LMFDB - Embedded newform 668.2.b.a.667.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(667,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("668.667");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.b (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:	$5.33400685502$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			667.12
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.00426214 + 1.41421i) q^{2} +3.24555i q^{3} +(-1.99996 + 0.0120551i) q^{4} +(-4.58987 + 0.0138330i) q^{6} +4.80685i q^{7} +(-0.0255725 - 2.82831i) q^{8} -7.53357 q^{9} +O(q^{10})$	\(q+(0.00426214 + 1.41421i) q^{2} +3.24555i q^{3} +(-1.99996 + 0.0120551i) q^{4} +(-4.58987 + 0.0138330i) q^{6} +4.80685i q^{7} +(-0.0255725 - 2.82831i) q^{8} -7.53357 q^{9} -6.45227i q^{11} +(-0.0391254 - 6.49097i) q^{12} +(-6.79788 + 0.0204875i) q^{14} +(3.99971 - 0.0482195i) q^{16} +(-0.0321091 - 10.6540i) q^{18} +7.38301i q^{19} -15.6009 q^{21} +(9.12484 - 0.0275004i) q^{22} +(9.17941 - 0.0829968i) q^{24} +5.00000 q^{25} -14.7139i q^{27} +(-0.0579470 - 9.61353i) q^{28} -6.97779 q^{29} +6.33639i q^{31} +(0.0852397 + 5.65621i) q^{32} +20.9411 q^{33} +(15.0669 - 0.0908178i) q^{36} +(-10.4411 + 0.0314674i) q^{38} +(-0.0664930 - 22.0628i) q^{42} +(0.0777826 + 12.9043i) q^{44} +0.477465i q^{47} +(0.156499 + 12.9812i) q^{48} -16.1058 q^{49} +(0.0213107 + 7.07104i) q^{50} +(20.8085 - 0.0627127i) q^{54} +(13.5953 - 0.122923i) q^{56} -23.9619 q^{57} +(-0.0297403 - 9.86804i) q^{58} +12.3659 q^{61} +(-8.96097 + 0.0270066i) q^{62} -36.2127i q^{63} +(-7.99869 + 0.144654i) q^{64} +(0.0892540 + 29.6151i) q^{66} +(0.192652 + 21.3073i) q^{72} +16.2277i q^{75} +(-0.0890028 - 14.7657i) q^{76} +31.0151 q^{77} +25.1539 q^{81} +(31.2012 - 0.188070i) q^{84} -22.6467i q^{87} +(-18.2490 + 0.165001i) q^{88} -9.94402 q^{89} -20.5651 q^{93} +(-0.675235 + 0.00203502i) q^{94} +(-18.3575 + 0.276649i) q^{96} +3.38310 q^{97} +(-0.0686453 - 22.7770i) q^{98} +48.6086i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 66 q^{9}+O(q^{10}) $ 22 * q - 66 * q^9	$ 22 q - 66 q^{9} + 110 q^{25} - 11 q^{42} - 33 q^{44} + 55 q^{48} - 154 q^{49} + 77 q^{54} - 99 q^{62} - 121 q^{72} + 198 q^{81} + 143 q^{84} + 165 q^{98}+O(q^{100}) $ 22 * q - 66 * q^9 + 110 * q^25 - 11 * q^42 - 33 * q^44 + 55 * q^48 - 154 * q^49 + 77 * q^54 - 99 * q^62 - 121 * q^72 + 198 * q^81 + 143 * q^84 + 165 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/668\mathbb{Z}\right)^\times$.

$n$	$5$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.00426214	+	1.41421i	0.00301379	+	0.999995i
$2$	0.00426214	+	1.41421i	0.00301379	+	0.999995i
$3$			3.24555i			1.87382i	0.349575	+	0.936908i	$0.386326\pi$
$3$			3.24555i			1.87382i	−0.349575	+	0.936908i	$0.613674\pi$
$4$	−1.99996	+	0.0120551i	−0.999982	+	0.00602755i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−4.58987	+	0.0138330i	−1.87381	+	0.00564728i
$7$			4.80685i			1.81682i	0.418081	+	0.908410i	$0.362703\pi$
$7$			4.80685i			1.81682i	−0.418081	+	0.908410i	$0.637297\pi$
$8$	−0.0255725	−	2.82831i	−0.00904125	−	0.999959i
$9$	−7.53357			−2.51119
$10$		0			0
$11$		−	6.45227i		−	1.94543i	−0.232001	−	0.972716i	$-0.574527\pi$
$11$		−	6.45227i		−	1.94543i	0.232001	−	0.972716i	$-0.425473\pi$
$12$	−0.0391254	−	6.49097i	−0.0112945	−	1.87378i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−6.79788	+	0.0204875i	−1.81681	+	0.00547551i
$15$		0			0
$16$	3.99971	−	0.0482195i	0.999927	−	0.0120549i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−0.0321091	−	10.6540i	−0.00756819	−	2.51118i
$19$			7.38301i			1.69378i	0.531769	+	0.846889i	$0.321527\pi$
$19$			7.38301i			1.69378i	−0.531769	+	0.846889i	$0.678473\pi$
$20$		0			0
$21$	−15.6009			−3.40439
$22$	9.12484	−	0.0275004i	1.94542	−	0.00586311i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	9.17941	−	0.0829968i	1.87374	−	0.0169416i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	14.7139i		−	2.83169i
$28$	−0.0579470	−	9.61353i	−0.0109510	−	1.81679i
$29$	−6.97779			−1.29574			−0.647872	−	0.761749i	$-0.724341\pi$
$29$	−6.97779			−1.29574			−0.647872	+	0.761749i	$0.724341\pi$
$30$		0			0
$31$			6.33639i			1.13805i	0.822320	+	0.569025i	$0.192679\pi$
$31$			6.33639i			1.13805i	−0.822320	+	0.569025i	$0.807321\pi$
$32$	0.0852397	+	5.65621i	0.0150684	+	0.999886i
$33$	20.9411			3.64538
$34$		0			0
$35$		0			0
$36$	15.0669	−	0.0908178i	2.51114	−	0.0151363i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−10.4411	+	0.0314674i	−1.69377	+	0.00510469i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.0664930	−	22.0628i	−0.0102601	−	3.40437i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	0.0777826	+	12.9043i	0.0117262	+	1.94540i
