LMFDB - Embedded newform 2205.2.g.b.2204.5

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			2204.5
Character	$\chi$	$=$	2205.2204
Dual form			2205.2.g.b.2204.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91314 q^{2} +1.66012 q^{4} +(-0.630583 - 2.14531i) q^{5} +0.650234 q^{8} +O(q^{10})$	\(q-1.91314 q^{2} +1.66012 q^{4} +(-0.630583 - 2.14531i) q^{5} +0.650234 q^{8} +(1.20640 + 4.10429i) q^{10} -3.22740i q^{11} +4.86146 q^{13} -4.56424 q^{16} +0.729142i q^{17} +7.87149i q^{19} +(-1.04685 - 3.56148i) q^{20} +6.17449i q^{22} +4.86820 q^{23} +(-4.20473 + 2.70560i) q^{25} -9.30068 q^{26} +7.75958i q^{29} -1.42896i q^{31} +7.43158 q^{32} -1.39495i q^{34} +3.16538i q^{37} -15.0593i q^{38} +(-0.410026 - 1.39495i) q^{40} -1.40264 q^{41} +6.42489i q^{43} -5.35789i q^{44} -9.31357 q^{46} -4.87049i q^{47} +(8.04426 - 5.17620i) q^{50} +8.07063 q^{52} -1.52147 q^{53} +(-6.92379 + 2.03515i) q^{55} -14.8452i q^{58} -6.30162 q^{59} +2.37224i q^{61} +2.73380i q^{62} -5.08921 q^{64} +(-3.06556 - 10.4294i) q^{65} +11.1259i q^{67} +1.21047i q^{68} +10.1351i q^{71} +13.8239 q^{73} -6.05583i q^{74} +13.0676i q^{76} +3.98037 q^{79} +(2.87813 + 9.79171i) q^{80} +2.68346 q^{82} -4.19208i q^{83} +(1.56424 - 0.459785i) q^{85} -12.2917i q^{86} -2.09857i q^{88} -11.2734 q^{89} +8.08181 q^{92} +9.31795i q^{94} +(16.8868 - 4.96363i) q^{95} -2.21388 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4}+O(q^{10}) $ 24 * q + 48 * q^4	$ 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

Display 100 coefficients 10 coefficients

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 $

$2$	−1.91314			−1.35280			−0.676399	−	0.736536i	$-0.736460\pi$
$2$	−1.91314			−1.35280			−0.676399	+	0.736536i	$0.736460\pi$
$3$		0			0
$3$		0			0
$4$	1.66012			0.830062
$5$	−0.630583	−	2.14531i	−0.282005	−	0.959413i
$5$	−0.630583	−	2.14531i	−0.282005	−	0.959413i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	0.650234			0.229892
$9$		0			0
$10$	1.20640	+	4.10429i	0.381496	+	1.29789i
$11$		−	3.22740i		−	0.973099i	−0.873653	−	0.486549i	$-0.838255\pi$
$11$		−	3.22740i		−	0.973099i	0.873653	−	0.486549i	$-0.161745\pi$
$12$		0			0
$13$	4.86146			1.34833			0.674164	−	0.738582i	$-0.264504\pi$
$13$	4.86146			1.34833			0.674164	+	0.738582i	$0.264504\pi$
$14$		0			0
$15$		0			0
$16$	−4.56424			−1.14106
$17$			0.729142i			0.176843i	0.996083	+	0.0884215i	$0.0281822\pi$
$17$			0.729142i			0.176843i	−0.996083	+	0.0884215i	$0.971818\pi$
$18$		0			0
$19$			7.87149i			1.80584i	0.429805	+	0.902922i	$0.358582\pi$
$19$			7.87149i			1.80584i	−0.429805	+	0.902922i	$0.641418\pi$
$20$	−1.04685	−	3.56148i	−0.234082	−	0.796372i
$21$		0			0
$22$			6.17449i			1.31641i
$23$	4.86820			1.01509			0.507545	−	0.861625i	$-0.330553\pi$
$23$	4.86820			1.01509			0.507545	+	0.861625i	$0.330553\pi$
$24$		0			0
$25$	−4.20473	+	2.70560i	−0.840946	+	0.541119i
$26$	−9.30068			−1.82401
$27$		0			0
$28$		0			0
$29$			7.75958i			1.44092i	0.693498	+	0.720459i	$0.256069\pi$
$29$			7.75958i			1.44092i	−0.693498	+	0.720459i	$0.743931\pi$
$30$		0			0
$31$		−	1.42896i		−	0.256648i	−0.991732	−	0.128324i	$-0.959040\pi$
$31$		−	1.42896i		−	0.256648i	0.991732	−	0.128324i	$-0.0409597\pi$
$32$	7.43158			1.31373
$33$		0			0
$34$		−	1.39495i		−	0.239233i
$35$		0			0
$36$		0			0
$37$			3.16538i			0.520385i	0.965557	+	0.260193i	$0.0837861\pi$
$37$			3.16538i			0.520385i	−0.965557	+	0.260193i	$0.916214\pi$
$38$		−	15.0593i		−	2.44294i
$39$		0			0
$40$	−0.410026	−	1.39495i	−0.0648309	−	0.220562i
$41$	−1.40264			−0.219056			−0.109528	−	0.993984i	$-0.534934\pi$
$41$	−1.40264			−0.219056			−0.109528	+	0.993984i	$0.534934\pi$
$42$		0			0
$43$			6.42489i			0.979787i	0.871782	+	0.489893i	$0.162964\pi$
$43$			6.42489i			0.979787i	−0.871782	+	0.489893i	$0.837036\pi$
$44$		−	5.35789i		−	0.807732i
$45$		0			0
$46$	−9.31357			−1.37321
$47$		−	4.87049i		−	0.710434i	−0.934784	−	0.355217i	$-0.884407\pi$
$47$		−	4.87049i		−	0.710434i	0.934784	−	0.355217i	$-0.115593\pi$
$48$		0			0
$49$		0			0
$50$	8.04426	−	5.17620i	1.13763	−	0.732025i
$51$		0			0
$52$	8.07063			1.11919
$53$	−1.52147			−0.208990			−0.104495	−	0.994525i	$-0.533323\pi$
$53$	−1.52147			−0.208990			−0.104495	+	0.994525i	$0.533323\pi$
$54$		0			0
$55$	−6.92379	+	2.03515i	−0.933603	+	0.274419i
$56$		0			0
$57$		0			0
$58$		−	14.8452i		−	1.94927i
$59$	−6.30162			−0.820402			−0.410201	−	0.911995i	$-0.634541\pi$
$59$	−6.30162			−0.820402			−0.410201	+	0.911995i	$0.634541\pi$
$60$		0			0
$61$			2.37224i			0.303735i	0.988401	+	0.151867i	$0.0485286\pi$
$61$			2.37224i			0.303735i	−0.988401	+	0.151867i	$0.951471\pi$
$62$			2.73380i			0.347193i
$63$		0			0
$64$	−5.08921			−0.636152
$65$	−3.06556	−	10.4294i	−0.380235	−	1.29360i
$66$		0			0
$67$			11.1259i			1.35924i	0.733562	+	0.679622i	$0.237856\pi$
$67$			11.1259i			1.35924i	−0.733562	+	0.679622i	$0.762144\pi$
$68$			1.21047i			0.146791i
$69$		0			0
$70$		0			0
$71$			10.1351i			1.20282i	0.798942	+	0.601408i	$0.205393\pi$
$71$			10.1351i			1.20282i	−0.798942	+	0.601408i	$0.794607\pi$
$72$		0			0
$73$	13.8239			1.61796			0.808982	−	0.587833i	$-0.200019\pi$
$73$	13.8239			1.61796			0.808982	+	0.587833i	$0.200019\pi$
$74$		−	6.05583i		−	0.703976i
$75$		0			0
$76$			13.0676i			1.49896i
$77$		0			0
$78$		0			0
$79$	3.98037			0.447827			0.223913	−	0.974609i	$-0.428117\pi$
$79$	3.98037			0.447827			0.223913	+	0.974609i	$0.428117\pi$
$80$	2.87813	+	9.79171i	0.321785	+	1.09475i
$81$		0			0
$82$	2.68346			0.296339
$83$		−	4.19208i		−	0.460141i	−0.973174	−	0.230070i	$-0.926104\pi$
$83$		−	4.19208i		−	0.460141i	0.973174	−	0.230070i	$-0.0738957\pi$
$84$		0			0
$85$	1.56424	−	0.459785i	0.169665	−	0.0498706i
$86$		−	12.2917i		−	1.32545i
$87$		0			0
$88$		−	2.09857i		−	0.223708i
