LMFDB - Embedded newform 2940.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2940,2,Mod(1,2940)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2940, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2940.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.a (trivial)

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:	yes
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2940.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -0.828427 q^{11} +1.00000 q^{15} -2.82843 q^{17} +1.41421 q^{19} +3.41421 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -5.41421 q^{31} +0.828427 q^{33} +3.17157 q^{37} -0.343146 q^{41} +5.65685 q^{43} -1.00000 q^{45} -0.828427 q^{47} +2.82843 q^{51} +0.585786 q^{53} +0.828427 q^{55} -1.41421 q^{57} -8.48528 q^{59} +7.07107 q^{61} +6.82843 q^{67} -3.41421 q^{69} +7.65685 q^{71} -7.31371 q^{73} -1.00000 q^{75} +0.343146 q^{79} +1.00000 q^{81} +4.82843 q^{83} +2.82843 q^{85} +2.00000 q^{87} +6.48528 q^{89} +5.41421 q^{93} -1.41421 q^{95} +14.8284 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{33} + 12 q^{37} - 12 q^{41} - 2 q^{45} + 4 q^{47} + 4 q^{53} - 4 q^{55} + 8 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 12 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 4 q^{89} + 8 q^{93} + 24 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 4 * q^33 + 12 * q^37 - 12 * q^41 - 2 * q^45 + 4 * q^47 + 4 * q^53 - 4 * q^55 + 8 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 + 12 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^93 + 24 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	1.41421	0.324443	0.162221	−	0.986754i	$-0.448134\pi$
$19$	1.41421	0.324443	0.162221	+	0.986754i	$0.448134\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.41421	0.711913	0.355956	−	0.934503i	$-0.384155\pi$
$23$	3.41421	0.711913	0.355956	+	0.934503i	$0.384155\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.41421	−0.972421	−0.486211	−	0.873842i	$-0.661621\pi$
$31$	−5.41421	−0.972421	−0.486211	+	0.873842i	$0.661621\pi$
$32$	0	0
$33$	0.828427	0.144211
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.17157	0.521403	0.260702	−	0.965419i	$-0.416046\pi$
$37$	3.17157	0.521403	0.260702	+	0.965419i	$0.416046\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.343146	−0.0535904	−0.0267952	−	0.999641i	$-0.508530\pi$
$41$	−0.343146	−0.0535904	−0.0267952	+	0.999641i	$0.508530\pi$
$42$	0	0
$43$	5.65685	0.862662	0.431331	−	0.902194i	$-0.358044\pi$
$43$	5.65685	0.862662	0.431331	+	0.902194i	$0.358044\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−0.828427	−0.120839	−0.0604193	−	0.998173i	$-0.519244\pi$
$47$	−0.828427	−0.120839	−0.0604193	+	0.998173i	$0.519244\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.82843	0.396059
$52$	0	0
$53$	0.585786	0.0804640	0.0402320	−	0.999190i	$-0.487190\pi$
$53$	0.585786	0.0804640	0.0402320	+	0.999190i	$0.487190\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	−1.41421	−0.187317
$58$	0	0
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	7.07107	0.905357	0.452679	−	0.891674i	$-0.350468\pi$
$61$	7.07107	0.905357	0.452679	+	0.891674i	$0.350468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	0	0
$69$	−3.41421	−0.411023
$70$	0	0
$71$	7.65685	0.908701	0.454351	−	0.890823i	$-0.349871\pi$
$71$	7.65685	0.908701	0.454351	+	0.890823i	$0.349871\pi$
$72$	0	0
$73$	−7.31371	−0.856005	−0.428002	−	0.903778i	$-0.640783\pi$
$73$	−7.31371	−0.856005	−0.428002	+	0.903778i	$0.640783\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0.343146	0.0386069	0.0193035	−	0.999814i	$-0.493855\pi$
$79$	0.343146	0.0386069	0.0193035	+	0.999814i	$0.493855\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.82843	0.529989	0.264994	−	0.964250i	$-0.414630\pi$
$83$	4.82843	0.529989	0.264994	+	0.964250i	$0.414630\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	6.48528	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$89$	6.48528	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.41421	0.561428
$94$	0	0
$95$	−1.41421	−0.145095
$96$	0	0
$97$	14.8284	1.50560	0.752799	−	0.658250i	$-0.228703\pi$
$97$	14.8284	1.50560	0.752799	+	0.658250i	$0.228703\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−17.3137	−1.72278	−0.861389	−	0.507946i	$-0.830405\pi$
