LMFDB - Embedded newform 115.4.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,4,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	115.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:	$6.78521965066$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.72015$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} +O(q^{10})$	\(q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} -15.0000 q^{10} +12.6008 q^{11} +3.72015 q^{12} +8.16046 q^{13} +76.8023 q^{14} +18.6008 q^{15} -71.0000 q^{16} -76.0411 q^{17} +39.4814 q^{18} -103.362 q^{19} +5.00000 q^{20} -95.2388 q^{21} -37.8023 q^{22} -23.0000 q^{23} +78.1232 q^{24} +25.0000 q^{25} -24.4814 q^{26} -149.403 q^{27} -25.6008 q^{28} -267.202 q^{29} -55.8023 q^{30} -63.7574 q^{31} +45.0000 q^{32} +46.8768 q^{33} +228.123 q^{34} -128.004 q^{35} -13.1605 q^{36} +112.164 q^{37} +310.086 q^{38} +30.3582 q^{39} +105.000 q^{40} -239.078 q^{41} +285.716 q^{42} +282.239 q^{43} +12.6008 q^{44} -65.8023 q^{45} +69.0000 q^{46} +577.291 q^{47} -264.131 q^{48} +312.399 q^{49} -75.0000 q^{50} -282.884 q^{51} +8.16046 q^{52} -2.31326 q^{53} +448.209 q^{54} +63.0038 q^{55} -537.616 q^{56} -384.522 q^{57} +801.605 q^{58} +272.888 q^{59} +18.6008 q^{60} +294.049 q^{61} +191.272 q^{62} +336.918 q^{63} +433.000 q^{64} +40.8023 q^{65} -140.630 q^{66} +426.732 q^{67} -76.0411 q^{68} -85.5635 q^{69} +384.011 q^{70} -1020.85 q^{71} -276.370 q^{72} -286.650 q^{73} -336.493 q^{74} +93.0038 q^{75} -103.362 q^{76} -322.589 q^{77} -91.0745 q^{78} -551.866 q^{79} -355.000 q^{80} -200.470 q^{81} +717.235 q^{82} +21.7021 q^{83} -95.2388 q^{84} -380.205 q^{85} -846.716 q^{86} -994.031 q^{87} +264.616 q^{88} -1049.63 q^{89} +197.407 q^{90} -208.914 q^{91} -23.0000 q^{92} -237.187 q^{93} -1731.87 q^{94} -516.810 q^{95} +167.407 q^{96} +1729.19 q^{97} -937.198 q^{98} -165.832 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9}+O(q^{10}) $ 2 * q - 6 * q^2 - 3 * q^3 + 2 * q^4 + 10 * q^5 + 9 * q^6 + q^7 + 42 * q^8 + 5 * q^9	\( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19} + 10 q^{20} - 274 q^{21} + 81 q^{22} - 46 q^{23} - 63 q^{24} + 50 q^{25} + 45 q^{26} - 90 q^{27} + q^{28} - 430 q^{29} + 45 q^{30} - 305 q^{31} + 90 q^{32} + 313 q^{33} + 237 q^{34} + 5 q^{35} + 5 q^{36} - 68 q^{37} + 213 q^{38} + 186 q^{39} + 210 q^{40} - 593 q^{41} + 822 q^{42} + 648 q^{43} - 27 q^{44} + 25 q^{45} + 138 q^{46} + 382 q^{47} + 213 q^{48} + 677 q^{49} - 150 q^{50} - 263 q^{51} - 15 q^{52} - 464 q^{53} + 270 q^{54} - 135 q^{55} + 21 q^{56} - 602 q^{57} + 1290 q^{58} - 18 q^{59} - 15 q^{60} - 7 q^{61} + 915 q^{62} + 820 q^{63} + 866 q^{64} - 75 q^{65} - 939 q^{66} + 60 q^{67} - 79 q^{68} + 69 q^{69} - 15 q^{70} - 1029 q^{71} + 105 q^{72} + 74 q^{73} + 204 q^{74} - 75 q^{75} - 71 q^{76} - 1376 q^{77} - 558 q^{78} + 692 q^{79} - 710 q^{80} - 1090 q^{81} + 1779 q^{82} - 1460 q^{83} - 274 q^{84} - 395 q^{85} - 1944 q^{86} + 100 q^{87} - 567 q^{88} - 220 q^{89} - 75 q^{90} - 825 q^{91} - 46 q^{92} + 1384 q^{93} - 1146 q^{94} - 355 q^{95} - 135 q^{96} + 1339 q^{97} - 2031 q^{98} - 885 q^{99}+O(q^{100}) \) 2 * q - 6 * q^2 - 3 * q^3 + 2 * q^4 + 10 * q^5 + 9 * q^6 + q^7 + 42 * q^8 + 5 * q^9 - 30 * q^10 - 27 * q^11 - 3 * q^12 - 15 * q^13 - 3 * q^14 - 15 * q^15 - 142 * q^16 - 79 * q^17 - 15 * q^18 - 71 * q^19 + 10 * q^20 - 274 * q^21 + 81 * q^22 - 46 * q^23 - 63 * q^24 + 50 * q^25 + 45 * q^26 - 90 * q^27 + q^28 - 430 * q^29 + 45 * q^30 - 305 * q^31 + 90 * q^32 + 313 * q^33 + 237 * q^34 + 5 * q^35 + 5 * q^36 - 68 * q^37 + 213 * q^38 + 186 * q^39 + 210 * q^40 - 593 * q^41 + 822 * q^42 + 648 * q^43 - 27 * q^44 + 25 * q^45 + 138 * q^46 + 382 * q^47 + 213 * q^48 + 677 * q^49 - 150 * q^50 - 263 * q^51 - 15 * q^52 - 464 * q^53 + 270 * q^54 - 135 * q^55 + 21 * q^56 - 602 * q^57 + 1290 * q^58 - 18 * q^59 - 15 * q^60 - 7 * q^61 + 915 * q^62 + 820 * q^63 + 866 * q^64 - 75 * q^65 - 939 * q^66 + 60 * q^67 - 79 * q^68 + 69 * q^69 - 15 * q^70 - 1029 * q^71 + 105 * q^72 + 74 * q^73 + 204 * q^74 - 75 * q^75 - 71 * q^76 - 1376 * q^77 - 558 * q^78 + 692 * q^79 - 710 * q^80 - 1090 * q^81 + 1779 * q^82 - 1460 * q^83 - 274 * q^84 - 395 * q^85 - 1944 * q^86 + 100 * q^87 - 567 * q^88 - 220 * q^89 - 75 * q^90 - 825 * q^91 - 46 * q^92 + 1384 * q^93 - 1146 * q^94 - 355 * q^95 - 135 * q^96 + 1339 * q^97 - 2031 * q^98 - 885 * 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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	3.72015	0.715944	0.357972	−	0.933732i	$-0.383468\pi$
$3$	3.72015	0.715944	0.357972	+	0.933732i	$0.383468\pi$
$4$	1.00000	0.125000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−11.1605	−0.759373
$7$	−25.6008	−1.38231	−0.691156	−	0.722706i	$-0.742898\pi$
$7$	−25.6008	−1.38231	−0.691156	+	0.722706i	$0.742898\pi$
$8$	21.0000	0.928078
$9$	−13.1605	−0.487424
$10$	−15.0000	−0.474342
$11$	12.6008	0.345389	0.172694	−	0.984975i	$-0.444753\pi$
$11$	12.6008	0.345389	0.172694	+	0.984975i	$0.444753\pi$
$12$	3.72015	0.0894930
$13$	8.16046	0.174100	0.0870502	−	0.996204i	$-0.472256\pi$
$13$	8.16046	0.174100	0.0870502	+	0.996204i	$0.472256\pi$
$14$	76.8023	1.46616
$15$	18.6008	0.320180
$16$	−71.0000	−1.10938
$17$	−76.0411	−1.08486	−0.542431	−	0.840100i	$-0.682496\pi$
$17$	−76.0411	−1.08486	−0.542431	+	0.840100i	$0.682496\pi$
$18$	39.4814	0.516992
$19$	−103.362	−1.24805	−0.624023	−	0.781406i	$-0.714503\pi$
$19$	−103.362	−1.24805	−0.624023	+	0.781406i	$0.714503\pi$
$20$	5.00000	0.0559017
$21$	−95.2388	−0.989657
$22$	−37.8023	−0.366340
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	78.1232	0.664451
$25$	25.0000	0.200000
$26$	−24.4814	−0.184661
$27$	−149.403	−1.06491
$28$	−25.6008	−0.172789
$29$	−267.202	−1.71097	−0.855484	−	0.517829i	$-0.826740\pi$
$29$	−267.202	−1.71097	−0.855484	+	0.517829i	$0.826740\pi$
$30$	−55.8023	−0.339602
$31$	−63.7574	−0.369392	−0.184696	−	0.982796i	$-0.559130\pi$
$31$	−63.7574	−0.369392	−0.184696	+	0.982796i	$0.559130\pi$
$32$	45.0000	0.248592
$33$	46.8768	0.247279
$34$	228.123	1.15067
$35$	−128.004	−0.618188
$36$	−13.1605	−0.0609281
$37$	112.164	0.498370	0.249185	−	0.968456i	$-0.419837\pi$
$37$	112.164	0.498370	0.249185	+	0.968456i	$0.419837\pi$
$38$	310.086	1.32375
$39$	30.3582	0.124646
