LMFDB - Embedded newform 2574.2.b.a.989.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2574,2,Mod(989,2574)] 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("2574.989"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2574, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2574.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$20.5534934803$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			989.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2574.989
Dual form			2574.2.b.a.989.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000i q^{5} +3.41421i q^{7} -1.00000 q^{8} -2.00000i q^{10} +(1.41421 - 3.00000i) q^{11} -1.00000i q^{13} -3.41421i q^{14} +1.00000 q^{16} -2.00000 q^{17} -6.82843i q^{19} +2.00000i q^{20} +(-1.41421 + 3.00000i) q^{22} -6.24264i q^{23} +1.00000 q^{25} +1.00000i q^{26} +3.41421i q^{28} -10.2426 q^{29} +1.17157 q^{31} -1.00000 q^{32} +2.00000 q^{34} -6.82843 q^{35} -6.24264 q^{37} +6.82843i q^{38} -2.00000i q^{40} -5.65685 q^{41} +6.00000i q^{43} +(1.41421 - 3.00000i) q^{44} +6.24264i q^{46} -13.3137i q^{47} -4.65685 q^{49} -1.00000 q^{50} -1.00000i q^{52} -10.4853i q^{53} +(6.00000 + 2.82843i) q^{55} -3.41421i q^{56} +10.2426 q^{58} +7.65685i q^{59} +2.82843i q^{61} -1.17157 q^{62} +1.00000 q^{64} +2.00000 q^{65} +5.41421 q^{67} -2.00000 q^{68} +6.82843 q^{70} -7.65685i q^{71} -13.8995i q^{73} +6.24264 q^{74} -6.82843i q^{76} +(10.2426 + 4.82843i) q^{77} +6.00000i q^{79} +2.00000i q^{80} +5.65685 q^{82} -8.00000 q^{83} -4.00000i q^{85} -6.00000i q^{86} +(-1.41421 + 3.00000i) q^{88} +9.17157i q^{89} +3.41421 q^{91} -6.24264i q^{92} +13.3137i q^{94} +13.6569 q^{95} +11.6569 q^{97} +4.65685 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 8 q^{17} + 4 q^{25} - 24 q^{29} + 16 q^{31} - 4 q^{32} + 8 q^{34} - 16 q^{35} - 8 q^{37} + 4 q^{49} - 4 q^{50} + 24 q^{55} + 24 q^{58} - 16 q^{62} + 4 q^{64}+ \cdots - 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 - 8 * q^17 + 4 * q^25 - 24 * q^29 + 16 * q^31 - 4 * q^32 + 8 * q^34 - 16 * q^35 - 8 * q^37 + 4 * q^49 - 4 * q^50 + 24 * q^55 + 24 * q^58 - 16 * q^62 + 4 * q^64 + 8 * q^65 + 16 * q^67 - 8 * q^68 + 16 * q^70 + 8 * q^74 + 24 * q^77 - 32 * q^83 + 8 * q^91 + 32 * q^95 + 24 * q^97 - 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2574\mathbb{Z}\right)^\times$.

$n$	$937$	$1145$	$1783$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$			3.41421i			1.29045i	0.763992	+	0.645226i	$0.223237\pi$
$7$			3.41421i			1.29045i	−0.763992	+	0.645226i	$0.776763\pi$
$8$	−1.00000			−0.353553
$9$		0			0
$10$		−	2.00000i		−	0.632456i
$11$	1.41421	−	3.00000i	0.426401	−	0.904534i
$11$	1.41421	−	3.00000i	0.426401	−	0.904534i
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$		−	3.41421i		−	0.912487i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$		−	6.82843i		−	1.56655i	−0.621676	−	0.783274i	$-0.713548\pi$
$19$		−	6.82843i		−	1.56655i	0.621676	−	0.783274i	$-0.286452\pi$
$20$			2.00000i			0.447214i
$21$		0			0
$22$	−1.41421	+	3.00000i	−0.301511	+	0.639602i
$23$		−	6.24264i		−	1.30168i	−0.759215	−	0.650840i	$-0.774417\pi$
$23$		−	6.24264i		−	1.30168i	0.759215	−	0.650840i	$-0.225583\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			1.00000i			0.196116i
$27$		0			0
$28$			3.41421i			0.645226i
$29$	−10.2426			−1.90201			−0.951005	−	0.309175i	$-0.899947\pi$
$29$	−10.2426			−1.90201			−0.951005	+	0.309175i	$0.899947\pi$
$30$		0			0
$31$	1.17157			0.210421			0.105210	−	0.994450i	$-0.466448\pi$
$31$	1.17157			0.210421			0.105210	+	0.994450i	$0.466448\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$	2.00000			0.342997
$35$	−6.82843			−1.15421
$36$		0			0
$37$	−6.24264			−1.02628			−0.513142	−	0.858304i	$-0.671519\pi$
$37$	−6.24264			−1.02628			−0.513142	+	0.858304i	$0.671519\pi$
$38$			6.82843i			1.10772i
$39$		0			0
$40$		−	2.00000i		−	0.316228i
$41$	−5.65685			−0.883452			−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685			−0.883452			−0.441726	+	0.897150i	$0.645634\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	1.41421	−	3.00000i	0.213201	−	0.452267i
$45$		0			0
$46$			6.24264i			0.920427i
$47$		−	13.3137i		−	1.94200i	−0.239071	−	0.971002i	$-0.576843\pi$
$47$		−	13.3137i		−	1.94200i	0.239071	−	0.971002i	$-0.423157\pi$
$48$		0			0
$49$	−4.65685			−0.665265
$50$	−1.00000			−0.141421
$51$		0			0
$52$		−	1.00000i		−	0.138675i
$53$		−	10.4853i		−	1.44026i	−0.693837	−	0.720132i	$-0.744081\pi$
$53$		−	10.4853i		−	1.44026i	0.693837	−	0.720132i	$-0.255919\pi$
$54$		0			0
$55$	6.00000	+	2.82843i	0.809040	+	0.381385i
$56$		−	3.41421i		−	0.456243i
$57$		0			0
$58$	10.2426			1.34492
$59$			7.65685i			0.996838i	0.866936	+	0.498419i	$0.166086\pi$
$59$			7.65685i			0.996838i	−0.866936	+	0.498419i	$0.833914\pi$
$60$		0			0
$61$			2.82843i			0.362143i	0.983470	+	0.181071i	$0.0579565\pi$
$61$			2.82843i			0.362143i	−0.983470	+	0.181071i	$0.942043\pi$
$62$	−1.17157			−0.148790
$63$		0			0
$64$	1.00000			0.125000
$65$	2.00000			0.248069
$66$		0			0
$67$	5.41421			0.661451			0.330726	−	0.943727i	$-0.392706\pi$
$67$	5.41421			0.661451			0.330726	+	0.943727i	$0.392706\pi$
$68$	−2.00000			−0.242536
$69$		0			0
$70$	6.82843			0.816153
$71$		−	7.65685i		−	0.908701i	−0.890823	−	0.454351i	$-0.849871\pi$
$71$		−	7.65685i		−	0.908701i	0.890823	−	0.454351i	$-0.150129\pi$
$72$		0			0
$73$		−	13.8995i		−	1.62681i	−0.581696	−	0.813406i	$-0.697611\pi$
$73$		−	13.8995i		−	1.62681i	0.581696	−	0.813406i	$-0.302389\pi$
$74$	6.24264			0.725692
$75$		0			0
$76$		−	6.82843i		−	0.783274i
$77$	10.2426	+	4.82843i	1.16726	+	0.550250i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$			2.00000i			0.223607i
$81$		0			0
