LMFDB - Embedded newform 668.2.b.a.667.3

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			667.3
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.28464 - 0.591360i) q^{2} -0.736748i q^{3} +(1.30059 + 1.51937i) q^{4} +(-0.435683 + 0.946454i) q^{6} -5.23983i q^{7} +(-0.772291 - 2.72095i) q^{8} +2.45720 q^{9} +O(q^{10})$	\(q+(-1.28464 - 0.591360i) q^{2} -0.736748i q^{3} +(1.30059 + 1.51937i) q^{4} +(-0.435683 + 0.946454i) q^{6} -5.23983i q^{7} +(-0.772291 - 2.72095i) q^{8} +2.45720 q^{9} -3.06230i q^{11} +(1.11939 - 0.958205i) q^{12} +(-3.09862 + 6.73128i) q^{14} +(-0.616947 + 3.95214i) q^{16} +(-3.15661 - 1.45309i) q^{18} +8.39002i q^{19} -3.86043 q^{21} +(-1.81092 + 3.93394i) q^{22} +(-2.00465 + 0.568984i) q^{24} +5.00000 q^{25} -4.02058i q^{27} +(7.96122 - 6.81485i) q^{28} -10.3057 q^{29} -10.9617i q^{31} +(3.12969 - 4.71222i) q^{32} -2.25614 q^{33} +(3.19581 + 3.73339i) q^{36} +(4.96152 - 10.7781i) q^{38} +(4.95926 + 2.28291i) q^{42} +(4.65275 - 3.98278i) q^{44} -3.40245i q^{47} +(2.91173 + 0.454534i) q^{48} -20.4558 q^{49} +(-6.42319 - 2.95680i) q^{50} +(-2.37761 + 5.16499i) q^{54} +(-14.2573 + 4.04667i) q^{56} +6.18133 q^{57} +(13.2390 + 6.09436i) q^{58} +7.68676 q^{61} +(-6.48231 + 14.0818i) q^{62} -12.8753i q^{63} +(-6.80713 + 4.20273i) q^{64} +(2.89832 + 1.33419i) q^{66} +(-1.89767 - 6.68592i) q^{72} -3.68374i q^{75} +(-12.7475 + 10.9119i) q^{76} -16.0459 q^{77} +4.40945 q^{81} +(-5.02083 - 5.86541i) q^{84} +7.59268i q^{87} +(-8.33236 + 2.36498i) q^{88} -14.4565 q^{89} -8.07602 q^{93} +(-2.01208 + 4.37092i) q^{94} +(-3.47172 - 2.30579i) q^{96} +12.4499 q^{97} +(26.2783 + 12.0967i) q^{98} -7.52468i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.28464	−	0.591360i	−0.908376	−	0.418155i
$2$	−1.28464	−	0.591360i	−0.908376	−	0.418155i
$3$		−	0.736748i		−	0.425362i	−0.977122	−	0.212681i	$-0.931781\pi$
$3$		−	0.736748i		−	0.425362i	0.977122	−	0.212681i	$-0.0682195\pi$
$4$	1.30059	+	1.51937i	0.650294	+	0.759683i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	−0.435683	+	0.946454i	−0.177867	+	0.386388i
$7$		−	5.23983i		−	1.98047i	−0.139411	−	0.990235i	$-0.544521\pi$
$7$		−	5.23983i		−	1.98047i	0.139411	−	0.990235i	$-0.455479\pi$
$8$	−0.772291	−	2.72095i	−0.273046	−	0.962001i
$9$	2.45720			0.819067
$10$		0			0
$11$		−	3.06230i		−	0.923317i	−0.887058	−	0.461659i	$-0.847254\pi$
$11$		−	3.06230i		−	0.923317i	0.887058	−	0.461659i	$-0.152746\pi$
$12$	1.11939	−	0.958205i	0.323140	−	0.276610i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−3.09862	+	6.73128i	−0.828142	+	1.79901i
$15$		0			0
$16$	−0.616947	+	3.95214i	−0.154237	+	0.988034i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−3.15661	−	1.45309i	−0.744021	−	0.342497i
$19$			8.39002i			1.92480i	0.271633	+	0.962401i	$0.412436\pi$
$19$			8.39002i			1.92480i	−0.271633	+	0.962401i	$0.587564\pi$
$20$		0			0
$21$	−3.86043			−0.842416
$22$	−1.81092	+	3.93394i	−0.386089	+	0.838719i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−2.00465	+	0.568984i	−0.409198	+	0.116143i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	4.02058i		−	0.773762i
$28$	7.96122	−	6.81485i	1.50453	−	1.28789i
$29$	−10.3057			−1.91371			−0.956857	−	0.290558i	$-0.906159\pi$
$29$	−10.3057			−1.91371			−0.956857	+	0.290558i	$0.906159\pi$
$30$		0			0
$31$		−	10.9617i		−	1.96878i	−0.175999	−	0.984390i	$-0.556316\pi$
$31$		−	10.9617i		−	1.96878i	0.175999	−	0.984390i	$-0.443684\pi$
$32$	3.12969	−	4.71222i	0.553256	−	0.833011i
$33$	−2.25614			−0.392744
$34$		0			0
$35$		0			0
$36$	3.19581	+	3.73339i	0.532634	+	0.622232i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	4.96152	−	10.7781i	0.804865	−	1.74844i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	4.95926	+	2.28291i	0.765230	+	0.352260i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	4.65275	−	3.98278i	0.701429	−	0.600427i
$45$		0			0
$46$		0			0
$47$		−	3.40245i		−	0.496299i	−0.968722	−	0.248150i	$-0.920178\pi$
$47$		−	3.40245i		−	0.496299i	0.968722	−	0.248150i	$-0.0798224\pi$
$48$	2.91173	+	0.454534i	0.420272	+	0.0656064i
$49$	−20.4558			−2.92226
$50$	−6.42319	−	2.95680i	−0.908376	−	0.418155i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−2.37761	+	5.16499i	−0.323552	+	0.702866i
$55$		0			0
$56$	−14.2573	+	4.04667i	−1.90521	+	0.540759i
$57$	6.18133			0.818737
$58$	13.2390	+	6.09436i	1.73837	+	0.800228i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	7.68676			0.984189			0.492094	−	0.870542i	$-0.336231\pi$
