LMFDB - Embedded newform 1875.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.35083$ of defining polynomial
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35083 q^{2} +1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} -3.48189 q^{7} -3.58831 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.35083 q^{2} +1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} -3.48189 q^{7} -3.58831 q^{8} +1.00000 q^{9} +2.93111 q^{11} +3.52640 q^{12} +1.87575 q^{13} +8.18532 q^{14} +1.38270 q^{16} -6.78566 q^{17} -2.35083 q^{18} -2.94950 q^{19} -3.48189 q^{21} -6.89053 q^{22} +5.49019 q^{23} -3.58831 q^{24} -4.40956 q^{26} +1.00000 q^{27} -12.2785 q^{28} +2.55593 q^{29} +0.418084 q^{31} +3.92613 q^{32} +2.93111 q^{33} +15.9519 q^{34} +3.52640 q^{36} -5.23959 q^{37} +6.93377 q^{38} +1.87575 q^{39} +1.67869 q^{41} +8.18532 q^{42} -10.9233 q^{43} +10.3363 q^{44} -12.9065 q^{46} +7.49178 q^{47} +1.38270 q^{48} +5.12353 q^{49} -6.78566 q^{51} +6.61463 q^{52} -3.70953 q^{53} -2.35083 q^{54} +12.4941 q^{56} -2.94950 q^{57} -6.00857 q^{58} -7.10854 q^{59} -6.43710 q^{61} -0.982844 q^{62} -3.48189 q^{63} -11.9951 q^{64} -6.89053 q^{66} -10.0415 q^{67} -23.9290 q^{68} +5.49019 q^{69} +0.728602 q^{71} -3.58831 q^{72} +3.59269 q^{73} +12.3174 q^{74} -10.4011 q^{76} -10.2058 q^{77} -4.40956 q^{78} +3.07265 q^{79} +1.00000 q^{81} -3.94632 q^{82} -10.1152 q^{83} -12.2785 q^{84} +25.6788 q^{86} +2.55593 q^{87} -10.5177 q^{88} +0.287512 q^{89} -6.53114 q^{91} +19.3606 q^{92} +0.418084 q^{93} -17.6119 q^{94} +3.92613 q^{96} +10.4090 q^{97} -12.0446 q^{98} +2.93111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	$ 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24} + 6 q^{26} + 8 q^{27} - 16 q^{28} + 2 q^{29} - 22 q^{31} + 2 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{36} - 28 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} + 6 q^{42} - 20 q^{43} + 22 q^{44} - 2 q^{46} - 10 q^{47} - 16 q^{51} - 16 q^{52} - 44 q^{53} - 4 q^{54} + 30 q^{56} - 14 q^{57} - 8 q^{58} + 14 q^{59} - 20 q^{61} - 16 q^{62} - 8 q^{63} + 6 q^{64} - 12 q^{66} - 16 q^{67} + 2 q^{68} - 14 q^{69} + 16 q^{71} - 12 q^{72} - 24 q^{73} + 26 q^{74} - 16 q^{76} - 46 q^{77} + 6 q^{78} - 30 q^{79} + 8 q^{81} - 16 q^{82} - 12 q^{83} - 16 q^{84} + 32 q^{86} + 2 q^{87} - 32 q^{88} + 16 q^{89} - 12 q^{91} + 2 q^{92} - 22 q^{93} + 14 q^{94} + 2 q^{96} - 16 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) $ 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.35083	−1.66229	−0.831144	−	0.556058i	$-0.812313\pi$
$2$	−2.35083	−1.66229	−0.831144	+	0.556058i	$0.812313\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.52640	1.76320
$5$	0	0
$5$	0	0
$6$	−2.35083	−0.959722
$7$	−3.48189	−1.31603	−0.658015	−	0.753005i	$-0.728604\pi$
$7$	−3.48189	−1.31603	−0.658015	+	0.753005i	$0.728604\pi$
$8$	−3.58831	−1.26866
$9$	1.00000	0.333333
$10$	0	0
$11$	2.93111	0.883762	0.441881	−	0.897074i	$-0.354311\pi$
$11$	2.93111	0.883762	0.441881	+	0.897074i	$0.354311\pi$
$12$	3.52640	1.01798
$13$	1.87575	0.520239	0.260119	−	0.965576i	$-0.416238\pi$
$13$	1.87575	0.520239	0.260119	+	0.965576i	$0.416238\pi$
$14$	8.18532	2.18762
$15$	0	0
$16$	1.38270	0.345674
$17$	−6.78566	−1.64576	−0.822882	−	0.568212i	$-0.807635\pi$
$17$	−6.78566	−1.64576	−0.822882	+	0.568212i	$0.807635\pi$
$18$	−2.35083	−0.554096
$19$	−2.94950	−0.676662	−0.338331	−	0.941027i	$-0.609862\pi$
$19$	−2.94950	−0.676662	−0.338331	+	0.941027i	$0.609862\pi$
$20$	0	0
$21$	−3.48189	−0.759810
$22$	−6.89053	−1.46907
$23$	5.49019	1.14478	0.572392	−	0.819980i	$-0.306015\pi$
$23$	5.49019	1.14478	0.572392	+	0.819980i	$0.306015\pi$
$24$	−3.58831	−0.732460
$25$	0	0
$26$	−4.40956	−0.864786
$27$	1.00000	0.192450
$28$	−12.2785	−2.32042
$29$	2.55593	0.474625	0.237313	−	0.971433i	$-0.423733\pi$
$29$	2.55593	0.474625	0.237313	+	0.971433i	$0.423733\pi$
$30$	0	0
$31$	0.418084	0.0750901	0.0375451	−	0.999295i	$-0.488046\pi$
$31$	0.418084	0.0750901	0.0375451	+	0.999295i	$0.488046\pi$
$32$	3.92613	0.694048
$33$	2.93111	0.510240
$34$	15.9519	2.73573
$35$	0	0
$36$	3.52640	0.587733
$37$	−5.23959	−0.861383	−0.430691	−	0.902499i	$-0.641730\pi$
$37$	−5.23959	−0.861383	−0.430691	+	0.902499i	$0.641730\pi$
$38$	6.93377	1.12481
$39$	1.87575	0.300360
$40$	0	0
$41$	1.67869	0.262167	0.131084	−	0.991371i	$-0.458154\pi$
$41$	1.67869	0.262167	0.131084	+	0.991371i	$0.458154\pi$
$42$	8.18532	1.26302
$43$	−10.9233	−1.66578	−0.832892	−	0.553436i	$-0.813316\pi$
$43$	−10.9233	−1.66578	−0.832892	+	0.553436i	$0.813316\pi$
$44$	10.3363	1.55825
$45$	0	0
$46$	−12.9065	−1.90296
$47$	7.49178	1.09279	0.546394	−	0.837528i	$-0.316000\pi$
$47$	7.49178	1.09279	0.546394	+	0.837528i	$0.316000\pi$
$48$	1.38270	0.199575
$49$	5.12353	0.731933
$50$	0	0
$51$	−6.78566	−0.950183
$52$	6.61463	0.917285
$53$	−3.70953	−0.509543	−0.254771	−	0.967001i	$-0.582000\pi$
$53$	−3.70953	−0.509543	−0.254771	+	0.967001i	$0.582000\pi$
$54$	−2.35083	−0.319907
$55$	0	0
$56$	12.4941	1.66959
$57$	−2.94950	−0.390671
$58$	−6.00857	−0.788964
$59$	−7.10854	−0.925453	−0.462727	−	0.886501i	$-0.653129\pi$
$59$	−7.10854	−0.925453	−0.462727	+	0.886501i	$0.653129\pi$
$60$	0	0
$61$	−6.43710	−0.824186	−0.412093	−	0.911142i	$-0.635202\pi$
$61$	−6.43710	−0.824186	−0.412093	+	0.911142i	$0.635202\pi$
$62$	−0.982844	−0.124821
$63$	−3.48189	−0.438676
$64$	−11.9951	−1.49938
$65$	0	0
$66$	−6.89053	−0.848166
$67$	−10.0415	−1.22677	−0.613383	−	0.789786i	$-0.710192\pi$
$67$	−10.0415	−1.22677	−0.613383	+	0.789786i	$0.710192\pi$
$68$	−23.9290	−2.90181
$69$	5.49019	0.660942
