LMFDB - Embedded newform 476.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,2,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("476.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	476.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:	$3.80087913621$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})$	$q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -0.605551 q^{13} +3.00000 q^{15} +1.00000 q^{17} -0.605551 q^{19} +2.30278 q^{21} -3.30278 q^{25} -1.60555 q^{27} +0.697224 q^{31} +1.30278 q^{35} +4.60555 q^{37} -1.39445 q^{39} -6.90833 q^{41} +3.69722 q^{43} +3.00000 q^{45} -2.60555 q^{47} +1.00000 q^{49} +2.30278 q^{51} -7.30278 q^{53} -1.39445 q^{57} -5.21110 q^{59} +2.90833 q^{61} +2.30278 q^{63} -0.788897 q^{65} -5.30278 q^{67} +13.8167 q^{71} +2.90833 q^{73} -7.60555 q^{75} +5.39445 q^{79} -10.6056 q^{81} +6.00000 q^{83} +1.30278 q^{85} -9.39445 q^{89} -0.605551 q^{91} +1.60555 q^{93} -0.788897 q^{95} -2.69722 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q + q^3 - q^5 + 2 * q^7 + q^9	$ 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{13} + 6 q^{15} + 2 q^{17} + 6 q^{19} + q^{21} - 3 q^{25} + 4 q^{27} + 5 q^{31} - q^{35} + 2 q^{37} - 10 q^{39} - 3 q^{41} + 11 q^{43} + 6 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 11 q^{53} - 10 q^{57} + 4 q^{59} - 5 q^{61} + q^{63} - 16 q^{65} - 7 q^{67} + 6 q^{71} - 5 q^{73} - 8 q^{75} + 18 q^{79} - 14 q^{81} + 12 q^{83} - q^{85} - 26 q^{89} + 6 q^{91} - 4 q^{93} - 16 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^13 + 6 * q^15 + 2 * q^17 + 6 * q^19 + q^21 - 3 * q^25 + 4 * q^27 + 5 * q^31 - q^35 + 2 * q^37 - 10 * q^39 - 3 * q^41 + 11 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 11 * q^53 - 10 * q^57 + 4 * q^59 - 5 * q^61 + q^63 - 16 * q^65 - 7 * q^67 + 6 * q^71 - 5 * q^73 - 8 * q^75 + 18 * q^79 - 14 * q^81 + 12 * q^83 - q^85 - 26 * q^89 + 6 * q^91 - 4 * q^93 - 16 * q^95 - 9 * q^97

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
$3$	2.30278	1.32951	0.664754	−	0.747062i	$-0.268536\pi$
$3$	2.30278	1.32951	0.664754	+	0.747062i	$0.268536\pi$
$4$	0	0
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.30278	0.767592
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−0.605551	−0.167950	−0.0839749	−	0.996468i	$-0.526762\pi$
$13$	−0.605551	−0.167950	−0.0839749	+	0.996468i	$0.526762\pi$
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	0	0
$21$	2.30278	0.502507
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	−1.60555	−0.308988
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0.697224	0.125225	0.0626126	−	0.998038i	$-0.480057\pi$
$31$	0.697224	0.125225	0.0626126	+	0.998038i	$0.480057\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.30278	0.220209
$36$	0	0
$37$	4.60555	0.757148	0.378574	−	0.925571i	$-0.376415\pi$
$37$	4.60555	0.757148	0.378574	+	0.925571i	$0.376415\pi$
$38$	0	0
$39$	−1.39445	−0.223290
$40$	0	0
$41$	−6.90833	−1.07890	−0.539450	−	0.842018i	$-0.681368\pi$
$41$	−6.90833	−1.07890	−0.539450	+	0.842018i	$0.681368\pi$
$42$	0	0
$43$	3.69722	0.563821	0.281911	−	0.959441i	$-0.409032\pi$
$43$	3.69722	0.563821	0.281911	+	0.959441i	$0.409032\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	−2.60555	−0.380059	−0.190029	−	0.981778i	$-0.560858\pi$
$47$	−2.60555	−0.380059	−0.190029	+	0.981778i	$0.560858\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.30278	0.322453
$52$	0	0
$53$	−7.30278	−1.00311	−0.501557	−	0.865125i	$-0.667239\pi$
$53$	−7.30278	−1.00311	−0.501557	+	0.865125i	$0.667239\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.39445	−0.184699
$58$	0	0
$59$	−5.21110	−0.678428	−0.339214	−	0.940709i	$-0.610161\pi$
$59$	−5.21110	−0.678428	−0.339214	+	0.940709i	$0.610161\pi$
$60$	0	0
$61$	2.90833	0.372373	0.186187	−	0.982514i	$-0.440387\pi$
$61$	2.90833	0.372373	0.186187	+	0.982514i	$0.440387\pi$
$62$	0	0
$63$	2.30278	0.290122
$64$	0	0
$65$	−0.788897	−0.0978507
$66$	0	0
$67$	−5.30278	−0.647837	−0.323919	−	0.946085i	$-0.605000\pi$
$67$	−5.30278	−0.647837	−0.323919	+	0.946085i	$0.605000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.8167	1.63974	0.819868	−	0.572553i	$-0.194047\pi$
$71$	13.8167	1.63974	0.819868	+	0.572553i	$0.194047\pi$
$72$	0	0
$73$	2.90833	0.340394	0.170197	−	0.985410i	$-0.445560\pi$
$73$	2.90833	0.340394	0.170197	+	0.985410i	$0.445560\pi$
$74$	0	0
$75$	−7.60555	−0.878213
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.39445	0.606923	0.303461	−	0.952844i	$-0.401858\pi$
