LMFDB - Embedded newform 1176.3.d.h.785.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(785,1176)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1176.785"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.12
Character	$\chi$	$=$	1176.785
Dual form			1176.3.d.h.785.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.348188 + 2.97973i) q^{3} +1.45894i q^{5} +(-8.75753 - 2.07501i) q^{9} -13.0814i q^{11} -8.32386 q^{13} +(-4.34724 - 0.507985i) q^{15} +9.11185i q^{17} +1.82391 q^{19} -2.01937i q^{23} +22.8715 q^{25} +(9.23223 - 25.3725i) q^{27} -39.5488i q^{29} +38.1505 q^{31} +(38.9790 + 4.55478i) q^{33} +5.48266 q^{37} +(2.89827 - 24.8028i) q^{39} -47.0794i q^{41} -56.0059 q^{43} +(3.02731 - 12.7767i) q^{45} +0.490303i q^{47} +(-27.1508 - 3.17264i) q^{51} -35.8982i q^{53} +19.0850 q^{55} +(-0.635064 + 5.43476i) q^{57} +41.1127i q^{59} +113.886 q^{61} -12.1440i q^{65} +109.748 q^{67} +(6.01716 + 0.703120i) q^{69} +57.7108i q^{71} +19.6158 q^{73} +(-7.96358 + 68.1508i) q^{75} -141.239 q^{79} +(72.3887 + 36.3439i) q^{81} +92.8411i q^{83} -13.2936 q^{85} +(117.845 + 13.7704i) q^{87} -103.629i q^{89} +(-13.2835 + 113.678i) q^{93} +2.66098i q^{95} -20.1233 q^{97} +(-27.1440 + 114.561i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 16 q^{15} - 152 q^{25} + 64 q^{37} - 16 q^{39} + 16 q^{43} + 248 q^{51} - 120 q^{57} - 416 q^{67} - 224 q^{79} - 344 q^{81} + 496 q^{85} - 464 q^{93}+O(q^{100}) $ 24 * q - 16 * q^15 - 152 * q^25 + 64 * q^37 - 16 * q^39 + 16 * q^43 + 248 * q^51 - 120 * q^57 - 416 * q^67 - 224 * q^79 - 344 * q^81 + 496 * q^85 - 464 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	−0.348188	+	2.97973i	−0.116063	+	0.993242i
$3$	−0.348188	+	2.97973i	−0.116063	+	0.993242i
$4$		0			0
$5$			1.45894i			0.291788i	0.989300	+	0.145894i	$0.0466058\pi$
$5$			1.45894i			0.291788i	−0.989300	+	0.145894i	$0.953394\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−8.75753	−	2.07501i	−0.973059	−	0.230557i
$10$		0			0
$11$		−	13.0814i		−	1.18922i	−0.804015	−	0.594609i	$-0.797307\pi$
$11$		−	13.0814i		−	1.18922i	0.804015	−	0.594609i	$-0.202693\pi$
$12$		0			0
$13$	−8.32386			−0.640297			−0.320148	−	0.947367i	$-0.603733\pi$
$13$	−8.32386			−0.640297			−0.320148	+	0.947367i	$0.603733\pi$
$14$		0			0
$15$	−4.34724	−	0.507985i	−0.289816	−	0.0338657i
$16$		0			0
$17$			9.11185i			0.535991i	0.963420	+	0.267995i	$0.0863612\pi$
$17$			9.11185i			0.535991i	−0.963420	+	0.267995i	$0.913639\pi$
$18$		0			0
$19$	1.82391			0.0959953			0.0479977	−	0.998847i	$-0.484716\pi$
$19$	1.82391			0.0959953			0.0479977	+	0.998847i	$0.484716\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.01937i		−	0.0877986i	−0.999036	−	0.0438993i	$-0.986022\pi$
$23$		−	2.01937i		−	0.0877986i	0.999036	−	0.0438993i	$-0.0139781\pi$
$24$		0			0
$25$	22.8715			0.914860
$26$		0			0
$27$	9.23223	−	25.3725i	0.341934	−	0.939724i
$28$		0			0
$29$		−	39.5488i		−	1.36375i	−0.731467	−	0.681877i	$-0.761164\pi$
$29$		−	39.5488i		−	1.36375i	0.731467	−	0.681877i	$-0.238836\pi$
$30$		0			0
$31$	38.1505			1.23066			0.615331	−	0.788269i	$-0.289023\pi$
$31$	38.1505			1.23066			0.615331	+	0.788269i	$0.289023\pi$
$32$		0			0
$33$	38.9790	+	4.55478i	1.18118	+	0.138024i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.48266			0.148180			0.0740900	−	0.997252i	$-0.476395\pi$
$37$	5.48266			0.148180			0.0740900	+	0.997252i	$0.476395\pi$
$38$		0			0
$39$	2.89827	−	24.8028i	0.0743146	−	0.635970i
$40$		0			0
$41$		−	47.0794i		−	1.14828i	−0.818758	−	0.574139i	$-0.805337\pi$
$41$		−	47.0794i		−	1.14828i	0.818758	−	0.574139i	$-0.194663\pi$
$42$		0			0
$43$	−56.0059			−1.30246			−0.651231	−	0.758880i	$-0.725747\pi$
$43$	−56.0059			−1.30246			−0.651231	+	0.758880i	$0.725747\pi$
$44$		0			0
$45$	3.02731	−	12.7767i	0.0672736	−	0.283927i
$46$		0			0
$47$			0.490303i			0.0104320i	0.999986	+	0.00521599i	$0.00166031\pi$
$47$			0.490303i			0.0104320i	−0.999986	+	0.00521599i	$0.998340\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−27.1508	−	3.17264i	−0.532369	−	0.0622085i
$52$		0			0
$53$		−	35.8982i		−	0.677325i	−0.940908	−	0.338663i	$-0.890025\pi$
$53$		−	35.8982i		−	0.677325i	0.940908	−	0.338663i	$-0.109975\pi$
$54$		0			0
$55$	19.0850			0.346999
$56$		0			0
$57$	−0.635064	+	5.43476i	−0.0111415	+	0.0953466i
$58$		0			0
$59$			41.1127i			0.696826i	0.937341	+	0.348413i	$0.113279\pi$
$59$			41.1127i			0.696826i	−0.937341	+	0.348413i	$0.886721\pi$
$60$		0			0
$61$	113.886			1.86698			0.933491	−	0.358600i	$-0.116746\pi$
$61$	113.886			1.86698			0.933491	+	0.358600i	$0.116746\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	12.1440i		−	0.186831i
$66$		0			0
$67$	109.748			1.63802			0.819012	−	0.573777i	$-0.194522\pi$
$67$	109.748			1.63802			0.819012	+	0.573777i	$0.194522\pi$
$68$		0			0
$69$	6.01716	+	0.703120i	0.0872052	+	0.0101901i
$70$		0			0
$71$			57.7108i			0.812828i	0.913689	+	0.406414i	$0.133221\pi$
$71$			57.7108i			0.812828i	−0.913689	+	0.406414i	$0.866779\pi$
$72$		0			0
$73$	19.6158			0.268709			0.134355	−	0.990933i	$-0.457104\pi$
$73$	19.6158			0.268709			0.134355	+	0.990933i	$0.457104\pi$
$74$		0			0
$75$	−7.96358	+	68.1508i	−0.106181	+	0.908677i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−141.239			−1.78784			−0.893920	−	0.448226i	$-0.852056\pi$
$79$	−141.239			−1.78784			−0.893920	+	0.448226i	$0.852056\pi$
$80$		0			0
$81$	72.3887	+	36.3439i	0.893687	+	0.448690i
$82$		0			0
