Properties

Label 2576.2.f.f
Level $2576$
Weight $2$
Character orbit 2576.f
Analytic conductor $20.569$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2576,2,Mod(321,2576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2576.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2576.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.5694635607\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 12 x^{10} - 37 x^{9} + 37 x^{8} - 152 x^{7} + 284 x^{6} - 40 x^{5} + 375 x^{4} + 37 x^{3} - 1992 x^{2} - 887 x + 3979 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 161)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + \beta_{3} q^{5} + \beta_{7} q^{7} + (\beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + \beta_{3} q^{5} + \beta_{7} q^{7} + (\beta_{4} - 2) q^{9} + (\beta_{8} - \beta_{7}) q^{11} - \beta_{10} q^{13} + ( - \beta_{11} + \beta_{6}) q^{15} + (\beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{3}) q^{17} + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{3}) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{6}) q^{21} + ( - \beta_{6} + \beta_1 + 1) q^{23} + ( - 2 \beta_{4} + 1) q^{25} - \beta_{10} q^{27} + ( - \beta_1 + 1) q^{29} + \beta_{5} q^{31} + (2 \beta_{9} + \beta_{8} + \beta_{7}) q^{33} + (\beta_{10} + \beta_{2} - 2 \beta_1 + 1) q^{35} + ( - \beta_{11} + \beta_{6}) q^{37} + ( - 4 \beta_{4} - \beta_1 - 1) q^{39} + (\beta_{10} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{41} + 2 \beta_{6} q^{43} + ( - \beta_{9} - 4 \beta_{3}) q^{45} + ( - \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{47} + (\beta_{10} - \beta_{5} - \beta_{4} - \beta_{2} - 1) q^{49} + ( - 2 \beta_{11} - \beta_{8} + \beta_{7}) q^{51} + ( - \beta_{11} + \beta_{8} - \beta_{7} - \beta_{6}) q^{53} + ( - 2 \beta_{10} - \beta_{4} - 2 \beta_{2} + \beta_1) q^{55} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6}) q^{57} + ( - \beta_{10} - 3 \beta_{5}) q^{59} + ( - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} + 2 \beta_{3}) q^{61} + ( - \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{3}) q^{63} + (\beta_{11} + 2 \beta_{8} - 2 \beta_{7} - \beta_{6}) q^{65} + ( - \beta_{11} - \beta_{6}) q^{67} + ( - \beta_{10} + 2 \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{69} + ( - 3 \beta_{4} + 3) q^{71} + ( - \beta_{10} - 2 \beta_{5}) q^{73} + (2 \beta_{10} - 3 \beta_{5}) q^{75} + ( - \beta_{10} + \beta_{5} + \beta_{4} + \beta_{2} + 8) q^{77} + (\beta_{11} + \beta_{8} - \beta_{7} - \beta_{6}) q^{79} + ( - \beta_{4} - \beta_1 - 7) q^{81} + (\beta_{8} + \beta_{7} - 2 \beta_{3}) q^{83} + ( - 5 \beta_{4} + \beta_1 + 6) q^{85} + (\beta_{10} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} - \beta_1) q^{87} + ( - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{3}) q^{89} + (\beta_{11} - \beta_{9} + 3 \beta_{8} + \beta_{3}) q^{91} + ( - \beta_{4} + 5) q^{93} + ( - 3 \beta_{4} + \beta_1) q^{95} + ( - 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{3}) q^{97} + ( - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 28 q^{9} + 8 q^{23} + 20 q^{25} + 16 q^{29} + 20 q^{35} + 8 q^{39} - 8 q^{49} + 48 q^{71} + 92 q^{77} - 76 q^{81} + 88 q^{85} + 64 q^{93} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 12 x^{10} - 37 x^{9} + 37 x^{8} - 152 x^{7} + 284 x^{6} - 40 x^{5} + 375 x^{4} + 37 x^{3} - 1992 x^{2} - 887 x + 3979 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3917751 \nu^{11} - 523537753 \nu^{10} - 274083757 \nu^{9} - 6901032636 \nu^{8} + 7981987520 \nu^{7} - 7216075440 \nu^{6} + \cdots + 47656458956 ) / 257814235068 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 207146886 \nu^{11} + 9506363755 \nu^{10} + 9052287305 \nu^{9} + 183368444379 \nu^{8} - 72578650976 \nu^{7} + \cdots + 5917533461663 ) / 9539126697516 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6955944633 \nu^{11} - 4302550030 \nu^{10} - 112251543742 \nu^{9} + 75307224765 \nu^{8} - 356952564178 \nu^{7} + \cdots - 10199110379113 ) / 9539126697516 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 400562257 \nu^{11} - 713713985 \nu^{10} - 6116579929 \nu^{9} - 