LMFDB - Embedded newform 1875.2.b.h.1249.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 $ x^16 + 20x^14 + 156x^12 + 610x^10 + 1286x^8 + 1440x^6 + 761x^4 + 130*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.2
Root			$1.35083i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35083i q^{2} -1.00000i q^{3} -3.52640 q^{4} -2.35083 q^{6} -3.48189i q^{7} +3.58831i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.35083i q^{2} -1.00000i q^{3} -3.52640 q^{4} -2.35083 q^{6} -3.48189i q^{7} +3.58831i q^{8} -1.00000 q^{9} +2.93111 q^{11} +3.52640i q^{12} -1.87575i q^{13} -8.18532 q^{14} +1.38270 q^{16} -6.78566i q^{17} +2.35083i q^{18} +2.94950 q^{19} -3.48189 q^{21} -6.89053i q^{22} -5.49019i q^{23} +3.58831 q^{24} -4.40956 q^{26} +1.00000i q^{27} +12.2785i q^{28} -2.55593 q^{29} +0.418084 q^{31} +3.92613i q^{32} -2.93111i q^{33} -15.9519 q^{34} +3.52640 q^{36} -5.23959i q^{37} -6.93377i q^{38} -1.87575 q^{39} +1.67869 q^{41} +8.18532i q^{42} +10.9233i q^{43} -10.3363 q^{44} -12.9065 q^{46} +7.49178i q^{47} -1.38270i q^{48} -5.12353 q^{49} -6.78566 q^{51} +6.61463i q^{52} +3.70953i q^{53} +2.35083 q^{54} +12.4941 q^{56} -2.94950i q^{57} +6.00857i q^{58} +7.10854 q^{59} -6.43710 q^{61} -0.982844i q^{62} +3.48189i q^{63} +11.9951 q^{64} -6.89053 q^{66} -10.0415i q^{67} +23.9290i q^{68} -5.49019 q^{69} +0.728602 q^{71} -3.58831i q^{72} -3.59269i q^{73} -12.3174 q^{74} -10.4011 q^{76} -10.2058i q^{77} +4.40956i q^{78} -3.07265 q^{79} +1.00000 q^{81} -3.94632i q^{82} +10.1152i q^{83} +12.2785 q^{84} +25.6788 q^{86} +2.55593i q^{87} +10.5177i q^{88} -0.287512 q^{89} -6.53114 q^{91} +19.3606i q^{92} -0.418084i q^{93} +17.6119 q^{94} +3.92613 q^{96} +10.4090i q^{97} +12.0446i q^{98} -2.93111 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.35083i	− 1.66229i	−0.556058	−	0.831144i	$-0.687687\pi$
$2$	− 2.35083i	− 1.66229i	0.556058	−	0.831144i	$-0.312313\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.52640	−1.76320
$5$	0	0
$5$	0	0
$6$	−2.35083	−0.959722
$7$	− 3.48189i	− 1.31603i	−0.753005	−	0.658015i	$-0.771396\pi$
$7$	− 3.48189i	− 1.31603i	0.753005	−	0.658015i	$-0.228604\pi$
$8$	3.58831i	1.26866i
$9$	−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	$0.354311\pi$
$12$	3.52640i	1.01798i
$13$	− 1.87575i	− 0.520239i	−0.965576	−	0.260119i	$-0.916238\pi$
$13$	− 1.87575i	− 0.520239i	0.965576	−	0.260119i	$-0.0837619\pi$
$14$	−8.18532	−2.18762
$15$	0	0
$16$	1.38270	0.345674
$17$	− 6.78566i	− 1.64576i	−0.568212	−	0.822882i	$-0.692365\pi$
$17$	− 6.78566i	− 1.64576i	0.568212	−	0.822882i	$-0.307635\pi$
$18$	2.35083i	0.554096i
$19$	2.94950	0.676662	0.338331	−	0.941027i	$-0.390138\pi$
$19$	2.94950	0.676662	0.338331	+	0.941027i	$0.390138\pi$
$20$	0	0
$21$	−3.48189	−0.759810
$22$	− 6.89053i	− 1.46907i
$23$	− 5.49019i	− 1.14478i	−0.819980	−	0.572392i	$-0.806015\pi$
$23$	− 5.49019i	− 1.14478i	0.819980	−	0.572392i	$-0.193985\pi$
$24$	3.58831	0.732460
$25$	0	0
$26$	−4.40956	−0.864786
$27$	1.00000i	0.192450i
$28$	12.2785i	2.32042i
$29$	−2.55593	−0.474625	−0.237313	−	0.971433i	$-0.576267\pi$
$29$	−2.55593	−0.474625	−0.237313	+	0.971433i	$0.576267\pi$
$30$	0	0
$31$	0.418084	0.0750901	0.0375451	−	0.999295i	$-0.488046\pi$
$31$	0.418084	0.0750901	0.0375451	+	0.999295i	$0.488046\pi$
$32$	3.92613i	0.694048i
$33$	− 2.93111i	− 0.510240i
$34$	−15.9519	−2.73573
$35$	0	0
$36$	3.52640	0.587733
$37$	− 5.23959i	− 0.861383i	−0.902499	−	0.430691i	$-0.858270\pi$
$37$	− 5.23959i	− 0.861383i	0.902499	−	0.430691i	$-0.141730\pi$
$38$	− 6.93377i	− 1.12481i
$39$	−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	$0.458154\pi$
$42$	8.18532i	1.26302i
$43$	10.9233i	1.66578i	0.553436	+	0.832892i	$0.313316\pi$
$43$	10.9233i	1.66578i	−0.553436	+	0.832892i	$0.686684\pi$
$44$	−10.3363	−1.55825
$45$	0	0
$46$	−12.9065	−1.90296
$47$	7.49178i	1.09279i	0.837528	+	0.546394i	$0.184000\pi$
$47$	7.49178i	1.09279i	−0.837528	+	0.546394i	$0.816000\pi$
$48$	− 1.38270i	− 0.199575i
$49$	−5.12353	−0.731933
$50$	0	0
$51$	−6.78566	−0.950183
$52$	6.61463i	0.917285i
$53$	3.70953i	0.509543i	0.967001	+	0.254771i	$0.0820002\pi$
$53$	3.70953i	0.509543i	−0.967001	+	0.254771i	$0.918000\pi$
$54$	2.35083	0.319907
$55$	0	0
$56$	12.4941	1.66959
$57$	− 2.94950i	− 0.390671i
$58$	6.00857i	0.788964i
$59$	7.10854	0.925453	0.462727	−	0.886501i	$-0.346871\pi$
$59$	7.10854	0.925453	0.462727	+	0.886501i	$0.346871\pi$
$60$	0	0
$61$	−6.43710	−0.824186	−0.412093	−	0.911142i	$-0.635202\pi$
$61$	−6.43710	−0.824186	−0.412093	+	0.911142i	$0.635202\pi$
$62$	− 0.982844i	− 0.124821i
$63$	3.48189i	0.438676i
$64$	11.9951	1.49938
$65$	0	0
$66$	−6.89053	−0.848166
$67$	− 10.0415i	− 1.22677i	−0.789786	−	0.613383i	$-0.789808\pi$
$67$	− 10.0415i	− 1.22677i	0.789786	−	0.613383i	$-0.210192\pi$
$68$	23.9290i	2.90181i
$69$	−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	$0.486234\pi$
$72$	− 3.58831i	− 0.422886i
$73$	− 3.59269i	− 0.420492i	−0.977648	−	0.210246i	$-0.932573\pi$
$73$	− 3.59269i	− 0.420492i	0.977648	−	0.210246i	$-0.0674266\pi$
$74$	−12.3174	−1.43187
$75$	0	0
$76$	−10.4011	−1.19309
