LMFDB - Embedded newform 1638.2.c.i.883.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,2,Mod(883,1638)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1638.883"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1638.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-6,0,0,0,0,0,-4,0,0,-10,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.0794958511$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.30647296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 8x^{3} + 25x^{2} - 10x + 2 $ x^6 - 2x^5 + 2x^4 + 8x^3 + 25x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.3
Root			$-1.27280 - 1.27280i$ of defining polynomial
Character	$\chi$	$=$	1638.883
Dual form			1638.2.c.i.883.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +2.54561i q^{5} -1.00000i q^{7} +1.00000i q^{8} +2.54561 q^{10} -3.33127i q^{11} +(-3.27280 + 1.51286i) q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.09122 q^{17} +4.54561i q^{19} -2.54561i q^{20} -3.33127 q^{22} +1.75994 q^{23} -1.48012 q^{25} +(1.51286 + 3.27280i) q^{26} +1.00000i q^{28} -7.57133 q^{29} +3.33127i q^{31} -1.00000i q^{32} -7.09122i q^{34} +2.54561 q^{35} -3.75994i q^{37} +4.54561 q^{38} -2.54561 q^{40} +4.24006i q^{41} +9.09122 q^{43} +3.33127i q^{44} -1.75994i q^{46} +11.3313i q^{47} -1.00000 q^{49} +1.48012i q^{50} +(3.27280 - 1.51286i) q^{52} +6.00000 q^{53} +8.48012 q^{55} +1.00000 q^{56} +7.57133i q^{58} +8.06549i q^{59} -0.785667 q^{61} +3.33127 q^{62} -1.00000 q^{64} +(-3.85116 - 8.33127i) q^{65} +5.75994i q^{67} -7.09122 q^{68} -2.54561i q^{70} +11.1427i q^{71} +9.33127i q^{73} -3.75994 q^{74} -4.54561i q^{76} -3.33127 q^{77} +4.85116 q^{79} +2.54561i q^{80} +4.24006 q^{82} +11.2082i q^{83} +18.0515i q^{85} -9.09122i q^{86} +3.33127 q^{88} -15.0912i q^{89} +(1.51286 + 3.27280i) q^{91} -1.75994 q^{92} +11.3313 q^{94} -11.5713 q^{95} -7.75994i q^{97} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{10} - 10 q^{13} - 6 q^{14} + 6 q^{16} + 4 q^{17} + 14 q^{22} + 6 q^{23} - 18 q^{25} + 4 q^{26} - 16 q^{29} - 4 q^{35} + 8 q^{38} + 4 q^{40} + 16 q^{43} - 6 q^{49} + 10 q^{52} + 36 q^{53}+ \cdots - 40 q^{95}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^10 - 10 * q^13 - 6 * q^14 + 6 * q^16 + 4 * q^17 + 14 * q^22 + 6 * q^23 - 18 * q^25 + 4 * q^26 - 16 * q^29 - 4 * q^35 + 8 * q^38 + 4 * q^40 + 16 * q^43 - 6 * q^49 + 10 * q^52 + 36 * q^53 + 60 * q^55 + 6 * q^56 + 10 * q^61 - 14 * q^62 - 6 * q^64 + 20 * q^65 - 4 * q^68 - 18 * q^74 + 14 * q^77 - 14 * q^79 + 30 * q^82 - 14 * q^88 + 4 * q^91 - 6 * q^92 + 34 * q^94 - 40 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times$.

$n$	$379$	$703$	$911$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			2.54561i			1.13843i	0.822189	+	0.569215i	$0.192753\pi$
$5$			2.54561i			1.13843i	−0.822189	+	0.569215i	$0.807247\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.54561			0.804992
$11$		−	3.33127i		−	1.00442i	−0.864747	−	0.502209i	$-0.832521\pi$
$11$		−	3.33127i		−	1.00442i	0.864747	−	0.502209i	$-0.167479\pi$
$12$		0			0
$13$	−3.27280	+	1.51286i	−0.907712	+	0.419593i
$13$	−3.27280	+	1.51286i	−0.907712	+	0.419593i
$14$	−1.00000			−0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$	7.09122			1.71987			0.859936	−	0.510402i	$-0.170503\pi$
$17$	7.09122			1.71987			0.859936	+	0.510402i	$0.170503\pi$
$18$		0			0
$19$			4.54561i			1.04283i	0.853302	+	0.521417i	$0.174596\pi$
$19$			4.54561i			1.04283i	−0.853302	+	0.521417i	$0.825404\pi$
$20$		−	2.54561i		−	0.569215i
$21$		0			0
$22$	−3.33127			−0.710230
$23$	1.75994			0.366973			0.183487	−	0.983022i	$-0.441262\pi$
$23$	1.75994			0.366973			0.183487	+	0.983022i	$0.441262\pi$
$24$		0			0
$25$	−1.48012			−0.296024
$26$	1.51286	+	3.27280i	0.296697	+	0.641850i
$27$		0			0
$28$			1.00000i			0.188982i
$29$	−7.57133			−1.40596			−0.702981	−	0.711209i	$-0.748148\pi$
$29$	−7.57133			−1.40596			−0.702981	+	0.711209i	$0.748148\pi$
$30$		0			0
$31$			3.33127i			0.598315i	0.954204	+	0.299157i	$0.0967055\pi$
$31$			3.33127i			0.598315i	−0.954204	+	0.299157i	$0.903294\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	7.09122i		−	1.21613i
$35$	2.54561			0.430286
$36$		0			0
$37$		−	3.75994i		−	0.618130i	−0.951041	−	0.309065i	$-0.899984\pi$
$37$		−	3.75994i		−	0.618130i	0.951041	−	0.309065i	$-0.100016\pi$
$38$	4.54561			0.737395
$39$		0			0
$40$	−2.54561			−0.402496
$41$			4.24006i			0.662186i	0.943598	+	0.331093i	$0.107417\pi$
$41$			4.24006i			0.662186i	−0.943598	+	0.331093i	$0.892583\pi$
$42$		0			0
$43$	9.09122			1.38640			0.693199	−	0.720747i	$-0.256201\pi$
$43$	9.09122			1.38640			0.693199	+	0.720747i	$0.256201\pi$
$44$			3.33127i			0.502209i
$45$		0			0
$46$		−	1.75994i		−	0.259489i
$47$			11.3313i			1.65284i	0.563057	+	0.826418i	$0.309625\pi$
$47$			11.3313i			1.65284i	−0.563057	+	0.826418i	$0.690375\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$			1.48012i			0.209320i
$51$		0			0
$52$	3.27280	−	1.51286i	0.453856	−	0.209796i
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	8.48012			1.14346
$56$	1.00000			0.133631
$57$		0			0
$58$			7.57133i			0.994165i
$59$			8.06549i			1.05004i	0.851091	+	0.525019i	$0.175942\pi$
$59$			8.06549i			1.05004i	−0.851091	+	0.525019i	$0.824058\pi$
$60$		0			0
$61$	−0.785667			−0.100594			−0.0502972	−	0.998734i	$-0.516017\pi$
$61$	−0.785667			−0.100594			−0.0502972	+	0.998734i	$0.516017\pi$
$62$	3.33127			0.423072
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−3.85116	−	8.33127i	−0.477677	−	1.03337i
$66$		0			0
$67$			5.75994i			0.703689i	0.936059	+	0.351844i	$0.114445\pi$
$67$			5.75994i			0.703689i	−0.936059	+	0.351844i	$0.885555\pi$
$68$	−7.09122			−0.859936
$69$		0			0
$70$		−	2.54561i		−	0.304258i
$71$			11.1427i			1.32239i	0.750213	+	0.661196i	$0.229951\pi$
$71$			11.1427i			1.32239i	−0.750213	+	0.661196i	$0.770049\pi$
$72$		0			0
