LMFDB - Embedded newform 7448.2.a.x.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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +O(q^{10})$	$q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} -3.24264 q^{11} +3.41421 q^{13} -3.41421 q^{15} +6.82843 q^{17} +1.00000 q^{19} +4.41421 q^{23} -4.00000 q^{25} -19.3137 q^{27} -9.41421 q^{29} -4.58579 q^{31} +11.0711 q^{33} -6.58579 q^{37} -11.6569 q^{39} +5.65685 q^{41} -3.24264 q^{43} +8.65685 q^{45} +1.24264 q^{47} -23.3137 q^{51} -5.41421 q^{53} -3.24264 q^{55} -3.41421 q^{57} +0.828427 q^{59} +5.00000 q^{61} +3.41421 q^{65} +3.17157 q^{67} -15.0711 q^{69} -7.75736 q^{71} +7.00000 q^{73} +13.6569 q^{75} +3.89949 q^{79} +39.9706 q^{81} -14.8995 q^{83} +6.82843 q^{85} +32.1421 q^{87} -13.5563 q^{89} +15.6569 q^{93} +1.00000 q^{95} +5.65685 q^{97} -28.0711 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10}) $ 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9	$ 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 6 * q^23 - 8 * q^25 - 16 * q^27 - 16 * q^29 - 12 * q^31 + 8 * q^33 - 16 * q^37 - 12 * q^39 + 2 * q^43 + 6 * q^45 - 6 * q^47 - 24 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 - 4 * q^59 + 10 * q^61 + 4 * q^65 + 12 * q^67 - 16 * q^69 - 24 * q^71 + 14 * q^73 + 16 * q^75 - 12 * q^79 + 46 * q^81 - 10 * q^83 + 8 * q^85 + 36 * q^87 + 4 * q^89 + 20 * q^93 + 2 * q^95 - 42 * 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
$3$	−3.41421	−1.97120	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−3.41421	−1.97120	−0.985599	+	0.169102i	$0.945913\pi$
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.65685	2.88562
$10$	0	0
$11$	−3.24264	−0.977693	−0.488846	−	0.872370i	$-0.662582\pi$
$11$	−3.24264	−0.977693	−0.488846	+	0.872370i	$0.662582\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	−3.41421	−0.881546
$16$	0	0
$17$	6.82843	1.65614	0.828068	−	0.560627i	$-0.189440\pi$
$17$	6.82843	1.65614	0.828068	+	0.560627i	$0.189440\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.41421	0.920427	0.460214	−	0.887808i	$-0.347773\pi$
$23$	4.41421	0.920427	0.460214	+	0.887808i	$0.347773\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−19.3137	−3.71692
$28$	0	0
$29$	−9.41421	−1.74818	−0.874088	−	0.485768i	$-0.838540\pi$
$29$	−9.41421	−1.74818	−0.874088	+	0.485768i	$0.838540\pi$
$30$	0	0
$31$	−4.58579	−0.823632	−0.411816	−	0.911267i	$-0.635105\pi$
$31$	−4.58579	−0.823632	−0.411816	+	0.911267i	$0.635105\pi$
$32$	0	0
$33$	11.0711	1.92723
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.58579	−1.08270	−0.541348	−	0.840798i	$-0.682086\pi$
$37$	−6.58579	−1.08270	−0.541348	+	0.840798i	$0.682086\pi$
$38$	0	0
$39$	−11.6569	−1.86659
$40$	0	0
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	−3.24264	−0.494498	−0.247249	−	0.968952i	$-0.579527\pi$
$43$	−3.24264	−0.494498	−0.247249	+	0.968952i	$0.579527\pi$
$44$	0	0
$45$	8.65685	1.29049
$46$	0	0
$47$	1.24264	0.181258	0.0906289	−	0.995885i	$-0.471112\pi$
$47$	1.24264	0.181258	0.0906289	+	0.995885i	$0.471112\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−23.3137	−3.26457
$52$	0	0
$53$	−5.41421	−0.743699	−0.371850	−	0.928293i	$-0.621276\pi$
$53$	−5.41421	−0.743699	−0.371850	+	0.928293i	$0.621276\pi$
$54$	0	0
$55$	−3.24264	−0.437238
$56$	0	0
$57$	−3.41421	−0.452224
$58$	0	0
$59$	0.828427	0.107852	0.0539260	−	0.998545i	$-0.482826\pi$
$59$	0.828427	0.107852	0.0539260	+	0.998545i	$0.482826\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	3.17157	0.387469	0.193735	−	0.981054i	$-0.437940\pi$
$67$	3.17157	0.387469	0.193735	+	0.981054i	$0.437940\pi$
$68$	0	0
$69$	−15.0711	−1.81434
$70$	0	0
$71$	−7.75736	−0.920629	−0.460315	−	0.887756i	$-0.652263\pi$
$71$	−7.75736	−0.920629	−0.460315	+	0.887756i	$0.652263\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	13.6569	1.57696
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.89949	0.438727	0.219364	−	0.975643i	$-0.429602\pi$
$79$	3.89949	0.438727	0.219364	+	0.975643i	$0.429602\pi$
$80$	0	0
$81$	39.9706	4.44117
$82$	0	0
$83$	−14.8995	−1.63543	−0.817716	−	0.575622i	$-0.804760\pi$
$83$	−14.8995	−1.63543	−0.817716	+	0.575622i	$0.804760\pi$
$84$	0	0
$85$	6.82843	0.740647
$86$	0	0
$87$	32.1421	3.44600
$88$	0	0
$89$	−13.5563	−1.43697	−0.718485	−	0.695542i	$-0.755164\pi$
$89$	−13.5563	−1.43697	−0.718485	+	0.695542i	$0.755164\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	15.6569	1.62354
