LMFDB - Embedded newform 960.2.h.d.191.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(191,960)]

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, 0, 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("960.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.h (of order $2$, degree $1$, not 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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.4
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	960.191
Dual form			960.2.h.d.191.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +1.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +1.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} -3.46410 q^{11} -4.00000 q^{13} +1.73205i q^{15} +6.00000i q^{17} +3.46410i q^{19} +6.00000i q^{21} +3.46410 q^{23} -1.00000 q^{25} +5.19615 q^{27} -6.00000i q^{29} +3.46410i q^{31} -6.00000 q^{33} -3.46410 q^{35} +4.00000 q^{37} -6.92820 q^{39} -12.0000i q^{41} +6.92820i q^{43} +3.00000i q^{45} -3.46410 q^{47} -5.00000 q^{49} +10.3923i q^{51} -6.00000i q^{53} -3.46410i q^{55} +6.00000i q^{57} -3.46410 q^{59} +10.0000 q^{61} +10.3923i q^{63} -4.00000i q^{65} +6.92820i q^{67} +6.00000 q^{69} +13.8564 q^{71} -2.00000 q^{73} -1.73205 q^{75} -12.0000i q^{77} +10.3923i q^{79} +9.00000 q^{81} +10.3923 q^{83} -6.00000 q^{85} -10.3923i q^{87} -13.8564i q^{91} +6.00000i q^{93} -3.46410 q^{95} +10.0000 q^{97} -10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 16 q^{13} - 4 q^{25} - 24 q^{33} + 16 q^{37} - 20 q^{49} + 40 q^{61} + 24 q^{69} - 8 q^{73} + 36 q^{81} - 24 q^{85} + 40 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 16 * q^13 - 4 * q^25 - 24 * q^33 + 16 * q^37 - 20 * q^49 + 40 * q^61 + 24 * q^69 - 8 * q^73 + 36 * q^81 - 24 * q^85 + 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
$3$	1.73205	1.00000
$3$	1.73205	1.00000
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.46410i	1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$	3.46410i	1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	1.73205i	0.447214i
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	3.46410i	0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$	3.46410i	0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	0	0
$21$	6.00000i	1.30931i
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.19615	1.00000
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	3.46410i	0.622171i	0.950382	+	0.311086i	$0.100693\pi$
$31$	3.46410i	0.622171i	−0.950382	+	0.311086i	$0.899307\pi$
$32$	0	0
$33$	−6.00000	−1.04447
$34$	0	0
$35$	−3.46410	−0.585540
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	−6.92820	−1.10940
$40$	0	0
$41$	− 12.0000i	− 1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	− 12.0000i	− 1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$	0	0
$43$	6.92820i	1.05654i	0.849076	+	0.528271i	$0.177159\pi$
$43$	6.92820i	1.05654i	−0.849076	+	0.528271i	$0.822841\pi$
$44$	0	0
$45$	3.00000i	0.447214i
$46$	0	0
$47$	−3.46410	−0.505291	−0.252646	−	0.967559i	$-0.581301\pi$
$47$	−3.46410	−0.505291	−0.252646	+	0.967559i	$0.581301\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	10.3923i	1.45521i
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	− 3.46410i	− 0.467099i
$56$	0	0
$57$	6.00000i	0.794719i
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	10.3923i	1.30931i
$64$	0	0
$65$	− 4.00000i	− 0.496139i
$66$	0	0
$67$	6.92820i	0.846415i	0.906033	+	0.423207i	$0.139096\pi$
$67$	6.92820i	0.846415i	−0.906033	+	0.423207i	$0.860904\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	13.8564	1.64445	0.822226	−	0.569160i	$-0.192732\pi$
$71$	13.8564	1.64445	0.822226	+	0.569160i	$0.192732\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	0	0
$75$	−1.73205	−0.200000
$76$	0	0
$77$	− 12.0000i	− 1.36753i
$78$	0	0
$79$	10.3923i	1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$	10.3923i	1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	10.3923	1.14070	0.570352	−	0.821401i	$-0.306807\pi$