$82$	5.65685			0.624695
$83$	−8.00000			−0.878114			−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000			−0.878114			−0.439057	+	0.898459i	$0.644687\pi$
$84$		0			0
$85$		−	4.00000i		−	0.433861i
$86$		−	6.00000i		−	0.646997i
$87$		0			0
$88$	−1.41421	+	3.00000i	−0.150756	+	0.319801i
$89$			9.17157i			0.972185i	0.873907	+	0.486092i	$0.161578\pi$
$89$			9.17157i			0.972185i	−0.873907	+	0.486092i	$0.838422\pi$
$90$		0			0
$91$	3.41421			0.357907
$92$		−	6.24264i		−	0.650840i
$93$		0			0
$94$			13.3137i			1.37320i
$95$	13.6569			1.40116
$96$		0			0
$97$	11.6569			1.18357			0.591787	−	0.806094i	$-0.298423\pi$
$97$	11.6569			1.18357			0.591787	+	0.806094i	$0.298423\pi$
$98$	4.65685			0.470413
$99$		0			0
$100$	1.00000			0.100000
$101$	−2.24264			−0.223151			−0.111576	−	0.993756i	$-0.535590\pi$
$101$	−2.24264			−0.223151			−0.111576	+	0.993756i	$0.535590\pi$
$102$		0			0
$103$	−2.34315			−0.230877			−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315			−0.230877			−0.115439	+	0.993315i	$0.536827\pi$
$104$			1.00000i			0.0980581i
$105$		0			0
$106$			10.4853i			1.01842i
$107$	5.89949			0.570326			0.285163	−	0.958479i	$-0.407952\pi$
$107$	5.89949			0.570326			0.285163	+	0.958479i	$0.407952\pi$
$108$		0			0
$109$		−	8.82843i		−	0.845610i	−0.906221	−	0.422805i	$-0.861046\pi$
$109$		−	8.82843i		−	0.845610i	0.906221	−	0.422805i	$-0.138954\pi$
$110$	−6.00000	−	2.82843i	−0.572078	−	0.269680i
$111$		0			0
$112$			3.41421i			0.322613i
$113$		−	9.89949i		−	0.931266i	−0.884978	−	0.465633i	$-0.845827\pi$
$113$		−	9.89949i		−	0.931266i	0.884978	−	0.465633i	$-0.154173\pi$
$114$		0			0
$115$	12.4853			1.16426
$116$	−10.2426			−0.951005
$117$		0			0
$118$		−	7.65685i		−	0.704871i
$119$		−	6.82843i		−	0.625961i
$120$		0			0
$121$	−7.00000	−	8.48528i	−0.636364	−	0.771389i
$122$		−	2.82843i		−	0.256074i
$123$		0			0
$124$	1.17157			0.105210
$125$			12.0000i			1.07331i
$126$		0			0
$127$			13.3137i			1.18140i	0.806891	+	0.590700i	$0.201148\pi$
$127$			13.3137i			1.18140i	−0.806891	+	0.590700i	$0.798852\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	−2.00000			−0.175412
$131$	1.89949			0.165960			0.0829798	−	0.996551i	$-0.473556\pi$
$131$	1.89949			0.165960			0.0829798	+	0.996551i	$0.473556\pi$
$132$		0			0
$133$	23.3137			2.02155
$134$	−5.41421			−0.467717
$135$		0			0
$136$	2.00000			0.171499
$137$		−	4.00000i		−	0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$		−	4.00000i		−	0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$		0			0
$139$		−	11.6569i		−	0.988721i	−0.869257	−	0.494361i	$-0.835402\pi$
$139$		−	11.6569i		−	0.988721i	0.869257	−	0.494361i	$-0.164598\pi$
$140$	−6.82843			−0.577107
$141$		0			0
$142$			7.65685i			0.642549i
$143$	−3.00000	−	1.41421i	−0.250873	−	0.118262i
$144$		0			0
$145$		−	20.4853i		−	1.70121i
$146$			13.8995i			1.15033i
$147$		0			0
$148$	−6.24264			−0.513142
$149$	−13.3137			−1.09070			−0.545351	−	0.838208i	$-0.683604\pi$
$149$	−13.3137			−1.09070			−0.545351	+	0.838208i	$0.683604\pi$
$150$		0			0
$151$		−	13.5563i		−	1.10320i	−0.834109	−	0.551600i	$-0.814017\pi$
$151$		−	13.5563i		−	1.10320i	0.834109	−	0.551600i	$-0.185983\pi$
$152$			6.82843i			0.553859i
$153$		0			0
$154$	−10.2426	−	4.82843i	−0.825376	−	0.389086i
$155$			2.34315i			0.188206i
$156$		0			0
$157$	18.4853			1.47529			0.737643	−	0.675191i	$-0.235939\pi$
$157$	18.4853			1.47529			0.737643	+	0.675191i	$0.235939\pi$
$158$		−	6.00000i		−	0.477334i
$159$		0			0
$160$		−	2.00000i		−	0.158114i
$161$	21.3137			1.67976
$162$		0			0
$163$	−12.7279			−0.996928			−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279			−0.996928			−0.498464	+	0.866910i	$0.666102\pi$
$164$	−5.65685			−0.441726
$165$		0			0
$166$	8.00000			0.620920
$167$	−3.31371			−0.256422			−0.128211	−	0.991747i	$-0.540924\pi$
$167$	−3.31371			−0.256422			−0.128211	+	0.991747i	$0.540924\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$			4.00000i			0.306786i
$171$		0			0
$172$			6.00000i			0.457496i
$173$	15.4142			1.17192			0.585960	−	0.810340i	$-0.300717\pi$
$173$	15.4142			1.17192			0.585960	+	0.810340i	$0.300717\pi$
$174$		0			0
$175$			3.41421i			0.258090i
$176$	1.41421	−	3.00000i	0.106600	−	0.226134i
$177$		0			0
$178$		−	9.17157i		−	0.687438i
$179$			14.1421i			1.05703i	0.848923	+	0.528516i	$0.177252\pi$
$179$			14.1421i			1.05703i	−0.848923	+	0.528516i	$0.822748\pi$
$180$		0			0
$181$	−3.17157			−0.235741			−0.117871	−	0.993029i	$-0.537607\pi$
$181$	−3.17157			−0.235741			−0.117871	+	0.993029i	$0.537607\pi$
$182$	−3.41421			−0.253078
$183$		0			0
$184$			6.24264i			0.460214i
$185$		−	12.4853i		−	0.917936i
$186$		0			0
$187$	−2.82843	+	6.00000i	−0.206835	+	0.438763i
$188$		−	13.3137i		−	0.971002i
$189$		0			0
$190$	−13.6569			−0.990772
$191$			9.07107i			0.656359i	0.944615	+	0.328180i	$0.106435\pi$
$191$			9.07107i			0.656359i	−0.944615	+	0.328180i	$0.893565\pi$
$192$		0			0
$193$		−	9.89949i		−	0.712581i	−0.934375	−	0.356291i	$-0.884041\pi$
$193$		−	9.89949i		−	0.712581i	0.934375	−	0.356291i	$-0.115959\pi$
$194$	−11.6569			−0.836913
$195$		0			0
$196$	−4.65685			−0.332632
$197$	−8.82843			−0.628999			−0.314500	−	0.949258i	$-0.601837\pi$
$197$	−8.82843			−0.628999			−0.314500	+	0.949258i	$0.601837\pi$
$198$		0			0
$199$	−27.3137			−1.93622			−0.968109	−	0.250530i	$-0.919395\pi$
$199$	−27.3137			−1.93622			−0.968109	+	0.250530i	$0.919395\pi$
$200$	−1.00000			−0.0707107
$201$		0			0
$202$	2.24264			0.157792
$203$		−	34.9706i		−	2.45445i
$204$		0			0
$205$		−	11.3137i		−	0.790184i
$206$	2.34315			0.163255
$207$		0			0
$208$		−	1.00000i		−	0.0693375i
$209$	−20.4853	−	9.65685i	−1.41700	−	0.667979i
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		−	10.4853i		−	0.720132i
