LMFDB - Embedded newform 75.4.b.b.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,4,Mod(49,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("75.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	75.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:	$4.42514325043$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	75.49
Dual form			75.4.b.b.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +52.0000 q^{11} +21.0000i q^{12} +22.0000i q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +9.00000i q^{18} +20.0000 q^{19} -72.0000 q^{21} -52.0000i q^{22} -168.000i q^{23} +45.0000 q^{24} +22.0000 q^{26} -27.0000i q^{27} +168.000i q^{28} -230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +156.000i q^{33} +14.0000 q^{34} -63.0000 q^{36} +34.0000i q^{37} -20.0000i q^{38} -66.0000 q^{39} +122.000 q^{41} +72.0000i q^{42} -188.000i q^{43} +364.000 q^{44} -168.000 q^{46} -256.000i q^{47} +123.000i q^{48} -233.000 q^{49} -42.0000 q^{51} +154.000i q^{52} -338.000i q^{53} -27.0000 q^{54} +360.000 q^{56} +60.0000i q^{57} +230.000i q^{58} -100.000 q^{59} +742.000 q^{61} +288.000i q^{62} -216.000i q^{63} +167.000 q^{64} +156.000 q^{66} +84.0000i q^{67} +98.0000i q^{68} +504.000 q^{69} -328.000 q^{71} +135.000i q^{72} -38.0000i q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +66.0000i q^{78} +240.000 q^{79} +81.0000 q^{81} -122.000i q^{82} +1212.00i q^{83} -504.000 q^{84} -188.000 q^{86} -690.000i q^{87} -780.000i q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -864.000i q^{93} -256.000 q^{94} +483.000 q^{96} -866.000i q^{97} +233.000i q^{98} -468.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) $ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9	$ 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 104 q^{11} + 48 q^{14} + 82 q^{16} + 40 q^{19} - 144 q^{21} + 90 q^{24} + 44 q^{26} - 460 q^{29} - 576 q^{31} + 28 q^{34} - 126 q^{36} - 132 q^{39} + 244 q^{41} + 728 q^{44} - 336 q^{46} - 466 q^{49} - 84 q^{51} - 54 q^{54} + 720 q^{56} - 200 q^{59} + 1484 q^{61} + 334 q^{64} + 312 q^{66} + 1008 q^{69} - 656 q^{71} + 68 q^{74} + 280 q^{76} + 480 q^{79} + 162 q^{81} - 1008 q^{84} - 376 q^{86} - 660 q^{89} - 1056 q^{91} - 512 q^{94} + 966 q^{96} - 936 q^{99}+O(q^{100}) $ 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 104 * q^11 + 48 * q^14 + 82 * q^16 + 40 * q^19 - 144 * q^21 + 90 * q^24 + 44 * q^26 - 460 * q^29 - 576 * q^31 + 28 * q^34 - 126 * q^36 - 132 * q^39 + 244 * q^41 + 728 * q^44 - 336 * q^46 - 466 * q^49 - 84 * q^51 - 54 * q^54 + 720 * q^56 - 200 * q^59 + 1484 * q^61 + 334 * q^64 + 312 * q^66 + 1008 * q^69 - 656 * q^71 + 68 * q^74 + 280 * q^76 + 480 * q^79 + 162 * q^81 - 1008 * q^84 - 376 * q^86 - 660 * q^89 - 1056 * q^91 - 512 * q^94 + 966 * q^96 - 936 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.353553i	−0.984251	−	0.176777i	$-0.943433\pi$
$2$	− 1.00000i	− 0.353553i	0.984251	−	0.176777i	$-0.0565670\pi$
$3$	3.00000i	0.577350i
$3$	3.00000i	0.577350i
$4$	7.00000	0.875000
$5$	0	0
$5$	0	0
$6$	3.00000	0.204124
$7$	24.0000i	1.29588i	0.761692	+	0.647939i	$0.224369\pi$
$7$	24.0000i	1.29588i	−0.761692	+	0.647939i	$0.775631\pi$
$8$	− 15.0000i	− 0.662913i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	52.0000	1.42533	0.712663	−	0.701506i	$-0.247489\pi$
$11$	52.0000	1.42533	0.712663	+	0.701506i	$0.247489\pi$
$12$	21.0000i	0.505181i
$13$	22.0000i	0.469362i	0.972072	+	0.234681i	$0.0754045\pi$
$13$	22.0000i	0.469362i	−0.972072	+	0.234681i	$0.924595\pi$
$14$	24.0000	0.458162
$15$	0	0
$16$	41.0000	0.640625
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	9.00000i	0.117851i
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	−72.0000	−0.748176
$22$	− 52.0000i	− 0.503929i
$23$	− 168.000i	− 1.52306i	−0.648129	−	0.761531i	$-0.724448\pi$
$23$	− 168.000i	− 1.52306i	0.648129	−	0.761531i	$-0.275552\pi$
$24$	45.0000	0.382733
$25$	0	0
$26$	22.0000	0.165944
$27$	− 27.0000i	− 0.192450i
$28$	168.000i	1.13389i
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	−288.000	−1.66859	−0.834296	−	0.551317i	$-0.814125\pi$
$31$	−288.000	−1.66859	−0.834296	+	0.551317i	$0.814125\pi$
$32$	− 161.000i	− 0.889408i
$33$	156.000i	0.822913i
$34$	14.0000	0.0706171
$35$	0	0
$36$	−63.0000	−0.291667
$37$	34.0000i	0.151069i	0.997143	+	0.0755347i	$0.0240664\pi$
$37$	34.0000i	0.151069i	−0.997143	+	0.0755347i	$0.975934\pi$
$38$	− 20.0000i	− 0.0853797i
$39$	−66.0000	−0.270986
$40$	0	0
$41$	122.000	0.464712	0.232356	−	0.972631i	$-0.425357\pi$
$41$	122.000	0.464712	0.232356	+	0.972631i	$0.425357\pi$
$42$	72.0000i	0.264520i
$43$	− 188.000i	− 0.666738i	−0.942796	−	0.333369i	$-0.891815\pi$
$43$	− 188.000i	− 0.666738i	0.942796	−	0.333369i	$-0.108185\pi$
$44$	364.000	1.24716
$45$	0	0
$46$	−168.000	−0.538484
$47$	− 256.000i	− 0.794499i	−0.917711	−	0.397249i	$-0.869965\pi$
$47$	− 256.000i	− 0.794499i	0.917711	−	0.397249i	$-0.130035\pi$
$48$	123.000i	0.369865i
$49$	−233.000	−0.679300
$50$	0	0
$51$	−42.0000	−0.115317
$52$	154.000i	0.410691i
$53$	− 338.000i	− 0.875998i	−0.898976	−	0.437999i	$-0.855687\pi$
$53$	− 338.000i	− 0.875998i	0.898976	−	0.437999i	$-0.144313\pi$
$54$	−27.0000	−0.0680414
$55$	0	0
$56$	360.000	0.859054
$57$	60.0000i	0.139424i
$58$	230.000i	0.520698i
$59$	−100.000	−0.220659	−0.110330	−	0.993895i	$-0.535191\pi$
$59$	−100.000	−0.220659	−0.110330	+	0.993895i	$0.535191\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	288.000i	0.589936i
$63$	− 216.000i	− 0.431959i
$64$	167.000	0.326172
$65$	0	0
$66$	156.000	0.290944
$67$	84.0000i	0.153168i	0.997063	+	0.0765838i	$0.0244013\pi$
$67$	84.0000i	0.153168i	−0.997063	+	0.0765838i	$0.975599\pi$
$68$	98.0000i	0.174768i
$69$	504.000	0.879340
$70$	0	0