$89$	−11.2734			−1.19498			−0.597491	−	0.801875i	$-0.703836\pi$
$89$	−11.2734			−1.19498			−0.597491	+	0.801875i	$0.703836\pi$
$90$		0			0
$91$		0			0
$92$	8.08181			0.842587
$93$		0			0
$94$			9.31795i			0.961073i
$95$	16.8868	−	4.96363i	1.73255	−	0.509258i
$96$		0			0
$97$	−2.21388			−0.224785			−0.112393	−	0.993664i	$-0.535851\pi$
$97$	−2.21388			−0.224785			−0.112393	+	0.993664i	$0.535851\pi$
$98$		0			0
$99$		0			0
$100$	−6.98037	+	4.49162i	−0.698037	+	0.449162i
$101$	11.7021			1.16440			0.582201	−	0.813045i	$-0.302192\pi$
$101$	11.7021			1.16440			0.582201	+	0.813045i	$0.302192\pi$
$102$		0			0
$103$	0.585261			0.0576674			0.0288337	−	0.999584i	$-0.490821\pi$
$103$	0.585261			0.0576674			0.0288337	+	0.999584i	$0.490821\pi$
$104$	3.16109			0.309970
$105$		0			0
$106$	2.91079			0.282721
$107$	−1.69215			−0.163586			−0.0817929	−	0.996649i	$-0.526065\pi$
$107$	−1.69215			−0.163586			−0.0817929	+	0.996649i	$0.526065\pi$
$108$		0			0
$109$	−13.4028			−1.28375			−0.641877	−	0.766808i	$-0.721844\pi$
$109$	−13.4028			−1.28375			−0.641877	+	0.766808i	$0.721844\pi$
$110$	13.2462	−	3.89353i	1.26298	−	0.371233i
$111$		0			0
$112$		0			0
$113$	−14.2129			−1.33704			−0.668520	−	0.743694i	$-0.733072\pi$
$113$	−14.2129			−1.33704			−0.668520	+	0.743694i	$0.733072\pi$
$114$		0			0
$115$	−3.06981	−	10.4438i	−0.286261	−	0.973891i
$116$			12.8819i			1.19605i
$117$		0			0
$118$	12.0559			1.10984
$119$		0			0
$120$		0			0
$121$	0.583868			0.0530789
$122$		−	4.53844i		−	0.410891i
$123$		0			0
$124$		−	2.37224i		−	0.213034i
$125$	8.45578	+	7.31436i	0.756308	+	0.654216i
$126$		0			0
$127$		−	2.08954i		−	0.185417i	−0.995693	−	0.0927084i	$-0.970448\pi$
$127$		−	2.08954i		−	0.185417i	0.995693	−	0.0927084i	$-0.0295524\pi$
$128$	−5.12676			−0.453146
$129$		0			0
$130$	5.86485	+	19.9529i	0.514382	+	1.74998i
$131$	11.2048			0.978969			0.489484	−	0.872012i	$-0.337185\pi$
$131$	11.2048			0.978969			0.489484	+	0.872012i	$0.337185\pi$
$132$		0			0
$133$		0			0
$134$		−	21.2855i		−	1.83878i
$135$		0			0
$136$			0.474113i			0.0406548i
$137$	7.03990			0.601459			0.300730	−	0.953709i	$-0.402770\pi$
$137$	7.03990			0.601459			0.300730	+	0.953709i	$0.402770\pi$
$138$		0			0
$139$		−	6.45903i		−	0.547848i	−0.961751	−	0.273924i	$-0.911678\pi$
$139$		−	6.45903i		−	0.547848i	0.961751	−	0.273924i	$-0.0883216\pi$
$140$		0			0
$141$		0			0
$142$		−	19.3899i		−	1.62717i
$143$		−	15.6899i		−	1.31206i
$144$		0			0
$145$	16.6467	−	4.89306i	1.38243	−	0.406346i
$146$	−26.4471			−2.18878
$147$		0			0
$148$			5.25492i			0.431952i
$149$			23.5515i			1.92942i	0.263320	+	0.964709i	$0.415182\pi$
$149$			23.5515i			1.92942i	−0.263320	+	0.964709i	$0.584818\pi$
$150$		0			0
$151$	18.1088			1.47368			0.736838	−	0.676069i	$-0.236318\pi$
$151$	18.1088			1.47368			0.736838	+	0.676069i	$0.236318\pi$
$152$			5.11831i			0.415150i
$153$		0			0
$154$		0			0
$155$	−3.06556	+	0.901075i	−0.246231	+	0.0723761i
$156$		0			0
$157$	−2.48454			−0.198288			−0.0991441	−	0.995073i	$-0.531610\pi$
$157$	−2.48454			−0.198288			−0.0991441	+	0.995073i	$0.531610\pi$
$158$	−7.61502			−0.605819
$159$		0			0
$160$	−4.68623	−	15.9431i	−0.370479	−	1.26041i
$161$		0			0
$162$		0			0
$163$		−	3.93771i		−	0.308425i	−0.988038	−	0.154212i	$-0.950716\pi$
$163$		−	3.93771i		−	0.308425i	0.988038	−	0.154212i	$-0.0492840\pi$
$164$	−2.32856			−0.181830
$165$		0			0
$166$			8.02006i			0.622478i
$167$		−	5.31832i		−	0.411544i	−0.978600	−	0.205772i	$-0.934029\pi$
$167$		−	5.31832i		−	0.411544i	0.978600	−	0.205772i	$-0.0659705\pi$
$168$		0			0
$169$	10.6338			0.817986
$170$	−2.99261	+	0.879635i	−0.229523	+	0.0674649i
$171$		0			0
$172$			10.6661i			0.813283i
$173$			7.45501i			0.566794i	0.959003	+	0.283397i	$0.0914614\pi$
$173$			7.45501i			0.566794i	−0.959003	+	0.283397i	$0.908539\pi$
$174$		0			0
$175$		0			0
$176$			14.7306i			1.11036i
$177$		0			0
$178$	21.5677			1.61657
$179$			13.3070i			0.994611i	0.867576	+	0.497305i	$0.165677\pi$
$179$			13.3070i			0.994611i	−0.867576	+	0.497305i	$0.834323\pi$
$180$		0			0
$181$		−	9.95814i		−	0.740183i	−0.928995	−	0.370091i	$-0.879326\pi$
$181$		−	9.95814i		−	0.740183i	0.928995	−	0.370091i	$-0.120674\pi$
$182$		0			0
$183$		0			0
$184$	3.16547			0.233361
$185$	6.79073	−	1.99604i	0.499264	−	0.146751i
$186$		0			0
$187$	2.35324			0.172086
$188$		−	8.08561i		−	0.589704i
$189$		0			0
$190$	−32.3069	+	9.49614i	−2.34379	+	0.688922i
$191$			24.6113i			1.78081i	0.455171	+	0.890404i	$0.349578\pi$
$191$			24.6113i			1.78081i	−0.455171	+	0.890404i	$0.650422\pi$
$192$		0			0
$193$		−	12.5143i		−	0.900797i	−0.892828	−	0.450399i	$-0.851282\pi$
$193$		−	12.5143i		−	0.900797i	0.892828	−	0.450399i	$-0.148718\pi$
$194$	4.23547			0.304089
$195$		0			0
$196$		0			0
$197$	15.6968			1.11835			0.559177	−	0.829049i	$-0.311117\pi$
$197$	15.6968			1.11835			0.559177	+	0.829049i	$0.311117\pi$
$198$		0			0
$199$			19.7958i			1.40328i	0.712529	+	0.701642i	$0.247550\pi$
$199$			19.7958i			1.40328i	−0.712529	+	0.701642i	$0.752450\pi$
$200$	−2.73406	+	1.75927i	−0.193327	+	0.124399i
$201$		0			0
$202$	−22.3878			−1.57520
$203$		0			0
$204$		0			0
$205$	0.884484	+	3.00911i	0.0617750	+	0.210165i
$206$	−1.11969			−0.0780124
$207$		0			0
$208$	−22.1889			−1.53852
$209$	25.4045			1.75726
$210$		0			0
$211$	11.5839			0.797466			0.398733	−	0.917067i	$-0.369450\pi$
$211$	11.5839			0.797466			0.398733	+	0.917067i	$0.369450\pi$
$212$	−2.52582			−0.173474
$213$		0			0
$214$	3.23732			0.221299
$215$	13.7834	−	4.05143i	0.940020	−	0.276305i
$216$		0			0
$217$		0			0
$218$	25.6415			1.73666
$219$		0			0
$220$	−11.4943	+	3.37859i	−0.774948	+	0.227785i
$221$			3.54470i			0.238442i
$222$		0			0
$223$	24.1321			1.61600			0.808002	−	0.589180i	$-0.200549\pi$
$223$	24.1321			1.61600			0.808002	+	0.589180i	$0.200549\pi$
