LMFDB - Embedded newform 9405.2.a.bq.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 $ x^14 - x^13 - 19x^12 + 15x^11 + 137x^10 - 80x^9 - 467x^8 + 193x^7 + 766x^6 - 240x^5 - 567x^4 + 142x^3 + 142x^2 - 25x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.0992940$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.0992940 q^{2} -1.99014 q^{4} +1.00000 q^{5} -0.183420 q^{7} +0.396197 q^{8} +O(q^{10})$	\(q-0.0992940 q^{2} -1.99014 q^{4} +1.00000 q^{5} -0.183420 q^{7} +0.396197 q^{8} -0.0992940 q^{10} +1.00000 q^{11} +2.10884 q^{13} +0.0182125 q^{14} +3.94094 q^{16} +0.306315 q^{17} -1.00000 q^{19} -1.99014 q^{20} -0.0992940 q^{22} +4.96912 q^{23} +1.00000 q^{25} -0.209396 q^{26} +0.365032 q^{28} -8.07721 q^{29} -7.74196 q^{31} -1.18371 q^{32} -0.0304152 q^{34} -0.183420 q^{35} -8.70551 q^{37} +0.0992940 q^{38} +0.396197 q^{40} +1.10346 q^{41} +10.9070 q^{43} -1.99014 q^{44} -0.493404 q^{46} -13.0773 q^{47} -6.96636 q^{49} -0.0992940 q^{50} -4.19690 q^{52} +4.19890 q^{53} +1.00000 q^{55} -0.0726705 q^{56} +0.802019 q^{58} -4.66823 q^{59} +5.15009 q^{61} +0.768730 q^{62} -7.76435 q^{64} +2.10884 q^{65} +3.41863 q^{67} -0.609609 q^{68} +0.0182125 q^{70} -10.3789 q^{71} +0.188594 q^{73} +0.864405 q^{74} +1.99014 q^{76} -0.183420 q^{77} +3.49139 q^{79} +3.94094 q^{80} -0.109567 q^{82} +15.9734 q^{83} +0.306315 q^{85} -1.08300 q^{86} +0.396197 q^{88} -0.481229 q^{89} -0.386804 q^{91} -9.88925 q^{92} +1.29850 q^{94} -1.00000 q^{95} +4.16047 q^{97} +0.691718 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

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 $

$2$	−0.0992940	−0.0702115	−0.0351057	−	0.999384i	$-0.511177\pi$
$2$	−0.0992940	−0.0702115	−0.0351057	+	0.999384i	$0.511177\pi$
$3$	0	0
$3$	0	0
$4$	−1.99014	−0.995070
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.183420	−0.0693263	−0.0346631	−	0.999399i	$-0.511036\pi$
$7$	−0.183420	−0.0693263	−0.0346631	+	0.999399i	$0.511036\pi$
$8$	0.396197	0.140077
$9$	0	0
$10$	−0.0992940	−0.0313995
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.10884	0.584888	0.292444	−	0.956283i	$-0.405531\pi$
$13$	2.10884	0.584888	0.292444	+	0.956283i	$0.405531\pi$
$14$	0.0182125	0.00486750
$15$	0	0
$16$	3.94094	0.985235
$17$	0.306315	0.0742922	0.0371461	−	0.999310i	$-0.488173\pi$
$17$	0.306315	0.0742922	0.0371461	+	0.999310i	$0.488173\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.99014	−0.445009
$21$	0	0
$22$	−0.0992940	−0.0211696
$23$	4.96912	1.03613	0.518067	−	0.855340i	$-0.326652\pi$
$23$	4.96912	1.03613	0.518067	+	0.855340i	$0.326652\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.209396	−0.0410659
$27$	0	0
$28$	0.365032	0.0689845
$29$	−8.07721	−1.49990	−0.749950	−	0.661494i	$-0.769923\pi$
$29$	−8.07721	−1.49990	−0.749950	+	0.661494i	$0.769923\pi$
$30$	0	0
$31$	−7.74196	−1.39050	−0.695248	−	0.718770i	$-0.744706\pi$
$31$	−7.74196	−1.39050	−0.695248	+	0.718770i	$0.744706\pi$
$32$	−1.18371	−0.209252
$33$	0	0
$34$	−0.0304152	−0.00521616
$35$	−0.183420	−0.0310036
$36$	0	0
$37$	−8.70551	−1.43118	−0.715588	−	0.698522i	$-0.753841\pi$
$37$	−8.70551	−1.43118	−0.715588	+	0.698522i	$0.753841\pi$
$38$	0.0992940	0.0161076
$39$	0	0
$40$	0.396197	0.0626443
$41$	1.10346	0.172331	0.0861655	−	0.996281i	$-0.472539\pi$
$41$	1.10346	0.172331	0.0861655	+	0.996281i	$0.472539\pi$
$42$	0	0
$43$	10.9070	1.66330	0.831652	−	0.555297i	$-0.187395\pi$
$43$	10.9070	1.66330	0.831652	+	0.555297i	$0.187395\pi$
$44$	−1.99014	−0.300025
$45$	0	0
$46$	−0.493404	−0.0727485
$47$	−13.0773	−1.90752	−0.953762	−	0.300562i	$-0.902826\pi$
$47$	−13.0773	−1.90752	−0.953762	+	0.300562i	$0.902826\pi$
$48$	0	0
$49$	−6.96636	−0.995194
$50$	−0.0992940	−0.0140423
$51$	0	0
$52$	−4.19690	−0.582005
$53$	4.19890	0.576764	0.288382	−	0.957516i	$-0.406883\pi$
$53$	4.19890	0.576764	0.288382	+	0.957516i	$0.406883\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−0.0726705	−0.00971100
$57$	0	0
$58$	0.802019	0.105310
$59$	−4.66823	−0.607751	−0.303876	−	0.952712i	$-0.598281\pi$
$59$	−4.66823	−0.607751	−0.303876	+	0.952712i	$0.598281\pi$
$60$	0	0
$61$	5.15009	0.659401	0.329700	−	0.944086i	$-0.393052\pi$
$61$	5.15009	0.659401	0.329700	+	0.944086i	$0.393052\pi$
$62$	0.768730	0.0976288
$63$	0	0
$64$	−7.76435	−0.970543
$65$	2.10884	0.261570
$66$	0	0
$67$	3.41863	0.417652	0.208826	−	0.977953i	$-0.433036\pi$
$67$	3.41863	0.417652	0.208826	+	0.977953i	$0.433036\pi$
$68$	−0.609609	−0.0739260
$69$	0	0
$70$	0.0182125	0.00217681
$71$	−10.3789	−1.23175	−0.615873	−	0.787845i	$-0.711197\pi$
$71$	−10.3789	−1.23175	−0.615873	+	0.787845i	$0.711197\pi$
$72$	0	0
$73$	0.188594	0.0220733	0.0110366	−	0.999939i	$-0.496487\pi$
$73$	0.188594	0.0220733	0.0110366	+	0.999939i	$0.496487\pi$
$74$	0.864405	0.100485
$75$	0	0
$76$	1.99014	0.228285
$77$	−0.183420	−0.0209027
$78$	0	0
$79$	3.49139	0.392812	0.196406	−	0.980523i	$-0.437073\pi$
$79$	3.49139	0.392812	0.196406	+	0.980523i	$0.437073\pi$
