LMFDB - Embedded newform 1176.3.f.d.97.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(97,1176)] 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("1176.97"); 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, 0, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 66 x^{14} + 136 x^{13} + 3441 x^{12} - 4512 x^{11} - 100322 x^{10} + \cdots + 13841287201 $ x^16 - 4x^15 - 66x^14 + 136x^13 + 3441x^12 - 4512x^11 - 100322x^10 + 24868x^9 + 2325060x^8 + 260012x^7 - 34273754x^6 - 26559312x^5 + 402103713x^4 + 286444088x^3 - 2633783418x^2 - 3060285788*x + 13841287201
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.7
Root			$3.82621 + 1.63171i$ of defining polynomial
Character	$\chi$	$=$	1176.97
Dual form			1176.3.f.d.97.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +5.38055i q^{5} -3.00000 q^{9} +20.7553 q^{11} +5.85486i q^{13} +9.31938 q^{15} +29.1939i q^{17} -29.3627i q^{19} -15.0506 q^{23} -3.95028 q^{25} +5.19615i q^{27} -40.8578 q^{29} +1.60464i q^{31} -35.9492i q^{33} +36.3508 q^{37} +10.1409 q^{39} +29.4486i q^{41} -26.6560 q^{43} -16.1416i q^{45} +48.6100i q^{47} +50.5653 q^{51} +19.4011 q^{53} +111.675i q^{55} -50.8576 q^{57} +33.0513i q^{59} +81.3118i q^{61} -31.5023 q^{65} +47.2759 q^{67} +26.0684i q^{69} +107.830 q^{71} -40.9454i q^{73} +6.84209i q^{75} -59.6662 q^{79} +9.00000 q^{81} +89.6080i q^{83} -157.079 q^{85} +70.7678i q^{87} +11.1776i q^{89} +2.77933 q^{93} +157.987 q^{95} +86.7164i q^{97} -62.2659 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) $ 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	5.38055i	1.07611i	0.842910	+	0.538055i	$0.180841\pi$
$5$	5.38055i	1.07611i	−0.842910	+	0.538055i	$0.819159\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	20.7553	1.88684	0.943422	−	0.331595i	$-0.107587\pi$
$11$	20.7553	1.88684	0.943422	+	0.331595i	$0.107587\pi$
$12$	0	0
$13$	5.85486i	0.450374i	0.974316	+	0.225187i	$0.0722992\pi$
$13$	5.85486i	0.450374i	−0.974316	+	0.225187i	$0.927701\pi$
$14$	0	0
$15$	9.31938	0.621292
$16$	0	0
$17$	29.1939i	1.71729i	0.512574	+	0.858643i	$0.328692\pi$
$17$	29.1939i	1.71729i	−0.512574	+	0.858643i	$0.671308\pi$
$18$	0	0
$19$	− 29.3627i	− 1.54540i	−0.634770	−	0.772701i	$-0.718905\pi$
$19$	− 29.3627i	− 1.54540i	0.634770	−	0.772701i	$-0.281095\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−15.0506	−0.654374	−0.327187	−	0.944960i	$-0.606101\pi$
$23$	−15.0506	−0.654374	−0.327187	+	0.944960i	$0.606101\pi$
$24$	0	0
$25$	−3.95028	−0.158011
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−40.8578	−1.40889	−0.704445	−	0.709758i	$-0.748804\pi$
$29$	−40.8578	−1.40889	−0.704445	+	0.709758i	$0.748804\pi$
$30$	0	0
$31$	1.60464i	0.0517627i	0.999665	+	0.0258814i	$0.00823922\pi$
$31$	1.60464i	0.0517627i	−0.999665	+	0.0258814i	$0.991761\pi$
$32$	0	0
$33$	− 35.9492i	− 1.08937i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	36.3508	0.982453	0.491227	−	0.871032i	$-0.336549\pi$
$37$	36.3508	0.982453	0.491227	+	0.871032i	$0.336549\pi$
$38$	0	0
$39$	10.1409	0.260023
$40$	0	0
$41$	29.4486i	0.718259i	0.933288	+	0.359129i	$0.116926\pi$
$41$	29.4486i	0.718259i	−0.933288	+	0.359129i	$0.883074\pi$
$42$	0	0
$43$	−26.6560	−0.619908	−0.309954	−	0.950752i	$-0.600314\pi$
$43$	−26.6560	−0.619908	−0.309954	+	0.950752i	$0.600314\pi$
$44$	0	0
$45$	− 16.1416i	− 0.358703i
$46$	0	0
$47$	48.6100i	1.03425i	0.855909	+	0.517127i	$0.172999\pi$
$47$	48.6100i	1.03425i	−0.855909	+	0.517127i	$0.827001\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	50.5653	0.991476
$52$	0	0
$53$	19.4011	0.366058	0.183029	−	0.983107i	$-0.441410\pi$
$53$	19.4011	0.366058	0.183029	+	0.983107i	$0.441410\pi$
$54$	0	0
$55$	111.675i	2.03045i
$56$	0	0
$57$	−50.8576	−0.892239
$58$	0	0
$59$	33.0513i	0.560192i	0.959972	+	0.280096i	$0.0903663\pi$
$59$	33.0513i	0.560192i	−0.959972	+	0.280096i	$0.909634\pi$
$60$	0	0
$61$	81.3118i	1.33298i	0.745514	+	0.666490i	$0.232204\pi$
$61$	81.3118i	1.33298i	−0.745514	+	0.666490i	$0.767796\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−31.5023	−0.484651
$66$	0	0
$67$	47.2759	0.705611	0.352805	−	0.935697i	$-0.385228\pi$
$67$	47.2759	0.705611	0.352805	+	0.935697i	$0.385228\pi$
$68$	0	0
$69$	26.0684i	0.377803i
$70$	0	0
$71$	107.830	1.51873	0.759367	−	0.650662i	$-0.225509\pi$
$71$	107.830	1.51873	0.759367	+	0.650662i	$0.225509\pi$
$72$	0	0
$73$	− 40.9454i	− 0.560896i	−0.959869	−	0.280448i	$-0.909517\pi$
$73$	− 40.9454i	− 0.560896i	0.959869	−	0.280448i	$-0.0904830\pi$
$74$	0	0
$75$	6.84209i	0.0912279i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−59.6662	−0.755269	−0.377634	−	0.925955i	$-0.623262\pi$
$79$	−59.6662	−0.755269	−0.377634	+	0.925955i	$0.623262\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	89.6080i	1.07961i	0.841789	+	0.539807i	$0.181503\pi$
