LMFDB - Embedded newform 9016.2.a.bk.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9016,2,Mod(1,9016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9016, 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("9016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9016 = 2^{3} \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9016.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:	$71.9931224624$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 14 x^{9} + 61 x^{8} + 71 x^{7} - 343 x^{6} - 152 x^{5} + 867 x^{4} + 102 x^{3} + \cdots + 243 $ x^11 - 4x^10 - 14x^9 + 61x^8 + 71x^7 - 343x^6 - 152x^5 + 867x^4 + 102x^3 - 918x^2 + 35x + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.58959$ of defining polynomial
Character	$\chi$	$=$	9016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58959 q^{3} -1.03689 q^{5} -0.473209 q^{9} +O(q^{10})$	$q+1.58959 q^{3} -1.03689 q^{5} -0.473209 q^{9} +0.432880 q^{11} -1.93020 q^{13} -1.64822 q^{15} +5.83238 q^{17} +3.09651 q^{19} -1.00000 q^{23} -3.92487 q^{25} -5.52097 q^{27} -7.12421 q^{29} +5.54366 q^{31} +0.688101 q^{33} -9.75279 q^{37} -3.06823 q^{39} -7.95534 q^{41} +6.55694 q^{43} +0.490663 q^{45} +6.54794 q^{47} +9.27108 q^{51} +14.2780 q^{53} -0.448847 q^{55} +4.92217 q^{57} -7.77986 q^{59} -1.28270 q^{61} +2.00140 q^{65} -3.35573 q^{67} -1.58959 q^{69} -14.9999 q^{71} -15.2997 q^{73} -6.23892 q^{75} +8.75304 q^{79} -7.35645 q^{81} -11.2950 q^{83} -6.04751 q^{85} -11.3246 q^{87} +6.39765 q^{89} +8.81214 q^{93} -3.21072 q^{95} -14.7223 q^{97} -0.204842 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 4 q^{3} - 3 q^{5} + 11 q^{9}+O(q^{10}) $ 11 * q - 4 * q^3 - 3 * q^5 + 11 * q^9	$ 11 q - 4 q^{3} - 3 q^{5} + 11 q^{9} + 13 q^{13} - 7 q^{17} - 8 q^{19} - 11 q^{23} + 6 q^{25} - 25 q^{27} - 3 q^{29} - 12 q^{31} + 2 q^{33} - q^{37} - 21 q^{39} - 12 q^{41} + 9 q^{43} - 19 q^{45} - 17 q^{47} + 19 q^{51} - 5 q^{53} - 21 q^{55} + 11 q^{57} - 33 q^{59} + 15 q^{61} - 9 q^{65} - 5 q^{67} + 4 q^{69} - 9 q^{71} - 5 q^{73} - 44 q^{75} + 11 q^{79} - 13 q^{81} - 51 q^{83} + 33 q^{85} - 4 q^{87} - 26 q^{89} + 6 q^{93} - 19 q^{95} - 21 q^{97} + 15 q^{99}+O(q^{100}) $ 11 * q - 4 * q^3 - 3 * q^5 + 11 * q^9 + 13 * q^13 - 7 * q^17 - 8 * q^19 - 11 * q^23 + 6 * q^25 - 25 * q^27 - 3 * q^29 - 12 * q^31 + 2 * q^33 - q^37 - 21 * q^39 - 12 * q^41 + 9 * q^43 - 19 * q^45 - 17 * q^47 + 19 * q^51 - 5 * q^53 - 21 * q^55 + 11 * q^57 - 33 * q^59 + 15 * q^61 - 9 * q^65 - 5 * q^67 + 4 * q^69 - 9 * q^71 - 5 * q^73 - 44 * q^75 + 11 * q^79 - 13 * q^81 - 51 * q^83 + 33 * q^85 - 4 * q^87 - 26 * q^89 + 6 * q^93 - 19 * q^95 - 21 * q^97 + 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.58959	0.917749	0.458875	−	0.888501i	$-0.348253\pi$
$3$	1.58959	0.917749	0.458875	+	0.888501i	$0.348253\pi$
$4$	0	0
$5$	−1.03689	−0.463709	−0.231855	−	0.972750i	$-0.574479\pi$
$5$	−1.03689	−0.463709	−0.231855	+	0.972750i	$0.574479\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.473209	−0.157736
$10$	0	0
$11$	0.432880	0.130518	0.0652591	−	0.997868i	$-0.479213\pi$
$11$	0.432880	0.130518	0.0652591	+	0.997868i	$0.479213\pi$
$12$	0	0
$13$	−1.93020	−0.535342	−0.267671	−	0.963510i	$-0.586254\pi$
$13$	−1.93020	−0.535342	−0.267671	+	0.963510i	$0.586254\pi$
$14$	0	0
$15$	−1.64822	−0.425569
$16$	0	0
$17$	5.83238	1.41456	0.707280	−	0.706934i	$-0.249922\pi$
$17$	5.83238	1.41456	0.707280	+	0.706934i	$0.249922\pi$
$18$	0	0
$19$	3.09651	0.710387	0.355194	−	0.934793i	$-0.384415\pi$
$19$	3.09651	0.710387	0.355194	+	0.934793i	$0.384415\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.92487	−0.784974
$26$	0	0
$27$	−5.52097	−1.06251
$28$	0	0
$29$	−7.12421	−1.32293	−0.661466	−	0.749975i	$-0.730066\pi$
$29$	−7.12421	−1.32293	−0.661466	+	0.749975i	$0.730066\pi$
$30$	0	0
$31$	5.54366	0.995671	0.497835	−	0.867272i	$-0.334128\pi$
$31$	5.54366	0.995671	0.497835	+	0.867272i	$0.334128\pi$
$32$	0	0
$33$	0.688101	0.119783
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.75279	−1.60335	−0.801674	−	0.597761i	$-0.796057\pi$
$37$	−9.75279	−1.60335	−0.801674	+	0.597761i	$0.796057\pi$
$38$	0	0
$39$	−3.06823	−0.491310
$40$	0	0
$41$	−7.95534	−1.24242	−0.621208	−	0.783646i	$-0.713358\pi$
$41$	−7.95534	−1.24242	−0.621208	+	0.783646i	$0.713358\pi$
$42$	0	0
$43$	6.55694	0.999924	0.499962	−	0.866047i	$-0.333347\pi$
$43$	6.55694	0.999924	0.499962	+	0.866047i	$0.333347\pi$
$44$	0	0
$45$	0.490663	0.0731438
$46$	0	0
$47$	6.54794	0.955116	0.477558	−	0.878600i	$-0.341522\pi$
$47$	6.54794	0.955116	0.477558	+	0.878600i	$0.341522\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	9.27108	1.29821
$52$	0	0
$53$	14.2780	1.96123	0.980614	−	0.195947i	$-0.0627782\pi$
$53$	14.2780	1.96123	0.980614	+	0.195947i	$0.0627782\pi$
$54$	0	0
