LMFDB - Embedded newform 1716.2.p.a.857.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1716,2,Mod(857,1716)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1716, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1716.857");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1716.p (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$13.7023289869$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			857.11
Character	$\chi$	$=$	1716.857
Dual form			1716.2.p.a.857.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.32610 - 1.11421i) q^{3} +3.23239 q^{5} +2.02885 q^{7} +(0.517092 + 2.95510i) q^{9} +O(q^{10})$	\(q+(-1.32610 - 1.11421i) q^{3} +3.23239 q^{5} +2.02885 q^{7} +(0.517092 + 2.95510i) q^{9} +(-2.30831 - 2.38153i) q^{11} +(-2.84425 + 2.21590i) q^{13} +(-4.28648 - 3.60154i) q^{15} +3.31128 q^{17} +5.39630 q^{19} +(-2.69046 - 2.26055i) q^{21} -6.32857i q^{23} +5.44833 q^{25} +(2.60687 - 4.49491i) q^{27} +2.60561 q^{29} -4.85032i q^{31} +(0.407542 + 5.73009i) q^{33} +6.55803 q^{35} -0.389277i q^{37} +(6.24074 + 0.230572i) q^{39} +7.10595i q^{41} +11.3138i q^{43} +(1.67144 + 9.55203i) q^{45} +6.31181 q^{47} -2.88377 q^{49} +(-4.39109 - 3.68944i) q^{51} -4.85394i q^{53} +(-7.46136 - 7.69803i) q^{55} +(-7.15605 - 6.01259i) q^{57} +1.91176 q^{59} -4.76982i q^{61} +(1.04910 + 5.99545i) q^{63} +(-9.19373 + 7.16265i) q^{65} -3.81290i q^{67} +(-7.05133 + 8.39233i) q^{69} +6.70382 q^{71} +5.34541 q^{73} +(-7.22504 - 6.07056i) q^{75} +(-4.68322 - 4.83177i) q^{77} -12.6883i q^{79} +(-8.46523 + 3.05612i) q^{81} +14.2905i q^{83} +10.7033 q^{85} +(-3.45530 - 2.90318i) q^{87} +11.9950 q^{89} +(-5.77056 + 4.49573i) q^{91} +(-5.40425 + 6.43202i) q^{93} +17.4429 q^{95} -6.10961i q^{97} +(5.84405 - 8.05277i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 8 q^{9}+O(q^{10}) $ 56 * q - 8 * q^9	$ 56 q - 8 q^{9} + 48 q^{25} - 12 q^{27} + 48 q^{49} - 16 q^{55} - 12 q^{69} + 44 q^{75} - 40 q^{81} - 24 q^{91}+O(q^{100}) $ 56 * q - 8 * q^9 + 48 * q^25 - 12 * q^27 + 48 * q^49 - 16 * q^55 - 12 * q^69 + 44 * q^75 - 40 * q^81 - 24 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1716\mathbb{Z}\right)^\times$.

$n$	$859$	$925$	$937$	$1145$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.32610	−	1.11421i	−0.765625	−	0.643287i
$3$	−1.32610	−	1.11421i	−0.765625	−	0.643287i
$4$		0			0
$5$	3.23239			1.44557			0.722784	−	0.691074i	$-0.242862\pi$
$5$	3.23239			1.44557			0.722784	+	0.691074i	$0.242862\pi$
$6$		0			0
$7$	2.02885			0.766833			0.383416	−	0.923576i	$-0.374747\pi$
$7$	2.02885			0.766833			0.383416	+	0.923576i	$0.374747\pi$
$8$		0			0
$9$	0.517092	+	2.95510i	0.172364	+	0.985033i
$10$		0			0
$11$	−2.30831	−	2.38153i	−0.695982	−	0.718059i
$11$	−2.30831	−	2.38153i	−0.695982	−	0.718059i
$12$		0			0
$13$	−2.84425	+	2.21590i	−0.788854	+	0.614581i
$13$	−2.84425	+	2.21590i	−0.788854	+	0.614581i
$14$		0			0
$15$	−4.28648	−	3.60154i	−1.10676	−	0.929915i
$16$		0			0
$17$	3.31128			0.803103			0.401552	−	0.915836i	$-0.368471\pi$
$17$	3.31128			0.803103			0.401552	+	0.915836i	$0.368471\pi$
$18$		0			0
$19$	5.39630			1.23800			0.618999	−	0.785392i	$-0.287539\pi$
$19$	5.39630			1.23800			0.618999	+	0.785392i	$0.287539\pi$
$20$		0			0
$21$	−2.69046	−	2.26055i	−0.587107	−	0.493293i
$22$		0			0
$23$		−	6.32857i		−	1.31960i	−0.751442	−	0.659799i	$-0.770641\pi$
$23$		−	6.32857i		−	1.31960i	0.751442	−	0.659799i	$-0.229359\pi$
$24$		0			0
$25$	5.44833			1.08967
$26$		0			0
$27$	2.60687	−	4.49491i	0.501693	−	0.865046i
$28$		0			0
$29$	2.60561			0.483849			0.241924	−	0.970295i	$-0.422221\pi$
$29$	2.60561			0.483849			0.241924	+	0.970295i	$0.422221\pi$
$30$		0			0
$31$		−	4.85032i		−	0.871143i	−0.900154	−	0.435572i	$-0.856546\pi$
$31$		−	4.85032i		−	0.871143i	0.900154	−	0.435572i	$-0.143454\pi$
$32$		0			0
$33$	0.407542	+	5.73009i	0.0709440	+	0.997480i
$34$		0			0
$35$	6.55803			1.10851
$36$		0			0
$37$		−	0.389277i		−	0.0639967i	−0.999488	−	0.0319983i	$-0.989813\pi$
$37$		−	0.389277i		−	0.0639967i	0.999488	−	0.0319983i	$-0.0101871\pi$
$38$		0			0
$39$	6.24074	+	0.230572i	0.999318	+	0.0369211i
$40$		0			0
$41$			7.10595i			1.10976i	0.831930	+	0.554881i	$0.187236\pi$
$41$			7.10595i			1.10976i	−0.831930	+	0.554881i	$0.812764\pi$
$42$		0			0
$43$			11.3138i			1.72534i	0.505771	+	0.862668i	$0.331208\pi$
$43$			11.3138i			1.72534i	−0.505771	+	0.862668i	$0.668792\pi$
$44$		0			0
$45$	1.67144	+	9.55203i	0.249164	+	1.42393i
$46$		0			0
$47$	6.31181			0.920672			0.460336	−	0.887745i	$-0.347729\pi$
$47$	6.31181			0.920672			0.460336	+	0.887745i	$0.347729\pi$
$48$		0			0
$49$	−2.88377			−0.411967
$50$		0			0
$51$	−4.39109	−	3.68944i	−0.614876	−	0.516626i
$52$		0			0
$53$		−	4.85394i		−	0.666740i	−0.942796	−	0.333370i	$-0.891814\pi$
$53$		−	4.85394i		−	0.666740i	0.942796	−	0.333370i	$-0.108186\pi$
$54$		0			0
$55$	−7.46136	−	7.69803i	−1.00609	−	1.03800i
$56$		0			0
$57$	−7.15605	−	6.01259i	−0.947842	−	0.796387i
$58$		0			0
$59$	1.91176			0.248890			0.124445	−	0.992226i	$-0.460285\pi$
$59$	1.91176			0.248890			0.124445	+	0.992226i	$0.460285\pi$
$60$		0			0
$61$		−	4.76982i		−	0.610713i	−0.952238	−	0.305357i	$-0.901224\pi$
$61$		−	4.76982i		−	0.610713i	0.952238	−	0.305357i	$-0.0987756\pi$
$62$		0			0
$63$	1.04910	+	5.99545i	0.132174	+	0.755356i
$64$		0			0
$65$	−9.19373	+	7.16265i	−1.14034	+	0.888418i
$66$		0			0
$67$		−	3.81290i		−	0.465820i	−0.972498	−	0.232910i	$-0.925175\pi$
$67$		−	3.81290i		−	0.465820i	0.972498	−	0.232910i	$-0.0748248\pi$
$68$		0			0
$69$	−7.05133	+	8.39233i	−0.848881	+	1.01032i
$70$		0			0
$71$	6.70382			0.795597			0.397799	−	0.917473i	$-0.369774\pi$
$71$	6.70382			0.795597			0.397799	+	0.917473i	$0.369774\pi$
$72$		0			0
$73$	5.34541			0.625633			0.312816	−	0.949814i	$-0.398728\pi$
$73$	5.34541			0.625633			0.312816	+	0.949814i	$0.398728\pi$
$74$		0			0