$101$	−17.3137	−1.72278	−0.861389	+	0.507946i	$0.830405\pi$
$102$	0	0
$103$	10.8284	1.06696	0.533478	−	0.845814i	$-0.320885\pi$
$103$	10.8284	1.06696	0.533478	+	0.845814i	$0.320885\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.2426	1.76358	0.881791	−	0.471640i	$-0.156338\pi$
$107$	18.2426	1.76358	0.881791	+	0.471640i	$0.156338\pi$
$108$	0	0
$109$	11.6569	1.11652	0.558262	−	0.829665i	$-0.311468\pi$
$109$	11.6569	1.11652	0.558262	+	0.829665i	$0.311468\pi$
$110$	0	0
$111$	−3.17157	−0.301032
$112$	0	0
$113$	−9.07107	−0.853334	−0.426667	−	0.904409i	$-0.640312\pi$
$113$	−9.07107	−0.853334	−0.426667	+	0.904409i	$0.640312\pi$
$114$	0	0
$115$	−3.41421	−0.318377
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0.343146	0.0309404
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.17157	0.813845	0.406923	−	0.913463i	$-0.366602\pi$
$127$	9.17157	0.813845	0.406923	+	0.913463i	$0.366602\pi$
$128$	0	0
$129$	−5.65685	−0.498058
$130$	0	0
$131$	1.65685	0.144760	0.0723800	−	0.997377i	$-0.476941\pi$
$131$	1.65685	0.144760	0.0723800	+	0.997377i	$0.476941\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	6.24264	0.533345	0.266672	−	0.963787i	$-0.414076\pi$
$137$	6.24264	0.533345	0.266672	+	0.963787i	$0.414076\pi$
$138$	0	0
$139$	13.4142	1.13778	0.568889	−	0.822414i	$-0.307373\pi$
$139$	13.4142	1.13778	0.568889	+	0.822414i	$0.307373\pi$
$140$	0	0
$141$	0.828427	0.0697661
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.65685	0.627274	0.313637	−	0.949543i	$-0.398453\pi$
$149$	7.65685	0.627274	0.313637	+	0.949543i	$0.398453\pi$
$150$	0	0
$151$	−1.31371	−0.106908	−0.0534540	−	0.998570i	$-0.517023\pi$
$151$	−1.31371	−0.106908	−0.0534540	+	0.998570i	$0.517023\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	5.41421	0.434880
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	0	0
$159$	−0.585786	−0.0464559
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−0.485281	−0.0380102	−0.0190051	−	0.999819i	$-0.506050\pi$
$163$	−0.485281	−0.0380102	−0.0190051	+	0.999819i	$0.506050\pi$
$164$	0	0
$165$	−0.828427	−0.0644930
$166$	0	0
$167$	11.3137	0.875481	0.437741	−	0.899101i	$-0.355779\pi$
$167$	11.3137	0.875481	0.437741	+	0.899101i	$0.355779\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	1.41421	0.108148
$172$	0	0
$173$	4.48528	0.341010	0.170505	−	0.985357i	$-0.445460\pi$
$173$	4.48528	0.341010	0.170505	+	0.985357i	$0.445460\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.48528	0.637793
$178$	0	0
$179$	7.65685	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$179$	7.65685	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$180$	0	0
$181$	4.92893	0.366365	0.183182	−	0.983079i	$-0.441360\pi$
$181$	4.92893	0.366365	0.183182	+	0.983079i	$0.441360\pi$
$182$	0	0
$183$	−7.07107	−0.522708
$184$	0	0
$185$	−3.17157	−0.233179
$186$	0	0
$187$	2.34315	0.171348
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.8284	1.50709	0.753546	−	0.657395i	$-0.228342\pi$
$191$	20.8284	1.50709	0.753546	+	0.657395i	$0.228342\pi$
$192$	0	0
$193$	17.7990	1.28120	0.640600	−	0.767875i	$-0.278686\pi$
$193$	17.7990	1.28120	0.640600	+	0.767875i	$0.278686\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.3848	−0.882379	−0.441189	−	0.897414i	$-0.645443\pi$
$197$	−12.3848	−0.882379	−0.441189	+	0.897414i	$0.645443\pi$
$198$	0	0
$199$	0.928932	0.0658503	0.0329251	−	0.999458i	$-0.489518\pi$
$199$	0.928932	0.0658503	0.0329251	+	0.999458i	$0.489518\pi$
$200$	0	0
$201$	−6.82843	−0.481640
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.343146	0.0239663
$206$	0	0
$207$	3.41421	0.237304
$208$	0	0
$209$	−1.17157	−0.0810394
$210$	0	0
$211$	26.6274	1.83311	0.916553	−	0.399912i	$-0.130959\pi$
$211$	26.6274	1.83311	0.916553	+	0.399912i	$0.130959\pi$
$212$	0	0
$213$	−7.65685	−0.524639
$214$	0	0
$215$	−5.65685	−0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	7.31371	0.494215
$220$	0	0
$221$	0	0
$222$	0	0
$223$	14.1421	0.947027	0.473514	−	0.880786i	$-0.342985\pi$
$223$	14.1421	0.947027	0.473514	+	0.880786i	$0.342985\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	14.3431	0.951988	0.475994	−	0.879449i	$-0.342088\pi$
$227$	14.3431	0.951988	0.475994	+	0.879449i	$0.342088\pi$
$228$	0	0
$229$	27.5563	1.82097	0.910487	−	0.413537i	$-0.135707\pi$