$55$	−0.448847	−0.0605225
$56$	0	0
$57$	4.92217	0.651957
$58$	0	0
$59$	−7.77986	−1.01285	−0.506426	−	0.862284i	$-0.669034\pi$
$59$	−7.77986	−1.01285	−0.506426	+	0.862284i	$0.669034\pi$
$60$	0	0
$61$	−1.28270	−0.164233	−0.0821165	−	0.996623i	$-0.526168\pi$
$61$	−1.28270	−0.164233	−0.0821165	+	0.996623i	$0.526168\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.00140	0.248243
$66$	0	0
$67$	−3.35573	−0.409968	−0.204984	−	0.978765i	$-0.565714\pi$
$67$	−3.35573	−0.409968	−0.204984	+	0.978765i	$0.565714\pi$
$68$	0	0
$69$	−1.58959	−0.191364
$70$	0	0
$71$	−14.9999	−1.78016	−0.890078	−	0.455808i	$-0.849350\pi$
$71$	−14.9999	−1.78016	−0.890078	+	0.455808i	$0.849350\pi$
$72$	0	0
$73$	−15.2997	−1.79069	−0.895346	−	0.445372i	$-0.853072\pi$
$73$	−15.2997	−1.79069	−0.895346	+	0.445372i	$0.853072\pi$
$74$	0	0
$75$	−6.23892	−0.720409
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.75304	0.984794	0.492397	−	0.870371i	$-0.336121\pi$
$79$	8.75304	0.984794	0.492397	+	0.870371i	$0.336121\pi$
$80$	0	0
$81$	−7.35645	−0.817383
$82$	0	0
$83$	−11.2950	−1.23979	−0.619895	−	0.784685i	$-0.712825\pi$
$83$	−11.2950	−1.23979	−0.619895	+	0.784685i	$0.712825\pi$
$84$	0	0
$85$	−6.04751	−0.655945
$86$	0	0
$87$	−11.3246	−1.21412
$88$	0	0
$89$	6.39765	0.678150	0.339075	−	0.940759i	$-0.389886\pi$
$89$	6.39765	0.678150	0.339075	+	0.940759i	$0.389886\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.81214	0.913776
$94$	0	0
$95$	−3.21072	−0.329413
$96$	0	0
$97$	−14.7223	−1.49482	−0.747411	−	0.664362i	$-0.768704\pi$
$97$	−14.7223	−1.49482	−0.747411	+	0.664362i	$0.768704\pi$
$98$	0	0
$99$	−0.204842	−0.0205874
$100$	0	0
$101$	13.7078	1.36398	0.681991	−	0.731361i	$-0.261114\pi$
$101$	13.7078	1.36398	0.681991	+	0.731361i	$0.261114\pi$
$102$	0	0
$103$	5.22502	0.514836	0.257418	−	0.966300i	$-0.417128\pi$
$103$	5.22502	0.514836	0.257418	+	0.966300i	$0.417128\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.60565	0.445245	0.222622	−	0.974905i	$-0.428538\pi$
$107$	4.60565	0.445245	0.222622	+	0.974905i	$0.428538\pi$
$108$	0	0
$109$	−3.68914	−0.353356	−0.176678	−	0.984269i	$-0.556535\pi$
$109$	−3.68914	−0.353356	−0.176678	+	0.984269i	$0.556535\pi$
$110$	0	0
$111$	−15.5029	−1.47147
$112$	0	0
$113$	4.72304	0.444306	0.222153	−	0.975012i	$-0.428692\pi$
$113$	4.72304	0.444306	0.222153	+	0.975012i	$0.428692\pi$
$114$	0	0
$115$	1.03689	0.0966901
$116$	0	0
$117$	0.913388	0.0844428
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8126	−0.982965
$122$	0	0
$123$	−12.6457	−1.14023
$124$	0	0
$125$	9.25407	0.827709
$126$	0	0
$127$	12.1445	1.07765	0.538823	−	0.842419i	$-0.318869\pi$
$127$	12.1445	1.07765	0.538823	+	0.842419i	$0.318869\pi$
$128$	0	0
$129$	10.4228	0.917679
$130$	0	0
$131$	−21.7203	−1.89771	−0.948857	−	0.315707i	$-0.897758\pi$
$131$	−21.7203	−1.89771	−0.948857	+	0.315707i	$0.897758\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	5.72462	0.492697
$136$	0	0
$137$	−17.5945	−1.50320	−0.751601	−	0.659618i	$-0.770718\pi$
$137$	−17.5945	−1.50320	−0.751601	+	0.659618i	$0.770718\pi$
$138$	0	0
$139$	−3.50739	−0.297493	−0.148746	−	0.988875i	$-0.547524\pi$
$139$	−3.50739	−0.297493	−0.148746	+	0.988875i	$0.547524\pi$
$140$	0	0
$141$	10.4085	0.876557
$142$	0	0
$143$	−0.835546	−0.0698719
$144$	0	0
$145$	7.38699	0.613456
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−9.71028	−0.795497	−0.397748	−	0.917495i	$-0.630208\pi$
$149$	−9.71028	−0.795497	−0.397748	+	0.917495i	$0.630208\pi$
$150$	0	0
$151$	15.9142	1.29508	0.647541	−	0.762031i	$-0.275798\pi$
$151$	15.9142	1.29508	0.647541	+	0.762031i	$0.275798\pi$
$152$	0	0
$153$	−2.75993	−0.223127
$154$	0	0
$155$	−5.74814	−0.461702
$156$	0	0
$157$	13.0190	1.03903	0.519515	−	0.854461i	$-0.326113\pi$
$157$	13.0190	1.03903	0.519515	+	0.854461i	$0.326113\pi$
$158$	0	0
$159$	22.6961	1.79992
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2.37327	−0.185889	−0.0929443	−	0.995671i	$-0.529628\pi$
$163$	−2.37327	−0.185889	−0.0929443	+	0.995671i	$0.529628\pi$
$164$	0	0
$165$	−0.713482	−0.0555445
$166$	0	0
$167$	1.57704	0.122035	0.0610174	−	0.998137i	$-0.480565\pi$
$167$	1.57704	0.122035	0.0610174	+	0.998137i	$0.480565\pi$
$168$	0	0
$169$	−9.27432	−0.713409
$170$	0	0
$171$	−1.46529	−0.112054
$172$	0	0
$173$	−8.11882	−0.617262	−0.308631	−	0.951182i	$-0.599871\pi$
$173$	−8.11882	−0.617262	−0.308631	+	0.951182i	$0.599871\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.3668	−0.929544
$178$	0	0
$179$	22.9545	1.71570	0.857849	−	0.513902i	$-0.171800\pi$
$179$	22.9545	1.71570	0.857849	+	0.513902i	$0.171800\pi$
$180$	0	0
$181$	7.44213	0.553169	0.276585	−	0.960990i	$-0.410797\pi$
$181$	7.44213	0.553169	0.276585	+	0.960990i	$0.410797\pi$
$182$	0	0
$183$	−2.03897	−0.150725
$184$	0	0
$185$	10.1125	0.743488
