LMFDB - Embedded newform 2376.2.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2376,2,Mod(1,2376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2376 = 2^{3} \cdot 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,2,0,-3,0,0,0,-3,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.9724555203$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	2376.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35194 q^{5} -2.73419 q^{7} -1.00000 q^{11} +5.52420 q^{13} +4.79001 q^{17} +3.52420 q^{19} -0.351939 q^{23} -3.17226 q^{25} -3.61033 q^{29} -7.69646 q^{31} -3.69646 q^{35} +7.82032 q^{37} +0.561931 q^{41} -5.38225 q^{43} +5.17226 q^{47} +0.475800 q^{49} +8.87614 q^{53} -1.35194 q^{55} +5.11644 q^{59} +4.99258 q^{61} +7.46838 q^{65} +7.69646 q^{67} +12.8761 q^{71} +4.87614 q^{73} +2.73419 q^{77} -5.20999 q^{79} +12.2281 q^{83} +6.47580 q^{85} +6.00000 q^{89} -15.1042 q^{91} +4.76450 q^{95} -13.9926 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{17} - 6 q^{19} + q^{23} + 5 q^{25} + 13 q^{29} + 8 q^{31} + 20 q^{35} + 11 q^{37} + 11 q^{41} - 13 q^{43} + q^{47} + 18 q^{49} + 8 q^{53} - 2 q^{55} + 7 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) $ 3 * q + 2 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^17 - 6 * q^19 + q^23 + 5 * q^25 + 13 * q^29 + 8 * q^31 + 20 * q^35 + 11 * q^37 + 11 * q^41 - 13 * q^43 + q^47 + 18 * q^49 + 8 * q^53 - 2 * q^55 + 7 * q^59 - 12 * q^61 + 12 * q^65 - 8 * q^67 + 20 * q^71 - 4 * q^73 + 3 * q^77 - 27 * q^79 + 16 * q^83 + 36 * q^85 + 18 * q^89 - 6 * q^91 + 8 * q^95 - 15 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.35194	0.604606	0.302303	−	0.953212i	$-0.402245\pi$
$5$	1.35194	0.604606	0.302303	+	0.953212i	$0.402245\pi$
$6$	0	0
$7$	−2.73419	−1.03343	−0.516714	−	0.856158i	$-0.672845\pi$
$7$	−2.73419	−1.03343	−0.516714	+	0.856158i	$0.672845\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.52420	1.53214	0.766069	−	0.642759i	$-0.222210\pi$
$13$	5.52420	1.53214	0.766069	+	0.642759i	$0.222210\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.79001	1.16175	0.580874	−	0.813994i	$-0.302711\pi$
$17$	4.79001	1.16175	0.580874	+	0.813994i	$0.302711\pi$
$18$	0	0
$19$	3.52420	0.808507	0.404253	−	0.914647i	$-0.367531\pi$
$19$	3.52420	0.808507	0.404253	+	0.914647i	$0.367531\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.351939	−0.0733844	−0.0366922	−	0.999327i	$-0.511682\pi$
$23$	−0.351939	−0.0733844	−0.0366922	+	0.999327i	$0.511682\pi$
$24$	0	0
$25$	−3.17226	−0.634452
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.61033	−0.670421	−0.335211	−	0.942143i	$-0.608808\pi$
$29$	−3.61033	−0.670421	−0.335211	+	0.942143i	$0.608808\pi$
$30$	0	0
$31$	−7.69646	−1.38233	−0.691163	−	0.722699i	$-0.742901\pi$
$31$	−7.69646	−1.38233	−0.691163	+	0.722699i	$0.742901\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.69646	−0.624816
$36$	0	0
$37$	7.82032	1.28565	0.642826	−	0.766012i	$-0.277762\pi$
$37$	7.82032	1.28565	0.642826	+	0.766012i	$0.277762\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.561931	0.0877588	0.0438794	−	0.999037i	$-0.486028\pi$
$41$	0.561931	0.0877588	0.0438794	+	0.999037i	$0.486028\pi$
$42$	0	0
$43$	−5.38225	−0.820786	−0.410393	−	0.911909i	$-0.634608\pi$
$43$	−5.38225	−0.820786	−0.410393	+	0.911909i	$0.634608\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.17226	0.754452	0.377226	−	0.926121i	$-0.376878\pi$
$47$	5.17226	0.754452	0.377226	+	0.926121i	$0.376878\pi$
$48$	0	0
$49$	0.475800	0.0679715
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.87614	1.21923	0.609616	−	0.792697i	$-0.291324\pi$
$53$	8.87614	1.21923	0.609616	+	0.792697i	$0.291324\pi$
$54$	0	0
$55$	−1.35194	−0.182295
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.11644	0.666104	0.333052	−	0.942908i	$-0.391922\pi$
$59$	5.11644	0.666104	0.333052	+	0.942908i	$0.391922\pi$
$60$	0	0
$61$	4.99258	0.639235	0.319617	−	0.947547i	$-0.396446\pi$
$61$	4.99258	0.639235	0.319617	+	0.947547i	$0.396446\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.46838	0.926339
$66$	0	0
$67$	7.69646	0.940272	0.470136	−	0.882594i	$-0.344205\pi$
$67$	7.69646	0.940272	0.470136	+	0.882594i	$0.344205\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.8761	1.52812	0.764058	−	0.645147i	$-0.223204\pi$
$71$	12.8761	1.52812	0.764058	+	0.645147i	$0.223204\pi$
$72$	0	0
$73$	4.87614	0.570709	0.285354	−	0.958422i	$-0.407889\pi$
$73$	4.87614	0.570709	0.285354	+	0.958422i	$0.407889\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.73419	0.311590
