LMFDB - Embedded newform 668.2.b.a.667.7

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			667.7
Character	$\chi$	$=$	668.667
Dual form			668.2.b.a.667.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.760993 - 1.19201i) q^{2} +3.04057i q^{3} +(-0.841781 + 1.81422i) q^{4} +(3.62439 - 2.31385i) q^{6} +5.23541i q^{7} +(2.80316 - 0.377198i) q^{8} -6.24504 q^{9} +O(q^{10})$	\(q+(-0.760993 - 1.19201i) q^{2} +3.04057i q^{3} +(-0.841781 + 1.81422i) q^{4} +(3.62439 - 2.31385i) q^{6} +5.23541i q^{7} +(2.80316 - 0.377198i) q^{8} -6.24504 q^{9} +4.08022i q^{11} +(-5.51626 - 2.55949i) q^{12} +(6.24067 - 3.98411i) q^{14} +(-2.58281 - 3.05436i) q^{16} +(4.75243 + 7.44416i) q^{18} -3.53796i q^{19} -15.9186 q^{21} +(4.86366 - 3.10502i) q^{22} +(1.14690 + 8.52320i) q^{24} +5.00000 q^{25} -9.86675i q^{27} +(-9.49820 - 4.40707i) q^{28} +9.00656 q^{29} -10.2811i q^{31} +(-1.67533 + 5.40308i) q^{32} -12.4062 q^{33} +(5.25695 - 11.3299i) q^{36} +(-4.21729 + 2.69236i) q^{38} +(12.1139 + 18.9752i) q^{42} +(-7.40243 - 3.43465i) q^{44} +13.6315i q^{47} +(9.28697 - 7.85321i) q^{48} -20.4095 q^{49} +(-3.80496 - 5.96006i) q^{50} +(-11.7613 + 7.50853i) q^{54} +(1.97479 + 14.6757i) q^{56} +10.7574 q^{57} +(-6.85392 - 10.7359i) q^{58} -15.3107 q^{61} +(-12.2552 + 7.82388i) q^{62} -32.6953i q^{63} +(7.71544 - 2.11470i) q^{64} +(9.44100 + 14.7883i) q^{66} +(-17.5059 + 2.35562i) q^{72} +15.2028i q^{75} +(6.41865 + 2.97819i) q^{76} -21.3616 q^{77} +11.2654 q^{81} +(13.4000 - 28.8799i) q^{84} +27.3850i q^{87} +(1.53905 + 11.4375i) q^{88} +18.6303 q^{89} +31.2605 q^{93} +(16.2489 - 10.3734i) q^{94} +(-16.4284 - 5.09394i) q^{96} +19.0568 q^{97} +(15.5315 + 24.3284i) q^{98} -25.4811i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.760993	−	1.19201i	−0.538103	−	0.842879i
$2$	−0.760993	−	1.19201i	−0.538103	−	0.842879i
$3$			3.04057i			1.75547i	0.479145	+	0.877736i	$0.340947\pi$
$3$			3.04057i			1.75547i	−0.479145	+	0.877736i	$0.659053\pi$
$4$	−0.841781	+	1.81422i	−0.420890	+	0.907112i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$	3.62439	−	2.31385i	1.47965	−	0.944624i
$7$			5.23541i			1.97880i	0.145217	+	0.989400i	$0.453612\pi$
$7$			5.23541i			1.97880i	−0.145217	+	0.989400i	$0.546388\pi$
$8$	2.80316	−	0.377198i	0.991068	−	0.133360i
$9$	−6.24504			−2.08168
$10$		0			0
$11$			4.08022i			1.23023i	0.788437	+	0.615116i	$0.210891\pi$
$11$			4.08022i			1.23023i	−0.788437	+	0.615116i	$0.789109\pi$
$12$	−5.51626	−	2.55949i	−1.59241	−	0.738861i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	6.24067	−	3.98411i	1.66789	−	1.06480i
$15$		0			0
$16$	−2.58281	−	3.05436i	−0.645703	−	0.763589i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	4.75243	+	7.44416i	1.12016	+	1.75460i
$19$		−	3.53796i		−	0.811664i	−0.913948	−	0.405832i	$-0.866982\pi$
$19$		−	3.53796i		−	0.811664i	0.913948	−	0.405832i	$-0.133018\pi$
$20$		0			0
$21$	−15.9186			−3.47373
$22$	4.86366	−	3.10502i	1.03694	−	0.661992i
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	1.14690	+	8.52320i	0.234109	+	1.73979i
$25$	5.00000			1.00000
$26$		0			0
$27$		−	9.86675i		−	1.89886i
$28$	−9.49820	−	4.40707i	−1.79499	−	0.832858i
$29$	9.00656			1.67248			0.836238	−	0.548367i	$-0.184750\pi$
$29$	9.00656			1.67248			0.836238	+	0.548367i	$0.184750\pi$
$30$		0			0
$31$		−	10.2811i		−	1.84655i	−0.384142	−	0.923274i	$-0.625503\pi$
$31$		−	10.2811i		−	1.84655i	0.384142	−	0.923274i	$-0.374497\pi$
$32$	−1.67533	+	5.40308i	−0.296159	+	0.955139i
$33$	−12.4062			−2.15964
$34$		0			0
$35$		0			0
$36$	5.25695	−	11.3299i	0.876159	−	1.88832i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−4.21729	+	2.69236i	−0.684134	+	0.436759i
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	12.1139	+	18.9752i	1.86922	+	2.92793i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$	−7.40243	−	3.43465i	−1.11596	−	0.517793i
$45$		0			0
$46$		0			0
$47$			13.6315i			1.98835i	0.107760	+	0.994177i	$0.465632\pi$
$47$			13.6315i			1.98835i	−0.107760	+	0.994177i	$0.534368\pi$
$48$	9.28697	−	7.85321i	1.34046	−	1.13351i
$49$	−20.4095			−2.91565
$50$	−3.80496	−	5.96006i	−0.538103	−	0.842879i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−11.7613	+	7.50853i	−1.60051	+	1.02178i
$55$		0			0
$56$	1.97479	+	14.6757i	0.263892	+	1.96112i
$57$	10.7574			1.42485
$58$	−6.85392	−	10.7359i	−0.899964	−	1.40970i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−15.3107			−1.96033			−0.980165	−	0.198184i	$-0.936496\pi$
