LMFDB - Embedded newform 6016.2.a.r.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, 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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
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} - 2 x^{13} - 30 x^{12} + 56 x^{11} + 331 x^{10} - 562 x^{9} - 1630 x^{8} + 2458 x^{7} + \cdots + 128 $ x^14 - 2x^13 - 30x^12 + 56x^11 + 331x^10 - 562x^9 - 1630x^8 + 2458x^7 + 3313x^6 - 4512x^5 - 1612x^4 + 2960x^3 - 256x^2 - 512*x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$-3.32999$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.32999 q^{3} +4.19635 q^{5} -3.58783 q^{7} +8.08881 q^{9} +O(q^{10})$	$q+3.32999 q^{3} +4.19635 q^{5} -3.58783 q^{7} +8.08881 q^{9} -3.64415 q^{11} +2.34285 q^{13} +13.9738 q^{15} -4.50124 q^{17} -0.900429 q^{19} -11.9474 q^{21} +5.94493 q^{23} +12.6093 q^{25} +16.9457 q^{27} +5.93076 q^{29} -2.66525 q^{31} -12.1350 q^{33} -15.0558 q^{35} +8.52350 q^{37} +7.80165 q^{39} -0.502777 q^{41} -4.63110 q^{43} +33.9435 q^{45} +1.00000 q^{47} +5.87250 q^{49} -14.9891 q^{51} +8.85250 q^{53} -15.2921 q^{55} -2.99842 q^{57} +5.61553 q^{59} +3.75806 q^{61} -29.0213 q^{63} +9.83140 q^{65} +9.66989 q^{67} +19.7965 q^{69} -15.5624 q^{71} -12.9563 q^{73} +41.9889 q^{75} +13.0746 q^{77} +8.82925 q^{79} +32.1624 q^{81} -10.8266 q^{83} -18.8888 q^{85} +19.7494 q^{87} +0.284535 q^{89} -8.40573 q^{91} -8.87524 q^{93} -3.77851 q^{95} -13.5774 q^{97} -29.4768 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) $ 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9	$ 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9} - 14 q^{11} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} + 6 q^{21} + 18 q^{23} + 22 q^{25} - 8 q^{27} + 22 q^{29} + 4 q^{31} + 2 q^{33} - 26 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} - 12 q^{43} + 30 q^{45} + 14 q^{47} + 34 q^{49} - 18 q^{51} + 20 q^{53} + 2 q^{55} - 4 q^{57} - 32 q^{59} + 12 q^{61} + 40 q^{65} - 16 q^{67} + 46 q^{69} + 16 q^{71} - 12 q^{73} + 16 q^{75} + 10 q^{77} + 16 q^{79} + 74 q^{81} - 14 q^{83} + 12 q^{85} + 14 q^{87} - 28 q^{91} - 16 q^{93} + 52 q^{95} + 28 q^{97} - 50 q^{99}+O(q^{100}) $ 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9 - 14 * q^11 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 + 6 * q^21 + 18 * q^23 + 22 * q^25 - 8 * q^27 + 22 * q^29 + 4 * q^31 + 2 * q^33 - 26 * q^35 + 6 * q^37 + 20 * q^39 + 16 * q^41 - 12 * q^43 + 30 * q^45 + 14 * q^47 + 34 * q^49 - 18 * q^51 + 20 * q^53 + 2 * q^55 - 4 * q^57 - 32 * q^59 + 12 * q^61 + 40 * q^65 - 16 * q^67 + 46 * q^69 + 16 * q^71 - 12 * q^73 + 16 * q^75 + 10 * q^77 + 16 * q^79 + 74 * q^81 - 14 * q^83 + 12 * q^85 + 14 * q^87 - 28 * q^91 - 16 * q^93 + 52 * q^95 + 28 * q^97 - 50 * 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
$3$	3.32999	1.92257	0.961284	−	0.275558i	$-0.0888627\pi$
$3$	3.32999	1.92257	0.961284	+	0.275558i	$0.0888627\pi$
$4$	0	0
$5$	4.19635	1.87666	0.938332	−	0.345735i	$-0.112371\pi$
$5$	4.19635	1.87666	0.938332	+	0.345735i	$0.112371\pi$
$6$	0	0
$7$	−3.58783	−1.35607	−0.678036	−	0.735029i	$-0.737168\pi$
$7$	−3.58783	−1.35607	−0.678036	+	0.735029i	$0.737168\pi$
$8$	0	0
$9$	8.08881	2.69627
$10$	0	0
$11$	−3.64415	−1.09875	−0.549376	−	0.835575i	$-0.685135\pi$
$11$	−3.64415	−1.09875	−0.549376	+	0.835575i	$0.685135\pi$
$12$	0	0
$13$	2.34285	0.649789	0.324894	−	0.945750i	$-0.394671\pi$
$13$	2.34285	0.649789	0.324894	+	0.945750i	$0.394671\pi$
$14$	0	0
$15$	13.9738	3.60802
$16$	0	0
$17$	−4.50124	−1.09171	−0.545855	−	0.837880i	$-0.683795\pi$
$17$	−4.50124	−1.09171	−0.545855	+	0.837880i	$0.683795\pi$
$18$	0	0
$19$	−0.900429	−0.206572	−0.103286	−	0.994652i	$-0.532936\pi$
$19$	−0.900429	−0.206572	−0.103286	+	0.994652i	$0.532936\pi$
$20$	0	0
$21$	−11.9474	−2.60714
$22$	0	0
$23$	5.94493	1.23960	0.619802	−	0.784759i	$-0.287213\pi$
$23$	5.94493	1.23960	0.619802	+	0.784759i	$0.287213\pi$
$24$	0	0
$25$	12.6093	2.52187
$26$	0	0
$27$	16.9457	3.26120
$28$	0	0
$29$	5.93076	1.10132	0.550658	−	0.834731i	$-0.314377\pi$
$29$	5.93076	1.10132	0.550658	+	0.834731i	$0.314377\pi$
$30$	0	0
$31$	−2.66525	−0.478693	−0.239346	−	0.970934i	$-0.576933\pi$
$31$	−2.66525	−0.478693	−0.239346	+	0.970934i	$0.576933\pi$
$32$	0	0
$33$	−12.1350	−2.11243
$34$	0	0
$35$	−15.0558	−2.54489
$36$	0	0
$37$	8.52350	1.40125	0.700627	−	0.713528i	$-0.252904\pi$
$37$	8.52350	1.40125	0.700627	+	0.713528i	$0.252904\pi$
$38$	0	0
$39$	7.80165	1.24926
$40$	0	0
$41$	−0.502777	−0.0785205	−0.0392603	−	0.999229i	$-0.512500\pi$
$41$	−0.502777	−0.0785205	−0.0392603	+	0.999229i	$0.512500\pi$
$42$	0	0
$43$	−4.63110	−0.706236	−0.353118	−	0.935579i	$-0.614879\pi$
$43$	−4.63110	−0.706236	−0.353118	+	0.935579i	$0.614879\pi$
$44$	0	0
$45$	33.9435	5.05999
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	5.87250	0.838929
