LMFDB - Embedded newform 476.2.a.b.1.1

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.61803 q^{3} +0.618034 q^{5} -1.00000 q^{7} -0.381966 q^{9} -0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} -8.47214 q^{19} +1.61803 q^{21} -7.70820 q^{23} -4.61803 q^{25} +5.47214 q^{27} -5.70820 q^{29} +6.32624 q^{31} +1.23607 q^{33} -0.618034 q^{35} +0.472136 q^{37} -2.00000 q^{39} -0.0901699 q^{41} -12.0902 q^{43} -0.236068 q^{45} -8.47214 q^{47} +1.00000 q^{49} -1.61803 q^{51} +10.7984 q^{53} -0.472136 q^{55} +13.7082 q^{57} -7.32624 q^{61} +0.381966 q^{63} +0.763932 q^{65} -13.0902 q^{67} +12.4721 q^{69} +10.9443 q^{71} +7.14590 q^{73} +7.47214 q^{75} +0.763932 q^{77} +2.94427 q^{79} -7.70820 q^{81} +15.4164 q^{83} +0.618034 q^{85} +9.23607 q^{87} +2.00000 q^{89} -1.23607 q^{91} -10.2361 q^{93} -5.23607 q^{95} +15.0902 q^{97} +0.291796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 + 2 * q^29 - 3 * q^31 - 2 * q^33 + q^35 - 8 * q^37 - 4 * q^39 + 11 * q^41 - 13 * q^43 + 4 * q^45 - 8 * q^47 + 2 * q^49 - q^51 - 3 * q^53 + 8 * q^55 + 14 * q^57 + q^61 + 3 * q^63 + 6 * q^65 - 15 * q^67 + 16 * q^69 + 4 * q^71 + 21 * q^73 + 6 * q^75 + 6 * q^77 - 12 * q^79 - 2 * q^81 + 4 * q^83 - q^85 + 14 * q^87 + 4 * q^89 + 2 * q^91 - 16 * q^93 - 6 * q^95 + 19 * q^97 + 14 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0.618034	0.276393	0.138197	−	0.990405i	$-0.455869\pi$
$5$	0.618034	0.276393	0.138197	+	0.990405i	$0.455869\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−0.763932	−0.230334	−0.115167	−	0.993346i	$-0.536740\pi$
$11$	−0.763932	−0.230334	−0.115167	+	0.993346i	$0.536740\pi$
$12$	0	0
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−8.47214	−1.94364	−0.971821	−	0.235722i	$-0.924255\pi$
$19$	−8.47214	−1.94364	−0.971821	+	0.235722i	$0.924255\pi$
$20$	0	0
$21$	1.61803	0.353084
$22$	0	0
$23$	−7.70820	−1.60727	−0.803636	−	0.595121i	$-0.797104\pi$
$23$	−7.70820	−1.60727	−0.803636	+	0.595121i	$0.797104\pi$
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−5.70820	−1.05999	−0.529993	−	0.848002i	$-0.677806\pi$
$29$	−5.70820	−1.05999	−0.529993	+	0.848002i	$0.677806\pi$
$30$	0	0
$31$	6.32624	1.13623	0.568113	−	0.822951i	$-0.307674\pi$
$31$	6.32624	1.13623	0.568113	+	0.822951i	$0.307674\pi$
$32$	0	0
$33$	1.23607	0.215172
$34$	0	0
$35$	−0.618034	−0.104467
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−0.0901699	−0.0140822	−0.00704109	−	0.999975i	$-0.502241\pi$
$41$	−0.0901699	−0.0140822	−0.00704109	+	0.999975i	$0.502241\pi$
$42$	0	0
$43$	−12.0902	−1.84373	−0.921867	−	0.387507i	$-0.873336\pi$
$43$	−12.0902	−1.84373	−0.921867	+	0.387507i	$0.873336\pi$
$44$	0	0
$45$	−0.236068	−0.0351909
$46$	0	0
$47$	−8.47214	−1.23579	−0.617894	−	0.786261i	$-0.712014\pi$
$47$	−8.47214	−1.23579	−0.617894	+	0.786261i	$0.712014\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	10.7984	1.48327	0.741635	−	0.670803i	$-0.234050\pi$
$53$	10.7984	1.48327	0.741635	+	0.670803i	$0.234050\pi$
$54$	0	0
$55$	−0.472136	−0.0636628
$56$	0	0
$57$	13.7082	1.81570
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−7.32624	−0.938029	−0.469014	−	0.883191i	$-0.655391\pi$
$61$	−7.32624	−0.938029	−0.469014	+	0.883191i	$0.655391\pi$
$62$	0	0
$63$	0.381966	0.0481232
$64$	0	0
$65$	0.763932	0.0947541
$66$	0	0
$67$	−13.0902	−1.59922	−0.799609	−	0.600520i	$-0.794960\pi$
$67$	−13.0902	−1.59922	−0.799609	+	0.600520i	$0.794960\pi$
$68$	0	0
$69$	12.4721	1.50147
$70$	0	0
$71$	10.9443	1.29885	0.649423	−	0.760427i	$-0.275010\pi$
$71$	10.9443	1.29885	0.649423	+	0.760427i	$0.275010\pi$
$72$	0	0
$73$	7.14590	0.836364	0.418182	−	0.908363i	$-0.362667\pi$
$73$	7.14590	0.836364	0.418182	+	0.908363i	$0.362667\pi$
$74$	0	0
$75$	7.47214	0.862808
$76$	0	0
$77$	0.763932	0.0870581
$78$	0	0
$79$	2.94427	0.331256	0.165628	−	0.986188i	$-0.447035\pi$
$79$	2.94427	0.331256	0.165628	+	0.986188i	$0.447035\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	15.4164	1.69217	0.846085	−	0.533048i	$-0.178953\pi$
$83$	15.4164	1.69217	0.846085	+	0.533048i	$0.178953\pi$
$84$	0	0
$85$	0.618034	0.0670352
$86$	0	0
$87$	9.23607	0.990210
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−1.23607	−0.129575
$92$	0	0
$93$	−10.2361	−1.06143
$94$	0	0
$95$	−5.23607	−0.537209
$96$	0	0
$97$	15.0902	1.53217	0.766087	−	0.642737i	$-0.222201\pi$
$97$	15.0902	1.53217	0.766087	+	0.642737i	$0.222201\pi$
$98$	0	0
$99$	0.291796	0.0293266
$100$	0	0