$75$	−7.22504	−	6.07056i	−0.834276	−	0.700968i
$76$		0			0
$77$	−4.68322	−	4.83177i	−0.533702	−	0.550631i
$78$		0			0
$79$		−	12.6883i		−	1.42755i	−0.700375	−	0.713775i	$-0.746984\pi$
$79$		−	12.6883i		−	1.42755i	0.700375	−	0.713775i	$-0.253016\pi$
$80$		0			0
$81$	−8.46523	+	3.05612i	−0.940581	+	0.339569i
$82$		0			0
$83$			14.2905i			1.56858i	0.620393	+	0.784291i	$0.286973\pi$
$83$			14.2905i			1.56858i	−0.620393	+	0.784291i	$0.713027\pi$
$84$		0			0
$85$	10.7033			1.16094
$86$		0			0
$87$	−3.45530	−	2.90318i	−0.370447	−	0.311254i
$88$		0			0
$89$	11.9950			1.27147			0.635735	−	0.771907i	$-0.280697\pi$
$89$	11.9950			1.27147			0.635735	+	0.771907i	$0.280697\pi$
$90$		0			0
$91$	−5.77056	+	4.49573i	−0.604919	+	0.471281i
$92$		0			0
$93$	−5.40425	+	6.43202i	−0.560395	+	0.666969i
$94$		0			0
$95$	17.4429			1.78961
$96$		0			0
$97$		−	6.10961i		−	0.620337i	−0.950682	−	0.310168i	$-0.899615\pi$
$97$		−	6.10961i		−	0.620337i	0.950682	−	0.310168i	$-0.100385\pi$
$98$		0			0
$99$	5.84405	−	8.05277i	0.587349	−	0.809333i
$100$		0			0
$101$	13.5725			1.35051			0.675257	−	0.737582i	$-0.264033\pi$
$101$	13.5725			1.35051			0.675257	+	0.737582i	$0.264033\pi$
$102$		0			0
$103$	−13.1201			−1.29276			−0.646381	−	0.763015i	$-0.723718\pi$
$103$	−13.1201			−1.29276			−0.646381	+	0.763015i	$0.723718\pi$
$104$		0			0
$105$	−8.69661	−	7.30699i	−0.848702	−	0.713089i
$106$		0			0
$107$	19.9551			1.92914			0.964569	−	0.263832i	$-0.0849865\pi$
$107$	19.9551			1.92914			0.964569	+	0.263832i	$0.0849865\pi$
$108$		0			0
$109$	−11.6650			−1.11730			−0.558652	−	0.829402i	$-0.688681\pi$
$109$	−11.6650			−1.11730			−0.558652	+	0.829402i	$0.688681\pi$
$110$		0			0
$111$	−0.433734	+	0.516220i	−0.0411682	+	0.0489975i
$112$		0			0
$113$		−	11.7730i		−	1.10751i	−0.832678	−	0.553757i	$-0.813194\pi$
$113$		−	11.7730i		−	1.10751i	0.832678	−	0.553757i	$-0.186806\pi$
$114$		0			0
$115$		−	20.4564i		−	1.90757i
$116$		0			0
$117$	−8.01895	−	7.25923i	−0.741352	−	0.671116i
$118$		0			0
$119$	6.71808			0.615846
$120$		0			0
$121$	−0.343384	+	10.9946i	−0.0312167	+	0.999513i
$122$		0			0
$123$	7.91748	−	9.42321i	0.713895	−	0.849662i
$124$		0			0
$125$	1.44917			0.129618
$126$		0			0
$127$		−	22.0915i		−	1.96031i	−0.198243	−	0.980153i	$-0.563524\pi$
$127$		−	22.0915i		−	1.96031i	0.198243	−	0.980153i	$-0.436476\pi$
$128$		0			0
$129$	12.6059	−	15.0032i	1.10989	−	1.32096i
$130$		0			0
$131$	−12.1820			−1.06435			−0.532173	−	0.846635i	$-0.678625\pi$
$131$	−12.1820			−1.06435			−0.532173	+	0.846635i	$0.678625\pi$
$132$		0			0
$133$	10.9483			0.949337
$134$		0			0
$135$	8.42642	−	14.5293i	0.725231	−	1.25048i
$136$		0			0
$137$	−12.2537			−1.04690			−0.523451	−	0.852056i	$-0.675356\pi$
$137$	−12.2537			−1.04690			−0.523451	+	0.852056i	$0.675356\pi$
$138$		0			0
$139$		−	4.33315i		−	0.367533i	−0.982970	−	0.183766i	$-0.941171\pi$
$139$		−	4.33315i		−	0.367533i	0.982970	−	0.183766i	$-0.0588290\pi$
$140$		0			0
$141$	−8.37010	−	7.03265i	−0.704890	−	0.592256i
$142$		0			0
$143$	11.8427	+	1.65869i	0.990334	+	0.138706i
$144$		0			0
$145$	8.42233			0.699436
$146$		0			0
$147$	3.82418	+	3.21311i	0.315413	+	0.265013i
$148$		0			0
$149$			9.64746i			0.790350i	0.918606	+	0.395175i	$0.129316\pi$
$149$			9.64746i			0.790350i	−0.918606	+	0.395175i	$0.870684\pi$
$150$		0			0
$151$	7.49451			0.609894			0.304947	−	0.952369i	$-0.401361\pi$
$151$	7.49451			0.609894			0.304947	+	0.952369i	$0.401361\pi$
$152$		0			0
$153$	1.71224	+	9.78516i	0.138426	+	0.791083i
$154$		0			0
$155$		−	15.6781i		−	1.25930i
$156$		0			0
$157$	12.3065			0.982169			0.491084	−	0.871112i	$-0.336601\pi$
$157$	12.3065			0.982169			0.491084	+	0.871112i	$0.336601\pi$
$158$		0			0
$159$	−5.40829	+	6.43682i	−0.428905	+	0.510473i
$160$		0			0
$161$		−	12.8397i		−	1.01191i
$162$		0			0
$163$			17.3994i			1.36283i	0.731898	+	0.681414i	$0.238635\pi$
$163$			17.3994i			1.36283i	−0.731898	+	0.681414i	$0.761365\pi$
$164$		0			0
$165$	1.31733	+	18.5219i	0.102554	+	1.44193i
$166$		0			0
$167$			2.05817i			0.159266i	0.996824	+	0.0796329i	$0.0253748\pi$
$167$			2.05817i			0.159266i	−0.996824	+	0.0796329i	$0.974625\pi$
$168$		0			0
$169$	3.17956	−	12.6052i	0.244581	−	0.969629i
$170$		0			0
$171$	2.79039	+	15.9466i	0.213386	+	1.21947i
$172$		0			0
$173$	0.798551			0.0607127			0.0303564	−	0.999539i	$-0.490336\pi$
$173$	0.798551			0.0607127			0.0303564	+	0.999539i	$0.490336\pi$
$174$		0			0
$175$	11.0538			0.835592
$176$		0			0
$177$	−2.53519	−	2.13010i	−0.190557	−	0.160108i
$178$		0			0
$179$		−	11.3402i		−	0.847606i	−0.905754	−	0.423803i	$-0.860695\pi$
$179$		−	11.3402i		−	0.847606i	0.905754	−	0.423803i	$-0.139305\pi$
$180$		0			0
$181$	−8.89415			−0.661097			−0.330548	−	0.943789i	$-0.607234\pi$
$181$	−8.89415			−0.661097			−0.330548	+	0.943789i	$0.607234\pi$
$182$		0			0
$183$	−5.31456	+	6.32527i	−0.392864	+	0.467577i
$184$		0			0
$185$		−	1.25829i		−	0.0925115i
$186$		0			0
$187$	−7.64347	−	7.88591i	−0.558946	−	0.576675i
$188$		0			0
$189$	5.28895	−	9.11949i	0.384714	−	0.663346i
$190$		0			0
$191$		−	19.7540i		−	1.42935i	−0.699458	−	0.714673i	$-0.746575\pi$
$191$		−	19.7540i		−	1.42935i	0.699458	−	0.714673i	$-0.253425\pi$
$192$		0			0
$193$	−21.5189			−1.54897			−0.774484	−	0.632594i	$-0.781990\pi$
$193$	−21.5189			−1.54897			−0.774484	+	0.632594i	$0.781990\pi$
$194$		0			0
$195$	20.1725	+	0.745298i	1.44458	+	0.0533719i
$196$		0			0
$197$			7.14023i			0.508720i	0.967110	+	0.254360i	$0.0818648\pi$
$197$			7.14023i			0.508720i	−0.967110	+	0.254360i	$0.918135\pi$
$198$		0			0
$199$	−25.6348			−1.81720			−0.908601	−	0.417665i	$-0.862848\pi$
$199$	−25.6348			−1.81720			−0.908601	+	0.417665i	$0.862848\pi$
$200$		0			0
$201$	−4.24836	+	5.05630i	−0.299656	+	0.356644i
$202$		0			0
$203$	5.28638			0.371031
$204$		0			0
$205$			22.9692i			1.60424i
$206$		0			0