$50$	0	0
$51$	−14.9891	−2.09889
$52$	0	0
$53$	8.85250	1.21598	0.607992	−	0.793943i	$-0.291975\pi$
$53$	8.85250	1.21598	0.607992	+	0.793943i	$0.291975\pi$
$54$	0	0
$55$	−15.2921	−2.06199
$56$	0	0
$57$	−2.99842	−0.397150
$58$	0	0
$59$	5.61553	0.731080	0.365540	−	0.930796i	$-0.380884\pi$
$59$	5.61553	0.731080	0.365540	+	0.930796i	$0.380884\pi$
$60$	0	0
$61$	3.75806	0.481170	0.240585	−	0.970628i	$-0.422661\pi$
$61$	3.75806	0.481170	0.240585	+	0.970628i	$0.422661\pi$
$62$	0	0
$63$	−29.0213	−3.65633
$64$	0	0
$65$	9.83140	1.21943
$66$	0	0
$67$	9.66989	1.18137	0.590683	−	0.806904i	$-0.298858\pi$
$67$	9.66989	1.18137	0.590683	+	0.806904i	$0.298858\pi$
$68$	0	0
$69$	19.7965	2.38322
$70$	0	0
$71$	−15.5624	−1.84692	−0.923460	−	0.383695i	$-0.874651\pi$
$71$	−15.5624	−1.84692	−0.923460	+	0.383695i	$0.874651\pi$
$72$	0	0
$73$	−12.9563	−1.51641	−0.758207	−	0.652014i	$-0.773924\pi$
$73$	−12.9563	−1.51641	−0.758207	+	0.652014i	$0.773924\pi$
$74$	0	0
$75$	41.9889	4.84846
$76$	0	0
$77$	13.0746	1.48999
$78$	0	0
$79$	8.82925	0.993368	0.496684	−	0.867931i	$-0.334551\pi$
$79$	8.82925	0.993368	0.496684	+	0.867931i	$0.334551\pi$
$80$	0	0
$81$	32.1624	3.57361
$82$	0	0
$83$	−10.8266	−1.18838	−0.594189	−	0.804326i	$-0.702527\pi$
$83$	−10.8266	−1.18838	−0.594189	+	0.804326i	$0.702527\pi$
$84$	0	0
$85$	−18.8888	−2.04877
$86$	0	0
$87$	19.7494	2.11735
$88$	0	0
$89$	0.284535	0.0301607	0.0150804	−	0.999886i	$-0.495200\pi$
$89$	0.284535	0.0301607	0.0150804	+	0.999886i	$0.495200\pi$
$90$	0	0
$91$	−8.40573	−0.881159
$92$	0	0
$93$	−8.87524	−0.920320
$94$	0	0
$95$	−3.77851	−0.387667
$96$	0	0
$97$	−13.5774	−1.37858	−0.689289	−	0.724487i	$-0.742077\pi$
$97$	−13.5774	−1.37858	−0.689289	+	0.724487i	$0.742077\pi$
$98$	0	0
$99$	−29.4768	−2.96253
$100$	0	0
$101$	5.83519	0.580623	0.290312	−	0.956932i	$-0.406241\pi$
$101$	5.83519	0.580623	0.290312	+	0.956932i	$0.406241\pi$
$102$	0	0
$103$	4.24069	0.417848	0.208924	−	0.977932i	$-0.433004\pi$
$103$	4.24069	0.417848	0.208924	+	0.977932i	$0.433004\pi$
$104$	0	0
$105$	−50.1355	−4.89273
$106$	0	0
$107$	14.2944	1.38189	0.690944	−	0.722909i	$-0.257195\pi$
$107$	14.2944	1.38189	0.690944	+	0.722909i	$0.257195\pi$
$108$	0	0
$109$	−4.62855	−0.443335	−0.221667	−	0.975122i	$-0.571150\pi$
$109$	−4.62855	−0.443335	−0.221667	+	0.975122i	$0.571150\pi$
$110$	0	0
$111$	28.3831	2.69401
$112$	0	0
$113$	−6.64779	−0.625372	−0.312686	−	0.949857i	$-0.601229\pi$
$113$	−6.64779	−0.625372	−0.312686	+	0.949857i	$0.601229\pi$
$114$	0	0
$115$	24.9470	2.32632
$116$	0	0
$117$	18.9508	1.75201
$118$	0	0
$119$	16.1497	1.48044
$120$	0	0
$121$	2.27983	0.207257
$122$	0	0
$123$	−1.67424	−0.150961
$124$	0	0
$125$	31.9314	2.85604
$126$	0	0
$127$	−13.8936	−1.23285	−0.616427	−	0.787412i	$-0.711420\pi$
$127$	−13.8936	−1.23285	−0.616427	+	0.787412i	$0.711420\pi$
$128$	0	0
$129$	−15.4215	−1.35779
$130$	0	0
$131$	−2.32207	−0.202880	−0.101440	−	0.994842i	$-0.532345\pi$
$131$	−2.32207	−0.202880	−0.101440	+	0.994842i	$0.532345\pi$
$132$	0	0
$133$	3.23058	0.280127
$134$	0	0
$135$	71.1100	6.12017
$136$	0	0
$137$	−4.02326	−0.343730	−0.171865	−	0.985120i	$-0.554979\pi$
$137$	−4.02326	−0.343730	−0.171865	+	0.985120i	$0.554979\pi$
$138$	0	0
$139$	9.92751	0.842040	0.421020	−	0.907051i	$-0.361672\pi$
$139$	9.92751	0.842040	0.421020	+	0.907051i	$0.361672\pi$
$140$	0	0
$141$	3.32999	0.280435
$142$	0	0
$143$	−8.53768	−0.713957
$144$	0	0
$145$	24.8875	2.06680
$146$	0	0
$147$	19.5554	1.61290
$148$	0	0
$149$	11.3211	0.927463	0.463731	−	0.885976i	$-0.346510\pi$
$149$	11.3211	0.927463	0.463731	+	0.885976i	$0.346510\pi$
$150$	0	0
$151$	−7.23756	−0.588984	−0.294492	−	0.955654i	$-0.595150\pi$
$151$	−7.23756	−0.588984	−0.294492	+	0.955654i	$0.595150\pi$
$152$	0	0
$153$	−36.4096	−2.94355
$154$	0	0
$155$	−11.1843	−0.898346
$156$	0	0
$157$	−10.3420	−0.825382	−0.412691	−	0.910871i	$-0.635411\pi$
$157$	−10.3420	−0.825382	−0.412691	+	0.910871i	$0.635411\pi$
$158$	0	0
$159$	29.4787	2.33781
$160$	0	0
$161$	−21.3294	−1.68099
$162$	0	0
$163$	−8.64360	−0.677019	−0.338510	−	0.940963i	$-0.609923\pi$
$163$	−8.64360	−0.677019	−0.338510	+	0.940963i	$0.609923\pi$
$164$	0	0
$165$	−50.9226	−3.96432
$166$	0	0
$167$	−10.8270	−0.837818	−0.418909	−	0.908028i	$-0.637587\pi$
$167$	−10.8270	−0.837818	−0.418909	+	0.908028i	$0.637587\pi$
$168$	0	0
$169$	−7.51107	−0.577775
$170$	0	0
$171$	−7.28340	−0.556975
$172$	0	0
$173$	−5.83163	−0.443371	−0.221685	−	0.975118i	$-0.571156\pi$
$173$	−5.83163	−0.443371	−0.221685	+	0.975118i	$0.571156\pi$
$174$	0	0
$175$	−45.2401	−3.41983
$176$	0	0
$177$	18.6996	1.40555
$178$	0	0
$179$	−21.4607	−1.60405	−0.802023	−	0.597293i	$-0.796243\pi$
$179$	−21.4607	−1.60405	−0.802023	+	0.597293i	$0.796243\pi$
$180$	0	0
$181$	−9.81282	−0.729381	−0.364691	−	0.931129i	$-0.618825\pi$