$40$	105.000	0.415049
$41$	−239.078	−0.910677	−0.455339	−	0.890318i	$-0.650482\pi$
$41$	−239.078	−0.910677	−0.455339	+	0.890318i	$0.650482\pi$
$42$	285.716	1.04969
$43$	282.239	1.00095	0.500477	−	0.865750i	$-0.333158\pi$
$43$	282.239	1.00095	0.500477	+	0.865750i	$0.333158\pi$
$44$	12.6008	0.0431736
$45$	−65.8023	−0.217983
$46$	69.0000	0.221163
$47$	577.291	1.79163	0.895815	−	0.444427i	$-0.146593\pi$
$47$	577.291	1.79163	0.895815	+	0.444427i	$0.146593\pi$
$48$	−264.131	−0.794250
$49$	312.399	0.910785
$50$	−75.0000	−0.212132
$51$	−282.884	−0.776701
$52$	8.16046	0.0217625
$53$	−2.31326	−0.00599529	−0.00299764	−	0.999996i	$-0.500954\pi$
$53$	−2.31326	−0.00599529	−0.00299764	+	0.999996i	$0.500954\pi$
$54$	448.209	1.12951
$55$	63.0038	0.154462
$56$	−537.616	−1.28289
$57$	−384.522	−0.893531
$58$	801.605	1.81476
$59$	272.888	0.602153	0.301077	−	0.953600i	$-0.402654\pi$
$59$	272.888	0.602153	0.301077	+	0.953600i	$0.402654\pi$
$60$	18.6008	0.0400225
$61$	294.049	0.617198	0.308599	−	0.951192i	$-0.400140\pi$
$61$	294.049	0.617198	0.308599	+	0.951192i	$0.400140\pi$
$62$	191.272	0.391800
$63$	336.918	0.673772
$64$	433.000	0.845703
$65$	40.8023	0.0778600
$66$	−140.630	−0.262279
$67$	426.732	0.778113	0.389056	−	0.921214i	$-0.372801\pi$
$67$	426.732	0.778113	0.389056	+	0.921214i	$0.372801\pi$
$68$	−76.0411	−0.135608
$69$	−85.5635	−0.149285
$70$	384.011	0.655688
$71$	−1020.85	−1.70638	−0.853191	−	0.521598i	$-0.825336\pi$
$71$	−1020.85	−1.70638	−0.853191	+	0.521598i	$0.825336\pi$
$72$	−276.370	−0.452368
$73$	−286.650	−0.459586	−0.229793	−	0.973240i	$-0.573805\pi$
$73$	−286.650	−0.459586	−0.229793	+	0.973240i	$0.573805\pi$
$74$	−336.493	−0.528601
$75$	93.0038	0.143189
$76$	−103.362	−0.156006
$77$	−322.589	−0.477435
$78$	−91.0745	−0.132207
$79$	−551.866	−0.785947	−0.392974	−	0.919550i	$-0.628554\pi$
$79$	−551.866	−0.785947	−0.392974	+	0.919550i	$0.628554\pi$
$80$	−355.000	−0.496128
$81$	−200.470	−0.274993
$82$	717.235	0.965919
$83$	21.7021	0.0287001	0.0143501	−	0.999897i	$-0.495432\pi$
$83$	21.7021	0.0287001	0.0143501	+	0.999897i	$0.495432\pi$
$84$	−95.2388	−0.123707
$85$	−380.205	−0.485165
$86$	−846.716	−1.06167
$87$	−994.031	−1.22496
$88$	264.616	0.320547
$89$	−1049.63	−1.25012	−0.625058	−	0.780578i	$-0.714925\pi$
$89$	−1049.63	−1.25012	−0.625058	+	0.780578i	$0.714925\pi$
$90$	197.407	0.231206
$91$	−208.914	−0.240661
$92$	−23.0000	−0.0260643
$93$	−237.187	−0.264464
$94$	−1731.87	−1.90031
$95$	−516.810	−0.558143
$96$	167.407	0.177978
$97$	1729.19	1.81003	0.905014	−	0.425381i	$-0.139860\pi$
$97$	1729.19	1.81003	0.905014	+	0.425381i	$0.139860\pi$
$98$	−937.198	−0.966033
$99$	−165.832	−0.168351
$100$	25.0000	0.0250000
$101$	855.874	0.843195	0.421597	−	0.906783i	$-0.361470\pi$
$101$	855.874	0.843195	0.421597	+	0.906783i	$0.361470\pi$
$102$	848.653	0.823816
$103$	−632.534	−0.605101	−0.302551	−	0.953133i	$-0.597838\pi$
$103$	−632.534	−0.605101	−0.302551	+	0.953133i	$0.597838\pi$
$104$	171.370	0.161579
$105$	−476.194	−0.442588
$106$	6.93977	0.00635896
$107$	1309.62	1.18323	0.591616	−	0.806220i	$-0.298490\pi$
$107$	1309.62	1.18323	0.591616	+	0.806220i	$0.298490\pi$
$108$	−149.403	−0.133114
$109$	1726.21	1.51689	0.758443	−	0.651739i	$-0.225960\pi$
$109$	1726.21	1.51689	0.758443	+	0.651739i	$0.225960\pi$
$110$	−189.011	−0.163832
$111$	417.268	0.356805
$112$	1817.65	1.53350
$113$	−1492.24	−1.24228	−0.621142	−	0.783698i	$-0.713331\pi$
$113$	−1492.24	−1.24228	−0.621142	+	0.783698i	$0.713331\pi$
$114$	1153.57	0.947732
$115$	−115.000	−0.0932505
$116$	−267.202	−0.213871
$117$	−107.395	−0.0848608
$118$	−818.665	−0.638680
$119$	1946.71	1.49962
$120$	390.616	0.297152
$121$	−1172.22	−0.880707
$122$	−882.146	−0.654637
$123$	−889.408	−0.651994
$124$	−63.7574	−0.0461741
$125$	125.000	0.0894427
$126$	−1010.75	−0.714644
$127$	−1368.29	−0.956034	−0.478017	−	0.878351i	$-0.658644\pi$
$127$	−1368.29	−0.956034	−0.478017	+	0.878351i	$0.658644\pi$
$128$	−1659.00	−1.14560
$129$	1049.97	0.716627
$130$	−122.407	−0.0825830
$131$	1984.24	1.32339	0.661694	−	0.749774i	$-0.269838\pi$
$131$	1984.24	1.32339	0.661694	+	0.749774i	$0.269838\pi$
$132$	46.8768	0.0309098
$133$	2646.15	1.72519
$134$	−1280.19	−0.825313
$135$	−747.015	−0.476243
$136$	−1596.86	−1.00684
$137$	273.331	0.170455	0.0852273	−	0.996362i	$-0.472838\pi$
$137$	273.331	0.170455	0.0852273	+	0.996362i	$0.472838\pi$
$138$	256.691	0.158340
$139$	−557.889	−0.340428	−0.170214	−	0.985407i	$-0.554446\pi$
$139$	−557.889	−0.340428	−0.170214	+	0.985407i	$0.554446\pi$
$140$	−128.004	−0.0772736
$141$	2147.61	1.28271
$142$	3062.56	1.80989
$143$	102.828	0.0601323
$144$	934.393	0.540736
$145$	−1336.01	−0.765168
$146$	859.949	0.487465
$147$	1162.17	0.652071
$148$	112.164	0.0622963
$149$	−1354.16	−0.744545	−0.372272	−	0.928124i	$-0.621421\pi$
$149$	−1354.16	−0.744545	−0.372272	+	0.928124i	$0.621421\pi$
$150$	−279.011	−0.151875
$151$	162.652	0.0876586	0.0438293	−	0.999039i	$-0.486044\pi$
$151$	162.652	0.0876586	0.0438293	+	0.999039i	$0.486044\pi$
$152$	−2170.60	−1.15828
$153$	1000.74	0.528789
$154$	967.768	0.506396
$155$	−318.787	−0.165197
$156$	30.3582	0.0155808
$157$	−2364.83	−1.20213	−0.601063	−	0.799201i	$-0.705256\pi$
$157$	−2364.83	−1.20213	−0.601063	+	0.799201i	$0.705256\pi$
$158$	1655.60	0.833623
$159$	−8.60567	−0.00429229
$160$	225.000	0.111174
$161$	588.818	0.288232
$162$	601.410	0.291674
$163$	471.162	0.226406	0.113203	−	0.993572i	$-0.463889\pi$
$163$	471.162	0.226406	0.113203	+	0.993572i	$0.463889\pi$
$164$	−239.078	−0.113835
$165$	234.384	0.110586
$166$	−65.1062	−0.0304411
$167$	−3735.73	−1.73102	−0.865508	−	0.500895i	$-0.833004\pi$
$167$	−3735.73	−1.73102	−0.865508	+	0.500895i	$0.833004\pi$
$168$	−2000.01	−0.918479
$169$	−2130.41	−0.969689
$170$	1140.62	0.514596
$171$	1360.29	0.608328
$172$	282.239	0.125119
$173$	−2709.33	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$173$	−2709.33	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$174$	2982.09	1.29926
$175$	−640.019	−0.276462
$176$	−894.654	−0.383165
$177$	1015.19	0.431108
$178$	3148.88	1.32595
$179$	3300.43	1.37813	0.689066	−	0.724699i	$-0.258021\pi$
$179$	3300.43	1.37813	0.689066	+	0.724699i	$0.258021\pi$
$180$	−65.8023	−0.0272479