$101$	−0.472136	−0.0469793	−0.0234896	−	0.999724i	$-0.507478\pi$
$101$	−0.472136	−0.0469793	−0.0234896	+	0.999724i	$0.507478\pi$
$102$	0	0
$103$	15.4164	1.51902	0.759512	−	0.650493i	$-0.225438\pi$
$103$	15.4164	1.51902	0.759512	+	0.650493i	$0.225438\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	10.7639	1.04059	0.520294	−	0.853987i	$-0.325822\pi$
$107$	10.7639	1.04059	0.520294	+	0.853987i	$0.325822\pi$
$108$	0	0
$109$	−3.52786	−0.337908	−0.168954	−	0.985624i	$-0.554039\pi$
$109$	−3.52786	−0.337908	−0.168954	+	0.985624i	$0.554039\pi$
$110$	0	0
$111$	−0.763932	−0.0725092
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−4.76393	−0.444239
$116$	0	0
$117$	−0.472136	−0.0436490
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.4164	−0.946946
$122$	0	0
$123$	0.145898	0.0131552
$124$	0	0
$125$	−5.94427	−0.531672
$126$	0	0
$127$	−7.85410	−0.696939	−0.348469	−	0.937320i	$-0.613298\pi$
$127$	−7.85410	−0.696939	−0.348469	+	0.937320i	$0.613298\pi$
$128$	0	0
$129$	19.5623	1.72236
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	8.47214	0.734627
$134$	0	0
$135$	3.38197	0.291073
$136$	0	0
$137$	2.38197	0.203505	0.101753	−	0.994810i	$-0.467555\pi$
$137$	2.38197	0.203505	0.101753	+	0.994810i	$0.467555\pi$
$138$	0	0
$139$	−14.0344	−1.19039	−0.595193	−	0.803583i	$-0.702924\pi$
$139$	−14.0344	−1.19039	−0.595193	+	0.803583i	$0.702924\pi$
$140$	0	0
$141$	13.7082	1.15444
$142$	0	0
$143$	−0.944272	−0.0789640
$144$	0	0
$145$	−3.52786	−0.292973
$146$	0	0
$147$	−1.61803	−0.133453
$148$	0	0
$149$	−13.6180	−1.11563	−0.557816	−	0.829964i	$-0.688361\pi$
$149$	−13.6180	−1.11563	−0.557816	+	0.829964i	$0.688361\pi$
$150$	0	0
$151$	−15.1459	−1.23256	−0.616278	−	0.787529i	$-0.711360\pi$
$151$	−15.1459	−1.23256	−0.616278	+	0.787529i	$0.711360\pi$
$152$	0	0
$153$	−0.381966	−0.0308801
$154$	0	0
$155$	3.90983	0.314045
$156$	0	0
$157$	−2.18034	−0.174010	−0.0870050	−	0.996208i	$-0.527730\pi$
$157$	−2.18034	−0.174010	−0.0870050	+	0.996208i	$0.527730\pi$
$158$	0	0
$159$	−17.4721	−1.38563
$160$	0	0
$161$	7.70820	0.607492
$162$	0	0
$163$	5.70820	0.447101	0.223551	−	0.974692i	$-0.428235\pi$
$163$	5.70820	0.447101	0.223551	+	0.974692i	$0.428235\pi$
$164$	0	0
$165$	0.763932	0.0594720
$166$	0	0
$167$	14.3820	1.11291	0.556455	−	0.830878i	$-0.312161\pi$
$167$	14.3820	1.11291	0.556455	+	0.830878i	$0.312161\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	3.23607	0.247468
$172$	0	0
$173$	−11.9098	−0.905488	−0.452744	−	0.891641i	$-0.649555\pi$
$173$	−11.9098	−0.905488	−0.452744	+	0.891641i	$0.649555\pi$
$174$	0	0
$175$	4.61803	0.349091
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.3820	−1.37393	−0.686966	−	0.726689i	$-0.741058\pi$
$179$	−18.3820	−1.37393	−0.686966	+	0.726689i	$0.741058\pi$
$180$	0	0
$181$	−12.4721	−0.927047	−0.463523	−	0.886085i	$-0.653415\pi$
$181$	−12.4721	−0.927047	−0.463523	+	0.886085i	$0.653415\pi$
$182$	0	0
$183$	11.8541	0.876280
$184$	0	0
$185$	0.291796	0.0214533
$186$	0	0
$187$	−0.763932	−0.0558642
$188$	0	0
$189$	−5.47214	−0.398039
$190$	0	0
$191$	20.7984	1.50492	0.752459	−	0.658639i	$-0.228868\pi$
$191$	20.7984	1.50492	0.752459	+	0.658639i	$0.228868\pi$
$192$	0	0
$193$	−14.1803	−1.02072	−0.510362	−	0.859960i	$-0.670488\pi$
$193$	−14.1803	−1.02072	−0.510362	+	0.859960i	$0.670488\pi$
$194$	0	0
$195$	−1.23607	−0.0885167
$196$	0	0
$197$	9.70820	0.691681	0.345840	−	0.938293i	$-0.387594\pi$
$197$	9.70820	0.691681	0.345840	+	0.938293i	$0.387594\pi$
$198$	0	0
$199$	−12.0902	−0.857049	−0.428525	−	0.903530i	$-0.640967\pi$
$199$	−12.0902	−0.857049	−0.428525	+	0.903530i	$0.640967\pi$
$200$	0	0
$201$	21.1803	1.49395
$202$	0	0
$203$	5.70820	0.400637
$204$	0	0
$205$	−0.0557281	−0.00389222
$206$	0	0
$207$	2.94427	0.204641
$208$	0	0
$209$	6.47214	0.447687
$210$	0	0
$211$	−2.29180	−0.157774	−0.0788869	−	0.996884i	$-0.525137\pi$
$211$	−2.29180	−0.157774	−0.0788869	+	0.996884i	$0.525137\pi$
$212$	0	0
$213$	−17.7082	−1.21335
$214$	0	0
$215$	−7.47214	−0.509595
$216$	0	0
$217$	−6.32624	−0.429453
$218$	0	0
$219$	−11.5623	−0.781308
$220$	0	0
$221$	1.23607	0.0831469
$222$	0	0
$223$	2.94427	0.197163	0.0985815	−	0.995129i	$-0.468569\pi$
$223$	2.94427	0.197163	0.0985815	+	0.995129i	$0.468569\pi$
$224$	0	0
$225$	1.76393	0.117595
$226$	0	0
$227$	25.7426	1.70860	0.854300	−	0.519781i	$-0.173986\pi$
$227$	25.7426	1.70860	0.854300	+	0.519781i	$0.173986\pi$
$228$	0	0
$229$	−19.2361	−1.27116	−0.635578	−	0.772037i	$-0.719238\pi$