$61$	7.68676			0.984189			0.492094	+	0.870542i	$0.336231\pi$
$62$	−6.48231	+	14.0818i	−0.823255	+	1.78839i
$63$		−	12.8753i		−	1.62214i
$64$	−6.80713	+	4.20273i	−0.850892	+	0.525341i
$65$		0			0
$66$	2.89832	+	1.33419i	0.356759	+	0.164228i
$67$		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	$0.5\pi$
$72$	−1.89767	−	6.68592i	−0.223643	−	0.787944i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		−	3.68374i		−	0.425362i
$76$	−12.7475	+	10.9119i	−1.46224	+	1.25169i
$77$	−16.0459			−1.82860
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	4.40945			0.489939
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−5.02083	−	5.86541i	−0.547818	−	0.639969i
$85$		0			0
$86$		0			0
$87$			7.59268i			0.814021i
$88$	−8.33236	+	2.36498i	−0.888232	+	0.252108i
$89$	−14.4565			−1.53238			−0.766191	−	0.642613i	$-0.777850\pi$
$89$	−14.4565			−1.53238			−0.766191	+	0.642613i	$0.777850\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−8.07602			−0.837444
$94$	−2.01208	+	4.37092i	−0.207530	+	0.450826i
$95$		0			0
$96$	−3.47172	−	2.30579i	−0.354331	−	0.235334i
$97$	12.4499			1.26409			0.632047	−	0.774930i	$-0.282215\pi$
$97$	12.4499			1.26409			0.632047	+	0.774930i	$0.282215\pi$
$98$	26.2783	+	12.0967i	2.65451	+	1.22196i
$99$		−	7.52468i		−	0.756259i
$100$	6.50294	+	7.59683i	0.650294	+	0.759683i
$101$		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
$107$		−	9.46067i		−	0.914597i	−0.889313	−	0.457299i	$-0.848817\pi$
$107$		−	9.46067i		−	0.914597i	0.889313	−	0.457299i	$-0.151183\pi$
$108$	6.10874	−	5.22912i	0.587814	−	0.503172i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	20.7085	+	3.23270i	1.95677	+	0.305461i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−7.94077	−	3.65539i	−0.743721	−	0.342359i
$115$		0			0
$116$	−13.4034	−	15.6581i	−1.24448	−	1.45382i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.62233			0.147485
$122$	−9.87470	−	4.54564i	−0.894013	−	0.411543i
$123$		0			0
$124$	16.6548	−	14.2567i	1.49565	−	1.28029i
$125$		0			0
$126$	−7.61395	+	16.5401i	−0.678304	+	1.47351i
$127$			21.3418i			1.89378i	0.321557	+	0.946890i	$0.395794\pi$
$127$			21.3418i			1.89378i	−0.321557	+	0.946890i	$0.604206\pi$
$128$	11.2300	−	1.37352i	0.992603	−	0.121403i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	−2.93431	−	3.42791i	−0.255399	−	0.298361i
$133$	43.9623			3.81201
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.72297			0.830689			0.415345	−	0.909664i	$-0.363661\pi$
$137$	9.72297			0.830689			0.415345	+	0.909664i	$0.363661\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−2.50675			−0.211107
$142$		0			0
$143$		0			0
$144$	−1.51596	+	9.71120i	−0.126330	+	0.809266i
$145$		0			0
$146$		0			0
$147$			15.0708i			1.24302i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	−2.17842	+	4.73227i	−0.177867	+	0.386388i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	22.8288	−	6.47953i	1.85166	−	0.525560i
$153$		0			0
$154$	20.6132	+	9.48891i	1.66106	+	0.764638i
$155$		0			0
$156$		0			0
$157$	17.3755			1.38672			0.693358	−	0.720594i	$-0.256131\pi$
$157$	17.3755			1.38672			0.693358	+	0.720594i	$0.256131\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−5.66454	−	2.60757i	−0.445049	−	0.204870i
$163$		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
$168$	2.98138	+	10.5040i	0.230018	+	0.810405i
$169$	13.0000			1.00000
$170$		0			0
$171$			20.6160i			1.57654i
$172$		0			0
$173$	−21.4911			−1.63394			−0.816970	−	0.576681i	$-0.804348\pi$
$173$	−21.4911			−1.63394			−0.816970	+	0.576681i	$0.804348\pi$
$174$	4.49001	−	9.75384i	0.340387	−	0.739437i
$175$		−	26.1991i		−	1.98047i
$176$	12.1026	+	1.88927i	0.912269	+	0.142409i
$177$		0			0
$178$	18.5713	+	8.54897i	1.39198	+	0.640772i
$179$			18.3651i			1.37268i	0.727283	+	0.686338i	$0.240783\pi$
$179$			18.3651i			1.37268i	−0.727283	+	0.686338i	$0.759217\pi$
$180$		0			0
$181$	11.6355			0.864862			0.432431	−	0.901667i	$-0.357656\pi$
$181$	11.6355			0.864862			0.432431	+	0.901667i	$0.357656\pi$
$182$		0			0
$183$		−	5.66321i		−	0.418636i
$184$		0			0
$185$		0			0
$186$	10.3748	+	4.77583i	0.760714	+	0.350181i
$187$		0			0
$188$	5.16957	−	4.42519i	0.377030	−	0.322740i
$189$	−21.0672			−1.53241
$190$		0			0
$191$			23.4234i			1.69486i	0.530911	+	0.847428i	$0.321850\pi$
$191$			23.4234i			1.69486i	−0.530911	+	0.847428i	$0.678150\pi$
$192$	3.09635	+	5.01514i	0.223460	+	0.361937i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−15.9936	−	7.36236i	−1.14827	−	0.528586i
