LMFDB - Embedded newform 968.2.w.a.835.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(43,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 11, 5]))

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

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

chi := DirichletCharacter("968.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.w (of order $22$, degree $10$, 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:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	20.0.5969915757478328440239161344.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2x^{18} + 4x^{16} - 8x^{14} + 16x^{12} - 32x^{10} + 64x^{8} - 128x^{6} + 256x^{4} - 512x^{2} + 1024 $ x^20 - 2x^18 + 4x^16 - 8x^14 + 16x^12 - 32x^10 + 64x^8 - 128x^6 + 256x^4 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{22}]$

Embedding invariants

Embedding label			835.1
Root			$-1.28641 + 0.587486i$ of defining polynomial
Character	$\chi$	$=$	968.835
Dual form			968.2.w.a.131.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.764582 - 1.18971i) q^{2} -1.74200 q^{3} +(-0.830830 + 1.81926i) q^{4} +(1.33190 + 2.07248i) q^{6} +(2.79964 - 0.402527i) q^{8} +0.0345563 q^{9} +O(q^{10})$	\(q+(-0.764582 - 1.18971i) q^{2} -1.74200 q^{3} +(-0.830830 + 1.81926i) q^{4} +(1.33190 + 2.07248i) q^{6} +(2.79964 - 0.402527i) q^{8} +0.0345563 q^{9} +(-0.972874 - 3.17073i) q^{11} +(1.44730 - 3.16915i) q^{12} +(-2.61944 - 3.02300i) q^{16} +(-1.44638 + 4.92592i) q^{17} +(-0.0264211 - 0.0411120i) q^{18} +(-0.217515 - 0.740788i) q^{19} +(-3.02841 + 3.58172i) q^{22} +(-4.87696 + 0.701201i) q^{24} +(-4.79746 - 1.40866i) q^{25} +5.16580 q^{27} +(-1.59372 + 5.42771i) q^{32} +(1.69475 + 5.52340i) q^{33} +(6.96630 - 2.04549i) q^{34} +(-0.0287104 + 0.0628670i) q^{36} +(-0.715016 + 0.825173i) q^{38} +(6.45709 + 10.0474i) q^{41} +(5.27005 - 0.757719i) q^{43} +(6.57668 + 0.864421i) q^{44} +(4.56306 + 5.26606i) q^{48} +(0.996204 + 6.92875i) q^{49} +(1.99215 + 6.78464i) q^{50} +(2.51959 - 8.58094i) q^{51} +(-3.94967 - 6.14581i) q^{54} +(0.378910 + 1.29045i) q^{57} +(8.72519 + 5.60734i) q^{59} +(7.67594 - 2.25386i) q^{64} +(5.27549 - 6.23935i) q^{66} +(11.7775 - 7.56897i) q^{67} +(-7.75985 - 6.72395i) q^{68} +(0.0967451 - 0.0139098i) q^{72} +(6.70281 - 5.80802i) q^{73} +(8.35717 + 2.45389i) q^{75} +(1.52841 + 0.219752i) q^{76} -9.10247 q^{81} +(7.01658 - 15.3642i) q^{82} +(2.13758 + 1.85223i) q^{83} +(-4.93085 - 5.69050i) q^{86} +(-4.00000 - 8.48528i) q^{88} +(7.20803 + 2.11647i) q^{89} +(2.77626 - 9.45506i) q^{96} +(0.108507 + 0.754683i) q^{97} +(7.48154 - 6.48279i) q^{98} +(-0.0336189 - 0.109569i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9	$ 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9} + 6 q^{11} - 36 q^{12} - 8 q^{16} - 4 q^{22} - 10 q^{25} + 8 q^{27} + 12 q^{33} + 16 q^{34} + 4 q^{36} - 24 q^{38} - 12 q^{44} - 16 q^{48} + 14 q^{49} + 22 q^{51} + 110 q^{57} + 12 q^{59} + 16 q^{64} + 168 q^{66} - 28 q^{67} - 20 q^{75} + 44 q^{76} + 220 q^{81} + 32 q^{82} + 24 q^{86} - 80 q^{88} - 36 q^{89} - 20 q^{97} + 6 q^{99}+O(q^{100}) $ 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9 + 6 * q^11 - 36 * q^12 - 8 * q^16 - 4 * q^22 - 10 * q^25 + 8 * q^27 + 12 * q^33 + 16 * q^34 + 4 * q^36 - 24 * q^38 - 12 * q^44 - 16 * q^48 + 14 * q^49 + 22 * q^51 + 110 * q^57 + 12 * q^59 + 16 * q^64 + 168 * q^66 - 28 * q^67 - 20 * q^75 + 44 * q^76 + 220 * q^81 + 32 * q^82 + 24 * q^86 - 80 * q^88 - 36 * q^89 - 20 * q^97 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/968\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$849$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{7}{22}\right)$

Coefficient data

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

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

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

$2$	−0.764582	−	1.18971i	−0.540641	−	0.841254i
$2$	−0.764582	−	1.18971i	−0.540641	−	0.841254i
$3$	−1.74200			−1.00574			−0.502871	−	0.864361i	$-0.667723\pi$
$3$	−1.74200			−1.00574			−0.502871	+	0.864361i	$0.667723\pi$
$4$	−0.830830	+	1.81926i	−0.415415	+	0.909632i
$5$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$5$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$6$	1.33190	+	2.07248i	0.543746	+	0.846085i
$7$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$7$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$8$	2.79964	−	0.402527i	0.989821	−	0.142315i
$9$	0.0345563			0.0115188
$10$		0			0
$11$	−0.972874	−	3.17073i	−0.293333	−	0.956010i
$11$	−0.972874	−	3.17073i	−0.293333	−	0.956010i
$12$	1.44730	−	3.16915i	0.417801	−	0.914856i
$13$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$13$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$14$		0			0
$15$		0			0
$16$	−2.61944	−	3.02300i	−0.654861	−	0.755750i
$17$	−1.44638	+	4.92592i	−0.350799	+	1.19471i	0.575463	+	0.817828i	$0.304822\pi$
$17$	−1.44638	+	4.92592i	−0.350799	+	1.19471i	−0.926261	+	0.376882i	$0.876996\pi$
$18$	−0.0264211	−	0.0411120i	−0.00622751	−	0.00969020i
$19$	−0.217515	−	0.740788i	−0.0499013	−	0.169948i	0.930772	−	0.365600i	$-0.119136\pi$
$19$	−0.217515	−	0.740788i	−0.0499013	−	0.169948i	−0.980674	+	0.195651i	$0.937318\pi$
$20$		0			0
$21$		0			0
$22$	−3.02841	+	3.58172i	−0.645660	+	0.763625i
$23$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$23$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$24$	−4.87696	+	0.701201i	−0.995506	+	0.143132i
$25$	−4.79746	−	1.40866i	−0.959493	−	0.281733i
$26$		0			0
$27$	5.16580			0.994158
$28$		0			0
$29$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$29$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$30$		0			0
$31$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$31$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$32$	−1.59372	+	5.42771i	−0.281733	+	0.959493i
$33$	1.69475	+	5.52340i	0.295017	+	0.961501i
$34$	6.96630	−	2.04549i	1.19471	−	0.350799i
$35$		0			0
$36$	−0.0287104	+	0.0628670i	−0.00478507	+	0.0104778i
