LMFDB - Embedded newform 9016.2.a.bk.1.10

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			1.10
Root			$-1.86046$ 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)

$f(q)$	$=$	$q+1.86046 q^{3} -3.75356 q^{5} +0.461318 q^{9} +O(q^{10})$	$q+1.86046 q^{3} -3.75356 q^{5} +0.461318 q^{9} +4.63115 q^{11} +3.79378 q^{13} -6.98336 q^{15} -2.49425 q^{17} -4.34759 q^{19} -1.00000 q^{23} +9.08922 q^{25} -4.72312 q^{27} +7.79584 q^{29} -9.67360 q^{31} +8.61607 q^{33} -2.76844 q^{37} +7.05818 q^{39} -2.22205 q^{41} -5.01807 q^{43} -1.73158 q^{45} +1.90487 q^{47} -4.64045 q^{51} -4.43238 q^{53} -17.3833 q^{55} -8.08852 q^{57} +4.41766 q^{59} +8.71798 q^{61} -14.2402 q^{65} -4.14452 q^{67} -1.86046 q^{69} +4.01111 q^{71} +14.5130 q^{73} +16.9101 q^{75} +16.7413 q^{79} -10.1711 q^{81} -16.3835 q^{83} +9.36230 q^{85} +14.5039 q^{87} -11.9874 q^{89} -17.9974 q^{93} +16.3189 q^{95} -3.38808 q^{97} +2.13643 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.86046	1.07414	0.537069	−	0.843538i	$-0.319531\pi$
$3$	1.86046	1.07414	0.537069	+	0.843538i	$0.319531\pi$
$4$	0	0
$5$	−3.75356	−1.67864	−0.839322	−	0.543635i	$-0.817048\pi$
$5$	−3.75356	−1.67864	−0.839322	+	0.543635i	$0.817048\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.461318	0.153773
$10$	0	0
$11$	4.63115	1.39634	0.698172	−	0.715931i	$-0.253997\pi$
$11$	4.63115	1.39634	0.698172	+	0.715931i	$0.253997\pi$
$12$	0	0
$13$	3.79378	1.05221	0.526103	−	0.850421i	$-0.323653\pi$
$13$	3.79378	1.05221	0.526103	+	0.850421i	$0.323653\pi$
$14$	0	0
$15$	−6.98336	−1.80309
$16$	0	0
$17$	−2.49425	−0.604944	−0.302472	−	0.953158i	$-0.597812\pi$
$17$	−2.49425	−0.604944	−0.302472	+	0.953158i	$0.597812\pi$
$18$	0	0
$19$	−4.34759	−0.997405	−0.498703	−	0.866773i	$-0.666190\pi$
$19$	−4.34759	−0.997405	−0.498703	+	0.866773i	$0.666190\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	9.08922	1.81784
$26$	0	0
$27$	−4.72312	−0.908965
$28$	0	0
$29$	7.79584	1.44765	0.723825	−	0.689983i	$-0.242382\pi$
$29$	7.79584	1.44765	0.723825	+	0.689983i	$0.242382\pi$
$30$	0	0
$31$	−9.67360	−1.73743	−0.868715	−	0.495312i	$-0.835054\pi$
$31$	−9.67360	−1.73743	−0.868715	+	0.495312i	$0.835054\pi$
$32$	0	0
$33$	8.61607	1.49987
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.76844	−0.455129	−0.227565	−	0.973763i	$-0.573076\pi$
$37$	−2.76844	−0.455129	−0.227565	+	0.973763i	$0.573076\pi$
$38$	0	0
$39$	7.05818	1.13021
$40$	0	0
$41$	−2.22205	−0.347026	−0.173513	−	0.984832i	$-0.555512\pi$
$41$	−2.22205	−0.347026	−0.173513	+	0.984832i	$0.555512\pi$
$42$	0	0
$43$	−5.01807	−0.765248	−0.382624	−	0.923904i	$-0.624980\pi$
$43$	−5.01807	−0.765248	−0.382624	+	0.923904i	$0.624980\pi$
$44$	0	0
$45$	−1.73158	−0.258129
$46$	0	0
$47$	1.90487	0.277854	0.138927	−	0.990303i	$-0.455635\pi$
$47$	1.90487	0.277854	0.138927	+	0.990303i	$0.455635\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.64045	−0.649793
$52$	0	0
$53$	−4.43238	−0.608834	−0.304417	−	0.952539i	$-0.598462\pi$
$53$	−4.43238	−0.608834	−0.304417	+	0.952539i	$0.598462\pi$
$54$	0	0
$55$	−17.3833	−2.34396
$56$	0	0
$57$	−8.08852	−1.07135
$58$	0	0
$59$	4.41766	0.575130	0.287565	−	0.957761i	$-0.407154\pi$
$59$	4.41766	0.575130	0.287565	+	0.957761i	$0.407154\pi$
$60$	0	0
$61$	8.71798	1.11622	0.558112	−	0.829766i	$-0.311526\pi$
$61$	8.71798	1.11622	0.558112	+	0.829766i	$0.311526\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−14.2402	−1.76628
$66$	0	0
$67$	−4.14452	−0.506334	−0.253167	−	0.967423i	$-0.581472\pi$
$67$	−4.14452	−0.506334	−0.253167	+	0.967423i	$0.581472\pi$
$68$	0	0
$69$	−1.86046	−0.223973
$70$	0	0
$71$	4.01111	0.476032	0.238016	−	0.971261i	$-0.423503\pi$
$71$	4.01111	0.476032	0.238016	+	0.971261i	$0.423503\pi$
$72$	0	0
$73$	14.5130	1.69862	0.849312	−	0.527892i	$-0.177017\pi$
$73$	14.5130	1.69862	0.849312	+	0.527892i	$0.177017\pi$
$74$	0	0
$75$	16.9101	1.95262
$76$	0	0
$77$	0	0
$78$	0	0
$79$	16.7413	1.88354	0.941770	−	0.336259i	$-0.109162\pi$
$79$	16.7413	1.88354	0.941770	+	0.336259i	$0.109162\pi$
$80$	0	0
$81$	−10.1711	−1.13013
$82$	0	0
$83$	−16.3835	−1.79832	−0.899159	−	0.437622i	$-0.855821\pi$
$83$	−16.3835	−1.79832	−0.899159	+	0.437622i	$0.855821\pi$
$84$	0	0
$85$	9.36230	1.01548
$86$	0	0
$87$	14.5039	1.55498
$88$	0	0
$89$	−11.9874	−1.27066	−0.635332	−	0.772239i	$-0.719137\pi$
$89$	−11.9874	−1.27066	−0.635332	+	0.772239i	$0.719137\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−17.9974	−1.86624
