LMFDB - Embedded newform 6096.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.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:	$48.6768050722$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.81509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 $ x^5 - x^4 - 5x^3 + 3x^2 + 5*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.71377$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.13449 q^{5} -0.967471 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.13449 q^{5} -0.967471 q^{7} +1.00000 q^{9} +6.54277 q^{11} +2.28226 q^{13} -4.13449 q^{15} -2.37575 q^{17} -2.94028 q^{19} -0.967471 q^{21} +1.52353 q^{23} +12.0940 q^{25} +1.00000 q^{27} +2.59455 q^{29} +4.67977 q^{31} +6.54277 q^{33} +4.00000 q^{35} -9.81176 q^{37} +2.28226 q^{39} +4.37325 q^{41} -1.55355 q^{43} -4.13449 q^{45} -11.3763 q^{47} -6.06400 q^{49} -2.37575 q^{51} -4.24599 q^{53} -27.0510 q^{55} -2.94028 q^{57} -3.66649 q^{59} -0.791268 q^{61} -0.967471 q^{63} -9.43600 q^{65} +14.3220 q^{67} +1.52353 q^{69} -6.43287 q^{71} -3.78528 q^{73} +12.0940 q^{75} -6.32994 q^{77} +14.7073 q^{79} +1.00000 q^{81} +3.28601 q^{83} +9.82253 q^{85} +2.59455 q^{87} -7.18063 q^{89} -2.20802 q^{91} +4.67977 q^{93} +12.1566 q^{95} +12.1936 q^{97} +6.54277 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 5 q^{5} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - 5 * q^5 + 5 * q^9	$ 5 q + 5 q^{3} - 5 q^{5} + 5 q^{9} + 16 q^{11} + 3 q^{13} - 5 q^{15} - 6 q^{17} + 8 q^{19} + 9 q^{23} + 4 q^{25} + 5 q^{27} - 17 q^{29} + 9 q^{31} + 16 q^{33} + 20 q^{35} - q^{37} + 3 q^{39} - 2 q^{41} + 4 q^{43} - 5 q^{45} + 8 q^{47} + 13 q^{49} - 6 q^{51} - 15 q^{53} - 20 q^{55} + 8 q^{57} + 19 q^{59} + q^{61} - 5 q^{65} + 2 q^{67} + 9 q^{69} + 13 q^{73} + 4 q^{75} + 2 q^{77} + 28 q^{79} + 5 q^{81} + q^{83} + 6 q^{85} - 17 q^{87} + q^{89} + 18 q^{91} + 9 q^{93} - 4 q^{95} + 28 q^{97} + 16 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 5 * q^5 + 5 * q^9 + 16 * q^11 + 3 * q^13 - 5 * q^15 - 6 * q^17 + 8 * q^19 + 9 * q^23 + 4 * q^25 + 5 * q^27 - 17 * q^29 + 9 * q^31 + 16 * q^33 + 20 * q^35 - q^37 + 3 * q^39 - 2 * q^41 + 4 * q^43 - 5 * q^45 + 8 * q^47 + 13 * q^49 - 6 * q^51 - 15 * q^53 - 20 * q^55 + 8 * q^57 + 19 * q^59 + q^61 - 5 * q^65 + 2 * q^67 + 9 * q^69 + 13 * q^73 + 4 * q^75 + 2 * q^77 + 28 * q^79 + 5 * q^81 + q^83 + 6 * q^85 - 17 * q^87 + q^89 + 18 * q^91 + 9 * q^93 - 4 * q^95 + 28 * q^97 + 16 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−4.13449	−1.84900	−0.924500	−	0.381181i	$-0.875518\pi$
$5$	−4.13449	−1.84900	−0.924500	+	0.381181i	$0.875518\pi$
$6$	0	0
$7$	−0.967471	−0.365670	−0.182835	−	0.983144i	$-0.558527\pi$
$7$	−0.967471	−0.365670	−0.182835	+	0.983144i	$0.558527\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.54277	1.97272	0.986360	−	0.164601i	$-0.0526335\pi$
$11$	6.54277	1.97272	0.986360	+	0.164601i	$0.0526335\pi$
$12$	0	0
$13$	2.28226	0.632986	0.316493	−	0.948595i	$-0.397495\pi$
$13$	2.28226	0.632986	0.316493	+	0.948595i	$0.397495\pi$
$14$	0	0
$15$	−4.13449	−1.06752
$16$	0	0
$17$	−2.37575	−0.576205	−0.288102	−	0.957600i	$-0.593024\pi$
$17$	−2.37575	−0.576205	−0.288102	+	0.957600i	$0.593024\pi$
$18$	0	0
$19$	−2.94028	−0.674547	−0.337274	−	0.941407i	$-0.609505\pi$
$19$	−2.94028	−0.674547	−0.337274	+	0.941407i	$0.609505\pi$
$20$	0	0
$21$	−0.967471	−0.211119
$22$	0	0
$23$	1.52353	0.317677	0.158839	−	0.987305i	$-0.449225\pi$
$23$	1.52353	0.317677	0.158839	+	0.987305i	$0.449225\pi$
$24$	0	0
$25$	12.0940	2.41880
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.59455	0.481796	0.240898	−	0.970550i	$-0.422558\pi$
$29$	2.59455	0.481796	0.240898	+	0.970550i	$0.422558\pi$
$30$	0	0
$31$	4.67977	0.840512	0.420256	−	0.907406i	$-0.361940\pi$
$31$	4.67977	0.840512	0.420256	+	0.907406i	$0.361940\pi$
$32$	0	0
$33$	6.54277	1.13895
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	−9.81176	−1.61304	−0.806522	−	0.591205i	$-0.798653\pi$
$37$	−9.81176	−1.61304	−0.806522	+	0.591205i	$0.798653\pi$
$38$	0	0
$39$	2.28226	0.365455
$40$	0	0
$41$	4.37325	0.682986	0.341493	−	0.939884i	$-0.389067\pi$
$41$	4.37325	0.682986	0.341493	+	0.939884i	$0.389067\pi$
$42$	0	0
$43$	−1.55355	−0.236914	−0.118457	−	0.992959i	$-0.537795\pi$
$43$	−1.55355	−0.236914	−0.118457	+	0.992959i	$0.537795\pi$
$44$	0	0
$45$	−4.13449	−0.616334
$46$	0	0
$47$	−11.3763	−1.65940	−0.829701	−	0.558208i	$-0.811489\pi$
$47$	−11.3763	−1.65940	−0.829701	+	0.558208i	$0.811489\pi$
$48$	0	0
$49$	−6.06400	−0.866286
$50$	0	0
$51$	−2.37575	−0.332672
$52$	0	0
$53$	−4.24599	−0.583231	−0.291616	−	0.956536i	$-0.594193\pi$
$53$	−4.24599	−0.583231	−0.291616	+	0.956536i	$0.594193\pi$
$54$	0	0
$55$	−27.0510	−3.64756
$56$	0	0
$57$	−2.94028	−0.389450
$58$	0	0