$79$	5.39445	0.606923	0.303461	+	0.952844i	$0.401858\pi$
$80$	0	0
$81$	−10.6056	−1.17839
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	1.30278	0.141306
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.39445	−0.995810	−0.497905	−	0.867232i	$-0.665897\pi$
$89$	−9.39445	−0.995810	−0.497905	+	0.867232i	$0.665897\pi$
$90$	0	0
$91$	−0.605551	−0.0634790
$92$	0	0
$93$	1.60555	0.166488
$94$	0	0
$95$	−0.788897	−0.0809392
$96$	0	0
$97$	−2.69722	−0.273862	−0.136931	−	0.990581i	$-0.543724\pi$
$97$	−2.69722	−0.273862	−0.136931	+	0.990581i	$0.543724\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.81665	−0.180764	−0.0903819	−	0.995907i	$-0.528809\pi$
$101$	−1.81665	−0.180764	−0.0903819	+	0.995907i	$0.528809\pi$
$102$	0	0
$103$	7.21110	0.710531	0.355266	−	0.934765i	$-0.384390\pi$
$103$	7.21110	0.710531	0.355266	+	0.934765i	$0.384390\pi$
$104$	0	0
$105$	3.00000	0.292770
$106$	0	0
$107$	−8.60555	−0.831930	−0.415965	−	0.909381i	$-0.636556\pi$
$107$	−8.60555	−0.831930	−0.415965	+	0.909381i	$0.636556\pi$
$108$	0	0
$109$	−20.4222	−1.95609	−0.978046	−	0.208388i	$-0.933178\pi$
$109$	−20.4222	−1.95609	−0.978046	+	0.208388i	$0.933178\pi$
$110$	0	0
$111$	10.6056	1.00663
$112$	0	0
$113$	−2.60555	−0.245110	−0.122555	−	0.992462i	$-0.539109\pi$
$113$	−2.60555	−0.245110	−0.122555	+	0.992462i	$0.539109\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.39445	−0.128917
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	−15.9083	−1.43441
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	4.09167	0.363077	0.181539	−	0.983384i	$-0.441892\pi$
$127$	4.09167	0.363077	0.181539	+	0.983384i	$0.441892\pi$
$128$	0	0
$129$	8.51388	0.749605
$130$	0	0
$131$	−6.78890	−0.593149	−0.296574	−	0.955010i	$-0.595844\pi$
$131$	−6.78890	−0.593149	−0.296574	+	0.955010i	$0.595844\pi$
$132$	0	0
$133$	−0.605551	−0.0525080
$134$	0	0
$135$	−2.09167	−0.180023
$136$	0	0
$137$	−16.3028	−1.39284	−0.696420	−	0.717634i	$-0.745225\pi$
$137$	−16.3028	−1.39284	−0.696420	+	0.717634i	$0.745225\pi$
$138$	0	0
$139$	17.9083	1.51896	0.759482	−	0.650528i	$-0.225452\pi$
$139$	17.9083	1.51896	0.759482	+	0.650528i	$0.225452\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.30278	0.189930
$148$	0	0
$149$	−14.7250	−1.20632	−0.603159	−	0.797621i	$-0.706091\pi$
$149$	−14.7250	−1.20632	−0.603159	+	0.797621i	$0.706091\pi$
$150$	0	0
$151$	14.1194	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$151$	14.1194	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$152$	0	0
$153$	2.30278	0.186168
$154$	0	0
$155$	0.908327	0.0729586
$156$	0	0
$157$	7.21110	0.575509	0.287754	−	0.957704i	$-0.407091\pi$
$157$	7.21110	0.575509	0.287754	+	0.957704i	$0.407091\pi$
$158$	0	0
$159$	−16.8167	−1.33365
$160$	0	0
$161$	0	0
$162$	0	0
$163$	5.39445	0.422526	0.211263	−	0.977429i	$-0.432242\pi$
$163$	5.39445	0.422526	0.211263	+	0.977429i	$0.432242\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.72498	0.675159	0.337580	−	0.941297i	$-0.390392\pi$
$167$	8.72498	0.675159	0.337580	+	0.941297i	$0.390392\pi$
$168$	0	0
$169$	−12.6333	−0.971793
$170$	0	0
$171$	−1.39445	−0.106636
$172$	0	0
$173$	13.6972	1.04138	0.520690	−	0.853746i	$-0.325675\pi$
$173$	13.6972	1.04138	0.520690	+	0.853746i	$0.325675\pi$
$174$	0	0
$175$	−3.30278	−0.249666
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	−0.908327	−0.0678915	−0.0339458	−	0.999424i	$-0.510807\pi$
$179$	−0.908327	−0.0678915	−0.0339458	+	0.999424i	$0.510807\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	6.69722	0.495073
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.60555	−0.116787
$190$	0	0
$191$	−11.7250	−0.848390	−0.424195	−	0.905571i	$-0.639443\pi$
$191$	−11.7250	−0.848390	−0.424195	+	0.905571i	$0.639443\pi$
$192$	0	0
$193$	9.81665	0.706618	0.353309	−	0.935507i	$-0.385056\pi$
$193$	9.81665	0.706618	0.353309	+	0.935507i	$0.385056\pi$
$194$	0	0
$195$	−1.81665	−0.130093
$196$	0	0
$197$	2.60555	0.185638	0.0928189	−	0.995683i	$-0.470412\pi$
$197$	2.60555	0.185638	0.0928189	+	0.995683i	$0.470412\pi$
$198$	0	0
$199$	−4.11943	−0.292019	−0.146009	−	0.989283i	$-0.546643\pi$
$199$	−4.11943	−0.292019	−0.146009	+	0.989283i	$0.546643\pi$
$200$	0	0
$201$	−12.2111	−0.861305
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	15.0278	1.03455	0.517277	−	0.855818i	$-0.326946\pi$
$211$	15.0278	1.03455	0.517277	+	0.855818i	$0.326946\pi$
$212$	0	0
$213$	31.8167	2.18004
$214$	0	0