$181$	−583.868	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$181$	−583.868	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$182$	626.742	0.255259
$183$	1093.91	0.441879
$184$	−483.000	−0.193518
$185$	560.821	0.222878
$186$	711.562	0.280507
$187$	−958.176	−0.374699
$188$	577.291	0.223954
$189$	3824.83	1.47204
$190$	1550.43	0.592000
$191$	−3058.98	−1.15885	−0.579424	−	0.815026i	$-0.696722\pi$
$191$	−3058.98	−1.15885	−0.579424	+	0.815026i	$0.696722\pi$
$192$	1610.83	0.605476
$193$	−1336.93	−0.498625	−0.249313	−	0.968423i	$-0.580205\pi$
$193$	−1336.93	−0.498625	−0.249313	+	0.968423i	$0.580205\pi$
$194$	−5187.57	−1.91983
$195$	151.791	0.0557434
$196$	312.399	0.113848
$197$	−2449.21	−0.885783	−0.442892	−	0.896575i	$-0.646047\pi$
$197$	−2449.21	−0.885783	−0.442892	+	0.896575i	$0.646047\pi$
$198$	497.496	0.178563
$199$	1138.66	0.405614	0.202807	−	0.979219i	$-0.434994\pi$
$199$	1138.66	0.405614	0.202807	+	0.979219i	$0.434994\pi$
$200$	525.000	0.185616
$201$	1587.51	0.557085
$202$	−2567.62	−0.894343
$203$	6840.56	2.36509
$204$	−282.884	−0.0970876
$205$	−1195.39	−0.407267
$206$	1897.60	0.641807
$207$	302.691	0.101635
$208$	−579.393	−0.193143
$209$	−1302.44	−0.431061
$210$	1428.58	0.469436
$211$	5596.81	1.82607	0.913034	−	0.407884i	$-0.133733\pi$
$211$	5596.81	1.82607	0.913034	+	0.407884i	$0.133733\pi$
$212$	−2.31326	−0.000749411 0
$213$	−3797.74	−1.22167
$214$	−3928.86	−1.25501
$215$	1411.19	0.447640
$216$	−3137.46	−0.988321
$217$	1632.24	0.510615
$218$	−5178.62	−1.60890
$219$	−1066.38	−0.329038
$220$	63.0038	0.0193078
$221$	−620.530	−0.188875
$222$	−1251.81	−0.378449
$223$	5580.60	1.67581	0.837903	−	0.545820i	$-0.183782\pi$
$223$	5580.60	1.67581	0.837903	+	0.545820i	$0.183782\pi$
$224$	−1152.03	−0.343632
$225$	−329.011	−0.0974849
$226$	4476.72	1.31764
$227$	−566.323	−0.165587	−0.0827934	−	0.996567i	$-0.526384\pi$
$227$	−566.323	−0.165587	−0.0827934	+	0.996567i	$0.526384\pi$
$228$	−384.522	−0.111691
$229$	−693.702	−0.200180	−0.100090	−	0.994978i	$-0.531913\pi$
$229$	−693.702	−0.200180	−0.100090	+	0.994978i	$0.531913\pi$
$230$	345.000	0.0989071
$231$	−1200.08	−0.341816
$232$	−5611.23	−1.58791
$233$	−2208.68	−0.621011	−0.310506	−	0.950572i	$-0.600498\pi$
$233$	−2208.68	−0.621011	−0.310506	+	0.950572i	$0.600498\pi$
$234$	322.186	0.0900084
$235$	2886.46	0.801241
$236$	272.888	0.0752691
$237$	−2053.03	−0.562694
$238$	−5840.13	−1.59059
$239$	−3876.79	−1.04924	−0.524621	−	0.851336i	$-0.675793\pi$
$239$	−3876.79	−1.04924	−0.524621	+	0.851336i	$0.675793\pi$
$240$	−1320.65	−0.355199
$241$	−2024.57	−0.541136	−0.270568	−	0.962701i	$-0.587211\pi$
$241$	−2024.57	−0.541136	−0.270568	+	0.962701i	$0.587211\pi$
$242$	3516.66	0.934131
$243$	3288.10	0.868033
$244$	294.049	0.0771498
$245$	1562.00	0.407315
$246$	2668.22	0.691544
$247$	−843.481	−0.217285
$248$	−1338.91	−0.342825
$249$	80.7350	0.0205477
$250$	−375.000	−0.0948683
$251$	−733.444	−0.184441	−0.0922203	−	0.995739i	$-0.529396\pi$
$251$	−733.444	−0.184441	−0.0922203	+	0.995739i	$0.529396\pi$
$252$	336.918	0.0842215
$253$	−289.818	−0.0720185
$254$	4104.88	1.01403
$255$	−1414.42	−0.347351
$256$	1513.00	0.369385
$257$	2367.83	0.574713	0.287356	−	0.957824i	$-0.407224\pi$
$257$	2367.83	0.574713	0.287356	+	0.957824i	$0.407224\pi$
$258$	−3149.91	−0.760097
$259$	−2871.49	−0.688903
$260$	40.8023	0.00973250
$261$	3516.50	0.833968
$262$	−5952.72	−1.40366
$263$	6632.76	1.55511	0.777554	−	0.628816i	$-0.216460\pi$
$263$	6632.76	1.55511	0.777554	+	0.628816i	$0.216460\pi$
$264$	984.412	0.229494
$265$	−11.5663	−0.00268117
$266$	−7938.44	−1.82984
$267$	−3904.78	−0.895013
$268$	426.732	0.0972641
$269$	−6092.97	−1.38102	−0.690511	−	0.723322i	$-0.742614\pi$
$269$	−6092.97	−1.38102	−0.690511	+	0.723322i	$0.742614\pi$
$270$	2241.05	0.505132
$271$	−1981.30	−0.444115	−0.222058	−	0.975034i	$-0.571277\pi$
$271$	−1981.30	−0.444115	−0.222058	+	0.975034i	$0.571277\pi$
$272$	5398.92	1.20352
$273$	−777.192	−0.172300
$274$	−819.994	−0.180794
$275$	315.019	0.0690777
$276$	−85.5635	−0.0186606
$277$	−4097.42	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$277$	−4097.42	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$278$	1673.67	0.361079
$279$	839.077	0.180051
$280$	−2688.08	−0.573727
$281$	−3866.40	−0.820819	−0.410409	−	0.911901i	$-0.634614\pi$
$281$	−3866.40	−0.820819	−0.410409	+	0.911901i	$0.634614\pi$
$282$	−6442.84	−1.36052
$283$	3982.00	0.836415	0.418208	−	0.908351i	$-0.362658\pi$
$283$	3982.00	0.836415	0.418208	+	0.908351i	$0.362658\pi$
$284$	−1020.85	−0.213298
$285$	−1922.61	−0.399599
$286$	−308.484	−0.0637799
$287$	6120.59	1.25884
$288$	−592.221	−0.121170
$289$	869.245	0.176927
$290$	4008.02	0.811583
$291$	6432.86	1.29588
$292$	−286.650	−0.0574483
$293$	−7491.70	−1.49375	−0.746877	−	0.664962i	$-0.768448\pi$
$293$	−7491.70	−1.49375	−0.746877	+	0.664962i	$0.768448\pi$
$294$	−3486.52	−0.691626
$295$	1364.44	0.269291
$296$	2355.45	0.462526
$297$	−1882.59	−0.367809
$298$	4062.48	0.789709
$299$	−187.691	−0.0363024
$300$	93.0038	0.0178986
$301$	−7225.53	−1.38363
$302$	−487.957	−0.0929760
$303$	3183.98	0.603680
$304$	7338.70	1.38455
$305$	1470.24	0.276019
$306$	−3002.21	−0.560865
$307$	−5813.01	−1.08067	−0.540335	−	0.841450i	$-0.681703\pi$
$307$	−5813.01	−1.08067	−0.540335	+	0.841450i	$0.681703\pi$
$308$	−322.589	−0.0596793
$309$	−2353.12	−0.433218
$310$	956.361	0.175218
$311$	4769.43	0.869613	0.434807	−	0.900524i	$-0.356817\pi$
$311$	4769.43	0.869613	0.434807	+	0.900524i	$0.356817\pi$
$312$	637.521	0.115681
$313$	7053.43	1.27375	0.636875	−	0.770967i	$-0.280227\pi$
$313$	7053.43	1.27375	0.636875	+	0.770967i	$0.280227\pi$
$314$	7094.49	1.27505
$315$	1684.59	0.301320
$316$	−551.866	−0.0982434
$317$	8845.36	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$317$	8845.36	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$318$	25.8170	0.00455266
$319$	−3366.94	−0.590949
$320$	2165.00	0.378210
$321$	4871.99	0.847127
$322$	−1766.45	−0.305716
$323$	7859.76	1.35396
$324$	−200.470	−0.0343741
$325$	204.011	0.0348201
$326$	−1413.48	−0.240140
$327$	6421.75	1.08601
$328$	−5020.64	−0.845179
$329$	−14779.1	−2.47659
$330$	−703.152	−0.117295
$331$	4264.07	0.708079	0.354040	−	0.935230i	$-0.384808\pi$
$331$	4264.07	0.708079	0.354040	+	0.935230i	$0.384808\pi$
