LMFDB - Embedded newform 2793.1.er.a.2180.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2793,1,Mod(59,2793)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2793, base_ring=CyclotomicField(126))

chi = DirichletCharacter(H, H._module([63, 39, 7]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2793.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2793 = 3 \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2793.er (of order $126$, degree $36$, 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:	$1.39388858028$
Analytic rank:	$0$
Dimension:	$36$
Coefficient field:	$\Q(\zeta_{63})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{36} - x^{33} + x^{27} - x^{24} + x^{18} - x^{12} + x^{9} - x^{3} + 1 $ x^36 - x^33 + x^27 - x^24 + x^18 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{126}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{126} - \cdots)$

Embedding invariants

Embedding label			2180.1
Root			$0.456211 + 0.889872i$ of defining polynomial
Character	$\chi$	$=$	2793.2180
Dual form			2793.1.er.a.2105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.980172 + 0.198146i) q^{3} +(-0.270840 + 0.962624i) q^{4} +(0.542546 - 0.840026i) q^{7} +(0.921476 - 0.388435i) q^{9} +(0.0747301 - 0.997204i) q^{12} +(0.636181 + 0.0317505i) q^{13} +(-0.853291 - 0.521435i) q^{16} +(0.222521 - 0.974928i) q^{19} +(-0.365341 + 0.930874i) q^{21} +(0.124344 - 0.992239i) q^{25} +(-0.826239 + 0.563320i) q^{27} +(0.661686 + 0.749781i) q^{28} -1.93815 q^{31} +(0.124344 + 0.992239i) q^{36} +(1.08453 - 1.59071i) q^{37} +(-0.629859 + 0.0949359i) q^{39} +(0.0459461 - 1.84238i) q^{43} +(0.939693 + 0.342020i) q^{48} +(-0.411287 - 0.911506i) q^{49} +(-0.202867 + 0.603804i) q^{52} +(-0.0249307 + 0.999689i) q^{57} +(0.0491382 + 0.491062i) q^{61} +(0.173648 - 0.984808i) q^{63} +(0.733052 - 0.680173i) q^{64} +(1.21863 + 1.45231i) q^{67} +(1.27406 + 0.653172i) q^{73} +(0.0747301 + 0.997204i) q^{75} +(0.878222 + 0.478254i) q^{76} +(0.623507 + 1.71307i) q^{79} +(0.698237 - 0.715867i) q^{81} +(-0.797133 - 0.603804i) q^{84} +(0.371829 - 0.517183i) q^{91} +(1.89973 - 0.384038i) q^{93} +(-1.37769 + 0.501437i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 3 q^{12} + 3 q^{13} + 6 q^{19} - 3 q^{21} - 3 q^{27} - 3 q^{43} - 36 q^{52} - 3 q^{61} - 3 q^{64} + 3 q^{67} + 3 q^{73} + 3 q^{75} - 3 q^{79} + 3 q^{91} + 6 q^{93}+O(q^{100}) $ 36 * q + 3 * q^12 + 3 * q^13 + 6 * q^19 - 3 * q^21 - 3 * q^27 - 3 * q^43 - 36 * q^52 - 3 * q^61 - 3 * q^64 + 3 * q^67 + 3 * q^73 + 3 * q^75 - 3 * q^79 + 3 * q^91 + 6 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2793\mathbb{Z}\right)^\times$.

$n$	$932$	$2110$	$2206$
$\chi(n)$	$-1$	$e\left(\frac{37}{42}\right)$	$e\left(\frac{7}{18}\right)$

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		−0.603804	−	0.797133i	$-0.706349\pi$
$2$		0			0		0.603804	+	0.797133i	$0.293651\pi$
$3$	−0.980172	+	0.198146i	−0.980172	+	0.198146i
$3$	−0.980172	+	0.198146i	−0.980172	+	0.198146i
$4$	−0.270840	+	0.962624i	−0.270840	+	0.962624i
$5$		0			0		0.749781	−	0.661686i	$-0.230159\pi$
$5$		0			0		−0.749781	+	0.661686i	$0.769841\pi$
$6$		0			0
$7$	0.542546	−	0.840026i	0.542546	−	0.840026i
$7$	0.542546	−	0.840026i	0.542546	−	0.840026i
$8$		0			0
$9$	0.921476	−	0.388435i	0.921476	−	0.388435i
$10$		0			0
$11$		0			0		0.955573	−	0.294755i	$-0.0952381\pi$
$11$		0			0		−0.955573	+	0.294755i	$0.904762\pi$
$12$	0.0747301	−	0.997204i	0.0747301	−	0.997204i
$13$	0.636181	+	0.0317505i	0.636181	+	0.0317505i	0.365341	−	0.930874i	$-0.380952\pi$
$13$	0.636181	+	0.0317505i	0.636181	+	0.0317505i	0.270840	+	0.962624i	$0.412698\pi$
$14$		0			0
$15$		0			0
$16$	−0.853291	−	0.521435i	−0.853291	−	0.521435i
$17$		0			0		0.962624	−	0.270840i	$-0.0873016\pi$
$17$		0			0		−0.962624	+	0.270840i	$0.912698\pi$
$18$		0			0
$19$	0.222521	−	0.974928i	0.222521	−	0.974928i
$19$	0.222521	−	0.974928i	0.222521	−	0.974928i
$20$		0			0
$21$	−0.365341	+	0.930874i	−0.365341	+	0.930874i
$22$		0			0
$23$		0			0		−0.698237	−	0.715867i	$-0.746032\pi$
$23$		0			0		0.698237	+	0.715867i	$0.253968\pi$
$24$		0			0
$25$	0.124344	−	0.992239i	0.124344	−	0.992239i
$26$		0			0
$27$	−0.826239	+	0.563320i	−0.826239	+	0.563320i
$28$	0.661686	+	0.749781i	0.661686	+	0.749781i
$29$		0			0		−0.0995678	−	0.995031i	$-0.531746\pi$
$29$		0			0		0.0995678	+	0.995031i	$0.468254\pi$
$30$		0			0
$31$	−1.93815			−1.93815			−0.969077	−	0.246757i	$-0.920635\pi$
$31$	−1.93815			−1.93815			−0.969077	+	0.246757i	$0.920635\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.124344	+	0.992239i	0.124344	+	0.992239i
$37$	1.08453	−	1.59071i	1.08453	−	1.59071i	0.318487	−	0.947927i	$-0.396825\pi$
$37$	1.08453	−	1.59071i	1.08453	−	1.59071i	0.766044	−	0.642788i	$-0.222222\pi$
$38$		0			0
$39$	−0.629859	+	0.0949359i	−0.629859	+	0.0949359i
$40$		0			0
$41$		0			0		−0.661686	−	0.749781i	$-0.730159\pi$
$41$		0			0		0.661686	+	0.749781i	$0.269841\pi$
$42$		0			0
$43$	0.0459461	−	1.84238i	0.0459461	−	1.84238i	−0.365341	−	0.930874i	$-0.619048\pi$
$43$	0.0459461	−	1.84238i	0.0459461	−	1.84238i	0.411287	−	0.911506i	$-0.365079\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.0498459	−	0.998757i	$-0.484127\pi$
$47$		0			0		−0.0498459	+	0.998757i	$0.515873\pi$
$48$	0.939693	+	0.342020i	0.939693	+	0.342020i
$49$	−0.411287	−	0.911506i	−0.411287	−	0.911506i
$50$		0			0
$51$		0			0
