35} -2.98703 q^{36} -2.70420 q^{37} +0.568917 q^{39} +3.74313 q^{40} -4.63992 q^{41} -0.113874 q^{42} +10.6643 q^{43} +0.445283 q^{44} +11.1808 q^{45} +1.13076 q^{46} -11.5865 q^{47} +0.113874 q^{48} +1.00000 q^{49} -9.01099 q^{50} -0.720001 q^{51} +4.99602 q^{52} -6.90540 q^{53} +0.681767 q^{54} -1.66675 q^{55} -1.00000 q^{56} -4.92240 q^{58} -1.70751 q^{59} -0.426244 q^{60} +8.89796 q^{61} +9.61879 q^{62} -2.98703 q^{63} +1.00000 q^{64} -18.7007 q^{65} -0.0507061 q^{66} +6.89550 q^{67} -6.32279 q^{68} -0.128764 q^{69} +3.74313 q^{70} -9.12052 q^{71} +2.98703 q^{72} -9.16441 q^{73} +2.70420 q^{74} +1.02612 q^{75} +0.445283 q^{77} -0.568917 q^{78} +2.88340 q^{79} -3.74313 q^{80} +8.88346 q^{81} +4.63992 q^{82} +6.23955 q^{83} +0.113874 q^{84} +23.6670 q^{85} -10.6643 q^{86} +0.560533 q^{87} -0.445283 q^{88} +5.56970 q^{89} -11.1808 q^{90} +4.99602 q^{91} -1.13076 q^{92} -1.09533 q^{93} +11.5865 q^{94} -0.113874 q^{96} -16.9366 q^{97} -1.00000 q^{98} -1.33007 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9}+O(q^{10}) $ 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 - 12 * q^8 + 21 * q^9	\( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{16} + 3 q^{17} - 21 q^{18} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} - 21 q^{27} + 12 q^{28} - 6 q^{29} - 6 q^{30} - 3 q^{31} - 12 q^{32} + 6 q^{33} - 3 q^{34} + 3 q^{35} + 21 q^{36} - 27 q^{37} + 27 q^{39} - 3 q^{40} + 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} - 12 q^{46} + 21 q^{47} - 3 q^{48} + 12 q^{49} - 33 q^{50} - 3 q^{51} + 6 q^{52} - 6 q^{53} + 21 q^{54} + 48 q^{55} - 12 q^{56} + 6 q^{58} + 27 q^{59} + 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} + 9 q^{65} - 6 q^{66} - 9 q^{67} + 3 q^{68} - 18 q^{69} - 3 q^{70} - 27 q^{71} - 21 q^{72} - 9 q^{73} + 27 q^{74} - 33 q^{75} + 3 q^{77} - 27 q^{78} + 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} - 3 q^{84} + 78 q^{85} - 30 q^{86} - 45 q^{87} - 3 q^{88} + 24 q^{89} - 30 q^{90} + 6 q^{91} + 12 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 18 q^{97} - 12 q^{98} + 48 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 - 12 * q^8 + 21 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 + 6 * q^13 - 12 * q^14 + 6 * q^15 + 12 * q^16 + 3 * q^17 - 21 * q^18 + 3 * q^20 - 3 * q^21 - 3 * q^22 + 12 * q^23 + 3 * q^24 + 33 * q^25 - 6 * q^26 - 21 * q^27 + 12 * q^28 - 6 * q^29 - 6 * q^30 - 3 * q^31 - 12 * q^32 + 6 * q^33 - 3 * q^34 + 3 * q^35 + 21 * q^36 - 27 * q^37 + 27 * q^39 - 3 * q^40 + 27 * q^41 + 3 * q^42 + 30 * q^43 + 3 * q^44 + 30 * q^45 - 12 * q^46 + 21 * q^47 - 3 * q^48 + 12 * q^49 - 33 * q^50 - 3 * q^51 + 6 * q^52 - 6 * q^53 + 21 * q^54 + 48 * q^55 - 12 * q^56 + 6 * q^58 + 27 * q^59 + 6 * q^60 + 18 * q^61 + 3 * q^62 + 21 * q^63 + 12 * q^64 + 9 * q^65 - 6 * q^66 - 9 * q^67 + 3 * q^68 - 18 * q^69 - 3 * q^70 - 27 * q^71 - 21 * q^72 - 9 * q^73 + 27 * q^74 - 33 * q^75 + 3 * q^77 - 27 * q^78 + 12 * q^79 + 3 * q^80 + 24 * q^81 - 27 * q^82 + 9 * q^83 - 3 * q^84 + 78 * q^85 - 30 * q^86 - 45 * q^87 - 3 * q^88 + 24 * q^89 - 30 * q^90 + 6 * q^91 + 12 * q^92 + 3 * q^93 - 21 * q^94 + 3 * q^96 + 18 * q^97 - 12 * q^98 + 48 * 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
$3$	0.113874	0.0657451	0.0328726	−	0.999460i	$-0.489534\pi$
$3$	0.113874	0.0657451	0.0328726	+	0.999460i	$0.489534\pi$
$4$	1.00000	0.500000
$5$	−3.74313	−1.67398	−0.836988	−	0.547221i	$-0.815686\pi$
$5$	−3.74313	−1.67398	−0.836988	+	0.547221i	$0.815686\pi$
$6$	−0.113874	−0.0464888
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.98703	−0.995678
$10$	3.74313	1.18368
$11$	0.445283	0.134258	0.0671289	−	0.997744i	$-0.478616\pi$
$11$	0.445283	0.134258	0.0671289	+	0.997744i	$0.478616\pi$
$12$	0.113874	0.0328726
$13$	4.99602	1.38565	0.692824	−	0.721107i	$-0.256366\pi$
$13$	4.99602	1.38565	0.692824	+	0.721107i	$0.256366\pi$
$14$	−1.00000	−0.267261
$15$	−0.426244	−0.110056
$16$	1.00000	0.250000
$17$	−6.32279	−1.53350	−0.766751	−	0.641945i	$-0.778128\pi$
$17$	−6.32279	−1.53350	−0.766751	+	0.641945i	$0.778128\pi$
$18$	2.98703	0.704050
$19$	0	0
$19$	0	0
$20$	−3.74313	−0.836988
$21$	0.113874	0.0248493
$22$	−0.445283	−0.0949346
$23$	−1.13076	−0.235780	−0.117890	−	0.993027i	$-0.537613\pi$
$23$	−1.13076	−0.235780	−0.117890	+	0.993027i	$0.537613\pi$
$24$	−0.113874	−0.0232444
$25$	9.01099	1.80220
$26$	−4.99602	−0.979801
$27$	−0.681767	−0.131206
$28$	1.00000	0.188982
$29$	4.92240	0.914067	0.457033	−	0.889450i	$-0.348912\pi$
$29$	4.92240	0.914067	0.457033	+	0.889450i	$0.348912\pi$
$30$	0.426244	0.0778212
$31$	−9.61879	−1.72759	−0.863793	−	0.503846i	$-0.831918\pi$
$31$	−9.61879	−1.72759	−0.863793	+	0.503846i	$0.831918\pi$
$32$	−1.00000	−0.176777
$33$	0.0507061	0.00882680
$34$	6.32279	1.08435
$35$	−3.74313	−0.632704
$36$	−2.98703	−0.497839
$37$	−2.70420	−0.444568	−0.222284	−	0.974982i	$-0.571351\pi$
$37$	−2.70420	−0.444568	−0.222284	+	0.974982i	$0.571351\pi$
$38$	0	0
$39$	0.568917	0.0910996
$40$	3.74313	0.591840
$41$	−4.63992	−0.724633	−0.362316	−	0.932055i	$-0.618014\pi$
$41$	−4.63992	−0.724633	−0.362316	+	0.932055i	$0.618014\pi$
$42$	−0.113874	−0.0175711
$43$	10.6643	1.62629	0.813146	−	0.582060i	$-0.197753\pi$
$43$	10.6643	1.62629	0.813146	+	0.582060i	$0.197753\pi$
$44$	0.445283	0.0671289
$45$	11.1808	1.66674
$46$	1.13076	0.166722
$47$	−11.5865	−1.69007	−0.845035	−	0.534711i	$-0.820420\pi$
$47$	−11.5865	−1.69007	−0.845035	+	0.534711i	$0.820420\pi$
$48$	0.113874	0.0164363
$49$	1.00000	0.142857
$50$	−9.01099	−1.27435
$51$	−0.720001	−0.100820
$52$	4.99602	0.692824
$53$	−6.90540	−0.948530	−0.474265	−	0.880382i	$-0.657286\pi$
$53$	−6.90540	−0.948530	−0.474265	+	0.880382i	$0.657286\pi$
$54$	0.681767	0.0927767
$55$	−1.66675	−0.224744
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−4.92240	−0.646343
$59$	−1.70751	−0.222299	−0.111149	−	0.993804i	$-0.535453\pi$
$59$	−1.70751	−0.222299	−0.111149	+	0.993804i	$0.535453\pi$
$60$	−0.426244	−0.0550279