$94$	0	0
$95$	16.3189	1.67429
$96$	0	0
$97$	−3.38808	−0.344007	−0.172004	−	0.985096i	$-0.555024\pi$
$97$	−3.38808	−0.344007	−0.172004	+	0.985096i	$0.555024\pi$
$98$	0	0
$99$	2.13643	0.214719
$100$	0	0
$101$	−4.64163	−0.461860	−0.230930	−	0.972970i	$-0.574177\pi$
$101$	−4.64163	−0.461860	−0.230930	+	0.972970i	$0.574177\pi$
$102$	0	0
$103$	6.27288	0.618085	0.309043	−	0.951048i	$-0.399991\pi$
$103$	6.27288	0.618085	0.309043	+	0.951048i	$0.399991\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.72543	0.456825	0.228412	−	0.973565i	$-0.426647\pi$
$107$	4.72543	0.456825	0.228412	+	0.973565i	$0.426647\pi$
$108$	0	0
$109$	5.95025	0.569930	0.284965	−	0.958538i	$-0.408018\pi$
$109$	5.95025	0.569930	0.284965	+	0.958538i	$0.408018\pi$
$110$	0	0
$111$	−5.15058	−0.488871
$112$	0	0
$113$	−12.6836	−1.19318	−0.596588	−	0.802548i	$-0.703477\pi$
$113$	−12.6836	−1.19318	−0.596588	+	0.802548i	$0.703477\pi$
$114$	0	0
$115$	3.75356	0.350021
$116$	0	0
$117$	1.75014	0.161800
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.4475	0.949774
$122$	0	0
$123$	−4.13403	−0.372753
$124$	0	0
$125$	−15.3491	−1.37287
$126$	0	0
$127$	1.35271	0.120034	0.0600170	−	0.998197i	$-0.480885\pi$
$127$	1.35271	0.120034	0.0600170	+	0.998197i	$0.480885\pi$
$128$	0	0
$129$	−9.33592	−0.821982
$130$	0	0
$131$	−17.1933	−1.50218	−0.751092	−	0.660197i	$-0.770473\pi$
$131$	−17.1933	−1.50218	−0.751092	+	0.660197i	$0.770473\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	17.7285	1.52583
$136$	0	0
$137$	−7.24627	−0.619091	−0.309545	−	0.950885i	$-0.600177\pi$
$137$	−7.24627	−0.619091	−0.309545	+	0.950885i	$0.600177\pi$
$138$	0	0
$139$	−19.2113	−1.62948	−0.814740	−	0.579826i	$-0.803120\pi$
$139$	−19.2113	−1.62948	−0.814740	+	0.579826i	$0.803120\pi$
$140$	0	0
$141$	3.54394	0.298454
$142$	0	0
$143$	17.5696	1.46924
$144$	0	0
$145$	−29.2621	−2.43009
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.5437	−1.27339	−0.636697	−	0.771114i	$-0.719700\pi$
$149$	−15.5437	−1.27339	−0.636697	+	0.771114i	$0.719700\pi$
$150$	0	0
$151$	−19.5890	−1.59413	−0.797064	−	0.603895i	$-0.793615\pi$
$151$	−19.5890	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	0	0
$153$	−1.15064	−0.0930237
$154$	0	0
$155$	36.3104	2.91653
$156$	0	0
$157$	−23.1552	−1.84799	−0.923995	−	0.382405i	$-0.875096\pi$
$157$	−23.1552	−1.84799	−0.923995	+	0.382405i	$0.875096\pi$
$158$	0	0
$159$	−8.24627	−0.653972
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0110	−0.940770	−0.470385	−	0.882461i	$-0.655885\pi$
$163$	−12.0110	−0.940770	−0.470385	+	0.882461i	$0.655885\pi$
$164$	0	0
$165$	−32.3409	−2.51774
$166$	0	0
$167$	14.2649	1.10385	0.551925	−	0.833894i	$-0.313893\pi$
$167$	14.2649	1.10385	0.551925	+	0.833894i	$0.313893\pi$
$168$	0	0
$169$	1.39277	0.107136
$170$	0	0
$171$	−2.00562	−0.153374
$172$	0	0
$173$	18.5970	1.41390	0.706951	−	0.707263i	$-0.250070\pi$
$173$	18.5970	1.41390	0.706951	+	0.707263i	$0.250070\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.21888	0.617769
$178$	0	0
$179$	1.20203	0.0898440	0.0449220	−	0.998990i	$-0.485696\pi$
$179$	1.20203	0.0898440	0.0449220	+	0.998990i	$0.485696\pi$
$180$	0	0
$181$	−3.06132	−0.227546	−0.113773	−	0.993507i	$-0.536294\pi$
$181$	−3.06132	−0.227546	−0.113773	+	0.993507i	$0.536294\pi$
$182$	0	0
$183$	16.2195	1.19898
$184$	0	0
$185$	10.3915	0.763999
$186$	0	0
$187$	−11.5512	−0.844709
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−10.1958	−0.737743	−0.368872	−	0.929480i	$-0.620256\pi$
$191$	−10.1958	−0.737743	−0.368872	+	0.929480i	$0.620256\pi$
$192$	0	0
$193$	−10.6408	−0.765941	−0.382970	−	0.923761i	$-0.625099\pi$
$193$	−10.6408	−0.765941	−0.382970	+	0.923761i	$0.625099\pi$
$194$	0	0
$195$	−26.4933	−1.89723
$196$	0	0
$197$	2.79402	0.199066	0.0995329	−	0.995034i	$-0.468265\pi$
$197$	2.79402	0.199066	0.0995329	+	0.995034i	$0.468265\pi$
$198$	0	0
$199$	5.28003	0.374291	0.187146	−	0.982332i	$-0.440076\pi$
$199$	5.28003	0.374291	0.187146	+	0.982332i	$0.440076\pi$
$200$	0	0
$201$	−7.71073	−0.543873
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.34059	0.582532
$206$	0	0
$207$	−0.461318	−0.0320638
$208$	0	0
$209$	−20.1343	−1.39272
$210$	0	0
$211$	10.9218	0.751888	0.375944	−	0.926642i	$-0.377319\pi$
$211$	10.9218	0.751888	0.375944	+	0.926642i	$0.377319\pi$
$212$	0	0
$213$	7.46252	0.511324
$214$	0	0
$215$	18.8356	1.28458
$216$	0	0
$217$	0	0
$218$	0	0
$219$	27.0010	1.82456