$71$	−328.000	−0.548260	−0.274130	−	0.961693i	$-0.588390\pi$
$71$	−328.000	−0.548260	−0.274130	+	0.961693i	$0.588390\pi$
$72$	135.000i	0.220971i
$73$	− 38.0000i	− 0.0609255i	−0.999536	−	0.0304628i	$-0.990302\pi$
$73$	− 38.0000i	− 0.0609255i	0.999536	−	0.0304628i	$-0.00969810\pi$
$74$	34.0000	0.0534111
$75$	0	0
$76$	140.000	0.211304
$77$	1248.00i	1.84705i
$78$	66.0000i	0.0958081i
$79$	240.000	0.341799	0.170899	−	0.985288i	$-0.445333\pi$
$79$	240.000	0.341799	0.170899	+	0.985288i	$0.445333\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 122.000i	− 0.164301i
$83$	1212.00i	1.60282i	0.598114	+	0.801411i	$0.295917\pi$
$83$	1212.00i	1.60282i	−0.598114	+	0.801411i	$0.704083\pi$
$84$	−504.000	−0.654654
$85$	0	0
$86$	−188.000	−0.235727
$87$	− 690.000i	− 0.850296i
$88$	− 780.000i	− 0.944867i
$89$	−330.000	−0.393033	−0.196516	−	0.980501i	$-0.562963\pi$
$89$	−330.000	−0.393033	−0.196516	+	0.980501i	$0.562963\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	− 1176.00i	− 1.33268i
$93$	− 864.000i	− 0.963362i
$94$	−256.000	−0.280898
$95$	0	0
$96$	483.000	0.513500
$97$	− 866.000i	− 0.906484i	−0.891387	−	0.453242i	$-0.850267\pi$
$97$	− 866.000i	− 0.906484i	0.891387	−	0.453242i	$-0.149733\pi$
$98$	233.000i	0.240169i
$99$	−468.000	−0.475109
$100$	0	0
$101$	−1218.00	−1.19996	−0.599978	−	0.800017i	$-0.704824\pi$
$101$	−1218.00	−1.19996	−0.599978	+	0.800017i	$0.704824\pi$
$102$	42.0000i	0.0407708i
$103$	− 88.0000i	− 0.0841835i	−0.999114	−	0.0420917i	$-0.986598\pi$
$103$	− 88.0000i	− 0.0841835i	0.999114	−	0.0420917i	$-0.0134022\pi$
$104$	330.000	0.311146
$105$	0	0
$106$	−338.000	−0.309712
$107$	− 36.0000i	− 0.0325257i	−0.999868	−	0.0162629i	$-0.994823\pi$
$107$	− 36.0000i	− 0.0325257i	0.999868	−	0.0162629i	$-0.00517686\pi$
$108$	− 189.000i	− 0.168394i
$109$	970.000	0.852378	0.426189	−	0.904634i	$-0.359856\pi$
$109$	970.000	0.852378	0.426189	+	0.904634i	$0.359856\pi$
$110$	0	0
$111$	−102.000	−0.0872199
$112$	984.000i	0.830172i
$113$	1042.00i	0.867461i	0.901043	+	0.433731i	$0.142803\pi$
$113$	1042.00i	0.867461i	−0.901043	+	0.433731i	$0.857197\pi$
$114$	60.0000	0.0492940
$115$	0	0
$116$	−1610.00	−1.28866
$117$	− 198.000i	− 0.156454i
$118$	100.000i	0.0780148i
$119$	−336.000	−0.258833
$120$	0	0
$121$	1373.00	1.03156
$122$	− 742.000i	− 0.550635i
$123$	366.000i	0.268302i
$124$	−2016.00	−1.46002
$125$	0	0
$126$	−216.000	−0.152721
$127$	− 1936.00i	− 1.35269i	−0.736583	−	0.676347i	$-0.763562\pi$
$127$	− 1936.00i	− 1.35269i	0.736583	−	0.676347i	$-0.236438\pi$
$128$	− 1455.00i	− 1.00473i
$129$	564.000	0.384941
$130$	0	0
$131$	732.000	0.488207	0.244104	−	0.969749i	$-0.421506\pi$
$131$	732.000	0.488207	0.244104	+	0.969749i	$0.421506\pi$
$132$	1092.00i	0.720048i
$133$	480.000i	0.312942i
$134$	84.0000	0.0541529
$135$	0	0
$136$	210.000	0.132407
$137$	2214.00i	1.38069i	0.723479	+	0.690346i	$0.242542\pi$
$137$	2214.00i	1.38069i	−0.723479	+	0.690346i	$0.757458\pi$
$138$	− 504.000i	− 0.310894i
$139$	−20.0000	−0.0122042	−0.00610208	−	0.999981i	$-0.501942\pi$
$139$	−20.0000	−0.0122042	−0.00610208	+	0.999981i	$0.501942\pi$
$140$	0	0
$141$	768.000	0.458704
$142$	328.000i	0.193839i
$143$	1144.00i	0.668994i
$144$	−369.000	−0.213542
$145$	0	0
$146$	−38.0000	−0.0215404
$147$	− 699.000i	− 0.392194i
$148$	238.000i	0.132186i
$149$	1330.00	0.731261	0.365630	−	0.930760i	$-0.380853\pi$
$149$	1330.00	0.731261	0.365630	+	0.930760i	$0.380853\pi$
$150$	0	0
$151$	−1208.00	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−1208.00	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 300.000i	− 0.160087i
$153$	− 126.000i	− 0.0665784i
$154$	1248.00	0.653031
$155$	0	0
$156$	−462.000	−0.237113
$157$	3514.00i	1.78629i	0.449768	+	0.893146i	$0.351507\pi$
$157$	3514.00i	1.78629i	−0.449768	+	0.893146i	$0.648493\pi$
$158$	− 240.000i	− 0.120844i
$159$	1014.00	0.505757
$160$	0	0
$161$	4032.00	1.97370
$162$	− 81.0000i	− 0.0392837i
$163$	− 2068.00i	− 0.993732i	−0.867827	−	0.496866i	$-0.834484\pi$
$163$	− 2068.00i	− 0.993732i	0.867827	−	0.496866i	$-0.165516\pi$
$164$	854.000	0.406623
$165$	0	0
$166$	1212.00	0.566683
$167$	24.0000i	0.0111208i	0.999985	+	0.00556041i	$0.00176994\pi$
$167$	24.0000i	0.0111208i	−0.999985	+	0.00556041i	$0.998230\pi$
$168$	1080.00i	0.495975i
$169$	1713.00	0.779700
$170$	0	0
$171$	−180.000	−0.0804967
$172$	− 1316.00i	− 0.583396i
$173$	− 618.000i	− 0.271593i	−0.990737	−	0.135797i	$-0.956641\pi$
$173$	− 618.000i	− 0.271593i	0.990737	−	0.135797i	$-0.0433594\pi$
$174$	−690.000	−0.300625
$175$	0	0
$176$	2132.00	0.913100
$177$	− 300.000i	− 0.127398i
$178$	330.000i	0.138958i
$179$	−3340.00	−1.39466	−0.697328	−	0.716752i	$-0.745628\pi$
$179$	−3340.00	−1.39466	−0.697328	+	0.716752i	$0.745628\pi$
$180$	0	0
$181$	−178.000	−0.0730974	−0.0365487	−	0.999332i	$-0.511636\pi$
$181$	−178.000	−0.0730974	−0.0365487	+	0.999332i	$0.511636\pi$
$182$	528.000i	0.215044i
$183$	2226.00i	0.899184i
$184$	−2520.00	−1.00966
$185$	0	0
$186$	−864.000	−0.340600
$187$	728.000i	0.284688i
$188$	− 1792.00i	− 0.695186i
$189$	648.000	0.249392
$190$	0	0
$191$	−1888.00	−0.715240	−0.357620	−	0.933867i	$-0.616412\pi$
$191$	−1888.00	−0.715240	−0.357620	+	0.933867i	$0.616412\pi$
$192$	501.000i	0.188315i
$193$	1922.00i	0.716832i	0.933562	+	0.358416i	$0.116683\pi$
$193$	1922.00i	0.716832i	−0.933562	+	0.358416i	$0.883317\pi$
$194$	−866.000	−0.320491
$195$	0	0
$196$	−1631.00	−0.594388
$197$	− 2526.00i	− 0.913554i	−0.889581	−	0.456777i	$-0.849004\pi$
$197$	− 2526.00i	− 0.913554i	0.889581	−	0.456777i	$-0.150996\pi$
$198$	468.000i	0.167976i
$199$	1160.00	0.413217	0.206609	−	0.978424i	$-0.433757\pi$
$199$	1160.00	0.413217	0.206609	+	0.978424i	$0.433757\pi$
$200$	0	0
$201$	−252.000	−0.0884314
$202$	1218.00i	0.424248i
