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Properties

Label 441.2.p.b.80.4
Level $441$
Weight $2$
Character 441.80
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 80.4
Root \(2.23256 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 441.80
Dual form 441.2.p.b.215.4

$q$-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 \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

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

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 0.583609 − 0.812035i \(-0.301640\pi\)
0.995047 − 0.0994033i \(-0.0316934\pi\)
\(3\) 0 0
\(4\) 2.32288 − 4.02334i 1.16144 − 2.01167i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 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 0.614468 − 0.788941i \(-0.289370\pi\)
−0.376009 + 0.926616i \(0.622704\pi\)
\(12\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.35425 0.501928
\(23\) −8.13935 + 4.69926i −1.69717 + 0.979863i −0.748749 + 0.662853i \(0.769345\pi\)
−0.948422 + 0.317009i \(0.897321\pi\)
\(24\) 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 \(0.190546\pi\)
−0.826115 + 0.563502i \(0.809454\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −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 0.862080 + 0.506772i \(0.169162\pi\)
0.00783774 + 0.999969i \(0.497505\pi\)
\(38\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 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 −0.879835 − 0.475280i \(-0.842347\pi\)
−0.851522 − 0.524320i \(-0.824320\pi\)
\(54\) 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 \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 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 −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.57205i 0.898637i 0.893372 + 0.449319i \(0.148333\pi\)
−0.893372 + 0.449319i \(0.851667\pi\)
\(72\) 0 0
\(73\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(74\) 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 0.548352 − 0.836247i \(-0.315255\pi\)
−0.998388 + 0.0567635i \(0.981922\pi\)
\(80\) 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 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.80.4 8
3.2 odd 2 inner 441.2.p.b.80.1 8
7.2 even 3 inner 441.2.p.b.215.1 8
7.3 odd 6 63.2.c.a.62.4 yes 4
7.4 even 3 63.2.c.a.62.4 yes 4
7.5 odd 6 inner 441.2.p.b.215.1 8
7.6 odd 2 CM 441.2.p.b.80.4 8
21.2 odd 6 inner 441.2.p.b.215.4 8
21.5 even 6 inner 441.2.p.b.215.4 8
21.11 odd 6 63.2.c.a.62.1 ✓ 4
21.17 even 6 63.2.c.a.62.1 ✓ 4
21.20 even 2 inner 441.2.p.b.80.1 8
28.3 even 6 1008.2.k.a.881.2 4
28.11 odd 6 1008.2.k.a.881.2 4
35.3 even 12 1575.2.g.d.1574.7 8
35.4 even 6 1575.2.b.a.251.1 4
35.17 even 12 1575.2.g.d.1574.2 8
35.18 odd 12 1575.2.g.d.1574.7 8
35.24 odd 6 1575.2.b.a.251.1 4
35.32 odd 12 1575.2.g.d.1574.2 8
56.3 even 6 4032.2.k.b.3905.1 4
56.11 odd 6 4032.2.k.b.3905.1 4
56.45 odd 6 4032.2.k.c.3905.4 4
56.53 even 6 4032.2.k.c.3905.4 4
63.4 even 3 567.2.o.f.188.1 8
63.11 odd 6 567.2.o.f.377.1 8
63.25 even 3 567.2.o.f.377.4 8
63.31 odd 6 567.2.o.f.188.1 8
63.32 odd 6 567.2.o.f.188.4 8
63.38 even 6 567.2.o.f.377.1 8
63.52 odd 6 567.2.o.f.377.4 8
63.59 even 6 567.2.o.f.188.4 8
84.11 even 6 1008.2.k.a.881.1 4
84.59 odd 6 1008.2.k.a.881.1 4
105.17 odd 12 1575.2.g.d.1574.8 8
105.32 even 12 1575.2.g.d.1574.8 8
105.38 odd 12 1575.2.g.d.1574.1 8
105.53 even 12 1575.2.g.d.1574.1 8
105.59 even 6 1575.2.b.a.251.4 4
105.74 odd 6 1575.2.b.a.251.4 4
168.11 even 6 4032.2.k.b.3905.2 4
168.53 odd 6 4032.2.k.c.3905.3 4
168.59 odd 6 4032.2.k.b.3905.2 4
168.101 even 6 4032.2.k.c.3905.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.c.a.62.1 ✓ 4 21.11 odd 6
63.2.c.a.62.1 ✓ 4 21.17 even 6
63.2.c.a.62.4 yes 4 7.3 odd 6
63.2.c.a.62.4 yes 4 7.4 even 3
441.2.p.b.80.1 8 3.2 odd 2 inner
441.2.p.b.80.1 8 21.20 even 2 inner
441.2.p.b.80.4 8 1.1 even 1 trivial
441.2.p.b.80.4 8 7.6 odd 2 CM
441.2.p.b.215.1 8 7.2 even 3 inner
441.2.p.b.215.1 8 7.5 odd 6 inner
441.2.p.b.215.4 8 21.2 odd 6 inner
441.2.p.b.215.4 8 21.5 even 6 inner
567.2.o.f.188.1 8 63.4 even 3
567.2.o.f.188.1 8 63.31 odd 6
567.2.o.f.188.4 8 63.32 odd 6
567.2.o.f.188.4 8 63.59 even 6
567.2.o.f.377.1 8 63.11 odd 6
567.2.o.f.377.1 8 63.38 even 6
567.2.o.f.377.4 8 63.25 even 3
567.2.o.f.377.4 8 63.52 odd 6
1008.2.k.a.881.1 4 84.11 even 6
1008.2.k.a.881.1 4 84.59 odd 6
1008.2.k.a.881.2 4 28.3 even 6
1008.2.k.a.881.2 4 28.11 odd 6
1575.2.b.a.251.1 4 35.4 even 6
1575.2.b.a.251.1 4 35.24 odd 6
1575.2.b.a.251.4 4 105.59 even 6
1575.2.b.a.251.4 4 105.74 odd 6
1575.2.g.d.1574.1 8 105.38 odd 12
1575.2.g.d.1574.1 8 105.53 even 12
1575.2.g.d.1574.2 8 35.17 even 12
1575.2.g.d.1574.2 8 35.32 odd 12
1575.2.g.d.1574.7 8 35.3 even 12
1575.2.g.d.1574.7 8 35.18 odd 12
1575.2.g.d.1574.8 8 105.17 odd 12
1575.2.g.d.1574.8 8 105.32 even 12
4032.2.k.b.3905.1 4 56.3 even 6
4032.2.k.b.3905.1 4 56.11 odd 6
4032.2.k.b.3905.2 4 168.11 even 6
4032.2.k.b.3905.2 4 168.59 odd 6
4032.2.k.c.3905.3 4 168.53 odd 6
4032.2.k.c.3905.3 4 168.101 even 6
4032.2.k.c.3905.4 4 56.45 odd 6
4032.2.k.c.3905.4 4 56.53 even 6