LMFDB - Embedded newform 6096.2.a.ba.1.4

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.71789.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 14x^{2} + 12x + 16 $ x^4 - x^3 - 14x^2 + 12x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3048)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.61563$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.61563 q^{5} -3.10631 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.61563 q^{5} -3.10631 q^{7} +1.00000 q^{9} -3.85354 q^{11} -6.23791 q^{13} +3.61563 q^{15} +4.34421 q^{17} +3.23791 q^{19} -3.10631 q^{21} -7.61563 q^{23} +8.07279 q^{25} +1.00000 q^{27} +1.61563 q^{29} -1.27142 q^{31} -3.85354 q^{33} -11.2313 q^{35} -5.74723 q^{37} -6.23791 q^{39} +3.85354 q^{41} -0.542838 q^{43} +3.61563 q^{45} -3.29007 q^{47} +2.64915 q^{49} +4.34421 q^{51} +0.860180 q^{53} -13.9330 q^{55} +3.23791 q^{57} +6.98514 q^{59} -0.290068 q^{61} -3.10631 q^{63} -22.5540 q^{65} +5.23126 q^{67} -7.61563 q^{69} -8.43187 q^{71} +3.88705 q^{73} +8.07279 q^{75} +11.9703 q^{77} -9.09809 q^{79} +1.00000 q^{81} +1.49068 q^{83} +15.7071 q^{85} +1.61563 q^{87} -13.1978 q^{89} +19.3769 q^{91} -1.27142 q^{93} +11.7071 q^{95} -7.70708 q^{97} -3.85354 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9} + 9 q^{11} - 14 q^{13} + q^{15} - q^{17} + 2 q^{19} - 5 q^{21} - 17 q^{23} + 9 q^{25} + 4 q^{27} - 7 q^{29} - 10 q^{31} + 9 q^{33} - 18 q^{35} - 6 q^{37} - 14 q^{39} - 9 q^{41} - 12 q^{43} + q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 5 q^{53} - 24 q^{55} + 2 q^{57} + 6 q^{61} - 5 q^{63} - q^{65} - 6 q^{67} - 17 q^{69} - 20 q^{71} + 7 q^{73} + 9 q^{75} - 8 q^{77} - 17 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 7 q^{87} - 10 q^{89} + 4 q^{91} - 10 q^{93} - 2 q^{95} + 18 q^{97} + 9 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9 + 9 * q^11 - 14 * q^13 + q^15 - q^17 + 2 * q^19 - 5 * q^21 - 17 * q^23 + 9 * q^25 + 4 * q^27 - 7 * q^29 - 10 * q^31 + 9 * q^33 - 18 * q^35 - 6 * q^37 - 14 * q^39 - 9 * q^41 - 12 * q^43 + q^45 - 6 * q^47 + 13 * q^49 - q^51 - 5 * q^53 - 24 * q^55 + 2 * q^57 + 6 * q^61 - 5 * q^63 - q^65 - 6 * q^67 - 17 * q^69 - 20 * q^71 + 7 * q^73 + 9 * q^75 - 8 * q^77 - 17 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 7 * q^87 - 10 * q^89 + 4 * q^91 - 10 * q^93 - 2 * q^95 + 18 * q^97 + 9 * 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
$5$	3.61563	1.61696	0.808480	−	0.588524i	$-0.200291\pi$
$5$	3.61563	1.61696	0.808480	+	0.588524i	$0.200291\pi$
$6$	0	0
$7$	−3.10631	−1.17407	−0.587037	−	0.809560i	$-0.699706\pi$
$7$	−3.10631	−1.17407	−0.587037	+	0.809560i	$0.699706\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.85354	−1.16189	−0.580943	−	0.813944i	$-0.697316\pi$
$11$	−3.85354	−1.16189	−0.580943	+	0.813944i	$0.697316\pi$
$12$	0	0
$13$	−6.23791	−1.73008	−0.865042	−	0.501700i	$-0.832708\pi$
$13$	−6.23791	−1.73008	−0.865042	+	0.501700i	$0.832708\pi$
$14$	0	0
$15$	3.61563	0.933552
$16$	0	0
$17$	4.34421	1.05363	0.526813	−	0.849981i	$-0.323387\pi$
$17$	4.34421	1.05363	0.526813	+	0.849981i	$0.323387\pi$
$18$	0	0
$19$	3.23791	0.742827	0.371413	−	0.928468i	$-0.378873\pi$
$19$	3.23791	0.742827	0.371413	+	0.928468i	$0.378873\pi$
$20$	0	0
$21$	−3.10631	−0.677852
$22$	0	0
$23$	−7.61563	−1.58797	−0.793985	−	0.607938i	$-0.791997\pi$
$23$	−7.61563	−1.58797	−0.793985	+	0.607938i	$0.791997\pi$
$24$	0	0
$25$	8.07279	1.61456
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.61563	0.300015	0.150008	−	0.988685i	$-0.452070\pi$
$29$	1.61563	0.300015	0.150008	+	0.988685i	$0.452070\pi$
$30$	0	0
$31$	−1.27142	−0.228354	−0.114177	−	0.993460i	$-0.536423\pi$
$31$	−1.27142	−0.228354	−0.114177	+	0.993460i	$0.536423\pi$
$32$	0	0
$33$	−3.85354	−0.670815
$34$	0	0
$35$	−11.2313	−1.89843
$36$	0	0
$37$	−5.74723	−0.944839	−0.472419	−	0.881374i	$-0.656619\pi$
$37$	−5.74723	−0.944839	−0.472419	+	0.881374i	$0.656619\pi$
$38$	0	0
$39$	−6.23791	−0.998864
$40$	0	0
$41$	3.85354	0.601821	0.300911	−	0.953652i	$-0.402709\pi$
$41$	3.85354	0.601821	0.300911	+	0.953652i	$0.402709\pi$
$42$	0	0
$43$	−0.542838	−0.0827820	−0.0413910	−	0.999143i	$-0.513179\pi$
$43$	−0.542838	−0.0827820	−0.0413910	+	0.999143i	$0.513179\pi$
$44$	0	0
$45$	3.61563	0.538987
$46$	0	0
$47$	−3.29007	−0.479906	−0.239953	−	0.970785i	$-0.577132\pi$
$47$	−3.29007	−0.479906	−0.239953	+	0.970785i	$0.577132\pi$
$48$	0	0
$49$	2.64915	0.378449
$50$	0	0
$51$	4.34421	0.608311
$52$	0	0
$53$	0.860180	0.118155	0.0590774	−	0.998253i	$-0.481184\pi$
$53$	0.860180	0.118155	0.0590774	+	0.998253i	$0.481184\pi$
$54$	0	0
$55$	−13.9330	−1.87872
$56$	0	0
$57$	3.23791	0.428871
$58$	0	0
$59$	6.98514	0.909387	0.454694	−	0.890648i	$-0.349749\pi$
$59$	6.98514	0.909387	0.454694	+	0.890648i	$0.349749\pi$
