LMFDB - Embedded newform 2028.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(1,2028)] 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("2028.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4,0,0,0,0,0,4,0,0,0,0,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{43})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 23x^{2} + 100 $ x^4 - 23*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.41269$ of defining polynomial
Character	$\chi$	$=$	2028.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.41269 q^{5} +4.14474 q^{7} +1.00000 q^{9} +3.46410 q^{11} +2.41269 q^{15} +6.17891 q^{17} -3.46410 q^{19} +4.14474 q^{21} +2.00000 q^{23} +0.821092 q^{25} +1.00000 q^{27} -8.17891 q^{29} -7.60885 q^{31} +3.46410 q^{33} +10.0000 q^{35} +1.05141 q^{37} -5.87680 q^{41} +0.821092 q^{43} +2.41269 q^{45} -10.3923 q^{47} +10.1789 q^{49} +6.17891 q^{51} -10.1789 q^{53} +8.35782 q^{55} -3.46410 q^{57} -1.36129 q^{59} +5.00000 q^{61} +4.14474 q^{63} -4.14474 q^{67} +2.00000 q^{69} -3.46410 q^{71} -11.3828 q^{73} +0.821092 q^{75} +14.3578 q^{77} +13.1789 q^{79} +1.00000 q^{81} -11.7536 q^{83} +14.9078 q^{85} -8.17891 q^{87} +6.92820 q^{89} -7.60885 q^{93} -8.35782 q^{95} -2.04193 q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9} + 2 q^{17} + 8 q^{23} + 26 q^{25} + 4 q^{27} - 10 q^{29} + 40 q^{35} + 26 q^{43} + 18 q^{49} + 2 q^{51} - 18 q^{53} - 12 q^{55} + 20 q^{61} + 8 q^{69} + 26 q^{75} + 12 q^{77} + 30 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 2 * q^17 + 8 * q^23 + 26 * q^25 + 4 * q^27 - 10 * q^29 + 40 * q^35 + 26 * q^43 + 18 * q^49 + 2 * q^51 - 18 * q^53 - 12 * q^55 + 20 * q^61 + 8 * q^69 + 26 * q^75 + 12 * q^77 + 30 * q^79 + 4 * q^81 - 10 * q^87 + 12 * q^95

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.41269	1.07899	0.539495	−	0.841989i	$-0.318615\pi$
$5$	2.41269	1.07899	0.539495	+	0.841989i	$0.318615\pi$
$6$	0	0
$7$	4.14474	1.56657	0.783283	−	0.621665i	$-0.213544\pi$
$7$	4.14474	1.56657	0.783283	+	0.621665i	$0.213544\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.41269	0.622955
$16$	0	0
$17$	6.17891	1.49861	0.749303	−	0.662228i	$-0.230389\pi$
$17$	6.17891	1.49861	0.749303	+	0.662228i	$0.230389\pi$
$18$	0	0
$19$	−3.46410	−0.794719	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	−3.46410	−0.794719	−0.397360	+	0.917663i	$0.630073\pi$
$20$	0	0
$21$	4.14474	0.904457
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	0.821092	0.164218
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.17891	−1.51879	−0.759393	−	0.650633i	$-0.774504\pi$
$29$	−8.17891	−1.51879	−0.759393	+	0.650633i	$0.774504\pi$
$30$	0	0
$31$	−7.60885	−1.36659	−0.683295	−	0.730143i	$-0.739454\pi$
$31$	−7.60885	−1.36659	−0.683295	+	0.730143i	$0.739454\pi$
$32$	0	0
$33$	3.46410	0.603023
$34$	0	0
$35$	10.0000	1.69031
$36$	0	0
$37$	1.05141	0.172850	0.0864252	−	0.996258i	$-0.472456\pi$
$37$	1.05141	0.172850	0.0864252	+	0.996258i	$0.472456\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.87680	−0.917801	−0.458901	−	0.888488i	$-0.651757\pi$
$41$	−5.87680	−0.917801	−0.458901	+	0.888488i	$0.651757\pi$
$42$	0	0
$43$	0.821092	0.125215	0.0626077	−	0.998038i	$-0.480058\pi$
$43$	0.821092	0.125215	0.0626077	+	0.998038i	$0.480058\pi$
$44$	0	0
$45$	2.41269	0.359663
$46$	0	0
$47$	−10.3923	−1.51587	−0.757937	−	0.652328i	$-0.773792\pi$
$47$	−10.3923	−1.51587	−0.757937	+	0.652328i	$0.773792\pi$
$48$	0	0
$49$	10.1789	1.45413
$50$	0	0
$51$	6.17891	0.865220
$52$	0	0
$53$	−10.1789	−1.39818	−0.699090	−	0.715033i	$-0.746411\pi$
$53$	−10.1789	−1.39818	−0.699090	+	0.715033i	$0.746411\pi$
$54$	0	0
$55$	8.35782	1.12697
$56$	0	0
$57$	−3.46410	−0.458831
$58$	0	0
$59$	−1.36129	−0.177224	−0.0886122	−	0.996066i	$-0.528243\pi$
$59$	−1.36129	−0.177224	−0.0886122	+	0.996066i	$0.528243\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	0	0
$63$	4.14474	0.522189
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.14474	−0.506361	−0.253181	−	0.967419i	$-0.581477\pi$
$67$	−4.14474	−0.506361	−0.253181	+	0.967419i	$0.581477\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	−11.3828	−1.33226	−0.666130	−	0.745836i	$-0.732050\pi$
$73$	−11.3828	−1.33226	−0.666130	+	0.745836i	$0.732050\pi$
$74$	0	0
$75$	0.821092	0.0948115
$76$	0	0
$77$	14.3578	1.63623
$78$	0	0
$79$	13.1789	1.48274	0.741372	−	0.671095i	$-0.234176\pi$
$79$	13.1789	1.48274	0.741372	+	0.671095i	$0.234176\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.7536	−1.29012	−0.645062	−	0.764130i	$-0.723169\pi$
$83$	−11.7536	−1.29012	−0.645062	+	0.764130i	$0.723169\pi$
$84$	0	0
$85$	14.9078	1.61698
$86$	0	0