$83$	89.6080i	1.07961i	−0.841789	+	0.539807i	$0.818497\pi$
$84$	0	0
$85$	−157.079	−1.84799
$86$	0	0
$87$	70.7678i	0.813424i
$88$	0	0
$89$	11.1776i	0.125591i	0.998026	+	0.0627953i	$0.0200015\pi$
$89$	11.1776i	0.125591i	−0.998026	+	0.0627953i	$0.979998\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.77933	0.0298852
$94$	0	0
$95$	157.987	1.66302
$96$	0	0
$97$	86.7164i	0.893983i	0.894538	+	0.446992i	$0.147505\pi$
$97$	86.7164i	0.893983i	−0.894538	+	0.446992i	$0.852495\pi$
$98$	0	0
$99$	−62.2659	−0.628948
$100$	0	0
$101$	− 108.291i	− 1.07219i	−0.844159	−	0.536093i	$-0.819900\pi$
$101$	− 108.291i	− 1.07219i	0.844159	−	0.536093i	$-0.180100\pi$
$102$	0	0
$103$	156.369i	1.51815i	0.651004	+	0.759075i	$0.274348\pi$
$103$	156.369i	1.51815i	−0.651004	+	0.759075i	$0.725652\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	156.354	1.46125	0.730626	−	0.682777i	$-0.239228\pi$
$107$	156.354	1.46125	0.730626	+	0.682777i	$0.239228\pi$
$108$	0	0
$109$	−146.779	−1.34660	−0.673299	−	0.739371i	$-0.735123\pi$
$109$	−146.779	−1.34660	−0.673299	+	0.739371i	$0.735123\pi$
$110$	0	0
$111$	− 62.9614i	− 0.567220i
$112$	0	0
$113$	−64.3443	−0.569419	−0.284709	−	0.958614i	$-0.591897\pi$
$113$	−64.3443	−0.569419	−0.284709	+	0.958614i	$0.591897\pi$
$114$	0	0
$115$	− 80.9805i	− 0.704178i
$116$	0	0
$117$	− 17.5646i	− 0.150125i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	309.782	2.56018
$122$	0	0
$123$	51.0065	0.414687
$124$	0	0
$125$	113.259i	0.906072i
$126$	0	0
$127$	−180.830	−1.42386	−0.711930	−	0.702250i	$-0.752179\pi$
$127$	−180.830	−1.42386	−0.711930	+	0.702250i	$0.752179\pi$
$128$	0	0
$129$	46.1696i	0.357904i
$130$	0	0
$131$	43.7377i	0.333875i	0.985967	+	0.166938i	$0.0533879\pi$
$131$	43.7377i	0.333875i	−0.985967	+	0.166938i	$0.946612\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−27.9581	−0.207097
$136$	0	0
$137$	205.397	1.49925	0.749623	−	0.661865i	$-0.230235\pi$
$137$	205.397	1.49925	0.749623	+	0.661865i	$0.230235\pi$
$138$	0	0
$139$	− 98.5007i	− 0.708638i	−0.935125	−	0.354319i	$-0.884713\pi$
$139$	− 98.5007i	− 0.708638i	0.935125	−	0.354319i	$-0.115287\pi$
$140$	0	0
$141$	84.1949	0.597127
$142$	0	0
$143$	121.519i	0.849785i
$144$	0	0
$145$	− 219.837i	− 1.51612i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	222.025	1.49010	0.745051	−	0.667008i	$-0.232425\pi$
$149$	222.025	1.49010	0.745051	+	0.667008i	$0.232425\pi$
$150$	0	0
$151$	146.094	0.967512	0.483756	−	0.875203i	$-0.339272\pi$
$151$	146.094	0.967512	0.483756	+	0.875203i	$0.339272\pi$
$152$	0	0
$153$	− 87.5816i	− 0.572429i
$154$	0	0
$155$	−8.63387	−0.0557024
$156$	0	0
$157$	26.1171i	0.166351i	0.996535	+	0.0831753i	$0.0265062\pi$
$157$	26.1171i	0.166351i	−0.996535	+	0.0831753i	$0.973494\pi$
$158$	0	0
$159$	− 33.6037i	− 0.211344i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−198.735	−1.21923	−0.609617	−	0.792696i	$-0.708677\pi$
$163$	−198.735	−1.21923	−0.609617	+	0.792696i	$0.708677\pi$
$164$	0	0
$165$	193.426	1.17228
$166$	0	0
$167$	111.349i	0.666758i	0.942793	+	0.333379i	$0.108189\pi$
$167$	111.349i	0.666758i	−0.942793	+	0.333379i	$0.891811\pi$
$168$	0	0
$169$	134.721	0.797164
$170$	0	0
$171$	88.0880i	0.515134i
$172$	0	0
$173$	− 231.165i	− 1.33622i	−0.744064	−	0.668108i	$-0.767104\pi$
$173$	− 231.165i	− 1.33622i	0.744064	−	0.668108i	$-0.232896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	57.2466	0.323427
$178$	0	0
$179$	130.241	0.727601	0.363800	−	0.931477i	$-0.381479\pi$
$179$	130.241	0.727601	0.363800	+	0.931477i	$0.381479\pi$
$180$	0	0
$181$	360.775i	1.99323i	0.0822038	+	0.996616i	$0.473804\pi$
$181$	360.775i	1.99323i	−0.0822038	+	0.996616i	$0.526196\pi$
$182$	0	0
$183$	140.836	0.769596
$184$	0	0
$185$	195.587i	1.05723i
$186$	0	0
$187$	605.927i	3.24025i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−196.737	−1.03004	−0.515019	−	0.857179i	$-0.672215\pi$
$191$	−196.737	−1.03004	−0.515019	+	0.857179i	$0.672215\pi$
$192$	0	0
$193$	−187.389	−0.970930	−0.485465	−	0.874256i	$-0.661350\pi$
$193$	−187.389	−0.970930	−0.485465	+	0.874256i	$0.661350\pi$
$194$	0	0
$195$	54.5636i	0.279814i
$196$	0	0
$197$	−347.588	−1.76440	−0.882202	−	0.470871i	$-0.843940\pi$
$197$	−347.588	−1.76440	−0.882202	+	0.470871i	$0.843940\pi$
$198$	0	0
$199$	− 58.6307i	− 0.294627i	−0.989090	−	0.147313i	$-0.952937\pi$
$199$	− 58.6307i	− 0.294627i	0.989090	−	0.147313i	$-0.0470626\pi$
$200$	0	0
$201$	− 81.8843i	− 0.407385i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−158.450	−0.772925
$206$	0	0
$207$	45.1518	0.218125
$208$	0	0
$209$	− 609.430i	− 2.91593i
$210$	0	0
$211$	320.844	1.52059	0.760293	−	0.649580i	$-0.225055\pi$
