LMFDB - Embedded newform 4620.2.m.b.1121.43

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1121,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4620.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.m (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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.43
Character	$\chi$	$=$	4620.1121
Dual form			4620.2.m.b.1121.44

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.49249 - 0.878897i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.45508 - 2.62350i) q^{9} +O(q^{10})$	\(q+(1.49249 - 0.878897i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.45508 - 2.62350i) q^{9} +(-1.95498 - 2.67919i) q^{11} +4.39367i q^{13} +(0.878897 + 1.49249i) q^{15} -4.59972 q^{17} +4.38128i q^{19} +(-0.878897 - 1.49249i) q^{21} -1.04181i q^{23} -1.00000 q^{25} +(-0.134086 - 5.19442i) q^{27} -0.197371 q^{29} -10.4887 q^{31} +(-5.27253 - 2.28044i) q^{33} +1.00000 q^{35} -9.08458 q^{37} +(3.86158 + 6.55753i) q^{39} -5.58243 q^{41} +3.88396i q^{43} +(2.62350 + 1.45508i) q^{45} -3.19088i q^{47} -1.00000 q^{49} +(-6.86506 + 4.04268i) q^{51} +1.86346i q^{53} +(2.67919 - 1.95498i) q^{55} +(3.85069 + 6.53904i) q^{57} +3.21957i q^{59} -8.75779i q^{61} +(-2.62350 - 1.45508i) q^{63} -4.39367 q^{65} -1.97450 q^{67} +(-0.915643 - 1.55489i) q^{69} +6.82276i q^{71} -8.08786i q^{73} +(-1.49249 + 0.878897i) q^{75} +(-2.67919 + 1.95498i) q^{77} +0.740791i q^{79} +(-4.76548 - 7.63480i) q^{81} +3.94465 q^{83} -4.59972i q^{85} +(-0.294575 + 0.173469i) q^{87} -14.8884i q^{89} +4.39367 q^{91} +(-15.6543 + 9.21847i) q^{93} -4.38128 q^{95} +3.91856 q^{97} +(-9.87350 + 1.23047i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) $ 48 * q - 4 * q^3 + 6 * q^9	$ 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) $ 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.49249	−	0.878897i	0.861692	−	0.507431i
$3$	1.49249	−	0.878897i	0.861692	−	0.507431i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.45508	−	2.62350i	0.485027	−	0.874499i
$10$		0			0
$11$	−1.95498	−	2.67919i	−0.589450	−	0.807805i
$11$	−1.95498	−	2.67919i	−0.589450	−	0.807805i
$12$		0			0
$13$			4.39367i			1.21858i	0.792946	+	0.609292i	$0.208546\pi$
$13$			4.39367i			1.21858i	−0.792946	+	0.609292i	$0.791454\pi$
$14$		0			0
$15$	0.878897	+	1.49249i	0.226930	+	0.385360i
$16$		0			0
$17$	−4.59972			−1.11560			−0.557798	−	0.829977i	$-0.688354\pi$
$17$	−4.59972			−1.11560			−0.557798	+	0.829977i	$0.688354\pi$
$18$		0			0
$19$			4.38128i			1.00513i	0.864538	+	0.502567i	$0.167611\pi$
$19$			4.38128i			1.00513i	−0.864538	+	0.502567i	$0.832389\pi$
$20$		0			0
$21$	−0.878897	−	1.49249i	−0.191791	−	0.325689i
$22$		0			0
$23$		−	1.04181i		−	0.217232i	−0.994084	−	0.108616i	$-0.965358\pi$
$23$		−	1.04181i		−	0.217232i	0.994084	−	0.108616i	$-0.0346419\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−0.134086	−	5.19442i	−0.0258048	−	0.999667i
$28$		0			0
$29$	−0.197371			−0.0366509			−0.0183254	−	0.999832i	$-0.505833\pi$
$29$	−0.197371			−0.0366509			−0.0183254	+	0.999832i	$0.505833\pi$
$30$		0			0
$31$	−10.4887			−1.88382			−0.941912	−	0.335860i	$-0.890973\pi$
$31$	−10.4887			−1.88382			−0.941912	+	0.335860i	$0.890973\pi$
$32$		0			0
$33$	−5.27253	−	2.28044i	−0.917830	−	0.396974i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−9.08458			−1.49350			−0.746748	−	0.665107i	$-0.768386\pi$
$37$	−9.08458			−1.49350			−0.746748	+	0.665107i	$0.768386\pi$
$38$		0			0
$39$	3.86158	+	6.55753i	0.618348	+	1.05005i
$40$		0			0
$41$	−5.58243			−0.871829			−0.435914	−	0.899988i	$-0.643575\pi$
$41$	−5.58243			−0.871829			−0.435914	+	0.899988i	$0.643575\pi$
$42$		0			0
$43$			3.88396i			0.592298i	0.955142	+	0.296149i	$0.0957025\pi$
$43$			3.88396i			0.592298i	−0.955142	+	0.296149i	$0.904297\pi$
$44$		0			0
$45$	2.62350	+	1.45508i	0.391088	+	0.216911i
$46$		0			0
$47$		−	3.19088i		−	0.465438i	−0.972544	−	0.232719i	$-0.925238\pi$
$47$		−	3.19088i		−	0.465438i	0.972544	−	0.232719i	$-0.0747622\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−6.86506	+	4.04268i	−0.961301	+	0.566089i
$52$		0			0
$53$			1.86346i			0.255967i	0.991776	+	0.127983i	$0.0408504\pi$
$53$			1.86346i			0.255967i	−0.991776	+	0.127983i	$0.959150\pi$
$54$		0			0
$55$	2.67919	−	1.95498i	0.361261	−	0.263610i
$56$		0			0
$57$	3.85069	+	6.53904i	0.510037	+	0.866117i
$58$		0			0
$59$			3.21957i			0.419152i	0.977792	+	0.209576i	$0.0672083\pi$
$59$			3.21957i			0.419152i	−0.977792	+	0.209576i	$0.932792\pi$
$60$		0			0
$61$		−	8.75779i		−	1.12132i	−0.828046	−	0.560660i	$-0.810547\pi$
$61$		−	8.75779i		−	1.12132i	0.828046	−	0.560660i	$-0.189453\pi$
$62$		0			0
$63$	−2.62350	−	1.45508i	−0.330530	−	0.183323i
$64$		0			0
$65$	−4.39367			−0.544968
$66$		0			0
$67$	−1.97450			−0.241224			−0.120612	−	0.992700i	$-0.538486\pi$
$67$	−1.97450			−0.241224			−0.120612	+	0.992700i	$0.538486\pi$
$68$		0			0
$69$	−0.915643	−	1.55489i	−0.110230	−	0.187187i
$70$		0			0
$71$			6.82276i			0.809712i	0.914380	+	0.404856i	$0.132678\pi$
$71$			6.82276i			0.809712i	−0.914380	+	0.404856i	$0.867322\pi$
$72$		0			0
$73$		−	8.08786i		−	0.946612i	−0.880898	−	0.473306i	$-0.843060\pi$
$73$		−	8.08786i		−	0.946612i	0.880898	−	0.473306i	$-0.156940\pi$
$74$		0			0
$75$	−1.49249	+	0.878897i	−0.172338	+	0.101486i
$76$		0			0
$77$	−2.67919	+	1.95498i	−0.305322	+	0.222791i
$78$		0			0
$79$			0.740791i			0.0833455i	0.999131	+	0.0416727i	$0.0132687\pi$
$79$			0.740791i			0.0833455i	−0.999131	+	0.0416727i	$0.986731\pi$
$80$		0			0
$81$	−4.76548	−	7.63480i	−0.529498	−	0.848311i
$82$		0			0
$83$	3.94465			0.432982			0.216491	−	0.976285i	$-0.430539\pi$