$229$	27.5563	1.82097	0.910487	+	0.413537i	$0.135707\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.75736	−0.377177	−0.188589	−	0.982056i	$-0.560391\pi$
$233$	−5.75736	−0.377177	−0.188589	+	0.982056i	$0.560391\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	0	0
$237$	−0.343146	−0.0222897
$238$	0	0
$239$	−13.7990	−0.892582	−0.446291	−	0.894888i	$-0.647255\pi$
$239$	−13.7990	−0.892582	−0.446291	+	0.894888i	$0.647255\pi$
$240$	0	0
$241$	−16.2426	−1.04628	−0.523140	−	0.852247i	$-0.675240\pi$
$241$	−16.2426	−1.04628	−0.523140	+	0.852247i	$0.675240\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4.82843	−0.305989
$250$	0	0
$251$	−9.17157	−0.578905	−0.289452	−	0.957192i	$-0.593473\pi$
$251$	−9.17157	−0.578905	−0.289452	+	0.957192i	$0.593473\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	0	0
$255$	−2.82843	−0.177123
$256$	0	0
$257$	−4.34315	−0.270918	−0.135459	−	0.990783i	$-0.543251\pi$
$257$	−4.34315	−0.270918	−0.135459	+	0.990783i	$0.543251\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−11.8995	−0.733754	−0.366877	−	0.930269i	$-0.619573\pi$
$263$	−11.8995	−0.733754	−0.366877	+	0.930269i	$0.619573\pi$
$264$	0	0
$265$	−0.585786	−0.0359846
$266$	0	0
$267$	−6.48528	−0.396893
$268$	0	0
$269$	7.17157	0.437259	0.218629	−	0.975808i	$-0.429841\pi$
$269$	7.17157	0.437259	0.218629	+	0.975808i	$0.429841\pi$
$270$	0	0
$271$	−8.72792	−0.530184	−0.265092	−	0.964223i	$-0.585402\pi$
$271$	−8.72792	−0.530184	−0.265092	+	0.964223i	$0.585402\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	17.3137	1.04028	0.520140	−	0.854081i	$-0.325880\pi$
$277$	17.3137	1.04028	0.520140	+	0.854081i	$0.325880\pi$
$278$	0	0
$279$	−5.41421	−0.324140
$280$	0	0
$281$	6.97056	0.415829	0.207914	−	0.978147i	$-0.433332\pi$
$281$	6.97056	0.415829	0.207914	+	0.978147i	$0.433332\pi$
$282$	0	0
$283$	−0.485281	−0.0288470	−0.0144235	−	0.999896i	$-0.504591\pi$
$283$	−0.485281	−0.0288470	−0.0144235	+	0.999896i	$0.504591\pi$
$284$	0	0
$285$	1.41421	0.0837708
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−14.8284	−0.869258
$292$	0	0
$293$	−28.6274	−1.67243	−0.836216	−	0.548401i	$-0.815237\pi$
$293$	−28.6274	−1.67243	−0.836216	+	0.548401i	$0.815237\pi$
$294$	0	0
$295$	8.48528	0.494032
$296$	0	0
$297$	0.828427	0.0480702
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.3137	0.994647
$304$	0	0
$305$	−7.07107	−0.404888
$306$	0	0
$307$	25.6569	1.46431	0.732157	−	0.681136i	$-0.238514\pi$
$307$	25.6569	1.46431	0.732157	+	0.681136i	$0.238514\pi$
$308$	0	0
$309$	−10.8284	−0.616008
$310$	0	0
$311$	−21.6569	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$311$	−21.6569	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$312$	0	0
$313$	15.3137	0.865582	0.432791	−	0.901494i	$-0.357529\pi$
$313$	15.3137	0.865582	0.432791	+	0.901494i	$0.357529\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2132	0.854459	0.427229	−	0.904143i	$-0.359490\pi$
$317$	15.2132	0.854459	0.427229	+	0.904143i	$0.359490\pi$
$318$	0	0
$319$	1.65685	0.0927660
$320$	0	0
$321$	−18.2426	−1.01820
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.6569	−0.644626
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.2843	−1.33478	−0.667392	−	0.744706i	$-0.732589\pi$
$331$	−24.2843	−1.33478	−0.667392	+	0.744706i	$0.732589\pi$
$332$	0	0
$333$	3.17157	0.173801
$334$	0	0
$335$	−6.82843	−0.373077
$336$	0	0
$337$	−25.7990	−1.40536	−0.702680	−	0.711506i	$-0.748014\pi$
$337$	−25.7990	−1.40536	−0.702680	+	0.711506i	$0.748014\pi$
$338$	0	0
$339$	9.07107	0.492673
$340$	0	0
$341$	4.48528	0.242892
$342$	0	0
$343$	0	0
$344$	0	0
$345$	3.41421	0.183815
$346$	0	0
$347$	21.0711	1.13115	0.565577	−	0.824695i	$-0.308653\pi$
$347$	21.0711	1.13115	0.565577	+	0.824695i	$0.308653\pi$
$348$	0	0
$349$	8.92893	0.477955	0.238977	−	0.971025i	$-0.423188\pi$
$349$	8.92893	0.477955	0.238977	+	0.971025i	$0.423188\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.9706	0.583904	0.291952	−	0.956433i	$-0.405695\pi$
$353$	10.9706	0.583904	0.291952	+	0.956433i	$0.405695\pi$
$354$	0	0
$355$	−7.65685	−0.406384
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.1421	0.640837	0.320419	−	0.947276i	$-0.396176\pi$
$359$	12.1421	0.640837	0.320419	+	0.947276i	$0.396176\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	10.3137	0.541329
$364$	0	0
$365$	7.31371	0.382817
$366$	0	0