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	5.65685	0.574367	0.287183	−	0.957876i	$-0.407281\pi$
$97$	5.65685	0.574367	0.287183	+	0.957876i	$0.407281\pi$
$98$	0	0
$99$	−28.0711	−2.82125
$100$	0	0
$101$	−1.48528	−0.147791	−0.0738955	−	0.997266i	$-0.523543\pi$
$101$	−1.48528	−0.147791	−0.0738955	+	0.997266i	$0.523543\pi$
$102$	0	0
$103$	−2.34315	−0.230877	−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315	−0.230877	−0.115439	+	0.993315i	$0.536827\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.0711	1.65032	0.825161	−	0.564897i	$-0.191084\pi$
$107$	17.0711	1.65032	0.825161	+	0.564897i	$0.191084\pi$
$108$	0	0
$109$	−19.3137	−1.84992	−0.924959	−	0.380067i	$-0.875901\pi$
$109$	−19.3137	−1.84992	−0.924959	+	0.380067i	$0.875901\pi$
$110$	0	0
$111$	22.4853	2.13421
$112$	0	0
$113$	11.0711	1.04148	0.520739	−	0.853716i	$-0.325656\pi$
$113$	11.0711	1.04148	0.520739	+	0.853716i	$0.325656\pi$
$114$	0	0
$115$	4.41421	0.411628
$116$	0	0
$117$	29.5563	2.73249
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.485281	−0.0441165
$122$	0	0
$123$	−19.3137	−1.74146
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−19.5563	−1.73535	−0.867673	−	0.497136i	$-0.834385\pi$
$127$	−19.5563	−1.73535	−0.867673	+	0.497136i	$0.834385\pi$
$128$	0	0
$129$	11.0711	0.974753
$130$	0	0
$131$	−1.65685	−0.144760	−0.0723800	−	0.997377i	$-0.523059\pi$
$131$	−1.65685	−0.144760	−0.0723800	+	0.997377i	$0.523059\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−19.3137	−1.66226
$136$	0	0
$137$	0.656854	0.0561188	0.0280594	−	0.999606i	$-0.491067\pi$
$137$	0.656854	0.0561188	0.0280594	+	0.999606i	$0.491067\pi$
$138$	0	0
$139$	18.5563	1.57393	0.786964	−	0.616998i	$-0.211651\pi$
$139$	18.5563	1.57393	0.786964	+	0.616998i	$0.211651\pi$
$140$	0	0
$141$	−4.24264	−0.357295
$142$	0	0
$143$	−11.0711	−0.925809
$144$	0	0
$145$	−9.41421	−0.781808
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.65685	−0.545351	−0.272675	−	0.962106i	$-0.587908\pi$
$149$	−6.65685	−0.545351	−0.272675	+	0.962106i	$0.587908\pi$
$150$	0	0
$151$	5.65685	0.460348	0.230174	−	0.973149i	$-0.426070\pi$
$151$	5.65685	0.460348	0.230174	+	0.973149i	$0.426070\pi$
$152$	0	0
$153$	59.1127	4.77898
$154$	0	0
$155$	−4.58579	−0.368339
$156$	0	0
$157$	1.68629	0.134581	0.0672904	−	0.997733i	$-0.478565\pi$
$157$	1.68629	0.134581	0.0672904	+	0.997733i	$0.478565\pi$
$158$	0	0
$159$	18.4853	1.46598
$160$	0	0
$161$	0	0
$162$	0	0
$163$	14.0711	1.10213	0.551066	−	0.834462i	$-0.314221\pi$
$163$	14.0711	1.10213	0.551066	+	0.834462i	$0.314221\pi$
$164$	0	0
$165$	11.0711	0.861881
$166$	0	0
$167$	−18.5858	−1.43821	−0.719106	−	0.694901i	$-0.755448\pi$
$167$	−18.5858	−1.43821	−0.719106	+	0.694901i	$0.755448\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	8.65685	0.662006
$172$	0	0
$173$	9.65685	0.734197	0.367099	−	0.930182i	$-0.380351\pi$
$173$	9.65685	0.734197	0.367099	+	0.930182i	$0.380351\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.82843	−0.212598
$178$	0	0
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	0	0
$181$	14.7279	1.09472	0.547359	−	0.836898i	$-0.315633\pi$
$181$	14.7279	1.09472	0.547359	+	0.836898i	$0.315633\pi$
$182$	0	0
$183$	−17.0711	−1.26193
$184$	0	0
$185$	−6.58579	−0.484197
$186$	0	0
$187$	−22.1421	−1.61919
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.24264	0.668774	0.334387	−	0.942436i	$-0.391471\pi$
$191$	9.24264	0.668774	0.334387	+	0.942436i	$0.391471\pi$
$192$	0	0
$193$	2.24264	0.161429	0.0807144	−	0.996737i	$-0.474280\pi$
$193$	2.24264	0.161429	0.0807144	+	0.996737i	$0.474280\pi$
$194$	0	0
$195$	−11.6569	−0.834765
$196$	0	0
$197$	−0.514719	−0.0366722	−0.0183361	−	0.999832i	$-0.505837\pi$
$197$	−0.514719	−0.0366722	−0.0183361	+	0.999832i	$0.505837\pi$
$198$	0	0
$199$	−2.07107	−0.146814	−0.0734071	−	0.997302i	$-0.523387\pi$
$199$	−2.07107	−0.146814	−0.0734071	+	0.997302i	$0.523387\pi$
$200$	0	0
$201$	−10.8284	−0.763778
$202$	0	0
$203$	0	0
$204$	0	0
$205$	5.65685	0.395092
$206$	0	0
$207$	38.2132	2.65600
$208$	0	0
$209$	−3.24264	−0.224298
$210$	0	0
$211$	12.1421	0.835899	0.417950	−	0.908470i	$-0.362749\pi$
$211$	12.1421	0.835899	0.417950	+	0.908470i	$0.362749\pi$
$212$	0	0
$213$	26.4853	1.81474
$214$	0	0
$215$	−3.24264	−0.221146
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−23.8995	−1.61498
$220$	0	0
$221$	23.3137	1.56825
$222$	0	0
$223$	16.2426	1.08769	0.543844	−	0.839186i	$-0.316968\pi$
