LMFDB - Embedded newform 2016.3.g.b.1135.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,3,Mod(1135,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2016.1135");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2016.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$54.9320212950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.292213762624.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 $ x^8 - x^7 - 2x^6 - 2x^5 + 24x^4 - 8x^3 - 32x^2 - 64x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.3
Root			$-1.05468 - 1.69931i$ of defining polynomial
Character	$\chi$	$=$	2016.1135
Dual form			2016.3.g.b.1135.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.88287i q^{5} -2.64575i q^{7} +O(q^{10})$	$q-4.88287i q^{5} -2.64575i q^{7} -21.4776 q^{11} +13.0760i q^{13} +0.234889 q^{17} -4.55872 q^{19} -10.9523i q^{23} +1.15761 q^{25} +34.6435i q^{29} -34.1079i q^{31} -12.9189 q^{35} +54.2370i q^{37} +37.8300 q^{41} +4.84714 q^{43} +72.3368i q^{47} -7.00000 q^{49} +21.6707i q^{53} +104.872i q^{55} +34.9007 q^{59} -63.6012i q^{61} +63.8485 q^{65} -18.4344 q^{67} +47.5244i q^{71} +55.9103 q^{73} +56.8243i q^{77} -95.0135i q^{79} +71.5156 q^{83} -1.14693i q^{85} +159.756 q^{89} +34.5959 q^{91} +22.2596i q^{95} -90.4794 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) $ 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 4.88287i	− 0.976573i	−0.872683	−	0.488287i	$-0.837622\pi$
$5$	− 4.88287i	− 0.976573i	0.872683	−	0.488287i	$-0.162378\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−21.4776	−1.95251	−0.976253	−	0.216632i	$-0.930493\pi$
$11$	−21.4776	−1.95251	−0.976253	+	0.216632i	$0.930493\pi$
$12$	0	0
$13$	13.0760i	1.00585i	0.864331	+	0.502924i	$0.167742\pi$
$13$	13.0760i	1.00585i	−0.864331	+	0.502924i	$0.832258\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.234889	0.0138170	0.00690851	−	0.999976i	$-0.497801\pi$
$17$	0.234889	0.0138170	0.00690851	+	0.999976i	$0.497801\pi$
$18$	0	0
$19$	−4.55872	−0.239933	−0.119966	−	0.992778i	$-0.538279\pi$
$19$	−4.55872	−0.239933	−0.119966	+	0.992778i	$0.538279\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 10.9523i	− 0.476187i	−0.971242	−	0.238094i	$-0.923478\pi$
$23$	− 10.9523i	− 0.476187i	0.971242	−	0.238094i	$-0.0765225\pi$
$24$	0	0
$25$	1.15761	0.0463043
$26$	0	0
$27$	0	0
$28$	0	0
$29$	34.6435i	1.19460i	0.802016	+	0.597302i	$0.203761\pi$
$29$	34.6435i	1.19460i	−0.802016	+	0.597302i	$0.796239\pi$
$30$	0	0
$31$	− 34.1079i	− 1.10025i	−0.835081	−	0.550127i	$-0.814579\pi$
$31$	− 34.1079i	− 1.10025i	0.835081	−	0.550127i	$-0.185421\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.9189	−0.369110
$36$	0	0
$37$	54.2370i	1.46586i	0.680302	+	0.732932i	$0.261849\pi$
$37$	54.2370i	1.46586i	−0.680302	+	0.732932i	$0.738151\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	37.8300	0.922682	0.461341	−	0.887223i	$-0.347368\pi$
$41$	37.8300	0.922682	0.461341	+	0.887223i	$0.347368\pi$
$42$	0	0
$43$	4.84714	0.112724	0.0563621	−	0.998410i	$-0.482050\pi$
$43$	4.84714	0.112724	0.0563621	+	0.998410i	$0.482050\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	72.3368i	1.53908i	0.638598	+	0.769541i	$0.279515\pi$
$47$	72.3368i	1.53908i	−0.638598	+	0.769541i	$0.720485\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	21.6707i	0.408881i	0.978879	+	0.204440i	$0.0655374\pi$
$53$	21.6707i	0.408881i	−0.978879	+	0.204440i	$0.934463\pi$
$54$	0	0
$55$	104.872i	1.90677i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	34.9007	0.591537	0.295768	−	0.955260i	$-0.404424\pi$
$59$	34.9007	0.591537	0.295768	+	0.955260i	$0.404424\pi$
$60$	0	0
$61$	− 63.6012i	− 1.04264i	−0.853360	−	0.521321i	$-0.825439\pi$
$61$	− 63.6012i	− 1.04264i	0.853360	−	0.521321i	$-0.174561\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	63.8485	0.982284
$66$	0	0
$67$	−18.4344	−0.275140	−0.137570	−	0.990492i	$-0.543929\pi$
$67$	−18.4344	−0.275140	−0.137570	+	0.990492i	$0.543929\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	47.5244i	0.669358i	0.942332	+	0.334679i	$0.108628\pi$
$71$	47.5244i	0.669358i	−0.942332	+	0.334679i	$0.891372\pi$
$72$	0	0
$73$	55.9103	0.765894	0.382947	−	0.923770i	$-0.374909\pi$
$73$	55.9103	0.765894	0.382947	+	0.923770i	$0.374909\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	56.8243i	0.737978i
$78$	0	0
$79$	− 95.0135i	− 1.20270i	−0.798985	−	0.601351i	$-0.794629\pi$
$79$	− 95.0135i	− 1.20270i	0.798985	−	0.601351i	$-0.205371\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	71.5156	0.861634	0.430817	−	0.902439i	$-0.358225\pi$
$83$	71.5156	0.861634	0.430817	+	0.902439i	$0.358225\pi$
$84$	0	0
$85$	− 1.14693i	− 0.0134933i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	159.756	1.79501	0.897504	−	0.441006i	$-0.145378\pi$
