LMFDB - Embedded newform 960.2.b.c.671.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(671,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.671");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 $ x^12 - 16x^8 - 24x^7 + 96x^5 + 304x^4 + 384x^3 + 288x^2 + 144*x + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			671.3
Root			$-1.50511 - 0.403293i$ of defining polynomial
Character	$\chi$	$=$	960.671
Dual form			960.2.b.c.671.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.10182 - 1.33641i) q^{3} -1.00000 q^{5} +4.67282i q^{7} +(-0.571993 + 2.94497i) q^{9} +O(q^{10})$	\(q+(-1.10182 - 1.33641i) q^{3} -1.00000 q^{5} +4.67282i q^{7} +(-0.571993 + 2.94497i) q^{9} -1.14399i q^{11} -2.42583i q^{13} +(1.10182 + 1.33641i) q^{15} -7.87137i q^{17} -2.42583 q^{19} +(6.24482 - 5.14860i) q^{21} -4.62947 q^{23} +1.00000 q^{25} +(4.56592 - 2.48040i) q^{27} +6.48963 q^{29} -3.14399i q^{31} +(-1.52884 + 1.26047i) q^{33} -4.67282i q^{35} -6.83310i q^{37} +(-3.24191 + 2.67282i) q^{39} -5.88993i q^{41} -6.61091 q^{43} +(0.571993 - 2.94497i) q^{45} +7.15040 q^{47} -14.8353 q^{49} +(-10.5194 + 8.67282i) q^{51} -9.05767 q^{53} +1.14399i q^{55} +(2.67282 + 3.24191i) q^{57} -6.48963i q^{59} +1.48266i q^{61} +(-13.7613 - 2.67282i) q^{63} +2.42583i q^{65} -13.9835 q^{67} +(5.10083 + 6.18687i) q^{69} +2.52093 q^{71} +5.63362 q^{73} +(-1.10182 - 1.33641i) q^{75} +5.34565 q^{77} +6.77761i q^{79} +(-8.34565 - 3.36900i) q^{81} -14.6728i q^{83} +7.87137i q^{85} +(-7.15040 - 8.67282i) q^{87} +6.73800i q^{89} +11.3355 q^{91} +(-4.20166 + 3.46410i) q^{93} +2.42583 q^{95} -3.34565 q^{97} +(3.36900 + 0.654353i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{5} - 4 q^{9}+O(q^{10}) $ 12 * q - 12 * q^5 - 4 * q^9	$ 12 q - 12 q^{5} - 4 q^{9} + 32 q^{21} + 12 q^{25} - 8 q^{29} + 16 q^{33} + 4 q^{45} - 12 q^{49} - 40 q^{53} - 8 q^{57} + 24 q^{69} - 24 q^{73} - 16 q^{77} - 20 q^{81} + 24 q^{93} + 40 q^{97}+O(q^{100}) $ 12 * q - 12 * q^5 - 4 * q^9 + 32 * q^21 + 12 * q^25 - 8 * q^29 + 16 * q^33 + 4 * q^45 - 12 * q^49 - 40 * q^53 - 8 * q^57 + 24 * q^69 - 24 * q^73 - 16 * q^77 - 20 * q^81 + 24 * q^93 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.10182	−	1.33641i	−0.636135	−	0.771578i
$3$	−1.10182	−	1.33641i	−0.636135	−	0.771578i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			4.67282i			1.76616i	0.469221	+	0.883081i	$0.344535\pi$
$7$			4.67282i			1.76616i	−0.469221	+	0.883081i	$0.655465\pi$
$8$		0			0
$9$	−0.571993	+	2.94497i	−0.190664	+	0.981655i
$10$		0			0
$11$		−	1.14399i		−	0.344925i	−0.985016	−	0.172462i	$-0.944828\pi$
$11$		−	1.14399i		−	0.344925i	0.985016	−	0.172462i	$-0.0551723\pi$
$12$		0			0
$13$		−	2.42583i		−	0.672804i	−0.941718	−	0.336402i	$-0.890790\pi$
$13$		−	2.42583i		−	0.672804i	0.941718	−	0.336402i	$-0.109210\pi$
$14$		0			0
$15$	1.10182	+	1.33641i	0.284488	+	0.345060i
$16$		0			0
$17$		−	7.87137i		−	1.90909i	−0.298069	−	0.954544i	$-0.596343\pi$
$17$		−	7.87137i		−	1.90909i	0.298069	−	0.954544i	$-0.403657\pi$
$18$		0			0
$19$	−2.42583			−0.556524			−0.278262	−	0.960505i	$-0.589758\pi$
$19$	−2.42583			−0.556524			−0.278262	+	0.960505i	$0.589758\pi$
$20$		0			0
$21$	6.24482	−	5.14860i	1.36273	−	1.12352i
$22$		0			0
$23$	−4.62947			−0.965310			−0.482655	−	0.875810i	$-0.660328\pi$
$23$	−4.62947			−0.965310			−0.482655	+	0.875810i	$0.660328\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.56592	−	2.48040i	0.878712	−	0.477353i
$28$		0			0
$29$	6.48963			1.20509			0.602547	−	0.798083i	$-0.294152\pi$
$29$	6.48963			1.20509			0.602547	+	0.798083i	$0.294152\pi$
$30$		0			0
$31$		−	3.14399i		−	0.564677i	−0.959315	−	0.282338i	$-0.908890\pi$
$31$		−	3.14399i		−	0.564677i	0.959315	−	0.282338i	$-0.0911101\pi$
$32$		0			0
$33$	−1.52884	+	1.26047i	−0.266136	+	0.219419i
$34$		0			0
$35$		−	4.67282i		−	0.789851i
$36$		0			0
$37$		−	6.83310i		−	1.12336i	−0.827356	−	0.561678i	$-0.810156\pi$
$37$		−	6.83310i		−	1.12336i	0.827356	−	0.561678i	$-0.189844\pi$
$38$		0			0
$39$	−3.24191	+	2.67282i	−0.519121	+	0.427994i
$40$		0			0
$41$		−	5.88993i		−	0.919853i	−0.887957	−	0.459926i	$-0.847876\pi$
$41$		−	5.88993i		−	0.919853i	0.887957	−	0.459926i	$-0.152124\pi$
$42$		0			0
$43$	−6.61091			−1.00815			−0.504077	−	0.863659i	$-0.668167\pi$
$43$	−6.61091			−1.00815			−0.504077	+	0.863659i	$0.668167\pi$
$44$		0			0
$45$	0.571993	−	2.94497i	0.0852677	−	0.439010i
$46$		0			0
$47$	7.15040			1.04299			0.521496	−	0.853254i	$-0.325374\pi$
$47$	7.15040			1.04299			0.521496	+	0.853254i	$0.325374\pi$
$48$		0			0
$49$	−14.8353			−2.11933
$50$		0			0
$51$	−10.5194	+	8.67282i	−1.47301	+	1.21444i
$52$		0			0
$53$	−9.05767			−1.24417			−0.622084	−	0.782951i	$-0.713714\pi$
$53$	−9.05767			−1.24417			−0.622084	+	0.782951i	$0.713714\pi$
$54$		0			0
$55$			1.14399i			0.154255i
$56$		0			0
$57$	2.67282	+	3.24191i	0.354024	+	0.429401i
$58$		0			0
$59$		−	6.48963i		−	0.844878i	−0.906391	−	0.422439i	$-0.861174\pi$
$59$		−	6.48963i		−	0.844878i	0.906391	−	0.422439i	$-0.138826\pi$
$60$		0			0
$61$			1.48266i			0.189835i	0.995485	+	0.0949175i	$0.0302587\pi$
$61$			1.48266i			0.189835i	−0.995485	+	0.0949175i	$0.969741\pi$
$62$		0			0
$63$	−13.7613	−	2.67282i	−1.73376	−	0.336744i
$64$		0			0
$65$			2.42583i			0.300887i
$66$		0			0
$67$	−13.9835			−1.70836			−0.854178	−	0.519980i	$-0.825939\pi$
$67$	−13.9835			−1.70836			−0.854178	+	0.519980i	$0.825939\pi$
$68$		0			0
$69$	5.10083	+	6.18687i	0.614068	+	0.744812i
$70$		0			0
$71$	2.52093			0.299179			0.149590	−	0.988748i	$-0.452205\pi$
$71$	2.52093			0.299179			0.149590	+	0.988748i	$0.452205\pi$
$72$		0			0
$73$	5.63362			0.659365			0.329683	−	0.944092i	$-0.393058\pi$
