LMFDB - Embedded newform 2700.2.j.i.1457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2700,2,Mod(593,2700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2700, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("2700.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.j (of order $4$, degree $2$, 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:	$21.5596085457$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 540)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1457.2
Root			$-0.323042 - 0.323042i$ of defining polynomial
Character	$\chi$	$=$	2700.1457
Dual form			2700.2.j.i.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.79129 - 2.79129i) q^{7} +O(q^{10})$	$q+(-2.79129 - 2.79129i) q^{7} +3.09557i q^{11} +(0.791288 - 0.791288i) q^{13} +(-2.51691 + 2.51691i) q^{17} +2.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +4.25290 q^{29} +8.58258 q^{31} +(-3.00000 - 3.00000i) q^{37} +7.99455i q^{41} +(-3.79129 + 3.79129i) q^{43} +(9.28672 - 9.28672i) q^{47} +8.58258i q^{49} +(9.21930 + 9.21930i) q^{53} +9.15188 q^{59} -12.1652 q^{61} +(7.00000 + 7.00000i) q^{67} -11.8711i q^{71} +(5.79129 - 5.79129i) q^{73} +(8.64064 - 8.64064i) q^{77} -12.1652i q^{79} +(-1.22474 - 1.22474i) q^{83} +16.7700 q^{89} -4.41742 q^{91} +(1.58258 + 1.58258i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7}+O(q^{10}) $ 8 * q - 4 * q^7	$ 8 q - 4 q^{7} - 12 q^{13} + 32 q^{31} - 24 q^{37} - 12 q^{43} - 24 q^{61} + 56 q^{67} + 28 q^{73} - 72 q^{91} - 24 q^{97}+O(q^{100}) $ 8 * q - 4 * q^7 - 12 * q^13 + 32 * q^31 - 24 * q^37 - 12 * q^43 - 24 * q^61 + 56 * q^67 + 28 * q^73 - 72 * q^91 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.79129	−	2.79129i	−1.05501	−	1.05501i	−0.998396	−	0.0566113i	$-0.981970\pi$
$7$	−2.79129	−	2.79129i	−1.05501	−	1.05501i	−0.0566113	−	0.998396i	$-0.518030\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.09557i			0.933351i	0.884429	+	0.466675i	$0.154548\pi$
$11$			3.09557i			0.933351i	−0.884429	+	0.466675i	$0.845452\pi$
$12$		0			0
$13$	0.791288	−	0.791288i	0.219464	−	0.219464i	−0.588809	−	0.808272i	$-0.700403\pi$
$13$	0.791288	−	0.791288i	0.219464	−	0.219464i	0.808272	+	0.588809i	$0.200403\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.51691	+	2.51691i	−0.610441	+	0.610441i	−0.943061	−	0.332620i	$-0.892067\pi$
$17$	−2.51691	+	2.51691i	−0.610441	+	0.610441i	0.332620	+	0.943061i	$0.392067\pi$
$18$		0			0
$19$			2.58258i			0.592483i	0.955113	+	0.296242i	$0.0957334\pi$
$19$			2.58258i			0.592483i	−0.955113	+	0.296242i	$0.904267\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.87083	−	1.87083i	−0.390095	−	0.390095i	0.484626	−	0.874721i	$-0.338956\pi$
$23$	−1.87083	−	1.87083i	−0.390095	−	0.390095i	−0.874721	+	0.484626i	$0.838956\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.25290			0.789743			0.394871	−	0.918736i	$-0.370789\pi$
$29$	4.25290			0.789743			0.394871	+	0.918736i	$0.370789\pi$
$30$		0			0
$31$	8.58258			1.54148			0.770738	−	0.637152i	$-0.219888\pi$
$31$	8.58258			1.54148			0.770738	+	0.637152i	$0.219888\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			7.99455i			1.24854i	0.781209	+	0.624270i	$0.214603\pi$
$41$			7.99455i			1.24854i	−0.781209	+	0.624270i	$0.785397\pi$
$42$		0			0
$43$	−3.79129	+	3.79129i	−0.578166	+	0.578166i	−0.934398	−	0.356232i	$-0.884061\pi$
$43$	−3.79129	+	3.79129i	−0.578166	+	0.578166i	0.356232	+	0.934398i	$0.384061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.28672	−	9.28672i	1.35461	−	1.35461i	0.474179	−	0.880428i	$-0.342745\pi$
$47$	9.28672	−	9.28672i	1.35461	−	1.35461i	0.880428	−	0.474179i	$-0.157255\pi$
$48$		0			0
$49$			8.58258i			1.22608i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.21930	+	9.21930i	1.26637	+	1.26637i	0.947951	+	0.318417i	$0.103151\pi$
$53$	9.21930	+	9.21930i	1.26637	+	1.26637i	0.318417	+	0.947951i	$0.396849\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.15188			1.19147			0.595736	−	0.803180i	$-0.296860\pi$
$59$	9.15188			1.19147			0.595736	+	0.803180i	$0.296860\pi$
$60$		0			0
$61$	−12.1652			−1.55759			−0.778794	−	0.627280i	$-0.784168\pi$
$61$	−12.1652			−1.55759			−0.778794	+	0.627280i	$0.784168\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	0.990766	−	0.135580i	$-0.0432899\pi$
$67$	7.00000	+	7.00000i	0.855186	+	0.855186i	−0.135580	+	0.990766i	$0.543290\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.8711i		−	1.40884i	−0.709786	−	0.704418i	$-0.751208\pi$
$71$		−	11.8711i		−	1.40884i	0.709786	−	0.704418i	$-0.248792\pi$
$72$		0			0
$73$	5.79129	−	5.79129i	0.677819	−	0.677819i	−0.281687	−	0.959506i	$-0.590894\pi$
$73$	5.79129	−	5.79129i	0.677819	−	0.677819i	0.959506	+	0.281687i	$0.0908941\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.64064	−	8.64064i	0.984692	−	0.984692i
$78$		0			0
$79$		−	12.1652i		−	1.36869i	−0.729160	−	0.684343i	$-0.760089\pi$
$79$		−	12.1652i		−	1.36869i	0.729160	−	0.684343i	$-0.239911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	0.636688	−	0.771121i	$-0.280304\pi$