$195$		0			0
$196$	−26.6046	−	31.0799i	−1.90033	−	2.21999i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−4.44980	+	9.66649i	−0.316233	+	0.686968i
$199$		−	2.96804i		−	0.210398i	−0.994451	−	0.105199i	$-0.966452\pi$
$199$		−	2.96804i		−	0.210398i	0.994451	−	0.105199i	$-0.0335480\pi$
$200$	−3.86145	−	13.6047i	−0.273046	−	0.962001i
$201$		0			0
$202$		0			0
$203$			53.9999i			3.79005i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	25.6927			1.77720
$210$		0			0
$211$			3.23093i			0.222426i	0.993797	+	0.111213i	$0.0354736\pi$
$211$			3.23093i			0.222426i	−0.993797	+	0.111213i	$0.964526\pi$
$212$		0			0
$213$		0			0
$214$	−5.59466	+	12.1535i	−0.382443	+	0.830798i
$215$		0			0
$216$	−10.9398	+	3.10506i	−0.744359	+	0.211273i
$217$	−57.4375			−3.89911
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	29.4068i		−	1.96923i	−0.174744	−	0.984614i	$-0.555910\pi$
$223$		−	29.4068i		−	1.96923i	0.174744	−	0.984614i	$-0.444090\pi$
$224$	−24.6912	−	16.3990i	−1.64975	−	1.09571i
$225$	12.2860			0.819067
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	8.03936	+	9.39170i	0.532419	+	0.621981i
$229$	27.9662			1.84806			0.924030	−	0.382319i	$-0.124875\pi$
$229$	27.9662			1.84806			0.924030	+	0.382319i	$0.124875\pi$
$230$		0			0
$231$			11.8218i			0.777817i
$232$	7.95897	+	28.0412i	0.522532	+	1.84100i
$233$	−0.740976			−0.0485430			−0.0242715	−	0.999705i	$-0.507727\pi$
$233$	−0.740976			−0.0485430			−0.0242715	+	0.999705i	$0.507727\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			30.8803i			1.99748i	0.0501540	+	0.998741i	$0.484029\pi$
$239$			30.8803i			1.99748i	−0.0501540	+	0.998741i	$0.515971\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−2.08411	−	0.959383i	−0.133972	−	0.0616715i
$243$		−	15.3104i		−	0.982163i
$244$	9.99730	+	11.6790i	0.640012	+	0.747672i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−29.8263	+	8.46563i	−1.89397	+	0.537568i
$249$		0			0
$250$		0			0
$251$		−	21.9783i		−	1.38726i	−0.720332	−	0.693629i	$-0.756011\pi$
$251$		−	21.9783i		−	1.38726i	0.720332	−	0.693629i	$-0.243989\pi$
$252$	19.5623	−	16.7455i	1.23231	−	1.05487i
$253$		0			0
$254$	12.6207	−	27.4165i	0.791893	−	1.72026i
$255$		0			0
$256$	−15.2388	−	4.87651i	−0.952422	−	0.304782i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−25.3231			−1.56746
$262$		0			0
$263$		−	26.4598i		−	1.63158i	−0.578345	−	0.815792i	$-0.696301\pi$
$263$		−	26.4598i		−	1.63158i	0.578345	−	0.815792i	$-0.303699\pi$
$264$	1.74240	+	6.13885i	0.107237	+	0.377820i
$265$		0			0
$266$	−56.4756	−	25.9975i	−3.46274	−	1.59401i
$267$			10.6508i			0.651816i
$268$		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
$274$	−12.4905	−	5.74977i	−0.754578	−	0.347357i
$275$		−	15.3115i		−	0.923317i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		−	26.9351i		−	1.61256i
$280$		0			0
$281$	26.1617			1.56068			0.780339	−	0.625357i	$-0.215047\pi$
$281$	26.1617			1.56068			0.780339	+	0.625357i	$0.215047\pi$
$282$	3.22027	+	1.48239i	0.191764	+	0.0882752i
$283$			33.4968i			1.99118i	0.0938371	+	0.995588i	$0.470087\pi$
$283$			33.4968i			1.99118i	−0.0938371	+	0.995588i	$0.529913\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	7.69027	−	11.5789i	0.453154	−	0.682292i
$289$	17.0000			1.00000
$290$		0			0
$291$		−	9.17242i		−	0.537697i
$292$		0			0
$293$	30.0903			1.75790			0.878948	−	0.476918i	$-0.158246\pi$
$293$	30.0903			1.75790			0.878948	+	0.476918i	$0.158246\pi$
$294$	8.91225	−	19.3605i	0.519773	−	1.12913i
$295$		0			0
$296$		0			0
$297$	−12.3122			−0.714428
$298$		0			0
$299$		0			0
$300$	5.59695	−	4.79103i	0.323140	−	0.276610i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−33.1585	−	5.17619i	−1.90177	−	0.296875i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−20.8691	−	24.3796i	−1.18913	−	1.38916i
$309$		0			0
$310$		0			0
$311$		−	25.8457i		−	1.46557i	−0.680458	−	0.732787i	$-0.738219\pi$
$311$		−	25.8457i		−	1.46557i	0.680458	−	0.732787i	$-0.261781\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−22.3212	−	10.2752i	−1.25966	−	0.579861i
$315$		0			0
$316$		0			0
$317$	34.7656			1.95263			0.976314	−	0.216359i	$-0.0694181\pi$
$317$	34.7656			1.95263			0.976314	+	0.216359i	$0.0694181\pi$
$318$		0			0
$319$			31.5590i			1.76697i
$320$		0			0
$321$	−6.97013			−0.389035
$322$		0			0
$323$		0			0
$324$	5.73487	+	6.69957i	0.318604	+	0.372198i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−17.8283			−0.982905