$207$	18.7016	−	3.27246i	1.29985	−	0.227451i
$208$		0			0
$209$	−12.4564	−	12.8515i	−0.861624	−	0.888955i
$210$		0			0
$211$			1.72408i			0.118690i	0.998238	+	0.0593452i	$0.0189013\pi$
$211$			1.72408i			0.118690i	−0.998238	+	0.0593452i	$0.981099\pi$
$212$		0			0
$213$	−8.88995	−	7.46944i	−0.609129	−	0.511797i
$214$		0			0
$215$			36.5705i			2.49409i
$216$		0			0
$217$		−	9.84057i		−	0.668021i
$218$		0			0
$219$	−7.08856	−	5.95589i	−0.479000	−	0.402461i
$220$		0			0
$221$	−9.41812	+	7.33747i	−0.633531	+	0.493572i
$222$		0			0
$223$			25.2457i			1.69058i	0.534311	+	0.845288i	$0.320571\pi$
$223$			25.2457i			1.69058i	−0.534311	+	0.845288i	$0.679429\pi$
$224$		0			0
$225$	2.81729	+	16.1004i	0.187819	+	1.07336i
$226$		0			0
$227$		−	7.54999i		−	0.501110i	−0.968102	−	0.250555i	$-0.919387\pi$
$227$		−	7.54999i		−	0.501110i	0.968102	−	0.250555i	$-0.0806131\pi$
$228$		0			0
$229$			25.6590i			1.69559i	0.530320	+	0.847797i	$0.322072\pi$
$229$			25.6590i			1.69559i	−0.530320	+	0.847797i	$0.677928\pi$
$230$		0			0
$231$	0.826842	+	11.6255i	0.0544022	+	0.764901i
$232$		0			0
$233$	10.7641			0.705179			0.352589	−	0.935778i	$-0.385301\pi$
$233$	10.7641			0.705179			0.352589	+	0.935778i	$0.385301\pi$
$234$		0			0
$235$	20.4022			1.33089
$236$		0			0
$237$	−14.1374	+	16.8260i	−0.918324	+	1.09297i
$238$		0			0
$239$		−	12.5893i		−	0.814333i	−0.913354	−	0.407167i	$-0.866517\pi$
$239$		−	12.5893i		−	0.814333i	0.913354	−	0.407167i	$-0.133483\pi$
$240$		0			0
$241$	17.8086			1.14715			0.573576	−	0.819152i	$-0.305556\pi$
$241$	17.8086			1.14715			0.573576	+	0.819152i	$0.305556\pi$
$242$		0			0
$243$	14.6309	+	5.37928i	0.938573	+	0.345081i
$244$		0			0
$245$	−9.32147			−0.595527
$246$		0			0
$247$	−15.3485	+	11.9577i	−0.976599	+	0.760849i
$248$		0			0
$249$	15.9225	−	18.9506i	1.00905	−	1.20095i
$250$		0			0
$251$		−	1.57229i		−	0.0992418i	−0.998768	−	0.0496209i	$-0.984199\pi$
$251$		−	1.57229i		−	0.0992418i	0.998768	−	0.0496209i	$-0.0158013\pi$
$252$		0			0
$253$	−15.0717	+	14.6083i	−0.947549	+	0.918418i
$254$		0			0
$255$	−14.1937	−	11.9257i	−0.888845	−	0.746817i
$256$		0			0
$257$			25.9849i			1.62089i	0.585813	+	0.810446i	$0.300775\pi$
$257$			25.9849i			1.62089i	−0.585813	+	0.810446i	$0.699225\pi$
$258$		0			0
$259$		−	0.789783i		−	0.0490747i
$260$		0			0
$261$	1.34734	+	7.69983i	0.0833982	+	0.476607i
$262$		0			0
$263$	−20.9494			−1.29179			−0.645897	−	0.763424i	$-0.723517\pi$
$263$	−20.9494			−1.29179			−0.645897	+	0.763424i	$0.723517\pi$
$264$		0			0
$265$		−	15.6898i		−	0.963818i
$266$		0			0
$267$	−15.9066	−	13.3649i	−0.973470	−	0.817921i
$268$		0			0
$269$		−	4.55754i		−	0.277878i	−0.990301	−	0.138939i	$-0.955631\pi$
$269$		−	4.55754i		−	0.277878i	0.990301	−	0.138939i	$-0.0443692\pi$
$270$		0			0
$271$	−20.7101			−1.25805			−0.629025	−	0.777385i	$-0.716546\pi$
$271$	−20.7101			−1.25805			−0.629025	+	0.777385i	$0.716546\pi$
$272$		0			0
$273$	12.6615	+	0.467796i	0.766310	+	0.0283123i
$274$		0			0
$275$	−12.5764	−	12.9754i	−0.758388	−	0.782444i
$276$		0			0
$277$			9.59988i			0.576801i	0.957510	+	0.288400i	$0.0931234\pi$
$277$			9.59988i			0.576801i	−0.957510	+	0.288400i	$0.906877\pi$
$278$		0			0
$279$	14.3332	−	2.50806i	0.858105	−	0.150154i
$280$		0			0
$281$		−	0.0433267i		−	0.00258465i	−0.999999	−	0.00129233i	$-0.999589\pi$
$281$		−	0.0433267i		−	0.00258465i	0.999999	−	0.00129233i	$-0.000411360\pi$
$282$		0			0
$283$		−	7.91493i		−	0.470494i	−0.971936	−	0.235247i	$-0.924410\pi$
$283$		−	7.91493i		−	0.470494i	0.971936	−	0.235247i	$-0.0755899\pi$
$284$		0			0
$285$	−23.1311	−	19.4350i	−1.37017	−	1.15123i
$286$		0			0
$287$			14.4169i			0.851002i
$288$		0			0
$289$	−6.03543			−0.355026
$290$		0			0
$291$	−6.80736	+	8.10196i	−0.399054	+	0.474945i
$292$		0			0
$293$			4.87552i			0.284831i	0.989807	+	0.142415i	$0.0454869\pi$
$293$			4.87552i			0.284831i	−0.989807	+	0.142415i	$0.954513\pi$
$294$		0			0
$295$	6.17956			0.359788
$296$		0			0
$297$	−16.7222	+	4.16731i	−0.970323	+	0.241812i
$298$		0			0
$299$	14.0235	+	18.0001i	0.811000	+	1.04097i
$300$		0			0
$301$			22.9540i			1.32304i
$302$		0			0
$303$	−17.9985	−	15.1226i	−1.03399	−	0.868768i
$304$		0			0
$305$		−	15.4179i		−	0.882827i
$306$		0			0
$307$	−24.5049			−1.39857			−0.699285	−	0.714843i	$-0.746498\pi$
$307$	−24.5049			−1.39857			−0.699285	+	0.714843i	$0.746498\pi$
$308$		0			0
$309$	17.3986	+	14.6185i	0.989772	+	0.831617i
$310$		0			0
$311$			14.9129i			0.845634i	0.906215	+	0.422817i	$0.138959\pi$
$311$			14.9129i			0.845634i	−0.906215	+	0.422817i	$0.861041\pi$
$312$		0			0
$313$	21.1737			1.19681			0.598403	−	0.801195i	$-0.295802\pi$
$313$	21.1737			1.19681			0.598403	+	0.801195i	$0.295802\pi$
$314$		0			0
$315$	3.39110	+	19.3796i	0.191067	+	1.09192i
$316$		0			0
$317$	33.0417			1.85580			0.927902	−	0.372823i	$-0.121610\pi$
$317$	33.0417			1.85580			0.927902	+	0.372823i	$0.121610\pi$
$318$		0			0
$319$	−6.01455	−	6.20533i	−0.336750	−	0.347432i
$320$		0			0
$321$	−26.4626	−	22.2341i	−1.47700	−	1.24099i
$322$		0			0
$323$	17.8687			0.994239
$324$		0			0
$325$	−15.4964	+	12.0730i	−0.859587	+	0.669687i
$326$		0			0
$327$	15.4690	+	12.9972i	0.855437	+	0.718747i
$328$		0			0
$329$	12.8057			0.706002
$330$		0			0
$331$			7.81589i			0.429600i	0.976658	+	0.214800i	$0.0689100\pi$
$331$			7.81589i			0.429600i	−0.976658	+	0.214800i	$0.931090\pi$
$332$		0			0
$333$	1.15035	−	0.201292i	0.0630389	−	0.0110307i
$334$		0			0
$335$		−	12.3248i		−	0.673375i
$336$		0			0
$337$			14.7011i			0.800822i	0.916336	+	0.400411i	$0.131133\pi$
$337$			14.7011i			0.800822i	−0.916336	+	0.400411i	$0.868867\pi$
$338$		0			0
$339$	−13.1176	+	15.6122i	−0.712449	+	0.847941i
$340$		0			0
$341$	−11.5512	+	11.1961i	−0.625532	+	0.606300i
$342$		0			0
$343$	−20.0527			−1.08274
$344$		0			0
$345$	−22.7926	+	27.1273i	−1.22711	+	1.46048i
$346$		0			0
