LMFDB - Embedded newform 2028.2.b.e.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(337,2028)] 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("2028.337"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.b (of order $2$, degree $1$, not minimal)

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:	no
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} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 156)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$3.08945 + 1.20635i$ of defining polynomial
Character	$\chi$	$=$	2028.337
Dual form			2028.2.b.e.337.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	2.41269i	1.07899i	0.841989	+	0.539495i	$0.181385\pi$
$5$	2.41269i	1.07899i	−0.841989	+	0.539495i	$0.818615\pi$
$6$	0	0
$7$	− 4.14474i	− 1.56657i	−0.621665	−	0.783283i	$-0.713544\pi$
$7$	− 4.14474i	− 1.56657i	0.621665	−	0.783283i	$-0.286456\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.41269i	0.622955i
$16$	0	0
$17$	−6.17891	−1.49861	−0.749303	−	0.662228i	$-0.769611\pi$
$17$	−6.17891	−1.49861	−0.749303	+	0.662228i	$0.769611\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	− 4.14474i	− 0.904457i
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	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
$31$	− 7.60885i	− 1.36659i	−0.730143	−	0.683295i	$-0.760546\pi$
$31$	− 7.60885i	− 1.36659i	0.730143	−	0.683295i	$-0.239454\pi$
$32$	0	0
$33$	− 3.46410i	− 0.603023i
$34$	0	0
$35$	10.0000	1.69031
$36$	0	0
$37$	− 1.05141i	− 0.172850i	−0.996258	−	0.0864252i	$-0.972456\pi$
$37$	− 1.05141i	− 0.172850i	0.996258	−	0.0864252i	$-0.0275444\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.87680i	− 0.917801i	−0.888488	−	0.458901i	$-0.848243\pi$
$41$	− 5.87680i	− 0.917801i	0.888488	−	0.458901i	$-0.151757\pi$
$42$	0	0
$43$	−0.821092	−0.125215	−0.0626077	−	0.998038i	$-0.519942\pi$
$43$	−0.821092	−0.125215	−0.0626077	+	0.998038i	$0.519942\pi$
$44$	0	0
$45$	2.41269i	0.359663i
$46$	0	0
$47$	10.3923i	1.51587i	0.652328	+	0.757937i	$0.273792\pi$
$47$	10.3923i	1.51587i	−0.652328	+	0.757937i	$0.726208\pi$
$48$	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
$57$	− 3.46410i	− 0.458831i
$58$	0	0
$59$	1.36129i	0.177224i	0.996066	+	0.0886122i	$0.0282432\pi$
$59$	1.36129i	0.177224i	−0.996066	+	0.0886122i	$0.971757\pi$
$60$	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
$63$	− 4.14474i	− 0.522189i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.14474i	− 0.506361i	−0.967419	−	0.253181i	$-0.918523\pi$
$67$	− 4.14474i	− 0.506361i	0.967419	−	0.253181i	$-0.0814767\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	− 3.46410i	− 0.411113i	−0.978645	−	0.205557i	$-0.934100\pi$
$71$	− 3.46410i	− 0.411113i	0.978645	−	0.205557i	$-0.0659005\pi$
$72$	0	0
$73$	11.3828i	1.33226i	0.745836	+	0.666130i	$0.232050\pi$
$73$	11.3828i	1.33226i	−0.745836	+	0.666130i	$0.767950\pi$
$74$	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
$83$	− 11.7536i	− 1.29012i	−0.764130	−	0.645062i	$-0.776831\pi$
$83$	− 11.7536i	− 1.29012i	0.764130	−	0.645062i	$-0.223169\pi$
$84$	0	0
$85$	− 14.9078i	− 1.61698i
$86$	0	0
$87$	−8.17891	−0.876871
$88$	0	0
$89$	− 6.92820i	− 0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$	− 6.92820i	− 0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 7.60885i	− 0.789001i
$94$	0	0
$95$	8.35782	0.857494
$96$	0	0
$97$	− 2.04193i	− 0.207326i	−0.994612	−	0.103663i	$-0.966944\pi$
$97$	− 2.04193i	− 0.207326i	0.994612	−	0.103663i	$-0.0330564\pi$
$98$	0	0
$99$	− 3.46410i	− 0.348155i
$100$	0	0
$101$	12.1789	1.21185	0.605923	−	0.795523i	$-0.292804\pi$
$101$	12.1789	1.21185	0.605923	+	0.795523i	$0.292804\pi$
$102$	0	0
$103$	3.17891	0.313227	0.156614	−	0.987660i	$-0.449942\pi$
$103$	3.17891	0.313227	0.156614	+	0.987660i	$0.449942\pi$
$104$	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
