LMFDB - Embedded newform 1875.2.b.c.1249.3

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:	$8$
Coefficient field:	8.0.6724000000.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 $ x^8 + 16x^6 + 86x^4 + 181*x^2 + 121
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.3
Root			$-1.70636i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.c.1249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.70636i q^{2} -1.00000i q^{3} -0.911672 q^{4} -1.70636 q^{6} -3.94243i q^{7} -1.85708i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.70636i q^{2} -1.00000i q^{3} -0.911672 q^{4} -1.70636 q^{6} -3.94243i q^{7} -1.85708i q^{8} -1.00000 q^{9} -5.90869 q^{11} +0.911672i q^{12} -3.29066i q^{13} -6.72721 q^{14} -4.99220 q^{16} +2.70636i q^{17} +1.70636i q^{18} +2.35813 q^{19} -3.94243 q^{21} +10.0824i q^{22} +0.584296i q^{23} -1.85708 q^{24} -5.61505 q^{26} +1.00000i q^{27} +3.59420i q^{28} +3.91167 q^{29} +2.70934 q^{31} +4.80433i q^{32} +5.90869i q^{33} +4.61803 q^{34} +0.911672 q^{36} -0.0208515i q^{37} -4.02383i q^{38} -3.29066 q^{39} +1.47214 q^{41} +6.72721i q^{42} -1.27279i q^{43} +5.38679 q^{44} +0.997020 q^{46} +5.43358i q^{47} +4.99220i q^{48} -8.54276 q^{49} +2.70636 q^{51} +3.00000i q^{52} -2.81554i q^{53} +1.70636 q^{54} -7.32142 q^{56} -2.35813i q^{57} -6.67473i q^{58} +4.69033 q^{59} +5.58132 q^{61} -4.62312i q^{62} +3.94243i q^{63} -1.78646 q^{64} +10.0824 q^{66} +6.03076i q^{67} -2.46731i q^{68} +0.584296 q^{69} -8.10138 q^{71} +1.85708i q^{72} -13.3166i q^{73} -0.0355801 q^{74} -2.14984 q^{76} +23.2946i q^{77} +5.61505i q^{78} -16.6648 q^{79} +1.00000 q^{81} -2.51200i q^{82} -0.781641i q^{83} +3.59420 q^{84} -2.17183 q^{86} -3.91167i q^{87} +10.9729i q^{88} +3.47327 q^{89} -12.9732 q^{91} -0.532686i q^{92} -2.70934i q^{93} +9.27165 q^{94} +4.80433 q^{96} -2.45443i q^{97} +14.5770i q^{98} +5.90869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 + 4 * q^6 - 8 * q^9	$ 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9} - 14 q^{11} - 32 q^{14} + 8 q^{16} - 10 q^{19} + 4 q^{21} - 30 q^{24} + 6 q^{26} + 40 q^{29} + 46 q^{31} + 28 q^{34} + 16 q^{36} - 2 q^{39} - 24 q^{41} + 58 q^{44} - 34 q^{46} - 16 q^{49} + 4 q^{51} - 4 q^{54} + 10 q^{56} + 30 q^{59} - 4 q^{61} - 46 q^{64} - 12 q^{66} - 2 q^{69} - 4 q^{71} + 38 q^{74} + 80 q^{76} - 70 q^{79} + 8 q^{81} - 18 q^{84} + 6 q^{86} + 70 q^{89} - 24 q^{91} + 18 q^{94} + 24 q^{96} + 14 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 + 4 * q^6 - 8 * q^9 - 14 * q^11 - 32 * q^14 + 8 * q^16 - 10 * q^19 + 4 * q^21 - 30 * q^24 + 6 * q^26 + 40 * q^29 + 46 * q^31 + 28 * q^34 + 16 * q^36 - 2 * q^39 - 24 * q^41 + 58 * q^44 - 34 * q^46 - 16 * q^49 + 4 * q^51 - 4 * q^54 + 10 * q^56 + 30 * q^59 - 4 * q^61 - 46 * q^64 - 12 * q^66 - 2 * q^69 - 4 * q^71 + 38 * q^74 + 80 * q^76 - 70 * q^79 + 8 * q^81 - 18 * q^84 + 6 * q^86 + 70 * q^89 - 24 * q^91 + 18 * q^94 + 24 * q^96 + 14 * 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 $

$2$	− 1.70636i	− 1.20658i	−0.797522	−	0.603290i	$-0.793856\pi$
$2$	− 1.70636i	− 1.20658i	0.797522	−	0.603290i	$-0.206144\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−0.911672	−0.455836
$5$	0	0
$5$	0	0
$6$	−1.70636	−0.696619
$7$	− 3.94243i	− 1.49010i	−0.667009	−	0.745049i	$-0.732426\pi$
$7$	− 3.94243i	− 1.49010i	0.667009	−	0.745049i	$-0.267574\pi$
$8$	− 1.85708i	− 0.656578i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−5.90869	−1.78154	−0.890769	−	0.454457i	$-0.849833\pi$
$11$	−5.90869	−1.78154	−0.890769	+	0.454457i	$0.849833\pi$
$12$	0.911672i	0.263177i
$13$	− 3.29066i	− 0.912664i	−0.889810	−	0.456332i	$-0.849163\pi$
$13$	− 3.29066i	− 0.912664i	0.889810	−	0.456332i	$-0.150837\pi$
$14$	−6.72721	−1.79792
$15$	0	0
$16$	−4.99220	−1.24805
$17$	2.70636i	0.656389i	0.944610	+	0.328195i	$0.106440\pi$
$17$	2.70636i	0.656389i	−0.944610	+	0.328195i	$0.893560\pi$
$18$	1.70636i	0.402193i
$19$	2.35813	0.540993	0.270497	−	0.962721i	$-0.412812\pi$
$19$	2.35813	0.540993	0.270497	+	0.962721i	$0.412812\pi$
$20$	0	0
$21$	−3.94243	−0.860309
$22$	10.0824i	2.14957i
$23$	0.584296i	0.121834i	0.998143	+	0.0609170i	$0.0194025\pi$
$23$	0.584296i	0.121834i	−0.998143	+	0.0609170i	$0.980597\pi$
$24$	−1.85708	−0.379075
$25$	0	0
$26$	−5.61505	−1.10120
$27$	1.00000i	0.192450i
$28$	3.59420i	0.679240i
$29$	3.91167	0.726379	0.363190	−	0.931715i	$-0.381688\pi$
$29$	3.91167	0.726379	0.363190	+	0.931715i	$0.381688\pi$
$30$	0	0
$31$	2.70934	0.486612	0.243306	−	0.969950i	$-0.421768\pi$
$31$	2.70934	0.486612	0.243306	+	0.969950i	$0.421768\pi$
$32$	4.80433i	0.849294i
$33$	5.90869i	1.02857i
$34$	4.61803	0.791986
$35$	0	0
$36$	0.911672	0.151945
$37$	− 0.0208515i	− 0.00342796i	−0.999999	−	0.00171398i	$-0.999454\pi$
$37$	− 0.0208515i	− 0.00342796i	0.999999	−	0.00171398i	$-0.000545577\pi$
$38$	− 4.02383i	− 0.652752i
$39$	−3.29066	−0.526927
$40$	0	0
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	6.72721i	1.03803i
$43$	− 1.27279i	− 0.194098i	−0.995280	−	0.0970491i	$-0.969060\pi$
$43$	− 1.27279i	− 0.194098i	0.995280	−	0.0970491i	$-0.0309404\pi$
$44$	5.38679	0.812089
$45$	0	0
$46$	0.997020	0.147003
$47$	5.43358i	0.792568i	0.918128	+	0.396284i	$0.129701\pi$
$47$	5.43358i	0.792568i	−0.918128	+	0.396284i	$0.870299\pi$
$48$	4.99220i	0.720562i
$49$	−8.54276	−1.22039
$50$	0	0
$51$	2.70636	0.378967
$52$	3.00000i	0.416025i
$53$	− 2.81554i	− 0.386744i	−0.981125	−	0.193372i	$-0.938058\pi$
$53$	− 2.81554i	− 0.386744i	0.981125	−	0.193372i	$-0.0619425\pi$
$54$	1.70636	0.232206
$55$	0	0
$56$	−7.32142	−0.978365
$57$	− 2.35813i	− 0.312343i
$58$	− 6.67473i	− 0.876435i
$59$	4.69033	0.610629	0.305315	−	0.952252i	$-0.401238\pi$
$59$	4.69033	0.610629	0.305315	+	0.952252i	$0.401238\pi$
$60$	0	0