$60$	0	0
$61$	−0.290068	−0.0371394	−0.0185697	−	0.999828i	$-0.505911\pi$
$61$	−0.290068	−0.0371394	−0.0185697	+	0.999828i	$0.505911\pi$
$62$	0	0
$63$	−3.10631	−0.391358
$64$	0	0
$65$	−22.5540	−2.79748
$66$	0	0
$67$	5.23126	0.639101	0.319550	−	0.947569i	$-0.396468\pi$
$67$	5.23126	0.639101	0.319550	+	0.947569i	$0.396468\pi$
$68$	0	0
$69$	−7.61563	−0.916814
$70$	0	0
$71$	−8.43187	−1.00068	−0.500339	−	0.865829i	$-0.666791\pi$
$71$	−8.43187	−1.00068	−0.500339	+	0.865829i	$0.666791\pi$
$72$	0	0
$73$	3.88705	0.454945	0.227472	−	0.973785i	$-0.426954\pi$
$73$	3.88705	0.454945	0.227472	+	0.973785i	$0.426954\pi$
$74$	0	0
$75$	8.07279	0.932166
$76$	0	0
$77$	11.9703	1.36414
$78$	0	0
$79$	−9.09809	−1.02361	−0.511807	−	0.859100i	$-0.671024\pi$
$79$	−9.09809	−1.02361	−0.511807	+	0.859100i	$0.671024\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.49068	0.163623	0.0818114	−	0.996648i	$-0.473929\pi$
$83$	1.49068	0.163623	0.0818114	+	0.996648i	$0.473929\pi$
$84$	0	0
$85$	15.7071	1.70367
$86$	0	0
$87$	1.61563	0.173214
$88$	0	0
$89$	−13.1978	−1.39896	−0.699479	−	0.714653i	$-0.746585\pi$
$89$	−13.1978	−1.39896	−0.699479	+	0.714653i	$0.746585\pi$
$90$	0	0
$91$	19.3769	2.03125
$92$	0	0
$93$	−1.27142	−0.131840
$94$	0	0
$95$	11.7071	1.20112
$96$	0	0
$97$	−7.70708	−0.782535	−0.391267	−	0.920277i	$-0.627963\pi$
$97$	−7.70708	−0.782535	−0.391267	+	0.920277i	$0.627963\pi$
$98$	0	0
$99$	−3.85354	−0.387295
$100$	0	0
$101$	−15.5486	−1.54714	−0.773572	−	0.633708i	$-0.781532\pi$
$101$	−15.5486	−1.54714	−0.773572	+	0.633708i	$0.781532\pi$
$102$	0	0
$103$	0.396375	0.0390560	0.0195280	−	0.999809i	$-0.493784\pi$
$103$	0.396375	0.0390560	0.0195280	+	0.999809i	$0.493784\pi$
$104$	0	0
$105$	−11.2313	−1.09606
$106$	0	0
$107$	−13.8387	−1.33784	−0.668918	−	0.743337i	$-0.733242\pi$
$107$	−13.8387	−1.33784	−0.668918	+	0.743337i	$0.733242\pi$
$108$	0	0
$109$	−11.3189	−1.08416	−0.542078	−	0.840328i	$-0.682362\pi$
$109$	−11.3189	−1.08416	−0.542078	+	0.840328i	$0.682362\pi$
$110$	0	0
$111$	−5.74723	−0.545503
$112$	0	0
$113$	−10.2921	−0.968195	−0.484097	−	0.875014i	$-0.660852\pi$
$113$	−10.2921	−0.968195	−0.484097	+	0.875014i	$0.660852\pi$
$114$	0	0
$115$	−27.5353	−2.56768
$116$	0	0
$117$	−6.23791	−0.576695
$118$	0	0
$119$	−13.4945	−1.23704
$120$	0	0
$121$	3.84975	0.349977
$122$	0	0
$123$	3.85354	0.347462
$124$	0	0
$125$	11.1101	0.993717
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	−0.542838	−0.0477942
$130$	0	0
$131$	−13.4439	−1.17460	−0.587299	−	0.809370i	$-0.699809\pi$
$131$	−13.4439	−1.17460	−0.587299	+	0.809370i	$0.699809\pi$
$132$	0	0
$133$	−10.0579	−0.872133
$134$	0	0
$135$	3.61563	0.311184
$136$	0	0
$137$	−12.0038	−1.02555	−0.512776	−	0.858522i	$-0.671383\pi$
$137$	−12.0038	−1.02555	−0.512776	+	0.858522i	$0.671383\pi$
$138$	0	0
$139$	−6.47581	−0.549271	−0.274636	−	0.961548i	$-0.588557\pi$
$139$	−6.47581	−0.549271	−0.274636	+	0.961548i	$0.588557\pi$
$140$	0	0
$141$	−3.29007	−0.277074
$142$	0	0
$143$	24.0380	2.01016
$144$	0	0
$145$	5.84153	0.485113
$146$	0	0
$147$	2.64915	0.218498
$148$	0	0
$149$	−16.4692	−1.34921	−0.674603	−	0.738180i	$-0.735685\pi$
$149$	−16.4692	−1.34921	−0.674603	+	0.738180i	$0.735685\pi$
$150$	0	0
$151$	15.4142	1.25439	0.627193	−	0.778864i	$-0.284204\pi$
$151$	15.4142	1.25439	0.627193	+	0.778864i	$0.284204\pi$
$152$	0	0
$153$	4.34421	0.351209
$154$	0	0
$155$	−4.59698	−0.369239
$156$	0	0
$157$	0.446958	0.0356711	0.0178356	−	0.999841i	$-0.494322\pi$
$157$	0.446958	0.0356711	0.0178356	+	0.999841i	$0.494322\pi$
$158$	0	0
$159$	0.860180	0.0682167
$160$	0	0
$161$	23.6565	1.86439
$162$	0	0
$163$	−5.21103	−0.408160	−0.204080	−	0.978954i	$-0.565420\pi$
$163$	−5.21103	−0.408160	−0.204080	+	0.978954i	$0.565420\pi$
$164$	0	0
$165$	−13.9330	−1.08468
$166$	0	0
$167$	−10.2126	−0.790276	−0.395138	−	0.918622i	$-0.629303\pi$
$167$	−10.2126	−0.790276	−0.395138	+	0.918622i	$0.629303\pi$
$168$	0	0
$169$	25.9115	1.99319
$170$	0	0
$171$	3.23791	0.247609
$172$	0	0
$173$	−2.14559	−0.163126	−0.0815630	−	0.996668i	$-0.525991\pi$
$173$	−2.14559	−0.163126	−0.0815630	+	0.996668i	$0.525991\pi$
$174$	0	0
$175$	−25.0766	−1.89561
$176$	0	0
$177$	6.98514	0.525035
$178$	0	0
$179$	−25.2453	−1.88692	−0.943459	−	0.331488i	$-0.892449\pi$
$179$	−25.2453	−1.88692	−0.943459	+	0.331488i	$0.892449\pi$
$180$	0	0
$181$	12.6214	0.938141	0.469071	−	0.883161i	$-0.344589\pi$
$181$	12.6214	0.938141	0.469071	+	0.883161i	$0.344589\pi$
$182$	0	0
$183$	−0.290068	−0.0214425
$184$	0	0
$185$	−20.7799	−1.52777
$186$	0	0
$187$	−16.7406	−1.22419
$188$	0	0
$189$	−3.10631	−0.225951
$190$	0	0
$191$	−24.0446	−1.73981	−0.869905	−	0.493220i	$-0.835820\pi$
$191$	−24.0446	−1.73981	−0.869905	+	0.493220i	$0.835820\pi$
$192$	0	0