$109$	− 7.60885i	− 0.728795i	−0.931243	−	0.364398i	$-0.881275\pi$
$109$	− 7.60885i	− 0.728795i	0.931243	−	0.364398i	$-0.118725\pi$
$110$	0	0
$111$	− 1.05141i	− 0.0997952i
$112$	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
$115$	− 4.82539i	− 0.449970i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.6100i	2.34766i
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	− 5.87680i	− 0.529893i
$124$	0	0
$125$	10.0824i	0.901800i
$126$	0	0
$127$	−2.82109	−0.250331	−0.125166	−	0.992136i	$-0.539946\pi$
$127$	−2.82109	−0.250331	−0.125166	+	0.992136i	$0.539946\pi$
$128$	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
$135$	2.41269i	0.207652i
$136$	0	0
$137$	− 12.8050i	− 1.09400i	−0.837131	−	0.547002i	$-0.815769\pi$
$137$	− 12.8050i	− 1.09400i	0.837131	−	0.547002i	$-0.184231\pi$
$138$	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
$141$	10.3923i	0.875190i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	− 19.7332i	− 1.63875i
$146$	0	0
$147$	−10.1789	−0.839542
$148$	0	0
$149$	5.87680i	0.481446i	0.970594	+	0.240723i	$0.0773845\pi$
$149$	5.87680i	0.481446i	−0.970594	+	0.240723i	$0.922615\pi$
$150$	0	0
$151$	− 20.0431i	− 1.63108i	−0.578699	−	0.815541i	$-0.696439\pi$
$151$	− 20.0431i	− 1.63108i	0.578699	−	0.815541i	$-0.303561\pi$
$152$	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
$161$	8.28949i	0.653303i
$162$	0	0
$163$	− 7.60885i	− 0.595971i	−0.954570	−	0.297985i	$-0.903685\pi$
$163$	− 7.60885i	− 0.595971i	0.954570	−	0.297985i	$-0.0963147\pi$
$164$	0	0
$165$	8.35782	0.650655
$166$	0	0
$167$	− 6.92820i	− 0.536120i	−0.963402	−	0.268060i	$-0.913617\pi$
$167$	− 6.92820i	− 0.536120i	0.963402	−	0.268060i	$-0.0863826\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 3.46410i	− 0.264906i
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	3.40322i	0.257259i
$176$	0	0
$177$	1.36129i	0.102321i
$178$	0	0
$179$	−8.35782	−0.624693	−0.312346	−	0.949968i	$-0.601115\pi$
$179$	−8.35782	−0.624693	−0.312346	+	0.949968i	$0.601115\pi$
$180$	0	0
$181$	−20.5367	−1.52648	−0.763241	−	0.646113i	$-0.776393\pi$
$181$	−20.5367	−1.52648	−0.763241	+	0.646113i	$0.776393\pi$
$182$	0	0
$183$	5.00000	0.369611
$184$	0	0
$185$	2.53673	0.186504
$186$	0	0
$187$	21.4044i	1.56524i
$188$	0	0
$189$	− 4.14474i	− 0.301486i
$190$	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
$193$	13.4856i	0.970718i	0.874315	+	0.485359i	$0.161311\pi$
$193$	13.4856i	0.970718i	−0.874315	+	0.485359i	$0.838689\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.65078i	0.687589i	0.939045	+	0.343795i	$0.111712\pi$
$197$	9.65078i	0.687589i	−0.939045	+	0.343795i	$0.888288\pi$
$198$	0	0
$199$	−21.5367	−1.52670	−0.763349	−	0.645986i	$-0.776446\pi$
$199$	−21.5367	−1.52670	−0.763349	+	0.645986i	$0.776446\pi$
$200$	0	0
$201$	− 4.14474i	− 0.292348i
$202$	0	0
$203$	33.8995i	2.37928i
$204$	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
$213$	− 3.46410i	− 0.237356i
$214$	0	0
$215$	− 1.98104i	− 0.135106i
$216$	0	0
$217$	−31.5367	−2.14085
$218$	0	0
$219$	11.3828i	0.769180i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	15.2177i	1.01905i	0.860455	+	0.509526i	$0.170179\pi$
$223$	15.2177i	1.01905i	−0.860455	+	0.509526i	$0.829821\pi$
$224$	0	0
$225$	−0.821092	−0.0547394
$226$	0	0
$227$	9.03102i	0.599410i	0.954032	+	0.299705i	$0.0968882\pi$
$227$	9.03102i	0.599410i	−0.954032	+	0.299705i	$0.903112\pi$
$228$	0	0
$229$	− 9.65078i	− 0.637741i	−0.947798	−	0.318871i	$-0.896696\pi$
$229$	− 9.65078i	− 0.637741i	0.947798	−	0.318871i	$-0.103304\pi$
$230$	0	0
$231$	−14.3578	−0.944675
$232$	0	0
$233$	22.7156	1.48815	0.744075	−	0.668096i	$-0.232890\pi$
$233$	22.7156	1.48815	0.744075	+	0.668096i	$0.232890\pi$
$234$	0	0
$235$	−25.0735	−1.63561
$236$	0	0
$237$	13.1789	0.856062
$238$	0	0
$239$	2.10282i	0.136020i	0.997685	+	0.0680099i	$0.0216650\pi$
$239$	2.10282i	0.136020i	−0.997685	+	0.0680099i	$0.978335\pi$