$213$		0			0
$214$	−5.89949			−0.403281
$215$	−12.0000			−0.818393
$216$		0			0
$217$			4.00000i			0.271538i
$218$			8.82843i			0.597937i
$219$		0			0
$220$	6.00000	+	2.82843i	0.404520	+	0.190693i
$221$			2.00000i			0.134535i
$222$		0			0
$223$	−14.1421			−0.947027			−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421			−0.947027			−0.473514	+	0.880786i	$0.657015\pi$
$224$		−	3.41421i		−	0.228122i
$225$		0			0
$226$			9.89949i			0.658505i
$227$	17.6569			1.17193			0.585963	−	0.810338i	$-0.300716\pi$
$227$	17.6569			1.17193			0.585963	+	0.810338i	$0.300716\pi$
$228$		0			0
$229$	−10.7279			−0.708921			−0.354461	−	0.935071i	$-0.615335\pi$
$229$	−10.7279			−0.708921			−0.354461	+	0.935071i	$0.615335\pi$
$230$	−12.4853			−0.823255
$231$		0			0
$232$	10.2426			0.672462
$233$	21.3137			1.39631			0.698154	−	0.715948i	$-0.254005\pi$
$233$	21.3137			1.39631			0.698154	+	0.715948i	$0.254005\pi$
$234$		0			0
$235$	26.6274			1.73698
$236$			7.65685i			0.498419i
$237$		0			0
$238$			6.82843i			0.442621i
$239$	−20.1421			−1.30289			−0.651443	−	0.758697i	$-0.725836\pi$
$239$	−20.1421			−1.30289			−0.651443	+	0.758697i	$0.725836\pi$
$240$		0			0
$241$			19.7574i			1.27268i	0.771407	+	0.636342i	$0.219553\pi$
$241$			19.7574i			1.27268i	−0.771407	+	0.636342i	$0.780447\pi$
$242$	7.00000	+	8.48528i	0.449977	+	0.545455i
$243$		0			0
$244$			2.82843i			0.181071i
$245$		−	9.31371i		−	0.595031i
$246$		0			0
$247$	−6.82843			−0.434482
$248$	−1.17157			−0.0743950
$249$		0			0
$250$		−	12.0000i		−	0.758947i
$251$		−	10.8284i		−	0.683484i	−0.939794	−	0.341742i	$-0.888983\pi$
$251$		−	10.8284i		−	0.683484i	0.939794	−	0.341742i	$-0.111017\pi$
$252$		0			0
$253$	−18.7279	−	8.82843i	−1.17741	−	0.555038i
$254$		−	13.3137i		−	0.835376i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	6.58579i		−	0.410810i	−0.978677	−	0.205405i	$-0.934149\pi$
$257$		−	6.58579i		−	0.410810i	0.978677	−	0.205405i	$-0.0658512\pi$
$258$		0			0
$259$		−	21.3137i		−	1.32437i
$260$	2.00000			0.124035
$261$		0			0
$262$	−1.89949			−0.117351
$263$	20.9706			1.29310			0.646550	−	0.762871i	$-0.276211\pi$
$263$	20.9706			1.29310			0.646550	+	0.762871i	$0.276211\pi$
$264$		0			0
$265$	20.9706			1.28821
$266$	−23.3137			−1.42946
$267$		0			0
$268$	5.41421			0.330726
$269$			29.3137i			1.78729i	0.448776	+	0.893644i	$0.351860\pi$
$269$			29.3137i			1.78729i	−0.448776	+	0.893644i	$0.648140\pi$
$270$		0			0
$271$		−	25.5563i		−	1.55244i	−0.630464	−	0.776219i	$-0.717135\pi$
$271$		−	25.5563i		−	1.55244i	0.630464	−	0.776219i	$-0.282865\pi$
$272$	−2.00000			−0.121268
$273$		0			0
$274$			4.00000i			0.241649i
$275$	1.41421	−	3.00000i	0.0852803	−	0.180907i
$276$		0			0
$277$		−	26.2843i		−	1.57927i	−0.613578	−	0.789634i	$-0.710270\pi$
$277$		−	26.2843i		−	1.57927i	0.613578	−	0.789634i	$-0.289730\pi$
$278$			11.6569i			0.699132i
$279$		0			0
$280$	6.82843			0.408077
$281$	4.68629			0.279561			0.139780	−	0.990183i	$-0.455360\pi$
$281$	4.68629			0.279561			0.139780	+	0.990183i	$0.455360\pi$
$282$		0			0
$283$			15.3137i			0.910305i	0.890413	+	0.455153i	$0.150415\pi$
$283$			15.3137i			0.910305i	−0.890413	+	0.455153i	$0.849585\pi$
$284$		−	7.65685i		−	0.454351i
$285$		0			0
$286$	3.00000	+	1.41421i	0.177394	+	0.0836242i
$287$		−	19.3137i		−	1.14005i
$288$		0			0
$289$	−13.0000			−0.764706
$290$			20.4853i			1.20294i
$291$		0			0
$292$		−	13.8995i		−	0.813406i
$293$	8.34315			0.487412			0.243706	−	0.969849i	$-0.421637\pi$
$293$	8.34315			0.487412			0.243706	+	0.969849i	$0.421637\pi$
$294$		0			0
$295$	−15.3137			−0.891599
$296$	6.24264			0.362846
$297$		0			0
$298$	13.3137			0.771242
$299$	−6.24264			−0.361021
$300$		0			0
$301$	−20.4853			−1.18075
$302$			13.5563i			0.780080i
$303$		0			0
$304$		−	6.82843i		−	0.391637i
$305$	−5.65685			−0.323911
$306$		0			0
$307$		−	2.82843i		−	0.161427i	−0.996737	−	0.0807134i	$-0.974280\pi$
$307$		−	2.82843i		−	0.161427i	0.996737	−	0.0807134i	$-0.0257199\pi$
$308$	10.2426	+	4.82843i	0.583629	+	0.275125i
$309$		0			0
$310$		−	2.34315i		−	0.133082i
$311$		−	25.0711i		−	1.42165i	−0.703369	−	0.710825i	$-0.748322\pi$
$311$		−	25.0711i		−	1.42165i	0.703369	−	0.710825i	$-0.251678\pi$
$312$		0			0
$313$	20.9706			1.18533			0.592663	−	0.805450i	$-0.298077\pi$
$313$	20.9706			1.18533			0.592663	+	0.805450i	$0.298077\pi$
$314$	−18.4853			−1.04318
$315$		0			0
$316$			6.00000i			0.337526i
$317$		−	23.7990i		−	1.33668i	−0.743854	−	0.668342i	$-0.767004\pi$
$317$		−	23.7990i		−	1.33668i	0.743854	−	0.668342i	$-0.232996\pi$
$318$		0			0
$319$	−14.4853	+	30.7279i	−0.811020	+	1.72043i
$320$			2.00000i			0.111803i
$321$		0			0
$322$	−21.3137			−1.18777
$323$			13.6569i			0.759888i
$324$		0			0
$325$		−	1.00000i		−	0.0554700i
$326$	12.7279			0.704934
$327$		0			0
$328$	5.65685			0.312348
$329$	45.4558			2.50606
$330$		0			0
$331$	9.89949			0.544125			0.272063	−	0.962280i	$-0.412294\pi$
$331$	9.89949			0.544125			0.272063	+	0.962280i	$0.412294\pi$
$332$	−8.00000			−0.439057
$333$		0			0
$334$	3.31371			0.181318
$335$			10.8284i			0.591620i
$336$		0			0
$337$		−	18.1421i		−	0.988265i	−0.869387	−	0.494133i	$-0.835486\pi$
$337$		−	18.1421i		−	0.988265i	0.869387	−	0.494133i	$-0.164514\pi$
$338$	1.00000			0.0543928
$339$		0			0
$340$		−	4.00000i		−	0.216930i
$341$	1.65685	−	3.51472i	0.0897237	−	0.190333i
$342$		0			0
$343$			8.00000i			0.431959i
$344$		−	6.00000i		−	0.323498i
$345$		0			0
$346$	−15.4142			−0.828673
$347$	27.7574			1.49009			0.745047	−	0.667012i	$-0.232427\pi$
$347$	27.7574			1.49009			0.745047	+	0.667012i	$0.232427\pi$
$348$		0			0
$349$		−	15.1716i		−	0.812116i	−0.913847	−	0.406058i	$-0.866903\pi$