$229$	−19.2361	−1.27116	−0.635578	+	0.772037i	$0.719238\pi$
$230$	0	0
$231$	−1.23607	−0.0813273
$232$	0	0
$233$	6.47214	0.424004	0.212002	−	0.977269i	$-0.432002\pi$
$233$	6.47214	0.424004	0.212002	+	0.977269i	$0.432002\pi$
$234$	0	0
$235$	−5.23607	−0.341563
$236$	0	0
$237$	−4.76393	−0.309451
$238$	0	0
$239$	11.9098	0.770383	0.385191	−	0.922837i	$-0.374135\pi$
$239$	11.9098	0.770383	0.385191	+	0.922837i	$0.374135\pi$
$240$	0	0
$241$	4.03444	0.259881	0.129941	−	0.991522i	$-0.458521\pi$
$241$	4.03444	0.259881	0.129941	+	0.991522i	$0.458521\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0.618034	0.0394847
$246$	0	0
$247$	−10.4721	−0.666326
$248$	0	0
$249$	−24.9443	−1.58078
$250$	0	0
$251$	−17.4164	−1.09931	−0.549657	−	0.835390i	$-0.685242\pi$
$251$	−17.4164	−1.09931	−0.549657	+	0.835390i	$0.685242\pi$
$252$	0	0
$253$	5.88854	0.370210
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	25.7082	1.60363	0.801817	−	0.597570i	$-0.203867\pi$
$257$	25.7082	1.60363	0.801817	+	0.597570i	$0.203867\pi$
$258$	0	0
$259$	−0.472136	−0.0293371
$260$	0	0
$261$	2.18034	0.134960
$262$	0	0
$263$	−16.9443	−1.04483	−0.522414	−	0.852692i	$-0.674969\pi$
$263$	−16.9443	−1.04483	−0.522414	+	0.852692i	$0.674969\pi$
$264$	0	0
$265$	6.67376	0.409966
$266$	0	0
$267$	−3.23607	−0.198044
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	3.70820	0.225257	0.112629	−	0.993637i	$-0.464073\pi$
$271$	3.70820	0.225257	0.112629	+	0.993637i	$0.464073\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	3.52786	0.212738
$276$	0	0
$277$	2.76393	0.166069	0.0830343	−	0.996547i	$-0.473539\pi$
$277$	2.76393	0.166069	0.0830343	+	0.996547i	$0.473539\pi$
$278$	0	0
$279$	−2.41641	−0.144667
$280$	0	0
$281$	−8.90983	−0.531516	−0.265758	−	0.964040i	$-0.585622\pi$
$281$	−8.90983	−0.531516	−0.265758	+	0.964040i	$0.585622\pi$
$282$	0	0
$283$	−4.43769	−0.263794	−0.131897	−	0.991263i	$-0.542107\pi$
$283$	−4.43769	−0.263794	−0.131897	+	0.991263i	$0.542107\pi$
$284$	0	0
$285$	8.47214	0.501846
$286$	0	0
$287$	0.0901699	0.00532256
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−24.4164	−1.43132
$292$	0	0
$293$	−6.29180	−0.367571	−0.183785	−	0.982966i	$-0.558835\pi$
$293$	−6.29180	−0.367571	−0.183785	+	0.982966i	$0.558835\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.18034	−0.242568
$298$	0	0
$299$	−9.52786	−0.551011
$300$	0	0
$301$	12.0902	0.696866
$302$	0	0
$303$	0.763932	0.0438867
$304$	0	0
$305$	−4.52786	−0.259265
$306$	0	0
$307$	1.05573	0.0602536	0.0301268	−	0.999546i	$-0.490409\pi$
$307$	1.05573	0.0602536	0.0301268	+	0.999546i	$0.490409\pi$
$308$	0	0
$309$	−24.9443	−1.41903
$310$	0	0
$311$	26.6180	1.50937	0.754685	−	0.656087i	$-0.227790\pi$
$311$	26.6180	1.50937	0.754685	+	0.656087i	$0.227790\pi$
$312$	0	0
$313$	−1.27051	−0.0718135	−0.0359067	−	0.999355i	$-0.511432\pi$
$313$	−1.27051	−0.0718135	−0.0359067	+	0.999355i	$0.511432\pi$
$314$	0	0
$315$	0.236068	0.0133009
$316$	0	0
$317$	−28.1803	−1.58277	−0.791383	−	0.611321i	$-0.790638\pi$
$317$	−28.1803	−1.58277	−0.791383	+	0.611321i	$0.790638\pi$
$318$	0	0
$319$	4.36068	0.244151
$320$	0	0
$321$	−17.4164	−0.972089
$322$	0	0
$323$	−8.47214	−0.471402
$324$	0	0
$325$	−5.70820	−0.316634
$326$	0	0
$327$	5.70820	0.315664
$328$	0	0
$329$	8.47214	0.467084
$330$	0	0
$331$	18.0344	0.991263	0.495631	−	0.868533i	$-0.334937\pi$
$331$	18.0344	0.991263	0.495631	+	0.868533i	$0.334937\pi$
$332$	0	0
$333$	−0.180340	−0.00988256
$334$	0	0
$335$	−8.09017	−0.442013
$336$	0	0
$337$	9.52786	0.519016	0.259508	−	0.965741i	$-0.416440\pi$
$337$	9.52786	0.519016	0.259508	+	0.965741i	$0.416440\pi$
$338$	0	0
$339$	−9.70820	−0.527277
$340$	0	0
$341$	−4.83282	−0.261712
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.70820	0.414996
$346$	0	0
$347$	30.9443	1.66118	0.830588	−	0.556888i	$-0.188005\pi$
$347$	30.9443	1.66118	0.830588	+	0.556888i	$0.188005\pi$
$348$	0	0
$349$	−15.5279	−0.831188	−0.415594	−	0.909550i	$-0.636426\pi$
$349$	−15.5279	−0.831188	−0.415594	+	0.909550i	$0.636426\pi$
$350$	0	0
$351$	6.76393	0.361032
$352$	0	0
$353$	−12.7639	−0.679356	−0.339678	−	0.940542i	$-0.610318\pi$
$353$	−12.7639	−0.679356	−0.339678	+	0.940542i	$0.610318\pi$
$354$	0	0
$355$	6.76393	0.358992
$356$	0	0
$357$	1.61803	0.0856354
$358$	0	0
$359$	19.5623	1.03246	0.516230	−	0.856450i	$-0.327335\pi$
$359$	19.5623	1.03246	0.516230	+	0.856450i	$0.327335\pi$
$360$	0	0
$361$	52.7771	2.77774
$362$	0	0
$363$	16.8541	0.884611
$364$	0	0
$365$	4.41641	0.231165