$37$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$37$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$38$	−0.715016	+	0.825173i	−0.115991	+	0.133861i
$39$		0			0
$40$		0			0
$41$	6.45709	+	10.0474i	1.00843	+	1.56914i	0.807782	+	0.589481i	$0.200668\pi$
$41$	6.45709	+	10.0474i	1.00843	+	1.56914i	0.200646	+	0.979664i	$0.435696\pi$
$42$		0			0
$43$	5.27005	−	0.757719i	0.803675	−	0.115551i	0.271778	−	0.962360i	$-0.412388\pi$
$43$	5.27005	−	0.757719i	0.803675	−	0.115551i	0.531897	+	0.846809i	$0.321479\pi$
$44$	6.57668	+	0.864421i	0.991472	+	0.130316i
$45$		0			0
$46$		0			0
$47$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$47$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$48$	4.56306	+	5.26606i	0.658622	+	0.760090i
$49$	0.996204	+	6.92875i	0.142315	+	0.989821i
$50$	1.99215	+	6.78464i	0.281733	+	0.959493i
$51$	2.51959	−	8.58094i	0.352813	−	1.20157i
$52$		0			0
$53$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$53$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$54$	−3.94967	−	6.14581i	−0.537482	−	0.836339i
$55$		0			0
$56$		0			0
$57$	0.378910	+	1.29045i	0.0501879	+	0.170924i
$58$		0			0
$59$	8.72519	+	5.60734i	1.13592	+	0.730013i	0.966788	−	0.255580i	$-0.0822664\pi$
$59$	8.72519	+	5.60734i	1.13592	+	0.730013i	0.169135	+	0.985593i	$0.445903\pi$
$60$		0			0
$61$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$61$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$62$		0			0
$63$		0			0
$64$	7.67594	−	2.25386i	0.959493	−	0.281733i
$65$		0			0
$66$	5.27549	−	6.23935i	0.649367	−	0.768011i
$67$	11.7775	−	7.56897i	1.43886	−	0.924697i	0.439203	−	0.898388i	$-0.355261\pi$
$67$	11.7775	−	7.56897i	1.43886	−	0.924697i	0.999654	−	0.0263091i	$-0.00837541\pi$
$68$	−7.75985	−	6.72395i	−0.941020	−	0.815398i
$69$		0			0
$70$		0			0
$71$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$71$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$72$	0.0967451	−	0.0139098i	0.0114015	−	0.00163929i
$73$	6.70281	−	5.80802i	0.784505	−	0.679777i	−0.167496	−	0.985873i	$-0.553568\pi$
$73$	6.70281	−	5.80802i	0.784505	−	0.679777i	0.952001	+	0.306096i	$0.0990227\pi$
$74$		0			0
$75$	8.35717	+	2.45389i	0.965003	+	0.283351i
$76$	1.52841	+	0.219752i	0.175320	+	0.0252072i
$77$		0			0
$78$		0			0
$79$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$79$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$80$		0			0
$81$	−9.10247			−1.01139
$82$	7.01658	−	15.3642i	0.774851	−	1.69669i
$83$	2.13758	+	1.85223i	0.234630	+	0.203308i	0.764238	−	0.644935i	$-0.223115\pi$
$83$	2.13758	+	1.85223i	0.234630	+	0.203308i	−0.529608	+	0.848243i	$0.677661\pi$
$84$		0			0
$85$		0			0
$86$	−4.93085	−	5.69050i	−0.531707	−	0.613623i
$87$		0			0
$88$	−4.00000	−	8.48528i	−0.426401	−	0.904534i
$89$	7.20803	+	2.11647i	0.764049	+	0.224345i	0.640463	−	0.767989i	$-0.278742\pi$
$89$	7.20803	+	2.11647i	0.764049	+	0.224345i	0.123586	+	0.992334i	$0.460561\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	2.77626	−	9.45506i	0.283351	−	0.965003i
$97$	0.108507	+	0.754683i	0.0110172	+	0.0766265i	0.994588	−	0.103895i	$-0.0331306\pi$
$97$	0.108507	+	0.754683i	0.0110172	+	0.0766265i	−0.983571	+	0.180521i	$0.942221\pi$
$98$	7.48154	−	6.48279i	0.755750	−	0.654861i
$99$	−0.0336189	−	0.109569i	−0.00337883	−	0.0110121i
$100$	6.54861	−	7.55750i	0.654861	−	0.755750i
$101$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$101$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$102$	−12.1353	+	3.56324i	−1.20157	+	0.352813i
$103$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$103$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.08947	−	0.444199i	−0.298670	−	0.0429423i	−0.00864929	−	0.999963i	$-0.502753\pi$
$107$	−3.08947	−	0.444199i	−0.298670	−	0.0429423i	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−4.29190	+	9.39795i	−0.412988	+	0.904318i
$109$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$109$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.02550	+	21.0428i	0.284615	+	1.97954i	0.152263	+	0.988340i	$0.451344\pi$
$113$	3.02550	+	21.0428i	0.284615	+	1.97954i	0.132352	+	0.991203i	$0.457747\pi$
$114$	1.24556	−	1.43745i	0.116657	−	0.134629i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	14.6677i		−	1.35027i
$119$		0			0
$120$		0			0
$121$	−9.10703	+	6.16944i	−0.827912	+	0.560858i
$122$		0			0
$123$	−11.2482	−	17.5026i	−1.01422	−	1.57816i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$127$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$128$	−8.55033	−	7.40890i	−0.755750	−	0.654861i
$129$	−9.18041	+	1.31994i	−0.808290	+	0.116215i
$130$		0			0
$131$	−1.96989	+	0.899617i	−0.172110	+	0.0785999i	−0.499607	−	0.866252i	$-0.666522\pi$
$131$	−1.96989	+	0.899617i	−0.172110	+	0.0785999i	0.327497	+	0.944852i	$0.393795\pi$
$132$	−11.4566	−	1.50582i	−0.997166	−	0.131065i
$133$		0			0
$134$	−18.0098	−	8.22480i	−1.55581	−	0.710514i
$135$		0			0
$136$	−2.06652	+	14.3730i	−0.177203	+	1.23247i
$137$	15.2262	+	17.5719i	1.30086	+	1.50127i	0.738200	+	0.674582i	$0.235676\pi$
$137$	15.2262	+	17.5719i	1.30086	+	1.50127i	0.562658	+	0.826689i	$0.309779\pi$
$138$		0			0
$139$	8.91354	+	13.8697i	0.756037	+	1.17642i	0.979449	+	0.201692i	$0.0646440\pi$
$139$	8.91354	+	13.8697i	0.756037	+	1.17642i	−0.223412	+	0.974724i	$0.571720\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.0905182	−	0.104464i	−0.00754318	−	0.00870530i
$145$		0			0
$146$	−12.0347	−	3.53371i	−0.996000	−	0.292452i
$147$	−1.73538	−	12.0699i	−0.143132	−	0.995506i
$148$		0			0
$149$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$149$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$150$	−3.47032	−	11.8188i	−0.283351	−	0.965003i
$151$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$151$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$152$	−0.907150	−	1.98638i	−0.0735796	−	0.161117i
$153$	−0.0499815	+	0.170221i	−0.00404076	+	0.0137616i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$157$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	6.95958	+	10.8293i	0.546797	+	0.850832i
