LMFDB - Embedded newform 7448.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.04436$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.04436 q^{3} +2.60221 q^{5} +6.26815 q^{9} +O(q^{10})$	$q-3.04436 q^{3} +2.60221 q^{5} +6.26815 q^{9} +3.68293 q^{11} -6.17208 q^{13} -7.92207 q^{15} +6.92207 q^{17} +1.00000 q^{19} -2.61920 q^{23} +1.77149 q^{25} -9.94945 q^{27} +3.12082 q^{29} -6.97542 q^{31} -11.2122 q^{33} +0.292486 q^{37} +18.7901 q^{39} -5.29265 q^{41} -1.96364 q^{43} +16.3111 q^{45} -7.99528 q^{47} -21.0733 q^{51} +1.79173 q^{53} +9.58376 q^{55} -3.04436 q^{57} -7.12345 q^{59} +2.14820 q^{61} -16.0610 q^{65} -7.63547 q^{67} +7.97378 q^{69} +3.29274 q^{71} -8.82925 q^{73} -5.39308 q^{75} -6.39094 q^{79} +11.4853 q^{81} -7.35997 q^{83} +18.0127 q^{85} -9.50092 q^{87} +13.5002 q^{89} +21.2357 q^{93} +2.60221 q^{95} +8.90409 q^{97} +23.0852 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * 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
$3$	−3.04436	−1.75766	−0.878832	−	0.477131i	$-0.841677\pi$
$3$	−3.04436	−1.75766	−0.878832	+	0.477131i	$0.841677\pi$
$4$	0	0
$5$	2.60221	1.16374	0.581872	−	0.813281i	$-0.302321\pi$
$5$	2.60221	1.16374	0.581872	+	0.813281i	$0.302321\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.26815	2.08938
$10$	0	0
$11$	3.68293	1.11045	0.555223	−	0.831702i	$-0.312633\pi$
$11$	3.68293	1.11045	0.555223	+	0.831702i	$0.312633\pi$
$12$	0	0
$13$	−6.17208	−1.71183	−0.855913	−	0.517119i	$-0.827004\pi$
$13$	−6.17208	−1.71183	−0.855913	+	0.517119i	$0.827004\pi$
$14$	0	0
$15$	−7.92207	−2.04547
$16$	0	0
$17$	6.92207	1.67885	0.839425	−	0.543476i	$-0.182892\pi$
$17$	6.92207	1.67885	0.839425	+	0.543476i	$0.182892\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.61920	−0.546140	−0.273070	−	0.961994i	$-0.588039\pi$
$23$	−2.61920	−0.546140	−0.273070	+	0.961994i	$0.588039\pi$
$24$	0	0
$25$	1.77149	0.354299
$26$	0	0
$27$	−9.94945	−1.91477
$28$	0	0
$29$	3.12082	0.579522	0.289761	−	0.957099i	$-0.406424\pi$
$29$	3.12082	0.579522	0.289761	+	0.957099i	$0.406424\pi$
$30$	0	0
$31$	−6.97542	−1.25282	−0.626411	−	0.779493i	$-0.715477\pi$
$31$	−6.97542	−1.25282	−0.626411	+	0.779493i	$0.715477\pi$
$32$	0	0
$33$	−11.2122	−1.95179
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.292486	0.0480843	0.0240422	−	0.999711i	$-0.492346\pi$
$37$	0.292486	0.0480843	0.0240422	+	0.999711i	$0.492346\pi$
$38$	0	0
$39$	18.7901	3.00882
$40$	0	0
$41$	−5.29265	−0.826573	−0.413287	−	0.910601i	$-0.635619\pi$
$41$	−5.29265	−0.826573	−0.413287	+	0.910601i	$0.635619\pi$
$42$	0	0
$43$	−1.96364	−0.299453	−0.149726	−	0.988727i	$-0.547839\pi$
$43$	−1.96364	−0.299453	−0.149726	+	0.988727i	$0.547839\pi$
$44$	0	0
$45$	16.3111	2.43151
$46$	0	0
$47$	−7.99528	−1.16623	−0.583116	−	0.812389i	$-0.698167\pi$
$47$	−7.99528	−1.16623	−0.583116	+	0.812389i	$0.698167\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−21.0733	−2.95085
$52$	0	0
$53$	1.79173	0.246113	0.123056	−	0.992400i	$-0.460730\pi$
$53$	1.79173	0.246113	0.123056	+	0.992400i	$0.460730\pi$
$54$	0	0
$55$	9.58376	1.29227
$56$	0	0
$57$	−3.04436	−0.403236
$58$	0	0
$59$	−7.12345	−0.927395	−0.463697	−	0.885994i	$-0.653477\pi$
$59$	−7.12345	−0.927395	−0.463697	+	0.885994i	$0.653477\pi$
$60$	0	0
$61$	2.14820	0.275049	0.137525	−	0.990498i	$-0.456085\pi$
$61$	2.14820	0.275049	0.137525	+	0.990498i	$0.456085\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−16.0610	−1.99213
$66$	0	0
$67$	−7.63547	−0.932821	−0.466410	−	0.884568i	$-0.654453\pi$
$67$	−7.63547	−0.932821	−0.466410	+	0.884568i	$0.654453\pi$
$68$	0	0
$69$	7.97378	0.959931
$70$	0	0
$71$	3.29274	0.390776	0.195388	−	0.980726i	$-0.437403\pi$
$71$	3.29274	0.390776	0.195388	+	0.980726i	$0.437403\pi$
$72$	0	0
$73$	−8.82925	−1.03339	−0.516693	−	0.856171i	$-0.672837\pi$
$73$	−8.82925	−1.03339	−0.516693	+	0.856171i	$0.672837\pi$
$74$	0	0
$75$	−5.39308	−0.622739
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−6.39094	−0.719037	−0.359519	−	0.933138i	$-0.617059\pi$
$79$	−6.39094	−0.719037	−0.359519	+	0.933138i	$0.617059\pi$
$80$	0	0
$81$	11.4853	1.27614
$82$	0	0
$83$	−7.35997	−0.807861	−0.403931	−	0.914790i	$-0.632356\pi$
$83$	−7.35997	−0.807861	−0.403931	+	0.914790i	$0.632356\pi$
$84$	0	0
$85$	18.0127	1.95375
$86$	0	0
$87$	−9.50092	−1.01861
$88$	0	0
$89$	13.5002	1.43102	0.715509	−	0.698603i	$-0.246195\pi$