$59$	−3.66649	−0.477337	−0.238668	−	0.971101i	$-0.576711\pi$
$59$	−3.66649	−0.477337	−0.238668	+	0.971101i	$0.576711\pi$
$60$	0	0
$61$	−0.791268	−0.101311	−0.0506557	−	0.998716i	$-0.516131\pi$
$61$	−0.791268	−0.101311	−0.0506557	+	0.998716i	$0.516131\pi$
$62$	0	0
$63$	−0.967471	−0.121890
$64$	0	0
$65$	−9.43600	−1.17039
$66$	0	0
$67$	14.3220	1.74971	0.874856	−	0.484384i	$-0.160956\pi$
$67$	14.3220	1.74971	0.874856	+	0.484384i	$0.160956\pi$
$68$	0	0
$69$	1.52353	0.183411
$70$	0	0
$71$	−6.43287	−0.763441	−0.381721	−	0.924278i	$-0.624668\pi$
$71$	−6.43287	−0.763441	−0.381721	+	0.924278i	$0.624668\pi$
$72$	0	0
$73$	−3.78528	−0.443033	−0.221517	−	0.975157i	$-0.571101\pi$
$73$	−3.78528	−0.443033	−0.221517	+	0.975157i	$0.571101\pi$
$74$	0	0
$75$	12.0940	1.39650
$76$	0	0
$77$	−6.32994	−0.721364
$78$	0	0
$79$	14.7073	1.65470	0.827350	−	0.561687i	$-0.189848\pi$
$79$	14.7073	1.65470	0.827350	+	0.561687i	$0.189848\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.28601	0.360687	0.180343	−	0.983604i	$-0.442279\pi$
$83$	3.28601	0.360687	0.180343	+	0.983604i	$0.442279\pi$
$84$	0	0
$85$	9.82253	1.06540
$86$	0	0
$87$	2.59455	0.278165
$88$	0	0
$89$	−7.18063	−0.761145	−0.380573	−	0.924751i	$-0.624273\pi$
$89$	−7.18063	−0.761145	−0.380573	+	0.924751i	$0.624273\pi$
$90$	0	0
$91$	−2.20802	−0.231464
$92$	0	0
$93$	4.67977	0.485270
$94$	0	0
$95$	12.1566	1.24724
$96$	0	0
$97$	12.1936	1.23807	0.619034	−	0.785364i	$-0.287524\pi$
$97$	12.1936	1.23807	0.619034	+	0.785364i	$0.287524\pi$
$98$	0	0
$99$	6.54277	0.657574
$100$	0	0
$101$	8.34252	0.830111	0.415056	−	0.909796i	$-0.363762\pi$
$101$	8.34252	0.830111	0.415056	+	0.909796i	$0.363762\pi$
$102$	0	0
$103$	14.0668	1.38605	0.693023	−	0.720915i	$-0.256278\pi$
$103$	14.0668	1.38605	0.693023	+	0.720915i	$0.256278\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	16.8382	1.62781	0.813907	−	0.580995i	$-0.197337\pi$
$107$	16.8382	1.62781	0.813907	+	0.580995i	$0.197337\pi$
$108$	0	0
$109$	1.57844	0.151187	0.0755934	−	0.997139i	$-0.475915\pi$
$109$	1.57844	0.151187	0.0755934	+	0.997139i	$0.475915\pi$
$110$	0	0
$111$	−9.81176	−0.931291
$112$	0	0
$113$	−7.95578	−0.748417	−0.374208	−	0.927345i	$-0.622086\pi$
$113$	−7.95578	−0.748417	−0.374208	+	0.927345i	$0.622086\pi$
$114$	0	0
$115$	−6.29901	−0.587385
$116$	0	0
$117$	2.28226	0.210995
$118$	0	0
$119$	2.29847	0.210701
$120$	0	0
$121$	31.8079	2.89163
$122$	0	0
$123$	4.37325	0.394322
$124$	0	0
$125$	−29.3302	−2.62337
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	−1.55355	−0.136782
$130$	0	0
$131$	9.12906	0.797610	0.398805	−	0.917036i	$-0.369425\pi$
$131$	9.12906	0.797610	0.398805	+	0.917036i	$0.369425\pi$
$132$	0	0
$133$	2.84464	0.246661
$134$	0	0
$135$	−4.13449	−0.355840
$136$	0	0
$137$	4.41142	0.376893	0.188446	−	0.982083i	$-0.439655\pi$
$137$	4.41142	0.376893	0.188446	+	0.982083i	$0.439655\pi$
$138$	0	0
$139$	3.79440	0.321837	0.160918	−	0.986968i	$-0.448554\pi$
$139$	3.79440	0.321837	0.160918	+	0.986968i	$0.448554\pi$
$140$	0	0
$141$	−11.3763	−0.958056
$142$	0	0
$143$	14.9323	1.24871
$144$	0	0
$145$	−10.7272	−0.890841
$146$	0	0
$147$	−6.06400	−0.500150
$148$	0	0
$149$	8.87021	0.726676	0.363338	−	0.931657i	$-0.381637\pi$
$149$	8.87021	0.726676	0.363338	+	0.931657i	$0.381637\pi$
$150$	0	0
$151$	16.0849	1.30897	0.654487	−	0.756074i	$-0.272885\pi$
$151$	16.0849	1.30897	0.654487	+	0.756074i	$0.272885\pi$
$152$	0	0
$153$	−2.37575	−0.192068
$154$	0	0
$155$	−19.3485	−1.55411
$156$	0	0
$157$	6.83832	0.545757	0.272879	−	0.962048i	$-0.412024\pi$
$157$	6.83832	0.545757	0.272879	+	0.962048i	$0.412024\pi$
$158$	0	0
$159$	−4.24599	−0.336729
$160$	0	0
$161$	−1.47397	−0.116165
$162$	0	0
$163$	−14.9961	−1.17459	−0.587294	−	0.809374i	$-0.699807\pi$
$163$	−14.9961	−1.17459	−0.587294	+	0.809374i	$0.699807\pi$
$164$	0	0
$165$	−27.0510	−2.10592
$166$	0	0
$167$	−16.5131	−1.27782	−0.638910	−	0.769281i	$-0.720614\pi$
$167$	−16.5131	−1.27782	−0.638910	+	0.769281i	$0.720614\pi$
$168$	0	0
$169$	−7.79127	−0.599328
$170$	0	0
$171$	−2.94028	−0.224849
$172$	0	0
$173$	13.2791	1.00959	0.504796	−	0.863239i	$-0.331568\pi$
$173$	13.2791	1.00959	0.504796	+	0.863239i	$0.331568\pi$
$174$	0	0
$175$	−11.7006	−0.884483
$176$	0	0
$177$	−3.66649	−0.275590
$178$	0	0
$179$	20.6583	1.54408	0.772039	−	0.635576i	$-0.219237\pi$
$179$	20.6583	1.54408	0.772039	+	0.635576i	$0.219237\pi$
$180$	0	0
$181$	−7.79062	−0.579072	−0.289536	−	0.957167i	$-0.593501\pi$
$181$	−7.79062	−0.579072	−0.289536	+	0.957167i	$0.593501\pi$
$182$	0	0
$183$	−0.791268	−0.0584922
$184$	0	0
$185$	40.5666	2.98252
$186$	0	0
$187$	−15.5440	−1.13669
$188$	0	0
$189$	−0.967471	−0.0703731
$190$	0	0
$191$	−7.10710	−0.514251	−0.257126	−	0.966378i	$-0.582775\pi$
$191$	−7.10710	−0.514251	−0.257126	+	0.966378i	$0.582775\pi$
$192$	0	0
