LMFDB - Embedded newform 4140.2.a.t.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(1,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, 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("4140.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.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:	$33.0580664368$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.14345904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 13x^{3} + 34x^{2} - 11x - 12 $ x^5 - 2x^4 - 13x^3 + 34x^2 - 11x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.430705$ of defining polynomial
Character	$\chi$	$=$	4140.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +5.21120 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +5.21120 q^{7} -2.67519 q^{11} +0.186219 q^{13} +1.99097 q^{17} -0.675190 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.02498 q^{29} +7.52757 q^{31} -5.21120 q^{35} +4.34979 q^{37} +6.60401 q^{41} -2.84336 q^{43} +13.5866 q^{47} +20.1566 q^{49} -12.9023 q^{53} +2.67519 q^{55} +6.74779 q^{59} +8.65714 q^{61} -0.186219 q^{65} -12.1226 q^{67} -0.181619 q^{71} +3.81378 q^{73} -13.9409 q^{77} -10.4904 q^{79} -16.8538 q^{83} -1.99097 q^{85} +12.4224 q^{89} +0.970423 q^{91} +0.675190 q^{95} -4.92941 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 + 4 * q^7	$ 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * 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
$7$	5.21120	1.96965	0.984823	−	0.173559i	$-0.0555268\pi$
$7$	5.21120	1.96965	0.984823	+	0.173559i	$0.0555268\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.67519	−0.806600	−0.403300	−	0.915068i	$-0.632137\pi$
$11$	−2.67519	−0.806600	−0.403300	+	0.915068i	$0.632137\pi$
$12$	0	0
$13$	0.186219	0.0516478	0.0258239	−	0.999667i	$-0.491779\pi$
$13$	0.186219	0.0516478	0.0258239	+	0.999667i	$0.491779\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.99097	0.482882	0.241441	−	0.970415i	$-0.422380\pi$
$17$	1.99097	0.482882	0.241441	+	0.970415i	$0.422380\pi$
$18$	0	0
$19$	−0.675190	−0.154899	−0.0774496	−	0.996996i	$-0.524678\pi$
$19$	−0.675190	−0.154899	−0.0774496	+	0.996996i	$0.524678\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5.02498	−0.933115	−0.466557	−	0.884491i	$-0.654506\pi$
$29$	−5.02498	−0.933115	−0.466557	+	0.884491i	$0.654506\pi$
$30$	0	0
$31$	7.52757	1.35199	0.675996	−	0.736905i	$-0.263714\pi$
$31$	7.52757	1.35199	0.675996	+	0.736905i	$0.263714\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.21120	−0.880853
$36$	0	0
$37$	4.34979	0.715101	0.357550	−	0.933894i	$-0.383612\pi$
$37$	4.34979	0.715101	0.357550	+	0.933894i	$0.383612\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.60401	1.03137	0.515687	−	0.856777i	$-0.327537\pi$
$41$	6.60401	1.03137	0.515687	+	0.856777i	$0.327537\pi$
$42$	0	0
$43$	−2.84336	−0.433608	−0.216804	−	0.976215i	$-0.569563\pi$
$43$	−2.84336	−0.433608	−0.216804	+	0.976215i	$0.569563\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	13.5866	1.98180	0.990901	−	0.134591i	$-0.0429721\pi$
$47$	13.5866	1.98180	0.990901	+	0.134591i	$0.0429721\pi$
$48$	0	0
$49$	20.1566	2.87951
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.9023	−1.77227	−0.886136	−	0.463425i	$-0.846620\pi$
$53$	−12.9023	−1.77227	−0.886136	+	0.463425i	$0.846620\pi$
$54$	0	0
$55$	2.67519	0.360723
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.74779	0.878488	0.439244	−	0.898368i	$-0.355246\pi$
$59$	6.74779	0.878488	0.439244	+	0.898368i	$0.355246\pi$
$60$	0	0
$61$	8.65714	1.10843	0.554217	−	0.832373i	$-0.313018\pi$
$61$	8.65714	1.10843	0.554217	+	0.832373i	$0.313018\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.186219	−0.0230976
$66$	0	0
$67$	−12.1226	−1.48101	−0.740503	−	0.672053i	$-0.765413\pi$
$67$	−12.1226	−1.48101	−0.740503	+	0.672053i	$0.765413\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.181619	−0.0215542	−0.0107771	−	0.999942i	$-0.503431\pi$
$71$	−0.181619	−0.0215542	−0.0107771	+	0.999942i	$0.503431\pi$
$72$	0	0
$73$	3.81378	0.446369	0.223185	−	0.974776i	$-0.428355\pi$
$73$	3.81378	0.446369	0.223185	+	0.974776i	$0.428355\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.9409	−1.58872
$78$	0	0
$79$	−10.4904	−1.18026	−0.590131	−	0.807308i	$-0.700924\pi$
$79$	−10.4904	−1.18026	−0.590131	+	0.807308i	$0.700924\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−16.8538	−1.84995	−0.924973	−	0.380033i	$-0.875913\pi$
$83$	−16.8538	−1.84995	−0.924973	+	0.380033i	$0.875913\pi$
$84$	0	0