$211$	320.844	1.52059	0.760293	+	0.649580i	$0.225055\pi$
$212$	0	0
$213$	− 186.767i	− 0.876842i
$214$	0	0
$215$	− 143.424i	− 0.667089i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−70.9195	−0.323833
$220$	0	0
$221$	−170.926	−0.773420
$222$	0	0
$223$	174.988i	0.784697i	0.919817	+	0.392349i	$0.128337\pi$
$223$	174.988i	0.784697i	−0.919817	+	0.392349i	$0.871663\pi$
$224$	0	0
$225$	11.8508	0.0526704
$226$	0	0
$227$	45.0702i	0.198547i	0.995060	+	0.0992735i	$0.0316519\pi$
$227$	45.0702i	0.198547i	−0.995060	+	0.0992735i	$0.968348\pi$
$228$	0	0
$229$	214.027i	0.934616i	0.884095	+	0.467308i	$0.154776\pi$
$229$	214.027i	0.934616i	−0.884095	+	0.467308i	$0.845224\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	51.1938	0.219716	0.109858	−	0.993947i	$-0.464960\pi$
$233$	51.1938	0.219716	0.109858	+	0.993947i	$0.464960\pi$
$234$	0	0
$235$	−261.548	−1.11297
$236$	0	0
$237$	103.345i	0.436055i
$238$	0	0
$239$	39.9405	0.167115	0.0835575	−	0.996503i	$-0.473372\pi$
$239$	39.9405	0.167115	0.0835575	+	0.996503i	$0.473372\pi$
$240$	0	0
$241$	− 384.101i	− 1.59378i	−0.604126	−	0.796889i	$-0.706477\pi$
$241$	− 384.101i	− 1.59378i	0.604126	−	0.796889i	$-0.293523\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	171.914	0.696009
$248$	0	0
$249$	155.206	0.623315
$250$	0	0
$251$	− 116.147i	− 0.462735i	−0.972866	−	0.231368i	$-0.925680\pi$
$251$	− 116.147i	− 0.462735i	0.972866	−	0.231368i	$-0.0743200\pi$
$252$	0	0
$253$	−312.380	−1.23470
$254$	0	0
$255$	272.069i	1.06694i
$256$	0	0
$257$	− 316.695i	− 1.23228i	−0.787638	−	0.616138i	$-0.788696\pi$
$257$	− 316.695i	− 1.23228i	0.787638	−	0.616138i	$-0.211304\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	122.574	0.469630
$262$	0	0
$263$	−283.962	−1.07970	−0.539852	−	0.841760i	$-0.681520\pi$
$263$	−283.962	−1.07970	−0.539852	+	0.841760i	$0.681520\pi$
$264$	0	0
$265$	104.388i	0.393919i
$266$	0	0
$267$	19.3601	0.0725098
$268$	0	0
$269$	110.316i	0.410097i	0.978752	+	0.205049i	$0.0657353\pi$
$269$	110.316i	0.410097i	−0.978752	+	0.205049i	$0.934265\pi$
$270$	0	0
$271$	− 87.8002i	− 0.323986i	−0.986792	−	0.161993i	$-0.948208\pi$
$271$	− 87.8002i	− 0.323986i	0.986792	−	0.161993i	$-0.0517922\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−81.9892	−0.298143
$276$	0	0
$277$	−214.641	−0.774876	−0.387438	−	0.921896i	$-0.626640\pi$
$277$	−214.641	−0.774876	−0.387438	+	0.921896i	$0.626640\pi$
$278$	0	0
$279$	− 4.81393i	− 0.0172542i
$280$	0	0
$281$	−245.946	−0.875251	−0.437626	−	0.899157i	$-0.644180\pi$
$281$	−245.946	−0.875251	−0.437626	+	0.899157i	$0.644180\pi$
$282$	0	0
$283$	− 224.163i	− 0.792096i	−0.918230	−	0.396048i	$-0.870381\pi$
$283$	− 224.163i	− 0.792096i	0.918230	−	0.396048i	$-0.129619\pi$
$284$	0	0
$285$	− 273.642i	− 0.960146i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−563.282	−1.94907
$290$	0	0
$291$	150.197	0.516141
$292$	0	0
$293$	− 67.3702i	− 0.229932i	−0.993369	−	0.114966i	$-0.963324\pi$
$293$	− 67.3702i	− 0.229932i	0.993369	−	0.114966i	$-0.0366760\pi$
$294$	0	0
$295$	−177.834	−0.602828
$296$	0	0
$297$	107.848i	0.363123i
$298$	0	0
$299$	− 88.1191i	− 0.294713i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−187.565	−0.619027
$304$	0	0
$305$	−437.502	−1.43443
$306$	0	0
$307$	163.788i	0.533512i	0.963764	+	0.266756i	$0.0859518\pi$
$307$	163.788i	0.533512i	−0.963764	+	0.266756i	$0.914048\pi$
$308$	0	0
$309$	270.840	0.876504
$310$	0	0
$311$	− 405.695i	− 1.30449i	−0.758010	−	0.652243i	$-0.773828\pi$
$311$	− 405.695i	− 1.30449i	0.758010	−	0.652243i	$-0.226172\pi$
$312$	0	0
$313$	− 137.914i	− 0.440620i	−0.975430	−	0.220310i	$-0.929293\pi$
$313$	− 137.914i	− 0.440620i	0.975430	−	0.220310i	$-0.0707068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	48.0560	0.151596	0.0757981	−	0.997123i	$-0.475850\pi$
$317$	48.0560	0.151596	0.0757981	+	0.997123i	$0.475850\pi$
$318$	0	0
$319$	−848.016	−2.65836
$320$	0	0
$321$	− 270.813i	− 0.843655i
$322$	0	0
$323$	857.209	2.65390
$324$	0	0
$325$	− 23.1283i	− 0.0711641i
$326$	0	0
$327$	254.229i	0.777458i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−395.676	−1.19540	−0.597698	−	0.801721i	$-0.703918\pi$
$331$	−395.676	−1.19540	−0.597698	+	0.801721i	$0.703918\pi$
$332$	0	0
$333$	−109.052	−0.327484
$334$	0	0
$335$	254.370i	0.759315i
$336$	0	0
$337$	346.264	1.02749	0.513745	−	0.857943i	$-0.328258\pi$
$337$	346.264	1.02749	0.513745	+	0.857943i	$0.328258\pi$
$338$	0	0
$339$	111.448i	0.328754i
$340$	0	0
$341$	33.3049i	0.0976682i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−140.262	−0.406557
$346$	0	0
$347$	538.234	1.55111	0.775554	−	0.631282i	$-0.217471\pi$
$347$	538.234	1.55111	0.775554	+	0.631282i	$0.217471\pi$
$348$	0	0
$349$	− 583.911i	− 1.67310i	−0.547892	−	0.836549i	$-0.684570\pi$
