LMFDB - Embedded newform 288.3.b.b.271.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [288,3,Mod(271,288)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("288.271"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$N$	$=$	$288 = 2^{5} \cdot 3^{2}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	288.b (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-32] 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:	$7.84743161358$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.4752.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} - 6x + 6$ x^4 + 3x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$-0.866025 - 1.99551i$ of defining polynomial
Character	$\chi$	$=$	288.271
Dual form			288.3.b.b.271.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.98203i q^{5} -2.13878i q^{7} -8.00000 q^{11} +11.6865i q^{13} -11.8564 q^{17} -14.9282 q^{19} -4.27756i q^{23} -38.7128 q^{25} -0.573084i q^{29} +57.4399i q^{31} +17.0718 q^{35} +27.6506i q^{37} +31.5692 q^{41} -28.7846 q^{43} +59.5787i q^{47} +44.4256 q^{49} -31.3550i q^{53} -63.8562i q^{55} -52.7846 q^{59} -59.5787i q^{61} -93.2820 q^{65} +84.7846 q^{67} -42.4685i q^{71} -5.42563 q^{73} +17.1102i q^{77} -44.6072i q^{79} +67.7128 q^{83} -94.6382i q^{85} +133.138 q^{89} +24.9948 q^{91} -119.157i q^{95} +97.1384 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 32 q^{11} + 8 q^{17} - 32 q^{19} - 44 q^{25} + 96 q^{35} - 40 q^{41} - 32 q^{43} - 44 q^{49} - 128 q^{59} - 96 q^{65} + 256 q^{67} + 200 q^{73} + 160 q^{83} + 200 q^{89} - 288 q^{91} + 56 q^{97}+O(q^{100})$ 4 * q - 32 * q^11 + 8 * q^17 - 32 * q^19 - 44 * q^25 + 96 * q^35 - 40 * q^41 - 32 * q^43 - 44 * q^49 - 128 * q^59 - 96 * q^65 + 256 * q^67 + 200 * q^73 + 160 * q^83 + 200 * q^89 - 288 * q^91 + 56 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$ .

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	7.98203i	1.59641i	0.602388	+	0.798203i	$0.294216\pi$
$5$	7.98203i	1.59641i	−0.602388	+	0.798203i	$0.705784\pi$
$6$	0	0
$7$	− 2.13878i	− 0.305540i	−0.988262	−	0.152770i	$-0.951181\pi$
$7$	− 2.13878i	− 0.305540i	0.988262	−	0.152770i	$-0.0488193\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.00000	−0.727273	−0.363636	−	0.931541i	$-0.618465\pi$
$11$	−8.00000	−0.727273	−0.363636	+	0.931541i	$0.618465\pi$
$12$	0	0
$13$	11.6865i	0.898962i	0.893290	+	0.449481i	$0.148391\pi$
$13$	11.6865i	0.898962i	−0.893290	+	0.449481i	$0.851609\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−11.8564	−0.697436	−0.348718	−	0.937228i	$-0.613383\pi$
$17$	−11.8564	−0.697436	−0.348718	+	0.937228i	$0.613383\pi$
$18$	0	0
$19$	−14.9282	−0.785695	−0.392847	−	0.919604i	$-0.628510\pi$
$19$	−14.9282	−0.785695	−0.392847	+	0.919604i	$0.628510\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.27756i	− 0.185981i	−0.995667	−	0.0929904i	$-0.970357\pi$
$23$	− 4.27756i	− 0.185981i	0.995667	−	0.0929904i	$-0.0296426\pi$
$24$	0	0
$25$	−38.7128	−1.54851
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 0.573084i	− 0.0197615i	−0.999951	−	0.00988076i	$-0.996855\pi$
$29$	− 0.573084i	− 0.0197615i	0.999951	−	0.00988076i	$-0.00314519\pi$
$30$	0	0
$31$	57.4399i	1.85290i	0.376417	+	0.926450i	$0.377156\pi$
$31$	57.4399i	1.85290i	−0.376417	+	0.926450i	$0.622844\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	17.0718	0.487766
$36$	0	0
$37$	27.6506i	0.747313i	0.927567	+	0.373656i	$0.121896\pi$
$37$	27.6506i	0.747313i	−0.927567	+	0.373656i	$0.878104\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	31.5692	0.769981	0.384990	−	0.922921i	$-0.374205\pi$
$41$	31.5692	0.769981	0.384990	+	0.922921i	$0.374205\pi$
$42$	0	0
$43$	−28.7846	−0.669410	−0.334705	−	0.942323i	$-0.608637\pi$
$43$	−28.7846	−0.669410	−0.334705	+	0.942323i	$0.608637\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	59.5787i	1.26763i	0.773484	+	0.633816i	$0.218512\pi$
$47$	59.5787i	1.26763i	−0.773484	+	0.633816i	$0.781488\pi$
$48$	0	0
$49$	44.4256	0.906645
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 31.3550i	− 0.591604i	−0.955249	−	0.295802i	$-0.904413\pi$
$53$	− 31.3550i	− 0.591604i	0.955249	−	0.295802i	$-0.0955869\pi$
$54$	0	0
$55$	− 63.8562i	− 1.16102i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−52.7846	−0.894654	−0.447327	−	0.894370i	$-0.647624\pi$
$59$	−52.7846	−0.894654	−0.447327	+	0.894370i	$0.647624\pi$
$60$	0	0
$61$	− 59.5787i	− 0.976700i	−0.872648	−	0.488350i	$-0.837599\pi$
$61$	− 59.5787i	− 0.976700i	0.872648	−	0.488350i	$-0.162401\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−93.2820	−1.43511
$66$	0	0
$67$	84.7846	1.26544	0.632721	−	0.774380i	$-0.281938\pi$
$67$	84.7846	1.26544	0.632721	+	0.774380i	$0.281938\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 42.4685i	− 0.598147i	−0.954230	−	0.299074i	$-0.903322\pi$
