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Properties

Label 1925.2.b.r
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
Analytic rank $0$
Dimension $14$
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] = [1925,2,Mod(1849,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 287x^{10} + 1510x^{8} + 4019x^{6} + 4803x^{4} + 1669x^{2} + 36 \) 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

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{10} q^{6} + \beta_{8} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{9} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{10} q^{6} + \beta_{8} q^{7} + (\beta_{3} - 2 \beta_1) q^{8} + ( - \beta_{7} - 1) q^{9} + q^{11} + ( - \beta_{13} - 2 \beta_{9} + \cdots - \beta_{6}) q^{12}+ \cdots + ( - \beta_{7} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 26 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 26 q^{4} + 6 q^{6} - 18 q^{9} + 14 q^{11} - 2 q^{14} + 42 q^{16} - 40 q^{19} - 36 q^{24} - 26 q^{26} - 8 q^{29} + 12 q^{31} + 14 q^{34} + 24 q^{36} + 20 q^{39} + 48 q^{41} - 26 q^{44} + 66 q^{46} - 14 q^{49} + 38 q^{51} - 18 q^{54} - 18 q^{59} + 36 q^{61} - 16 q^{64} + 6 q^{66} + 30 q^{69} + 36 q^{71} + 48 q^{74} + 176 q^{76} - 50 q^{79} - 2 q^{81} + 18 q^{84} - 82 q^{86} + 32 q^{89} - 6 q^{91} + 62 q^{94} + 34 q^{96} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 27x^{12} + 287x^{10} + 1510x^{8} + 4019x^{6} + 4803x^{4} + 1669x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{12} - 26\nu^{10} - 241\nu^{8} - 929\nu^{6} - 1170\nu^{4} + 387\nu^{2} + 364 ) / 160 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{12} + 68\nu^{10} + 553\nu^{8} + 1907\nu^{6} + 2460\nu^{4} + 589\nu^{2} - 12 ) / 160 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{13} - 26\nu^{11} - 241\nu^{9} - 929\nu^{7} - 1170\nu^{5} + 387\nu^{3} + 364\nu ) / 160 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{12} - 20\nu^{10} - 139\nu^{8} - 369\nu^{6} - 188\nu^{4} + 297\nu^{2} + 36 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{13} + 36\nu^{11} + 491\nu^{9} + 3169\nu^{7} + 9740\nu^{5} + 12183\nu^{3} + 3436\nu ) / 480 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{13} + 18\nu^{11} + 89\nu^{9} - 95\nu^{7} - 1750\nu^{5} - 3675\nu^{3} - 1484\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{12} + 66\nu^{10} + 535\nu^{8} + 1923\nu^{6} + 2826\nu^{4} + 1051\nu^{2} + 12 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{12} + 68\nu^{10} + 573\nu^{8} + 2187\nu^{6} + 3660\nu^{4} + 2209\nu^{2} + 348 ) / 40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -13\nu^{13} - 348\nu^{11} - 3623\nu^{9} - 18397\nu^{7} - 46340\nu^{5} - 50499\nu^{3} - 13708\nu ) / 480 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9\nu^{13} + 224\nu^{11} + 2159\nu^{9} + 10121\nu^{7} + 23720\nu^{5} + 25307\nu^{3} + 8444\nu ) / 160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{4} - 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 2\beta_{12} - \beta_{9} + \beta_{8} - \beta_{6} - 9\beta_{3} + 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{11} + 11\beta_{10} + 12\beta_{7} + 14\beta_{5} + 15\beta_{4} + 47\beta_{2} - 143 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -12\beta_{13} - 26\beta_{12} + 10\beta_{9} - 22\beta_{8} + 14\beta_{6} + 70\beta_{3} - 249\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 96\beta_{11} - 94\beta_{10} - 108\beta_{7} - 144\beta_{5} - 150\beta_{4} - 319\beta_{2} + 928 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 110\beta_{13} + 252\beta_{12} - 76\beta_{9} + 278\beta_{8} - 136\beta_{6} - 523\beta_{3} + 1674\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -769\beta_{11} + 735\beta_{10} + 885\beta_{7} + 1305\beta_{5} + 1287\beta_{4} + 2197\beta_{2} - 6199 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -919\beta_{13} - 2190\beta_{12} + 517\beta_{9} - 2831\beta_{8} + 1137\beta_{6} + 3851\beta_{3} - 11478\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 5907\beta_{11} - 5505\beta_{10} - 6960\beta_{7} - 11062\beta_{5} - 10237\beta_{4} - 15329\beta_{2} + 42279 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7362\beta_{13} + 18022\beta_{12} - 3246\beta_{9} + 25876\beta_{8} - 8782\beta_{6} - 28196\beta_{3} + 79897\beta_1 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