$203$	− 5520.00i	− 1.90851i
$204$	−294.000	−0.100903
$205$	0	0
$206$	−88.0000	−0.0297634
$207$	1512.00i	0.507687i
$208$	902.000i	0.300685i
$209$	1040.00	0.344202
$210$	0	0
$211$	−4468.00	−1.45777	−0.728886	−	0.684635i	$-0.759961\pi$
$211$	−4468.00	−1.45777	−0.728886	+	0.684635i	$0.759961\pi$
$212$	− 2366.00i	− 0.766498i
$213$	− 984.000i	− 0.316538i
$214$	−36.0000	−0.0114996
$215$	0	0
$216$	−405.000	−0.127578
$217$	− 6912.00i	− 2.16229i
$218$	− 970.000i	− 0.301361i
$219$	114.000	0.0351754
$220$	0	0
$221$	−308.000	−0.0937481
$222$	102.000i	0.0308369i
$223$	6032.00i	1.81136i	0.423965	+	0.905678i	$0.360638\pi$
$223$	6032.00i	1.81136i	−0.423965	+	0.905678i	$0.639362\pi$
$224$	3864.00	1.15256
$225$	0	0
$226$	1042.00	0.306694
$227$	− 2636.00i	− 0.770738i	−0.922763	−	0.385369i	$-0.874074\pi$
$227$	− 2636.00i	− 0.770738i	0.922763	−	0.385369i	$-0.125926\pi$
$228$	420.000i	0.121996i
$229$	−4830.00	−1.39378	−0.696889	−	0.717179i	$-0.745433\pi$
$229$	−4830.00	−1.39378	−0.696889	+	0.717179i	$0.745433\pi$
$230$	0	0
$231$	−3744.00	−1.06639
$232$	3450.00i	0.976309i
$233$	2682.00i	0.754093i	0.926194	+	0.377046i	$0.123060\pi$
$233$	2682.00i	0.754093i	−0.926194	+	0.377046i	$0.876940\pi$
$234$	−198.000	−0.0553148
$235$	0	0
$236$	−700.000	−0.193077
$237$	720.000i	0.197338i
$238$	336.000i	0.0915111i
$239$	−2320.00	−0.627901	−0.313950	−	0.949439i	$-0.601653\pi$
$239$	−2320.00	−0.627901	−0.313950	+	0.949439i	$0.601653\pi$
$240$	0	0
$241$	2002.00	0.535104	0.267552	−	0.963543i	$-0.413785\pi$
$241$	2002.00	0.535104	0.267552	+	0.963543i	$0.413785\pi$
$242$	− 1373.00i	− 0.364710i
$243$	243.000i	0.0641500i
$244$	5194.00	1.36275
$245$	0	0
$246$	366.000	0.0948590
$247$	440.000i	0.113346i
$248$	4320.00i	1.10613i
$249$	−3636.00	−0.925390
$250$	0	0
$251$	132.000	0.0331943	0.0165971	−	0.999862i	$-0.494717\pi$
$251$	132.000	0.0331943	0.0165971	+	0.999862i	$0.494717\pi$
$252$	− 1512.00i	− 0.377964i
$253$	− 8736.00i	− 2.17086i
$254$	−1936.00	−0.478250
$255$	0	0
$256$	−119.000	−0.0290527
$257$	7614.00i	1.84805i	0.382335	+	0.924024i	$0.375120\pi$
$257$	7614.00i	1.84805i	−0.382335	+	0.924024i	$0.624880\pi$
$258$	− 564.000i	− 0.136097i
$259$	−816.000	−0.195767
$260$	0	0
$261$	2070.00	0.490919
$262$	− 732.000i	− 0.172607i
$263$	− 4888.00i	− 1.14603i	−0.819543	−	0.573017i	$-0.805773\pi$
$263$	− 4888.00i	− 1.14603i	0.819543	−	0.573017i	$-0.194227\pi$
$264$	2340.00	0.545519
$265$	0	0
$266$	480.000	0.110642
$267$	− 990.000i	− 0.226918i
$268$	588.000i	0.134022i
$269$	−1270.00	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$269$	−1270.00	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$270$	0	0
$271$	1072.00	0.240293	0.120146	−	0.992756i	$-0.461664\pi$
$271$	1072.00	0.240293	0.120146	+	0.992756i	$0.461664\pi$
$272$	574.000i	0.127955i
$273$	− 1584.00i	− 0.351165i
$274$	2214.00	0.488148
$275$	0	0
$276$	3528.00	0.769423
$277$	5394.00i	1.17001i	0.811028	+	0.585007i	$0.198908\pi$
$277$	5394.00i	1.17001i	−0.811028	+	0.585007i	$0.801092\pi$
$278$	20.0000i	0.00431482i
$279$	2592.00	0.556197
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	− 768.000i	− 0.162176i
$283$	2772.00i	0.582255i	0.956684	+	0.291128i	$0.0940305\pi$
$283$	2772.00i	0.582255i	−0.956684	+	0.291128i	$0.905970\pi$
$284$	−2296.00	−0.479727
$285$	0	0
$286$	1144.00	0.236525
$287$	2928.00i	0.602210i
$288$	1449.00i	0.296469i
$289$	4717.00	0.960106
$290$	0	0
$291$	2598.00	0.523359
$292$	− 266.000i	− 0.0533098i
$293$	4542.00i	0.905619i	0.891607	+	0.452810i	$0.149578\pi$
$293$	4542.00i	0.905619i	−0.891607	+	0.452810i	$0.850422\pi$
$294$	−699.000	−0.138662
$295$	0	0
$296$	510.000	0.100146
$297$	− 1404.00i	− 0.274304i
$298$	− 1330.00i	− 0.258540i
$299$	3696.00	0.714867
$300$	0	0
$301$	4512.00	0.864011
$302$	1208.00i	0.230174i
$303$	− 3654.00i	− 0.692795i
$304$	820.000	0.154705
$305$	0	0
$306$	−126.000	−0.0235390
$307$	− 5116.00i	− 0.951093i	−0.879691	−	0.475546i	$-0.842250\pi$
$307$	− 5116.00i	− 0.951093i	0.879691	−	0.475546i	$-0.157750\pi$
$308$	8736.00i	1.61617i
$309$	264.000	0.0486034
$310$	0	0
$311$	−2808.00	−0.511984	−0.255992	−	0.966679i	$-0.582402\pi$
$311$	−2808.00	−0.511984	−0.255992	+	0.966679i	$0.582402\pi$
$312$	990.000i	0.179640i
$313$	− 7318.00i	− 1.32153i	−0.750594	−	0.660763i	$-0.770233\pi$
$313$	− 7318.00i	− 1.32153i	0.750594	−	0.660763i	$-0.229767\pi$
$314$	3514.00	0.631549
$315$	0	0
$316$	1680.00	0.299074
$317$	− 2246.00i	− 0.397943i	−0.980005	−	0.198971i	$-0.936240\pi$
$317$	− 2246.00i	− 0.397943i	0.980005	−	0.198971i	$-0.0637601\pi$
$318$	− 1014.00i	− 0.178812i
$319$	−11960.0	−2.09916
$320$	0	0
$321$	108.000	0.0187787
$322$	− 4032.00i	− 0.697809i
$323$	280.000i	0.0482341i
$324$	567.000	0.0972222
$325$	0	0
$326$	−2068.00	−0.351337
$327$	2910.00i	0.492120i
$328$	− 1830.00i	− 0.308064i
$329$	6144.00	1.02957
$330$	0	0
$331$	1332.00	0.221188	0.110594	−	0.993866i	$-0.464725\pi$
$331$	1332.00	0.221188	0.110594	+	0.993866i	$0.464725\pi$
$332$	8484.00i	1.40247i
$333$	− 306.000i	− 0.0503564i
$334$	24.0000	0.00393180
$335$	0	0
$336$	−2952.00	−0.479300
$337$	11534.0i	1.86438i	0.361966	+	0.932191i	$0.382106\pi$
$337$	11534.0i	1.86438i	−0.361966	+	0.932191i	$0.617894\pi$
$338$	− 1713.00i	− 0.275665i
$339$	−3126.00	−0.500829
$340$	0	0
$341$	−14976.0	−2.37829
$342$	180.000i	0.0284599i
$343$	2640.00i	0.415588i
$344$	−2820.00	−0.441989
$345$	0	0
$346$	−618.000	−0.0960228
$347$	− 11956.0i	− 1.84966i	−0.380382	−	0.924830i	$-0.624207\pi$
$347$	− 11956.0i	− 1.84966i	0.380382	−	0.924830i	$-0.375793\pi$
$348$	− 4830.00i	− 0.744009i
$349$	−4870.00	−0.746949	−0.373474	−	0.927640i	$-0.621834\pi$
$349$	−4870.00	−0.746949	−0.373474	+	0.927640i	$0.621834\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	− 8372.00i	− 1.26770i