$83$	3.94465			0.432982			0.216491	+	0.976285i	$0.430539\pi$
$84$		0			0
$85$		−	4.59972i		−	0.498910i
$86$		0			0
$87$	−0.294575	+	0.173469i	−0.0315818	+	0.0185978i
$88$		0			0
$89$		−	14.8884i		−	1.57817i	−0.614285	−	0.789084i	$-0.710555\pi$
$89$		−	14.8884i		−	1.57817i	0.614285	−	0.789084i	$-0.289445\pi$
$90$		0			0
$91$	4.39367			0.460582
$92$		0			0
$93$	−15.6543	+	9.21847i	−1.62328	+	0.955911i
$94$		0			0
$95$	−4.38128			−0.449510
$96$		0			0
$97$	3.91856			0.397869			0.198935	−	0.980013i	$-0.436252\pi$
$97$	3.91856			0.397869			0.198935	+	0.980013i	$0.436252\pi$
$98$		0			0
$99$	−9.87350	+	1.23047i	−0.992324	+	0.123667i
$100$		0			0
$101$	8.18413			0.814352			0.407176	−	0.913350i	$-0.366514\pi$
$101$	8.18413			0.814352			0.407176	+	0.913350i	$0.366514\pi$
$102$		0			0
$103$	0.874025			0.0861202			0.0430601	−	0.999072i	$-0.486289\pi$
$103$	0.874025			0.0861202			0.0430601	+	0.999072i	$0.486289\pi$
$104$		0			0
$105$	1.49249	−	0.878897i	0.145653	−	0.0857716i
$106$		0			0
$107$	−7.69199			−0.743613			−0.371806	−	0.928310i	$-0.621261\pi$
$107$	−7.69199			−0.743613			−0.371806	+	0.928310i	$0.621261\pi$
$108$		0			0
$109$			18.3281i			1.75551i	0.479110	+	0.877755i	$0.340959\pi$
$109$			18.3281i			1.75551i	−0.479110	+	0.877755i	$0.659041\pi$
$110$		0			0
$111$	−13.5587	+	7.98441i	−1.28693	+	0.757847i
$112$		0			0
$113$		−	10.8682i		−	1.02239i	−0.859465	−	0.511195i	$-0.829203\pi$
$113$		−	10.8682i		−	1.02239i	0.859465	−	0.511195i	$-0.170797\pi$
$114$		0			0
$115$	1.04181			0.0971492
$116$		0			0
$117$	11.5268	+	6.39314i	1.06565	+	0.591046i
$118$		0			0
$119$			4.59972i			0.421656i
$120$		0			0
$121$	−3.35607	+	10.4755i	−0.305098	+	0.952321i
$122$		0			0
$123$	−8.33174	+	4.90638i	−0.751248	+	0.442393i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			4.73853i			0.420477i	0.977650	+	0.210238i	$0.0674240\pi$
$127$			4.73853i			0.420477i	−0.977650	+	0.210238i	$0.932576\pi$
$128$		0			0
$129$	3.41360	+	5.79679i	0.300551	+	0.510379i
$130$		0			0
$131$	−12.5088			−1.09290			−0.546450	−	0.837491i	$-0.684021\pi$
$131$	−12.5088			−1.09290			−0.546450	+	0.837491i	$0.684021\pi$
$132$		0			0
$133$	4.38128			0.379905
$134$		0			0
$135$	5.19442	−	0.134086i	0.447065	−	0.0115402i
$136$		0			0
$137$			8.28011i			0.707418i	0.935356	+	0.353709i	$0.115080\pi$
$137$			8.28011i			0.707418i	−0.935356	+	0.353709i	$0.884920\pi$
$138$		0			0
$139$			7.21397i			0.611881i	0.952051	+	0.305940i	$0.0989708\pi$
$139$			7.21397i			0.611881i	−0.952051	+	0.305940i	$0.901029\pi$
$140$		0			0
$141$	−2.80446	−	4.76237i	−0.236178	−	0.401064i
$142$		0			0
$143$	11.7715	−	8.58956i	0.984379	−	0.718295i
$144$		0			0
$145$		−	0.197371i		−	0.0163908i
$146$		0			0
$147$	−1.49249	+	0.878897i	−0.123099	+	0.0724902i
$148$		0			0
$149$	−21.1248			−1.73061			−0.865304	−	0.501247i	$-0.832875\pi$
$149$	−21.1248			−1.73061			−0.865304	+	0.501247i	$0.832875\pi$
$150$		0			0
$151$			16.2137i			1.31945i	0.751506	+	0.659726i	$0.229328\pi$
$151$			16.2137i			1.31945i	−0.751506	+	0.659726i	$0.770672\pi$
$152$		0			0
$153$	−6.69296	+	12.0674i	−0.541094	+	0.975588i
$154$		0			0
$155$		−	10.4887i		−	0.842472i
$156$		0			0
$157$	−0.261869			−0.0208994			−0.0104497	−	0.999945i	$-0.503326\pi$
$157$	−0.261869			−0.0208994			−0.0104497	+	0.999945i	$0.503326\pi$
$158$		0			0
$159$	1.63779	+	2.78121i	0.129885	+	0.220564i
$160$		0			0
$161$	−1.04181			−0.0821061
$162$		0			0
$163$	4.97178			0.389420			0.194710	−	0.980861i	$-0.437623\pi$
$163$	4.97178			0.389420			0.194710	+	0.980861i	$0.437623\pi$
$164$		0			0
$165$	2.28044	−	5.27253i	0.177532	−	0.410466i
$166$		0			0
$167$	11.6228			0.899402			0.449701	−	0.893179i	$-0.351531\pi$
$167$	11.6228			0.899402			0.449701	+	0.893179i	$0.351531\pi$
$168$		0			0
$169$	−6.30434			−0.484949
$170$		0			0
$171$	11.4943	+	6.37511i	0.878989	+	0.487517i
$172$		0			0
$173$	−15.5802			−1.18454			−0.592272	−	0.805738i	$-0.701769\pi$
$173$	−15.5802			−1.18454			−0.592272	+	0.805738i	$0.701769\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	2.82967	+	4.80518i	0.212691	+	0.361180i
$178$		0			0
$179$			8.32132i			0.621965i	0.950416	+	0.310982i	$0.100658\pi$
$179$			8.32132i			0.621965i	−0.950416	+	0.310982i	$0.899342\pi$
$180$		0			0
$181$	15.3565			1.14144			0.570721	−	0.821144i	$-0.306664\pi$
$181$	15.3565			1.14144			0.570721	+	0.821144i	$0.306664\pi$
$182$		0			0
$183$	−7.69719	−	13.0710i	−0.568993	−	0.966233i
$184$		0			0
$185$		−	9.08458i		−	0.667912i
$186$		0			0
$187$	8.99238	+	12.3235i	0.657588	+	0.901184i
$188$		0			0
$189$	−5.19442	+	0.134086i	−0.377839	+	0.00975329i
$190$		0			0
$191$			4.80881i			0.347954i	0.984750	+	0.173977i	$0.0556618\pi$
$191$			4.80881i			0.347954i	−0.984750	+	0.173977i	$0.944338\pi$
$192$		0			0
$193$		−	8.74072i		−	0.629171i	−0.949229	−	0.314585i	$-0.898135\pi$
$193$		−	8.74072i		−	0.629171i	0.949229	−	0.314585i	$-0.101865\pi$
$194$		0			0
$195$	−6.55753	+	3.86158i	−0.469594	+	0.276534i
$196$		0			0
$197$	12.0474			0.858340			0.429170	−	0.903224i	$-0.358806\pi$
$197$	12.0474			0.858340			0.429170	+	0.903224i	$0.358806\pi$
$198$		0			0
$199$	−20.7069			−1.46787			−0.733935	−	0.679219i	$-0.762318\pi$
$199$	−20.7069			−1.46787			−0.733935	+	0.679219i	$0.762318\pi$
$200$		0			0
$201$	−2.94693	+	1.73538i	−0.207860	+	0.122404i
$202$		0			0
$203$			0.197371i			0.0138527i
$204$		0			0
$205$		−	5.58243i		−	0.389894i
$206$		0			0
$207$	−2.73318	−	1.51592i	−0.189969	−	0.105363i
$208$		0			0
$209$	11.7383	−	8.56533i	0.811953	−	0.592476i
$210$		0			0
$211$		−	6.57673i		−	0.452761i	−0.974039	−	0.226380i	$-0.927311\pi$
$211$		−	6.57673i		−	0.452761i	0.974039	−	0.226380i	$-0.0726893\pi$