$349$	− 583.911i	− 1.67310i	0.547892	−	0.836549i	$-0.315430\pi$
$350$	0	0
$351$	−30.4227	−0.0866744
$352$	0	0
$353$	448.776i	1.27132i	0.771969	+	0.635660i	$0.219272\pi$
$353$	448.776i	1.27132i	−0.771969	+	0.635660i	$0.780728\pi$
$354$	0	0
$355$	580.185i	1.63432i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−170.770	−0.475682	−0.237841	−	0.971304i	$-0.576440\pi$
$359$	−170.770	−0.475682	−0.237841	+	0.971304i	$0.576440\pi$
$360$	0	0
$361$	−501.165	−1.38827
$362$	0	0
$363$	− 536.558i	− 1.47812i
$364$	0	0
$365$	220.309	0.603585
$366$	0	0
$367$	− 103.574i	− 0.282217i	−0.989994	−	0.141109i	$-0.954933\pi$
$367$	− 103.574i	− 0.282217i	0.989994	−	0.141109i	$-0.0450667\pi$
$368$	0	0
$369$	− 88.3458i	− 0.239420i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	572.808	1.53568	0.767840	−	0.640642i	$-0.221332\pi$
$373$	572.808	1.53568	0.767840	+	0.640642i	$0.221332\pi$
$374$	0	0
$375$	196.170	0.523121
$376$	0	0
$377$	− 239.217i	− 0.634527i
$378$	0	0
$379$	−46.7351	−0.123312	−0.0616558	−	0.998097i	$-0.519638\pi$
$379$	−46.7351	−0.123312	−0.0616558	+	0.998097i	$0.519638\pi$
$380$	0	0
$381$	313.207i	0.822066i
$382$	0	0
$383$	− 207.343i	− 0.541367i	−0.962668	−	0.270683i	$-0.912750\pi$
$383$	− 207.343i	− 0.541367i	0.962668	−	0.270683i	$-0.0872496\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	79.9681	0.206636
$388$	0	0
$389$	629.880	1.61923	0.809614	−	0.586962i	$-0.199676\pi$
$389$	629.880	1.61923	0.809614	+	0.586962i	$0.199676\pi$
$390$	0	0
$391$	− 439.385i	− 1.12375i
$392$	0	0
$393$	75.7559	0.192763
$394$	0	0
$395$	− 321.037i	− 0.812752i
$396$	0	0
$397$	189.131i	0.476401i	0.971216	+	0.238201i	$0.0765576\pi$
$397$	189.131i	0.476401i	−0.971216	+	0.238201i	$0.923442\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−290.598	−0.724683	−0.362341	−	0.932045i	$-0.618023\pi$
$401$	−290.598	−0.724683	−0.362341	+	0.932045i	$0.618023\pi$
$402$	0	0
$403$	−9.39497	−0.0233126
$404$	0	0
$405$	48.4249i	0.119568i
$406$	0	0
$407$	754.471	1.85374
$408$	0	0
$409$	− 491.079i	− 1.20068i	−0.799744	−	0.600341i	$-0.795032\pi$
$409$	− 491.079i	− 1.20068i	0.799744	−	0.600341i	$-0.204968\pi$
$410$	0	0
$411$	− 355.758i	− 0.865590i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−482.140	−1.16178
$416$	0	0
$417$	−170.608	−0.409132
$418$	0	0
$419$	− 385.645i	− 0.920395i	−0.887817	−	0.460197i	$-0.847779\pi$
$419$	− 385.645i	− 0.920395i	0.887817	−	0.460197i	$-0.152221\pi$
$420$	0	0
$421$	−254.510	−0.604538	−0.302269	−	0.953223i	$-0.597744\pi$
$421$	−254.510	−0.604538	−0.302269	+	0.953223i	$0.597744\pi$
$422$	0	0
$423$	− 145.830i	− 0.344751i
$424$	0	0
$425$	− 115.324i	− 0.271351i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	210.477	0.490623
$430$	0	0
$431$	370.546	0.859735	0.429867	−	0.902892i	$-0.358560\pi$
$431$	370.546	0.859735	0.429867	+	0.902892i	$0.358560\pi$
$432$	0	0
$433$	− 359.620i	− 0.830531i	−0.909700	−	0.415266i	$-0.863689\pi$
$433$	− 359.620i	− 0.830531i	0.909700	−	0.415266i	$-0.136311\pi$
$434$	0	0
$435$	−380.770	−0.875333
$436$	0	0
$437$	441.926i	1.01127i
$438$	0	0
$439$	− 492.973i	− 1.12295i	−0.827495	−	0.561473i	$-0.810235\pi$
$439$	− 492.973i	− 1.12295i	0.827495	−	0.561473i	$-0.189765\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	499.038	1.12650	0.563249	−	0.826287i	$-0.309551\pi$
$443$	499.038	1.12650	0.563249	+	0.826287i	$0.309551\pi$
$444$	0	0
$445$	−60.1414	−0.135149
$446$	0	0
$447$	− 384.559i	− 0.860310i
$448$	0	0
$449$	73.9706	0.164745	0.0823727	−	0.996602i	$-0.473750\pi$
$449$	73.9706	0.164745	0.0823727	+	0.996602i	$0.473750\pi$
$450$	0	0
$451$	611.214i	1.35524i
$452$	0	0
$453$	− 253.043i	− 0.558593i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.4346	0.0228327	0.0114164	−	0.999935i	$-0.496366\pi$
$457$	10.4346	0.0228327	0.0114164	+	0.999935i	$0.496366\pi$
$458$	0	0
$459$	−151.696	−0.330492
$460$	0	0
$461$	− 331.513i	− 0.719117i	−0.933123	−	0.359559i	$-0.882927\pi$
$461$	− 331.513i	− 0.719117i	0.933123	−	0.359559i	$-0.117073\pi$
$462$	0	0
$463$	139.451	0.301190	0.150595	−	0.988596i	$-0.451881\pi$
$463$	139.451	0.301190	0.150595	+	0.988596i	$0.451881\pi$
$464$	0	0
$465$	14.9543i	0.0321598i
$466$	0	0
$467$	− 879.066i	− 1.88237i	−0.337894	−	0.941184i	$-0.609715\pi$
$467$	− 879.066i	− 1.88237i	0.337894	−	0.941184i	$-0.390285\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	45.2361	0.0960426
$472$	0	0
$473$	−553.254	−1.16967
$474$	0	0
$475$	115.991i	0.244191i
$476$	0	0
$477$	−58.2033	−0.122019
$478$	0	0
$479$	46.3016i	0.0966630i	0.998831	+	0.0483315i	$0.0153904\pi$
$479$	46.3016i	0.0966630i	−0.998831	+	0.0483315i	$0.984610\pi$
$480$	0	0
$481$	212.829i	0.442471i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−466.582	−0.962024
