LMFDB - Embedded newform 416.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,2,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	416.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:	$3.32177672409$
Analytic rank:	$0$
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:	$ 2 $
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	$\chi$	$=$	416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{3} +3.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} -1.00000 q^{13} -6.70820 q^{15} -3.00000 q^{17} +4.47214 q^{19} +5.00000 q^{21} +8.94427 q^{23} +4.00000 q^{25} +2.23607 q^{27} +10.0000 q^{29} -10.0000 q^{33} -6.70820 q^{35} +3.00000 q^{37} +2.23607 q^{39} -6.70820 q^{43} +6.00000 q^{45} -2.23607 q^{47} -2.00000 q^{49} +6.70820 q^{51} +4.00000 q^{53} +13.4164 q^{55} -10.0000 q^{57} +4.47214 q^{59} -4.47214 q^{63} -3.00000 q^{65} -13.4164 q^{67} -20.0000 q^{69} -6.70820 q^{71} +14.0000 q^{73} -8.94427 q^{75} -10.0000 q^{77} +8.94427 q^{79} -11.0000 q^{81} -17.8885 q^{83} -9.00000 q^{85} -22.3607 q^{87} -10.0000 q^{89} +2.23607 q^{91} +13.4164 q^{95} -2.00000 q^{97} +8.94427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 4 q^{9} - 2 q^{13} - 6 q^{17} + 10 q^{21} + 8 q^{25} + 20 q^{29} - 20 q^{33} + 6 q^{37} + 12 q^{45} - 4 q^{49} + 8 q^{53} - 20 q^{57} - 6 q^{65} - 40 q^{69} + 28 q^{73} - 20 q^{77} - 22 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 4 * q^9 - 2 * q^13 - 6 * q^17 + 10 * q^21 + 8 * q^25 + 20 * q^29 - 20 * q^33 + 6 * q^37 + 12 * q^45 - 4 * q^49 + 8 * q^53 - 20 * q^57 - 6 * q^65 - 40 * q^69 + 28 * q^73 - 20 * q^77 - 22 * q^81 - 18 * q^85 - 20 * q^89 - 4 * q^97

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
$3$	−2.23607	−1.29099	−0.645497	−	0.763763i	$-0.723350\pi$
$3$	−2.23607	−1.29099	−0.645497	+	0.763763i	$0.723350\pi$
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−6.70820	−1.73205
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	5.00000	1.09109
$22$	0	0
$23$	8.94427	1.86501	0.932505	−	0.361158i	$-0.117618\pi$
$23$	8.94427	1.86501	0.932505	+	0.361158i	$0.117618\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−10.0000	−1.74078
$34$	0	0
$35$	−6.70820	−1.13389
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	0	0
$39$	2.23607	0.358057
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−6.70820	−1.02299	−0.511496	−	0.859286i	$-0.670908\pi$
$43$	−6.70820	−1.02299	−0.511496	+	0.859286i	$0.670908\pi$
$44$	0	0
$45$	6.00000	0.894427
$46$	0	0
$47$	−2.23607	−0.326164	−0.163082	−	0.986613i	$-0.552144\pi$
$47$	−2.23607	−0.326164	−0.163082	+	0.986613i	$0.552144\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	6.70820	0.939336
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	13.4164	1.80907
$56$	0	0
$57$	−10.0000	−1.32453
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	−4.47214	−0.563436
$64$	0	0
$65$	−3.00000	−0.372104
$66$	0	0
$67$	−13.4164	−1.63908	−0.819538	−	0.573025i	$-0.805770\pi$
$67$	−13.4164	−1.63908	−0.819538	+	0.573025i	$0.805770\pi$
$68$	0	0
$69$	−20.0000	−2.40772
$70$	0	0
$71$	−6.70820	−0.796117	−0.398059	−	0.917360i	$-0.630316\pi$
$71$	−6.70820	−0.796117	−0.398059	+	0.917360i	$0.630316\pi$
$72$	0	0
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	−8.94427	−1.03280
$76$	0	0
$77$	−10.0000	−1.13961
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	−17.8885	−1.96352	−0.981761	−	0.190117i	$-0.939113\pi$
$83$	−17.8885	−1.96352	−0.981761	+	0.190117i	$0.939113\pi$
$84$	0	0
$85$	−9.00000	−0.976187
$86$	0	0
$87$	−22.3607	−2.39732
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	2.23607	0.234404
$92$	0	0
$93$	0	0
$94$	0	0
$95$	13.4164	1.37649
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	8.94427	0.898933
$100$	0	0
