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Properties

Label 1480.4.a.b
Level $1480$
Weight $4$
Character orbit 1480.a
Self dual yes
Analytic conductor $87.323$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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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] = [1480,4,Mod(1,1480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1480.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1480 = 2^{3} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1480.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(87.3228268085\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 171 x^{10} + 145 x^{9} + 10638 x^{8} - 6560 x^{7} - 290008 x^{6} + 96492 x^{5} + \cdots - 1271192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 5 q^{5} + (\beta_{5} + 1) q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{4} - 1) q^{11} + (\beta_{11} - \beta_{6} - \beta_{4} + \beta_1) q^{13} + 5 \beta_1 q^{15} + ( - \beta_{11} - \beta_{9} - \beta_{7} + \cdots - 6) q^{17}+ \cdots + (10 \beta_{11} - 2 \beta_{10} + \cdots - 150) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} - 60 q^{5} + 10 q^{7} + 19 q^{9} - 12 q^{11} + 8 q^{13} + 5 q^{15} - 61 q^{17} + 146 q^{19} + 75 q^{21} + 14 q^{23} + 300 q^{25} - 25 q^{27} - 25 q^{29} - 197 q^{31} + 89 q^{33} - 50 q^{35}+ \cdots - 1815 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 171 x^{10} + 145 x^{9} + 10638 x^{8} - 6560 x^{7} - 290008 x^{6} + 96492 x^{5} + \cdots - 1271192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 29 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 131648768569 \nu^{11} - 1062622194891 \nu^{10} - 22714674426471 \nu^{9} + \cdots + 11\!\cdots\!84 ) / 13\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 80958559597 \nu^{11} + 4847556603 \nu^{10} - 12722688808053 \nu^{9} + 646696777429 \nu^{8} + \cdots - 22\!\cdots\!32 ) / 31\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1186647266413 \nu^{11} + 2294183906847 \nu^{10} + 201038488662717 \nu^{9} + \cdots + 29\!\cdots\!16 ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1506940368221 \nu^{11} + 1079009417043 \nu^{10} + 254556792171873 \nu^{9} + \cdots + 64\!\cdots\!96 ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 984738909841 \nu^{11} + 806673668724 \nu^{10} + 162180718168590 \nu^{9} + \cdots + 88\!\cdots\!30 ) / 20\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1002220769099 \nu^{11} - 228661805565 \nu^{10} - 167593817090133 \nu^{9} + \cdots - 25\!\cdots\!84 ) / 20\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 822701273535 \nu^{11} - 1442462909863 \nu^{10} - 142409273511535 \nu^{9} + \cdots - 16\!\cdots\!56 ) / 13\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4454510405725 \nu^{11} - 5120200226133 \nu^{10} - 741803581078749 \nu^{9} + \cdots - 59\!\cdots\!68 ) / 40\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2264877700040 \nu^{11} + 4631700847179 \nu^{10} + 391308218023800 \nu^{9} + \cdots + 61\!\cdots\!94 ) / 20\!\cdots\!22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 48\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{11} - 2 \beta_{10} - 3 \beta_{8} - 8 \beta_{7} + 4 \beta_{6} + 12 \beta_{5} + 2 \beta_{4} + \cdots + 1361 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17 \beta_{11} + 92 \beta_{10} - 85 \beta_{9} - 13 \beta_{8} + 76 \beta_{7} - 189 \beta_{6} + 118 \beta_{5} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 362 \beta_{11} - 135 \beta_{10} + 74 \beta_{9} - 348 \beta_{8} - 660 \beta_{7} + 522 \beta_{6} + \cdots + 70708 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1653 \beta_{11} + 6696 \beta_{10} - 6633 \beta_{9} - 1343 \beta_{8} + 4675 \beta_{7} - 14272 \beta_{6} + \cdots - 23977 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 26433 \beta_{11} - 7027 \beta_{10} + 9622 \beta_{9} - 28531 \beta_{8} - 43172 \beta_{7} + \cdots + 3889876 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 125485 \beta_{11} + 451778 \beta_{10} - 487887 \beta_{9} - 103661 \beta_{8} + 272799 \beta_{7} + \cdots - 3039599 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1797455 \beta_{11} - 358935 \beta_{10} + 921152 \beta_{9} - 2056323 \beta_{8} - 2660454 \beta_{7} + \cdots + 222235380 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8855151 \beta_{11} + 29501102 \beta_{10} - 34318925 \beta_{9} - 7125665 \beta_{8} + 15839865 \beta_{7} + \cdots - 287073391 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
7.68384
6.86958
6.76522
4.49937
2.31294
−0.351375
−0.660006
−1.03437
−4.81587
−5.39432
−6.78486
−8.09016
0 −7.68384 0 −5.00000 0 25.3268 0 32.0414 0
1.2 0 −6.86958 0 −5.00000 0 −6.66079 0 20.1912 0
1.3 0 −6.76522 0 −5.00000 0 −4.20097 0 18.7681 0
1.4 0 −4.49937 0 −5.00000 0 −31.5535 0 −6.75565 0
1.5 0 −2.31294 0 −5.00000 0 −16.4158 0 −21.6503 0
1.6 0 0.351375 0 −5.00000 0 21.1138 0 −26.8765 0
1.7 0 0.660006 0 −5.00000 0 3.12170 0 −26.5644 0
1.8 0 1.03437 0 −5.00000 0 27.3624 0 −25.9301 0
1.9 0 4.81587 0 −5.00000 0 −12.2733 0 −3.80744 0
1.10 0 5.39432 0 −5.00000 0 10.1597 0 2.09866 0
1.11 0 6.78486 0 −5.00000 0 −23.2812 0 19.0343 0
1.12 0 8.09016 0 −5.00000 0 17.3013 0 38.4507 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(37\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1480.4.a.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1480.4.a.b ✓ 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + T_{3}^{11} - 171 T_{3}^{10} - 145 T_{3}^{9} + 10638 T_{3}^{8} + 6560 T_{3}^{7} + \cdots - 1271192 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1480))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + T^{11} + \cdots - 1271192 \) Copy content Toggle raw display
$5$ \( (T + 5)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 33250748227500 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 9709488122880 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 85\!\cdots\!60 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 51\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( (T - 37)^{12} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots - 14\!\cdots\!08 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 64\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 49\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 87\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 22\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 68\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 23\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 66\!\cdots\!40 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 23\!\cdots\!40 \) Copy content Toggle raw display
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