LMFDB - Embedded newform 2100.2.x.a.1357.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1357,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 1, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2100.1357");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.x (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1357.3
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2100.1357
Dual form			2100.2.x.a.1693.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{3} +(-2.63896 + 0.189469i) q^{7} +1.00000i q^{9} +O(q^{10})$	$q+(0.707107 + 0.707107i) q^{3} +(-2.63896 + 0.189469i) q^{7} +1.00000i q^{9} -3.00000 q^{11} +(-1.41421 - 1.41421i) q^{13} +(4.24264 - 4.24264i) q^{17} +3.46410 q^{19} +(-2.00000 - 1.73205i) q^{21} +(3.67423 - 3.67423i) q^{23} +(-0.707107 + 0.707107i) q^{27} +9.00000i q^{29} -6.92820i q^{31} +(-2.12132 - 2.12132i) q^{33} +(3.67423 + 3.67423i) q^{37} -2.00000i q^{39} +(8.57321 - 8.57321i) q^{43} +(4.24264 - 4.24264i) q^{47} +(6.92820 - 1.00000i) q^{49} +6.00000 q^{51} +(2.44949 + 2.44949i) q^{57} +10.3923 q^{59} +6.92820i q^{61} +(-0.189469 - 2.63896i) q^{63} +(8.57321 + 8.57321i) q^{67} +5.19615 q^{69} +3.00000 q^{71} +(1.41421 + 1.41421i) q^{73} +(7.91688 - 0.568406i) q^{77} +1.00000i q^{79} -1.00000 q^{81} +(-8.48528 - 8.48528i) q^{83} +(-6.36396 + 6.36396i) q^{87} -10.3923 q^{89} +(4.00000 + 3.46410i) q^{91} +(4.89898 - 4.89898i) q^{93} +(2.82843 - 2.82843i) q^{97} -3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{11} - 16 q^{21} + 48 q^{51} + 24 q^{71} - 8 q^{81} + 32 q^{91}+O(q^{100}) $ 8 * q - 24 * q^11 - 16 * q^21 + 48 * q^51 + 24 * q^71 - 8 * q^81 + 32 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$-1$

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
$3$	0.707107	+	0.707107i	0.408248	+	0.408248i
$3$	0.707107	+	0.707107i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.63896	+	0.189469i	−0.997433	+	0.0716124i
$7$	−2.63896	+	0.189469i	−0.997433	+	0.0716124i
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$	−1.41421	−	1.41421i	−0.392232	−	0.392232i	0.483250	−	0.875482i	$-0.339456\pi$
$13$	−1.41421	−	1.41421i	−0.392232	−	0.392232i	−0.875482	+	0.483250i	$0.839456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.0294245	−	0.999567i	$-0.490633\pi$
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.999567	−	0.0294245i	$-0.00936746\pi$
$18$		0			0
$19$	3.46410			0.794719			0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410			0.794719			0.397360	+	0.917663i	$0.369927\pi$
$20$		0			0
$21$	−2.00000	−	1.73205i	−0.436436	−	0.377964i
$22$		0			0
$23$	3.67423	−	3.67423i	0.766131	−	0.766131i	−0.211292	−	0.977423i	$-0.567767\pi$
$23$	3.67423	−	3.67423i	0.766131	−	0.766131i	0.977423	+	0.211292i	$0.0677671\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.707107	+	0.707107i	−0.136083	+	0.136083i
$28$		0			0
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$		0			0
$31$		−	6.92820i		−	1.24434i	−0.782881	−	0.622171i	$-0.786251\pi$
$31$		−	6.92820i		−	1.24434i	0.782881	−	0.622171i	$-0.213749\pi$
$32$		0			0
$33$	−2.12132	−	2.12132i	−0.369274	−	0.369274i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	0.941382	−	0.337342i	$-0.109528\pi$
$37$	3.67423	+	3.67423i	0.604040	+	0.604040i	−0.337342	+	0.941382i	$0.609528\pi$
$38$		0			0
$39$		−	2.00000i		−	0.320256i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	8.57321	−	8.57321i	1.30740	−	1.30740i	0.384120	−	0.923283i	$-0.374505\pi$
$43$	8.57321	−	8.57321i	1.30740	−	1.30740i	0.923283	−	0.384120i	$-0.125495\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.24264	−	4.24264i	0.618853	−	0.618853i	−0.326384	−	0.945237i	$-0.605830\pi$
$47$	4.24264	−	4.24264i	0.618853	−	0.618853i	0.945237	+	0.326384i	$0.105830\pi$
$48$		0			0
$49$	6.92820	−	1.00000i	0.989743	−	0.142857i
$50$		0			0
$51$	6.00000			0.840168
$52$		0			0
$53$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$53$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.44949	+	2.44949i	0.324443	+	0.324443i
$58$		0			0
$59$	10.3923			1.35296			0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923			1.35296			0.676481	+	0.736460i	$0.263504\pi$
$60$		0			0
$61$			6.92820i			0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$			6.92820i			0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$		0			0
$63$	−0.189469	−	2.63896i	−0.0238708	−	0.332478i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	8.57321	+	8.57321i	1.04738	+	1.04738i	0.998820	+	0.0485648i	$0.0154647\pi$
$67$	8.57321	+	8.57321i	1.04738	+	1.04738i	0.0485648	+	0.998820i	$0.484535\pi$
$68$		0			0
$69$	5.19615			0.625543
$70$		0			0
$71$	3.00000			0.356034			0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000			0.356034			0.178017	+	0.984027i	$0.443032\pi$
$72$		0			0
$73$	1.41421	+	1.41421i	0.165521	+	0.165521i	0.785007	−	0.619486i	$-0.212659\pi$
$73$	1.41421	+	1.41421i	0.165521	+	0.165521i	−0.619486	+	0.785007i	$0.712659\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	7.91688	−	0.568406i	0.902212	−	0.0647759i
$78$		0			0
$79$			1.00000i			0.112509i	0.998416	+	0.0562544i	$0.0179158\pi$
$79$			1.00000i			0.112509i	−0.998416	+	0.0562544i	$0.982084\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−8.48528	−	8.48528i	−0.931381	−	0.931381i	0.0664117	−	0.997792i	$-0.478845\pi$
$83$	−8.48528	−	8.48528i	−0.931381	−	0.931381i	−0.997792	+	0.0664117i	$0.978845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−6.36396	+	6.36396i	−0.682288	+	0.682288i
