LMFDB - Embedded newform 2205.2.g.b.2204.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,2,Mod(2204,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2205.2204");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2205.g (of order $2$, degree $1$, 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:	$17.6070136457$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 315)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2204.16
Character	$\chi$	$=$	2205.2204
Dual form			2205.2.g.b.2204.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.31841 q^{2} -0.261802 q^{4} +(2.19538 - 0.424645i) q^{5} -2.98198 q^{8} +(2.89440 - 0.559855i) q^{10} -2.40972i q^{11} -1.69332 q^{13} -3.40786 q^{16} +0.960462i q^{17} -4.11804i q^{19} +(-0.574754 + 0.111173i) q^{20} -3.17699i q^{22} +5.77140 q^{23} +(4.63935 - 1.86451i) q^{25} -2.23249 q^{26} -5.56553i q^{29} -8.75434i q^{31} +1.47101 q^{32} +1.26628i q^{34} -3.97309i q^{37} -5.42925i q^{38} +(-6.54656 + 1.26628i) q^{40} -1.87474 q^{41} +10.2706i q^{43} +0.630869i q^{44} +7.60905 q^{46} -10.1075i q^{47} +(6.11656 - 2.45818i) q^{50} +0.443316 q^{52} +12.7086 q^{53} +(-1.02327 - 5.29024i) q^{55} -7.33764i q^{58} -6.77371 q^{59} -2.29191i q^{61} -11.5418i q^{62} +8.75510 q^{64} +(-3.71748 + 0.719061i) q^{65} -4.79743i q^{67} -0.251451i q^{68} +2.24602i q^{71} -14.6987 q^{73} -5.23814i q^{74} +1.07811i q^{76} -1.78541 q^{79} +(-7.48152 + 1.44713i) q^{80} -2.47166 q^{82} +9.62086i q^{83} +(0.407855 + 2.10858i) q^{85} +13.5408i q^{86} +7.18572i q^{88} -0.441699 q^{89} -1.51096 q^{92} -13.3258i q^{94} +(-1.74870 - 9.04065i) q^{95} +12.4926 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} - 24 q^{25} + 48 q^{46} + 48 q^{64} + 120 q^{79} - 72 q^{85}+O(q^{100}) $ 24 * q + 48 * q^4 - 24 * q^25 + 48 * q^46 + 48 * q^64 + 120 * q^79 - 72 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.31841			0.932255			0.466127	−	0.884718i	$-0.345649\pi$
$2$	1.31841			0.932255			0.466127	+	0.884718i	$0.345649\pi$
$3$		0			0
$3$		0			0
$4$	−0.261802			−0.130901
$5$	2.19538	−	0.424645i	0.981802	−	0.189907i
$5$	2.19538	−	0.424645i	0.981802	−	0.189907i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−2.98198			−1.05429
$9$		0			0
$10$	2.89440	−	0.559855i	0.915290	−	0.177042i
$11$		−	2.40972i		−	0.726557i	−0.931681	−	0.363279i	$-0.881657\pi$
$11$		−	2.40972i		−	0.726557i	0.931681	−	0.363279i	$-0.118343\pi$
$12$		0			0
$13$	−1.69332			−0.469643			−0.234822	−	0.972038i	$-0.575451\pi$
$13$	−1.69332			−0.469643			−0.234822	+	0.972038i	$0.575451\pi$
$14$		0			0
$15$		0			0
$16$	−3.40786			−0.851964
$17$			0.960462i			0.232946i	0.993194	+	0.116473i	$0.0371589\pi$
$17$			0.960462i			0.232946i	−0.993194	+	0.116473i	$0.962841\pi$
$18$		0			0
$19$		−	4.11804i		−	0.944743i	−0.881399	−	0.472372i	$-0.843398\pi$
$19$		−	4.11804i		−	0.944743i	0.881399	−	0.472372i	$-0.156602\pi$
$20$	−0.574754	+	0.111173i	−0.128519	+	0.0248590i
$21$		0			0
$22$		−	3.17699i		−	0.677336i
$23$	5.77140			1.20342			0.601710	−	0.798715i	$-0.294486\pi$
$23$	5.77140			1.20342			0.601710	+	0.798715i	$0.294486\pi$
$24$		0			0
$25$	4.63935	−	1.86451i	0.927871	−	0.372902i
$26$	−2.23249			−0.437827
$27$		0			0
$28$		0			0
$29$		−	5.56553i		−	1.03349i	−0.856138	−	0.516747i	$-0.827143\pi$
$29$		−	5.56553i		−	1.03349i	0.856138	−	0.516747i	$-0.172857\pi$
$30$		0			0
$31$		−	8.75434i		−	1.57233i	−0.618019	−	0.786163i	$-0.712065\pi$
$31$		−	8.75434i		−	1.57233i	0.618019	−	0.786163i	$-0.287935\pi$
$32$	1.47101			0.260041
$33$		0			0
$34$			1.26628i			0.217165i
$35$		0			0
$36$		0			0
$37$		−	3.97309i		−	0.653171i	−0.945168	−	0.326586i	$-0.894102\pi$
$37$		−	3.97309i		−	0.653171i	0.945168	−	0.326586i	$-0.105898\pi$
$38$		−	5.42925i		−	0.880741i
$39$		0			0
$40$	−6.54656	+	1.26628i	−1.03510	+	0.200216i
$41$	−1.87474			−0.292784			−0.146392	−	0.989227i	$-0.546766\pi$
$41$	−1.87474			−0.292784			−0.146392	+	0.989227i	$0.546766\pi$
$42$		0			0
$43$			10.2706i			1.56625i	0.621867	+	0.783123i	$0.286375\pi$
$43$			10.2706i			1.56625i	−0.621867	+	0.783123i	$0.713625\pi$
$44$			0.630869i			0.0951071i
$45$		0			0
$46$	7.60905			1.12189
$47$		−	10.1075i		−	1.47433i	−0.675711	−	0.737167i	$-0.736163\pi$
$47$		−	10.1075i		−	1.47433i	0.675711	−	0.737167i	$-0.263837\pi$
$48$		0			0
$49$		0			0
$50$	6.11656	−	2.45818i	0.865012	−	0.347640i
$51$		0			0
$52$	0.443316			0.0614769
$53$	12.7086			1.74566			0.872830	−	0.488025i	$-0.162282\pi$
$53$	12.7086			1.74566			0.872830	+	0.488025i	$0.162282\pi$
$54$		0			0
$55$	−1.02327	−	5.29024i	−0.137978	−	0.713335i
$56$		0			0
$57$		0			0
$58$		−	7.33764i		−	0.963479i
$59$	−6.77371			−0.881863			−0.440931	−	0.897541i	$-0.645352\pi$
$59$	−6.77371			−0.881863			−0.440931	+	0.897541i	$0.645352\pi$
$60$		0			0
$61$		−	2.29191i		−	0.293448i	−0.989177	−	0.146724i	$-0.953127\pi$
$61$		−	2.29191i		−	0.293448i	0.989177	−	0.146724i	$-0.0468730\pi$
$62$		−	11.5418i		−	1.46581i
$63$		0			0
$64$	8.75510			1.09439
$65$	−3.71748	+	0.719061i	−0.461097	+	0.0891885i
$66$		0			0
$67$		−	4.79743i		−	0.586100i	−0.956097	−	0.293050i	$-0.905330\pi$
$67$		−	4.79743i		−	0.586100i	0.956097	−	0.293050i	$-0.0946702\pi$
$68$		−	0.251451i		−	0.0304929i
$69$		0			0
$70$		0			0
$71$			2.24602i			0.266554i	0.991079	+	0.133277i	$0.0425500\pi$
$71$			2.24602i			0.266554i	−0.991079	+	0.133277i	$0.957450\pi$
$72$		0			0
$73$	−14.6987			−1.72036			−0.860178	−	0.509993i	$-0.829648\pi$
$73$	−14.6987			−1.72036			−0.860178	+	0.509993i	$0.829648\pi$
$74$		−	5.23814i		−	0.608922i
$75$		0			0
$76$			1.07811i			0.123668i
$77$		0			0
$78$		0			0
$79$	−1.78541			−0.200874			−0.100437	−	0.994943i	$-0.532024\pi$
$79$	−1.78541			−0.200874			−0.100437	+	0.994943i	$0.532024\pi$
$80$	−7.48152	+	1.44713i	−0.836460	+	0.161794i
$81$		0			0