$73$	5.63362			0.659365			0.329683	+	0.944092i	$0.393058\pi$
$74$		0			0
$75$	−1.10182	−	1.33641i	−0.127227	−	0.154316i
$76$		0			0
$77$	5.34565			0.609193
$78$		0			0
$79$			6.77761i			0.762540i	0.924464	+	0.381270i	$0.124513\pi$
$79$			6.77761i			0.762540i	−0.924464	+	0.381270i	$0.875487\pi$
$80$		0			0
$81$	−8.34565	−	3.36900i	−0.927294	−	0.374333i
$82$		0			0
$83$		−	14.6728i		−	1.61055i	−0.592900	−	0.805276i	$-0.702017\pi$
$83$		−	14.6728i		−	1.61055i	0.592900	−	0.805276i	$-0.297983\pi$
$84$		0			0
$85$			7.87137i			0.853770i
$86$		0			0
$87$	−7.15040	−	8.67282i	−0.766603	−	0.929824i
$88$		0			0
$89$			6.73800i			0.714227i	0.934061	+	0.357113i	$0.116239\pi$
$89$			6.73800i			0.714227i	−0.934061	+	0.357113i	$0.883761\pi$
$90$		0			0
$91$	11.3355			1.18828
$92$		0			0
$93$	−4.20166	+	3.46410i	−0.435692	+	0.359211i
$94$		0			0
$95$	2.42583			0.248885
$96$		0			0
$97$	−3.34565			−0.339699			−0.169850	−	0.985470i	$-0.554328\pi$
$97$	−3.34565			−0.339699			−0.169850	+	0.985470i	$0.554328\pi$
$98$		0			0
$99$	3.36900	+	0.654353i	0.338597	+	0.0657649i
$100$		0			0
$101$	4.20166			0.418081			0.209040	−	0.977907i	$-0.432966\pi$
$101$	4.20166			0.418081			0.209040	+	0.977907i	$0.432966\pi$
$102$		0			0
$103$		−	6.96080i		−	0.685868i	−0.939360	−	0.342934i	$-0.888579\pi$
$103$		−	6.96080i		−	0.685868i	0.939360	−	0.342934i	$-0.111421\pi$
$104$		0			0
$105$	−6.24482	+	5.14860i	−0.609432	+	0.502452i
$106$		0			0
$107$			2.67282i			0.258392i	0.991619	+	0.129196i	$0.0412396\pi$
$107$			2.67282i			0.258392i	−0.991619	+	0.129196i	$0.958760\pi$
$108$		0			0
$109$		−	8.41086i		−	0.805614i	−0.915285	−	0.402807i	$-0.868034\pi$
$109$		−	8.41086i		−	0.805614i	0.915285	−	0.402807i	$-0.131966\pi$
$110$		0			0
$111$	−9.13184	+	7.52884i	−0.866756	+	0.714606i
$112$		0			0
$113$		−	17.1303i		−	1.61148i	−0.592267	−	0.805742i	$-0.701767\pi$
$113$		−	17.1303i		−	1.61148i	0.592267	−	0.805742i	$-0.298233\pi$
$114$		0			0
$115$	4.62947			0.431700
$116$		0			0
$117$	7.14399	+	1.38756i	0.660462	+	0.128280i
$118$		0			0
$119$	36.7815			3.37176
$120$		0			0
$121$	9.69129			0.881027
$122$		0			0
$123$	−7.87137	+	6.48963i	−0.709738	+	0.585151i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	2.96080i		−	0.262728i	−0.991334	−	0.131364i	$-0.958064\pi$
$127$		−	2.96080i		−	0.262728i	0.991334	−	0.131364i	$-0.0419357\pi$
$128$		0			0
$129$	7.28402	+	8.83490i	0.641322	+	0.777869i
$130$		0			0
$131$			11.8353i			1.03405i	0.855969	+	0.517027i	$0.172961\pi$
$131$			11.8353i			1.03405i	−0.855969	+	0.517027i	$0.827039\pi$
$132$		0			0
$133$		−	11.3355i		−	0.982910i
$134$		0			0
$135$	−4.56592	+	2.48040i	−0.392972	+	0.213479i
$136$		0			0
$137$			5.35044i			0.457119i	0.973530	+	0.228560i	$0.0734016\pi$
$137$			5.35044i			0.457119i	−0.973530	+	0.228560i	$0.926598\pi$
$138$		0			0
$139$	18.1686			1.54104			0.770519	−	0.637417i	$-0.219997\pi$
$139$	18.1686			1.54104			0.770519	+	0.637417i	$0.219997\pi$
$140$		0			0
$141$	−7.87844	−	9.55588i	−0.663484	−	0.804750i
$142$		0			0
$143$	−2.77512			−0.232067
$144$		0			0
$145$	−6.48963			−0.538935
$146$		0			0
$147$	16.3458	+	19.8260i	1.34818	+	1.63522i
$148$		0			0
$149$	−8.28797			−0.678977			−0.339489	−	0.940610i	$-0.610254\pi$
$149$	−8.28797			−0.678977			−0.339489	+	0.940610i	$0.610254\pi$
$150$		0			0
$151$		−	9.83528i		−	0.800384i	−0.916431	−	0.400192i	$-0.868943\pi$
$151$		−	9.83528i		−	0.800384i	0.916431	−	0.400192i	$-0.131057\pi$
$152$		0			0
$153$	23.1809	+	4.50237i	1.87407	+	0.363995i
$154$		0			0
$155$			3.14399i			0.252531i
$156$		0			0
$157$			13.7613i			1.09827i	0.835733	+	0.549136i	$0.185043\pi$
$157$			13.7613i			1.09827i	−0.835733	+	0.549136i	$0.814957\pi$
$158$		0			0
$159$	9.97991	+	12.1048i	0.791458	+	0.959972i
$160$		0			0
$161$		−	21.6327i		−	1.70489i
$162$		0			0
$163$	9.57623			0.750068			0.375034	−	0.927011i	$-0.377631\pi$
$163$	9.57623			0.750068			0.375034	+	0.927011i	$0.377631\pi$
$164$		0			0
$165$	1.52884	−	1.26047i	0.119020	−	0.0981271i
$166$		0			0
$167$	−16.4093			−1.26979			−0.634896	−	0.772598i	$-0.718957\pi$
$167$	−16.4093			−1.26979			−0.634896	+	0.772598i	$0.718957\pi$
$168$		0			0
$169$	7.11535			0.547335
$170$		0			0
$171$	1.38756	−	7.14399i	0.106109	−	0.546314i
$172$		0			0
$173$	−21.2672			−1.61692			−0.808459	−	0.588552i	$-0.799698\pi$
$173$	−21.2672			−1.61692			−0.808459	+	0.588552i	$0.799698\pi$
$174$		0			0
$175$			4.67282i			0.353232i
$176$		0			0
$177$	−8.67282	+	7.15040i	−0.651889	+	0.537457i
$178$		0			0
$179$		−	16.2017i		−	1.21097i	−0.795857	−	0.605484i	$-0.792979\pi$
$179$		−	16.2017i		−	1.21097i	0.795857	−	0.605484i	$-0.207021\pi$
$180$		0			0
$181$			13.6662i			1.01580i	0.861416	+	0.507901i	$0.169578\pi$
$181$			13.6662i			1.01580i	−0.861416	+	0.507901i	$0.830422\pi$
$182$		0			0
$183$	1.98144	−	1.63362i	0.146472	−	0.120761i
$184$		0			0
$185$			6.83310i			0.502380i
$186$		0			0
$187$	−9.00475			−0.658492
$188$		0			0
$189$	11.5905	+	21.3357i	0.843082	+	1.55195i
$190$		0			0
$191$	6.48382			0.469152			0.234576	−	0.972098i	$-0.424630\pi$
$191$	6.48382			0.469152			0.234576	+	0.972098i	$0.424630\pi$
$192$		0			0
$193$	11.9216			0.858135			0.429068	−	0.903272i	$-0.358842\pi$
$193$	11.9216			0.858135			0.429068	+	0.903272i	$0.358842\pi$
$194$		0			0
$195$	3.24191	−	2.67282i	0.232158	−	0.191405i
$196$		0			0
$197$	−24.3249			−1.73308			−0.866539	−	0.499109i	$-0.833661\pi$
$197$	−24.3249			−1.73308			−0.866539	+	0.499109i	$0.833661\pi$
$198$		0			0
$199$			15.1809i			1.07615i	0.842898	+	0.538074i	$0.180848\pi$
$199$			15.1809i			1.07615i	−0.842898	+	0.538074i	$0.819152\pi$
$200$		0			0
$201$	15.4073	+	18.6877i	1.08675	+	1.31813i