$89$	13.5002	1.43102	0.715509	+	0.698603i	$0.246195\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	21.2357	2.20204
$94$	0	0
$95$	2.60221	0.266981
$96$	0	0
$97$	8.90409	0.904073	0.452037	−	0.891999i	$-0.350698\pi$
$97$	8.90409	0.904073	0.452037	+	0.891999i	$0.350698\pi$
$98$	0	0
$99$	23.0852	2.32015
$100$	0	0
$101$	−14.8128	−1.47392	−0.736962	−	0.675934i	$-0.763741\pi$
$101$	−14.8128	−1.47392	−0.736962	+	0.675934i	$0.763741\pi$
$102$	0	0
$103$	−16.8373	−1.65902	−0.829512	−	0.558489i	$-0.811381\pi$
$103$	−16.8373	−1.65902	−0.829512	+	0.558489i	$0.811381\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.1650	−1.27271	−0.636353	−	0.771398i	$-0.719558\pi$
$107$	−13.1650	−1.27271	−0.636353	+	0.771398i	$0.719558\pi$
$108$	0	0
$109$	10.0004	0.957866	0.478933	−	0.877852i	$-0.341024\pi$
$109$	10.0004	0.957866	0.478933	+	0.877852i	$0.341024\pi$
$110$	0	0
$111$	−0.890433	−0.0845162
$112$	0	0
$113$	−17.6013	−1.65579	−0.827895	−	0.560883i	$-0.810462\pi$
$113$	−17.6013	−1.65579	−0.827895	+	0.560883i	$0.810462\pi$
$114$	0	0
$115$	−6.81569	−0.635567
$116$	0	0
$117$	−38.6876	−3.57667
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2.56398	0.233089
$122$	0	0
$123$	16.1128	1.45284
$124$	0	0
$125$	−8.40125	−0.751430
$126$	0	0
$127$	13.8533	1.22928	0.614640	−	0.788807i	$-0.289301\pi$
$127$	13.8533	1.22928	0.614640	+	0.788807i	$0.289301\pi$
$128$	0	0
$129$	5.97805	0.526338
$130$	0	0
$131$	−5.67842	−0.496126	−0.248063	−	0.968744i	$-0.579794\pi$
$131$	−5.67842	−0.496126	−0.248063	+	0.968744i	$0.579794\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−25.8906	−2.22830
$136$	0	0
$137$	20.8928	1.78499	0.892495	−	0.451057i	$-0.148953\pi$
$137$	20.8928	1.78499	0.892495	+	0.451057i	$0.148953\pi$
$138$	0	0
$139$	18.3740	1.55847	0.779233	−	0.626735i	$-0.215609\pi$
$139$	18.3740	1.55847	0.779233	+	0.626735i	$0.215609\pi$
$140$	0	0
$141$	24.3406	2.04984
$142$	0	0
$143$	−22.7313	−1.90089
$144$	0	0
$145$	8.12104	0.674415
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.86236	−0.316417	−0.158208	−	0.987406i	$-0.550572\pi$
$149$	−3.86236	−0.316417	−0.158208	+	0.987406i	$0.550572\pi$
$150$	0	0
$151$	21.4781	1.74786	0.873932	−	0.486047i	$-0.161562\pi$
$151$	21.4781	1.74786	0.873932	+	0.486047i	$0.161562\pi$
$152$	0	0
$153$	43.3886	3.50776
$154$	0	0
$155$	−18.1515	−1.45796
$156$	0	0
$157$	1.22729	0.0979485	0.0489742	−	0.998800i	$-0.484405\pi$
$157$	1.22729	0.0979485	0.0489742	+	0.998800i	$0.484405\pi$
$158$	0	0
$159$	−5.45467	−0.432584
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.6255	−1.06723	−0.533615	−	0.845727i	$-0.679167\pi$
$163$	−13.6255	−1.06723	−0.533615	+	0.845727i	$0.679167\pi$
$164$	0	0
$165$	−29.1764	−2.27138
$166$	0	0
$167$	23.8703	1.84714	0.923570	−	0.383430i	$-0.125257\pi$
$167$	23.8703	1.84714	0.923570	+	0.383430i	$0.125257\pi$
$168$	0	0
$169$	25.0946	1.93035
$170$	0	0
$171$	6.26815	0.479338
$172$	0	0
$173$	−2.40276	−0.182678	−0.0913392	−	0.995820i	$-0.529115\pi$
$173$	−2.40276	−0.182678	−0.0913392	+	0.995820i	$0.529115\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	21.6864	1.63005
$178$	0	0
$179$	−4.56753	−0.341393	−0.170697	−	0.985324i	$-0.554602\pi$
$179$	−4.56753	−0.341393	−0.170697	+	0.985324i	$0.554602\pi$
$180$	0	0
$181$	−18.6789	−1.38839	−0.694196	−	0.719786i	$-0.744240\pi$
$181$	−18.6789	−1.38839	−0.694196	+	0.719786i	$0.744240\pi$
$182$	0	0
$183$	−6.53991	−0.483444
$184$	0	0
$185$	0.761109	0.0559578
$186$	0	0
$187$	25.4935	1.86427
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.89439	−0.137074	−0.0685368	−	0.997649i	$-0.521833\pi$
$191$	−1.89439	−0.137074	−0.0685368	+	0.997649i	$0.521833\pi$
$192$	0	0
$193$	26.7270	1.92385	0.961927	−	0.273306i	$-0.0881173\pi$
$193$	26.7270	1.92385	0.961927	+	0.273306i	$0.0881173\pi$
$194$	0	0
$195$	48.8957	3.50149
$196$	0	0
$197$	−13.5172	−0.963060	−0.481530	−	0.876430i	$-0.659919\pi$
$197$	−13.5172	−0.963060	−0.481530	+	0.876430i	$0.659919\pi$
$198$	0	0
$199$	10.8090	0.766228	0.383114	−	0.923701i	$-0.374852\pi$
$199$	10.8090	0.766228	0.383114	+	0.923701i	$0.374852\pi$
$200$	0	0
$201$	23.2451	1.63959
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−13.7726	−0.961919
$206$	0	0
$207$	−16.4175	−1.14110
$208$	0	0
$209$	3.68293	0.254754
$210$	0	0
$211$	−9.11248	−0.627329	−0.313664	−	0.949534i	$-0.601557\pi$
$211$	−9.11248	−0.627329	−0.313664	+	0.949534i	$0.601557\pi$
$212$	0	0
$213$	−10.0243	−0.686853
