LMFDB - Embedded newform 4016.2.a.j.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.20631$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.20631 q^{3} -0.868402 q^{5} +2.87616 q^{7} +1.86782 q^{9} +O(q^{10})$	$q+2.20631 q^{3} -0.868402 q^{5} +2.87616 q^{7} +1.86782 q^{9} -5.76405 q^{11} -5.68088 q^{13} -1.91597 q^{15} -1.64248 q^{17} +5.32894 q^{19} +6.34570 q^{21} +0.134275 q^{23} -4.24588 q^{25} -2.49794 q^{27} +4.69611 q^{29} -9.83184 q^{31} -12.7173 q^{33} -2.49766 q^{35} -4.73348 q^{37} -12.5338 q^{39} +10.6117 q^{41} -6.78447 q^{43} -1.62202 q^{45} -5.57841 q^{47} +1.27227 q^{49} -3.62384 q^{51} -2.13457 q^{53} +5.00551 q^{55} +11.7573 q^{57} +0.764585 q^{59} +11.2665 q^{61} +5.37214 q^{63} +4.93329 q^{65} -5.53676 q^{67} +0.296253 q^{69} -6.03997 q^{71} +5.41518 q^{73} -9.36774 q^{75} -16.5783 q^{77} -4.59972 q^{79} -11.1147 q^{81} +1.93557 q^{83} +1.42634 q^{85} +10.3611 q^{87} -6.47279 q^{89} -16.3391 q^{91} -21.6921 q^{93} -4.62766 q^{95} -6.88404 q^{97} -10.7662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	2.20631	1.27382	0.636908	−	0.770940i	$-0.280213\pi$
$3$	2.20631	1.27382	0.636908	+	0.770940i	$0.280213\pi$
$4$	0	0
$5$	−0.868402	−0.388361	−0.194181	−	0.980966i	$-0.562205\pi$
$5$	−0.868402	−0.388361	−0.194181	+	0.980966i	$0.562205\pi$
$6$	0	0
$7$	2.87616	1.08708	0.543542	−	0.839382i	$-0.317083\pi$
$7$	2.87616	1.08708	0.543542	+	0.839382i	$0.317083\pi$
$8$	0	0
$9$	1.86782	0.622607
$10$	0	0
$11$	−5.76405	−1.73793	−0.868963	−	0.494877i	$-0.835213\pi$
$11$	−5.76405	−1.73793	−0.868963	+	0.494877i	$0.835213\pi$
$12$	0	0
$13$	−5.68088	−1.57559	−0.787797	−	0.615935i	$-0.788778\pi$
$13$	−5.68088	−1.57559	−0.787797	+	0.615935i	$0.788778\pi$
$14$	0	0
$15$	−1.91597	−0.494701
$16$	0	0
$17$	−1.64248	−0.398361	−0.199181	−	0.979963i	$-0.563828\pi$
$17$	−1.64248	−0.398361	−0.199181	+	0.979963i	$0.563828\pi$
$18$	0	0
$19$	5.32894	1.22254	0.611271	−	0.791421i	$-0.290659\pi$
$19$	5.32894	1.22254	0.611271	+	0.791421i	$0.290659\pi$
$20$	0	0
$21$	6.34570	1.38475
$22$	0	0
$23$	0.134275	0.0279983	0.0139991	−	0.999902i	$-0.495544\pi$
$23$	0.134275	0.0279983	0.0139991	+	0.999902i	$0.495544\pi$
$24$	0	0
$25$	−4.24588	−0.849175
$26$	0	0
$27$	−2.49794	−0.480729
$28$	0	0
$29$	4.69611	0.872045	0.436023	−	0.899936i	$-0.356387\pi$
$29$	4.69611	0.872045	0.436023	+	0.899936i	$0.356387\pi$
$30$	0	0
$31$	−9.83184	−1.76585	−0.882925	−	0.469514i	$-0.844429\pi$
$31$	−9.83184	−1.76585	−0.882925	+	0.469514i	$0.844429\pi$
$32$	0	0
$33$	−12.7173	−2.21380
$34$	0	0
$35$	−2.49766	−0.422182
$36$	0	0
$37$	−4.73348	−0.778179	−0.389089	−	0.921200i	$-0.627210\pi$
$37$	−4.73348	−0.778179	−0.389089	+	0.921200i	$0.627210\pi$
$38$	0	0
$39$	−12.5338	−2.00702
$40$	0	0
$41$	10.6117	1.65727	0.828633	−	0.559793i	$-0.189119\pi$
$41$	10.6117	1.65727	0.828633	+	0.559793i	$0.189119\pi$
$42$	0	0
$43$	−6.78447	−1.03462	−0.517311	−	0.855798i	$-0.673067\pi$
$43$	−6.78447	−1.03462	−0.517311	+	0.855798i	$0.673067\pi$
$44$	0	0
$45$	−1.62202	−0.241796
$46$	0	0
$47$	−5.57841	−0.813695	−0.406847	−	0.913496i	$-0.633372\pi$
$47$	−5.57841	−0.813695	−0.406847	+	0.913496i	$0.633372\pi$
$48$	0	0
$49$	1.27227	0.181753
$50$	0	0
$51$	−3.62384	−0.507439
$52$	0	0
$53$	−2.13457	−0.293206	−0.146603	−	0.989195i	$-0.546834\pi$
$53$	−2.13457	−0.293206	−0.146603	+	0.989195i	$0.546834\pi$
$54$	0	0
$55$	5.00551	0.674943
$56$	0	0
$57$	11.7573	1.55729
$58$	0	0
$59$	0.764585	0.0995405	0.0497702	−	0.998761i	$-0.484151\pi$
$59$	0.764585	0.0995405	0.0497702	+	0.998761i	$0.484151\pi$
$60$	0	0
$61$	11.2665	1.44253	0.721264	−	0.692660i	$-0.243562\pi$
$61$	11.2665	1.44253	0.721264	+	0.692660i	$0.243562\pi$
$62$	0	0
$63$	5.37214	0.676826
$64$	0	0
$65$	4.93329	0.611900
$66$	0	0
$67$	−5.53676	−0.676423	−0.338212	−	0.941070i	$-0.609822\pi$
$67$	−5.53676	−0.676423	−0.338212	+	0.941070i	$0.609822\pi$
$68$	0	0
$69$	0.296253	0.0356646
$70$	0	0
$71$	−6.03997	−0.716812	−0.358406	−	0.933566i	$-0.616680\pi$
$71$	−6.03997	−0.716812	−0.358406	+	0.933566i	$0.616680\pi$
$72$	0	0
$73$	5.41518	0.633799	0.316900	−	0.948459i	$-0.397358\pi$
$73$	5.41518	0.633799	0.316900	+	0.948459i	$0.397358\pi$
$74$	0	0
$75$	−9.36774	−1.08169
$76$	0	0
$77$	−16.5783	−1.88927
$78$	0	0
$79$	−4.59972	−0.517509	−0.258754	−	0.965943i	$-0.583312\pi$
$79$	−4.59972	−0.517509	−0.258754	+	0.965943i	$0.583312\pi$
$80$	0	0
$81$	−11.1147	−1.23497
$82$	0	0
$83$	1.93557	0.212456	0.106228	−	0.994342i	$-0.466123\pi$