$78$	0	0
$79$	−5.20999	−0.586170	−0.293085	−	0.956086i	$-0.594682\pi$
$79$	−5.20999	−0.586170	−0.293085	+	0.956086i	$0.594682\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.2281	1.34221	0.671103	−	0.741364i	$-0.265821\pi$
$83$	12.2281	1.34221	0.671103	+	0.741364i	$0.265821\pi$
$84$	0	0
$85$	6.47580	0.702399
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−15.1042	−1.58335
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.76450	0.488828
$96$	0	0
$97$	−13.9926	−1.42073	−0.710366	−	0.703833i	$-0.751470\pi$
$97$	−13.9926	−1.42073	−0.710366	+	0.703833i	$0.751470\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	19.3068	1.92110	0.960549	−	0.278112i	$-0.0897086\pi$
$101$	19.3068	1.92110	0.960549	+	0.278112i	$0.0897086\pi$
$102$	0	0
$103$	12.2887	1.21084	0.605421	−	0.795905i	$-0.293005\pi$
$103$	12.2887	1.21084	0.605421	+	0.795905i	$0.293005\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4003	1.39213	0.696067	−	0.717977i	$-0.254932\pi$
$107$	14.4003	1.39213	0.696067	+	0.717977i	$0.254932\pi$
$108$	0	0
$109$	−13.9245	−1.33373	−0.666864	−	0.745179i	$-0.732364\pi$
$109$	−13.9245	−1.33373	−0.666864	+	0.745179i	$0.732364\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.86872	−0.552083	−0.276041	−	0.961146i	$-0.589023\pi$
$113$	−5.86872	−0.552083	−0.276041	+	0.961146i	$0.589023\pi$
$114$	0	0
$115$	−0.475800	−0.0443686
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−13.0968	−1.20058
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.0484	−0.988199
$126$	0	0
$127$	−1.88356	−0.167139	−0.0835693	−	0.996502i	$-0.526632\pi$
$127$	−1.88356	−0.167139	−0.0835693	+	0.996502i	$0.526632\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.87614	−0.251289	−0.125645	−	0.992075i	$-0.540100\pi$
$131$	−2.87614	−0.251289	−0.125645	+	0.992075i	$0.540100\pi$
$132$	0	0
$133$	−9.63583	−0.835533
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.53162	0.387162	0.193581	−	0.981084i	$-0.437990\pi$
$137$	4.53162	0.387162	0.193581	+	0.981084i	$0.437990\pi$
$138$	0	0
$139$	2.25839	0.191554	0.0957771	−	0.995403i	$-0.469466\pi$
$139$	2.25839	0.191554	0.0957771	+	0.995403i	$0.469466\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−5.52420	−0.461957
$144$	0	0
$145$	−4.88095	−0.405341
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.58744	0.211971	0.105985	−	0.994368i	$-0.466200\pi$
$149$	2.58744	0.211971	0.105985	+	0.994368i	$0.466200\pi$
$150$	0	0
$151$	−6.75970	−0.550096	−0.275048	−	0.961430i	$-0.588694\pi$
$151$	−6.75970	−0.550096	−0.275048	+	0.961430i	$0.588694\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.4051	−0.835761
$156$	0	0
$157$	22.5800	1.80208	0.901041	−	0.433734i	$-0.142804\pi$
$157$	22.5800	1.80208	0.901041	+	0.433734i	$0.142804\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.962269	0.0758374
$162$	0	0
$163$	−22.0000	−1.72317	−0.861586	−	0.507611i	$-0.830529\pi$
$163$	−22.0000	−1.72317	−0.861586	+	0.507611i	$0.830529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.228078	0.0176492	0.00882460	−	0.999961i	$-0.497191\pi$
$167$	0.228078	0.0176492	0.00882460	+	0.999961i	$0.497191\pi$
$168$	0	0
$169$	17.5168	1.34744
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.46357	−0.263331	−0.131665	−	0.991294i	$-0.542032\pi$
$173$	−3.46357	−0.263331	−0.131665	+	0.991294i	$0.542032\pi$
$174$	0	0
$175$	8.67357	0.655660
$176$	0	0
$177$	0	0
$178$	0	0
$179$	16.5800	1.23925	0.619624	−	0.784899i	$-0.287285\pi$
$179$	16.5800	1.23925	0.619624	+	0.784899i	$0.287285\pi$
$180$	0	0
$181$	15.3445	1.14055	0.570275	−	0.821454i	$-0.306837\pi$
$181$	15.3445	1.14055	0.570275	+	0.821454i	$0.306837\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.5726	0.777313
$186$	0	0
$187$	−4.79001	−0.350280
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.76450	0.127675	0.0638375	−	0.997960i	$-0.479666\pi$
$191$	1.76450	0.127675	0.0638375	+	0.997960i	$0.479666\pi$
$192$	0	0
$193$	3.24030	0.233242	0.116621	−	0.993176i	$-0.462794\pi$
$193$	3.24030	0.233242	0.116621	+	0.993176i	$0.462794\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.8942	−1.84489	−0.922444	−	0.386132i	$-0.873811\pi$
$197$	−25.8942	−1.84489	−0.922444	+	0.386132i	$0.873811\pi$
$198$	0	0
$199$	−2.28870	−0.162242	−0.0811209	−	0.996704i	$-0.525850\pi$
$199$	−2.28870	−0.162242	−0.0811209	+	0.996704i	$0.525850\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.87133	0.692832
$204$	0	0
$205$	0.759696	0.0530595