$83$	10.3923	1.14070	0.570352	+	0.821401i	$0.306807\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	− 10.3923i	− 1.11417i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 13.8564i	− 1.45255i
$92$	0	0
$93$	6.00000i	0.622171i
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−10.3923	−1.04447
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	− 10.3923i	− 1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$	− 10.3923i	− 1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$	0	0
$105$	−6.00000	−0.585540
$106$	0	0
$107$	3.46410	0.334887	0.167444	−	0.985882i	$-0.446449\pi$
$107$	3.46410	0.334887	0.167444	+	0.985882i	$0.446449\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	6.92820	0.657596
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	3.46410i	0.323029i
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	−20.7846	−1.90532
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 20.7846i	− 1.87409i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	10.3923i	0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$	10.3923i	0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	12.0000i	1.05654i
$130$	0	0
$131$	17.3205	1.51330	0.756650	−	0.653820i	$-0.226835\pi$
$131$	17.3205	1.51330	0.756650	+	0.653820i	$0.226835\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	5.19615i	0.447214i
$136$	0	0
$137$	− 18.0000i	− 1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$	− 18.0000i	− 1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$	0	0
$139$	− 10.3923i	− 0.881464i	−0.897639	−	0.440732i	$-0.854719\pi$
$139$	− 10.3923i	− 0.881464i	0.897639	−	0.440732i	$-0.145281\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	6.00000	0.498273
$146$	0	0
$147$	−8.66025	−0.714286
$148$	0	0
$149$	6.00000i	0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$	6.00000i	0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$	0	0
$151$	− 3.46410i	− 0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$	− 3.46410i	− 0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$	0	0
$153$	18.0000i	1.45521i
$154$	0	0
$155$	−3.46410	−0.278243
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	0	0
$159$	− 10.3923i	− 0.824163i
$160$	0	0
$161$	12.0000i	0.945732i
$162$	0	0
$163$	13.8564i	1.08532i	0.839953	+	0.542659i	$0.182582\pi$
$163$	13.8564i	1.08532i	−0.839953	+	0.542659i	$0.817418\pi$
$164$	0	0
$165$	− 6.00000i	− 0.467099i
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	10.3923i	0.794719i
$172$	0	0
$173$	6.00000i	0.456172i	0.973641	+	0.228086i	$0.0732467\pi$
$173$	6.00000i	0.456172i	−0.973641	+	0.228086i	$0.926753\pi$
$174$	0	0
$175$	− 3.46410i	− 0.261861i
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	−24.2487	−1.81243	−0.906217	−	0.422813i	$-0.861043\pi$
$179$	−24.2487	−1.81243	−0.906217	+	0.422813i	$0.861043\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	17.3205	1.28037
$184$	0	0
$185$	4.00000i	0.294086i
$186$	0	0
$187$	− 20.7846i	− 1.51992i
$188$	0	0
$189$	18.0000i	1.30931i
$190$	0	0
$191$	−20.7846	−1.50392	−0.751961	−	0.659208i	$-0.770892\pi$
$191$	−20.7846	−1.50392	−0.751961	+	0.659208i	$0.770892\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	− 6.92820i	− 0.496139i
$196$	0	0
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	− 24.2487i	− 1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$	− 24.2487i	− 1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$	0	0
$201$	12.0000i	0.846415i
$202$	0	0
$203$	20.7846	1.45879
$204$	0	0
$205$	12.0000	0.838116
$206$	0	0
$207$	10.3923	0.722315
$208$	0	0
$209$	− 12.0000i	− 0.830057i
$210$	0	0
$211$	− 17.3205i	− 1.19239i	−0.802839	−	0.596196i	$-0.796678\pi$
$211$	− 17.3205i	− 1.19239i	0.802839	−	0.596196i	$-0.203322\pi$
$212$	0	0
$213$	24.0000	1.64445
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	−3.46410	−0.234082
$220$	0	0
$221$	− 24.0000i	− 1.61441i
$222$	0	0
$223$	− 17.3205i	− 1.15987i	−0.814664	−	0.579934i	$-0.803079\pi$
$223$	− 17.3205i	− 1.15987i	0.814664	−	0.579934i	$-0.196921\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−3.46410	−0.229920	−0.114960	−	0.993370i	$-0.536674\pi$