$486$	0	0
$487$	−421.048	−0.864575	−0.432288	−	0.901736i	$-0.642293\pi$
$487$	−421.048	−0.864575	−0.432288	+	0.901736i	$0.642293\pi$
$488$	0	0
$489$	344.220i	0.703925i
$490$	0	0
$491$	−31.6874	−0.0645364	−0.0322682	−	0.999479i	$-0.510273\pi$
$491$	−31.6874	−0.0645364	−0.0322682	+	0.999479i	$0.510273\pi$
$492$	0	0
$493$	− 1192.80i	− 2.41947i
$494$	0	0
$495$	− 335.024i	− 0.676817i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	361.358	0.724165	0.362083	−	0.932146i	$-0.382066\pi$
$499$	361.358	0.724165	0.362083	+	0.932146i	$0.382066\pi$
$500$	0	0
$501$	192.861	0.384953
$502$	0	0
$503$	− 425.369i	− 0.845664i	−0.906208	−	0.422832i	$-0.861036\pi$
$503$	− 425.369i	− 0.845664i	0.906208	−	0.422832i	$-0.138964\pi$
$504$	0	0
$505$	582.664	1.15379
$506$	0	0
$507$	− 233.343i	− 0.460243i
$508$	0	0
$509$	911.230i	1.79023i	0.445831	+	0.895117i	$0.352908\pi$
$509$	911.230i	1.79023i	−0.445831	+	0.895117i	$0.647092\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	152.573	0.297413
$514$	0	0
$515$	−841.353	−1.63369
$516$	0	0
$517$	1008.91i	1.95148i
$518$	0	0
$519$	−400.390	−0.771465
$520$	0	0
$521$	− 552.635i	− 1.06072i	−0.847772	−	0.530360i	$-0.822057\pi$
$521$	− 552.635i	− 1.06072i	0.847772	−	0.530360i	$-0.177943\pi$
$522$	0	0
$523$	473.434i	0.905227i	0.891707	+	0.452613i	$0.149508\pi$
$523$	473.434i	0.905227i	−0.891707	+	0.452613i	$0.850492\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−46.8458	−0.0888914
$528$	0	0
$529$	−302.479	−0.571795
$530$	0	0
$531$	− 99.1540i	− 0.186731i
$532$	0	0
$533$	−172.417	−0.323485
$534$	0	0
$535$	841.270i	1.57247i
$536$	0	0
$537$	− 225.583i	− 0.420081i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	105.655	0.195297	0.0976483	−	0.995221i	$-0.468868\pi$
$541$	105.655	0.195297	0.0976483	+	0.995221i	$0.468868\pi$
$542$	0	0
$543$	624.880	1.15079
$544$	0	0
$545$	− 789.752i	− 1.44909i
$546$	0	0
$547$	−447.104	−0.817374	−0.408687	−	0.912675i	$-0.634013\pi$
$547$	−447.104	−0.817374	−0.408687	+	0.912675i	$0.634013\pi$
$548$	0	0
$549$	− 243.935i	− 0.444327i
$550$	0	0
$551$	1199.69i	2.17730i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	338.767	0.610390
$556$	0	0
$557$	585.387	1.05096	0.525482	−	0.850805i	$-0.323885\pi$
$557$	585.387	1.05096	0.525482	+	0.850805i	$0.323885\pi$
$558$	0	0
$559$	− 156.067i	− 0.279190i
$560$	0	0
$561$	1049.50	1.87076
$562$	0	0
$563$	− 38.2112i	− 0.0678706i	−0.999424	−	0.0339353i	$-0.989196\pi$
$563$	− 38.2112i	− 0.0678706i	0.999424	−	0.0339353i	$-0.0108040\pi$
$564$	0	0
$565$	− 346.208i	− 0.612757i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−695.422	−1.22218	−0.611091	−	0.791560i	$-0.709269\pi$
$569$	−695.422	−1.22218	−0.611091	+	0.791560i	$0.709269\pi$
$570$	0	0
$571$	−387.295	−0.678275	−0.339137	−	0.940737i	$-0.610135\pi$
$571$	−387.295	−0.678275	−0.339137	+	0.940737i	$0.610135\pi$
$572$	0	0
$573$	340.759i	0.594692i
$574$	0	0
$575$	59.4541	0.103399
$576$	0	0
$577$	− 686.904i	− 1.19048i	−0.803550	−	0.595238i	$-0.797058\pi$
$577$	− 686.904i	− 1.19048i	0.803550	−	0.595238i	$-0.202942\pi$
$578$	0	0
$579$	324.568i	0.560567i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	402.675	0.690695
$584$	0	0
$585$	94.5070	0.161550
$586$	0	0
$587$	− 20.4465i	− 0.0348322i	−0.999848	−	0.0174161i	$-0.994456\pi$
$587$	− 20.4465i	− 0.0348322i	0.999848	−	0.0174161i	$-0.00554400\pi$
$588$	0	0
$589$	47.1166	0.0799943
$590$	0	0
$591$	602.039i	1.01868i
$592$	0	0
$593$	895.808i	1.51064i	0.655357	+	0.755319i	$0.272518\pi$
$593$	895.808i	1.51064i	−0.655357	+	0.755319i	$0.727482\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−101.551	−0.170103
$598$	0	0
$599$	447.572	0.747199	0.373599	−	0.927590i	$-0.378124\pi$
$599$	447.572	0.747199	0.373599	+	0.927590i	$0.378124\pi$
$600$	0	0
$601$	− 33.3613i	− 0.0555096i	−0.999615	−	0.0277548i	$-0.991164\pi$
$601$	− 33.3613i	− 0.0555096i	0.999615	−	0.0277548i	$-0.00883576\pi$
$602$	0	0
$603$	−141.828	−0.235204
$604$	0	0
$605$	1666.80i	2.75503i
$606$	0	0
$607$	− 162.882i	− 0.268339i	−0.990958	−	0.134170i	$-0.957163\pi$
$607$	− 162.882i	− 0.268339i	0.990958	−	0.134170i	$-0.0428367\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−284.604	−0.465801
$612$	0	0
$613$	1105.86	1.80402	0.902009	−	0.431718i	$-0.142092\pi$
$613$	1105.86	1.80402	0.902009	+	0.431718i	$0.142092\pi$
$614$	0	0
$615$	274.443i	0.446248i
$616$	0	0
$617$	354.579	0.574683	0.287341	−	0.957828i	$-0.407229\pi$
$617$	354.579	0.574683	0.287341	+	0.957828i	$0.407229\pi$
$618$	0	0
$619$	− 623.125i	− 1.00666i	−0.864093	−	0.503332i	$-0.832107\pi$
$619$	− 623.125i	− 1.00666i	0.864093	−	0.503332i	$-0.167893\pi$
$620$	0	0
$621$	− 78.2052i	− 0.125934i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−708.152	−1.13304
