LMFDB - Embedded newform 900.3.f.h.199.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(199,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("900.199");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.f (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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 14 x^{13} + 23 x^{12} - 26 x^{11} + 18 x^{10} - 10 x^{9} + 9 x^{8} + \cdots + 256 $ x^16 - 4x^15 + 8x^14 - 14x^13 + 23x^12 - 26x^11 + 18x^10 - 10x^9 + 9x^8 - 20x^7 + 72x^6 - 208x^5 + 368x^4 - 448x^3 + 512x^2 - 512*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 300)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.3
Root			$0.717516 + 1.21868i$ of defining polynomial
Character	$\chi$	$=$	900.199
Dual form			900.3.f.h.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.92737 - 0.534079i) q^{2} +(3.42952 + 2.05874i) q^{4} -11.9716 q^{7} +(-5.51043 - 5.79958i) q^{8} +O(q^{10})$	\(q+(-1.92737 - 0.534079i) q^{2} +(3.42952 + 2.05874i) q^{4} -11.9716 q^{7} +(-5.51043 - 5.79958i) q^{8} -14.5382i q^{11} +22.4802i q^{13} +(23.0738 + 6.39379i) q^{14} +(7.52322 + 14.1209i) q^{16} +12.6890i q^{17} +8.76336i q^{19} +(-7.76455 + 28.0205i) q^{22} +4.99653 q^{23} +(12.0062 - 43.3278i) q^{26} +(-41.0570 - 24.6464i) q^{28} +2.74712 q^{29} -16.3466i q^{31} +(-6.95833 - 31.2343i) q^{32} +(6.77695 - 24.4565i) q^{34} -32.4872i q^{37} +(4.68032 - 16.8902i) q^{38} -42.7586 q^{41} -16.5435 q^{43} +(29.9303 - 49.8591i) q^{44} +(-9.63018 - 2.66854i) q^{46} +48.5912 q^{47} +94.3200 q^{49} +(-46.2809 + 77.0964i) q^{52} -94.1066i q^{53} +(65.9689 + 69.4305i) q^{56} +(-5.29471 - 1.46718i) q^{58} +43.2650i q^{59} +56.7678 q^{61} +(-8.73038 + 31.5060i) q^{62} +(-3.27028 + 63.9164i) q^{64} +61.1106 q^{67} +(-26.1234 + 43.5173i) q^{68} -39.6643i q^{71} -99.5452i q^{73} +(-17.3507 + 62.6149i) q^{74} +(-18.0414 + 30.0541i) q^{76} +174.046i q^{77} +10.7780i q^{79} +(82.4118 + 22.8365i) q^{82} -140.263 q^{83} +(31.8855 + 8.83554i) q^{86} +(-84.3156 + 80.1118i) q^{88} +54.8723 q^{89} -269.125i q^{91} +(17.1357 + 10.2865i) q^{92} +(-93.6533 - 25.9515i) q^{94} -14.1601i q^{97} +(-181.790 - 50.3743i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{4}+O(q^{10}) $ 16 * q + 16 * q^4	$ 16 q + 16 q^{4} + 44 q^{14} + 80 q^{16} + 132 q^{26} - 64 q^{29} - 248 q^{34} + 32 q^{41} + 80 q^{44} - 152 q^{46} - 32 q^{49} + 344 q^{56} + 272 q^{61} - 32 q^{64} - 216 q^{74} + 240 q^{76} - 428 q^{86} + 256 q^{89} - 24 q^{94}+O(q^{100}) $ 16 * q + 16 * q^4 + 44 * q^14 + 80 * q^16 + 132 * q^26 - 64 * q^29 - 248 * q^34 + 32 * q^41 + 80 * q^44 - 152 * q^46 - 32 * q^49 + 344 * q^56 + 272 * q^61 - 32 * q^64 - 216 * q^74 + 240 * q^76 - 428 * q^86 + 256 * q^89 - 24 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	−1.92737	−	0.534079i	−0.963686	−	0.267039i
$2$	−1.92737	−	0.534079i	−0.963686	−	0.267039i
$3$		0			0
$3$		0			0
$4$	3.42952	+	2.05874i	0.857380	+	0.514684i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−11.9716			−1.71023			−0.855117	−	0.518436i	$-0.826515\pi$
$7$	−11.9716			−1.71023			−0.855117	+	0.518436i	$0.826515\pi$
$8$	−5.51043	−	5.79958i	−0.688804	−	0.724948i
$9$		0			0
$10$		0			0
$11$		−	14.5382i		−	1.32166i	−0.750537	−	0.660828i	$-0.770205\pi$
$11$		−	14.5382i		−	1.32166i	0.750537	−	0.660828i	$-0.229795\pi$
$12$		0			0
$13$			22.4802i			1.72925i	0.502418	+	0.864625i	$0.332444\pi$
$13$			22.4802i			1.72925i	−0.502418	+	0.864625i	$0.667556\pi$
$14$	23.0738	+	6.39379i	1.64813	+	0.456699i
$15$		0			0
$16$	7.52322	+	14.1209i	0.470201	+	0.882559i
$17$			12.6890i			0.746414i	0.927748	+	0.373207i	$0.121742\pi$
$17$			12.6890i			0.746414i	−0.927748	+	0.373207i	$0.878258\pi$
$18$		0			0
$19$			8.76336i			0.461229i	0.973045	+	0.230615i	$0.0740737\pi$
$19$			8.76336i			0.461229i	−0.973045	+	0.230615i	$0.925926\pi$
$20$		0			0
$21$		0			0
$22$	−7.76455	+	28.0205i	−0.352934	+	1.27366i
$23$	4.99653			0.217241			0.108620	−	0.994083i	$-0.465357\pi$
$23$	4.99653			0.217241			0.108620	+	0.994083i	$0.465357\pi$
$24$		0			0
$25$		0			0
$26$	12.0062	−	43.3278i	0.461778	−	1.66645i
$27$		0			0
$28$	−41.0570	−	24.6464i	−1.46632	−	0.880229i
$29$	2.74712			0.0947282			0.0473641	−	0.998878i	$-0.484918\pi$
$29$	2.74712			0.0947282			0.0473641	+	0.998878i	$0.484918\pi$
$30$		0			0
$31$		−	16.3466i		−	0.527310i	−0.964617	−	0.263655i	$-0.915072\pi$
$31$		−	16.3466i		−	0.527310i	0.964617	−	0.263655i	$-0.0849281\pi$
$32$	−6.95833	−	31.2343i	−0.217448	−	0.976072i
$33$		0			0
$34$	6.77695	−	24.4565i	0.199322	−	0.719309i
$35$		0			0
$36$		0			0
$37$		−	32.4872i		−	0.878032i	−0.898479	−	0.439016i	$-0.855327\pi$
$37$		−	32.4872i		−	0.878032i	0.898479	−	0.439016i	$-0.144673\pi$
$38$	4.68032	−	16.8902i	0.123166	−	0.444480i
$39$		0			0
$40$		0			0
$41$	−42.7586			−1.04289			−0.521447	−	0.853284i	$-0.674607\pi$
$41$	−42.7586			−1.04289			−0.521447	+	0.853284i	$0.674607\pi$
$42$		0			0
$43$	−16.5435			−0.384733			−0.192367	−	0.981323i	$-0.561616\pi$
$43$	−16.5435			−0.384733			−0.192367	+	0.981323i	$0.561616\pi$
$44$	29.9303	−	49.8591i	0.680235	−	1.13316i
$45$		0			0
$46$	−9.63018	−	2.66854i	−0.209352	−	0.0580118i
$47$	48.5912			1.03386			0.516928	−	0.856029i	$-0.327076\pi$
$47$	48.5912			1.03386			0.516928	+	0.856029i	$0.327076\pi$
$48$		0			0
$49$	94.3200			1.92490
$50$		0			0
$51$		0			0
$52$	−46.2809	+	77.0964i	−0.890017	+	1.48262i
$53$		−	94.1066i		−	1.77560i	−0.460233	−	0.887798i	$-0.652234\pi$
$53$		−	94.1066i		−	1.77560i	0.460233	−	0.887798i	$-0.347766\pi$
$54$		0			0
$55$		0			0
$56$	65.9689	+	69.4305i	1.17802	+	1.23983i
$57$		0			0
$58$	−5.29471	−	1.46718i	−0.0912882	−	0.0252961i
$59$			43.2650i			0.733305i	0.930358	+	0.366653i	$0.119496\pi$
$59$			43.2650i			0.733305i	−0.930358	+	0.366653i	$0.880504\pi$
$60$		0			0
$61$	56.7678			0.930620			0.465310	−	0.885148i	$-0.345943\pi$
$61$	56.7678			0.930620			0.465310	+	0.885148i	$0.345943\pi$
$62$	−8.73038	+	31.5060i	−0.140813	+	0.508161i
$63$		0			0
$64$	−3.27028	+	63.9164i	−0.0510981	+	0.998694i
$65$		0			0
$66$		0			0
$67$	61.1106			0.912098			0.456049	−	0.889955i	$-0.349264\pi$
$67$	61.1106			0.912098			0.456049	+	0.889955i	$0.349264\pi$