$77$	− 10.2058i	− 1.16306i
$78$	4.40956i	0.499285i
$79$	−3.07265	−0.345700	−0.172850	−	0.984948i	$-0.555298\pi$
$79$	−3.07265	−0.345700	−0.172850	+	0.984948i	$0.555298\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.94632i	− 0.435798i
$83$	10.1152i	1.11029i	0.831753	+	0.555146i	$0.187338\pi$
$83$	10.1152i	1.11029i	−0.831753	+	0.555146i	$0.812662\pi$
$84$	12.2785	1.33970
$85$	0	0
$86$	25.6788	2.76901
$87$	2.55593i	0.274025i
$88$	10.5177i	1.12119i
$89$	−0.287512	−0.0304762	−0.0152381	−	0.999884i	$-0.504851\pi$
$89$	−0.287512	−0.0304762	−0.0152381	+	0.999884i	$0.504851\pi$
$90$	0	0
$91$	−6.53114	−0.684649
$92$	19.3606i	2.01848i
$93$	− 0.418084i	− 0.0433533i
$94$	17.6119	1.81653
$95$	0	0
$96$	3.92613	0.400709
$97$	10.4090i	1.05688i	0.848971	+	0.528439i	$0.177222\pi$
$97$	10.4090i	1.05688i	−0.848971	+	0.528439i	$0.822778\pi$
$98$	12.0446i	1.21668i
$99$	−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	$0.375590\pi$
$102$	15.9519i	1.57948i
$103$	− 2.98602i	− 0.294221i	−0.989120	−	0.147111i	$-0.953003\pi$
$103$	− 2.98602i	− 0.294221i	0.989120	−	0.147111i	$-0.0469974\pi$
$104$	6.73076	0.660005
$105$	0	0
$106$	8.72047	0.847007
$107$	7.07213i	0.683689i	0.939757	+	0.341844i	$0.111052\pi$
$107$	7.07213i	0.683689i	−0.939757	+	0.341844i	$0.888948\pi$
$108$	− 3.52640i	− 0.339328i
$109$	13.3022	1.27412	0.637058	−	0.770816i	$-0.280151\pi$
$109$	13.3022	1.27412	0.637058	+	0.770816i	$0.280151\pi$
$110$	0	0
$111$	−5.23959	−0.497320
$112$	− 4.81440i	− 0.454918i
$113$	10.0233i	0.942911i	0.881890	+	0.471456i	$0.156271\pi$
$113$	10.0233i	0.942911i	−0.881890	+	0.471456i	$0.843729\pi$
$114$	−6.93377	−0.649407
$115$	0	0
$116$	9.01325	0.836859
$117$	1.87575i	0.173413i
$118$	− 16.7110i	− 1.53837i
$119$	−23.6269	−2.16587
$120$	0	0
$121$	−2.40861	−0.218965
$122$	15.1325i	1.37003i
$123$	− 1.67869i	− 0.151362i
$124$	−1.47433	−0.132399
$125$	0	0
$126$	8.18532	0.729206
$127$	− 10.5730i	− 0.938205i	−0.883144	−	0.469103i	$-0.844577\pi$
$127$	− 10.5730i	− 0.938205i	0.883144	−	0.469103i	$-0.155423\pi$
$128$	− 20.3461i	− 1.79836i
$129$	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	$0.629015\pi$
$132$	10.3363i	0.899656i
$133$	− 10.2698i	− 0.890507i
$134$	−23.6059	−2.03924
$135$	0	0
$136$	24.3490	2.08791
$137$	19.6646i	1.68006i	0.542541	+	0.840029i	$0.317462\pi$
$137$	19.6646i	1.68006i	−0.542541	+	0.840029i	$0.682538\pi$
$138$	12.9065i	1.09868i
$139$	11.6520	0.988310	0.494155	−	0.869374i	$-0.335478\pi$
$139$	11.6520	0.988310	0.494155	+	0.869374i	$0.335478\pi$
$140$	0	0
$141$	7.49178	0.630922
$142$	− 1.71282i	− 0.143737i
$143$	− 5.49801i	− 0.459767i
$144$	−1.38270	−0.115225
$145$	0	0
$146$	−8.44580	−0.698979
$147$	5.12353i	0.422582i
$148$	18.4769i	1.51879i
$149$	7.33020	0.600513	0.300257	−	0.953858i	$-0.402928\pi$
$149$	7.33020	0.600513	0.300257	+	0.953858i	$0.402928\pi$
$150$	0	0
$151$	−16.7358	−1.36194	−0.680968	−	0.732313i	$-0.738441\pi$
$151$	−16.7358	−1.36194	−0.680968	+	0.732313i	$0.738441\pi$
$152$	10.5837i	0.858453i
$153$	6.78566i	0.548588i
$154$	−23.9921	−1.93333
$155$	0	0
$156$	6.61463	0.529595
$157$	− 7.88635i	− 0.629399i	−0.949191	−	0.314700i	$-0.898096\pi$
$157$	− 7.88635i	− 0.629399i	0.949191	−	0.314700i	$-0.101904\pi$
$158$	7.22328i	0.574653i
$159$	3.70953	0.294185
$160$	0	0
$161$	−19.1162	−1.50657
$162$	− 2.35083i	− 0.184699i
$163$	− 9.93992i	− 0.778555i	−0.921121	−	0.389277i	$-0.872725\pi$
$163$	− 9.93992i	− 0.778555i	0.921121	−	0.389277i	$-0.127275\pi$
$164$	−5.91973	−0.462254
$165$	0	0
$166$	23.7792	1.84563
$167$	− 5.57767i	− 0.431613i	−0.976436	−	0.215806i	$-0.930762\pi$
$167$	− 5.57767i	− 0.431613i	0.976436	−	0.215806i	$-0.0692380\pi$
$168$	− 12.4941i	− 0.963939i
$169$	9.48157	0.729352
$170$	0	0
$171$	−2.94950	−0.225554
$172$	− 38.5198i	− 2.93711i
$173$	16.5682i	1.25966i	0.776734	+	0.629828i	$0.216875\pi$
$173$	16.5682i	1.25966i	−0.776734	+	0.629828i	$0.783125\pi$
$174$	6.00857	0.455508
$175$	0	0
$176$	4.05284	0.305494
$177$	− 7.10854i	− 0.534311i
$178$	0.675892i	0.0506602i
$179$	5.36021	0.400641	0.200321	−	0.979730i	$-0.435802\pi$
$179$	5.36021	0.400641	0.200321	+	0.979730i	$0.435802\pi$
$180$	0	0
$181$	−0.327745	−0.0243611	−0.0121805	−	0.999926i	$-0.503877\pi$
$181$	−0.327745	−0.0243611	−0.0121805	+	0.999926i	$0.503877\pi$
$182$	15.3536i	1.13808i
$183$	6.43710i	0.475844i
$184$	19.7005	1.45234
$185$	0	0
$186$	−0.982844	−0.0720656
$187$	− 19.8895i	− 1.45446i
$188$	− 26.4190i	− 1.92680i
$189$	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	$0.540065\pi$
$192$	− 11.9951i	− 0.865668i
$193$	− 24.3134i	− 1.75012i	−0.484018	−	0.875058i	$-0.660823\pi$
$193$	− 24.3134i	− 1.75012i	0.484018	−	0.875058i	$-0.339177\pi$
$194$	24.4699	1.75684
$195$	0	0
$196$	18.0676	1.29055
$197$	− 14.9561i	− 1.06558i	−0.846248	−	0.532789i	$-0.821144\pi$
$197$	− 14.9561i	− 1.06558i	0.846248	−	0.532789i	$-0.178856\pi$
$198$	6.89053i	0.489689i
$199$	11.3251	0.802817	0.401408	−	0.915899i	$-0.368521\pi$
$199$	11.3251	0.802817	0.401408	+	0.915899i	$0.368521\pi$
$200$	0	0
$201$	−10.0415	−0.708274
$202$	− 18.0013i	− 1.26657i
$203$	8.89948i	0.624621i