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.52420	−0.243774
$210$	0	0
$211$	−25.5955	−1.76207	−0.881033	−	0.473054i	$-0.843151\pi$
$211$	−25.5955	−1.76207	−0.881033	+	0.473054i	$0.843151\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.27648	−0.496252
$216$	0	0
$217$	21.0436	1.42853
$218$	0	0
$219$	0	0
$220$	0	0
$221$	26.4610	1.77996
$222$	0	0
$223$	6.05582	0.405528	0.202764	−	0.979228i	$-0.435008\pi$
$223$	6.05582	0.405528	0.202764	+	0.979228i	$0.435008\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.06804	−0.0708885	−0.0354443	−	0.999372i	$-0.511285\pi$
$227$	−1.06804	−0.0708885	−0.0354443	+	0.999372i	$0.511285\pi$
$228$	0	0
$229$	17.5120	1.15722	0.578612	−	0.815603i	$-0.303595\pi$
$229$	17.5120	1.15722	0.578612	+	0.815603i	$0.303595\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.4003	−0.681349	−0.340674	−	0.940181i	$-0.610655\pi$
$233$	−10.4003	−0.681349	−0.340674	+	0.940181i	$0.610655\pi$
$234$	0	0
$235$	6.99258	0.456146
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−23.3371	−1.50955	−0.754776	−	0.655983i	$-0.772254\pi$
$239$	−23.3371	−1.50955	−0.754776	+	0.655983i	$0.772254\pi$
$240$	0	0
$241$	−0.359358	−0.0231483	−0.0115741	−	0.999933i	$-0.503684\pi$
$241$	−0.359358	−0.0231483	−0.0115741	+	0.999933i	$0.503684\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.643253	0.0410959
$246$	0	0
$247$	19.4684	1.23874
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.9852	1.57705	0.788525	−	0.615003i	$-0.210845\pi$
$251$	24.9852	1.57705	0.788525	+	0.615003i	$0.210845\pi$
$252$	0	0
$253$	0.351939	0.0221262
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−13.6965	−0.854362	−0.427181	−	0.904166i	$-0.640493\pi$
$257$	−13.6965	−0.854362	−0.427181	+	0.904166i	$0.640493\pi$
$258$	0	0
$259$	−21.3823	−1.32863
$260$	0	0
$261$	0	0
$262$	0	0
$263$	13.2813	0.818959	0.409480	−	0.912319i	$-0.365710\pi$
$263$	13.2813	0.818959	0.409480	+	0.912319i	$0.365710\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.6965	−1.20091	−0.600457	−	0.799657i	$-0.705015\pi$
$269$	−19.6965	−1.20091	−0.600457	+	0.799657i	$0.705015\pi$
$270$	0	0
$271$	−15.5545	−0.944869	−0.472435	−	0.881366i	$-0.656625\pi$
$271$	−15.5545	−0.944869	−0.472435	+	0.881366i	$0.656625\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.17226	0.191294
$276$	0	0
$277$	−23.5046	−1.41225	−0.706126	−	0.708086i	$-0.749559\pi$
$277$	−23.5046	−1.41225	−0.706126	+	0.708086i	$0.749559\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.31902	0.496271	0.248135	−	0.968725i	$-0.420182\pi$
$281$	8.31902	0.496271	0.248135	+	0.968725i	$0.420182\pi$
$282$	0	0
$283$	−8.40034	−0.499348	−0.249674	−	0.968330i	$-0.580324\pi$
$283$	−8.40034	−0.499348	−0.249674	+	0.968330i	$0.580324\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.53643	−0.0906923
$288$	0	0
$289$	5.94418	0.349658
$290$	0	0
$291$	0	0
$292$	0	0
$293$	29.5242	1.72482	0.862411	−	0.506209i	$-0.168953\pi$
$293$	29.5242	1.72482	0.862411	+	0.506209i	$0.168953\pi$
$294$	0	0
$295$	6.91712	0.402730
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.94418	−0.112435
$300$	0	0
$301$	14.7161	0.848222
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.74967	0.386485
$306$	0	0
$307$	−7.15417	−0.408310	−0.204155	−	0.978939i	$-0.565445\pi$
$307$	−7.15417	−0.408310	−0.204155	+	0.978939i	$0.565445\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.2132	1.08948	0.544741	−	0.838604i	$-0.316628\pi$
$311$	19.2132	1.08948	0.544741	+	0.838604i	$0.316628\pi$
$312$	0	0
$313$	8.70388	0.491972	0.245986	−	0.969273i	$-0.420888\pi$
$313$	8.70388	0.491972	0.245986	+	0.969273i	$0.420888\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.2207	−1.52886	−0.764432	−	0.644704i	$-0.776981\pi$
$317$	−27.2207	−1.52886	−0.764432	+	0.644704i	$0.776981\pi$
$318$	0	0
$319$	3.61033	0.202140
$320$	0	0
$321$	0	0
$322$	0	0
$323$	16.8809	0.939281
$324$	0	0
$325$	−17.5242	−0.972068
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.1419	−0.779671
$330$	0	0
$331$	−7.00742	−0.385163	−0.192581	−	0.981281i	$-0.561686\pi$
$331$	−7.00742	−0.385163	−0.192581	+	0.981281i	$0.561686\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.4051	0.568494
$336$	0	0
$337$	−29.7933	−1.62294	−0.811471	−	0.584393i	$-0.801333\pi$
$337$	−29.7933	−1.62294	−0.811471	+	0.584393i	$0.801333\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.69646	0.416787
$342$	0	0
$343$	17.8384	0.963183
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−32.1574	−1.72630	−0.863150	−	0.504947i	$-0.831512\pi$