$52$	−0.202867	+	0.603804i	−0.202867	+	0.603804i
$53$		0			0		−0.962624	−	0.270840i	$-0.912698\pi$
$53$		0			0		0.962624	+	0.270840i	$0.0873016\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$58$		0			0
$59$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$59$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$60$		0			0
$61$	0.0491382	+	0.491062i	0.0491382	+	0.491062i	0.988831	+	0.149042i	$0.0476190\pi$
$61$	0.0491382	+	0.491062i	0.0491382	+	0.491062i	−0.939693	+	0.342020i	$0.888889\pi$
$62$		0			0
$63$	0.173648	−	0.984808i	0.173648	−	0.984808i
$64$	0.733052	−	0.680173i	0.733052	−	0.680173i
$65$		0			0
$66$		0			0
$67$	1.21863	+	1.45231i	1.21863	+	1.45231i	0.853291	+	0.521435i	$0.174603\pi$
$67$	1.21863	+	1.45231i	1.21863	+	1.45231i	0.365341	+	0.930874i	$0.380952\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.911506	−	0.411287i	$-0.134921\pi$
$71$		0			0		−0.911506	+	0.411287i	$0.865079\pi$
$72$		0			0
$73$	1.27406	+	0.653172i	1.27406	+	0.653172i	0.955573	−	0.294755i	$-0.0952381\pi$
$73$	1.27406	+	0.653172i	1.27406	+	0.653172i	0.318487	+	0.947927i	$0.396825\pi$
$74$		0			0
$75$	0.0747301	+	0.997204i	0.0747301	+	0.997204i
$76$	0.878222	+	0.478254i	0.878222	+	0.478254i
$77$		0			0
$78$		0			0
$79$	0.623507	+	1.71307i	0.623507	+	1.71307i	0.698237	+	0.715867i	$0.253968\pi$
$79$	0.623507	+	1.71307i	0.623507	+	1.71307i	−0.0747301	+	0.997204i	$0.523810\pi$
$80$		0			0
$81$	0.698237	−	0.715867i	0.698237	−	0.715867i
$82$		0			0
$83$		0			0		−0.680173	−	0.733052i	$-0.738095\pi$
$83$		0			0		0.680173	+	0.733052i	$0.261905\pi$
$84$	−0.797133	−	0.603804i	−0.797133	−	0.603804i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.797133	−	0.603804i	$-0.793651\pi$
$89$		0			0		0.797133	+	0.603804i	$0.206349\pi$
$90$		0			0
$91$	0.371829	−	0.517183i	0.371829	−	0.517183i
$92$		0			0
$93$	1.89973	−	0.384038i	1.89973	−	0.384038i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.37769	+	0.501437i	−1.37769	+	0.501437i	−0.921476	−	0.388435i	$-0.873016\pi$
$97$	−1.37769	+	0.501437i	−1.37769	+	0.501437i	−0.456211	+	0.889872i	$0.650794\pi$
$98$		0			0
$99$		0			0
$100$	0.921476	+	0.388435i	0.921476	+	0.388435i
$101$		0			0		−0.521435	−	0.853291i	$-0.674603\pi$
$101$		0			0		0.521435	+	0.853291i	$0.325397\pi$
$102$		0			0
$103$	1.24543	−	1.56172i	1.24543	−	1.56172i	0.583744	−	0.811938i	$-0.301587\pi$
$103$	1.24543	−	1.56172i	1.24543	−	1.56172i	0.661686	−	0.749781i	$-0.269841\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.294755	−	0.955573i	$-0.404762\pi$
$107$		0			0		−0.294755	+	0.955573i	$0.595238\pi$
$108$	−0.318487	−	0.947927i	−0.318487	−	0.947927i
$109$	1.84730	−	0.231497i	1.84730	−	0.231497i	0.878222	−	0.478254i	$-0.158730\pi$
$109$	1.84730	−	0.231497i	1.84730	−	0.231497i	0.969077	+	0.246757i	$0.0793651\pi$
$110$		0			0
$111$	−0.747834	+	1.77407i	−0.747834	+	1.77407i
$112$	−0.900969	+	0.433884i	−0.900969	+	0.433884i
$113$		0			0		0.974928	−	0.222521i	$-0.0714286\pi$
$113$		0			0		−0.974928	+	0.222521i	$0.928571\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.598559	−	0.217858i	0.598559	−	0.217858i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.826239	−	0.563320i	0.826239	−	0.563320i
$122$		0			0
$123$		0			0
$124$	0.524931	−	1.86571i	0.524931	−	1.86571i
$125$		0			0
$126$		0			0
$127$	−1.58316	+	1.13822i	−1.58316	+	1.13822i	−0.661686	+	0.749781i	$0.730159\pi$
$127$	−1.58316	+	1.13822i	−1.58316	+	1.13822i	−0.921476	+	0.388435i	$0.873016\pi$
$128$		0			0
$129$	0.320025	+	1.81495i	0.320025	+	1.81495i
$130$		0			0
$131$		0			0		−0.999689	−	0.0249307i	$-0.992063\pi$
$131$		0			0		0.999689	+	0.0249307i	$0.00793651\pi$
$132$		0			0
$133$	−0.698237	−	0.715867i	−0.698237	−	0.715867i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.661686	−	0.749781i	$-0.269841\pi$
$137$		0			0		−0.661686	+	0.749781i	$0.730159\pi$
$138$		0			0
$139$	0.717172	−	1.31695i	0.717172	−	1.31695i	−0.222521	−	0.974928i	$-0.571429\pi$
$139$	0.717172	−	1.31695i	0.717172	−	1.31695i	0.939693	−	0.342020i	$-0.111111\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$145$		0			0
$146$		0			0
$147$	0.583744	+	0.811938i	0.583744	+	0.811938i
$148$	1.23753	+	1.47483i	1.23753	+	1.47483i
$149$		0			0		−0.124344	−	0.992239i	$-0.539683\pi$
$149$		0			0		0.124344	+	0.992239i	$0.460317\pi$
$150$		0			0
$151$	−0.332083	−	0.487076i	−0.332083	−	0.487076i	0.623490	−	0.781831i	$-0.285714\pi$
$151$	−0.332083	−	0.487076i	−0.332083	−	0.487076i	−0.955573	+	0.294755i	$0.904762\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.0792036	−	0.632030i	0.0792036	−	0.632030i
$157$	−0.498757	+	0.816180i	−0.498757	+	0.816180i	−0.998757	−	0.0498459i	$-0.984127\pi$
$157$	−0.498757	+	0.816180i	−0.498757	+	0.816180i	0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0310881	+	0.0389832i	0.0310881	+	0.0389832i	0.797133	−	0.603804i	$-0.206349\pi$
$163$	0.0310881	+	0.0389832i	0.0310881	+	0.0389832i	−0.766044	+	0.642788i	$0.777778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.411287	−	0.911506i	$-0.365079\pi$
$167$		0			0		−0.411287	+	0.911506i	$0.634921\pi$
$168$		0			0
$169$	−0.591312	−	0.0591697i	−0.591312	−	0.0591697i
$170$		0			0
$171$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$172$	1.76108	+	0.543220i	1.76108	+	0.543220i
$173$		0			0		−0.411287	−	0.911506i	$-0.634921\pi$
$173$		0			0		0.411287	+	0.911506i	$0.365079\pi$
$174$		0			0