$204$	23.9290	1.67536
$205$	0	0
$206$	−7.01963	−0.489081
$207$	5.49019i	0.381595i
$208$	− 2.59359i	− 0.179833i
$209$	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	$0.416398\pi$
$212$	− 13.0813i	− 0.898426i
$213$	− 0.728602i	− 0.0499230i
$214$	16.6254	1.13649
$215$	0	0
$216$	−3.58831	−0.244153
$217$	− 1.45572i	− 0.0988208i
$218$	− 31.2711i	− 2.11795i
$219$	−3.59269	−0.242771
$220$	0	0
$221$	−12.7282	−0.856190
$222$	12.3174i	0.826688i
$223$	− 7.79539i	− 0.522018i	−0.965336	−	0.261009i	$-0.915945\pi$
$223$	− 7.79539i	− 0.522018i	0.965336	−	0.261009i	$-0.0840553\pi$
$224$	13.6703	0.913387
$225$	0	0
$226$	23.5630	1.56739
$227$	12.8319i	0.851686i	0.904797	+	0.425843i	$0.140022\pi$
$227$	12.8319i	0.851686i	−0.904797	+	0.425843i	$0.859978\pi$
$228$	10.4011i	0.688831i
$229$	14.5033	0.958405	0.479203	−	0.877704i	$-0.340926\pi$
$229$	14.5033	0.958405	0.479203	+	0.877704i	$0.340926\pi$
$230$	0	0
$231$	−10.2058	−0.671491
$232$	− 9.17148i	− 0.602137i
$233$	11.3970i	0.746646i	0.927702	+	0.373323i	$0.121782\pi$
$233$	11.3970i	0.746646i	−0.927702	+	0.373323i	$0.878218\pi$
$234$	4.40956	0.288262
$235$	0	0
$236$	−25.0676	−1.63176
$237$	3.07265i	0.199590i
$238$	55.5428i	3.60031i
$239$	−6.90072	−0.446371	−0.223185	−	0.974776i	$-0.571646\pi$
$239$	−6.90072	−0.446371	−0.223185	+	0.974776i	$0.571646\pi$
$240$	0	0
$241$	−28.4467	−1.83242	−0.916208	−	0.400704i	$-0.868766\pi$
$241$	−28.4467	−1.83242	−0.916208	+	0.400704i	$0.868766\pi$
$242$	5.66224i	0.363982i
$243$	− 1.00000i	− 0.0641500i
$244$	22.6998	1.45320
$245$	0	0
$246$	−3.94632	−0.251608
$247$	− 5.53252i	− 0.352026i
$248$	1.50021i	0.0952637i
$249$	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	$0.627179\pi$
$252$	− 12.2785i	− 0.773474i
$253$	− 16.0923i	− 1.01172i
$254$	−24.8554	−1.55957
$255$	0	0
$256$	−23.8400	−1.49000
$257$	− 20.8274i	− 1.29918i	−0.760285	−	0.649590i	$-0.774941\pi$
$257$	− 20.8274i	− 1.29918i	0.760285	−	0.649590i	$-0.225059\pi$
$258$	− 25.6788i	− 1.59869i
$259$	−18.2437	−1.13361
$260$	0	0
$261$	2.55593	0.158208
$262$	21.2188i	1.31090i
$263$	2.23517i	0.137826i	0.997623	+	0.0689131i	$0.0219531\pi$
$263$	2.23517i	0.137826i	−0.997623	+	0.0689131i	$0.978047\pi$
$264$	10.5177	0.647320
$265$	0	0
$266$	−24.1426	−1.48028
$267$	0.287512i	0.0175955i
$268$	35.4104i	2.16303i
$269$	19.3036	1.17696	0.588482	−	0.808511i	$-0.299726\pi$
$269$	19.3036	1.17696	0.588482	+	0.808511i	$0.299726\pi$
$270$	0	0
$271$	−11.9411	−0.725372	−0.362686	−	0.931911i	$-0.618140\pi$
$271$	−11.9411	−0.725372	−0.362686	+	0.931911i	$0.618140\pi$
$272$	− 9.38252i	− 0.568899i
$273$	6.53114i	0.395282i
$274$	46.2281	2.79274
$275$	0	0
$276$	19.3606	1.16537
$277$	− 26.1466i	− 1.57100i	−0.618863	−	0.785499i	$-0.712406\pi$
$277$	− 26.1466i	− 1.57100i	0.618863	−	0.785499i	$-0.287594\pi$
$278$	− 27.3919i	− 1.64285i
$279$	−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	$0.402376\pi$
$282$	− 17.6119i	− 1.04877i
$283$	− 31.8638i	− 1.89410i	−0.321081	−	0.947052i	$-0.604046\pi$
$283$	− 31.8638i	− 1.89410i	0.321081	−	0.947052i	$-0.395954\pi$
$284$	−2.56934	−0.152462
$285$	0	0
$286$	−12.9249	−0.764265
$287$	− 5.84501i	− 0.345020i
$288$	− 3.92613i	− 0.231349i
$289$	−29.0452	−1.70854
$290$	0	0
$291$	10.4090	0.610189
$292$	12.6693i	0.741412i
$293$	16.6235i	0.971153i	0.874194	+	0.485576i	$0.161390\pi$
$293$	16.6235i	0.971153i	−0.874194	+	0.485576i	$0.838610\pi$
$294$	12.0446	0.702453
$295$	0	0
$296$	18.8012	1.09280
$297$	2.93111i	0.170080i
$298$	− 17.2320i	− 0.998225i
$299$	−10.2982	−0.595561
$300$	0	0
$301$	38.0336	2.19222
$302$	39.3429i	2.26393i
$303$	− 7.65744i	− 0.439908i
$304$	4.07827	0.233905
$305$	0	0
$306$	15.9519	0.911911
$307$	14.1643i	0.808402i	0.914670	+	0.404201i	$0.132450\pi$
$307$	14.1643i	0.808402i	−0.914670	+	0.404201i	$0.867550\pi$
$308$	35.9897i	2.05070i
$309$	−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	$0.359675\pi$
$312$	− 6.73076i	− 0.381054i
$313$	− 3.57476i	− 0.202057i	−0.994884	−	0.101029i	$-0.967787\pi$
$313$	− 3.57476i	− 0.202057i	0.994884	−	0.101029i	$-0.0322134\pi$
$314$	−18.5395	−1.04624
$315$	0	0
$316$	10.8354	0.609539
$317$	− 12.7820i	− 0.717907i	−0.933355	−	0.358954i	$-0.883134\pi$
$317$	− 12.7820i	− 0.717907i	0.933355	−	0.358954i	$-0.116866\pi$
$318$	− 8.72047i	− 0.489020i
$319$	−7.49172	−0.419456
$320$	0	0
$321$	7.07213	0.394728
$322$	44.9390i	2.50435i
$323$	− 20.0143i	− 1.11363i
$324$	−3.52640	−0.195911
$325$	0	0
$326$	−23.3671	−1.29418
$327$	− 13.3022i	− 0.735612i
$328$	6.02366i	0.332601i
$329$	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	$0.458725\pi$
$332$	− 35.6704i	− 1.95767i
$333$	5.23959i	0.287128i
$334$	−13.1121	−0.717465
$335$	0	0
$336$	−4.81440	−0.262647
$337$	17.4048i	0.948098i	0.880499	+	0.474049i	$0.157208\pi$
$337$	17.4048i	0.948098i	−0.880499	+	0.474049i	$0.842792\pi$
$338$	− 22.2896i	− 1.21239i
$339$	10.0233	0.544390
$340$	0	0
$341$	1.22545	0.0663618
$342$	6.93377i	0.374936i
$343$	− 6.53364i	− 0.352783i
$344$	−39.1961	−2.11331
$345$	0	0
$346$	38.9490	2.09391
$347$	− 14.0106i	− 0.752130i	−0.926593	−	0.376065i	$-0.877277\pi$