$227$	−3.46410	−0.229920	−0.114960	+	0.993370i	$0.536674\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	− 20.7846i	− 1.36753i
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	− 3.46410i	− 0.225973i
$236$	0	0
$237$	18.0000i	1.16923i
$238$	0	0
$239$	−13.8564	−0.896296	−0.448148	−	0.893959i	$-0.647916\pi$
$239$	−13.8564	−0.896296	−0.448148	+	0.893959i	$0.647916\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	15.5885	1.00000
$244$	0	0
$245$	− 5.00000i	− 0.319438i
$246$	0	0
$247$	− 13.8564i	− 0.881662i
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	−3.46410	−0.218652	−0.109326	−	0.994006i	$-0.534869\pi$
$251$	−3.46410	−0.218652	−0.109326	+	0.994006i	$0.534869\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	−10.3923	−0.650791
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	13.8564i	0.860995i
$260$	0	0
$261$	− 18.0000i	− 1.11417i
$262$	0	0
$263$	3.46410	0.213606	0.106803	−	0.994280i	$-0.465939\pi$
$263$	3.46410	0.213606	0.106803	+	0.994280i	$0.465939\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.00000i	− 0.365826i	−0.983129	−	0.182913i	$-0.941447\pi$
$269$	− 6.00000i	− 0.365826i	0.983129	−	0.182913i	$-0.0585527\pi$
$270$	0	0
$271$	10.3923i	0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$	10.3923i	0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$	0	0
$273$	− 24.0000i	− 1.45255i
$274$	0	0
$275$	3.46410	0.208893
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	0	0
$279$	10.3923i	0.622171i
$280$	0	0
$281$	12.0000i	0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$	12.0000i	0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$	0	0
$283$	− 6.92820i	− 0.411839i	−0.978569	−	0.205919i	$-0.933982\pi$
$283$	− 6.92820i	− 0.411839i	0.978569	−	0.205919i	$-0.0660185\pi$
$284$	0	0
$285$	−6.00000	−0.355409
$286$	0	0
$287$	41.5692	2.45375
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	17.3205	1.01535
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	− 3.46410i	− 0.201688i
$296$	0	0
$297$	−18.0000	−1.04447
$298$	0	0
$299$	−13.8564	−0.801337
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	10.3923i	0.597022i
$304$	0	0
$305$	10.0000i	0.572598i
$306$	0	0
$307$	− 6.92820i	− 0.395413i	−0.980261	−	0.197707i	$-0.936651\pi$
$307$	− 6.92820i	− 0.395413i	0.980261	−	0.197707i	$-0.0633494\pi$
$308$	0	0
$309$	− 18.0000i	− 1.02398i
$310$	0	0
$311$	13.8564	0.785725	0.392862	−	0.919597i	$-0.371485\pi$
$311$	13.8564	0.785725	0.392862	+	0.919597i	$0.371485\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	−10.3923	−0.585540
$316$	0	0
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	20.7846i	1.16371i
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−17.3205	−0.957826
$328$	0	0
$329$	− 12.0000i	− 0.661581i
$330$	0	0
$331$	10.3923i	0.571213i	0.958347	+	0.285606i	$0.0921950\pi$
$331$	10.3923i	0.571213i	−0.958347	+	0.285606i	$0.907805\pi$
$332$	0	0
$333$	12.0000	0.657596
$334$	0	0
$335$	−6.92820	−0.378528
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	− 10.3923i	− 0.564433i
$340$	0	0
$341$	− 12.0000i	− 0.649836i
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	6.00000i	0.323029i
$346$	0	0
$347$	10.3923	0.557888	0.278944	−	0.960307i	$-0.410016\pi$
$347$	10.3923	0.557888	0.278944	+	0.960307i	$0.410016\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	0	0
$351$	−20.7846	−1.10940
$352$	0	0
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	13.8564i	0.735422i
$356$	0	0
$357$	−36.0000	−1.90532
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	7.00000	0.368421
$362$	0	0
$363$	1.73205	0.0909091
$364$	0	0
$365$	− 2.00000i	− 0.104685i
$366$	0	0
$367$	− 3.46410i	− 0.180825i	−0.995904	−	0.0904123i	$-0.971182\pi$
$367$	− 3.46410i	− 0.180825i	0.995904	−	0.0904123i	$-0.0288185\pi$
$368$	0	0
$369$	− 36.0000i	− 1.87409i
$370$	0	0
$371$	20.7846	1.07908
$372$	0	0
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	0	0
$375$	− 1.73205i	− 0.0894427i
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	− 24.2487i	− 1.24557i	−0.782392	−	0.622786i	$-0.786001\pi$
$379$	− 24.2487i	− 1.24557i	0.782392	−	0.622786i	$-0.213999\pi$
