LMFDB - Embedded newform 4016.2.a.h.1.6

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.540607$ 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)

$f(q)$	$=$	$q+0.540607 q^{3} -2.60958 q^{5} -0.631745 q^{7} -2.70774 q^{9} +O(q^{10})$	$q+0.540607 q^{3} -2.60958 q^{5} -0.631745 q^{7} -2.70774 q^{9} -0.859025 q^{11} +4.66986 q^{13} -1.41076 q^{15} +4.67223 q^{17} +3.41887 q^{19} -0.341526 q^{21} +0.487573 q^{23} +1.80990 q^{25} -3.08565 q^{27} -4.94156 q^{29} -3.02500 q^{31} -0.464395 q^{33} +1.64859 q^{35} +1.07480 q^{37} +2.52456 q^{39} -4.56721 q^{41} -6.99645 q^{43} +7.06607 q^{45} +11.3815 q^{47} -6.60090 q^{49} +2.52584 q^{51} +2.74259 q^{53} +2.24169 q^{55} +1.84827 q^{57} +7.12976 q^{59} -4.59355 q^{61} +1.71060 q^{63} -12.1864 q^{65} -1.62717 q^{67} +0.263586 q^{69} +8.13012 q^{71} -7.67381 q^{73} +0.978444 q^{75} +0.542684 q^{77} +12.9477 q^{79} +6.45511 q^{81} -4.56138 q^{83} -12.1925 q^{85} -2.67145 q^{87} -10.7681 q^{89} -2.95016 q^{91} -1.63534 q^{93} -8.92182 q^{95} -14.7143 q^{97} +2.32602 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.540607	0.312120	0.156060	−	0.987748i	$-0.450121\pi$
$3$	0.540607	0.312120	0.156060	+	0.987748i	$0.450121\pi$
$4$	0	0
$5$	−2.60958	−1.16704	−0.583519	−	0.812099i	$-0.698325\pi$
$5$	−2.60958	−1.16704	−0.583519	+	0.812099i	$0.698325\pi$
$6$	0	0
$7$	−0.631745	−0.238777	−0.119389	−	0.992848i	$-0.538093\pi$
$7$	−0.631745	−0.238777	−0.119389	+	0.992848i	$0.538093\pi$
$8$	0	0
$9$	−2.70774	−0.902581
$10$	0	0
$11$	−0.859025	−0.259006	−0.129503	−	0.991579i	$-0.541338\pi$
$11$	−0.859025	−0.259006	−0.129503	+	0.991579i	$0.541338\pi$
$12$	0	0
$13$	4.66986	1.29519	0.647594	−	0.761986i	$-0.275775\pi$
$13$	4.66986	1.29519	0.647594	+	0.761986i	$0.275775\pi$
$14$	0	0
$15$	−1.41076	−0.364256
$16$	0	0
$17$	4.67223	1.13318	0.566591	−	0.823999i	$-0.308262\pi$
$17$	4.67223	1.13318	0.566591	+	0.823999i	$0.308262\pi$
$18$	0	0
$19$	3.41887	0.784343	0.392172	−	0.919892i	$-0.371724\pi$
$19$	3.41887	0.784343	0.392172	+	0.919892i	$0.371724\pi$
$20$	0	0
$21$	−0.341526	−0.0745271
$22$	0	0
$23$	0.487573	0.101666	0.0508331	−	0.998707i	$-0.483812\pi$
$23$	0.487573	0.101666	0.0508331	+	0.998707i	$0.483812\pi$
$24$	0	0
$25$	1.80990	0.361980
$26$	0	0
$27$	−3.08565	−0.593833
$28$	0	0
$29$	−4.94156	−0.917625	−0.458812	−	0.888533i	$-0.651725\pi$
$29$	−4.94156	−0.917625	−0.458812	+	0.888533i	$0.651725\pi$
$30$	0	0
$31$	−3.02500	−0.543306	−0.271653	−	0.962395i	$-0.587570\pi$
$31$	−3.02500	−0.543306	−0.271653	+	0.962395i	$0.587570\pi$
$32$	0	0
$33$	−0.464395	−0.0808408
$34$	0	0
$35$	1.64859	0.278662
$36$	0	0
$37$	1.07480	0.176696	0.0883478	−	0.996090i	$-0.471841\pi$
$37$	1.07480	0.176696	0.0883478	+	0.996090i	$0.471841\pi$
$38$	0	0
$39$	2.52456	0.404254
$40$	0	0
$41$	−4.56721	−0.713278	−0.356639	−	0.934242i	$-0.616077\pi$
$41$	−4.56721	−0.713278	−0.356639	+	0.934242i	$0.616077\pi$
$42$	0	0
$43$	−6.99645	−1.06695	−0.533475	−	0.845816i	$-0.679114\pi$
$43$	−6.99645	−1.06695	−0.533475	+	0.845816i	$0.679114\pi$
$44$	0	0
$45$	7.06607	1.05335
$46$	0	0
$47$	11.3815	1.66016	0.830082	−	0.557641i	$-0.188293\pi$
$47$	11.3815	1.66016	0.830082	+	0.557641i	$0.188293\pi$
$48$	0	0
$49$	−6.60090	−0.942986
$50$	0	0
$51$	2.52584	0.353688
$52$	0	0
$53$	2.74259	0.376724	0.188362	−	0.982100i	$-0.439682\pi$
$53$	2.74259	0.376724	0.188362	+	0.982100i	$0.439682\pi$
$54$	0	0
$55$	2.24169	0.302270
$56$	0	0
$57$	1.84827	0.244809
$58$	0	0
$59$	7.12976	0.928216	0.464108	−	0.885779i	$-0.346375\pi$
$59$	7.12976	0.928216	0.464108	+	0.885779i	$0.346375\pi$
$60$	0	0
$61$	−4.59355	−0.588143	−0.294072	−	0.955783i	$-0.595010\pi$
$61$	−4.59355	−0.588143	−0.294072	+	0.955783i	$0.595010\pi$
$62$	0	0
$63$	1.71060	0.215516
$64$	0	0
$65$	−12.1864	−1.51153
$66$	0	0
$67$	−1.62717	−0.198791	−0.0993955	−	0.995048i	$-0.531691\pi$
$67$	−1.62717	−0.198791	−0.0993955	+	0.995048i	$0.531691\pi$
$68$	0	0
$69$	0.263586	0.0317320
$70$	0	0
$71$	8.13012	0.964868	0.482434	−	0.875932i	$-0.339753\pi$
$71$	8.13012	0.964868	0.482434	+	0.875932i	$0.339753\pi$
$72$	0	0
$73$	−7.67381	−0.898151	−0.449076	−	0.893494i	$-0.648247\pi$
$73$	−7.67381	−0.898151	−0.449076	+	0.893494i	$0.648247\pi$
$74$	0	0
$75$	0.978444	0.112981
$76$	0	0
$77$	0.542684	0.0618446
$78$	0	0
$79$	12.9477	1.45672	0.728362	−	0.685192i	$-0.240282\pi$
$79$	12.9477	1.45672	0.728362	+	0.685192i	$0.240282\pi$
$80$	0	0
$81$	6.45511	0.717234
$82$	0	0
$83$	−4.56138	−0.500676	−0.250338	−	0.968159i	$-0.580542\pi$