$82$	−2.47166			−0.272950
$83$			9.62086i			1.05603i	0.849236	+	0.528013i	$0.177063\pi$
$83$			9.62086i			1.05603i	−0.849236	+	0.528013i	$0.822937\pi$
$84$		0			0
$85$	0.407855	+	2.10858i	0.0442381	+	0.228707i
$86$			13.5408i			1.46014i
$87$		0			0
$88$			7.18572i			0.766000i
$89$	−0.441699			−0.0468201			−0.0234100	−	0.999726i	$-0.507452\pi$
$89$	−0.441699			−0.0468201			−0.0234100	+	0.999726i	$0.507452\pi$
$90$		0			0
$91$		0			0
$92$	−1.51096			−0.157529
$93$		0			0
$94$		−	13.3258i		−	1.37445i
$95$	−1.74870	−	9.04065i	−0.179413	−	0.927551i
$96$		0			0
$97$	12.4926			1.26843			0.634215	−	0.773157i	$-0.281323\pi$
$97$	12.4926			1.26843			0.634215	+	0.773157i	$0.281323\pi$
$98$		0			0
$99$		0			0
$100$	−1.21459	+	0.488133i	−0.121459	+	0.0488133i
$101$	−9.48931			−0.944222			−0.472111	−	0.881539i	$-0.656508\pi$
$101$	−9.48931			−0.944222			−0.472111	+	0.881539i	$0.656508\pi$
$102$		0			0
$103$	8.08266			0.796408			0.398204	−	0.917297i	$-0.369634\pi$
$103$	8.08266			0.796408			0.398204	+	0.917297i	$0.369634\pi$
$104$	5.04945			0.495139
$105$		0			0
$106$	16.7551			1.62740
$107$	−5.42623			−0.524574			−0.262287	−	0.964990i	$-0.584477\pi$
$107$	−5.42623			−0.524574			−0.262287	+	0.964990i	$0.584477\pi$
$108$		0			0
$109$	17.3642			1.66318			0.831592	−	0.555387i	$-0.187430\pi$
$109$	17.3642			1.66318			0.831592	+	0.555387i	$0.187430\pi$
$110$	−1.34909	−	6.97469i	−0.128631	−	0.665010i
$111$		0			0
$112$		0			0
$113$	−5.92400			−0.557283			−0.278642	−	0.960395i	$-0.589884\pi$
$113$	−5.92400			−0.557283			−0.278642	+	0.960395i	$0.589884\pi$
$114$		0			0
$115$	12.6704	−	2.45079i	1.18152	−	0.228538i
$116$			1.45707i			0.135285i
$117$		0			0
$118$	−8.93051			−0.822121
$119$		0			0
$120$		0			0
$121$	5.19326			0.472115
$122$		−	3.02166i		−	0.273569i
$123$		0			0
$124$			2.29191i			0.205819i
$125$	9.39337	−	6.06338i	0.840169	−	0.542325i
$126$		0			0
$127$		−	5.01325i		−	0.444854i	−0.974949	−	0.222427i	$-0.928602\pi$
$127$		−	5.01325i		−	0.444854i	0.974949	−	0.222427i	$-0.0713979\pi$
$128$	8.60077			0.760208
$129$		0			0
$130$	−4.90116	+	0.948015i	−0.429860	+	0.0831464i
$131$	−17.0548			−1.49008			−0.745042	−	0.667018i	$-0.767570\pi$
$131$	−17.0548			−1.49008			−0.745042	+	0.667018i	$0.767570\pi$
$132$		0			0
$133$		0			0
$134$		−	6.32497i		−	0.546394i
$135$		0			0
$136$		−	2.86408i		−	0.245592i
$137$	−9.91917			−0.847452			−0.423726	−	0.905790i	$-0.639278\pi$
$137$	−9.91917			−0.847452			−0.423726	+	0.905790i	$0.639278\pi$
$138$		0			0
$139$		−	15.1104i		−	1.28164i	−0.767689	−	0.640822i	$-0.778593\pi$
$139$		−	15.1104i		−	1.28164i	0.767689	−	0.640822i	$-0.221407\pi$
$140$		0			0
$141$		0			0
$142$			2.96117i			0.248496i
$143$			4.08043i			0.341223i
$144$		0			0
$145$	−2.36337	−	12.2184i	−0.196268	−	1.01469i
$146$	−19.3789			−1.60381
$147$		0			0
$148$			1.04016i			0.0855008i
$149$			2.33734i			0.191483i	0.995406	+	0.0957413i	$0.0305222\pi$
$149$			2.33734i			0.191483i	−0.995406	+	0.0957413i	$0.969478\pi$
$150$		0			0
$151$	10.0303			0.816255			0.408127	−	0.912925i	$-0.366182\pi$
$151$	10.0303			0.816255			0.408127	+	0.912925i	$0.366182\pi$
$152$			12.2799i			0.996031i
$153$		0			0
$154$		0			0
$155$	−3.71748	−	19.2191i	−0.298595	−	1.54371i
$156$		0			0
$157$	20.7530			1.65627			0.828137	−	0.560526i	$-0.189401\pi$
$157$	20.7530			1.65627			0.828137	+	0.560526i	$0.189401\pi$
$158$	−2.35389			−0.187266
$159$		0			0
$160$	3.22943	−	0.624657i	0.255308	−	0.0493835i
$161$		0			0
$162$		0			0
$163$			15.6200i			1.22345i	0.791069	+	0.611727i	$0.209525\pi$
$163$			15.6200i			1.22345i	−0.791069	+	0.611727i	$0.790475\pi$
$164$	0.490810			0.0383258
$165$		0			0
$166$			12.6842i			0.984486i
$167$			14.7858i			1.14416i	0.820198	+	0.572080i	$0.193863\pi$
$167$			14.7858i			1.14416i	−0.820198	+	0.572080i	$0.806137\pi$
$168$		0			0
$169$	−10.1327			−0.779435
$170$	0.537719	+	2.77996i	0.0412412	+	0.213213i
$171$		0			0
$172$		−	2.68886i		−	0.205023i
$173$			6.86351i			0.521823i	0.965363	+	0.260912i	$0.0840231\pi$
$173$			6.86351i			0.521823i	−0.965363	+	0.260912i	$0.915977\pi$
$174$		0			0
$175$		0			0
$176$			8.21197i			0.619000i
$177$		0			0
$178$	−0.582340			−0.0436482
$179$		−	11.7079i		−	0.875087i	−0.899197	−	0.437543i	$-0.855849\pi$
$179$		−	11.7079i		−	0.875087i	0.899197	−	0.437543i	$-0.144151\pi$
$180$		0			0
$181$			17.3498i			1.28960i	0.764351	+	0.644800i	$0.223059\pi$
$181$			17.3498i			1.28960i	−0.764351	+	0.644800i	$0.776941\pi$
$182$		0			0
$183$		0			0
$184$	−17.2102			−1.26875
$185$	−1.68715	−	8.72242i	−0.124042	−	0.641285i
$186$		0			0
$187$	2.31444			0.169249
$188$			2.64617i			0.192992i
$189$		0			0
$190$	−2.30550	−	11.9193i	−0.167259	−	0.864714i
$191$		−	18.8998i		−	1.36754i	−0.729698	−	0.683770i	$-0.760339\pi$
$191$		−	18.8998i		−	1.36754i	0.729698	−	0.683770i	$-0.239661\pi$
$192$		0			0
$193$			23.3221i			1.67876i	0.543542	+	0.839382i	$0.317083\pi$
$193$			23.3221i			1.67876i	−0.543542	+	0.839382i	$0.682917\pi$
$194$	16.4703			1.18250
$195$		0			0
$196$		0			0
$197$	0.842929			0.0600562			0.0300281	−	0.999549i	$-0.490440\pi$
$197$	0.842929			0.0600562			0.0300281	+	0.999549i	$0.490440\pi$
$198$		0			0
$199$		−	0.854422i		−	0.0605684i	−0.999541	−	0.0302842i	$-0.990359\pi$
$199$		−	0.854422i		−	0.0605684i	0.999541	−	0.0302842i	$-0.00964123\pi$
$200$	−13.8344	+	5.55992i	−0.978243	+	0.393146i
$201$		0			0
$202$	−12.5108			−0.880255
$203$		0			0
$204$		0			0
$205$	−4.11575	+	0.796096i	−0.287456	+	0.0556018i
$206$	10.6562			0.742455
$207$		0			0
$208$	5.77060			0.400119
$209$	−9.92331			−0.686410
$210$		0			0
$211$	16.1933			1.11479			0.557395	−	0.830247i	$-0.311801\pi$
$211$	16.1933			1.11479			0.557395	+	0.830247i	$0.311801\pi$
$212$	−3.32714			−0.228509
$213$		0			0
$214$	−7.15399			−0.489036