$240$	0	0
$241$	5.87680i	0.378558i	0.981923	+	0.189279i	$0.0606150\pi$
$241$	5.87680i	0.378558i	−0.981923	+	0.189279i	$0.939385\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	− 24.5586i	− 1.56899i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 11.7536i	− 0.744854i
$250$	0	0
$251$	−20.7156	−1.30756	−0.653780	−	0.756685i	$-0.726818\pi$
$251$	−20.7156	−1.30756	−0.653780	+	0.756685i	$0.726818\pi$
$252$	0	0
$253$	6.92820i	0.435572i
$254$	0	0
$255$	− 14.9078i	− 0.933564i
$256$	0	0
$257$	22.1789	1.38348	0.691741	−	0.722146i	$-0.256844\pi$
$257$	22.1789	1.38348	0.691741	+	0.722146i	$0.256844\pi$
$258$	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
$265$	− 24.5586i	− 1.50862i
$266$	0	0
$267$	− 6.92820i	− 0.423999i
$268$	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
$271$	14.5370i	0.883063i	0.897246	+	0.441531i	$0.145565\pi$
$271$	14.5370i	0.883063i	−0.897246	+	0.441531i	$0.854435\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.84434i	0.171520i
$276$	0	0
$277$	−0.178908	−0.0107496	−0.00537478	−	0.999986i	$-0.501711\pi$
$277$	−0.178908	−0.0107496	−0.00537478	+	0.999986i	$0.501711\pi$
$278$	0	0
$279$	− 7.60885i	− 0.455530i
$280$	0	0
$281$	− 11.4437i	− 0.682675i	−0.939941	−	0.341337i	$-0.889120\pi$
$281$	− 11.4437i	− 0.682675i	0.939941	−	0.341337i	$-0.110880\pi$
$282$	0	0
$283$	−27.1789	−1.61562	−0.807809	−	0.589444i	$-0.799347\pi$
$283$	−27.1789	−1.61562	−0.807809	+	0.589444i	$0.799347\pi$
$284$	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
$291$	− 2.04193i	− 0.119700i
$292$	0	0
$293$	32.8481i	1.91901i	0.281700	+	0.959503i	$0.409102\pi$
$293$	32.8481i	1.91901i	−0.281700	+	0.959503i	$0.590898\pi$
$294$	0	0
$295$	−3.28437	−0.191223
$296$	0	0
$297$	− 3.46410i	− 0.201008i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3.40322i	0.196158i
$302$	0	0
$303$	12.1789	0.699660
$304$	0	0
$305$	12.0635i	0.690752i
$306$	0	0
$307$	− 11.0729i	− 0.631967i	−0.948765	−	0.315983i	$-0.897666\pi$
$307$	− 11.0729i	− 0.631967i	0.948765	−	0.315983i	$-0.102334\pi$
$308$	0	0
$309$	3.17891	0.180842
$310$	0	0
$311$	11.6422	0.660168	0.330084	−	0.943952i	$-0.392923\pi$
$311$	11.6422	0.660168	0.330084	+	0.943952i	$0.392923\pi$
$312$	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
$317$	− 3.77398i	− 0.211968i	−0.994368	−	0.105984i	$-0.966201\pi$
$317$	− 3.77398i	− 0.211968i	0.994368	−	0.105984i	$-0.0337992\pi$
$318$	0	0
$319$	28.3326i	1.58632i
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	21.4044i	1.19097i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 7.60885i	− 0.420770i
$328$	0	0
$329$	43.0735	2.37472
$330$	0	0
$331$	− 20.1040i	− 1.10501i	−0.833508	−	0.552507i	$-0.813671\pi$
$331$	− 20.1040i	− 1.10501i	0.833508	−	0.552507i	$-0.186329\pi$
$332$	0	0
$333$	− 1.05141i	− 0.0576168i
$334$	0	0
$335$	10.0000	0.546358
$336$	0	0
$337$	−25.3578	−1.38133	−0.690664	−	0.723176i	$-0.742682\pi$
$337$	−25.3578	−1.38133	−0.690664	+	0.723176i	$0.742682\pi$
$338$	0	0
$339$	12.1789	0.661468
$340$	0	0
$341$	−26.3578	−1.42736
$342$	0	0
$343$	13.1758i	0.711424i
$344$	0	0
$345$	− 4.82539i	− 0.259790i
$346$	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
$349$	− 10.3314i	− 0.553028i	−0.961010	−	0.276514i	$-0.910821\pi$
$349$	− 10.3314i	− 0.553028i	0.961010	−	0.276514i	$-0.0891792\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 18.9917i	− 1.01082i	−0.862878	−	0.505412i	$-0.831340\pi$
$353$	− 18.9917i	− 1.01082i	0.862878	−	0.505412i	$-0.168660\pi$
$354$	0	0
$355$	8.35782	0.443587
$356$	0	0
$357$	25.6100i	1.35542i
$358$	0	0
$359$	18.6818i	0.985987i	0.870033	+	0.492994i	$0.164097\pi$
$359$	18.6818i	0.985987i	−0.870033	+	0.492994i	$0.835903\pi$
$360$	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
$369$	− 5.87680i	− 0.305934i
$370$	0	0
$371$	42.1890i	2.19034i