$70$	0	0
$71$	0.728602	0.0864691	0.0432346	−	0.999065i	$-0.486234\pi$
$71$	0.728602	0.0864691	0.0432346	+	0.999065i	$0.486234\pi$
$72$	−3.58831	−0.422886
$73$	3.59269	0.420492	0.210246	−	0.977648i	$-0.432573\pi$
$73$	3.59269	0.420492	0.210246	+	0.977648i	$0.432573\pi$
$74$	12.3174	1.43187
$75$	0	0
$76$	−10.4011	−1.19309
$77$	−10.2058	−1.16306
$78$	−4.40956	−0.499285
$79$	3.07265	0.345700	0.172850	−	0.984948i	$-0.444702\pi$
$79$	3.07265	0.345700	0.172850	+	0.984948i	$0.444702\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.94632	−0.435798
$83$	−10.1152	−1.11029	−0.555146	−	0.831753i	$-0.687338\pi$
$83$	−10.1152	−1.11029	−0.555146	+	0.831753i	$0.687338\pi$
$84$	−12.2785	−1.33970
$85$	0	0
$86$	25.6788	2.76901
$87$	2.55593	0.274025
$88$	−10.5177	−1.12119
$89$	0.287512	0.0304762	0.0152381	−	0.999884i	$-0.495149\pi$
$89$	0.287512	0.0304762	0.0152381	+	0.999884i	$0.495149\pi$
$90$	0	0
$91$	−6.53114	−0.684649
$92$	19.3606	2.01848
$93$	0.418084	0.0433533
$94$	−17.6119	−1.81653
$95$	0	0
$96$	3.92613	0.400709
$97$	10.4090	1.05688	0.528439	−	0.848971i	$-0.322778\pi$
$97$	10.4090	1.05688	0.528439	+	0.848971i	$0.322778\pi$
$98$	−12.0446	−1.21668
$99$	2.93111	0.294587
$100$	0	0
$101$	7.65744	0.761943	0.380972	−	0.924587i	$-0.375590\pi$
$101$	7.65744	0.761943	0.380972	+	0.924587i	$0.375590\pi$
$102$	15.9519	1.57948
$103$	2.98602	0.294221	0.147111	−	0.989120i	$-0.453003\pi$
$103$	2.98602	0.294221	0.147111	+	0.989120i	$0.453003\pi$
$104$	−6.73076	−0.660005
$105$	0	0
$106$	8.72047	0.847007
$107$	7.07213	0.683689	0.341844	−	0.939757i	$-0.388948\pi$
$107$	7.07213	0.683689	0.341844	+	0.939757i	$0.388948\pi$
$108$	3.52640	0.339328
$109$	−13.3022	−1.27412	−0.637058	−	0.770816i	$-0.719849\pi$
$109$	−13.3022	−1.27412	−0.637058	+	0.770816i	$0.719849\pi$
$110$	0	0
$111$	−5.23959	−0.497320
$112$	−4.81440	−0.454918
$113$	−10.0233	−0.942911	−0.471456	−	0.881890i	$-0.656271\pi$
$113$	−10.0233	−0.942911	−0.471456	+	0.881890i	$0.656271\pi$
$114$	6.93377	0.649407
$115$	0	0
$116$	9.01325	0.836859
$117$	1.87575	0.173413
$118$	16.7110	1.53837
$119$	23.6269	2.16587
$120$	0	0
$121$	−2.40861	−0.218965
$122$	15.1325	1.37003
$123$	1.67869	0.151362
$124$	1.47433	0.132399
$125$	0	0
$126$	8.18532	0.729206
$127$	−10.5730	−0.938205	−0.469103	−	0.883144i	$-0.655423\pi$
$127$	−10.5730	−0.938205	−0.469103	+	0.883144i	$0.655423\pi$
$128$	20.3461	1.79836
$129$	−10.9233	−0.961741
$130$	0	0
$131$	−9.02608	−0.788612	−0.394306	−	0.918979i	$-0.629015\pi$
$131$	−9.02608	−0.788612	−0.394306	+	0.918979i	$0.629015\pi$
$132$	10.3363	0.899656
$133$	10.2698	0.890507
$134$	23.6059	2.03924
$135$	0	0
$136$	24.3490	2.08791
$137$	19.6646	1.68006	0.840029	−	0.542541i	$-0.182538\pi$
$137$	19.6646	1.68006	0.840029	+	0.542541i	$0.182538\pi$
$138$	−12.9065	−1.09868
$139$	−11.6520	−0.988310	−0.494155	−	0.869374i	$-0.664522\pi$
$139$	−11.6520	−0.988310	−0.494155	+	0.869374i	$0.664522\pi$
$140$	0	0
$141$	7.49178	0.630922
$142$	−1.71282	−0.143737
$143$	5.49801	0.459767
$144$	1.38270	0.115225
$145$	0	0
$146$	−8.44580	−0.698979
$147$	5.12353	0.422582
$148$	−18.4769	−1.51879
$149$	−7.33020	−0.600513	−0.300257	−	0.953858i	$-0.597072\pi$
$149$	−7.33020	−0.600513	−0.300257	+	0.953858i	$0.597072\pi$
$150$	0	0
$151$	−16.7358	−1.36194	−0.680968	−	0.732313i	$-0.738441\pi$
$151$	−16.7358	−1.36194	−0.680968	+	0.732313i	$0.738441\pi$
$152$	10.5837	0.858453
$153$	−6.78566	−0.548588
$154$	23.9921	1.93333
$155$	0	0
$156$	6.61463	0.529595
$157$	−7.88635	−0.629399	−0.314700	−	0.949191i	$-0.601904\pi$
$157$	−7.88635	−0.629399	−0.314700	+	0.949191i	$0.601904\pi$
$158$	−7.22328	−0.574653
$159$	−3.70953	−0.294185
$160$	0	0
$161$	−19.1162	−1.50657
$162$	−2.35083	−0.184699
$163$	9.93992	0.778555	0.389277	−	0.921121i	$-0.372725\pi$
$163$	9.93992	0.778555	0.389277	+	0.921121i	$0.372725\pi$
$164$	5.91973	0.462254
$165$	0	0
$166$	23.7792	1.84563
$167$	−5.57767	−0.431613	−0.215806	−	0.976436i	$-0.569238\pi$
$167$	−5.57767	−0.431613	−0.215806	+	0.976436i	$0.569238\pi$
$168$	12.4941	0.963939
$169$	−9.48157	−0.729352
$170$	0	0
$171$	−2.94950	−0.225554
$172$	−38.5198	−2.93711
$173$	−16.5682	−1.25966	−0.629828	−	0.776734i	$-0.716875\pi$
$173$	−16.5682	−1.25966	−0.629828	+	0.776734i	$0.716875\pi$
$174$	−6.00857	−0.455508
$175$	0	0
$176$	4.05284	0.305494
$177$	−7.10854	−0.534311
$178$	−0.675892	−0.0506602
$179$	−5.36021	−0.400641	−0.200321	−	0.979730i	$-0.564198\pi$
$179$	−5.36021	−0.400641	−0.200321	+	0.979730i	$0.564198\pi$
$180$	0	0
$181$	−0.327745	−0.0243611	−0.0121805	−	0.999926i	$-0.503877\pi$
$181$	−0.327745	−0.0243611	−0.0121805	+	0.999926i	$0.503877\pi$
$182$	15.3536	1.13808
$183$	−6.43710	−0.475844
$184$	−19.7005	−1.45234
$185$	0	0
$186$	−0.982844	−0.0720656
$187$	−19.8895	−1.45446
$188$	26.4190	1.92680
$189$	−3.48189	−0.253270
$190$	0	0
$191$	−3.46992	−0.251074	−0.125537	−	0.992089i	$-0.540065\pi$
$191$	−3.46992	−0.251074	−0.125537	+	0.992089i	$0.540065\pi$
$192$	−11.9951	−0.865668
$193$	24.3134	1.75012	0.875058	−	0.484018i	$-0.160823\pi$
$193$	24.3134	1.75012	0.875058	+	0.484018i	$0.160823\pi$
$194$	−24.4699	−1.75684
$195$	0	0
$196$	18.0676	1.29055
$197$	−14.9561	−1.06558	−0.532789	−	0.846248i	$-0.678856\pi$
$197$	−14.9561	−1.06558	−0.532789	+	0.846248i	$0.678856\pi$
$198$	−6.89053	−0.489689
$199$	−11.3251	−0.802817	−0.401408	−	0.915899i	$-0.631479\pi$
$199$	−11.3251	−0.802817	−0.401408	+	0.915899i	$0.631479\pi$
$200$	0	0
$201$	−10.0415	−0.708274
$202$	−18.0013	−1.26657
$203$	−8.89948	−0.624621
$204$	−23.9290	−1.67536
$205$	0	0
$206$	−7.01963	−0.489081