$68$	−26.1234	+	43.5173i	−0.384167	+	0.639961i
$69$		0			0
$70$		0			0
$71$		−	39.6643i		−	0.558652i	−0.960196	−	0.279326i	$-0.909889\pi$
$71$		−	39.6643i		−	0.558652i	0.960196	−	0.279326i	$-0.0901110\pi$
$72$		0			0
$73$		−	99.5452i		−	1.36363i	−0.731523	−	0.681817i	$-0.761190\pi$
$73$		−	99.5452i		−	1.36363i	0.731523	−	0.681817i	$-0.238810\pi$
$74$	−17.3507	+	62.6149i	−0.234469	+	0.846147i
$75$		0			0
$76$	−18.0414	+	30.0541i	−0.237387	+	0.395449i
$77$			174.046i			2.26034i
$78$		0			0
$79$			10.7780i			0.136430i	0.997671	+	0.0682151i	$0.0217304\pi$
$79$			10.7780i			0.136430i	−0.997671	+	0.0682151i	$0.978270\pi$
$80$		0			0
$81$		0			0
$82$	82.4118	+	22.8365i	1.00502	+	0.278494i
$83$	−140.263			−1.68991			−0.844955	−	0.534837i	$-0.820373\pi$
$83$	−140.263			−1.68991			−0.844955	+	0.534837i	$0.820373\pi$
$84$		0			0
$85$		0			0
$86$	31.8855	+	8.83554i	0.370762	+	0.102739i
$87$		0			0
$88$	−84.3156	+	80.1118i	−0.958131	+	0.910362i
$89$	54.8723			0.616543			0.308271	−	0.951298i	$-0.400249\pi$
$89$	54.8723			0.616543			0.308271	+	0.951298i	$0.400249\pi$
$90$		0			0
$91$		−	269.125i		−	2.95742i
$92$	17.1357	+	10.2865i	0.186258	+	0.111810i
$93$		0			0
$94$	−93.6533	−	25.9515i	−0.996312	−	0.276080i
$95$		0			0
$96$		0			0
$97$		−	14.1601i		−	0.145980i	−0.997333	−	0.0729902i	$-0.976746\pi$
$97$		−	14.1601i		−	0.145980i	0.997333	−	0.0729902i	$-0.0232542\pi$
$98$	−181.790	−	50.3743i	−1.85500	−	0.514023i
$99$		0			0
$100$		0			0
$101$	163.410			1.61792			0.808962	−	0.587861i	$-0.200030\pi$
$101$	163.410			1.61792			0.808962	+	0.587861i	$0.200030\pi$
$102$		0			0
$103$	169.591			1.64651			0.823255	−	0.567672i	$-0.192156\pi$
$103$	169.591			1.64651			0.823255	+	0.567672i	$0.192156\pi$
$104$	130.376	−	123.876i	1.25362	−	1.19111i
$105$		0			0
$106$	−50.2603	+	181.378i	−0.474154	+	1.71112i
$107$	8.14840			0.0761532			0.0380766	−	0.999275i	$-0.487877\pi$
$107$	8.14840			0.0761532			0.0380766	+	0.999275i	$0.487877\pi$
$108$		0			0
$109$	25.2322			0.231488			0.115744	−	0.993279i	$-0.463075\pi$
$109$	25.2322			0.231488			0.115744	+	0.993279i	$0.463075\pi$
$110$		0			0
$111$		0			0
$112$	−90.0652	−	169.051i	−0.804153	−	1.50938i
$113$		−	97.8142i		−	0.865613i	−0.901487	−	0.432806i	$-0.857523\pi$
$113$		−	97.8142i		−	0.865613i	0.901487	−	0.432806i	$-0.142477\pi$
$114$		0			0
$115$		0			0
$116$	9.42129	+	5.65559i	0.0812180	+	0.0487551i
$117$		0			0
$118$	23.1069	−	83.3877i	0.195821	−	0.706676i
$119$		−	151.909i		−	1.27654i
$120$		0			0
$121$	−90.3597			−0.746774
$122$	−109.413	−	30.3185i	−0.896825	−	0.248512i
$123$		0			0
$124$	33.6534	−	56.0611i	0.271398	−	0.452105i
$125$		0			0
$126$		0			0
$127$	167.563			1.31939			0.659695	−	0.751533i	$-0.270685\pi$
$127$	167.563			1.31939			0.659695	+	0.751533i	$0.270685\pi$
$128$	40.4394	−	121.444i	0.315933	−	0.948782i
$129$		0			0
$130$		0			0
$131$		−	82.0465i		−	0.626309i	−0.949702	−	0.313155i	$-0.898614\pi$
$131$		−	82.0465i		−	0.626309i	0.949702	−	0.313155i	$-0.101386\pi$
$132$		0			0
$133$		−	104.912i		−	0.788810i
$134$	−117.783	−	32.6378i	−0.878976	−	0.243566i
$135$		0			0
$136$	73.5911	−	69.9221i	0.541111	−	0.514133i
$137$		−	254.459i		−	1.85737i	−0.370874	−	0.928683i	$-0.620942\pi$
$137$		−	254.459i		−	1.85737i	0.370874	−	0.928683i	$-0.379058\pi$
$138$		0			0
$139$			78.9483i			0.567974i	0.958828	+	0.283987i	$0.0916572\pi$
$139$			78.9483i			0.567974i	−0.958828	+	0.283987i	$0.908343\pi$
$140$		0			0
$141$		0			0
$142$	−21.1838	+	76.4478i	−0.149182	+	0.538365i
$143$	326.823			2.28547
$144$		0			0
$145$		0			0
$146$	−53.1650	+	191.861i	−0.364144	+	1.31411i
$147$		0			0
$148$	66.8825	−	111.415i	0.451909	−	0.752807i
$149$	−32.3433			−0.217069			−0.108534	−	0.994093i	$-0.534616\pi$
$149$	−32.3433			−0.217069			−0.108534	+	0.994093i	$0.534616\pi$
$150$		0			0
$151$			38.7953i			0.256922i	0.991715	+	0.128461i	$0.0410038\pi$
$151$			38.7953i			0.256922i	−0.991715	+	0.128461i	$0.958996\pi$
$152$	50.8238	−	48.2899i	0.334367	−	0.317697i
$153$		0			0
$154$	92.9543	−	335.452i	0.603600	−	2.17826i
$155$		0			0
$156$		0			0
$157$		−	44.2021i		−	0.281542i	−0.990042	−	0.140771i	$-0.955042\pi$
$157$		−	44.2021i		−	0.281542i	0.990042	−	0.140771i	$-0.0449581\pi$
$158$	5.75629	−	20.7732i	0.0364322	−	0.131476i
$159$		0			0
$160$		0			0
$161$	−59.8167			−0.371532
$162$		0			0
$163$	52.9366			0.324764			0.162382	−	0.986728i	$-0.448082\pi$
$163$	52.9366			0.324764			0.162382	+	0.986728i	$0.448082\pi$
$164$	−146.642	−	88.0287i	−0.894156	−	0.536760i
$165$		0			0
$166$	270.338	+	74.9112i	1.62854	+	0.451272i
$167$	179.273			1.07349			0.536745	−	0.843744i	$-0.319654\pi$
$167$	179.273			1.07349			0.536745	+	0.843744i	$0.319654\pi$
$168$		0			0
$169$	−336.361			−1.99030
$170$		0			0
$171$		0			0
$172$	−56.7364	−	34.0587i	−0.329863	−	0.198016i
$173$			177.276i			1.02471i	0.858772	+	0.512357i	$0.171228\pi$
$173$			177.276i			1.02471i	−0.858772	+	0.512357i	$0.828772\pi$
$174$		0			0
$175$		0			0
$176$	205.293	−	109.374i	1.16644	−	0.621444i
$177$		0			0
$178$	−105.759	−	29.3061i	−0.594154	−	0.164641i
$179$			102.849i			0.574573i	0.957845	+	0.287286i	$0.0927532\pi$
$179$			102.849i			0.574573i	−0.957845	+	0.287286i	$0.907247\pi$
$180$		0			0
$181$	−115.413			−0.637640			−0.318820	−	0.947815i	$-0.603286\pi$
$181$	−115.413			−0.637640			−0.318820	+	0.947815i	$0.603286\pi$
$182$	−143.734	+	518.704i	−0.789747	+	2.85002i
$183$		0			0
$184$	−27.5331	−	28.9778i	−0.149636	−	0.157488i
$185$		0			0
$186$		0			0
$187$	184.476			0.986503
$188$	166.645	+	100.036i	0.886407	+	0.532109i
$189$		0			0
$190$		0			0
$191$		−	191.305i		−	1.00160i	−0.865563	−	0.500799i	$-0.833040\pi$
$191$		−	191.305i		−	1.00160i	0.865563	−	0.500799i	$-0.166960\pi$
$192$		0			0
$193$			160.332i			0.830734i	0.909654	+	0.415367i	$0.136347\pi$
$193$			160.332i			0.830734i	−0.909654	+	0.415367i	$0.863653\pi$
$194$	−7.56261	+	27.2918i	−0.0389825	+	0.140679i
$195$		0			0
$196$	323.472	+	194.180i	1.65037	+	0.990714i
$197$		−	355.081i		−	1.80244i	−0.433362	−	0.901220i	$-0.642673\pi$
$197$		−	355.081i		−	1.80244i	0.433362	−	0.901220i	$-0.357327\pi$
$198$		0			0
$199$		−	88.2032i		−	0.443232i	−0.975134	−	0.221616i	$-0.928867\pi$