$202$		0			0
$203$			30.3249i			2.12839i
$204$		0			0
$205$			5.88993i			0.411371i
$206$		0			0
$207$	2.64802	−	13.6336i	0.184050	−	0.947602i
$208$		0			0
$209$			2.77512i			0.191959i
$210$		0			0
$211$	2.87022			0.197594			0.0987970	−	0.995108i	$-0.468501\pi$
$211$	2.87022			0.197594			0.0987970	+	0.995108i	$0.468501\pi$
$212$		0			0
$213$	−2.77761	−	3.36900i	−0.190318	−	0.230840i
$214$		0			0
$215$	6.61091			0.450860
$216$		0			0
$217$	14.6913			0.997310
$218$		0			0
$219$	−6.20723	−	7.52884i	−0.419445	−	0.508752i
$220$		0			0
$221$	−19.0946			−1.28444
$222$		0			0
$223$		−	13.6521i		−	0.914212i	−0.889412	−	0.457106i	$-0.848886\pi$
$223$		−	13.6521i		−	0.914212i	0.889412	−	0.457106i	$-0.151114\pi$
$224$		0			0
$225$	−0.571993	+	2.94497i	−0.0381329	+	0.196331i
$226$		0			0
$227$		−	9.32718i		−	0.619066i	−0.950889	−	0.309533i	$-0.899827\pi$
$227$		−	9.32718i		−	0.619066i	0.950889	−	0.309533i	$-0.100173\pi$
$228$		0			0
$229$			6.73800i			0.445260i	0.974903	+	0.222630i	$0.0714641\pi$
$229$			6.73800i			0.445260i	−0.974903	+	0.222630i	$0.928536\pi$
$230$		0			0
$231$	−5.88993	−	7.14399i	−0.387529	−	0.470040i
$232$		0			0
$233$		−	15.6884i		−	1.02778i	−0.857857	−	0.513889i	$-0.828204\pi$
$233$		−	15.6884i		−	1.02778i	0.857857	−	0.513889i	$-0.171796\pi$
$234$		0			0
$235$	−7.15040			−0.466440
$236$		0			0
$237$	9.05767	−	7.46769i	0.588359	−	0.485079i
$238$		0			0
$239$	−9.00475			−0.582469			−0.291234	−	0.956652i	$-0.594066\pi$
$239$	−9.00475			−0.582469			−0.291234	+	0.956652i	$0.594066\pi$
$240$		0			0
$241$	7.43196			0.478735			0.239367	−	0.970929i	$-0.423060\pi$
$241$	7.43196			0.478735			0.239367	+	0.970929i	$0.423060\pi$
$242$		0			0
$243$	4.69301	+	14.8652i	0.301057	+	0.953606i
$244$		0			0
$245$	14.8353			0.947791
$246$		0			0
$247$			5.88465i			0.374431i
$248$		0			0
$249$	−19.6089	+	16.1668i	−1.24267	+	1.02453i
$250$		0			0
$251$			3.43196i			0.216623i	0.994117	+	0.108312i	$0.0345444\pi$
$251$			3.43196i			0.216623i	−0.994117	+	0.108312i	$0.965456\pi$
$252$		0			0
$253$			5.29605i			0.332960i
$254$		0			0
$255$	10.5194	−	8.67282i	0.658750	−	0.543113i
$256$		0			0
$257$		−	1.38756i		−	0.0865535i	−0.999063	−	0.0432768i	$-0.986220\pi$
$257$		−	1.38756i		−	0.0865535i	0.999063	−	0.0432768i	$-0.0137797\pi$
$258$		0			0
$259$	31.9299			1.98403
$260$		0			0
$261$	−3.71203	+	19.1118i	−0.229769	+	1.18299i
$262$		0			0
$263$	−6.07142			−0.374380			−0.187190	−	0.982324i	$-0.559938\pi$
$263$	−6.07142			−0.374380			−0.187190	+	0.982324i	$0.559938\pi$
$264$		0			0
$265$	9.05767			0.556409
$266$		0			0
$267$	9.00475	−	7.42405i	0.551082	−	0.454345i
$268$		0			0
$269$	2.40332			0.146533			0.0732666	−	0.997312i	$-0.476658\pi$
$269$	2.40332			0.146533			0.0732666	+	0.997312i	$0.476658\pi$
$270$		0			0
$271$		−	19.5473i		−	1.18741i	−0.804681	−	0.593707i	$-0.797664\pi$
$271$		−	19.5473i		−	1.18741i	0.804681	−	0.593707i	$-0.202336\pi$
$272$		0			0
$273$	−12.4896	−	15.1489i	−0.755907	−	0.916851i
$274$		0			0
$275$		−	1.14399i		−	0.0689850i
$276$		0			0
$277$			18.8032i			1.12977i	0.825169	+	0.564886i	$0.191080\pi$
$277$			18.8032i			1.12977i	−0.825169	+	0.564886i	$0.808920\pi$
$278$		0			0
$279$	9.25893	+	1.79834i	0.554318	+	0.107664i
$280$		0			0
$281$			9.85282i			0.587770i	0.955841	+	0.293885i	$0.0949482\pi$
$281$			9.85282i			0.587770i	−0.955841	+	0.293885i	$0.905052\pi$
$282$		0			0
$283$	−2.64802			−0.157409			−0.0787043	−	0.996898i	$-0.525078\pi$
$283$	−2.64802			−0.157409			−0.0787043	+	0.996898i	$0.525078\pi$
$284$		0			0
$285$	−2.67282	−	3.24191i	−0.158324	−	0.192034i
$286$		0			0
$287$	27.5226			1.62461
$288$		0			0
$289$	−44.9585			−2.64462
$290$		0			0
$291$	3.68630	+	4.47116i	0.216094	+	0.262104i
$292$		0			0
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$			6.48963i			0.377841i
$296$		0			0
$297$	−2.83754	−	5.22335i	−0.164651	−	0.303090i
$298$		0			0
$299$			11.2303i			0.649465i
$300$		0			0
$301$		−	30.8916i		−	1.78056i
$302$		0			0
$303$	−4.62947	−	5.61515i	−0.265956	−	0.322582i
$304$		0			0
$305$		−	1.48266i		−	0.0848968i
$306$		0			0
$307$	0.127093			0.00725359			0.00362680	−	0.999993i	$-0.498846\pi$
$307$	0.127093			0.00725359			0.00362680	+	0.999993i	$0.498846\pi$
$308$		0			0
$309$	−9.30249	+	7.66953i	−0.529200	+	0.436304i
$310$		0			0
$311$	−14.3008			−0.810924			−0.405462	−	0.914112i	$-0.632889\pi$
$311$	−14.3008			−0.810924			−0.405462	+	0.914112i	$0.632889\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$	13.7613	+	2.67282i	0.775362	+	0.150597i
$316$		0			0
$317$	1.63362			0.0917533			0.0458766	−	0.998947i	$-0.485392\pi$
$317$	1.63362			0.0917533			0.0458766	+	0.998947i	$0.485392\pi$
$318$		0			0
$319$		−	7.42405i		−	0.415667i
$320$		0			0
$321$	3.57199	−	2.94497i	0.199369	−	0.164372i
$322$		0			0
$323$			19.0946i			1.06245i
$324$		0			0
$325$		−	2.42583i		−	0.134561i
$326$		0			0
$327$	−11.2404	+	9.26724i	−0.621594	+	0.512480i
$328$		0			0
$329$			33.4125i			1.84209i
$330$		0			0
$331$	−34.3557			−1.88836			−0.944180	−	0.329429i	$-0.893144\pi$
$331$	−34.3557			−1.88836			−0.944180	+	0.329429i	$0.893144\pi$
$332$		0			0
$333$	20.1233	+	3.90849i	1.10275	+	0.214184i
$334$		0			0
$335$	13.9835			0.764000
$336$		0			0
$337$	16.2880			0.887262			0.443631	−	0.896209i	$-0.353690\pi$
$337$	16.2880			0.887262			0.443631	+	0.896209i	$0.353690\pi$
$338$		0			0
$339$	−22.8931	+	18.8745i	−1.24339	+	1.02512i
$340$		0			0
$341$	−3.59668			−0.194771
$342$		0			0
$343$		−	36.6129i		−	1.97691i
$344$		0			0
$345$	−5.10083	−	6.18687i	−0.274619	−	0.333090i
$346$		0			0
$347$			3.98153i			0.213740i	0.994273	+	0.106870i	$0.0340828\pi$
$347$			3.98153i			0.213740i	−0.994273	+	0.106870i	$0.965917\pi$