$186$	0	0
$187$	2.52472	0.184626
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.247730	0.0179251	0.00896257	−	0.999960i	$-0.497147\pi$
$191$	0.247730	0.0179251	0.00896257	+	0.999960i	$0.497147\pi$
$192$	0	0
$193$	−21.5572	−1.55172	−0.775860	−	0.630905i	$-0.782684\pi$
$193$	−21.5572	−1.55172	−0.775860	+	0.630905i	$0.782684\pi$
$194$	0	0
$195$	3.18140	0.227825
$196$	0	0
$197$	21.1000	1.50331	0.751655	−	0.659557i	$-0.229256\pi$
$197$	21.1000	1.50331	0.751655	+	0.659557i	$0.229256\pi$
$198$	0	0
$199$	2.35099	0.166657	0.0833285	−	0.996522i	$-0.473445\pi$
$199$	2.35099	0.166657	0.0833285	+	0.996522i	$0.473445\pi$
$200$	0	0
$201$	−5.33424	−0.376248
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.24878	0.576120
$206$	0	0
$207$	0.473209	0.0328903
$208$	0	0
$209$	1.34042	0.0927185
$210$	0	0
$211$	12.9437	0.891080	0.445540	−	0.895262i	$-0.353012\pi$
$211$	12.9437	0.891080	0.445540	+	0.895262i	$0.353012\pi$
$212$	0	0
$213$	−23.8436	−1.63374
$214$	0	0
$215$	−6.79880	−0.463674
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−24.3202	−1.64341
$220$	0	0
$221$	−11.2577	−0.757273
$222$	0	0
$223$	−29.5064	−1.97590	−0.987949	−	0.154781i	$-0.950533\pi$
$223$	−29.5064	−1.97590	−0.987949	+	0.154781i	$0.950533\pi$
$224$	0	0
$225$	1.85728	0.123819
$226$	0	0
$227$	−7.68703	−0.510206	−0.255103	−	0.966914i	$-0.582109\pi$
$227$	−7.68703	−0.510206	−0.255103	+	0.966914i	$0.582109\pi$
$228$	0	0
$229$	−23.2356	−1.53545	−0.767724	−	0.640780i	$-0.778611\pi$
$229$	−23.2356	−1.53545	−0.767724	+	0.640780i	$0.778611\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.8486	−1.36584	−0.682918	−	0.730495i	$-0.739289\pi$
$233$	−20.8486	−1.36584	−0.682918	+	0.730495i	$0.739289\pi$
$234$	0	0
$235$	−6.78947	−0.442896
$236$	0	0
$237$	13.9137	0.903794
$238$	0	0
$239$	−1.51241	−0.0978299	−0.0489150	−	0.998803i	$-0.515576\pi$
$239$	−1.51241	−0.0978299	−0.0489150	+	0.998803i	$0.515576\pi$
$240$	0	0
$241$	0.926696	0.0596937	0.0298469	−	0.999554i	$-0.490498\pi$
$241$	0.926696	0.0596937	0.0298469	+	0.999554i	$0.490498\pi$
$242$	0	0
$243$	4.86919	0.312359
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.97688	−0.380300
$248$	0	0
$249$	−17.9544	−1.13782
$250$	0	0
$251$	−2.48517	−0.156862	−0.0784312	−	0.996920i	$-0.524991\pi$
$251$	−2.48517	−0.156862	−0.0784312	+	0.996920i	$0.524991\pi$
$252$	0	0
$253$	−0.432880	−0.0272149
$254$	0	0
$255$	−9.61306	−0.601993
$256$	0	0
$257$	−28.1558	−1.75631	−0.878155	−	0.478376i	$-0.841225\pi$
$257$	−28.1558	−1.75631	−0.878155	+	0.478376i	$0.841225\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.37124	0.208674
$262$	0	0
$263$	−6.94402	−0.428187	−0.214093	−	0.976813i	$-0.568680\pi$
$263$	−6.94402	−0.428187	−0.214093	+	0.976813i	$0.568680\pi$
$264$	0	0
$265$	−14.8046	−0.909440
$266$	0	0
$267$	10.1696	0.622371
$268$	0	0
$269$	−7.77693	−0.474168	−0.237084	−	0.971489i	$-0.576192\pi$
$269$	−7.77693	−0.474168	−0.237084	+	0.971489i	$0.576192\pi$
$270$	0	0
$271$	−30.8660	−1.87497	−0.937487	−	0.348021i	$-0.886854\pi$
$271$	−30.8660	−1.87497	−0.937487	+	0.348021i	$0.886854\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.69900	−0.102453
$276$	0	0
$277$	−29.4236	−1.76790	−0.883948	−	0.467586i	$-0.845124\pi$
$277$	−29.4236	−1.76790	−0.883948	+	0.467586i	$0.845124\pi$
$278$	0	0
$279$	−2.62331	−0.157053
$280$	0	0
$281$	−24.3006	−1.44965	−0.724826	−	0.688932i	$-0.758080\pi$
$281$	−24.3006	−1.44965	−0.724826	+	0.688932i	$0.758080\pi$
$282$	0	0
$283$	6.14663	0.365380	0.182690	−	0.983171i	$-0.441520\pi$
$283$	6.14663	0.365380	0.182690	+	0.983171i	$0.441520\pi$
$284$	0	0
$285$	−5.10373	−0.302319
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0167	1.00098
$290$	0	0
$291$	−23.4024	−1.37187
$292$	0	0
$293$	−23.1258	−1.35102	−0.675511	−	0.737350i	$-0.736077\pi$
$293$	−23.1258	−1.35102	−0.675511	+	0.737350i	$0.736077\pi$
$294$	0	0
$295$	8.06683	0.469669
$296$	0	0
$297$	−2.38992	−0.138677
$298$	0	0
$299$	1.93020	0.111627
$300$	0	0
$301$	0	0
$302$	0	0
$303$	21.7898	1.25179
$304$	0	0
$305$	1.33001	0.0761564
$306$	0	0
$307$	7.81450	0.445997	0.222999	−	0.974819i	$-0.428415\pi$
$307$	7.81450	0.445997	0.222999	+	0.974819i	$0.428415\pi$
$308$	0	0
$309$	8.30562	0.472490
$310$	0	0
$311$	−9.97148	−0.565431	−0.282715	−	0.959204i	$-0.591235\pi$
$311$	−9.97148	−0.565431	−0.282715	+	0.959204i	$0.591235\pi$
$312$	0	0
$313$	−7.86801	−0.444726	−0.222363	−	0.974964i	$-0.571377\pi$
$313$	−7.86801	−0.444726	−0.222363	+	0.974964i	$0.571377\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.5552	−1.21066	−0.605329	−	0.795975i	$-0.706959\pi$
$317$	−21.5552	−1.21066	−0.605329	+	0.795975i	$0.706959\pi$
$318$	0	0
$319$	−3.08393	−0.172667
$320$	0	0
$321$	7.32108	0.408623
$322$	0	0
$323$	18.0600	1.00489
$324$	0	0
$325$	7.57579	0.420229