$45$		0			0
$46$		0			0
$47$			0.477465i			0.0696454i	0.999394	+	0.0348227i	$0.0110867\pi$
$47$			0.477465i			0.0696454i	−0.999394	+	0.0348227i	$0.988913\pi$
$48$	0.156499	+	12.9812i	0.0225886	+	1.87368i
$49$	−16.1058			−2.30083
$50$	0.0213107	+	7.07104i	0.00301379	+	0.999995i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	20.8085	−	0.0627127i	2.83168	−	0.00853411i
$55$		0			0
$56$	13.5953	−	0.122923i	1.81675	−	0.0164263i
$57$	−23.9619			−3.17383
$58$	−0.0297403	−	9.86804i	−0.00390509	−	1.29574i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	12.3659			1.58330			0.791649	−	0.610977i	$-0.209223\pi$
$61$	12.3659			1.58330			0.791649	+	0.610977i	$0.209223\pi$
$62$	−8.96097	+	0.0270066i	−1.13804	+	0.00342984i
$63$		−	36.2127i		−	4.56238i
$64$	−7.99869	+	0.144654i	−0.999837	+	0.0180818i
$65$		0			0
$66$	0.0892540	+	29.6151i	0.0109864	+	3.64537i
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	0.192652	+	21.3073i	0.0227043	+	2.51109i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			16.2277i			1.87382i
$76$	−0.0890028	−	14.7657i	−0.0102093	−	1.69375i
$77$	31.0151			3.53450
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	25.1539			2.79488
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	31.2012	−	0.188070i	3.40432	−	0.0205201i
$85$		0			0
$86$		0			0
$87$		−	22.6467i		−	2.42799i
$88$	−18.2490	+	0.165001i	−1.94535	+	0.0175891i
$89$	−9.94402			−1.05406			−0.527032	−	0.849845i	$-0.676695\pi$
$89$	−9.94402			−1.05406			−0.527032	+	0.849845i	$0.676695\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−20.5651			−2.13250
$94$	−0.675235	+	0.00203502i	−0.0696451	+	0.000209896i
$95$		0			0
$96$	−18.3575	+	0.276649i	−1.87360	+	0.0282354i
$97$	3.38310			0.343501			0.171751	−	0.985140i	$-0.445058\pi$
$97$	3.38310			0.343501			0.171751	+	0.985140i	$0.445058\pi$
$98$	−0.0686453	−	22.7770i	−0.00693422	−	2.30082i
$99$			48.6086i			4.88535i
$100$	−9.99982	+	0.0602755i	−0.999982	+	0.00602755i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.98803i		−	0.192190i	−0.995372	−	0.0960950i	$-0.969365\pi$
$107$		−	1.98803i		−	0.192190i	0.995372	−	0.0960950i	$-0.0306353\pi$
$108$	0.177377	+	29.4273i	0.0170682	+	2.83164i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	0.231784	+	19.2260i	0.0219015	+	1.81669i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−0.102129	−	33.8871i	−0.00956525	−	3.17382i
$115$		0			0
$116$	13.9553	−	0.0841179i	1.29572	−	0.00781015i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−30.6317			−2.78470
$122$	0.0527054	+	17.4880i	0.00477172	+	1.58329i
$123$		0			0
$124$	−0.0763858	−	12.6726i	−0.00685965	−	1.13803i
$125$		0			0
$126$	51.2123	−	0.154344i	4.56236	−	0.0137500i
$127$			19.4532i			1.72620i	0.505037	+	0.863098i	$0.331479\pi$
$127$			19.4532i			1.72620i	−0.505037	+	0.863098i	$0.668521\pi$
$128$	−0.238662	−	11.3112i	−0.0210950	−	0.999777i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−41.8815	+	0.252447i	−3.64532	+	0.0219727i
$133$	−35.4890			−3.07729
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−22.4606			−1.91894			−0.959471	−	0.281807i	$-0.909066\pi$
$137$	−22.4606			−1.91894			−0.959471	+	0.281807i	$0.909066\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−1.54963			−0.130503
$142$		0			0
$143$		0			0
$144$	−30.1321	+	0.363265i	−2.51101	+	0.0302721i
$145$		0			0
$146$		0			0
$147$		−	52.2722i		−	4.31134i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−22.9494	+	0.0691648i	−1.87381	+	0.00564728i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	20.8814	−	0.188802i	1.69371	−	0.0153139i
$153$		0			0
$154$	0.132191	+	43.8618i	0.0106522	+	3.53448i
$155$		0			0
$156$		0			0
$157$	−9.20812			−0.734888			−0.367444	−	0.930046i	$-0.619767\pi$
$157$	−9.20812			−0.734888			−0.367444	+	0.930046i	$0.619767\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.107210	+	35.5729i	0.00842318	+	2.79487i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			12.9228i			1.00000i
$167$			12.9228i			1.00000i
$168$	0.398953	+	44.1241i	0.0307799	+	3.40425i
$169$	13.0000			1.00000
$170$		0			0
$171$		−	55.6204i		−	4.25340i
$172$		0			0
$173$	25.5385			1.94165			0.970827	−	0.239779i	$-0.0770750\pi$
$173$	25.5385			1.94165			0.970827	+	0.239779i	$0.0770750\pi$
$174$	32.0272	−	0.0965236i	2.42798	−	0.00731743i
$175$			24.0343i			1.81682i
$176$	−0.311125	−	25.8072i	−0.0234519	−	1.94529i
$177$		0			0
$178$	−0.0423828	−	14.0629i	−0.00317672	−	1.05406i
$179$			23.1040i			1.72687i	0.504459	+	0.863435i	$0.331692\pi$
$179$			23.1040i			1.72687i	−0.504459	+	0.863435i	$0.668308\pi$
$180$		0			0
$181$	−4.32899			−0.321771			−0.160886	−	0.986973i	$-0.551435\pi$