$214$	0	0
$215$	−5.10981	−0.348486
$216$	0	0
$217$	0	0
$218$	0	0
$219$	26.8794	1.81634
$220$	0	0
$221$	−42.7236	−2.87390
$222$	0	0
$223$	−26.2526	−1.75801	−0.879003	−	0.476817i	$-0.841790\pi$
$223$	−26.2526	−1.75801	−0.879003	+	0.476817i	$0.841790\pi$
$224$	0	0
$225$	11.1040	0.740267
$226$	0	0
$227$	22.4173	1.48789	0.743945	−	0.668241i	$-0.232953\pi$
$227$	22.4173	1.48789	0.743945	+	0.668241i	$0.232953\pi$
$228$	0	0
$229$	−15.0486	−0.994441	−0.497221	−	0.867624i	$-0.665646\pi$
$229$	−15.0486	−0.994441	−0.497221	+	0.867624i	$0.665646\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.82983	−0.316413	−0.158206	−	0.987406i	$-0.550571\pi$
$233$	−4.82983	−0.316413	−0.158206	+	0.987406i	$0.550571\pi$
$234$	0	0
$235$	−20.8054	−1.35720
$236$	0	0
$237$	19.4564	1.26383
$238$	0	0
$239$	−7.09173	−0.458726	−0.229363	−	0.973341i	$-0.573664\pi$
$239$	−7.09173	−0.458726	−0.229363	+	0.973341i	$0.573664\pi$
$240$	0	0
$241$	−18.4405	−1.18786	−0.593930	−	0.804517i	$-0.702424\pi$
$241$	−18.4405	−1.18786	−0.593930	+	0.804517i	$0.702424\pi$
$242$	0	0
$243$	−5.11707	−0.328260
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.17208	−0.392720
$248$	0	0
$249$	22.4064	1.41995
$250$	0	0
$251$	−14.6808	−0.926641	−0.463321	−	0.886191i	$-0.653342\pi$
$251$	−14.6808	−0.926641	−0.463321	+	0.886191i	$0.653342\pi$
$252$	0	0
$253$	−9.64631	−0.606459
$254$	0	0
$255$	−54.8372	−3.43404
$256$	0	0
$257$	−10.0101	−0.624415	−0.312208	−	0.950014i	$-0.601068\pi$
$257$	−10.0101	−0.624415	−0.312208	+	0.950014i	$0.601068\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	19.5618	1.21085
$262$	0	0
$263$	−11.2916	−0.696268	−0.348134	−	0.937445i	$-0.613185\pi$
$263$	−11.2916	−0.696268	−0.348134	+	0.937445i	$0.613185\pi$
$264$	0	0
$265$	4.66245	0.286412
$266$	0	0
$267$	−41.0995	−2.51525
$268$	0	0
$269$	25.9497	1.58218	0.791089	−	0.611701i	$-0.209514\pi$
$269$	25.9497	1.58218	0.791089	+	0.611701i	$0.209514\pi$
$270$	0	0
$271$	20.2412	1.22956	0.614781	−	0.788698i	$-0.289244\pi$
$271$	20.2412	1.22956	0.614781	+	0.788698i	$0.289244\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.52429	0.393430
$276$	0	0
$277$	13.0873	0.786340	0.393170	−	0.919466i	$-0.371378\pi$
$277$	13.0873	0.786340	0.393170	+	0.919466i	$0.371378\pi$
$278$	0	0
$279$	−43.7230	−2.61763
$280$	0	0
$281$	21.1772	1.26333	0.631664	−	0.775242i	$-0.282372\pi$
$281$	21.1772	1.26333	0.631664	+	0.775242i	$0.282372\pi$
$282$	0	0
$283$	−7.32367	−0.435347	−0.217673	−	0.976022i	$-0.569847\pi$
$283$	−7.32367	−0.435347	−0.217673	+	0.976022i	$0.569847\pi$
$284$	0	0
$285$	−7.92207	−0.469263
$286$	0	0
$287$	0	0
$288$	0	0
$289$	30.9151	1.81854
$290$	0	0
$291$	−27.1073	−1.58906
$292$	0	0
$293$	−17.5488	−1.02521	−0.512606	−	0.858624i	$-0.671320\pi$
$293$	−17.5488	−1.02521	−0.512606	+	0.858624i	$0.671320\pi$
$294$	0	0
$295$	−18.5367	−1.07925
$296$	0	0
$297$	−36.6431	−2.12625
$298$	0	0
$299$	16.1659	0.934897
$300$	0	0
$301$	0	0
$302$	0	0
$303$	45.0954	2.59067
$304$	0	0
$305$	5.59007	0.320087
$306$	0	0
$307$	−32.5131	−1.85562	−0.927810	−	0.373053i	$-0.878311\pi$
$307$	−32.5131	−1.85562	−0.927810	+	0.373053i	$0.878311\pi$
$308$	0	0
$309$	51.2587	2.91601
$310$	0	0
$311$	8.72637	0.494827	0.247414	−	0.968910i	$-0.420419\pi$
$311$	8.72637	0.494827	0.247414	+	0.968910i	$0.420419\pi$
$312$	0	0
$313$	−10.6345	−0.601096	−0.300548	−	0.953767i	$-0.597170\pi$
$313$	−10.6345	−0.601096	−0.300548	+	0.953767i	$0.597170\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.2316	−0.799326	−0.399663	−	0.916662i	$-0.630873\pi$
$317$	−14.2316	−0.799326	−0.399663	+	0.916662i	$0.630873\pi$
$318$	0	0
$319$	11.4938	0.643528
$320$	0	0
$321$	40.0790	2.23699
$322$	0	0
$323$	6.92207	0.385155
$324$	0	0
$325$	−10.9338	−0.606498
$326$	0	0
$327$	−30.4449	−1.68361
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.8929	1.64306	0.821532	−	0.570163i	$-0.193120\pi$
$331$	29.8929	1.64306	0.821532	+	0.570163i	$0.193120\pi$
$332$	0	0
$333$	1.83335	0.100467
$334$	0	0
$335$	−19.8691	−1.08556
$336$	0	0
$337$	17.6020	0.958841	0.479421	−	0.877585i	$-0.340847\pi$
$337$	17.6020	0.958841	0.479421	+	0.877585i	$0.340847\pi$
$338$	0	0
$339$	53.5847	2.91032
$340$	0	0
$341$	−25.6900	−1.39119
$342$	0	0
$343$	0	0
$344$	0	0
$345$	20.7495	1.11711
$346$	0	0
$347$	−1.55959	−0.0837234	−0.0418617	−	0.999123i	$-0.513329\pi$
$347$	−1.55959	−0.0837234	−0.0418617	+	0.999123i	$0.513329\pi$