$87$	−8.17891	−0.876871
$88$	0	0
$89$	6.92820	0.734388	0.367194	−	0.930144i	$-0.380318\pi$
$89$	6.92820	0.734388	0.367194	+	0.930144i	$0.380318\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.60885	−0.789001
$94$	0	0
$95$	−8.35782	−0.857494
$96$	0	0
$97$	−2.04193	−0.207326	−0.103663	−	0.994612i	$-0.533056\pi$
$97$	−2.04193	−0.207326	−0.103663	+	0.994612i	$0.533056\pi$
$98$	0	0
$99$	3.46410	0.348155
$100$	0	0
$101$	−12.1789	−1.21185	−0.605923	−	0.795523i	$-0.707196\pi$
$101$	−12.1789	−1.21185	−0.605923	+	0.795523i	$0.707196\pi$
$102$	0	0
$103$	−3.17891	−0.313227	−0.156614	−	0.987660i	$-0.550058\pi$
$103$	−3.17891	−0.313227	−0.156614	+	0.987660i	$0.550058\pi$
$104$	0	0
$105$	10.0000	0.975900
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−7.60885	−0.728795	−0.364398	−	0.931243i	$-0.618725\pi$
$109$	−7.60885	−0.728795	−0.364398	+	0.931243i	$0.618725\pi$
$110$	0	0
$111$	1.05141	0.0997952
$112$	0	0
$113$	12.1789	1.14570	0.572848	−	0.819662i	$-0.305839\pi$
$113$	12.1789	1.14570	0.572848	+	0.819662i	$0.305839\pi$
$114$	0	0
$115$	4.82539	0.449970
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.6100	2.34766
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.87680	−0.529893
$124$	0	0
$125$	−10.0824	−0.901800
$126$	0	0
$127$	2.82109	0.250331	0.125166	−	0.992136i	$-0.460054\pi$
$127$	2.82109	0.250331	0.125166	+	0.992136i	$0.460054\pi$
$128$	0	0
$129$	0.821092	0.0722931
$130$	0	0
$131$	6.35782	0.555485	0.277743	−	0.960656i	$-0.410414\pi$
$131$	6.35782	0.555485	0.277743	+	0.960656i	$0.410414\pi$
$132$	0	0
$133$	−14.3578	−1.24498
$134$	0	0
$135$	2.41269	0.207652
$136$	0	0
$137$	12.8050	1.09400	0.547002	−	0.837131i	$-0.315769\pi$
$137$	12.8050	1.09400	0.547002	+	0.837131i	$0.315769\pi$
$138$	0	0
$139$	17.1789	1.45710	0.728548	−	0.684995i	$-0.240196\pi$
$139$	17.1789	1.45710	0.728548	+	0.684995i	$0.240196\pi$
$140$	0	0
$141$	−10.3923	−0.875190
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−19.7332	−1.63875
$146$	0	0
$147$	10.1789	0.839542
$148$	0	0
$149$	5.87680	0.481446	0.240723	−	0.970594i	$-0.422615\pi$
$149$	5.87680	0.481446	0.240723	+	0.970594i	$0.422615\pi$
$150$	0	0
$151$	20.0431	1.63108	0.815541	−	0.578699i	$-0.196439\pi$
$151$	20.0431	1.63108	0.815541	+	0.578699i	$0.196439\pi$
$152$	0	0
$153$	6.17891	0.499535
$154$	0	0
$155$	−18.3578	−1.47454
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	0	0
$159$	−10.1789	−0.807240
$160$	0	0
$161$	8.28949	0.653303
$162$	0	0
$163$	7.60885	0.595971	0.297985	−	0.954570i	$-0.403685\pi$
$163$	7.60885	0.595971	0.297985	+	0.954570i	$0.403685\pi$
$164$	0	0
$165$	8.35782	0.650655
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.46410	−0.264906
$172$	0	0
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	3.40322	0.257259
$176$	0	0
$177$	−1.36129	−0.102321
$178$	0	0
$179$	8.35782	0.624693	0.312346	−	0.949968i	$-0.398885\pi$
$179$	8.35782	0.624693	0.312346	+	0.949968i	$0.398885\pi$
$180$	0	0
$181$	20.5367	1.52648	0.763241	−	0.646113i	$-0.223607\pi$
$181$	20.5367	1.52648	0.763241	+	0.646113i	$0.223607\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	2.53673	0.186504
$186$	0	0
$187$	21.4044	1.56524
$188$	0	0
$189$	4.14474	0.301486
$190$	0	0
$191$	−2.35782	−0.170606	−0.0853028	−	0.996355i	$-0.527186\pi$
$191$	−2.35782	−0.170606	−0.0853028	+	0.996355i	$0.527186\pi$
$192$	0	0
$193$	−13.4856	−0.970718	−0.485359	−	0.874315i	$-0.661311\pi$
$193$	−13.4856	−0.970718	−0.485359	+	0.874315i	$0.661311\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.65078	0.687589	0.343795	−	0.939045i	$-0.388288\pi$
$197$	9.65078	0.687589	0.343795	+	0.939045i	$0.388288\pi$
$198$	0	0
$199$	21.5367	1.52670	0.763349	−	0.645986i	$-0.223554\pi$
$199$	21.5367	1.52670	0.763349	+	0.645986i	$0.223554\pi$
$200$	0	0
$201$	−4.14474	−0.292348
$202$	0	0
$203$	−33.8995	−2.37928
$204$	0	0
$205$	−14.1789	−0.990298
$206$	0	0
$207$	2.00000	0.139010
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−15.5367	−1.06959	−0.534796	−	0.844981i	$-0.679612\pi$
$211$	−15.5367	−1.06959	−0.534796	+	0.844981i	$0.679612\pi$
$212$	0	0
$213$	−3.46410	−0.237356
$214$	0	0
$215$	1.98104	0.135106
$216$	0	0
$217$	−31.5367	−2.14085
$218$	0	0
$219$	−11.3828	−0.769180
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.2177	1.01905	0.509526	−	0.860455i	$-0.329821\pi$
$223$	15.2177	1.01905	0.509526	+	0.860455i	$0.329821\pi$
$224$	0	0
$225$	0.821092	0.0547394
$226$	0	0
$227$	9.03102	0.599410	0.299705	−	0.954032i	$-0.403112\pi$
$227$	9.03102	0.599410	0.299705	+	0.954032i	$0.403112\pi$
$228$	0	0
$229$	9.65078	0.637741	0.318871	−	0.947798i	$-0.396696\pi$