$61$	5.58132	0.714614	0.357307	−	0.933987i	$-0.383695\pi$
$61$	5.58132	0.714614	0.357307	+	0.933987i	$0.383695\pi$
$62$	− 4.62312i	− 0.587137i
$63$	3.94243i	0.496700i
$64$	−1.78646	−0.223308
$65$	0	0
$66$	10.0824	1.24105
$67$	6.03076i	0.736774i	0.929672	+	0.368387i	$0.120090\pi$
$67$	6.03076i	0.736774i	−0.929672	+	0.368387i	$0.879910\pi$
$68$	− 2.46731i	− 0.299206i
$69$	0.584296	0.0703409
$70$	0	0
$71$	−8.10138	−0.961457	−0.480728	−	0.876870i	$-0.659628\pi$
$71$	−8.10138	−0.961457	−0.480728	+	0.876870i	$0.659628\pi$
$72$	1.85708i	0.218859i
$73$	− 13.3166i	− 1.55859i	−0.626658	−	0.779295i	$-0.715578\pi$
$73$	− 13.3166i	− 1.55859i	0.626658	−	0.779295i	$-0.284422\pi$
$74$	−0.0355801	−0.00413611
$75$	0	0
$76$	−2.14984	−0.246604
$77$	23.2946i	2.65467i
$78$	5.61505i	0.635780i
$79$	−16.6648	−1.87494	−0.937469	−	0.348067i	$-0.886838\pi$
$79$	−16.6648	−1.87494	−0.937469	+	0.348067i	$0.886838\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.51200i	− 0.277404i
$83$	− 0.781641i	− 0.0857962i	−0.999079	−	0.0428981i	$-0.986341\pi$
$83$	− 0.781641i	− 0.0857962i	0.999079	−	0.0428981i	$-0.0136591\pi$
$84$	3.59420	0.392160
$85$	0	0
$86$	−2.17183	−0.234195
$87$	− 3.91167i	− 0.419375i
$88$	10.9729i	1.16972i
$89$	3.47327	0.368166	0.184083	−	0.982911i	$-0.441068\pi$
$89$	3.47327	0.368166	0.184083	+	0.982911i	$0.441068\pi$
$90$	0	0
$91$	−12.9732	−1.35996
$92$	− 0.532686i	− 0.0555363i
$93$	− 2.70934i	− 0.280946i
$94$	9.27165	0.956297
$95$	0	0
$96$	4.80433	0.490340
$97$	− 2.45443i	− 0.249209i	−0.992206	−	0.124605i	$-0.960234\pi$
$97$	− 2.45443i	− 0.249209i	0.992206	−	0.124605i	$-0.0397663\pi$
$98$	14.5770i	1.47250i
$99$	5.90869	0.593846
$100$	0	0
$101$	−6.87495	−0.684083	−0.342042	−	0.939685i	$-0.611118\pi$
$101$	−6.87495	−0.684083	−0.342042	+	0.939685i	$0.611118\pi$
$102$	− 4.61803i	− 0.457254i
$103$	− 11.7532i	− 1.15807i	−0.815302	−	0.579036i	$-0.803429\pi$
$103$	− 11.7532i	− 1.15807i	0.815302	−	0.579036i	$-0.196571\pi$
$104$	−6.11102	−0.599235
$105$	0	0
$106$	−4.80433	−0.466638
$107$	5.66780i	0.547927i	0.961740	+	0.273964i	$0.0883348\pi$
$107$	5.66780i	0.547927i	−0.961740	+	0.273964i	$0.911665\pi$
$108$	− 0.911672i	− 0.0877257i
$109$	1.36128	0.130387	0.0651934	−	0.997873i	$-0.479234\pi$
$109$	1.36128	0.130387	0.0651934	+	0.997873i	$0.479234\pi$
$110$	0	0
$111$	−0.0208515	−0.00197913
$112$	19.6814i	1.85972i
$113$	10.6857i	1.00522i	0.864512	+	0.502612i	$0.167627\pi$
$113$	10.6857i	1.00522i	−0.864512	+	0.502612i	$0.832373\pi$
$114$	−4.02383	−0.376866
$115$	0	0
$116$	−3.56616	−0.331110
$117$	3.29066i	0.304221i
$118$	− 8.00341i	− 0.736773i
$119$	10.6696	0.978085
$120$	0	0
$121$	23.9126	2.17388
$122$	− 9.52375i	− 0.862239i
$123$	− 1.47214i	− 0.132738i
$124$	−2.47003	−0.221815
$125$	0	0
$126$	6.72721	0.599308
$127$	− 13.5428i	− 1.20173i	−0.799352	−	0.600863i	$-0.794824\pi$
$127$	− 13.5428i	− 1.20173i	0.799352	−	0.600863i	$-0.205176\pi$
$128$	12.6570i	1.11873i
$129$	−1.27279	−0.112063
$130$	0	0
$131$	4.59236	0.401236	0.200618	−	0.979670i	$-0.435705\pi$
$131$	4.59236	0.401236	0.200618	+	0.979670i	$0.435705\pi$
$132$	− 5.38679i	− 0.468860i
$133$	− 9.29678i	− 0.806133i
$134$	10.2907	0.888977
$135$	0	0
$136$	5.02594	0.430970
$137$	15.2620i	1.30392i	0.758253	+	0.651961i	$0.226053\pi$
$137$	15.2620i	1.30392i	−0.758253	+	0.651961i	$0.773947\pi$
$138$	− 0.997020i	− 0.0848720i
$139$	−18.2382	−1.54694	−0.773471	−	0.633832i	$-0.781481\pi$
$139$	−18.2382	−1.54694	−0.773471	+	0.633832i	$0.781481\pi$
$140$	0	0
$141$	5.43358	0.457590
$142$	13.8239i	1.16007i
$143$	19.4435i	1.62595i
$144$	4.99220	0.416017
$145$	0	0
$146$	−22.7229	−1.88056
$147$	8.54276i	0.704595i
$148$	0.0190097i	0.00156259i
$149$	−14.7323	−1.20692	−0.603458	−	0.797394i	$-0.706211\pi$
$149$	−14.7323	−1.20692	−0.603458	+	0.797394i	$0.706211\pi$
$150$	0	0
$151$	−17.4354	−1.41887	−0.709437	−	0.704769i	$-0.751051\pi$
$151$	−17.4354	−1.41887	−0.709437	+	0.704769i	$0.751051\pi$
$152$	− 4.37925i	− 0.355204i
$153$	− 2.70636i	− 0.218796i
$154$	39.7490	3.20307
$155$	0	0
$156$	3.00000	0.240192
$157$	17.9105i	1.42942i	0.699423	+	0.714708i	$0.253440\pi$
$157$	17.9105i	1.42942i	−0.699423	+	0.714708i	$0.746560\pi$
$158$	28.4362i	2.26226i
$159$	−2.81554	−0.223287
$160$	0	0
$161$	2.30354	0.181545
$162$	− 1.70636i	− 0.134064i
$163$	− 21.7598i	− 1.70436i	−0.523249	−	0.852180i	$-0.675280\pi$
$163$	− 21.7598i	− 1.70436i	0.523249	−	0.852180i	$-0.324720\pi$
$164$	−1.34210	−0.104801
$165$	0	0
$166$	−1.33376	−0.103520
$167$	− 22.6030i	− 1.74907i	−0.484962	−	0.874535i	$-0.661167\pi$
$167$	− 22.6030i	− 1.74907i	0.484962	−	0.874535i	$-0.338833\pi$
$168$	7.32142i	0.564860i
$169$	2.17157	0.167044
$170$	0	0
$171$	−2.35813	−0.180331
$172$	1.16036i	0.0884769i
$173$	− 12.9117i	− 0.981656i	−0.871256	−	0.490828i	$-0.836694\pi$
$173$	− 12.9117i	− 0.981656i	0.871256	−	0.490828i	$-0.163306\pi$
$174$	−6.67473	−0.506010
$175$	0	0
$176$	29.4974	2.22345
$177$	− 4.69033i	− 0.352547i
$178$	− 5.92666i	− 0.444222i
$179$	6.43060	0.480645	0.240323	−	0.970693i	$-0.422747\pi$
$179$	6.43060	0.480645	0.240323	+	0.970693i	$0.422747\pi$
$180$	0	0
$181$	14.7071	1.09317	0.546584	−	0.837404i	$-0.315928\pi$
$181$	14.7071	1.09317	0.546584	+	0.837404i	$0.315928\pi$
$182$	22.1370i	1.64090i
$183$	− 5.58132i	− 0.412583i
$184$	1.08508	0.0799935
$185$	0	0
$186$	−4.62312	−0.338984
$187$	− 15.9911i	− 1.16938i
$188$	− 4.95364i	− 0.361281i
$189$	3.94243	0.286770
$190$	0	0
$191$	6.71303	0.485737	0.242869	−	0.970059i	$-0.421912\pi$
$191$	6.71303	0.485737	0.242869	+	0.970059i	$0.421912\pi$
$192$	1.78646i	0.128927i
$193$	− 4.82817i	− 0.347539i	−0.984786	−	0.173769i	$-0.944405\pi$
$193$	− 4.82817i	− 0.347539i	0.984786	−	0.173769i	$-0.0555948\pi$
$194$	−4.18814	−0.300691
$195$	0	0
$196$	7.78819	0.556299