$366$	0	0
$367$	−19.5066	−1.01824	−0.509118	−	0.860697i	$-0.670028\pi$
$367$	−19.5066	−1.01824	−0.509118	+	0.860697i	$0.670028\pi$
$368$	0	0
$369$	0.0344419	0.00179297
$370$	0	0
$371$	−10.7984	−0.560624
$372$	0	0
$373$	−16.8541	−0.872672	−0.436336	−	0.899784i	$-0.643724\pi$
$373$	−16.8541	−0.872672	−0.436336	+	0.899784i	$0.643724\pi$
$374$	0	0
$375$	9.61803	0.496673
$376$	0	0
$377$	−7.05573	−0.363388
$378$	0	0
$379$	10.6525	0.547181	0.273590	−	0.961846i	$-0.411789\pi$
$379$	10.6525	0.547181	0.273590	+	0.961846i	$0.411789\pi$
$380$	0	0
$381$	12.7082	0.651061
$382$	0	0
$383$	−30.6525	−1.56627	−0.783134	−	0.621853i	$-0.786380\pi$
$383$	−30.6525	−1.56627	−0.783134	+	0.621853i	$0.786380\pi$
$384$	0	0
$385$	0.472136	0.0240623
$386$	0	0
$387$	4.61803	0.234748
$388$	0	0
$389$	−21.5623	−1.09325	−0.546626	−	0.837377i	$-0.684088\pi$
$389$	−21.5623	−1.09325	−0.546626	+	0.837377i	$0.684088\pi$
$390$	0	0
$391$	−7.70820	−0.389821
$392$	0	0
$393$	12.9443	0.652952
$394$	0	0
$395$	1.81966	0.0915570
$396$	0	0
$397$	−6.79837	−0.341201	−0.170600	−	0.985340i	$-0.554571\pi$
$397$	−6.79837	−0.341201	−0.170600	+	0.985340i	$0.554571\pi$
$398$	0	0
$399$	−13.7082	−0.686269
$400$	0	0
$401$	−10.4721	−0.522954	−0.261477	−	0.965210i	$-0.584209\pi$
$401$	−10.4721	−0.522954	−0.261477	+	0.965210i	$0.584209\pi$
$402$	0	0
$403$	7.81966	0.389525
$404$	0	0
$405$	−4.76393	−0.236722
$406$	0	0
$407$	−0.360680	−0.0178782
$408$	0	0
$409$	−8.18034	−0.404492	−0.202246	−	0.979335i	$-0.564824\pi$
$409$	−8.18034	−0.404492	−0.202246	+	0.979335i	$0.564824\pi$
$410$	0	0
$411$	−3.85410	−0.190109
$412$	0	0
$413$	0	0
$414$	0	0
$415$	9.52786	0.467704
$416$	0	0
$417$	22.7082	1.11203
$418$	0	0
$419$	37.4508	1.82959	0.914797	−	0.403914i	$-0.132351\pi$
$419$	37.4508	1.82959	0.914797	+	0.403914i	$0.132351\pi$
$420$	0	0
$421$	−29.2148	−1.42384	−0.711921	−	0.702260i	$-0.752174\pi$
$421$	−29.2148	−1.42384	−0.711921	+	0.702260i	$0.752174\pi$
$422$	0	0
$423$	3.23607	0.157343
$424$	0	0
$425$	−4.61803	−0.224008
$426$	0	0
$427$	7.32624	0.354542
$428$	0	0
$429$	1.52786	0.0737660
$430$	0	0
$431$	5.05573	0.243526	0.121763	−	0.992559i	$-0.461145\pi$
$431$	5.05573	0.243526	0.121763	+	0.992559i	$0.461145\pi$
$432$	0	0
$433$	16.9443	0.814290	0.407145	−	0.913364i	$-0.366524\pi$
$433$	16.9443	0.814290	0.407145	+	0.913364i	$0.366524\pi$
$434$	0	0
$435$	5.70820	0.273687
$436$	0	0
$437$	65.3050	3.12396
$438$	0	0
$439$	−3.32624	−0.158753	−0.0793763	−	0.996845i	$-0.525293\pi$
$439$	−3.32624	−0.158753	−0.0793763	+	0.996845i	$0.525293\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	0	0
$443$	25.8885	1.23000	0.615001	−	0.788526i	$-0.289156\pi$
$443$	25.8885	1.23000	0.615001	+	0.788526i	$0.289156\pi$
$444$	0	0
$445$	1.23607	0.0585952
$446$	0	0
$447$	22.0344	1.04219
$448$	0	0
$449$	17.7082	0.835702	0.417851	−	0.908516i	$-0.362783\pi$
$449$	17.7082	0.835702	0.417851	+	0.908516i	$0.362783\pi$
$450$	0	0
$451$	0.0688837	0.00324361
$452$	0	0
$453$	24.5066	1.15142
$454$	0	0
$455$	−0.763932	−0.0358137
$456$	0	0
$457$	0.618034	0.0289104	0.0144552	−	0.999896i	$-0.495399\pi$
$457$	0.618034	0.0289104	0.0144552	+	0.999896i	$0.495399\pi$
$458$	0	0
$459$	5.47214	0.255417
$460$	0	0
$461$	−12.9443	−0.602875	−0.301437	−	0.953486i	$-0.597466\pi$
$461$	−12.9443	−0.602875	−0.301437	+	0.953486i	$0.597466\pi$
$462$	0	0
$463$	−13.0344	−0.605762	−0.302881	−	0.953028i	$-0.597948\pi$
$463$	−13.0344	−0.605762	−0.302881	+	0.953028i	$0.597948\pi$
$464$	0	0
$465$	−6.32624	−0.293372
$466$	0	0
$467$	10.4721	0.484593	0.242296	−	0.970202i	$-0.422099\pi$
$467$	10.4721	0.484593	0.242296	+	0.970202i	$0.422099\pi$
$468$	0	0
$469$	13.0902	0.604448
$470$	0	0
$471$	3.52786	0.162555
$472$	0	0
$473$	9.23607	0.424675
$474$	0	0
$475$	39.1246	1.79516
$476$	0	0
$477$	−4.12461	−0.188853
$478$	0	0
$479$	−24.2705	−1.10895	−0.554474	−	0.832201i	$-0.687080\pi$
$479$	−24.2705	−1.10895	−0.554474	+	0.832201i	$0.687080\pi$
$480$	0	0
$481$	0.583592	0.0266095
$482$	0	0
$483$	−12.4721	−0.567502
$484$	0	0
$485$	9.32624	0.423483
$486$	0	0
$487$	−22.3607	−1.01326	−0.506630	−	0.862164i	$-0.669109\pi$
$487$	−22.3607	−1.01326	−0.506630	+	0.862164i	$0.669109\pi$
$488$	0	0
$489$	−9.23607	−0.417669
$490$	0	0
$491$	−1.09017	−0.0491987	−0.0245993	−	0.999697i	$-0.507831\pi$
$491$	−1.09017	−0.0491987	−0.0245993	+	0.999697i	$0.507831\pi$
$492$	0	0
$493$	−5.70820	−0.257085
$494$	0	0
$495$	0.180340	0.00810568
$496$	0	0
$497$	−10.9443	−0.490918
