LMFDB - Embedded newform 1200.2.o.b.1199.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1200,2,Mod(1199,1200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1200.1199"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1200.o (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,24,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.58204824255$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1199.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.1199
Dual form			1200.2.o.b.1199.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -3.46410 q^{7} -3.00000 q^{9} -3.46410 q^{11} +4.00000i q^{13} +6.00000 q^{17} -3.46410i q^{19} -6.00000i q^{21} -3.46410i q^{23} -5.19615i q^{27} -6.00000i q^{29} -3.46410i q^{31} -6.00000i q^{33} +4.00000i q^{37} -6.92820 q^{39} -12.0000i q^{41} -6.92820 q^{43} -3.46410i q^{47} +5.00000 q^{49} +10.3923i q^{51} -6.00000 q^{53} +6.00000 q^{57} +3.46410 q^{59} -10.0000 q^{61} +10.3923 q^{63} +6.92820 q^{67} +6.00000 q^{69} -13.8564 q^{71} -2.00000i q^{73} +12.0000 q^{77} +10.3923i q^{79} +9.00000 q^{81} +10.3923i q^{83} +10.3923 q^{87} -13.8564i q^{91} +6.00000 q^{93} -10.0000i q^{97} +10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 12 q^{9} + 24 q^{17} + 20 q^{49} - 24 q^{53} + 24 q^{57} - 40 q^{61} + 24 q^{69} + 48 q^{77} + 36 q^{81} + 24 q^{93}+O(q^{100})$ 4 * q - 12 * q^9 + 24 * q^17 + 20 * q^49 - 24 * q^53 + 24 * q^57 - 40 * q^61 + 24 * q^69 + 48 * q^77 + 36 * q^81 + 24 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$ .

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	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
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	− 3.46410i	− 0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$	− 3.46410i	− 0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$	0	0
$21$	− 6.00000i	− 1.30931i
$22$	0	0
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	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	$-0.899307\pi$
$31$	− 3.46410i	− 0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$	0	0
$33$	− 6.00000i	− 1.04447i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	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
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.46410i	− 0.505291i	−0.967559	−	0.252646i	$-0.918699\pi$
$47$	− 3.46410i	− 0.505291i	0.967559	−	0.252646i	$-0.0813007\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	10.3923i	1.45521i
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	3.46410	0.450988	0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410	0.450988	0.225494	+	0.974245i	$0.427600\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	10.3923	1.30931
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.92820	0.846415	0.423207	−	0.906033i	$-0.360904\pi$
$67$	6.92820	0.846415	0.423207	+	0.906033i	$0.360904\pi$
$68$	0	0
$69$	6.00000	0.722315
$70$	0	0
$71$	−13.8564	−1.64445	−0.822226	−	0.569160i	$-0.807268\pi$
$71$	−13.8564	−1.64445	−0.822226	+	0.569160i	$0.807268\pi$
$72$	0	0
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.0000	1.36753
$78$	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
$83$	10.3923i	1.14070i	0.821401	+	0.570352i	$0.193193\pi$
$83$	10.3923i	1.14070i	−0.821401	+	0.570352i	$0.806807\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	10.3923	1.11417
$88$	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
$93$	6.00000	0.622171
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	0	0
$99$	10.3923	1.04447
$100$	0	0
$101$	− 6.00000i	− 0.597022i	−0.954406	−	0.298511i	$-0.903510\pi$
$101$	− 6.00000i	− 0.597022i	0.954406	−	0.298511i	$-0.0964900\pi$
$102$	0	0
$103$	−10.3923	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$103$	−10.3923	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 3.46410i	− 0.334887i	−0.985882	−	0.167444i	$-0.946449\pi$