$193$	22.3655	1.60990	0.804952	−	0.593339i	$-0.202191\pi$
$193$	22.3655	1.60990	0.804952	+	0.593339i	$0.202191\pi$
$194$	0	0
$195$	−9.43600	−0.675726
$196$	0	0
$197$	−9.36592	−0.667294	−0.333647	−	0.942698i	$-0.608279\pi$
$197$	−9.36592	−0.667294	−0.333647	+	0.942698i	$0.608279\pi$
$198$	0	0
$199$	9.62395	0.682224	0.341112	−	0.940023i	$-0.389197\pi$
$199$	9.62395	0.682224	0.341112	+	0.940023i	$0.389197\pi$
$200$	0	0
$201$	14.3220	1.01020
$202$	0	0
$203$	−2.51015	−0.176178
$204$	0	0
$205$	−18.0811	−1.26284
$206$	0	0
$207$	1.52353	0.105892
$208$	0	0
$209$	−19.2376	−1.33069
$210$	0	0
$211$	26.3273	1.81245	0.906225	−	0.422797i	$-0.138952\pi$
$211$	26.3273	1.81245	0.906225	+	0.422797i	$0.138952\pi$
$212$	0	0
$213$	−6.43287	−0.440773
$214$	0	0
$215$	6.42313	0.438054
$216$	0	0
$217$	−4.52754	−0.307350
$218$	0	0
$219$	−3.78528	−0.255785
$220$	0	0
$221$	−5.42210	−0.364730
$222$	0	0
$223$	5.71989	0.383032	0.191516	−	0.981489i	$-0.438660\pi$
$223$	5.71989	0.383032	0.191516	+	0.981489i	$0.438660\pi$
$224$	0	0
$225$	12.0940	0.806268
$226$	0	0
$227$	−9.50301	−0.630737	−0.315369	−	0.948969i	$-0.602128\pi$
$227$	−9.50301	−0.630737	−0.315369	+	0.948969i	$0.602128\pi$
$228$	0	0
$229$	19.7400	1.30446	0.652229	−	0.758022i	$-0.273834\pi$
$229$	19.7400	1.30446	0.652229	+	0.758022i	$0.273834\pi$
$230$	0	0
$231$	−6.32994	−0.416480
$232$	0	0
$233$	22.7039	1.48738	0.743692	−	0.668522i	$-0.233073\pi$
$233$	22.7039	1.48738	0.743692	+	0.668522i	$0.233073\pi$
$234$	0	0
$235$	47.0352	3.06824
$236$	0	0
$237$	14.7073	0.955341
$238$	0	0
$239$	−8.01257	−0.518290	−0.259145	−	0.965838i	$-0.583441\pi$
$239$	−8.01257	−0.518290	−0.259145	+	0.965838i	$0.583441\pi$
$240$	0	0
$241$	13.1361	0.846170	0.423085	−	0.906090i	$-0.360947\pi$
$241$	13.1361	0.846170	0.423085	+	0.906090i	$0.360947\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	25.0716	1.60176
$246$	0	0
$247$	−6.71050	−0.426979
$248$	0	0
$249$	3.28601	0.208243
$250$	0	0
$251$	9.38715	0.592512	0.296256	−	0.955109i	$-0.404262\pi$
$251$	9.38715	0.592512	0.296256	+	0.955109i	$0.404262\pi$
$252$	0	0
$253$	9.96809	0.626688
$254$	0	0
$255$	9.82253	0.615111
$256$	0	0
$257$	−27.1711	−1.69488	−0.847442	−	0.530887i	$-0.821859\pi$
$257$	−27.1711	−1.69488	−0.847442	+	0.530887i	$0.821859\pi$
$258$	0	0
$259$	9.49259	0.589841
$260$	0	0
$261$	2.59455	0.160599
$262$	0	0
$263$	0.652677	0.0402458	0.0201229	−	0.999798i	$-0.493594\pi$
$263$	0.652677	0.0402458	0.0201229	+	0.999798i	$0.493594\pi$
$264$	0	0
$265$	17.5550	1.07839
$266$	0	0
$267$	−7.18063	−0.439447
$268$	0	0
$269$	−26.3984	−1.60954	−0.804768	−	0.593589i	$-0.797711\pi$
$269$	−26.3984	−1.60954	−0.804768	+	0.593589i	$0.797711\pi$
$270$	0	0
$271$	10.0760	0.612074	0.306037	−	0.952020i	$-0.400997\pi$
$271$	10.0760	0.612074	0.306037	+	0.952020i	$0.400997\pi$
$272$	0	0
$273$	−2.20802	−0.133636
$274$	0	0
$275$	79.1285	4.77163
$276$	0	0
$277$	−5.10952	−0.307001	−0.153501	−	0.988149i	$-0.549055\pi$
$277$	−5.10952	−0.307001	−0.153501	+	0.988149i	$0.549055\pi$
$278$	0	0
$279$	4.67977	0.280171
$280$	0	0
$281$	28.8753	1.72255	0.861277	−	0.508137i	$-0.169665\pi$
$281$	28.8753	1.72255	0.861277	+	0.508137i	$0.169665\pi$
$282$	0	0
$283$	−17.4253	−1.03582	−0.517912	−	0.855434i	$-0.673291\pi$
$283$	−17.4253	−1.03582	−0.517912	+	0.855434i	$0.673291\pi$
$284$	0	0
$285$	12.1566	0.720093
$286$	0	0
$287$	−4.23099	−0.249747
$288$	0	0
$289$	−11.3558	−0.667988
$290$	0	0
$291$	12.1936	0.714799
$292$	0	0
$293$	−0.405367	−0.0236818	−0.0118409	−	0.999930i	$-0.503769\pi$
$293$	−0.405367	−0.0236818	−0.0118409	+	0.999930i	$0.503769\pi$
$294$	0	0
$295$	15.1591	0.882596
$296$	0	0
$297$	6.54277	0.379650
$298$	0	0
$299$	3.47709	0.201085
$300$	0	0
$301$	1.50301	0.0866322
$302$	0	0
$303$	8.34252	0.479265
$304$	0	0
$305$	3.27149	0.187325
$306$	0	0
$307$	25.1406	1.43485	0.717424	−	0.696636i	$-0.245321\pi$
$307$	25.1406	1.43485	0.717424	+	0.696636i	$0.245321\pi$
$308$	0	0
$309$	14.0668	0.800234
$310$	0	0
$311$	−30.3282	−1.71975	−0.859877	−	0.510501i	$-0.829460\pi$
$311$	−30.3282	−1.71975	−0.859877	+	0.510501i	$0.829460\pi$
$312$	0	0
$313$	14.6214	0.826450	0.413225	−	0.910629i	$-0.364402\pi$
$313$	14.6214	0.826450	0.413225	+	0.910629i	$0.364402\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	−7.11545	−0.399644	−0.199822	−	0.979832i	$-0.564036\pi$
$317$	−7.11545	−0.399644	−0.199822	+	0.979832i	$0.564036\pi$
$318$	0	0
$319$	16.9756	0.950449
$320$	0	0
$321$	16.8382	0.939819
$322$	0	0
$323$	6.98539	0.388677
$324$	0	0
$325$	27.6018	1.53107
$326$	0	0
$327$	1.57844	0.0872877
$328$	0	0
$329$	11.0062	0.606793
$330$	0	0
$331$	6.68736	0.367571	0.183785	−	0.982966i	$-0.441165\pi$
$331$	6.68736	0.367571	0.183785	+	0.982966i	$0.441165\pi$
$332$	0	0
$333$	−9.81176	−0.537681
$334$	0	0