$372$	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
$375$	10.0824i	0.520654i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	6.24756i	0.320916i	0.987043	+	0.160458i	$0.0512971\pi$
$379$	6.24756i	0.320916i	−0.987043	+	0.160458i	$0.948703\pi$
$380$	0	0
$381$	−2.82109	−0.144529
$382$	0	0
$383$	− 19.3016i	− 0.986263i	−0.869955	−	0.493132i	$-0.835852\pi$
$383$	− 19.3016i	− 0.986263i	0.869955	−	0.493132i	$-0.164148\pi$
$384$	0	0
$385$	− 34.6410i	− 1.76547i
$386$	0	0
$387$	−0.821092	−0.0417384
$388$	0	0
$389$	−6.53673	−0.331425	−0.165713	−	0.986174i	$-0.552992\pi$
$389$	−6.53673	−0.331425	−0.165713	+	0.986174i	$0.552992\pi$
$390$	0	0
$391$	12.3578	0.624962
$392$	0	0
$393$	6.35782	0.320709
$394$	0	0
$395$	31.7967i	1.59986i
$396$	0	0
$397$	21.4653i	1.07731i	0.842526	+	0.538655i	$0.181067\pi$
$397$	21.4653i	1.07731i	−0.842526	+	0.538655i	$0.818933\pi$
$398$	0	0
$399$	−14.3578	−0.718790
$400$	0	0
$401$	− 10.7022i	− 0.534442i	−0.963635	−	0.267221i	$-0.913895\pi$
$401$	− 10.7022i	− 0.534442i	0.963635	−	0.267221i	$-0.0861053\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.41269i	0.119888i
$406$	0	0
$407$	−3.64218	−0.180536
$408$	0	0
$409$	− 0.990521i	− 0.0489781i	−0.999700	−	0.0244891i	$-0.992204\pi$
$409$	− 0.990521i	− 0.0489781i	0.999700	−	0.0244891i	$-0.00779589\pi$
$410$	0	0
$411$	− 12.8050i	− 0.631624i
$412$	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
$421$	32.1674i	1.56774i	0.620922	+	0.783872i	$0.286758\pi$
$421$	32.1674i	1.56774i	−0.620922	+	0.783872i	$0.713242\pi$
$422$	0	0
$423$	10.3923i	0.505291i
$424$	0	0
$425$	5.07345	0.246098
$426$	0	0
$427$	− 20.7237i	− 1.00289i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.6818i	0.899870i	0.893061	+	0.449935i	$0.148553\pi$
$431$	18.6818i	0.899870i	−0.893061	+	0.449935i	$0.851447\pi$
$432$	0	0
$433$	−2.64218	−0.126975	−0.0634876	−	0.997983i	$-0.520222\pi$
$433$	−2.64218	−0.126975	−0.0634876	+	0.997983i	$0.520222\pi$
$434$	0	0
$435$	− 19.7332i	− 0.946135i
$436$	0	0
$437$	6.92820i	0.331421i
$438$	0	0
$439$	23.8945	1.14042	0.570212	−	0.821497i	$-0.306861\pi$
$439$	23.8945	1.14042	0.570212	+	0.821497i	$0.306861\pi$
$440$	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
$447$	5.87680i	0.277963i
$448$	0	0
$449$	− 11.0121i	− 0.519691i	−0.965650	−	0.259846i	$-0.916328\pi$
$449$	− 11.0121i	− 0.519691i	0.965650	−	0.259846i	$-0.0836717\pi$
$450$	0	0
$451$	−20.3578	−0.958612
$452$	0	0
$453$	− 20.0431i	− 0.941706i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.29897i	− 0.341431i	−0.985320	−	0.170716i	$-0.945392\pi$
$457$	− 7.29897i	− 0.341431i	0.985320	−	0.170716i	$-0.0546080\pi$
$458$	0	0
$459$	−6.17891	−0.288407
$460$	0	0
$461$	− 24.5586i	− 1.14381i	−0.820321	−	0.571904i	$-0.806205\pi$
$461$	− 24.5586i	− 1.14381i	0.820321	−	0.571904i	$-0.193795\pi$
$462$	0	0
$463$	− 1.42217i	− 0.0660940i	−0.999454	−	0.0330470i	$-0.989479\pi$
$463$	− 1.42217i	− 0.0660940i	0.999454	−	0.0330470i	$-0.0105211\pi$
$464$	0	0
$465$	18.3578	0.851323
$466$	0	0
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	−17.1789	−0.793248
$470$	0	0
$471$	7.00000	0.322543
$472$	0	0
$473$	2.84434i	0.130783i
$474$	0	0
$475$	2.84434i	0.130507i
$476$	0	0
$477$	−10.1789	−0.466060
$478$	0	0
$479$	6.92820i	0.316558i	0.987394	+	0.158279i	$0.0505945\pi$
$479$	6.92820i	0.316558i	−0.987394	+	0.158279i	$0.949406\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	8.28949i	0.377185i
$484$	0	0
$485$	4.92655	0.223703
$486$	0	0
$487$	24.2487i	1.09881i	0.835555	+	0.549407i	$0.185146\pi$
$487$	24.2487i	1.09881i	−0.835555	+	0.549407i	$0.814854\pi$
$488$	0	0
$489$	− 7.60885i	− 0.344084i
$490$	0	0
$491$	18.7156	0.844625	0.422312	−	0.906450i	$-0.361219\pi$
$491$	18.7156	0.844625	0.422312	+	0.906450i	$0.361219\pi$
$492$	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
$499$	− 1.36129i	− 0.0609395i	−0.999536	−	0.0304698i	$-0.990300\pi$