$348$	0	0
$349$	27.9765	1.49755	0.748773	−	0.662826i	$-0.230643\pi$
$349$	27.9765	1.49755	0.748773	+	0.662826i	$0.230643\pi$
$350$	0	0
$351$	61.4088	3.27776
$352$	0	0
$353$	4.89276	0.260415	0.130208	−	0.991487i	$-0.458436\pi$
$353$	4.89276	0.260415	0.130208	+	0.991487i	$0.458436\pi$
$354$	0	0
$355$	8.56840	0.454763
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−37.7566	−1.99272	−0.996360	−	0.0852473i	$-0.972832\pi$
$359$	−37.7566	−1.99272	−0.996360	+	0.0852473i	$0.972832\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.80568	−0.409692
$364$	0	0
$365$	−22.9756	−1.20260
$366$	0	0
$367$	−14.1666	−0.739493	−0.369746	−	0.929133i	$-0.620555\pi$
$367$	−14.1666	−0.739493	−0.369746	+	0.929133i	$0.620555\pi$
$368$	0	0
$369$	−33.1751	−1.72703
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.80652	−0.0935380	−0.0467690	−	0.998906i	$-0.514892\pi$
$373$	−1.80652	−0.0935380	−0.0467690	+	0.998906i	$0.514892\pi$
$374$	0	0
$375$	25.5765	1.32076
$376$	0	0
$377$	−19.2620	−0.992042
$378$	0	0
$379$	−35.3424	−1.81542	−0.907710	−	0.419599i	$-0.862171\pi$
$379$	−35.3424	−1.81542	−0.907710	+	0.419599i	$0.862171\pi$
$380$	0	0
$381$	−42.1745	−2.16066
$382$	0	0
$383$	−19.8292	−1.01323	−0.506614	−	0.862173i	$-0.669103\pi$
$383$	−19.8292	−1.01323	−0.506614	+	0.862173i	$0.669103\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.3084	−0.625672
$388$	0	0
$389$	−22.2057	−1.12587	−0.562936	−	0.826501i	$-0.690328\pi$
$389$	−22.2057	−1.12587	−0.562936	+	0.826501i	$0.690328\pi$
$390$	0	0
$391$	−18.1303	−0.916887
$392$	0	0
$393$	17.2872	0.872023
$394$	0	0
$395$	−16.6306	−0.836775
$396$	0	0
$397$	0.875987	0.0439645	0.0219823	−	0.999758i	$-0.493002\pi$
$397$	0.875987	0.0439645	0.0219823	+	0.999758i	$0.493002\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.6864	−1.93191	−0.965953	−	0.258717i	$-0.916700\pi$
$401$	−38.6864	−1.93191	−0.965953	+	0.258717i	$0.916700\pi$
$402$	0	0
$403$	43.0528	2.14461
$404$	0	0
$405$	29.8871	1.48510
$406$	0	0
$407$	1.07720	0.0533950
$408$	0	0
$409$	11.6808	0.577580	0.288790	−	0.957392i	$-0.406747\pi$
$409$	11.6808	0.577580	0.288790	+	0.957392i	$0.406747\pi$
$410$	0	0
$411$	−63.6052	−3.13741
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−19.1522	−0.940143
$416$	0	0
$417$	−55.9373	−2.73926
$418$	0	0
$419$	11.2773	0.550932	0.275466	−	0.961311i	$-0.411168\pi$
$419$	11.2773	0.550932	0.275466	+	0.961311i	$0.411168\pi$
$420$	0	0
$421$	−5.72033	−0.278792	−0.139396	−	0.990237i	$-0.544516\pi$
$421$	−5.72033	−0.278792	−0.139396	+	0.990237i	$0.544516\pi$
$422$	0	0
$423$	−50.1157	−2.43671
$424$	0	0
$425$	12.2624	0.594815
$426$	0	0
$427$	0	0
$428$	0	0
$429$	69.2025	3.34113
$430$	0	0
$431$	15.3845	0.741045	0.370522	−	0.928824i	$-0.379179\pi$
$431$	15.3845	0.741045	0.370522	+	0.928824i	$0.379179\pi$
$432$	0	0
$433$	−20.1376	−0.967749	−0.483875	−	0.875137i	$-0.660771\pi$
$433$	−20.1376	−0.967749	−0.483875	+	0.875137i	$0.660771\pi$
$434$	0	0
$435$	−24.7234	−1.18540
$436$	0	0
$437$	−2.61920	−0.125293
$438$	0	0
$439$	−10.7327	−0.512245	−0.256122	−	0.966644i	$-0.582445\pi$
$439$	−10.7327	−0.512245	−0.256122	+	0.966644i	$0.582445\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	35.1006	1.66768	0.833840	−	0.552006i	$-0.186137\pi$
$443$	35.1006	1.66768	0.833840	+	0.552006i	$0.186137\pi$
$444$	0	0
$445$	35.1304	1.66534
$446$	0	0
$447$	11.7584	0.556154
$448$	0	0
$449$	−35.3230	−1.66699	−0.833497	−	0.552524i	$-0.813665\pi$
$449$	−35.3230	−1.66699	−0.833497	+	0.552524i	$0.813665\pi$
$450$	0	0
$451$	−19.4925	−0.917864
$452$	0	0
$453$	−65.3872	−3.07216
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.6444	1.43348	0.716741	−	0.697339i	$-0.245633\pi$
$457$	30.6444	1.43348	0.716741	+	0.697339i	$0.245633\pi$
$458$	0	0
$459$	−68.8709	−3.21462
$460$	0	0
$461$	−2.47914	−0.115465	−0.0577325	−	0.998332i	$-0.518387\pi$
$461$	−2.47914	−0.115465	−0.0577325	+	0.998332i	$0.518387\pi$
$462$	0	0
$463$	−6.46751	−0.300571	−0.150285	−	0.988643i	$-0.548019\pi$
$463$	−6.46751	−0.300571	−0.150285	+	0.988643i	$0.548019\pi$
$464$	0	0
$465$	55.2598	2.56261
$466$	0	0
$467$	−6.61582	−0.306144	−0.153072	−	0.988215i	$-0.548917\pi$
$467$	−6.61582	−0.306144	−0.153072	+	0.988215i	$0.548917\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−3.73632	−0.172161
$472$	0	0
$473$	−7.23196	−0.332526
$474$	0	0
$475$	1.77149	0.0812817
$476$	0	0