$349$		−	15.1716i		−	0.812116i	0.913847	−	0.406058i	$-0.133097\pi$
$350$		−	3.41421i		−	0.182497i
$351$		0			0
$352$	−1.41421	+	3.00000i	−0.0753778	+	0.159901i
$353$		−	23.3137i		−	1.24086i	−0.784260	−	0.620432i	$-0.786957\pi$
$353$		−	23.3137i		−	1.24086i	0.784260	−	0.620432i	$-0.213043\pi$
$354$		0			0
$355$	15.3137			0.812767
$356$			9.17157i			0.486092i
$357$		0			0
$358$		−	14.1421i		−	0.747435i
$359$	−8.00000			−0.422224			−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000			−0.422224			−0.211112	+	0.977462i	$0.567708\pi$
$360$		0			0
$361$	−27.6274			−1.45407
$362$	3.17157			0.166694
$363$		0			0
$364$	3.41421			0.178953
$365$	27.7990			1.45507
$366$		0			0
$367$	−4.14214			−0.216218			−0.108109	−	0.994139i	$-0.534480\pi$
$367$	−4.14214			−0.216218			−0.108109	+	0.994139i	$0.534480\pi$
$368$		−	6.24264i		−	0.325420i
$369$		0			0
$370$			12.4853i			0.649079i
$371$	35.7990			1.85859
$372$		0			0
$373$		−	1.31371i		−	0.0680212i	−0.999421	−	0.0340106i	$-0.989172\pi$
$373$		−	1.31371i		−	0.0680212i	0.999421	−	0.0340106i	$-0.0108280\pi$
$374$	2.82843	−	6.00000i	0.146254	−	0.310253i
$375$		0			0
$376$			13.3137i			0.686602i
$377$			10.2426i			0.527523i
$378$		0			0
$379$	26.3848			1.35529			0.677647	−	0.735387i	$-0.263000\pi$
$379$	26.3848			1.35529			0.677647	+	0.735387i	$0.263000\pi$
$380$	13.6569			0.700582
$381$		0			0
$382$		−	9.07107i		−	0.464116i
$383$			15.6569i			0.800028i	0.916509	+	0.400014i	$0.130995\pi$
$383$			15.6569i			0.800028i	−0.916509	+	0.400014i	$0.869005\pi$
$384$		0			0
$385$	−9.65685	+	20.4853i	−0.492159	+	1.04403i
$386$			9.89949i			0.503871i
$387$		0			0
$388$	11.6569			0.591787
$389$		−	2.00000i		−	0.101404i	−0.998714	−	0.0507020i	$-0.983854\pi$
$389$		−	2.00000i		−	0.101404i	0.998714	−	0.0507020i	$-0.0161459\pi$
$390$		0			0
$391$			12.4853i			0.631408i
$392$	4.65685			0.235207
$393$		0			0
$394$	8.82843			0.444770
$395$	−12.0000			−0.603786
$396$		0			0
$397$	−1.75736			−0.0881993			−0.0440997	−	0.999027i	$-0.514042\pi$
$397$	−1.75736			−0.0881993			−0.0440997	+	0.999027i	$0.514042\pi$
$398$	27.3137			1.36911
$399$		0			0
$400$	1.00000			0.0500000
$401$			2.82843i			0.141245i	0.997503	+	0.0706225i	$0.0224986\pi$
$401$			2.82843i			0.141245i	−0.997503	+	0.0706225i	$0.977501\pi$
$402$		0			0
$403$		−	1.17157i		−	0.0583602i
$404$	−2.24264			−0.111576
$405$		0			0
$406$			34.9706i			1.73556i
$407$	−8.82843	+	18.7279i	−0.437609	+	0.928309i
$408$		0			0
$409$		−	12.2426i		−	0.605360i	−0.953092	−	0.302680i	$-0.902119\pi$
$409$		−	12.2426i		−	0.605360i	0.953092	−	0.302680i	$-0.0978813\pi$
$410$			11.3137i			0.558744i
$411$		0			0
$412$	−2.34315			−0.115439
$413$	−26.1421			−1.28637
$414$		0			0
$415$		−	16.0000i		−	0.785409i
$416$			1.00000i			0.0490290i
$417$		0			0
$418$	20.4853	+	9.65685i	1.00197	+	0.472332i
$419$			9.17157i			0.448061i	0.974582	+	0.224030i	$0.0719215\pi$
$419$			9.17157i			0.448061i	−0.974582	+	0.224030i	$0.928079\pi$
$420$		0			0
$421$	15.8995			0.774894			0.387447	−	0.921892i	$-0.373357\pi$
$421$	15.8995			0.774894			0.387447	+	0.921892i	$0.373357\pi$
$422$		−	12.0000i		−	0.584151i
$423$		0			0
$424$			10.4853i			0.509210i
$425$	−2.00000			−0.0970143
$426$		0			0
$427$	−9.65685			−0.467328
$428$	5.89949			0.285163
$429$		0			0
$430$	12.0000			0.578691
$431$	−14.4853			−0.697731			−0.348866	−	0.937173i	$-0.613433\pi$
$431$	−14.4853			−0.697731			−0.348866	+	0.937173i	$0.613433\pi$
$432$		0			0
$433$	−39.6569			−1.90579			−0.952893	−	0.303305i	$-0.901910\pi$
$433$	−39.6569			−1.90579			−0.952893	+	0.303305i	$0.901910\pi$
$434$		−	4.00000i		−	0.192006i
$435$		0			0
$436$		−	8.82843i		−	0.422805i
$437$	−42.6274			−2.03915
$438$		0			0
$439$			17.7990i			0.849499i	0.905311	+	0.424750i	$0.139638\pi$
$439$			17.7990i			0.849499i	−0.905311	+	0.424750i	$0.860362\pi$
$440$	−6.00000	−	2.82843i	−0.286039	−	0.134840i
$441$		0			0
$442$		−	2.00000i		−	0.0951303i
$443$			26.1421i			1.24205i	0.783791	+	0.621025i	$0.213284\pi$
$443$			26.1421i			1.24205i	−0.783791	+	0.621025i	$0.786716\pi$
$444$		0			0
$445$	−18.3431			−0.869549
$446$	14.1421			0.669650
$447$		0			0
$448$			3.41421i			0.161306i
$449$		−	23.5147i		−	1.10973i	−0.831941	−	0.554864i	$-0.812770\pi$
$449$		−	23.5147i		−	1.10973i	0.831941	−	0.554864i	$-0.187230\pi$
$450$		0			0
$451$	−8.00000	+	16.9706i	−0.376705	+	0.799113i
$452$		−	9.89949i		−	0.465633i
$453$		0			0
$454$	−17.6569			−0.828677
$455$			6.82843i			0.320122i
$456$		0			0
$457$		−	12.2426i		−	0.572687i	−0.958127	−	0.286343i	$-0.907560\pi$
$457$		−	12.2426i		−	0.572687i	0.958127	−	0.286343i	$-0.0924398\pi$
$458$	10.7279			0.501283
$459$		0			0
$460$	12.4853			0.582129
$461$	15.1716			0.706611			0.353305	−	0.935508i	$-0.385058\pi$
$461$	15.1716			0.706611			0.353305	+	0.935508i	$0.385058\pi$
$462$		0			0
$463$	−2.82843			−0.131448			−0.0657241	−	0.997838i	$-0.520936\pi$
$463$	−2.82843			−0.131448			−0.0657241	+	0.997838i	$0.520936\pi$
$464$	−10.2426			−0.475503
$465$		0			0
$466$	−21.3137			−0.987338
$467$			29.6569i			1.37236i	0.727434	+	0.686178i	$0.240713\pi$
$467$			29.6569i			1.37236i	−0.727434	+	0.686178i	$0.759287\pi$
$468$		0			0
$469$			18.4853i			0.853571i
$470$	−26.6274			−1.22823
$471$		0			0
$472$		−	7.65685i		−	0.352435i
$473$	18.0000	+	8.48528i	0.827641	+	0.390154i
$474$		0			0
$475$		−	6.82843i		−	0.313310i
$476$		−	6.82843i		−	0.312980i
$477$		0			0
$478$	20.1421			0.921280
$479$	−22.6274			−1.03387			−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274			−1.03387			−0.516937	+	0.856024i	$0.672928\pi$
$480$		0			0
$481$			6.24264i			0.284640i
$482$		−	19.7574i		−	0.899923i
$483$		0			0
$484$	−7.00000	−	8.48528i	−0.318182	−	0.385695i