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	−0.771121	+	0.636688i	$0.780304\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	16.7700			1.77762			0.888810	−	0.458276i	$-0.151533\pi$
$89$	16.7700			1.77762			0.888810	+	0.458276i	$0.151533\pi$
$90$		0			0
$91$	−4.41742			−0.463072
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.58258	+	1.58258i	0.160686	+	0.160686i	0.782871	−	0.622184i	$-0.213755\pi$
$97$	1.58258	+	1.58258i	0.160686	+	0.160686i	−0.622184	+	0.782871i	$0.713755\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			15.9891i			1.59098i	0.605970	+	0.795488i	$0.292785\pi$
$101$			15.9891i			1.59098i	−0.605970	+	0.795488i	$0.707215\pi$
$102$		0			0
$103$	−5.58258	+	5.58258i	−0.550068	+	0.550068i	−0.926460	−	0.376393i	$-0.877164\pi$
$103$	−5.58258	+	5.58258i	−0.550068	+	0.550068i	0.376393	+	0.926460i	$0.377164\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.05630	−	6.05630i	0.585485	−	0.585485i	−0.350920	−	0.936405i	$-0.614131\pi$
$107$	6.05630	−	6.05630i	0.585485	−	0.585485i	0.936405	+	0.350920i	$0.114131\pi$
$108$		0			0
$109$		−	11.7477i		−	1.12523i	−0.826720	−	0.562614i	$-0.809796\pi$
$109$		−	11.7477i		−	1.12523i	0.826720	−	0.562614i	$-0.190204\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.93280	+	9.93280i	0.934400	+	0.934400i	0.997977	−	0.0635773i	$-0.0202509\pi$
$113$	9.93280	+	9.93280i	0.934400	+	0.934400i	−0.0635773	+	0.997977i	$0.520251\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.0509			1.28804
$120$		0			0
$121$	1.41742			0.128857
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	2.58258	+	2.58258i	0.229167	+	0.229167i	0.812344	−	0.583178i	$-0.198191\pi$
$127$	2.58258	+	2.58258i	0.229167	+	0.229167i	−0.583178	+	0.812344i	$0.698191\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			5.03383i			0.439807i	0.975522	+	0.219904i	$0.0705743\pi$
$131$			5.03383i			0.439807i	−0.975522	+	0.219904i	$0.929426\pi$
$132$		0			0
$133$	7.20871	−	7.20871i	0.625075	−	0.625075i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.22474	−	1.22474i	0.104637	−	0.104637i	−0.652850	−	0.757487i	$-0.726427\pi$
$137$	1.22474	−	1.22474i	0.104637	−	0.104637i	0.757487	+	0.652850i	$0.226427\pi$
$138$		0			0
$139$		−	7.16515i		−	0.607740i	−0.952713	−	0.303870i	$-0.901721\pi$
$139$		−	7.16515i		−	0.607740i	0.952713	−	0.303870i	$-0.0982789\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.44949	+	2.44949i	0.204837	+	0.204837i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.511238			0.0418823			0.0209411	−	0.999781i	$-0.493334\pi$
$149$	0.511238			0.0418823			0.0209411	+	0.999781i	$0.493334\pi$
$150$		0			0
$151$	1.16515			0.0948187			0.0474093	−	0.998876i	$-0.484903\pi$
$151$	1.16515			0.0948187			0.0474093	+	0.998876i	$0.484903\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	17.3739	+	17.3739i	1.38659	+	1.38659i	0.832380	+	0.554205i	$0.186978\pi$
$157$	17.3739	+	17.3739i	1.38659	+	1.38659i	0.554205	+	0.832380i	$0.313022\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			10.4440i			0.823106i
$162$		0			0
$163$	−0.582576	+	0.582576i	−0.0456309	+	0.0456309i	−0.729554	−	0.683923i	$-0.760272\pi$
$163$	−0.582576	+	0.582576i	−0.0456309	+	0.0456309i	0.683923	+	0.729554i	$0.260272\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.02815	−	3.02815i	0.234325	−	0.234325i	−0.580170	−	0.814495i	$-0.697014\pi$
$167$	3.02815	−	3.02815i	0.234325	−	0.234325i	0.814495	+	0.580170i	$0.197014\pi$
$168$		0			0
$169$			11.7477i			0.903671i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.35959	−	1.35959i	−0.103368	−	0.103368i	0.653532	−	0.756899i	$-0.273287\pi$
$173$	−1.35959	−	1.35959i	−0.103368	−	0.103368i	−0.756899	+	0.653532i	$0.773287\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.16867			−0.386325			−0.193162	−	0.981167i	$-0.561874\pi$
$179$	−5.16867			−0.386325			−0.193162	+	0.981167i	$0.561874\pi$
$180$		0			0
$181$	−14.5826			−1.08391			−0.541957	−	0.840406i	$-0.682317\pi$
$181$	−14.5826			−1.08391			−0.541957	+	0.840406i	$0.682317\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−7.79129	−	7.79129i	−0.569755	−	0.569755i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3.74166i			0.270737i	0.990795	+	0.135368i	$0.0432218\pi$
$191$			3.74166i			0.270737i	−0.990795	+	0.135368i	$0.956778\pi$
$192$		0			0
$193$	−11.2087	+	11.2087i	−0.806821	+	0.806821i	−0.984151	−	0.177331i	$-0.943254\pi$
$193$	−11.2087	+	11.2087i	−0.806821	+	0.806821i	0.177331	+	0.984151i	$0.443254\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.12372	+	6.12372i	−0.436297	+	0.436297i	−0.890764	−	0.454467i	$-0.849830\pi$
$197$	−6.12372	+	6.12372i	−0.436297	+	0.436297i	0.454467	+	0.890764i	$0.349830\pi$
$198$		0			0
$199$		−	4.41742i		−	0.313143i	−0.987667	−	0.156571i	$-0.949956\pi$
$199$		−	4.41742i		−	0.313143i	0.987667	−	0.156571i	$-0.0500441\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.8711	−	11.8711i	−0.833185	−	0.833185i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−7.99455			−0.552995
$210$		0			0
$211$	7.83485			0.539373			0.269687	−	0.962948i	$-0.413080\pi$
$211$	7.83485			0.539373			0.269687	+	0.962948i	$0.413080\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−23.9564	−	23.9564i	−1.62627	−	1.62627i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3.98320i			0.267939i