$223$	16.2426	1.08769	0.543844	+	0.839186i	$0.316968\pi$
$224$	0	0
$225$	−34.6274	−2.30849
$226$	0	0
$227$	−1.41421	−0.0938647	−0.0469323	−	0.998898i	$-0.514945\pi$
$227$	−1.41421	−0.0938647	−0.0469323	+	0.998898i	$0.514945\pi$
$228$	0	0
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.34315	−0.415553	−0.207777	−	0.978176i	$-0.566623\pi$
$233$	−6.34315	−0.415553	−0.207777	+	0.978176i	$0.566623\pi$
$234$	0	0
$235$	1.24264	0.0810609
$236$	0	0
$237$	−13.3137	−0.864818
$238$	0	0
$239$	−4.34315	−0.280935	−0.140467	−	0.990085i	$-0.544861\pi$
$239$	−4.34315	−0.280935	−0.140467	+	0.990085i	$0.544861\pi$
$240$	0	0
$241$	−8.72792	−0.562215	−0.281107	−	0.959676i	$-0.590702\pi$
$241$	−8.72792	−0.562215	−0.281107	+	0.959676i	$0.590702\pi$
$242$	0	0
$243$	−78.5269	−5.03750
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.41421	0.217241
$248$	0	0
$249$	50.8701	3.22376
$250$	0	0
$251$	22.2132	1.40208	0.701042	−	0.713120i	$-0.252718\pi$
$251$	22.2132	1.40208	0.701042	+	0.713120i	$0.252718\pi$
$252$	0	0
$253$	−14.3137	−0.899895
$254$	0	0
$255$	−23.3137	−1.45996
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−81.4975	−5.04457
$262$	0	0
$263$	−21.1716	−1.30550	−0.652748	−	0.757575i	$-0.726384\pi$
$263$	−21.1716	−1.30550	−0.652748	+	0.757575i	$0.726384\pi$
$264$	0	0
$265$	−5.41421	−0.332592
$266$	0	0
$267$	46.2843	2.83255
$268$	0	0
$269$	16.0000	0.975537	0.487769	−	0.872973i	$-0.337811\pi$
$269$	16.0000	0.975537	0.487769	+	0.872973i	$0.337811\pi$
$270$	0	0
$271$	−2.27208	−0.138019	−0.0690095	−	0.997616i	$-0.521984\pi$
$271$	−2.27208	−0.138019	−0.0690095	+	0.997616i	$0.521984\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.9706	0.782154
$276$	0	0
$277$	−27.9706	−1.68059	−0.840294	−	0.542131i	$-0.817618\pi$
$277$	−27.9706	−1.68059	−0.840294	+	0.542131i	$0.817618\pi$
$278$	0	0
$279$	−39.6985	−2.37669
$280$	0	0
$281$	−17.8995	−1.06779	−0.533897	−	0.845549i	$-0.679273\pi$
$281$	−17.8995	−1.06779	−0.533897	+	0.845549i	$0.679273\pi$
$282$	0	0
$283$	−14.5563	−0.865285	−0.432643	−	0.901566i	$-0.642419\pi$
$283$	−14.5563	−0.865285	−0.432643	+	0.901566i	$0.642419\pi$
$284$	0	0
$285$	−3.41421	−0.202241
$286$	0	0
$287$	0	0
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	−19.3137	−1.13219
$292$	0	0
$293$	−9.65685	−0.564159	−0.282080	−	0.959391i	$-0.591024\pi$
$293$	−9.65685	−0.564159	−0.282080	+	0.959391i	$0.591024\pi$
$294$	0	0
$295$	0.828427	0.0482329
$296$	0	0
$297$	62.6274	3.63401
$298$	0	0
$299$	15.0711	0.871582
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.07107	0.291325
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	−18.6274	−1.06312	−0.531561	−	0.847020i	$-0.678395\pi$
$307$	−18.6274	−1.06312	−0.531561	+	0.847020i	$0.678395\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−1.14214	−0.0645573	−0.0322787	−	0.999479i	$-0.510276\pi$
$313$	−1.14214	−0.0645573	−0.0322787	+	0.999479i	$0.510276\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.2426	−0.575284	−0.287642	−	0.957738i	$-0.592871\pi$
$317$	−10.2426	−0.575284	−0.287642	+	0.957738i	$0.592871\pi$
$318$	0	0
$319$	30.5269	1.70918
$320$	0	0
$321$	−58.2843	−3.25311
$322$	0	0
$323$	6.82843	0.379944
$324$	0	0
$325$	−13.6569	−0.757546
$326$	0	0
$327$	65.9411	3.64655
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.4853	−1.01604	−0.508021	−	0.861344i	$-0.669623\pi$
$331$	−18.4853	−1.01604	−0.508021	+	0.861344i	$0.669623\pi$
$332$	0	0
$333$	−57.0122	−3.12425
$334$	0	0
$335$	3.17157	0.173282
$336$	0	0
$337$	5.07107	0.276239	0.138119	−	0.990416i	$-0.455894\pi$
$337$	5.07107	0.276239	0.138119	+	0.990416i	$0.455894\pi$
$338$	0	0
$339$	−37.7990	−2.05296
$340$	0	0
$341$	14.8701	0.805259
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−15.0711	−0.811399
$346$	0	0
$347$	12.4142	0.666430	0.333215	−	0.942851i	$-0.391867\pi$
$347$	12.4142	0.666430	0.333215	+	0.942851i	$0.391867\pi$
$348$	0	0
$349$	−5.65685	−0.302804	−0.151402	−	0.988472i	$-0.548379\pi$
$349$	−5.65685	−0.302804	−0.151402	+	0.988472i	$0.548379\pi$
$350$	0	0
$351$	−65.9411	−3.51968
$352$	0	0
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−7.75736	−0.411718
$356$	0	0
$357$	0	0
$358$	0	0
$359$	36.6985	1.93687	0.968436	−	0.249262i	$-0.0801882\pi$
$359$	36.6985	1.93687	0.968436	+	0.249262i	$0.0801882\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.65685	0.0869623
$364$	0	0
$365$	7.00000	0.366397
$366$	0	0
$367$	−16.6274	−0.867944	−0.433972	−	0.900926i	$-0.642888\pi$