$80$	3.94094	0.440611
$81$	0	0
$82$	−0.109567	−0.0120996
$83$	15.9734	1.75330	0.876652	−	0.481125i	$-0.159772\pi$
$83$	15.9734	1.75330	0.876652	+	0.481125i	$0.159772\pi$
$84$	0	0
$85$	0.306315	0.0332245
$86$	−1.08300	−0.116783
$87$	0	0
$88$	0.396197	0.0422348
$89$	−0.481229	−0.0510102	−0.0255051	−	0.999675i	$-0.508119\pi$
$89$	−0.481229	−0.0510102	−0.0255051	+	0.999675i	$0.508119\pi$
$90$	0	0
$91$	−0.386804	−0.0405481
$92$	−9.88925	−1.03103
$93$	0	0
$94$	1.29850	0.133930
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.16047	0.422432	0.211216	−	0.977439i	$-0.432258\pi$
$97$	4.16047	0.422432	0.211216	+	0.977439i	$0.432258\pi$
$98$	0.691718	0.0698740
$99$	0	0
$100$	−1.99014	−0.199014
$101$	5.47619	0.544901	0.272450	−	0.962170i	$-0.412166\pi$
$101$	5.47619	0.544901	0.272450	+	0.962170i	$0.412166\pi$
$102$	0	0
$103$	−2.35268	−0.231816	−0.115908	−	0.993260i	$-0.536978\pi$
$103$	−2.35268	−0.231816	−0.115908	+	0.993260i	$0.536978\pi$
$104$	0.835518	0.0819293
$105$	0	0
$106$	−0.416926	−0.0404954
$107$	7.10310	0.686682	0.343341	−	0.939211i	$-0.388441\pi$
$107$	7.10310	0.686682	0.343341	+	0.939211i	$0.388441\pi$
$108$	0	0
$109$	−10.7358	−1.02830	−0.514151	−	0.857700i	$-0.671893\pi$
$109$	−10.7358	−1.02830	−0.514151	+	0.857700i	$0.671893\pi$
$110$	−0.0992940	−0.00946731
$111$	0	0
$112$	−0.722848	−0.0683027
$113$	−14.8159	−1.39376	−0.696881	−	0.717187i	$-0.745429\pi$
$113$	−14.8159	−1.39376	−0.696881	+	0.717187i	$0.745429\pi$
$114$	0	0
$115$	4.96912	0.463373
$116$	16.0748	1.49251
$117$	0	0
$118$	0.463527	0.0426711
$119$	−0.0561842	−0.00515040
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.511373	−0.0462975
$123$	0	0
$124$	15.4076	1.38364
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.83303	0.428862	0.214431	−	0.976739i	$-0.431210\pi$
$127$	4.83303	0.428862	0.214431	+	0.976739i	$0.431210\pi$
$128$	3.13837	0.277395
$129$	0	0
$130$	−0.209396	−0.0183652
$131$	−3.02604	−0.264386	−0.132193	−	0.991224i	$-0.542202\pi$
$131$	−3.02604	−0.264386	−0.132193	+	0.991224i	$0.542202\pi$
$132$	0	0
$133$	0.183420	0.0159045
$134$	−0.339449	−0.0293239
$135$	0	0
$136$	0.121361	0.0104066
$137$	−0.501217	−0.0428219	−0.0214109	−	0.999771i	$-0.506816\pi$
$137$	−0.501217	−0.0428219	−0.0214109	+	0.999771i	$0.506816\pi$
$138$	0	0
$139$	14.6951	1.24642	0.623210	−	0.782055i	$-0.285828\pi$
$139$	14.6951	1.24642	0.623210	+	0.782055i	$0.285828\pi$
$140$	0.365032	0.0308508
$141$	0	0
$142$	1.03056	0.0864827
$143$	2.10884	0.176350
$144$	0	0
$145$	−8.07721	−0.670776
$146$	−0.0187263	−0.00154980
$147$	0	0
$148$	17.3252	1.42412
$149$	3.29575	0.269998	0.134999	−	0.990846i	$-0.456897\pi$
$149$	3.29575	0.269998	0.134999	+	0.990846i	$0.456897\pi$
$150$	0	0
$151$	−8.70356	−0.708286	−0.354143	−	0.935191i	$-0.615227\pi$
$151$	−8.70356	−0.708286	−0.354143	+	0.935191i	$0.615227\pi$
$152$	−0.396197	−0.0321358
$153$	0	0
$154$	0.0182125	0.00146761
$155$	−7.74196	−0.621849
$156$	0	0
$157$	10.0628	0.803097	0.401548	−	0.915838i	$-0.368472\pi$
$157$	10.0628	0.803097	0.401548	+	0.915838i	$0.368472\pi$
$158$	−0.346674	−0.0275799
$159$	0	0
$160$	−1.18371	−0.0935802
$161$	−0.911436	−0.0718313
$162$	0	0
$163$	6.06126	0.474755	0.237377	−	0.971418i	$-0.423712\pi$
$163$	6.06126	0.474755	0.237377	+	0.971418i	$0.423712\pi$
$164$	−2.19603	−0.171481
$165$	0	0
$166$	−1.58606	−0.123102
$167$	−3.47793	−0.269131	−0.134565	−	0.990905i	$-0.542964\pi$
$167$	−3.47793	−0.269131	−0.134565	+	0.990905i	$0.542964\pi$
$168$	0	0
$169$	−8.55277	−0.657906
$170$	−0.0304152	−0.00233274
$171$	0	0
$172$	−21.7065	−1.65510
$173$	−1.56344	−0.118866	−0.0594332	−	0.998232i	$-0.518929\pi$
$173$	−1.56344	−0.118866	−0.0594332	+	0.998232i	$0.518929\pi$
$174$	0	0
$175$	−0.183420	−0.0138653
$176$	3.94094	0.297060
$177$	0	0
$178$	0.0477832	0.00358150
$179$	11.9269	0.891460	0.445730	−	0.895167i	$-0.352944\pi$
$179$	11.9269	0.891460	0.445730	+	0.895167i	$0.352944\pi$
$180$	0	0
$181$	3.78989	0.281701	0.140850	−	0.990031i	$-0.455016\pi$
$181$	3.78989	0.281701	0.140850	+	0.990031i	$0.455016\pi$
$182$	0.0384074	0.00284694
$183$	0	0
$184$	1.96875	0.145138
$185$	−8.70551	−0.640042
$186$	0	0
$187$	0.306315	0.0223999
$188$	26.0257	1.89812
$189$	0	0
$190$	0.0992940	0.00720355
$191$	−25.2549	−1.82738	−0.913692	−	0.406408i	$-0.866781\pi$
$191$	−25.2549	−1.82738	−0.913692	+	0.406408i	$0.866781\pi$
$192$	0	0
$193$	−21.1604	−1.52316	−0.761581	−	0.648070i	$-0.775576\pi$
$193$	−21.1604	−1.52316	−0.761581	+	0.648070i	$0.775576\pi$
$194$	−0.413110	−0.0296596
$195$	0	0
$196$	13.8640	0.990288
$197$	10.1579	0.723724	0.361862	−	0.932232i	$-0.382141\pi$
$197$	10.1579	0.723724	0.361862	+	0.932232i	$0.382141\pi$
$198$	0	0
$199$	2.76107	0.195727	0.0978636	−	0.995200i	$-0.468799\pi$
$199$	2.76107	0.195727	0.0978636	+	0.995200i	$0.468799\pi$
$200$	0.396197	0.0280154
$201$	0	0
$202$	−0.543753	−0.0382583
$203$	1.48152	0.103982
$204$	0	0
$205$	1.10346	0.0770687
$206$	0.233607	0.0162762
$207$	0	0