$332$	21.7021	0.00358752
$333$	−1476.13	−0.242918
$334$	11207.2	1.83602
$335$	2133.66	0.347983
$336$	6761.95	1.09790
$337$	−1290.58	−0.208613	−0.104306	−	0.994545i	$-0.533262\pi$
$337$	−1290.58	−0.208613	−0.104306	+	0.994545i	$0.533262\pi$
$338$	6391.22	1.02851
$339$	−5551.36	−0.889405
$340$	−380.205	−0.0606457
$341$	−803.392	−0.127584
$342$	−4080.87	−0.645229
$343$	783.403	0.123323
$344$	5927.01	0.928963
$345$	−427.818	−0.0667621
$346$	8127.98	1.26290
$347$	−1805.09	−0.279257	−0.139629	−	0.990204i	$-0.544591\pi$
$347$	−1805.09	−0.279257	−0.139629	+	0.990204i	$0.544591\pi$
$348$	−994.031	−0.153120
$349$	−3586.81	−0.550136	−0.275068	−	0.961425i	$-0.588700\pi$
$349$	−3586.81	−0.550136	−0.275068	+	0.961425i	$0.588700\pi$
$350$	1920.06	0.293233
$351$	−1219.20	−0.185402
$352$	567.034	0.0858609
$353$	−2292.90	−0.345719	−0.172859	−	0.984947i	$-0.555301\pi$
$353$	−2292.90	−0.345719	−0.172859	+	0.984947i	$0.555301\pi$
$354$	−3045.56	−0.457259
$355$	−5104.27	−0.763118
$356$	−1049.63	−0.156264
$357$	7242.06	1.07364
$358$	−9901.28	−1.46173
$359$	−3077.18	−0.452388	−0.226194	−	0.974082i	$-0.572628\pi$
$359$	−3077.18	−0.452388	−0.226194	+	0.974082i	$0.572628\pi$
$360$	−1381.85	−0.202305
$361$	3824.70	0.557618
$362$	1751.60	0.254316
$363$	−4360.84	−0.630537
$364$	−208.914	−0.0300826
$365$	−1433.25	−0.205533
$366$	−3281.72	−0.468684
$367$	−1378.56	−0.196076	−0.0980382	−	0.995183i	$-0.531257\pi$
$367$	−1378.56	−0.196076	−0.0980382	+	0.995183i	$0.531257\pi$
$368$	1633.00	0.231321
$369$	3146.38	0.443886
$370$	−1682.46	−0.236398
$371$	59.2211	0.00828736
$372$	−237.187	−0.0330580
$373$	11182.3	1.55227	0.776135	−	0.630567i	$-0.217178\pi$
$373$	11182.3	1.55227	0.776135	+	0.630567i	$0.217178\pi$
$374$	2874.53	0.397429
$375$	465.019	0.0640360
$376$	12123.1	1.66277
$377$	−2180.49	−0.297880
$378$	−11474.5	−1.56133
$379$	12021.9	1.62935	0.814677	−	0.579915i	$-0.196914\pi$
$379$	12021.9	1.62935	0.814677	+	0.579915i	$0.196914\pi$
$380$	−516.810	−0.0697679
$381$	−5090.26	−0.684467
$382$	9176.93	1.22914
$383$	−13274.6	−1.77102	−0.885512	−	0.464617i	$-0.846192\pi$
$383$	−13274.6	−1.77102	−0.885512	+	0.464617i	$0.846192\pi$
$384$	−6171.73	−0.820182
$385$	−1612.95	−0.213515
$386$	4010.80	0.528872
$387$	−3714.39	−0.487889
$388$	1729.19	0.226254
$389$	11016.4	1.43587	0.717934	−	0.696112i	$-0.245088\pi$
$389$	11016.4	1.43587	0.717934	+	0.696112i	$0.245088\pi$
$390$	−455.372	−0.0591248
$391$	1748.94	0.226210
$392$	6560.38	0.845279
$393$	7381.67	0.947471
$394$	7347.64	0.939515
$395$	−2759.33	−0.351486
$396$	−165.832	−0.0210439
$397$	−1893.18	−0.239335	−0.119668	−	0.992814i	$-0.538183\pi$
$397$	−1893.18	−0.239335	−0.119668	+	0.992814i	$0.538183\pi$
$398$	−3415.97	−0.430219
$399$	9844.07	1.23514
$400$	−1775.00	−0.221875
$401$	11281.8	1.40495	0.702475	−	0.711709i	$-0.252078\pi$
$401$	11281.8	1.40495	0.702475	+	0.711709i	$0.252078\pi$
$402$	−4762.52	−0.590878
$403$	−520.290	−0.0643113
$404$	855.874	0.105399
$405$	−1002.35	−0.122981
$406$	−20521.7	−2.50856
$407$	1413.36	0.172131
$408$	−5940.57	−0.720839
$409$	−14677.6	−1.77447	−0.887236	−	0.461316i	$-0.847377\pi$
$409$	−14677.6	−1.77447	−0.887236	+	0.461316i	$0.847377\pi$
$410$	3586.17	0.431972
$411$	1016.83	0.122036
$412$	−632.534	−0.0756376
$413$	−6986.15	−0.832363
$414$	−908.072	−0.107800
$415$	108.510	0.0128351
$416$	367.221	0.0432800
$417$	−2075.43	−0.243728
$418$	3907.32	0.457209
$419$	2196.50	0.256100	0.128050	−	0.991768i	$-0.459128\pi$
$419$	2196.50	0.256100	0.128050	+	0.991768i	$0.459128\pi$
$420$	−476.194	−0.0553235
$421$	−6571.28	−0.760723	−0.380362	−	0.924838i	$-0.624200\pi$
$421$	−6571.28	−0.760723	−0.380362	+	0.924838i	$0.624200\pi$
$422$	−16790.4	−1.93684
$423$	−7597.42	−0.873284
$424$	−48.5784	−0.00556409
$425$	−1901.03	−0.216973
$426$	11393.2	1.29578
$427$	−7527.87	−0.853160
$428$	1309.62	0.147904
$429$	382.536	0.0430513
$430$	−4233.58	−0.474794
$431$	−583.607	−0.0652235	−0.0326118	−	0.999468i	$-0.510382\pi$
$431$	−583.607	−0.0652235	−0.0326118	+	0.999468i	$0.510382\pi$
$432$	10607.6	1.18139
$433$	−2677.60	−0.297176	−0.148588	−	0.988899i	$-0.547473\pi$
$433$	−2677.60	−0.297176	−0.148588	+	0.988899i	$0.547473\pi$
$434$	−4896.71	−0.541589
$435$	−4970.15	−0.547817
$436$	1726.21	0.189611
$437$	2377.33	0.260236
$438$	3199.14	0.348997
$439$	−5509.18	−0.598949	−0.299475	−	0.954104i	$-0.596811\pi$
$439$	−5509.18	−0.598949	−0.299475	+	0.954104i	$0.596811\pi$
$440$	1323.08	0.143353
$441$	−4111.32	−0.443939
$442$	1861.59	0.200332
$443$	−13054.6	−1.40009	−0.700047	−	0.714097i	$-0.746838\pi$
$443$	−13054.6	−1.40009	−0.700047	+	0.714097i	$0.746838\pi$
$444$	417.268	0.0446006
$445$	−5248.14	−0.559069
$446$	−16741.8	−1.77746
$447$	−5037.68	−0.533052
$448$	−11085.1	−1.16903
$449$	−11819.1	−1.24227	−0.621135	−	0.783704i	$-0.713328\pi$
$449$	−11819.1	−1.24227	−0.621135	+	0.783704i	$0.713328\pi$
$450$	987.034	0.103398
$451$	−3012.57	−0.314537
$452$	−1492.24	−0.155285
$453$	605.091	0.0627587
$454$	1698.97	0.175631
$455$	−1044.57	−0.107627
$456$	−8074.97	−0.829266
$457$	−4144.87	−0.424264	−0.212132	−	0.977241i	$-0.568041\pi$
$457$	−4144.87	−0.424264	−0.212132	+	0.977241i	$0.568041\pi$
$458$	2081.11	0.212323
$459$	11360.8	1.15528
$460$	−115.000	−0.0116563
$461$	−6061.38	−0.612379	−0.306189	−	0.951971i	$-0.599054\pi$
$461$	−6061.38	−0.612379	−0.306189	+	0.951971i	$0.599054\pi$
$462$	3600.24	0.362551
$463$	−5729.13	−0.575066	−0.287533	−	0.957771i	$-0.592835\pi$
$463$	−5729.13	−0.575066	−0.287533	+	0.957771i	$0.592835\pi$
$464$	18971.3	1.89811
$465$	−1185.94	−0.118272
$466$	6626.05	0.658682
$467$	8958.05	0.887643	0.443821	−	0.896115i	$-0.353622\pi$
$467$	8958.05	0.887643	0.443821	+	0.896115i	$0.353622\pi$
$468$	−107.395	−0.0106076
$469$	−10924.7	−1.07559
$470$	−8659.37	−0.849845
$471$	−8797.53	−0.860655
$472$	5730.65	0.558845
$473$	3556.42	0.345718
$474$	6159.08	0.596827
$475$	−2584.05	−0.249609
$476$	1946.71	0.187452
$477$	30.4435	0.00292225
$478$	11630.4	1.11289
$479$	6356.60	0.606347	0.303174	−	0.952935i	$-0.401954\pi$
$479$	6356.60	0.606347	0.303174	+	0.952935i	$0.401954\pi$
$480$	837.034	0.0795942
$481$	915.312	0.0867664
$482$	6073.70	0.573961
$483$	2190.49	0.206358
$484$	−1172.22	−0.110088