$107$	− 3.46410i	− 0.334887i	0.985882	−	0.167444i	$-0.0535512\pi$
$108$	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
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 12.0000i	− 1.10940i
$118$	0	0
$119$	−20.7846	−1.90532
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.7846	1.87409
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.3923	−0.922168	−0.461084	−	0.887357i	$-0.652539\pi$
$127$	−10.3923	−0.922168	−0.461084	+	0.887357i	$0.652539\pi$
$128$	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
$133$	12.0000i	1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	10.3923i	0.881464i	0.897639	+	0.440732i	$0.145281\pi$
$139$	10.3923i	0.881464i	−0.897639	+	0.440732i	$0.854719\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	− 13.8564i	− 1.15873i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	8.66025i	0.714286i
$148$	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	$0.0450164\pi$
$151$	3.46410i	0.281905i	−0.990016	+	0.140952i	$0.954984\pi$
$152$	0	0
$153$	−18.0000	−1.45521
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0000i	1.27694i	0.769647	+	0.638470i	$0.220432\pi$
$157$	16.0000i	1.27694i	−0.769647	+	0.638470i	$0.779568\pi$
$158$	0	0
$159$	− 10.3923i	− 0.824163i
$160$	0	0
$161$	12.0000i	0.945732i
$162$	0	0
$163$	−13.8564	−1.08532	−0.542659	−	0.839953i	$-0.682582\pi$
$163$	−13.8564	−1.08532	−0.542659	+	0.839953i	$0.682582\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.46410i	− 0.268060i	−0.990977	−	0.134030i	$-0.957208\pi$
$167$	− 3.46410i	− 0.268060i	0.990977	−	0.134030i	$-0.0427919\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	10.3923i	0.794719i
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.00000i	0.450988i
$178$	0	0
$179$	24.2487	1.81243	0.906217	−	0.422813i	$-0.138957\pi$
$179$	24.2487	1.81243	0.906217	+	0.422813i	$0.138957\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 17.3205i	− 1.28037i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.7846	−1.51992
$188$	0	0
$189$	18.0000i	1.30931i
$190$	0	0
$191$	20.7846	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$191$	20.7846	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	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
$203$	20.7846i	1.45879i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10.3923i	0.722315i
$208$	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
$213$	− 24.0000i	− 1.64445i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0000i	0.814613i
$218$	0	0
$219$	3.46410	0.234082
$220$	0	0
$221$	24.0000i	1.61441i
$222$	0	0
$223$	−17.3205	−1.15987	−0.579934	−	0.814664i	$-0.696921\pi$
$223$	−17.3205	−1.15987	−0.579934	+	0.814664i	$0.696921\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.46410i	0.229920i	0.993370	+	0.114960i	$0.0366741\pi$
$227$	3.46410i	0.229920i	−0.993370	+	0.114960i	$0.963326\pi$
$228$	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
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−18.0000	−1.16923
$238$	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
$243$	15.5885i	1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.8564	0.881662
$248$	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
$253$	12.0000i	0.754434i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	− 13.8564i	− 0.860995i
$260$	0	0
$261$	18.0000i	1.11417i
$262$	0	0
$263$	− 3.46410i	− 0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$263$	− 3.46410i	− 0.213606i	0.994280	−	0.106803i	$-0.0340614\pi$
$264$	0	0
$265$	0	0
$266$	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	$-0.897780\pi$
$271$	− 10.3923i	− 0.631288i	0.948878	−	0.315644i	$-0.102220\pi$
$272$	0	0