$181$	−4.32899			−0.321771			−0.160886	+	0.986973i	$0.551435\pi$
$182$		0			0
$183$			40.1342i			2.96681i
$184$		0			0
$185$		0			0
$186$	−0.0876511	−	29.0832i	−0.00642689	−	2.13249i
$187$		0			0
$188$	−0.00575589	−	0.954913i	−0.000419791	−	0.0696442i
$189$	70.7275			5.14467
$190$		0			0
$191$			3.61814i			0.261799i	0.991396	+	0.130900i	$0.0417865\pi$
$191$			3.61814i			0.261799i	−0.991396	+	0.130900i	$0.958213\pi$
$192$	−0.469481	−	25.9601i	−0.0338819	−	1.87351i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	0.0144192	+	4.78440i	0.00103524	+	0.343500i
$195$		0			0
$196$	32.2111	−	0.194157i	2.30079	−	0.0138684i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−68.7426	+	0.207176i	−4.88532	+	0.0147234i
$199$			19.2604i			1.36533i	0.730731	+	0.682665i	$0.239179\pi$
$199$			19.2604i			1.36533i	−0.730731	+	0.682665i	$0.760821\pi$
$200$	−0.127863	−	14.1416i	−0.00904125	−	0.999959i
$201$		0			0
$202$		0			0
$203$		−	33.5412i		−	2.35413i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	47.6371			3.29513
$210$		0			0
$211$		−	27.6046i		−	1.90038i	−0.311675	−	0.950189i	$-0.600890\pi$
$211$		−	27.6046i		−	1.90038i	0.311675	−	0.950189i	$-0.399110\pi$
$212$		0			0
$213$		0			0
$214$	2.81148	−	0.00847325i	0.192189	−	0.000579219i
$215$		0			0
$216$	−41.6155	+	0.376272i	−2.83158	+	0.0256020i
$217$	−30.4581			−2.06763
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	9.35088i		−	0.626181i	−0.949723	−	0.313091i	$-0.898636\pi$
$223$		−	9.35088i		−	0.626181i	0.949723	−	0.313091i	$-0.101364\pi$
$224$	−27.1886	+	0.409734i	−1.81661	+	0.0273765i
$225$	−37.6678			−2.51119
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	47.9229	−	0.288863i	3.17377	−	0.0191304i
$229$	29.7825			1.96808			0.984041	−	0.177941i	$-0.0569436\pi$
$229$	29.7825			1.96808			0.984041	+	0.177941i	$0.0569436\pi$
$230$		0			0
$231$			100.661i			6.62300i
$232$	0.178440	+	19.7354i	0.0117151	+	1.29569i
$233$	27.4539			1.79856			0.899281	−	0.437371i	$-0.144090\pi$
$233$	27.4539			1.79856			0.899281	+	0.437371i	$0.144090\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			2.85979i			0.184984i	0.995713	+	0.0924922i	$0.0294833\pi$
$239$			2.85979i			0.184984i	−0.995713	+	0.0924922i	$0.970517\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−0.130557	−	43.3196i	−0.00839250	−	2.78469i
$243$			37.4966i			2.40541i
$244$	−24.7314	+	0.149073i	−1.58327	+	0.00954340i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	17.9213	−	0.162038i	1.13800	−	0.0102894i
$249$		0			0
$250$		0			0
$251$			30.8291i			1.94592i	0.230978	+	0.972959i	$0.425808\pi$
$251$			30.8291i			1.94592i	−0.230978	+	0.972959i	$0.574192\pi$
$252$	0.436548	+	72.4242i	0.0274999	+	4.56229i
$253$		0			0
$254$	−27.5109	+	0.0829124i	−1.72619	+	0.00520239i
$255$		0			0
$256$	15.9953	−	0.385728i	0.999709	−	0.0241080i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	52.5677			3.25386
$262$		0			0
$263$		−	22.3331i		−	1.37712i	−0.725181	−	0.688558i	$-0.758244\pi$
$263$		−	22.3331i		−	1.37712i	0.725181	−	0.688558i	$-0.241756\pi$
$264$	−0.535517	−	59.2280i	−0.0329588	−	3.64523i
$265$		0			0
$266$	−0.151259	−	50.1888i	−0.00927429	−	3.07728i
$267$		−	32.2738i		−	1.97512i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	−0.0957304	−	31.7640i	−0.00578328	−	1.91893i
$275$		−	32.2613i		−	1.94543i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		−	47.7356i		−	2.85786i
$280$		0			0
$281$	−24.4756			−1.46009			−0.730045	−	0.683399i	$-0.760501\pi$
$281$	−24.4756			−1.46009			−0.730045	+	0.683399i	$0.760501\pi$
$282$	−0.00660476	−	2.19150i	−0.000393308	−	0.130502i
$283$			19.5497i			1.16211i	0.813865	+	0.581054i	$0.197359\pi$
$283$			19.5497i			1.16211i	−0.813865	+	0.581054i	$0.802641\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.642159	−	42.6115i	−0.0378396	−	2.51090i
$289$	17.0000			1.00000
$290$		0			0
$291$			10.9800i			0.643659i
$292$		0			0
$293$	27.3516			1.59790			0.798948	−	0.601400i	$-0.205390\pi$
$293$	27.3516			1.59790			0.798948	+	0.601400i	$0.205390\pi$
$294$	73.9237	−	0.222791i	4.31132	−	0.0129935i
$295$		0			0
$296$		0			0
$297$	−94.9380			−5.50886
$298$		0			0
$299$		0			0
$300$	−0.195627	−	32.4549i	−0.0112945	−	1.87378i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.356005	+	29.5299i	0.0204183	+	1.69366i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−62.0290	+	0.373890i	−3.53443	+	0.0213043i
$309$		0			0
$310$		0			0
$311$			25.8457i			1.46557i	0.680458	+	0.732787i	$0.261781\pi$
$311$			25.8457i			1.46557i	−0.680458	+	0.732787i	$0.738219\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−0.0392463	−	13.0222i	−0.00221479	−	0.734884i
$315$		0			0
$316$		0			0