$626$	0	0
$627$	−1055.56	−1.68352
$628$	0	0
$629$	1061.22i	1.68715i
$630$	0	0
$631$	−425.556	−0.674415	−0.337207	−	0.941430i	$-0.609482\pi$
$631$	−425.556	−0.674415	−0.337207	+	0.941430i	$0.609482\pi$
$632$	0	0
$633$	− 555.718i	− 0.877911i
$634$	0	0
$635$	− 972.966i	− 1.53223i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−323.490	−0.506245
$640$	0	0
$641$	−666.589	−1.03992	−0.519960	−	0.854190i	$-0.674053\pi$
$641$	−666.589	−1.03992	−0.519960	+	0.854190i	$0.674053\pi$
$642$	0	0
$643$	409.903i	0.637486i	0.947841	+	0.318743i	$0.103261\pi$
$643$	409.903i	0.637486i	−0.947841	+	0.318743i	$0.896739\pi$
$644$	0	0
$645$	−248.418	−0.385144
$646$	0	0
$647$	− 177.591i	− 0.274484i	−0.990538	−	0.137242i	$-0.956176\pi$
$647$	− 177.591i	− 0.274484i	0.990538	−	0.137242i	$-0.0438238\pi$
$648$	0	0
$649$	685.990i	1.05700i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	245.999	0.376721	0.188361	−	0.982100i	$-0.439683\pi$
$653$	245.999	0.376721	0.188361	+	0.982100i	$0.439683\pi$
$654$	0	0
$655$	−235.333	−0.359286
$656$	0	0
$657$	122.836i	0.186965i
$658$	0	0
$659$	−1273.21	−1.93204	−0.966019	−	0.258471i	$-0.916781\pi$
$659$	−1273.21	−1.93204	−0.966019	+	0.258471i	$0.916781\pi$
$660$	0	0
$661$	983.291i	1.48758i	0.668413	+	0.743790i	$0.266974\pi$
$661$	983.291i	1.48758i	−0.668413	+	0.743790i	$0.733026\pi$
$662$	0	0
$663$	296.052i	0.446535i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	614.935	0.921942
$668$	0	0
$669$	303.087	0.453045
$670$	0	0
$671$	1687.65i	2.51513i
$672$	0	0
$673$	−421.581	−0.626420	−0.313210	−	0.949684i	$-0.601404\pi$
$673$	−421.581	−0.626420	−0.313210	+	0.949684i	$0.601404\pi$
$674$	0	0
$675$	− 20.5263i	− 0.0304093i
$676$	0	0
$677$	123.355i	0.182209i	0.995841	+	0.0911045i	$0.0290397\pi$
$677$	123.355i	0.182209i	−0.995841	+	0.0911045i	$0.970960\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	78.0638	0.114631
$682$	0	0
$683$	984.909	1.44203	0.721017	−	0.692918i	$-0.243675\pi$
$683$	984.909	1.44203	0.721017	+	0.692918i	$0.243675\pi$
$684$	0	0
$685$	1105.15i	1.61335i
$686$	0	0
$687$	370.706	0.539601
$688$	0	0
$689$	113.591i	0.164863i
$690$	0	0
$691$	− 916.472i	− 1.32630i	−0.748487	−	0.663149i	$-0.769219\pi$
$691$	− 916.472i	− 1.32630i	0.748487	−	0.663149i	$-0.230781\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	529.987	0.762572
$696$	0	0
$697$	−859.719	−1.23346
$698$	0	0
$699$	− 88.6703i	− 0.126853i
$700$	0	0
$701$	−1289.76	−1.83988	−0.919942	−	0.392054i	$-0.871765\pi$
$701$	−1289.76	−1.83988	−0.919942	+	0.392054i	$0.871765\pi$
$702$	0	0
$703$	− 1067.35i	− 1.51829i
$704$	0	0
$705$	453.015i	0.642574i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−411.444	−0.580316	−0.290158	−	0.956979i	$-0.593708\pi$
$709$	−411.444	−0.580316	−0.290158	+	0.956979i	$0.593708\pi$
$710$	0	0
$711$	178.999	0.251756
$712$	0	0
$713$	− 24.1509i	− 0.0338722i
$714$	0	0
$715$	−653.840	−0.914461
$716$	0	0
$717$	− 69.1790i	− 0.0964839i
$718$	0	0
$719$	− 1281.10i	− 1.78178i	−0.454223	−	0.890888i	$-0.650083\pi$
$719$	− 1281.10i	− 1.78178i	0.454223	−	0.890888i	$-0.349917\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−665.282	−0.920168
$724$	0	0
$725$	161.400	0.222621
$726$	0	0
$727$	− 887.557i	− 1.22085i	−0.792075	−	0.610424i	$-0.790999\pi$
$727$	− 887.557i	− 1.22085i	0.792075	−	0.610424i	$-0.209001\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 778.193i	− 1.06456i
$732$	0	0
$733$	− 512.057i	− 0.698577i	−0.937015	−	0.349289i	$-0.886423\pi$
$733$	− 512.057i	− 0.698577i	0.937015	−	0.349289i	$-0.113577\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	981.225	1.33138
$738$	0	0
$739$	1082.65	1.46502	0.732509	−	0.680757i	$-0.238349\pi$
$739$	1082.65	1.46502	0.732509	+	0.680757i	$0.238349\pi$
$740$	0	0
$741$	− 297.764i	− 0.401841i
$742$	0	0
$743$	79.2533	0.106667	0.0533333	−	0.998577i	$-0.483015\pi$
$743$	79.2533	0.106667	0.0533333	+	0.998577i	$0.483015\pi$
$744$	0	0
$745$	1194.62i	1.60351i
$746$	0	0
$747$	− 268.824i	− 0.359871i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	470.166	0.626053	0.313027	−	0.949744i	$-0.398657\pi$
$751$	470.166	0.626053	0.313027	+	0.949744i	$0.398657\pi$
$752$	0	0
$753$	−201.172	−0.267160
$754$	0	0
$755$	786.067i	1.04115i
$756$	0	0
$757$	1472.65	1.94537	0.972687	−	0.232122i	$-0.0745667\pi$
$757$	1472.65	1.94537	0.972687	+	0.232122i	$0.0745667\pi$
$758$	0	0
$759$	541.057i	0.712855i
$760$	0	0
$761$	− 91.6228i	− 0.120398i	−0.998186	−	0.0601990i	$-0.980826\pi$
$761$	− 91.6228i	− 0.120398i	0.998186	−	0.0601990i	$-0.0191735\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	471.237	0.615996
$766$	0	0
$767$	−193.511	−0.252296
$768$	0	0
$769$	− 188.544i	− 0.245181i	−0.992457	−	0.122590i	$-0.960880\pi$
