LMFDB - Embedded newform 1875.2.b.c.1249.8

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			1249.8
Root			$2.70636i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.c.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70636i q^{2} -1.00000i q^{3} -5.32440 q^{4} +2.70636 q^{6} +0.470294i q^{7} -8.99702i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.70636i q^{2} -1.00000i q^{3} -5.32440 q^{4} +2.70636 q^{6} +0.470294i q^{7} -8.99702i q^{8} -1.00000 q^{9} -3.18148 q^{11} +5.32440i q^{12} -0.563444i q^{13} -1.27279 q^{14} +13.7004 q^{16} -1.70636i q^{17} -2.70636i q^{18} -3.74010 q^{19} +0.470294 q^{21} -8.61023i q^{22} +2.26981i q^{23} -8.99702 q^{24} +1.52488 q^{26} +1.00000i q^{27} -2.50403i q^{28} +8.32440 q^{29} +5.43656 q^{31} +19.0842i q^{32} +3.18148i q^{33} +4.61803 q^{34} +5.32440 q^{36} +1.02085i q^{37} -10.1221i q^{38} -0.563444 q^{39} +1.47214 q^{41} +1.27279i q^{42} -6.72721i q^{43} +16.9395 q^{44} -6.14292 q^{46} -4.43358i q^{47} -13.7004i q^{48} +6.77882 q^{49} -1.70636 q^{51} +3.00000i q^{52} +7.05161i q^{53} -2.70636 q^{54} +4.23125 q^{56} +3.74010i q^{57} +22.5288i q^{58} +12.8720 q^{59} +0.126888 q^{61} +14.7133i q^{62} -0.470294i q^{63} -24.2480 q^{64} -8.61023 q^{66} -2.79469i q^{67} +9.08535i q^{68} +2.26981 q^{69} +16.0456 q^{71} +8.99702i q^{72} +9.78873i q^{73} -2.76279 q^{74} +19.9138 q^{76} -1.49623i q^{77} -1.52488i q^{78} +4.75499 q^{79} +1.00000 q^{81} +3.98413i q^{82} -11.6905i q^{83} -2.50403 q^{84} +18.2063 q^{86} -8.32440i q^{87} +28.6238i q^{88} +6.20049 q^{89} +0.264985 q^{91} -12.0853i q^{92} -5.43656i q^{93} +11.9989 q^{94} +19.0842 q^{96} +8.45443i q^{97} +18.3460i q^{98} +3.18148 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.70636i	1.91369i	0.290604	+	0.956844i	$0.406144\pi$
$2$	2.70636i	1.91369i	−0.290604	+	0.956844i	$0.593856\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.32440	−2.66220
$5$	0	0
$5$	0	0
$6$	2.70636	1.10487
$7$	0.470294i	0.177754i	0.996043	+	0.0888772i	$0.0283279\pi$
$7$	0.470294i	0.177754i	−0.996043	+	0.0888772i	$0.971672\pi$
$8$	− 8.99702i	− 3.18093i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.18148	−0.959252	−0.479626	−	0.877473i	$-0.659228\pi$
$11$	−3.18148	−0.959252	−0.479626	+	0.877473i	$0.659228\pi$
$12$	5.32440i	1.53702i
$13$	− 0.563444i	− 0.156271i	−0.996943	−	0.0781356i	$-0.975103\pi$
$13$	− 0.563444i	− 0.156271i	0.996943	−	0.0781356i	$-0.0248967\pi$
$14$	−1.27279	−0.340166
$15$	0	0
$16$	13.7004	3.42510
$17$	− 1.70636i	− 0.413854i	−0.978356	−	0.206927i	$-0.933654\pi$
$17$	− 1.70636i	− 0.413854i	0.978356	−	0.206927i	$-0.0663462\pi$
$18$	− 2.70636i	− 0.637896i
$19$	−3.74010	−0.858038	−0.429019	−	0.903295i	$-0.641141\pi$
$19$	−3.74010	−0.858038	−0.429019	+	0.903295i	$0.641141\pi$
$20$	0	0
$21$	0.470294	0.102627
$22$	− 8.61023i	− 1.83571i
$23$	2.26981i	0.473287i	0.971597	+	0.236644i	$0.0760474\pi$
$23$	2.26981i	0.473287i	−0.971597	+	0.236644i	$0.923953\pi$
$24$	−8.99702	−1.83651
$25$	0	0
$26$	1.52488	0.299054
$27$	1.00000i	0.192450i
$28$	− 2.50403i	− 0.473218i
$29$	8.32440	1.54580	0.772901	−	0.634527i	$-0.218805\pi$
$29$	8.32440	1.54580	0.772901	+	0.634527i	$0.218805\pi$
$30$	0	0
$31$	5.43656	0.976434	0.488217	−	0.872722i	$-0.337647\pi$
$31$	5.43656	0.976434	0.488217	+	0.872722i	$0.337647\pi$
$32$	19.0842i	3.37364i
$33$	3.18148i	0.553824i
$34$	4.61803	0.791986
$35$	0	0
$36$	5.32440	0.887399
$37$	1.02085i	0.167827i	0.996473	+	0.0839135i	$0.0267419\pi$
$37$	1.02085i	0.167827i	−0.996473	+	0.0839135i	$0.973258\pi$
$38$	− 10.1221i	− 1.64202i
$39$	−0.563444	−0.0902233
$40$	0	0
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	1.27279i	0.196395i
$43$	− 6.72721i	− 1.02589i	−0.858421	−	0.512945i	$-0.828554\pi$
$43$	− 6.72721i	− 1.02589i	0.858421	−	0.512945i	$-0.171446\pi$
$44$	16.9395	2.55372
$45$	0	0
$46$	−6.14292	−0.905724
$47$	− 4.43358i	− 0.646703i	−0.946279	−	0.323352i	$-0.895190\pi$
$47$	− 4.43358i	− 0.646703i	0.946279	−	0.323352i	$-0.104810\pi$
$48$	− 13.7004i	− 1.97748i
$49$	6.77882	0.968403
$50$	0	0
$51$	−1.70636	−0.238938
$52$	3.00000i	0.416025i
$53$	7.05161i	0.968613i	0.874898	+	0.484307i	$0.160928\pi$
$53$	7.05161i	0.968613i	−0.874898	+	0.484307i	$0.839072\pi$
$54$	−2.70636	−0.368289
$55$	0	0
$56$	4.23125	0.565424
$57$	3.74010i	0.495388i
$58$	22.5288i	2.95818i
$59$	12.8720	1.67579	0.837894	−	0.545833i	$-0.183787\pi$
$59$	12.8720	1.67579	0.837894	+	0.545833i	$0.183787\pi$
$60$	0	0
$61$	0.126888	0.0162464	0.00812319	−	0.999967i	$-0.497414\pi$
$61$	0.126888	0.0162464	0.00812319	+	0.999967i	$0.497414\pi$
$62$	14.7133i	1.86859i
$63$	− 0.470294i	− 0.0592515i
$64$	−24.2480	−3.03100
$65$	0	0
$66$	−8.61023	−1.05985
$67$	− 2.79469i	− 0.341426i	−0.985321	−	0.170713i	$-0.945393\pi$
$67$	− 2.79469i	− 0.341426i	0.985321	−	0.170713i	$-0.0546071\pi$
$68$	9.08535i	1.10176i
$69$	2.26981	0.273253
$70$	0	0
$71$	16.0456	1.90427	0.952134	−	0.305681i	$-0.0988839\pi$
$71$	16.0456	1.90427	0.952134	+	0.305681i	$0.0988839\pi$
$72$	8.99702i	1.06031i
$73$	9.78873i	1.14568i	0.819666	+	0.572842i	$0.194159\pi$
$73$	9.78873i	1.14568i	−0.819666	+	0.572842i	$0.805841\pi$
$74$	−2.76279	−0.321168
$75$	0	0
$76$	19.9138	2.28427
$77$	− 1.49623i	− 0.170511i
$78$	− 1.52488i	− 0.172659i
$79$	4.75499	0.534978	0.267489	−	0.963561i	$-0.413806\pi$
$79$	4.75499	0.534978	0.267489	+	0.963561i	$0.413806\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.98413i	0.439974i