$175$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.563320	−	0.826239i	$-0.309524\pi$
$179$		0			0		−0.563320	+	0.826239i	$0.690476\pi$
$180$		0			0
$181$	0.0908557	−	0.725011i	0.0908557	−	0.725011i	−0.878222	−	0.478254i	$-0.841270\pi$
$181$	0.0908557	−	0.725011i	0.0908557	−	0.725011i	0.969077	−	0.246757i	$-0.0793651\pi$
$182$		0			0
$183$	−0.145466	−	0.471589i	−0.145466	−	0.471589i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.0249307	+	0.999689i	0.0249307	+	0.999689i
$190$		0			0
$191$		0			0		−0.988831	−	0.149042i	$-0.952381\pi$
$191$		0			0		0.988831	+	0.149042i	$0.0476190\pi$
$192$	−0.583744	+	0.811938i	−0.583744	+	0.811938i
$193$	0.396169	+	1.95974i	0.396169	+	1.95974i	0.222521	+	0.974928i	$0.428571\pi$
$193$	0.396169	+	1.95974i	0.396169	+	1.95974i	0.173648	+	0.984808i	$0.444444\pi$
$194$		0			0
$195$		0			0
$196$	0.988831	−	0.149042i	0.988831	−	0.149042i
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$	−1.98386	+	0.0494744i	−1.98386	+	0.0494744i	−0.995031	−	0.0995678i	$-0.968254\pi$
$199$	−1.98386	+	0.0494744i	−1.98386	+	0.0494744i	−0.988831	+	0.149042i	$0.952381\pi$
$200$		0			0
$201$	−1.48224	−	1.18205i	−1.48224	−	1.18205i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.526292	−	0.358820i	−0.526292	−	0.358820i
$209$		0			0
$210$		0			0
$211$	−0.167279	+	0.108040i	−0.167279	+	0.108040i	−0.623490	−	0.781831i	$-0.714286\pi$
$211$	−0.167279	+	0.108040i	−0.167279	+	0.108040i	0.456211	+	0.889872i	$0.349206\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.05154	+	1.62810i	−1.05154	+	1.62810i
$218$		0			0
$219$	−1.37822	−	0.387771i	−1.37822	−	0.387771i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.480172	+	1.06417i	0.480172	+	1.06417i	0.980172	+	0.198146i	$0.0634921\pi$
$223$	0.480172	+	1.06417i	0.480172	+	1.06417i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.270840	−	0.962624i	−0.270840	−	0.962624i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−0.955573	−	0.294755i	−0.955573	−	0.294755i
$229$	−1.65672	−	0.650213i	−1.65672	−	0.650213i	−0.661686	−	0.749781i	$-0.730159\pi$
$229$	−1.65672	−	0.650213i	−1.65672	−	0.650213i	−0.995031	+	0.0995678i	$0.968254\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.270840	−	0.962624i	$-0.587302\pi$
$233$		0			0		0.270840	+	0.962624i	$0.412698\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.950582	−	1.55556i	−0.950582	−	1.55556i
$238$		0			0
$239$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$239$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$240$		0			0
$241$	−1.22641	−	1.25738i	−1.22641	−	1.25738i	−0.955573	−	0.294755i	$-0.904762\pi$
$241$	−1.22641	−	1.25738i	−1.22641	−	1.25738i	−0.270840	−	0.962624i	$-0.587302\pi$
$242$		0			0
$243$	−0.542546	+	0.840026i	−0.542546	+	0.840026i
$244$	−0.486017	−	0.0856979i	−0.486017	−	0.0856979i
$245$		0			0
$246$		0			0
$247$	0.172518	−	0.613166i	0.172518	−	0.613166i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.198146	−	0.980172i	$-0.436508\pi$
$251$		0			0		−0.198146	+	0.980172i	$0.563492\pi$
$252$	0.900969	+	0.433884i	0.900969	+	0.433884i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.456211	+	0.889872i	0.456211	+	0.889872i
$257$		0			0		−0.583744	−	0.811938i	$-0.698413\pi$
$257$		0			0		0.583744	+	0.811938i	$0.301587\pi$
$258$		0			0
$259$	−0.747834	−	1.77407i	−0.747834	−	1.77407i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.72808	+	0.779741i	−1.72808	+	0.779741i
$269$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$269$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$270$		0			0
$271$	−0.242026	−	0.174005i	−0.242026	−	0.174005i	0.456211	−	0.889872i	$-0.349206\pi$
$271$	−0.242026	−	0.174005i	−0.242026	−	0.174005i	−0.698237	+	0.715867i	$0.746032\pi$
$272$		0			0
$273$	−0.261979	+	0.580605i	−0.261979	+	0.580605i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.63402	+	1.11406i	1.63402	+	1.11406i	0.900969	+	0.433884i	$0.142857\pi$
$277$	1.63402	+	1.11406i	1.63402	+	1.11406i	0.733052	+	0.680173i	$0.238095\pi$
$278$		0			0
$279$	−1.78596	+	0.752847i	−1.78596	+	0.752847i
$280$		0			0
$281$		0			0		0.889872	−	0.456211i	$-0.150794\pi$
$281$		0			0		−0.889872	+	0.456211i	$0.849206\pi$
$282$		0			0
$283$	0.337071	−	0.799627i	0.337071	−	0.799627i	−0.661686	−	0.749781i	$-0.730159\pi$
$283$	0.337071	−	0.799627i	0.337071	−	0.799627i	0.998757	−	0.0498459i	$-0.0158730\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.853291	−	0.521435i	0.853291	−	0.521435i
$290$		0			0
$291$	1.25101	−	0.764478i	1.25101	−	0.764478i
$292$	−0.973826	+	1.04954i	−0.973826	+	1.04954i
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.980172	−	0.198146i	−0.980172	−	0.198146i
$301$	−1.52272	−	1.03817i	−1.52272	−	1.03817i
$302$		0			0
$303$		0			0
$304$	−0.698237	+	0.715867i	−0.698237	+	0.715867i
$305$		0			0
$306$		0			0
$307$	1.76108	+	0.742355i	1.76108	+	0.742355i	0.995031	+	0.0995678i	$0.0317460\pi$
$307$	1.76108	+	0.742355i	1.76108	+	0.742355i	0.766044	+	0.642788i	$0.222222\pi$
$308$		0			0
$309$	−0.911287	+	1.77753i	−0.911287	+	1.77753i
$310$		0			0
$311$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$311$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$312$		0			0
$313$	1.96411	−	0.346325i	1.96411	−	0.346325i	0.969077	−	0.246757i	$-0.0793651\pi$
$313$	1.96411	−	0.346325i	1.96411	−	0.346325i	0.995031	−	0.0995678i	$-0.0317460\pi$