$769$	− 188.544i	− 0.245181i	0.992457	−	0.122590i	$-0.0391201\pi$
$770$	0	0
$771$	−548.532	−0.711455
$772$	0	0
$773$	− 390.417i	− 0.505068i	−0.967588	−	0.252534i	$-0.918736\pi$
$773$	− 390.417i	− 0.505068i	0.967588	−	0.252534i	$-0.0812639\pi$
$774$	0	0
$775$	− 6.33880i	− 0.00817910i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	864.689	1.11000
$780$	0	0
$781$	2238.05	2.86562
$782$	0	0
$783$	− 212.304i	− 0.271141i
$784$	0	0
$785$	−140.524	−0.179012
$786$	0	0
$787$	1541.80i	1.95908i	0.201251	+	0.979540i	$0.435499\pi$
$787$	1541.80i	1.95908i	−0.201251	+	0.979540i	$0.564501\pi$
$788$	0	0
$789$	491.836i	0.623367i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−476.069	−0.600339
$794$	0	0
$795$	180.806	0.227429
$796$	0	0
$797$	− 279.016i	− 0.350083i	−0.984561	−	0.175041i	$-0.943994\pi$
$797$	− 279.016i	− 0.350083i	0.984561	−	0.175041i	$-0.0560059\pi$
$798$	0	0
$799$	−1419.11	−1.77611
$800$	0	0
$801$	− 33.5327i	− 0.0418635i
$802$	0	0
$803$	− 849.833i	− 1.05832i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	191.073	0.236770
$808$	0	0
$809$	591.859	0.731594	0.365797	−	0.930695i	$-0.380797\pi$
$809$	591.859	0.731594	0.365797	+	0.930695i	$0.380797\pi$
$810$	0	0
$811$	− 460.758i	− 0.568136i	−0.958804	−	0.284068i	$-0.908316\pi$
$811$	− 460.758i	− 0.568136i	0.958804	−	0.284068i	$-0.0916841\pi$
$812$	0	0
$813$	−152.074	−0.187053
$814$	0	0
$815$	− 1069.30i	− 1.31203i
$816$	0	0
$817$	782.692i	0.958008i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	426.019	0.518903	0.259452	−	0.965756i	$-0.416458\pi$
$821$	426.019	0.518903	0.259452	+	0.965756i	$0.416458\pi$
$822$	0	0
$823$	1295.16	1.57371	0.786855	−	0.617138i	$-0.211708\pi$
$823$	1295.16	1.57371	0.786855	+	0.617138i	$0.211708\pi$
$824$	0	0
$825$	142.010i	0.172133i
$826$	0	0
$827$	1427.32	1.72590	0.862951	−	0.505287i	$-0.168613\pi$
$827$	1427.32	1.72590	0.862951	+	0.505287i	$0.168613\pi$
$828$	0	0
$829$	684.451i	0.825634i	0.910814	+	0.412817i	$0.135455\pi$
$829$	684.451i	0.825634i	−0.910814	+	0.412817i	$0.864545\pi$
$830$	0	0
$831$	371.768i	0.447375i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−599.116	−0.717504
$836$	0	0
$837$	−8.33798	−0.00996174
$838$	0	0
$839$	878.029i	1.04652i	0.852173	+	0.523259i	$0.175284\pi$
$839$	878.029i	1.04652i	−0.852173	+	0.523259i	$0.824716\pi$
$840$	0	0
$841$	828.363	0.984973
$842$	0	0
$843$	425.990i	0.505327i
$844$	0	0
$845$	724.871i	0.857835i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−388.262	−0.457317
$850$	0	0
$851$	−547.101	−0.642892
$852$	0	0
$853$	1651.39i	1.93598i	0.250995	+	0.967988i	$0.419242\pi$
$853$	1651.39i	1.93598i	−0.250995	+	0.967988i	$0.580758\pi$
$854$	0	0
$855$	−473.961	−0.554341
$856$	0	0
$857$	509.814i	0.594882i	0.954740	+	0.297441i	$0.0961331\pi$
$857$	509.814i	0.594882i	−0.954740	+	0.297441i	$0.903867\pi$
$858$	0	0
$859$	633.039i	0.736949i	0.929638	+	0.368474i	$0.120120\pi$
$859$	633.039i	0.736949i	−0.929638	+	0.368474i	$0.879880\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	938.863	1.08791	0.543953	−	0.839116i	$-0.316927\pi$
$863$	938.863	1.08791	0.543953	+	0.839116i	$0.316927\pi$
$864$	0	0
$865$	1243.80	1.43792
$866$	0	0
$867$	975.633i	1.12530i
$868$	0	0
$869$	−1238.39	−1.42507
$870$	0	0
$871$	276.794i	0.317789i
$872$	0	0
$873$	− 260.149i	− 0.297994i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	124.596	0.142071	0.0710353	−	0.997474i	$-0.477370\pi$
$877$	124.596	0.142071	0.0710353	+	0.997474i	$0.477370\pi$
$878$	0	0
$879$	−116.689	−0.132751
$880$	0	0
$881$	310.270i	0.352180i	0.984374	+	0.176090i	$0.0563450\pi$
$881$	310.270i	0.352180i	−0.984374	+	0.176090i	$0.943655\pi$
$882$	0	0
$883$	277.720	0.314519	0.157259	−	0.987557i	$-0.449734\pi$
$883$	277.720	0.314519	0.157259	+	0.987557i	$0.449734\pi$
$884$	0	0
$885$	308.018i	0.348043i
$886$	0	0
$887$	69.2888i	0.0781159i	0.999237	+	0.0390580i	$0.0124357\pi$
$887$	69.2888i	0.0781159i	−0.999237	+	0.0390580i	$0.987564\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	186.798	0.209649
$892$	0	0
$893$	1427.32	1.59834
$894$	0	0
$895$	700.766i	0.782978i
$896$	0	0
$897$	−152.627	−0.170153
$898$	0	0
$899$	− 65.5623i	− 0.0729281i
$900$	0	0
$901$	566.393i	0.628627i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1941.17	−2.14493
$906$	0	0
$907$	−250.336	−0.276005	−0.138002	−	0.990432i	$-0.544068\pi$
$907$	−250.336	−0.276005	−0.138002	+	0.990432i	$0.544068\pi$
$908$	0	0
$909$	324.872i	0.357395i
$910$	0	0
$911$	−779.364	−0.855503	−0.427752	−	0.903896i	$-0.640694\pi$
$911$	−779.364	−0.855503	−0.427752	+	0.903896i	$0.640694\pi$
$912$	0	0
$913$	1859.84i	2.03706i
$914$	0	0
$915$	757.775i	0.828170i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1774.85	1.93129	0.965644	−	0.259870i	$-0.0836798\pi$
