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Properties

Label 3750.2.c.k
Level $3750$
Weight $2$
Character orbit 3750.c
Analytic conductor $29.944$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3750,2,Mod(1249,3750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3750.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.c (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: \(29.9439007580\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 150)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{6} q^{3} - q^{4} + q^{6} + ( - \beta_{9} - \beta_{6}) q^{7} + \beta_{6} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{6} q^{3} - q^{4} + q^{6} + ( - \beta_{9} - \beta_{6}) q^{7} + \beta_{6} q^{8} - q^{9} + ( - \beta_{3} + 1) q^{11} - \beta_{6} q^{12} + ( - \beta_{14} + \beta_{6} + \beta_{5}) q^{13} + \beta_{8} q^{14} + q^{16} + ( - \beta_{15} - \beta_{13} + \cdots + \beta_{5}) q^{17}+ \cdots + (\beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{6} - 16 q^{9} + 12 q^{11} - 8 q^{14} + 16 q^{16} - 20 q^{19} + 8 q^{21} - 16 q^{24} + 4 q^{26} - 20 q^{29} + 32 q^{31} - 28 q^{34} + 16 q^{36} - 4 q^{39} + 12 q^{41} - 12 q^{44} + 24 q^{46} - 52 q^{49} + 28 q^{51} - 16 q^{54} + 8 q^{56} + 32 q^{61} - 16 q^{64} + 12 q^{66} - 24 q^{69} + 12 q^{71} + 12 q^{74} + 20 q^{76} - 20 q^{79} + 16 q^{81} - 8 q^{84} + 4 q^{86} - 40 q^{89} + 12 q^{91} - 28 q^{94} + 16 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 24 x^{14} + 94 x^{13} + 262 x^{12} - 936 x^{11} - 1584 x^{10} + 4642 x^{9} + \cdots + 11105 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 71\!\cdots\!18 \nu^{15} + \cdots - 20\!\cdots\!45 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 706683448390084 \nu^{15} + \cdots + 11\!\cdots\!75 ) / 13\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 71\!\cdots\!92 \nu^{15} + \cdots - 16\!\cdots\!40 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10\!\cdots\!26 \nu^{15} + \cdots - 18\!\cdots\!05 ) / 75\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!24 \nu^{15} + \cdots + 36\!\cdots\!45 ) / 27\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 68829028972 \nu^{15} - 306672362945 \nu^{14} - 1464283156713 \nu^{13} + \cdots + 945642292769660 ) / 376857420945125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!10 \nu^{15} + \cdots + 18\!\cdots\!40 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!89 \nu^{15} + \cdots - 80\!\cdots\!20 ) / 60\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 90\!\cdots\!14 \nu^{15} + \cdots + 15\!\cdots\!20 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!78 \nu^{15} + \cdots - 44\!\cdots\!65 ) / 75\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 50\!\cdots\!12 \nu^{15} + \cdots + 77\!\cdots\!60 ) / 12\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!03 \nu^{15} + \cdots + 27\!\cdots\!90 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!97 \nu^{15} + \cdots - 26\!\cdots\!35 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!67 \nu^{15} + \cdots - 34\!\cdots\!35 ) / 30\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 91\!\cdots\!96 \nu^{15} + \cdots + 11\!\cdots\!05 ) / 12\!\cdots\!55 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + 2\beta_{8} + 4\beta_{7} + 5\beta_{5} - 2\beta_{4} + 2\beta_{3} - 10\beta_{2} + 4\beta _1 + 10 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{14} + 2 \beta_{13} - 4 \beta_{12} - \beta_{11} + 4 \beta_{10} + 6 \beta_{9} + \cdots + 45 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 12 \beta_{15} + 6 \beta_{14} - 6 \beta_{13} - 3 \beta_{12} + 12 \beta_{11} + 25 \beta_{10} + \cdots + 100 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23 \beta_{15} - 4 \beta_{14} + 14 \beta_{13} - 38 \beta_{12} - 30 \beta_{11} + 74 \beta_{10} + \cdots + 330 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 200 \beta_{15} + 80 \beta_{14} - 155 \beta_{13} - 65 \beta_{12} + 105 \beta_{11} + 298 \beta_{10} + \cdots + 935 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 449 \beta_{15} + 63 \beta_{14} - 133 \beta_{13} - 424 \beta_{12} - 445 \beta_{11} + 999 \beta_{10} + \cdots + 2970 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2702 \beta_{15} + 1001 \beta_{14} - 2331 \beta_{13} - 1008 \beta_{12} + 322 \beta_{11} + 3079 \beta_{10} + \cdots + 9200 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7934 \beta_{15} + 1906 \beta_{14} - 5116 \beta_{13} - 4938 \beta_{12} - 4907 \beta_{11} + \cdots + 29065 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 35562 \beta_{15} + 11550 \beta_{14} - 29820 \beta_{13} - 14895 \beta_{12} - 6144 \beta_{11} + \cdots + 91380 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61058 \beta_{15} + 16844 \beta_{14} - 46679 \beta_{13} - 29862 \beta_{12} - 25002 \beta_{11} + \cdots + 140950 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 462792 \beta_{15} + 132011 \beta_{14} - 366696 \beta_{13} - 208208 \beta_{12} - 156398 \beta_{11} + \cdots + 861390 