$83$	−4.56138	−0.500676	−0.250338	+	0.968159i	$0.580542\pi$
$84$	0	0
$85$	−12.1925	−1.32247
$86$	0	0
$87$	−2.67145	−0.286409
$88$	0	0
$89$	−10.7681	−1.14142	−0.570710	−	0.821151i	$-0.693332\pi$
$89$	−10.7681	−1.14142	−0.570710	+	0.821151i	$0.693332\pi$
$90$	0	0
$91$	−2.95016	−0.309261
$92$	0	0
$93$	−1.63534	−0.169577
$94$	0	0
$95$	−8.92182	−0.915359
$96$	0	0
$97$	−14.7143	−1.49401	−0.747005	−	0.664819i	$-0.768509\pi$
$97$	−14.7143	−1.49401	−0.747005	+	0.664819i	$0.768509\pi$
$98$	0	0
$99$	2.32602	0.233774
$100$	0	0
$101$	−5.17913	−0.515343	−0.257671	−	0.966233i	$-0.582955\pi$
$101$	−5.17913	−0.515343	−0.257671	+	0.966233i	$0.582955\pi$
$102$	0	0
$103$	−8.16575	−0.804595	−0.402298	−	0.915509i	$-0.631788\pi$
$103$	−8.16575	−0.804595	−0.402298	+	0.915509i	$0.631788\pi$
$104$	0	0
$105$	0.891238	0.0869760
$106$	0	0
$107$	−3.43335	−0.331915	−0.165957	−	0.986133i	$-0.553071\pi$
$107$	−3.43335	−0.331915	−0.165957	+	0.986133i	$0.553071\pi$
$108$	0	0
$109$	−9.89055	−0.947343	−0.473671	−	0.880702i	$-0.657072\pi$
$109$	−9.89055	−0.947343	−0.473671	+	0.880702i	$0.657072\pi$
$110$	0	0
$111$	0.581043	0.0551502
$112$	0	0
$113$	5.59681	0.526504	0.263252	−	0.964727i	$-0.415205\pi$
$113$	5.59681	0.526504	0.263252	+	0.964727i	$0.415205\pi$
$114$	0	0
$115$	−1.27236	−0.118648
$116$	0	0
$117$	−12.6448	−1.16901
$118$	0	0
$119$	−2.95165	−0.270578
$120$	0	0
$121$	−10.2621	−0.932916
$122$	0	0
$123$	−2.46907	−0.222628
$124$	0	0
$125$	8.32482	0.744595
$126$	0	0
$127$	−19.6602	−1.74456	−0.872281	−	0.489004i	$-0.837360\pi$
$127$	−19.6602	−1.74456	−0.872281	+	0.489004i	$0.837360\pi$
$128$	0	0
$129$	−3.78233	−0.333016
$130$	0	0
$131$	6.27985	0.548673	0.274337	−	0.961634i	$-0.411542\pi$
$131$	6.27985	0.548673	0.274337	+	0.961634i	$0.411542\pi$
$132$	0	0
$133$	−2.15985	−0.187283
$134$	0	0
$135$	8.05224	0.693027
$136$	0	0
$137$	−18.8915	−1.61401	−0.807005	−	0.590544i	$-0.798913\pi$
$137$	−18.8915	−1.61401	−0.807005	+	0.590544i	$0.798913\pi$
$138$	0	0
$139$	−21.6128	−1.83318	−0.916588	−	0.399833i	$-0.869068\pi$
$139$	−21.6128	−1.83318	−0.916588	+	0.399833i	$0.869068\pi$
$140$	0	0
$141$	6.15293	0.518170
$142$	0	0
$143$	−4.01153	−0.335461
$144$	0	0
$145$	12.8954	1.07090
$146$	0	0
$147$	−3.56850	−0.294325
$148$	0	0
$149$	−1.76689	−0.144750	−0.0723748	−	0.997378i	$-0.523058\pi$
$149$	−1.76689	−0.144750	−0.0723748	+	0.997378i	$0.523058\pi$
$150$	0	0
$151$	−10.4675	−0.851836	−0.425918	−	0.904762i	$-0.640049\pi$
$151$	−10.4675	−0.851836	−0.425918	+	0.904762i	$0.640049\pi$
$152$	0	0
$153$	−12.6512	−1.02279
$154$	0	0
$155$	7.89398	0.634060
$156$	0	0
$157$	−17.0849	−1.36352	−0.681762	−	0.731574i	$-0.738786\pi$
$157$	−17.0849	−1.36352	−0.681762	+	0.731574i	$0.738786\pi$
$158$	0	0
$159$	1.48266	0.117583
$160$	0	0
$161$	−0.308022	−0.0242755
$162$	0	0
$163$	−2.40052	−0.188023	−0.0940115	−	0.995571i	$-0.529969\pi$
$163$	−2.40052	−0.188023	−0.0940115	+	0.995571i	$0.529969\pi$
$164$	0	0
$165$	1.21188	0.0943444
$166$	0	0
$167$	−8.41619	−0.651264	−0.325632	−	0.945497i	$-0.605577\pi$
$167$	−8.41619	−0.651264	−0.325632	+	0.945497i	$0.605577\pi$
$168$	0	0
$169$	8.80763	0.677510
$170$	0	0
$171$	−9.25743	−0.707933
$172$	0	0
$173$	−7.66938	−0.583092	−0.291546	−	0.956557i	$-0.594170\pi$
$173$	−7.66938	−0.583092	−0.291546	+	0.956557i	$0.594170\pi$
$174$	0	0
$175$	−1.14339	−0.0864324
$176$	0	0
$177$	3.85440	0.289715
$178$	0	0
$179$	20.8290	1.55683	0.778415	−	0.627750i	$-0.216024\pi$
$179$	20.8290	1.55683	0.778415	+	0.627750i	$0.216024\pi$
$180$	0	0
$181$	12.9124	0.959774	0.479887	−	0.877330i	$-0.340678\pi$
$181$	12.9124	0.959774	0.479887	+	0.877330i	$0.340678\pi$
$182$	0	0
$183$	−2.48331	−0.183571
$184$	0	0
$185$	−2.80477	−0.206211
$186$	0	0
$187$	−4.01356	−0.293500
$188$	0	0
$189$	1.94934	0.141794
$190$	0	0
$191$	15.7928	1.14272	0.571362	−	0.820698i	$-0.306415\pi$
$191$	15.7928	1.14272	0.571362	+	0.820698i	$0.306415\pi$
$192$	0	0
$193$	1.06016	0.0763117	0.0381559	−	0.999272i	$-0.487852\pi$
$193$	1.06016	0.0763117	0.0381559	+	0.999272i	$0.487852\pi$
$194$	0	0
$195$	−6.58805	−0.471780
$196$	0	0
$197$	−16.8773	−1.20246	−0.601228	−	0.799078i	$-0.705321\pi$
$197$	−16.8773	−1.20246	−0.601228	+	0.799078i	$0.705321\pi$
$198$	0	0
$199$	0.845321	0.0599232	0.0299616	−	0.999551i	$-0.490461\pi$
$199$	0.845321	0.0599232	0.0299616	+	0.999551i	$0.490461\pi$
$200$	0	0
$201$	−0.879663	−0.0620466
$202$	0	0
$203$	3.12181	0.219108
$204$	0	0
$205$	11.9185	0.832423