$477$	11.2308	0.514224
$478$	0	0
$479$	10.5627	0.482621	0.241310	−	0.970448i	$-0.422423\pi$
$479$	10.5627	0.482621	0.241310	+	0.970448i	$0.422423\pi$
$480$	0	0
$481$	−1.80524	−0.0823121
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.1703	1.05211
$486$	0	0
$487$	−25.1801	−1.14102	−0.570510	−	0.821291i	$-0.693254\pi$
$487$	−25.1801	−1.14102	−0.570510	+	0.821291i	$0.693254\pi$
$488$	0	0
$489$	41.4810	1.87583
$490$	0	0
$491$	−10.3251	−0.465966	−0.232983	−	0.972481i	$-0.574849\pi$
$491$	−10.3251	−0.465966	−0.232983	+	0.972481i	$0.574849\pi$
$492$	0	0
$493$	21.6026	0.972931
$494$	0	0
$495$	60.0725	2.70006
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.0441	−1.03160	−0.515798	−	0.856710i	$-0.672504\pi$
$499$	−23.0441	−1.03160	−0.515798	+	0.856710i	$0.672504\pi$
$500$	0	0
$501$	−72.6699	−3.24665
$502$	0	0
$503$	−5.09453	−0.227154	−0.113577	−	0.993529i	$-0.536231\pi$
$503$	−5.09453	−0.227154	−0.113577	+	0.993529i	$0.536231\pi$
$504$	0	0
$505$	−38.5459	−1.71527
$506$	0	0
$507$	−76.3970	−3.39291
$508$	0	0
$509$	−39.2110	−1.73800	−0.868999	−	0.494815i	$-0.835236\pi$
$509$	−39.2110	−1.73800	−0.868999	+	0.494815i	$0.835236\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−9.94945	−0.439279
$514$	0	0
$515$	−43.8141	−1.93068
$516$	0	0
$517$	−29.4461	−1.29504
$518$	0	0
$519$	7.31487	0.321087
$520$	0	0
$521$	8.68475	0.380486	0.190243	−	0.981737i	$-0.439072\pi$
$521$	8.68475	0.380486	0.190243	+	0.981737i	$0.439072\pi$
$522$	0	0
$523$	33.5727	1.46803	0.734017	−	0.679131i	$-0.237643\pi$
$523$	33.5727	1.46803	0.734017	+	0.679131i	$0.237643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−48.2843	−2.10330
$528$	0	0
$529$	−16.1398	−0.701731
$530$	0	0
$531$	−44.6509	−1.93768
$532$	0	0
$533$	32.6667	1.41495
$534$	0	0
$535$	−34.2580	−1.48110
$536$	0	0
$537$	13.9052	0.600055
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.5514	−1.74344	−0.871720	−	0.490004i	$-0.836995\pi$
$541$	−40.5514	−1.74344	−0.871720	+	0.490004i	$0.836995\pi$
$542$	0	0
$543$	56.8654	2.44033
$544$	0	0
$545$	26.0232	1.11471
$546$	0	0
$547$	−12.7094	−0.543414	−0.271707	−	0.962380i	$-0.587588\pi$
$547$	−12.7094	−0.543414	−0.271707	+	0.962380i	$0.587588\pi$
$548$	0	0
$549$	13.4653	0.574683
$550$	0	0
$551$	3.12082	0.132952
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.31709	−0.0983551
$556$	0	0
$557$	−16.8513	−0.714011	−0.357005	−	0.934102i	$-0.616202\pi$
$557$	−16.8513	−0.714011	−0.357005	+	0.934102i	$0.616202\pi$
$558$	0	0
$559$	12.1198	0.512611
$560$	0	0
$561$	−77.6116	−3.27676
$562$	0	0
$563$	−5.30235	−0.223467	−0.111734	−	0.993738i	$-0.535640\pi$
$563$	−5.30235	−0.223467	−0.111734	+	0.993738i	$0.535640\pi$
$564$	0	0
$565$	−45.8022	−1.92691
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12.4621	0.522438	0.261219	−	0.965280i	$-0.415876\pi$
$569$	12.4621	0.522438	0.261219	+	0.965280i	$0.415876\pi$
$570$	0	0
$571$	15.5137	0.649230	0.324615	−	0.945846i	$-0.394765\pi$
$571$	15.5137	0.649230	0.324615	+	0.945846i	$0.394765\pi$
$572$	0	0
$573$	5.76723	0.240929
$574$	0	0
$575$	−4.63989	−0.193497
$576$	0	0
$577$	−11.8485	−0.493260	−0.246630	−	0.969110i	$-0.579323\pi$
$577$	−11.8485	−0.493260	−0.246630	+	0.969110i	$0.579323\pi$
$578$	0	0
$579$	−81.3668	−3.38149
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.59881	0.273295
$584$	0	0
$585$	−100.673	−4.16232
$586$	0	0
$587$	−21.7714	−0.898603	−0.449302	−	0.893380i	$-0.648327\pi$
$587$	−21.7714	−0.898603	−0.449302	+	0.893380i	$0.648327\pi$
$588$	0	0
$589$	−6.97542	−0.287417
$590$	0	0
$591$	41.1513	1.69274
$592$	0	0
$593$	−28.0822	−1.15320	−0.576599	−	0.817028i	$-0.695620\pi$
$593$	−28.0822	−1.15320	−0.576599	+	0.817028i	$0.695620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−32.9065	−1.34677
$598$	0	0
$599$	29.0002	1.18492	0.592459	−	0.805601i	$-0.298157\pi$
$599$	29.0002	1.18492	0.592459	+	0.805601i	$0.298157\pi$
$600$	0	0
$601$	5.11828	0.208779	0.104390	−	0.994536i	$-0.466711\pi$
$601$	5.11828	0.208779	0.104390	+	0.994536i	$0.466711\pi$
$602$	0	0
$603$	−47.8603	−1.94902
$604$	0	0
$605$	6.67200	0.271256
$606$	0	0
$607$	1.89786	0.0770317	0.0385159	−	0.999258i	$-0.487737\pi$
$607$	1.89786	0.0770317	0.0385159	+	0.999258i	$0.487737\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	49.3475	1.99639
$612$	0	0
$613$	−30.0880	−1.21524	−0.607621	−	0.794227i	$-0.707876\pi$
$613$	−30.0880	−1.21524	−0.607621	+	0.794227i	$0.707876\pi$