$347$	−27.2046			−1.46042			−0.730209	−	0.683224i	$-0.760577\pi$
$347$	−27.2046			−1.46042			−0.730209	+	0.683224i	$0.760577\pi$
$348$		0			0
$349$	10.6371			0.569389			0.284694	−	0.958618i	$-0.408108\pi$
$349$	10.6371			0.569389			0.284694	+	0.958618i	$0.408108\pi$
$350$		0			0
$351$	2.54567	+	18.5612i	0.135878	+	0.990726i
$352$		0			0
$353$	−4.98389			−0.265266			−0.132633	−	0.991165i	$-0.542343\pi$
$353$	−4.98389			−0.265266			−0.132633	+	0.991165i	$0.542343\pi$
$354$		0			0
$355$	21.6693			1.15009
$356$		0			0
$357$	−8.90886	−	7.48533i	−0.471507	−	0.396165i
$358$		0			0
$359$			15.7833i			0.833008i	0.909134	+	0.416504i	$0.136745\pi$
$359$			15.7833i			0.833008i	−0.909134	+	0.416504i	$0.863255\pi$
$360$		0			0
$361$	10.1201			0.532637
$362$		0			0
$363$	12.7056	−	14.1974i	0.666874	−	0.745171i
$364$		0			0
$365$	17.2784			0.904395
$366$		0			0
$367$	1.48662			0.0776007			0.0388004	−	0.999247i	$-0.487646\pi$
$367$	1.48662			0.0776007			0.0388004	+	0.999247i	$0.487646\pi$
$368$		0			0
$369$	−20.9988	+	3.67443i	−1.09315	+	0.191283i
$370$		0			0
$371$		−	9.84792i		−	0.511278i
$372$		0			0
$373$			13.8235i			0.715756i	0.933768	+	0.357878i	$0.116500\pi$
$373$			13.8235i			0.715756i	−0.933768	+	0.357878i	$0.883500\pi$
$374$		0			0
$375$	−1.92175	−	1.61468i	−0.0992389	−	0.0833816i
$376$		0			0
$377$	−7.41101	+	5.77377i	−0.381686	+	0.297364i
$378$		0			0
$379$			8.06538i			0.414291i	0.978310	+	0.207145i	$0.0664174\pi$
$379$			8.06538i			0.414291i	−0.978310	+	0.207145i	$0.933583\pi$
$380$		0			0
$381$	−24.6145	+	29.2956i	−1.26104	+	1.50086i
$382$		0			0
$383$	6.67151			0.340898			0.170449	−	0.985366i	$-0.445478\pi$
$383$	6.67151			0.340898			0.170449	+	0.985366i	$0.445478\pi$
$384$		0			0
$385$	−15.1380	−	15.6181i	−0.771503	−	0.795974i
$386$		0			0
$387$	−33.4334	+	5.85027i	−1.69951	+	0.297386i
$388$		0			0
$389$		−	3.11862i		−	0.158121i	−0.996870	−	0.0790603i	$-0.974808\pi$
$389$		−	3.11862i		−	0.158121i	0.996870	−	0.0790603i	$-0.0251920\pi$
$390$		0			0
$391$		−	20.9557i		−	1.05977i
$392$		0			0
$393$	16.1546	+	13.5733i	0.814891	+	0.684680i
$394$		0			0
$395$		−	41.0136i		−	2.06362i
$396$		0			0
$397$			26.8022i			1.34516i	0.740023	+	0.672582i	$0.234815\pi$
$397$			26.8022i			1.34516i	−0.740023	+	0.672582i	$0.765185\pi$
$398$		0			0
$399$	−14.5185	−	12.1986i	−0.726836	−	0.610696i
$400$		0			0
$401$	2.46698			0.123195			0.0615976	−	0.998101i	$-0.480380\pi$
$401$	2.46698			0.123195			0.0615976	+	0.998101i	$0.480380\pi$
$402$		0			0
$403$	10.7478	+	13.7955i	0.535388	+	0.687205i
$404$		0			0
$405$	−27.3629	+	9.87856i	−1.35967	+	0.490869i
$406$		0			0
$407$	−0.927074	+	0.898572i	−0.0459534	+	0.0445406i
$408$		0			0
$409$	10.2009			0.504403			0.252201	−	0.967675i	$-0.418845\pi$
$409$	10.2009			0.504403			0.252201	+	0.967675i	$0.418845\pi$
$410$		0			0
$411$	16.2496	+	13.6531i	0.801534	+	0.673458i
$412$		0			0
$413$	3.87868			0.190857
$414$		0			0
$415$			46.1923i			2.26749i
$416$		0			0
$417$	−4.82802	+	5.74619i	−0.236429	+	0.281392i
$418$		0			0
$419$			6.93756i			0.338922i	0.985537	+	0.169461i	$0.0542027\pi$
$419$			6.93756i			0.338922i	−0.985537	+	0.169461i	$0.945797\pi$
$420$		0			0
$421$		−	22.4702i		−	1.09513i	−0.836763	−	0.547565i	$-0.815555\pi$
$421$		−	22.4702i		−	1.09513i	0.836763	−	0.547565i	$-0.184445\pi$
$422$		0			0
$423$	3.26379	+	18.6520i	0.158691	+	0.906893i
$424$		0			0
$425$	18.0409			0.875114
$426$		0			0
$427$		−	9.67725i		−	0.468315i
$428$		0			0
$429$	−13.8565	−	15.3948i	−0.668996	−	0.743266i
$430$		0			0
$431$			10.8525i			0.522747i	0.965238	+	0.261374i	$0.0841755\pi$
$431$			10.8525i			0.522747i	−0.965238	+	0.261374i	$0.915825\pi$
$432$		0			0
$433$	−15.6172			−0.750516			−0.375258	−	0.926920i	$-0.622446\pi$
$433$	−15.6172			−0.750516			−0.375258	+	0.926920i	$0.622446\pi$
$434$		0			0
$435$	−11.1689	−	9.38421i	−0.535506	−	0.449938i
$436$		0			0
$437$		−	34.1509i		−	1.63366i
$438$		0			0
$439$		−	15.6354i		−	0.746238i	−0.927783	−	0.373119i	$-0.878288\pi$
$439$		−	15.6354i		−	0.746238i	0.927783	−	0.373119i	$-0.121712\pi$
$440$		0			0
$441$	−1.49118	−	8.52183i	−0.0710084	−	0.405802i
$442$		0			0
$443$			24.1092i			1.14546i	0.819742	+	0.572732i	$0.194117\pi$
$443$			24.1092i			1.14546i	−0.819742	+	0.572732i	$0.805883\pi$
$444$		0			0
$445$	38.7726			1.83800
$446$		0			0
$447$	10.7492	−	12.7935i	0.508422	−	0.605112i
$448$		0			0
$449$	−14.4569			−0.682264			−0.341132	−	0.940015i	$-0.610810\pi$
$449$	−14.4569			−0.682264			−0.341132	+	0.940015i	$0.610810\pi$
$450$		0			0
$451$	16.9230	−	16.4027i	0.796875	−	0.772375i
$452$		0			0
$453$	−9.93848	−	8.35042i	−0.466951	−	0.392337i
$454$		0			0
$455$	−18.6527	+	14.5319i	−0.874452	+	0.681268i
$456$		0			0
$457$	5.75041			0.268993			0.134496	−	0.990914i	$-0.457058\pi$
$457$	5.75041			0.268993			0.134496	+	0.990914i	$0.457058\pi$
$458$		0			0
$459$	8.63208	−	14.8839i	0.402911	−	0.694721i
$460$		0			0
$461$			8.52057i			0.396842i	0.980117	+	0.198421i	$0.0635814\pi$
$461$			8.52057i			0.396842i	−0.980117	+	0.198421i	$0.936419\pi$
$462$		0			0
$463$		−	42.2798i		−	1.96491i	−0.186500	−	0.982455i	$-0.559714\pi$
$463$		−	42.2798i		−	1.96491i	0.186500	−	0.982455i	$-0.440286\pi$
$464$		0			0
$465$	−17.4686	+	20.7908i	−0.810089	+	0.964149i
$466$		0			0
$467$			21.4578i			0.992950i	0.868051	+	0.496475i	$0.165373\pi$
$467$			21.4578i			0.992950i	−0.868051	+	0.496475i	$0.834627\pi$
$468$		0			0
$469$		−	7.73581i		−	0.357206i
$470$		0			0
$471$	−16.3197	−	13.7120i	−0.751973	−	0.631816i
$472$		0			0
$473$	26.9441	−	26.1157i	1.23889	−	1.20080i
$474$		0			0
$475$	29.4008			1.34900
$476$		0			0
$477$	14.3439	−	2.50994i	0.656761	−	0.114922i
$478$		0			0
$479$			29.4231i			1.34438i	0.740381	+	0.672188i	$0.234645\pi$
$479$			29.4231i			1.34438i	−0.740381	+	0.672188i	$0.765355\pi$
$480$		0			0
$481$	0.862599	+	1.10720i	0.0393311	+	0.0504840i