$347$	−32.1574	−1.72630	−0.863150	+	0.504947i	$0.831512\pi$
$348$	0	0
$349$	−0.415175	−0.0222238	−0.0111119	−	0.999938i	$-0.503537\pi$
$349$	−0.415175	−0.0222238	−0.0111119	+	0.999938i	$0.503537\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	26.0362	1.38577	0.692883	−	0.721050i	$-0.256340\pi$
$353$	26.0362	1.38577	0.692883	+	0.721050i	$0.256340\pi$
$354$	0	0
$355$	17.4078	0.923908
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.4003	−0.971133	−0.485566	−	0.874200i	$-0.661387\pi$
$359$	−18.4003	−0.971133	−0.485566	+	0.874200i	$0.661387\pi$
$360$	0	0
$361$	−6.58002	−0.346317
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.59224	0.345054
$366$	0	0
$367$	−3.81290	−0.199032	−0.0995160	−	0.995036i	$-0.531729\pi$
$367$	−3.81290	−0.199032	−0.0995160	+	0.995036i	$0.531729\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−24.2691	−1.25999
$372$	0	0
$373$	11.1797	0.578862	0.289431	−	0.957199i	$-0.406534\pi$
$373$	11.1797	0.578862	0.289431	+	0.957199i	$0.406534\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−19.9442	−1.02718
$378$	0	0
$379$	−18.5726	−0.954010	−0.477005	−	0.878900i	$-0.658278\pi$
$379$	−18.5726	−0.954010	−0.477005	+	0.878900i	$0.658278\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.8761	−1.06672	−0.533360	−	0.845888i	$-0.679071\pi$
$383$	−20.8761	−1.06672	−0.533360	+	0.845888i	$0.679071\pi$
$384$	0	0
$385$	3.69646	0.188389
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8.59224	−0.435644	−0.217822	−	0.975989i	$-0.569895\pi$
$389$	−8.59224	−0.435644	−0.217822	+	0.975989i	$0.569895\pi$
$390$	0	0
$391$	−1.68579	−0.0852542
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7.04359	−0.354402
$396$	0	0
$397$	20.3297	1.02032	0.510159	−	0.860080i	$-0.329587\pi$
$397$	20.3297	1.02032	0.510159	+	0.860080i	$0.329587\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.17226	−0.208353	−0.104176	−	0.994559i	$-0.533221\pi$
$401$	−4.17226	−0.208353	−0.104176	+	0.994559i	$0.533221\pi$
$402$	0	0
$403$	−42.5168	−2.11791
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.82032	−0.387639
$408$	0	0
$409$	24.5774	1.21527	0.607637	−	0.794215i	$-0.292117\pi$
$409$	24.5774	1.21527	0.607637	+	0.794215i	$0.292117\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.9893	−0.688370
$414$	0	0
$415$	16.5316	0.811505
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.1239	−0.738849	−0.369424	−	0.929261i	$-0.620445\pi$
$419$	−15.1239	−0.738849	−0.369424	+	0.929261i	$0.620445\pi$
$420$	0	0
$421$	29.0362	1.41514	0.707568	−	0.706645i	$-0.249792\pi$
$421$	29.0362	1.41514	0.707568	+	0.706645i	$0.249792\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−15.1952	−0.737073
$426$	0	0
$427$	−13.6507	−0.660602
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.52420	−0.0734181	−0.0367091	−	0.999326i	$-0.511687\pi$
$431$	−1.52420	−0.0734181	−0.0367091	+	0.999326i	$0.511687\pi$
$432$	0	0
$433$	−21.8565	−1.05036	−0.525178	−	0.850992i	$-0.676001\pi$
$433$	−21.8565	−1.05036	−0.525178	+	0.850992i	$0.676001\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.24030	−0.0593318
$438$	0	0
$439$	−1.30679	−0.0623697	−0.0311848	−	0.999514i	$-0.509928\pi$
$439$	−1.30679	−0.0623697	−0.0311848	+	0.999514i	$0.509928\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−31.3929	−1.49152	−0.745761	−	0.666213i	$-0.767914\pi$
$443$	−31.3929	−1.49152	−0.745761	+	0.666213i	$0.767914\pi$
$444$	0	0
$445$	8.11164	0.384528
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.75709	−0.271694	−0.135847	−	0.990730i	$-0.543376\pi$
$449$	−5.75709	−0.271694	−0.135847	+	0.990730i	$0.543376\pi$
$450$	0	0
$451$	−0.561931	−0.0264603
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−20.4200	−0.957304
$456$	0	0
$457$	−38.7449	−1.81241	−0.906204	−	0.422841i	$-0.861033\pi$
$457$	−38.7449	−1.81241	−0.906204	+	0.422841i	$0.861033\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−0.303540	−0.0141373	−0.00706863	−	0.999975i	$-0.502250\pi$
$461$	−0.303540	−0.0141373	−0.00706863	+	0.999975i	$0.502250\pi$
$462$	0	0
$463$	12.9516	0.601912	0.300956	−	0.953638i	$-0.402694\pi$
$463$	12.9516	0.601912	0.300956	+	0.953638i	$0.402694\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−15.9926	−0.740048	−0.370024	−	0.929022i	$-0.620651\pi$
$467$	−15.9926	−0.740048	−0.370024	+	0.929022i	$0.620651\pi$
$468$	0	0
$469$	−21.0436	−0.971703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.38225	0.247476
$474$	0	0
$475$	−11.1797	−0.512959