$353$	10722.0i	1.61664i	0.588742	+	0.808321i	$0.299623\pi$
$353$	10722.0i	1.61664i	−0.588742	+	0.808321i	$0.700377\pi$
$354$	−300.000	−0.0450419
$355$	0	0
$356$	−2310.00	−0.343904
$357$	− 1008.00i	− 0.149437i
$358$	3340.00i	0.493085i
$359$	−120.000	−0.0176417	−0.00882083	−	0.999961i	$-0.502808\pi$
$359$	−120.000	−0.0176417	−0.00882083	+	0.999961i	$0.502808\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	178.000i	0.0258438i
$363$	4119.00i	0.595569i
$364$	−3696.00	−0.532206
$365$	0	0
$366$	2226.00	0.317910
$367$	− 3936.00i	− 0.559830i	−0.960025	−	0.279915i	$-0.909694\pi$
$367$	− 3936.00i	− 0.559830i	0.960025	−	0.279915i	$-0.0903063\pi$
$368$	− 6888.00i	− 0.975711i
$369$	−1098.00	−0.154904
$370$	0	0
$371$	8112.00	1.13519
$372$	− 6048.00i	− 0.842941i
$373$	3022.00i	0.419499i	0.977755	+	0.209750i	$0.0672649\pi$
$373$	3022.00i	0.419499i	−0.977755	+	0.209750i	$0.932735\pi$
$374$	728.000	0.100652
$375$	0	0
$376$	−3840.00	−0.526683
$377$	− 5060.00i	− 0.691255i
$378$	− 648.000i	− 0.0881733i
$379$	13340.0	1.80799	0.903997	−	0.427539i	$-0.140619\pi$
$379$	13340.0	1.80799	0.903997	+	0.427539i	$0.140619\pi$
$380$	0	0
$381$	5808.00	0.780979
$382$	1888.00i	0.252876i
$383$	− 1008.00i	− 0.134481i	−0.997737	−	0.0672407i	$-0.978580\pi$
$383$	− 1008.00i	− 0.134481i	0.997737	−	0.0672407i	$-0.0214195\pi$
$384$	4365.00	0.580079
$385$	0	0
$386$	1922.00	0.253438
$387$	1692.00i	0.222246i
$388$	− 6062.00i	− 0.793174i
$389$	−9630.00	−1.25517	−0.627584	−	0.778549i	$-0.715956\pi$
$389$	−9630.00	−1.25517	−0.627584	+	0.778549i	$0.715956\pi$
$390$	0	0
$391$	2352.00	0.304209
$392$	3495.00i	0.450317i
$393$	2196.00i	0.281867i
$394$	−2526.00	−0.322990
$395$	0	0
$396$	−3276.00	−0.415720
$397$	− 7126.00i	− 0.900866i	−0.892810	−	0.450433i	$-0.851270\pi$
$397$	− 7126.00i	− 0.900866i	0.892810	−	0.450433i	$-0.148730\pi$
$398$	− 1160.00i	− 0.146094i
$399$	−1440.00	−0.180677
$400$	0	0
$401$	−8718.00	−1.08568	−0.542838	−	0.839837i	$-0.682650\pi$
$401$	−8718.00	−1.08568	−0.542838	+	0.839837i	$0.682650\pi$
$402$	252.000i	0.0312652i
$403$	− 6336.00i	− 0.783173i
$404$	−8526.00	−1.04996
$405$	0	0
$406$	−5520.00	−0.674761
$407$	1768.00i	0.215323i
$408$	630.000i	0.0764452i
$409$	10870.0	1.31415	0.657074	−	0.753826i	$-0.271794\pi$
$409$	10870.0	1.31415	0.657074	+	0.753826i	$0.271794\pi$
$410$	0	0
$411$	−6642.00	−0.797143
$412$	− 616.000i	− 0.0736605i
$413$	− 2400.00i	− 0.285947i
$414$	1512.00	0.179495
$415$	0	0
$416$	3542.00	0.417454
$417$	− 60.0000i	− 0.00704607i
$418$	− 1040.00i	− 0.121694i
$419$	9700.00	1.13097	0.565484	−	0.824759i	$-0.308689\pi$
$419$	9700.00	1.13097	0.565484	+	0.824759i	$0.308689\pi$
$420$	0	0
$421$	862.000	0.0997893	0.0498947	−	0.998754i	$-0.484111\pi$
$421$	862.000	0.0997893	0.0498947	+	0.998754i	$0.484111\pi$
$422$	4468.00i	0.515400i
$423$	2304.00i	0.264833i
$424$	−5070.00	−0.580710
$425$	0	0
$426$	−984.000	−0.111913
$427$	17808.0i	2.01824i
$428$	− 252.000i	− 0.0284600i
$429$	−3432.00	−0.386244
$430$	0	0
$431$	15792.0	1.76490	0.882452	−	0.470402i	$-0.155891\pi$
$431$	15792.0	1.76490	0.882452	+	0.470402i	$0.155891\pi$
$432$	− 1107.00i	− 0.123288i
$433$	11602.0i	1.28766i	0.765169	+	0.643830i	$0.222655\pi$
$433$	11602.0i	1.28766i	−0.765169	+	0.643830i	$0.777345\pi$
$434$	−6912.00	−0.764485
$435$	0	0
$436$	6790.00	0.745830
$437$	− 3360.00i	− 0.367805i
$438$	− 114.000i	− 0.0124364i
$439$	440.000	0.0478361	0.0239181	−	0.999714i	$-0.492386\pi$
$439$	440.000	0.0478361	0.0239181	+	0.999714i	$0.492386\pi$
$440$	0	0
$441$	2097.00	0.226433
$442$	308.000i	0.0331449i
$443$	− 10188.0i	− 1.09266i	−0.837571	−	0.546328i	$-0.816025\pi$
$443$	− 10188.0i	− 1.09266i	0.837571	−	0.546328i	$-0.183975\pi$
$444$	−714.000	−0.0763174
$445$	0	0
$446$	6032.00	0.640411
$447$	3990.00i	0.422194i
$448$	4008.00i	0.422679i
$449$	13310.0	1.39897	0.699485	−	0.714647i	$-0.253413\pi$
$449$	13310.0	1.39897	0.699485	+	0.714647i	$0.253413\pi$
$450$	0	0
$451$	6344.00	0.662367
$452$	7294.00i	0.759029i
$453$	− 3624.00i	− 0.375873i
$454$	−2636.00	−0.272497
$455$	0	0
$456$	900.000	0.0924262
$457$	− 3226.00i	− 0.330210i	−0.986276	−	0.165105i	$-0.947204\pi$
$457$	− 3226.00i	− 0.330210i	0.986276	−	0.165105i	$-0.0527963\pi$
$458$	4830.00i	0.492775i
$459$	378.000	0.0384391
$460$	0	0
$461$	6582.00	0.664977	0.332488	−	0.943107i	$-0.392112\pi$
$461$	6582.00	0.664977	0.332488	+	0.943107i	$0.392112\pi$
$462$	3744.00i	0.377027i
$463$	15072.0i	1.51286i	0.654073	+	0.756431i	$0.273059\pi$
$463$	15072.0i	1.51286i	−0.654073	+	0.756431i	$0.726941\pi$
$464$	−9430.00	−0.943484
$465$	0	0
$466$	2682.00	0.266612
$467$	− 476.000i	− 0.0471663i	−0.999722	−	0.0235831i	$-0.992493\pi$
$467$	− 476.000i	− 0.0471663i	0.999722	−	0.0235831i	$-0.00750744\pi$
$468$	− 1386.00i	− 0.136897i
$469$	−2016.00	−0.198487
$470$	0	0
$471$	−10542.0	−1.03132
$472$	1500.00i	0.146278i
$473$	− 9776.00i	− 0.950319i
$474$	720.000	0.0697694
$475$	0	0
$476$	−2352.00	−0.226478
$477$	3042.00i	0.291999i
$478$	2320.00i	0.221997i
$479$	19680.0	1.87725	0.938624	−	0.344941i	$-0.112101\pi$
$479$	19680.0	1.87725	0.938624	+	0.344941i	$0.112101\pi$
$480$	0	0
$481$	−748.000	−0.0709062
$482$	− 2002.00i	− 0.189188i
$483$	12096.0i	1.13952i
$484$	9611.00	0.902611
$485$	0	0
$486$	243.000	0.0226805
$487$	5944.00i	0.553077i	0.961003	+	0.276538i	$0.0891873\pi$
$487$	5944.00i	0.553077i	−0.961003	+	0.276538i	$0.910813\pi$
$488$	− 11130.0i	− 1.03244i
$489$	6204.00	0.573731
$490$	0	0
$491$	10772.0	0.990089	0.495044	−	0.868868i	$-0.335152\pi$
$491$	10772.0	0.990089	0.495044	+	0.868868i	$0.335152\pi$
$492$	2562.00i	0.234764i
$493$	− 3220.00i	− 0.294161i
$494$	440.000	0.0400740
$495$	0	0
$496$	−11808.0	−1.06894
$497$	− 7872.00i	− 0.710478i