$83$	− 11.6905i	− 1.28320i	−0.767040	−	0.641599i	$-0.778271\pi$
$83$	− 11.6905i	− 1.28320i	0.767040	−	0.641599i	$-0.221729\pi$
$84$	−2.50403	−0.273212
$85$	0	0
$86$	18.2063	1.96323
$87$	− 8.32440i	− 0.892469i
$88$	28.6238i	3.05131i
$89$	6.20049	0.657250	0.328625	−	0.944460i	$-0.393415\pi$
$89$	6.20049	0.657250	0.328625	+	0.944460i	$0.393415\pi$
$90$	0	0
$91$	0.264985	0.0277779
$92$	− 12.0853i	− 1.25998i
$93$	− 5.43656i	− 0.563745i
$94$	11.9989	1.23759
$95$	0	0
$96$	19.0842	1.94777
$97$	8.45443i	0.858417i	0.903205	+	0.429209i	$0.141207\pi$
$97$	8.45443i	0.858417i	−0.903205	+	0.429209i	$0.858793\pi$
$98$	18.3460i	1.85322i
$99$	3.18148	0.319751
$100$	0	0
$101$	−5.83325	−0.580430	−0.290215	−	0.956961i	$-0.593727\pi$
$101$	−5.83325	−0.580430	−0.290215	+	0.956961i	$0.593727\pi$
$102$	− 4.61803i	− 0.457254i
$103$	14.0794i	1.38728i	0.720320	+	0.693642i	$0.243995\pi$
$103$	14.0794i	1.38728i	−0.720320	+	0.693642i	$0.756005\pi$
$104$	−5.06932	−0.497088
$105$	0	0
$106$	−19.0842	−1.85362
$107$	− 8.61207i	− 0.832561i	−0.909236	−	0.416280i	$-0.863334\pi$
$107$	− 8.61207i	− 0.832561i	0.909236	−	0.416280i	$-0.136666\pi$
$108$	− 5.32440i	− 0.512340i
$109$	−16.6875	−1.59837	−0.799187	−	0.601082i	$-0.794736\pi$
$109$	−16.6875	−1.59837	−0.799187	+	0.601082i	$0.794736\pi$
$110$	0	0
$111$	1.02085	0.0968949
$112$	6.44322i	0.608827i
$113$	− 11.7758i	− 1.10778i	−0.832590	−	0.553889i	$-0.813143\pi$
$113$	− 11.7758i	− 1.10778i	0.832590	−	0.553889i	$-0.186857\pi$
$114$	−10.1221	−0.948018
$115$	0	0
$116$	−44.3224	−4.11523
$117$	0.563444i	0.0520904i
$118$	34.8362i	3.20693i
$119$	0.802492	0.0735643
$120$	0	0
$121$	−0.878197	−0.0798361
$122$	0.343406i	0.0310905i
$123$	− 1.47214i	− 0.132738i
$124$	−28.9464	−2.59946
$125$	0	0
$126$	1.27279	0.113389
$127$	1.77882i	0.157845i	0.996881	+	0.0789225i	$0.0251480\pi$
$127$	1.77882i	0.157845i	−0.996881	+	0.0789225i	$0.974852\pi$
$128$	− 27.4554i	− 2.42674i
$129$	−6.72721	−0.592298
$130$	0	0
$131$	−5.91860	−0.517110	−0.258555	−	0.965996i	$-0.583246\pi$
$131$	−5.91860	−0.517110	−0.258555	+	0.965996i	$0.583246\pi$
$132$	− 16.9395i	− 1.47439i
$133$	− 1.75895i	− 0.152520i
$134$	7.56344	0.653382
$135$	0	0
$136$	−15.3522	−1.31644
$137$	− 5.11611i	− 0.437098i	−0.975826	−	0.218549i	$-0.929868\pi$
$137$	− 5.11611i	− 0.437098i	0.975826	−	0.218549i	$-0.0701324\pi$
$138$	6.14292i	0.522920i
$139$	8.23817	0.698753	0.349376	−	0.936982i	$-0.386393\pi$
$139$	8.23817	0.698753	0.349376	+	0.936982i	$0.386393\pi$
$140$	0	0
$141$	−4.43358	−0.373374
$142$	43.4253i	3.64417i
$143$	1.79259i	0.149904i
$144$	−13.7004	−1.14170
$145$	0	0
$146$	−26.4918	−2.19248
$147$	− 6.77882i	− 0.559108i
$148$	− 5.43542i	− 0.446789i
$149$	10.0585	0.824027	0.412014	−	0.911178i	$-0.364826\pi$
$149$	10.0585	0.824027	0.412014	+	0.911178i	$0.364826\pi$
$150$	0	0
$151$	−11.9810	−0.974999	−0.487500	−	0.873123i	$-0.662091\pi$
$151$	−11.9810	−0.974999	−0.487500	+	0.873123i	$0.662091\pi$
$152$	33.6498i	2.72936i
$153$	1.70636i	0.137951i
$154$	4.04934	0.326305
$155$	0	0
$156$	3.00000	0.240192
$157$	19.5960i	1.56393i	0.623319	+	0.781967i	$0.285784\pi$
$157$	19.5960i	1.56393i	−0.623319	+	0.781967i	$0.714216\pi$
$158$	12.8687i	1.02378i
$159$	7.05161	0.559229
$160$	0	0
$161$	−1.06748	−0.0841290
$162$	2.70636i	0.212632i
$163$	− 11.8927i	− 0.931505i	−0.884915	−	0.465753i	$-0.845784\pi$
$163$	− 11.8927i	− 0.931505i	0.884915	−	0.465753i	$-0.154216\pi$
$164$	−7.83824	−0.612063
$165$	0	0
$166$	31.6387	2.45564
$167$	− 5.99378i	− 0.463812i	−0.972738	−	0.231906i	$-0.925504\pi$
$167$	− 5.99378i	− 0.463812i	0.972738	−	0.231906i	$-0.0744962\pi$
$168$	− 4.23125i	− 0.326448i
$169$	12.6825	0.975579
$170$	0	0
$171$	3.74010	0.286013
$172$	35.8184i	2.73112i
$173$	− 17.3244i	− 1.31715i	−0.752515	−	0.658575i	$-0.771160\pi$
$173$	− 17.3244i	− 1.31715i	0.752515	−	0.658575i	$-0.228840\pi$
$174$	22.5288	1.70791
$175$	0	0
$176$	−43.5875	−3.28553
$177$	− 12.8720i	− 0.967517i
$178$	16.7808i	1.25777i
$179$	−10.5765	−0.790524	−0.395262	−	0.918568i	$-0.629346\pi$
$179$	−10.5765	−0.790524	−0.395262	+	0.918568i	$0.629346\pi$
$180$	0	0
$181$	11.9799	0.890455	0.445228	−	0.895417i	$-0.353123\pi$
$181$	11.9799	0.890455	0.445228	+	0.895417i	$0.353123\pi$
$182$	0.717144i	0.0531583i
$183$	− 0.126888i	− 0.00937986i
$184$	20.4215	1.50549
$185$	0	0
$186$	14.7133	1.07883
$187$	5.42875i	0.396990i
$188$	23.6061i	1.72165i
$189$	−0.470294	−0.0342089
$190$	0	0
$191$	18.2657	1.32166	0.660829	−	0.750536i	$-0.270205\pi$
$191$	18.2657	1.32166	0.660829	+	0.750536i	$0.270205\pi$
$192$	24.2480i	1.74995i
$193$	− 25.2063i	− 1.81439i	−0.420713	−	0.907194i	$-0.638220\pi$
$193$	− 25.2063i	− 1.81439i	0.420713	−	0.907194i	$-0.361780\pi$
$194$	−22.8807	−1.64274
$195$	0	0
$196$	−36.0931	−2.57808
$197$	12.0923i	0.861539i	0.902462	+	0.430769i	$0.141758\pi$
$197$	12.0923i	0.861539i	−0.902462	+	0.430769i	$0.858242\pi$
$198$	8.61023i	0.611903i
$199$	0.544434	0.0385939	0.0192970	−	0.999814i	$-0.493857\pi$
$199$	0.544434	0.0385939	0.0192970	+	0.999814i	$0.493857\pi$
$200$	0	0
$201$	−2.79469	−0.197122
$202$	− 15.7869i	− 1.11076i
$203$	3.91492i	0.274773i
$204$	9.08535	0.636102
$205$	0	0
$206$	−38.1039	−2.65483
$207$	− 2.26981i	− 0.157762i
$208$	− 7.71941i	− 0.535245i
$209$	11.8990	0.823074
$210$	0	0
$211$	−3.12207	−0.214932	−0.107466	−	0.994209i	$-0.534274\pi$
$211$	−3.12207	−0.214932	−0.107466	+	0.994209i	$0.534274\pi$
$212$	− 37.5456i	− 2.57864i
$213$	− 16.0456i	− 1.09943i
$214$	23.3074	1.59326
$215$	0	0