$197$	− 14.3841i	− 1.02482i	−0.858740	−	0.512411i	$-0.828752\pi$
$197$	− 14.3841i	− 1.02482i	0.858740	−	0.512411i	$-0.171248\pi$
$198$	− 10.0824i	− 0.716523i
$199$	8.72608	0.618575	0.309288	−	0.950969i	$-0.399909\pi$
$199$	8.72608	0.618575	0.309288	+	0.950969i	$0.399909\pi$
$200$	0	0
$201$	6.03076	0.425377
$202$	11.7312i	0.825402i
$203$	− 15.4215i	− 1.08238i
$204$	−2.46731	−0.172747
$205$	0	0
$206$	−20.0551	−1.39731
$207$	− 0.584296i	− 0.0406114i
$208$	16.4276i	1.13905i
$209$	−13.9335	−0.963800
$210$	0	0
$211$	2.97617	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$211$	2.97617	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$212$	2.56685i	0.176292i
$213$	8.10138i	0.555097i
$214$	9.67132	0.661118
$215$	0	0
$216$	1.85708	0.126358
$217$	− 10.6814i	− 0.725100i
$218$	− 2.32283i	− 0.157322i
$219$	−13.3166	−0.899852
$220$	0	0
$221$	8.90571	0.599063
$222$	0.0355801i	0.00238798i
$223$	0.304683i	0.0204031i	0.999948	+	0.0102015i	$0.00324731\pi$
$223$	0.304683i	0.0204031i	−0.999948	+	0.0102015i	$0.996753\pi$
$224$	18.9408	1.26553
$225$	0	0
$226$	18.2336	1.21288
$227$	− 9.57221i	− 0.635330i	−0.948203	−	0.317665i	$-0.897101\pi$
$227$	− 9.57221i	− 0.635330i	0.948203	−	0.317665i	$-0.102899\pi$
$228$	2.14984i	0.142377i
$229$	−12.1292	−0.801517	−0.400759	−	0.916184i	$-0.631253\pi$
$229$	−12.1292	−0.801517	−0.400759	+	0.916184i	$0.631253\pi$
$230$	0	0
$231$	23.2946	1.53267
$232$	− 7.26430i	− 0.476924i
$233$	− 20.1002i	− 1.31681i	−0.752664	−	0.658405i	$-0.771231\pi$
$233$	− 20.1002i	− 1.31681i	0.752664	−	0.658405i	$-0.228769\pi$
$234$	5.61505	0.367068
$235$	0	0
$236$	−4.27604	−0.278347
$237$	16.6648i	1.08250i
$238$	− 18.2063i	− 1.18014i
$239$	−17.6637	−1.14257	−0.571284	−	0.820752i	$-0.693555\pi$
$239$	−17.6637	−1.14257	−0.571284	+	0.820752i	$0.693555\pi$
$240$	0	0
$241$	29.5387	1.90276	0.951379	−	0.308024i	$-0.0996676\pi$
$241$	29.5387	1.90276	0.951379	+	0.308024i	$0.0996676\pi$
$242$	− 40.8036i	− 2.62296i
$243$	− 1.00000i	− 0.0641500i
$244$	−5.08833	−0.325747
$245$	0	0
$246$	−2.51200	−0.160159
$247$	− 7.75981i	− 0.493745i
$248$	− 5.03147i	− 0.319499i
$249$	−0.781641	−0.0495345
$250$	0	0
$251$	1.89396	0.119546	0.0597729	−	0.998212i	$-0.480962\pi$
$251$	1.89396	0.119546	0.0597729	+	0.998212i	$0.480962\pi$
$252$	− 3.59420i	− 0.226413i
$253$	− 3.45242i	− 0.217052i
$254$	−23.1088	−1.44998
$255$	0	0
$256$	18.0245	1.12653
$257$	22.1211i	1.37988i	0.723869	+	0.689938i	$0.242362\pi$
$257$	22.1211i	1.37988i	−0.723869	+	0.689938i	$0.757638\pi$
$258$	2.17183i	0.135213i
$259$	−0.0822054	−0.00510799
$260$	0	0
$261$	−3.91167	−0.242126
$262$	− 7.83623i	− 0.484124i
$263$	19.7153i	1.21570i	0.794053	+	0.607849i	$0.207967\pi$
$263$	19.7153i	1.21570i	−0.794053	+	0.607849i	$0.792033\pi$
$264$	10.9729	0.675337
$265$	0	0
$266$	−15.8637	−0.972664
$267$	− 3.47327i	− 0.212561i
$268$	− 5.49807i	− 0.335848i
$269$	−18.1647	−1.10752	−0.553762	−	0.832675i	$-0.686808\pi$
$269$	−18.1647	−1.10752	−0.553762	+	0.832675i	$0.686808\pi$
$270$	0	0
$271$	22.5695	1.37100	0.685500	−	0.728073i	$-0.259584\pi$
$271$	22.5695	1.37100	0.685500	+	0.728073i	$0.259584\pi$
$272$	− 13.5107i	− 0.819206i
$273$	12.9732i	0.785173i
$274$	26.0425	1.57329
$275$	0	0
$276$	−0.532686	−0.0320639
$277$	8.25491i	0.495990i	0.968761	+	0.247995i	$0.0797716\pi$
$277$	8.25491i	0.495990i	−0.968761	+	0.247995i	$0.920228\pi$
$278$	31.1209i	1.86651i
$279$	−2.70934	−0.162204
$280$	0	0
$281$	1.28129	0.0764355	0.0382177	−	0.999269i	$-0.487832\pi$
$281$	1.28129	0.0764355	0.0382177	+	0.999269i	$0.487832\pi$
$282$	− 9.27165i	− 0.552119i
$283$	− 2.67132i	− 0.158794i	−0.996843	−	0.0793968i	$-0.974701\pi$
$283$	− 2.67132i	− 0.158794i	0.996843	−	0.0793968i	$-0.0252994\pi$
$284$	7.38580	0.438266
$285$	0	0
$286$	33.1776	1.96183
$287$	− 5.80379i	− 0.342587i
$288$	− 4.80433i	− 0.283098i
$289$	9.67560	0.569153
$290$	0	0
$291$	−2.45443	−0.143881
$292$	12.1404i	0.710461i
$293$	− 17.6605i	− 1.03174i	−0.856667	−	0.515870i	$-0.827469\pi$
$293$	− 17.6605i	− 1.03174i	0.856667	−	0.515870i	$-0.172531\pi$
$294$	14.5770	0.850150
$295$	0	0
$296$	−0.0387229	−0.00225072
$297$	− 5.90869i	− 0.342857i
$298$	25.1386i	1.45624i
$299$	1.92272	0.111194
$300$	0	0
$301$	−5.01787	−0.289225
$302$	29.7511i	1.71199i
$303$	6.87495i	0.394956i
$304$	−11.7723	−0.675186
$305$	0	0
$306$	−4.61803	−0.263995
$307$	− 28.5593i	− 1.62997i	−0.579484	−	0.814983i	$-0.696746\pi$
$307$	− 28.5593i	− 1.62997i	0.579484	−	0.814983i	$-0.303254\pi$
$308$	− 21.2370i	− 1.21009i
$309$	−11.7532	−0.668613
$310$	0	0
$311$	−29.3283	−1.66306	−0.831529	−	0.555482i	$-0.812534\pi$
$311$	−29.3283	−1.66306	−0.831529	+	0.555482i	$0.812534\pi$
$312$	6.11102i	0.345968i
$313$	17.4933i	0.988777i	0.869241	+	0.494388i	$0.164608\pi$
$313$	17.4933i	0.988777i	−0.869241	+	0.494388i	$0.835392\pi$
$314$	30.5619	1.72471
$315$	0	0
$316$	15.1928	0.854665
$317$	3.91763i	0.220036i	0.993930	+	0.110018i	$0.0350909\pi$
$317$	3.91763i	0.220036i	−0.993930	+	0.110018i	$0.964909\pi$
$318$	4.80433i	0.269414i
$319$	−23.1129	−1.29407
$320$	0	0
$321$	5.66780	0.316346
$322$	− 3.93068i	− 0.219048i
$323$	6.38197i	0.355102i
$324$	−0.911672	−0.0506484
$325$	0	0
$326$	−37.1301	−2.05645
$327$	− 1.36128i	− 0.0752788i
$328$	− 2.73388i	− 0.150953i
$329$	21.4215	1.18101
$330$	0	0
$331$	−17.5534	−0.964820	−0.482410	−	0.875945i	$-0.660238\pi$
$331$	−17.5534	−0.964820	−0.482410	+	0.875945i	$0.660238\pi$
$332$	0.712600i	0.0391090i
$333$	0.0208515i	0.00114265i
$334$	−38.5689	−2.11039
$335$	0	0
$336$	19.6814	1.07371
$337$	− 14.0476i	− 0.765221i	−0.923910	−	0.382611i	$-0.875025\pi$
$337$	− 14.0476i	− 0.765221i	0.923910	−	0.382611i	$-0.124975\pi$
$338$	− 3.70549i	− 0.201552i
$339$	10.6857	0.580366
$340$	0	0
$341$	−16.0087	−0.866918
$342$	4.02383i	0.217584i
$343$	6.08221i	0.328408i