$347$	− 14.0106i	− 0.752130i	0.926593	−	0.376065i	$-0.122723\pi$
$348$	− 9.01325i	− 0.483161i
$349$	−35.9459	−1.92414	−0.962069	−	0.272806i	$-0.912048\pi$
$349$	−35.9459	−1.92414	−0.962069	+	0.272806i	$0.912048\pi$
$350$	0	0
$351$	1.87575	0.100120
$352$	11.5079i	0.613373i
$353$	8.80198i	0.468482i	0.972179	+	0.234241i	$0.0752605\pi$
$353$	8.80198i	0.468482i	−0.972179	+	0.234241i	$0.924740\pi$
$354$	−16.7110	−0.888178
$355$	0	0
$356$	1.01388	0.0537357
$357$	23.6269i	1.25047i
$358$	− 12.6009i	− 0.665981i
$359$	−6.02962	−0.318231	−0.159116	−	0.987260i	$-0.550864\pi$
$359$	−6.02962	−0.318231	−0.159116	+	0.987260i	$0.550864\pi$
$360$	0	0
$361$	−10.3004	−0.542129
$362$	0.770472i	0.0404951i
$363$	2.40861i	0.126419i
$364$	23.0314	1.20717
$365$	0	0
$366$	15.1325	0.790989
$367$	− 28.7662i	− 1.50158i	−0.660539	−	0.750792i	$-0.729672\pi$
$367$	− 28.7662i	− 1.50158i	0.660539	−	0.750792i	$-0.270328\pi$
$368$	− 7.59128i	− 0.395723i
$369$	−1.67869	−0.0873891
$370$	0	0
$371$	12.9162	0.670574
$372$	1.47433i	0.0764405i
$373$	33.6000i	1.73974i	0.493278	+	0.869872i	$0.335799\pi$
$373$	33.6000i	1.73974i	−0.493278	+	0.869872i	$0.664201\pi$
$374$	−46.7568	−2.41774
$375$	0	0
$376$	−26.8828	−1.38637
$377$	4.79429i	0.246918i
$378$	− 8.18532i	− 0.421008i
$379$	−4.63403	−0.238034	−0.119017	−	0.992892i	$-0.537974\pi$
$379$	−4.63403	−0.238034	−0.119017	+	0.992892i	$0.537974\pi$
$380$	0	0
$381$	−10.5730	−0.541673
$382$	8.15718i	0.417358i
$383$	− 12.9058i	− 0.659453i	−0.944076	−	0.329727i	$-0.893043\pi$
$383$	− 12.9058i	− 0.659453i	0.944076	−	0.329727i	$-0.106957\pi$
$384$	−20.3461	−1.03828
$385$	0	0
$386$	−57.1567	−2.90920
$387$	− 10.9233i	− 0.555261i
$388$	− 36.7064i	− 1.86349i
$389$	17.5246	0.888532	0.444266	−	0.895895i	$-0.353465\pi$
$389$	17.5246	0.888532	0.444266	+	0.895895i	$0.353465\pi$
$390$	0	0
$391$	−37.2546	−1.88405
$392$	− 18.3848i	− 0.928573i
$393$	9.02608i	0.455305i
$394$	−35.1593	−1.77130
$395$	0	0
$396$	10.3363	0.519416
$397$	27.0176i	1.35597i	0.735074	+	0.677987i	$0.237147\pi$
$397$	27.0176i	1.35597i	−0.735074	+	0.677987i	$0.762853\pi$
$398$	− 26.6234i	− 1.33451i
$399$	−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	$0.630637\pi$
$402$	23.6059i	1.17735i
$403$	− 0.784220i	− 0.0390648i
$404$	−27.0032	−1.34346
$405$	0	0
$406$	20.9212	1.03830
$407$	− 15.3578i	− 0.761258i
$408$	− 24.3490i	− 1.20546i
$409$	30.9962	1.53266	0.766331	−	0.642446i	$-0.222080\pi$
$409$	30.9962	1.53266	0.766331	+	0.642446i	$0.222080\pi$
$410$	0	0
$411$	19.6646	0.969982
$412$	10.5299i	0.518771i
$413$	− 24.7511i	− 1.21792i
$414$	12.9065	0.634321
$415$	0	0
$416$	7.36442	0.361070
$417$	− 11.6520i	− 0.570601i
$418$	− 20.3236i	− 0.994061i
$419$	−25.9153	−1.26605	−0.633024	−	0.774132i	$-0.718186\pi$
$419$	−25.9153	−1.26605	−0.633024	+	0.774132i	$0.718186\pi$
$420$	0	0
$421$	3.92646	0.191364	0.0956821	−	0.995412i	$-0.469497\pi$
$421$	3.92646	0.191364	0.0956821	+	0.995412i	$0.469497\pi$
$422$	− 17.7319i	− 0.863176i
$423$	− 7.49178i	− 0.364263i
$424$	−13.3109	−0.646436
$425$	0	0
$426$	−1.71282	−0.0829863
$427$	22.4132i	1.08465i
$428$	− 24.9392i	− 1.20548i
$429$	−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	$0.825892\pi$
$432$	1.38270i	0.0665251i
$433$	10.0959i	0.485178i	0.970129	+	0.242589i	$0.0779967\pi$
$433$	10.0959i	0.485178i	−0.970129	+	0.242589i	$0.922003\pi$
$434$	−3.42215	−0.164269
$435$	0	0
$436$	−46.9088	−2.24652
$437$	− 16.1933i	− 0.774632i
$438$	8.44580i	0.403556i
$439$	−7.10560	−0.339132	−0.169566	−	0.985519i	$-0.554237\pi$
$439$	−7.10560	−0.339132	−0.169566	+	0.985519i	$0.554237\pi$
$440$	0	0
$441$	5.12353	0.243978
$442$	29.9218i	1.42323i
$443$	20.6841i	0.982733i	0.870953	+	0.491366i	$0.163502\pi$
$443$	20.6841i	0.982733i	−0.870953	+	0.491366i	$0.836498\pi$
$444$	18.4769	0.876874
$445$	0	0
$446$	−18.3256	−0.867744
$447$	− 7.33020i	− 0.346706i
$448$	− 41.7654i	− 1.97323i
$449$	−19.4940	−0.919980	−0.459990	−	0.887924i	$-0.652147\pi$
$449$	−19.4940	−0.919980	−0.459990	+	0.887924i	$0.652147\pi$
$450$	0	0
$451$	4.92042	0.231694
$452$	− 35.3461i	− 1.66254i
$453$	16.7358i	0.786314i
$454$	30.1657	1.41575
$455$	0	0
$456$	10.5837	0.495628
$457$	4.34194i	0.203107i	0.994830	+	0.101554i	$0.0323813\pi$
$457$	4.34194i	0.203107i	−0.994830	+	0.101554i	$0.967619\pi$
$458$	− 34.0948i	− 1.59315i
$459$	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	$0.411471\pi$
$462$	23.9921i	1.11621i
$463$	− 9.07870i	− 0.421923i	−0.977494	−	0.210962i	$-0.932341\pi$
$463$	− 9.07870i	− 0.421923i	0.977494	−	0.210962i	$-0.0676595\pi$
$464$	−3.53409	−0.164066
$465$	0	0
$466$	26.7925	1.24114
$467$	34.9181i	1.61582i	0.589307	+	0.807910i	$0.299401\pi$
$467$	34.9181i	1.61582i	−0.589307	+	0.807910i	$0.700599\pi$
$468$	− 6.61463i	− 0.305762i
$469$	−34.9634	−1.61446
$470$	0	0
$471$	−7.88635	−0.363384
$472$	25.5076i	1.17408i
$473$	32.0173i	1.47216i
$474$	7.22328	0.331776
$475$	0	0
$476$	83.3179	3.81887
$477$	− 3.70953i	− 0.169848i
$478$	16.2224i	0.741996i
$479$	8.64649	0.395068	0.197534	−	0.980296i	$-0.436707\pi$
$479$	8.64649	0.395068	0.197534	+	0.980296i	$0.436707\pi$
$480$	0	0