$181$	−9.81282	−0.729381	−0.364691	+	0.931129i	$0.618825\pi$
$182$	0	0
$183$	12.5143	0.925083
$184$	0	0
$185$	35.7676	2.62968
$186$	0	0
$187$	16.4032	1.19952
$188$	0	0
$189$	−60.7982	−4.42242
$190$	0	0
$191$	10.1661	0.735595	0.367797	−	0.929906i	$-0.380112\pi$
$191$	10.1661	0.735595	0.367797	+	0.929906i	$0.380112\pi$
$192$	0	0
$193$	11.1120	0.799861	0.399931	−	0.916545i	$-0.369034\pi$
$193$	11.1120	0.799861	0.399931	+	0.916545i	$0.369034\pi$
$194$	0	0
$195$	32.7384	2.34445
$196$	0	0
$197$	8.91500	0.635167	0.317584	−	0.948230i	$-0.397129\pi$
$197$	8.91500	0.635167	0.317584	+	0.948230i	$0.397129\pi$
$198$	0	0
$199$	−1.72094	−0.121994	−0.0609972	−	0.998138i	$-0.519428\pi$
$199$	−1.72094	−0.121994	−0.0609972	+	0.998138i	$0.519428\pi$
$200$	0	0
$201$	32.2006	2.27126
$202$	0	0
$203$	−21.2786	−1.49346
$204$	0	0
$205$	−2.10983	−0.147357
$206$	0	0
$207$	48.0874	3.34231
$208$	0	0
$209$	3.28130	0.226972
$210$	0	0
$211$	15.5992	1.07389	0.536947	−	0.843616i	$-0.319578\pi$
$211$	15.5992	1.07389	0.536947	+	0.843616i	$0.319578\pi$
$212$	0	0
$213$	−51.8227	−3.55083
$214$	0	0
$215$	−19.4337	−1.32537
$216$	0	0
$217$	9.56245	0.649142
$218$	0	0
$219$	−43.1441	−2.91541
$220$	0	0
$221$	−10.5457	−0.709381
$222$	0	0
$223$	2.16789	0.145173	0.0725863	−	0.997362i	$-0.476875\pi$
$223$	2.16789	0.145173	0.0725863	+	0.997362i	$0.476875\pi$
$224$	0	0
$225$	101.995	6.79964
$226$	0	0
$227$	19.8280	1.31603	0.658014	−	0.753006i	$-0.271397\pi$
$227$	19.8280	1.31603	0.658014	+	0.753006i	$0.271397\pi$
$228$	0	0
$229$	18.6220	1.23058	0.615288	−	0.788303i	$-0.289040\pi$
$229$	18.6220	1.23058	0.615288	+	0.788303i	$0.289040\pi$
$230$	0	0
$231$	43.5382	2.86460
$232$	0	0
$233$	−13.6156	−0.891988	−0.445994	−	0.895036i	$-0.647150\pi$
$233$	−13.6156	−0.891988	−0.445994	+	0.895036i	$0.647150\pi$
$234$	0	0
$235$	4.19635	0.273740
$236$	0	0
$237$	29.4013	1.90982
$238$	0	0
$239$	−4.08256	−0.264079	−0.132039	−	0.991244i	$-0.542153\pi$
$239$	−4.08256	−0.264079	−0.132039	+	0.991244i	$0.542153\pi$
$240$	0	0
$241$	−21.5972	−1.39120	−0.695599	−	0.718431i	$-0.744861\pi$
$241$	−21.5972	−1.39120	−0.695599	+	0.718431i	$0.744861\pi$
$242$	0	0
$243$	56.2635	3.60930
$244$	0	0
$245$	24.6431	1.57439
$246$	0	0
$247$	−2.10957	−0.134228
$248$	0	0
$249$	−36.0525	−2.28474
$250$	0	0
$251$	−2.68997	−0.169789	−0.0848947	−	0.996390i	$-0.527055\pi$
$251$	−2.68997	−0.169789	−0.0848947	+	0.996390i	$0.527055\pi$
$252$	0	0
$253$	−21.6642	−1.36202
$254$	0	0
$255$	−62.8993	−3.93891
$256$	0	0
$257$	−15.9879	−0.997296	−0.498648	−	0.866805i	$-0.666170\pi$
$257$	−15.9879	−0.997296	−0.498648	+	0.866805i	$0.666170\pi$
$258$	0	0
$259$	−30.5808	−1.90020
$260$	0	0
$261$	47.9728	2.96944
$262$	0	0
$263$	6.83547	0.421493	0.210746	−	0.977541i	$-0.432411\pi$
$263$	6.83547	0.421493	0.210746	+	0.977541i	$0.432411\pi$
$264$	0	0
$265$	37.1482	2.28199
$266$	0	0
$267$	0.947499	0.0579860
$268$	0	0
$269$	−27.3947	−1.67028	−0.835141	−	0.550036i	$-0.814614\pi$
$269$	−27.3947	−1.67028	−0.835141	+	0.550036i	$0.814614\pi$
$270$	0	0
$271$	−7.95148	−0.483018	−0.241509	−	0.970399i	$-0.577642\pi$
$271$	−7.95148	−0.483018	−0.241509	+	0.970399i	$0.577642\pi$
$272$	0	0
$273$	−27.9910	−1.69409
$274$	0	0
$275$	−45.9503	−2.77091
$276$	0	0
$277$	4.98689	0.299633	0.149817	−	0.988714i	$-0.452132\pi$
$277$	4.98689	0.299633	0.149817	+	0.988714i	$0.452132\pi$
$278$	0	0
$279$	−21.5587	−1.29069
$280$	0	0
$281$	7.41563	0.442379	0.221190	−	0.975231i	$-0.429006\pi$
$281$	7.41563	0.442379	0.221190	+	0.975231i	$0.429006\pi$
$282$	0	0
$283$	18.6834	1.11061	0.555306	−	0.831646i	$-0.312601\pi$
$283$	18.6834	1.11061	0.555306	+	0.831646i	$0.312601\pi$
$284$	0	0
$285$	−12.5824	−0.745317
$286$	0	0
$287$	1.80388	0.106479
$288$	0	0
$289$	3.26112	0.191831
$290$	0	0
$291$	−45.2126	−2.65041
$292$	0	0
$293$	−31.8686	−1.86178	−0.930892	−	0.365295i	$-0.880968\pi$
$293$	−31.8686	−1.86178	−0.930892	+	0.365295i	$0.880968\pi$
$294$	0	0
$295$	23.5647	1.37199
$296$	0	0
$297$	−61.7526	−3.58325
$298$	0	0
$299$	13.9280	0.805480
$300$	0	0
$301$	16.6156	0.957707
$302$	0	0
$303$	19.4311	1.11629
$304$	0	0
$305$	15.7701	0.902995
$306$	0	0
$307$	−24.4064	−1.39294	−0.696472	−	0.717584i	$-0.745248\pi$
$307$	−24.4064	−1.39294	−0.696472	+	0.717584i	$0.745248\pi$
$308$	0	0
$309$	14.1214	0.803341
$310$	0	0
$311$	6.59172	0.373782	0.186891	−	0.982381i	$-0.440159\pi$
$311$	6.59172	0.373782	0.186891	+	0.982381i	$0.440159\pi$
$312$	0	0
$313$	12.6158	0.713085	0.356542	−	0.934279i	$-0.383956\pi$
$313$	12.6158	0.713085	0.356542	+	0.934279i	$0.383956\pi$
$314$	0	0
$315$	−121.783	−6.86171
$316$	0	0
$317$	−22.5473	−1.26638	−0.633191	−	0.773995i	$-0.718255\pi$
$317$	−22.5473	−1.26638	−0.633191	+	0.773995i	$0.718255\pi$
$318$	0	0
$319$	−21.6126	−1.21007
$320$	0	0
$321$	47.6000	2.65677
$322$	0	0
$323$	4.05304	0.225517
$324$	0	0
$325$	29.5417	1.63868