$224$		0			0
$225$		0			0
$226$	27.1914			1.80874
$227$			10.1888i			0.676255i	0.941100	+	0.338128i	$0.109794\pi$
$227$			10.1888i			0.676255i	−0.941100	+	0.338128i	$0.890206\pi$
$228$		0			0
$229$		−	26.2548i		−	1.73497i	−0.497467	−	0.867483i	$-0.665736\pi$
$229$		−	26.2548i		−	1.73497i	0.497467	−	0.867483i	$-0.334264\pi$
$230$	5.87298	+	19.9805i	0.387253	+	1.31748i
$231$		0			0
$232$			5.04554i			0.331256i
$233$	−18.3933			−1.20499			−0.602494	−	0.798124i	$-0.705826\pi$
$233$	−18.3933			−1.20499			−0.602494	+	0.798124i	$0.705826\pi$
$234$		0			0
$235$	−10.4487	+	3.07125i	−0.681599	+	0.200346i
$236$	−10.4615			−0.680984
$237$		0			0
$238$		0			0
$239$			0.961317i			0.0621824i	0.999517	+	0.0310912i	$0.00989824\pi$
$239$			0.961317i			0.0621824i	−0.999517	+	0.0310912i	$0.990102\pi$
$240$		0			0
$241$		−	6.51054i		−	0.419381i	−0.977768	−	0.209690i	$-0.932754\pi$
$241$		−	6.51054i		−	0.419381i	0.977768	−	0.209690i	$-0.0672456\pi$
$242$	−1.11702			−0.0718050
$243$		0			0
$244$			3.93821i			0.252118i
$245$		0			0
$246$		0			0
$247$			38.2670i			2.43487i
$248$		−	0.929155i		−	0.0590014i
$249$		0			0
$250$	−16.1771	−	13.9934i	−1.02313	−	0.885022i
$251$	2.02506			0.127820			0.0639102	−	0.997956i	$-0.479643\pi$
$251$	2.02506			0.127820			0.0639102	+	0.997956i	$0.479643\pi$
$252$		0			0
$253$		−	15.7116i		−	0.987783i
$254$			3.99759i			0.250831i
$255$		0			0
$256$	19.9867			1.24917
$257$		−	18.1902i		−	1.13467i	−0.823486	−	0.567337i	$-0.807974\pi$
$257$		−	18.1902i		−	1.13467i	0.823486	−	0.567337i	$-0.192026\pi$
$258$		0			0
$259$		0			0
$260$	−5.08920	−	17.3140i	−0.315619	−	1.07377i
$261$		0			0
$262$	−21.4364			−1.32435
$263$	−19.5607			−1.20616			−0.603082	−	0.797679i	$-0.706061\pi$
$263$	−19.5607			−1.20616			−0.603082	+	0.797679i	$0.706061\pi$
$264$		0			0
$265$	0.959411	+	3.26402i	0.0589362	+	0.200507i
$266$		0			0
$267$		0			0
$268$			18.4704i			1.12826i
$269$	4.61602			0.281444			0.140722	−	0.990049i	$-0.455058\pi$
$269$	4.61602			0.281444			0.140722	+	0.990049i	$0.455058\pi$
$270$		0			0
$271$			10.7809i			0.654894i	0.944870	+	0.327447i	$0.106188\pi$
$271$			10.7809i			0.654894i	−0.944870	+	0.327447i	$0.893812\pi$
$272$		−	3.32798i		−	0.201788i
$273$		0			0
$274$	−13.4684			−0.813653
$275$	8.73205	+	13.5704i	0.526562	+	0.818323i
$276$		0			0
$277$			20.5068i			1.23213i	0.787694	+	0.616067i	$0.211275\pi$
$277$			20.5068i			1.23213i	−0.787694	+	0.616067i	$0.788725\pi$
$278$			12.3571i			0.741127i
$279$		0			0
$280$		0			0
$281$		−	0.714666i		−	0.0426334i	−0.999773	−	0.0213167i	$-0.993214\pi$
$281$		−	0.714666i		−	0.0426334i	0.999773	−	0.0213167i	$-0.00678583\pi$
$282$		0			0
$283$	28.0219			1.66573			0.832866	−	0.553475i	$-0.186699\pi$
$283$	28.0219			1.66573			0.832866	+	0.553475i	$0.186699\pi$
$284$			16.8255i			0.998411i
$285$		0			0
$286$			30.0171i			1.77495i
$287$		0			0
$288$		0			0
$289$	16.4684			0.968727
$290$	−31.8476	+	9.36113i	−1.87015	+	0.549704i
$291$		0			0
$292$	22.9494			1.34301
$293$		−	9.79171i		−	0.572038i	−0.958224	−	0.286019i	$-0.907668\pi$
$293$		−	9.79171i		−	0.572038i	0.958224	−	0.286019i	$-0.0923321\pi$
$294$		0			0
$295$	3.97370	+	13.5190i	0.231358	+	0.787104i
$296$			2.05824i			0.119633i
$297$		0			0
$298$		−	45.0575i		−	2.61011i
$299$	23.6666			1.36867
$300$		0			0
$301$		0			0
$302$	−34.6448			−1.99359
$303$		0			0
$304$		−	35.9273i		−	2.06057i
$305$	5.08920	−	1.49590i	0.291407	−	0.0856548i
$306$		0			0
$307$	25.4778			1.45410			0.727048	−	0.686586i	$-0.240892\pi$
$307$	25.4778			1.45410			0.727048	+	0.686586i	$0.240892\pi$
$308$		0			0
$309$		0			0
$310$	5.86485	−	1.72389i	0.333101	−	0.0979102i
$311$	10.1084			0.573198			0.286599	−	0.958051i	$-0.407475\pi$
$311$	10.1084			0.573198			0.286599	+	0.958051i	$0.407475\pi$
$312$		0			0
$313$	23.4751			1.32689			0.663445	−	0.748225i	$-0.269094\pi$
$313$	23.4751			1.32689			0.663445	+	0.748225i	$0.269094\pi$
$314$	4.75329			0.268244
$315$		0			0
$316$	6.60790			0.371724
$317$	31.4312			1.76535			0.882677	−	0.469980i	$-0.155739\pi$
$317$	31.4312			1.76535			0.882677	+	0.469980i	$0.155739\pi$
$318$		0			0
$319$	25.0433			1.40215
$320$	3.20917	+	10.9180i	0.179398	+	0.610332i
$321$		0			0
$322$		0			0
$323$	−5.73943			−0.319351
$324$		0			0
$325$	−20.4411	+	13.1532i	−1.13387	+	0.729606i
$326$			7.53340i			0.417236i
$327$		0			0
$328$	−0.912047			−0.0503593
$329$		0			0
$330$		0			0
$331$	−1.17843			−0.0647722			−0.0323861	−	0.999475i	$-0.510311\pi$
$331$	−1.17843			−0.0647722			−0.0323861	+	0.999475i	$0.510311\pi$
$332$		−	6.95937i		−	0.381945i
$333$		0			0
$334$			10.1747i			0.556736i
$335$	23.8685	−	7.01580i	1.30408	−	0.383314i
$336$		0			0
$337$		−	3.92868i		−	0.214009i	−0.994259	−	0.107004i	$-0.965874\pi$
$337$		−	3.92868i		−	0.214009i	0.994259	−	0.107004i	$-0.0341259\pi$
$338$	−20.3440			−1.10657
$339$		0			0
$340$	2.59683	−	0.763299i	0.140833	−	0.0413957i
$341$	−4.61182			−0.249744
$342$		0			0
$343$		0			0
$344$			4.17768i			0.225245i
$345$		0			0
$346$		−	14.2625i		−	0.766758i
$347$	−4.83065			−0.259323			−0.129661	−	0.991558i	$-0.541389\pi$
$347$	−4.83065			−0.259323			−0.129661	+	0.991558i	$0.541389\pi$
$348$		0			0
$349$		−	16.0461i		−	0.858927i	−0.903084	−	0.429463i	$-0.858703\pi$
$349$		−	16.0461i		−	0.858927i	0.903084	−	0.429463i	$-0.141297\pi$
$350$		0			0
$351$		0			0
$352$		−	23.9847i		−	1.27839i
$353$			5.88094i			0.313011i	0.987677	+	0.156505i	$0.0500229\pi$
$353$			5.88094i			0.313011i	−0.987677	+	0.156505i	$0.949977\pi$
$354$		0			0
$355$	21.7430	−	6.39103i	1.15400	−	0.339200i
$356$	−18.7153			−0.991909
$357$		0			0
$358$		−	25.4582i		−	1.34551i
$359$		−	8.08631i		−	0.426779i	−0.976967	−	0.213389i	$-0.931550\pi$
$359$		−	8.08631i		−	0.426779i	0.976967	−	0.213389i	$-0.0684503\pi$