$215$	4.36134	+	22.5477i	0.297441	+	1.53774i
$216$		0			0
$217$		0			0
$218$	22.8930			1.55051
$219$		0			0
$220$	0.267895	+	1.38500i	0.0180615	+	0.0933764i
$221$		−	1.62637i		−	0.109402i
$222$		0			0
$223$	−10.0027			−0.669832			−0.334916	−	0.942248i	$-0.608708\pi$
$223$	−10.0027			−0.669832			−0.334916	+	0.942248i	$0.608708\pi$
$224$		0			0
$225$		0			0
$226$	−7.81025			−0.519530
$227$		−	4.67827i		−	0.310508i	−0.987875	−	0.155254i	$-0.950380\pi$
$227$		−	4.67827i		−	0.310508i	0.987875	−	0.155254i	$-0.0496195\pi$
$228$		0			0
$229$		−	14.2559i		−	0.942060i	−0.882117	−	0.471030i	$-0.843882\pi$
$229$		−	14.2559i		−	0.942060i	0.882117	−	0.471030i	$-0.156118\pi$
$230$	16.7047	−	3.23114i	1.10148	−	0.213055i
$231$		0			0
$232$			16.5963i			1.08960i
$233$	−22.3049			−1.46124			−0.730621	−	0.682783i	$-0.760769\pi$
$233$	−22.3049			−1.46124			−0.730621	+	0.682783i	$0.760769\pi$
$234$		0			0
$235$	−4.29211	−	22.1898i	−0.279986	−	1.44750i
$236$	1.77337			0.115437
$237$		0			0
$238$		0			0
$239$			6.39734i			0.413810i	0.978361	+	0.206905i	$0.0663391\pi$
$239$			6.39734i			0.413810i	−0.978361	+	0.206905i	$0.933661\pi$
$240$		0			0
$241$		−	7.16886i		−	0.461787i	−0.972979	−	0.230893i	$-0.925835\pi$
$241$		−	7.16886i		−	0.461787i	0.972979	−	0.230893i	$-0.0741649\pi$
$242$	6.84683			0.440131
$243$		0			0
$244$			0.600026i			0.0384127i
$245$		0			0
$246$		0			0
$247$			6.97318i			0.443693i
$248$			26.1052i			1.65768i
$249$		0			0
$250$	12.3843	−	7.99400i	0.783251	−	0.505585i
$251$	−16.3470			−1.03181			−0.515906	−	0.856645i	$-0.672545\pi$
$251$	−16.3470			−1.03181			−0.515906	+	0.856645i	$0.672545\pi$
$252$		0			0
$253$		−	13.9074i		−	0.874353i
$254$		−	6.60950i		−	0.414717i
$255$		0			0
$256$	−6.17089			−0.385681
$257$			12.2379i			0.763381i	0.924290	+	0.381690i	$0.124658\pi$
$257$			12.2379i			0.763381i	−0.924290	+	0.381690i	$0.875342\pi$
$258$		0			0
$259$		0			0
$260$	0.973245	−	0.188252i	0.0603581	−	0.0116749i
$261$		0			0
$262$	−22.4852			−1.38914
$263$	9.42141			0.580949			0.290474	−	0.956883i	$-0.406187\pi$
$263$	9.42141			0.580949			0.290474	+	0.956883i	$0.406187\pi$
$264$		0			0
$265$	27.9001	−	5.39664i	1.71389	−	0.331513i
$266$		0			0
$267$		0			0
$268$			1.25598i			0.0767211i
$269$	−7.63200			−0.465331			−0.232666	−	0.972557i	$-0.574745\pi$
$269$	−7.63200			−0.465331			−0.232666	+	0.972557i	$0.574745\pi$
$270$		0			0
$271$			5.44912i			0.331011i	0.986209	+	0.165505i	$0.0529255\pi$
$271$			5.44912i			0.331011i	−0.986209	+	0.165505i	$0.947074\pi$
$272$		−	3.27312i		−	0.198462i
$273$		0			0
$274$	−13.0775			−0.790041
$275$	−4.49294	−	11.1795i	−0.270935	−	0.674151i
$276$		0			0
$277$			17.0562i			1.02481i	0.858745	+	0.512403i	$0.171245\pi$
$277$			17.0562i			1.02481i	−0.858745	+	0.512403i	$0.828755\pi$
$278$		−	19.9216i		−	1.19482i
$279$		0			0
$280$		0			0
$281$			25.3828i			1.51421i	0.653292	+	0.757106i	$0.273387\pi$
$281$			25.3828i			1.51421i	−0.653292	+	0.757106i	$0.726613\pi$
$282$		0			0
$283$	−9.57998			−0.569471			−0.284735	−	0.958606i	$-0.591906\pi$
$283$	−9.57998			−0.569471			−0.284735	+	0.958606i	$0.591906\pi$
$284$		−	0.588014i		−	0.0348922i
$285$		0			0
$286$			5.37967i			0.318107i
$287$		0			0
$288$		0			0
$289$	16.0775			0.945736
$290$	−3.11589	−	16.1089i	−0.182971	−	0.945946i
$291$		0			0
$292$	3.84816			0.225197
$293$		−	1.44713i		−	0.0845421i	−0.999106	−	0.0422710i	$-0.986541\pi$
$293$		−	1.44713i		−	0.0845421i	0.999106	−	0.0422710i	$-0.0134593\pi$
$294$		0			0
$295$	−14.8709	+	2.87642i	−0.865815	+	0.167472i
$296$			11.8476i			0.688631i
$297$		0			0
$298$			3.08157i			0.178511i
$299$	−9.77284			−0.565178
$300$		0			0
$301$		0			0
$302$	13.2240			0.760957
$303$		0			0
$304$			14.0337i			0.804887i
$305$	−0.973245	−	5.03159i	−0.0557279	−	0.288108i
$306$		0			0
$307$	7.69871			0.439388			0.219694	−	0.975569i	$-0.429494\pi$
$307$	7.69871			0.439388			0.219694	+	0.975569i	$0.429494\pi$
$308$		0			0
$309$		0			0
$310$	−4.90116	−	25.3386i	−0.278367	−	1.43913i
$311$	33.1750			1.88118			0.940592	−	0.339539i	$-0.110271\pi$
$311$	33.1750			1.88118			0.940592	+	0.339539i	$0.110271\pi$
$312$		0			0
$313$	8.45647			0.477988			0.238994	−	0.971021i	$-0.423182\pi$
$313$	8.45647			0.477988			0.238994	+	0.971021i	$0.423182\pi$
$314$	27.3610			1.54407
$315$		0			0
$316$	0.467423			0.0262946
$317$	−5.94166			−0.333717			−0.166859	−	0.985981i	$-0.553362\pi$
$317$	−5.94166			−0.333717			−0.166859	+	0.985981i	$0.553362\pi$
$318$		0			0
$319$	−13.4114			−0.750892
$320$	19.2207	−	3.71781i	1.07447	−	0.207832i
$321$		0			0
$322$		0			0
$323$	3.95522			0.220074
$324$		0			0
$325$	−7.85593	+	3.15722i	−0.435768	+	0.175131i
$326$			20.5936i			1.14057i
$327$		0			0
$328$	5.59042			0.308679
$329$		0			0
$330$		0			0
$331$	26.5102			1.45713			0.728566	−	0.684975i	$-0.240187\pi$
$331$	26.5102			1.45713			0.728566	+	0.684975i	$0.240187\pi$
$332$		−	2.51876i		−	0.138235i
$333$		0			0
$334$			19.4937i			1.06665i
$335$	−2.03720	−	10.5322i	−0.111304	−	0.575434i
$336$		0			0
$337$			3.42106i			0.186357i	0.995649	+	0.0931784i	$0.0297027\pi$
$337$			3.42106i			0.186357i	−0.995649	+	0.0931784i	$0.970297\pi$
$338$	−13.3590			−0.726632
$339$		0			0
$340$	−0.106777	−	0.552030i	−0.00579082	−	0.0299380i
$341$	−21.0955			−1.14238
$342$		0			0
$343$		0			0
$344$		−	30.6266i		−	1.65127i
$345$		0			0
$346$			9.04891i			0.486472i
$347$	−13.3989			−0.719292			−0.359646	−	0.933089i	$-0.617102\pi$
$347$	−13.3989			−0.719292			−0.359646	+	0.933089i	$0.617102\pi$
$348$		0			0
$349$			15.2584i			0.816762i	0.912812	+	0.408381i	$0.133906\pi$
$349$			15.2584i			0.816762i	−0.912812	+	0.408381i	$0.866094\pi$
$350$		0			0
$351$		0			0
$352$		−	3.54472i		−	0.188934i
$353$			36.9218i			1.96515i	0.185874	+	0.982574i	$0.440488\pi$
$353$			36.9218i			1.96515i	−0.185874	+	0.982574i	$0.559512\pi$