$83$			92.8411i			1.11857i	0.828976	+	0.559284i	$0.188924\pi$
$83$			92.8411i			1.11857i	−0.828976	+	0.559284i	$0.811076\pi$
$84$		0			0
$85$	−13.2936			−0.156396
$86$		0			0
$87$	117.845	+	13.7704i	1.35454	+	0.158281i
$88$		0			0
$89$		−	103.629i		−	1.16437i	−0.813057	−	0.582184i	$-0.802199\pi$
$89$		−	103.629i		−	1.16437i	0.813057	−	0.582184i	$-0.197801\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−13.2835	+	113.678i	−0.142834	+	1.22234i
$94$		0			0
$95$			2.66098i			0.0280103i
$96$		0			0
$97$	−20.1233			−0.207457			−0.103729	−	0.994606i	$-0.533077\pi$
$97$	−20.1233			−0.207457			−0.103729	+	0.994606i	$0.533077\pi$
$98$		0			0
$99$	−27.1440	+	114.561i	−0.274182	+	1.15718i
$100$		0			0
$101$		−	123.738i		−	1.22513i	−0.790419	−	0.612566i	$-0.790137\pi$
$101$		−	123.738i		−	1.22513i	0.790419	−	0.612566i	$-0.209863\pi$
$102$		0			0
$103$	106.401			1.03302			0.516509	−	0.856282i	$-0.327231\pi$
$103$	106.401			1.03302			0.516509	+	0.856282i	$0.327231\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	51.1822i		−	0.478338i	−0.970978	−	0.239169i	$-0.923125\pi$
$107$		−	51.1822i		−	0.478338i	0.970978	−	0.239169i	$-0.0768750\pi$
$108$		0			0
$109$	176.670			1.62083			0.810415	−	0.585856i	$-0.199242\pi$
$109$	176.670			1.62083			0.810415	+	0.585856i	$0.199242\pi$
$110$		0			0
$111$	−1.90900	+	16.3368i	−0.0171982	+	0.147179i
$112$		0			0
$113$		−	123.258i		−	1.09077i	−0.838184	−	0.545387i	$-0.816383\pi$
$113$		−	123.258i		−	1.09077i	0.838184	−	0.545387i	$-0.183617\pi$
$114$		0			0
$115$	2.94613			0.0256186
$116$		0			0
$117$	72.8965	+	17.2721i	0.623047	+	0.147625i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−50.1229			−0.414239
$122$		0			0
$123$	140.284	+	16.3925i	1.14052	+	0.133272i
$124$		0			0
$125$			69.8416i			0.558733i
$126$		0			0
$127$	−93.2274			−0.734074			−0.367037	−	0.930206i	$-0.619628\pi$
$127$	−93.2274			−0.734074			−0.367037	+	0.930206i	$0.619628\pi$
$128$		0			0
$129$	19.5006	−	166.882i	0.151167	−	1.29366i
$130$		0			0
$131$		−	120.272i		−	0.918110i	−0.888408	−	0.459055i	$-0.848188\pi$
$131$		−	120.272i		−	0.918110i	0.888408	−	0.459055i	$-0.151812\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	37.0170	+	13.4693i	0.274200	+	0.0997723i
$136$		0			0
$137$			223.391i			1.63059i	0.579043	+	0.815297i	$0.303426\pi$
$137$			223.391i			1.63059i	−0.579043	+	0.815297i	$0.696574\pi$
$138$		0			0
$139$	191.153			1.37520			0.687601	−	0.726088i	$-0.258664\pi$
$139$	191.153			1.37520			0.687601	+	0.726088i	$0.258664\pi$
$140$		0			0
$141$	−1.46097	−	0.170718i	−0.0103615	−	0.00121076i
$142$		0			0
$143$			108.888i			0.761452i
$144$		0			0
$145$	57.6994			0.397927
$146$		0			0
$147$		0			0
$148$		0			0
$149$			28.5518i			0.191623i	0.995399	+	0.0958115i	$0.0305446\pi$
$149$			28.5518i			0.191623i	−0.995399	+	0.0958115i	$0.969455\pi$
$150$		0			0
$151$	−58.7547			−0.389104			−0.194552	−	0.980892i	$-0.562325\pi$
$151$	−58.7547			−0.389104			−0.194552	+	0.980892i	$0.562325\pi$
$152$		0			0
$153$	18.9072	−	79.7973i	0.123576	−	0.521551i
$154$		0			0
$155$			55.6593i			0.359092i
$156$		0			0
$157$	122.759			0.781907			0.390953	−	0.920410i	$-0.372145\pi$
$157$	122.759			0.781907			0.390953	+	0.920410i	$0.372145\pi$
$158$		0			0
$159$	106.967	+	12.4993i	0.672748	+	0.0786122i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−9.47536			−0.0581310			−0.0290655	−	0.999578i	$-0.509253\pi$
$163$	−9.47536			−0.0581310			−0.0290655	+	0.999578i	$0.509253\pi$
$164$		0			0
$165$	−6.64515	+	56.8679i	−0.0402737	+	0.344654i
$166$		0			0
$167$			231.313i			1.38511i	0.721367	+	0.692553i	$0.243514\pi$
$167$			231.313i			1.38511i	−0.721367	+	0.692553i	$0.756486\pi$
$168$		0			0
$169$	−99.7134			−0.590020
$170$		0			0
$171$	−15.9730	−	3.78463i	−0.0934091	−	0.0221324i
$172$		0			0
$173$		−	223.231i		−	1.29035i	−0.764033	−	0.645177i	$-0.776784\pi$
$173$		−	223.231i		−	1.29035i	0.764033	−	0.645177i	$-0.223216\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−122.505	−	14.3150i	−0.692117	−	0.0808755i
$178$		0			0
$179$		−	279.763i		−	1.56292i	−0.623953	−	0.781462i	$-0.714474\pi$
$179$		−	279.763i		−	1.56292i	0.623953	−	0.781462i	$-0.285526\pi$
$180$		0			0
$181$	12.6281			0.0697684			0.0348842	−	0.999391i	$-0.488894\pi$
$181$	12.6281			0.0697684			0.0348842	+	0.999391i	$0.488894\pi$
$182$		0			0
$183$	−39.6537	+	339.349i	−0.216687	+	1.85437i
$184$		0			0
$185$			7.99887i			0.0432371i
$186$		0			0
$187$	119.196			0.637410
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	150.027i		−	0.785479i	−0.919650	−	0.392740i	$-0.871527\pi$
$191$		−	150.027i		−	0.785479i	0.919650	−	0.392740i	$-0.128473\pi$
$192$		0			0
$193$	175.379			0.908698			0.454349	−	0.890824i	$-0.349872\pi$
$193$	175.379			0.908698			0.454349	+	0.890824i	$0.349872\pi$
$194$		0			0
$195$	36.1858	+	4.22840i	0.185568	+	0.0216841i
$196$		0			0
$197$		−	283.134i		−	1.43723i	−0.695408	−	0.718615i	$-0.744776\pi$
$197$		−	283.134i		−	1.43723i	0.695408	−	0.718615i	$-0.255224\pi$
$198$		0			0
$199$	13.2136			0.0664002			0.0332001	−	0.999449i	$-0.489430\pi$
$199$	13.2136			0.0664002			0.0332001	+	0.999449i	$0.489430\pi$
$200$		0			0
$201$	−38.2128	+	327.018i	−0.190113	+	1.62695i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	68.6860			0.335054
$206$		0			0
$207$	−4.19021	+	17.6847i	−0.0202425	+	0.0854332i
$208$		0			0
$209$		−	23.8593i		−	0.114159i
$210$		0			0
