LMFDB - Embedded newform 3420.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, 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("3420.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.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:	$27.3088374913$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	3420.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -4.82843 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.82843 q^{7} +2.00000 q^{11} +2.24264 q^{13} +4.82843 q^{17} -1.00000 q^{19} +6.00000 q^{23} +1.00000 q^{25} -10.4853 q^{29} -1.17157 q^{31} +4.82843 q^{35} -10.2426 q^{37} +7.65685 q^{41} -0.828427 q^{43} -0.828427 q^{47} +16.3137 q^{49} +5.07107 q^{53} -2.00000 q^{55} -2.34315 q^{59} -2.34315 q^{61} -2.24264 q^{65} -0.585786 q^{67} -10.8284 q^{71} +14.4853 q^{73} -9.65685 q^{77} -9.65685 q^{79} -9.31371 q^{83} -4.82843 q^{85} -10.4853 q^{89} -10.8284 q^{91} +1.00000 q^{95} -1.75736 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 4 * q^7	$ 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.4853	−1.94707	−0.973534	−	0.228543i	$-0.926604\pi$
$29$	−10.4853	−1.94707	−0.973534	+	0.228543i	$0.926604\pi$
$30$	0	0
$31$	−1.17157	−0.210421	−0.105210	−	0.994450i	$-0.533552\pi$
$31$	−1.17157	−0.210421	−0.105210	+	0.994450i	$0.533552\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.82843	0.816153
$36$	0	0
$37$	−10.2426	−1.68388	−0.841940	−	0.539571i	$-0.818586\pi$
$37$	−10.2426	−1.68388	−0.841940	+	0.539571i	$0.818586\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	−0.828427	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$43$	−0.828427	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$44$	0	0
$45$	0	0
$46$	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
$49$	16.3137	2.33053
$50$	0	0
$51$	0	0
$52$	0	0
$53$	5.07107	0.696565	0.348282	−	0.937390i	$-0.386765\pi$
$53$	5.07107	0.696565	0.348282	+	0.937390i	$0.386765\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.34315	−0.305052	−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315	−0.305052	−0.152526	+	0.988299i	$0.548741\pi$
$60$	0	0
$61$	−2.34315	−0.300009	−0.150005	−	0.988685i	$-0.547929\pi$
$61$	−2.34315	−0.300009	−0.150005	+	0.988685i	$0.547929\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.24264	−0.278165
$66$	0	0
$67$	−0.585786	−0.0715652	−0.0357826	−	0.999360i	$-0.511392\pi$
$67$	−0.585786	−0.0715652	−0.0357826	+	0.999360i	$0.511392\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.8284	−1.28510	−0.642549	−	0.766245i	$-0.722123\pi$
$71$	−10.8284	−1.28510	−0.642549	+	0.766245i	$0.722123\pi$
$72$	0	0
$73$	14.4853	1.69537	0.847687	−	0.530497i	$-0.177995\pi$
$73$	14.4853	1.69537	0.847687	+	0.530497i	$0.177995\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.65685	−1.10050
$78$	0	0
$79$	−9.65685	−1.08648	−0.543240	−	0.839577i	$-0.682803\pi$
$79$	−9.65685	−1.08648	−0.543240	+	0.839577i	$0.682803\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.31371	−1.02231	−0.511156	−	0.859488i	$-0.670783\pi$
$83$	−9.31371	−1.02231	−0.511156	+	0.859488i	$0.670783\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.4853	−1.11144	−0.555719	−	0.831370i	$-0.687557\pi$
$89$	−10.4853	−1.11144	−0.555719	+	0.831370i	$0.687557\pi$
$90$	0	0
$91$	−10.8284	−1.13513
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−1.75736	−0.178433	−0.0892164	−	0.996012i	$-0.528436\pi$
$97$	−1.75736	−0.178433	−0.0892164	+	0.996012i	$0.528436\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.00000	−0.398015	−0.199007	−	0.979998i	$-0.563772\pi$
$101$	−4.00000	−0.398015	−0.199007	+	0.979998i	$0.563772\pi$
$102$	0	0
$103$	−15.8995	−1.56662	−0.783312	−	0.621629i	$-0.786471\pi$
$103$	−15.8995	−1.56662	−0.783312	+	0.621629i	$0.786471\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.58579	−0.443325	−0.221662	−	0.975123i	$-0.571148\pi$
$107$	−4.58579	−0.443325	−0.221662	+	0.975123i	$0.571148\pi$
$108$	0	0
$109$	−8.82843	−0.845610	−0.422805	−	0.906221i	$-0.638954\pi$