$215$	4.81665	0.328493
$216$	0	0
$217$	0.697224	0.0473307
$218$	0	0
$219$	6.69722	0.452556
$220$	0	0
$221$	−0.605551	−0.0407338
$222$	0	0
$223$	27.0278	1.80991	0.904956	−	0.425505i	$-0.139903\pi$
$223$	27.0278	1.80991	0.904956	+	0.425505i	$0.139903\pi$
$224$	0	0
$225$	−7.60555	−0.507037
$226$	0	0
$227$	−12.5139	−0.830575	−0.415288	−	0.909690i	$-0.636319\pi$
$227$	−12.5139	−0.830575	−0.415288	+	0.909690i	$0.636319\pi$
$228$	0	0
$229$	25.2111	1.66600	0.832998	−	0.553276i	$-0.186622\pi$
$229$	25.2111	1.66600	0.832998	+	0.553276i	$0.186622\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.81665	0.512086	0.256043	−	0.966665i	$-0.417581\pi$
$233$	7.81665	0.512086	0.256043	+	0.966665i	$0.417581\pi$
$234$	0	0
$235$	−3.39445	−0.221429
$236$	0	0
$237$	12.4222	0.806909
$238$	0	0
$239$	7.69722	0.497892	0.248946	−	0.968517i	$-0.419916\pi$
$239$	7.69722	0.497892	0.248946	+	0.968517i	$0.419916\pi$
$240$	0	0
$241$	26.5139	1.70791	0.853955	−	0.520348i	$-0.174198\pi$
$241$	26.5139	1.70791	0.853955	+	0.520348i	$0.174198\pi$
$242$	0	0
$243$	−19.6056	−1.25770
$244$	0	0
$245$	1.30278	0.0832313
$246$	0	0
$247$	0.366692	0.0233321
$248$	0	0
$249$	13.8167	0.875595
$250$	0	0
$251$	25.8167	1.62953	0.814766	−	0.579789i	$-0.196865\pi$
$251$	25.8167	1.62953	0.814766	+	0.579789i	$0.196865\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	−11.2111	−0.699329	−0.349665	−	0.936875i	$-0.613704\pi$
$257$	−11.2111	−0.699329	−0.349665	+	0.936875i	$0.613704\pi$
$258$	0	0
$259$	4.60555	0.286175
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.4222	0.642661	0.321330	−	0.946967i	$-0.395870\pi$
$263$	10.4222	0.642661	0.321330	+	0.946967i	$0.395870\pi$
$264$	0	0
$265$	−9.51388	−0.584433
$266$	0	0
$267$	−21.6333	−1.32394
$268$	0	0
$269$	0.788897	0.0480999	0.0240500	−	0.999711i	$-0.492344\pi$
$269$	0.788897	0.0480999	0.0240500	+	0.999711i	$0.492344\pi$
$270$	0	0
$271$	13.2111	0.802517	0.401259	−	0.915965i	$-0.368573\pi$
$271$	13.2111	0.802517	0.401259	+	0.915965i	$0.368573\pi$
$272$	0	0
$273$	−1.39445	−0.0843959
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.42221	0.385873	0.192936	−	0.981211i	$-0.438199\pi$
$277$	6.42221	0.385873	0.192936	+	0.981211i	$0.438199\pi$
$278$	0	0
$279$	1.60555	0.0961218
$280$	0	0
$281$	7.30278	0.435647	0.217824	−	0.975988i	$-0.430104\pi$
$281$	7.30278	0.435647	0.217824	+	0.975988i	$0.430104\pi$
$282$	0	0
$283$	8.51388	0.506098	0.253049	−	0.967454i	$-0.418567\pi$
$283$	8.51388	0.506098	0.253049	+	0.967454i	$0.418567\pi$
$284$	0	0
$285$	−1.81665	−0.107609
$286$	0	0
$287$	−6.90833	−0.407786
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.21110	−0.364101
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	−6.78890	−0.395265
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.69722	0.213104
$302$	0	0
$303$	−4.18335	−0.240327
$304$	0	0
$305$	3.78890	0.216952
$306$	0	0
$307$	13.2111	0.753997	0.376999	−	0.926214i	$-0.376956\pi$
$307$	13.2111	0.753997	0.376999	+	0.926214i	$0.376956\pi$
$308$	0	0
$309$	16.6056	0.944657
$310$	0	0
$311$	−27.1194	−1.53780	−0.768901	−	0.639368i	$-0.779196\pi$
$311$	−27.1194	−1.53780	−0.768901	+	0.639368i	$0.779196\pi$
$312$	0	0
$313$	−6.33053	−0.357823	−0.178911	−	0.983865i	$-0.557258\pi$
$313$	−6.33053	−0.357823	−0.178911	+	0.983865i	$0.557258\pi$
$314$	0	0
$315$	3.00000	0.169031
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−19.8167	−1.10606
$322$	0	0
$323$	−0.605551	−0.0336938
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−47.0278	−2.60064
$328$	0	0
$329$	−2.60555	−0.143649
$330$	0	0
$331$	9.30278	0.511327	0.255663	−	0.966766i	$-0.417706\pi$
$331$	9.30278	0.511327	0.255663	+	0.966766i	$0.417706\pi$
$332$	0	0
$333$	10.6056	0.581181
$334$	0	0
$335$	−6.90833	−0.377442
$336$	0	0
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.0000	−1.61048	−0.805242	−	0.592946i	$-0.797965\pi$
$347$	−30.0000	−1.61048	−0.805242	+	0.592946i	$0.797965\pi$
$348$	0	0
$349$	14.7889	0.791632	0.395816	−	0.918330i	$-0.370462\pi$
$349$	14.7889	0.791632	0.395816	+	0.918330i	$0.370462\pi$
$350$	0	0
$351$	0.972244	0.0518945
$352$	0	0
$353$	27.6333	1.47077	0.735386	−	0.677648i	$-0.237001\pi$
$353$	27.6333	1.47077	0.735386	+	0.677648i	$0.237001\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	0	0
$357$	2.30278	0.121876
$358$	0	0
$359$	−3.90833	−0.206274	−0.103137	−	0.994667i	$-0.532888\pi$
$359$	−3.90833	−0.206274	−0.103137	+	0.994667i	$0.532888\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	0	0