$348$		0			0
$349$			17.6291i			0.943664i	0.881689	+	0.471832i	$0.156407\pi$
$349$			17.6291i			0.943664i	−0.881689	+	0.471832i	$0.843593\pi$
$350$		0			0
$351$	−6.01702	−	11.0761i	−0.321165	−	0.591201i
$352$		0			0
$353$		−	5.35044i		−	0.284775i	−0.989811	−	0.142388i	$-0.954522\pi$
$353$		−	5.35044i		−	0.284775i	0.989811	−	0.142388i	$-0.0454780\pi$
$354$		0			0
$355$	−2.52093			−0.133797
$356$		0			0
$357$	−40.5266	−	49.1553i	−2.14489	−	2.60157i
$358$		0			0
$359$	7.81698			0.412564			0.206282	−	0.978493i	$-0.433864\pi$
$359$	7.81698			0.412564			0.206282	+	0.978493i	$0.433864\pi$
$360$		0			0
$361$	−13.1153			−0.690282
$362$		0			0
$363$	−10.6780	−	12.9516i	−0.560452	−	0.679781i
$364$		0			0
$365$	−5.63362			−0.294877
$366$		0			0
$367$			18.7882i			0.980735i	0.871516	+	0.490367i	$0.163137\pi$
$367$			18.7882i			0.980735i	−0.871516	+	0.490367i	$0.836863\pi$
$368$		0			0
$369$	17.3456	+	3.36900i	0.902978	+	0.175383i
$370$		0			0
$371$		−	42.3249i		−	2.19740i
$372$		0			0
$373$		−	10.7960i		−	0.558995i	−0.960146	−	0.279498i	$-0.909832\pi$
$373$		−	10.7960i		−	0.558995i	0.960146	−	0.279498i	$-0.0901679\pi$
$374$		0			0
$375$	1.10182	+	1.33641i	0.0568976	+	0.0690120i
$376$		0			0
$377$		−	15.7427i		−	0.810793i
$378$		0			0
$379$	−4.31217			−0.221501			−0.110751	−	0.993848i	$-0.535326\pi$
$379$	−4.31217			−0.221501			−0.110751	+	0.993848i	$0.535326\pi$
$380$		0			0
$381$	−3.95684	+	3.26226i	−0.202715	+	0.167131i
$382$		0			0
$383$	−13.6342			−0.696676			−0.348338	−	0.937369i	$-0.613254\pi$
$383$	−13.6342			−0.696676			−0.348338	+	0.937369i	$0.613254\pi$
$384$		0			0
$385$	−5.34565			−0.272439
$386$		0			0
$387$	3.78140	−	19.4689i	0.192219	−	0.989660i
$388$		0			0
$389$	33.2672			1.68672			0.843358	−	0.537352i	$-0.180575\pi$
$389$	33.2672			1.68672			0.843358	+	0.537352i	$0.180575\pi$
$390$		0			0
$391$			36.4403i			1.84286i
$392$		0			0
$393$	15.8168	−	13.0403i	0.797853	−	0.657798i
$394$		0			0
$395$		−	6.77761i		−	0.341018i
$396$		0			0
$397$			35.9879i			1.80618i	0.429452	+	0.903090i	$0.358707\pi$
$397$			35.9879i			1.80618i	−0.429452	+	0.903090i	$0.641293\pi$
$398$		0			0
$399$	−15.1489	+	12.4896i	−0.758392	+	0.625264i
$400$		0			0
$401$			32.5645i			1.62619i	0.582129	+	0.813096i	$0.302220\pi$
$401$			32.5645i			1.62619i	−0.582129	+	0.813096i	$0.697780\pi$
$402$		0			0
$403$	−7.62678			−0.379917
$404$		0			0
$405$	8.34565	+	3.36900i	0.414699	+	0.167407i
$406$		0			0
$407$	−7.81698			−0.387473
$408$		0			0
$409$	5.14399			0.254354			0.127177	−	0.991880i	$-0.459408\pi$
$409$	5.14399			0.254354			0.127177	+	0.991880i	$0.459408\pi$
$410$		0			0
$411$	7.15040	−	5.89522i	0.352703	−	0.290790i
$412$		0			0
$413$	30.3249			1.49219
$414$		0			0
$415$			14.6728i			0.720261i
$416$		0			0
$417$	−20.0185	−	24.2807i	−0.980309	−	1.18903i
$418$		0			0
$419$		−	19.4689i		−	0.951118i	−0.879684	−	0.475559i	$-0.842246\pi$
$419$		−	19.4689i		−	0.951118i	0.879684	−	0.475559i	$-0.157754\pi$
$420$		0			0
$421$		−	2.37143i		−	0.115577i	−0.998329	−	0.0577883i	$-0.981595\pi$
$421$		−	2.37143i		−	0.115577i	0.998329	−	0.0577883i	$-0.0184048\pi$
$422$		0			0
$423$	−4.08998	+	21.0577i	−0.198862	+	1.02386i
$424$		0			0
$425$		−	7.87137i		−	0.381818i
$426$		0			0
$427$	−6.92820			−0.335279
$428$		0			0
$429$	3.05767	+	3.70870i	0.147626	+	0.179058i
$430$		0			0
$431$	−28.9646			−1.39517			−0.697587	−	0.716500i	$-0.745743\pi$
$431$	−28.9646			−1.39517			−0.697587	+	0.716500i	$0.745743\pi$
$432$		0			0
$433$	21.4610			1.03135			0.515675	−	0.856784i	$-0.327541\pi$
$433$	21.4610			1.03135			0.515675	+	0.856784i	$0.327541\pi$
$434$		0			0
$435$	7.15040	+	8.67282i	0.342835	+	0.415830i
$436$		0			0
$437$	11.2303			0.537218
$438$		0			0
$439$		−	31.1809i		−	1.48818i	−0.668077	−	0.744092i	$-0.732882\pi$
$439$		−	31.1809i		−	1.48818i	0.668077	−	0.744092i	$-0.267118\pi$
$440$		0			0
$441$	8.48568	−	43.6894i	0.404080	−	2.08045i
$442$		0			0
$443$		−	0.594417i		−	0.0282416i	−0.999900	−	0.0141208i	$-0.995505\pi$
$443$		−	0.594417i		−	0.0282416i	0.999900	−	0.0141208i	$-0.00449494\pi$
$444$		0			0
$445$		−	6.73800i		−	0.319412i
$446$		0			0
$447$	9.13184	+	11.0761i	0.431921	+	0.523884i
$448$		0			0
$449$			5.88993i			0.277963i	0.990295	+	0.138982i	$0.0443829\pi$
$449$			5.88993i			0.277963i	−0.990295	+	0.138982i	$0.955617\pi$
$450$		0			0
$451$	−6.73800			−0.317280
$452$		0			0
$453$	−13.1440	+	10.8367i	−0.617558	+	0.509152i
$454$		0			0
$455$	−11.3355			−0.531415
$456$		0			0
$457$	24.2880			1.13614			0.568072	−	0.822979i	$-0.307690\pi$
$457$	24.2880			1.13614			0.568072	+	0.822979i	$0.307690\pi$
$458$		0			0
$459$	−19.5241	−	35.9401i	−0.911309	−	1.67754i
$460$		0			0
$461$	−35.2963			−1.64391			−0.821956	−	0.569551i	$-0.807117\pi$
$461$	−35.2963			−1.64391			−0.821956	+	0.569551i	$0.807117\pi$
$462$		0			0
$463$			27.3641i			1.27172i	0.771805	+	0.635859i	$0.219354\pi$
$463$			27.3641i			1.27172i	−0.771805	+	0.635859i	$0.780646\pi$
$464$		0			0
$465$	4.20166	−	3.46410i	0.194847	−	0.160644i
$466$		0			0
$467$		−	14.6728i		−	0.678977i	−0.940610	−	0.339489i	$-0.889746\pi$
$467$		−	14.6728i		−	0.678977i	0.940610	−	0.339489i	$-0.110254\pi$
$468$		0			0
$469$		−	65.3424i		−	3.01723i
$470$		0			0
$471$	18.3908	−	15.1625i	0.847402	−	0.698649i
$472$		0			0
$473$			7.56279i			0.347738i
$474$		0			0
$475$	−2.42583			−0.111305
$476$		0			0
$477$	5.18093	−	26.6745i	0.237218	−	1.22134i
$478$		0			0
$479$	−7.81698			−0.357167			−0.178583	−	0.983925i	$-0.557151\pi$
$479$	−7.81698			−0.357167			−0.178583	+	0.983925i	$0.557151\pi$
$480$		0			0
$481$	−16.5759			−0.755798
$482$		0			0
$483$	−28.9102	+	23.8353i	−1.31546	+	1.08454i