$335$	−59.2142	−3.23522
$336$	0	0
$337$	−13.0075	−0.708566	−0.354283	−	0.935138i	$-0.615275\pi$
$337$	−13.0075	−0.708566	−0.354283	+	0.935138i	$0.615275\pi$
$338$	0	0
$339$	−7.95578	−0.432099
$340$	0	0
$341$	30.6187	1.65810
$342$	0	0
$343$	12.6390	0.682444
$344$	0	0
$345$	−6.29901	−0.339127
$346$	0	0
$347$	6.25915	0.336009	0.168004	−	0.985786i	$-0.446268\pi$
$347$	6.25915	0.336009	0.168004	+	0.985786i	$0.446268\pi$
$348$	0	0
$349$	14.7050	0.787141	0.393571	−	0.919294i	$-0.371240\pi$
$349$	14.7050	0.787141	0.393571	+	0.919294i	$0.371240\pi$
$350$	0	0
$351$	2.28226	0.121818
$352$	0	0
$353$	−14.0618	−0.748436	−0.374218	−	0.927341i	$-0.622089\pi$
$353$	−14.0618	−0.748436	−0.374218	+	0.927341i	$0.622089\pi$
$354$	0	0
$355$	26.5967	1.41160
$356$	0	0
$357$	2.29847	0.121648
$358$	0	0
$359$	−10.9074	−0.575672	−0.287836	−	0.957680i	$-0.592936\pi$
$359$	−10.9074	−0.575672	−0.287836	+	0.957680i	$0.592936\pi$
$360$	0	0
$361$	−10.3547	−0.544986
$362$	0	0
$363$	31.8079	1.66948
$364$	0	0
$365$	15.6502	0.819169
$366$	0	0
$367$	−34.2092	−1.78570	−0.892852	−	0.450349i	$-0.851299\pi$
$367$	−34.2092	−1.78570	−0.892852	+	0.450349i	$0.851299\pi$
$368$	0	0
$369$	4.37325	0.227662
$370$	0	0
$371$	4.10787	0.213270
$372$	0	0
$373$	27.2740	1.41220	0.706098	−	0.708114i	$-0.250454\pi$
$373$	27.2740	1.41220	0.706098	+	0.708114i	$0.250454\pi$
$374$	0	0
$375$	−29.3302	−1.51460
$376$	0	0
$377$	5.92145	0.304970
$378$	0	0
$379$	−7.60197	−0.390487	−0.195243	−	0.980755i	$-0.562550\pi$
$379$	−7.60197	−0.390487	−0.195243	+	0.980755i	$0.562550\pi$
$380$	0	0
$381$	1.00000	0.0512316
$382$	0	0
$383$	−13.6041	−0.695136	−0.347568	−	0.937655i	$-0.612992\pi$
$383$	−13.6041	−0.695136	−0.347568	+	0.937655i	$0.612992\pi$
$384$	0	0
$385$	26.1711	1.33380
$386$	0	0
$387$	−1.55355	−0.0789713
$388$	0	0
$389$	0.253158	0.0128356	0.00641780	−	0.999979i	$-0.497957\pi$
$389$	0.253158	0.0128356	0.00641780	+	0.999979i	$0.497957\pi$
$390$	0	0
$391$	−3.61952	−0.183047
$392$	0	0
$393$	9.12906	0.460500
$394$	0	0
$395$	−60.8072	−3.05954
$396$	0	0
$397$	30.3282	1.52213	0.761064	−	0.648677i	$-0.224677\pi$
$397$	30.3282	1.52213	0.761064	+	0.648677i	$0.224677\pi$
$398$	0	0
$399$	2.84464	0.142410
$400$	0	0
$401$	−26.1009	−1.30342	−0.651708	−	0.758470i	$-0.725947\pi$
$401$	−26.1009	−1.30342	−0.651708	+	0.758470i	$0.725947\pi$
$402$	0	0
$403$	10.6805	0.532033
$404$	0	0
$405$	−4.13449	−0.205445
$406$	0	0
$407$	−64.1961	−3.18208
$408$	0	0
$409$	−20.2917	−1.00336	−0.501679	−	0.865054i	$-0.667284\pi$
$409$	−20.2917	−1.00336	−0.501679	+	0.865054i	$0.667284\pi$
$410$	0	0
$411$	4.41142	0.217599
$412$	0	0
$413$	3.54722	0.174547
$414$	0	0
$415$	−13.5860	−0.666910
$416$	0	0
$417$	3.79440	0.185813
$418$	0	0
$419$	−14.8913	−0.727488	−0.363744	−	0.931499i	$-0.618502\pi$
$419$	−14.8913	−0.727488	−0.363744	+	0.931499i	$0.618502\pi$
$420$	0	0
$421$	36.5814	1.78287	0.891433	−	0.453152i	$-0.149700\pi$
$421$	36.5814	1.78287	0.891433	+	0.453152i	$0.149700\pi$
$422$	0	0
$423$	−11.3763	−0.553134
$424$	0	0
$425$	−28.7324	−1.39373
$426$	0	0
$427$	0.765529	0.0370465
$428$	0	0
$429$	14.9323	0.720940
$430$	0	0
$431$	28.0009	1.34875	0.674377	−	0.738387i	$-0.264412\pi$
$431$	28.0009	1.34875	0.674377	+	0.738387i	$0.264412\pi$
$432$	0	0
$433$	28.4444	1.36695	0.683475	−	0.729974i	$-0.260468\pi$
$433$	28.4444	1.36695	0.683475	+	0.729974i	$0.260468\pi$
$434$	0	0
$435$	−10.7272	−0.514328
$436$	0	0
$437$	−4.47960	−0.214288
$438$	0	0
$439$	25.3271	1.20879	0.604397	−	0.796683i	$-0.293414\pi$
$439$	25.3271	1.20879	0.604397	+	0.796683i	$0.293414\pi$
$440$	0	0
$441$	−6.06400	−0.288762
$442$	0	0
$443$	12.5092	0.594331	0.297165	−	0.954826i	$-0.403959\pi$
$443$	12.5092	0.594331	0.297165	+	0.954826i	$0.403959\pi$
$444$	0	0
$445$	29.6883	1.40736
$446$	0	0
$447$	8.87021	0.419547
$448$	0	0
$449$	−0.745131	−0.0351649	−0.0175825	−	0.999845i	$-0.505597\pi$
$449$	−0.745131	−0.0351649	−0.0175825	+	0.999845i	$0.505597\pi$
$450$	0	0
$451$	28.6132	1.34734
$452$	0	0
$453$	16.0849	0.755736
$454$	0	0
$455$	9.12906	0.427977
$456$	0	0
$457$	−31.4303	−1.47025	−0.735125	−	0.677932i	$-0.762876\pi$
$457$	−31.4303	−1.47025	−0.735125	+	0.677932i	$0.762876\pi$
$458$	0	0
$459$	−2.37575	−0.110891
$460$	0	0
$461$	−5.65191	−0.263236	−0.131618	−	0.991301i	$-0.542017\pi$
$461$	−5.65191	−0.263236	−0.131618	+	0.991301i	$0.542017\pi$
$462$	0	0
$463$	−28.3746	−1.31868	−0.659340	−	0.751845i	$-0.729164\pi$
$463$	−28.3746	−1.31868	−0.659340	+	0.751845i	$0.729164\pi$
$464$	0	0
$465$	−19.3485	−0.897264
$466$	0	0
$467$	17.6407	0.816314	0.408157	−	0.912912i	$-0.366172\pi$
$467$	17.6407	0.816314	0.408157	+	0.912912i	$0.366172\pi$
$468$	0	0
$469$	−13.8561	−0.639816
$470$	0	0
$471$	6.83832	0.315093
$472$	0	0
$473$	−10.1645	−0.467365
$474$	0	0