$216$	8.99702	0.612170
$217$	2.55678i	0.173566i
$218$	− 45.1625i	− 3.05879i
$219$	9.78873	0.661461
$220$	0	0
$221$	−0.961440	−0.0646734
$222$	2.76279i	0.185427i
$223$	− 0.339125i	− 0.0227095i	−0.999936	−	0.0113547i	$-0.996386\pi$
$223$	− 0.339125i	− 0.0227095i	0.999936	−	0.0113547i	$-0.00361440\pi$
$224$	−8.97519	−0.599680
$225$	0	0
$226$	31.8697	2.11994
$227$	− 1.78847i	− 0.118705i	−0.998237	−	0.0593524i	$-0.981096\pi$
$227$	− 1.78847i	− 0.118705i	0.998237	−	0.0593524i	$-0.0189036\pi$
$228$	− 19.9138i	− 1.31882i
$229$	27.9833	1.84919	0.924593	−	0.380957i	$-0.124405\pi$
$229$	27.9833	1.84919	0.924593	+	0.380957i	$0.124405\pi$
$230$	0	0
$231$	−1.49623	−0.0984448
$232$	− 74.8948i	− 4.91708i
$233$	6.77400i	0.443780i	0.975072	+	0.221890i	$0.0712225\pi$
$233$	6.77400i	0.443780i	−0.975072	+	0.221890i	$0.928777\pi$
$234$	−1.52488	−0.0996848
$235$	0	0
$236$	−68.5355	−4.46128
$237$	− 4.75499i	− 0.308870i
$238$	2.17183i	0.140779i
$239$	6.48334	0.419373	0.209686	−	0.977769i	$-0.432756\pi$
$239$	6.48334	0.419373	0.209686	+	0.977769i	$0.432756\pi$
$240$	0	0
$241$	−7.44857	−0.479804	−0.239902	−	0.970797i	$-0.577115\pi$
$241$	−7.44857	−0.479804	−0.239902	+	0.970797i	$0.577115\pi$
$242$	− 2.37672i	− 0.152781i
$243$	− 1.00000i	− 0.0641500i
$244$	−0.675604	−0.0432511
$245$	0	0
$246$	3.98413	0.254019
$247$	2.10734i	0.134087i
$248$	− 48.9128i	− 3.10597i
$249$	−11.6905	−0.740855
$250$	0	0
$251$	−4.60217	−0.290486	−0.145243	−	0.989396i	$-0.546396\pi$
$251$	−4.60217	−0.290486	−0.145243	+	0.989396i	$0.546396\pi$
$252$	2.50403i	0.157739i
$253$	− 7.22134i	− 0.454002i
$254$	−4.81414	−0.302066
$255$	0	0
$256$	25.8083	1.61302
$257$	− 5.79485i	− 0.361473i	−0.983532	−	0.180736i	$-0.942152\pi$
$257$	− 5.79485i	− 0.361473i	0.983532	−	0.180736i	$-0.0578481\pi$
$258$	− 18.2063i	− 1.13347i
$259$	−0.480101	−0.0298320
$260$	0	0
$261$	−8.32440	−0.515267
$262$	− 16.0179i	− 0.989587i
$263$	− 14.2989i	− 0.881707i	−0.897579	−	0.440854i	$-0.854676\pi$
$263$	− 14.2989i	− 0.881707i	0.897579	−	0.440854i	$-0.145324\pi$
$264$	28.6238	1.76167
$265$	0	0
$266$	4.76035	0.291876
$267$	− 6.20049i	− 0.379464i
$268$	14.8800i	0.908943i
$269$	19.2205	1.17189	0.585946	−	0.810350i	$-0.300723\pi$
$269$	19.2205	1.17189	0.585946	+	0.810350i	$0.300723\pi$
$270$	0	0
$271$	−23.2433	−1.41193	−0.705964	−	0.708248i	$-0.749486\pi$
$271$	−23.2433	−1.41193	−0.705964	+	0.708248i	$0.749486\pi$
$272$	− 23.3778i	− 1.41749i
$273$	− 0.264985i	− 0.0160376i
$274$	13.8460	0.836470
$275$	0	0
$276$	−12.0853	−0.727452
$277$	21.8910i	1.31530i	0.753323	+	0.657651i	$0.228450\pi$
$277$	21.8910i	1.31530i	−0.753323	+	0.657651i	$0.771550\pi$
$278$	22.2955i	1.33719i
$279$	−5.43656	−0.325478
$280$	0	0
$281$	27.1138	1.61748	0.808738	−	0.588169i	$-0.200151\pi$
$281$	27.1138	1.61748	0.808738	+	0.588169i	$0.200151\pi$
$282$	− 11.9989i	− 0.714522i
$283$	− 16.3074i	− 0.969374i	−0.874688	−	0.484687i	$-0.838934\pi$
$283$	− 16.3074i	− 0.969374i	0.874688	−	0.484687i	$-0.161066\pi$
$284$	−85.4334	−5.06954
$285$	0	0
$286$	−4.85139	−0.286868
$287$	0.692337i	0.0408674i
$288$	− 19.0842i	− 1.12455i
$289$	14.0883	0.828725
$290$	0	0
$291$	8.45443	0.495607
$292$	− 52.1191i	− 3.05004i
$293$	− 5.46407i	− 0.319214i	−0.987181	−	0.159607i	$-0.948977\pi$
$293$	− 5.46407i	− 0.319214i	0.987181	−	0.159607i	$-0.0510228\pi$
$294$	18.3460	1.06996
$295$	0	0
$296$	9.18462	0.533845
$297$	− 3.18148i	− 0.184608i
$298$	27.2220i	1.57693i
$299$	1.27891	0.0739612
$300$	0	0
$301$	3.16377	0.182357
$302$	− 32.4249i	− 1.86584i
$303$	5.83325i	0.335111i
$304$	−51.2409	−2.93887
$305$	0	0
$306$	−4.61803	−0.263995
$307$	− 21.4194i	− 1.22247i	−0.791450	−	0.611235i	$-0.790673\pi$
$307$	− 21.4194i	− 1.22247i	0.791450	−	0.611235i	$-0.209327\pi$
$308$	7.96652i	0.453935i
$309$	14.0794	0.800948
$310$	0	0
$311$	−2.85200	−0.161722	−0.0808610	−	0.996725i	$-0.525767\pi$
$311$	−2.85200	−0.161722	−0.0808610	+	0.996725i	$0.525767\pi$
$312$	5.06932i	0.286994i
$313$	− 14.4375i	− 0.816057i	−0.912969	−	0.408029i	$-0.866216\pi$
$313$	− 14.4375i	− 0.816057i	0.912969	−	0.408029i	$-0.133784\pi$
$314$	−53.0340	−2.99288
$315$	0	0
$316$	−25.3175	−1.42422
$317$	22.6102i	1.26992i	0.772546	+	0.634959i	$0.218983\pi$
$317$	22.6102i	1.26992i	−0.772546	+	0.634959i	$0.781017\pi$
$318$	19.0842i	1.07019i
$319$	−26.4839	−1.48281
$320$	0	0
$321$	−8.61207	−0.480679
$322$	− 2.88898i	− 0.160996i
$323$	6.38197i	0.355102i
$324$	−5.32440	−0.295800
$325$	0	0
$326$	32.1859	1.78261
$327$	16.6875i	0.922822i
$328$	− 13.2448i	− 0.731324i
$329$	2.08508	0.114954
$330$	0	0
$331$	3.86645	0.212519	0.106260	−	0.994338i	$-0.466113\pi$
$331$	3.86645	0.212519	0.106260	+	0.994338i	$0.466113\pi$
$332$	62.2448i	3.41613i
$333$	− 1.02085i	− 0.0559423i
$334$	16.2213	0.887592
$335$	0	0
$336$	6.44322	0.351506
$337$	17.4853i	0.952484i	0.879314	+	0.476242i	$0.158001\pi$
$337$	17.4853i	0.952484i	−0.879314	+	0.476242i	$0.841999\pi$
$338$	34.3235i	1.86695i
$339$	−11.7758	−0.639576
$340$	0	0
$341$	−17.2963	−0.936646
$342$	10.1221i	0.547339i
$343$	6.48010i	0.349893i
$344$	−60.5249	−3.26328
$345$	0	0
$346$	46.8861	2.52061
$347$	− 23.0941i	− 1.23976i	−0.784698	−	0.619879i	$-0.787182\pi$
$347$	− 23.0941i	− 1.23976i	0.784698	−	0.619879i	$-0.212818\pi$
$348$	44.3224i	2.37593i
$349$	9.32650	0.499236	0.249618	−	0.968344i	$-0.419695\pi$
$349$	9.32650	0.499236	0.249618	+	0.968344i	$0.419695\pi$
$350$	0	0
$351$	0.563444	0.0300744
$352$	− 60.7160i	− 3.23617i
$353$	− 31.9471i	− 1.70037i	−0.526483	−	0.850186i	$-0.676490\pi$