$380$	0	0
$381$	18.0000i	0.922168i
$382$	0	0
$383$	17.3205	0.885037	0.442518	−	0.896759i	$-0.354085\pi$
$383$	17.3205	0.885037	0.442518	+	0.896759i	$0.354085\pi$
$384$	0	0
$385$	12.0000	0.611577
$386$	0	0
$387$	20.7846i	1.05654i
$388$	0	0
$389$	6.00000i	0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$	6.00000i	0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$	0	0
$391$	20.7846i	1.05112i
$392$	0	0
$393$	30.0000	1.51330
$394$	0	0
$395$	−10.3923	−0.522894
$396$	0	0
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	0	0
$399$	−20.7846	−1.04053
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 13.8564i	− 0.690237i
$404$	0	0
$405$	9.00000i	0.447214i
$406$	0	0
$407$	−13.8564	−0.686837
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	− 31.1769i	− 1.53784i
$412$	0	0
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	10.3923i	0.510138i
$416$	0	0
$417$	− 18.0000i	− 0.881464i
$418$	0	0
$419$	−10.3923	−0.507697	−0.253849	−	0.967244i	$-0.581697\pi$
$419$	−10.3923	−0.507697	−0.253849	+	0.967244i	$0.581697\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−10.3923	−0.505291
$424$	0	0
$425$	− 6.00000i	− 0.291043i
$426$	0	0
$427$	34.6410i	1.67640i
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	10.3923	0.498273
$436$	0	0
$437$	12.0000i	0.574038i
$438$	0	0
$439$	3.46410i	0.165333i	0.996577	+	0.0826663i	$0.0263436\pi$
$439$	3.46410i	0.165333i	−0.996577	+	0.0826663i	$0.973656\pi$
$440$	0	0
$441$	−15.0000	−0.714286
$442$	0	0
$443$	17.3205	0.822922	0.411461	−	0.911427i	$-0.365019\pi$
$443$	17.3205	0.822922	0.411461	+	0.911427i	$0.365019\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.3923i	0.491539i
$448$	0	0
$449$	36.0000i	1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$	36.0000i	1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$	0	0
$451$	41.5692i	1.95742i
$452$	0	0
$453$	− 6.00000i	− 0.281905i
$454$	0	0
$455$	13.8564	0.649598
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	0	0
$459$	31.1769i	1.45521i
$460$	0	0
$461$	18.0000i	0.838344i	0.907907	+	0.419172i	$0.137680\pi$
$461$	18.0000i	0.838344i	−0.907907	+	0.419172i	$0.862320\pi$
$462$	0	0
$463$	10.3923i	0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$	10.3923i	0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	0	0
$465$	−6.00000	−0.278243
$466$	0	0
$467$	−24.2487	−1.12210	−0.561048	−	0.827783i	$-0.689602\pi$
$467$	−24.2487	−1.12210	−0.561048	+	0.827783i	$0.689602\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	27.7128	1.27694
$472$	0	0
$473$	− 24.0000i	− 1.10352i
$474$	0	0
$475$	− 3.46410i	− 0.158944i
$476$	0	0
$477$	− 18.0000i	− 0.824163i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	20.7846i	0.945732i
$484$	0	0
$485$	10.0000i	0.454077i
$486$	0	0
$487$	17.3205i	0.784867i	0.919780	+	0.392434i	$0.128367\pi$
$487$	17.3205i	0.784867i	−0.919780	+	0.392434i	$0.871633\pi$
$488$	0	0
$489$	24.0000i	1.08532i
$490$	0	0
$491$	−31.1769	−1.40699	−0.703497	−	0.710698i	$-0.748379\pi$
$491$	−31.1769	−1.40699	−0.703497	+	0.710698i	$0.748379\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	− 10.3923i	− 0.467099i
$496$	0	0
$497$	48.0000i	2.15309i
$498$	0	0
$499$	3.46410i	0.155074i	0.996989	+	0.0775372i	$0.0247057\pi$
$499$	3.46410i	0.155074i	−0.996989	+	0.0775372i	$0.975294\pi$
$500$	0	0
$501$	−6.00000	−0.268060
$502$	0	0
$503$	−10.3923	−0.463370	−0.231685	−	0.972791i	$-0.574424\pi$
$503$	−10.3923	−0.463370	−0.231685	+	0.972791i	$0.574424\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	5.19615	0.230769
$508$	0	0
$509$	− 6.00000i	− 0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$	− 6.00000i	− 0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$	0	0
$511$	− 6.92820i	− 0.306486i
$512$	0	0
$513$	18.0000i	0.794719i
$514$	0	0
$515$	10.3923	0.457940
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	10.3923i	0.456172i
$520$	0	0
$521$	− 12.0000i	− 0.525730i	−0.964833	−	0.262865i	$-0.915333\pi$
$521$	− 12.0000i	− 0.525730i	0.964833	−	0.262865i	$-0.0846673\pi$
$522$	0	0
$523$	34.6410i	1.51475i	0.652983	+	0.757373i	$0.273517\pi$