$212$		0			0
$213$	5.99650	+	10.1829i	0.410873	+	0.697723i
$214$		0			0
$215$	−3.88396			−0.264884
$216$		0			0
$217$			10.4887i			0.712018i
$218$		0			0
$219$	−7.10839	−	12.0711i	−0.480341	−	0.815688i
$220$		0			0
$221$		−	20.2097i		−	1.35945i
$222$		0			0
$223$	29.0724			1.94683			0.973415	−	0.229047i	$-0.0735611\pi$
$223$	29.0724			1.94683			0.973415	+	0.229047i	$0.0735611\pi$
$224$		0			0
$225$	−1.45508	+	2.62350i	−0.0970053	+	0.174900i
$226$		0			0
$227$	−12.5683			−0.834184			−0.417092	−	0.908864i	$-0.636951\pi$
$227$	−12.5683			−0.834184			−0.417092	+	0.908864i	$0.636951\pi$
$228$		0			0
$229$	−2.57211			−0.169970			−0.0849850	−	0.996382i	$-0.527084\pi$
$229$	−2.57211			−0.169970			−0.0849850	+	0.996382i	$0.527084\pi$
$230$		0			0
$231$	−2.28044	+	5.27253i	−0.150042	+	0.346907i
$232$		0			0
$233$	−21.5128			−1.40935			−0.704676	−	0.709529i	$-0.748908\pi$
$233$	−21.5128			−1.40935			−0.704676	+	0.709529i	$0.748908\pi$
$234$		0			0
$235$	3.19088			0.208150
$236$		0			0
$237$	0.651079	+	1.10563i	0.0422921	+	0.0718182i
$238$		0			0
$239$	13.8612			0.896605			0.448302	−	0.893882i	$-0.352029\pi$
$239$	13.8612			0.896605			0.448302	+	0.893882i	$0.352029\pi$
$240$		0			0
$241$		−	28.3568i		−	1.82662i	−0.407261	−	0.913312i	$-0.633516\pi$
$241$		−	28.3568i		−	1.82662i	0.407261	−	0.913312i	$-0.366484\pi$
$242$		0			0
$243$	−13.8227	−	7.20653i	−0.886724	−	0.462299i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	−19.2499			−1.22484
$248$		0			0
$249$	5.88737	−	3.46694i	0.373097	−	0.219709i
$250$		0			0
$251$			5.44040i			0.343395i	0.985150	+	0.171697i	$0.0549251\pi$
$251$			5.44040i			0.343395i	−0.985150	+	0.171697i	$0.945075\pi$
$252$		0			0
$253$	−2.79120	+	2.03672i	−0.175481	+	0.128048i
$254$		0			0
$255$	−4.04268	−	6.86506i	−0.253163	−	0.429907i
$256$		0			0
$257$		−	13.8635i		−	0.864781i	−0.901687	−	0.432390i	$-0.857670\pi$
$257$		−	13.8635i		−	0.864781i	0.901687	−	0.432390i	$-0.142330\pi$
$258$		0			0
$259$			9.08458i			0.564488i
$260$		0			0
$261$	−0.287191	+	0.517802i	−0.0177767	+	0.0320512i
$262$		0			0
$263$	−17.1868			−1.05979			−0.529893	−	0.848064i	$-0.677768\pi$
$263$	−17.1868			−1.05979			−0.529893	+	0.848064i	$0.677768\pi$
$264$		0			0
$265$	−1.86346			−0.114472
$266$		0			0
$267$	−13.0854	−	22.2209i	−0.800812	−	1.35990i
$268$		0			0
$269$		−	19.5046i		−	1.18922i	−0.804015	−	0.594609i	$-0.797307\pi$
$269$		−	19.5046i		−	1.18922i	0.804015	−	0.594609i	$-0.202693\pi$
$270$		0			0
$271$		−	7.04098i		−	0.427709i	−0.976865	−	0.213855i	$-0.931398\pi$
$271$		−	7.04098i		−	0.427709i	0.976865	−	0.213855i	$-0.0686019\pi$
$272$		0			0
$273$	6.55753	−	3.86158i	0.396880	−	0.233714i
$274$		0			0
$275$	1.95498	+	2.67919i	0.117890	+	0.161561i
$276$		0			0
$277$			5.29628i			0.318223i	0.987261	+	0.159111i	$0.0508629\pi$
$277$			5.29628i			0.318223i	−0.987261	+	0.159111i	$0.949137\pi$
$278$		0			0
$279$	−15.2619	+	27.5170i	−0.913705	+	1.64740i
$280$		0			0
$281$	−14.3165			−0.854051			−0.427025	−	0.904240i	$-0.640438\pi$
$281$	−14.3165			−0.854051			−0.427025	+	0.904240i	$0.640438\pi$
$282$		0			0
$283$			22.2257i			1.32118i	0.750746	+	0.660591i	$0.229694\pi$
$283$			22.2257i			1.32118i	−0.750746	+	0.660591i	$0.770306\pi$
$284$		0			0
$285$	−6.53904	+	3.85069i	−0.387339	+	0.228095i
$286$		0			0
$287$			5.58243i			0.329520i
$288$		0			0
$289$	4.15744			0.244555
$290$		0			0
$291$	5.84843	−	3.44401i	0.342841	−	0.201891i
$292$		0			0
$293$	−11.6280			−0.679315			−0.339658	−	0.940549i	$-0.610311\pi$
$293$	−11.6280			−0.679315			−0.339658	+	0.940549i	$0.610311\pi$
$294$		0			0
$295$	−3.21957			−0.187450
$296$		0			0
$297$	−13.6547	+	10.5143i	−0.792325	+	0.610099i
$298$		0			0
$299$	4.57737			0.264716
$300$		0			0
$301$	3.88396			0.223868
$302$		0			0
$303$	12.2148	−	7.19301i	0.701720	−	0.413228i
$304$		0			0
$305$	8.75779			0.501470
$306$		0			0
$307$		−	20.2197i		−	1.15400i	−0.816744	−	0.577001i	$-0.804223\pi$
$307$		−	20.2197i		−	1.15400i	0.816744	−	0.577001i	$-0.195777\pi$
$308$		0			0
$309$	1.30448	−	0.768178i	0.0742091	−	0.0437001i
$310$		0			0
$311$		−	24.9013i		−	1.41202i	−0.708201	−	0.706011i	$-0.750493\pi$
$311$		−	24.9013i		−	1.41202i	0.708201	−	0.706011i	$-0.249507\pi$
$312$		0			0
$313$	−10.5481			−0.596212			−0.298106	−	0.954533i	$-0.596355\pi$
$313$	−10.5481			−0.596212			−0.298106	+	0.954533i	$0.596355\pi$
$314$		0			0
$315$	1.45508	−	2.62350i	0.0819845	−	0.147817i
$316$		0			0
$317$			31.9506i			1.79453i	0.441495	+	0.897264i	$0.354448\pi$
$317$			31.9506i			1.79453i	−0.441495	+	0.897264i	$0.645552\pi$
$318$		0			0
$319$	0.385857	+	0.528794i	0.0216039	+	0.0296068i
$320$		0			0
$321$	−11.4803	+	6.76047i	−0.640765	+	0.377333i
$322$		0			0
$323$		−	20.1527i		−	1.12132i
$324$		0			0
$325$		−	4.39367i		−	0.243717i
$326$		0			0
$327$	16.1085	+	27.3545i	0.890801	+	1.51271i
$328$		0			0
$329$	−3.19088			−0.175919
$330$		0			0
$331$	16.2867			0.895198			0.447599	−	0.894234i	$-0.352279\pi$
$331$	16.2867			0.895198			0.447599	+	0.894234i	$0.352279\pi$
$332$		0			0
$333$	−13.2188	+	23.8334i	−0.724385	+	1.30606i
$334$		0			0
$335$		−	1.97450i		−	0.107878i
$336$		0			0
$337$		−	18.6865i		−	1.01792i	−0.860791	−	0.508959i	$-0.830031\pi$
$337$		−	18.6865i		−	1.01792i	0.860791	−	0.508959i	$-0.169969\pi$
$338$		0			0
$339$	−9.55200	−	16.2207i	−0.518793	−	0.880986i
$340$		0			0
$341$	20.5052	+	28.1011i	1.11042	+	1.52176i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	1.55489	−	0.915643i	0.0837127	−	0.0492966i
$346$		0			0
$347$	−9.53784			−0.512018			−0.256009	−	0.966674i	$-0.582408\pi$
$347$	−9.53784			−0.512018			−0.256009	+	0.966674i	$0.582408\pi$