$109$	−8.82843	−0.845610	−0.422805	+	0.906221i	$0.638954\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.07107	−0.477046	−0.238523	−	0.971137i	$-0.576663\pi$
$113$	−5.07107	−0.477046	−0.238523	+	0.971137i	$0.576663\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−23.3137	−2.13716
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−21.5563	−1.91282	−0.956408	−	0.292033i	$-0.905668\pi$
$127$	−21.5563	−1.91282	−0.956408	+	0.292033i	$0.905668\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	4.82843	0.418678
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.1421	1.72086	0.860429	−	0.509570i	$-0.170195\pi$
$137$	20.1421	1.72086	0.860429	+	0.509570i	$0.170195\pi$
$138$	0	0
$139$	−17.3137	−1.46853	−0.734265	−	0.678863i	$-0.762473\pi$
$139$	−17.3137	−1.46853	−0.734265	+	0.678863i	$0.762473\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.48528	0.375078
$144$	0	0
$145$	10.4853	0.870755
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	18.1421	1.47639	0.738193	−	0.674590i	$-0.235679\pi$
$151$	18.1421	1.47639	0.738193	+	0.674590i	$0.235679\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.17157	0.0941030
$156$	0	0
$157$	3.17157	0.253119	0.126560	−	0.991959i	$-0.459607\pi$
$157$	3.17157	0.253119	0.126560	+	0.991959i	$0.459607\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−28.9706	−2.28320
$162$	0	0
$163$	−2.48528	−0.194662	−0.0973311	−	0.995252i	$-0.531031\pi$
$163$	−2.48528	−0.194662	−0.0973311	+	0.995252i	$0.531031\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.7279	0.830152	0.415076	−	0.909787i	$-0.363755\pi$
$167$	10.7279	0.830152	0.415076	+	0.909787i	$0.363755\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.89949	0.296473	0.148237	−	0.988952i	$-0.452640\pi$
$173$	3.89949	0.296473	0.148237	+	0.988952i	$0.452640\pi$
$174$	0	0
$175$	−4.82843	−0.364995
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.6569	−1.61871	−0.809355	−	0.587320i	$-0.800183\pi$
$179$	−21.6569	−1.61871	−0.809355	+	0.587320i	$0.800183\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.2426	0.753054
$186$	0	0
$187$	9.65685	0.706179
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	0.585786	0.0421658	0.0210829	−	0.999778i	$-0.493289\pi$
$193$	0.585786	0.0421658	0.0210829	+	0.999778i	$0.493289\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.31371	0.378586	0.189293	−	0.981921i	$-0.439380\pi$
$197$	5.31371	0.378586	0.189293	+	0.981921i	$0.439380\pi$
$198$	0	0
$199$	−10.3431	−0.733206	−0.366603	−	0.930377i	$-0.619479\pi$
$199$	−10.3431	−0.733206	−0.366603	+	0.930377i	$0.619479\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	50.6274	3.55335
$204$	0	0
$205$	−7.65685	−0.534778
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−11.5147	−0.792706	−0.396353	−	0.918098i	$-0.629724\pi$
$211$	−11.5147	−0.792706	−0.396353	+	0.918098i	$0.629724\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.828427	0.0564983
$216$	0	0
$217$	5.65685	0.384012
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.8284	0.728399
$222$	0	0
$223$	19.2132	1.28661	0.643306	−	0.765609i	$-0.277562\pi$
$223$	19.2132	1.28661	0.643306	+	0.765609i	$0.277562\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.5858	−1.36633	−0.683163	−	0.730266i	$-0.739396\pi$
$227$	−20.5858	−1.36633	−0.683163	+	0.730266i	$0.739396\pi$
$228$	0	0
$229$	−21.6569	−1.43113	−0.715563	−	0.698549i	$-0.753830\pi$
$229$	−21.6569	−1.43113	−0.715563	+	0.698549i	$0.753830\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	0.828427	0.0540406
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.3431	−0.669042	−0.334521	−	0.942388i	$-0.608575\pi$
$239$	−10.3431	−0.669042	−0.334521	+	0.942388i	$0.608575\pi$
$240$	0	0
$241$	−30.2843	−1.95078	−0.975391	−	0.220484i	$-0.929236\pi$
$241$	−30.2843	−1.95078	−0.975391	+	0.220484i	$0.929236\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.3137	−1.04224
$246$	0	0
$247$	−2.24264	−0.142696
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.6569	0.862013	0.431006	−	0.902349i	$-0.358159\pi$
$251$	13.6569	0.862013	0.431006	+	0.902349i	$0.358159\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.24264	0.139892	0.0699460	−	0.997551i	$-0.477717\pi$