$482$		0			0
$483$	−14.3061	+	17.0268i	−0.650949	+	0.774745i
$484$		0			0
$485$		−	19.7486i		−	0.896738i
$486$		0			0
$487$			25.9569i			1.17622i	0.808781	+	0.588109i	$0.200128\pi$
$487$			25.9569i			1.17622i	−0.808781	+	0.588109i	$0.799872\pi$
$488$		0			0
$489$	19.3865	−	23.0734i	0.876690	−	1.04342i
$490$		0			0
$491$	3.30305			0.149065			0.0745323	−	0.997219i	$-0.476254\pi$
$491$	3.30305			0.149065			0.0745323	+	0.997219i	$0.476254\pi$
$492$		0			0
$493$	8.62789			0.388581
$494$		0			0
$495$	18.8902	−	26.0297i	0.849053	−	1.16995i
$496$		0			0
$497$	13.6010			0.610090
$498$		0			0
$499$		−	7.15642i		−	0.320365i	−0.987087	−	0.160183i	$-0.948792\pi$
$499$		−	7.15642i		−	0.320365i	0.987087	−	0.160183i	$-0.0512083\pi$
$500$		0			0
$501$	2.29322	−	2.72934i	0.102454	−	0.121938i
$502$		0			0
$503$	0.691393			0.0308277			0.0154138	−	0.999881i	$-0.495093\pi$
$503$	0.691393			0.0308277			0.0154138	+	0.999881i	$0.495093\pi$
$504$		0			0
$505$	43.8716			1.95226
$506$		0			0
$507$	−18.2612	+	13.1731i	−0.811007	+	0.585036i
$508$		0			0
$509$	−0.813287			−0.0360483			−0.0180242	−	0.999838i	$-0.505738\pi$
$509$	−0.813287			−0.0360483			−0.0180242	+	0.999838i	$0.505738\pi$
$510$		0			0
$511$	10.8450			0.479756
$512$		0			0
$513$	14.0675	−	24.2559i	0.621094	−	1.07092i
$514$		0			0
$515$	−42.4093			−1.86878
$516$		0			0
$517$	−14.5696	−	15.0318i	−0.640772	−	0.661097i
$518$		0			0
$519$	−1.05896	−	0.889750i	−0.0464832	−	0.0390557i
$520$		0			0
$521$		−	29.2399i		−	1.28102i	−0.767948	−	0.640512i	$-0.778722\pi$
$521$		−	29.2399i		−	1.28102i	0.767948	−	0.640512i	$-0.221278\pi$
$522$		0			0
$523$		−	5.51035i		−	0.240951i	−0.992716	−	0.120475i	$-0.961558\pi$
$523$		−	5.51035i		−	0.240951i	0.992716	−	0.120475i	$-0.0384419\pi$
$524$		0			0
$525$	−14.6585	−	12.3162i	−0.639750	−	0.537525i
$526$		0			0
$527$		−	16.0608i		−	0.699618i
$528$		0			0
$529$	−17.0508			−0.741341
$530$		0			0
$531$	0.988558	+	5.64945i	0.0428997	+	0.245165i
$532$		0			0
$533$	−15.7461	−	20.2111i	−0.682038	−	0.875440i
$534$		0			0
$535$	64.5028			2.78870
$536$		0			0
$537$	−12.6353	+	15.0383i	−0.545254	+	0.648949i
$538$		0			0
$539$	6.65665	+	6.86779i	0.286722	+	0.295817i
$540$		0			0
$541$	−16.9758			−0.729845			−0.364922	−	0.931038i	$-0.618905\pi$
$541$	−16.9758			−0.729845			−0.364922	+	0.931038i	$0.618905\pi$
$542$		0			0
$543$	11.7945	+	9.90991i	0.506152	+	0.425275i
$544$		0			0
$545$	−37.7058			−1.61514
$546$		0			0
$547$			18.2479i			0.780226i	0.920767	+	0.390113i	$0.127564\pi$
$547$			18.2479i			0.780226i	−0.920767	+	0.390113i	$0.872436\pi$
$548$		0			0
$549$	14.0953	−	2.46644i	0.601573	−	0.105265i
$550$		0			0
$551$	14.0606			0.599004
$552$		0			0
$553$		−	25.7427i		−	1.09469i
$554$		0			0
$555$	−1.40200	+	1.66862i	−0.0595114	+	0.0708292i
$556$		0			0
$557$			10.7688i			0.456289i	0.973627	+	0.228144i	$0.0732658\pi$
$557$			10.7688i			0.456289i	−0.973627	+	0.228144i	$0.926734\pi$
$558$		0			0
$559$	−25.0702	−	32.1793i	−1.06036	−	1.36104i
$560$		0			0
$561$	1.34949	+	18.9739i	0.0569754	+	0.801079i
$562$		0			0
$563$	−30.0885			−1.26808			−0.634039	−	0.773301i	$-0.718604\pi$
$563$	−30.0885			−1.26808			−0.634039	+	0.773301i	$0.718604\pi$
$564$		0			0
$565$		−	38.0550i		−	1.60099i
$566$		0			0
$567$	−17.1747	+	6.20040i	−0.721269	+	0.260392i
$568$		0			0
$569$	−13.3413			−0.559298			−0.279649	−	0.960102i	$-0.590218\pi$
$569$	−13.3413			−0.559298			−0.279649	+	0.960102i	$0.590218\pi$
$570$		0			0
$571$			42.9523i			1.79750i	0.438462	+	0.898750i	$0.355523\pi$
$571$			42.9523i			1.79750i	−0.438462	+	0.898750i	$0.644477\pi$
$572$		0			0
$573$	−22.0100	+	26.1958i	−0.919480	+	1.09434i
$574$		0			0
$575$		−	34.4802i		−	1.43792i
$576$		0			0
$577$			11.0627i			0.460545i	0.973126	+	0.230272i	$0.0739617\pi$
$577$			11.0627i			0.460545i	−0.973126	+	0.230272i	$0.926038\pi$
$578$		0			0
$579$	28.5363	+	23.9765i	1.18593	+	0.996430i
$580$		0			0
$581$			28.9932i			1.20284i
$582$		0			0
$583$	−11.5598	+	11.2044i	−0.478759	+	0.464040i
$584$		0			0
$585$	−25.9204	−	23.4646i	−1.07168	−	0.970144i
$586$		0			0
$587$	−11.3239			−0.467388			−0.233694	−	0.972310i	$-0.575081\pi$
$587$	−11.3239			−0.467388			−0.233694	+	0.972310i	$0.575081\pi$
$588$		0			0
$589$		−	26.1738i		−	1.07847i
$590$		0			0
$591$	7.95568	−	9.46867i	0.327253	−	0.389489i
$592$		0			0
$593$			37.7063i			1.54841i	0.632933	+	0.774207i	$0.281851\pi$
$593$			37.7063i			1.54841i	−0.632933	+	0.774207i	$0.718149\pi$
$594$		0			0
$595$	21.7155			0.890247
$596$		0			0
$597$	33.9943	+	28.5624i	1.39130	+	1.16898i
$598$		0			0
$599$		−	28.5127i		−	1.16500i	−0.812832	−	0.582499i	$-0.802075\pi$
$599$		−	28.5127i		−	1.16500i	0.812832	−	0.582499i	$-0.197925\pi$
$600$		0			0
$601$		−	28.4906i		−	1.16215i	−0.813848	−	0.581077i	$-0.802631\pi$
$601$		−	28.4906i		−	1.16215i	0.813848	−	0.581077i	$-0.197369\pi$
$602$		0			0
$603$	11.2675	−	1.97162i	0.458849	−	0.0802907i
$604$		0			0
$605$	−1.10995	+	35.5389i	−0.0451259	+	1.44486i
$606$		0			0
$607$			40.9822i			1.66342i	0.555213	+	0.831708i	$0.312637\pi$
$607$			40.9822i			1.66342i	−0.555213	+	0.831708i	$0.687363\pi$
$608$		0			0
$609$	−7.01028	−	5.89012i	−0.284071	−	0.238680i
$610$		0			0
$611$	−17.9524	+	13.9864i	−0.726276	+	0.565827i
$612$		0			0
$613$	−1.15888			−0.0468067			−0.0234034	−	0.999726i	$-0.507450\pi$
$613$	−1.15888			−0.0468067			−0.0234034	+	0.999726i	$0.507450\pi$
$614$		0			0
$615$	25.5924	−	30.4595i	1.03198	−	1.22824i
$616$		0			0
$617$	−10.1986			−0.410580			−0.205290	−	0.978701i	$-0.565814\pi$
$617$	−10.1986			−0.410580			−0.205290	+	0.978701i	$0.565814\pi$
$618$		0			0
$619$		−	35.4661i		−	1.42550i	−0.701417	−	0.712751i	$-0.747449\pi$
$619$		−	35.4661i		−	1.42550i	0.701417	−	0.712751i	$-0.252551\pi$
$620$		0			0
$621$	−28.4464	−	16.4978i	−1.14151	−	0.662033i
$622$		0			0