$222$		0			0
$223$	6.16515	−	6.16515i	0.412849	−	0.412849i	−0.469881	−	0.882730i	$-0.655703\pi$
$223$	6.16515	−	6.16515i	0.412849	−	0.412849i	0.882730	+	0.469881i	$0.155703\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.6463	+	10.6463i	−0.706620	+	0.706620i	−0.965823	−	0.259203i	$-0.916540\pi$
$227$	−10.6463	+	10.6463i	−0.706620	+	0.706620i	0.259203	+	0.965823i	$0.416540\pi$
$228$		0			0
$229$			15.0000i			0.991228i	0.868543	+	0.495614i	$0.165057\pi$
$229$			15.0000i			0.991228i	−0.868543	+	0.495614i	$0.834943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.83723	−	6.83723i	−0.447922	−	0.447922i	0.446741	−	0.894663i	$-0.352584\pi$
$233$	−6.83723	−	6.83723i	−0.447922	−	0.447922i	−0.894663	+	0.446741i	$0.852584\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	29.7984			1.92750			0.963750	−	0.266808i	$-0.0859691\pi$
$239$	29.7984			1.92750			0.963750	+	0.266808i	$0.0859691\pi$
$240$		0			0
$241$	23.7477			1.52973			0.764863	−	0.644193i	$-0.222807\pi$
$241$	23.7477			1.52973			0.764863	+	0.644193i	$0.222807\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.04356	+	2.04356i	0.130029	+	0.130029i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			17.5510i			1.10781i	0.832581	+	0.553904i	$0.186862\pi$
$251$			17.5510i			1.10781i	−0.832581	+	0.553904i	$0.813138\pi$
$252$		0			0
$253$	5.79129	−	5.79129i	0.364095	−	0.364095i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.87083	+	1.87083i	−0.116699	+	0.116699i	−0.763045	−	0.646346i	$-0.776296\pi$
$257$	−1.87083	+	1.87083i	−0.116699	+	0.116699i	0.646346	+	0.763045i	$0.276296\pi$
$258$		0			0
$259$			16.7477i			1.04065i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.42157	+	9.42157i	0.580959	+	0.580959i	0.935167	−	0.354208i	$-0.115250\pi$
$263$	9.42157	+	9.42157i	0.580959	+	0.580959i	−0.354208	+	0.935167i	$0.615250\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.134846			−0.00822168			−0.00411084	−	0.999992i	$-0.501309\pi$
$269$	−0.134846			−0.00822168			−0.00411084	+	0.999992i	$0.501309\pi$
$270$		0			0
$271$	−13.7477			−0.835115			−0.417557	−	0.908651i	$-0.637114\pi$
$271$	−13.7477			−0.835115			−0.417557	+	0.908651i	$0.637114\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.79129	+	3.79129i	0.227796	+	0.227796i	0.811772	−	0.583975i	$-0.198503\pi$
$277$	3.79129	+	3.79129i	0.227796	+	0.227796i	−0.583975	+	0.811772i	$0.698503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	11.7362i		−	0.700124i	−0.936727	−	0.350062i	$-0.886161\pi$
$281$		−	11.7362i		−	0.700124i	0.936727	−	0.350062i	$-0.113839\pi$
$282$		0			0
$283$	−10.2087	+	10.2087i	−0.606845	+	0.606845i	−0.942120	−	0.335275i	$-0.891171\pi$
$283$	−10.2087	+	10.2087i	−0.606845	+	0.606845i	0.335275	+	0.942120i	$0.391171\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.3151	−	22.3151i	1.31722	−	1.31722i
$288$		0			0
$289$			4.33030i			0.254724i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.9610	+	12.9610i	0.757187	+	0.757187i	0.975810	−	0.218622i	$-0.0701563\pi$
$293$	12.9610	+	12.9610i	0.757187	+	0.757187i	−0.218622	+	0.975810i	$0.570156\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.96073			−0.171223
$300$		0			0
$301$	21.1652			1.21994
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−15.0000	−	15.0000i	−0.856095	−	0.856095i	0.134780	−	0.990876i	$-0.456967\pi$
$307$	−15.0000	−	15.0000i	−0.856095	−	0.856095i	−0.990876	+	0.134780i	$0.956967\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	30.0400i		−	1.70341i	−0.524022	−	0.851705i	$-0.675569\pi$
$311$		−	30.0400i		−	1.70341i	0.524022	−	0.851705i	$-0.324431\pi$
$312$		0			0
$313$	9.16515	−	9.16515i	0.518045	−	0.518045i	−0.398934	−	0.916979i	$-0.630620\pi$
$313$	9.16515	−	9.16515i	0.518045	−	0.518045i	0.916979	+	0.398934i	$0.130620\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.22474	−	1.22474i	0.0687885	−	0.0687885i	−0.671876	−	0.740664i	$-0.734511\pi$
$317$	1.22474	−	1.22474i	0.0687885	−	0.0687885i	0.740664	+	0.671876i	$0.234511\pi$
$318$		0			0
$319$			13.1652i			0.737107i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.50012	−	6.50012i	−0.361676	−	0.361676i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−51.8438			−2.85824
$330$		0			0
$331$	−0.417424			−0.0229437			−0.0114719	−	0.999934i	$-0.503652\pi$
$331$	−0.417424			−0.0229437			−0.0114719	+	0.999934i	$0.503652\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.95644	−	3.95644i	−0.215521	−	0.215521i	0.591087	−	0.806608i	$-0.298699\pi$
$337$	−3.95644	−	3.95644i	−0.215521	−	0.215521i	−0.806608	+	0.591087i	$0.798699\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			26.5680i			1.43874i
$342$		0			0
$343$	4.41742	−	4.41742i	0.238518	−	0.238518i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.9891	+	15.9891i	−0.858340	+	0.858340i	−0.991143	−	0.132802i	$-0.957602\pi$
$347$	−15.9891	+	15.9891i	−0.858340	+	0.858340i	0.132802	+	0.991143i	$0.457602\pi$
$348$		0			0
$349$		−	27.7477i		−	1.48530i	−0.669679	−	0.742651i	$-0.733568\pi$