$353$	− 31.9471i	− 1.70037i	0.526483	−	0.850186i	$-0.323510\pi$
$354$	34.8362	1.85152
$355$	0	0
$356$	−33.0139	−1.74973
$357$	− 0.802492i	− 0.0424724i
$358$	− 28.6238i	− 1.51282i
$359$	9.58954	0.506117	0.253058	−	0.967451i	$-0.418564\pi$
$359$	9.58954	0.506117	0.253058	+	0.967451i	$0.418564\pi$
$360$	0	0
$361$	−5.01165	−0.263771
$362$	32.4218i	1.70405i
$363$	0.878197i	0.0460934i
$364$	−1.41088	−0.0739503
$365$	0	0
$366$	0.343406	0.0179501
$367$	1.06423i	0.0555525i	0.999614	+	0.0277763i	$0.00884260\pi$
$367$	1.06423i	0.0555525i	−0.999614	+	0.0277763i	$0.991157\pi$
$368$	31.0973i	1.62106i
$369$	−1.47214	−0.0766363
$370$	0	0
$371$	−3.31633	−0.172175
$372$	28.9464i	1.50080i
$373$	4.12022i	0.213337i	0.994295	+	0.106669i	$0.0340184\pi$
$373$	4.12022i	0.213337i	−0.994295	+	0.106669i	$0.965982\pi$
$374$	−14.6922	−0.759714
$375$	0	0
$376$	−39.8890	−2.05712
$377$	− 4.69033i	− 0.241564i
$378$	− 1.27279i	− 0.0654651i
$379$	−10.3563	−0.531967	−0.265984	−	0.963978i	$-0.585697\pi$
$379$	−10.3563	−0.531967	−0.265984	+	0.963978i	$0.585697\pi$
$380$	0	0
$381$	1.77882	0.0911319
$382$	49.4336i	2.52924i
$383$	− 1.19935i	− 0.0612839i	−0.999530	−	0.0306420i	$-0.990245\pi$
$383$	− 1.19935i	− 0.0612839i	0.999530	−	0.0306420i	$-0.00975516\pi$
$384$	−27.4554	−1.40108
$385$	0	0
$386$	68.2173	3.47217
$387$	6.72721i	0.341963i
$388$	− 45.0147i	− 2.28528i
$389$	−12.6043	−0.639062	−0.319531	−	0.947576i	$-0.603525\pi$
$389$	−12.6043	−0.639062	−0.319531	+	0.947576i	$0.603525\pi$
$390$	0	0
$391$	3.87311	0.195872
$392$	− 60.9892i	− 3.08042i
$393$	5.91860i	0.298554i
$394$	−32.7261	−1.64872
$395$	0	0
$396$	−16.9395	−0.851239
$397$	− 1.31335i	− 0.0659152i	−0.999457	−	0.0329576i	$-0.989507\pi$
$397$	− 1.31335i	− 0.0659152i	0.999457	−	0.0329576i	$-0.0104926\pi$
$398$	1.47344i	0.0738567i
$399$	−1.75895	−0.0880575
$400$	0	0
$401$	16.1042	0.804205	0.402102	−	0.915595i	$-0.368280\pi$
$401$	16.1042	0.804205	0.402102	+	0.915595i	$0.368280\pi$
$402$	− 7.56344i	− 0.377230i
$403$	− 3.06320i	− 0.152589i
$404$	31.0585	1.54522
$405$	0	0
$406$	−10.5952	−0.525830
$407$	− 3.24782i	− 0.160988i
$408$	15.3522i	0.760046i
$409$	−21.1600	−1.04630	−0.523148	−	0.852242i	$-0.675242\pi$
$409$	−21.1600	−1.04630	−0.523148	+	0.852242i	$0.675242\pi$
$410$	0	0
$411$	−5.11611	−0.252359
$412$	− 74.9642i	− 3.69322i
$413$	6.05361i	0.297879i
$414$	6.14292	0.301908
$415$	0	0
$416$	10.7529	0.527204
$417$	− 8.23817i	− 0.403425i
$418$	32.2031i	1.57511i
$419$	6.67094	0.325897	0.162948	−	0.986635i	$-0.447900\pi$
$419$	6.67094	0.325897	0.162948	+	0.986635i	$0.447900\pi$
$420$	0	0
$421$	27.2980	1.33042	0.665212	−	0.746655i	$-0.268341\pi$
$421$	27.2980	1.33042	0.665212	+	0.746655i	$0.268341\pi$
$422$	− 8.44944i	− 0.411312i
$423$	4.43358i	0.215568i
$424$	63.4435	3.08109
$425$	0	0
$426$	43.4253	2.10396
$427$	0.0596749i	0.00288787i
$428$	45.8541i	2.21644i
$429$	1.79259	0.0865468
$430$	0	0
$431$	12.2974	0.592346	0.296173	−	0.955134i	$-0.404290\pi$
$431$	12.2974	0.592346	0.296173	+	0.955134i	$0.404290\pi$
$432$	13.7004i	0.659161i
$433$	33.8452i	1.62649i	0.581918	+	0.813247i	$0.302302\pi$
$433$	33.8452i	1.62649i	−0.581918	+	0.813247i	$0.697698\pi$
$434$	−6.91957	−0.332150
$435$	0	0
$436$	88.8509	4.25519
$437$	− 8.48930i	− 0.406098i
$438$	26.4918i	1.26583i
$439$	20.9654	1.00062	0.500312	−	0.865845i	$-0.333219\pi$
$439$	20.9654	1.00062	0.500312	+	0.865845i	$0.333219\pi$
$440$	0	0
$441$	−6.77882	−0.322801
$442$	− 2.60200i	− 0.123765i
$443$	− 6.04847i	− 0.287371i	−0.989623	−	0.143686i	$-0.954105\pi$
$443$	− 6.04847i	− 0.287371i	0.989623	−	0.143686i	$-0.0458954\pi$
$444$	−5.43542	−0.257954
$445$	0	0
$446$	0.917795	0.0434588
$447$	− 10.0585i	− 0.475752i
$448$	− 11.4037i	− 0.538773i
$449$	−15.5896	−0.735721	−0.367860	−	0.929881i	$-0.619910\pi$
$449$	−15.5896	−0.735721	−0.367860	+	0.929881i	$0.619910\pi$
$450$	0	0
$451$	−4.68357	−0.220541
$452$	62.6993i	2.94912i
$453$	11.9810i	0.562916i
$454$	4.84024	0.227164
$455$	0	0
$456$	33.6498	1.57579
$457$	− 2.50193i	− 0.117035i	−0.998286	−	0.0585176i	$-0.981363\pi$
$457$	− 2.50193i	− 0.117035i	0.998286	−	0.0585176i	$-0.0186374\pi$
$458$	75.7328i	3.53876i
$459$	1.70636	0.0796462
$460$	0	0
$461$	−0.153963	−0.00717078	−0.00358539	−	0.999994i	$-0.501141\pi$
$461$	−0.153963	−0.00717078	−0.00358539	+	0.999994i	$0.501141\pi$
$462$	− 4.04934i	− 0.188392i
$463$	11.6327i	0.540616i	0.962774	+	0.270308i	$0.0871255\pi$
$463$	11.6327i	0.540616i	−0.962774	+	0.270308i	$0.912875\pi$
$464$	114.048	5.29453
$465$	0	0
$466$	−18.3329	−0.849255
$467$	0.470294i	0.0217626i	0.999941	+	0.0108813i	$0.00346370\pi$
$467$	0.470294i	0.0217626i	−0.999941	+	0.0108813i	$0.996536\pi$
$468$	− 3.00000i	− 0.138675i
$469$	1.31433	0.0606900
$470$	0	0
$471$	19.5960	0.902938
$472$	− 115.809i	− 5.33056i
$473$	21.4025i	0.984087i
$474$	12.8687	0.591080
$475$	0	0
$476$	−4.27279	−0.195843
$477$	− 7.05161i	− 0.322871i
$478$	17.5463i	0.802548i
$479$	−12.4114	−0.567092	−0.283546	−	0.958959i	$-0.591511\pi$
$479$	−12.4114	−0.567092	−0.283546	+	0.958959i	$0.591511\pi$
$480$	0	0
$481$	0.575193	0.0262265
$482$	− 20.1585i	− 0.918196i
$483$	1.06748i	0.0485719i
$484$	4.67587	0.212539
$485$	0	0
$486$	2.70636	0.122763
$487$	34.5646i	1.56627i	0.621852	+	0.783135i	$0.286381\pi$
$487$	34.5646i	1.56627i	−0.621852	+	0.783135i	$0.713619\pi$
$488$	− 1.14162i	− 0.0516786i
$489$	−11.8927	−0.537805
$490$	0	0
$491$	−24.8658	−1.12218	−0.561088	−	0.827756i	$-0.689617\pi$