$499$	− 1.36129i	− 0.0609395i	0.999536	−	0.0304698i	$-0.00970033\pi$
$500$	0	0
$501$	− 6.92820i	− 0.309529i
$502$	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
$505$	29.3840i	1.30757i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1663i	0.627910i	0.949438	+	0.313955i	$0.101654\pi$
$509$	14.1663i	0.627910i	−0.949438	+	0.313955i	$0.898346\pi$
$510$	0	0
$511$	47.1789	2.08707
$512$	0	0
$513$	− 3.46410i	− 0.152944i
$514$	0	0
$515$	7.66973i	0.337969i
$516$	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
$525$	3.40322i	0.148528i
$526$	0	0
$527$	47.0144i	2.04798i
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	1.36129i	0.0590748i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 24.1269i	− 1.04310i
$536$	0	0
$537$	−8.35782	−0.360666
$538$	0	0
$539$	35.2608i	1.51879i
$540$	0	0
$541$	3.09334i	0.132993i	0.997787	+	0.0664965i	$0.0211821\pi$
$541$	3.09334i	0.132993i	−0.997787	+	0.0664965i	$0.978818\pi$
$542$	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
$551$	28.3326i	1.20701i
$552$	0	0
$553$	− 54.6232i	− 2.32282i
$554$	0	0
$555$	2.53673	0.107678
$556$	0	0
$557$	32.1065i	1.36040i	0.733027	+	0.680199i	$0.238107\pi$
$557$	32.1065i	1.36040i	−0.733027	+	0.680199i	$0.761893\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	21.4044i	0.903693i
$562$	0	0
$563$	−18.3578	−0.773690	−0.386845	−	0.922145i	$-0.626435\pi$
$563$	−18.3578	−0.773690	−0.386845	+	0.922145i	$0.626435\pi$
$564$	0	0
$565$	29.3840i	1.23619i
$566$	0	0
$567$	− 4.14474i	− 0.174063i
$568$	0	0
$569$	−30.7156	−1.28767	−0.643833	−	0.765166i	$-0.722657\pi$
$569$	−30.7156	−1.28767	−0.643833	+	0.765166i	$0.722657\pi$
$570$	0	0
$571$	−7.64218	−0.319815	−0.159908	−	0.987132i	$-0.551120\pi$
$571$	−7.64218	−0.319815	−0.159908	+	0.987132i	$0.551120\pi$
$572$	0	0
$573$	−2.35782	−0.0984992
$574$	0	0
$575$	1.64218	0.0684838
$576$	0	0
$577$	− 43.2404i	− 1.80012i	−0.435765	−	0.900060i	$-0.643522\pi$
$577$	− 43.2404i	− 1.80012i	0.435765	−	0.900060i	$-0.356478\pi$
$578$	0	0
$579$	13.4856i	0.560444i
$580$	0	0
$581$	−48.7156	−2.02107
$582$	0	0
$583$	35.2608i	1.46035i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.1459i	0.914059i	0.889452	+	0.457029i	$0.151087\pi$
$587$	22.1459i	0.914059i	−0.889452	+	0.457029i	$0.848913\pi$
$588$	0	0
$589$	−26.3578	−1.08605
$590$	0	0
$591$	9.65078i	0.396980i
$592$	0	0
$593$	− 32.8481i	− 1.34891i	−0.738316	−	0.674454i	$-0.764379\pi$
$593$	− 32.8481i	− 1.34891i	0.738316	−	0.674454i	$-0.235621\pi$
$594$	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
$603$	− 4.14474i	− 0.168787i
$604$	0	0
$605$	− 2.41269i	− 0.0980900i
$606$	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
$609$	33.8995i	1.37368i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	2.47358i	0.0999070i	0.998752	+	0.0499535i	$0.0159073\pi$
$613$	2.47358i	0.0999070i	−0.998752	+	0.0499535i	$0.984093\pi$
$614$	0	0
$615$	14.1789	0.571749
$616$	0	0
$617$	2.41269i	0.0971314i	0.998820	+	0.0485657i	$0.0154650\pi$
$617$	2.41269i	0.0971314i	−0.998820	+	0.0485657i	$0.984535\pi$
$618$	0	0
$619$	10.3314i	0.415255i	0.978208	+	0.207627i	$0.0665742\pi$
$619$	10.3314i	0.415255i	−0.978208	+	0.207627i	$0.933426\pi$
$620$	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
$629$	6.49655i	0.259034i
$630$	0	0
$631$	6.24756i	0.248711i	0.992238	+	0.124356i	$0.0396864\pi$
$631$	6.24756i	0.248711i	−0.992238	+	0.124356i	$0.960314\pi$
$632$	0	0
$633$	−15.5367	−0.617529
$634$	0	0
$635$	− 6.80643i	− 0.270105i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 3.46410i	− 0.137038i
$640$	0	0
$641$	−0.536725	−0.0211994	−0.0105997	−	0.999944i	$-0.503374\pi$
$641$	−0.536725	−0.0211994	−0.0105997	+	0.999944i	$0.503374\pi$
$642$	0	0
$643$	− 15.8983i	− 0.626969i	−0.949593	−	0.313485i	$-0.898504\pi$
$643$	− 15.8983i	− 0.626969i	0.949593	−	0.313485i	$-0.101496\pi$
$644$	0	0