$85$	−1.99097	−0.215951
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.4224	1.31677	0.658385	−	0.752681i	$-0.271240\pi$
$89$	12.4224	1.31677	0.658385	+	0.752681i	$0.271240\pi$
$90$	0	0
$91$	0.970423	0.101728
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.675190	0.0692730
$96$	0	0
$97$	−4.92941	−0.500506	−0.250253	−	0.968180i	$-0.580514\pi$
$97$	−4.92941	−0.500506	−0.250253	+	0.968180i	$0.580514\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	11.5912	1.15336	0.576681	−	0.816969i	$-0.304347\pi$
$101$	11.5912	1.15336	0.576681	+	0.816969i	$0.304347\pi$
$102$	0	0
$103$	12.1937	1.20148	0.600742	−	0.799443i	$-0.294872\pi$
$103$	12.1937	1.20148	0.600742	+	0.799443i	$0.294872\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.57001	0.925168	0.462584	−	0.886576i	$-0.346922\pi$
$107$	9.57001	0.925168	0.462584	+	0.886576i	$0.346922\pi$
$108$	0	0
$109$	−0.675190	−0.0646715	−0.0323357	−	0.999477i	$-0.510295\pi$
$109$	−0.675190	−0.0646715	−0.0323357	+	0.999477i	$0.510295\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.21562	−0.302500	−0.151250	−	0.988496i	$-0.548330\pi$
$113$	−3.21562	−0.302500	−0.151250	+	0.988496i	$0.548330\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	10.3754	0.951107
$120$	0	0
$121$	−3.84336	−0.349396
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.2542	1.08739	0.543693	−	0.839284i	$-0.317025\pi$
$127$	12.2542	1.08739	0.543693	+	0.839284i	$0.317025\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.42239	0.561127	0.280563	−	0.959835i	$-0.409479\pi$
$131$	6.42239	0.561127	0.280563	+	0.959835i	$0.409479\pi$
$132$	0	0
$133$	−3.51855	−0.305097
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.1885	1.29764	0.648822	−	0.760940i	$-0.275262\pi$
$137$	15.1885	1.29764	0.648822	+	0.760940i	$0.275262\pi$
$138$	0	0
$139$	−15.1566	−1.28556	−0.642781	−	0.766050i	$-0.722220\pi$
$139$	−15.1566	−1.28556	−0.642781	+	0.766050i	$0.722220\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.498171	−0.0416592
$144$	0	0
$145$	5.02498	0.417302
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.02557	−0.165941	−0.0829706	−	0.996552i	$-0.526441\pi$
$149$	−2.02557	−0.165941	−0.0829706	+	0.996552i	$0.526441\pi$
$150$	0	0
$151$	3.15664	0.256884	0.128442	−	0.991717i	$-0.459002\pi$
$151$	3.15664	0.256884	0.128442	+	0.991717i	$0.459002\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−7.52757	−0.604629
$156$	0	0
$157$	−10.1891	−0.813182	−0.406591	−	0.913610i	$-0.633283\pi$
$157$	−10.1891	−0.813182	−0.406591	+	0.913610i	$0.633283\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.21120	−0.410700
$162$	0	0
$163$	11.9819	0.938499	0.469249	−	0.883066i	$-0.344525\pi$
$163$	11.9819	0.938499	0.469249	+	0.883066i	$0.344525\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.3042	−1.10689	−0.553445	−	0.832886i	$-0.686687\pi$
$167$	−14.3042	−1.10689	−0.553445	+	0.832886i	$0.686687\pi$
$168$	0	0
$169$	−12.9653	−0.997333
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.1051	1.60459	0.802296	−	0.596927i	$-0.203612\pi$
$173$	21.1051	1.60459	0.802296	+	0.596927i	$0.203612\pi$
$174$	0	0
$175$	5.21120	0.393929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.2798	−0.917835	−0.458917	−	0.888479i	$-0.651763\pi$
$179$	−12.2798	−0.917835	−0.458917	+	0.888479i	$0.651763\pi$
$180$	0	0
$181$	9.42758	0.700747	0.350373	−	0.936610i	$-0.386055\pi$
$181$	9.42758	0.700747	0.350373	+	0.936610i	$0.386055\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.34979	−0.319803
$186$	0	0
$187$	−5.32623	−0.389493
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.2847	−0.744175	−0.372087	−	0.928198i	$-0.621358\pi$
$191$	−10.2847	−0.744175	−0.372087	+	0.928198i	$0.621358\pi$
$192$	0	0
$193$	2.35439	0.169472	0.0847362	−	0.996403i	$-0.472995\pi$
$193$	2.35439	0.169472	0.0847362	+	0.996403i	$0.472995\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.7948	1.05409	0.527044	−	0.849838i	$-0.323300\pi$
$197$	14.7948	1.05409	0.527044	+	0.849838i	$0.323300\pi$
$198$	0	0
$199$	25.1220	1.78085	0.890424	−	0.455131i	$-0.150408\pi$
$199$	25.1220	1.78085	0.890424	+	0.455131i	$0.150408\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−26.1861	−1.83791
$204$	0	0
$205$	−6.60401	−0.461244
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.80626	0.124942
$210$	0	0
$211$	15.9985	1.10138	0.550691	−	0.834709i	$-0.314364\pi$
$211$	15.9985	1.10138	0.550691	+	0.834709i	$0.314364\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.84336	0.193915