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$	4.00000	+	3.46410i	0.419314	+	0.363137i
$92$		0			0
$93$	4.89898	−	4.89898i	0.508001	−	0.508001i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.82843	−	2.82843i	0.287183	−	0.287183i	−0.548782	−	0.835965i	$-0.684908\pi$
$97$	2.82843	−	2.82843i	0.287183	−	0.287183i	0.835965	+	0.548782i	$0.184908\pi$
$98$		0			0
$99$		−	3.00000i		−	0.301511i
$100$		0			0
$101$		−	10.3923i		−	1.03407i	−0.855963	−	0.517036i	$-0.827035\pi$
$101$		−	10.3923i		−	1.03407i	0.855963	−	0.517036i	$-0.172965\pi$
$102$		0			0
$103$	−9.89949	−	9.89949i	−0.975426	−	0.975426i	0.0242790	−	0.999705i	$-0.492271\pi$
$103$	−9.89949	−	9.89949i	−0.975426	−	0.975426i	−0.999705	+	0.0242790i	$0.992271\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.34847	−	7.34847i	−0.710403	−	0.710403i	0.256216	−	0.966620i	$-0.417524\pi$
$107$	−7.34847	−	7.34847i	−0.710403	−	0.710403i	−0.966620	+	0.256216i	$0.917524\pi$
$108$		0			0
$109$			17.0000i			1.62830i	0.580651	+	0.814152i	$0.302798\pi$
$109$			17.0000i			1.62830i	−0.580651	+	0.814152i	$0.697202\pi$
$110$		0			0
$111$			5.19615i			0.493197i
$112$		0			0
$113$	3.67423	−	3.67423i	0.345643	−	0.345643i	−0.512841	−	0.858484i	$-0.671407\pi$
$113$	3.67423	−	3.67423i	0.345643	−	0.345643i	0.858484	+	0.512841i	$0.171407\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.41421	−	1.41421i	0.130744	−	0.130744i
$118$		0			0
$119$	−10.3923	+	12.0000i	−0.952661	+	1.10004i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.67423	−	3.67423i	−0.326036	−	0.326036i	0.525041	−	0.851077i	$-0.324050\pi$
$127$	−3.67423	−	3.67423i	−0.326036	−	0.326036i	−0.851077	+	0.525041i	$0.824050\pi$
$128$		0			0
$129$	12.1244			1.06749
$130$		0			0
$131$		−	10.3923i		−	0.907980i	−0.891007	−	0.453990i	$-0.850000\pi$
$131$		−	10.3923i		−	0.907980i	0.891007	−	0.453990i	$-0.150000\pi$
$132$		0			0
$133$	−9.14162	+	0.656339i	−0.792679	+	0.0569118i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$137$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$138$		0			0
$139$	10.3923			0.881464			0.440732	−	0.897639i	$-0.354719\pi$
$139$	10.3923			0.881464			0.440732	+	0.897639i	$0.354719\pi$
$140$		0			0
$141$	6.00000			0.505291
$142$		0			0
$143$	4.24264	+	4.24264i	0.354787	+	0.354787i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	5.60609	+	4.19187i	0.462382	+	0.345740i
$148$		0			0
$149$			15.0000i			1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$149$			15.0000i			1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$150$		0			0
$151$	5.00000			0.406894			0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000			0.406894			0.203447	+	0.979086i	$0.434786\pi$
$152$		0			0
$153$	4.24264	+	4.24264i	0.342997	+	0.342997i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	11.3137	−	11.3137i	0.902932	−	0.902932i	−0.0927566	−	0.995689i	$-0.529568\pi$
$157$	11.3137	−	11.3137i	0.902932	−	0.902932i	0.995689	+	0.0927566i	$0.0295678\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−9.00000	+	10.3923i	−0.709299	+	0.819028i
$162$		0			0
$163$	2.44949	−	2.44949i	0.191859	−	0.191859i	−0.604640	−	0.796499i	$-0.706683\pi$
$163$	2.44949	−	2.44949i	0.191859	−	0.191859i	0.796499	+	0.604640i	$0.206683\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.24264	−	4.24264i	0.328305	−	0.328305i	−0.523636	−	0.851942i	$-0.675425\pi$
$167$	4.24264	−	4.24264i	0.328305	−	0.328305i	0.851942	+	0.523636i	$0.175425\pi$
$168$		0			0
$169$		−	9.00000i		−	0.692308i
$170$		0			0
$171$			3.46410i			0.264906i
$172$		0			0
$173$	−12.7279	−	12.7279i	−0.967686	−	0.967686i	0.0318080	−	0.999494i	$-0.489873\pi$
$173$	−12.7279	−	12.7279i	−0.967686	−	0.967686i	−0.999494	+	0.0318080i	$0.989873\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.34847	+	7.34847i	0.552345	+	0.552345i
$178$		0			0
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$		0			0
$181$			3.46410i			0.257485i	0.991678	+	0.128742i	$0.0410940\pi$
$181$			3.46410i			0.257485i	−0.991678	+	0.128742i	$0.958906\pi$
$182$		0			0
$183$	−4.89898	+	4.89898i	−0.362143	+	0.362143i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.7279	+	12.7279i	−0.930758	+	0.930758i
$188$		0			0
$189$	1.73205	−	2.00000i	0.125988	−	0.145479i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	13.4722	−	13.4722i	0.969750	−	0.969750i	−0.0298060	−	0.999556i	$-0.509489\pi$
$193$	13.4722	−	13.4722i	0.969750	−	0.969750i	0.999556	+	0.0298060i	$0.00948895\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.0227	−	11.0227i	−0.785335	−	0.785335i	0.195390	−	0.980726i	$-0.437403\pi$
$197$	−11.0227	−	11.0227i	−0.785335	−	0.785335i	−0.980726	+	0.195390i	$0.937403\pi$
$198$		0			0
$199$	6.92820			0.491127			0.245564	−	0.969380i	$-0.421027\pi$
$199$	6.92820			0.491127			0.245564	+	0.969380i	$0.421027\pi$
$200$		0			0
$201$			12.1244i			0.855186i
$202$		0			0
$203$	−1.70522	−	23.7506i	−0.119683	−	1.66697i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	3.67423	+	3.67423i	0.255377	+	0.255377i
$208$		0			0
$209$	−10.3923			−0.718851
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$	2.12132	+	2.12132i	0.145350	+	0.145350i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.31268	+	18.2832i	0.0891104	+	1.24115i
$218$		0			0
$219$			2.00000i			0.135147i
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$	9.89949	+	9.89949i	0.662919	+	0.662919i	0.956067	−	0.293148i	$-0.0947028\pi$