$363$	−25.3305	−1.32951
$364$	0	0
$365$	3.78890	0.198320
$366$	0	0
$367$	−33.3305	−1.73984	−0.869920	−	0.493193i	$-0.835830\pi$
$367$	−33.3305	−1.73984	−0.869920	+	0.493193i	$0.835830\pi$
$368$	0	0
$369$	−15.9083	−0.828154
$370$	0	0
$371$	−7.30278	−0.379141
$372$	0	0
$373$	15.9361	0.825139	0.412570	−	0.910926i	$-0.364631\pi$
$373$	15.9361	0.825139	0.412570	+	0.910926i	$0.364631\pi$
$374$	0	0
$375$	−24.9083	−1.28626
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−21.2111	−1.08954	−0.544771	−	0.838585i	$-0.683383\pi$
$379$	−21.2111	−1.08954	−0.544771	+	0.838585i	$0.683383\pi$
$380$	0	0
$381$	9.42221	0.482714
$382$	0	0
$383$	−10.4222	−0.532550	−0.266275	−	0.963897i	$-0.585793\pi$
$383$	−10.4222	−0.532550	−0.266275	+	0.963897i	$0.585793\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.51388	0.432785
$388$	0	0
$389$	15.1194	0.766586	0.383293	−	0.923627i	$-0.374790\pi$
$389$	15.1194	0.766586	0.383293	+	0.923627i	$0.374790\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−15.6333	−0.788596
$394$	0	0
$395$	7.02776	0.353605
$396$	0	0
$397$	24.9361	1.25151	0.625753	−	0.780021i	$-0.284792\pi$
$397$	24.9361	1.25151	0.625753	+	0.780021i	$0.284792\pi$
$398$	0	0
$399$	−1.39445	−0.0698098
$400$	0	0
$401$	−9.39445	−0.469136	−0.234568	−	0.972100i	$-0.575368\pi$
$401$	−9.39445	−0.469136	−0.234568	+	0.972100i	$0.575368\pi$
$402$	0	0
$403$	−0.422205	−0.0210315
$404$	0	0
$405$	−13.8167	−0.686555
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−19.3944	−0.958994	−0.479497	−	0.877544i	$-0.659181\pi$
$409$	−19.3944	−0.958994	−0.479497	+	0.877544i	$0.659181\pi$
$410$	0	0
$411$	−37.5416	−1.85179
$412$	0	0
$413$	−5.21110	−0.256422
$414$	0	0
$415$	7.81665	0.383704
$416$	0	0
$417$	41.2389	2.01948
$418$	0	0
$419$	0.513878	0.0251046	0.0125523	−	0.999921i	$-0.496004\pi$
$419$	0.513878	0.0251046	0.0125523	+	0.999921i	$0.496004\pi$
$420$	0	0
$421$	15.9361	0.776677	0.388339	−	0.921517i	$-0.373049\pi$
$421$	15.9361	0.776677	0.388339	+	0.921517i	$0.373049\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−3.30278	−0.160208
$426$	0	0
$427$	2.90833	0.140744
$428$	0	0
$429$	0	0
$430$	0	0
$431$	27.6333	1.33105	0.665525	−	0.746376i	$-0.268208\pi$
$431$	27.6333	1.33105	0.665525	+	0.746376i	$0.268208\pi$
$432$	0	0
$433$	−17.8167	−0.856214	−0.428107	−	0.903728i	$-0.640819\pi$
$433$	−17.8167	−0.856214	−0.428107	+	0.903728i	$0.640819\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	2.90833	0.138807	0.0694034	−	0.997589i	$-0.477890\pi$
$439$	2.90833	0.138807	0.0694034	+	0.997589i	$0.477890\pi$
$440$	0	0
$441$	2.30278	0.109656
$442$	0	0
$443$	−6.78890	−0.322550	−0.161275	−	0.986909i	$-0.551561\pi$
$443$	−6.78890	−0.322550	−0.161275	+	0.986909i	$0.551561\pi$
$444$	0	0
$445$	−12.2389	−0.580178
$446$	0	0
$447$	−33.9083	−1.60381
$448$	0	0
$449$	18.2389	0.860745	0.430372	−	0.902651i	$-0.358382\pi$
$449$	18.2389	0.860745	0.430372	+	0.902651i	$0.358382\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	32.5139	1.52764
$454$	0	0
$455$	−0.788897	−0.0369841
$456$	0	0
$457$	−13.9083	−0.650604	−0.325302	−	0.945610i	$-0.605466\pi$
$457$	−13.9083	−0.650604	−0.325302	+	0.945610i	$0.605466\pi$
$458$	0	0
$459$	−1.60555	−0.0749407
$460$	0	0
$461$	−10.1833	−0.474286	−0.237143	−	0.971475i	$-0.576211\pi$
$461$	−10.1833	−0.474286	−0.237143	+	0.971475i	$0.576211\pi$
$462$	0	0
$463$	−20.3028	−0.943550	−0.471775	−	0.881719i	$-0.656387\pi$
$463$	−20.3028	−0.943550	−0.471775	+	0.881719i	$0.656387\pi$
$464$	0	0
$465$	2.09167	0.0969990
$466$	0	0
$467$	29.4500	1.36278	0.681391	−	0.731920i	$-0.261375\pi$
$467$	29.4500	1.36278	0.681391	+	0.731920i	$0.261375\pi$
$468$	0	0
$469$	−5.30278	−0.244859
$470$	0	0
$471$	16.6056	0.765143
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	−16.8167	−0.769982
$478$	0	0
$479$	−3.51388	−0.160553	−0.0802766	−	0.996773i	$-0.525580\pi$
$479$	−3.51388	−0.160553	−0.0802766	+	0.996773i	$0.525580\pi$
$480$	0	0
$481$	−2.78890	−0.127163
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.51388	−0.159557
$486$	0	0
$487$	−9.21110	−0.417395	−0.208697	−	0.977980i	$-0.566922\pi$
$487$	−9.21110	−0.417395	−0.208697	+	0.977980i	$0.566922\pi$
$488$	0	0
$489$	12.4222	0.561752
$490$	0	0
$491$	14.3305	0.646728	0.323364	−	0.946275i	$-0.395186\pi$
$491$	14.3305	0.646728	0.323364	+	0.946275i	$0.395186\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.8167	0.619762
$498$	0	0