$257$	2.24264	0.139892	0.0699460	+	0.997551i	$0.477717\pi$
$258$	0	0
$259$	49.4558	3.07304
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.65685	−0.225491	−0.112746	−	0.993624i	$-0.535965\pi$
$263$	−3.65685	−0.225491	−0.112746	+	0.993624i	$0.535965\pi$
$264$	0	0
$265$	−5.07107	−0.311513
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.4853	1.37095	0.685476	−	0.728095i	$-0.259594\pi$
$269$	22.4853	1.37095	0.685476	+	0.728095i	$0.259594\pi$
$270$	0	0
$271$	23.6569	1.43705	0.718526	−	0.695500i	$-0.244817\pi$
$271$	23.6569	1.43705	0.718526	+	0.695500i	$0.244817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.00000	0.120605
$276$	0	0
$277$	16.8284	1.01112	0.505561	−	0.862791i	$-0.331286\pi$
$277$	16.8284	1.01112	0.505561	+	0.862791i	$0.331286\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.9706	−1.13169	−0.565844	−	0.824512i	$-0.691450\pi$
$281$	−18.9706	−1.13169	−0.565844	+	0.824512i	$0.691450\pi$
$282$	0	0
$283$	−2.68629	−0.159683	−0.0798417	−	0.996808i	$-0.525441\pi$
$283$	−2.68629	−0.159683	−0.0798417	+	0.996808i	$0.525441\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.9706	−2.18230
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	23.4142	1.36787	0.683936	−	0.729542i	$-0.260267\pi$
$293$	23.4142	1.36787	0.683936	+	0.729542i	$0.260267\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.4558	0.778172
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.34315	0.134168
$306$	0	0
$307$	7.89949	0.450848	0.225424	−	0.974261i	$-0.427623\pi$
$307$	7.89949	0.450848	0.225424	+	0.974261i	$0.427623\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5.75736	0.323366	0.161683	−	0.986843i	$-0.448308\pi$
$317$	5.75736	0.323366	0.161683	+	0.986843i	$0.448308\pi$
$318$	0	0
$319$	−20.9706	−1.17413
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.82843	−0.268661
$324$	0	0
$325$	2.24264	0.124399
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	1.17157	0.0643955	0.0321977	−	0.999482i	$-0.489749\pi$
$331$	1.17157	0.0643955	0.0321977	+	0.999482i	$0.489749\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.585786	0.0320049
$336$	0	0
$337$	17.7574	0.967305	0.483652	−	0.875260i	$-0.339310\pi$
$337$	17.7574	0.967305	0.483652	+	0.875260i	$0.339310\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.34315	−0.126888
$342$	0	0
$343$	−44.9706	−2.42818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−22.9706	−1.23312	−0.616562	−	0.787306i	$-0.711475\pi$
$347$	−22.9706	−1.23312	−0.616562	+	0.787306i	$0.711475\pi$
$348$	0	0
$349$	−0.343146	−0.0183682	−0.00918409	−	0.999958i	$-0.502923\pi$
$349$	−0.343146	−0.0183682	−0.00918409	+	0.999958i	$0.502923\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.85786	0.205333	0.102667	−	0.994716i	$-0.467262\pi$
$353$	3.85786	0.205333	0.102667	+	0.994716i	$0.467262\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.3137	1.12489	0.562447	−	0.826833i	$-0.309860\pi$
$359$	21.3137	1.12489	0.562447	+	0.826833i	$0.309860\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.4853	−0.758194
$366$	0	0
$367$	−28.1421	−1.46901	−0.734504	−	0.678605i	$-0.762585\pi$
$367$	−28.1421	−1.46901	−0.734504	+	0.678605i	$0.762585\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.4853	−1.27121
$372$	0	0
$373$	9.55635	0.494809	0.247405	−	0.968912i	$-0.420422\pi$
$373$	9.55635	0.494809	0.247405	+	0.968912i	$0.420422\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.5147	−1.21107
$378$	0	0
$379$	22.3431	1.14769	0.573845	−	0.818964i	$-0.305451\pi$
$379$	22.3431	1.14769	0.573845	+	0.818964i	$0.305451\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.92893	−0.149661	−0.0748307	−	0.997196i	$-0.523842\pi$
$383$	−2.92893	−0.149661	−0.0748307	+	0.997196i	$0.523842\pi$
$384$	0	0
$385$	9.65685	0.492159
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	28.9706	1.46510
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9.65685	0.485889
$396$	0	0
$397$	−32.1421	−1.61317	−0.806584	−	0.591120i	$-0.798686\pi$