$614$	0	0
$615$	41.9288	1.69073
$616$	0	0
$617$	−14.3309	−0.576941	−0.288470	−	0.957489i	$-0.593147\pi$
$617$	−14.3309	−0.576941	−0.288470	+	0.957489i	$0.593147\pi$
$618$	0	0
$619$	−16.3218	−0.656027	−0.328013	−	0.944673i	$-0.606379\pi$
$619$	−16.3218	−0.656027	−0.328013	+	0.944673i	$0.606379\pi$
$620$	0	0
$621$	26.0596	1.04573
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.7193	−1.22877
$626$	0	0
$627$	−11.2122	−0.447771
$628$	0	0
$629$	2.02461	0.0807264
$630$	0	0
$631$	12.0842	0.481065	0.240533	−	0.970641i	$-0.422678\pi$
$631$	12.0842	0.481065	0.240533	+	0.970641i	$0.422678\pi$
$632$	0	0
$633$	27.7417	1.10263
$634$	0	0
$635$	36.0492	1.43057
$636$	0	0
$637$	0	0
$638$	0	0
$639$	20.6394	0.816482
$640$	0	0
$641$	−8.04908	−0.317919	−0.158960	−	0.987285i	$-0.550814\pi$
$641$	−8.04908	−0.317919	−0.158960	+	0.987285i	$0.550814\pi$
$642$	0	0
$643$	−4.82685	−0.190352	−0.0951762	−	0.995460i	$-0.530341\pi$
$643$	−4.82685	−0.190352	−0.0951762	+	0.995460i	$0.530341\pi$
$644$	0	0
$645$	15.5561	0.612522
$646$	0	0
$647$	26.1071	1.02637	0.513187	−	0.858277i	$-0.328465\pi$
$647$	26.1071	1.02637	0.513187	+	0.858277i	$0.328465\pi$
$648$	0	0
$649$	−26.2352	−1.02982
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.5954	−1.54949	−0.774743	−	0.632276i	$-0.782121\pi$
$653$	−39.5954	−1.54949	−0.774743	+	0.632276i	$0.782121\pi$
$654$	0	0
$655$	−14.7764	−0.577364
$656$	0	0
$657$	−55.3431	−2.15914
$658$	0	0
$659$	34.5877	1.34735	0.673674	−	0.739029i	$-0.264715\pi$
$659$	34.5877	1.34735	0.673674	+	0.739029i	$0.264715\pi$
$660$	0	0
$661$	16.4839	0.641149	0.320574	−	0.947223i	$-0.396124\pi$
$661$	16.4839	0.641149	0.320574	+	0.947223i	$0.396124\pi$
$662$	0	0
$663$	130.066	5.05135
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.17404	−0.316500
$668$	0	0
$669$	79.9225	3.08998
$670$	0	0
$671$	7.91168	0.305427
$672$	0	0
$673$	−23.4114	−0.902442	−0.451221	−	0.892412i	$-0.649011\pi$
$673$	−23.4114	−0.902442	−0.451221	+	0.892412i	$0.649011\pi$
$674$	0	0
$675$	−17.6254	−0.678402
$676$	0	0
$677$	−35.5410	−1.36595	−0.682977	−	0.730440i	$-0.739315\pi$
$677$	−35.5410	−1.36595	−0.682977	+	0.730440i	$0.739315\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−68.2465	−2.61521
$682$	0	0
$683$	−16.7886	−0.642397	−0.321198	−	0.947012i	$-0.604086\pi$
$683$	−16.7886	−0.642397	−0.321198	+	0.947012i	$0.604086\pi$
$684$	0	0
$685$	54.3674	2.07727
$686$	0	0
$687$	45.8135	1.74789
$688$	0	0
$689$	−11.0587	−0.421302
$690$	0	0
$691$	40.5064	1.54094	0.770468	−	0.637478i	$-0.220022\pi$
$691$	40.5064	1.54094	0.770468	+	0.637478i	$0.220022\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	47.8131	1.81365
$696$	0	0
$697$	−36.6361	−1.38769
$698$	0	0
$699$	14.7038	0.556148
$700$	0	0
$701$	−1.84233	−0.0695838	−0.0347919	−	0.999395i	$-0.511077\pi$
$701$	−1.84233	−0.0695838	−0.0347919	+	0.999395i	$0.511077\pi$
$702$	0	0
$703$	0.292486	0.0110313
$704$	0	0
$705$	63.3392	2.38549
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.60227	0.135286	0.0676430	−	0.997710i	$-0.478452\pi$
$709$	3.60227	0.135286	0.0676430	+	0.997710i	$0.478452\pi$
$710$	0	0
$711$	−40.0594	−1.50235
$712$	0	0
$713$	18.2700	0.684216
$714$	0	0
$715$	−59.1517	−2.21215
$716$	0	0
$717$	21.5898	0.806286
$718$	0	0
$719$	−36.7155	−1.36926	−0.684629	−	0.728892i	$-0.740036\pi$
$719$	−36.7155	−1.36926	−0.684629	+	0.728892i	$0.740036\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	56.1397	2.08786
$724$	0	0
$725$	5.52852	0.205324
$726$	0	0
$727$	26.0217	0.965090	0.482545	−	0.875871i	$-0.339713\pi$
$727$	26.0217	0.965090	0.482545	+	0.875871i	$0.339713\pi$
$728$	0	0
$729$	−18.8777	−0.699173
$730$	0	0
$731$	−13.5925	−0.502736
$732$	0	0
$733$	16.8036	0.620656	0.310328	−	0.950630i	$-0.399561\pi$
$733$	16.8036	0.620656	0.310328	+	0.950630i	$0.399561\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−28.1209	−1.03585
$738$	0	0
$739$	0.614452	0.0226030	0.0113015	−	0.999936i	$-0.496403\pi$
$739$	0.614452	0.0226030	0.0113015	+	0.999936i	$0.496403\pi$
$740$	0	0
$741$	18.7901	0.690270
$742$	0	0
$743$	4.75699	0.174517	0.0872585	−	0.996186i	$-0.472189\pi$
$743$	4.75699	0.174517	0.0872585	+	0.996186i	$0.472189\pi$
$744$	0	0
$745$	−10.0507	−0.368228
$746$	0	0
$747$	−46.1334	−1.68793
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.0603	0.914463	0.457232	−	0.889348i	$-0.348841\pi$