$314$		0			0
$315$		0			0
$316$	−1.81791	+	0.136234i	−1.81791	+	0.136234i
$317$		0			0		−0.246757	−	0.969077i	$-0.579365\pi$
$317$		0			0		0.246757	+	0.969077i	$0.420635\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.500000	+	0.866025i	0.500000	+	0.866025i
$325$	0.110609	−	0.627296i	0.110609	−	0.627296i
$326$		0			0
$327$	−1.76480	+	0.592942i	−1.76480	+	0.592942i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.523962	−	1.08802i	−0.523962	−	1.08802i	−0.980172	−	0.198146i	$-0.936508\pi$
$331$	−0.523962	−	1.08802i	−0.523962	−	1.08802i	0.456211	−	0.889872i	$-0.349206\pi$
$332$		0			0
$333$	0.381481	−	1.88708i	0.381481	−	1.88708i
$334$		0			0
$335$		0			0
$336$	0.797133	−	0.603804i	0.797133	−	0.603804i
$337$	−1.04255	−	0.0259995i	−1.04255	−	0.0259995i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.04255	−	0.0259995i	−1.04255	−	0.0259995i	−0.542546	+	0.840026i	$0.682540\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.988831	−	0.149042i	−0.988831	−	0.149042i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.995031	−	0.0995678i	$-0.968254\pi$
$347$		0			0		0.995031	+	0.0995678i	$0.0317460\pi$
$348$		0			0
$349$	1.56392	+	0.613792i	1.56392	+	0.613792i	0.980172	−	0.198146i	$-0.0634921\pi$
$349$	1.56392	+	0.613792i	1.56392	+	0.613792i	0.583744	+	0.811938i	$0.301587\pi$
$350$		0			0
$351$	−0.543524	+	0.332140i	−0.543524	+	0.332140i
$352$		0			0
$353$		0			0		−0.930874	−	0.365341i	$-0.880952\pi$
$353$		0			0		0.930874	+	0.365341i	$0.119048\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.0249307	−	0.999689i	$-0.492063\pi$
$359$		0			0		−0.0249307	+	0.999689i	$0.507937\pi$
$360$		0			0
$361$	−0.900969	−	0.433884i	−0.900969	−	0.433884i
$362$		0			0
$363$	−0.698237	+	0.715867i	−0.698237	+	0.715867i
$364$	0.397146	+	0.498006i	0.397146	+	0.498006i
$365$		0			0
$366$		0			0
$367$	−0.393217	−	0.0492765i	−0.393217	−	0.0492765i	−0.0747301	−	0.997204i	$-0.523810\pi$
$367$	−0.393217	−	0.0492765i	−0.393217	−	0.0492765i	−0.318487	+	0.947927i	$0.603175\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.144838	+	1.93274i	−0.144838	+	1.93274i
$373$	−1.17809	−	0.680173i	−1.17809	−	0.680173i	−0.222521	−	0.974928i	$-0.571429\pi$
$373$	−1.17809	−	0.680173i	−1.17809	−	0.680173i	−0.955573	+	0.294755i	$0.904762\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.757392	−	0.172870i	−0.757392	−	0.172870i	−0.173648	−	0.984808i	$-0.555556\pi$
$379$	−0.757392	−	0.172870i	−0.757392	−	0.172870i	−0.583744	+	0.811938i	$0.698413\pi$
$380$		0			0
$381$	1.32624	−	1.42935i	1.32624	−	1.42935i
$382$		0			0
$383$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$383$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.673306	−	1.71556i	−0.673306	−	1.71556i
$388$	−0.109562	−	1.46200i	−0.109562	−	1.46200i
$389$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$389$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.321765	+	1.59168i	0.321765	+	1.59168i	0.733052	+	0.680173i	$0.238095\pi$
$397$	0.321765	+	1.59168i	0.321765	+	1.59168i	−0.411287	+	0.911506i	$0.634921\pi$
$398$		0			0
$399$	0.826239	+	0.563320i	0.826239	+	0.563320i
$400$	−0.623490	+	0.781831i	−0.623490	+	0.781831i
$401$		0			0		−0.889872	−	0.456211i	$-0.849206\pi$
$401$		0			0		0.889872	+	0.456211i	$0.150794\pi$
$402$		0			0
$403$	−1.23302	−	0.0615374i	−1.23302	−	0.0615374i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.679643	+	1.50624i	−0.679643	+	1.50624i	0.173648	+	0.984808i	$0.444444\pi$
$409$	−0.679643	+	1.50624i	−0.679643	+	1.50624i	−0.853291	+	0.521435i	$0.825397\pi$
$410$		0			0
$411$		0			0
$412$	1.16604	+	1.62186i	1.16604	+	1.62186i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.442004	+	1.43294i	−0.442004	+	1.43294i
$418$		0			0
$419$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$419$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$420$		0			0
$421$	1.50522	−	0.423503i	1.50522	−	0.423503i	0.583744	−	0.811938i	$-0.301587\pi$
$421$	1.50522	−	0.423503i	1.50522	−	0.423503i	0.921476	+	0.388435i	$0.126984\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.439165	+	0.225147i	0.439165	+	0.225147i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.947927	−	0.318487i	$-0.103175\pi$
$431$		0			0		−0.947927	+	0.318487i	$0.896825\pi$
$432$	0.998757	−	0.0498459i	0.998757	−	0.0498459i
$433$	0.0227473	+	0.912138i	0.0227473	+	0.912138i	0.900969	+	0.433884i	$0.142857\pi$
$433$	0.0227473	+	0.912138i	0.0227473	+	0.912138i	−0.878222	+	0.478254i	$0.841270\pi$
$434$		0			0
$435$		0			0
$436$	−0.277479	+	1.84095i	−0.277479	+	1.84095i
$437$		0			0
$438$		0			0
$439$	−1.56265	+	1.18366i	−1.56265	+	1.18366i	−0.661686	+	0.749781i	$0.730159\pi$
$439$	−1.56265	+	1.18366i	−1.56265	+	1.18366i	−0.900969	+	0.433884i	$0.857143\pi$
$440$		0			0
$441$	−0.733052	−	0.680173i	−0.733052	−	0.680173i
$442$		0			0
$443$		0			0		−0.456211	−	0.889872i	$-0.650794\pi$
$443$		0			0		0.456211	+	0.889872i	$0.349206\pi$
$444$	−1.50522	−	1.20037i	−1.50522	−	1.20037i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$449$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$449$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.422011	+	0.411618i	0.422011	+	0.411618i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.510189	+	1.29994i	−0.510189	+	1.29994i	0.411287	+	0.911506i	$0.365079\pi$