$273$	24.0000	1.45255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	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
$283$	6.92820	0.411839	0.205919	−	0.978569i	$-0.433982\pi$
$283$	6.92820	0.411839	0.205919	+	0.978569i	$0.433982\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.5692i	2.45375i
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	17.3205	1.01535
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	18.0000i	1.04447i
$298$	0	0
$299$	13.8564	0.801337
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	10.3923	0.597022
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.92820	−0.395413	−0.197707	−	0.980261i	$-0.563349\pi$
$307$	−6.92820	−0.395413	−0.197707	+	0.980261i	$0.563349\pi$
$308$	0	0
$309$	− 18.0000i	− 1.02398i
$310$	0	0
$311$	−13.8564	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$311$	−13.8564	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	20.7846i	1.16371i
$320$	0	0
$321$	6.00000	0.334887
$322$	0	0
$323$	− 20.7846i	− 1.15649i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 17.3205i	− 0.957826i
$328$	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
$333$	− 12.0000i	− 0.657596i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34.0000i	− 1.85210i	−0.377403	−	0.926049i	$-0.623183\pi$
$337$	− 34.0000i	− 1.85210i	0.377403	−	0.926049i	$-0.376817\pi$
$338$	0	0
$339$	10.3923i	0.564433i
$340$	0	0
$341$	12.0000i	0.649836i
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 10.3923i	− 0.557888i	−0.960307	−	0.278944i	$-0.910016\pi$
$347$	− 10.3923i	− 0.557888i	0.960307	−	0.278944i	$-0.0899844\pi$
$348$	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
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 36.0000i	− 1.90532i
$358$	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
$363$	1.73205i	0.0909091i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.46410	0.180825	0.0904123	−	0.995904i	$-0.471182\pi$
$367$	3.46410	0.180825	0.0904123	+	0.995904i	$0.471182\pi$
$368$	0	0
$369$	36.0000i	1.87409i
$370$	0	0
$371$	20.7846	1.07908
$372$	0	0
$373$	− 16.0000i	− 0.828449i	−0.910175	−	0.414224i	$-0.864053\pi$
$373$	− 16.0000i	− 0.828449i	0.910175	−	0.414224i	$-0.135947\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000	1.23606
$378$	0	0
$379$	24.2487i	1.24557i	0.782392	+	0.622786i	$0.213999\pi$
$379$	24.2487i	1.24557i	−0.782392	+	0.622786i	$0.786001\pi$
$380$	0	0
$381$	− 18.0000i	− 0.922168i
$382$	0	0
$383$	− 17.3205i	− 0.885037i	−0.896759	−	0.442518i	$-0.854085\pi$
$383$	− 17.3205i	− 0.885037i	0.896759	−	0.442518i	$-0.145915\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20.7846	1.05654
$388$	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
$393$	30.0000i	1.51330i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	16.0000i	0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$	16.0000i	0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$	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
$403$	13.8564	0.690237
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 13.8564i	− 0.686837i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	− 31.1769i	− 1.53784i
$412$	0	0
$413$	−12.0000	−0.590481
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.0000	−0.881464
$418$	0	0
$419$	10.3923	0.507697	0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923	0.507697	0.253849	+	0.967244i	$0.418303\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	10.3923i	0.505291i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.6410	1.67640
$428$	0	0
$429$	24.0000	1.15873
$430$	0	0
$431$	−6.92820	−0.333720	−0.166860	−	0.985981i	$-0.553363\pi$
$431$	−6.92820	−0.333720	−0.166860	+	0.985981i	$0.553363\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.0000	−0.574038
$438$	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