$397$	−32.1421	−1.61317	−0.806584	+	0.591120i	$0.798686\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	−2.62742	−0.130881
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.4853	−1.01542
$408$	0	0
$409$	9.51472	0.470473	0.235236	−	0.971938i	$-0.424414\pi$
$409$	9.51472	0.470473	0.235236	+	0.971938i	$0.424414\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	11.3137	0.556711
$414$	0	0
$415$	9.31371	0.457192
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−9.65685	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$419$	−9.65685	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$420$	0	0
$421$	−8.34315	−0.406620	−0.203310	−	0.979114i	$-0.565170\pi$
$421$	−8.34315	−0.406620	−0.203310	+	0.979114i	$0.565170\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.82843	0.234213
$426$	0	0
$427$	11.3137	0.547509
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−33.4558	−1.61151	−0.805756	−	0.592248i	$-0.798241\pi$
$431$	−33.4558	−1.61151	−0.805756	+	0.592248i	$0.798241\pi$
$432$	0	0
$433$	−15.2132	−0.731100	−0.365550	−	0.930792i	$-0.619119\pi$
$433$	−15.2132	−0.731100	−0.365550	+	0.930792i	$0.619119\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−29.6569	−1.41544	−0.707722	−	0.706491i	$-0.750277\pi$
$439$	−29.6569	−1.41544	−0.707722	+	0.706491i	$0.750277\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.2843	−1.62889	−0.814447	−	0.580237i	$-0.802960\pi$
$443$	−34.2843	−1.62889	−0.814447	+	0.580237i	$0.802960\pi$
$444$	0	0
$445$	10.4853	0.497050
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.5147	0.637799	0.318900	−	0.947789i	$-0.396687\pi$
$449$	13.5147	0.637799	0.318900	+	0.947789i	$0.396687\pi$
$450$	0	0
$451$	15.3137	0.721094
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.8284	0.507644
$456$	0	0
$457$	18.9706	0.887405	0.443703	−	0.896174i	$-0.353665\pi$
$457$	18.9706	0.887405	0.443703	+	0.896174i	$0.353665\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.31371	0.247484	0.123742	−	0.992314i	$-0.460510\pi$
$461$	5.31371	0.247484	0.123742	+	0.992314i	$0.460510\pi$
$462$	0	0
$463$	−9.31371	−0.432845	−0.216422	−	0.976300i	$-0.569439\pi$
$463$	−9.31371	−0.432845	−0.216422	+	0.976300i	$0.569439\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.31371	−0.430987	−0.215494	−	0.976505i	$-0.569136\pi$
$467$	−9.31371	−0.430987	−0.215494	+	0.976505i	$0.569136\pi$
$468$	0	0
$469$	2.82843	0.130605
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.65685	−0.0761822
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0000	0.456912	0.228456	−	0.973554i	$-0.426632\pi$
$479$	10.0000	0.456912	0.228456	+	0.973554i	$0.426632\pi$
$480$	0	0
$481$	−22.9706	−1.04737
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.75736	0.0797976
$486$	0	0
$487$	24.3848	1.10498	0.552490	−	0.833520i	$-0.313678\pi$
$487$	24.3848	1.10498	0.552490	+	0.833520i	$0.313678\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.3137	−1.59369	−0.796843	−	0.604187i	$-0.793498\pi$
$491$	−35.3137	−1.59369	−0.796843	+	0.604187i	$0.793498\pi$
$492$	0	0
$493$	−50.6274	−2.28014
$494$	0	0
$495$	0	0
$496$	0	0
$497$	52.2843	2.34527
$498$	0	0
$499$	−26.9706	−1.20737	−0.603684	−	0.797224i	$-0.706301\pi$
$499$	−26.9706	−1.20737	−0.603684	+	0.797224i	$0.706301\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.97056	−0.132451	−0.0662254	−	0.997805i	$-0.521096\pi$
$503$	−2.97056	−0.132451	−0.0662254	+	0.997805i	$0.521096\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.7990	1.49811	0.749057	−	0.662506i	$-0.230507\pi$
$509$	33.7990	1.49811	0.749057	+	0.662506i	$0.230507\pi$
$510$	0	0
$511$	−69.9411	−3.09401
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15.8995	0.700615
$516$	0	0
$517$	−1.65685	−0.0728684
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−28.3431	−1.24174	−0.620868	−	0.783915i	$-0.713220\pi$
$521$	−28.3431	−1.24174	−0.620868	+	0.783915i	$0.713220\pi$
$522$	0	0
$523$	−6.72792	−0.294191	−0.147096	−	0.989122i	$-0.546993\pi$
$523$	−6.72792	−0.294191	−0.147096	+	0.989122i	$0.546993\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.65685	−0.246416
