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Properties

Label 1075.2.b.k
Level $1075$
Weight $2$
Character orbit 1075.b
Analytic conductor $8.584$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1075,2,Mod(474,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1075.474");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1075.b (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: \(8.58391821729\)
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: \( x^{12} + 19x^{10} + 121x^{8} + 339x^{6} + 437x^{4} + 247x^{2} + 49 \) 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 215)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_1) q^{2} + ( - \beta_{10} - \beta_{7}) q^{3} + ( - \beta_{4} - \beta_{3} - 2) q^{4} + (\beta_{5} + \beta_{4}) q^{6} + ( - \beta_{10} - \beta_{9} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - 2 \beta_{7} - \beta_1) q^{8} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_1) q^{2} + ( - \beta_{10} - \beta_{7}) q^{3} + ( - \beta_{4} - \beta_{3} - 2) q^{4} + (\beta_{5} + \beta_{4}) q^{6} + ( - \beta_{10} - \beta_{9} - \beta_1) q^{7} + (\beta_{11} - \beta_{9} - 2 \beta_{7} - \beta_1) q^{8} + ( - \beta_{6} + \beta_{4} + \cdots - \beta_{2}) q^{9}+ \cdots + ( - 2 \beta_{6} + 6 \beta_{5} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 14 q^{4} - 16 q^{9} + 8 q^{14} + 10 q^{16} - 12 q^{19} - 24 q^{21} - 64 q^{26} + 20 q^{29} - 28 q^{34} - 82 q^{36} - 16 q^{39} - 12 q^{41} - 8 q^{44} - 16 q^{46} - 40 q^{49} - 16 q^{51} + 10 q^{54} - 70 q^{56} + 40 q^{59} - 16 q^{61} - 34 q^{64} - 70 q^{66} + 84 q^{69} + 16 q^{71} - 90 q^{74} - 32 q^{76} + 32 q^{79} + 92 q^{81} - 74 q^{84} + 6 q^{86} + 48 q^{91} - 24 q^{94} - 46 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 19x^{10} + 121x^{8} + 339x^{6} + 437x^{4} + 247x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 24\nu^{8} + 241\nu^{6} + 1242\nu^{4} + 2570\nu^{2} + 1168 ) / 151 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -22\nu^{10} - 377\nu^{8} - 1980\nu^{6} - 4070\nu^{4} - 3237\nu^{2} - 1083 ) / 151 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -22\nu^{10} - 377\nu^{8} - 1980\nu^{6} - 4070\nu^{4} - 3086\nu^{2} - 630 ) / 151 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 51\nu^{10} + 922\nu^{8} + 5345\nu^{6} + 12757\nu^{4} + 12384\nu^{2} + 4000 ) / 151 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54\nu^{10} + 994\nu^{8} + 5917\nu^{6} + 14369\nu^{4} + 13299\nu^{2} + 3427 ) / 151 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -90\nu^{11} - 1556\nu^{9} - 8251\nu^{7} - 16650\nu^{5} - 10840\nu^{3} - 628\nu ) / 1057 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -116\nu^{11} - 2029\nu^{9} - 10893\nu^{7} - 21460\nu^{5} - 10918\nu^{3} + 1469\nu ) / 1057 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 54\nu^{11} + 994\nu^{9} + 5917\nu^{7} + 14369\nu^{5} + 13299\nu^{3} + 3125\nu ) / 151 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 383\nu^{11} + 6927\nu^{9} + 40057\nu^{7} + 94109\nu^{5} + 86766\nu^{3} + 25903\nu ) / 1057 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 390\nu^{11} + 7095\nu^{9} + 41744\nu^{7} + 102803\nu^{5} + 104756\nu^{3} + 34079\nu ) / 1057 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{9} - 3\beta_{8} + 4\beta_{7} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{6} + 3\beta_{5} - 10\beta_{4} + 12\beta_{3} - \beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{11} + 4\beta_{10} + 10\beta_{9} + 37\beta_{8} - 45\beta_{7} + 36\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} - 41\beta_{5} + 95\beta_{4} - 123\beta_{3} + 17\beta_{2} - 144 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 140\beta_{11} - 58\beta_{10} - 96\beta_{9} - 382\beta_{8} + 449\beta_{7} - 308\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -286\beta_{6} + 440\beta_{5} - 903\beta_{4} + 1212\beta_{3} - 198\beta_{2} + 1292 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -1410\beta_{11} + 638\beta_{10} + 926\beta_{9} + 3767\beta_{8} - 4361\beta_{7} + 2835\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2841\beta_{6} - 4405\beta_{5} + 8627\beta_{4} - 11779\beta_{3} + 2048\beta_{2} - 12118 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 13827\beta_{11} - 6453\beta_{10} - 8938\beta_{9} - 36590\beta_{8} + 42065\beta_{7} - 26842\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).