$326$	0	0
$327$	−15.4130	−0.852341
$328$	0	0
$329$	−3.58783	−0.197803
$330$	0	0
$331$	4.26007	0.234155	0.117077	−	0.993123i	$-0.462647\pi$
$331$	4.26007	0.234155	0.117077	+	0.993123i	$0.462647\pi$
$332$	0	0
$333$	68.9450	3.77816
$334$	0	0
$335$	40.5782	2.21703
$336$	0	0
$337$	−10.9812	−0.598184	−0.299092	−	0.954224i	$-0.596684\pi$
$337$	−10.9812	−0.598184	−0.299092	+	0.954224i	$0.596684\pi$
$338$	0	0
$339$	−22.1371	−1.20232
$340$	0	0
$341$	9.71257	0.525965
$342$	0	0
$343$	4.04527	0.218424
$344$	0	0
$345$	83.0731	4.47251
$346$	0	0
$347$	−18.5012	−0.993198	−0.496599	−	0.867980i	$-0.665418\pi$
$347$	−18.5012	−0.993198	−0.496599	+	0.867980i	$0.665418\pi$
$348$	0	0
$349$	12.2442	0.655419	0.327710	−	0.944778i	$-0.393723\pi$
$349$	12.2442	0.655419	0.327710	+	0.944778i	$0.393723\pi$
$350$	0	0
$351$	39.7011	2.11909
$352$	0	0
$353$	−3.30274	−0.175787	−0.0878935	−	0.996130i	$-0.528014\pi$
$353$	−3.30274	−0.175787	−0.0878935	+	0.996130i	$0.528014\pi$
$354$	0	0
$355$	−65.3053	−3.46605
$356$	0	0
$357$	53.7781	2.84624
$358$	0	0
$359$	−14.8973	−0.786250	−0.393125	−	0.919485i	$-0.628606\pi$
$359$	−14.8973	−0.786250	−0.393125	+	0.919485i	$0.628606\pi$
$360$	0	0
$361$	−18.1892	−0.957328
$362$	0	0
$363$	7.59180	0.398466
$364$	0	0
$365$	−54.3689	−2.84580
$366$	0	0
$367$	12.8149	0.668932	0.334466	−	0.942408i	$-0.391444\pi$
$367$	12.8149	0.668932	0.334466	+	0.942408i	$0.391444\pi$
$368$	0	0
$369$	−4.06687	−0.211713
$370$	0	0
$371$	−31.7612	−1.64896
$372$	0	0
$373$	23.8839	1.23666	0.618332	−	0.785917i	$-0.287809\pi$
$373$	23.8839	1.23666	0.618332	+	0.785917i	$0.287809\pi$
$374$	0	0
$375$	106.331	5.49092
$376$	0	0
$377$	13.8949	0.715622
$378$	0	0
$379$	26.9368	1.38365	0.691824	−	0.722066i	$-0.256807\pi$
$379$	26.9368	1.38365	0.691824	+	0.722066i	$0.256807\pi$
$380$	0	0
$381$	−46.2653	−2.37025
$382$	0	0
$383$	−22.4440	−1.14683	−0.573417	−	0.819263i	$-0.694383\pi$
$383$	−22.4440	−1.14683	−0.573417	+	0.819263i	$0.694383\pi$
$384$	0	0
$385$	54.8655	2.79620
$386$	0	0
$387$	−37.4601	−1.90420
$388$	0	0
$389$	34.8681	1.76788	0.883942	−	0.467597i	$-0.154880\pi$
$389$	34.8681	1.76788	0.883942	+	0.467597i	$0.154880\pi$
$390$	0	0
$391$	−26.7595	−1.35329
$392$	0	0
$393$	−7.73246	−0.390051
$394$	0	0
$395$	37.0506	1.86422
$396$	0	0
$397$	7.99040	0.401027	0.200513	−	0.979691i	$-0.435739\pi$
$397$	7.99040	0.401027	0.200513	+	0.979691i	$0.435739\pi$
$398$	0	0
$399$	10.7578	0.538563
$400$	0	0
$401$	−29.8591	−1.49109	−0.745547	−	0.666453i	$-0.767812\pi$
$401$	−29.8591	−1.49109	−0.745547	+	0.666453i	$0.767812\pi$
$402$	0	0
$403$	−6.24427	−0.311049
$404$	0	0
$405$	134.965	6.70646
$406$	0	0
$407$	−31.0609	−1.53963
$408$	0	0
$409$	−2.85221	−0.141033	−0.0705164	−	0.997511i	$-0.522465\pi$
$409$	−2.85221	−0.141033	−0.0705164	+	0.997511i	$0.522465\pi$
$410$	0	0
$411$	−13.3974	−0.660845
$412$	0	0
$413$	−20.1476	−0.991396
$414$	0	0
$415$	−45.4323	−2.23019
$416$	0	0
$417$	33.0585	1.61888
$418$	0	0
$419$	−35.5131	−1.73493	−0.867463	−	0.497501i	$-0.834251\pi$
$419$	−35.5131	−1.73493	−0.867463	+	0.497501i	$0.834251\pi$
$420$	0	0
$421$	−39.8879	−1.94402	−0.972008	−	0.234950i	$-0.924507\pi$
$421$	−39.8879	−1.94402	−0.972008	+	0.234950i	$0.924507\pi$
$422$	0	0
$423$	8.08881	0.393292
$424$	0	0
$425$	−56.7576	−2.75315
$426$	0	0
$427$	−13.4833	−0.652501
$428$	0	0
$429$	−28.4304	−1.37263
$430$	0	0
$431$	−8.45181	−0.407110	−0.203555	−	0.979064i	$-0.565249\pi$
$431$	−8.45181	−0.407110	−0.203555	+	0.979064i	$0.565249\pi$
$432$	0	0
$433$	−32.8941	−1.58079	−0.790395	−	0.612598i	$-0.790125\pi$
$433$	−32.8941	−1.58079	−0.790395	+	0.612598i	$0.790125\pi$
$434$	0	0
$435$	82.8752	3.97356
$436$	0	0
$437$	−5.35298	−0.256068
$438$	0	0
$439$	−4.77694	−0.227991	−0.113995	−	0.993481i	$-0.536365\pi$
$439$	−4.77694	−0.227991	−0.113995	+	0.993481i	$0.536365\pi$
$440$	0	0
$441$	47.5016	2.26198
$442$	0	0
$443$	22.7650	1.08160	0.540798	−	0.841152i	$-0.318122\pi$
$443$	22.7650	1.08160	0.540798	+	0.841152i	$0.318122\pi$
$444$	0	0
$445$	1.19401	0.0566015
$446$	0	0
$447$	37.6992	1.78311
$448$	0	0
$449$	−21.6480	−1.02163	−0.510816	−	0.859690i	$-0.670657\pi$
$449$	−21.6480	−1.02163	−0.510816	+	0.859690i	$0.670657\pi$
$450$	0	0
$451$	1.83219	0.0862746
$452$	0	0
$453$	−24.1010	−1.13236
$454$	0	0
$455$	−35.2734	−1.65364
$456$	0	0
$457$	23.7335	1.11020	0.555102	−	0.831782i	$-0.312679\pi$
$457$	23.7335	1.11020	0.555102	+	0.831782i	$0.312679\pi$
$458$	0	0
$459$	−76.2765	−3.56028
$460$	0	0
$461$	−33.5484	−1.56251	−0.781253	−	0.624214i	$-0.785419\pi$
$461$	−33.5484	−1.56251	−0.781253	+	0.624214i	$0.785419\pi$
$462$	0	0
$463$	−1.67349	−0.0777739	−0.0388869	−	0.999244i	$-0.512381\pi$
$463$	−1.67349	−0.0777739	−0.0388869	+	0.999244i	$0.512381\pi$
$464$	0	0
$465$	−37.2436	−1.72713
$466$	0	0
$467$	−22.8604	−1.05785	−0.528926	−	0.848668i	$-0.677405\pi$