$71$	− 42.4685i	− 0.598147i	0.954230	−	0.299074i	$-0.0966776\pi$
$72$	0	0
$73$	−5.42563	−0.0743236	−0.0371618	−	0.999309i	$-0.511832\pi$
$73$	−5.42563	−0.0743236	−0.0371618	+	0.999309i	$0.511832\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.1102i	0.222211i
$78$	0	0
$79$	− 44.6072i	− 0.564649i	−0.959319	−	0.282324i	$-0.908895\pi$
$79$	− 44.6072i	− 0.564649i	0.959319	−	0.282324i	$-0.0911054\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	67.7128	0.815817	0.407909	−	0.913023i	$-0.366258\pi$
$83$	67.7128	0.815817	0.407909	+	0.913023i	$0.366258\pi$
$84$	0	0
$85$	− 94.6382i	− 1.11339i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	133.138	1.49594	0.747969	−	0.663734i	$-0.231029\pi$
$89$	133.138	1.49594	0.747969	+	0.663734i	$0.231029\pi$
$90$	0	0
$91$	24.9948	0.274669
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 119.157i	− 1.25429i
$96$	0	0
$97$	97.1384	1.00143	0.500714	−	0.865613i	$-0.333071\pi$
$97$	97.1384	1.00143	0.500714	+	0.865613i	$0.333071\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	62.1370i	0.615218i	0.951513	+	0.307609i	$0.0995288\pi$
$101$	62.1370i	0.615218i	−0.951513	+	0.307609i	$0.900471\pi$
$102$	0	0
$103$	27.8041i	0.269943i	0.990849	+	0.134971i	$0.0430943\pi$
$103$	27.8041i	0.269943i	−0.990849	+	0.134971i	$0.956906\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	37.7795	0.353079	0.176540	−	0.984294i	$-0.443510\pi$
$107$	37.7795	0.353079	0.176540	+	0.984294i	$0.443510\pi$
$108$	0	0
$109$	141.691i	1.29992i	0.759968	+	0.649960i	$0.225214\pi$
$109$	141.691i	1.29992i	−0.759968	+	0.649960i	$0.774786\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	58.2872	0.515816	0.257908	−	0.966170i	$-0.416967\pi$
$113$	58.2872	0.515816	0.257908	+	0.966170i	$0.416967\pi$
$114$	0	0
$115$	34.1436	0.296901
$116$	0	0
$117$	0	0
$118$	0	0
$119$	25.3582i	0.213094i
$120$	0	0
$121$	−57.0000	−0.471074
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 109.456i	− 0.875649i
$126$	0	0
$127$	− 185.152i	− 1.45789i	−0.684571	−	0.728946i	$-0.740010\pi$
$127$	− 185.152i	− 1.45789i	0.684571	−	0.728946i	$-0.259990\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	125.359	0.956939	0.478469	−	0.878104i	$-0.341192\pi$
$131$	125.359	0.956939	0.478469	+	0.878104i	$0.341192\pi$
$132$	0	0
$133$	31.9281i	0.240061i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−99.5692	−0.726783	−0.363391	−	0.931637i	$-0.618381\pi$
$137$	−99.5692	−0.726783	−0.363391	+	0.931637i	$0.618381\pi$
$138$	0	0
$139$	−177.492	−1.27692	−0.638461	−	0.769654i	$-0.720429\pi$
$139$	−177.492	−1.27692	−0.638461	+	0.769654i	$0.720429\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 93.4920i	− 0.653790i
$144$	0	0
$145$	4.57437	0.0315474
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 87.8023i	− 0.589277i	−0.955609	−	0.294639i	$-0.904801\pi$
$149$	− 87.8023i	− 0.589277i	0.955609	−	0.294639i	$-0.0951993\pi$
$150$	0	0
$151$	219.066i	1.45077i	0.688345	+	0.725383i	$0.258337\pi$
$151$	219.066i	1.45077i	−0.688345	+	0.725383i	$0.741663\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−458.487	−2.95798
$156$	0	0
$157$	− 253.440i	− 1.61427i	−0.590370	−	0.807133i	$-0.701018\pi$
$157$	− 253.440i	− 1.61427i	0.590370	−	0.807133i	$-0.298982\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.14875	−0.0568245
$162$	0	0
$163$	102.354	0.627938	0.313969	−	0.949433i	$-0.398341\pi$
$163$	102.354	0.627938	0.313969	+	0.949433i	$0.398341\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	281.090i	1.68318i	0.540120	+	0.841588i	$0.318379\pi$
$167$	281.090i	1.68318i	−0.540120	+	0.841588i	$0.681621\pi$
$168$	0	0
$169$	32.4256	0.191868
$170$	0	0
$171$	0	0
$172$	0	0
$173$	242.858i	1.40381i	0.712273	+	0.701903i	$0.247666\pi$
$173$	242.858i	1.40381i	−0.712273	+	0.701903i	$0.752334\pi$
$174$	0	0
$175$	82.7981i	0.473132i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	318.354	1.77851	0.889257	−	0.457409i	$-0.151222\pi$
$179$	318.354	1.77851	0.889257	+	0.457409i	$0.151222\pi$
$180$	0	0
$181$	79.5132i	0.439299i	0.975579	+	0.219650i	$0.0704914\pi$
$181$	79.5132i	0.439299i	−0.975579	+	0.219650i	$0.929509\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−220.708	−1.19301
$186$	0	0
$187$	94.8513	0.507226
$188$	0	0
$189$	0	0
$190$	0	0
$191$	352.887i	1.84758i	0.382902	+	0.923789i	$0.374925\pi$
$191$	352.887i	1.84758i	−0.382902	+	0.923789i	$0.625075\pi$
$192$	0	0
$193$	−284.277	−1.47294	−0.736469	−	0.676472i	$-0.763508\pi$
$193$	−284.277	−1.47294	−0.736469	+	0.676472i	$0.763508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 75.8087i	− 0.384816i	−0.981315	−	0.192408i	$-0.938370\pi$
$197$	− 75.8087i	− 0.384816i	0.981315	−	0.192408i	$-0.0616297\pi$