$73$			9.33127i			1.09214i	0.837739	+	0.546072i	$0.183877\pi$
$73$			9.33127i			1.09214i	−0.837739	+	0.546072i	$0.816123\pi$
$74$	−3.75994			−0.437084
$75$		0			0
$76$		−	4.54561i		−	0.521417i
$77$	−3.33127			−0.379634
$78$		0			0
$79$	4.85116			0.545798			0.272899	−	0.962043i	$-0.412018\pi$
$79$	4.85116			0.545798			0.272899	+	0.962043i	$0.412018\pi$
$80$			2.54561i			0.284608i
$81$		0			0
$82$	4.24006			0.468236
$83$			11.2082i			1.23026i	0.788428	+	0.615128i	$0.210895\pi$
$83$			11.2082i			1.23026i	−0.788428	+	0.615128i	$0.789105\pi$
$84$		0			0
$85$			18.0515i			1.95795i
$86$		−	9.09122i		−	0.980331i
$87$		0			0
$88$	3.33127			0.355115
$89$		−	15.0912i		−	1.59967i	−0.600223	−	0.799833i	$-0.704921\pi$
$89$		−	15.0912i		−	1.59967i	0.600223	−	0.799833i	$-0.295079\pi$
$90$		0			0
$91$	1.51286	+	3.27280i	0.158591	+	0.343083i
$92$	−1.75994			−0.183487
$93$		0			0
$94$	11.3313			1.16873
$95$	−11.5713			−1.18719
$96$		0			0
$97$		−	7.75994i		−	0.787903i	−0.919131	−	0.393951i	$-0.871108\pi$
$97$		−	7.75994i		−	0.787903i	0.919131	−	0.393951i	$-0.128892\pi$
$98$			1.00000i			0.101015i
$99$		0			0
$100$	1.48012			0.148012
$101$	3.69445			0.367612			0.183806	−	0.982963i	$-0.441158\pi$
$101$	3.69445			0.367612			0.183806	+	0.982963i	$0.441158\pi$
$102$		0			0
$103$	13.0912			1.28992			0.644958	−	0.764218i	$-0.276875\pi$
$103$	13.0912			1.28992			0.644958	+	0.764218i	$0.276875\pi$
$104$	−1.51286	−	3.27280i	−0.148348	−	0.320925i
$105$		0			0
$106$		−	6.00000i		−	0.582772i
$107$	3.51988			0.340280			0.170140	−	0.985420i	$-0.445578\pi$
$107$	3.51988			0.340280			0.170140	+	0.985420i	$0.445578\pi$
$108$		0			0
$109$			13.1427i			1.25884i	0.777066	+	0.629420i	$0.216707\pi$
$109$			13.1427i			1.25884i	−0.777066	+	0.629420i	$0.783293\pi$
$110$		−	8.48012i		−	0.808548i
$111$		0			0
$112$		−	1.00000i		−	0.0944911i
$113$	1.75994			0.165561			0.0827806	−	0.996568i	$-0.473620\pi$
$113$	1.75994			0.165561			0.0827806	+	0.996568i	$0.473620\pi$
$114$		0			0
$115$			4.48012i			0.417773i
$116$	7.57133			0.702981
$117$		0			0
$118$	8.06549			0.742488
$119$		−	7.09122i		−	0.650051i
$120$		0			0
$121$	−0.0973913			−0.00885375
$122$			0.785667i			0.0711309i
$123$		0			0
$124$		−	3.33127i		−	0.299157i
$125$			8.96024i			0.801428i
$126$		0			0
$127$	−12.9026			−1.14492			−0.572461	−	0.819932i	$-0.694011\pi$
$127$	−12.9026			−1.14492			−0.572461	+	0.819932i	$0.694011\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−8.33127	+	3.85116i	−0.730701	+	0.337769i
$131$	16.7795			1.46603			0.733015	−	0.680212i	$-0.238112\pi$
$131$	16.7795			1.46603			0.733015	+	0.680212i	$0.238112\pi$
$132$		0			0
$133$	4.54561			0.394154
$134$	5.75994			0.497583
$135$		0			0
$136$			7.09122i			0.608067i
$137$		−	11.1427i		−	0.951982i	−0.879450	−	0.475991i	$-0.842089\pi$
$137$		−	11.1427i		−	0.951982i	0.879450	−	0.475991i	$-0.157911\pi$
$138$		0			0
$139$	4.54561			0.385553			0.192777	−	0.981243i	$-0.438251\pi$
$139$	4.54561			0.385553			0.192777	+	0.981243i	$0.438251\pi$
$140$	−2.54561			−0.215143
$141$		0			0
$142$	11.1427			0.935072
$143$	5.03976	+	10.9026i	0.421446	+	0.911722i
$144$		0			0
$145$		−	19.2736i		−	1.60059i
$146$	9.33127			0.772262
$147$		0			0
$148$			3.75994i			0.309065i
$149$		−	5.81139i		−	0.476088i	−0.971254	−	0.238044i	$-0.923494\pi$
$149$		−	5.81139i		−	0.476088i	0.971254	−	0.238044i	$-0.0765062\pi$
$150$		0			0
$151$		−	3.09122i		−	0.251560i	−0.992058	−	0.125780i	$-0.959857\pi$
$151$		−	3.09122i		−	0.251560i	0.992058	−	0.125780i	$-0.0401433\pi$
$152$	−4.54561			−0.368697
$153$		0			0
$154$			3.33127i			0.268442i
$155$	−8.48012			−0.681140
$156$		0			0
$157$	−7.92834			−0.632750			−0.316375	−	0.948634i	$-0.602466\pi$
$157$	−7.92834			−0.632750			−0.316375	+	0.948634i	$0.602466\pi$
$158$		−	4.85116i		−	0.385937i
$159$		0			0
$160$	2.54561			0.201248
$161$		−	1.75994i		−	0.138703i
$162$		0			0
$163$		−	10.0515i		−	0.787291i	−0.919262	−	0.393645i	$-0.871214\pi$
$163$		−	10.0515i		−	0.787291i	0.919262	−	0.393645i	$-0.128786\pi$
$164$		−	4.24006i		−	0.331093i
$165$		0			0
$166$	11.2082			0.869922
$167$		−	24.2339i		−	1.87527i	−0.347616	−	0.937637i	$-0.613009\pi$
$167$		−	24.2339i		−	1.87527i	0.347616	−	0.937637i	$-0.386991\pi$
$168$		0			0
$169$	8.42249	−	9.90261i	0.647884	−	0.761739i
$170$	18.0515			1.38448
$171$		0			0
$172$	−9.09122			−0.693199
$173$	−20.5971			−1.56597			−0.782983	−	0.622043i	$-0.786303\pi$
$173$	−20.5971			−1.56597			−0.782983	+	0.622043i	$0.786303\pi$
$174$		0			0
$175$			1.48012i			0.111886i
$176$		−	3.33127i		−	0.251104i
$177$		0			0
$178$	−15.0912			−1.13113
$179$	−17.3251			−1.29494			−0.647469	−	0.762091i	$-0.724173\pi$
$179$	−17.3251			−1.29494			−0.647469	+	0.762091i	$0.724173\pi$
$180$		0			0
$181$	−16.9166			−1.25740			−0.628702	−	0.777646i	$-0.716414\pi$
$181$	−16.9166			−1.25740			−0.628702	+	0.777646i	$0.716414\pi$
$182$	3.27280	−	1.51286i	0.242596	−	0.112141i
$183$		0			0
$184$			1.75994i			0.129745i
$185$	9.57133			0.703698
$186$		0			0
$187$		−	23.6228i		−	1.72747i
$188$		−	11.3313i		−	0.826418i
$189$		0			0
$190$			11.5713i			0.839473i
$191$	−4.42867			−0.320447			−0.160224	−	0.987081i	$-0.551221\pi$
$191$	−4.42867			−0.320447			−0.160224	+	0.987081i	$0.551221\pi$
$192$		0			0
$193$		−	2.48012i		−	0.178523i	−0.996008	−	0.0892614i	$-0.971549\pi$
$193$		−	2.48012i		−	0.178523i	0.996008	−	0.0892614i	$-0.0284507\pi$
$194$	−7.75994			−0.557131
$195$		0			0
$196$	1.00000			0.0714286
$197$		−	8.85116i		−	0.630619i	−0.948989	−	0.315309i	$-0.897892\pi$
$197$		−	8.85116i		−	0.630619i	0.948989	−	0.315309i	$-0.102108\pi$
$198$		0			0
$199$	1.09122			0.0773542			0.0386771	−	0.999252i	$-0.487686\pi$
$199$	1.09122			0.0773542			0.0386771	+	0.999252i	$0.487686\pi$
$200$		−	1.48012i		−	0.104660i
$201$		0			0
$202$		−	3.69445i		−	0.259941i
$203$			7.57133i			0.531403i
$204$		0			0
$205$	−10.7935			−0.753853
$206$		−	13.0912i		−	0.912108i