$89$	159.756	1.79501	0.897504	+	0.441006i	$0.145378\pi$
$90$	0	0
$91$	34.5959	0.380175
$92$	0	0
$93$	0	0
$94$	0	0
$95$	22.2596i	0.234312i
$96$	0	0
$97$	−90.4794	−0.932777	−0.466389	−	0.884580i	$-0.654445\pi$
$97$	−90.4794	−0.932777	−0.466389	+	0.884580i	$0.654445\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 181.147i	− 1.79353i	−0.442503	−	0.896767i	$-0.645909\pi$
$101$	− 181.147i	− 1.79353i	0.442503	−	0.896767i	$-0.354091\pi$
$102$	0	0
$103$	39.3003i	0.381556i	0.981633	+	0.190778i	$0.0611010\pi$
$103$	39.3003i	0.381556i	−0.981633	+	0.190778i	$0.938899\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	38.4498	0.359344	0.179672	−	0.983727i	$-0.442496\pi$
$107$	38.4498	0.359344	0.179672	+	0.983727i	$0.442496\pi$
$108$	0	0
$109$	− 27.8786i	− 0.255767i	−0.991789	−	0.127883i	$-0.959182\pi$
$109$	− 27.8786i	− 0.255767i	0.991789	−	0.127883i	$-0.0408183\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−82.4419	−0.729574	−0.364787	−	0.931091i	$-0.618858\pi$
$113$	−82.4419	−0.729574	−0.364787	+	0.931091i	$0.618858\pi$
$114$	0	0
$115$	−53.4786	−0.465032
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 0.621458i	− 0.00522234i
$120$	0	0
$121$	340.286	2.81228
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 127.724i	− 1.02179i
$126$	0	0
$127$	25.1408i	0.197959i	0.995089	+	0.0989796i	$0.0315579\pi$
$127$	25.1408i	0.197959i	−0.995089	+	0.0989796i	$0.968442\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−126.398	−0.964872	−0.482436	−	0.875931i	$-0.660248\pi$
$131$	−126.398	−0.964872	−0.482436	+	0.875931i	$0.660248\pi$
$132$	0	0
$133$	12.0612i	0.0906861i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−34.9456	−0.255078	−0.127539	−	0.991834i	$-0.540708\pi$
$137$	−34.9456	−0.255078	−0.127539	+	0.991834i	$0.540708\pi$
$138$	0	0
$139$	119.148	0.857177	0.428589	−	0.903500i	$-0.359011\pi$
$139$	119.148	0.857177	0.428589	+	0.903500i	$0.359011\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 280.841i	− 1.96392i
$144$	0	0
$145$	169.160	1.16662
$146$	0	0
$147$	0	0
$148$	0	0
$149$	121.932i	0.818334i	0.912460	+	0.409167i	$0.134181\pi$
$149$	121.932i	0.818334i	−0.912460	+	0.409167i	$0.865819\pi$
$150$	0	0
$151$	220.404i	1.45963i	0.683645	+	0.729815i	$0.260394\pi$
$151$	220.404i	1.45963i	−0.683645	+	0.729815i	$0.739606\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−166.544	−1.07448
$156$	0	0
$157$	− 6.77014i	− 0.0431219i	−0.999768	−	0.0215610i	$-0.993136\pi$
$157$	− 6.77014i	− 0.0431219i	0.999768	−	0.0215610i	$-0.00686360\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−28.9771	−0.179982
$162$	0	0
$163$	207.243	1.27143	0.635715	−	0.771924i	$-0.280706\pi$
$163$	207.243	1.27143	0.635715	+	0.771924i	$0.280706\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	165.529i	0.991193i	0.868553	+	0.495596i	$0.165050\pi$
$167$	165.529i	0.991193i	−0.868553	+	0.495596i	$0.834950\pi$
$168$	0	0
$169$	−1.98237	−0.0117300
$170$	0	0
$171$	0	0
$172$	0	0
$173$	88.8530i	0.513601i	0.966464	+	0.256800i	$0.0826683\pi$
$173$	88.8530i	0.513601i	−0.966464	+	0.256800i	$0.917332\pi$
$174$	0	0
$175$	− 3.06274i	− 0.0175014i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	80.3791	0.449045	0.224523	−	0.974469i	$-0.427918\pi$
$179$	80.3791	0.449045	0.224523	+	0.974469i	$0.427918\pi$
$180$	0	0
$181$	276.353i	1.52681i	0.645919	+	0.763406i	$0.276474\pi$
$181$	276.353i	1.52681i	−0.645919	+	0.763406i	$0.723526\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	264.832	1.43152
$186$	0	0
$187$	−5.04485	−0.0269778
$188$	0	0
$189$	0	0
$190$	0	0
$191$	203.015i	1.06290i	0.847088	+	0.531452i	$0.178353\pi$
$191$	203.015i	1.06290i	−0.847088	+	0.531452i	$0.821647\pi$
$192$	0	0
$193$	87.3328	0.452502	0.226251	−	0.974069i	$-0.427353\pi$
$193$	87.3328	0.452502	0.226251	+	0.974069i	$0.427353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.6639i	− 0.109969i	−0.998487	−	0.0549845i	$-0.982489\pi$
$197$	− 21.6639i	− 0.109969i	0.998487	−	0.0549845i	$-0.0175109\pi$
$198$	0	0
$199$	181.933i	0.914235i	0.889406	+	0.457118i	$0.151118\pi$
$199$	181.933i	0.914235i	−0.889406	+	0.457118i	$0.848882\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	91.6582	0.451518
$204$	0	0
$205$	− 184.719i	− 0.901067i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	97.9103	0.468470
$210$	0	0
$211$	21.4204	0.101519	0.0507594	−	0.998711i	$-0.483836\pi$
$211$	21.4204	0.101519	0.0507594	+	0.998711i	$0.483836\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 23.6680i	− 0.110084i
$216$	0	0
$217$	−90.2410	−0.415857
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.07142i	0.0138978i
$222$	0	0
$223$	195.958i	0.878735i	0.898307	+	0.439367i	$0.144797\pi$