$498$	0	0
$499$	17.0557	0.763519	0.381760	−	0.924262i	$-0.375318\pi$
$499$	17.0557	0.763519	0.381760	+	0.924262i	$0.375318\pi$
$500$	0	0
$501$	−23.2705	−1.03965
$502$	0	0
$503$	−10.6180	−0.473435	−0.236717	−	0.971579i	$-0.576072\pi$
$503$	−10.6180	−0.473435	−0.236717	+	0.971579i	$0.576072\pi$
$504$	0	0
$505$	−0.291796	−0.0129848
$506$	0	0
$507$	18.5623	0.824381
$508$	0	0
$509$	−0.763932	−0.0338607	−0.0169303	−	0.999857i	$-0.505389\pi$
$509$	−0.763932	−0.0338607	−0.0169303	+	0.999857i	$0.505389\pi$
$510$	0	0
$511$	−7.14590	−0.316116
$512$	0	0
$513$	−46.3607	−2.04687
$514$	0	0
$515$	9.52786	0.419848
$516$	0	0
$517$	6.47214	0.284644
$518$	0	0
$519$	19.2705	0.845881
$520$	0	0
$521$	−26.7426	−1.17162	−0.585808	−	0.810450i	$-0.699223\pi$
$521$	−26.7426	−1.17162	−0.585808	+	0.810450i	$0.699223\pi$
$522$	0	0
$523$	34.1803	1.49460	0.747301	−	0.664486i	$-0.231349\pi$
$523$	34.1803	1.49460	0.747301	+	0.664486i	$0.231349\pi$
$524$	0	0
$525$	−7.47214	−0.326111
$526$	0	0
$527$	6.32624	0.275575
$528$	0	0
$529$	36.4164	1.58332
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−0.111456	−0.00482770
$534$	0	0
$535$	6.65248	0.287612
$536$	0	0
$537$	29.7426	1.28349
$538$	0	0
$539$	−0.763932	−0.0329049
$540$	0	0
$541$	−5.23607	−0.225116	−0.112558	−	0.993645i	$-0.535904\pi$
$541$	−5.23607	−0.225116	−0.112558	+	0.993645i	$0.535904\pi$
$542$	0	0
$543$	20.1803	0.866021
$544$	0	0
$545$	−2.18034	−0.0933955
$546$	0	0
$547$	−40.6525	−1.73817	−0.869087	−	0.494659i	$-0.835293\pi$
$547$	−40.6525	−1.73817	−0.869087	+	0.494659i	$0.835293\pi$
$548$	0	0
$549$	2.79837	0.119432
$550$	0	0
$551$	48.3607	2.06023
$552$	0	0
$553$	−2.94427	−0.125203
$554$	0	0
$555$	−0.472136	−0.0200411
$556$	0	0
$557$	34.3607	1.45591	0.727954	−	0.685626i	$-0.240471\pi$
$557$	34.3607	1.45591	0.727954	+	0.685626i	$0.240471\pi$
$558$	0	0
$559$	−14.9443	−0.632075
$560$	0	0
$561$	1.23607	0.0521868
$562$	0	0
$563$	−29.1246	−1.22746	−0.613728	−	0.789518i	$-0.710331\pi$
$563$	−29.1246	−1.22746	−0.613728	+	0.789518i	$0.710331\pi$
$564$	0	0
$565$	3.70820	0.156005
$566$	0	0
$567$	7.70820	0.323714
$568$	0	0
$569$	−43.5066	−1.82389	−0.911945	−	0.410312i	$-0.865420\pi$
$569$	−43.5066	−1.82389	−0.911945	+	0.410312i	$0.865420\pi$
$570$	0	0
$571$	8.47214	0.354548	0.177274	−	0.984162i	$-0.443272\pi$
$571$	8.47214	0.354548	0.177274	+	0.984162i	$0.443272\pi$
$572$	0	0
$573$	−33.6525	−1.40585
$574$	0	0
$575$	35.5967	1.48449
$576$	0	0
$577$	17.1246	0.712907	0.356453	−	0.934313i	$-0.383986\pi$
$577$	17.1246	0.712907	0.356453	+	0.934313i	$0.383986\pi$
$578$	0	0
$579$	22.9443	0.953531
$580$	0	0
$581$	−15.4164	−0.639580
$582$	0	0
$583$	−8.24922	−0.341648
$584$	0	0
$585$	−0.291796	−0.0120643
$586$	0	0
$587$	−42.8328	−1.76790	−0.883950	−	0.467582i	$-0.845125\pi$
$587$	−42.8328	−1.76790	−0.883950	+	0.467582i	$0.845125\pi$
$588$	0	0
$589$	−53.5967	−2.20842
$590$	0	0
$591$	−15.7082	−0.646149
$592$	0	0
$593$	−22.3607	−0.918243	−0.459122	−	0.888373i	$-0.651836\pi$
$593$	−22.3607	−0.918243	−0.459122	+	0.888373i	$0.651836\pi$
$594$	0	0
$595$	−0.618034	−0.0253369
$596$	0	0
$597$	19.5623	0.800632
$598$	0	0
$599$	36.0902	1.47460	0.737302	−	0.675563i	$-0.236099\pi$
$599$	36.0902	1.47460	0.737302	+	0.675563i	$0.236099\pi$
$600$	0	0
$601$	48.2492	1.96813	0.984063	−	0.177818i	$-0.0569037\pi$
$601$	48.2492	1.96813	0.984063	+	0.177818i	$0.0569037\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	0	0
$605$	−6.43769	−0.261729
$606$	0	0
$607$	−39.8541	−1.61763	−0.808814	−	0.588064i	$-0.799890\pi$
$607$	−39.8541	−1.61763	−0.808814	+	0.588064i	$0.799890\pi$
$608$	0	0
$609$	−9.23607	−0.374264
$610$	0	0
$611$	−10.4721	−0.423657
$612$	0	0
$613$	−17.5623	−0.709335	−0.354667	−	0.934993i	$-0.615406\pi$
$613$	−17.5623	−0.709335	−0.354667	+	0.934993i	$0.615406\pi$
$614$	0	0
$615$	0.0901699	0.00363600
$616$	0	0
$617$	−35.0132	−1.40958	−0.704788	−	0.709418i	$-0.748958\pi$
$617$	−35.0132	−1.40958	−0.704788	+	0.709418i	$0.748958\pi$
$618$	0	0
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	0	0
$621$	−42.1803	−1.69264
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	−10.4721	−0.418217
$628$	0	0
$629$	0.472136	0.0188253
$630$	0	0
$631$	−37.0902	−1.47654	−0.738268	−	0.674507i	$-0.764356\pi$
$631$	−37.0902	−1.47654	−0.738268	+	0.674507i	$0.764356\pi$
$632$	0	0
$633$	3.70820	0.147388
$634$	0	0
$635$	−4.85410	−0.192629
$636$	0	0
$637$	1.23607	0.0489748
$638$	0	0
$639$	−4.18034	−0.165372