$211$	316.623			1.50058			0.750292	−	0.661107i	$-0.229913\pi$
$211$	316.623			1.50058			0.750292	+	0.661107i	$0.229913\pi$
$212$		0			0
$213$	−171.962	−	20.0942i	−0.807334	−	0.0943389i
$214$		0			0
$215$		−	81.7092i		−	0.380043i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.82997	+	58.4496i	−0.0311871	+	0.266893i
$220$		0			0
$221$		−	75.8457i		−	0.343193i
$222$		0			0
$223$	−386.857			−1.73479			−0.867393	−	0.497624i	$-0.834206\pi$
$223$	−386.857			−1.73479			−0.867393	+	0.497624i	$0.834206\pi$
$224$		0			0
$225$	−200.298	−	47.4586i	−0.890213	−	0.210927i
$226$		0			0
$227$			402.121i			1.77146i	0.464202	+	0.885729i	$0.346341\pi$
$227$			402.121i			1.77146i	−0.464202	+	0.885729i	$0.653659\pi$
$228$		0			0
$229$	−302.264			−1.31993			−0.659965	−	0.751296i	$-0.729429\pi$
$229$	−302.264			−1.31993			−0.659965	+	0.751296i	$0.729429\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.89166i			0.0252861i	0.999920	+	0.0126431i	$0.00402452\pi$
$233$			5.89166i			0.0252861i	−0.999920	+	0.0126431i	$0.995975\pi$
$234$		0			0
$235$	−0.715322			−0.00304392
$236$		0			0
$237$	49.1779	−	420.855i	0.207501	−	1.77576i
$238$		0			0
$239$		−	140.052i		−	0.585992i	−0.956114	−	0.292996i	$-0.905348\pi$
$239$		−	140.052i		−	0.585992i	0.956114	−	0.292996i	$-0.0946523\pi$
$240$		0			0
$241$	292.339			1.21303			0.606513	−	0.795073i	$-0.292568\pi$
$241$	292.339			1.21303			0.606513	+	0.795073i	$0.292568\pi$
$242$		0			0
$243$	−133.500	+	203.044i	−0.549382	+	0.835571i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−15.1820			−0.0614655
$248$		0			0
$249$	−276.641	−	32.3262i	−1.11101	−	0.129824i
$250$		0			0
$251$		−	60.3015i		−	0.240245i	−0.992759	−	0.120122i	$-0.961671\pi$
$251$		−	60.3015i		−	0.240245i	0.992759	−	0.120122i	$-0.0383287\pi$
$252$		0			0
$253$	−26.4161			−0.104412
$254$		0			0
$255$	4.62868	−	39.6114i	0.0181517	−	0.155339i
$256$		0			0
$257$			38.9687i			0.151629i	0.997122	+	0.0758146i	$0.0241557\pi$
$257$			38.9687i			0.151629i	−0.997122	+	0.0758146i	$0.975844\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−82.0642	+	346.350i	−0.314422	+	1.32701i
$262$		0			0
$263$		−	213.219i		−	0.810718i	−0.914158	−	0.405359i	$-0.867146\pi$
$263$		−	213.219i		−	0.810718i	0.914158	−	0.405359i	$-0.132854\pi$
$264$		0			0
$265$	52.3734			0.197635
$266$		0			0
$267$	308.785	+	36.0823i	1.15650	+	0.135140i
$268$		0			0
$269$		−	350.822i		−	1.30417i	−0.758146	−	0.652085i	$-0.773895\pi$
$269$		−	350.822i		−	1.30417i	0.758146	−	0.652085i	$-0.226105\pi$
$270$		0			0
$271$	57.5664			0.212422			0.106211	−	0.994344i	$-0.466128\pi$
$271$	57.5664			0.212422			0.106211	+	0.994344i	$0.466128\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	299.191i		−	1.08797i
$276$		0			0
$277$	−383.047			−1.38284			−0.691421	−	0.722452i	$-0.743015\pi$
$277$	−383.047			−1.38284			−0.691421	+	0.722452i	$0.743015\pi$
$278$		0			0
$279$	−334.104	−	79.1626i	−1.19751	−	0.283737i
$280$		0			0
$281$		−	514.562i		−	1.83118i	−0.402111	−	0.915591i	$-0.631724\pi$
$281$		−	514.562i		−	1.83118i	0.402111	−	0.915591i	$-0.368276\pi$
$282$		0			0
$283$	−400.373			−1.41475			−0.707374	−	0.706840i	$-0.750120\pi$
$283$	−400.373			−1.41475			−0.707374	+	0.706840i	$0.750120\pi$
$284$		0			0
$285$	−7.92898	−	0.926520i	−0.0278210	−	0.00325095i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	205.974			0.712714
$290$		0			0
$291$	7.00670	−	59.9620i	0.0240780	−	0.206055i
$292$		0			0
$293$		−	281.798i		−	0.961770i	−0.876784	−	0.480885i	$-0.840316\pi$
$293$		−	281.798i		−	0.961770i	0.876784	−	0.480885i	$-0.159684\pi$
$294$		0			0
$295$	−59.9810			−0.203325
$296$		0			0
$297$	−331.908	−	120.770i	−1.11754	−	0.406634i
$298$		0			0
$299$			16.8089i			0.0562172i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	368.707	+	43.0842i	1.21685	+	0.142192i
$304$		0			0
$305$			166.153i			0.544763i
$306$		0			0
$307$	360.665			1.17480			0.587402	−	0.809296i	$-0.300151\pi$
$307$	360.665			1.17480			0.587402	+	0.809296i	$0.300151\pi$
$308$		0			0
$309$	−37.0475	+	317.046i	−0.119895	+	1.02604i
$310$		0			0
$311$			439.152i			1.41207i	0.708179	+	0.706033i	$0.249517\pi$
$311$			439.152i			1.41207i	−0.708179	+	0.706033i	$0.750483\pi$
$312$		0			0
$313$	−395.788			−1.26450			−0.632249	−	0.774765i	$-0.717868\pi$
$313$	−395.788			−1.26450			−0.632249	+	0.774765i	$0.717868\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	147.015i		−	0.463771i	−0.972743	−	0.231885i	$-0.925511\pi$
$317$		−	147.015i		−	0.463771i	0.972743	−	0.231885i	$-0.0744894\pi$
$318$		0			0
$319$	−517.354			−1.62180
$320$		0			0
$321$	152.509	+	17.8210i	0.475106	+	0.0555172i
$322$		0			0
$323$			16.6192i			0.0514526i
$324$		0			0
$325$	−190.379			−0.585782
$326$		0			0
$327$	−61.5145	+	526.430i	−0.188118	+	1.60988i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	98.8293			0.298578			0.149289	−	0.988794i	$-0.452302\pi$
$331$	98.8293			0.298578			0.149289	+	0.988794i	$0.452302\pi$
$332$		0			0
$333$	−48.0146	−	11.3766i	−0.144188	−	0.0341639i
$334$		0			0
$335$			160.115i			0.477955i
$336$		0			0
$337$	64.9025			0.192589			0.0962945	−	0.995353i	$-0.469301\pi$
$337$	64.9025			0.192589			0.0962945	+	0.995353i	$0.469301\pi$
$338$		0			0
$339$	367.274	+	42.9168i	1.08340	+	0.126598i
$340$		0			0
$341$		−	499.062i		−	1.46352i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.02581	+	8.77867i	−0.00297336	+	0.0254454i
$346$		0			0
$347$			71.8394i			0.207030i	0.994628	+	0.103515i	$0.0330090\pi$