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.1716	0.743783
$534$	0	0
$535$	4.58579	0.198261
$536$	0	0
$537$	0	0
$538$	0	0
$539$	32.6274	1.40536
$540$	0	0
$541$	11.3137	0.486414	0.243207	−	0.969974i	$-0.421801\pi$
$541$	11.3137	0.486414	0.243207	+	0.969974i	$0.421801\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.82843	0.378168
$546$	0	0
$547$	36.3848	1.55570	0.777850	−	0.628450i	$-0.216310\pi$
$547$	36.3848	1.55570	0.777850	+	0.628450i	$0.216310\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	10.4853	0.446688
$552$	0	0
$553$	46.6274	1.98280
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−41.7990	−1.77108	−0.885540	−	0.464563i	$-0.846211\pi$
$557$	−41.7990	−1.77108	−0.885540	+	0.464563i	$0.846211\pi$
$558$	0	0
$559$	−1.85786	−0.0785793
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.4142	0.481052	0.240526	−	0.970643i	$-0.422680\pi$
$563$	11.4142	0.481052	0.240526	+	0.970643i	$0.422680\pi$
$564$	0	0
$565$	5.07107	0.213341
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−0.828427	−0.0347295	−0.0173647	−	0.999849i	$-0.505528\pi$
$569$	−0.828427	−0.0347295	−0.0173647	+	0.999849i	$0.505528\pi$
$570$	0	0
$571$	−14.6863	−0.614602	−0.307301	−	0.951612i	$-0.599426\pi$
$571$	−14.6863	−0.614602	−0.307301	+	0.951612i	$0.599426\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	44.9706	1.86569
$582$	0	0
$583$	10.1421	0.420044
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.9706	0.617901	0.308951	−	0.951078i	$-0.400022\pi$
$587$	14.9706	0.617901	0.308951	+	0.951078i	$0.400022\pi$
$588$	0	0
$589$	1.17157	0.0482738
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8.62742	−0.354286	−0.177143	−	0.984185i	$-0.556685\pi$
$593$	−8.62742	−0.354286	−0.177143	+	0.984185i	$0.556685\pi$
$594$	0	0
$595$	23.3137	0.955769
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.68629	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$599$	4.68629	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$600$	0	0
$601$	−18.0000	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	5.07107	0.205828	0.102914	−	0.994690i	$-0.467183\pi$
$607$	5.07107	0.205828	0.102914	+	0.994690i	$0.467183\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.85786	−0.0751611
$612$	0	0
$613$	17.5147	0.707413	0.353706	−	0.935356i	$-0.384921\pi$
$613$	17.5147	0.707413	0.353706	+	0.935356i	$0.384921\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.20101	0.249643	0.124822	−	0.992179i	$-0.460164\pi$
$617$	6.20101	0.249643	0.124822	+	0.992179i	$0.460164\pi$
$618$	0	0
$619$	18.0000	0.723481	0.361741	−	0.932279i	$-0.382183\pi$
$619$	18.0000	0.723481	0.361741	+	0.932279i	$0.382183\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	50.6274	2.02834
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−49.4558	−1.97193
$630$	0	0
$631$	−3.65685	−0.145577	−0.0727885	−	0.997347i	$-0.523190\pi$
$631$	−3.65685	−0.145577	−0.0727885	+	0.997347i	$0.523190\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	21.5563	0.855438
$636$	0	0
$637$	36.5858	1.44958
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.3137	0.841841	0.420920	−	0.907098i	$-0.361707\pi$
$641$	21.3137	0.841841	0.420920	+	0.907098i	$0.361707\pi$
$642$	0	0
$643$	24.6274	0.971211	0.485605	−	0.874178i	$-0.338599\pi$
$643$	24.6274	0.971211	0.485605	+	0.874178i	$0.338599\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.68629	−0.262865	−0.131433	−	0.991325i	$-0.541958\pi$
$647$	−6.68629	−0.262865	−0.131433	+	0.991325i	$0.541958\pi$
$648$	0	0
$649$	−4.68629	−0.183953
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−13.7990	−0.539996	−0.269998	−	0.962861i	$-0.587023\pi$
$653$	−13.7990	−0.539996	−0.269998	+	0.962861i	$0.587023\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.6274	1.34889	0.674446	−	0.738324i	$-0.264382\pi$
$659$	34.6274	1.34889	0.674446	+	0.738324i	$0.264382\pi$
$660$	0	0
$661$	−12.6274	−0.491150	−0.245575	−	0.969378i	$-0.578977\pi$
$661$	−12.6274	−0.491150	−0.245575	+	0.969378i	$0.578977\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.82843	−0.187238
$666$	0	0
$667$	−62.9117	−2.43595