$326$	0	0
$327$	−5.86422	−0.324292
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2.12100	−0.116581	−0.0582904	−	0.998300i	$-0.518565\pi$
$331$	−2.12100	−0.116581	−0.0582904	+	0.998300i	$0.518565\pi$
$332$	0	0
$333$	4.61510	0.252906
$334$	0	0
$335$	3.47951	0.190106
$336$	0	0
$337$	23.6242	1.28689	0.643445	−	0.765492i	$-0.277504\pi$
$337$	23.6242	1.28689	0.643445	+	0.765492i	$0.277504\pi$
$338$	0	0
$339$	7.50768	0.407761
$340$	0	0
$341$	2.39974	0.129953
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.64822	0.0887373
$346$	0	0
$347$	−11.0968	−0.595706	−0.297853	−	0.954612i	$-0.596270\pi$
$347$	−11.0968	−0.595706	−0.297853	+	0.954612i	$0.596270\pi$
$348$	0	0
$349$	3.28541	0.175864	0.0879320	−	0.996126i	$-0.471974\pi$
$349$	3.28541	0.175864	0.0879320	+	0.996126i	$0.471974\pi$
$350$	0	0
$351$	10.6566	0.568807
$352$	0	0
$353$	−9.37017	−0.498724	−0.249362	−	0.968410i	$-0.580221\pi$
$353$	−9.37017	−0.498724	−0.249362	+	0.968410i	$0.580221\pi$
$354$	0	0
$355$	15.5531	0.825475
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.09548	0.427263	0.213632	−	0.976914i	$-0.431471\pi$
$359$	8.09548	0.427263	0.213632	+	0.976914i	$0.431471\pi$
$360$	0	0
$361$	−9.41165	−0.495350
$362$	0	0
$363$	−17.1876	−0.902115
$364$	0	0
$365$	15.8640	0.830361
$366$	0	0
$367$	8.93681	0.466497	0.233249	−	0.972417i	$-0.425064\pi$
$367$	8.93681	0.466497	0.233249	+	0.972417i	$0.425064\pi$
$368$	0	0
$369$	3.76454	0.195974
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.9493	0.981160	0.490580	−	0.871396i	$-0.336785\pi$
$373$	18.9493	0.981160	0.490580	+	0.871396i	$0.336785\pi$
$374$	0	0
$375$	14.7102	0.759629
$376$	0	0
$377$	13.7512	0.708221
$378$	0	0
$379$	−34.9466	−1.79509	−0.897544	−	0.440926i	$-0.854650\pi$
$379$	−34.9466	−1.79509	−0.897544	+	0.440926i	$0.854650\pi$
$380$	0	0
$381$	19.3047	0.989009
$382$	0	0
$383$	5.30937	0.271296	0.135648	−	0.990757i	$-0.456688\pi$
$383$	5.30937	0.271296	0.135648	+	0.990757i	$0.456688\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3.10280	−0.157724
$388$	0	0
$389$	17.4248	0.883473	0.441736	−	0.897145i	$-0.354363\pi$
$389$	17.4248	0.883473	0.441736	+	0.897145i	$0.354363\pi$
$390$	0	0
$391$	−5.83238	−0.294956
$392$	0	0
$393$	−34.5264	−1.74163
$394$	0	0
$395$	−9.07591	−0.456658
$396$	0	0
$397$	28.7071	1.44077	0.720383	−	0.693576i	$-0.243966\pi$
$397$	28.7071	1.44077	0.720383	+	0.693576i	$0.243966\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.7456	1.18580	0.592899	−	0.805277i	$-0.297983\pi$
$401$	23.7456	1.18580	0.592899	+	0.805277i	$0.297983\pi$
$402$	0	0
$403$	−10.7004	−0.533024
$404$	0	0
$405$	7.62780	0.379028
$406$	0	0
$407$	−4.22179	−0.209266
$408$	0	0
$409$	28.5648	1.41244	0.706219	−	0.707993i	$-0.250399\pi$
$409$	28.5648	1.41244	0.706219	+	0.707993i	$0.250399\pi$
$410$	0	0
$411$	−27.9680	−1.37956
$412$	0	0
$413$	0	0
$414$	0	0
$415$	11.7116	0.574902
$416$	0	0
$417$	−5.57531	−0.273024
$418$	0	0
$419$	7.15965	0.349772	0.174886	−	0.984589i	$-0.444044\pi$
$419$	7.15965	0.349772	0.174886	+	0.984589i	$0.444044\pi$
$420$	0	0
$421$	3.99764	0.194833	0.0974164	−	0.995244i	$-0.468942\pi$
$421$	3.99764	0.194833	0.0974164	+	0.995244i	$0.468942\pi$
$422$	0	0
$423$	−3.09854	−0.150656
$424$	0	0
$425$	−22.8913	−1.11039
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.32817	−0.0641249
$430$	0	0
$431$	35.4763	1.70883	0.854417	−	0.519587i	$-0.173914\pi$
$431$	35.4763	1.70883	0.854417	+	0.519587i	$0.173914\pi$
$432$	0	0
$433$	8.44736	0.405954	0.202977	−	0.979183i	$-0.434938\pi$
$433$	8.44736	0.405954	0.202977	+	0.979183i	$0.434938\pi$
$434$	0	0
$435$	11.7423	0.562999
$436$	0	0
$437$	−3.09651	−0.148126
$438$	0	0
$439$	−22.3967	−1.06894	−0.534469	−	0.845188i	$-0.679488\pi$
$439$	−22.3967	−1.06894	−0.534469	+	0.845188i	$0.679488\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.7103	−0.841443	−0.420721	−	0.907190i	$-0.638223\pi$
$443$	−17.7103	−0.841443	−0.420721	+	0.907190i	$0.638223\pi$
$444$	0	0
$445$	−6.63363	−0.314464
$446$	0	0
$447$	−15.4353	−0.730067
$448$	0	0
$449$	32.2556	1.52224	0.761118	−	0.648613i	$-0.224651\pi$
$449$	32.2556	1.52224	0.761118	+	0.648613i	$0.224651\pi$
$450$	0	0
$451$	−3.44371	−0.162158
$452$	0	0
$453$	25.2971	1.18856
$454$	0	0
$455$	0	0
$456$	0	0
$457$	21.1304	0.988440	0.494220	−	0.869337i	$-0.335454\pi$
$457$	21.1304	0.988440	0.494220	+	0.869337i	$0.335454\pi$
$458$	0	0
$459$	−32.2004	−1.50299
$460$	0	0
$461$	−27.3547	−1.27403	−0.637017	−	0.770850i	$-0.719832\pi$
$461$	−27.3547	−1.27403	−0.637017	+	0.770850i	$0.719832\pi$
$462$	0	0
$463$	−18.5650	−0.862787	−0.431393	−	0.902164i	$-0.641978\pi$
$463$	−18.5650	−0.862787	−0.431393	+	0.902164i	$0.641978\pi$
$464$	0	0
$465$	−9.13718	−0.423727
$466$	0	0
$467$	−32.0104	−1.48127	−0.740633	−	0.671910i	$-0.765474\pi$