$208$	8.31083	0.576253
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−24.8808	−1.71286	−0.856432	−	0.516259i	$-0.827324\pi$
$211$	−24.8808	−1.71286	−0.856432	+	0.516259i	$0.827324\pi$
$212$	−8.35641	−0.573920
$213$	0	0
$214$	−0.705295	−0.0482130
$215$	10.9070	0.743852
$216$	0	0
$217$	1.42003	0.0963979
$218$	1.06600	0.0721986
$219$	0	0
$220$	−1.99014	−0.134175
$221$	0.645970	0.0434526
$222$	0	0
$223$	−28.0254	−1.87672	−0.938358	−	0.345664i	$-0.887654\pi$
$223$	−28.0254	−1.87672	−0.938358	+	0.345664i	$0.887654\pi$
$224$	0.217115	0.0145066
$225$	0	0
$226$	1.47113	0.0978581
$227$	−15.5412	−1.03150	−0.515752	−	0.856738i	$-0.672488\pi$
$227$	−15.5412	−1.03150	−0.515752	+	0.856738i	$0.672488\pi$
$228$	0	0
$229$	13.5651	0.896406	0.448203	−	0.893932i	$-0.352064\pi$
$229$	13.5651	0.896406	0.448203	+	0.893932i	$0.352064\pi$
$230$	−0.493404	−0.0325341
$231$	0	0
$232$	−3.20017	−0.210101
$233$	−13.5212	−0.885804	−0.442902	−	0.896570i	$-0.646051\pi$
$233$	−13.5212	−0.885804	−0.442902	+	0.896570i	$0.646051\pi$
$234$	0	0
$235$	−13.0773	−0.853071
$236$	9.29043	0.604755
$237$	0	0
$238$	0.00557876	0.000361617 0
$239$	22.6819	1.46717	0.733586	−	0.679596i	$-0.237845\pi$
$239$	22.6819	1.46717	0.733586	+	0.679596i	$0.237845\pi$
$240$	0	0
$241$	−14.8538	−0.956818	−0.478409	−	0.878137i	$-0.658786\pi$
$241$	−14.8538	−0.956818	−0.478409	+	0.878137i	$0.658786\pi$
$242$	−0.0992940	−0.00638286
$243$	0	0
$244$	−10.2494	−0.656150
$245$	−6.96636	−0.445064
$246$	0	0
$247$	−2.10884	−0.134183
$248$	−3.06734	−0.194776
$249$	0	0
$250$	−0.0992940	−0.00627991
$251$	−6.02045	−0.380008	−0.190004	−	0.981783i	$-0.560850\pi$
$251$	−6.02045	−0.380008	−0.190004	+	0.981783i	$0.560850\pi$
$252$	0	0
$253$	4.96912	0.312406
$254$	−0.479891	−0.0301110
$255$	0	0
$256$	15.2171	0.951067
$257$	−0.572943	−0.0357392	−0.0178696	−	0.999840i	$-0.505688\pi$
$257$	−0.572943	−0.0357392	−0.0178696	+	0.999840i	$0.505688\pi$
$258$	0	0
$259$	1.59676	0.0992181
$260$	−4.19690	−0.260281
$261$	0	0
$262$	0.300467	0.0185629
$263$	2.90164	0.178923	0.0894614	−	0.995990i	$-0.471485\pi$
$263$	2.90164	0.178923	0.0894614	+	0.995990i	$0.471485\pi$
$264$	0	0
$265$	4.19890	0.257937
$266$	−0.0182125	−0.00111668
$267$	0	0
$268$	−6.80355	−0.415593
$269$	11.3086	0.689497	0.344749	−	0.938695i	$-0.387964\pi$
$269$	11.3086	0.689497	0.344749	+	0.938695i	$0.387964\pi$
$270$	0	0
$271$	−13.8264	−0.839892	−0.419946	−	0.907549i	$-0.637951\pi$
$271$	−13.8264	−0.839892	−0.419946	+	0.907549i	$0.637951\pi$
$272$	1.20717	0.0731953
$273$	0	0
$274$	0.0497678	0.00300659
$275$	1.00000	0.0603023
$276$	0	0
$277$	9.99115	0.600310	0.300155	−	0.953890i	$-0.402962\pi$
$277$	9.99115	0.600310	0.300155	+	0.953890i	$0.402962\pi$
$278$	−1.45913	−0.0875130
$279$	0	0
$280$	−0.0726705	−0.00434289
$281$	−26.4029	−1.57506	−0.787532	−	0.616274i	$-0.788641\pi$
$281$	−26.4029	−1.57506	−0.787532	+	0.616274i	$0.788641\pi$
$282$	0	0
$283$	−26.2982	−1.56327	−0.781633	−	0.623739i	$-0.785613\pi$
$283$	−26.2982	−1.56327	−0.781633	+	0.623739i	$0.785613\pi$
$284$	20.6554	1.22567
$285$	0	0
$286$	−0.209396	−0.0123818
$287$	−0.202396	−0.0119471
$288$	0	0
$289$	−16.9062	−0.994481
$290$	0.802019	0.0470962
$291$	0	0
$292$	−0.375329	−0.0219644
$293$	13.8325	0.808103	0.404052	−	0.914736i	$-0.367602\pi$
$293$	13.8325	0.808103	0.404052	+	0.914736i	$0.367602\pi$
$294$	0	0
$295$	−4.66823	−0.271795
$296$	−3.44910	−0.200475
$297$	0	0
$298$	−0.327248	−0.0189570
$299$	10.4791	0.606022
$300$	0	0
$301$	−2.00057	−0.115311
$302$	0.864212	0.0497298
$303$	0	0
$304$	−3.94094	−0.226028
$305$	5.15009	0.294893
$306$	0	0
$307$	−21.2681	−1.21383	−0.606917	−	0.794765i	$-0.707594\pi$
$307$	−21.2681	−1.21383	−0.606917	+	0.794765i	$0.707594\pi$
$308$	0.365032	0.0207996
$309$	0	0
$310$	0.768730	0.0436609
$311$	−4.23297	−0.240030	−0.120015	−	0.992772i	$-0.538294\pi$
$311$	−4.23297	−0.240030	−0.120015	+	0.992772i	$0.538294\pi$
$312$	0	0
$313$	−14.3314	−0.810060	−0.405030	−	0.914303i	$-0.632739\pi$
$313$	−14.3314	−0.810060	−0.405030	+	0.914303i	$0.632739\pi$
$314$	−0.999173	−0.0563866
$315$	0	0
$316$	−6.94835	−0.390875
$317$	17.3836	0.976358	0.488179	−	0.872743i	$-0.337661\pi$
$317$	17.3836	0.976358	0.488179	+	0.872743i	$0.337661\pi$
$318$	0	0
$319$	−8.07721	−0.452237
$320$	−7.76435	−0.434040
$321$	0	0
$322$	0.0905002	0.00504338
$323$	−0.306315	−0.0170438
$324$	0	0
$325$	2.10884	0.116978
$326$	−0.601847	−0.0333332
$327$	0	0
$328$	0.437186	0.0241396
$329$	2.39864	0.132242
$330$	0	0
$331$	−6.16836	−0.339044	−0.169522	−	0.985526i	$-0.554222\pi$
$331$	−6.16836	−0.339044	−0.169522	+	0.985526i	$0.554222\pi$
$332$	−31.7892	−1.74466
$333$	0	0
$334$	0.345338	0.0188961
$335$	3.41863	0.186779
$336$	0	0
$337$	−8.33440	−0.454004	−0.227002	−	0.973894i	$-0.572892\pi$
$337$	−8.33440	−0.454004	−0.227002	+	0.973894i	$0.572892\pi$
$338$	0.849239	0.0461925
$339$	0	0
$340$	−0.609609	−0.0330607
$341$	−7.74196	−0.419250
$342$	0	0
$343$	2.56171	0.138319