$443$	17.3205i	0.822922i	0.911427	+	0.411461i	$0.134981\pi$
$443$	17.3205i	0.822922i	−0.911427	+	0.411461i	$0.865019\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−10.3923	−0.491539
$448$	0	0
$449$	− 36.0000i	− 1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$449$	− 36.0000i	− 1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$450$	0	0
$451$	41.5692i	1.95742i
$452$	0	0
$453$	−6.00000	−0.281905
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	0	0
$459$	− 31.1769i	− 1.45521i
$460$	0	0
$461$	− 18.0000i	− 0.838344i	−0.907907	−	0.419172i	$-0.862320\pi$
$461$	− 18.0000i	− 0.838344i	0.907907	−	0.419172i	$-0.137680\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.2487i	1.12210i	0.827783	+	0.561048i	$0.189602\pi$
$467$	24.2487i	1.12210i	−0.827783	+	0.561048i	$0.810398\pi$
$468$	0	0
$469$	−24.0000	−1.10822
$470$	0	0
$471$	−27.7128	−1.27694
$472$	0	0
$473$	24.0000	1.10352
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.0000	0.824163
$478$	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
$483$	−20.7846	−0.945732
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−17.3205	−0.784867	−0.392434	−	0.919780i	$-0.628367\pi$
$487$	−17.3205	−0.784867	−0.392434	+	0.919780i	$0.628367\pi$
$488$	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
$493$	− 36.0000i	− 1.62136i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	48.0000	2.15309
$498$	0	0
$499$	− 3.46410i	− 0.155074i	−0.996989	−	0.0775372i	$-0.975294\pi$
$499$	− 3.46410i	− 0.155074i	0.996989	−	0.0775372i	$-0.0247057\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	10.3923i	0.463370i	0.972791	+	0.231685i	$0.0744239\pi$
$503$	10.3923i	0.463370i	−0.972791	+	0.231685i	$0.925576\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 5.19615i	− 0.230769i
$508$	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
$513$	−18.0000	−0.794719
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	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
$523$	−34.6410	−1.51475	−0.757373	−	0.652983i	$-0.773517\pi$
$523$	−34.6410	−1.51475	−0.757373	+	0.652983i	$0.773517\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 20.7846i	− 0.905392i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	−10.3923	−0.450988
$532$	0	0
$533$	48.0000	2.07911
$534$	0	0
$535$	0	0
$536$	0	0
$537$	42.0000i	1.81243i
$538$	0	0
$539$	−17.3205	−0.746047
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	− 3.46410i	− 0.148659i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−20.7846	−0.888686	−0.444343	−	0.895857i	$-0.646563\pi$
$547$	−20.7846	−0.888686	−0.444343	+	0.895857i	$0.646563\pi$
$548$	0	0
$549$	30.0000	1.28037
$550$	0	0
$551$	−20.7846	−0.885454
$552$	0	0
$553$	− 36.0000i	− 1.53088i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	− 27.7128i	− 1.17213i
$560$	0	0
$561$	− 36.0000i	− 1.51992i
$562$	0	0
$563$	3.46410i	0.145994i	0.997332	+	0.0729972i	$0.0232564\pi$
$563$	3.46410i	0.145994i	−0.997332	+	0.0729972i	$0.976744\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−31.1769	−1.30931
$568$	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
$573$	36.0000i	1.50392i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	0	0
$579$	−3.46410	−0.143963
$580$	0	0
$581$	− 36.0000i	− 1.49353i
$582$	0	0
$583$	20.7846	0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 31.1769i	− 1.28681i	−0.765526	−	0.643404i	$-0.777521\pi$
$587$	− 31.1769i	− 1.28681i	0.765526	−	0.643404i	$-0.222479\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	− 10.3923i	− 0.427482i
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	42.0000	1.71895
$598$	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
$603$	−20.7846	−0.846415
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.1769	1.26543	0.632716	−	0.774384i	$-0.281940\pi$