$317$	2.67824			0.150425			0.0752125	−	0.997168i	$-0.476036\pi$
$317$	2.67824			0.150425			0.0752125	+	0.997168i	$0.476036\pi$
$318$		0			0
$319$			45.0226i			2.52078i
$320$		0			0
$321$	6.45224			0.360129
$322$		0			0
$323$		0			0
$324$	−50.3070	+	0.303233i	−2.79483	+	0.0168463i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.29510			−0.126533
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	−18.2756	+	0.0550790i	−0.999995	+	0.00301379i
$335$		0			0
$336$	−62.3989	+	0.752265i	−3.40414	+	0.0410394i
$337$	32.1441			1.75100			0.875501	−	0.483216i	$-0.160531\pi$
$337$	32.1441			1.75100			0.875501	+	0.483216i	$0.160531\pi$
$338$	0.0554078	+	18.3847i	0.00301379	+	0.999995i
$339$		0			0
$340$		0			0
$341$	40.8841			2.21400
$342$	78.6588	−	0.237062i	4.25338	−	0.0128188i
$343$		−	43.7704i		−	2.36338i
$344$		0			0
$345$		0			0
$346$	0.108849	+	36.1167i	0.00585173	+	1.94165i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	0.273009	+	45.2927i	0.0146348	+	2.42794i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−33.9894	+	0.102437i	−1.81681	+	0.00547551i
$351$		0			0
$352$	36.4954	−	0.549989i	1.94521	−	0.0293145i
$353$	−6.29588			−0.335096			−0.167548	−	0.985864i	$-0.553585\pi$
$353$	−6.29588			−0.335096			−0.167548	+	0.985864i	$0.553585\pi$
$354$		0			0
$355$		0			0
$356$	19.8877	−	0.119876i	1.05404	−	0.00635342i
$357$		0			0
$358$	−32.6738	+	0.0984723i	−1.72686	+	0.00520442i
$359$			26.2455i			1.38518i	0.721331	+	0.692591i	$0.243531\pi$
$359$			26.2455i			1.38518i	−0.721331	+	0.692591i	$0.756469\pi$
$360$		0			0
$361$	−35.5088			−1.86889
$362$	−0.0184507	−	6.12209i	−0.000969749	−	0.321770i
$363$		−	99.4167i		−	5.21802i
$364$		0			0
$365$		0			0
$366$	−56.7581	+	0.171058i	−2.96680	+	0.00894133i
$367$		−	30.9644i		−	1.61633i	−0.588956	−	0.808165i	$-0.700461\pi$
$367$		−	30.9644i		−	1.61633i	0.588956	−	0.808165i	$-0.299539\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	41.1294	−	0.247914i	2.13246	−	0.0128537i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	1.35042	−	0.0122100i	0.0696426	−	0.000629682i
$377$		0			0
$378$	0.301451	+	100.023i	0.0155049	+	5.14465i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−63.1364			−3.23457
$382$	−5.11679	+	0.0154210i	−0.261798	+	0.000789007i
$383$			18.3966i			0.940023i	0.882660	+	0.470011i	$0.155750\pi$
$383$			18.3966i			0.940023i	−0.882660	+	0.470011i	$0.844250\pi$
$384$	36.7110	−	0.774590i	1.87340	−	0.0395281i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−6.76607	+	0.0407835i	−0.343495	+	0.00207047i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.411867	+	45.5523i	0.0208024	+	2.30074i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.585981	−	97.2154i	−0.0294466	−	4.88526i
$397$	26.8975			1.34995			0.674974	−	0.737842i	$-0.264155\pi$
$397$	26.8975			1.34995			0.674974	+	0.737842i	$0.264155\pi$
$398$	−27.2381	+	0.0820903i	−1.36532	+	0.00411481i
$399$		−	115.181i		−	5.76628i
$400$	19.9985	−	0.241097i	0.999927	−	0.0120549i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	47.4342	−	0.142957i	2.35412	−	0.00709485i
$407$		0			0
$408$		0			0
$409$	35.3825			1.74955			0.874776	−	0.484527i	$-0.161008\pi$
$409$	35.3825			1.74955			0.874776	+	0.484527i	$0.161008\pi$
$410$		0			0
$411$		−	72.8971i		−	3.59575i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	0.203036	+	67.3688i	0.00993082	+	3.29511i
$419$		−	30.3228i		−	1.48137i	−0.671854	−	0.740684i	$-0.734502\pi$
$419$		−	30.3228i		−	1.48137i	0.671854	−	0.740684i	$-0.265498\pi$
$420$		0			0
$421$	38.6316			1.88279			0.941395	−	0.337307i	$-0.109516\pi$
$421$	38.6316			1.88279			0.941395	+	0.337307i	$0.109516\pi$
$422$	39.0386	−	0.117655i	1.90037	−	0.00572733i
$423$		−	3.59702i		−	0.174893i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			59.4413i			2.87657i
$428$	0.0239659	+	3.97598i	0.00115843	+	0.192186i
$429$		0			0
$430$		0			0
$431$			16.6135i			0.800243i	0.916462	+	0.400122i	$0.131032\pi$
$431$			16.6135i			0.800243i	−0.916462	+	0.400122i	$0.868968\pi$
$432$	−0.709497	−	58.8513i	−0.0341357	−	2.83149i
$433$	−4.94273			−0.237532			−0.118766	−	0.992922i	$-0.537894\pi$
$433$	−4.94273			−0.237532			−0.118766	+	0.992922i	$0.537894\pi$
$434$	−0.129817	−	43.0741i	−0.00623140	−	2.06762i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	121.334			5.77783
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	13.2241	−	0.0398547i	0.626179	−	0.00188718i
$447$		0			0
$448$	−0.695331	−	38.4485i	−0.0328513	−	1.81652i
$449$	−41.2113			−1.94488			−0.972440	−	0.233153i	$-0.925096\pi$
$449$	−41.2113			−1.94488			−0.972440	+	0.233153i	$0.925096\pi$