$207$	5.49019	0.381595
$208$	2.59359	0.179833
$209$	−8.64530	−0.598008
$210$	0	0
$211$	7.54283	0.519270	0.259635	−	0.965707i	$-0.416398\pi$
$211$	7.54283	0.519270	0.259635	+	0.965707i	$0.416398\pi$
$212$	−13.0813	−0.898426
$213$	0.728602	0.0499230
$214$	−16.6254	−1.13649
$215$	0	0
$216$	−3.58831	−0.244153
$217$	−1.45572	−0.0988208
$218$	31.2711	2.11795
$219$	3.59269	0.242771
$220$	0	0
$221$	−12.7282	−0.856190
$222$	12.3174	0.826688
$223$	7.79539	0.522018	0.261009	−	0.965336i	$-0.415945\pi$
$223$	7.79539	0.522018	0.261009	+	0.965336i	$0.415945\pi$
$224$	−13.6703	−0.913387
$225$	0	0
$226$	23.5630	1.56739
$227$	12.8319	0.851686	0.425843	−	0.904797i	$-0.359978\pi$
$227$	12.8319	0.851686	0.425843	+	0.904797i	$0.359978\pi$
$228$	−10.4011	−0.688831
$229$	−14.5033	−0.958405	−0.479203	−	0.877704i	$-0.659074\pi$
$229$	−14.5033	−0.958405	−0.479203	+	0.877704i	$0.659074\pi$
$230$	0	0
$231$	−10.2058	−0.671491
$232$	−9.17148	−0.602137
$233$	−11.3970	−0.746646	−0.373323	−	0.927702i	$-0.621782\pi$
$233$	−11.3970	−0.746646	−0.373323	+	0.927702i	$0.621782\pi$
$234$	−4.40956	−0.288262
$235$	0	0
$236$	−25.0676	−1.63176
$237$	3.07265	0.199590
$238$	−55.5428	−3.60031
$239$	6.90072	0.446371	0.223185	−	0.974776i	$-0.428354\pi$
$239$	6.90072	0.446371	0.223185	+	0.974776i	$0.428354\pi$
$240$	0	0
$241$	−28.4467	−1.83242	−0.916208	−	0.400704i	$-0.868766\pi$
$241$	−28.4467	−1.83242	−0.916208	+	0.400704i	$0.868766\pi$
$242$	5.66224	0.363982
$243$	1.00000	0.0641500
$244$	−22.6998	−1.45320
$245$	0	0
$246$	−3.94632	−0.251608
$247$	−5.53252	−0.352026
$248$	−1.50021	−0.0952637
$249$	−10.1152	−0.641028
$250$	0	0
$251$	−12.3258	−0.777999	−0.389000	−	0.921238i	$-0.627179\pi$
$251$	−12.3258	−0.777999	−0.389000	+	0.921238i	$0.627179\pi$
$252$	−12.2785	−0.773474
$253$	16.0923	1.01172
$254$	24.8554	1.55957
$255$	0	0
$256$	−23.8400	−1.49000
$257$	−20.8274	−1.29918	−0.649590	−	0.760285i	$-0.725059\pi$
$257$	−20.8274	−1.29918	−0.649590	+	0.760285i	$0.725059\pi$
$258$	25.6788	1.59869
$259$	18.2437	1.13361
$260$	0	0
$261$	2.55593	0.158208
$262$	21.2188	1.31090
$263$	−2.23517	−0.137826	−0.0689131	−	0.997623i	$-0.521953\pi$
$263$	−2.23517	−0.137826	−0.0689131	+	0.997623i	$0.521953\pi$
$264$	−10.5177	−0.647320
$265$	0	0
$266$	−24.1426	−1.48028
$267$	0.287512	0.0175955
$268$	−35.4104	−2.16303
$269$	−19.3036	−1.17696	−0.588482	−	0.808511i	$-0.700274\pi$
$269$	−19.3036	−1.17696	−0.588482	+	0.808511i	$0.700274\pi$
$270$	0	0
$271$	−11.9411	−0.725372	−0.362686	−	0.931911i	$-0.618140\pi$
$271$	−11.9411	−0.725372	−0.362686	+	0.931911i	$0.618140\pi$
$272$	−9.38252	−0.568899
$273$	−6.53114	−0.395282
$274$	−46.2281	−2.79274
$275$	0	0
$276$	19.3606	1.16537
$277$	−26.1466	−1.57100	−0.785499	−	0.618863i	$-0.787594\pi$
$277$	−26.1466	−1.57100	−0.785499	+	0.618863i	$0.787594\pi$
$278$	27.3919	1.64285
$279$	0.418084	0.0250300
$280$	0	0
$281$	10.1219	0.603820	0.301910	−	0.953336i	$-0.402376\pi$
$281$	10.1219	0.603820	0.301910	+	0.953336i	$0.402376\pi$
$282$	−17.6119	−1.04877
$283$	31.8638	1.89410	0.947052	−	0.321081i	$-0.104046\pi$
$283$	31.8638	1.89410	0.947052	+	0.321081i	$0.104046\pi$
$284$	2.56934	0.152462
$285$	0	0
$286$	−12.9249	−0.764265
$287$	−5.84501	−0.345020
$288$	3.92613	0.231349
$289$	29.0452	1.70854
$290$	0	0
$291$	10.4090	0.610189
$292$	12.6693	0.741412
$293$	−16.6235	−0.971153	−0.485576	−	0.874194i	$-0.661390\pi$
$293$	−16.6235	−0.971153	−0.485576	+	0.874194i	$0.661390\pi$
$294$	−12.0446	−0.702453
$295$	0	0
$296$	18.8012	1.09280
$297$	2.93111	0.170080
$298$	17.2320	0.998225
$299$	10.2982	0.595561
$300$	0	0
$301$	38.0336	2.19222
$302$	39.3429	2.26393
$303$	7.65744	0.439908
$304$	−4.07827	−0.233905
$305$	0	0
$306$	15.9519	0.911911
$307$	14.1643	0.808402	0.404201	−	0.914670i	$-0.367550\pi$
$307$	14.1643	0.808402	0.404201	+	0.914670i	$0.367550\pi$
$308$	−35.9897	−2.05070
$309$	2.98602	0.169869
$310$	0	0
$311$	15.0500	0.853406	0.426703	−	0.904392i	$-0.359675\pi$
$311$	15.0500	0.853406	0.426703	+	0.904392i	$0.359675\pi$
$312$	−6.73076	−0.381054
$313$	3.57476	0.202057	0.101029	−	0.994884i	$-0.467787\pi$
$313$	3.57476	0.202057	0.101029	+	0.994884i	$0.467787\pi$
$314$	18.5395	1.04624
$315$	0	0
$316$	10.8354	0.609539
$317$	−12.7820	−0.717907	−0.358954	−	0.933355i	$-0.616866\pi$
$317$	−12.7820	−0.717907	−0.358954	+	0.933355i	$0.616866\pi$
$318$	8.72047	0.489020
$319$	7.49172	0.419456
$320$	0	0
$321$	7.07213	0.394728
$322$	44.9390	2.50435
$323$	20.0143	1.11363
$324$	3.52640	0.195911
$325$	0	0
$326$	−23.3671	−1.29418
$327$	−13.3022	−0.735612
$328$	−6.02366	−0.332601
$329$	−26.0855	−1.43814
$330$	0	0
$331$	4.70504	0.258612	0.129306	−	0.991605i	$-0.458725\pi$
$331$	4.70504	0.258612	0.129306	+	0.991605i	$0.458725\pi$
$332$	−35.6704	−1.95767
$333$	−5.23959	−0.287128
$334$	13.1121	0.717465
$335$	0	0
$336$	−4.81440	−0.262647
$337$	17.4048	0.948098	0.474049	−	0.880499i	$-0.342792\pi$
$337$	17.4048	0.948098	0.474049	+	0.880499i	$0.342792\pi$
$338$	22.2896	1.21239
$339$	−10.0233	−0.544390
$340$	0	0
$341$	1.22545	0.0663618
$342$	6.93377	0.374936
$343$	6.53364	0.352783
$344$	39.1961	2.11331
$345$	0	0
$346$	38.9490	2.09391
$347$	−14.0106	−0.752130	−0.376065	−	0.926593i	$-0.622723\pi$
$347$	−14.0106	−0.752130	−0.376065	+	0.926593i	$0.622723\pi$
$348$	9.01325	0.483161
$349$	35.9459	1.92414	0.962069	−	0.272806i	$-0.0879518\pi$
$349$	35.9459	1.92414	0.962069	+	0.272806i	$0.0879518\pi$
$350$	0	0
$351$	1.87575	0.100120
$352$	11.5079	0.613373
$353$	−8.80198	−0.468482	−0.234241	−	0.972179i	$-0.575260\pi$
$353$	−8.80198	−0.468482	−0.234241	+	0.972179i	$0.575260\pi$