$491$	−24.8658	−1.12218	−0.561088	+	0.827756i	$0.689617\pi$
$492$	7.83824i	0.353375i
$493$	− 14.2044i	− 0.639736i
$494$	−5.70322	−0.256600
$495$	0	0
$496$	74.4830	3.34439
$497$	7.54618i	0.338492i
$498$	− 31.6387i	− 1.41776i
$499$	−12.6037	−0.564220	−0.282110	−	0.959382i	$-0.591034\pi$
$499$	−12.6037	−0.564220	−0.282110	+	0.959382i	$0.591034\pi$
$500$	0	0
$501$	−5.99378	−0.267782
$502$	− 12.4551i	− 0.555900i
$503$	− 29.9927i	− 1.33731i	−0.743573	−	0.668655i	$-0.766870\pi$
$503$	− 29.9927i	− 1.33731i	0.743573	−	0.668655i	$-0.233130\pi$
$504$	−4.23125	−0.188475
$505$	0	0
$506$	19.5436	0.868817
$507$	− 12.6825i	− 0.563251i
$508$	− 9.47116i	− 0.420215i
$509$	0.466989	0.0206989	0.0103495	−	0.999946i	$-0.496706\pi$
$509$	0.466989	0.0206989	0.0103495	+	0.999946i	$0.496706\pi$
$510$	0	0
$511$	−4.60358	−0.203651
$512$	14.9358i	0.660074i
$513$	− 3.74010i	− 0.165129i
$514$	15.6830	0.691746
$515$	0	0
$516$	35.8184	1.57681
$517$	14.1053i	0.620351i
$518$	− 1.29933i	− 0.0570891i
$519$	−17.3244	−0.760457
$520$	0	0
$521$	−17.2819	−0.757133	−0.378566	−	0.925574i	$-0.623583\pi$
$521$	−17.2819	−0.757133	−0.378566	+	0.925574i	$0.623583\pi$
$522$	− 22.5288i	− 0.986060i
$523$	− 40.6479i	− 1.77741i	−0.458480	−	0.888705i	$-0.651606\pi$
$523$	− 40.6479i	− 1.77741i	0.458480	−	0.888705i	$-0.348394\pi$
$524$	31.5130	1.37665
$525$	0	0
$526$	38.6980	1.68731
$527$	− 9.27673i	− 0.404101i
$528$	43.5875i	1.89690i
$529$	17.8480	0.775999
$530$	0	0
$531$	−12.8720	−0.558596
$532$	9.36533i	0.406039i
$533$	− 0.829466i	− 0.0359282i
$534$	16.7808	0.726175
$535$	0	0
$536$	−25.1439	−1.08605
$537$	10.5765i	0.456409i
$538$	52.0175i	2.24264i
$539$	−21.5667	−0.928943
$540$	0	0
$541$	−1.98099	−0.0851694	−0.0425847	−	0.999093i	$-0.513559\pi$
$541$	−1.98099	−0.0851694	−0.0425847	+	0.999093i	$0.513559\pi$
$542$	− 62.9047i	− 2.70199i
$543$	− 11.9799i	− 0.514105i
$544$	32.5646	1.39619
$545$	0	0
$546$	0.717144	0.0306909
$547$	35.0543i	1.49881i	0.662111	+	0.749406i	$0.269661\pi$
$547$	35.0543i	1.49881i	−0.662111	+	0.749406i	$0.730339\pi$
$548$	27.2402i	1.16364i
$549$	−0.126888	−0.00541546
$550$	0	0
$551$	−31.1341	−1.32636
$552$	− 20.4215i	− 0.869196i
$553$	2.23625i	0.0950948i
$554$	−59.2449	−2.51708
$555$	0	0
$556$	−43.8633	−1.86022
$557$	19.2383i	0.815154i	0.913171	+	0.407577i	$0.133626\pi$
$557$	19.2383i	0.815154i	−0.913171	+	0.407577i	$0.866374\pi$
$558$	− 14.7133i	− 0.622863i
$559$	−3.79041	−0.160317
$560$	0	0
$561$	5.42875	0.229202
$562$	73.3798i	3.09534i
$563$	29.1741i	1.22954i	0.788706	+	0.614771i	$0.210752\pi$
$563$	29.1741i	1.22954i	−0.788706	+	0.614771i	$0.789248\pi$
$564$	23.6061	0.993997
$565$	0	0
$566$	44.1337	1.85508
$567$	0.470294i	0.0197505i
$568$	− 144.363i	− 6.05734i
$569$	32.9112	1.37971	0.689855	−	0.723947i	$-0.257674\pi$
$569$	32.9112	1.37971	0.689855	+	0.723947i	$0.257674\pi$
$570$	0	0
$571$	26.6349	1.11464	0.557319	−	0.830299i	$-0.311830\pi$
$571$	26.6349	1.11464	0.557319	+	0.830299i	$0.311830\pi$
$572$	− 9.54443i	− 0.399073i
$573$	− 18.2657i	− 0.763060i
$574$	−1.87371	−0.0782073
$575$	0	0
$576$	24.2480	1.01033
$577$	34.5156i	1.43690i	0.695577	+	0.718452i	$0.255149\pi$
$577$	34.5156i	1.43690i	−0.695577	+	0.718452i	$0.744851\pi$
$578$	38.1281i	1.58592i
$579$	−25.2063	−1.04754
$580$	0	0
$581$	5.49797	0.228094
$582$	22.8807i	0.948437i
$583$	− 22.4345i	− 0.929144i
$584$	88.0694	3.64434
$585$	0	0
$586$	14.7878	0.610877
$587$	− 22.2755i	− 0.919408i	−0.888072	−	0.459704i	$-0.847955\pi$
$587$	− 22.2755i	− 0.919408i	0.888072	−	0.459704i	$-0.152045\pi$
$588$	36.0931i	1.48846i
$589$	−20.3333	−0.837818
$590$	0	0
$591$	12.0923	0.497410
$592$	13.9861i	0.574824i
$593$	30.9158i	1.26956i	0.772693	+	0.634779i	$0.218909\pi$
$593$	30.9158i	1.26956i	−0.772693	+	0.634779i	$0.781091\pi$
$594$	8.61023	0.353282
$595$	0	0
$596$	−53.5556	−2.19372
$597$	− 0.544434i	− 0.0222822i
$598$	3.46119i	0.141539i
$599$	24.0276	0.981742	0.490871	−	0.871232i	$-0.336679\pi$
$599$	24.0276	0.981742	0.490871	+	0.871232i	$0.336679\pi$
$600$	0	0
$601$	−32.0387	−1.30688	−0.653442	−	0.756977i	$-0.726676\pi$
$601$	−32.0387	−1.30688	−0.653442	+	0.756977i	$0.726676\pi$
$602$	8.56231i	0.348974i
$603$	2.79469i	0.113809i
$604$	63.7915	2.59564
$605$	0	0
$606$	−15.7869	−0.641299
$607$	45.5915i	1.85050i	0.379356	+	0.925251i	$0.376145\pi$
$607$	45.5915i	1.85050i	−0.379356	+	0.925251i	$0.623855\pi$
$608$	− 71.3769i	− 2.89471i
$609$	3.91492	0.158640
$610$	0	0
$611$	−2.49807	−0.101061
$612$	− 9.08535i	− 0.367253i
$613$	10.8564i	0.438485i	0.975670	+	0.219242i	$0.0703585\pi$
$613$	10.8564i	0.438485i	−0.975670	+	0.219242i	$0.929641\pi$
$614$	57.9686	2.33942
$615$	0	0
$616$	−13.4616	−0.542384
$617$	− 16.1054i	− 0.648380i	−0.945992	−	0.324190i	$-0.894908\pi$
$617$	− 16.1054i	− 0.648380i	0.945992	−	0.324190i	$-0.105092\pi$
$618$	38.1039i	1.53276i
$619$	−38.3581	−1.54174	−0.770872	−	0.636991i	$-0.780179\pi$
$619$	−38.3581	−1.54174	−0.770872	+	0.636991i	$0.780179\pi$
$620$	0	0
$621$	−2.26981	−0.0910842
$622$	− 7.71854i	− 0.309485i
$623$	2.91605i	0.116829i
$624$	−7.71941	−0.309024
$625$	0	0
$626$	39.0732	1.56168
$627$	− 11.8990i	− 0.475202i
$628$	− 104.337i	− 4.16350i
$629$	1.74194	0.0694558
$630$	0	0
$631$	22.7292	0.904833	0.452417	−	0.891807i	$-0.350562\pi$
$631$	22.7292	0.904833	0.452417	+	0.891807i	$0.350562\pi$
$632$	− 42.7808i	− 1.70173i
$633$	3.12207i	0.124091i
$634$	−61.1915	−2.43022
$635$	0	0
$636$	−37.5456	−1.48878
$637$	− 3.81949i	− 0.151334i