$450$	−0.160546	−	53.2701i	−0.00756819	−	2.51118i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	0.612766	+	67.7717i	0.0286954	+	3.17370i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	0.126937	+	42.1186i	0.00593138	+	1.96807i
$459$		0			0
$460$		0			0
$461$	−41.0067			−1.90987			−0.954936	−	0.296812i	$-0.904077\pi$
$461$	−41.0067			−1.90987			−0.954936	+	0.296812i	$0.904077\pi$
$462$	−142.355	+	0.429031i	−6.62297	+	0.0199603i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−27.9091	+	0.336466i	−1.29565	+	0.0156200i
$465$		0			0
$466$	0.117012	+	38.8255i	0.00542048	+	1.79855i
$467$		−	26.8531i		−	1.24261i	−0.783568	−	0.621306i	$-0.786602\pi$
$467$		−	26.8531i		−	1.24261i	0.783568	−	0.621306i	$-0.213398\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	29.8854i		−	1.37704i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			36.9150i			1.69378i
$476$		0			0
$477$		0			0
$478$	−4.04434	+	0.0121888i	−0.184984	+	0.000557504i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	61.2623	−	0.369268i	2.78465	−	0.0167849i
$485$		0			0
$486$	−53.0279	+	0.159816i	−2.40539	+	0.00724938i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−0.316228	−	34.9748i	−0.0143150	−	1.58323i
$489$		0			0
$490$		0			0
$491$		−	25.8457i		−	1.16640i	−0.812329	−	0.583200i	$-0.801800\pi$
$491$		−	25.8457i		−	1.16640i	0.812329	−	0.583200i	$-0.198200\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.305538	+	25.3437i	0.0137190	+	1.13797i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−41.9417			−1.87382
$502$	−43.5988	+	0.131398i	−1.94591	+	0.00586458i
$503$			41.8063i			1.86405i	0.362391	+	0.932026i	$0.381961\pi$
$503$			41.8063i			1.86405i	−0.362391	+	0.932026i	$0.618039\pi$
$504$	−102.421	+	0.926051i	−4.56219	+	0.0412496i
$505$		0			0
$506$		0			0
$507$			42.1921i			1.87382i
$508$	−0.234511	−	38.9058i	−0.0104047	−	1.72616i
$509$	31.6942			1.40482			0.702409	−	0.711773i	$-0.252108\pi$
$509$	31.6942			1.40482			0.702409	+	0.711773i	$0.252108\pi$
$510$		0			0
$511$		0			0
$512$	0.613674	+	22.6191i	0.0271208	+	0.999632i
$513$	108.633			4.79626
$514$		0			0
$515$		0			0
$516$		0			0
$517$	3.08073			0.135490
$518$		0			0
$519$			82.8863i			3.63831i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.224051	+	74.3416i	0.00980643	+	3.25384i
$523$		−	2.25901i		−	0.0987795i	−0.998780	−	0.0493898i	$-0.984272\pi$
$523$		−	2.25901i		−	0.0987795i	0.998780	−	0.0493898i	$-0.0157276\pi$
$524$		0			0
$525$	−78.0043			−3.40439
$526$	31.5836	−	0.0951866i	1.37711	−	0.00415033i
$527$		0			0
$528$	83.7584	−	1.00977i	3.64512	−	0.0439446i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	70.9768	−	0.427824i	3.07723	−	0.0185485i
$533$		0			0
$534$	45.6418	−	0.137555i	1.97511	−	0.00595260i
$535$		0			0
$536$		0			0
$537$	−74.9850			−3.23584
$538$		0			0
$539$			103.919i			4.47611i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	14.0499i		−	0.602940i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	44.9205	−	0.270765i	1.91891	−	0.0115665i
$549$	−93.1597			−3.97596
$550$	45.6242	−	0.137502i	1.94542	−	0.00586311i
$551$		−	51.5171i		−	2.19470i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.2271			0.560451			0.280226	−	0.959934i	$-0.409591\pi$
$557$	13.2271			0.560451			0.280226	+	0.959934i	$0.409591\pi$
$558$	67.5081	−	0.203456i	2.85785	−	0.00861297i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.104318	−	34.6135i	−0.00440040	−	1.46008i
$563$		−	24.0589i		−	1.01396i	−0.861957	−	0.506981i	$-0.830762\pi$
$563$		−	24.0589i		−	1.01396i	0.861957	−	0.506981i	$-0.169238\pi$
$564$	3.09921	−	0.0186810i	0.130500	−	0.000786612i
$565$		0			0
$566$	−27.6473	+	0.0833234i	−1.16210	+	0.00350234i
$567$			120.911i			5.07780i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	−11.7428			−0.490564
$574$		0			0
$575$		0			0
$576$	60.2587	−	1.08976i	2.51078	−	0.0454067i
$577$	−45.7337			−1.90392			−0.951960	−	0.306223i	$-0.900935\pi$
$577$	−45.7337			−1.90392			−0.951960	+	0.306223i	$0.900935\pi$
$578$	0.0724563	+	24.0415i	0.00301379	+	0.999995i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−15.5280	+	0.0467983i	−0.643656	+	0.00193985i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.116576	+	38.6808i	0.00481572	+	1.59789i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.630146	+	104.542i	0.0259868	+	4.31126i
$589$	−46.7816			−1.92760
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.404639	−	134.262i	−0.0166025	−	5.50884i
$595$		0			0
$596$		0			0
$597$	−62.5104			−2.55838
$598$		0			0
$599$			10.4445i			0.426751i	0.976970	+	0.213375i	$0.0684457\pi$
$599$			10.4445i			0.426751i	−0.976970	+	0.213375i	$0.931554\pi$
$600$	45.8971	−	0.414984i	1.87374	−	0.0169416i