$354$	16.7110	0.888178
$355$	0	0
$356$	1.01388	0.0537357
$357$	23.6269	1.25047
$358$	12.6009	0.665981
$359$	6.02962	0.318231	0.159116	−	0.987260i	$-0.449136\pi$
$359$	6.02962	0.318231	0.159116	+	0.987260i	$0.449136\pi$
$360$	0	0
$361$	−10.3004	−0.542129
$362$	0.770472	0.0404951
$363$	−2.40861	−0.126419
$364$	−23.0314	−1.20717
$365$	0	0
$366$	15.1325	0.790989
$367$	−28.7662	−1.50158	−0.750792	−	0.660539i	$-0.770328\pi$
$367$	−28.7662	−1.50158	−0.750792	+	0.660539i	$0.770328\pi$
$368$	7.59128	0.395723
$369$	1.67869	0.0873891
$370$	0	0
$371$	12.9162	0.670574
$372$	1.47433	0.0764405
$373$	−33.6000	−1.73974	−0.869872	−	0.493278i	$-0.835799\pi$
$373$	−33.6000	−1.73974	−0.869872	+	0.493278i	$0.835799\pi$
$374$	46.7568	2.41774
$375$	0	0
$376$	−26.8828	−1.38637
$377$	4.79429	0.246918
$378$	8.18532	0.421008
$379$	4.63403	0.238034	0.119017	−	0.992892i	$-0.462026\pi$
$379$	4.63403	0.238034	0.119017	+	0.992892i	$0.462026\pi$
$380$	0	0
$381$	−10.5730	−0.541673
$382$	8.15718	0.417358
$383$	12.9058	0.659453	0.329727	−	0.944076i	$-0.393043\pi$
$383$	12.9058	0.659453	0.329727	+	0.944076i	$0.393043\pi$
$384$	20.3461	1.03828
$385$	0	0
$386$	−57.1567	−2.90920
$387$	−10.9233	−0.555261
$388$	36.7064	1.86349
$389$	−17.5246	−0.888532	−0.444266	−	0.895895i	$-0.646535\pi$
$389$	−17.5246	−0.888532	−0.444266	+	0.895895i	$0.646535\pi$
$390$	0	0
$391$	−37.2546	−1.88405
$392$	−18.3848	−0.928573
$393$	−9.02608	−0.455305
$394$	35.1593	1.77130
$395$	0	0
$396$	10.3363	0.519416
$397$	27.0176	1.35597	0.677987	−	0.735074i	$-0.262853\pi$
$397$	27.0176	1.35597	0.677987	+	0.735074i	$0.262853\pi$
$398$	26.6234	1.33451
$399$	10.2698	0.514134
$400$	0	0
$401$	−15.9792	−0.797965	−0.398983	−	0.916958i	$-0.630637\pi$
$401$	−15.9792	−0.797965	−0.398983	+	0.916958i	$0.630637\pi$
$402$	23.6059	1.17735
$403$	0.784220	0.0390648
$404$	27.0032	1.34346
$405$	0	0
$406$	20.9212	1.03830
$407$	−15.3578	−0.761258
$408$	24.3490	1.20546
$409$	−30.9962	−1.53266	−0.766331	−	0.642446i	$-0.777920\pi$
$409$	−30.9962	−1.53266	−0.766331	+	0.642446i	$0.777920\pi$
$410$	0	0
$411$	19.6646	0.969982
$412$	10.5299	0.518771
$413$	24.7511	1.21792
$414$	−12.9065	−0.634321
$415$	0	0
$416$	7.36442	0.361070
$417$	−11.6520	−0.570601
$418$	20.3236	0.994061
$419$	25.9153	1.26605	0.633024	−	0.774132i	$-0.281814\pi$
$419$	25.9153	1.26605	0.633024	+	0.774132i	$0.281814\pi$
$420$	0	0
$421$	3.92646	0.191364	0.0956821	−	0.995412i	$-0.469497\pi$
$421$	3.92646	0.191364	0.0956821	+	0.995412i	$0.469497\pi$
$422$	−17.7319	−0.863176
$423$	7.49178	0.364263
$424$	13.3109	0.646436
$425$	0	0
$426$	−1.71282	−0.0829863
$427$	22.4132	1.08465
$428$	24.9392	1.20548
$429$	5.49801	0.265447
$430$	0	0
$431$	−35.4632	−1.70820	−0.854101	−	0.520108i	$-0.825892\pi$
$431$	−35.4632	−1.70820	−0.854101	+	0.520108i	$0.825892\pi$
$432$	1.38270	0.0665251
$433$	−10.0959	−0.485178	−0.242589	−	0.970129i	$-0.577997\pi$
$433$	−10.0959	−0.485178	−0.242589	+	0.970129i	$0.577997\pi$
$434$	3.42215	0.164269
$435$	0	0
$436$	−46.9088	−2.24652
$437$	−16.1933	−0.774632
$438$	−8.44580	−0.403556
$439$	7.10560	0.339132	0.169566	−	0.985519i	$-0.445763\pi$
$439$	7.10560	0.339132	0.169566	+	0.985519i	$0.445763\pi$
$440$	0	0
$441$	5.12353	0.243978
$442$	29.9218	1.42323
$443$	−20.6841	−0.982733	−0.491366	−	0.870953i	$-0.663502\pi$
$443$	−20.6841	−0.982733	−0.491366	+	0.870953i	$0.663502\pi$
$444$	−18.4769	−0.876874
$445$	0	0
$446$	−18.3256	−0.867744
$447$	−7.33020	−0.346706
$448$	41.7654	1.97323
$449$	19.4940	0.919980	0.459990	−	0.887924i	$-0.347853\pi$
$449$	19.4940	0.919980	0.459990	+	0.887924i	$0.347853\pi$
$450$	0	0
$451$	4.92042	0.231694
$452$	−35.3461	−1.66254
$453$	−16.7358	−0.786314
$454$	−30.1657	−1.41575
$455$	0	0
$456$	10.5837	0.495628
$457$	4.34194	0.203107	0.101554	−	0.994830i	$-0.467619\pi$
$457$	4.34194	0.203107	0.101554	+	0.994830i	$0.467619\pi$
$458$	34.0948	1.59315
$459$	−6.78566	−0.316728
$460$	0	0
$461$	11.7897	0.549102	0.274551	−	0.961573i	$-0.411471\pi$
$461$	11.7897	0.549102	0.274551	+	0.961573i	$0.411471\pi$
$462$	23.9921	1.11621
$463$	9.07870	0.421923	0.210962	−	0.977494i	$-0.432341\pi$
$463$	9.07870	0.421923	0.210962	+	0.977494i	$0.432341\pi$
$464$	3.53409	0.164066
$465$	0	0
$466$	26.7925	1.24114
$467$	34.9181	1.61582	0.807910	−	0.589307i	$-0.200599\pi$
$467$	34.9181	1.61582	0.807910	+	0.589307i	$0.200599\pi$
$468$	6.61463	0.305762
$469$	34.9634	1.61446
$470$	0	0
$471$	−7.88635	−0.363384
$472$	25.5076	1.17408
$473$	−32.0173	−1.47216
$474$	−7.22328	−0.331776
$475$	0	0
$476$	83.3179	3.81887
$477$	−3.70953	−0.169848
$478$	−16.2224	−0.741996
$479$	−8.64649	−0.395068	−0.197534	−	0.980296i	$-0.563293\pi$
$479$	−8.64649	−0.395068	−0.197534	+	0.980296i	$0.563293\pi$
$480$	0	0
$481$	−9.82814	−0.448125
$482$	66.8734	3.04600
$483$	−19.1162	−0.869819
$484$	−8.49373	−0.386079
$485$	0	0
$486$	−2.35083	−0.106636
$487$	−2.84462	−0.128902	−0.0644510	−	0.997921i	$-0.520530\pi$
$487$	−2.84462	−0.128902	−0.0644510	+	0.997921i	$0.520530\pi$
$488$	23.0983	1.04561
$489$	9.93992	0.449499
$490$	0	0
$491$	36.8041	1.66095	0.830473	−	0.557059i	$-0.188070\pi$
$491$	36.8041	1.66095	0.830473	+	0.557059i	$0.188070\pi$
$492$	5.91973	0.266882
$493$	−17.3437	−0.781121
$494$	13.0060	0.585168
$495$	0	0
$496$	0.578084	0.0259567
$497$	−2.53691	−0.113796
$498$	23.7792	1.06557
$499$	5.85775	0.262229	0.131114	−	0.991367i	$-0.458144\pi$
$499$	5.85775	0.262229	0.131114	+	0.991367i	$0.458144\pi$
$500$	0	0
$501$	−5.57767	−0.249192
$502$	28.9759	1.29326
$503$	30.2874	1.35045	0.675223	−	0.737613i	$-0.264047\pi$