$467$	−22.8604	−1.05785	−0.528926	+	0.848668i	$0.677405\pi$
$468$	0	0
$469$	−34.6939	−1.60202
$470$	0	0
$471$	−34.4388	−1.58685
$472$	0	0
$473$	16.8764	0.775979
$474$	0	0
$475$	−11.3538	−0.520949
$476$	0	0
$477$	71.6062	3.27862
$478$	0	0
$479$	−13.7034	−0.626122	−0.313061	−	0.949733i	$-0.601354\pi$
$479$	−13.7034	−0.626122	−0.313061	+	0.949733i	$0.601354\pi$
$480$	0	0
$481$	19.9692	0.910519
$482$	0	0
$483$	−71.0265	−3.23182
$484$	0	0
$485$	−56.9755	−2.58713
$486$	0	0
$487$	−20.6492	−0.935705	−0.467853	−	0.883807i	$-0.654972\pi$
$487$	−20.6492	−0.935705	−0.467853	+	0.883807i	$0.654972\pi$
$488$	0	0
$489$	−28.7831	−1.30162
$490$	0	0
$491$	−14.6077	−0.659238	−0.329619	−	0.944114i	$-0.606920\pi$
$491$	−14.6077	−0.659238	−0.329619	+	0.944114i	$0.606920\pi$
$492$	0	0
$493$	−26.6958	−1.20232
$494$	0	0
$495$	−123.695	−5.55968
$496$	0	0
$497$	55.8353	2.50455
$498$	0	0
$499$	−29.9561	−1.34102	−0.670510	−	0.741900i	$-0.733925\pi$
$499$	−29.9561	−1.34102	−0.670510	+	0.741900i	$0.733925\pi$
$500$	0	0
$501$	−36.0538	−1.61076
$502$	0	0
$503$	−9.47615	−0.422521	−0.211260	−	0.977430i	$-0.567757\pi$
$503$	−9.47615	−0.422521	−0.211260	+	0.977430i	$0.567757\pi$
$504$	0	0
$505$	24.4865	1.08964
$506$	0	0
$507$	−25.0118	−1.11081
$508$	0	0
$509$	12.5776	0.557494	0.278747	−	0.960365i	$-0.410081\pi$
$509$	12.5776	0.557494	0.278747	+	0.960365i	$0.410081\pi$
$510$	0	0
$511$	46.4848	2.05637
$512$	0	0
$513$	−15.2584	−0.673674
$514$	0	0
$515$	17.7954	0.784160
$516$	0	0
$517$	−3.64415	−0.160270
$518$	0	0
$519$	−19.4192	−0.852410
$520$	0	0
$521$	−11.6649	−0.511048	−0.255524	−	0.966803i	$-0.582248\pi$
$521$	−11.6649	−0.511048	−0.255524	+	0.966803i	$0.582248\pi$
$522$	0	0
$523$	37.4430	1.63727	0.818633	−	0.574317i	$-0.194732\pi$
$523$	37.4430	1.63727	0.818633	+	0.574317i	$0.194732\pi$
$524$	0	0
$525$	−150.649	−6.57486
$526$	0	0
$527$	11.9969	0.522594
$528$	0	0
$529$	12.3422	0.536616
$530$	0	0
$531$	45.4230	1.97119
$532$	0	0
$533$	−1.17793	−0.0510217
$534$	0	0
$535$	59.9841	2.59334
$536$	0	0
$537$	−71.4638	−3.08389
$538$	0	0
$539$	−21.4003	−0.921775
$540$	0	0
$541$	16.1040	0.692366	0.346183	−	0.938167i	$-0.387478\pi$
$541$	16.1040	0.692366	0.346183	+	0.938167i	$0.387478\pi$
$542$	0	0
$543$	−32.6766	−1.40229
$544$	0	0
$545$	−19.4230	−0.831990
$546$	0	0
$547$	25.5700	1.09329	0.546647	−	0.837363i	$-0.315904\pi$
$547$	25.5700	1.09329	0.546647	+	0.837363i	$0.315904\pi$
$548$	0	0
$549$	30.3982	1.29736
$550$	0	0
$551$	−5.34023	−0.227501
$552$	0	0
$553$	−31.6778	−1.34708
$554$	0	0
$555$	119.106	5.05575
$556$	0	0
$557$	27.4519	1.16317	0.581587	−	0.813484i	$-0.302432\pi$
$557$	27.4519	1.16317	0.581587	+	0.813484i	$0.302432\pi$
$558$	0	0
$559$	−10.8500	−0.458904
$560$	0	0
$561$	54.6224	2.30616
$562$	0	0
$563$	13.5034	0.569099	0.284549	−	0.958661i	$-0.408156\pi$
$563$	13.5034	0.569099	0.284549	+	0.958661i	$0.408156\pi$
$564$	0	0
$565$	−27.8965	−1.17361
$566$	0	0
$567$	−115.393	−4.84606
$568$	0	0
$569$	7.51579	0.315078	0.157539	−	0.987513i	$-0.449644\pi$
$569$	7.51579	0.315078	0.157539	+	0.987513i	$0.449644\pi$
$570$	0	0
$571$	7.65878	0.320510	0.160255	−	0.987076i	$-0.448768\pi$
$571$	7.65878	0.320510	0.160255	+	0.987076i	$0.448768\pi$
$572$	0	0
$573$	33.8530	1.41423
$574$	0	0
$575$	74.9616	3.12612
$576$	0	0
$577$	21.0544	0.876507	0.438253	−	0.898851i	$-0.355597\pi$
$577$	21.0544	0.876507	0.438253	+	0.898851i	$0.355597\pi$
$578$	0	0
$579$	37.0029	1.53779
$580$	0	0
$581$	38.8441	1.61152
$582$	0	0
$583$	−32.2598	−1.33607
$584$	0	0
$585$	79.5243	3.28793
$586$	0	0
$587$	5.48943	0.226573	0.113287	−	0.993562i	$-0.463862\pi$
$587$	5.48943	0.226573	0.113287	+	0.993562i	$0.463862\pi$
$588$	0	0
$589$	2.39987	0.0988848
$590$	0	0
$591$	29.6868	1.22115
$592$	0	0
$593$	−7.53654	−0.309489	−0.154744	−	0.987955i	$-0.549455\pi$
$593$	−7.53654	−0.309489	−0.154744	+	0.987955i	$0.549455\pi$
$594$	0	0
$595$	67.7696	2.77828
$596$	0	0
$597$	−5.73072	−0.234543
$598$	0	0
$599$	31.6067	1.29141	0.645707	−	0.763585i	$-0.276563\pi$
$599$	31.6067	1.29141	0.645707	+	0.763585i	$0.276563\pi$
$600$	0	0
$601$	−40.2630	−1.64236	−0.821180	−	0.570669i	$-0.806684\pi$
$601$	−40.2630	−1.64236	−0.821180	+	0.570669i	$0.806684\pi$
$602$	0	0
$603$	78.2180	3.18528
$604$	0	0
$605$	9.56696	0.388952
$606$	0	0
$607$	−22.2754	−0.904129	−0.452064	−	0.891985i	$-0.649312\pi$
$607$	−22.2754	−0.904129	−0.452064	+	0.891985i	$0.649312\pi$
$608$	0	0
$609$	−70.8573	−2.87128
$610$	0	0
$611$	2.34285	0.0947814
$612$	0	0
$613$	−24.8796	−1.00488	−0.502439	−	0.864613i	$-0.667564\pi$
$613$	−24.8796	−1.00488	−0.502439	+	0.864613i	$0.667564\pi$
$614$	0	0
$615$	−7.02569	−0.283303
$616$	0	0
$617$	29.4227	1.18451	0.592256	−	0.805750i	$-0.298237\pi$
$617$	29.4227	1.18451	0.592256	+	0.805750i	$0.298237\pi$
$618$	0	0
$619$	−41.9590	−1.68647	−0.843237	−	0.537541i	$-0.819353\pi$