$216$	0	0
$217$	39.2277	2.66295
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.370757	0.0249398
$222$	0	0
$223$	4.01805	0.269069	0.134534	−	0.990909i	$-0.457046\pi$
$223$	4.01805	0.269069	0.134534	+	0.990909i	$0.457046\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.5676	1.76335	0.881677	−	0.471854i	$-0.156415\pi$
$227$	26.5676	1.76335	0.881677	+	0.471854i	$0.156415\pi$
$228$	0	0
$229$	16.2132	1.07140	0.535700	−	0.844409i	$-0.320048\pi$
$229$	16.2132	1.07140	0.535700	+	0.844409i	$0.320048\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−16.4723	−1.07914	−0.539570	−	0.841941i	$-0.681413\pi$
$233$	−16.4723	−1.07914	−0.539570	+	0.841941i	$0.681413\pi$
$234$	0	0
$235$	−13.5866	−0.886289
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.3407	−1.70384	−0.851918	−	0.523674i	$-0.824561\pi$
$239$	−26.3407	−1.70384	−0.851918	+	0.523674i	$0.824561\pi$
$240$	0	0
$241$	−5.01153	−0.322821	−0.161410	−	0.986887i	$-0.551604\pi$
$241$	−5.01153	−0.322821	−0.161410	+	0.986887i	$0.551604\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−20.1566	−1.28776
$246$	0	0
$247$	−0.125733	−0.00800021
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.9848	1.26143	0.630715	−	0.776015i	$-0.282762\pi$
$251$	19.9848	1.26143	0.630715	+	0.776015i	$0.282762\pi$
$252$	0	0
$253$	2.67519	0.168188
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.04905	−0.439708	−0.219854	−	0.975533i	$-0.570558\pi$
$257$	−7.04905	−0.439708	−0.219854	+	0.975533i	$0.570558\pi$
$258$	0	0
$259$	22.6676	1.40850
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.6211	0.839916	0.419958	−	0.907544i	$-0.362045\pi$
$263$	13.6211	0.839916	0.419958	+	0.907544i	$0.362045\pi$
$264$	0	0
$265$	12.9023	0.792584
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.23015	0.135974	0.0679872	−	0.997686i	$-0.478342\pi$
$269$	2.23015	0.135974	0.0679872	+	0.997686i	$0.478342\pi$
$270$	0	0
$271$	19.0206	1.15542	0.577708	−	0.816243i	$-0.303947\pi$
$271$	19.0206	1.15542	0.577708	+	0.816243i	$0.303947\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.67519	−0.161320
$276$	0	0
$277$	30.6084	1.83908	0.919539	−	0.392999i	$-0.128562\pi$
$277$	30.6084	1.83908	0.919539	+	0.392999i	$0.128562\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.41087	−0.322785	−0.161393	−	0.986890i	$-0.551599\pi$
$281$	−5.41087	−0.322785	−0.161393	+	0.986890i	$0.551599\pi$
$282$	0	0
$283$	−20.0572	−1.19227	−0.596137	−	0.802882i	$-0.703299\pi$
$283$	−20.0572	−1.19227	−0.596137	+	0.802882i	$0.703299\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	34.4148	2.03144
$288$	0	0
$289$	−13.0360	−0.766825
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.6394	1.03051	0.515253	−	0.857038i	$-0.327698\pi$
$293$	17.6394	1.03051	0.515253	+	0.857038i	$0.327698\pi$
$294$	0	0
$295$	−6.74779	−0.392872
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.186219	−0.0107693
$300$	0	0
$301$	−14.8173	−0.854055
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.65714	−0.495706
$306$	0	0
$307$	13.6227	0.777486	0.388743	−	0.921346i	$-0.372909\pi$
$307$	13.6227	0.777486	0.388743	+	0.921346i	$0.372909\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.67752	0.0951232	0.0475616	−	0.998868i	$-0.484855\pi$
$311$	1.67752	0.0951232	0.0475616	+	0.998868i	$0.484855\pi$
$312$	0	0
$313$	−4.15866	−0.235061	−0.117531	−	0.993069i	$-0.537498\pi$
$313$	−4.15866	−0.235061	−0.117531	+	0.993069i	$0.537498\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6.51335	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$317$	−6.51335	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$318$	0	0
$319$	13.4428	0.752651
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.34429	−0.0747981
$324$	0	0
$325$	0.186219	0.0103296
$326$	0	0
$327$	0	0
$328$	0	0
$329$	70.8022	3.90345
$330$	0	0
$331$	−5.25039	−0.288588	−0.144294	−	0.989535i	$-0.546091\pi$
$331$	−5.25039	−0.288588	−0.144294	+	0.989535i	$0.546091\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.1226	0.662326
$336$	0	0
$337$	−15.8408	−0.862902	−0.431451	−	0.902136i	$-0.641998\pi$
$337$	−15.8408	−0.862902	−0.431451	+	0.902136i	$0.641998\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−20.1377	−1.09052
$342$	0	0
$343$	68.5614	3.70197
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.7228	0.736679	0.368340	−	0.929691i	$-0.379926\pi$
$347$	13.7228	0.736679	0.368340	+	0.929691i	$0.379926\pi$
$348$	0	0
$349$	13.0897	0.700678	0.350339	−	0.936623i	$-0.386066\pi$