$207$		0			0
$208$	−3.27280	+	1.51286i	−0.226928	+	0.104898i
$209$	15.1427			1.04744
$210$		0			0
$211$	15.0398			1.03538			0.517690	−	0.855568i	$-0.326792\pi$
$211$	15.0398			1.03538			0.517690	+	0.855568i	$0.326792\pi$
$212$	−6.00000			−0.412082
$213$		0			0
$214$		−	3.51988i		−	0.240614i
$215$			23.1427i			1.57832i
$216$		0			0
$217$	3.33127			0.226142
$218$	13.1427			0.890134
$219$		0			0
$220$	−8.48012			−0.571729
$221$	−23.2082	+	10.7280i	−1.56115	+	0.721646i
$222$		0			0
$223$			12.9026i			0.864023i	0.901868	+	0.432011i	$0.142196\pi$
$223$			12.9026i			0.864023i	−0.901868	+	0.432011i	$0.857804\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$		−	1.75994i		−	0.117070i
$227$			9.02573i			0.599058i	0.954087	+	0.299529i	$0.0968296\pi$
$227$			9.02573i			0.599058i	−0.954087	+	0.299529i	$0.903170\pi$
$228$		0			0
$229$		−	14.1684i		−	0.936274i	−0.883656	−	0.468137i	$-0.844925\pi$
$229$		−	14.1684i		−	0.936274i	0.883656	−	0.468137i	$-0.155075\pi$
$230$	4.48012			0.295410
$231$		0			0
$232$		−	7.57133i		−	0.497082i
$233$	7.38273			0.483659			0.241829	−	0.970319i	$-0.422253\pi$
$233$	7.38273			0.483659			0.241829	+	0.970319i	$0.422253\pi$
$234$		0			0
$235$	−28.8450			−1.88164
$236$		−	8.06549i		−	0.525019i
$237$		0			0
$238$	−7.09122			−0.459655
$239$			6.13098i			0.396580i	0.980143	+	0.198290i	$0.0635388\pi$
$239$			6.13098i			0.396580i	−0.980143	+	0.198290i	$0.936461\pi$
$240$		0			0
$241$			27.3251i			1.76016i	0.474821	+	0.880082i	$0.342513\pi$
$241$			27.3251i			1.76016i	−0.474821	+	0.880082i	$0.657487\pi$
$242$			0.0973913i			0.00626055i
$243$		0			0
$244$	0.785667			0.0502972
$245$		−	2.54561i		−	0.162633i
$246$		0			0
$247$	−6.87688	−	14.8769i	−0.437566	−	0.946593i
$248$	−3.33127			−0.211536
$249$		0			0
$250$	8.96024			0.566695
$251$	−16.4880			−1.04071			−0.520356	−	0.853949i	$-0.674201\pi$
$251$	−16.4880			−1.04071			−0.520356	+	0.853949i	$0.674201\pi$
$252$		0			0
$253$		−	5.86285i		−	0.368594i
$254$			12.9026i			0.809582i
$255$		0			0
$256$	1.00000			0.0625000
$257$	1.14267			0.0712777			0.0356388	−	0.999365i	$-0.488653\pi$
$257$	1.14267			0.0712777			0.0356388	+	0.999365i	$0.488653\pi$
$258$		0			0
$259$	−3.75994			−0.233631
$260$	3.85116	+	8.33127i	0.238839	+	0.516684i
$261$		0			0
$262$		−	16.7795i		−	1.03664i
$263$	10.6111			0.654308			0.327154	−	0.944971i	$-0.393910\pi$
$263$	10.6111			0.654308			0.327154	+	0.944971i	$0.393910\pi$
$264$		0			0
$265$			15.2736i			0.938253i
$266$		−	4.54561i		−	0.278709i
$267$		0			0
$268$		−	5.75994i		−	0.351844i
$269$	12.0593			0.735269			0.367635	−	0.929970i	$-0.380168\pi$
$269$	12.0593			0.735269			0.367635	+	0.929970i	$0.380168\pi$
$270$		0			0
$271$			1.38273i			0.0839947i	0.999118	+	0.0419974i	$0.0133721\pi$
$271$			1.38273i			0.0839947i	−0.999118	+	0.0419974i	$0.986628\pi$
$272$	7.09122			0.429968
$273$		0			0
$274$	−11.1427			−0.673153
$275$			4.93068i			0.297331i
$276$		0			0
$277$	1.14267			0.0686563			0.0343281	−	0.999411i	$-0.489071\pi$
$277$	1.14267			0.0686563			0.0343281	+	0.999411i	$0.489071\pi$
$278$		−	4.54561i		−	0.272627i
$279$		0			0
$280$			2.54561i			0.152129i
$281$		−	7.14267i		−	0.426096i	−0.977042	−	0.213048i	$-0.931661\pi$
$281$		−	7.14267i		−	0.426096i	0.977042	−	0.213048i	$-0.0683390\pi$
$282$		0			0
$283$	−4.83712			−0.287537			−0.143768	−	0.989611i	$-0.545922\pi$
$283$	−4.83712			−0.287537			−0.143768	+	0.989611i	$0.545922\pi$
$284$		−	11.1427i		−	0.661196i
$285$		0			0
$286$	10.9026	−	5.03976i	0.644685	−	0.298007i
$287$	4.24006			0.250283
$288$		0			0
$289$	33.2853			1.95796
$290$	−19.2736			−1.13179
$291$		0			0
$292$		−	9.33127i		−	0.546072i
$293$			7.63682i			0.446148i	0.974802	+	0.223074i	$0.0716092\pi$
$293$			7.63682i			0.446148i	−0.974802	+	0.223074i	$0.928391\pi$
$294$		0			0
$295$	−20.5316			−1.19539
$296$	3.75994			0.218542
$297$		0			0
$298$	−5.81139			−0.336645
$299$	−5.75994	+	2.66255i	−0.333106	+	0.153979i
$300$		0			0
$301$		−	9.09122i		−	0.524009i
$302$	−3.09122			−0.177879
$303$		0			0
$304$			4.54561i			0.260708i
$305$		−	2.00000i		−	0.114520i
$306$		0			0
$307$		−	14.3508i		−	0.819045i	−0.912300	−	0.409522i	$-0.865695\pi$
$307$		−	14.3508i		−	0.819045i	0.912300	−	0.409522i	$-0.134305\pi$
$308$	3.33127			0.189817
$309$		0			0
$310$			8.48012i			0.481638i
$311$	−2.18243			−0.123754			−0.0618771	−	0.998084i	$-0.519709\pi$
$311$	−2.18243			−0.123754			−0.0618771	+	0.998084i	$0.519709\pi$
$312$		0			0
$313$	15.5713			0.880144			0.440072	−	0.897963i	$-0.354953\pi$
$313$	15.5713			0.880144			0.440072	+	0.897963i	$0.354953\pi$
$314$			7.92834i			0.447422i
$315$		0			0
$316$	−4.85116			−0.272899
$317$			14.1886i			0.796912i	0.917188	+	0.398456i	$0.130454\pi$
$317$			14.1886i			0.796912i	−0.917188	+	0.398456i	$0.869546\pi$
$318$		0			0
$319$			25.2222i			1.41217i
$320$		−	2.54561i		−	0.142304i
$321$		0			0
$322$	−1.75994			−0.0980777
$323$			32.2339i			1.79354i
$324$		0			0
$325$	4.84414	−	2.23922i	0.268704	−	0.124209i
$326$	−10.0515			−0.556698
$327$		0			0
$328$	−4.24006			−0.234118
$329$	11.3313			0.624713
$330$		0			0
$331$			2.85116i			0.156714i	0.996925	+	0.0783569i	$0.0249674\pi$
$331$			2.85116i			0.156714i	−0.996925	+	0.0783569i	$0.975033\pi$
$332$		−	11.2082i		−	0.615128i
$333$		0			0
$334$	−24.2339			−1.32602
$335$	−14.6625			−0.801101
$336$		0			0
$337$	−32.5254			−1.77177			−0.885886	−	0.463904i	$-0.846448\pi$
$337$	−32.5254			−1.77177			−0.885886	+	0.463904i	$0.846448\pi$
$338$	−9.90261	−	8.42249i	−0.538631	−	0.458123i
$339$		0			0
$340$		−	18.0515i		−	0.978977i
$341$	11.0974			0.600957
$342$		0			0
$343$			1.00000i			0.0539949i
$344$			9.09122i			0.490165i
$345$		0			0
$346$			20.5971i			1.10730i
$347$	−5.94855			−0.319335			−0.159667	−	0.987171i	$-0.551042\pi$
$347$	−5.94855			−0.319335			−0.159667	+	0.987171i	$0.551042\pi$
$348$		0			0