$484$		0			0
$485$	3.34565			0.151918
$486$		0			0
$487$			12.3064i			0.557658i	0.960341	+	0.278829i	$0.0899463\pi$
$487$			12.3064i			0.557658i	−0.960341	+	0.278829i	$0.910054\pi$
$488$		0			0
$489$	−10.5513	−	12.7978i	−0.477145	−	0.578736i
$490$		0			0
$491$			20.5680i			0.928223i	0.885777	+	0.464111i	$0.153626\pi$
$491$			20.5680i			0.928223i	−0.885777	+	0.464111i	$0.846374\pi$
$492$		0			0
$493$		−	51.0823i		−	2.30063i
$494$		0			0
$495$	−3.36900	−	0.654353i	−0.151425	−	0.0294110i
$496$		0			0
$497$			11.7799i			0.528399i
$498$		0			0
$499$	9.54423			0.427259			0.213629	−	0.976915i	$-0.431472\pi$
$499$	9.54423			0.427259			0.213629	+	0.976915i	$0.431472\pi$
$500$		0			0
$501$	18.0801	+	21.9296i	0.807759	+	0.979743i
$502$		0			0
$503$	24.3351			1.08505			0.542524	−	0.840040i	$-0.317469\pi$
$503$	24.3351			1.08505			0.542524	+	0.840040i	$0.317469\pi$
$504$		0			0
$505$	−4.20166			−0.186971
$506$		0			0
$507$	−7.83982	−	9.50904i	−0.348179	−	0.422311i
$508$		0			0
$509$	18.4896			0.819539			0.409769	−	0.912189i	$-0.365609\pi$
$509$	18.4896			0.819539			0.409769	+	0.912189i	$0.365609\pi$
$510$		0			0
$511$			26.3249i			1.16455i
$512$		0			0
$513$	−11.0761	+	6.01702i	−0.489024	+	0.265658i
$514$		0			0
$515$			6.96080i			0.306729i
$516$		0			0
$517$		−	8.17996i		−	0.359754i
$518$		0			0
$519$	23.4326	+	28.4218i	1.02858	+	1.24758i
$520$		0			0
$521$		−	7.92577i		−	0.347234i	−0.984813	−	0.173617i	$-0.944454\pi$
$521$		−	7.92577i		−	0.347234i	0.984813	−	0.173617i	$-0.0555455\pi$
$522$		0			0
$523$	37.0988			1.62222			0.811109	−	0.584894i	$-0.198864\pi$
$523$	37.0988			1.62222			0.811109	+	0.584894i	$0.198864\pi$
$524$		0			0
$525$	6.24482	−	5.14860i	0.272546	−	0.224703i
$526$		0			0
$527$	−24.7475			−1.07802
$528$		0			0
$529$	−1.56804			−0.0681757
$530$		0			0
$531$	19.1118	+	3.71203i	0.829379	+	0.161088i
$532$		0			0
$533$	−14.2880			−0.618881
$534$		0			0
$535$		−	2.67282i		−	0.115556i
$536$		0			0
$537$	−21.6521	+	17.8513i	−0.934357	+	0.770340i
$538$		0			0
$539$			16.9714i			0.731008i
$540$		0			0
$541$		−	33.3718i		−	1.43477i	−0.696678	−	0.717384i	$-0.745339\pi$
$541$		−	33.3718i		−	1.43477i	0.696678	−	0.717384i	$-0.254661\pi$
$542$		0			0
$543$	18.2637	−	15.0577i	0.783769	−	0.646187i
$544$		0			0
$545$			8.41086i			0.360282i
$546$		0			0
$547$	−22.9882			−0.982906			−0.491453	−	0.870904i	$-0.663534\pi$
$547$	−22.9882			−0.982906			−0.491453	+	0.870904i	$0.663534\pi$
$548$		0			0
$549$	−4.36638	−	0.848071i	−0.186353	−	0.0361948i
$550$		0			0
$551$	−15.7427			−0.670664
$552$		0			0
$553$	−31.6706			−1.34677
$554$		0			0
$555$	9.13184	−	7.52884i	0.387625	−	0.319581i
$556$		0			0
$557$	40.3619			1.71019			0.855093	−	0.518474i	$-0.173500\pi$
$557$	40.3619			1.71019			0.855093	+	0.518474i	$0.173500\pi$
$558$		0			0
$559$			16.0369i			0.678290i
$560$		0			0
$561$	9.92159	+	12.0340i	0.418890	+	0.508078i
$562$		0			0
$563$		−	4.42179i		−	0.186356i	−0.995649	−	0.0931782i	$-0.970297\pi$
$563$		−	4.42179i		−	0.186356i	0.995649	−	0.0931782i	$-0.0297026\pi$
$564$		0			0
$565$			17.1303i			0.720677i
$566$		0			0
$567$	15.7427	−	38.9977i	0.661133	−	1.63775i
$568$		0			0
$569$		−	40.1505i		−	1.68320i	−0.540103	−	0.841599i	$-0.681615\pi$
$569$		−	40.1505i		−	1.68320i	0.540103	−	0.841599i	$-0.318385\pi$
$570$		0			0
$571$	0.0951004			0.00397983			0.00198991	−	0.999998i	$-0.499367\pi$
$571$	0.0951004			0.00397983			0.00198991	+	0.999998i	$0.499367\pi$
$572$		0			0
$573$	−7.14399	−	8.66505i	−0.298444	−	0.361988i
$574$		0			0
$575$	−4.62947			−0.193062
$576$		0			0
$577$	16.1153			0.670891			0.335445	−	0.942060i	$-0.391113\pi$
$577$	16.1153			0.670891			0.335445	+	0.942060i	$0.391113\pi$
$578$		0			0
$579$	−13.1354	−	15.9322i	−0.545890	−	0.662118i
$580$		0			0
$581$	68.5635			2.84449
$582$		0			0
$583$			10.3619i			0.429144i
$584$		0			0
$585$	−7.14399	−	1.38756i	−0.295367	−	0.0573685i
$586$		0			0
$587$			24.3849i			1.00647i	0.864149	+	0.503235i	$0.167857\pi$
$587$			24.3849i			1.00647i	−0.864149	+	0.503235i	$0.832143\pi$
$588$		0			0
$589$			7.62678i			0.314256i
$590$		0			0
$591$	26.8016	+	32.5081i	1.10247	+	1.33720i
$592$		0			0
$593$		−	2.46653i		−	0.101288i	−0.998717	−	0.0506442i	$-0.983873\pi$
$593$		−	2.46653i		−	0.101288i	0.998717	−	0.0506442i	$-0.0161274\pi$
$594$		0			0
$595$	−36.7815			−1.50790
$596$		0			0
$597$	20.2880	−	16.7266i	0.830331	−	0.684575i
$598$		0			0
$599$	−32.5645			−1.33055			−0.665274	−	0.746599i	$-0.731685\pi$
$599$	−32.5645			−1.33055			−0.665274	+	0.746599i	$0.731685\pi$
$600$		0			0
$601$	−28.8145			−1.17537			−0.587685	−	0.809090i	$-0.699961\pi$
$601$	−28.8145			−1.17537			−0.587685	+	0.809090i	$0.699961\pi$
$602$		0			0
$603$	7.99847	−	41.1809i	0.325723	−	1.67702i
$604$		0			0
$605$	−9.69129			−0.394007
$606$		0			0
$607$		−	39.4011i		−	1.59924i	−0.600507	−	0.799620i	$-0.705034\pi$
$607$		−	39.4011i		−	1.59924i	0.600507	−	0.799620i	$-0.294966\pi$
$608$		0			0
$609$	40.5266	−	33.4125i	1.64222	−	1.35394i
$610$		0			0
$611$		−	17.3456i		−	0.701730i
$612$		0			0
$613$			29.0597i			1.17371i	0.809692	+	0.586854i	$0.199634\pi$
$613$			29.0597i			1.17371i	−0.809692	+	0.586854i	$0.800366\pi$
$614$		0			0
$615$	7.87137	−	6.48963i	0.317404	−	0.261687i
$616$		0			0
$617$			33.6978i			1.35662i	0.734774	+	0.678312i	$0.237288\pi$
$617$			33.6978i			1.35662i	−0.734774	+	0.678312i	$0.762712\pi$
$618$		0			0
$619$	34.1015			1.37066			0.685328	−	0.728234i	$-0.259659\pi$
$619$	34.1015			1.37066			0.685328	+	0.728234i	$0.259659\pi$
$620$		0			0
$621$	−21.1378	+	11.4829i	−0.848230	+	0.460794i
$622$		0			0
$623$	−31.4855			−1.26144