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.9852	1.55282	0.776411	−	0.630227i	$-0.217038\pi$
$479$	33.9852	1.55282	0.776411	+	0.630227i	$0.217038\pi$
$480$	0	0
$481$	43.2010	1.96980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−18.9171	−0.858982
$486$	0	0
$487$	40.9219	1.85435	0.927175	−	0.374629i	$-0.122230\pi$
$487$	40.9219	1.85435	0.927175	+	0.374629i	$0.122230\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.05101	−0.182820	−0.0914098	−	0.995813i	$-0.529137\pi$
$491$	−4.05101	−0.182820	−0.0914098	+	0.995813i	$0.529137\pi$
$492$	0	0
$493$	−17.2935	−0.778861
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−35.2058	−1.57920
$498$	0	0
$499$	−21.5094	−0.962891	−0.481446	−	0.876476i	$-0.659888\pi$
$499$	−21.5094	−0.962891	−0.481446	+	0.876476i	$0.659888\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.93937	0.175648	0.0878240	−	0.996136i	$-0.472009\pi$
$503$	3.93937	0.175648	0.0878240	+	0.996136i	$0.472009\pi$
$504$	0	0
$505$	26.1016	1.16151
$506$	0	0
$507$	0	0
$508$	0	0
$509$	36.2132	1.60512	0.802562	−	0.596569i	$-0.203470\pi$
$509$	36.2132	1.60512	0.802562	+	0.596569i	$0.203470\pi$
$510$	0	0
$511$	−13.3323	−0.589786
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.6136	0.732082
$516$	0	0
$517$	−5.17226	−0.227476
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.52420	−0.329641	−0.164821	−	0.986324i	$-0.552705\pi$
$521$	−7.52420	−0.329641	−0.164821	+	0.986324i	$0.552705\pi$
$522$	0	0
$523$	−13.0377	−0.570100	−0.285050	−	0.958513i	$-0.592010\pi$
$523$	−13.0377	−0.570100	−0.285050	+	0.958513i	$0.592010\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.8661	−1.60591
$528$	0	0
$529$	−22.8761	−0.994615
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.10422	0.134459
$534$	0	0
$535$	19.4684	0.841692
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.475800	−0.0204942
$540$	0	0
$541$	−25.2355	−1.08496	−0.542479	−	0.840069i	$-0.682514\pi$
$541$	−25.2355	−1.08496	−0.542479	+	0.840069i	$0.682514\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−18.8251	−0.806380
$546$	0	0
$547$	−25.1855	−1.07686	−0.538428	−	0.842672i	$-0.680982\pi$
$547$	−25.1855	−1.07686	−0.538428	+	0.842672i	$0.680982\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.7235	−0.542040
$552$	0	0
$553$	14.2451	0.605764
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.1468	−1.19262	−0.596308	−	0.802756i	$-0.703366\pi$
$557$	−28.1468	−1.19262	−0.596308	+	0.802756i	$0.703366\pi$
$558$	0	0
$559$	−29.7326	−1.25756
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.4200	−0.607730	−0.303865	−	0.952715i	$-0.598277\pi$
$563$	−14.4200	−0.607730	−0.303865	+	0.952715i	$0.598277\pi$
$564$	0	0
$565$	−7.93415	−0.333792
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.88095	0.204620	0.102310	−	0.994753i	$-0.467377\pi$
$569$	4.88095	0.204620	0.102310	+	0.994753i	$0.467377\pi$
$570$	0	0
$571$	39.7678	1.66423	0.832114	−	0.554604i	$-0.187130\pi$
$571$	39.7678	1.66423	0.832114	+	0.554604i	$0.187130\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.11644	0.0465589
$576$	0	0
$577$	21.1116	0.878889	0.439444	−	0.898270i	$-0.355175\pi$
$577$	21.1116	0.878889	0.439444	+	0.898270i	$0.355175\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−33.4339	−1.38707
$582$	0	0
$583$	−8.87614	−0.367612
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.4636	−0.762073	−0.381037	−	0.924560i	$-0.624433\pi$
$587$	−18.4636	−0.762073	−0.381037	+	0.924560i	$0.624433\pi$
$588$	0	0
$589$	−27.1239	−1.11762
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.16159	−0.253026	−0.126513	−	0.991965i	$-0.540379\pi$
$593$	−6.16159	−0.253026	−0.126513	+	0.991965i	$0.540379\pi$
$594$	0	0
$595$	−17.7061	−0.725878
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.30354	0.216697	0.108348	−	0.994113i	$-0.465444\pi$
$599$	5.30354	0.216697	0.108348	+	0.994113i	$0.465444\pi$
$600$	0	0
$601$	21.2765	0.867886	0.433943	−	0.900940i	$-0.357122\pi$
$601$	21.2765	0.867886	0.433943	+	0.900940i	$0.357122\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.35194	0.0549641
$606$	0	0
$607$	11.2913	0.458300	0.229150	−	0.973391i	$-0.426405\pi$
$607$	11.2913	0.458300	0.229150	+	0.973391i	$0.426405\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.5726	1.15592
$612$	0	0
$613$	−24.3100	−0.981873	−0.490937	−	0.871195i	$-0.663345\pi$
$613$	−24.3100	−0.981873	−0.490937	+	0.871195i	$0.663345\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	30.8661	1.24262	0.621311	−	0.783564i	$-0.286600\pi$