$498$	3636.00i	0.327175i
$499$	−8140.00	−0.730253	−0.365127	−	0.930958i	$-0.618974\pi$
$499$	−8140.00	−0.730253	−0.365127	+	0.930958i	$0.618974\pi$
$500$	0	0
$501$	−72.0000	−0.00642060
$502$	− 132.000i	− 0.0117360i
$503$	− 13768.0i	− 1.22045i	−0.792229	−	0.610223i	$-0.791080\pi$
$503$	− 13768.0i	− 1.22045i	0.792229	−	0.610223i	$-0.208920\pi$
$504$	−3240.00	−0.286351
$505$	0	0
$506$	−8736.00	−0.767515
$507$	5139.00i	0.450160i
$508$	− 13552.0i	− 1.18361i
$509$	−22150.0	−1.92884	−0.964422	−	0.264368i	$-0.914837\pi$
$509$	−22150.0	−1.92884	−0.964422	+	0.264368i	$0.914837\pi$
$510$	0	0
$511$	912.000	0.0789521
$512$	− 11521.0i	− 0.994455i
$513$	− 540.000i	− 0.0464748i
$514$	7614.00	0.653384
$515$	0	0
$516$	3948.00	0.336824
$517$	− 13312.0i	− 1.13242i
$518$	816.000i	0.0692143i
$519$	1854.00	0.156805
$520$	0	0
$521$	1562.00	0.131348	0.0656741	−	0.997841i	$-0.479080\pi$
$521$	1562.00	0.131348	0.0656741	+	0.997841i	$0.479080\pi$
$522$	− 2070.00i	− 0.173566i
$523$	− 668.000i	− 0.0558501i	−0.999610	−	0.0279250i	$-0.991110\pi$
$523$	− 668.000i	− 0.0558501i	0.999610	−	0.0279250i	$-0.00888997\pi$
$524$	5124.00	0.427181
$525$	0	0
$526$	−4888.00	−0.405184
$527$	− 4032.00i	− 0.333276i
$528$	6396.00i	0.527178i
$529$	−16057.0	−1.31972
$530$	0	0
$531$	900.000	0.0735531
$532$	3360.00i	0.273824i
$533$	2684.00i	0.218118i
$534$	−990.000	−0.0802275
$535$	0	0
$536$	1260.00	0.101537
$537$	− 10020.0i	− 0.805205i
$538$	1270.00i	0.101772i
$539$	−12116.0	−0.968225
$540$	0	0
$541$	−6138.00	−0.487788	−0.243894	−	0.969802i	$-0.578425\pi$
$541$	−6138.00	−0.487788	−0.243894	+	0.969802i	$0.578425\pi$
$542$	− 1072.00i	− 0.0849564i
$543$	− 534.000i	− 0.0422028i
$544$	2254.00	0.177646
$545$	0	0
$546$	−1584.00	−0.124156
$547$	10484.0i	0.819494i	0.912199	+	0.409747i	$0.134383\pi$
$547$	10484.0i	0.819494i	−0.912199	+	0.409747i	$0.865617\pi$
$548$	15498.0i	1.20811i
$549$	−6678.00	−0.519144
$550$	0	0
$551$	−4600.00	−0.355656
$552$	− 7560.00i	− 0.582926i
$553$	5760.00i	0.442930i
$554$	5394.00	0.413663
$555$	0	0
$556$	−140.000	−0.0106786
$557$	− 3606.00i	− 0.274311i	−0.990550	−	0.137155i	$-0.956204\pi$
$557$	− 3606.00i	− 0.274311i	0.990550	−	0.137155i	$-0.0437960\pi$
$558$	− 2592.00i	− 0.196645i
$559$	4136.00	0.312941
$560$	0	0
$561$	−2184.00	−0.164365
$562$	− 2442.00i	− 0.183291i
$563$	12252.0i	0.917159i	0.888654	+	0.458579i	$0.151641\pi$
$563$	12252.0i	0.917159i	−0.888654	+	0.458579i	$0.848359\pi$
$564$	5376.00	0.401366
$565$	0	0
$566$	2772.00	0.205858
$567$	1944.00i	0.143986i
$568$	4920.00i	0.363448i
$569$	14550.0	1.07200	0.536000	−	0.844218i	$-0.319935\pi$
$569$	14550.0	1.07200	0.536000	+	0.844218i	$0.319935\pi$
$570$	0	0
$571$	−25468.0	−1.86655	−0.933277	−	0.359157i	$-0.883064\pi$
$571$	−25468.0	−1.86655	−0.933277	+	0.359157i	$0.883064\pi$
$572$	8008.00i	0.585369i
$573$	− 5664.00i	− 0.412944i
$574$	2928.00	0.212914
$575$	0	0
$576$	−1503.00	−0.108724
$577$	− 12866.0i	− 0.928282i	−0.885761	−	0.464141i	$-0.846363\pi$
$577$	− 12866.0i	− 0.928282i	0.885761	−	0.464141i	$-0.153637\pi$
$578$	− 4717.00i	− 0.339449i
$579$	−5766.00	−0.413863
$580$	0	0
$581$	−29088.0	−2.07706
$582$	− 2598.00i	− 0.185035i
$583$	− 17576.0i	− 1.24858i
$584$	−570.000	−0.0403883
$585$	0	0
$586$	4542.00	0.320185
$587$	14844.0i	1.04374i	0.853024	+	0.521872i	$0.174766\pi$
$587$	14844.0i	1.04374i	−0.853024	+	0.521872i	$0.825234\pi$
$588$	− 4893.00i	− 0.343170i
$589$	−5760.00	−0.402948
$590$	0	0
$591$	7578.00	0.527440
$592$	1394.00i	0.0967788i
$593$	20402.0i	1.41283i	0.707797	+	0.706416i	$0.249689\pi$
$593$	20402.0i	1.41283i	−0.707797	+	0.706416i	$0.750311\pi$
$594$	−1404.00	−0.0969812
$595$	0	0
$596$	9310.00	0.639853
$597$	3480.00i	0.238571i
$598$	− 3696.00i	− 0.252744i
$599$	−10760.0	−0.733959	−0.366980	−	0.930229i	$-0.619608\pi$
$599$	−10760.0	−0.733959	−0.366980	+	0.930229i	$0.619608\pi$
$600$	0	0
$601$	14282.0	0.969343	0.484671	−	0.874696i	$-0.338939\pi$
$601$	14282.0	0.969343	0.484671	+	0.874696i	$0.338939\pi$
$602$	− 4512.00i	− 0.305474i
$603$	− 756.000i	− 0.0510559i
$604$	−8456.00	−0.569652
$605$	0	0
$606$	−3654.00	−0.244940
$607$	− 11056.0i	− 0.739290i	−0.929173	−	0.369645i	$-0.879479\pi$
$607$	− 11056.0i	− 0.739290i	0.929173	−	0.369645i	$-0.120521\pi$
$608$	− 3220.00i	− 0.214783i
$609$	16560.0	1.10188
$610$	0	0
$611$	5632.00	0.372907
$612$	− 882.000i	− 0.0582561i
$613$	− 16418.0i	− 1.08176i	−0.841101	−	0.540878i	$-0.818092\pi$
$613$	− 16418.0i	− 1.08176i	0.841101	−	0.540878i	$-0.181908\pi$
$614$	−5116.00	−0.336262
$615$	0	0
$616$	18720.0	1.22443
$617$	10374.0i	0.676891i	0.940986	+	0.338445i	$0.109901\pi$
$617$	10374.0i	0.676891i	−0.940986	+	0.338445i	$0.890099\pi$
$618$	− 264.000i	− 0.0171839i
$619$	5260.00	0.341546	0.170773	−	0.985310i	$-0.445373\pi$
$619$	5260.00	0.341546	0.170773	+	0.985310i	$0.445373\pi$
$620$	0	0
$621$	−4536.00	−0.293113
$622$	2808.00i	0.181014i
$623$	− 7920.00i	− 0.509323i
$624$	−2706.00	−0.173600
$625$	0	0
$626$	−7318.00	−0.467230
$627$	3120.00i	0.198725i
$628$	24598.0i	1.56300i
$629$	−476.000	−0.0301739
$630$	0	0
$631$	21352.0	1.34708	0.673542	−	0.739149i	$-0.264772\pi$
$631$	21352.0	1.34708	0.673542	+	0.739149i	$0.264772\pi$
$632$	− 3600.00i	− 0.226583i
$633$	− 13404.0i	− 0.841645i
$634$	−2246.00	−0.140694
$635$	0	0
$636$	7098.00	0.442538
$637$	− 5126.00i	− 0.318838i
$638$	11960.0i	0.742164i
$639$	2952.00	0.182753
$640$	0	0
$641$	−29118.0	−1.79422	−0.897108	−	0.441812i	$-0.854336\pi$
$641$	−29118.0	−1.79422	−0.897108	+	0.441812i	$0.854336\pi$
$642$	− 108.000i	− 0.00663928i
$643$	5772.00i	0.354005i	0.984210	+	0.177003i	$0.0566401\pi$
$643$	5772.00i	0.354005i	−0.984210	+	0.177003i	$0.943360\pi$
$644$	28224.0	1.72699
$645$	0	0
$646$	280.000	0.0170533