$348$		0			0
$349$			1.47674i			0.0790479i	0.999219	+	0.0395239i	$0.0125841\pi$
$349$			1.47674i			0.0790479i	−0.999219	+	0.0395239i	$0.987416\pi$
$350$		0			0
$351$	22.8226	−	0.589128i	1.21818	−	0.0314453i
$352$		0			0
$353$		−	15.1774i		−	0.807810i	−0.914801	−	0.403905i	$-0.867653\pi$
$353$		−	15.1774i		−	0.807810i	0.914801	−	0.403905i	$-0.132347\pi$
$354$		0			0
$355$	−6.82276			−0.362114
$356$		0			0
$357$	4.04268	+	6.86506i	0.213961	+	0.363337i
$358$		0			0
$359$	19.0332			1.00454			0.502268	−	0.864712i	$-0.332499\pi$
$359$	19.0332			1.00454			0.502268	+	0.864712i	$0.332499\pi$
$360$		0			0
$361$	−0.195613			−0.0102954
$362$		0			0
$363$	4.19799	+	18.5843i	0.220337	+	0.975424i
$364$		0			0
$365$	8.08786			0.423338
$366$		0			0
$367$	−27.4033			−1.43044			−0.715222	−	0.698898i	$-0.753674\pi$
$367$	−27.4033			−1.43044			−0.715222	+	0.698898i	$0.753674\pi$
$368$		0			0
$369$	−8.12288	+	14.6455i	−0.422860	+	0.762414i
$370$		0			0
$371$	1.86346			0.0967463
$372$		0			0
$373$		−	12.6088i		−	0.652859i	−0.945222	−	0.326430i	$-0.894154\pi$
$373$		−	12.6088i		−	0.652859i	0.945222	−	0.326430i	$-0.105846\pi$
$374$		0			0
$375$	−0.878897	−	1.49249i	−0.0453860	−	0.0770721i
$376$		0			0
$377$		−	0.867183i		−	0.0446622i
$378$		0			0
$379$	5.72317			0.293979			0.146990	−	0.989138i	$-0.453042\pi$
$379$	5.72317			0.293979			0.146990	+	0.989138i	$0.453042\pi$
$380$		0			0
$381$	4.16468	+	7.07223i	0.213363	+	0.362321i
$382$		0			0
$383$			4.45622i			0.227702i	0.993498	+	0.113851i	$0.0363187\pi$
$383$			4.45622i			0.227702i	−0.993498	+	0.113851i	$0.963681\pi$
$384$		0			0
$385$	−1.95498	−	2.67919i	−0.0996352	−	0.136544i
$386$		0			0
$387$	10.1896	+	5.65147i	0.517964	+	0.287280i
$388$		0			0
$389$			18.9842i			0.962536i	0.876573	+	0.481268i	$0.159824\pi$
$389$			18.9842i			0.962536i	−0.876573	+	0.481268i	$0.840176\pi$
$390$		0			0
$391$			4.79203i			0.242344i
$392$		0			0
$393$	−18.6693	+	10.9940i	−0.941744	+	0.554572i
$394$		0			0
$395$	−0.740791			−0.0372732
$396$		0			0
$397$	25.8390			1.29682			0.648412	−	0.761290i	$-0.275434\pi$
$397$	25.8390			1.29682			0.648412	+	0.761290i	$0.275434\pi$
$398$		0			0
$399$	6.53904	−	3.85069i	0.327361	−	0.192776i
$400$		0			0
$401$			2.30502i			0.115107i	0.998342	+	0.0575536i	$0.0183300\pi$
$401$			2.30502i			0.115107i	−0.998342	+	0.0575536i	$0.981670\pi$
$402$		0			0
$403$		−	46.0838i		−	2.29560i
$404$		0			0
$405$	7.63480	−	4.76548i	0.379376	−	0.236799i
$406$		0			0
$407$	17.7602	+	24.3393i	0.880341	+	1.20645i
$408$		0			0
$409$		−	27.7907i		−	1.37416i	−0.726581	−	0.687080i	$-0.758892\pi$
$409$		−	27.7907i		−	1.37416i	0.726581	−	0.687080i	$-0.241108\pi$
$410$		0			0
$411$	7.27736	+	12.3580i	0.358966	+	0.609576i
$412$		0			0
$413$	3.21957			0.158424
$414$		0			0
$415$			3.94465i			0.193635i
$416$		0			0
$417$	6.34033	+	10.7668i	0.310487	+	0.527253i
$418$		0			0
$419$			28.6773i			1.40098i	0.713663	+	0.700489i	$0.247035\pi$
$419$			28.6773i			1.40098i	−0.713663	+	0.700489i	$0.752965\pi$
$420$		0			0
$421$	21.1478			1.03068			0.515340	−	0.856986i	$-0.327666\pi$
$421$	21.1478			1.03068			0.515340	+	0.856986i	$0.327666\pi$
$422$		0			0
$423$	−8.37127	−	4.64299i	−0.407025	−	0.225750i
$424$		0			0
$425$	4.59972			0.223119
$426$		0			0
$427$	−8.75779			−0.423819
$428$		0			0
$429$	10.0195	−	23.1658i	0.483746	−	1.11845i
$430$		0			0
$431$	18.8644			0.908667			0.454334	−	0.890832i	$-0.349877\pi$
$431$	18.8644			0.908667			0.454334	+	0.890832i	$0.349877\pi$
$432$		0			0
$433$	−16.3578			−0.786108			−0.393054	−	0.919515i	$-0.628581\pi$
$433$	−16.3578			−0.786108			−0.393054	+	0.919515i	$0.628581\pi$
$434$		0			0
$435$	−0.173469	−	0.294575i	−0.00831719	−	0.0141238i
$436$		0			0
$437$	4.56446			0.218348
$438$		0			0
$439$		−	7.02596i		−	0.335331i	−0.985844	−	0.167665i	$-0.946377\pi$
$439$		−	7.02596i		−	0.335331i	0.985844	−	0.167665i	$-0.0536228\pi$
$440$		0			0
$441$	−1.45508	+	2.62350i	−0.0692895	+	0.124928i
$442$		0			0
$443$		−	4.90721i		−	0.233149i	−0.993182	−	0.116574i	$-0.962809\pi$
$443$		−	4.90721i		−	0.233149i	0.993182	−	0.116574i	$-0.0371913\pi$
$444$		0			0
$445$	14.8884			0.705778
$446$		0			0
$447$	−31.5286	+	18.5665i	−1.49125	+	0.878165i
$448$		0			0
$449$			6.57828i			0.310448i	0.987879	+	0.155224i	$0.0496100\pi$
$449$			6.57828i			0.310448i	−0.987879	+	0.155224i	$0.950390\pi$
$450$		0			0
$451$	10.9136	+	14.9564i	0.513899	+	0.704268i
$452$		0			0
$453$	14.2502	+	24.1989i	0.669532	+	1.13696i
$454$		0			0
$455$			4.39367i			0.205978i
$456$		0			0
$457$			6.40274i			0.299508i	0.988723	+	0.149754i	$0.0478481\pi$
$457$			6.40274i			0.299508i	−0.988723	+	0.149754i	$0.952152\pi$
$458$		0			0
$459$	0.616756	+	23.8929i	0.0287877	+	1.11522i
$460$		0			0
$461$	16.1117			0.750399			0.375199	−	0.926944i	$-0.377574\pi$
$461$	16.1117			0.750399			0.375199	+	0.926944i	$0.377574\pi$
$462$		0			0
$463$	3.41187			0.158563			0.0792815	−	0.996852i	$-0.474737\pi$
$463$	3.41187			0.158563			0.0792815	+	0.996852i	$0.474737\pi$
$464$		0			0
$465$	−9.21847	−	15.6543i	−0.427497	−	0.725951i
$466$		0			0
$467$			13.0458i			0.603686i	0.953358	+	0.301843i	$0.0976019\pi$
$467$			13.0458i			0.603686i	−0.953358	+	0.301843i	$0.902398\pi$
$468$		0			0
$469$			1.97450i			0.0911740i
$470$		0			0
$471$	−0.390838	+	0.230156i	−0.0180089	+	0.0106050i
$472$		0			0
$473$	10.4058	−	7.59308i	0.478461	−	0.349130i
$474$		0			0
$475$		−	4.38128i		−	0.201027i
$476$		0			0
$477$	4.88880	+	2.71149i	0.223843	+	0.124151i
$478$		0			0
$479$	30.5850			1.39746			0.698732	−	0.715384i	$-0.253748\pi$
$479$	30.5850			1.39746			0.698732	+	0.715384i	$0.253748\pi$
$480$		0			0