$640$	0	0
$641$	21.0557	0.831651	0.415826	−	0.909444i	$-0.363493\pi$
$641$	21.0557	0.831651	0.415826	+	0.909444i	$0.363493\pi$
$642$	0	0
$643$	3.14590	0.124062	0.0620311	−	0.998074i	$-0.480242\pi$
$643$	3.14590	0.124062	0.0620311	+	0.998074i	$0.480242\pi$
$644$	0	0
$645$	12.0902	0.476050
$646$	0	0
$647$	−20.7639	−0.816314	−0.408157	−	0.912912i	$-0.633828\pi$
$647$	−20.7639	−0.816314	−0.408157	+	0.912912i	$0.633828\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	10.2361	0.401183
$652$	0	0
$653$	−32.4721	−1.27073	−0.635366	−	0.772211i	$-0.719151\pi$
$653$	−32.4721	−1.27073	−0.635366	+	0.772211i	$0.719151\pi$
$654$	0	0
$655$	−4.94427	−0.193189
$656$	0	0
$657$	−2.72949	−0.106488
$658$	0	0
$659$	1.74265	0.0678838	0.0339419	−	0.999424i	$-0.489194\pi$
$659$	1.74265	0.0678838	0.0339419	+	0.999424i	$0.489194\pi$
$660$	0	0
$661$	30.8328	1.19926	0.599629	−	0.800278i	$-0.295315\pi$
$661$	30.8328	1.19926	0.599629	+	0.800278i	$0.295315\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	5.23607	0.203046
$666$	0	0
$667$	44.0000	1.70369
$668$	0	0
$669$	−4.76393	−0.184184
$670$	0	0
$671$	5.59675	0.216060
$672$	0	0
$673$	−14.4721	−0.557860	−0.278930	−	0.960311i	$-0.589980\pi$
$673$	−14.4721	−0.557860	−0.278930	+	0.960311i	$0.589980\pi$
$674$	0	0
$675$	−25.2705	−0.972662
$676$	0	0
$677$	−33.7771	−1.29816	−0.649079	−	0.760721i	$-0.724846\pi$
$677$	−33.7771	−1.29816	−0.649079	+	0.760721i	$0.724846\pi$
$678$	0	0
$679$	−15.0902	−0.579108
$680$	0	0
$681$	−41.6525	−1.59613
$682$	0	0
$683$	28.1803	1.07829	0.539145	−	0.842213i	$-0.318747\pi$
$683$	28.1803	1.07829	0.539145	+	0.842213i	$0.318747\pi$
$684$	0	0
$685$	1.47214	0.0562474
$686$	0	0
$687$	31.1246	1.18748
$688$	0	0
$689$	13.3475	0.508500
$690$	0	0
$691$	8.49342	0.323105	0.161553	−	0.986864i	$-0.448350\pi$
$691$	8.49342	0.323105	0.161553	+	0.986864i	$0.448350\pi$
$692$	0	0
$693$	−0.291796	−0.0110844
$694$	0	0
$695$	−8.67376	−0.329015
$696$	0	0
$697$	−0.0901699	−0.00341543
$698$	0	0
$699$	−10.4721	−0.396093
$700$	0	0
$701$	−45.4164	−1.71535	−0.857677	−	0.514189i	$-0.828093\pi$
$701$	−45.4164	−1.71535	−0.857677	+	0.514189i	$0.828093\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	8.47214	0.319079
$706$	0	0
$707$	0.472136	0.0177565
$708$	0	0
$709$	40.6525	1.52674	0.763368	−	0.645964i	$-0.223544\pi$
$709$	40.6525	1.52674	0.763368	+	0.645964i	$0.223544\pi$
$710$	0	0
$711$	−1.12461	−0.0421762
$712$	0	0
$713$	−48.7639	−1.82622
$714$	0	0
$715$	−0.583592	−0.0218251
$716$	0	0
$717$	−19.2705	−0.719670
$718$	0	0
$719$	−39.1459	−1.45990	−0.729948	−	0.683503i	$-0.760456\pi$
$719$	−39.1459	−1.45990	−0.729948	+	0.683503i	$0.760456\pi$
$720$	0	0
$721$	−15.4164	−0.574137
$722$	0	0
$723$	−6.52786	−0.242774
$724$	0	0
$725$	26.3607	0.979011
$726$	0	0
$727$	16.4721	0.610918	0.305459	−	0.952205i	$-0.401190\pi$
$727$	16.4721	0.610918	0.305459	+	0.952205i	$0.401190\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−12.0902	−0.447171
$732$	0	0
$733$	36.0000	1.32969	0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000	1.32969	0.664845	+	0.746981i	$0.268498\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	10.0000	0.368355
$738$	0	0
$739$	−1.43769	−0.0528864	−0.0264432	−	0.999650i	$-0.508418\pi$
$739$	−1.43769	−0.0528864	−0.0264432	+	0.999650i	$0.508418\pi$
$740$	0	0
$741$	16.9443	0.622463
$742$	0	0
$743$	4.65248	0.170683	0.0853414	−	0.996352i	$-0.472802\pi$
$743$	4.65248	0.170683	0.0853414	+	0.996352i	$0.472802\pi$
$744$	0	0
$745$	−8.41641	−0.308353
$746$	0	0
$747$	−5.88854	−0.215451
$748$	0	0
$749$	−10.7639	−0.393306
$750$	0	0
$751$	10.9443	0.399362	0.199681	−	0.979861i	$-0.436009\pi$
$751$	10.9443	0.399362	0.199681	+	0.979861i	$0.436009\pi$
$752$	0	0
$753$	28.1803	1.02695
$754$	0	0
$755$	−9.36068	−0.340670
$756$	0	0
$757$	−9.43769	−0.343019	−0.171509	−	0.985182i	$-0.554864\pi$
$757$	−9.43769	−0.343019	−0.171509	+	0.985182i	$0.554864\pi$
$758$	0	0
$759$	−9.52786	−0.345840
$760$	0	0
$761$	30.1803	1.09404	0.547018	−	0.837121i	$-0.315763\pi$
$761$	30.1803	1.09404	0.547018	+	0.837121i	$0.315763\pi$
$762$	0	0
$763$	3.52786	0.127717
$764$	0	0
$765$	−0.236068	−0.00853506
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−7.12461	−0.256920	−0.128460	−	0.991715i	$-0.541003\pi$
$769$	−7.12461	−0.256920	−0.128460	+	0.991715i	$0.541003\pi$
$770$	0	0
$771$	−41.5967	−1.49807
$772$	0	0
$773$	5.41641	0.194815	0.0974073	−	0.995245i	$-0.468945\pi$