$61$	−15.3107			−1.96033			−0.980165	+	0.198184i	$0.936496\pi$
$62$	−12.2552	+	7.82388i	−1.55642	+	0.993633i
$63$		−	32.6953i		−	4.11923i
$64$	7.71544	−	2.11470i	0.964430	−	0.264337i
$65$		0			0
$66$	9.44100	+	14.7883i	1.16211	+	1.82031i
$67$		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	$0.5\pi$
$72$	−17.5059	+	2.35562i	−2.06309	+	0.277612i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$			15.2028i			1.75547i
$76$	6.41865	+	2.97819i	0.736269	+	0.341621i
$77$	−21.3616			−2.43438
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	11.2654			1.25171
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	13.4000	−	28.8799i	1.46206	−	3.15106i
$85$		0			0
$86$		0			0
$87$			27.3850i			2.93598i
$88$	1.53905	+	11.4375i	0.164063	+	1.21924i
$89$	18.6303			1.97481			0.987403	−	0.158226i	$-0.0505775\pi$
$89$	18.6303			1.97481			0.987403	+	0.158226i	$0.0505775\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	31.2605			3.24156
$94$	16.2489	−	10.3734i	1.67594	−	1.06994i
$95$		0			0
$96$	−16.4284	−	5.09394i	−1.67672	−	0.519898i
$97$	19.0568			1.93493			0.967463	−	0.253012i	$-0.0814213\pi$
$97$	19.0568			1.93493			0.967463	+	0.253012i	$0.0814213\pi$
$98$	15.5315	+	24.3284i	1.56892	+	2.45754i
$99$		−	25.4811i		−	2.56095i
$100$	−4.20890	+	9.07112i	−0.420890	+	0.907112i
$101$		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
$107$			3.89406i			0.376453i	0.982126	+	0.188226i	$0.0602738\pi$
$107$			3.89406i			0.376453i	−0.982126	+	0.188226i	$0.939726\pi$
$108$	17.9005	+	8.30564i	1.72248	+	0.799211i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	15.9908	−	13.5221i	1.51099	−	1.27772i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$	−8.18630	−	12.8229i	−0.766717	−	1.20098i
$115$		0			0
$116$	−7.58155	+	16.3399i	−0.703929	+	1.51712i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.64818			−0.513471
$122$	11.6513	+	18.2505i	1.05486	+	1.65232i
$123$		0			0
$124$	18.6523	+	8.65447i	1.67503	+	0.777194i
$125$		0			0
$126$	−38.9732	+	24.8809i	−3.47201	+	2.21657i
$127$			2.27313i			0.201708i	0.994901	+	0.100854i	$0.0321575\pi$
$127$			2.27313i			0.201708i	−0.994901	+	0.100854i	$0.967843\pi$
$128$	−8.39214	−	7.58762i	−0.741767	−	0.670658i
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$	10.4433	−	22.5076i	0.908970	−	1.95903i
$133$	18.5227			1.60612
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−15.3313			−1.30984			−0.654919	−	0.755699i	$-0.727297\pi$
$137$	−15.3313			−1.30984			−0.654919	+	0.755699i	$0.727297\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−41.4474			−3.49050
$142$		0			0
$143$		0			0
$144$	16.1298	+	19.0746i	1.34415	+	1.58955i
$145$		0			0
$146$		0			0
$147$		−	62.0565i		−	5.11834i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$	18.1219	−	11.5692i	1.47965	−	0.944624i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−1.33451	−	9.91748i	−0.108243	−	0.804414i
$153$		0			0
$154$	16.2560	+	25.4633i	1.30995	+	2.05189i
$155$		0			0
$156$		0			0
$157$	4.85429			0.387414			0.193707	−	0.981059i	$-0.437949\pi$
$157$	4.85429			0.387414			0.193707	+	0.981059i	$0.437949\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−8.57288	−	13.4285i	−0.673549	−	1.05504i
$163$		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
$168$	−44.6225	+	6.00448i	−3.44270	+	0.463255i
$169$	13.0000			1.00000
$170$		0			0
$171$			22.0947i			1.68962i
$172$		0			0
$173$	4.87147			0.370371			0.185186	−	0.982704i	$-0.440711\pi$
$173$	4.87147			0.370371			0.185186	+	0.982704i	$0.440711\pi$
$174$	32.6433	−	20.8398i	2.47468	−	1.57986i
$175$			26.1771i			1.97880i
$176$	12.4624	−	10.5384i	0.939392	−	0.794364i
$177$		0			0
$178$	−14.1775	−	22.2075i	−1.06265	−	1.66452i
$179$			16.6491i			1.24441i	0.782855	+	0.622204i	$0.213763\pi$
$179$			16.6491i			1.24441i	−0.782855	+	0.622204i	$0.786237\pi$
$180$		0			0
$181$	−25.6703			−1.90806			−0.954031	−	0.299709i	$-0.903110\pi$
$181$	−25.6703			−1.90806			−0.954031	+	0.299709i	$0.903110\pi$
$182$		0			0
$183$		−	46.5531i		−	3.44130i
$184$		0			0
$185$		0			0
$186$	−23.7890	−	37.2629i	−1.74429	−	2.73225i
$187$		0			0
$188$	−24.7305	−	11.4747i	−1.80366	−	0.836879i
$189$	51.6565			3.75746
$190$		0			0
$191$			11.7713i			0.851738i	0.904785	+	0.425869i	$0.140032\pi$
$191$			11.7713i			0.851738i	−0.904785	+	0.425869i	$0.859968\pi$