$481$		−	39.9147i		−	1.81995i
$482$		0			0
$483$	−1.55489	+	0.915643i	−0.0707502	+	0.0416632i
$484$		0			0
$485$			3.91856i			0.177933i
$486$		0			0
$487$	−19.6780			−0.891694			−0.445847	−	0.895109i	$-0.647097\pi$
$487$	−19.6780			−0.891694			−0.445847	+	0.895109i	$0.647097\pi$
$488$		0			0
$489$	7.42036	−	4.36968i	0.335560	−	0.197604i
$490$		0			0
$491$	−28.9315			−1.30566			−0.652831	−	0.757504i	$-0.726419\pi$
$491$	−28.9315			−1.30566			−0.652831	+	0.757504i	$0.726419\pi$
$492$		0			0
$493$	0.907852			0.0408876
$494$		0			0
$495$	−1.23047	−	9.87350i	−0.0553054	−	0.443781i
$496$		0			0
$497$	6.82276			0.306042
$498$		0			0
$499$	−7.27991			−0.325894			−0.162947	−	0.986635i	$-0.552100\pi$
$499$	−7.27991			−0.325894			−0.162947	+	0.986635i	$0.552100\pi$
$500$		0			0
$501$	17.3470	−	10.2153i	0.775007	−	0.456385i
$502$		0			0
$503$	−34.5074			−1.53861			−0.769304	−	0.638883i	$-0.779397\pi$
$503$	−34.5074			−1.53861			−0.769304	+	0.638883i	$0.779397\pi$
$504$		0			0
$505$			8.18413i			0.364189i
$506$		0			0
$507$	−9.40919	+	5.54087i	−0.417877	+	0.246079i
$508$		0			0
$509$			38.9983i			1.72857i	0.503003	+	0.864285i	$0.332228\pi$
$509$			38.9983i			1.72857i	−0.503003	+	0.864285i	$0.667772\pi$
$510$		0			0
$511$	−8.08786			−0.357786
$512$		0			0
$513$	22.7582	−	0.587466i	1.00480	−	0.0259373i
$514$		0			0
$515$			0.874025i			0.0385141i
$516$		0			0
$517$	−8.54897	+	6.23812i	−0.375983	+	0.274352i
$518$		0			0
$519$	−23.2534	+	13.6934i	−1.02071	+	0.601074i
$520$		0			0
$521$			1.16653i			0.0511064i	0.999673	+	0.0255532i	$0.00813473\pi$
$521$			1.16653i			0.0511064i	−0.999673	+	0.0255532i	$0.991865\pi$
$522$		0			0
$523$		−	16.0000i		−	0.699633i	−0.936818	−	0.349817i	$-0.886244\pi$
$523$		−	16.0000i		−	0.699633i	0.936818	−	0.349817i	$-0.113756\pi$
$524$		0			0
$525$	0.878897	+	1.49249i	0.0383582	+	0.0651378i
$526$		0			0
$527$	48.2450			2.10159
$528$		0			0
$529$	21.9146			0.952810
$530$		0			0
$531$	8.44652	+	4.68473i	0.366548	+	0.203300i
$532$		0			0
$533$		−	24.5274i		−	1.06240i
$534$		0			0
$535$		−	7.69199i		−	0.332554i
$536$		0			0
$537$	7.31358	+	12.4195i	0.315604	+	0.535942i
$538$		0			0
$539$	1.95498	+	2.67919i	0.0842071	+	0.115401i
$540$		0			0
$541$			18.3166i			0.787493i	0.919219	+	0.393747i	$0.128821\pi$
$541$			18.3166i			0.787493i	−0.919219	+	0.393747i	$0.871179\pi$
$542$		0			0
$543$	22.9195	−	13.4968i	0.983571	−	0.579203i
$544$		0			0
$545$	−18.3281			−0.785088
$546$		0			0
$547$			36.6362i			1.56645i	0.621738	+	0.783225i	$0.286427\pi$
$547$			36.6362i			1.56645i	−0.621738	+	0.783225i	$0.713573\pi$
$548$		0			0
$549$	−22.9760	−	12.7433i	−0.980594	−	0.543870i
$550$		0			0
$551$		−	0.864738i		−	0.0368391i
$552$		0			0
$553$	0.740791			0.0315016
$554$		0			0
$555$	−7.98441	−	13.5587i	−0.338919	−	0.575534i
$556$		0			0
$557$	−37.9939			−1.60985			−0.804927	−	0.593374i	$-0.797796\pi$
$557$	−37.9939			−1.60985			−0.804927	+	0.593374i	$0.797796\pi$
$558$		0			0
$559$	−17.0648			−0.721766
$560$		0			0
$561$	24.2522	+	10.4894i	1.02393	+	0.442862i
$562$		0			0
$563$	17.2863			0.728529			0.364265	−	0.931295i	$-0.381320\pi$
$563$	17.2863			0.728529			0.364265	+	0.931295i	$0.381320\pi$
$564$		0			0
$565$	10.8682			0.457227
$566$		0			0
$567$	−7.63480	+	4.76548i	−0.320631	+	0.200132i
$568$		0			0
$569$	−23.3520			−0.978965			−0.489483	−	0.872013i	$-0.662814\pi$
$569$	−23.3520			−0.978965			−0.489483	+	0.872013i	$0.662814\pi$
$570$		0			0
$571$		−	19.2576i		−	0.805906i	−0.915221	−	0.402953i	$-0.867984\pi$
$571$		−	19.2576i		−	0.805906i	0.915221	−	0.402953i	$-0.132016\pi$
$572$		0			0
$573$	4.22645	+	7.17713i	0.176563	+	0.299829i
$574$		0			0
$575$			1.04181i			0.0434465i
$576$		0			0
$577$	−10.5947			−0.441064			−0.220532	−	0.975380i	$-0.570779\pi$
$577$	−10.5947			−0.441064			−0.220532	+	0.975380i	$0.570779\pi$
$578$		0			0
$579$	−7.68219	−	13.0455i	−0.319261	−	0.542152i
$580$		0			0
$581$		−	3.94465i		−	0.163652i
$582$		0			0
$583$	4.99257	−	3.64304i	0.206771	−	0.150879i
$584$		0			0
$585$	−6.39314	+	11.5268i	−0.264324	+	0.476574i
$586$		0			0
$587$			17.5848i			0.725803i	0.931828	+	0.362901i	$0.118214\pi$
$587$			17.5848i			0.725803i	−0.931828	+	0.362901i	$0.881786\pi$
$588$		0			0
$589$		−	45.9539i		−	1.89350i
$590$		0			0
$591$	17.9807	−	10.5884i	0.739625	−	0.435549i
$592$		0			0
$593$	−39.1488			−1.60765			−0.803825	−	0.594865i	$-0.797205\pi$
$593$	−39.1488			−1.60765			−0.803825	+	0.594865i	$0.797205\pi$
$594$		0			0
$595$	−4.59972			−0.188570
$596$		0			0
$597$	−30.9049	+	18.1992i	−1.26485	+	0.744844i
$598$		0			0
$599$		−	1.58269i		−	0.0646668i	−0.999477	−	0.0323334i	$-0.989706\pi$
$599$		−	1.58269i		−	0.0646668i	0.999477	−	0.0323334i	$-0.0102938\pi$
$600$		0			0
$601$			4.15185i			0.169358i	0.996408	+	0.0846788i	$0.0269864\pi$
$601$			4.15185i			0.169358i	−0.996408	+	0.0846788i	$0.973014\pi$
$602$		0			0
$603$	−2.87306	+	5.18010i	−0.117000	+	0.210950i
$604$		0			0
$605$	−10.4755	−	3.35607i	−0.425891	−	0.136444i
$606$		0			0
$607$			21.8871i			0.888371i	0.895935	+	0.444185i	$0.146507\pi$
$607$			21.8871i			0.888371i	−0.895935	+	0.444185i	$0.853493\pi$
$608$		0			0
$609$	0.173469	+	0.294575i	0.00702931	+	0.0119368i
$610$		0			0
$611$	14.0197			0.567176
$612$		0			0
$613$			18.0647i			0.729626i	0.931081	+	0.364813i	$0.118867\pi$
$613$			18.0647i			0.729626i	−0.931081	+	0.364813i	$0.881133\pi$
$614$		0			0
$615$	−4.90638	−	8.33174i	−0.197844	−	0.335968i
$616$		0			0
$617$			16.0774i			0.647250i	0.946185	+	0.323625i	$0.104902\pi$
$617$			16.0774i			0.647250i	−0.946185	+	0.323625i	$0.895098\pi$
$618$		0			0
$619$	3.81867			0.153485			0.0767426	−	0.997051i	$-0.475548\pi$
