LMFDB - Embedded newform 476.2.a.b.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:	$1$
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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+0.618034 q^{3} -1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} -5.23607 q^{11} -3.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} -0.618034 q^{21} +5.70820 q^{23} -2.38197 q^{25} -3.47214 q^{27} +7.70820 q^{29} -9.32624 q^{31} -3.23607 q^{33} +1.61803 q^{35} -8.47214 q^{37} -2.00000 q^{39} +11.0902 q^{41} -0.909830 q^{43} +4.23607 q^{45} +0.472136 q^{47} +1.00000 q^{49} +0.618034 q^{51} -13.7984 q^{53} +8.47214 q^{55} +0.291796 q^{57} +8.32624 q^{61} +2.61803 q^{63} +5.23607 q^{65} -1.90983 q^{67} +3.52786 q^{69} -6.94427 q^{71} +13.8541 q^{73} -1.47214 q^{75} +5.23607 q^{77} -14.9443 q^{79} +5.70820 q^{81} -11.4164 q^{83} -1.61803 q^{85} +4.76393 q^{87} +2.00000 q^{89} +3.23607 q^{91} -5.76393 q^{93} -0.763932 q^{95} +3.90983 q^{97} +13.7082 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	0	0
$13$	−3.23607	−0.897524	−0.448762	−	0.893651i	$-0.648135\pi$
$13$	−3.23607	−0.897524	−0.448762	+	0.893651i	$0.648135\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.472136	0.108315	0.0541577	−	0.998532i	$-0.482753\pi$
$19$	0.472136	0.108315	0.0541577	+	0.998532i	$0.482753\pi$
$20$	0	0
$21$	−0.618034	−0.134866
$22$	0	0
$23$	5.70820	1.19024	0.595121	−	0.803636i	$-0.297104\pi$
$23$	5.70820	1.19024	0.595121	+	0.803636i	$0.297104\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	7.70820	1.43138	0.715689	−	0.698419i	$-0.246113\pi$
$29$	7.70820	1.43138	0.715689	+	0.698419i	$0.246113\pi$
$30$	0	0
$31$	−9.32624	−1.67504	−0.837521	−	0.546405i	$-0.815996\pi$
$31$	−9.32624	−1.67504	−0.837521	+	0.546405i	$0.815996\pi$
$32$	0	0
$33$	−3.23607	−0.563327
$34$	0	0
$35$	1.61803	0.273498
$36$	0	0
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	11.0902	1.73199	0.865997	−	0.500050i	$-0.166685\pi$
$41$	11.0902	1.73199	0.865997	+	0.500050i	$0.166685\pi$
$42$	0	0
$43$	−0.909830	−0.138748	−0.0693739	−	0.997591i	$-0.522100\pi$
$43$	−0.909830	−0.138748	−0.0693739	+	0.997591i	$0.522100\pi$
$44$	0	0
$45$	4.23607	0.631476
$46$	0	0
$47$	0.472136	0.0688681	0.0344341	−	0.999407i	$-0.489037\pi$
$47$	0.472136	0.0688681	0.0344341	+	0.999407i	$0.489037\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.618034	0.0865421
$52$	0	0
$53$	−13.7984	−1.89535	−0.947676	−	0.319233i	$-0.896575\pi$
$53$	−13.7984	−1.89535	−0.947676	+	0.319233i	$0.896575\pi$
$54$	0	0
$55$	8.47214	1.14238
$56$	0	0
$57$	0.291796	0.0386493
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.32624	1.06607	0.533033	−	0.846095i	$-0.321052\pi$
$61$	8.32624	1.06607	0.533033	+	0.846095i	$0.321052\pi$
$62$	0	0
$63$	2.61803	0.329841
$64$	0	0
$65$	5.23607	0.649454
$66$	0	0
$67$	−1.90983	−0.233323	−0.116661	−	0.993172i	$-0.537219\pi$
$67$	−1.90983	−0.233323	−0.116661	+	0.993172i	$0.537219\pi$
$68$	0	0
$69$	3.52786	0.424705
$70$	0	0
$71$	−6.94427	−0.824133	−0.412067	−	0.911154i	$-0.635193\pi$
$71$	−6.94427	−0.824133	−0.412067	+	0.911154i	$0.635193\pi$
$72$	0	0
$73$	13.8541	1.62150	0.810750	−	0.585393i	$-0.199060\pi$
$73$	13.8541	1.62150	0.810750	+	0.585393i	$0.199060\pi$
$74$	0	0
$75$	−1.47214	−0.169988
$76$	0	0
$77$	5.23607	0.596705
$78$	0	0
$79$	−14.9443	−1.68136	−0.840681	−	0.541531i	$-0.817845\pi$
$79$	−14.9443	−1.68136	−0.840681	+	0.541531i	$0.817845\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−11.4164	−1.25311	−0.626557	−	0.779376i	$-0.715536\pi$
$83$	−11.4164	−1.25311	−0.626557	+	0.779376i	$0.715536\pi$
$84$	0	0
$85$	−1.61803	−0.175500
$86$	0	0
$87$	4.76393	0.510747
$88$	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
$91$	3.23607	0.339232
$92$	0	0
$93$	−5.76393	−0.597692
$94$	0	0
$95$	−0.763932	−0.0783778
$96$	0	0
$97$	3.90983	0.396983	0.198492	−	0.980103i	$-0.436396\pi$
$97$	3.90983	0.396983	0.198492	+	0.980103i	$0.436396\pi$
$98$	0	0
$99$	13.7082	1.37773
$100$	0	0
$101$	8.47214	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$101$	8.47214	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$102$	0	0
$103$	−11.4164	−1.12489	−0.562446	−	0.826834i	$-0.690140\pi$
$103$	−11.4164	−1.12489	−0.562446	+	0.826834i	$0.690140\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	15.2361	1.47293	0.736463	−	0.676478i	$-0.236494\pi$