$163$	3.41360	−	23.7421i	0.267374	−	1.85963i	−0.205714	−	0.978612i	$-0.565952\pi$
$163$	3.41360	−	23.7421i	0.267374	−	1.85963i	0.473088	−	0.881015i	$-0.343139\pi$
$164$	−23.6437	+	3.39945i	−1.84626	+	0.265452i
$165$		0			0
$166$	0.569259	−	3.95929i	0.0441831	−	0.307300i
$167$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$167$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$168$		0			0
$169$	8.51319	+	9.82474i	0.654861	+	0.755750i
$170$		0			0
$171$	−0.00751650	−	0.0255989i	−0.000574801	−	0.00195759i
$172$	−3.00002	+	10.2171i	−0.228750	+	0.779050i
$173$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$173$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$174$		0			0
$175$		0			0
$176$	−7.03672	+	11.2465i	−0.530412	+	0.847740i
$177$	−15.1993	−	9.76797i	−1.14245	−	0.734205i
$178$	−2.99314	−	10.1937i	−0.224345	−	0.764049i
$179$	−17.5179	+	20.2167i	−1.30935	+	1.51107i	−0.636658	+	0.771147i	$0.719684\pi$
$179$	−17.5179	+	20.2167i	−1.30935	+	1.51107i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$181$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	17.0259	−	0.206222i	1.24506	−	0.0150804i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$191$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$192$	−13.3715	+	3.92622i	−0.965003	+	0.283351i
$193$	2.35820	−	0.339058i	0.169747	−	0.0244059i	−0.0569173	−	0.998379i	$-0.518127\pi$
$193$	2.35820	−	0.339058i	0.169747	−	0.0244059i	0.226664	+	0.973973i	$0.427218\pi$
$194$	0.814893	−	0.706109i	0.0585059	−	0.0506957i
$195$		0			0
$196$	−13.4329	−	3.94426i	−0.959493	−	0.281733i
$197$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$197$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$198$	−0.104651	+	0.123771i	−0.00743720	+	0.00879602i
$199$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$199$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$200$	−13.9982	−	2.01264i	−0.989821	−	0.142315i
$201$	−20.5165	+	13.1851i	−1.44712	+	0.930007i
$202$		0			0
$203$		0			0
$204$	13.5176	+	11.7131i	0.946424	+	0.820081i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.13722	+	1.41037i	−0.147835	+	0.0975576i
$210$		0			0
$211$	−15.9006	+	7.26156i	−1.09464	+	0.499906i	−0.879124	−	0.476594i	$-0.841871\pi$
$211$	−15.9006	+	7.26156i	−1.09464	+	0.499906i	−0.215518	+	0.976500i	$0.569144\pi$
$212$		0			0
$213$		0			0
$214$	1.83368	+	4.01520i	0.125348	+	0.274474i
$215$		0			0
$216$	14.4624	−	2.07937i	0.984039	−	0.141483i
$217$		0			0
$218$		0			0
$219$	−11.6763	+	10.1176i	−0.789010	+	0.683681i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$223$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$224$		0			0
$225$	−0.165783	−	0.0486781i	−0.0110522	−	0.00324521i
$226$	22.7217	−	19.6884i	1.51142	−	1.30966i
$227$	24.2751	+	3.49023i	1.61119	+	0.231655i	0.888286	−	0.459290i	$-0.151896\pi$
$227$	24.2751	+	3.49023i	1.61119	+	0.231655i	0.722908	+	0.690945i	$0.242805\pi$
$228$	−2.66248	−	0.382807i	−0.176327	−	0.0253520i
$229$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$229$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	30.4997i		−	1.99810i	−0.0435600	−	0.999051i	$-0.513870\pi$
$233$		−	30.4997i		−	1.99810i	0.0435600	−	0.999051i	$-0.486130\pi$
$234$		0			0
$235$		0			0
$236$	−17.4504	+	11.2147i	−1.13592	+	0.730013i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			16.6006i			1.06934i	0.845061	+	0.534669i	$0.179564\pi$
$241$			16.6006i			1.06934i	−0.845061	+	0.534669i	$0.820436\pi$
$242$	14.3029	+	6.11771i	0.919427	+	0.393261i
$243$	0.359102			0.0230364
$244$		0			0
$245$		0			0
$246$	−12.2229	+	26.7643i	−0.779301	+	1.70643i
$247$		0			0
$248$		0			0
$249$	−3.72366	−	3.22657i	−0.235978	−	0.204476i
$250$		0			0
$251$	−27.4426			−1.73216			−0.866080	−	0.499906i	$-0.833368\pi$
$251$	−27.4426			−1.73216			−0.866080	+	0.499906i	$0.833368\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.27704	+	15.8371i	−0.142315	+	0.989821i
$257$	−2.72120	+	18.9263i	−0.169744	+	1.18059i	0.709670	+	0.704534i	$0.248844\pi$
$257$	−2.72120	+	18.9263i	−0.169744	+	1.18059i	−0.879414	+	0.476058i	$0.842065\pi$
$258$	8.58953	+	9.91285i	0.534761	+	0.617147i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	2.57642	+	1.65577i	0.159172	+	0.102294i
$263$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$263$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$264$	6.96799	+	14.7813i	0.428850	+	0.909729i
$265$		0			0
$266$		0			0
$267$	−12.5564	−	3.68688i	−0.768437	−	0.225633i
$268$	3.98482	+	27.7150i	0.243411	+	1.69296i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$271$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$272$	18.6797	−	8.53075i	1.13263	−	0.517253i
$273$		0			0
$274$	9.26389	−	31.5499i	0.559652	−	1.90600i
$275$	0.200844	+	16.5819i	0.0121114	+	0.999927i
$276$		0			0
$277$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$277$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$278$	9.68587	−	21.2091i	0.580920	−	1.27204i
$279$		0			0
$280$		0			0
$281$	−29.2226	−	13.3455i	−1.74327	−	0.796127i	−0.990444	−	0.137912i	$-0.955961\pi$
$281$	−29.2226	−	13.3455i	−1.74327	−	0.796127i	−0.752830	−	0.658215i	$-0.771312\pi$
$282$		0			0
$283$	−9.81443	−	15.2716i	−0.583407	−	0.907799i	0.416592	−	0.909094i	$-0.363224\pi$
$283$	−9.81443	−	15.2716i	−0.583407	−	0.907799i	−0.999999	+	0.00129416i	$0.999588\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.0550730	+	0.187562i	−0.00324521	+	0.0110522i
$289$	−7.87133	−	5.05860i	−0.463019	−	0.297564i
$290$		0			0
$291$	−0.189019	−	1.31466i	−0.0110805	−	0.0770665i
$292$	4.99742	+	17.0197i	0.292452	+	0.996000i
$293$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$293$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$294$	−13.0328	+	11.2930i	−0.760090	+	0.658622i
$295$		0			0
$296$		0			0
$297$	−5.02567	−	16.3793i	−0.291619	−	0.950425i
$298$		0			0
$299$		0			0
$300$	−11.4077	+	13.1651i	−0.658622	+	0.760090i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.66963	+	2.59800i	−0.0957599	+	0.149005i
$305$		0			0