$475$	−35.5598	−1.63160
$476$	0	0
$477$	−4.24599	−0.194410
$478$	0	0
$479$	30.9778	1.41541	0.707707	−	0.706506i	$-0.249730\pi$
$479$	30.9778	1.41541	0.707707	+	0.706506i	$0.249730\pi$
$480$	0	0
$481$	−22.3930	−1.02103
$482$	0	0
$483$	−1.47397	−0.0670678
$484$	0	0
$485$	−50.4142	−2.28919
$486$	0	0
$487$	−6.99711	−0.317069	−0.158535	−	0.987353i	$-0.550677\pi$
$487$	−6.99711	−0.317069	−0.158535	+	0.987353i	$0.550677\pi$
$488$	0	0
$489$	−14.9961	−0.678149
$490$	0	0
$491$	13.0495	0.588915	0.294457	−	0.955665i	$-0.404861\pi$
$491$	13.0495	0.588915	0.294457	+	0.955665i	$0.404861\pi$
$492$	0	0
$493$	−6.16401	−0.277613
$494$	0	0
$495$	−27.0510	−1.21585
$496$	0	0
$497$	6.22362	0.279167
$498$	0	0
$499$	7.13316	0.319324	0.159662	−	0.987172i	$-0.448960\pi$
$499$	7.13316	0.319324	0.159662	+	0.987172i	$0.448960\pi$
$500$	0	0
$501$	−16.5131	−0.737750
$502$	0	0
$503$	7.32651	0.326673	0.163336	−	0.986570i	$-0.447774\pi$
$503$	7.32651	0.326673	0.163336	+	0.986570i	$0.447774\pi$
$504$	0	0
$505$	−34.4921	−1.53488
$506$	0	0
$507$	−7.79127	−0.346022
$508$	0	0
$509$	1.50803	0.0668421	0.0334211	−	0.999441i	$-0.489360\pi$
$509$	1.50803	0.0668421	0.0334211	+	0.999441i	$0.489360\pi$
$510$	0	0
$511$	3.66214	0.162004
$512$	0	0
$513$	−2.94028	−0.129817
$514$	0	0
$515$	−58.1592	−2.56280
$516$	0	0
$517$	−74.4325	−3.27354
$518$	0	0
$519$	13.2791	0.582888
$520$	0	0
$521$	6.01622	0.263575	0.131788	−	0.991278i	$-0.457928\pi$
$521$	6.01622	0.263575	0.131788	+	0.991278i	$0.457928\pi$
$522$	0	0
$523$	−23.9339	−1.04656	−0.523278	−	0.852162i	$-0.675291\pi$
$523$	−23.9339	−1.04656	−0.523278	+	0.852162i	$0.675291\pi$
$524$	0	0
$525$	−11.7006	−0.510657
$526$	0	0
$527$	−11.1180	−0.484307
$528$	0	0
$529$	−20.6789	−0.899081
$530$	0	0
$531$	−3.66649	−0.159112
$532$	0	0
$533$	9.98090	0.432321
$534$	0	0
$535$	−69.6176	−3.00983
$536$	0	0
$537$	20.6583	0.891473
$538$	0	0
$539$	−39.6754	−1.70894
$540$	0	0
$541$	14.9989	0.644855	0.322427	−	0.946594i	$-0.395501\pi$
$541$	14.9989	0.644855	0.322427	+	0.946594i	$0.395501\pi$
$542$	0	0
$543$	−7.79062	−0.334327
$544$	0	0
$545$	−6.52603	−0.279545
$546$	0	0
$547$	−14.9122	−0.637599	−0.318799	−	0.947822i	$-0.603280\pi$
$547$	−14.9122	−0.637599	−0.318799	+	0.947822i	$0.603280\pi$
$548$	0	0
$549$	−0.791268	−0.0337705
$550$	0	0
$551$	−7.62871	−0.324994
$552$	0	0
$553$	−14.2289	−0.605073
$554$	0	0
$555$	40.5666	1.72196
$556$	0	0
$557$	−7.80482	−0.330701	−0.165351	−	0.986235i	$-0.552876\pi$
$557$	−7.80482	−0.330701	−0.165351	+	0.986235i	$0.552876\pi$
$558$	0	0
$559$	−3.54561	−0.149963
$560$	0	0
$561$	−15.5440	−0.656269
$562$	0	0
$563$	37.3614	1.57459	0.787297	−	0.616574i	$-0.211480\pi$
$563$	37.3614	1.57459	0.787297	+	0.616574i	$0.211480\pi$
$564$	0	0
$565$	32.8931	1.38382
$566$	0	0
$567$	−0.967471	−0.0406300
$568$	0	0
$569$	−17.6673	−0.740653	−0.370326	−	0.928902i	$-0.620754\pi$
$569$	−17.6673	−0.740653	−0.370326	+	0.928902i	$0.620754\pi$
$570$	0	0
$571$	27.4651	1.14938	0.574690	−	0.818371i	$-0.305123\pi$
$571$	27.4651	1.14938	0.574690	+	0.818371i	$0.305123\pi$
$572$	0	0
$573$	−7.10710	−0.296903
$574$	0	0
$575$	18.4256	0.768399
$576$	0	0
$577$	−27.0627	−1.12664	−0.563318	−	0.826240i	$-0.690475\pi$
$577$	−27.0627	−1.12664	−0.563318	+	0.826240i	$0.690475\pi$
$578$	0	0
$579$	22.3655	0.929479
$580$	0	0
$581$	−3.17912	−0.131892
$582$	0	0
$583$	−27.7805	−1.15055
$584$	0	0
$585$	−9.43600	−0.390131
$586$	0	0
$587$	7.89556	0.325885	0.162942	−	0.986636i	$-0.447902\pi$
$587$	7.89556	0.325885	0.162942	+	0.986636i	$0.447902\pi$
$588$	0	0
$589$	−13.7599	−0.566965
$590$	0	0
$591$	−9.36592	−0.385263
$592$	0	0
$593$	−7.65294	−0.314269	−0.157134	−	0.987577i	$-0.550226\pi$
$593$	−7.65294	−0.314269	−0.157134	+	0.987577i	$0.550226\pi$
$594$	0	0
$595$	−9.50301	−0.389586
$596$	0	0
$597$	9.62395	0.393882
$598$	0	0
$599$	18.8580	0.770518	0.385259	−	0.922809i	$-0.374112\pi$
$599$	18.8580	0.770518	0.385259	+	0.922809i	$0.374112\pi$
$600$	0	0
$601$	−12.1847	−0.497024	−0.248512	−	0.968629i	$-0.579942\pi$
$601$	−12.1847	−0.497024	−0.248512	+	0.968629i	$0.579942\pi$
$602$	0	0
$603$	14.3220	0.583237
$604$	0	0
$605$	−131.509	−5.34662
$606$	0	0
$607$	34.5344	1.40171	0.700855	−	0.713304i	$-0.252802\pi$
$607$	34.5344	1.40171	0.700855	+	0.713304i	$0.252802\pi$
$608$	0	0
$609$	−2.51015	−0.101717
$610$	0	0
$611$	−25.9637	−1.05038
$612$	0	0
$613$	25.5621	1.03245	0.516223	−	0.856454i	$-0.327338\pi$
$613$	25.5621	1.03245	0.516223	+	0.856454i	$0.327338\pi$
$614$	0	0
$615$	−18.0811	−0.729102
$616$	0	0
$617$	41.7910	1.68244	0.841222	−	0.540690i	$-0.181837\pi$
$617$	41.7910	1.68244	0.841222	+	0.540690i	$0.181837\pi$
$618$	0	0
$619$	19.5859	0.787225	0.393613	−	0.919276i	$-0.371225\pi$
$619$	19.5859	0.787225	0.393613	+	0.919276i	$0.371225\pi$
$620$	0	0
$621$	1.52353	0.0611370