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	−7.64205	+	16.6012i	−0.418155	+	0.908376i
$335$		0			0
$336$	2.38168	−	15.2570i	0.129931	−	0.832335i
$337$	−25.8438			−1.40780			−0.703900	−	0.710299i	$-0.748560\pi$
$337$	−25.8438			−1.40780			−0.703900	+	0.710299i	$0.748560\pi$
$338$	−16.7003	−	7.68768i	−0.908376	−	0.418155i
$339$		0			0
$340$		0			0
$341$	−33.5680			−1.81781
$342$	12.1915	−	26.4840i	0.659238	−	1.43209i
$343$			70.5061i			3.80697i
$344$		0			0
$345$		0			0
$346$	27.6083	+	12.7090i	1.48423	+	0.683239i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−11.5361	+	9.87494i	−0.618398	+	0.529353i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−15.4931	+	33.6564i	−0.828142	+	1.79901i
$351$		0			0
$352$	−14.4302	−	9.58403i	−0.769134	−	0.510831i
$353$	14.7318			0.784096			0.392048	−	0.919945i	$-0.371767\pi$
$353$	14.7318			0.784096			0.392048	+	0.919945i	$0.371767\pi$
$354$		0			0
$355$		0			0
$356$	−18.8019	−	21.9647i	−0.996498	−	1.16412i
$357$		0			0
$358$	10.8604	−	23.5926i	0.573990	−	1.24691i
$359$			13.9616i			0.736865i	0.929654	+	0.368433i	$0.120105\pi$
$359$			13.9616i			0.736865i	−0.929654	+	0.368433i	$0.879895\pi$
$360$		0			0
$361$	−51.3924			−2.70486
$362$	−14.9474	−	6.88078i	−0.785620	−	0.361646i
$363$		−	1.19525i		−	0.0627344i
$364$		0			0
$365$		0			0
$366$	−3.34899	+	7.27517i	−0.175055	+	0.380279i
$367$		−	23.3527i		−	1.21900i	−0.792786	−	0.609501i	$-0.791370\pi$
$367$		−	23.3527i		−	1.21900i	0.792786	−	0.609501i	$-0.208630\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−10.5036	−	12.2704i	−0.544584	−	0.636192i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−9.25791	+	2.62768i	−0.477440	+	0.135512i
$377$		0			0
$378$	27.0637	+	12.4583i	1.39201	+	0.640785i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	15.7235			0.805542
$382$	13.8516	−	30.0905i	0.708711	−	1.53957i
$383$			3.20185i			0.163607i	0.996648	+	0.0818034i	$0.0260679\pi$
$383$			3.20185i			0.163607i	−0.996648	+	0.0818034i	$0.973932\pi$
$384$	−1.01194	−	8.27370i	−0.0516402	−	0.422215i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	16.1921	+	18.9159i	0.822032	+	0.960310i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	15.7978	+	55.6592i	0.797911	+	2.81122i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	11.4328	−	9.78651i	0.574517	−	0.491790i
$397$	−32.9314			−1.65278			−0.826390	−	0.563097i	$-0.809610\pi$
$397$	−32.9314			−1.65278			−0.826390	+	0.563097i	$0.809610\pi$
$398$	−1.75518	+	3.81285i	−0.0879791	+	0.191121i
$399$		−	32.3891i		−	1.62148i
$400$	−3.08473	+	19.7607i	−0.154237	+	0.988034i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	31.9334	−	69.3703i	1.58483	−	3.44279i
$407$		0			0
$408$		0			0
$409$	−8.35952			−0.413352			−0.206676	−	0.978409i	$-0.566265\pi$
$409$	−8.35952			−0.413352			−0.206676	+	0.978409i	$0.566265\pi$
$410$		0			0
$411$		−	7.16338i		−	0.353343i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−33.0058	−	15.1936i	−1.61437	−	0.743146i
$419$			37.6160i			1.83766i	0.394651	+	0.918831i	$0.370865\pi$
$419$			37.6160i			1.83766i	−0.394651	+	0.918831i	$0.629135\pi$
$420$		0			0
$421$	25.0155			1.21918			0.609589	−	0.792717i	$-0.291334\pi$
$421$	25.0155			1.21918			0.609589	+	0.792717i	$0.291334\pi$
$422$	1.91064	−	4.15057i	0.0930086	−	0.202047i
$423$		−	8.36052i		−	0.406502i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	40.2773i		−	1.94916i
$428$	14.3742	−	12.3044i	0.694804	−	0.594757i
$429$		0			0
$430$		0			0
$431$		−	35.3008i		−	1.70038i	−0.526475	−	0.850190i	$-0.676487\pi$
$431$		−	35.3008i		−	1.70038i	0.526475	−	0.850190i	$-0.323513\pi$
$432$	15.8899	+	2.48049i	0.764503	+	0.119342i
$433$	−6.89945			−0.331566			−0.165783	−	0.986162i	$-0.553015\pi$
$433$	−6.89945			−0.331566			−0.165783	+	0.986162i	$0.553015\pi$
$434$	73.7863	+	33.9662i	3.54186	+	1.63043i
$435$		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
$441$	−50.2641			−2.39353
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	−17.3900	+	37.7771i	−0.823442	+	1.78880i
$447$		0			0
$448$	22.0216	+	35.6682i	1.04042	+	1.68516i
$449$	36.7582			1.73473			0.867363	−	0.497677i	$-0.165813\pi$
$449$	36.7582			1.73473			0.867363	+	0.497677i	$0.165813\pi$
$450$	−15.7831	−	7.26545i	−0.744021	−	0.342497i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	−4.77378	−	16.8191i	−0.223553	−	0.787626i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−35.9265	−	16.5381i	−1.67873	−	0.772775i