$623$	24.3361			0.975006
$624$		0			0
$625$	−22.5574			−0.902294
$626$		0			0
$627$	2.19922	+	30.9213i	0.0878285	+	1.23488i
$628$		0			0
$629$		−	1.28900i		−	0.0513959i
$630$		0			0
$631$			16.2675i			0.647600i	0.946126	+	0.323800i	$0.104961\pi$
$631$			16.2675i			0.647600i	−0.946126	+	0.323800i	$0.895039\pi$
$632$		0			0
$633$	1.92098	−	2.28630i	0.0763520	−	0.0908724i
$634$		0			0
$635$		−	71.4084i		−	2.83375i
$636$		0			0
$637$	8.20218	−	6.39016i	0.324982	−	0.253187i
$638$		0			0
$639$	3.46649	+	19.8105i	0.137132	+	0.783690i
$640$		0			0
$641$			7.10667i			0.280696i	0.990102	+	0.140348i	$0.0448222\pi$
$641$			7.10667i			0.280696i	−0.990102	+	0.140348i	$0.955178\pi$
$642$		0			0
$643$			4.92907i			0.194383i	0.995266	+	0.0971917i	$0.0309860\pi$
$643$			4.92907i			0.194383i	−0.995266	+	0.0971917i	$0.969014\pi$
$644$		0			0
$645$	40.7471	−	48.4962i	1.60441	−	1.90954i
$646$		0			0
$647$		−	25.4362i		−	0.999999i	−0.866026	−	0.500000i	$-0.833333\pi$
$647$		−	25.4362i		−	0.999999i	0.866026	−	0.500000i	$-0.166667\pi$
$648$		0			0
$649$	−4.41295	−	4.55292i	−0.173223	−	0.178718i
$650$		0			0
$651$	−10.9644	+	13.0496i	−0.429729	+	0.511454i
$652$		0			0
$653$			12.1687i			0.476197i	0.971241	+	0.238098i	$0.0765241\pi$
$653$			12.1687i			0.476197i	−0.971241	+	0.238098i	$0.923476\pi$
$654$		0			0
$655$	−39.3770			−1.53859
$656$		0			0
$657$	2.76407	+	15.7962i	0.107837	+	0.616269i
$658$		0			0
$659$	9.11996			0.355263			0.177632	−	0.984097i	$-0.443156\pi$
$659$	9.11996			0.355263			0.177632	+	0.984097i	$0.443156\pi$
$660$		0			0
$661$		−	10.8712i		−	0.422841i	−0.977395	−	0.211420i	$-0.932191\pi$
$661$		−	10.8712i		−	0.422841i	0.977395	−	0.211420i	$-0.0678089\pi$
$662$		0			0
$663$	20.6648	+	0.763488i	0.802555	+	0.0296514i
$664$		0			0
$665$	35.3891			1.37233
$666$		0			0
$667$		−	16.4898i		−	0.638486i
$668$		0			0
$669$	28.1289	−	33.4783i	1.08753	−	1.29435i
$670$		0			0
$671$	−11.3595	+	11.0102i	−0.438528	+	0.425046i
$672$		0			0
$673$			7.88371i			0.303895i	0.988389	+	0.151947i	$0.0485544\pi$
$673$			7.88371i			0.303895i	−0.988389	+	0.151947i	$0.951446\pi$
$674$		0			0
$675$	14.2031	−	24.4898i	0.546677	−	0.942611i
$676$		0			0
$677$	2.85925			0.109890			0.0549449	−	0.998489i	$-0.482502\pi$
$677$	2.85925			0.109890			0.0549449	+	0.998489i	$0.482502\pi$
$678$		0			0
$679$		−	12.3955i		−	0.475694i
$680$		0			0
$681$	−8.41224	+	10.0121i	−0.322358	+	0.383663i
$682$		0			0
$683$	−38.8874			−1.48799			−0.743993	−	0.668187i	$-0.767071\pi$
$683$	−38.8874			−1.48799			−0.743993	+	0.668187i	$0.767071\pi$
$684$		0			0
$685$	−39.6086			−1.51337
$686$		0			0
$687$	28.5894	−	34.0265i	1.09075	−	1.29819i
$688$		0			0
$689$	10.7559	+	13.8058i	0.409766	+	0.525961i
$690$		0			0
$691$			11.4727i			0.436441i	0.975899	+	0.218221i	$0.0700252\pi$
$691$			11.4727i			0.436441i	−0.975899	+	0.218221i	$0.929975\pi$
$692$		0			0
$693$	11.8567	−	16.3378i	0.450399	−	0.620623i
$694$		0			0
$695$		−	14.0064i		−	0.531293i
$696$		0			0
$697$			23.5298i			0.891253i
$698$		0			0
$699$	−14.2743	−	11.9934i	−0.539903	−	0.453632i
$700$		0			0
$701$	−2.37987			−0.0898864			−0.0449432	−	0.998990i	$-0.514311\pi$
$701$	−2.37987			−0.0898864			−0.0449432	+	0.998990i	$0.514311\pi$
$702$		0			0
$703$		−	2.10066i		−	0.0792277i
$704$		0			0
$705$	−27.0554	−	22.7323i	−1.01897	−	0.856147i
$706$		0			0
$707$	27.5365			1.03562
$708$		0			0
$709$			45.7374i			1.71770i	0.512223	+	0.858852i	$0.328822\pi$
$709$			45.7374i			1.71770i	−0.512223	+	0.858852i	$0.671178\pi$
$710$		0			0
$711$	37.4953	−	6.56104i	1.40618	−	0.246058i
$712$		0			0
$713$	−30.6956			−1.14956
$714$		0			0
$715$	38.2801	+	5.36151i	1.43159	+	0.200509i
$716$		0			0
$717$	−14.0271	+	16.6947i	−0.523850	+	0.623474i
$718$		0			0
$719$		−	32.7560i		−	1.22159i	−0.791788	−	0.610797i	$-0.790849\pi$
$719$		−	32.7560i		−	1.22159i	0.791788	−	0.610797i	$-0.209151\pi$
$720$		0			0
$721$	−26.6187			−0.991333
$722$		0			0
$723$	−23.6160	−	19.8424i	−0.878289	−	0.737948i
$724$		0			0
$725$	14.1962			0.527234
$726$		0			0
$727$	−20.9576			−0.777274			−0.388637	−	0.921391i	$-0.627054\pi$
$727$	−20.9576			−0.777274			−0.388637	+	0.921391i	$0.627054\pi$
$728$		0			0
$729$	−13.4084	−	23.4353i	−0.496609	−	0.867974i
$730$		0			0
$731$			37.4631i			1.38562i
$732$		0			0
$733$	−30.8539			−1.13962			−0.569808	−	0.821778i	$-0.692982\pi$
$733$	−30.8539			−1.13962			−0.569808	+	0.821778i	$0.692982\pi$
$734$		0			0
$735$	12.3612	+	10.3860i	0.455950	+	0.383095i
$736$		0			0
$737$	−9.08055	+	8.80137i	−0.334486	+	0.324203i
$738$		0			0
$739$	−12.2773			−0.451628			−0.225814	−	0.974170i	$-0.572504\pi$
$739$	−12.2773			−0.451628			−0.225814	+	0.974170i	$0.572504\pi$
$740$		0			0
$741$	33.6769	+	1.24424i	1.23715	+	0.0457082i
$742$		0			0
$743$			21.6385i			0.793839i	0.917853	+	0.396920i	$0.129921\pi$
$743$			21.6385i			0.793839i	−0.917853	+	0.396920i	$0.870079\pi$
$744$		0			0
$745$			31.1843i			1.14250i
$746$		0			0
$747$	−42.2297	+	7.38948i	−1.54511	+	0.270367i
$748$		0			0
$749$	40.4860			1.47933
$750$		0			0
$751$	42.0795			1.53550			0.767752	−	0.640747i	$-0.221375\pi$
$751$	42.0795			1.53550			0.767752	+	0.640747i	$0.221375\pi$
$752$		0			0
$753$	−1.75185	+	2.08501i	−0.0638410	+	0.0759821i
$754$		0			0
$755$	24.2252			0.881644
$756$		0			0
$757$	13.7262			0.498886			0.249443	−	0.968389i	$-0.419752\pi$
$757$	13.7262			0.498886			0.249443	+	0.968389i	$0.419752\pi$
$758$		0			0
$759$	36.2633	−	2.57916i	1.31627	−	0.0936176i
$760$		0			0
$761$			26.2802i			0.952657i	0.879267	+	0.476329i	$0.158033\pi$
$761$			26.2802i			0.952657i	−0.879267	+	0.476329i	$0.841967\pi$
$762$		0			0
$763$	−23.6665			−0.856786
$764$		0			0
$765$	5.53461	+	31.6294i	0.200104	+	1.14356i
$766$		0			0
$767$	−5.43754	+	4.23628i	−0.196338	+	0.152963i
$768$		0			0
$769$	17.5947			0.634480			0.317240	−	0.948345i	$-0.397244\pi$