$347$			71.8394i			0.207030i	−0.994628	+	0.103515i	$0.966991\pi$
$348$		0			0
$349$	−306.624			−0.878579			−0.439289	−	0.898346i	$-0.644770\pi$
$349$	−306.624			−0.878579			−0.439289	+	0.898346i	$0.644770\pi$
$350$		0			0
$351$	−76.8478	+	211.198i	−0.218939	+	0.601702i
$352$		0			0
$353$			407.628i			1.15475i	0.816478	+	0.577377i	$0.195924\pi$
$353$			407.628i			1.15475i	−0.816478	+	0.577377i	$0.804076\pi$
$354$		0			0
$355$	−84.1965			−0.237173
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	263.566i		−	0.734167i	−0.930188	−	0.367084i	$-0.880356\pi$
$359$		−	263.566i		−	0.734167i	0.930188	−	0.367084i	$-0.119644\pi$
$360$		0			0
$361$	−357.673			−0.990785
$362$		0			0
$363$	17.4522	−	149.352i	0.0480776	−	0.411439i
$364$		0			0
$365$			28.6182i			0.0784060i
$366$		0			0
$367$	−237.850			−0.648092			−0.324046	−	0.946041i	$-0.605043\pi$
$367$	−237.850			−0.648092			−0.324046	+	0.946041i	$0.605043\pi$
$368$		0			0
$369$	−97.6902	+	412.299i	−0.264743	+	1.11734i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	257.234			0.689635			0.344817	−	0.938670i	$-0.387941\pi$
$373$	257.234			0.689635			0.344817	+	0.938670i	$0.387941\pi$
$374$		0			0
$375$	−208.109	−	24.3180i	−0.554957	−	0.0648480i
$376$		0			0
$377$			329.199i			0.873207i
$378$		0			0
$379$	308.987			0.815269			0.407635	−	0.913145i	$-0.366354\pi$
$379$	308.987			0.815269			0.407635	+	0.913145i	$0.366354\pi$
$380$		0			0
$381$	32.4607	−	277.792i	0.0851986	−	0.729113i
$382$		0			0
$383$		−	571.802i		−	1.49296i	−0.665410	−	0.746478i	$-0.731743\pi$
$383$		−	571.802i		−	1.49296i	0.665410	−	0.746478i	$-0.268257\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	490.473	+	116.213i	1.26737	+	0.300291i
$388$		0			0
$389$		−	384.614i		−	0.988724i	−0.869256	−	0.494362i	$-0.835402\pi$
$389$		−	384.614i		−	0.988724i	0.869256	−	0.494362i	$-0.164598\pi$
$390$		0			0
$391$	18.4002			0.0470592
$392$		0			0
$393$	358.379	+	41.8774i	0.911905	+	0.106558i
$394$		0			0
$395$		−	206.060i		−	0.521670i
$396$		0			0
$397$	−387.737			−0.976667			−0.488334	−	0.872657i	$-0.662395\pi$
$397$	−387.737			−0.976667			−0.488334	+	0.872657i	$0.662395\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			24.5893i			0.0613199i	0.999530	+	0.0306600i	$0.00976090\pi$
$401$			24.5893i			0.0613199i	−0.999530	+	0.0306600i	$0.990239\pi$
$402$		0			0
$403$	−317.559			−0.787989
$404$		0			0
$405$	−53.0236	+	105.611i	−0.130922	+	0.260767i
$406$		0			0
$407$		−	71.7209i		−	0.176218i
$408$		0			0
$409$	−400.893			−0.980178			−0.490089	−	0.871672i	$-0.663036\pi$
$409$	−400.893			−0.980178			−0.490089	+	0.871672i	$0.663036\pi$
$410$		0			0
$411$	−665.645	−	77.7822i	−1.61957	−	0.189251i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−135.450			−0.326384
$416$		0			0
$417$	−66.5572	+	569.584i	−0.159610	+	1.36591i
$418$		0			0
$419$			565.750i			1.35024i	0.737709	+	0.675119i	$0.235908\pi$
$419$			565.750i			1.35024i	−0.737709	+	0.675119i	$0.764092\pi$
$420$		0			0
$421$	434.911			1.03304			0.516521	−	0.856275i	$-0.327227\pi$
$421$	434.911			1.03304			0.516521	+	0.856275i	$0.327227\pi$
$422$		0			0
$423$	1.01738	−	4.29384i	0.00240516	−	0.0101509i
$424$		0			0
$425$			208.402i			0.490357i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−324.455	−	37.9134i	−0.756306	−	0.0883762i
$430$		0			0
$431$			408.152i			0.946988i	0.880797	+	0.473494i	$0.157007\pi$
$431$			408.152i			0.946988i	−0.880797	+	0.473494i	$0.842993\pi$
$432$		0			0
$433$	−213.334			−0.492689			−0.246345	−	0.969182i	$-0.579230\pi$
$433$	−213.334			−0.492689			−0.246345	+	0.969182i	$0.579230\pi$
$434$		0			0
$435$	−20.0902	+	171.928i	−0.0461844	+	0.395237i
$436$		0			0
$437$		−	3.68315i		−	0.00842826i
$438$		0			0
$439$	−347.083			−0.790622			−0.395311	−	0.918547i	$-0.629363\pi$
$439$	−347.083			−0.790622			−0.395311	+	0.918547i	$0.629363\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			53.8146i			0.121478i	0.998154	+	0.0607388i	$0.0193457\pi$
$443$			53.8146i			0.121478i	−0.998154	+	0.0607388i	$0.980654\pi$
$444$		0			0
$445$	151.188			0.339748
$446$		0			0
$447$	−85.0767	−	9.94141i	−0.190328	−	0.0222403i
$448$		0			0
$449$		−	487.920i		−	1.08668i	−0.839512	−	0.543341i	$-0.817159\pi$
$449$		−	487.920i		−	1.08668i	0.839512	−	0.543341i	$-0.182841\pi$
$450$		0			0
$451$	−615.864			−1.36555
$452$		0			0
$453$	20.4577	−	175.073i	0.0451604	−	0.386474i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	392.453			0.858760			0.429380	−	0.903124i	$-0.358732\pi$
$457$	392.453			0.858760			0.429380	+	0.903124i	$0.358732\pi$
$458$		0			0
$459$	231.191	+	84.1226i	0.503683	+	0.183274i
$460$		0			0
$461$			387.463i			0.840483i	0.907412	+	0.420242i	$0.138055\pi$
$461$			387.463i			0.840483i	−0.907412	+	0.420242i	$0.861945\pi$
$462$		0			0
$463$	−312.748			−0.675481			−0.337741	−	0.941239i	$-0.609663\pi$
$463$	−312.748			−0.675481			−0.337741	+	0.941239i	$0.609663\pi$
$464$		0			0
$465$	−165.849	−	19.3799i	−0.356665	−	0.0416772i
$466$		0			0
$467$		−	87.6804i		−	0.187753i	−0.995584	−	0.0938763i	$-0.970074\pi$
$467$		−	87.6804i		−	0.187753i	0.995584	−	0.0938763i	$-0.0299258\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−42.7433	+	365.789i	−0.0907502	+	0.776623i
$472$		0			0
$473$			732.635i			1.54891i
$474$		0			0
$475$	41.7156			0.0878223
$476$		0			0
$477$	−74.4892	+	314.380i	−0.156162	+	0.659077i
$478$		0			0