$220$	0	0
$221$	−9.46262	−0.636525
$222$	0	0
$223$	12.8994	0.863810	0.431905	−	0.901919i	$-0.357842\pi$
$223$	12.8994	0.863810	0.431905	+	0.901919i	$0.357842\pi$
$224$	0	0
$225$	4.19302	0.279535
$226$	0	0
$227$	−24.7303	−1.64141	−0.820704	−	0.571354i	$-0.806418\pi$
$227$	−24.7303	−1.64141	−0.820704	+	0.571354i	$0.806418\pi$
$228$	0	0
$229$	18.7160	1.23679	0.618393	−	0.785869i	$-0.287784\pi$
$229$	18.7160	1.23679	0.618393	+	0.785869i	$0.287784\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.45965	−0.357673	−0.178837	−	0.983879i	$-0.557233\pi$
$233$	−5.45965	−0.357673	−0.178837	+	0.983879i	$0.557233\pi$
$234$	0	0
$235$	−7.15005	−0.466418
$236$	0	0
$237$	31.1465	2.02318
$238$	0	0
$239$	−1.82054	−0.117761	−0.0588806	−	0.998265i	$-0.518753\pi$
$239$	−1.82054	−0.117761	−0.0588806	+	0.998265i	$0.518753\pi$
$240$	0	0
$241$	−23.7097	−1.52728	−0.763639	−	0.645644i	$-0.776589\pi$
$241$	−23.7097	−1.52728	−0.763639	+	0.645644i	$0.776589\pi$
$242$	0	0
$243$	−4.75365	−0.304947
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−16.4938	−1.04947
$248$	0	0
$249$	−30.4808	−1.93164
$250$	0	0
$251$	4.56984	0.288446	0.144223	−	0.989545i	$-0.453932\pi$
$251$	4.56984	0.288446	0.144223	+	0.989545i	$0.453932\pi$
$252$	0	0
$253$	−4.63115	−0.291158
$254$	0	0
$255$	17.4182	1.09077
$256$	0	0
$257$	22.6537	1.41310	0.706549	−	0.707664i	$-0.250251\pi$
$257$	22.6537	1.41310	0.706549	+	0.707664i	$0.250251\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	3.59636	0.222609
$262$	0	0
$263$	13.1079	0.808265	0.404133	−	0.914700i	$-0.367573\pi$
$263$	13.1079	0.808265	0.404133	+	0.914700i	$0.367573\pi$
$264$	0	0
$265$	16.6372	1.02201
$266$	0	0
$267$	−22.3021	−1.36487
$268$	0	0
$269$	3.30121	0.201278	0.100639	−	0.994923i	$-0.467911\pi$
$269$	3.30121	0.201278	0.100639	+	0.994923i	$0.467911\pi$
$270$	0	0
$271$	−10.6622	−0.647684	−0.323842	−	0.946111i	$-0.604975\pi$
$271$	−10.6622	−0.647684	−0.323842	+	0.946111i	$0.604975\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	42.0935	2.53833
$276$	0	0
$277$	13.3989	0.805064	0.402532	−	0.915406i	$-0.368130\pi$
$277$	13.3989	0.805064	0.402532	+	0.915406i	$0.368130\pi$
$278$	0	0
$279$	−4.46260	−0.267169
$280$	0	0
$281$	−0.791612	−0.0472236	−0.0236118	−	0.999721i	$-0.507517\pi$
$281$	−0.791612	−0.0472236	−0.0236118	+	0.999721i	$0.507517\pi$
$282$	0	0
$283$	5.65465	0.336134	0.168067	−	0.985776i	$-0.446247\pi$
$283$	5.65465	0.336134	0.168067	+	0.985776i	$0.446247\pi$
$284$	0	0
$285$	30.3608	1.79842
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−10.7787	−0.634043
$290$	0	0
$291$	−6.30339	−0.369511
$292$	0	0
$293$	−7.33432	−0.428476	−0.214238	−	0.976782i	$-0.568727\pi$
$293$	−7.33432	−0.428476	−0.214238	+	0.976782i	$0.568727\pi$
$294$	0	0
$295$	−16.5819	−0.965438
$296$	0	0
$297$	−21.8735	−1.26923
$298$	0	0
$299$	−3.79378	−0.219400
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.63558	−0.496101
$304$	0	0
$305$	−32.7235	−1.87374
$306$	0	0
$307$	−22.8249	−1.30269	−0.651343	−	0.758784i	$-0.725794\pi$
$307$	−22.8249	−1.30269	−0.651343	+	0.758784i	$0.725794\pi$
$308$	0	0
$309$	11.6705	0.663909
$310$	0	0
$311$	−16.0316	−0.909066	−0.454533	−	0.890730i	$-0.650194\pi$
$311$	−16.0316	−0.909066	−0.454533	+	0.890730i	$0.650194\pi$
$312$	0	0
$313$	−15.0991	−0.853452	−0.426726	−	0.904381i	$-0.640333\pi$
$313$	−15.0991	−0.853452	−0.426726	+	0.904381i	$0.640333\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.4597	−0.755971	−0.377985	−	0.925812i	$-0.623383\pi$
$317$	−13.4597	−0.755971	−0.377985	+	0.925812i	$0.623383\pi$
$318$	0	0
$319$	36.1037	2.02142
$320$	0	0
$321$	8.79148	0.490693
$322$	0	0
$323$	10.8440	0.603374
$324$	0	0
$325$	34.4825	1.91274
$326$	0	0
$327$	11.0702	0.612184
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−25.1795	−1.38399	−0.691996	−	0.721901i	$-0.743268\pi$
$331$	−25.1795	−1.38399	−0.691996	+	0.721901i	$0.743268\pi$
$332$	0	0
$333$	−1.27713	−0.0699864
$334$	0	0
$335$	15.5567	0.849955
$336$	0	0
$337$	24.3045	1.32395	0.661976	−	0.749525i	$-0.269718\pi$
$337$	24.3045	1.32395	0.661976	+	0.749525i	$0.269718\pi$
$338$	0	0
$339$	−23.5974	−1.28164
$340$	0	0
$341$	−44.7999	−2.42605
$342$	0	0
$343$	0	0
$344$	0	0
$345$	6.98336	0.375971
$346$	0	0
$347$	−18.4368	−0.989739	−0.494869	−	0.868967i	$-0.664784\pi$
$347$	−18.4368	−0.989739	−0.494869	+	0.868967i	$0.664784\pi$
$348$	0	0
$349$	−30.2982	−1.62182	−0.810912	−	0.585169i	$-0.801028\pi$