$601$	−43.0562			−1.75630			−0.878149	−	0.478388i	$-0.841221\pi$
$601$	−43.0562			−1.75630			−0.878149	+	0.478388i	$0.841221\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−41.7599	+	0.629325i	−1.69359	+	0.0255225i
$609$	108.860			4.41121
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−45.6314			−1.84304			−0.921518	−	0.388336i	$-0.873050\pi$
$613$	−45.6314			−1.84304			−0.921518	+	0.388336i	$0.873050\pi$
$614$		0			0
$615$		0			0
$616$	−0.793134	−	87.7203i	−0.0319563	−	3.53435i
$617$	48.1626			1.93895			0.969477	−	0.245181i	$-0.0788475\pi$
$617$	48.1626			1.93895			0.969477	+	0.245181i	$0.0788475\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−36.5512	+	0.110158i	−1.46557	+	0.00441693i
$623$		−	47.7994i		−	1.91504i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$			154.608i			6.17447i
$628$	18.4159	−	0.111005i	0.734874	−	0.00442957i
$629$		0			0
$630$		0			0
$631$		−	48.2974i		−	1.92269i	−0.275346	−	0.961345i	$-0.588792\pi$
$631$		−	48.2974i		−	1.92269i	0.275346	−	0.961345i	$-0.411208\pi$
$632$		0			0
$633$	89.5919			3.56096
$634$	0.0114150	+	3.78759i	0.000453349	+	0.150424i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−63.6712	+	0.191892i	−2.52077	+	0.00759709i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	0.0275003	+	9.12480i	0.00108535	+	0.360127i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.643250	−	71.1432i	−0.0252692	−	2.79477i
$649$		0			0
$650$		0			0
$651$		−	98.8532i		−	3.87436i
$652$		0			0
$653$	27.2834			1.06768			0.533841	−	0.845585i	$-0.320748\pi$
$653$	27.2834			1.06768			0.533841	+	0.845585i	$0.320748\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−0.00978205	−	3.24575i	−0.000381344	−	0.126533i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.155786	−	25.8452i	−0.00602755	−	0.999982i
$669$	30.3487			1.17335
$670$		0			0
$671$		−	79.7884i		−	3.08020i
$672$	−1.32981	−	88.2418i	−0.0512986	−	3.40400i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	0.137003	+	45.4585i	0.00527715	+	1.75099i
$675$		−	73.5695i		−	2.83169i
$676$	−25.9995	+	0.156716i	−0.999982	+	0.00602755i
$677$	6.00000			0.230599			0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000			0.230599			0.115299	+	0.993331i	$0.463217\pi$
$678$		0			0
$679$			16.2620i			0.624080i
$680$		0			0
$681$		0			0
$682$	0.174254	+	57.8186i	0.00667252	+	2.21399i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	0.670509	+	111.239i	0.0256376	+	4.25332i
$685$		0			0
$686$	61.9004	−	0.186555i	2.36337	−	0.00712271i
$687$			96.6604i			3.68783i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−51.0760	+	0.307869i	−1.94162	+	0.0117034i
$693$	−233.654			−8.87579
$694$		0			0
$695$		0			0
$696$	−64.0521	+	0.579134i	−2.42789	+	0.0219520i
$697$		0			0
$698$		0			0
$699$			89.1028i			3.37018i
$700$	−0.289735	−	48.0676i	−0.0109510	−	1.81679i
$701$	31.7414			1.19885			0.599427	−	0.800429i	$-0.295395\pi$
$701$	31.7414			1.19885			0.599427	+	0.800429i	$0.295395\pi$
$702$		0			0
$703$		0			0
$704$	0.933347	+	51.6097i	0.0351768	+	1.94511i
$705$		0			0
$706$	−0.0268339	−	8.90367i	−0.00100991	−	0.335094i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	0.254294	+	28.1248i	0.00953006	+	1.05402i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.278520	−	46.2071i	−0.0104088	−	1.72684i
$717$	−9.28158			−0.346627
$718$	−37.1165	+	0.111862i	−1.38518	+	0.00417464i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−0.151343	−	50.2168i	−0.00563242	−	1.86888i
$723$		0			0
$724$	8.65782	−	0.0521863i	0.321765	−	0.00193949i
$725$	−34.8890			−1.29574
$726$	140.596	−	0.423728i	5.21800	−	0.0157260i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−46.2350			−1.71241
$730$		0			0
$731$		0			0
$732$	−0.483822	−	80.2670i	−0.0178826	−	2.96676i
$733$	50.0738			1.84952			0.924759	−	0.380552i	$-0.124266\pi$
$733$	50.0738			1.84952			0.924759	+	0.380552i	$0.124266\pi$
$734$	43.7901	−	0.131975i	1.61632	−	0.00487127i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			25.8457i			0.948187i	0.880475	+	0.474093i	$0.157224\pi$
$743$			25.8457i			0.948187i	−0.880475	+	0.474093i	$0.842776\pi$
$744$	0.525900	+	58.1644i	0.0192804	+	2.13241i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.55616			0.349174
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	0.0230231	+	1.90972i	0.000839567	+	0.0696404i
$753$	−100.057			−3.64629
$754$		0			0
$755$		0			0
$756$	−141.453	+	0.852627i	−5.14458	+	0.0310097i
$757$	51.3822			1.86752			0.933759	−	0.357901i	$-0.116508\pi$
$757$	51.3822			1.86752			0.933759	+	0.357901i	$0.116508\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.58754			0.166298			0.0831491	−	0.996537i	$-0.473502\pi$