$349$		−	32.9619i		−	1.76441i	−0.470864	−	0.882206i	$-0.656058\pi$
$349$		−	32.9619i		−	1.76441i	0.470864	−	0.882206i	$-0.343942\pi$
$350$	1.48012			0.0791157
$351$		0			0
$352$	−3.33127			−0.177558
$353$		−	0.851156i		−	0.0453025i	−0.999743	−	0.0226512i	$-0.992789\pi$
$353$		−	0.851156i		−	0.0453025i	0.999743	−	0.0226512i	$-0.00721073\pi$
$354$		0			0
$355$	−28.3649			−1.50545
$356$			15.0912i			0.799833i
$357$		0			0
$358$			17.3251i			0.915660i
$359$		−	27.9362i		−	1.47442i	−0.675666	−	0.737208i	$-0.736144\pi$
$359$		−	27.9362i		−	1.47442i	0.675666	−	0.737208i	$-0.263856\pi$
$360$		0			0
$361$	−1.66255			−0.0875026
$362$			16.9166i			0.889119i
$363$		0			0
$364$	−1.51286	−	3.27280i	−0.0792956	−	0.171542i
$365$	−23.7538			−1.24333
$366$		0			0
$367$	−7.14267			−0.372844			−0.186422	−	0.982470i	$-0.559689\pi$
$367$	−7.14267			−0.372844			−0.186422	+	0.982470i	$0.559689\pi$
$368$	1.75994			0.0917433
$369$		0			0
$370$		−	9.57133i		−	0.497590i
$371$		−	6.00000i		−	0.311504i
$372$		0			0
$373$	−33.1427			−1.71606			−0.858031	−	0.513598i	$-0.828312\pi$
$373$	−33.1427			−1.71606			−0.858031	+	0.513598i	$0.828312\pi$
$374$	−23.6228			−1.22151
$375$		0			0
$376$	−11.3313			−0.584366
$377$	24.7795	−	11.4544i	1.27621	−	0.589931i
$378$		0			0
$379$			35.2736i			1.81189i	0.423400	+	0.905943i	$0.360836\pi$
$379$			35.2736i			1.81189i	−0.423400	+	0.905943i	$0.639164\pi$
$380$	11.5713			0.593597
$381$		0			0
$382$			4.42867i			0.226590i
$383$		−	6.95406i		−	0.355336i	−0.984090	−	0.177668i	$-0.943145\pi$
$383$		−	6.95406i		−	0.355336i	0.984090	−	0.177668i	$-0.0568553\pi$
$384$		0			0
$385$		−	8.48012i		−	0.432187i
$386$	−2.48012			−0.126235
$387$		0			0
$388$			7.75994i			0.393951i
$389$	5.27365			0.267384			0.133692	−	0.991023i	$-0.457317\pi$
$389$	5.27365			0.267384			0.133692	+	0.991023i	$0.457317\pi$
$390$		0			0
$391$	12.4801			0.631147
$392$		−	1.00000i		−	0.0505076i
$393$		0			0
$394$	−8.85116			−0.445915
$395$			12.3491i			0.621353i
$396$		0			0
$397$		−	18.6766i		−	0.937351i	−0.883370	−	0.468675i	$-0.844731\pi$
$397$		−	18.6766i		−	0.937351i	0.883370	−	0.468675i	$-0.155269\pi$
$398$		−	1.09122i		−	0.0546977i
$399$		0			0
$400$	−1.48012			−0.0740059
$401$		−	21.5199i		−	1.07465i	−0.843375	−	0.537326i	$-0.819435\pi$
$401$		−	21.5199i		−	1.07465i	0.843375	−	0.537326i	$-0.180565\pi$
$402$		0			0
$403$	−5.03976	−	10.9026i	−0.251048	−	0.543098i
$404$	−3.69445			−0.183806
$405$		0			0
$406$	7.57133			0.375759
$407$	−12.5254			−0.620861
$408$		0			0
$409$		−	19.3251i		−	0.955565i	−0.878478	−	0.477782i	$-0.841441\pi$
$409$		−	19.3251i		−	0.955565i	0.878478	−	0.477782i	$-0.158559\pi$
$410$			10.7935i			0.533054i
$411$		0			0
$412$	−13.0912			−0.644958
$413$	8.06549			0.396877
$414$		0			0
$415$	−28.5316			−1.40056
$416$	1.51286	+	3.27280i	0.0741742	+	0.160462i
$417$		0			0
$418$		−	15.1427i		−	0.740652i
$419$	−12.8371			−0.627134			−0.313567	−	0.949566i	$-0.601524\pi$
$419$	−12.8371			−0.627134			−0.313567	+	0.949566i	$0.601524\pi$
$420$		0			0
$421$			21.3313i			1.03962i	0.854281	+	0.519811i	$0.173998\pi$
$421$			21.3313i			1.03962i	−0.854281	+	0.519811i	$0.826002\pi$
$422$		−	15.0398i		−	0.732124i
$423$		0			0
$424$			6.00000i			0.291386i
$425$	−10.4958			−0.509123
$426$		0			0
$427$			0.785667i			0.0380211i
$428$	−3.51988			−0.170140
$429$		0			0
$430$	23.1427			1.11604
$431$			11.5713i			0.557372i	0.960382	+	0.278686i	$0.0898988\pi$
$431$			11.5713i			0.557372i	−0.960382	+	0.278686i	$0.910101\pi$
$432$		0			0
$433$	29.3766			1.41175			0.705873	−	0.708338i	$-0.250555\pi$
$433$	29.3766			1.41175			0.705873	+	0.708338i	$0.250555\pi$
$434$		−	3.33127i		−	0.159906i
$435$		0			0
$436$		−	13.1427i		−	0.629420i
$437$			8.00000i			0.382692i
$438$		0			0
$439$	0.102905			0.00491140			0.00245570	−	0.999997i	$-0.499218\pi$
$439$	0.102905			0.00491140			0.00245570	+	0.999997i	$0.499218\pi$
$440$			8.48012i			0.404274i
$441$		0			0
$442$	10.7280	+	23.2082i	0.510281	+	1.10390i
$443$	−27.1427			−1.28959			−0.644794	−	0.764357i	$-0.723057\pi$
$443$	−27.1427			−1.28959			−0.644794	+	0.764357i	$0.723057\pi$
$444$		0			0
$445$	38.4163			1.82111
$446$	12.9026			0.610956
$447$		0			0
$448$			1.00000i			0.0472456i
$449$			9.70231i			0.457880i	0.973440	+	0.228940i	$0.0735260\pi$
$449$			9.70231i			0.457880i	−0.973440	+	0.228940i	$0.926474\pi$
$450$		0			0
$451$	14.1248			0.665111
$452$	−1.75994			−0.0827806
$453$		0			0
$454$	9.02573			0.423598
$455$	−8.33127	+	3.85116i	−0.390576	+	0.180545i
$456$		0			0
$457$			36.1701i			1.69196i	0.533211	+	0.845982i	$0.320985\pi$
$457$			36.1701i			1.69196i	−0.533211	+	0.845982i	$0.679015\pi$
$458$	−14.1684			−0.662046
$459$		0			0
$460$		−	4.48012i		−	0.208887i
$461$		−	20.9743i		−	0.976869i	−0.872600	−	0.488435i	$-0.837568\pi$
$461$		−	20.9743i		−	0.976869i	0.872600	−	0.488435i	$-0.162432\pi$
$462$		0			0
$463$			24.8450i			1.15464i	0.816517	+	0.577322i	$0.195902\pi$
$463$			24.8450i			1.15464i	−0.816517	+	0.577322i	$0.804098\pi$
$464$	−7.57133			−0.351490
$465$		0			0
$466$		−	7.38273i		−	0.341998i
$467$	−1.27196			−0.0588594			−0.0294297	−	0.999567i	$-0.509369\pi$
$467$	−1.27196			−0.0588594			−0.0294297	+	0.999567i	$0.509369\pi$
$468$		0			0
$469$	5.75994			0.265969
$470$			28.8450i			1.33052i
$471$		0			0
$472$	−8.06549			−0.371244
$473$		−	30.2853i		−	1.39252i
$474$		0			0
$475$		−	6.72804i		−	0.308704i
$476$			7.09122i			0.325025i
$477$		0			0
$478$	6.13098			0.280424
$479$		−	22.0515i		−	1.00756i	−0.863833	−	0.503778i	$-0.831943\pi$
$479$		−	22.0515i		−	1.00756i	0.863833	−	0.503778i	$-0.168057\pi$
$480$		0			0
$481$	5.68828	+	12.3055i	0.259363	+	0.561085i
$482$	27.3251			1.24462
$483$		0			0
$484$	0.0973913			0.00442688
$485$	19.7538			0.896972
$486$		0			0
$487$		−	4.42867i		−	0.200682i	−0.994953	−	0.100341i	$-0.968007\pi$
$487$		−	4.42867i		−	0.200682i	0.994953	−	0.100341i	$-0.0319934\pi$