$229$	9.65078	0.637741	0.318871	+	0.947798i	$0.396696\pi$
$230$	0	0
$231$	14.3578	0.944675
$232$	0	0
$233$	−22.7156	−1.48815	−0.744075	−	0.668096i	$-0.767110\pi$
$233$	−22.7156	−1.48815	−0.744075	+	0.668096i	$0.767110\pi$
$234$	0	0
$235$	−25.0735	−1.63561
$236$	0	0
$237$	13.1789	0.856062
$238$	0	0
$239$	2.10282	0.136020	0.0680099	−	0.997685i	$-0.478335\pi$
$239$	2.10282	0.136020	0.0680099	+	0.997685i	$0.478335\pi$
$240$	0	0
$241$	−5.87680	−0.378558	−0.189279	−	0.981923i	$-0.560615\pi$
$241$	−5.87680	−0.378558	−0.189279	+	0.981923i	$0.560615\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	24.5586	1.56899
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−11.7536	−0.744854
$250$	0	0
$251$	20.7156	1.30756	0.653780	−	0.756685i	$-0.273182\pi$
$251$	20.7156	1.30756	0.653780	+	0.756685i	$0.273182\pi$
$252$	0	0
$253$	6.92820	0.435572
$254$	0	0
$255$	14.9078	0.933564
$256$	0	0
$257$	−22.1789	−1.38348	−0.691741	−	0.722146i	$-0.743156\pi$
$257$	−22.1789	−1.38348	−0.691741	+	0.722146i	$0.743156\pi$
$258$	0	0
$259$	4.35782	0.270782
$260$	0	0
$261$	−8.17891	−0.506262
$262$	0	0
$263$	32.3578	1.99527	0.997634	−	0.0687455i	$-0.0218997\pi$
$263$	32.3578	1.99527	0.997634	+	0.0687455i	$0.0218997\pi$
$264$	0	0
$265$	−24.5586	−1.50862
$266$	0	0
$267$	6.92820	0.423999
$268$	0	0
$269$	−12.3578	−0.753469	−0.376735	−	0.926321i	$-0.622953\pi$
$269$	−12.3578	−0.753469	−0.376735	+	0.926321i	$0.622953\pi$
$270$	0	0
$271$	−14.5370	−0.883063	−0.441531	−	0.897246i	$-0.645565\pi$
$271$	−14.5370	−0.883063	−0.441531	+	0.897246i	$0.645565\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.84434	0.171520
$276$	0	0
$277$	0.178908	0.0107496	0.00537478	−	0.999986i	$-0.498289\pi$
$277$	0.178908	0.0107496	0.00537478	+	0.999986i	$0.498289\pi$
$278$	0	0
$279$	−7.60885	−0.455530
$280$	0	0
$281$	11.4437	0.682675	0.341337	−	0.939941i	$-0.389120\pi$
$281$	11.4437	0.682675	0.341337	+	0.939941i	$0.389120\pi$
$282$	0	0
$283$	27.1789	1.61562	0.807809	−	0.589444i	$-0.200653\pi$
$283$	27.1789	1.61562	0.807809	+	0.589444i	$0.200653\pi$
$284$	0	0
$285$	−8.35782	−0.495074
$286$	0	0
$287$	−24.3578	−1.43780
$288$	0	0
$289$	21.1789	1.24582
$290$	0	0
$291$	−2.04193	−0.119700
$292$	0	0
$293$	−32.8481	−1.91901	−0.959503	−	0.281700i	$-0.909102\pi$
$293$	−32.8481	−1.91901	−0.959503	+	0.281700i	$0.909102\pi$
$294$	0	0
$295$	−3.28437	−0.191223
$296$	0	0
$297$	3.46410	0.201008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.40322	0.196158
$302$	0	0
$303$	−12.1789	−0.699660
$304$	0	0
$305$	12.0635	0.690752
$306$	0	0
$307$	11.0729	0.631967	0.315983	−	0.948765i	$-0.397666\pi$
$307$	11.0729	0.631967	0.315983	+	0.948765i	$0.397666\pi$
$308$	0	0
$309$	−3.17891	−0.180842
$310$	0	0
$311$	−11.6422	−0.660168	−0.330084	−	0.943952i	$-0.607077\pi$
$311$	−11.6422	−0.660168	−0.330084	+	0.943952i	$0.607077\pi$
$312$	0	0
$313$	27.8945	1.57669	0.788346	−	0.615232i	$-0.210938\pi$
$313$	27.8945	1.57669	0.788346	+	0.615232i	$0.210938\pi$
$314$	0	0
$315$	10.0000	0.563436
$316$	0	0
$317$	−3.77398	−0.211968	−0.105984	−	0.994368i	$-0.533799\pi$
$317$	−3.77398	−0.211968	−0.105984	+	0.994368i	$0.533799\pi$
$318$	0	0
$319$	−28.3326	−1.58632
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	−21.4044	−1.19097
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−7.60885	−0.420770
$328$	0	0
$329$	−43.0735	−2.37472
$330$	0	0
$331$	−20.1040	−1.10501	−0.552507	−	0.833508i	$-0.686329\pi$
$331$	−20.1040	−1.10501	−0.552507	+	0.833508i	$0.686329\pi$
$332$	0	0
$333$	1.05141	0.0576168
$334$	0	0
$335$	−10.0000	−0.546358
$336$	0	0
$337$	25.3578	1.38133	0.690664	−	0.723176i	$-0.257318\pi$
$337$	25.3578	1.38133	0.690664	+	0.723176i	$0.257318\pi$
$338$	0	0
$339$	12.1789	0.661468
$340$	0	0
$341$	−26.3578	−1.42736
$342$	0	0
$343$	13.1758	0.711424
$344$	0	0
$345$	4.82539	0.259790
$346$	0	0
$347$	20.3578	1.09286	0.546432	−	0.837503i	$-0.315986\pi$
$347$	20.3578	1.09286	0.546432	+	0.837503i	$0.315986\pi$
$348$	0	0
$349$	10.3314	0.553028	0.276514	−	0.961010i	$-0.410821\pi$
$349$	10.3314	0.553028	0.276514	+	0.961010i	$0.410821\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−18.9917	−1.01082	−0.505412	−	0.862878i	$-0.668660\pi$
$353$	−18.9917	−1.01082	−0.505412	+	0.862878i	$0.668660\pi$
$354$	0	0
$355$	−8.35782	−0.443587
$356$	0	0
$357$	25.6100	1.35542
$358$	0	0
$359$	−18.6818	−0.985987	−0.492994	−	0.870033i	$-0.664097\pi$
$359$	−18.6818	−0.985987	−0.492994	+	0.870033i	$0.664097\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−27.4633	−1.43749
$366$	0	0
$367$	12.8211	0.669256	0.334628	−	0.942350i	$-0.391389\pi$
$367$	12.8211	0.669256	0.334628	+	0.942350i	$0.391389\pi$