$306$	0.240729	−	0.0706845i	0.0137616	−	0.00404076i
$307$	13.7972	−	21.4688i	0.787447	−	1.22529i	−0.182794	−	0.983151i	$-0.558514\pi$
$307$	13.7972	−	21.4688i	0.787447	−	1.22529i	0.970241	−	0.242140i	$-0.0778494\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$311$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$312$		0			0
$313$	−0.0312887	+	0.00918721i	−0.00176854	+	0.000519292i	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−0.0312887	+	0.00918721i	−0.00176854	+	0.000519292i	0.280848	+	0.959752i	$0.409384\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$317$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	5.38185	+	0.773793i	0.300385	+	0.0431889i
$322$		0			0
$323$	3.96367			0.220544
$324$	7.56261	−	16.5598i	0.420145	−	0.919989i
$325$		0			0
$326$	−30.8563	+	14.0916i	−1.70897	+	0.780461i
$327$		0			0
$328$	22.1219	+	25.5300i	1.22148	+	1.40966i
$329$		0			0
$330$		0			0
$331$	−11.8542	−	3.48071i	−0.651566	−	0.191317i	−0.0607875	−	0.998151i	$-0.519361\pi$
$331$	−11.8542	−	3.48071i	−0.651566	−	0.191317i	−0.590779	+	0.806834i	$0.701179\pi$
$332$	−5.14566	+	2.34994i	−0.282405	+	0.128970i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	33.5957	−	4.83033i	1.83007	−	0.263125i	0.860769	−	0.508995i	$-0.169983\pi$
$337$	33.5957	−	4.83033i	1.83007	−	0.263125i	0.969303	+	0.245871i	$0.0790739\pi$
$338$	5.17959	−	17.6401i	0.281733	−	0.959493i
$339$	−5.27042	−	36.6566i	−0.286250	−	1.99091i
$340$		0			0
$341$		0			0
$342$	−0.0247083	+	0.0285149i	−0.00133607	+	0.00154191i
$343$		0			0
$344$	14.4492	−	4.24268i	0.779050	−	0.228750i
$345$		0			0
$346$		0			0
$347$	−7.41907	+	6.42866i	−0.398276	+	0.345108i	−0.830856	−	0.556487i	$-0.812149\pi$
$347$	−7.41907	+	6.42866i	−0.398276	+	0.345108i	0.432580	+	0.901596i	$0.357603\pi$
$348$		0			0
$349$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$349$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$350$		0			0
$351$		0			0
$352$	18.7603	−	0.227229i	0.999927	−	0.0121114i
$353$	−15.5660	−	34.0847i	−0.828493	−	1.81415i	−0.482419	−	0.875941i	$-0.660242\pi$
$353$	−15.5660	−	34.0847i	−0.828493	−	1.81415i	−0.346075	−	0.938207i	$-0.612486\pi$
$354$			25.5511i			1.35803i
$355$		0			0
$356$	−9.83906	+	11.3549i	−0.521469	+	0.601807i
$357$		0			0
$358$	37.4460	+	5.38392i	1.97908	+	0.284549i
$359$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$359$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$360$		0			0
$361$	15.4824	−	9.94991i	0.814861	−	0.523680i
$362$		0			0
$363$	15.8644	−	10.7472i	0.832667	−	0.564079i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$367$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$368$		0			0
$369$	0.223133	+	0.347202i	0.0116158	+	0.0180746i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$373$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$374$	−13.2630	−	20.0982i	−0.685815	−	1.03925i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	23.9385	+	27.6265i	1.22964	+	1.41908i	0.875033	+	0.484063i	$0.160839\pi$
$379$	23.9385	+	27.6265i	1.22964	+	1.41908i	0.354606	+	0.935016i	$0.384615\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$383$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$384$	14.8947	+	12.9063i	0.760090	+	0.658622i
$385$		0			0
$386$	−2.20642	−	2.54634i	−0.112304	−	0.129605i
$387$	0.182113	−	0.0261839i	0.00925734	−	0.00133100i
$388$	−1.46312	−	0.429610i	−0.0742786	−	0.0218102i
$389$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$389$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$390$		0			0
$391$		0			0
$392$	5.57802	+	18.9970i	0.281733	+	0.959493i
$393$	3.43154	−	1.56713i	0.173098	−	0.0790513i
$394$		0			0
$395$		0			0
$396$	0.227266	+	0.0298712i	0.0114205	+	0.00150108i
$397$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$397$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$	8.30830	+	18.1926i	0.415415	+	0.909632i
$401$	−17.0244	+	19.6472i	−0.850158	+	0.981135i	−0.999971	−	0.00757925i	$-0.997587\pi$
$401$	−17.0244	+	19.6472i	−0.850158	+	0.981135i	0.149813	+	0.988714i	$0.452133\pi$
$402$	31.3730	+	14.3276i	1.56474	+	0.714595i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	3.59988	−	25.0377i	0.178221	−	1.23955i
$409$	7.42877	−	25.3001i	0.367329	−	1.25101i	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	7.42877	−	25.3001i	0.367329	−	1.25101i	0.911244	−	0.411867i	$-0.135123\pi$
$410$		0			0
$411$	−26.5239	−	30.6103i	−1.30833	−	1.50989i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−15.5274	−	24.1611i	−0.760378	−	1.18317i
$418$	3.31202	+	1.46433i	0.161996	+	0.0716228i
$419$	−23.2439	−	14.9379i	−1.13554	−	0.729766i	−0.168829	−	0.985645i	$-0.553999\pi$
$419$	−23.2439	−	14.9379i	−1.13554	−	0.729766i	−0.966709	+	0.255879i	$0.917635\pi$
$420$		0			0
$421$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$421$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$422$	20.7965	+	13.3651i	1.01236	+	0.650602i
$423$		0			0
$424$		0			0
$425$	13.8779	−	21.5944i	0.673178	−	1.04748i
$426$		0			0
$427$		0			0
$428$	3.37494	−	5.25151i	0.163134	−	0.253841i
$429$		0			0
$430$		0			0
$431$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$431$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$432$	−13.5315	−	15.6162i	−0.651035	−	0.751334i
$433$	−21.8694	−	6.42143i	−1.05098	−	0.308594i	−0.289764	−	0.957098i	$-0.593577\pi$
$433$	−21.8694	−	6.42143i	−1.05098	−	0.308594i	−0.761212	+	0.648504i	$0.775395\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	20.9645	+	6.15572i	1.00172	+	0.294132i
$439$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$439$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$440$		0			0
$441$	0.0344251	+	0.239432i	0.00163929	+	0.0114015i
$442$		0			0
$443$	−28.4373	+	18.2755i	−1.35110	+	0.868297i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−28.4373	+	18.2755i	−1.35110	+	0.868297i	−0.353357	+	0.935489i	$0.614960\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	25.3464	+	29.2513i	1.19617	+	1.38045i	0.905891	+	0.423511i	$0.139202\pi$