$349$	−30.2982	−1.62182	−0.810912	+	0.585169i	$0.801028\pi$
$350$	0	0
$351$	−17.9185	−0.956418
$352$	0	0
$353$	7.21754	0.384151	0.192075	−	0.981380i	$-0.438478\pi$
$353$	7.21754	0.384151	0.192075	+	0.981380i	$0.438478\pi$
$354$	0	0
$355$	−15.0560	−0.799087
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.3413	−1.97080	−0.985400	−	0.170253i	$-0.945542\pi$
$359$	−37.3413	−1.97080	−0.985400	+	0.170253i	$0.945542\pi$
$360$	0	0
$361$	−0.0984806	−0.00518319
$362$	0	0
$363$	19.4372	1.02019
$364$	0	0
$365$	−54.4756	−2.85138
$366$	0	0
$367$	−12.8555	−0.671053	−0.335527	−	0.942031i	$-0.608914\pi$
$367$	−12.8555	−0.671053	−0.335527	+	0.942031i	$0.608914\pi$
$368$	0	0
$369$	−1.02507	−0.0533630
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.148138	−0.00767030	−0.00383515	−	0.999993i	$-0.501221\pi$
$373$	−0.148138	−0.00767030	−0.00383515	+	0.999993i	$0.501221\pi$
$374$	0	0
$375$	−28.5565	−1.47465
$376$	0	0
$377$	29.5757	1.52323
$378$	0	0
$379$	27.2252	1.39846	0.699232	−	0.714895i	$-0.253525\pi$
$379$	27.2252	1.39846	0.699232	+	0.714895i	$0.253525\pi$
$380$	0	0
$381$	2.51667	0.128933
$382$	0	0
$383$	−32.1177	−1.64114	−0.820568	−	0.571548i	$-0.806343\pi$
$383$	−32.1177	−1.64114	−0.820568	+	0.571548i	$0.806343\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.31492	−0.117674
$388$	0	0
$389$	−26.8167	−1.35966	−0.679831	−	0.733369i	$-0.737947\pi$
$389$	−26.8167	−1.35966	−0.679831	+	0.733369i	$0.737947\pi$
$390$	0	0
$391$	2.49425	0.126139
$392$	0	0
$393$	−31.9875	−1.61355
$394$	0	0
$395$	−62.8393	−3.16179
$396$	0	0
$397$	5.08117	0.255017	0.127508	−	0.991837i	$-0.459302\pi$
$397$	5.08117	0.255017	0.127508	+	0.991837i	$0.459302\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.68353	0.134009	0.0670046	−	0.997753i	$-0.478656\pi$
$401$	2.68353	0.134009	0.0670046	+	0.997753i	$0.478656\pi$
$402$	0	0
$403$	−36.6995	−1.82813
$404$	0	0
$405$	38.1780	1.89708
$406$	0	0
$407$	−12.8211	−0.635516
$408$	0	0
$409$	30.4404	1.50518	0.752591	−	0.658489i	$-0.228804\pi$
$409$	30.4404	1.50518	0.752591	+	0.658489i	$0.228804\pi$
$410$	0	0
$411$	−13.4814	−0.664989
$412$	0	0
$413$	0	0
$414$	0	0
$415$	61.4963	3.01873
$416$	0	0
$417$	−35.7419	−1.75029
$418$	0	0
$419$	18.7299	0.915017	0.457508	−	0.889205i	$-0.348742\pi$
$419$	18.7299	0.915017	0.457508	+	0.889205i	$0.348742\pi$
$420$	0	0
$421$	−8.51512	−0.415002	−0.207501	−	0.978235i	$-0.566533\pi$
$421$	−8.51512	−0.415002	−0.207501	+	0.978235i	$0.566533\pi$
$422$	0	0
$423$	0.878751	0.0427264
$424$	0	0
$425$	−22.6707	−1.09969
$426$	0	0
$427$	0	0
$428$	0	0
$429$	32.6875	1.57817
$430$	0	0
$431$	−8.38414	−0.403850	−0.201925	−	0.979401i	$-0.564720\pi$
$431$	−8.38414	−0.403850	−0.201925	+	0.979401i	$0.564720\pi$
$432$	0	0
$433$	−23.6401	−1.13607	−0.568035	−	0.823005i	$-0.692296\pi$
$433$	−23.6401	−1.13607	−0.568035	+	0.823005i	$0.692296\pi$
$434$	0	0
$435$	−54.4411	−2.61025
$436$	0	0
$437$	4.34759	0.207973
$438$	0	0
$439$	−21.0114	−1.00282	−0.501409	−	0.865210i	$-0.667185\pi$
$439$	−21.0114	−1.00282	−0.501409	+	0.865210i	$0.667185\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.53129	0.0727536	0.0363768	−	0.999338i	$-0.488418\pi$
$443$	1.53129	0.0727536	0.0363768	+	0.999338i	$0.488418\pi$
$444$	0	0
$445$	44.9955	2.13299
$446$	0	0
$447$	−28.9185	−1.36780
$448$	0	0
$449$	21.2897	1.00472	0.502360	−	0.864658i	$-0.332465\pi$
$449$	21.2897	1.00472	0.502360	+	0.864658i	$0.332465\pi$
$450$	0	0
$451$	−10.2906	−0.484567
$452$	0	0
$453$	−36.4445	−1.71231
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.6723	1.71546	0.857729	−	0.514102i	$-0.171875\pi$
$457$	36.6723	1.71546	0.857729	+	0.514102i	$0.171875\pi$
$458$	0	0
$459$	11.7806	0.549873
$460$	0	0
$461$	−19.0746	−0.888392	−0.444196	−	0.895930i	$-0.646511\pi$
$461$	−19.0746	−0.888392	−0.444196	+	0.895930i	$0.646511\pi$
$462$	0	0
$463$	35.4714	1.64850	0.824248	−	0.566229i	$-0.191598\pi$
$463$	35.4714	1.64850	0.824248	+	0.566229i	$0.191598\pi$
$464$	0	0
$465$	67.5542	3.13275
$466$	0	0
$467$	−4.57801	−0.211845	−0.105922	−	0.994374i	$-0.533780\pi$
$467$	−4.57801	−0.211845	−0.105922	+	0.994374i	$0.533780\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−43.0794	−1.98500
$472$	0	0
$473$	−23.2394	−1.06855
$474$	0	0
$475$	−39.5162	−1.81313
$476$	0	0
$477$	−2.04473	−0.0936220
$478$	0	0
$479$	−13.3952	−0.612042	−0.306021	−	0.952025i	$-0.598998\pi$
$479$	−13.3952	−0.612042	−0.306021	+	0.952025i	$0.598998\pi$