$367$	−16.6274	−0.867944	−0.433972	+	0.900926i	$0.642888\pi$
$368$	0	0
$369$	48.9706	2.54931
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.34315	−0.328436	−0.164218	−	0.986424i	$-0.552510\pi$
$373$	−6.34315	−0.328436	−0.164218	+	0.986424i	$0.552510\pi$
$374$	0	0
$375$	30.7279	1.58678
$376$	0	0
$377$	−32.1421	−1.65540
$378$	0	0
$379$	12.5858	0.646488	0.323244	−	0.946316i	$-0.395226\pi$
$379$	12.5858	0.646488	0.323244	+	0.946316i	$0.395226\pi$
$380$	0	0
$381$	66.7696	3.42071
$382$	0	0
$383$	3.75736	0.191992	0.0959960	−	0.995382i	$-0.469396\pi$
$383$	3.75736	0.191992	0.0959960	+	0.995382i	$0.469396\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.0711	−1.42693
$388$	0	0
$389$	−15.3137	−0.776436	−0.388218	−	0.921568i	$-0.626909\pi$
$389$	−15.3137	−0.776436	−0.388218	+	0.921568i	$0.626909\pi$
$390$	0	0
$391$	30.1421	1.52435
$392$	0	0
$393$	5.65685	0.285351
$394$	0	0
$395$	3.89949	0.196205
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.62742	−0.430833	−0.215416	−	0.976522i	$-0.569111\pi$
$401$	−8.62742	−0.430833	−0.215416	+	0.976522i	$0.569111\pi$
$402$	0	0
$403$	−15.6569	−0.779923
$404$	0	0
$405$	39.9706	1.98615
$406$	0	0
$407$	21.3553	1.05854
$408$	0	0
$409$	29.3137	1.44947	0.724735	−	0.689028i	$-0.241962\pi$
$409$	29.3137	1.44947	0.724735	+	0.689028i	$0.241962\pi$
$410$	0	0
$411$	−2.24264	−0.110621
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−14.8995	−0.731387
$416$	0	0
$417$	−63.3553	−3.10252
$418$	0	0
$419$	30.6985	1.49972	0.749860	−	0.661597i	$-0.230121\pi$
$419$	30.6985	1.49972	0.749860	+	0.661597i	$0.230121\pi$
$420$	0	0
$421$	−24.5858	−1.19824	−0.599119	−	0.800660i	$-0.704482\pi$
$421$	−24.5858	−1.19824	−0.599119	+	0.800660i	$0.704482\pi$
$422$	0	0
$423$	10.7574	0.523041
$424$	0	0
$425$	−27.3137	−1.32491
$426$	0	0
$427$	0	0
$428$	0	0
$429$	37.7990	1.82495
$430$	0	0
$431$	−13.8995	−0.669515	−0.334758	−	0.942304i	$-0.608654\pi$
$431$	−13.8995	−0.669515	−0.334758	+	0.942304i	$0.608654\pi$
$432$	0	0
$433$	−12.0000	−0.576683	−0.288342	−	0.957528i	$-0.593104\pi$
$433$	−12.0000	−0.576683	−0.288342	+	0.957528i	$0.593104\pi$
$434$	0	0
$435$	32.1421	1.54110
$436$	0	0
$437$	4.41421	0.211160
$438$	0	0
$439$	−26.4853	−1.26407	−0.632037	−	0.774938i	$-0.717781\pi$
$439$	−26.4853	−1.26407	−0.632037	+	0.774938i	$0.717781\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.6863	−0.792790	−0.396395	−	0.918080i	$-0.629739\pi$
$443$	−16.6863	−0.792790	−0.396395	+	0.918080i	$0.629739\pi$
$444$	0	0
$445$	−13.5563	−0.642633
$446$	0	0
$447$	22.7279	1.07499
$448$	0	0
$449$	30.6274	1.44540	0.722699	−	0.691163i	$-0.242901\pi$
$449$	30.6274	1.44540	0.722699	+	0.691163i	$0.242901\pi$
$450$	0	0
$451$	−18.3431	−0.863745
$452$	0	0
$453$	−19.3137	−0.907437
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.82843	0.272642	0.136321	−	0.990665i	$-0.456472\pi$
$457$	5.82843	0.272642	0.136321	+	0.990665i	$0.456472\pi$
$458$	0	0
$459$	−131.882	−6.15574
$460$	0	0
$461$	−10.7990	−0.502959	−0.251480	−	0.967863i	$-0.580917\pi$
$461$	−10.7990	−0.502959	−0.251480	+	0.967863i	$0.580917\pi$
$462$	0	0
$463$	−27.7279	−1.28863	−0.644313	−	0.764762i	$-0.722857\pi$
$463$	−27.7279	−1.28863	−0.644313	+	0.764762i	$0.722857\pi$
$464$	0	0
$465$	15.6569	0.726069
$466$	0	0
$467$	7.72792	0.357606	0.178803	−	0.983885i	$-0.442778\pi$
$467$	7.72792	0.357606	0.178803	+	0.983885i	$0.442778\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−5.75736	−0.265285
$472$	0	0
$473$	10.5147	0.483467
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−46.8701	−2.14603
$478$	0	0
$479$	9.92893	0.453664	0.226832	−	0.973934i	$-0.427163\pi$
$479$	9.92893	0.453664	0.226832	+	0.973934i	$0.427163\pi$
$480$	0	0
$481$	−22.4853	−1.02524
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.65685	0.256865
$486$	0	0
$487$	−11.8579	−0.537331	−0.268666	−	0.963234i	$-0.586583\pi$
$487$	−11.8579	−0.537331	−0.268666	+	0.963234i	$0.586583\pi$
$488$	0	0
$489$	−48.0416	−2.17252
$490$	0	0
$491$	−18.6985	−0.843851	−0.421925	−	0.906631i	$-0.638646\pi$
$491$	−18.6985	−0.843851	−0.421925	+	0.906631i	$0.638646\pi$
$492$	0	0
$493$	−64.2843	−2.89522
$494$	0	0
$495$	−28.0711	−1.26170
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.757359	−0.0339041	−0.0169520	−	0.999856i	$-0.505396\pi$
$499$	−0.757359	−0.0339041	−0.0169520	+	0.999856i	$0.505396\pi$
$500$	0	0
$501$	63.4558	2.83500
$502$	0	0