$638$	− 71.6750i	− 2.83764i
$639$	−16.0456	−0.634756
$640$	0	0
$641$	26.9930	1.06616	0.533080	−	0.846065i	$-0.321035\pi$
$641$	26.9930	1.06616	0.533080	+	0.846065i	$0.321035\pi$
$642$	− 23.3074i	− 0.919869i
$643$	25.9062i	1.02164i	0.859687	+	0.510821i	$0.170659\pi$
$643$	25.9062i	1.02164i	−0.859687	+	0.510821i	$0.829341\pi$
$644$	5.68367	0.223968
$645$	0	0
$646$	−17.2719	−0.679554
$647$	− 21.6623i	− 0.851632i	−0.904810	−	0.425816i	$-0.859987\pi$
$647$	− 21.6623i	− 0.851632i	0.904810	−	0.425816i	$-0.140013\pi$
$648$	− 8.99702i	− 0.353436i
$649$	−40.9519	−1.60750
$650$	0	0
$651$	2.55678	0.100208
$652$	63.3212i	2.47985i
$653$	− 11.1775i	− 0.437411i	−0.975791	−	0.218705i	$-0.929817\pi$
$653$	− 11.1775i	− 0.437411i	0.975791	−	0.218705i	$-0.0701833\pi$
$654$	−45.1625	−1.76599
$655$	0	0
$656$	20.1689	0.787461
$657$	− 9.78873i	− 0.381895i
$658$	5.64299i	0.219987i
$659$	−13.4744	−0.524888	−0.262444	−	0.964947i	$-0.584529\pi$
$659$	−13.4744	−0.524888	−0.262444	+	0.964947i	$0.584529\pi$
$660$	0	0
$661$	−23.2622	−0.904793	−0.452397	−	0.891817i	$-0.649431\pi$
$661$	−23.2622	−0.904793	−0.452397	+	0.891817i	$0.649431\pi$
$662$	10.4640i	0.406695i
$663$	0.961440i	0.0373392i
$664$	−105.180	−4.08176
$665$	0	0
$666$	2.76279	0.107056
$667$	18.8948i	0.731608i
$668$	31.9132i	1.23476i
$669$	−0.339125	−0.0131113
$670$	0	0
$671$	−0.403693	−0.0155844
$672$	8.97519i	0.346226i
$673$	− 18.8392i	− 0.726198i	−0.931751	−	0.363099i	$-0.881719\pi$
$673$	− 18.8392i	− 0.726198i	0.931751	−	0.363099i	$-0.118281\pi$
$674$	−47.3215	−1.82276
$675$	0	0
$676$	−67.5268	−2.59719
$677$	− 42.2440i	− 1.62357i	−0.583957	−	0.811785i	$-0.698496\pi$
$677$	− 42.2440i	− 1.62357i	0.583957	−	0.811785i	$-0.301504\pi$
$678$	− 31.8697i	− 1.22395i
$679$	−3.97607	−0.152587
$680$	0	0
$681$	−1.78847	−0.0685342
$682$	− 46.8100i	− 1.79245i
$683$	− 48.9502i	− 1.87303i	−0.350633	−	0.936513i	$-0.614034\pi$
$683$	− 48.9502i	− 1.87303i	0.350633	−	0.936513i	$-0.385966\pi$
$684$	−19.9138	−0.761422
$685$	0	0
$686$	−17.5375	−0.669585
$687$	− 27.9833i	− 1.06763i
$688$	− 92.1655i	− 3.51378i
$689$	3.97319	0.151366
$690$	0	0
$691$	−14.1917	−0.539878	−0.269939	−	0.962877i	$-0.587004\pi$
$691$	−14.1917	−0.539878	−0.269939	+	0.962877i	$0.587004\pi$
$692$	92.2419i	3.50651i
$693$	1.49623i	0.0568371i
$694$	62.5010	2.37251
$695$	0	0
$696$	−74.8948	−2.83888
$697$	− 2.51200i	− 0.0951487i
$698$	25.2409i	0.955382i
$699$	6.77400	0.256216
$700$	0	0
$701$	−9.00786	−0.340222	−0.170111	−	0.985425i	$-0.554413\pi$
$701$	−9.00786	−0.340222	−0.170111	+	0.985425i	$0.554413\pi$
$702$	1.52488i	0.0575530i
$703$	− 3.81809i	− 0.144002i
$704$	77.1444	2.90749
$705$	0	0
$706$	86.4604	3.25398
$707$	− 2.74334i	− 0.103174i
$708$	68.5355i	2.57572i
$709$	−49.6994	−1.86650	−0.933249	−	0.359229i	$-0.883040\pi$
$709$	−49.6994	−1.86650	−0.933249	+	0.359229i	$0.883040\pi$
$710$	0	0
$711$	−4.75499	−0.178326
$712$	− 55.7859i	− 2.09067i
$713$	12.3399i	0.462134i
$714$	2.17183	0.0812789
$715$	0	0
$716$	56.3134	2.10453
$717$	− 6.48334i	− 0.242125i
$718$	25.9528i	0.968549i
$719$	2.13706	0.0796988	0.0398494	−	0.999206i	$-0.487312\pi$
$719$	2.13706	0.0796988	0.0398494	+	0.999206i	$0.487312\pi$
$720$	0	0
$721$	−6.62145	−0.246596
$722$	− 13.5633i	− 0.504775i
$723$	7.44857i	0.277015i
$724$	−63.7855	−2.37057
$725$	0	0
$726$	−2.37672	−0.0882083
$727$	− 41.4634i	− 1.53779i	−0.639375	−	0.768895i	$-0.720807\pi$
$727$	− 41.4634i	− 1.53779i	0.639375	−	0.768895i	$-0.279193\pi$
$728$	− 2.38407i	− 0.0883596i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−11.4791	−0.424568
$732$	0.675604i	0.0249710i
$733$	− 32.5210i	− 1.20119i	−0.799553	−	0.600596i	$-0.794930\pi$
$733$	− 32.5210i	− 1.20119i	0.799553	−	0.600596i	$-0.205070\pi$
$734$	−2.88020	−0.106310
$735$	0	0
$736$	−43.3175	−1.59670
$737$	8.89125i	0.327513i
$738$	− 3.98413i	− 0.146658i
$739$	35.3175	1.29917	0.649587	−	0.760287i	$-0.274942\pi$
$739$	35.3175	1.29917	0.649587	+	0.760287i	$0.274942\pi$
$740$	0	0
$741$	2.10734	0.0774150
$742$	− 8.97519i	− 0.329490i
$743$	36.0897i	1.32400i	0.749503	+	0.662001i	$0.230292\pi$
$743$	36.0897i	1.32400i	−0.749503	+	0.662001i	$0.769708\pi$
$744$	−48.9128	−1.79323
$745$	0	0
$746$	−11.1508	−0.408261
$747$	11.6905i	0.427733i
$748$	− 28.9048i	− 1.05687i
$749$	4.05021	0.147991
$750$	0	0
$751$	−4.62810	−0.168882	−0.0844409	−	0.996428i	$-0.526910\pi$
$751$	−4.62810	−0.168882	−0.0844409	+	0.996428i	$0.526910\pi$
$752$	− 60.7418i	− 2.21502i
$753$	4.60217i	0.167712i
$754$	12.6937	0.462279
$755$	0	0
$756$	2.50403	0.0910708
$757$	14.6243i	0.531530i	0.964038	+	0.265765i	$0.0856245\pi$
$757$	14.6243i	0.531530i	−0.964038	+	0.265765i	$0.914375\pi$
$758$	− 28.0279i	− 1.01802i
$759$	−7.22134	−0.262118
$760$	0	0
$761$	12.1893	0.441861	0.220930	−	0.975290i	$-0.429091\pi$
$761$	12.1893	0.441861	0.220930	+	0.975290i	$0.429091\pi$
$762$	4.81414i	0.174398i
$763$	− 7.84804i	− 0.284118i
$764$	−97.2538	−3.51852
$765$	0	0
$766$	3.24587	0.117278
$767$	− 7.25264i	− 0.261878i
$768$	− 25.8083i	− 0.931276i
$769$	−13.7961	−0.497500	−0.248750	−	0.968568i	$-0.580020\pi$
$769$	−13.7961	−0.497500	−0.248750	+	0.968568i	$0.580020\pi$
$770$	0	0
$771$	−5.79485	−0.208697
$772$	134.208i	4.83026i
$773$	− 22.6539i	− 0.814803i	−0.913249	−	0.407402i	$-0.866435\pi$
$773$	− 22.6539i	− 0.814803i	0.913249	−	0.407402i	$-0.133565\pi$
$774$	−18.2063	−0.654411
$775$	0	0
$776$	76.0647	2.73056
$777$	0.480101i	0.0172235i
$778$	− 34.1117i	− 1.22296i
$779$	−5.50594	−0.197271