$192$	6.42988	+	23.4593i	0.464036	+	1.69303i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$	−14.5021	−	22.7159i	−1.04119	−	1.63091i
$195$		0			0
$196$	17.1803	−	37.0274i	1.22717	−	2.64482i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−30.3738	+	19.3909i	−2.15857	+	1.37805i
$199$		−	26.7544i		−	1.89657i	−0.317417	−	0.948286i	$-0.602816\pi$
$199$		−	26.7544i		−	1.89657i	0.317417	−	0.948286i	$-0.397184\pi$
$200$	14.0158	−	1.88599i	0.991068	−	0.133360i
$201$		0			0
$202$		0			0
$203$			47.1530i			3.30949i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	14.4356			0.998535
$210$		0			0
$211$			18.3271i			1.26169i	0.775908	+	0.630845i	$0.217292\pi$
$211$			18.3271i			1.26169i	−0.775908	+	0.630845i	$0.782708\pi$
$212$		0			0
$213$		0			0
$214$	4.64176	−	2.96335i	0.317304	−	0.202570i
$215$		0			0
$216$	−3.72172	−	27.6581i	−0.253231	−	1.88190i
$217$	53.8260			3.65395
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			15.3131i			1.02544i	0.858555	+	0.512722i	$0.171363\pi$
$223$			15.3131i			1.02544i	−0.858555	+	0.512722i	$0.828637\pi$
$224$	−28.2874	−	8.77102i	−1.89003	−	0.586038i
$225$	−31.2252			−2.08168
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−9.05537	+	19.5163i	−0.599707	+	1.29250i
$229$	−30.0934			−1.98862			−0.994312	−	0.106503i	$-0.966034\pi$
$229$	−30.0934			−1.98862			−0.994312	+	0.106503i	$0.966034\pi$
$230$		0			0
$231$		−	64.9514i		−	4.27349i
$232$	25.2469	−	3.39726i	1.65754	−	0.223041i
$233$	15.8768			1.04013			0.520063	−	0.854128i	$-0.325908\pi$
$233$	15.8768			1.04013			0.520063	+	0.854128i	$0.325908\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	21.3943i		−	1.38388i	−0.721955	−	0.691940i	$-0.756756\pi$
$239$		−	21.3943i		−	1.38388i	0.721955	−	0.691940i	$-0.243244\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	4.29822	+	6.73269i	0.276300	+	0.432794i
$243$			4.65291i			0.298484i
$244$	12.8882	−	27.7770i	0.825084	−	1.77824i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−3.87803	−	28.8197i	−0.246255	−	1.83005i
$249$		0			0
$250$		0			0
$251$			27.5184i			1.73695i	0.495736	+	0.868473i	$0.334898\pi$
$251$			27.5184i			1.73695i	−0.495736	+	0.868473i	$0.665102\pi$
$252$	59.3167	+	27.5223i	3.73660	+	1.73374i
$253$		0			0
$254$	2.70960	−	1.72984i	0.170015	−	0.108540i
$255$		0			0
$256$	−2.65818	+	15.7776i	−0.166136	+	0.986103i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−56.2463			−3.48156
$262$		0			0
$263$			3.15089i			0.194292i	0.995270	+	0.0971460i	$0.0309714\pi$
$263$			3.15089i			0.194292i	−0.995270	+	0.0971460i	$0.969029\pi$
$264$	−34.7765	+	4.67959i	−2.14035	+	0.288009i
$265$		0			0
$266$	−14.0956	−	22.0792i	−0.864258	−	1.35376i
$267$			56.6466i			3.46671i
$268$		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
$274$	11.6670	+	18.2750i	0.704828	+	1.10404i
$275$			20.4011i			1.23023i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$			64.2062i			3.84392i
$280$		0			0
$281$	−1.28741			−0.0768004			−0.0384002	−	0.999262i	$-0.512226\pi$
$281$	−1.28741			−0.0768004			−0.0384002	+	0.999262i	$0.512226\pi$
$282$	31.5411	+	49.4057i	1.87825	+	2.94207i
$283$			16.7869i			0.997879i	0.866637	+	0.498939i	$0.166277\pi$
$283$			16.7869i			0.997879i	−0.866637	+	0.498939i	$0.833723\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	10.4625	−	33.7425i	0.616507	−	1.98829i
$289$	17.0000			1.00000
$290$		0			0
$291$			57.9435i			3.39671i
$292$		0			0
$293$	34.1407			1.99452			0.997259	−	0.0739860i	$-0.0235720\pi$
$293$	34.1407			1.99452			0.997259	+	0.0739860i	$0.0235720\pi$
$294$	−73.9721	+	47.2246i	−4.31414	+	2.75419i
$295$		0			0
$296$		0			0
$297$	40.2585			2.33604
$298$		0			0
$299$		0			0
$300$	−27.5813	−	12.7974i	−1.59241	−	0.738861i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−10.8062	+	9.13788i	−0.619777	+	0.524093i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	17.9818	−	38.7547i	1.02461	−	2.20826i
$309$		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			$\pi$
$314$	−3.69408	−	5.78637i	−0.208469	−	0.326544i
$315$		0			0
$316$		0			0
$317$	−28.5891			−1.60573			−0.802863	−	0.596164i	$-0.796691\pi$
$317$	−28.5891			−1.60573			−0.802863	+	0.596164i	$0.796691\pi$
$318$		0			0
$319$			36.7487i			2.05753i
$320$		0			0
$321$	−11.8401			−0.660852
$322$		0			0
$323$		0			0
$324$	−9.48299	+	20.4379i	−0.526833	+	1.13544i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−71.3663			−3.93455
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$		0			0