$607$	31.1769	1.26543	0.632716	+	0.774384i	$0.281940\pi$
$608$	0	0
$609$	−36.0000	−1.45879
$610$	0	0
$611$	13.8564	0.560570
$612$	0	0
$613$	32.0000i	1.29247i	0.763139	+	0.646234i	$0.223657\pi$
$613$	32.0000i	1.29247i	−0.763139	+	0.646234i	$0.776343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	− 45.0333i	− 1.81004i	−0.425367	−	0.905021i	$-0.639855\pi$
$619$	− 45.0333i	− 1.81004i	0.425367	−	0.905021i	$-0.360145\pi$
$620$	0	0
$621$	−18.0000	−0.722315
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−20.7846	−0.830057
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	− 3.46410i	− 0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$	− 3.46410i	− 0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$	0	0
$633$	30.0000	1.19239
$634$	0	0
$635$	0	0
$636$	0	0
$637$	20.0000i	0.792429i
$638$	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
$643$	41.5692	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 45.0333i	− 1.77044i	−0.465170	−	0.885221i	$-0.654007\pi$
$647$	− 45.0333i	− 1.77044i	0.465170	−	0.885221i	$-0.345993\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	−20.7846	−0.814613
$652$	0	0
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6.00000i	0.234082i
$658$	0	0
$659$	−31.1769	−1.21448	−0.607240	−	0.794518i	$-0.707723\pi$
$659$	−31.1769	−1.21448	−0.607240	+	0.794518i	$0.707723\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	−41.5692	−1.61441
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.7846	−0.804783
$668$	0	0
$669$	− 30.0000i	− 1.15987i
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.0000	1.15299	0.576497	−	0.817099i	$-0.304419\pi$
$677$	30.0000	1.15299	0.576497	+	0.817099i	$0.304419\pi$
$678$	0	0
$679$	34.6410i	1.32940i
$680$	0	0
$681$	−6.00000	−0.229920
$682$	0	0
$683$	− 17.3205i	− 0.662751i	−0.943499	−	0.331375i	$-0.892487\pi$
$683$	− 17.3205i	− 0.662751i	0.943499	−	0.331375i	$-0.107513\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	24.2487i	0.925146i
$688$	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
$693$	−36.0000	−1.36753
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 72.0000i	− 2.72719i
$698$	0	0
$699$	10.3923i	0.393073i
$700$	0	0
$701$	− 42.0000i	− 1.58632i	−0.609015	−	0.793159i	$-0.708435\pi$
$701$	− 42.0000i	− 1.58632i	0.609015	−	0.793159i	$-0.291565\pi$
$702$	0	0
$703$	13.8564	0.522604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.7846i	0.781686i
$708$	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
$713$	−12.0000	−0.449404
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.0000i	− 0.896296i
$718$	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
$723$	− 45.0333i	− 1.67481i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−3.46410	−0.128476	−0.0642382	−	0.997935i	$-0.520462\pi$
$727$	−3.46410	−0.128476	−0.0642382	+	0.997935i	$0.520462\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−41.5692	−1.53749
$732$	0	0
$733$	28.0000i	1.03420i	0.855924	+	0.517102i	$0.172989\pi$
$733$	28.0000i	1.03420i	−0.855924	+	0.517102i	$0.827011\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.0000	−0.884051
$738$	0	0
$739$	10.3923i	0.382287i	0.981562	+	0.191144i	$0.0612196\pi$
$739$	10.3923i	0.382287i	−0.981562	+	0.191144i	$0.938780\pi$
$740$	0	0
$741$	24.0000i	0.881662i
$742$	0	0
$743$	38.1051i	1.39794i	0.715150	+	0.698971i	$0.246358\pi$
$743$	38.1051i	1.39794i	−0.715150	+	0.698971i	$0.753642\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 31.1769i	− 1.14070i
$748$	0	0
$749$	12.0000i	0.438470i
$750$	0	0
$751$	51.9615i	1.89610i	0.318117	+	0.948051i	$0.396950\pi$
$751$	51.9615i	1.89610i	−0.318117	+	0.948051i	$0.603050\pi$
$752$	0	0
$753$	− 6.00000i	− 0.218652i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	20.0000i	0.726912i	0.931611	+	0.363456i	$0.118403\pi$