$459$		0			0
$460$		0			0
$461$	42.9365			1.99975			0.999876	−	0.0157523i	$-0.00501433\pi$
$461$	42.9365			1.99975			0.999876	+	0.0157523i	$0.00501433\pi$
$462$	6.99094	−	15.1867i	0.325248	−	0.706550i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	6.35805	−	40.7294i	0.295165	−	1.89081i
$465$		0			0
$466$	0.951886	+	0.438183i	0.0440953	+	0.0202985i
$467$			40.8996i			1.89261i	0.323276	+	0.946305i	$0.395216\pi$
$467$			40.8996i			1.89261i	−0.323276	+	0.946305i	$0.604784\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		−	12.8014i		−	0.589856i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			41.9501i			1.92480i
$476$		0			0
$477$		0			0
$478$	18.2614	−	39.6700i	0.835257	−	1.81447i
$479$		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
$484$	2.10999	+	2.46492i	0.0959084	+	0.112042i
$485$		0			0
$486$	−9.05396	+	19.6683i	−0.410696	+	0.892173i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−5.93641	−	20.9153i	−0.268729	−	0.946791i
$489$		0			0
$490$		0			0
$491$			25.8457i			1.16640i	0.812329	+	0.583200i	$0.198200\pi$
$491$			25.8457i			1.16640i	−0.812329	+	0.583200i	$0.801800\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	43.3222	+	6.76279i	1.94522	+	0.303658i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	−9.52088			−0.425362
$502$	−12.9971	+	28.2342i	−0.580088	+	1.26015i
$503$			22.0394i			0.982686i	0.870966	+	0.491343i	$0.163494\pi$
$503$			22.0394i			0.982686i	−0.870966	+	0.491343i	$0.836506\pi$
$504$	−35.0331	+	9.94349i	−1.56050	+	0.442918i
$505$		0			0
$506$		0			0
$507$		−	9.57773i		−	0.425362i
$508$	−32.4260	+	27.7569i	−1.43867	+	1.23151i
$509$	−45.0274			−1.99581			−0.997903	−	0.0647334i	$-0.979380\pi$
$509$	−45.0274			−1.99581			−0.997903	+	0.0647334i	$0.979380\pi$
$510$		0			0
$511$		0			0
$512$	16.6925	+	15.2761i	0.737711	+	0.675116i
$513$	33.7328			1.48934
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−10.4193			−0.458242
$518$		0			0
$519$			15.8335i			0.695015i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	32.5310	+	14.9751i	1.42384	+	0.655441i
$523$		−	40.6159i		−	1.77601i	−0.459835	−	0.888005i	$-0.652091\pi$
$523$		−	40.6159i		−	1.77601i	0.459835	−	0.888005i	$-0.347909\pi$
$524$		0			0
$525$	−19.3022			−0.842416
$526$	−15.6473	+	33.9913i	−0.682254	+	1.48209i
$527$		0			0
$528$	1.39192	−	8.91658i	0.0605755	−	0.388044i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	57.1767	+	66.7948i	2.47893	+	2.89592i
$533$		0			0
$534$	6.29844	−	13.6824i	0.272560	−	0.592094i
$535$		0			0
$536$		0			0
$537$	13.5305			0.583884
$538$		0			0
$539$			62.6418i			2.69817i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	8.57245i		−	0.367879i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	12.6456	+	14.7728i	0.540192	+	0.631061i
$549$	18.8879			0.806117
$550$	−9.05460	+	19.6697i	−0.386089	+	0.838719i
$551$		−	86.4647i		−	3.68352i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	25.5815			1.08392			0.541961	−	0.840404i	$-0.317682\pi$
$557$	25.5815			1.08392			0.541961	+	0.840404i	$0.317682\pi$
$558$	−15.9284	+	34.6019i	−0.674301	+	1.46481i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−33.6083	−	15.4710i	−1.41768	−	0.652605i
$563$		−	11.5602i		−	0.487206i	−0.969875	−	0.243603i	$-0.921671\pi$
$563$		−	11.5602i		−	0.487206i	0.969875	−	0.243603i	$-0.0783293\pi$
$564$	−3.26025	−	3.80867i	−0.137281	−	0.160374i
$565$		0			0
$566$	19.8086	−	43.0312i	0.832619	−	1.80874i
$567$		−	23.1048i		−	0.970309i
$568$		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
$573$	17.2571			0.720927
$574$		0			0
$575$		0			0
$576$	−16.7265	+	10.3270i	−0.696938	+	0.430290i
$577$	−5.61644			−0.233815			−0.116908	−	0.993143i	$-0.537298\pi$
$577$	−5.61644			−0.233815			−0.116908	+	0.993143i	$0.537298\pi$
$578$	−21.8388	−	10.0531i	−0.908376	−	0.418155i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−5.42420	+	11.7832i	−0.224840	+	0.488431i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−38.6552	−	17.7942i	−1.59683	−	0.735072i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−22.8980	+	19.6009i	−0.944299	+	0.808326i
$589$	91.9689			3.78951
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	15.8167	+	7.28095i	0.648969	+	0.298741i
$595$		0			0
$596$		0			0
$597$	−2.18670			−0.0894955
$598$		0			0
$599$		−	34.6408i		−	1.41539i	−0.706520	−	0.707693i	$-0.749736\pi$
$599$		−	34.6408i		−	1.41539i	0.706520	−	0.707693i	$-0.250264\pi$