$334$	−15.4042	+	9.83419i	−0.842879	+	0.538103i
$335$		0			0
$336$	41.1148	+	48.6211i	2.24299	+	2.65250i
$337$	−22.1353			−1.20579			−0.602894	−	0.797821i	$-0.705986\pi$
$337$	−22.1353			−1.20579			−0.602894	+	0.797821i	$0.705986\pi$
$338$	−9.89290	−	15.4961i	−0.538103	−	0.842879i
$339$		0			0
$340$		0			0
$341$	41.9493			2.27168
$342$	26.3371	−	16.8139i	1.42415	−	0.909192i
$343$		−	70.2044i		−	3.79068i
$344$		0			0
$345$		0			0
$346$	−3.70715	−	5.80685i	−0.199298	−	0.312178i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	−49.6826	−	23.0522i	−2.66326	−	1.23573i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	31.2033	−	19.9205i	1.66789	−	1.06480i
$351$		0			0
$352$	−22.0457	−	6.83570i	−1.17504	−	0.364344i
$353$	−4.39605			−0.233978			−0.116989	−	0.993133i	$-0.537324\pi$
$353$	−4.39605			−0.233978			−0.116989	+	0.993133i	$0.537324\pi$
$354$		0			0
$355$		0			0
$356$	−15.6826	+	33.7995i	−0.831177	+	1.79137i
$357$		0			0
$358$	19.8459	−	12.6698i	1.04889	−	0.669620i
$359$		−	7.30092i		−	0.385328i	−0.981265	−	0.192664i	$-0.938287\pi$
$359$		−	7.30092i		−	0.385328i	0.981265	−	0.192664i	$-0.0617127\pi$
$360$		0			0
$361$	6.48284			0.341202
$362$	19.5349	+	30.5993i	1.02673	+	1.60827i
$363$		−	17.1737i		−	0.901383i
$364$		0			0
$365$		0			0
$366$	−55.4918	+	35.4265i	−2.90060	+	1.85178i
$367$		−	26.7426i		−	1.39595i	−0.716122	−	0.697975i	$-0.754085\pi$
$367$		−	26.7426i		−	1.39595i	0.716122	−	0.697975i	$-0.245915\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−26.3145	+	56.7135i	−1.36434	+	2.94046i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	5.14177	+	38.2112i	0.265166	+	1.97059i
$377$		0			0
$378$	−39.3102	−	61.5751i	−2.02190	−	3.16708i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$	−6.91160			−0.354092
$382$	14.0315	−	8.95784i	0.717913	−	0.458323i
$383$			7.91812i			0.404597i	0.979324	+	0.202299i	$0.0648411\pi$
$383$			7.91812i			0.404597i	−0.979324	+	0.202299i	$0.935159\pi$
$384$	23.0707	−	25.5168i	1.17732	−	1.30215i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−16.0417	+	34.5733i	−0.814392	+	1.75519i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	−57.2112	+	7.69844i	−2.88960	+	0.388830i
$393$		0			0
$394$		0			0
$395$		0			0
$396$	46.2284	+	21.4495i	2.32307	+	1.07788i
$397$	4.60702			0.231220			0.115610	−	0.993295i	$-0.463118\pi$
$397$	4.60702			0.231220			0.115610	+	0.993295i	$0.463118\pi$
$398$	−31.8916	+	20.3599i	−1.59858	+	1.02055i
$399$			56.3194i			2.81950i
$400$	−12.9141	−	15.2718i	−0.645703	−	0.763589i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	56.2069	−	35.8831i	2.78950	−	1.78085i
$407$		0			0
$408$		0			0
$409$	32.5253			1.60827			0.804136	−	0.594445i	$-0.202628\pi$
$409$	32.5253			1.60827			0.804136	+	0.594445i	$0.202628\pi$
$410$		0			0
$411$		−	46.6157i		−	2.29938i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−10.9854	−	17.2075i	−0.537314	−	0.841644i
$419$			40.3795i			1.97267i	0.164756	+	0.986334i	$0.447316\pi$
$419$			40.3795i			1.97267i	−0.164756	+	0.986334i	$0.552684\pi$
$420$		0			0
$421$	−33.1670			−1.61646			−0.808231	−	0.588866i	$-0.799575\pi$
$421$	−33.1670			−1.61646			−0.808231	+	0.588866i	$0.799575\pi$
$422$	21.8461	−	13.9468i	1.06345	−	0.678920i
$423$		−	85.1290i		−	4.13912i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	80.1576i		−	3.87910i
$428$	−7.06469	−	3.27794i	−0.341484	−	0.158445i
$429$		0			0
$430$		0			0
$431$			39.6377i			1.90928i	0.297763	+	0.954640i	$0.403759\pi$
$431$			39.6377i			1.90928i	−0.297763	+	0.954640i	$0.596241\pi$
$432$	−30.1366	+	25.4840i	−1.44995	+	1.22610i
$433$	41.6056			1.99944			0.999718	−	0.0237503i	$-0.00756065\pi$
$433$	41.6056			1.99944			0.999718	+	0.0237503i	$0.00756065\pi$
$434$	−40.9612	−	64.1612i	−1.96620	−	3.07984i
$435$		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
$441$	127.458			6.06944
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$	18.2534	−	11.6532i	0.864326	−	0.551795i
$447$		0			0
$448$	11.0713	+	40.3935i	0.523070	+	1.90841i
$449$	15.6453			0.738345			0.369173	−	0.929361i	$-0.379641\pi$
$449$	15.6453			0.738345			0.369173	+	0.929361i	$0.379641\pi$
$450$	23.7621	+	37.2208i	1.12016	+	1.75460i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$	30.1547	−	4.05767i	1.41213	−	0.190018i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	22.9008	+	35.8716i	1.07008	+	1.67617i
$459$		0			0
$460$		0			0