$83$	1.93557	0.212456	0.106228	+	0.994342i	$0.466123\pi$
$84$	0	0
$85$	1.42634	0.154708
$86$	0	0
$87$	10.3611	1.11083
$88$	0	0
$89$	−6.47279	−0.686115	−0.343057	−	0.939314i	$-0.611463\pi$
$89$	−6.47279	−0.686115	−0.343057	+	0.939314i	$0.611463\pi$
$90$	0	0
$91$	−16.3391	−1.71280
$92$	0	0
$93$	−21.6921	−2.24937
$94$	0	0
$95$	−4.62766	−0.474788
$96$	0	0
$97$	−6.88404	−0.698968	−0.349484	−	0.936942i	$-0.613643\pi$
$97$	−6.88404	−0.698968	−0.349484	+	0.936942i	$0.613643\pi$
$98$	0	0
$99$	−10.7662	−1.08204
$100$	0	0
$101$	−15.8366	−1.57580	−0.787898	−	0.615806i	$-0.788831\pi$
$101$	−15.8366	−1.57580	−0.787898	+	0.615806i	$0.788831\pi$
$102$	0	0
$103$	2.82676	0.278529	0.139265	−	0.990255i	$-0.455526\pi$
$103$	2.82676	0.278529	0.139265	+	0.990255i	$0.455526\pi$
$104$	0	0
$105$	−5.51062	−0.537782
$106$	0	0
$107$	−15.8485	−1.53213	−0.766067	−	0.642761i	$-0.777789\pi$
$107$	−15.8485	−1.53213	−0.766067	+	0.642761i	$0.777789\pi$
$108$	0	0
$109$	−11.3491	−1.08705	−0.543524	−	0.839394i	$-0.682910\pi$
$109$	−11.3491	−1.08705	−0.543524	+	0.839394i	$0.682910\pi$
$110$	0	0
$111$	−10.4435	−0.991256
$112$	0	0
$113$	17.4781	1.64420	0.822100	−	0.569342i	$-0.192802\pi$
$113$	17.4781	1.64420	0.822100	+	0.569342i	$0.192802\pi$
$114$	0	0
$115$	−0.116605	−0.0108734
$116$	0	0
$117$	−10.6109	−0.980975
$118$	0	0
$119$	−4.72404	−0.433052
$120$	0	0
$121$	22.2242	2.02039
$122$	0	0
$123$	23.4127	2.11105
$124$	0	0
$125$	8.02914	0.718148
$126$	0	0
$127$	5.48123	0.486381	0.243190	−	0.969979i	$-0.421806\pi$
$127$	5.48123	0.486381	0.243190	+	0.969979i	$0.421806\pi$
$128$	0	0
$129$	−14.9687	−1.31792
$130$	0	0
$131$	11.4620	1.00144	0.500719	−	0.865610i	$-0.333069\pi$
$131$	11.4620	1.00144	0.500719	+	0.865610i	$0.333069\pi$
$132$	0	0
$133$	15.3269	1.32901
$134$	0	0
$135$	2.16922	0.186697
$136$	0	0
$137$	21.4707	1.83436	0.917182	−	0.398468i	$-0.130458\pi$
$137$	21.4707	1.83436	0.917182	+	0.398468i	$0.130458\pi$
$138$	0	0
$139$	−14.2718	−1.21052	−0.605258	−	0.796029i	$-0.706930\pi$
$139$	−14.2718	−1.21052	−0.605258	+	0.796029i	$0.706930\pi$
$140$	0	0
$141$	−12.3077	−1.03650
$142$	0	0
$143$	32.7449	2.73826
$144$	0	0
$145$	−4.07811	−0.338669
$146$	0	0
$147$	2.80703	0.231520
$148$	0	0
$149$	−6.36815	−0.521699	−0.260849	−	0.965379i	$-0.584003\pi$
$149$	−6.36815	−0.521699	−0.260849	+	0.965379i	$0.584003\pi$
$150$	0	0
$151$	−11.9538	−0.972790	−0.486395	−	0.873739i	$-0.661688\pi$
$151$	−11.9538	−0.972790	−0.486395	+	0.873739i	$0.661688\pi$
$152$	0	0
$153$	−3.06787	−0.248022
$154$	0	0
$155$	8.53799	0.685788
$156$	0	0
$157$	−3.83434	−0.306013	−0.153007	−	0.988225i	$-0.548896\pi$
$157$	−3.83434	−0.306013	−0.153007	+	0.988225i	$0.548896\pi$
$158$	0	0
$159$	−4.70953	−0.373490
$160$	0	0
$161$	0.386196	0.0304365
$162$	0	0
$163$	−11.3482	−0.888860	−0.444430	−	0.895814i	$-0.646594\pi$
$163$	−11.3482	−0.888860	−0.444430	+	0.895814i	$0.646594\pi$
$164$	0	0
$165$	11.0437	0.859753
$166$	0	0
$167$	18.4831	1.43026	0.715131	−	0.698990i	$-0.246367\pi$
$167$	18.4831	1.43026	0.715131	+	0.698990i	$0.246367\pi$
$168$	0	0
$169$	19.2724	1.48250
$170$	0	0
$171$	9.95350	0.761163
$172$	0	0
$173$	−8.40818	−0.639262	−0.319631	−	0.947542i	$-0.603559\pi$
$173$	−8.40818	−0.639262	−0.319631	+	0.947542i	$0.603559\pi$
$174$	0	0
$175$	−12.2118	−0.923125
$176$	0	0
$177$	1.68691	0.126796
$178$	0	0
$179$	15.5631	1.16324	0.581621	−	0.813460i	$-0.302419\pi$
$179$	15.5631	1.16324	0.581621	+	0.813460i	$0.302419\pi$
$180$	0	0
$181$	−11.3818	−0.846001	−0.423001	−	0.906129i	$-0.639023\pi$
$181$	−11.3818	−0.846001	−0.423001	+	0.906129i	$0.639023\pi$
$182$	0	0
$183$	24.8574	1.83751
$184$	0	0
$185$	4.11056	0.302215
$186$	0	0
$187$	9.46736	0.692322
$188$	0	0
$189$	−7.18447	−0.522593
$190$	0	0
$191$	9.37352	0.678244	0.339122	−	0.940742i	$-0.389870\pi$
$191$	9.37352	0.678244	0.339122	+	0.940742i	$0.389870\pi$
$192$	0	0
$193$	24.6014	1.77085	0.885425	−	0.464782i	$-0.153867\pi$
$193$	24.6014	1.77085	0.885425	+	0.464782i	$0.153867\pi$
$194$	0	0
$195$	10.8844	0.779448
$196$	0	0
$197$	1.86868	0.133138	0.0665691	−	0.997782i	$-0.478795\pi$
$197$	1.86868	0.133138	0.0665691	+	0.997782i	$0.478795\pi$
$198$	0	0
$199$	−20.6736	−1.46551	−0.732755	−	0.680492i	$-0.761766\pi$
$199$	−20.6736	−1.46551	−0.732755	+	0.680492i	$0.761766\pi$
$200$	0	0
$201$	−12.2158	−0.861638
$202$	0	0
$203$	13.5067	0.947987
$204$	0	0
$205$	−9.21521	−0.643618
$206$	0	0
$207$	0.250801	0.0174319
$208$	0	0
$209$	−30.7163	−2.12469
$210$	0	0