$198$	0	0
$199$	104.186i	0.523547i	0.965129	+	0.261774i	$0.0843074\pi$
$199$	104.186i	0.523547i	−0.965129	+	0.261774i	$0.915693\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.22570	−0.00603793
$204$	0	0
$205$	251.986i	1.22920i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	119.426	0.571414
$210$	0	0
$211$	−136.918	−0.648900	−0.324450	−	0.945903i	$-0.605179\pi$
$211$	−136.918	−0.648900	−0.324450	+	0.945903i	$0.605179\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 229.760i	− 1.06865i
$216$	0	0
$217$	122.851	0.566135
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 138.560i	− 0.626968i
$222$	0	0
$223$	53.1624i	0.238396i	0.992870	+	0.119198i	$0.0380324\pi$
$223$	53.1624i	0.238396i	−0.992870	+	0.119198i	$0.961968\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−119.846	−0.527956	−0.263978	−	0.964529i	$-0.585035\pi$
$227$	−119.846	−0.527956	−0.263978	+	0.964529i	$0.585035\pi$
$228$	0	0
$229$	214.103i	0.934946i	0.884007	+	0.467473i	$0.154836\pi$
$229$	214.103i	0.934946i	−0.884007	+	0.467473i	$0.845164\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	127.436	0.546935	0.273468	−	0.961881i	$-0.411829\pi$
$233$	127.436	0.546935	0.273468	+	0.961881i	$0.411829\pi$
$234$	0	0
$235$	−475.559	−2.02365
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 319.281i	− 1.33590i	−0.744204	−	0.667952i	$-0.767171\pi$
$239$	− 319.281i	− 1.33590i	0.744204	−	0.667952i	$-0.232829\pi$
$240$	0	0
$241$	−247.415	−1.02662	−0.513310	−	0.858203i	$-0.671581\pi$
$241$	−247.415	−1.02662	−0.513310	+	0.858203i	$0.671581\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	354.607i	1.44737i
$246$	0	0
$247$	− 174.459i	− 0.706310i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−214.851	−0.855981	−0.427991	−	0.903783i	$-0.640778\pi$
$251$	−214.851	−0.855981	−0.427991	+	0.903783i	$0.640778\pi$
$252$	0	0
$253$	34.2205i	0.135259i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	84.2769	0.327926	0.163963	−	0.986467i	$-0.447572\pi$
$257$	84.2769	0.327926	0.163963	+	0.986467i	$0.447572\pi$
$258$	0	0
$259$	59.1384	0.228334
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 277.120i	− 1.05369i	−0.849962	−	0.526844i	$-0.823375\pi$
$263$	− 277.120i	− 1.05369i	0.849962	−	0.526844i	$-0.176625\pi$
$264$	0	0
$265$	250.277	0.944441
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 123.701i	− 0.459855i	−0.973208	−	0.229927i	$-0.926151\pi$
$269$	− 123.701i	− 0.459855i	0.973208	−	0.229927i	$-0.0738489\pi$
$270$	0	0
$271$	197.985i	0.730572i	0.930895	+	0.365286i	$0.119029\pi$
$271$	197.985i	0.730572i	−0.930895	+	0.365286i	$0.880971\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	309.703	1.12619
$276$	0	0
$277$	− 247.709i	− 0.894256i	−0.894470	−	0.447128i	$-0.852447\pi$
$277$	− 247.709i	− 0.894256i	0.894470	−	0.447128i	$-0.147553\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−443.128	−1.57697	−0.788484	−	0.615055i	$-0.789134\pi$
$281$	−443.128	−1.57697	−0.788484	+	0.615055i	$0.789134\pi$
$282$	0	0
$283$	294.620	1.04106	0.520531	−	0.853843i	$-0.325734\pi$
$283$	294.620	1.04106	0.520531	+	0.853843i	$0.325734\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 67.5196i	− 0.235260i
$288$	0	0
$289$	−148.426	−0.513583
$290$	0	0
$291$	0	0
$292$	0	0
$293$	66.7217i	0.227719i	0.993497	+	0.113859i	$0.0363214\pi$
$293$	66.7217i	0.227719i	−0.993497	+	0.113859i	$0.963679\pi$
$294$	0	0
$295$	− 421.328i	− 1.42823i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	49.9897	0.167190
$300$	0	0
$301$	61.5639i	0.204531i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	475.559	1.55921
$306$	0	0
$307$	524.210	1.70753	0.853763	−	0.520662i	$-0.174315\pi$
$307$	524.210	1.70753	0.853763	+	0.520662i	$0.174315\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	362.057i	1.16417i	0.813128	+	0.582085i	$0.197763\pi$
$311$	362.057i	1.16417i	−0.813128	+	0.582085i	$0.802237\pi$
$312$	0	0
$313$	252.277	0.805996	0.402998	−	0.915201i	$-0.367968\pi$
$313$	252.277	0.805996	0.402998	+	0.915201i	$0.367968\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	80.3934i	0.253607i	0.991928	+	0.126803i	$0.0404717\pi$
$317$	80.3934i	0.253607i	−0.991928	+	0.126803i	$0.959528\pi$
$318$	0	0
$319$	4.58467i	0.0143720i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	176.995	0.547972
$324$	0	0
$325$	− 452.417i	− 1.39205i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	127.426	0.387312
$330$	0	0
$331$	−172.056	−0.519808	−0.259904	−	0.965635i	$-0.583691\pi$
$331$	−172.056	−0.519808	−0.259904	+	0.965635i	$0.583691\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	676.753i	2.02016i
$336$	0	0
$337$	564.277	1.67441	0.837206	−	0.546888i	$-0.184187\pi$