$367$	29.4558	1.53758	0.768791	−	0.639500i	$-0.220858\pi$
$367$	29.4558	1.53758	0.768791	+	0.639500i	$0.220858\pi$
$368$	0	0
$369$	−0.343146	−0.0178635
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13.3137	−0.689358	−0.344679	−	0.938721i	$-0.612012\pi$
$373$	−13.3137	−0.689358	−0.344679	+	0.938721i	$0.612012\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	0	0
$381$	−9.17157	−0.469874
$382$	0	0
$383$	−25.7990	−1.31827	−0.659133	−	0.752026i	$-0.729077\pi$
$383$	−25.7990	−1.31827	−0.659133	+	0.752026i	$0.729077\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.65685	0.287554
$388$	0	0
$389$	−17.7990	−0.902445	−0.451222	−	0.892412i	$-0.649012\pi$
$389$	−17.7990	−0.902445	−0.451222	+	0.892412i	$0.649012\pi$
$390$	0	0
$391$	−9.65685	−0.488368
$392$	0	0
$393$	−1.65685	−0.0835772
$394$	0	0
$395$	−0.343146	−0.0172655
$396$	0	0
$397$	5.17157	0.259554	0.129777	−	0.991543i	$-0.458574\pi$
$397$	5.17157	0.259554	0.129777	+	0.991543i	$0.458574\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.1421	−0.806100	−0.403050	−	0.915178i	$-0.632050\pi$
$401$	−16.1421	−0.806100	−0.403050	+	0.915178i	$0.632050\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−2.62742	−0.130236
$408$	0	0
$409$	10.5858	0.523433	0.261717	−	0.965145i	$-0.415711\pi$
$409$	10.5858	0.523433	0.261717	+	0.965145i	$0.415711\pi$
$410$	0	0
$411$	−6.24264	−0.307927
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.82843	−0.237018
$416$	0	0
$417$	−13.4142	−0.656897
$418$	0	0
$419$	−5.65685	−0.276355	−0.138178	−	0.990407i	$-0.544125\pi$
$419$	−5.65685	−0.276355	−0.138178	+	0.990407i	$0.544125\pi$
$420$	0	0
$421$	16.2843	0.793647	0.396823	−	0.917895i	$-0.370113\pi$
$421$	16.2843	0.793647	0.396823	+	0.917895i	$0.370113\pi$
$422$	0	0
$423$	−0.828427	−0.0402795
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.62742	−0.222895	−0.111447	−	0.993770i	$-0.535549\pi$
$431$	−4.62742	−0.222895	−0.111447	+	0.993770i	$0.535549\pi$
$432$	0	0
$433$	18.3431	0.881515	0.440758	−	0.897626i	$-0.354710\pi$
$433$	18.3431	0.881515	0.440758	+	0.897626i	$0.354710\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	4.82843	0.230975
$438$	0	0
$439$	−9.89949	−0.472477	−0.236239	−	0.971695i	$-0.575915\pi$
$439$	−9.89949	−0.472477	−0.236239	+	0.971695i	$0.575915\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.44365	−0.306147	−0.153074	−	0.988215i	$-0.548917\pi$
$443$	−6.44365	−0.306147	−0.153074	+	0.988215i	$0.548917\pi$
$444$	0	0
$445$	−6.48528	−0.307432
$446$	0	0
$447$	−7.65685	−0.362157
$448$	0	0
$449$	−12.3431	−0.582509	−0.291255	−	0.956646i	$-0.594073\pi$
$449$	−12.3431	−0.582509	−0.291255	+	0.956646i	$0.594073\pi$
$450$	0	0
$451$	0.284271	0.0133858
$452$	0	0
$453$	1.31371	0.0617234
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.1421	−1.31643	−0.658217	−	0.752828i	$-0.728689\pi$
$457$	−28.1421	−1.31643	−0.658217	+	0.752828i	$0.728689\pi$
$458$	0	0
$459$	2.82843	0.132020
$460$	0	0
$461$	−30.4853	−1.41984	−0.709921	−	0.704282i	$-0.751269\pi$
$461$	−30.4853	−1.41984	−0.709921	+	0.704282i	$0.751269\pi$
$462$	0	0
$463$	11.7990	0.548346	0.274173	−	0.961680i	$-0.411596\pi$
$463$	11.7990	0.548346	0.274173	+	0.961680i	$0.411596\pi$
$464$	0	0
$465$	−5.41421	−0.251078
$466$	0	0
$467$	−4.97056	−0.230010	−0.115005	−	0.993365i	$-0.536688\pi$
$467$	−4.97056	−0.230010	−0.115005	+	0.993365i	$0.536688\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.34315	−0.292277
$472$	0	0
$473$	−4.68629	−0.215476
$474$	0	0
$475$	1.41421	0.0648886
$476$	0	0
$477$	0.585786	0.0268213
$478$	0	0
$479$	20.9706	0.958169	0.479085	−	0.877769i	$-0.340969\pi$
$479$	20.9706	0.958169	0.479085	+	0.877769i	$0.340969\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14.8284	−0.673324
$486$	0	0
$487$	−3.02944	−0.137277	−0.0686385	−	0.997642i	$-0.521865\pi$
$487$	−3.02944	−0.137277	−0.0686385	+	0.997642i	$0.521865\pi$
$488$	0	0
$489$	0.485281	0.0219452
$490$	0	0
$491$	25.3137	1.14239	0.571196	−	0.820814i	$-0.306480\pi$
$491$	25.3137	1.14239	0.571196	+	0.820814i	$0.306480\pi$
$492$	0	0
$493$	5.65685	0.254772
$494$	0	0
$495$	0.828427	0.0372350
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−38.2843	−1.71384	−0.856920	−	0.515450i	$-0.827625\pi$
$499$	−38.2843	−1.71384	−0.856920	+	0.515450i	$0.827625\pi$
$500$	0	0
$501$	−11.3137	−0.505459