$645$	− 1.98104i	− 0.0780035i
$646$	0	0
$647$	13.6422	0.536330	0.268165	−	0.963373i	$-0.413583\pi$
$647$	13.6422	0.536330	0.268165	+	0.963373i	$0.413583\pi$
$648$	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
$655$	15.3395i	0.599363i
$656$	0	0
$657$	11.3828i	0.444086i
$658$	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
$661$	− 12.8659i	− 0.500425i	−0.968191	−	0.250212i	$-0.919500\pi$
$661$	− 12.8659i	− 0.500425i	0.968191	−	0.250212i	$-0.0805005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 34.6410i	− 1.34332i
$666$	0	0
$667$	16.3578	0.633377
$668$	0	0
$669$	15.2177i	0.588350i
$670$	0	0
$671$	− 17.3205i	− 0.668651i
$672$	0	0
$673$	−5.35782	−0.206529	−0.103264	−	0.994654i	$-0.532929\pi$
$673$	−5.35782	−0.206529	−0.103264	+	0.994654i	$0.532929\pi$
$674$	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
$681$	9.03102i	0.346069i
$682$	0	0
$683$	− 17.9403i	− 0.686465i	−0.939250	−	0.343233i	$-0.888478\pi$
$683$	− 17.9403i	− 0.686465i	0.939250	−	0.343233i	$-0.111522\pi$
$684$	0	0
$685$	30.8945	1.18042
$686$	0	0
$687$	− 9.65078i	− 0.368200i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.22860i	0.313031i	0.987676	+	0.156515i	$0.0500261\pi$
$691$	8.22860i	0.313031i	−0.987676	+	0.156515i	$0.949974\pi$
$692$	0	0
$693$	−14.3578	−0.545408
$694$	0	0
$695$	41.4474i	1.57219i
$696$	0	0
$697$	36.3122i	1.37542i
$698$	0	0
$699$	22.7156	0.859184
$700$	0	0
$701$	48.3578	1.82645	0.913225	−	0.407456i	$-0.133584\pi$
$701$	48.3578	1.82645	0.913225	+	0.407456i	$0.133584\pi$
$702$	0	0
$703$	−3.64218	−0.137368
$704$	0	0
$705$	−25.0735	−0.944321
$706$	0	0
$707$	− 50.4785i	− 1.89844i
$708$	0	0
$709$	29.4449i	1.10583i	0.833239	+	0.552913i	$0.186484\pi$
$709$	29.4449i	1.10583i	−0.833239	+	0.552913i	$0.813516\pi$
$710$	0	0
$711$	13.1789	0.494248
$712$	0	0
$713$	15.2177i	0.569907i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.10282i	0.0785311i
$718$	0	0
$719$	−30.3578	−1.13216	−0.566078	−	0.824352i	$-0.691540\pi$
$719$	−30.3578	−1.13216	−0.566078	+	0.824352i	$0.691540\pi$
$720$	0	0
$721$	− 13.1758i	− 0.490691i
$722$	0	0
$723$	5.87680i	0.218560i
$724$	0	0
$725$	6.71563	0.249412
$726$	0	0
$727$	17.5367	0.650401	0.325201	−	0.945645i	$-0.394568\pi$
$727$	17.5367	0.650401	0.325201	+	0.945645i	$0.394568\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.07345	0.187648
$732$	0	0
$733$	10.6413i	0.393045i	0.980499	+	0.196523i	$0.0629649\pi$
$733$	10.6413i	0.393045i	−0.980499	+	0.196523i	$0.937035\pi$
$734$	0	0
$735$	− 24.5586i	− 0.905857i
$736$	0	0
$737$	−14.3578	−0.528877
$738$	0	0
$739$	15.2177i	0.559792i	0.960030	+	0.279896i	$0.0903000\pi$
$739$	15.2177i	0.559792i	−0.960030	+	0.279896i	$0.909700\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.36129i	0.0499407i	0.999688	+	0.0249704i	$0.00794914\pi$
$743$	1.36129i	0.0499407i	−0.999688	+	0.0249704i	$0.992051\pi$
$744$	0	0
$745$	−14.1789	−0.519475
$746$	0	0
$747$	− 11.7536i	− 0.430041i
$748$	0	0
$749$	41.4474i	1.51446i
$750$	0	0
$751$	−19.6422	−0.716753	−0.358377	−	0.933577i	$-0.616670\pi$
$751$	−19.6422	−0.716753	−0.358377	+	0.933577i	$0.616670\pi$
$752$	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
$759$	6.92820i	0.251478i
$760$	0	0
$761$	− 48.4974i	− 1.75803i	−0.476794	−	0.879015i	$-0.658201\pi$
$761$	− 48.4974i	− 1.75803i	0.476794	−	0.879015i	$-0.341799\pi$
$762$	0	0
$763$	−31.5367	−1.14171
$764$	0	0
$765$	− 14.9078i	− 0.538993i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.92820i	0.249837i	0.992167	+	0.124919i	$0.0398670\pi$
$769$	6.92820i	0.249837i	−0.992167	+	0.124919i	$0.960133\pi$
$770$	0	0
$771$	22.1789	0.798754
$772$	0	0
$773$	− 31.7967i	− 1.14365i	−0.820377	−	0.571823i	$-0.806236\pi$
$773$	− 31.7967i	− 1.14365i	0.820377	−	0.571823i	$-0.193764\pi$
$774$	0	0
$775$	6.24756i	0.224419i
$776$	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