$206$	0	0
$207$	−1.32022	−0.0917619
$208$	0	0
$209$	−2.93690	−0.203149
$210$	0	0
$211$	17.2476	1.18738	0.593688	−	0.804695i	$-0.297671\pi$
$211$	17.2476	1.18738	0.593688	+	0.804695i	$0.297671\pi$
$212$	0	0
$213$	4.39520	0.301154
$214$	0	0
$215$	18.2578	1.24517
$216$	0	0
$217$	1.91103	0.129729
$218$	0	0
$219$	−4.14852	−0.280331
$220$	0	0
$221$	21.8187	1.46768
$222$	0	0
$223$	−21.3994	−1.43301	−0.716506	−	0.697581i	$-0.754260\pi$
$223$	−21.3994	−1.43301	−0.716506	+	0.697581i	$0.754260\pi$
$224$	0	0
$225$	−4.90074	−0.326716
$226$	0	0
$227$	7.54044	0.500477	0.250238	−	0.968184i	$-0.419491\pi$
$227$	7.54044	0.500477	0.250238	+	0.968184i	$0.419491\pi$
$228$	0	0
$229$	−13.2203	−0.873619	−0.436810	−	0.899554i	$-0.643892\pi$
$229$	−13.2203	−0.873619	−0.436810	+	0.899554i	$0.643892\pi$
$230$	0	0
$231$	0.293379	0.0193029
$232$	0	0
$233$	−12.8654	−0.842843	−0.421422	−	0.906865i	$-0.638469\pi$
$233$	−12.8654	−0.842843	−0.421422	+	0.906865i	$0.638469\pi$
$234$	0	0
$235$	−29.7010	−1.93748
$236$	0	0
$237$	6.99960	0.454673
$238$	0	0
$239$	0.180371	0.0116672	0.00583362	−	0.999983i	$-0.498143\pi$
$239$	0.180371	0.0116672	0.00583362	+	0.999983i	$0.498143\pi$
$240$	0	0
$241$	−14.1173	−0.909375	−0.454688	−	0.890651i	$-0.650249\pi$
$241$	−14.1173	−0.909375	−0.454688	+	0.890651i	$0.650249\pi$
$242$	0	0
$243$	12.7466	0.817696
$244$	0	0
$245$	17.2256	1.10050
$246$	0	0
$247$	15.9657	1.01587
$248$	0	0
$249$	−2.46591	−0.156271
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.418838	−0.0263321
$254$	0	0
$255$	−6.59138	−0.412768
$256$	0	0
$257$	−12.1136	−0.755627	−0.377814	−	0.925882i	$-0.623324\pi$
$257$	−12.1136	−0.755627	−0.377814	+	0.925882i	$0.623324\pi$
$258$	0	0
$259$	−0.678997	−0.0421908
$260$	0	0
$261$	13.3805	0.828231
$262$	0	0
$263$	−12.2262	−0.753897	−0.376949	−	0.926234i	$-0.623027\pi$
$263$	−12.2262	−0.753897	−0.376949	+	0.926234i	$0.623027\pi$
$264$	0	0
$265$	−7.15700	−0.439651
$266$	0	0
$267$	−5.82134	−0.356260
$268$	0	0
$269$	10.3461	0.630813	0.315407	−	0.948957i	$-0.397859\pi$
$269$	10.3461	0.630813	0.315407	+	0.948957i	$0.397859\pi$
$270$	0	0
$271$	−2.60296	−0.158118	−0.0790592	−	0.996870i	$-0.525192\pi$
$271$	−2.60296	−0.158118	−0.0790592	+	0.996870i	$0.525192\pi$
$272$	0	0
$273$	−1.59488	−0.0965265
$274$	0	0
$275$	−1.55475	−0.0937548
$276$	0	0
$277$	32.4680	1.95081	0.975407	−	0.220412i	$-0.0707402\pi$
$277$	32.4680	1.95081	0.975407	+	0.220412i	$0.0707402\pi$
$278$	0	0
$279$	8.19093	0.490378
$280$	0	0
$281$	−17.9337	−1.06984	−0.534918	−	0.844904i	$-0.679657\pi$
$281$	−17.9337	−1.06984	−0.534918	+	0.844904i	$0.679657\pi$
$282$	0	0
$283$	−28.9824	−1.72282	−0.861412	−	0.507907i	$-0.830419\pi$
$283$	−28.9824	−1.72282	−0.861412	+	0.507907i	$0.830419\pi$
$284$	0	0
$285$	−4.82320	−0.285702
$286$	0	0
$287$	2.88531	0.170314
$288$	0	0
$289$	4.82970	0.284100
$290$	0	0
$291$	−7.95466	−0.466310
$292$	0	0
$293$	14.7745	0.863137	0.431568	−	0.902080i	$-0.357960\pi$
$293$	14.7745	0.863137	0.431568	+	0.902080i	$0.357960\pi$
$294$	0	0
$295$	−18.6057	−1.08326
$296$	0	0
$297$	2.65065	0.153806
$298$	0	0
$299$	2.27690	0.131677
$300$	0	0
$301$	4.41997	0.254763
$302$	0	0
$303$	−2.79988	−0.160849
$304$	0	0
$305$	11.9872	0.686386
$306$	0	0
$307$	26.6735	1.52234	0.761168	−	0.648555i	$-0.224626\pi$
$307$	26.6735	1.52234	0.761168	+	0.648555i	$0.224626\pi$
$308$	0	0
$309$	−4.41447	−0.251130
$310$	0	0
$311$	−14.2433	−0.807662	−0.403831	−	0.914834i	$-0.632322\pi$
$311$	−14.2433	−0.807662	−0.403831	+	0.914834i	$0.632322\pi$
$312$	0	0
$313$	11.5136	0.650790	0.325395	−	0.945578i	$-0.394503\pi$
$313$	11.5136	0.650790	0.325395	+	0.945578i	$0.394503\pi$
$314$	0	0
$315$	−4.46395	−0.251515
$316$	0	0
$317$	30.9253	1.73694	0.868468	−	0.495745i	$-0.165105\pi$
$317$	30.9253	1.73694	0.868468	+	0.495745i	$0.165105\pi$
$318$	0	0
$319$	4.24492	0.237670
$320$	0	0
$321$	−1.85610	−0.103597
$322$	0	0
$323$	15.9737	0.888803
$324$	0	0
$325$	8.45198	0.468831
$326$	0	0
$327$	−5.34690	−0.295684
$328$	0	0
$329$	−7.19021	−0.396409
$330$	0	0
$331$	−24.0454	−1.32165	−0.660827	−	0.750538i	$-0.729794\pi$
$331$	−24.0454	−1.32165	−0.660827	+	0.750538i	$0.729794\pi$
$332$	0	0
$333$	−2.91028	−0.159482
$334$	0	0
$335$	4.24624	0.231997
$336$	0	0
$337$	−1.85036	−0.100796	−0.0503979	−	0.998729i	$-0.516049\pi$
$337$	−1.85036	−0.100796	−0.0503979	+	0.998729i	$0.516049\pi$
$338$	0	0
$339$	3.02568	0.164332
$340$	0	0
$341$	2.59855	0.140719
$342$	0	0
$343$	8.59229	0.463940
$344$	0	0