$481$	−9.82814	−0.448125
$482$	66.8734i	3.04600i
$483$	19.1162i	0.869819i
$484$	8.49373	0.386079
$485$	0	0
$486$	−2.35083	−0.106636
$487$	− 2.84462i	− 0.128902i	−0.997921	−	0.0644510i	$-0.979470\pi$
$487$	− 2.84462i	− 0.128902i	0.997921	−	0.0644510i	$-0.0205296\pi$
$488$	− 23.0983i	− 1.04561i
$489$	−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	$0.188070\pi$
$492$	5.91973i	0.266882i
$493$	17.3437i	0.781121i
$494$	−13.0060	−0.585168
$495$	0	0
$496$	0.578084	0.0259567
$497$	− 2.53691i	− 0.113796i
$498$	− 23.7792i	− 1.06557i
$499$	−5.85775	−0.262229	−0.131114	−	0.991367i	$-0.541856\pi$
$499$	−5.85775	−0.262229	−0.131114	+	0.991367i	$0.541856\pi$
$500$	0	0
$501$	−5.57767	−0.249192
$502$	28.9759i	1.29326i
$503$	− 30.2874i	− 1.35045i	−0.737613	−	0.675223i	$-0.764047\pi$
$503$	− 30.2874i	− 1.35045i	0.737613	−	0.675223i	$-0.235953\pi$
$504$	−12.4941	−0.556530
$505$	0	0
$506$	−37.8304	−1.68177
$507$	− 9.48157i	− 0.421091i
$508$	37.2848i	1.65424i
$509$	−15.1430	−0.671201	−0.335601	−	0.942004i	$-0.608939\pi$
$509$	−15.1430	−0.671201	−0.335601	+	0.942004i	$0.608939\pi$
$510$	0	0
$511$	−12.5093	−0.553380
$512$	15.3517i	0.678457i
$513$	2.94950i	0.130224i
$514$	−48.9618	−2.15961
$515$	0	0
$516$	−38.5198	−1.69574
$517$	21.9592i	0.965765i
$518$	42.8877i	1.88438i
$519$	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	$0.172860\pi$
$522$	− 6.00857i	− 0.262988i
$523$	− 38.7976i	− 1.69650i	−0.529594	−	0.848251i	$-0.677656\pi$
$523$	− 38.7976i	− 1.69650i	0.529594	−	0.848251i	$-0.322344\pi$
$524$	31.8296	1.39048
$525$	0	0
$526$	5.25449	0.229107
$527$	− 2.83698i	− 0.123581i
$528$	− 4.05284i	− 0.176377i
$529$	−7.14224	−0.310532
$530$	0	0
$531$	−7.10854	−0.308484
$532$	36.2155i	1.57014i
$533$	− 3.14880i	− 0.136390i
$534$	0.675892	0.0292487
$535$	0	0
$536$	36.0320	1.55635
$537$	− 5.36021i	− 0.231310i
$538$	− 45.3796i	− 1.95645i
$539$	−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	$0.401058\pi$
$542$	28.0716i	1.20578i
$543$	0.327745i	0.0140649i
$544$	26.6414	1.14224
$545$	0	0
$546$	15.3536	0.657073
$547$	− 39.3229i	− 1.68133i	−0.541559	−	0.840663i	$-0.682166\pi$
$547$	− 39.3229i	− 1.68133i	0.541559	−	0.840663i	$-0.317834\pi$
$548$	− 69.3452i	− 2.96228i
$549$	6.43710	0.274729
$550$	0	0
$551$	−7.53873	−0.321161
$552$	− 19.7005i	− 0.838509i
$553$	10.6986i	0.454952i
$554$	−61.4662	−2.61145
$555$	0	0
$556$	−41.0896	−1.74259
$557$	− 35.3849i	− 1.49931i	−0.661830	−	0.749654i	$-0.730220\pi$
$557$	− 35.3849i	− 1.49931i	0.661830	−	0.749654i	$-0.269780\pi$
$558$	0.982844i	0.0416071i
$559$	20.4893	0.866605
$560$	0	0
$561$	−19.8895	−0.839735
$562$	− 23.7948i	− 1.00372i
$563$	15.0386i	0.633800i	0.948459	+	0.316900i	$0.102642\pi$
$563$	15.0386i	0.633800i	−0.948459	+	0.316900i	$0.897358\pi$
$564$	−26.4190	−1.11244
$565$	0	0
$566$	−74.9063	−3.14855
$567$	− 3.48189i	− 0.146225i
$568$	2.61445i	0.109700i
$569$	2.72409	0.114200	0.0570998	−	0.998368i	$-0.481815\pi$
$569$	2.72409	0.114200	0.0570998	+	0.998368i	$0.481815\pi$
$570$	0	0
$571$	12.6236	0.528282	0.264141	−	0.964484i	$-0.414912\pi$
$571$	12.6236	0.528282	0.264141	+	0.964484i	$0.414912\pi$
$572$	19.3882i	0.810661i
$573$	3.46992i	0.144958i
$574$	−13.7406	−0.573522
$575$	0	0
$576$	−11.9951	−0.499794
$577$	23.5844i	0.981831i	0.871207	+	0.490915i	$0.163338\pi$
$577$	23.5844i	0.981831i	−0.871207	+	0.490915i	$0.836662\pi$
$578$	68.2803i	2.84009i
$579$	−24.3134	−1.01043
$580$	0	0
$581$	35.2201	1.46118
$582$	− 24.4699i	− 1.01431i
$583$	10.8730i	0.450315i
$584$	12.8917	0.533461
$585$	0	0
$586$	39.0789	1.61434
$587$	22.6772i	0.935988i	0.883732	+	0.467994i	$0.155023\pi$
$587$	22.6772i	0.935988i	−0.883732	+	0.467994i	$0.844977\pi$
$588$	− 18.0676i	− 0.745097i
$589$	1.23314	0.0508106
$590$	0	0
$591$	−14.9561	−0.615212
$592$	− 7.24477i	− 0.297758i
$593$	− 8.74287i	− 0.359027i	−0.983756	−	0.179513i	$-0.942548\pi$
$593$	− 8.74287i	− 0.359027i	0.983756	−	0.179513i	$-0.0574523\pi$
$594$	6.89053	0.282722
$595$	0	0
$596$	−25.8492	−1.05882
$597$	− 11.3251i	− 0.463506i
$598$	24.2094i	0.989994i
$599$	−16.3209	−0.666854	−0.333427	−	0.942776i	$-0.608205\pi$
$599$	−16.3209	−0.666854	−0.333427	+	0.942776i	$0.608205\pi$
$600$	0	0
$601$	36.2713	1.47954	0.739768	−	0.672862i	$-0.234935\pi$
$601$	36.2713	1.47954	0.739768	+	0.672862i	$0.234935\pi$
$602$	− 89.4105i	− 3.64410i
$603$	10.0415i	0.408922i
$604$	59.0170	2.40137
$605$	0	0
$606$	−18.0013	−0.731254
$607$	16.6820i	0.677102i	0.940948	+	0.338551i	$0.109937\pi$
$607$	16.6820i	0.677102i	−0.940948	+	0.338551i	$0.890063\pi$
$608$	11.5801i	0.469636i
$609$	8.89948	0.360625
$610$	0	0
$611$	14.0527	0.568511
$612$	− 23.9290i	− 0.967271i
$613$	24.7967i	1.00153i	0.865583	+	0.500765i	$0.166948\pi$
$613$	24.7967i	1.00153i	−0.865583	+	0.500765i	$0.833052\pi$
$614$	33.2980	1.34380
$615$	0	0
$616$	36.6215	1.47552
$617$	− 27.9669i	− 1.12591i	−0.826489	−	0.562953i	$-0.809665\pi$
$617$	− 27.9669i	− 1.12591i	0.826489	−	0.562953i	$-0.190335\pi$
$618$	7.01963i	0.282371i
$619$	−27.2686	−1.09602	−0.548009	−	0.836473i	$-0.684614\pi$
$619$	−27.2686	−1.09602	−0.548009	+	0.836473i	$0.684614\pi$