$751$	25.0603	0.914463	0.457232	+	0.889348i	$0.348841\pi$
$752$	0	0
$753$	44.6936	1.62872
$754$	0	0
$755$	55.8906	2.03407
$756$	0	0
$757$	−13.3559	−0.485429	−0.242715	−	0.970098i	$-0.578038\pi$
$757$	−13.3559	−0.485429	−0.242715	+	0.970098i	$0.578038\pi$
$758$	0	0
$759$	29.3669	1.06595
$760$	0	0
$761$	45.7947	1.66005	0.830027	−	0.557723i	$-0.188325\pi$
$761$	45.7947	1.66005	0.830027	+	0.557723i	$0.188325\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	112.906	4.08214
$766$	0	0
$767$	43.9665	1.58754
$768$	0	0
$769$	25.8371	0.931708	0.465854	−	0.884862i	$-0.345747\pi$
$769$	25.8371	0.931708	0.465854	+	0.884862i	$0.345747\pi$
$770$	0	0
$771$	30.4745	1.09751
$772$	0	0
$773$	15.7479	0.566412	0.283206	−	0.959059i	$-0.408602\pi$
$773$	15.7479	0.566412	0.283206	+	0.959059i	$0.408602\pi$
$774$	0	0
$775$	−12.3569	−0.443873
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.29265	−0.189629
$780$	0	0
$781$	12.1269	0.433936
$782$	0	0
$783$	−31.0505	−1.10965
$784$	0	0
$785$	3.19367	0.113987
$786$	0	0
$787$	1.22949	0.0438267	0.0219133	−	0.999760i	$-0.493024\pi$
$787$	1.22949	0.0438267	0.0219133	+	0.999760i	$0.493024\pi$
$788$	0	0
$789$	34.3756	1.22380
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−13.2589	−0.470836
$794$	0	0
$795$	−14.1942	−0.503416
$796$	0	0
$797$	47.3799	1.67828	0.839141	−	0.543913i	$-0.183058\pi$
$797$	47.3799	1.67828	0.839141	+	0.543913i	$0.183058\pi$
$798$	0	0
$799$	−55.3440	−1.95793
$800$	0	0
$801$	84.6214	2.98995
$802$	0	0
$803$	−32.5175	−1.14752
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−79.0002	−2.78094
$808$	0	0
$809$	−19.8717	−0.698653	−0.349326	−	0.937001i	$-0.613590\pi$
$809$	−19.8717	−0.698653	−0.349326	+	0.937001i	$0.613590\pi$
$810$	0	0
$811$	23.5779	0.827931	0.413966	−	0.910293i	$-0.364143\pi$
$811$	23.5779	0.827931	0.413966	+	0.910293i	$0.364143\pi$
$812$	0	0
$813$	−61.6214	−2.16116
$814$	0	0
$815$	−35.4564	−1.24198
$816$	0	0
$817$	−1.96364	−0.0686992
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.4995	−0.994638	−0.497319	−	0.867568i	$-0.665682\pi$
$821$	−28.4995	−0.994638	−0.497319	+	0.867568i	$0.665682\pi$
$822$	0	0
$823$	−21.0443	−0.733558	−0.366779	−	0.930308i	$-0.619540\pi$
$823$	−21.0443	−0.733558	−0.366779	+	0.930308i	$0.619540\pi$
$824$	0	0
$825$	−19.8623	−0.691517
$826$	0	0
$827$	−0.283949	−0.00987387	−0.00493693	−	0.999988i	$-0.501571\pi$
$827$	−0.283949	−0.00987387	−0.00493693	+	0.999988i	$0.501571\pi$
$828$	0	0
$829$	−47.7341	−1.65787	−0.828937	−	0.559342i	$-0.811054\pi$
$829$	−47.7341	−1.65787	−0.828937	+	0.559342i	$0.811054\pi$
$830$	0	0
$831$	−39.8425	−1.38212
$832$	0	0
$833$	0	0
$834$	0	0
$835$	62.1156	2.14960
$836$	0	0
$837$	69.4016	2.39887
$838$	0	0
$839$	24.9296	0.860664	0.430332	−	0.902671i	$-0.358397\pi$
$839$	24.9296	0.860664	0.430332	+	0.902671i	$0.358397\pi$
$840$	0	0
$841$	−19.2605	−0.664154
$842$	0	0
$843$	−64.4712	−2.22051
$844$	0	0
$845$	65.3013	2.24643
$846$	0	0
$847$	0	0
$848$	0	0
$849$	22.2959	0.765194
$850$	0	0
$851$	−0.766077	−0.0262608
$852$	0	0
$853$	−30.1668	−1.03289	−0.516447	−	0.856319i	$-0.672746\pi$
$853$	−30.1668	−1.03289	−0.516447	+	0.856319i	$0.672746\pi$
$854$	0	0
$855$	16.3111	0.557826
$856$	0	0
$857$	30.7276	1.04963	0.524817	−	0.851215i	$-0.324134\pi$
$857$	30.7276	1.04963	0.524817	+	0.851215i	$0.324134\pi$
$858$	0	0
$859$	29.1043	0.993025	0.496513	−	0.868029i	$-0.334614\pi$
$859$	29.1043	0.993025	0.496513	+	0.868029i	$0.334614\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	25.8047	0.878401	0.439201	−	0.898389i	$-0.355262\pi$
$863$	25.8047	0.878401	0.439201	+	0.898389i	$0.355262\pi$
$864$	0	0
$865$	−6.25248	−0.212591
$866$	0	0
$867$	−94.1169	−3.19638
$868$	0	0
$869$	−23.5374	−0.798451
$870$	0	0
$871$	47.1267	1.59683
$872$	0	0
$873$	55.8122	1.88896
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.4888	−1.02953	−0.514766	−	0.857331i	$-0.672121\pi$
$877$	−30.4888	−1.02953	−0.514766	+	0.857331i	$0.672121\pi$
$878$	0	0
$879$	53.4249	1.80198
$880$	0	0
$881$	12.0111	0.404663	0.202331	−	0.979317i	$-0.435148\pi$
$881$	12.0111	0.404663	0.202331	+	0.979317i	$0.435148\pi$
$882$	0	0
$883$	−29.4398	−0.990729	−0.495365	−	0.868685i	$-0.664966\pi$
$883$	−29.4398	−0.990729	−0.495365	+	0.868685i	$0.664966\pi$
$884$	0	0
$885$	56.4325	1.89696
$886$	0	0
$887$	−14.9000	−0.500295	−0.250147	−	0.968208i	$-0.580479\pi$