$345$	−0.687848	−0.0370325
$346$	0	0
$347$	−7.33569	−0.393800	−0.196900	−	0.980424i	$-0.563087\pi$
$347$	−7.33569	−0.393800	−0.196900	+	0.980424i	$0.563087\pi$
$348$	0	0
$349$	−4.14958	−0.222122	−0.111061	−	0.993814i	$-0.535425\pi$
$349$	−4.14958	−0.222122	−0.111061	+	0.993814i	$0.535425\pi$
$350$	0	0
$351$	−14.4096	−0.769125
$352$	0	0
$353$	−23.5578	−1.25385	−0.626927	−	0.779078i	$-0.715688\pi$
$353$	−23.5578	−1.25385	−0.626927	+	0.779078i	$0.715688\pi$
$354$	0	0
$355$	−21.2162	−1.12604
$356$	0	0
$357$	−1.59569	−0.0844527
$358$	0	0
$359$	5.65372	0.298392	0.149196	−	0.988808i	$-0.452331\pi$
$359$	5.65372	0.298392	0.149196	+	0.988808i	$0.452331\pi$
$360$	0	0
$361$	−7.31131	−0.384806
$362$	0	0
$363$	−5.54776	−0.291182
$364$	0	0
$365$	20.0254	1.04818
$366$	0	0
$367$	9.02449	0.471075	0.235537	−	0.971865i	$-0.424315\pi$
$367$	9.02449	0.471075	0.235537	+	0.971865i	$0.424315\pi$
$368$	0	0
$369$	12.3668	0.643791
$370$	0	0
$371$	−1.73262	−0.0899530
$372$	0	0
$373$	−5.81268	−0.300969	−0.150484	−	0.988612i	$-0.548083\pi$
$373$	−5.81268	−0.300969	−0.150484	+	0.988612i	$0.548083\pi$
$374$	0	0
$375$	4.50046	0.232403
$376$	0	0
$377$	−23.0764	−1.18850
$378$	0	0
$379$	22.2042	1.14055	0.570276	−	0.821453i	$-0.306836\pi$
$379$	22.2042	1.14055	0.570276	+	0.821453i	$0.306836\pi$
$380$	0	0
$381$	−10.6285	−0.544513
$382$	0	0
$383$	3.26064	0.166611	0.0833055	−	0.996524i	$-0.473452\pi$
$383$	3.26064	0.166611	0.0833055	+	0.996524i	$0.473452\pi$
$384$	0	0
$385$	−1.41618	−0.0721751
$386$	0	0
$387$	18.9446	0.963008
$388$	0	0
$389$	29.9983	1.52098	0.760488	−	0.649352i	$-0.224960\pi$
$389$	29.9983	1.52098	0.760488	+	0.649352i	$0.224960\pi$
$390$	0	0
$391$	2.27805	0.115206
$392$	0	0
$393$	3.39493	0.171252
$394$	0	0
$395$	−33.7879	−1.70005
$396$	0	0
$397$	8.60873	0.432060	0.216030	−	0.976387i	$-0.430689\pi$
$397$	8.60873	0.432060	0.216030	+	0.976387i	$0.430689\pi$
$398$	0	0
$399$	−1.16763	−0.0584548
$400$	0	0
$401$	−35.7580	−1.78567	−0.892834	−	0.450385i	$-0.851287\pi$
$401$	−35.7580	−1.78567	−0.892834	+	0.450385i	$0.851287\pi$
$402$	0	0
$403$	−14.1263	−0.703684
$404$	0	0
$405$	−16.8451	−0.837040
$406$	0	0
$407$	−0.923277	−0.0457652
$408$	0	0
$409$	17.6286	0.871679	0.435839	−	0.900024i	$-0.356452\pi$
$409$	17.6286	0.871679	0.435839	+	0.900024i	$0.356452\pi$
$410$	0	0
$411$	−10.2129	−0.503765
$412$	0	0
$413$	−4.50419	−0.221637
$414$	0	0
$415$	11.9033	0.584308
$416$	0	0
$417$	−11.6841	−0.572171
$418$	0	0
$419$	−1.55057	−0.0757501	−0.0378750	−	0.999282i	$-0.512059\pi$
$419$	−1.55057	−0.0757501	−0.0378750	+	0.999282i	$0.512059\pi$
$420$	0	0
$421$	22.7835	1.11040	0.555199	−	0.831718i	$-0.312642\pi$
$421$	22.7835	1.11040	0.555199	+	0.831718i	$0.312642\pi$
$422$	0	0
$423$	−30.8182	−1.49843
$424$	0	0
$425$	8.45625	0.410188
$426$	0	0
$427$	2.90195	0.140435
$428$	0	0
$429$	−2.16866	−0.104704
$430$	0	0
$431$	21.7462	1.04748	0.523739	−	0.851879i	$-0.324537\pi$
$431$	21.7462	1.04748	0.523739	+	0.851879i	$0.324537\pi$
$432$	0	0
$433$	0.685686	0.0329520	0.0164760	−	0.999864i	$-0.494755\pi$
$433$	0.685686	0.0329520	0.0164760	+	0.999864i	$0.494755\pi$
$434$	0	0
$435$	6.97134	0.334250
$436$	0	0
$437$	1.66695	0.0797411
$438$	0	0
$439$	23.9255	1.14190	0.570950	−	0.820985i	$-0.306575\pi$
$439$	23.9255	1.14190	0.570950	+	0.820985i	$0.306575\pi$
$440$	0	0
$441$	17.8735	0.851121
$442$	0	0
$443$	21.5409	1.02344	0.511719	−	0.859153i	$-0.329009\pi$
$443$	21.5409	1.02344	0.511719	+	0.859153i	$0.329009\pi$
$444$	0	0
$445$	28.1003	1.33208
$446$	0	0
$447$	−0.955196	−0.0451792
$448$	0	0
$449$	8.17136	0.385630	0.192815	−	0.981235i	$-0.438238\pi$
$449$	8.17136	0.385630	0.192815	+	0.981235i	$0.438238\pi$
$450$	0	0
$451$	3.92334	0.184743
$452$	0	0
$453$	−5.65883	−0.265875
$454$	0	0
$455$	7.69868	0.360920
$456$	0	0
$457$	−10.6133	−0.496469	−0.248234	−	0.968700i	$-0.579850\pi$
$457$	−10.6133	−0.496469	−0.248234	+	0.968700i	$0.579850\pi$
$458$	0	0
$459$	−14.4168	−0.672921
$460$	0	0
$461$	6.54383	0.304777	0.152388	−	0.988321i	$-0.451304\pi$
$461$	6.54383	0.304777	0.152388	+	0.988321i	$0.451304\pi$
$462$	0	0
$463$	18.2518	0.848234	0.424117	−	0.905607i	$-0.360584\pi$
$463$	18.2518	0.848234	0.424117	+	0.905607i	$0.360584\pi$
$464$	0	0
$465$	4.26754	0.197903
$466$	0	0
$467$	21.0350	0.973386	0.486693	−	0.873573i	$-0.338203\pi$
$467$	21.0350	0.973386	0.486693	+	0.873573i	$0.338203\pi$
$468$	0	0
$469$	1.02796	0.0474667
$470$	0	0
$471$	−9.23623	−0.425583
$472$	0	0
$473$	6.01013	0.276346