$199$		−	88.2032i		−	0.443232i	0.975134	−	0.221616i	$-0.0711332\pi$
$200$		0			0
$201$		0			0
$202$	−314.952	−	87.2740i	−1.55917	−	0.432049i
$203$	−32.8875			−0.162007
$204$		0			0
$205$		0			0
$206$	−326.864	−	90.5747i	−1.58672	−	0.439683i
$207$		0			0
$208$	−317.442	+	169.124i	−1.52617	+	0.813095i
$209$	127.404			0.609586
$210$		0			0
$211$		−	190.584i		−	0.903243i	−0.892210	−	0.451622i	$-0.850846\pi$
$211$		−	190.584i		−	0.903243i	0.892210	−	0.451622i	$-0.149154\pi$
$212$	193.741	−	322.741i	0.913871	−	1.52236i
$213$		0			0
$214$	−15.7050	−	4.35188i	−0.0733878	−	0.0203359i
$215$		0			0
$216$		0			0
$217$			195.696i			0.901824i
$218$	−48.6318	−	13.4760i	−0.223082	−	0.0618164i
$219$		0			0
$220$		0			0
$221$	−285.253			−1.29074
$222$		0			0
$223$	79.2869			0.355547			0.177773	−	0.984071i	$-0.443111\pi$
$223$	79.2869			0.355547			0.177773	+	0.984071i	$0.443111\pi$
$224$	83.3026	+	373.926i	0.371887	+	1.66931i
$225$		0			0
$226$	−52.2405	+	188.524i	−0.231153	+	0.834178i
$227$	−353.645			−1.55791			−0.778953	−	0.627082i	$-0.784249\pi$
$227$	−353.645			−1.55791			−0.778953	+	0.627082i	$0.784249\pi$
$228$		0			0
$229$	22.7911			0.0995244			0.0497622	−	0.998761i	$-0.484154\pi$
$229$	22.7911			0.0995244			0.0497622	+	0.998761i	$0.484154\pi$
$230$		0			0
$231$		0			0
$232$	−15.1378	−	15.9321i	−0.0652491	−	0.0686730i
$233$			189.710i			0.814205i	0.913382	+	0.407103i	$0.133461\pi$
$233$			189.710i			0.814205i	−0.913382	+	0.407103i	$0.866539\pi$
$234$		0			0
$235$		0			0
$236$	−89.0712	+	148.378i	−0.377420	+	0.628721i
$237$		0			0
$238$	−81.1311	+	292.784i	−0.340887	+	1.23019i
$239$			267.778i			1.12041i	0.828355	+	0.560204i	$0.189277\pi$
$239$			267.778i			1.12041i	−0.828355	+	0.560204i	$0.810723\pi$
$240$		0			0
$241$	−301.663			−1.25171			−0.625857	−	0.779938i	$-0.715251\pi$
$241$	−301.663			−1.25171			−0.625857	+	0.779938i	$0.715251\pi$
$242$	174.157	+	48.2592i	0.719656	+	0.199418i
$243$		0			0
$244$	194.686	+	116.870i	0.797895	+	0.478975i
$245$		0			0
$246$		0			0
$247$	−197.002			−0.797580
$248$	−94.8035	+	90.0769i	−0.382272	+	0.363213i
$249$		0			0
$250$		0			0
$251$		−	63.1891i		−	0.251749i	−0.992046	−	0.125875i	$-0.959826\pi$
$251$		−	63.1891i		−	0.251749i	0.992046	−	0.125875i	$-0.0401737\pi$
$252$		0			0
$253$		−	72.6407i		−	0.287117i
$254$	−322.955	−	89.4916i	−1.27148	−	0.352329i
$255$		0			0
$256$	−142.802	+	212.470i	−0.557822	+	0.829961i
$257$			150.719i			0.586456i	0.956043	+	0.293228i	$0.0947295\pi$
$257$			150.719i			0.586456i	−0.956043	+	0.293228i	$0.905271\pi$
$258$		0			0
$259$			388.925i			1.50164i
$260$		0			0
$261$		0			0
$262$	−43.8193	+	158.134i	−0.167249	+	0.603565i
$263$	203.755			0.774735			0.387368	−	0.921925i	$-0.373384\pi$
$263$	203.755			0.774735			0.387368	+	0.921925i	$0.373384\pi$
$264$		0			0
$265$		0			0
$266$	−56.0311	+	202.204i	−0.210643	+	0.760165i
$267$		0			0
$268$	209.580	+	125.810i	0.782014	+	0.469442i
$269$	76.3986			0.284010			0.142005	−	0.989866i	$-0.454645\pi$
$269$	76.3986			0.284010			0.142005	+	0.989866i	$0.454645\pi$
$270$		0			0
$271$			169.216i			0.624414i	0.950014	+	0.312207i	$0.101068\pi$
$271$			169.216i			0.624414i	−0.950014	+	0.312207i	$0.898932\pi$
$272$	−179.181	+	95.4624i	−0.658755	+	0.350965i
$273$		0			0
$274$	−135.901	+	490.437i	−0.495990	+	1.78992i
$275$		0			0
$276$		0			0
$277$			273.891i			0.988774i	0.869242	+	0.494387i	$0.164607\pi$
$277$			273.891i			0.988774i	−0.869242	+	0.494387i	$0.835393\pi$
$278$	42.1646	−	152.163i	0.151671	−	0.547348i
$279$		0			0
$280$		0			0
$281$	311.672			1.10915			0.554577	−	0.832133i	$-0.312880\pi$
$281$	311.672			1.10915			0.554577	+	0.832133i	$0.312880\pi$
$282$		0			0
$283$	264.566			0.934861			0.467431	−	0.884030i	$-0.345180\pi$
$283$	264.566			0.934861			0.467431	+	0.884030i	$0.345180\pi$
$284$	81.6583	−	136.029i	0.287529	−	0.478977i
$285$		0			0
$286$	−629.909	−	174.549i	−2.20248	−	0.610311i
$287$	511.891			1.78359
$288$		0			0
$289$	127.988			0.442866
$290$		0			0
$291$		0			0
$292$	204.937	−	341.392i	0.701840	−	1.16915i
$293$		−	121.281i		−	0.413927i	−0.978349	−	0.206964i	$-0.933642\pi$
$293$		−	121.281i		−	0.413927i	0.978349	−	0.206964i	$-0.0663582\pi$
$294$		0			0
$295$		0			0
$296$	−188.412	+	179.018i	−0.636527	+	0.604792i
$297$		0			0
$298$	62.3375	+	17.2739i	0.209186	+	0.0579659i
$299$			112.323i			0.375663i
$300$		0			0
$301$	198.053			0.657984
$302$	20.7197	−	74.7729i	0.0686083	−	0.247592i
$303$		0			0
$304$	−123.747	+	65.9286i	−0.407062	+	0.216870i
$305$		0			0
$306$		0			0
$307$	161.768			0.526932			0.263466	−	0.964669i	$-0.415134\pi$
$307$	161.768			0.526932			0.263466	+	0.964669i	$0.415134\pi$
$308$	−358.315	+	596.895i	−1.16336	+	1.93797i
$309$		0			0
$310$		0			0
$311$			26.3813i			0.0848273i	0.999100	+	0.0424137i	$0.0135047\pi$
$311$			26.3813i			0.0848273i	−0.999100	+	0.0424137i	$0.986495\pi$
$312$		0			0
$313$			5.39902i			0.0172493i	0.999963	+	0.00862463i	$0.00274534\pi$
$313$			5.39902i			0.0172493i	−0.999963	+	0.00862463i	$0.997255\pi$
$314$	−23.6074	+	85.1938i	−0.0751827	+	0.271318i
$315$		0			0
$316$	−22.1890	+	36.9633i	−0.0702184	+	0.116972i
$317$			270.157i			0.852231i	0.904669	+	0.426116i	$0.140118\pi$
$317$			270.157i			0.852231i	−0.904669	+	0.426116i	$0.859882\pi$
$318$		0			0
$319$		−	39.9382i		−	0.125198i
$320$		0			0
$321$		0			0
$322$	115.289	+	31.9468i	0.358040	+	0.0992137i
$323$	−111.199			−0.344268
$324$		0			0
$325$		0			0
$326$	−102.028	−	28.2723i	−0.312971	−	0.0867248i
$327$		0			0
$328$	235.619	+	247.982i	0.718349	+	0.756043i
$329$	−581.716			−1.76813
$330$		0			0
$331$		−	480.728i		−	1.45235i	−0.687510	−	0.726174i	$-0.741296\pi$
$331$		−	480.728i		−	1.45235i	0.687510	−	0.726174i	$-0.258704\pi$
$332$	−481.033	−	288.763i	−1.44890	−	0.869770i
$333$		0			0
$334$	−345.526	−	95.7459i	−1.03451	−	0.286664i
$335$		0			0
$336$		0			0
$337$		−	568.382i		−	1.68659i	−0.537448	−	0.843297i	$-0.680612\pi$
$337$		−	568.382i		−	1.68659i	0.537448	−	0.843297i	$-0.319388\pi$
$338$	648.293	+	179.643i	1.91803	+	0.531489i
$339$		0			0
$340$		0			0
$341$	−237.651			−0.696923
$342$		0			0
$343$	−542.554			−1.58179
$344$	91.1620	+	95.9455i	0.265006	+	0.278911i