$61$	8.89796	1.13927	0.569634	−	0.821899i	$-0.307085\pi$
$61$	8.89796	1.13927	0.569634	+	0.821899i	$0.307085\pi$
$62$	9.61879	1.22159
$63$	−2.98703	−0.376331
$64$	1.00000	0.125000
$65$	−18.7007	−2.31954
$66$	−0.0507061	−0.00624149
$67$	6.89550	0.842419	0.421210	−	0.906963i	$-0.361606\pi$
$67$	6.89550	0.842419	0.421210	+	0.906963i	$0.361606\pi$
$68$	−6.32279	−0.766751
$69$	−0.128764	−0.0155014
$70$	3.74313	0.447389
$71$	−9.12052	−1.08241	−0.541203	−	0.840892i	$-0.682031\pi$
$71$	−9.12052	−1.08241	−0.541203	+	0.840892i	$0.682031\pi$
$72$	2.98703	0.352025
$73$	−9.16441	−1.07261	−0.536306	−	0.844023i	$-0.680181\pi$
$73$	−9.16441	−1.07261	−0.536306	+	0.844023i	$0.680181\pi$
$74$	2.70420	0.314357
$75$	1.02612	0.118486
$76$	0	0
$77$	0.445283	0.0507447
$78$	−0.568917	−0.0644171
$79$	2.88340	0.324408	0.162204	−	0.986757i	$-0.448140\pi$
$79$	2.88340	0.324408	0.162204	+	0.986757i	$0.448140\pi$
$80$	−3.74313	−0.418494
$81$	8.88346	0.987051
$82$	4.63992	0.512393
$83$	6.23955	0.684880	0.342440	−	0.939540i	$-0.388747\pi$
$83$	6.23955	0.684880	0.342440	+	0.939540i	$0.388747\pi$
$84$	0.113874	0.0124247
$85$	23.6670	2.56705
$86$	−10.6643	−1.14996
$87$	0.560533	0.0600955
$88$	−0.445283	−0.0474673
$89$	5.56970	0.590387	0.295194	−	0.955437i	$-0.404616\pi$
$89$	5.56970	0.590387	0.295194	+	0.955437i	$0.404616\pi$
$90$	−11.1808	−1.17856
$91$	4.99602	0.523726
$92$	−1.13076	−0.117890
$93$	−1.09533	−0.113580
$94$	11.5865	1.19506
$95$	0	0
$96$	−0.113874	−0.0116222
$97$	−16.9366	−1.71966	−0.859828	−	0.510584i	$-0.829429\pi$
$97$	−16.9366	−1.71966	−0.859828	+	0.510584i	$0.829429\pi$
$98$	−1.00000	−0.101015
$99$	−1.33007	−0.133677
$100$	9.01099	0.901099
$101$	−3.39170	−0.337486	−0.168743	−	0.985660i	$-0.553971\pi$
$101$	−3.39170	−0.337486	−0.168743	+	0.985660i	$0.553971\pi$
$102$	0.720001	0.0712907
$103$	4.09997	0.403982	0.201991	−	0.979387i	$-0.435259\pi$
$103$	4.09997	0.403982	0.201991	+	0.979387i	$0.435259\pi$
$104$	−4.99602	−0.489900
$105$	−0.426244	−0.0415972
$106$	6.90540	0.670712
$107$	−2.46812	−0.238602	−0.119301	−	0.992858i	$-0.538065\pi$
$107$	−2.46812	−0.238602	−0.119301	+	0.992858i	$0.538065\pi$
$108$	−0.681767	−0.0656030
$109$	−5.50948	−0.527713	−0.263856	−	0.964562i	$-0.584994\pi$
$109$	−5.50948	−0.527713	−0.263856	+	0.964562i	$0.584994\pi$
$110$	1.66675	0.158918
$111$	−0.307938	−0.0292282
$112$	1.00000	0.0944911
$113$	11.2670	1.05991	0.529955	−	0.848026i	$-0.322209\pi$
$113$	11.2670	1.05991	0.529955	+	0.848026i	$0.322209\pi$
$114$	0	0
$115$	4.23259	0.394691
$116$	4.92240	0.457033
$117$	−14.9233	−1.37966
$118$	1.70751	0.157189
$119$	−6.32279	−0.579609
$120$	0.426244	0.0389106
$121$	−10.8017	−0.981975
$122$	−8.89796	−0.805584
$123$	−0.528365	−0.0476411
$124$	−9.61879	−0.863793
$125$	−15.0136	−1.34286
$126$	2.98703	0.266106
$127$	−12.5296	−1.11182	−0.555911	−	0.831242i	$-0.687630\pi$
$127$	−12.5296	−1.11182	−0.555911	+	0.831242i	$0.687630\pi$
$128$	−1.00000	−0.0883883
$129$	1.21439	0.106921
$130$	18.7007	1.64016
$131$	−8.01072	−0.699900	−0.349950	−	0.936768i	$-0.613801\pi$
$131$	−8.01072	−0.699900	−0.349950	+	0.936768i	$0.613801\pi$
$132$	0.0507061	0.00441340
$133$	0	0
$134$	−6.89550	−0.595680
$135$	2.55194	0.219636
$136$	6.32279	0.542175
$137$	12.4110	1.06034	0.530172	−	0.847890i	$-0.322127\pi$
$137$	12.4110	1.06034	0.530172	+	0.847890i	$0.322127\pi$
$138$	0.128764	0.0109611
$139$	8.20900	0.696279	0.348139	−	0.937443i	$-0.386814\pi$
$139$	8.20900	0.696279	0.348139	+	0.937443i	$0.386814\pi$
$140$	−3.74313	−0.316352
$141$	−1.31940	−0.111114
$142$	9.12052	0.765377
$143$	2.22464	0.186034
$144$	−2.98703	−0.248919
$145$	−18.4252	−1.53013
$146$	9.16441	0.758452
$147$	0.113874	0.00939216
$148$	−2.70420	−0.222284
$149$	17.1739	1.40694	0.703471	−	0.710724i	$-0.251632\pi$
$149$	17.1739	1.40694	0.703471	+	0.710724i	$0.251632\pi$
$150$	−1.02612	−0.0837821
$151$	−2.27289	−0.184965	−0.0924825	−	0.995714i	$-0.529480\pi$
$151$	−2.27289	−0.184965	−0.0924825	+	0.995714i	$0.529480\pi$
$152$	0	0
$153$	18.8864	1.52687
$154$	−0.445283	−0.0358819
$155$	36.0044	2.89194
$156$	0.568917	0.0455498
$157$	3.22494	0.257378	0.128689	−	0.991685i	$-0.458923\pi$
$157$	3.22494	0.257378	0.128689	+	0.991685i	$0.458923\pi$
$158$	−2.88340	−0.229391
$159$	−0.786345	−0.0623612
$160$	3.74313	0.295920
$161$	−1.13076	−0.0891166
$162$	−8.88346	−0.697951
$163$	−11.3760	−0.891035	−0.445517	−	0.895273i	$-0.646980\pi$
$163$	−11.3760	−0.891035	−0.445517	+	0.895273i	$0.646980\pi$
$164$	−4.63992	−0.362316
$165$	−0.189799	−0.0147759
$166$	−6.23955	−0.484283
$167$	22.1814	1.71645	0.858223	−	0.513277i	$-0.171569\pi$
$167$	22.1814	1.71645	0.858223	+	0.513277i	$0.171569\pi$
$168$	−0.113874	−0.00878556
$169$	11.9603	0.920020
$170$	−23.6670	−1.81518
$171$	0	0
$172$	10.6643	0.813146
$173$	20.2068	1.53629	0.768147	−	0.640274i	$-0.221179\pi$
$173$	20.2068	1.53629	0.768147	+	0.640274i	$0.221179\pi$
$174$	−0.560533	−0.0424939
$175$	9.01099	0.681167
$176$	0.445283	0.0335644
$177$	−0.194441	−0.0146151
$178$	−5.56970	−0.417467
$179$	0.722972	0.0540374	0.0270187	−	0.999635i	$-0.491399\pi$
$179$	0.722972	0.0540374	0.0270187	+	0.999635i	$0.491399\pi$
$180$	11.1808	0.833370
$181$	−8.58680	−0.638252	−0.319126	−	0.947712i	$-0.603389\pi$
$181$	−8.58680	−0.638252	−0.319126	+	0.947712i	$0.603389\pi$
$182$	−4.99602	−0.370330
$183$	1.01325	0.0749013
$184$	1.13076	0.0833609
$185$	10.1222	0.744196
$186$	1.09533	0.0803135
$187$	−2.81543	−0.205885
$188$	−11.5865	−0.845035
$189$	−0.681767	−0.0495912
$190$	0	0
$191$	13.5182	0.978140	0.489070	−	0.872245i	$-0.337336\pi$
$191$	13.5182	0.978140	0.489070	+	0.872245i	$0.337336\pi$
$192$	0.113874	0.00821814
$193$	−9.06521	−0.652528	−0.326264	−	0.945279i	$-0.605790\pi$
$193$	−9.06521	−0.652528	−0.326264	+	0.945279i	$0.605790\pi$
$194$	16.9366	1.21598
$195$	−2.12953	−0.152499
$196$	1.00000	0.0714286
$197$	21.1319	1.50559	0.752793	−	0.658258i	$-0.228706\pi$
$197$	21.1319	1.50559	0.752793	+	0.658258i	$0.228706\pi$