$474$	0	0
$475$	6.18781	0.283916
$476$	0	0
$477$	−7.42623	−0.340024
$478$	0	0
$479$	11.1818	0.510909	0.255454	−	0.966821i	$-0.417775\pi$
$479$	11.1818	0.510909	0.255454	+	0.966821i	$0.417775\pi$
$480$	0	0
$481$	5.01916	0.228854
$482$	0	0
$483$	−0.166519	−0.00757687
$484$	0	0
$485$	38.3981	1.74357
$486$	0	0
$487$	−8.94485	−0.405330	−0.202665	−	0.979248i	$-0.564960\pi$
$487$	−8.94485	−0.405330	−0.202665	+	0.979248i	$0.564960\pi$
$488$	0	0
$489$	−1.29774	−0.0586857
$490$	0	0
$491$	−30.6880	−1.38493	−0.692464	−	0.721452i	$-0.743475\pi$
$491$	−30.6880	−1.38493	−0.692464	+	0.721452i	$0.743475\pi$
$492$	0	0
$493$	−23.0881	−1.03984
$494$	0	0
$495$	−6.06993	−0.272823
$496$	0	0
$497$	−5.13616	−0.230388
$498$	0	0
$499$	−19.9194	−0.891716	−0.445858	−	0.895104i	$-0.647101\pi$
$499$	−19.9194	−0.891716	−0.445858	+	0.895104i	$0.647101\pi$
$500$	0	0
$501$	−4.54986	−0.203273
$502$	0	0
$503$	−21.9000	−0.976471	−0.488236	−	0.872712i	$-0.662359\pi$
$503$	−21.9000	−0.976471	−0.488236	+	0.872712i	$0.662359\pi$
$504$	0	0
$505$	13.5153	0.601425
$506$	0	0
$507$	4.76147	0.211464
$508$	0	0
$509$	3.49398	0.154868	0.0774339	−	0.996997i	$-0.475327\pi$
$509$	3.49398	0.154868	0.0774339	+	0.996997i	$0.475327\pi$
$510$	0	0
$511$	4.84789	0.214458
$512$	0	0
$513$	−10.5494	−0.465769
$514$	0	0
$515$	21.3092	0.938994
$516$	0	0
$517$	−9.77700	−0.429992
$518$	0	0
$519$	−4.14612	−0.181995
$520$	0	0
$521$	0.764019	0.0334723	0.0167361	−	0.999860i	$-0.494672\pi$
$521$	0.764019	0.0334723	0.0167361	+	0.999860i	$0.494672\pi$
$522$	0	0
$523$	3.03314	0.132630	0.0663150	−	0.997799i	$-0.478876\pi$
$523$	3.03314	0.132630	0.0663150	+	0.997799i	$0.478876\pi$
$524$	0	0
$525$	−0.618127	−0.0269773
$526$	0	0
$527$	−14.1335	−0.615665
$528$	0	0
$529$	−22.7623	−0.989664
$530$	0	0
$531$	−19.3056	−0.837790
$532$	0	0
$533$	−21.3282	−0.923829
$534$	0	0
$535$	8.95960	0.387357
$536$	0	0
$537$	11.2603	0.485917
$538$	0	0
$539$	5.67033	0.244239
$540$	0	0
$541$	11.0520	0.475165	0.237582	−	0.971367i	$-0.423645\pi$
$541$	11.0520	0.475165	0.237582	+	0.971367i	$0.423645\pi$
$542$	0	0
$543$	6.98056	0.299565
$544$	0	0
$545$	25.8102	1.10559
$546$	0	0
$547$	8.85309	0.378531	0.189265	−	0.981926i	$-0.439389\pi$
$547$	8.85309	0.378531	0.189265	+	0.981926i	$0.439389\pi$
$548$	0	0
$549$	12.4381	0.530847
$550$	0	0
$551$	−16.8946	−0.719733
$552$	0	0
$553$	−8.17961	−0.347832
$554$	0	0
$555$	−1.51628	−0.0643624
$556$	0	0
$557$	4.18154	0.177178	0.0885888	−	0.996068i	$-0.471764\pi$
$557$	4.18154	0.177178	0.0885888	+	0.996068i	$0.471764\pi$
$558$	0	0
$559$	−32.6725	−1.38190
$560$	0	0
$561$	−2.16976	−0.0916073
$562$	0	0
$563$	−27.8182	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$563$	−27.8182	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$564$	0	0
$565$	−14.6053	−0.614451
$566$	0	0
$567$	−4.07798	−0.171259
$568$	0	0
$569$	23.7218	0.994470	0.497235	−	0.867616i	$-0.334349\pi$
$569$	23.7218	0.994470	0.497235	+	0.867616i	$0.334349\pi$
$570$	0	0
$571$	11.7126	0.490157	0.245079	−	0.969503i	$-0.421186\pi$
$571$	11.7126	0.490157	0.245079	+	0.969503i	$0.421186\pi$
$572$	0	0
$573$	8.53768	0.356667
$574$	0	0
$575$	0.882458	0.0368010
$576$	0	0
$577$	47.7616	1.98834	0.994171	−	0.107812i	$-0.0343844\pi$
$577$	47.7616	1.98834	0.994171	+	0.107812i	$0.0343844\pi$
$578$	0	0
$579$	0.573128	0.0238184
$580$	0	0
$581$	2.88162	0.119550
$582$	0	0
$583$	−2.35595	−0.0975736
$584$	0	0
$585$	32.9976	1.36428
$586$	0	0
$587$	−7.29049	−0.300911	−0.150455	−	0.988617i	$-0.548074\pi$
$587$	−7.29049	−0.300911	−0.150455	+	0.988617i	$0.548074\pi$
$588$	0	0
$589$	−10.3421	−0.426139
$590$	0	0
$591$	−9.12397	−0.375310
$592$	0	0
$593$	−32.6302	−1.33996	−0.669981	−	0.742378i	$-0.733698\pi$
$593$	−32.6302	−1.33996	−0.669981	+	0.742378i	$0.733698\pi$
$594$	0	0
$595$	7.70257	0.315775
$596$	0	0
$597$	0.456987	0.0187032
$598$	0	0
$599$	1.60745	0.0656785	0.0328393	−	0.999461i	$-0.489545\pi$
$599$	1.60745	0.0656785	0.0328393	+	0.999461i	$0.489545\pi$
$600$	0	0
$601$	10.5749	0.431361	0.215680	−	0.976464i	$-0.430803\pi$
$601$	10.5749	0.431361	0.215680	+	0.976464i	$0.430803\pi$
$602$	0	0
$603$	4.40597	0.179425
$604$	0	0
$605$	26.7797	1.08875
$606$	0	0
$607$	−14.2908	−0.580046	−0.290023	−	0.957020i	$-0.593663\pi$
$607$	−14.2908	−0.580046	−0.290023	+	0.957020i	$0.593663\pi$
$608$	0	0
$609$	1.68767	0.0683879
$610$	0	0
$611$	53.1501	2.15022
$612$	0	0
$613$	−39.8312	−1.60877	−0.804383	−	0.594111i	$-0.797504\pi$