$488$		−	0.785667i		−	0.0355655i
$489$		0			0
$490$	−2.54561			−0.114999
$491$	31.0274			1.40025			0.700124	−	0.714022i	$-0.253128\pi$
$491$	31.0274			1.40025			0.700124	+	0.714022i	$0.253128\pi$
$492$		0			0
$493$	−53.6900			−2.41807
$494$	−14.8769	+	6.87688i	−0.669343	+	0.309406i
$495$		0			0
$496$			3.33127i			0.149579i
$497$	11.1427			0.499817
$498$		0			0
$499$		−	23.0850i		−	1.03343i	−0.856158	−	0.516714i	$-0.827155\pi$
$499$		−	23.0850i		−	1.03343i	0.856158	−	0.516714i	$-0.172845\pi$
$500$		−	8.96024i		−	0.400714i
$501$		0			0
$502$			16.4880i			0.735895i
$503$	17.7023			0.789307			0.394654	−	0.918830i	$-0.370865\pi$
$503$	17.7023			0.789307			0.394654	+	0.918830i	$0.370865\pi$
$504$		0			0
$505$			9.40462i			0.418500i
$506$	−5.86285			−0.260635
$507$		0			0
$508$	12.9026			0.572461
$509$		−	24.7280i		−	1.09605i	−0.836462	−	0.548026i	$-0.815380\pi$
$509$		−	24.7280i		−	1.09605i	0.836462	−	0.548026i	$-0.184620\pi$
$510$		0			0
$511$	9.33127			0.412791
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	1.14267i		−	0.0504009i
$515$			33.3251i			1.46848i
$516$		0			0
$517$	37.7476			1.66014
$518$			3.75994i			0.165202i
$519$		0			0
$520$	8.33127	−	3.85116i	0.365351	−	0.168884i
$521$	20.4287			0.894996			0.447498	−	0.894285i	$-0.352315\pi$
$521$	20.4287			0.894996			0.447498	+	0.894285i	$0.352315\pi$
$522$		0			0
$523$	25.8254			1.12927			0.564634	−	0.825342i	$-0.309017\pi$
$523$	25.8254			1.12927			0.564634	+	0.825342i	$0.309017\pi$
$524$	−16.7795			−0.733015
$525$		0			0
$526$		−	10.6111i		−	0.462666i
$527$			23.6228i			1.02902i
$528$		0			0
$529$	−19.9026			−0.865331
$530$	15.2736			0.663445
$531$		0			0
$532$	−4.54561			−0.197077
$533$	−6.41463	−	13.8769i	−0.277848	−	0.601075i
$534$		0			0
$535$			8.96024i			0.387385i
$536$	−5.75994			−0.248792
$537$		0			0
$538$		−	12.0593i		−	0.519914i
$539$			3.33127i			0.143488i
$540$		0			0
$541$		−	28.3134i		−	1.21729i	−0.793443	−	0.608644i	$-0.791714\pi$
$541$		−	28.3134i		−	1.21729i	0.793443	−	0.608644i	$-0.208286\pi$
$542$	1.38273			0.0593932
$543$		0			0
$544$		−	7.09122i		−	0.304033i
$545$	−33.4561			−1.43310
$546$		0			0
$547$	25.5713			1.09335			0.546676	−	0.837344i	$-0.315893\pi$
$547$	25.5713			1.09335			0.546676	+	0.837344i	$0.315893\pi$
$548$			11.1427i			0.475991i
$549$		0			0
$550$	4.93068			0.210245
$551$		−	34.4163i		−	1.46618i
$552$		0			0
$553$		−	4.85116i		−	0.206292i
$554$		−	1.14267i		−	0.0485473i
$555$		0			0
$556$	−4.54561			−0.192777
$557$			3.99382i			0.169224i	0.996414	+	0.0846119i	$0.0269650\pi$
$557$			3.99382i			0.169224i	−0.996414	+	0.0846119i	$0.973035\pi$
$558$		0			0
$559$	−29.7538	+	13.7538i	−1.25845	+	0.581722i
$560$	2.54561			0.107572
$561$		0			0
$562$	−7.14267			−0.301295
$563$	−43.4997			−1.83329			−0.916646	−	0.399699i	$-0.869114\pi$
$563$	−43.4997			−1.83329			−0.916646	+	0.399699i	$0.869114\pi$
$564$		0			0
$565$			4.48012i			0.188480i
$566$			4.83712i			0.203319i
$567$		0			0
$568$	−11.1427			−0.467536
$569$	−41.2675			−1.73002			−0.865011	−	0.501753i	$-0.832689\pi$
$569$	−41.2675			−1.73002			−0.865011	+	0.501753i	$0.832689\pi$
$570$		0			0
$571$	13.9362			0.583212			0.291606	−	0.956539i	$-0.405810\pi$
$571$	13.9362			0.583212			0.291606	+	0.956539i	$0.405810\pi$
$572$	−5.03976	−	10.9026i	−0.210723	−	0.455861i
$573$		0			0
$574$		−	4.24006i		−	0.176977i
$575$	−2.60492			−0.108633
$576$		0			0
$577$		−	32.4163i		−	1.34951i	−0.738042	−	0.674754i	$-0.764250\pi$
$577$		−	32.4163i		−	1.34951i	0.738042	−	0.674754i	$-0.235750\pi$
$578$		−	33.2853i		−	1.38449i
$579$		0			0
$580$			19.2736i			0.800295i
$581$	11.2082			0.464993
$582$		0			0
$583$		−	19.9876i		−	0.827804i
$584$	−9.33127			−0.386131
$585$		0			0
$586$	7.63682			0.315474
$587$		−	4.03742i		−	0.166642i	−0.996523	−	0.0833210i	$-0.973447\pi$
$587$		−	4.03742i		−	0.166642i	0.996523	−	0.0833210i	$-0.0265527\pi$
$588$		0			0
$589$	−15.1427			−0.623943
$590$			20.5316i			0.845271i
$591$		0			0
$592$		−	3.75994i		−	0.154533i
$593$			20.4163i			0.838398i	0.907894	+	0.419199i	$0.137689\pi$
$593$			20.4163i			0.838398i	−0.907894	+	0.419199i	$0.862311\pi$
$594$		0			0
$595$	18.0515			0.740037
$596$			5.81139i			0.238044i
$597$		0			0
$598$	2.66255	+	5.75994i	0.108880	+	0.235541i
$599$	8.10908			0.331328			0.165664	−	0.986182i	$-0.447023\pi$
$599$	8.10908			0.331328			0.165664	+	0.986182i	$0.447023\pi$
$600$		0			0
$601$	−7.70231			−0.314184			−0.157092	−	0.987584i	$-0.550212\pi$
$601$	−7.70231			−0.314184			−0.157092	+	0.987584i	$0.550212\pi$
$602$	−9.09122			−0.370530
$603$		0			0
$604$			3.09122i			0.125780i
$605$		−	0.247920i		−	0.0100794i
$606$		0			0
$607$	−33.7023			−1.36793			−0.683967	−	0.729513i	$-0.739747\pi$
$607$	−33.7023			−1.36793			−0.683967	+	0.729513i	$0.739747\pi$
$608$	4.54561			0.184349
$609$		0			0
$610$	−2.00000			−0.0809776
$611$	−17.1427	−	37.0850i	−0.693518	−	1.50030i
$612$		0			0
$613$		−	23.2675i		−	0.939764i	−0.882729	−	0.469882i	$-0.844296\pi$
$613$		−	23.2675i		−	0.939764i	0.882729	−	0.469882i	$-0.155704\pi$
$614$	−14.3508			−0.579152
$615$		0			0
$616$		−	3.33127i		−	0.134221i
$617$		−	1.51988i		−	0.0611881i	−0.999532	−	0.0305941i	$-0.990260\pi$
$617$		−	1.51988i		−	0.0611881i	0.999532	−	0.0305941i	$-0.00973991\pi$
$618$		0			0
$619$		−	0.779491i		−	0.0313304i	−0.999877	−	0.0156652i	$-0.995013\pi$
$619$		−	0.779491i		−	0.0313304i	0.999877	−	0.0156652i	$-0.00498659\pi$
$620$	8.48012			0.340570
$621$		0			0
$622$			2.18243i			0.0875075i
$623$	−15.0912			−0.604617
$624$		0			0
$625$	−30.2098			−1.20839
$626$		−	15.5713i		−	0.622356i
$627$		0			0
$628$	7.92834			0.316375
$629$		−	26.6625i		−	1.06311i
$630$		0			0
$631$		−	32.4678i		−	1.29252i	−0.763117	−	0.646261i	$-0.776332\pi$
$631$		−	32.4678i		−	1.29252i	0.763117	−	0.646261i	$-0.223668\pi$
$632$			4.85116i			0.192969i
$633$		0			0
$634$	14.1886			0.563502
$635$		−	32.8450i		−	1.30341i