$223$	9.89949	+	9.89949i	0.662919	+	0.662919i	−0.293148	+	0.956067i	$0.594703\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.24264	−	4.24264i	0.281594	−	0.281594i	−0.552151	−	0.833744i	$-0.686193\pi$
$227$	4.24264	−	4.24264i	0.281594	−	0.281594i	0.833744	+	0.552151i	$0.186193\pi$
$228$		0			0
$229$	6.92820			0.457829			0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820			0.457829			0.228914	+	0.973447i	$0.426482\pi$
$230$		0			0
$231$	6.00000	+	5.19615i	0.394771	+	0.341882i
$232$		0			0
$233$	−18.3712	+	18.3712i	−1.20354	+	1.20354i	−0.230452	+	0.973084i	$0.574020\pi$
$233$	−18.3712	+	18.3712i	−1.20354	+	1.20354i	−0.973084	+	0.230452i	$0.925980\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.707107	+	0.707107i	−0.0459315	+	0.0459315i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	17.3205i		−	1.11571i	−0.829938	−	0.557856i	$-0.811624\pi$
$241$		−	17.3205i		−	1.11571i	0.829938	−	0.557856i	$-0.188376\pi$
$242$		0			0
$243$	−0.707107	−	0.707107i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−4.89898	−	4.89898i	−0.311715	−	0.311715i
$248$		0			0
$249$		−	12.0000i		−	0.760469i
$250$		0			0
$251$		−	31.1769i		−	1.96787i	−0.178529	−	0.983935i	$-0.557134\pi$
$251$		−	31.1769i		−	1.96787i	0.178529	−	0.983935i	$-0.442866\pi$
$252$		0			0
$253$	−11.0227	+	11.0227i	−0.692991	+	0.692991i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.48528	−	8.48528i	0.529297	−	0.529297i	−0.391066	−	0.920363i	$-0.627893\pi$
$257$	8.48528	−	8.48528i	0.529297	−	0.529297i	0.920363	+	0.391066i	$0.127893\pi$
$258$		0			0
$259$	−10.3923	−	9.00000i	−0.645746	−	0.559233i
$260$		0			0
$261$	−9.00000			−0.557086
$262$		0			0
$263$	−11.0227	+	11.0227i	−0.679689	+	0.679689i	−0.959930	−	0.280241i	$-0.909586\pi$
$263$	−11.0227	+	11.0227i	−0.679689	+	0.679689i	0.280241	+	0.959930i	$0.409586\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−7.34847	−	7.34847i	−0.449719	−	0.449719i
$268$		0			0
$269$	−20.7846			−1.26726			−0.633630	−	0.773636i	$-0.718436\pi$
$269$	−20.7846			−1.26726			−0.633630	+	0.773636i	$0.718436\pi$
$270$		0			0
$271$			31.1769i			1.89386i	0.321436	+	0.946931i	$0.395835\pi$
$271$			31.1769i			1.89386i	−0.321436	+	0.946931i	$0.604165\pi$
$272$		0			0
$273$	0.378937	+	5.27792i	0.0229343	+	0.319434i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	9.79796	+	9.79796i	0.588702	+	0.588702i	0.937280	−	0.348578i	$-0.113335\pi$
$277$	9.79796	+	9.79796i	0.588702	+	0.588702i	−0.348578	+	0.937280i	$0.613335\pi$
$278$		0			0
$279$	6.92820			0.414781
$280$		0			0
$281$	3.00000			0.178965			0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000			0.178965			0.0894825	+	0.995988i	$0.471479\pi$
$282$		0			0
$283$	11.3137	+	11.3137i	0.672530	+	0.672530i	0.958299	−	0.285769i	$-0.0922488\pi$
$283$	11.3137	+	11.3137i	0.672530	+	0.672530i	−0.285769	+	0.958299i	$0.592249\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	19.0000i		−	1.11765i
$290$		0			0
$291$	4.00000			0.234484
$292$		0			0
$293$	12.7279	+	12.7279i	0.743573	+	0.743573i	0.973264	−	0.229691i	$-0.0737714\pi$
$293$	12.7279	+	12.7279i	0.743573	+	0.743573i	−0.229691	+	0.973264i	$0.573771\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	2.12132	−	2.12132i	0.123091	−	0.123091i
$298$		0			0
$299$	−10.3923			−0.601003
$300$		0			0
$301$	−21.0000	+	24.2487i	−1.21042	+	1.39767i
$302$		0			0
$303$	7.34847	−	7.34847i	0.422159	−	0.422159i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.07107	+	7.07107i	−0.403567	+	0.403567i	−0.879488	−	0.475921i	$-0.842115\pi$
$307$	−7.07107	+	7.07107i	−0.403567	+	0.403567i	0.475921	+	0.879488i	$0.342115\pi$
$308$		0			0
$309$		−	14.0000i		−	0.796432i
$310$		0			0
$311$		−	31.1769i		−	1.76788i	−0.467600	−	0.883940i	$-0.654881\pi$
$311$		−	31.1769i		−	1.76788i	0.467600	−	0.883940i	$-0.345119\pi$
$312$		0			0
$313$	9.89949	+	9.89949i	0.559553	+	0.559553i	0.929180	−	0.369627i	$-0.120515\pi$
$313$	9.89949	+	9.89949i	0.559553	+	0.559553i	−0.369627	+	0.929180i	$0.620515\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.67423	−	3.67423i	−0.206366	−	0.206366i	0.596355	−	0.802721i	$-0.296615\pi$
$317$	−3.67423	−	3.67423i	−0.206366	−	0.206366i	−0.802721	+	0.596355i	$0.796615\pi$
$318$		0			0
$319$		−	27.0000i		−	1.51171i
$320$		0			0
$321$		−	10.3923i		−	0.580042i
$322$		0			0
$323$	14.6969	−	14.6969i	0.817760	−	0.817760i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−12.0208	+	12.0208i	−0.664753	+	0.664753i
$328$		0			0
$329$	−10.3923	+	12.0000i	−0.572946	+	0.661581i
$330$		0			0
$331$	19.0000			1.04433			0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000			1.04433			0.522167	+	0.852843i	$0.325124\pi$
$332$		0			0
$333$	−3.67423	+	3.67423i	−0.201347	+	0.201347i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	4.89898	+	4.89898i	0.266864	+	0.266864i	0.827835	−	0.560971i	$-0.189572\pi$
$337$	4.89898	+	4.89898i	0.266864	+	0.266864i	−0.560971	+	0.827835i	$0.689572\pi$
$338$		0			0
$339$	5.19615			0.282216
$340$		0			0
$341$			20.7846i			1.12555i
$342$		0			0
$343$	−18.0938	+	3.95164i	−0.976972	+	0.213368i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−25.7196	−	25.7196i	−1.38070	−	1.38070i	−0.843371	−	0.537332i	$-0.819432\pi$
$347$	−25.7196	−	25.7196i	−1.38070	−	1.38070i	−0.537332	−	0.843371i	$-0.680568\pi$
$348$		0			0
$349$	10.3923			0.556287			0.278144	−	0.960539i	$-0.410281\pi$
$349$	10.3923			0.556287			0.278144	+	0.960539i	$0.410281\pi$
$350$		0			0
$351$	2.00000			0.106752
$352$		0			0
$353$	25.4558	+	25.4558i	1.35488	+	1.35488i	0.880112	+	0.474766i	$0.157467\pi$