$503$	−21.7279	−0.968800	−0.484400	−	0.874847i	$-0.660962\pi$
$503$	−21.7279	−0.968800	−0.484400	+	0.874847i	$0.660962\pi$
$504$	0	0
$505$	−1.48528	−0.0660942
$506$	0	0
$507$	4.58579	0.203662
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−19.3137	−0.852721
$514$	0	0
$515$	−2.34315	−0.103251
$516$	0	0
$517$	−4.02944	−0.177214
$518$	0	0
$519$	−32.9706	−1.44725
$520$	0	0
$521$	21.8995	0.959434	0.479717	−	0.877423i	$-0.340739\pi$
$521$	21.8995	0.959434	0.479717	+	0.877423i	$0.340739\pi$
$522$	0	0
$523$	−7.31371	−0.319806	−0.159903	−	0.987133i	$-0.551118\pi$
$523$	−7.31371	−0.319806	−0.159903	+	0.987133i	$0.551118\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.3137	−1.36405
$528$	0	0
$529$	−3.51472	−0.152814
$530$	0	0
$531$	7.17157	0.311220
$532$	0	0
$533$	19.3137	0.836570
$534$	0	0
$535$	17.0711	0.738047
$536$	0	0
$537$	−22.1421	−0.955504
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−29.1421	−1.25292	−0.626459	−	0.779454i	$-0.715496\pi$
$541$	−29.1421	−1.25292	−0.626459	+	0.779454i	$0.715496\pi$
$542$	0	0
$543$	−50.2843	−2.15790
$544$	0	0
$545$	−19.3137	−0.827308
$546$	0	0
$547$	−21.5563	−0.921683	−0.460841	−	0.887482i	$-0.652452\pi$
$547$	−21.5563	−0.921683	−0.460841	+	0.887482i	$0.652452\pi$
$548$	0	0
$549$	43.2843	1.84733
$550$	0	0
$551$	−9.41421	−0.401059
$552$	0	0
$553$	0	0
$554$	0	0
$555$	22.4853	0.954447
$556$	0	0
$557$	35.4853	1.50356	0.751780	−	0.659414i	$-0.229196\pi$
$557$	35.4853	1.50356	0.751780	+	0.659414i	$0.229196\pi$
$558$	0	0
$559$	−11.0711	−0.468256
$560$	0	0
$561$	75.5980	3.19175
$562$	0	0
$563$	−42.8701	−1.80676	−0.903379	−	0.428844i	$-0.858921\pi$
$563$	−42.8701	−1.80676	−0.903379	+	0.428844i	$0.858921\pi$
$564$	0	0
$565$	11.0711	0.465763
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.31371	−0.222762	−0.111381	−	0.993778i	$-0.535527\pi$
$569$	−5.31371	−0.222762	−0.111381	+	0.993778i	$0.535527\pi$
$570$	0	0
$571$	−42.6985	−1.78688	−0.893438	−	0.449187i	$-0.851714\pi$
$571$	−42.6985	−1.78688	−0.893438	+	0.449187i	$0.851714\pi$
$572$	0	0
$573$	−31.5563	−1.31829
$574$	0	0
$575$	−17.6569	−0.736342
$576$	0	0
$577$	−28.1716	−1.17280	−0.586399	−	0.810022i	$-0.699455\pi$
$577$	−28.1716	−1.17280	−0.586399	+	0.810022i	$0.699455\pi$
$578$	0	0
$579$	−7.65685	−0.318208
$580$	0	0
$581$	0	0
$582$	0	0
$583$	17.5563	0.727110
$584$	0	0
$585$	29.5563	1.22200
$586$	0	0
$587$	34.0000	1.40333	0.701665	−	0.712507i	$-0.252440\pi$
$587$	34.0000	1.40333	0.701665	+	0.712507i	$0.252440\pi$
$588$	0	0
$589$	−4.58579	−0.188954
$590$	0	0
$591$	1.75736	0.0722881
$592$	0	0
$593$	−4.31371	−0.177143	−0.0885714	−	0.996070i	$-0.528230\pi$
$593$	−4.31371	−0.177143	−0.0885714	+	0.996070i	$0.528230\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	7.07107	0.289400
$598$	0	0
$599$	21.2132	0.866748	0.433374	−	0.901214i	$-0.357323\pi$
$599$	21.2132	0.866748	0.433374	+	0.901214i	$0.357323\pi$
$600$	0	0
$601$	16.5269	0.674147	0.337073	−	0.941478i	$-0.390563\pi$
$601$	16.5269	0.674147	0.337073	+	0.941478i	$0.390563\pi$
$602$	0	0
$603$	27.4558	1.11809
$604$	0	0
$605$	−0.485281	−0.0197295
$606$	0	0
$607$	−15.8995	−0.645341	−0.322670	−	0.946511i	$-0.604581\pi$
$607$	−15.8995	−0.645341	−0.322670	+	0.946511i	$0.604581\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.24264	0.171639
$612$	0	0
$613$	−32.4853	−1.31207	−0.656034	−	0.754731i	$-0.727767\pi$
$613$	−32.4853	−1.31207	−0.656034	+	0.754731i	$0.727767\pi$
$614$	0	0
$615$	−19.3137	−0.778804
$616$	0	0
$617$	27.2843	1.09842	0.549212	−	0.835683i	$-0.314928\pi$
$617$	27.2843	1.09842	0.549212	+	0.835683i	$0.314928\pi$
$618$	0	0
$619$	−9.10051	−0.365780	−0.182890	−	0.983133i	$-0.558545\pi$
$619$	−9.10051	−0.365780	−0.182890	+	0.983133i	$0.558545\pi$
$620$	0	0
$621$	−85.2548	−3.42116
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	11.0711	0.442136
$628$	0	0
$629$	−44.9706	−1.79309
$630$	0	0
$631$	−17.3848	−0.692077	−0.346039	−	0.938220i	$-0.612473\pi$
$631$	−17.3848	−0.692077	−0.346039	+	0.938220i	$0.612473\pi$
$632$	0	0
$633$	−41.4558	−1.64772
$634$	0	0
$635$	−19.5563	−0.776070
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−67.1543	−2.65658
$640$	0	0
$641$	10.6863	0.422083	0.211042	−	0.977477i	$-0.432314\pi$
$641$	10.6863	0.422083	0.211042	+	0.977477i	$0.432314\pi$
$642$	0	0
$643$	−27.5147	−1.08507	−0.542537	−	0.840032i	$-0.682536\pi$
$643$	−27.5147	−1.08507	−0.542537	+	0.840032i	$0.682536\pi$
$644$	0	0
$645$	11.0711	0.435923
$646$	0	0