$107$	15.2361	1.47293	0.736463	+	0.676478i	$0.236494\pi$
$108$	0	0
$109$	−12.4721	−1.19461	−0.597307	−	0.802013i	$-0.703763\pi$
$109$	−12.4721	−1.19461	−0.597307	+	0.802013i	$0.703763\pi$
$110$	0	0
$111$	−5.23607	−0.496986
$112$	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
$115$	−9.23607	−0.861268
$116$	0	0
$117$	8.47214	0.783249
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	6.85410	0.618014
$124$	0	0
$125$	11.9443	1.06833
$126$	0	0
$127$	−1.14590	−0.101682	−0.0508410	−	0.998707i	$-0.516190\pi$
$127$	−1.14590	−0.101682	−0.0508410	+	0.998707i	$0.516190\pi$
$128$	0	0
$129$	−0.562306	−0.0495083
$130$	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
$133$	−0.472136	−0.0409394
$134$	0	0
$135$	5.61803	0.483523
$136$	0	0
$137$	4.61803	0.394545	0.197273	−	0.980349i	$-0.436792\pi$
$137$	4.61803	0.394545	0.197273	+	0.980349i	$0.436792\pi$
$138$	0	0
$139$	15.0344	1.27520	0.637602	−	0.770366i	$-0.279926\pi$
$139$	15.0344	1.27520	0.637602	+	0.770366i	$0.279926\pi$
$140$	0	0
$141$	0.291796	0.0245737
$142$	0	0
$143$	16.9443	1.41695
$144$	0	0
$145$	−12.4721	−1.03575
$146$	0	0
$147$	0.618034	0.0509746
$148$	0	0
$149$	−11.3820	−0.932447	−0.466223	−	0.884667i	$-0.654386\pi$
$149$	−11.3820	−0.932447	−0.466223	+	0.884667i	$0.654386\pi$
$150$	0	0
$151$	−21.8541	−1.77846	−0.889231	−	0.457459i	$-0.848760\pi$
$151$	−21.8541	−1.77846	−0.889231	+	0.457459i	$0.848760\pi$
$152$	0	0
$153$	−2.61803	−0.211656
$154$	0	0
$155$	15.0902	1.21207
$156$	0	0
$157$	20.1803	1.61057	0.805283	−	0.592890i	$-0.202013\pi$
$157$	20.1803	1.61057	0.805283	+	0.592890i	$0.202013\pi$
$158$	0	0
$159$	−8.52786	−0.676304
$160$	0	0
$161$	−5.70820	−0.449869
$162$	0	0
$163$	−7.70820	−0.603753	−0.301877	−	0.953347i	$-0.597613\pi$
$163$	−7.70820	−0.603753	−0.301877	+	0.953347i	$0.597613\pi$
$164$	0	0
$165$	5.23607	0.407627
$166$	0	0
$167$	16.6180	1.28594	0.642971	−	0.765890i	$-0.277702\pi$
$167$	16.6180	1.28594	0.642971	+	0.765890i	$0.277702\pi$
$168$	0	0
$169$	−2.52786	−0.194451
$170$	0	0
$171$	−1.23607	−0.0945245
$172$	0	0
$173$	−23.0902	−1.75551	−0.877757	−	0.479107i	$-0.840961\pi$
$173$	−23.0902	−1.75551	−0.877757	+	0.479107i	$0.840961\pi$
$174$	0	0
$175$	2.38197	0.180060
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.6180	−1.54106	−0.770532	−	0.637401i	$-0.780009\pi$
$179$	−20.6180	−1.54106	−0.770532	+	0.637401i	$0.780009\pi$
$180$	0	0
$181$	−3.52786	−0.262224	−0.131112	−	0.991368i	$-0.541855\pi$
$181$	−3.52786	−0.262224	−0.131112	+	0.991368i	$0.541855\pi$
$182$	0	0
$183$	5.14590	0.380396
$184$	0	0
$185$	13.7082	1.00785
$186$	0	0
$187$	−5.23607	−0.382899
$188$	0	0
$189$	3.47214	0.252561
$190$	0	0
$191$	−3.79837	−0.274841	−0.137420	−	0.990513i	$-0.543881\pi$
$191$	−3.79837	−0.274841	−0.137420	+	0.990513i	$0.543881\pi$
$192$	0	0
$193$	8.18034	0.588834	0.294417	−	0.955677i	$-0.404875\pi$
$193$	8.18034	0.588834	0.294417	+	0.955677i	$0.404875\pi$
$194$	0	0
$195$	3.23607	0.231740
$196$	0	0
$197$	−3.70820	−0.264199	−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820	−0.264199	−0.132099	+	0.991236i	$0.542172\pi$
$198$	0	0
$199$	−0.909830	−0.0644961	−0.0322481	−	0.999480i	$-0.510267\pi$
$199$	−0.909830	−0.0644961	−0.0322481	+	0.999480i	$0.510267\pi$
$200$	0	0
$201$	−1.18034	−0.0832548
$202$	0	0
$203$	−7.70820	−0.541010
$204$	0	0
$205$	−17.9443	−1.25328
$206$	0	0
$207$	−14.9443	−1.03870
$208$	0	0
$209$	−2.47214	−0.171001
$210$	0	0
$211$	−15.7082	−1.08140	−0.540699	−	0.841216i	$-0.681840\pi$
$211$	−15.7082	−1.08140	−0.540699	+	0.841216i	$0.681840\pi$
$212$	0	0
$213$	−4.29180	−0.294069
$214$	0	0
$215$	1.47214	0.100399
$216$	0	0
$217$	9.32624	0.633106
$218$	0	0
$219$	8.56231	0.578587
$220$	0	0
$221$	−3.23607	−0.217681
$222$	0	0
$223$	−14.9443	−1.00074	−0.500371	−	0.865811i	$-0.666803\pi$
$223$	−14.9443	−1.00074	−0.500371	+	0.865811i	$0.666803\pi$
$224$	0	0
$225$	6.23607	0.415738
$226$	0	0
$227$	−16.7426	−1.11125	−0.555624	−	0.831434i	$-0.687521\pi$
$227$	−16.7426	−1.11125	−0.555624	+	0.831434i	$0.687521\pi$
$228$	0	0
$229$	−14.7639	−0.975628	−0.487814	−	0.872948i	$-0.662206\pi$
$229$	−14.7639	−0.975628	−0.487814	+	0.872948i	$0.662206\pi$
$230$	0	0
$231$	3.23607	0.212918
$232$	0	0