$344$	−2.36367	−0.127440
$345$	0	0
$346$	−22.0320	−1.18445
$347$	− 1.03050i	− 0.0553199i	−0.999617	−	0.0276599i	$-0.991194\pi$
$347$	− 1.03050i	− 0.0553199i	0.999617	−	0.0276599i	$-0.00880555\pi$
$348$	3.56616i	0.191166i
$349$	−21.5626	−1.15422	−0.577109	−	0.816667i	$-0.695819\pi$
$349$	−21.5626	−1.15422	−0.577109	+	0.816667i	$0.695819\pi$
$350$	0	0
$351$	3.29066	0.175642
$352$	− 28.3873i	− 1.51305i
$353$	− 7.15625i	− 0.380888i	−0.981698	−	0.190444i	$-0.939007\pi$
$353$	− 7.15625i	− 0.380888i	0.981698	−	0.190444i	$-0.0609928\pi$
$354$	−8.00341	−0.425376
$355$	0	0
$356$	−3.16649	−0.167823
$357$	− 10.6696i	− 0.564697i
$358$	− 10.9729i	− 0.579937i
$359$	−27.1518	−1.43302	−0.716510	−	0.697577i	$-0.754262\pi$
$359$	−27.1518	−1.43302	−0.716510	+	0.697577i	$0.754262\pi$
$360$	0	0
$361$	−13.4392	−0.707326
$362$	− 25.0956i	− 1.31899i
$363$	− 23.9126i	− 1.25509i
$364$	11.8273	0.619918
$365$	0	0
$366$	−9.52375	−0.497814
$367$	21.4423i	1.11928i	0.828735	+	0.559641i	$0.189061\pi$
$367$	21.4423i	1.11928i	−0.828735	+	0.559641i	$0.810939\pi$
$368$	− 2.91692i	− 0.152055i
$369$	−1.47214	−0.0766363
$370$	0	0
$371$	−11.1001	−0.576287
$372$	2.47003i	0.128065i
$373$	− 6.39073i	− 0.330900i	−0.986218	−	0.165450i	$-0.947092\pi$
$373$	− 6.39073i	− 0.330900i	0.986218	−	0.165450i	$-0.0529076\pi$
$374$	−27.2865	−1.41095
$375$	0	0
$376$	10.0906	0.520383
$377$	− 12.8720i	− 0.662940i
$378$	− 6.72721i	− 0.346011i
$379$	0.154667	0.00794469	0.00397235	−	0.999992i	$-0.498736\pi$
$379$	0.154667	0.00794469	0.00397235	+	0.999992i	$0.498736\pi$
$380$	0	0
$381$	−13.5428	−0.693816
$382$	− 11.4549i	− 0.586081i
$383$	4.25508i	0.217424i	0.994073	+	0.108712i	$0.0346727\pi$
$383$	4.25508i	0.217424i	−0.994073	+	0.108712i	$0.965327\pi$
$384$	12.6570	0.645901
$385$	0	0
$386$	−8.23860	−0.419334
$387$	1.27279i	0.0646994i
$388$	2.23763i	0.113599i
$389$	20.3682	1.03271	0.516354	−	0.856375i	$-0.327289\pi$
$389$	20.3682	1.03271	0.516354	+	0.856375i	$0.327289\pi$
$390$	0	0
$391$	−1.58132	−0.0799706
$392$	15.8646i	0.801283i
$393$	− 4.59236i	− 0.231654i
$394$	−24.5444	−1.23653
$395$	0	0
$396$	−5.38679	−0.270696
$397$	− 1.95716i	− 0.0982270i	−0.998793	−	0.0491135i	$-0.984360\pi$
$397$	− 1.95716i	− 0.0982270i	0.998793	−	0.0491135i	$-0.0156396\pi$
$398$	− 14.8898i	− 0.746360i
$399$	−9.29678	−0.465421
$400$	0	0
$401$	−32.8337	−1.63964	−0.819818	−	0.572624i	$-0.805925\pi$
$401$	−32.8337	−1.63964	−0.819818	+	0.572624i	$0.805925\pi$
$402$	− 10.2907i	− 0.513251i
$403$	− 8.91552i	− 0.444114i
$404$	6.26770	0.311830
$405$	0	0
$406$	−26.3147	−1.30597
$407$	0.123205i	0.00610704i
$408$	− 5.02594i	− 0.248821i
$409$	39.5764	1.95693	0.978464	−	0.206417i	$-0.0661804\pi$
$409$	39.5764	1.95693	0.978464	+	0.206417i	$0.0661804\pi$
$410$	0	0
$411$	15.2620	0.752819
$412$	10.7150i	0.527891i
$413$	− 18.4913i	− 0.909898i
$414$	−0.997020	−0.0490009
$415$	0	0
$416$	15.8094	0.775121
$417$	18.2382i	0.893127i
$418$	23.7756i	1.16290i
$419$	−19.5595	−0.955544	−0.477772	−	0.878484i	$-0.658555\pi$
$419$	−19.5595	−0.955544	−0.477772	+	0.878484i	$0.658555\pi$
$420$	0	0
$421$	−40.3325	−1.96568	−0.982842	−	0.184451i	$-0.940949\pi$
$421$	−40.3325	−1.96568	−0.982842	+	0.184451i	$0.940949\pi$
$422$	− 5.07842i	− 0.247214i
$423$	− 5.43358i	− 0.264189i
$424$	−5.22869	−0.253928
$425$	0	0
$426$	13.8239	0.669769
$427$	− 22.0039i	− 1.06485i
$428$	− 5.16718i	− 0.249765i
$429$	19.4435	0.938740
$430$	0	0
$431$	13.7370	0.661689	0.330844	−	0.943685i	$-0.392666\pi$
$431$	13.7370	0.661689	0.330844	+	0.943685i	$0.392666\pi$
$432$	− 4.99220i	− 0.240187i
$433$	12.4253i	0.597124i	0.954390	+	0.298562i	$0.0965070\pi$
$433$	12.4253i	0.597124i	−0.954390	+	0.298562i	$0.903493\pi$
$434$	−18.2263	−0.874892
$435$	0	0
$436$	−1.24104	−0.0594349
$437$	1.37785i	0.0659114i
$438$	22.7229i	1.08574i
$439$	−10.9654	−0.523349	−0.261675	−	0.965156i	$-0.584275\pi$
$439$	−10.9654	−0.523349	−0.261675	+	0.965156i	$0.584275\pi$
$440$	0	0
$441$	8.54276	0.406798
$442$	− 15.1964i	− 0.722818i
$443$	− 8.13187i	− 0.386357i	−0.981164	−	0.193178i	$-0.938120\pi$
$443$	− 8.13187i	− 0.386357i	0.981164	−	0.193178i	$-0.0618796\pi$
$444$	0.0190097	0.000902160 0
$445$	0	0
$446$	0.519899	0.0246180
$447$	14.7323i	0.696814i
$448$	7.04300i	0.332751i
$449$	32.9503	1.55502	0.777511	−	0.628869i	$-0.216482\pi$
$449$	32.9503	1.55502	0.777511	+	0.628869i	$0.216482\pi$
$450$	0	0
$451$	−8.69840	−0.409592
$452$	− 9.74183i	− 0.458217i
$453$	17.4354i	0.819187i
$454$	−16.3337	−0.766577
$455$	0	0
$456$	−4.37925	−0.205077
$457$	− 22.8800i	− 1.07028i	−0.844762	−	0.535142i	$-0.820258\pi$
$457$	− 22.8800i	− 1.07028i	0.844762	−	0.535142i	$-0.179742\pi$
$458$	20.6967i	0.967095i
$459$	−2.70636	−0.126322
$460$	0	0
$461$	12.0425	0.560875	0.280438	−	0.959872i	$-0.409520\pi$
$461$	12.0425	0.560875	0.280438	+	0.959872i	$0.409520\pi$
$462$	− 39.7490i	− 1.84929i
$463$	27.2002i	1.26410i	0.774928	+	0.632049i	$0.217786\pi$
$463$	27.2002i	1.26410i	−0.774928	+	0.632049i	$0.782214\pi$
$464$	−19.5278	−0.906557
$465$	0	0
$466$	−34.2983	−1.58884
$467$	− 3.94243i	− 0.182434i	−0.995831	−	0.0912170i	$-0.970924\pi$
$467$	− 3.94243i	− 0.182434i	0.995831	−	0.0912170i	$-0.0290757\pi$
$468$	− 3.00000i	− 0.138675i
$469$	23.7758	1.09787
$470$	0	0
$471$	17.9105	0.825274
$472$	− 8.71033i	− 0.400926i
$473$	7.52050i	0.345793i
$474$	28.4362	1.30612
$475$	0	0
$476$	−9.72721	−0.445846
$477$	2.81554i	0.128915i
$478$	30.1406i	1.37860i
$479$	−19.9493	−0.911505	−0.455752	−	0.890107i	$-0.650630\pi$
$479$	−19.9493	−0.911505	−0.455752	+	0.890107i	$0.650630\pi$
$480$	0	0
$481$	−0.0686150	−0.00312857
$482$	− 50.4038i	− 2.29583i
$483$	− 2.30354i	− 0.104815i
$484$	−21.8005	−0.990931
$485$	0	0
$486$	−1.70636	−0.0774022
$487$	− 11.0023i	− 0.498560i	−0.968431	−	0.249280i	$-0.919806\pi$