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	15.0000	1.46385
$106$	0	0
$107$	17.8885	1.72935	0.864675	−	0.502331i	$-0.167524\pi$
$107$	17.8885	1.72935	0.864675	+	0.502331i	$0.167524\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	0	0
$111$	−6.70820	−0.636715
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	26.8328	2.50217
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	6.70820	0.614940
$120$	0	0
$121$	9.00000	0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	15.0000	1.32068
$130$	0	0
$131$	−15.6525	−1.36756	−0.683782	−	0.729687i	$-0.739666\pi$
$131$	−15.6525	−1.36756	−0.683782	+	0.729687i	$0.739666\pi$
$132$	0	0
$133$	−10.0000	−0.867110
$134$	0	0
$135$	6.70820	0.577350
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	11.1803	0.948304	0.474152	−	0.880443i	$-0.342755\pi$
$139$	11.1803	0.948304	0.474152	+	0.880443i	$0.342755\pi$
$140$	0	0
$141$	5.00000	0.421076
$142$	0	0
$143$	−4.47214	−0.373979
$144$	0	0
$145$	30.0000	2.49136
$146$	0	0
$147$	4.47214	0.368856
$148$	0	0
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	0	0
$151$	−6.70820	−0.545906	−0.272953	−	0.962027i	$-0.588000\pi$
$151$	−6.70820	−0.545906	−0.272953	+	0.962027i	$0.588000\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.0000	1.75579	0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000	1.75579	0.877896	+	0.478852i	$0.158947\pi$
$158$	0	0
$159$	−8.94427	−0.709327
$160$	0	0
$161$	−20.0000	−1.57622
$162$	0	0
$163$	−8.94427	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$164$	0	0
$165$	−30.0000	−2.33550
$166$	0	0
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	8.94427	0.683986
$172$	0	0
$173$	4.00000	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$173$	4.00000	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$174$	0	0
$175$	−8.94427	−0.676123
$176$	0	0
$177$	−10.0000	−0.751646
$178$	0	0
$179$	11.1803	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$179$	11.1803	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.00000	0.661693
$186$	0	0
$187$	−13.4164	−0.981105
$188$	0	0
$189$	−5.00000	−0.363696
$190$	0	0
$191$	−4.47214	−0.323592	−0.161796	−	0.986824i	$-0.551729\pi$
$191$	−4.47214	−0.323592	−0.161796	+	0.986824i	$0.551729\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	6.70820	0.480384
$196$	0	0
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	30.0000	2.11604
$202$	0	0
$203$	−22.3607	−1.56941
$204$	0	0
$205$	0	0
$206$	0	0
$207$	17.8885	1.24334
$208$	0	0
$209$	20.0000	1.38343
$210$	0	0
$211$	−15.6525	−1.07756	−0.538780	−	0.842446i	$-0.681115\pi$
$211$	−15.6525	−1.07756	−0.538780	+	0.842446i	$0.681115\pi$
$212$	0	0
$213$	15.0000	1.02778
$214$	0	0
$215$	−20.1246	−1.37249
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−31.3050	−2.11539
$220$	0	0
$221$	3.00000	0.201802
$222$	0	0
$223$	2.23607	0.149738	0.0748691	−	0.997193i	$-0.476146\pi$
$223$	2.23607	0.149738	0.0748691	+	0.997193i	$0.476146\pi$
$224$	0	0
$225$	8.00000	0.533333
$226$	0	0
$227$	17.8885	1.18730	0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885	1.18730	0.593652	+	0.804722i	$0.297686\pi$
$228$	0	0
$229$	5.00000	0.330409	0.165205	−	0.986259i	$-0.447172\pi$
$229$	5.00000	0.330409	0.165205	+	0.986259i	$0.447172\pi$
$230$	0	0
$231$	22.3607	1.47122
$232$	0	0
$233$	1.00000	0.0655122	0.0327561	−	0.999463i	$-0.489572\pi$
$233$	1.00000	0.0655122	0.0327561	+	0.999463i	$0.489572\pi$
$234$	0	0
$235$	−6.70820	−0.437595
$236$	0	0
$237$	−20.0000	−1.29914
$238$	0	0
$239$	15.6525	1.01247	0.506237	−	0.862394i	$-0.331036\pi$
$239$	15.6525	1.01247	0.506237	+	0.862394i	$0.331036\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	17.8885	1.14755
$244$	0	0
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−4.47214	−0.284555
$248$	0	0
$249$	40.0000	2.53490
$250$	0	0