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.68629	−0.180912
$672$	0	0
$673$	10.2426	0.394825	0.197412	−	0.980321i	$-0.436746\pi$
$673$	10.2426	0.394825	0.197412	+	0.980321i	$0.436746\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−25.5563	−0.982210	−0.491105	−	0.871100i	$-0.663407\pi$
$677$	−25.5563	−0.982210	−0.491105	+	0.871100i	$0.663407\pi$
$678$	0	0
$679$	8.48528	0.325635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	38.7279	1.48188	0.740941	−	0.671570i	$-0.234380\pi$
$683$	38.7279	1.48188	0.740941	+	0.671570i	$0.234380\pi$
$684$	0	0
$685$	−20.1421	−0.769591
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.3726	0.433261
$690$	0	0
$691$	−15.6569	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$691$	−15.6569	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.3137	0.656746
$696$	0	0
$697$	36.9706	1.40036
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−49.6569	−1.87551	−0.937757	−	0.347293i	$-0.887101\pi$
$701$	−49.6569	−1.87551	−0.937757	+	0.347293i	$0.887101\pi$
$702$	0	0
$703$	10.2426	0.386309
$704$	0	0
$705$	0	0
$706$	0	0
$707$	19.3137	0.726367
$708$	0	0
$709$	18.9706	0.712454	0.356227	−	0.934399i	$-0.384063\pi$
$709$	18.9706	0.712454	0.356227	+	0.934399i	$0.384063\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.02944	−0.263254
$714$	0	0
$715$	−4.48528	−0.167740
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−38.2843	−1.42776	−0.713881	−	0.700267i	$-0.753064\pi$
$719$	−38.2843	−1.42776	−0.713881	+	0.700267i	$0.753064\pi$
$720$	0	0
$721$	76.7696	2.85905
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.4853	−0.389414
$726$	0	0
$727$	−29.5147	−1.09464	−0.547320	−	0.836923i	$-0.684352\pi$
$727$	−29.5147	−1.09464	−0.547320	+	0.836923i	$0.684352\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	39.6569	1.46476	0.732380	−	0.680896i	$-0.238410\pi$
$733$	39.6569	1.46476	0.732380	+	0.680896i	$0.238410\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.17157	−0.0431554
$738$	0	0
$739$	34.6274	1.27379	0.636895	−	0.770951i	$-0.280218\pi$
$739$	34.6274	1.27379	0.636895	+	0.770951i	$0.280218\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.0416	−0.515137	−0.257569	−	0.966260i	$-0.582921\pi$
$743$	−14.0416	−0.515137	−0.257569	+	0.966260i	$0.582921\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	22.1421	0.809056
$750$	0	0
$751$	14.1421	0.516054	0.258027	−	0.966138i	$-0.416928\pi$
$751$	14.1421	0.516054	0.258027	+	0.966138i	$0.416928\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−18.1421	−0.660260
$756$	0	0
$757$	19.6569	0.714441	0.357220	−	0.934020i	$-0.383725\pi$
$757$	19.6569	0.714441	0.357220	+	0.934020i	$0.383725\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.6274	−1.11024	−0.555121	−	0.831769i	$-0.687328\pi$
$761$	−30.6274	−1.11024	−0.555121	+	0.831769i	$0.687328\pi$
$762$	0	0
$763$	42.6274	1.54322
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.25483	−0.189741
$768$	0	0
$769$	38.6274	1.39294	0.696470	−	0.717586i	$-0.254753\pi$
$769$	38.6274	1.39294	0.696470	+	0.717586i	$0.254753\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.100505	0.00361492	0.00180746	−	0.999998i	$-0.499425\pi$
$773$	0.100505	0.00361492	0.00180746	+	0.999998i	$0.499425\pi$
$774$	0	0
$775$	−1.17157	−0.0420841
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.65685	−0.274335
$780$	0	0
$781$	−21.6569	−0.774943
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.17157	−0.113198
$786$	0	0
$787$	−41.8406	−1.49146	−0.745729	−	0.666250i	$-0.767898\pi$
$787$	−41.8406	−1.49146	−0.745729	+	0.666250i	$0.767898\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	24.4853	0.870596
$792$	0	0
$793$	−5.25483	−0.186605
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−11.8995	−0.421502	−0.210751	−	0.977540i	$-0.567591\pi$
$797$	−11.8995	−0.421502	−0.210751	+	0.977540i	$0.567591\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	28.9706	1.02235
$804$	0	0
$805$	28.9706	1.02108
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.2548	1.38013	0.690063	−	0.723749i	$-0.257583\pi$
