Properties

Label 740.2.h.b
Level $740$
Weight $2$
Character orbit 740.h
Analytic conductor $5.909$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(221,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("740.221");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.h (of order \(2\), degree \(1\), 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: \(5.90892974957\)
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: \( x^{12} + 21x^{10} + 146x^{8} + 386x^{6} + 297x^{4} + 77x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 1) q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 1) q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + ( - \beta_{3} - \beta_{2} + 1) q^{9} + (\beta_{8} - \beta_{3}) q^{11} + (\beta_{10} + \beta_{9} - 2 \beta_{6}) q^{13} + (\beta_{6} + \beta_1) q^{15} + (\beta_{10} - \beta_{9}) q^{17} + ( - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{19}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 2 q^{7} + 6 q^{9} - 10 q^{11} + 18 q^{21} - 12 q^{25} - 18 q^{27} - 26 q^{33} - 6 q^{37} - 2 q^{41} - 22 q^{47} + 46 q^{49} + 26 q^{53} + 20 q^{63} - 20 q^{65} - 28 q^{67} - 26 q^{71} - 14 q^{73} + 6 q^{75} - 50 q^{77} + 4 q^{81} + 58 q^{83} - 4 q^{85} + 4 q^{95} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 146x^{8} + 386x^{6} + 297x^{4} + 77x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{10} - 103\nu^{8} - 687\nu^{6} - 1627\nu^{4} - 722\nu^{2} - 26 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{10} - 103\nu^{8} - 687\nu^{6} - 1627\nu^{4} - 728\nu^{2} - 44 ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{10} + 41\nu^{8} + 271\nu^{6} + 629\nu^{4} + 251\nu^{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{11} - 61\nu^{9} - 397\nu^{7} - 887\nu^{5} - 262\nu^{3} + 22\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11\nu^{11} + 226\nu^{9} + 1503\nu^{7} + 3559\nu^{5} + 1640\nu^{3} + 119\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -14\nu^{11} - 286\nu^{9} - 1881\nu^{7} - 4333\nu^{5} - 1667\nu^{3} + \nu ) / 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\nu^{10} + 349\nu^{8} + 2319\nu^{6} + 5491\nu^{4} + 2552\nu^{2} + 182 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\nu^{11} + 389\nu^{9} + 2568\nu^{7} + 5960\nu^{5} + 2389\nu^{3} + 13\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\nu^{11} + 575\nu^{9} + 3822\nu^{7} + 9050\nu^{5} + 4192\nu^{3} + 307\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -14\nu^{10} - 287\nu^{8} - 1901\nu^{6} - 4461\nu^{4} - 1953\nu^{2} - 104 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} - \beta_{9} - \beta_{7} + 3\beta_{6} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{8} + 3\beta_{4} + 10\beta_{3} - 8\beta_{2} + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{10} + 11\beta_{9} + 15\beta_{7} - 38\beta_{6} - 3\beta_{5} + 44\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{11} - 29\beta_{8} - 47\beta_{4} - 96\beta_{3} + 67\beta_{2} - 173 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -157\beta_{10} - 114\beta_{9} - 176\beta_{7} + 411\beta_{6} + 47\beta_{5} - 359\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 176\beta_{11} + 333\beta_{8} + 556\beta_{4} + 923\beta_{3} - 602\beta_{2} + 1543 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1636\beta_{10} + 1158\beta_{9} + 1890\beta_{7} - 4214\beta_{6} - 556\beta_{5} + 3161\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -1890\beta_{11} - 3526\beta_{8} - 5972\beta_{4} - 8933\beta_{3} + 5653\beta_{2} - 14421 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -16541\beta_{10} - 11625\beta_{9} - 19487\beta_{7} + 42269\beta_{6} + 5972\beta_{5} - 29244\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
221.1
2.17849i
2.17849i
0.702234i
0.702234i
0.261643i
0.261643i
0.687519i
0.687519i
2.31377i
2.31377i
3.14108i
3.14108i
0 −3.17849 0 1.00000i 0 −0.931661 0 7.10280 0
221.2 0 −3.17849 0 1.00000i 0 −0.931661 0 7.10280 0
221.3 0 −1.70223 0 1.00000i 0 2.52593 0 −0.102398 0
221.4 0 −1.70223 0 1.00000i 0 2.52593 0 −0.102398 0
221.5 0 −1.26164 0 1.00000i 0 −4.16052 0 −1.40826 0
221.6 0 −1.26164 0 1.00000i 0 −4.16052 0 −1.40826 0
221.7 0 −0.312481 0 1.00000i 0 0.636191 0 −2.90236 0
221.8 0 −0.312481 0 1.00000i 0 0.636191 0 −2.90236 0
221.9 0 1.31377 0 1.00000i 0 −3.98494 0 −1.27402 0
221.10 0 1.31377 0 1.00000i 0 −3.98494 0 −1.27402 0
221.11 0 2.14108 0 1.00000i 0 4.91501 0 1.58424 0
221.12 0 2.14108 0 1.00000i 0 4.91501 0 1.58424 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 221.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.h.b 12
3.b odd 2 1 6660.2.n.f 12
4.b odd 2 1 2960.2.p.i 12
5.b even 2 1 3700.2.h.g 12
5.c odd 4 1 3700.2.e.e 12
5.c odd 4 1 3700.2.e.f 12
37.b even 2 1 inner 740.2.h.b 12
111.d odd 2 1 6660.2.n.f 12
148.b odd 2 1 2960.2.p.i 12
185.d even 2 1 3700.2.h.g 12
185.h odd 4 1 3700.2.e.e 12
185.h odd 4 1 3700.2.e.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.h.b 12 1.a even 1 1 trivial
740.2.h.b 12 37.b even 2 1 inner
2960.2.p.i 12 4.b odd 2 1
2960.2.p.i 12 148.b odd 2 1
3700.2.e.e 12 5.c odd 4 1
3700.2.e.e 12 185.h odd 4 1
3700.2.e.f 12 5.c odd 4 1
3700.2.e.f 12 185.h odd 4 1
3700.2.h.g 12 5.b even 2 1
3700.2.h.g 12 185.d even 2 1
6660.2.n.f 12 3.b odd 2 1
6660.2.n.f 12 111.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3T_{3}^{5} - 6T_{3}^{4} - 18T_{3}^{3} + 4T_{3}^{2} + 22T_{3} + 6 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 3 T^{5} - 6 T^{4} + \cdots + 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} - 32 T^{4} + \cdots - 122)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 5 T^{5} + \cdots + 192)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 100 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$17$ \( T^{12} + 132 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$19$ \( T^{12} + 124 T^{10} + \cdots + 3111696 \) Copy content Toggle raw display
$23$ \( T^{12} + 148 T^{10} + \cdots + 1327104 \) Copy content Toggle raw display
$29$ \( T^{12} + 224 T^{10} + \cdots + 21233664 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 118200384 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2565726409 \) Copy content Toggle raw display
$41$ \( (T^{6} + T^{5} + \cdots + 11472)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 376 T^{10} + \cdots + 69755904 \) Copy content Toggle raw display
$47$ \( (T^{6} + 11 T^{5} + \cdots + 125406)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 13 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 118200384 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 5957369856 \) Copy content Toggle raw display
$67$ \( (T^{6} + 14 T^{5} + \cdots - 527528)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + 13 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 7 T^{5} + \cdots - 12096)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1536953616 \) Copy content Toggle raw display
$83$ \( (T^{6} - 29 T^{5} + \cdots - 39654)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 191102976 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 267545493504 \) Copy content Toggle raw display
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