$251$	−8.94427	−0.564557	−0.282279	−	0.959332i	$-0.591090\pi$
$251$	−8.94427	−0.564557	−0.282279	+	0.959332i	$0.591090\pi$
$252$	0	0
$253$	40.0000	2.51478
$254$	0	0
$255$	20.1246	1.26025
$256$	0	0
$257$	−23.0000	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$257$	−23.0000	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	20.0000	1.23797
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	22.3607	1.36845
$268$	0	0
$269$	−16.0000	−0.975537	−0.487769	−	0.872973i	$-0.662189\pi$
$269$	−16.0000	−0.975537	−0.487769	+	0.872973i	$0.662189\pi$
$270$	0	0
$271$	−20.1246	−1.22248	−0.611242	−	0.791444i	$-0.709330\pi$
$271$	−20.1246	−1.22248	−0.611242	+	0.791444i	$0.709330\pi$
$272$	0	0
$273$	−5.00000	−0.302614
$274$	0	0
$275$	17.8885	1.07872
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−17.8885	−1.06336	−0.531682	−	0.846944i	$-0.678440\pi$
$283$	−17.8885	−1.06336	−0.531682	+	0.846944i	$0.678440\pi$
$284$	0	0
$285$	−30.0000	−1.77705
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	4.47214	0.262161
$292$	0	0
$293$	31.0000	1.81104	0.905520	−	0.424304i	$-0.139481\pi$
$293$	31.0000	1.81104	0.905520	+	0.424304i	$0.139481\pi$
$294$	0	0
$295$	13.4164	0.781133
$296$	0	0
$297$	10.0000	0.580259
$298$	0	0
$299$	−8.94427	−0.517261
$300$	0	0
$301$	15.0000	0.864586
$302$	0	0
$303$	26.8328	1.54150
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.47214	0.255238	0.127619	−	0.991823i	$-0.459266\pi$
$307$	4.47214	0.255238	0.127619	+	0.991823i	$0.459266\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−4.47214	−0.253592	−0.126796	−	0.991929i	$-0.540469\pi$
$311$	−4.47214	−0.253592	−0.126796	+	0.991929i	$0.540469\pi$
$312$	0	0
$313$	−29.0000	−1.63918	−0.819588	−	0.572953i	$-0.805798\pi$
$313$	−29.0000	−1.63918	−0.819588	+	0.572953i	$0.805798\pi$
$314$	0	0
$315$	−13.4164	−0.755929
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	44.7214	2.50392
$320$	0	0
$321$	−40.0000	−2.23258
$322$	0	0
$323$	−13.4164	−0.746509
$324$	0	0
$325$	−4.00000	−0.221880
$326$	0	0
$327$	−24.5967	−1.36020
$328$	0	0
$329$	5.00000	0.275659
$330$	0	0
$331$	26.8328	1.47486	0.737432	−	0.675421i	$-0.236038\pi$
$331$	26.8328	1.47486	0.737432	+	0.675421i	$0.236038\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−40.2492	−2.19905
$336$	0	0
$337$	3.00000	0.163420	0.0817102	−	0.996656i	$-0.473962\pi$
$337$	3.00000	0.163420	0.0817102	+	0.996656i	$0.473962\pi$
$338$	0	0
$339$	31.3050	1.70025
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.1246	1.08663
$344$	0	0
$345$	−60.0000	−3.23029
$346$	0	0
$347$	11.1803	0.600192	0.300096	−	0.953909i	$-0.402981\pi$
$347$	11.1803	0.600192	0.300096	+	0.953909i	$0.402981\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	4.00000	0.212899	0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000	0.212899	0.106449	+	0.994318i	$0.466052\pi$
$354$	0	0
$355$	−20.1246	−1.06810
$356$	0	0
$357$	−15.0000	−0.793884
$358$	0	0
$359$	−8.94427	−0.472061	−0.236030	−	0.971746i	$-0.575846\pi$
$359$	−8.94427	−0.472061	−0.236030	+	0.971746i	$0.575846\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.1246	−1.05627
$364$	0	0
$365$	42.0000	2.19838
$366$	0	0
$367$	4.47214	0.233444	0.116722	−	0.993165i	$-0.462761\pi$
$367$	4.47214	0.233444	0.116722	+	0.993165i	$0.462761\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.94427	−0.464363
$372$	0	0
$373$	4.00000	0.207112	0.103556	−	0.994624i	$-0.466978\pi$
$373$	4.00000	0.207112	0.103556	+	0.994624i	$0.466978\pi$
$374$	0	0
$375$	6.70820	0.346410
$376$	0	0
$377$	−10.0000	−0.515026
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.6525	−0.799804	−0.399902	−	0.916558i	$-0.630956\pi$
$383$	−15.6525	−0.799804	−0.399902	+	0.916558i	$0.630956\pi$
$384$	0	0
$385$	−30.0000	−1.52894
$386$	0	0