$780$	0	0
$781$	−51.0489	−1.82667
$782$	10.4820i	0.374837i
$783$	8.32440i	0.297490i
$784$	92.8726	3.31688
$785$	0	0
$786$	−16.0179	−0.571339
$787$	− 47.6857i	− 1.69981i	−0.526936	−	0.849905i	$-0.676659\pi$
$787$	− 47.6857i	− 1.69981i	0.526936	−	0.849905i	$-0.323341\pi$
$788$	− 64.3841i	− 2.29359i
$789$	−14.2989	−0.509054
$790$	0	0
$791$	5.53811	0.196913
$792$	− 28.6238i	− 1.01710i
$793$	− 0.0714945i	− 0.00253884i
$794$	3.55440	0.126141
$795$	0	0
$796$	−2.89878	−0.102745
$797$	27.2131i	0.963938i	0.876188	+	0.481969i	$0.160078\pi$
$797$	27.2131i	0.963938i	−0.876188	+	0.481969i	$0.839922\pi$
$798$	− 4.76035i	− 0.168515i
$799$	−7.56529	−0.267641
$800$	0	0
$801$	−6.20049	−0.219083
$802$	43.5838i	1.53900i
$803$	− 31.1426i	− 1.09900i
$804$	14.8800	0.524778
$805$	0	0
$806$	8.29012	0.292007
$807$	− 19.2205i	− 0.676592i
$808$	52.4819i	1.84631i
$809$	19.6155	0.689644	0.344822	−	0.938668i	$-0.387939\pi$
$809$	19.6155	0.689644	0.344822	+	0.938668i	$0.387939\pi$
$810$	0	0
$811$	21.0710	0.739903	0.369951	−	0.929051i	$-0.379374\pi$
$811$	21.0710	0.739903	0.369951	+	0.929051i	$0.379374\pi$
$812$	− 20.8446i	− 0.731501i
$813$	23.2433i	0.815177i
$814$	8.78977	0.308081
$815$	0	0
$816$	−23.3778	−0.818388
$817$	25.1605i	0.880253i
$818$	− 57.2667i	− 2.00228i
$819$	−0.264985	−0.00925931
$820$	0	0
$821$	41.4342	1.44606	0.723031	−	0.690815i	$-0.242748\pi$
$821$	41.4342	1.44606	0.723031	+	0.690815i	$0.242748\pi$
$822$	− 13.8460i	− 0.482936i
$823$	− 6.04093i	− 0.210574i	−0.994442	−	0.105287i	$-0.966424\pi$
$823$	− 6.04093i	− 0.210574i	0.994442	−	0.105287i	$-0.0335760\pi$
$824$	126.673	4.41285
$825$	0	0
$826$	−16.3833	−0.570047
$827$	− 2.99361i	− 0.104098i	−0.998645	−	0.0520491i	$-0.983425\pi$
$827$	− 2.99361i	− 0.104098i	0.998645	−	0.0520491i	$-0.0165752\pi$
$828$	12.0853i	0.419995i
$829$	−26.8530	−0.932642	−0.466321	−	0.884616i	$-0.654421\pi$
$829$	−26.8530	−0.932642	−0.466321	+	0.884616i	$0.654421\pi$
$830$	0	0
$831$	21.8910	0.759390
$832$	13.6624i	0.473658i
$833$	− 11.5671i	− 0.400777i
$834$	22.2955	0.772029
$835$	0	0
$836$	−63.3552	−2.19119
$837$	5.43656i	0.187915i
$838$	18.0540i	0.623665i
$839$	−12.4595	−0.430150	−0.215075	−	0.976598i	$-0.569000\pi$
$839$	−12.4595	−0.430150	−0.215075	+	0.976598i	$0.569000\pi$
$840$	0	0
$841$	40.2956	1.38950
$842$	73.8783i	2.54601i
$843$	− 27.1138i	− 0.933850i
$844$	16.6231	0.572191
$845$	0	0
$846$	−11.9989	−0.412529
$847$	− 0.413011i	− 0.0141912i
$848$	96.6099i	3.31760i
$849$	−16.3074	−0.559668
$850$	0	0
$851$	−2.31714	−0.0794304
$852$	85.4334i	2.92690i
$853$	5.69736i	0.195074i	0.995232	+	0.0975369i	$0.0310964\pi$
$853$	5.69736i	0.195074i	−0.995232	+	0.0975369i	$0.968904\pi$
$854$	−0.161502	−0.00552648
$855$	0	0
$856$	−77.4830	−2.64831
$857$	− 7.41982i	− 0.253456i	−0.991937	−	0.126728i	$-0.959552\pi$
$857$	− 7.41982i	− 0.253456i	0.991937	−	0.126728i	$-0.0404476\pi$
$858$	4.85139i	0.165624i
$859$	5.68253	0.193885	0.0969427	−	0.995290i	$-0.469094\pi$
$859$	5.68253	0.193885	0.0969427	+	0.995290i	$0.469094\pi$
$860$	0	0
$861$	0.692337	0.0235948
$862$	33.2813i	1.13356i
$863$	34.5747i	1.17693i	0.808521	+	0.588467i	$0.200268\pi$
$863$	34.5747i	1.17693i	−0.808521	+	0.588467i	$0.799732\pi$
$864$	−19.0842	−0.649258
$865$	0	0
$866$	−91.5973	−3.11260
$867$	− 14.0883i	− 0.478465i
$868$	− 13.6133i	− 0.462066i
$869$	−15.1279	−0.513179
$870$	0	0
$871$	−1.57465	−0.0533550
$872$	150.138i	5.08431i
$873$	− 8.45443i	− 0.286139i
$874$	22.9751	0.777145
$875$	0	0
$876$	−52.1191	−1.76094
$877$	− 5.30013i	− 0.178973i	−0.995988	−	0.0894863i	$-0.971477\pi$
$877$	− 5.30013i	− 0.178973i	0.995988	−	0.0894863i	$-0.0285225\pi$
$878$	56.7399i	1.91488i
$879$	−5.46407	−0.184299
$880$	0	0
$881$	7.61288	0.256484	0.128242	−	0.991743i	$-0.459067\pi$
$881$	7.61288	0.256484	0.128242	+	0.991743i	$0.459067\pi$
$882$	− 18.3460i	− 0.617740i
$883$	52.8334i	1.77799i	0.457921	+	0.888993i	$0.348594\pi$
$883$	52.8334i	1.77799i	−0.457921	+	0.888993i	$0.651406\pi$
$884$	5.11909	0.172174
$885$	0	0
$886$	16.3693	0.549939
$887$	7.70778i	0.258802i	0.991592	+	0.129401i	$0.0413054\pi$
$887$	7.70778i	0.258802i	−0.991592	+	0.129401i	$0.958695\pi$
$888$	− 9.18462i	− 0.308216i
$889$	−0.836570	−0.0280577
$890$	0	0
$891$	−3.18148	−0.106584
$892$	1.80563i	0.0604571i
$893$	16.5820i	0.554896i
$894$	27.2220	0.910441
$895$	0	0
$896$	12.9121	0.431363
$897$	− 1.27891i	− 0.0427015i
$898$	− 42.1912i	− 1.40794i
$899$	45.2560	1.50937
$900$	0	0
$901$	12.0326	0.400864
$902$	− 12.6754i	− 0.422046i
$903$	− 3.16377i	− 0.105284i
$904$	−105.947	−3.52376
$905$	0	0
$906$	−32.4249	−1.07725
$907$	− 11.3735i	− 0.377650i	−0.982011	−	0.188825i	$-0.939532\pi$
$907$	− 11.3735i	− 0.377650i	0.982011	−	0.188825i	$-0.0604678\pi$
$908$	9.52251i	0.316015i
$909$	5.83325	0.193477
$910$	0	0
$911$	−18.5896	−0.615902	−0.307951	−	0.951402i	$-0.599643\pi$
$911$	−18.5896	−0.615902	−0.307951	+	0.951402i	$0.599643\pi$
$912$	51.2409i	1.69676i
$913$	37.1931i	1.23091i
$914$	6.77112	0.223969
$915$	0	0
$916$	−148.994	−4.92290
$917$	− 2.78348i	− 0.0919187i
$918$	4.61803i	0.152418i
$919$	−7.64799	−0.252284	−0.126142	−	0.992012i	$-0.540259\pi$
$919$	−7.64799	−0.252284	−0.126142	+	0.992012i	$0.540259\pi$
$920$	0	0
$921$	−21.4194	−0.705793
$922$	− 0.416680i	− 0.0137226i
$923$	− 9.04083i	− 0.297582i
$924$	7.96652	0.262079
$925$	0	0
$926$	−31.4822	−1.03457
$927$	− 14.0794i	− 0.462428i
$928$	158.865i	5.21498i
$929$	30.9487	1.01539	0.507696	−	0.861536i	$-0.330497\pi$