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1700180 \beta_{15} + 487561 \beta_{14} - 1364491 \beta_{13} - 739638 \beta_{12} - 531674 \beta_{11} + \cdots + 2534505 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 5921409 \beta_{15} + 1561560 \beta_{14} - 4520750 \beta_{13} - 2739425 \beta_{12} - 2409953 \beta_{11} + \cdots + 7256500 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22036443 \beta_{15} + 6329388 \beta_{14} - 17845173 \beta_{13} - 9191079 \beta_{12} + \cdots + 19468025 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 74060127 \beta_{15} + 19028436 \beta_{14} - 55930176 \beta_{13} - 34220578 \beta_{12} + \cdots + 48033630 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3750\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
−2.79002 − 0.809017i
−0.705457 + 0.309017i
0.543374 − 0.809017i
3.42137 + 0.309017i
2.32349 + 0.309017i
−1.16141 − 0.809017i
−1.80334 + 0.309017i
2.17199 − 0.809017i
2.17199 + 0.809017i
−1.80334 − 0.309017i
−1.16141 + 0.809017i
2.32349 − 0.309017i
3.42137 − 0.309017i
0.543374 + 0.809017i
−0.705457 − 0.309017i
−2.79002 + 0.809017i
− 1.00000i 1.00000i −1.00000 0 1.00000 − 4.63137i 1.00000i −1.00000 0
1249.2 − 1.00000i 1.00000i −1.00000 0 1.00000 − 3.23143i 1.00000i −1.00000 0
1249.3 − 1.00000i 1.00000i −1.00000 0 1.00000 − 2.70913i 1.00000i −1.00000 0
1249.4 − 1.00000i 1.00000i −1.00000 0 1.00000 − 2.61995i 1.00000i −1.00000 0
1249.5 − 1.00000i 1.00000i −1.00000 0 1.00000 0.329315i 1.00000i −1.00000 0
1249.6 − 1.00000i 1.00000i −1.00000 0 1.00000 0.533559i 1.00000i −1.00000 0
1249.7 − 1.00000i 1.00000i −1.00000 0 1.00000 3.52206i 1.00000i −1.00000 0
1249.8 − 1.00000i 1.00000i −1.00000 0 1.00000 4.80694i 1.00000i −1.00000 0
1249.9 1.00000i − 1.00000i −1.00000 0 1.00000 − 4.80694i − 1.00000i −1.00000 0
1249.10 1.00000i − 1.00000i −1.00000 0 1.00000 − 3.52206i − 1.00000i −1.00000 0
1249.11 1.00000i − 1.00000i −1.00000 0 1.00000 − 0.533559i − 1.00000i −1.00000 0
1249.12 1.00000i − 1.00000i −1.00000 0 1.00000 − 0.329315i − 1.00000i −1.00000 0
1249.13 1.00000i − 1.00000i −1.00000 0 1.00000 2.61995i − 1.00000i −1.00000 0
1249.14 1.00000i − 1.00000i −1.00000 0 1.00000 2.70913i − 1.00000i −1.00000 0
1249.15 1.00000i − 1.00000i −1.00000 0 1.00000 3.23143i − 1.00000i −1.00000 0
1249.16 1.00000i − 1.00000i −1.00000 0 1.00000 4.63137i − 1.00000i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.16
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 3750.2.c.k 16
5.b even 2 1 inner 3750.2.c.k 16
5.c odd 4 1 3750.2.a.u 8
5.c odd 4 1 3750.2.a.v 8
25.d even 5 1 150.2.h.b ✓ 16
25.d even 5 1 750.2.h.d 16
25.e even 10 1 150.2.h.b ✓ 16
25.e even 10 1 750.2.h.d 16
25.f odd 20 2 750.2.g.f 16
25.f odd 20 2 750.2.g.g 16
75.h odd 10 1 450.2.l.c 16
75.j odd 10 1 450.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b ✓ 16 25.d even 5 1
150.2.h.b ✓ 16 25.e even 10 1
450.2.l.c 16 75.h odd 10 1
450.2.l.c 16 75.j odd 10 1
750.2.g.f 16 25.f odd 20 2
750.2.g.g 16 25.f odd 20 2
750.2.h.d 16 25.d even 5 1
750.2.h.d 16 25.e even 10 1
3750.2.a.u 8 5.c odd 4 1
3750.2.a.v 8 5.c odd 4 1
3750.2.c.k 16 1.a even 1 1 trivial
3750.2.c.k 16 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 82 T_{7}^{14} + 2683 T_{7}^{12} + 44874 T_{7}^{10} + 407105 T_{7}^{8} + 1927704 T_{7}^{6} + \cdots + 99856 \) acting on \(S_{2}^{\mathrm{new}}(3750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 82 T^{14} + \cdots + 99856 \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} + \cdots - 724)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 118 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{16} + 162 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + \cdots - 1600)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 20533743616 \) Copy content Toggle raw display
$29$ \( (T^{8} + 10 T^{7} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 16 T^{7} + \cdots - 14324)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 142 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$41$ \( (T^{8} - 6 T^{7} + \cdots + 279376)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 15083769856 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 172199901270016 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 111534721 \) Copy content Toggle raw display
$59$ \( (T^{8} - 225 T^{6} + \cdots - 40000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 16 T^{7} + \cdots - 3184)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 201983672713216 \) Copy content Toggle raw display
$71$ \( (T^{8} - 6 T^{7} + \cdots + 1843456)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 4778526048256 \) Copy content Toggle raw display
$79$ \( (T^{8} + 10 T^{7} + \cdots - 10529600)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 25573127056 \) Copy content Toggle raw display
$89$ \( (T^{8} + 20 T^{7} + \cdots - 190000)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 91700905971481 \) Copy content Toggle raw display
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