$349$		−	27.7477i		−	1.48530i	0.669679	−	0.742651i	$-0.266432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.79796	+	9.79796i	0.521493	+	0.521493i	0.918022	−	0.396529i	$-0.129785\pi$
$353$	9.79796	+	9.79796i	0.521493	+	0.521493i	−0.396529	+	0.918022i	$0.629785\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	14.8318			0.782792			0.391396	−	0.920222i	$-0.371992\pi$
$359$	14.8318			0.782792			0.391396	+	0.920222i	$0.371992\pi$
$360$		0			0
$361$	12.3303			0.648963
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−13.2087	−	13.2087i	−0.689489	−	0.689489i	0.272630	−	0.962119i	$-0.412107\pi$
$367$	−13.2087	−	13.2087i	−0.689489	−	0.689489i	−0.962119	+	0.272630i	$0.912107\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	51.4674i		−	2.67206i
$372$		0			0
$373$	−20.7477	+	20.7477i	−1.07428	+	1.07428i	−0.0772661	+	0.997011i	$0.524619\pi$
$373$	−20.7477	+	20.7477i	−1.07428	+	1.07428i	−0.997011	+	0.0772661i	$0.975381\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.36526	−	3.36526i	0.173320	−	0.173320i
$378$		0			0
$379$			1.00000i			0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$			1.00000i			0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.10125	+	5.10125i	0.260662	+	0.260662i	0.825323	−	0.564661i	$-0.190993\pi$
$383$	5.10125	+	5.10125i	0.260662	+	0.260662i	−0.564661	+	0.825323i	$0.690993\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	17.0397			0.863948			0.431974	−	0.901886i	$-0.357817\pi$
$389$	17.0397			0.863948			0.431974	+	0.901886i	$0.357817\pi$
$390$		0			0
$391$	9.41742			0.476260
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.95644	+	5.95644i	0.298945	+	0.298945i	0.840601	−	0.541655i	$-0.182202\pi$
$397$	5.95644	+	5.95644i	0.298945	+	0.298945i	−0.541655	+	0.840601i	$0.682202\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.44949i			0.122322i	0.998128	+	0.0611608i	$0.0194803\pi$
$401$			2.44949i			0.122322i	−0.998128	+	0.0611608i	$0.980520\pi$
$402$		0			0
$403$	6.79129	−	6.79129i	0.338298	−	0.338298i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.28672	−	9.28672i	0.460326	−	0.460326i
$408$		0			0
$409$		−	6.16515i		−	0.304847i	−0.988315	−	0.152424i	$-0.951292\pi$
$409$		−	6.16515i		−	0.304847i	0.988315	−	0.152424i	$-0.0487078\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−25.5455	−	25.5455i	−1.25701	−	1.25701i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	18.0622			0.882396			0.441198	−	0.897410i	$-0.354554\pi$
$419$	18.0622			0.882396			0.441198	+	0.897410i	$0.354554\pi$
$420$		0			0
$421$	−20.5826			−1.00313			−0.501567	−	0.865119i	$-0.667243\pi$
$421$	−20.5826			−1.00313			−0.501567	+	0.865119i	$0.667243\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	33.9564	+	33.9564i	1.64327	+	1.64327i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	13.0284i		−	0.627555i	−0.949497	−	0.313777i	$-0.898405\pi$
$431$		−	13.0284i		−	0.627555i	0.949497	−	0.313777i	$-0.101595\pi$
$432$		0			0
$433$	−1.20871	+	1.20871i	−0.0580870	+	0.0580870i	−0.735554	−	0.677467i	$-0.763078\pi$
$433$	−1.20871	+	1.20871i	−0.0580870	+	0.0580870i	0.677467	+	0.735554i	$0.263078\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.83156	−	4.83156i	0.231125	−	0.231125i
$438$		0			0
$439$			7.83485i			0.373937i	0.982366	+	0.186968i	$0.0598662\pi$
$439$			7.83485i			0.373937i	−0.982366	+	0.186968i	$0.940134\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.713507	−	0.713507i	−0.0338997	−	0.0338997i	0.689954	−	0.723853i	$-0.257631\pi$
$443$	−0.713507	−	0.713507i	−0.0338997	−	0.0338997i	−0.723853	+	0.689954i	$0.757631\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−38.8154			−1.83181			−0.915907	−	0.401391i	$-0.868527\pi$
$449$	−38.8154			−1.83181			−0.915907	+	0.401391i	$0.868527\pi$
$450$		0			0
$451$	−24.7477			−1.16532
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.74773	−	4.74773i	−0.222089	−	0.222089i	0.587288	−	0.809378i	$-0.300195\pi$
$457$	−4.74773	−	4.74773i	−0.222089	−	0.222089i	−0.809378	+	0.587288i	$0.800195\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	3.74166i		−	0.174266i	−0.996197	−	0.0871332i	$-0.972229\pi$
$461$		−	3.74166i		−	0.174266i	0.996197	−	0.0871332i	$-0.0277706\pi$
$462$		0			0
$463$	23.7477	−	23.7477i	1.10365	−	1.10365i	0.109684	−	0.993967i	$-0.465016\pi$
$463$	23.7477	−	23.7477i	1.10365	−	1.10365i	0.993967	−	0.109684i	$-0.0349839\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.8261	+	12.8261i	−0.593522	+	0.593522i	−0.938581	−	0.345059i	$-0.887859\pi$
$467$	−12.8261	+	12.8261i	−0.593522	+	0.593522i	0.345059	+	0.938581i	$0.387859\pi$
$468$		0			0
$469$		−	39.0780i		−	1.80446i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.7362	−	11.7362i	−0.539632	−	0.539632i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−35.3435			−1.61488			−0.807442	−	0.589946i	$-0.799149\pi$
$479$	−35.3435			−1.61488			−0.807442	+	0.589946i	$0.799149\pi$
$480$		0			0
$481$	−4.74773			−0.216478
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	5.37386	+	5.37386i	0.243513	+	0.243513i	0.818302	−	0.574789i	$-0.194916\pi$