$467$	−32.0104	−1.48127	−0.740633	+	0.671910i	$0.765474\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.6949	0.953569
$472$	0	0
$473$	2.83837	0.130508
$474$	0	0
$475$	−12.1534	−0.557635
$476$	0	0
$477$	−6.75645	−0.309357
$478$	0	0
$479$	−26.6950	−1.21973	−0.609864	−	0.792506i	$-0.708776\pi$
$479$	−26.6950	−1.21973	−0.609864	+	0.792506i	$0.708776\pi$
$480$	0	0
$481$	18.8249	0.858339
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.2653	0.693163
$486$	0	0
$487$	4.06065	0.184006	0.0920028	−	0.995759i	$-0.470673\pi$
$487$	4.06065	0.184006	0.0920028	+	0.995759i	$0.470673\pi$
$488$	0	0
$489$	−3.77252	−0.170599
$490$	0	0
$491$	23.4990	1.06050	0.530248	−	0.847843i	$-0.322099\pi$
$491$	23.4990	1.06050	0.530248	+	0.847843i	$0.322099\pi$
$492$	0	0
$493$	−41.5511	−1.87137
$494$	0	0
$495$	0.212398	0.00954659
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3.93183	0.176013	0.0880065	−	0.996120i	$-0.471950\pi$
$499$	3.93183	0.176013	0.0880065	+	0.996120i	$0.471950\pi$
$500$	0	0
$501$	2.50684	0.111997
$502$	0	0
$503$	4.62716	0.206315	0.103157	−	0.994665i	$-0.467105\pi$
$503$	4.62716	0.206315	0.103157	+	0.994665i	$0.467105\pi$
$504$	0	0
$505$	−14.2135	−0.632491
$506$	0	0
$507$	−14.7423	−0.654731
$508$	0	0
$509$	9.87634	0.437761	0.218881	−	0.975752i	$-0.429759\pi$
$509$	9.87634	0.437761	0.218881	+	0.975752i	$0.429759\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−17.0957	−0.754795
$514$	0	0
$515$	−5.41774	−0.238734
$516$	0	0
$517$	2.83447	0.124660
$518$	0	0
$519$	−12.9056	−0.566492
$520$	0	0
$521$	1.56186	0.0684264	0.0342132	−	0.999415i	$-0.489107\pi$
$521$	1.56186	0.0684264	0.0342132	+	0.999415i	$0.489107\pi$
$522$	0	0
$523$	−36.7989	−1.60911	−0.804553	−	0.593881i	$-0.797595\pi$
$523$	−36.7989	−1.60911	−0.804553	+	0.593881i	$0.797595\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	32.3327	1.40844
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.68150	0.159763
$532$	0	0
$533$	15.3554	0.665117
$534$	0	0
$535$	−4.77553	−0.206464
$536$	0	0
$537$	36.4882	1.57458
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.60180	−0.283834	−0.141917	−	0.989879i	$-0.545327\pi$
$541$	−6.60180	−0.283834	−0.141917	+	0.989879i	$0.545327\pi$
$542$	0	0
$543$	11.8299	0.507671
$544$	0	0
$545$	3.82522	0.163854
$546$	0	0
$547$	2.10625	0.0900569	0.0450285	−	0.998986i	$-0.485662\pi$
$547$	2.10625	0.0900569	0.0450285	+	0.998986i	$0.485662\pi$
$548$	0	0
$549$	0.606985	0.0259055
$550$	0	0
$551$	−22.0602	−0.939794
$552$	0	0
$553$	0	0
$554$	0	0
$555$	16.0748	0.682335
$556$	0	0
$557$	−4.58027	−0.194072	−0.0970362	−	0.995281i	$-0.530936\pi$
$557$	−4.58027	−0.194072	−0.0970362	+	0.995281i	$0.530936\pi$
$558$	0	0
$559$	−12.6562	−0.535301
$560$	0	0
$561$	4.01327	0.169440
$562$	0	0
$563$	−1.35266	−0.0570079	−0.0285040	−	0.999594i	$-0.509074\pi$
$563$	−1.35266	−0.0570079	−0.0285040	+	0.999594i	$0.509074\pi$
$564$	0	0
$565$	−4.89725	−0.206029
$566$	0	0
$567$	0	0
$568$	0	0
$569$	29.4636	1.23518	0.617589	−	0.786501i	$-0.288110\pi$
$569$	29.4636	1.23518	0.617589	+	0.786501i	$0.288110\pi$
$570$	0	0
$571$	22.3580	0.935655	0.467827	−	0.883820i	$-0.345037\pi$
$571$	22.3580	0.935655	0.467827	+	0.883820i	$0.345037\pi$
$572$	0	0
$573$	0.393789	0.0164508
$574$	0	0
$575$	3.92487	0.163678
$576$	0	0
$577$	−6.08310	−0.253243	−0.126621	−	0.991951i	$-0.540413\pi$
$577$	−6.08310	−0.253243	−0.126621	+	0.991951i	$0.540413\pi$
$578$	0	0
$579$	−34.2670	−1.42409
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.18064	0.255976
$584$	0	0
$585$	−0.947079	−0.0391569
$586$	0	0
$587$	37.3574	1.54191	0.770953	−	0.636893i	$-0.219781\pi$
$587$	37.3574	1.54191	0.770953	+	0.636893i	$0.219781\pi$
$588$	0	0
$589$	17.1660	0.707312
$590$	0	0
$591$	33.5402	1.37966
$592$	0	0
$593$	2.02521	0.0831653	0.0415827	−	0.999135i	$-0.486760\pi$
$593$	2.02521	0.0831653	0.0415827	+	0.999135i	$0.486760\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.73710	0.152949
$598$	0	0
$599$	−27.6626	−1.13026	−0.565132	−	0.825001i	$-0.691175\pi$
$599$	−27.6626	−1.13026	−0.565132	+	0.825001i	$0.691175\pi$
$600$	0	0
$601$	−17.6238	−0.718888	−0.359444	−	0.933167i	$-0.617034\pi$
$601$	−17.6238	−0.718888	−0.359444	+	0.933167i	$0.617034\pi$
$602$	0	0
$603$	1.58796	0.0646668
$604$	0	0
$605$	11.2114	0.455810
$606$	0	0
$607$	3.68985	0.149766	0.0748831	−	0.997192i	$-0.476142\pi$
$607$	3.68985	0.149766	0.0748831	+	0.997192i	$0.476142\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.6389	−0.511313
$612$	0	0
$613$	9.15142	0.369623	0.184811	−	0.982774i	$-0.440833\pi$
$613$	9.15142	0.369623	0.184811	+	0.982774i	$0.440833\pi$
$614$	0	0
$615$	13.1122	0.528734
$616$	0	0
$617$	−9.61886	−0.387241	−0.193620	−	0.981077i	$-0.562023\pi$
$617$	−9.61886	−0.387241	−0.193620	+	0.981077i	$0.562023\pi$