$647$	14264.0i	0.866732i	0.901218	+	0.433366i	$0.142674\pi$
$647$	14264.0i	0.866732i	−0.901218	+	0.433366i	$0.857326\pi$
$648$	− 1215.00i	− 0.0736570i
$649$	−5200.00	−0.314511
$650$	0	0
$651$	20736.0	1.24840
$652$	− 14476.0i	− 0.869515i
$653$	6902.00i	0.413623i	0.978381	+	0.206812i	$0.0663088\pi$
$653$	6902.00i	0.413623i	−0.978381	+	0.206812i	$0.933691\pi$
$654$	2910.00	0.173991
$655$	0	0
$656$	5002.00	0.297706
$657$	342.000i	0.0203085i
$658$	− 6144.00i	− 0.364009i
$659$	−20140.0	−1.19051	−0.595253	−	0.803539i	$-0.702948\pi$
$659$	−20140.0	−1.19051	−0.595253	+	0.803539i	$0.702948\pi$
$660$	0	0
$661$	−3218.00	−0.189358	−0.0946790	−	0.995508i	$-0.530182\pi$
$661$	−3218.00	−0.189358	−0.0946790	+	0.995508i	$0.530182\pi$
$662$	− 1332.00i	− 0.0782019i
$663$	− 924.000i	− 0.0541255i
$664$	18180.0	1.06253
$665$	0	0
$666$	−306.000	−0.0178037
$667$	38640.0i	2.24310i
$668$	168.000i	0.00973071i
$669$	−18096.0	−1.04579
$670$	0	0
$671$	38584.0	2.21985
$672$	11592.0i	0.665433i
$673$	− 7518.00i	− 0.430606i	−0.976547	−	0.215303i	$-0.930926\pi$
$673$	− 7518.00i	− 0.430606i	0.976547	−	0.215303i	$-0.0690739\pi$
$674$	11534.0	0.659159
$675$	0	0
$676$	11991.0	0.682237
$677$	18114.0i	1.02833i	0.857692	+	0.514164i	$0.171898\pi$
$677$	18114.0i	1.02833i	−0.857692	+	0.514164i	$0.828102\pi$
$678$	3126.00i	0.177070i
$679$	20784.0	1.17469
$680$	0	0
$681$	7908.00	0.444986
$682$	14976.0i	0.840851i
$683$	− 23868.0i	− 1.33716i	−0.743638	−	0.668582i	$-0.766901\pi$
$683$	− 23868.0i	− 1.33716i	0.743638	−	0.668582i	$-0.233099\pi$
$684$	−1260.00	−0.0704347
$685$	0	0
$686$	2640.00	0.146932
$687$	− 14490.0i	− 0.804699i
$688$	− 7708.00i	− 0.427129i
$689$	7436.00	0.411160
$690$	0	0
$691$	172.000	0.00946916	0.00473458	−	0.999989i	$-0.498493\pi$
$691$	172.000	0.00946916	0.00473458	+	0.999989i	$0.498493\pi$
$692$	− 4326.00i	− 0.237644i
$693$	− 11232.0i	− 0.615683i
$694$	−11956.0	−0.653953
$695$	0	0
$696$	−10350.0	−0.563672
$697$	1708.00i	0.0928194i
$698$	4870.00i	0.264086i
$699$	−8046.00	−0.435376
$700$	0	0
$701$	−22138.0	−1.19278	−0.596391	−	0.802694i	$-0.703399\pi$
$701$	−22138.0	−1.19278	−0.596391	+	0.802694i	$0.703399\pi$
$702$	− 594.000i	− 0.0319360i
$703$	680.000i	0.0364818i
$704$	8684.00	0.464901
$705$	0	0
$706$	10722.0	0.571569
$707$	− 29232.0i	− 1.55500i
$708$	− 2100.00i	− 0.111473i
$709$	−3070.00	−0.162618	−0.0813091	−	0.996689i	$-0.525910\pi$
$709$	−3070.00	−0.162618	−0.0813091	+	0.996689i	$0.525910\pi$
$710$	0	0
$711$	−2160.00	−0.113933
$712$	4950.00i	0.260546i
$713$	48384.0i	2.54137i
$714$	−1008.00	−0.0528340
$715$	0	0
$716$	−23380.0	−1.22032
$717$	− 6960.00i	− 0.362519i
$718$	120.000i	0.00623727i
$719$	−15600.0	−0.809154	−0.404577	−	0.914504i	$-0.632581\pi$
$719$	−15600.0	−0.809154	−0.404577	+	0.914504i	$0.632581\pi$
$720$	0	0
$721$	2112.00	0.109092
$722$	6459.00i	0.332935i
$723$	6006.00i	0.308943i
$724$	−1246.00	−0.0639603
$725$	0	0
$726$	4119.00	0.210565
$727$	− 20696.0i	− 1.05581i	−0.849304	−	0.527904i	$-0.822978\pi$
$727$	− 20696.0i	− 1.05581i	0.849304	−	0.527904i	$-0.177022\pi$
$728$	7920.00i	0.403207i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	2632.00	0.133171
$732$	15582.0i	0.786786i
$733$	− 30778.0i	− 1.55090i	−0.631408	−	0.775451i	$-0.717522\pi$
$733$	− 30778.0i	− 1.55090i	0.631408	−	0.775451i	$-0.282478\pi$
$734$	−3936.00	−0.197930
$735$	0	0
$736$	−27048.0	−1.35462
$737$	4368.00i	0.218314i
$738$	1098.00i	0.0547669i
$739$	−11740.0	−0.584388	−0.292194	−	0.956359i	$-0.594385\pi$
$739$	−11740.0	−0.584388	−0.292194	+	0.956359i	$0.594385\pi$
$740$	0	0
$741$	−1320.00	−0.0654405
$742$	− 8112.00i	− 0.401349i
$743$	2632.00i	0.129958i	0.997887	+	0.0649789i	$0.0206980\pi$
$743$	2632.00i	0.129958i	−0.997887	+	0.0649789i	$0.979302\pi$
$744$	−12960.0	−0.638625
$745$	0	0
$746$	3022.00	0.148315
$747$	− 10908.0i	− 0.534274i
$748$	5096.00i	0.249102i
$749$	864.000	0.0421494
$750$	0	0
$751$	−20528.0	−0.997440	−0.498720	−	0.866763i	$-0.666196\pi$
$751$	−20528.0	−0.997440	−0.498720	+	0.866763i	$0.666196\pi$
$752$	− 10496.0i	− 0.508976i
$753$	396.000i	0.0191647i
$754$	−5060.00	−0.244396
$755$	0	0
$756$	4536.00	0.218218
$757$	− 21646.0i	− 1.03928i	−0.854384	−	0.519642i	$-0.826066\pi$
$757$	− 21646.0i	− 1.03928i	0.854384	−	0.519642i	$-0.173934\pi$
$758$	− 13340.0i	− 0.639222i
$759$	26208.0	1.25335
$760$	0	0
$761$	18282.0	0.870857	0.435428	−	0.900223i	$-0.356597\pi$
$761$	18282.0	0.870857	0.435428	+	0.900223i	$0.356597\pi$
$762$	− 5808.00i	− 0.276118i
$763$	23280.0i	1.10458i
$764$	−13216.0	−0.625835
$765$	0	0
$766$	−1008.00	−0.0475464
$767$	− 2200.00i	− 0.103569i
$768$	− 357.000i	− 0.0167736i
$769$	24190.0	1.13435	0.567174	−	0.823598i	$-0.308037\pi$
$769$	24190.0	1.13435	0.567174	+	0.823598i	$0.308037\pi$
$770$	0	0
$771$	−22842.0	−1.06697
$772$	13454.0i	0.627228i
$773$	− 25698.0i	− 1.19572i	−0.801600	−	0.597861i	$-0.796018\pi$
$773$	− 25698.0i	− 1.19572i	0.801600	−	0.597861i	$-0.203982\pi$
$774$	1692.00	0.0785758
$775$	0	0
$776$	−12990.0	−0.600920
$777$	− 2448.00i	− 0.113026i
$778$	9630.00i	0.443769i
$779$	2440.00	0.112223
$780$	0	0
$781$	−17056.0	−0.781449
$782$	− 2352.00i	− 0.107554i
$783$	6210.00i	0.283432i
$784$	−9553.00	−0.435177
$785$	0	0
$786$	2196.00	0.0996549
$787$	− 33436.0i	− 1.51444i	−0.653160	−	0.757220i	$-0.726557\pi$
$787$	− 33436.0i	− 1.51444i	0.653160	−	0.757220i	$-0.273443\pi$
$788$	− 17682.0i	− 0.799359i
$789$	14664.0	0.661663
$790$	0	0
$791$	−25008.0	−1.12412
$792$	7020.00i	0.314956i
$793$	16324.0i	0.730999i
$794$	−7126.00	−0.318504
$795$	0	0
$796$	8120.00	0.361565
$797$	37594.0i	1.67083i	0.549623	+	0.835413i	$0.314771\pi$
$797$	37594.0i	1.67083i	−0.549623	+	0.835413i	$0.685229\pi$
$798$	1440.00i	0.0638790i
$799$	3584.00	0.158689
$800$	0	0