$620$	0	0
$621$	5.49019	0.220314
$622$	− 35.3799i	− 1.41861i
$623$	1.00108i	0.0401076i
$624$	−2.59359	−0.103827
$625$	0	0
$626$	−8.40365	−0.335878
$627$	− 8.64530i	− 0.345260i
$628$	27.8104i	1.10976i
$629$	−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	$0.181863\pi$
$632$	− 11.0256i	− 0.438575i
$633$	− 7.54283i	− 0.299801i
$634$	−30.0482	−1.19337
$635$	0	0
$636$	−13.0813	−0.518707
$637$	9.61045i	0.380780i
$638$	17.6118i	0.697256i
$639$	−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	$0.139574\pi$
$642$	− 16.6254i	− 0.656151i
$643$	− 46.6710i	− 1.84052i	−0.391304	−	0.920261i	$-0.627976\pi$
$643$	− 46.6710i	− 1.84052i	0.391304	−	0.920261i	$-0.372024\pi$
$644$	67.4115	2.65639
$645$	0	0
$646$	−47.0502	−1.85117
$647$	12.1264i	0.476740i	0.971174	+	0.238370i	$0.0766131\pi$
$647$	12.1264i	0.476740i	−0.971174	+	0.238370i	$0.923387\pi$
$648$	3.58831i	0.140962i
$649$	20.8359	0.817880
$650$	0	0
$651$	−1.45572	−0.0570542
$652$	35.0521i	1.37275i
$653$	− 1.72017i	− 0.0673154i	−0.999433	−	0.0336577i	$-0.989284\pi$
$653$	− 1.72017i	− 0.0673154i	0.999433	−	0.0336577i	$-0.0107156\pi$
$654$	−31.2711	−1.22280
$655$	0	0
$656$	2.32112	0.0906246
$657$	3.59269i	0.140164i
$658$	− 61.3226i	− 2.39060i
$659$	−5.47947	−0.213450	−0.106725	−	0.994289i	$-0.534036\pi$
$659$	−5.47947	−0.213450	−0.106725	+	0.994289i	$0.534036\pi$
$660$	0	0
$661$	0.797448	0.0310171	0.0155086	−	0.999880i	$-0.495063\pi$
$661$	0.797448	0.0310171	0.0155086	+	0.999880i	$0.495063\pi$
$662$	− 11.0607i	− 0.429888i
$663$	12.7282i	0.494322i
$664$	−36.2966	−1.40858
$665$	0	0
$666$	12.3174	0.477289
$667$	14.0326i	0.543344i
$668$	19.6691i	0.761020i
$669$	−7.79539	−0.301387
$670$	0	0
$671$	−18.8678	−0.728384
$672$	− 13.6703i	− 0.527344i
$673$	− 51.3280i	− 1.97855i	−0.146074	−	0.989274i	$-0.546664\pi$
$673$	− 51.3280i	− 1.97855i	0.146074	−	0.989274i	$-0.453336\pi$
$674$	40.9156	1.57601
$675$	0	0
$676$	−33.4358	−1.28599
$677$	− 15.0929i	− 0.580069i	−0.957016	−	0.290034i	$-0.906333\pi$
$677$	− 15.0929i	− 0.580069i	0.957016	−	0.290034i	$-0.0936667\pi$
$678$	− 23.5630i	− 0.904933i
$679$	36.2431	1.39088
$680$	0	0
$681$	12.8319	0.491721
$682$	− 2.88082i	− 0.110312i
$683$	− 10.8602i	− 0.415555i	−0.978176	−	0.207777i	$-0.933377\pi$
$683$	− 10.8602i	− 0.415555i	0.978176	−	0.207777i	$-0.0666229\pi$
$684$	10.4011	0.397697
$685$	0	0
$686$	−15.3595	−0.586428
$687$	− 14.5033i	− 0.553336i
$688$	15.1036i	0.575819i
$689$	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	$0.448644\pi$
$692$	− 58.4261i	− 2.22103i
$693$	10.2058i	0.387686i
$694$	−32.9366	−1.25026
$695$	0	0
$696$	−9.17148	−0.347644
$697$	− 11.3910i	− 0.431466i
$698$	84.5026i	3.19847i
$699$	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	$0.124719\pi$
$702$	− 4.40956i	− 0.166428i
$703$	− 15.4542i	− 0.582865i
$704$	35.1588	1.32510
$705$	0	0
$706$	20.6920	0.778752
$707$	− 26.6623i	− 1.00274i
$708$	25.0676i	0.942097i
$709$	−24.7834	−0.930762	−0.465381	−	0.885111i	$-0.654083\pi$
$709$	−24.7834	−0.930762	−0.465381	+	0.885111i	$0.654083\pi$
$710$	0	0
$711$	3.07265	0.115233
$712$	− 1.03168i	− 0.0386639i
$713$	− 2.29536i	− 0.0859620i
$714$	55.5428	2.07864
$715$	0	0
$716$	−18.9023	−0.706410
$717$	6.90072i	0.257712i
$718$	14.1746i	0.528992i
$719$	18.2171	0.679382	0.339691	−	0.940537i	$-0.389678\pi$
$719$	18.2171	0.679382	0.339691	+	0.940537i	$0.389678\pi$
$720$	0	0
$721$	−10.3970	−0.387204
$722$	24.2146i	0.901174i
$723$	28.4467i	1.05795i
$724$	1.15576	0.0429534
$725$	0	0
$726$	5.66224	0.210145
$727$	− 16.6699i	− 0.618253i	−0.951021	−	0.309127i	$-0.899963\pi$
$727$	− 16.6699i	− 0.618253i	0.951021	−	0.309127i	$-0.100037\pi$
$728$	− 23.4357i	− 0.868586i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	74.1216	2.74149
$732$	− 22.6998i	− 0.839008i
$733$	20.0644i	0.741094i	0.928814	+	0.370547i	$0.120830\pi$
$733$	20.0644i	0.741094i	−0.928814	+	0.370547i	$0.879170\pi$
$734$	−67.6245	−2.49606
$735$	0	0
$736$	21.5552	0.794535
$737$	− 29.4328i	− 1.08417i
$738$	3.94632i	0.145266i
$739$	34.0571	1.25281	0.626406	−	0.779497i	$-0.284525\pi$
$739$	34.0571	1.25281	0.626406	+	0.779497i	$0.284525\pi$
$740$	0	0
$741$	−5.53252	−0.203242
$742$	− 30.3637i	− 1.11469i
$743$	− 38.5355i	− 1.41373i	−0.707348	−	0.706865i	$-0.750109\pi$
$743$	− 38.5355i	− 1.41373i	0.707348	−	0.706865i	$-0.249891\pi$
$744$	1.50021	0.0550005
$745$	0	0
$746$	78.9880	2.89195
$747$	− 10.1152i	− 0.370097i
$748$	70.1383i	2.56451i
$749$	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	$0.728373\pi$
$752$	10.3589i	0.377749i
$753$	12.3258i	0.449178i
$754$	11.2706	0.410449
$755$	0	0
$756$	−12.2785	−0.446566
$757$	− 35.0131i	− 1.27257i	−0.771453	−	0.636287i	$-0.780470\pi$
$757$	− 35.0131i	− 1.27257i	0.771453	−	0.636287i	$-0.219530\pi$
$758$	10.8938i	0.395681i
$759$	−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	$0.307579\pi$
$762$	24.8554i	0.900416i
$763$	− 46.3166i	− 1.67678i
$764$	12.2363	0.442694
$765$	0	0
$766$	−30.3392	−1.09620
$767$	− 13.3338i	− 0.481456i
$768$	23.8400i	0.860253i
$769$	−2.39815	−0.0864793	−0.0432397	−	0.999065i	$-0.513768\pi$