$211$	23.2599	1.60128	0.800638	−	0.599148i	$-0.204494\pi$
$211$	23.2599	1.60128	0.800638	+	0.599148i	$0.204494\pi$
$212$	0	0
$213$	−13.3261	−0.913087
$214$	0	0
$215$	5.89165	0.401807
$216$	0	0
$217$	−28.2779	−1.91963
$218$	0	0
$219$	11.9476	0.807343
$220$	0	0
$221$	9.33076	0.627655
$222$	0	0
$223$	0.356655	0.0238834	0.0119417	−	0.999929i	$-0.496199\pi$
$223$	0.356655	0.0238834	0.0119417	+	0.999929i	$0.496199\pi$
$224$	0	0
$225$	−7.93054	−0.528702
$226$	0	0
$227$	10.0079	0.664246	0.332123	−	0.943236i	$-0.392235\pi$
$227$	10.0079	0.664246	0.332123	+	0.943236i	$0.392235\pi$
$228$	0	0
$229$	−23.6986	−1.56605	−0.783023	−	0.621992i	$-0.786324\pi$
$229$	−23.6986	−1.56605	−0.783023	+	0.621992i	$0.786324\pi$
$230$	0	0
$231$	−36.5769	−2.40659
$232$	0	0
$233$	18.5031	1.21218	0.606088	−	0.795398i	$-0.292738\pi$
$233$	18.5031	1.21218	0.606088	+	0.795398i	$0.292738\pi$
$234$	0	0
$235$	4.84431	0.316008
$236$	0	0
$237$	−10.1484	−0.659211
$238$	0	0
$239$	−0.138909	−0.00898527	−0.00449263	−	0.999990i	$-0.501430\pi$
$239$	−0.138909	−0.00898527	−0.00449263	+	0.999990i	$0.501430\pi$
$240$	0	0
$241$	1.53407	0.0988179	0.0494090	−	0.998779i	$-0.484266\pi$
$241$	1.53407	0.0988179	0.0494090	+	0.998779i	$0.484266\pi$
$242$	0	0
$243$	−17.0287	−1.09239
$244$	0	0
$245$	−1.10484	−0.0705858
$246$	0	0
$247$	−30.2731	−1.92623
$248$	0	0
$249$	4.27047	0.270630
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−0.773967	−0.0486589
$254$	0	0
$255$	3.14695	0.197070
$256$	0	0
$257$	−20.8770	−1.30227	−0.651135	−	0.758962i	$-0.725707\pi$
$257$	−20.8770	−1.30227	−0.651135	+	0.758962i	$0.725707\pi$
$258$	0	0
$259$	−13.6142	−0.845946
$260$	0	0
$261$	8.77149	0.542942
$262$	0	0
$263$	4.13670	0.255080	0.127540	−	0.991833i	$-0.459292\pi$
$263$	4.13670	0.255080	0.127540	+	0.991833i	$0.459292\pi$
$264$	0	0
$265$	1.85367	0.113870
$266$	0	0
$267$	−14.2810	−0.873984
$268$	0	0
$269$	29.8214	1.81824	0.909122	−	0.416529i	$-0.136754\pi$
$269$	29.8214	1.81824	0.909122	+	0.416529i	$0.136754\pi$
$270$	0	0
$271$	2.63990	0.160362	0.0801812	−	0.996780i	$-0.474450\pi$
$271$	2.63990	0.160362	0.0801812	+	0.996780i	$0.474450\pi$
$272$	0	0
$273$	−36.0492	−2.18180
$274$	0	0
$275$	24.4734	1.47580
$276$	0	0
$277$	1.77318	0.106540	0.0532699	−	0.998580i	$-0.483036\pi$
$277$	1.77318	0.106540	0.0532699	+	0.998580i	$0.483036\pi$
$278$	0	0
$279$	−18.3641	−1.09943
$280$	0	0
$281$	29.3479	1.75075	0.875376	−	0.483443i	$-0.160614\pi$
$281$	29.3479	1.75075	0.875376	+	0.483443i	$0.160614\pi$
$282$	0	0
$283$	−12.1349	−0.721342	−0.360671	−	0.932693i	$-0.617452\pi$
$283$	−12.1349	−0.721342	−0.360671	+	0.932693i	$0.617452\pi$
$284$	0	0
$285$	−10.2101	−0.604793
$286$	0	0
$287$	30.5208	1.80159
$288$	0	0
$289$	−14.3022	−0.841308
$290$	0	0
$291$	−15.1884	−0.890357
$292$	0	0
$293$	13.1424	0.767785	0.383892	−	0.923378i	$-0.374583\pi$
$293$	13.1424	0.767785	0.383892	+	0.923378i	$0.374583\pi$
$294$	0	0
$295$	−0.663967	−0.0386577
$296$	0	0
$297$	14.3983	0.835472
$298$	0	0
$299$	−0.762800	−0.0441139
$300$	0	0
$301$	−19.5132	−1.12472
$302$	0	0
$303$	−34.9404	−2.00727
$304$	0	0
$305$	−9.78386	−0.560222
$306$	0	0
$307$	11.6465	0.664701	0.332350	−	0.943156i	$-0.392158\pi$
$307$	11.6465	0.664701	0.332350	+	0.943156i	$0.392158\pi$
$308$	0	0
$309$	6.23673	0.354795
$310$	0	0
$311$	−14.0514	−0.796784	−0.398392	−	0.917215i	$-0.630432\pi$
$311$	−14.0514	−0.796784	−0.398392	+	0.917215i	$0.630432\pi$
$312$	0	0
$313$	−11.5790	−0.654482	−0.327241	−	0.944941i	$-0.606119\pi$
$313$	−11.5790	−0.654482	−0.327241	+	0.944941i	$0.606119\pi$
$314$	0	0
$315$	−4.66518	−0.262853
$316$	0	0
$317$	−10.6759	−0.599618	−0.299809	−	0.953999i	$-0.596923\pi$
$317$	−10.6759	−0.599618	−0.299809	+	0.953999i	$0.596923\pi$
$318$	0	0
$319$	−27.0686	−1.51555
$320$	0	0
$321$	−34.9668	−1.95166
$322$	0	0
$323$	−8.75270	−0.487013
$324$	0	0
$325$	24.1203	1.33796
$326$	0	0
$327$	−25.0397	−1.38470
$328$	0	0
$329$	−16.0444	−0.884555
$330$	0	0
$331$	−31.8128	−1.74859	−0.874294	−	0.485397i	$-0.838675\pi$
$331$	−31.8128	−1.74859	−0.874294	+	0.485397i	$0.838675\pi$
$332$	0	0
$333$	−8.84128	−0.484499
$334$	0	0
$335$	4.80814	0.262697
$336$	0	0
$337$	15.9358	0.868077	0.434038	−	0.900894i	$-0.357088\pi$
$337$	15.9358	0.868077	0.434038	+	0.900894i	$0.357088\pi$
$338$	0	0
$339$	38.5622	2.09441
$340$	0	0
$341$	56.6712	3.06892
$342$	0	0
$343$	−16.4738	−0.889504
$344$	0	0
$345$	−0.257266	−0.0138508
$346$	0	0
$347$	−31.0159	−1.66502	−0.832510	−	0.554011i	$-0.813097\pi$
