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Properties

Label 588.3.m.a.325.1
Level $588$
Weight $3$
Character 588.325
Analytic conductor $16.022$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,3,Mod(313,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.313"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 325.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.325
Dual form 588.3.m.a.313.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{3} +(-1.50000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{9} +(1.50000 - 2.59808i) q^{11} -6.92820i q^{13} +3.00000 q^{15} +(15.0000 + 8.66025i) q^{17} +(-9.00000 + 5.19615i) q^{19} +(-18.0000 - 31.1769i) q^{23} +(-11.0000 + 19.0526i) q^{25} -5.19615i q^{27} -51.0000 q^{29} +(10.5000 + 6.06218i) q^{31} +(-4.50000 + 2.59808i) q^{33} +(-11.0000 - 19.0526i) q^{37} +(-6.00000 + 10.3923i) q^{39} -24.2487i q^{41} +10.0000 q^{43} +(-4.50000 - 2.59808i) q^{45} +(-78.0000 + 45.0333i) q^{47} +(-15.0000 - 25.9808i) q^{51} +(-25.5000 + 44.1673i) q^{53} +5.19615i q^{55} +18.0000 q^{57} +(-64.5000 - 37.2391i) q^{59} +(-60.0000 + 34.6410i) q^{61} +(6.00000 + 10.3923i) q^{65} +(-34.0000 + 58.8897i) q^{67} +62.3538i q^{69} +(18.0000 + 10.3923i) q^{73} +(33.0000 - 19.0526i) q^{75} +(-62.5000 - 108.253i) q^{79} +(-4.50000 + 7.79423i) q^{81} -154.153i q^{83} -30.0000 q^{85} +(76.5000 + 44.1673i) q^{87} +(-63.0000 + 36.3731i) q^{89} +(-10.5000 - 18.1865i) q^{93} +(9.00000 - 15.5885i) q^{95} +147.224i q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 3 q^{5} + 3 q^{9} + 3 q^{11} + 6 q^{15} + 30 q^{17} - 18 q^{19} - 36 q^{23} - 22 q^{25} - 102 q^{29} + 21 q^{31} - 9 q^{33} - 22 q^{37} - 12 q^{39} + 20 q^{43} - 9 q^{45} - 156 q^{47} - 30 q^{51}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.50000 − 0.866025i −0.500000 − 0.288675i
\(4\) 0 0
\(5\) −1.50000 + 0.866025i −0.300000 + 0.173205i −0.642443 − 0.766334i \(-0.722079\pi\)
0.342443 + 0.939539i \(0.388746\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.50000 − 2.59808i 0.136364 − 0.236189i −0.789754 − 0.613424i \(-0.789792\pi\)
0.926118 + 0.377235i \(0.123125\pi\)
\(12\) 0 0
\(13\) − 6.92820i − 0.532939i −0.963843 − 0.266469i \(-0.914143\pi\)
0.963843 − 0.266469i \(-0.0858571\pi\)
\(14\) 0 0
\(15\) 3.00000 0.200000
\(16\) 0 0
\(17\) 15.0000 + 8.66025i 0.882353 + 0.509427i 0.871433 − 0.490514i \(-0.163191\pi\)
0.0109194 + 0.999940i \(0.496524\pi\)
\(18\) 0 0
\(19\) −9.00000 + 5.19615i −0.473684 + 0.273482i −0.717781 − 0.696269i \(-0.754842\pi\)
0.244096 + 0.969751i \(0.421509\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −18.0000 − 31.1769i −0.782609 − 1.35552i −0.930417 − 0.366502i \(-0.880555\pi\)
0.147809 − 0.989016i \(-0.452778\pi\)
\(24\) 0 0
\(25\) −11.0000 + 19.0526i −0.440000 + 0.762102i
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) −51.0000 −1.75862 −0.879310 − 0.476249i \(-0.841996\pi\)
−0.879310 + 0.476249i \(0.841996\pi\)
\(30\) 0 0
\(31\) 10.5000 + 6.06218i 0.338710 + 0.195554i 0.659701 − 0.751528i \(-0.270683\pi\)
−0.320992 + 0.947082i \(0.604016\pi\)
\(32\) 0 0
\(33\) −4.50000 + 2.59808i −0.136364 + 0.0787296i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11.0000 − 19.0526i −0.297297 − 0.514934i 0.678219 − 0.734859i \(-0.262752\pi\)
−0.975517 + 0.219925i \(0.929419\pi\)
\(38\) 0 0
\(39\) −6.00000 + 10.3923i −0.153846 + 0.266469i
\(40\) 0 0
\(41\) − 24.2487i − 0.591432i −0.955276 − 0.295716i \(-0.904442\pi\)
0.955276 − 0.295716i \(-0.0955582\pi\)
\(42\) 0 0
\(43\) 10.0000 0.232558 0.116279 − 0.993217i \(-0.462903\pi\)
0.116279 + 0.993217i \(0.462903\pi\)
\(44\) 0 0
\(45\) −4.50000 − 2.59808i −0.100000 − 0.0577350i