$523$	34.6410i	1.51475i	−0.652983	+	0.757373i	$0.726483\pi$
$524$	0	0
$525$	− 6.00000i	− 0.261861i
$526$	0	0
$527$	−20.7846	−0.905392
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−10.3923	−0.450988
$532$	0	0
$533$	48.0000i	2.07911i
$534$	0	0
$535$	3.46410i	0.149766i
$536$	0	0
$537$	−42.0000	−1.81243
$538$	0	0
$539$	17.3205	0.746047
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	3.46410	0.148659
$544$	0	0
$545$	− 10.0000i	− 0.428353i
$546$	0	0
$547$	− 20.7846i	− 0.888686i	−0.895857	−	0.444343i	$-0.853437\pi$
$547$	− 20.7846i	− 0.888686i	0.895857	−	0.444343i	$-0.146563\pi$
$548$	0	0
$549$	30.0000	1.28037
$550$	0	0
$551$	20.7846	0.885454
$552$	0	0
$553$	−36.0000	−1.53088
$554$	0	0
$555$	6.92820i	0.294086i
$556$	0	0
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	− 27.7128i	− 1.17213i
$560$	0	0
$561$	− 36.0000i	− 1.51992i
$562$	0	0
$563$	3.46410	0.145994	0.0729972	−	0.997332i	$-0.476744\pi$
$563$	3.46410	0.145994	0.0729972	+	0.997332i	$0.476744\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	31.1769i	1.30931i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	24.2487i	1.01478i	0.861717	+	0.507388i	$0.169389\pi$
$571$	24.2487i	1.01478i	−0.861717	+	0.507388i	$0.830611\pi$
$572$	0	0
$573$	−36.0000	−1.50392
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	3.46410	0.143963
$580$	0	0
$581$	36.0000i	1.49353i
$582$	0	0
$583$	20.7846i	0.860811i
$584$	0	0
$585$	− 12.0000i	− 0.496139i
$586$	0	0
$587$	31.1769	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	10.3923i	0.427482i
$592$	0	0
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	− 20.7846i	− 0.852086i
$596$	0	0
$597$	− 42.0000i	− 1.71895i
$598$	0	0
$599$	−20.7846	−0.849236	−0.424618	−	0.905373i	$-0.639592\pi$
$599$	−20.7846	−0.849236	−0.424618	+	0.905373i	$0.639592\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	20.7846i	0.846415i
$604$	0	0
$605$	1.00000i	0.0406558i
$606$	0	0
$607$	− 31.1769i	− 1.26543i	−0.774384	−	0.632716i	$-0.781940\pi$
$607$	− 31.1769i	− 1.26543i	0.774384	−	0.632716i	$-0.218060\pi$
$608$	0	0
$609$	36.0000	1.45879
$610$	0	0
$611$	13.8564	0.560570
$612$	0	0
$613$	−32.0000	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$613$	−32.0000	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$614$	0	0
$615$	20.7846	0.838116
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	0	0
$619$	45.0333i	1.81004i	0.425367	+	0.905021i	$0.360145\pi$
$619$	45.0333i	1.81004i	−0.425367	+	0.905021i	$0.639855\pi$
$620$	0	0
$621$	18.0000	0.722315
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 20.7846i	− 0.830057i
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	3.46410i	0.137904i	0.997620	+	0.0689519i	$0.0219655\pi$
$631$	3.46410i	0.137904i	−0.997620	+	0.0689519i	$0.978035\pi$
$632$	0	0
$633$	− 30.0000i	− 1.19239i
$634$	0	0
$635$	−10.3923	−0.412406
$636$	0	0
$637$	20.0000	0.792429
$638$	0	0
$639$	41.5692	1.64445
$640$	0	0
$641$	− 24.0000i	− 0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$	− 24.0000i	− 0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$	0	0
$643$	− 41.5692i	− 1.63933i	−0.572843	−	0.819665i	$-0.694160\pi$
$643$	− 41.5692i	− 1.63933i	0.572843	−	0.819665i	$-0.305840\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−45.0333	−1.77044	−0.885221	−	0.465170i	$-0.845993\pi$
$647$	−45.0333	−1.77044	−0.885221	+	0.465170i	$0.845993\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	−20.7846	−0.814613
$652$	0	0
$653$	− 42.0000i	− 1.64359i	−0.569785	−	0.821794i	$-0.692974\pi$
$653$	− 42.0000i	− 1.64359i	0.569785	−	0.821794i	$-0.307026\pi$
$654$	0	0
$655$	17.3205i	0.676768i
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	31.1769	1.21448	0.607240	−	0.794518i	$-0.292277\pi$
$659$	31.1769	1.21448	0.607240	+	0.794518i	$0.292277\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	− 41.5692i	− 1.61441i
$664$	0	0
$665$	− 12.0000i	− 0.465340i
$666$	0	0
$667$	− 20.7846i	− 0.804783i
$668$	0	0
$669$	− 30.0000i	− 1.15987i
$670$	0	0
$671$	−34.6410	−1.33730
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	0	0
$675$	−5.19615	−0.200000