$600$	−10.0233	+	2.84492i	−0.409198	+	0.116143i
$601$	−23.5401			−0.960219			−0.480110	−	0.877208i	$-0.659403\pi$
$601$	−23.5401			−0.960219			−0.480110	+	0.877208i	$0.659403\pi$
$602$		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	$0.5\pi$
$608$	39.5356	+	26.2581i	1.60338	+	1.06491i
$609$	39.7844			1.61214
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−36.4477			−1.47211			−0.736055	−	0.676922i	$-0.763314\pi$
$613$	−36.4477			−1.47211			−0.736055	+	0.676922i	$0.763314\pi$
$614$		0			0
$615$		0			0
$616$	12.3921	+	43.6601i	0.499292	+	1.75912i
$617$	47.1022			1.89626			0.948131	−	0.317879i	$-0.102971\pi$
$617$	47.1022			1.89626			0.948131	+	0.317879i	$0.102971\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−15.2841	+	33.2024i	−0.612837	+	1.33129i
$623$			75.7494i			3.03483i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		−	18.9291i		−	0.755954i
$628$	22.5983	+	26.3997i	0.901772	+	1.05346i
$629$		0			0
$630$		0			0
$631$		−	20.5659i		−	0.818714i	−0.912374	−	0.409357i	$-0.865753\pi$
$631$		−	20.5659i		−	0.818714i	0.912374	−	0.409357i	$-0.134247\pi$
$632$		0			0
$633$	2.38038			0.0946117
$634$	−44.6611	−	20.5590i	−1.77372	−	0.816500i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	18.6627	−	40.5419i	0.738865	−	1.60507i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	8.95409	+	4.12185i	0.353390	+	0.162677i
$643$		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	$0.5\pi$
$648$	−3.40538	−	11.9979i	−0.133776	−	0.471322i
$649$		0			0
$650$		0			0
$651$			42.3170i			1.65853i
$652$		0			0
$653$	50.6445			1.98187			0.990937	−	0.134330i	$-0.0428881\pi$
$653$	50.6445			1.98187			0.990937	+	0.134330i	$0.0428881\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	22.9029	+	10.5429i	0.892847	+	0.411006i
$659$		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
$668$	19.6345	−	16.8073i	0.759683	−	0.650294i
$669$	−21.6654			−0.837634
$670$		0			0
$671$		−	23.5391i		−	0.908719i
$672$	−12.0820	+	18.1912i	−0.466071	+	0.701742i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	33.1999	+	15.2830i	1.27881	+	0.588678i
$675$		−	20.1029i		−	0.773762i
$676$	16.9076	+	19.7518i	0.650294	+	0.759683i
$677$	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
$679$		−	65.2352i		−	2.50350i
$680$		0			0
$681$		0			0
$682$	43.1227	+	19.8508i	1.65125	+	0.760125i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−31.3232	+	26.8129i	−1.19767	+	1.02522i
$685$		0			0
$686$	41.6945	−	90.5748i	1.59190	−	3.45816i
$687$		−	20.6041i		−	0.786094i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−27.9511	−	32.6529i	−1.06254	−	1.24128i
$693$	−39.4281			−1.49775
$694$		0			0
$695$		0			0
$696$	20.6593	−	5.86376i	0.783089	−	0.222265i
$697$		0			0
$698$		0			0
$699$			0.545913i			0.0206483i
$700$	39.8061	−	34.0743i	1.50453	−	1.28789i
$701$	−46.4708			−1.75518			−0.877589	−	0.479413i	$-0.840850\pi$
$701$	−46.4708			−1.75518			−0.877589	+	0.479413i	$0.840850\pi$
$702$		0			0
$703$		0			0
$704$	12.8700	+	20.8455i	0.485057	+	0.785643i
$705$		0			0
$706$	−18.9251	−	8.71181i	−0.712254	−	0.327873i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	11.1646	+	39.3353i	0.418411	+	1.47415i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−27.9034	+	23.8855i	−1.04280	+	0.892642i
$717$	22.7510			0.849653
$718$	8.25633	−	17.9356i	0.308124	−	0.669351i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	66.0206	+	30.3914i	2.45703	+	1.13105i
$723$		0			0
$724$	15.1330	+	17.6786i	0.562414	+	0.657021i
$725$	−51.5283			−1.91371
$726$	−0.706824	+	1.53546i	−0.0262327	+	0.0569864i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	1.94843			0.0721642
$730$		0			0
$731$		0			0
$732$	8.60448	−	7.36549i	0.318031	−	0.272236i
$733$	−53.8509			−1.98903			−0.994514	−	0.104602i	$-0.966643\pi$
$733$	−53.8509			−1.98903			−0.994514	+	0.104602i	$0.966643\pi$
$734$	−13.8099	+	29.9998i	−0.509731	+	1.10731i
$735$		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	$-0.842776\pi$
$743$		−	25.8457i		−	0.948187i	0.880475	−	0.474093i	$-0.157224\pi$
$744$	6.23703	+	21.9744i	0.228661	+	0.805622i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−49.5723			−1.81133
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	13.4470	+	2.09913i	0.490360	+	0.0765475i
$753$	−16.1925			−0.590087
$754$		0			0
$755$		0			0
$756$	−27.3997	−	32.0087i	−0.996517	−	1.16415i
$757$	−18.7642			−0.681997			−0.340998	−	0.940064i	$-0.610765\pi$