$809$	39.2548	1.38013	0.690063	+	0.723749i	$0.257583\pi$
$810$	0	0
$811$	19.7990	0.695237	0.347618	−	0.937636i	$-0.386991\pi$
$811$	19.7990	0.695237	0.347618	+	0.937636i	$0.386991\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.48528	0.0870556
$816$	0	0
$817$	0.828427	0.0289830
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.0000	−0.349002	−0.174501	−	0.984657i	$-0.555831\pi$
$821$	−10.0000	−0.349002	−0.174501	+	0.984657i	$0.555831\pi$
$822$	0	0
$823$	−15.1716	−0.528848	−0.264424	−	0.964407i	$-0.585182\pi$
$823$	−15.1716	−0.528848	−0.264424	+	0.964407i	$0.585182\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.0711	1.01090	0.505450	−	0.862856i	$-0.331326\pi$
$827$	29.0711	1.01090	0.505450	+	0.862856i	$0.331326\pi$
$828$	0	0
$829$	−40.4264	−1.40407	−0.702034	−	0.712144i	$-0.747724\pi$
$829$	−40.4264	−1.40407	−0.702034	+	0.712144i	$0.747724\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	78.7696	2.72920
$834$	0	0
$835$	−10.7279	−0.371255
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−43.5980	−1.50517	−0.752585	−	0.658495i	$-0.771193\pi$
$839$	−43.5980	−1.50517	−0.752585	+	0.658495i	$0.771193\pi$
$840$	0	0
$841$	80.9411	2.79107
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.97056	0.274196
$846$	0	0
$847$	33.7990	1.16135
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−61.4558	−2.10668
$852$	0	0
$853$	42.0000	1.43805	0.719026	−	0.694983i	$-0.244588\pi$
$853$	42.0000	1.43805	0.719026	+	0.694983i	$0.244588\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.2132	0.929585	0.464793	−	0.885420i	$-0.346129\pi$
$857$	27.2132	0.929585	0.464793	+	0.885420i	$0.346129\pi$
$858$	0	0
$859$	−24.2843	−0.828569	−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843	−0.828569	−0.414284	+	0.910148i	$0.635968\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.89949	0.132740	0.0663702	−	0.997795i	$-0.478858\pi$
$863$	3.89949	0.132740	0.0663702	+	0.997795i	$0.478858\pi$
$864$	0	0
$865$	−3.89949	−0.132587
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.3137	−0.655173
$870$	0	0
$871$	−1.31371	−0.0445133
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.82843	0.163231
$876$	0	0
$877$	−37.0711	−1.25180	−0.625901	−	0.779903i	$-0.715268\pi$
$877$	−37.0711	−1.25180	−0.625901	+	0.779903i	$0.715268\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44.2843	−1.49198	−0.745988	−	0.665960i	$-0.768022\pi$
$881$	−44.2843	−1.49198	−0.745988	+	0.665960i	$0.768022\pi$
$882$	0	0
$883$	−39.1716	−1.31823	−0.659114	−	0.752043i	$-0.729069\pi$
$883$	−39.1716	−1.31823	−0.659114	+	0.752043i	$0.729069\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.0711	1.64765	0.823823	−	0.566848i	$-0.191837\pi$
$887$	49.0711	1.64765	0.823823	+	0.566848i	$0.191837\pi$
$888$	0	0
$889$	104.083	3.49084
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.828427	0.0277223
$894$	0	0
$895$	21.6569	0.723909
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.2843	0.409703
$900$	0	0
$901$	24.4853	0.815723
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18.0000	0.598340
$906$	0	0
$907$	−20.3848	−0.676865	−0.338433	−	0.940991i	$-0.609897\pi$
$907$	−20.3848	−0.676865	−0.338433	+	0.940991i	$0.609897\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−4.20101	−0.139186	−0.0695928	−	0.997575i	$-0.522170\pi$
$911$	−4.20101	−0.139186	−0.0695928	+	0.997575i	$0.522170\pi$
$912$	0	0
$913$	−18.6274	−0.616478
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−27.3137	−0.901978
$918$	0	0
$919$	35.3137	1.16489	0.582446	−	0.812869i	$-0.302096\pi$
$919$	35.3137	1.16489	0.582446	+	0.812869i	$0.302096\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.2843	−0.799327
$924$	0	0
$925$	−10.2426	−0.336776
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.68629	−0.219370	−0.109685	−	0.993966i	$-0.534984\pi$
$929$	−6.68629	−0.219370	−0.109685	+	0.993966i	$0.534984\pi$
$930$	0	0
$931$	−16.3137	−0.534660
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9.65685	−0.315813
$936$	0	0
$937$	−40.1421	−1.31139	−0.655693	−	0.755027i	$-0.727624\pi$