$785$	16.8889i	0.602789i
$786$	0	0
$787$	− 33.9604i	− 1.21056i	−0.796014	−	0.605278i	$-0.793062\pi$
$787$	− 33.9604i	− 1.21056i	0.796014	−	0.605278i	$-0.206938\pi$
$788$	0	0
$789$	32.3578	1.15197
$790$	0	0
$791$	− 50.4785i	− 1.79481i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 24.5586i	− 0.871003i
$796$	0	0
$797$	17.0735	0.604773	0.302386	−	0.953185i	$-0.402217\pi$
$797$	17.0735	0.604773	0.302386	+	0.953185i	$0.402217\pi$
$798$	0	0
$799$	− 64.2131i	− 2.27170i
$800$	0	0
$801$	− 6.92820i	− 0.244796i
$802$	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
$811$	− 0.680643i	− 0.0239006i	−0.999929	−	0.0119503i	$-0.996196\pi$
$811$	− 0.680643i	− 0.0239006i	0.999929	−	0.0119503i	$-0.00380399\pi$
$812$	0	0
$813$	14.5370i	0.509837i
$814$	0	0
$815$	18.3578	0.643046
$816$	0	0
$817$	2.84434i	0.0995110i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 31.7967i	− 1.10971i	−0.831947	−	0.554856i	$-0.812773\pi$
$821$	− 31.7967i	− 1.10971i	0.831947	−	0.554856i	$-0.187227\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	0	0
$825$	2.84434i	0.0990274i
$826$	0	0
$827$	− 35.8805i	− 1.24769i	−0.781549	−	0.623844i	$-0.785570\pi$
$827$	− 35.8805i	− 1.24769i	0.781549	−	0.623844i	$-0.214430\pi$
$828$	0	0
$829$	23.3578	0.811251	0.405625	−	0.914039i	$-0.367054\pi$
$829$	23.3578	0.811251	0.405625	+	0.914039i	$0.367054\pi$
$830$	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
$837$	− 7.60885i	− 0.263000i
$838$	0	0
$839$	− 15.2177i	− 0.525373i	−0.964881	−	0.262687i	$-0.915391\pi$
$839$	− 15.2177i	− 0.525373i	0.964881	−	0.262687i	$-0.0846085\pi$
$840$	0	0
$841$	37.8945	1.30671
$842$	0	0
$843$	− 11.4437i	− 0.394142i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.14474i	0.142415i
$848$	0	0
$849$	−27.1789	−0.932778
$850$	0	0
$851$	2.10282i	0.0720836i
$852$	0	0
$853$	24.6195i	0.842955i	0.906839	+	0.421477i	$0.138488\pi$
$853$	24.6195i	0.842955i	−0.906839	+	0.421477i	$0.861512\pi$
$854$	0	0
$855$	8.35782	0.285831
$856$	0	0
$857$	−20.1789	−0.689298	−0.344649	−	0.938732i	$-0.612002\pi$
$857$	−20.1789	−0.689298	−0.344649	+	0.938732i	$0.612002\pi$
$858$	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
$863$	− 20.0431i	− 0.682274i	−0.940014	−	0.341137i	$-0.889188\pi$
$863$	− 20.0431i	− 0.682274i	0.940014	−	0.341137i	$-0.110812\pi$
$864$	0	0
$865$	43.4285i	1.47661i
$866$	0	0
$867$	21.1789	0.719273
$868$	0	0
$869$	− 45.6531i	− 1.54867i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 2.04193i	− 0.0691088i
$874$	0	0
$875$	41.7891	1.41273
$876$	0	0
$877$	12.8050i	0.432394i	0.976350	+	0.216197i	$0.0693654\pi$
$877$	12.8050i	0.432394i	−0.976350	+	0.216197i	$0.930635\pi$
$878$	0	0
$879$	32.8481i	1.10794i
$880$	0	0
$881$	−37.8211	−1.27422	−0.637112	−	0.770771i	$-0.719871\pi$
$881$	−37.8211	−1.27422	−0.637112	+	0.770771i	$0.719871\pi$
$882$	0	0
$883$	−17.8945	−0.602199	−0.301100	−	0.953593i	$-0.597354\pi$
$883$	−17.8945	−0.602199	−0.301100	+	0.953593i	$0.597354\pi$
$884$	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
$889$	11.6927i	0.392161i
$890$	0	0
$891$	− 3.46410i	− 0.116052i
$892$	0	0
$893$	36.0000	1.20469
$894$	0	0
$895$	− 20.1649i	− 0.674037i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	62.2321i	2.07556i
$900$	0	0
$901$	62.8945	2.09532
$902$	0	0
$903$	3.40322i	0.113252i
$904$	0	0
$905$	− 49.5488i	− 1.64706i
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	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
$915$	12.0635i	0.398806i
$916$	0	0
$917$	− 26.3515i	− 0.870204i
$918$	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
$921$	− 11.0729i	− 0.364866i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0.863302i	0.0283852i
$926$	0	0
$927$	3.17891	0.104409
$928$	0	0
$929$	− 31.4868i	− 1.03305i	−0.856273	−	0.516524i	$-0.827226\pi$
$929$	− 31.4868i	− 1.03305i	0.856273	−	0.516524i	$-0.172774\pi$
$930$	0	0
