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Properties

Label 1452.3.f.c
Level $1452$
Weight $3$
Character orbit 1452.f
Analytic conductor $39.564$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1452,3,Mod(241,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.241"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1452.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.5641343851\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 122x^{6} - 84x^{5} + 5617x^{4} + 5796x^{3} - 106140x^{2} - 56448x + 902232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + (\beta_{3} + 1) q^{5} + (\beta_{6} + 3 \beta_{4} - \beta_{2}) q^{7} + 3 q^{9} + ( - \beta_{6} - 3 \beta_{4} + \cdots + \beta_1) q^{13} + ( - \beta_{7} + \beta_{5} + \beta_{3} + 1) q^{15}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{5} + \cdots + 29) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 24 q^{9} + 12 q^{15} + 20 q^{23} + 68 q^{25} + 96 q^{31} - 24 q^{37} + 24 q^{45} + 248 q^{47} - 24 q^{49} + 76 q^{53} + 28 q^{59} + 44 q^{67} - 36 q^{69} - 372 q^{71} + 192 q^{75} + 72 q^{81}+ \cdots + 244 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 122x^{6} - 84x^{5} + 5617x^{4} + 5796x^{3} - 106140x^{2} - 56448x + 902232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1247 \nu^{7} + 1698660 \nu^{6} - 5007887 \nu^{5} - 125025246 \nu^{4} + 155034982 \nu^{3} + \cdots - 44389436328 ) / 3386282328 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5099 \nu^{7} - 11837 \nu^{6} - 499567 \nu^{5} + 660961 \nu^{4} + 18791390 \nu^{3} + \cdots + 342590304 ) / 188126796 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5099 \nu^{7} + 11837 \nu^{6} + 499567 \nu^{5} - 660961 \nu^{4} - 18791390 \nu^{3} + \cdots - 342590304 ) / 188126796 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 120037 \nu^{7} + 293472 \nu^{6} + 15622667 \nu^{5} - 40371930 \nu^{4} - 616795462 \nu^{3} + \cdots - 25480030968 ) / 3386282328 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 727 \nu^{7} - 4893 \nu^{6} - 61949 \nu^{5} + 374703 \nu^{4} + 2005228 \nu^{3} - 9764328 \nu^{2} + \cdots + 92736000 ) / 7040088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107401 \nu^{7} + 512340 \nu^{6} + 12540770 \nu^{5} - 60451626 \nu^{4} - 497676709 \nu^{3} + \cdots - 33447159540 ) / 564380388 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1929895 \nu^{7} - 11851269 \nu^{6} - 160376105 \nu^{5} + 895138755 \nu^{4} + \cdots + 255591770832 ) / 3386282328 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + 7\beta_{5} + 2\beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{7} - 6\beta_{6} + 31\beta_{5} + 39\beta_{4} + 35\beta_{3} + 73\beta_{2} + 3\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -53\beta_{7} - 56\beta_{6} + 371\beta_{5} + 348\beta_{4} + 95\beta_{3} - 140\beta_{2} + 156\beta _1 + 886 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -357\beta_{7} - 610\beta_{6} + 1689\beta_{5} + 4175\beta_{4} + 1093\beta_{3} + 3349\beta_{2} + 515\beta _1 + 2604 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1815 \beta_{7} - 5404 \beta_{6} + 11949 \beta_{5} + 29566 \beta_{4} + 4713 \beta_{3} - 1790 \beta_{2} + \cdots + 21350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 9443 \beta_{7} - 42938 \beta_{6} + 51323 \beta_{5} + 294679 \beta_{4} + 20343 \beta_{3} + \cdots + 107072 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).

\(n\) \(485\) \(727\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
241.1
−5.29611 + 1.41421i
−5.29611 − 1.41421i
3.56406 − 1.41421i
3.56406 + 1.41421i
−5.75775 − 1.41421i
−5.75775 + 1.41421i
7.48980 + 1.41421i
7.48980 − 1.41421i
0 −1.73205 0 −4.29611 0 − 11.2666i 0 3.00000 0
241.2 0 −1.73205 0 −4.29611 0 11.2666i 0 3.00000 0
241.3 0 −1.73205 0 4.56406 0 − 5.84996i 0 3.00000 0
241.4 0 −1.73205 0 4.56406 0 5.84996i 0 3.00000 0
241.5 0 1.73205 0 −4.75775 0 − 6.84413i 0 3.00000 0
241.6 0 1.73205 0 −4.75775 0 6.84413i 0 3.00000 0
241.7 0 1.73205 0 8.48980 0 − 0.0133011i 0 3.00000 0
241.8 0 1.73205 0 8.48980 0 0.0133011i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 241.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1452.3.f.c ✓ 8
3.b odd 2 1 4356.3.f.h 8
11.b odd 2 1 inner 1452.3.f.c ✓ 8
33.d even 2 1 4356.3.f.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1452.3.f.c ✓ 8 1.a even 1 1 trivial
1452.3.f.c ✓ 8 11.b odd 2 1 inner
4356.3.f.h 8 3.b odd 2 1
4356.3.f.h 8 33.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4T_{5}^{3} - 59T_{5}^{2} + 84T_{5} + 792 \) acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{3} + \cdots + 792)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 208 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 416 T^{6} + \cdots + 30492484 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 17915287104 \) Copy content Toggle raw display
$19$ \( T^{8} + 1796 T^{6} + \cdots + 47997184 \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + \cdots + 404568)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 34199844624 \) Copy content Toggle raw display
$31$ \( (T^{4} - 48 T^{3} + \cdots - 1296108)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} + \cdots + 49221)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 217980669456 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 2776742318736 \) Copy content Toggle raw display
$47$ \( (T^{4} - 124 T^{3} + \cdots + 264528)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 38 T^{3} + \cdots - 6076908)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 14 T^{3} + \cdots + 24555672)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 8820436686084 \) Copy content Toggle raw display
$67$ \( (T^{4} - 22 T^{3} + \cdots + 129006)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 186 T^{3} + \cdots - 82623024)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20638049439744 \) Copy content Toggle raw display
$89$ \( (T^{4} - 76 T^{3} + \cdots - 38609568)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 122 T^{3} + \cdots + 3120549)^{2} \) Copy content Toggle raw display
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