$676$	0	0
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	34.6410i	1.32940i
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	−17.3205	−0.662751	−0.331375	−	0.943499i	$-0.607513\pi$
$683$	−17.3205	−0.662751	−0.331375	+	0.943499i	$0.607513\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	24.2487	0.925146
$688$	0	0
$689$	24.0000i	0.914327i
$690$	0	0
$691$	− 31.1769i	− 1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$	− 31.1769i	− 1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$	0	0
$693$	− 36.0000i	− 1.36753i
$694$	0	0
$695$	10.3923	0.394203
$696$	0	0
$697$	72.0000	2.72719
$698$	0	0
$699$	− 10.3923i	− 0.393073i
$700$	0	0
$701$	42.0000i	1.58632i	0.609015	+	0.793159i	$0.291565\pi$
$701$	42.0000i	1.58632i	−0.609015	+	0.793159i	$0.708435\pi$
$702$	0	0
$703$	13.8564i	0.522604i
$704$	0	0
$705$	− 6.00000i	− 0.225973i
$706$	0	0
$707$	−20.7846	−0.781686
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	31.1769i	1.16923i
$712$	0	0
$713$	12.0000i	0.449404i
$714$	0	0
$715$	13.8564i	0.518200i
$716$	0	0
$717$	−24.0000	−0.896296
$718$	0	0
$719$	27.7128	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$719$	27.7128	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$720$	0	0
$721$	36.0000	1.34071
$722$	0	0
$723$	−45.0333	−1.67481
$724$	0	0
$725$	6.00000i	0.222834i
$726$	0	0
$727$	3.46410i	0.128476i	0.997935	+	0.0642382i	$0.0204617\pi$
$727$	3.46410i	0.128476i	−0.997935	+	0.0642382i	$0.979538\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	−41.5692	−1.53749
$732$	0	0
$733$	−28.0000	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$733$	−28.0000	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$734$	0	0
$735$	− 8.66025i	− 0.319438i
$736$	0	0
$737$	− 24.0000i	− 0.884051i
$738$	0	0
$739$	− 10.3923i	− 0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$	− 10.3923i	− 0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$	0	0
$741$	− 24.0000i	− 0.881662i
$742$	0	0
$743$	−38.1051	−1.39794	−0.698971	−	0.715150i	$-0.746358\pi$
$743$	−38.1051	−1.39794	−0.698971	+	0.715150i	$0.746358\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	0	0
$747$	31.1769	1.14070
$748$	0	0
$749$	12.0000i	0.438470i
$750$	0	0
$751$	− 51.9615i	− 1.89610i	−0.318117	−	0.948051i	$-0.603050\pi$
$751$	− 51.9615i	− 1.89610i	0.318117	−	0.948051i	$-0.396950\pi$
$752$	0	0
$753$	−6.00000	−0.218652
$754$	0	0
$755$	3.46410	0.126072
$756$	0	0
$757$	20.0000	0.726912	0.363456	−	0.931611i	$-0.381597\pi$
$757$	20.0000	0.726912	0.363456	+	0.931611i	$0.381597\pi$
$758$	0	0
$759$	−20.7846	−0.754434
$760$	0	0
$761$	− 12.0000i	− 0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$	− 12.0000i	− 0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$	0	0
$763$	− 34.6410i	− 1.25409i
$764$	0	0
$765$	−18.0000	−0.650791
$766$	0	0
$767$	13.8564	0.500326
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	− 31.1769i	− 1.12281i
$772$	0	0
$773$	54.0000i	1.94225i	0.238581	+	0.971123i	$0.423318\pi$
$773$	54.0000i	1.94225i	−0.238581	+	0.971123i	$0.576682\pi$
$774$	0	0
$775$	− 3.46410i	− 0.124434i
$776$	0	0
$777$	24.0000i	0.860995i
$778$	0	0
$779$	41.5692	1.48937
$780$	0	0
$781$	−48.0000	−1.71758
$782$	0	0
$783$	− 31.1769i	− 1.11417i
$784$	0	0
$785$	16.0000i	0.571064i
$786$	0	0
$787$	34.6410i	1.23482i	0.786642	+	0.617409i	$0.211818\pi$
$787$	34.6410i	1.23482i	−0.786642	+	0.617409i	$0.788182\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	−40.0000	−1.42044
$794$	0	0
$795$	10.3923	0.368577
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	− 20.7846i	− 0.735307i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.92820	0.244491
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	− 10.3923i	− 0.365826i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	24.2487i	0.851487i	0.904844	+	0.425744i	$0.139987\pi$
$811$	24.2487i	0.851487i	−0.904844	+	0.425744i	$0.860013\pi$
$812$	0	0
$813$	18.0000i	0.631288i
$814$	0	0
$815$	−13.8564	−0.485369
$816$	0	0
$817$	−24.0000	−0.839654
$818$	0	0
$819$	− 41.5692i	− 1.45255i