$801$	2970.00	0.131011
$802$	8718.00i	0.383844i
$803$	− 1976.00i	− 0.0868388i
$804$	−1764.00	−0.0773775
$805$	0	0
$806$	−6336.00	−0.276893
$807$	− 3810.00i	− 0.166194i
$808$	18270.0i	0.795466i
$809$	−4730.00	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$809$	−4730.00	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$810$	0	0
$811$	−8748.00	−0.378772	−0.189386	−	0.981903i	$-0.560650\pi$
$811$	−8748.00	−0.378772	−0.189386	+	0.981903i	$0.560650\pi$
$812$	− 38640.0i	− 1.66995i
$813$	3216.00i	0.138733i
$814$	1768.00	0.0761282
$815$	0	0
$816$	−1722.00	−0.0738751
$817$	− 3760.00i	− 0.161011i
$818$	− 10870.0i	− 0.464622i
$819$	4752.00	0.202745
$820$	0	0
$821$	44142.0	1.87645	0.938226	−	0.346024i	$-0.112468\pi$
$821$	44142.0	1.87645	0.938226	+	0.346024i	$0.112468\pi$
$822$	6642.00i	0.281833i
$823$	3992.00i	0.169079i	0.996420	+	0.0845397i	$0.0269420\pi$
$823$	3992.00i	0.169079i	−0.996420	+	0.0845397i	$0.973058\pi$
$824$	−1320.00	−0.0558063
$825$	0	0
$826$	−2400.00	−0.101098
$827$	14444.0i	0.607336i	0.952778	+	0.303668i	$0.0982114\pi$
$827$	14444.0i	0.607336i	−0.952778	+	0.303668i	$0.901789\pi$
$828$	10584.0i	0.444226i
$829$	−42150.0	−1.76590	−0.882949	−	0.469468i	$-0.844446\pi$
$829$	−42150.0	−1.76590	−0.882949	+	0.469468i	$0.844446\pi$
$830$	0	0
$831$	−16182.0	−0.675508
$832$	3674.00i	0.153093i
$833$	− 3262.00i	− 0.135680i
$834$	−60.0000	−0.00249116
$835$	0	0
$836$	7280.00	0.301177
$837$	7776.00i	0.321121i
$838$	− 9700.00i	− 0.399858i
$839$	−13400.0	−0.551394	−0.275697	−	0.961245i	$-0.588909\pi$
$839$	−13400.0	−0.551394	−0.275697	+	0.961245i	$0.588909\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	− 862.000i	− 0.0352809i
$843$	7326.00i	0.299313i
$844$	−31276.0	−1.27555
$845$	0	0
$846$	2304.00	0.0936326
$847$	32952.0i	1.33677i
$848$	− 13858.0i	− 0.561186i
$849$	−8316.00	−0.336165
$850$	0	0
$851$	5712.00	0.230088
$852$	− 6888.00i	− 0.276971i
$853$	− 8658.00i	− 0.347531i	−0.984787	−	0.173766i	$-0.944406\pi$
$853$	− 8658.00i	− 0.347531i	0.984787	−	0.173766i	$-0.0555935\pi$
$854$	17808.0	0.713556
$855$	0	0
$856$	−540.000	−0.0215617
$857$	− 42826.0i	− 1.70701i	−0.521084	−	0.853505i	$-0.674472\pi$
$857$	− 42826.0i	− 1.70701i	0.521084	−	0.853505i	$-0.325528\pi$
$858$	3432.00i	0.136558i
$859$	35900.0	1.42595	0.712976	−	0.701189i	$-0.247347\pi$
$859$	35900.0	1.42595	0.712976	+	0.701189i	$0.247347\pi$
$860$	0	0
$861$	−8784.00	−0.347686
$862$	− 15792.0i	− 0.623988i
$863$	− 3088.00i	− 0.121804i	−0.998144	−	0.0609019i	$-0.980602\pi$
$863$	− 3088.00i	− 0.121804i	0.998144	−	0.0609019i	$-0.0193977\pi$
$864$	−4347.00	−0.171167
$865$	0	0
$866$	11602.0	0.455256
$867$	14151.0i	0.554317i
$868$	− 48384.0i	− 1.89200i
$869$	12480.0	0.487175
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	− 14550.0i	− 0.565052i
$873$	7794.00i	0.302161i
$874$	−3360.00	−0.130039
$875$	0	0
$876$	798.000	0.0307784
$877$	35274.0i	1.35817i	0.734058	+	0.679087i	$0.237624\pi$
$877$	35274.0i	1.35817i	−0.734058	+	0.679087i	$0.762376\pi$
$878$	− 440.000i	− 0.0169126i
$879$	−13626.0	−0.522860
$880$	0	0
$881$	25042.0	0.957646	0.478823	−	0.877911i	$-0.341064\pi$
$881$	25042.0	0.957646	0.478823	+	0.877911i	$0.341064\pi$
$882$	− 2097.00i	− 0.0800563i
$883$	12572.0i	0.479141i	0.970879	+	0.239570i	$0.0770066\pi$
$883$	12572.0i	0.479141i	−0.970879	+	0.239570i	$0.922993\pi$
$884$	−2156.00	−0.0820296
$885$	0	0
$886$	−10188.0	−0.386312
$887$	21864.0i	0.827645i	0.910358	+	0.413823i	$0.135807\pi$
$887$	21864.0i	0.827645i	−0.910358	+	0.413823i	$0.864193\pi$
$888$	1530.00i	0.0578192i
$889$	46464.0	1.75293
$890$	0	0
$891$	4212.00	0.158370
$892$	42224.0i	1.58494i
$893$	− 5120.00i	− 0.191864i
$894$	3990.00	0.149268
$895$	0	0
$896$	34920.0	1.30200
$897$	11088.0i	0.412729i
$898$	− 13310.0i	− 0.494611i
$899$	66240.0	2.45743
$900$	0	0
$901$	4732.00	0.174968
$902$	− 6344.00i	− 0.234182i
$903$	13536.0i	0.498837i
$904$	15630.0	0.575051
$905$	0	0
$906$	−3624.00	−0.132891
$907$	− 31236.0i	− 1.14352i	−0.820420	−	0.571761i	$-0.806260\pi$
$907$	− 31236.0i	− 1.14352i	0.820420	−	0.571761i	$-0.193740\pi$
$908$	− 18452.0i	− 0.674396i
$909$	10962.0	0.399985
$910$	0	0
$911$	8272.00	0.300838	0.150419	−	0.988622i	$-0.451938\pi$
$911$	8272.00	0.300838	0.150419	+	0.988622i	$0.451938\pi$
$912$	2460.00i	0.0893188i
$913$	63024.0i	2.28455i
$914$	−3226.00	−0.116747
$915$	0	0
$916$	−33810.0	−1.21956
$917$	17568.0i	0.632657i
$918$	− 378.000i	− 0.0135903i
$919$	−20200.0	−0.725067	−0.362533	−	0.931971i	$-0.618088\pi$
$919$	−20200.0	−0.725067	−0.362533	+	0.931971i	$0.618088\pi$
$920$	0	0
$921$	15348.0	0.549114
$922$	− 6582.00i	− 0.235105i
$923$	− 7216.00i	− 0.257332i
$924$	−26208.0	−0.933095
$925$	0	0
$926$	15072.0	0.534878
$927$	792.000i	0.0280612i
$928$	37030.0i	1.30988i
$929$	−31010.0	−1.09516	−0.547581	−	0.836753i	$-0.684451\pi$
$929$	−31010.0	−1.09516	−0.547581	+	0.836753i	$0.684451\pi$
$930$	0	0
$931$	−4660.00	−0.164044
$932$	18774.0i	0.659831i
$933$	− 8424.00i	− 0.295594i
$934$	−476.000	−0.0166758
$935$	0	0
$936$	−2970.00	−0.103715
$937$	39174.0i	1.36580i	0.730510	+	0.682902i	$0.239283\pi$
$937$	39174.0i	1.36580i	−0.730510	+	0.682902i	$0.760717\pi$
$938$	2016.00i	0.0701756i
$939$	21954.0	0.762984
$940$	0	0
$941$	−4138.00	−0.143353	−0.0716764	−	0.997428i	$-0.522835\pi$
$941$	−4138.00	−0.143353	−0.0716764	+	0.997428i	$0.522835\pi$
$942$	10542.0i	0.364625i
$943$	− 20496.0i	− 0.707785i
$944$	−4100.00	−0.141360
$945$	0	0
$946$	−9776.00	−0.335989
$947$	− 23676.0i	− 0.812425i	−0.913779	−	0.406213i	$-0.866849\pi$
$947$	− 23676.0i	− 0.812425i	0.913779	−	0.406213i	$-0.133151\pi$
$948$	5040.00i	0.172670i
$949$	836.000	0.0285961
$950$	0	0
$951$	6738.00	0.229752
$952$	5040.00i	0.171583i