$773$	5.41641	0.194815	0.0974073	+	0.995245i	$0.468945\pi$
$774$	0	0
$775$	−29.2148	−1.04943
$776$	0	0
$777$	0.763932	0.0274059
$778$	0	0
$779$	0.763932	0.0273707
$780$	0	0
$781$	−8.36068	−0.299169
$782$	0	0
$783$	−31.2361	−1.11629
$784$	0	0
$785$	−1.34752	−0.0480952
$786$	0	0
$787$	−12.0000	−0.427754	−0.213877	−	0.976861i	$-0.568609\pi$
$787$	−12.0000	−0.427754	−0.213877	+	0.976861i	$0.568609\pi$
$788$	0	0
$789$	27.4164	0.976050
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−9.05573	−0.321578
$794$	0	0
$795$	−10.7984	−0.382979
$796$	0	0
$797$	15.4164	0.546077	0.273039	−	0.962003i	$-0.411971\pi$
$797$	15.4164	0.546077	0.273039	+	0.962003i	$0.411971\pi$
$798$	0	0
$799$	−8.47214	−0.299723
$800$	0	0
$801$	−0.763932	−0.0269922
$802$	0	0
$803$	−5.45898	−0.192643
$804$	0	0
$805$	4.76393	0.167907
$806$	0	0
$807$	16.1803	0.569575
$808$	0	0
$809$	−53.4853	−1.88044	−0.940221	−	0.340564i	$-0.889382\pi$
$809$	−53.4853	−1.88044	−0.940221	+	0.340564i	$0.889382\pi$
$810$	0	0
$811$	−2.96556	−0.104135	−0.0520674	−	0.998644i	$-0.516581\pi$
$811$	−2.96556	−0.104135	−0.0520674	+	0.998644i	$0.516581\pi$
$812$	0	0
$813$	−6.00000	−0.210429
$814$	0	0
$815$	3.52786	0.123576
$816$	0	0
$817$	102.430	3.58356
$818$	0	0
$819$	0.472136	0.0164978
$820$	0	0
$821$	14.3607	0.501191	0.250596	−	0.968092i	$-0.419374\pi$
$821$	14.3607	0.501191	0.250596	+	0.968092i	$0.419374\pi$
$822$	0	0
$823$	47.4853	1.65523	0.827617	−	0.561294i	$-0.189696\pi$
$823$	47.4853	1.65523	0.827617	+	0.561294i	$0.189696\pi$
$824$	0	0
$825$	−5.70820	−0.198734
$826$	0	0
$827$	27.5279	0.957238	0.478619	−	0.878023i	$-0.341138\pi$
$827$	27.5279	0.957238	0.478619	+	0.878023i	$0.341138\pi$
$828$	0	0
$829$	−6.18034	−0.214652	−0.107326	−	0.994224i	$-0.534229\pi$
$829$	−6.18034	−0.214652	−0.107326	+	0.994224i	$0.534229\pi$
$830$	0	0
$831$	−4.47214	−0.155137
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	8.88854	0.307601
$836$	0	0
$837$	34.6180	1.19657
$838$	0	0
$839$	−13.5279	−0.467034	−0.233517	−	0.972353i	$-0.575023\pi$
$839$	−13.5279	−0.467034	−0.233517	+	0.972353i	$0.575023\pi$
$840$	0	0
$841$	3.58359	0.123572
$842$	0	0
$843$	14.4164	0.496527
$844$	0	0
$845$	−7.09017	−0.243909
$846$	0	0
$847$	10.4164	0.357912
$848$	0	0
$849$	7.18034	0.246429
$850$	0	0
$851$	−3.63932	−0.124754
$852$	0	0
$853$	−38.3607	−1.31344	−0.656722	−	0.754132i	$-0.728058\pi$
$853$	−38.3607	−1.31344	−0.656722	+	0.754132i	$0.728058\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−15.0902	−0.515470	−0.257735	−	0.966216i	$-0.582976\pi$
$857$	−15.0902	−0.515470	−0.257735	+	0.966216i	$0.582976\pi$
$858$	0	0
$859$	14.9443	0.509892	0.254946	−	0.966955i	$-0.417942\pi$
$859$	14.9443	0.509892	0.254946	+	0.966955i	$0.417942\pi$
$860$	0	0
$861$	−0.145898	−0.00497219
$862$	0	0
$863$	−46.3262	−1.57696	−0.788482	−	0.615058i	$-0.789133\pi$
$863$	−46.3262	−1.57696	−0.788482	+	0.615058i	$0.789133\pi$
$864$	0	0
$865$	−7.36068	−0.250271
$866$	0	0
$867$	−1.61803	−0.0549513
$868$	0	0
$869$	−2.24922	−0.0762997
$870$	0	0
$871$	−16.1803	−0.548250
$872$	0	0
$873$	−5.76393	−0.195080
$874$	0	0
$875$	5.94427	0.200953
$876$	0	0
$877$	18.2918	0.617670	0.308835	−	0.951116i	$-0.400061\pi$
$877$	18.2918	0.617670	0.308835	+	0.951116i	$0.400061\pi$
$878$	0	0
$879$	10.1803	0.343374
$880$	0	0
$881$	−19.6738	−0.662826	−0.331413	−	0.943486i	$-0.607525\pi$
$881$	−19.6738	−0.662826	−0.331413	+	0.943486i	$0.607525\pi$
$882$	0	0
$883$	14.9787	0.504074	0.252037	−	0.967718i	$-0.418900\pi$
$883$	14.9787	0.504074	0.252037	+	0.967718i	$0.418900\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.5066	−0.554237	−0.277118	−	0.960836i	$-0.589379\pi$
$887$	−16.5066	−0.554237	−0.277118	+	0.960836i	$0.589379\pi$
$888$	0	0
$889$	7.85410	0.263418
$890$	0	0
$891$	5.88854	0.197274
$892$	0	0
$893$	71.7771	2.40193
$894$	0	0
$895$	−11.3607	−0.379746
$896$	0	0
$897$	15.4164	0.514739
$898$	0	0
$899$	−36.1115	−1.20438
$900$	0	0
$901$	10.7984	0.359746
$902$	0	0
$903$	−19.5623	−0.650993
$904$	0	0
$905$	−7.70820	−0.256229
$906$	0	0
$907$	−31.3050	−1.03946	−0.519732	−	0.854329i	$-0.673968\pi$
$907$	−31.3050	−1.03946	−0.519732	+	0.854329i	$0.673968\pi$
$908$	0	0
$909$	0.180340	0.00598150
$910$	0	0
$911$	−14.4721	−0.479483	−0.239742	−	0.970837i	$-0.577063\pi$
$911$	−14.4721	−0.479483	−0.239742	+	0.970837i	$0.577063\pi$
$912$	0	0
$913$	−11.7771	−0.389765
$914$	0	0
$915$	7.32624	0.242198
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	−30.1591	−0.994855	−0.497428	−	0.867505i	$-0.665722\pi$