$485$	8645.96	0.809469
$486$	−9864.31	−0.920688
$487$	13991.2	1.30185	0.650926	−	0.759141i	$-0.274381\pi$
$487$	13991.2	1.30185	0.650926	+	0.759141i	$0.274381\pi$
$488$	6175.02	0.572808
$489$	1752.79	0.162094
$490$	−4685.99	−0.432023
$491$	−5301.96	−0.487320	−0.243660	−	0.969861i	$-0.578348\pi$
$491$	−5301.96	−0.487320	−0.243660	+	0.969861i	$0.578348\pi$
$492$	−889.408	−0.0814992
$493$	20318.3	1.85617
$494$	2530.44	0.230466
$495$	−829.159	−0.0752888
$496$	4526.78	0.409795
$497$	26134.7	2.35875
$498$	−242.205	−0.0217941
$499$	−8013.45	−0.718900	−0.359450	−	0.933164i	$-0.617036\pi$
$499$	−8013.45	−0.718900	−0.359450	+	0.933164i	$0.617036\pi$
$500$	125.000	0.0111803
$501$	−13897.5	−1.23931
$502$	2200.33	0.195629
$503$	−11881.9	−1.05325	−0.526627	−	0.850097i	$-0.676544\pi$
$503$	−11881.9	−1.05325	−0.526627	+	0.850097i	$0.676544\pi$
$504$	7075.27	0.625313
$505$	4279.37	0.377088
$506$	869.453	0.0763871
$507$	−7925.44	−0.694243
$508$	−1368.29	−0.119504
$509$	−12113.8	−1.05488	−0.527441	−	0.849592i	$-0.676848\pi$
$509$	−12113.8	−1.05488	−0.527441	+	0.849592i	$0.676848\pi$
$510$	4243.27	0.368422
$511$	7338.45	0.635291
$512$	8733.00	0.753804
$513$	15442.6	1.32906
$514$	−7103.49	−0.609575
$515$	−3162.67	−0.270609
$516$	1049.97	0.0895783
$517$	7274.31	0.618808
$518$	8614.48	0.730692
$519$	−10079.1	−0.852454
$520$	856.848	0.0722602
$521$	−15327.6	−1.28890	−0.644449	−	0.764647i	$-0.722913\pi$
$521$	−15327.6	−1.28890	−0.644449	+	0.764647i	$0.722913\pi$
$522$	−10549.5	−0.884556
$523$	14545.8	1.21614	0.608070	−	0.793883i	$-0.291944\pi$
$523$	14545.8	1.21614	0.608070	+	0.793883i	$0.291944\pi$
$524$	1984.24	0.165423
$525$	−2380.97	−0.197931
$526$	−19898.3	−1.64944
$527$	4848.18	0.400740
$528$	−3328.25	−0.274325
$529$	529.000	0.0434783
$530$	34.6989	0.00284381
$531$	−3591.34	−0.293504
$532$	2646.15	0.215648
$533$	−1950.99	−0.158549
$534$	11714.3	0.949304
$535$	6548.10	0.529157
$536$	8961.36	0.722149
$537$	12278.1	0.986665
$538$	18278.9	1.46479
$539$	3936.47	0.314575
$540$	−747.015	−0.0595304
$541$	−5468.93	−0.434617	−0.217308	−	0.976103i	$-0.569728\pi$
$541$	−5468.93	−0.434617	−0.217308	+	0.976103i	$0.569728\pi$
$542$	5943.89	0.471055
$543$	−2172.08	−0.171663
$544$	−3421.85	−0.269688
$545$	8631.03	0.678372
$546$	2331.58	0.182751
$547$	−19566.3	−1.52943	−0.764713	−	0.644371i	$-0.777119\pi$
$547$	−19566.3	−1.52943	−0.764713	+	0.644371i	$0.777119\pi$
$548$	273.331	0.0213068
$549$	−3869.82	−0.300837
$550$	−945.057	−0.0732680
$551$	27618.5	2.13537
$552$	−1796.83	−0.138548
$553$	14128.2	1.08642
$554$	12292.3	0.942686
$555$	2086.34	0.159568
$556$	−557.889	−0.0425536
$557$	−9803.11	−0.745729	−0.372865	−	0.927886i	$-0.621624\pi$
$557$	−9803.11	−0.745729	−0.372865	+	0.927886i	$0.621624\pi$
$558$	−2517.23	−0.190973
$559$	2303.20	0.174266
$560$	9088.27	0.685803
$561$	−3564.56	−0.268264
$562$	11599.2	0.870610
$563$	−4909.73	−0.367532	−0.183766	−	0.982970i	$-0.558829\pi$
$563$	−4909.73	−0.367532	−0.183766	+	0.982970i	$0.558829\pi$
$564$	2147.61	0.160338
$565$	−7461.19	−0.555566
$566$	−11946.0	−0.887153
$567$	5132.18	0.380126
$568$	−21438.0	−1.58366
$569$	11498.2	0.847155	0.423578	−	0.905860i	$-0.360774\pi$
$569$	11498.2	0.847155	0.423578	+	0.905860i	$0.360774\pi$
$570$	5767.84	0.423839
$571$	14766.6	1.08225	0.541124	−	0.840943i	$-0.317999\pi$
$571$	14766.6	1.08225	0.541124	+	0.840943i	$0.317999\pi$
$572$	102.828	0.00751653
$573$	−11379.9	−0.829670
$574$	−18361.8	−1.33520
$575$	−575.000	−0.0417029
$576$	−5698.48	−0.412216
$577$	13364.0	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$577$	13364.0	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$578$	−2607.73	−0.187660
$579$	−4973.60	−0.356988
$580$	−1336.01	−0.0956460
$581$	−555.590	−0.0396725
$582$	−19298.6	−1.37449
$583$	−29.1488	−0.00207070
$584$	−6019.64	−0.426532
$585$	−536.977	−0.0379509
$586$	22475.1	1.58437
$587$	12524.7	0.880664	0.440332	−	0.897835i	$-0.354861\pi$
$587$	12524.7	0.880664	0.440332	+	0.897835i	$0.354861\pi$
$588$	1162.17	0.0815089
$589$	6590.09	0.461019
$590$	−4093.32	−0.285626
$591$	−9111.45	−0.634171
$592$	−7963.66	−0.552879
$593$	−17938.6	−1.24224	−0.621121	−	0.783714i	$-0.713323\pi$
$593$	−17938.6	−1.24224	−0.621121	+	0.783714i	$0.713323\pi$
$594$	5647.78	0.390120
$595$	9733.55	0.670650
$596$	−1354.16	−0.0930681
$597$	4235.98	0.290397
$598$	563.072	0.0385045
$599$	−26735.8	−1.82370	−0.911848	−	0.410528i	$-0.865344\pi$
$599$	−26735.8	−1.82370	−0.911848	+	0.410528i	$0.865344\pi$
$600$	1953.08	0.132890
$601$	−21043.6	−1.42826	−0.714131	−	0.700012i	$-0.753178\pi$
$601$	−21043.6	−1.42826	−0.714131	+	0.700012i	$0.753178\pi$
$602$	21676.6	1.46756
$603$	−5615.98	−0.379271
$604$	162.652	0.0109573
$605$	−5861.10	−0.393864
$606$	−9551.95	−0.640299
$607$	5098.83	0.340947	0.170474	−	0.985362i	$-0.445470\pi$
$607$	5098.83	0.340947	0.170474	+	0.985362i	$0.445470\pi$
$608$	−4651.29	−0.310254
$609$	25447.9	1.69327
$610$	−4410.73	−0.292763
$611$	4710.96	0.311923
$612$	1000.74	0.0660986
$613$	−12784.8	−0.842369	−0.421185	−	0.906975i	$-0.638386\pi$
$613$	−12784.8	−0.842369	−0.421185	+	0.906975i	$0.638386\pi$
$614$	17439.0	1.14622
$615$	−4447.04	−0.291580
$616$	−6774.37	−0.443096
$617$	−8940.73	−0.583372	−0.291686	−	0.956514i	$-0.594216\pi$
$617$	−8940.73	−0.583372	−0.291686	+	0.956514i	$0.594216\pi$
$618$	7059.37	0.459498
$619$	1333.26	0.0865725	0.0432863	−	0.999063i	$-0.486217\pi$
$619$	1333.26	0.0865725	0.0432863	+	0.999063i	$0.486217\pi$
$620$	−318.787	−0.0206497
$621$	3436.27	0.222050
$622$	−14308.3	−0.922364
$623$	26871.3	1.72805
$624$	−2155.43	−0.138279
$625$	625.000	0.0400000
$626$	−21160.3	−1.35102
$627$	−4845.28	−0.308615
$628$	−2364.83	−0.150266
$629$	−8529.09	−0.540663
$630$	−5053.77	−0.319598
$631$	−16667.9	−1.05157	−0.525784	−	0.850618i	$-0.676228\pi$
$631$	−16667.9	−1.05157	−0.525784	+	0.850618i	$0.676228\pi$
$632$	−11589.2	−0.729420
$633$	20821.0	1.30736
$634$	−26536.1	−1.66228
$635$	−6841.46	−0.427551
$636$	−8.60567	−0.000536536 0
$637$	2549.32	0.158568
$638$	10100.8	0.626796
$639$	13434.9	0.831733
$640$	−8295.00	−0.512326
$641$	27431.9	1.69032	0.845158	−	0.534516i	$-0.179506\pi$
$641$	27431.9	1.69032	0.845158	+	0.534516i	$0.179506\pi$
$642$	−14616.0	−0.898514