$347$	−31.0159	−1.66502	−0.832510	+	0.554011i	$0.813097\pi$
$348$	0	0
$349$	16.7797	0.898196	0.449098	−	0.893483i	$-0.351745\pi$
$349$	16.7797	0.898196	0.449098	+	0.893483i	$0.351745\pi$
$350$	0	0
$351$	14.1905	0.757434
$352$	0	0
$353$	−12.2625	−0.652670	−0.326335	−	0.945254i	$-0.605814\pi$
$353$	−12.2625	−0.652670	−0.326335	+	0.945254i	$0.605814\pi$
$354$	0	0
$355$	5.24512	0.278382
$356$	0	0
$357$	−10.4227	−0.551629
$358$	0	0
$359$	−28.5038	−1.50437	−0.752187	−	0.658950i	$-0.771001\pi$
$359$	−28.5038	−1.50437	−0.752187	+	0.658950i	$0.771001\pi$
$360$	0	0
$361$	9.39758	0.494609
$362$	0	0
$363$	49.0337	2.57360
$364$	0	0
$365$	−4.70256	−0.246143
$366$	0	0
$367$	−10.1919	−0.532014	−0.266007	−	0.963971i	$-0.585704\pi$
$367$	−10.1919	−0.532014	−0.266007	+	0.963971i	$0.585704\pi$
$368$	0	0
$369$	19.8207	1.03182
$370$	0	0
$371$	−6.13935	−0.318739
$372$	0	0
$373$	13.2723	0.687214	0.343607	−	0.939113i	$-0.388351\pi$
$373$	13.2723	0.687214	0.343607	+	0.939113i	$0.388351\pi$
$374$	0	0
$375$	17.7148	0.914789
$376$	0	0
$377$	−26.6780	−1.37399
$378$	0	0
$379$	35.7638	1.83707	0.918533	−	0.395345i	$-0.129375\pi$
$379$	35.7638	1.83707	0.918533	+	0.395345i	$0.129375\pi$
$380$	0	0
$381$	12.0933	0.619560
$382$	0	0
$383$	−11.7973	−0.602814	−0.301407	−	0.953496i	$-0.597456\pi$
$383$	−11.7973	−0.602814	−0.301407	+	0.953496i	$0.597456\pi$
$384$	0	0
$385$	14.3966	0.733720
$386$	0	0
$387$	−12.6722	−0.644163
$388$	0	0
$389$	−10.9599	−0.555689	−0.277845	−	0.960626i	$-0.589620\pi$
$389$	−10.9599	−0.555689	−0.277845	+	0.960626i	$0.589620\pi$
$390$	0	0
$391$	−0.220544	−0.0111534
$392$	0	0
$393$	25.2887	1.27565
$394$	0	0
$395$	3.99441	0.200980
$396$	0	0
$397$	−31.6558	−1.58876	−0.794380	−	0.607421i	$-0.792204\pi$
$397$	−31.6558	−1.58876	−0.794380	+	0.607421i	$0.792204\pi$
$398$	0	0
$399$	33.8158	1.69291
$400$	0	0
$401$	−4.11300	−0.205393	−0.102697	−	0.994713i	$-0.532747\pi$
$401$	−4.11300	−0.205393	−0.102697	+	0.994713i	$0.532747\pi$
$402$	0	0
$403$	55.8535	2.78226
$404$	0	0
$405$	9.65204	0.479614
$406$	0	0
$407$	27.2840	1.35242
$408$	0	0
$409$	31.3939	1.55233	0.776163	−	0.630532i	$-0.217163\pi$
$409$	31.3939	1.55233	0.776163	+	0.630532i	$0.217163\pi$
$410$	0	0
$411$	47.3711	2.33664
$412$	0	0
$413$	2.19906	0.108209
$414$	0	0
$415$	−1.68085	−0.0825098
$416$	0	0
$417$	−31.4880	−1.54198
$418$	0	0
$419$	−14.7806	−0.722080	−0.361040	−	0.932550i	$-0.617578\pi$
$419$	−14.7806	−0.722080	−0.361040	+	0.932550i	$0.617578\pi$
$420$	0	0
$421$	−40.3621	−1.96713	−0.983563	−	0.180563i	$-0.942208\pi$
$421$	−40.3621	−1.96713	−0.983563	+	0.180563i	$0.942208\pi$
$422$	0	0
$423$	−10.4195	−0.506612
$424$	0	0
$425$	6.97379	0.338278
$426$	0	0
$427$	32.4042	1.56815
$428$	0	0
$429$	72.2455	3.48805
$430$	0	0
$431$	1.85214	0.0892145	0.0446072	−	0.999005i	$-0.485796\pi$
$431$	1.85214	0.0892145	0.0446072	+	0.999005i	$0.485796\pi$
$432$	0	0
$433$	10.1415	0.487367	0.243684	−	0.969855i	$-0.421644\pi$
$433$	10.1415	0.487367	0.243684	+	0.969855i	$0.421644\pi$
$434$	0	0
$435$	−8.99760	−0.431402
$436$	0	0
$437$	0.715543	0.0342290
$438$	0	0
$439$	9.44411	0.450743	0.225371	−	0.974273i	$-0.427640\pi$
$439$	9.44411	0.450743	0.225371	+	0.974273i	$0.427640\pi$
$440$	0	0
$441$	2.37637	0.113161
$442$	0	0
$443$	−7.34198	−0.348828	−0.174414	−	0.984672i	$-0.555803\pi$
$443$	−7.34198	−0.348828	−0.174414	+	0.984672i	$0.555803\pi$
$444$	0	0
$445$	5.62099	0.266460
$446$	0	0
$447$	−14.0501	−0.664548
$448$	0	0
$449$	38.9291	1.83718	0.918590	−	0.395213i	$-0.129329\pi$
$449$	38.9291	1.83718	0.918590	+	0.395213i	$0.129329\pi$
$450$	0	0
$451$	−61.1662	−2.88020
$452$	0	0
$453$	−26.3739	−1.23916
$454$	0	0
$455$	14.1889	0.665187
$456$	0	0
$457$	27.8922	1.30474	0.652371	−	0.757900i	$-0.273774\pi$
$457$	27.8922	1.30474	0.652371	+	0.757900i	$0.273774\pi$
$458$	0	0
$459$	4.10283	0.191504
$460$	0	0
$461$	−10.1469	−0.472588	−0.236294	−	0.971682i	$-0.575933\pi$
$461$	−10.1469	−0.472588	−0.236294	+	0.971682i	$0.575933\pi$
$462$	0	0
$463$	−17.2475	−0.801561	−0.400780	−	0.916174i	$-0.631261\pi$
$463$	−17.2475	−0.801561	−0.400780	+	0.916174i	$0.631261\pi$
$464$	0	0
$465$	18.8375	0.873568
$466$	0	0
$467$	0.356787	0.0165101	0.00825507	−	0.999966i	$-0.497372\pi$
$467$	0.356787	0.0165101	0.00825507	+	0.999966i	$0.497372\pi$
$468$	0	0
$469$	−15.9246	−0.735329
$470$	0	0
$471$	−8.45975	−0.389805
$472$	0	0
$473$	39.1060	1.79810
$474$	0	0
$475$	−22.6260	−1.03815
$476$	0	0
$477$	−3.98699	−0.182552