$647$	−27.9289	−1.09800	−0.549000	−	0.835822i	$-0.684991\pi$
$647$	−27.9289	−1.09800	−0.549000	+	0.835822i	$0.684991\pi$
$648$	0	0
$649$	−2.68629	−0.105446
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.1421	−1.17955	−0.589776	−	0.807567i	$-0.700784\pi$
$653$	−30.1421	−1.17955	−0.589776	+	0.807567i	$0.700784\pi$
$654$	0	0
$655$	−1.65685	−0.0647387
$656$	0	0
$657$	60.5980	2.36415
$658$	0	0
$659$	24.3848	0.949896	0.474948	−	0.880014i	$-0.342467\pi$
$659$	24.3848	0.949896	0.474948	+	0.880014i	$0.342467\pi$
$660$	0	0
$661$	−8.87006	−0.345005	−0.172503	−	0.985009i	$-0.555185\pi$
$661$	−8.87006	−0.345005	−0.172503	+	0.985009i	$0.555185\pi$
$662$	0	0
$663$	−79.5980	−3.09133
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−41.5563	−1.60907
$668$	0	0
$669$	−55.4558	−2.14405
$670$	0	0
$671$	−16.2132	−0.625904
$672$	0	0
$673$	−40.6274	−1.56607	−0.783036	−	0.621976i	$-0.786330\pi$
$673$	−40.6274	−1.56607	−0.783036	+	0.621976i	$0.786330\pi$
$674$	0	0
$675$	77.2548	2.97354
$676$	0	0
$677$	−11.3137	−0.434821	−0.217411	−	0.976080i	$-0.569761\pi$
$677$	−11.3137	−0.434821	−0.217411	+	0.976080i	$0.569761\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.82843	0.185026
$682$	0	0
$683$	−24.1421	−0.923773	−0.461887	−	0.886939i	$-0.652827\pi$
$683$	−24.1421	−0.923773	−0.461887	+	0.886939i	$0.652827\pi$
$684$	0	0
$685$	0.656854	0.0250971
$686$	0	0
$687$	54.6274	2.08417
$688$	0	0
$689$	−18.4853	−0.704233
$690$	0	0
$691$	16.6274	0.632537	0.316268	−	0.948670i	$-0.397570\pi$
$691$	16.6274	0.632537	0.316268	+	0.948670i	$0.397570\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.5563	0.703882
$696$	0	0
$697$	38.6274	1.46312
$698$	0	0
$699$	21.6569	0.819137
$700$	0	0
$701$	−20.7990	−0.785567	−0.392784	−	0.919631i	$-0.628488\pi$
$701$	−20.7990	−0.785567	−0.392784	+	0.919631i	$0.628488\pi$
$702$	0	0
$703$	−6.58579	−0.248388
$704$	0	0
$705$	−4.24264	−0.159787
$706$	0	0
$707$	0	0
$708$	0	0
$709$	52.5980	1.97536	0.987679	−	0.156492i	$-0.0500184\pi$
$709$	52.5980	1.97536	0.987679	+	0.156492i	$0.0500184\pi$
$710$	0	0
$711$	33.7574	1.26600
$712$	0	0
$713$	−20.2426	−0.758093
$714$	0	0
$715$	−11.0711	−0.414034
$716$	0	0
$717$	14.8284	0.553778
$718$	0	0
$719$	−3.31371	−0.123580	−0.0617902	−	0.998089i	$-0.519681\pi$
$719$	−3.31371	−0.123580	−0.0617902	+	0.998089i	$0.519681\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	29.7990	1.10824
$724$	0	0
$725$	37.6569	1.39854
$726$	0	0
$727$	−3.24264	−0.120263	−0.0601314	−	0.998190i	$-0.519152\pi$
$727$	−3.24264	−0.120263	−0.0601314	+	0.998190i	$0.519152\pi$
$728$	0	0
$729$	148.196	5.48874
$730$	0	0
$731$	−22.1421	−0.818956
$732$	0	0
$733$	51.3137	1.89532	0.947658	−	0.319289i	$-0.103444\pi$
$733$	51.3137	1.89532	0.947658	+	0.319289i	$0.103444\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.2843	−0.378826
$738$	0	0
$739$	44.6274	1.64165	0.820823	−	0.571183i	$-0.193515\pi$
$739$	44.6274	1.64165	0.820823	+	0.571183i	$0.193515\pi$
$740$	0	0
$741$	−11.6569	−0.428225
$742$	0	0
$743$	21.2721	0.780397	0.390198	−	0.920731i	$-0.372406\pi$
$743$	21.2721	0.780397	0.390198	+	0.920731i	$0.372406\pi$
$744$	0	0
$745$	−6.65685	−0.243888
$746$	0	0
$747$	−128.983	−4.71923
$748$	0	0
$749$	0	0
$750$	0	0
$751$	5.51472	0.201235	0.100617	−	0.994925i	$-0.467918\pi$
$751$	5.51472	0.201235	0.100617	+	0.994925i	$0.467918\pi$
$752$	0	0
$753$	−75.8406	−2.76379
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	−31.6863	−1.15166	−0.575829	−	0.817570i	$-0.695321\pi$
$757$	−31.6863	−1.15166	−0.575829	+	0.817570i	$0.695321\pi$
$758$	0	0
$759$	48.8701	1.77387
$760$	0	0
$761$	13.1421	0.476402	0.238201	−	0.971216i	$-0.423442\pi$
$761$	13.1421	0.476402	0.238201	+	0.971216i	$0.423442\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	59.1127	2.13722
$766$	0	0
$767$	2.82843	0.102129
$768$	0	0
$769$	3.54416	0.127806	0.0639028	−	0.997956i	$-0.479645\pi$
$769$	3.54416	0.127806	0.0639028	+	0.997956i	$0.479645\pi$
$770$	0	0
$771$	47.7990	1.72144
$772$	0	0
$773$	0.928932	0.0334114	0.0167057	−	0.999860i	$-0.494682\pi$
$773$	0.928932	0.0334114	0.0167057	+	0.999860i	$0.494682\pi$
$774$	0	0
$775$	18.3431	0.658905
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.65685	0.202678
$780$	0	0
$781$	25.1543	0.900093
$782$	0	0
$783$	181.823	6.49784
$784$	0	0
$785$	1.68629	0.0601863
$786$	0	0
$787$	−11.7574	−0.419105	−0.209552	−	0.977797i	$-0.567201\pi$
$787$	−11.7574	−0.419105	−0.209552	+	0.977797i	$0.567201\pi$