$368$	0	0
$369$	−5.87680	−0.305934
$370$	0	0
$371$	−42.1890	−2.19034
$372$	0	0
$373$	−11.3578	−0.588085	−0.294043	−	0.955792i	$-0.595001\pi$
$373$	−11.3578	−0.588085	−0.294043	+	0.955792i	$0.595001\pi$
$374$	0	0
$375$	−10.0824	−0.520654
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.24756	0.320916	0.160458	−	0.987043i	$-0.448703\pi$
$379$	6.24756	0.320916	0.160458	+	0.987043i	$0.448703\pi$
$380$	0	0
$381$	2.82109	0.144529
$382$	0	0
$383$	−19.3016	−0.986263	−0.493132	−	0.869955i	$-0.664148\pi$
$383$	−19.3016	−0.986263	−0.493132	+	0.869955i	$0.664148\pi$
$384$	0	0
$385$	34.6410	1.76547
$386$	0	0
$387$	0.821092	0.0417384
$388$	0	0
$389$	6.53673	0.331425	0.165713	−	0.986174i	$-0.447008\pi$
$389$	6.53673	0.331425	0.165713	+	0.986174i	$0.447008\pi$
$390$	0	0
$391$	12.3578	0.624962
$392$	0	0
$393$	6.35782	0.320709
$394$	0	0
$395$	31.7967	1.59986
$396$	0	0
$397$	−21.4653	−1.07731	−0.538655	−	0.842526i	$-0.681067\pi$
$397$	−21.4653	−1.07731	−0.538655	+	0.842526i	$0.681067\pi$
$398$	0	0
$399$	−14.3578	−0.718790
$400$	0	0
$401$	10.7022	0.534442	0.267221	−	0.963635i	$-0.413895\pi$
$401$	10.7022	0.534442	0.267221	+	0.963635i	$0.413895\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.41269	0.119888
$406$	0	0
$407$	3.64218	0.180536
$408$	0	0
$409$	−0.990521	−0.0489781	−0.0244891	−	0.999700i	$-0.507796\pi$
$409$	−0.990521	−0.0489781	−0.0244891	+	0.999700i	$0.507796\pi$
$410$	0	0
$411$	12.8050	0.631624
$412$	0	0
$413$	−5.64218	−0.277634
$414$	0	0
$415$	−28.3578	−1.39203
$416$	0	0
$417$	17.1789	0.841255
$418$	0	0
$419$	−12.7156	−0.621199	−0.310600	−	0.950541i	$-0.600530\pi$
$419$	−12.7156	−0.621199	−0.310600	+	0.950541i	$0.600530\pi$
$420$	0	0
$421$	32.1674	1.56774	0.783872	−	0.620922i	$-0.213242\pi$
$421$	32.1674	1.56774	0.783872	+	0.620922i	$0.213242\pi$
$422$	0	0
$423$	−10.3923	−0.505291
$424$	0	0
$425$	5.07345	0.246098
$426$	0	0
$427$	20.7237	1.00289
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.6818	0.899870	0.449935	−	0.893061i	$-0.351447\pi$
$431$	18.6818	0.899870	0.449935	+	0.893061i	$0.351447\pi$
$432$	0	0
$433$	2.64218	0.126975	0.0634876	−	0.997983i	$-0.479778\pi$
$433$	2.64218	0.126975	0.0634876	+	0.997983i	$0.479778\pi$
$434$	0	0
$435$	−19.7332	−0.946135
$436$	0	0
$437$	−6.92820	−0.331421
$438$	0	0
$439$	−23.8945	−1.14042	−0.570212	−	0.821497i	$-0.693139\pi$
$439$	−23.8945	−1.14042	−0.570212	+	0.821497i	$0.693139\pi$
$440$	0	0
$441$	10.1789	0.484710
$442$	0	0
$443$	8.00000	0.380091	0.190046	−	0.981775i	$-0.439136\pi$
$443$	8.00000	0.380091	0.190046	+	0.981775i	$0.439136\pi$
$444$	0	0
$445$	16.7156	0.792397
$446$	0	0
$447$	5.87680	0.277963
$448$	0	0
$449$	11.0121	0.519691	0.259846	−	0.965650i	$-0.416328\pi$
$449$	11.0121	0.519691	0.259846	+	0.965650i	$0.416328\pi$
$450$	0	0
$451$	−20.3578	−0.958612
$452$	0	0
$453$	20.0431	0.941706
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.29897	−0.341431	−0.170716	−	0.985320i	$-0.554608\pi$
$457$	−7.29897	−0.341431	−0.170716	+	0.985320i	$0.554608\pi$
$458$	0	0
$459$	6.17891	0.288407
$460$	0	0
$461$	−24.5586	−1.14381	−0.571904	−	0.820321i	$-0.693795\pi$
$461$	−24.5586	−1.14381	−0.571904	+	0.820321i	$0.693795\pi$
$462$	0	0
$463$	1.42217	0.0660940	0.0330470	−	0.999454i	$-0.489479\pi$
$463$	1.42217	0.0660940	0.0330470	+	0.999454i	$0.489479\pi$
$464$	0	0
$465$	−18.3578	−0.851323
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−17.1789	−0.793248
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	2.84434	0.130783
$474$	0	0
$475$	−2.84434	−0.130507
$476$	0	0
$477$	−10.1789	−0.466060
$478$	0	0
$479$	−6.92820	−0.316558	−0.158279	−	0.987394i	$-0.550594\pi$
$479$	−6.92820	−0.316558	−0.158279	+	0.987394i	$0.550594\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	8.28949	0.377185
$484$	0	0
$485$	−4.92655	−0.223703
$486$	0	0
$487$	24.2487	1.09881	0.549407	−	0.835555i	$-0.314854\pi$
$487$	24.2487	1.09881	0.549407	+	0.835555i	$0.314854\pi$
$488$	0	0
$489$	7.60885	0.344084
$490$	0	0
$491$	−18.7156	−0.844625	−0.422312	−	0.906450i	$-0.638781\pi$
$491$	−18.7156	−0.844625	−0.422312	+	0.906450i	$0.638781\pi$
$492$	0	0
$493$	−50.5367	−2.27606
$494$	0	0
$495$	8.35782	0.375656
$496$	0	0
$497$	−14.3578	−0.644036
$498$	0	0
$499$	−1.36129	−0.0609395	−0.0304698	−	0.999536i	$-0.509700\pi$
$499$	−1.36129	−0.0609395	−0.0304698	+	0.999536i	$0.509700\pi$
$500$	0	0
$501$	6.92820	0.309529
$502$	0	0
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−29.3840	−1.30757
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1663	0.627910	0.313955	−	0.949438i	$-0.398346\pi$
$509$	14.1663	0.627910	0.313955	+	0.949438i	$0.398346\pi$