$360$		0			0
$361$	−42.9604			−2.26107
$362$			19.0514i			1.00132i
$363$		0			0
$364$		0			0
$365$	−8.71711	−	29.6566i	−0.456275	−	1.55230i
$366$		0			0
$367$	−18.5697			−0.969329			−0.484665	−	0.874700i	$-0.661058\pi$
$367$	−18.5697			−0.969329			−0.484665	+	0.874700i	$0.661058\pi$
$368$	−22.2196			−1.15828
$369$		0			0
$370$	−12.9916	+	3.81870i	−0.675404	+	0.198525i
$371$		0			0
$372$		0			0
$373$			2.64346i			0.136873i	0.997655	+	0.0684365i	$0.0218011\pi$
$373$			2.64346i			0.136873i	−0.997655	+	0.0684365i	$0.978199\pi$
$374$	−4.50208			−0.232797
$375$		0			0
$376$		−	3.16696i		−	0.163323i
$377$			37.7229i			1.94283i
$378$		0			0
$379$	−23.8582			−1.22551			−0.612756	−	0.790272i	$-0.709939\pi$
$379$	−23.8582			−1.22551			−0.612756	+	0.790272i	$0.709939\pi$
$380$	28.0342	−	8.24023i	1.43812	−	0.422715i
$381$		0			0
$382$		−	47.0849i		−	2.40907i
$383$		−	11.1150i		−	0.567952i	−0.958831	−	0.283976i	$-0.908346\pi$
$383$		−	11.1150i		−	0.567952i	0.958831	−	0.283976i	$-0.0916536\pi$
$384$		0			0
$385$		0			0
$386$			23.9416i			1.21860i
$387$		0			0
$388$	−3.67531			−0.186586
$389$		−	12.5813i		−	0.637896i	−0.947772	−	0.318948i	$-0.896670\pi$
$389$		−	12.5813i		−	0.637896i	0.947772	−	0.318948i	$-0.103330\pi$
$390$		0			0
$391$			3.54961i			0.179512i
$392$		0			0
$393$		0			0
$394$	−30.0303			−1.51291
$395$	−2.50995	−	8.53914i	−0.126289	−	0.429651i
$396$		0			0
$397$	−28.6309			−1.43694			−0.718471	−	0.695557i	$-0.755158\pi$
$397$	−28.6309			−1.43694			−0.718471	+	0.695557i	$0.755158\pi$
$398$		−	37.8722i		−	1.89836i
$399$		0			0
$400$	19.1914	−	12.3490i	0.959569	−	0.617449i
$401$		−	11.8928i		−	0.593897i	−0.954894	−	0.296948i	$-0.904031\pi$
$401$		−	11.8928i		−	0.593897i	0.954894	−	0.296948i	$-0.0959689\pi$
$402$		0			0
$403$		−	6.94681i		−	0.346045i
$404$	19.4269			0.966525
$405$		0			0
$406$		0			0
$407$	10.2160			0.506386
$408$		0			0
$409$		−	14.4170i		−	0.712877i	−0.934319	−	0.356438i	$-0.883991\pi$
$409$		−	14.4170i		−	0.712877i	0.934319	−	0.356438i	$-0.116009\pi$
$410$	−1.69215	−	5.75686i	−0.0835691	−	0.284311i
$411$		0			0
$412$	0.971605			0.0478675
$413$		0			0
$414$		0			0
$415$	−8.99333	+	2.64346i	−0.441465	+	0.129762i
$416$	36.1283			1.77134
$417$		0			0
$418$	−48.6024			−2.37722
$419$	−34.9400			−1.70693			−0.853466	−	0.521149i	$-0.825504\pi$
$419$	−34.9400			−1.70693			−0.853466	+	0.521149i	$0.825504\pi$
$420$		0			0
$421$	23.8515			1.16245			0.581226	−	0.813742i	$-0.302573\pi$
$421$	23.8515			1.16245			0.581226	+	0.813742i	$0.302573\pi$
$422$	−22.1616			−1.07881
$423$		0			0
$424$	−0.989309			−0.0480451
$425$	−1.97276	−	3.06585i	−0.0956931	−	0.148715i
$426$		0			0
$427$		0			0
$428$	−2.80917			−0.135786
$429$		0			0
$430$	−26.3696	+	7.75097i	−1.27166	+	0.373785i
$431$		−	8.15855i		−	0.392984i	−0.980505	−	0.196492i	$-0.937045\pi$
$431$		−	8.15855i		−	0.392984i	0.980505	−	0.196492i	$-0.0629549\pi$
$432$		0			0
$433$	−25.2667			−1.21424			−0.607121	−	0.794610i	$-0.707676\pi$
$433$	−25.2667			−1.21424			−0.607121	+	0.794610i	$0.707676\pi$
$434$		0			0
$435$		0			0
$436$	−22.2503			−1.06560
$437$			38.3200i			1.83309i
$438$		0			0
$439$			0.794714i			0.0379296i	0.999820	+	0.0189648i	$0.00603705\pi$
$439$			0.794714i			0.0379296i	−0.999820	+	0.0189648i	$0.993963\pi$
$440$	−4.50208	+	1.32332i	−0.214628	+	0.0630868i
$441$		0			0
$442$		−	6.78152i		−	0.322564i
$443$	22.6489			1.07608			0.538040	−	0.842919i	$-0.319165\pi$
$443$	22.6489			1.07608			0.538040	+	0.842919i	$0.319165\pi$
$444$		0			0
$445$	7.10884	+	24.1851i	0.336992	+	1.14648i
$446$	−46.1682			−2.18613
$447$		0			0
$448$		0			0
$449$		−	2.15664i		−	0.101778i	−0.998704	−	0.0508891i	$-0.983794\pi$
$449$		−	2.15664i		−	0.101778i	0.998704	−	0.0508891i	$-0.0162055\pi$
$450$		0			0
$451$			4.52690i			0.213163i
$452$	−23.5952			−1.10983
$453$		0			0
$454$		−	19.4927i		−	0.914836i
$455$		0			0
$456$		0			0
$457$			5.47333i			0.256032i	0.991772	+	0.128016i	$0.0408608\pi$
$457$			5.47333i			0.256032i	−0.991772	+	0.128016i	$0.959139\pi$
$458$			50.2292i			2.34706i
$459$		0			0
$460$	−5.09626	−	17.3380i	−0.237614	−	0.808389i
$461$	−10.1084			−0.470797			−0.235399	−	0.971899i	$-0.575640\pi$
$461$	−10.1084			−0.470797			−0.235399	+	0.971899i	$0.575640\pi$
$462$		0			0
$463$		−	12.0455i		−	0.559800i	−0.960029	−	0.279900i	$-0.909699\pi$
$463$		−	12.0455i		−	0.559800i	0.960029	−	0.279900i	$-0.0903014\pi$
$464$		−	35.4165i		−	1.64417i
$465$		0			0
$466$	35.1891			1.63010
$467$			19.9327i			0.922376i	0.887302	+	0.461188i	$0.152577\pi$
$467$			19.9327i			0.922376i	−0.887302	+	0.461188i	$0.847423\pi$
$468$		0			0
$469$		0			0
$470$	19.9899	−	5.87574i	0.922066	−	0.271028i
$471$		0			0
$472$	−4.09753			−0.188604
$473$	20.7357			0.953429
$474$		0			0
$475$	−21.2971	−	33.0975i	−0.977176	−	1.51862i
$476$		0			0
$477$		0			0
$478$		−	1.83914i		−	0.0841202i
$479$	−19.3858			−0.885760			−0.442880	−	0.896581i	$-0.646043\pi$
$479$	−19.3858			−0.885760			−0.442880	+	0.896581i	$0.646043\pi$
$480$		0			0
$481$			15.3884i			0.701650i
$482$			12.4556i			0.567337i
$483$		0			0
$484$	0.969293			0.0440588
$485$	1.39603	+	4.74946i	0.0633906	+	0.215662i
$486$		0			0
$487$			1.84816i			0.0837483i	0.999123	+	0.0418742i	$0.0133329\pi$
$487$			1.84816i			0.0837483i	−0.999123	+	0.0418742i	$0.986667\pi$
$488$			1.54251i			0.0698262i
$489$		0			0
$490$		0			0
$491$			25.4836i			1.15006i	0.818133	+	0.575030i	$0.195009\pi$
$491$			25.4836i			1.15006i	−0.818133	+	0.575030i	$0.804991\pi$
$492$		0			0
$493$	−5.65783			−0.254816
$494$		−	73.2102i		−	3.29388i
$495$		0			0
$496$			6.52209i			0.292851i
$497$		0			0
$498$		0			0
$499$	−26.2284			−1.17414			−0.587072	−	0.809535i	$-0.699719\pi$
$499$	−26.2284			−1.17414			−0.587072	+	0.809535i	$0.699719\pi$