\(n\) \(302\) \(476\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
474.1
− 1.72658i
− 3.10292i
− 0.840555i
− 0.673596i
− 1.96109i
− 1.17673i
1.17673i
1.96109i
0.673596i
0.840555i
3.10292i
1.72658i
− 2.72658i − 0.195967i −5.43426 0 −0.534322 2.90692i 9.36379i 2.96160 0
474.2 − 2.10292i − 0.742586i −2.42225 0 −1.56160 2.60406i 0.887964i 2.44857 0
474.3 − 1.84056i 0.291442i −1.38764 0 0.536415 − 5.13977i − 1.12707i 2.91506 0
474.4 − 1.67360i 2.56351i −0.800923 0 4.29028 0.173417i − 2.00677i −3.57157 0
474.5 − 0.961086i − 3.34672i 1.07631 0 −3.21649 − 1.04792i − 2.95660i −8.20055 0
474.6 − 0.176734i 2.74829i 1.96877 0 0.485715 4.38443i − 0.701414i −4.55310 0
474.7 0.176734i − 2.74829i 1.96877 0 0.485715 − 4.38443i 0.701414i −4.55310 0
474.8 0.961086i 3.34672i 1.07631 0 −3.21649 1.04792i 2.95660i −8.20055 0
474.9 1.67360i − 2.56351i −0.800923 0 4.29028 − 0.173417i 2.00677i −3.57157 0
474.10 1.84056i − 0.291442i −1.38764 0 0.536415 5.13977i 1.12707i 2.91506 0
474.11 2.10292i 0.742586i −2.42225 0 −1.56160 − 2.60406i − 0.887964i 2.44857 0
474.12 2.72658i 0.195967i −5.43426 0 −0.534322 − 2.90692i − 9.36379i 2.96160 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 474.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1075.2.b.k 12
5.b even 2 1 inner 1075.2.b.k 12
5.c odd 4 1 215.2.a.d ✓ 6
5.c odd 4 1 1075.2.a.p 6
15.e even 4 1 1935.2.a.z 6
15.e even 4 1 9675.2.a.cl 6
20.e even 4 1 3440.2.a.x 6
215.g even 4 1 9245.2.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.a.d ✓ 6 5.c odd 4 1
1075.2.a.p 6 5.c odd 4 1
1075.2.b.k 12 1.a even 1 1 trivial
1075.2.b.k 12 5.b even 2 1 inner
1935.2.a.z 6 15.e even 4 1
3440.2.a.x 6 20.e even 4 1
9245.2.a.n 6 215.g even 4 1
9675.2.a.cl 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1075, [\chi])\):

\( T_{2}^{12} + 19T_{2}^{10} + 133T_{2}^{8} + 427T_{2}^{6} + 617T_{2}^{4} + 307T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{12} + 26T_{3}^{10} + 225T_{3}^{8} + 698T_{3}^{6} + 390T_{3}^{4} + 40T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{12} + 62T_{7}^{10} + 1329T_{7}^{8} + 11774T_{7}^{6} + 40818T_{7}^{4} + 33172T_{7}^{2} + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 19 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} + 26 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 62 T^{10} + \cdots + 961 \) Copy content Toggle raw display
$11$ \( (T^{6} - 41 T^{4} + \cdots - 93)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 76 T^{10} + \cdots + 200704 \) Copy content Toggle raw display
$17$ \( T^{12} + 156 T^{10} + \cdots + 1806336 \) Copy content Toggle raw display
$19$ \( (T^{6} + 6 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 192 T^{10} + \cdots + 35426304 \) Copy content Toggle raw display
$29$ \( (T^{6} - 10 T^{5} + \cdots + 5952)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 97 T^{4} + \cdots - 10133)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 888814969 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots - 10911)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$47$ \( T^{12} + 156 T^{10} + \cdots + 19501056 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 291999744 \) Copy content Toggle raw display
$59$ \( (T^{6} - 20 T^{5} + \cdots + 6987)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 8 T^{5} + \cdots + 6848)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1036324864 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} + \cdots + 192)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 102677689 \) Copy content Toggle raw display
$79$ \( (T^{6} - 16 T^{5} + \cdots + 194267)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 10394210304 \) Copy content Toggle raw display
$89$ \( (T^{6} - 264 T^{4} + \cdots + 265152)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 138674176 \) Copy content Toggle raw display
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