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	45.0333i	1.56976i	0.619646	+	0.784881i	$0.287276\pi$
$823$	45.0333i	1.56976i	−0.619646	+	0.784881i	$0.712724\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	38.1051	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$827$	38.1051	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	−6.92820	−0.240337
$832$	0	0
$833$	− 30.0000i	− 1.03944i
$834$	0	0
$835$	− 3.46410i	− 0.119880i
$836$	0	0
$837$	18.0000i	0.622171i
$838$	0	0
$839$	48.4974	1.67432	0.837158	−	0.546960i	$-0.184215\pi$
$839$	48.4974	1.67432	0.837158	+	0.546960i	$0.184215\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	20.7846i	0.715860i
$844$	0	0
$845$	3.00000i	0.103203i
$846$	0	0
$847$	3.46410i	0.119028i
$848$	0	0
$849$	− 12.0000i	− 0.411839i
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	−8.00000	−0.273915	−0.136957	−	0.990577i	$-0.543732\pi$
$853$	−8.00000	−0.273915	−0.136957	+	0.990577i	$0.543732\pi$
$854$	0	0
$855$	−10.3923	−0.355409
$856$	0	0
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	0	0
$859$	− 51.9615i	− 1.77290i	−0.462820	−	0.886452i	$-0.653162\pi$
$859$	− 51.9615i	− 1.77290i	0.462820	−	0.886452i	$-0.346838\pi$
$860$	0	0
$861$	72.0000	2.45375
$862$	0	0
$863$	−51.9615	−1.76879	−0.884395	−	0.466738i	$-0.845429\pi$
$863$	−51.9615	−1.76879	−0.884395	+	0.466738i	$0.845429\pi$
$864$	0	0
$865$	−6.00000	−0.204006
$866$	0	0
$867$	−32.9090	−1.11765
$868$	0	0
$869$	− 36.0000i	− 1.22122i
$870$	0	0
$871$	− 27.7128i	− 0.939013i
$872$	0	0
$873$	30.0000	1.01535
$874$	0	0
$875$	3.46410	0.117108
$876$	0	0
$877$	−32.0000	−1.08056	−0.540282	−	0.841484i	$-0.681682\pi$
$877$	−32.0000	−1.08056	−0.540282	+	0.841484i	$0.681682\pi$
$878$	0	0
$879$	− 10.3923i	− 0.350524i
$880$	0	0
$881$	48.0000i	1.61716i	0.588386	+	0.808581i	$0.299764\pi$
$881$	48.0000i	1.61716i	−0.588386	+	0.808581i	$0.700236\pi$
$882$	0	0
$883$	− 13.8564i	− 0.466305i	−0.972440	−	0.233153i	$-0.925096\pi$
$883$	− 13.8564i	− 0.466305i	0.972440	−	0.233153i	$-0.0749042\pi$
$884$	0	0
$885$	− 6.00000i	− 0.201688i
$886$	0	0
$887$	24.2487	0.814192	0.407096	−	0.913385i	$-0.366541\pi$
$887$	24.2487	0.814192	0.407096	+	0.913385i	$0.366541\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	−31.1769	−1.04447
$892$	0	0
$893$	− 12.0000i	− 0.401565i
$894$	0	0
$895$	− 24.2487i	− 0.810545i
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	20.7846	0.693206
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	−41.5692	−1.38334
$904$	0	0
$905$	2.00000i	0.0664822i
$906$	0	0
$907$	41.5692i	1.38028i	0.723674	+	0.690142i	$0.242452\pi$
$907$	41.5692i	1.38028i	−0.723674	+	0.690142i	$0.757548\pi$
$908$	0	0
$909$	18.0000i	0.597022i
$910$	0	0
$911$	−34.6410	−1.14771	−0.573854	−	0.818958i	$-0.694552\pi$
$911$	−34.6410	−1.14771	−0.573854	+	0.818958i	$0.694552\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	17.3205i	0.572598i
$916$	0	0
$917$	60.0000i	1.98137i
$918$	0	0
$919$	− 45.0333i	− 1.48551i	−0.669562	−	0.742756i	$-0.733518\pi$
$919$	− 45.0333i	− 1.48551i	0.669562	−	0.742756i	$-0.266482\pi$
$920$	0	0
$921$	− 12.0000i	− 0.395413i
$922$	0	0
$923$	−55.4256	−1.82436
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	− 31.1769i	− 1.02398i
$928$	0	0
$929$	− 12.0000i	− 0.393707i	−0.980433	−	0.196854i	$-0.936928\pi$
$929$	− 12.0000i	− 0.393707i	0.980433	−	0.196854i	$-0.0630724\pi$
$930$	0	0
$931$	− 17.3205i	− 0.567657i
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	20.7846	0.679729
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	0	0
$939$	24.2487	0.791327
$940$	0	0
$941$	− 54.0000i	− 1.76035i	−0.474650	−	0.880175i	$-0.657425\pi$
$941$	− 54.0000i	− 1.76035i	0.474650	−	0.880175i	$-0.342575\pi$
$942$	0	0
$943$	− 41.5692i	− 1.35368i
$944$	0	0
$945$	−18.0000	−0.585540
$946$	0	0
$947$	−31.1769	−1.01311	−0.506557	−	0.862207i	$-0.669082\pi$
$947$	−31.1769	−1.01311	−0.506557	+	0.862207i	$0.669082\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	31.1769i	1.01098i
$952$	0	0
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	− 20.7846i	− 0.672574i
$956$	0	0
$957$	36.0000i	1.16371i
$958$	0	0