$354$		0			0
$355$	0.953762	+	4.93087i	0.0506204	+	0.261703i
$356$	0.115638			0.00612880
$357$		0			0
$358$		−	15.4357i		−	0.815804i
$359$		−	8.90399i		−	0.469935i	−0.972003	−	0.234967i	$-0.924502\pi$
$359$		−	8.90399i		−	0.469935i	0.972003	−	0.234967i	$-0.0754983\pi$
$360$		0			0
$361$	2.04174			0.107460
$362$			22.8741i			1.20224i
$363$		0			0
$364$		0			0
$365$	−32.2693	+	6.24174i	−1.68905	+	0.326708i
$366$		0			0
$367$	14.4538			0.754482			0.377241	−	0.926115i	$-0.376873\pi$
$367$	14.4538			0.754482			0.377241	+	0.926115i	$0.376873\pi$
$368$	−19.6681			−1.02527
$369$		0			0
$370$	−2.22435	−	11.4997i	−0.115638	−	0.597841i
$371$		0			0
$372$		0			0
$373$			21.1214i			1.09362i	0.837255	+	0.546812i	$0.184159\pi$
$373$			21.1214i			1.09362i	−0.837255	+	0.546812i	$0.815841\pi$
$374$	3.05138			0.157783
$375$		0			0
$376$			30.1404i			1.55437i
$377$			9.42425i			0.485374i
$378$		0			0
$379$	−0.0133979			−0.000688205			−0.000344103	−	1.00000i	$-0.500110\pi$
$379$	−0.0133979			−0.000688205			−0.000344103		1.00000i	$0.500110\pi$
$380$	0.457815	+	2.36686i	0.0234854	+	0.121417i
$381$		0			0
$382$		−	24.9176i		−	1.27489i
$383$		−	17.2225i		−	0.880028i	−0.897991	−	0.440014i	$-0.854973\pi$
$383$		−	17.2225i		−	0.880028i	0.897991	−	0.440014i	$-0.145027\pi$
$384$		0			0
$385$		0			0
$386$			30.7481i			1.56504i
$387$		0			0
$388$	−3.27059			−0.166039
$389$			25.7587i			1.30602i	0.757351	+	0.653008i	$0.226493\pi$
$389$			25.7587i			1.30602i	−0.757351	+	0.653008i	$0.773507\pi$
$390$		0			0
$391$			5.54321i			0.280332i
$392$		0			0
$393$		0			0
$394$	1.11132			0.0559877
$395$	−3.91964	+	0.758163i	−0.197218	+	0.0381473i
$396$		0			0
$397$	−15.2480			−0.765273			−0.382637	−	0.923899i	$-0.624984\pi$
$397$	−15.2480			−0.765273			−0.382637	+	0.923899i	$0.624984\pi$
$398$		−	1.12648i		−	0.0564652i
$399$		0			0
$400$	−15.8102	+	6.35398i	−0.790512	+	0.317699i
$401$			13.1221i			0.655285i	0.944802	+	0.327643i	$0.106254\pi$
$401$			13.1221i			0.655285i	−0.944802	+	0.327643i	$0.893746\pi$
$402$		0			0
$403$			14.8239i			0.738432i
$404$	2.48432			0.123600
$405$		0			0
$406$		0			0
$407$	−9.57401			−0.474566
$408$		0			0
$409$			32.8734i			1.62549i	0.582622	+	0.812744i	$0.302027\pi$
$409$			32.8734i			1.62549i	−0.582622	+	0.812744i	$0.697973\pi$
$410$	−5.42623	+	1.04958i	−0.267983	+	0.0518350i
$411$		0			0
$412$	−2.11606			−0.104251
$413$		0			0
$414$		0			0
$415$	4.08545	+	21.1214i	0.200547	+	1.03681i
$416$	−2.49090			−0.122126
$417$		0			0
$418$	−13.0830			−0.639909
$419$	9.33114			0.455856			0.227928	−	0.973678i	$-0.426805\pi$
$419$	9.33114			0.455856			0.227928	+	0.973678i	$0.426805\pi$
$420$		0			0
$421$	−13.0720			−0.637093			−0.318546	−	0.947907i	$-0.603195\pi$
$421$	−13.0720			−0.637093			−0.318546	+	0.947907i	$0.603195\pi$
$422$	21.3493			1.03927
$423$		0			0
$424$	−37.8967			−1.84043
$425$	1.79079	+	4.45592i	0.0868661	+	0.216144i
$426$		0			0
$427$		0			0
$428$	1.42060			0.0686673
$429$		0			0
$430$	5.75002	+	29.7271i	0.277291	+	1.43357i
$431$			5.98424i			0.288251i	0.989559	+	0.144125i	$0.0460369\pi$
$431$			5.98424i			0.288251i	−0.989559	+	0.144125i	$0.953963\pi$
$432$		0			0
$433$	−19.4335			−0.933916			−0.466958	−	0.884280i	$-0.654650\pi$
$433$	−19.4335			−0.933916			−0.466958	+	0.884280i	$0.654650\pi$
$434$		0			0
$435$		0			0
$436$	−4.54597			−0.217713
$437$		−	23.7668i		−	1.13692i
$438$		0			0
$439$		−	27.8485i		−	1.32914i	−0.747228	−	0.664568i	$-0.768616\pi$
$439$		−	27.8485i		−	1.32914i	0.747228	−	0.664568i	$-0.231384\pi$
$440$	3.05138	+	15.7754i	0.145469	+	0.752061i
$441$		0			0
$442$		−	2.14422i		−	0.101990i
$443$	23.4307			1.11323			0.556614	−	0.830771i	$-0.312100\pi$
$443$	23.4307			1.11323			0.556614	+	0.830771i	$0.312100\pi$
$444$		0			0
$445$	−0.969697	+	0.187565i	−0.0459680	+	0.00889145i
$446$	−13.1877			−0.624454
$447$		0			0
$448$		0			0
$449$			15.7868i			0.745025i	0.928027	+	0.372513i	$0.121504\pi$
$449$			15.7868i			0.745025i	−0.928027	+	0.372513i	$0.878496\pi$
$450$		0			0
$451$			4.51758i			0.212725i
$452$	1.55092			0.0729490
$453$		0			0
$454$		−	6.16786i		−	0.289472i
$455$		0			0
$456$		0			0
$457$		−	26.7149i		−	1.24967i	−0.780756	−	0.624836i	$-0.785166\pi$
$457$		−	26.7149i		−	1.24967i	0.780756	−	0.624836i	$-0.214834\pi$
$458$		−	18.7951i		−	0.878240i
$459$		0			0
$460$	−3.31714	+	0.641623i	−0.154662	+	0.0299158i
$461$	−33.1750			−1.54512			−0.772558	−	0.634944i	$-0.781023\pi$
$461$	−33.1750			−1.54512			−0.772558	+	0.634944i	$0.781023\pi$
$462$		0			0
$463$			9.01458i			0.418943i	0.977815	+	0.209472i	$0.0671744\pi$
$463$			9.01458i			0.418943i	−0.977815	+	0.209472i	$0.932826\pi$
$464$			18.9665i			0.880499i
$465$		0			0
$466$	−29.4069			−1.36225
$467$		−	32.4692i		−	1.50250i	−0.660020	−	0.751248i	$-0.729452\pi$
$467$		−	32.4692i		−	1.50250i	0.660020	−	0.751248i	$-0.270548\pi$
$468$		0			0
$469$		0			0
$470$	−5.65874	−	29.2552i	−0.261018	−	1.34944i
$471$		0			0
$472$	20.1991			0.929737
$473$	24.7492			1.13797
$474$		0			0
$475$	−7.67812	−	19.1050i	−0.352297	−	0.876600i
$476$		0			0
$477$		0			0
$478$			8.43430i			0.385776i
$479$	4.16605			0.190352			0.0951759	−	0.995460i	$-0.469659\pi$
$479$	4.16605			0.190352			0.0951759	+	0.995460i	$0.469659\pi$
$480$		0			0
$481$			6.72772i			0.306758i
$482$		−	9.45148i		−	0.430503i
$483$		0			0
$484$	−1.35961			−0.0618003
$485$	27.4259	−	5.30491i	1.24535	−	0.240883i
$486$		0			0
$487$		−	20.6333i		−	0.934983i	−0.883998	−	0.467491i	$-0.845158\pi$
$487$		−	20.6333i		−	0.934983i	0.883998	−	0.467491i	$-0.154842\pi$
$488$			6.83441i			0.309379i
$489$		0			0
$490$		0			0
$491$			33.6376i			1.51805i	0.651064	+	0.759023i	$0.274323\pi$
$491$			33.6376i			1.51805i	−0.651064	+	0.759023i	$0.725677\pi$
$492$		0			0
$493$	5.34548			0.240748