$500$	14.0376	+	12.1427i	0.627782	+	0.543039i
$501$		0			0
$502$	−3.87422			−0.172915
$503$		−	4.25713i		−	0.189816i	−0.995486	−	0.0949081i	$-0.969744\pi$
$503$		−	4.25713i		−	0.189816i	0.995486	−	0.0949081i	$-0.0302557\pi$
$504$		0			0
$505$	−7.37914	−	25.1046i	−0.328367	−	1.11714i
$506$			30.0587i			1.33627i
$507$		0			0
$508$		−	3.46889i		−	0.153907i
$509$	19.1192			0.847443			0.423722	−	0.905792i	$-0.360723\pi$
$509$	19.1192			0.847443			0.423722	+	0.905792i	$0.360723\pi$
$510$		0			0
$511$		0			0
$512$	−27.9839			−1.23672
$513$		0			0
$514$			34.8005i			1.53498i
$515$	−0.369055	−	1.25557i	−0.0162625	−	0.0553269i
$516$		0			0
$517$	−15.7190			−0.691322
$518$		0			0
$519$		0			0
$520$	−1.99333	−	6.78152i	−0.0874132	−	0.297389i
$521$	−22.8135			−0.999478			−0.499739	−	0.866176i	$-0.666571\pi$
$521$	−22.8135			−0.999478			−0.499739	+	0.866176i	$0.666571\pi$
$522$		0			0
$523$	3.12936			0.136838			0.0684188	−	0.997657i	$-0.478205\pi$
$523$	3.12936			0.136838			0.0684188	+	0.997657i	$0.478205\pi$
$524$	18.6014			0.812604
$525$		0			0
$526$	37.4224			1.63169
$527$	1.04191			0.0453864
$528$		0			0
$529$	0.699384			0.0304080
$530$	−1.83549	−	6.24455i	−0.0797287	−	0.271246i
$531$		0			0
$532$		0			0
$533$	−6.81890			−0.295359
$534$		0			0
$535$	1.06704	+	3.63018i	0.0461321	+	0.156946i
$536$			7.23443i			0.312480i
$537$		0			0
$538$	−8.83112			−0.380737
$539$		0			0
$540$		0			0
$541$	−22.4617			−0.965703			−0.482852	−	0.875702i	$-0.660399\pi$
$541$	−22.4617			−0.965703			−0.482852	+	0.875702i	$0.660399\pi$
$542$		−	20.6254i		−	0.885939i
$543$		0			0
$544$			5.41868i			0.232324i
$545$	8.45157	+	28.7532i	0.362026	+	1.23165i
$546$		0			0
$547$		−	10.4567i		−	0.447097i	−0.974693	−	0.223549i	$-0.928236\pi$
$547$		−	10.4567i		−	0.447097i	0.974693	−	0.223549i	$-0.0717642\pi$
$548$	11.6871			0.499248
$549$		0			0
$550$	−16.7057	−	25.9621i	−0.712332	−	1.10703i
$551$	−61.0794			−2.60207
$552$		0			0
$553$		0			0
$554$		−	39.2324i		−	1.66683i
$555$		0			0
$556$		−	10.7228i		−	0.454747i
$557$	31.2478			1.32401			0.662006	−	0.749499i	$-0.269705\pi$
$557$	31.2478			1.32401			0.662006	+	0.749499i	$0.269705\pi$
$558$		0			0
$559$			31.2344i			1.32107i
$560$		0			0
$561$		0			0
$562$			1.36726i			0.0576744i
$563$		−	29.8708i		−	1.25890i	−0.777040	−	0.629452i	$-0.783280\pi$
$563$		−	29.8708i		−	1.25890i	0.777040	−	0.629452i	$-0.216720\pi$
$564$		0			0
$565$	8.96243	+	30.4912i	0.377052	+	1.28277i
$566$	−53.6100			−2.25340
$567$		0			0
$568$			6.59019i			0.276518i
$569$		−	28.8927i		−	1.21125i	−0.795752	−	0.605623i	$-0.792924\pi$
$569$		−	28.8927i		−	1.21125i	0.795752	−	0.605623i	$-0.207076\pi$
$570$		0			0
$571$	−7.37687			−0.308712			−0.154356	−	0.988015i	$-0.549330\pi$
$571$	−7.37687			−0.308712			−0.154356	+	0.988015i	$0.549330\pi$
$572$		−	26.0472i		−	1.08909i
$573$		0			0
$574$		0			0
$575$	−20.4695	+	13.1714i	−0.853636	+	0.549285i
$576$		0			0
$577$	−18.0757			−0.752502			−0.376251	−	0.926518i	$-0.622787\pi$
$577$	−18.0757			−0.752502			−0.376251	+	0.926518i	$0.622787\pi$
$578$	−31.5063			−1.31049
$579$		0			0
$580$	27.6356	−	8.12308i	1.14751	−	0.337292i
$581$		0			0
$582$		0			0
$583$			4.91039i			0.203367i
$584$	8.98876			0.371958
$585$		0			0
$586$			18.7330i			0.773852i
$587$		−	35.0876i		−	1.44822i	−0.689684	−	0.724111i	$-0.742250\pi$
$587$		−	35.0876i		−	1.44822i	0.689684	−	0.724111i	$-0.257750\pi$
$588$		0			0
$589$	11.2480			0.463466
$590$	−7.60226	−	25.8637i	−0.312980	−	1.06479i
$591$		0			0
$592$		−	14.4475i		−	0.593790i
$593$		−	7.05792i		−	0.289834i	−0.989444	−	0.144917i	$-0.953709\pi$
$593$		−	7.05792i		−	0.289834i	0.989444	−	0.144917i	$-0.0462915\pi$
$594$		0			0
$595$		0			0
$596$			39.0984i			1.60153i
$597$		0			0
$598$	−45.2776			−1.85154
$599$			4.88666i			0.199664i	0.995004	+	0.0998318i	$0.0318305\pi$
$599$			4.88666i			0.199664i	−0.995004	+	0.0998318i	$0.968170\pi$
$600$		0			0
$601$			36.1905i			1.47624i	0.674669	+	0.738121i	$0.264286\pi$
$601$			36.1905i			1.47624i	−0.674669	+	0.738121i	$0.735714\pi$
$602$		0			0
$603$		0			0
$604$	30.0629			1.22324
$605$	−0.368177	−	1.25258i	−0.0149685	−	0.0509246i
$606$		0			0
$607$	16.9254			0.686982			0.343491	−	0.939156i	$-0.388391\pi$
$607$	16.9254			0.686982			0.343491	+	0.939156i	$0.388391\pi$
$608$			58.4976i			2.37239i
$609$		0			0
$610$	−9.73638	+	2.86187i	−0.394214	+	0.115874i
$611$		−	23.6777i		−	0.957897i
$612$		0			0
$613$		−	18.6055i		−	0.751469i	−0.926727	−	0.375735i	$-0.877390\pi$
$613$		−	18.6055i		−	0.751469i	0.926727	−	0.375735i	$-0.122610\pi$
$614$	−48.7428			−1.96710
$615$		0			0
$616$		0			0
$617$	−44.6022			−1.79562			−0.897809	−	0.440384i	$-0.854842\pi$
$617$	−44.6022			−1.79562			−0.897809	+	0.440384i	$0.854842\pi$
$618$		0			0
$619$		−	11.3181i		−	0.454912i	−0.973788	−	0.227456i	$-0.926959\pi$
$619$		−	11.3181i		−	0.454912i	0.973788	−	0.227456i	$-0.0730408\pi$
$620$	−5.08920	+	1.49590i	−0.204387	+	0.0600766i
$621$		0			0
$622$	−19.3389			−0.775420
$623$		0			0
$624$		0			0
$625$	10.3595	−	22.7526i	0.414380	−	0.910104i
$626$	−44.9112			−1.79501
$627$		0			0
$628$	−4.12465			−0.164591
$629$	−2.30801			−0.0920265
$630$		0			0
$631$	−6.29394			−0.250558			−0.125279	−	0.992122i	$-0.539983\pi$
$631$	−6.29394			−0.250558			−0.125279	+	0.992122i	$0.539983\pi$
$632$	2.58817			0.102952
$633$		0			0
$634$	−60.1325			−2.38817
$635$	−4.48272	+	1.31763i	−0.177891	+	0.0522885i
$636$		0			0
$637$		0			0
$638$	−47.9114			−1.89683
$639$		0			0
$640$	3.23285	+	10.9985i	0.127789	+	0.434754i
$641$			0.590103i			0.0233077i	0.999932	+	0.0116538i	$0.00370961\pi$
$641$			0.590103i			0.0233077i	−0.999932	+	0.0116538i	$0.996290\pi$
$642$		0			0
$643$	28.7107			1.13224			0.566119	−	0.824323i	$-0.308444\pi$
$643$	28.7107			1.13224			0.566119	+	0.824323i	$0.308444\pi$