$510$	0	0
$511$	−47.1789	−2.08707
$512$	0	0
$513$	−3.46410	−0.152944
$514$	0	0
$515$	−7.66973	−0.337969
$516$	0	0
$517$	−36.0000	−1.58328
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	−38.8945	−1.70400	−0.852000	−	0.523541i	$-0.824611\pi$
$521$	−38.8945	−1.70400	−0.852000	+	0.523541i	$0.824611\pi$
$522$	0	0
$523$	24.3578	1.06509	0.532546	−	0.846401i	$-0.321235\pi$
$523$	24.3578	1.06509	0.532546	+	0.846401i	$0.321235\pi$
$524$	0	0
$525$	3.40322	0.148528
$526$	0	0
$527$	−47.0144	−2.04798
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−1.36129	−0.0590748
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−24.1269	−1.04310
$536$	0	0
$537$	8.35782	0.360666
$538$	0	0
$539$	35.2608	1.51879
$540$	0	0
$541$	−3.09334	−0.132993	−0.0664965	−	0.997787i	$-0.521182\pi$
$541$	−3.09334	−0.132993	−0.0664965	+	0.997787i	$0.521182\pi$
$542$	0	0
$543$	20.5367	0.881315
$544$	0	0
$545$	−18.3578	−0.786362
$546$	0	0
$547$	13.5367	0.578789	0.289394	−	0.957210i	$-0.406546\pi$
$547$	13.5367	0.578789	0.289394	+	0.957210i	$0.406546\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	28.3326	1.20701
$552$	0	0
$553$	54.6232	2.32282
$554$	0	0
$555$	2.53673	0.107678
$556$	0	0
$557$	−32.1065	−1.36040	−0.680199	−	0.733027i	$-0.738107\pi$
$557$	−32.1065	−1.36040	−0.680199	+	0.733027i	$0.738107\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	21.4044	0.903693
$562$	0	0
$563$	18.3578	0.773690	0.386845	−	0.922145i	$-0.373565\pi$
$563$	18.3578	0.773690	0.386845	+	0.922145i	$0.373565\pi$
$564$	0	0
$565$	29.3840	1.23619
$566$	0	0
$567$	4.14474	0.174063
$568$	0	0
$569$	30.7156	1.28767	0.643833	−	0.765166i	$-0.277343\pi$
$569$	30.7156	1.28767	0.643833	+	0.765166i	$0.277343\pi$
$570$	0	0
$571$	7.64218	0.319815	0.159908	−	0.987132i	$-0.448880\pi$
$571$	7.64218	0.319815	0.159908	+	0.987132i	$0.448880\pi$
$572$	0	0
$573$	−2.35782	−0.0984992
$574$	0	0
$575$	1.64218	0.0684838
$576$	0	0
$577$	−43.2404	−1.80012	−0.900060	−	0.435765i	$-0.856478\pi$
$577$	−43.2404	−1.80012	−0.900060	+	0.435765i	$0.856478\pi$
$578$	0	0
$579$	−13.4856	−0.560444
$580$	0	0
$581$	−48.7156	−2.02107
$582$	0	0
$583$	−35.2608	−1.46035
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.1459	0.914059	0.457029	−	0.889452i	$-0.348913\pi$
$587$	22.1459	0.914059	0.457029	+	0.889452i	$0.348913\pi$
$588$	0	0
$589$	26.3578	1.08605
$590$	0	0
$591$	9.65078	0.396980
$592$	0	0
$593$	32.8481	1.34891	0.674454	−	0.738316i	$-0.264379\pi$
$593$	32.8481	1.34891	0.674454	+	0.738316i	$0.264379\pi$
$594$	0	0
$595$	61.7891	2.53311
$596$	0	0
$597$	21.5367	0.881439
$598$	0	0
$599$	9.64218	0.393969	0.196984	−	0.980407i	$-0.436885\pi$
$599$	9.64218	0.393969	0.196984	+	0.980407i	$0.436885\pi$
$600$	0	0
$601$	−0.178908	−0.00729782	−0.00364891	−	0.999993i	$-0.501161\pi$
$601$	−0.178908	−0.00729782	−0.00364891	+	0.999993i	$0.501161\pi$
$602$	0	0
$603$	−4.14474	−0.168787
$604$	0	0
$605$	2.41269	0.0980900
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−33.8995	−1.37368
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.47358	0.0999070	0.0499535	−	0.998752i	$-0.484093\pi$
$613$	2.47358	0.0999070	0.0499535	+	0.998752i	$0.484093\pi$
$614$	0	0
$615$	−14.1789	−0.571749
$616$	0	0
$617$	2.41269	0.0971314	0.0485657	−	0.998820i	$-0.484535\pi$
$617$	2.41269	0.0971314	0.0485657	+	0.998820i	$0.484535\pi$
$618$	0	0
$619$	−10.3314	−0.415255	−0.207627	−	0.978208i	$-0.566574\pi$
$619$	−10.3314	−0.415255	−0.207627	+	0.978208i	$0.566574\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	28.7156	1.15047
$624$	0	0
$625$	−28.4313	−1.13725
$626$	0	0
$627$	−12.0000	−0.479234
$628$	0	0
$629$	6.49655	0.259034
$630$	0	0
$631$	−6.24756	−0.248711	−0.124356	−	0.992238i	$-0.539686\pi$
$631$	−6.24756	−0.248711	−0.124356	+	0.992238i	$0.539686\pi$
$632$	0	0
$633$	−15.5367	−0.617529
$634$	0	0
$635$	6.80643	0.270105
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.46410	−0.137038
$640$	0	0
$641$	0.536725	0.0211994	0.0105997	−	0.999944i	$-0.496626\pi$
$641$	0.536725	0.0211994	0.0105997	+	0.999944i	$0.496626\pi$
$642$	0	0
$643$	−15.8983	−0.626969	−0.313485	−	0.949593i	$-0.601496\pi$
$643$	−15.8983	−0.626969	−0.313485	+	0.949593i	$0.601496\pi$
$644$	0	0
$645$	1.98104	0.0780035
$646$	0	0
$647$	−13.6422	−0.536330	−0.268165	−	0.963373i	$-0.586417\pi$
$647$	−13.6422	−0.536330	−0.268165	+	0.963373i	$0.586417\pi$
$648$	0	0
$649$	−4.71563	−0.185105
$650$	0	0
$651$	−31.5367	−1.23602
$652$	0	0
$653$	0.357817	0.0140024	0.00700122	−	0.999975i	$-0.497771\pi$
$653$	0.357817	0.0140024	0.00700122	+	0.999975i	$0.497771\pi$
$654$	0	0
$655$	15.3395	0.599363
$656$	0	0
$657$	−11.3828	−0.444086