$449$	25.3464	+	29.2513i	1.19617	+	1.38045i	0.290278	+	0.956942i	$0.406252\pi$
$450$	0.0688413	+	0.234452i	0.00324521	+	0.0110522i
$451$	25.5757	−	30.2486i	1.20431	−	1.42435i
$452$	−40.7961	−	11.9788i	−1.91889	−	0.563437i
$453$		0			0
$454$	−14.4079	−	31.5489i	−0.676197	−	1.48066i
$455$		0			0
$456$	1.58025	+	3.46027i	0.0740021	+	0.162042i
$457$		−	33.9411i		−	1.58770i	−0.608114	−	0.793849i	$-0.708074\pi$
$457$		−	33.9411i		−	1.58770i	0.608114	−	0.793849i	$-0.291926\pi$
$458$		0			0
$459$	−7.47170	+	25.4463i	−0.348749	+	1.18773i
$460$		0			0
$461$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$461$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$462$		0			0
$463$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$463$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$464$		0			0
$465$		0			0
$466$	−36.2859	+	23.3195i	−1.68091	+	1.08026i
$467$	33.6451	+	9.87908i	1.55691	+	0.457149i	0.943156	−	0.332350i	$-0.107841\pi$
$467$	33.6451	+	9.87908i	1.55691	+	0.457149i	0.613752	+	0.789499i	$0.289660\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	26.6845	+	12.1864i	1.22825	+	0.560924i
$473$	−7.52962	−	15.9727i	−0.346212	−	0.734427i
$474$		0			0
$475$			3.86031i			0.177123i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$479$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$480$		0			0
$481$		0			0
$482$	19.7499	−	12.6925i	0.899585	−	0.578128i
$483$		0			0
$484$	−3.65745	−	21.6938i	−0.166248	−	0.986084i
$485$		0			0
$486$	−0.274562	−	0.427227i	−0.0124544	−	0.0193794i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−5.94649	+	41.3587i	−0.268910	+	1.87031i
$490$		0			0
$491$	33.3130	+	28.8659i	1.50339	+	1.30270i	0.819423	+	0.573189i	$0.194294\pi$
$491$	33.3130	+	28.8659i	1.50339	+	1.30270i	0.683971	+	0.729509i	$0.260251\pi$
$492$	41.1872	−	5.92183i	1.85686	−	0.266977i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.991649	+	6.89707i	−0.0444368	+	0.309065i
$499$	3.25840	−	22.6626i	0.145866	−	1.01452i	−0.777028	−	0.629466i	$-0.783274\pi$
$499$	3.25840	−	22.6626i	0.145866	−	1.01452i	0.922894	−	0.385054i	$-0.125817\pi$
$500$		0			0
$501$		0			0
$502$	20.9821	+	32.6488i	0.936476	+	1.45719i
$503$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$503$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−14.8300	−	17.1147i	−0.658622	−	0.760090i
$508$		0			0
$509$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$509$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$510$		0			0
$511$		0			0
$512$	20.5826	−	9.39977i	0.909632	−	0.415415i
$513$	−1.12364	−	3.82676i	−0.0496098	−	0.168955i
$514$	24.5975	−	11.2333i	1.08495	−	0.495479i
$515$		0			0
$516$	5.22604	−	17.7982i	0.230063	−	0.783524i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.49249	−	5.45779i	−0.109198	−	0.239110i	0.847142	−	0.531366i	$-0.178321\pi$
$521$	−2.49249	−	5.45779i	−0.109198	−	0.239110i	−0.956340	+	0.292256i	$0.905594\pi$
$522$		0			0
$523$	−12.4791	−	5.69902i	−0.545674	−	0.249201i	0.123451	−	0.992351i	$-0.460604\pi$
$523$	−12.4791	−	5.69902i	−0.545674	−	0.249201i	−0.669125	+	0.743150i	$0.733331\pi$
$524$		−	4.33117i		−	0.189208i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	12.2579	−	19.5914i	0.533459	−	0.852608i
$529$	−3.27324	+	22.7659i	−0.142315	+	0.989821i
$530$		0			0
$531$	0.301510	+	0.193769i	0.0130844	+	0.00840884i
$532$		0			0
$533$		0			0
$534$	5.21404	+	17.7574i	0.225633	+	0.768437i
$535$		0			0
$536$	29.9262	−	25.9312i	1.29261	−	1.12006i
$537$	30.5161	−	35.2175i	1.31687	−	1.51975i
$538$		0			0
$539$	21.0000	−	9.89949i	0.904534	−	0.426401i
$540$		0			0
$541$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$541$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$542$		0			0
$543$		0			0
$544$	−24.4313	−	15.7011i	−1.04748	−	0.673178i
$545$		0			0
$546$		0			0
$547$	−3.64130	−	3.15520i	−0.155691	−	0.134907i	0.573532	−	0.819183i	$-0.305573\pi$
$547$	−3.64130	−	3.15520i	−0.155691	−	0.134907i	−0.729223	+	0.684276i	$0.760118\pi$
$548$	−44.6183	+	13.1011i	−1.90600	+	0.559652i
$549$		0			0
$550$	19.5741	−	12.9172i	0.834644	−	0.550790i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−32.6384	+	4.69269i	−1.38417	+	0.199014i
$557$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$557$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−29.6591	+	0.359238i	−1.25221	+	0.0151670i
$562$	6.46574	+	44.9702i	0.272741	+	1.89695i
$563$	−14.5635	−	2.09392i	−0.613780	−	0.0882482i	−0.171588	−	0.985169i	$-0.554890\pi$
$563$	−14.5635	−	2.09392i	−0.613780	−	0.0882482i	−0.442192	+	0.896921i	$0.645799\pi$
$564$		0			0
$565$		0			0
$566$	−10.6648	+	23.3527i	−0.448276	+	0.981587i
$567$		0			0
$568$		0			0
$569$	8.98542	+	4.10350i	0.376688	+	0.172028i	0.594757	−	0.803906i	$-0.297248\pi$
$569$	8.98542	+	4.10350i	0.376688	+	0.172028i	−0.218068	+	0.975933i	$0.569976\pi$
$570$		0			0
$571$	−3.14326	−	10.7050i	−0.131541	−	0.447989i	0.867210	−	0.497942i	$-0.165911\pi$
$571$	−3.14326	−	10.7050i	−0.131541	−	0.447989i	−0.998752	+	0.0499535i	$0.984093\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.265252	−	0.0778850i	0.0110522	−	0.00324521i
$577$	18.6929	+	40.9317i	0.778194	+	1.70401i	0.707719	+	0.706494i	$0.249724\pi$
$577$	18.6929	+	40.9317i	0.778194	+	1.70401i	0.0704748	+	0.997514i	$0.477549\pi$
$578$			13.2323i			0.550392i
$579$	−4.10798	+	0.590638i	−0.170722	+	0.0245461i
$580$		0			0
$581$		0			0
$582$	−1.41954	+	1.23004i	−0.0588419	+	0.0509868i
$583$		0			0
$584$	16.4276	−	18.9584i	0.679777	−	0.784505i
$585$		0			0
$586$		0			0
$587$	16.2392	−	10.4363i	0.670264	−	0.430752i	−0.160757	−	0.986994i	$-0.551394\pi$
$587$	16.2392	−	10.4363i	0.670264	−	0.430752i	0.831021	+	0.556242i	$0.187757\pi$
$588$	23.4001	+	6.87088i	0.965003	+	0.283351i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	43.4825	+	19.8578i	1.78561	+	0.815462i	0.972336	+	0.233585i	$0.0750458\pi$
$593$	43.4825	+	19.8578i	1.78561	+	0.815462i	0.813277	+	0.581877i	$0.197681\pi$