$502$	0	0
$503$	−4.82843	−0.215289	−0.107644	−	0.994189i	$-0.534331\pi$
$503$	−4.82843	−0.215289	−0.107644	+	0.994189i	$0.534331\pi$
$504$	0	0
$505$	17.3137	0.770450
$506$	0	0
$507$	13.0000	0.577350
$508$	0	0
$509$	3.17157	0.140577	0.0702887	−	0.997527i	$-0.477608\pi$
$509$	3.17157	0.140577	0.0702887	+	0.997527i	$0.477608\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.41421	−0.0624391
$514$	0	0
$515$	−10.8284	−0.477158
$516$	0	0
$517$	0.686292	0.0301831
$518$	0	0
$519$	−4.48528	−0.196882
$520$	0	0
$521$	−40.1421	−1.75866	−0.879329	−	0.476214i	$-0.842009\pi$
$521$	−40.1421	−1.75866	−0.879329	+	0.476214i	$0.842009\pi$
$522$	0	0
$523$	−6.14214	−0.268577	−0.134288	−	0.990942i	$-0.542875\pi$
$523$	−6.14214	−0.268577	−0.134288	+	0.990942i	$0.542875\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.3137	0.667076
$528$	0	0
$529$	−11.3431	−0.493180
$530$	0	0
$531$	−8.48528	−0.368230
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−18.2426	−0.788698
$536$	0	0
$537$	−7.65685	−0.330418
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.9706	−1.76146	−0.880731	−	0.473617i	$-0.842948\pi$
$541$	−40.9706	−1.76146	−0.880731	+	0.473617i	$0.842948\pi$
$542$	0	0
$543$	−4.92893	−0.211521
$544$	0	0
$545$	−11.6569	−0.499325
$546$	0	0
$547$	−1.17157	−0.0500928	−0.0250464	−	0.999686i	$-0.507973\pi$
$547$	−1.17157	−0.0500928	−0.0250464	+	0.999686i	$0.507973\pi$
$548$	0	0
$549$	7.07107	0.301786
$550$	0	0
$551$	−2.82843	−0.120495
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.17157	0.134626
$556$	0	0
$557$	−2.92893	−0.124103	−0.0620514	−	0.998073i	$-0.519764\pi$
$557$	−2.92893	−0.124103	−0.0620514	+	0.998073i	$0.519764\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−2.34315	−0.0989277
$562$	0	0
$563$	42.7696	1.80252	0.901261	−	0.433277i	$-0.142643\pi$
$563$	42.7696	1.80252	0.901261	+	0.433277i	$0.142643\pi$
$564$	0	0
$565$	9.07107	0.381623
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.48528	0.104188	0.0520942	−	0.998642i	$-0.483410\pi$
$569$	2.48528	0.104188	0.0520942	+	0.998642i	$0.483410\pi$
$570$	0	0
$571$	−13.6569	−0.571522	−0.285761	−	0.958301i	$-0.592246\pi$
$571$	−13.6569	−0.571522	−0.285761	+	0.958301i	$0.592246\pi$
$572$	0	0
$573$	−20.8284	−0.870120
$574$	0	0
$575$	3.41421	0.142383
$576$	0	0
$577$	34.3431	1.42972	0.714862	−	0.699266i	$-0.246490\pi$
$577$	34.3431	1.42972	0.714862	+	0.699266i	$0.246490\pi$
$578$	0	0
$579$	−17.7990	−0.739701
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.485281	−0.0200983
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.8284	0.859681	0.429841	−	0.902905i	$-0.358570\pi$
$587$	20.8284	0.859681	0.429841	+	0.902905i	$0.358570\pi$
$588$	0	0
$589$	−7.65685	−0.315495
$590$	0	0
$591$	12.3848	0.509442
$592$	0	0
$593$	−26.9706	−1.10755	−0.553774	−	0.832667i	$-0.686813\pi$
$593$	−26.9706	−1.10755	−0.553774	+	0.832667i	$0.686813\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.928932	−0.0380187
$598$	0	0
$599$	34.7696	1.42065	0.710323	−	0.703876i	$-0.248549\pi$
$599$	34.7696	1.42065	0.710323	+	0.703876i	$0.248549\pi$
$600$	0	0
$601$	−3.27208	−0.133471	−0.0667354	−	0.997771i	$-0.521258\pi$
$601$	−3.27208	−0.133471	−0.0667354	+	0.997771i	$0.521258\pi$
$602$	0	0
$603$	6.82843	0.278075
$604$	0	0
$605$	10.3137	0.419312
$606$	0	0
$607$	3.51472	0.142658	0.0713290	−	0.997453i	$-0.477276\pi$
$607$	3.51472	0.142658	0.0713290	+	0.997453i	$0.477276\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	−0.343146	−0.0138370
$616$	0	0
$617$	−35.4142	−1.42572	−0.712861	−	0.701305i	$-0.752601\pi$
$617$	−35.4142	−1.42572	−0.712861	+	0.701305i	$0.752601\pi$
$618$	0	0
$619$	11.5563	0.464489	0.232244	−	0.972657i	$-0.425393\pi$
$619$	11.5563	0.464489	0.232244	+	0.972657i	$0.425393\pi$
$620$	0	0
$621$	−3.41421	−0.137008
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.17157	0.0467881
$628$	0	0
$629$	−8.97056	−0.357680
$630$	0	0
$631$	49.5980	1.97446	0.987232	−	0.159288i	$-0.0509198\pi$
$631$	49.5980	1.97446	0.987232	+	0.159288i	$0.0509198\pi$
$632$	0	0
$633$	−26.6274	−1.05834
$634$	0	0
$635$	−9.17157	−0.363963
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.65685	0.302900
$640$	0	0
$641$	−39.9411	−1.57758	−0.788790	−	0.614663i	$-0.789292\pi$