$487$	5.37386	+	5.37386i	0.243513	+	0.243513i	−0.574789	+	0.818302i	$0.694916\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			2.71918i			0.122715i	0.998116	+	0.0613575i	$0.0195430\pi$
$491$			2.71918i			0.122715i	−0.998116	+	0.0613575i	$0.980457\pi$
$492$		0			0
$493$	−10.7042	+	10.7042i	−0.482091	+	0.482091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−33.1355	+	33.1355i	−1.48633	+	1.48633i
$498$		0			0
$499$			9.41742i			0.421582i	0.977531	+	0.210791i	$0.0676039\pi$
$499$			9.41742i			0.421582i	−0.977531	+	0.210791i	$0.932396\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.73598	−	1.73598i	−0.0774037	−	0.0774037i	0.667345	−	0.744749i	$-0.267431\pi$
$503$	−1.73598	−	1.73598i	−0.0774037	−	0.0774037i	−0.744749	+	0.667345i	$0.767431\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−22.1803			−0.983122			−0.491561	−	0.870843i	$-0.663574\pi$
$509$	−22.1803			−0.983122			−0.491561	+	0.870843i	$0.663574\pi$
$510$		0			0
$511$	−32.3303			−1.43021
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	28.7477	+	28.7477i	1.26432	+	1.26432i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	15.2082i		−	0.666282i	−0.942877	−	0.333141i	$-0.891891\pi$
$521$		−	15.2082i		−	0.666282i	0.942877	−	0.333141i	$-0.108109\pi$
$522$		0			0
$523$	27.9564	−	27.9564i	1.22245	−	1.22245i	0.255691	−	0.966758i	$-0.417697\pi$
$523$	27.9564	−	27.9564i	1.22245	−	1.22245i	0.966758	−	0.255691i	$-0.0823030\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.6016	+	21.6016i	−0.940980	+	0.940980i
$528$		0			0
$529$		−	16.0000i		−	0.695652i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.32599	+	6.32599i	0.274009	+	0.274009i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−26.5680			−1.14436
$540$		0			0
$541$	−22.8348			−0.981747			−0.490873	−	0.871231i	$-0.663322\pi$
$541$	−22.8348			−0.981747			−0.490873	+	0.871231i	$0.663322\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.53901	+	2.53901i	0.108560	+	0.108560i	0.759301	−	0.650740i	$-0.225541\pi$
$547$	2.53901	+	2.53901i	0.108560	+	0.108560i	−0.650740	+	0.759301i	$0.725541\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			10.9834i			0.467910i
$552$		0			0
$553$	−33.9564	+	33.9564i	−1.44397	+	1.44397i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.77548	+	8.77548i	−0.371829	+	0.371829i	−0.868143	−	0.496314i	$-0.834686\pi$
$557$	−8.77548	+	8.77548i	−0.371829	+	0.371829i	0.496314	+	0.868143i	$0.334686\pi$
$558$		0			0
$559$			6.00000i			0.253773i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.9219	−	25.9219i	−1.09248	−	1.09248i	−0.995263	−	0.0972149i	$-0.969007\pi$
$563$	−25.9219	−	25.9219i	−1.09248	−	1.09248i	−0.0972149	−	0.995263i	$-0.530993\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.7478			−1.20517			−0.602585	−	0.798055i	$-0.705863\pi$
$569$	−28.7478			−1.20517			−0.602585	+	0.798055i	$0.705863\pi$
$570$		0			0
$571$	30.9129			1.29366			0.646832	−	0.762633i	$-0.276094\pi$
$571$	30.9129			1.29366			0.646832	+	0.762633i	$0.276094\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−9.37386	−	9.37386i	−0.390239	−	0.390239i	0.484534	−	0.874773i	$-0.338989\pi$
$577$	−9.37386	−	9.37386i	−0.390239	−	0.390239i	−0.874773	+	0.484534i	$0.838989\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			6.83723i			0.283656i
$582$		0			0
$583$	−28.5390	+	28.5390i	−1.18197	+	1.18197i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.202268	+	0.202268i	−0.00834851	+	0.00834851i	−0.711269	−	0.702920i	$-0.751879\pi$
$587$	−0.202268	+	0.202268i	−0.00834851	+	0.00834851i	0.702920	+	0.711269i	$0.251879\pi$
$588$		0			0
$589$			22.1652i			0.913299i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.2531	+	14.2531i	0.585306	+	0.585306i	0.936356	−	0.351051i	$-0.114176\pi$
$593$	14.2531	+	14.2531i	0.585306	+	0.585306i	−0.351051	+	0.936356i	$0.614176\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−29.0175			−1.18562			−0.592811	−	0.805342i	$-0.701982\pi$
$599$	−29.0175			−1.18562			−0.592811	+	0.805342i	$0.701982\pi$
$600$		0			0
$601$	−11.7477			−0.479200			−0.239600	−	0.970872i	$-0.577016\pi$
$601$	−11.7477			−0.479200			−0.239600	+	0.970872i	$0.577016\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	16.1216	+	16.1216i	0.654355	+	0.654355i	0.954039	−	0.299684i	$-0.0968811\pi$
$607$	16.1216	+	16.1216i	0.654355	+	0.654355i	−0.299684	+	0.954039i	$0.596881\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	14.6969i		−	0.594574i
$612$		0			0
$613$	−12.3303	+	12.3303i	−0.498016	+	0.498016i	−0.910820	−	0.412804i	$-0.864550\pi$
$613$	−12.3303	+	12.3303i	−0.498016	+	0.498016i	0.412804	+	0.910820i	$0.364550\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.2702	+	23.2702i	−0.936821	+	0.936821i	−0.998119	−	0.0612984i	$-0.980476\pi$
$617$	−23.2702	+	23.2702i	−0.936821	+	0.936821i	0.0612984	+	0.998119i	$0.480476\pi$
$618$		0			0
$619$		−	16.7477i		−	0.673148i	−0.941657	−	0.336574i	$-0.890732\pi$
$619$		−	16.7477i		−	0.673148i	0.941657	−	0.336574i	$-0.109268\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−46.8100	−	46.8100i	−1.87540	−	1.87540i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	15.1015			0.602135
$630$		0			0