$198$	1.33007	0.0945242
$199$	13.0468	0.924860	0.462430	−	0.886656i	$-0.346978\pi$
$199$	13.0468	0.924860	0.462430	+	0.886656i	$0.346978\pi$
$200$	−9.01099	−0.637173
$201$	0.785217	0.0553850
$202$	3.39170	0.238639
$203$	4.92240	0.345485
$204$	−0.720001	−0.0504101
$205$	17.3678	1.21302
$206$	−4.09997	−0.285658
$207$	3.37762	0.234761
$208$	4.99602	0.346412
$209$	0	0
$210$	0.426244	0.0294137
$211$	22.1678	1.52610	0.763049	−	0.646341i	$-0.223702\pi$
$211$	22.1678	1.52610	0.763049	+	0.646341i	$0.223702\pi$
$212$	−6.90540	−0.474265
$213$	−1.03859	−0.0711630
$214$	2.46812	0.168717
$215$	−39.9178	−2.72237
$216$	0.681767	0.0463884
$217$	−9.61879	−0.652966
$218$	5.50948	0.373149
$219$	−1.04359	−0.0705191
$220$	−1.66675	−0.112372
$221$	−31.5888	−2.12489
$222$	0.307938	0.0206674
$223$	23.4644	1.57129	0.785645	−	0.618678i	$-0.212331\pi$
$223$	23.4644	1.57129	0.785645	+	0.618678i	$0.212331\pi$
$224$	−1.00000	−0.0668153
$225$	−26.9161	−1.79441
$226$	−11.2670	−0.749469
$227$	4.09986	0.272118	0.136059	−	0.990701i	$-0.456556\pi$
$227$	4.09986	0.272118	0.136059	+	0.990701i	$0.456556\pi$
$228$	0	0
$229$	10.1573	0.671212	0.335606	−	0.942003i	$-0.391059\pi$
$229$	10.1573	0.671212	0.335606	+	0.942003i	$0.391059\pi$
$230$	−4.23259	−0.279088
$231$	0.0507061	0.00333622
$232$	−4.92240	−0.323171
$233$	−13.6127	−0.891794	−0.445897	−	0.895084i	$-0.647115\pi$
$233$	−13.6127	−0.891794	−0.445897	+	0.895084i	$0.647115\pi$
$234$	14.9233	0.975566
$235$	43.3699	2.82914
$236$	−1.70751	−0.111149
$237$	0.328344	0.0213283
$238$	6.32279	0.409846
$239$	1.42078	0.0919027	0.0459513	−	0.998944i	$-0.485368\pi$
$239$	1.42078	0.0919027	0.0459513	+	0.998944i	$0.485368\pi$
$240$	−0.426244	−0.0275140
$241$	−13.6334	−0.878204	−0.439102	−	0.898437i	$-0.644703\pi$
$241$	−13.6334	−0.878204	−0.439102	+	0.898437i	$0.644703\pi$
$242$	10.8017	0.694361
$243$	3.05690	0.196100
$244$	8.89796	0.569634
$245$	−3.74313	−0.239140
$246$	0.528365	0.0336873
$247$	0	0
$248$	9.61879	0.610794
$249$	0.710523	0.0450275
$250$	15.0136	0.949546
$251$	10.5187	0.663933	0.331967	−	0.943291i	$-0.392288\pi$
$251$	10.5187	0.663933	0.331967	+	0.943291i	$0.392288\pi$
$252$	−2.98703	−0.188165
$253$	−0.503509	−0.0316553
$254$	12.5296	0.786177
$255$	2.69505	0.168771
$256$	1.00000	0.0625000
$257$	−2.16644	−0.135139	−0.0675694	−	0.997715i	$-0.521524\pi$
$257$	−2.16644	−0.135139	−0.0675694	+	0.997715i	$0.521524\pi$
$258$	−1.21439	−0.0756044
$259$	−2.70420	−0.168031
$260$	−18.7007	−1.15977
$261$	−14.7034	−0.910116
$262$	8.01072	0.494904
$263$	11.0041	0.678543	0.339271	−	0.940689i	$-0.389820\pi$
$263$	11.0041	0.678543	0.339271	+	0.940689i	$0.389820\pi$
$264$	−0.0507061	−0.00312074
$265$	25.8478	1.58782
$266$	0	0
$267$	0.634244	0.0388151
$268$	6.89550	0.421210
$269$	−2.86743	−0.174830	−0.0874151	−	0.996172i	$-0.527861\pi$
$269$	−2.86743	−0.174830	−0.0874151	+	0.996172i	$0.527861\pi$
$270$	−2.55194	−0.155306
$271$	−2.12512	−0.129092	−0.0645458	−	0.997915i	$-0.520560\pi$
$271$	−2.12512	−0.129092	−0.0645458	+	0.997915i	$0.520560\pi$
$272$	−6.32279	−0.383375
$273$	0.568917	0.0344324
$274$	−12.4110	−0.749777
$275$	4.01244	0.241959
$276$	−0.128764	−0.00775070
$277$	13.4784	0.809836	0.404918	−	0.914353i	$-0.367300\pi$
$277$	13.4784	0.809836	0.404918	+	0.914353i	$0.367300\pi$
$278$	−8.20900	−0.492343
$279$	28.7317	1.72012
$280$	3.74313	0.223695
$281$	−29.7072	−1.77218	−0.886091	−	0.463512i	$-0.846589\pi$
$281$	−29.7072	−1.77218	−0.886091	+	0.463512i	$0.846589\pi$
$282$	1.31940	0.0785694
$283$	24.3926	1.44999	0.724994	−	0.688755i	$-0.241843\pi$
$283$	24.3926	1.44999	0.724994	+	0.688755i	$0.241843\pi$
$284$	−9.12052	−0.541203
$285$	0	0
$286$	−2.22464	−0.131546
$287$	−4.63992	−0.273886
$288$	2.98703	0.176013
$289$	22.9777	1.35163
$290$	18.4252	1.08196
$291$	−1.92864	−0.113059
$292$	−9.16441	−0.536306
$293$	6.08565	0.355528	0.177764	−	0.984073i	$-0.443114\pi$
$293$	6.08565	0.355528	0.177764	+	0.984073i	$0.443114\pi$
$294$	−0.113874	−0.00664126
$295$	6.39143	0.372123
$296$	2.70420	0.157178
$297$	−0.303579	−0.0176154
$298$	−17.1739	−0.994858
$299$	−5.64932	−0.326708
$300$	1.02612	0.0592429
$301$	10.6643	0.614680
$302$	2.27289	0.130790
$303$	−0.386226	−0.0221881
$304$	0	0
$305$	−33.3062	−1.90711
$306$	−18.8864	−1.07966
$307$	7.69160	0.438983	0.219491	−	0.975614i	$-0.429560\pi$
$307$	7.69160	0.438983	0.219491	+	0.975614i	$0.429560\pi$
$308$	0.445283	0.0253723
$309$	0.466879	0.0265598
$310$	−36.0044	−2.04491
$311$	25.6831	1.45636	0.728179	−	0.685387i	$-0.240367\pi$
$311$	25.6831	1.45636	0.728179	+	0.685387i	$0.240367\pi$
$312$	−0.568917	−0.0322086
$313$	8.35591	0.472304	0.236152	−	0.971716i	$-0.424114\pi$
$313$	8.35591	0.472304	0.236152	+	0.971716i	$0.424114\pi$
$314$	−3.22494	−0.181994
$315$	11.1808	0.629969
$316$	2.88340	0.162204
$317$	−3.64558	−0.204756	−0.102378	−	0.994746i	$-0.532645\pi$
$317$	−3.64558	−0.204756	−0.102378	+	0.994746i	$0.532645\pi$
$318$	0.786345	0.0440961
$319$	2.19186	0.122721
$320$	−3.74313	−0.209247
$321$	−0.281055	−0.0156870
$322$	1.13076	0.0630149
$323$	0	0
$324$	8.88346	0.493526
$325$	45.0191	2.49721
$326$	11.3760	0.630057
$327$	−0.627386	−0.0346945
$328$	4.63992	0.256196
$329$	−11.5865	−0.638786
$330$	0.189799	0.0104481
$331$	−18.4604	−1.01468	−0.507339	−	0.861747i	$-0.669371\pi$
$331$	−18.4604	−1.01468	−0.507339	+	0.861747i	$0.669371\pi$
$332$	6.23955	0.342440
$333$	8.07754	0.442646
$334$	−22.1814	−1.21371
$335$	−25.8107	−1.41019
$336$	0.113874	0.00621233
$337$	32.4569	1.76804	0.884020	−	0.467449i	$-0.154827\pi$
$337$	32.4569	1.76804	0.884020	+	0.467449i	$0.154827\pi$
$338$	−11.9603	−0.650552
$339$	1.28302	0.0696839
$340$	23.6670	1.28352
$341$	−4.28308	−0.231942
$342$	0	0
$343$	1.00000	0.0539949
$344$	−10.6643	−0.574981
$345$	0.481981	0.0259490
$346$	−20.2068	−1.08632
$347$	−12.2197	−0.655986	−0.327993	−	0.944680i	$-0.606372\pi$
$347$	−12.2197	−0.655986	−0.327993	+	0.944680i	$0.606372\pi$
$348$	0.560533	0.0300477