$641$	−39.9411	−1.57758	−0.788790	+	0.614663i	$0.789292\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	5.65685	0.222738
$646$	0	0
$647$	27.3137	1.07381	0.536906	−	0.843642i	$-0.319593\pi$
$647$	27.3137	1.07381	0.536906	+	0.843642i	$0.319593\pi$
$648$	0	0
$649$	7.02944	0.275930
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−45.5563	−1.78276	−0.891379	−	0.453259i	$-0.850261\pi$
$653$	−45.5563	−1.78276	−0.891379	+	0.453259i	$0.850261\pi$
$654$	0	0
$655$	−1.65685	−0.0647387
$656$	0	0
$657$	−7.31371	−0.285335
$658$	0	0
$659$	−17.3137	−0.674446	−0.337223	−	0.941425i	$-0.609488\pi$
$659$	−17.3137	−0.674446	−0.337223	+	0.941425i	$0.609488\pi$
$660$	0	0
$661$	16.0416	0.623947	0.311974	−	0.950091i	$-0.399010\pi$
$661$	16.0416	0.623947	0.311974	+	0.950091i	$0.399010\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.82843	−0.264398
$668$	0	0
$669$	−14.1421	−0.546767
$670$	0	0
$671$	−5.85786	−0.226140
$672$	0	0
$673$	−24.1421	−0.930611	−0.465305	−	0.885150i	$-0.654056\pi$
$673$	−24.1421	−0.930611	−0.465305	+	0.885150i	$0.654056\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	32.6274	1.25397	0.626987	−	0.779030i	$-0.284288\pi$
$677$	32.6274	1.25397	0.626987	+	0.779030i	$0.284288\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.3431	−0.549631
$682$	0	0
$683$	−40.5858	−1.55297	−0.776486	−	0.630135i	$-0.783000\pi$
$683$	−40.5858	−1.55297	−0.776486	+	0.630135i	$0.783000\pi$
$684$	0	0
$685$	−6.24264	−0.238519
$686$	0	0
$687$	−27.5563	−1.05134
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−19.5563	−0.743959	−0.371979	−	0.928241i	$-0.621321\pi$
$691$	−19.5563	−0.743959	−0.371979	+	0.928241i	$0.621321\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.4142	−0.508830
$696$	0	0
$697$	0.970563	0.0367627
$698$	0	0
$699$	5.75736	0.217763
$700$	0	0
$701$	38.2843	1.44598	0.722988	−	0.690860i	$-0.242768\pi$
$701$	38.2843	1.44598	0.722988	+	0.690860i	$0.242768\pi$
$702$	0	0
$703$	4.48528	0.169166
$704$	0	0
$705$	−0.828427	−0.0312004
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.62742	0.0986747	0.0493374	−	0.998782i	$-0.484289\pi$
$709$	2.62742	0.0986747	0.0493374	+	0.998782i	$0.484289\pi$
$710$	0	0
$711$	0.343146	0.0128690
$712$	0	0
$713$	−18.4853	−0.692279
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.7990	0.515333
$718$	0	0
$719$	−30.8284	−1.14971	−0.574853	−	0.818257i	$-0.694941\pi$
$719$	−30.8284	−1.14971	−0.574853	+	0.818257i	$0.694941\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.2426	0.604070
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−8.28427	−0.307247	−0.153623	−	0.988129i	$-0.549094\pi$
$727$	−8.28427	−0.307247	−0.153623	+	0.988129i	$0.549094\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.0000	−0.591781
$732$	0	0
$733$	16.9706	0.626822	0.313411	−	0.949618i	$-0.398528\pi$
$733$	16.9706	0.626822	0.313411	+	0.949618i	$0.398528\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.65685	−0.208373
$738$	0	0
$739$	22.6274	0.832363	0.416181	−	0.909282i	$-0.363368\pi$
$739$	22.6274	0.832363	0.416181	+	0.909282i	$0.363368\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.8701	−0.765648	−0.382824	−	0.923821i	$-0.625048\pi$
$743$	−20.8701	−0.765648	−0.382824	+	0.923821i	$0.625048\pi$
$744$	0	0
$745$	−7.65685	−0.280525
$746$	0	0
$747$	4.82843	0.176663
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	9.17157	0.334231
$754$	0	0
$755$	1.31371	0.0478107
$756$	0	0
$757$	−39.4558	−1.43405	−0.717024	−	0.697049i	$-0.754496\pi$
$757$	−39.4558	−1.43405	−0.717024	+	0.697049i	$0.754496\pi$
$758$	0	0
$759$	2.82843	0.102665
$760$	0	0
$761$	−25.5147	−0.924908	−0.462454	−	0.886643i	$-0.653031\pi$
$761$	−25.5147	−0.924908	−0.462454	+	0.886643i	$0.653031\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.82843	0.102262
$766$	0	0
$767$	0	0
$768$	0	0
$769$	16.4437	0.592973	0.296487	−	0.955037i	$-0.404185\pi$
$769$	16.4437	0.592973	0.296487	+	0.955037i	$0.404185\pi$
$770$	0	0
$771$	4.34315	0.156415
$772$	0	0
$773$	44.7696	1.61025	0.805124	−	0.593106i	$-0.202098\pi$
$773$	44.7696	1.61025	0.805124	+	0.593106i	$0.202098\pi$
$774$	0	0
$775$	−5.41421	−0.194484
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.485281	−0.0173870
$780$	0	0
$781$	−6.34315	−0.226976
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−6.34315	−0.226397
$786$	0	0
$787$	8.48528	0.302468	0.151234	−	0.988498i	$-0.451675\pi$