$193$	14.7928	1.06481	0.532403	−	0.846491i	$-0.321289\pi$
$193$	14.7928	1.06481	0.532403	+	0.846491i	$0.321289\pi$
$194$	0	0
$195$	−22.5540	−1.61512
$196$	0	0
$197$	10.9731	0.781803	0.390902	−	0.920433i	$-0.372163\pi$
$197$	10.9731	0.781803	0.390902	+	0.920433i	$0.372163\pi$
$198$	0	0
$199$	−11.0977	−0.786694	−0.393347	−	0.919390i	$-0.628683\pi$
$199$	−11.0977	−0.786694	−0.393347	+	0.919390i	$0.628683\pi$
$200$	0	0
$201$	5.23126	0.368985
$202$	0	0
$203$	−5.01865	−0.352240
$204$	0	0
$205$	13.9330	0.973121
$206$	0	0
$207$	−7.61563	−0.529323
$208$	0	0
$209$	−12.4774	−0.863079
$210$	0	0
$211$	−2.61849	−0.180264	−0.0901321	−	0.995930i	$-0.528729\pi$
$211$	−2.61849	−0.180264	−0.0901321	+	0.995930i	$0.528729\pi$
$212$	0	0
$213$	−8.43187	−0.577742
$214$	0	0
$215$	−1.96270	−0.133855
$216$	0	0
$217$	3.94942	0.268104
$218$	0	0
$219$	3.88705	0.262663
$220$	0	0
$221$	−27.0988	−1.82286
$222$	0	0
$223$	20.4998	1.37277	0.686385	−	0.727238i	$-0.259197\pi$
$223$	20.4998	1.37277	0.686385	+	0.727238i	$0.259197\pi$
$224$	0	0
$225$	8.07279	0.538186
$226$	0	0
$227$	−4.42523	−0.293713	−0.146856	−	0.989158i	$-0.546916\pi$
$227$	−4.42523	−0.293713	−0.146856	+	0.989158i	$0.546916\pi$
$228$	0	0
$229$	−15.3025	−1.01122	−0.505608	−	0.862763i	$-0.668732\pi$
$229$	−15.3025	−1.01122	−0.505608	+	0.862763i	$0.668732\pi$
$230$	0	0
$231$	11.9703	0.787586
$232$	0	0
$233$	24.9719	1.63596	0.817980	−	0.575246i	$-0.195094\pi$
$233$	24.9719	1.63596	0.817980	+	0.575246i	$0.195094\pi$
$234$	0	0
$235$	−11.8957	−0.775988
$236$	0	0
$237$	−9.09809	−0.590984
$238$	0	0
$239$	26.0111	1.68252	0.841260	−	0.540630i	$-0.181814\pi$
$239$	26.0111	1.68252	0.841260	+	0.540630i	$0.181814\pi$
$240$	0	0
$241$	8.00000	0.515325	0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000	0.515325	0.257663	+	0.966235i	$0.417048\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	9.57833	0.611937
$246$	0	0
$247$	−20.1978	−1.28515
$248$	0	0
$249$	1.49068	0.0944677
$250$	0	0
$251$	26.8425	1.69428	0.847140	−	0.531369i	$-0.178322\pi$
$251$	26.8425	1.69428	0.847140	+	0.531369i	$0.178322\pi$
$252$	0	0
$253$	29.3471	1.84504
$254$	0	0
$255$	15.7071	0.983615
$256$	0	0
$257$	24.2370	1.51186	0.755932	−	0.654650i	$-0.227184\pi$
$257$	24.2370	1.51186	0.755932	+	0.654650i	$0.227184\pi$
$258$	0	0
$259$	17.8527	1.10931
$260$	0	0
$261$	1.61563	0.100005
$262$	0	0
$263$	22.8274	1.40760	0.703798	−	0.710400i	$-0.251486\pi$
$263$	22.8274	1.40760	0.703798	+	0.710400i	$0.251486\pi$
$264$	0	0
$265$	3.11009	0.191051
$266$	0	0
$267$	−13.1978	−0.807689
$268$	0	0
$269$	17.8385	1.08763	0.543815	−	0.839205i	$-0.316979\pi$
$269$	17.8385	1.08763	0.543815	+	0.839205i	$0.316979\pi$
$270$	0	0
$271$	−15.8788	−0.964570	−0.482285	−	0.876014i	$-0.660193\pi$
$271$	−15.8788	−0.964570	−0.482285	+	0.876014i	$0.660193\pi$
$272$	0	0
$273$	19.3769	1.17274
$274$	0	0
$275$	−31.1088	−1.87593
$276$	0	0
$277$	10.5801	0.635699	0.317849	−	0.948141i	$-0.397039\pi$
$277$	10.5801	0.635699	0.317849	+	0.948141i	$0.397039\pi$
$278$	0	0
$279$	−1.27142	−0.0761179
$280$	0	0
$281$	−28.8878	−1.72330	−0.861650	−	0.507504i	$-0.830568\pi$
$281$	−28.8878	−1.72330	−0.861650	+	0.507504i	$0.830568\pi$
$282$	0	0
$283$	22.3955	1.33127	0.665637	−	0.746276i	$-0.268160\pi$
$283$	22.3955	1.33127	0.665637	+	0.746276i	$0.268160\pi$
$284$	0	0
$285$	11.7071	0.693467
$286$	0	0
$287$	−11.9703	−0.706583
$288$	0	0
$289$	1.87219	0.110129
$290$	0	0
$291$	−7.70708	−0.451797
$292$	0	0
$293$	33.0169	1.92887	0.964434	−	0.264324i	$-0.0851488\pi$
$293$	33.0169	1.92887	0.964434	+	0.264324i	$0.0851488\pi$
$294$	0	0
$295$	25.2557	1.47044
$296$	0	0
$297$	−3.85354	−0.223605
$298$	0	0
$299$	47.5056	2.74732
$300$	0	0
$301$	1.68622	0.0971922
$302$	0	0
$303$	−15.5486	−0.893244
$304$	0	0
$305$	−1.04878	−0.0600529
$306$	0	0
$307$	−5.45716	−0.311457	−0.155728	−	0.987800i	$-0.549772\pi$
$307$	−5.45716	−0.311457	−0.155728	+	0.987800i	$0.549772\pi$
$308$	0	0
$309$	0.396375	0.0225490
$310$	0	0
$311$	22.4084	1.27066	0.635331	−	0.772240i	$-0.280864\pi$
$311$	22.4084	1.27066	0.635331	+	0.772240i	$0.280864\pi$
$312$	0	0
$313$	−28.8340	−1.62979	−0.814897	−	0.579605i	$-0.803207\pi$
$313$	−28.8340	−1.62979	−0.814897	+	0.579605i	$0.803207\pi$
$314$	0	0
$315$	−11.2313	−0.632810
$316$	0	0
$317$	31.4647	1.76724	0.883618	−	0.468209i	$-0.155101\pi$
$317$	31.4647	1.76724	0.883618	+	0.468209i	$0.155101\pi$
$318$	0	0
$319$	−6.22590	−0.348583
$320$	0	0
$321$	−13.8387	−0.772399
$322$	0	0
$323$	14.0662	0.782662
$324$	0	0
$325$	−50.3573	−2.79332
$326$	0	0
$327$	−11.3189	−0.625938
$328$	0	0
$329$	10.2200	0.563445
$330$	0	0
$331$	−12.7555	−0.701103	−0.350552	−	0.936543i	$-0.614006\pi$
$331$	−12.7555	−0.701103	−0.350552	+	0.936543i	$0.614006\pi$
$332$	0	0
$333$	−5.74723	−0.314946