$618$	0	0
$619$	−36.3719	−1.46191	−0.730954	−	0.682427i	$-0.760925\pi$
$619$	−36.3719	−1.46191	−0.730954	+	0.682427i	$0.760925\pi$
$620$	0	0
$621$	5.52097	0.221549
$622$	0	0
$623$	0	0
$624$	0	0
$625$	10.0289	0.401157
$626$	0	0
$627$	2.13071	0.0850923
$628$	0	0
$629$	−56.8819	−2.26803
$630$	0	0
$631$	−20.7741	−0.827006	−0.413503	−	0.910503i	$-0.635695\pi$
$631$	−20.7741	−0.827006	−0.413503	+	0.910503i	$0.635695\pi$
$632$	0	0
$633$	20.5751	0.817788
$634$	0	0
$635$	−12.5924	−0.499715
$636$	0	0
$637$	0	0
$638$	0	0
$639$	7.09806	0.280795
$640$	0	0
$641$	−5.53828	−0.218749	−0.109375	−	0.994001i	$-0.534885\pi$
$641$	−5.53828	−0.218749	−0.109375	+	0.994001i	$0.534885\pi$
$642$	0	0
$643$	−11.4185	−0.450302	−0.225151	−	0.974324i	$-0.572288\pi$
$643$	−11.4185	−0.450302	−0.225151	+	0.974324i	$0.572288\pi$
$644$	0	0
$645$	−10.8073	−0.425537
$646$	0	0
$647$	9.04340	0.355533	0.177766	−	0.984073i	$-0.443113\pi$
$647$	9.04340	0.355533	0.177766	+	0.984073i	$0.443113\pi$
$648$	0	0
$649$	−3.36774	−0.132196
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.7518	0.499018	0.249509	−	0.968372i	$-0.419731\pi$
$653$	12.7518	0.499018	0.249509	+	0.968372i	$0.419731\pi$
$654$	0	0
$655$	22.5215	0.879988
$656$	0	0
$657$	7.23994	0.282457
$658$	0	0
$659$	−12.5108	−0.487351	−0.243676	−	0.969857i	$-0.578353\pi$
$659$	−12.5108	−0.487351	−0.243676	+	0.969857i	$0.578353\pi$
$660$	0	0
$661$	42.7084	1.66116	0.830582	−	0.556896i	$-0.188008\pi$
$661$	42.7084	1.66116	0.830582	+	0.556896i	$0.188008\pi$
$662$	0	0
$663$	−17.8951	−0.694987
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.12421	0.275851
$668$	0	0
$669$	−46.9031	−1.81338
$670$	0	0
$671$	−0.555256	−0.0214354
$672$	0	0
$673$	34.0733	1.31343	0.656714	−	0.754139i	$-0.271946\pi$
$673$	34.0733	1.31343	0.656714	+	0.754139i	$0.271946\pi$
$674$	0	0
$675$	21.6691	0.834043
$676$	0	0
$677$	21.4607	0.824800	0.412400	−	0.911003i	$-0.364691\pi$
$677$	21.4607	0.824800	0.412400	+	0.911003i	$0.364691\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.2192	−0.468241
$682$	0	0
$683$	21.1578	0.809581	0.404791	−	0.914409i	$-0.367344\pi$
$683$	21.1578	0.809581	0.404791	+	0.914409i	$0.367344\pi$
$684$	0	0
$685$	18.2435	0.697049
$686$	0	0
$687$	−36.9350	−1.40916
$688$	0	0
$689$	−27.5594	−1.04993
$690$	0	0
$691$	51.2282	1.94881	0.974407	−	0.224793i	$-0.0721705\pi$
$691$	51.2282	1.94881	0.974407	+	0.224793i	$0.0721705\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.63676	0.137950
$696$	0	0
$697$	−46.3986	−1.75747
$698$	0	0
$699$	−33.1407	−1.25349
$700$	0	0
$701$	11.3805	0.429835	0.214918	−	0.976632i	$-0.431052\pi$
$701$	11.3805	0.429835	0.214918	+	0.976632i	$0.431052\pi$
$702$	0	0
$703$	−30.1996	−1.13900
$704$	0	0
$705$	−10.7925	−0.406468
$706$	0	0
$707$	0	0
$708$	0	0
$709$	21.7210	0.815748	0.407874	−	0.913038i	$-0.366270\pi$
$709$	21.7210	0.815748	0.407874	+	0.913038i	$0.366270\pi$
$710$	0	0
$711$	−4.14201	−0.155338
$712$	0	0
$713$	−5.54366	−0.207612
$714$	0	0
$715$	0.866366	0.0324002
$716$	0	0
$717$	−2.40412	−0.0897834
$718$	0	0
$719$	−9.55987	−0.356523	−0.178261	−	0.983983i	$-0.557047\pi$
$719$	−9.55987	−0.356523	−0.178261	+	0.983983i	$0.557047\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.47307	0.0547839
$724$	0	0
$725$	27.9616	1.03847
$726$	0	0
$727$	21.2414	0.787800	0.393900	−	0.919153i	$-0.371126\pi$
$727$	21.2414	0.787800	0.393900	+	0.919153i	$0.371126\pi$
$728$	0	0
$729$	29.8094	1.10405
$730$	0	0
$731$	38.2426	1.41445
$732$	0	0
$733$	17.7647	0.656154	0.328077	−	0.944651i	$-0.393600\pi$
$733$	17.7647	0.656154	0.328077	+	0.944651i	$0.393600\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.45263	−0.0535083
$738$	0	0
$739$	42.7688	1.57328	0.786638	−	0.617414i	$-0.211820\pi$
$739$	42.7688	1.57328	0.786638	+	0.617414i	$0.211820\pi$
$740$	0	0
$741$	−9.50078	−0.349020
$742$	0	0
$743$	−30.8875	−1.13315	−0.566577	−	0.824009i	$-0.691733\pi$
$743$	−30.8875	−1.13315	−0.566577	+	0.824009i	$0.691733\pi$
$744$	0	0
$745$	10.0684	0.368879
$746$	0	0
$747$	5.34490	0.195560
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17.0237	0.621205	0.310603	−	0.950540i	$-0.399469\pi$
$751$	17.0237	0.621205	0.310603	+	0.950540i	$0.399469\pi$
$752$	0	0
$753$	−3.95040	−0.143960
$754$	0	0
$755$	−16.5012	−0.600542
$756$	0	0
$757$	−18.8049	−0.683476	−0.341738	−	0.939795i	$-0.611015\pi$
$757$	−18.8049	−0.683476	−0.341738	+	0.939795i	$0.611015\pi$
$758$	0	0
$759$	−0.688101	−0.0249765
$760$	0	0
$761$	−22.1008	−0.801152	−0.400576	−	0.916264i	$-0.631190\pi$
$761$	−22.1008	−0.801152	−0.400576	+	0.916264i	$0.631190\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.86173	0.103466
$766$	0	0
$767$	15.0167	0.542222
$768$	0	0
$769$	−24.4982	−0.883429	−0.441715	−	0.897156i	$-0.645630\pi$