$787$	8.48528	0.302468	0.151234	+	0.988498i	$0.451675\pi$
$788$	0	0
$789$	11.8995	0.423633
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0.585786	0.0207757
$796$	0	0
$797$	−13.4558	−0.476630	−0.238315	−	0.971188i	$-0.576595\pi$
$797$	−13.4558	−0.476630	−0.238315	+	0.971188i	$0.576595\pi$
$798$	0	0
$799$	2.34315	0.0828945
$800$	0	0
$801$	6.48528	0.229146
$802$	0	0
$803$	6.05887	0.213813
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.17157	−0.252451
$808$	0	0
$809$	−32.1421	−1.13006	−0.565029	−	0.825071i	$-0.691135\pi$
$809$	−32.1421	−1.13006	−0.565029	+	0.825071i	$0.691135\pi$
$810$	0	0
$811$	−32.2426	−1.13219	−0.566096	−	0.824339i	$-0.691547\pi$
$811$	−32.2426	−1.13219	−0.566096	+	0.824339i	$0.691547\pi$
$812$	0	0
$813$	8.72792	0.306102
$814$	0	0
$815$	0.485281	0.0169987
$816$	0	0
$817$	8.00000	0.279885
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.1716	0.669093	0.334546	−	0.942379i	$-0.391417\pi$
$821$	19.1716	0.669093	0.334546	+	0.942379i	$0.391417\pi$
$822$	0	0
$823$	−24.7696	−0.863412	−0.431706	−	0.902014i	$-0.642088\pi$
$823$	−24.7696	−0.863412	−0.431706	+	0.902014i	$0.642088\pi$
$824$	0	0
$825$	0.828427	0.0288421
$826$	0	0
$827$	−27.0122	−0.939306	−0.469653	−	0.882851i	$-0.655621\pi$
$827$	−27.0122	−0.939306	−0.469653	+	0.882851i	$0.655621\pi$
$828$	0	0
$829$	46.3848	1.61101	0.805505	−	0.592589i	$-0.201894\pi$
$829$	46.3848	1.61101	0.805505	+	0.592589i	$0.201894\pi$
$830$	0	0
$831$	−17.3137	−0.600606
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−11.3137	−0.391527
$836$	0	0
$837$	5.41421	0.187143
$838$	0	0
$839$	18.1421	0.626336	0.313168	−	0.949698i	$-0.398610\pi$
$839$	18.1421	0.626336	0.313168	+	0.949698i	$0.398610\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−6.97056	−0.240079
$844$	0	0
$845$	13.0000	0.447214
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.485281	0.0166548
$850$	0	0
$851$	10.8284	0.371194
$852$	0	0
$853$	−7.79899	−0.267032	−0.133516	−	0.991047i	$-0.542627\pi$
$853$	−7.79899	−0.267032	−0.133516	+	0.991047i	$0.542627\pi$
$854$	0	0
$855$	−1.41421	−0.0483651
$856$	0	0
$857$	−5.45584	−0.186368	−0.0931840	−	0.995649i	$-0.529704\pi$
$857$	−5.45584	−0.186368	−0.0931840	+	0.995649i	$0.529704\pi$
$858$	0	0
$859$	−45.2132	−1.54265	−0.771327	−	0.636439i	$-0.780407\pi$
$859$	−45.2132	−1.54265	−0.771327	+	0.636439i	$0.780407\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.78680	−0.162944	−0.0814722	−	0.996676i	$-0.525962\pi$
$863$	−4.78680	−0.162944	−0.0814722	+	0.996676i	$0.525962\pi$
$864$	0	0
$865$	−4.48528	−0.152504
$866$	0	0
$867$	9.00000	0.305656
$868$	0	0
$869$	−0.284271	−0.00964324
$870$	0	0
$871$	0	0
$872$	0	0
$873$	14.8284	0.501866
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.2843	0.617416	0.308708	−	0.951157i	$-0.400103\pi$
$877$	18.2843	0.617416	0.308708	+	0.951157i	$0.400103\pi$
$878$	0	0
$879$	28.6274	0.965579
$880$	0	0
$881$	20.6274	0.694955	0.347478	−	0.937688i	$-0.387038\pi$
$881$	20.6274	0.694955	0.347478	+	0.937688i	$0.387038\pi$
$882$	0	0
$883$	26.3431	0.886517	0.443259	−	0.896394i	$-0.353822\pi$
$883$	26.3431	0.886517	0.443259	+	0.896394i	$0.353822\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	0	0
$887$	−33.5147	−1.12531	−0.562657	−	0.826690i	$-0.690221\pi$
$887$	−33.5147	−1.12531	−0.562657	+	0.826690i	$0.690221\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	−1.17157	−0.0392052
$894$	0	0
$895$	−7.65685	−0.255940
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10.8284	0.361148
$900$	0	0
$901$	−1.65685	−0.0551978
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.92893	−0.163843
$906$	0	0
$907$	20.9706	0.696316	0.348158	−	0.937436i	$-0.386807\pi$
$907$	20.9706	0.696316	0.348158	+	0.937436i	$0.386807\pi$
$908$	0	0
$909$	−17.3137	−0.574259
$910$	0	0
$911$	25.7990	0.854759	0.427379	−	0.904072i	$-0.359437\pi$
$911$	25.7990	0.854759	0.427379	+	0.904072i	$0.359437\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	0	0
$915$	7.07107	0.233762
$916$	0	0
$917$	0	0
$918$	0	0
$919$	20.6863	0.682378	0.341189	−	0.939995i	$-0.389170\pi$
$919$	20.6863	0.682378	0.341189	+	0.939995i	$0.389170\pi$
$920$	0	0
$921$	−25.6569	−0.845422
$922$	0	0
$923$	0	0
$924$	0	0
$925$	3.17157	0.104281
$926$	0	0
$927$	10.8284	0.355652
$928$	0	0