$487$	− 11.0023i	− 0.498560i	0.968431	−	0.249280i	$-0.0801940\pi$
$488$	− 10.3650i	− 0.469200i
$489$	−21.7598	−0.984013
$490$	0	0
$491$	−33.2933	−1.50251	−0.751253	−	0.660014i	$-0.770550\pi$
$491$	−33.2933	−1.50251	−0.751253	+	0.660014i	$0.770550\pi$
$492$	1.34210i	0.0605068i
$493$	10.5864i	0.476788i
$494$	−13.2411	−0.595743
$495$	0	0
$496$	−13.5256	−0.607316
$497$	31.9391i	1.43267i
$498$	1.33376i	0.0597673i
$499$	41.1448	1.84189	0.920946	−	0.389690i	$-0.127418\pi$
$499$	41.1448	1.84189	0.920946	+	0.389690i	$0.127418\pi$
$500$	0	0
$501$	−22.6030	−1.00983
$502$	− 3.23179i	− 0.144242i
$503$	− 32.0761i	− 1.43020i	−0.699020	−	0.715102i	$-0.746380\pi$
$503$	− 32.0761i	− 1.43020i	0.699020	−	0.715102i	$-0.253620\pi$
$504$	7.32142	0.326122
$505$	0	0
$506$	−5.89108	−0.261891
$507$	− 2.17157i	− 0.0964429i
$508$	12.3465i	0.547790i
$509$	27.0953	1.20098	0.600489	−	0.799633i	$-0.294972\pi$
$509$	27.0953	1.20098	0.600489	+	0.799633i	$0.294972\pi$
$510$	0	0
$511$	−52.4997	−2.32245
$512$	− 5.44234i	− 0.240520i
$513$	2.35813i	0.104114i
$514$	37.7466	1.66493
$515$	0	0
$516$	1.16036	0.0510822
$517$	− 32.1053i	− 1.41199i
$518$	0.140272i	0.00616321i
$519$	−12.9117	−0.566759
$520$	0	0
$521$	25.9556	1.13714	0.568569	−	0.822636i	$-0.307497\pi$
$521$	25.9556	1.13714	0.568569	+	0.822636i	$0.307497\pi$
$522$	6.67473i	0.292145i
$523$	1.79382i	0.0784381i	0.999231	+	0.0392190i	$0.0124870\pi$
$523$	1.79382i	0.0784381i	−0.999231	+	0.0392190i	$0.987513\pi$
$524$	−4.18673	−0.182898
$525$	0	0
$526$	33.6414	1.46684
$527$	7.33246i	0.319407i
$528$	− 29.4974i	− 1.28371i
$529$	22.6586	0.985156
$530$	0	0
$531$	−4.69033	−0.203543
$532$	8.47561i	0.367464i
$533$	− 4.84430i	− 0.209830i
$534$	−5.92666	−0.256472
$535$	0	0
$536$	11.1996	0.483750
$537$	− 6.43060i	− 0.277501i
$538$	30.9956i	1.33632i
$539$	50.4765	2.17418
$540$	0	0
$541$	−7.43542	−0.319674	−0.159837	−	0.987143i	$-0.551097\pi$
$541$	−7.43542	−0.319674	−0.159837	+	0.987143i	$0.551097\pi$
$542$	− 38.5117i	− 1.65422i
$543$	− 14.7071i	− 0.631141i
$544$	−13.0023	−0.557468
$545$	0	0
$546$	22.1370	0.947374
$547$	19.4868i	0.833194i	0.909091	+	0.416597i	$0.136777\pi$
$547$	19.4868i	0.833194i	−0.909091	+	0.416597i	$0.863223\pi$
$548$	− 13.9139i	− 0.594374i
$549$	−5.58132	−0.238205
$550$	0	0
$551$	9.22425	0.392966
$552$	− 1.08508i	− 0.0461843i
$553$	65.6999i	2.79384i
$554$	14.0859	0.598451
$555$	0	0
$556$	16.6272	0.705152
$557$	− 26.3285i	− 1.11557i	−0.829984	−	0.557787i	$-0.811650\pi$
$557$	− 26.3285i	− 1.11557i	0.829984	−	0.557787i	$-0.188350\pi$
$558$	4.62312i	0.195712i
$559$	−4.18830	−0.177146
$560$	0	0
$561$	−15.9911	−0.675143
$562$	− 2.18635i	− 0.0922255i
$563$	− 36.7708i	− 1.54971i	−0.632141	−	0.774853i	$-0.717824\pi$
$563$	− 36.7708i	− 1.54971i	0.632141	−	0.774853i	$-0.282176\pi$
$564$	−4.95364	−0.208586
$565$	0	0
$566$	−4.55824	−0.191597
$567$	− 3.94243i	− 0.165567i
$568$	15.0449i	0.631271i
$569$	36.2823	1.52103	0.760516	−	0.649320i	$-0.224946\pi$
$569$	36.2823	1.52103	0.760516	+	0.649320i	$0.224946\pi$
$570$	0	0
$571$	−3.36442	−0.140797	−0.0703983	−	0.997519i	$-0.522427\pi$
$571$	−3.36442	−0.140797	−0.0703983	+	0.997519i	$0.522427\pi$
$572$	− 17.7261i	− 0.741164i
$573$	− 6.71303i	− 0.280441i
$574$	−9.90337	−0.413359
$575$	0	0
$576$	1.78646	0.0744359
$577$	− 6.88442i	− 0.286602i	−0.989679	−	0.143301i	$-0.954228\pi$
$577$	− 6.88442i	− 0.286602i	0.989679	−	0.143301i	$-0.0457717\pi$
$578$	− 16.5101i	− 0.686729i
$579$	−4.82817	−0.200652
$580$	0	0
$581$	−3.08156	−0.127845
$582$	4.18814i	0.173604i
$583$	16.6362i	0.689000i
$584$	−24.7300	−1.02334
$585$	0	0
$586$	−30.1353	−1.24488
$587$	− 14.7377i	− 0.608288i	−0.952626	−	0.304144i	$-0.901630\pi$
$587$	− 14.7377i	− 0.608288i	0.952626	−	0.304144i	$-0.0983705\pi$
$588$	− 7.78819i	− 0.321180i
$589$	6.38899	0.263254
$590$	0	0
$591$	−14.3841	−0.591682
$592$	0.104095i	0.00427826i
$593$	− 5.82561i	− 0.239229i	−0.992820	−	0.119615i	$-0.961834\pi$
$593$	− 5.82561i	− 0.239229i	0.992820	−	0.119615i	$-0.0381659\pi$
$594$	−10.0824	−0.413685
$595$	0	0
$596$	13.4310	0.550156
$597$	− 8.72608i	− 0.357134i
$598$	− 3.28085i	− 0.134164i
$599$	27.1527	1.10943	0.554715	−	0.832040i	$-0.312827\pi$
$599$	27.1527	1.10943	0.554715	+	0.832040i	$0.312827\pi$
$600$	0	0
$601$	20.9140	0.853101	0.426551	−	0.904464i	$-0.359729\pi$
$601$	20.9140	0.853101	0.426551	+	0.904464i	$0.359729\pi$
$602$	8.56231i	0.348974i
$603$	− 6.03076i	− 0.245591i
$604$	15.8954	0.646774
$605$	0	0
$606$	11.7312	0.476546
$607$	1.46424i	0.0594318i	0.999558	+	0.0297159i	$0.00946025\pi$
$607$	1.46424i	0.0594318i	−0.999558	+	0.0297159i	$0.990540\pi$
$608$	11.3293i	0.459462i
$609$	−15.4215	−0.624910
$610$	0	0
$611$	17.8800	0.723349
$612$	2.46731i	0.0997353i
$613$	− 34.7105i	− 1.40194i	−0.713189	−	0.700971i	$-0.752750\pi$
$613$	− 34.7105i	− 1.40194i	0.713189	−	0.700971i	$-0.247250\pi$
$614$	−48.7326	−1.96669
$615$	0	0
$616$	43.2600	1.74299
$617$	41.9038i	1.68698i	0.537142	+	0.843492i	$0.319504\pi$
$617$	41.9038i	1.68698i	−0.537142	+	0.843492i	$0.680496\pi$
$618$	20.0551i	0.806736i
$619$	18.7614	0.754083	0.377042	−	0.926196i	$-0.376941\pi$
$619$	18.7614	0.754083	0.377042	+	0.926196i	$0.376941\pi$
$620$	0	0
$621$	−0.584296	−0.0234470
$622$	50.0448i	2.00661i
$623$	− 13.6931i	− 0.548604i
$624$	16.4276	0.657631
$625$	0	0
$626$	29.8498	1.19304
$627$	13.9335i	0.556450i
$628$	− 16.3285i	− 0.651579i
$629$	0.0564316	0.00225007
$630$	0	0
$631$	9.88888	0.393670	0.196835	−	0.980437i	$-0.436934\pi$
$631$	9.88888	0.393670	0.196835	+	0.980437i	$0.436934\pi$
$632$	30.9479i	1.23104i
$633$	− 2.97617i	− 0.118292i
$634$	6.68490	0.265491
$635$	0	0
$636$	2.56685	0.101782
$637$	28.1113i	1.11381i
$638$	39.4389i	1.56140i
$639$	8.10138	0.320486
$640$	0	0