$457$	−0.510189	+	1.29994i	−0.510189	+	1.29994i	−0.921476	+	0.388435i	$0.873016\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.246757	−	0.969077i	$-0.420635\pi$
$461$		0			0		−0.246757	+	0.969077i	$0.579365\pi$
$462$		0			0
$463$	−0.0988957	−	1.31967i	−0.0988957	−	1.31967i	−0.797133	−	0.603804i	$-0.793651\pi$
$463$	−0.0988957	−	1.31967i	−0.0988957	−	1.31967i	0.698237	−	0.715867i	$-0.253968\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.433884	−	0.900969i	$-0.357143\pi$
$467$		0			0		−0.433884	+	0.900969i	$0.642857\pi$
$468$	0.0476011	+	0.635192i	0.0476011	+	0.635192i
$469$	1.88114	−	0.235738i	1.88114	−	0.235738i
$470$		0			0
$471$	0.327145	−	0.898823i	0.327145	−	0.898823i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.0498459	−	0.998757i	$-0.484127\pi$
$479$		0			0		−0.0498459	+	0.998757i	$0.515873\pi$
$480$		0			0
$481$	0.740465	−	0.977549i	0.740465	−	0.977549i
$482$		0			0
$483$		0			0
$484$	0.318487	+	0.947927i	0.318487	+	0.947927i
$485$		0			0
$486$		0			0
$487$	0.880843	+	0.702449i	0.880843	+	0.702449i	0.955573	−	0.294755i	$-0.0952381\pi$
$487$	0.880843	+	0.702449i	0.880843	+	0.702449i	−0.0747301	+	0.997204i	$0.523810\pi$
$488$		0			0
$489$	−0.0381960	−	0.0320503i	−0.0381960	−	0.0320503i
$490$		0			0
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.65381	+	1.01062i	1.65381	+	1.01062i
$497$		0			0
$498$		0			0
$499$	−0.541008	+	0.0270006i	−0.541008	+	0.0270006i	−0.318487	−	0.947927i	$-0.603175\pi$
$499$	−0.541008	+	0.0270006i	−0.541008	+	0.0270006i	−0.222521	+	0.974928i	$0.571429\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.840026	−	0.542546i	$-0.817460\pi$
$503$		0			0		0.840026	+	0.542546i	$0.182540\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.591312	−	0.0591697i	0.591312	−	0.0591697i
$508$	−0.666890	−	1.83227i	−0.666890	−	1.83227i
$509$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$509$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$	1.23992	−	0.715867i	1.23992	−	0.715867i
$512$		0			0
$513$	0.365341	+	0.930874i	0.365341	+	0.930874i
$514$		0			0
$515$		0			0
$516$	−1.83379	−	0.183499i	−1.83379	−	0.183499i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−0.358423	+	0.406142i	−0.358423	+	0.406142i	−0.900969	−	0.433884i	$-0.857143\pi$
$523$	−0.358423	+	0.406142i	−0.358423	+	0.406142i	0.542546	+	0.840026i	$0.317460\pi$
$524$		0			0
$525$	0.878222	+	0.478254i	0.878222	+	0.478254i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.0249307	+	0.999689i	−0.0249307	+	0.999689i
$530$		0			0
$531$		0			0
$532$	0.878222	−	0.478254i	0.878222	−	0.478254i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.248378	+	0.0123960i	0.248378	+	0.0123960i	0.173648	−	0.984808i	$-0.444444\pi$
$541$	0.248378	+	0.0123960i	0.248378	+	0.0123960i	0.0747301	+	0.997204i	$0.476190\pi$
$542$		0			0
$543$	0.0546039	+	0.728639i	0.0546039	+	0.728639i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.62337	+	0.0404844i	−1.62337	+	0.0404844i	−0.826239	−	0.563320i	$-0.809524\pi$
$547$	−1.62337	+	0.0404844i	−1.62337	+	0.0404844i	−0.797133	+	0.603804i	$0.793651\pi$
$548$		0			0
$549$	0.236025	+	0.433415i	0.236025	+	0.433415i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.77730	+	0.405658i	1.77730	+	0.405658i
$554$		0			0
$555$		0			0
$556$	1.07349	+	1.04705i	1.07349	+	1.04705i
$557$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$557$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$558$		0			0
$559$	0.0877265	−	1.17063i	0.0877265	−	1.17063i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.222521	−	0.974928i	$-0.571429\pi$
$563$		0			0		0.222521	+	0.974928i	$0.428571\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.222521	−	0.974928i	−0.222521	−	0.974928i
$568$		0			0
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−0.0614710	+	0.820274i	−0.0614710	+	0.820274i	0.878222	+	0.478254i	$0.158730\pi$
$571$	−0.0614710	+	0.820274i	−0.0614710	+	0.820274i	−0.939693	+	0.342020i	$0.888889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.411287	−	0.911506i	0.411287	−	0.911506i
$577$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.900969	−	0.433884i	$-0.142857\pi$
$577$	1.52446	+	0.347948i	1.52446	+	0.347948i	0.623490	+	0.781831i	$0.285714\pi$
$578$		0			0
$579$	−0.776628	−	1.84238i	−0.776628	−	1.84238i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.342020	−	0.939693i	$-0.611111\pi$
$587$		0			0		0.342020	+	0.939693i	$0.388889\pi$
$588$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$589$	−0.431280	+	1.88956i	−0.431280	+	1.88956i
$590$		0			0
$591$		0			0
$592$	−1.75488	+	0.791830i	−1.75488	+	0.791830i
$593$		0			0		−0.521435	−	0.853291i	$-0.674603\pi$
$593$		0			0		0.521435	+	0.853291i	$0.325397\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.93472	−	0.441588i	1.93472	−	0.441588i
$598$		0			0
$599$		0			0		0.889872	−	0.456211i	$-0.150794\pi$
$599$		0			0		−0.889872	+	0.456211i	$0.849206\pi$
$600$		0			0
$601$	−0.331867	+	0.102367i	−0.331867	+	0.102367i	−0.456211	−	0.889872i	$-0.650794\pi$
$601$	−0.331867	+	0.102367i	−0.331867	+	0.102367i	0.124344	+	0.992239i	$0.460317\pi$
$602$		0			0
$603$	1.68707	+	0.864909i	1.68707	+	0.864909i
$604$	0.558813	−	0.187751i	0.558813	−	0.187751i
$605$		0			0
$606$		0			0
$607$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$607$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.896546	−	1.38812i	−0.896546	−	1.38812i	−0.921476	−	0.388435i	$-0.873016\pi$