$769$	−2.39815	−0.0864793	−0.0432397	+	0.999065i	$0.513768\pi$
$770$	0	0
$771$	−20.8274	−0.750082
$772$	85.7388i	3.08581i
$773$	− 10.9226i	− 0.392858i	−0.980518	−	0.196429i	$-0.937065\pi$
$773$	− 10.9226i	− 0.392858i	0.980518	−	0.196429i	$-0.0629345\pi$
$774$	−25.6788	−0.923004
$775$	0	0
$776$	−37.3508	−1.34082
$777$	18.2437i	0.654487i
$778$	− 41.1973i	− 1.47700i
$779$	4.95130	0.177399
$780$	0	0
$781$	2.13561	0.0764181
$782$	87.5792i	3.13183i
$783$	− 2.55593i	− 0.0913417i
$784$	−7.08430	−0.253011
$785$	0	0
$786$	21.2188	0.756848
$787$	− 11.5724i	− 0.412511i	−0.978498	−	0.206256i	$-0.933872\pi$
$787$	− 11.5724i	− 0.412511i	0.978498	−	0.206256i	$-0.0661278\pi$
$788$	52.7412i	1.87883i
$789$	2.23517	0.0795740
$790$	0	0
$791$	34.8999	1.24090
$792$	− 10.5177i	− 0.373731i
$793$	12.0744i	0.428773i
$794$	63.5137	2.25402
$795$	0	0
$796$	−39.9369	−1.41553
$797$	− 21.8923i	− 0.775467i	−0.921772	−	0.387733i	$-0.873258\pi$
$797$	− 21.8923i	− 0.775467i	0.921772	−	0.387733i	$-0.126742\pi$
$798$	24.1426i	0.854639i
$799$	50.8367	1.79847
$800$	0	0
$801$	0.287512	0.0101587
$802$	37.5645i	1.32645i
$803$	− 10.5306i	− 0.371615i
$804$	35.4104	1.24883
$805$	0	0
$806$	−1.84357	−0.0649369
$807$	− 19.3036i	− 0.679520i
$808$	27.4772i	0.966646i
$809$	21.5498	0.757652	0.378826	−	0.925468i	$-0.376328\pi$
$809$	21.5498	0.757652	0.378826	+	0.925468i	$0.376328\pi$
$810$	0	0
$811$	−43.9615	−1.54370	−0.771848	−	0.635807i	$-0.780667\pi$
$811$	−43.9615	−1.54370	−0.771848	+	0.635807i	$0.780667\pi$
$812$	− 31.3831i	− 1.10133i
$813$	11.9411i	0.418794i
$814$	−36.1036	−1.26543
$815$	0	0
$816$	−9.38252	−0.328454
$817$	32.2182i	1.12717i
$818$	− 72.8667i	− 2.54772i
$819$	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	$0.301218\pi$
$822$	− 46.2281i	− 1.61239i
$823$	55.9860i	1.95155i	0.218781	+	0.975774i	$0.429792\pi$
$823$	55.9860i	1.95155i	−0.218781	+	0.975774i	$0.570208\pi$
$824$	10.7148	0.373266
$825$	0	0
$826$	−58.1857	−2.02454
$827$	40.8100i	1.41910i	0.704653	+	0.709552i	$0.251103\pi$
$827$	40.8100i	1.41910i	−0.704653	+	0.709552i	$0.748897\pi$
$828$	− 19.3606i	− 0.672828i
$829$	−0.122914	−0.00426896	−0.00213448	−	0.999998i	$-0.500679\pi$
$829$	−0.122914	−0.00426896	−0.00213448	+	0.999998i	$0.500679\pi$
$830$	0	0
$831$	−26.1466	−0.907016
$832$	− 22.4997i	− 0.780036i
$833$	34.7666i	1.20459i
$834$	−27.3919	−0.948503
$835$	0	0
$836$	−30.4868	−1.05441
$837$	0.418084i	0.0144511i
$838$	60.9226i	2.10454i
$839$	−14.8083	−0.511241	−0.255620	−	0.966777i	$-0.582280\pi$
$839$	−14.8083	−0.511241	−0.255620	+	0.966777i	$0.582280\pi$
$840$	0	0
$841$	−22.4672	−0.774731
$842$	− 9.23045i	− 0.318102i
$843$	− 10.1219i	− 0.348616i
$844$	−26.5990	−0.915577
$845$	0	0
$846$	−17.6119	−0.605509
$847$	8.38651i	0.288164i
$848$	5.12916i	0.176136i
$849$	−31.8638	−1.09356
$850$	0	0
$851$	−28.7664	−0.986098
$852$	2.56934i	0.0880242i
$853$	7.26474i	0.248740i	0.992236	+	0.124370i	$0.0396910\pi$
$853$	7.26474i	0.248740i	−0.992236	+	0.124370i	$0.960309\pi$
$854$	52.6897	1.80301
$855$	0	0
$856$	−25.3770	−0.867367
$857$	26.8175i	0.916068i	0.888935	+	0.458034i	$0.151446\pi$
$857$	26.8175i	0.916068i	−0.888935	+	0.458034i	$0.848554\pi$
$858$	12.9249i	0.441249i
$859$	20.6038	0.702991	0.351496	−	0.936190i	$-0.385673\pi$
$859$	20.6038	0.702991	0.351496	+	0.936190i	$0.385673\pi$
$860$	0	0
$861$	−5.84501	−0.199197
$862$	83.3679i	2.83952i
$863$	13.7488i	0.468015i	0.972235	+	0.234008i	$0.0751840\pi$
$863$	13.7488i	0.468015i	−0.972235	+	0.234008i	$0.924816\pi$
$864$	−3.92613	−0.133570
$865$	0	0
$866$	23.7338	0.806506
$867$	29.0452i	0.986427i
$868$	5.13346i	0.174241i
$869$	−9.00627	−0.305517
$870$	0	0
$871$	−18.8353	−0.638211
$872$	47.7323i	1.61642i
$873$	− 10.4090i	− 0.352293i
$874$	−38.0678	−1.28766
$875$	0	0
$876$	12.6693	0.428055
$877$	− 22.0430i	− 0.744339i	−0.928165	−	0.372170i	$-0.878614\pi$
$877$	− 22.0430i	− 0.744339i	0.928165	−	0.372170i	$-0.121386\pi$
$878$	16.7041i	0.563735i
$879$	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	$0.548659\pi$
$882$	− 12.0446i	− 0.405561i
$883$	− 1.60120i	− 0.0538848i	−0.999637	−	0.0269424i	$-0.991423\pi$
$883$	− 1.60120i	− 0.0538848i	0.999637	−	0.0269424i	$-0.00857708\pi$
$884$	44.8847	1.50963
$885$	0	0
$886$	48.6249	1.63358
$887$	− 17.4253i	− 0.585083i	−0.956253	−	0.292541i	$-0.905499\pi$
$887$	− 17.4253i	− 0.585083i	0.956253	−	0.292541i	$-0.0945009\pi$
$888$	− 18.8012i	− 0.630929i
$889$	−36.8141	−1.23471
$890$	0	0
$891$	2.93111	0.0981958
$892$	27.4897i	0.920423i
$893$	22.0970i	0.739448i
$894$	−17.2320	−0.576326
$895$	0	0
$896$	−70.8427	−2.36669
$897$	10.2982i	0.343847i
$898$	45.8272i	1.52927i
$899$	−1.06860	−0.0356397
$900$	0	0
$901$	25.1716	0.838588
$902$	− 11.5671i	− 0.385141i
$903$	− 38.0336i	− 1.26568i
$904$	−35.9666	−1.19623
$905$	0	0
$906$	39.3429	1.30708
$907$	20.8690i	0.692942i	0.938061	+	0.346471i	$0.112620\pi$
$907$	20.8690i	0.692942i	−0.938061	+	0.346471i	$0.887380\pi$
$908$	− 45.2506i	− 1.50169i
$909$	−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	$0.196947\pi$
$912$	− 4.07827i	− 0.135045i
$913$	29.6489i	0.981234i
$914$	10.2072	0.337623