$233$	−2.47214	−0.161955	−0.0809775	−	0.996716i	$-0.525804\pi$
$233$	−2.47214	−0.161955	−0.0809775	+	0.996716i	$0.525804\pi$
$234$	0	0
$235$	−0.763932	−0.0498334
$236$	0	0
$237$	−9.23607	−0.599947
$238$	0	0
$239$	23.0902	1.49358	0.746789	−	0.665061i	$-0.231594\pi$
$239$	23.0902	1.49358	0.746789	+	0.665061i	$0.231594\pi$
$240$	0	0
$241$	−25.0344	−1.61261	−0.806305	−	0.591500i	$-0.798536\pi$
$241$	−25.0344	−1.61261	−0.806305	+	0.591500i	$0.798536\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	−1.61803	−0.103372
$246$	0	0
$247$	−1.52786	−0.0972157
$248$	0	0
$249$	−7.05573	−0.447139
$250$	0	0
$251$	9.41641	0.594358	0.297179	−	0.954822i	$-0.403954\pi$
$251$	9.41641	0.594358	0.297179	+	0.954822i	$0.403954\pi$
$252$	0	0
$253$	−29.8885	−1.87908
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	12.2918	0.766741	0.383371	−	0.923595i	$-0.374763\pi$
$257$	12.2918	0.766741	0.383371	+	0.923595i	$0.374763\pi$
$258$	0	0
$259$	8.47214	0.526433
$260$	0	0
$261$	−20.1803	−1.24913
$262$	0	0
$263$	0.944272	0.0582263	0.0291132	−	0.999576i	$-0.490732\pi$
$263$	0.944272	0.0582263	0.0291132	+	0.999576i	$0.490732\pi$
$264$	0	0
$265$	22.3262	1.37149
$266$	0	0
$267$	1.23607	0.0756461
$268$	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
$271$	−9.70820	−0.589731	−0.294866	−	0.955539i	$-0.595275\pi$
$271$	−9.70820	−0.589731	−0.294866	+	0.955539i	$0.595275\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	12.4721	0.752098
$276$	0	0
$277$	7.23607	0.434773	0.217387	−	0.976086i	$-0.430247\pi$
$277$	7.23607	0.434773	0.217387	+	0.976086i	$0.430247\pi$
$278$	0	0
$279$	24.4164	1.46177
$280$	0	0
$281$	−20.0902	−1.19848	−0.599240	−	0.800570i	$-0.704530\pi$
$281$	−20.0902	−1.19848	−0.599240	+	0.800570i	$0.704530\pi$
$282$	0	0
$283$	−24.5623	−1.46008	−0.730039	−	0.683406i	$-0.760498\pi$
$283$	−24.5623	−1.46008	−0.730039	+	0.683406i	$0.760498\pi$
$284$	0	0
$285$	−0.472136	−0.0279669
$286$	0	0
$287$	−11.0902	−0.654632
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.41641	0.141652
$292$	0	0
$293$	−19.7082	−1.15137	−0.575683	−	0.817673i	$-0.695264\pi$
$293$	−19.7082	−1.15137	−0.575683	+	0.817673i	$0.695264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.1803	1.05493
$298$	0	0
$299$	−18.4721	−1.06827
$300$	0	0
$301$	0.909830	0.0524417
$302$	0	0
$303$	5.23607	0.300804
$304$	0	0
$305$	−13.4721	−0.771412
$306$	0	0
$307$	18.9443	1.08121	0.540603	−	0.841278i	$-0.318196\pi$
$307$	18.9443	1.08121	0.540603	+	0.841278i	$0.318196\pi$
$308$	0	0
$309$	−7.05573	−0.401386
$310$	0	0
$311$	24.3820	1.38257	0.691287	−	0.722580i	$-0.257044\pi$
$311$	24.3820	1.38257	0.691287	+	0.722580i	$0.257044\pi$
$312$	0	0
$313$	32.2705	1.82404	0.912019	−	0.410149i	$-0.134523\pi$
$313$	32.2705	1.82404	0.912019	+	0.410149i	$0.134523\pi$
$314$	0	0
$315$	−4.23607	−0.238675
$316$	0	0
$317$	−5.81966	−0.326865	−0.163432	−	0.986555i	$-0.552257\pi$
$317$	−5.81966	−0.326865	−0.163432	+	0.986555i	$0.552257\pi$
$318$	0	0
$319$	−40.3607	−2.25976
$320$	0	0
$321$	9.41641	0.525573
$322$	0	0
$323$	0.472136	0.0262703
$324$	0	0
$325$	7.70820	0.427574
$326$	0	0
$327$	−7.70820	−0.426265
$328$	0	0
$329$	−0.472136	−0.0260297
$330$	0	0
$331$	−11.0344	−0.606508	−0.303254	−	0.952910i	$-0.598073\pi$
$331$	−11.0344	−0.606508	−0.303254	+	0.952910i	$0.598073\pi$
$332$	0	0
$333$	22.1803	1.21548
$334$	0	0
$335$	3.09017	0.168834
$336$	0	0
$337$	18.4721	1.00624	0.503121	−	0.864216i	$-0.332185\pi$
$337$	18.4721	1.00624	0.503121	+	0.864216i	$0.332185\pi$
$338$	0	0
$339$	3.70820	0.201402
$340$	0	0
$341$	48.8328	2.64445
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.70820	−0.307319
$346$	0	0
$347$	13.0557	0.700868	0.350434	−	0.936587i	$-0.386034\pi$
$347$	13.0557	0.700868	0.350434	+	0.936587i	$0.386034\pi$
$348$	0	0
$349$	−24.4721	−1.30996	−0.654982	−	0.755645i	$-0.727324\pi$
$349$	−24.4721	−1.30996	−0.654982	+	0.755645i	$0.727324\pi$
$350$	0	0
$351$	11.2361	0.599737
$352$	0	0
$353$	−17.2361	−0.917383	−0.458692	−	0.888595i	$-0.651682\pi$
$353$	−17.2361	−0.917383	−0.458692	+	0.888595i	$0.651682\pi$
$354$	0	0
$355$	11.2361	0.596349
$356$	0	0
$357$	−0.618034	−0.0327098
$358$	0	0
$359$	−0.562306	−0.0296774	−0.0148387	−	0.999890i	$-0.504723\pi$
$359$	−0.562306	−0.0296774	−0.0148387	+	0.999890i	$0.504723\pi$
$360$	0	0