$613$	−0.896546	−	1.38812i	−0.896546	−	1.38812i	0.0249307	−	0.999689i	$-0.492063\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$617$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$618$		0			0
$619$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$619$	1.11334	−	0.642788i	1.11334	−	0.642788i	0.939693	+	0.342020i	$0.111111\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.586956	+	0.247423i	0.586956	+	0.247423i
$625$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$626$		0			0
$627$		0			0
$628$	−0.650591	−	0.701170i	−0.650591	−	0.701170i
$629$		0			0
$630$		0			0
$631$	−1.95060	−	0.394323i	−1.95060	−	0.394323i	−0.995031	−	0.0995678i	$-0.968254\pi$
$631$	−1.95060	−	0.394323i	−1.95060	−	0.394323i	−0.955573	−	0.294755i	$-0.904762\pi$
$632$		0			0
$633$	0.142555	−	0.139044i	0.142555	−	0.139044i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.232712	−	0.592942i	−0.232712	−	0.592942i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.0995678	−	0.995031i	$-0.468254\pi$
$641$		0			0		−0.0995678	+	0.995031i	$0.531746\pi$
$642$		0			0
$643$	−0.498757	−	0.915871i	−0.498757	−	0.915871i	−0.998757	−	0.0498459i	$-0.984127\pi$
$643$	−0.498757	−	0.915871i	−0.498757	−	0.915871i	0.500000	−	0.866025i	$-0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.149042	−	0.988831i	$-0.547619\pi$
$647$		0			0		0.149042	+	0.988831i	$0.452381\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.708087	−	1.80418i	0.708087	−	1.80418i
$652$	−0.0459461	+	0.0193679i	−0.0459461	+	0.0193679i
$653$		0			0		0.988831	−	0.149042i	$-0.0476190\pi$
$653$		0			0		−0.988831	+	0.149042i	$0.952381\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	1.42773	+	0.106994i	1.42773	+	0.106994i
$658$		0			0
$659$		0			0		−0.962624	−	0.270840i	$-0.912698\pi$
$659$		0			0		0.962624	+	0.270840i	$0.0873016\pi$
$660$		0			0
$661$	−0.921476	+	0.388435i	−0.921476	+	0.388435i	−0.797133	−	0.603804i	$-0.793651\pi$
$661$	−0.921476	+	0.388435i	−0.921476	+	0.388435i	−0.124344	+	0.992239i	$0.539683\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.681513	−	0.947927i	−0.681513	−	0.947927i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.269547	−	0.290503i	0.269547	−	0.290503i	−0.583744	−	0.811938i	$-0.698413\pi$
$673$	0.269547	−	0.290503i	0.269547	−	0.290503i	0.853291	+	0.521435i	$0.174603\pi$
$674$		0			0
$675$	0.456211	+	0.889872i	0.456211	+	0.889872i
$676$	0.217109	−	0.553186i	0.217109	−	0.553186i
$677$		0			0		−0.955573	−	0.294755i	$-0.904762\pi$
$677$		0			0		0.955573	+	0.294755i	$0.0952381\pi$
$678$		0			0
$679$	−0.326239	+	1.42935i	−0.326239	+	1.42935i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.930874	−	0.365341i	$-0.119048\pi$
$683$		0			0		−0.930874	+	0.365341i	$0.880952\pi$
$684$	0.995031	+	0.0995678i	0.995031	+	0.0995678i
$685$		0			0
$686$		0			0
$687$	1.75271	+	0.309049i	1.75271	+	0.309049i
$688$	−0.999887	+	1.54813i	−0.999887	+	1.54813i
$689$		0			0
$690$		0			0
$691$	−1.61232	+	0.632789i	−1.61232	+	0.632789i	−0.988831	−	0.149042i	$-0.952381\pi$
$691$	−1.61232	+	0.632789i	−1.61232	+	0.632789i	−0.623490	+	0.781831i	$0.714286\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.826239	−	0.563320i	0.826239	−	0.563320i
$701$		0			0		−0.542546	−	0.840026i	$-0.682540\pi$
$701$		0			0		0.542546	+	0.840026i	$0.317460\pi$
$702$		0			0
$703$	−1.30950	−	1.41131i	−1.30950	−	1.41131i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.884207	−	0.225147i	0.884207	−	0.225147i	0.222521	−	0.974928i	$-0.428571\pi$
$709$	0.884207	−	0.225147i	0.884207	−	0.225147i	0.661686	+	0.749781i	$0.269841\pi$
$710$		0			0
$711$	1.23996	+	1.33636i	1.23996	+	1.33636i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.521435	−	0.853291i	$-0.325397\pi$
$719$		0			0		−0.521435	+	0.853291i	$0.674603\pi$
$720$		0			0
$721$	−0.636181	−	1.89350i	−0.636181	−	1.89350i
$722$		0			0
$723$	1.45124	+	0.989440i	1.45124	+	0.989440i
$724$	0.673306	+	0.283822i	0.673306	+	0.283822i
$725$		0			0
$726$		0			0
$727$	−0.135540	−	0.670477i	−0.135540	−	0.670477i	−0.988831	−	0.149042i	$-0.952381\pi$
$727$	−0.135540	−	0.670477i	−0.135540	−	0.670477i	0.853291	−	0.521435i	$-0.174603\pi$
$728$		0			0
$729$	0.365341	−	0.930874i	0.365341	−	0.930874i
$730$		0			0
$731$		0			0
$732$	0.493361	−	0.0123037i	0.493361	−	0.0123037i
$733$	1.32624	+	0.302705i	1.32624	+	0.302705i	0.826239	−	0.563320i	$-0.190476\pi$
$733$	1.32624	+	0.302705i	1.32624	+	0.302705i	0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.14434	−	1.59168i	1.14434	−	1.59168i	0.411287	−	0.911506i	$-0.365079\pi$
$739$	1.14434	−	1.59168i	1.14434	−	1.59168i	0.733052	−	0.680173i	$-0.238095\pi$
$740$		0			0
$741$	−0.0476011	+	0.635192i	−0.0476011	+	0.635192i
$742$		0			0
$743$		0			0		−0.947927	−	0.318487i	$-0.896825\pi$
$743$		0			0		0.947927	+	0.318487i	$0.103175\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0747470	+	0.0659646i	0.0747470	+	0.0659646i	0.698237	−	0.715867i	$-0.253968\pi$
$751$	0.0747470	+	0.0659646i	0.0747470	+	0.0659646i	−0.623490	+	0.781831i	$0.714286\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.969077	−	0.246757i	−0.969077	−	0.246757i
$757$	0.996206	−	1.38564i	0.996206	−	1.38564i	0.0747301	−	0.997204i	$-0.476190\pi$
$757$	0.996206	−	1.38564i	0.996206	−	1.38564i	0.921476	−	0.388435i	$-0.126984\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.997204	−	0.0747301i	$-0.0238095\pi$