$485$			23.3137i			1.05862i
$486$		0			0
$487$	−32.4853			−1.47205			−0.736024	−	0.676955i	$-0.763299\pi$
$487$	−32.4853			−1.47205			−0.736024	+	0.676955i	$0.763299\pi$
$488$		−	2.82843i		−	0.128037i
$489$		0			0
$490$			9.31371i			0.420750i
$491$	−5.89949			−0.266240			−0.133120	−	0.991100i	$-0.542500\pi$
$491$	−5.89949			−0.266240			−0.133120	+	0.991100i	$0.542500\pi$
$492$		0			0
$493$	20.4853			0.922611
$494$	6.82843			0.307225
$495$		0			0
$496$	1.17157			0.0526052
$497$	26.1421			1.17264
$498$		0			0
$499$	24.0416			1.07625			0.538126	−	0.842865i	$-0.319133\pi$
$499$	24.0416			1.07625			0.538126	+	0.842865i	$0.319133\pi$
$500$			12.0000i			0.536656i
$501$		0			0
$502$			10.8284i			0.483296i
$503$	−26.1421			−1.16562			−0.582810	−	0.812608i	$-0.698047\pi$
$503$	−26.1421			−1.16562			−0.582810	+	0.812608i	$0.698047\pi$
$504$		0			0
$505$		−	4.48528i		−	0.199592i
$506$	18.7279	+	8.82843i	0.832558	+	0.392471i
$507$		0			0
$508$			13.3137i			0.590700i
$509$			26.1421i			1.15873i	0.815068	+	0.579365i	$0.196699\pi$
$509$			26.1421i			1.15873i	−0.815068	+	0.579365i	$0.803301\pi$
$510$		0			0
$511$	47.4558			2.09932
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			6.58579i			0.290487i
$515$		−	4.68629i		−	0.206503i
$516$		0			0
$517$	−39.9411	−	18.8284i	−1.75661	−	0.828073i
$518$			21.3137i			0.936471i
$519$		0			0
$520$	−2.00000			−0.0877058
$521$			30.3848i			1.33118i	0.746317	+	0.665591i	$0.231820\pi$
$521$			30.3848i			1.33118i	−0.746317	+	0.665591i	$0.768180\pi$
$522$		0			0
$523$		−	28.9706i		−	1.26679i	−0.773827	−	0.633397i	$-0.781660\pi$
$523$		−	28.9706i		−	1.26679i	0.773827	−	0.633397i	$-0.218340\pi$
$524$	1.89949			0.0829798
$525$		0			0
$526$	−20.9706			−0.914360
$527$	−2.34315			−0.102069
$528$		0			0
$529$	−15.9706			−0.694372
$530$	−20.9706			−0.910903
$531$		0			0
$532$	23.3137			1.01078
$533$			5.65685i			0.245026i
$534$		0			0
$535$			11.7990i			0.510115i
$536$	−5.41421			−0.233858
$537$		0			0
$538$		−	29.3137i		−	1.26380i
$539$	−6.58579	+	13.9706i	−0.283670	+	0.601755i
$540$		0			0
$541$			21.3137i			0.916348i	0.888863	+	0.458174i	$0.151496\pi$
$541$			21.3137i			0.916348i	−0.888863	+	0.458174i	$0.848504\pi$
$542$			25.5563i			1.09774i
$543$		0			0
$544$	2.00000			0.0857493
$545$	17.6569			0.756337
$546$		0			0
$547$		−	12.3431i		−	0.527755i	−0.964556	−	0.263877i	$-0.914999\pi$
$547$		−	12.3431i		−	0.527755i	0.964556	−	0.263877i	$-0.0850014\pi$
$548$		−	4.00000i		−	0.170872i
$549$		0			0
$550$	−1.41421	+	3.00000i	−0.0603023	+	0.127920i
$551$			69.9411i			2.97959i
$552$		0			0
$553$	−20.4853			−0.871123
$554$			26.2843i			1.11671i
$555$		0			0
$556$		−	11.6569i		−	0.494361i
$557$	−45.5980			−1.93205			−0.966024	−	0.258452i	$-0.916788\pi$
$557$	−45.5980			−1.93205			−0.966024	+	0.258452i	$0.916788\pi$
$558$		0			0
$559$	6.00000			0.253773
$560$	−6.82843			−0.288554
$561$		0			0
$562$	−4.68629			−0.197679
$563$	19.5563			0.824202			0.412101	−	0.911138i	$-0.364795\pi$
$563$	19.5563			0.824202			0.412101	+	0.911138i	$0.364795\pi$
$564$		0			0
$565$	19.7990			0.832950
$566$		−	15.3137i		−	0.643683i
$567$		0			0
$568$			7.65685i			0.321274i
$569$	−18.2843			−0.766517			−0.383258	−	0.923641i	$-0.625198\pi$
$569$	−18.2843			−0.766517			−0.383258	+	0.923641i	$0.625198\pi$
$570$		0			0
$571$			16.9706i			0.710196i	0.934829	+	0.355098i	$0.115552\pi$
$571$			16.9706i			0.710196i	−0.934829	+	0.355098i	$0.884448\pi$
$572$	−3.00000	−	1.41421i	−0.125436	−	0.0591312i
$573$		0			0
$574$			19.3137i			0.806139i
$575$		−	6.24264i		−	0.260336i
$576$		0			0
$577$	−32.8284			−1.36667			−0.683333	−	0.730107i	$-0.739470\pi$
$577$	−32.8284			−1.36667			−0.683333	+	0.730107i	$0.739470\pi$
$578$	13.0000			0.540729
$579$		0			0
$580$		−	20.4853i		−	0.850605i
$581$		−	27.3137i		−	1.13316i
$582$		0			0
$583$	−31.4558	−	14.8284i	−1.30277	−	0.614131i
$584$			13.8995i			0.575165i
$585$		0			0
$586$	−8.34315			−0.344652
$587$			10.3431i			0.426907i	0.976953	+	0.213454i	$0.0684712\pi$
$587$			10.3431i			0.426907i	−0.976953	+	0.213454i	$0.931529\pi$
$588$		0			0
$589$		−	8.00000i		−	0.329634i
$590$	15.3137			0.630455
$591$		0			0
$592$	−6.24264			−0.256571
$593$	20.9706			0.861158			0.430579	−	0.902553i	$-0.358309\pi$
$593$	20.9706			0.861158			0.430579	+	0.902553i	$0.358309\pi$
$594$		0			0
$595$	13.6569			0.559876
$596$	−13.3137			−0.545351
$597$		0			0
$598$	6.24264			0.255281
$599$			38.0416i			1.55434i	0.629291	+	0.777169i	$0.283345\pi$
$599$			38.0416i			1.55434i	−0.629291	+	0.777169i	$0.716655\pi$
$600$		0			0
$601$			1.85786i			0.0757839i	0.999282	+	0.0378919i	$0.0120643\pi$
$601$			1.85786i			0.0757839i	−0.999282	+	0.0378919i	$0.987936\pi$
$602$	20.4853			0.834918
$603$		0			0
$604$		−	13.5563i		−	0.551600i
$605$	16.9706	−	14.0000i	0.689951	−	0.569181i
$606$		0			0
$607$		−	36.8284i		−	1.49482i	−0.664363	−	0.747410i	$-0.731297\pi$
$607$		−	36.8284i		−	1.49482i	0.664363	−	0.747410i	$-0.268703\pi$
$608$			6.82843i			0.276929i
$609$		0			0
$610$	5.65685			0.229039
$611$	−13.3137			−0.538615
$612$		0			0
$613$			17.7990i			0.718894i	0.933165	+	0.359447i	$0.117035\pi$
$613$			17.7990i			0.718894i	−0.933165	+	0.359447i	$0.882965\pi$
$614$			2.82843i			0.114146i
$615$		0			0
$616$	−10.2426	−	4.82843i	−0.412688	−	0.194543i
$617$			40.9706i			1.64941i	0.565561	+	0.824706i	$0.308660\pi$
$617$			40.9706i			1.64941i	−0.565561	+	0.824706i	$0.691340\pi$
$618$		0			0
$619$	26.1005			1.04907			0.524534	−	0.851390i	$-0.324240\pi$
$619$	26.1005			1.04907			0.524534	+	0.851390i	$0.324240\pi$
$620$			2.34315i			0.0941030i
$621$		0			0
$622$			25.0711i			1.00526i
$623$	−31.3137			−1.25456
$624$		0			0
$625$	−19.0000			−0.760000