$349$	−22.3460	−1.19615	−0.598076	−	0.801439i	$-0.704068\pi$
$349$	−22.3460	−1.19615	−0.598076	+	0.801439i	$0.704068\pi$
$350$	−9.01099	−0.481658
$351$	−3.40612	−0.181805
$352$	−0.445283	−0.0237336
$353$	4.51862	0.240502	0.120251	−	0.992744i	$-0.461630\pi$
$353$	4.51862	0.240502	0.120251	+	0.992744i	$0.461630\pi$
$354$	0.194441	0.0103344
$355$	34.1393	1.81192
$356$	5.56970	0.295194
$357$	−0.720001	−0.0381065
$358$	−0.722972	−0.0382102
$359$	1.21213	0.0639736	0.0319868	−	0.999488i	$-0.489817\pi$
$359$	1.21213	0.0639736	0.0319868	+	0.999488i	$0.489817\pi$
$360$	−11.1808	−0.589282
$361$	0	0
$362$	8.58680	0.451312
$363$	−1.23003	−0.0645601
$364$	4.99602	0.261863
$365$	34.3035	1.79553
$366$	−1.01325	−0.0529632
$367$	25.7736	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$367$	25.7736	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$368$	−1.13076	−0.0589451
$369$	13.8596	0.721501
$370$	−10.1222	−0.526226
$371$	−6.90540	−0.358511
$372$	−1.09533	−0.0567902
$373$	−13.9643	−0.723044	−0.361522	−	0.932364i	$-0.617743\pi$
$373$	−13.9643	−0.723044	−0.361522	+	0.932364i	$0.617743\pi$
$374$	2.81543	0.145582
$375$	−1.70966	−0.0882865
$376$	11.5865	0.597530
$377$	24.5924	1.26657
$378$	0.681767	0.0350663
$379$	33.1820	1.70445	0.852224	−	0.523178i	$-0.175254\pi$
$379$	33.1820	1.70445	0.852224	+	0.523178i	$0.175254\pi$
$380$	0	0
$381$	−1.42679	−0.0730969
$382$	−13.5182	−0.691649
$383$	−21.9371	−1.12094	−0.560468	−	0.828176i	$-0.689379\pi$
$383$	−21.9371	−1.12094	−0.560468	+	0.828176i	$0.689379\pi$
$384$	−0.113874	−0.00581110
$385$	−1.66675	−0.0849454
$386$	9.06521	0.461407
$387$	−31.8546	−1.61926
$388$	−16.9366	−0.859828
$389$	1.11469	0.0565171	0.0282586	−	0.999601i	$-0.491004\pi$
$389$	1.11469	0.0565171	0.0282586	+	0.999601i	$0.491004\pi$
$390$	2.12953	0.107833
$391$	7.14957	0.361569
$392$	−1.00000	−0.0505076
$393$	−0.912212	−0.0460150
$394$	−21.1319	−1.06461
$395$	−10.7929	−0.543052
$396$	−1.33007	−0.0668387
$397$	−20.8096	−1.04440	−0.522201	−	0.852823i	$-0.674889\pi$
$397$	−20.8096	−1.04440	−0.522201	+	0.852823i	$0.674889\pi$
$398$	−13.0468	−0.653975
$399$	0	0
$400$	9.01099	0.450549
$401$	14.6596	0.732067	0.366034	−	0.930602i	$-0.380716\pi$
$401$	14.6596	0.732067	0.366034	+	0.930602i	$0.380716\pi$
$402$	−0.785217	−0.0391631
$403$	−48.0557	−2.39383
$404$	−3.39170	−0.168743
$405$	−33.2519	−1.65230
$406$	−4.92240	−0.244295
$407$	−1.20413	−0.0596867
$408$	0.720001	0.0356454
$409$	1.42751	0.0705859	0.0352929	−	0.999377i	$-0.488764\pi$
$409$	1.42751	0.0705859	0.0352929	+	0.999377i	$0.488764\pi$
$410$	−17.3678	−0.857734
$411$	1.41329	0.0697125
$412$	4.09997	0.201991
$413$	−1.70751	−0.0840211
$414$	−3.37762	−0.166001
$415$	−23.3554	−1.14647
$416$	−4.99602	−0.244950
$417$	0.934791	0.0457769
$418$	0	0
$419$	24.0982	1.17727	0.588636	−	0.808398i	$-0.299665\pi$
$419$	24.0982	1.17727	0.588636	+	0.808398i	$0.299665\pi$
$420$	−0.426244	−0.0207986
$421$	−34.5275	−1.68277	−0.841385	−	0.540437i	$-0.818259\pi$
$421$	−34.5275	−1.68277	−0.841385	+	0.540437i	$0.818259\pi$
$422$	−22.1678	−1.07911
$423$	34.6094	1.68276
$424$	6.90540	0.335356
$425$	−56.9746	−2.76367
$426$	1.03859	0.0503198
$427$	8.89796	0.430603
$428$	−2.46812	−0.119301
$429$	0.253329	0.0122308
$430$	39.9178	1.92501
$431$	20.9254	1.00794	0.503970	−	0.863721i	$-0.331872\pi$
$431$	20.9254	1.00794	0.503970	+	0.863721i	$0.331872\pi$
$432$	−0.681767	−0.0328015
$433$	3.53190	0.169732	0.0848661	−	0.996392i	$-0.472954\pi$
$433$	3.53190	0.169732	0.0848661	+	0.996392i	$0.472954\pi$
$434$	9.61879	0.461717
$435$	−2.09815	−0.100598
$436$	−5.50948	−0.263856
$437$	0	0
$438$	1.04359	0.0498645
$439$	−11.6834	−0.557620	−0.278810	−	0.960346i	$-0.589940\pi$
$439$	−11.6834	−0.557620	−0.278810	+	0.960346i	$0.589940\pi$
$440$	1.66675	0.0794591
$441$	−2.98703	−0.142240
$442$	31.5888	1.50253
$443$	23.8626	1.13375	0.566874	−	0.823804i	$-0.308153\pi$
$443$	23.8626	1.13375	0.566874	+	0.823804i	$0.308153\pi$
$444$	−0.307938	−0.0146141
$445$	−20.8481	−0.988295
$446$	−23.4644	−1.11107
$447$	1.95566	0.0924996
$448$	1.00000	0.0472456
$449$	7.98821	0.376987	0.188494	−	0.982074i	$-0.439640\pi$
$449$	7.98821	0.376987	0.188494	+	0.982074i	$0.439640\pi$
$450$	26.9161	1.26884
$451$	−2.06607	−0.0972876
$452$	11.2670	0.529955
$453$	−0.258823	−0.0121605
$454$	−4.09986	−0.192416
$455$	−18.7007	−0.876704
$456$	0	0
$457$	−14.3870	−0.672996	−0.336498	−	0.941684i	$-0.609243\pi$
$457$	−14.3870	−0.672996	−0.336498	+	0.941684i	$0.609243\pi$
$458$	−10.1573	−0.474618
$459$	4.31067	0.201205
$460$	4.23259	0.197345
$461$	11.9026	0.554357	0.277179	−	0.960818i	$-0.410601\pi$
$461$	11.9026	0.554357	0.277179	+	0.960818i	$0.410601\pi$
$462$	−0.0507061	−0.00235906
$463$	15.4030	0.715840	0.357920	−	0.933752i	$-0.383486\pi$
$463$	15.4030	0.715840	0.357920	+	0.933752i	$0.383486\pi$
$464$	4.92240	0.228517
$465$	4.09996	0.190131
$466$	13.6127	0.630594
$467$	−16.8900	−0.781574	−0.390787	−	0.920481i	$-0.627797\pi$
$467$	−16.8900	−0.781574	−0.390787	+	0.920481i	$0.627797\pi$
$468$	−14.9233	−0.689829
$469$	6.89550	0.318405
$470$	−43.3699	−2.00050
$471$	0.367236	0.0169214
$472$	1.70751	0.0785946
$473$	4.74863	0.218342
$474$	−0.328344	−0.0150814
$475$	0	0
$476$	−6.32279	−0.289805
$477$	20.6267	0.944430
$478$	−1.42078	−0.0649850
$479$	−10.5626	−0.482620	−0.241310	−	0.970448i	$-0.577577\pi$
$479$	−10.5626	−0.482620	−0.241310	+	0.970448i	$0.577577\pi$
$480$	0.426244	0.0194553
$481$	−13.5103	−0.616015
$482$	13.6334	0.620984
$483$	−0.128764	−0.00585898
$484$	−10.8017	−0.490987
$485$	63.3960	2.87866
$486$	−3.05690	−0.138664
$487$	30.5440	1.38408	0.692041	−	0.721859i	$-0.256712\pi$
$487$	30.5440	1.38408	0.692041	+	0.721859i	$0.256712\pi$
$488$	−8.89796	−0.402792
$489$	−1.29543	−0.0585812
$490$	3.74313	0.169097
$491$	0.102404	0.00462141	0.00231071	−	0.999997i	$-0.499264\pi$
$491$	0.102404	0.00462141	0.00231071	+	0.999997i	$0.499264\pi$
$492$	−0.528365	−0.0238205
$493$	−31.1233	−1.40172