− 2.71503i
− 2.60926i
− 2.27174i
− 2.13935i
− 1.59433i
− 0.719607i
− 0.151896i
0.151896i
0.719607i
1.59433i
2.13935i
2.27174i
2.60926i
2.71503i
− 2.71503i − 0.525975i −5.37138 0 −1.42804 − 1.00000i 9.15341i 2.72335 0
1849.2 − 2.60926i 2.47161i −4.80822 0 6.44906 1.00000i 7.32737i −3.10886 0
1849.3 − 2.27174i − 1.44691i −3.16080 0 −3.28700 1.00000i 2.63702i 0.906454 0
1849.4 − 2.13935i 2.65419i −2.57680 0 5.67823 − 1.00000i 1.23398i −4.04471 0
1849.5 − 1.59433i − 3.08438i −0.541890 0 −4.91751 − 1.00000i − 2.32471i −6.51337 0
1849.6 − 0.719607i 0.234508i 1.48217 0 0.168754 1.00000i − 2.50579i 2.94501 0
1849.7 − 0.151896i 2.21537i 1.97693 0 0.336506 − 1.00000i − 0.604080i −1.90787 0
1849.8 0.151896i − 2.21537i 1.97693 0 0.336506 1.00000i 0.604080i −1.90787 0
1849.9 0.719607i − 0.234508i 1.48217 0 0.168754 − 1.00000i 2.50579i 2.94501 0
1849.10 1.59433i 3.08438i −0.541890 0 −4.91751 1.00000i 2.32471i −6.51337 0
1849.11 2.13935i − 2.65419i −2.57680 0 5.67823 1.00000i − 1.23398i −4.04471 0
1849.12 2.27174i 1.44691i −3.16080 0 −3.28700 − 1.00000i − 2.63702i 0.906454 0
1849.13 2.60926i − 2.47161i −4.80822 0 6.44906 − 1.00000i − 7.32737i −3.10886 0
1849.14 2.71503i 0.525975i −5.37138 0 −1.42804 1.00000i − 9.15341i 2.72335 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.r 14
5.b even 2 1 inner 1925.2.b.r 14
5.c odd 4 1 1925.2.a.bb ✓ 7
5.c odd 4 1 1925.2.a.bd yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1925.2.a.bb ✓ 7 5.c odd 4 1
1925.2.a.bd yes 7 5.c odd 4 1
1925.2.b.r 14 1.a even 1 1 trivial
1925.2.b.r 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 \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{14} + 27T_{2}^{12} + 287T_{2}^{10} + 1510T_{2}^{8} + 4019T_{2}^{6} + 4803T_{2}^{4} + 1669T_{2}^{2} + 36 \) Copy content Toggle raw display
\( T_{3}^{14} + 30T_{3}^{12} + 347T_{3}^{10} + 1932T_{3}^{8} + 5203T_{3}^{6} + 5758T_{3}^{4} + 1465T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{19}^{7} + 20T_{19}^{6} + 146T_{19}^{5} + 445T_{19}^{4} + 344T_{19}^{3} - 680T_{19}^{2} - 640T_{19} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 27 T^{12} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{14} + 30 T^{12} + \cdots + 64 \) 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
$13$ \( T^{14} + 115 T^{12} + \cdots + 1373584 \) Copy content Toggle raw display
$17$ \( T^{14} + 152 T^{12} + \cdots + 5349969 \) Copy content Toggle raw display
$19$ \( (T^{7} + 20 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 191 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$29$ \( (T^{7} + 4 T^{6} + \cdots - 960)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 6 T^{6} + \cdots + 132280)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + 129 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( (T^{7} - 24 T^{6} + \cdots + 972)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 996538624 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 2781507600 \) Copy content Toggle raw display
$53$ \( T^{14} + 335 T^{12} + \cdots + 23648769 \) Copy content Toggle raw display
$59$ \( (T^{7} + 9 T^{6} + \cdots + 107496)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 18 T^{6} + \cdots - 175300)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 2529219483904 \) Copy content Toggle raw display
$71$ \( (T^{7} - 18 T^{6} + \cdots - 4405392)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 9382581610000 \) Copy content Toggle raw display
$79$ \( (T^{7} + 25 T^{6} + \cdots - 4652962)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 315 T^{12} + \cdots + 33177600 \) Copy content Toggle raw display
$89$ \( (T^{7} - 16 T^{6} + \cdots - 38784)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 21667840000 \) Copy content Toggle raw display
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