$345$		0			0
$346$	94.6791	−	341.676i	0.273639	−	0.987503i
$347$	−370.184			−1.06681			−0.533406	−	0.845859i	$-0.679088\pi$
$347$	−370.184			−1.06681			−0.533406	+	0.845859i	$0.679088\pi$
$348$		0			0
$349$	488.570			1.39991			0.699957	−	0.714185i	$-0.253203\pi$
$349$	488.570			1.39991			0.699957	+	0.714185i	$0.253203\pi$
$350$		0			0
$351$		0			0
$352$	−454.091	+	101.162i	−1.29003	+	0.287391i
$353$			649.728i			1.84059i	0.391226	+	0.920295i	$0.372051\pi$
$353$			649.728i			1.84059i	−0.391226	+	0.920295i	$0.627949\pi$
$354$		0			0
$355$		0			0
$356$	188.186	+	112.968i	0.528612	+	0.317325i
$357$		0			0
$358$	54.9292	−	198.227i	0.153434	−	0.553708i
$359$		−	405.910i		−	1.13067i	−0.824862	−	0.565334i	$-0.808747\pi$
$359$		−	405.910i		−	1.13067i	0.824862	−	0.565334i	$-0.191253\pi$
$360$		0			0
$361$	284.204			0.787268
$362$	222.443	+	61.6395i	0.614484	+	0.170275i
$363$		0			0
$364$	554.058	−	922.970i	1.52214	−	2.53563i
$365$		0			0
$366$		0			0
$367$	46.2347			0.125980			0.0629900	−	0.998014i	$-0.479936\pi$
$367$	46.2347			0.125980			0.0629900	+	0.998014i	$0.479936\pi$
$368$	37.5900	+	70.5558i	0.102147	+	0.191728i
$369$		0			0
$370$		0			0
$371$			1126.61i			3.03668i
$372$		0			0
$373$			138.262i			0.370676i	0.982675	+	0.185338i	$0.0593380\pi$
$373$			138.262i			0.370676i	−0.982675	+	0.185338i	$0.940662\pi$
$374$	−355.554	−	98.5247i	−0.950679	−	0.263435i
$375$		0			0
$376$	−267.759	−	281.809i	−0.712124	−	0.749491i
$377$			61.7559i			0.163809i
$378$		0			0
$379$			254.516i			0.671546i	0.941943	+	0.335773i	$0.108998\pi$
$379$			254.516i			0.671546i	−0.941943	+	0.335773i	$0.891002\pi$
$380$		0			0
$381$		0			0
$382$	−102.172	+	368.716i	−0.267466	+	0.965226i
$383$	−62.7205			−0.163761			−0.0818805	−	0.996642i	$-0.526093\pi$
$383$	−62.7205			−0.163761			−0.0818805	+	0.996642i	$0.526093\pi$
$384$		0			0
$385$		0			0
$386$	85.6298	−	309.019i	0.221839	−	0.800567i
$387$		0			0
$388$	29.1519	−	48.5623i	0.0751338	−	0.125161i
$389$	110.130			0.283112			0.141556	−	0.989930i	$-0.454790\pi$
$389$	110.130			0.283112			0.141556	+	0.989930i	$0.454790\pi$
$390$		0			0
$391$			63.4012i			0.162152i
$392$	−519.744	−	547.016i	−1.32588	−	1.39545i
$393$		0			0
$394$	−189.641	+	684.372i	−0.481322	+	1.73698i
$395$		0			0
$396$		0			0
$397$		−	292.953i		−	0.737916i	−0.929446	−	0.368958i	$-0.879715\pi$
$397$		−	292.953i		−	0.737916i	0.929446	−	0.368958i	$-0.120285\pi$
$398$	−47.1074	+	170.000i	−0.118360	+	0.427136i
$399$		0			0
$400$		0			0
$401$	−518.103			−1.29203			−0.646014	−	0.763325i	$-0.723565\pi$
$401$	−518.103			−1.29203			−0.646014	+	0.763325i	$0.723565\pi$
$402$		0			0
$403$	367.476			0.911851
$404$	560.419	+	336.419i	1.38718	+	0.832720i
$405$		0			0
$406$	63.3864	+	17.5645i	0.156124	+	0.0432623i
$407$	−472.306			−1.16046
$408$		0			0
$409$	181.984			0.444948			0.222474	−	0.974939i	$-0.428587\pi$
$409$	181.984			0.444948			0.222474	+	0.974939i	$0.428587\pi$
$410$		0			0
$411$		0			0
$412$	581.614	+	349.142i	1.41168	+	0.847432i
$413$		−	517.953i		−	1.25412i
$414$		0			0
$415$		0			0
$416$	702.155	−	156.425i	1.68787	−	0.376022i
$417$		0			0
$418$	−245.554	−	68.0435i	−0.587450	−	0.162784i
$419$			163.347i			0.389849i	0.980818	+	0.194925i	$0.0624462\pi$
$419$			163.347i			0.389849i	−0.980818	+	0.194925i	$0.937554\pi$
$420$		0			0
$421$	−467.206			−1.10975			−0.554876	−	0.831933i	$-0.687234\pi$
$421$	−467.206			−1.10975			−0.554876	+	0.831933i	$0.687234\pi$
$422$	−101.787	+	367.327i	−0.241201	+	0.870442i
$423$		0			0
$424$	−545.779	+	518.568i	−1.28721	+	1.22304i
$425$		0			0
$426$		0			0
$427$	−679.603			−1.59158
$428$	27.9451	+	16.7754i	0.0652923	+	0.0391948i
$429$		0			0
$430$		0			0
$431$		−	685.527i		−	1.59055i	−0.606248	−	0.795275i	$-0.707326\pi$
$431$		−	685.527i		−	1.59055i	0.606248	−	0.795275i	$-0.292674\pi$
$432$		0			0
$433$			592.777i			1.36900i	0.729013	+	0.684500i	$0.239979\pi$
$433$			592.777i			1.36900i	−0.729013	+	0.684500i	$0.760021\pi$
$434$	104.517	−	377.178i	0.240822	−	0.869074i
$435$		0			0
$436$	86.5343	+	51.9464i	0.198473	+	0.119143i
$437$			43.7864i			0.100198i
$438$		0			0
$439$		−	464.439i		−	1.05795i	−0.848638	−	0.528974i	$-0.822577\pi$
$439$		−	464.439i		−	1.05795i	0.848638	−	0.528974i	$-0.177423\pi$
$440$		0			0
$441$		0			0
$442$	549.788	+	152.347i	1.24386	+	0.344677i
$443$	−54.2868			−0.122544			−0.0612718	−	0.998121i	$-0.519516\pi$
$443$	−54.2868			−0.122544			−0.0612718	+	0.998121i	$0.519516\pi$
$444$		0			0
$445$		0			0
$446$	−152.815	−	42.3454i	−0.342635	−	0.0949449i
$447$		0			0
$448$	39.1506	−	765.184i	0.0873897	−	1.70800i
$449$	−428.051			−0.953343			−0.476671	−	0.879082i	$-0.658157\pi$
$449$	−428.051			−0.953343			−0.476671	+	0.879082i	$0.658157\pi$
$450$		0			0
$451$			621.634i			1.37835i
$452$	201.374	−	335.456i	0.445517	−	0.742159i
$453$		0			0
$454$	681.605	+	188.874i	1.50133	+	0.416022i
$455$		0			0
$456$		0			0
$457$			66.5848i			0.145700i	0.997343	+	0.0728498i	$0.0232094\pi$
$457$			66.5848i			0.145700i	−0.997343	+	0.0728498i	$0.976791\pi$
$458$	−43.9269	−	12.1722i	−0.0959103	−	0.0265769i
$459$		0			0
$460$		0			0
$461$	238.626			0.517627			0.258814	−	0.965927i	$-0.416668\pi$
$461$	238.626			0.517627			0.258814	+	0.965927i	$0.416668\pi$
$462$		0			0
$463$	−386.958			−0.835762			−0.417881	−	0.908502i	$-0.637227\pi$
$463$	−386.958			−0.835762			−0.417881	+	0.908502i	$0.637227\pi$
$464$	20.6672	+	38.7919i	0.0445413	+	0.0836032i
$465$		0			0
$466$	101.320	−	365.641i	0.217425	−	0.784638i
$467$	235.964			0.505276			0.252638	−	0.967561i	$-0.418702\pi$
$467$	235.964			0.505276			0.252638	+	0.967561i	$0.418702\pi$
$468$		0			0
$469$	−731.593			−1.55990
$470$		0			0
$471$		0			0
$472$	250.919	−	238.409i	0.531608	−	0.505104i
$473$			240.513i			0.508485i
$474$		0			0
$475$		0			0
$476$	312.740	−	520.974i	0.657016	−	1.09448i
$477$		0			0
$478$	143.014	−	516.107i	0.299193	−	1.07972i
$479$			529.496i			1.10542i	0.833374	+	0.552710i	$0.186406\pi$
$479$			529.496i			1.10542i	−0.833374	+	0.552710i	$0.813594\pi$
$480$		0			0
$481$	730.320			1.51834
$482$	581.417	+	161.112i	1.20626	+	0.334257i
$483$		0			0
$484$	−309.890	−	186.027i	−0.640270	−	0.384353i