$644$		0			0
$645$		0			0
$646$	10.9804			0.432017
$647$			11.9284i			0.468954i	0.972122	+	0.234477i	$0.0753378\pi$
$647$			11.9284i			0.468954i	−0.972122	+	0.234477i	$0.924662\pi$
$648$		0			0
$649$			20.3379i			0.798332i
$650$	39.1069	−	25.1639i	1.53390	−	0.987009i
$651$		0			0
$652$		−	6.53708i		−	0.256012i
$653$	−15.9929			−0.625852			−0.312926	−	0.949777i	$-0.601309\pi$
$653$	−15.9929			−0.625852			−0.312926	+	0.949777i	$0.601309\pi$
$654$		0			0
$655$	−7.06556	−	24.0378i	−0.276074	−	0.939235i
$656$	6.40200			0.249956
$657$		0			0
$658$		0			0
$659$		−	27.6958i		−	1.07887i	−0.842026	−	0.539437i	$-0.818637\pi$
$659$		−	27.6958i		−	1.07887i	0.842026	−	0.539437i	$-0.181363\pi$
$660$		0			0
$661$		−	41.7412i		−	1.62355i	−0.583973	−	0.811773i	$-0.698503\pi$
$661$		−	41.7412i		−	1.62355i	0.583973	−	0.811773i	$-0.301497\pi$
$662$	2.25450			0.0876237
$663$		0			0
$664$		−	2.72583i		−	0.105783i
$665$		0			0
$666$		0			0
$667$			37.7752i			1.46266i
$668$		−	8.82907i		−	0.341607i
$669$		0			0
$670$	−45.6639	+	13.4222i	−1.76415	+	0.518547i
$671$	7.65618			0.295564
$672$		0			0
$673$		−	7.91950i		−	0.305274i	−0.988282	−	0.152637i	$-0.951223\pi$
$673$		−	7.91950i		−	0.305274i	0.988282	−	0.152637i	$-0.0487766\pi$
$674$			7.51613i			0.289511i
$675$		0			0
$676$	17.6535			0.678979
$677$		−	10.6672i		−	0.409974i	−0.978765	−	0.204987i	$-0.934285\pi$
$677$		−	10.6672i		−	0.409974i	0.978765	−	0.204987i	$-0.0657152\pi$
$678$		0			0
$679$		0			0
$680$	1.01712	−	0.298967i	0.0390048	−	0.0114649i
$681$		0			0
$682$	8.82307			0.337853
$683$	2.20926			0.0845349			0.0422674	−	0.999106i	$-0.486542\pi$
$683$	2.20926			0.0845349			0.0422674	+	0.999106i	$0.486542\pi$
$684$		0			0
$685$	−4.43924	−	15.1028i	−0.169615	−	0.577048i
$686$		0			0
$687$		0			0
$688$		−	29.3247i		−	1.11799i
$689$	−7.39655			−0.281786
$690$		0			0
$691$			9.26644i			0.352512i	0.984344	+	0.176256i	$0.0563987\pi$
$691$			9.26644i			0.352512i	−0.984344	+	0.176256i	$0.943601\pi$
$692$			12.3762i			0.470474i
$693$		0			0
$694$	9.24172			0.350811
$695$	−13.8566	+	4.07295i	−0.525612	+	0.154496i
$696$		0			0
$697$		−	1.02273i		−	0.0387385i
$698$			30.6985i			1.16195i
$699$		0			0
$700$		0			0
$701$			6.99882i			0.264342i	0.991227	+	0.132171i	$0.0421948\pi$
$701$			6.99882i			0.264342i	−0.991227	+	0.132171i	$0.957805\pi$
$702$		0			0
$703$	−24.9163			−0.939734
$704$			16.4249i			0.619038i
$705$		0			0
$706$		−	11.2511i		−	0.423441i
$707$		0			0
$708$		0			0
$709$	−31.1611			−1.17028			−0.585139	−	0.810933i	$-0.698960\pi$
$709$	−31.1611			−1.17028			−0.585139	+	0.810933i	$0.698960\pi$
$710$	−41.5974	+	12.2270i	−1.56112	+	0.458870i
$711$		0			0
$712$	−7.33037			−0.274717
$713$		−	6.95644i		−	0.260521i
$714$		0			0
$715$	−33.6597	+	9.89379i	−1.25880	+	0.370007i
$716$			22.0912i			0.825588i
$717$		0			0
$718$			15.4703i			0.577345i
$719$	9.06160			0.337941			0.168970	−	0.985621i	$-0.445956\pi$
$719$	9.06160			0.337941			0.168970	+	0.985621i	$0.445956\pi$
$720$		0			0
$721$		0			0
$722$	82.1894			3.05877
$723$		0			0
$724$		−	16.5317i		−	0.614397i
$725$	−20.9943	−	32.6269i	−0.779708	−	1.21173i
$726$		0			0
$727$	17.0567			0.632599			0.316300	−	0.948659i	$-0.397559\pi$
$727$	17.0567			0.632599			0.316300	+	0.948659i	$0.397559\pi$
$728$		0			0
$729$		0			0
$730$	16.6771	+	56.7373i	0.617247	+	2.09994i
$731$	−4.68466			−0.173268
$732$		0			0
$733$	15.9660			0.589718			0.294859	−	0.955541i	$-0.404727\pi$
$733$	15.9660			0.589718			0.294859	+	0.955541i	$0.404727\pi$
$734$	35.5265			1.31131
$735$		0			0
$736$	36.1784			1.33355
$737$	35.9078			1.32268
$738$		0			0
$739$	−13.8515			−0.509536			−0.254768	−	0.967002i	$-0.581999\pi$
$739$	−13.8515			−0.509536			−0.254768	+	0.967002i	$0.581999\pi$
$740$	11.2734	−	3.31366i	0.414420	−	0.121813i
$741$		0			0
$742$		0			0
$743$	27.6801			1.01548			0.507741	−	0.861510i	$-0.330481\pi$
$743$	27.6801			1.01548			0.507741	+	0.861510i	$0.330481\pi$
$744$		0			0
$745$	50.5254	−	14.8512i	1.85111	−	0.544106i
$746$		−	5.05732i		−	0.185161i
$747$		0			0
$748$	3.90666			0.142842
$749$		0			0
$750$		0			0
$751$	33.3332			1.21635			0.608173	−	0.793805i	$-0.291903\pi$
$751$	33.3332			1.21635			0.608173	+	0.793805i	$0.291903\pi$
$752$			22.2301i			0.810647i
$753$		0			0
$754$		−	72.1693i		−	2.62825i
$755$	−11.4191	−	38.8491i	−0.415585	−	1.41386i
$756$		0			0
$757$		−	10.1442i		−	0.368697i	−0.982861	−	0.184348i	$-0.940983\pi$
$757$		−	10.1442i		−	0.368697i	0.982861	−	0.184348i	$-0.0590175\pi$
$758$	45.6442			1.65787
$759$		0			0
$760$	10.9804	−	3.22752i	0.398300	−	0.117074i
$761$	−14.7011			−0.532916			−0.266458	−	0.963847i	$-0.585853\pi$
$761$	−14.7011			−0.532916			−0.266458	+	0.963847i	$0.585853\pi$
$762$		0			0
$763$		0			0
$764$			40.8577i			1.47818i
$765$		0			0
$766$			21.2647i			0.768324i
$767$	−30.6351			−1.10617
$768$		0			0
$769$			22.4992i			0.811340i	0.914020	+	0.405670i	$0.132962\pi$
$769$			22.4992i			0.811340i	−0.914020	+	0.405670i	$0.867038\pi$
$770$		0			0
$771$		0			0
$772$		−	20.7752i		−	0.747717i
$773$		−	21.2531i		−	0.764421i	−0.924075	−	0.382211i	$-0.875163\pi$
$773$		−	21.2531i		−	0.764421i	0.924075	−	0.382211i	$-0.124837\pi$
$774$		0			0
$775$	3.86618	+	6.00837i	0.138877	+	0.215827i
$776$	−1.43954			−0.0516764
$777$		0			0
$778$			24.0698i			0.862945i
$779$		−	11.0409i		−	0.395581i
$780$		0			0
$781$	32.7101			1.17046
$782$		−	6.79092i		−	0.242843i
$783$		0			0
$784$		0			0
$785$	1.56671	+	5.33012i	0.0559183	+	0.190240i
$786$		0			0
$787$	15.7190			0.560323			0.280162	−	0.959953i	$-0.409612\pi$
$787$	15.7190			0.560323			0.280162	+	0.959953i	$0.409612\pi$
$788$	26.0587			0.928302
$789$		0			0
$790$	4.80190	+	16.3366i	0.170844	+	0.581230i
$791$		0			0
$792$		0			0
$793$			11.5326i			0.409533i