$643$	9618.00	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$643$	9618.00	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$644$	588.818	0.0360290
$645$	5249.86	0.320485
$646$	−23579.3	−1.43609
$647$	−12548.6	−0.762496	−0.381248	−	0.924473i	$-0.624506\pi$
$647$	−12548.6	−0.762496	−0.381248	+	0.924473i	$0.624506\pi$
$648$	−4209.87	−0.255215
$649$	3438.60	0.207977
$650$	−612.034	−0.0369323
$651$	6072.18	0.365572
$652$	471.162	0.0283008
$653$	−8829.79	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$653$	−8829.79	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$654$	−19265.3	−1.15188
$655$	9921.19	0.591837
$656$	16974.6	1.01028
$657$	3772.44	0.224014
$658$	44337.3	2.62682
$659$	5935.56	0.350860	0.175430	−	0.984492i	$-0.443868\pi$
$659$	5935.56	0.350860	0.175430	+	0.984492i	$0.443868\pi$
$660$	234.384	0.0138233
$661$	−1182.03	−0.0695549	−0.0347775	−	0.999395i	$-0.511072\pi$
$661$	−1182.03	−0.0695549	−0.0347775	+	0.999395i	$0.511072\pi$
$662$	−12792.2	−0.751032
$663$	−2308.47	−0.135224
$664$	455.743	0.0266360
$665$	13230.7	0.771527
$666$	4428.40	0.257653
$667$	6145.64	0.356762
$668$	−3735.73	−0.216377
$669$	20760.7	1.19978
$670$	−6400.97	−0.369091
$671$	3705.24	0.213173
$672$	−4285.74	−0.246021
$673$	−6053.25	−0.346710	−0.173355	−	0.984859i	$-0.555461\pi$
$673$	−6053.25	−0.346710	−0.173355	+	0.984859i	$0.555461\pi$
$674$	3871.74	0.221267
$675$	−3735.08	−0.212982
$676$	−2130.41	−0.121211
$677$	16694.2	0.947728	0.473864	−	0.880598i	$-0.342859\pi$
$677$	16694.2	0.947728	0.473864	+	0.880598i	$0.342859\pi$
$678$	16654.1	0.943357
$679$	−44268.6	−2.50202
$680$	−7984.31	−0.450271
$681$	−2106.81	−0.118551
$682$	2410.18	0.135323
$683$	−4150.03	−0.232499	−0.116249	−	0.993220i	$-0.537087\pi$
$683$	−4150.03	−0.232499	−0.116249	+	0.993220i	$0.537087\pi$
$684$	1360.29	0.0760410
$685$	1366.66	0.0762296
$686$	−2350.21	−0.130804
$687$	−2580.68	−0.143317
$688$	−20039.0	−1.11043
$689$	−18.8772	−0.00104378
$690$	1283.45	0.0708119
$691$	33382.4	1.83781	0.918904	−	0.394481i	$-0.129076\pi$
$691$	33382.4	1.83781	0.918904	+	0.394481i	$0.129076\pi$
$692$	−2709.33	−0.148834
$693$	4245.42	0.232713
$694$	5415.27	0.296197
$695$	−2789.45	−0.152244
$696$	−20874.6	−1.13686
$697$	18179.8	0.987960
$698$	10760.4	0.583508
$699$	−8216.64	−0.444609
$700$	−640.019	−0.0345578
$701$	−29528.7	−1.59099	−0.795494	−	0.605961i	$-0.792789\pi$
$701$	−29528.7	−1.59099	−0.795494	+	0.605961i	$0.792789\pi$
$702$	3657.59	0.196648
$703$	−11593.5	−0.621989
$704$	5456.13	0.292096
$705$	10738.1	0.573644
$706$	6878.69	0.366690
$707$	−21911.0	−1.16556
$708$	1015.19	0.0538885
$709$	27016.3	1.43106	0.715529	−	0.698583i	$-0.246186\pi$
$709$	27016.3	1.43106	0.715529	+	0.698583i	$0.246186\pi$
$710$	15312.8	0.809408
$711$	7262.82	0.383090
$712$	−22042.2	−1.16020
$713$	1466.42	0.0770237
$714$	−21726.2	−1.13877
$715$	514.140	0.0268920
$716$	3300.43	0.172266
$717$	−14422.3	−0.751198
$718$	9231.54	0.479830
$719$	24752.9	1.28390	0.641952	−	0.766745i	$-0.278125\pi$
$719$	24752.9	1.28390	0.641952	+	0.766745i	$0.278125\pi$
$720$	4671.96	0.241825
$721$	16193.4	0.836438
$722$	−11474.1	−0.591443
$723$	−7531.69	−0.387423
$724$	−583.868	−0.0299714
$725$	−6680.04	−0.342194
$726$	13082.5	0.668785
$727$	−17574.5	−0.896565	−0.448283	−	0.893892i	$-0.647964\pi$
$727$	−17574.5	−0.896565	−0.448283	+	0.893892i	$0.647964\pi$
$728$	−4387.19	−0.223352
$729$	17644.9	0.896456
$730$	4299.74	0.218001
$731$	−21461.7	−1.08590
$732$	1093.91	0.0552349
$733$	27739.6	1.39780	0.698898	−	0.715221i	$-0.253674\pi$
$733$	27739.6	1.39780	0.698898	+	0.715221i	$0.253674\pi$
$734$	4135.67	0.207970
$735$	5810.87	0.291615
$736$	−1035.00	−0.0518351
$737$	5377.15	0.268751
$738$	−9439.14	−0.470812
$739$	−3140.78	−0.156340	−0.0781701	−	0.996940i	$-0.524908\pi$
$739$	−3140.78	−0.156340	−0.0781701	+	0.996940i	$0.524908\pi$
$740$	560.821	0.0278597
$741$	−3137.88	−0.155564
$742$	−177.663	−0.00879007
$743$	−35881.5	−1.77169	−0.885844	−	0.463984i	$-0.846420\pi$
$743$	−35881.5	−1.77169	−0.885844	+	0.463984i	$0.846420\pi$
$744$	−4980.93	−0.245443
$745$	−6770.80	−0.332970
$746$	−33546.9	−1.64643
$747$	−285.609	−0.0139891
$748$	−958.176	−0.0468374
$749$	−33527.3	−1.63559
$750$	−1395.06	−0.0679204
$751$	−7009.03	−0.340563	−0.170282	−	0.985395i	$-0.554468\pi$
$751$	−7009.03	−0.340563	−0.170282	+	0.985395i	$0.554468\pi$
$752$	−40987.7	−1.98759
$753$	−2728.52	−0.132049
$754$	6541.46	0.315950
$755$	813.261	0.0392021
$756$	3824.83	0.184005
$757$	29998.3	1.44030	0.720150	−	0.693818i	$-0.244073\pi$
$757$	29998.3	1.44030	0.720150	+	0.693818i	$0.244073\pi$
$758$	−36065.8	−1.72819
$759$	−1078.17	−0.0515612
$760$	−10853.0	−0.518000
$761$	5283.49	0.251677	0.125839	−	0.992051i	$-0.459838\pi$
$761$	5283.49	0.251677	0.125839	+	0.992051i	$0.459838\pi$
$762$	15270.8	0.725987
$763$	−44192.2	−2.09681
$764$	−3058.98	−0.144856
$765$	5003.68	0.236481
$766$	39823.9	1.87845
$767$	2226.89	0.104835
$768$	5628.59	0.264459
$769$	−15216.2	−0.713538	−0.356769	−	0.934193i	$-0.616122\pi$
$769$	−15216.2	−0.713538	−0.356769	+	0.934193i	$0.616122\pi$
$770$	4838.84	0.226467
$771$	8808.69	0.411462
$772$	−1336.93	−0.0623281
$773$	15651.3	0.728250	0.364125	−	0.931350i	$-0.381368\pi$
$773$	15651.3	0.728250	0.364125	+	0.931350i	$0.381368\pi$
$774$	11143.2	0.517485
$775$	−1593.93	−0.0738785
$776$	36313.0	1.67985
$777$	−10682.4	−0.493216
$778$	−33049.1	−1.52297
$779$	24711.6	1.13657
$780$	151.791	0.00696793
$781$	−12863.6	−0.589365
$782$	−5246.83	−0.239931
$783$	39920.7	1.82203
$784$	−22180.3	−1.01040
$785$	−11824.1	−0.537607
$786$	−22145.0	−1.00494
$787$	9401.09	0.425810	0.212905	−	0.977073i	$-0.431707\pi$
$787$	9401.09	0.425810	0.212905	+	0.977073i	$0.431707\pi$
$788$	−2449.21	−0.110723
$789$	24674.9	1.11337
$790$	8278.00	0.372807
$791$	38202.5	1.71722
$792$	−3482.47	−0.156243
$793$	2399.57	0.107454
$794$	5679.55	0.253854
$795$	−43.0283	−0.00191957
$796$	1138.66	0.0507018
$797$	37388.5	1.66169	0.830845	−	0.556504i	$-0.187857\pi$
$797$	37388.5	1.66169	0.830845	+	0.556504i	$0.187857\pi$
$798$	−29532.2	−1.31006
$799$	−43897.9	−1.94367
$800$	1125.00	0.0497184
$801$	13813.6	0.609337
$802$	−33845.3	−1.49017
$803$	−3612.00	−0.158736
$804$	1587.51	0.0696356
$805$	2944.09	0.128901