$636$		0			0
$637$	3.27280	−	1.51286i	0.129673	−	0.0599418i
$638$	25.2222			0.998556
$639$		0			0
$640$	−2.54561			−0.100624
$641$	42.2277			1.66789			0.833947	−	0.551844i	$-0.186076\pi$
$641$	42.2277			1.66789			0.833947	+	0.551844i	$0.186076\pi$
$642$		0			0
$643$		−	6.84330i		−	0.269873i	−0.990854	−	0.134937i	$-0.956917\pi$
$643$		−	6.84330i		−	0.269873i	0.990854	−	0.134937i	$-0.0430831\pi$
$644$			1.75994i			0.0693514i
$645$		0			0
$646$	32.2339			1.26823
$647$	3.75376			0.147576			0.0737879	−	0.997274i	$-0.476491\pi$
$647$	3.75376			0.147576			0.0737879	+	0.997274i	$0.476491\pi$
$648$		0			0
$649$	26.8684			1.05468
$650$	−2.23922	−	4.84414i	−0.0878293	−	0.190003i
$651$		0			0
$652$			10.0515i			0.393645i
$653$	−23.8333			−0.932669			−0.466334	−	0.884609i	$-0.654426\pi$
$653$	−23.8333			−0.932669			−0.466334	+	0.884609i	$0.654426\pi$
$654$		0			0
$655$			42.7140i			1.66897i
$656$			4.24006i			0.165547i
$657$		0			0
$658$		−	11.3313i		−	0.441739i
$659$	21.0912			0.821597			0.410799	−	0.911726i	$-0.365250\pi$
$659$	21.0912			0.821597			0.410799	+	0.911726i	$0.365250\pi$
$660$		0			0
$661$			34.4023i			1.33809i	0.743220	+	0.669047i	$0.233297\pi$
$661$			34.4023i			1.33809i	−0.743220	+	0.669047i	$0.766703\pi$
$662$	2.85116			0.110813
$663$		0			0
$664$	−11.2082			−0.434961
$665$			11.5713i			0.448717i
$666$		0			0
$667$	−13.3251			−0.515950
$668$			24.2339i			0.937637i
$669$		0			0
$670$			14.6625i			0.566464i
$671$			2.61727i			0.101039i
$672$		0			0
$673$	10.7997			0.416298			0.208149	−	0.978097i	$-0.433256\pi$
$673$	10.7997			0.416298			0.208149	+	0.978097i	$0.433256\pi$
$674$			32.5254i			1.25283i
$675$		0			0
$676$	−8.42249	+	9.90261i	−0.323942	+	0.380870i
$677$	33.7336			1.29649			0.648243	−	0.761434i	$-0.275504\pi$
$677$	33.7336			1.29649			0.648243	+	0.761434i	$0.275504\pi$
$678$		0			0
$679$	−7.75994			−0.297799
$680$	−18.0515			−0.692242
$681$		0			0
$682$		−	11.0974i		−	0.424941i
$683$			21.8628i			0.836559i	0.908318	+	0.418279i	$0.137367\pi$
$683$			21.8628i			0.836559i	−0.908318	+	0.418279i	$0.862633\pi$
$684$		0			0
$685$	28.3649			1.08377
$686$	1.00000			0.0381802
$687$		0			0
$688$	9.09122			0.346599
$689$	−19.6368	+	9.07718i	−0.748103	+	0.345813i
$690$		0			0
$691$		−	10.7404i		−	0.408584i	−0.978910	−	0.204292i	$-0.934511\pi$
$691$		−	10.7404i		−	0.408584i	0.978910	−	0.204292i	$-0.0654892\pi$
$692$	20.5971			0.782983
$693$		0			0
$694$			5.94855i			0.225804i
$695$			11.5713i			0.438926i
$696$		0			0
$697$			30.0672i			1.13888i
$698$	−32.9619			−1.24763
$699$		0			0
$700$		−	1.48012i		−	0.0559432i
$701$	11.1941			0.422796			0.211398	−	0.977400i	$-0.432198\pi$
$701$	11.1941			0.422796			0.211398	+	0.977400i	$0.432198\pi$
$702$		0			0
$703$	17.0912			0.644607
$704$			3.33127i			0.125552i
$705$		0			0
$706$	−0.851156			−0.0320337
$707$		−	3.69445i		−	0.138944i
$708$		0			0
$709$		−	5.20030i		−	0.195301i	−0.995221	−	0.0976506i	$-0.968867\pi$
$709$		−	5.20030i		−	0.195301i	0.995221	−	0.0976506i	$-0.0311328\pi$
$710$			28.3649i			1.06451i
$711$		0			0
$712$	15.0912			0.565567
$713$			5.86285i			0.219565i
$714$		0			0
$715$	−27.7538	+	12.8293i	−1.03793	+	0.479787i
$716$	17.3251			0.647469
$717$		0			0
$718$	−27.9362			−1.04257
$719$	38.0515			1.41908			0.709540	−	0.704665i	$-0.248903\pi$
$719$	38.0515			1.41908			0.709540	+	0.704665i	$0.248903\pi$
$720$		0			0
$721$		−	13.0912i		−	0.487542i
$722$			1.66255i			0.0618737i
$723$		0			0
$724$	16.9166			0.628702
$725$	11.2065			0.416198
$726$		0			0
$727$	−28.6111			−1.06113			−0.530563	−	0.847645i	$-0.678020\pi$
$727$	−28.6111			−1.06113			−0.530563	+	0.847645i	$0.678020\pi$
$728$	−3.27280	+	1.51286i	−0.121298	+	0.0560704i
$729$		0			0
$730$			23.7538i			0.879166i
$731$	64.4678			2.38443
$732$		0			0
$733$			8.21985i			0.303607i	0.988411	+	0.151803i	$0.0485081\pi$
$733$			8.21985i			0.303607i	−0.988411	+	0.151803i	$0.951492\pi$
$734$			7.14267i			0.263641i
$735$		0			0
$736$		−	1.75994i		−	0.0648723i
$737$	19.1879			0.706797
$738$		0			0
$739$		−	1.58369i		−	0.0582568i	−0.999576	−	0.0291284i	$-0.990727\pi$
$739$		−	1.58369i		−	0.0582568i	0.999576	−	0.0291284i	$-0.00927317\pi$
$740$	−9.57133			−0.351849
$741$		0			0
$742$	−6.00000			−0.220267
$743$			11.9485i			0.438350i	0.975686	+	0.219175i	$0.0703365\pi$
$743$			11.9485i			0.438350i	−0.975686	+	0.219175i	$0.929663\pi$
$744$		0			0
$745$	14.7935			0.541993
$746$			33.1427i			1.21344i
$747$		0			0
$748$			23.6228i			0.863735i
$749$		−	3.51988i		−	0.128614i
$750$		0			0
$751$	27.5652			1.00587			0.502933	−	0.864325i	$-0.332254\pi$
$751$	27.5652			1.00587			0.502933	+	0.864325i	$0.332254\pi$
$752$			11.3313i			0.413209i
$753$		0			0
$754$	−11.4544	−	24.7795i	−0.417144	−	0.902416i
$755$	7.86902			0.286383
$756$		0			0
$757$	−38.2339			−1.38963			−0.694817	−	0.719187i	$-0.744515\pi$
$757$	−38.2339			−1.38963			−0.694817	+	0.719187i	$0.744515\pi$
$758$	35.2736			1.28120
$759$		0			0
$760$		−	11.5713i		−	0.419736i
$761$		−	51.7476i		−	1.87585i	−0.346840	−	0.937924i	$-0.612745\pi$
$761$		−	51.7476i		−	1.87585i	0.346840	−	0.937924i	$-0.387255\pi$
$762$		0			0
$763$	13.1427			0.475797
$764$	4.42867			0.160224
$765$		0			0
$766$	−6.95406			−0.251260
$767$	−12.2020	−	26.3968i	−0.440588	−	0.953132i
$768$		0			0
$769$		−	24.2401i		−	0.874119i	−0.899433	−	0.437059i	$-0.856020\pi$
$769$		−	24.2401i		−	0.874119i	0.899433	−	0.437059i	$-0.143980\pi$
$770$	−8.48012			−0.305602
$771$		0			0
$772$			2.48012i			0.0892614i
$773$			16.5971i			0.596955i	0.954417	+	0.298477i	$0.0964787\pi$
$773$			16.5971i			0.596955i	−0.954417	+	0.298477i	$0.903521\pi$
$774$		0			0
$775$		−	4.93068i		−	0.177115i
$776$	7.75994			0.278566
$777$		0			0
$778$		−	5.27365i		−	0.189069i
$779$	−19.2736			−0.690550
$780$		0			0
$781$	37.1193			1.32823
$782$		−	12.4801i		−	0.446288i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$		−	20.1824i		−	0.720342i