$478$	0	0
$479$	14.5555	0.665058	0.332529	−	0.943093i	$-0.392098\pi$
$479$	14.5555	0.665058	0.332529	+	0.943093i	$0.392098\pi$
$480$	0	0
$481$	26.8903	1.22609
$482$	0	0
$483$	0.852068	0.0387705
$484$	0	0
$485$	5.97812	0.271452
$486$	0	0
$487$	−37.4731	−1.69807	−0.849035	−	0.528337i	$-0.822816\pi$
$487$	−37.4731	−1.69807	−0.849035	+	0.528337i	$0.822816\pi$
$488$	0	0
$489$	−25.0377	−1.13224
$490$	0	0
$491$	−16.8620	−0.760970	−0.380485	−	0.924787i	$-0.624243\pi$
$491$	−16.8620	−0.760970	−0.380485	+	0.924787i	$0.624243\pi$
$492$	0	0
$493$	−7.71329	−0.347389
$494$	0	0
$495$	9.34940	0.420224
$496$	0	0
$497$	−17.3719	−0.779236
$498$	0	0
$499$	−6.95261	−0.311241	−0.155621	−	0.987817i	$-0.549738\pi$
$499$	−6.95261	−0.311241	−0.155621	+	0.987817i	$0.549738\pi$
$500$	0	0
$501$	40.7794	1.82189
$502$	0	0
$503$	2.41383	0.107627	0.0538137	−	0.998551i	$-0.482862\pi$
$503$	2.41383	0.107627	0.0538137	+	0.998551i	$0.482862\pi$
$504$	0	0
$505$	13.7525	0.611978
$506$	0	0
$507$	42.5210	1.88843
$508$	0	0
$509$	−26.4307	−1.17152	−0.585760	−	0.810484i	$-0.699204\pi$
$509$	−26.4307	−1.17152	−0.585760	+	0.810484i	$0.699204\pi$
$510$	0	0
$511$	15.5749	0.688993
$512$	0	0
$513$	−13.3114	−0.587712
$514$	0	0
$515$	−2.45477	−0.108170
$516$	0	0
$517$	32.1542	1.41414
$518$	0	0
$519$	−18.5511	−0.814302
$520$	0	0
$521$	27.8875	1.22178	0.610888	−	0.791717i	$-0.290813\pi$
$521$	27.8875	1.22178	0.610888	+	0.791717i	$0.290813\pi$
$522$	0	0
$523$	18.5236	0.809979	0.404989	−	0.914321i	$-0.367275\pi$
$523$	18.5236	0.809979	0.404989	+	0.914321i	$0.367275\pi$
$524$	0	0
$525$	−26.9431	−1.17589
$526$	0	0
$527$	16.1486	0.703446
$528$	0	0
$529$	−22.9820	−0.999216
$530$	0	0
$531$	1.42811	0.0619746
$532$	0	0
$533$	−60.2837	−2.61118
$534$	0	0
$535$	13.7629	0.595022
$536$	0	0
$537$	34.3371	1.48176
$538$	0	0
$539$	−7.33342	−0.315873
$540$	0	0
$541$	−11.4224	−0.491089	−0.245544	−	0.969385i	$-0.578967\pi$
$541$	−11.4224	−0.491089	−0.245544	+	0.969385i	$0.578967\pi$
$542$	0	0
$543$	−25.1118	−1.07765
$544$	0	0
$545$	9.85560	0.422168
$546$	0	0
$547$	−7.70905	−0.329615	−0.164808	−	0.986326i	$-0.552700\pi$
$547$	−7.70905	−0.329615	−0.164808	+	0.986326i	$0.552700\pi$
$548$	0	0
$549$	21.0438	0.898128
$550$	0	0
$551$	25.0253	1.06611
$552$	0	0
$553$	−13.2295	−0.562576
$554$	0	0
$555$	9.06919	0.384966
$556$	0	0
$557$	42.7334	1.81067	0.905335	−	0.424698i	$-0.139620\pi$
$557$	42.7334	1.81067	0.905335	+	0.424698i	$0.139620\pi$
$558$	0	0
$559$	38.5418	1.63014
$560$	0	0
$561$	20.8880	0.881891
$562$	0	0
$563$	2.01896	0.0850892	0.0425446	−	0.999095i	$-0.486454\pi$
$563$	2.01896	0.0850892	0.0425446	+	0.999095i	$0.486454\pi$
$564$	0	0
$565$	−15.1780	−0.638544
$566$	0	0
$567$	−31.9676	−1.34251
$568$	0	0
$569$	−36.9568	−1.54931	−0.774655	−	0.632385i	$-0.782076\pi$
$569$	−36.9568	−1.54931	−0.774655	+	0.632385i	$0.782076\pi$
$570$	0	0
$571$	11.0265	0.461444	0.230722	−	0.973020i	$-0.425891\pi$
$571$	11.0265	0.461444	0.230722	+	0.973020i	$0.425891\pi$
$572$	0	0
$573$	20.6809	0.863958
$574$	0	0
$575$	−0.570115	−0.0237754
$576$	0	0
$577$	25.9567	1.08059	0.540297	−	0.841475i	$-0.318312\pi$
$577$	25.9567	1.08059	0.540297	+	0.841475i	$0.318312\pi$
$578$	0	0
$579$	54.2785	2.25574
$580$	0	0
$581$	5.56700	0.230958
$582$	0	0
$583$	12.3038	0.509570
$584$	0	0
$585$	9.21451	0.380973
$586$	0	0
$587$	−34.0627	−1.40592	−0.702959	−	0.711230i	$-0.748138\pi$
$587$	−34.0627	−1.40592	−0.702959	+	0.711230i	$0.748138\pi$
$588$	0	0
$589$	−52.3932	−2.15883
$590$	0	0
$591$	4.12290	0.169593
$592$	0	0
$593$	−1.97273	−0.0810102	−0.0405051	−	0.999179i	$-0.512897\pi$
$593$	−1.97273	−0.0810102	−0.0405051	+	0.999179i	$0.512897\pi$
$594$	0	0
$595$	4.10237	0.168181
$596$	0	0
$597$	−45.6124	−1.86679
$598$	0	0
$599$	−11.0734	−0.452445	−0.226222	−	0.974076i	$-0.572638\pi$
$599$	−11.0734	−0.452445	−0.226222	+	0.974076i	$0.572638\pi$
$600$	0	0
$601$	18.5384	0.756197	0.378099	−	0.925765i	$-0.376578\pi$
$601$	18.5384	0.756197	0.378099	+	0.925765i	$0.376578\pi$
$602$	0	0
$603$	−10.3417	−0.421146
$604$	0	0
$605$	−19.2996	−0.784640
$606$	0	0
$607$	−21.1364	−0.857900	−0.428950	−	0.903328i	$-0.641116\pi$
$607$	−21.1364	−0.857900	−0.428950	+	0.903328i	$0.641116\pi$
$608$	0	0
$609$	29.8001	1.20756
$610$	0	0
$611$	31.6903	1.28205
$612$	0	0
$613$	25.6324	1.03528	0.517641	−	0.855598i	$-0.326810\pi$
$613$	25.6324	1.03528	0.517641	+	0.855598i	$0.326810\pi$
$614$	0	0
$615$	−20.3316	−0.819851
$616$	0	0