\(46\) 0 0
\(47\) −78.0000 + 45.0333i −1.65957 + 0.958156i −0.686664 + 0.726975i \(0.740925\pi\)
−0.972911 + 0.231180i \(0.925741\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −15.0000 − 25.9808i −0.294118 − 0.509427i
\(52\) 0 0
\(53\) −25.5000 + 44.1673i −0.481132 + 0.833345i −0.999766 − 0.0216515i \(-0.993108\pi\)
0.518634 + 0.854997i \(0.326441\pi\)
\(54\) 0 0
\(55\) 5.19615i 0.0944755i
\(56\) 0 0
\(57\) 18.0000 0.315789
\(58\) 0 0
\(59\) −64.5000 − 37.2391i −1.09322 − 0.631171i −0.158788 − 0.987313i \(-0.550759\pi\)
−0.934432 + 0.356142i \(0.884092\pi\)
\(60\) 0 0
\(61\) −60.0000 + 34.6410i −0.983607 + 0.567886i −0.903357 − 0.428889i \(-0.858905\pi\)
−0.0802495 + 0.996775i \(0.525572\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 + 10.3923i 0.0923077 + 0.159882i
\(66\) 0 0
\(67\) −34.0000 + 58.8897i −0.507463 + 0.878951i 0.492500 + 0.870312i \(0.336083\pi\)
−0.999963 + 0.00863871i \(0.997250\pi\)
\(68\) 0 0
\(69\) 62.3538i 0.903679i
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 18.0000 + 10.3923i 0.246575 + 0.142360i 0.618195 − 0.786025i \(-0.287864\pi\)
−0.371620 + 0.928385i \(0.621197\pi\)
\(74\) 0 0
\(75\) 33.0000 − 19.0526i 0.440000 − 0.254034i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −62.5000 − 108.253i −0.791139 − 1.37029i −0.925262 − 0.379329i \(-0.876155\pi\)
0.134123 − 0.990965i \(-0.457178\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) − 154.153i − 1.85726i −0.371008 − 0.928630i \(-0.620988\pi\)
0.371008 − 0.928630i \(-0.379012\pi\)
\(84\) 0 0
\(85\) −30.0000 −0.352941
\(86\) 0 0
\(87\) 76.5000 + 44.1673i 0.879310 + 0.507670i
\(88\) 0 0
\(89\) −63.0000 + 36.3731i −0.707865 + 0.408686i −0.810270 − 0.586057i \(-0.800680\pi\)
0.102405 + 0.994743i \(0.467346\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −10.5000 − 18.1865i −0.112903 − 0.195554i
\(94\) 0 0
\(95\) 9.00000 − 15.5885i 0.0947368 − 0.164089i
\(96\) 0 0
\(97\) 147.224i 1.51778i 0.651221 + 0.758888i \(0.274257\pi\)
−0.651221 + 0.758888i \(0.725743\pi\)
\(98\) 0 0
\(99\) 9.00000 0.0909091
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 588.3.m.a.325.1 2
3.2 odd 2 1764.3.z.e.325.1 2
7.2 even 3 84.3.m.a.61.1 ✓ 2
7.3 odd 6 588.3.d.a.97.1 2
7.4 even 3 588.3.d.a.97.2 2
7.5 odd 6 inner 588.3.m.a.313.1 2
7.6 odd 2 84.3.m.a.73.1 yes 2
21.2 odd 6 252.3.z.b.145.1 2
21.5 even 6 1764.3.z.e.901.1 2
21.11 odd 6 1764.3.d.c.685.2 2
21.17 even 6 1764.3.d.c.685.1 2
21.20 even 2 252.3.z.b.73.1 2
28.3 even 6 2352.3.f.c.97.2 2
28.11 odd 6 2352.3.f.c.97.1 2
28.23 odd 6 336.3.bh.b.145.1 2
28.27 even 2 336.3.bh.b.241.1 2
35.2 odd 12 2100.3.be.c.649.1 4
35.9 even 6 2100.3.bd.b.901.1 2
35.13 even 4 2100.3.be.c.1249.1 4
35.23 odd 12 2100.3.be.c.649.2 4
35.27 even 4 2100.3.be.c.1249.2 4
35.34 odd 2 2100.3.bd.b.1501.1 2
84.23 even 6 1008.3.cg.b.145.1 2
84.83 odd 2 1008.3.cg.b.577.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.3.m.a.61.1 ✓ 2 7.2 even 3
84.3.m.a.73.1 yes 2 7.6 odd 2
252.3.z.b.73.1 2 21.20 even 2
252.3.z.b.145.1 2 21.2 odd 6
336.3.bh.b.145.1 2 28.23 odd 6
336.3.bh.b.241.1 2 28.27 even 2
588.3.d.a.97.1 2 7.3 odd 6
588.3.d.a.97.2 2 7.4 even 3
588.3.m.a.313.1 2 7.5 odd 6 inner
588.3.m.a.325.1 2 1.1 even 1 trivial
1008.3.cg.b.145.1 2 84.23 even 6
1008.3.cg.b.577.1 2 84.83 odd 2
1764.3.d.c.685.1 2 21.17 even 6
1764.3.d.c.685.2 2 21.11 odd 6
1764.3.z.e.325.1 2 3.2 odd 2
1764.3.z.e.901.1 2 21.5 even 6
2100.3.bd.b.901.1 2 35.9 even 6
2100.3.bd.b.1501.1 2 35.34 odd 2
2100.3.be.c.649.1 4 35.2 odd 12
2100.3.be.c.649.2 4 35.23 odd 12
2100.3.be.c.1249.1 4 35.13 even 4
2100.3.be.c.1249.2 4 35.27 even 4
2352.3.f.c.97.1 2 28.11 odd 6
2352.3.f.c.97.2 2 28.3 even 6