$622$	0	0
$623$	6.94705	0.278328
$624$	0	0
$625$	60.7953	2.43181
$626$	0	0
$627$	−19.2376	−0.768276
$628$	0	0
$629$	23.3103	0.929443
$630$	0	0
$631$	−3.52816	−0.140454	−0.0702269	−	0.997531i	$-0.522372\pi$
$631$	−3.52816	−0.140454	−0.0702269	+	0.997531i	$0.522372\pi$
$632$	0	0
$633$	26.3273	1.04642
$634$	0	0
$635$	−4.13449	−0.164072
$636$	0	0
$637$	−13.8397	−0.548347
$638$	0	0
$639$	−6.43287	−0.254480
$640$	0	0
$641$	4.70906	0.185997	0.0929984	−	0.995666i	$-0.470355\pi$
$641$	4.70906	0.185997	0.0929984	+	0.995666i	$0.470355\pi$
$642$	0	0
$643$	−9.96706	−0.393063	−0.196531	−	0.980498i	$-0.562968\pi$
$643$	−9.96706	−0.393063	−0.196531	+	0.980498i	$0.562968\pi$
$644$	0	0
$645$	6.42313	0.252911
$646$	0	0
$647$	30.3043	1.19139	0.595693	−	0.803212i	$-0.296877\pi$
$647$	30.3043	1.19139	0.595693	+	0.803212i	$0.296877\pi$
$648$	0	0
$649$	−23.9890	−0.941652
$650$	0	0
$651$	−4.52754	−0.177448
$652$	0	0
$653$	−6.17310	−0.241572	−0.120786	−	0.992679i	$-0.538542\pi$
$653$	−6.17310	−0.241572	−0.120786	+	0.992679i	$0.538542\pi$
$654$	0	0
$655$	−37.7440	−1.47478
$656$	0	0
$657$	−3.78528	−0.147678
$658$	0	0
$659$	14.0718	0.548158	0.274079	−	0.961707i	$-0.411627\pi$
$659$	14.0718	0.548158	0.274079	+	0.961707i	$0.411627\pi$
$660$	0	0
$661$	−23.0627	−0.897033	−0.448517	−	0.893774i	$-0.648048\pi$
$661$	−23.0627	−0.897033	−0.448517	+	0.893774i	$0.648048\pi$
$662$	0	0
$663$	−5.42210	−0.210577
$664$	0	0
$665$	−11.7611	−0.456077
$666$	0	0
$667$	3.95287	0.153056
$668$	0	0
$669$	5.71989	0.221144
$670$	0	0
$671$	−5.17709	−0.199859
$672$	0	0
$673$	−23.0552	−0.888714	−0.444357	−	0.895850i	$-0.646568\pi$
$673$	−23.0552	−0.888714	−0.444357	+	0.895850i	$0.646568\pi$
$674$	0	0
$675$	12.0940	0.465499
$676$	0	0
$677$	−0.732075	−0.0281359	−0.0140680	−	0.999901i	$-0.504478\pi$
$677$	−0.732075	−0.0281359	−0.0140680	+	0.999901i	$0.504478\pi$
$678$	0	0
$679$	−11.7969	−0.452724
$680$	0	0
$681$	−9.50301	−0.364156
$682$	0	0
$683$	35.1152	1.34365	0.671824	−	0.740711i	$-0.265511\pi$
$683$	35.1152	1.34365	0.671824	+	0.740711i	$0.265511\pi$
$684$	0	0
$685$	−18.2390	−0.696875
$686$	0	0
$687$	19.7400	0.753129
$688$	0	0
$689$	−9.69047	−0.369177
$690$	0	0
$691$	3.36991	0.128197	0.0640987	−	0.997944i	$-0.479583\pi$
$691$	3.36991	0.128197	0.0640987	+	0.997944i	$0.479583\pi$
$692$	0	0
$693$	−6.32994	−0.240455
$694$	0	0
$695$	−15.6879	−0.595077
$696$	0	0
$697$	−10.3898	−0.393540
$698$	0	0
$699$	22.7039	0.858742
$700$	0	0
$701$	−14.7054	−0.555417	−0.277709	−	0.960665i	$-0.589575\pi$
$701$	−14.7054	−0.555417	−0.277709	+	0.960665i	$0.589575\pi$
$702$	0	0
$703$	28.8493	1.08807
$704$	0	0
$705$	47.0352	1.77145
$706$	0	0
$707$	−8.07114	−0.303546
$708$	0	0
$709$	49.3869	1.85476	0.927382	−	0.374115i	$-0.122053\pi$
$709$	49.3869	1.85476	0.927382	+	0.374115i	$0.122053\pi$
$710$	0	0
$711$	14.7073	0.551566
$712$	0	0
$713$	7.12976	0.267011
$714$	0	0
$715$	−61.7376	−2.30886
$716$	0	0
$717$	−8.01257	−0.299235
$718$	0	0
$719$	18.0040	0.671437	0.335718	−	0.941962i	$-0.391021\pi$
$719$	18.0040	0.671437	0.335718	+	0.941962i	$0.391021\pi$
$720$	0	0
$721$	−13.6093	−0.506835
$722$	0	0
$723$	13.1361	0.488536
$724$	0	0
$725$	31.3786	1.16537
$726$	0	0
$727$	32.8062	1.21672	0.608358	−	0.793663i	$-0.291828\pi$
$727$	32.8062	1.21672	0.608358	+	0.793663i	$0.291828\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.69085	0.136511
$732$	0	0
$733$	7.99902	0.295451	0.147725	−	0.989028i	$-0.452805\pi$
$733$	7.99902	0.295451	0.147725	+	0.989028i	$0.452805\pi$
$734$	0	0
$735$	25.0716	0.924778
$736$	0	0
$737$	93.7056	3.45169
$738$	0	0
$739$	14.5624	0.535688	0.267844	−	0.963462i	$-0.413689\pi$
$739$	14.5624	0.535688	0.267844	+	0.963462i	$0.413689\pi$
$740$	0	0
$741$	−6.71050	−0.246516
$742$	0	0
$743$	28.9373	1.06161	0.530804	−	0.847495i	$-0.321890\pi$
$743$	28.9373	1.06161	0.530804	+	0.847495i	$0.321890\pi$
$744$	0	0
$745$	−36.6738	−1.34362
$746$	0	0
$747$	3.28601	0.120229
$748$	0	0
$749$	−16.2905	−0.595242
$750$	0	0
$751$	38.4252	1.40215	0.701077	−	0.713085i	$-0.252703\pi$
$751$	38.4252	1.40215	0.701077	+	0.713085i	$0.252703\pi$
$752$	0	0
$753$	9.38715	0.342087
$754$	0	0
$755$	−66.5030	−2.42029
$756$	0	0
$757$	13.2476	0.481494	0.240747	−	0.970588i	$-0.422608\pi$
$757$	13.2476	0.481494	0.240747	+	0.970588i	$0.422608\pi$
$758$	0	0
$759$	9.96809	0.361819
$760$	0	0
$761$	−14.1547	−0.513107	−0.256553	−	0.966530i	$-0.582587\pi$
$761$	−14.1547	−0.513107	−0.256553	+	0.966530i	$0.582587\pi$
$762$	0	0
$763$	−1.52709	−0.0552844
$764$	0	0
$765$	9.82253	0.355134
$766$	0	0
$767$	−8.36791	−0.302148
$768$	0	0
$769$	−43.6959	−1.57571	−0.787857	−	0.615858i	$-0.788810\pi$
$769$	−43.6959	−1.57571	−0.787857	+	0.615858i	$0.788810\pi$
$770$	0	0
$771$	−27.1711	−0.978542
$772$	0	0