$344$	4.32133	0.232990
$345$	0	0
$346$	0.155241	0.00834578
$347$	−24.4694	−1.31359	−0.656793	−	0.754071i	$-0.728087\pi$
$347$	−24.4694	−1.31359	−0.656793	+	0.754071i	$0.728087\pi$
$348$	0	0
$349$	−25.4591	−1.36280	−0.681398	−	0.731914i	$-0.738627\pi$
$349$	−25.4591	−1.36280	−0.681398	+	0.731914i	$0.738627\pi$
$350$	0.0182125	0.000973500 0
$351$	0	0
$352$	−1.18371	−0.0630918
$353$	35.7338	1.90192	0.950959	−	0.309317i	$-0.100101\pi$
$353$	35.7338	1.90192	0.950959	+	0.309317i	$0.100101\pi$
$354$	0	0
$355$	−10.3789	−0.550854
$356$	0.957714	0.0507587
$357$	0	0
$358$	−1.18427	−0.0625908
$359$	−1.45867	−0.0769858	−0.0384929	−	0.999259i	$-0.512256\pi$
$359$	−1.45867	−0.0769858	−0.0384929	+	0.999259i	$0.512256\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.376314	−0.0197786
$363$	0	0
$364$	0.769795	0.0403482
$365$	0.188594	0.00987146
$366$	0	0
$367$	−24.2427	−1.26546	−0.632728	−	0.774374i	$-0.718065\pi$
$367$	−24.2427	−1.26546	−0.632728	+	0.774374i	$0.718065\pi$
$368$	19.5830	1.02084
$369$	0	0
$370$	0.864405	0.0449383
$371$	−0.770163	−0.0399849
$372$	0	0
$373$	−13.7681	−0.712887	−0.356443	−	0.934317i	$-0.616011\pi$
$373$	−13.7681	−0.712887	−0.356443	+	0.934317i	$0.616011\pi$
$374$	−0.0304152	−0.00157273
$375$	0	0
$376$	−5.18120	−0.267200
$377$	−17.0336	−0.877274
$378$	0	0
$379$	−28.3316	−1.45530	−0.727649	−	0.685950i	$-0.759387\pi$
$379$	−28.3316	−1.45530	−0.727649	+	0.685950i	$0.759387\pi$
$380$	1.99014	0.102092
$381$	0	0
$382$	2.50766	0.128303
$383$	7.06825	0.361171	0.180585	−	0.983559i	$-0.442201\pi$
$383$	7.06825	0.361171	0.180585	+	0.983559i	$0.442201\pi$
$384$	0	0
$385$	−0.183420	−0.00934795
$386$	2.10110	0.106943
$387$	0	0
$388$	−8.27992	−0.420349
$389$	3.08794	0.156565	0.0782823	−	0.996931i	$-0.475056\pi$
$389$	3.08794	0.156565	0.0782823	+	0.996931i	$0.475056\pi$
$390$	0	0
$391$	1.52211	0.0769766
$392$	−2.76005	−0.139404
$393$	0	0
$394$	−1.00862	−0.0508137
$395$	3.49139	0.175671
$396$	0	0
$397$	28.7830	1.44458	0.722288	−	0.691592i	$-0.243090\pi$
$397$	28.7830	1.44458	0.722288	+	0.691592i	$0.243090\pi$
$398$	−0.274158	−0.0137423
$399$	0	0
$400$	3.94094	0.197047
$401$	−16.6037	−0.829148	−0.414574	−	0.910016i	$-0.636069\pi$
$401$	−16.6037	−0.829148	−0.414574	+	0.910016i	$0.636069\pi$
$402$	0	0
$403$	−16.3266	−0.813285
$404$	−10.8984	−0.542215
$405$	0	0
$406$	−0.147106	−0.00730076
$407$	−8.70551	−0.431516
$408$	0	0
$409$	−26.2462	−1.29779	−0.648895	−	0.760878i	$-0.724769\pi$
$409$	−26.2462	−1.29779	−0.648895	+	0.760878i	$0.724769\pi$
$410$	−0.109567	−0.00541111
$411$	0	0
$412$	4.68216	0.230673
$413$	0.856247	0.0421331
$414$	0	0
$415$	15.9734	0.784101
$416$	−2.49625	−0.122389
$417$	0	0
$418$	0.0992940	0.00485663
$419$	34.7784	1.69904	0.849518	−	0.527559i	$-0.176893\pi$
$419$	34.7784	1.69904	0.849518	+	0.527559i	$0.176893\pi$
$420$	0	0
$421$	2.96158	0.144339	0.0721694	−	0.997392i	$-0.477008\pi$
$421$	2.96158	0.144339	0.0721694	+	0.997392i	$0.477008\pi$
$422$	2.47051	0.120263
$423$	0	0
$424$	1.66359	0.0807912
$425$	0.306315	0.0148584
$426$	0	0
$427$	−0.944629	−0.0457138
$428$	−14.1362	−0.683297
$429$	0	0
$430$	−1.08300	−0.0522270
$431$	31.4579	1.51527	0.757637	−	0.652676i	$-0.226354\pi$
$431$	31.4579	1.51527	0.757637	+	0.652676i	$0.226354\pi$
$432$	0	0
$433$	17.6134	0.846448	0.423224	−	0.906025i	$-0.360898\pi$
$433$	17.6134	0.846448	0.423224	+	0.906025i	$0.360898\pi$
$434$	−0.141000	−0.00676824
$435$	0	0
$436$	21.3657	1.02323
$437$	−4.96912	−0.237705
$438$	0	0
$439$	8.26782	0.394602	0.197301	−	0.980343i	$-0.436782\pi$
$439$	8.26782	0.394602	0.197301	+	0.980343i	$0.436782\pi$
$440$	0.396197	0.0188880
$441$	0	0
$442$	−0.0641410	−0.00305087
$443$	32.6647	1.55195	0.775974	−	0.630764i	$-0.217259\pi$
$443$	32.6647	1.55195	0.775974	+	0.630764i	$0.217259\pi$
$444$	0	0
$445$	−0.481229	−0.0228125
$446$	2.78275	0.131767
$447$	0	0
$448$	1.42414	0.0672841
$449$	−24.0081	−1.13301	−0.566506	−	0.824058i	$-0.691705\pi$
$449$	−24.0081	−1.13301	−0.566506	+	0.824058i	$0.691705\pi$
$450$	0	0
$451$	1.10346	0.0519597
$452$	29.4857	1.38689
$453$	0	0
$454$	1.54315	0.0724235
$455$	−0.386804	−0.0181337
$456$	0	0
$457$	−1.05482	−0.0493424	−0.0246712	−	0.999696i	$-0.507854\pi$
$457$	−1.05482	−0.0493424	−0.0246712	+	0.999696i	$0.507854\pi$
$458$	−1.34693	−0.0629380
$459$	0	0
$460$	−9.88925	−0.461089
$461$	−6.23002	−0.290161	−0.145080	−	0.989420i	$-0.546344\pi$
$461$	−6.23002	−0.290161	−0.145080	+	0.989420i	$0.546344\pi$
$462$	0	0
$463$	−6.98856	−0.324786	−0.162393	−	0.986726i	$-0.551921\pi$
$463$	−6.98856	−0.324786	−0.162393	+	0.986726i	$0.551921\pi$
$464$	−31.8318	−1.47776
$465$	0	0
$466$	1.34258	0.0621936
$467$	−32.7086	−1.51357	−0.756787	−	0.653662i	$-0.773232\pi$
$467$	−32.7086	−1.51357	−0.756787	+	0.653662i	$0.773232\pi$
$468$	0	0
$469$	−0.627044	−0.0289542
$470$	1.29850	0.0598954
$471$	0	0
$472$	−1.84954	−0.0851319
$473$	10.9070	0.501505
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0.111815	0.00512501