$494$			9.19348i			0.413634i
$495$		0			0
$496$			29.8335i			1.33956i
$497$		0			0
$498$		0			0
$499$	26.8361			1.20135			0.600675	−	0.799493i	$-0.294899\pi$
$499$	26.8361			1.20135			0.600675	+	0.799493i	$0.294899\pi$
$500$	−2.45921	+	1.58741i	−0.109979	+	0.0709909i
$501$		0			0
$502$	−21.5520			−0.961911
$503$			22.8321i			1.01803i	0.860757	+	0.509017i	$0.169991\pi$
$503$			22.8321i			1.01803i	−0.860757	+	0.509017i	$0.830009\pi$
$504$		0			0
$505$	−20.8326	+	4.02959i	−0.927039	+	0.179314i
$506$		−	18.3357i		−	0.815120i
$507$		0			0
$508$			1.31248i			0.0582319i
$509$	15.3556			0.680627			0.340313	−	0.940312i	$-0.389467\pi$
$509$	15.3556			0.680627			0.340313	+	0.940312i	$0.389467\pi$
$510$		0			0
$511$		0			0
$512$	−25.3373			−1.11976
$513$		0			0
$514$			16.1346i			0.711665i
$515$	17.7445	−	3.43226i	0.781915	−	0.151243i
$516$		0			0
$517$	−24.3563			−1.07119
$518$		0			0
$519$		0			0
$520$	11.0854	−	2.14422i	0.486129	−	0.0940304i
$521$	18.6383			0.816558			0.408279	−	0.912857i	$-0.366129\pi$
$521$	18.6383			0.816558			0.408279	+	0.912857i	$0.366129\pi$
$522$		0			0
$523$	−9.19603			−0.402114			−0.201057	−	0.979580i	$-0.564438\pi$
$523$	−9.19603			−0.402114			−0.201057	+	0.979580i	$0.564438\pi$
$524$	4.46498			0.195054
$525$		0			0
$526$	12.4213			0.541592
$527$	8.40821			0.366267
$528$		0			0
$529$	10.3090			0.448218
$530$	36.7838	−	7.11496i	1.59778	−	0.309054i
$531$		0			0
$532$		0			0
$533$	3.17453			0.137504
$534$		0			0
$535$	−11.9126	+	2.30422i	−0.515028	+	0.0996202i
$536$			14.3058i			0.617918i
$537$		0			0
$538$	−10.0621			−0.433807
$539$		0			0
$540$		0			0
$541$	−8.99207			−0.386599			−0.193300	−	0.981140i	$-0.561919\pi$
$541$	−8.99207			−0.386599			−0.193300	+	0.981140i	$0.561919\pi$
$542$			7.18416i			0.308586i
$543$		0			0
$544$			1.41285i			0.0605755i
$545$	38.1209	−	7.37359i	1.63292	−	0.315850i
$546$		0			0
$547$		−	12.8673i		−	0.550165i	−0.961421	−	0.275082i	$-0.911295\pi$
$547$		−	12.8673i		−	0.550165i	0.961421	−	0.275082i	$-0.0887051\pi$
$548$	2.59686			0.110932
$549$		0			0
$550$	−5.92353	−	14.7392i	−0.252580	−	0.628481i
$551$	−22.9191			−0.976386
$552$		0			0
$553$		0			0
$554$			22.4870i			0.955380i
$555$		0			0
$556$			3.95593i			0.167769i
$557$	−6.82455			−0.289165			−0.144583	−	0.989493i	$-0.546184\pi$
$557$	−6.82455			−0.289165			−0.144583	+	0.989493i	$0.546184\pi$
$558$		0			0
$559$		−	17.3914i		−	0.735577i
$560$		0			0
$561$		0			0
$562$			33.4649i			1.41163i
$563$		−	8.68613i		−	0.366077i	−0.983106	−	0.183038i	$-0.941407\pi$
$563$		−	8.68613i		−	0.366077i	0.983106	−	0.183038i	$-0.0585933\pi$
$564$		0			0
$565$	−13.0054	+	2.51560i	−0.547142	+	0.105832i
$566$	−12.6303			−0.530892
$567$		0			0
$568$		−	6.69759i		−	0.281025i
$569$		−	32.9583i		−	1.38168i	−0.723006	−	0.690842i	$-0.757240\pi$
$569$		−	32.9583i		−	1.38168i	0.723006	−	0.690842i	$-0.242760\pi$
$570$		0			0
$571$	8.76408			0.366765			0.183383	−	0.983042i	$-0.441295\pi$
$571$	8.76408			0.366765			0.183383	+	0.983042i	$0.441295\pi$
$572$		−	1.06827i		−	0.0446665i
$573$		0			0
$574$		0			0
$575$	26.7755	−	10.7608i	1.11662	−	0.448757i
$576$		0			0
$577$	−35.5577			−1.48029			−0.740143	−	0.672449i	$-0.765242\pi$
$577$	−35.5577			−1.48029			−0.740143	+	0.672449i	$0.765242\pi$
$578$	21.1967			0.881667
$579$		0			0
$580$	0.618737	+	3.19881i	0.0256916	+	0.132824i
$581$		0			0
$582$		0			0
$583$		−	30.6241i		−	1.26832i
$584$	43.8313			1.81375
$585$		0			0
$586$		−	1.90790i		−	0.0788148i
$587$		−	31.4362i		−	1.29751i	−0.760997	−	0.648755i	$-0.775290\pi$
$587$		−	31.4362i		−	1.29751i	0.760997	−	0.648755i	$-0.224710\pi$
$588$		0			0
$589$	−36.0507			−1.48544
$590$	−19.6058	+	3.79230i	−0.807160	+	0.156126i
$591$		0			0
$592$			13.5397i			0.556478i
$593$		−	12.9889i		−	0.533391i	−0.963781	−	0.266695i	$-0.914068\pi$
$593$		−	12.9889i		−	0.533391i	0.963781	−	0.266695i	$-0.0859317\pi$
$594$		0			0
$595$		0			0
$596$		−	0.611922i		−	0.0250653i
$597$		0			0
$598$	−12.8846			−0.526890
$599$			14.6761i			0.599648i	0.953995	+	0.299824i	$0.0969280\pi$
$599$			14.6761i			0.599648i	−0.953995	+	0.299824i	$0.903072\pi$
$600$		0			0
$601$			9.55020i			0.389561i	0.980847	+	0.194780i	$0.0623994\pi$
$601$			9.55020i			0.389561i	−0.980847	+	0.194780i	$0.937601\pi$
$602$		0			0
$603$		0			0
$604$	−2.62596			−0.106849
$605$	11.4012	−	2.20529i	0.463523	−	0.0896578i
$606$		0			0
$607$	27.2507			1.10607			0.553035	−	0.833158i	$-0.313469\pi$
$607$	27.2507			1.10607			0.553035	+	0.833158i	$0.313469\pi$
$608$		−	6.05769i		−	0.245672i
$609$		0			0
$610$	−1.28313	−	6.63369i	−0.0519526	−	0.268590i
$611$			17.1153i			0.692411i
$612$		0			0
$613$		−	48.4764i		−	1.95794i	−0.203997	−	0.978971i	$-0.565393\pi$
$613$		−	48.4764i		−	1.95794i	0.203997	−	0.978971i	$-0.434607\pi$
$614$	10.1500			0.409622
$615$		0			0
$616$		0			0
$617$	8.42587			0.339213			0.169606	−	0.985512i	$-0.445750\pi$
$617$	8.42587			0.339213			0.169606	+	0.985512i	$0.445750\pi$
$618$		0			0
$619$		−	17.3082i		−	0.695675i	−0.937555	−	0.347837i	$-0.886916\pi$
$619$		−	17.3082i		−	0.695675i	0.937555	−	0.347837i	$-0.113084\pi$
$620$	0.973245	+	5.03159i	0.0390865	+	0.202074i
$621$		0			0
$622$	43.7382			1.75374
$623$		0			0
$624$		0			0
$625$	18.0472	−	17.3002i	0.721888	−	0.692010i
$626$	11.1491			0.445606
$627$		0			0
$628$	−5.43319			−0.216808
$629$	3.81600			0.152154
$630$		0			0
$631$	16.3945			0.652653			0.326327	−	0.945257i	$-0.394189\pi$
$631$	16.3945			0.652653			0.326327	+	0.945257i	$0.394189\pi$
$632$	5.32404			0.211779
$633$		0			0
$634$	−7.83353			−0.311109
$635$	−2.12885	−	11.0060i	−0.0844808	−	0.436758i
$636$		0			0
$637$		0			0
$638$	−17.6816			−0.700023
$639$		0			0
$640$	18.8819	−	3.65227i	0.746373	−	0.144369i
$641$			14.9979i			0.592380i	0.955129	+	0.296190i	$0.0957161\pi$