$387$	−13.4164	−0.681994
$388$	0	0
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	−26.8328	−1.35699
$392$	0	0
$393$	35.0000	1.76552
$394$	0	0
$395$	26.8328	1.35011
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	22.3607	1.11943
$400$	0	0
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−33.0000	−1.63978
$406$	0	0
$407$	13.4164	0.665027
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	26.8328	1.32357
$412$	0	0
$413$	−10.0000	−0.492068
$414$	0	0
$415$	−53.6656	−2.63434
$416$	0	0
$417$	−25.0000	−1.22426
$418$	0	0
$419$	−2.23607	−0.109239	−0.0546195	−	0.998507i	$-0.517395\pi$
$419$	−2.23607	−0.109239	−0.0546195	+	0.998507i	$0.517395\pi$
$420$	0	0
$421$	15.0000	0.731055	0.365528	−	0.930800i	$-0.380889\pi$
$421$	15.0000	0.731055	0.365528	+	0.930800i	$0.380889\pi$
$422$	0	0
$423$	−4.47214	−0.217443
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	10.0000	0.482805
$430$	0	0
$431$	15.6525	0.753953	0.376977	−	0.926223i	$-0.376964\pi$
$431$	15.6525	0.753953	0.376977	+	0.926223i	$0.376964\pi$
$432$	0	0
$433$	−21.0000	−1.00920	−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000	−1.00920	−0.504598	+	0.863355i	$0.668359\pi$
$434$	0	0
$435$	−67.0820	−3.21634
$436$	0	0
$437$	40.0000	1.91346
$438$	0	0
$439$	−4.47214	−0.213443	−0.106722	−	0.994289i	$-0.534035\pi$
$439$	−4.47214	−0.213443	−0.106722	+	0.994289i	$0.534035\pi$
$440$	0	0
$441$	−4.00000	−0.190476
$442$	0	0
$443$	15.6525	0.743672	0.371836	−	0.928299i	$-0.378728\pi$
$443$	15.6525	0.743672	0.371836	+	0.928299i	$0.378728\pi$
$444$	0	0
$445$	−30.0000	−1.42214
$446$	0	0
$447$	−31.3050	−1.48067
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	15.0000	0.704761
$454$	0	0
$455$	6.70820	0.314485
$456$	0	0
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	0	0
$459$	−6.70820	−0.313112
$460$	0	0
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	−8.94427	−0.415676	−0.207838	−	0.978163i	$-0.566643\pi$
$463$	−8.94427	−0.415676	−0.207838	+	0.978163i	$0.566643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	30.0000	1.38527
$470$	0	0
$471$	−49.1935	−2.26672
$472$	0	0
$473$	−30.0000	−1.37940
$474$	0	0
$475$	17.8885	0.820783
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	−20.1246	−0.919517	−0.459758	−	0.888044i	$-0.652064\pi$
$479$	−20.1246	−0.919517	−0.459758	+	0.888044i	$0.652064\pi$
$480$	0	0
$481$	−3.00000	−0.136788
$482$	0	0
$483$	44.7214	2.03489
$484$	0	0
$485$	−6.00000	−0.272446
$486$	0	0
$487$	26.8328	1.21591	0.607955	−	0.793971i	$-0.291990\pi$
$487$	26.8328	1.21591	0.607955	+	0.793971i	$0.291990\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	11.1803	0.504562	0.252281	−	0.967654i	$-0.418819\pi$
$491$	11.1803	0.504562	0.252281	+	0.967654i	$0.418819\pi$
$492$	0	0
$493$	−30.0000	−1.35113
$494$	0	0
$495$	26.8328	1.20605
$496$	0	0
$497$	15.0000	0.672842
$498$	0	0
$499$	35.7771	1.60160	0.800801	−	0.598930i	$-0.204407\pi$
$499$	35.7771	1.60160	0.800801	+	0.598930i	$0.204407\pi$
$500$	0	0
$501$	−20.0000	−0.893534
$502$	0	0
$503$	31.3050	1.39582	0.697909	−	0.716186i	$-0.254114\pi$
$503$	31.3050	1.39582	0.697909	+	0.716186i	$0.254114\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	−2.23607	−0.0993073
$508$	0	0
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−31.3050	−1.38485
$512$	0	0
$513$	10.0000	0.441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	−8.94427	−0.392610
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	17.8885	0.782211	0.391106	−	0.920346i	$-0.372093\pi$
$523$	17.8885	0.782211	0.391106	+	0.920346i	$0.372093\pi$
$524$	0	0
$525$	20.0000	0.872872
$526$	0	0
$527$	0	0
$528$	0	0
$529$	57.0000	2.47826
$530$	0	0
$531$	8.94427	0.388148
$532$	0	0
$533$	0	0
$534$	0	0
$535$	53.6656	2.32017
$536$	0	0
$537$	−25.0000	−1.07883
$538$	0	0