$594$	−15.6442	+	18.5024i	−0.641888	+	0.759164i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$599$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$600$	24.3848	+	3.50601i	0.995506	+	0.143132i
$601$	−3.02311	−	1.38061i	−0.123315	−	0.0563161i	0.352802	−	0.935698i	$-0.385229\pi$
$601$	−3.02311	−	1.38061i	−0.123315	−	0.0563161i	−0.476117	+	0.879382i	$0.657956\pi$
$602$		0			0
$603$	0.406988	−	0.261555i	0.0165738	−	0.0106514i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$607$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$608$	4.36744			0.177123
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.268151	−	0.232355i	−0.0108394	−	0.00939237i
$613$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$613$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$614$	−36.0908			−1.45651
$615$		0			0
$616$		0			0
$617$	−20.1402	+	44.1008i	−0.810813	+	1.77543i	−0.206935	+	0.978355i	$0.566349\pi$
$617$	−20.1402	+	44.1008i	−0.810813	+	1.77543i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−6.37713	+	44.3539i	−0.256319	+	1.78274i	0.302201	+	0.953244i	$0.402279\pi$
$619$	−6.37713	+	44.3539i	−0.256319	+	1.78274i	−0.558520	+	0.829491i	$0.688631\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	21.0313	+	13.5160i	0.841254	+	0.540641i
$626$	0.0348529	+	0.0302002i	0.00139300	+	0.00120704i
$627$	3.72303	−	2.45687i	0.148684	−	0.0981178i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$631$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$632$		0			0
$633$	27.6988	−	12.6496i	1.10093	−	0.502777i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	13.4304	−	29.4085i	0.530469	−	1.16157i	−0.434852	−	0.900502i	$-0.643199\pi$
$641$	13.4304	−	29.4085i	0.530469	−	1.16157i	0.965322	−	0.261064i	$-0.0840734\pi$
$642$	−3.19427	−	6.99448i	−0.126068	−	0.276050i
$643$	10.1599	−	11.7252i	0.400669	−	0.462397i	−0.519183	−	0.854663i	$-0.673764\pi$
$643$	10.1599	−	11.7252i	0.400669	−	0.462397i	0.919851	+	0.392267i	$0.128309\pi$
$644$		0			0
$645$		0			0
$646$	−3.03055	−	4.71562i	−0.119235	−	0.185534i
$647$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$647$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$648$	−25.4836	+	3.66399i	−1.00109	+	0.143935i
$649$	9.29083	−	33.1204i	0.364697	−	1.30009i
$650$		0			0
$651$		0			0
$652$	40.3571	+	25.9359i	1.58051	+	1.01573i
$653$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$653$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$654$		0			0
$655$		0			0
$656$	13.4594	−	45.8384i	0.525500	−	1.78969i
$657$	0.231624	−	0.200704i	0.00903652	−	0.00783019i
$658$		0			0
$659$	22.2011	+	34.5455i	0.864831	+	1.34570i	0.937369	+	0.348338i	$0.113254\pi$
$659$	22.2011	+	34.5455i	0.864831	+	1.34570i	−0.0725382	+	0.997366i	$0.523110\pi$
$660$		0			0
$661$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$661$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$662$	4.92247	+	16.7644i	0.191317	+	0.651566i
$663$		0			0
$664$	6.73003	+	4.32513i	0.261176	+	0.167847i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	28.2826	+	24.5070i	1.09021	+	0.944676i	0.998692	−	0.0511275i	$-0.0162815\pi$
$673$	28.2826	+	24.5070i	1.09021	+	0.944676i	0.0915216	+	0.995803i	$0.470827\pi$
$674$	−31.4333	−	36.2760i	−1.21077	−	1.39730i
$675$	−24.7827	−	7.27687i	−0.953888	−	0.280087i
$676$	−24.9468	+	7.32505i	−0.959493	+	0.281733i
$677$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$677$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$678$	−39.5811	+	34.2972i	−1.52010	+	1.31718i
$679$		0			0
$680$		0			0
$681$	−42.2871	−	6.07997i	−1.62045	−	0.232985i
$682$		0			0
$683$	7.42221	+	51.6226i	0.284003	+	1.97528i	0.207652	+	0.978203i	$0.433418\pi$
$683$	7.42221	+	51.6226i	0.284003	+	1.97528i	0.0763508	+	0.997081i	$0.475673\pi$
$684$	0.0528160	+	0.00759380i	0.00201947	+	0.000290356i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−16.0952	−	13.9465i	−0.613623	−	0.531707i
$689$		0			0
$690$		0			0
$691$	−7.12839	−	8.22660i	−0.271177	−	0.312955i	0.603785	−	0.797148i	$-0.293659\pi$
$691$	−7.12839	−	8.22660i	−0.271177	−	0.312955i	−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$	13.3207	+	3.91132i	0.505648	+	0.148472i
$695$		0			0
$696$		0			0
$697$	−58.8322	+	17.2747i	−2.22843	+	0.654326i
$698$		0			0
$699$			53.1304i			2.00958i
$700$		0			0
$701$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$701$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$702$		0			0
$703$		0			0
$704$	−14.6141	−	22.1456i	−0.550790	−	0.834644i
$705$		0			0
$706$	−28.6496	+	44.5796i	−1.07824	+	1.67778i
$707$		0			0
$708$	30.3985	−	19.5359i	1.14245	−	0.734205i
$709$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$709$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$710$		0			0
$711$		0			0
$712$	21.0318	+	3.02392i	0.788200	+	0.113326i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−22.2252	−	48.6664i	−0.830594	−	1.81875i
$717$		0			0
$718$		0			0
$719$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$719$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$720$		0			0
$721$		0			0
$722$	−23.6751	−	10.8120i	−0.881095	−	0.402382i
$723$		−	28.9182i		−	1.07548i
$724$		0			0
$725$		0			0
$726$	−24.9157	−	10.6570i	−0.924707	−	0.395519i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	26.6819			0.988217
$730$		0			0
$731$	−3.89003	+	27.0558i	−0.143878	+	1.00069i
$732$		0			0
$733$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$733$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−35.4572	−	29.9797i	−1.30608	−	1.10432i
$738$	0.242467	−	0.530928i	0.00892532	−	0.0195437i
$739$	−1.89922	−	0.867346i	−0.0698640	−	0.0319058i	0.380177	−	0.924914i	$-0.375863\pi$
$739$	−1.89922	−	0.867346i	−0.0698640	−	0.0319058i	−0.450041	+	0.893008i	$0.648590\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$743$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.0738669	+	0.0640060i	0.00270265	+	0.00234186i
$748$	−13.7704	+	31.1459i	−0.503497	+	1.13881i
$749$		0			0
$750$		0			0
$751$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$751$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$752$		0			0