$769$	−24.4982	−0.883429	−0.441715	+	0.897156i	$0.645630\pi$
$770$	0	0
$771$	−44.7561	−1.61185
$772$	0	0
$773$	29.5602	1.06321	0.531604	−	0.846993i	$-0.321590\pi$
$773$	29.5602	1.06321	0.531604	+	0.846993i	$0.321590\pi$
$774$	0	0
$775$	−21.7581	−0.781575
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.6338	−0.882596
$780$	0	0
$781$	−6.49314	−0.232343
$782$	0	0
$783$	39.3326	1.40563
$784$	0	0
$785$	−13.4992	−0.481808
$786$	0	0
$787$	−31.5636	−1.12512	−0.562561	−	0.826756i	$-0.690184\pi$
$787$	−31.5636	−1.12512	−0.562561	+	0.826756i	$0.690184\pi$
$788$	0	0
$789$	−11.0381	−0.392968
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2.47587	0.0879208
$794$	0	0
$795$	−23.5332	−0.834638
$796$	0	0
$797$	2.00294	0.0709479	0.0354740	−	0.999371i	$-0.488706\pi$
$797$	2.00294	0.0709479	0.0354740	+	0.999371i	$0.488706\pi$
$798$	0	0
$799$	38.1901	1.35107
$800$	0	0
$801$	−3.02742	−0.106969
$802$	0	0
$803$	−6.62292	−0.233718
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.3621	−0.435167
$808$	0	0
$809$	18.2073	0.640136	0.320068	−	0.947395i	$-0.396294\pi$
$809$	18.2073	0.640136	0.320068	+	0.947395i	$0.396294\pi$
$810$	0	0
$811$	−39.8278	−1.39854	−0.699271	−	0.714857i	$-0.746492\pi$
$811$	−39.8278	−1.39854	−0.699271	+	0.714857i	$0.746492\pi$
$812$	0	0
$813$	−49.0642	−1.72076
$814$	0	0
$815$	2.46081	0.0861983
$816$	0	0
$817$	20.3036	0.710333
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−21.5347	−0.751568	−0.375784	−	0.926707i	$-0.622627\pi$
$821$	−21.5347	−0.751568	−0.375784	+	0.926707i	$0.622627\pi$
$822$	0	0
$823$	23.5511	0.820941	0.410471	−	0.911874i	$-0.365364\pi$
$823$	23.5511	0.820941	0.410471	+	0.911874i	$0.365364\pi$
$824$	0	0
$825$	−2.70071	−0.0940265
$826$	0	0
$827$	−20.8830	−0.726173	−0.363087	−	0.931755i	$-0.618277\pi$
$827$	−20.8830	−0.726173	−0.363087	+	0.931755i	$0.618277\pi$
$828$	0	0
$829$	−29.9236	−1.03929	−0.519645	−	0.854383i	$-0.673936\pi$
$829$	−29.9236	−1.03929	−0.519645	+	0.854383i	$0.673936\pi$
$830$	0	0
$831$	−46.7715	−1.62248
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1.63521	−0.0565887
$836$	0	0
$837$	−30.6064	−1.05791
$838$	0	0
$839$	−35.2800	−1.21800	−0.609001	−	0.793170i	$-0.708429\pi$
$839$	−35.2800	−1.21800	−0.609001	+	0.793170i	$0.708429\pi$
$840$	0	0
$841$	21.7544	0.750151
$842$	0	0
$843$	−38.6279	−1.33042
$844$	0	0
$845$	9.61641	0.330815
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.77062	0.335327
$850$	0	0
$851$	9.75279	0.334321
$852$	0	0
$853$	40.4145	1.38377	0.691883	−	0.722010i	$-0.256781\pi$
$853$	40.4145	1.38377	0.691883	+	0.722010i	$0.256781\pi$
$854$	0	0
$855$	1.51934	0.0519604
$856$	0	0
$857$	−0.717313	−0.0245029	−0.0122515	−	0.999925i	$-0.503900\pi$
$857$	−0.717313	−0.0245029	−0.0122515	+	0.999925i	$0.503900\pi$
$858$	0	0
$859$	39.4600	1.34636	0.673179	−	0.739480i	$-0.264929\pi$
$859$	39.4600	1.34636	0.673179	+	0.739480i	$0.264929\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.40621	0.0819084	0.0409542	−	0.999161i	$-0.486960\pi$
$863$	2.40621	0.0819084	0.0409542	+	0.999161i	$0.486960\pi$
$864$	0	0
$865$	8.41829	0.286230
$866$	0	0
$867$	27.0495	0.918648
$868$	0	0
$869$	3.78902	0.128534
$870$	0	0
$871$	6.47725	0.219473
$872$	0	0
$873$	6.96672	0.235788
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.59028	0.0536998	0.0268499	−	0.999639i	$-0.491452\pi$
$877$	1.59028	0.0536998	0.0268499	+	0.999639i	$0.491452\pi$
$878$	0	0
$879$	−36.7605	−1.23990
$880$	0	0
$881$	−15.7246	−0.529777	−0.264888	−	0.964279i	$-0.585335\pi$
$881$	−15.7246	−0.529777	−0.264888	+	0.964279i	$0.585335\pi$
$882$	0	0
$883$	21.7017	0.730320	0.365160	−	0.930945i	$-0.381014\pi$
$883$	21.7017	0.730320	0.365160	+	0.930945i	$0.381014\pi$
$884$	0	0
$885$	12.8229	0.431038
$886$	0	0
$887$	4.51533	0.151610	0.0758050	−	0.997123i	$-0.475847\pi$
$887$	4.51533	0.151610	0.0758050	+	0.997123i	$0.475847\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3.18446	−0.106683
$892$	0	0
$893$	20.2757	0.678502
$894$	0	0
$895$	−23.8012	−0.795586
$896$	0	0
$897$	3.06823	0.102445
$898$	0	0
$899$	−39.4942	−1.31721
$900$	0	0
$901$	83.2745	2.77428
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.71664	−0.256510
$906$	0	0
$907$	37.2651	1.23737	0.618684	−	0.785640i	$-0.287666\pi$
$907$	37.2651	1.23737	0.618684	+	0.785640i	$0.287666\pi$
$908$	0	0
$909$	−6.48667	−0.215149
$910$	0	0
$911$	−41.2066	−1.36523	−0.682617	−	0.730776i	$-0.739158\pi$
$911$	−41.2066	−1.36523	−0.682617	+	0.730776i	$0.739158\pi$
$912$	0	0
$913$	−4.88939	−0.161815
$914$	0	0
$915$	2.11418	0.0698925
$916$	0	0
$917$	0	0
$918$	0	0
$919$	27.7423	0.915134	0.457567	−	0.889175i	$-0.348721\pi$
$919$	27.7423	0.915134	0.457567	+	0.889175i	$0.348721\pi$
$920$	0	0
$921$	12.4218	0.409314
$922$	0	0
$923$	28.9528	0.952992
$924$	0	0
$925$	38.2784	1.25859