$658$	0	0
$659$	−8.71563	−0.339513	−0.169756	−	0.985486i	$-0.554298\pi$
$659$	−8.71563	−0.339513	−0.169756	+	0.985486i	$0.554298\pi$
$660$	0	0
$661$	12.8659	0.500425	0.250212	−	0.968191i	$-0.419500\pi$
$661$	12.8659	0.500425	0.250212	+	0.968191i	$0.419500\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−34.6410	−1.34332
$666$	0	0
$667$	−16.3578	−0.633377
$668$	0	0
$669$	15.2177	0.588350
$670$	0	0
$671$	17.3205	0.668651
$672$	0	0
$673$	5.35782	0.206529	0.103264	−	0.994654i	$-0.467071\pi$
$673$	5.35782	0.206529	0.103264	+	0.994654i	$0.467071\pi$
$674$	0	0
$675$	0.821092	0.0316038
$676$	0	0
$677$	−3.64218	−0.139980	−0.0699902	−	0.997548i	$-0.522297\pi$
$677$	−3.64218	−0.139980	−0.0699902	+	0.997548i	$0.522297\pi$
$678$	0	0
$679$	−8.46327	−0.324791
$680$	0	0
$681$	9.03102	0.346069
$682$	0	0
$683$	17.9403	0.686465	0.343233	−	0.939250i	$-0.388478\pi$
$683$	17.9403	0.686465	0.343233	+	0.939250i	$0.388478\pi$
$684$	0	0
$685$	30.8945	1.18042
$686$	0	0
$687$	9.65078	0.368200
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.22860	0.313031	0.156515	−	0.987676i	$-0.449974\pi$
$691$	8.22860	0.313031	0.156515	+	0.987676i	$0.449974\pi$
$692$	0	0
$693$	14.3578	0.545408
$694$	0	0
$695$	41.4474	1.57219
$696$	0	0
$697$	−36.3122	−1.37542
$698$	0	0
$699$	−22.7156	−0.859184
$700$	0	0
$701$	−48.3578	−1.82645	−0.913225	−	0.407456i	$-0.866416\pi$
$701$	−48.3578	−1.82645	−0.913225	+	0.407456i	$0.866416\pi$
$702$	0	0
$703$	−3.64218	−0.137368
$704$	0	0
$705$	−25.0735	−0.944321
$706$	0	0
$707$	−50.4785	−1.89844
$708$	0	0
$709$	−29.4449	−1.10583	−0.552913	−	0.833239i	$-0.686484\pi$
$709$	−29.4449	−1.10583	−0.552913	+	0.833239i	$0.686484\pi$
$710$	0	0
$711$	13.1789	0.494248
$712$	0	0
$713$	−15.2177	−0.569907
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.10282	0.0785311
$718$	0	0
$719$	30.3578	1.13216	0.566078	−	0.824352i	$-0.308460\pi$
$719$	30.3578	1.13216	0.566078	+	0.824352i	$0.308460\pi$
$720$	0	0
$721$	−13.1758	−0.490691
$722$	0	0
$723$	−5.87680	−0.218560
$724$	0	0
$725$	−6.71563	−0.249412
$726$	0	0
$727$	−17.5367	−0.650401	−0.325201	−	0.945645i	$-0.605432\pi$
$727$	−17.5367	−0.650401	−0.325201	+	0.945645i	$0.605432\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.07345	0.187648
$732$	0	0
$733$	10.6413	0.393045	0.196523	−	0.980499i	$-0.437035\pi$
$733$	10.6413	0.393045	0.196523	+	0.980499i	$0.437035\pi$
$734$	0	0
$735$	24.5586	0.905857
$736$	0	0
$737$	−14.3578	−0.528877
$738$	0	0
$739$	−15.2177	−0.559792	−0.279896	−	0.960030i	$-0.590300\pi$
$739$	−15.2177	−0.559792	−0.279896	+	0.960030i	$0.590300\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.36129	0.0499407	0.0249704	−	0.999688i	$-0.492051\pi$
$743$	1.36129	0.0499407	0.0249704	+	0.999688i	$0.492051\pi$
$744$	0	0
$745$	14.1789	0.519475
$746$	0	0
$747$	−11.7536	−0.430041
$748$	0	0
$749$	−41.4474	−1.51446
$750$	0	0
$751$	19.6422	0.716753	0.358377	−	0.933577i	$-0.383330\pi$
$751$	19.6422	0.716753	0.358377	+	0.933577i	$0.383330\pi$
$752$	0	0
$753$	20.7156	0.754920
$754$	0	0
$755$	48.3578	1.75992
$756$	0	0
$757$	−38.7156	−1.40714	−0.703572	−	0.710624i	$-0.748413\pi$
$757$	−38.7156	−1.40714	−0.703572	+	0.710624i	$0.748413\pi$
$758$	0	0
$759$	6.92820	0.251478
$760$	0	0
$761$	48.4974	1.75803	0.879015	−	0.476794i	$-0.158201\pi$
$761$	48.4974	1.75803	0.879015	+	0.476794i	$0.158201\pi$
$762$	0	0
$763$	−31.5367	−1.14171
$764$	0	0
$765$	14.9078	0.538993
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.92820	0.249837	0.124919	−	0.992167i	$-0.460133\pi$
$769$	6.92820	0.249837	0.124919	+	0.992167i	$0.460133\pi$
$770$	0	0
$771$	−22.1789	−0.798754
$772$	0	0
$773$	−31.7967	−1.14365	−0.571823	−	0.820377i	$-0.693764\pi$
$773$	−31.7967	−1.14365	−0.571823	+	0.820377i	$0.693764\pi$
$774$	0	0
$775$	−6.24756	−0.224419
$776$	0	0
$777$	4.35782	0.156336
$778$	0	0
$779$	20.3578	0.729394
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	−8.17891	−0.292290
$784$	0	0
$785$	16.8889	0.602789
$786$	0	0
$787$	33.9604	1.21056	0.605278	−	0.796014i	$-0.293062\pi$
$787$	33.9604	1.21056	0.605278	+	0.796014i	$0.293062\pi$
$788$	0	0
$789$	32.3578	1.15197
$790$	0	0
$791$	50.4785	1.79481
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−24.5586	−0.871003
$796$	0	0
$797$	−17.0735	−0.604773	−0.302386	−	0.953185i	$-0.597783\pi$
$797$	−17.0735	−0.604773	−0.302386	+	0.953185i	$0.597783\pi$
$798$	0	0
$799$	−64.2131	−2.27170
$800$	0	0
$801$	6.92820	0.244796
$802$	0	0
$803$	−39.4313	−1.39150
$804$	0	0
$805$	20.0000	0.704907
$806$	0	0
$807$	−12.3578	−0.435016
$808$	0	0
$809$	46.5367	1.63614	0.818072	−	0.575116i	$-0.195043\pi$
