↑

Properties

Label 1925.2.b.q
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
Analytic rank $0$
Dimension $14$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1925,2,Mod(1849,1925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1925.1849"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,0,-26,0,-2,0,0,-26,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 283x^{10} + 1442x^{8} + 3659x^{6} + 4315x^{4} + 2225x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{6} q^{3} + (\beta_{8} - \beta_{7} - 2) q^{4} + (\beta_{11} - \beta_{5} - \beta_{2}) q^{6} - \beta_{3} q^{7} + (\beta_{10} + \beta_{9} + \cdots - 2 \beta_1) q^{8} + ( - \beta_{12} - \beta_{5} - 2) q^{9}+ \cdots + (\beta_{12} + \beta_{5} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 26 q^{4} - 2 q^{6} - 26 q^{9} - 14 q^{11} - 2 q^{14} + 58 q^{16} - 36 q^{19} + 44 q^{24} + 26 q^{26} + 4 q^{29} + 48 q^{31} - 66 q^{34} + 88 q^{36} - 20 q^{39} - 20 q^{41} + 26 q^{44} + 6 q^{46}+ \cdots + 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 27x^{12} + 283x^{10} + 1442x^{8} + 3659x^{6} + 4315x^{4} + 2225x^{2} + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} + 12\nu^{10} - 17\nu^{8} - 663\nu^{6} - 2396\nu^{4} - 2105\nu^{2} - 400 ) / 160 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + 28\nu^{11} + 295\nu^{9} + 1425\nu^{7} + 2996\nu^{5} + 1919\nu^{3} + 120\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{13} + 34\nu^{11} + 437\nu^{9} + 2673\nu^{7} + 7938\nu^{5} + 10213\nu^{3} + 4000\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{12} - 27\nu^{10} - 278\nu^{8} - 1357\nu^{6} - 3159\nu^{4} - 3090\nu^{2} - 1000 ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{13} + 79\nu^{11} + 800\nu^{9} + 3855\nu^{7} + 8763\nu^{5} + 7892\nu^{3} + 2040\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{12} + 81\nu^{10} + 834\nu^{8} + 4031\nu^{6} + 8997\nu^{4} + 7710\nu^{2} + 1920 ) / 80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{12} + 81\nu^{10} + 834\nu^{8} + 4031\nu^{6} + 8997\nu^{4} + 7790\nu^{2} + 2240 ) / 80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{13} + 184\nu^{11} + 1861\nu^{9} + 8999\nu^{7} + 20848\nu^{5} + 20125\nu^{3} + 6440\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -2\nu^{13} - 53\nu^{11} - 539\nu^{9} - 2606\nu^{7} - 5961\nu^{5} - 5471\nu^{3} - 1400\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{12} - 194\nu^{10} - 2071\nu^{8} - 10519\nu^{6} - 25138\nu^{4} - 23735\nu^{2} - 6800 ) / 160 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -13\nu^{12} - 336\nu^{10} - 3319\nu^{8} - 15461\nu^{6} - 33432\nu^{4} - 27615\nu^{2} - 6640 ) / 160 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\nu^{13} + 346\nu^{11} + 3529\nu^{9} + 17061\nu^{7} + 38842\nu^{5} + 35545\nu^{3} + 10120\nu ) / 160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 7\beta_{8} + 9\beta_{7} + \beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{13} - 7\beta_{10} - 10\beta_{9} - 12\beta_{3} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{12} + 45\beta_{8} - 71\beta_{7} - 3\beta_{5} - 12\beta_{2} - 159 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -26\beta_{13} + 42\beta_{10} + 86\beta_{9} - 6\beta_{6} + 108\beta_{3} - 275\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 112\beta_{12} - 6\beta_{11} - 291\beta_{8} + 541\beta_{7} + 50\beta_{5} + 114\beta_{2} + 1092 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 250\beta_{13} - 235\beta_{10} - 703\beta_{9} + 114\beta_{6} + 4\beta_{4} - 897\beta_{3} + 1924\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -957\beta_{12} + 118\beta_{11} + 1909\beta_{8} - 4065\beta_{7} - 582\beta_{5} - 1007\beta_{2} - 7660 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -2156\beta_{13} + 1209\beta_{10} + 5604\beta_{9} - 1450\beta_{6} - 68\beta_{4} + 7240\beta_{3} - 13634\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7828\beta_{12} - 1518\beta_{11} - 12687\beta_{8} + 30363\beta_{7} + 5845\beta_{5} + 8622\beta_{2} + 54551 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17676\beta_{13} - 5324\beta_{10} - 44036\beta_{9} + 15520\beta_{6} + 724\beta_{4} - 57812\beta_{3} + 97601\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