$887$	−14.9000	−0.500295	−0.250147	+	0.968208i	$0.580479\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	42.2995	1.41709
$892$	0	0
$893$	−7.99528	−0.267552
$894$	0	0
$895$	−11.8857	−0.397294
$896$	0	0
$897$	−49.2148	−1.64324
$898$	0	0
$899$	−21.7690	−0.726038
$900$	0	0
$901$	12.4025	0.413186
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−48.6064	−1.61573
$906$	0	0
$907$	15.9921	0.531009	0.265505	−	0.964110i	$-0.414461\pi$
$907$	15.9921	0.531009	0.265505	+	0.964110i	$0.414461\pi$
$908$	0	0
$909$	−92.8487	−3.07960
$910$	0	0
$911$	7.54020	0.249818	0.124909	−	0.992168i	$-0.460136\pi$
$911$	7.54020	0.249818	0.124909	+	0.992168i	$0.460136\pi$
$912$	0	0
$913$	−27.1062	−0.897086
$914$	0	0
$915$	−17.0182	−0.562605
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.87076	0.160672	0.0803358	−	0.996768i	$-0.474401\pi$
$919$	4.87076	0.160672	0.0803358	+	0.996768i	$0.474401\pi$
$920$	0	0
$921$	98.9817	3.26156
$922$	0	0
$923$	−20.3230	−0.668941
$924$	0	0
$925$	0.518137	0.0170362
$926$	0	0
$927$	−105.539	−3.46634
$928$	0	0
$929$	49.7616	1.63263	0.816313	−	0.577610i	$-0.196014\pi$
$929$	49.7616	1.63263	0.816313	+	0.577610i	$0.196014\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−26.5663	−0.869740
$934$	0	0
$935$	66.3395	2.16953
$936$	0	0
$937$	−19.8753	−0.649299	−0.324649	−	0.945834i	$-0.605246\pi$
$937$	−19.8753	−0.649299	−0.324649	+	0.945834i	$0.605246\pi$
$938$	0	0
$939$	32.3752	1.05652
$940$	0	0
$941$	−17.1505	−0.559090	−0.279545	−	0.960133i	$-0.590184\pi$
$941$	−17.1505	−0.559090	−0.279545	+	0.960133i	$0.590184\pi$
$942$	0	0
$943$	13.8625	0.451425
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−13.8371	−0.449645	−0.224823	−	0.974400i	$-0.572180\pi$
$947$	−13.8371	−0.449645	−0.224823	+	0.974400i	$0.572180\pi$
$948$	0	0
$949$	54.4948	1.76898
$950$	0	0
$951$	43.3262	1.40495
$952$	0	0
$953$	19.7888	0.641023	0.320511	−	0.947245i	$-0.396145\pi$
$953$	19.7888	0.641023	0.320511	+	0.947245i	$0.396145\pi$
$954$	0	0
$955$	−4.92961	−0.159518
$956$	0	0
$957$	−34.9912	−1.13111
$958$	0	0
$959$	0	0
$960$	0	0
$961$	17.6564	0.569562
$962$	0	0
$963$	−82.5201	−2.65917
$964$	0	0
$965$	69.5493	2.23887
$966$	0	0
$967$	−13.5184	−0.434721	−0.217361	−	0.976091i	$-0.569745\pi$
$967$	−13.5184	−0.434721	−0.217361	+	0.976091i	$0.569745\pi$
$968$	0	0
$969$	−21.0733	−0.676972
$970$	0	0
$971$	26.7127	0.857250	0.428625	−	0.903482i	$-0.358998\pi$
$971$	26.7127	0.857250	0.428625	+	0.903482i	$0.358998\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	33.2865	1.06602
$976$	0	0
$977$	46.5330	1.48872	0.744361	−	0.667778i	$-0.232754\pi$
$977$	46.5330	1.48872	0.744361	+	0.667778i	$0.232754\pi$
$978$	0	0
$979$	49.7203	1.58907
$980$	0	0
$981$	62.6841	2.00135
$982$	0	0
$983$	28.4260	0.906648	0.453324	−	0.891346i	$-0.350238\pi$
$983$	28.4260	0.906648	0.453324	+	0.891346i	$0.350238\pi$
$984$	0	0
$985$	−35.1746	−1.12076
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.14317	0.163543
$990$	0	0
$991$	7.12413	0.226305	0.113153	−	0.993578i	$-0.463905\pi$
$991$	7.12413	0.226305	0.113153	+	0.993578i	$0.463905\pi$
$992$	0	0
$993$	−91.0049	−2.88795
$994$	0	0
$995$	28.1272	0.891693
$996$	0	0
$997$	−27.2902	−0.864290	−0.432145	−	0.901804i	$-0.642243\pi$
$997$	−27.2902	−0.864290	−0.432145	+	0.901804i	$0.642243\pi$
$998$	0	0
$999$	−2.91007	−0.0920706

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bm.1.1	✓	6
7.6	odd	2		7448.2.a.bn.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.1	✓	6	1.1	even	1	trivial
7448.2.a.bn.1.6	yes	6	7.6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-3.04436 q^{3} +2.60221 q^{5} +6.26815 q^{9} +O(q^{10})\)	\(q-3.04436 q^{3} +2.60221 q^{5} +6.26815 q^{9} +3.68293 q^{11} -6.17208 q^{13} -7.92207 q^{15} +6.92207 q^{17} +1.00000 q^{19} -2.61920 q^{23} +1.77149 q^{25} -9.94945 q^{27} +3.12082 q^{29} -6.97542 q^{31} -11.2122 q^{33} +0.292486 q^{37} +18.7901 q^{39} -5.29265 q^{41} -1.96364 q^{43} +16.3111 q^{45} -7.99528 q^{47} -21.0733 q^{51} +1.79173 q^{53} +9.58376 q^{55} -3.04436 q^{57} -7.12345 q^{59} +2.14820 q^{61} -16.0610 q^{65} -7.63547 q^{67} +7.97378 q^{69} +3.29274 q^{71} -8.82925 q^{73} -5.39308 q^{75} -6.39094 q^{79} +11.4853 q^{81} -7.35997 q^{83} +18.0127 q^{85} -9.50092 q^{87} +13.5002 q^{89} +21.2357 q^{93} +2.60221 q^{95} +8.90409 q^{97} +23.0852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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