$617$	30.8661	1.24262	0.621311	+	0.783564i	$0.286600\pi$
$618$	0	0
$619$	45.0288	1.80986	0.904929	−	0.425562i	$-0.139924\pi$
$619$	45.0288	1.80986	0.904929	+	0.425562i	$0.139924\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−16.4051	−0.657258
$624$	0	0
$625$	0.924538	0.0369815
$626$	0	0
$627$	0	0
$628$	0	0
$629$	37.4594	1.49360
$630$	0	0
$631$	18.1164	0.721204	0.360602	−	0.932720i	$-0.382571\pi$
$631$	18.1164	0.721204	0.360602	+	0.932720i	$0.382571\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.54645	−0.101053
$636$	0	0
$637$	2.62842	0.104142
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.74486	−0.345401	−0.172701	−	0.984974i	$-0.555249\pi$
$641$	−8.74486	−0.345401	−0.172701	+	0.984974i	$0.555249\pi$
$642$	0	0
$643$	24.8613	0.980434	0.490217	−	0.871600i	$-0.336918\pi$
$643$	24.8613	0.980434	0.490217	+	0.871600i	$0.336918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.9516	0.509180	0.254590	−	0.967049i	$-0.418059\pi$
$647$	12.9516	0.509180	0.254590	+	0.967049i	$0.418059\pi$
$648$	0	0
$649$	−5.11644	−0.200838
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7.28128	0.284939	0.142469	−	0.989799i	$-0.454496\pi$
$653$	7.28128	0.284939	0.142469	+	0.989799i	$0.454496\pi$
$654$	0	0
$655$	−3.88836	−0.151931
$656$	0	0
$657$	0	0
$658$	0	0
$659$	22.1164	0.861534	0.430767	−	0.902463i	$-0.358243\pi$
$659$	22.1164	0.861534	0.430767	+	0.902463i	$0.358243\pi$
$660$	0	0
$661$	−34.6864	−1.34915	−0.674573	−	0.738208i	$-0.735672\pi$
$661$	−34.6864	−1.34915	−0.674573	+	0.738208i	$0.735672\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13.0271	−0.505168
$666$	0	0
$667$	1.27062	0.0491985
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.99258	−0.192736
$672$	0	0
$673$	2.63322	0.101503	0.0507517	−	0.998711i	$-0.483838\pi$
$673$	2.63322	0.101503	0.0507517	+	0.998711i	$0.483838\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29.5497	−1.13569	−0.567844	−	0.823136i	$-0.692222\pi$
$677$	−29.5497	−1.13569	−0.567844	+	0.823136i	$0.692222\pi$
$678$	0	0
$679$	38.2584	1.46822
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32.3175	−1.23659	−0.618297	−	0.785945i	$-0.712177\pi$
$683$	−32.3175	−1.23659	−0.618297	+	0.785945i	$0.712177\pi$
$684$	0	0
$685$	6.12647	0.234080
$686$	0	0
$687$	0	0
$688$	0	0
$689$	49.0336	1.86803
$690$	0	0
$691$	−30.9729	−1.17827	−0.589133	−	0.808036i	$-0.700531\pi$
$691$	−30.9729	−1.17827	−0.589133	+	0.808036i	$0.700531\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.05321	0.115815
$696$	0	0
$697$	2.69165	0.101954
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.0107	−0.680254	−0.340127	−	0.940380i	$-0.610470\pi$
$701$	−18.0107	−0.680254	−0.340127	+	0.940380i	$0.610470\pi$
$702$	0	0
$703$	27.5604	1.03946
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−52.7885	−1.98531
$708$	0	0
$709$	16.1797	0.607641	0.303820	−	0.952729i	$-0.401738\pi$
$709$	16.1797	0.607641	0.303820	+	0.952729i	$0.401738\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.70869	0.101441
$714$	0	0
$715$	−7.46838	−0.279302
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.3955	1.13356	0.566781	−	0.823868i	$-0.308189\pi$
$719$	30.3955	1.13356	0.566781	+	0.823868i	$0.308189\pi$
$720$	0	0
$721$	−33.5997	−1.25132
$722$	0	0
$723$	0	0
$724$	0	0
$725$	11.4529	0.425350
$726$	0	0
$727$	17.2255	0.638857	0.319429	−	0.947610i	$-0.396509\pi$
$727$	17.2255	0.638857	0.319429	+	0.947610i	$0.396509\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.7810	−0.953546
$732$	0	0
$733$	−3.71130	−0.137080	−0.0685399	−	0.997648i	$-0.521834\pi$
$733$	−3.71130	−0.137080	−0.0685399	+	0.997648i	$0.521834\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.69646	−0.283503
$738$	0	0
$739$	39.5397	1.45449	0.727245	−	0.686378i	$-0.240800\pi$
$739$	39.5397	1.45449	0.727245	+	0.686378i	$0.240800\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	23.0484	0.845564	0.422782	−	0.906231i	$-0.361054\pi$
$743$	23.0484	0.845564	0.422782	+	0.906231i	$0.361054\pi$
$744$	0	0
$745$	3.49806	0.128159
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−39.3733	−1.43867
$750$	0	0
$751$	8.24291	0.300788	0.150394	−	0.988626i	$-0.451946\pi$
$751$	8.24291	0.300788	0.150394	+	0.988626i	$0.451946\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−9.13870	−0.332591
$756$	0	0
$757$	4.73744	0.172185	0.0860926	−	0.996287i	$-0.472562\pi$
$757$	4.73744	0.172185	0.0860926	+	0.996287i	$0.472562\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−43.9197	−1.59209	−0.796044	−	0.605238i	$-0.793078\pi$