$619$	−41.9590	−1.68647	−0.843237	+	0.537541i	$0.819353\pi$
$620$	0	0
$621$	100.741	4.04259
$622$	0	0
$623$	−1.02086	−0.0409001
$624$	0	0
$625$	70.9488	2.83795
$626$	0	0
$627$	10.9267	0.436369
$628$	0	0
$629$	−38.3663	−1.52976
$630$	0	0
$631$	−7.31068	−0.291034	−0.145517	−	0.989356i	$-0.546484\pi$
$631$	−7.31068	−0.291034	−0.145517	+	0.989356i	$0.546484\pi$
$632$	0	0
$633$	51.9451	2.06463
$634$	0	0
$635$	−58.3022	−2.31365
$636$	0	0
$637$	13.7584	0.545126
$638$	0	0
$639$	−125.882	−4.97980
$640$	0	0
$641$	12.1695	0.480667	0.240333	−	0.970690i	$-0.422743\pi$
$641$	12.1695	0.480667	0.240333	+	0.970690i	$0.422743\pi$
$642$	0	0
$643$	42.0908	1.65990	0.829949	−	0.557839i	$-0.188369\pi$
$643$	42.0908	1.65990	0.829949	+	0.557839i	$0.188369\pi$
$644$	0	0
$645$	−64.7140	−2.54811
$646$	0	0
$647$	−31.9553	−1.25629	−0.628146	−	0.778096i	$-0.716186\pi$
$647$	−31.9553	−1.25629	−0.628146	+	0.778096i	$0.716186\pi$
$648$	0	0
$649$	−20.4638	−0.803276
$650$	0	0
$651$	31.8428	1.24802
$652$	0	0
$653$	−3.43707	−0.134503	−0.0672514	−	0.997736i	$-0.521423\pi$
$653$	−3.43707	−0.134503	−0.0672514	+	0.997736i	$0.521423\pi$
$654$	0	0
$655$	−9.74421	−0.380738
$656$	0	0
$657$	−104.801	−4.08866
$658$	0	0
$659$	17.9608	0.699653	0.349826	−	0.936815i	$-0.386241\pi$
$659$	17.9608	0.699653	0.349826	+	0.936815i	$0.386241\pi$
$660$	0	0
$661$	31.8475	1.23872	0.619362	−	0.785105i	$-0.287391\pi$
$661$	31.8475	1.23872	0.619362	+	0.785105i	$0.287391\pi$
$662$	0	0
$663$	−35.1170	−1.36383
$664$	0	0
$665$	13.5566	0.525704
$666$	0	0
$667$	35.2580	1.36519
$668$	0	0
$669$	7.21904	0.279104
$670$	0	0
$671$	−13.6949	−0.528687
$672$	0	0
$673$	25.8884	0.997925	0.498962	−	0.866624i	$-0.333715\pi$
$673$	25.8884	0.997925	0.498962	+	0.866624i	$0.333715\pi$
$674$	0	0
$675$	213.674	8.22431
$676$	0	0
$677$	30.4174	1.16904	0.584518	−	0.811381i	$-0.301284\pi$
$677$	30.4174	1.16904	0.584518	+	0.811381i	$0.301284\pi$
$678$	0	0
$679$	48.7134	1.86945
$680$	0	0
$681$	66.0269	2.53015
$682$	0	0
$683$	−12.8316	−0.490986	−0.245493	−	0.969398i	$-0.578950\pi$
$683$	−12.8316	−0.490986	−0.245493	+	0.969398i	$0.578950\pi$
$684$	0	0
$685$	−16.8830	−0.645066
$686$	0	0
$687$	62.0110	2.36587
$688$	0	0
$689$	20.7400	0.790133
$690$	0	0
$691$	3.64178	0.138540	0.0692699	−	0.997598i	$-0.477933\pi$
$691$	3.64178	0.138540	0.0692699	+	0.997598i	$0.477933\pi$
$692$	0	0
$693$	105.758	4.01741
$694$	0	0
$695$	41.6593	1.58023
$696$	0	0
$697$	2.26312	0.0857216
$698$	0	0
$699$	−45.3398	−1.71491
$700$	0	0
$701$	11.9053	0.449656	0.224828	−	0.974398i	$-0.427818\pi$
$701$	11.9053	0.449656	0.224828	+	0.974398i	$0.427818\pi$
$702$	0	0
$703$	−7.67480	−0.289461
$704$	0	0
$705$	13.9738	0.526283
$706$	0	0
$707$	−20.9357	−0.787367
$708$	0	0
$709$	6.74060	0.253149	0.126574	−	0.991957i	$-0.459602\pi$
$709$	6.74060	0.253149	0.126574	+	0.991957i	$0.459602\pi$
$710$	0	0
$711$	71.4181	2.67839
$712$	0	0
$713$	−15.8447	−0.593389
$714$	0	0
$715$	−35.8271	−1.33986
$716$	0	0
$717$	−13.5949	−0.507710
$718$	0	0
$719$	22.4153	0.835949	0.417974	−	0.908459i	$-0.362740\pi$
$719$	22.4153	0.835949	0.417974	+	0.908459i	$0.362740\pi$
$720$	0	0
$721$	−15.2149	−0.566631
$722$	0	0
$723$	−71.9183	−2.67467
$724$	0	0
$725$	74.7830	2.77737
$726$	0	0
$727$	3.43850	0.127527	0.0637635	−	0.997965i	$-0.479690\pi$
$727$	3.43850	0.127527	0.0637635	+	0.997965i	$0.479690\pi$
$728$	0	0
$729$	90.8693	3.36553
$730$	0	0
$731$	20.8457	0.771005
$732$	0	0
$733$	24.5289	0.905996	0.452998	−	0.891512i	$-0.350355\pi$
$733$	24.5289	0.905996	0.452998	+	0.891512i	$0.350355\pi$
$734$	0	0
$735$	82.0611	3.02687
$736$	0	0
$737$	−35.2385	−1.29803
$738$	0	0
$739$	42.8735	1.57713	0.788564	−	0.614952i	$-0.210825\pi$
$739$	42.8735	1.57713	0.788564	+	0.614952i	$0.210825\pi$
$740$	0	0
$741$	−7.02482	−0.258063
$742$	0	0
$743$	17.2573	0.633109	0.316555	−	0.948574i	$-0.397474\pi$
$743$	17.2573	0.633109	0.316555	+	0.948574i	$0.397474\pi$
$744$	0	0
$745$	47.5074	1.74054
$746$	0	0
$747$	−87.5746	−3.20419
$748$	0	0
$749$	−51.2857	−1.87394
$750$	0	0
$751$	22.9425	0.837183	0.418591	−	0.908175i	$-0.362524\pi$
$751$	22.9425	0.837183	0.418591	+	0.908175i	$0.362524\pi$
$752$	0	0
$753$	−8.95756	−0.326432
$754$	0	0
$755$	−30.3713	−1.10533
$756$	0	0
$757$	8.63677	0.313909	0.156954	−	0.987606i	$-0.449832\pi$
$757$	8.63677	0.313909	0.156954	+	0.987606i	$0.449832\pi$
$758$	0	0
$759$	−72.1415	−2.61857
$760$	0	0
$761$	50.0832	1.81552	0.907758	−	0.419495i	$-0.137793\pi$
$761$	50.0832	1.81552	0.907758	+	0.419495i	$0.137793\pi$
$762$	0	0
$763$	16.6064	0.601193
$764$	0	0
$765$	−152.788	−5.52405
$766$	0	0
$767$	13.1563	0.475047
$768$	0	0
$769$	−16.1186	−0.581253	−0.290627	−	0.956837i	$-0.593864\pi$
$769$	−16.1186	−0.581253	−0.290627	+	0.956837i	$0.593864\pi$
$770$	0	0
$771$	−53.2394	−1.91737
$772$	0	0