$353$	25.4558	+	25.4558i	1.35488	+	1.35488i	0.474766	+	0.880112i	$0.342533\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−15.8338	+	1.13681i	−0.838011	+	0.0601665i
$358$		0			0
$359$		−	27.0000i		−	1.42501i	−0.701669	−	0.712503i	$-0.747562\pi$
$359$		−	27.0000i		−	1.42501i	0.701669	−	0.712503i	$-0.252438\pi$
$360$		0			0
$361$	−7.00000			−0.368421
$362$		0			0
$363$	−1.41421	−	1.41421i	−0.0742270	−	0.0742270i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−22.6274	+	22.6274i	−1.18114	+	1.18114i	−0.201693	+	0.979449i	$0.564644\pi$
$367$	−22.6274	+	22.6274i	−1.18114	+	1.18114i	−0.979449	+	0.201693i	$0.935356\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	6.12372	−	6.12372i	0.317074	−	0.317074i	−0.530568	−	0.847642i	$-0.678021\pi$
$373$	6.12372	−	6.12372i	0.317074	−	0.317074i	0.847642	+	0.530568i	$0.178021\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.7279	−	12.7279i	0.655521	−	0.655521i
$378$		0			0
$379$			1.00000i			0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$			1.00000i			0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$		0			0
$381$		−	5.19615i		−	0.266207i
$382$		0			0
$383$	4.24264	+	4.24264i	0.216789	+	0.216789i	0.807144	−	0.590355i	$-0.201012\pi$
$383$	4.24264	+	4.24264i	0.216789	+	0.216789i	−0.590355	+	0.807144i	$0.701012\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.57321	+	8.57321i	0.435801	+	0.435801i
$388$		0			0
$389$		−	3.00000i		−	0.152106i	−0.997104	−	0.0760530i	$-0.975768\pi$
$389$		−	3.00000i		−	0.152106i	0.997104	−	0.0760530i	$-0.0242318\pi$
$390$		0			0
$391$		−	31.1769i		−	1.57668i
$392$		0			0
$393$	7.34847	−	7.34847i	0.370681	−	0.370681i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	5.65685	−	5.65685i	0.283909	−	0.283909i	−0.550757	−	0.834666i	$-0.685661\pi$
$397$	5.65685	−	5.65685i	0.283909	−	0.283909i	0.834666	+	0.550757i	$0.185661\pi$
$398$		0			0
$399$	−6.92820	−	6.00000i	−0.346844	−	0.300376i
$400$		0			0
$401$	−27.0000			−1.34832			−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000			−1.34832			−0.674158	+	0.738587i	$0.735493\pi$
$402$		0			0
$403$	−9.79796	+	9.79796i	−0.488071	+	0.488071i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−11.0227	−	11.0227i	−0.546375	−	0.546375i
$408$		0			0
$409$	38.1051			1.88418			0.942088	−	0.335365i	$-0.108860\pi$
$409$	38.1051			1.88418			0.942088	+	0.335365i	$0.108860\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−27.4249	+	1.96902i	−1.34949	+	0.0968890i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.34847	+	7.34847i	0.359856	+	0.359856i
$418$		0			0
$419$	−20.7846			−1.01539			−0.507697	−	0.861536i	$-0.669503\pi$
$419$	−20.7846			−1.01539			−0.507697	+	0.861536i	$0.669503\pi$
$420$		0			0
$421$	−17.0000			−0.828529			−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000			−0.828529			−0.414265	+	0.910156i	$0.635961\pi$
$422$		0			0
$423$	4.24264	+	4.24264i	0.206284	+	0.206284i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.31268	−	18.2832i	−0.0635249	−	0.884788i
$428$		0			0
$429$			6.00000i			0.289683i
$430$		0			0
$431$	−12.0000			−0.578020			−0.289010	−	0.957326i	$-0.593326\pi$
$431$	−12.0000			−0.578020			−0.289010	+	0.957326i	$0.593326\pi$
$432$		0			0
$433$	5.65685	+	5.65685i	0.271851	+	0.271851i	0.829845	−	0.557994i	$-0.188429\pi$
$433$	5.65685	+	5.65685i	0.271851	+	0.271851i	−0.557994	+	0.829845i	$0.688429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.7279	−	12.7279i	0.608859	−	0.608859i
$438$		0			0
$439$	−38.1051			−1.81866			−0.909329	−	0.416078i	$-0.863404\pi$
$439$	−38.1051			−1.81866			−0.909329	+	0.416078i	$0.863404\pi$
$440$		0			0
$441$	1.00000	+	6.92820i	0.0476190	+	0.329914i
$442$		0			0
$443$	−7.34847	+	7.34847i	−0.349136	+	0.349136i	−0.859788	−	0.510651i	$-0.829404\pi$
$443$	−7.34847	+	7.34847i	−0.349136	+	0.349136i	0.510651	+	0.859788i	$0.329404\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−10.6066	+	10.6066i	−0.501675	+	0.501675i
$448$		0			0
$449$			27.0000i			1.27421i	0.770778	+	0.637104i	$0.219868\pi$
$449$			27.0000i			1.27421i	−0.770778	+	0.637104i	$0.780132\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3.53553	+	3.53553i	0.166114	+	0.166114i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.67423	−	3.67423i	−0.171873	−	0.171873i	0.615929	−	0.787802i	$-0.288781\pi$
$457$	−3.67423	−	3.67423i	−0.171873	−	0.171873i	−0.787802	+	0.615929i	$0.788781\pi$
$458$		0			0
$459$			6.00000i			0.280056i
$460$		0			0
$461$			31.1769i			1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$			31.1769i			1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$		0			0
$463$	−7.34847	+	7.34847i	−0.341512	+	0.341512i	−0.856936	−	0.515423i	$-0.827635\pi$
$463$	−7.34847	+	7.34847i	−0.341512	+	0.341512i	0.515423	+	0.856936i	$0.327635\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.48528	+	8.48528i	−0.392652	+	0.392652i	−0.875632	−	0.482980i	$-0.839555\pi$
$467$	−8.48528	+	8.48528i	−0.392652	+	0.392652i	0.482980	+	0.875632i	$0.339555\pi$
$468$		0			0
$469$	−24.2487	−	21.0000i	−1.11970	−	0.969690i
$470$		0			0
$471$	16.0000			0.737241
$472$		0			0
$473$	−25.7196	+	25.7196i	−1.18259	+	1.18259i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	10.3923			0.474837			0.237418	−	0.971408i	$-0.423699\pi$
$479$	10.3923			0.474837			0.237418	+	0.971408i	$0.423699\pi$
$480$		0			0
$481$		−	10.3923i		−	0.473848i
$482$		0			0