$461$	−6.78005			−0.315778			−0.157889	−	0.987457i	$-0.550469\pi$
$461$	−6.78005			−0.315778			−0.157889	+	0.987457i	$0.550469\pi$
$462$	−77.4228	+	49.4275i	−3.60203	+	2.29958i
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$	−23.2622	−	27.5092i	−1.07992	−	1.27708i
$465$		0			0
$466$	−12.0822	−	18.9254i	−0.559695	−	0.876701i
$467$		−	35.3065i		−	1.63379i	−0.576786	−	0.816896i	$-0.695693\pi$
$467$		−	35.3065i		−	1.63379i	0.576786	−	0.816896i	$-0.304307\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$			14.7598i			0.680095i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	17.6898i		−	0.811664i
$476$		0			0
$477$		0			0
$478$	−25.5022	+	16.2809i	−1.16644	+	0.744670i
$479$		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
$484$	4.75453	−	10.2471i	0.216115	−	0.465775i
$485$		0			0
$486$	5.54632	−	3.54083i	0.251586	−	0.160615i
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	−42.9183	+	5.77516i	−1.94282	+	0.261429i
$489$		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
$496$	−31.4023	+	26.5543i	−1.41000	+	1.19232i
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		0			0
$501$	39.2928			1.75547
$502$	32.8023	−	20.9413i	1.46404	−	0.934656i
$503$			15.0925i			0.672942i	0.941694	+	0.336471i	$0.109233\pi$
$503$			15.0925i			0.672942i	−0.941694	+	0.336471i	$0.890767\pi$
$504$	−12.3326	−	91.6504i	−0.549339	−	4.08243i
$505$		0			0
$506$		0			0
$507$			39.5274i			1.75547i
$508$	−4.12397	−	1.91348i	−0.182971	−	0.0848968i
$509$	−16.0481			−0.711320			−0.355660	−	0.934615i	$-0.615744\pi$
$509$	−16.0481			−0.711320			−0.355660	+	0.934615i	$0.615744\pi$
$510$		0			0
$511$		0			0
$512$	20.8300	−	8.83809i	0.920564	−	0.390592i
$513$	−34.9082			−1.54123
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−55.6194			−2.44614
$518$		0			0
$519$			14.8120i			0.650176i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	42.8030	+	67.0462i	1.87344	+	2.93453i
$523$		−	22.7975i		−	0.996863i	−0.866929	−	0.498432i	$-0.833909\pi$
$523$		−	22.7975i		−	0.996863i	0.866929	−	0.498432i	$-0.166091\pi$
$524$		0			0
$525$	−79.5931			−3.47373
$526$	3.75589	−	2.39780i	0.163765	−	0.104549i
$527$		0			0
$528$	32.0428	+	37.8929i	1.39448	+	1.64907i
$529$	−23.0000			−1.00000
$530$		0			0
$531$		0			0
$532$	−15.5920	+	33.6043i	−0.676000	+	1.45693i
$533$		0			0
$534$	67.5234	−	43.1076i	2.92202	−	1.86545i
$535$		0			0
$536$		0			0
$537$	−50.6225			−2.18452
$538$		0			0
$539$		−	83.2753i		−	3.58692i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		−	78.0524i		−	3.34955i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	12.9056	−	27.8143i	0.551298	−	1.18817i
$549$	95.6157			4.08078
$550$	24.3183	−	15.5251i	1.03694	−	0.661992i
$551$		−	31.8648i		−	1.35749i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	46.7106			1.97919			0.989597	−	0.143869i	$-0.0459543\pi$
$557$	46.7106			1.97919			0.989597	+	0.143869i	$0.0459543\pi$
$558$	76.5344	−	48.8604i	3.23996	−	2.06843i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.979709	+	1.53461i	0.0413265	+	0.0647335i
$563$		−	43.9120i		−	1.85067i	−0.379151	−	0.925335i	$-0.623784\pi$
$563$		−	43.9120i		−	1.85067i	0.379151	−	0.925335i	$-0.376216\pi$
$564$	34.8896	−	75.1948i	1.46912	−	3.16627i
$565$		0			0
$566$	20.0102	−	12.7747i	0.841091	−	0.536961i
$567$			58.9790i			2.47688i
$568$		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
$573$	−35.7913			−1.49520
$574$		0			0
$575$		0			0
$576$	−48.1832	+	13.2064i	−2.00763	+	0.550265i
$577$	−30.5200			−1.27057			−0.635283	−	0.772280i	$-0.719116\pi$
$577$	−30.5200			−1.27057			−0.635283	+	0.772280i	$0.719116\pi$
$578$	−12.9369	−	20.2642i	−0.538103	−	0.842879i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	69.0693	−	44.0946i	2.86301	−	1.82778i
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−25.9808	−	40.6960i	−1.07326	−	1.68114i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	112.584	+	52.2380i	4.64290	+	2.15426i
$589$	−36.3743			−1.49878
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−30.6364	−	47.9886i	−1.25703	−	1.96900i
$595$		0			0
$596$		0			0
$597$	81.3486			3.32938
$598$		0			0
$599$			23.4943i			0.959953i	0.877281	+	0.479976i	$0.159355\pi$
$599$			23.4943i			0.959953i	−0.877281	+	0.479976i	$0.840645\pi$
$600$	5.73448	+	42.6160i	0.234109	+	1.73979i
$601$	34.7039			1.41560			0.707800	−	0.706413i	$-0.249688\pi$
$601$	34.7039			1.41560			0.707800	+	0.706413i	$0.249688\pi$