$631$	16.1652			0.643525			0.321762	−	0.946820i	$-0.395725\pi$
$631$	16.1652			0.643525			0.321762	+	0.946820i	$0.395725\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	6.79129	+	6.79129i	0.269081	+	0.269081i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			29.6636i			1.17164i	0.810441	+	0.585820i	$0.199228\pi$
$641$			29.6636i			1.17164i	−0.810441	+	0.585820i	$0.800772\pi$
$642$		0			0
$643$	23.5390	−	23.5390i	0.928288	−	0.928288i	−0.0693072	−	0.997595i	$-0.522079\pi$
$643$	23.5390	−	23.5390i	0.928288	−	0.928288i	0.997595	+	0.0693072i	$0.0220789\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.9330	+	19.9330i	−0.783648	+	0.783648i	−0.980444	−	0.196796i	$-0.936946\pi$
$647$	−19.9330	+	19.9330i	−0.783648	+	0.783648i	0.196796	+	0.980444i	$0.436946\pi$
$648$		0			0
$649$			28.3303i			1.11206i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.0849	+	29.0849i	1.13818	+	1.13818i	0.988776	+	0.149404i	$0.0477355\pi$
$653$	29.0849	+	29.0849i	1.13818	+	1.13818i	0.149404	+	0.988776i	$0.452264\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.42157			0.367012			0.183506	−	0.983019i	$-0.441255\pi$
$659$	9.42157			0.367012			0.183506	+	0.983019i	$0.441255\pi$
$660$		0			0
$661$	23.4955			0.913867			0.456934	−	0.889501i	$-0.348948\pi$
$661$	23.4955			0.913867			0.456934	+	0.889501i	$0.348948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−7.95644	−	7.95644i	−0.308075	−	0.308075i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	37.6581i		−	1.45378i
$672$		0			0
$673$	29.9564	−	29.9564i	1.15474	−	1.15474i	0.169145	−	0.985591i	$-0.445900\pi$
$673$	29.9564	−	29.9564i	1.15474	−	1.15474i	0.985591	−	0.169145i	$-0.0541005\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.5396	+	13.5396i	−0.520370	+	0.520370i	−0.917683	−	0.397313i	$-0.869943\pi$
$677$	−13.5396	+	13.5396i	−0.520370	+	0.520370i	0.397313	+	0.917683i	$0.369943\pi$
$678$		0			0
$679$		−	8.83485i		−	0.339050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.0002	−	10.0002i	−0.382648	−	0.382648i	0.489407	−	0.872055i	$-0.337213\pi$
$683$	−10.0002	−	10.0002i	−0.382648	−	0.382648i	−0.872055	+	0.489407i	$0.837213\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	14.5902			0.555844
$690$		0			0
$691$	−33.6606			−1.28051			−0.640255	−	0.768163i	$-0.721171\pi$
$691$	−33.6606			−1.28051			−0.640255	+	0.768163i	$0.721171\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−20.1216	−	20.1216i	−0.762160	−	0.762160i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	2.71918i		−	0.102702i	−0.998681	−	0.0513510i	$-0.983647\pi$
$701$		−	2.71918i		−	0.102702i	0.998681	−	0.0513510i	$-0.0163527\pi$
$702$		0			0
$703$	7.74773	−	7.74773i	0.292211	−	0.292211i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	44.6302	−	44.6302i	1.67849	−	1.67849i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.0565	−	16.0565i	−0.601322	−	0.601322i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7.07878			−0.263994			−0.131997	−	0.991250i	$-0.542139\pi$
$719$	−7.07878			−0.263994			−0.131997	+	0.991250i	$0.542139\pi$
$720$		0			0
$721$	31.1652			1.16065
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−24.3303	−	24.3303i	−0.902361	−	0.902361i	0.0932790	−	0.995640i	$-0.470265\pi$
$727$	−24.3303	−	24.3303i	−0.902361	−	0.902361i	−0.995640	+	0.0932790i	$0.970265\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	19.0847i		−	0.705872i
$732$		0			0
$733$	21.9129	−	21.9129i	0.809371	−	0.809371i	−0.175168	−	0.984539i	$-0.556047\pi$
$733$	21.9129	−	21.9129i	0.809371	−	0.809371i	0.984539	+	0.175168i	$0.0560468\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.6690	+	21.6690i	−0.798188	+	0.798188i
$738$		0			0
$739$		−	20.5826i		−	0.757142i	−0.925572	−	0.378571i	$-0.876416\pi$
$739$		−	20.5826i		−	0.757142i	0.925572	−	0.378571i	$-0.123584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.0677	−	10.0677i	−0.369346	−	0.369346i	0.497892	−	0.867239i	$-0.334107\pi$
$743$	−10.0677	−	10.0677i	−0.369346	−	0.369346i	−0.867239	+	0.497892i	$0.834107\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−33.8098			−1.23538
$750$		0			0
$751$	15.0000			0.547358			0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000			0.547358			0.273679	+	0.961821i	$0.411759\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	0.379860	−	0.925044i	$-0.375972\pi$
$757$	−15.0000	−	15.0000i	−0.545184	−	0.545184i	−0.925044	+	0.379860i	$0.875972\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	12.0059i		−	0.435214i	−0.976037	−	0.217607i	$-0.930175\pi$
$761$		−	12.0059i		−	0.435214i	0.976037	−	0.217607i	$-0.0698250\pi$
$762$		0			0
$763$	−32.7913	+	32.7913i	−1.18712	+	1.18712i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.24177	−	7.24177i	0.261485	−	0.261485i
$768$		0			0
$769$			23.0000i			0.829401i	0.909958	+	0.414701i	$0.136114\pi$
$769$			23.0000i			0.829401i	−0.909958	+	0.414701i	$0.863886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	30.5119	+	30.5119i	1.09744	+	1.09744i	0.994709	+	0.102728i	$0.0327571\pi$
$773$	30.5119	+	30.5119i	1.09744	+	1.09744i	0.102728	+	0.994709i	$0.467243\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−20.6465			−0.739739