$761$	4.58754			0.166298			0.0831491	+	0.996537i	$0.473502\pi$
$762$	−0.269096	−	89.2879i	−0.00974832	−	3.23456i
$763$		0			0
$764$	−0.0436170	−	7.23614i	−0.00157801	−	0.261794i
$765$		0			0
$766$	−26.0166	+	0.0784089i	−0.940019	+	0.00283303i
$767$		0			0
$768$	1.25190	+	51.9136i	0.0451740	+	1.87327i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$			31.6820i			1.13805i
$776$	−0.0865143	−	9.56845i	−0.00310568	−	0.343487i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			102.671i			3.66915i
$784$	−64.4186	+	0.776615i	−2.30067	+	0.0277362i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	72.4830			2.58046
$790$		0			0
$791$		0			0
$792$	137.480	−	1.24304i	4.88515	−	0.0441696i
$793$		0			0
$794$	0.114641	+	38.0386i	0.00406845	+	1.34994i
$795$		0			0
$796$	−0.232185	−	38.5200i	−0.00822959	−	1.36531i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	162.890	−	0.490918i	5.76625	−	0.0173783i
$799$		0			0
$800$	0.426198	+	28.2811i	0.0150684	+	0.999886i
$801$	74.9140			2.64695
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−34.7720			−1.22252			−0.611259	−	0.791430i	$-0.709337\pi$
$809$	−34.7720			−1.22252			−0.611259	+	0.791430i	$0.709337\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	0.404342	+	67.0812i	0.0141896	+	2.35409i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.150805	+	50.0382i	0.00527278	+	1.74954i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	103.092	−	0.310697i	3.59573	−	0.0108368i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	104.706			3.64538
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−95.2725	+	0.574270i	−3.29507	+	0.0198615i
$837$	93.2331			3.22261
$838$	42.8828	−	0.129240i	1.48136	−	0.00446453i
$839$		−	36.0868i		−	1.24585i	−0.782280	−	0.622927i	$-0.785943\pi$
$839$		−	36.0868i		−	1.24585i	0.782280	−	0.622927i	$-0.214057\pi$
$840$		0			0
$841$	19.6896			0.678951
$842$	0.164653	+	54.6331i	0.00567432	+	1.88278i
$843$		−	79.4366i		−	2.73594i
$844$	0.332776	+	55.2082i	0.0114546	+	1.90034i
$845$		0			0
$846$	5.08693	−	0.0153310i	0.174892	−	0.000527090i
$847$		−	147.242i		−	5.05930i
$848$		0			0
$849$	−63.4494			−2.17758
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.85042			0.200315			0.100157	−	0.994972i	$-0.468065\pi$
$853$	5.85042			0.200315			0.100157	+	0.994972i	$0.468065\pi$
$854$	−84.0623	+	0.253347i	−2.87655	+	0.00866935i
$855$		0			0
$856$	−5.62276	+	0.0508389i	−0.192182	+	0.00173764i
$857$	13.1915			0.450615			0.225307	−	0.974288i	$-0.427661\pi$
$857$	13.1915			0.450615			0.225307	+	0.974288i	$0.427661\pi$
$858$		0			0
$859$			25.1915i			0.859523i	0.902943	+	0.429761i	$0.141402\pi$
$859$			25.1915i			0.859523i	−0.902943	+	0.429761i	$0.858598\pi$
$860$		0			0
$861$		0			0
$862$	−23.4949	+	0.0708090i	−0.800240	+	0.00241176i
$863$			45.4552i			1.54731i	0.633606	+	0.773656i	$0.281574\pi$
$863$			45.4552i			1.54731i	−0.633606	+	0.773656i	$0.718426\pi$
$864$	83.2249	−	1.25421i	2.83137	−	0.0426690i
$865$		0			0
$866$	−0.0210666	−	6.99004i	−0.000715872	−	0.237531i
$867$			55.1743i			1.87382i
$868$	60.9151	−	0.367175i	2.06759	−	0.0124627i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−25.4868			−0.862597
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.5304			1.06471			0.532353	−	0.846523i	$-0.321308\pi$
$877$	31.5304			1.06471			0.532353	+	0.846523i	$0.321308\pi$
$878$		0			0
$879$			88.7708i			2.99416i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.517144	+	171.592i	0.0174131	+	5.77780i
$883$		−	15.5778i		−	0.524234i	−0.965036	−	0.262117i	$-0.915579\pi$
$883$		−	15.5778i		−	0.524234i	0.965036	−	0.262117i	$-0.0844207\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−93.5088			−3.13619
$890$		0			0
$891$		−	162.300i		−	5.43725i
$892$	0.112726	+	18.7014i	0.00377434	+	0.626170i
$893$	−3.52513			−0.117964
$894$		0			0
$895$		0			0
$896$	54.3712	−	1.14721i	1.81641	−	0.0383258i
$897$		0			0
$898$	−0.175648	−	58.2813i	−0.00586145	−	1.94487i
$899$		−	44.2140i		−	1.47462i
$900$	75.3343	−	0.454089i	2.51114	−	0.0151363i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			28.4384i			0.944282i	0.881523	+	0.472141i	$0.156519\pi$
$907$			28.4384i			0.944282i	−0.881523	+	0.472141i	$0.843481\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			43.8206i			1.45184i	0.687779	+	0.725920i	$0.258586\pi$
$911$			43.8206i			1.45184i	−0.687779	+	0.725920i	$0.741414\pi$
$912$	−95.8406	+	1.15543i	−3.17360	+	0.0382601i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−59.5639	+	0.359031i	−1.96805	+	0.0118627i
$917$		0			0
$918$		0			0
$919$		−	60.4965i		−	1.99559i	−0.0663387	−	0.997797i	$-0.521132\pi$
$919$		−	60.4965i		−	1.99559i	0.0663387	−	0.997797i	$-0.478868\pi$