$483$	−13.7124	+	0.984508i	−0.623937	+	0.0447967i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−15.9217	−	15.9217i	−0.721480	−	0.721480i	0.247426	−	0.968907i	$-0.420415\pi$
$487$	−15.9217	−	15.9217i	−0.721480	−	0.721480i	−0.968907	+	0.247426i	$0.920415\pi$
$488$		0			0
$489$	3.46410			0.156652
$490$		0			0
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		0			0
$493$	38.1838	+	38.1838i	1.71971	+	1.71971i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−7.91688	+	0.568406i	−0.355120	+	0.0254965i
$498$		0			0
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$	6.00000			0.268060
$502$		0			0
$503$	25.4558	+	25.4558i	1.13502	+	1.13502i	0.989330	+	0.145690i	$0.0465401\pi$
$503$	25.4558	+	25.4558i	1.13502	+	1.13502i	0.145690	+	0.989330i	$0.453460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	6.36396	−	6.36396i	0.282633	−	0.282633i
$508$		0			0
$509$	−31.1769			−1.38189			−0.690946	−	0.722906i	$-0.742806\pi$
$509$	−31.1769			−1.38189			−0.690946	+	0.722906i	$0.742806\pi$
$510$		0			0
$511$	−4.00000	−	3.46410i	−0.176950	−	0.153243i
$512$		0			0
$513$	−2.44949	+	2.44949i	−0.108148	+	0.108148i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−12.7279	+	12.7279i	−0.559773	+	0.559773i
$518$		0			0
$519$		−	18.0000i		−	0.790112i
$520$		0			0
$521$		−	20.7846i		−	0.910590i	−0.890341	−	0.455295i	$-0.849534\pi$
$521$		−	20.7846i		−	0.910590i	0.890341	−	0.455295i	$-0.150466\pi$
$522$		0			0
$523$	−31.1127	−	31.1127i	−1.36046	−	1.36046i	−0.873325	−	0.487139i	$-0.838041\pi$
$523$	−31.1127	−	31.1127i	−1.36046	−	1.36046i	−0.487139	−	0.873325i	$-0.661959\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−29.3939	−	29.3939i	−1.28042	−	1.28042i
$528$		0			0
$529$		−	4.00000i		−	0.173913i
$530$		0			0
$531$			10.3923i			0.450988i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−8.48528	+	8.48528i	−0.366167	+	0.366167i
$538$		0			0
$539$	−20.7846	+	3.00000i	−0.895257	+	0.129219i
$540$		0			0
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−2.44949	+	2.44949i	−0.105118	+	0.105118i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−11.0227	−	11.0227i	−0.471297	−	0.471297i	0.431037	−	0.902334i	$-0.358148\pi$
$547$	−11.0227	−	11.0227i	−0.471297	−	0.471297i	−0.902334	+	0.431037i	$0.858148\pi$
$548$		0			0
$549$	−6.92820			−0.295689
$550$		0			0
$551$			31.1769i			1.32818i
$552$		0			0
$553$	−0.189469	−	2.63896i	−0.00805703	−	0.112220i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.7196	+	25.7196i	1.08978	+	1.08978i	0.995551	+	0.0942253i	$0.0300374\pi$
$557$	25.7196	+	25.7196i	1.08978	+	1.08978i	0.0942253	+	0.995551i	$0.469963\pi$
$558$		0			0
$559$	−24.2487			−1.02561
$560$		0			0
$561$	−18.0000			−0.759961
$562$		0			0
$563$	16.9706	+	16.9706i	0.715224	+	0.715224i	0.967623	−	0.252399i	$-0.0812196\pi$
$563$	16.9706	+	16.9706i	0.715224	+	0.715224i	−0.252399	+	0.967623i	$0.581220\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.63896	−	0.189469i	0.110826	−	0.00795694i
$568$		0			0
$569$		−	39.0000i		−	1.63497i	−0.575953	−	0.817483i	$-0.695369\pi$
$569$		−	39.0000i		−	1.63497i	0.575953	−	0.817483i	$-0.304631\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−7.07107	+	7.07107i	−0.294372	+	0.294372i	−0.838805	−	0.544432i	$-0.816745\pi$
$577$	−7.07107	+	7.07107i	−0.294372	+	0.294372i	0.544432	+	0.838805i	$0.316745\pi$
$578$		0			0
$579$	19.0526			0.791797
$580$		0			0
$581$	24.0000	+	20.7846i	0.995688	+	0.862291i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12.7279	−	12.7279i	0.525338	−	0.525338i	−0.393841	−	0.919179i	$-0.628854\pi$
$587$	12.7279	−	12.7279i	0.525338	−	0.525338i	0.919179	+	0.393841i	$0.128854\pi$
$588$		0			0
$589$		−	24.0000i		−	0.988903i
$590$		0			0
$591$		−	15.5885i		−	0.641223i
$592$		0			0
$593$	−25.4558	−	25.4558i	−1.04535	−	1.04535i	−0.998922	−	0.0464244i	$-0.985217\pi$
$593$	−25.4558	−	25.4558i	−1.04535	−	1.04535i	−0.0464244	−	0.998922i	$-0.514783\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	4.89898	+	4.89898i	0.200502	+	0.200502i
$598$		0			0
$599$		−	39.0000i		−	1.59350i	−0.604311	−	0.796748i	$-0.706552\pi$
$599$		−	39.0000i		−	1.59350i	0.604311	−	0.796748i	$-0.293448\pi$
$600$		0			0
$601$			27.7128i			1.13043i	0.824944	+	0.565215i	$0.191207\pi$
$601$			27.7128i			1.13043i	−0.824944	+	0.565215i	$0.808793\pi$
$602$		0			0
$603$	−8.57321	+	8.57321i	−0.349128	+	0.349128i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−15.5563	+	15.5563i	−0.631413	+	0.631413i	−0.948422	−	0.317010i	$-0.897321\pi$
$607$	−15.5563	+	15.5563i	−0.631413	+	0.631413i	0.317010	+	0.948422i	$0.397321\pi$
$608$		0			0
$609$	15.5885	−	18.0000i	0.631676	−	0.729397i
$610$		0			0
$611$	−12.0000			−0.485468
$612$		0			0
$613$	−11.0227	+	11.0227i	−0.445203	+	0.445203i	−0.893756	−	0.448553i	$-0.851939\pi$
$613$	−11.0227	+	11.0227i	−0.445203	+	0.445203i	0.448553	+	0.893756i	$0.351939\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.7196	+	25.7196i	1.03543	+	1.03543i	0.999349	+	0.0360851i	$0.0114887\pi$
$617$	25.7196	+	25.7196i	1.03543	+	1.03543i	0.0360851	+	0.999349i	$0.488511\pi$
$618$		0			0
$619$	24.2487			0.974638			0.487319	−	0.873224i	$-0.337975\pi$
$619$	24.2487			0.974638			0.487319	+	0.873224i	$0.337975\pi$
$620$		0			0
$621$			5.19615i			0.208514i
$622$		0			0
$623$	27.4249	−	1.96902i	1.09875	−	0.0788870i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−7.34847	−	7.34847i	−0.293470	−	0.293470i
$628$		0			0
$629$	31.1769			1.24310
$630$		0			0
$631$	35.0000			1.39333			0.696664	−	0.717398i	$-0.254667\pi$