$480$	0	0
$481$	−10.5029	−0.478889
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.7174	0.577466
$486$	0	0
$487$	17.6903	0.801625	0.400813	−	0.916160i	$-0.368728\pi$
$487$	17.6903	0.801625	0.400813	+	0.916160i	$0.368728\pi$
$488$	0	0
$489$	−22.3459	−1.01052
$490$	0	0
$491$	−20.9660	−0.946182	−0.473091	−	0.881013i	$-0.656862\pi$
$491$	−20.9660	−0.946182	−0.473091	+	0.881013i	$0.656862\pi$
$492$	0	0
$493$	−19.4447	−0.875747
$494$	0	0
$495$	−8.01922	−0.360437
$496$	0	0
$497$	0	0
$498$	0	0
$499$	34.6067	1.54921	0.774604	−	0.632446i	$-0.217949\pi$
$499$	34.6067	1.54921	0.774604	+	0.632446i	$0.217949\pi$
$500$	0	0
$501$	26.5393	1.18569
$502$	0	0
$503$	−22.2215	−0.990808	−0.495404	−	0.868663i	$-0.664980\pi$
$503$	−22.2215	−0.990808	−0.495404	+	0.868663i	$0.664980\pi$
$504$	0	0
$505$	17.4227	0.775298
$506$	0	0
$507$	2.59119	0.115079
$508$	0	0
$509$	−26.6175	−1.17980	−0.589900	−	0.807476i	$-0.700833\pi$
$509$	−26.6175	−1.17980	−0.589900	+	0.807476i	$0.700833\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	20.5342	0.906606
$514$	0	0
$515$	−23.5456	−1.03754
$516$	0	0
$517$	8.82174	0.387980
$518$	0	0
$519$	34.5989	1.51873
$520$	0	0
$521$	28.8901	1.26570	0.632850	−	0.774275i	$-0.281885\pi$
$521$	28.8901	1.26570	0.632850	+	0.774275i	$0.281885\pi$
$522$	0	0
$523$	−0.774406	−0.0338624	−0.0169312	−	0.999857i	$-0.505390\pi$
$523$	−0.774406	−0.0338624	−0.0169312	+	0.999857i	$0.505390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.1283	1.05105
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.03794	0.0884393
$532$	0	0
$533$	−8.42996	−0.365142
$534$	0	0
$535$	−17.7372	−0.766845
$536$	0	0
$537$	2.23633	0.0965049
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.9512	−1.02974	−0.514872	−	0.857267i	$-0.672161\pi$
$541$	−23.9512	−1.02974	−0.514872	+	0.857267i	$0.672161\pi$
$542$	0	0
$543$	−5.69548	−0.244416
$544$	0	0
$545$	−22.3346	−0.956710
$546$	0	0
$547$	15.3138	0.654771	0.327386	−	0.944891i	$-0.393832\pi$
$547$	15.3138	0.654771	0.327386	+	0.944891i	$0.393832\pi$
$548$	0	0
$549$	4.02176	0.171645
$550$	0	0
$551$	−33.8931	−1.44389
$552$	0	0
$553$	0	0
$554$	0	0
$555$	19.3330	0.820641
$556$	0	0
$557$	−25.9256	−1.09850	−0.549251	−	0.835657i	$-0.685087\pi$
$557$	−25.9256	−1.09850	−0.549251	+	0.835657i	$0.685087\pi$
$558$	0	0
$559$	−19.0374	−0.805198
$560$	0	0
$561$	−21.4906	−0.907334
$562$	0	0
$563$	−19.2984	−0.813329	−0.406665	−	0.913578i	$-0.633308\pi$
$563$	−19.2984	−0.813329	−0.406665	+	0.913578i	$0.633308\pi$
$564$	0	0
$565$	47.6088	2.00292
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.2488	−0.974641	−0.487321	−	0.873223i	$-0.662026\pi$
$569$	−23.2488	−0.974641	−0.487321	+	0.873223i	$0.662026\pi$
$570$	0	0
$571$	29.7010	1.24295	0.621474	−	0.783435i	$-0.286534\pi$
$571$	29.7010	1.24295	0.621474	+	0.783435i	$0.286534\pi$
$572$	0	0
$573$	−18.9689	−0.792438
$574$	0	0
$575$	−9.08922	−0.379047
$576$	0	0
$577$	−36.1387	−1.50447	−0.752236	−	0.658894i	$-0.771025\pi$
$577$	−36.1387	−1.50447	−0.752236	+	0.658894i	$0.771025\pi$
$578$	0	0
$579$	−19.7968	−0.822726
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−20.5270	−0.850141
$584$	0	0
$585$	−6.56925	−0.271605
$586$	0	0
$587$	15.1920	0.627042	0.313521	−	0.949581i	$-0.398491\pi$
$587$	15.1920	0.627042	0.313521	+	0.949581i	$0.398491\pi$
$588$	0	0
$589$	42.0568	1.73292
$590$	0	0
$591$	5.19817	0.213824
$592$	0	0
$593$	−36.8191	−1.51198	−0.755990	−	0.654583i	$-0.772844\pi$
$593$	−36.8191	−1.51198	−0.755990	+	0.654583i	$0.772844\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9.82329	0.402040
$598$	0	0
$599$	23.1135	0.944392	0.472196	−	0.881494i	$-0.343462\pi$
$599$	23.1135	0.944392	0.472196	+	0.881494i	$0.343462\pi$
$600$	0	0
$601$	−41.1975	−1.68048	−0.840240	−	0.542215i	$-0.817586\pi$
$601$	−41.1975	−1.68048	−0.840240	+	0.542215i	$0.817586\pi$
$602$	0	0
$603$	−1.91194	−0.0778603
$604$	0	0
$605$	−39.2154	−1.59433
$606$	0	0
$607$	18.1137	0.735211	0.367606	−	0.929982i	$-0.380178\pi$
$607$	18.1137	0.735211	0.367606	+	0.929982i	$0.380178\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.22667	0.292360
$612$	0	0
$613$	−25.1292	−1.01496	−0.507479	−	0.861664i	$-0.669423\pi$
$613$	−25.1292	−1.01496	−0.507479	+	0.861664i	$0.669423\pi$
$614$	0	0
$615$	15.5173	0.625720
$616$	0	0
$617$	−3.69215	−0.148640	−0.0743202	−	0.997234i	$-0.523679\pi$
$617$	−3.69215	−0.148640	−0.0743202	+	0.997234i	$0.523679\pi$