$920$		0			0
$921$		0			0
$922$	−0.174776	−	57.9920i	−0.00575595	−	1.90986i
$923$		0			0
$924$	−1.21348	−	201.318i	−0.0399204	−	6.62288i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.594785	−	39.4679i	−0.0195248	−	1.29560i
$929$	27.2493			0.894020			0.447010	−	0.894529i	$-0.352489\pi$
$929$	27.2493			0.894020			0.447010	+	0.894529i	$0.352489\pi$
$930$		0			0
$931$		−	118.909i		−	3.89710i
$932$	−54.9067	+	0.330959i	−1.79853	+	0.0108409i
$933$	−83.8834			−2.74622
$934$	37.9758	−	0.114452i	1.24261	−	0.00374497i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$	42.2641	−	0.127376i	1.37704	−	0.00415012i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	34.9295i		−	1.13506i	−0.823354	−	0.567528i	$-0.807900\pi$
$947$		−	34.9295i		−	1.13506i	0.823354	−	0.567528i	$-0.192100\pi$
$948$		0			0
$949$		0			0
$950$	−52.2055	+	0.157337i	−1.69377	+	0.00510469i
$951$			8.69235i			0.281869i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.0344750	−	5.71948i	−0.00111500	−	0.184981i
$957$	−146.123			−4.72348
$958$		0			0
$959$		−	107.965i		−	3.48637i
$960$		0			0
$961$	−9.14987			−0.295157
$962$		0			0
$963$			14.9769i			0.482625i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	61.5456i		−	1.97917i	−0.143945	−	0.989586i	$-0.545979\pi$
$967$		−	61.5456i		−	1.97917i	0.143945	−	0.989586i	$-0.454021\pi$
$968$	0.783330	+	86.6361i	0.0251772	+	2.78459i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−0.452024	−	74.9918i	−0.0144987	−	2.40536i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	49.4602	−	0.596280i	1.58318	−	0.0190864i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$			64.1615i			2.05061i
$980$		0			0
$981$		0			0
$982$	36.5512	−	0.110158i	1.16639	−	0.00351528i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	7.44887i		−	0.237100i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−35.8400	+	0.540112i	−1.13792	+	0.0171486i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.23515			0.102458			0.0512292	−	0.998687i	$-0.483686\pi$
$997$	3.23515			0.102458			0.0512292	+	0.998687i	$0.483686\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.b.a.667.12	yes	22
4.3	odd	2	inner	668.2.b.a.667.11	✓	22
167.166	odd	2	CM	668.2.b.a.667.12	yes	22
668.667	even	2	inner	668.2.b.a.667.11	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.11	✓	22	4.3	odd	2	inner
668.2.b.a.667.11	✓	22	668.667	even	2	inner
668.2.b.a.667.12	yes	22	1.1	even	1	trivial
668.2.b.a.667.12	yes	22	167.166	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:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.00426214 + 1.41421i) q^{2} +3.24555i q^{3} +(-1.99996 + 0.0120551i) q^{4} +(-4.58987 + 0.0138330i) q^{6} +4.80685i q^{7} +(-0.0255725 - 2.82831i) q^{8} -7.53357 q^{9} +O(q^{10})\)	\(q+(0.00426214 + 1.41421i) q^{2} +3.24555i q^{3} +(-1.99996 + 0.0120551i) q^{4} +(-4.58987 + 0.0138330i) q^{6} +4.80685i q^{7} +(-0.0255725 - 2.82831i) q^{8} -7.53357 q^{9} -6.45227i q^{11} +(-0.0391254 - 6.49097i) q^{12} +(-6.79788 + 0.0204875i) q^{14} +(3.99971 - 0.0482195i) q^{16} +(-0.0321091 - 10.6540i) q^{18} +7.38301i q^{19} -15.6009 q^{21} +(9.12484 - 0.0275004i) q^{22} +(9.17941 - 0.0829968i) q^{24} +5.00000 q^{25} -14.7139i q^{27} +(-0.0579470 - 9.61353i) q^{28} -6.97779 q^{29} +6.33639i q^{31} +(0.0852397 + 5.65621i) q^{32} +20.9411 q^{33} +(15.0669 - 0.0908178i) q^{36} +(-10.4411 + 0.0314674i) q^{38} +(-0.0664930 - 22.0628i) q^{42} +(0.0777826 + 12.9043i) q^{44} +0.477465i q^{47} +(0.156499 + 12.9812i) q^{48} -16.1058 q^{49} +(0.0213107 + 7.07104i) q^{50} +(20.8085 - 0.0627127i) q^{54} +(13.5953 - 0.122923i) q^{56} -23.9619 q^{57} +(-0.0297403 - 9.86804i) q^{58} +12.3659 q^{61} +(-8.96097 + 0.0270066i) q^{62} -36.2127i q^{63} +(-7.99869 + 0.144654i) q^{64} +(0.0892540 + 29.6151i) q^{66} +(0.192652 + 21.3073i) q^{72} +16.2277i q^{75} +(-0.0890028 - 14.7657i) q^{76} +31.0151 q^{77} +25.1539 q^{81} +(31.2012 - 0.188070i) q^{84} -22.6467i q^{87} +(-18.2490 + 0.165001i) q^{88} -9.94402 q^{89} -20.5651 q^{93} +(-0.675235 + 0.00203502i) q^{94} +(-18.3575 + 0.276649i) q^{96} +3.38310 q^{97} +(-0.0686453 - 22.7770i) q^{98} +48.6086i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 22 * q - 66 * q^9	\( 22 q - 66 q^{9} + 110 q^{25} - 11 q^{42} - 33 q^{44} + 55 q^{48} - 154 q^{49} + 77 q^{54} - 99 q^{62} - 121 q^{72} + 198 q^{81} + 143 q^{84} + 165 q^{98}+O(q^{100}) \) 22 * q - 66 * q^9 + 110 * q^25 - 11 * q^42 - 33 * q^44 + 55 * q^48 - 154 * q^49 + 77 * q^54 - 99 * q^62 - 121 * q^72 + 198 * q^81 + 143 * q^84 + 165 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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