$485$		0			0
$486$		0			0
$487$	−880.801			−1.80863			−0.904314	−	0.426869i	$-0.859617\pi$
$487$	−880.801			−1.80863			−0.904314	+	0.426869i	$0.859617\pi$
$488$	−312.815	−	329.230i	−0.641015	−	0.674651i
$489$		0			0
$490$		0			0
$491$		−	86.4466i		−	0.176062i	−0.996118	−	0.0880312i	$-0.971942\pi$
$491$		−	86.4466i		−	0.176062i	0.996118	−	0.0880312i	$-0.0280575\pi$
$492$		0			0
$493$			34.8583i			0.0707065i
$494$	379.697	+	105.215i	0.768617	+	0.212985i
$495$		0			0
$496$	230.830	−	122.979i	0.465383	−	0.247942i
$497$			474.846i			0.955425i
$498$		0			0
$499$		−	874.536i		−	1.75258i	−0.481786	−	0.876289i	$-0.660012\pi$
$499$		−	874.536i		−	1.75258i	0.481786	−	0.876289i	$-0.339988\pi$
$500$		0			0
$501$		0			0
$502$	−33.7479	+	121.789i	−0.0672269	+	0.242607i
$503$	17.5479			0.0348865			0.0174433	−	0.999848i	$-0.494447\pi$
$503$	17.5479			0.0348865			0.0174433	+	0.999848i	$0.494447\pi$
$504$		0			0
$505$		0			0
$506$	−38.7958	+	140.006i	−0.0766716	+	0.276691i
$507$		0			0
$508$	574.659	+	344.967i	1.13122	+	0.679069i
$509$	609.132			1.19672			0.598362	−	0.801226i	$-0.295819\pi$
$509$	609.132			1.19672			0.598362	+	0.801226i	$0.295819\pi$
$510$		0			0
$511$			1191.72i			2.33213i
$512$	388.709	−	333.241i	0.759197	−	0.650861i
$513$		0			0
$514$	80.4959	−	290.492i	0.156607	−	0.565159i
$515$		0			0
$516$		0			0
$517$		−	706.430i		−	1.36640i
$518$	207.716	−	749.602i	0.400997	−	1.44711i
$519$		0			0
$520$		0			0
$521$	−433.724			−0.832484			−0.416242	−	0.909254i	$-0.636653\pi$
$521$	−433.724			−0.832484			−0.416242	+	0.909254i	$0.636653\pi$
$522$		0			0
$523$	473.223			0.904823			0.452412	−	0.891809i	$-0.350564\pi$
$523$	473.223			0.904823			0.452412	+	0.891809i	$0.350564\pi$
$524$	168.912	−	281.380i	0.322351	−	0.536985i
$525$		0			0
$526$	−392.712	−	108.821i	−0.746601	−	0.206885i
$527$	207.423			0.393592
$528$		0			0
$529$	−504.035			−0.952807
$530$		0			0
$531$		0			0
$532$	215.985	−	359.797i	0.405988	−	0.676310i
$533$		−	961.224i		−	1.80342i
$534$		0			0
$535$		0			0
$536$	−336.746	−	354.416i	−0.628257	−	0.661223i
$537$		0			0
$538$	−147.249	−	40.8029i	−0.273696	−	0.0758418i
$539$		−	1371.24i		−	2.54405i
$540$		0			0
$541$	294.889			0.545081			0.272540	−	0.962144i	$-0.412136\pi$
$541$	294.889			0.545081			0.272540	+	0.962144i	$0.412136\pi$
$542$	90.3747	−	326.142i	0.166743	−	0.601739i
$543$		0			0
$544$	396.333	−	88.2946i	0.728554	−	0.162306i
$545$		0			0
$546$		0			0
$547$	966.695			1.76727			0.883634	−	0.468179i	$-0.155090\pi$
$547$	966.695			1.76727			0.883634	+	0.468179i	$0.155090\pi$
$548$	523.864	−	872.673i	0.955956	−	1.59247i
$549$		0			0
$550$		0			0
$551$			24.0740i			0.0436914i
$552$		0			0
$553$		−	129.030i		−	0.233327i
$554$	146.279	−	527.889i	0.264042	−	0.952868i
$555$		0			0
$556$	−162.534	+	270.755i	−0.292327	+	0.486969i
$557$			74.2603i			0.133322i	0.997776	+	0.0666609i	$0.0212346\pi$
$557$			74.2603i			0.133322i	−0.997776	+	0.0666609i	$0.978765\pi$
$558$		0			0
$559$		−	371.903i		−	0.665300i
$560$		0			0
$561$		0			0
$562$	−600.708	−	166.457i	−1.06887	−	0.296187i
$563$	−663.688			−1.17884			−0.589421	−	0.807826i	$-0.700644\pi$
$563$	−663.688			−1.17884			−0.589421	+	0.807826i	$0.700644\pi$
$564$		0			0
$565$		0			0
$566$	−509.916	−	141.299i	−0.900912	−	0.249645i
$567$		0			0
$568$	−230.036	+	218.567i	−0.404993	+	0.384802i
$569$	−667.450			−1.17302			−0.586511	−	0.809941i	$-0.699499\pi$
$569$	−667.450			−1.17302			−0.586511	+	0.809941i	$0.699499\pi$
$570$		0			0
$571$			185.898i			0.325565i	0.986662	+	0.162782i	$0.0520469\pi$
$571$			185.898i			0.325565i	−0.986662	+	0.162782i	$0.947953\pi$
$572$	1120.84	+	672.841i	1.95952	+	1.17630i
$573$		0			0
$574$	−986.603	−	273.390i	−1.71882	−	0.476289i
$575$		0			0
$576$		0			0
$577$			664.331i			1.15135i	0.817678	+	0.575676i	$0.195261\pi$
$577$			664.331i			1.15135i	−0.817678	+	0.575676i	$0.804739\pi$
$578$	−246.681	−	68.3557i	−0.426783	−	0.118263i
$579$		0			0
$580$		0			0
$581$	1679.17			2.89014
$582$		0			0
$583$	−1368.14			−2.34673
$584$	−577.321	+	548.537i	−0.988563	+	0.939276i
$585$		0			0
$586$	−64.7734	+	233.753i	−0.110535	+	0.398896i
$587$	763.083			1.29997			0.649986	−	0.759946i	$-0.274775\pi$
$587$	763.083			1.29997			0.649986	+	0.759946i	$0.274775\pi$
$588$		0			0
$589$	143.251			0.243211
$590$		0			0
$591$		0			0
$592$	458.750	−	244.408i	0.774915	−	0.412852i
$593$		−	286.193i		−	0.482618i	−0.970448	−	0.241309i	$-0.922423\pi$
$593$		−	286.193i		−	0.482618i	0.970448	−	0.241309i	$-0.0775768\pi$
$594$		0			0
$595$		0			0
$596$	−110.922	−	66.5862i	−0.186111	−	0.111722i
$597$		0			0
$598$	59.9895	−	216.489i	0.100317	−	0.362021i
$599$			604.151i			1.00860i	0.863529	+	0.504300i	$0.168249\pi$
$599$			604.151i			1.00860i	−0.863529	+	0.504300i	$0.831751\pi$
$600$		0			0
$601$	275.562			0.458505			0.229253	−	0.973367i	$-0.426372\pi$
$601$	275.562			0.458505			0.229253	+	0.973367i	$0.426372\pi$
$602$	−381.722	−	105.776i	−0.634089	−	0.175707i
$603$		0			0
$604$	−79.8692	+	133.049i	−0.132234	+	0.220280i
$605$		0			0
$606$		0			0
$607$	−52.1487			−0.0859121			−0.0429561	−	0.999077i	$-0.513678\pi$
$607$	−52.1487			−0.0859121			−0.0429561	+	0.999077i	$0.513678\pi$
$608$	273.717	−	60.9784i	0.450193	−	0.100293i
$609$		0			0
$610$		0			0
$611$			1092.34i			1.78779i
$612$		0			0
$613$		−	898.128i		−	1.46513i	−0.680695	−	0.732567i	$-0.738322\pi$
$613$		−	898.128i		−	1.46513i	0.680695	−	0.732567i	$-0.261678\pi$
$614$	−311.787	−	86.3968i	−0.507796	−	0.140711i
$615$		0			0
$616$	1009.39	−	959.070i	1.63863	−	1.55693i
$617$		−	636.868i		−	1.03220i	−0.856528	−	0.516101i	$-0.827383\pi$
$617$		−	636.868i		−	1.03220i	0.856528	−	0.516101i	$-0.172617\pi$
$618$		0			0
$619$			190.559i			0.307849i	0.988083	+	0.153925i	$0.0491913\pi$
$619$			190.559i			0.307849i	−0.988083	+	0.153925i	$0.950809\pi$
$620$		0			0
$621$		0			0
$622$	14.0897	−	50.8466i	0.0226522	−	0.0817469i
$623$	−656.911			−1.05443
$624$		0			0
$625$		0			0
$626$	2.88350	−	10.4059i	0.00460623	−	0.0166229i
$627$		0			0
$628$	91.0004	−	151.592i	0.144905	−	0.241388i
$629$	412.231			0.655376
$630$		0			0
$631$			578.160i			0.916261i	0.888885	+	0.458130i	$0.151481\pi$