$477$	0	0
$478$	−2.25218	−0.103012
$479$	4.08815	0.186792	0.0933962	−	0.995629i	$-0.470228\pi$
$479$	4.08815	0.186792	0.0933962	+	0.995629i	$0.470228\pi$
$480$	0	0
$481$	−18.3586	−0.837078
$482$	1.47490	0.0671796
$483$	0	0
$484$	−1.99014	−0.0904609
$485$	4.16047	0.188917
$486$	0	0
$487$	−36.9289	−1.67341	−0.836704	−	0.547655i	$-0.815521\pi$
$487$	−36.9289	−1.67341	−0.836704	+	0.547655i	$0.815521\pi$
$488$	2.04045	0.0923668
$489$	0	0
$490$	0.691718	0.0312486
$491$	18.5713	0.838109	0.419054	−	0.907961i	$-0.362362\pi$
$491$	18.5713	0.838109	0.419054	+	0.907961i	$0.362362\pi$
$492$	0	0
$493$	−2.47417	−0.111431
$494$	0.209396	0.00942116
$495$	0	0
$496$	−30.5106	−1.36997
$497$	1.90369	0.0853924
$498$	0	0
$499$	30.7149	1.37499	0.687495	−	0.726189i	$-0.258710\pi$
$499$	30.7149	1.37499	0.687495	+	0.726189i	$0.258710\pi$
$500$	−1.99014	−0.0890018
$501$	0	0
$502$	0.597795	0.0266809
$503$	29.8474	1.33083	0.665416	−	0.746473i	$-0.268254\pi$
$503$	29.8474	1.33083	0.665416	+	0.746473i	$0.268254\pi$
$504$	0	0
$505$	5.47619	0.243687
$506$	−0.493404	−0.0219345
$507$	0	0
$508$	−9.61841	−0.426748
$509$	−2.49273	−0.110489	−0.0552443	−	0.998473i	$-0.517594\pi$
$509$	−2.49273	−0.110489	−0.0552443	+	0.998473i	$0.517594\pi$
$510$	0	0
$511$	−0.0345919	−0.00153026
$512$	−7.78770	−0.344171
$513$	0	0
$514$	0.0568898	0.00250930
$515$	−2.35268	−0.103671
$516$	0	0
$517$	−13.0773	−0.575140
$518$	−0.158549	−0.00696625
$519$	0	0
$520$	0.835518	0.0366399
$521$	−21.9913	−0.963454	−0.481727	−	0.876321i	$-0.659990\pi$
$521$	−21.9913	−0.963454	−0.481727	+	0.876321i	$0.659990\pi$
$522$	0	0
$523$	−37.9067	−1.65754	−0.828772	−	0.559586i	$-0.810960\pi$
$523$	−37.9067	−1.65754	−0.828772	+	0.559586i	$0.810960\pi$
$524$	6.02224	0.263083
$525$	0	0
$526$	−0.288115	−0.0125624
$527$	−2.37147	−0.103303
$528$	0	0
$529$	1.69217	0.0735725
$530$	−0.416926	−0.0181101
$531$	0	0
$532$	−0.365032	−0.0158261
$533$	2.32702	0.100794
$534$	0	0
$535$	7.10310	0.307094
$536$	1.35445	0.0585033
$537$	0	0
$538$	−1.12288	−0.0484106
$539$	−6.96636	−0.300062
$540$	0	0
$541$	28.1663	1.21096	0.605482	−	0.795859i	$-0.292980\pi$
$541$	28.1663	1.21096	0.605482	+	0.795859i	$0.292980\pi$
$542$	1.37288	0.0589701
$543$	0	0
$544$	−0.362586	−0.0155458
$545$	−10.7358	−0.459871
$546$	0	0
$547$	−0.0757006	−0.00323672	−0.00161836	−	0.999999i	$-0.500515\pi$
$547$	−0.0757006	−0.00323672	−0.00161836	+	0.999999i	$0.500515\pi$
$548$	0.997492	0.0426108
$549$	0	0
$550$	−0.0992940	−0.00423391
$551$	8.07721	0.344101
$552$	0	0
$553$	−0.640390	−0.0272322
$554$	−0.992061	−0.0421486
$555$	0	0
$556$	−29.2453	−1.24028
$557$	10.2405	0.433906	0.216953	−	0.976182i	$-0.430388\pi$
$557$	10.2405	0.433906	0.216953	+	0.976182i	$0.430388\pi$
$558$	0	0
$559$	23.0012	0.972847
$560$	−0.722848	−0.0305459
$561$	0	0
$562$	2.62165	0.110588
$563$	−16.6290	−0.700829	−0.350414	−	0.936595i	$-0.613959\pi$
$563$	−16.6290	−0.700829	−0.350414	+	0.936595i	$0.613959\pi$
$564$	0	0
$565$	−14.8159	−0.623309
$566$	2.61125	0.109759
$567$	0	0
$568$	−4.11208	−0.172539
$569$	−18.7314	−0.785260	−0.392630	−	0.919697i	$-0.628435\pi$
$569$	−18.7314	−0.785260	−0.392630	+	0.919697i	$0.628435\pi$
$570$	0	0
$571$	7.30548	0.305725	0.152862	−	0.988247i	$-0.451151\pi$
$571$	7.30548	0.305725	0.152862	+	0.988247i	$0.451151\pi$
$572$	−4.19690	−0.175481
$573$	0	0
$574$	0.0200967	0.000838821 0
$575$	4.96912	0.207227
$576$	0	0
$577$	−30.0600	−1.25142	−0.625708	−	0.780057i	$-0.715190\pi$
$577$	−30.0600	−1.25142	−0.625708	+	0.780057i	$0.715190\pi$
$578$	1.67868	0.0698240
$579$	0	0
$580$	16.0748	0.667469
$581$	−2.92983	−0.121550
$582$	0	0
$583$	4.19890	0.173901
$584$	0.0747204	0.00309195
$585$	0	0
$586$	−1.37348	−0.0567381
$587$	−2.94668	−0.121623	−0.0608113	−	0.998149i	$-0.519369\pi$
$587$	−2.94668	−0.121623	−0.0608113	+	0.998149i	$0.519369\pi$
$588$	0	0
$589$	7.74196	0.319002
$590$	0.463527	0.0190831
$591$	0	0
$592$	−34.3079	−1.41005
$593$	−8.10854	−0.332978	−0.166489	−	0.986043i	$-0.553243\pi$
$593$	−8.10854	−0.332978	−0.166489	+	0.986043i	$0.553243\pi$
$594$	0	0
$595$	−0.0561842	−0.00230333
$596$	−6.55900	−0.268667
$597$	0	0
$598$	−1.04051	−0.0425497
$599$	17.5424	0.716764	0.358382	−	0.933575i	$-0.383329\pi$
$599$	17.5424	0.716764	0.358382	+	0.933575i	$0.383329\pi$
$600$	0	0
$601$	8.05540	0.328586	0.164293	−	0.986412i	$-0.447466\pi$
$601$	8.05540	0.328586	0.164293	+	0.986412i	$0.447466\pi$
$602$	0.198644	0.00809613
$603$	0	0
$604$	17.3213	0.704794
$605$	1.00000	0.0406558
$606$	0	0
$607$	6.34672	0.257606	0.128803	−	0.991670i	$-0.458887\pi$
$607$	6.34672	0.257606	0.128803	+	0.991670i	$0.458887\pi$
$608$	1.18371	0.0480056
$609$	0	0
$610$	−0.511373	−0.0207049
$611$	−27.5781	−1.11569
$612$	0	0
$613$	0.0844105	0.00340931	0.00170465	−	0.999999i	$-0.499457\pi$
$613$	0.0844105	0.00340931	0.00170465	+	0.999999i	$0.499457\pi$
$614$	2.11179	0.0852251
$615$	0	0
$616$	−0.0726705	−0.00292798