$499$	30.4222	1.36188	0.680942	−	0.732337i	$-0.261570\pi$
$499$	30.4222	1.36188	0.680942	+	0.732337i	$0.261570\pi$
$500$	0	0
$501$	20.0917	0.897630
$502$	0	0
$503$	−40.1472	−1.79007	−0.895037	−	0.445991i	$-0.852851\pi$
$503$	−40.1472	−1.79007	−0.895037	+	0.445991i	$0.852851\pi$
$504$	0	0
$505$	−2.36669	−0.105316
$506$	0	0
$507$	−29.0917	−1.29201
$508$	0	0
$509$	25.0278	1.10934	0.554668	−	0.832072i	$-0.312845\pi$
$509$	25.0278	1.10934	0.554668	+	0.832072i	$0.312845\pi$
$510$	0	0
$511$	2.90833	0.128657
$512$	0	0
$513$	0.972244	0.0429256
$514$	0	0
$515$	9.39445	0.413969
$516$	0	0
$517$	0	0
$518$	0	0
$519$	31.5416	1.38452
$520$	0	0
$521$	−36.3583	−1.59289	−0.796443	−	0.604714i	$-0.793287\pi$
$521$	−36.3583	−1.59289	−0.796443	+	0.604714i	$0.793287\pi$
$522$	0	0
$523$	25.2111	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$523$	25.2111	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$524$	0	0
$525$	−7.60555	−0.331933
$526$	0	0
$527$	0.697224	0.0303716
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	4.18335	0.181201
$534$	0	0
$535$	−11.2111	−0.484698
$536$	0	0
$537$	−2.09167	−0.0902624
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−30.0555	−1.29219	−0.646094	−	0.763258i	$-0.723598\pi$
$541$	−30.0555	−1.29219	−0.646094	+	0.763258i	$0.723598\pi$
$542$	0	0
$543$	32.2389	1.38350
$544$	0	0
$545$	−26.6056	−1.13966
$546$	0	0
$547$	−16.7889	−0.717841	−0.358921	−	0.933368i	$-0.616855\pi$
$547$	−16.7889	−0.717841	−0.358921	+	0.933368i	$0.616855\pi$
$548$	0	0
$549$	6.69722	0.285831
$550$	0	0
$551$	0	0
$552$	0	0
$553$	5.39445	0.229395
$554$	0	0
$555$	13.8167	0.586484
$556$	0	0
$557$	−9.63331	−0.408176	−0.204088	−	0.978953i	$-0.565423\pi$
$557$	−9.63331	−0.408176	−0.204088	+	0.978953i	$0.565423\pi$
$558$	0	0
$559$	−2.23886	−0.0946936
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.6333	0.658865	0.329433	−	0.944179i	$-0.393143\pi$
$563$	15.6333	0.658865	0.329433	+	0.944179i	$0.393143\pi$
$564$	0	0
$565$	−3.39445	−0.142806
$566$	0	0
$567$	−10.6056	−0.445391
$568$	0	0
$569$	30.1194	1.26267	0.631336	−	0.775509i	$-0.282507\pi$
$569$	30.1194	1.26267	0.631336	+	0.775509i	$0.282507\pi$
$570$	0	0
$571$	27.0278	1.13108	0.565538	−	0.824722i	$-0.308668\pi$
$571$	27.0278	1.13108	0.565538	+	0.824722i	$0.308668\pi$
$572$	0	0
$573$	−27.0000	−1.12794
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−22.2389	−0.925816	−0.462908	−	0.886406i	$-0.653194\pi$
$577$	−22.2389	−0.925816	−0.462908	+	0.886406i	$0.653194\pi$
$578$	0	0
$579$	22.6056	0.939455
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−1.81665	−0.0751094
$586$	0	0
$587$	−3.39445	−0.140104	−0.0700519	−	0.997543i	$-0.522317\pi$
$587$	−3.39445	−0.140104	−0.0700519	+	0.997543i	$0.522317\pi$
$588$	0	0
$589$	−0.422205	−0.0173967
$590$	0	0
$591$	6.00000	0.246807
$592$	0	0
$593$	22.1833	0.910961	0.455480	−	0.890246i	$-0.349468\pi$
$593$	22.1833	0.910961	0.455480	+	0.890246i	$0.349468\pi$
$594$	0	0
$595$	1.30278	0.0534086
$596$	0	0
$597$	−9.48612	−0.388241
$598$	0	0
$599$	−38.9638	−1.59202	−0.796010	−	0.605284i	$-0.793060\pi$
$599$	−38.9638	−1.59202	−0.796010	+	0.605284i	$0.793060\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	−12.2111	−0.497275
$604$	0	0
$605$	−14.3305	−0.582619
$606$	0	0
$607$	−3.48612	−0.141497	−0.0707487	−	0.997494i	$-0.522539\pi$
$607$	−3.48612	−0.141497	−0.0707487	+	0.997494i	$0.522539\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.57779	0.0638307
$612$	0	0
$613$	−32.1472	−1.29841	−0.649206	−	0.760612i	$-0.724899\pi$
$613$	−32.1472	−1.29841	−0.649206	+	0.760612i	$0.724899\pi$
$614$	0	0
$615$	−20.7250	−0.835712
$616$	0	0
$617$	−24.0000	−0.966204	−0.483102	−	0.875564i	$-0.660490\pi$
$617$	−24.0000	−0.966204	−0.483102	+	0.875564i	$0.660490\pi$
$618$	0	0
$619$	−0.366692	−0.0147386	−0.00736930	−	0.999973i	$-0.502346\pi$
$619$	−0.366692	−0.0147386	−0.00736930	+	0.999973i	$0.502346\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.39445	−0.376381
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.60555	0.183635
$630$	0	0
$631$	17.1194	0.681514	0.340757	−	0.940151i	$-0.389317\pi$
$631$	17.1194	0.681514	0.340757	+	0.940151i	$0.389317\pi$
$632$	0	0
$633$	34.6056	1.37545
$634$	0	0
$635$	5.33053	0.211536
$636$	0	0
$637$	−0.605551	−0.0239928
$638$	0	0
$639$	31.8167	1.25865
$640$	0	0
$641$	44.8444	1.77125	0.885624	−	0.464403i	$-0.153731\pi$
$641$	44.8444	1.77125	0.885624	+	0.464403i	$0.153731\pi$
$642$	0	0