$953$	18922.0i	0.643173i	0.946880	+	0.321586i	$0.104216\pi$
$953$	18922.0i	0.643173i	−0.946880	+	0.321586i	$0.895784\pi$
$954$	3042.00	0.103237
$955$	0	0
$956$	−16240.0	−0.549413
$957$	− 35880.0i	− 1.21195i
$958$	− 19680.0i	− 0.663708i
$959$	−53136.0	−1.78921
$960$	0	0
$961$	53153.0	1.78420
$962$	748.000i	0.0250691i
$963$	324.000i	0.0108419i
$964$	14014.0	0.468216
$965$	0	0
$966$	12096.0	0.402880
$967$	− 39656.0i	− 1.31877i	−0.751805	−	0.659385i	$-0.770817\pi$
$967$	− 39656.0i	− 1.31877i	0.751805	−	0.659385i	$-0.229183\pi$
$968$	− 20595.0i	− 0.683831i
$969$	−840.000	−0.0278480
$970$	0	0
$971$	−33228.0	−1.09818	−0.549092	−	0.835762i	$-0.685026\pi$
$971$	−33228.0	−1.09818	−0.549092	+	0.835762i	$0.685026\pi$
$972$	1701.00i	0.0561313i
$973$	− 480.000i	− 0.0158151i
$974$	5944.00	0.195542
$975$	0	0
$976$	30422.0	0.997730
$977$	974.000i	0.0318946i	0.999873	+	0.0159473i	$0.00507640\pi$
$977$	974.000i	0.0318946i	−0.999873	+	0.0159473i	$0.994924\pi$
$978$	− 6204.00i	− 0.202845i
$979$	−17160.0	−0.560200
$980$	0	0
$981$	−8730.00	−0.284126
$982$	− 10772.0i	− 0.350049i
$983$	− 13608.0i	− 0.441534i	−0.975327	−	0.220767i	$-0.929144\pi$
$983$	− 13608.0i	− 0.441534i	0.975327	−	0.220767i	$-0.0708560\pi$
$984$	5490.00	0.177861
$985$	0	0
$986$	−3220.00	−0.104002
$987$	18432.0i	0.594425i
$988$	3080.00i	0.0991780i
$989$	−31584.0	−1.01548
$990$	0	0
$991$	13472.0	0.431839	0.215919	−	0.976411i	$-0.430725\pi$
$991$	13472.0	0.431839	0.215919	+	0.976411i	$0.430725\pi$
$992$	46368.0i	1.48406i
$993$	3996.00i	0.127703i
$994$	−7872.00	−0.251192
$995$	0	0
$996$	−25452.0	−0.809716
$997$	3234.00i	0.102730i	0.998680	+	0.0513650i	$0.0163572\pi$
$997$	3234.00i	0.102730i	−0.998680	+	0.0513650i	$0.983643\pi$
$998$	8140.00i	0.258184i
$999$	918.000	0.0290733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.4.b.b.49.1		2
3.2	odd	2		225.4.b.e.199.2		2
4.3	odd	2		1200.4.f.b.49.1		2
5.2	odd	4		15.4.a.a.1.1	✓	1
5.3	odd	4		75.4.a.b.1.1		1
5.4	even	2	inner	75.4.b.b.49.2		2
15.2	even	4		45.4.a.c.1.1		1
15.8	even	4		225.4.a.f.1.1		1
15.14	odd	2		225.4.b.e.199.1		2
20.3	even	4		1200.4.a.t.1.1		1
20.7	even	4		240.4.a.e.1.1		1
20.19	odd	2		1200.4.f.b.49.2		2
35.27	even	4		735.4.a.e.1.1		1
40.27	even	4		960.4.a.ba.1.1		1
40.37	odd	4		960.4.a.b.1.1		1
45.2	even	12		405.4.e.i.271.1		2
45.7	odd	12		405.4.e.g.271.1		2
45.22	odd	12		405.4.e.g.136.1		2
45.32	even	12		405.4.e.i.136.1		2
55.32	even	4		1815.4.a.e.1.1		1
60.47	odd	4		720.4.a.n.1.1		1
105.62	odd	4		2205.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.a.1.1	✓	1	5.2	odd	4
45.4.a.c.1.1		1	15.2	even	4
75.4.a.b.1.1		1	5.3	odd	4
75.4.b.b.49.1		2	1.1	even	1	trivial
75.4.b.b.49.2		2	5.4	even	2	inner
225.4.a.f.1.1		1	15.8	even	4
225.4.b.e.199.1		2	15.14	odd	2
225.4.b.e.199.2		2	3.2	odd	2
240.4.a.e.1.1		1	20.7	even	4
405.4.e.g.136.1		2	45.22	odd	12
405.4.e.g.271.1		2	45.7	odd	12
405.4.e.i.136.1		2	45.32	even	12
405.4.e.i.271.1		2	45.2	even	12
720.4.a.n.1.1		1	60.47	odd	4
735.4.a.e.1.1		1	35.27	even	4
960.4.a.b.1.1		1	40.37	odd	4
960.4.a.ba.1.1		1	40.27	even	4
1200.4.a.t.1.1		1	20.3	even	4
1200.4.f.b.49.1		2	4.3	odd	2
1200.4.f.b.49.2		2	20.19	odd	2
1815.4.a.e.1.1		1	55.32	even	4
2205.4.a.l.1.1		1	105.62	odd	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +52.0000 q^{11} +21.0000i q^{12} +22.0000i q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +9.00000i q^{18} +20.0000 q^{19} -72.0000 q^{21} -52.0000i q^{22} -168.000i q^{23} +45.0000 q^{24} +22.0000 q^{26} -27.0000i q^{27} +168.000i q^{28} -230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +156.000i q^{33} +14.0000 q^{34} -63.0000 q^{36} +34.0000i q^{37} -20.0000i q^{38} -66.0000 q^{39} +122.000 q^{41} +72.0000i q^{42} -188.000i q^{43} +364.000 q^{44} -168.000 q^{46} -256.000i q^{47} +123.000i q^{48} -233.000 q^{49} -42.0000 q^{51} +154.000i q^{52} -338.000i q^{53} -27.0000 q^{54} +360.000 q^{56} +60.0000i q^{57} +230.000i q^{58} -100.000 q^{59} +742.000 q^{61} +288.000i q^{62} -216.000i q^{63} +167.000 q^{64} +156.000 q^{66} +84.0000i q^{67} +98.0000i q^{68} +504.000 q^{69} -328.000 q^{71} +135.000i q^{72} -38.0000i q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +66.0000i q^{78} +240.000 q^{79} +81.0000 q^{81} -122.000i q^{82} +1212.00i q^{83} -504.000 q^{84} -188.000 q^{86} -690.000i q^{87} -780.000i q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -864.000i q^{93} -256.000 q^{94} +483.000 q^{96} -866.000i q^{97} +233.000i q^{98} -468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \) 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9	\( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 104 q^{11} + 48 q^{14} + 82 q^{16} + 40 q^{19} - 144 q^{21} + 90 q^{24} + 44 q^{26} - 460 q^{29} - 576 q^{31} + 28 q^{34} - 126 q^{36} - 132 q^{39} + 244 q^{41} + 728 q^{44} - 336 q^{46} - 466 q^{49} - 84 q^{51} - 54 q^{54} + 720 q^{56} - 200 q^{59} + 1484 q^{61} + 334 q^{64} + 312 q^{66} + 1008 q^{69} - 656 q^{71} + 68 q^{74} + 280 q^{76} + 480 q^{79} + 162 q^{81} - 1008 q^{84} - 376 q^{86} - 660 q^{89} - 1056 q^{91} - 512 q^{94} + 966 q^{96} - 936 q^{99}+O(q^{100}) \) 2 * q + 14 * q^4 + 6 * q^6 - 18 * q^9 + 104 * q^11 + 48 * q^14 + 82 * q^16 + 40 * q^19 - 144 * q^21 + 90 * q^24 + 44 * q^26 - 460 * q^29 - 576 * q^31 + 28 * q^34 - 126 * q^36 - 132 * q^39 + 244 * q^41 + 728 * q^44 - 336 * q^46 - 466 * q^49 - 84 * q^51 - 54 * q^54 + 720 * q^56 - 200 * q^59 + 1484 * q^61 + 334 * q^64 + 312 * q^66 + 1008 * q^69 - 656 * q^71 + 68 * q^74 + 280 * q^76 + 480 * q^79 + 162 * q^81 - 1008 * q^84 - 376 * q^86 - 660 * q^89 - 1056 * q^91 - 512 * q^94 + 966 * q^96 - 936 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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