$619$	3.81867			0.153485			0.0767426	+	0.997051i	$0.475548\pi$
$620$		0			0
$621$	−5.41160	+	0.139692i	−0.217160	+	0.00560563i
$622$		0			0
$623$	−14.8884			−0.596492
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	9.99125	−	23.1004i	0.399012	−	0.922543i
$628$		0			0
$629$	41.7865			1.66614
$630$		0			0
$631$	34.0859			1.35694			0.678470	−	0.734628i	$-0.262643\pi$
$631$	34.0859			1.35694			0.678470	+	0.734628i	$0.262643\pi$
$632$		0			0
$633$	−5.78027	−	9.81574i	−0.229745	−	0.390140i
$634$		0			0
$635$	−4.73853			−0.188043
$636$		0			0
$637$		−	4.39367i		−	0.174084i
$638$		0			0
$639$	17.8995	+	9.92766i	0.708093	+	0.392732i
$640$		0			0
$641$			43.4941i			1.71791i	0.512048	+	0.858957i	$0.328887\pi$
$641$			43.4941i			1.71791i	−0.512048	+	0.858957i	$0.671113\pi$
$642$		0			0
$643$	31.8219			1.25493			0.627466	−	0.778644i	$-0.284092\pi$
$643$	31.8219			1.25493			0.627466	+	0.778644i	$0.284092\pi$
$644$		0			0
$645$	−5.79679	+	3.41360i	−0.228248	+	0.134410i
$646$		0			0
$647$		−	10.1058i		−	0.397301i	−0.980070	−	0.198650i	$-0.936344\pi$
$647$		−	10.1058i		−	0.397301i	0.980070	−	0.198650i	$-0.0636559\pi$
$648$		0			0
$649$	8.62582	−	6.29420i	0.338593	−	0.247069i
$650$		0			0
$651$	9.21847	+	15.6543i	0.361301	+	0.613541i
$652$		0			0
$653$		−	42.4304i		−	1.66043i	−0.557442	−	0.830216i	$-0.688217\pi$
$653$		−	42.4304i		−	1.66043i	0.557442	−	0.830216i	$-0.311783\pi$
$654$		0			0
$655$		−	12.5088i		−	0.488760i
$656$		0			0
$657$	−21.2185	−	11.7685i	−0.827812	−	0.459132i
$658$		0			0
$659$	6.05092			0.235710			0.117855	−	0.993031i	$-0.462398\pi$
$659$	6.05092			0.235710			0.117855	+	0.993031i	$0.462398\pi$
$660$		0			0
$661$	−18.3486			−0.713677			−0.356838	−	0.934166i	$-0.616145\pi$
$661$	−18.3486			−0.713677			−0.356838	+	0.934166i	$0.616145\pi$
$662$		0			0
$663$	−17.7622	−	30.1628i	−0.689827	−	1.17143i
$664$		0			0
$665$			4.38128i			0.169899i
$666$		0			0
$667$			0.205623i			0.00796175i
$668$		0			0
$669$	43.3904	−	25.5516i	1.67757	−	0.987883i
$670$		0			0
$671$	−23.4637	+	17.1213i	−0.905808	+	0.660962i
$672$		0			0
$673$			4.37548i			0.168662i	0.996438	+	0.0843311i	$0.0268754\pi$
$673$			4.37548i			0.168662i	−0.996438	+	0.0843311i	$0.973125\pi$
$674$		0			0
$675$	0.134086	+	5.19442i	0.00516095	+	0.199933i
$676$		0			0
$677$	27.1026			1.04164			0.520819	−	0.853667i	$-0.325627\pi$
$677$	27.1026			1.04164			0.520819	+	0.853667i	$0.325627\pi$
$678$		0			0
$679$		−	3.91856i		−	0.150381i
$680$		0			0
$681$	−18.7580	+	11.0462i	−0.718810	+	0.423291i
$682$		0			0
$683$		−	48.7141i		−	1.86399i	−0.362468	−	0.931996i	$-0.618066\pi$
$683$		−	48.7141i		−	1.86399i	0.362468	−	0.931996i	$-0.381934\pi$
$684$		0			0
$685$	−8.28011			−0.316367
$686$		0			0
$687$	−3.83886	+	2.26062i	−0.146462	+	0.0862481i
$688$		0			0
$689$	−8.18745			−0.311917
$690$		0			0
$691$	−15.8075			−0.601344			−0.300672	−	0.953728i	$-0.597211\pi$
$691$	−15.8075			−0.601344			−0.300672	+	0.953728i	$0.597211\pi$
$692$		0			0
$693$	1.23047	+	9.87350i	0.0467416	+	0.375063i
$694$		0			0
$695$	−7.21397			−0.273641
$696$		0			0
$697$	25.6776			0.972609
$698$		0			0
$699$	−32.1078	+	18.9076i	−1.21443	+	0.715150i
$700$		0			0
$701$	45.0635			1.70202			0.851012	−	0.525146i	$-0.175989\pi$
$701$	45.0635			1.70202			0.851012	+	0.525146i	$0.175989\pi$
$702$		0			0
$703$		−	39.8021i		−	1.50116i
$704$		0			0
$705$	4.76237	−	2.80446i	0.179361	−	0.105622i
$706$		0			0
$707$		−	8.18413i		−	0.307796i
$708$		0			0
$709$	25.0309			0.940055			0.470028	−	0.882652i	$-0.344244\pi$
$709$	25.0309			0.940055			0.470028	+	0.882652i	$0.344244\pi$
$710$		0			0
$711$	1.94346	+	1.07791i	0.0728856	+	0.0404248i
$712$		0			0
$713$			10.9272i			0.409227i
$714$		0			0
$715$	8.58956	+	11.7715i	0.321231	+	0.440228i
$716$		0			0
$717$	20.6877	−	12.1825i	0.772597	−	0.454965i
$718$		0			0
$719$		−	30.7208i		−	1.14569i	−0.819662	−	0.572847i	$-0.805839\pi$
$719$		−	30.7208i		−	1.14569i	0.819662	−	0.572847i	$-0.194161\pi$
$720$		0			0
$721$		−	0.874025i		−	0.0325504i
$722$		0			0
$723$	−24.9227	−	42.3224i	−0.926886	−	1.57399i
$724$		0			0
$725$	0.197371			0.00733018
$726$		0			0
$727$	40.2045			1.49110			0.745551	−	0.666448i	$-0.232186\pi$
$727$	40.2045			1.49110			0.745551	+	0.666448i	$0.232186\pi$
$728$		0			0
$729$	−26.9640	+	1.39299i	−0.998668	+	0.0515924i
$730$		0			0
$731$		−	17.8651i		−	0.660766i
$732$		0			0
$733$		−	43.7680i		−	1.61661i	−0.588766	−	0.808304i	$-0.700386\pi$
$733$		−	43.7680i		−	1.61661i	0.588766	−	0.808304i	$-0.299614\pi$
$734$		0			0
$735$	−0.878897	−	1.49249i	−0.0324186	−	0.0550515i
$736$		0			0
$737$	3.86012	+	5.29005i	0.142189	+	0.194862i
$738$		0			0
$739$			27.1645i			0.999262i	0.866238	+	0.499631i	$0.166531\pi$
$739$			27.1645i			0.999262i	−0.866238	+	0.499631i	$0.833469\pi$
$740$		0			0
$741$	−28.7304	+	16.9187i	−1.05544	+	0.621523i
$742$		0			0
$743$	−21.2406			−0.779242			−0.389621	−	0.920975i	$-0.627394\pi$
$743$	−21.2406			−0.779242			−0.389621	+	0.920975i	$0.627394\pi$
$744$		0			0
$745$		−	21.1248i		−	0.773952i
$746$		0			0
$747$	5.73978	−	10.3488i	0.210008	−	0.378642i
$748$		0			0
$749$			7.69199i			0.281059i
$750$		0			0
$751$	−46.7691			−1.70663			−0.853314	−	0.521397i	$-0.825411\pi$
$751$	−46.7691			−1.70663			−0.853314	+	0.521397i	$0.825411\pi$
$752$		0			0
$753$	4.78155	+	8.11976i	0.174249	+	0.295901i
$754$		0			0
$755$	−16.2137			−0.590077
$756$		0			0
$757$	12.4711			0.453269			0.226635	−	0.973980i	$-0.427228\pi$
$757$	12.4711			0.453269			0.226635	+	0.973980i	$0.427228\pi$
$758$		0			0
$759$	−2.37578	+	5.49297i	−0.0862355	+	0.199382i
$760$		0			0
$761$	26.6610			0.966460			0.483230	−	0.875493i	$-0.339463\pi$