$334$	0	0
$335$	18.9143	1.03340
$336$	0	0
$337$	24.4492	1.33184	0.665918	−	0.746025i	$-0.268040\pi$
$337$	24.4492	1.33184	0.665918	+	0.746025i	$0.268040\pi$
$338$	0	0
$339$	−10.2921	−0.558987
$340$	0	0
$341$	4.89946	0.265321
$342$	0	0
$343$	13.5151	0.729746
$344$	0	0
$345$	−27.5353	−1.48245
$346$	0	0
$347$	−2.40500	−0.129107	−0.0645536	−	0.997914i	$-0.520562\pi$
$347$	−2.40500	−0.129107	−0.0645536	+	0.997914i	$0.520562\pi$
$348$	0	0
$349$	−33.5151	−1.79402	−0.897011	−	0.442008i	$-0.854266\pi$
$349$	−33.5151	−1.79402	−0.897011	+	0.442008i	$0.854266\pi$
$350$	0	0
$351$	−6.23791	−0.332955
$352$	0	0
$353$	−1.86840	−0.0994450	−0.0497225	−	0.998763i	$-0.515834\pi$
$353$	−1.86840	−0.0994450	−0.0497225	+	0.998763i	$0.515834\pi$
$354$	0	0
$355$	−30.4865	−1.61806
$356$	0	0
$357$	−13.4945	−0.714203
$358$	0	0
$359$	−31.9570	−1.68663	−0.843313	−	0.537423i	$-0.819398\pi$
$359$	−31.9570	−1.68663	−0.843313	+	0.537423i	$0.819398\pi$
$360$	0	0
$361$	−8.51597	−0.448209
$362$	0	0
$363$	3.84975	0.202060
$364$	0	0
$365$	14.0541	0.735628
$366$	0	0
$367$	7.15381	0.373426	0.186713	−	0.982415i	$-0.440217\pi$
$367$	7.15381	0.373426	0.186713	+	0.982415i	$0.440217\pi$
$368$	0	0
$369$	3.85354	0.200607
$370$	0	0
$371$	−2.67198	−0.138722
$372$	0	0
$373$	−10.8804	−0.563366	−0.281683	−	0.959507i	$-0.590893\pi$
$373$	−10.8804	−0.563366	−0.281683	+	0.959507i	$0.590893\pi$
$374$	0	0
$375$	11.1101	0.573723
$376$	0	0
$377$	−10.0782	−0.519052
$378$	0	0
$379$	16.7422	0.859987	0.429994	−	0.902832i	$-0.358516\pi$
$379$	16.7422	0.859987	0.429994	+	0.902832i	$0.358516\pi$
$380$	0	0
$381$	1.00000	0.0512316
$382$	0	0
$383$	33.6938	1.72167	0.860836	−	0.508882i	$-0.169941\pi$
$383$	33.6938	1.72167	0.860836	+	0.508882i	$0.169941\pi$
$384$	0	0
$385$	43.2801	2.20576
$386$	0	0
$387$	−0.542838	−0.0275940
$388$	0	0
$389$	5.29742	0.268590	0.134295	−	0.990941i	$-0.457123\pi$
$389$	5.29742	0.268590	0.134295	+	0.990941i	$0.457123\pi$
$390$	0	0
$391$	−33.0839	−1.67313
$392$	0	0
$393$	−13.4439	−0.678154
$394$	0	0
$395$	−32.8953	−1.65514
$396$	0	0
$397$	−7.00822	−0.351733	−0.175866	−	0.984414i	$-0.556273\pi$
$397$	−7.00822	−0.351733	−0.175866	+	0.984414i	$0.556273\pi$
$398$	0	0
$399$	−10.0579	−0.503526
$400$	0	0
$401$	12.0276	0.600629	0.300314	−	0.953840i	$-0.402908\pi$
$401$	12.0276	0.600629	0.300314	+	0.953840i	$0.402908\pi$
$402$	0	0
$403$	7.93099	0.395071
$404$	0	0
$405$	3.61563	0.179662
$406$	0	0
$407$	22.1472	1.09779
$408$	0	0
$409$	−4.47581	−0.221315	−0.110657	−	0.993859i	$-0.535296\pi$
$409$	−4.47581	−0.221315	−0.110657	+	0.993859i	$0.535296\pi$
$410$	0	0
$411$	−12.0038	−0.592103
$412$	0	0
$413$	−21.6980	−1.06769
$414$	0	0
$415$	5.38973	0.264572
$416$	0	0
$417$	−6.47581	−0.317122
$418$	0	0
$419$	−12.5644	−0.613811	−0.306906	−	0.951740i	$-0.599294\pi$
$419$	−12.5644	−0.613811	−0.306906	+	0.951740i	$0.599294\pi$
$420$	0	0
$421$	37.3604	1.82083	0.910417	−	0.413691i	$-0.135761\pi$
$421$	37.3604	1.82083	0.910417	+	0.413691i	$0.135761\pi$
$422$	0	0
$423$	−3.29007	−0.159969
$424$	0	0
$425$	35.0699	1.70114
$426$	0	0
$427$	0.901041	0.0436044
$428$	0	0
$429$	24.0380	1.16057
$430$	0	0
$431$	11.8030	0.568528	0.284264	−	0.958746i	$-0.408251\pi$
$431$	11.8030	0.568528	0.284264	+	0.958746i	$0.408251\pi$
$432$	0	0
$433$	−20.1522	−0.968454	−0.484227	−	0.874942i	$-0.660899\pi$
$433$	−20.1522	−0.968454	−0.484227	+	0.874942i	$0.660899\pi$
$434$	0	0
$435$	5.84153	0.280080
$436$	0	0
$437$	−24.6587	−1.17959
$438$	0	0
$439$	−0.880409	−0.0420196	−0.0210098	−	0.999779i	$-0.506688\pi$
$439$	−0.880409	−0.0420196	−0.0210098	+	0.999779i	$0.506688\pi$
$440$	0	0
$441$	2.64915	0.126150
$442$	0	0
$443$	27.3629	1.30005	0.650024	−	0.759913i	$-0.274759\pi$
$443$	27.3629	1.30005	0.650024	+	0.759913i	$0.274759\pi$
$444$	0	0
$445$	−47.7182	−2.26206
$446$	0	0
$447$	−16.4692	−0.778965
$448$	0	0
$449$	−15.0848	−0.711896	−0.355948	−	0.934506i	$-0.615842\pi$
$449$	−15.0848	−0.711896	−0.355948	+	0.934506i	$0.615842\pi$
$450$	0	0
$451$	−14.8498	−0.699248
$452$	0	0
$453$	15.4142	0.724220
$454$	0	0
$455$	70.0596	3.28444
$456$	0	0
$457$	1.14804	0.0537032	0.0268516	−	0.999639i	$-0.491452\pi$
$457$	1.14804	0.0537032	0.0268516	+	0.999639i	$0.491452\pi$
$458$	0	0
$459$	4.34421	0.202770
$460$	0	0
$461$	−36.6081	−1.70501	−0.852505	−	0.522719i	$-0.824918\pi$
$461$	−36.6081	−1.70501	−0.852505	+	0.522719i	$0.824918\pi$
$462$	0	0
$463$	1.44674	0.0672355	0.0336177	−	0.999435i	$-0.489297\pi$
$463$	1.44674	0.0672355	0.0336177	+	0.999435i	$0.489297\pi$
$464$	0	0
$465$	−4.59698	−0.213180
$466$	0	0
$467$	26.3620	1.21989	0.609944	−	0.792445i	$-0.291192\pi$
$467$	26.3620	1.21989	0.609944	+	0.792445i	$0.291192\pi$
$468$	0	0
$469$	−16.2499	−0.750351
$470$	0	0
$471$	0.446958	0.0205947