$641$			14.9979i			0.592380i	−0.955129	+	0.296190i	$0.904284\pi$
$642$		0			0
$643$	26.5806			1.04824			0.524119	−	0.851645i	$-0.324395\pi$
$643$	26.5806			1.04824			0.524119	+	0.851645i	$0.324395\pi$
$644$		0			0
$645$		0			0
$646$	5.21459			0.205165
$647$			23.0964i			0.908015i	0.890998	+	0.454007i	$0.150006\pi$
$647$			23.0964i			0.908015i	−0.890998	+	0.454007i	$0.849994\pi$
$648$		0			0
$649$			16.3227i			0.640724i
$650$	−10.3573	+	4.16250i	−0.406247	+	0.163267i
$651$		0			0
$652$		−	4.08936i		−	0.160152i
$653$	21.1568			0.827928			0.413964	−	0.910293i	$-0.364144\pi$
$653$	21.1568			0.827928			0.413964	+	0.910293i	$0.364144\pi$
$654$		0			0
$655$	−37.4417	+	7.24222i	−1.46297	+	0.282977i
$656$	6.38883			0.249442
$657$		0			0
$658$		0			0
$659$		−	34.2144i		−	1.33281i	−0.745592	−	0.666403i	$-0.767833\pi$
$659$		−	34.2144i		−	1.33281i	0.745592	−	0.666403i	$-0.232167\pi$
$660$		0			0
$661$			0.217449i			0.00845780i	0.999991	+	0.00422890i	$0.00134610\pi$
$661$			0.217449i			0.00845780i	−0.999991	+	0.00422890i	$0.998654\pi$
$662$	34.9512			1.35842
$663$		0			0
$664$		−	28.6892i		−	1.11336i
$665$		0			0
$666$		0			0
$667$		−	32.1209i		−	1.24373i
$668$		−	3.87095i		−	0.149772i
$669$		0			0
$670$	−2.68586	−	13.8857i	−0.103764	−	0.536451i
$671$	−5.52284			−0.213207
$672$		0			0
$673$			29.8280i			1.14979i	0.818229	+	0.574893i	$0.194956\pi$
$673$			29.8280i			1.14979i	−0.818229	+	0.574893i	$0.805044\pi$
$674$			4.51034i			0.173732i
$675$		0			0
$676$	2.65275			0.102029
$677$		−	42.1158i		−	1.61864i	−0.587367	−	0.809321i	$-0.699835\pi$
$677$		−	42.1158i		−	1.61864i	0.587367	−	0.809321i	$-0.300165\pi$
$678$		0			0
$679$		0			0
$680$	−1.21621	−	6.28772i	−0.0466397	−	0.241123i
$681$		0			0
$682$	−27.8124			−1.06499
$683$	−23.3181			−0.892242			−0.446121	−	0.894973i	$-0.647195\pi$
$683$	−23.3181			−0.892242			−0.446121	+	0.894973i	$0.647195\pi$
$684$		0			0
$685$	−21.7763	+	4.21212i	−0.832030	+	0.160937i
$686$		0			0
$687$		0			0
$688$		−	35.0006i		−	1.33438i
$689$	−21.5198			−0.819838
$690$		0			0
$691$		−	5.38432i		−	0.204829i	−0.994742	−	0.102415i	$-0.967343\pi$
$691$		−	5.38432i		−	0.204829i	0.994742	−	0.102415i	$-0.0326568\pi$
$692$		−	1.79688i		−	0.0683073i
$693$		0			0
$694$	−17.6652			−0.670563
$695$	−6.41654	−	33.1729i	−0.243393	−	1.25832i
$696$		0			0
$697$		−	1.80061i		−	0.0682031i
$698$			20.1167i			0.761430i
$699$		0			0
$700$		0			0
$701$			21.9593i			0.829391i	0.909960	+	0.414696i	$0.136112\pi$
$701$			21.9593i			0.829391i	−0.909960	+	0.414696i	$0.863888\pi$
$702$		0			0
$703$	−16.3613			−0.617079
$704$		−	21.0973i		−	0.795135i
$705$		0			0
$706$			48.6779i			1.83202i
$707$		0			0
$708$		0			0
$709$	−27.3011			−1.02531			−0.512657	−	0.858594i	$-0.671339\pi$
$709$	−27.3011			−1.02531			−0.512657	+	0.858594i	$0.671339\pi$
$710$	1.25745	+	6.50089i	0.0471911	+	0.243974i
$711$		0			0
$712$	1.31714			0.0493618
$713$		−	50.5248i		−	1.89217i
$714$		0			0
$715$	1.73273	+	8.95808i	0.0648006	+	0.335013i
$716$			3.06514i			0.114550i
$717$		0			0
$718$		−	11.7391i		−	0.438099i
$719$	32.6003			1.21579			0.607893	−	0.794019i	$-0.292015\pi$
$719$	32.6003			1.21579			0.607893	+	0.794019i	$0.292015\pi$
$720$		0			0
$721$		0			0
$722$	2.69185			0.100180
$723$		0			0
$724$		−	4.54221i		−	0.168810i
$725$	−10.3770	−	25.8205i	−0.385392	−	0.958948i
$726$		0			0
$727$	4.18319			0.155146			0.0775729	−	0.996987i	$-0.475283\pi$
$727$	4.18319			0.155146			0.0775729	+	0.996987i	$0.475283\pi$
$728$		0			0
$729$		0			0
$730$	−42.5440	+	8.22915i	−1.57462	+	0.304575i
$731$	−9.86449			−0.364851
$732$		0			0
$733$	−0.649470			−0.0239887			−0.0119944	−	0.999928i	$-0.503818\pi$
$733$	−0.649470			−0.0239887			−0.0119944	+	0.999928i	$0.503818\pi$
$734$	19.0560			0.703370
$735$		0			0
$736$	8.48979			0.312938
$737$	−11.5605			−0.425835
$738$		0			0
$739$	23.0720			0.848719			0.424360	−	0.905494i	$-0.360499\pi$
$739$	23.0720			0.848719			0.424360	+	0.905494i	$0.360499\pi$
$740$	0.441699	+	2.28355i	0.0162372	+	0.0839449i
$741$		0			0
$742$		0			0
$743$	−18.5599			−0.680897			−0.340448	−	0.940263i	$-0.610579\pi$
$743$	−18.5599			−0.680897			−0.340448	+	0.940263i	$0.610579\pi$
$744$		0			0
$745$	0.992541	+	5.13135i	0.0363639	+	0.187998i
$746$			27.8466i			1.01954i
$747$		0			0
$748$	−0.605926			−0.0221549
$749$		0			0
$750$		0			0
$751$	22.1764			0.809227			0.404613	−	0.914488i	$-0.367406\pi$
$751$	22.1764			0.809227			0.404613	+	0.914488i	$0.367406\pi$
$752$			34.4450i			1.25608i
$753$		0			0
$754$			12.4250i			0.452492i
$755$	22.0203	−	4.25931i	0.801400	−	0.155012i
$756$		0			0
$757$		−	22.4056i		−	0.814347i	−0.913351	−	0.407173i	$-0.866515\pi$
$757$		−	22.4056i		−	0.814347i	0.913351	−	0.407173i	$-0.133485\pi$
$758$	−0.0176639			−0.000641582
$759$		0			0
$760$	5.21459	+	26.9590i	0.189153	+	0.977906i
$761$	14.0305			0.508607			0.254303	−	0.967125i	$-0.418154\pi$
$761$	14.0305			0.508607			0.254303	+	0.967125i	$0.418154\pi$
$762$		0			0
$763$		0			0
$764$			4.94800i			0.179012i
$765$		0			0
$766$		−	22.7063i		−	0.820410i
$767$	11.4701			0.414161
$768$		0			0
$769$			40.4788i			1.45970i	0.683606	+	0.729851i	$0.260411\pi$
$769$			40.4788i			1.45970i	−0.683606	+	0.729851i	$0.739589\pi$
$770$		0			0
$771$		0			0
$772$		−	6.10579i		−	0.219752i
$773$		−	31.3121i		−	1.12622i	−0.826382	−	0.563110i	$-0.809605\pi$
$773$		−	31.3121i		−	1.12622i	0.826382	−	0.563110i	$-0.190395\pi$
$774$		0			0
$775$	−16.3225	−	40.6145i	−0.586323	−	1.45891i
$776$	−37.2526			−1.33729
$777$		0			0
$778$			33.9604i			1.21754i
$779$			7.72024i			0.276606i
$780$		0			0
$781$	5.41228			0.193667
$782$			7.30821i			0.261341i
$783$		0			0
$784$		0			0
$785$	45.5607	−	8.81267i	1.62613	−	0.314538i
$786$		0			0
$787$	24.3563			0.868208			0.434104	−	0.900863i	$-0.357065\pi$