$631$			578.160i			0.916261i	−0.888885	+	0.458130i	$0.848519\pi$
$632$	62.5078	−	59.3913i	0.0989047	−	0.0939736i
$633$		0			0
$634$	144.285	−	520.693i	0.227579	−	0.821283i
$635$		0			0
$636$		0			0
$637$			2120.34i			3.32863i
$638$	−21.3301	+	76.9757i	−0.0334328	+	0.120652i
$639$		0			0
$640$		0			0
$641$	35.3085			0.0550834			0.0275417	−	0.999621i	$-0.491232\pi$
$641$	35.3085			0.0550834			0.0275417	+	0.999621i	$0.491232\pi$
$642$		0			0
$643$	−1045.67			−1.62623			−0.813117	−	0.582100i	$-0.802231\pi$
$643$	−1045.67			−1.62623			−0.813117	+	0.582100i	$0.802231\pi$
$644$	−205.143	−	123.147i	−0.318544	−	0.191222i
$645$		0			0
$646$	214.321	+	59.3888i	0.331766	+	0.0919331i
$647$	−2.71164			−0.00419110			−0.00209555	−	0.999998i	$-0.500667\pi$
$647$	−2.71164			−0.00419110			−0.00209555	+	0.999998i	$0.500667\pi$
$648$		0			0
$649$	628.996			0.969177
$650$		0			0
$651$		0			0
$652$	181.547	+	108.982i	0.278446	+	0.167151i
$653$			206.765i			0.316639i	0.987388	+	0.158319i	$0.0506076\pi$
$653$			206.765i			0.316639i	−0.987388	+	0.158319i	$0.949392\pi$
$654$		0			0
$655$		0			0
$656$	−321.682	−	603.792i	−0.490370	−	0.920415i
$657$		0			0
$658$	1121.18	+	310.682i	1.70393	+	0.472161i
$659$		−	708.330i		−	1.07486i	−0.843309	−	0.537428i	$-0.819396\pi$
$659$		−	708.330i		−	1.07486i	0.843309	−	0.537428i	$-0.180604\pi$
$660$		0			0
$661$	1229.66			1.86031			0.930155	−	0.367167i	$-0.119672\pi$
$661$	1229.66			1.86031			0.930155	+	0.367167i	$0.119672\pi$
$662$	−256.746	+	926.540i	−0.387834	+	1.39961i
$663$		0			0
$664$	772.907	+	813.464i	1.16402	+	1.22510i
$665$		0			0
$666$		0			0
$667$	13.7261			0.0205788
$668$	614.820	+	369.076i	0.920390	+	0.552508i
$669$		0			0
$670$		0			0
$671$		−	825.303i		−	1.22996i
$672$		0			0
$673$		−	753.492i		−	1.11960i	−0.828627	−	0.559801i	$-0.810878\pi$
$673$		−	753.492i		−	1.11960i	0.828627	−	0.559801i	$-0.189122\pi$
$674$	−303.561	+	1095.48i	−0.450387	+	1.62535i
$675$		0			0
$676$	−1153.56	−	692.479i	−1.70645	−	1.02438i
$677$			332.246i			0.490762i	0.969427	+	0.245381i	$0.0789131\pi$
$677$			332.246i			0.490762i	−0.969427	+	0.245381i	$0.921087\pi$
$678$		0			0
$679$			169.520i			0.249661i
$680$		0			0
$681$		0			0
$682$	458.041	+	126.924i	0.671615	+	0.186106i
$683$	1120.62			1.64074			0.820368	−	0.571835i	$-0.193768\pi$
$683$	1120.62			1.64074			0.820368	+	0.571835i	$0.193768\pi$
$684$		0			0
$685$		0			0
$686$	1045.70	+	289.767i	1.52435	+	0.422400i
$687$		0			0
$688$	−124.461	−	233.610i	−0.180902	−	0.339550i
$689$	2115.54			3.07045
$690$		0			0
$691$			331.115i			0.479182i	0.970874	+	0.239591i	$0.0770134\pi$
$691$			331.115i			0.479182i	−0.970874	+	0.239591i	$0.922987\pi$
$692$	−364.964	+	607.970i	−0.527404	+	0.878570i
$693$		0			0
$694$	713.482	+	197.707i	1.02807	+	0.284881i
$695$		0			0
$696$		0			0
$697$		−	542.566i		−	0.778431i
$698$	−941.655	−	260.935i	−1.34908	−	0.373832i
$699$		0			0
$700$		0			0
$701$	564.971			0.805949			0.402975	−	0.915211i	$-0.367976\pi$
$701$	564.971			0.805949			0.402975	+	0.915211i	$0.367976\pi$
$702$		0			0
$703$	284.697			0.404974
$704$	929.230	+	47.5440i	1.31993	+	0.0675341i
$705$		0			0
$706$	347.006	−	1252.27i	0.491510	−	1.77375i
$707$	−1956.29			−2.76703
$708$		0			0
$709$	−1.56083			−0.00220146			−0.00110073	−	0.999999i	$-0.500350\pi$
$709$	−1.56083			−0.00220146			−0.00110073	+	0.999999i	$0.500350\pi$
$710$		0			0
$711$		0			0
$712$	−302.370	−	318.236i	−0.424677	−	0.446961i
$713$		−	81.6765i		−	0.114553i
$714$		0			0
$715$		0			0
$716$	−211.738	+	352.721i	−0.295723	+	0.492627i
$717$		0			0
$718$	−216.788	+	782.339i	−0.301933	+	1.08961i
$719$			75.0325i			0.104357i	0.998638	+	0.0521784i	$0.0166164\pi$
$719$			75.0325i			0.104357i	−0.998638	+	0.0521784i	$0.983384\pi$
$720$		0			0
$721$	−2030.28			−2.81592
$722$	−547.766	−	151.787i	−0.758678	−	0.210231i
$723$		0			0
$724$	−395.810	−	237.604i	−0.546699	−	0.328183i
$725$		0			0
$726$		0			0
$727$	−1229.26			−1.69087			−0.845433	−	0.534082i	$-0.820657\pi$
$727$	−1229.26			−1.69087			−0.845433	+	0.534082i	$0.820657\pi$
$728$	−1560.81	+	1483.00i	−2.14397	+	2.03708i
$729$		0			0
$730$		0			0
$731$		−	209.922i		−	0.287170i
$732$		0			0
$733$		−	691.736i		−	0.943705i	−0.881678	−	0.471852i	$-0.843586\pi$
$733$		−	691.736i		−	0.943705i	0.881678	−	0.471852i	$-0.156414\pi$
$734$	−89.1114	−	24.6930i	−0.121405	−	0.0336416i
$735$		0			0
$736$	−34.7676	−	156.063i	−0.0472385	−	0.212042i
$737$		−	888.438i		−	1.20548i
$738$		0			0
$739$		−	71.4311i		−	0.0966591i	−0.998831	−	0.0483296i	$-0.984610\pi$
$739$		−	71.4311i		−	0.0966591i	0.998831	−	0.0483296i	$-0.0153898\pi$
$740$		0			0
$741$		0			0
$742$	601.698	−	2171.40i	0.810914	−	2.92641i
$743$	−1006.92			−1.35521			−0.677605	−	0.735426i	$-0.736982\pi$
$743$	−1006.92			−1.35521			−0.677605	+	0.735426i	$0.736982\pi$
$744$		0			0
$745$		0			0
$746$	73.8428	−	266.482i	0.0989850	−	0.357215i
$747$		0			0
$748$	632.664	+	379.787i	0.845808	+	0.507737i
$749$	−97.5496			−0.130240
$750$		0			0
$751$		−	1110.14i		−	1.47822i	−0.673587	−	0.739108i	$-0.735247\pi$
$751$		−	1110.14i		−	1.47822i	0.673587	−	0.739108i	$-0.264753\pi$
$752$	365.562	+	686.154i	0.486120	+	0.912439i
$753$		0			0
$754$	32.9825	−	119.026i	0.0437433	−	0.157860i
$755$		0			0
$756$		0			0
$757$			326.752i			0.431641i	0.976433	+	0.215821i	$0.0692426\pi$
$757$			326.752i			0.431641i	−0.976433	+	0.215821i	$0.930757\pi$
$758$	135.932	−	490.547i	0.179329	−	0.647160i
$759$		0			0
$760$		0			0
$761$	−162.162			−0.213091			−0.106546	−	0.994308i	$-0.533979\pi$
$761$	−162.162			−0.213091			−0.106546	+	0.994308i	$0.533979\pi$
$762$		0			0
$763$	−302.070			−0.395898
$764$	393.847	−	656.085i	0.515507	−	0.858750i
$765$		0			0
$766$	120.886	+	33.4977i	0.157814	+	0.0437306i
$767$	−972.608			−1.26807
$768$		0			0
$769$	154.694			0.201162			0.100581	−	0.994929i	$-0.467930\pi$
$769$	154.694			0.201162			0.100581	+	0.994929i	$0.467930\pi$
$770$		0			0
$771$		0			0
$772$	−330.081	+	549.861i	−0.427566	+	0.712255i
$773$		−	208.302i		−	0.269472i	−0.990882	−	0.134736i	$-0.956981\pi$
$773$		−	208.302i		−	0.269472i	0.990882	−	0.134736i	$-0.0430187\pi$
$774$		0			0
$775$		0			0