$349$	13.0897	0.700678	0.350339	+	0.936623i	$0.386066\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−24.7085	−1.31510	−0.657551	−	0.753410i	$-0.728408\pi$
$353$	−24.7085	−1.31510	−0.657551	+	0.753410i	$0.728408\pi$
$354$	0	0
$355$	0.181619	0.00963934
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.3148	−1.59995	−0.799977	−	0.600031i	$-0.795155\pi$
$359$	−30.3148	−1.59995	−0.799977	+	0.600031i	$0.795155\pi$
$360$	0	0
$361$	−18.5441	−0.976006
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.81378	−0.199622
$366$	0	0
$367$	−9.46374	−0.494003	−0.247002	−	0.969015i	$-0.579445\pi$
$367$	−9.46374	−0.494003	−0.247002	+	0.969015i	$0.579445\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−67.2366	−3.49075
$372$	0	0
$373$	2.70100	0.139852	0.0699262	−	0.997552i	$-0.477724\pi$
$373$	2.70100	0.139852	0.0699262	+	0.997552i	$0.477724\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.935746	−0.0481934
$378$	0	0
$379$	33.1551	1.70306	0.851530	−	0.524305i	$-0.175675\pi$
$379$	33.1551	1.70306	0.851530	+	0.524305i	$0.175675\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.14504	−0.211801	−0.105901	−	0.994377i	$-0.533773\pi$
$383$	−4.14504	−0.211801	−0.105901	+	0.994377i	$0.533773\pi$
$384$	0	0
$385$	13.9409	0.710496
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14.5084	0.735607	0.367804	−	0.929903i	$-0.380110\pi$
$389$	14.5084	0.735607	0.367804	+	0.929903i	$0.380110\pi$
$390$	0	0
$391$	−1.99097	−0.100688
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.4904	0.527829
$396$	0	0
$397$	−31.1719	−1.56447	−0.782237	−	0.622981i	$-0.785921\pi$
$397$	−31.1719	−1.56447	−0.782237	+	0.622981i	$0.785921\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.0680	−1.20190	−0.600949	−	0.799287i	$-0.705211\pi$
$401$	−24.0680	−1.20190	−0.600949	+	0.799287i	$0.705211\pi$
$402$	0	0
$403$	1.40178	0.0698275
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.6365	−0.576800
$408$	0	0
$409$	6.01654	0.297499	0.148749	−	0.988875i	$-0.452475\pi$
$409$	6.01654	0.297499	0.148749	+	0.988875i	$0.452475\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	35.1641	1.73031
$414$	0	0
$415$	16.8538	0.827321
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30.7390	1.50170	0.750849	−	0.660474i	$-0.229645\pi$
$419$	30.7390	1.50170	0.750849	+	0.660474i	$0.229645\pi$
$420$	0	0
$421$	−9.76525	−0.475929	−0.237965	−	0.971274i	$-0.576480\pi$
$421$	−9.76525	−0.475929	−0.237965	+	0.971274i	$0.576480\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.99097	0.0965764
$426$	0	0
$427$	45.1140	2.18322
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.1168	−1.06533	−0.532664	−	0.846327i	$-0.678809\pi$
$431$	−22.1168	−1.06533	−0.532664	+	0.846327i	$0.678809\pi$
$432$	0	0
$433$	−1.68069	−0.0807688	−0.0403844	−	0.999184i	$-0.512858\pi$
$433$	−1.68069	−0.0807688	−0.0403844	+	0.999184i	$0.512858\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.675190	0.0322987
$438$	0	0
$439$	−3.91656	−0.186927	−0.0934635	−	0.995623i	$-0.529794\pi$
$439$	−3.91656	−0.186927	−0.0934635	+	0.995623i	$0.529794\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.136265	−0.00647416	−0.00323708	−	0.999995i	$-0.501030\pi$
$443$	−0.136265	−0.00647416	−0.00323708	+	0.999995i	$0.501030\pi$
$444$	0	0
$445$	−12.4224	−0.588878
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.5334	1.20500	0.602498	−	0.798120i	$-0.294172\pi$
$449$	25.5334	1.20500	0.602498	+	0.798120i	$0.294172\pi$
$450$	0	0
$451$	−17.6670	−0.831906
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.970423	−0.0454941
$456$	0	0
$457$	−16.2166	−0.758582	−0.379291	−	0.925277i	$-0.623832\pi$
$457$	−16.2166	−0.758582	−0.379291	+	0.925277i	$0.623832\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.1923	−0.661002	−0.330501	−	0.943806i	$-0.607218\pi$
$461$	−14.1923	−0.661002	−0.330501	+	0.943806i	$0.607218\pi$
$462$	0	0
$463$	−17.2901	−0.803541	−0.401770	−	0.915740i	$-0.631605\pi$
$463$	−17.2901	−0.803541	−0.401770	+	0.915740i	$0.631605\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.9244	1.01454	0.507270	−	0.861787i	$-0.330655\pi$
$467$	21.9244	1.01454	0.507270	+	0.861787i	$0.330655\pi$
$468$	0	0
$469$	−63.1730	−2.91706
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.60652	0.349748
$474$	0	0
$475$	−0.675190	−0.0309798
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−14.0244	−0.640790	−0.320395	−	0.947284i	$-0.603816\pi$
$479$	−14.0244	−0.640790	−0.320395	+	0.947284i	$0.603816\pi$