$757$	20.0000i	0.726912i	−0.931611	+	0.363456i	$0.881597\pi$
$758$	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
$763$	34.6410	1.25409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8564i	0.500326i
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	− 31.1769i	− 1.12281i
$772$	0	0
$773$	54.0000	1.94225	0.971123	−	0.238581i	$-0.0766824\pi$
$773$	54.0000	1.94225	0.971123	+	0.238581i	$0.0766824\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.0000	0.860995
$778$	0	0
$779$	−41.5692	−1.48937
$780$	0	0
$781$	48.0000	1.71758
$782$	0	0
$783$	−31.1769	−1.11417
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.6410	1.23482	0.617409	−	0.786642i	$-0.288182\pi$
$787$	34.6410	1.23482	0.617409	+	0.786642i	$0.288182\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	−20.7846	−0.739016
$792$	0	0
$793$	− 40.0000i	− 1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	− 20.7846i	− 0.735307i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.92820i	0.244491i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	10.3923	0.365826
$808$	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
$813$	18.0000	0.631288
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.0000i	0.839654i
$818$	0	0
$819$	41.5692i	1.45255i
$820$	0	0
$821$	− 6.00000i	− 0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$	− 6.00000i	− 0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$	0	0
$823$	45.0333	1.56976	0.784881	−	0.619646i	$-0.212724\pi$
$823$	45.0333	1.56976	0.784881	+	0.619646i	$0.212724\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 38.1051i	− 1.32504i	−0.749042	−	0.662522i	$-0.769486\pi$
$827$	− 38.1051i	− 1.32504i	0.749042	−	0.662522i	$-0.230514\pi$
$828$	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
$833$	30.0000	1.03944
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−18.0000	−0.622171
$838$	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
$843$	−20.7846	−0.715860
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	12.0000i	0.411839i
$850$	0	0
$851$	13.8564	0.474991
$852$	0	0
$853$	8.00000i	0.273915i	0.990577	+	0.136957i	$0.0437323\pi$
$853$	8.00000i	0.273915i	−0.990577	+	0.136957i	$0.956268\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	51.9615i	1.77290i	0.462820	+	0.886452i	$0.346838\pi$
$859$	51.9615i	1.77290i	−0.462820	+	0.886452i	$0.653162\pi$
$860$	0	0
$861$	−72.0000	−2.45375
$862$	0	0
$863$	51.9615i	1.76879i	0.466738	+	0.884395i	$0.345429\pi$
$863$	51.9615i	1.76879i	−0.466738	+	0.884395i	$0.654571\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	32.9090i	1.11765i
$868$	0	0
$869$	− 36.0000i	− 1.22122i
$870$	0	0
$871$	27.7128i	0.939013i
$872$	0	0
$873$	30.0000i	1.01535i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	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
$883$	13.8564	0.466305	0.233153	−	0.972440i	$-0.425096\pi$
$883$	13.8564	0.466305	0.233153	+	0.972440i	$0.425096\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.2487i	0.814192i	0.913385	+	0.407096i	$0.133459\pi$
$887$	24.2487i	0.814192i	−0.913385	+	0.407096i	$0.866541\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	−31.1769	−1.04447
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.0000i	0.801337i
$898$	0	0
$899$	−20.7846	−0.693206
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	41.5692i	1.38334i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	41.5692	1.38028	0.690142	−	0.723674i	$-0.257548\pi$
$907$	41.5692	1.38028	0.690142	+	0.723674i	$0.257548\pi$
$908$	0	0
$909$	18.0000i	0.597022i
$910$	0	0
$911$	34.6410	1.14771	0.573854	−	0.818958i	$-0.305448\pi$
$911$	34.6410	1.14771	0.573854	+	0.818958i	$0.305448\pi$
$912$	0	0
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−60.0000	−1.98137