$776$	−82.1226	+	78.0283i	−0.105828	+	0.100552i
$777$		0			0
$778$	−212.262	−	58.8183i	−0.272831	−	0.0756019i
$779$		−	374.709i		−	0.481013i
$780$		0			0
$781$	−576.648			−0.738346
$782$	33.8612	−	122.198i	0.0433008	−	0.156263i
$783$		0			0
$784$	709.590	+	1331.89i	0.905089	+	1.69884i
$785$		0			0
$786$		0			0
$787$	−377.158			−0.479235			−0.239617	−	0.970867i	$-0.577022\pi$
$787$	−377.158			−0.479235			−0.239617	+	0.970867i	$0.577022\pi$
$788$	731.017	−	1217.76i	0.927687	−	1.54538i
$789$		0			0
$790$		0			0
$791$			1171.00i			1.48040i
$792$		0			0
$793$			1276.15i			1.60927i
$794$	−156.460	+	564.629i	−0.197053	+	0.711119i
$795$		0			0
$796$	181.587	−	302.495i	0.228124	−	0.380018i
$797$		−	321.141i		−	0.402937i	−0.979495	−	0.201468i	$-0.935429\pi$
$797$		−	321.141i		−	0.402937i	0.979495	−	0.201468i	$-0.0645713\pi$
$798$		0			0
$799$			616.576i			0.771685i
$800$		0			0
$801$		0			0
$802$	998.577	+	276.708i	1.24511	+	0.345022i
$803$	−1447.21			−1.80225
$804$		0			0
$805$		0			0
$806$	−708.263	−	196.261i	−0.878738	−	0.243500i
$807$		0			0
$808$	−900.462	−	947.712i	−1.11443	−	1.17291i
$809$	−861.938			−1.06544			−0.532718	−	0.846293i	$-0.678829\pi$
$809$	−861.938			−1.06544			−0.532718	+	0.846293i	$0.678829\pi$
$810$		0			0
$811$		−	1011.44i		−	1.24715i	−0.781765	−	0.623574i	$-0.785680\pi$
$811$		−	1011.44i		−	1.24715i	0.781765	−	0.623574i	$-0.214320\pi$
$812$	−112.788	−	67.7066i	−0.138902	−	0.0833825i
$813$		0			0
$814$	910.308	+	252.248i	1.11832	+	0.309887i
$815$		0			0
$816$		0			0
$817$		−	144.977i		−	0.177450i
$818$	−350.750	−	97.1935i	−0.428790	−	0.118818i
$819$		0			0
$820$		0			0
$821$	−68.0368			−0.0828707			−0.0414353	−	0.999141i	$-0.513193\pi$
$821$	−68.0368			−0.0828707			−0.0414353	+	0.999141i	$0.513193\pi$
$822$		0			0
$823$	980.340			1.19118			0.595589	−	0.803289i	$-0.296919\pi$
$823$	980.340			1.19118			0.595589	+	0.803289i	$0.296919\pi$
$824$	−934.517	−	983.554i	−1.13412	−	1.19363i
$825$		0			0
$826$	−276.628	+	998.288i	−0.334900	+	1.20858i
$827$	1183.88			1.43153			0.715766	−	0.698341i	$-0.246078\pi$
$827$	1183.88			1.43153			0.715766	+	0.698341i	$0.246078\pi$
$828$		0			0
$829$	−98.7892			−0.119167			−0.0595833	−	0.998223i	$-0.518977\pi$
$829$	−98.7892			−0.119167			−0.0595833	+	0.998223i	$0.518977\pi$
$830$		0			0
$831$		0			0
$832$	−1436.86	−	73.5167i	−1.72699	−	0.0883614i
$833$			1196.83i			1.43677i
$834$		0			0
$835$		0			0
$836$	436.933	+	262.290i	0.522647	+	0.313744i
$837$		0			0
$838$	87.2401	−	314.830i	0.104105	−	0.375692i
$839$		−	407.965i		−	0.486251i	−0.969995	−	0.243125i	$-0.921827\pi$
$839$		−	407.965i		−	0.486251i	0.969995	−	0.243125i	$-0.0781727\pi$
$840$		0			0
$841$	−833.453			−0.991027
$842$	900.479	+	249.525i	1.06945	+	0.296347i
$843$		0			0
$844$	392.363	−	653.613i	0.464885	−	0.774423i
$845$		0			0
$846$		0			0
$847$	1081.75			1.27716
$848$	1328.87	−	707.984i	1.56707	−	0.834887i
$849$		0			0
$850$		0			0
$851$		−	162.323i		−	0.190744i
$852$		0			0
$853$			907.020i			1.06333i	0.846955	+	0.531665i	$0.178433\pi$
$853$			907.020i			1.06333i	−0.846955	+	0.531665i	$0.821567\pi$
$854$	1309.85	+	362.962i	1.53378	+	0.425014i
$855$		0			0
$856$	−44.9012	−	47.2573i	−0.0524546	−	0.0552071i
$857$			693.549i			0.809275i	0.914477	+	0.404638i	$0.132602\pi$
$857$			693.549i			0.809275i	−0.914477	+	0.404638i	$0.867398\pi$
$858$		0			0
$859$		−	397.401i		−	0.462632i	−0.972879	−	0.231316i	$-0.925697\pi$
$859$		−	397.401i		−	0.462632i	0.972879	−	0.231316i	$-0.0743031\pi$
$860$		0			0
$861$		0			0
$862$	−366.126	+	1321.27i	−0.424740	+	1.53279i
$863$	1193.06			1.38246			0.691229	−	0.722636i	$-0.257070\pi$
$863$	1193.06			1.38246			0.691229	+	0.722636i	$0.257070\pi$
$864$		0			0
$865$		0			0
$866$	316.590	−	1142.50i	0.365577	−	1.31929i
$867$		0			0
$868$	−402.886	+	671.142i	−0.464154	+	0.773206i
$869$	156.693			0.180314
$870$		0			0
$871$			1373.78i			1.57724i
$872$	−139.040	−	146.336i	−0.159450	−	0.167817i
$873$		0			0
$874$	23.3854	−	84.3927i	0.0267567	−	0.0965591i
$875$		0			0
$876$		0			0
$877$		−	136.545i		−	0.155695i	−0.996965	−	0.0778477i	$-0.975195\pi$
$877$		−	136.545i		−	0.155695i	0.996965	−	0.0778477i	$-0.0248048\pi$
$878$	−248.047	+	895.146i	−0.282514	+	1.01953i
$879$		0			0
$880$		0			0
$881$	836.578			0.949578			0.474789	−	0.880100i	$-0.342524\pi$
$881$	836.578			0.949578			0.474789	+	0.880100i	$0.342524\pi$
$882$		0			0
$883$	632.625			0.716450			0.358225	−	0.933635i	$-0.383382\pi$
$883$	632.625			0.716450			0.358225	+	0.933635i	$0.383382\pi$
$884$	−978.280	−	587.260i	−1.10665	−	0.664321i
$885$		0			0
$886$	104.631	+	28.9934i	0.118093	+	0.0327239i
$887$	−290.957			−0.328024			−0.164012	−	0.986458i	$-0.552444\pi$
$887$	−290.957			−0.328024			−0.164012	+	0.986458i	$0.552444\pi$
$888$		0			0
$889$	−2006.00			−2.25646
$890$		0			0
$891$		0			0
$892$	271.916	+	163.231i	0.304838	+	0.182994i
$893$			425.822i			0.476844i
$894$		0			0
$895$		0			0
$896$	−484.126	+	1453.88i	−0.540319	+	1.62264i
$897$		0			0
$898$	825.013	+	228.613i	0.918723	+	0.254580i
$899$		−	44.9061i		−	0.0499511i
$900$		0			0
$901$	1194.12			1.32533
$902$	332.001	−	1198.12i	0.368073	−	1.32829i
$903$		0			0
$904$	−567.282	+	538.999i	−0.627524	+	0.596237i
$905$		0			0
$906$		0			0
$907$	473.341			0.521875			0.260938	−	0.965356i	$-0.415968\pi$
$907$	473.341			0.521875			0.260938	+	0.965356i	$0.415968\pi$
$908$	−1212.83	−	728.061i	−1.33572	−	0.801830i
$909$		0			0
$910$		0			0
$911$		−	1176.67i		−	1.29163i	−0.763495	−	0.645813i	$-0.776518\pi$
$911$		−	1176.67i		−	1.29163i	0.763495	−	0.645813i	$-0.223482\pi$
$912$		0			0
$913$			2039.17i			2.23348i
$914$	35.5615	−	128.334i	0.0389075	−	0.140409i
$915$		0			0
$916$	78.1625	+	46.9208i	0.0853303	+	0.0512236i
$917$			982.230i			1.07113i
$918$		0			0
$919$			1491.24i			1.62267i	0.584580	+	0.811336i	$0.301259\pi$
$919$			1491.24i			1.62267i	−0.584580	+	0.811336i	$0.698741\pi$
$920$		0			0
$921$		0			0
$922$	−459.921	−	127.445i	−0.498830	−	0.138227i
$923$	891.663			0.966048
$924$		0			0
$925$		0			0
$926$	745.812	+	206.666i	0.805412	+	0.223181i
$927$		0			0
$928$	−19.1154	−	85.8043i	−0.0205984	−	0.0924615i