$613$	−39.8312	−1.60877	−0.804383	+	0.594111i	$0.797504\pi$
$614$	0	0
$615$	6.44322	0.259816
$616$	0	0
$617$	23.3095	0.938406	0.469203	−	0.883090i	$-0.344541\pi$
$617$	23.3095	0.938406	0.469203	+	0.883090i	$0.344541\pi$
$618$	0	0
$619$	−37.3950	−1.50303	−0.751516	−	0.659715i	$-0.770677\pi$
$619$	−37.3950	−1.50303	−0.751516	+	0.659715i	$0.770677\pi$
$620$	0	0
$621$	−1.50448	−0.0603727
$622$	0	0
$623$	6.80271	0.272545
$624$	0	0
$625$	−30.7738	−1.23095
$626$	0	0
$627$	−1.58771	−0.0634069
$628$	0	0
$629$	5.02170	0.200228
$630$	0	0
$631$	−10.3584	−0.412360	−0.206180	−	0.978514i	$-0.566103\pi$
$631$	−10.3584	−0.412360	−0.206180	+	0.978514i	$0.566103\pi$
$632$	0	0
$633$	9.32420	0.370604
$634$	0	0
$635$	51.3049	2.03597
$636$	0	0
$637$	−30.8253	−1.22134
$638$	0	0
$639$	−22.0143	−0.870872
$640$	0	0
$641$	−2.35144	−0.0928763	−0.0464381	−	0.998921i	$-0.514787\pi$
$641$	−2.35144	−0.0928763	−0.0464381	+	0.998921i	$0.514787\pi$
$642$	0	0
$643$	21.9708	0.866445	0.433223	−	0.901287i	$-0.357376\pi$
$643$	21.9708	0.866445	0.433223	+	0.901287i	$0.357376\pi$
$644$	0	0
$645$	9.87030	0.388643
$646$	0	0
$647$	17.7477	0.697732	0.348866	−	0.937173i	$-0.386567\pi$
$647$	17.7477	0.697732	0.348866	+	0.937173i	$0.386567\pi$
$648$	0	0
$649$	−6.12464	−0.240413
$650$	0	0
$651$	1.03312	0.0404910
$652$	0	0
$653$	18.1838	0.711586	0.355793	−	0.934565i	$-0.384211\pi$
$653$	18.1838	0.711586	0.355793	+	0.934565i	$0.384211\pi$
$654$	0	0
$655$	−16.3878	−0.640323
$656$	0	0
$657$	20.7787	0.810654
$658$	0	0
$659$	1.90777	0.0743160	0.0371580	−	0.999309i	$-0.488170\pi$
$659$	1.90777	0.0743160	0.0371580	+	0.999309i	$0.488170\pi$
$660$	0	0
$661$	19.8959	0.773862	0.386931	−	0.922109i	$-0.373535\pi$
$661$	19.8959	0.773862	0.386931	+	0.922109i	$0.373535\pi$
$662$	0	0
$663$	11.7953	0.458093
$664$	0	0
$665$	5.63631	0.218567
$666$	0	0
$667$	−2.40937	−0.0932914
$668$	0	0
$669$	−11.5687	−0.447271
$670$	0	0
$671$	3.94597	0.152332
$672$	0	0
$673$	29.1461	1.12350	0.561749	−	0.827308i	$-0.310129\pi$
$673$	29.1461	1.12350	0.561749	+	0.827308i	$0.310129\pi$
$674$	0	0
$675$	−5.58471	−0.214956
$676$	0	0
$677$	5.84506	0.224644	0.112322	−	0.993672i	$-0.464171\pi$
$677$	5.84506	0.224644	0.112322	+	0.993672i	$0.464171\pi$
$678$	0	0
$679$	9.29567	0.356735
$680$	0	0
$681$	4.07642	0.156209
$682$	0	0
$683$	−8.54970	−0.327145	−0.163573	−	0.986531i	$-0.552302\pi$
$683$	−8.54970	−0.327145	−0.163573	+	0.986531i	$0.552302\pi$
$684$	0	0
$685$	49.2989	1.88361
$686$	0	0
$687$	−7.14697	−0.272674
$688$	0	0
$689$	12.8075	0.487928
$690$	0	0
$691$	−40.3965	−1.53676	−0.768378	−	0.639997i	$-0.778936\pi$
$691$	−40.3965	−1.53676	−0.768378	+	0.639997i	$0.778936\pi$
$692$	0	0
$693$	−1.46945	−0.0558198
$694$	0	0
$695$	56.4004	2.13939
$696$	0	0
$697$	−21.3390	−0.808273
$698$	0	0
$699$	−6.95516	−0.263068
$700$	0	0
$701$	−8.90835	−0.336464	−0.168232	−	0.985747i	$-0.553806\pi$
$701$	−8.90835	−0.336464	−0.168232	+	0.985747i	$0.553806\pi$
$702$	0	0
$703$	3.67460	0.138590
$704$	0	0
$705$	−16.0566	−0.604725
$706$	0	0
$707$	3.27189	0.123052
$708$	0	0
$709$	−25.7686	−0.967759	−0.483880	−	0.875135i	$-0.660773\pi$
$709$	−25.7686	−0.967759	−0.483880	+	0.875135i	$0.660773\pi$
$710$	0	0
$711$	−35.0589	−1.31481
$712$	0	0
$713$	−1.47491	−0.0552358
$714$	0	0
$715$	10.4684	0.391496
$716$	0	0
$717$	0.0975099	0.00364158
$718$	0	0
$719$	12.3119	0.459155	0.229578	−	0.973290i	$-0.426266\pi$
$719$	12.3119	0.459155	0.229578	+	0.973290i	$0.426266\pi$
$720$	0	0
$721$	5.15867	0.192119
$722$	0	0
$723$	−7.63192	−0.283834
$724$	0	0
$725$	−8.94372	−0.332161
$726$	0	0
$727$	20.0383	0.743178	0.371589	−	0.928397i	$-0.378813\pi$
$727$	20.0383	0.743178	0.371589	+	0.928397i	$0.378813\pi$
$728$	0	0
$729$	−12.4744	−0.462015
$730$	0	0
$731$	−32.6890	−1.20905
$732$	0	0
$733$	5.42412	0.200345	0.100172	−	0.994970i	$-0.468061\pi$
$733$	5.42412	0.200345	0.100172	+	0.994970i	$0.468061\pi$
$734$	0	0
$735$	9.31227	0.343488
$736$	0	0
$737$	1.39778	0.0514880
$738$	0	0
$739$	−15.4857	−0.569651	−0.284826	−	0.958579i	$-0.591936\pi$
$739$	−15.4857	−0.569651	−0.284826	+	0.958579i	$0.591936\pi$
$740$	0	0
$741$	8.63116	0.317074
$742$	0	0
$743$	−23.9745	−0.879538	−0.439769	−	0.898111i	$-0.644940\pi$
$743$	−23.9745	−0.879538	−0.439769	+	0.898111i	$0.644940\pi$
$744$	0	0
$745$	4.61085	0.168928
$746$	0	0
$747$	12.3510	0.451901
$748$	0	0
$749$	2.16900	0.0792536
$750$	0	0
$751$	−22.3515	−0.815618	−0.407809	−	0.913067i	$-0.633707\pi$
$751$	−22.3515	−0.815618	−0.407809	+	0.913067i	$0.633707\pi$