$479$		−	804.721i		−	1.68000i	−0.542585	−	0.840001i	$-0.682554\pi$
$479$		−	804.721i		−	1.68000i	0.542585	−	0.840001i	$-0.317446\pi$
$480$		0			0
$481$	−45.6369			−0.0948792
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	29.3587i		−	0.0605334i
$486$		0			0
$487$	−450.876			−0.925824			−0.462912	−	0.886404i	$-0.653195\pi$
$487$	−450.876			−0.925824			−0.462912	+	0.886404i	$0.653195\pi$
$488$		0			0
$489$	3.29921	−	28.2340i	0.00674684	−	0.0577382i
$490$		0			0
$491$		−	201.845i		−	0.411089i	−0.978648	−	0.205544i	$-0.934104\pi$
$491$		−	201.845i		−	0.411089i	0.978648	−	0.205544i	$-0.0658965\pi$
$492$		0			0
$493$	360.363			0.730959
$494$		0			0
$495$	−167.137	−	39.6015i	−0.337651	−	0.0800030i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−397.986			−0.797568			−0.398784	−	0.917045i	$-0.630568\pi$
$499$	−397.986			−0.797568			−0.398784	+	0.917045i	$0.630568\pi$
$500$		0			0
$501$	−689.249	−	80.5403i	−1.37575	−	0.160759i
$502$		0			0
$503$			170.620i			0.339205i	0.985513	+	0.169602i	$0.0542484\pi$
$503$			170.620i			0.339205i	−0.985513	+	0.169602i	$0.945752\pi$
$504$		0			0
$505$	180.527			0.357479
$506$		0			0
$507$	34.7190	−	297.118i	0.0684793	−	0.586032i
$508$		0			0
$509$		−	419.794i		−	0.824742i	−0.911016	−	0.412371i	$-0.864701\pi$
$509$		−	419.794i		−	0.824742i	0.911016	−	0.412371i	$-0.135299\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	16.8388	−	46.2773i	0.0328241	−	0.0902091i
$514$		0			0
$515$			155.232i			0.301422i
$516$		0			0
$517$	6.41384			0.0124059
$518$		0			0
$519$	665.168	+	77.7264i	1.28163	+	0.149762i
$520$		0			0
$521$			619.274i			1.18863i	0.804234	+	0.594313i	$0.202576\pi$
$521$			619.274i			1.18863i	−0.804234	+	0.594313i	$0.797424\pi$
$522$		0			0
$523$	−195.361			−0.373540			−0.186770	−	0.982404i	$-0.559802\pi$
$523$	−195.361			−0.373540			−0.186770	+	0.982404i	$0.559802\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			347.621i			0.659623i
$528$		0			0
$529$	524.922			0.992291
$530$		0			0
$531$	85.3093	−	360.046i	0.160658	−	0.678053i
$532$		0			0
$533$			391.882i			0.735239i
$534$		0			0
$535$	74.6717			0.139573
$536$		0			0
$537$	833.618	+	97.4103i	1.55236	+	0.181397i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	869.320			1.60688			0.803438	−	0.595388i	$-0.203002\pi$
$541$	869.320			1.60688			0.803438	+	0.595388i	$0.203002\pi$
$542$		0			0
$543$	−4.39694	+	37.6282i	−0.00809750	+	0.0692969i
$544$		0			0
$545$			257.752i			0.472939i
$546$		0			0
$547$	−470.259			−0.859706			−0.429853	−	0.902899i	$-0.641435\pi$
$547$	−470.259			−0.859706			−0.429853	+	0.902899i	$0.641435\pi$
$548$		0			0
$549$	−997.360	−	236.314i	−1.81668	−	0.430445i
$550$		0			0
$551$		−	72.1336i		−	0.130914i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−23.8344	−	2.78511i	−0.0429449	−	0.00501822i
$556$		0			0
$557$		−	1005.33i		−	1.80491i	−0.430785	−	0.902455i	$-0.641763\pi$
$557$		−	1005.33i		−	1.80491i	0.430785	−	0.902455i	$-0.358237\pi$
$558$		0			0
$559$	466.185			0.833962
$560$		0			0
$561$	−41.5025	+	355.170i	−0.0739795	+	0.633102i
$562$		0			0
$563$			430.731i			0.765065i	0.923942	+	0.382532i	$0.124948\pi$
$563$			430.731i			0.765065i	−0.923942	+	0.382532i	$0.875052\pi$
$564$		0			0
$565$	179.825			0.318275
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	940.339i		−	1.65262i	−0.563218	−	0.826309i	$-0.690437\pi$
$569$		−	940.339i		−	1.65262i	0.563218	−	0.826309i	$-0.309563\pi$
$570$		0			0
$571$	189.006			0.331009			0.165504	−	0.986209i	$-0.447075\pi$
$571$	189.006			0.331009			0.165504	+	0.986209i	$0.447075\pi$
$572$		0			0
$573$	447.038	+	52.2374i	0.780171	+	0.0911648i
$574$		0			0
$575$		−	46.1860i		−	0.0803234i
$576$		0			0
$577$	935.939			1.62208			0.811039	−	0.584992i	$-0.198903\pi$
$577$	935.939			1.62208			0.811039	+	0.584992i	$0.198903\pi$
$578$		0			0
$579$	−61.0648	+	522.581i	−0.105466	+	0.902557i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−469.599			−0.805487
$584$		0			0
$585$	−25.1989	+	106.352i	−0.0430751	+	0.181797i
$586$		0			0
$587$		−	614.569i		−	1.04697i	−0.852036	−	0.523483i	$-0.824632\pi$
$587$		−	614.569i		−	1.04697i	0.852036	−	0.523483i	$-0.175368\pi$
$588$		0			0
$589$	69.5831			0.118138
$590$		0			0
$591$	843.663	+	98.5840i	1.42752	+	0.166809i
$592$		0			0
$593$			611.212i			1.03071i	0.856976	+	0.515356i	$0.172340\pi$
$593$			611.212i			1.03071i	−0.856976	+	0.515356i	$0.827660\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.60083	+	39.3730i	−0.00770658	+	0.0659514i
$598$		0			0
$599$			46.3571i			0.0773908i	0.999251	+	0.0386954i	$0.0123202\pi$
$599$			46.3571i			0.0773908i	−0.999251	+	0.0386954i	$0.987680\pi$
$600$		0			0
$601$	−265.660			−0.442030			−0.221015	−	0.975270i	$-0.570937\pi$
$601$	−265.660			−0.442030			−0.221015	+	0.975270i	$0.570937\pi$
$602$		0			0
$603$	−961.118	−	227.727i	−1.59389	−	0.377657i
$604$		0			0
$605$		−	73.1262i		−	0.120870i
$606$		0			0
$607$	−51.9098			−0.0855185			−0.0427593	−	0.999085i	$-0.513615\pi$
$607$	−51.9098			−0.0855185			−0.0427593	+	0.999085i	$0.513615\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	4.08121i		−	0.00667956i
$612$		0			0
$613$	272.533			0.444590			0.222295	−	0.974979i	$-0.428645\pi$
$613$	272.533			0.444590			0.222295	+	0.974979i	$0.428645\pi$
$614$		0			0
$615$	−23.9156	+	204.665i	−0.0388872	+	0.332789i
$616$		0			0
$617$			612.948i			0.993433i	0.867913	+	0.496716i	$0.165461\pi$