$472$	0	0
$473$	2.09185	0.0961832
$474$	0	0
$475$	26.1389	1.19934
$476$	0	0
$477$	0.860180	0.0393849
$478$	0	0
$479$	9.84025	0.449613	0.224806	−	0.974403i	$-0.427825\pi$
$479$	9.84025	0.449613	0.224806	+	0.974403i	$0.427825\pi$
$480$	0	0
$481$	35.8507	1.63465
$482$	0	0
$483$	23.6565	1.07641
$484$	0	0
$485$	−27.8659	−1.26533
$486$	0	0
$487$	−36.5337	−1.65550	−0.827751	−	0.561096i	$-0.810380\pi$
$487$	−36.5337	−1.65550	−0.827751	+	0.561096i	$0.810380\pi$
$488$	0	0
$489$	−5.21103	−0.235651
$490$	0	0
$491$	31.3189	1.41340	0.706702	−	0.707512i	$-0.250182\pi$
$491$	31.3189	1.41340	0.706702	+	0.707512i	$0.250182\pi$
$492$	0	0
$493$	7.01865	0.316104
$494$	0	0
$495$	−13.9330	−0.626241
$496$	0	0
$497$	26.1920	1.17487
$498$	0	0
$499$	−37.7014	−1.68774	−0.843872	−	0.536544i	$-0.819730\pi$
$499$	−37.7014	−1.68774	−0.843872	+	0.536544i	$0.819730\pi$
$500$	0	0
$501$	−10.2126	−0.456266
$502$	0	0
$503$	3.74186	0.166842	0.0834208	−	0.996514i	$-0.473415\pi$
$503$	3.74186	0.166842	0.0834208	+	0.996514i	$0.473415\pi$
$504$	0	0
$505$	−56.2180	−2.50167
$506$	0	0
$507$	25.9115	1.15077
$508$	0	0
$509$	−2.76874	−0.122722	−0.0613610	−	0.998116i	$-0.519544\pi$
$509$	−2.76874	−0.122722	−0.0613610	+	0.998116i	$0.519544\pi$
$510$	0	0
$511$	−12.0744	−0.534139
$512$	0	0
$513$	3.23791	0.142957
$514$	0	0
$515$	1.43315	0.0631520
$516$	0	0
$517$	12.6784	0.557595
$518$	0	0
$519$	−2.14559	−0.0941809
$520$	0	0
$521$	−34.8200	−1.52549	−0.762746	−	0.646698i	$-0.776150\pi$
$521$	−34.8200	−1.52549	−0.762746	+	0.646698i	$0.776150\pi$
$522$	0	0
$523$	−22.2826	−0.974348	−0.487174	−	0.873305i	$-0.661972\pi$
$523$	−22.2826	−0.974348	−0.487174	+	0.873305i	$0.661972\pi$
$524$	0	0
$525$	−25.0766	−1.09443
$526$	0	0
$527$	−5.52331	−0.240599
$528$	0	0
$529$	34.9978	1.52165
$530$	0	0
$531$	6.98514	0.303129
$532$	0	0
$533$	−24.0380	−1.04120
$534$	0	0
$535$	−50.0356	−2.16323
$536$	0	0
$537$	−25.2453	−1.08941
$538$	0	0
$539$	−10.2086	−0.439715
$540$	0	0
$541$	4.77608	0.205340	0.102670	−	0.994715i	$-0.467261\pi$
$541$	4.77608	0.205340	0.102670	+	0.994715i	$0.467261\pi$
$542$	0	0
$543$	12.6214	0.541636
$544$	0	0
$545$	−40.9251	−1.75304
$546$	0	0
$547$	32.7795	1.40155	0.700774	−	0.713383i	$-0.252838\pi$
$547$	32.7795	1.40155	0.700774	+	0.713383i	$0.252838\pi$
$548$	0	0
$549$	−0.290068	−0.0123798
$550$	0	0
$551$	5.23126	0.222859
$552$	0	0
$553$	28.2614	1.20180
$554$	0	0
$555$	−20.7799	−0.882056
$556$	0	0
$557$	40.6008	1.72031	0.860155	−	0.510033i	$-0.170367\pi$
$557$	40.6008	1.72031	0.860155	+	0.510033i	$0.170367\pi$
$558$	0	0
$559$	3.38617	0.143220
$560$	0	0
$561$	−16.7406	−0.706788
$562$	0	0
$563$	9.79473	0.412799	0.206399	−	0.978468i	$-0.433825\pi$
$563$	9.79473	0.412799	0.206399	+	0.978468i	$0.433825\pi$
$564$	0	0
$565$	−37.2123	−1.56553
$566$	0	0
$567$	−3.10631	−0.130453
$568$	0	0
$569$	−9.11453	−0.382101	−0.191050	−	0.981580i	$-0.561189\pi$
$569$	−9.11453	−0.382101	−0.191050	+	0.981580i	$0.561189\pi$
$570$	0	0
$571$	38.4741	1.61009	0.805045	−	0.593214i	$-0.202141\pi$
$571$	38.4741	1.61009	0.805045	+	0.593214i	$0.202141\pi$
$572$	0	0
$573$	−24.0446	−1.00448
$574$	0	0
$575$	−61.4794	−2.56387
$576$	0	0
$577$	46.2304	1.92460	0.962298	−	0.271997i	$-0.0876840\pi$
$577$	46.2304	1.92460	0.962298	+	0.271997i	$0.0876840\pi$
$578$	0	0
$579$	14.7928	0.614766
$580$	0	0
$581$	−4.63050	−0.192105
$582$	0	0
$583$	−3.31473	−0.137282
$584$	0	0
$585$	−22.5540	−0.932492
$586$	0	0
$587$	−18.0886	−0.746596	−0.373298	−	0.927712i	$-0.621773\pi$
$587$	−18.0886	−0.746596	−0.373298	+	0.927712i	$0.621773\pi$
$588$	0	0
$589$	−4.11673	−0.169627
$590$	0	0
$591$	10.9731	0.451374
$592$	0	0
$593$	17.3566	0.712751	0.356376	−	0.934343i	$-0.384012\pi$
$593$	17.3566	0.712751	0.356376	+	0.934343i	$0.384012\pi$
$594$	0	0
$595$	−48.7910	−2.00024
$596$	0	0
$597$	−11.0977	−0.454198
$598$	0	0
$599$	38.2370	1.56232	0.781161	−	0.624329i	$-0.214628\pi$
$599$	38.2370	1.56232	0.781161	+	0.624329i	$0.214628\pi$
$600$	0	0
$601$	13.0692	0.533105	0.266553	−	0.963820i	$-0.414115\pi$
$601$	13.0692	0.533105	0.266553	+	0.963820i	$0.414115\pi$
$602$	0	0
$603$	5.23126	0.213034
$604$	0	0
$605$	13.9193	0.565900
$606$	0	0
$607$	14.4654	0.587132	0.293566	−	0.955939i	$-0.405158\pi$
$607$	14.4654	0.587132	0.293566	+	0.955939i	$0.405158\pi$
$608$	0	0
$609$	−5.01865	−0.203366
$610$	0	0
$611$	20.5231	0.830277
$612$	0	0
$613$	−39.5558	−1.59764	−0.798821	−	0.601568i	$-0.794543\pi$
$613$	−39.5558	−1.59764	−0.798821	+	0.601568i	$0.794543\pi$
$614$	0	0
$615$	13.9330	0.561832
$616$	0	0
$617$	0.296487	0.0119361	0.00596807	−	0.999982i	$-0.498100\pi$
$617$	0.296487	0.0119361	0.00596807	+	0.999982i	$0.498100\pi$
$618$	0	0
$619$	5.97027	0.239965	0.119983	−	0.992776i	$-0.461716\pi$