$617$	3.68285	0.148266	0.0741330	−	0.997248i	$-0.476381\pi$
$617$	3.68285	0.148266	0.0741330	+	0.997248i	$0.476381\pi$
$618$	0	0
$619$	−23.9316	−0.961893	−0.480947	−	0.876750i	$-0.659707\pi$
$619$	−23.9316	−0.961893	−0.480947	+	0.876750i	$0.659707\pi$
$620$	15.4076	0.618783
$621$	0	0
$622$	0.420309	0.0168529
$623$	0.0882671	0.00353635
$624$	0	0
$625$	1.00000	0.0400000
$626$	1.42302	0.0568755
$627$	0	0
$628$	−20.0263	−0.799138
$629$	−2.66662	−0.106325
$630$	0	0
$631$	−32.0815	−1.27714	−0.638572	−	0.769562i	$-0.720475\pi$
$631$	−32.0815	−1.27714	−0.638572	+	0.769562i	$0.720475\pi$
$632$	1.38328	0.0550238
$633$	0	0
$634$	−1.72608	−0.0685516
$635$	4.83303	0.191793
$636$	0	0
$637$	−14.6910	−0.582077
$638$	0.802019	0.0317522
$639$	0	0
$640$	3.13837	0.124055
$641$	29.2638	1.15585	0.577925	−	0.816090i	$-0.303863\pi$
$641$	29.2638	1.15585	0.577925	+	0.816090i	$0.303863\pi$
$642$	0	0
$643$	−6.81763	−0.268861	−0.134431	−	0.990923i	$-0.542921\pi$
$643$	−6.81763	−0.268861	−0.134431	+	0.990923i	$0.542921\pi$
$644$	1.81389	0.0714772
$645$	0	0
$646$	0.0304152	0.00119667
$647$	−19.0977	−0.750808	−0.375404	−	0.926861i	$-0.622496\pi$
$647$	−19.0977	−0.750808	−0.375404	+	0.926861i	$0.622496\pi$
$648$	0	0
$649$	−4.66823	−0.183244
$650$	−0.209396	−0.00821317
$651$	0	0
$652$	−12.0628	−0.472414
$653$	−5.36472	−0.209938	−0.104969	−	0.994476i	$-0.533474\pi$
$653$	−5.36472	−0.209938	−0.104969	+	0.994476i	$0.533474\pi$
$654$	0	0
$655$	−3.02604	−0.118237
$656$	4.34866	0.169787
$657$	0	0
$658$	−0.238171	−0.00928487
$659$	−9.42829	−0.367274	−0.183637	−	0.982994i	$-0.558787\pi$
$659$	−9.42829	−0.367274	−0.183637	+	0.982994i	$0.558787\pi$
$660$	0	0
$661$	24.5456	0.954712	0.477356	−	0.878710i	$-0.341595\pi$
$661$	24.5456	0.954712	0.477356	+	0.878710i	$0.341595\pi$
$662$	0.612481	0.0238048
$663$	0	0
$664$	6.32860	0.245597
$665$	0.183420	0.00711272
$666$	0	0
$667$	−40.1366	−1.55410
$668$	6.92158	0.267804
$669$	0	0
$670$	−0.339449	−0.0131141
$671$	5.15009	0.198817
$672$	0	0
$673$	2.84822	0.109791	0.0548954	−	0.998492i	$-0.482517\pi$
$673$	2.84822	0.109791	0.0548954	+	0.998492i	$0.482517\pi$
$674$	0.827556	0.0318763
$675$	0	0
$676$	17.0212	0.654662
$677$	−4.53877	−0.174439	−0.0872195	−	0.996189i	$-0.527798\pi$
$677$	−4.53877	−0.174439	−0.0872195	+	0.996189i	$0.527798\pi$
$678$	0	0
$679$	−0.763114	−0.0292856
$680$	0.121361	0.00465398
$681$	0	0
$682$	0.768730	0.0294362
$683$	−5.61534	−0.214865	−0.107433	−	0.994212i	$-0.534263\pi$
$683$	−5.61534	−0.214865	−0.107433	+	0.994212i	$0.534263\pi$
$684$	0	0
$685$	−0.501217	−0.0191505
$686$	−0.254362	−0.00971160
$687$	0	0
$688$	42.9839	1.63875
$689$	8.85483	0.337342
$690$	0	0
$691$	−30.6758	−1.16696	−0.583481	−	0.812127i	$-0.698310\pi$
$691$	−30.6758	−1.16696	−0.583481	+	0.812127i	$0.698310\pi$
$692$	3.11147	0.118280
$693$	0	0
$694$	2.42966	0.0922288
$695$	14.6951	0.557416
$696$	0	0
$697$	0.338005	0.0128028
$698$	2.52794	0.0956839
$699$	0	0
$700$	0.365032	0.0137969
$701$	−17.2341	−0.650924	−0.325462	−	0.945555i	$-0.605520\pi$
$701$	−17.2341	−0.650924	−0.325462	+	0.945555i	$0.605520\pi$
$702$	0	0
$703$	8.70551	0.328334
$704$	−7.76435	−0.292630
$705$	0	0
$706$	−3.54815	−0.133536
$707$	−1.00444	−0.0377759
$708$	0	0
$709$	−34.3335	−1.28942	−0.644710	−	0.764427i	$-0.723022\pi$
$709$	−34.3335	−1.28942	−0.644710	+	0.764427i	$0.723022\pi$
$710$	1.03056	0.0386763
$711$	0	0
$712$	−0.190662	−0.00714535
$713$	−38.4707	−1.44074
$714$	0	0
$715$	2.10884	0.0788663
$716$	−23.7363	−0.887066
$717$	0	0
$718$	0.144837	0.00540528
$719$	−33.4812	−1.24864	−0.624320	−	0.781169i	$-0.714624\pi$
$719$	−33.4812	−1.24864	−0.624320	+	0.781169i	$0.714624\pi$
$720$	0	0
$721$	0.431528	0.0160710
$722$	−0.0992940	−0.00369534
$723$	0	0
$724$	−7.54242	−0.280312
$725$	−8.07721	−0.299980
$726$	0	0
$727$	2.94701	0.109299	0.0546493	−	0.998506i	$-0.482596\pi$
$727$	2.94701	0.109299	0.0546493	+	0.998506i	$0.482596\pi$
$728$	−0.153251	−0.00567985
$729$	0	0
$730$	−0.0187263	−0.000693090 0
$731$	3.34098	0.123571
$732$	0	0
$733$	10.0207	0.370123	0.185062	−	0.982727i	$-0.440752\pi$
$733$	10.0207	0.370123	0.185062	+	0.982727i	$0.440752\pi$
$734$	2.40715	0.0888496
$735$	0	0
$736$	−5.88198	−0.216813
$737$	3.41863	0.125927
$738$	0	0
$739$	−7.19928	−0.264830	−0.132415	−	0.991194i	$-0.542273\pi$
$739$	−7.19928	−0.264830	−0.132415	+	0.991194i	$0.542273\pi$
$740$	17.3252	0.636886
$741$	0	0
$742$	0.0764726	0.00280740
$743$	28.0229	1.02806	0.514031	−	0.857772i	$-0.328152\pi$
$743$	28.0229	1.02806	0.514031	+	0.857772i	$0.328152\pi$
$744$	0	0
$745$	3.29575	0.120747
$746$	1.36709	0.0500528
$747$	0	0
$748$	−0.609609	−0.0222895
$749$	−1.30285	−0.0476051
$750$	0	0
$751$	9.86247	0.359887	0.179943	−	0.983677i	$-0.442409\pi$
$751$	9.86247	0.359887	0.179943	+	0.983677i	$0.442409\pi$
$752$	−51.5370	−1.87936
$753$	0	0
$754$	1.69133	0.0615947
$755$	−8.70356	−0.316755
$756$	0	0
$757$	32.4114	1.17801	0.589006	−	0.808128i	$-0.299519\pi$