$929$	30.9487	1.01539	0.507696	+	0.861536i	$0.330497\pi$
$930$	0	0
$931$	−25.3535	−0.830927
$932$	− 36.0675i	− 1.18143i
$933$	2.85200i	0.0933702i
$934$	−1.27279	−0.0416468
$935$	0	0
$936$	5.06932	0.165696
$937$	31.3694i	1.02480i	0.858748	+	0.512398i	$0.171243\pi$
$937$	31.3694i	1.02480i	−0.858748	+	0.512398i	$0.828757\pi$
$938$	3.55704i	0.116142i
$939$	−14.4375	−0.471151
$940$	0	0
$941$	−3.61810	−0.117947	−0.0589733	−	0.998260i	$-0.518783\pi$
$941$	−3.61810	−0.117947	−0.0589733	+	0.998260i	$0.518783\pi$
$942$	53.0340i	1.72794i
$943$	3.34146i	0.108813i
$944$	176.351	5.73974
$945$	0	0
$946$	−57.9229	−1.88323
$947$	− 11.3216i	− 0.367902i	−0.982935	−	0.183951i	$-0.941111\pi$
$947$	− 11.3216i	− 0.367902i	0.982935	−	0.183951i	$-0.0588888\pi$
$948$	25.3175i	0.822273i
$949$	5.51540	0.179038
$950$	0	0
$951$	22.6102	0.733187
$952$	− 7.22004i	− 0.234003i
$953$	17.2182i	0.557752i	0.960327	+	0.278876i	$0.0899618\pi$
$953$	17.2182i	0.557752i	−0.960327	+	0.278876i	$0.910038\pi$
$954$	19.0842	0.617874
$955$	0	0
$956$	−34.5199	−1.11645
$957$	26.4839i	0.856102i
$958$	− 33.5898i	− 1.08524i
$959$	2.40608	0.0776962
$960$	0	0
$961$	−1.44386	−0.0465762
$962$	1.55668i	0.0501894i
$963$	8.61207i	0.277520i
$964$	39.6591	1.27733
$965$	0	0
$966$	−2.88898	−0.0929514
$967$	− 29.9064i	− 0.961723i	−0.876796	−	0.480862i	$-0.840324\pi$
$967$	− 29.9064i	− 0.961723i	0.876796	−	0.480862i	$-0.159676\pi$
$968$	7.90115i	0.253953i
$969$	6.38197	0.205018
$970$	0	0
$971$	−25.1927	−0.808472	−0.404236	−	0.914655i	$-0.632463\pi$
$971$	−25.1927	−0.808472	−0.404236	+	0.914655i	$0.632463\pi$
$972$	5.32440i	0.170780i
$973$	3.87436i	0.124206i
$974$	−93.5443	−2.99735
$975$	0	0
$976$	1.73842	0.0556455
$977$	57.0948i	1.82663i	0.407259	+	0.913313i	$0.366485\pi$
$977$	57.0948i	1.82663i	−0.407259	+	0.913313i	$0.633515\pi$
$978$	− 32.1859i	− 1.02919i
$979$	−19.7267	−0.630469
$980$	0	0
$981$	16.6875	0.532791
$982$	− 67.2957i	− 2.14749i
$983$	26.4168i	0.842566i	0.906929	+	0.421283i	$0.138420\pi$
$983$	26.4168i	0.842566i	−0.906929	+	0.421283i	$0.861580\pi$
$984$	−13.2448	−0.422230
$985$	0	0
$986$	38.4423	1.22425
$987$	− 2.08508i	− 0.0663690i
$988$	− 11.2203i	− 0.356965i
$989$	15.2695	0.485541
$990$	0	0
$991$	−22.5276	−0.715612	−0.357806	−	0.933796i	$-0.616475\pi$
$991$	−22.5276	−0.715612	−0.357806	+	0.933796i	$0.616475\pi$
$992$	103.752i	3.29414i
$993$	− 3.86645i	− 0.122698i
$994$	−20.4227	−0.647768
$995$	0	0
$996$	62.2448	1.97230
$997$	− 24.1043i	− 0.763392i	−0.924288	−	0.381696i	$-0.875340\pi$
$997$	− 24.1043i	− 0.763392i	0.924288	−	0.381696i	$-0.124660\pi$
$998$	− 34.1103i	− 1.07974i
$999$	−1.02085	−0.0322983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.c.1249.8		8
5.2	odd	4		1875.2.a.e.1.1		4
5.3	odd	4		1875.2.a.h.1.4		4
5.4	even	2	inner	1875.2.b.c.1249.1		8
15.2	even	4		5625.2.a.n.1.4		4
15.8	even	4		5625.2.a.i.1.1		4
25.3	odd	20		75.2.g.b.16.1	✓	8
25.4	even	10		375.2.i.b.49.1		16
25.6	even	5		375.2.i.b.199.1		16
25.8	odd	20		75.2.g.b.61.1	yes	8
25.17	odd	20		375.2.g.b.301.2		8
25.19	even	10		375.2.i.b.199.4		16
25.21	even	5		375.2.i.b.49.4		16
25.22	odd	20		375.2.g.b.76.2		8
75.8	even	20		225.2.h.c.136.2		8
75.53	even	20		225.2.h.c.91.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.b.16.1	✓	8	25.3	odd	20
75.2.g.b.61.1	yes	8	25.8	odd	20
225.2.h.c.91.2		8	75.53	even	20
225.2.h.c.136.2		8	75.8	even	20
375.2.g.b.76.2		8	25.22	odd	20
375.2.g.b.301.2		8	25.17	odd	20
375.2.i.b.49.1		16	25.4	even	10
375.2.i.b.49.4		16	25.21	even	5
375.2.i.b.199.1		16	25.6	even	5
375.2.i.b.199.4		16	25.19	even	10
1875.2.a.e.1.1		4	5.2	odd	4
1875.2.a.h.1.4		4	5.3	odd	4
1875.2.b.c.1249.1		8	5.4	even	2	inner
1875.2.b.c.1249.8		8	1.1	even	1	trivial
5625.2.a.i.1.1		4	15.8	even	4
5625.2.a.n.1.4		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)

\(f(q)\)	\(=\)	\(q+2.70636i q^{2} -1.00000i q^{3} -5.32440 q^{4} +2.70636 q^{6} +0.470294i q^{7} -8.99702i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.70636i q^{2} -1.00000i q^{3} -5.32440 q^{4} +2.70636 q^{6} +0.470294i q^{7} -8.99702i q^{8} -1.00000 q^{9} -3.18148 q^{11} +5.32440i q^{12} -0.563444i q^{13} -1.27279 q^{14} +13.7004 q^{16} -1.70636i q^{17} -2.70636i q^{18} -3.74010 q^{19} +0.470294 q^{21} -8.61023i q^{22} +2.26981i q^{23} -8.99702 q^{24} +1.52488 q^{26} +1.00000i q^{27} -2.50403i q^{28} +8.32440 q^{29} +5.43656 q^{31} +19.0842i q^{32} +3.18148i q^{33} +4.61803 q^{34} +5.32440 q^{36} +1.02085i q^{37} -10.1221i q^{38} -0.563444 q^{39} +1.47214 q^{41} +1.27279i q^{42} -6.72721i q^{43} +16.9395 q^{44} -6.14292 q^{46} -4.43358i q^{47} -13.7004i q^{48} +6.77882 q^{49} -1.70636 q^{51} +3.00000i q^{52} +7.05161i q^{53} -2.70636 q^{54} +4.23125 q^{56} +3.74010i q^{57} +22.5288i q^{58} +12.8720 q^{59} +0.126888 q^{61} +14.7133i q^{62} -0.470294i q^{63} -24.2480 q^{64} -8.61023 q^{66} -2.79469i q^{67} +9.08535i q^{68} +2.26981 q^{69} +16.0456 q^{71} +8.99702i q^{72} +9.78873i q^{73} -2.76279 q^{74} +19.9138 q^{76} -1.49623i q^{77} -1.52488i q^{78} +4.75499 q^{79} +1.00000 q^{81} +3.98413i q^{82} -11.6905i q^{83} -2.50403 q^{84} +18.2063 q^{86} -8.32440i q^{87} +28.6238i q^{88} +6.20049 q^{89} +0.264985 q^{91} -12.0853i q^{92} -5.43656i q^{93} +11.9989 q^{94} +19.0842 q^{96} +8.45443i q^{97} +18.3460i q^{98} +3.18148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \) 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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