$619$	5.97027	0.239965	0.119983	+	0.992776i	$0.461716\pi$
$620$	0	0
$621$	−7.61563	−0.305605
$622$	0	0
$623$	40.9963	1.64248
$624$	0	0
$625$	−0.193965	−0.00775862
$626$	0	0
$627$	−12.4774	−0.498299
$628$	0	0
$629$	−24.9672	−0.995507
$630$	0	0
$631$	8.70906	0.346702	0.173351	−	0.984860i	$-0.444540\pi$
$631$	8.70906	0.346702	0.173351	+	0.984860i	$0.444540\pi$
$632$	0	0
$633$	−2.61849	−0.104076
$634$	0	0
$635$	3.61563	0.143482
$636$	0	0
$637$	−16.5251	−0.654749
$638$	0	0
$639$	−8.43187	−0.333560
$640$	0	0
$641$	−18.5369	−0.732164	−0.366082	−	0.930583i	$-0.619301\pi$
$641$	−18.5369	−0.732164	−0.366082	+	0.930583i	$0.619301\pi$
$642$	0	0
$643$	0.561183	0.0221309	0.0110655	−	0.999939i	$-0.496478\pi$
$643$	0.561183	0.0221309	0.0110655	+	0.999939i	$0.496478\pi$
$644$	0	0
$645$	−1.96270	−0.0772813
$646$	0	0
$647$	−13.6942	−0.538374	−0.269187	−	0.963088i	$-0.586755\pi$
$647$	−13.6942	−0.538374	−0.269187	+	0.963088i	$0.586755\pi$
$648$	0	0
$649$	−26.9175	−1.05660
$650$	0	0
$651$	3.94942	0.154790
$652$	0	0
$653$	−21.6856	−0.848622	−0.424311	−	0.905517i	$-0.639484\pi$
$653$	−21.6856	−0.848622	−0.424311	+	0.905517i	$0.639484\pi$
$654$	0	0
$655$	−48.6081	−1.89928
$656$	0	0
$657$	3.88705	0.151648
$658$	0	0
$659$	−6.03351	−0.235032	−0.117516	−	0.993071i	$-0.537493\pi$
$659$	−6.03351	−0.235032	−0.117516	+	0.993071i	$0.537493\pi$
$660$	0	0
$661$	−31.7976	−1.23679	−0.618393	−	0.785869i	$-0.712216\pi$
$661$	−31.7976	−1.23679	−0.618393	+	0.785869i	$0.712216\pi$
$662$	0	0
$663$	−27.0988	−1.05243
$664$	0	0
$665$	−36.3658	−1.41020
$666$	0	0
$667$	−12.3041	−0.476415
$668$	0	0
$669$	20.4998	0.792569
$670$	0	0
$671$	1.11779	0.0431517
$672$	0	0
$673$	12.3480	0.475980	0.237990	−	0.971268i	$-0.423511\pi$
$673$	12.3480	0.475980	0.237990	+	0.971268i	$0.423511\pi$
$674$	0	0
$675$	8.07279	0.310722
$676$	0	0
$677$	−2.46275	−0.0946512	−0.0473256	−	0.998880i	$-0.515070\pi$
$677$	−2.46275	−0.0946512	−0.0473256	+	0.998880i	$0.515070\pi$
$678$	0	0
$679$	23.9405	0.918754
$680$	0	0
$681$	−4.42523	−0.169575
$682$	0	0
$683$	47.0709	1.80112	0.900558	−	0.434735i	$-0.143158\pi$
$683$	47.0709	1.80112	0.900558	+	0.434735i	$0.143158\pi$
$684$	0	0
$685$	−43.4013	−1.65828
$686$	0	0
$687$	−15.3025	−0.583826
$688$	0	0
$689$	−5.36572	−0.204418
$690$	0	0
$691$	−28.5038	−1.08434	−0.542168	−	0.840270i	$-0.682396\pi$
$691$	−28.5038	−1.08434	−0.542168	+	0.840270i	$0.682396\pi$
$692$	0	0
$693$	11.9703	0.454713
$694$	0	0
$695$	−23.4142	−0.888149
$696$	0	0
$697$	16.7406	0.634095
$698$	0	0
$699$	24.9719	0.944522
$700$	0	0
$701$	28.0649	1.06000	0.529998	−	0.847999i	$-0.322193\pi$
$701$	28.0649	1.06000	0.529998	+	0.847999i	$0.322193\pi$
$702$	0	0
$703$	−18.6090	−0.701851
$704$	0	0
$705$	−11.8957	−0.448017
$706$	0	0
$707$	48.2987	1.81646
$708$	0	0
$709$	−44.2117	−1.66041	−0.830203	−	0.557461i	$-0.811776\pi$
$709$	−44.2117	−1.66041	−0.830203	+	0.557461i	$0.811776\pi$
$710$	0	0
$711$	−9.09809	−0.341205
$712$	0	0
$713$	9.68266	0.362618
$714$	0	0
$715$	86.9126	3.25035
$716$	0	0
$717$	26.0111	0.971404
$718$	0	0
$719$	−8.77876	−0.327393	−0.163696	−	0.986511i	$-0.552342\pi$
$719$	−8.77876	−0.327393	−0.163696	+	0.986511i	$0.552342\pi$
$720$	0	0
$721$	−1.23126	−0.0458547
$722$	0	0
$723$	8.00000	0.297523
$724$	0	0
$725$	13.0427	0.484392
$726$	0	0
$727$	−31.5256	−1.16922	−0.584610	−	0.811315i	$-0.698752\pi$
$727$	−31.5256	−1.16922	−0.584610	+	0.811315i	$0.698752\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.35820	−0.0872213
$732$	0	0
$733$	46.8451	1.73027	0.865133	−	0.501543i	$-0.167234\pi$
$733$	46.8451	1.73027	0.865133	+	0.501543i	$0.167234\pi$
$734$	0	0
$735$	9.57833	0.353302
$736$	0	0
$737$	−20.1589	−0.742562
$738$	0	0
$739$	4.96341	0.182582	0.0912910	−	0.995824i	$-0.470901\pi$
$739$	4.96341	0.182582	0.0912910	+	0.995824i	$0.470901\pi$
$740$	0	0
$741$	−20.1978	−0.741983
$742$	0	0
$743$	−34.6085	−1.26966	−0.634832	−	0.772650i	$-0.718931\pi$
$743$	−34.6085	−1.26966	−0.634832	+	0.772650i	$0.718931\pi$
$744$	0	0
$745$	−59.5465	−2.18161
$746$	0	0
$747$	1.49068	0.0545410
$748$	0	0
$749$	42.9872	1.57072
$750$	0	0
$751$	−13.5183	−0.493288	−0.246644	−	0.969106i	$-0.579328\pi$
$751$	−13.5183	−0.493288	−0.246644	+	0.969106i	$0.579328\pi$
$752$	0	0
$753$	26.8425	0.978194
$754$	0	0
$755$	55.7319	2.02829
$756$	0	0
$757$	−13.3234	−0.484248	−0.242124	−	0.970245i	$-0.577844\pi$
$757$	−13.3234	−0.484248	−0.242124	+	0.970245i	$0.577844\pi$
$758$	0	0
$759$	29.3471	1.06523
$760$	0	0
$761$	−3.26139	−0.118225	−0.0591127	−	0.998251i	$-0.518827\pi$
$761$	−3.26139	−0.118225	−0.0591127	+	0.998251i	$0.518827\pi$
$762$	0	0
$763$	35.1600	1.27288
$764$	0	0
$765$	15.7071	0.567891
$766$	0	0
$767$	−43.5726	−1.57332
$768$	0	0
$769$	23.9742	0.864533	0.432267	−	0.901746i	$-0.357714\pi$