$503$	30.2874	1.35045	0.675223	+	0.737613i	$0.264047\pi$
$504$	12.4941	0.556530
$505$	0	0
$506$	−37.8304	−1.68177
$507$	−9.48157	−0.421091
$508$	−37.2848	−1.65424
$509$	15.1430	0.671201	0.335601	−	0.942004i	$-0.391061\pi$
$509$	15.1430	0.671201	0.335601	+	0.942004i	$0.391061\pi$
$510$	0	0
$511$	−12.5093	−0.553380
$512$	15.3517	0.678457
$513$	−2.94950	−0.130224
$514$	48.9618	2.15961
$515$	0	0
$516$	−38.5198	−1.69574
$517$	21.9592	0.965765
$518$	−42.8877	−1.88438
$519$	−16.5682	−0.727263
$520$	0	0
$521$	39.0832	1.71227	0.856134	−	0.516754i	$-0.172860\pi$
$521$	39.0832	1.71227	0.856134	+	0.516754i	$0.172860\pi$
$522$	−6.00857	−0.262988
$523$	38.7976	1.69650	0.848251	−	0.529594i	$-0.177656\pi$
$523$	38.7976	1.69650	0.848251	+	0.529594i	$0.177656\pi$
$524$	−31.8296	−1.39048
$525$	0	0
$526$	5.25449	0.229107
$527$	−2.83698	−0.123581
$528$	4.05284	0.176377
$529$	7.14224	0.310532
$530$	0	0
$531$	−7.10854	−0.308484
$532$	36.2155	1.57014
$533$	3.14880	0.136390
$534$	−0.675892	−0.0292487
$535$	0	0
$536$	36.0320	1.55635
$537$	−5.36021	−0.231310
$538$	45.3796	1.95645
$539$	15.0176	0.646855
$540$	0	0
$541$	14.2280	0.611710	0.305855	−	0.952078i	$-0.401058\pi$
$541$	14.2280	0.611710	0.305855	+	0.952078i	$0.401058\pi$
$542$	28.0716	1.20578
$543$	−0.327745	−0.0140649
$544$	−26.6414	−1.14224
$545$	0	0
$546$	15.3536	0.657073
$547$	−39.3229	−1.68133	−0.840663	−	0.541559i	$-0.817834\pi$
$547$	−39.3229	−1.68133	−0.840663	+	0.541559i	$0.817834\pi$
$548$	69.3452	2.96228
$549$	−6.43710	−0.274729
$550$	0	0
$551$	−7.53873	−0.321161
$552$	−19.7005	−0.838509
$553$	−10.6986	−0.454952
$554$	61.4662	2.61145
$555$	0	0
$556$	−41.0896	−1.74259
$557$	−35.3849	−1.49931	−0.749654	−	0.661830i	$-0.769780\pi$
$557$	−35.3849	−1.49931	−0.749654	+	0.661830i	$0.769780\pi$
$558$	−0.982844	−0.0416071
$559$	−20.4893	−0.866605
$560$	0	0
$561$	−19.8895	−0.839735
$562$	−23.7948	−1.00372
$563$	−15.0386	−0.633800	−0.316900	−	0.948459i	$-0.602642\pi$
$563$	−15.0386	−0.633800	−0.316900	+	0.948459i	$0.602642\pi$
$564$	26.4190	1.11244
$565$	0	0
$566$	−74.9063	−3.14855
$567$	−3.48189	−0.146225
$568$	−2.61445	−0.109700
$569$	−2.72409	−0.114200	−0.0570998	−	0.998368i	$-0.518185\pi$
$569$	−2.72409	−0.114200	−0.0570998	+	0.998368i	$0.518185\pi$
$570$	0	0
$571$	12.6236	0.528282	0.264141	−	0.964484i	$-0.414912\pi$
$571$	12.6236	0.528282	0.264141	+	0.964484i	$0.414912\pi$
$572$	19.3882	0.810661
$573$	−3.46992	−0.144958
$574$	13.7406	0.573522
$575$	0	0
$576$	−11.9951	−0.499794
$577$	23.5844	0.981831	0.490915	−	0.871207i	$-0.336662\pi$
$577$	23.5844	0.981831	0.490915	+	0.871207i	$0.336662\pi$
$578$	−68.2803	−2.84009
$579$	24.3134	1.01043
$580$	0	0
$581$	35.2201	1.46118
$582$	−24.4699	−1.01431
$583$	−10.8730	−0.450315
$584$	−12.8917	−0.533461
$585$	0	0
$586$	39.0789	1.61434
$587$	22.6772	0.935988	0.467994	−	0.883732i	$-0.344977\pi$
$587$	22.6772	0.935988	0.467994	+	0.883732i	$0.344977\pi$
$588$	18.0676	0.745097
$589$	−1.23314	−0.0508106
$590$	0	0
$591$	−14.9561	−0.615212
$592$	−7.24477	−0.297758
$593$	8.74287	0.359027	0.179513	−	0.983756i	$-0.442548\pi$
$593$	8.74287	0.359027	0.179513	+	0.983756i	$0.442548\pi$
$594$	−6.89053	−0.282722
$595$	0	0
$596$	−25.8492	−1.05882
$597$	−11.3251	−0.463506
$598$	−24.2094	−0.989994
$599$	16.3209	0.666854	0.333427	−	0.942776i	$-0.391795\pi$
$599$	16.3209	0.666854	0.333427	+	0.942776i	$0.391795\pi$
$600$	0	0
$601$	36.2713	1.47954	0.739768	−	0.672862i	$-0.234935\pi$
$601$	36.2713	1.47954	0.739768	+	0.672862i	$0.234935\pi$
$602$	−89.4105	−3.64410
$603$	−10.0415	−0.408922
$604$	−59.0170	−2.40137
$605$	0	0
$606$	−18.0013	−0.731254
$607$	16.6820	0.677102	0.338551	−	0.940948i	$-0.390063\pi$
$607$	16.6820	0.677102	0.338551	+	0.940948i	$0.390063\pi$
$608$	−11.5801	−0.469636
$609$	−8.89948	−0.360625
$610$	0	0
$611$	14.0527	0.568511
$612$	−23.9290	−0.967271
$613$	−24.7967	−1.00153	−0.500765	−	0.865583i	$-0.666948\pi$
$613$	−24.7967	−1.00153	−0.500765	+	0.865583i	$0.666948\pi$
$614$	−33.2980	−1.34380
$615$	0	0
$616$	36.6215	1.47552
$617$	−27.9669	−1.12591	−0.562953	−	0.826489i	$-0.690335\pi$
$617$	−27.9669	−1.12591	−0.562953	+	0.826489i	$0.690335\pi$
$618$	−7.01963	−0.282371
$619$	27.2686	1.09602	0.548009	−	0.836473i	$-0.315386\pi$
$619$	27.2686	1.09602	0.548009	+	0.836473i	$0.315386\pi$
$620$	0	0
$621$	5.49019	0.220314
$622$	−35.3799	−1.41861
$623$	−1.00108	−0.0401076
$624$	2.59359	0.103827
$625$	0	0
$626$	−8.40365	−0.335878
$627$	−8.64530	−0.345260
$628$	−27.8104	−1.10976
$629$	35.5541	1.41763
$630$	0	0
$631$	42.2603	1.68235	0.841177	−	0.540759i	$-0.181863\pi$
$631$	42.2603	1.68235	0.841177	+	0.540759i	$0.181863\pi$
$632$	−11.0256	−0.438575
$633$	7.54283	0.299801
$634$	30.0482	1.19337
$635$	0	0
$636$	−13.0813	−0.518707
$637$	9.61045	0.380780
$638$	−17.6118	−0.697256
$639$	0.728602	0.0288230
$640$	0	0
$641$	45.8456	1.81079	0.905396	−	0.424569i	$-0.139574\pi$
$641$	45.8456	1.81079	0.905396	+	0.424569i	$0.139574\pi$
$642$	−16.6254	−0.656151
$643$	46.6710	1.84052	0.920261	−	0.391304i	$-0.127976\pi$
$643$	46.6710	1.84052	0.920261	+	0.391304i	$0.127976\pi$
$644$	−67.4115	−2.65639
$645$	0	0
$646$	−47.0502	−1.85117
$647$	12.1264	0.476740	0.238370	−	0.971174i	$-0.423387\pi$
$647$	12.1264	0.476740	0.238370	+	0.971174i	$0.423387\pi$
$648$	−3.58831	−0.140962
$649$	−20.8359	−0.817880
$650$	0	0
$651$	−1.45572	−0.0570542
$652$	35.0521	1.37275
$653$	1.72017	0.0673154	0.0336577	−	0.999433i	$-0.489284\pi$
$653$	1.72017	0.0673154	0.0336577	+	0.999433i	$0.489284\pi$
$654$	31.2711	1.22280
$655$	0	0
$656$	2.32112	0.0906246