$618$	0	0
$619$	30.2579	1.21617	0.608084	−	0.793873i	$-0.291938\pi$
$619$	30.2579	1.21617	0.608084	+	0.793873i	$0.291938\pi$
$620$	0	0
$621$	4.72312	0.189532
$622$	0	0
$623$	0	0
$624$	0	0
$625$	12.1678	0.486712
$626$	0	0
$627$	−37.4591	−1.49597
$628$	0	0
$629$	6.90517	0.275327
$630$	0	0
$631$	−1.91255	−0.0761375	−0.0380688	−	0.999275i	$-0.512121\pi$
$631$	−1.91255	−0.0761375	−0.0380688	+	0.999275i	$0.512121\pi$
$632$	0	0
$633$	20.3196	0.807631
$634$	0	0
$635$	−5.07749	−0.201494
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.85040	0.0732006
$640$	0	0
$641$	−10.8645	−0.429123	−0.214561	−	0.976711i	$-0.568832\pi$
$641$	−10.8645	−0.429123	−0.214561	+	0.976711i	$0.568832\pi$
$642$	0	0
$643$	2.10169	0.0828824	0.0414412	−	0.999141i	$-0.486805\pi$
$643$	2.10169	0.0828824	0.0414412	+	0.999141i	$0.486805\pi$
$644$	0	0
$645$	35.0430	1.37982
$646$	0	0
$647$	9.85197	0.387321	0.193660	−	0.981069i	$-0.437964\pi$
$647$	9.85197	0.387321	0.193660	+	0.981069i	$0.437964\pi$
$648$	0	0
$649$	20.4588	0.803079
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.8311	0.502119	0.251060	−	0.967972i	$-0.419221\pi$
$653$	12.8311	0.502119	0.251060	+	0.967972i	$0.419221\pi$
$654$	0	0
$655$	64.5361	2.52163
$656$	0	0
$657$	6.69513	0.261202
$658$	0	0
$659$	−39.9798	−1.55739	−0.778697	−	0.627401i	$-0.784119\pi$
$659$	−39.9798	−1.55739	−0.778697	+	0.627401i	$0.784119\pi$
$660$	0	0
$661$	35.5994	1.38465	0.692327	−	0.721584i	$-0.256585\pi$
$661$	35.5994	1.38465	0.692327	+	0.721584i	$0.256585\pi$
$662$	0	0
$663$	−17.6048	−0.683716
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.79584	−0.301856
$668$	0	0
$669$	23.9989	0.927851
$670$	0	0
$671$	40.3743	1.55863
$672$	0	0
$673$	19.8845	0.766490	0.383245	−	0.923647i	$-0.374807\pi$
$673$	19.8845	0.766490	0.383245	+	0.923647i	$0.374807\pi$
$674$	0	0
$675$	−42.9295	−1.65236
$676$	0	0
$677$	23.6621	0.909410	0.454705	−	0.890642i	$-0.349745\pi$
$677$	23.6621	0.909410	0.454705	+	0.890642i	$0.349745\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−46.0098	−1.76310
$682$	0	0
$683$	−0.581705	−0.0222583	−0.0111292	−	0.999938i	$-0.503543\pi$
$683$	−0.581705	−0.0222583	−0.0111292	+	0.999938i	$0.503543\pi$
$684$	0	0
$685$	27.1993	1.03923
$686$	0	0
$687$	34.8204	1.32848
$688$	0	0
$689$	−16.8155	−0.640618
$690$	0	0
$691$	−3.65831	−0.139169	−0.0695844	−	0.997576i	$-0.522167\pi$
$691$	−3.65831	−0.139169	−0.0695844	+	0.997576i	$0.522167\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	72.1107	2.73532
$696$	0	0
$697$	5.54233	0.209931
$698$	0	0
$699$	−10.1575	−0.384190
$700$	0	0
$701$	36.1036	1.36361	0.681807	−	0.731532i	$-0.261195\pi$
$701$	36.1036	1.36361	0.681807	+	0.731532i	$0.261195\pi$
$702$	0	0
$703$	12.0360	0.453948
$704$	0	0
$705$	−13.3024	−0.500997
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.7822	0.893159	0.446579	−	0.894744i	$-0.352642\pi$
$709$	23.7822	0.893159	0.446579	+	0.894744i	$0.352642\pi$
$710$	0	0
$711$	7.72304	0.289637
$712$	0	0
$713$	9.67360	0.362279
$714$	0	0
$715$	−65.9484	−2.46633
$716$	0	0
$717$	−3.38705	−0.126492
$718$	0	0
$719$	−2.79102	−0.104087	−0.0520437	−	0.998645i	$-0.516574\pi$
$719$	−2.79102	−0.104087	−0.0520437	+	0.998645i	$0.516574\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−44.1110	−1.64051
$724$	0	0
$725$	70.8580	2.63160
$726$	0	0
$727$	−2.01642	−0.0747850	−0.0373925	−	0.999301i	$-0.511905\pi$
$727$	−2.01642	−0.0747850	−0.0373925	+	0.999301i	$0.511905\pi$
$728$	0	0
$729$	21.6694	0.802571
$730$	0	0
$731$	12.5163	0.462932
$732$	0	0
$733$	37.5084	1.38541	0.692703	−	0.721223i	$-0.256420\pi$
$733$	37.5084	1.38541	0.692703	+	0.721223i	$0.256420\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.1939	−0.707016
$738$	0	0
$739$	−15.7371	−0.578898	−0.289449	−	0.957193i	$-0.593472\pi$
$739$	−15.7371	−0.578898	−0.289449	+	0.957193i	$0.593472\pi$
$740$	0	0
$741$	−30.6861	−1.12728
$742$	0	0
$743$	−38.1243	−1.39865	−0.699323	−	0.714806i	$-0.746515\pi$
$743$	−38.1243	−1.39865	−0.699323	+	0.714806i	$0.746515\pi$
$744$	0	0
$745$	58.3444	2.13757
$746$	0	0
$747$	−7.55798	−0.276532
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−19.0751	−0.696060	−0.348030	−	0.937483i	$-0.613149\pi$
$751$	−19.0751	−0.696060	−0.348030	+	0.937483i	$0.613149\pi$
$752$	0	0
$753$	8.50202	0.309831
$754$	0	0
$755$	73.5284	2.67597
$756$	0	0
$757$	14.7021	0.534359	0.267179	−	0.963647i	$-0.413908\pi$