$617$			612.948i			0.993433i	−0.867913	+	0.496716i	$0.834539\pi$
$618$		0			0
$619$	−314.074			−0.507390			−0.253695	−	0.967284i	$-0.581646\pi$
$619$	−314.074			−0.507390			−0.253695	+	0.967284i	$0.581646\pi$
$620$		0			0
$621$	−51.2365	−	18.6433i	−0.0825064	−	0.0300213i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	469.893			0.751828
$626$		0			0
$627$	71.0942	+	8.30752i	0.113388	+	0.0132496i
$628$		0			0
$629$			49.9572i			0.0794232i
$630$		0			0
$631$	−278.241			−0.440953			−0.220476	−	0.975392i	$-0.570761\pi$
$631$	−278.241			−0.440953			−0.220476	+	0.975392i	$0.570761\pi$
$632$		0			0
$633$	−110.244	+	943.450i	−0.174162	+	1.49044i
$634$		0			0
$635$		−	136.013i		−	0.214194i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	119.750	−	505.404i	0.187403	−	0.790929i
$640$		0			0
$641$		−	2.60040i		−	0.00405678i	−0.999998	−	0.00202839i	$-0.999354\pi$
$641$		−	2.60040i		−	0.00405678i	0.999998	−	0.00202839i	$-0.000645657\pi$
$642$		0			0
$643$	−574.117			−0.892873			−0.446437	−	0.894815i	$-0.647307\pi$
$643$	−574.117			−0.892873			−0.446437	+	0.894815i	$0.647307\pi$
$644$		0			0
$645$	243.471	+	28.4502i	0.377474	+	0.0441088i
$646$		0			0
$647$		−	251.669i		−	0.388978i	−0.980905	−	0.194489i	$-0.937695\pi$
$647$		−	251.669i		−	0.388978i	0.980905	−	0.194489i	$-0.0623048\pi$
$648$		0			0
$649$	537.812			0.828678
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1151.50i			1.76340i	0.471807	+	0.881702i	$0.343602\pi$
$653$			1151.50i			1.76340i	−0.471807	+	0.881702i	$0.656398\pi$
$654$		0			0
$655$	175.470			0.267893
$656$		0			0
$657$	−171.786	−	40.7029i	−0.261470	−	0.0619526i
$658$		0			0
$659$		−	711.265i		−	1.07931i	−0.841886	−	0.539655i	$-0.818555\pi$
$659$		−	711.265i		−	1.07931i	0.841886	−	0.539655i	$-0.181445\pi$
$660$		0			0
$661$	458.664			0.693894			0.346947	−	0.937885i	$-0.387218\pi$
$661$	458.664			0.693894			0.346947	+	0.937885i	$0.387218\pi$
$662$		0			0
$663$	225.999	+	26.4086i	0.340874	+	0.0398319i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−79.8636			−0.119736
$668$		0			0
$669$	134.699	−	1152.73i	0.201344	−	1.72306i
$670$		0			0
$671$		−	1489.79i		−	2.22025i
$672$		0			0
$673$	496.236			0.737349			0.368675	−	0.929559i	$-0.379812\pi$
$673$	496.236			0.737349			0.368675	+	0.929559i	$0.379812\pi$
$674$		0			0
$675$	211.155	−	580.308i	0.312822	−	0.859716i
$676$		0			0
$677$		−	682.765i		−	1.00852i	−0.863553	−	0.504258i	$-0.831766\pi$
$677$		−	682.765i		−	1.00852i	0.863553	−	0.504258i	$-0.168234\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1198.21	−	140.014i	−1.75949	−	0.205600i
$682$		0			0
$683$			117.270i			0.171698i	0.996308	+	0.0858492i	$0.0273603\pi$
$683$			117.270i			0.171698i	−0.996308	+	0.0858492i	$0.972640\pi$
$684$		0			0
$685$	−325.914			−0.475788
$686$		0			0
$687$	105.245	−	900.664i	0.153195	−	1.31101i
$688$		0			0
$689$			298.812i			0.433689i
$690$		0			0
$691$	1021.64			1.47850			0.739251	−	0.673430i	$-0.235180\pi$
$691$	1021.64			1.47850			0.739251	+	0.673430i	$0.235180\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			278.881i			0.401267i
$696$		0			0
$697$	428.980			0.615466
$698$		0			0
$699$	−17.5555	−	2.05141i	−0.0251152	−	0.00293477i
$700$		0			0
$701$			502.876i			0.717370i	0.933459	+	0.358685i	$0.116775\pi$
$701$			502.876i			0.717370i	−0.933459	+	0.358685i	$0.883225\pi$
$702$		0			0
$703$	9.99989			0.0142246
$704$		0			0
$705$	0.249067	−	2.13146i	0.000353286	−	0.00302335i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1173.40			−1.65501			−0.827506	−	0.561457i	$-0.810241\pi$
$709$	−1173.40			−1.65501			−0.827506	+	0.561457i	$0.810241\pi$
$710$		0			0
$711$	1236.91	+	293.073i	1.73967	+	0.412198i
$712$		0			0
$713$		−	77.0399i		−	0.108050i
$714$		0			0
$715$	−158.861			−0.222183
$716$		0			0
$717$	417.317	+	48.7645i	0.582032	+	0.0680118i
$718$		0			0
$719$			1087.04i			1.51188i	0.654642	+	0.755939i	$0.272819\pi$
$719$			1087.04i			1.51188i	−0.654642	+	0.755939i	$0.727181\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−101.789	+	871.091i	−0.140787	+	1.20483i
$724$		0			0
$725$		−	904.541i		−	1.24764i
$726$		0			0
$727$	1295.30			1.78170			0.890851	−	0.454296i	$-0.150109\pi$
$727$	1295.30			1.78170			0.890851	+	0.454296i	$0.150109\pi$
$728$		0			0
$729$	−558.532	−	468.490i	−0.766162	−	0.642648i
$730$		0			0
$731$		−	510.317i		−	0.698108i
$732$		0			0
$733$	−15.2622			−0.0208215			−0.0104107	−	0.999946i	$-0.503314\pi$
$733$	−15.2622			−0.0208215			−0.0104107	+	0.999946i	$0.503314\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	1435.65i		−	1.94797i
$738$		0			0
$739$	127.599			0.172665			0.0863323	−	0.996266i	$-0.472485\pi$
$739$	127.599			0.172665			0.0863323	+	0.996266i	$0.472485\pi$
$740$		0			0
$741$	5.28618	−	45.2382i	0.00713385	−	0.0610501i
$742$		0			0
$743$		−	1167.88i		−	1.57185i	−0.618322	−	0.785925i	$-0.712187\pi$
$743$		−	1167.88i		−	1.57185i	0.618322	−	0.785925i	$-0.287813\pi$
$744$		0			0
$745$	−41.6554			−0.0559133
$746$		0			0
$747$	192.646	−	813.059i	0.257893	−	1.08843i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−691.664			−0.920991			−0.460495	−	0.887662i	$-0.652328\pi$
$751$	−691.664			−0.920991			−0.460495	+	0.887662i	$0.652328\pi$
$752$		0			0
$753$	179.682	+	20.9962i	0.238621	+	0.0278835i
$754$		0			0
$755$		−	85.7195i		−	0.113536i
$756$		0			0
$757$	−1000.78			−1.32203			−0.661015	−	0.750373i	$-0.729874\pi$
$757$	−1000.78			−1.32203			−0.661015	+	0.750373i	$0.729874\pi$
