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Properties

Label 441.2.p.b.215.4
Level $441$
Weight $2$
Character 441.215
Analytic conductor $3.521$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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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] = [441,2,Mod(80,441)] 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("441.80"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.p (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 215.4
Root \(2.23256 + 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 441.215
Dual form 441.2.p.b.80.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.23256 + 1.28897i) q^{2} +(2.32288 + 4.02334i) q^{4} +6.82058i q^{8} +(0.790881 - 0.456615i) q^{11} +(-4.14575 + 7.18065i) q^{16} +2.35425 q^{22} +(-8.13935 - 4.69926i) q^{23} +(2.50000 + 4.33013i) q^{25} -6.06910i q^{29} +(-6.69767 + 3.86690i) q^{32} +(5.29150 - 9.16515i) q^{37} +5.29150 q^{43} +(3.67423 + 2.12132i) q^{44} +(-12.1144 - 20.9827i) q^{46} +12.8897i q^{50} +(-12.6045 + 7.27719i) q^{53} +(7.82288 - 13.5496i) q^{58} -3.35425 q^{64} +(2.00000 + 3.46410i) q^{67} -7.57205i q^{71} +(23.6272 - 13.6412i) q^{74} +(-4.00000 + 6.92820i) q^{79} +(11.8136 + 6.82058i) q^{86} +(3.11438 + 5.39426i) q^{88} -43.6631i q^{92} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} - 12 q^{16} + 40 q^{22} + 20 q^{25} - 44 q^{46} + 52 q^{58} - 48 q^{64} + 16 q^{67} - 32 q^{79} - 28 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23256 + 1.28897i 1.57866 + 0.911438i 0.995047 + 0.0994033i \(0.0316934\pi\)
0.583609 + 0.812035i \(0.301640\pi\)
\(3\) 0 0
\(4\) 2.32288 + 4.02334i 1.16144 + 2.01167i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 6.82058i 2.41144i
\(9\) 0 0
\(10\) 0 0
\(11\) 0.790881 − 0.456615i 0.238459 − 0.137675i −0.376009 − 0.926616i \(-0.622704\pi\)
0.614468 + 0.788941i \(0.289370\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.14575 + 7.18065i −1.03644 + 1.79516i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.35425 0.501928
\(23\) −8.13935 − 4.69926i −1.69717 − 0.979863i −0.948422 − 0.317009i \(-0.897321\pi\)
−0.748749 − 0.662853i \(-0.769345\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 6.06910i − 1.12700i −0.826115 − 0.563502i \(-0.809454\pi\)
0.826115 − 0.563502i \(-0.190546\pi\)
\(30\) 0 0
\(31\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) −6.69767 + 3.86690i −1.18399 + 0.683578i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.29150 − 9.16515i 0.869918 − 1.50674i 0.00783774 − 0.999969i \(-0.497505\pi\)
0.862080 − 0.506772i \(-0.169162\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.29150 0.806947 0.403473 − 0.914991i \(-0.367803\pi\)
0.403473 + 0.914991i \(0.367803\pi\)
\(44\) 3.67423 + 2.12132i 0.553912 + 0.319801i
\(45\) 0 0
\(46\) −12.1144 − 20.9827i −1.78617 − 3.09373i
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 12.8897i 1.82288i
\(51\) 0 0
\(52\) 0 0
\(53\) −12.6045 + 7.27719i −1.73136 + 0.999599i −0.851522 + 0.524320i \(0.824320\pi\)
−0.879835 + 0.475280i \(0.842347\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 7.82288 − 13.5496i 1.02719 − 1.77915i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −3.35425 −0.419281
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 − 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 7.57205i − 0.898637i −0.893372 − 0.449319i \(-0.851667\pi\)
0.893372 − 0.449319i \(-0.148333\pi\)
\(72\) 0 0