$631$	35.0000			1.39333			0.696664	+	0.717398i	$0.254667\pi$
$632$		0			0
$633$	11.3137	+	11.3137i	0.449680	+	0.449680i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−11.2122	−	8.38375i	−0.444242	−	0.332176i
$638$		0			0
$639$			3.00000i			0.118678i
$640$		0			0
$641$	9.00000			0.355479			0.177739	−	0.984078i	$-0.443122\pi$
$641$	9.00000			0.355479			0.177739	+	0.984078i	$0.443122\pi$
$642$		0			0
$643$	−2.82843	−	2.82843i	−0.111542	−	0.111542i	0.649133	−	0.760675i	$-0.275132\pi$
$643$	−2.82843	−	2.82843i	−0.111542	−	0.111542i	−0.760675	+	0.649133i	$0.775132\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$		0			0
$649$	−31.1769			−1.22380
$650$		0			0
$651$	−12.0000	+	13.8564i	−0.470317	+	0.543075i
$652$		0			0
$653$	−29.3939	+	29.3939i	−1.15027	+	1.15027i	−0.163773	+	0.986498i	$0.552367\pi$
$653$	−29.3939	+	29.3939i	−1.15027	+	1.15027i	−0.986498	+	0.163773i	$0.947633\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.41421	+	1.41421i	−0.0551737	+	0.0551737i
$658$		0			0
$659$			24.0000i			0.934907i	0.884018	+	0.467454i	$0.154829\pi$
$659$			24.0000i			0.934907i	−0.884018	+	0.467454i	$0.845171\pi$
$660$		0			0
$661$		−	31.1769i		−	1.21264i	−0.795220	−	0.606321i	$-0.792645\pi$
$661$		−	31.1769i		−	1.21264i	0.795220	−	0.606321i	$-0.207355\pi$
$662$		0			0
$663$	−8.48528	−	8.48528i	−0.329541	−	0.329541i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	33.0681	+	33.0681i	1.28040	+	1.28040i
$668$		0			0
$669$			14.0000i			0.541271i
$670$		0			0
$671$		−	20.7846i		−	0.802381i
$672$		0			0
$673$	−14.6969	+	14.6969i	−0.566525	+	0.566525i	−0.931153	−	0.364628i	$-0.881196\pi$
$673$	−14.6969	+	14.6969i	−0.566525	+	0.566525i	0.364628	+	0.931153i	$0.381196\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−8.48528	+	8.48528i	−0.326116	+	0.326116i	−0.851107	−	0.524992i	$-0.824068\pi$
$677$	−8.48528	+	8.48528i	−0.326116	+	0.326116i	0.524992	+	0.851107i	$0.324068\pi$
$678$		0			0
$679$	−6.92820	+	8.00000i	−0.265880	+	0.307012i
$680$		0			0
$681$	6.00000			0.229920
$682$		0			0
$683$	25.7196	−	25.7196i	0.984135	−	0.984135i	−0.0157413	−	0.999876i	$-0.505011\pi$
$683$	25.7196	−	25.7196i	0.984135	−	0.984135i	0.999876	+	0.0157413i	$0.00501083\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.89898	+	4.89898i	0.186908	+	0.186908i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			20.7846i			0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$691$			20.7846i			0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$692$		0			0
$693$	0.568406	+	7.91688i	0.0215920	+	0.300737i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−25.9808			−0.982683
$700$		0			0
$701$	30.0000			1.13308			0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000			1.13308			0.566542	+	0.824033i	$0.308281\pi$
$702$		0			0
$703$	12.7279	+	12.7279i	0.480043	+	0.480043i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.96902	+	27.4249i	0.0740525	+	1.03142i
$708$		0			0
$709$			22.0000i			0.826227i	0.910679	+	0.413114i	$0.135559\pi$
$709$			22.0000i			0.826227i	−0.910679	+	0.413114i	$0.864441\pi$
$710$		0			0
$711$	−1.00000			−0.0375029
$712$		0			0
$713$	−25.4558	−	25.4558i	−0.953329	−	0.953329i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	28.0000	+	24.2487i	1.04277	+	0.903069i
$722$		0			0
$723$	12.2474	−	12.2474i	0.455488	−	0.455488i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	9.89949	−	9.89949i	0.367152	−	0.367152i	−0.499286	−	0.866437i	$-0.666404\pi$
$727$	9.89949	−	9.89949i	0.367152	−	0.367152i	0.866437	+	0.499286i	$0.166404\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$		−	72.7461i		−	2.69061i
$732$		0			0
$733$	−2.82843	−	2.82843i	−0.104470	−	0.104470i	0.652940	−	0.757410i	$-0.273536\pi$
$733$	−2.82843	−	2.82843i	−0.104470	−	0.104470i	−0.757410	+	0.652940i	$0.773536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−25.7196	−	25.7196i	−0.947395	−	0.947395i
$738$		0			0
$739$			11.0000i			0.404642i	0.979319	+	0.202321i	$0.0648484\pi$
$739$			11.0000i			0.404642i	−0.979319	+	0.202321i	$0.935152\pi$
$740$		0			0
$741$		−	6.92820i		−	0.254514i
$742$		0			0
$743$	−7.34847	+	7.34847i	−0.269589	+	0.269589i	−0.828935	−	0.559345i	$-0.811053\pi$
$743$	−7.34847	+	7.34847i	−0.269589	+	0.269589i	0.559345	+	0.828935i	$0.311053\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	8.48528	−	8.48528i	0.310460	−	0.310460i
$748$		0			0
$749$	20.7846	+	18.0000i	0.759453	+	0.657706i
$750$		0			0
$751$	−40.0000			−1.45962			−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000			−1.45962			−0.729810	+	0.683650i	$0.760392\pi$
$752$		0			0
$753$	22.0454	−	22.0454i	0.803379	−	0.803379i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.22474	+	1.22474i	0.0445141	+	0.0445141i	0.729013	−	0.684499i	$-0.239979\pi$
$757$	1.22474	+	1.22474i	0.0445141	+	0.0445141i	−0.684499	+	0.729013i	$0.739979\pi$
$758$		0			0
$759$	−15.5885			−0.565825
$760$		0			0
$761$			41.5692i			1.50688i	0.657515	+	0.753442i	$0.271608\pi$
$761$			41.5692i			1.50688i	−0.657515	+	0.753442i	$0.728392\pi$
$762$		0			0
$763$	−3.22097	−	44.8623i	−0.116607	−	1.62412i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.6969	−	14.6969i	−0.530676	−	0.530676i
$768$		0			0
$769$	13.8564			0.499675			0.249837	−	0.968288i	$-0.419623\pi$
$769$	13.8564			0.499675			0.249837	+	0.968288i	$0.419623\pi$
$770$		0			0
$771$	12.0000			0.432169
$772$		0			0
$773$	−16.9706	−	16.9706i	−0.610389	−	0.610389i	0.332659	−	0.943047i	$-0.392054\pi$