$788$	0	0
$789$	72.2843	2.57339
$790$	0	0
$791$	0	0
$792$	0	0
$793$	17.0711	0.606211
$794$	0	0
$795$	18.4853	0.655605
$796$	0	0
$797$	45.4975	1.61160	0.805802	−	0.592186i	$-0.201735\pi$
$797$	45.4975	1.61160	0.805802	+	0.592186i	$0.201735\pi$
$798$	0	0
$799$	8.48528	0.300188
$800$	0	0
$801$	−117.355	−4.14655
$802$	0	0
$803$	−22.6985	−0.801012
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−54.6274	−1.92298
$808$	0	0
$809$	−27.9706	−0.983393	−0.491696	−	0.870767i	$-0.663623\pi$
$809$	−27.9706	−0.983393	−0.491696	+	0.870767i	$0.663623\pi$
$810$	0	0
$811$	−32.4264	−1.13865	−0.569323	−	0.822114i	$-0.692794\pi$
$811$	−32.4264	−1.13865	−0.569323	+	0.822114i	$0.692794\pi$
$812$	0	0
$813$	7.75736	0.272062
$814$	0	0
$815$	14.0711	0.492888
$816$	0	0
$817$	−3.24264	−0.113446
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8.02944	0.280229	0.140115	−	0.990135i	$-0.455253\pi$
$821$	8.02944	0.280229	0.140115	+	0.990135i	$0.455253\pi$
$822$	0	0
$823$	9.72792	0.339094	0.169547	−	0.985522i	$-0.445770\pi$
$823$	9.72792	0.339094	0.169547	+	0.985522i	$0.445770\pi$
$824$	0	0
$825$	−44.2843	−1.54178
$826$	0	0
$827$	−0.686292	−0.0238647	−0.0119323	−	0.999929i	$-0.503798\pi$
$827$	−0.686292	−0.0238647	−0.0119323	+	0.999929i	$0.503798\pi$
$828$	0	0
$829$	−39.3137	−1.36542	−0.682711	−	0.730689i	$-0.739199\pi$
$829$	−39.3137	−1.36542	−0.682711	+	0.730689i	$0.739199\pi$
$830$	0	0
$831$	95.4975	3.31277
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−18.5858	−0.643188
$836$	0	0
$837$	88.5685	3.06138
$838$	0	0
$839$	−30.4853	−1.05247	−0.526234	−	0.850340i	$-0.676397\pi$
$839$	−30.4853	−1.05247	−0.526234	+	0.850340i	$0.676397\pi$
$840$	0	0
$841$	59.6274	2.05612
$842$	0	0
$843$	61.1127	2.10483
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	0	0
$848$	0	0
$849$	49.6985	1.70565
$850$	0	0
$851$	−29.0711	−0.996543
$852$	0	0
$853$	52.4558	1.79605	0.898027	−	0.439940i	$-0.145000\pi$
$853$	52.4558	1.79605	0.898027	+	0.439940i	$0.145000\pi$
$854$	0	0
$855$	8.65685	0.296058
$856$	0	0
$857$	−7.02944	−0.240121	−0.120061	−	0.992767i	$-0.538309\pi$
$857$	−7.02944	−0.240121	−0.120061	+	0.992767i	$0.538309\pi$
$858$	0	0
$859$	27.1838	0.927498	0.463749	−	0.885967i	$-0.346504\pi$
$859$	27.1838	0.927498	0.463749	+	0.885967i	$0.346504\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.8701	−0.846587	−0.423293	−	0.905993i	$-0.639126\pi$
$863$	−24.8701	−0.846587	−0.423293	+	0.905993i	$0.639126\pi$
$864$	0	0
$865$	9.65685	0.328343
$866$	0	0
$867$	−101.154	−3.43538
$868$	0	0
$869$	−12.6447	−0.428941
$870$	0	0
$871$	10.8284	0.366907
$872$	0	0
$873$	48.9706	1.65740
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36.8701	1.24501	0.622507	−	0.782614i	$-0.286114\pi$
$877$	36.8701	1.24501	0.622507	+	0.782614i	$0.286114\pi$
$878$	0	0
$879$	32.9706	1.11207
$880$	0	0
$881$	−7.31371	−0.246405	−0.123203	−	0.992382i	$-0.539316\pi$
$881$	−7.31371	−0.246405	−0.123203	+	0.992382i	$0.539316\pi$
$882$	0	0
$883$	−25.1716	−0.847091	−0.423545	−	0.905875i	$-0.639215\pi$
$883$	−25.1716	−0.847091	−0.423545	+	0.905875i	$0.639215\pi$
$884$	0	0
$885$	−2.82843	−0.0950765
$886$	0	0
$887$	28.2843	0.949693	0.474846	−	0.880069i	$-0.342504\pi$
$887$	28.2843	0.949693	0.474846	+	0.880069i	$0.342504\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−129.610	−4.34210
$892$	0	0
$893$	1.24264	0.0415834
$894$	0	0
$895$	6.48528	0.216779
$896$	0	0
$897$	−51.4558	−1.71806
$898$	0	0
$899$	43.1716	1.43985
$900$	0	0
$901$	−36.9706	−1.23167
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14.7279	0.489573
$906$	0	0
$907$	−49.5563	−1.64549	−0.822746	−	0.568410i	$-0.807559\pi$
$907$	−49.5563	−1.64549	−0.822746	+	0.568410i	$0.807559\pi$
$908$	0	0
$909$	−12.8579	−0.426468
$910$	0	0
$911$	−25.5563	−0.846720	−0.423360	−	0.905962i	$-0.639149\pi$
$911$	−25.5563	−0.846720	−0.423360	+	0.905962i	$0.639149\pi$
$912$	0	0
$913$	48.3137	1.59895
$914$	0	0
$915$	−17.0711	−0.564352
$916$	0	0
$917$	0	0
$918$	0	0
$919$	5.58579	0.184258	0.0921290	−	0.995747i	$-0.470633\pi$
$919$	5.58579	0.184258	0.0921290	+	0.995747i	$0.470633\pi$
$920$	0	0
$921$	63.5980	2.09562
$922$	0	0
$923$	−26.4853	−0.871774
$924$	0	0
$925$	26.3431	0.866157
$926$	0	0
$927$	−20.2843	−0.666223
$928$	0	0
$929$	35.6274	1.16890	0.584449	−	0.811431i	$-0.301311\pi$
$929$	35.6274	1.16890	0.584449	+	0.811431i	$0.301311\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	27.3137	0.894211