$223$	195.958i	0.878735i	−0.898307	+	0.439367i	$0.855203\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−27.2652	−0.120111	−0.0600554	−	0.998195i	$-0.519128\pi$
$227$	−27.2652	−0.120111	−0.0600554	+	0.998195i	$0.519128\pi$
$228$	0	0
$229$	176.347i	0.770076i	0.922901	+	0.385038i	$0.125812\pi$
$229$	176.347i	0.770076i	−0.922901	+	0.385038i	$0.874188\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−71.8366	−0.308312	−0.154156	−	0.988047i	$-0.549266\pi$
$233$	−71.8366	−0.308312	−0.154156	+	0.988047i	$0.549266\pi$
$234$	0	0
$235$	353.211	1.50303
$236$	0	0
$237$	0	0
$238$	0	0
$239$	71.0926i	0.297459i	0.988878	+	0.148729i	$0.0475183\pi$
$239$	71.0926i	0.297459i	−0.988878	+	0.148729i	$0.952482\pi$
$240$	0	0
$241$	56.1113	0.232827	0.116413	−	0.993201i	$-0.462860\pi$
$241$	56.1113	0.232827	0.116413	+	0.993201i	$0.462860\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	34.1801i	0.139510i
$246$	0	0
$247$	− 59.6100i	− 0.241336i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	368.953	1.46993	0.734966	−	0.678104i	$-0.237198\pi$
$251$	368.953	1.46993	0.734966	+	0.678104i	$0.237198\pi$
$252$	0	0
$253$	235.229i	0.929759i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−23.7428	−0.0923845	−0.0461923	−	0.998933i	$-0.514709\pi$
$257$	−23.7428	−0.0923845	−0.0461923	+	0.998933i	$0.514709\pi$
$258$	0	0
$259$	143.498	0.554044
$260$	0	0
$261$	0	0
$262$	0	0
$263$	73.9707i	0.281257i	0.990062	+	0.140629i	$0.0449124\pi$
$263$	73.9707i	0.281257i	−0.990062	+	0.140629i	$0.955088\pi$
$264$	0	0
$265$	105.815	0.399302
$266$	0	0
$267$	0	0
$268$	0	0
$269$	335.593i	1.24756i	0.781601	+	0.623779i	$0.214403\pi$
$269$	335.593i	1.24756i	−0.781601	+	0.623779i	$0.785597\pi$
$270$	0	0
$271$	187.276i	0.691054i	0.938409	+	0.345527i	$0.112300\pi$
$271$	187.276i	0.691054i	−0.938409	+	0.345527i	$0.887700\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−24.8626	−0.0904095
$276$	0	0
$277$	− 132.592i	− 0.478670i	−0.970937	−	0.239335i	$-0.923071\pi$
$277$	− 132.592i	− 0.478670i	0.970937	−	0.239335i	$-0.0769294\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−331.520	−1.17979	−0.589894	−	0.807481i	$-0.700830\pi$
$281$	−331.520	−1.17979	−0.589894	+	0.807481i	$0.700830\pi$
$282$	0	0
$283$	66.7158	0.235745	0.117873	−	0.993029i	$-0.462393\pi$
$283$	66.7158	0.235745	0.117873	+	0.993029i	$0.462393\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 100.089i	− 0.348741i
$288$	0	0
$289$	−288.945	−0.999809
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 289.215i	− 0.987082i	−0.869723	−	0.493541i	$-0.835702\pi$
$293$	− 289.215i	− 0.987082i	0.869723	−	0.493541i	$-0.164298\pi$
$294$	0	0
$295$	− 170.415i	− 0.577679i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	143.213	0.478972
$300$	0	0
$301$	− 12.8243i	− 0.0426058i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−310.556	−1.01822
$306$	0	0
$307$	−0.693177	−0.00225790	−0.00112895	−	0.999999i	$-0.500359\pi$
$307$	−0.693177	−0.00225790	−0.00112895	+	0.999999i	$0.500359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	62.1583i	0.199866i	0.994994	+	0.0999330i	$0.0318628\pi$
$311$	62.1583i	0.199866i	−0.994994	+	0.0999330i	$0.968137\pi$
$312$	0	0
$313$	213.594	0.682408	0.341204	−	0.939989i	$-0.389165\pi$
$313$	213.594	0.682408	0.341204	+	0.939989i	$0.389165\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 23.4577i	− 0.0739990i	−0.999315	−	0.0369995i	$-0.988220\pi$
$317$	− 23.4577i	− 0.0739990i	0.999315	−	0.0369995i	$-0.0117800\pi$
$318$	0	0
$319$	− 744.059i	− 2.33247i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.07079	−0.00331515
$324$	0	0
$325$	15.1369i	0.0465751i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	191.385	0.581718
$330$	0	0
$331$	−507.406	−1.53295	−0.766474	−	0.642275i	$-0.777991\pi$
$331$	−507.406	−1.53295	−0.766474	+	0.642275i	$0.777991\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	90.0126i	0.268694i
$336$	0	0
$337$	−342.726	−1.01699	−0.508495	−	0.861065i	$-0.669798\pi$
$337$	−342.726	−1.01699	−0.508495	+	0.861065i	$0.669798\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	732.555i	2.14825i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	136.745	0.394079	0.197039	−	0.980396i	$-0.436867\pi$
$347$	136.745	0.394079	0.197039	+	0.980396i	$0.436867\pi$
$348$	0	0
$349$	82.0565i	0.235119i	0.993066	+	0.117559i	$0.0375071\pi$
$349$	82.0565i	0.235119i	−0.993066	+	0.117559i	$0.962493\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−507.367	−1.43730	−0.718651	−	0.695371i	$-0.755240\pi$
$353$	−507.367	−1.43730	−0.718651	+	0.695371i	$0.755240\pi$
$354$	0	0
$355$	232.055	0.653677
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 560.809i	− 1.56214i	−0.624442	−	0.781071i	$-0.714674\pi$