$602$		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	$0.5\pi$
$608$	19.1159	+	5.92724i	0.775251	+	0.240381i
$609$	−143.372			−5.80972
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−48.7838			−1.97036			−0.985180	−	0.171521i	$-0.945132\pi$
$613$	−48.7838			−1.97036			−0.985180	+	0.171521i	$0.945132\pi$
$614$		0			0
$615$		0			0
$616$	−59.8801	+	8.05757i	−2.41264	+	0.324649i
$617$	−49.6433			−1.99856			−0.999282	−	0.0378837i	$-0.987938\pi$
$617$	−49.6433			−1.99856			−0.999282	+	0.0378837i	$0.987938\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−30.8084	+	19.6684i	−1.23530	+	0.788630i
$623$			97.5372i			3.90774i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$			43.8925i			1.75290i
$628$	−4.08625	+	8.80676i	−0.163059	+	0.351428i
$629$		0			0
$630$		0			0
$631$		−	21.1736i		−	0.842909i	−0.906849	−	0.421455i	$-0.861520\pi$
$631$		−	21.1736i		−	0.842909i	0.906849	−	0.421455i	$-0.138480\pi$
$632$		0			0
$633$	−55.7248			−2.21486
$634$	21.7561	+	34.0786i	0.864046	+	1.35343i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	43.8049	−	27.9655i	1.73425	−	1.10717i
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$	9.01025	+	14.1136i	0.355606	+	0.557018i
$643$		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	$0.5\pi$
$648$	31.5787	−	4.24929i	1.24053	−	0.166928i
$649$		0			0
$650$		0			0
$651$			163.662i			6.41440i
$652$		0			0
$653$	46.3165			1.81251			0.906253	−	0.422735i	$-0.138930\pi$
$653$	46.3165			1.81251			0.906253	+	0.422735i	$0.138930\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	54.3093	+	85.0695i	2.11720	+	3.31635i
$659$		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
$668$	23.4449	+	10.8782i	0.907112	+	0.420890i
$669$	−46.5606			−1.80014
$670$		0			0
$671$		−	62.4709i		−	2.41166i
$672$	26.6689	−	86.0095i	1.02877	−	3.31789i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	16.8448	+	26.3856i	0.648838	+	1.01633i
$675$		−	49.3338i		−	1.89886i
$676$	−10.9431	+	23.5849i	−0.420890	+	0.907112i
$677$	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
$679$			99.7703i			3.82883i
$680$		0			0
$681$		0			0
$682$	−31.9231	−	50.0040i	−1.22240	−	1.91475i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	−40.0847	−	18.5989i	−1.53268	−	0.711146i
$685$		0			0
$686$	−83.6844	+	53.4250i	−3.19509	+	2.03978i
$687$		−	91.5008i		−	3.49097i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	−4.10071	+	8.83793i	−0.155886	+	0.335968i
$693$	133.404			5.06760
$694$		0			0
$695$		0			0
$696$	10.3296	+	76.7647i	0.391542	+	2.90976i
$697$		0			0
$698$		0			0
$699$			48.2746i			1.82591i
$700$	−47.4910	−	22.0353i	−1.79499	−	0.832858i
$701$	11.2463			0.424765			0.212383	−	0.977187i	$-0.431878\pi$
$701$	11.2463			0.424765			0.212383	+	0.977187i	$0.431878\pi$
$702$		0			0
$703$		0			0
$704$	8.62843	+	31.4807i	0.325196	+	1.18647i
$705$		0			0
$706$	3.34536	+	5.24014i	0.125904	+	0.197215i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	52.2237	−	7.02731i	1.95717	−	0.263360i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−30.2051	−	14.0149i	−1.12882	−	0.523760i
$717$	65.0507			2.42936
$718$	−8.70277	+	5.55594i	−0.324785	+	0.207346i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−4.93339	−	7.72762i	−0.183602	−	0.287592i
$723$		0			0
$724$	21.6088	−	46.5717i	0.803085	−	1.73082i
$725$	45.0328			1.67248
$726$	−20.4712	+	13.0690i	−0.759757	+	0.485037i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	19.6487			0.727730
$730$		0			0
$731$		0			0
$732$	84.4577	+	39.1875i	3.12165	+	1.44841i
$733$	13.2701			0.490143			0.245072	−	0.969505i	$-0.421189\pi$
$733$	13.2701			0.490143			0.245072	+	0.969505i	$0.421189\pi$
$734$	−31.8774	+	20.3509i	−1.17662	+	0.751165i
$735$		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	$-0.157224\pi$
$744$	87.6283	−	11.7914i	3.21261	−	0.432294i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−20.3870			−0.744924
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	41.6353	−	35.2075i	1.51828	−	1.28389i
$753$	−83.6715			−3.04916
$754$		0			0
$755$		0			0
$756$	−43.4835	+	93.7164i	−1.58148	+	3.40843i
$757$	39.2595			1.42691			0.713456	−	0.700700i	$-0.247129\pi$
$757$	39.2595			1.42691			0.713456	+	0.700700i	$0.247129\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	38.5480			1.39736			0.698681	−	0.715433i	$-0.253770\pi$
$761$	38.5480			1.39736			0.698681	+	0.715433i	$0.253770\pi$