$643$	−23.1472	−0.912836	−0.456418	−	0.889766i	$-0.650868\pi$
$643$	−23.1472	−0.912836	−0.456418	+	0.889766i	$0.650868\pi$
$644$	0	0
$645$	11.0917	0.436734
$646$	0	0
$647$	−10.4222	−0.409739	−0.204870	−	0.978789i	$-0.565677\pi$
$647$	−10.4222	−0.409739	−0.204870	+	0.978789i	$0.565677\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1.60555	0.0629265
$652$	0	0
$653$	−44.0555	−1.72403	−0.862013	−	0.506887i	$-0.830796\pi$
$653$	−44.0555	−1.72403	−0.862013	+	0.506887i	$0.830796\pi$
$654$	0	0
$655$	−8.84441	−0.345580
$656$	0	0
$657$	6.69722	0.261284
$658$	0	0
$659$	34.6972	1.35161	0.675806	−	0.737080i	$-0.263796\pi$
$659$	34.6972	1.35161	0.675806	+	0.737080i	$0.263796\pi$
$660$	0	0
$661$	−35.8167	−1.39311	−0.696553	−	0.717505i	$-0.745284\pi$
$661$	−35.8167	−1.39311	−0.696553	+	0.717505i	$0.745284\pi$
$662$	0	0
$663$	−1.39445	−0.0541559
$664$	0	0
$665$	−0.788897	−0.0305921
$666$	0	0
$667$	0	0
$668$	0	0
$669$	62.2389	2.40629
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−41.8167	−1.61191	−0.805957	−	0.591974i	$-0.798349\pi$
$673$	−41.8167	−1.61191	−0.805957	+	0.591974i	$0.798349\pi$
$674$	0	0
$675$	5.30278	0.204104
$676$	0	0
$677$	19.5778	0.752436	0.376218	−	0.926531i	$-0.377224\pi$
$677$	19.5778	0.752436	0.376218	+	0.926531i	$0.377224\pi$
$678$	0	0
$679$	−2.69722	−0.103510
$680$	0	0
$681$	−28.8167	−1.10426
$682$	0	0
$683$	−23.4500	−0.897288	−0.448644	−	0.893711i	$-0.648093\pi$
$683$	−23.4500	−0.897288	−0.448644	+	0.893711i	$0.648093\pi$
$684$	0	0
$685$	−21.2389	−0.811495
$686$	0	0
$687$	58.0555	2.21496
$688$	0	0
$689$	4.42221	0.168473
$690$	0	0
$691$	47.5139	1.80751	0.903757	−	0.428047i	$-0.140798\pi$
$691$	47.5139	1.80751	0.903757	+	0.428047i	$0.140798\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.3305	0.884978
$696$	0	0
$697$	−6.90833	−0.261672
$698$	0	0
$699$	18.0000	0.680823
$700$	0	0
$701$	38.8444	1.46713	0.733567	−	0.679618i	$-0.237854\pi$
$701$	38.8444	1.46713	0.733567	+	0.679618i	$0.237854\pi$
$702$	0	0
$703$	−2.78890	−0.105185
$704$	0	0
$705$	−7.81665	−0.294392
$706$	0	0
$707$	−1.81665	−0.0683223
$708$	0	0
$709$	33.8167	1.27001	0.635006	−	0.772508i	$-0.280998\pi$
$709$	33.8167	1.27001	0.635006	+	0.772508i	$0.280998\pi$
$710$	0	0
$711$	12.4222	0.465869
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.7250	0.661952
$718$	0	0
$719$	−25.6972	−0.958345	−0.479172	−	0.877721i	$-0.659063\pi$
$719$	−25.6972	−0.958345	−0.479172	+	0.877721i	$0.659063\pi$
$720$	0	0
$721$	7.21110	0.268555
$722$	0	0
$723$	61.0555	2.27068
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.42221	0.238186	0.119093	−	0.992883i	$-0.462001\pi$
$727$	6.42221	0.238186	0.119093	+	0.992883i	$0.462001\pi$
$728$	0	0
$729$	−13.3305	−0.493723
$730$	0	0
$731$	3.69722	0.136747
$732$	0	0
$733$	20.2389	0.747539	0.373770	−	0.927522i	$-0.378065\pi$
$733$	20.2389	0.747539	0.373770	+	0.927522i	$0.378065\pi$
$734$	0	0
$735$	3.00000	0.110657
$736$	0	0
$737$	0	0
$738$	0	0
$739$	50.3583	1.85246	0.926230	−	0.376959i	$-0.123030\pi$
$739$	50.3583	1.85246	0.926230	+	0.376959i	$0.123030\pi$
$740$	0	0
$741$	0.844410	0.0310202
$742$	0	0
$743$	−19.8167	−0.727003	−0.363501	−	0.931594i	$-0.618419\pi$
$743$	−19.8167	−0.727003	−0.363501	+	0.931594i	$0.618419\pi$
$744$	0	0
$745$	−19.1833	−0.702823
$746$	0	0
$747$	13.8167	0.505525
$748$	0	0
$749$	−8.60555	−0.314440
$750$	0	0
$751$	11.3944	0.415789	0.207895	−	0.978151i	$-0.433339\pi$
$751$	11.3944	0.415789	0.207895	+	0.978151i	$0.433339\pi$
$752$	0	0
$753$	59.4500	2.16648
$754$	0	0
$755$	18.3944	0.669443
$756$	0	0
$757$	−13.2750	−0.482489	−0.241244	−	0.970464i	$-0.577556\pi$
$757$	−13.2750	−0.482489	−0.241244	+	0.970464i	$0.577556\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	16.1833	0.586646	0.293323	−	0.956013i	$-0.405239\pi$
$761$	16.1833	0.586646	0.293323	+	0.956013i	$0.405239\pi$
$762$	0	0
$763$	−20.4222	−0.739333
$764$	0	0
$765$	3.00000	0.108465
$766$	0	0
$767$	3.15559	0.113942
$768$	0	0
$769$	33.2666	1.19962	0.599812	−	0.800141i	$-0.295242\pi$
$769$	33.2666	1.19962	0.599812	+	0.800141i	$0.295242\pi$
$770$	0	0
$771$	−25.8167	−0.929764
$772$	0	0
$773$	4.18335	0.150465	0.0752323	−	0.997166i	$-0.476030\pi$
$773$	4.18335	0.150465	0.0752323	+	0.997166i	$0.476030\pi$
$774$	0	0
$775$	−2.30278	−0.0827181
$776$	0	0
$777$	10.6056	0.380472
$778$	0	0
$779$	4.18335	0.149884
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.39445	0.335302
$786$	0	0
$787$	−26.4222	−0.941850	−0.470925	−	0.882173i	$-0.656080\pi$