$494$	0	0
$495$	4.97863	0.223773
$496$	−9.61879	−0.431897
$497$	−9.12052	−0.409111
$498$	−0.710523	−0.0318393
$499$	0.647893	0.0290037	0.0145018	−	0.999895i	$-0.495384\pi$
$499$	0.647893	0.0290037	0.0145018	+	0.999895i	$0.495384\pi$
$500$	−15.0136	−0.671430
$501$	2.52588	0.112848
$502$	−10.5187	−0.469472
$503$	−36.7180	−1.63718	−0.818588	−	0.574382i	$-0.805243\pi$
$503$	−36.7180	−1.63718	−0.818588	+	0.574382i	$0.805243\pi$
$504$	2.98703	0.133053
$505$	12.6955	0.564944
$506$	0.503509	0.0223837
$507$	1.36196	0.0604868
$508$	−12.5296	−0.555911
$509$	−8.71458	−0.386267	−0.193133	−	0.981173i	$-0.561865\pi$
$509$	−8.71458	−0.386267	−0.193133	+	0.981173i	$0.561865\pi$
$510$	−2.69505	−0.119339
$511$	−9.16441	−0.405409
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	2.16644	0.0955576
$515$	−15.3467	−0.676256
$516$	1.21439	0.0534604
$517$	−5.15928	−0.226905
$518$	2.70420	0.118816
$519$	2.30103	0.101004
$520$	18.7007	0.820082
$521$	6.01674	0.263598	0.131799	−	0.991276i	$-0.457925\pi$
$521$	6.01674	0.263598	0.131799	+	0.991276i	$0.457925\pi$
$522$	14.7034	0.643549
$523$	0.583321	0.0255068	0.0127534	−	0.999919i	$-0.495940\pi$
$523$	0.583321	0.0255068	0.0127534	+	0.999919i	$0.495940\pi$
$524$	−8.01072	−0.349950
$525$	1.02612	0.0447834
$526$	−11.0041	−0.479802
$527$	60.8176	2.64926
$528$	0.0507061	0.00220670
$529$	−21.7214	−0.944408
$530$	−25.8478	−1.12276
$531$	5.10039	0.221338
$532$	0	0
$533$	−23.1811	−1.00409
$534$	−0.634244	−0.0274464
$535$	9.23850	0.399415
$536$	−6.89550	−0.297840
$537$	0.0823276	0.00355270
$538$	2.86743	0.123624
$539$	0.445283	0.0191797
$540$	2.55194	0.109818
$541$	32.2006	1.38441	0.692205	−	0.721701i	$-0.256639\pi$
$541$	32.2006	1.38441	0.692205	+	0.721701i	$0.256639\pi$
$542$	2.12512	0.0912816
$543$	−0.977813	−0.0419620
$544$	6.32279	0.271087
$545$	20.6227	0.883379
$546$	−0.568917	−0.0243474
$547$	−21.3254	−0.911809	−0.455904	−	0.890029i	$-0.650684\pi$
$547$	−21.3254	−0.911809	−0.455904	+	0.890029i	$0.650684\pi$
$548$	12.4110	0.530172
$549$	−26.5785	−1.13434
$550$	−4.01244	−0.171091
$551$	0	0
$552$	0.128764	0.00548057
$553$	2.88340	0.122615
$554$	−13.4784	−0.572641
$555$	1.15265	0.0489273
$556$	8.20900	0.348139
$557$	20.0222	0.848370	0.424185	−	0.905576i	$-0.360561\pi$
$557$	20.0222	0.848370	0.424185	+	0.905576i	$0.360561\pi$
$558$	−28.7317	−1.21631
$559$	53.2791	2.25347
$560$	−3.74313	−0.158176
$561$	−0.320604	−0.0135359
$562$	29.7072	1.25312
$563$	−29.5407	−1.24499	−0.622497	−	0.782622i	$-0.713882\pi$
$563$	−29.5407	−1.24499	−0.622497	+	0.782622i	$0.713882\pi$
$564$	−1.31940	−0.0555569
$565$	−42.1738	−1.77426
$566$	−24.3926	−1.02530
$567$	8.88346	0.373070
$568$	9.12052	0.382689
$569$	29.9858	1.25707	0.628535	−	0.777781i	$-0.283655\pi$
$569$	29.9858	1.25707	0.628535	+	0.777781i	$0.283655\pi$
$570$	0	0
$571$	18.9205	0.791797	0.395898	−	0.918294i	$-0.370433\pi$
$571$	18.9205	0.791797	0.395898	+	0.918294i	$0.370433\pi$
$572$	2.22464	0.0930170
$573$	1.53937	0.0643079
$574$	4.63992	0.193666
$575$	−10.1893	−0.424923
$576$	−2.98703	−0.124460
$577$	21.8827	0.910988	0.455494	−	0.890239i	$-0.349463\pi$
$577$	21.8827	0.910988	0.455494	+	0.890239i	$0.349463\pi$
$578$	−22.9777	−0.955745
$579$	−1.03229	−0.0429006
$580$	−18.4252	−0.765063
$581$	6.23955	0.258860
$582$	1.92864	0.0799448
$583$	−3.07486	−0.127348
$584$	9.16441	0.379226
$585$	55.8597	2.30952
$586$	−6.08565	−0.251396
$587$	18.1580	0.749461	0.374730	−	0.927134i	$-0.377735\pi$
$587$	18.1580	0.749461	0.374730	+	0.927134i	$0.377735\pi$
$588$	0.113874	0.00469608
$589$	0	0
$590$	−6.39143	−0.263131
$591$	2.40637	0.0989849
$592$	−2.70420	−0.111142
$593$	−0.348178	−0.0142979	−0.00714897	−	0.999974i	$-0.502276\pi$
$593$	−0.348178	−0.0142979	−0.00714897	+	0.999974i	$0.502276\pi$
$594$	0.303579	0.0124560
$595$	23.6670	0.970252
$596$	17.1739	0.703471
$597$	1.48569	0.0608050
$598$	5.64932	0.231018
$599$	17.2882	0.706378	0.353189	−	0.935552i	$-0.385097\pi$
$599$	17.2882	0.706378	0.353189	+	0.935552i	$0.385097\pi$
$600$	−1.02612	−0.0418910
$601$	44.0965	1.79873	0.899367	−	0.437194i	$-0.144028\pi$
$601$	44.0965	1.79873	0.899367	+	0.437194i	$0.144028\pi$
$602$	−10.6643	−0.434645
$603$	−20.5971	−0.838778
$604$	−2.27289	−0.0924825
$605$	40.4322	1.64380
$606$	0.386226	0.0156893
$607$	−0.516886	−0.0209797	−0.0104899	−	0.999945i	$-0.503339\pi$
$607$	−0.516886	−0.0209797	−0.0104899	+	0.999945i	$0.503339\pi$
$608$	0	0
$609$	0.560533	0.0227139
$610$	33.3062	1.34853
$611$	−57.8866	−2.34184
$612$	18.8864	0.763437
$613$	4.84485	0.195682	0.0978409	−	0.995202i	$-0.468806\pi$
$613$	4.84485	0.195682	0.0978409	+	0.995202i	$0.468806\pi$
$614$	−7.69160	−0.310408
$615$	1.97774	0.0797501
$616$	−0.445283	−0.0179410
$617$	28.4271	1.14443	0.572215	−	0.820103i	$-0.306084\pi$
$617$	28.4271	1.14443	0.572215	+	0.820103i	$0.306084\pi$
$618$	−0.466879	−0.0187806
$619$	3.51552	0.141301	0.0706503	−	0.997501i	$-0.477493\pi$
$619$	3.51552	0.141301	0.0706503	+	0.997501i	$0.477493\pi$
$620$	36.0044	1.44597
$621$	0.770916	0.0309358
$622$	−25.6831	−1.02980
$623$	5.56970	0.223145
$624$	0.568917	0.0227749
$625$	11.1430	0.445719
$626$	−8.35591	−0.333969
$627$	0	0
$628$	3.22494	0.128689
$629$	17.0981	0.681746
$630$	−11.1808	−0.445455
$631$	−9.48090	−0.377429	−0.188714	−	0.982032i	$-0.560432\pi$
$631$	−9.48090	−0.377429	−0.188714	+	0.982032i	$0.560432\pi$
$632$	−2.88340	−0.114696
$633$	2.52434	0.100333
$634$	3.64558	0.144785
$635$	46.8999	1.86116
$636$	−0.786345	−0.0311806
$637$	4.99602	0.197950
$638$	−2.19186	−0.0867766
$639$	27.2433	1.07773
$640$	3.74313	0.147960
$641$	30.8841	1.21985	0.609924	−	0.792460i	$-0.291200\pi$
$641$	30.8841	1.21985	0.609924	+	0.792460i	$0.291200\pi$
$642$	0.281055	0.0110924
$643$	−37.1896	−1.46661	−0.733307	−	0.679898i	$-0.762024\pi$
$643$	−37.1896	−1.46661	−0.733307	+	0.679898i	$0.762024\pi$
$644$	−1.13076	−0.0445583
$645$	−4.54560	−0.178983
$646$	0	0
$647$	−40.5380	−1.59371	−0.796856	−	0.604169i	$-0.793505\pi$