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	3.70870	−	3.05767i	0.148111	−	0.122112i
$628$		0			0
$629$	−53.7859			−2.14458
$630$		0			0
$631$		−	18.2386i		−	0.726067i	−0.931776	−	0.363034i	$-0.881741\pi$
$631$		−	18.2386i		−	0.726067i	0.931776	−	0.363034i	$-0.118259\pi$
$632$		0			0
$633$	−3.16246	−	3.83579i	−0.125696	−	0.152459i
$634$		0			0
$635$			2.96080i			0.117496i
$636$		0			0
$637$			35.9879i			1.42589i
$638$		0			0
$639$	−1.44196	+	7.42405i	−0.0570429	+	0.293691i
$640$		0			0
$641$		−	25.4868i		−	1.00667i	−0.864092	−	0.503333i	$-0.832107\pi$
$641$		−	25.4868i		−	1.00667i	0.864092	−	0.503333i	$-0.167893\pi$
$642$		0			0
$643$	−28.8375			−1.13724			−0.568619	−	0.822601i	$-0.692522\pi$
$643$	−28.8375			−1.13724			−0.568619	+	0.822601i	$0.692522\pi$
$644$		0			0
$645$	−7.28402	−	8.83490i	−0.286808	−	0.347874i
$646$		0			0
$647$	−28.2980			−1.11251			−0.556254	−	0.831012i	$-0.687762\pi$
$647$	−28.2980			−1.11251			−0.556254	+	0.831012i	$0.687762\pi$
$648$		0			0
$649$	−7.42405			−0.291420
$650$		0			0
$651$	−16.1871	−	19.6336i	−0.634424	−	0.769502i
$652$		0			0
$653$	15.9216			0.623060			0.311530	−	0.950236i	$-0.399159\pi$
$653$	15.9216			0.623060			0.311530	+	0.950236i	$0.399159\pi$
$654$		0			0
$655$		−	11.8353i		−	0.462443i
$656$		0			0
$657$	−3.22239	+	16.5908i	−0.125718	+	0.647270i
$658$		0			0
$659$			15.8722i			0.618294i	0.951014	+	0.309147i	$0.100044\pi$
$659$			15.8722i			0.618294i	−0.951014	+	0.309147i	$0.899956\pi$
$660$		0			0
$661$			4.36657i			0.169840i	0.996388	+	0.0849199i	$0.0270634\pi$
$661$			4.36657i			0.169840i	−0.996388	+	0.0849199i	$0.972937\pi$
$662$		0			0
$663$	21.0388	+	25.5183i	0.817079	+	0.991048i
$664$		0			0
$665$			11.3355i			0.439571i
$666$		0			0
$667$	−30.0435			−1.16329
$668$		0			0
$669$	−18.2448	+	15.0421i	−0.705385	+	0.581562i
$670$		0			0
$671$	1.69614			0.0654788
$672$		0			0
$673$	−4.07841			−0.157211			−0.0786055	−	0.996906i	$-0.525047\pi$
$673$	−4.07841			−0.157211			−0.0786055	+	0.996906i	$0.525047\pi$
$674$		0			0
$675$	4.56592	−	2.48040i	0.175742	−	0.0954706i
$676$		0			0
$677$	−25.3042			−0.972519			−0.486259	−	0.873815i	$-0.661639\pi$
$677$	−25.3042			−0.972519			−0.486259	+	0.873815i	$0.661639\pi$
$678$		0			0
$679$		−	15.6336i		−	0.599963i
$680$		0			0
$681$	−12.4649	+	10.2769i	−0.477658	+	0.393810i
$682$		0			0
$683$		−	38.3434i		−	1.46717i	−0.679598	−	0.733584i	$-0.737846\pi$
$683$		−	38.3434i		−	1.46717i	0.679598	−	0.733584i	$-0.262154\pi$
$684$		0			0
$685$		−	5.35044i		−	0.204430i
$686$		0			0
$687$	9.00475	−	7.42405i	0.343553	−	0.283245i
$688$		0			0
$689$			21.9724i			0.837081i
$690$		0			0
$691$	0.902468			0.0343315			0.0171657	−	0.999853i	$-0.494536\pi$
$691$	0.902468			0.0343315			0.0171657	+	0.999853i	$0.494536\pi$
$692$		0			0
$693$	−3.05767	+	15.7427i	−0.116151	+	0.598018i
$694$		0			0
$695$	−18.1686			−0.689173
$696$		0			0
$697$	−46.3619			−1.75608
$698$		0			0
$699$	−20.9661	+	17.2857i	−0.793011	+	0.653806i
$700$		0			0
$701$	2.73276			0.103215			0.0516074	−	0.998667i	$-0.483566\pi$
$701$	2.73276			0.103215			0.0516074	+	0.998667i	$0.483566\pi$
$702$		0			0
$703$			16.5759i			0.625174i
$704$		0			0
$705$	7.87844	+	9.55588i	0.296719	+	0.359895i
$706$		0			0
$707$			19.6336i			0.738398i
$708$		0			0
$709$			0.888774i			0.0333786i	0.999861	+	0.0166893i	$0.00531262\pi$
$709$			0.888774i			0.0333786i	−0.999861	+	0.0166893i	$0.994687\pi$
$710$		0			0
$711$	−19.9598	−	3.87675i	−0.748552	−	0.145389i
$712$		0			0
$713$			14.5550i			0.545088i
$714$		0			0
$715$	2.77512			0.103783
$716$		0			0
$717$	9.92159	+	12.0340i	0.370529	+	0.449420i
$718$		0			0
$719$	24.7475			0.922926			0.461463	−	0.887159i	$-0.347325\pi$
$719$	24.7475			0.922926			0.461463	+	0.887159i	$0.347325\pi$
$720$		0			0
$721$	32.5266			1.21135
$722$		0			0
$723$	−8.18867	−	9.93216i	−0.304540	−	0.369381i
$724$		0			0
$725$	6.48963			0.241019
$726$		0			0
$727$			21.0392i			0.780301i	0.920751	+	0.390150i	$0.127577\pi$
$727$			21.0392i			0.780301i	−0.920751	+	0.390150i	$0.872423\pi$
$728$		0			0
$729$	14.6952	−	22.6506i	0.544268	−	0.838911i
$730$		0			0
$731$			52.0369i			1.92466i
$732$		0			0
$733$			24.4622i			0.903532i	0.892137	+	0.451766i	$0.149206\pi$
$733$			24.4622i			0.903532i	−0.892137	+	0.451766i	$0.850794\pi$
$734$		0			0
$735$	−16.3458	−	19.8260i	−0.602923	−	0.731295i
$736$		0			0
$737$			15.9969i			0.589255i
$738$		0			0
$739$	−19.0574			−0.701036			−0.350518	−	0.936556i	$-0.613994\pi$
$739$	−19.0574			−0.701036			−0.350518	+	0.936556i	$0.613994\pi$
$740$		0			0
$741$	7.86432	−	6.48382i	0.288903	−	0.238189i
$742$		0			0
$743$	−18.6761			−0.685159			−0.342579	−	0.939489i	$-0.611301\pi$
$743$	−18.6761			−0.685159			−0.342579	+	0.939489i	$0.611301\pi$
$744$		0			0
$745$	8.28797			0.303648
$746$		0			0
$747$	43.2110	+	8.39276i	1.58101	+	0.307075i
$748$		0			0
$749$	−12.4896			−0.456361
$750$		0			0
$751$		−	41.4320i		−	1.51187i	−0.654645	−	0.755937i	$-0.727182\pi$
$751$		−	41.4320i		−	1.51187i	0.654645	−	0.755937i	$-0.272818\pi$
$752$		0			0
$753$	4.58651	−	3.78140i	0.167142	−	0.137802i
$754$		0			0
$755$			9.83528i			0.357943i
$756$		0			0
$757$			19.6919i			0.715716i	0.933776	+	0.357858i	$0.116493\pi$
$757$			19.6919i			0.715716i	−0.933776	+	0.357858i	$0.883507\pi$
$758$		0			0
$759$	7.07770	−	5.83528i	0.256904	−	0.211807i
$760$		0			0
$761$			24.7475i			0.897096i	0.893759	+	0.448548i	$0.148059\pi$
$761$			24.7475i			0.897096i	−0.893759	+	0.448548i	$0.851941\pi$
$762$		0			0
$763$	39.3025			1.42285
$764$		0			0
$765$	−23.1809	−	4.50237i	−0.838108	−	0.162784i
$766$		0			0
$767$	−15.7427			−0.568438
$768$		0			0
$769$	26.6498			0.961017			0.480509	−	0.876990i	$-0.340452\pi$