$959$	62.3538	2.01351
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	10.3923	0.334887
$964$	0	0
$965$	2.00000i	0.0643823i
$966$	0	0
$967$	− 51.9615i	− 1.67097i	−0.549513	−	0.835485i	$-0.685187\pi$
$967$	− 51.9615i	− 1.67097i	0.549513	−	0.835485i	$-0.314813\pi$
$968$	0	0
$969$	−36.0000	−1.15649
$970$	0	0
$971$	−3.46410	−0.111168	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410	−0.111168	−0.0555842	+	0.998454i	$0.517702\pi$
$972$	0	0
$973$	36.0000	1.15411
$974$	0	0
$975$	6.92820	0.221880
$976$	0	0
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−30.0000	−0.957826
$982$	0	0
$983$	58.8897	1.87829	0.939145	−	0.343520i	$-0.111619\pi$
$983$	58.8897	1.87829	0.939145	+	0.343520i	$0.111619\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	− 20.7846i	− 0.661581i
$988$	0	0
$989$	24.0000i	0.763156i
$990$	0	0
$991$	− 17.3205i	− 0.550204i	−0.961415	−	0.275102i	$-0.911288\pi$
$991$	− 17.3205i	− 0.550204i	0.961415	−	0.275102i	$-0.0887116\pi$
$992$	0	0
$993$	18.0000i	0.571213i
$994$	0	0
$995$	24.2487	0.768736
$996$	0	0
$997$	52.0000	1.64686	0.823428	−	0.567420i	$-0.192059\pi$
$997$	52.0000	1.64686	0.823428	+	0.567420i	$0.192059\pi$
$998$	0	0
$999$	20.7846	0.657596

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.h.d.191.4		4
3.2	odd	2	inner	960.2.h.d.191.1		4
4.3	odd	2	inner	960.2.h.d.191.2		4
8.3	odd	2		240.2.h.b.191.3	yes	4
8.5	even	2		240.2.h.b.191.1	✓	4
12.11	even	2	inner	960.2.h.d.191.3		4
24.5	odd	2		240.2.h.b.191.4	yes	4
24.11	even	2		240.2.h.b.191.2	yes	4
40.3	even	4		1200.2.o.b.1199.3		4
40.13	odd	4		1200.2.o.b.1199.2		4
40.19	odd	2		1200.2.h.m.1151.2		4
40.27	even	4		1200.2.o.a.1199.2		4
40.29	even	2		1200.2.h.m.1151.3		4
40.37	odd	4		1200.2.o.a.1199.3		4
120.29	odd	2		1200.2.h.m.1151.1		4
120.53	even	4		1200.2.o.a.1199.4		4
120.59	even	2		1200.2.h.m.1151.4		4
120.77	even	4		1200.2.o.b.1199.1		4
120.83	odd	4		1200.2.o.a.1199.1		4
120.107	odd	4		1200.2.o.b.1199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.h.b.191.1	✓	4	8.5	even	2
240.2.h.b.191.2	yes	4	24.11	even	2
240.2.h.b.191.3	yes	4	8.3	odd	2
240.2.h.b.191.4	yes	4	24.5	odd	2
960.2.h.d.191.1		4	3.2	odd	2	inner
960.2.h.d.191.2		4	4.3	odd	2	inner
960.2.h.d.191.3		4	12.11	even	2	inner
960.2.h.d.191.4		4	1.1	even	1	trivial
1200.2.h.m.1151.1		4	120.29	odd	2
1200.2.h.m.1151.2		4	40.19	odd	2
1200.2.h.m.1151.3		4	40.29	even	2
1200.2.h.m.1151.4		4	120.59	even	2
1200.2.o.a.1199.1		4	120.83	odd	4
1200.2.o.a.1199.2		4	40.27	even	4
1200.2.o.a.1199.3		4	40.37	odd	4
1200.2.o.a.1199.4		4	120.53	even	4
1200.2.o.b.1199.1		4	120.77	even	4
1200.2.o.b.1199.2		4	40.13	odd	4
1200.2.o.b.1199.3		4	40.3	even	4
1200.2.o.b.1199.4		4	120.107	odd	4

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.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +1.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +1.00000i q^{5} +3.46410i q^{7} +3.00000 q^{9} -3.46410 q^{11} -4.00000 q^{13} +1.73205i q^{15} +6.00000i q^{17} +3.46410i q^{19} +6.00000i q^{21} +3.46410 q^{23} -1.00000 q^{25} +5.19615 q^{27} -6.00000i q^{29} +3.46410i q^{31} -6.00000 q^{33} -3.46410 q^{35} +4.00000 q^{37} -6.92820 q^{39} -12.0000i q^{41} +6.92820i q^{43} +3.00000i q^{45} -3.46410 q^{47} -5.00000 q^{49} +10.3923i q^{51} -6.00000i q^{53} -3.46410i q^{55} +6.00000i q^{57} -3.46410 q^{59} +10.0000 q^{61} +10.3923i q^{63} -4.00000i q^{65} +6.92820i q^{67} +6.00000 q^{69} +13.8564 q^{71} -2.00000 q^{73} -1.73205 q^{75} -12.0000i q^{77} +10.3923i q^{79} +9.00000 q^{81} +10.3923 q^{83} -6.00000 q^{85} -10.3923i q^{87} -13.8564i q^{91} +6.00000i q^{93} -3.46410 q^{95} +10.0000 q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 16 q^{13} - 4 q^{25} - 24 q^{33} + 16 q^{37} - 20 q^{49} + 40 q^{61} + 24 q^{69} - 8 q^{73} + 36 q^{81} - 24 q^{85} + 40 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 16 * q^13 - 4 * q^25 - 24 * q^33 + 16 * q^37 - 20 * q^49 + 40 * q^61 + 24 * q^69 - 8 * q^73 + 36 * q^81 - 24 * q^85 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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