$769$	17.5947			0.634480			0.317240	+	0.948345i	$0.397244\pi$
$770$		0			0
$771$	28.9525	−	34.4586i	1.04270	−	1.24100i
$772$		0			0
$773$	22.7176			0.817097			0.408548	−	0.912737i	$-0.366035\pi$
$773$	22.7176			0.817097			0.408548	+	0.912737i	$0.366035\pi$
$774$		0			0
$775$		−	26.4261i		−	0.949255i
$776$		0			0
$777$	−0.879981	+	1.04733i	−0.0315691	+	0.0375729i
$778$		0			0
$779$			38.3459i			1.37388i
$780$		0			0
$781$	−15.4745	−	15.9654i	−0.553722	−	0.571286i
$782$		0			0
$783$	6.79248	−	11.7120i	0.242744	−	0.418552i
$784$		0			0
$785$	39.7795			1.41979
$786$		0			0
$787$	−30.2887			−1.07968			−0.539839	−	0.841769i	$-0.681515\pi$
$787$	−30.2887			−1.07968			−0.539839	+	0.841769i	$0.681515\pi$
$788$		0			0
$789$	27.7810	+	23.3419i	0.989031	+	0.830995i
$790$		0			0
$791$		−	23.8857i		−	0.849278i
$792$		0			0
$793$	10.5695	+	13.5666i	0.375332	+	0.481763i
$794$		0			0
$795$	−17.4817	+	20.8063i	−0.620012	+	0.737924i
$796$		0			0
$797$			28.0647i			0.994103i	0.867721	+	0.497052i	$0.165584\pi$
$797$			28.0647i			0.994103i	−0.867721	+	0.497052i	$0.834416\pi$
$798$		0			0
$799$	20.9002			0.739395
$800$		0			0
$801$	6.20254	+	35.4465i	0.219156	+	1.25244i
$802$		0			0
$803$	−12.3389	−	12.7303i	−0.435430	−	0.449241i
$804$		0			0
$805$		−	41.5029i		−	1.46279i
$806$		0			0
$807$	−5.07804	+	6.04376i	−0.178755	+	0.212750i
$808$		0			0
$809$	−34.6138			−1.21696			−0.608479	−	0.793570i	$-0.708220\pi$
$809$	−34.6138			−1.21696			−0.608479	+	0.793570i	$0.708220\pi$
$810$		0			0
$811$	28.9659			1.01713			0.508565	−	0.861023i	$-0.330176\pi$
$811$	28.9659			1.01713			0.508565	+	0.861023i	$0.330176\pi$
$812$		0			0
$813$	27.4637	+	23.0753i	0.963195	+	0.809287i
$814$		0			0
$815$			56.2417i			1.97006i
$816$		0			0
$817$			61.0526i			2.13596i
$818$		0			0
$819$	−16.2692	−	14.7279i	−0.568493	−	0.514634i
$820$		0			0
$821$		−	43.1923i		−	1.50742i	−0.657207	−	0.753710i	$-0.728262\pi$
$821$		−	43.1923i		−	1.50742i	0.657207	−	0.753710i	$-0.271738\pi$
$822$		0			0
$823$	20.0676			0.699512			0.349756	−	0.936841i	$-0.386265\pi$
$823$	20.0676			0.699512			0.349756	+	0.936841i	$0.386265\pi$
$824$		0			0
$825$	2.22043	+	31.2194i	0.0773053	+	1.08692i
$826$		0			0
$827$			23.8081i			0.827889i	0.910302	+	0.413944i	$0.135849\pi$
$827$			23.8081i			0.827889i	−0.910302	+	0.413944i	$0.864151\pi$
$828$		0			0
$829$	−17.4653			−0.606595			−0.303298	−	0.952896i	$-0.598088\pi$
$829$	−17.4653			−0.606595			−0.303298	+	0.952896i	$0.598088\pi$
$830$		0			0
$831$	10.6962	−	12.7304i	0.371048	−	0.441613i
$832$		0			0
$833$	−9.54897			−0.330852
$834$		0			0
$835$			6.65279i			0.230229i
$836$		0			0
$837$	−21.8018	−	12.6442i	−0.753579	−	0.437046i
$838$		0			0
$839$	26.6611			0.920443			0.460221	−	0.887804i	$-0.347770\pi$
$839$	26.6611			0.920443			0.460221	+	0.887804i	$0.347770\pi$
$840$		0			0
$841$	−22.2108			−0.765890
$842$		0			0
$843$	−0.0482748	+	0.0574556i	−0.00166267	+	0.00197888i
$844$		0			0
$845$	10.2776	−	40.7448i	0.353559	−	1.40166i
$846$		0			0
$847$	−0.696674	+	22.3065i	−0.0239380	+	0.766459i
$848$		0			0
$849$	−8.81886	+	10.4960i	−0.302662	+	0.360222i
$850$		0			0
$851$	−2.46357			−0.0844499
$852$		0			0
$853$	−3.64853			−0.124923			−0.0624616	−	0.998047i	$-0.519895\pi$
$853$	−3.64853			−0.124923			−0.0624616	+	0.998047i	$0.519895\pi$
$854$		0			0
$855$	9.01961	+	51.5457i	0.308464	+	1.76282i
$856$		0			0
$857$	13.6714			0.467007			0.233504	−	0.972356i	$-0.424981\pi$
$857$	13.6714			0.467007			0.233504	+	0.972356i	$0.424981\pi$
$858$		0			0
$859$	−34.7748			−1.18650			−0.593250	−	0.805018i	$-0.702156\pi$
$859$	−34.7748			−1.18650			−0.593250	+	0.805018i	$0.702156\pi$
$860$		0			0
$861$	16.0634	−	19.1183i	0.547438	−	0.651549i
$862$		0			0
$863$	−9.91386			−0.337472			−0.168736	−	0.985661i	$-0.553968\pi$
$863$	−9.91386			−0.337472			−0.168736	+	0.985661i	$0.553968\pi$
$864$		0			0
$865$	2.58123			0.0877643
$866$		0			0
$867$	8.00360	+	6.72471i	0.271816	+	0.228383i
$868$		0			0
$869$	−30.2177	+	29.2887i	−1.02506	+	0.993550i
$870$		0			0
$871$	8.44902	+	10.8449i	0.286284	+	0.367464i
$872$		0			0
$873$	18.0545	−	3.15923i	0.611052	−	0.106924i
$874$		0			0
$875$	2.94016			0.0993954
$876$		0			0
$877$	−58.5056			−1.97559			−0.987796	−	0.155753i	$-0.950220\pi$
$877$	−58.5056			−1.97559			−0.987796	+	0.155753i	$0.950220\pi$
$878$		0			0
$879$	5.43233	−	6.46544i	0.183228	−	0.218074i
$880$		0			0
$881$		−	4.28930i		−	0.144510i	−0.997386	−	0.0722552i	$-0.976980\pi$
$881$		−	4.28930i		−	0.144510i	0.997386	−	0.0722552i	$-0.0230196\pi$
$882$		0			0
$883$	−42.5088			−1.43053			−0.715267	−	0.698851i	$-0.753695\pi$
$883$	−42.5088			−1.43053			−0.715267	+	0.698851i	$0.753695\pi$
$884$		0			0
$885$	−8.19473	−	6.88530i	−0.275463	−	0.231447i
$886$		0			0
$887$	20.6815			0.694416			0.347208	−	0.937788i	$-0.387130\pi$
$887$	20.6815			0.694416			0.347208	+	0.937788i	$0.387130\pi$
$888$		0			0
$889$		−	44.8204i		−	1.50323i
$890$		0			0
$891$	26.8186	+	13.1057i	0.898458	+	0.439059i
$892$		0			0
$893$	34.0605			1.13979
$894$		0			0
$895$		−	36.6559i		−	1.22527i
$896$		0			0
$897$	1.45919	−	39.4950i	0.0487210	−	1.31870i
$898$		0			0
$899$		−	12.6380i		−	0.421502i
$900$		0			0
$901$		−	16.0728i		−	0.535461i
$902$		0			0
$903$	25.5754	−	30.4393i	0.851097	−	1.01296i
$904$		0			0
$905$	−28.7493			−0.955660
$906$		0			0
$907$	14.4513			0.479849			0.239924	−	0.970792i	$-0.422877\pi$
$907$	14.4513			0.479849			0.239924	+	0.970792i	$0.422877\pi$
$908$		0			0
$909$	7.01823	+	40.1081i	0.232780	+	1.33030i
$910$		0			0
$911$			6.86224i			0.227356i	0.993518	+	0.113678i	$0.0362632\pi$
$911$			6.86224i			0.227356i	−0.993518	+	0.113678i	$0.963737\pi$
$912$		0			0
$913$	34.0332	−	32.9868i	1.12633	−	1.09171i
$914$		0			0
$915$	−17.1787	+	20.4457i	−0.567911	+	0.675915i
$916$		0			0
$917$	−24.7155			−0.816176
$918$		0			0