$758$		0			0
$759$	9.19778	−	78.7129i	0.0121183	−	0.103706i
$760$		0			0
$761$			837.540i			1.10058i	0.834974	+	0.550289i	$0.185482\pi$
$761$			837.540i			1.10058i	−0.834974	+	0.550289i	$0.814518\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	116.419	+	27.5844i	0.152182	+	0.0360580i
$766$		0			0
$767$		−	342.217i		−	0.446176i
$768$		0			0
$769$	−488.742			−0.635555			−0.317778	−	0.948165i	$-0.602937\pi$
$769$	−488.742			−0.635555			−0.317778	+	0.948165i	$0.602937\pi$
$770$		0			0
$771$	−116.116	−	13.5684i	−0.150605	−	0.0175985i
$772$		0			0
$773$			370.163i			0.478865i	0.970913	+	0.239433i	$0.0769614\pi$
$773$			370.163i			0.478865i	−0.970913	+	0.239433i	$0.923039\pi$
$774$		0			0
$775$	872.559			1.12588
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	85.8686i		−	0.110229i
$780$		0			0
$781$	754.937			0.966629
$782$		0			0
$783$	−1003.45	−	365.124i	−1.28155	−	0.466314i
$784$		0			0
$785$			179.098i			0.228151i
$786$		0			0
$787$	363.011			0.461259			0.230630	−	0.973042i	$-0.425921\pi$
$787$	363.011			0.461259			0.230630	+	0.973042i	$0.425921\pi$
$788$		0			0
$789$	635.334	+	74.2403i	0.805239	+	0.0940941i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−947.971			−1.19542
$794$		0			0
$795$	−18.2358	+	156.058i	−0.0229381	+	0.196300i
$796$		0			0
$797$		−	306.686i		−	0.384801i	−0.981316	−	0.192401i	$-0.938373\pi$
$797$		−	306.686i		−	0.384801i	0.981316	−	0.192401i	$-0.0616273\pi$
$798$		0			0
$799$	−4.46756			−0.00559144
$800$		0			0
$801$	−215.030	+	907.531i	−0.268453	+	1.13300i
$802$		0			0
$803$		−	256.601i		−	0.319553i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	1045.35	+	122.152i	1.29536	+	0.151365i
$808$		0			0
$809$			473.236i			0.584965i	0.956271	+	0.292482i	$0.0944813\pi$
$809$			473.236i			0.584965i	−0.956271	+	0.292482i	$0.905519\pi$
$810$		0			0
$811$	−449.984			−0.554851			−0.277425	−	0.960747i	$-0.589481\pi$
$811$	−449.984			−0.554851			−0.277425	+	0.960747i	$0.589481\pi$
$812$		0			0
$813$	−20.0439	+	171.532i	−0.0246543	+	0.210987i
$814$		0			0
$815$		−	13.8240i		−	0.0169619i
$816$		0			0
$817$	−102.150			−0.125030
$818$		0			0
$819$		0			0
$820$		0			0
$821$			970.034i			1.18153i	0.806845	+	0.590763i	$0.201173\pi$
$821$			970.034i			1.18153i	−0.806845	+	0.590763i	$0.798827\pi$
$822$		0			0
$823$	509.539			0.619124			0.309562	−	0.950879i	$-0.399818\pi$
$823$	509.539			0.619124			0.309562	+	0.950879i	$0.399818\pi$
$824$		0			0
$825$	891.507	+	104.175i	1.08061	+	0.126272i
$826$		0			0
$827$			319.896i			0.386815i	0.981119	+	0.193408i	$0.0619539\pi$
$827$			319.896i			0.386815i	−0.981119	+	0.193408i	$0.938046\pi$
$828$		0			0
$829$	1044.16			1.25955			0.629774	−	0.776779i	$-0.283148\pi$
$829$	1044.16			1.25955			0.629774	+	0.776779i	$0.283148\pi$
$830$		0			0
$831$	133.372	−	1141.37i	0.160496	−	1.37350i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−337.471			−0.404157
$836$		0			0
$837$	352.214	−	967.975i	0.420805	−	1.15648i
$838$		0			0
$839$			344.169i			0.410214i	0.978740	+	0.205107i	$0.0657542\pi$
$839$			344.169i			0.410214i	−0.978740	+	0.205107i	$0.934246\pi$
$840$		0			0
$841$	−723.111			−0.859823
$842$		0			0
$843$	1533.25	+	179.164i	1.81881	+	0.212532i
$844$		0			0
$845$		−	145.476i		−	0.172161i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	139.405	−	1193.00i	0.164199	−	1.40519i
$850$		0			0
$851$		−	11.0715i		−	0.0130100i
$852$		0			0
$853$	1496.03			1.75384			0.876921	−	0.480635i	$-0.159594\pi$
$853$	1496.03			1.75384			0.876921	+	0.480635i	$0.159594\pi$
$854$		0			0
$855$	5.52155	−	23.3036i	0.00645795	−	0.0272557i
$856$		0			0
$857$			197.170i			0.230070i	0.993361	+	0.115035i	$0.0366981\pi$
$857$			197.170i			0.230070i	−0.993361	+	0.115035i	$0.963302\pi$
$858$		0			0
$859$	1368.44			1.59306			0.796530	−	0.604600i	$-0.206667\pi$
$859$	1368.44			1.59306			0.796530	+	0.604600i	$0.206667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			325.682i			0.377384i	0.982036	+	0.188692i	$0.0604247\pi$
$863$			325.682i			0.377384i	−0.982036	+	0.188692i	$0.939575\pi$
$864$		0			0
$865$	325.681			0.376510
$866$		0			0
$867$	−71.7178	+	613.747i	−0.0827195	+	0.707897i
$868$		0			0
$869$			1847.61i			2.12613i
$870$		0			0
$871$	−913.523			−1.04882
$872$		0			0
$873$	176.231	+	41.7561i	0.201868	+	0.0478306i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	637.824			0.727279			0.363640	−	0.931540i	$-0.381534\pi$
$877$	637.824			0.727279			0.363640	+	0.931540i	$0.381534\pi$
$878$		0			0
$879$	839.682	+	98.1188i	0.955270	+	0.111626i
$880$		0			0
$881$			705.868i			0.801212i	0.916250	+	0.400606i	$0.131200\pi$
$881$			705.868i			0.801212i	−0.916250	+	0.400606i	$0.868800\pi$
$882$		0			0
$883$	405.418			0.459137			0.229569	−	0.973292i	$-0.426268\pi$
$883$	405.418			0.459137			0.229569	+	0.973292i	$0.426268\pi$
$884$		0			0
$885$	20.8847	−	178.727i	0.0235985	−	0.201951i
$886$		0			0
$887$		−	574.456i		−	0.647640i	−0.946119	−	0.323820i	$-0.895033\pi$
$887$		−	574.456i		−	0.647640i	0.946119	−	0.323820i	$-0.104967\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	475.429	−	946.945i	0.533590	−	1.06279i
$892$		0			0
$893$			0.894269i			0.00100142i
$894$		0			0
$895$	408.158			0.456042
$896$		0			0
$897$	−50.0860	−	5.85267i	−0.0558372	−	0.00652471i
$898$		0			0
$899$		−	1508.81i		−	1.67832i
$900$		0			0
$901$	327.099			0.363040
$902$		0			0
$903$		0			0
$904$		0			0
$905$			18.4236i			0.0203576i