$657$	3.59269	0.140164
$658$	61.3226	2.39060
$659$	5.47947	0.213450	0.106725	−	0.994289i	$-0.465964\pi$
$659$	5.47947	0.213450	0.106725	+	0.994289i	$0.465964\pi$
$660$	0	0
$661$	0.797448	0.0310171	0.0155086	−	0.999880i	$-0.495063\pi$
$661$	0.797448	0.0310171	0.0155086	+	0.999880i	$0.495063\pi$
$662$	−11.0607	−0.429888
$663$	−12.7282	−0.494322
$664$	36.2966	1.40858
$665$	0	0
$666$	12.3174	0.477289
$667$	14.0326	0.543344
$668$	−19.6691	−0.761020
$669$	7.79539	0.301387
$670$	0	0
$671$	−18.8678	−0.728384
$672$	−13.6703	−0.527344
$673$	51.3280	1.97855	0.989274	−	0.146074i	$-0.0466639\pi$
$673$	51.3280	1.97855	0.989274	+	0.146074i	$0.0466639\pi$
$674$	−40.9156	−1.57601
$675$	0	0
$676$	−33.4358	−1.28599
$677$	−15.0929	−0.580069	−0.290034	−	0.957016i	$-0.593667\pi$
$677$	−15.0929	−0.580069	−0.290034	+	0.957016i	$0.593667\pi$
$678$	23.5630	0.904933
$679$	−36.2431	−1.39088
$680$	0	0
$681$	12.8319	0.491721
$682$	−2.88082	−0.110312
$683$	10.8602	0.415555	0.207777	−	0.978176i	$-0.433377\pi$
$683$	10.8602	0.415555	0.207777	+	0.978176i	$0.433377\pi$
$684$	−10.4011	−0.397697
$685$	0	0
$686$	−15.3595	−0.586428
$687$	−14.5033	−0.553336
$688$	−15.1036	−0.575819
$689$	−6.95814	−0.265084
$690$	0	0
$691$	8.44552	0.321283	0.160641	−	0.987013i	$-0.448644\pi$
$691$	8.44552	0.321283	0.160641	+	0.987013i	$0.448644\pi$
$692$	−58.4261	−2.22103
$693$	−10.2058	−0.387686
$694$	32.9366	1.25026
$695$	0	0
$696$	−9.17148	−0.347644
$697$	−11.3910	−0.431466
$698$	−84.5026	−3.19847
$699$	−11.3970	−0.431076
$700$	0	0
$701$	48.9399	1.84843	0.924216	−	0.381869i	$-0.124719\pi$
$701$	48.9399	1.84843	0.924216	+	0.381869i	$0.124719\pi$
$702$	−4.40956	−0.166428
$703$	15.4542	0.582865
$704$	−35.1588	−1.32510
$705$	0	0
$706$	20.6920	0.778752
$707$	−26.6623	−1.00274
$708$	−25.0676	−0.942097
$709$	24.7834	0.930762	0.465381	−	0.885111i	$-0.345917\pi$
$709$	24.7834	0.930762	0.465381	+	0.885111i	$0.345917\pi$
$710$	0	0
$711$	3.07265	0.115233
$712$	−1.03168	−0.0386639
$713$	2.29536	0.0859620
$714$	−55.5428	−2.07864
$715$	0	0
$716$	−18.9023	−0.706410
$717$	6.90072	0.257712
$718$	−14.1746	−0.528992
$719$	−18.2171	−0.679382	−0.339691	−	0.940537i	$-0.610322\pi$
$719$	−18.2171	−0.679382	−0.339691	+	0.940537i	$0.610322\pi$
$720$	0	0
$721$	−10.3970	−0.387204
$722$	24.2146	0.901174
$723$	−28.4467	−1.05795
$724$	−1.15576	−0.0429534
$725$	0	0
$726$	5.66224	0.210145
$727$	−16.6699	−0.618253	−0.309127	−	0.951021i	$-0.600037\pi$
$727$	−16.6699	−0.618253	−0.309127	+	0.951021i	$0.600037\pi$
$728$	23.4357	0.868586
$729$	1.00000	0.0370370
$730$	0	0
$731$	74.1216	2.74149
$732$	−22.6998	−0.839008
$733$	−20.0644	−0.741094	−0.370547	−	0.928814i	$-0.620830\pi$
$733$	−20.0644	−0.741094	−0.370547	+	0.928814i	$0.620830\pi$
$734$	67.6245	2.49606
$735$	0	0
$736$	21.5552	0.794535
$737$	−29.4328	−1.08417
$738$	−3.94632	−0.145266
$739$	−34.0571	−1.25281	−0.626406	−	0.779497i	$-0.715475\pi$
$739$	−34.0571	−1.25281	−0.626406	+	0.779497i	$0.715475\pi$
$740$	0	0
$741$	−5.53252	−0.203242
$742$	−30.3637	−1.11469
$743$	38.5355	1.41373	0.706865	−	0.707348i	$-0.250109\pi$
$743$	38.5355	1.41373	0.706865	+	0.707348i	$0.250109\pi$
$744$	−1.50021	−0.0550005
$745$	0	0
$746$	78.9880	2.89195
$747$	−10.1152	−0.370097
$748$	−70.1383	−2.56451
$749$	−24.6244	−0.899754
$750$	0	0
$751$	−36.0351	−1.31494	−0.657470	−	0.753481i	$-0.728373\pi$
$751$	−36.0351	−1.31494	−0.657470	+	0.753481i	$0.728373\pi$
$752$	10.3589	0.377749
$753$	−12.3258	−0.449178
$754$	−11.2706	−0.410449
$755$	0	0
$756$	−12.2785	−0.446566
$757$	−35.0131	−1.27257	−0.636287	−	0.771453i	$-0.719530\pi$
$757$	−35.0131	−1.27257	−0.636287	+	0.771453i	$0.719530\pi$
$758$	−10.8938	−0.395681
$759$	16.0923	0.584115
$760$	0	0
$761$	31.3577	1.13672	0.568358	−	0.822781i	$-0.307579\pi$
$761$	31.3577	1.13672	0.568358	+	0.822781i	$0.307579\pi$
$762$	24.8554	0.900416
$763$	46.3166	1.67678
$764$	−12.2363	−0.442694
$765$	0	0
$766$	−30.3392	−1.09620
$767$	−13.3338	−0.481456
$768$	−23.8400	−0.860253
$769$	2.39815	0.0864793	0.0432397	−	0.999065i	$-0.486232\pi$
$769$	2.39815	0.0864793	0.0432397	+	0.999065i	$0.486232\pi$
$770$	0	0
$771$	−20.8274	−0.750082
$772$	85.7388	3.08581
$773$	10.9226	0.392858	0.196429	−	0.980518i	$-0.437065\pi$
$773$	10.9226	0.392858	0.196429	+	0.980518i	$0.437065\pi$
$774$	25.6788	0.923004
$775$	0	0
$776$	−37.3508	−1.34082
$777$	18.2437	0.654487
$778$	41.1973	1.47700
$779$	−4.95130	−0.177399
$780$	0	0
$781$	2.13561	0.0764181
$782$	87.5792	3.13183
$783$	2.55593	0.0913417
$784$	7.08430	0.253011
$785$	0	0
$786$	21.2188	0.756848
$787$	−11.5724	−0.412511	−0.206256	−	0.978498i	$-0.566128\pi$
$787$	−11.5724	−0.412511	−0.206256	+	0.978498i	$0.566128\pi$
$788$	−52.7412	−1.87883
$789$	−2.23517	−0.0795740
$790$	0	0
$791$	34.8999	1.24090
$792$	−10.5177	−0.373731
$793$	−12.0744	−0.428773
$794$	−63.5137	−2.25402
$795$	0	0
$796$	−39.9369	−1.41553
$797$	−21.8923	−0.775467	−0.387733	−	0.921772i	$-0.626742\pi$
$797$	−21.8923	−0.775467	−0.387733	+	0.921772i	$0.626742\pi$
$798$	−24.1426	−0.854639
$799$	−50.8367	−1.79847
$800$	0	0
$801$	0.287512	0.0101587
$802$	37.5645	1.32645
$803$	10.5306	0.371615
$804$	−35.4104	−1.24883
$805$	0	0
$806$	−1.84357	−0.0649369
$807$	−19.3036	−0.679520
$808$	−27.4772	−0.966646
$809$	−21.5498	−0.757652	−0.378826	−	0.925468i	$-0.623672\pi$
$809$	−21.5498	−0.757652	−0.378826	+	0.925468i	$0.623672\pi$
$810$	0	0
$811$	−43.9615	−1.54370	−0.771848	−	0.635807i	$-0.780667\pi$
$811$	−43.9615	−1.54370	−0.771848	+	0.635807i	$0.780667\pi$
$812$	−31.3831	−1.10133
$813$	−11.9411	−0.418794