$786$		0			0
$787$		−	8.31172i		−	0.296281i	−0.988966	−	0.148140i	$-0.952671\pi$
$787$		−	8.31172i		−	0.296281i	0.988966	−	0.148140i	$-0.0473288\pi$
$788$			8.85116i			0.315309i
$789$		0			0
$790$	12.3491			0.439363
$791$		−	1.75994i		−	0.0625763i
$792$		0			0
$793$	2.57133	−	1.18861i	0.0913107	−	0.0422087i
$794$	−18.6766			−0.662807
$795$		0			0
$796$	−1.09122			−0.0386771
$797$	−24.3055			−0.860947			−0.430473	−	0.902603i	$-0.641653\pi$
$797$	−24.3055			−0.860947			−0.430473	+	0.902603i	$0.641653\pi$
$798$		0			0
$799$			80.3525i			2.84267i
$800$			1.48012i			0.0523301i
$801$		0			0
$802$	−21.5199			−0.759893
$803$	31.0850			1.09697
$804$		0			0
$805$	4.48012			0.157903
$806$	−10.9026	+	5.03976i	−0.384028	+	0.177518i
$807$		0			0
$808$			3.69445i			0.129970i
$809$	15.9876			0.562096			0.281048	−	0.959694i	$-0.409318\pi$
$809$	15.9876			0.562096			0.281048	+	0.959694i	$0.409318\pi$
$810$		0			0
$811$		−	19.7163i		−	0.692335i	−0.938173	−	0.346167i	$-0.887483\pi$
$811$		−	19.7163i		−	0.692335i	0.938173	−	0.346167i	$-0.112517\pi$
$812$		−	7.57133i		−	0.265702i
$813$		0			0
$814$			12.5254i			0.439015i
$815$	25.5871			0.896275
$816$		0			0
$817$			41.3251i			1.44578i
$818$	−19.3251			−0.675686
$819$		0			0
$820$	10.7935			0.376926
$821$			15.9485i			0.556608i	0.960493	+	0.278304i	$0.0897723\pi$
$821$			15.9485i			0.556608i	−0.960493	+	0.278304i	$0.910228\pi$
$822$		0			0
$823$	23.9938			0.836372			0.418186	−	0.908361i	$-0.362666\pi$
$823$	23.9938			0.836372			0.418186	+	0.908361i	$0.362666\pi$
$824$			13.0912i			0.456054i
$825$		0			0
$826$		−	8.06549i		−	0.280634i
$827$			49.4280i			1.71878i	0.511321	+	0.859390i	$0.329156\pi$
$827$			49.4280i			1.71878i	−0.511321	+	0.859390i	$0.670844\pi$
$828$		0			0
$829$	−11.7678			−0.408713			−0.204356	−	0.978897i	$-0.565510\pi$
$829$	−11.7678			−0.408713			−0.204356	+	0.978897i	$0.565510\pi$
$830$			28.5316i			0.990345i
$831$		0			0
$832$	3.27280	−	1.51286i	0.113464	−	0.0524491i
$833$	−7.09122			−0.245696
$834$		0			0
$835$	61.6900			2.13487
$836$	−15.1427			−0.523720
$837$		0			0
$838$			12.8371i			0.443451i
$839$			16.6687i			0.575468i	0.957710	+	0.287734i	$0.0929020\pi$
$839$			16.6687i			0.575468i	−0.957710	+	0.287734i	$0.907098\pi$
$840$		0			0
$841$	28.3251			0.976728
$842$	21.3313			0.735124
$843$		0			0
$844$	−15.0398			−0.517690
$845$	25.2082	+	21.4404i	0.867187	+	0.737571i
$846$		0			0
$847$			0.0973913i			0.00334640i
$848$	6.00000			0.206041
$849$		0			0
$850$			10.4958i			0.360004i
$851$		−	6.61727i		−	0.226837i
$852$		0			0
$853$			23.1286i			0.791909i	0.918270	+	0.395955i	$0.129586\pi$
$853$			23.1286i			0.791909i	−0.918270	+	0.395955i	$0.870414\pi$
$854$	0.785667			0.0268850
$855$		0			0
$856$			3.51988i			0.120307i
$857$	−13.6509			−0.466304			−0.233152	−	0.972440i	$-0.574904\pi$
$857$	−13.6509			−0.466304			−0.233152	+	0.972440i	$0.574904\pi$
$858$		0			0
$859$	−12.8371			−0.437997			−0.218998	−	0.975725i	$-0.570279\pi$
$859$	−12.8371			−0.437997			−0.218998	+	0.975725i	$0.570279\pi$
$860$		−	23.1427i		−	0.789158i
$861$		0			0
$862$	11.5713			0.394121
$863$		−	39.6228i		−	1.34878i	−0.738377	−	0.674388i	$-0.764408\pi$
$863$		−	39.6228i		−	1.34878i	0.738377	−	0.674388i	$-0.235592\pi$
$864$		0			0
$865$		−	52.4320i		−	1.78274i
$866$		−	29.3766i		−	0.998256i
$867$		0			0
$868$	−3.33127			−0.113071
$869$		−	16.1605i		−	0.548209i
$870$		0			0
$871$	−8.71400	−	18.8512i	−0.295263	−	0.638747i
$872$	−13.1427			−0.445067
$873$		0			0
$874$	8.00000			0.270604
$875$	8.96024			0.302911
$876$		0			0
$877$		−	47.3704i		−	1.59958i	−0.600277	−	0.799792i	$-0.704943\pi$
$877$		−	47.3704i		−	1.59958i	0.600277	−	0.799792i	$-0.295057\pi$
$878$		−	0.102905i		−	0.00347288i
$879$		0			0
$880$	8.48012			0.285865
$881$	43.8052			1.47584			0.737918	−	0.674891i	$-0.235809\pi$
$881$	43.8052			1.47584			0.737918	+	0.674891i	$0.235809\pi$
$882$		0			0
$883$	−1.81757			−0.0611661			−0.0305830	−	0.999532i	$-0.509736\pi$
$883$	−1.81757			−0.0611661			−0.0305830	+	0.999532i	$0.509736\pi$
$884$	23.2082	−	10.7280i	0.780575	−	0.360823i
$885$		0			0
$886$			27.1427i			0.911876i
$887$	−7.27365			−0.244225			−0.122113	−	0.992516i	$-0.538967\pi$
$887$	−7.27365			−0.244225			−0.122113	+	0.992516i	$0.538967\pi$
$888$		0			0
$889$			12.9026i			0.432740i
$890$		−	38.4163i		−	1.28772i
$891$		0			0
$892$		−	12.9026i		−	0.432011i
$893$	−51.5075			−1.72363
$894$		0			0
$895$		−	44.1029i		−	1.47420i
$896$	1.00000			0.0334077
$897$		0			0
$898$	9.70231			0.323770
$899$		−	25.2222i		−	0.841207i
$900$		0			0
$901$	42.5473			1.41746
$902$		−	14.1248i		−	0.470304i
$903$		0			0
$904$			1.75994i			0.0585348i
$905$		−	43.0631i		−	1.43147i
$906$		0			0
$907$	−12.4678			−0.413985			−0.206993	−	0.978342i	$-0.566368\pi$
$907$	−12.4678			−0.413985			−0.206993	+	0.978342i	$0.566368\pi$
$908$		−	9.02573i		−	0.299529i
$909$		0			0
$910$	3.85116	+	8.33127i	0.127665	+	0.276179i
$911$	−1.92047			−0.0636282			−0.0318141	−	0.999494i	$-0.510128\pi$
$911$	−1.92047			−0.0636282			−0.0318141	+	0.999494i	$0.510128\pi$
$912$		0			0
$913$	37.3375			1.23569
$914$	36.1701			1.19640
$915$		0			0
$916$			14.1684i			0.468137i
$917$		−	16.7795i		−	0.554108i
$918$		0			0
$919$	−13.2284			−0.436364			−0.218182	−	0.975908i	$-0.570013\pi$
$919$	−13.2284			−0.436364			−0.218182	+	0.975908i	$0.570013\pi$
$920$	−4.48012			−0.147705
$921$		0			0
$922$	−20.9743			−0.690751
$923$	−16.8573	−	36.4678i	−0.554866	−	1.20035i
$924$		0			0
$925$			5.56516i			0.182981i
$926$	24.8450			0.816457
$927$		0			0
$928$			7.57133i			0.248541i
$929$			1.30320i			0.0427567i	0.999771	+	0.0213783i	$0.00680545\pi$
$929$			1.30320i			0.0427567i	−0.999771	+	0.0213783i	$0.993195\pi$
$930$		0			0
$931$		−	4.54561i		−	0.148976i
$932$	−7.38273			−0.241829
$933$		0			0
$934$			1.27196i			0.0416199i