$937$	−40.1421	−1.31139	−0.655693	+	0.755027i	$0.727624\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	30.2843	0.987239	0.493620	−	0.869678i	$-0.335674\pi$
$941$	30.2843	0.987239	0.493620	+	0.869678i	$0.335674\pi$
$942$	0	0
$943$	45.9411	1.49605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.34315	0.271116	0.135558	−	0.990769i	$-0.456717\pi$
$947$	8.34315	0.271116	0.135558	+	0.990769i	$0.456717\pi$
$948$	0	0
$949$	32.4853	1.05452
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.75736	−0.186499	−0.0932496	−	0.995643i	$-0.529725\pi$
$953$	−5.75736	−0.186499	−0.0932496	+	0.995643i	$0.529725\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−97.2548	−3.14052
$960$	0	0
$961$	−29.6274	−0.955723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.585786	−0.0188571
$966$	0	0
$967$	20.6274	0.663333	0.331667	−	0.943397i	$-0.392389\pi$
$967$	20.6274	0.663333	0.331667	+	0.943397i	$0.392389\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.1127	1.12682	0.563410	−	0.826177i	$-0.309489\pi$
$971$	35.1127	1.12682	0.563410	+	0.826177i	$0.309489\pi$
$972$	0	0
$973$	83.5980	2.68003
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.27208	−0.296640	−0.148320	−	0.988939i	$-0.547387\pi$
$977$	−9.27208	−0.296640	−0.148320	+	0.988939i	$0.547387\pi$
$978$	0	0
$979$	−20.9706	−0.670222
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.4142	0.746797	0.373399	−	0.927671i	$-0.378192\pi$
$983$	23.4142	0.746797	0.373399	+	0.927671i	$0.378192\pi$
$984$	0	0
$985$	−5.31371	−0.169309
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.97056	−0.158055
$990$	0	0
$991$	−51.1127	−1.62365	−0.811824	−	0.583902i	$-0.801525\pi$
$991$	−51.1127	−1.62365	−0.811824	+	0.583902i	$0.801525\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10.3431	0.327900
$996$	0	0
$997$	−13.5147	−0.428015	−0.214008	−	0.976832i	$-0.568652\pi$
$997$	−13.5147	−0.428015	−0.214008	+	0.976832i	$0.568652\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.a.g.1.1		2
3.2	odd	2		380.2.a.c.1.2	✓	2
12.11	even	2		1520.2.a.o.1.1		2
15.2	even	4		1900.2.c.d.1749.3		4
15.8	even	4		1900.2.c.d.1749.2		4
15.14	odd	2		1900.2.a.e.1.1		2
24.5	odd	2		6080.2.a.bl.1.1		2
24.11	even	2		6080.2.a.y.1.2		2
57.56	even	2		7220.2.a.m.1.1		2
60.59	even	2		7600.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.a.c.1.2	✓	2	3.2	odd	2
1520.2.a.o.1.1		2	12.11	even	2
1900.2.a.e.1.1		2	15.14	odd	2
1900.2.c.d.1749.2		4	15.8	even	4
1900.2.c.d.1749.3		4	15.2	even	4
3420.2.a.g.1.1		2	1.1	even	1	trivial
6080.2.a.y.1.2		2	24.11	even	2
6080.2.a.bl.1.1		2	24.5	odd	2
7220.2.a.m.1.1		2	57.56	even	2
7600.2.a.u.1.2		2	60.59	even	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.82843 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.82843 q^{7} +2.00000 q^{11} +2.24264 q^{13} +4.82843 q^{17} -1.00000 q^{19} +6.00000 q^{23} +1.00000 q^{25} -10.4853 q^{29} -1.17157 q^{31} +4.82843 q^{35} -10.2426 q^{37} +7.65685 q^{41} -0.828427 q^{43} -0.828427 q^{47} +16.3137 q^{49} +5.07107 q^{53} -2.00000 q^{55} -2.34315 q^{59} -2.34315 q^{61} -2.24264 q^{65} -0.585786 q^{67} -10.8284 q^{71} +14.4853 q^{73} -9.65685 q^{77} -9.65685 q^{79} -9.31371 q^{83} -4.82843 q^{85} -10.4853 q^{89} -10.8284 q^{91} +1.00000 q^{95} -1.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 4 * q^7	\( 2 q - 2 q^{5} - 4 q^{7} + 4 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 4 q^{29} - 8 q^{31} + 4 q^{35} - 12 q^{37} + 4 q^{41} + 4 q^{43} + 4 q^{47} + 10 q^{49} - 4 q^{53} - 4 q^{55} - 16 q^{59} - 16 q^{61} + 4 q^{65} - 4 q^{67} - 16 q^{71} + 12 q^{73} - 8 q^{77} - 8 q^{79} + 4 q^{83} - 4 q^{85} - 4 q^{89} - 16 q^{91} + 2 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 4 * q^7 + 4 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 4 * q^29 - 8 * q^31 + 4 * q^35 - 12 * q^37 + 4 * q^41 + 4 * q^43 + 4 * q^47 + 10 * q^49 - 4 * q^53 - 4 * q^55 - 16 * q^59 - 16 * q^61 + 4 * q^65 - 4 * q^67 - 16 * q^71 + 12 * q^73 - 8 * q^77 - 8 * q^79 + 4 * q^83 - 4 * q^85 - 4 * q^89 - 16 * q^91 + 2 * q^95 - 12 * q^97

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