$773$	−16.9706	−	16.9706i	−0.610389	−	0.610389i	−0.943047	+	0.332659i	$0.892054\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.984508	−	13.7124i	−0.0353190	−	0.491931i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−9.00000			−0.322045
$782$		0			0
$783$	−6.36396	−	6.36396i	−0.227429	−	0.227429i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	26.8701	−	26.8701i	0.957814	−	0.957814i	−0.0413314	−	0.999145i	$-0.513160\pi$
$787$	26.8701	−	26.8701i	0.957814	−	0.957814i	0.999145	+	0.0413314i	$0.0131599\pi$
$788$		0			0
$789$	−15.5885			−0.554964
$790$		0			0
$791$	−9.00000	+	10.3923i	−0.320003	+	0.369508i
$792$		0			0
$793$	9.79796	−	9.79796i	0.347936	−	0.347936i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.2132	+	21.2132i	−0.751410	+	0.751410i	−0.974742	−	0.223332i	$-0.928307\pi$
$797$	−21.2132	+	21.2132i	−0.751410	+	0.751410i	0.223332	+	0.974742i	$0.428307\pi$
$798$		0			0
$799$		−	36.0000i		−	1.27359i
$800$		0			0
$801$		−	10.3923i		−	0.367194i
$802$		0			0
$803$	−4.24264	−	4.24264i	−0.149720	−	0.149720i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−14.6969	−	14.6969i	−0.517357	−	0.517357i
$808$		0			0
$809$			51.0000i			1.79306i	0.442978	+	0.896532i	$0.353922\pi$
$809$			51.0000i			1.79306i	−0.442978	+	0.896532i	$0.646078\pi$
$810$		0			0
$811$			27.7128i			0.973128i	0.873645	+	0.486564i	$0.161750\pi$
$811$			27.7128i			0.973128i	−0.873645	+	0.486564i	$0.838250\pi$
$812$		0			0
$813$	−22.0454	+	22.0454i	−0.773166	+	0.773166i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	29.6985	−	29.6985i	1.03902	−	1.03902i
$818$		0			0
$819$	−3.46410	+	4.00000i	−0.121046	+	0.139771i
$820$		0			0
$821$	30.0000			1.04701			0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000			1.04701			0.523504	+	0.852023i	$0.324625\pi$
$822$		0			0
$823$	−1.22474	+	1.22474i	−0.0426919	+	0.0426919i	−0.728130	−	0.685439i	$-0.759611\pi$
$823$	−1.22474	+	1.22474i	−0.0426919	+	0.0426919i	0.685439	+	0.728130i	$0.259611\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	18.3712	+	18.3712i	0.638828	+	0.638828i	0.950266	−	0.311438i	$-0.100811\pi$
$827$	18.3712	+	18.3712i	0.638828	+	0.638828i	−0.311438	+	0.950266i	$0.600811\pi$
$828$		0			0
$829$	10.3923			0.360940			0.180470	−	0.983581i	$-0.442238\pi$
$829$	10.3923			0.360940			0.180470	+	0.983581i	$0.442238\pi$
$830$		0			0
$831$			13.8564i			0.480673i
$832$		0			0
$833$	25.1512	−	33.6365i	0.871439	−	1.16544i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.89898	+	4.89898i	0.169334	+	0.169334i
$838$		0			0
$839$	−10.3923			−0.358782			−0.179391	−	0.983778i	$-0.557413\pi$
$839$	−10.3923			−0.358782			−0.179391	+	0.983778i	$0.557413\pi$
$840$		0			0
$841$	−52.0000			−1.79310
$842$		0			0
$843$	2.12132	+	2.12132i	0.0730622	+	0.0730622i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	5.27792	−	0.378937i	0.181351	−	0.0130204i
$848$		0			0
$849$			16.0000i			0.549119i
$850$		0			0
$851$	27.0000			0.925548
$852$		0			0
$853$	2.82843	+	2.82843i	0.0968435	+	0.0968435i	0.753869	−	0.657025i	$-0.228185\pi$
$853$	2.82843	+	2.82843i	0.0968435	+	0.0968435i	−0.657025	+	0.753869i	$0.728185\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	8.48528	−	8.48528i	0.289852	−	0.289852i	−0.547170	−	0.837022i	$-0.684295\pi$
$857$	8.48528	−	8.48528i	0.289852	−	0.289852i	0.837022	+	0.547170i	$0.184295\pi$
$858$		0			0
$859$	41.5692			1.41832			0.709162	−	0.705046i	$-0.249074\pi$
$859$	41.5692			1.41832			0.709162	+	0.705046i	$0.249074\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.3712	+	18.3712i	−0.625362	+	0.625362i	−0.946898	−	0.321536i	$-0.895801\pi$
$863$	−18.3712	+	18.3712i	−0.625362	+	0.625362i	0.321536	+	0.946898i	$0.395801\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	13.4350	−	13.4350i	0.456278	−	0.456278i
$868$		0			0
$869$		−	3.00000i		−	0.101768i
$870$		0			0
$871$		−	24.2487i		−	0.821636i
$872$		0			0
$873$	2.82843	+	2.82843i	0.0957278	+	0.0957278i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	9.79796	+	9.79796i	0.330854	+	0.330854i	0.852911	−	0.522057i	$-0.174835\pi$
$877$	9.79796	+	9.79796i	0.330854	+	0.330854i	−0.522057	+	0.852911i	$0.674835\pi$
$878$		0			0
$879$			18.0000i			0.607125i
$880$		0			0
$881$		−	20.7846i		−	0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$		−	20.7846i		−	0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$		0			0
$883$	−13.4722	+	13.4722i	−0.453375	+	0.453375i	−0.896473	−	0.443098i	$-0.853879\pi$
$883$	−13.4722	+	13.4722i	−0.453375	+	0.453375i	0.443098	+	0.896473i	$0.353879\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−25.4558	+	25.4558i	−0.854724	+	0.854724i	−0.990711	−	0.135987i	$-0.956579\pi$
$887$	−25.4558	+	25.4558i	−0.854724	+	0.854724i	0.135987	+	0.990711i	$0.456579\pi$
$888$		0			0
$889$	10.3923	+	9.00000i	0.348547	+	0.301850i
$890$		0			0
$891$	3.00000			0.100504
$892$		0			0
$893$	14.6969	−	14.6969i	0.491814	−	0.491814i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−7.34847	−	7.34847i	−0.245358	−	0.245358i
$898$		0			0
$899$	62.3538			2.07962
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−31.9957	+	2.29719i	−1.06475	+	0.0764456i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	7.34847	+	7.34847i	0.244002	+	0.244002i	0.818503	−	0.574502i	$-0.194804\pi$
$907$	7.34847	+	7.34847i	0.244002	+	0.244002i	−0.574502	+	0.818503i	$0.694804\pi$
$908$		0			0
$909$	10.3923			0.344691
$910$		0			0
$911$	−51.0000			−1.68971			−0.844853	−	0.534999i	$-0.820312\pi$