$359$	− 560.809i	− 1.56214i	0.624442	−	0.781071i	$-0.285326\pi$
$360$	0	0
$361$	−340.218	−0.942432
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 273.003i	− 0.747952i
$366$	0	0
$367$	26.9431i	0.0734145i	0.999326	+	0.0367072i	$0.0116869\pi$
$367$	26.9431i	0.0734145i	−0.999326	+	0.0367072i	$0.988313\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	57.3352	0.154542
$372$	0	0
$373$	538.034i	1.44245i	0.692701	+	0.721225i	$0.256421\pi$
$373$	538.034i	1.44245i	−0.692701	+	0.721225i	$0.743579\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−453.000	−1.20159
$378$	0	0
$379$	182.132	0.480560	0.240280	−	0.970704i	$-0.422761\pi$
$379$	182.132	0.480560	0.240280	+	0.970704i	$0.422761\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	333.271i	0.870160i	0.900392	+	0.435080i	$0.143280\pi$
$383$	333.271i	0.870160i	−0.900392	+	0.435080i	$0.856720\pi$
$384$	0	0
$385$	277.466	0.720690
$386$	0	0
$387$	0	0
$388$	0	0
$389$	109.639i	0.281847i	0.990020	+	0.140924i	$0.0450072\pi$
$389$	109.639i	0.281847i	−0.990020	+	0.140924i	$0.954993\pi$
$390$	0	0
$391$	− 2.57258i	− 0.00657948i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−463.938	−1.17453
$396$	0	0
$397$	310.938i	0.783219i	0.920131	+	0.391610i	$0.128082\pi$
$397$	310.938i	0.783219i	−0.920131	+	0.391610i	$0.871918\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	423.903	1.05711	0.528557	−	0.848898i	$-0.322733\pi$
$401$	423.903	1.05711	0.528557	+	0.848898i	$0.322733\pi$
$402$	0	0
$403$	445.995	1.10669
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1164.88i	− 2.86211i
$408$	0	0
$409$	444.543	1.08690	0.543451	−	0.839441i	$-0.317117\pi$
$409$	444.543	1.08690	0.543451	+	0.839441i	$0.317117\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 92.3385i	− 0.223580i
$414$	0	0
$415$	− 349.201i	− 0.841449i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	457.129	1.09100	0.545500	−	0.838111i	$-0.316340\pi$
$419$	457.129	1.09100	0.545500	+	0.838111i	$0.316340\pi$
$420$	0	0
$421$	− 25.4812i	− 0.0605255i	−0.999542	−	0.0302628i	$-0.990366\pi$
$421$	− 25.4812i	− 0.0605255i	0.999542	−	0.0302628i	$-0.00963441\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.271910	0.000639787 0
$426$	0	0
$427$	−168.273	−0.394082
$428$	0	0
$429$	0	0
$430$	0	0
$431$	124.595i	0.289084i	0.989499	+	0.144542i	$0.0461709\pi$
$431$	124.595i	0.289084i	−0.989499	+	0.144542i	$0.953829\pi$
$432$	0	0
$433$	−272.271	−0.628802	−0.314401	−	0.949290i	$-0.601804\pi$
$433$	−272.271	−0.628802	−0.314401	+	0.949290i	$0.601804\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	49.9285i	0.114253i
$438$	0	0
$439$	255.069i	0.581023i	0.956871	+	0.290512i	$0.0938255\pi$
$439$	255.069i	0.581023i	−0.956871	+	0.290512i	$0.906174\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	131.274	0.296330	0.148165	−	0.988963i	$-0.452663\pi$
$443$	131.274	0.296330	0.148165	+	0.988963i	$0.452663\pi$
$444$	0	0
$445$	− 780.066i	− 1.75296i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	642.824	1.43168	0.715839	−	0.698265i	$-0.246044\pi$
$449$	642.824	1.43168	0.715839	+	0.698265i	$0.246044\pi$
$450$	0	0
$451$	−812.496	−1.80154
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 168.927i	− 0.371269i
$456$	0	0
$457$	693.088	1.51660	0.758302	−	0.651903i	$-0.226029\pi$
$457$	693.088	1.51660	0.758302	+	0.651903i	$0.226029\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 258.699i	− 0.561170i	−0.959829	−	0.280585i	$-0.909472\pi$
$461$	− 258.699i	− 0.561170i	0.959829	−	0.280585i	$-0.0905285\pi$
$462$	0	0
$463$	637.226i	1.37630i	0.725569	+	0.688150i	$0.241577\pi$
$463$	637.226i	1.37630i	−0.725569	+	0.688150i	$0.758423\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−199.483	−0.427159	−0.213580	−	0.976926i	$-0.568512\pi$
$467$	−199.483	−0.427159	−0.213580	+	0.976926i	$0.568512\pi$
$468$	0	0
$469$	48.7727i	0.103993i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−104.105	−0.220095
$474$	0	0
$475$	−5.27721	−0.0111099
$476$	0	0
$477$	0	0
$478$	0	0
$479$	674.160i	1.40743i	0.710481	+	0.703716i	$0.248477\pi$
$479$	674.160i	1.40743i	−0.710481	+	0.703716i	$0.751523\pi$
$480$	0	0
$481$	−709.204	−1.47444
$482$	0	0
$483$	0	0
$484$	0	0
$485$	441.799i	0.910926i
$486$	0	0
$487$	− 401.718i	− 0.824883i	−0.910984	−	0.412442i	$-0.864676\pi$
$487$	− 401.718i	− 0.824883i	0.910984	−	0.412442i	$-0.135324\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	428.880	0.873482	0.436741	−	0.899587i	$-0.356133\pi$
$491$	428.880	0.873482	0.436741	+	0.899587i	$0.356133\pi$
$492$	0	0
$493$	8.13739i	0.0165059i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	125.738	0.252993
$498$	0	0
$499$	−182.619	−0.365970	−0.182985	−	0.983116i	$-0.558576\pi$