$626$	−20.9706			−0.838152
$627$		0			0
$628$	18.4853			0.737643
$629$	12.4853			0.497821
$630$		0			0
$631$	19.3137			0.768867			0.384433	−	0.923153i	$-0.374397\pi$
$631$	19.3137			0.768867			0.384433	+	0.923153i	$0.374397\pi$
$632$		−	6.00000i		−	0.238667i
$633$		0			0
$634$			23.7990i			0.945179i
$635$	−26.6274			−1.05668
$636$		0			0
$637$			4.65685i			0.184511i
$638$	14.4853	−	30.7279i	0.573478	−	1.21653i
$639$		0			0
$640$		−	2.00000i		−	0.0790569i
$641$		−	48.7279i		−	1.92464i	−0.271924	−	0.962319i	$-0.587660\pi$
$641$		−	48.7279i		−	1.92464i	0.271924	−	0.962319i	$-0.412340\pi$
$642$		0			0
$643$	11.5563			0.455738			0.227869	−	0.973692i	$-0.426824\pi$
$643$	11.5563			0.455738			0.227869	+	0.973692i	$0.426824\pi$
$644$	21.3137			0.839878
$645$		0			0
$646$		−	13.6569i		−	0.537322i
$647$			16.8701i			0.663230i	0.943415	+	0.331615i	$0.107594\pi$
$647$			16.8701i			0.663230i	−0.943415	+	0.331615i	$0.892406\pi$
$648$		0			0
$649$	22.9706	+	10.8284i	0.901673	+	0.425053i
$650$			1.00000i			0.0392232i
$651$		0			0
$652$	−12.7279			−0.498464
$653$		−	41.7990i		−	1.63572i	−0.575417	−	0.817860i	$-0.695160\pi$
$653$		−	41.7990i		−	1.63572i	0.575417	−	0.817860i	$-0.304840\pi$
$654$		0			0
$655$			3.79899i			0.148439i
$656$	−5.65685			−0.220863
$657$		0			0
$658$	−45.4558			−1.77205
$659$	14.8701			0.579255			0.289628	−	0.957139i	$-0.406469\pi$
$659$	14.8701			0.579255			0.289628	+	0.957139i	$0.406469\pi$
$660$		0			0
$661$	−43.4142			−1.68862			−0.844309	−	0.535857i	$-0.819989\pi$
$661$	−43.4142			−1.68862			−0.844309	+	0.535857i	$0.819989\pi$
$662$	−9.89949			−0.384755
$663$		0			0
$664$	8.00000			0.310460
$665$			46.6274i			1.80813i
$666$		0			0
$667$			63.9411i			2.47581i
$668$	−3.31371			−0.128211
$669$		0			0
$670$		−	10.8284i		−	0.418339i
$671$	8.48528	+	4.00000i	0.327571	+	0.154418i
$672$		0			0
$673$			32.0000i			1.23351i	0.787155	+	0.616755i	$0.211553\pi$
$673$			32.0000i			1.23351i	−0.787155	+	0.616755i	$0.788447\pi$
$674$			18.1421i			0.698809i
$675$		0			0
$676$	−1.00000			−0.0384615
$677$	38.5269			1.48071			0.740355	−	0.672216i	$-0.234658\pi$
$677$	38.5269			1.48071			0.740355	+	0.672216i	$0.234658\pi$
$678$		0			0
$679$			39.7990i			1.52735i
$680$			4.00000i			0.153393i
$681$		0			0
$682$	−1.65685	+	3.51472i	−0.0634442	+	0.134586i
$683$			6.62742i			0.253591i	0.991929	+	0.126796i	$0.0404692\pi$
$683$			6.62742i			0.253591i	−0.991929	+	0.126796i	$0.959531\pi$
$684$		0			0
$685$	8.00000			0.305664
$686$		−	8.00000i		−	0.305441i
$687$		0			0
$688$			6.00000i			0.228748i
$689$	−10.4853			−0.399457
$690$		0			0
$691$	−25.2132			−0.959155			−0.479578	−	0.877499i	$-0.659210\pi$
$691$	−25.2132			−0.959155			−0.479578	+	0.877499i	$0.659210\pi$
$692$	15.4142			0.585960
$693$		0			0
$694$	−27.7574			−1.05365
$695$	23.3137			0.884339
$696$		0			0
$697$	11.3137			0.428537
$698$			15.1716i			0.574253i
$699$		0			0
$700$			3.41421i			0.129045i
$701$	−14.2426			−0.537937			−0.268969	−	0.963149i	$-0.586683\pi$
$701$	−14.2426			−0.537937			−0.268969	+	0.963149i	$0.586683\pi$
$702$		0			0
$703$			42.6274i			1.60772i
$704$	1.41421	−	3.00000i	0.0533002	−	0.113067i
$705$		0			0
$706$			23.3137i			0.877423i
$707$		−	7.65685i		−	0.287966i
$708$		0			0
$709$	−18.7279			−0.703342			−0.351671	−	0.936124i	$-0.614386\pi$
$709$	−18.7279			−0.703342			−0.351671	+	0.936124i	$0.614386\pi$
$710$	−15.3137			−0.574713
$711$		0			0
$712$		−	9.17157i		−	0.343719i
$713$		−	7.31371i		−	0.273901i
$714$		0			0
$715$	2.82843	−	6.00000i	0.105777	−	0.224387i
$716$			14.1421i			0.528516i
$717$		0			0
$718$	8.00000			0.298557
$719$		−	29.7574i		−	1.10976i	−0.831930	−	0.554881i	$-0.812764\pi$
$719$		−	29.7574i		−	1.10976i	0.831930	−	0.554881i	$-0.187236\pi$
$720$		0			0
$721$		−	8.00000i		−	0.297936i
$722$	27.6274			1.02819
$723$		0			0
$724$	−3.17157			−0.117871
$725$	−10.2426			−0.380402
$726$		0			0
$727$	49.1127			1.82149			0.910745	−	0.412970i	$-0.135509\pi$
$727$	49.1127			1.82149			0.910745	+	0.412970i	$0.135509\pi$
$728$	−3.41421			−0.126539
$729$		0			0
$730$	−27.7990			−1.02889
$731$		−	12.0000i		−	0.443836i
$732$		0			0
$733$		−	6.00000i		−	0.221615i	−0.993842	−	0.110808i	$-0.964656\pi$
$733$		−	6.00000i		−	0.221615i	0.993842	−	0.110808i	$-0.0353437\pi$
$734$	4.14214			0.152889
$735$		0			0
$736$			6.24264i			0.230107i
$737$	7.65685	−	16.2426i	0.282044	−	0.598305i
$738$		0			0
$739$			16.2843i			0.599027i	0.954092	+	0.299513i	$0.0968243\pi$
$739$			16.2843i			0.599027i	−0.954092	+	0.299513i	$0.903176\pi$
$740$		−	12.4853i		−	0.458968i
$741$		0			0
$742$	−35.7990			−1.31422
$743$	9.51472			0.349061			0.174531	−	0.984652i	$-0.444159\pi$
$743$	9.51472			0.349061			0.174531	+	0.984652i	$0.444159\pi$
$744$		0			0
$745$		−	26.6274i		−	0.975553i
$746$			1.31371i			0.0480983i
$747$		0			0
$748$	−2.82843	+	6.00000i	−0.103418	+	0.219382i
$749$			20.1421i			0.735978i
$750$		0			0
$751$	−29.7990			−1.08738			−0.543690	−	0.839286i	$-0.682973\pi$
$751$	−29.7990			−1.08738			−0.543690	+	0.839286i	$0.682973\pi$
$752$		−	13.3137i		−	0.485501i
$753$		0			0
$754$		−	10.2426i		−	0.373015i
$755$	27.1127			0.986732
$756$		0			0
$757$	34.4853			1.25339			0.626694	−	0.779265i	$-0.284407\pi$
$757$	34.4853			1.25339			0.626694	+	0.779265i	$0.284407\pi$
$758$	−26.3848			−0.958338
$759$		0			0
$760$	−13.6569			−0.495386
$761$	24.6274			0.892743			0.446372	−	0.894848i	$-0.352716\pi$
$761$	24.6274			0.892743			0.446372	+	0.894848i	$0.352716\pi$
$762$		0			0
$763$	30.1421			1.09122
$764$			9.07107i			0.328180i
$765$		0			0
$766$		−	15.6569i		−	0.565705i
$767$	7.65685			0.276473
$768$		0			0
$769$		−	15.5563i		−	0.560976i	−0.959857	−	0.280488i	$-0.909504\pi$