$753$	47.8049			1.74211
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$757$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$758$	14.5647	−	49.6027i	0.529012	−	1.80165i
$759$		0			0
$760$		0			0
$761$	40.6745	−	35.2446i	1.47445	−	1.27762i	0.593117	−	0.805116i	$-0.297897\pi$
$761$	40.6745	−	35.2446i	1.47445	−	1.27762i	0.881331	−	0.472500i	$-0.156648\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	3.96659	−	27.5883i	0.143132	−	0.995506i
$769$	−49.4580	+	7.11098i	−1.78350	+	0.256429i	−0.953513	−	0.301353i	$-0.902562\pi$
$769$	−49.4580	+	7.11098i	−1.78350	+	0.256429i	−0.829987	+	0.557782i	$0.811653\pi$
$770$		0			0
$771$	4.74032	−	32.9696i	0.170718	−	1.18737i
$772$	−1.34243	+	4.57188i	−0.0483150	+	0.164546i
$773$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$773$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$774$	−0.170392	−	0.196643i	−0.00612461	−	0.00706817i
$775$		0			0
$776$	0.607561	+	2.06916i	0.0218102	+	0.0742786i
$777$		0			0
$778$		0			0
$779$	6.03850	−	6.96880i	0.216352	−	0.249683i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	18.3361	−	21.1610i	0.654861	−	0.755750i
$785$		0			0
$786$	−4.48813	−	2.88434i	−0.160086	−	0.102881i
$787$	−13.7625	+	21.4148i	−0.490579	+	0.763356i	−0.994976	−	0.100116i	$-0.968079\pi$
$787$	−13.7625	+	21.4148i	−0.490579	+	0.763356i	0.504397	+	0.863472i	$0.331715\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.138225	−	0.293220i	−0.00491162	−	0.0104191i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$797$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$798$		0			0
$799$		0			0
$800$	15.2916	−	23.7942i	0.540641	−	0.841254i
$801$	0.249083	+	0.0731372i	0.00880090	+	0.00258418i
$802$	36.3911	+	5.23225i	1.28501	+	0.184757i
$803$	−24.9366	−	15.6023i	−0.879995	−	0.550594i
$804$	−6.94154	−	48.2795i	−0.244809	−	1.70269i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.6350	−	11.8148i	−0.479379	−	0.415385i	0.381363	−	0.924425i	$-0.375455\pi$
$809$	−13.6350	−	11.8148i	−0.479379	−	0.415385i	−0.860742	+	0.509041i	$0.830000\pi$
$810$		0			0
$811$	−0.850593	−	0.388453i	−0.0298684	−	0.0136404i	0.400425	−	0.916330i	$-0.368863\pi$
$811$	−0.850593	−	0.388453i	−0.0298684	−	0.0136404i	−0.430293	+	0.902689i	$0.641590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−32.5401	+	14.8606i	−1.13913	+	0.520223i
$817$	−1.70762	−	3.73917i	−0.0597421	−	0.130817i
$818$	−35.7797	+	10.5059i	−1.25101	+	0.367329i
$819$		0			0
$820$		0			0
$821$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$821$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$822$	−16.1377	+	54.9599i	−0.562866	+	1.91695i
$823$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$823$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$824$		0			0
$825$	−0.349870	−	28.8856i	−0.0121809	−	1.00567i
$826$		0			0
$827$	−18.4956	+	28.7797i	−0.643156	+	1.00077i	0.354678	+	0.934988i	$0.384590\pi$
$827$	−18.4956	+	28.7797i	−0.643156	+	1.00077i	−0.997834	+	0.0657814i	$0.979046\pi$
$828$		0			0
$829$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$829$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−35.5713	−	5.11439i	−1.23247	−	0.177203i
$834$	−16.8728	+	36.9462i	−0.584256	+	1.27934i
$835$		0			0
$836$	−0.790174	−	5.05995i	−0.0273288	−	0.175002i
$837$		0			0
$838$			39.0748i			1.34982i
$839$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$839$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$840$		0			0
$841$	−24.3964	+	15.6786i	−0.841254	+	0.540641i
$842$		0			0
$843$	50.9057	+	23.2479i	1.75329	+	0.800699i
$844$		−	34.9605i		−	1.20339i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	17.0967	+	26.6030i	0.586758	+	0.913013i
$850$	−36.3020			−1.24515
$851$		0			0
$852$		0			0
$853$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$853$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$854$		0			0
$855$		0			0
$856$	−8.82820			−0.301742
$857$	33.5876	−	15.3390i	1.14733	−	0.523969i	0.251278	−	0.967915i	$-0.419149\pi$
$857$	33.5876	−	15.3390i	1.14733	−	0.523969i	0.896053	+	0.443946i	$0.146422\pi$
$858$		0			0
$859$	−24.1112	+	52.7961i	−0.822663	+	1.80138i	−0.284099	+	0.958795i	$0.591695\pi$
$859$	−24.1112	+	52.7961i	−0.822663	+	1.80138i	−0.538563	+	0.842585i	$0.681033\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$863$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$864$	−8.23283	+	28.0385i	−0.280087	+	0.953888i
$865$		0			0
$866$	9.08128	+	30.9280i	0.308594	+	1.05098i
$867$	13.7118	+	8.81206i	0.465678	+	0.299273i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.00374960	+	0.0260790i	0.000126905	+	0.000882642i
$874$		0			0
$875$		0			0
$876$	−8.70550	−	29.6482i	−0.294132	−	1.00172i
$877$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$877$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−17.2709	+	5.07119i	−0.581871	+	0.170853i	−0.559404	−	0.828895i	$-0.688970\pi$
$881$	−17.2709	+	5.07119i	−0.581871	+	0.170853i	−0.0224663	+	0.999748i	$0.507152\pi$
$882$	0.258534	−	0.224021i	0.00870530	−	0.00754318i
$883$	6.15440	−	13.4763i	0.207112	−	0.453512i	−0.777359	−	0.629057i	$-0.783441\pi$
$883$	6.15440	−	13.4763i	0.207112	−	0.453512i	0.984472	+	0.175544i	$0.0561685\pi$
$884$		0			0
$885$		0			0
$886$	43.4853	+	19.8591i	1.46092	+	0.667178i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	8.85556	+	28.8615i	0.296673	+	0.966896i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	15.4212	−	52.5199i	0.514613	−	1.75261i
$899$		0			0
$900$	0.226295	−	0.261159i	0.00754318	−	0.00870530i
$901$		0			0
$902$	−55.5418	−	7.30026i	−1.84934	−	0.243072i
$903$		0			0
$904$	16.9406	+	57.6945i	0.563437	+	1.91889i
$905$		0			0
$906$		0			0
$907$	−48.2621	−	31.0162i	−1.60252	−	1.02987i	−0.965976	−	0.258631i	$-0.916729\pi$
$907$	−48.2621	−	31.0162i	−1.60252	−	1.02987i	−0.636540	−	0.771244i	$-0.719635\pi$
$908$	−26.5181	+	41.2630i	−0.880035	+	1.36936i
$909$		0			0
$910$		0			0
$911$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$911$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$912$	2.90849	−	4.52571i	0.0963099	−	0.149861i