$361$	−18.7771	−0.988268
$362$	0	0
$363$	10.1459	0.532522
$364$	0	0
$365$	−22.4164	−1.17333
$366$	0	0
$367$	18.5066	0.966035	0.483018	−	0.875611i	$-0.339541\pi$
$367$	18.5066	0.966035	0.483018	+	0.875611i	$0.339541\pi$
$368$	0	0
$369$	−29.0344	−1.51147
$370$	0	0
$371$	13.7984	0.716376
$372$	0	0
$373$	−10.1459	−0.525335	−0.262667	−	0.964886i	$-0.584602\pi$
$373$	−10.1459	−0.525335	−0.262667	+	0.964886i	$0.584602\pi$
$374$	0	0
$375$	7.38197	0.381203
$376$	0	0
$377$	−24.9443	−1.28470
$378$	0	0
$379$	−20.6525	−1.06085	−0.530423	−	0.847733i	$-0.677967\pi$
$379$	−20.6525	−1.06085	−0.530423	+	0.847733i	$0.677967\pi$
$380$	0	0
$381$	−0.708204	−0.0362824
$382$	0	0
$383$	0.652476	0.0333400	0.0166700	−	0.999861i	$-0.494694\pi$
$383$	0.652476	0.0333400	0.0166700	+	0.999861i	$0.494694\pi$
$384$	0	0
$385$	−8.47214	−0.431780
$386$	0	0
$387$	2.38197	0.121082
$388$	0	0
$389$	−1.43769	−0.0728940	−0.0364470	−	0.999336i	$-0.511604\pi$
$389$	−1.43769	−0.0728940	−0.0364470	+	0.999336i	$0.511604\pi$
$390$	0	0
$391$	5.70820	0.288676
$392$	0	0
$393$	−4.94427	−0.249406
$394$	0	0
$395$	24.1803	1.21664
$396$	0	0
$397$	17.7984	0.893275	0.446637	−	0.894715i	$-0.352621\pi$
$397$	17.7984	0.893275	0.446637	+	0.894715i	$0.352621\pi$
$398$	0	0
$399$	−0.291796	−0.0146081
$400$	0	0
$401$	−1.52786	−0.0762979	−0.0381489	−	0.999272i	$-0.512146\pi$
$401$	−1.52786	−0.0762979	−0.0381489	+	0.999272i	$0.512146\pi$
$402$	0	0
$403$	30.1803	1.50339
$404$	0	0
$405$	−9.23607	−0.458944
$406$	0	0
$407$	44.3607	2.19888
$408$	0	0
$409$	14.1803	0.701173	0.350586	−	0.936530i	$-0.385982\pi$
$409$	14.1803	0.701173	0.350586	+	0.936530i	$0.385982\pi$
$410$	0	0
$411$	2.85410	0.140782
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.4721	0.906761
$416$	0	0
$417$	9.29180	0.455021
$418$	0	0
$419$	−18.4508	−0.901383	−0.450691	−	0.892680i	$-0.648823\pi$
$419$	−18.4508	−0.901383	−0.450691	+	0.892680i	$0.648823\pi$
$420$	0	0
$421$	22.2148	1.08268	0.541341	−	0.840803i	$-0.317917\pi$
$421$	22.2148	1.08268	0.541341	+	0.840803i	$0.317917\pi$
$422$	0	0
$423$	−1.23607	−0.0600997
$424$	0	0
$425$	−2.38197	−0.115542
$426$	0	0
$427$	−8.32624	−0.402935
$428$	0	0
$429$	10.4721	0.505599
$430$	0	0
$431$	22.9443	1.10519	0.552593	−	0.833451i	$-0.313638\pi$
$431$	22.9443	1.10519	0.552593	+	0.833451i	$0.313638\pi$
$432$	0	0
$433$	−0.944272	−0.0453788	−0.0226894	−	0.999743i	$-0.507223\pi$
$433$	−0.944272	−0.0453788	−0.0226894	+	0.999743i	$0.507223\pi$
$434$	0	0
$435$	−7.70820	−0.369580
$436$	0	0
$437$	2.69505	0.128922
$438$	0	0
$439$	12.3262	0.588299	0.294150	−	0.955759i	$-0.404964\pi$
$439$	12.3262	0.588299	0.294150	+	0.955759i	$0.404964\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	0	0
$443$	−9.88854	−0.469819	−0.234909	−	0.972017i	$-0.575479\pi$
$443$	−9.88854	−0.469819	−0.234909	+	0.972017i	$0.575479\pi$
$444$	0	0
$445$	−3.23607	−0.153404
$446$	0	0
$447$	−7.03444	−0.332718
$448$	0	0
$449$	4.29180	0.202542	0.101271	−	0.994859i	$-0.467709\pi$
$449$	4.29180	0.202542	0.101271	+	0.994859i	$0.467709\pi$
$450$	0	0
$451$	−58.0689	−2.73436
$452$	0	0
$453$	−13.5066	−0.634594
$454$	0	0
$455$	−5.23607	−0.245471
$456$	0	0
$457$	−1.61803	−0.0756884	−0.0378442	−	0.999284i	$-0.512049\pi$
$457$	−1.61803	−0.0756884	−0.0378442	+	0.999284i	$0.512049\pi$
$458$	0	0
$459$	−3.47214	−0.162065
$460$	0	0
$461$	4.94427	0.230278	0.115139	−	0.993349i	$-0.463269\pi$
$461$	4.94427	0.230278	0.115139	+	0.993349i	$0.463269\pi$
$462$	0	0
$463$	16.0344	0.745184	0.372592	−	0.927995i	$-0.378469\pi$
$463$	16.0344	0.745184	0.372592	+	0.927995i	$0.378469\pi$
$464$	0	0
$465$	9.32624	0.432494
$466$	0	0
$467$	1.52786	0.0707011	0.0353506	−	0.999375i	$-0.488745\pi$
$467$	1.52786	0.0707011	0.0353506	+	0.999375i	$0.488745\pi$
$468$	0	0
$469$	1.90983	0.0881878
$470$	0	0
$471$	12.4721	0.574686
$472$	0	0
$473$	4.76393	0.219046
$474$	0	0
$475$	−1.12461	−0.0516007
$476$	0	0
$477$	36.1246	1.65403
$478$	0	0
$479$	9.27051	0.423580	0.211790	−	0.977315i	$-0.432071\pi$
$479$	9.27051	0.423580	0.211790	+	0.977315i	$0.432071\pi$
$480$	0	0
$481$	27.4164	1.25008
$482$	0	0
$483$	−3.52786	−0.160523
$484$	0	0
$485$	−6.32624	−0.287260
$486$	0	0
$487$	22.3607	1.01326	0.506630	−	0.862164i	$-0.330891\pi$
$487$	22.3607	1.01326	0.506630	+	0.862164i	$0.330891\pi$
$488$	0	0