$499$	−182.619	−0.365970	−0.182985	+	0.983116i	$0.558576\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	380.158i	0.755781i	0.925850	+	0.377891i	$0.123350\pi$
$503$	380.158i	0.755781i	−0.925850	+	0.377891i	$0.876650\pi$
$504$	0	0
$505$	−884.516	−1.75152
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 289.538i	− 0.568836i	−0.958700	−	0.284418i	$-0.908200\pi$
$509$	− 289.538i	− 0.568836i	0.958700	−	0.284418i	$-0.0918004\pi$
$510$	0	0
$511$	− 147.925i	− 0.289481i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	191.898	0.372617
$516$	0	0
$517$	− 1553.62i	− 3.00507i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	738.899	1.41823	0.709116	−	0.705092i	$-0.249094\pi$
$521$	738.899	1.41823	0.709116	+	0.705092i	$0.249094\pi$
$522$	0	0
$523$	−647.126	−1.23734	−0.618668	−	0.785653i	$-0.712327\pi$
$523$	−647.126	−1.23734	−0.618668	+	0.785653i	$0.712327\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.01157i	− 0.0152022i
$528$	0	0
$529$	409.047	0.773246
$530$	0	0
$531$	0	0
$532$	0	0
$533$	494.666i	0.928078i
$534$	0	0
$535$	− 187.745i	− 0.350926i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	150.343	0.278930
$540$	0	0
$541$	− 178.722i	− 0.330355i	−0.986264	−	0.165178i	$-0.947180\pi$
$541$	− 178.722i	− 0.330355i	0.986264	−	0.165178i	$-0.0528198\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−136.127	−0.249775
$546$	0	0
$547$	−452.236	−0.826758	−0.413379	−	0.910559i	$-0.635651\pi$
$547$	−452.236	−0.826758	−0.413379	+	0.910559i	$0.635651\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 157.930i	− 0.286625i
$552$	0	0
$553$	−251.382	−0.454579
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 854.108i	− 1.53341i	−0.642001	−	0.766704i	$-0.721895\pi$
$557$	− 854.108i	− 1.53341i	0.642001	−	0.766704i	$-0.278105\pi$
$558$	0	0
$559$	63.3814i	0.113383i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	249.654	0.443436	0.221718	−	0.975111i	$-0.428834\pi$
$563$	249.654	0.443436	0.221718	+	0.975111i	$0.428834\pi$
$564$	0	0
$565$	402.553i	0.712483i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−104.353	−0.183396	−0.0916982	−	0.995787i	$-0.529229\pi$
$569$	−104.353	−0.183396	−0.0916982	+	0.995787i	$0.529229\pi$
$570$	0	0
$571$	649.705	1.13784	0.568919	−	0.822394i	$-0.307362\pi$
$571$	649.705	1.13784	0.568919	+	0.822394i	$0.307362\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 12.6785i	− 0.0220495i
$576$	0	0
$577$	−346.022	−0.599692	−0.299846	−	0.953988i	$-0.596935\pi$
$577$	−346.022	−0.599692	−0.299846	+	0.953988i	$0.596935\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 189.213i	− 0.325667i
$582$	0	0
$583$	− 465.434i	− 0.798342i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1153.54	−1.96514	−0.982572	−	0.185885i	$-0.940485\pi$
$587$	−1153.54	−1.96514	−0.982572	+	0.185885i	$0.940485\pi$
$588$	0	0
$589$	155.488i	0.263987i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−880.135	−1.48421	−0.742104	−	0.670285i	$-0.766172\pi$
$593$	−880.135	−1.48421	−0.742104	+	0.670285i	$0.766172\pi$
$594$	0	0
$595$	−3.03450	−0.00510000
$596$	0	0
$597$	0	0
$598$	0	0
$599$	554.939i	0.926442i	0.886243	+	0.463221i	$0.153306\pi$
$599$	554.939i	0.926442i	−0.886243	+	0.463221i	$0.846694\pi$
$600$	0	0
$601$	−666.057	−1.10825	−0.554124	−	0.832434i	$-0.686946\pi$
$601$	−666.057	−1.10825	−0.554124	+	0.832434i	$0.686946\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1661.57i	− 2.74640i
$606$	0	0
$607$	− 192.927i	− 0.317836i	−0.987292	−	0.158918i	$-0.949199\pi$
$607$	− 192.927i	− 0.317836i	0.987292	−	0.158918i	$-0.0508006\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−945.878	−1.54808
$612$	0	0
$613$	− 608.234i	− 0.992226i	−0.868258	−	0.496113i	$-0.834760\pi$
$613$	− 608.234i	− 0.992226i	0.868258	−	0.496113i	$-0.165240\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.884056	−0.00143283	−0.000716415	−	1.00000i	$-0.500228\pi$
$617$	−0.884056	−0.00143283	−0.000716415		1.00000i	$0.500228\pi$
$618$	0	0
$619$	−358.525	−0.579200	−0.289600	−	0.957148i	$-0.593522\pi$
$619$	−358.525	−0.579200	−0.289600	+	0.957148i	$0.593522\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 422.674i	− 0.678449i
$624$	0	0
$625$	−594.720	−0.951552
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.7397i	0.0202539i
$630$	0	0
$631$	− 390.515i	− 0.618883i	−0.950918	−	0.309442i	$-0.899858\pi$
$631$	− 390.515i	− 0.618883i	0.950918	−	0.309442i	$-0.100142\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	122.759	0.193322
$636$	0	0
$637$	− 91.5322i	− 0.143693i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	431.936	0.673848	0.336924	−	0.941532i	$-0.390614\pi$
$641$	431.936	0.673848	0.336924	+	0.941532i	$0.390614\pi$
$642$	0	0
$643$	−49.9370	−0.0776625	−0.0388313	−	0.999246i	$-0.512363\pi$