$337$	564.277	1.67441	0.837206	+	0.546888i	$0.184187\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 459.519i	− 1.34756i
$342$	0	0
$343$	− 199.817i	− 0.582556i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−286.123	−0.824562	−0.412281	−	0.911057i	$-0.635268\pi$
$347$	−286.123	−0.824562	−0.412281	+	0.911057i	$0.635268\pi$
$348$	0	0
$349$	− 421.021i	− 1.20636i	−0.797603	−	0.603182i	$-0.793899\pi$
$349$	− 421.021i	− 1.20636i	0.797603	−	0.603182i	$-0.206101\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−429.138	−1.21569	−0.607845	−	0.794056i	$-0.707966\pi$
$353$	−429.138	−1.21569	−0.607845	+	0.794056i	$0.707966\pi$
$354$	0	0
$355$	338.985	0.954886
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 263.673i	− 0.734465i	−0.930129	−	0.367233i	$-0.880305\pi$
$359$	− 263.673i	− 0.734465i	0.930129	−	0.367233i	$-0.119695\pi$
$360$	0	0
$361$	−138.149	−0.382684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 43.3075i	− 0.118651i
$366$	0	0
$367$	− 129.544i	− 0.352981i	−0.984302	−	0.176491i	$-0.943525\pi$
$367$	− 129.544i	− 0.352981i	0.984302	−	0.176491i	$-0.0564745\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−67.0615	−0.180759
$372$	0	0
$373$	302.478i	0.810933i	0.914110	+	0.405467i	$0.132891\pi$
$373$	302.478i	0.810933i	−0.914110	+	0.405467i	$0.867109\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.69735	0.0177649
$378$	0	0
$379$	116.210	0.306623	0.153312	−	0.988178i	$-0.451006\pi$
$379$	116.210	0.306623	0.153312	+	0.988178i	$0.451006\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 566.151i	− 1.47820i	−0.673595	−	0.739101i	$-0.735251\pi$
$383$	− 566.151i	− 1.47820i	0.673595	−	0.739101i	$-0.264749\pi$
$384$	0	0
$385$	−136.574	−0.354739
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 350.104i	− 0.900011i	−0.893026	−	0.450006i	$-0.851422\pi$
$389$	− 350.104i	− 0.900011i	0.893026	−	0.450006i	$-0.148578\pi$
$390$	0	0
$391$	50.7165i	0.129710i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	356.056	0.901408
$396$	0	0
$397$	− 544.149i	− 1.37065i	−0.728236	−	0.685326i	$-0.759660\pi$
$397$	− 544.149i	− 1.37065i	0.728236	−	0.685326i	$-0.240340\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	296.431	0.739229	0.369614	−	0.929185i	$-0.379490\pi$
$401$	296.431	0.739229	0.369614	+	0.929185i	$0.379490\pi$
$402$	0	0
$403$	−671.272	−1.66569
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 221.205i	− 0.543500i
$408$	0	0
$409$	−247.415	−0.604927	−0.302464	−	0.953161i	$-0.597809\pi$
$409$	−247.415	−0.604927	−0.302464	+	0.953161i	$0.597809\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	112.895i	0.273353i
$414$	0	0
$415$	540.486i	1.30238i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−92.1333	−0.219889	−0.109944	−	0.993938i	$-0.535067\pi$
$419$	−92.1333	−0.219889	−0.109944	+	0.993938i	$0.535067\pi$
$420$	0	0
$421$	445.540i	1.05829i	0.848531	+	0.529145i	$0.177487\pi$
$421$	445.540i	1.05829i	−0.848531	+	0.529145i	$0.822513\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	458.995	1.07999
$426$	0	0
$427$	−127.426	−0.298421
$428$	0	0
$429$	0	0
$430$	0	0
$431$	186.677i	0.433125i	0.976269	+	0.216563i	$0.0694845\pi$
$431$	186.677i	0.433125i	−0.976269	+	0.216563i	$0.930515\pi$
$432$	0	0
$433$	291.128	0.672351	0.336176	−	0.941799i	$-0.390866\pi$
$433$	291.128	0.672351	0.336176	+	0.941799i	$0.390866\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	63.8562i	0.146124i
$438$	0	0
$439$	87.6899i	0.199749i	0.995000	+	0.0998746i	$0.0318442\pi$
$439$	87.6899i	0.199749i	−0.995000	+	0.0998746i	$0.968156\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.7077	0.0467441	0.0233721	−	0.999727i	$-0.492560\pi$
$443$	20.7077	0.0467441	0.0233721	+	0.999727i	$0.492560\pi$
$444$	0	0
$445$	1062.72i	2.38812i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−584.410	−1.30158	−0.650791	−	0.759257i	$-0.725563\pi$
$449$	−584.410	−1.30158	−0.650791	+	0.759257i	$0.725563\pi$
$450$	0	0
$451$	−252.554	−0.559986
$452$	0	0
$453$	0	0
$454$	0	0
$455$	199.510i	0.438483i
$456$	0	0
$457$	−269.692	−0.590136	−0.295068	−	0.955476i	$-0.595342\pi$
$457$	−269.692	−0.590136	−0.295068	+	0.955476i	$0.595342\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 44.4948i	− 0.0965181i	−0.998835	−	0.0482590i	$-0.984633\pi$
$461$	− 44.4948i	− 0.0965181i	0.998835	−	0.0482590i	$-0.0153673\pi$
$462$	0	0
$463$	− 611.065i	− 1.31980i	−0.751355	−	0.659898i	$-0.770600\pi$
$463$	− 611.065i	− 1.31980i	0.751355	−	0.659898i	$-0.229400\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	146.410	0.313512	0.156756	−	0.987637i	$-0.449896\pi$
$467$	146.410	0.313512	0.156756	+	0.987637i	$0.449896\pi$
$468$	0	0
$469$	− 181.336i	− 0.386643i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	230.277	0.486843
$474$	0	0
$475$	577.913	1.21666
$476$	0	0