$761$	26.6610			0.966460			0.483230	+	0.875493i	$0.339463\pi$
$762$		0			0
$763$	18.3281			0.663520
$764$		0			0
$765$	−12.0674	−	6.69296i	−0.436296	−	0.241985i
$766$		0			0
$767$	−14.1457			−0.510772
$768$		0			0
$769$		−	18.3239i		−	0.660777i	−0.943845	−	0.330388i	$-0.892820\pi$
$769$		−	18.3239i		−	0.660777i	0.943845	−	0.330388i	$-0.107180\pi$
$770$		0			0
$771$	−12.1846	−	20.6912i	−0.438817	−	0.745175i
$772$		0			0
$773$		−	22.1885i		−	0.798063i	−0.916937	−	0.399032i	$-0.869346\pi$
$773$		−	22.1885i		−	0.798063i	0.916937	−	0.399032i	$-0.130654\pi$
$774$		0			0
$775$	10.4887			0.376765
$776$		0			0
$777$	7.98441	+	13.5587i	0.286439	+	0.486415i
$778$		0			0
$779$		−	24.4582i		−	0.876305i
$780$		0			0
$781$	18.2794	−	13.3384i	0.654090	−	0.477285i
$782$		0			0
$783$	0.0264646	+	1.02523i	0.000945767	+	0.0366387i
$784$		0			0
$785$		−	0.261869i		−	0.00934651i
$786$		0			0
$787$			21.1068i			0.752375i	0.926544	+	0.376187i	$0.122765\pi$
$787$			21.1068i			0.752375i	−0.926544	+	0.376187i	$0.877235\pi$
$788$		0			0
$789$	−25.6513	+	15.1055i	−0.913209	+	0.537769i
$790$		0			0
$791$	−10.8682			−0.386427
$792$		0			0
$793$	38.4788			1.36642
$794$		0			0
$795$	−2.78121	+	1.63779i	−0.0986394	+	0.0580866i
$796$		0			0
$797$		−	46.8384i		−	1.65910i	−0.558433	−	0.829550i	$-0.688597\pi$
$797$		−	46.8384i		−	1.65910i	0.558433	−	0.829550i	$-0.311403\pi$
$798$		0			0
$799$			14.6772i			0.519241i
$800$		0			0
$801$	−39.0597	−	21.6638i	−1.38011	−	0.765454i
$802$		0			0
$803$	−21.6689	+	15.8116i	−0.764678	+	0.557980i
$804$		0			0
$805$		−	1.04181i		−	0.0367190i
$806$		0			0
$807$	−17.1426	−	29.1106i	−0.603447	−	1.02474i
$808$		0			0
$809$	23.6559			0.831698			0.415849	−	0.909434i	$-0.363485\pi$
$809$	23.6559			0.831698			0.415849	+	0.909434i	$0.363485\pi$
$810$		0			0
$811$			4.23151i			0.148588i	0.997236	+	0.0742942i	$0.0236704\pi$
$811$			4.23151i			0.148588i	−0.997236	+	0.0742942i	$0.976330\pi$
$812$		0			0
$813$	−6.18830	−	10.5086i	−0.217033	−	0.368554i
$814$		0			0
$815$			4.97178i			0.174154i
$816$		0			0
$817$	−17.0167			−0.595339
$818$		0			0
$819$	6.39314	−	11.5268i	0.223394	−	0.402779i
$820$		0			0
$821$	−18.0626			−0.630391			−0.315195	−	0.949027i	$-0.602070\pi$
$821$	−18.0626			−0.630391			−0.315195	+	0.949027i	$0.602070\pi$
$822$		0			0
$823$	29.9376			1.04356			0.521779	−	0.853080i	$-0.325268\pi$
$823$	29.9376			1.04356			0.521779	+	0.853080i	$0.325268\pi$
$824$		0			0
$825$	5.27253	+	2.28044i	0.183566	+	0.0793948i
$826$		0			0
$827$	5.48296			0.190661			0.0953306	−	0.995446i	$-0.469609\pi$
$827$	5.48296			0.190661			0.0953306	+	0.995446i	$0.469609\pi$
$828$		0			0
$829$	−16.5010			−0.573102			−0.286551	−	0.958065i	$-0.592509\pi$
$829$	−16.5010			−0.573102			−0.286551	+	0.958065i	$0.592509\pi$
$830$		0			0
$831$	4.65489	+	7.90467i	0.161476	+	0.274210i
$832$		0			0
$833$	4.59972			0.159371
$834$		0			0
$835$			11.6228i			0.402225i
$836$		0			0
$837$	1.40638	+	54.4827i	0.0486116	+	1.88320i
$838$		0			0
$839$			31.9456i			1.10288i	0.834213	+	0.551442i	$0.185922\pi$
$839$			31.9456i			1.10288i	−0.834213	+	0.551442i	$0.814078\pi$
$840$		0			0
$841$	−28.9610			−0.998657
$842$		0			0
$843$	−21.3673	+	12.5827i	−0.735929	+	0.433372i
$844$		0			0
$845$		−	6.30434i		−	0.216876i
$846$		0			0
$847$	10.4755	+	3.35607i	0.359944	+	0.115316i
$848$		0			0
$849$	19.5341	+	33.1718i	0.670409	+	1.13845i
$850$		0			0
$851$			9.46440i			0.324435i
$852$		0			0
$853$		−	36.6213i		−	1.25389i	−0.779064	−	0.626944i	$-0.784305\pi$
$853$		−	36.6213i		−	1.25389i	0.779064	−	0.626944i	$-0.215695\pi$
$854$		0			0
$855$	−6.37511	+	11.4943i	−0.218024	+	0.393096i
$856$		0			0
$857$	47.4696			1.62153			0.810765	−	0.585371i	$-0.199051\pi$
$857$	47.4696			1.62153			0.810765	+	0.585371i	$0.199051\pi$
$858$		0			0
$859$	16.0636			0.548083			0.274041	−	0.961718i	$-0.411639\pi$
$859$	16.0636			0.548083			0.274041	+	0.961718i	$0.411639\pi$
$860$		0			0
$861$	4.90638	+	8.33174i	0.167209	+	0.283945i
$862$		0			0
$863$			17.1766i			0.584698i	0.956312	+	0.292349i	$0.0944369\pi$
$863$			17.1766i			0.584698i	−0.956312	+	0.292349i	$0.905563\pi$
$864$		0			0
$865$		−	15.5802i		−	0.529744i
$866$		0			0
$867$	6.20495	−	3.65396i	0.210731	−	0.124095i
$868$		0			0
$869$	1.98472	−	1.44823i	0.0673269	−	0.0491280i
$870$		0			0
$871$		−	8.67530i		−	0.293951i
$872$		0			0
$873$	5.70182	−	10.2803i	0.192977	−	0.347937i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$		−	58.6042i		−	1.97892i	−0.144796	−	0.989462i	$-0.546253\pi$
$877$		−	58.6042i		−	1.97892i	0.144796	−	0.989462i	$-0.453747\pi$
$878$		0			0
$879$	−17.3547	+	10.2198i	−0.585361	+	0.344706i
$880$		0			0
$881$			3.94767i			0.133000i	0.997786	+	0.0665001i	$0.0211833\pi$
$881$			3.94767i			0.133000i	−0.997786	+	0.0665001i	$0.978817\pi$
$882$		0			0
$883$	2.53612			0.0853472			0.0426736	−	0.999089i	$-0.486412\pi$
$883$	2.53612			0.0853472			0.0426736	+	0.999089i	$0.486412\pi$
$884$		0			0
$885$	−4.80518	+	2.82967i	−0.161525	+	0.0951182i
$886$		0			0
$887$	−35.7616			−1.20076			−0.600378	−	0.799716i	$-0.704983\pi$
$887$	−35.7616			−1.20076			−0.600378	+	0.799716i	$0.704983\pi$
$888$		0			0
$889$	4.73853			0.158925
$890$		0			0
$891$	−11.1386	+	27.6935i	−0.373157	+	0.927768i
$892$		0			0
$893$	13.9801			0.467828
$894$		0			0
$895$	−8.32132			−0.278151
$896$		0			0
$897$	6.83170	−	4.02303i	0.228104	−	0.134325i
$898$		0			0
$899$	2.07016			0.0690438
$900$		0			0
$901$		−	8.57142i		−	0.285555i
$902$		0			0
$903$	5.79679	−	3.41360i	0.192905	−	0.113597i
$904$		0			0
$905$			15.3565i			0.510468i
$906$		0			0
$907$	−33.7081			−1.11926			−0.559630	−	0.828742i	$-0.689057\pi$