$806$	1560.87	0.0682125
$807$	−22666.8	−0.988734
$808$	17973.4	0.782550
$809$	24390.6	1.05999	0.529993	−	0.848002i	$-0.322195\pi$
$809$	24390.6	1.05999	0.529993	+	0.848002i	$0.322195\pi$
$810$	3007.05	0.130441
$811$	23750.8	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$811$	23750.8	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$812$	6840.56	0.295636
$813$	−7370.73	−0.317962
$814$	−4240.07	−0.182573
$815$	2355.81	0.101252
$816$	20084.8	0.861653
$817$	−29172.8	−1.24924
$818$	44032.7	1.88211
$819$	2749.40	0.117304
$820$	−1195.39	−0.0509084
$821$	3041.59	0.129296	0.0646481	−	0.997908i	$-0.479408\pi$
$821$	3041.59	0.129296	0.0646481	+	0.997908i	$0.479408\pi$
$822$	−3050.50	−0.129439
$823$	28411.6	1.20336	0.601681	−	0.798737i	$-0.294498\pi$
$823$	28411.6	1.20336	0.601681	+	0.798737i	$0.294498\pi$
$824$	−13283.2	−0.561581
$825$	1171.92	0.0494558
$826$	20958.4	0.882854
$827$	−24351.5	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$827$	−24351.5	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$828$	302.691	0.0127044
$829$	−47240.8	−1.97918	−0.989590	−	0.143914i	$-0.954031\pi$
$829$	−47240.8	−1.97918	−0.989590	+	0.143914i	$0.954031\pi$
$830$	−325.531	−0.0136137
$831$	−15243.0	−0.636311
$832$	3533.48	0.147237
$833$	−23755.2	−0.988077
$834$	6226.30	0.258512
$835$	−18678.7	−0.774134
$836$	−1302.44	−0.0538826
$837$	9525.55	0.393371
$838$	−6589.49	−0.271635
$839$	2646.33	0.108893	0.0544466	−	0.998517i	$-0.482661\pi$
$839$	2646.33	0.108893	0.0544466	+	0.998517i	$0.482661\pi$
$840$	−10000.1	−0.410756
$841$	47007.7	1.92741
$842$	19713.8	0.806869
$843$	−14383.6	−0.587660
$844$	5596.81	0.228258
$845$	−10652.0	−0.433658
$846$	22792.3	0.926258
$847$	30009.7	1.21741
$848$	164.241	0.00665102
$849$	14813.7	0.598827
$850$	5703.08	0.230134
$851$	−2579.78	−0.103917
$852$	−3797.74	−0.152709
$853$	−40826.4	−1.63877	−0.819385	−	0.573244i	$-0.805685\pi$
$853$	−40826.4	−1.63877	−0.819385	+	0.573244i	$0.805685\pi$
$854$	22583.6	0.904913
$855$	6801.46	0.272053
$856$	27502.0	1.09813
$857$	1516.61	0.0604510	0.0302255	−	0.999543i	$-0.490377\pi$
$857$	1516.61	0.0604510	0.0302255	+	0.999543i	$0.490377\pi$
$858$	−1147.61	−0.0456628
$859$	6514.74	0.258766	0.129383	−	0.991595i	$-0.458700\pi$
$859$	6514.74	0.258766	0.129383	+	0.991595i	$0.458700\pi$
$860$	1411.19	0.0559550
$861$	22769.5	0.901258
$862$	1750.82	0.0691800
$863$	−7120.18	−0.280850	−0.140425	−	0.990091i	$-0.544847\pi$
$863$	−7120.18	−0.280850	−0.140425	+	0.990091i	$0.544847\pi$
$864$	−6723.14	−0.264729
$865$	−13546.6	−0.532485
$866$	8032.79	0.315202
$867$	3233.72	0.126670
$868$	1632.24	0.0638269
$869$	−6953.94	−0.271457
$870$	14910.5	0.581048
$871$	3482.33	0.135470
$872$	36250.3	1.40779
$873$	−22757.0	−0.882252
$874$	−7131.98	−0.276021
$875$	−3200.10	−0.123638
$876$	−1066.38	−0.0411297
$877$	−34013.4	−1.30964	−0.654818	−	0.755787i	$-0.727255\pi$
$877$	−34013.4	−1.30964	−0.654818	+	0.755787i	$0.727255\pi$
$878$	16527.5	0.635282
$879$	−27870.3	−1.06944
$880$	−4473.27	−0.171357
$881$	6704.11	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$881$	6704.11	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$882$	12334.0	0.470868
$883$	47808.5	1.82207	0.911033	−	0.412333i	$-0.135286\pi$
$883$	47808.5	1.82207	0.911033	+	0.412333i	$0.135286\pi$
$884$	−620.530	−0.0236094
$885$	5075.93	0.192797
$886$	39163.7	1.48502
$887$	−38955.7	−1.47464	−0.737320	−	0.675544i	$-0.763909\pi$
$887$	−38955.7	−1.47464	−0.737320	+	0.675544i	$0.763909\pi$
$888$	8762.64	0.331143
$889$	35029.3	1.32154
$890$	15744.4	0.592982
$891$	−2526.07	−0.0949794
$892$	5580.60	0.209476
$893$	−59670.0	−2.23604
$894$	15113.1	0.565387
$895$	16502.1	0.616319
$896$	42471.7	1.58357
$897$	−698.238	−0.0259905
$898$	35457.4	1.31763
$899$	17036.1	0.632019
$900$	−329.011	−0.0121856
$901$	175.903	0.00650407
$902$	9037.71	0.333617
$903$	−26880.1	−0.990601
$904$	−31337.0	−1.15294
$905$	−2919.34	−0.107229
$906$	−1815.27	−0.0665656
$907$	7725.66	0.282830	0.141415	−	0.989950i	$-0.454835\pi$
$907$	7725.66	0.282830	0.141415	+	0.989950i	$0.454835\pi$
$908$	−566.323	−0.0206983
$909$	−11263.7	−0.410994
$910$	3133.71	0.114155
$911$	−13150.7	−0.478269	−0.239135	−	0.970986i	$-0.576864\pi$
$911$	−13150.7	−0.478269	−0.239135	+	0.970986i	$0.576864\pi$
$912$	27301.1	0.991260
$913$	273.463	0.00991270
$914$	12434.6	0.450000
$915$	5469.53	0.197614
$916$	−693.702	−0.0250224
$917$	−50798.0	−1.82933
$918$	−34082.3	−1.22536
$919$	33083.7	1.18752	0.593759	−	0.804643i	$-0.297643\pi$
$919$	33083.7	1.18752	0.593759	+	0.804643i	$0.297643\pi$
$920$	−2415.00	−0.0865437
$921$	−21625.3	−0.773699
$922$	18184.1	0.649526
$923$	−8330.65	−0.297082
$924$	−1200.08	−0.0427270
$925$	2804.11	0.0996740
$926$	17187.4	0.609949
$927$	8324.44	0.294941
$928$	−12024.1	−0.425333
$929$	−1310.12	−0.0462686	−0.0231343	−	0.999732i	$-0.507365\pi$
$929$	−1310.12	−0.0462686	−0.0231343	+	0.999732i	$0.507365\pi$
$930$	3557.81	0.125446
$931$	−32290.2	−1.13670
$932$	−2208.68	−0.0776264
$933$	17743.0	0.622594
$934$	−26874.1	−0.941487
$935$	−4790.88	−0.167571
$936$	−2255.30	−0.0787574
$937$	40623.1	1.41633	0.708163	−	0.706049i	$-0.249524\pi$
$937$	40623.1	1.41633	0.708163	+	0.706049i	$0.249524\pi$
$938$	32774.0	1.14084
$939$	26239.8	0.911933
$940$	2886.46	0.100155
$941$	−10434.0	−0.361466	−0.180733	−	0.983532i	$-0.557847\pi$
$941$	−10434.0	−0.361466	−0.180733	+	0.983532i	$0.557847\pi$
$942$	26392.6	0.912863
$943$	5498.80	0.189889
$944$	−19375.1	−0.668013
$945$	19124.2	0.658317
$946$	−10669.3	−0.366689
$947$	−51333.4	−1.76147	−0.880734	−	0.473612i	$-0.842950\pi$
$947$	−51333.4	−1.76147	−0.880734	+	0.473612i	$0.842950\pi$
$948$	−2053.03	−0.0703367
$949$	−2339.19	−0.0800141
$950$	7752.15	0.264750
$951$	32906.1	1.12203
$952$	40880.9	1.39176
$953$	40263.6	1.36859	0.684294	−	0.729206i	$-0.260110\pi$
$953$	40263.6	1.36859	0.684294	+	0.729206i	$0.260110\pi$
$954$	−91.3306	−0.00309951
$955$	−15294.9	−0.518252
$956$	−3876.79	−0.131155
$957$	−12525.5	−0.423086
$958$	−19069.8	−0.643128
$959$	−6997.49	−0.235621
$960$	8054.13	0.270777
$961$	−25726.0	−0.863549
$962$	−2745.94	−0.0920297
$963$	−17235.2	−0.576736
$964$	−2024.57	−0.0676420
$965$	−6684.67	−0.222992
$966$	−6571.48	−0.218876