$769$	23.9742	0.864533	0.432267	+	0.901746i	$0.357714\pi$
$770$	0	0
$771$	24.2370	0.872875
$772$	0	0
$773$	−28.6017	−1.02873	−0.514366	−	0.857571i	$-0.671973\pi$
$773$	−28.6017	−1.02873	−0.514366	+	0.857571i	$0.671973\pi$
$774$	0	0
$775$	−10.2639	−0.368690
$776$	0	0
$777$	17.8527	0.640461
$778$	0	0
$779$	12.4774	0.447049
$780$	0	0
$781$	32.4925	1.16267
$782$	0	0
$783$	1.61563	0.0577380
$784$	0	0
$785$	1.61603	0.0576787
$786$	0	0
$787$	−5.40214	−0.192566	−0.0962828	−	0.995354i	$-0.530695\pi$
$787$	−5.40214	−0.192566	−0.0962828	+	0.995354i	$0.530695\pi$
$788$	0	0
$789$	22.8274	0.812676
$790$	0	0
$791$	31.9703	1.13673
$792$	0	0
$793$	1.80942	0.0642543
$794$	0	0
$795$	3.11009	0.110304
$796$	0	0
$797$	−42.2541	−1.49672	−0.748359	−	0.663294i	$-0.769158\pi$
$797$	−42.2541	−1.49672	−0.748359	+	0.663294i	$0.769158\pi$
$798$	0	0
$799$	−14.2928	−0.505641
$800$	0	0
$801$	−13.1978	−0.466320
$802$	0	0
$803$	−14.9789	−0.528594
$804$	0	0
$805$	85.5332	3.01465
$806$	0	0
$807$	17.8385	0.627943
$808$	0	0
$809$	23.4414	0.824157	0.412078	−	0.911148i	$-0.364803\pi$
$809$	23.4414	0.824157	0.412078	+	0.911148i	$0.364803\pi$
$810$	0	0
$811$	47.7875	1.67805	0.839023	−	0.544095i	$-0.183127\pi$
$811$	47.7875	1.67805	0.839023	+	0.544095i	$0.183127\pi$
$812$	0	0
$813$	−15.8788	−0.556895
$814$	0	0
$815$	−18.8412	−0.659978
$816$	0	0
$817$	−1.75766	−0.0614927
$818$	0	0
$819$	19.3769	0.677082
$820$	0	0
$821$	34.5748	1.20667	0.603335	−	0.797488i	$-0.293838\pi$
$821$	34.5748	1.20667	0.603335	+	0.797488i	$0.293838\pi$
$822$	0	0
$823$	−9.60204	−0.334706	−0.167353	−	0.985897i	$-0.553522\pi$
$823$	−9.60204	−0.334706	−0.167353	+	0.985897i	$0.553522\pi$
$824$	0	0
$825$	−31.1088	−1.08307
$826$	0	0
$827$	−24.2164	−0.842087	−0.421043	−	0.907041i	$-0.638336\pi$
$827$	−24.2164	−0.842087	−0.421043	+	0.907041i	$0.638336\pi$
$828$	0	0
$829$	−39.3935	−1.36819	−0.684097	−	0.729391i	$-0.739803\pi$
$829$	−39.3935	−1.36819	−0.684097	+	0.729391i	$0.739803\pi$
$830$	0	0
$831$	10.5801	0.367021
$832$	0	0
$833$	11.5085	0.398744
$834$	0	0
$835$	−36.9251	−1.27784
$836$	0	0
$837$	−1.27142	−0.0439467
$838$	0	0
$839$	−23.0564	−0.795994	−0.397997	−	0.917387i	$-0.630294\pi$
$839$	−23.0564	−0.795994	−0.397997	+	0.917387i	$0.630294\pi$
$840$	0	0
$841$	−26.3897	−0.909991
$842$	0	0
$843$	−28.8878	−0.994947
$844$	0	0
$845$	93.6863	3.22291
$846$	0	0
$847$	−11.9585	−0.410899
$848$	0	0
$849$	22.3955	0.768612
$850$	0	0
$851$	43.7688	1.50037
$852$	0	0
$853$	0.330223	0.0113066	0.00565331	−	0.999984i	$-0.498200\pi$
$853$	0.330223	0.0113066	0.00565331	+	0.999984i	$0.498200\pi$
$854$	0	0
$855$	11.7071	0.400374
$856$	0	0
$857$	−9.22177	−0.315010	−0.157505	−	0.987518i	$-0.550345\pi$
$857$	−9.22177	−0.315010	−0.157505	+	0.987518i	$0.550345\pi$
$858$	0	0
$859$	25.4945	0.869860	0.434930	−	0.900464i	$-0.356773\pi$
$859$	25.4945	0.869860	0.434930	+	0.900464i	$0.356773\pi$
$860$	0	0
$861$	−11.9703	−0.407946
$862$	0	0
$863$	28.5952	0.973393	0.486697	−	0.873571i	$-0.338202\pi$
$863$	28.5952	0.973393	0.486697	+	0.873571i	$0.338202\pi$
$864$	0	0
$865$	−7.75766	−0.263768
$866$	0	0
$867$	1.87219	0.0635828
$868$	0	0
$869$	35.0598	1.18932
$870$	0	0
$871$	−32.6321	−1.10570
$872$	0	0
$873$	−7.70708	−0.260845
$874$	0	0
$875$	−34.5114	−1.16670
$876$	0	0
$877$	58.2285	1.96624	0.983118	−	0.182974i	$-0.0585725\pi$
$877$	58.2285	1.96624	0.983118	+	0.182974i	$0.0585725\pi$
$878$	0	0
$879$	33.0169	1.11363
$880$	0	0
$881$	−7.78360	−0.262236	−0.131118	−	0.991367i	$-0.541857\pi$
$881$	−7.78360	−0.262236	−0.131118	+	0.991367i	$0.541857\pi$
$882$	0	0
$883$	−17.4543	−0.587384	−0.293692	−	0.955900i	$-0.594884\pi$
$883$	−17.4543	−0.587384	−0.293692	+	0.955900i	$0.594884\pi$
$884$	0	0
$885$	25.2557	0.848960
$886$	0	0
$887$	49.0001	1.64526	0.822630	−	0.568576i	$-0.192506\pi$
$887$	49.0001	1.64526	0.822630	+	0.568576i	$0.192506\pi$
$888$	0	0
$889$	−3.10631	−0.104182
$890$	0	0
$891$	−3.85354	−0.129098
$892$	0	0
$893$	−10.6529	−0.356487
$894$	0	0
$895$	−91.2775	−3.05107
$896$	0	0
$897$	47.5056	1.58617
$898$	0	0
$899$	−2.05414	−0.0685096
$900$	0	0
$901$	3.73680	0.124491
$902$	0	0
$903$	1.68622	0.0561139
$904$	0	0
$905$	45.6343	1.51694
$906$	0	0
$907$	2.56791	0.0852659	0.0426330	−	0.999091i	$-0.486425\pi$
$907$	2.56791	0.0852659	0.0426330	+	0.999091i	$0.486425\pi$
$908$	0	0
$909$	−15.5486	−0.515715
$910$	0	0
$911$	−36.9963	−1.22574	−0.612871	−	0.790183i	$-0.709985\pi$
$911$	−36.9963	−1.22574	−0.612871	+	0.790183i	$0.709985\pi$
$912$	0	0
$913$	−5.74437	−0.190111
$914$	0	0
$915$	−1.04878	−0.0346716
$916$	0	0
$917$	41.7608	1.37906
$918$	0	0
$919$	28.5711	0.942474	0.471237	−	0.882007i	$-0.343808\pi$
$919$	28.5711	0.942474	0.471237	+	0.882007i	$0.343808\pi$
$920$	0	0
$921$	−5.45716	−0.179820