$761$		0			0		−0.997204	+	0.0747301i	$0.976190\pi$
$762$		0			0
$763$	0.807782	−	1.67738i	0.807782	−	1.67738i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.623490	−	0.781831i	−0.623490	−	0.781831i
$769$	−1.95433	−	0.244909i	−1.95433	−	0.244909i	−0.955573	−	0.294755i	$-0.904762\pi$
$769$	−1.95433	−	0.244909i	−1.95433	−	0.244909i	−0.998757	+	0.0498459i	$0.984127\pi$
$770$		0			0
$771$		0			0
$772$	−1.99379	−	0.149414i	−1.99379	−	0.149414i
$773$		0			0		0.542546	−	0.840026i	$-0.317460\pi$
$773$		0			0		−0.542546	+	0.840026i	$0.682540\pi$
$774$		0			0
$775$	−0.240997	+	1.92311i	−0.240997	+	1.92311i
$776$		0			0
$777$	1.08453	+	1.59071i	1.08453	+	1.59071i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.124344	+	0.992239i	−0.124344	+	0.992239i
$785$		0			0
$786$		0			0
$787$	0.813387	+	0.122598i	0.813387	+	0.122598i	0.542546	−	0.840026i	$-0.317460\pi$
$787$	0.813387	+	0.122598i	0.813387	+	0.122598i	0.270840	+	0.962624i	$0.412698\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.0156693	+	0.313965i	0.0156693	+	0.313965i
$794$		0			0
$795$		0			0
$796$	0.489685	−	1.92311i	0.489685	−	1.92311i
$797$		0			0		0.222521	−	0.974928i	$-0.428571\pi$
$797$		0			0		−0.222521	+	0.974928i	$0.571429\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.53932	−	1.10669i	1.53932	−	1.10669i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.733052	−	0.680173i	$-0.238095\pi$
$809$		0			0		−0.733052	+	0.680173i	$0.761905\pi$
$810$		0			0
$811$	0.535631	−	1.90375i	0.535631	−	1.90375i	0.124344	−	0.992239i	$-0.460317\pi$
$811$	0.535631	−	1.90375i	0.535631	−	1.90375i	0.411287	−	0.911506i	$-0.365079\pi$
$812$		0			0
$813$	0.271706	+	0.122598i	0.271706	+	0.122598i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.78596	−	0.454762i	−1.78596	−	0.454762i
$818$		0			0
$819$	0.141740	−	0.621003i	0.141740	−	0.621003i
$820$		0			0
$821$		0			0		0.995031	−	0.0995678i	$-0.0317460\pi$
$821$		0			0		−0.995031	+	0.0995678i	$0.968254\pi$
$822$		0			0
$823$	0.608674	+	1.81163i	0.608674	+	1.81163i	0.583744	+	0.811938i	$0.301587\pi$
$823$	0.608674	+	1.81163i	0.608674	+	1.81163i	0.0249307	+	0.999689i	$0.492063\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.947927	−	0.318487i	$-0.103175\pi$
$827$		0			0		−0.947927	+	0.318487i	$0.896825\pi$
$828$		0			0
$829$	−0.741114	−	0.356902i	−0.741114	−	0.356902i	0.0249307	−	0.999689i	$-0.492063\pi$
$829$	−0.741114	−	0.356902i	−0.741114	−	0.356902i	−0.766044	+	0.642788i	$0.777778\pi$
$830$		0			0
$831$	−1.82237	−	0.768193i	−1.82237	−	0.768193i
$832$	0.487950	−	0.409439i	0.487950	−	0.409439i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.60138	−	1.09180i	1.60138	−	1.09180i
$838$		0			0
$839$		0			0		0.878222	−	0.478254i	$-0.158730\pi$
$839$		0			0		−0.878222	+	0.478254i	$0.841270\pi$
$840$		0			0
$841$	−0.980172	+	0.198146i	−0.980172	+	0.198146i
$842$		0			0
$843$		0			0
$844$	−0.0586963	−	0.190289i	−0.0586963	−	0.190289i
$845$		0			0
$846$		0			0
$847$	−0.0249307	−	0.999689i	−0.0249307	−	0.999689i
$848$		0			0
$849$	−0.171945	+	0.850562i	−0.171945	+	0.850562i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.556135	+	0.542439i	0.556135	+	0.542439i	0.921476	−	0.388435i	$-0.126984\pi$
$853$	0.556135	+	0.542439i	0.556135	+	0.542439i	−0.365341	+	0.930874i	$0.619048\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.998757	−	0.0498459i	$-0.0158730\pi$
$857$		0			0		−0.998757	+	0.0498459i	$0.984127\pi$
$858$		0			0
$859$	−0.556135	+	0.542439i	−0.556135	+	0.542439i	−0.921476	−	0.388435i	$-0.873016\pi$
$859$	−0.556135	+	0.542439i	−0.556135	+	0.542439i	0.365341	+	0.930874i	$0.380952\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.733052	+	0.680173i	−0.733052	+	0.680173i
$868$	−1.28245	−	1.45319i	−1.28245	−	1.45319i
$869$		0			0
$870$		0			0
$871$	0.729160	+	0.962624i	0.729160	+	0.962624i
$872$		0			0
$873$	−1.07473	+	0.997204i	−1.07473	+	0.997204i
$874$		0			0
$875$		0			0
$876$	0.746556	−	1.22169i	0.746556	−	1.22169i
$877$	0.0945006	+	0.0317505i	0.0945006	+	0.0317505i	0.365341	−	0.930874i	$-0.380952\pi$
$877$	0.0945006	+	0.0317505i	0.0945006	+	0.0317505i	−0.270840	+	0.962624i	$0.587302\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$881$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$882$		0			0
$883$	1.60366	+	0.583685i	1.60366	+	0.583685i	0.980172	−	0.198146i	$-0.0634921\pi$
$883$	1.60366	+	0.583685i	1.60366	+	0.583685i	0.623490	+	0.781831i	$0.285714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.0249307	−	0.999689i	$-0.507937\pi$
$887$		0			0		0.0249307	+	0.999689i	$0.492063\pi$
$888$		0			0
$889$	0.0971923	+	1.94743i	0.0971923	+	1.94743i
$890$		0			0
$891$		0			0
$892$	−1.15445	+	0.174005i	−1.15445	+	0.174005i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$		0			0
$903$	1.69824	+	0.715867i	1.69824	+	0.715867i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.250399	−	0.161725i	0.250399	−	0.161725i	−0.411287	−	0.911506i	$-0.634921\pi$
$907$	0.250399	−	0.161725i	0.250399	−	0.161725i	0.661686	+	0.749781i	$0.269841\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.433884	−	0.900969i	$-0.642857\pi$
$911$		0			0		0.433884	+	0.900969i	$0.357143\pi$
$912$	0.542546	−	0.840026i	0.542546	−	0.840026i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.07462	−	1.41869i	1.07462	−	1.41869i
$917$		0			0
$918$		0			0