$647$	−40.5380	−1.59371	−0.796856	+	0.604169i	$0.793505\pi$
$648$	−8.88346	−0.348975
$649$	−0.760325	−0.0298454
$650$	−45.0191	−1.76580
$651$	−1.09533	−0.0429294
$652$	−11.3760	−0.445517
$653$	10.1633	0.397719	0.198860	−	0.980028i	$-0.436276\pi$
$653$	10.1633	0.397719	0.198860	+	0.980028i	$0.436276\pi$
$654$	0.627386	0.0245327
$655$	29.9851	1.17162
$656$	−4.63992	−0.181158
$657$	27.3744	1.06798
$658$	11.5865	0.451690
$659$	−12.4208	−0.483846	−0.241923	−	0.970295i	$-0.577778\pi$
$659$	−12.4208	−0.483846	−0.241923	+	0.970295i	$0.577778\pi$
$660$	−0.189799	−0.00738793
$661$	−12.8554	−0.500018	−0.250009	−	0.968243i	$-0.580434\pi$
$661$	−12.8554	−0.500018	−0.250009	+	0.968243i	$0.580434\pi$
$662$	18.4604	0.717486
$663$	−3.59714	−0.139701
$664$	−6.23955	−0.242142
$665$	0	0
$666$	−8.07754	−0.312998
$667$	−5.56607	−0.215519
$668$	22.1814	0.858223
$669$	2.67198	0.103305
$670$	25.8107	0.997155
$671$	3.96211	0.152956
$672$	−0.113874	−0.00439278
$673$	10.7901	0.415929	0.207964	−	0.978136i	$-0.433316\pi$
$673$	10.7901	0.415929	0.207964	+	0.978136i	$0.433316\pi$
$674$	−32.4569	−1.25019
$675$	−6.14339	−0.236459
$676$	11.9603	0.460010
$677$	1.88112	0.0722973	0.0361487	−	0.999346i	$-0.488491\pi$
$677$	1.88112	0.0722973	0.0361487	+	0.999346i	$0.488491\pi$
$678$	−1.28302	−0.0492740
$679$	−16.9366	−0.649969
$680$	−23.6670	−0.907588
$681$	0.466868	0.0178904
$682$	4.28308	0.164008
$683$	26.3309	1.00752	0.503762	−	0.863843i	$-0.331949\pi$
$683$	26.3309	1.00752	0.503762	+	0.863843i	$0.331949\pi$
$684$	0	0
$685$	−46.4560	−1.77499
$686$	−1.00000	−0.0381802
$687$	1.15665	0.0441289
$688$	10.6643	0.406573
$689$	−34.4996	−1.31433
$690$	−0.481981	−0.0183487
$691$	43.5511	1.65676	0.828381	−	0.560166i	$-0.189262\pi$
$691$	43.5511	1.65676	0.828381	+	0.560166i	$0.189262\pi$
$692$	20.2068	0.768147
$693$	−1.33007	−0.0505253
$694$	12.2197	0.463852
$695$	−30.7273	−1.16555
$696$	−0.560533	−0.0212470
$697$	29.3372	1.11123
$698$	22.3460	0.845808
$699$	−1.55013	−0.0586311
$700$	9.01099	0.340583
$701$	−3.18573	−0.120323	−0.0601616	−	0.998189i	$-0.519162\pi$
$701$	−3.18573	−0.120323	−0.0601616	+	0.998189i	$0.519162\pi$
$702$	3.40612	0.128556
$703$	0	0
$704$	0.445283	0.0167822
$705$	4.93870	0.186002
$706$	−4.51862	−0.170060
$707$	−3.39170	−0.127558
$708$	−0.194441	−0.00730754
$709$	18.1777	0.682679	0.341340	−	0.939940i	$-0.389119\pi$
$709$	18.1777	0.682679	0.341340	+	0.939940i	$0.389119\pi$
$710$	−34.1393	−1.28122
$711$	−8.61282	−0.323006
$712$	−5.56970	−0.208733
$713$	10.8766	0.407331
$714$	0.720001	0.0269454
$715$	−8.32712	−0.311417
$716$	0.722972	0.0270187
$717$	0.161790	0.00604215
$718$	−1.21213	−0.0452361
$719$	−33.7321	−1.25800	−0.628998	−	0.777407i	$-0.716535\pi$
$719$	−33.7321	−1.25800	−0.628998	+	0.777407i	$0.716535\pi$
$720$	11.1808	0.416685
$721$	4.09997	0.152691
$722$	0	0
$723$	−1.55249	−0.0577376
$724$	−8.58680	−0.319126
$725$	44.3557	1.64733
$726$	1.23003	0.0456509
$727$	−27.6738	−1.02636	−0.513182	−	0.858280i	$-0.671533\pi$
$727$	−27.6738	−1.02636	−0.513182	+	0.858280i	$0.671533\pi$
$728$	−4.99602	−0.185165
$729$	−26.3023	−0.974159
$730$	−34.3035	−1.26963
$731$	−67.4282	−2.49392
$732$	1.01325	0.0374506
$733$	−31.3510	−1.15797	−0.578987	−	0.815337i	$-0.696552\pi$
$733$	−31.3510	−1.15797	−0.578987	+	0.815337i	$0.696552\pi$
$734$	−25.7736	−0.951323
$735$	−0.426244	−0.0157223
$736$	1.13076	0.0416805
$737$	3.07045	0.113101
$738$	−13.8596	−0.510178
$739$	−4.29034	−0.157823	−0.0789114	−	0.996882i	$-0.525144\pi$
$739$	−4.29034	−0.157823	−0.0789114	+	0.996882i	$0.525144\pi$
$740$	10.1222	0.372098
$741$	0	0
$742$	6.90540	0.253505
$743$	−33.3490	−1.22346	−0.611728	−	0.791068i	$-0.709525\pi$
$743$	−33.3490	−1.22346	−0.611728	+	0.791068i	$0.709525\pi$
$744$	1.09533	0.0401567
$745$	−64.2841	−2.35519
$746$	13.9643	0.511269
$747$	−18.6378	−0.681920
$748$	−2.81543	−0.102942
$749$	−2.46812	−0.0901833
$750$	1.70966	0.0624280
$751$	33.1631	1.21014	0.605069	−	0.796173i	$-0.293145\pi$
$751$	33.1631	1.21014	0.605069	+	0.796173i	$0.293145\pi$
$752$	−11.5865	−0.422517
$753$	1.19780	0.0436504
$754$	−24.5924	−0.895604
$755$	8.50770	0.309627
$756$	−0.681767	−0.0247956
$757$	−24.5646	−0.892814	−0.446407	−	0.894830i	$-0.647297\pi$
$757$	−24.5646	−0.892814	−0.446407	+	0.894830i	$0.647297\pi$
$758$	−33.1820	−1.20523
$759$	−0.0573365	−0.00208118
$760$	0	0
$761$	30.3490	1.10015	0.550075	−	0.835115i	$-0.314599\pi$
$761$	30.3490	1.10015	0.550075	+	0.835115i	$0.314599\pi$
$762$	1.42679	0.0516873
$763$	−5.50948	−0.199457
$764$	13.5182	0.489070
$765$	−70.6941	−2.55595
$766$	21.9371	0.792621
$767$	−8.53077	−0.308028
$768$	0.113874	0.00410907
$769$	−11.2790	−0.406730	−0.203365	−	0.979103i	$-0.565188\pi$
$769$	−11.2790	−0.406730	−0.203365	+	0.979103i	$0.565188\pi$
$770$	1.66675	0.0600655
$771$	−0.246701	−0.00888472
$772$	−9.06521	−0.326264
$773$	−15.6025	−0.561184	−0.280592	−	0.959827i	$-0.590531\pi$
$773$	−15.6025	−0.561184	−0.280592	+	0.959827i	$0.590531\pi$
$774$	31.8546	1.14499
$775$	−86.6748	−3.11345
$776$	16.9366	0.607990
$777$	−0.307938	−0.0110472
$778$	−1.11469	−0.0399636
$779$	0	0
$780$	−2.12953	−0.0762493
$781$	−4.06121	−0.145322
$782$	−7.14957	−0.255668
$783$	−3.35593	−0.119931
$784$	1.00000	0.0357143
$785$	−12.0713	−0.430845
$786$	0.912212	0.0325375
$787$	−17.5434	−0.625355	−0.312678	−	0.949859i	$-0.601226\pi$
$787$	−17.5434	−0.625355	−0.312678	+	0.949859i	$0.601226\pi$
$788$	21.1319	0.752793
$789$	1.25308	0.0446109
$790$	10.7929	0.383996
$791$	11.2670	0.400608
$792$	1.33007	0.0472621
$793$	44.4544	1.57862
$794$	20.8096	0.738503
$795$	2.94339	0.104391
$796$	13.0468	0.462430
$797$	25.2604	0.894769	0.447384	−	0.894342i	$-0.352356\pi$
$797$	25.2604	0.894769	0.447384	+	0.894342i	$0.352356\pi$
$798$	0	0
$799$	73.2592	2.59173
$800$	−9.01099	−0.318587
$801$	−16.6369	−0.587836
$802$	−14.6596	−0.517650
$803$	−4.08075	−0.144007
$804$	0.785217	0.0276925
$805$	4.23259	0.149179
$806$	48.0557	1.69269