$780$		0			0
$781$	36.7477			1.31494
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21.7042	−	21.7042i	−0.773670	−	0.773670i	0.205076	−	0.978746i	$-0.434256\pi$
$787$	−21.7042	−	21.7042i	−0.773670	−	0.773670i	−0.978746	+	0.205076i	$0.934256\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	55.4506i		−	1.97160i
$792$		0			0
$793$	−9.62614	+	9.62614i	−0.341834	+	0.341834i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.59001	−	4.59001i	0.162586	−	0.162586i	−0.621125	−	0.783711i	$-0.713324\pi$
$797$	4.59001	−	4.59001i	0.162586	−	0.162586i	0.783711	+	0.621125i	$0.213324\pi$
$798$		0			0
$799$			46.7477i			1.65382i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.9274	+	17.9274i	0.632643	+	0.632643i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−36.0963			−1.26908			−0.634539	−	0.772891i	$-0.718810\pi$
$809$	−36.0963			−1.26908			−0.634539	+	0.772891i	$0.718810\pi$
$810$		0			0
$811$	20.3303			0.713893			0.356947	−	0.934125i	$-0.383818\pi$
$811$	20.3303			0.713893			0.356947	+	0.934125i	$0.383818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.79129	−	9.79129i	−0.342554	−	0.342554i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			28.5062i			0.994875i	0.867500	+	0.497437i	$0.165726\pi$
$821$			28.5062i			0.994875i	−0.867500	+	0.497437i	$0.834274\pi$
$822$		0			0
$823$	−30.1216	+	30.1216i	−1.04997	+	1.04997i	−0.0512888	+	0.998684i	$0.516333\pi$
$823$	−30.1216	+	30.1216i	−1.04997	+	1.04997i	−0.998684	+	0.0512888i	$0.983667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.06198	−	8.06198i	0.280342	−	0.280342i	−0.552903	−	0.833246i	$-0.686480\pi$
$827$	8.06198	−	8.06198i	0.280342	−	0.280342i	0.833246	+	0.552903i	$0.186480\pi$
$828$		0			0
$829$			31.0780i			1.07938i	0.841862	+	0.539692i	$0.181459\pi$
$829$			31.0780i			1.07938i	−0.841862	+	0.539692i	$0.818541\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.6016	−	21.6016i	−0.748451	−	0.748451i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	14.4272			0.498084			0.249042	−	0.968493i	$-0.419884\pi$
$839$	14.4272			0.498084			0.249042	+	0.968493i	$0.419884\pi$
$840$		0			0
$841$	−10.9129			−0.376306
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−3.95644	−	3.95644i	−0.135945	−	0.135945i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			11.2250i			0.384787i
$852$		0			0
$853$	20.3303	−	20.3303i	0.696096	−	0.696096i	−0.267470	−	0.963566i	$-0.586188\pi$
$853$	20.3303	−	20.3303i	0.696096	−	0.696096i	0.963566	+	0.267470i	$0.0861876\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	29.0849	−	29.0849i	0.993521	−	0.993521i	−0.00645765	−	0.999979i	$-0.502056\pi$
$857$	29.0849	−	29.0849i	0.993521	−	0.993521i	0.999979	+	0.00645765i	$0.00205555\pi$
$858$		0			0
$859$			0.912878i			0.0311470i	0.999879	+	0.0155735i	$0.00495740\pi$
$859$			0.912878i			0.0311470i	−0.999879	+	0.0155735i	$0.995043\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.6633	+	19.6633i	0.669348	+	0.669348i	0.957565	−	0.288217i	$-0.0930625\pi$
$863$	19.6633	+	19.6633i	0.669348	+	0.669348i	−0.288217	+	0.957565i	$0.593063\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	37.6581			1.27746
$870$		0			0
$871$	11.0780			0.375365
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−21.7913	−	21.7913i	−0.735840	−	0.735840i	0.235930	−	0.971770i	$-0.424186\pi$
$877$	−21.7913	−	21.7913i	−0.735840	−	0.735840i	−0.971770	+	0.235930i	$0.924186\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	45.2763i		−	1.52540i	−0.646754	−	0.762698i	$-0.723874\pi$
$881$		−	45.2763i		−	1.52540i	0.646754	−	0.762698i	$-0.276126\pi$
$882$		0			0
$883$	13.2087	−	13.2087i	0.444509	−	0.444509i	−0.449015	−	0.893524i	$-0.648225\pi$
$883$	13.2087	−	13.2087i	0.444509	−	0.444509i	0.893524	+	0.449015i	$0.148225\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.59569	−	9.59569i	0.322192	−	0.322192i	−0.527416	−	0.849607i	$-0.676839\pi$
$887$	9.59569	−	9.59569i	0.322192	−	0.322192i	0.849607	+	0.527416i	$0.176839\pi$
$888$		0			0
$889$		−	14.4174i		−	0.483545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.9837	+	23.9837i	0.802583	+	0.802583i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36.5008			1.21737
$900$		0			0
$901$	−46.4083			−1.54609
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−8.74773	−	8.74773i	−0.290464	−	0.290464i	0.546800	−	0.837263i	$-0.315846\pi$
$907$	−8.74773	−	8.74773i	−0.290464	−	0.290464i	−0.837263	+	0.546800i	$0.815846\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			27.7253i			0.918581i	0.888286	+	0.459290i	$0.151896\pi$
$911$			27.7253i			0.918581i	−0.888286	+	0.459290i	$0.848104\pi$
$912$		0			0
$913$	3.79129	−	3.79129i	0.125473	−	0.125473i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.0509	−	14.0509i	0.464000	−	0.464000i
$918$		0			0
$919$		−	16.0000i		−	0.527791i	−0.964551	−	0.263896i	$-0.914993\pi$
$919$		−	16.0000i		−	0.527791i	0.964551	−	0.263896i	$-0.0850075\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−9.39342	−	9.39342i	−0.309188	−	0.309188i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19.3544			0.634996			0.317498	−	0.948259i	$-0.397157\pi$