$787$	24.3563			0.868208			0.434104	+	0.900863i	$0.357065\pi$
$788$	−0.220681			−0.00786142
$789$		0			0
$790$	−5.16768	+	0.999568i	−0.183858	+	0.0355630i
$791$		0			0
$792$		0			0
$793$			3.88094i			0.137816i
$794$	−20.1030			−0.713429
$795$		0			0
$796$			0.223690i			0.00792847i
$797$			42.7862i			1.51557i	0.652507	+	0.757783i	$0.273717\pi$
$797$			42.7862i			1.51557i	−0.652507	+	0.757783i	$0.726283\pi$
$798$		0			0
$799$	9.70789			0.343441
$800$	6.82455	−	2.74272i	0.241284	−	0.0969697i
$801$		0			0
$802$			17.3002i			0.610893i
$803$			35.4198i			1.24994i
$804$		0			0
$805$		0			0
$806$			19.5440i			0.688407i
$807$		0			0
$808$	28.2969			0.995482
$809$		−	38.1157i		−	1.34008i	−0.742326	−	0.670038i	$-0.766278\pi$
$809$		−	38.1157i		−	1.34008i	0.742326	−	0.670038i	$-0.233722\pi$
$810$		0			0
$811$		−	43.0019i		−	1.51000i	−0.655725	−	0.755000i	$-0.727637\pi$
$811$		−	43.0019i		−	1.51000i	0.655725	−	0.755000i	$-0.272363\pi$
$812$		0			0
$813$		0			0
$814$	−12.6224			−0.442417
$815$	6.63296	+	34.2918i	0.232342	+	1.20119i
$816$		0			0
$817$	42.2946			1.47970
$818$			43.3406i			1.51537i
$819$		0			0
$820$	1.07751	−	0.208420i	0.0376284	−	0.00727833i
$821$		−	7.79326i		−	0.271987i	−0.990710	−	0.135993i	$-0.956577\pi$
$821$		−	7.79326i		−	0.271987i	0.990710	−	0.135993i	$-0.0434226\pi$
$822$		0			0
$823$			23.0457i			0.803322i	0.915788	+	0.401661i	$0.131567\pi$
$823$			23.0457i			0.803322i	−0.915788	+	0.401661i	$0.868433\pi$
$824$	−24.1023			−0.839644
$825$		0			0
$826$		0			0
$827$	−0.555383			−0.0193126			−0.00965628	−	0.999953i	$-0.503074\pi$
$827$	−0.555383			−0.0193126			−0.00965628	+	0.999953i	$0.503074\pi$
$828$		0			0
$829$			24.1007i			0.837052i	0.908205	+	0.418526i	$0.137453\pi$
$829$			24.1007i			0.837052i	−0.908205	+	0.418526i	$0.862547\pi$
$830$	5.38628	+	27.8466i	0.186961	+	0.966570i
$831$		0			0
$832$	−14.8252			−0.513972
$833$		0			0
$834$		0			0
$835$	6.27871	+	32.4604i	0.217284	+	1.12334i
$836$	2.59795			0.0898518
$837$		0			0
$838$	12.3022			0.424974
$839$	−45.5904			−1.57395			−0.786977	−	0.616982i	$-0.788355\pi$
$839$	−45.5904			−1.57395			−0.786977	+	0.616982i	$0.788355\pi$
$840$		0			0
$841$	−1.97516			−0.0681090
$842$	−17.2343			−0.593933
$843$		0			0
$844$	−4.23943			−0.145927
$845$	−22.2450	+	4.30278i	−0.765251	+	0.148020i
$846$		0			0
$847$		0			0
$848$	−43.3090			−1.48724
$849$		0			0
$850$	2.36099	+	5.87472i	0.0809813	+	0.201501i
$851$		−	22.9302i		−	0.786039i
$852$		0			0
$853$	−0.440934			−0.0150973			−0.00754864	−	0.999972i	$-0.502403\pi$
$853$	−0.440934			−0.0150973			−0.00754864	+	0.999972i	$0.502403\pi$
$854$		0			0
$855$		0			0
$856$	16.1809			0.553052
$857$		−	8.35261i		−	0.285320i	−0.989772	−	0.142660i	$-0.954434\pi$
$857$		−	8.35261i		−	0.285320i	0.989772	−	0.142660i	$-0.0455655\pi$
$858$		0			0
$859$		−	2.42151i		−	0.0826209i	−0.999146	−	0.0413104i	$-0.986847\pi$
$859$		−	2.42151i		−	0.0826209i	0.999146	−	0.0413104i	$-0.0131533\pi$
$860$	−1.14181	−	5.90305i	−0.0389353	−	0.201292i
$861$		0			0
$862$			7.88967i			0.268723i
$863$	15.2655			0.519644			0.259822	−	0.965657i	$-0.416336\pi$
$863$	15.2655			0.519644			0.259822	+	0.965657i	$0.416336\pi$
$864$		0			0
$865$	2.91455	+	15.0680i	0.0990978	+	0.512327i
$866$	−25.6213			−0.870647
$867$		0			0
$868$		0			0
$869$			4.30233i			0.145946i
$870$		0			0
$871$			8.12361i			0.275258i
$872$	−51.7795			−1.75348
$873$		0			0
$874$		−	31.3344i		−	1.05990i
$875$		0			0
$876$		0			0
$877$			34.6255i			1.16922i	0.811315	+	0.584609i	$0.198752\pi$
$877$			34.6255i			1.16922i	−0.811315	+	0.584609i	$0.801248\pi$
$878$		−	36.7157i		−	1.23909i
$879$		0			0
$880$	3.48717	+	18.0284i	0.117552	+	0.607736i
$881$	18.2805			0.615887			0.307943	−	0.951405i	$-0.400359\pi$
$881$	18.2805			0.615887			0.307943	+	0.951405i	$0.400359\pi$
$882$		0			0
$883$		−	15.7048i		−	0.528508i	−0.964453	−	0.264254i	$-0.914874\pi$
$883$		−	15.7048i		−	0.528508i	0.964453	−	0.264254i	$-0.0851257\pi$
$884$			0.425788i			0.0143208i
$885$		0			0
$886$	30.8913			1.03781
$887$			31.6876i			1.06397i	0.846755	+	0.531983i	$0.178553\pi$
$887$			31.6876i			1.06397i	−0.846755	+	0.531983i	$0.821447\pi$
$888$		0			0
$889$		0			0
$890$	−1.27845	+	0.247287i	−0.0428539	+	0.00828909i
$891$		0			0
$892$	2.61873			0.0876817
$893$	−41.6232			−1.39287
$894$		0			0
$895$	−4.97168	−	25.7032i	−0.166185	−	0.859162i
$896$		0			0
$897$		0			0
$898$			20.8134i			0.694554i
$899$	−48.7226			−1.62499
$900$		0			0
$901$			12.2061i			0.406645i
$902$			5.95601i			0.198314i
$903$		0			0
$904$	17.6652			0.587537
$905$	7.36749	+	38.0893i	0.244904	+	1.26613i
$906$		0			0
$907$			3.21264i			0.106674i	0.998577	+	0.0533369i	$0.0169857\pi$
$907$			3.21264i			0.106674i	−0.998577	+	0.0533369i	$0.983014\pi$
$908$			1.22478i			0.0406458i
$909$		0			0
$910$		0			0
$911$		−	43.0454i		−	1.42616i	−0.701084	−	0.713079i	$-0.747300\pi$
$911$		−	43.0454i		−	1.42616i	0.701084	−	0.713079i	$-0.252700\pi$
$912$		0			0
$913$	23.1836			0.767264
$914$		−	35.2212i		−	1.16501i
$915$		0			0
$916$			3.73224i			0.123317i
$917$		0			0
$918$		0			0
$919$	35.8575			1.18283			0.591414	−	0.806368i	$-0.298570\pi$
$919$	35.8575			1.18283			0.591414	+	0.806368i	$0.298570\pi$
$920$	−37.7828	+	7.30821i	−1.24566	+	0.240944i
$921$		0			0
$922$	−43.7382			−1.44044
$923$		−	3.80324i		−	0.125185i
$924$		0			0
$925$	−7.40786	−	18.4325i	−0.243569	−	0.606058i
$926$			11.8849i			0.390562i
$927$		0			0
$928$		−	8.18697i		−	0.268750i
$929$	−6.89137			−0.226098			−0.113049	−	0.993589i	$-0.536062\pi$
$929$	−6.89137			−0.226098			−0.113049	+	0.993589i	$0.536062\pi$
$930$		0			0
$931$		0			0
$932$	5.83947			0.191278
$933$		0			0
$934$		−	42.8076i		−	1.40071i
$935$	5.08107	−	0.982816i	0.166169	−	0.0321415i
$936$		0			0