− 2.78276i
− 2.62008i
− 2.47176i
− 2.02168i
− 1.06204i
− 0.815390i
− 0.633891i
0.633891i
0.815390i
1.06204i
2.02168i
2.47176i
2.62008i
2.78276i
− 2.78276i − 3.17748i −5.74378 0 −8.84217 − 1.00000i 10.4180i −7.09637 0
1849.2 − 2.62008i 2.13745i −4.86484 0 5.60029 1.00000i 7.50611i −1.56868 0
1849.3 − 2.47176i 1.33124i −4.10959 0 3.29051 − 1.00000i 5.21439i 1.22779 0
1849.4 − 2.02168i − 2.31630i −2.08720 0 −4.68282 1.00000i 0.176288i −2.36524 0
1849.5 − 1.06204i 1.17047i 0.872076 0 1.24309 1.00000i − 3.05025i 1.63000 0
1849.6 − 0.815390i 3.26294i 1.33514 0 2.66057 − 1.00000i − 2.71944i −7.64680 0
1849.7 − 0.633891i − 0.425088i 1.59818 0 −0.269459 − 1.00000i − 2.28085i 2.81930 0
1849.8 0.633891i 0.425088i 1.59818 0 −0.269459 1.00000i 2.28085i 2.81930 0
1849.9 0.815390i − 3.26294i 1.33514 0 2.66057 1.00000i 2.71944i −7.64680 0
1849.10 1.06204i − 1.17047i 0.872076 0 1.24309 − 1.00000i 3.05025i 1.63000 0
1849.11 2.02168i 2.31630i −2.08720 0 −4.68282 − 1.00000i − 0.176288i −2.36524 0
1849.12 2.47176i − 1.33124i −4.10959 0 3.29051 1.00000i − 5.21439i 1.22779 0
1849.13 2.62008i − 2.13745i −4.86484 0 5.60029 − 1.00000i − 7.50611i −1.56868 0
1849.14 2.78276i 3.17748i −5.74378 0 −8.84217 1.00000i − 10.4180i −7.09637 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.14
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 1925.2.b.q 14
5.b even 2 1 inner 1925.2.b.q 14
5.c odd 4 1 1925.2.a.ba ✓ 7
5.c odd 4 1 1925.2.a.bc yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.2.a.ba ✓ 7 5.c odd 4 1
1925.2.a.bc yes 7 5.c odd 4 1
1925.2.b.q 14 1.a even 1 1 trivial
1925.2.b.q 14 5.b 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 \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{14} + 27T_{2}^{12} + 283T_{2}^{10} + 1442T_{2}^{8} + 3659T_{2}^{6} + 4315T_{2}^{4} + 2225T_{2}^{2} + 400 \) Copy content Toggle raw display
\( T_{3}^{14} + 34T_{3}^{12} + 443T_{3}^{10} + 2792T_{3}^{8} + 8899T_{3}^{6} + 13626T_{3}^{4} + 8585T_{3}^{2} + 1156 \) Copy content Toggle raw display
\( T_{19}^{7} + 18T_{19}^{6} + 52T_{19}^{5} - 593T_{19}^{4} - 2400T_{19}^{3} + 8504T_{19}^{2} + 24448T_{19} - 64912 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 27 T^{12} + \cdots + 400 \) Copy content Toggle raw display
$3$ \( T^{14} + 34 T^{12} + \cdots + 1156 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$11$ \( (T + 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 107 T^{12} + \cdots + 1752976 \) Copy content Toggle raw display
$17$ \( T^{14} + 160 T^{12} + \cdots + 259081 \) Copy content Toggle raw display
$19$ \( (T^{7} + 18 T^{6} + \cdots - 64912)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 199 T^{12} + \cdots + 82591744 \) Copy content Toggle raw display
$29$ \( (T^{7} - 2 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 24 T^{6} + \cdots - 9680)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 305 T^{12} + \cdots + 4376464 \) Copy content Toggle raw display
$41$ \( (T^{7} + 10 T^{6} + \cdots - 1836)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + 270 T^{12} + \cdots + 12588304 \) Copy content Toggle raw display
$47$ \( T^{14} + 151 T^{12} + \cdots + 462400 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 10274457769 \) Copy content Toggle raw display
$59$ \( (T^{7} + T^{6} - 279 T^{5} + \cdots - 9112)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 28 T^{6} + \cdots + 59060)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 16404486400 \) Copy content Toggle raw display
$71$ \( (T^{7} + 10 T^{6} + \cdots - 358928)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 329349904 \) Copy content Toggle raw display
$79$ \( (T^{7} + 19 T^{6} + \cdots - 777320)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 137694429184 \) Copy content Toggle raw display
$89$ \( (T^{7} - 2 T^{6} + \cdots + 702976)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 17163096064 \) Copy content Toggle raw display
show more
show less