$773$	−27.4422	−0.987029	−0.493514	−	0.869738i	$-0.664288\pi$
$773$	−27.4422	−0.987029	−0.493514	+	0.869738i	$0.664288\pi$
$774$	0	0
$775$	56.5973	2.03303
$776$	0	0
$777$	9.49259	0.340545
$778$	0	0
$779$	−12.8586	−0.460706
$780$	0	0
$781$	−42.0888	−1.50606
$782$	0	0
$783$	2.59455	0.0927217
$784$	0	0
$785$	−28.2730	−1.00911
$786$	0	0
$787$	−35.8638	−1.27841	−0.639204	−	0.769037i	$-0.720736\pi$
$787$	−35.8638	−1.27841	−0.639204	+	0.769037i	$0.720736\pi$
$788$	0	0
$789$	0.652677	0.0232359
$790$	0	0
$791$	7.69699	0.273673
$792$	0	0
$793$	−1.80588	−0.0641288
$794$	0	0
$795$	17.5550	0.622612
$796$	0	0
$797$	−4.75192	−0.168322	−0.0841608	−	0.996452i	$-0.526821\pi$
$797$	−4.75192	−0.168322	−0.0841608	+	0.996452i	$0.526821\pi$
$798$	0	0
$799$	27.0272	0.956155
$800$	0	0
$801$	−7.18063	−0.253715
$802$	0	0
$803$	−24.7662	−0.873981
$804$	0	0
$805$	6.09410	0.214789
$806$	0	0
$807$	−26.3984	−0.929267
$808$	0	0
$809$	−17.3059	−0.608441	−0.304221	−	0.952602i	$-0.598396\pi$
$809$	−17.3059	−0.608441	−0.304221	+	0.952602i	$0.598396\pi$
$810$	0	0
$811$	−1.95966	−0.0688128	−0.0344064	−	0.999408i	$-0.510954\pi$
$811$	−1.95966	−0.0688128	−0.0344064	+	0.999408i	$0.510954\pi$
$812$	0	0
$813$	10.0760	0.353381
$814$	0	0
$815$	62.0014	2.17181
$816$	0	0
$817$	4.56787	0.159810
$818$	0	0
$819$	−2.20802	−0.0771546
$820$	0	0
$821$	17.7854	0.620716	0.310358	−	0.950620i	$-0.399551\pi$
$821$	17.7854	0.620716	0.310358	+	0.950620i	$0.399551\pi$
$822$	0	0
$823$	−28.7192	−1.00109	−0.500544	−	0.865711i	$-0.666867\pi$
$823$	−28.7192	−1.00109	−0.500544	+	0.865711i	$0.666867\pi$
$824$	0	0
$825$	79.1285	2.75490
$826$	0	0
$827$	−33.0749	−1.15013	−0.575063	−	0.818109i	$-0.695022\pi$
$827$	−33.0749	−1.15013	−0.575063	+	0.818109i	$0.695022\pi$
$828$	0	0
$829$	41.8288	1.45278	0.726388	−	0.687285i	$-0.241198\pi$
$829$	41.8288	1.45278	0.726388	+	0.687285i	$0.241198\pi$
$830$	0	0
$831$	−5.10952	−0.177247
$832$	0	0
$833$	14.4066	0.499158
$834$	0	0
$835$	68.2732	2.36269
$836$	0	0
$837$	4.67977	0.161757
$838$	0	0
$839$	−42.8105	−1.47798	−0.738991	−	0.673716i	$-0.764697\pi$
$839$	−42.8105	−1.47798	−0.738991	+	0.673716i	$0.764697\pi$
$840$	0	0
$841$	−22.2683	−0.767873
$842$	0	0
$843$	28.8753	0.994516
$844$	0	0
$845$	32.2129	1.10816
$846$	0	0
$847$	−30.7732	−1.05738
$848$	0	0
$849$	−17.4253	−0.598033
$850$	0	0
$851$	−14.9485	−0.512427
$852$	0	0
$853$	−4.09685	−0.140273	−0.0701367	−	0.997537i	$-0.522344\pi$
$853$	−4.09685	−0.140273	−0.0701367	+	0.997537i	$0.522344\pi$
$854$	0	0
$855$	12.1566	0.415746
$856$	0	0
$857$	−16.7221	−0.571215	−0.285608	−	0.958347i	$-0.592195\pi$
$857$	−16.7221	−0.571215	−0.285608	+	0.958347i	$0.592195\pi$
$858$	0	0
$859$	44.8146	1.52906	0.764528	−	0.644591i	$-0.222972\pi$
$859$	44.8146	1.52906	0.764528	+	0.644591i	$0.222972\pi$
$860$	0	0
$861$	−4.23099	−0.144192
$862$	0	0
$863$	−41.7931	−1.42265	−0.711327	−	0.702861i	$-0.751906\pi$
$863$	−41.7931	−1.42265	−0.711327	+	0.702861i	$0.751906\pi$
$864$	0	0
$865$	−54.9024	−1.86674
$866$	0	0
$867$	−11.3558	−0.385663
$868$	0	0
$869$	96.2265	3.26426
$870$	0	0
$871$	32.6866	1.10754
$872$	0	0
$873$	12.1936	0.412689
$874$	0	0
$875$	28.3761	0.959287
$876$	0	0
$877$	−33.4985	−1.13117	−0.565583	−	0.824692i	$-0.691349\pi$
$877$	−33.4985	−1.13117	−0.565583	+	0.824692i	$0.691349\pi$
$878$	0	0
$879$	−0.405367	−0.0136727
$880$	0	0
$881$	−17.6316	−0.594024	−0.297012	−	0.954874i	$-0.595990\pi$
$881$	−17.6316	−0.594024	−0.297012	+	0.954874i	$0.595990\pi$
$882$	0	0
$883$	−42.9645	−1.44587	−0.722936	−	0.690915i	$-0.757208\pi$
$883$	−42.9645	−1.44587	−0.722936	+	0.690915i	$0.757208\pi$
$884$	0	0
$885$	15.1591	0.509567
$886$	0	0
$887$	−2.21154	−0.0742563	−0.0371282	−	0.999311i	$-0.511821\pi$
$887$	−2.21154	−0.0742563	−0.0371282	+	0.999311i	$0.511821\pi$
$888$	0	0
$889$	−0.967471	−0.0324479
$890$	0	0
$891$	6.54277	0.219191
$892$	0	0
$893$	33.4495	1.11934
$894$	0	0
$895$	−85.4118	−2.85500
$896$	0	0
$897$	3.47709	0.116097
$898$	0	0
$899$	12.1419	0.404955
$900$	0	0
$901$	10.0874	0.336061
$902$	0	0
$903$	1.50301	0.0500171
$904$	0	0
$905$	32.2102	1.07070
$906$	0	0
$907$	8.99937	0.298819	0.149410	−	0.988775i	$-0.452263\pi$
$907$	8.99937	0.298819	0.149410	+	0.988775i	$0.452263\pi$
$908$	0	0
$909$	8.34252	0.276704
$910$	0	0
$911$	−15.9672	−0.529016	−0.264508	−	0.964383i	$-0.585210\pi$
$911$	−15.9672	−0.529016	−0.264508	+	0.964383i	$0.585210\pi$
$912$	0	0
$913$	21.4996	0.711534
$914$	0	0
$915$	3.27149	0.108152
$916$	0	0
$917$	−8.83210	−0.291662
$918$	0	0
$919$	−12.4596	−0.411003	−0.205501	−	0.978657i	$-0.565882\pi$
$919$	−12.4596	−0.411003	−0.205501	+	0.978657i	$0.565882\pi$
$920$	0	0
$921$	25.1406	0.828410
$922$	0	0
$923$	−14.6815	−0.483248
$924$	0	0
$925$	−118.664	−3.90164
$926$	0	0
$927$	14.0668	0.462015
$928$	0	0