$769$	26.6498			0.961017			0.480509	+	0.876990i	$0.340452\pi$
$770$		0			0
$771$	−1.85435	+	1.52884i	−0.0667828	+	0.0550597i
$772$		0			0
$773$	−19.7490			−0.710321			−0.355161	−	0.934805i	$-0.615574\pi$
$773$	−19.7490			−0.710321			−0.355161	+	0.934805i	$0.615574\pi$
$774$		0			0
$775$		−	3.14399i		−	0.112935i
$776$		0			0
$777$	−35.1809	−	42.6715i	−1.26211	−	1.53083i
$778$		0			0
$779$			14.2880i			0.511920i
$780$		0			0
$781$		−	2.88391i		−	0.103194i
$782$		0			0
$783$	29.6311	−	16.0969i	1.05893	−	0.575255i
$784$		0			0
$785$		−	13.7613i		−	0.491162i
$786$		0			0
$787$	−2.83822			−0.101172			−0.0505859	−	0.998720i	$-0.516109\pi$
$787$	−2.83822			−0.101172			−0.0505859	+	0.998720i	$0.516109\pi$
$788$		0			0
$789$	6.68960	+	8.11392i	0.238156	+	0.288863i
$790$		0			0
$791$	80.0469			2.84614
$792$		0			0
$793$	3.59668			0.127722
$794$		0			0
$795$	−9.97991	−	12.1048i	−0.353951	−	0.429312i
$796$		0			0
$797$	22.5971			0.800430			0.400215	−	0.916421i	$-0.368936\pi$
$797$	22.5971			0.800430			0.400215	+	0.916421i	$0.368936\pi$
$798$		0			0
$799$		−	56.2835i		−	1.99117i
$800$		0			0
$801$	−19.8432	−	3.85409i	−0.701125	−	0.136178i
$802$		0			0
$803$		−	6.44479i		−	0.227432i
$804$		0			0
$805$			21.6327i			0.762452i
$806$		0			0
$807$	−2.64802	−	3.21183i	−0.0932148	−	0.113062i
$808$		0			0
$809$		−	34.2606i		−	1.20454i	−0.798293	−	0.602270i	$-0.794263\pi$
$809$		−	34.2606i		−	1.20454i	0.798293	−	0.602270i	$-0.205737\pi$
$810$		0			0
$811$	32.2792			1.13348			0.566738	−	0.823898i	$-0.308205\pi$
$811$	32.2792			1.13348			0.566738	+	0.823898i	$0.308205\pi$
$812$		0			0
$813$	−26.1233	+	21.5376i	−0.916183	+	0.755356i
$814$		0			0
$815$	−9.57623			−0.335441
$816$		0			0
$817$	16.0369			0.561062
$818$		0			0
$819$	−6.48382	+	33.3826i	−0.226563	+	1.16648i
$820$		0			0
$821$	28.3619			0.989836			0.494918	−	0.868940i	$-0.335198\pi$
$821$	28.3619			0.989836			0.494918	+	0.868940i	$0.335198\pi$
$822$		0			0
$823$		−	16.5002i		−	0.575161i	−0.957757	−	0.287580i	$-0.907149\pi$
$823$		−	16.5002i		−	0.575161i	0.957757	−	0.287580i	$-0.0928509\pi$
$824$		0			0
$825$	−1.52884	+	1.26047i	−0.0532273	+	0.0438838i
$826$		0			0
$827$			14.8824i			0.517511i	0.965943	+	0.258756i	$0.0833125\pi$
$827$			14.8824i			0.517511i	−0.965943	+	0.258756i	$0.916688\pi$
$828$		0			0
$829$		−	52.5650i		−	1.82566i	−0.408342	−	0.912829i	$-0.633893\pi$
$829$		−	52.5650i		−	1.82566i	0.408342	−	0.912829i	$-0.366107\pi$
$830$		0			0
$831$	25.1288	−	20.7177i	0.871708	−	0.718688i
$832$		0			0
$833$			116.774i			4.04598i
$834$		0			0
$835$	16.4093			0.567868
$836$		0			0
$837$	−7.79834	−	14.3552i	−0.269550	−	0.496188i
$838$		0			0
$839$	−52.7785			−1.82212			−0.911058	−	0.412279i	$-0.864733\pi$
$839$	−52.7785			−1.82212			−0.911058	+	0.412279i	$0.864733\pi$
$840$		0			0
$841$	13.1153			0.452253
$842$		0			0
$843$	13.1674	−	10.8560i	0.453510	−	0.373901i
$844$		0			0
$845$	−7.11535			−0.244775
$846$		0			0
$847$			45.2857i			1.55604i
$848$		0			0
$849$	2.91764	+	3.53885i	0.100133	+	0.121453i
$850$		0			0
$851$			31.6336i			1.08439i
$852$		0			0
$853$		−	35.1805i		−	1.20456i	−0.798286	−	0.602279i	$-0.794260\pi$
$853$		−	35.1805i		−	1.20456i	0.798286	−	0.602279i	$-0.205740\pi$
$854$		0			0
$855$	−1.38756	+	7.14399i	−0.0474535	+	0.244319i
$856$		0			0
$857$		−	44.6529i		−	1.52531i	−0.646803	−	0.762657i	$-0.723894\pi$
$857$		−	44.6529i		−	1.52531i	0.646803	−	0.762657i	$-0.276106\pi$
$858$		0			0
$859$	17.1710			0.585867			0.292934	−	0.956133i	$-0.405368\pi$
$859$	17.1710			0.585867			0.292934	+	0.956133i	$0.405368\pi$
$860$		0			0
$861$	−30.3249	−	36.7815i	−1.03347	−	1.25351i
$862$		0			0
$863$	42.8530			1.45873			0.729366	−	0.684124i	$-0.239815\pi$
$863$	42.8530			1.45873			0.729366	+	0.684124i	$0.239815\pi$
$864$		0			0
$865$	21.2672			0.723108
$866$		0			0
$867$	49.5361	+	60.0831i	1.68234	+	2.04053i
$868$		0			0
$869$	7.75349			0.263019
$870$		0			0
$871$			33.9216i			1.14939i
$872$		0			0
$873$	1.91369	−	9.85282i	0.0647685	−	0.333467i
$874$		0			0
$875$		−	4.67282i		−	0.157970i
$876$		0			0
$877$			31.3904i			1.05998i	0.848004	+	0.529989i	$0.177804\pi$
$877$			31.3904i			1.05998i	−0.848004	+	0.529989i	$0.822196\pi$
$878$		0			0
$879$	−19.8327	−	24.0554i	−0.668941	−	0.811369i
$880$		0			0
$881$			31.1458i			1.04933i	0.851309	+	0.524664i	$0.175809\pi$
$881$			31.1458i			1.04933i	−0.851309	+	0.524664i	$0.824191\pi$
$882$		0			0
$883$	54.5377			1.83534			0.917670	−	0.397343i	$-0.130068\pi$
$883$	54.5377			1.83534			0.917670	+	0.397343i	$0.130068\pi$
$884$		0			0
$885$	8.67282	−	7.15040i	0.291534	−	0.240358i
$886$		0			0
$887$	43.1072			1.44740			0.723698	−	0.690117i	$-0.242441\pi$
$887$	43.1072			1.44740			0.723698	+	0.690117i	$0.242441\pi$
$888$		0			0
$889$	13.8353			0.464020
$890$		0			0
$891$	−3.85409	+	9.54731i	−0.129117	+	0.319847i
$892$		0			0
$893$	−17.3456			−0.580450
$894$		0			0
$895$			16.2017i			0.541562i
$896$		0			0
$897$	15.0083	−	12.3737i	0.501113	−	0.413147i
$898$		0			0
$899$		−	20.4033i		−	0.680489i
$900$		0			0
$901$			71.2963i			2.37523i
$902$		0			0
$903$	−41.2839	+	34.0369i	−1.37384	+	1.13268i
$904$		0			0
$905$		−	13.6662i		−	0.454280i
$906$		0			0
$907$	−19.0254			−0.631727			−0.315863	−	0.948805i	$-0.602294\pi$
$907$	−19.0254			−0.631727			−0.315863	+	0.948805i	$0.602294\pi$
$908$		0			0
$909$	−2.40332	+	12.3737i	−0.0797132	+	0.410411i
$910$		0			0
$911$	28.9646			0.959639			0.479819	−	0.877367i	$-0.340702\pi$
$911$	28.9646			0.959639			0.479819	+	0.877367i	$0.340702\pi$
$912$		0			0
$913$	−16.7855			−0.555519
$914$		0			0
$915$	−1.98144	+	1.63362i	−0.0655045	+	0.0540058i
$916$		0			0
$917$	−55.3042			−1.82631