$489$	−4.76393	−0.215432
$490$	0	0
$491$	10.0902	0.455363	0.227681	−	0.973736i	$-0.426885\pi$
$491$	10.0902	0.455363	0.227681	+	0.973736i	$0.426885\pi$
$492$	0	0
$493$	7.70820	0.347160
$494$	0	0
$495$	−22.1803	−0.996932
$496$	0	0
$497$	6.94427	0.311493
$498$	0	0
$499$	34.9443	1.56432	0.782160	−	0.623077i	$-0.214118\pi$
$499$	34.9443	1.56432	0.782160	+	0.623077i	$0.214118\pi$
$500$	0	0
$501$	10.2705	0.458853
$502$	0	0
$503$	−8.38197	−0.373733	−0.186867	−	0.982385i	$-0.559833\pi$
$503$	−8.38197	−0.373733	−0.186867	+	0.982385i	$0.559833\pi$
$504$	0	0
$505$	−13.7082	−0.610007
$506$	0	0
$507$	−1.56231	−0.0693844
$508$	0	0
$509$	−5.23607	−0.232085	−0.116042	−	0.993244i	$-0.537021\pi$
$509$	−5.23607	−0.232085	−0.116042	+	0.993244i	$0.537021\pi$
$510$	0	0
$511$	−13.8541	−0.612869
$512$	0	0
$513$	−1.63932	−0.0723778
$514$	0	0
$515$	18.4721	0.813980
$516$	0	0
$517$	−2.47214	−0.108724
$518$	0	0
$519$	−14.2705	−0.626406
$520$	0	0
$521$	15.7426	0.689698	0.344849	−	0.938658i	$-0.387930\pi$
$521$	15.7426	0.689698	0.344849	+	0.938658i	$0.387930\pi$
$522$	0	0
$523$	11.8197	0.516838	0.258419	−	0.966033i	$-0.416799\pi$
$523$	11.8197	0.516838	0.258419	+	0.966033i	$0.416799\pi$
$524$	0	0
$525$	1.47214	0.0642493
$526$	0	0
$527$	−9.32624	−0.406257
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−35.8885	−1.55451
$534$	0	0
$535$	−24.6525	−1.06582
$536$	0	0
$537$	−12.7426	−0.549886
$538$	0	0
$539$	−5.23607	−0.225533
$540$	0	0
$541$	−0.763932	−0.0328440	−0.0164220	−	0.999865i	$-0.505228\pi$
$541$	−0.763932	−0.0328440	−0.0164220	+	0.999865i	$0.505228\pi$
$542$	0	0
$543$	−2.18034	−0.0935673
$544$	0	0
$545$	20.1803	0.864431
$546$	0	0
$547$	−9.34752	−0.399671	−0.199836	−	0.979829i	$-0.564041\pi$
$547$	−9.34752	−0.399671	−0.199836	+	0.979829i	$0.564041\pi$
$548$	0	0
$549$	−21.7984	−0.930332
$550$	0	0
$551$	3.63932	0.155040
$552$	0	0
$553$	14.9443	0.635495
$554$	0	0
$555$	8.47214	0.359622
$556$	0	0
$557$	−10.3607	−0.438996	−0.219498	−	0.975613i	$-0.570442\pi$
$557$	−10.3607	−0.438996	−0.219498	+	0.975613i	$0.570442\pi$
$558$	0	0
$559$	2.94427	0.124529
$560$	0	0
$561$	−3.23607	−0.136627
$562$	0	0
$563$	11.1246	0.468846	0.234423	−	0.972135i	$-0.424680\pi$
$563$	11.1246	0.468846	0.234423	+	0.972135i	$0.424680\pi$
$564$	0	0
$565$	−9.70820	−0.408427
$566$	0	0
$567$	−5.70820	−0.239722
$568$	0	0
$569$	−5.49342	−0.230296	−0.115148	−	0.993348i	$-0.536734\pi$
$569$	−5.49342	−0.230296	−0.115148	+	0.993348i	$0.536734\pi$
$570$	0	0
$571$	−0.472136	−0.0197583	−0.00987914	−	0.999951i	$-0.503145\pi$
$571$	−0.472136	−0.0197583	−0.00987914	+	0.999951i	$0.503145\pi$
$572$	0	0
$573$	−2.34752	−0.0980692
$574$	0	0
$575$	−13.5967	−0.567024
$576$	0	0
$577$	−23.1246	−0.962690	−0.481345	−	0.876531i	$-0.659852\pi$
$577$	−23.1246	−0.962690	−0.481345	+	0.876531i	$0.659852\pi$
$578$	0	0
$579$	5.05573	0.210109
$580$	0	0
$581$	11.4164	0.473632
$582$	0	0
$583$	72.2492	2.99226
$584$	0	0
$585$	−13.7082	−0.566764
$586$	0	0
$587$	10.8328	0.447118	0.223559	−	0.974690i	$-0.428232\pi$
$587$	10.8328	0.447118	0.223559	+	0.974690i	$0.428232\pi$
$588$	0	0
$589$	−4.40325	−0.181433
$590$	0	0
$591$	−2.29180	−0.0942719
$592$	0	0
$593$	22.3607	0.918243	0.459122	−	0.888373i	$-0.348164\pi$
$593$	22.3607	0.918243	0.459122	+	0.888373i	$0.348164\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	0	0
$597$	−0.562306	−0.0230136
$598$	0	0
$599$	24.9098	1.01779	0.508894	−	0.860829i	$-0.330055\pi$
$599$	24.9098	1.01779	0.508894	+	0.860829i	$0.330055\pi$
$600$	0	0
$601$	−32.2492	−1.31547	−0.657737	−	0.753248i	$-0.728486\pi$
$601$	−32.2492	−1.31547	−0.657737	+	0.753248i	$0.728486\pi$
$602$	0	0
$603$	5.00000	0.203616
$604$	0	0
$605$	−26.5623	−1.07991
$606$	0	0
$607$	−33.1459	−1.34535	−0.672675	−	0.739938i	$-0.734855\pi$
$607$	−33.1459	−1.34535	−0.672675	+	0.739938i	$0.734855\pi$
$608$	0	0
$609$	−4.76393	−0.193044
$610$	0	0
$611$	−1.52786	−0.0618108
$612$	0	0
$613$	2.56231	0.103491	0.0517453	−	0.998660i	$-0.483522\pi$
$613$	2.56231	0.103491	0.0517453	+	0.998660i	$0.483522\pi$
$614$	0	0
$615$	−11.0902	−0.447199
$616$	0	0
$617$	41.0132	1.65113	0.825564	−	0.564309i	$-0.190857\pi$
$617$	41.0132	1.65113	0.825564	+	0.564309i	$0.190857\pi$
$618$	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
$621$	−19.8197	−0.795336