$480$	0	0
$481$	0.810013	0.0369334
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.92941	0.223833
$486$	0	0
$487$	0.835839	0.0378755	0.0189377	−	0.999821i	$-0.493972\pi$
$487$	0.835839	0.0378755	0.0189377	+	0.999821i	$0.493972\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.84044	−0.128187	−0.0640936	−	0.997944i	$-0.520416\pi$
$491$	−2.84044	−0.128187	−0.0640936	+	0.997944i	$0.520416\pi$
$492$	0	0
$493$	−10.0046	−0.450585
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.946452	−0.0424542
$498$	0	0
$499$	−15.5944	−0.698101	−0.349050	−	0.937104i	$-0.613496\pi$
$499$	−15.5944	−0.698101	−0.349050	+	0.937104i	$0.613496\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.7666	0.925935	0.462968	−	0.886375i	$-0.346785\pi$
$503$	20.7666	0.925935	0.462968	+	0.886375i	$0.346785\pi$
$504$	0	0
$505$	−11.5912	−0.515800
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.65104	−0.117505	−0.0587527	−	0.998273i	$-0.518712\pi$
$509$	−2.65104	−0.117505	−0.0587527	+	0.998273i	$0.518712\pi$
$510$	0	0
$511$	19.8744	0.879190
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.1937	−0.537320
$516$	0	0
$517$	−36.3466	−1.59852
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−41.8383	−1.83297	−0.916484	−	0.400072i	$-0.868985\pi$
$521$	−41.8383	−1.83297	−0.916484	+	0.400072i	$0.868985\pi$
$522$	0	0
$523$	−21.8396	−0.954979	−0.477489	−	0.878638i	$-0.658453\pi$
$523$	−21.8396	−0.954979	−0.477489	+	0.878638i	$0.658453\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.9872	0.652853
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.22979	0.0532682
$534$	0	0
$535$	−9.57001	−0.413748
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−53.9226	−2.32261
$540$	0	0
$541$	−28.0967	−1.20797	−0.603985	−	0.796995i	$-0.706421\pi$
$541$	−28.0967	−1.20797	−0.603985	+	0.796995i	$0.706421\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.675190	0.0289220
$546$	0	0
$547$	14.5375	0.621579	0.310789	−	0.950479i	$-0.399407\pi$
$547$	14.5375	0.621579	0.310789	+	0.950479i	$0.399407\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.39281	0.144539
$552$	0	0
$553$	−54.6675	−2.32470
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.6699	−0.494470	−0.247235	−	0.968956i	$-0.579522\pi$
$557$	−11.6699	−0.494470	−0.247235	+	0.968956i	$0.579522\pi$
$558$	0	0
$559$	−0.529487	−0.0223949
$560$	0	0
$561$	0	0
$562$	0	0
$563$	42.8564	1.80618	0.903091	−	0.429448i	$-0.141292\pi$
$563$	42.8564	1.80618	0.903091	+	0.429448i	$0.141292\pi$
$564$	0	0
$565$	3.21562	0.135282
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−17.5544	−0.735919	−0.367959	−	0.929842i	$-0.619943\pi$
$569$	−17.5544	−0.735919	−0.367959	+	0.929842i	$0.619943\pi$
$570$	0	0
$571$	10.6890	0.447322	0.223661	−	0.974667i	$-0.428199\pi$
$571$	10.6890	0.447322	0.223661	+	0.974667i	$0.428199\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−29.5547	−1.23038	−0.615190	−	0.788379i	$-0.710921\pi$
$577$	−29.5547	−1.23038	−0.615190	+	0.788379i	$0.710921\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−87.8285	−3.64374
$582$	0	0
$583$	34.5162	1.42952
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.8779	−0.861722	−0.430861	−	0.902418i	$-0.641790\pi$
$587$	−20.8779	−0.861722	−0.430861	+	0.902418i	$0.641790\pi$
$588$	0	0
$589$	−5.08254	−0.209423
$590$	0	0
$591$	0	0
$592$	0	0
$593$	23.1002	0.948611	0.474306	−	0.880360i	$-0.342699\pi$
$593$	23.1002	0.948611	0.474306	+	0.880360i	$0.342699\pi$
$594$	0	0
$595$	−10.3754	−0.425348
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.363238	0.0148415	0.00742075	−	0.999972i	$-0.497638\pi$
$599$	0.363238	0.0148415	0.00742075	+	0.999972i	$0.497638\pi$
$600$	0	0
$601$	−9.24000	−0.376908	−0.188454	−	0.982082i	$-0.560348\pi$
$601$	−9.24000	−0.376908	−0.188454	+	0.982082i	$0.560348\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3.84336	0.156255
$606$	0	0
$607$	25.8317	1.04848	0.524238	−	0.851572i	$-0.324350\pi$
$607$	25.8317	1.04848	0.524238	+	0.851572i	$0.324350\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.53007	0.102356
$612$	0	0
$613$	−25.1039	−1.01394	−0.506969	−	0.861964i	$-0.669234\pi$
$613$	−25.1039	−1.01394	−0.506969	+	0.861964i	$0.669234\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.9862	−1.44875	−0.724374	−	0.689407i	$-0.757871\pi$
$617$	−35.9862	−1.44875	−0.724374	+	0.689407i	$0.757871\pi$
$618$	0	0
$619$	−7.03591	−0.282797	−0.141399	−	0.989953i	$-0.545160\pi$