$617$	−21.0447	−0.847228	−0.423614	−	0.905843i	$-0.639239\pi$
$617$	−21.0447	−0.847228	−0.423614	+	0.905843i	$0.639239\pi$
$618$	0	0
$619$	20.7393	0.833585	0.416792	−	0.909002i	$-0.363154\pi$
$619$	20.7393	0.833585	0.416792	+	0.909002i	$0.363154\pi$
$620$	0	0
$621$	−0.335411	−0.0134596
$622$	0	0
$623$	−18.6168	−0.745865
$624$	0	0
$625$	14.2569	0.570274
$626$	0	0
$627$	−67.7697	−2.70646
$628$	0	0
$629$	7.77466	0.309996
$630$	0	0
$631$	16.4562	0.655110	0.327555	−	0.944832i	$-0.393775\pi$
$631$	16.4562	0.655110	0.327555	+	0.944832i	$0.393775\pi$
$632$	0	0
$633$	51.3186	2.03973
$634$	0	0
$635$	−4.75992	−0.188892
$636$	0	0
$637$	−7.22762	−0.286369
$638$	0	0
$639$	−11.2816	−0.446292
$640$	0	0
$641$	−31.9484	−1.26189	−0.630943	−	0.775829i	$-0.717332\pi$
$641$	−31.9484	−1.26189	−0.630943	+	0.775829i	$0.717332\pi$
$642$	0	0
$643$	−3.75171	−0.147953	−0.0739765	−	0.997260i	$-0.523569\pi$
$643$	−3.75171	−0.147953	−0.0739765	+	0.997260i	$0.523569\pi$
$644$	0	0
$645$	12.9988	0.511828
$646$	0	0
$647$	17.6752	0.694885	0.347442	−	0.937701i	$-0.387050\pi$
$647$	17.6752	0.694885	0.347442	+	0.937701i	$0.387050\pi$
$648$	0	0
$649$	−4.40710	−0.172994
$650$	0	0
$651$	−62.3899	−2.44525
$652$	0	0
$653$	−14.3584	−0.561886	−0.280943	−	0.959724i	$-0.590647\pi$
$653$	−14.3584	−0.561886	−0.280943	+	0.959724i	$0.590647\pi$
$654$	0	0
$655$	−9.95361	−0.388920
$656$	0	0
$657$	10.1146	0.394608
$658$	0	0
$659$	2.49505	0.0971934	0.0485967	−	0.998818i	$-0.484525\pi$
$659$	2.49505	0.0971934	0.0485967	+	0.998818i	$0.484525\pi$
$660$	0	0
$661$	−10.4850	−0.407819	−0.203909	−	0.978990i	$-0.565365\pi$
$661$	−10.4850	−0.407819	−0.203909	+	0.978990i	$0.565365\pi$
$662$	0	0
$663$	20.5866	0.799517
$664$	0	0
$665$	−13.3099	−0.516135
$666$	0	0
$667$	0.630570	0.0244158
$668$	0	0
$669$	0.786894	0.0304231
$670$	0	0
$671$	−64.9407	−2.50701
$672$	0	0
$673$	5.04767	0.194573	0.0972866	−	0.995256i	$-0.468984\pi$
$673$	5.04767	0.194573	0.0972866	+	0.995256i	$0.468984\pi$
$674$	0	0
$675$	10.6060	0.408224
$676$	0	0
$677$	−3.64771	−0.140193	−0.0700964	−	0.997540i	$-0.522331\pi$
$677$	−3.64771	−0.140193	−0.0700964	+	0.997540i	$0.522331\pi$
$678$	0	0
$679$	−19.7996	−0.759838
$680$	0	0
$681$	22.0805	0.846128
$682$	0	0
$683$	−3.15785	−0.120832	−0.0604160	−	0.998173i	$-0.519243\pi$
$683$	−3.15785	−0.120832	−0.0604160	+	0.998173i	$0.519243\pi$
$684$	0	0
$685$	−18.6452	−0.712396
$686$	0	0
$687$	−52.2865	−1.99486
$688$	0	0
$689$	12.1262	0.461973
$690$	0	0
$691$	−23.0458	−0.876704	−0.438352	−	0.898803i	$-0.644438\pi$
$691$	−23.0458	−0.876704	−0.438352	+	0.898803i	$0.644438\pi$
$692$	0	0
$693$	−30.9653	−1.17627
$694$	0	0
$695$	12.3937	0.470118
$696$	0	0
$697$	−17.4295	−0.660190
$698$	0	0
$699$	40.8236	1.54409
$700$	0	0
$701$	9.37374	0.354041	0.177021	−	0.984207i	$-0.443354\pi$
$701$	9.37374	0.354041	0.177021	+	0.984207i	$0.443354\pi$
$702$	0	0
$703$	−25.2244	−0.951356
$704$	0	0
$705$	10.6881	0.402536
$706$	0	0
$707$	−45.5484	−1.71302
$708$	0	0
$709$	9.44917	0.354871	0.177435	−	0.984132i	$-0.443220\pi$
$709$	9.44917	0.354871	0.177435	+	0.984132i	$0.443220\pi$
$710$	0	0
$711$	−8.59145	−0.322205
$712$	0	0
$713$	−1.32017	−0.0494407
$714$	0	0
$715$	−28.4357	−1.06344
$716$	0	0
$717$	−0.306477	−0.0114456
$718$	0	0
$719$	28.7991	1.07403	0.537013	−	0.843574i	$-0.319553\pi$
$719$	28.7991	1.07403	0.537013	+	0.843574i	$0.319553\pi$
$720$	0	0
$721$	8.13021	0.302785
$722$	0	0
$723$	3.38463	0.125876
$724$	0	0
$725$	−19.9391	−0.740520
$726$	0	0
$727$	45.3939	1.68357	0.841784	−	0.539815i	$-0.181506\pi$
$727$	45.3939	1.68357	0.841784	+	0.539815i	$0.181506\pi$
$728$	0	0
$729$	−4.22654	−0.156539
$730$	0	0
$731$	11.1434	0.412153
$732$	0	0
$733$	−26.8510	−0.991764	−0.495882	−	0.868390i	$-0.665155\pi$
$733$	−26.8510	−0.991764	−0.495882	+	0.868390i	$0.665155\pi$
$734$	0	0
$735$	−2.43763	−0.0899133
$736$	0	0
$737$	31.9142	1.17557
$738$	0	0
$739$	27.8940	1.02610	0.513050	−	0.858359i	$-0.328516\pi$
$739$	27.8940	1.02610	0.513050	+	0.858359i	$0.328516\pi$
$740$	0	0
$741$	−66.7919	−2.45366
$742$	0	0
$743$	32.8167	1.20393	0.601963	−	0.798524i	$-0.294385\pi$
$743$	32.8167	1.20393	0.601963	+	0.798524i	$0.294385\pi$
$744$	0	0
$745$	5.53011	0.202608
$746$	0	0
$747$	3.61530	0.132277
$748$	0	0
$749$	−45.5828	−1.66556
$750$	0	0
$751$	−21.3418	−0.778772	−0.389386	−	0.921075i	$-0.627313\pi$
$751$	−21.3418	−0.778772	−0.389386	+	0.921075i	$0.627313\pi$
$752$	0	0
$753$	2.20631	0.0804025
$754$	0	0
$755$	10.3808	0.377794
$756$	0	0