$926$	0	0
$927$	−2.47252	−0.0812083
$928$	0	0
$929$	−17.1817	−0.563712	−0.281856	−	0.959457i	$-0.590950\pi$
$929$	−17.1817	−0.563712	−0.281856	+	0.959457i	$0.590950\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.8505	−0.518924
$934$	0	0
$935$	−2.61785	−0.0856127
$936$	0	0
$937$	−1.04087	−0.0340039	−0.0170020	−	0.999855i	$-0.505412\pi$
$937$	−1.04087	−0.0340039	−0.0170020	+	0.999855i	$0.505412\pi$
$938$	0	0
$939$	−12.5069	−0.408147
$940$	0	0
$941$	−46.7458	−1.52387	−0.761934	−	0.647654i	$-0.775750\pi$
$941$	−46.7458	−1.52387	−0.761934	+	0.647654i	$0.775750\pi$
$942$	0	0
$943$	7.95534	0.259062
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0.931516	0.0302702	0.0151351	−	0.999885i	$-0.495182\pi$
$947$	0.931516	0.0302702	0.0151351	+	0.999885i	$0.495182\pi$
$948$	0	0
$949$	29.5315	0.958632
$950$	0	0
$951$	−34.2638	−1.11108
$952$	0	0
$953$	45.9904	1.48978	0.744888	−	0.667190i	$-0.232503\pi$
$953$	45.9904	1.48978	0.744888	+	0.667190i	$0.232503\pi$
$954$	0	0
$955$	−0.256868	−0.00831205
$956$	0	0
$957$	−4.90218	−0.158465
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−0.267842	−0.00864008
$962$	0	0
$963$	−2.17943	−0.0702312
$964$	0	0
$965$	22.3523	0.719547
$966$	0	0
$967$	29.4981	0.948594	0.474297	−	0.880365i	$-0.342702\pi$
$967$	29.4981	0.948594	0.474297	+	0.880365i	$0.342702\pi$
$968$	0	0
$969$	28.7080	0.922233
$970$	0	0
$971$	13.6407	0.437750	0.218875	−	0.975753i	$-0.429761\pi$
$971$	13.6407	0.437750	0.218875	+	0.975753i	$0.429761\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	12.0424	0.385665
$976$	0	0
$977$	2.14232	0.0685390	0.0342695	−	0.999413i	$-0.489090\pi$
$977$	2.14232	0.0685390	0.0342695	+	0.999413i	$0.489090\pi$
$978$	0	0
$979$	2.76942	0.0885109
$980$	0	0
$981$	1.74573	0.0557370
$982$	0	0
$983$	−36.2218	−1.15530	−0.577648	−	0.816286i	$-0.696029\pi$
$983$	−36.2218	−1.15530	−0.577648	+	0.816286i	$0.696029\pi$
$984$	0	0
$985$	−21.8782	−0.697099
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.55694	−0.208499
$990$	0	0
$991$	−27.3659	−0.869307	−0.434653	−	0.900598i	$-0.643129\pi$
$991$	−27.3659	−0.869307	−0.434653	+	0.900598i	$0.643129\pi$
$992$	0	0
$993$	−3.37152	−0.106992
$994$	0	0
$995$	−2.43771	−0.0772805
$996$	0	0
$997$	−26.3282	−0.833823	−0.416911	−	0.908947i	$-0.636887\pi$
$997$	−26.3282	−0.833823	−0.416911	+	0.908947i	$0.636887\pi$
$998$	0	0
$999$	53.8449	1.70358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bk.1.8		11
7.2	even	3		1288.2.q.d.921.4	yes	22
7.4	even	3		1288.2.q.d.737.4	✓	22
7.6	odd	2		9016.2.a.br.1.4		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.d.737.4	✓	22	7.4	even	3
1288.2.q.d.921.4	yes	22	7.2	even	3
9016.2.a.bk.1.8		11	1.1	even	1	trivial
9016.2.a.br.1.4		11	7.6	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:	\( N \)	\(=\)	\( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.58959 q^{3} -1.03689 q^{5} -0.473209 q^{9} +O(q^{10})\)	\(q+1.58959 q^{3} -1.03689 q^{5} -0.473209 q^{9} +0.432880 q^{11} -1.93020 q^{13} -1.64822 q^{15} +5.83238 q^{17} +3.09651 q^{19} -1.00000 q^{23} -3.92487 q^{25} -5.52097 q^{27} -7.12421 q^{29} +5.54366 q^{31} +0.688101 q^{33} -9.75279 q^{37} -3.06823 q^{39} -7.95534 q^{41} +6.55694 q^{43} +0.490663 q^{45} +6.54794 q^{47} +9.27108 q^{51} +14.2780 q^{53} -0.448847 q^{55} +4.92217 q^{57} -7.77986 q^{59} -1.28270 q^{61} +2.00140 q^{65} -3.35573 q^{67} -1.58959 q^{69} -14.9999 q^{71} -15.2997 q^{73} -6.23892 q^{75} +8.75304 q^{79} -7.35645 q^{81} -11.2950 q^{83} -6.04751 q^{85} -11.3246 q^{87} +6.39765 q^{89} +8.81214 q^{93} -3.21072 q^{95} -14.7223 q^{97} -0.204842 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 4 q^{3} - 3 q^{5} + 11 q^{9}+O(q^{10}) \) 11 * q - 4 * q^3 - 3 * q^5 + 11 * q^9	\( 11 q - 4 q^{3} - 3 q^{5} + 11 q^{9} + 13 q^{13} - 7 q^{17} - 8 q^{19} - 11 q^{23} + 6 q^{25} - 25 q^{27} - 3 q^{29} - 12 q^{31} + 2 q^{33} - q^{37} - 21 q^{39} - 12 q^{41} + 9 q^{43} - 19 q^{45} - 17 q^{47} + 19 q^{51} - 5 q^{53} - 21 q^{55} + 11 q^{57} - 33 q^{59} + 15 q^{61} - 9 q^{65} - 5 q^{67} + 4 q^{69} - 9 q^{71} - 5 q^{73} - 44 q^{75} + 11 q^{79} - 13 q^{81} - 51 q^{83} + 33 q^{85} - 4 q^{87} - 26 q^{89} + 6 q^{93} - 19 q^{95} - 21 q^{97} + 15 q^{99}+O(q^{100}) \) 11 * q - 4 * q^3 - 3 * q^5 + 11 * q^9 + 13 * q^13 - 7 * q^17 - 8 * q^19 - 11 * q^23 + 6 * q^25 - 25 * q^27 - 3 * q^29 - 12 * q^31 + 2 * q^33 - q^37 - 21 * q^39 - 12 * q^41 + 9 * q^43 - 19 * q^45 - 17 * q^47 + 19 * q^51 - 5 * q^53 - 21 * q^55 + 11 * q^57 - 33 * q^59 + 15 * q^61 - 9 * q^65 - 5 * q^67 + 4 * q^69 - 9 * q^71 - 5 * q^73 - 44 * q^75 + 11 * q^79 - 13 * q^81 - 51 * q^83 + 33 * q^85 - 4 * q^87 - 26 * q^89 + 6 * q^93 - 19 * q^95 - 21 * q^97 + 15 * q^99

Introduction

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