$919$	−30.1591	−0.994855	−0.497428	+	0.867505i	$0.665722\pi$
$920$	0	0
$921$	−1.70820	−0.0562872
$922$	0	0
$923$	13.5279	0.445275
$924$	0	0
$925$	−2.18034	−0.0716891
$926$	0	0
$927$	−5.88854	−0.193405
$928$	0	0
$929$	21.2705	0.697863	0.348931	−	0.937148i	$-0.386545\pi$
$929$	21.2705	0.697863	0.348931	+	0.937148i	$0.386545\pi$
$930$	0	0
$931$	−8.47214	−0.277663
$932$	0	0
$933$	−43.0689	−1.41001
$934$	0	0
$935$	−0.472136	−0.0154405
$936$	0	0
$937$	−5.30495	−0.173305	−0.0866526	−	0.996239i	$-0.527617\pi$
$937$	−5.30495	−0.173305	−0.0866526	+	0.996239i	$0.527617\pi$
$938$	0	0
$939$	2.05573	0.0670862
$940$	0	0
$941$	45.9787	1.49886	0.749432	−	0.662082i	$-0.230327\pi$
$941$	45.9787	1.49886	0.749432	+	0.662082i	$0.230327\pi$
$942$	0	0
$943$	0.695048	0.0226339
$944$	0	0
$945$	−3.38197	−0.110015
$946$	0	0
$947$	6.87539	0.223420	0.111710	−	0.993741i	$-0.464367\pi$
$947$	6.87539	0.223420	0.111710	+	0.993741i	$0.464367\pi$
$948$	0	0
$949$	8.83282	0.286725
$950$	0	0
$951$	45.5967	1.47858
$952$	0	0
$953$	32.3262	1.04715	0.523575	−	0.851980i	$-0.324598\pi$
$953$	32.3262	1.04715	0.523575	+	0.851980i	$0.324598\pi$
$954$	0	0
$955$	12.8541	0.415949
$956$	0	0
$957$	−7.05573	−0.228079
$958$	0	0
$959$	−2.38197	−0.0769177
$960$	0	0
$961$	9.02129	0.291009
$962$	0	0
$963$	−4.11146	−0.132490
$964$	0	0
$965$	−8.76393	−0.282121
$966$	0	0
$967$	37.7984	1.21551	0.607757	−	0.794123i	$-0.292070\pi$
$967$	37.7984	1.21551	0.607757	+	0.794123i	$0.292070\pi$
$968$	0	0
$969$	13.7082	0.440371
$970$	0	0
$971$	21.1246	0.677921	0.338961	−	0.940801i	$-0.389925\pi$
$971$	21.1246	0.677921	0.338961	+	0.940801i	$0.389925\pi$
$972$	0	0
$973$	14.0344	0.449924
$974$	0	0
$975$	9.23607	0.295791
$976$	0	0
$977$	−2.21478	−0.0708571	−0.0354286	−	0.999372i	$-0.511280\pi$
$977$	−2.21478	−0.0708571	−0.0354286	+	0.999372i	$0.511280\pi$
$978$	0	0
$979$	−1.52786	−0.0488307
$980$	0	0
$981$	1.34752	0.0430231
$982$	0	0
$983$	6.90983	0.220389	0.110195	−	0.993910i	$-0.464853\pi$
$983$	6.90983	0.220389	0.110195	+	0.993910i	$0.464853\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−13.7082	−0.436337
$988$	0	0
$989$	93.1935	2.96338
$990$	0	0
$991$	−28.2492	−0.897366	−0.448683	−	0.893691i	$-0.648107\pi$
$991$	−28.2492	−0.897366	−0.448683	+	0.893691i	$0.648107\pi$
$992$	0	0
$993$	−29.1803	−0.926010
$994$	0	0
$995$	−7.47214	−0.236883
$996$	0	0
$997$	29.0344	0.919530	0.459765	−	0.888041i	$-0.347934\pi$
$997$	29.0344	0.919530	0.459765	+	0.888041i	$0.347934\pi$
$998$	0	0
$999$	2.58359	0.0817412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.2.a.b.1.1	✓	2
3.2	odd	2		4284.2.a.m.1.1		2
4.3	odd	2		1904.2.a.j.1.2		2
7.6	odd	2		3332.2.a.l.1.2		2
8.3	odd	2		7616.2.a.p.1.1		2
8.5	even	2		7616.2.a.u.1.2		2
17.16	even	2		8092.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.2.a.b.1.1	✓	2	1.1	even	1	trivial
1904.2.a.j.1.2		2	4.3	odd	2
3332.2.a.l.1.2		2	7.6	odd	2
4284.2.a.m.1.1		2	3.2	odd	2
7616.2.a.p.1.1		2	8.3	odd	2
7616.2.a.u.1.2		2	8.5	even	2
8092.2.a.m.1.2		2	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} +0.618034 q^{5} -1.00000 q^{7} -0.381966 q^{9} -0.763932 q^{11} +1.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} -8.47214 q^{19} +1.61803 q^{21} -7.70820 q^{23} -4.61803 q^{25} +5.47214 q^{27} -5.70820 q^{29} +6.32624 q^{31} +1.23607 q^{33} -0.618034 q^{35} +0.472136 q^{37} -2.00000 q^{39} -0.0901699 q^{41} -12.0902 q^{43} -0.236068 q^{45} -8.47214 q^{47} +1.00000 q^{49} -1.61803 q^{51} +10.7984 q^{53} -0.472136 q^{55} +13.7082 q^{57} -7.32624 q^{61} +0.381966 q^{63} +0.763932 q^{65} -13.0902 q^{67} +12.4721 q^{69} +10.9443 q^{71} +7.14590 q^{73} +7.47214 q^{75} +0.763932 q^{77} +2.94427 q^{79} -7.70820 q^{81} +15.4164 q^{83} +0.618034 q^{85} +9.23607 q^{87} +2.00000 q^{89} -1.23607 q^{91} -10.2361 q^{93} -5.23607 q^{95} +15.0902 q^{97} +0.291796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 + 2 * q^17 - 8 * q^19 + q^21 - 2 * q^23 - 7 * q^25 + 2 * q^27 + 2 * q^29 - 3 * q^31 - 2 * q^33 + q^35 - 8 * q^37 - 4 * q^39 + 11 * q^41 - 13 * q^43 + 4 * q^45 - 8 * q^47 + 2 * q^49 - q^51 - 3 * q^53 + 8 * q^55 + 14 * q^57 + q^61 + 3 * q^63 + 6 * q^65 - 15 * q^67 + 16 * q^69 + 4 * q^71 + 21 * q^73 + 6 * q^75 + 6 * q^77 - 12 * q^79 - 2 * q^81 + 4 * q^83 - q^85 + 14 * q^87 + 4 * q^89 + 2 * q^91 - 16 * q^93 - 6 * q^95 + 19 * q^97 + 14 * q^99

Introduction

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