$807$	−0.326525	−0.0114942
$808$	3.39170	0.119319
$809$	46.8105	1.64577	0.822884	−	0.568209i	$-0.192363\pi$
$809$	46.8105	1.64577	0.822884	+	0.568209i	$0.192363\pi$
$810$	33.2519	1.16835
$811$	−29.7233	−1.04373	−0.521863	−	0.853029i	$-0.674763\pi$
$811$	−29.7233	−1.04373	−0.521863	+	0.853029i	$0.674763\pi$
$812$	4.92240	0.172742
$813$	−0.241995	−0.00848715
$814$	1.20413	0.0422049
$815$	42.5817	1.49157
$816$	−0.720001	−0.0252051
$817$	0	0
$818$	−1.42751	−0.0499118
$819$	−14.9233	−0.521462
$820$	17.3678	0.606509
$821$	48.5993	1.69613	0.848064	−	0.529893i	$-0.177768\pi$
$821$	48.5993	1.69613	0.848064	+	0.529893i	$0.177768\pi$
$822$	−1.41329	−0.0492942
$823$	−17.5409	−0.611439	−0.305719	−	0.952122i	$-0.598897\pi$
$823$	−17.5409	−0.611439	−0.305719	+	0.952122i	$0.598897\pi$
$824$	−4.09997	−0.142829
$825$	0.456912	0.0159076
$826$	1.70751	0.0594119
$827$	27.4591	0.954847	0.477423	−	0.878673i	$-0.341571\pi$
$827$	27.4591	0.954847	0.477423	+	0.878673i	$0.341571\pi$
$828$	3.37762	0.117381
$829$	−25.9442	−0.901078	−0.450539	−	0.892757i	$-0.648768\pi$
$829$	−25.9442	−0.901078	−0.450539	+	0.892757i	$0.648768\pi$
$830$	23.3554	0.810679
$831$	1.53483	0.0532428
$832$	4.99602	0.173206
$833$	−6.32279	−0.219072
$834$	−0.934791	−0.0323692
$835$	−83.0276	−2.87329
$836$	0	0
$837$	6.55778	0.226670
$838$	−24.0982	−0.832457
$839$	3.49364	0.120614	0.0603069	−	0.998180i	$-0.480792\pi$
$839$	3.49364	0.120614	0.0603069	+	0.998180i	$0.480792\pi$
$840$	0.426244	0.0147068
$841$	−4.76997	−0.164482
$842$	34.5275	1.18990
$843$	−3.38287	−0.116512
$844$	22.1678	0.763049
$845$	−44.7687	−1.54009
$846$	−34.6094	−1.18989
$847$	−10.8017	−0.371152
$848$	−6.90540	−0.237133
$849$	2.77768	0.0953296
$850$	56.9746	1.95421
$851$	3.05781	0.104820
$852$	−1.03859	−0.0355815
$853$	54.3551	1.86108	0.930542	−	0.366186i	$-0.119337\pi$
$853$	54.3551	1.86108	0.930542	+	0.366186i	$0.119337\pi$
$854$	−8.89796	−0.304482
$855$	0	0
$856$	2.46812	0.0843587
$857$	7.25425	0.247800	0.123900	−	0.992295i	$-0.460460\pi$
$857$	7.25425	0.247800	0.123900	+	0.992295i	$0.460460\pi$
$858$	−0.253329	−0.00864850
$859$	37.4752	1.27864	0.639319	−	0.768942i	$-0.279216\pi$
$859$	37.4752	1.27864	0.639319	+	0.768942i	$0.279216\pi$
$860$	−39.9178	−1.36119
$861$	−0.528365	−0.0180066
$862$	−20.9254	−0.712722
$863$	−18.7367	−0.637805	−0.318903	−	0.947787i	$-0.603314\pi$
$863$	−18.7367	−0.637805	−0.318903	+	0.947787i	$0.603314\pi$
$864$	0.681767	0.0231942
$865$	−75.6366	−2.57172
$866$	−3.53190	−0.120019
$867$	2.61656	0.0888630
$868$	−9.61879	−0.326483
$869$	1.28393	0.0435543
$870$	2.09815	0.0711338
$871$	34.4501	1.16730
$872$	5.50948	0.186575
$873$	50.5903	1.71222
$874$	0	0
$875$	−15.0136	−0.507553
$876$	−1.04359	−0.0352595
$877$	11.0211	0.372155	0.186077	−	0.982535i	$-0.440422\pi$
$877$	11.0211	0.372155	0.186077	+	0.982535i	$0.440422\pi$
$878$	11.6834	0.394297
$879$	0.692997	0.0233742
$880$	−1.66675	−0.0561861
$881$	−8.53481	−0.287545	−0.143773	−	0.989611i	$-0.545923\pi$
$881$	−8.53481	−0.287545	−0.143773	+	0.989611i	$0.545923\pi$
$882$	2.98703	0.100579
$883$	−27.8602	−0.937571	−0.468785	−	0.883312i	$-0.655308\pi$
$883$	−27.8602	−0.937571	−0.468785	+	0.883312i	$0.655308\pi$
$884$	−31.5888	−1.06245
$885$	0.727817	0.0244653
$886$	−23.8626	−0.801681
$887$	−44.8715	−1.50664	−0.753319	−	0.657655i	$-0.771548\pi$
$887$	−44.8715	−1.50664	−0.753319	+	0.657655i	$0.771548\pi$
$888$	0.307938	0.0103337
$889$	−12.5296	−0.420229
$890$	20.8481	0.698830
$891$	3.95565	0.132519
$892$	23.4644	0.785645
$893$	0	0
$894$	−1.95566	−0.0654071
$895$	−2.70617	−0.0904574
$896$	−1.00000	−0.0334077
$897$	−0.643310	−0.0214795
$898$	−7.98821	−0.266570
$899$	−47.3476	−1.57913
$900$	−26.9161	−0.897204
$901$	43.6614	1.45457
$902$	2.06607	0.0687927
$903$	1.21439	0.0404122
$904$	−11.2670	−0.374735
$905$	32.1415	1.06842
$906$	0.258823	0.00859881
$907$	16.5663	0.550075	0.275037	−	0.961434i	$-0.411310\pi$
$907$	16.5663	0.550075	0.275037	+	0.961434i	$0.411310\pi$
$908$	4.09986	0.136059
$909$	10.1311	0.336028
$910$	18.7007	0.619924
$911$	20.6558	0.684358	0.342179	−	0.939635i	$-0.388835\pi$
$911$	20.6558	0.684358	0.342179	+	0.939635i	$0.388835\pi$
$912$	0	0
$913$	2.77837	0.0919505
$914$	14.3870	0.475880
$915$	−3.79271	−0.125383
$916$	10.1573	0.335606
$917$	−8.01072	−0.264537
$918$	−4.31067	−0.142273
$919$	29.4278	0.970733	0.485367	−	0.874311i	$-0.338686\pi$
$919$	29.4278	0.970733	0.485367	+	0.874311i	$0.338686\pi$
$920$	−4.23259	−0.139544
$921$	0.875873	0.0288610
$922$	−11.9026	−0.391990
$923$	−45.5664	−1.49983
$924$	0.0507061	0.00166811
$925$	−24.3675	−0.801199
$926$	−15.4030	−0.506175
$927$	−12.2467	−0.402236
$928$	−4.92240	−0.161586
$929$	−19.5739	−0.642199	−0.321099	−	0.947046i	$-0.604052\pi$
$929$	−19.5739	−0.642199	−0.321099	+	0.947046i	$0.604052\pi$
$930$	−4.09996	−0.134443
$931$	0	0
$932$	−13.6127	−0.445897
$933$	2.92464	0.0957484
$934$	16.8900	0.552656
$935$	10.5385	0.344646
$936$	14.9233	0.487783
$937$	6.72879	0.219820	0.109910	−	0.993942i	$-0.464944\pi$
$937$	6.72879	0.219820	0.109910	+	0.993942i	$0.464944\pi$
$938$	−6.89550	−0.225146
$939$	0.951520	0.0310517
$940$	43.3699	1.41457
$941$	−16.5806	−0.540511	−0.270255	−	0.962789i	$-0.587108\pi$
$941$	−16.5806	−0.540511	−0.270255	+	0.962789i	$0.587108\pi$
$942$	−0.367236	−0.0119652
$943$	5.24664	0.170854
$944$	−1.70751	−0.0555747
$945$	2.55194	0.0830146
$946$	−4.74863	−0.154391
$947$	47.9906	1.55949	0.779743	−	0.626100i	$-0.215350\pi$
$947$	47.9906	1.55949	0.779743	+	0.626100i	$0.215350\pi$
$948$	0.328344	0.0106641
$949$	−45.7856	−1.48626
$950$	0	0
$951$	−0.415137	−0.0134617
$952$	6.32279	0.204923
$953$	−34.9315	−1.13154	−0.565772	−	0.824562i	$-0.691422\pi$
$953$	−34.9315	−1.13154	−0.565772	+	0.824562i	$0.691422\pi$
$954$	−20.6267	−0.667813
$955$	−50.6002	−1.63738
$956$	1.42078	0.0459513
$957$	0.249596	0.00806828
$958$	10.5626	0.341264
$959$	12.4110	0.400773
$960$	−0.426244	−0.0137570
$961$	61.5212	1.98455
$962$	13.5103	0.435588