$911$	−51.0000			−1.68971			−0.844853	+	0.534999i	$0.820312\pi$
$912$		0			0
$913$	25.4558	+	25.4558i	0.842465	+	0.842465i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.96902	+	27.4249i	0.0650226	+	0.905649i
$918$		0			0
$919$			55.0000i			1.81428i	0.420826	+	0.907141i	$0.361740\pi$
$919$			55.0000i			1.81428i	−0.420826	+	0.907141i	$0.638260\pi$
$920$		0			0
$921$	−10.0000			−0.329511
$922$		0			0
$923$	−4.24264	−	4.24264i	−0.139648	−	0.139648i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	9.89949	−	9.89949i	0.325142	−	0.325142i
$928$		0			0
$929$	−31.1769			−1.02288			−0.511441	−	0.859319i	$-0.670888\pi$
$929$	−31.1769			−1.02288			−0.511441	+	0.859319i	$0.670888\pi$
$930$		0			0
$931$	24.0000	−	3.46410i	0.786568	−	0.113531i
$932$		0			0
$933$	22.0454	−	22.0454i	0.721734	−	0.721734i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−5.65685	+	5.65685i	−0.184801	+	0.184801i	−0.793444	−	0.608643i	$-0.791714\pi$
$937$	−5.65685	+	5.65685i	−0.184801	+	0.184801i	0.608643	+	0.793444i	$0.291714\pi$
$938$		0			0
$939$			14.0000i			0.456873i
$940$		0			0
$941$			20.7846i			0.677559i	0.940866	+	0.338779i	$0.110014\pi$
$941$			20.7846i			0.677559i	−0.940866	+	0.338779i	$0.889986\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	22.0454	+	22.0454i	0.716379	+	0.716379i	0.967862	−	0.251482i	$-0.0809181\pi$
$947$	22.0454	+	22.0454i	0.716379	+	0.716379i	−0.251482	+	0.967862i	$0.580918\pi$
$948$		0			0
$949$		−	4.00000i		−	0.129845i
$950$		0			0
$951$		−	5.19615i		−	0.168497i
$952$		0			0
$953$	40.4166	−	40.4166i	1.30922	−	1.30922i	0.387244	−	0.921977i	$-0.373427\pi$
$953$	40.4166	−	40.4166i	1.30922	−	1.30922i	0.921977	−	0.387244i	$-0.126573\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	19.0919	−	19.0919i	0.617153	−	0.617153i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−17.0000			−0.548387
$962$		0			0
$963$	7.34847	−	7.34847i	0.236801	−	0.236801i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−22.0454	−	22.0454i	−0.708933	−	0.708933i	0.257378	−	0.966311i	$-0.417141\pi$
$967$	−22.0454	−	22.0454i	−0.708933	−	0.708933i	−0.966311	+	0.257378i	$0.917141\pi$
$968$		0			0
$969$	20.7846			0.667698
$970$		0			0
$971$		−	10.3923i		−	0.333505i	−0.985999	−	0.166752i	$-0.946672\pi$
$971$		−	10.3923i		−	0.333505i	0.985999	−	0.166752i	$-0.0533281\pi$
$972$		0			0
$973$	−27.4249	+	1.96902i	−0.879201	+	0.0631238i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11.0227	−	11.0227i	−0.352648	−	0.352648i	0.508446	−	0.861094i	$-0.330220\pi$
$977$	−11.0227	−	11.0227i	−0.352648	−	0.352648i	−0.861094	+	0.508446i	$0.830220\pi$
$978$		0			0
$979$	31.1769			0.996419
$980$		0			0
$981$	−17.0000			−0.542768
$982$		0			0
$983$	33.9411	+	33.9411i	1.08255	+	1.08255i	0.996271	+	0.0862831i	$0.0274990\pi$
$983$	33.9411	+	33.9411i	1.08255	+	1.08255i	0.0862831	+	0.996271i	$0.472501\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−15.8338	+	1.13681i	−0.503994	+	0.0361851i
$988$		0			0
$989$		−	63.0000i		−	2.00328i
$990$		0			0
$991$	55.0000			1.74713			0.873566	−	0.486705i	$-0.161801\pi$
$991$	55.0000			1.74713			0.873566	+	0.486705i	$0.161801\pi$
$992$		0			0
$993$	13.4350	+	13.4350i	0.426348	+	0.426348i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	7.07107	−	7.07107i	0.223943	−	0.223943i	−0.586214	−	0.810157i	$-0.699382\pi$
$997$	7.07107	−	7.07107i	0.223943	−	0.223943i	0.810157	+	0.586214i	$0.199382\pi$
$998$		0			0
$999$	−5.19615			−0.164399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.x.a.1357.3	yes	8
5.2	odd	4	inner	2100.2.x.a.1693.4	yes	8
5.3	odd	4	inner	2100.2.x.a.1693.1	yes	8
5.4	even	2	inner	2100.2.x.a.1357.2	yes	8
7.6	odd	2	inner	2100.2.x.a.1357.1	✓	8
35.13	even	4	inner	2100.2.x.a.1693.3	yes	8
35.27	even	4	inner	2100.2.x.a.1693.2	yes	8
35.34	odd	2	inner	2100.2.x.a.1357.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2100.2.x.a.1357.1	✓	8	7.6	odd	2	inner
2100.2.x.a.1357.2	yes	8	5.4	even	2	inner
2100.2.x.a.1357.3	yes	8	1.1	even	1	trivial
2100.2.x.a.1357.4	yes	8	35.34	odd	2	inner
2100.2.x.a.1693.1	yes	8	5.3	odd	4	inner
2100.2.x.a.1693.2	yes	8	35.27	even	4	inner
2100.2.x.a.1693.3	yes	8	35.13	even	4	inner
2100.2.x.a.1693.4	yes	8	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{3} +(-2.63896 + 0.189469i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{3} +(-2.63896 + 0.189469i) q^{7} +1.00000i q^{9} -3.00000 q^{11} +(-1.41421 - 1.41421i) q^{13} +(4.24264 - 4.24264i) q^{17} +3.46410 q^{19} +(-2.00000 - 1.73205i) q^{21} +(3.67423 - 3.67423i) q^{23} +(-0.707107 + 0.707107i) q^{27} +9.00000i q^{29} -6.92820i q^{31} +(-2.12132 - 2.12132i) q^{33} +(3.67423 + 3.67423i) q^{37} -2.00000i q^{39} +(8.57321 - 8.57321i) q^{43} +(4.24264 - 4.24264i) q^{47} +(6.92820 - 1.00000i) q^{49} +6.00000 q^{51} +(2.44949 + 2.44949i) q^{57} +10.3923 q^{59} +6.92820i q^{61} +(-0.189469 - 2.63896i) q^{63} +(8.57321 + 8.57321i) q^{67} +5.19615 q^{69} +3.00000 q^{71} +(1.41421 + 1.41421i) q^{73} +(7.91688 - 0.568406i) q^{77} +1.00000i q^{79} -1.00000 q^{81} +(-8.48528 - 8.48528i) q^{83} +(-6.36396 + 6.36396i) q^{87} -10.3923 q^{89} +(4.00000 + 3.46410i) q^{91} +(4.89898 - 4.89898i) q^{93} +(2.82843 - 2.82843i) q^{97} -3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{11} - 16 q^{21} + 48 q^{51} + 24 q^{71} - 8 q^{81} + 32 q^{91}+O(q^{100}) \) 8 * q - 24 * q^11 - 16 * q^21 + 48 * q^51 + 24 * q^71 - 8 * q^81 + 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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