$929$	19.3544			0.634996			0.317498	+	0.948259i	$0.397157\pi$
$930$		0			0
$931$	−22.1652			−0.726433
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	37.0780	+	37.0780i	1.21129	+	1.21129i	0.970604	+	0.240683i	$0.0773713\pi$
$937$	37.0780	+	37.0780i	1.21129	+	1.21129i	0.240683	+	0.970604i	$0.422629\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			38.5739i			1.25747i	0.777618	+	0.628737i	$0.216428\pi$
$941$			38.5739i			1.25747i	−0.777618	+	0.628737i	$0.783572\pi$
$942$		0			0
$943$	14.9564	−	14.9564i	0.487049	−	0.487049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.28105	+	7.28105i	−0.236602	+	0.236602i	−0.815442	−	0.578839i	$-0.803506\pi$
$947$	−7.28105	+	7.28105i	−0.236602	+	0.236602i	0.578839	+	0.815442i	$0.303506\pi$
$948$		0			0
$949$		−	9.16515i		−	0.297513i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.3092	+	10.3092i	0.333948	+	0.333948i	0.854084	−	0.520136i	$-0.174119\pi$
$953$	10.3092	+	10.3092i	0.333948	+	0.333948i	−0.520136	+	0.854084i	$0.674119\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−6.83723			−0.220786
$960$		0			0
$961$	42.6606			1.37615
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	27.7477	+	27.7477i	0.892307	+	0.892307i	0.994740	−	0.102433i	$-0.0326628\pi$
$967$	27.7477	+	27.7477i	0.892307	+	0.892307i	−0.102433	+	0.994740i	$0.532663\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	13.8093i		−	0.443162i	−0.975142	−	0.221581i	$-0.928878\pi$
$971$		−	13.8093i		−	0.443162i	0.975142	−	0.221581i	$-0.0711217\pi$
$972$		0			0
$973$	−20.0000	+	20.0000i	−0.641171	+	0.641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.4724	+	23.4724i	−0.750949	+	0.750949i	−0.974656	−	0.223707i	$-0.928184\pi$
$977$	−23.4724	+	23.4724i	−0.750949	+	0.750949i	0.223707	+	0.974656i	$0.428184\pi$
$978$		0			0
$979$			51.9129i			1.65914i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−38.2368	−	38.2368i	−1.21956	−	1.21956i	−0.967785	−	0.251779i	$-0.918984\pi$
$983$	−38.2368	−	38.2368i	−1.21956	−	1.21956i	−0.251779	−	0.967785i	$-0.581016\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.1857			0.451079
$990$		0			0
$991$	13.0000			0.412959			0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000			0.412959			0.206479	+	0.978451i	$0.433799\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	32.9564	+	32.9564i	1.04374	+	1.04374i	0.998999	+	0.0447423i	$0.0142467\pi$
$997$	32.9564	+	32.9564i	1.04374	+	1.04374i	0.0447423	+	0.998999i	$0.485753\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.i.1457.2		8
3.2	odd	2	inner	2700.2.j.i.1457.1		8
5.2	odd	4		540.2.j.b.53.3	yes	8
5.3	odd	4	inner	2700.2.j.i.593.2		8
5.4	even	2		540.2.j.b.377.1	yes	8
15.2	even	4		540.2.j.b.53.2	✓	8
15.8	even	4	inner	2700.2.j.i.593.1		8
15.14	odd	2		540.2.j.b.377.4	yes	8
20.7	even	4		2160.2.w.c.593.3		8
20.19	odd	2		2160.2.w.c.1457.1		8
45.2	even	12		1620.2.x.c.53.2		16
45.4	even	6		1620.2.x.c.917.2		16
45.7	odd	12		1620.2.x.c.53.3		16
45.14	odd	6		1620.2.x.c.917.3		16
45.22	odd	12		1620.2.x.c.593.1		16
45.29	odd	6		1620.2.x.c.377.1		16
45.32	even	12		1620.2.x.c.593.4		16
45.34	even	6		1620.2.x.c.377.4		16
60.47	odd	4		2160.2.w.c.593.2		8
60.59	even	2		2160.2.w.c.1457.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.b.53.2	✓	8	15.2	even	4
540.2.j.b.53.3	yes	8	5.2	odd	4
540.2.j.b.377.1	yes	8	5.4	even	2
540.2.j.b.377.4	yes	8	15.14	odd	2
1620.2.x.c.53.2		16	45.2	even	12
1620.2.x.c.53.3		16	45.7	odd	12
1620.2.x.c.377.1		16	45.29	odd	6
1620.2.x.c.377.4		16	45.34	even	6
1620.2.x.c.593.1		16	45.22	odd	12
1620.2.x.c.593.4		16	45.32	even	12
1620.2.x.c.917.2		16	45.4	even	6
1620.2.x.c.917.3		16	45.14	odd	6
2160.2.w.c.593.2		8	60.47	odd	4
2160.2.w.c.593.3		8	20.7	even	4
2160.2.w.c.1457.1		8	20.19	odd	2
2160.2.w.c.1457.4		8	60.59	even	2
2700.2.j.i.593.1		8	15.8	even	4	inner
2700.2.j.i.593.2		8	5.3	odd	4	inner
2700.2.j.i.1457.1		8	3.2	odd	2	inner
2700.2.j.i.1457.2		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.79129 - 2.79129i) q^{7} +O(q^{10})\)	\(q+(-2.79129 - 2.79129i) q^{7} +3.09557i q^{11} +(0.791288 - 0.791288i) q^{13} +(-2.51691 + 2.51691i) q^{17} +2.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +4.25290 q^{29} +8.58258 q^{31} +(-3.00000 - 3.00000i) q^{37} +7.99455i q^{41} +(-3.79129 + 3.79129i) q^{43} +(9.28672 - 9.28672i) q^{47} +8.58258i q^{49} +(9.21930 + 9.21930i) q^{53} +9.15188 q^{59} -12.1652 q^{61} +(7.00000 + 7.00000i) q^{67} -11.8711i q^{71} +(5.79129 - 5.79129i) q^{73} +(8.64064 - 8.64064i) q^{77} -12.1652i q^{79} +(-1.22474 - 1.22474i) q^{83} +16.7700 q^{89} -4.41742 q^{91} +(1.58258 + 1.58258i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7}+O(q^{10}) \) 8 * q - 4 * q^7	\( 8 q - 4 q^{7} - 12 q^{13} + 32 q^{31} - 24 q^{37} - 12 q^{43} - 24 q^{61} + 56 q^{67} + 28 q^{73} - 72 q^{91} - 24 q^{97}+O(q^{100}) \) 8 * q - 4 * q^7 - 12 * q^13 + 32 * q^31 - 24 * q^37 - 12 * q^43 - 24 * q^61 + 56 * q^67 + 28 * q^73 - 72 * q^91 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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