$918$		0			0
$919$			2.93442i			0.0967976i	0.998828	+	0.0483988i	$0.0154118\pi$
$919$			2.93442i			0.0967976i	−0.998828	+	0.0483988i	$0.984588\pi$
$920$		0			0
$921$	−0.140034	−	0.169849i	−0.00461427	−	0.00559671i
$922$		0			0
$923$		−	6.11535i		−	0.201289i
$924$		0			0
$925$		−	6.83310i		−	0.224671i
$926$		0			0
$927$	20.4993	+	3.98153i	0.673286	+	0.130771i
$928$		0			0
$929$			2.03584i			0.0667937i	0.999442	+	0.0333969i	$0.0106325\pi$
$929$			2.03584i			0.0667937i	−0.999442	+	0.0333969i	$0.989367\pi$
$930$		0			0
$931$	35.9879			1.17945
$932$		0			0
$933$	15.7569	+	19.1118i	0.515857	+	0.625691i
$934$		0			0
$935$	9.00475			0.294487
$936$		0			0
$937$	14.3664			0.469329			0.234665	−	0.972076i	$-0.424601\pi$
$937$	14.3664			0.469329			0.234665	+	0.972076i	$0.424601\pi$
$938$		0			0
$939$	15.4255	+	18.7098i	0.503391	+	0.610570i
$940$		0			0
$941$	12.9344			0.421650			0.210825	−	0.977524i	$-0.432385\pi$
$941$	12.9344			0.421650			0.210825	+	0.977524i	$0.432385\pi$
$942$		0			0
$943$			27.2672i			0.887944i
$944$		0			0
$945$	−11.5905	−	21.3357i	−0.377038	−	0.694052i
$946$		0			0
$947$		−	41.9401i		−	1.36287i	−0.731879	−	0.681434i	$-0.761356\pi$
$947$		−	41.9401i		−	1.36287i	0.731879	−	0.681434i	$-0.238644\pi$
$948$		0			0
$949$		−	13.6662i		−	0.443624i
$950$		0			0
$951$	−1.79995	−	2.18319i	−0.0583675	−	0.0707948i
$952$		0			0
$953$			19.9054i			0.644800i	0.946604	+	0.322400i	$0.104490\pi$
$953$			19.9054i			0.644800i	−0.946604	+	0.322400i	$0.895510\pi$
$954$		0			0
$955$	−6.48382			−0.209811
$956$		0			0
$957$	−9.92159	+	8.17996i	−0.320720	+	0.264420i
$958$		0			0
$959$	−25.0017			−0.807346
$960$		0			0
$961$	21.1153			0.681140
$962$		0			0
$963$	−7.87137	−	1.52884i	−0.253651	−	0.0492661i
$964$		0			0
$965$	−11.9216			−0.383770
$966$		0			0
$967$			33.8458i			1.08841i	0.838953	+	0.544205i	$0.183168\pi$
$967$			33.8458i			1.08841i	−0.838953	+	0.544205i	$0.816832\pi$
$968$		0			0
$969$	25.5183	−	21.0388i	0.819765	−	0.675864i
$970$		0			0
$971$			4.41123i			0.141563i	0.997492	+	0.0707815i	$0.0225493\pi$
$971$			4.41123i			0.141563i	−0.997492	+	0.0707815i	$0.977451\pi$
$972$		0			0
$973$			84.8986i			2.72172i
$974$		0			0
$975$	−3.24191	+	2.67282i	−0.103824	+	0.0855989i
$976$		0			0
$977$			35.3940i			1.13235i	0.824284	+	0.566177i	$0.191578\pi$
$977$			35.3940i			1.13235i	−0.824284	+	0.566177i	$0.808422\pi$
$978$		0			0
$979$	7.70818			0.246355
$980$		0			0
$981$	24.7697	+	4.81096i	0.790836	+	0.153602i
$982$		0			0
$983$	36.3692			1.16000			0.579998	−	0.814618i	$-0.303053\pi$
$983$	36.3692			1.16000			0.579998	+	0.814618i	$0.303053\pi$
$984$		0			0
$985$	24.3249			0.775056
$986$		0			0
$987$	44.6529	−	36.8145i	1.42132	−	1.17182i
$988$		0			0
$989$	30.6050			0.973182
$990$		0			0
$991$		−	15.1440i		−	0.481065i	−0.970641	−	0.240532i	$-0.922678\pi$
$991$		−	15.1440i		−	0.481065i	0.970641	−	0.240532i	$-0.0773220\pi$
$992$		0			0
$993$	37.8538	+	45.9134i	1.20125	+	1.45702i
$994$		0			0
$995$		−	15.1809i		−	0.481268i
$996$		0			0
$997$		−	33.7211i		−	1.06796i	−0.845497	−	0.533979i	$-0.820696\pi$
$997$		−	33.7211i		−	1.06796i	0.845497	−	0.533979i	$-0.179304\pi$
$998$		0			0
$999$	−16.9488	−	31.1994i	−0.536237	−	0.987105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.b.c.671.3	✓	12
3.2	odd	2		960.2.b.d.671.4	yes	12
4.3	odd	2	inner	960.2.b.c.671.10	yes	12
8.3	odd	2		960.2.b.d.671.3	yes	12
8.5	even	2		960.2.b.d.671.10	yes	12
12.11	even	2		960.2.b.d.671.9	yes	12
24.5	odd	2	inner	960.2.b.c.671.9	yes	12
24.11	even	2	inner	960.2.b.c.671.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.b.c.671.3	✓	12	1.1	even	1	trivial
960.2.b.c.671.4	yes	12	24.11	even	2	inner
960.2.b.c.671.9	yes	12	24.5	odd	2	inner
960.2.b.c.671.10	yes	12	4.3	odd	2	inner
960.2.b.d.671.3	yes	12	8.3	odd	2
960.2.b.d.671.4	yes	12	3.2	odd	2
960.2.b.d.671.9	yes	12	12.11	even	2
960.2.b.d.671.10	yes	12	8.5	even	2

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.10182 - 1.33641i) q^{3} -1.00000 q^{5} +4.67282i q^{7} +(-0.571993 + 2.94497i) q^{9} +O(q^{10})\)	\(q+(-1.10182 - 1.33641i) q^{3} -1.00000 q^{5} +4.67282i q^{7} +(-0.571993 + 2.94497i) q^{9} -1.14399i q^{11} -2.42583i q^{13} +(1.10182 + 1.33641i) q^{15} -7.87137i q^{17} -2.42583 q^{19} +(6.24482 - 5.14860i) q^{21} -4.62947 q^{23} +1.00000 q^{25} +(4.56592 - 2.48040i) q^{27} +6.48963 q^{29} -3.14399i q^{31} +(-1.52884 + 1.26047i) q^{33} -4.67282i q^{35} -6.83310i q^{37} +(-3.24191 + 2.67282i) q^{39} -5.88993i q^{41} -6.61091 q^{43} +(0.571993 - 2.94497i) q^{45} +7.15040 q^{47} -14.8353 q^{49} +(-10.5194 + 8.67282i) q^{51} -9.05767 q^{53} +1.14399i q^{55} +(2.67282 + 3.24191i) q^{57} -6.48963i q^{59} +1.48266i q^{61} +(-13.7613 - 2.67282i) q^{63} +2.42583i q^{65} -13.9835 q^{67} +(5.10083 + 6.18687i) q^{69} +2.52093 q^{71} +5.63362 q^{73} +(-1.10182 - 1.33641i) q^{75} +5.34565 q^{77} +6.77761i q^{79} +(-8.34565 - 3.36900i) q^{81} -14.6728i q^{83} +7.87137i q^{85} +(-7.15040 - 8.67282i) q^{87} +6.73800i q^{89} +11.3355 q^{91} +(-4.20166 + 3.46410i) q^{93} +2.42583 q^{95} -3.34565 q^{97} +(3.36900 + 0.654353i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{5} - 4 q^{9}+O(q^{10}) \) 12 * q - 12 * q^5 - 4 * q^9	\( 12 q - 12 q^{5} - 4 q^{9} + 32 q^{21} + 12 q^{25} - 8 q^{29} + 16 q^{33} + 4 q^{45} - 12 q^{49} - 40 q^{53} - 8 q^{57} + 24 q^{69} - 24 q^{73} - 16 q^{77} - 20 q^{81} + 24 q^{93} + 40 q^{97}+O(q^{100}) \) 12 * q - 12 * q^5 - 4 * q^9 + 32 * q^21 + 12 * q^25 - 8 * q^29 + 16 * q^33 + 4 * q^45 - 12 * q^49 - 40 * q^53 - 8 * q^57 + 24 * q^69 - 24 * q^73 - 16 * q^77 - 20 * q^81 + 24 * q^93 + 40 * q^97

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