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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [745,2,Mod(1,745)] 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("745.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(745, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 745 = 5 \cdot 149 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 745.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.94885495059\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 13 x^{12} + 66 x^{11} + 41 x^{10} - 393 x^{9} + 83 x^{8} + 1012 x^{7} - 577 x^{6} + \cdots + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{5} + 1) q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + ( - \beta_{11} + \beta_{6} + \cdots + \beta_1) q^{6} + ( - \beta_{10} + 1) q^{7} + (\beta_{3} + \beta_1) q^{8} + (\beta_{12} + \beta_{8} - \beta_{6} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{13} - 4 \beta_{12} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 4 q^{2} + 10 q^{3} + 14 q^{4} + 14 q^{5} - q^{6} + 11 q^{7} + 6 q^{8} + 18 q^{9} + 4 q^{10} + 11 q^{11} + 10 q^{12} + 5 q^{13} + 3 q^{14} + 10 q^{15} + 10 q^{16} + 5 q^{17} + 5 q^{18} - 5 q^{19}+ \cdots - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 13 x^{12} + 66 x^{11} + 41 x^{10} - 393 x^{9} + 83 x^{8} + 1012 x^{7} - 577 x^{6} + \cdots + 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 119 \nu^{13} - 175 \nu^{12} + 2794 \nu^{11} + 2592 \nu^{10} - 24571 \nu^{9} - 11520 \nu^{8} + \cdots + 5069 ) / 1828 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 75 \nu^{13} - 51 \nu^{12} - 1254 \nu^{11} + 348 \nu^{10} + 7667 \nu^{9} + 2414 \nu^{8} - 21699 \nu^{7} + \cdots + 3979 ) / 914 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 118 \nu^{13} - 1305 \nu^{12} + 440 \nu^{11} + 20034 \nu^{10} - 25460 \nu^{9} - 104121 \nu^{8} + \cdots - 1789 ) / 1828 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77 \nu^{13} - 290 \nu^{12} - 1324 \nu^{11} + 5366 \nu^{10} + 8103 \nu^{9} - 37305 \nu^{8} + \cdots - 4866 ) / 914 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 201 \nu^{13} + 484 \nu^{12} + 2922 \nu^{11} - 7550 \nu^{10} - 13199 \nu^{9} + 41223 \nu^{8} + \cdots - 7940 ) / 1828 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 222 \nu^{13} + 23 \nu^{12} + 5028 \nu^{11} - 1286 \nu^{10} - 42912 \nu^{9} + 17039 \nu^{8} + \cdots + 159 ) / 1828 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 345 \nu^{13} - 588 \nu^{12} + 7962 \nu^{11} + 9550 \nu^{10} - 68355 \nu^{9} - 53057 \nu^{8} + \cdots - 1760 ) / 1828 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 466 \nu^{13} - 1761 \nu^{12} - 6256 \nu^{11} + 28522 \nu^{10} + 22984 \nu^{9} - 164797 \nu^{8} + \cdots - 6213 ) / 1828 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 467 \nu^{13} + 281 \nu^{12} + 9490 \nu^{11} - 5896 \nu^{10} - 73015 \nu^{9} + 49156 \nu^{8} + \cdots + 16805 ) / 1828 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 555 \nu^{13} + 1200 \nu^{12} + 8914 \nu^{11} - 19210 \nu^{10} - 50155 \nu^{9} + 110005 \nu^{8} + \cdots + 8852 ) / 914 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + \beta_{11} - \beta_{6} - \beta_{4} + \beta_{3} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{12} + 12 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} - 12 \beta_{6} + \beta_{5} - 10 \beta_{4} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2 \beta_{13} + 10 \beta_{12} + 15 \beta_{11} + 14 \beta_{9} + 12 \beta_{8} + 12 \beta_{7} - 13 \beta_{6} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{13} + 76 \beta_{12} + 106 \beta_{11} + 2 \beta_{10} + 31 \beta_{9} + 26 \beta_{8} + 5 \beta_{7} + \cdots + 416 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 31 \beta_{13} + 74 \beta_{12} + 155 \beta_{11} + 4 \beta_{10} + 139 \beta_{9} + 105 \beta_{8} + \cdots + 171 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 35 \beta_{13} + 524 \beta_{12} + 838 \beta_{11} + 35 \beta_{10} + 331 \beta_{9} + 242 \beta_{8} + \cdots + 2440 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 326 \beta_{13} + 499 \beta_{12} + 1381 \beta_{11} + 70 \beta_{10} + 1210 \beta_{9} + 820 \beta_{8} + \cdots + 1550 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 408 \beta_{13} + 3464 \beta_{12} + 6299 \beta_{11} + 396 \beta_{10} + 3022 \beta_{9} + 1985 \beta_{8} + \cdots + 14802 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2911 \beta_{13} + 3283 \beta_{12} + 11400 \beta_{11} + 804 \beta_{10} + 9849 \beta_{9} + 6094 \beta_{8} + \cdots + 12906 \) 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