$619$	−7.03591	−0.282797	−0.141399	+	0.989953i	$0.545160\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	64.7355	2.59357
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.66031	0.345309
$630$	0	0
$631$	25.4259	1.01219	0.506095	−	0.862478i	$-0.331089\pi$
$631$	25.4259	1.01219	0.506095	+	0.862478i	$0.331089\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.2542	−0.486294
$636$	0	0
$637$	3.75353	0.148720
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.6750	1.60657	0.803283	−	0.595597i	$-0.203084\pi$
$641$	40.6750	1.60657	0.803283	+	0.595597i	$0.203084\pi$
$642$	0	0
$643$	−29.1097	−1.14797	−0.573987	−	0.818864i	$-0.694604\pi$
$643$	−29.1097	−1.14797	−0.573987	+	0.818864i	$0.694604\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.8457	1.09473	0.547363	−	0.836895i	$-0.315632\pi$
$647$	27.8457	1.09473	0.547363	+	0.836895i	$0.315632\pi$
$648$	0	0
$649$	−18.0516	−0.708589
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.86492	−0.229512	−0.114756	−	0.993394i	$-0.536609\pi$
$653$	−5.86492	−0.229512	−0.114756	+	0.993394i	$0.536609\pi$
$654$	0	0
$655$	−6.42239	−0.250944
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.3928	0.599619	0.299809	−	0.953999i	$-0.403077\pi$
$659$	15.3928	0.599619	0.299809	+	0.953999i	$0.403077\pi$
$660$	0	0
$661$	−37.5816	−1.46176	−0.730878	−	0.682508i	$-0.760889\pi$
$661$	−37.5816	−1.46176	−0.730878	+	0.682508i	$0.760889\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.51855	0.136443
$666$	0	0
$667$	5.02498	0.194568
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−23.1595	−0.894062
$672$	0	0
$673$	8.51335	0.328166	0.164083	−	0.986447i	$-0.447534\pi$
$673$	8.51335	0.328166	0.164083	+	0.986447i	$0.447534\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−10.6432	−0.409052	−0.204526	−	0.978861i	$-0.565565\pi$
$677$	−10.6432	−0.409052	−0.204526	+	0.978861i	$0.565565\pi$
$678$	0	0
$679$	−25.6881	−0.985820
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.11421	0.348745	0.174373	−	0.984680i	$-0.444210\pi$
$683$	9.11421	0.348745	0.174373	+	0.984680i	$0.444210\pi$
$684$	0	0
$685$	−15.1885	−0.580324
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.40266	−0.0915340
$690$	0	0
$691$	−16.2345	−0.617589	−0.308795	−	0.951129i	$-0.599926\pi$
$691$	−16.2345	−0.617589	−0.308795	+	0.951129i	$0.599926\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.1566	0.574921
$696$	0	0
$697$	13.1484	0.498032
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.3108	0.653820	0.326910	−	0.945056i	$-0.393993\pi$
$701$	17.3108	0.653820	0.326910	+	0.945056i	$0.393993\pi$
$702$	0	0
$703$	−2.93693	−0.110769
$704$	0	0
$705$	0	0
$706$	0	0
$707$	60.4038	2.27172
$708$	0	0
$709$	−23.8425	−0.895422	−0.447711	−	0.894178i	$-0.647761\pi$
$709$	−23.8425	−0.895422	−0.447711	+	0.894178i	$0.647761\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.52757	−0.281910
$714$	0	0
$715$	0.498171	0.0186305
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−39.4205	−1.47014	−0.735068	−	0.677994i	$-0.762850\pi$
$719$	−39.4205	−1.47014	−0.735068	+	0.677994i	$0.762850\pi$
$720$	0	0
$721$	63.5440	2.36650
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−5.02498	−0.186623
$726$	0	0
$727$	11.4523	0.424741	0.212371	−	0.977189i	$-0.431882\pi$
$727$	11.4523	0.424741	0.212371	+	0.977189i	$0.431882\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.66105	−0.209382
$732$	0	0
$733$	−38.0726	−1.40624	−0.703122	−	0.711069i	$-0.748211\pi$
$733$	−38.0726	−1.40624	−0.703122	+	0.711069i	$0.748211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.4301	1.19458
$738$	0	0
$739$	13.2992	0.489217	0.244609	−	0.969622i	$-0.421340\pi$
$739$	13.2992	0.489217	0.244609	+	0.969622i	$0.421340\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.7449	0.614310	0.307155	−	0.951660i	$-0.400623\pi$
$743$	16.7449	0.614310	0.307155	+	0.951660i	$0.400623\pi$
$744$	0	0
$745$	2.02557	0.0742112
$746$	0	0
$747$	0	0
$748$	0	0
$749$	49.8712	1.82225
$750$	0	0
$751$	−31.5791	−1.15234	−0.576169	−	0.817330i	$-0.695453\pi$
$751$	−31.5791	−1.15234	−0.576169	+	0.817330i	$0.695453\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.15664	−0.114882
$756$	0	0
$757$	−17.4896	−0.635669	−0.317835	−	0.948146i	$-0.602956\pi$
$757$	−17.4896	−0.635669	−0.317835	+	0.948146i	$0.602956\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−32.0162	−1.16058	−0.580292	−	0.814408i	$-0.697062\pi$
$761$	−32.0162	−1.16058	−0.580292	+	0.814408i	$0.697062\pi$
$762$	0	0
$763$	−3.51855	−0.127380