$757$	41.0423	1.49171	0.745855	−	0.666109i	$-0.232041\pi$
$757$	41.0423	1.49171	0.745855	+	0.666109i	$0.232041\pi$
$758$	0	0
$759$	−1.70761	−0.0619825
$760$	0	0
$761$	−13.7663	−0.499028	−0.249514	−	0.968371i	$-0.580271\pi$
$761$	−13.7663	−0.499028	−0.249514	+	0.968371i	$0.580271\pi$
$762$	0	0
$763$	−32.6418	−1.18171
$764$	0	0
$765$	2.66414	0.0963223
$766$	0	0
$767$	−4.34352	−0.156835
$768$	0	0
$769$	52.4960	1.89306	0.946528	−	0.322623i	$-0.104565\pi$
$769$	52.4960	1.89306	0.946528	+	0.322623i	$0.104565\pi$
$770$	0	0
$771$	−46.0612	−1.65885
$772$	0	0
$773$	14.4377	0.519290	0.259645	−	0.965704i	$-0.416394\pi$
$773$	14.4377	0.519290	0.259645	+	0.965704i	$0.416394\pi$
$774$	0	0
$775$	41.7448	1.49952
$776$	0	0
$777$	−30.0372	−1.07758
$778$	0	0
$779$	56.5490	2.02608
$780$	0	0
$781$	34.8147	1.24577
$782$	0	0
$783$	−11.7306	−0.419218
$784$	0	0
$785$	3.32975	0.118844
$786$	0	0
$787$	−36.8239	−1.31263	−0.656315	−	0.754487i	$-0.727886\pi$
$787$	−36.8239	−1.31263	−0.656315	+	0.754487i	$0.727886\pi$
$788$	0	0
$789$	9.12687	0.324925
$790$	0	0
$791$	50.2697	1.78739
$792$	0	0
$793$	−64.0037	−2.27284
$794$	0	0
$795$	4.08977	0.145049
$796$	0	0
$797$	6.88953	0.244040	0.122020	−	0.992528i	$-0.461063\pi$
$797$	6.88953	0.244040	0.122020	+	0.992528i	$0.461063\pi$
$798$	0	0
$799$	9.16246	0.324144
$800$	0	0
$801$	−12.0900	−0.427180
$802$	0	0
$803$	−31.2134	−1.10150
$804$	0	0
$805$	−0.335373	−0.0118203
$806$	0	0
$807$	65.7954	2.31611
$808$	0	0
$809$	−30.0534	−1.05662	−0.528311	−	0.849051i	$-0.677175\pi$
$809$	−30.0534	−1.05662	−0.528311	+	0.849051i	$0.677175\pi$
$810$	0	0
$811$	−45.1070	−1.58392	−0.791960	−	0.610572i	$-0.790939\pi$
$811$	−45.1070	−1.58392	−0.791960	+	0.610572i	$0.790939\pi$
$812$	0	0
$813$	5.82444	0.204272
$814$	0	0
$815$	9.85481	0.345199
$816$	0	0
$817$	−36.1540	−1.26487
$818$	0	0
$819$	−30.5185	−1.06640
$820$	0	0
$821$	−13.9888	−0.488211	−0.244106	−	0.969749i	$-0.578494\pi$
$821$	−13.9888	−0.488211	−0.244106	+	0.969749i	$0.578494\pi$
$822$	0	0
$823$	27.4149	0.955623	0.477811	−	0.878463i	$-0.341430\pi$
$823$	27.4149	0.955623	0.477811	+	0.878463i	$0.341430\pi$
$824$	0	0
$825$	53.9961	1.87990
$826$	0	0
$827$	−2.12745	−0.0739788	−0.0369894	−	0.999316i	$-0.511777\pi$
$827$	−2.12745	−0.0739788	−0.0369894	+	0.999316i	$0.511777\pi$
$828$	0	0
$829$	5.36055	0.186180	0.0930898	−	0.995658i	$-0.470326\pi$
$829$	5.36055	0.186180	0.0930898	+	0.995658i	$0.470326\pi$
$830$	0	0
$831$	3.91218	0.135712
$832$	0	0
$833$	−2.08968	−0.0724033
$834$	0	0
$835$	−16.0507	−0.555459
$836$	0	0
$837$	24.5594	0.848896
$838$	0	0
$839$	−29.3472	−1.01318	−0.506590	−	0.862187i	$-0.669094\pi$
$839$	−29.3472	−1.01318	−0.506590	+	0.862187i	$0.669094\pi$
$840$	0	0
$841$	−6.94657	−0.239537
$842$	0	0
$843$	64.7508	2.23013
$844$	0	0
$845$	−16.7362	−0.575744
$846$	0	0
$847$	63.9204	2.19633
$848$	0	0
$849$	−26.7733	−0.918857
$850$	0	0
$851$	−0.635587	−0.0217876
$852$	0	0
$853$	6.09494	0.208687	0.104343	−	0.994541i	$-0.466726\pi$
$853$	6.09494	0.208687	0.104343	+	0.994541i	$0.466726\pi$
$854$	0	0
$855$	−8.64364	−0.295606
$856$	0	0
$857$	39.3665	1.34473	0.672367	−	0.740218i	$-0.265278\pi$
$857$	39.3665	1.34473	0.672367	+	0.740218i	$0.265278\pi$
$858$	0	0
$859$	−45.3023	−1.54569	−0.772847	−	0.634593i	$-0.781168\pi$
$859$	−45.3023	−1.54569	−0.772847	+	0.634593i	$0.781168\pi$
$860$	0	0
$861$	67.3385	2.29489
$862$	0	0
$863$	22.0267	0.749798	0.374899	−	0.927066i	$-0.377677\pi$
$863$	22.0267	0.749798	0.374899	+	0.927066i	$0.377677\pi$
$864$	0	0
$865$	7.30168	0.248265
$866$	0	0
$867$	−31.5552	−1.07167
$868$	0	0
$869$	26.5130	0.899392
$870$	0	0
$871$	31.4537	1.06577
$872$	0	0
$873$	−12.8582	−0.435183
$874$	0	0
$875$	23.0931	0.780688
$876$	0	0
$877$	−26.6906	−0.901279	−0.450639	−	0.892706i	$-0.648804\pi$
$877$	−26.6906	−0.901279	−0.450639	+	0.892706i	$0.648804\pi$
$878$	0	0
$879$	28.9962	0.978017
$880$	0	0
$881$	−28.8995	−0.973648	−0.486824	−	0.873500i	$-0.661845\pi$
$881$	−28.8995	−0.973648	−0.486824	+	0.873500i	$0.661845\pi$
$882$	0	0
$883$	−9.80820	−0.330072	−0.165036	−	0.986288i	$-0.552774\pi$
$883$	−9.80820	−0.330072	−0.165036	+	0.986288i	$0.552774\pi$
$884$	0	0
$885$	−1.46492	−0.0492428
$886$	0	0
$887$	13.1169	0.440423	0.220212	−	0.975452i	$-0.429325\pi$
$887$	13.1169	0.440423	0.220212	+	0.975452i	$0.429325\pi$
$888$	0	0
$889$	15.7649	0.528737
$890$	0	0
$891$	64.0657	2.14628
$892$	0	0
$893$	−29.7270	−0.994776
$894$	0	0
$895$	−13.5150	−0.451758
$896$	0	0