$622$	0	0
$623$	−2.00000	−0.0801283
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−1.52786	−0.0610170
$628$	0	0
$629$	−8.47214	−0.337806
$630$	0	0
$631$	−25.9098	−1.03145	−0.515727	−	0.856753i	$-0.672478\pi$
$631$	−25.9098	−1.03145	−0.515727	+	0.856753i	$0.672478\pi$
$632$	0	0
$633$	−9.70820	−0.385866
$634$	0	0
$635$	1.85410	0.0735778
$636$	0	0
$637$	−3.23607	−0.128218
$638$	0	0
$639$	18.1803	0.719203
$640$	0	0
$641$	38.9443	1.53821	0.769103	−	0.639125i	$-0.220703\pi$
$641$	38.9443	1.53821	0.769103	+	0.639125i	$0.220703\pi$
$642$	0	0
$643$	9.85410	0.388608	0.194304	−	0.980941i	$-0.437755\pi$
$643$	9.85410	0.388608	0.194304	+	0.980941i	$0.437755\pi$
$644$	0	0
$645$	0.909830	0.0358245
$646$	0	0
$647$	−25.2361	−0.992132	−0.496066	−	0.868285i	$-0.665223\pi$
$647$	−25.2361	−0.992132	−0.496066	+	0.868285i	$0.665223\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	5.76393	0.225906
$652$	0	0
$653$	−23.5279	−0.920716	−0.460358	−	0.887733i	$-0.652279\pi$
$653$	−23.5279	−0.920716	−0.460358	+	0.887733i	$0.652279\pi$
$654$	0	0
$655$	12.9443	0.505775
$656$	0	0
$657$	−36.2705	−1.41505
$658$	0	0
$659$	−40.7426	−1.58711	−0.793554	−	0.608500i	$-0.791772\pi$
$659$	−40.7426	−1.58711	−0.793554	+	0.608500i	$0.791772\pi$
$660$	0	0
$661$	−22.8328	−0.888094	−0.444047	−	0.896004i	$-0.646458\pi$
$661$	−22.8328	−0.888094	−0.444047	+	0.896004i	$0.646458\pi$
$662$	0	0
$663$	−2.00000	−0.0776736
$664$	0	0
$665$	0.763932	0.0296240
$666$	0	0
$667$	44.0000	1.70369
$668$	0	0
$669$	−9.23607	−0.357087
$670$	0	0
$671$	−43.5967	−1.68303
$672$	0	0
$673$	−5.52786	−0.213083	−0.106542	−	0.994308i	$-0.533978\pi$
$673$	−5.52786	−0.213083	−0.106542	+	0.994308i	$0.533978\pi$
$674$	0	0
$675$	8.27051	0.318332
$676$	0	0
$677$	37.7771	1.45189	0.725946	−	0.687752i	$-0.241402\pi$
$677$	37.7771	1.45189	0.725946	+	0.687752i	$0.241402\pi$
$678$	0	0
$679$	−3.90983	−0.150046
$680$	0	0
$681$	−10.3475	−0.396518
$682$	0	0
$683$	5.81966	0.222683	0.111342	−	0.993782i	$-0.464485\pi$
$683$	5.81966	0.222683	0.111342	+	0.993782i	$0.464485\pi$
$684$	0	0
$685$	−7.47214	−0.285496
$686$	0	0
$687$	−9.12461	−0.348126
$688$	0	0
$689$	44.6525	1.70112
$690$	0	0
$691$	46.5066	1.76919	0.884597	−	0.466357i	$-0.154434\pi$
$691$	46.5066	1.76919	0.884597	+	0.466357i	$0.154434\pi$
$692$	0	0
$693$	−13.7082	−0.520732
$694$	0	0
$695$	−24.3262	−0.922747
$696$	0	0
$697$	11.0902	0.420070
$698$	0	0
$699$	−1.52786	−0.0577891
$700$	0	0
$701$	−18.5836	−0.701893	−0.350946	−	0.936396i	$-0.614140\pi$
$701$	−18.5836	−0.701893	−0.350946	+	0.936396i	$0.614140\pi$
$702$	0	0
$703$	−4.00000	−0.150863
$704$	0	0
$705$	−0.472136	−0.0177817
$706$	0	0
$707$	−8.47214	−0.318627
$708$	0	0
$709$	9.34752	0.351054	0.175527	−	0.984475i	$-0.443837\pi$
$709$	9.34752	0.351054	0.175527	+	0.984475i	$0.443837\pi$
$710$	0	0
$711$	39.1246	1.46729
$712$	0	0
$713$	−53.2361	−1.99371
$714$	0	0
$715$	−27.4164	−1.02532
$716$	0	0
$717$	14.2705	0.532942
$718$	0	0
$719$	−45.8541	−1.71007	−0.855035	−	0.518571i	$-0.826464\pi$
$719$	−45.8541	−1.71007	−0.855035	+	0.518571i	$0.826464\pi$
$720$	0	0
$721$	11.4164	0.425169
$722$	0	0
$723$	−15.4721	−0.575415
$724$	0	0
$725$	−18.3607	−0.681899
$726$	0	0
$727$	7.52786	0.279193	0.139597	−	0.990208i	$-0.455419\pi$
$727$	7.52786	0.279193	0.139597	+	0.990208i	$0.455419\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	−0.909830	−0.0336513
$732$	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
$739$	−21.5623	−0.793182	−0.396591	−	0.917995i	$-0.629807\pi$
$739$	−21.5623	−0.793182	−0.396591	+	0.917995i	$0.629807\pi$
$740$	0	0
$741$	−0.944272	−0.0346887
$742$	0	0
$743$	−26.6525	−0.977785	−0.488892	−	0.872344i	$-0.662599\pi$
$743$	−26.6525	−0.977785	−0.488892	+	0.872344i	$0.662599\pi$
$744$	0	0
$745$	18.4164	0.674725
$746$	0	0
$747$	29.8885	1.09356
$748$	0	0
$749$	−15.2361	−0.556714
$750$	0	0
$751$	−6.94427	−0.253400	−0.126700	−	0.991941i	$-0.540439\pi$
$751$	−6.94427	−0.253400	−0.126700	+	0.991941i	$0.540439\pi$
$752$	0	0
$753$	5.81966	0.212080
$754$	0	0
$755$	35.3607	1.28691
$756$	0	0
$757$	−29.5623	−1.07446	−0.537230	−	0.843436i	$-0.680529\pi$
$757$	−29.5623	−1.07446	−0.537230	+	0.843436i	$0.680529\pi$
$758$	0	0
$759$	−18.4721	−0.670496
$760$	0	0
$761$	7.81966	0.283462	0.141731	−	0.989905i	$-0.454733\pi$