$967$	−8353.12	−0.277785	−0.138893	−	0.990307i	$-0.544354\pi$
$967$	−8353.12	−0.277785	−0.138893	+	0.990307i	$0.544354\pi$
$968$	−24616.6	−0.817364
$969$	29239.5	0.969358
$970$	−25937.9	−0.858572
$971$	−28020.1	−0.926063	−0.463032	−	0.886342i	$-0.653238\pi$
$971$	−28020.1	−0.926063	−0.463032	+	0.886342i	$0.653238\pi$
$972$	3288.10	0.108504
$973$	14282.4	0.470578
$974$	−41973.6	−1.38082
$975$	758.954	0.0249292
$976$	−20877.5	−0.684704
$977$	−50115.3	−1.64107	−0.820537	−	0.571593i	$-0.806326\pi$
$977$	−50115.3	−1.64107	−0.820537	+	0.571593i	$0.806326\pi$
$978$	−5258.38	−0.171927
$979$	−13226.1	−0.431776
$980$	1562.00	0.0509144
$981$	−22717.7	−0.739367
$982$	15905.9	0.516881
$983$	−38113.0	−1.23664	−0.618320	−	0.785926i	$-0.712186\pi$
$983$	−38113.0	−1.23664	−0.618320	+	0.785926i	$0.712186\pi$
$984$	−18677.6	−0.605101
$985$	−12246.1	−0.396134
$986$	−60954.9	−1.96876
$987$	−54980.5	−1.77310
$988$	−843.481	−0.0271606
$989$	−6491.49	−0.208713
$990$	2487.48	0.0798558
$991$	−5245.02	−0.168127	−0.0840634	−	0.996460i	$-0.526790\pi$
$991$	−5245.02	−0.168127	−0.0840634	+	0.996460i	$0.526790\pi$
$992$	−2869.08	−0.0918281
$993$	15863.0	0.506945
$994$	−78404.0	−2.50183
$995$	5693.29	0.181396
$996$	80.7350	0.00256846
$997$	21896.1	0.695544	0.347772	−	0.937579i	$-0.386938\pi$
$997$	21896.1	0.695544	0.347772	+	0.937579i	$0.386938\pi$
$998$	24040.3	0.762509
$999$	−16757.7	−0.530721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.4.a.c.1.2	✓	2
3.2	odd	2		1035.4.a.g.1.1		2
4.3	odd	2		1840.4.a.h.1.1		2
5.2	odd	4		575.4.b.f.24.1		4
5.3	odd	4		575.4.b.f.24.4		4
5.4	even	2		575.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.c.1.2	✓	2	1.1	even	1	trivial
575.4.a.h.1.1		2	5.4	even	2
575.4.b.f.24.1		4	5.2	odd	4
575.4.b.f.24.4		4	5.3	odd	4
1035.4.a.g.1.1		2	3.2	odd	2
1840.4.a.h.1.1		2	4.3	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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} +O(q^{10})\)	\(q-3.00000 q^{2} +3.72015 q^{3} +1.00000 q^{4} +5.00000 q^{5} -11.1605 q^{6} -25.6008 q^{7} +21.0000 q^{8} -13.1605 q^{9} -15.0000 q^{10} +12.6008 q^{11} +3.72015 q^{12} +8.16046 q^{13} +76.8023 q^{14} +18.6008 q^{15} -71.0000 q^{16} -76.0411 q^{17} +39.4814 q^{18} -103.362 q^{19} +5.00000 q^{20} -95.2388 q^{21} -37.8023 q^{22} -23.0000 q^{23} +78.1232 q^{24} +25.0000 q^{25} -24.4814 q^{26} -149.403 q^{27} -25.6008 q^{28} -267.202 q^{29} -55.8023 q^{30} -63.7574 q^{31} +45.0000 q^{32} +46.8768 q^{33} +228.123 q^{34} -128.004 q^{35} -13.1605 q^{36} +112.164 q^{37} +310.086 q^{38} +30.3582 q^{39} +105.000 q^{40} -239.078 q^{41} +285.716 q^{42} +282.239 q^{43} +12.6008 q^{44} -65.8023 q^{45} +69.0000 q^{46} +577.291 q^{47} -264.131 q^{48} +312.399 q^{49} -75.0000 q^{50} -282.884 q^{51} +8.16046 q^{52} -2.31326 q^{53} +448.209 q^{54} +63.0038 q^{55} -537.616 q^{56} -384.522 q^{57} +801.605 q^{58} +272.888 q^{59} +18.6008 q^{60} +294.049 q^{61} +191.272 q^{62} +336.918 q^{63} +433.000 q^{64} +40.8023 q^{65} -140.630 q^{66} +426.732 q^{67} -76.0411 q^{68} -85.5635 q^{69} +384.011 q^{70} -1020.85 q^{71} -276.370 q^{72} -286.650 q^{73} -336.493 q^{74} +93.0038 q^{75} -103.362 q^{76} -322.589 q^{77} -91.0745 q^{78} -551.866 q^{79} -355.000 q^{80} -200.470 q^{81} +717.235 q^{82} +21.7021 q^{83} -95.2388 q^{84} -380.205 q^{85} -846.716 q^{86} -994.031 q^{87} +264.616 q^{88} -1049.63 q^{89} +197.407 q^{90} -208.914 q^{91} -23.0000 q^{92} -237.187 q^{93} -1731.87 q^{94} -516.810 q^{95} +167.407 q^{96} +1729.19 q^{97} -937.198 q^{98} -165.832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9}+O(q^{10}) \) 2 * q - 6 * q^2 - 3 * q^3 + 2 * q^4 + 10 * q^5 + 9 * q^6 + q^7 + 42 * q^8 + 5 * q^9	\( 2 q - 6 q^{2} - 3 q^{3} + 2 q^{4} + 10 q^{5} + 9 q^{6} + q^{7} + 42 q^{8} + 5 q^{9} - 30 q^{10} - 27 q^{11} - 3 q^{12} - 15 q^{13} - 3 q^{14} - 15 q^{15} - 142 q^{16} - 79 q^{17} - 15 q^{18} - 71 q^{19} + 10 q^{20} - 274 q^{21} + 81 q^{22} - 46 q^{23} - 63 q^{24} + 50 q^{25} + 45 q^{26} - 90 q^{27} + q^{28} - 430 q^{29} + 45 q^{30} - 305 q^{31} + 90 q^{32} + 313 q^{33} + 237 q^{34} + 5 q^{35} + 5 q^{36} - 68 q^{37} + 213 q^{38} + 186 q^{39} + 210 q^{40} - 593 q^{41} + 822 q^{42} + 648 q^{43} - 27 q^{44} + 25 q^{45} + 138 q^{46} + 382 q^{47} + 213 q^{48} + 677 q^{49} - 150 q^{50} - 263 q^{51} - 15 q^{52} - 464 q^{53} + 270 q^{54} - 135 q^{55} + 21 q^{56} - 602 q^{57} + 1290 q^{58} - 18 q^{59} - 15 q^{60} - 7 q^{61} + 915 q^{62} + 820 q^{63} + 866 q^{64} - 75 q^{65} - 939 q^{66} + 60 q^{67} - 79 q^{68} + 69 q^{69} - 15 q^{70} - 1029 q^{71} + 105 q^{72} + 74 q^{73} + 204 q^{74} - 75 q^{75} - 71 q^{76} - 1376 q^{77} - 558 q^{78} + 692 q^{79} - 710 q^{80} - 1090 q^{81} + 1779 q^{82} - 1460 q^{83} - 274 q^{84} - 395 q^{85} - 1944 q^{86} + 100 q^{87} - 567 q^{88} - 220 q^{89} - 75 q^{90} - 825 q^{91} - 46 q^{92} + 1384 q^{93} - 1146 q^{94} - 355 q^{95} - 135 q^{96} + 1339 q^{97} - 2031 q^{98} - 885 q^{99}+O(q^{100}) \) 2 * q - 6 * q^2 - 3 * q^3 + 2 * q^4 + 10 * q^5 + 9 * q^6 + q^7 + 42 * q^8 + 5 * q^9 - 30 * q^10 - 27 * q^11 - 3 * q^12 - 15 * q^13 - 3 * q^14 - 15 * q^15 - 142 * q^16 - 79 * q^17 - 15 * q^18 - 71 * q^19 + 10 * q^20 - 274 * q^21 + 81 * q^22 - 46 * q^23 - 63 * q^24 + 50 * q^25 + 45 * q^26 - 90 * q^27 + q^28 - 430 * q^29 + 45 * q^30 - 305 * q^31 + 90 * q^32 + 313 * q^33 + 237 * q^34 + 5 * q^35 + 5 * q^36 - 68 * q^37 + 213 * q^38 + 186 * q^39 + 210 * q^40 - 593 * q^41 + 822 * q^42 + 648 * q^43 - 27 * q^44 + 25 * q^45 + 138 * q^46 + 382 * q^47 + 213 * q^48 + 677 * q^49 - 150 * q^50 - 263 * q^51 - 15 * q^52 - 464 * q^53 + 270 * q^54 - 135 * q^55 + 21 * q^56 - 602 * q^57 + 1290 * q^58 - 18 * q^59 - 15 * q^60 - 7 * q^61 + 915 * q^62 + 820 * q^63 + 866 * q^64 - 75 * q^65 - 939 * q^66 + 60 * q^67 - 79 * q^68 + 69 * q^69 - 15 * q^70 - 1029 * q^71 + 105 * q^72 + 74 * q^73 + 204 * q^74 - 75 * q^75 - 71 * q^76 - 1376 * q^77 - 558 * q^78 + 692 * q^79 - 710 * q^80 - 1090 * q^81 + 1779 * q^82 - 1460 * q^83 - 274 * q^84 - 395 * q^85 - 1944 * q^86 + 100 * q^87 - 567 * q^88 - 220 * q^89 - 75 * q^90 - 825 * q^91 - 46 * q^92 + 1384 * q^93 - 1146 * q^94 - 355 * q^95 - 135 * q^96 + 1339 * q^97 - 2031 * q^98 - 885 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more