$757$	32.4114	1.17801	0.589006	+	0.808128i	$0.299519\pi$
$758$	2.81316	0.102179
$759$	0	0
$760$	−0.396197	−0.0143716
$761$	44.3520	1.60776	0.803879	−	0.594793i	$-0.202766\pi$
$761$	44.3520	1.60776	0.803879	+	0.594793i	$0.202766\pi$
$762$	0	0
$763$	1.96916	0.0712883
$764$	50.2609	1.81838
$765$	0	0
$766$	−0.701835	−0.0253583
$767$	−9.84457	−0.355467
$768$	0	0
$769$	−19.6302	−0.707881	−0.353941	−	0.935268i	$-0.615159\pi$
$769$	−19.6302	−0.707881	−0.353941	+	0.935268i	$0.615159\pi$
$770$	0.0182125	0.000656333 0
$771$	0	0
$772$	42.1122	1.51565
$773$	−3.82171	−0.137457	−0.0687287	−	0.997635i	$-0.521894\pi$
$773$	−3.82171	−0.137457	−0.0687287	+	0.997635i	$0.521894\pi$
$774$	0	0
$775$	−7.74196	−0.278099
$776$	1.64837	0.0591729
$777$	0	0
$778$	−0.306614	−0.0109926
$779$	−1.10346	−0.0395354
$780$	0	0
$781$	−10.3789	−0.371385
$782$	−0.151137	−0.00540464
$783$	0	0
$784$	−27.4540	−0.980500
$785$	10.0628	0.359156
$786$	0	0
$787$	11.0430	0.393640	0.196820	−	0.980440i	$-0.436939\pi$
$787$	11.0430	0.393640	0.196820	+	0.980440i	$0.436939\pi$
$788$	−20.2157	−0.720156
$789$	0	0
$790$	−0.346674	−0.0123341
$791$	2.71753	0.0966243
$792$	0	0
$793$	10.8607	0.385676
$794$	−2.85798	−0.101426
$795$	0	0
$796$	−5.49492	−0.194762
$797$	19.8719	0.703898	0.351949	−	0.936019i	$-0.385519\pi$
$797$	19.8719	0.703898	0.351949	+	0.936019i	$0.385519\pi$
$798$	0	0
$799$	−4.00578	−0.141714
$800$	−1.18371	−0.0418503
$801$	0	0
$802$	1.64865	0.0582157
$803$	0.188594	0.00665534
$804$	0	0
$805$	−0.911436	−0.0321239
$806$	1.62113	0.0571019
$807$	0	0
$808$	2.16965	0.0763280
$809$	−23.3416	−0.820648	−0.410324	−	0.911940i	$-0.634584\pi$
$809$	−23.3416	−0.820648	−0.410324	+	0.911940i	$0.634584\pi$
$810$	0	0
$811$	−31.4447	−1.10417	−0.552087	−	0.833787i	$-0.686168\pi$
$811$	−31.4447	−1.10417	−0.552087	+	0.833787i	$0.686168\pi$
$812$	−2.94844	−0.103470
$813$	0	0
$814$	0.864405	0.0302974
$815$	6.06126	0.212317
$816$	0	0
$817$	−10.9070	−0.381588
$818$	2.60609	0.0911198
$819$	0	0
$820$	−2.19603	−0.0766888
$821$	−42.1611	−1.47143	−0.735715	−	0.677291i	$-0.763154\pi$
$821$	−42.1611	−1.47143	−0.735715	+	0.677291i	$0.763154\pi$
$822$	0	0
$823$	−8.42174	−0.293563	−0.146782	−	0.989169i	$-0.546891\pi$
$823$	−8.42174	−0.293563	−0.146782	+	0.989169i	$0.546891\pi$
$824$	−0.932124	−0.0324721
$825$	0	0
$826$	−0.0850202	−0.00295823
$827$	1.56943	0.0545743	0.0272871	−	0.999628i	$-0.491313\pi$
$827$	1.56943	0.0545743	0.0272871	+	0.999628i	$0.491313\pi$
$828$	0	0
$829$	33.2980	1.15649	0.578244	−	0.815864i	$-0.303738\pi$
$829$	33.2980	1.15649	0.578244	+	0.815864i	$0.303738\pi$
$830$	−1.58606	−0.0550529
$831$	0	0
$832$	−16.3738	−0.567660
$833$	−2.13390	−0.0739351
$834$	0	0
$835$	−3.47793	−0.120359
$836$	1.99014	0.0688305
$837$	0	0
$838$	−3.45329	−0.119292
$839$	13.5476	0.467715	0.233858	−	0.972271i	$-0.424865\pi$
$839$	13.5476	0.467715	0.233858	+	0.972271i	$0.424865\pi$
$840$	0	0
$841$	36.2413	1.24970
$842$	−0.294068	−0.0101342
$843$	0	0
$844$	49.5163	1.70442
$845$	−8.55277	−0.294224
$846$	0	0
$847$	−0.183420	−0.00630239
$848$	16.5476	0.568248
$849$	0	0
$850$	−0.0304152	−0.00104323
$851$	−43.2587	−1.48289
$852$	0	0
$853$	−13.6885	−0.468685	−0.234342	−	0.972154i	$-0.575294\pi$
$853$	−13.6885	−0.468685	−0.234342	+	0.972154i	$0.575294\pi$
$854$	0.0937960	0.00320963
$855$	0	0
$856$	2.81423	0.0961883
$857$	−11.5973	−0.396155	−0.198078	−	0.980186i	$-0.563470\pi$
$857$	−11.5973	−0.396155	−0.198078	+	0.980186i	$0.563470\pi$
$858$	0	0
$859$	−53.9029	−1.83914	−0.919572	−	0.392921i	$-0.871465\pi$
$859$	−53.9029	−1.83914	−0.919572	+	0.392921i	$0.871465\pi$
$860$	−21.7065	−0.740185
$861$	0	0
$862$	−3.12358	−0.106390
$863$	−2.62404	−0.0893235	−0.0446617	−	0.999002i	$-0.514221\pi$
$863$	−2.62404	−0.0893235	−0.0446617	+	0.999002i	$0.514221\pi$
$864$	0	0
$865$	−1.56344	−0.0531586
$866$	−1.74891	−0.0594303
$867$	0	0
$868$	−2.82606	−0.0959227
$869$	3.49139	0.118437
$870$	0	0
$871$	7.20935	0.244280
$872$	−4.25349	−0.144041
$873$	0	0
$874$	0.493404	0.0166896
$875$	−0.183420	−0.00620073
$876$	0	0
$877$	28.4606	0.961047	0.480524	−	0.876982i	$-0.340447\pi$
$877$	28.4606	0.961047	0.480524	+	0.876982i	$0.340447\pi$
$878$	−0.820945	−0.0277056
$879$	0	0
$880$	3.94094	0.132849
$881$	−21.9511	−0.739550	−0.369775	−	0.929121i	$-0.620565\pi$
$881$	−21.9511	−0.739550	−0.369775	+	0.929121i	$0.620565\pi$
$882$	0	0
$883$	25.7551	0.866728	0.433364	−	0.901219i	$-0.357327\pi$
$883$	25.7551	0.866728	0.433364	+	0.901219i	$0.357327\pi$
$884$	−1.28557	−0.0432384
$885$	0	0
$886$	−3.24341	−0.108965
$887$	−6.45676	−0.216797	−0.108398	−	0.994108i	$-0.534572\pi$
$887$	−6.45676	−0.216797	−0.108398	+	0.994108i	$0.534572\pi$
$888$	0	0
$889$	−0.886475	−0.0297314
$890$	0.0477832	0.00160170
$891$	0	0
$892$	55.7744	1.86747
$893$	13.0773	0.437616
$894$	0	0
$895$	11.9269	0.398673
$896$	−0.575639	−0.0192308
$897$	0	0
$898$	2.38386	0.0795504
$899$	62.5334	2.08561
$900$	0	0
$901$	1.28618	0.0428490
$902$	−0.109567	−0.00364817