$919$	0.146497	−	1.95486i	0.146497	−	1.95486i	−0.124344	−	0.992239i	$-0.539683\pi$
$919$	0.146497	−	1.95486i	0.146497	−	1.95486i	0.270840	−	0.962624i	$-0.412698\pi$
$920$		0			0
$921$	−1.87325	−	0.378686i	−1.87325	−	0.378686i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.44352	−	1.27391i	−1.44352	−	1.27391i
$926$		0			0
$927$	0.541008	−	1.92286i	0.541008	−	1.92286i
$928$		0			0
$929$		0			0		−0.603804	−	0.797133i	$-0.706349\pi$
$929$		0			0		0.603804	+	0.797133i	$0.293651\pi$
$930$		0			0
$931$	−0.980172	+	0.198146i	−0.980172	+	0.198146i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.905442	−	0.799058i	−0.905442	−	0.799058i	0.0747301	−	0.997204i	$-0.476190\pi$
$937$	−0.905442	−	0.799058i	−0.905442	−	0.799058i	−0.980172	+	0.198146i	$0.936508\pi$
$938$		0			0
$939$	−1.85654	+	0.728639i	−1.85654	+	0.728639i
$940$		0			0
$941$		0			0		−0.980172	−	0.198146i	$-0.936508\pi$
$941$		0			0		0.980172	+	0.198146i	$0.0634921\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.853291	−	0.521435i	$-0.825397\pi$
$947$		0			0		0.853291	+	0.521435i	$0.174603\pi$
$948$	1.75488	−	0.493745i	1.75488	−	0.493745i
$949$	0.789795	+	0.455988i	0.789795	+	0.455988i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.811938	−	0.583744i	$-0.801587\pi$
$953$		0			0		0.811938	+	0.583744i	$0.198413\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.75644			2.75644
$962$		0			0
$963$		0			0
$964$	1.54255	−	0.840026i	1.54255	−	0.840026i
$965$		0			0
$966$		0			0
$967$	0.113454	+	0.905340i	0.113454	+	0.905340i	0.939693	+	0.342020i	$0.111111\pi$
$967$	0.113454	+	0.905340i	0.113454	+	0.905340i	−0.826239	+	0.563320i	$0.809524\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.456211	−	0.889872i	$-0.349206\pi$
$971$		0			0		−0.456211	+	0.889872i	$0.650794\pi$
$972$	−0.661686	−	0.749781i	−0.661686	−	0.749781i
$973$	−0.717172	−	1.31695i	−0.717172	−	1.31695i
$974$		0			0
$975$	0.0158802	+	0.636775i	0.0158802	+	0.636775i
$976$	0.214128	−	0.444641i	0.214128	−	0.444641i
$977$		0			0		0.149042	−	0.988831i	$-0.452381\pi$
$977$		0			0		−0.149042	+	0.988831i	$0.547619\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	1.61232	−	0.930874i	1.61232	−	0.930874i
$982$		0			0
$983$		0			0		0.318487	−	0.947927i	$-0.396825\pi$
$983$		0			0		−0.318487	+	0.947927i	$0.603175\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.543524	+	0.332140i	0.543524	+	0.332140i
$989$		0			0
$990$		0			0
$991$	1.01442	+	1.33922i	1.01442	+	1.33922i	0.939693	+	0.342020i	$0.111111\pi$
$991$	1.01442	+	1.33922i	1.01442	+	1.33922i	0.0747301	+	0.997204i	$0.476190\pi$
$992$		0			0
$993$	0.729160	+	0.962624i	0.729160	+	0.962624i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.79532	+	0.810077i	1.79532	+	0.810077i	0.969077	+	0.246757i	$0.0793651\pi$
$997$	1.79532	+	0.810077i	1.79532	+	0.810077i	0.826239	+	0.563320i	$0.190476\pi$
$998$		0			0
$999$			1.92525i			1.92525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2793.1.er.a.2180.1	yes	36
3.2	odd	2	CM	2793.1.er.a.2180.1	yes	36
19.15	odd	18		2793.1.ew.a.1592.1	yes	36
49.47	odd	42		2793.1.ew.a.2693.1	yes	36
57.53	even	18		2793.1.ew.a.1592.1	yes	36
147.47	even	42		2793.1.ew.a.2693.1	yes	36
931.243	even	126	inner	2793.1.er.a.2105.1	✓	36
2793.2105	odd	126	inner	2793.1.er.a.2105.1	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.1.er.a.2105.1	✓	36	931.243	even	126	inner
2793.1.er.a.2105.1	✓	36	2793.2105	odd	126	inner
2793.1.er.a.2180.1	yes	36	1.1	even	1	trivial
2793.1.er.a.2180.1	yes	36	3.2	odd	2	CM
2793.1.ew.a.1592.1	yes	36	19.15	odd	18
2793.1.ew.a.1592.1	yes	36	57.53	even	18
2793.1.ew.a.2693.1	yes	36	49.47	odd	42
2793.1.ew.a.2693.1	yes	36	147.47	even	42

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:	\( N \)	\(=\)	\( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2793.er (of order \(126\), degree \(36\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.980172 + 0.198146i) q^{3} +(-0.270840 + 0.962624i) q^{4} +(0.542546 - 0.840026i) q^{7} +(0.921476 - 0.388435i) q^{9} +(0.0747301 - 0.997204i) q^{12} +(0.636181 + 0.0317505i) q^{13} +(-0.853291 - 0.521435i) q^{16} +(0.222521 - 0.974928i) q^{19} +(-0.365341 + 0.930874i) q^{21} +(0.124344 - 0.992239i) q^{25} +(-0.826239 + 0.563320i) q^{27} +(0.661686 + 0.749781i) q^{28} -1.93815 q^{31} +(0.124344 + 0.992239i) q^{36} +(1.08453 - 1.59071i) q^{37} +(-0.629859 + 0.0949359i) q^{39} +(0.0459461 - 1.84238i) q^{43} +(0.939693 + 0.342020i) q^{48} +(-0.411287 - 0.911506i) q^{49} +(-0.202867 + 0.603804i) q^{52} +(-0.0249307 + 0.999689i) q^{57} +(0.0491382 + 0.491062i) q^{61} +(0.173648 - 0.984808i) q^{63} +(0.733052 - 0.680173i) q^{64} +(1.21863 + 1.45231i) q^{67} +(1.27406 + 0.653172i) q^{73} +(0.0747301 + 0.997204i) q^{75} +(0.878222 + 0.478254i) q^{76} +(0.623507 + 1.71307i) q^{79} +(0.698237 - 0.715867i) q^{81} +(-0.797133 - 0.603804i) q^{84} +(0.371829 - 0.517183i) q^{91} +(1.89973 - 0.384038i) q^{93} +(-1.37769 + 0.501437i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 3 q^{12} + 3 q^{13} + 6 q^{19} - 3 q^{21} - 3 q^{27} - 3 q^{43} - 36 q^{52} - 3 q^{61} - 3 q^{64} + 3 q^{67} + 3 q^{73} + 3 q^{75} - 3 q^{79} + 3 q^{91} + 6 q^{93}+O(q^{100}) \) 36 * q + 3 * q^12 + 3 * q^13 + 6 * q^19 - 3 * q^21 - 3 * q^27 - 3 * q^43 - 36 * q^52 - 3 * q^61 - 3 * q^64 + 3 * q^67 + 3 * q^73 + 3 * q^75 - 3 * q^79 + 3 * q^91 + 6 * q^93

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