$643$	−49.9370	−0.0776625	−0.0388313	+	0.999246i	$0.512363\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	224.141i	0.346431i	0.984884	+	0.173216i	$0.0554157\pi$
$647$	224.141i	0.346431i	−0.984884	+	0.173216i	$0.944584\pi$
$648$	0	0
$649$	−749.582	−1.15498
$650$	0	0
$651$	0	0
$652$	0	0
$653$	80.7637i	0.123681i	0.998086	+	0.0618405i	$0.0196970\pi$
$653$	80.7637i	0.123681i	−0.998086	+	0.0618405i	$0.980303\pi$
$654$	0	0
$655$	617.186i	0.942268i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−940.466	−1.42711	−0.713555	−	0.700599i	$-0.752916\pi$
$659$	−940.466	−1.42711	−0.713555	+	0.700599i	$0.752916\pi$
$660$	0	0
$661$	− 119.930i	− 0.181437i	−0.995877	−	0.0907184i	$-0.971084\pi$
$661$	− 119.930i	− 0.181437i	0.995877	−	0.0907184i	$-0.0289163\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	58.8935	0.0885616
$666$	0	0
$667$	379.426	0.568855
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1366.00i	2.03577i
$672$	0	0
$673$	1085.06	1.61227	0.806136	−	0.591731i	$-0.201555\pi$
$673$	1085.06	1.61227	0.806136	+	0.591731i	$0.201555\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	949.901i	1.40310i	0.712618	+	0.701552i	$0.247509\pi$
$677$	949.901i	1.40310i	−0.712618	+	0.701552i	$0.752491\pi$
$678$	0	0
$679$	239.386i	0.352557i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−893.785	−1.30862	−0.654308	−	0.756228i	$-0.727040\pi$
$683$	−893.785	−1.30862	−0.654308	+	0.756228i	$0.727040\pi$
$684$	0	0
$685$	170.635i	0.249102i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−283.366	−0.411272
$690$	0	0
$691$	−1208.56	−1.74901	−0.874504	−	0.485019i	$-0.838813\pi$
$691$	−1208.56	−1.74901	−0.874504	+	0.485019i	$0.838813\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 581.782i	− 0.837096i
$696$	0	0
$697$	8.88585	0.0127487
$698$	0	0
$699$	0	0
$700$	0	0
$701$	219.477i	0.313091i	0.987671	+	0.156546i	$0.0500358\pi$
$701$	219.477i	0.313091i	−0.987671	+	0.156546i	$0.949964\pi$
$702$	0	0
$703$	− 247.251i	− 0.351709i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−479.270	−0.677892
$708$	0	0
$709$	− 1265.13i	− 1.78439i	−0.451651	−	0.892195i	$-0.649165\pi$
$709$	− 1265.13i	− 1.78439i	0.451651	−	0.892195i	$-0.350835\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−373.560	−0.523927
$714$	0	0
$715$	−1371.31	−1.91792
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1163.47i	− 1.61818i	−0.587687	−	0.809089i	$-0.699961\pi$
$719$	− 1163.47i	− 1.61818i	0.587687	−	0.809089i	$-0.300039\pi$
$720$	0	0
$721$	103.979	0.144215
$722$	0	0
$723$	0	0
$724$	0	0
$725$	40.1036i	0.0553153i
$726$	0	0
$727$	1303.68i	1.79324i	0.442803	+	0.896619i	$0.353984\pi$
$727$	1303.68i	1.79324i	−0.442803	+	0.896619i	$0.646016\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.13854	0.00155751
$732$	0	0
$733$	1256.12i	1.71367i	0.515589	+	0.856836i	$0.327573\pi$
$733$	1256.12i	1.71367i	−0.515589	+	0.856836i	$0.672427\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	395.925	0.537212
$738$	0	0
$739$	−687.168	−0.929862	−0.464931	−	0.885347i	$-0.653921\pi$
$739$	−687.168	−0.929862	−0.464931	+	0.885347i	$0.653921\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	362.628i	0.488059i	0.969768	+	0.244030i	$0.0784694\pi$
$743$	362.628i	0.488059i	−0.969768	+	0.244030i	$0.921531\pi$
$744$	0	0
$745$	595.376	0.799163
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 101.729i	− 0.135819i
$750$	0	0
$751$	261.366i	0.348024i	0.984744	+	0.174012i	$0.0556732\pi$
$751$	261.366i	0.348024i	−0.984744	+	0.174012i	$0.944327\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1076.20	1.42544
$756$	0	0
$757$	1395.34i	1.84325i	0.388081	+	0.921625i	$0.373138\pi$
$757$	1395.34i	1.84325i	−0.388081	+	0.921625i	$0.626862\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	319.500	0.419843	0.209921	−	0.977718i	$-0.432679\pi$
$761$	319.500	0.419843	0.209921	+	0.977718i	$0.432679\pi$
$762$	0	0
$763$	−73.7598	−0.0966708
$764$	0	0
$765$	0	0
$766$	0	0
$767$	456.362i	0.594996i
$768$	0	0
$769$	634.936	0.825664	0.412832	−	0.910807i	$-0.364540\pi$
$769$	634.936	0.825664	0.412832	+	0.910807i	$0.364540\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	96.1663i	0.124407i	0.998063	+	0.0622033i	$0.0198127\pi$
$773$	96.1663i	0.124407i	−0.998063	+	0.0622033i	$0.980187\pi$
$774$	0	0
$775$	− 39.4836i	− 0.0509465i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−172.456	−0.221382
$780$	0	0
$781$	− 1020.71i	− 1.30693i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−33.0577	−0.0421117
$786$	0	0
$787$	1319.25	1.67630	0.838148	−	0.545442i	$-0.183638\pi$
$787$	1319.25	1.67630	0.838148	+	0.545442i	$0.183638\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	218.121i	0.275753i
$792$	0	0
$793$	831.651	1.04874