$787$	−26.4222	−0.941850	−0.470925	+	0.882173i	$0.656080\pi$
$788$	0	0
$789$	24.0000	0.854423
$790$	0	0
$791$	−2.60555	−0.0926427
$792$	0	0
$793$	−1.76114	−0.0625400
$794$	0	0
$795$	−21.9083	−0.777008
$796$	0	0
$797$	−28.4222	−1.00677	−0.503383	−	0.864063i	$-0.667912\pi$
$797$	−28.4222	−1.00677	−0.503383	+	0.864063i	$0.667912\pi$
$798$	0	0
$799$	−2.60555	−0.0921778
$800$	0	0
$801$	−21.6333	−0.764375
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.81665	0.0639492
$808$	0	0
$809$	33.3944	1.17409	0.587043	−	0.809556i	$-0.300292\pi$
$809$	33.3944	1.17409	0.587043	+	0.809556i	$0.300292\pi$
$810$	0	0
$811$	14.9083	0.523502	0.261751	−	0.965135i	$-0.415700\pi$
$811$	14.9083	0.523502	0.261751	+	0.965135i	$0.415700\pi$
$812$	0	0
$813$	30.4222	1.06695
$814$	0	0
$815$	7.02776	0.246172
$816$	0	0
$817$	−2.23886	−0.0783278
$818$	0	0
$819$	−1.39445	−0.0487260
$820$	0	0
$821$	−3.39445	−0.118467	−0.0592335	−	0.998244i	$-0.518866\pi$
$821$	−3.39445	−0.118467	−0.0592335	+	0.998244i	$0.518866\pi$
$822$	0	0
$823$	−49.8722	−1.73843	−0.869217	−	0.494430i	$-0.835377\pi$
$823$	−49.8722	−1.73843	−0.869217	+	0.494430i	$0.835377\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.6333	−0.543623	−0.271812	−	0.962350i	$-0.587623\pi$
$827$	−15.6333	−0.543623	−0.271812	+	0.962350i	$0.587623\pi$
$828$	0	0
$829$	−19.3944	−0.673597	−0.336799	−	0.941577i	$-0.609344\pi$
$829$	−19.3944	−0.673597	−0.336799	+	0.941577i	$0.609344\pi$
$830$	0	0
$831$	14.7889	0.513021
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	11.3667	0.393361
$836$	0	0
$837$	−1.11943	−0.0386931
$838$	0	0
$839$	−10.4222	−0.359814	−0.179907	−	0.983684i	$-0.557580\pi$
$839$	−10.4222	−0.359814	−0.179907	+	0.983684i	$0.557580\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	16.8167	0.579196
$844$	0	0
$845$	−16.4584	−0.566185
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	19.6056	0.672861
$850$	0	0
$851$	0	0
$852$	0	0
$853$	12.4222	0.425328	0.212664	−	0.977125i	$-0.431786\pi$
$853$	12.4222	0.425328	0.212664	+	0.977125i	$0.431786\pi$
$854$	0	0
$855$	−1.81665	−0.0621282
$856$	0	0
$857$	19.5416	0.667530	0.333765	−	0.942656i	$-0.391681\pi$
$857$	19.5416	0.667530	0.333765	+	0.942656i	$0.391681\pi$
$858$	0	0
$859$	−27.4500	−0.936581	−0.468290	−	0.883575i	$-0.655130\pi$
$859$	−27.4500	−0.936581	−0.468290	+	0.883575i	$0.655130\pi$
$860$	0	0
$861$	−15.9083	−0.542154
$862$	0	0
$863$	8.09167	0.275444	0.137722	−	0.990471i	$-0.456022\pi$
$863$	8.09167	0.275444	0.137722	+	0.990471i	$0.456022\pi$
$864$	0	0
$865$	17.8444	0.606728
$866$	0	0
$867$	2.30278	0.0782064
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3.21110	0.108804
$872$	0	0
$873$	−6.21110	−0.210214
$874$	0	0
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−25.3944	−0.857510	−0.428755	−	0.903421i	$-0.641048\pi$
$877$	−25.3944	−0.857510	−0.428755	+	0.903421i	$0.641048\pi$
$878$	0	0
$879$	49.8167	1.68027
$880$	0	0
$881$	−13.3028	−0.448182	−0.224091	−	0.974568i	$-0.571941\pi$
$881$	−13.3028	−0.448182	−0.224091	+	0.974568i	$0.571941\pi$
$882$	0	0
$883$	−54.3305	−1.82837	−0.914184	−	0.405299i	$-0.867167\pi$
$883$	−54.3305	−1.82837	−0.914184	+	0.405299i	$0.867167\pi$
$884$	0	0
$885$	−15.6333	−0.525508
$886$	0	0
$887$	−38.0917	−1.27899	−0.639497	−	0.768794i	$-0.720857\pi$
$887$	−38.0917	−1.27899	−0.639497	+	0.768794i	$0.720857\pi$
$888$	0	0
$889$	4.09167	0.137230
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.57779	0.0527989
$894$	0	0
$895$	−1.18335	−0.0395549
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−7.30278	−0.243291
$902$	0	0
$903$	8.51388	0.283324
$904$	0	0
$905$	18.2389	0.606280
$906$	0	0
$907$	34.8444	1.15699	0.578495	−	0.815686i	$-0.303640\pi$
$907$	34.8444	1.15699	0.578495	+	0.815686i	$0.303640\pi$
$908$	0	0
$909$	−4.18335	−0.138753
$910$	0	0
$911$	16.1833	0.536178	0.268089	−	0.963394i	$-0.413608\pi$
$911$	16.1833	0.536178	0.268089	+	0.963394i	$0.413608\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	8.72498	0.288439
$916$	0	0
$917$	−6.78890	−0.224189
$918$	0	0
$919$	−15.3305	−0.505708	−0.252854	−	0.967505i	$-0.581369\pi$
$919$	−15.3305	−0.505708	−0.252854	+	0.967505i	$0.581369\pi$
$920$	0	0
$921$	30.4222	1.00245
$922$	0	0
$923$	−8.36669	−0.275393
$924$	0	0
$925$	−15.2111	−0.500138
$926$	0	0
$927$	16.6056	0.545398
$928$	0	0
$929$	7.06392	0.231760	0.115880	−	0.993263i	$-0.463031\pi$
$929$	7.06392	0.231760	0.115880	+	0.993263i	$0.463031\pi$
$930$	0	0