$761$	7.81966	0.283462	0.141731	+	0.989905i	$0.454733\pi$
$762$	0	0
$763$	12.4721	0.451522
$764$	0	0
$765$	4.23607	0.153155
$766$	0	0
$767$	0	0
$768$	0	0
$769$	33.1246	1.19450	0.597252	−	0.802054i	$-0.296259\pi$
$769$	33.1246	1.19450	0.597252	+	0.802054i	$0.296259\pi$
$770$	0	0
$771$	7.59675	0.273590
$772$	0	0
$773$	−21.4164	−0.770295	−0.385147	−	0.922855i	$-0.625849\pi$
$773$	−21.4164	−0.770295	−0.385147	+	0.922855i	$0.625849\pi$
$774$	0	0
$775$	22.2148	0.797979
$776$	0	0
$777$	5.23607	0.187843
$778$	0	0
$779$	5.23607	0.187602
$780$	0	0
$781$	36.3607	1.30109
$782$	0	0
$783$	−26.7639	−0.956465
$784$	0	0
$785$	−32.6525	−1.16542
$786$	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
$789$	0.583592	0.0207764
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	−26.9443	−0.956819
$794$	0	0
$795$	13.7984	0.489378
$796$	0	0
$797$	−11.4164	−0.404390	−0.202195	−	0.979345i	$-0.564807\pi$
$797$	−11.4164	−0.404390	−0.202195	+	0.979345i	$0.564807\pi$
$798$	0	0
$799$	0.472136	0.0167030
$800$	0	0
$801$	−5.23607	−0.185007
$802$	0	0
$803$	−72.5410	−2.55992
$804$	0	0
$805$	9.23607	0.325529
$806$	0	0
$807$	−6.18034	−0.217558
$808$	0	0
$809$	31.4853	1.10696	0.553482	−	0.832861i	$-0.313299\pi$
$809$	31.4853	1.10696	0.553482	+	0.832861i	$0.313299\pi$
$810$	0	0
$811$	−32.0344	−1.12488	−0.562441	−	0.826838i	$-0.690138\pi$
$811$	−32.0344	−1.12488	−0.562441	+	0.826838i	$0.690138\pi$
$812$	0	0
$813$	−6.00000	−0.210429
$814$	0	0
$815$	12.4721	0.436880
$816$	0	0
$817$	−0.429563	−0.0150285
$818$	0	0
$819$	−8.47214	−0.296040
$820$	0	0
$821$	−30.3607	−1.05960	−0.529798	−	0.848124i	$-0.677732\pi$
$821$	−30.3607	−1.05960	−0.529798	+	0.848124i	$0.677732\pi$
$822$	0	0
$823$	−37.4853	−1.30666	−0.653328	−	0.757075i	$-0.726628\pi$
$823$	−37.4853	−1.30666	−0.653328	+	0.757075i	$0.726628\pi$
$824$	0	0
$825$	7.70820	0.268365
$826$	0	0
$827$	36.4721	1.26826	0.634130	−	0.773226i	$-0.281358\pi$
$827$	36.4721	1.26826	0.634130	+	0.773226i	$0.281358\pi$
$828$	0	0
$829$	16.1803	0.561966	0.280983	−	0.959713i	$-0.409339\pi$
$829$	16.1803	0.561966	0.280983	+	0.959713i	$0.409339\pi$
$830$	0	0
$831$	4.47214	0.155137
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−26.8885	−0.930516
$836$	0	0
$837$	32.3820	1.11928
$838$	0	0
$839$	−22.4721	−0.775824	−0.387912	−	0.921696i	$-0.626804\pi$
$839$	−22.4721	−0.775824	−0.387912	+	0.921696i	$0.626804\pi$
$840$	0	0
$841$	30.4164	1.04884
$842$	0	0
$843$	−12.4164	−0.427644
$844$	0	0
$845$	4.09017	0.140706
$846$	0	0
$847$	−16.4164	−0.564074
$848$	0	0
$849$	−15.1803	−0.520988
$850$	0	0
$851$	−48.3607	−1.65778
$852$	0	0
$853$	6.36068	0.217786	0.108893	−	0.994054i	$-0.465269\pi$
$853$	6.36068	0.217786	0.108893	+	0.994054i	$0.465269\pi$
$854$	0	0
$855$	2.00000	0.0683986
$856$	0	0
$857$	−3.90983	−0.133557	−0.0667786	−	0.997768i	$-0.521272\pi$
$857$	−3.90983	−0.133557	−0.0667786	+	0.997768i	$0.521272\pi$
$858$	0	0
$859$	−2.94427	−0.100457	−0.0502286	−	0.998738i	$-0.515995\pi$
$859$	−2.94427	−0.100457	−0.0502286	+	0.998738i	$0.515995\pi$
$860$	0	0
$861$	−6.85410	−0.233587
$862$	0	0
$863$	−30.6738	−1.04415	−0.522074	−	0.852900i	$-0.674841\pi$
$863$	−30.6738	−1.04415	−0.522074	+	0.852900i	$0.674841\pi$
$864$	0	0
$865$	37.3607	1.27030
$866$	0	0
$867$	0.618034	0.0209895
$868$	0	0
$869$	78.2492	2.65442
$870$	0	0
$871$	6.18034	0.209413
$872$	0	0
$873$	−10.2361	−0.346438
$874$	0	0
$875$	−11.9443	−0.403790
$876$	0	0
$877$	31.7082	1.07071	0.535355	−	0.844627i	$-0.320178\pi$
$877$	31.7082	1.07071	0.535355	+	0.844627i	$0.320178\pi$
$878$	0	0
$879$	−12.1803	−0.410833
$880$	0	0
$881$	−35.3262	−1.19017	−0.595086	−	0.803662i	$-0.702882\pi$
$881$	−35.3262	−1.19017	−0.595086	+	0.803662i	$0.702882\pi$
$882$	0	0
$883$	−31.9787	−1.07617	−0.538085	−	0.842891i	$-0.680852\pi$
$883$	−31.9787	−1.07617	−0.538085	+	0.842891i	$0.680852\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21.5066	0.722120	0.361060	−	0.932543i	$-0.382415\pi$
$887$	21.5066	0.722120	0.361060	+	0.932543i	$0.382415\pi$
$888$	0	0
$889$	1.14590	0.0384322
$890$	0	0
$891$	−29.8885	−1.00130
$892$	0	0
$893$	0.222912	0.00745948
$894$	0	0
$895$	33.3607	1.11512
$896$	0	0
$897$	−11.4164	−0.381183
$898$	0	0
$899$	−71.8885	−2.39762
$900$	0	0
$901$	−13.7984	−0.459690
$902$	0	0
$903$	0.562306	0.0187124