$641$	−1.81266	−0.0715959	−0.0357979	−	0.999359i	$-0.511397\pi$
$641$	−1.81266	−0.0715959	−0.0357979	+	0.999359i	$0.511397\pi$
$642$	− 9.67132i	− 0.381697i
$643$	− 45.7391i	− 1.80377i	−0.431974	−	0.901886i	$-0.642183\pi$
$643$	− 45.7391i	− 1.80377i	0.431974	−	0.901886i	$-0.357817\pi$
$644$	−2.10008	−0.0827546
$645$	0	0
$646$	10.8899	0.428459
$647$	− 25.6771i	− 1.00947i	−0.863274	−	0.504736i	$-0.831590\pi$
$647$	− 25.6771i	− 1.00947i	0.863274	−	0.504736i	$-0.168410\pi$
$648$	− 1.85708i	− 0.0729531i
$649$	−27.7137	−1.08786
$650$	0	0
$651$	−10.6814	−0.418637
$652$	19.8378i	0.776909i
$653$	− 35.9684i	− 1.40755i	−0.710422	−	0.703775i	$-0.751496\pi$
$653$	− 35.9684i	− 1.40755i	0.710422	−	0.703775i	$-0.248504\pi$
$654$	−2.32283	−0.0908299
$655$	0	0
$656$	−7.34919	−0.286938
$657$	13.3166i	0.519530i
$658$	− 36.5528i	− 1.42498i
$659$	32.0924	1.25014	0.625072	−	0.780567i	$-0.285070\pi$
$659$	32.0924	1.25014	0.625072	+	0.780567i	$0.285070\pi$
$660$	0	0
$661$	16.2064	0.630357	0.315179	−	0.949032i	$-0.397936\pi$
$661$	16.2064	0.630357	0.315179	+	0.949032i	$0.397936\pi$
$662$	29.9524i	1.16413i
$663$	− 8.90571i	− 0.345869i
$664$	−1.45157	−0.0563319
$665$	0	0
$666$	0.0355801	0.00137870
$667$	2.28557i	0.0884977i
$668$	20.6065i	0.797289i
$669$	0.304683	0.0117797
$670$	0	0
$671$	−32.9783	−1.27311
$672$	− 18.9408i	− 0.730655i
$673$	− 34.1608i	− 1.31680i	−0.752667	−	0.658401i	$-0.771233\pi$
$673$	− 34.1608i	− 1.31680i	0.752667	−	0.658401i	$-0.228767\pi$
$674$	−23.9703	−0.923301
$675$	0	0
$676$	−1.97976	−0.0761446
$677$	9.17514i	0.352629i	0.984334	+	0.176315i	$0.0564176\pi$
$677$	9.17514i	0.352629i	−0.984334	+	0.176315i	$0.943582\pi$
$678$	− 18.2336i	− 0.700258i
$679$	−9.67641	−0.371346
$680$	0	0
$681$	−9.57221	−0.366808
$682$	27.3166i	1.04601i
$683$	− 19.5007i	− 0.746173i	−0.927797	−	0.373087i	$-0.878299\pi$
$683$	− 19.5007i	− 0.746173i	0.927797	−	0.373087i	$-0.121701\pi$
$684$	2.14984	0.0822014
$685$	0	0
$686$	10.3784	0.396251
$687$	12.1292i	0.462756i
$688$	6.35400i	0.242244i
$689$	−9.26498	−0.352968
$690$	0	0
$691$	29.0458	1.10495	0.552477	−	0.833528i	$-0.313683\pi$
$691$	29.0458	1.10495	0.552477	+	0.833528i	$0.313683\pi$
$692$	11.7712i	0.447474i
$693$	− 23.2946i	− 0.884889i
$694$	−1.75840	−0.0667479
$695$	0	0
$696$	−7.26430	−0.275352
$697$	3.98413i	0.150910i
$698$	36.7936i	1.39266i
$699$	−20.1002	−0.760261
$700$	0	0
$701$	−20.4085	−0.770820	−0.385410	−	0.922745i	$-0.625940\pi$
$701$	−20.4085	−0.770820	−0.385410	+	0.922745i	$0.625940\pi$
$702$	− 5.61505i	− 0.211927i
$703$	− 0.0491705i	− 0.00185450i
$704$	10.5557	0.397831
$705$	0	0
$706$	−12.2111	−0.459573
$707$	27.1040i	1.01935i
$708$	4.27604i	0.160704i
$709$	−16.4810	−0.618956	−0.309478	−	0.950907i	$-0.600154\pi$
$709$	−16.4810	−0.618956	−0.309478	+	0.950907i	$0.600154\pi$
$710$	0	0
$711$	16.6648	0.624980
$712$	− 6.45016i	− 0.241730i
$713$	1.58306i	0.0592859i
$714$	−18.2063	−0.681353
$715$	0	0
$716$	−5.86259	−0.219095
$717$	17.6637i	0.659662i
$718$	46.3309i	1.72905i
$719$	−31.0813	−1.15914	−0.579569	−	0.814923i	$-0.696779\pi$
$719$	−31.0813	−1.15914	−0.579569	+	0.814923i	$0.696779\pi$
$720$	0	0
$721$	−46.3360	−1.72564
$722$	22.9321i	0.853446i
$723$	− 29.5387i	− 1.09856i
$724$	−13.4080	−0.498305
$725$	0	0
$726$	−40.8036	−1.51436
$727$	− 0.953049i	− 0.0353466i	−0.999844	−	0.0176733i	$-0.994374\pi$
$727$	− 0.953049i	− 0.0353466i	0.999844	−	0.0176733i	$-0.00562589\pi$
$728$	24.0923i	0.892919i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	3.44462	0.127404
$732$	5.08833i	0.188070i
$733$	15.3751i	0.567893i	0.958840	+	0.283947i	$0.0916439\pi$
$733$	15.3751i	0.567893i	−0.958840	+	0.283947i	$0.908356\pi$
$734$	36.5884	1.35050
$735$	0	0
$736$	−2.80715	−0.103473
$737$	− 35.6339i	− 1.31259i
$738$	2.51200i	0.0924679i
$739$	−5.19285	−0.191022	−0.0955110	−	0.995428i	$-0.530449\pi$
$739$	−5.19285	−0.191022	−0.0955110	+	0.995428i	$0.530449\pi$
$740$	0	0
$741$	−7.75981	−0.285064
$742$	18.9408i	0.695337i
$743$	42.3399i	1.55330i	0.629932	+	0.776650i	$0.283083\pi$
$743$	42.3399i	1.55330i	−0.629932	+	0.776650i	$0.716917\pi$
$744$	−5.03147	−0.184463
$745$	0	0
$746$	−10.9049	−0.399257
$747$	0.781641i	0.0285987i
$748$	14.5786i	0.533046i
$749$	22.3449	0.816465
$750$	0	0
$751$	22.2461	0.811773	0.405887	−	0.913923i	$-0.366963\pi$
$751$	22.2461	0.811773	0.405887	+	0.913923i	$0.366963\pi$
$752$	− 27.1255i	− 0.989165i
$753$	− 1.89396i	− 0.0690199i
$754$	−21.9642	−0.799891
$755$	0	0
$756$	−3.59420	−0.130720
$757$	− 9.27680i	− 0.337171i	−0.985687	−	0.168585i	$-0.946080\pi$
$757$	− 9.27680i	− 0.337171i	0.985687	−	0.168585i	$-0.0539199\pi$
$758$	− 0.263917i	− 0.00958591i
$759$	−3.45242	−0.125315
$760$	0	0
$761$	−17.4122	−0.631191	−0.315596	−	0.948894i	$-0.602204\pi$
$761$	−17.4122	−0.631191	−0.315596	+	0.948894i	$0.602204\pi$
$762$	23.1088i	0.837145i
$763$	− 5.36674i	− 0.194289i
$764$	−6.12008	−0.221417
$765$	0	0
$766$	7.26070	0.262340
$767$	− 15.4343i	− 0.557300i
$768$	− 18.0245i	− 0.650404i
$769$	23.1912	0.836297	0.418148	−	0.908379i	$-0.362679\pi$
$769$	23.1912	0.836297	0.418148	+	0.908379i	$0.362679\pi$
$770$	0	0
$771$	22.1211	0.796672
$772$	4.40170i	0.158421i
$773$	5.50798i	0.198108i	0.995082	+	0.0990541i	$0.0315817\pi$
$773$	5.50798i	0.198108i	−0.995082	+	0.0990541i	$0.968418\pi$
$774$	2.17183	0.0780650
$775$	0	0
$776$	−4.55807	−0.163625
$777$	0.0822054i	0.00294910i
$778$	− 34.7555i	− 1.24605i
$779$	3.47149	0.124379
$780$	0	0
$781$	47.8685	1.71287
$782$	2.69830i	0.0964909i
$783$	3.91167i	0.139792i
$784$	42.6471	1.52311
$785$	0	0
$786$	−7.83623	−0.279509
$787$	25.7971i	0.919568i	0.888031	+	0.459784i	$0.152073\pi$
$787$	25.7971i	0.919568i	−0.888031	+	0.459784i	$0.847927\pi$
$788$	13.1136i	0.467151i
$789$	19.7153	0.701883
$790$	0	0
$791$	42.1275	1.49788
$792$	− 10.9729i	− 0.389906i