$907$	−33.7081			−1.11926			−0.559630	+	0.828742i	$0.689057\pi$
$908$		0			0
$909$	11.9086	−	21.4711i	0.394982	−	0.712150i
$910$		0			0
$911$		−	8.96589i		−	0.297053i	−0.988908	−	0.148527i	$-0.952547\pi$
$911$		−	8.96589i		−	0.297053i	0.988908	−	0.148527i	$-0.0474531\pi$
$912$		0			0
$913$	−7.71173	−	10.5685i	−0.255221	−	0.349765i
$914$		0			0
$915$	13.0710	−	7.69719i	0.432112	−	0.254461i
$916$		0			0
$917$			12.5088i			0.413078i
$918$		0			0
$919$		−	3.38846i		−	0.111775i	−0.998437	−	0.0558875i	$-0.982201\pi$
$919$		−	3.38846i		−	0.111775i	0.998437	−	0.0558875i	$-0.0177988\pi$
$920$		0			0
$921$	−17.7711	−	30.1778i	−0.585576	−	0.994394i
$922$		0			0
$923$	−29.9769			−0.986703
$924$		0			0
$925$	9.08458			0.298699
$926$		0			0
$927$	1.27178	−	2.29300i	0.0417706	−	0.0753121i
$928$		0			0
$929$			22.9924i			0.754356i	0.926141	+	0.377178i	$0.123106\pi$
$929$			22.9924i			0.754356i	−0.926141	+	0.377178i	$0.876894\pi$
$930$		0			0
$931$		−	4.38128i		−	0.143591i
$932$		0			0
$933$	−21.8856	−	37.1650i	−0.716504	−	1.21673i
$934$		0			0
$935$	−12.3235	+	8.99238i	−0.403022	+	0.294082i
$936$		0			0
$937$			43.9509i			1.43581i	0.696139	+	0.717907i	$0.254900\pi$
$937$			43.9509i			1.43581i	−0.696139	+	0.717907i	$0.745100\pi$
$938$		0			0
$939$	−15.7429	+	9.27066i	−0.513751	+	0.302537i
$940$		0			0
$941$	41.0136			1.33701			0.668503	−	0.743709i	$-0.266935\pi$
$941$	41.0136			1.33701			0.668503	+	0.743709i	$0.266935\pi$
$942$		0			0
$943$			5.81583i			0.189389i
$944$		0			0
$945$	−0.134086	−	5.19442i	−0.00436180	−	0.168975i
$946$		0			0
$947$		−	38.0914i		−	1.23780i	−0.785469	−	0.618901i	$-0.787578\pi$
$947$		−	38.0914i		−	1.23780i	0.785469	−	0.618901i	$-0.212422\pi$
$948$		0			0
$949$	35.5354			1.15353
$950$		0			0
$951$	28.0813	+	47.6862i	0.910599	+	1.54633i
$952$		0			0
$953$	−10.4511			−0.338544			−0.169272	−	0.985569i	$-0.554142\pi$
$953$	−10.4511			−0.338544			−0.169272	+	0.985569i	$0.554142\pi$
$954$		0			0
$955$	−4.80881			−0.155610
$956$		0			0
$957$	1.04064	+	0.450093i	0.0336393	+	0.0145494i
$958$		0			0
$959$	8.28011			0.267379
$960$		0			0
$961$	79.0125			2.54879
$962$		0			0
$963$	−11.1925	+	20.1799i	−0.360672	+	0.650289i
$964$		0			0
$965$	8.74072			0.281374
$966$		0			0
$967$		−	20.0590i		−	0.645053i	−0.946560	−	0.322527i	$-0.895468\pi$
$967$		−	20.0590i		−	0.645053i	0.946560	−	0.322527i	$-0.104532\pi$
$968$		0			0
$969$	−17.7121	−	30.0777i	−0.568995	−	0.966236i
$970$		0			0
$971$			30.4025i			0.975662i	0.872938	+	0.487831i	$0.162212\pi$
$971$			30.4025i			0.975662i	−0.872938	+	0.487831i	$0.837788\pi$
$972$		0			0
$973$	7.21397			0.231269
$974$		0			0
$975$	−3.86158	−	6.55753i	−0.123670	−	0.210009i
$976$		0			0
$977$		−	31.0047i		−	0.991928i	−0.868343	−	0.495964i	$-0.834815\pi$
$977$		−	31.0047i		−	0.991928i	0.868343	−	0.495964i	$-0.165185\pi$
$978$		0			0
$979$	−39.8888	+	29.1066i	−1.27485	+	0.930251i
$980$		0			0
$981$	48.0836	+	26.6688i	1.53519	+	0.851469i
$982$		0			0
$983$		−	7.79198i		−	0.248526i	−0.992249	−	0.124263i	$-0.960343\pi$
$983$		−	7.79198i		−	0.248526i	0.992249	−	0.124263i	$-0.0396566\pi$
$984$		0			0
$985$			12.0474i			0.383862i
$986$		0			0
$987$	−4.76237	+	2.80446i	−0.151588	+	0.0892668i
$988$		0			0
$989$	4.04634			0.128666
$990$		0			0
$991$	−57.3948			−1.82321			−0.911603	−	0.411071i	$-0.865155\pi$
$991$	−57.3948			−1.82321			−0.911603	+	0.411071i	$0.865155\pi$
$992$		0			0
$993$	24.3078	−	14.3143i	0.771385	−	0.454252i
$994$		0			0
$995$		−	20.7069i		−	0.656452i
$996$		0			0
$997$		−	56.0440i		−	1.77493i	−0.460875	−	0.887465i	$-0.652464\pi$
$997$		−	56.0440i		−	1.77493i	0.460875	−	0.887465i	$-0.347536\pi$
$998$		0			0
$999$	1.21811	+	47.1891i	0.0385393	+	1.49300i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.m.b.1121.43	yes	48
3.2	odd	2		4620.2.m.a.1121.44	yes	48
11.10	odd	2		4620.2.m.a.1121.43	✓	48
33.32	even	2	inner	4620.2.m.b.1121.44	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.m.a.1121.43	✓	48	11.10	odd	2
4620.2.m.a.1121.44	yes	48	3.2	odd	2
4620.2.m.b.1121.43	yes	48	1.1	even	1	trivial
4620.2.m.b.1121.44	yes	48	33.32	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.49249 - 0.878897i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.45508 - 2.62350i) q^{9} +O(q^{10})\)	\(q+(1.49249 - 0.878897i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.45508 - 2.62350i) q^{9} +(-1.95498 - 2.67919i) q^{11} +4.39367i q^{13} +(0.878897 + 1.49249i) q^{15} -4.59972 q^{17} +4.38128i q^{19} +(-0.878897 - 1.49249i) q^{21} -1.04181i q^{23} -1.00000 q^{25} +(-0.134086 - 5.19442i) q^{27} -0.197371 q^{29} -10.4887 q^{31} +(-5.27253 - 2.28044i) q^{33} +1.00000 q^{35} -9.08458 q^{37} +(3.86158 + 6.55753i) q^{39} -5.58243 q^{41} +3.88396i q^{43} +(2.62350 + 1.45508i) q^{45} -3.19088i q^{47} -1.00000 q^{49} +(-6.86506 + 4.04268i) q^{51} +1.86346i q^{53} +(2.67919 - 1.95498i) q^{55} +(3.85069 + 6.53904i) q^{57} +3.21957i q^{59} -8.75779i q^{61} +(-2.62350 - 1.45508i) q^{63} -4.39367 q^{65} -1.97450 q^{67} +(-0.915643 - 1.55489i) q^{69} +6.82276i q^{71} -8.08786i q^{73} +(-1.49249 + 0.878897i) q^{75} +(-2.67919 + 1.95498i) q^{77} +0.740791i q^{79} +(-4.76548 - 7.63480i) q^{81} +3.94465 q^{83} -4.59972i q^{85} +(-0.294575 + 0.173469i) q^{87} -14.8884i q^{89} +4.39367 q^{91} +(-15.6543 + 9.21847i) q^{93} -4.38128 q^{95} +3.91856 q^{97} +(-9.87350 + 1.23047i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) 48 * q - 4 * q^3 + 6 * q^9	\( 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) \) 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

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