$809$	46.5367	1.63614	0.818072	+	0.575116i	$0.195043\pi$
$810$	0	0
$811$	−0.680643	−0.0239006	−0.0119503	−	0.999929i	$-0.503804\pi$
$811$	−0.680643	−0.0239006	−0.0119503	+	0.999929i	$0.503804\pi$
$812$	0	0
$813$	−14.5370	−0.509837
$814$	0	0
$815$	18.3578	0.643046
$816$	0	0
$817$	−2.84434	−0.0995110
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−31.7967	−1.10971	−0.554856	−	0.831947i	$-0.687227\pi$
$821$	−31.7967	−1.10971	−0.554856	+	0.831947i	$0.687227\pi$
$822$	0	0
$823$	24.0000	0.836587	0.418294	−	0.908312i	$-0.362628\pi$
$823$	24.0000	0.836587	0.418294	+	0.908312i	$0.362628\pi$
$824$	0	0
$825$	2.84434	0.0990274
$826$	0	0
$827$	35.8805	1.24769	0.623844	−	0.781549i	$-0.285570\pi$
$827$	35.8805	1.24769	0.623844	+	0.781549i	$0.285570\pi$
$828$	0	0
$829$	−23.3578	−0.811251	−0.405625	−	0.914039i	$-0.632946\pi$
$829$	−23.3578	−0.811251	−0.405625	+	0.914039i	$0.632946\pi$
$830$	0	0
$831$	0.178908	0.00620626
$832$	0	0
$833$	62.8945	2.17917
$834$	0	0
$835$	16.7156	0.578468
$836$	0	0
$837$	−7.60885	−0.263000
$838$	0	0
$839$	15.2177	0.525373	0.262687	−	0.964881i	$-0.415391\pi$
$839$	15.2177	0.525373	0.262687	+	0.964881i	$0.415391\pi$
$840$	0	0
$841$	37.8945	1.30671
$842$	0	0
$843$	11.4437	0.394142
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.14474	0.142415
$848$	0	0
$849$	27.1789	0.932778
$850$	0	0
$851$	2.10282	0.0720836
$852$	0	0
$853$	−24.6195	−0.842955	−0.421477	−	0.906839i	$-0.638488\pi$
$853$	−24.6195	−0.842955	−0.421477	+	0.906839i	$0.638488\pi$
$854$	0	0
$855$	−8.35782	−0.285831
$856$	0	0
$857$	20.1789	0.689298	0.344649	−	0.938732i	$-0.387998\pi$
$857$	20.1789	0.689298	0.344649	+	0.938732i	$0.387998\pi$
$858$	0	0
$859$	54.2524	1.85107	0.925533	−	0.378666i	$-0.123617\pi$
$859$	54.2524	1.85107	0.925533	+	0.378666i	$0.123617\pi$
$860$	0	0
$861$	−24.3578	−0.830112
$862$	0	0
$863$	−20.0431	−0.682274	−0.341137	−	0.940014i	$-0.610812\pi$
$863$	−20.0431	−0.682274	−0.341137	+	0.940014i	$0.610812\pi$
$864$	0	0
$865$	−43.4285	−1.47661
$866$	0	0
$867$	21.1789	0.719273
$868$	0	0
$869$	45.6531	1.54867
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.04193	−0.0691088
$874$	0	0
$875$	−41.7891	−1.41273
$876$	0	0
$877$	12.8050	0.432394	0.216197	−	0.976350i	$-0.430635\pi$
$877$	12.8050	0.432394	0.216197	+	0.976350i	$0.430635\pi$
$878$	0	0
$879$	−32.8481	−1.10794
$880$	0	0
$881$	37.8211	1.27422	0.637112	−	0.770771i	$-0.280129\pi$
$881$	37.8211	1.27422	0.637112	+	0.770771i	$0.280129\pi$
$882$	0	0
$883$	17.8945	0.602199	0.301100	−	0.953593i	$-0.402646\pi$
$883$	17.8945	0.602199	0.301100	+	0.953593i	$0.402646\pi$
$884$	0	0
$885$	−3.28437	−0.110403
$886$	0	0
$887$	21.6422	0.726673	0.363337	−	0.931658i	$-0.381637\pi$
$887$	21.6422	0.726673	0.363337	+	0.931658i	$0.381637\pi$
$888$	0	0
$889$	11.6927	0.392161
$890$	0	0
$891$	3.46410	0.116052
$892$	0	0
$893$	36.0000	1.20469
$894$	0	0
$895$	20.1649	0.674037
$896$	0	0
$897$	0	0
$898$	0	0
$899$	62.2321	2.07556
$900$	0	0
$901$	−62.8945	−2.09532
$902$	0	0
$903$	3.40322	0.113252
$904$	0	0
$905$	49.5488	1.64706
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−12.1789	−0.403949
$910$	0	0
$911$	4.71563	0.156236	0.0781180	−	0.996944i	$-0.475109\pi$
$911$	4.71563	0.156236	0.0781180	+	0.996944i	$0.475109\pi$
$912$	0	0
$913$	−40.7156	−1.34749
$914$	0	0
$915$	12.0635	0.398806
$916$	0	0
$917$	26.3515	0.870204
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	11.0729	0.364866
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.863302	0.0283852
$926$	0	0
$927$	−3.17891	−0.104409
$928$	0	0
$929$	−31.4868	−1.03305	−0.516524	−	0.856273i	$-0.672774\pi$
$929$	−31.4868	−1.03305	−0.516524	+	0.856273i	$0.672774\pi$
$930$	0	0
$931$	−35.2608	−1.15563
$932$	0	0
$933$	−11.6422	−0.381148
$934$	0	0
$935$	51.6422	1.68888
$936$	0	0
$937$	48.8945	1.59732	0.798658	−	0.601786i	$-0.205544\pi$
$937$	48.8945	1.59732	0.798658	+	0.601786i	$0.205544\pi$
$938$	0	0
$939$	27.8945	0.910304
$940$	0	0
$941$	−37.2418	−1.21405	−0.607024	−	0.794683i	$-0.707637\pi$
$941$	−37.2418	−1.21405	−0.607024	+	0.794683i	$0.707637\pi$
$942$	0	0
$943$	−11.7536	−0.382750
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	−30.4354	−0.989017	−0.494509	−	0.869173i	$-0.664652\pi$
$947$	−30.4354	−0.989017	−0.494509	+	0.869173i	$0.664652\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−3.77398	−0.122380
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	−5.68869	−0.184082
$956$	0	0
$957$	−28.3326	−0.915862
$958$	0	0
$959$	53.0735	1.71383
$960$	0	0
$961$	26.8945	0.867566
$962$	0	0
$963$	−10.0000	−0.322245
$964$	0	0
$965$	−32.5367	−1.04739
$966$	0	0
$967$	−35.3825	−1.13783	−0.568913	−	0.822398i	$-0.692636\pi$