$752$	0	0
$753$	−0.540607	−0.0197008
$754$	0	0
$755$	27.3158	0.994126
$756$	0	0
$757$	24.6193	0.894803	0.447401	−	0.894333i	$-0.352350\pi$
$757$	24.6193	0.894803	0.447401	+	0.894333i	$0.352350\pi$
$758$	0	0
$759$	−0.226427	−0.00821877
$760$	0	0
$761$	−27.5996	−1.00048	−0.500242	−	0.865885i	$-0.666756\pi$
$761$	−27.5996	−1.00048	−0.500242	+	0.865885i	$0.666756\pi$
$762$	0	0
$763$	6.24830	0.226204
$764$	0	0
$765$	33.0143	1.19363
$766$	0	0
$767$	33.2950	1.20221
$768$	0	0
$769$	−11.5356	−0.415985	−0.207992	−	0.978130i	$-0.566693\pi$
$769$	−11.5356	−0.415985	−0.207992	+	0.978130i	$0.566693\pi$
$770$	0	0
$771$	−6.54872	−0.235846
$772$	0	0
$773$	−7.37491	−0.265257	−0.132629	−	0.991166i	$-0.542342\pi$
$773$	−7.37491	−0.265257	−0.132629	+	0.991166i	$0.542342\pi$
$774$	0	0
$775$	−5.47494	−0.196666
$776$	0	0
$777$	−0.367071	−0.0131686
$778$	0	0
$779$	−15.6147	−0.559455
$780$	0	0
$781$	−6.98397	−0.249906
$782$	0	0
$783$	15.2479	0.544916
$784$	0	0
$785$	44.5844	1.59129
$786$	0	0
$787$	14.2397	0.507591	0.253796	−	0.967258i	$-0.418321\pi$
$787$	14.2397	0.507591	0.253796	+	0.967258i	$0.418321\pi$
$788$	0	0
$789$	−6.60955	−0.235306
$790$	0	0
$791$	−3.53576	−0.125717
$792$	0	0
$793$	−21.4512	−0.761756
$794$	0	0
$795$	−3.86913	−0.137224
$796$	0	0
$797$	−55.4832	−1.96532	−0.982658	−	0.185429i	$-0.940633\pi$
$797$	−55.4832	−1.96532	−0.982658	+	0.185429i	$0.940633\pi$
$798$	0	0
$799$	53.1770	1.88127
$800$	0	0
$801$	29.1574	1.03022
$802$	0	0
$803$	6.59199	0.232626
$804$	0	0
$805$	0.803807	0.0283305
$806$	0	0
$807$	5.59318	0.196889
$808$	0	0
$809$	−28.7148	−1.00956	−0.504779	−	0.863249i	$-0.668426\pi$
$809$	−28.7148	−1.00956	−0.504779	+	0.863249i	$0.668426\pi$
$810$	0	0
$811$	38.6207	1.35615	0.678077	−	0.734991i	$-0.262813\pi$
$811$	38.6207	1.35615	0.678077	+	0.734991i	$0.262813\pi$
$812$	0	0
$813$	−1.40718	−0.0493519
$814$	0	0
$815$	6.26434	0.219430
$816$	0	0
$817$	−23.9200	−0.836854
$818$	0	0
$819$	7.98828	0.279133
$820$	0	0
$821$	−20.1838	−0.704421	−0.352211	−	0.935921i	$-0.614570\pi$
$821$	−20.1838	−0.704421	−0.352211	+	0.935921i	$0.614570\pi$
$822$	0	0
$823$	11.1826	0.389803	0.194901	−	0.980823i	$-0.437561\pi$
$823$	11.1826	0.389803	0.194901	+	0.980823i	$0.437561\pi$
$824$	0	0
$825$	−0.840508	−0.0292627
$826$	0	0
$827$	43.6393	1.51749	0.758743	−	0.651390i	$-0.225814\pi$
$827$	43.6393	1.51749	0.758743	+	0.651390i	$0.225814\pi$
$828$	0	0
$829$	−34.7942	−1.20845	−0.604227	−	0.796812i	$-0.706518\pi$
$829$	−34.7942	−1.20845	−0.604227	+	0.796812i	$0.706518\pi$
$830$	0	0
$831$	17.5524	0.608888
$832$	0	0
$833$	−30.8409	−1.06857
$834$	0	0
$835$	21.9627	0.760051
$836$	0	0
$837$	9.33409	0.322633
$838$	0	0
$839$	6.89990	0.238211	0.119106	−	0.992882i	$-0.461997\pi$
$839$	6.89990	0.238211	0.119106	+	0.992882i	$0.461997\pi$
$840$	0	0
$841$	−4.58097	−0.157964
$842$	0	0
$843$	−9.69510	−0.333917
$844$	0	0
$845$	−22.9842	−0.790681
$846$	0	0
$847$	6.48301	0.222759
$848$	0	0
$849$	−15.6681	−0.537727
$850$	0	0
$851$	0.524043	0.0179640
$852$	0	0
$853$	−12.9664	−0.443963	−0.221981	−	0.975051i	$-0.571252\pi$
$853$	−12.9664	−0.443963	−0.221981	+	0.975051i	$0.571252\pi$
$854$	0	0
$855$	24.1580	0.826186
$856$	0	0
$857$	−37.0411	−1.26530	−0.632649	−	0.774438i	$-0.718033\pi$
$857$	−37.0411	−1.26530	−0.632649	+	0.774438i	$0.718033\pi$
$858$	0	0
$859$	−40.2211	−1.37233	−0.686164	−	0.727447i	$-0.740707\pi$
$859$	−40.2211	−1.37233	−0.686164	+	0.727447i	$0.740707\pi$
$860$	0	0
$861$	1.55982	0.0531585
$862$	0	0
$863$	−29.6044	−1.00775	−0.503873	−	0.863778i	$-0.668092\pi$
$863$	−29.6044	−1.00775	−0.503873	+	0.863778i	$0.668092\pi$
$864$	0	0
$865$	20.0138	0.680491
$866$	0	0
$867$	2.61097	0.0886733
$868$	0	0
$869$	−11.1224	−0.377300
$870$	0	0
$871$	−7.59869	−0.257472
$872$	0	0
$873$	39.8425	1.34847
$874$	0	0
$875$	−5.25916	−0.177792
$876$	0	0
$877$	−34.8435	−1.17658	−0.588291	−	0.808649i	$-0.700199\pi$
$877$	−34.8435	−1.17658	−0.588291	+	0.808649i	$0.700199\pi$
$878$	0	0
$879$	7.98721	0.269402
$880$	0	0
$881$	−16.1133	−0.542872	−0.271436	−	0.962457i	$-0.587499\pi$
$881$	−16.1133	−0.542872	−0.271436	+	0.962457i	$0.587499\pi$
$882$	0	0
$883$	27.1499	0.913667	0.456833	−	0.889552i	$-0.348984\pi$
$883$	27.1499	0.913667	0.456833	+	0.889552i	$0.348984\pi$
$884$	0	0
$885$	−10.0584	−0.338108
$886$	0	0
$887$	−20.5308	−0.689356	−0.344678	−	0.938721i	$-0.612012\pi$
$887$	−20.5308	−0.689356	−0.344678	+	0.938721i	$0.612012\pi$
$888$	0	0
$889$	12.4202	0.416562