$929$	15.8579	0.520280	0.260140	−	0.965571i	$-0.416231\pi$
$929$	15.8579	0.520280	0.260140	+	0.965571i	$0.416231\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.6569	0.709014
$934$	0	0
$935$	−2.34315	−0.0766291
$936$	0	0
$937$	29.4558	0.962280	0.481140	−	0.876644i	$-0.340223\pi$
$937$	29.4558	0.962280	0.481140	+	0.876644i	$0.340223\pi$
$938$	0	0
$939$	−15.3137	−0.499744
$940$	0	0
$941$	−60.6274	−1.97640	−0.988199	−	0.153178i	$-0.951049\pi$
$941$	−60.6274	−1.97640	−0.988199	+	0.153178i	$0.951049\pi$
$942$	0	0
$943$	−1.17157	−0.0381517
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.7279	0.348611	0.174305	−	0.984692i	$-0.444232\pi$
$947$	10.7279	0.348611	0.174305	+	0.984692i	$0.444232\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−15.2132	−0.493322
$952$	0	0
$953$	−16.1005	−0.521547	−0.260773	−	0.965400i	$-0.583977\pi$
$953$	−16.1005	−0.521547	−0.260773	+	0.965400i	$0.583977\pi$
$954$	0	0
$955$	−20.8284	−0.673992
$956$	0	0
$957$	−1.65685	−0.0535585
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.68629	−0.0543965
$962$	0	0
$963$	18.2426	0.587861
$964$	0	0
$965$	−17.7990	−0.572970
$966$	0	0
$967$	−13.1716	−0.423569	−0.211785	−	0.977316i	$-0.567928\pi$
$967$	−13.1716	−0.423569	−0.211785	+	0.977316i	$0.567928\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	33.4558	1.07365	0.536825	−	0.843694i	$-0.319624\pi$
$971$	33.4558	1.07365	0.536825	+	0.843694i	$0.319624\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.87006	0.283778	0.141889	−	0.989883i	$-0.454682\pi$
$977$	8.87006	0.283778	0.141889	+	0.989883i	$0.454682\pi$
$978$	0	0
$979$	−5.37258	−0.171708
$980$	0	0
$981$	11.6569	0.372175
$982$	0	0
$983$	43.3137	1.38149	0.690746	−	0.723097i	$-0.257282\pi$
$983$	43.3137	1.38149	0.690746	+	0.723097i	$0.257282\pi$
$984$	0	0
$985$	12.3848	0.394612
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.3137	0.614140
$990$	0	0
$991$	−28.2843	−0.898479	−0.449240	−	0.893411i	$-0.648305\pi$
$991$	−28.2843	−0.898479	−0.449240	+	0.893411i	$0.648305\pi$
$992$	0	0
$993$	24.2843	0.770638
$994$	0	0
$995$	−0.928932	−0.0294491
$996$	0	0
$997$	34.8284	1.10303	0.551514	−	0.834166i	$-0.314050\pi$
$997$	34.8284	1.10303	0.551514	+	0.834166i	$0.314050\pi$
$998$	0	0
$999$	−3.17157	−0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.a.n.1.1	✓	2
3.2	odd	2		8820.2.a.bi.1.2		2
7.2	even	3		2940.2.q.s.361.2		4
7.3	odd	6		2940.2.q.o.961.2		4
7.4	even	3		2940.2.q.s.961.2		4
7.5	odd	6		2940.2.q.o.361.2		4
7.6	odd	2		2940.2.a.t.1.1	yes	2
21.20	even	2		8820.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.a.n.1.1	✓	2	1.1	even	1	trivial
2940.2.a.t.1.1	yes	2	7.6	odd	2
2940.2.q.o.361.2		4	7.5	odd	6
2940.2.q.o.961.2		4	7.3	odd	6
2940.2.q.s.361.2		4	7.2	even	3
2940.2.q.s.961.2		4	7.4	even	3
8820.2.a.bd.1.2		2	21.20	even	2
8820.2.a.bi.1.2		2	3.2	odd	2

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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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -0.828427 q^{11} +1.00000 q^{15} -2.82843 q^{17} +1.41421 q^{19} +3.41421 q^{23} +1.00000 q^{25} -1.00000 q^{27} -2.00000 q^{29} -5.41421 q^{31} +0.828427 q^{33} +3.17157 q^{37} -0.343146 q^{41} +5.65685 q^{43} -1.00000 q^{45} -0.828427 q^{47} +2.82843 q^{51} +0.585786 q^{53} +0.828427 q^{55} -1.41421 q^{57} -8.48528 q^{59} +7.07107 q^{61} +6.82843 q^{67} -3.41421 q^{69} +7.65685 q^{71} -7.31371 q^{73} -1.00000 q^{75} +0.343146 q^{79} +1.00000 q^{81} +4.82843 q^{83} +2.82843 q^{85} +2.00000 q^{87} +6.48528 q^{89} +5.41421 q^{93} -1.41421 q^{95} +14.8284 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 2 q^{15} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 4 q^{33} + 12 q^{37} - 12 q^{41} - 2 q^{45} + 4 q^{47} + 4 q^{53} - 4 q^{55} + 8 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 12 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{87} - 4 q^{89} + 8 q^{93} + 24 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 + 2 * q^15 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 4 * q^33 + 12 * q^37 - 12 * q^41 - 2 * q^45 + 4 * q^47 + 4 * q^53 - 4 * q^55 + 8 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 + 12 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^93 + 24 * q^97 + 4 * q^99

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