$794$	54.7750			1.94389
$795$		0			0
$796$			32.8634i			1.16481i
$797$		−	39.5812i		−	1.40204i	−0.713143	−	0.701019i	$-0.752729\pi$
$797$		−	39.5812i		−	1.40204i	0.713143	−	0.701019i	$-0.247271\pi$
$798$		0			0
$799$	3.55128			0.125635
$800$	−31.2478	+	20.1068i	−1.10478	+	0.710884i
$801$		0			0
$802$			22.7526i			0.803422i
$803$		−	44.6153i		−	1.57444i
$804$		0			0
$805$		0			0
$806$			13.2903i			0.468130i
$807$		0			0
$808$	7.60909			0.267687
$809$			44.0002i			1.54697i	0.633817	+	0.773483i	$0.281487\pi$
$809$			44.0002i			1.54697i	−0.633817	+	0.773483i	$0.718513\pi$
$810$		0			0
$811$			44.9306i			1.57773i	0.614567	+	0.788864i	$0.289331\pi$
$811$			44.9306i			1.57773i	−0.614567	+	0.788864i	$0.710669\pi$
$812$		0			0
$813$		0			0
$814$	−19.5446			−0.685038
$815$	−8.44761	+	2.48305i	−0.295907	+	0.0869775i
$816$		0			0
$817$	−50.5735			−1.76934
$818$			27.5819i			0.964378i
$819$		0			0
$820$	1.46835	+	4.99549i	0.0512771	+	0.174450i
$821$		−	15.1296i		−	0.528027i	−0.964519	−	0.264014i	$-0.914954\pi$
$821$		−	15.1296i		−	0.528027i	0.964519	−	0.264014i	$-0.0850464\pi$
$822$		0			0
$823$			31.5796i			1.10079i	0.834903	+	0.550397i	$0.185524\pi$
$823$			31.5796i			1.10079i	−0.834903	+	0.550397i	$0.814476\pi$
$824$	0.380556			0.0132573
$825$		0			0
$826$		0			0
$827$	48.6368			1.69127			0.845633	−	0.533765i	$-0.179223\pi$
$827$	48.6368			1.69127			0.845633	+	0.533765i	$0.179223\pi$
$828$		0			0
$829$			46.7538i			1.62383i	0.583778	+	0.811914i	$0.301574\pi$
$829$			46.7538i			1.62383i	−0.583778	+	0.811914i	$0.698426\pi$
$830$	17.2055	−	5.05732i	0.597213	−	0.175542i
$831$		0			0
$832$	−24.7410			−0.857741
$833$		0			0
$834$		0			0
$835$	−11.4095	+	3.35364i	−0.394841	+	0.116058i
$836$	42.1746			1.45864
$837$		0			0
$838$	66.8453			2.30913
$839$	−29.1067			−1.00488			−0.502438	−	0.864613i	$-0.667563\pi$
$839$	−29.1067			−1.00488			−0.502438	+	0.864613i	$0.667563\pi$
$840$		0			0
$841$	−31.2110			−1.07624
$842$	−45.6314			−1.57256
$843$		0			0
$844$	19.2306			0.661946
$845$	−6.70551	−	22.8129i	−0.230676	−	0.784786i
$846$		0			0
$847$		0			0
$848$	6.94434			0.238469
$849$		0			0
$850$	3.77418	+	5.86541i	0.129453	+	0.201182i
$851$			15.4097i			0.528238i
$852$		0			0
$853$	−28.4915			−0.975531			−0.487766	−	0.872975i	$-0.662188\pi$
$853$	−28.4915			−0.975531			−0.487766	+	0.872975i	$0.662188\pi$
$854$		0			0
$855$		0			0
$856$	−1.10029			−0.0376071
$857$		−	40.5552i		−	1.38534i	−0.721255	−	0.692670i	$-0.756434\pi$
$857$		−	40.5552i		−	1.38534i	0.721255	−	0.692670i	$-0.243566\pi$
$858$		0			0
$859$		−	37.7225i		−	1.28707i	−0.765415	−	0.643537i	$-0.777466\pi$
$859$		−	37.7225i		−	1.28707i	0.765415	−	0.643537i	$-0.222534\pi$
$860$	22.8821	−	6.72587i	0.780274	−	0.229350i
$861$		0			0
$862$			15.6085i			0.531627i
$863$	25.6790			0.874124			0.437062	−	0.899431i	$-0.356019\pi$
$863$	25.6790			0.874124			0.437062	+	0.899431i	$0.356019\pi$
$864$		0			0
$865$	15.9933	−	4.70100i	0.543789	−	0.159839i
$866$	48.3389			1.64262
$867$		0			0
$868$		0			0
$869$		−	12.8463i		−	0.435779i
$870$		0			0
$871$			54.0881i			1.83271i
$872$	−8.71494			−0.295125
$873$		0			0
$874$		−	73.3117i		−	2.47981i
$875$		0			0
$876$		0			0
$877$		−	19.0454i		−	0.643117i	−0.946890	−	0.321559i	$-0.895793\pi$
$877$		−	19.0454i		−	0.643117i	0.946890	−	0.321559i	$-0.104207\pi$
$878$		−	1.52040i		−	0.0513111i
$879$		0			0
$880$	31.6018	−	9.28889i	1.06530	−	0.313128i
$881$	−30.7115			−1.03470			−0.517349	−	0.855774i	$-0.673081\pi$
$881$	−30.7115			−1.03470			−0.517349	+	0.855774i	$0.673081\pi$
$882$		0			0
$883$			36.8466i			1.23999i	0.784607	+	0.619993i	$0.212865\pi$
$883$			36.8466i			1.23999i	−0.784607	+	0.619993i	$0.787135\pi$
$884$			5.88463i			0.197922i
$885$		0			0
$886$	−43.3306			−1.45572
$887$			33.8772i			1.13748i	0.822516	+	0.568742i	$0.192570\pi$
$887$			33.8772i			1.13748i	−0.822516	+	0.568742i	$0.807430\pi$
$888$		0			0
$889$		0			0
$890$	−13.6002	−	46.2695i	−0.455881	−	1.55096i
$891$		0			0
$892$	40.0622			1.34138
$893$	38.3380			1.28293
$894$		0			0
$895$	28.5476	−	8.39116i	0.954242	−	0.280486i
$896$		0			0
$897$		0			0
$898$			4.12597i			0.137685i
$899$	11.0881			0.369809
$900$		0			0
$901$		−	1.10937i		−	0.0369583i
$902$		−	8.66061i		−	0.288367i
$903$		0			0
$904$	−9.24172			−0.307375
$905$	−21.3633	+	6.27944i	−0.710141	+	0.208736i
$906$		0			0
$907$			1.06681i			0.0354230i	0.999843	+	0.0177115i	$0.00563803\pi$
$907$			1.06681i			0.0354230i	−0.999843	+	0.0177115i	$0.994362\pi$
$908$			16.9147i			0.561333i
$909$		0			0
$910$		0			0
$911$		−	9.61157i		−	0.318445i	−0.987243	−	0.159223i	$-0.949101\pi$
$911$		−	9.61157i		−	0.318445i	0.987243	−	0.159223i	$-0.0508988\pi$
$912$		0			0
$913$	−13.5295			−0.447763
$914$		−	10.4713i		−	0.346359i
$915$		0			0
$916$		−	43.5862i		−	1.44013i
$917$		0			0
$918$		0			0
$919$	−6.83188			−0.225363			−0.112681	−	0.993631i	$-0.535944\pi$
$919$	−6.83188			−0.225363			−0.112681	+	0.993631i	$0.535944\pi$
$920$	−1.99609	−	6.79092i	−0.0658092	−	0.223890i
$921$		0			0
$922$	19.3389			0.636894
$923$			49.2714i			1.62179i
$924$		0			0
$925$	−8.56424	−	13.3096i	−0.281590	−	0.437616i
$926$			23.0447i			0.757296i
$927$		0			0
$928$			57.6659i			1.89298i
$929$	53.8766			1.76764			0.883818	−	0.467832i	$-0.154965\pi$
$929$	53.8766			1.76764			0.883818	+	0.467832i	$0.154965\pi$
$930$		0			0
$931$		0			0
$932$	−30.5352			−1.00021
$933$		0			0
$934$		−	38.1342i		−	1.24779i
$935$	−1.48391	−	5.04843i	−0.0485291	−	0.165101i
$936$		0			0
$937$	21.6036			0.705759			0.352880	−	0.935669i	$-0.385203\pi$
$937$	21.6036			0.705759			0.352880	+	0.935669i	$0.385203\pi$
$938$		0			0
$939$		0			0
$940$	−17.3462	+	5.09865i	−0.565769	+	0.166300i
$941$	58.2013			1.89731			0.948654	−	0.316316i	$-0.102446\pi$