−2.42473
−2.26868
−1.89798
−0.969203
−0.919145
−0.196294
0.300609
0.520823
1.05463
1.54196
1.93569
2.31963
2.33365
2.66903
−2.42473 3.19350 3.87934 1.00000 −7.74338 0.643383 −4.55689 7.19842 −2.42473
1.2 −2.26868 0.577918 3.14689 1.00000 −1.31111 0.691095 −2.60192 −2.66601 −2.26868
1.3 −1.89798 −1.28055 1.60231 1.00000 2.43045 2.68302 0.754802 −1.36020 −1.89798
1.4 −0.969203 −0.661708 −1.06065 1.00000 0.641329 −3.11113 2.96639 −2.56214 −0.969203
1.5 −0.919145 2.30734 −1.15517 1.00000 −2.12078 2.90743 2.90006 2.32383 −0.919145
1.6 −0.196294 1.23819 −1.96147 1.00000 −0.243049 −0.0991844 0.777611 −1.46689 −0.196294
1.7 0.300609 −2.59163 −1.90963 1.00000 −0.779069 2.74620 −1.17527 3.71656 0.300609
1.8 0.520823 2.75352 −1.72874 1.00000 1.43410 −3.40756 −1.94202 4.58188 0.520823
1.9 1.05463 3.16770 −0.887749 1.00000 3.34076 3.61098 −3.04552 7.03432 1.05463
1.10 1.54196 −1.92842 0.377645 1.00000 −2.97355 2.19042 −2.50161 0.718804 1.54196
1.11 1.93569 1.63442 1.74689 1.00000 3.16373 0.469425 −0.489946 −0.328662 1.93569
1.12 2.31963 0.582505 3.38067 1.00000 1.35119 4.98506 3.20263 −2.66069 2.31963
1.13 2.33365 2.62063 3.44593 1.00000 6.11563 −2.92562 3.37430 3.86768 2.33365
1.14 2.66903 −1.61341 5.12374 1.00000 −4.30625 −0.383513 8.33738 −0.396900 2.66903
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(149\) \( +1 \)

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 745.2.a.e ✓ 14
3.b odd 2 1 6705.2.a.bc 14
5.b even 2 1 3725.2.a.g 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
745.2.a.e ✓ 14 1.a even 1 1 trivial
3725.2.a.g 14 5.b even 2 1
6705.2.a.bc 14 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} - 4 T_{2}^{13} - 13 T_{2}^{12} + 66 T_{2}^{11} + 41 T_{2}^{10} - 393 T_{2}^{9} + 83 T_{2}^{8} + \cdots + 13 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(745))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 4 T^{13} + \cdots + 13 \) Copy content Toggle raw display
$3$ \( T^{14} - 10 T^{13} + \cdots - 784 \) Copy content Toggle raw display
$5$ \( (T - 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} - 11 T^{13} + \cdots - 208 \) Copy content Toggle raw display
$11$ \( T^{14} - 11 T^{13} + \cdots - 196 \) Copy content Toggle raw display
$13$ \( T^{14} - 5 T^{13} + \cdots + 24784 \) Copy content Toggle raw display
$17$ \( T^{14} - 5 T^{13} + \cdots - 208816 \) Copy content Toggle raw display
$19$ \( T^{14} + 5 T^{13} + \cdots - 1510720 \) Copy content Toggle raw display
$23$ \( T^{14} - 31 T^{13} + \cdots + 1266224 \) Copy content Toggle raw display
$29$ \( T^{14} - 17 T^{13} + \cdots + 10633280 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 324646912 \) Copy content Toggle raw display
$37$ \( T^{14} - 13 T^{13} + \cdots - 40751824 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 6301267184 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 86312011328 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 16401641408 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 9886270448 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 20867821060 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 214826329088 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 4726150290644 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 32226469348 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 15062780608 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 1092750580 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 2287140880496 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 175758320 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 2836506248192 \) Copy content Toggle raw display
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