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 818.575i	− 1.02707i	−0.858068	−	0.513535i	$-0.828336\pi$
$797$	− 818.575i	− 1.02707i	0.858068	−	0.513535i	$-0.171664\pi$
$798$	0	0
$799$	16.9911i	0.0212655i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1200.82	−1.49541
$804$	0	0
$805$	141.491i	0.175765i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1232.72	−1.52376	−0.761881	−	0.647717i	$-0.775724\pi$
$809$	−1232.72	−1.52376	−0.761881	+	0.647717i	$0.775724\pi$
$810$	0	0
$811$	1009.05	1.24421	0.622103	−	0.782935i	$-0.286278\pi$
$811$	1009.05	1.24421	0.622103	+	0.782935i	$0.286278\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1011.94i	− 1.24164i
$816$	0	0
$817$	−22.0968	−0.0270462
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 939.093i	− 1.14384i	−0.820309	−	0.571920i	$-0.806199\pi$
$821$	− 939.093i	− 1.14384i	0.820309	−	0.571920i	$-0.193801\pi$
$822$	0	0
$823$	− 911.100i	− 1.10705i	−0.832833	−	0.553524i	$-0.813283\pi$
$823$	− 911.100i	− 1.10705i	0.832833	−	0.553524i	$-0.186717\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−65.6564	−0.0793910	−0.0396955	−	0.999212i	$-0.512639\pi$
$827$	−65.6564	−0.0793910	−0.0396955	+	0.999212i	$0.512639\pi$
$828$	0	0
$829$	− 1515.94i	− 1.82864i	−0.404997	−	0.914318i	$-0.632728\pi$
$829$	− 1515.94i	− 1.82864i	0.404997	−	0.914318i	$-0.367272\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.64422	−0.00197386
$834$	0	0
$835$	808.257	0.967972
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 869.972i	− 1.03692i	−0.855103	−	0.518458i	$-0.826506\pi$
$839$	− 869.972i	− 1.03692i	0.855103	−	0.518458i	$-0.173494\pi$
$840$	0	0
$841$	−359.174	−0.427080
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.67964i	0.0114552i
$846$	0	0
$847$	− 900.313i	− 1.06294i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	594.020	0.698026
$852$	0	0
$853$	1643.91i	1.92721i	0.267322	+	0.963607i	$0.413861\pi$
$853$	1643.91i	1.92721i	−0.267322	+	0.963607i	$0.586139\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−286.059	−0.333791	−0.166895	−	0.985975i	$-0.553374\pi$
$857$	−286.059	−0.333791	−0.166895	+	0.985975i	$0.553374\pi$
$858$	0	0
$859$	−719.782	−0.837930	−0.418965	−	0.908002i	$-0.637607\pi$
$859$	−719.782	−0.837930	−0.418965	+	0.908002i	$0.637607\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1120.47i	− 1.29835i	−0.760641	−	0.649173i	$-0.775115\pi$
$863$	− 1120.47i	− 1.29835i	0.760641	−	0.649173i	$-0.224885\pi$
$864$	0	0
$865$	433.857	0.501569
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2040.66i	2.34828i
$870$	0	0
$871$	− 241.048i	− 0.276749i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−337.926	−0.386201
$876$	0	0
$877$	− 145.400i	− 0.165792i	−0.996558	−	0.0828960i	$-0.973583\pi$
$877$	− 145.400i	− 0.165792i	0.996558	−	0.0828960i	$-0.0264169\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−476.080	−0.540386	−0.270193	−	0.962806i	$-0.587087\pi$
$881$	−476.080	−0.540386	−0.270193	+	0.962806i	$0.587087\pi$
$882$	0	0
$883$	1101.22	1.24714	0.623568	−	0.781769i	$-0.285682\pi$
$883$	1101.22	1.24714	0.623568	+	0.781769i	$0.285682\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1491.49i	1.68150i	0.541427	+	0.840748i	$0.317884\pi$
$887$	1491.49i	1.68150i	−0.541427	+	0.840748i	$0.682116\pi$
$888$	0	0
$889$	66.5164	0.0748216
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 329.764i	− 0.369276i
$894$	0	0
$895$	− 392.481i	− 0.438526i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1181.62	1.31437
$900$	0	0
$901$	5.09021i	0.00564951i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1349.40	1.49104
$906$	0	0
$907$	−1155.46	−1.27394	−0.636969	−	0.770889i	$-0.719812\pi$
$907$	−1155.46	−1.27394	−0.636969	+	0.770889i	$0.719812\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	944.690i	1.03698i	0.855083	+	0.518491i	$0.173506\pi$
$911$	944.690i	1.03698i	−0.855083	+	0.518491i	$0.826494\pi$
$912$	0	0
$913$	−1535.98	−1.68235
$914$	0	0
$915$	0	0
$916$	0	0
$917$	334.418i	0.364687i
$918$	0	0
$919$	149.150i	0.162296i	0.996702	+	0.0811478i	$0.0258586\pi$
$919$	149.150i	0.162296i	−0.996702	+	0.0811478i	$0.974141\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−621.430	−0.673272
$924$	0	0
$925$	62.7851i	0.0678758i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	58.8399	0.0633368	0.0316684	−	0.999498i	$-0.489918\pi$
$929$	58.8399	0.0633368	0.0316684	+	0.999498i	$0.489918\pi$
$930$	0	0
$931$	31.9111	0.0342761
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.6333i	0.0263458i
$936$	0	0
$937$	−1700.18	−1.81449	−0.907246	−	0.420601i	$-0.861819\pi$
$937$	−1700.18	−1.81449	−0.907246	+	0.420601i	$0.861819\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 56.6116i	− 0.0601611i	−0.999547	−	0.0300805i	$-0.990424\pi$