$929$	−1217.24			−1.31027			−0.655134	−	0.755513i	$-0.727388\pi$
$929$	−1217.24			−1.31027			−0.655134	+	0.755513i	$0.727388\pi$
$930$		0			0
$931$			826.560i			0.887819i
$932$	−390.562	+	650.613i	−0.419058	+	0.698083i
$933$		0			0
$934$	−454.790	−	126.023i	−0.486927	−	0.134928i
$935$		0			0
$936$		0			0
$937$		−	468.840i		−	0.500363i	−0.968199	−	0.250182i	$-0.919510\pi$
$937$		−	468.840i		−	0.500363i	0.968199	−	0.250182i	$-0.0804903\pi$
$938$	1410.05	+	390.728i	1.50325	+	0.416555i
$939$		0			0
$940$		0			0
$941$	−358.033			−0.380481			−0.190241	−	0.981737i	$-0.560927\pi$
$941$	−358.033			−0.380481			−0.190241	+	0.981737i	$0.560927\pi$
$942$		0			0
$943$	−213.645			−0.226559
$944$	−610.943	+	325.492i	−0.647185	+	0.344801i
$945$		0			0
$946$	128.453	−	463.559i	0.135785	−	0.490020i
$947$	1148.77			1.21306			0.606532	−	0.795059i	$-0.292560\pi$
$947$	1148.77			1.21306			0.606532	+	0.795059i	$0.292560\pi$
$948$		0			0
$949$	2237.80			2.35806
$950$		0			0
$951$		0			0
$952$	−881.006	+	837.082i	−0.925427	+	0.879288i
$953$			911.785i			0.956753i	0.878155	+	0.478376i	$0.158774\pi$
$953$			911.785i			0.956753i	−0.878155	+	0.478376i	$0.841226\pi$
$954$		0			0
$955$		0			0
$956$	−551.283	+	918.348i	−0.576656	+	0.960615i
$957$		0			0
$958$	282.792	−	1020.54i	0.295190	−	1.06528i
$959$			3046.29i			3.17653i
$960$		0			0
$961$	693.788			0.721944
$962$	−1407.60	−	390.048i	−1.46320	−	0.405455i
$963$		0			0
$964$	−1034.56	−	621.045i	−1.07320	−	0.644237i
$965$		0			0
$966$		0			0
$967$	−311.565			−0.322197			−0.161099	−	0.986938i	$-0.551504\pi$
$967$	−311.565			−0.322197			−0.161099	+	0.986938i	$0.551504\pi$
$968$	497.921	+	524.048i	0.514381	+	0.541372i
$969$		0			0
$970$		0			0
$971$			370.530i			0.381596i	0.981629	+	0.190798i	$0.0611075\pi$
$971$			370.530i			0.381596i	−0.981629	+	0.190798i	$0.938892\pi$
$972$		0			0
$973$		−	945.141i		−	0.971368i
$974$	1697.63	+	470.417i	1.74295	+	0.482974i
$975$		0			0
$976$	427.077	+	801.615i	0.437578	+	0.821327i
$977$		−	1402.53i		−	1.43555i	−0.696278	−	0.717773i	$-0.745162\pi$
$977$		−	1402.53i		−	1.43555i	0.696278	−	0.717773i	$-0.254838\pi$
$978$		0			0
$979$		−	797.746i		−	0.814858i
$980$		0			0
$981$		0			0
$982$	−46.1693	+	166.615i	−0.0470156	+	0.169669i
$983$	988.944			1.00605			0.503023	−	0.864273i	$-0.332221\pi$
$983$	988.944			1.00605			0.503023	+	0.864273i	$0.332221\pi$
$984$		0			0
$985$		0			0
$986$	18.6171	−	67.1849i	0.0188814	−	0.0681388i
$987$		0			0
$988$	−675.624	−	405.576i	−0.683830	−	0.410502i
$989$	−82.6603			−0.0835797
$990$		0			0
$991$			90.1483i			0.0909670i	0.998965	+	0.0454835i	$0.0144828\pi$
$991$			90.1483i			0.0909670i	−0.998965	+	0.0454835i	$0.985517\pi$
$992$	−510.575	+	113.745i	−0.514693	+	0.114663i
$993$		0			0
$994$	253.605	−	915.205i	0.255136	−	0.920729i
$995$		0			0
$996$		0			0
$997$		−	655.605i		−	0.657578i	−0.944403	−	0.328789i	$-0.893360\pi$
$997$		−	655.605i		−	0.657578i	0.944403	−	0.328789i	$-0.106640\pi$
$998$	−467.071	+	1685.56i	−0.468007	+	1.68893i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.f.h.199.3		16
3.2	odd	2		300.3.f.c.199.14		16
4.3	odd	2	inner	900.3.f.h.199.13		16
5.2	odd	4		900.3.c.t.451.5		8
5.3	odd	4		900.3.c.n.451.4		8
5.4	even	2	inner	900.3.f.h.199.14		16
12.11	even	2		300.3.f.c.199.4		16
15.2	even	4		300.3.c.e.151.4	yes	8
15.8	even	4		300.3.c.g.151.5	yes	8
15.14	odd	2		300.3.f.c.199.3		16
20.3	even	4		900.3.c.n.451.3		8
20.7	even	4		900.3.c.t.451.6		8
20.19	odd	2	inner	900.3.f.h.199.4		16
60.23	odd	4		300.3.c.g.151.6	yes	8
60.47	odd	4		300.3.c.e.151.3	✓	8
60.59	even	2		300.3.f.c.199.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
300.3.c.e.151.3	✓	8	60.47	odd	4
300.3.c.e.151.4	yes	8	15.2	even	4
300.3.c.g.151.5	yes	8	15.8	even	4
300.3.c.g.151.6	yes	8	60.23	odd	4
300.3.f.c.199.3		16	15.14	odd	2
300.3.f.c.199.4		16	12.11	even	2
300.3.f.c.199.13		16	60.59	even	2
300.3.f.c.199.14		16	3.2	odd	2
900.3.c.n.451.3		8	20.3	even	4
900.3.c.n.451.4		8	5.3	odd	4
900.3.c.t.451.5		8	5.2	odd	4
900.3.c.t.451.6		8	20.7	even	4
900.3.f.h.199.3		16	1.1	even	1	trivial
900.3.f.h.199.4		16	20.19	odd	2	inner
900.3.f.h.199.13		16	4.3	odd	2	inner
900.3.f.h.199.14		16	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.92737 - 0.534079i) q^{2} +(3.42952 + 2.05874i) q^{4} -11.9716 q^{7} +(-5.51043 - 5.79958i) q^{8} +O(q^{10})\)	\(q+(-1.92737 - 0.534079i) q^{2} +(3.42952 + 2.05874i) q^{4} -11.9716 q^{7} +(-5.51043 - 5.79958i) q^{8} -14.5382i q^{11} +22.4802i q^{13} +(23.0738 + 6.39379i) q^{14} +(7.52322 + 14.1209i) q^{16} +12.6890i q^{17} +8.76336i q^{19} +(-7.76455 + 28.0205i) q^{22} +4.99653 q^{23} +(12.0062 - 43.3278i) q^{26} +(-41.0570 - 24.6464i) q^{28} +2.74712 q^{29} -16.3466i q^{31} +(-6.95833 - 31.2343i) q^{32} +(6.77695 - 24.4565i) q^{34} -32.4872i q^{37} +(4.68032 - 16.8902i) q^{38} -42.7586 q^{41} -16.5435 q^{43} +(29.9303 - 49.8591i) q^{44} +(-9.63018 - 2.66854i) q^{46} +48.5912 q^{47} +94.3200 q^{49} +(-46.2809 + 77.0964i) q^{52} -94.1066i q^{53} +(65.9689 + 69.4305i) q^{56} +(-5.29471 - 1.46718i) q^{58} +43.2650i q^{59} +56.7678 q^{61} +(-8.73038 + 31.5060i) q^{62} +(-3.27028 + 63.9164i) q^{64} +61.1106 q^{67} +(-26.1234 + 43.5173i) q^{68} -39.6643i q^{71} -99.5452i q^{73} +(-17.3507 + 62.6149i) q^{74} +(-18.0414 + 30.0541i) q^{76} +174.046i q^{77} +10.7780i q^{79} +(82.4118 + 22.8365i) q^{82} -140.263 q^{83} +(31.8855 + 8.83554i) q^{86} +(-84.3156 + 80.1118i) q^{88} +54.8723 q^{89} -269.125i q^{91} +(17.1357 + 10.2865i) q^{92} +(-93.6533 - 25.9515i) q^{94} -14.1601i q^{97} +(-181.790 - 50.3743i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{4}+O(q^{10}) \) 16 * q + 16 * q^4	\( 16 q + 16 q^{4} + 44 q^{14} + 80 q^{16} + 132 q^{26} - 64 q^{29} - 248 q^{34} + 32 q^{41} + 80 q^{44} - 152 q^{46} - 32 q^{49} + 344 q^{56} + 272 q^{61} - 32 q^{64} - 216 q^{74} + 240 q^{76} - 428 q^{86} + 256 q^{89} - 24 q^{94}+O(q^{100}) \) 16 * q + 16 * q^4 + 44 * q^14 + 80 * q^16 + 132 * q^26 - 64 * q^29 - 248 * q^34 + 32 * q^41 + 80 * q^44 - 152 * q^46 - 32 * q^49 + 344 * q^56 + 272 * q^61 - 32 * q^64 - 216 * q^74 + 240 * q^76 - 428 * q^86 + 256 * q^89 - 24 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more