$761$	−43.9197	−1.59209	−0.796044	+	0.605238i	$0.793078\pi$
$762$	0	0
$763$	38.0723	1.37831
$764$	0	0
$765$	0	0
$766$	0	0
$767$	28.2643	1.02056
$768$	0	0
$769$	−44.2084	−1.59420	−0.797098	−	0.603849i	$-0.793633\pi$
$769$	−44.2084	−1.59420	−0.797098	+	0.603849i	$0.793633\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.89578	0.104154	0.0520770	−	0.998643i	$-0.483416\pi$
$773$	2.89578	0.104154	0.0520770	+	0.998643i	$0.483416\pi$
$774$	0	0
$775$	24.4152	0.877019
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.98036	0.0709536
$780$	0	0
$781$	−12.8761	−0.460744
$782$	0	0
$783$	0	0
$784$	0	0
$785$	30.5268	1.08955
$786$	0	0
$787$	−35.9900	−1.28290	−0.641452	−	0.767163i	$-0.721668\pi$
$787$	−35.9900	−1.28290	−0.641452	+	0.767163i	$0.721668\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.0462	0.570537
$792$	0	0
$793$	27.5800	0.979395
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.99519	−0.141517	−0.0707585	−	0.997493i	$-0.522542\pi$
$797$	−3.99519	−0.141517	−0.0707585	+	0.997493i	$0.522542\pi$
$798$	0	0
$799$	24.7752	0.876483
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4.87614	−0.172075
$804$	0	0
$805$	1.30093	0.0458517
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.9278	−1.36863	−0.684314	−	0.729187i	$-0.739898\pi$
$809$	−38.9278	−1.36863	−0.684314	+	0.729187i	$0.739898\pi$
$810$	0	0
$811$	21.1903	0.744094	0.372047	−	0.928214i	$-0.378656\pi$
$811$	21.1903	0.744094	0.372047	+	0.928214i	$0.378656\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−29.7427	−1.04184
$816$	0	0
$817$	−18.9681	−0.663611
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.61514	0.265770	0.132885	−	0.991131i	$-0.457576\pi$
$821$	7.61514	0.265770	0.132885	+	0.991131i	$0.457576\pi$
$822$	0	0
$823$	−37.7885	−1.31722	−0.658611	−	0.752483i	$-0.728856\pi$
$823$	−37.7885	−1.31722	−0.658611	+	0.752483i	$0.728856\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.9171	1.35328	0.676640	−	0.736314i	$-0.263435\pi$
$827$	38.9171	1.35328	0.676640	+	0.736314i	$0.263435\pi$
$828$	0	0
$829$	−20.1797	−0.700869	−0.350435	−	0.936587i	$-0.613966\pi$
$829$	−20.1797	−0.700869	−0.350435	+	0.936587i	$0.613966\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.27909	0.0789657
$834$	0	0
$835$	0.308348	0.0106708
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−16.8687	−0.582373	−0.291186	−	0.956666i	$-0.594050\pi$
$839$	−16.8687	−0.582373	−0.291186	+	0.956666i	$0.594050\pi$
$840$	0	0
$841$	−15.9655	−0.550535
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23.6816	0.814673
$846$	0	0
$847$	−2.73419	−0.0939479
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.75228	−0.0943469
$852$	0	0
$853$	−13.7161	−0.469630	−0.234815	−	0.972040i	$-0.575449\pi$
$853$	−13.7161	−0.469630	−0.234815	+	0.972040i	$0.575449\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21.0032	0.717457	0.358729	−	0.933442i	$-0.383210\pi$
$857$	21.0032	0.717457	0.358729	+	0.933442i	$0.383210\pi$
$858$	0	0
$859$	−29.5604	−1.00859	−0.504293	−	0.863532i	$-0.668247\pi$
$859$	−29.5604	−1.00859	−0.504293	+	0.863532i	$0.668247\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.8055	0.742267	0.371134	−	0.928579i	$-0.378969\pi$
$863$	21.8055	0.742267	0.371134	+	0.928579i	$0.378969\pi$
$864$	0	0
$865$	−4.68254	−0.159211
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.20999	0.176737
$870$	0	0
$871$	42.5168	1.44063
$872$	0	0
$873$	0	0
$874$	0	0
$875$	30.2084	1.02123
$876$	0	0
$877$	−34.1574	−1.15341	−0.576707	−	0.816951i	$-0.695663\pi$
$877$	−34.1574	−1.15341	−0.576707	+	0.816951i	$0.695663\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	45.5094	1.53325	0.766625	−	0.642095i	$-0.221935\pi$
$881$	45.5094	1.53325	0.766625	+	0.642095i	$0.221935\pi$
$882$	0	0
$883$	1.12386	0.0378209	0.0189105	−	0.999821i	$-0.493980\pi$
$883$	1.12386	0.0378209	0.0189105	+	0.999821i	$0.493980\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.86872	0.129899	0.0649495	−	0.997889i	$-0.479311\pi$
$887$	3.86872	0.129899	0.0649495	+	0.997889i	$0.479311\pi$
$888$	0	0
$889$	5.15001	0.172726
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.2281	0.609979
$894$	0	0
$895$	22.4152	0.749257
$896$	0	0
$897$	0	0
$898$	0	0
$899$	27.7868	0.926740
$900$	0	0
$901$	42.5168	1.41644
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.7449	0.689582
$906$	0	0
$907$	8.36417	0.277728	0.138864	−	0.990311i	$-0.455655\pi$
$907$	8.36417	0.277728	0.138864	+	0.990311i	$0.455655\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44.2568	1.46629	0.733147	−	0.680070i	$-0.238051\pi$