$935$	60.1343			1.96660
$936$		0			0
$937$	14.9602			0.488730			0.244365	−	0.969683i	$-0.421421\pi$
$937$	14.9602			0.488730			0.244365	+	0.969683i	$0.421421\pi$
$938$		−	5.75994i		−	0.188069i
$939$		0			0
$940$	28.8450			0.940820
$941$			9.93451i			0.323856i	0.986803	+	0.161928i	$0.0517712\pi$
$941$			9.93451i			0.323856i	−0.986803	+	0.161928i	$0.948229\pi$
$942$		0			0
$943$			7.46225i			0.243004i
$944$			8.06549i			0.262509i
$945$		0			0
$946$	−30.2853			−0.984661
$947$			40.4958i			1.31594i	0.753045	+	0.657969i	$0.228584\pi$
$947$			40.4958i			1.31594i	−0.753045	+	0.657969i	$0.771416\pi$
$948$		0			0
$949$	−14.1169	−	30.5394i	−0.458255	−	0.991352i
$950$	−6.72804			−0.218286
$951$		0			0
$952$	7.09122			0.229828
$953$	18.9602			0.614182			0.307091	−	0.951680i	$-0.400644\pi$
$953$	18.9602			0.614182			0.307091	+	0.951680i	$0.400644\pi$
$954$		0			0
$955$		−	11.2736i		−	0.364807i
$956$		−	6.13098i		−	0.198290i
$957$		0			0
$958$	−22.0515			−0.712450
$959$	−11.1427			−0.359816
$960$		0			0
$961$	19.9026			0.642020
$962$	12.3055	−	5.68828i	0.396747	−	0.183397i
$963$		0			0
$964$		−	27.3251i		−	0.880082i
$965$	6.31341			0.203236
$966$		0			0
$967$			4.63448i			0.149035i	0.997220	+	0.0745174i	$0.0237416\pi$
$967$			4.63448i			0.149035i	−0.997220	+	0.0745174i	$0.976258\pi$
$968$		−	0.0973913i		−	0.00313027i
$969$		0			0
$970$		−	19.7538i		−	0.634255i
$971$	−23.7336			−0.761646			−0.380823	−	0.924648i	$-0.624359\pi$
$971$	−23.7336			−0.761646			−0.380823	+	0.924648i	$0.624359\pi$
$972$		0			0
$973$		−	4.54561i		−	0.145725i
$974$	−4.42867			−0.141904
$975$		0			0
$976$	−0.785667			−0.0251486
$977$		−	4.74208i		−	0.151712i	−0.997119	−	0.0758562i	$-0.975831\pi$
$977$		−	4.74208i		−	0.151712i	0.997119	−	0.0758562i	$-0.0241690\pi$
$978$		0			0
$979$	−50.2730			−1.60673
$980$			2.54561i			0.0813165i
$981$		0			0
$982$		−	31.0274i		−	0.990124i
$983$			23.7414i			0.757233i	0.925554	+	0.378617i	$0.123600\pi$
$983$			23.7414i			0.757233i	−0.925554	+	0.378617i	$0.876400\pi$
$984$		0			0
$985$	22.5316			0.717916
$986$			53.6900i			1.70984i
$987$		0			0
$988$	6.87688	+	14.8769i	0.218783	+	0.473297i
$989$	16.0000			0.508770
$990$		0			0
$991$	40.8512			1.29768			0.648840	−	0.760925i	$-0.275254\pi$
$991$	40.8512			1.29768			0.648840	+	0.760925i	$0.275254\pi$
$992$	3.33127			0.105768
$993$		0			0
$994$		−	11.1427i		−	0.353424i
$995$			2.77781i			0.0880624i
$996$		0			0
$997$	−5.87688			−0.186123			−0.0930614	−	0.995660i	$-0.529665\pi$
$997$	−5.87688			−0.186123			−0.0930614	+	0.995660i	$0.529665\pi$
$998$	−23.0850			−0.730744
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.2.c.i.883.3		6
3.2	odd	2		182.2.d.b.155.6	yes	6
12.11	even	2		1456.2.k.b.337.1		6
13.12	even	2	inner	1638.2.c.i.883.4		6
21.2	odd	6		1274.2.n.m.753.4		12
21.5	even	6		1274.2.n.n.753.6		12
21.11	odd	6		1274.2.n.m.961.1		12
21.17	even	6		1274.2.n.n.961.3		12
21.20	even	2		1274.2.d.l.883.4		6
39.5	even	4		2366.2.a.bc.1.3		3
39.8	even	4		2366.2.a.x.1.3		3
39.38	odd	2		182.2.d.b.155.3	✓	6
156.155	even	2		1456.2.k.b.337.2		6
273.38	even	6		1274.2.n.n.961.6		12
273.116	odd	6		1274.2.n.m.961.4		12
273.194	even	6		1274.2.n.n.753.3		12
273.233	odd	6		1274.2.n.m.753.1		12
273.272	even	2		1274.2.d.l.883.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.2.d.b.155.3	✓	6	39.38	odd	2
182.2.d.b.155.6	yes	6	3.2	odd	2
1274.2.d.l.883.1		6	273.272	even	2
1274.2.d.l.883.4		6	21.20	even	2
1274.2.n.m.753.1		12	273.233	odd	6
1274.2.n.m.753.4		12	21.2	odd	6
1274.2.n.m.961.1		12	21.11	odd	6
1274.2.n.m.961.4		12	273.116	odd	6
1274.2.n.n.753.3		12	273.194	even	6
1274.2.n.n.753.6		12	21.5	even	6
1274.2.n.n.961.3		12	21.17	even	6
1274.2.n.n.961.6		12	273.38	even	6
1456.2.k.b.337.1		6	12.11	even	2
1456.2.k.b.337.2		6	156.155	even	2
1638.2.c.i.883.3		6	1.1	even	1	trivial
1638.2.c.i.883.4		6	13.12	even	2	inner
2366.2.a.x.1.3		3	39.8	even	4
2366.2.a.bc.1.3		3	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1638.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +2.54561i q^{5} -1.00000i q^{7} +1.00000i q^{8} +2.54561 q^{10} -3.33127i q^{11} +(-3.27280 + 1.51286i) q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.09122 q^{17} +4.54561i q^{19} -2.54561i q^{20} -3.33127 q^{22} +1.75994 q^{23} -1.48012 q^{25} +(1.51286 + 3.27280i) q^{26} +1.00000i q^{28} -7.57133 q^{29} +3.33127i q^{31} -1.00000i q^{32} -7.09122i q^{34} +2.54561 q^{35} -3.75994i q^{37} +4.54561 q^{38} -2.54561 q^{40} +4.24006i q^{41} +9.09122 q^{43} +3.33127i q^{44} -1.75994i q^{46} +11.3313i q^{47} -1.00000 q^{49} +1.48012i q^{50} +(3.27280 - 1.51286i) q^{52} +6.00000 q^{53} +8.48012 q^{55} +1.00000 q^{56} +7.57133i q^{58} +8.06549i q^{59} -0.785667 q^{61} +3.33127 q^{62} -1.00000 q^{64} +(-3.85116 - 8.33127i) q^{65} +5.75994i q^{67} -7.09122 q^{68} -2.54561i q^{70} +11.1427i q^{71} +9.33127i q^{73} -3.75994 q^{74} -4.54561i q^{76} -3.33127 q^{77} +4.85116 q^{79} +2.54561i q^{80} +4.24006 q^{82} +11.2082i q^{83} +18.0515i q^{85} -9.09122i q^{86} +3.33127 q^{88} -15.0912i q^{89} +(1.51286 + 3.27280i) q^{91} -1.75994 q^{92} +11.3313 q^{94} -11.5713 q^{95} -7.75994i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{10} - 10 q^{13} - 6 q^{14} + 6 q^{16} + 4 q^{17} + 14 q^{22} + 6 q^{23} - 18 q^{25} + 4 q^{26} - 16 q^{29} - 4 q^{35} + 8 q^{38} + 4 q^{40} + 16 q^{43} - 6 q^{49} + 10 q^{52} + 36 q^{53}+ \cdots - 40 q^{95}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^10 - 10 * q^13 - 6 * q^14 + 6 * q^16 + 4 * q^17 + 14 * q^22 + 6 * q^23 - 18 * q^25 + 4 * q^26 - 16 * q^29 - 4 * q^35 + 8 * q^38 + 4 * q^40 + 16 * q^43 - 6 * q^49 + 10 * q^52 + 36 * q^53 + 60 * q^55 + 6 * q^56 + 10 * q^61 - 14 * q^62 - 6 * q^64 + 20 * q^65 - 4 * q^68 - 18 * q^74 + 14 * q^77 - 14 * q^79 + 30 * q^82 - 14 * q^88 + 4 * q^91 - 6 * q^92 + 34 * q^94 - 40 * q^95

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