\(73\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 23.6272 − 13.6412i 2.74660 − 1.58575i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −4.00000 + 6.92820i −0.450035 + 0.779484i −0.998388 − 0.0567635i \(-0.981922\pi\)
0.548352 + 0.836247i \(0.315255\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 11.8136 + 6.82058i 1.27389 + 0.735482i
\(87\) 0 0
\(88\) 3.11438 + 5.39426i 0.331994 + 0.575030i
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 43.6631i − 4.55220i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
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 441.2.p.b.215.4 8
3.2 odd 2 inner 441.2.p.b.215.1 8
7.2 even 3 63.2.c.a.62.1 ✓ 4
7.3 odd 6 inner 441.2.p.b.80.1 8
7.4 even 3 inner 441.2.p.b.80.1 8
7.5 odd 6 63.2.c.a.62.1 ✓ 4
7.6 odd 2 CM 441.2.p.b.215.4 8
21.2 odd 6 63.2.c.a.62.4 yes 4
21.5 even 6 63.2.c.a.62.4 yes 4
21.11 odd 6 inner 441.2.p.b.80.4 8
21.17 even 6 inner 441.2.p.b.80.4 8
21.20 even 2 inner 441.2.p.b.215.1 8
28.19 even 6 1008.2.k.a.881.1 4
28.23 odd 6 1008.2.k.a.881.1 4
35.2 odd 12 1575.2.g.d.1574.8 8
35.9 even 6 1575.2.b.a.251.4 4
35.12 even 12 1575.2.g.d.1574.8 8
35.19 odd 6 1575.2.b.a.251.4 4
35.23 odd 12 1575.2.g.d.1574.1 8
35.33 even 12 1575.2.g.d.1574.1 8
56.5 odd 6 4032.2.k.c.3905.3 4
56.19 even 6 4032.2.k.b.3905.2 4
56.37 even 6 4032.2.k.c.3905.3 4
56.51 odd 6 4032.2.k.b.3905.2 4
63.2 odd 6 567.2.o.f.377.4 8
63.5 even 6 567.2.o.f.188.1 8
63.16 even 3 567.2.o.f.377.1 8
63.23 odd 6 567.2.o.f.188.1 8
63.40 odd 6 567.2.o.f.188.4 8
63.47 even 6 567.2.o.f.377.4 8
63.58 even 3 567.2.o.f.188.4 8
63.61 odd 6 567.2.o.f.377.1 8
84.23 even 6 1008.2.k.a.881.2 4
84.47 odd 6 1008.2.k.a.881.2 4
105.2 even 12 1575.2.g.d.1574.2 8
105.23 even 12 1575.2.g.d.1574.7 8
105.44 odd 6 1575.2.b.a.251.1 4
105.47 odd 12 1575.2.g.d.1574.2 8
105.68 odd 12 1575.2.g.d.1574.7 8
105.89 even 6 1575.2.b.a.251.1 4
168.5 even 6 4032.2.k.c.3905.4 4
168.107 even 6 4032.2.k.b.3905.1 4
168.131 odd 6 4032.2.k.b.3905.1 4
168.149 odd 6 4032.2.k.c.3905.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 7.2 even 3
63.2.c.a.62.1 ✓ 4 7.5 odd 6
63.2.c.a.62.4 yes 4 21.2 odd 6
63.2.c.a.62.4 yes 4 21.5 even 6
441.2.p.b.80.1 8 7.3 odd 6 inner
441.2.p.b.80.1 8 7.4 even 3 inner
441.2.p.b.80.4 8 21.11 odd 6 inner
441.2.p.b.80.4 8 21.17 even 6 inner
441.2.p.b.215.1 8 3.2 odd 2 inner
441.2.p.b.215.1 8 21.20 even 2 inner
441.2.p.b.215.4 8 1.1 even 1 trivial
441.2.p.b.215.4 8 7.6 odd 2 CM
567.2.o.f.188.1 8 63.5 even 6
567.2.o.f.188.1 8 63.23 odd 6
567.2.o.f.188.4 8 63.40 odd 6
567.2.o.f.188.4 8 63.58 even 3
567.2.o.f.377.1 8 63.16 even 3
567.2.o.f.377.1 8 63.61 odd 6
567.2.o.f.377.4 8 63.2 odd 6
567.2.o.f.377.4 8 63.47 even 6
1008.2.k.a.881.1 4 28.19 even 6
1008.2.k.a.881.1 4 28.23 odd 6
1008.2.k.a.881.2 4 84.23 even 6
1008.2.k.a.881.2 4 84.47 odd 6
1575.2.b.a.251.1 4 105.44 odd 6
1575.2.b.a.251.1 4 105.89 even 6
1575.2.b.a.251.4 4 35.9 even 6
1575.2.b.a.251.4 4 35.19 odd 6
1575.2.g.d.1574.1 8 35.23 odd 12
1575.2.g.d.1574.1 8 35.33 even 12
1575.2.g.d.1574.2 8 105.2 even 12
1575.2.g.d.1574.2 8 105.47 odd 12
1575.2.g.d.1574.7 8 105.23 even 12
1575.2.g.d.1574.7 8 105.68 odd 12
1575.2.g.d.1574.8 8 35.2 odd 12
1575.2.g.d.1574.8 8 35.12 even 12
4032.2.k.b.3905.1 4 168.107 even 6
4032.2.k.b.3905.1 4 168.131 odd 6
4032.2.k.b.3905.2 4 56.19 even 6
4032.2.k.b.3905.2 4 56.51 odd 6
4032.2.k.c.3905.3 4 56.5 odd 6
4032.2.k.c.3905.3 4 56.37 even 6
4032.2.k.c.3905.4 4 168.5 even 6
4032.2.k.c.3905.4 4 168.149 odd 6