$814$	36.1036	1.26543
$815$	0	0
$816$	−9.38252	−0.328454
$817$	32.2182	1.12717
$818$	72.8667	2.54772
$819$	−6.53114	−0.228216
$820$	0	0
$821$	33.5061	1.16937	0.584686	−	0.811260i	$-0.301218\pi$
$821$	33.5061	1.16937	0.584686	+	0.811260i	$0.301218\pi$
$822$	−46.2281	−1.61239
$823$	−55.9860	−1.95155	−0.975774	−	0.218781i	$-0.929792\pi$
$823$	−55.9860	−1.95155	−0.975774	+	0.218781i	$0.929792\pi$
$824$	−10.7148	−0.373266
$825$	0	0
$826$	−58.1857	−2.02454
$827$	40.8100	1.41910	0.709552	−	0.704653i	$-0.248897\pi$
$827$	40.8100	1.41910	0.709552	+	0.704653i	$0.248897\pi$
$828$	19.3606	0.672828
$829$	0.122914	0.00426896	0.00213448	−	0.999998i	$-0.499321\pi$
$829$	0.122914	0.00426896	0.00213448	+	0.999998i	$0.499321\pi$
$830$	0	0
$831$	−26.1466	−0.907016
$832$	−22.4997	−0.780036
$833$	−34.7666	−1.20459
$834$	27.3919	0.948503
$835$	0	0
$836$	−30.4868	−1.05441
$837$	0.418084	0.0144511
$838$	−60.9226	−2.10454
$839$	14.8083	0.511241	0.255620	−	0.966777i	$-0.417720\pi$
$839$	14.8083	0.511241	0.255620	+	0.966777i	$0.417720\pi$
$840$	0	0
$841$	−22.4672	−0.774731
$842$	−9.23045	−0.318102
$843$	10.1219	0.348616
$844$	26.5990	0.915577
$845$	0	0
$846$	−17.6119	−0.605509
$847$	8.38651	0.288164
$848$	−5.12916	−0.176136
$849$	31.8638	1.09356
$850$	0	0
$851$	−28.7664	−0.986098
$852$	2.56934	0.0880242
$853$	−7.26474	−0.248740	−0.124370	−	0.992236i	$-0.539691\pi$
$853$	−7.26474	−0.248740	−0.124370	+	0.992236i	$0.539691\pi$
$854$	−52.6897	−1.80301
$855$	0	0
$856$	−25.3770	−0.867367
$857$	26.8175	0.916068	0.458034	−	0.888935i	$-0.348554\pi$
$857$	26.8175	0.916068	0.458034	+	0.888935i	$0.348554\pi$
$858$	−12.9249	−0.441249
$859$	−20.6038	−0.702991	−0.351496	−	0.936190i	$-0.614327\pi$
$859$	−20.6038	−0.702991	−0.351496	+	0.936190i	$0.614327\pi$
$860$	0	0
$861$	−5.84501	−0.199197
$862$	83.3679	2.83952
$863$	−13.7488	−0.468015	−0.234008	−	0.972235i	$-0.575184\pi$
$863$	−13.7488	−0.468015	−0.234008	+	0.972235i	$0.575184\pi$
$864$	3.92613	0.133570
$865$	0	0
$866$	23.7338	0.806506
$867$	29.0452	0.986427
$868$	−5.13346	−0.174241
$869$	9.00627	0.305517
$870$	0	0
$871$	−18.8353	−0.638211
$872$	47.7323	1.61642
$873$	10.4090	0.352293
$874$	38.0678	1.28766
$875$	0	0
$876$	12.6693	0.428055
$877$	−22.0430	−0.744339	−0.372170	−	0.928165i	$-0.621386\pi$
$877$	−22.0430	−0.744339	−0.372170	+	0.928165i	$0.621386\pi$
$878$	−16.7041	−0.563735
$879$	−16.6235	−0.560695
$880$	0	0
$881$	−9.03929	−0.304541	−0.152271	−	0.988339i	$-0.548659\pi$
$881$	−9.03929	−0.304541	−0.152271	+	0.988339i	$0.548659\pi$
$882$	−12.0446	−0.405561
$883$	1.60120	0.0538848	0.0269424	−	0.999637i	$-0.491423\pi$
$883$	1.60120	0.0538848	0.0269424	+	0.999637i	$0.491423\pi$
$884$	−44.8847	−1.50963
$885$	0	0
$886$	48.6249	1.63358
$887$	−17.4253	−0.585083	−0.292541	−	0.956253i	$-0.594501\pi$
$887$	−17.4253	−0.585083	−0.292541	+	0.956253i	$0.594501\pi$
$888$	18.8012	0.630929
$889$	36.8141	1.23471
$890$	0	0
$891$	2.93111	0.0981958
$892$	27.4897	0.920423
$893$	−22.0970	−0.739448
$894$	17.2320	0.576326
$895$	0	0
$896$	−70.8427	−2.36669
$897$	10.2982	0.343847
$898$	−45.8272	−1.52927
$899$	1.06860	0.0356397
$900$	0	0
$901$	25.1716	0.838588
$902$	−11.5671	−0.385141
$903$	38.0336	1.26568
$904$	35.9666	1.19623
$905$	0	0
$906$	39.3429	1.30708
$907$	20.8690	0.692942	0.346471	−	0.938061i	$-0.387380\pi$
$907$	20.8690	0.692942	0.346471	+	0.938061i	$0.387380\pi$
$908$	45.2506	1.50169
$909$	7.65744	0.253981
$910$	0	0
$911$	49.1748	1.62923	0.814617	−	0.580000i	$-0.196947\pi$
$911$	49.1748	1.62923	0.814617	+	0.580000i	$0.196947\pi$
$912$	−4.07827	−0.135045
$913$	−29.6489	−0.981234
$914$	−10.2072	−0.337623
$915$	0	0
$916$	−51.1445	−1.68986
$917$	31.4278	1.03784
$918$	15.9519	0.526492
$919$	−11.1959	−0.369318	−0.184659	−	0.982803i	$-0.559118\pi$
$919$	−11.1959	−0.369318	−0.184659	+	0.982803i	$0.559118\pi$
$920$	0	0
$921$	14.1643	0.466731
$922$	−27.7156	−0.912765
$923$	1.36667	0.0449846
$924$	−35.9897	−1.18397
$925$	0	0
$926$	−21.3425	−0.701357
$927$	2.98602	0.0980738
$928$	10.0349	0.329413
$929$	48.2290	1.58234	0.791171	−	0.611596i	$-0.209472\pi$
$929$	48.2290	1.58234	0.791171	+	0.611596i	$0.209472\pi$
$930$	0	0
$931$	−15.1119	−0.495271
$932$	−40.1906	−1.31649
$933$	15.0500	0.492714
$934$	−82.0866	−2.68596
$935$	0	0
$936$	−6.73076	−0.220002
$937$	−50.5397	−1.65106	−0.825529	−	0.564359i	$-0.809123\pi$
$937$	−50.5397	−1.65106	−0.825529	+	0.564359i	$0.809123\pi$
$938$	−82.1930	−2.68370
$939$	3.57476	0.116658
$940$	0	0
$941$	−38.1644	−1.24412	−0.622061	−	0.782969i	$-0.713705\pi$
$941$	−38.1644	−1.24412	−0.622061	+	0.782969i	$0.713705\pi$
$942$	18.5395	0.604049
$943$	9.21634	0.300125
$944$	−9.82896	−0.319906
$945$	0	0
$946$	75.2672	2.44715
$947$	−24.5190	−0.796759	−0.398380	−	0.917221i	$-0.630427\pi$
$947$	−24.5190	−0.796759	−0.398380	+	0.917221i	$0.630427\pi$
$948$	10.8354	0.351917
$949$	6.73897	0.218756
$950$	0	0
$951$	−12.7820	−0.414484
$952$	−84.7806	−2.74775
$953$	−13.0019	−0.421173	−0.210586	−	0.977575i	$-0.567537\pi$
$953$	−13.0019	−0.421173	−0.210586	+	0.977575i	$0.567537\pi$
$954$	8.72047	0.282336
$955$	0	0
$956$	24.3347	0.787041
$957$	7.49172	0.242173
$958$	20.3264	0.656717
$959$	−68.4698	−2.21101
$960$	0	0
$961$	−30.8252	−0.994361
$962$	23.1043	0.744912
$963$	7.07213	0.227896
$964$	−100.315	−3.23091
$965$	0	0
$966$	44.9390	1.44589
$967$	43.3212	1.39312	0.696559	−	0.717500i	$-0.254714\pi$
$967$	43.3212	1.39312	0.696559	+	0.717500i	$0.254714\pi$
$968$	8.64284	0.277791
$969$	20.0143	0.642952
$970$	0	0
$971$	−29.5324	−0.947741	−0.473870	−	0.880595i	$-0.657143\pi$