$931$	−0.605551	−0.0198461
$932$	0	0
$933$	−62.4500	−2.04452
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.2111	−0.692937	−0.346468	−	0.938062i	$-0.612619\pi$
$937$	−21.2111	−0.692937	−0.346468	+	0.938062i	$0.612619\pi$
$938$	0	0
$939$	−14.5778	−0.475728
$940$	0	0
$941$	−32.7250	−1.06680	−0.533402	−	0.845862i	$-0.679087\pi$
$941$	−32.7250	−1.06680	−0.533402	+	0.845862i	$0.679087\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.09167	−0.0680421
$946$	0	0
$947$	−8.84441	−0.287405	−0.143702	−	0.989621i	$-0.545901\pi$
$947$	−8.84441	−0.287405	−0.143702	+	0.989621i	$0.545901\pi$
$948$	0	0
$949$	−1.76114	−0.0571691
$950$	0	0
$951$	−13.8167	−0.448036
$952$	0	0
$953$	−0.513878	−0.0166461	−0.00832307	−	0.999965i	$-0.502649\pi$
$953$	−0.513878	−0.0166461	−0.00832307	+	0.999965i	$0.502649\pi$
$954$	0	0
$955$	−15.2750	−0.494288
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.3028	−0.526444
$960$	0	0
$961$	−30.5139	−0.984319
$962$	0	0
$963$	−19.8167	−0.638583
$964$	0	0
$965$	12.7889	0.411689
$966$	0	0
$967$	−1.51388	−0.0486830	−0.0243415	−	0.999704i	$-0.507749\pi$
$967$	−1.51388	−0.0486830	−0.0243415	+	0.999704i	$0.507749\pi$
$968$	0	0
$969$	−1.39445	−0.0447961
$970$	0	0
$971$	−20.8444	−0.668929	−0.334464	−	0.942408i	$-0.608555\pi$
$971$	−20.8444	−0.668929	−0.334464	+	0.942408i	$0.608555\pi$
$972$	0	0
$973$	17.9083	0.574115
$974$	0	0
$975$	4.60555	0.147496
$976$	0	0
$977$	28.1472	0.900508	0.450254	−	0.892900i	$-0.351333\pi$
$977$	28.1472	0.900508	0.450254	+	0.892900i	$0.351333\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−47.0278	−1.50148
$982$	0	0
$983$	33.9083	1.08151	0.540754	−	0.841181i	$-0.318139\pi$
$983$	33.9083	1.08151	0.540754	+	0.841181i	$0.318139\pi$
$984$	0	0
$985$	3.39445	0.108156
$986$	0	0
$987$	−6.00000	−0.190982
$988$	0	0
$989$	0	0
$990$	0	0
$991$	14.0000	0.444725	0.222362	−	0.974964i	$-0.428623\pi$
$991$	14.0000	0.444725	0.222362	+	0.974964i	$0.428623\pi$
$992$	0	0
$993$	21.4222	0.679813
$994$	0	0
$995$	−5.36669	−0.170136
$996$	0	0
$997$	−47.1472	−1.49317	−0.746583	−	0.665292i	$-0.768307\pi$
$997$	−47.1472	−1.49317	−0.746583	+	0.665292i	$0.768307\pi$
$998$	0	0
$999$	−7.39445	−0.233950

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.2.a.d.1.2	✓	2
3.2	odd	2		4284.2.a.n.1.1		2
4.3	odd	2		1904.2.a.h.1.1		2
7.6	odd	2		3332.2.a.j.1.1		2
8.3	odd	2		7616.2.a.v.1.2		2
8.5	even	2		7616.2.a.q.1.1		2
17.16	even	2		8092.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.d.1.2	✓	2	1.1	even	1	trivial
1904.2.a.h.1.1		2	4.3	odd	2
3332.2.a.j.1.1		2	7.6	odd	2
4284.2.a.n.1.1		2	3.2	odd	2
7616.2.a.q.1.1		2	8.5	even	2
7616.2.a.v.1.2		2	8.3	odd	2
8092.2.a.k.1.1		2	17.16	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} +O(q^{10})\)	\(q+2.30278 q^{3} +1.30278 q^{5} +1.00000 q^{7} +2.30278 q^{9} -0.605551 q^{13} +3.00000 q^{15} +1.00000 q^{17} -0.605551 q^{19} +2.30278 q^{21} -3.30278 q^{25} -1.60555 q^{27} +0.697224 q^{31} +1.30278 q^{35} +4.60555 q^{37} -1.39445 q^{39} -6.90833 q^{41} +3.69722 q^{43} +3.00000 q^{45} -2.60555 q^{47} +1.00000 q^{49} +2.30278 q^{51} -7.30278 q^{53} -1.39445 q^{57} -5.21110 q^{59} +2.90833 q^{61} +2.30278 q^{63} -0.788897 q^{65} -5.30278 q^{67} +13.8167 q^{71} +2.90833 q^{73} -7.60555 q^{75} +5.39445 q^{79} -10.6056 q^{81} +6.00000 q^{83} +1.30278 q^{85} -9.39445 q^{89} -0.605551 q^{91} +1.60555 q^{93} -0.788897 q^{95} -2.69722 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q + q^3 - q^5 + 2 * q^7 + q^9	\( 2 q + q^{3} - q^{5} + 2 q^{7} + q^{9} + 6 q^{13} + 6 q^{15} + 2 q^{17} + 6 q^{19} + q^{21} - 3 q^{25} + 4 q^{27} + 5 q^{31} - q^{35} + 2 q^{37} - 10 q^{39} - 3 q^{41} + 11 q^{43} + 6 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 11 q^{53} - 10 q^{57} + 4 q^{59} - 5 q^{61} + q^{63} - 16 q^{65} - 7 q^{67} + 6 q^{71} - 5 q^{73} - 8 q^{75} + 18 q^{79} - 14 q^{81} + 12 q^{83} - q^{85} - 26 q^{89} + 6 q^{91} - 4 q^{93} - 16 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + q^3 - q^5 + 2 * q^7 + q^9 + 6 * q^13 + 6 * q^15 + 2 * q^17 + 6 * q^19 + q^21 - 3 * q^25 + 4 * q^27 + 5 * q^31 - q^35 + 2 * q^37 - 10 * q^39 - 3 * q^41 + 11 * q^43 + 6 * q^45 + 2 * q^47 + 2 * q^49 + q^51 - 11 * q^53 - 10 * q^57 + 4 * q^59 - 5 * q^61 + q^63 - 16 * q^65 - 7 * q^67 + 6 * q^71 - 5 * q^73 - 8 * q^75 + 18 * q^79 - 14 * q^81 + 12 * q^83 - q^85 - 26 * q^89 + 6 * q^91 - 4 * q^93 - 16 * q^95 - 9 * q^97

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