$762$	5.25968	+	8.23871i	0.190538	+	0.298457i
$763$		0			0
$764$	−21.3557	−	9.90882i	−0.772622	−	0.358488i
$765$		0			0
$766$	9.43849	−	6.02563i	0.341026	−	0.217715i
$767$		0			0
$768$	−47.9730	−	8.08236i	−1.73108	−	0.291647i
$769$		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
$775$		−	51.4057i		−	1.84655i
$776$	53.4193	−	7.18820i	1.91764	−	0.258041i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	88.8655i		−	3.17579i
$784$	52.7140	+	62.3380i	1.88264	+	2.22636i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$	−9.58048			−0.341074
$790$		0			0
$791$		0			0
$792$	−9.61144	−	71.4277i	−0.341528	−	2.53807i
$793$		0			0
$794$	−3.50591	−	5.49162i	−0.124420	−	0.194890i
$795$		0			0
$796$	48.5385	+	22.5214i	1.72040	+	0.798249i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$	67.1334	−	42.8587i	2.37650	−	1.51718i
$799$		0			0
$800$	−8.37663	+	27.0154i	−0.296159	+	0.955139i
$801$	−116.347			−4.11091
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.5079			0.931968			0.465984	−	0.884793i	$-0.345700\pi$
$809$	26.5079			0.931968			0.465984	+	0.884793i	$0.345700\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−85.5461	−	39.6925i	−3.00208	−	1.39293i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−24.7515	−	38.7705i	−0.865416	−	1.35558i
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$	−55.5664	+	35.4742i	−1.93810	+	1.23731i
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$	−62.0309			−2.15964
$826$		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
$836$	−12.1516	+	26.1895i	−0.420274	+	0.905782i
$837$	−101.442			−3.50633
$838$	48.1328	−	30.7285i	1.66272	−	1.06150i
$839$		−	57.8811i		−	1.99828i	−0.0415081	−	0.999138i	$-0.513216\pi$
$839$		−	57.8811i		−	1.99828i	0.0415081	−	0.999138i	$-0.486784\pi$
$840$		0			0
$841$	52.1181			1.79718
$842$	25.2399	+	39.5355i	0.869823	+	1.36248i
$843$		−	3.91445i		−	0.134821i
$844$	−33.2495	−	15.4274i	−1.14449	−	0.531034i
$845$		0			0
$846$	−101.475	+	64.7826i	−3.48877	+	2.22727i
$847$		−	29.5705i		−	1.01606i
$848$		0			0
$849$	−51.0417			−1.75175
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−50.4362			−1.72690			−0.863451	−	0.504432i	$-0.831702\pi$
$853$	−50.4362			−1.72690			−0.863451	+	0.504432i	$0.831702\pi$
$854$	−95.5488	+	60.9994i	−3.26961	+	2.08736i
$855$		0			0
$856$	1.46883	+	10.9157i	0.0502036	+	0.373090i
$857$	54.5857			1.86461			0.932306	−	0.361670i	$-0.117793\pi$
$857$	54.5857			1.86461			0.932306	+	0.361670i	$0.117793\pi$
$858$		0			0
$859$			9.25948i			0.315929i	0.987445	+	0.157965i	$0.0504932\pi$
$859$			9.25948i			0.315929i	−0.987445	+	0.157965i	$0.949507\pi$
$860$		0			0
$861$		0			0
$862$	47.2485	−	30.1640i	1.60929	−	1.02739i
$863$			14.9798i			0.509919i	0.966952	+	0.254960i	$0.0820622\pi$
$863$			14.9798i			0.509919i	−0.966952	+	0.254960i	$0.917938\pi$
$864$	53.3109	+	16.5300i	1.81367	+	0.562363i
$865$		0			0
$866$	−31.6615	−	49.5943i	−1.07590	−	1.68528i
$867$			51.6896i			1.75547i
$868$	−45.3097	+	97.6524i	−1.53791	+	3.31454i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−119.011			−4.02790
$874$		0			0
$875$		0			0
$876$		0			0
$877$	45.1406			1.52429			0.762144	−	0.647407i	$-0.224147\pi$
$877$	45.1406			1.52429			0.762144	+	0.647407i	$0.224147\pi$
$878$		0			0
$879$			103.807i			3.50132i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−96.9948	−	151.932i	−3.26599	−	5.11581i
$883$			1.21134i			0.0407650i	0.999792	+	0.0203825i	$0.00648840\pi$
$883$			1.21134i			0.0407650i	−0.999792	+	0.0203825i	$0.993512\pi$
$884$		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
$889$	−11.9008			−0.399139
$890$		0			0
$891$			45.9653i			1.53989i
$892$	−27.7815	−	12.8903i	−0.930192	−	0.431600i
$893$	48.2276			1.61387
$894$		0			0
$895$		0			0
$896$	39.7243	−	43.9363i	1.32710	−	1.46781i
$897$		0			0
$898$	−11.9059	−	18.6493i	−0.397306	−	0.622336i
$899$		−	92.5977i		−	3.08831i
$900$	26.2848	−	56.6495i	0.876159	−	1.88832i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	21.5046i		−	0.714049i	−0.934095	−	0.357024i	$-0.883791\pi$
$907$		−	21.5046i		−	0.714049i	0.934095	−	0.357024i	$-0.116209\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	34.8592i		−	1.15494i	−0.816413	−	0.577469i	$-0.804040\pi$
$911$		−	34.8592i		−	1.15494i	0.816413	−	0.577469i	$-0.195960\pi$
$912$	−27.7843	−	32.8569i	−0.920031	−	1.08800i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	25.3320	−	54.5961i	0.836993	−	1.80390i