$971$	−29.5324	−0.947741	−0.473870	+	0.880595i	$0.657143\pi$
$972$	3.52640	0.113109
$973$	40.5709	1.30064
$974$	6.68721	0.214272
$975$	0	0
$976$	−8.90056	−0.284900
$977$	60.6153	1.93926	0.969628	−	0.244583i	$-0.0786512\pi$
$977$	60.6153	1.93926	0.969628	+	0.244583i	$0.0786512\pi$
$978$	−23.3671	−0.747196
$979$	0.842729	0.0269337
$980$	0	0
$981$	−13.3022	−0.424706
$982$	−86.5202	−2.76097
$983$	40.2796	1.28472	0.642359	−	0.766404i	$-0.277956\pi$
$983$	40.2796	1.28472	0.642359	+	0.766404i	$0.277956\pi$
$984$	−6.02366	−0.192027
$985$	0	0
$986$	40.7721	1.29845
$987$	−26.0855	−0.830311
$988$	−19.5099	−0.620692
$989$	−59.9709	−1.90696
$990$	0	0
$991$	38.1495	1.21186	0.605930	−	0.795518i	$-0.292801\pi$
$991$	38.1495	1.21186	0.605930	+	0.795518i	$0.292801\pi$
$992$	1.64145	0.0521161
$993$	4.70504	0.149310
$994$	5.96384	0.189162
$995$	0	0
$996$	−35.6704	−1.13026
$997$	−26.5777	−0.841724	−0.420862	−	0.907125i	$-0.638272\pi$
$997$	−26.5777	−0.841724	−0.420862	+	0.907125i	$0.638272\pi$
$998$	−13.7706	−0.435900
$999$	−5.23959	−0.165773

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.a.m.1.2		8
3.2	odd	2		5625.2.a.bd.1.7		8
5.2	odd	4		1875.2.b.h.1249.2		16
5.3	odd	4		1875.2.b.h.1249.15		16
5.4	even	2		1875.2.a.p.1.7		8
15.14	odd	2		5625.2.a.t.1.2		8
25.2	odd	20		75.2.i.a.4.1	✓	16
25.9	even	10		375.2.g.d.151.4		16
25.11	even	5		375.2.g.e.226.1		16
25.12	odd	20		375.2.i.c.349.4		16
25.13	odd	20		75.2.i.a.19.1	yes	16
25.14	even	10		375.2.g.d.226.4		16
25.16	even	5		375.2.g.e.151.1		16
25.23	odd	20		375.2.i.c.274.4		16
75.2	even	20		225.2.m.b.154.4		16
75.38	even	20		225.2.m.b.19.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.1	✓	16	25.2	odd	20
75.2.i.a.19.1	yes	16	25.13	odd	20
225.2.m.b.19.4		16	75.38	even	20
225.2.m.b.154.4		16	75.2	even	20
375.2.g.d.151.4		16	25.9	even	10
375.2.g.d.226.4		16	25.14	even	10
375.2.g.e.151.1		16	25.16	even	5
375.2.g.e.226.1		16	25.11	even	5
375.2.i.c.274.4		16	25.23	odd	20
375.2.i.c.349.4		16	25.12	odd	20
1875.2.a.m.1.2		8	1.1	even	1	trivial
1875.2.a.p.1.7		8	5.4	even	2
1875.2.b.h.1249.2		16	5.2	odd	4
1875.2.b.h.1249.15		16	5.3	odd	4
5625.2.a.t.1.2		8	15.14	odd	2
5625.2.a.bd.1.7		8	3.2	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35083 q^{2} +1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} -3.48189 q^{7} -3.58831 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.35083 q^{2} +1.00000 q^{3} +3.52640 q^{4} -2.35083 q^{6} -3.48189 q^{7} -3.58831 q^{8} +1.00000 q^{9} +2.93111 q^{11} +3.52640 q^{12} +1.87575 q^{13} +8.18532 q^{14} +1.38270 q^{16} -6.78566 q^{17} -2.35083 q^{18} -2.94950 q^{19} -3.48189 q^{21} -6.89053 q^{22} +5.49019 q^{23} -3.58831 q^{24} -4.40956 q^{26} +1.00000 q^{27} -12.2785 q^{28} +2.55593 q^{29} +0.418084 q^{31} +3.92613 q^{32} +2.93111 q^{33} +15.9519 q^{34} +3.52640 q^{36} -5.23959 q^{37} +6.93377 q^{38} +1.87575 q^{39} +1.67869 q^{41} +8.18532 q^{42} -10.9233 q^{43} +10.3363 q^{44} -12.9065 q^{46} +7.49178 q^{47} +1.38270 q^{48} +5.12353 q^{49} -6.78566 q^{51} +6.61463 q^{52} -3.70953 q^{53} -2.35083 q^{54} +12.4941 q^{56} -2.94950 q^{57} -6.00857 q^{58} -7.10854 q^{59} -6.43710 q^{61} -0.982844 q^{62} -3.48189 q^{63} -11.9951 q^{64} -6.89053 q^{66} -10.0415 q^{67} -23.9290 q^{68} +5.49019 q^{69} +0.728602 q^{71} -3.58831 q^{72} +3.59269 q^{73} +12.3174 q^{74} -10.4011 q^{76} -10.2058 q^{77} -4.40956 q^{78} +3.07265 q^{79} +1.00000 q^{81} -3.94632 q^{82} -10.1152 q^{83} -12.2785 q^{84} +25.6788 q^{86} +2.55593 q^{87} -10.5177 q^{88} +0.287512 q^{89} -6.53114 q^{91} +19.3606 q^{92} +0.418084 q^{93} -17.6119 q^{94} +3.92613 q^{96} +10.4090 q^{97} -12.0446 q^{98} +2.93111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9	\( 8 q - 4 q^{2} + 8 q^{3} + 4 q^{4} - 4 q^{6} - 8 q^{7} - 12 q^{8} + 8 q^{9} + 2 q^{11} + 4 q^{12} - 16 q^{13} + 6 q^{14} - 16 q^{17} - 4 q^{18} - 14 q^{19} - 8 q^{21} - 12 q^{22} - 14 q^{23} - 12 q^{24} + 6 q^{26} + 8 q^{27} - 16 q^{28} + 2 q^{29} - 22 q^{31} + 2 q^{32} + 2 q^{33} - 12 q^{34} + 4 q^{36} - 28 q^{37} + 16 q^{38} - 16 q^{39} + 8 q^{41} + 6 q^{42} - 20 q^{43} + 22 q^{44} - 2 q^{46} - 10 q^{47} - 16 q^{51} - 16 q^{52} - 44 q^{53} - 4 q^{54} + 30 q^{56} - 14 q^{57} - 8 q^{58} + 14 q^{59} - 20 q^{61} - 16 q^{62} - 8 q^{63} + 6 q^{64} - 12 q^{66} - 16 q^{67} + 2 q^{68} - 14 q^{69} + 16 q^{71} - 12 q^{72} - 24 q^{73} + 26 q^{74} - 16 q^{76} - 46 q^{77} + 6 q^{78} - 30 q^{79} + 8 q^{81} - 16 q^{82} - 12 q^{83} - 16 q^{84} + 32 q^{86} + 2 q^{87} - 32 q^{88} + 16 q^{89} - 12 q^{91} + 2 q^{92} - 22 q^{93} + 14 q^{94} + 2 q^{96} - 16 q^{97} - 4 q^{98} + 2 q^{99}+O(q^{100}) \) 8 * q - 4 * q^2 + 8 * q^3 + 4 * q^4 - 4 * q^6 - 8 * q^7 - 12 * q^8 + 8 * q^9 + 2 * q^11 + 4 * q^12 - 16 * q^13 + 6 * q^14 - 16 * q^17 - 4 * q^18 - 14 * q^19 - 8 * q^21 - 12 * q^22 - 14 * q^23 - 12 * q^24 + 6 * q^26 + 8 * q^27 - 16 * q^28 + 2 * q^29 - 22 * q^31 + 2 * q^32 + 2 * q^33 - 12 * q^34 + 4 * q^36 - 28 * q^37 + 16 * q^38 - 16 * q^39 + 8 * q^41 + 6 * q^42 - 20 * q^43 + 22 * q^44 - 2 * q^46 - 10 * q^47 - 16 * q^51 - 16 * q^52 - 44 * q^53 - 4 * q^54 + 30 * q^56 - 14 * q^57 - 8 * q^58 + 14 * q^59 - 20 * q^61 - 16 * q^62 - 8 * q^63 + 6 * q^64 - 12 * q^66 - 16 * q^67 + 2 * q^68 - 14 * q^69 + 16 * q^71 - 12 * q^72 - 24 * q^73 + 26 * q^74 - 16 * q^76 - 46 * q^77 + 6 * q^78 - 30 * q^79 + 8 * q^81 - 16 * q^82 - 12 * q^83 - 16 * q^84 + 32 * q^86 + 2 * q^87 - 32 * q^88 + 16 * q^89 - 12 * q^91 + 2 * q^92 - 22 * q^93 + 14 * q^94 + 2 * q^96 - 16 * q^97 - 4 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more