$915$	0	0
$916$	−51.1445	−1.68986
$917$	31.4278i	1.03784i
$918$	− 15.9519i	− 0.526492i
$919$	11.1959	0.369318	0.184659	−	0.982803i	$-0.440882\pi$
$919$	11.1959	0.369318	0.184659	+	0.982803i	$0.440882\pi$
$920$	0	0
$921$	14.1643	0.466731
$922$	− 27.7156i	− 0.912765i
$923$	− 1.36667i	− 0.0449846i
$924$	35.9897	1.18397
$925$	0	0
$926$	−21.3425	−0.701357
$927$	2.98602i	0.0980738i
$928$	− 10.0349i	− 0.329413i
$929$	−48.2290	−1.58234	−0.791171	−	0.611596i	$-0.790528\pi$
$929$	−48.2290	−1.58234	−0.791171	+	0.611596i	$0.790528\pi$
$930$	0	0
$931$	−15.1119	−0.495271
$932$	− 40.1906i	− 1.31649i
$933$	− 15.0500i	− 0.492714i
$934$	82.0866	2.68596
$935$	0	0
$936$	−6.73076	−0.220002
$937$	− 50.5397i	− 1.65106i	−0.564359	−	0.825529i	$-0.690877\pi$
$937$	− 50.5397i	− 1.65106i	0.564359	−	0.825529i	$-0.309123\pi$
$938$	82.1930i	2.68370i
$939$	−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	$0.713705\pi$
$942$	18.5395i	0.604049i
$943$	− 9.21634i	− 0.300125i
$944$	9.82896	0.319906
$945$	0	0
$946$	75.2672	2.44715
$947$	− 24.5190i	− 0.796759i	−0.917221	−	0.398380i	$-0.869573\pi$
$947$	− 24.5190i	− 0.796759i	0.917221	−	0.398380i	$-0.130427\pi$
$948$	− 10.8354i	− 0.351917i
$949$	−6.73897	−0.218756
$950$	0	0
$951$	−12.7820	−0.414484
$952$	− 84.7806i	− 2.74775i
$953$	13.0019i	0.421173i	0.977575	+	0.210586i	$0.0675373\pi$
$953$	13.0019i	0.421173i	−0.977575	+	0.210586i	$0.932463\pi$
$954$	−8.72047	−0.282336
$955$	0	0
$956$	24.3347	0.787041
$957$	7.49172i	0.242173i
$958$	− 20.3264i	− 0.656717i
$959$	68.4698	2.21101
$960$	0	0
$961$	−30.8252	−0.994361
$962$	23.1043i	0.744912i
$963$	− 7.07213i	− 0.227896i
$964$	100.315	3.23091
$965$	0	0
$966$	44.9390	1.44589
$967$	43.3212i	1.39312i	0.717500	+	0.696559i	$0.245286\pi$
$967$	43.3212i	1.39312i	−0.717500	+	0.696559i	$0.754714\pi$
$968$	− 8.64284i	− 0.277791i
$969$	−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	$0.657143\pi$
$972$	3.52640i	0.113109i
$973$	− 40.5709i	− 1.30064i
$974$	−6.68721	−0.214272
$975$	0	0
$976$	−8.90056	−0.284900
$977$	60.6153i	1.93926i	0.244583	+	0.969628i	$0.421349\pi$
$977$	60.6153i	1.93926i	−0.244583	+	0.969628i	$0.578651\pi$
$978$	23.3671i	0.747196i
$979$	−0.842729	−0.0269337
$980$	0	0
$981$	−13.3022	−0.424706
$982$	− 86.5202i	− 2.76097i
$983$	− 40.2796i	− 1.28472i	−0.766404	−	0.642359i	$-0.777956\pi$
$983$	− 40.2796i	− 1.28472i	0.766404	−	0.642359i	$-0.222044\pi$
$984$	6.02366	0.192027
$985$	0	0
$986$	40.7721	1.29845
$987$	− 26.0855i	− 0.830311i
$988$	19.5099i	0.620692i
$989$	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	$0.292801\pi$
$992$	1.64145i	0.0521161i
$993$	− 4.70504i	− 0.149310i
$994$	−5.96384	−0.189162
$995$	0	0
$996$	−35.6704	−1.13026
$997$	− 26.5777i	− 0.841724i	−0.907125	−	0.420862i	$-0.861728\pi$
$997$	− 26.5777i	− 0.841724i	0.907125	−	0.420862i	$-0.138272\pi$
$998$	13.7706i	0.435900i
$999$	5.23959	0.165773

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.1	✓	16	25.11	even	5
75.2.i.a.19.1	yes	16	25.9	even	10
225.2.m.b.19.4		16	75.59	odd	10
225.2.m.b.154.4		16	75.11	odd	10
375.2.g.d.151.4		16	25.12	odd	20
375.2.g.d.226.4		16	25.2	odd	20
375.2.g.e.151.1		16	25.13	odd	20
375.2.g.e.226.1		16	25.23	odd	20
375.2.i.c.274.4		16	25.14	even	10
375.2.i.c.349.4		16	25.16	even	5
1875.2.a.m.1.2		8	5.3	odd	4
1875.2.a.p.1.7		8	5.2	odd	4
1875.2.b.h.1249.2		16	1.1	even	1	trivial
1875.2.b.h.1249.15		16	5.4	even	2	inner
5625.2.a.t.1.2		8	15.2	even	4
5625.2.a.bd.1.7		8	15.8	even	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.35083i q^{2} -1.00000i q^{3} -3.52640 q^{4} -2.35083 q^{6} -3.48189i q^{7} +3.58831i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.35083i q^{2} -1.00000i q^{3} -3.52640 q^{4} -2.35083 q^{6} -3.48189i q^{7} +3.58831i q^{8} -1.00000 q^{9} +2.93111 q^{11} +3.52640i q^{12} -1.87575i q^{13} -8.18532 q^{14} +1.38270 q^{16} -6.78566i q^{17} +2.35083i q^{18} +2.94950 q^{19} -3.48189 q^{21} -6.89053i q^{22} -5.49019i q^{23} +3.58831 q^{24} -4.40956 q^{26} +1.00000i q^{27} +12.2785i q^{28} -2.55593 q^{29} +0.418084 q^{31} +3.92613i q^{32} -2.93111i q^{33} -15.9519 q^{34} +3.52640 q^{36} -5.23959i q^{37} -6.93377i q^{38} -1.87575 q^{39} +1.67869 q^{41} +8.18532i q^{42} +10.9233i q^{43} -10.3363 q^{44} -12.9065 q^{46} +7.49178i q^{47} -1.38270i q^{48} -5.12353 q^{49} -6.78566 q^{51} +6.61463i q^{52} +3.70953i q^{53} +2.35083 q^{54} +12.4941 q^{56} -2.94950i q^{57} +6.00857i q^{58} +7.10854 q^{59} -6.43710 q^{61} -0.982844i q^{62} +3.48189i q^{63} +11.9951 q^{64} -6.89053 q^{66} -10.0415i q^{67} +23.9290i q^{68} -5.49019 q^{69} +0.728602 q^{71} -3.58831i q^{72} -3.59269i q^{73} -12.3174 q^{74} -10.4011 q^{76} -10.2058i q^{77} +4.40956i q^{78} -3.07265 q^{79} +1.00000 q^{81} -3.94632i q^{82} +10.1152i q^{83} +12.2785 q^{84} +25.6788 q^{86} +2.55593i q^{87} +10.5177i q^{88} -0.287512 q^{89} -6.53114 q^{91} +19.3606i q^{92} -0.418084i q^{93} +17.6119 q^{94} +3.92613 q^{96} +10.4090i q^{97} +12.0446i q^{98} -2.93111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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