153932980 \nu^{8} - 6098931592 \nu^{7} + 54253030504 \nu^{6} + \cdots + 253359430808 ) / 257814235068 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15225232238 \nu^{11} - 1843082365 \nu^{10} - 203247617453 \nu^{9} + 365551271593 \nu^{8} - 356375747624 \nu^{7} + \cdots + 36537272992759 ) / 9539126697516 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 423978509 \nu^{11} - 351014413 \nu^{10} - 4919740121 \nu^{9} + 5853603706 \nu^{8} + 2707665718 \nu^{7} + 43342480388 \nu^{6} + \cdots + 1002011497882 ) / 257814235068 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8130225630 \nu^{11} - 6053183734 \nu^{10} - 97811936088 \nu^{9} + 126447804237 \nu^{8} + 78783860678 \nu^{7} + \cdots + 24201854901774 ) / 4769563348758 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10041116024 \nu^{11} + 2270069274 \nu^{10} + 113474930991 \nu^{9} - 231345642646 \nu^{8} - 61447568558 \nu^{7} + \cdots - 24743720692728 ) / 4769563348758 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 21444321649 \nu^{11} - 38229559648 \nu^{10} - 312753756348 \nu^{9} - 62475883387 \nu^{8} - 154346515584 \nu^{7} + \cdots + 38785880097207 ) / 9539126697516 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 48304093989 \nu^{11} + 24409492465 \nu^{10} + 585832049137 \nu^{9} - 966557495736 \nu^{8} - 112667565608 \nu^{7} + \cdots - 135029319325400 ) / 9539126697516 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1463340231 \nu^{11} - 18490865 \nu^{10} + 18138797139 \nu^{9} - 38057840928 \nu^{8} + 23300246644 \nu^{7} - 244208446912 \nu^{6} + \cdots - 3087019385112 ) / 257814235068 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} - 2\beta_{7} + \beta_{5} + \beta_{4} + 2\beta_{2} - 3\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{11} + \beta_{10} + 3 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} - \beta_{5} + 3 \beta_{4} - 12 \beta_{3} - 2 \beta_{2} - 5 \beta _1 - 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} - \beta_{10} - 2 \beta_{9} - 4 \beta_{7} - \beta_{6} - 7 \beta_{5} + 3 \beta_{3} - 4 \beta_{2} + 4 \beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30 \beta_{11} + 7 \beta_{10} - 3 \beta_{9} - 82 \beta_{8} - 34 \beta_{7} + 12 \beta_{6} + 17 \beta_{5} - 11 \beta_{4} + 92 \beta_{3} + 66 \beta_{2} + 33 \beta _1 + 168 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 98 \beta_{11} + 57 \beta_{10} + 79 \beta_{9} + 50 \beta_{8} + 290 \beta_{7} - 16 \beta_{6} + 167 \beta_{5} + 143 \beta_{4} - 264 \beta_{3} + 158 \beta_{2} - 321 \beta _1 - 592 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 56 \beta_{11} - 34 \beta_{10} + 5 \beta_{9} + 118 \beta_{8} - 4 \beta_{7} - 24 \beta_{6} - 85 \beta_{5} - 5 \beta_{4} - 84 \beta_{3} - 100 \beta_{2} - 20 \beta _1 - 140 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 598 \beta_{11} - 417 \beta_{10} - 807 \beta_{9} - 2042 \beta_{8} - 3434 \beta_{7} - 88 \beta_{6} - 1175 \beta_{5} - 1895 \beta_{4} + 3992 \beta_{3} - 934 \beta_{2} + 4165 \beta _1 + 7832 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5950 \beta_{11} + 3737 \beta_{10} - 285 \beta_{9} - 8670 \beta_{8} + 5298 \beta_{7} + 1868 \beta_{6} + 8975 \beta_{5} + 4123 \beta_{4} + 2692 \beta_{3} + 10942 \beta_{2} - 3237 \beta _1 + 4440 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 21 \beta_{11} - 87 \beta_{10} + 2518 \beta_{9} + 8802 \beta_{8} + 8654 \beta_{7} - 107 \beta_{6} + 17 \beta_{5} + 4334 \beta_{4} - 12615 \beta_{3} - 118 \beta_{2} - 11518 \beta _1 - 25164 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 71298 \beta_{11} - 43457 \beta_{10} - 6739 \beta_{9} + 58766 \beta_{8} - 103282 \beta_{7} - 24516 \beta_{6} - 105967 \beta_{5} - 76555 \beta_{4} + 31340 \beta_{3} - 130398 \beta_{2} + \cdots + 60656 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 87322 \beta_{11} + 55353 \beta_{10} - 120329 \beta_{9} - 483446 \beta_{8} - 279478 \beta_{7} + 30024 \beta_{6} + 130863 \beta_{5} - 107241 \beta_{4} + 560240 \beta_{3} + 161222 \beta_{2} + \cdots + 1134728 ) / 8 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2576\mathbb{Z}\right)^\times\).