$922$	0	0
$923$	52.5972	1.73126
$924$	0	0
$925$	−46.3962	−1.52550
$926$	0	0
$927$	0.396375	0.0130187
$928$	0	0
$929$	11.1900	0.367132	0.183566	−	0.983007i	$-0.441236\pi$
$929$	11.1900	0.367132	0.183566	+	0.983007i	$0.441236\pi$
$930$	0	0
$931$	8.57768	0.281122
$932$	0	0
$933$	22.4084	0.733618
$934$	0	0
$935$	−60.5278	−1.97947
$936$	0	0
$937$	−28.3152	−0.925017	−0.462508	−	0.886615i	$-0.653051\pi$
$937$	−28.3152	−0.925017	−0.462508	+	0.886615i	$0.653051\pi$
$938$	0	0
$939$	−28.8340	−0.940962
$940$	0	0
$941$	18.3367	0.597759	0.298880	−	0.954291i	$-0.403387\pi$
$941$	18.3367	0.597759	0.298880	+	0.954291i	$0.403387\pi$
$942$	0	0
$943$	−29.3471	−0.955674
$944$	0	0
$945$	−11.2313	−0.365353
$946$	0	0
$947$	−37.9816	−1.23424	−0.617118	−	0.786871i	$-0.711700\pi$
$947$	−37.9816	−1.23424	−0.617118	+	0.786871i	$0.711700\pi$
$948$	0	0
$949$	−24.2471	−0.787093
$950$	0	0
$951$	31.4647	1.02031
$952$	0	0
$953$	36.6711	1.18789	0.593947	−	0.804504i	$-0.297569\pi$
$953$	36.6711	1.18789	0.593947	+	0.804504i	$0.297569\pi$
$954$	0	0
$955$	−86.9366	−2.81320
$956$	0	0
$957$	−6.22590	−0.201255
$958$	0	0
$959$	37.2874	1.20407
$960$	0	0
$961$	−29.3835	−0.947855
$962$	0	0
$963$	−13.8387	−0.445945
$964$	0	0
$965$	53.4851	1.72175
$966$	0	0
$967$	31.9403	1.02713	0.513566	−	0.858050i	$-0.328324\pi$
$967$	31.9403	1.02713	0.513566	+	0.858050i	$0.328324\pi$
$968$	0	0
$969$	14.0662	0.451870
$970$	0	0
$971$	16.8557	0.540927	0.270463	−	0.962730i	$-0.412823\pi$
$971$	16.8557	0.540927	0.270463	+	0.962730i	$0.412823\pi$
$972$	0	0
$973$	20.1159	0.644885
$974$	0	0
$975$	−50.3573	−1.61273
$976$	0	0
$977$	−20.3871	−0.652240	−0.326120	−	0.945328i	$-0.605741\pi$
$977$	−20.3871	−0.652240	−0.326120	+	0.945328i	$0.605741\pi$
$978$	0	0
$979$	50.8580	1.62543
$980$	0	0
$981$	−11.3189	−0.361385
$982$	0	0
$983$	−38.5049	−1.22812	−0.614058	−	0.789261i	$-0.710464\pi$
$983$	−38.5049	−1.22812	−0.614058	+	0.789261i	$0.710464\pi$
$984$	0	0
$985$	39.6748	1.26414
$986$	0	0
$987$	10.2200	0.325305
$988$	0	0
$989$	4.13405	0.131455
$990$	0	0
$991$	18.3822	0.583930	0.291965	−	0.956429i	$-0.405691\pi$
$991$	18.3822	0.583930	0.291965	+	0.956429i	$0.405691\pi$
$992$	0	0
$993$	−12.7555	−0.404782
$994$	0	0
$995$	−40.1251	−1.27205
$996$	0	0
$997$	52.6918	1.66877	0.834383	−	0.551185i	$-0.185824\pi$
$997$	52.6918	1.66877	0.834383	+	0.551185i	$0.185824\pi$
$998$	0	0
$999$	−5.74723	−0.181834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.ba.1.4		4
4.3	odd	2		3048.2.a.h.1.4	✓	4
12.11	even	2		9144.2.a.s.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3048.2.a.h.1.4	✓	4	4.3	odd	2
6096.2.a.ba.1.4		4	1.1	even	1	trivial
9144.2.a.s.1.1		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.61563 q^{5} -3.10631 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.61563 q^{5} -3.10631 q^{7} +1.00000 q^{9} -3.85354 q^{11} -6.23791 q^{13} +3.61563 q^{15} +4.34421 q^{17} +3.23791 q^{19} -3.10631 q^{21} -7.61563 q^{23} +8.07279 q^{25} +1.00000 q^{27} +1.61563 q^{29} -1.27142 q^{31} -3.85354 q^{33} -11.2313 q^{35} -5.74723 q^{37} -6.23791 q^{39} +3.85354 q^{41} -0.542838 q^{43} +3.61563 q^{45} -3.29007 q^{47} +2.64915 q^{49} +4.34421 q^{51} +0.860180 q^{53} -13.9330 q^{55} +3.23791 q^{57} +6.98514 q^{59} -0.290068 q^{61} -3.10631 q^{63} -22.5540 q^{65} +5.23126 q^{67} -7.61563 q^{69} -8.43187 q^{71} +3.88705 q^{73} +8.07279 q^{75} +11.9703 q^{77} -9.09809 q^{79} +1.00000 q^{81} +1.49068 q^{83} +15.7071 q^{85} +1.61563 q^{87} -13.1978 q^{89} +19.3769 q^{91} -1.27142 q^{93} +11.7071 q^{95} -7.70708 q^{97} -3.85354 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + q^{5} - 5 q^{7} + 4 q^{9} + 9 q^{11} - 14 q^{13} + q^{15} - q^{17} + 2 q^{19} - 5 q^{21} - 17 q^{23} + 9 q^{25} + 4 q^{27} - 7 q^{29} - 10 q^{31} + 9 q^{33} - 18 q^{35} - 6 q^{37} - 14 q^{39} - 9 q^{41} - 12 q^{43} + q^{45} - 6 q^{47} + 13 q^{49} - q^{51} - 5 q^{53} - 24 q^{55} + 2 q^{57} + 6 q^{61} - 5 q^{63} - q^{65} - 6 q^{67} - 17 q^{69} - 20 q^{71} + 7 q^{73} + 9 q^{75} - 8 q^{77} - 17 q^{79} + 4 q^{81} + 12 q^{83} + 14 q^{85} - 7 q^{87} - 10 q^{89} + 4 q^{91} - 10 q^{93} - 2 q^{95} + 18 q^{97} + 9 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + q^5 - 5 * q^7 + 4 * q^9 + 9 * q^11 - 14 * q^13 + q^15 - q^17 + 2 * q^19 - 5 * q^21 - 17 * q^23 + 9 * q^25 + 4 * q^27 - 7 * q^29 - 10 * q^31 + 9 * q^33 - 18 * q^35 - 6 * q^37 - 14 * q^39 - 9 * q^41 - 12 * q^43 + q^45 - 6 * q^47 + 13 * q^49 - q^51 - 5 * q^53 - 24 * q^55 + 2 * q^57 + 6 * q^61 - 5 * q^63 - q^65 - 6 * q^67 - 17 * q^69 - 20 * q^71 + 7 * q^73 + 9 * q^75 - 8 * q^77 - 17 * q^79 + 4 * q^81 + 12 * q^83 + 14 * q^85 - 7 * q^87 - 10 * q^89 + 4 * q^91 - 10 * q^93 - 2 * q^95 + 18 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more