$773$	−2.25194	−0.0809967	−0.0404984	−	0.999180i	$-0.512895\pi$
$773$	−2.25194	−0.0809967	−0.0404984	+	0.999180i	$0.512895\pi$
$774$	0	0
$775$	−33.6070	−1.20720
$776$	0	0
$777$	−101.834	−3.65327
$778$	0	0
$779$	0.452714	0.0162202
$780$	0	0
$781$	56.7118	2.02931
$782$	0	0
$783$	100.501	3.59161
$784$	0	0
$785$	−43.3987	−1.54897
$786$	0	0
$787$	−9.08893	−0.323985	−0.161993	−	0.986792i	$-0.551792\pi$
$787$	−9.08893	−0.323985	−0.161993	+	0.986792i	$0.551792\pi$
$788$	0	0
$789$	22.7620	0.810349
$790$	0	0
$791$	23.8511	0.848049
$792$	0	0
$793$	8.80455	0.312659
$794$	0	0
$795$	123.703	4.38729
$796$	0	0
$797$	−41.8713	−1.48316	−0.741579	−	0.670865i	$-0.765923\pi$
$797$	−41.8713	−1.48316	−0.741579	+	0.670865i	$0.765923\pi$
$798$	0	0
$799$	−4.50124	−0.159242
$800$	0	0
$801$	2.30155	0.0813214
$802$	0	0
$803$	47.2145	1.66616
$804$	0	0
$805$	−89.5055	−3.15465
$806$	0	0
$807$	−91.2238	−3.21123
$808$	0	0
$809$	54.7320	1.92427	0.962137	−	0.272567i	$-0.0878726\pi$
$809$	54.7320	1.92427	0.962137	+	0.272567i	$0.0878726\pi$
$810$	0	0
$811$	−27.9260	−0.980614	−0.490307	−	0.871550i	$-0.663115\pi$
$811$	−27.9260	−0.980614	−0.490307	+	0.871550i	$0.663115\pi$
$812$	0	0
$813$	−26.4783	−0.928635
$814$	0	0
$815$	−36.2716	−1.27054
$816$	0	0
$817$	4.16998	0.145889
$818$	0	0
$819$	−67.9923	−2.37584
$820$	0	0
$821$	26.0850	0.910374	0.455187	−	0.890396i	$-0.349572\pi$
$821$	26.0850	0.910374	0.455187	+	0.890396i	$0.349572\pi$
$822$	0	0
$823$	35.5150	1.23798	0.618988	−	0.785401i	$-0.287543\pi$
$823$	35.5150	1.23798	0.618988	+	0.785401i	$0.287543\pi$
$824$	0	0
$825$	−153.014	−5.32726
$826$	0	0
$827$	24.9840	0.868780	0.434390	−	0.900725i	$-0.356964\pi$
$827$	24.9840	0.868780	0.434390	+	0.900725i	$0.356964\pi$
$828$	0	0
$829$	20.9270	0.726824	0.363412	−	0.931629i	$-0.381612\pi$
$829$	20.9270	0.726824	0.363412	+	0.931629i	$0.381612\pi$
$830$	0	0
$831$	16.6063	0.576066
$832$	0	0
$833$	−26.4335	−0.915867
$834$	0	0
$835$	−45.4338	−1.57230
$836$	0	0
$837$	−45.1645	−1.56111
$838$	0	0
$839$	−7.96161	−0.274865	−0.137433	−	0.990511i	$-0.543885\pi$
$839$	−7.96161	−0.274865	−0.137433	+	0.990511i	$0.543885\pi$
$840$	0	0
$841$	6.17395	0.212895
$842$	0	0
$843$	24.6940	0.850505
$844$	0	0
$845$	−31.5191	−1.08429
$846$	0	0
$847$	−8.17963	−0.281055
$848$	0	0
$849$	62.2154	2.13523
$850$	0	0
$851$	50.6716	1.73700
$852$	0	0
$853$	−47.2038	−1.61623	−0.808114	−	0.589027i	$-0.799511\pi$
$853$	−47.2038	−1.61623	−0.808114	+	0.589027i	$0.799511\pi$
$854$	0	0
$855$	−30.5637	−1.04526
$856$	0	0
$857$	16.9090	0.577600	0.288800	−	0.957389i	$-0.406744\pi$
$857$	16.9090	0.577600	0.288800	+	0.957389i	$0.406744\pi$
$858$	0	0
$859$	−21.9721	−0.749677	−0.374839	−	0.927090i	$-0.622302\pi$
$859$	−21.9721	−0.749677	−0.374839	+	0.927090i	$0.622302\pi$
$860$	0	0
$861$	6.00688	0.204714
$862$	0	0
$863$	−24.7187	−0.841433	−0.420717	−	0.907192i	$-0.638221\pi$
$863$	−24.7187	−0.841433	−0.420717	+	0.907192i	$0.638221\pi$
$864$	0	0
$865$	−24.4715	−0.832058
$866$	0	0
$867$	10.8595	0.368808
$868$	0	0
$869$	−32.1751	−1.09147
$870$	0	0
$871$	22.6551	0.767638
$872$	0	0
$873$	−109.825	−3.71702
$874$	0	0
$875$	−114.565	−3.87299
$876$	0	0
$877$	3.62987	0.122572	0.0612860	−	0.998120i	$-0.480480\pi$
$877$	3.62987	0.122572	0.0612860	+	0.998120i	$0.480480\pi$
$878$	0	0
$879$	−106.122	−3.57941
$880$	0	0
$881$	35.1582	1.18451	0.592254	−	0.805751i	$-0.298238\pi$
$881$	35.1582	1.18451	0.592254	+	0.805751i	$0.298238\pi$
$882$	0	0
$883$	−3.76733	−0.126781	−0.0633904	−	0.997989i	$-0.520191\pi$
$883$	−3.76733	−0.126781	−0.0633904	+	0.997989i	$0.520191\pi$
$884$	0	0
$885$	78.4702	2.63775
$886$	0	0
$887$	−22.1232	−0.742823	−0.371411	−	0.928468i	$-0.621126\pi$
$887$	−22.1232	−0.742823	−0.371411	+	0.928468i	$0.621126\pi$
$888$	0	0
$889$	49.8477	1.67184
$890$	0	0
$891$	−117.205	−3.92651
$892$	0	0
$893$	−0.900429	−0.0301317
$894$	0	0
$895$	−90.0565	−3.01026
$896$	0	0
$897$	46.3802	1.54859
$898$	0	0
$899$	−15.8070	−0.527192
$900$	0	0
$901$	−39.8472	−1.32750
$902$	0	0
$903$	55.3297	1.84126
$904$	0	0
$905$	−41.1780	−1.36880
$906$	0	0
$907$	−21.3229	−0.708015	−0.354007	−	0.935243i	$-0.615181\pi$
$907$	−21.3229	−0.708015	−0.354007	+	0.935243i	$0.615181\pi$
$908$	0	0
$909$	47.1998	1.56552
$910$	0	0
$911$	13.2126	0.437752	0.218876	−	0.975753i	$-0.429761\pi$
$911$	13.2126	0.437752	0.218876	+	0.975753i	$0.429761\pi$
$912$	0	0
$913$	39.4539	1.30573
$914$	0	0
$915$	52.5143	1.73607
$916$	0	0
$917$	8.33118	0.275120
$918$	0	0
$919$	2.87997	0.0950014	0.0475007	−	0.998871i	$-0.484874\pi$
$919$	2.87997	0.0950014	0.0475007	+	0.998871i	$0.484874\pi$
$920$	0	0
$921$	−81.2729	−2.67803
$922$	0	0
$923$	−36.4604	−1.20011
$924$	0	0
$925$	107.476	3.53378
$926$	0	0
$927$	34.3021	1.12663
$928$	0	0
$929$	54.0348	1.77282	0.886411	−	0.462898i	$-0.153190\pi$
$929$	54.0348	1.77282	0.886411	+	0.462898i	$0.153190\pi$
$930$	0	0