$477$	0	0
$478$	0	0
$479$	191.876i	0.400576i	0.979737	+	0.200288i	$0.0641878\pi$
$479$	191.876i	0.400576i	−0.979737	+	0.200288i	$0.935812\pi$
$480$	0	0
$481$	−323.138	−0.671805
$482$	0	0
$483$	0	0
$484$	0	0
$485$	775.362i	1.59868i
$486$	0	0
$487$	610.758i	1.25412i	0.778969	+	0.627062i	$0.215743\pi$
$487$	610.758i	1.25412i	−0.778969	+	0.627062i	$0.784257\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−142.354	−0.289926	−0.144963	−	0.989437i	$-0.546306\pi$
$491$	−142.354	−0.289926	−0.144963	+	0.989437i	$0.546306\pi$
$492$	0	0
$493$	6.79472i	0.0137824i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−90.8306	−0.182758
$498$	0	0
$499$	−91.3693	−0.183105	−0.0915524	−	0.995800i	$-0.529183\pi$
$499$	−91.3693	−0.183105	−0.0915524	+	0.995800i	$0.529183\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	230.067i	0.457389i	0.973498	+	0.228695i	$0.0734457\pi$
$503$	230.067i	0.457389i	−0.973498	+	0.228695i	$0.926554\pi$
$504$	0	0
$505$	−495.979	−0.982137
$506$	0	0
$507$	0	0
$508$	0	0
$509$	527.387i	1.03612i	0.855343	+	0.518062i	$0.173346\pi$
$509$	527.387i	1.03612i	−0.855343	+	0.518062i	$0.826654\pi$
$510$	0	0
$511$	11.6042i	0.0227088i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−221.933	−0.430939
$516$	0	0
$517$	− 476.630i	− 0.921914i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−191.856	−0.368246	−0.184123	−	0.982903i	$-0.558945\pi$
$521$	−191.856	−0.368246	−0.184123	+	0.982903i	$0.558945\pi$
$522$	0	0
$523$	−105.492	−0.201706	−0.100853	−	0.994901i	$-0.532157\pi$
$523$	−105.492	−0.201706	−0.100853	+	0.994901i	$0.532157\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 681.031i	− 1.29228i
$528$	0	0
$529$	510.703	0.965411
$530$	0	0
$531$	0	0
$532$	0	0
$533$	368.934i	0.692184i
$534$	0	0
$535$	301.557i	0.563658i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−355.405	−0.659378
$540$	0	0
$541$	459.744i	0.849804i	0.905239	+	0.424902i	$0.139691\pi$
$541$	459.744i	0.849804i	−0.905239	+	0.424902i	$0.860309\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1130.98	−2.07520
$546$	0	0
$547$	−67.3693	−0.123161	−0.0615807	−	0.998102i	$-0.519614\pi$
$547$	−67.3693	−0.123161	−0.0615807	+	0.998102i	$0.519614\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.55511i	0.0155265i
$552$	0	0
$553$	−95.4050	−0.172523
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 476.671i	− 0.855782i	−0.903830	−	0.427891i	$-0.859257\pi$
$557$	− 476.671i	− 0.855782i	0.903830	−	0.427891i	$-0.140743\pi$
$558$	0	0
$559$	− 336.391i	− 0.601774i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−910.123	−1.61656	−0.808280	−	0.588799i	$-0.799601\pi$
$563$	−910.123	−1.61656	−0.808280	+	0.588799i	$0.799601\pi$
$564$	0	0
$565$	465.250i	0.823452i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	124.123	0.218142	0.109071	−	0.994034i	$-0.465212\pi$
$569$	124.123	0.218142	0.109071	+	0.994034i	$0.465212\pi$
$570$	0	0
$571$	−945.031	−1.65504	−0.827522	−	0.561433i	$-0.810250\pi$
$571$	−945.031	−1.65504	−0.827522	+	0.561433i	$0.810250\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	165.596i	0.287994i
$576$	0	0
$577$	215.682	0.373799	0.186899	−	0.982379i	$-0.440156\pi$
$577$	215.682	0.373799	0.186899	+	0.982379i	$0.440156\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 144.823i	− 0.249265i
$582$	0	0
$583$	250.840i	0.430258i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	900.785	1.53456	0.767278	−	0.641314i	$-0.221611\pi$
$587$	900.785	1.53456	0.767278	+	0.641314i	$0.221611\pi$
$588$	0	0
$589$	− 857.475i	− 1.45581i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	508.585	0.857647	0.428824	−	0.903388i	$-0.358928\pi$
$593$	508.585	0.857647	0.428824	+	0.903388i	$0.358928\pi$
$594$	0	0
$595$	−202.410	−0.340185
$596$	0	0
$597$	0	0
$598$	0	0
$599$	846.934i	1.41391i	0.707257	+	0.706957i	$0.249933\pi$
$599$	846.934i	1.41391i	−0.707257	+	0.706957i	$0.750067\pi$
$600$	0	0
$601$	−406.000	−0.675541	−0.337770	−	0.941229i	$-0.609673\pi$
$601$	−406.000	−0.675541	−0.337770	+	0.941229i	$0.609673\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 454.976i	− 0.752026i
$606$	0	0
$607$	− 771.156i	− 1.27044i	−0.772332	−	0.635219i	$-0.780910\pi$
$607$	− 771.156i	− 1.27044i	0.772332	−	0.635219i	$-0.219090\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−696.267	−1.13955
$612$	0	0
$613$	336.699i	0.549264i	0.961549	+	0.274632i	$0.0885560\pi$
$613$	336.699i	0.549264i	−0.961549	+	0.274632i	$0.911444\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	908.831	1.47298	0.736492	−	0.676447i	$-0.236481\pi$
$617$	908.831	1.47298	0.736492	+	0.676447i	$0.236481\pi$
$618$	0	0
$619$	1047.77	1.69268	0.846340	−	0.532643i	$-0.178801\pi$