$904$	0	0
$905$	5.70820	0.189747
$906$	0	0
$907$	31.3050	1.03946	0.519732	−	0.854329i	$-0.326032\pi$
$907$	31.3050	1.03946	0.519732	+	0.854329i	$0.326032\pi$
$908$	0	0
$909$	−22.1803	−0.735675
$910$	0	0
$911$	−5.52786	−0.183146	−0.0915732	−	0.995798i	$-0.529190\pi$
$911$	−5.52786	−0.183146	−0.0915732	+	0.995798i	$0.529190\pi$
$912$	0	0
$913$	59.7771	1.97833
$914$	0	0
$915$	−8.32624	−0.275257
$916$	0	0
$917$	8.00000	0.264183
$918$	0	0
$919$	39.1591	1.29174	0.645869	−	0.763448i	$-0.276495\pi$
$919$	39.1591	1.29174	0.645869	+	0.763448i	$0.276495\pi$
$920$	0	0
$921$	11.7082	0.385798
$922$	0	0
$923$	22.4721	0.739679
$924$	0	0
$925$	20.1803	0.663525
$926$	0	0
$927$	29.8885	0.981669
$928$	0	0
$929$	−12.2705	−0.402582	−0.201291	−	0.979531i	$-0.564514\pi$
$929$	−12.2705	−0.402582	−0.201291	+	0.979531i	$0.564514\pi$
$930$	0	0
$931$	0.472136	0.0154736
$932$	0	0
$933$	15.0689	0.493333
$934$	0	0
$935$	8.47214	0.277068
$936$	0	0
$937$	57.3050	1.87207	0.936036	−	0.351905i	$-0.114466\pi$
$937$	57.3050	1.87207	0.936036	+	0.351905i	$0.114466\pi$
$938$	0	0
$939$	19.9443	0.650857
$940$	0	0
$941$	−0.978714	−0.0319052	−0.0159526	−	0.999873i	$-0.505078\pi$
$941$	−0.978714	−0.0319052	−0.0159526	+	0.999873i	$0.505078\pi$
$942$	0	0
$943$	63.3050	2.06149
$944$	0	0
$945$	−5.61803	−0.182755
$946$	0	0
$947$	47.1246	1.53134	0.765672	−	0.643231i	$-0.222407\pi$
$947$	47.1246	1.53134	0.765672	+	0.643231i	$0.222407\pi$
$948$	0	0
$949$	−44.8328	−1.45533
$950$	0	0
$951$	−3.59675	−0.116633
$952$	0	0
$953$	16.6738	0.540116	0.270058	−	0.962844i	$-0.412957\pi$
$953$	16.6738	0.540116	0.270058	+	0.962844i	$0.412957\pi$
$954$	0	0
$955$	6.14590	0.198877
$956$	0	0
$957$	−24.9443	−0.806334
$958$	0	0
$959$	−4.61803	−0.149124
$960$	0	0
$961$	55.9787	1.80576
$962$	0	0
$963$	−39.8885	−1.28539
$964$	0	0
$965$	−13.2361	−0.426084
$966$	0	0
$967$	13.2016	0.424536	0.212268	−	0.977212i	$-0.431915\pi$
$967$	13.2016	0.424536	0.212268	+	0.977212i	$0.431915\pi$
$968$	0	0
$969$	0.291796	0.00937384
$970$	0	0
$971$	−19.1246	−0.613738	−0.306869	−	0.951752i	$-0.599281\pi$
$971$	−19.1246	−0.613738	−0.306869	+	0.951752i	$0.599281\pi$
$972$	0	0
$973$	−15.0344	−0.481982
$974$	0	0
$975$	4.76393	0.152568
$976$	0	0
$977$	49.2148	1.57452	0.787260	−	0.616621i	$-0.211499\pi$
$977$	49.2148	1.57452	0.787260	+	0.616621i	$0.211499\pi$
$978$	0	0
$979$	−10.4721	−0.334691
$980$	0	0
$981$	32.6525	1.04251
$982$	0	0
$983$	18.0902	0.576987	0.288493	−	0.957482i	$-0.406846\pi$
$983$	18.0902	0.576987	0.288493	+	0.957482i	$0.406846\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	−0.291796	−0.00928797
$988$	0	0
$989$	−5.19350	−0.165144
$990$	0	0
$991$	52.2492	1.65975	0.829876	−	0.557948i	$-0.188411\pi$
$991$	52.2492	1.65975	0.829876	+	0.557948i	$0.188411\pi$
$992$	0	0
$993$	−6.81966	−0.216415
$994$	0	0
$995$	1.47214	0.0466698
$996$	0	0
$997$	−0.0344419	−0.00109078	−0.000545392	−	1.00000i	$-0.500174\pi$
$997$	−0.0344419	−0.00109078	−0.000545392		1.00000i	$0.500174\pi$
$998$	0	0
$999$	29.4164	0.930694

Twists

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

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

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} -1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} -5.23607 q^{11} -3.23607 q^{13} -1.00000 q^{15} +1.00000 q^{17} +0.472136 q^{19} -0.618034 q^{21} +5.70820 q^{23} -2.38197 q^{25} -3.47214 q^{27} +7.70820 q^{29} -9.32624 q^{31} -3.23607 q^{33} +1.61803 q^{35} -8.47214 q^{37} -2.00000 q^{39} +11.0902 q^{41} -0.909830 q^{43} +4.23607 q^{45} +0.472136 q^{47} +1.00000 q^{49} +0.618034 q^{51} -13.7984 q^{53} +8.47214 q^{55} +0.291796 q^{57} +8.32624 q^{61} +2.61803 q^{63} +5.23607 q^{65} -1.90983 q^{67} +3.52786 q^{69} -6.94427 q^{71} +13.8541 q^{73} -1.47214 q^{75} +5.23607 q^{77} -14.9443 q^{79} +5.70820 q^{81} -11.4164 q^{83} -1.61803 q^{85} +4.76393 q^{87} +2.00000 q^{89} +3.23607 q^{91} -5.76393 q^{93} -0.763932 q^{95} +3.90983 q^{97} +13.7082 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} - 2 q^{23} - 7 q^{25} + 2 q^{27} + 2 q^{29} - 3 q^{31} - 2 q^{33} + q^{35} - 8 q^{37} - 4 q^{39}+ \cdots + 14 q^{99}+O(q^{100}) \) 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

L-functions

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