$941$	− 56.6116i	− 0.0601611i	0.999547	−	0.0300805i	$-0.00957637\pi$
$942$	0	0
$943$	− 414.325i	− 0.439369i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	242.533	0.256107	0.128053	−	0.991767i	$-0.459127\pi$
$947$	242.533	0.256107	0.128053	+	0.991767i	$0.459127\pi$
$948$	0	0
$949$	731.084i	0.770373i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	364.070	0.382025	0.191013	−	0.981588i	$-0.438823\pi$
$953$	364.070	0.382025	0.191013	+	0.981588i	$0.438823\pi$
$954$	0	0
$955$	991.294	1.03800
$956$	0	0
$957$	0	0
$958$	0	0
$959$	92.4575i	0.0964103i
$960$	0	0
$961$	−202.348	−0.210560
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 426.435i	− 0.441901i
$966$	0	0
$967$	1221.99i	1.26369i	0.775093	+	0.631847i	$0.217703\pi$
$967$	1221.99i	1.26369i	−0.775093	+	0.631847i	$0.782297\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1088.53	−1.12104	−0.560521	−	0.828140i	$-0.689399\pi$
$971$	−1088.53	−1.12104	−0.560521	+	0.828140i	$0.689399\pi$
$972$	0	0
$973$	− 315.235i	− 0.323982i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1061.51	1.08650	0.543252	−	0.839570i	$-0.317193\pi$
$977$	1061.51	1.08650	0.543252	+	0.839570i	$0.317193\pi$
$978$	0	0
$979$	−3431.17	−3.50477
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1322.61i	− 1.34549i	−0.739877	−	0.672743i	$-0.765116\pi$
$983$	− 1322.61i	− 1.34549i	0.739877	−	0.672743i	$-0.234884\pi$
$984$	0	0
$985$	−105.782	−0.107393
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 53.0874i	− 0.0536778i
$990$	0	0
$991$	− 675.806i	− 0.681944i	−0.940073	−	0.340972i	$-0.889244\pi$
$991$	− 675.806i	− 0.681944i	0.940073	−	0.340972i	$-0.110756\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	888.354	0.892818
$996$	0	0
$997$	409.220i	0.410452i	0.978715	+	0.205226i	$0.0657929\pi$
$997$	409.220i	0.410452i	−0.978715	+	0.205226i	$0.934207\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.3.g.b.1135.3		8
3.2	odd	2		224.3.g.b.15.6		8
4.3	odd	2		504.3.g.b.379.6		8
8.3	odd	2	inner	2016.3.g.b.1135.6		8
8.5	even	2		504.3.g.b.379.5		8
12.11	even	2		56.3.g.b.43.3	✓	8
21.20	even	2		1568.3.g.m.687.3		8
24.5	odd	2		56.3.g.b.43.4	yes	8
24.11	even	2		224.3.g.b.15.5		8
48.5	odd	4		1792.3.d.j.1023.5		16
48.11	even	4		1792.3.d.j.1023.11		16
48.29	odd	4		1792.3.d.j.1023.12		16
48.35	even	4		1792.3.d.j.1023.6		16
84.11	even	6		392.3.k.o.275.8		16
84.23	even	6		392.3.k.o.67.3		16
84.47	odd	6		392.3.k.n.67.3		16
84.59	odd	6		392.3.k.n.275.8		16
84.83	odd	2		392.3.g.m.99.3		8
168.5	even	6		392.3.k.n.67.8		16
168.53	odd	6		392.3.k.o.275.3		16
168.83	odd	2		1568.3.g.m.687.4		8
168.101	even	6		392.3.k.n.275.3		16
168.125	even	2		392.3.g.m.99.4		8
168.149	odd	6		392.3.k.o.67.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.g.b.43.3	✓	8	12.11	even	2
56.3.g.b.43.4	yes	8	24.5	odd	2
224.3.g.b.15.5		8	24.11	even	2
224.3.g.b.15.6		8	3.2	odd	2
392.3.g.m.99.3		8	84.83	odd	2
392.3.g.m.99.4		8	168.125	even	2
392.3.k.n.67.3		16	84.47	odd	6
392.3.k.n.67.8		16	168.5	even	6
392.3.k.n.275.3		16	168.101	even	6
392.3.k.n.275.8		16	84.59	odd	6
392.3.k.o.67.3		16	84.23	even	6
392.3.k.o.67.8		16	168.149	odd	6
392.3.k.o.275.3		16	168.53	odd	6
392.3.k.o.275.8		16	84.11	even	6
504.3.g.b.379.5		8	8.5	even	2
504.3.g.b.379.6		8	4.3	odd	2
1568.3.g.m.687.3		8	21.20	even	2
1568.3.g.m.687.4		8	168.83	odd	2
1792.3.d.j.1023.5		16	48.5	odd	4
1792.3.d.j.1023.6		16	48.35	even	4
1792.3.d.j.1023.11		16	48.11	even	4
1792.3.d.j.1023.12		16	48.29	odd	4
2016.3.g.b.1135.3		8	1.1	even	1	trivial
2016.3.g.b.1135.6		8	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2016.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.88287i q^{5} -2.64575i q^{7} +O(q^{10})\)	\(q-4.88287i q^{5} -2.64575i q^{7} -21.4776 q^{11} +13.0760i q^{13} +0.234889 q^{17} -4.55872 q^{19} -10.9523i q^{23} +1.15761 q^{25} +34.6435i q^{29} -34.1079i q^{31} -12.9189 q^{35} +54.2370i q^{37} +37.8300 q^{41} +4.84714 q^{43} +72.3368i q^{47} -7.00000 q^{49} +21.6707i q^{53} +104.872i q^{55} +34.9007 q^{59} -63.6012i q^{61} +63.8485 q^{65} -18.4344 q^{67} +47.5244i q^{71} +55.9103 q^{73} +56.8243i q^{77} -95.0135i q^{79} +71.5156 q^{83} -1.14693i q^{85} +159.756 q^{89} +34.5959 q^{91} +22.2596i q^{95} -90.4794 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 32 q^{11} + 80 q^{17} - 56 q^{19} - 16 q^{25} + 56 q^{35} - 128 q^{41} - 56 q^{49} + 104 q^{59} + 72 q^{65} - 304 q^{67} - 112 q^{73} + 72 q^{83} + 512 q^{89} + 56 q^{91} + 64 q^{97}+O(q^{100}) \) 8 * q - 32 * q^11 + 80 * q^17 - 56 * q^19 - 16 * q^25 + 56 * q^35 - 128 * q^41 - 56 * q^49 + 104 * q^59 + 72 * q^65 - 304 * q^67 - 112 * q^73 + 72 * q^83 + 512 * q^89 + 56 * q^91 + 64 * q^97

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