$911$	44.2568	1.46629	0.733147	+	0.680070i	$0.238051\pi$
$912$	0	0
$913$	−12.2281	−0.404690
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.86391	0.259689
$918$	0	0
$919$	−36.7045	−1.21077	−0.605385	−	0.795933i	$-0.706981\pi$
$919$	−36.7045	−1.21077	−0.605385	+	0.795933i	$0.706981\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	71.1304	2.34128
$924$	0	0
$925$	−24.8081	−0.815685
$926$	0	0
$927$	0	0
$928$	0	0
$929$	56.9581	1.86873	0.934367	−	0.356311i	$-0.115966\pi$
$929$	56.9581	1.86873	0.934367	+	0.356311i	$0.115966\pi$
$930$	0	0
$931$	1.67682	0.0549554
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.47580	−0.211781
$936$	0	0
$937$	−37.6062	−1.22854	−0.614270	−	0.789096i	$-0.710549\pi$
$937$	−37.6062	−1.22854	−0.614270	+	0.789096i	$0.710549\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.1755	−0.690302	−0.345151	−	0.938547i	$-0.612172\pi$
$941$	−21.1755	−0.690302	−0.345151	+	0.938547i	$0.612172\pi$
$942$	0	0
$943$	−0.197765	−0.00644013
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−53.1090	−1.72581	−0.862906	−	0.505365i	$-0.831358\pi$
$947$	−53.1090	−1.72581	−0.862906	+	0.505365i	$0.831358\pi$
$948$	0	0
$949$	26.9368	0.874404
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−47.8597	−1.55033	−0.775165	−	0.631759i	$-0.782333\pi$
$953$	−47.8597	−1.55033	−0.775165	+	0.631759i	$0.782333\pi$
$954$	0	0
$955$	2.38550	0.0771930
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12.3903	−0.400104
$960$	0	0
$961$	28.2355	0.910822
$962$	0	0
$963$	0	0
$964$	0	0
$965$	4.38069	0.141020
$966$	0	0
$967$	−8.90164	−0.286258	−0.143129	−	0.989704i	$-0.545716\pi$
$967$	−8.90164	−0.286258	−0.143129	+	0.989704i	$0.545716\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.5290	−0.434167	−0.217083	−	0.976153i	$-0.569654\pi$
$971$	−13.5290	−0.434167	−0.217083	+	0.976153i	$0.569654\pi$
$972$	0	0
$973$	−6.17487	−0.197957
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−47.2961	−1.51314	−0.756568	−	0.653914i	$-0.773126\pi$
$977$	−47.2961	−1.51314	−0.756568	+	0.653914i	$0.773126\pi$
$978$	0	0
$979$	−6.00000	−0.191761
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−44.3148	−1.41342	−0.706712	−	0.707501i	$-0.749822\pi$
$983$	−44.3148	−1.41342	−0.706712	+	0.707501i	$0.749822\pi$
$984$	0	0
$985$	−35.0074	−1.11543
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.89423	0.0602329
$990$	0	0
$991$	−3.77673	−0.119972	−0.0599859	−	0.998199i	$-0.519106\pi$
$991$	−3.77673	−0.119972	−0.0599859	+	0.998199i	$0.519106\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.09419	−0.0980923
$996$	0	0
$997$	42.2542	1.33820	0.669102	−	0.743170i	$-0.266679\pi$
$997$	42.2542	1.33820	0.669102	+	0.743170i	$0.266679\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2376.2.a.p.1.2	yes	3
3.2	odd	2		2376.2.a.k.1.2	✓	3
4.3	odd	2		4752.2.a.br.1.2		3
12.11	even	2		4752.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2376.2.a.k.1.2	✓	3	3.2	odd	2
2376.2.a.p.1.2	yes	3	1.1	even	1	trivial
4752.2.a.bj.1.2		3	12.11	even	2
4752.2.a.br.1.2		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2376 = 2^{3} \cdot 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2376.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35194 q^{5} -2.73419 q^{7} -1.00000 q^{11} +5.52420 q^{13} +4.79001 q^{17} +3.52420 q^{19} -0.351939 q^{23} -3.17226 q^{25} -3.61033 q^{29} -7.69646 q^{31} -3.69646 q^{35} +7.82032 q^{37} +0.561931 q^{41} -5.38225 q^{43} +5.17226 q^{47} +0.475800 q^{49} +8.87614 q^{53} -1.35194 q^{55} +5.11644 q^{59} +4.99258 q^{61} +7.46838 q^{65} +7.69646 q^{67} +12.8761 q^{71} +4.87614 q^{73} +2.73419 q^{77} -5.20999 q^{79} +12.2281 q^{83} +6.47580 q^{85} +6.00000 q^{89} -15.1042 q^{91} +4.76450 q^{95} -13.9926 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{5} - 3 q^{7} - 3 q^{11} + 3 q^{17} - 6 q^{19} + q^{23} + 5 q^{25} + 13 q^{29} + 8 q^{31} + 20 q^{35} + 11 q^{37} + 11 q^{41} - 13 q^{43} + q^{47} + 18 q^{49} + 8 q^{53} - 2 q^{55} + 7 q^{59}+ \cdots - 15 q^{97}+O(q^{100}) \) 3 * q + 2 * q^5 - 3 * q^7 - 3 * q^11 + 3 * q^17 - 6 * q^19 + q^23 + 5 * q^25 + 13 * q^29 + 8 * q^31 + 20 * q^35 + 11 * q^37 + 11 * q^41 - 13 * q^43 + q^47 + 18 * q^49 + 8 * q^53 - 2 * q^55 + 7 * q^59 - 12 * q^61 + 12 * q^65 - 8 * q^67 + 20 * q^71 - 4 * q^73 + 3 * q^77 - 27 * q^79 + 16 * q^83 + 36 * q^85 + 18 * q^89 - 6 * q^91 + 8 * q^95 - 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more