$757$	−18.7642			−0.681997			−0.340998	+	0.940064i	$0.610765\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−55.0746			−1.99645			−0.998227	−	0.0595193i	$-0.981043\pi$
$761$	−55.0746			−1.99645			−0.998227	+	0.0595193i	$0.981043\pi$
$762$	−20.1991	−	9.29827i	−0.731735	−	0.336841i
$763$		0			0
$764$	−35.5887	+	30.4641i	−1.28755	+	1.10215i
$765$		0			0
$766$	1.89344	−	4.11321i	0.0684129	−	0.148616i
$767$		0			0
$768$	−3.59276	+	11.2271i	−0.129643	+	0.405124i
$769$		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
$775$		−	54.8085i		−	1.96878i
$776$	−9.61492	−	33.8755i	−0.345156	−	1.21606i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			41.4348i			1.48076i
$784$	12.6201	−	80.8441i	0.450719	−	2.88729i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	−19.4942			−0.694013
$790$		0			0
$791$		0			0
$792$	−20.4743	+	5.81124i	−0.727522	+	0.206494i
$793$		0			0
$794$	42.3049	+	19.4743i	1.50135	+	0.691118i
$795$		0			0
$796$	4.50953	−	3.86019i	0.159836	−	0.136821i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	−19.1536	+	41.6083i	−0.678031	+	1.47292i
$799$		0			0
$800$	15.6484	−	23.5611i	0.553256	−	0.833011i
$801$	−35.5224			−1.25512
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	56.7956			1.99683			0.998413	−	0.0563179i	$-0.0179360\pi$
$809$	56.7956			1.99683			0.998413	+	0.0563179i	$0.0179360\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−82.0457	+	70.2316i	−2.87924	+	2.46465i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	10.7389	+	4.94348i	0.375479	+	0.172845i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−4.23614	+	9.20235i	−0.147752	+	0.320969i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−11.2807			−0.392744
$826$		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
$836$	33.4156	+	39.0367i	1.15570	+	1.35011i
$837$	−44.0725			−1.52337
$838$	22.2446	−	48.3229i	0.768427	−	1.66929i
$839$			39.7213i			1.37133i	0.727916	+	0.685666i	$0.240489\pi$
$839$			39.7213i			1.37133i	−0.727916	+	0.685666i	$0.759511\pi$
$840$		0			0
$841$	77.2068			2.66230
$842$	−32.1358	−	14.7931i	−1.10747	−	0.509805i
$843$		−	19.2746i		−	0.663853i
$844$	−4.90896	+	4.20211i	−0.168974	+	0.144642i
$845$		0			0
$846$	−4.94408	+	10.7402i	−0.169981	+	0.369257i
$847$		−	8.50075i		−	0.292089i
$848$		0			0
$849$	24.6787			0.846970
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−47.7543			−1.63508			−0.817538	−	0.575874i	$-0.804662\pi$
$853$	−47.7543			−1.63508			−0.817538	+	0.575874i	$0.804662\pi$
$854$	−23.8184	+	51.7417i	−0.815048	+	1.77057i
$855$		0			0
$856$	−25.7420	+	7.30639i	−0.879843	+	0.249727i
$857$	−28.7283			−0.981339			−0.490669	−	0.871346i	$-0.663248\pi$
$857$	−28.7283			−0.981339			−0.490669	+	0.871346i	$0.663248\pi$
$858$		0			0
$859$			7.42268i			0.253259i	0.991950	+	0.126629i	$0.0404159\pi$
$859$			7.42268i			0.253259i	−0.991950	+	0.126629i	$0.959584\pi$
$860$		0			0
$861$		0			0
$862$	−20.8755	+	45.3488i	−0.711022	+	1.54459i
$863$			57.9009i			1.97097i	0.169767	+	0.985484i	$0.445699\pi$
$863$			57.9009i			1.97097i	−0.169767	+	0.985484i	$0.554301\pi$
$864$	−18.9459	−	12.5832i	−0.644552	−	0.428088i
$865$		0			0
$866$	8.86329	+	4.08006i	0.301187	+	0.138646i
$867$		−	12.5247i		−	0.425362i
$868$	−74.7024	−	87.2685i	−2.53557	−	2.96209i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	30.5919			1.03538
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−44.3787			−1.49856			−0.749282	−	0.662251i	$-0.769601\pi$
$877$	−44.3787			−1.49856			−0.749282	+	0.662251i	$0.769601\pi$
$878$		0			0
$879$		−	22.1690i		−	0.747742i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	64.5711	+	29.7241i	2.17422	+	1.00086i
$883$		−	17.9023i		−	0.602462i	−0.953551	−	0.301231i	$-0.902603\pi$
$883$		−	17.9023i		−	0.602462i	0.953551	−	0.301231i	$-0.0973975\pi$
$884$		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
$889$	111.827			3.75057
$890$		0			0
$891$		−	13.5030i		−	0.452369i
$892$	44.6798	−	38.2462i	1.49599	−	1.28058i
$893$	28.5467			0.955277
$894$		0			0
$895$		0			0
$896$	−7.19700	−	58.8434i	−0.240435	−	1.96582i
$897$		0			0
$898$	−47.2209	−	21.7373i	−1.57578	−	0.725383i
$899$			112.968i			3.76768i
$900$	15.9790	+	18.6669i	0.532634	+	0.622232i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$			56.6034i			1.87949i	0.341881	+	0.939743i	$0.388936\pi$
$907$			56.6034i			1.87949i	−0.341881	+	0.939743i	$0.611064\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			53.7426i			1.78057i	0.455403	+	0.890285i	$0.349495\pi$