$903$	0	0
$904$	−5.87001	−0.195234
$905$	3.78989	0.125980
$906$	0	0
$907$	44.3080	1.47122	0.735612	−	0.677403i	$-0.236895\pi$
$907$	44.3080	1.47122	0.735612	+	0.677403i	$0.236895\pi$
$908$	30.9291	1.02642
$909$	0	0
$910$	0.0384074	0.00127319
$911$	28.3311	0.938652	0.469326	−	0.883025i	$-0.344497\pi$
$911$	28.3311	0.938652	0.469326	+	0.883025i	$0.344497\pi$
$912$	0	0
$913$	15.9734	0.528641
$914$	0.104737	0.00346440
$915$	0	0
$916$	−26.9964	−0.891987
$917$	0.555036	0.0183289
$918$	0	0
$919$	−55.1852	−1.82039	−0.910196	−	0.414178i	$-0.864069\pi$
$919$	−55.1852	−1.82039	−0.910196	+	0.414178i	$0.864069\pi$
$920$	1.96875	0.0649078
$921$	0	0
$922$	0.618603	0.0203726
$923$	−21.8874	−0.720434
$924$	0	0
$925$	−8.70551	−0.286235
$926$	0.693922	0.0228037
$927$	0	0
$928$	9.56105	0.313857
$929$	−3.94530	−0.129441	−0.0647205	−	0.997903i	$-0.520616\pi$
$929$	−3.94530	−0.129441	−0.0647205	+	0.997903i	$0.520616\pi$
$930$	0	0
$931$	6.96636	0.228313
$932$	26.9091	0.881437
$933$	0	0
$934$	3.24777	0.106270
$935$	0.306315	0.0100176
$936$	0	0
$937$	−38.8962	−1.27069	−0.635343	−	0.772230i	$-0.719141\pi$
$937$	−38.8962	−1.27069	−0.635343	+	0.772230i	$0.719141\pi$
$938$	0.0622618	0.00203292
$939$	0	0
$940$	26.0257	0.848866
$941$	42.8565	1.39708	0.698541	−	0.715570i	$-0.253833\pi$
$941$	42.8565	1.39708	0.698541	+	0.715570i	$0.253833\pi$
$942$	0	0
$943$	5.48321	0.178558
$944$	−18.3972	−0.598778
$945$	0	0
$946$	−1.08300	−0.0352114
$947$	23.2050	0.754063	0.377031	−	0.926200i	$-0.376945\pi$
$947$	23.2050	0.754063	0.377031	+	0.926200i	$0.376945\pi$
$948$	0	0
$949$	0.397715	0.0129104
$950$	0.0992940	0.00322152
$951$	0	0
$952$	−0.0222600	−0.000721452 0
$953$	−1.67327	−0.0542024	−0.0271012	−	0.999633i	$-0.508628\pi$
$953$	−1.67327	−0.0542024	−0.0271012	+	0.999633i	$0.508628\pi$
$954$	0	0
$955$	−25.2549	−0.817231
$956$	−45.1403	−1.45994
$957$	0	0
$958$	−0.405929	−0.0131150
$959$	0.0919332	0.00296868
$960$	0	0
$961$	28.9379	0.933480
$962$	1.82290	0.0587725
$963$	0	0
$964$	29.5612	0.952102
$965$	−21.1604	−0.681178
$966$	0	0
$967$	−8.64000	−0.277844	−0.138922	−	0.990303i	$-0.544364\pi$
$967$	−8.64000	−0.277844	−0.138922	+	0.990303i	$0.544364\pi$
$968$	0.396197	0.0127343
$969$	0	0
$970$	−0.413110	−0.0132642
$971$	15.1821	0.487217	0.243609	−	0.969874i	$-0.421669\pi$
$971$	15.1821	0.487217	0.243609	+	0.969874i	$0.421669\pi$
$972$	0	0
$973$	−2.69537	−0.0864096
$974$	3.66682	0.117492
$975$	0	0
$976$	20.2962	0.649665
$977$	−11.0878	−0.354731	−0.177366	−	0.984145i	$-0.556758\pi$
$977$	−11.0878	−0.354731	−0.177366	+	0.984145i	$0.556758\pi$
$978$	0	0
$979$	−0.481229	−0.0153802
$980$	13.8640	0.442870
$981$	0	0
$982$	−1.84401	−0.0588449
$983$	3.17688	0.101327	0.0506634	−	0.998716i	$-0.483866\pi$
$983$	3.17688	0.101327	0.0506634	+	0.998716i	$0.483866\pi$
$984$	0	0
$985$	10.1579	0.323659
$986$	0.245670	0.00782373
$987$	0	0
$988$	4.19690	0.133521
$989$	54.1983	1.72341
$990$	0	0
$991$	−27.6571	−0.878556	−0.439278	−	0.898351i	$-0.644766\pi$
$991$	−27.6571	−0.878556	−0.439278	+	0.898351i	$0.644766\pi$
$992$	9.16420	0.290964
$993$	0	0
$994$	−0.189025	−0.00599552
$995$	2.76107	0.0875319
$996$	0	0
$997$	−12.1174	−0.383762	−0.191881	−	0.981418i	$-0.561459\pi$
$997$	−12.1174	−0.383762	−0.191881	+	0.981418i	$0.561459\pi$
$998$	−3.04981	−0.0965401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bq.1.7	✓	14
3.2	odd	2		9405.2.a.br.1.8	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9405.2.a.bq.1.7	✓	14	1.1	even	1	trivial
9405.2.a.br.1.8	yes	14	3.2	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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0992940 q^{2} -1.99014 q^{4} +1.00000 q^{5} -0.183420 q^{7} +0.396197 q^{8} +O(q^{10})\)	\(q-0.0992940 q^{2} -1.99014 q^{4} +1.00000 q^{5} -0.183420 q^{7} +0.396197 q^{8} -0.0992940 q^{10} +1.00000 q^{11} +2.10884 q^{13} +0.0182125 q^{14} +3.94094 q^{16} +0.306315 q^{17} -1.00000 q^{19} -1.99014 q^{20} -0.0992940 q^{22} +4.96912 q^{23} +1.00000 q^{25} -0.209396 q^{26} +0.365032 q^{28} -8.07721 q^{29} -7.74196 q^{31} -1.18371 q^{32} -0.0304152 q^{34} -0.183420 q^{35} -8.70551 q^{37} +0.0992940 q^{38} +0.396197 q^{40} +1.10346 q^{41} +10.9070 q^{43} -1.99014 q^{44} -0.493404 q^{46} -13.0773 q^{47} -6.96636 q^{49} -0.0992940 q^{50} -4.19690 q^{52} +4.19890 q^{53} +1.00000 q^{55} -0.0726705 q^{56} +0.802019 q^{58} -4.66823 q^{59} +5.15009 q^{61} +0.768730 q^{62} -7.76435 q^{64} +2.10884 q^{65} +3.41863 q^{67} -0.609609 q^{68} +0.0182125 q^{70} -10.3789 q^{71} +0.188594 q^{73} +0.864405 q^{74} +1.99014 q^{76} -0.183420 q^{77} +3.49139 q^{79} +3.94094 q^{80} -0.109567 q^{82} +15.9734 q^{83} +0.306315 q^{85} -1.08300 q^{86} +0.396197 q^{88} -0.481229 q^{89} -0.386804 q^{91} -9.88925 q^{92} +1.29850 q^{94} -1.00000 q^{95} +4.16047 q^{97} +0.691718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

Introduction

L-functions

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