$918$	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
$923$	− 55.4256i	− 1.82436i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	31.1769	1.02398
$928$	0	0
$929$	12.0000i	0.393707i	0.980433	+	0.196854i	$0.0630724\pi$
$929$	12.0000i	0.393707i	−0.980433	+	0.196854i	$0.936928\pi$
$930$	0	0
$931$	− 17.3205i	− 0.567657i
$932$	0	0
$933$	− 24.0000i	− 0.785725i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	0	0
$939$	−24.2487	−0.791327
$940$	0	0
$941$	54.0000i	1.76035i	0.474650	+	0.880175i	$0.342575\pi$
$941$	54.0000i	1.76035i	−0.474650	+	0.880175i	$0.657425\pi$
$942$	0	0
$943$	−41.5692	−1.35368
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.1769i	1.01311i	0.862207	+	0.506557i	$0.169082\pi$
$947$	31.1769i	1.01311i	−0.862207	+	0.506557i	$0.830918\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	− 31.1769i	− 1.01098i
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−36.0000	−1.16371
$958$	0	0
$959$	62.3538	2.01351
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	10.3923i	0.334887i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	51.9615	1.67097	0.835485	−	0.549513i	$-0.185187\pi$
$967$	51.9615	1.67097	0.835485	+	0.549513i	$0.185187\pi$
$968$	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
$973$	− 36.0000i	− 1.15411i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.0000	0.959785	0.479893	−	0.877327i	$-0.340676\pi$
$977$	30.0000	0.959785	0.479893	+	0.877327i	$0.340676\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	30.0000	0.957826
$982$	0	0
$983$	− 58.8897i	− 1.87829i	−0.343520	−	0.939145i	$-0.611619\pi$
$983$	− 58.8897i	− 1.87829i	0.343520	−	0.939145i	$-0.388381\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−20.7846	−0.661581
$988$	0	0
$989$	24.0000i	0.763156i
$990$	0	0
$991$	17.3205i	0.550204i	0.961415	+	0.275102i	$0.0887116\pi$
$991$	17.3205i	0.550204i	−0.961415	+	0.275102i	$0.911288\pi$
$992$	0	0
$993$	−18.0000	−0.571213
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.0000i	1.64686i	0.567420	+	0.823428i	$0.307941\pi$
$997$	52.0000i	1.64686i	−0.567420	+	0.823428i	$0.692059\pi$
$998$	0	0
$999$	20.7846	0.657596

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.2.o.b.1199.3		4
3.2	odd	2		1200.2.o.a.1199.1		4
4.3	odd	2	inner	1200.2.o.b.1199.2		4
5.2	odd	4		240.2.h.b.191.3	yes	4
5.3	odd	4		1200.2.h.m.1151.2		4
5.4	even	2		1200.2.o.a.1199.2		4
12.11	even	2		1200.2.o.a.1199.4		4
15.2	even	4		240.2.h.b.191.2	yes	4
15.8	even	4		1200.2.h.m.1151.4		4
15.14	odd	2	inner	1200.2.o.b.1199.4		4
20.3	even	4		1200.2.h.m.1151.3		4
20.7	even	4		240.2.h.b.191.1	✓	4
20.19	odd	2		1200.2.o.a.1199.3		4
40.27	even	4		960.2.h.d.191.4		4
40.37	odd	4		960.2.h.d.191.2		4
60.23	odd	4		1200.2.h.m.1151.1		4
60.47	odd	4		240.2.h.b.191.4	yes	4
60.59	even	2	inner	1200.2.o.b.1199.1		4
120.77	even	4		960.2.h.d.191.3		4
120.107	odd	4		960.2.h.d.191.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.h.b.191.1	✓	4	20.7	even	4
240.2.h.b.191.2	yes	4	15.2	even	4
240.2.h.b.191.3	yes	4	5.2	odd	4
240.2.h.b.191.4	yes	4	60.47	odd	4
960.2.h.d.191.1		4	120.107	odd	4
960.2.h.d.191.2		4	40.37	odd	4
960.2.h.d.191.3		4	120.77	even	4
960.2.h.d.191.4		4	40.27	even	4
1200.2.h.m.1151.1		4	60.23	odd	4
1200.2.h.m.1151.2		4	5.3	odd	4
1200.2.h.m.1151.3		4	20.3	even	4
1200.2.h.m.1151.4		4	15.8	even	4
1200.2.o.a.1199.1		4	3.2	odd	2
1200.2.o.a.1199.2		4	5.4	even	2
1200.2.o.a.1199.3		4	20.19	odd	2
1200.2.o.a.1199.4		4	12.11	even	2
1200.2.o.b.1199.1		4	60.59	even	2	inner
1200.2.o.b.1199.2		4	4.3	odd	2	inner
1200.2.o.b.1199.3		4	1.1	even	1	trivial
1200.2.o.b.1199.4		4	15.14	odd	2	inner

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1200.2.o.b.1199.3

Level	$1200$
Weight	$2$
Character	1200.1199
Analytic conductor	$9.582$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$