$937$	−13.8533			−0.452566			−0.226283	−	0.974062i	$-0.572657\pi$
$937$	−13.8533			−0.452566			−0.226283	+	0.974062i	$0.572657\pi$
$938$		0			0
$939$		0			0
$940$	1.12368	+	5.80934i	0.0366505	+	0.189480i
$941$	30.4006			0.991032			0.495516	−	0.868599i	$-0.334979\pi$
$941$	30.4006			0.991032			0.495516	+	0.868599i	$0.334979\pi$
$942$		0			0
$943$	−10.8198			−0.352342
$944$	23.0838			0.751315
$945$		0			0
$946$	32.6295			1.06088
$947$	−26.7755			−0.870088			−0.435044	−	0.900409i	$-0.643267\pi$
$947$	−26.7755			−0.870088			−0.435044	+	0.900409i	$0.643267\pi$
$948$		0			0
$949$	24.8897			0.807954
$950$	−10.1229	−	25.1882i	−0.328430	−	0.817214i
$951$		0			0
$952$		0			0
$953$	23.9462			0.775693			0.387846	−	0.921724i	$-0.373219\pi$
$953$	23.9462			0.775693			0.387846	+	0.921724i	$0.373219\pi$
$954$		0			0
$955$	−8.02568	−	41.4921i	−0.259705	−	1.34265i
$956$		−	1.67484i		−	0.0541682i
$957$		0			0
$958$	5.49255			0.177456
$959$		0			0
$960$		0			0
$961$	−45.6384			−1.47221
$962$			8.86987i			0.285976i
$963$		0			0
$964$			1.87682i			0.0604484i
$965$	9.90362	+	51.2008i	0.318809	+	1.64821i
$966$		0			0
$967$		−	12.8640i		−	0.413678i	−0.978375	−	0.206839i	$-0.933682\pi$
$967$		−	12.8640i		−	0.413678i	0.978375	−	0.206839i	$-0.0663177\pi$
$968$	−15.4862			−0.497745
$969$		0			0
$970$	36.1585	−	6.99403i	1.16098	−	0.224565i
$971$	−34.9755			−1.12242			−0.561209	−	0.827674i	$-0.689664\pi$
$971$	−34.9755			−1.12242			−0.561209	+	0.827674i	$0.689664\pi$
$972$		0			0
$973$		0			0
$974$		−	27.2031i		−	0.871642i
$975$		0			0
$976$			7.81048i			0.250007i
$977$	8.83328			0.282602			0.141301	−	0.989967i	$-0.454872\pi$
$977$	8.83328			0.282602			0.141301	+	0.989967i	$0.454872\pi$
$978$		0			0
$979$			1.06437i			0.0340174i
$980$		0			0
$981$		0			0
$982$			44.3481i			1.41521i
$983$		−	51.3192i		−	1.63683i	−0.574629	−	0.818414i	$-0.694854\pi$
$983$		−	51.3192i		−	1.63683i	0.574629	−	0.818414i	$-0.305146\pi$
$984$		0			0
$985$	1.85055	−	0.357945i	0.0589633	−	0.0114051i
$986$	7.04753			0.224439
$987$		0			0
$988$		−	1.82559i		−	0.0580798i
$989$			59.2755i			1.88485i
$990$		0			0
$991$	−18.0854			−0.574503			−0.287252	−	0.957855i	$-0.592742\pi$
$991$	−18.0854			−0.574503			−0.287252	+	0.957855i	$0.592742\pi$
$992$		−	12.8777i		−	0.408869i
$993$		0			0
$994$		0			0
$995$	−0.362826	−	1.87578i	−0.0115024	−	0.0594662i
$996$		0			0
$997$	33.2250			1.05225			0.526124	−	0.850408i	$-0.323645\pi$
$997$	33.2250			1.05225			0.526124	+	0.850408i	$0.323645\pi$
$998$	35.3809			1.11996
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.g.b.2204.16		24
3.2	odd	2	inner	2205.2.g.b.2204.10		24
5.4	even	2	inner	2205.2.g.b.2204.12		24
7.4	even	3		315.2.bb.b.89.5	✓	24
7.5	odd	6		315.2.bb.b.269.6	yes	24
7.6	odd	2	inner	2205.2.g.b.2204.15		24
15.14	odd	2	inner	2205.2.g.b.2204.14		24
21.5	even	6		315.2.bb.b.269.7	yes	24
21.11	odd	6		315.2.bb.b.89.8	yes	24
21.20	even	2	inner	2205.2.g.b.2204.9		24
35.4	even	6		315.2.bb.b.89.7	yes	24
35.12	even	12		1575.2.bk.i.1151.7		24
35.18	odd	12		1575.2.bk.i.26.8		24
35.19	odd	6		315.2.bb.b.269.8	yes	24
35.32	odd	12		1575.2.bk.i.26.6		24
35.33	even	12		1575.2.bk.i.1151.5		24
35.34	odd	2	inner	2205.2.g.b.2204.11		24
105.32	even	12		1575.2.bk.i.26.7		24
105.47	odd	12		1575.2.bk.i.1151.6		24
105.53	even	12		1575.2.bk.i.26.5		24
105.68	odd	12		1575.2.bk.i.1151.8		24
105.74	odd	6		315.2.bb.b.89.6	yes	24
105.89	even	6		315.2.bb.b.269.5	yes	24
105.104	even	2	inner	2205.2.g.b.2204.13		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bb.b.89.5	✓	24	7.4	even	3
315.2.bb.b.89.6	yes	24	105.74	odd	6
315.2.bb.b.89.7	yes	24	35.4	even	6
315.2.bb.b.89.8	yes	24	21.11	odd	6
315.2.bb.b.269.5	yes	24	105.89	even	6
315.2.bb.b.269.6	yes	24	7.5	odd	6
315.2.bb.b.269.7	yes	24	21.5	even	6
315.2.bb.b.269.8	yes	24	35.19	odd	6
1575.2.bk.i.26.5		24	105.53	even	12
1575.2.bk.i.26.6		24	35.32	odd	12
1575.2.bk.i.26.7		24	105.32	even	12
1575.2.bk.i.26.8		24	35.18	odd	12
1575.2.bk.i.1151.5		24	35.33	even	12
1575.2.bk.i.1151.6		24	105.47	odd	12
1575.2.bk.i.1151.7		24	35.12	even	12
1575.2.bk.i.1151.8		24	105.68	odd	12
2205.2.g.b.2204.9		24	21.20	even	2	inner
2205.2.g.b.2204.10		24	3.2	odd	2	inner
2205.2.g.b.2204.11		24	35.34	odd	2	inner
2205.2.g.b.2204.12		24	5.4	even	2	inner
2205.2.g.b.2204.13		24	105.104	even	2	inner
2205.2.g.b.2204.14		24	15.14	odd	2	inner
2205.2.g.b.2204.15		24	7.6	odd	2	inner
2205.2.g.b.2204.16		24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.31841 q^{2} -0.261802 q^{4} +(2.19538 - 0.424645i) q^{5} -2.98198 q^{8} +(2.89440 - 0.559855i) q^{10} -2.40972i q^{11} -1.69332 q^{13} -3.40786 q^{16} +0.960462i q^{17} -4.11804i q^{19} +(-0.574754 + 0.111173i) q^{20} -3.17699i q^{22} +5.77140 q^{23} +(4.63935 - 1.86451i) q^{25} -2.23249 q^{26} -5.56553i q^{29} -8.75434i q^{31} +1.47101 q^{32} +1.26628i q^{34} -3.97309i q^{37} -5.42925i q^{38} +(-6.54656 + 1.26628i) q^{40} -1.87474 q^{41} +10.2706i q^{43} +0.630869i q^{44} +7.60905 q^{46} -10.1075i q^{47} +(6.11656 - 2.45818i) q^{50} +0.443316 q^{52} +12.7086 q^{53} +(-1.02327 - 5.29024i) q^{55} -7.33764i q^{58} -6.77371 q^{59} -2.29191i q^{61} -11.5418i q^{62} +8.75510 q^{64} +(-3.71748 + 0.719061i) q^{65} -4.79743i q^{67} -0.251451i q^{68} +2.24602i q^{71} -14.6987 q^{73} -5.23814i q^{74} +1.07811i q^{76} -1.78541 q^{79} +(-7.48152 + 1.44713i) q^{80} -2.47166 q^{82} +9.62086i q^{83} +(0.407855 + 2.10858i) q^{85} +13.5408i q^{86} +7.18572i q^{88} -0.441699 q^{89} -1.51096 q^{92} -13.3258i q^{94} +(-1.74870 - 9.04065i) q^{95} +12.4926 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} - 24 q^{25} + 48 q^{46} + 48 q^{64} + 120 q^{79} - 72 q^{85}+O(q^{100}) \) 24 * q + 48 * q^4 - 24 * q^25 + 48 * q^46 + 48 * q^64 + 120 * q^79 - 72 * q^85

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