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.25657	0.0453720
$768$	0	0
$769$	−26.3527	−0.950303	−0.475151	−	0.879904i	$-0.657607\pi$
$769$	−26.3527	−0.950303	−0.475151	+	0.879904i	$0.657607\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−36.3312	−1.30674	−0.653371	−	0.757038i	$-0.726646\pi$
$773$	−36.3312	−1.30674	−0.653371	+	0.757038i	$0.726646\pi$
$774$	0	0
$775$	7.52757	0.270398
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.45896	−0.159759
$780$	0	0
$781$	0.485865	0.0173856
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.1891	0.363666
$786$	0	0
$787$	−39.8105	−1.41909	−0.709545	−	0.704660i	$-0.751099\pi$
$787$	−39.8105	−1.41909	−0.709545	+	0.704660i	$0.751099\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.7572	−0.595819
$792$	0	0
$793$	1.61212	0.0572482
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−0.00926351	−0.000328130 0	−0.000164065	−	1.00000i	$-0.500052\pi$
$797$	−0.00926351	−0.000328130 0	−0.000164065		1.00000i	$0.500052\pi$
$798$	0	0
$799$	27.0505	0.956977
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.2026	−0.360042
$804$	0	0
$805$	5.21120	0.183671
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.9905	0.562196	0.281098	−	0.959679i	$-0.409301\pi$
$809$	15.9905	0.562196	0.281098	+	0.959679i	$0.409301\pi$
$810$	0	0
$811$	−1.70627	−0.0599154	−0.0299577	−	0.999551i	$-0.509537\pi$
$811$	−1.70627	−0.0599154	−0.0299577	+	0.999551i	$0.509537\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.9819	−0.419709
$816$	0	0
$817$	1.91981	0.0671655
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.9166	−0.834694	−0.417347	−	0.908747i	$-0.637040\pi$
$821$	−23.9166	−0.834694	−0.417347	+	0.908747i	$0.637040\pi$
$822$	0	0
$823$	12.1759	0.424426	0.212213	−	0.977223i	$-0.431933\pi$
$823$	12.1759	0.424426	0.212213	+	0.977223i	$0.431933\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	41.5339	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$827$	41.5339	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$828$	0	0
$829$	26.8628	0.932982	0.466491	−	0.884526i	$-0.345518\pi$
$829$	26.8628	0.932982	0.466491	+	0.884526i	$0.345518\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	40.1312	1.39046
$834$	0	0
$835$	14.3042	0.495016
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.7058	1.40532	0.702659	−	0.711526i	$-0.251996\pi$
$839$	40.7058	1.40532	0.702659	+	0.711526i	$0.251996\pi$
$840$	0	0
$841$	−3.74961	−0.129297
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.9653	0.446021
$846$	0	0
$847$	−20.0285	−0.688187
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.34979	−0.149109
$852$	0	0
$853$	37.6727	1.28989	0.644944	−	0.764230i	$-0.276881\pi$
$853$	37.6727	1.28989	0.644944	+	0.764230i	$0.276881\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−52.9539	−1.80887	−0.904435	−	0.426611i	$-0.859707\pi$
$857$	−52.9539	−1.80887	−0.904435	+	0.426611i	$0.859707\pi$
$858$	0	0
$859$	−40.4099	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$859$	−40.4099	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.9280	0.371993	0.185997	−	0.982550i	$-0.440449\pi$
$863$	10.9280	0.371993	0.185997	+	0.982550i	$0.440449\pi$
$864$	0	0
$865$	−21.1051	−0.717595
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28.0638	0.951999
$870$	0	0
$871$	−2.25745	−0.0764908
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.21120	−0.176171
$876$	0	0
$877$	−12.8860	−0.435131	−0.217565	−	0.976046i	$-0.569812\pi$
$877$	−12.8860	−0.435131	−0.217565	+	0.976046i	$0.569812\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.7123	−0.798887	−0.399444	−	0.916758i	$-0.630797\pi$
$881$	−23.7123	−0.798887	−0.399444	+	0.916758i	$0.630797\pi$
$882$	0	0
$883$	51.5545	1.73495	0.867473	−	0.497484i	$-0.165743\pi$
$883$	51.5545	1.73495	0.867473	+	0.497484i	$0.165743\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.07720	−0.0697457	−0.0348729	−	0.999392i	$-0.511103\pi$
$887$	−2.07720	−0.0697457	−0.0348729	+	0.999392i	$0.511103\pi$
$888$	0	0
$889$	63.8592	2.14177
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.17351	−0.306980
$894$	0	0
$895$	12.2798	0.410468
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.8259	−1.26156
$900$	0	0
$901$	−25.6882	−0.855799
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.42758	−0.313384
$906$	0	0
$907$	−17.2916	−0.574159	−0.287080	−	0.957907i	$-0.592684\pi$
$907$	−17.2916	−0.574159	−0.287080	+	0.957907i	$0.592684\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−53.3171	−1.76648	−0.883238	−	0.468926i	$-0.844641\pi$