$769$		−	15.5563i		−	0.560976i	0.959857	−	0.280488i	$-0.0904963\pi$
$770$	9.65685	−	20.4853i	0.348009	−	0.738238i
$771$		0			0
$772$		−	9.89949i		−	0.356291i
$773$		−	32.4853i		−	1.16841i	−0.811605	−	0.584207i	$-0.801406\pi$
$773$		−	32.4853i		−	1.16841i	0.811605	−	0.584207i	$-0.198594\pi$
$774$		0			0
$775$	1.17157			0.0420841
$776$	−11.6569			−0.418457
$777$		0			0
$778$			2.00000i			0.0717035i
$779$			38.6274i			1.38397i
$780$		0			0
$781$	−22.9706	−	10.8284i	−0.821951	−	0.387472i
$782$		−	12.4853i		−	0.446473i
$783$		0			0
$784$	−4.65685			−0.166316
$785$			36.9706i			1.31954i
$786$		0			0
$787$			37.6569i			1.34232i	0.741312	+	0.671161i	$0.234204\pi$
$787$			37.6569i			1.34232i	−0.741312	+	0.671161i	$0.765796\pi$
$788$	−8.82843			−0.314500
$789$		0			0
$790$	12.0000			0.426941
$791$	33.7990			1.20175
$792$		0			0
$793$	2.82843			0.100440
$794$	1.75736			0.0623663
$795$		0			0
$796$	−27.3137			−0.968109
$797$			19.1716i			0.679092i	0.940589	+	0.339546i	$0.110273\pi$
$797$			19.1716i			0.679092i	−0.940589	+	0.339546i	$0.889727\pi$
$798$		0			0
$799$			26.6274i			0.942010i
$800$	−1.00000			−0.0353553
$801$		0			0
$802$		−	2.82843i		−	0.0998752i
$803$	−41.6985	−	19.6569i	−1.47151	−	0.693675i
$804$		0			0
$805$			42.6274i			1.50242i
$806$			1.17157i			0.0412669i
$807$		0			0
$808$	2.24264			0.0788958
$809$	−1.51472			−0.0532547			−0.0266273	−	0.999645i	$-0.508477\pi$
$809$	−1.51472			−0.0532547			−0.0266273	+	0.999645i	$0.508477\pi$
$810$		0			0
$811$		−	16.6863i		−	0.585935i	−0.956122	−	0.292967i	$-0.905357\pi$
$811$		−	16.6863i		−	0.585935i	0.956122	−	0.292967i	$-0.0946428\pi$
$812$		−	34.9706i		−	1.22723i
$813$		0			0
$814$	8.82843	−	18.7279i	0.309436	−	0.656413i
$815$		−	25.4558i		−	0.891679i
$816$		0			0
$817$	40.9706			1.43338
$818$			12.2426i			0.428054i
$819$		0			0
$820$		−	11.3137i		−	0.395092i
$821$	22.6863			0.791757			0.395879	−	0.918303i	$-0.370440\pi$
$821$	22.6863			0.791757			0.395879	+	0.918303i	$0.370440\pi$
$822$		0			0
$823$	−25.7990			−0.899296			−0.449648	−	0.893206i	$-0.648451\pi$
$823$	−25.7990			−0.899296			−0.449648	+	0.893206i	$0.648451\pi$
$824$	2.34315			0.0816274
$825$		0			0
$826$	26.1421			0.909601
$827$	−46.6274			−1.62139			−0.810697	−	0.585466i	$-0.800912\pi$
$827$	−46.6274			−1.62139			−0.810697	+	0.585466i	$0.800912\pi$
$828$		0			0
$829$	9.02944			0.313605			0.156803	−	0.987630i	$-0.449881\pi$
$829$	9.02944			0.313605			0.156803	+	0.987630i	$0.449881\pi$
$830$			16.0000i			0.555368i
$831$		0			0
$832$		−	1.00000i		−	0.0346688i
$833$	9.31371			0.322701
$834$		0			0
$835$		−	6.62742i		−	0.229351i
$836$	−20.4853	−	9.65685i	−0.708498	−	0.333989i
$837$		0			0
$838$		−	9.17157i		−	0.316827i
$839$		−	11.8579i		−	0.409379i	−0.978827	−	0.204689i	$-0.934382\pi$
$839$		−	11.8579i		−	0.409379i	0.978827	−	0.204689i	$-0.0656185\pi$
$840$		0			0
$841$	75.9117			2.61764
$842$	−15.8995			−0.547933
$843$		0			0
$844$			12.0000i			0.413057i
$845$		−	2.00000i		−	0.0688021i
$846$		0			0
$847$	28.9706	−	23.8995i	0.995440	−	0.821196i
$848$		−	10.4853i		−	0.360066i
$849$		0			0
$850$	2.00000			0.0685994
$851$			38.9706i			1.33589i
$852$		0			0
$853$		−	44.6274i		−	1.52801i	−0.645208	−	0.764007i	$-0.723229\pi$
$853$		−	44.6274i		−	1.52801i	0.645208	−	0.764007i	$-0.276771\pi$
$854$	9.65685			0.330451
$855$		0			0
$856$	−5.89949			−0.201641
$857$	−1.51472			−0.0517418			−0.0258709	−	0.999665i	$-0.508236\pi$
$857$	−1.51472			−0.0517418			−0.0258709	+	0.999665i	$0.508236\pi$
$858$		0			0
$859$	11.1127			0.379160			0.189580	−	0.981865i	$-0.439287\pi$
$859$	11.1127			0.379160			0.189580	+	0.981865i	$0.439287\pi$
$860$	−12.0000			−0.409197
$861$		0			0
$862$	14.4853			0.493371
$863$		−	1.79899i		−	0.0612383i	−0.999531	−	0.0306192i	$-0.990252\pi$
$863$		−	1.79899i		−	0.0612383i	0.999531	−	0.0306192i	$-0.00974791\pi$
$864$		0			0
$865$			30.8284i			1.04820i
$866$	39.6569			1.34759
$867$		0			0
$868$			4.00000i			0.135769i
$869$	18.0000	+	8.48528i	0.610608	+	0.287843i
$870$		0			0
$871$		−	5.41421i		−	0.183454i
$872$			8.82843i			0.298968i
$873$		0			0
$874$	42.6274			1.44189
$875$	−40.9706			−1.38506
$876$		0			0
$877$			20.6274i			0.696538i	0.937395	+	0.348269i	$0.113230\pi$
$877$			20.6274i			0.696538i	−0.937395	+	0.348269i	$0.886770\pi$
$878$		−	17.7990i		−	0.600687i
$879$		0			0
$880$	6.00000	+	2.82843i	0.202260	+	0.0953463i
$881$		−	48.0416i		−	1.61856i	−0.587421	−	0.809282i	$-0.699857\pi$
$881$		−	48.0416i		−	1.61856i	0.587421	−	0.809282i	$-0.300143\pi$
$882$		0			0
$883$	−51.5980			−1.73641			−0.868205	−	0.496205i	$-0.834726\pi$
$883$	−51.5980			−1.73641			−0.868205	+	0.496205i	$0.834726\pi$
$884$			2.00000i			0.0672673i
$885$		0			0
$886$		−	26.1421i		−	0.878262i
$887$	15.1127			0.507435			0.253717	−	0.967278i	$-0.418347\pi$
$887$	15.1127			0.507435			0.253717	+	0.967278i	$0.418347\pi$
$888$		0			0
$889$	−45.4558			−1.52454
$890$	18.3431			0.614864
$891$		0			0
$892$	−14.1421			−0.473514
$893$	−90.9117			−3.04224
$894$		0			0
$895$	−28.2843			−0.945439
$896$		−	3.41421i		−	0.114061i
$897$		0			0
$898$			23.5147i			0.784696i
$899$	−12.0000			−0.400222
$900$		0			0
$901$			20.9706i			0.698631i
$902$	8.00000	−	16.9706i	0.266371	−	0.565058i
$903$		0			0
$904$			9.89949i			0.329252i
$905$		−	6.34315i		−	0.210853i
$906$		0			0
$907$	−25.9411			−0.861361			−0.430680	−	0.902504i	$-0.641726\pi$
$907$	−25.9411			−0.861361			−0.430680	+	0.902504i	$0.641726\pi$
$908$	17.6569			0.585963
$909$		0			0
$910$		−	6.82843i		−	0.226360i
$911$			48.1838i			1.59640i	0.602393	+	0.798200i	$0.294214\pi$
$911$			48.1838i			1.59640i	−0.602393	+	0.798200i	$0.705786\pi$
$912$		0			0