$757$	14.7021	0.534359	0.267179	+	0.963647i	$0.413908\pi$
$758$	0	0
$759$	−8.61607	−0.312744
$760$	0	0
$761$	32.0440	1.16159	0.580797	−	0.814048i	$-0.302741\pi$
$761$	32.0440	1.16159	0.580797	+	0.814048i	$0.302741\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	4.31900	0.156154
$766$	0	0
$767$	16.7596	0.605155
$768$	0	0
$769$	−47.6676	−1.71894	−0.859468	−	0.511189i	$-0.829205\pi$
$769$	−47.6676	−1.71894	−0.859468	+	0.511189i	$0.829205\pi$
$770$	0	0
$771$	42.1463	1.51786
$772$	0	0
$773$	9.50198	0.341762	0.170881	−	0.985292i	$-0.445339\pi$
$773$	9.50198	0.341762	0.170881	+	0.985292i	$0.445339\pi$
$774$	0	0
$775$	−87.9255	−3.15838
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.66055	0.346125
$780$	0	0
$781$	18.5761	0.664704
$782$	0	0
$783$	−36.8207	−1.31586
$784$	0	0
$785$	86.9146	3.10212
$786$	0	0
$787$	40.6798	1.45008	0.725039	−	0.688708i	$-0.241822\pi$
$787$	40.6798	1.45008	0.725039	+	0.688708i	$0.241822\pi$
$788$	0	0
$789$	24.3867	0.868188
$790$	0	0
$791$	0	0
$792$	0	0
$793$	33.0741	1.17450
$794$	0	0
$795$	30.9529	1.09779
$796$	0	0
$797$	−34.8515	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$797$	−34.8515	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$798$	0	0
$799$	−4.75122	−0.168086
$800$	0	0
$801$	−5.53001	−0.195393
$802$	0	0
$803$	67.2120	2.37186
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6.14177	0.216201
$808$	0	0
$809$	27.7741	0.976486	0.488243	−	0.872708i	$-0.337638\pi$
$809$	27.7741	0.976486	0.488243	+	0.872708i	$0.337638\pi$
$810$	0	0
$811$	46.9519	1.64870	0.824352	−	0.566077i	$-0.191540\pi$
$811$	46.9519	1.64870	0.824352	+	0.566077i	$0.191540\pi$
$812$	0	0
$813$	−19.8367	−0.695702
$814$	0	0
$815$	45.0838	1.57922
$816$	0	0
$817$	21.8165	0.763263
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.95554	0.138049	0.0690246	−	0.997615i	$-0.478011\pi$
$821$	3.95554	0.138049	0.0690246	+	0.997615i	$0.478011\pi$
$822$	0	0
$823$	−12.7875	−0.445745	−0.222872	−	0.974848i	$-0.571543\pi$
$823$	−12.7875	−0.445745	−0.222872	+	0.974848i	$0.571543\pi$
$824$	0	0
$825$	78.3133	2.72652
$826$	0	0
$827$	4.28811	0.149112	0.0745560	−	0.997217i	$-0.476246\pi$
$827$	4.28811	0.149112	0.0745560	+	0.997217i	$0.476246\pi$
$828$	0	0
$829$	20.9583	0.727912	0.363956	−	0.931416i	$-0.381426\pi$
$829$	20.9583	0.727912	0.363956	+	0.931416i	$0.381426\pi$
$830$	0	0
$831$	24.9282	0.864750
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−53.5441	−1.85297
$836$	0	0
$837$	45.6896	1.57926
$838$	0	0
$839$	−0.525841	−0.0181541	−0.00907703	−	0.999959i	$-0.502889\pi$
$839$	−0.525841	−0.0181541	−0.00907703	+	0.999959i	$0.502889\pi$
$840$	0	0
$841$	31.7750	1.09569
$842$	0	0
$843$	−1.47276	−0.0507247
$844$	0	0
$845$	−5.22784	−0.179843
$846$	0	0
$847$	0	0
$848$	0	0
$849$	10.5203	0.361054
$850$	0	0
$851$	2.76844	0.0949010
$852$	0	0
$853$	−28.5072	−0.976068	−0.488034	−	0.872825i	$-0.662286\pi$
$853$	−28.5072	−0.976068	−0.488034	+	0.872825i	$0.662286\pi$
$854$	0	0
$855$	7.52822	0.257460
$856$	0	0
$857$	−30.8207	−1.05281	−0.526407	−	0.850233i	$-0.676461\pi$
$857$	−30.8207	−1.05281	−0.526407	+	0.850233i	$0.676461\pi$
$858$	0	0
$859$	21.1246	0.720761	0.360380	−	0.932805i	$-0.382647\pi$
$859$	21.1246	0.720761	0.360380	+	0.932805i	$0.382647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	27.1486	0.924149	0.462074	−	0.886841i	$-0.347105\pi$
$863$	27.1486	0.924149	0.462074	+	0.886841i	$0.347105\pi$
$864$	0	0
$865$	−69.8048	−2.37344
$866$	0	0
$867$	−20.0534	−0.681050
$868$	0	0
$869$	77.5312	2.63007
$870$	0	0
$871$	−15.7234	−0.532768
$872$	0	0
$873$	−1.56298	−0.0528989
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.6210	1.00023	0.500115	−	0.865959i	$-0.333291\pi$
$877$	29.6210	1.00023	0.500115	+	0.865959i	$0.333291\pi$
$878$	0	0
$879$	−13.6452	−0.460242
$880$	0	0
$881$	8.47152	0.285413	0.142706	−	0.989765i	$-0.454420\pi$
$881$	8.47152	0.285413	0.142706	+	0.989765i	$0.454420\pi$
$882$	0	0
$883$	0.267843	0.00901362	0.00450681	−	0.999990i	$-0.498565\pi$
$883$	0.267843	0.00901362	0.00450681	+	0.999990i	$0.498565\pi$
$884$	0	0
$885$	−30.8501	−1.03701
$886$	0	0
$887$	21.5183	0.722515	0.361258	−	0.932466i	$-0.382348\pi$
$887$	21.5183	0.722515	0.361258	+	0.932466i	$0.382348\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−47.1040	−1.57804
$892$	0	0
$893$	−8.28160	−0.277133
$894$	0	0
$895$	−4.51190	−0.150816
$896$	0	0
$897$	−7.05818	−0.235666
$898$	0	0
$899$	−75.4138	−2.51519