$539$	−8.94427	−0.385257
$540$	0	0
$541$	33.0000	1.41878	0.709390	−	0.704816i	$-0.248970\pi$
$541$	33.0000	1.41878	0.709390	+	0.704816i	$0.248970\pi$
$542$	0	0
$543$	17.8885	0.767671
$544$	0	0
$545$	33.0000	1.41356
$546$	0	0
$547$	−38.0132	−1.62533	−0.812663	−	0.582735i	$-0.801983\pi$
$547$	−38.0132	−1.62533	−0.812663	+	0.582735i	$0.801983\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	44.7214	1.90519
$552$	0	0
$553$	−20.0000	−0.850487
$554$	0	0
$555$	−20.1246	−0.854242
$556$	0	0
$557$	−7.00000	−0.296600	−0.148300	−	0.988942i	$-0.547380\pi$
$557$	−7.00000	−0.296600	−0.148300	+	0.988942i	$0.547380\pi$
$558$	0	0
$559$	6.70820	0.283727
$560$	0	0
$561$	30.0000	1.26660
$562$	0	0
$563$	−15.6525	−0.659673	−0.329837	−	0.944038i	$-0.606994\pi$
$563$	−15.6525	−0.659673	−0.329837	+	0.944038i	$0.606994\pi$
$564$	0	0
$565$	−42.0000	−1.76695
$566$	0	0
$567$	24.5967	1.03297
$568$	0	0
$569$	−9.00000	−0.377300	−0.188650	−	0.982044i	$-0.560411\pi$
$569$	−9.00000	−0.377300	−0.188650	+	0.982044i	$0.560411\pi$
$570$	0	0
$571$	24.5967	1.02934	0.514671	−	0.857388i	$-0.327914\pi$
$571$	24.5967	1.02934	0.514671	+	0.857388i	$0.327914\pi$
$572$	0	0
$573$	10.0000	0.417756
$574$	0	0
$575$	35.7771	1.49201
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−35.7771	−1.48685
$580$	0	0
$581$	40.0000	1.65948
$582$	0	0
$583$	17.8885	0.740868
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	−44.7214	−1.84585	−0.922924	−	0.384982i	$-0.874208\pi$
$587$	−44.7214	−1.84585	−0.922924	+	0.384982i	$0.874208\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	15.6525	0.643857
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	20.1246	0.825029
$596$	0	0
$597$	30.0000	1.22782
$598$	0	0
$599$	−31.3050	−1.27909	−0.639543	−	0.768755i	$-0.720876\pi$
$599$	−31.3050	−1.27909	−0.639543	+	0.768755i	$0.720876\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	−26.8328	−1.09272
$604$	0	0
$605$	27.0000	1.09771
$606$	0	0
$607$	−13.4164	−0.544555	−0.272278	−	0.962219i	$-0.587777\pi$
$607$	−13.4164	−0.544555	−0.272278	+	0.962219i	$0.587777\pi$
$608$	0	0
$609$	50.0000	2.02610
$610$	0	0
$611$	2.23607	0.0904616
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.00000	−0.322068	−0.161034	−	0.986949i	$-0.551483\pi$
$617$	−8.00000	−0.322068	−0.161034	+	0.986949i	$0.551483\pi$
$618$	0	0
$619$	−26.8328	−1.07850	−0.539251	−	0.842145i	$-0.681293\pi$
$619$	−26.8328	−1.07850	−0.539251	+	0.842145i	$0.681293\pi$
$620$	0	0
$621$	20.0000	0.802572
$622$	0	0
$623$	22.3607	0.895862
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	−44.7214	−1.78600
$628$	0	0
$629$	−9.00000	−0.358854
$630$	0	0
$631$	−33.5410	−1.33525	−0.667623	−	0.744499i	$-0.732688\pi$
$631$	−33.5410	−1.33525	−0.667623	+	0.744499i	$0.732688\pi$
$632$	0	0
$633$	35.0000	1.39113
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−13.4164	−0.530745
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	22.3607	0.881819	0.440910	−	0.897552i	$-0.354656\pi$
$643$	22.3607	0.881819	0.440910	+	0.897552i	$0.354656\pi$
$644$	0	0
$645$	45.0000	1.77187
$646$	0	0
$647$	−40.2492	−1.58236	−0.791180	−	0.611583i	$-0.790533\pi$
$647$	−40.2492	−1.58236	−0.791180	+	0.611583i	$0.790533\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−46.9574	−1.83478
$656$	0	0
$657$	28.0000	1.09238
$658$	0	0
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	−6.70820	−0.260525
$664$	0	0
$665$	−30.0000	−1.16335
$666$	0	0
$667$	89.4427	3.46324
$668$	0	0
$669$	−5.00000	−0.193311
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	8.94427	0.344265
$676$	0	0
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	0	0
$679$	4.47214	0.171625
$680$	0	0
$681$	−40.0000	−1.53280
$682$	0	0
$683$	26.8328	1.02673	0.513365	−	0.858171i	$-0.328399\pi$
$683$	26.8328	1.02673	0.513365	+	0.858171i	$0.328399\pi$