$919$	1774.85	1.93129	0.965644	+	0.259870i	$0.0836798\pi$
$920$	0	0
$921$	283.690	0.308024
$922$	0	0
$923$	631.330i	0.683998i
$924$	0	0
$925$	−143.596	−0.155239
$926$	0	0
$927$	− 469.108i	− 0.506050i
$928$	0	0
$929$	449.632i	0.483996i	0.970277	+	0.241998i	$0.0778027\pi$
$929$	449.632i	0.483996i	−0.970277	+	0.241998i	$0.922197\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−702.685	−0.753145
$934$	0	0
$935$	−3260.22	−3.48686
$936$	0	0
$937$	− 1864.83i	− 1.99021i	−0.0988089	−	0.995106i	$-0.531503\pi$
$937$	− 1864.83i	− 1.99021i	0.0988089	−	0.995106i	$-0.468497\pi$
$938$	0	0
$939$	−238.874	−0.254392
$940$	0	0
$941$	− 905.229i	− 0.961986i	−0.876724	−	0.480993i	$-0.840276\pi$
$941$	− 905.229i	− 0.961986i	0.876724	−	0.480993i	$-0.159724\pi$
$942$	0	0
$943$	− 443.219i	− 0.470010i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	520.433	0.549560	0.274780	−	0.961507i	$-0.411395\pi$
$947$	520.433	0.549560	0.274780	+	0.961507i	$0.411395\pi$
$948$	0	0
$949$	239.729	0.252613
$950$	0	0
$951$	− 83.2354i	− 0.0875241i
$952$	0	0
$953$	612.431	0.642635	0.321318	−	0.946971i	$-0.395874\pi$
$953$	612.431	0.642635	0.321318	+	0.946971i	$0.395874\pi$
$954$	0	0
$955$	− 1058.55i	− 1.10843i
$956$	0	0
$957$	1468.81i	1.53480i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	958.425	0.997321
$962$	0	0
$963$	−469.062	−0.487084
$964$	0	0
$965$	− 1008.26i	− 1.04483i
$966$	0	0
$967$	−173.989	−0.179927	−0.0899634	−	0.995945i	$-0.528675\pi$
$967$	−173.989	−0.179927	−0.0899634	+	0.995945i	$0.528675\pi$
$968$	0	0
$969$	− 1484.73i	− 1.53223i
$970$	0	0
$971$	− 160.363i	− 0.165152i	−0.996585	−	0.0825760i	$-0.973685\pi$
$971$	− 160.363i	− 0.165152i	0.996585	−	0.0825760i	$-0.0263147\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−40.0595	−0.0410866
$976$	0	0
$977$	−344.662	−0.352776	−0.176388	−	0.984321i	$-0.556441\pi$
$977$	−344.662	−0.352776	−0.176388	+	0.984321i	$0.556441\pi$
$978$	0	0
$979$	231.993i	0.236970i
$980$	0	0
$981$	440.337	0.448866
$982$	0	0
$983$	− 747.406i	− 0.760331i	−0.924918	−	0.380166i	$-0.875867\pi$
$983$	− 747.406i	− 0.760331i	0.924918	−	0.380166i	$-0.124133\pi$
$984$	0	0
$985$	− 1870.21i	− 1.89869i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	401.190	0.405652
$990$	0	0
$991$	842.526	0.850177	0.425089	−	0.905152i	$-0.360243\pi$
$991$	842.526	0.850177	0.425089	+	0.905152i	$0.360243\pi$
$992$	0	0
$993$	685.332i	0.690163i
$994$	0	0
$995$	315.465	0.317051
$996$	0	0
$997$	− 874.010i	− 0.876640i	−0.898819	−	0.438320i	$-0.855574\pi$
$997$	− 874.010i	− 0.876640i	0.898819	−	0.438320i	$-0.144426\pi$
$998$	0	0
$999$	188.884i	0.189073i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.f.d.97.7	✓	16
3.2	odd	2		3528.3.f.j.2449.5		16
4.3	odd	2		2352.3.f.m.97.15		16
7.2	even	3		1176.3.z.h.913.7		16
7.3	odd	6		1176.3.z.h.313.7		16
7.4	even	3		1176.3.z.g.313.2		16
7.5	odd	6		1176.3.z.g.913.2		16
7.6	odd	2	inner	1176.3.f.d.97.10	yes	16
21.20	even	2		3528.3.f.j.2449.12		16
28.27	even	2		2352.3.f.m.97.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.3.f.d.97.7	✓	16	1.1	even	1	trivial
1176.3.f.d.97.10	yes	16	7.6	odd	2	inner
1176.3.z.g.313.2		16	7.4	even	3
1176.3.z.g.913.2		16	7.5	odd	6
1176.3.z.h.313.7		16	7.3	odd	6
1176.3.z.h.913.7		16	7.2	even	3
2352.3.f.m.97.2		16	28.27	even	2
2352.3.f.m.97.15		16	4.3	odd	2
3528.3.f.j.2449.5		16	3.2	odd	2
3528.3.f.j.2449.12		16	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +5.38055i q^{5} -3.00000 q^{9} +20.7553 q^{11} +5.85486i q^{13} +9.31938 q^{15} +29.1939i q^{17} -29.3627i q^{19} -15.0506 q^{23} -3.95028 q^{25} +5.19615i q^{27} -40.8578 q^{29} +1.60464i q^{31} -35.9492i q^{33} +36.3508 q^{37} +10.1409 q^{39} +29.4486i q^{41} -26.6560 q^{43} -16.1416i q^{45} +48.6100i q^{47} +50.5653 q^{51} +19.4011 q^{53} +111.675i q^{55} -50.8576 q^{57} +33.0513i q^{59} +81.3118i q^{61} -31.5023 q^{65} +47.2759 q^{67} +26.0684i q^{69} +107.830 q^{71} -40.9454i q^{73} +6.84209i q^{75} -59.6662 q^{79} +9.00000 q^{81} +89.6080i q^{83} -157.079 q^{85} +70.7678i q^{87} +11.1776i q^{89} +2.77933 q^{93} +157.987 q^{95} +86.7164i q^{97} -62.2659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 112 q^{23} - 256 q^{25} + 112 q^{29} - 128 q^{37} - 112 q^{43} + 112 q^{53} + 192 q^{57} - 112 q^{65} - 224 q^{67} + 160 q^{71} + 304 q^{79} + 144 q^{81} - 416 q^{85} - 192 q^{93} + 912 q^{95}+O(q^{100}) \) 16 * q - 48 * q^9 - 112 * q^23 - 256 * q^25 + 112 * q^29 - 128 * q^37 - 112 * q^43 + 112 * q^53 + 192 * q^57 - 112 * q^65 - 224 * q^67 + 160 * q^71 + 304 * q^79 + 144 * q^81 - 416 * q^85 - 192 * q^93 + 912 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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