$906$		0			0
$907$	194.253			0.214171			0.107086	−	0.994250i	$-0.465848\pi$
$907$	194.253			0.214171			0.107086	+	0.994250i	$0.465848\pi$
$908$		0			0
$909$	−256.758	+	1083.64i	−0.282462	+	1.19213i
$910$		0			0
$911$			811.661i			0.890956i	0.895293	+	0.445478i	$0.146966\pi$
$911$			811.661i			0.890956i	−0.895293	+	0.445478i	$0.853034\pi$
$912$		0			0
$913$	1214.49			1.33022
$914$		0			0
$915$	−495.089	−	57.8524i	−0.541081	−	0.0632266i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−484.566			−0.527276			−0.263638	−	0.964622i	$-0.584922\pi$
$919$	−484.566			−0.527276			−0.263638	+	0.964622i	$0.584922\pi$
$920$		0			0
$921$	−125.579	+	1074.68i	−0.136351	+	1.16686i
$922$		0			0
$923$		−	480.376i		−	0.520451i
$924$		0			0
$925$	125.397			0.135564
$926$		0			0
$927$	−931.809	−	220.783i	−1.00519	−	0.238169i
$928$		0			0
$929$			1634.85i			1.75980i	0.475161	+	0.879899i	$0.342390\pi$
$929$			1634.85i			1.75980i	−0.475161	+	0.879899i	$0.657610\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−1308.55	−	152.908i	−1.40252	−	0.163888i
$934$		0			0
$935$			173.899i			0.185988i
$936$		0			0
$937$	1004.00			1.07150			0.535752	−	0.844375i	$-0.320028\pi$
$937$	1004.00			1.07150			0.535752	+	0.844375i	$0.320028\pi$
$938$		0			0
$939$	137.809	−	1179.34i	0.146761	−	1.25595i
$940$		0			0
$941$			375.042i			0.398557i	0.979943	+	0.199279i	$0.0638599\pi$
$941$			375.042i			0.398557i	−0.979943	+	0.199279i	$0.936140\pi$
$942$		0			0
$943$	−95.0706			−0.100817
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1371.62i			1.44838i	0.689598	+	0.724192i	$0.257787\pi$
$947$			1371.62i			1.44838i	−0.689598	+	0.724192i	$0.742213\pi$
$948$		0			0
$949$	−163.279			−0.172054
$950$		0			0
$951$	438.065	+	51.1890i	0.460637	+	0.0538265i
$952$		0			0
$953$			436.692i			0.458228i	0.973400	+	0.229114i	$0.0735829\pi$
$953$			436.692i			0.458228i	−0.973400	+	0.229114i	$0.926417\pi$
$954$		0			0
$955$	218.880			0.229193
$956$		0			0
$957$	180.136	−	1541.57i	0.188230	−	1.61084i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	494.460			0.514527
$962$		0			0
$963$	−106.204	+	448.230i	−0.110284	+	0.465451i
$964$		0			0
$965$			255.867i			0.265147i
$966$		0			0
$967$	−608.569			−0.629338			−0.314669	−	0.949202i	$-0.601893\pi$
$967$	−608.569			−0.629338			−0.314669	+	0.949202i	$0.601893\pi$
$968$		0			0
$969$	−49.5207	−	5.78661i	−0.0511049	−	0.00597173i
$970$		0			0
$971$			596.115i			0.613919i	0.951723	+	0.306959i	$0.0993116\pi$
$971$			596.115i			0.613919i	−0.951723	+	0.306959i	$0.900688\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	66.2877	−	567.278i	0.0679874	−	0.581823i
$976$		0			0
$977$			1243.69i			1.27297i	0.771289	+	0.636485i	$0.219612\pi$
$977$			1243.69i			1.27297i	−0.771289	+	0.636485i	$0.780388\pi$
$978$		0			0
$979$	−1355.61			−1.38469
$980$		0			0
$981$	−1547.20	−	366.593i	−1.57716	−	0.373693i
$982$		0			0
$983$		−	692.875i		−	0.704857i	−0.935839	−	0.352429i	$-0.885356\pi$
$983$		−	692.875i		−	0.704857i	0.935839	−	0.352429i	$-0.114644\pi$
$984$		0			0
$985$	413.076			0.419366
$986$		0			0
$987$		0			0
$988$		0			0
$989$			113.096i			0.114354i
$990$		0			0
$991$	1400.47			1.41319			0.706595	−	0.707618i	$-0.250230\pi$
$991$	1400.47			1.41319			0.706595	+	0.707618i	$0.250230\pi$
$992$		0			0
$993$	−34.4112	+	294.484i	−0.0346538	+	0.296560i
$994$		0			0
$995$			19.2779i			0.0193748i
$996$		0			0
$997$	−392.565			−0.393746			−0.196873	−	0.980429i	$-0.563079\pi$
$997$	−392.565			−0.393746			−0.196873	+	0.980429i	$0.563079\pi$
$998$		0			0
$999$	50.6172	−	139.109i	0.0506678	−	0.139248i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.d.h.785.12	yes	24
3.2	odd	2	inner	1176.3.d.h.785.11	✓	24
7.6	odd	2	inner	1176.3.d.h.785.13	yes	24
21.20	even	2	inner	1176.3.d.h.785.14	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.d.h.785.11	✓	24	3.2	odd	2	inner
1176.3.d.h.785.12	yes	24	1.1	even	1	trivial
1176.3.d.h.785.13	yes	24	7.6	odd	2	inner
1176.3.d.h.785.14	yes	24	21.20	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.348188 + 2.97973i) q^{3} +1.45894i q^{5} +(-8.75753 - 2.07501i) q^{9} -13.0814i q^{11} -8.32386 q^{13} +(-4.34724 - 0.507985i) q^{15} +9.11185i q^{17} +1.82391 q^{19} -2.01937i q^{23} +22.8715 q^{25} +(9.23223 - 25.3725i) q^{27} -39.5488i q^{29} +38.1505 q^{31} +(38.9790 + 4.55478i) q^{33} +5.48266 q^{37} +(2.89827 - 24.8028i) q^{39} -47.0794i q^{41} -56.0059 q^{43} +(3.02731 - 12.7767i) q^{45} +0.490303i q^{47} +(-27.1508 - 3.17264i) q^{51} -35.8982i q^{53} +19.0850 q^{55} +(-0.635064 + 5.43476i) q^{57} +41.1127i q^{59} +113.886 q^{61} -12.1440i q^{65} +109.748 q^{67} +(6.01716 + 0.703120i) q^{69} +57.7108i q^{71} +19.6158 q^{73} +(-7.96358 + 68.1508i) q^{75} -141.239 q^{79} +(72.3887 + 36.3439i) q^{81} +92.8411i q^{83} -13.2936 q^{85} +(117.845 + 13.7704i) q^{87} -103.629i q^{89} +(-13.2835 + 113.678i) q^{93} +2.66098i q^{95} -20.1233 q^{97} +(-27.1440 + 114.561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 16 q^{15} - 152 q^{25} + 64 q^{37} - 16 q^{39} + 16 q^{43} + 248 q^{51} - 120 q^{57} - 416 q^{67} - 224 q^{79} - 344 q^{81} + 496 q^{85} - 464 q^{93}+O(q^{100}) \) 24 * q - 16 * q^15 - 152 * q^25 + 64 * q^37 - 16 * q^39 + 16 * q^43 + 248 * q^51 - 120 * q^57 - 416 * q^67 - 224 * q^79 - 344 * q^81 + 496 * q^85 - 464 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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