$913$	3.79330	−	8.57968i	0.125540	−	0.283946i
$914$	−40.3802	+	25.9508i	−1.33566	+	0.858375i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	35.9865	−	10.5666i	1.18773	−	0.348749i
$919$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$919$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$920$		0			0
$921$	−24.0347	+	37.3987i	−0.791969	+	1.23233i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	14.8226	−	32.4569i	0.486313	−	1.06488i	−0.494366	−	0.869254i	$-0.664600\pi$
$929$	14.8226	−	32.4569i	0.486313	−	1.06488i	0.980679	−	0.195623i	$-0.0626730\pi$
$930$		0			0
$931$	4.91604	−	2.24508i	0.161117	−	0.0735796i
$932$	55.4870	+	25.3401i	1.81754	+	0.830041i
$933$		0			0
$934$	−13.9711	−	47.5813i	−0.457149	−	1.55691i
$935$		0			0
$936$		0			0
$937$	−35.7217	+	16.3136i	−1.16698	+	0.532941i	−0.902178	−	0.431363i	$-0.858033\pi$
$937$	−35.7217	+	16.3136i	−1.16698	+	0.532941i	−0.264799	+	0.964304i	$0.585306\pi$
$938$		0			0
$939$	0.0545049	−	0.0160041i	0.00177870	−	0.000522274i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−5.90416	−	41.0643i	−0.192164	−	1.33653i
$945$		0			0
$946$	−13.2459	+	21.1705i	−0.430663	+	0.688313i
$947$	−37.6300	+	43.4273i	−1.22281	+	1.41120i	−0.340688	+	0.940176i	$0.610660\pi$
$947$	−37.6300	+	43.4273i	−1.22281	+	1.41120i	−0.882122	+	0.471022i	$0.843885\pi$
$948$		0			0
$949$		0			0
$950$	4.59265	−	2.95152i	0.149005	−	0.0957599i
$951$		0			0
$952$		0			0
$953$	−31.2674	−	4.49557i	−1.01285	−	0.145626i	−0.384148	−	0.923272i	$-0.625505\pi$
$953$	−31.2674	−	4.49557i	−1.01285	−	0.145626i	−0.628703	+	0.777646i	$0.716414\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.3007	+	23.4282i	−0.654861	+	0.755750i
$962$		0			0
$963$	−0.106761	−	0.0153498i	−0.00344031	−	0.000494642i
$964$	−30.2009	−	13.7923i	−0.972705	−	0.444219i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−23.0130	+	20.9380i	−0.739667	+	0.672974i
$969$	−6.90470			−0.221811
$970$		0			0
$971$	50.7335			1.62812			0.814058	−	0.580784i	$-0.197254\pi$
$971$	50.7335			1.62812			0.814058	+	0.580784i	$0.197254\pi$
$972$	−0.298352	+	0.653301i	−0.00956966	+	0.0209546i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	60.7381			1.94319			0.971593	−	0.236659i	$-0.0760526\pi$
$977$	60.7381			1.94319			0.971593	+	0.236659i	$0.0760526\pi$
$978$	53.7516	−	24.5475i	1.71879	−	0.784943i
$979$	−0.301762	−	24.9137i	−0.00964435	−	0.796247i
$980$		0			0
$981$		0			0
$982$	8.87158	−	61.7032i	0.283103	−	1.96903i
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$	−38.5363	−	44.4732i	−1.22849	−	1.41775i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$991$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$992$		0			0
$993$	20.6500	+	6.06339i	0.655308	+	0.192416i
$994$		0			0
$995$		0			0
$996$	8.96372	−	4.09359i	0.284026	−	0.129710i
$997$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$997$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$998$	−29.4533	+	13.4509i	−0.932329	+	0.425780i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.w.a.835.1	yes	20
8.3	odd	2	CM	968.2.w.a.835.1	yes	20
121.10	odd	22	inner	968.2.w.a.131.1	✓	20
968.131	even	22	inner	968.2.w.a.131.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.w.a.131.1	✓	20	121.10	odd	22	inner
968.2.w.a.131.1	✓	20	968.131	even	22	inner
968.2.w.a.835.1	yes	20	1.1	even	1	trivial
968.2.w.a.835.1	yes	20	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.w (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.764582 - 1.18971i) q^{2} -1.74200 q^{3} +(-0.830830 + 1.81926i) q^{4} +(1.33190 + 2.07248i) q^{6} +(2.79964 - 0.402527i) q^{8} +0.0345563 q^{9} +O(q^{10})\)	\(q+(-0.764582 - 1.18971i) q^{2} -1.74200 q^{3} +(-0.830830 + 1.81926i) q^{4} +(1.33190 + 2.07248i) q^{6} +(2.79964 - 0.402527i) q^{8} +0.0345563 q^{9} +(-0.972874 - 3.17073i) q^{11} +(1.44730 - 3.16915i) q^{12} +(-2.61944 - 3.02300i) q^{16} +(-1.44638 + 4.92592i) q^{17} +(-0.0264211 - 0.0411120i) q^{18} +(-0.217515 - 0.740788i) q^{19} +(-3.02841 + 3.58172i) q^{22} +(-4.87696 + 0.701201i) q^{24} +(-4.79746 - 1.40866i) q^{25} +5.16580 q^{27} +(-1.59372 + 5.42771i) q^{32} +(1.69475 + 5.52340i) q^{33} +(6.96630 - 2.04549i) q^{34} +(-0.0287104 + 0.0628670i) q^{36} +(-0.715016 + 0.825173i) q^{38} +(6.45709 + 10.0474i) q^{41} +(5.27005 - 0.757719i) q^{43} +(6.57668 + 0.864421i) q^{44} +(4.56306 + 5.26606i) q^{48} +(0.996204 + 6.92875i) q^{49} +(1.99215 + 6.78464i) q^{50} +(2.51959 - 8.58094i) q^{51} +(-3.94967 - 6.14581i) q^{54} +(0.378910 + 1.29045i) q^{57} +(8.72519 + 5.60734i) q^{59} +(7.67594 - 2.25386i) q^{64} +(5.27549 - 6.23935i) q^{66} +(11.7775 - 7.56897i) q^{67} +(-7.75985 - 6.72395i) q^{68} +(0.0967451 - 0.0139098i) q^{72} +(6.70281 - 5.80802i) q^{73} +(8.35717 + 2.45389i) q^{75} +(1.52841 + 0.219752i) q^{76} -9.10247 q^{81} +(7.01658 - 15.3642i) q^{82} +(2.13758 + 1.85223i) q^{83} +(-4.93085 - 5.69050i) q^{86} +(-4.00000 - 8.48528i) q^{88} +(7.20803 + 2.11647i) q^{89} +(2.77626 - 9.45506i) q^{96} +(0.108507 + 0.754683i) q^{97} +(7.48154 - 6.48279i) q^{98} +(-0.0336189 - 0.109569i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9	\( 20 q - 4 q^{3} + 4 q^{4} + 64 q^{9} + 6 q^{11} - 36 q^{12} - 8 q^{16} - 4 q^{22} - 10 q^{25} + 8 q^{27} + 12 q^{33} + 16 q^{34} + 4 q^{36} - 24 q^{38} - 12 q^{44} - 16 q^{48} + 14 q^{49} + 22 q^{51} + 110 q^{57} + 12 q^{59} + 16 q^{64} + 168 q^{66} - 28 q^{67} - 20 q^{75} + 44 q^{76} + 220 q^{81} + 32 q^{82} + 24 q^{86} - 80 q^{88} - 36 q^{89} - 20 q^{97} + 6 q^{99}+O(q^{100}) \) 20 * q - 4 * q^3 + 4 * q^4 + 64 * q^9 + 6 * q^11 - 36 * q^12 - 8 * q^16 - 4 * q^22 - 10 * q^25 + 8 * q^27 + 12 * q^33 + 16 * q^34 + 4 * q^36 - 24 * q^38 - 12 * q^44 - 16 * q^48 + 14 * q^49 + 22 * q^51 + 110 * q^57 + 12 * q^59 + 16 * q^64 + 168 * q^66 - 28 * q^67 - 20 * q^75 + 44 * q^76 + 220 * q^81 + 32 * q^82 + 24 * q^86 - 80 * q^88 - 36 * q^89 - 20 * q^97 + 6 * q^99

Introduction

L-functions

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