$931$	−5.28777	−0.173300
$932$	0	0
$933$	21.9504	0.718622
$934$	0	0
$935$	68.8334	2.25109
$936$	0	0
$937$	1.75380	0.0572943	0.0286471	−	0.999590i	$-0.490880\pi$
$937$	1.75380	0.0572943	0.0286471	+	0.999590i	$0.490880\pi$
$938$	0	0
$939$	42.0103	1.37095
$940$	0	0
$941$	−10.1057	−0.329437	−0.164719	−	0.986341i	$-0.552672\pi$
$941$	−10.1057	−0.329437	−0.164719	+	0.986341i	$0.552672\pi$
$942$	0	0
$943$	−2.98897	−0.0973343
$944$	0	0
$945$	−255.130	−8.29939
$946$	0	0
$947$	−6.85665	−0.222811	−0.111406	−	0.993775i	$-0.535535\pi$
$947$	−6.85665	−0.222811	−0.111406	+	0.993775i	$0.535535\pi$
$948$	0	0
$949$	−30.3545	−0.985349
$950$	0	0
$951$	−75.0822	−2.43471
$952$	0	0
$953$	−3.14175	−0.101771	−0.0508856	−	0.998704i	$-0.516204\pi$
$953$	−3.14175	−0.101771	−0.0508856	+	0.998704i	$0.516204\pi$
$954$	0	0
$955$	42.6606	1.38046
$956$	0	0
$957$	−71.9696	−2.32645
$958$	0	0
$959$	14.4348	0.466123
$960$	0	0
$961$	−23.8964	−0.770853
$962$	0	0
$963$	115.624	3.72594
$964$	0	0
$965$	46.6299	1.50107
$966$	0	0
$967$	−1.22637	−0.0394373	−0.0197187	−	0.999806i	$-0.506277\pi$
$967$	−1.22637	−0.0394373	−0.0197187	+	0.999806i	$0.506277\pi$
$968$	0	0
$969$	13.4966	0.433572
$970$	0	0
$971$	−57.4809	−1.84465	−0.922326	−	0.386414i	$-0.873714\pi$
$971$	−57.4809	−1.84465	−0.922326	+	0.386414i	$0.873714\pi$
$972$	0	0
$973$	−35.6182	−1.14187
$974$	0	0
$975$	98.3736	3.15048
$976$	0	0
$977$	48.2913	1.54497	0.772487	−	0.635030i	$-0.219012\pi$
$977$	48.2913	1.54497	0.772487	+	0.635030i	$0.219012\pi$
$978$	0	0
$979$	−1.03689	−0.0331391
$980$	0	0
$981$	−37.4395	−1.19535
$982$	0	0
$983$	−11.4426	−0.364961	−0.182481	−	0.983209i	$-0.558413\pi$
$983$	−11.4426	−0.364961	−0.182481	+	0.983209i	$0.558413\pi$
$984$	0	0
$985$	37.4104	1.19200
$986$	0	0
$987$	−11.9474	−0.380290
$988$	0	0
$989$	−27.5316	−0.875453
$990$	0	0
$991$	−10.8980	−0.346186	−0.173093	−	0.984905i	$-0.555376\pi$
$991$	−10.8980	−0.346186	−0.173093	+	0.984905i	$0.555376\pi$
$992$	0	0
$993$	14.1860	0.450178
$994$	0	0
$995$	−7.22168	−0.228943
$996$	0	0
$997$	−44.8760	−1.42124	−0.710619	−	0.703577i	$-0.751585\pi$
$997$	−44.8760	−1.42124	−0.710619	+	0.703577i	$0.751585\pi$
$998$	0	0
$999$	144.436	4.56977

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.r.1.14	yes	14
4.3	odd	2		6016.2.a.t.1.1	yes	14
8.3	odd	2		6016.2.a.q.1.14	✓	14
8.5	even	2		6016.2.a.s.1.1	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.q.1.14	✓	14	8.3	odd	2
6016.2.a.r.1.14	yes	14	1.1	even	1	trivial
6016.2.a.s.1.1	yes	14	8.5	even	2
6016.2.a.t.1.1	yes	14	4.3	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+3.32999 q^{3} +4.19635 q^{5} -3.58783 q^{7} +8.08881 q^{9} +O(q^{10})\)	\(q+3.32999 q^{3} +4.19635 q^{5} -3.58783 q^{7} +8.08881 q^{9} -3.64415 q^{11} +2.34285 q^{13} +13.9738 q^{15} -4.50124 q^{17} -0.900429 q^{19} -11.9474 q^{21} +5.94493 q^{23} +12.6093 q^{25} +16.9457 q^{27} +5.93076 q^{29} -2.66525 q^{31} -12.1350 q^{33} -15.0558 q^{35} +8.52350 q^{37} +7.80165 q^{39} -0.502777 q^{41} -4.63110 q^{43} +33.9435 q^{45} +1.00000 q^{47} +5.87250 q^{49} -14.9891 q^{51} +8.85250 q^{53} -15.2921 q^{55} -2.99842 q^{57} +5.61553 q^{59} +3.75806 q^{61} -29.0213 q^{63} +9.83140 q^{65} +9.66989 q^{67} +19.7965 q^{69} -15.5624 q^{71} -12.9563 q^{73} +41.9889 q^{75} +13.0746 q^{77} +8.82925 q^{79} +32.1624 q^{81} -10.8266 q^{83} -18.8888 q^{85} +19.7494 q^{87} +0.284535 q^{89} -8.40573 q^{91} -8.87524 q^{93} -3.77851 q^{95} -13.5774 q^{97} -29.4768 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) \) 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9	\( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9} - 14 q^{11} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} + 6 q^{21} + 18 q^{23} + 22 q^{25} - 8 q^{27} + 22 q^{29} + 4 q^{31} + 2 q^{33} - 26 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} - 12 q^{43} + 30 q^{45} + 14 q^{47} + 34 q^{49} - 18 q^{51} + 20 q^{53} + 2 q^{55} - 4 q^{57} - 32 q^{59} + 12 q^{61} + 40 q^{65} - 16 q^{67} + 46 q^{69} + 16 q^{71} - 12 q^{73} + 16 q^{75} + 10 q^{77} + 16 q^{79} + 74 q^{81} - 14 q^{83} + 12 q^{85} + 14 q^{87} - 28 q^{91} - 16 q^{93} + 52 q^{95} + 28 q^{97} - 50 q^{99}+O(q^{100}) \) 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9 - 14 * q^11 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 + 6 * q^21 + 18 * q^23 + 22 * q^25 - 8 * q^27 + 22 * q^29 + 4 * q^31 + 2 * q^33 - 26 * q^35 + 6 * q^37 + 20 * q^39 + 16 * q^41 - 12 * q^43 + 30 * q^45 + 14 * q^47 + 34 * q^49 - 18 * q^51 + 20 * q^53 + 2 * q^55 - 4 * q^57 - 32 * q^59 + 12 * q^61 + 40 * q^65 - 16 * q^67 + 46 * q^69 + 16 * q^71 - 12 * q^73 + 16 * q^75 + 10 * q^77 + 16 * q^79 + 74 * q^81 - 14 * q^83 + 12 * q^85 + 14 * q^87 - 28 * q^91 - 16 * q^93 + 52 * q^95 + 28 * q^97 - 50 * q^99

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