$684$	0	0
$685$	−36.0000	−1.37549
$686$	0	0
$687$	−11.1803	−0.426557
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	−17.8885	−0.680512	−0.340256	−	0.940333i	$-0.610514\pi$
$691$	−17.8885	−0.680512	−0.340256	+	0.940333i	$0.610514\pi$
$692$	0	0
$693$	−20.0000	−0.759737
$694$	0	0
$695$	33.5410	1.27228
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−2.23607	−0.0845759
$700$	0	0
$701$	−20.0000	−0.755390	−0.377695	−	0.925930i	$-0.623283\pi$
$701$	−20.0000	−0.755390	−0.377695	+	0.925930i	$0.623283\pi$
$702$	0	0
$703$	13.4164	0.506009
$704$	0	0
$705$	15.0000	0.564933
$706$	0	0
$707$	26.8328	1.00915
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	17.8885	0.670873
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−13.4164	−0.501745
$716$	0	0
$717$	−35.0000	−1.30710
$718$	0	0
$719$	−40.2492	−1.50104	−0.750521	−	0.660846i	$-0.770198\pi$
$719$	−40.2492	−1.50104	−0.750521	+	0.660846i	$0.770198\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−22.3607	−0.831603
$724$	0	0
$725$	40.0000	1.48556
$726$	0	0
$727$	31.3050	1.16104	0.580518	−	0.814247i	$-0.302850\pi$
$727$	31.3050	1.16104	0.580518	+	0.814247i	$0.302850\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	0	0
$731$	20.1246	0.744336
$732$	0	0
$733$	9.00000	0.332423	0.166211	−	0.986090i	$-0.446847\pi$
$733$	9.00000	0.332423	0.166211	+	0.986090i	$0.446847\pi$
$734$	0	0
$735$	13.4164	0.494872
$736$	0	0
$737$	−60.0000	−2.21013
$738$	0	0
$739$	−26.8328	−0.987061	−0.493531	−	0.869728i	$-0.664294\pi$
$739$	−26.8328	−0.987061	−0.493531	+	0.869728i	$0.664294\pi$
$740$	0	0
$741$	10.0000	0.367359
$742$	0	0
$743$	−29.0689	−1.06643	−0.533217	−	0.845979i	$-0.679017\pi$
$743$	−29.0689	−1.06643	−0.533217	+	0.845979i	$0.679017\pi$
$744$	0	0
$745$	42.0000	1.53876
$746$	0	0
$747$	−35.7771	−1.30902
$748$	0	0
$749$	−40.0000	−1.46157
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	20.0000	0.728841
$754$	0	0
$755$	−20.1246	−0.732410
$756$	0	0
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	0	0
$759$	−89.4427	−3.24657
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	−24.5967	−0.890462
$764$	0	0
$765$	−18.0000	−0.650791
$766$	0	0
$767$	−4.47214	−0.161479
$768$	0	0
$769$	0	0		−	1.00000i	$-0.5\pi$
$769$	0	0			1.00000i	$0.5\pi$
$770$	0	0
$771$	51.4296	1.85219
$772$	0	0
$773$	−9.00000	−0.323708	−0.161854	−	0.986815i	$-0.551747\pi$
$773$	−9.00000	−0.323708	−0.161854	+	0.986815i	$0.551747\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	15.0000	0.538122
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−30.0000	−1.07348
$782$	0	0
$783$	22.3607	0.799106
$784$	0	0
$785$	66.0000	2.35564
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	20.0000	0.712019
$790$	0	0
$791$	31.3050	1.11308
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−26.8328	−0.951662
$796$	0	0
$797$	−18.0000	−0.637593	−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000	−0.637593	−0.318796	+	0.947823i	$0.603279\pi$
$798$	0	0
$799$	6.70820	0.237319
$800$	0	0
$801$	−20.0000	−0.706665
$802$	0	0
$803$	62.6099	2.20946
$804$	0	0
$805$	−60.0000	−2.11472
$806$	0	0
$807$	35.7771	1.25941
$808$	0	0
$809$	35.0000	1.23053	0.615267	−	0.788319i	$-0.289048\pi$
$809$	35.0000	1.23053	0.615267	+	0.788319i	$0.289048\pi$
$810$	0	0
$811$	−8.94427	−0.314076	−0.157038	−	0.987593i	$-0.550194\pi$
$811$	−8.94427	−0.314076	−0.157038	+	0.987593i	$0.550194\pi$
$812$	0	0
$813$	45.0000	1.57822
$814$	0	0
$815$	−26.8328	−0.939913
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	4.47214	0.156269
$820$	0	0
$821$	−45.0000	−1.57051	−0.785255	−	0.619172i	$-0.787468\pi$
$821$	−45.0000	−1.57051	−0.785255	+	0.619172i	$0.787468\pi$
$822$	0	0
$823$	−31.3050	−1.09122	−0.545611	−	0.838039i	$-0.683702\pi$