$911$			53.7426i			1.78057i	−0.455403	+	0.890285i	$0.650505\pi$
$912$	−3.81355	+	24.4295i	−0.126279	+	0.808940i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	36.3725	+	42.4909i	1.20178	+	1.40394i
$917$		0			0
$918$		0			0
$919$		−	56.9128i		−	1.87738i	−0.344763	−	0.938690i	$-0.612041\pi$
$919$		−	56.9128i		−	1.87738i	0.344763	−	0.938690i	$-0.387959\pi$
$920$		0			0
$921$		0			0
$922$	−55.1578	−	25.3909i	−1.81653	−	0.836205i
$923$		0			0
$924$	−17.9616	+	15.3753i	−0.590895	+	0.505810i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−32.2535	+	48.5626i	−1.05877	+	1.59415i
$929$	60.9216			1.99877			0.999387	−	0.0350138i	$-0.0111475\pi$
$929$	60.9216			1.99877			0.999387	+	0.0350138i	$0.0111475\pi$
$930$		0			0
$931$		−	171.625i		−	5.62477i
$932$	−0.963704	−	1.12581i	−0.0315672	−	0.0368773i
$933$	−19.0418			−0.623400
$934$	24.1864	−	52.5412i	0.791403	−	1.71920i
$935$		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			$\pi$
$942$	−7.57021	+	16.4451i	−0.246651	+	0.535811i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	55.1299i		−	1.79148i	−0.444576	−	0.895741i	$-0.646646\pi$
$947$		−	55.1299i		−	1.79148i	0.444576	−	0.895741i	$-0.353354\pi$
$948$		0			0
$949$		0			0
$950$	24.8076	−	53.8907i	0.804865	−	1.74844i
$951$		−	25.6135i		−	0.830573i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−46.9185	+	40.1626i	−1.51745	+	1.29895i
$957$	23.2511			0.751600
$958$		0			0
$959$		−	50.9467i		−	1.64515i
$960$		0			0
$961$	−89.1590			−2.87610
$962$		0			0
$963$		−	23.2468i		−	0.749117i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			17.4236i			0.560304i	0.959956	+	0.280152i	$0.0903849\pi$
$967$			17.4236i			0.560304i	−0.959956	+	0.280152i	$0.909615\pi$
$968$	−1.25291	−	4.41429i	−0.0402702	−	0.141881i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	23.2621	−	19.9125i	0.746132	−	0.638694i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−4.74232	+	30.3791i	−0.151798	+	0.972412i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$			44.2700i			1.41487i
$980$		0			0
$981$		0			0
$982$	15.2841	−	33.2024i	0.487735	−	1.05953i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			13.1350i			0.418090i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	−51.6540	−	34.3067i	−1.64002	−	1.08924i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	36.8186			1.16606			0.583028	−	0.812452i	$-0.301868\pi$
$997$	36.8186			1.16606			0.583028	+	0.812452i	$0.301868\pi$
$998$		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.667.3	✓	22
4.3	odd	2	inner	668.2.b.a.667.4	yes	22
167.166	odd	2	CM	668.2.b.a.667.3	✓	22
668.667	even	2	inner	668.2.b.a.667.4	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.3	✓	22	1.1	even	1	trivial
668.2.b.a.667.3	✓	22	167.166	odd	2	CM
668.2.b.a.667.4	yes	22	4.3	odd	2	inner
668.2.b.a.667.4	yes	22	668.667	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.28464 - 0.591360i) q^{2} -0.736748i q^{3} +(1.30059 + 1.51937i) q^{4} +(-0.435683 + 0.946454i) q^{6} -5.23983i q^{7} +(-0.772291 - 2.72095i) q^{8} +2.45720 q^{9} +O(q^{10})\)	\(q+(-1.28464 - 0.591360i) q^{2} -0.736748i q^{3} +(1.30059 + 1.51937i) q^{4} +(-0.435683 + 0.946454i) q^{6} -5.23983i q^{7} +(-0.772291 - 2.72095i) q^{8} +2.45720 q^{9} -3.06230i q^{11} +(1.11939 - 0.958205i) q^{12} +(-3.09862 + 6.73128i) q^{14} +(-0.616947 + 3.95214i) q^{16} +(-3.15661 - 1.45309i) q^{18} +8.39002i q^{19} -3.86043 q^{21} +(-1.81092 + 3.93394i) q^{22} +(-2.00465 + 0.568984i) q^{24} +5.00000 q^{25} -4.02058i q^{27} +(7.96122 - 6.81485i) q^{28} -10.3057 q^{29} -10.9617i q^{31} +(3.12969 - 4.71222i) q^{32} -2.25614 q^{33} +(3.19581 + 3.73339i) q^{36} +(4.96152 - 10.7781i) q^{38} +(4.95926 + 2.28291i) q^{42} +(4.65275 - 3.98278i) q^{44} -3.40245i q^{47} +(2.91173 + 0.454534i) q^{48} -20.4558 q^{49} +(-6.42319 - 2.95680i) q^{50} +(-2.37761 + 5.16499i) q^{54} +(-14.2573 + 4.04667i) q^{56} +6.18133 q^{57} +(13.2390 + 6.09436i) q^{58} +7.68676 q^{61} +(-6.48231 + 14.0818i) q^{62} -12.8753i q^{63} +(-6.80713 + 4.20273i) q^{64} +(2.89832 + 1.33419i) q^{66} +(-1.89767 - 6.68592i) q^{72} -3.68374i q^{75} +(-12.7475 + 10.9119i) q^{76} -16.0459 q^{77} +4.40945 q^{81} +(-5.02083 - 5.86541i) q^{84} +7.59268i q^{87} +(-8.33236 + 2.36498i) q^{88} -14.4565 q^{89} -8.07602 q^{93} +(-2.01208 + 4.37092i) q^{94} +(-3.47172 - 2.30579i) q^{96} +12.4499 q^{97} +(26.2783 + 12.0967i) q^{98} -7.52468i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more