$967$	−35.3825	−1.13783	−0.568913	+	0.822398i	$0.692636\pi$
$968$	0	0
$969$	−21.4044	−0.687607
$970$	0	0
$971$	4.71563	0.151332	0.0756659	−	0.997133i	$-0.475892\pi$
$971$	4.71563	0.151332	0.0756659	+	0.997133i	$0.475892\pi$
$972$	0	0
$973$	71.2022	2.28264
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−17.0106	−0.544218	−0.272109	−	0.962266i	$-0.587721\pi$
$977$	−17.0106	−0.544218	−0.272109	+	0.962266i	$0.587721\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	0	0
$981$	−7.60885	−0.242932
$982$	0	0
$983$	24.8685	0.793181	0.396590	−	0.917996i	$-0.370193\pi$
$983$	24.8685	0.793181	0.396590	+	0.917996i	$0.370193\pi$
$984$	0	0
$985$	23.2844	0.741902
$986$	0	0
$987$	−43.0735	−1.37104
$988$	0	0
$989$	1.64218	0.0522184
$990$	0	0
$991$	−45.0735	−1.43181	−0.715903	−	0.698200i	$-0.753985\pi$
$991$	−45.0735	−1.43181	−0.715903	+	0.698200i	$0.753985\pi$
$992$	0	0
$993$	−20.1040	−0.637980
$994$	0	0
$995$	51.9615	1.64729
$996$	0	0
$997$	35.3578	1.11979	0.559897	−	0.828562i	$-0.310841\pi$
$997$	35.3578	1.11979	0.559897	+	0.828562i	$0.310841\pi$
$998$	0	0
$999$	1.05141	0.0332651

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.a.m.1.3		4
3.2	odd	2		6084.2.a.bd.1.2		4
4.3	odd	2		8112.2.a.cr.1.3		4
13.2	odd	12		2028.2.q.f.1837.1		4
13.3	even	3		2028.2.i.n.529.3		8
13.4	even	6		2028.2.i.n.2005.2		8
13.5	odd	4		2028.2.b.e.337.2		4
13.6	odd	12		156.2.q.b.49.1	✓	4
13.7	odd	12		2028.2.q.f.361.2		4
13.8	odd	4		2028.2.b.e.337.3		4
13.9	even	3		2028.2.i.n.2005.3		8
13.10	even	6		2028.2.i.n.529.2		8
13.11	odd	12		156.2.q.b.121.2	yes	4
13.12	even	2	inner	2028.2.a.m.1.2		4
39.5	even	4		6084.2.b.o.4393.3		4
39.8	even	4		6084.2.b.o.4393.2		4
39.11	even	12		468.2.t.d.433.1		4
39.32	even	12		468.2.t.d.361.2		4
39.38	odd	2		6084.2.a.bd.1.3		4
52.11	even	12		624.2.bv.f.433.2		4
52.19	even	12		624.2.bv.f.49.1		4
52.51	odd	2		8112.2.a.cr.1.2		4
65.19	odd	12		3900.2.cd.i.2701.2		4
65.24	odd	12		3900.2.cd.i.901.2		4
65.32	even	12		3900.2.bw.j.49.2		8
65.37	even	12		3900.2.bw.j.2149.3		8
65.58	even	12		3900.2.bw.j.49.3		8
65.63	even	12		3900.2.bw.j.2149.2		8
156.11	odd	12		1872.2.by.j.433.1		4
156.71	odd	12		1872.2.by.j.1297.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.b.49.1	✓	4	13.6	odd	12
156.2.q.b.121.2	yes	4	13.11	odd	12
468.2.t.d.361.2		4	39.32	even	12
468.2.t.d.433.1		4	39.11	even	12
624.2.bv.f.49.1		4	52.19	even	12
624.2.bv.f.433.2		4	52.11	even	12
1872.2.by.j.433.1		4	156.11	odd	12
1872.2.by.j.1297.2		4	156.71	odd	12
2028.2.a.m.1.2		4	13.12	even	2	inner
2028.2.a.m.1.3		4	1.1	even	1	trivial
2028.2.b.e.337.2		4	13.5	odd	4
2028.2.b.e.337.3		4	13.8	odd	4
2028.2.i.n.529.2		8	13.10	even	6
2028.2.i.n.529.3		8	13.3	even	3
2028.2.i.n.2005.2		8	13.4	even	6
2028.2.i.n.2005.3		8	13.9	even	3
2028.2.q.f.361.2		4	13.7	odd	12
2028.2.q.f.1837.1		4	13.2	odd	12
3900.2.bw.j.49.2		8	65.32	even	12
3900.2.bw.j.49.3		8	65.58	even	12
3900.2.bw.j.2149.2		8	65.63	even	12
3900.2.bw.j.2149.3		8	65.37	even	12
3900.2.cd.i.901.2		4	65.24	odd	12
3900.2.cd.i.2701.2		4	65.19	odd	12
6084.2.a.bd.1.2		4	3.2	odd	2
6084.2.a.bd.1.3		4	39.38	odd	2
6084.2.b.o.4393.2		4	39.8	even	4
6084.2.b.o.4393.3		4	39.5	even	4
8112.2.a.cr.1.2		4	52.51	odd	2
8112.2.a.cr.1.3		4	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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.41269 q^{5} +4.14474 q^{7} +1.00000 q^{9} +3.46410 q^{11} +2.41269 q^{15} +6.17891 q^{17} -3.46410 q^{19} +4.14474 q^{21} +2.00000 q^{23} +0.821092 q^{25} +1.00000 q^{27} -8.17891 q^{29} -7.60885 q^{31} +3.46410 q^{33} +10.0000 q^{35} +1.05141 q^{37} -5.87680 q^{41} +0.821092 q^{43} +2.41269 q^{45} -10.3923 q^{47} +10.1789 q^{49} +6.17891 q^{51} -10.1789 q^{53} +8.35782 q^{55} -3.46410 q^{57} -1.36129 q^{59} +5.00000 q^{61} +4.14474 q^{63} -4.14474 q^{67} +2.00000 q^{69} -3.46410 q^{71} -11.3828 q^{73} +0.821092 q^{75} +14.3578 q^{77} +13.1789 q^{79} +1.00000 q^{81} -11.7536 q^{83} +14.9078 q^{85} -8.17891 q^{87} +6.92820 q^{89} -7.60885 q^{93} -8.35782 q^{95} -2.04193 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9} + 2 q^{17} + 8 q^{23} + 26 q^{25} + 4 q^{27} - 10 q^{29} + 40 q^{35} + 26 q^{43} + 18 q^{49} + 2 q^{51} - 18 q^{53} - 12 q^{55} + 20 q^{61} + 8 q^{69} + 26 q^{75} + 12 q^{77} + 30 q^{79}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 2 * q^17 + 8 * q^23 + 26 * q^25 + 4 * q^27 - 10 * q^29 + 40 * q^35 + 26 * q^43 + 18 * q^49 + 2 * q^51 - 18 * q^53 - 12 * q^55 + 20 * q^61 + 8 * q^69 + 26 * q^75 + 12 * q^77 + 30 * q^79 + 4 * q^81 - 10 * q^87 + 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more