$890$	0	0
$891$	−5.54510	−0.185768
$892$	0	0
$893$	38.9120	1.30214
$894$	0	0
$895$	−54.3548	−1.81688
$896$	0	0
$897$	1.23091	0.0410989
$898$	0	0
$899$	14.9482	0.498551
$900$	0	0
$901$	12.8140	0.426896
$902$	0	0
$903$	2.38947	0.0795166
$904$	0	0
$905$	−33.6960	−1.12009
$906$	0	0
$907$	4.74897	0.157687	0.0788435	−	0.996887i	$-0.474877\pi$
$907$	4.74897	0.157687	0.0788435	+	0.996887i	$0.474877\pi$
$908$	0	0
$909$	14.0238	0.465139
$910$	0	0
$911$	49.6937	1.64642	0.823212	−	0.567734i	$-0.192180\pi$
$911$	49.6937	1.64642	0.823212	+	0.567734i	$0.192180\pi$
$912$	0	0
$913$	3.91833	0.129678
$914$	0	0
$915$	6.48038	0.214235
$916$	0	0
$917$	−3.96726	−0.131011
$918$	0	0
$919$	−38.8669	−1.28210	−0.641050	−	0.767499i	$-0.721501\pi$
$919$	−38.8669	−1.28210	−0.641050	+	0.767499i	$0.721501\pi$
$920$	0	0
$921$	14.4199	0.475151
$922$	0	0
$923$	37.9666	1.24968
$924$	0	0
$925$	1.94527	0.0639602
$926$	0	0
$927$	22.1108	0.726212
$928$	0	0
$929$	39.8245	1.30660	0.653300	−	0.757099i	$-0.273384\pi$
$929$	39.8245	1.30660	0.653300	+	0.757099i	$0.273384\pi$
$930$	0	0
$931$	−22.5676	−0.739624
$932$	0	0
$933$	−7.70002	−0.252087
$934$	0	0
$935$	10.4737	0.342526
$936$	0	0
$937$	−3.57012	−0.116631	−0.0583153	−	0.998298i	$-0.518573\pi$
$937$	−3.57012	−0.116631	−0.0583153	+	0.998298i	$0.518573\pi$
$938$	0	0
$939$	6.22436	0.203124
$940$	0	0
$941$	17.6601	0.575703	0.287852	−	0.957675i	$-0.407059\pi$
$941$	17.6601	0.575703	0.287852	+	0.957675i	$0.407059\pi$
$942$	0	0
$943$	−2.22685	−0.0725162
$944$	0	0
$945$	−5.08696	−0.165479
$946$	0	0
$947$	30.4892	0.990766	0.495383	−	0.868675i	$-0.335028\pi$
$947$	30.4892	0.990766	0.495383	+	0.868675i	$0.335028\pi$
$948$	0	0
$949$	−35.8356	−1.16327
$950$	0	0
$951$	16.7184	0.542132
$952$	0	0
$953$	−50.7554	−1.64413	−0.822064	−	0.569395i	$-0.807177\pi$
$953$	−50.7554	−1.64413	−0.822064	+	0.569395i	$0.807177\pi$
$954$	0	0
$955$	−41.2124	−1.33360
$956$	0	0
$957$	2.29484	0.0741816
$958$	0	0
$959$	11.9346	0.385389
$960$	0	0
$961$	−21.8494	−0.704818
$962$	0	0
$963$	9.29663	0.299580
$964$	0	0
$965$	−2.76656	−0.0890587
$966$	0	0
$967$	38.6082	1.24156	0.620778	−	0.783986i	$-0.286817\pi$
$967$	38.6082	1.24156	0.620778	+	0.783986i	$0.286817\pi$
$968$	0	0
$969$	8.63553	0.277413
$970$	0	0
$971$	−6.35123	−0.203821	−0.101910	−	0.994794i	$-0.532495\pi$
$971$	−6.35123	−0.203821	−0.101910	+	0.994794i	$0.532495\pi$
$972$	0	0
$973$	13.6538	0.437720
$974$	0	0
$975$	4.56920	0.146332
$976$	0	0
$977$	−26.5796	−0.850355	−0.425178	−	0.905110i	$-0.639788\pi$
$977$	−26.5796	−0.850355	−0.425178	+	0.905110i	$0.639788\pi$
$978$	0	0
$979$	9.25010	0.295634
$980$	0	0
$981$	26.7811	0.855054
$982$	0	0
$983$	−53.4075	−1.70344	−0.851718	−	0.524001i	$-0.824439\pi$
$983$	−53.4075	−1.70344	−0.851718	+	0.524001i	$0.824439\pi$
$984$	0	0
$985$	44.0425	1.40331
$986$	0	0
$987$	−3.88708	−0.123727
$988$	0	0
$989$	−3.41128	−0.108473
$990$	0	0
$991$	−48.6988	−1.54697	−0.773485	−	0.633815i	$-0.781488\pi$
$991$	−48.6988	−1.54697	−0.773485	+	0.633815i	$0.781488\pi$
$992$	0	0
$993$	−12.9991	−0.412514
$994$	0	0
$995$	−2.20593	−0.0699327
$996$	0	0
$997$	−33.8680	−1.07261	−0.536305	−	0.844025i	$-0.680180\pi$
$997$	−33.8680	−1.07261	−0.536305	+	0.844025i	$0.680180\pi$
$998$	0	0
$999$	−3.31645	−0.104928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.6		9
4.3	odd	2		2008.2.a.a.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.4	✓	9	4.3	odd	2
4016.2.a.h.1.6		9	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.540607 q^{3} -2.60958 q^{5} -0.631745 q^{7} -2.70774 q^{9} +O(q^{10})\)	\(q+0.540607 q^{3} -2.60958 q^{5} -0.631745 q^{7} -2.70774 q^{9} -0.859025 q^{11} +4.66986 q^{13} -1.41076 q^{15} +4.67223 q^{17} +3.41887 q^{19} -0.341526 q^{21} +0.487573 q^{23} +1.80990 q^{25} -3.08565 q^{27} -4.94156 q^{29} -3.02500 q^{31} -0.464395 q^{33} +1.64859 q^{35} +1.07480 q^{37} +2.52456 q^{39} -4.56721 q^{41} -6.99645 q^{43} +7.06607 q^{45} +11.3815 q^{47} -6.60090 q^{49} +2.52584 q^{51} +2.74259 q^{53} +2.24169 q^{55} +1.84827 q^{57} +7.12976 q^{59} -4.59355 q^{61} +1.71060 q^{63} -12.1864 q^{65} -1.62717 q^{67} +0.263586 q^{69} +8.13012 q^{71} -7.67381 q^{73} +0.978444 q^{75} +0.542684 q^{77} +12.9477 q^{79} +6.45511 q^{81} -4.56138 q^{83} -12.1925 q^{85} -2.67145 q^{87} -10.7681 q^{89} -2.95016 q^{91} -1.63534 q^{93} -8.92182 q^{95} -14.7143 q^{97} +2.32602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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