$931$	35.2608i	1.15563i
$932$	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
$941$	− 37.2418i	− 1.21405i	−0.794683	−	0.607024i	$-0.792363\pi$
$941$	− 37.2418i	− 1.21405i	0.794683	−	0.607024i	$-0.207637\pi$
$942$	0	0
$943$	11.7536i	0.382750i
$944$	0	0
$945$	10.0000	0.325300
$946$	0	0
$947$	30.4354i	0.989017i	0.869173	+	0.494509i	$0.164652\pi$
$947$	30.4354i	0.989017i	−0.869173	+	0.494509i	$0.835348\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 3.77398i	− 0.122380i
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	− 5.68869i	− 0.184082i
$956$	0	0
$957$	28.3326i	0.915862i
$958$	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
$967$	− 35.3825i	− 1.13783i	−0.822398	−	0.568913i	$-0.807364\pi$
$967$	− 35.3825i	− 1.13783i	0.822398	−	0.568913i	$-0.192636\pi$
$968$	0	0
$969$	21.4044i	0.687607i
$970$	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
$973$	− 71.2022i	− 2.28264i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 17.0106i	− 0.544218i	−0.962266	−	0.272109i	$-0.912279\pi$
$977$	− 17.0106i	− 0.544218i	0.962266	−	0.272109i	$-0.0877212\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	− 7.60885i	− 0.242932i
$982$	0	0
$983$	− 24.8685i	− 0.793181i	−0.917996	−	0.396590i	$-0.870193\pi$
$983$	− 24.8685i	− 0.793181i	0.917996	−	0.396590i	$-0.129807\pi$
$984$	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
$993$	− 20.1040i	− 0.637980i
$994$	0	0
$995$	− 51.9615i	− 1.64729i
$996$	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
$999$	− 1.05141i	− 0.0332651i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.b.e.337.3		4
3.2	odd	2		6084.2.b.o.4393.2		4
13.2	odd	12		2028.2.i.n.529.3		8
13.3	even	3		156.2.q.b.121.2	yes	4
13.4	even	6		156.2.q.b.49.1	✓	4
13.5	odd	4		2028.2.a.m.1.3		4
13.6	odd	12		2028.2.i.n.2005.3		8
13.7	odd	12		2028.2.i.n.2005.2		8
13.8	odd	4		2028.2.a.m.1.2		4
13.9	even	3		2028.2.q.f.361.2		4
13.10	even	6		2028.2.q.f.1837.1		4
13.11	odd	12		2028.2.i.n.529.2		8
13.12	even	2	inner	2028.2.b.e.337.2		4
39.5	even	4		6084.2.a.bd.1.2		4
39.8	even	4		6084.2.a.bd.1.3		4
39.17	odd	6		468.2.t.d.361.2		4
39.29	odd	6		468.2.t.d.433.1		4
39.38	odd	2		6084.2.b.o.4393.3		4
52.3	odd	6		624.2.bv.f.433.2		4
52.31	even	4		8112.2.a.cr.1.3		4
52.43	odd	6		624.2.bv.f.49.1		4
52.47	even	4		8112.2.a.cr.1.2		4
65.3	odd	12		3900.2.bw.j.2149.2		8
65.4	even	6		3900.2.cd.i.2701.2		4
65.17	odd	12		3900.2.bw.j.49.2		8
65.29	even	6		3900.2.cd.i.901.2		4
65.42	odd	12		3900.2.bw.j.2149.3		8
65.43	odd	12		3900.2.bw.j.49.3		8
156.95	even	6		1872.2.by.j.1297.2		4
156.107	even	6		1872.2.by.j.433.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.q.b.49.1	✓	4	13.4	even	6
156.2.q.b.121.2	yes	4	13.3	even	3
468.2.t.d.361.2		4	39.17	odd	6
468.2.t.d.433.1		4	39.29	odd	6
624.2.bv.f.49.1		4	52.43	odd	6
624.2.bv.f.433.2		4	52.3	odd	6
1872.2.by.j.433.1		4	156.107	even	6
1872.2.by.j.1297.2		4	156.95	even	6
2028.2.a.m.1.2		4	13.8	odd	4
2028.2.a.m.1.3		4	13.5	odd	4
2028.2.b.e.337.2		4	13.12	even	2	inner
2028.2.b.e.337.3		4	1.1	even	1	trivial
2028.2.i.n.529.2		8	13.11	odd	12
2028.2.i.n.529.3		8	13.2	odd	12
2028.2.i.n.2005.2		8	13.7	odd	12
2028.2.i.n.2005.3		8	13.6	odd	12
2028.2.q.f.361.2		4	13.9	even	3
2028.2.q.f.1837.1		4	13.10	even	6
3900.2.bw.j.49.2		8	65.17	odd	12
3900.2.bw.j.49.3		8	65.43	odd	12
3900.2.bw.j.2149.2		8	65.3	odd	12
3900.2.bw.j.2149.3		8	65.42	odd	12
3900.2.cd.i.901.2		4	65.29	even	6
3900.2.cd.i.2701.2		4	65.4	even	6
6084.2.a.bd.1.2		4	39.5	even	4
6084.2.a.bd.1.3		4	39.8	even	4
6084.2.b.o.4393.2		4	3.2	odd	2
6084.2.b.o.4393.3		4	39.38	odd	2
8112.2.a.cr.1.2		4	52.47	even	4
8112.2.a.cr.1.3		4	52.31	even	4

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.b (of order \(2\), degree \(1\), not minimal)

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

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