$793$	− 18.3662i	− 0.652203i
$794$	−3.33962	−0.118519
$795$	0	0
$796$	−7.95532	−0.281969
$797$	3.70988i	0.131411i	0.997839	+	0.0657054i	$0.0209298\pi$
$797$	3.70988i	0.131411i	−0.997839	+	0.0657054i	$0.979070\pi$
$798$	15.8637i	0.561568i
$799$	−14.7052	−0.520233
$800$	0	0
$801$	−3.47327	−0.122722
$802$	56.0261i	1.97835i
$803$	78.6837i	2.77669i
$804$	−5.49807	−0.193902
$805$	0	0
$806$	−15.2131	−0.535859
$807$	18.1647i	0.639429i
$808$	12.7674i	0.449154i
$809$	−37.5040	−1.31857	−0.659286	−	0.751893i	$-0.729141\pi$
$809$	−37.5040	−1.31857	−0.659286	+	0.751893i	$0.729141\pi$
$810$	0	0
$811$	45.6159	1.60179	0.800896	−	0.598804i	$-0.204357\pi$
$811$	45.6159	1.60179	0.800896	+	0.598804i	$0.204357\pi$
$812$	14.0593i	0.493386i
$813$	− 22.5695i	− 0.791547i
$814$	0.210232	0.00736863
$815$	0	0
$816$	−13.5107	−0.472969
$817$	− 3.00140i	− 0.105006i
$818$	− 67.5317i	− 2.36119i
$819$	12.9732	0.453320
$820$	0	0
$821$	45.0511	1.57229	0.786147	−	0.618039i	$-0.212073\pi$
$821$	45.0511	1.57229	0.786147	+	0.618039i	$0.212073\pi$
$822$	− 26.0425i	− 0.908337i
$823$	41.4573i	1.44511i	0.691312	+	0.722556i	$0.257033\pi$
$823$	41.4573i	1.44511i	−0.691312	+	0.722556i	$0.742967\pi$
$824$	−21.8266	−0.760364
$825$	0	0
$826$	−31.5529	−1.09786
$827$	− 38.6933i	− 1.34550i	−0.739871	−	0.672749i	$-0.765113\pi$
$827$	− 38.6933i	− 1.34550i	0.739871	−	0.672749i	$-0.234887\pi$
$828$	0.532686i	0.0185121i
$829$	−24.1258	−0.837922	−0.418961	−	0.908004i	$-0.637606\pi$
$829$	−24.1258	−0.837922	−0.418961	+	0.908004i	$0.637606\pi$
$830$	0	0
$831$	8.25491	0.286360
$832$	5.87864i	0.203805i
$833$	− 23.1198i	− 0.801053i
$834$	31.1209	1.07763
$835$	0	0
$836$	12.7028	0.439334
$837$	2.70934i	0.0936486i
$838$	33.3756i	1.15294i
$839$	17.7858	0.614032	0.307016	−	0.951704i	$-0.400669\pi$
$839$	17.7858	0.614032	0.307016	+	0.951704i	$0.400669\pi$
$840$	0	0
$841$	−13.6988	−0.472373
$842$	68.8218i	2.37175i
$843$	− 1.28129i	− 0.0441300i
$844$	−2.71329	−0.0933953
$845$	0	0
$846$	−9.27165	−0.318766
$847$	− 94.2739i	− 3.23929i
$848$	14.0557i	0.482676i
$849$	−2.67132	−0.0916796
$850$	0	0
$851$	0.0121834	0.000417642 0
$852$	− 7.38580i	− 0.253033i
$853$	− 12.8433i	− 0.439745i	−0.975529	−	0.219872i	$-0.929436\pi$
$853$	− 12.8433i	− 0.439745i	0.975529	−	0.219872i	$-0.0705641\pi$
$854$	−37.5467	−1.28482
$855$	0	0
$856$	10.5256	0.359757
$857$	− 15.6015i	− 0.532936i	−0.963844	−	0.266468i	$-0.914143\pi$
$857$	− 15.6015i	− 0.532936i	0.963844	−	0.266468i	$-0.0858566\pi$
$858$	− 33.1776i	− 1.13267i
$859$	−4.82843	−0.164744	−0.0823719	−	0.996602i	$-0.526250\pi$
$859$	−4.82843	−0.164744	−0.0823719	+	0.996602i	$0.526250\pi$
$860$	0	0
$861$	−5.80379	−0.197793
$862$	− 23.4403i	− 0.798381i
$863$	13.1548i	0.447796i	0.974613	+	0.223898i	$0.0718782\pi$
$863$	13.1548i	0.447796i	−0.974613	+	0.223898i	$0.928122\pi$
$864$	−4.80433	−0.163447
$865$	0	0
$866$	21.2021	0.720478
$867$	− 9.67560i	− 0.328601i
$868$	9.73792i	0.330527i
$869$	98.4673	3.34027
$870$	0	0
$871$	19.8452	0.672428
$872$	− 2.52800i	− 0.0856090i
$873$	2.45443i	0.0830698i
$874$	2.35111	0.0795274
$875$	0	0
$876$	12.1404	0.410185
$877$	6.25253i	0.211133i	0.994412	+	0.105567i	$0.0336656\pi$
$877$	6.25253i	0.211133i	−0.994412	+	0.105567i	$0.966334\pi$
$878$	18.7109i	0.631463i
$879$	−17.6605	−0.595675
$880$	0	0
$881$	−16.7801	−0.565335	−0.282667	−	0.959218i	$-0.591219\pi$
$881$	−16.7801	−0.565335	−0.282667	+	0.959218i	$0.591219\pi$
$882$	− 14.5770i	− 0.490834i
$883$	− 16.2367i	− 0.546407i	−0.961956	−	0.273203i	$-0.911917\pi$
$883$	− 16.2367i	− 0.546407i	0.961956	−	0.273203i	$-0.0880832\pi$
$884$	−8.11909	−0.273074
$885$	0	0
$886$	−13.8759	−0.466171
$887$	57.6873i	1.93695i	0.249108	+	0.968476i	$0.419863\pi$
$887$	57.6873i	1.93695i	−0.249108	+	0.968476i	$0.580137\pi$
$888$	0.0387229i	0.00129945i
$889$	−53.3914	−1.79069
$890$	0	0
$891$	−5.90869	−0.197949
$892$	− 0.277771i	− 0.00930046i
$893$	12.8131i	0.428774i
$894$	25.1386	0.840762
$895$	0	0
$896$	49.8994	1.66702
$897$	− 1.92272i	− 0.0641977i
$898$	− 56.2252i	− 1.87626i
$899$	10.5981	0.353465
$900$	0	0
$901$	7.61988	0.253855
$902$	14.8426i	0.494205i
$903$	5.01787i	0.166984i
$904$	19.8442	0.660007
$905$	0	0
$906$	29.7511	0.988415
$907$	9.00465i	0.298995i	0.988762	+	0.149497i	$0.0477655\pi$
$907$	9.00465i	0.298995i	−0.988762	+	0.149497i	$0.952234\pi$
$908$	8.72672i	0.289606i
$909$	6.87495	0.228028
$910$	0	0
$911$	29.9503	0.992298	0.496149	−	0.868237i	$-0.334747\pi$
$911$	29.9503	0.992298	0.496149	+	0.868237i	$0.334747\pi$
$912$	11.7723i	0.389819i
$913$	4.61847i	0.152849i
$914$	−39.0416	−1.29138
$915$	0	0
$916$	11.0578	0.365360
$917$	− 18.1051i	− 0.597882i
$918$	4.61803i	0.152418i
$919$	−8.93560	−0.294758	−0.147379	−	0.989080i	$-0.547084\pi$
$919$	−8.93560	−0.294758	−0.147379	+	0.989080i	$0.547084\pi$
$920$	0	0
$921$	−28.5593	−0.941062
$922$	− 20.5489i	− 0.676741i
$923$	26.6589i	0.877487i
$924$	−21.2370	−0.698647
$925$	0	0
$926$	46.4133	1.52524
$927$	11.7532i	0.386024i
$928$	18.7930i	0.616910i
$929$	41.4596	1.36025	0.680123	−	0.733098i	$-0.261926\pi$
$929$	41.4596	1.36025	0.680123	+	0.733098i	$0.261926\pi$
$930$	0	0
$931$	−20.1450	−0.660225
$932$	18.3248i	0.600250i
$933$	29.3283i	0.960167i
$934$	−6.72721	−0.220121
$935$	0	0
$936$	6.11102	0.199745
$937$	20.8585i	0.681417i	0.940169	+	0.340709i	$0.110667\pi$
$937$	20.8585i	0.681417i	−0.940169	+	0.340709i	$0.889333\pi$
$938$	− 40.5702i	− 1.32466i
$939$	17.4933	0.570871
$940$	0	0
$941$	3.67382	0.119763	0.0598816	−	0.998205i	$-0.480928\pi$
$941$	3.67382	0.119763	0.0598816	+	0.998205i	$0.480928\pi$
$942$	− 30.5619i	− 0.995759i
$943$	0.860163i	0.0280107i
$944$	−23.4151	−0.762096
$945$	0	0
$946$	12.8327	0.417227
$947$	− 31.6997i	− 1.03010i	−0.857160	−	0.515051i	$-0.827773\pi$