$934$	0	0
$935$	−22.1421	−0.724125
$936$	0	0
$937$	−30.5147	−0.996872	−0.498436	−	0.866926i	$-0.666092\pi$
$937$	−30.5147	−0.996872	−0.498436	+	0.866926i	$0.666092\pi$
$938$	0	0
$939$	3.89949	0.127255
$940$	0	0
$941$	−27.4142	−0.893678	−0.446839	−	0.894614i	$-0.647450\pi$
$941$	−27.4142	−0.893678	−0.446839	+	0.894614i	$0.647450\pi$
$942$	0	0
$943$	24.9706	0.813153
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.0000	−0.844886	−0.422443	−	0.906389i	$-0.638827\pi$
$947$	−26.0000	−0.844886	−0.422443	+	0.906389i	$0.638827\pi$
$948$	0	0
$949$	23.8995	0.775810
$950$	0	0
$951$	34.9706	1.13400
$952$	0	0
$953$	−59.1543	−1.91620	−0.958098	−	0.286439i	$-0.907528\pi$
$953$	−59.1543	−1.91620	−0.958098	+	0.286439i	$0.907528\pi$
$954$	0	0
$955$	9.24264	0.299085
$956$	0	0
$957$	−104.225	−3.36913
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.97056	−0.321631
$962$	0	0
$963$	147.782	4.76220
$964$	0	0
$965$	2.24264	0.0721932
$966$	0	0
$967$	16.9706	0.545737	0.272868	−	0.962051i	$-0.412028\pi$
$967$	16.9706	0.545737	0.272868	+	0.962051i	$0.412028\pi$
$968$	0	0
$969$	−23.3137	−0.748944
$970$	0	0
$971$	−48.8701	−1.56831	−0.784157	−	0.620562i	$-0.786905\pi$
$971$	−48.8701	−1.56831	−0.784157	+	0.620562i	$0.786905\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	46.6274	1.49327
$976$	0	0
$977$	17.3137	0.553915	0.276957	−	0.960882i	$-0.410674\pi$
$977$	17.3137	0.553915	0.276957	+	0.960882i	$0.410674\pi$
$978$	0	0
$979$	43.9584	1.40492
$980$	0	0
$981$	−167.196	−5.33816
$982$	0	0
$983$	49.9411	1.59287	0.796437	−	0.604721i	$-0.206715\pi$
$983$	49.9411	1.59287	0.796437	+	0.604721i	$0.206715\pi$
$984$	0	0
$985$	−0.514719	−0.0164003
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−14.3137	−0.455149
$990$	0	0
$991$	−43.4558	−1.38042	−0.690210	−	0.723609i	$-0.742482\pi$
$991$	−43.4558	−1.38042	−0.690210	+	0.723609i	$0.742482\pi$
$992$	0	0
$993$	63.1127	2.00282
$994$	0	0
$995$	−2.07107	−0.0656573
$996$	0	0
$997$	−19.5980	−0.620674	−0.310337	−	0.950627i	$-0.600442\pi$
$997$	−19.5980	−0.620674	−0.310337	+	0.950627i	$0.600442\pi$
$998$	0	0
$999$	127.196	4.02430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.x.1.1		2
7.3	odd	6		1064.2.q.k.457.1	yes	4
7.5	odd	6		1064.2.q.k.305.1	✓	4
7.6	odd	2		7448.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.q.k.305.1	✓	4	7.5	odd	6
1064.2.q.k.457.1	yes	4	7.3	odd	6
7448.2.a.x.1.1		2	1.1	even	1	trivial
7448.2.a.bd.1.2		2	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)

\(f(q)\)	\(=\)	\(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} +O(q^{10})\)	\(q-3.41421 q^{3} +1.00000 q^{5} +8.65685 q^{9} -3.24264 q^{11} +3.41421 q^{13} -3.41421 q^{15} +6.82843 q^{17} +1.00000 q^{19} +4.41421 q^{23} -4.00000 q^{25} -19.3137 q^{27} -9.41421 q^{29} -4.58579 q^{31} +11.0711 q^{33} -6.58579 q^{37} -11.6569 q^{39} +5.65685 q^{41} -3.24264 q^{43} +8.65685 q^{45} +1.24264 q^{47} -23.3137 q^{51} -5.41421 q^{53} -3.24264 q^{55} -3.41421 q^{57} +0.828427 q^{59} +5.00000 q^{61} +3.41421 q^{65} +3.17157 q^{67} -15.0711 q^{69} -7.75736 q^{71} +7.00000 q^{73} +13.6569 q^{75} +3.89949 q^{79} +39.9706 q^{81} -14.8995 q^{83} +6.82843 q^{85} +32.1421 q^{87} -13.5563 q^{89} +15.6569 q^{93} +1.00000 q^{95} +5.65685 q^{97} -28.0711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10}) \) 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9	\( 2 q - 4 q^{3} + 2 q^{5} + 6 q^{9} + 2 q^{11} + 4 q^{13} - 4 q^{15} + 8 q^{17} + 2 q^{19} + 6 q^{23} - 8 q^{25} - 16 q^{27} - 16 q^{29} - 12 q^{31} + 8 q^{33} - 16 q^{37} - 12 q^{39} + 2 q^{43} + 6 q^{45} - 6 q^{47} - 24 q^{51} - 8 q^{53} + 2 q^{55} - 4 q^{57} - 4 q^{59} + 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{69} - 24 q^{71} + 14 q^{73} + 16 q^{75} - 12 q^{79} + 46 q^{81} - 10 q^{83} + 8 q^{85} + 36 q^{87} + 4 q^{89} + 20 q^{93} + 2 q^{95} - 42 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 + 2 * q^5 + 6 * q^9 + 2 * q^11 + 4 * q^13 - 4 * q^15 + 8 * q^17 + 2 * q^19 + 6 * q^23 - 8 * q^25 - 16 * q^27 - 16 * q^29 - 12 * q^31 + 8 * q^33 - 16 * q^37 - 12 * q^39 + 2 * q^43 + 6 * q^45 - 6 * q^47 - 24 * q^51 - 8 * q^53 + 2 * q^55 - 4 * q^57 - 4 * q^59 + 10 * q^61 + 4 * q^65 + 12 * q^67 - 16 * q^69 - 24 * q^71 + 14 * q^73 + 16 * q^75 - 12 * q^79 + 46 * q^81 - 10 * q^83 + 8 * q^85 + 36 * q^87 + 4 * q^89 + 20 * q^93 + 2 * q^95 - 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more