$911$	−53.3171	−1.76648	−0.883238	+	0.468926i	$0.844641\pi$
$912$	0	0
$913$	45.0871	1.49217
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.4683	1.10522
$918$	0	0
$919$	41.5186	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$919$	41.5186	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.0338209	−0.00111323
$924$	0	0
$925$	4.34979	0.143020
$926$	0	0
$927$	0	0
$928$	0	0
$929$	47.4474	1.55670	0.778349	−	0.627832i	$-0.216058\pi$
$929$	47.4474	1.55670	0.778349	+	0.627832i	$0.216058\pi$
$930$	0	0
$931$	−13.6095	−0.446034
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5.32623	0.174187
$936$	0	0
$937$	−37.0293	−1.20970	−0.604848	−	0.796341i	$-0.706766\pi$
$937$	−37.0293	−1.20970	−0.604848	+	0.796341i	$0.706766\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.3469	−0.989280	−0.494640	−	0.869098i	$-0.664700\pi$
$941$	−30.3469	−0.989280	−0.494640	+	0.869098i	$0.664700\pi$
$942$	0	0
$943$	−6.60401	−0.215056
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.3584	1.50645	0.753223	−	0.657765i	$-0.228498\pi$
$947$	46.3584	1.50645	0.753223	+	0.657765i	$0.228498\pi$
$948$	0	0
$949$	0.710198	0.0230540
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.9226	1.16365	0.581824	−	0.813315i	$-0.302339\pi$
$953$	35.9226	1.16365	0.581824	+	0.813315i	$0.302339\pi$
$954$	0	0
$955$	10.2847	0.332805
$956$	0	0
$957$	0	0
$958$	0	0
$959$	79.1505	2.55590
$960$	0	0
$961$	25.6644	0.827883
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.35439	−0.0757904
$966$	0	0
$967$	−45.0531	−1.44881	−0.724405	−	0.689375i	$-0.757885\pi$
$967$	−45.0531	−1.44881	−0.724405	+	0.689375i	$0.757885\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26.0231	−0.835119	−0.417560	−	0.908649i	$-0.637115\pi$
$971$	−26.0231	−0.835119	−0.417560	+	0.908649i	$0.637115\pi$
$972$	0	0
$973$	−78.9838	−2.53210
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.2710	−0.520556	−0.260278	−	0.965534i	$-0.583814\pi$
$977$	−16.2710	−0.520556	−0.260278	+	0.965534i	$0.583814\pi$
$978$	0	0
$979$	−33.2323	−1.06211
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40.9840	−1.30719	−0.653593	−	0.756847i	$-0.726739\pi$
$983$	−40.9840	−1.30719	−0.653593	+	0.756847i	$0.726739\pi$
$984$	0	0
$985$	−14.7948	−0.471402
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.84336	0.0904135
$990$	0	0
$991$	36.3250	1.15390	0.576951	−	0.816779i	$-0.304242\pi$
$991$	36.3250	1.15390	0.576951	+	0.816779i	$0.304242\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−25.1220	−0.796420
$996$	0	0
$997$	20.3014	0.642952	0.321476	−	0.946918i	$-0.395821\pi$
$997$	20.3014	0.642952	0.321476	+	0.946918i	$0.395821\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.a.t.1.5	✓	5
3.2	odd	2		4140.2.a.u.1.5	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.2.a.t.1.5	✓	5	1.1	even	1	trivial
4140.2.a.u.1.5	yes	5	3.2	odd	2

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +5.21120 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +5.21120 q^{7} -2.67519 q^{11} +0.186219 q^{13} +1.99097 q^{17} -0.675190 q^{19} -1.00000 q^{23} +1.00000 q^{25} -5.02498 q^{29} +7.52757 q^{31} -5.21120 q^{35} +4.34979 q^{37} +6.60401 q^{41} -2.84336 q^{43} +13.5866 q^{47} +20.1566 q^{49} -12.9023 q^{53} +2.67519 q^{55} +6.74779 q^{59} +8.65714 q^{61} -0.186219 q^{65} -12.1226 q^{67} -0.181619 q^{71} +3.81378 q^{73} -13.9409 q^{77} -10.4904 q^{79} -16.8538 q^{83} -1.99097 q^{85} +12.4224 q^{89} +0.970423 q^{91} +0.675190 q^{95} -4.92941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 4 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 + 4 * q^7	\( 5 q - 5 q^{5} + 4 q^{7} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 6 q^{19} - 5 q^{23} + 5 q^{25} - 2 q^{29} + 8 q^{31} - 4 q^{35} + 8 q^{37} - 2 q^{41} + 18 q^{43} + 4 q^{47} + 13 q^{49} + 2 q^{53} + 4 q^{55} - 6 q^{59} + 10 q^{61} - 2 q^{65} + 16 q^{67} - 10 q^{71} + 18 q^{73} + 16 q^{77} + 14 q^{79} - 8 q^{83} + 2 q^{85} + 18 q^{89} + 36 q^{91} - 6 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 4 * q^7 - 4 * q^11 + 2 * q^13 - 2 * q^17 + 6 * q^19 - 5 * q^23 + 5 * q^25 - 2 * q^29 + 8 * q^31 - 4 * q^35 + 8 * q^37 - 2 * q^41 + 18 * q^43 + 4 * q^47 + 13 * q^49 + 2 * q^53 + 4 * q^55 - 6 * q^59 + 10 * q^61 - 2 * q^65 + 16 * q^67 - 10 * q^71 + 18 * q^73 + 16 * q^77 + 14 * q^79 - 8 * q^83 + 2 * q^85 + 18 * q^89 + 36 * q^91 - 6 * q^95 + 6 * q^97

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