$619$	1047.77	1.69268	0.846340	+	0.532643i	$0.178801\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 284.754i	− 0.457068i
$624$	0	0
$625$	−94.1384	−0.150622
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 327.836i	− 0.521202i
$630$	0	0
$631$	610.758i	0.967921i	0.875090	+	0.483961i	$0.160802\pi$
$631$	610.758i	0.967921i	−0.875090	+	0.483961i	$0.839198\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1477.89	2.32739
$636$	0	0
$637$	519.180i	0.815040i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−9.60015	−0.0149768	−0.00748842	−	0.999972i	$-0.502384\pi$
$641$	−9.60015	−0.0149768	−0.00748842	+	0.999972i	$0.502384\pi$
$642$	0	0
$643$	−86.1999	−0.134059	−0.0670295	−	0.997751i	$-0.521352\pi$
$643$	−86.1999	−0.134059	−0.0670295	+	0.997751i	$0.521352\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	352.580i	0.544946i	0.962163	+	0.272473i	$0.0878416\pi$
$647$	352.580i	0.544946i	−0.962163	+	0.272473i	$0.912158\pi$
$648$	0	0
$649$	422.277	0.650658
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 319.322i	− 0.489008i	−0.969648	−	0.244504i	$-0.921375\pi$
$653$	− 319.322i	− 0.489008i	0.969648	−	0.244504i	$-0.0786252\pi$
$654$	0	0
$655$	1000.62i	1.52766i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	275.328	0.417797	0.208898	−	0.977937i	$-0.433012\pi$
$659$	275.328	0.417797	0.208898	+	0.977937i	$0.433012\pi$
$660$	0	0
$661$	133.668i	0.202221i	0.994875	+	0.101111i	$0.0322396\pi$
$661$	133.668i	0.202221i	−0.994875	+	0.101111i	$0.967760\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−254.851	−0.383235
$666$	0	0
$667$	−2.45140	−0.00367526
$668$	0	0
$669$	0	0
$670$	0	0
$671$	476.630i	0.710327i
$672$	0	0
$673$	187.703	0.278904	0.139452	−	0.990229i	$-0.455466\pi$
$673$	187.703	0.278904	0.139452	+	0.990229i	$0.455466\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1169.84i	− 1.72797i	−0.503515	−	0.863986i	$-0.667960\pi$
$677$	− 1169.84i	− 1.72797i	0.503515	−	0.863986i	$-0.332040\pi$
$678$	0	0
$679$	− 207.758i	− 0.305976i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	89.4566	0.130976	0.0654880	−	0.997853i	$-0.479140\pi$
$683$	89.4566	0.130976	0.0654880	+	0.997853i	$0.479140\pi$
$684$	0	0
$685$	− 794.765i	− 1.16024i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	366.431	0.531830
$690$	0	0
$691$	−139.103	−0.201306	−0.100653	−	0.994922i	$-0.532093\pi$
$691$	−139.103	−0.201306	−0.100653	+	0.994922i	$0.532093\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1416.75i	− 2.03849i
$696$	0	0
$697$	−374.297	−0.537012
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1218.34i	1.73801i	0.494804	+	0.869004i	$0.335240\pi$
$701$	1218.34i	1.73801i	−0.494804	+	0.869004i	$0.664760\pi$
$702$	0	0
$703$	− 412.773i	− 0.587160i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	132.897	0.187974
$708$	0	0
$709$	− 1080.13i	− 1.52346i	−0.647895	−	0.761730i	$-0.724351\pi$
$709$	− 1080.13i	− 1.52346i	0.647895	−	0.761730i	$-0.275649\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	245.703	0.344604
$714$	0	0
$715$	746.256	1.04372
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 20.7736i	− 0.0288923i	−0.999896	−	0.0144461i	$-0.995401\pi$
$719$	− 20.7736i	− 0.0288923i	0.999896	−	0.0144461i	$-0.00459851\pi$
$720$	0	0
$721$	59.4669	0.0824783
$722$	0	0
$723$	0	0
$724$	0	0
$725$	22.1857i	0.0306010i
$726$	0	0
$727$	− 1031.17i	− 1.41838i	−0.705015	−	0.709192i	$-0.749060\pi$
$727$	− 1031.17i	− 1.41838i	0.705015	−	0.709192i	$-0.250940\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	341.282	0.466870
$732$	0	0
$733$	− 881.072i	− 1.20201i	−0.799246	−	0.601004i	$-0.794767\pi$
$733$	− 881.072i	− 1.20201i	0.799246	−	0.601004i	$-0.205233\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−678.277	−0.920321
$738$	0	0
$739$	671.195	0.908247	0.454124	−	0.890939i	$-0.349952\pi$
$739$	671.195	0.908247	0.454124	+	0.890939i	$0.349952\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 254.197i	− 0.342122i	−0.985260	−	0.171061i	$-0.945281\pi$
$743$	− 254.197i	− 0.342122i	0.985260	−	0.171061i	$-0.0547195\pi$
$744$	0	0
$745$	700.841	0.940726
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 80.8019i	− 0.107880i
$750$	0	0
$751$	728.994i	0.970698i	0.874320	+	0.485349i	$0.161307\pi$
$751$	728.994i	0.970698i	−0.874320	+	0.485349i	$0.838693\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1748.59	−2.31601
$756$	0	0
$757$	372.679i	0.492311i	0.969230	+	0.246155i	$0.0791674\pi$
$757$	372.679i	0.492311i	−0.969230	+	0.246155i	$0.920833\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1257.80	−1.65283	−0.826416	−	0.563060i	$-0.809624\pi$
$761$	−1257.80	−1.65283	−0.826416	+	0.563060i	$0.809624\pi$
$762$	0	0
$763$	303.046	0.397177
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 616.868i	− 0.804260i
$768$	0	0