$823$	−31.3050	−1.09122	−0.545611	+	0.838039i	$0.683702\pi$
$824$	0	0
$825$	−40.0000	−1.39262
$826$	0	0
$827$	22.3607	0.777557	0.388779	−	0.921331i	$-0.372897\pi$
$827$	22.3607	0.777557	0.388779	+	0.921331i	$0.372897\pi$
$828$	0	0
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	62.6099	2.17191
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	26.8328	0.928588
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.8328	0.926372	0.463186	−	0.886261i	$-0.346706\pi$
$839$	26.8328	0.926372	0.463186	+	0.886261i	$0.346706\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	22.3607	0.770143
$844$	0	0
$845$	3.00000	0.103203
$846$	0	0
$847$	−20.1246	−0.691490
$848$	0	0
$849$	40.0000	1.37280
$850$	0	0
$851$	26.8328	0.919817
$852$	0	0
$853$	−19.0000	−0.650548	−0.325274	−	0.945620i	$-0.605456\pi$
$853$	−19.0000	−0.650548	−0.325274	+	0.945620i	$0.605456\pi$
$854$	0	0
$855$	26.8328	0.917663
$856$	0	0
$857$	−2.00000	−0.0683187	−0.0341593	−	0.999416i	$-0.510875\pi$
$857$	−2.00000	−0.0683187	−0.0341593	+	0.999416i	$0.510875\pi$
$858$	0	0
$859$	35.7771	1.22070	0.610349	−	0.792132i	$-0.291029\pi$
$859$	35.7771	1.22070	0.610349	+	0.792132i	$0.291029\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.0689	−0.989516	−0.494758	−	0.869031i	$-0.664743\pi$
$863$	−29.0689	−0.989516	−0.494758	+	0.869031i	$0.664743\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	0	0
$867$	17.8885	0.607527
$868$	0	0
$869$	40.0000	1.35691
$870$	0	0
$871$	13.4164	0.454598
$872$	0	0
$873$	−4.00000	−0.135379
$874$	0	0
$875$	6.70820	0.226779
$876$	0	0
$877$	−3.00000	−0.101303	−0.0506514	−	0.998716i	$-0.516130\pi$
$877$	−3.00000	−0.101303	−0.0506514	+	0.998716i	$0.516130\pi$
$878$	0	0
$879$	−69.3181	−2.33804
$880$	0	0
$881$	5.00000	0.168454	0.0842271	−	0.996447i	$-0.473158\pi$
$881$	5.00000	0.168454	0.0842271	+	0.996447i	$0.473158\pi$
$882$	0	0
$883$	−2.23607	−0.0752497	−0.0376248	−	0.999292i	$-0.511979\pi$
$883$	−2.23607	−0.0752497	−0.0376248	+	0.999292i	$0.511979\pi$
$884$	0	0
$885$	−30.0000	−1.00844
$886$	0	0
$887$	17.8885	0.600639	0.300319	−	0.953839i	$-0.402907\pi$
$887$	17.8885	0.600639	0.300319	+	0.953839i	$0.402907\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−49.1935	−1.64804
$892$	0	0
$893$	−10.0000	−0.334637
$894$	0	0
$895$	33.5410	1.12115
$896$	0	0
$897$	20.0000	0.667781
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	−33.5410	−1.11618
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	−33.5410	−1.11371	−0.556856	−	0.830609i	$-0.687992\pi$
$907$	−33.5410	−1.11371	−0.556856	+	0.830609i	$0.687992\pi$
$908$	0	0
$909$	−24.0000	−0.796030
$910$	0	0
$911$	−4.47214	−0.148168	−0.0740842	−	0.997252i	$-0.523603\pi$
$911$	−4.47214	−0.148168	−0.0740842	+	0.997252i	$0.523603\pi$
$912$	0	0
$913$	−80.0000	−2.64761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	35.0000	1.15580
$918$	0	0
$919$	−35.7771	−1.18018	−0.590089	−	0.807338i	$-0.700907\pi$
$919$	−35.7771	−1.18018	−0.590089	+	0.807338i	$0.700907\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	0	0
$923$	6.70820	0.220803
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	−8.94427	−0.293137
$932$	0	0
$933$	10.0000	0.327385
$934$	0	0
$935$	−40.2492	−1.31629
$936$	0	0
$937$	−58.0000	−1.89478	−0.947389	−	0.320085i	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	0	0
$939$	64.8460	2.11617
$940$	0	0
$941$	−3.00000	−0.0977972	−0.0488986	−	0.998804i	$-0.515571\pi$
$941$	−3.00000	−0.0977972	−0.0488986	+	0.998804i	$0.515571\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	4.47214	0.145325	0.0726624	−	0.997357i	$-0.476850\pi$
$947$	4.47214	0.145325	0.0726624	+	0.997357i	$0.476850\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	40.2492	1.30517
$952$	0	0
$953$	−9.00000	−0.291539	−0.145769	−	0.989319i	$-0.546566\pi$