$929$	19.7470	0.647880	0.323940	−	0.946078i	$-0.394992\pi$
$929$	19.7470	0.647880	0.323940	+	0.946078i	$0.394992\pi$
$930$	0	0
$931$	17.8299	0.584351
$932$	0	0
$933$	−30.3282	−0.992901
$934$	0	0
$935$	64.2666	2.10174
$936$	0	0
$937$	17.5737	0.574109	0.287055	−	0.957914i	$-0.407324\pi$
$937$	17.5737	0.574109	0.287055	+	0.957914i	$0.407324\pi$
$938$	0	0
$939$	14.6214	0.477151
$940$	0	0
$941$	11.8957	0.387790	0.193895	−	0.981022i	$-0.437888\pi$
$941$	11.8957	0.387790	0.193895	+	0.981022i	$0.437888\pi$
$942$	0	0
$943$	6.66275	0.216969
$944$	0	0
$945$	4.00000	0.130120
$946$	0	0
$947$	−14.8940	−0.483990	−0.241995	−	0.970278i	$-0.577802\pi$
$947$	−14.8940	−0.483990	−0.241995	+	0.970278i	$0.577802\pi$
$948$	0	0
$949$	−8.63900	−0.280434
$950$	0	0
$951$	−7.11545	−0.230734
$952$	0	0
$953$	−17.6862	−0.572912	−0.286456	−	0.958093i	$-0.592477\pi$
$953$	−17.6862	−0.572912	−0.286456	+	0.958093i	$0.592477\pi$
$954$	0	0
$955$	29.3842	0.950851
$956$	0	0
$957$	16.9756	0.548742
$958$	0	0
$959$	−4.26792	−0.137818
$960$	0	0
$961$	−9.09972	−0.293539
$962$	0	0
$963$	16.8382	0.542605
$964$	0	0
$965$	−92.4700	−2.97672
$966$	0	0
$967$	−14.4381	−0.464297	−0.232149	−	0.972680i	$-0.574576\pi$
$967$	−14.4381	−0.464297	−0.232149	+	0.972680i	$0.574576\pi$
$968$	0	0
$969$	6.98539	0.224403
$970$	0	0
$971$	−7.53167	−0.241703	−0.120851	−	0.992671i	$-0.538562\pi$
$971$	−7.53167	−0.241703	−0.120851	+	0.992671i	$0.538562\pi$
$972$	0	0
$973$	−3.67097	−0.117686
$974$	0	0
$975$	27.6018	0.883964
$976$	0	0
$977$	3.00952	0.0962829	0.0481415	−	0.998841i	$-0.484670\pi$
$977$	3.00952	0.0962829	0.0481415	+	0.998841i	$0.484670\pi$
$978$	0	0
$979$	−46.9812	−1.50153
$980$	0	0
$981$	1.57844	0.0503956
$982$	0	0
$983$	−49.2460	−1.57070	−0.785352	−	0.619050i	$-0.787518\pi$
$983$	−49.2460	−1.57070	−0.785352	+	0.619050i	$0.787518\pi$
$984$	0	0
$985$	38.7233	1.23383
$986$	0	0
$987$	11.0062	0.350332
$988$	0	0
$989$	−2.36687	−0.0752621
$990$	0	0
$991$	17.6163	0.559600	0.279800	−	0.960058i	$-0.409732\pi$
$991$	17.6163	0.559600	0.279800	+	0.960058i	$0.409732\pi$
$992$	0	0
$993$	6.68736	0.212217
$994$	0	0
$995$	−39.7901	−1.26143
$996$	0	0
$997$	12.3492	0.391102	0.195551	−	0.980694i	$-0.437350\pi$
$997$	12.3492	0.391102	0.195551	+	0.980694i	$0.437350\pi$
$998$	0	0
$999$	−9.81176	−0.310430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.be.1.1		5
4.3	odd	2		381.2.a.c.1.5	✓	5
12.11	even	2		1143.2.a.h.1.1		5
20.19	odd	2		9525.2.a.k.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.c.1.5	✓	5	4.3	odd	2
1143.2.a.h.1.1		5	12.11	even	2
6096.2.a.be.1.1		5	1.1	even	1	trivial
9525.2.a.k.1.1		5	20.19	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:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.13449 q^{5} -0.967471 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.13449 q^{5} -0.967471 q^{7} +1.00000 q^{9} +6.54277 q^{11} +2.28226 q^{13} -4.13449 q^{15} -2.37575 q^{17} -2.94028 q^{19} -0.967471 q^{21} +1.52353 q^{23} +12.0940 q^{25} +1.00000 q^{27} +2.59455 q^{29} +4.67977 q^{31} +6.54277 q^{33} +4.00000 q^{35} -9.81176 q^{37} +2.28226 q^{39} +4.37325 q^{41} -1.55355 q^{43} -4.13449 q^{45} -11.3763 q^{47} -6.06400 q^{49} -2.37575 q^{51} -4.24599 q^{53} -27.0510 q^{55} -2.94028 q^{57} -3.66649 q^{59} -0.791268 q^{61} -0.967471 q^{63} -9.43600 q^{65} +14.3220 q^{67} +1.52353 q^{69} -6.43287 q^{71} -3.78528 q^{73} +12.0940 q^{75} -6.32994 q^{77} +14.7073 q^{79} +1.00000 q^{81} +3.28601 q^{83} +9.82253 q^{85} +2.59455 q^{87} -7.18063 q^{89} -2.20802 q^{91} +4.67977 q^{93} +12.1566 q^{95} +12.1936 q^{97} +6.54277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - 5 * q^5 + 5 * q^9	\( 5 q + 5 q^{3} - 5 q^{5} + 5 q^{9} + 16 q^{11} + 3 q^{13} - 5 q^{15} - 6 q^{17} + 8 q^{19} + 9 q^{23} + 4 q^{25} + 5 q^{27} - 17 q^{29} + 9 q^{31} + 16 q^{33} + 20 q^{35} - q^{37} + 3 q^{39} - 2 q^{41} + 4 q^{43} - 5 q^{45} + 8 q^{47} + 13 q^{49} - 6 q^{51} - 15 q^{53} - 20 q^{55} + 8 q^{57} + 19 q^{59} + q^{61} - 5 q^{65} + 2 q^{67} + 9 q^{69} + 13 q^{73} + 4 q^{75} + 2 q^{77} + 28 q^{79} + 5 q^{81} + q^{83} + 6 q^{85} - 17 q^{87} + q^{89} + 18 q^{91} + 9 q^{93} - 4 q^{95} + 28 q^{97} + 16 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 5 * q^5 + 5 * q^9 + 16 * q^11 + 3 * q^13 - 5 * q^15 - 6 * q^17 + 8 * q^19 + 9 * q^23 + 4 * q^25 + 5 * q^27 - 17 * q^29 + 9 * q^31 + 16 * q^33 + 20 * q^35 - q^37 + 3 * q^39 - 2 * q^41 + 4 * q^43 - 5 * q^45 + 8 * q^47 + 13 * q^49 - 6 * q^51 - 15 * q^53 - 20 * q^55 + 8 * q^57 + 19 * q^59 + q^61 - 5 * q^65 + 2 * q^67 + 9 * q^69 + 13 * q^73 + 4 * q^75 + 2 * q^77 + 28 * q^79 + 5 * q^81 + q^83 + 6 * q^85 - 17 * q^87 + q^89 + 18 * q^91 + 9 * q^93 - 4 * q^95 + 28 * q^97 + 16 * q^99

Introduction

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