$769$	247.703	0.322110	0.161055	−	0.986945i	$-0.448510\pi$
$769$	247.703	0.322110	0.161055	+	0.986945i	$0.448510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	587.805i	0.760420i	0.924900	+	0.380210i	$0.124148\pi$
$773$	587.805i	0.760420i	−0.924900	+	0.380210i	$0.875852\pi$
$774$	0	0
$775$	− 2223.66i	− 2.86924i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−471.272	−0.604970
$780$	0	0
$781$	339.748i	0.435016i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2022.96	2.57702
$786$	0	0
$787$	31.0821	0.0394944	0.0197472	−	0.999805i	$-0.493714\pi$
$787$	31.0821	0.0394944	0.0197472	+	0.999805i	$0.493714\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 124.663i	− 0.157602i
$792$	0	0
$793$	696.267	0.878016
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 490.260i	− 0.615132i	−0.951527	−	0.307566i	$-0.900486\pi$
$797$	− 490.260i	− 0.615132i	0.951527	−	0.307566i	$-0.0995145\pi$
$798$	0	0
$799$	− 706.389i	− 0.884092i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	43.4050	0.0540536
$804$	0	0
$805$	− 73.0256i	− 0.0907150i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−676.102	−0.835726	−0.417863	−	0.908510i	$-0.637221\pi$
$809$	−676.102	−0.835726	−0.417863	+	0.908510i	$0.637221\pi$
$810$	0	0
$811$	74.1793	0.0914665	0.0457332	−	0.998954i	$-0.485438\pi$
$811$	74.1793	0.0914665	0.0457332	+	0.998954i	$0.485438\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	816.991i	1.00244i
$816$	0	0
$817$	429.703	0.525952
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1130.58i	1.37708i	0.725198	+	0.688540i	$0.241748\pi$
$821$	1130.58i	1.37708i	−0.725198	+	0.688540i	$0.758252\pi$
$822$	0	0
$823$	82.1839i	0.0998589i	0.998753	+	0.0499295i	$0.0158997\pi$
$823$	82.1839i	0.0998589i	−0.998753	+	0.0499295i	$0.984100\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1504.57	1.81931	0.909655	−	0.415365i	$-0.136346\pi$
$827$	1504.57	1.81931	0.909655	+	0.415365i	$0.136346\pi$
$828$	0	0
$829$	1409.65i	1.70042i	0.526445	+	0.850209i	$0.323525\pi$
$829$	1409.65i	1.70042i	−0.526445	+	0.850209i	$0.676475\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−526.728	−0.632327
$834$	0	0
$835$	−2243.67	−2.68703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 288.110i	− 0.343397i	−0.985150	−	0.171698i	$-0.945075\pi$
$839$	− 288.110i	− 0.343397i	0.985150	−	0.171698i	$-0.0549254\pi$
$840$	0	0
$841$	840.672	0.999609
$842$	0	0
$843$	0	0
$844$	0	0
$845$	258.822i	0.306299i
$846$	0	0
$847$	121.910i	0.143932i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	118.277	0.138986
$852$	0	0
$853$	− 645.132i	− 0.756310i	−0.925742	−	0.378155i	$-0.876559\pi$
$853$	− 645.132i	− 0.756310i	0.925742	−	0.378155i	$-0.123441\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	995.549	1.16167	0.580833	−	0.814022i	$-0.302727\pi$
$857$	995.549	1.16167	0.580833	+	0.814022i	$0.302727\pi$
$858$	0	0
$859$	774.354	0.901460	0.450730	−	0.892660i	$-0.351164\pi$
$859$	774.354	0.901460	0.450730	+	0.892660i	$0.351164\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1007.33i	1.16724i	0.812025	+	0.583622i	$0.198365\pi$
$863$	1007.33i	1.16724i	−0.812025	+	0.583622i	$0.801635\pi$
$864$	0	0
$865$	−1938.50	−2.24104
$866$	0	0
$867$	0	0
$868$	0	0
$869$	356.858i	0.410654i
$870$	0	0
$871$	990.836i	1.13758i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−234.102	−0.267546
$876$	0	0
$877$	681.645i	0.777246i	0.921397	+	0.388623i	$0.127049\pi$
$877$	681.645i	0.777246i	−0.921397	+	0.388623i	$0.872951\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	679.108	0.770837	0.385419	−	0.922742i	$-0.374057\pi$
$881$	679.108	0.770837	0.385419	+	0.922742i	$0.374057\pi$
$882$	0	0
$883$	1059.44	1.19982	0.599910	−	0.800068i	$-0.295203\pi$
$883$	1059.44	1.19982	0.599910	+	0.800068i	$0.295203\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 856.411i	− 0.965514i	−0.875754	−	0.482757i	$-0.839635\pi$
$887$	− 856.411i	− 0.965514i	0.875754	−	0.482757i	$-0.160365\pi$
$888$	0	0
$889$	−396.000	−0.445444
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 889.403i	− 0.995972i
$894$	0	0
$895$	2541.11i	2.83923i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.9179	0.0366161
$900$	0	0
$901$	371.758i	0.412606i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−634.677	−0.701300
$906$	0	0
$907$	761.492	0.839573	0.419786	−	0.907623i	$-0.362105\pi$
$907$	761.492	0.839573	0.419786	+	0.907623i	$0.362105\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 897.344i	− 0.985009i	−0.870310	−	0.492505i	$-0.836081\pi$
$911$	− 897.344i	− 0.985009i	0.870310	−	0.492505i	$-0.163919\pi$
$912$	0	0
$913$	−541.703	−0.593321
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 268.115i	− 0.292383i
$918$	0	0
$919$	− 62.6388i	− 0.0681598i	−0.999419	−	0.0340799i	$-0.989150\pi$