$941$	58.2013			1.89731			0.948654	+	0.316316i	$0.102446\pi$
$942$		0			0
$943$	−6.82835			−0.222362
$944$	28.7621			0.936127
$945$		0			0
$946$	−39.6704			−1.28980
$947$	20.4695			0.665168			0.332584	−	0.943074i	$-0.392079\pi$
$947$	20.4695			0.665168			0.332584	+	0.943074i	$0.392079\pi$
$948$		0			0
$949$	67.2043			2.18155
$950$	40.7444	+	63.3203i	1.32192	+	2.05438i
$951$		0			0
$952$		0			0
$953$	−10.4745			−0.339303			−0.169651	−	0.985504i	$-0.554264\pi$
$953$	−10.4745			−0.339303			−0.169651	+	0.985504i	$0.554264\pi$
$954$		0			0
$955$	52.7988	−	15.5194i	1.70853	−	0.502197i
$956$			1.59590i			0.0516152i
$957$		0			0
$958$	37.0879			1.19825
$959$		0			0
$960$		0			0
$961$	28.9581			0.934132
$962$		−	29.4402i		−	0.949190i
$963$		0			0
$964$		−	10.8083i		−	0.348112i
$965$	−26.8470	+	7.89129i	−0.864236	+	0.254030i
$966$		0			0
$967$			43.1242i			1.38678i	0.720562	+	0.693391i	$0.243884\pi$
$967$			43.1242i			1.38678i	−0.720562	+	0.693391i	$0.756116\pi$
$968$	0.379651			0.0122024
$969$		0			0
$970$	−2.67081	−	9.08640i	−0.0857547	−	0.291747i
$971$	8.35144			0.268010			0.134005	−	0.990981i	$-0.457216\pi$
$971$	8.35144			0.268010			0.134005	+	0.990981i	$0.457216\pi$
$972$		0			0
$973$		0			0
$974$		−	3.53581i		−	0.113295i
$975$		0			0
$976$		−	10.8275i		−	0.346579i
$977$	−26.8088			−0.857690			−0.428845	−	0.903378i	$-0.641079\pi$
$977$	−26.8088			−0.857690			−0.428845	+	0.903378i	$0.641079\pi$
$978$		0			0
$979$			36.3840i			1.16284i
$980$		0			0
$981$		0			0
$982$		−	48.7538i		−	1.55580i
$983$			29.4574i			0.939546i	0.882787	+	0.469773i	$0.155664\pi$
$983$			29.4574i			0.939546i	−0.882787	+	0.469773i	$0.844336\pi$
$984$		0			0
$985$	−9.89816	−	33.6746i	−0.315382	−	1.07296i
$986$	10.8243			0.344714
$987$		0			0
$988$			63.5279i			2.02109i
$989$			31.2777i			0.994572i
$990$		0			0
$991$	−5.00667			−0.159042			−0.0795211	−	0.996833i	$-0.525339\pi$
$991$	−5.00667			−0.159042			−0.0795211	+	0.996833i	$0.525339\pi$
$992$		−	10.6194i		−	0.337166i
$993$		0			0
$994$		0			0
$995$	42.4681	−	12.4829i	1.34633	−	0.395734i
$996$		0			0
$997$	−8.87919			−0.281207			−0.140603	−	0.990066i	$-0.544904\pi$
$997$	−8.87919			−0.281207			−0.140603	+	0.990066i	$0.544904\pi$
$998$	50.1787			1.58838
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.2.g.b.2204.5		24
3.2	odd	2	inner	2205.2.g.b.2204.19		24
5.4	even	2	inner	2205.2.g.b.2204.17		24
7.4	even	3		315.2.bb.b.89.10	yes	24
7.5	odd	6		315.2.bb.b.269.9	yes	24
7.6	odd	2	inner	2205.2.g.b.2204.6		24
15.14	odd	2	inner	2205.2.g.b.2204.7		24
21.5	even	6		315.2.bb.b.269.4	yes	24
21.11	odd	6		315.2.bb.b.89.3	✓	24
21.20	even	2	inner	2205.2.g.b.2204.20		24
35.4	even	6		315.2.bb.b.89.4	yes	24
35.12	even	12		1575.2.bk.i.1151.3		24
35.18	odd	12		1575.2.bk.i.26.4		24
35.19	odd	6		315.2.bb.b.269.3	yes	24
35.32	odd	12		1575.2.bk.i.26.10		24
35.33	even	12		1575.2.bk.i.1151.9		24
35.34	odd	2	inner	2205.2.g.b.2204.18		24
105.32	even	12		1575.2.bk.i.26.3		24
105.47	odd	12		1575.2.bk.i.1151.10		24
105.53	even	12		1575.2.bk.i.26.9		24
105.68	odd	12		1575.2.bk.i.1151.4		24
105.74	odd	6		315.2.bb.b.89.9	yes	24
105.89	even	6		315.2.bb.b.269.10	yes	24
105.104	even	2	inner	2205.2.g.b.2204.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bb.b.89.3	✓	24	21.11	odd	6
315.2.bb.b.89.4	yes	24	35.4	even	6
315.2.bb.b.89.9	yes	24	105.74	odd	6
315.2.bb.b.89.10	yes	24	7.4	even	3
315.2.bb.b.269.3	yes	24	35.19	odd	6
315.2.bb.b.269.4	yes	24	21.5	even	6
315.2.bb.b.269.9	yes	24	7.5	odd	6
315.2.bb.b.269.10	yes	24	105.89	even	6
1575.2.bk.i.26.3		24	105.32	even	12
1575.2.bk.i.26.4		24	35.18	odd	12
1575.2.bk.i.26.9		24	105.53	even	12
1575.2.bk.i.26.10		24	35.32	odd	12
1575.2.bk.i.1151.3		24	35.12	even	12
1575.2.bk.i.1151.4		24	105.68	odd	12
1575.2.bk.i.1151.9		24	35.33	even	12
1575.2.bk.i.1151.10		24	105.47	odd	12
2205.2.g.b.2204.5		24	1.1	even	1	trivial
2205.2.g.b.2204.6		24	7.6	odd	2	inner
2205.2.g.b.2204.7		24	15.14	odd	2	inner
2205.2.g.b.2204.8		24	105.104	even	2	inner
2205.2.g.b.2204.17		24	5.4	even	2	inner
2205.2.g.b.2204.18		24	35.34	odd	2	inner
2205.2.g.b.2204.19		24	3.2	odd	2	inner
2205.2.g.b.2204.20		24	21.20	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:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2205.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.91314 q^{2} +1.66012 q^{4} +(-0.630583 - 2.14531i) q^{5} +0.650234 q^{8} +O(q^{10})\)	\(q-1.91314 q^{2} +1.66012 q^{4} +(-0.630583 - 2.14531i) q^{5} +0.650234 q^{8} +(1.20640 + 4.10429i) q^{10} -3.22740i q^{11} +4.86146 q^{13} -4.56424 q^{16} +0.729142i q^{17} +7.87149i q^{19} +(-1.04685 - 3.56148i) q^{20} +6.17449i q^{22} +4.86820 q^{23} +(-4.20473 + 2.70560i) q^{25} -9.30068 q^{26} +7.75958i q^{29} -1.42896i q^{31} +7.43158 q^{32} -1.39495i q^{34} +3.16538i q^{37} -15.0593i q^{38} +(-0.410026 - 1.39495i) q^{40} -1.40264 q^{41} +6.42489i q^{43} -5.35789i q^{44} -9.31357 q^{46} -4.87049i q^{47} +(8.04426 - 5.17620i) q^{50} +8.07063 q^{52} -1.52147 q^{53} +(-6.92379 + 2.03515i) q^{55} -14.8452i q^{58} -6.30162 q^{59} +2.37224i q^{61} +2.73380i q^{62} -5.08921 q^{64} +(-3.06556 - 10.4294i) q^{65} +11.1259i q^{67} +1.21047i q^{68} +10.1351i q^{71} +13.8239 q^{73} -6.05583i q^{74} +13.0676i q^{76} +3.98037 q^{79} +(2.87813 + 9.79171i) q^{80} +2.68346 q^{82} -4.19208i q^{83} +(1.56424 - 0.459785i) q^{85} -12.2917i q^{86} -2.09857i q^{88} -11.2734 q^{89} +8.08181 q^{92} +9.31795i q^{94} +(16.8868 - 4.96363i) q^{95} -2.21388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4}+O(q^{10}) \) 24 * q + 48 * q^4	\( 24 q + 48 q^{4} - 24 q^{25} + 48 q^{46} + 48 q^{64} + 120 q^{79} - 72 q^{85}+O(q^{100}) \) 24 * q + 48 * q^4 - 24 * q^25 + 48 * q^46 + 48 * q^64 + 120 * q^79 - 72 * q^85

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