$917$		0			0
$918$		0			0
$919$		−	12.5907i		−	0.415330i	−0.978200	−	0.207665i	$-0.933414\pi$
$919$		−	12.5907i		−	0.415330i	0.978200	−	0.207665i	$-0.0665863\pi$
$920$		0			0
$921$		0			0
$922$	5.15957	+	8.08189i	0.169921	+	0.266163i
$923$		0			0
$924$	117.836	+	54.6748i	3.87653	+	1.79867i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−15.0889	+	48.6632i	−0.495318	+	1.59745i
$929$	52.4045			1.71934			0.859668	−	0.510854i	$-0.170671\pi$
$929$	52.4045			1.71934			0.859668	+	0.510854i	$0.170671\pi$
$930$		0			0
$931$			72.2081i			2.36653i
$932$	−13.3648	+	28.8041i	−0.437779	+	0.943510i
$933$	78.5855			2.57277
$934$	−42.0858	+	26.8680i	−1.37709	+	0.879148i
$935$		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			$\pi$
$942$	17.5938	−	11.2321i	0.573238	−	0.365961i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$			15.4235i			0.501195i	0.968091	+	0.250598i	$0.0806271\pi$
$947$			15.4235i			0.501195i	−0.968091	+	0.250598i	$0.919373\pi$
$948$		0			0
$949$		0			0
$950$	−21.0864	+	13.4618i	−0.684134	+	0.436759i
$951$		−	86.9271i		−	2.81880i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	38.8140	+	18.0093i	1.25533	+	0.582462i
$957$	−111.737			−3.61194
$958$		0			0
$959$		−	80.2655i		−	2.59191i
$960$		0			0
$961$	−74.7019			−2.40974
$962$		0			0
$963$		−	24.3185i		−	0.783654i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			46.9354i			1.50934i	0.656104	+	0.754670i	$0.272203\pi$
$967$			46.9354i			1.50934i	−0.656104	+	0.754670i	$0.727797\pi$
$968$	−15.8328	+	2.13048i	−0.508884	+	0.0684764i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−8.44142	−	3.91673i	−0.270759	−	0.125629i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	39.5446	+	46.7642i	1.26579	+	1.49689i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$			76.0156i			2.42947i
$980$		0			0
$981$		0			0
$982$	30.8084	−	19.6684i	0.983134	−	0.627643i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	216.994i		−	6.90700i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	55.5499	+	17.2243i	1.76371	+	0.546871i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−20.8723			−0.661033			−0.330517	−	0.943800i	$-0.607223\pi$
$997$	−20.8723			−0.661033			−0.330517	+	0.943800i	$0.607223\pi$
$998$		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.667.7	✓	22
4.3	odd	2	inner	668.2.b.a.667.8	yes	22
167.166	odd	2	CM	668.2.b.a.667.7	✓	22
668.667	even	2	inner	668.2.b.a.667.8	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.b.a.667.7	✓	22	1.1	even	1	trivial
668.2.b.a.667.7	✓	22	167.166	odd	2	CM
668.2.b.a.667.8	yes	22	4.3	odd	2	inner
668.2.b.a.667.8	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)

\(f(q)\)	\(=\)	\(q+(-0.760993 - 1.19201i) q^{2} +3.04057i q^{3} +(-0.841781 + 1.81422i) q^{4} +(3.62439 - 2.31385i) q^{6} +5.23541i q^{7} +(2.80316 - 0.377198i) q^{8} -6.24504 q^{9} +O(q^{10})\)	\(q+(-0.760993 - 1.19201i) q^{2} +3.04057i q^{3} +(-0.841781 + 1.81422i) q^{4} +(3.62439 - 2.31385i) q^{6} +5.23541i q^{7} +(2.80316 - 0.377198i) q^{8} -6.24504 q^{9} +4.08022i q^{11} +(-5.51626 - 2.55949i) q^{12} +(6.24067 - 3.98411i) q^{14} +(-2.58281 - 3.05436i) q^{16} +(4.75243 + 7.44416i) q^{18} -3.53796i q^{19} -15.9186 q^{21} +(4.86366 - 3.10502i) q^{22} +(1.14690 + 8.52320i) q^{24} +5.00000 q^{25} -9.86675i q^{27} +(-9.49820 - 4.40707i) q^{28} +9.00656 q^{29} -10.2811i q^{31} +(-1.67533 + 5.40308i) q^{32} -12.4062 q^{33} +(5.25695 - 11.3299i) q^{36} +(-4.21729 + 2.69236i) q^{38} +(12.1139 + 18.9752i) q^{42} +(-7.40243 - 3.43465i) q^{44} +13.6315i q^{47} +(9.28697 - 7.85321i) q^{48} -20.4095 q^{49} +(-3.80496 - 5.96006i) q^{50} +(-11.7613 + 7.50853i) q^{54} +(1.97479 + 14.6757i) q^{56} +10.7574 q^{57} +(-6.85392 - 10.7359i) q^{58} -15.3107 q^{61} +(-12.2552 + 7.82388i) q^{62} -32.6953i q^{63} +(7.71544 - 2.11470i) q^{64} +(9.44100 + 14.7883i) q^{66} +(-17.5059 + 2.35562i) q^{72} +15.2028i q^{75} +(6.41865 + 2.97819i) q^{76} -21.3616 q^{77} +11.2654 q^{81} +(13.4000 - 28.8799i) q^{84} +27.3850i q^{87} +(1.53905 + 11.4375i) q^{88} +18.6303 q^{89} +31.2605 q^{93} +(16.2489 - 10.3734i) q^{94} +(-16.4284 - 5.09394i) q^{96} +19.0568 q^{97} +(15.5315 + 24.3284i) q^{98} -25.4811i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 66 q^{9}+O(q^{10}) \) 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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