$963$	7.37237	0.237571
$964$	−13.6334	−0.439102
$965$	33.9322	1.09232
$966$	0.128764	0.00414292
$967$	−39.7838	−1.27936	−0.639680	−	0.768641i	$-0.720933\pi$
$967$	−39.7838	−1.27936	−0.639680	+	0.768641i	$0.720933\pi$
$968$	10.8017	0.347181
$969$	0	0
$970$	−63.3960	−2.03552
$971$	−6.61679	−0.212343	−0.106172	−	0.994348i	$-0.533859\pi$
$971$	−6.61679	−0.212343	−0.106172	+	0.994348i	$0.533859\pi$
$972$	3.05690	0.0980500
$973$	8.20900	0.263169
$974$	−30.5440	−0.978693
$975$	5.12650	0.164179
$976$	8.89796	0.284817
$977$	−3.38549	−0.108311	−0.0541557	−	0.998533i	$-0.517247\pi$
$977$	−3.38549	−0.108311	−0.0541557	+	0.998533i	$0.517247\pi$
$978$	1.29543	0.0414232
$979$	2.48009	0.0792641
$980$	−3.74313	−0.119570
$981$	16.4570	0.525432
$982$	−0.102404	−0.00326783
$983$	−60.1249	−1.91769	−0.958843	−	0.283936i	$-0.908360\pi$
$983$	−60.1249	−1.91769	−0.958843	+	0.283936i	$0.908360\pi$
$984$	0.528365	0.0168437
$985$	−79.0993	−2.52031
$986$	31.1233	0.991168
$987$	−1.31940	−0.0419971
$988$	0	0
$989$	−12.0588	−0.383447
$990$	−4.97863	−0.158231
$991$	11.1279	0.353489	0.176744	−	0.984257i	$-0.443443\pi$
$991$	11.1279	0.353489	0.176744	+	0.984257i	$0.443443\pi$
$992$	9.61879	0.305397
$993$	−2.10216	−0.0667101
$994$	9.12052	0.289285
$995$	−48.8356	−1.54819
$996$	0.710523	0.0225138
$997$	15.4889	0.490539	0.245270	−	0.969455i	$-0.421123\pi$
$997$	15.4889	0.490539	0.245270	+	0.969455i	$0.421123\pi$
$998$	−0.647893	−0.0205087
$999$	1.84363	0.0583300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bl.1.8		12
19.6	even	9		266.2.u.d.169.3	yes	24
19.16	even	9		266.2.u.d.85.3	✓	24
19.18	odd	2		5054.2.a.bm.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.u.d.85.3	✓	24	19.16	even	9
266.2.u.d.169.3	yes	24	19.6	even	9
5054.2.a.bl.1.8		12	1.1	even	1	trivial
5054.2.a.bm.1.5		12	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.113874 q^{3} +1.00000 q^{4} -3.74313 q^{5} -0.113874 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.98703 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.113874 q^{3} +1.00000 q^{4} -3.74313 q^{5} -0.113874 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.98703 q^{9} +3.74313 q^{10} +0.445283 q^{11} +0.113874 q^{12} +4.99602 q^{13} -1.00000 q^{14} -0.426244 q^{15} +1.00000 q^{16} -6.32279 q^{17} +2.98703 q^{18} -3.74313 q^{20} +0.113874 q^{21} -0.445283 q^{22} -1.13076 q^{23} -0.113874 q^{24} +9.01099 q^{25} -4.99602 q^{26} -0.681767 q^{27} +1.00000 q^{28} +4.92240 q^{29} +0.426244 q^{30} -9.61879 q^{31} -1.00000 q^{32} +0.0507061 q^{33} +6.32279 q^{34} -3.74313 q^{35} -2.98703 q^{36} -2.70420 q^{37} +0.568917 q^{39} +3.74313 q^{40} -4.63992 q^{41} -0.113874 q^{42} +10.6643 q^{43} +0.445283 q^{44} +11.1808 q^{45} +1.13076 q^{46} -11.5865 q^{47} +0.113874 q^{48} +1.00000 q^{49} -9.01099 q^{50} -0.720001 q^{51} +4.99602 q^{52} -6.90540 q^{53} +0.681767 q^{54} -1.66675 q^{55} -1.00000 q^{56} -4.92240 q^{58} -1.70751 q^{59} -0.426244 q^{60} +8.89796 q^{61} +9.61879 q^{62} -2.98703 q^{63} +1.00000 q^{64} -18.7007 q^{65} -0.0507061 q^{66} +6.89550 q^{67} -6.32279 q^{68} -0.128764 q^{69} +3.74313 q^{70} -9.12052 q^{71} +2.98703 q^{72} -9.16441 q^{73} +2.70420 q^{74} +1.02612 q^{75} +0.445283 q^{77} -0.568917 q^{78} +2.88340 q^{79} -3.74313 q^{80} +8.88346 q^{81} +4.63992 q^{82} +6.23955 q^{83} +0.113874 q^{84} +23.6670 q^{85} -10.6643 q^{86} +0.560533 q^{87} -0.445283 q^{88} +5.56970 q^{89} -11.1808 q^{90} +4.99602 q^{91} -1.13076 q^{92} -1.09533 q^{93} +11.5865 q^{94} -0.113874 q^{96} -16.9366 q^{97} -1.00000 q^{98} -1.33007 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9}+O(q^{10}) \) 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 - 12 * q^8 + 21 * q^9	\( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} + 12 q^{7} - 12 q^{8} + 21 q^{9} - 3 q^{10} + 3 q^{11} - 3 q^{12} + 6 q^{13} - 12 q^{14} + 6 q^{15} + 12 q^{16} + 3 q^{17} - 21 q^{18} + 3 q^{20} - 3 q^{21} - 3 q^{22} + 12 q^{23} + 3 q^{24} + 33 q^{25} - 6 q^{26} - 21 q^{27} + 12 q^{28} - 6 q^{29} - 6 q^{30} - 3 q^{31} - 12 q^{32} + 6 q^{33} - 3 q^{34} + 3 q^{35} + 21 q^{36} - 27 q^{37} + 27 q^{39} - 3 q^{40} + 27 q^{41} + 3 q^{42} + 30 q^{43} + 3 q^{44} + 30 q^{45} - 12 q^{46} + 21 q^{47} - 3 q^{48} + 12 q^{49} - 33 q^{50} - 3 q^{51} + 6 q^{52} - 6 q^{53} + 21 q^{54} + 48 q^{55} - 12 q^{56} + 6 q^{58} + 27 q^{59} + 6 q^{60} + 18 q^{61} + 3 q^{62} + 21 q^{63} + 12 q^{64} + 9 q^{65} - 6 q^{66} - 9 q^{67} + 3 q^{68} - 18 q^{69} - 3 q^{70} - 27 q^{71} - 21 q^{72} - 9 q^{73} + 27 q^{74} - 33 q^{75} + 3 q^{77} - 27 q^{78} + 12 q^{79} + 3 q^{80} + 24 q^{81} - 27 q^{82} + 9 q^{83} - 3 q^{84} + 78 q^{85} - 30 q^{86} - 45 q^{87} - 3 q^{88} + 24 q^{89} - 30 q^{90} + 6 q^{91} + 12 q^{92} + 3 q^{93} - 21 q^{94} + 3 q^{96} + 18 q^{97} - 12 q^{98} + 48 q^{99}+O(q^{100}) \) 12 * q - 12 * q^2 - 3 * q^3 + 12 * q^4 + 3 * q^5 + 3 * q^6 + 12 * q^7 - 12 * q^8 + 21 * q^9 - 3 * q^10 + 3 * q^11 - 3 * q^12 + 6 * q^13 - 12 * q^14 + 6 * q^15 + 12 * q^16 + 3 * q^17 - 21 * q^18 + 3 * q^20 - 3 * q^21 - 3 * q^22 + 12 * q^23 + 3 * q^24 + 33 * q^25 - 6 * q^26 - 21 * q^27 + 12 * q^28 - 6 * q^29 - 6 * q^30 - 3 * q^31 - 12 * q^32 + 6 * q^33 - 3 * q^34 + 3 * q^35 + 21 * q^36 - 27 * q^37 + 27 * q^39 - 3 * q^40 + 27 * q^41 + 3 * q^42 + 30 * q^43 + 3 * q^44 + 30 * q^45 - 12 * q^46 + 21 * q^47 - 3 * q^48 + 12 * q^49 - 33 * q^50 - 3 * q^51 + 6 * q^52 - 6 * q^53 + 21 * q^54 + 48 * q^55 - 12 * q^56 + 6 * q^58 + 27 * q^59 + 6 * q^60 + 18 * q^61 + 3 * q^62 + 21 * q^63 + 12 * q^64 + 9 * q^65 - 6 * q^66 - 9 * q^67 + 3 * q^68 - 18 * q^69 - 3 * q^70 - 27 * q^71 - 21 * q^72 - 9 * q^73 + 27 * q^74 - 33 * q^75 + 3 * q^77 - 27 * q^78 + 12 * q^79 + 3 * q^80 + 24 * q^81 - 27 * q^82 + 9 * q^83 - 3 * q^84 + 78 * q^85 - 30 * q^86 - 45 * q^87 - 3 * q^88 + 24 * q^89 - 30 * q^90 + 6 * q^91 + 12 * q^92 + 3 * q^93 - 21 * q^94 + 3 * q^96 + 18 * q^97 - 12 * q^98 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more