$900$	0	0
$901$	11.0554	0.368310
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.4909	0.381969
$906$	0	0
$907$	11.1965	0.371773	0.185886	−	0.982571i	$-0.440484\pi$
$907$	11.1965	0.371773	0.185886	+	0.982571i	$0.440484\pi$
$908$	0	0
$909$	−2.14127	−0.0710214
$910$	0	0
$911$	−53.7908	−1.78217	−0.891085	−	0.453836i	$-0.850055\pi$
$911$	−53.7908	−1.78217	−0.891085	+	0.453836i	$0.850055\pi$
$912$	0	0
$913$	−75.8742	−2.51107
$914$	0	0
$915$	−60.8808	−2.01266
$916$	0	0
$917$	0	0
$918$	0	0
$919$	17.7992	0.587142	0.293571	−	0.955937i	$-0.405156\pi$
$919$	17.7992	0.587142	0.293571	+	0.955937i	$0.405156\pi$
$920$	0	0
$921$	−42.4648	−1.39926
$922$	0	0
$923$	15.2173	0.500883
$924$	0	0
$925$	−25.1630	−0.827353
$926$	0	0
$927$	2.89379	0.0950446
$928$	0	0
$929$	−4.30667	−0.141297	−0.0706487	−	0.997501i	$-0.522507\pi$
$929$	−4.30667	−0.141297	−0.0706487	+	0.997501i	$0.522507\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−29.8261	−0.976463
$934$	0	0
$935$	43.3582	1.41796
$936$	0	0
$937$	26.2696	0.858190	0.429095	−	0.903259i	$-0.358833\pi$
$937$	26.2696	0.858190	0.429095	+	0.903259i	$0.358833\pi$
$938$	0	0
$939$	−28.0913	−0.916725
$940$	0	0
$941$	27.1577	0.885317	0.442658	−	0.896690i	$-0.354035\pi$
$941$	27.1577	0.885317	0.442658	+	0.896690i	$0.354035\pi$
$942$	0	0
$943$	2.22205	0.0723598
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.9381	−1.26532	−0.632659	−	0.774431i	$-0.718036\pi$
$947$	−38.9381	−1.26532	−0.632659	+	0.774431i	$0.718036\pi$
$948$	0	0
$949$	55.0593	1.78730
$950$	0	0
$951$	−25.0412	−0.812017
$952$	0	0
$953$	−46.9335	−1.52032	−0.760162	−	0.649733i	$-0.774881\pi$
$953$	−46.9335	−1.52032	−0.760162	+	0.649733i	$0.774881\pi$
$954$	0	0
$955$	38.2706	1.23841
$956$	0	0
$957$	67.1695	2.17128
$958$	0	0
$959$	0	0
$960$	0	0
$961$	62.5785	2.01866
$962$	0	0
$963$	2.17992	0.0702471
$964$	0	0
$965$	39.9408	1.28574
$966$	0	0
$967$	6.42326	0.206558	0.103279	−	0.994652i	$-0.467067\pi$
$967$	6.42326	0.206558	0.103279	+	0.994652i	$0.467067\pi$
$968$	0	0
$969$	20.1748	0.648107
$970$	0	0
$971$	29.7539	0.954848	0.477424	−	0.878673i	$-0.341571\pi$
$971$	29.7539	0.954848	0.477424	+	0.878673i	$0.341571\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	64.1534	2.05455
$976$	0	0
$977$	37.0324	1.18477	0.592385	−	0.805655i	$-0.298186\pi$
$977$	37.0324	1.18477	0.592385	+	0.805655i	$0.298186\pi$
$978$	0	0
$979$	−55.5155	−1.77428
$980$	0	0
$981$	2.74496	0.0876397
$982$	0	0
$983$	−21.4005	−0.682570	−0.341285	−	0.939960i	$-0.610862\pi$
$983$	−21.4005	−0.682570	−0.341285	+	0.939960i	$0.610862\pi$
$984$	0	0
$985$	−10.4875	−0.334160
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.01807	0.159565
$990$	0	0
$991$	−41.7727	−1.32695	−0.663477	−	0.748197i	$-0.730920\pi$
$991$	−41.7727	−1.32695	−0.663477	+	0.748197i	$0.730920\pi$
$992$	0	0
$993$	−46.8455	−1.48660
$994$	0	0
$995$	−19.8189	−0.628301
$996$	0	0
$997$	37.1370	1.17614	0.588071	−	0.808809i	$-0.299888\pi$
$997$	37.1370	1.17614	0.588071	+	0.808809i	$0.299888\pi$
$998$	0	0
$999$	13.0757	0.413696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9016.2.a.bk.1.10		11
7.2	even	3		1288.2.q.d.921.2	yes	22
7.4	even	3		1288.2.q.d.737.2	✓	22
7.6	odd	2		9016.2.a.br.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.q.d.737.2	✓	22	7.4	even	3
1288.2.q.d.921.2	yes	22	7.2	even	3
9016.2.a.bk.1.10		11	1.1	even	1	trivial
9016.2.a.br.1.2		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)

\(f(q)\)	\(=\)	\(q+1.86046 q^{3} -3.75356 q^{5} +0.461318 q^{9} +O(q^{10})\)	\(q+1.86046 q^{3} -3.75356 q^{5} +0.461318 q^{9} +4.63115 q^{11} +3.79378 q^{13} -6.98336 q^{15} -2.49425 q^{17} -4.34759 q^{19} -1.00000 q^{23} +9.08922 q^{25} -4.72312 q^{27} +7.79584 q^{29} -9.67360 q^{31} +8.61607 q^{33} -2.76844 q^{37} +7.05818 q^{39} -2.22205 q^{41} -5.01807 q^{43} -1.73158 q^{45} +1.90487 q^{47} -4.64045 q^{51} -4.43238 q^{53} -17.3833 q^{55} -8.08852 q^{57} +4.41766 q^{59} +8.71798 q^{61} -14.2402 q^{65} -4.14452 q^{67} -1.86046 q^{69} +4.01111 q^{71} +14.5130 q^{73} +16.9101 q^{75} +16.7413 q^{79} -10.1711 q^{81} -16.3835 q^{83} +9.36230 q^{85} +14.5039 q^{87} -11.9874 q^{89} -17.9974 q^{93} +16.3189 q^{95} -3.38808 q^{97} +2.13643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 4 q^{3} - 3 q^{5} + 11 q^{9}+O(q^{10}) \) 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

L-functions

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