$913$	−11.3137	+	24.0000i	−0.374429	+	0.794284i
$914$			12.2426i			0.404951i
$915$		0			0
$916$	−10.7279			−0.354461
$917$			6.48528i			0.214163i
$918$		0			0
$919$		−	24.6274i		−	0.812384i	−0.913788	−	0.406192i	$-0.866856\pi$
$919$		−	24.6274i		−	0.812384i	0.913788	−	0.406192i	$-0.133144\pi$
$920$	−12.4853			−0.411628
$921$		0			0
$922$	−15.1716			−0.499649
$923$	−7.65685			−0.252028
$924$		0			0
$925$	−6.24264			−0.205257
$926$	2.82843			0.0929479
$927$		0			0
$928$	10.2426			0.336231
$929$		−	31.5980i		−	1.03670i	−0.855170	−	0.518348i	$-0.826547\pi$
$929$		−	31.5980i		−	1.03670i	0.855170	−	0.518348i	$-0.173453\pi$
$930$		0			0
$931$			31.7990i			1.04217i
$932$	21.3137			0.698154
$933$		0			0
$934$		−	29.6569i		−	0.970402i
$935$	−12.0000	−	5.65685i	−0.392442	−	0.184999i
$936$		0			0
$937$		−	14.6274i		−	0.477857i	−0.971037	−	0.238928i	$-0.923204\pi$
$937$		−	14.6274i		−	0.477857i	0.971037	−	0.238928i	$-0.0767961\pi$
$938$		−	18.4853i		−	0.603566i
$939$		0			0
$940$	26.6274			0.868491
$941$	6.68629			0.217967			0.108983	−	0.994044i	$-0.465240\pi$
$941$	6.68629			0.217967			0.108983	+	0.994044i	$0.465240\pi$
$942$		0			0
$943$			35.3137i			1.14997i
$944$			7.65685i			0.249209i
$945$		0			0
$946$	−18.0000	−	8.48528i	−0.585230	−	0.275880i
$947$			32.6274i			1.06025i	0.847920	+	0.530124i	$0.177855\pi$
$947$			32.6274i			1.06025i	−0.847920	+	0.530124i	$0.822145\pi$
$948$		0			0
$949$	−13.8995			−0.451197
$950$			6.82843i			0.221543i
$951$		0			0
$952$			6.82843i			0.221311i
$953$	26.7696			0.867151			0.433575	−	0.901117i	$-0.357252\pi$
$953$	26.7696			0.867151			0.433575	+	0.901117i	$0.357252\pi$
$954$		0			0
$955$	−18.1421			−0.587066
$956$	−20.1421			−0.651443
$957$		0			0
$958$	22.6274			0.731059
$959$	13.6569			0.441003
$960$		0			0
$961$	−29.6274			−0.955723
$962$		−	6.24264i		−	0.201271i
$963$		0			0
$964$			19.7574i			0.636342i
$965$	19.7990			0.637352
$966$		0			0
$967$			10.7279i			0.344987i	0.985011	+	0.172493i	$0.0551823\pi$
$967$			10.7279i			0.344987i	−0.985011	+	0.172493i	$0.944818\pi$
$968$	7.00000	+	8.48528i	0.224989	+	0.272727i
$969$		0			0
$970$		−	23.3137i		−	0.748558i
$971$		−	6.82843i		−	0.219135i	−0.993979	−	0.109567i	$-0.965053\pi$
$971$		−	6.82843i		−	0.219135i	0.993979	−	0.109567i	$-0.0349465\pi$
$972$		0			0
$973$	39.7990			1.27590
$974$	32.4853			1.04090
$975$		0			0
$976$			2.82843i			0.0905357i
$977$			14.1421i			0.452447i	0.974075	+	0.226224i	$0.0726380\pi$
$977$			14.1421i			0.452447i	−0.974075	+	0.226224i	$0.927362\pi$
$978$		0			0
$979$	27.5147	+	12.9706i	0.879374	+	0.414541i
$980$		−	9.31371i		−	0.297516i
$981$		0			0
$982$	5.89949			0.188260
$983$		−	37.5980i		−	1.19919i	−0.800304	−	0.599595i	$-0.795328\pi$
$983$		−	37.5980i		−	1.19919i	0.800304	−	0.599595i	$-0.204672\pi$
$984$		0			0
$985$		−	17.6569i		−	0.562594i
$986$	−20.4853			−0.652384
$987$		0			0
$988$	−6.82843			−0.217241
$989$	37.4558			1.19103
$990$		0			0
$991$	23.5980			0.749615			0.374807	−	0.927103i	$-0.377709\pi$
$991$	23.5980			0.749615			0.374807	+	0.927103i	$0.377709\pi$
$992$	−1.17157			−0.0371975
$993$		0			0
$994$	−26.1421			−0.829178
$995$		−	54.6274i		−	1.73181i
$996$		0			0
$997$			14.9706i			0.474122i	0.971495	+	0.237061i	$0.0761842\pi$
$997$			14.9706i			0.474122i	−0.971495	+	0.237061i	$0.923816\pi$
$998$	−24.0416			−0.761025
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2574.2.b.a.989.4	yes	4
3.2	odd	2		2574.2.b.b.989.2	yes	4
11.10	odd	2		2574.2.b.b.989.3	yes	4
33.32	even	2	inner	2574.2.b.a.989.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2574.2.b.a.989.1	✓	4	33.32	even	2	inner
2574.2.b.a.989.4	yes	4	1.1	even	1	trivial
2574.2.b.b.989.2	yes	4	3.2	odd	2
2574.2.b.b.989.3	yes	4	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2574 = 2 \cdot 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2574.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.00000i q^{5} +3.41421i q^{7} -1.00000 q^{8} -2.00000i q^{10} +(1.41421 - 3.00000i) q^{11} -1.00000i q^{13} -3.41421i q^{14} +1.00000 q^{16} -2.00000 q^{17} -6.82843i q^{19} +2.00000i q^{20} +(-1.41421 + 3.00000i) q^{22} -6.24264i q^{23} +1.00000 q^{25} +1.00000i q^{26} +3.41421i q^{28} -10.2426 q^{29} +1.17157 q^{31} -1.00000 q^{32} +2.00000 q^{34} -6.82843 q^{35} -6.24264 q^{37} +6.82843i q^{38} -2.00000i q^{40} -5.65685 q^{41} +6.00000i q^{43} +(1.41421 - 3.00000i) q^{44} +6.24264i q^{46} -13.3137i q^{47} -4.65685 q^{49} -1.00000 q^{50} -1.00000i q^{52} -10.4853i q^{53} +(6.00000 + 2.82843i) q^{55} -3.41421i q^{56} +10.2426 q^{58} +7.65685i q^{59} +2.82843i q^{61} -1.17157 q^{62} +1.00000 q^{64} +2.00000 q^{65} +5.41421 q^{67} -2.00000 q^{68} +6.82843 q^{70} -7.65685i q^{71} -13.8995i q^{73} +6.24264 q^{74} -6.82843i q^{76} +(10.2426 + 4.82843i) q^{77} +6.00000i q^{79} +2.00000i q^{80} +5.65685 q^{82} -8.00000 q^{83} -4.00000i q^{85} -6.00000i q^{86} +(-1.41421 + 3.00000i) q^{88} +9.17157i q^{89} +3.41421 q^{91} -6.24264i q^{92} +13.3137i q^{94} +13.6569 q^{95} +11.6569 q^{97} +4.65685 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} - 8 q^{17} + 4 q^{25} - 24 q^{29} + 16 q^{31} - 4 q^{32} + 8 q^{34} - 16 q^{35} - 8 q^{37} + 4 q^{49} - 4 q^{50} + 24 q^{55} + 24 q^{58} - 16 q^{62} + 4 q^{64}+ \cdots - 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 - 4 * q^8 + 4 * q^16 - 8 * q^17 + 4 * q^25 - 24 * q^29 + 16 * q^31 - 4 * q^32 + 8 * q^34 - 16 * q^35 - 8 * q^37 + 4 * q^49 - 4 * q^50 + 24 * q^55 + 24 * q^58 - 16 * q^62 + 4 * q^64 + 8 * q^65 + 16 * q^67 - 8 * q^68 + 16 * q^70 + 8 * q^74 + 24 * q^77 - 32 * q^83 + 8 * q^91 + 32 * q^95 + 24 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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