$919$		−	23.6670i		−	0.780702i	−0.920666	−	0.390351i	$-0.872354\pi$
$919$		−	23.6670i		−	0.780702i	0.920666	−	0.390351i	$-0.127646\pi$
$920$		0			0
$921$	32.4960	+	27.3035i	1.07078	+	0.899681i
$922$		0			0
$923$	−19.0674	+	14.8550i	−0.627610	+	0.488959i
$924$		0			0
$925$		−	2.12091i		−	0.0697350i
$926$		0			0
$927$	−6.78430	−	38.7712i	−0.222826	−	1.27341i
$928$		0			0
$929$	5.45468			0.178962			0.0894812	−	0.995989i	$-0.471479\pi$
$929$	5.45468			0.178962			0.0894812	+	0.995989i	$0.471479\pi$
$930$		0			0
$931$	−15.5617			−0.510015
$932$		0			0
$933$	16.6161	−	19.7760i	0.543985	−	0.647439i
$934$		0			0
$935$	−24.7066	−	25.4903i	−0.807994	−	0.833623i
$936$		0			0
$937$		−	13.0757i		−	0.427165i	−0.976925	−	0.213582i	$-0.931487\pi$
$937$		−	13.0757i		−	0.427165i	0.976925	−	0.213582i	$-0.0685132\pi$
$938$		0			0
$939$	−28.0784	−	23.5918i	−0.916305	−	0.769890i
$940$		0			0
$941$		−	53.1504i		−	1.73265i	−0.499478	−	0.866327i	$-0.666475\pi$
$941$		−	53.1504i		−	1.73265i	0.499478	−	0.866327i	$-0.333525\pi$
$942$		0			0
$943$	44.9705			1.46444
$944$		0			0
$945$	17.0959	−	29.4777i	0.556131	−	0.958911i
$946$		0			0
$947$	31.0132			1.00780			0.503898	−	0.863763i	$-0.331899\pi$
$947$	31.0132			1.00780			0.503898	+	0.863763i	$0.331899\pi$
$948$		0			0
$949$	−15.2037	+	11.8449i	−0.493533	+	0.384502i
$950$		0			0
$951$	−43.8166	−	36.8152i	−1.42085	−	1.19381i
$952$		0			0
$953$	55.7116			1.80468			0.902338	−	0.431029i	$-0.141849\pi$
$953$	55.7116			1.80468			0.902338	+	0.431029i	$0.141849\pi$
$954$		0			0
$955$		−	63.8525i		−	2.06622i
$956$		0			0
$957$	1.06190	+	14.9304i	0.0343262	+	0.482630i
$958$		0			0
$959$	−24.8608			−0.802798
$960$		0			0
$961$	7.47439			0.241109
$962$		0			0
$963$	10.3187	+	58.9695i	0.332514	+	1.90026i
$964$		0			0
$965$	−69.5576			−2.23914
$966$		0			0
$967$	31.5039			1.01310			0.506548	−	0.862212i	$-0.330921\pi$
$967$	31.5039			1.01310			0.506548	+	0.862212i	$0.330921\pi$
$968$		0			0
$969$	−23.6957	−	19.9094i	−0.761215	−	0.639581i
$970$		0			0
$971$			9.28501i			0.297970i	0.988839	+	0.148985i	$0.0476006\pi$
$971$			9.28501i			0.297970i	−0.988839	+	0.148985i	$0.952399\pi$
$972$		0			0
$973$		−	8.79130i		−	0.281836i
$974$		0			0
$975$	34.0016	+	1.25623i	1.08892	+	0.0402316i
$976$		0			0
$977$	−16.6509			−0.532709			−0.266354	−	0.963875i	$-0.585819\pi$
$977$	−16.6509			−0.532709			−0.266354	+	0.963875i	$0.585819\pi$
$978$		0			0
$979$	−27.6883	−	28.5665i	−0.884922	−	0.912991i
$980$		0			0
$981$	−6.03188	−	34.4712i	−0.192583	−	1.10058i
$982$		0			0
$983$	−19.4058			−0.618950			−0.309475	−	0.950908i	$-0.600153\pi$
$983$	−19.4058			−0.618950			−0.309475	+	0.950908i	$0.600153\pi$
$984$		0			0
$985$			23.0800i			0.735389i
$986$		0			0
$987$	−16.9817	−	14.2682i	−0.540533	−	0.454162i
$988$		0			0
$989$	71.6001			2.27675
$990$		0			0
$991$	23.1537			0.735501			0.367750	−	0.929925i	$-0.380128\pi$
$991$	23.1537			0.735501			0.367750	+	0.929925i	$0.380128\pi$
$992$		0			0
$993$	8.70851	−	10.3647i	0.276356	−	0.328913i
$994$		0			0
$995$	−82.8616			−2.62689
$996$		0			0
$997$			60.6359i			1.92036i	0.279386	+	0.960179i	$0.409869\pi$
$997$			60.6359i			1.92036i	−0.279386	+	0.960179i	$0.590131\pi$
$998$		0			0
$999$	−1.74976	−	1.01479i	−0.0553601	−	0.0321067i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1716.2.p.a.857.11	yes	56
3.2	odd	2	inner	1716.2.p.a.857.13	yes	56
11.10	odd	2	inner	1716.2.p.a.857.12	yes	56
13.12	even	2	inner	1716.2.p.a.857.9	✓	56
33.32	even	2	inner	1716.2.p.a.857.14	yes	56
39.38	odd	2	inner	1716.2.p.a.857.15	yes	56
143.142	odd	2	inner	1716.2.p.a.857.10	yes	56
429.428	even	2	inner	1716.2.p.a.857.16	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.p.a.857.9	✓	56	13.12	even	2	inner
1716.2.p.a.857.10	yes	56	143.142	odd	2	inner
1716.2.p.a.857.11	yes	56	1.1	even	1	trivial
1716.2.p.a.857.12	yes	56	11.10	odd	2	inner
1716.2.p.a.857.13	yes	56	3.2	odd	2	inner
1716.2.p.a.857.14	yes	56	33.32	even	2	inner
1716.2.p.a.857.15	yes	56	39.38	odd	2	inner
1716.2.p.a.857.16	yes	56	429.428	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1716 = 2^{2} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1716.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.32610 - 1.11421i) q^{3} +3.23239 q^{5} +2.02885 q^{7} +(0.517092 + 2.95510i) q^{9} +O(q^{10})\)	\(q+(-1.32610 - 1.11421i) q^{3} +3.23239 q^{5} +2.02885 q^{7} +(0.517092 + 2.95510i) q^{9} +(-2.30831 - 2.38153i) q^{11} +(-2.84425 + 2.21590i) q^{13} +(-4.28648 - 3.60154i) q^{15} +3.31128 q^{17} +5.39630 q^{19} +(-2.69046 - 2.26055i) q^{21} -6.32857i q^{23} +5.44833 q^{25} +(2.60687 - 4.49491i) q^{27} +2.60561 q^{29} -4.85032i q^{31} +(0.407542 + 5.73009i) q^{33} +6.55803 q^{35} -0.389277i q^{37} +(6.24074 + 0.230572i) q^{39} +7.10595i q^{41} +11.3138i q^{43} +(1.67144 + 9.55203i) q^{45} +6.31181 q^{47} -2.88377 q^{49} +(-4.39109 - 3.68944i) q^{51} -4.85394i q^{53} +(-7.46136 - 7.69803i) q^{55} +(-7.15605 - 6.01259i) q^{57} +1.91176 q^{59} -4.76982i q^{61} +(1.04910 + 5.99545i) q^{63} +(-9.19373 + 7.16265i) q^{65} -3.81290i q^{67} +(-7.05133 + 8.39233i) q^{69} +6.70382 q^{71} +5.34541 q^{73} +(-7.22504 - 6.07056i) q^{75} +(-4.68322 - 4.83177i) q^{77} -12.6883i q^{79} +(-8.46523 + 3.05612i) q^{81} +14.2905i q^{83} +10.7033 q^{85} +(-3.45530 - 2.90318i) q^{87} +11.9950 q^{89} +(-5.77056 + 4.49573i) q^{91} +(-5.40425 + 6.43202i) q^{93} +17.4429 q^{95} -6.10961i q^{97} +(5.84405 - 8.05277i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 8 q^{9}+O(q^{10}) \) 56 * q - 8 * q^9	\( 56 q - 8 q^{9} + 48 q^{25} - 12 q^{27} + 48 q^{49} - 16 q^{55} - 12 q^{69} + 44 q^{75} - 40 q^{81} - 24 q^{91}+O(q^{100}) \) 56 * q - 8 * q^9 + 48 * q^25 - 12 * q^27 + 48 * q^49 - 16 * q^55 - 12 * q^69 + 44 * q^75 - 40 * q^81 - 24 * q^91

Introduction

L-functions

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