$919$	− 62.6388i	− 0.0681598i	0.999419	−	0.0340799i	$-0.0108501\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	496.308	0.537712
$924$	0	0
$925$	− 1070.43i	− 1.15722i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−253.313	−0.272673	−0.136336	−	0.990663i	$-0.543533\pi$
$929$	−253.313	−0.272673	−0.136336	+	0.990663i	$0.543533\pi$
$930$	0	0
$931$	−663.195	−0.712347
$932$	0	0
$933$	0	0
$934$	0	0
$935$	757.106i	0.809739i
$936$	0	0
$937$	−30.5538	−0.0326081	−0.0163040	−	0.999867i	$-0.505190\pi$
$937$	−30.5538	−0.0326081	−0.0163040	+	0.999867i	$0.505190\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	388.745i	0.413119i	0.978434	+	0.206559i	$0.0662267\pi$
$941$	388.745i	0.413119i	−0.978434	+	0.206559i	$0.933773\pi$
$942$	0	0
$943$	− 135.039i	− 0.143202i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−263.615	−0.278369	−0.139184	−	0.990266i	$-0.544448\pi$
$947$	−263.615	−0.278369	−0.139184	+	0.990266i	$0.544448\pi$
$948$	0	0
$949$	− 63.4066i	− 0.0668141i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1292.41	−1.35615	−0.678075	−	0.734993i	$-0.737185\pi$
$953$	−1292.41	−1.35615	−0.678075	+	0.734993i	$0.737185\pi$
$954$	0	0
$955$	−2816.76	−2.94949
$956$	0	0
$957$	0	0
$958$	0	0
$959$	212.957i	0.222061i
$960$	0	0
$961$	−2338.34	−2.43324
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2269.11i	− 2.35141i
$966$	0	0
$967$	1110.00i	1.14788i	0.818896	+	0.573942i	$0.194587\pi$
$967$	1110.00i	1.14788i	−0.818896	+	0.573942i	$0.805413\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1341.66	−1.38173	−0.690866	−	0.722983i	$-0.742770\pi$
$971$	−1341.66	−1.38173	−0.690866	+	0.722983i	$0.742770\pi$
$972$	0	0
$973$	379.617i	0.390151i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	920.431	0.942099	0.471050	−	0.882107i	$-0.343875\pi$
$977$	920.431	0.942099	0.471050	+	0.882107i	$0.343875\pi$
$978$	0	0
$979$	−1065.11	−1.08795
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1697.84i	1.72720i	0.504176	+	0.863601i	$0.331796\pi$
$983$	1697.84i	1.72720i	−0.504176	+	0.863601i	$0.668204\pi$
$984$	0	0
$985$	605.108	0.614322
$986$	0	0
$987$	0	0
$988$	0	0
$989$	123.128i	0.124497i
$990$	0	0
$991$	− 284.765i	− 0.287351i	−0.989625	−	0.143675i	$-0.954108\pi$
$991$	− 284.765i	− 0.287351i	0.989625	−	0.143675i	$-0.0458921\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−831.615	−0.835794
$996$	0	0
$997$	1093.80i	1.09710i	0.836119	+	0.548548i	$0.184819\pi$
$997$	1093.80i	1.09710i	−0.836119	+	0.548548i	$0.815181\pi$
$998$	0	0
$999$	0	0

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	288.3.b.b.271.4		4
3.2	odd	2		96.3.b.a.79.3		4
4.3	odd	2		72.3.b.b.19.1		4
8.3	odd	2	inner	288.3.b.b.271.1		4
8.5	even	2		72.3.b.b.19.2		4
12.11	even	2		24.3.b.a.19.4	yes	4
15.2	even	4		2400.3.p.a.1999.3		8
15.8	even	4		2400.3.p.a.1999.6		8
15.14	odd	2		2400.3.g.a.751.2		4
16.3	odd	4		2304.3.g.z.1279.7		8
16.5	even	4		2304.3.g.z.1279.2		8
16.11	odd	4		2304.3.g.z.1279.1		8
16.13	even	4		2304.3.g.z.1279.8		8
24.5	odd	2		24.3.b.a.19.3	✓	4
24.11	even	2		96.3.b.a.79.4		4
48.5	odd	4		768.3.g.h.511.4		8
48.11	even	4		768.3.g.h.511.8		8
48.29	odd	4		768.3.g.h.511.5		8
48.35	even	4		768.3.g.h.511.1		8
60.23	odd	4		600.3.p.a.499.5		8
60.47	odd	4		600.3.p.a.499.4		8
60.59	even	2		600.3.g.a.451.1		4
120.29	odd	2		600.3.g.a.451.2		4
120.53	even	4		600.3.p.a.499.3		8
120.59	even	2		2400.3.g.a.751.1		4
120.77	even	4		600.3.p.a.499.6		8
120.83	odd	4		2400.3.p.a.1999.7		8
120.107	odd	4		2400.3.p.a.1999.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.3	✓	4	24.5	odd	2
24.3.b.a.19.4	yes	4	12.11	even	2
72.3.b.b.19.1		4	4.3	odd	2
72.3.b.b.19.2		4	8.5	even	2
96.3.b.a.79.3		4	3.2	odd	2
96.3.b.a.79.4		4	24.11	even	2
288.3.b.b.271.1		4	8.3	odd	2	inner
288.3.b.b.271.4		4	1.1	even	1	trivial
600.3.g.a.451.1		4	60.59	even	2
600.3.g.a.451.2		4	120.29	odd	2
600.3.p.a.499.3		8	120.53	even	4
600.3.p.a.499.4		8	60.47	odd	4
600.3.p.a.499.5		8	60.23	odd	4
600.3.p.a.499.6		8	120.77	even	4
768.3.g.h.511.1		8	48.35	even	4
768.3.g.h.511.4		8	48.5	odd	4
768.3.g.h.511.5		8	48.29	odd	4
768.3.g.h.511.8		8	48.11	even	4
2304.3.g.z.1279.1		8	16.11	odd	4
2304.3.g.z.1279.2		8	16.5	even	4
2304.3.g.z.1279.7		8	16.3	odd	4
2304.3.g.z.1279.8		8	16.13	even	4
2400.3.g.a.751.1		4	120.59	even	2
2400.3.g.a.751.2		4	15.14	odd	2
2400.3.p.a.1999.2		8	120.107	odd	4
2400.3.p.a.1999.3		8	15.2	even	4
2400.3.p.a.1999.6		8	15.8	even	4
2400.3.p.a.1999.7		8	120.83	odd	4

Overview	Random
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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	288.3.b.b.271.4

Level	$288$
Weight	$3$
Character	288.271
Analytic conductor	$7.847$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$