$897$	−1.68298	−0.0561929
$898$	0	0
$899$	−46.1714	−1.53990
$900$	0	0
$901$	3.50600	0.116802
$902$	0	0
$903$	−43.0522	−1.43269
$904$	0	0
$905$	9.88397	0.328554
$906$	0	0
$907$	−32.8257	−1.08996	−0.544980	−	0.838449i	$-0.683463\pi$
$907$	−32.8257	−1.08996	−0.544980	+	0.838449i	$0.683463\pi$
$908$	0	0
$909$	−29.5798	−0.981102
$910$	0	0
$911$	40.9224	1.35582	0.677910	−	0.735145i	$-0.262886\pi$
$911$	40.9224	1.35582	0.677910	+	0.735145i	$0.262886\pi$
$912$	0	0
$913$	−11.1567	−0.369233
$914$	0	0
$915$	−21.5863	−0.713620
$916$	0	0
$917$	32.9664	1.08865
$918$	0	0
$919$	1.53169	0.0505258	0.0252629	−	0.999681i	$-0.491958\pi$
$919$	1.53169	0.0505258	0.0252629	+	0.999681i	$0.491958\pi$
$920$	0	0
$921$	25.6958	0.846707
$922$	0	0
$923$	34.3124	1.12940
$924$	0	0
$925$	20.0978	0.660810
$926$	0	0
$927$	5.27989	0.173414
$928$	0	0
$929$	38.3196	1.25723	0.628613	−	0.777718i	$-0.283623\pi$
$929$	38.3196	1.25723	0.628613	+	0.777718i	$0.283623\pi$
$930$	0	0
$931$	6.77985	0.222201
$932$	0	0
$933$	−31.0019	−1.01496
$934$	0	0
$935$	−8.22148	−0.268871
$936$	0	0
$937$	−31.6028	−1.03242	−0.516209	−	0.856463i	$-0.672657\pi$
$937$	−31.6028	−1.03242	−0.516209	+	0.856463i	$0.672657\pi$
$938$	0	0
$939$	−25.5469	−0.833690
$940$	0	0
$941$	53.0158	1.72827	0.864134	−	0.503263i	$-0.167867\pi$
$941$	53.0158	1.72827	0.864134	+	0.503263i	$0.167867\pi$
$942$	0	0
$943$	1.42488	0.0464005
$944$	0	0
$945$	6.23901	0.202955
$946$	0	0
$947$	32.5135	1.05655	0.528274	−	0.849074i	$-0.322839\pi$
$947$	32.5135	1.05655	0.528274	+	0.849074i	$0.322839\pi$
$948$	0	0
$949$	−30.7630	−0.998610
$950$	0	0
$951$	−23.5544	−0.763803
$952$	0	0
$953$	11.9201	0.386129	0.193065	−	0.981186i	$-0.438157\pi$
$953$	11.9201	0.386129	0.193065	+	0.981186i	$0.438157\pi$
$954$	0	0
$955$	−8.13999	−0.263404
$956$	0	0
$957$	−59.7218	−1.93053
$958$	0	0
$959$	61.7530	1.99411
$960$	0	0
$961$	65.6650	2.11823
$962$	0	0
$963$	−29.6022	−0.953917
$964$	0	0
$965$	−21.3639	−0.687730
$966$	0	0
$967$	38.9795	1.25350	0.626748	−	0.779222i	$-0.284385\pi$
$967$	38.9795	1.25350	0.626748	+	0.779222i	$0.284385\pi$
$968$	0	0
$969$	−19.3112	−0.620365
$970$	0	0
$971$	53.8195	1.72715	0.863574	−	0.504222i	$-0.168221\pi$
$971$	53.8195	1.72715	0.863574	+	0.504222i	$0.168221\pi$
$972$	0	0
$973$	−41.0479	−1.31593
$974$	0	0
$975$	53.2170	1.70431
$976$	0	0
$977$	−40.5437	−1.29711	−0.648554	−	0.761168i	$-0.724626\pi$
$977$	−40.5437	−1.29711	−0.648554	+	0.761168i	$0.724626\pi$
$978$	0	0
$979$	37.3095	1.19242
$980$	0	0
$981$	−21.1981	−0.676804
$982$	0	0
$983$	−1.96849	−0.0627851	−0.0313926	−	0.999507i	$-0.509994\pi$
$983$	−1.96849	−0.0627851	−0.0313926	+	0.999507i	$0.509994\pi$
$984$	0	0
$985$	−1.62277	−0.0517057
$986$	0	0
$987$	−35.3989	−1.12676
$988$	0	0
$989$	−0.910984	−0.0289676
$990$	0	0
$991$	−34.2204	−1.08705	−0.543524	−	0.839394i	$-0.682910\pi$
$991$	−34.2204	−1.08705	−0.543524	+	0.839394i	$0.682910\pi$
$992$	0	0
$993$	−70.1890	−2.22738
$994$	0	0
$995$	17.9530	0.569148
$996$	0	0
$997$	−7.83214	−0.248046	−0.124023	−	0.992279i	$-0.539580\pi$
$997$	−7.83214	−0.248046	−0.124023	+	0.992279i	$0.539580\pi$
$998$	0	0
$999$	11.8240	0.374093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.13		14
4.3	odd	2		1004.2.a.b.1.2	✓	14
12.11	even	2		9036.2.a.m.1.10		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.2	✓	14	4.3	odd	2
4016.2.a.j.1.13		14	1.1	even	1	trivial
9036.2.a.m.1.10		14	12.11	even	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.20631 q^{3} -0.868402 q^{5} +2.87616 q^{7} +1.86782 q^{9} +O(q^{10})\)	\(q+2.20631 q^{3} -0.868402 q^{5} +2.87616 q^{7} +1.86782 q^{9} -5.76405 q^{11} -5.68088 q^{13} -1.91597 q^{15} -1.64248 q^{17} +5.32894 q^{19} +6.34570 q^{21} +0.134275 q^{23} -4.24588 q^{25} -2.49794 q^{27} +4.69611 q^{29} -9.83184 q^{31} -12.7173 q^{33} -2.49766 q^{35} -4.73348 q^{37} -12.5338 q^{39} +10.6117 q^{41} -6.78447 q^{43} -1.62202 q^{45} -5.57841 q^{47} +1.27227 q^{49} -3.62384 q^{51} -2.13457 q^{53} +5.00551 q^{55} +11.7573 q^{57} +0.764585 q^{59} +11.2665 q^{61} +5.37214 q^{63} +4.93329 q^{65} -5.53676 q^{67} +0.296253 q^{69} -6.03997 q^{71} +5.41518 q^{73} -9.36774 q^{75} -16.5783 q^{77} -4.59972 q^{79} -11.1147 q^{81} +1.93557 q^{83} +1.42634 q^{85} +10.3611 q^{87} -6.47279 q^{89} -16.3391 q^{91} -21.6921 q^{93} -4.62766 q^{95} -6.88404 q^{97} -10.7662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

Introduction

L-functions

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