\(n\) \(645\) \(1473\) \(1569\) \(2255\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
1.56715 0.516030i
0.0276182 + 2.02175i
−1.19108 0.505501i
−0.602791 + 3.39856i
1.81018 0.517196i
−1.11108 1.95526i
1.81018 + 0.517196i
−1.11108 + 1.95526i
−1.19108 + 0.505501i
−0.602791 3.39856i
1.56715 + 0.516030i
0.0276182 2.02175i
0 2.77152i 0 −3.37085 0 −2.11370 + 1.59132i 0 −4.68133 0
321.2 0 2.77152i 0 3.37085 0 2.11370 1.59132i 0 −4.68133 0
321.3 0 2.37530i 0 −2.69892 0 1.50452 2.17633i 0 −2.64207 0
321.4 0 2.37530i 0 2.69892 0 −1.50452 + 2.17633i 0 −2.64207 0
321.5 0 1.63603i 0 −1.16327 0 −1.66394 2.05701i 0 0.323404 0
321.6 0 1.63603i 0 1.16327 0 1.66394 + 2.05701i 0 0.323404 0
321.7 0 1.63603i 0 −1.16327 0 −1.66394 + 2.05701i 0 0.323404 0
321.8 0 1.63603i 0 1.16327 0 1.66394 2.05701i 0 0.323404 0
321.9 0 2.37530i 0 −2.69892 0 1.50452 + 2.17633i 0 −2.64207 0
321.10 0 2.37530i 0 2.69892 0 −1.50452 2.17633i 0 −2.64207 0
321.11 0 2.77152i 0 −3.37085 0 −2.11370 1.59132i 0 −4.68133 0
321.12 0 2.77152i 0 3.37085 0 2.11370 + 1.59132i 0 −4.68133 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2576.2.f.f 12
4.b odd 2 1 161.2.c.b 12
7.b odd 2 1 inner 2576.2.f.f 12
12.b even 2 1 1449.2.h.g 12
23.b odd 2 1 inner 2576.2.f.f 12
28.d even 2 1 161.2.c.b 12
84.h odd 2 1 1449.2.h.g 12
92.b even 2 1 161.2.c.b 12
161.c even 2 1 inner 2576.2.f.f 12
276.h odd 2 1 1449.2.h.g 12
644.h odd 2 1 161.2.c.b 12
1932.b even 2 1 1449.2.h.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.c.b 12 4.b odd 2 1
161.2.c.b 12 28.d even 2 1
161.2.c.b 12 92.b even 2 1
161.2.c.b 12 644.h odd 2 1
1449.2.h.g 12 12.b even 2 1
1449.2.h.g 12 84.h odd 2 1
1449.2.h.g 12 276.h odd 2 1
1449.2.h.g 12 1932.b even 2 1
2576.2.f.f 12 1.a even 1 1 trivial
2576.2.f.f 12 7.b odd 2 1 inner
2576.2.f.f 12 23.b odd 2 1 inner
2576.2.f.f 12 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2576, [\chi])\):

\( T_{3}^{6} + 16T_{3}^{4} + 79T_{3}^{2} + 116 \) Copy content Toggle raw display
\( T_{5}^{6} - 20T_{5}^{4} + 108T_{5}^{2} - 112 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 16 T^{4} + 79 T^{2} + 116)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} - 20 T^{4} + 108 T^{2} - 112)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 4 T^{10} + 131 T^{8} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 46 T^{4} + 684 T^{2} + 3248)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 52 T^{4} + 679 T^{2} + 464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 82 T^{4} + 2180 T^{2} + \cdots - 18928)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 58 T^{4} + 332 T^{2} - 448)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 4 T^{5} + 25 T^{4} - 216 T^{3} + \cdots + 12167)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 4 T^{2} - 7 T + 26)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 16 T^{4} + 79 T^{2} + 116)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 132 T^{4} + 4052 T^{2} + \cdots + 12992)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 156 T^{4} + 4439 T^{2} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 172 T^{4} + 7936 T^{2} + \cdots + 51968)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 136 T^{4} + 5583 T^{2} + \cdots + 61364)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 202 T^{4} + 3412 T^{2} + \cdots + 12992)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 184 T^{4} + 9088 T^{2} + \cdots + 90944)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 244 T^{4} + 3776 T^{2} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 144 T^{4} + 5060 T^{2} + \cdots + 3248)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 12 T^{2} - 9 T + 216)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + 108 T^{4} + 2463 T^{2} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 166 T^{4} + 1452 T^{2} + \cdots + 3248)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 78 T^{4} + 524 T^{2} - 448)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 276 T^{4} + 20480 T^{2} + \cdots - 458752)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 260 T^{4} + 16684 T^{2} + \cdots - 314608)^{2} \) Copy content Toggle raw display
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