$947$	− 31.6997i	− 1.03010i	0.857160	−	0.515051i	$-0.172227\pi$
$948$	− 15.1928i	− 0.493441i
$949$	−43.8204	−1.42247
$950$	0	0
$951$	3.91763	0.127038
$952$	− 19.8144i	− 0.642189i
$953$	− 25.6214i	− 0.829960i	−0.909831	−	0.414980i	$-0.863789\pi$
$953$	− 25.6214i	− 0.829960i	0.909831	−	0.414980i	$-0.136211\pi$
$954$	4.80433	0.155546
$955$	0	0
$956$	16.1035	0.520824
$957$	23.1129i	0.747133i
$958$	34.0407i	1.09980i
$959$	60.1694	1.94297
$960$	0	0
$961$	−23.6595	−0.763209
$962$	0.117082i	0.00377488i
$963$	− 5.66780i	− 0.182642i
$964$	−26.9296	−0.867345
$965$	0	0
$966$	−3.93068	−0.126468
$967$	− 29.0166i	− 0.933112i	−0.884492	−	0.466556i	$-0.845495\pi$
$967$	− 29.0166i	− 0.933112i	0.884492	−	0.466556i	$-0.154505\pi$
$968$	− 44.4077i	− 1.42732i
$969$	6.38197	0.205018
$970$	0	0
$971$	−3.77287	−0.121077	−0.0605386	−	0.998166i	$-0.519282\pi$
$971$	−3.77287	−0.121077	−0.0605386	+	0.998166i	$0.519282\pi$
$972$	0.911672i	0.0292419i
$973$	71.9027i	2.30510i
$974$	−18.7739	−0.601553
$975$	0	0
$976$	−27.8630	−0.891874
$977$	36.7167i	1.17467i	0.809343	+	0.587336i	$0.199823\pi$
$977$	36.7167i	1.17467i	−0.809343	+	0.587336i	$0.800177\pi$
$978$	37.1301i	1.18729i
$979$	−20.5225	−0.655902
$980$	0	0
$981$	−1.36128	−0.0434622
$982$	56.8104i	1.81289i
$983$	− 23.5627i	− 0.751534i	−0.926714	−	0.375767i	$-0.877379\pi$
$983$	− 23.5627i	− 0.751534i	0.926714	−	0.375767i	$-0.122621\pi$
$984$	−2.73388	−0.0871528
$985$	0	0
$986$	18.0642	0.575282
$987$	− 21.4215i	− 0.681854i
$988$	7.07440i	0.225067i
$989$	0.743684	0.0236478
$990$	0	0
$991$	29.1375	0.925583	0.462791	−	0.886467i	$-0.346848\pi$
$991$	29.1375	0.925583	0.462791	+	0.886467i	$0.346848\pi$
$992$	13.0166i	0.413277i
$993$	17.5534i	0.557039i
$994$	54.4997	1.72863
$995$	0	0
$996$	0.712600	0.0225796
$997$	43.9240i	1.39109i	0.718484	+	0.695544i	$0.244836\pi$
$997$	43.9240i	1.39109i	−0.718484	+	0.695544i	$0.755164\pi$
$998$	− 70.2078i	− 2.22239i
$999$	0.0208515	0.000659711 0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.c.1249.3		8
5.2	odd	4		1875.2.a.e.1.4		4
5.3	odd	4		1875.2.a.h.1.1		4
5.4	even	2	inner	1875.2.b.c.1249.6		8
15.2	even	4		5625.2.a.n.1.1		4
15.8	even	4		5625.2.a.i.1.4		4
25.3	odd	20		75.2.g.b.16.2	✓	8
25.4	even	10		375.2.i.b.49.3		16
25.6	even	5		375.2.i.b.199.3		16
25.8	odd	20		75.2.g.b.61.2	yes	8
25.17	odd	20		375.2.g.b.301.1		8
25.19	even	10		375.2.i.b.199.2		16
25.21	even	5		375.2.i.b.49.2		16
25.22	odd	20		375.2.g.b.76.1		8
75.8	even	20		225.2.h.c.136.1		8
75.53	even	20		225.2.h.c.91.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.16.2	✓	8	25.3	odd	20
75.2.g.b.61.2	yes	8	25.8	odd	20
225.2.h.c.91.1		8	75.53	even	20
225.2.h.c.136.1		8	75.8	even	20
375.2.g.b.76.1		8	25.22	odd	20
375.2.g.b.301.1		8	25.17	odd	20
375.2.i.b.49.2		16	25.21	even	5
375.2.i.b.49.3		16	25.4	even	10
375.2.i.b.199.2		16	25.19	even	10
375.2.i.b.199.3		16	25.6	even	5
1875.2.a.e.1.4		4	5.2	odd	4
1875.2.a.h.1.1		4	5.3	odd	4
1875.2.b.c.1249.3		8	1.1	even	1	trivial
1875.2.b.c.1249.6		8	5.4	even	2	inner
5625.2.a.i.1.4		4	15.8	even	4
5625.2.a.n.1.1		4	15.2	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)

\(f(q)\)	\(=\)	\(q-1.70636i q^{2} -1.00000i q^{3} -0.911672 q^{4} -1.70636 q^{6} -3.94243i q^{7} -1.85708i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.70636i q^{2} -1.00000i q^{3} -0.911672 q^{4} -1.70636 q^{6} -3.94243i q^{7} -1.85708i q^{8} -1.00000 q^{9} -5.90869 q^{11} +0.911672i q^{12} -3.29066i q^{13} -6.72721 q^{14} -4.99220 q^{16} +2.70636i q^{17} +1.70636i q^{18} +2.35813 q^{19} -3.94243 q^{21} +10.0824i q^{22} +0.584296i q^{23} -1.85708 q^{24} -5.61505 q^{26} +1.00000i q^{27} +3.59420i q^{28} +3.91167 q^{29} +2.70934 q^{31} +4.80433i q^{32} +5.90869i q^{33} +4.61803 q^{34} +0.911672 q^{36} -0.0208515i q^{37} -4.02383i q^{38} -3.29066 q^{39} +1.47214 q^{41} +6.72721i q^{42} -1.27279i q^{43} +5.38679 q^{44} +0.997020 q^{46} +5.43358i q^{47} +4.99220i q^{48} -8.54276 q^{49} +2.70636 q^{51} +3.00000i q^{52} -2.81554i q^{53} +1.70636 q^{54} -7.32142 q^{56} -2.35813i q^{57} -6.67473i q^{58} +4.69033 q^{59} +5.58132 q^{61} -4.62312i q^{62} +3.94243i q^{63} -1.78646 q^{64} +10.0824 q^{66} +6.03076i q^{67} -2.46731i q^{68} +0.584296 q^{69} -8.10138 q^{71} +1.85708i q^{72} -13.3166i q^{73} -0.0355801 q^{74} -2.14984 q^{76} +23.2946i q^{77} +5.61505i q^{78} -16.6648 q^{79} +1.00000 q^{81} -2.51200i q^{82} -0.781641i q^{83} +3.59420 q^{84} -2.17183 q^{86} -3.91167i q^{87} +10.9729i q^{88} +3.47327 q^{89} -12.9732 q^{91} -0.532686i q^{92} -2.70934i q^{93} +9.27165 q^{94} +4.80433 q^{96} -2.45443i q^{97} +14.5770i q^{98} +5.90869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 + 4 * q^6 - 8 * q^9	\( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9} - 14 q^{11} - 32 q^{14} + 8 q^{16} - 10 q^{19} + 4 q^{21} - 30 q^{24} + 6 q^{26} + 40 q^{29} + 46 q^{31} + 28 q^{34} + 16 q^{36} - 2 q^{39} - 24 q^{41} + 58 q^{44} - 34 q^{46} - 16 q^{49} + 4 q^{51} - 4 q^{54} + 10 q^{56} + 30 q^{59} - 4 q^{61} - 46 q^{64} - 12 q^{66} - 2 q^{69} - 4 q^{71} + 38 q^{74} + 80 q^{76} - 70 q^{79} + 8 q^{81} - 18 q^{84} + 6 q^{86} + 70 q^{89} - 24 q^{91} + 18 q^{94} + 24 q^{96} + 14 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 + 4 * q^6 - 8 * q^9 - 14 * q^11 - 32 * q^14 + 8 * q^16 - 10 * q^19 + 4 * q^21 - 30 * q^24 + 6 * q^26 + 40 * q^29 + 46 * q^31 + 28 * q^34 + 16 * q^36 - 2 * q^39 - 24 * q^41 + 58 * q^44 - 34 * q^46 - 16 * q^49 + 4 * q^51 - 4 * q^54 + 10 * q^56 + 30 * q^59 - 4 * q^61 - 46 * q^64 - 12 * q^66 - 2 * q^69 - 4 * q^71 + 38 * q^74 + 80 * q^76 - 70 * q^79 + 8 * q^81 - 18 * q^84 + 6 * q^86 + 70 * q^89 - 24 * q^91 + 18 * q^94 + 24 * q^96 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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