$953$	−9.00000	−0.291539	−0.145769	+	0.989319i	$0.546566\pi$
$954$	0	0
$955$	−13.4164	−0.434145
$956$	0	0
$957$	−100.000	−3.23254
$958$	0	0
$959$	26.8328	0.866477
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	35.7771	1.15290
$964$	0	0
$965$	48.0000	1.54517
$966$	0	0
$967$	38.0132	1.22242	0.611210	−	0.791468i	$-0.290683\pi$
$967$	38.0132	1.22242	0.611210	+	0.791468i	$0.290683\pi$
$968$	0	0
$969$	30.0000	0.963739
$970$	0	0
$971$	−55.9017	−1.79397	−0.896985	−	0.442060i	$-0.854248\pi$
$971$	−55.9017	−1.79397	−0.896985	+	0.442060i	$0.854248\pi$
$972$	0	0
$973$	−25.0000	−0.801463
$974$	0	0
$975$	8.94427	0.286446
$976$	0	0
$977$	58.0000	1.85558	0.927792	−	0.373097i	$-0.121704\pi$
$977$	58.0000	1.85558	0.927792	+	0.373097i	$0.121704\pi$
$978$	0	0
$979$	−44.7214	−1.42930
$980$	0	0
$981$	22.0000	0.702406
$982$	0	0
$983$	−2.23607	−0.0713195	−0.0356597	−	0.999364i	$-0.511353\pi$
$983$	−2.23607	−0.0713195	−0.0356597	+	0.999364i	$0.511353\pi$
$984$	0	0
$985$	−21.0000	−0.669116
$986$	0	0
$987$	−11.1803	−0.355874
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	31.3050	0.994435	0.497217	−	0.867626i	$-0.334355\pi$
$991$	31.3050	0.994435	0.497217	+	0.867626i	$0.334355\pi$
$992$	0	0
$993$	−60.0000	−1.90404
$994$	0	0
$995$	−40.2492	−1.27599
$996$	0	0
$997$	22.0000	0.696747	0.348373	−	0.937356i	$-0.386734\pi$
$997$	22.0000	0.696747	0.348373	+	0.937356i	$0.386734\pi$
$998$	0	0
$999$	6.70820	0.212238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.2.a.d.1.1	✓	2
3.2	odd	2		3744.2.a.q.1.1		2
4.3	odd	2	inner	416.2.a.d.1.2	yes	2
8.3	odd	2		832.2.a.m.1.1		2
8.5	even	2		832.2.a.m.1.2		2
12.11	even	2		3744.2.a.q.1.2		2
13.12	even	2		5408.2.a.q.1.1		2
16.3	odd	4		3328.2.b.x.1665.3		4
16.5	even	4		3328.2.b.x.1665.4		4
16.11	odd	4		3328.2.b.x.1665.2		4
16.13	even	4		3328.2.b.x.1665.1		4
24.5	odd	2		7488.2.a.cw.1.1		2
24.11	even	2		7488.2.a.cw.1.2		2
52.51	odd	2		5408.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.a.d.1.1	✓	2	1.1	even	1	trivial
416.2.a.d.1.2	yes	2	4.3	odd	2	inner
832.2.a.m.1.1		2	8.3	odd	2
832.2.a.m.1.2		2	8.5	even	2
3328.2.b.x.1665.1		4	16.13	even	4
3328.2.b.x.1665.2		4	16.11	odd	4
3328.2.b.x.1665.3		4	16.3	odd	4
3328.2.b.x.1665.4		4	16.5	even	4
3744.2.a.q.1.1		2	3.2	odd	2
3744.2.a.q.1.2		2	12.11	even	2
5408.2.a.q.1.1		2	13.12	even	2
5408.2.a.q.1.2		2	52.51	odd	2
7488.2.a.cw.1.1		2	24.5	odd	2
7488.2.a.cw.1.2		2	24.11	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{3} +3.00000 q^{5} -2.23607 q^{7} +2.00000 q^{9} +4.47214 q^{11} -1.00000 q^{13} -6.70820 q^{15} -3.00000 q^{17} +4.47214 q^{19} +5.00000 q^{21} +8.94427 q^{23} +4.00000 q^{25} +2.23607 q^{27} +10.0000 q^{29} -10.0000 q^{33} -6.70820 q^{35} +3.00000 q^{37} +2.23607 q^{39} -6.70820 q^{43} +6.00000 q^{45} -2.23607 q^{47} -2.00000 q^{49} +6.70820 q^{51} +4.00000 q^{53} +13.4164 q^{55} -10.0000 q^{57} +4.47214 q^{59} -4.47214 q^{63} -3.00000 q^{65} -13.4164 q^{67} -20.0000 q^{69} -6.70820 q^{71} +14.0000 q^{73} -8.94427 q^{75} -10.0000 q^{77} +8.94427 q^{79} -11.0000 q^{81} -17.8885 q^{83} -9.00000 q^{85} -22.3607 q^{87} -10.0000 q^{89} +2.23607 q^{91} +13.4164 q^{95} -2.00000 q^{97} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 4 q^{9} - 2 q^{13} - 6 q^{17} + 10 q^{21} + 8 q^{25} + 20 q^{29} - 20 q^{33} + 6 q^{37} + 12 q^{45} - 4 q^{49} + 8 q^{53} - 20 q^{57} - 6 q^{65} - 40 q^{69} + 28 q^{73} - 20 q^{77} - 22 q^{81}+ \cdots - 4 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 4 * q^9 - 2 * q^13 - 6 * q^17 + 10 * q^21 + 8 * q^25 + 20 * q^29 - 20 * q^33 + 6 * q^37 + 12 * q^45 - 4 * q^49 + 8 * q^53 - 20 * q^57 - 6 * q^65 - 40 * q^69 + 28 * q^73 - 20 * q^77 - 22 * q^81 - 18 * q^85 - 20 * q^89 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more