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Properties

Label 645.2.i.g
Level $645$
Weight $2$
Character orbit 645.i
Analytic conductor $5.150$
Analytic rank $0$
Dimension $14$
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] = [645,2,Mod(436,645)] 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("645.436"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(645, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 645 = 3 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 645.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,2,7,26,-7] 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: \(5.15035093037\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
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: \( x^{14} - x^{13} + 14 x^{12} - 7 x^{11} + 131 x^{10} - 59 x^{9} + 627 x^{8} - 130 x^{7} + 2078 x^{6} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{43}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{9} + 1) q^{3} + ( - \beta_{3} + 2) q^{4} + (\beta_{9} - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{13} - \beta_{12} + \cdots + \beta_1) q^{7} + ( - \beta_{6} + \beta_{5} - \beta_{3} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{13} + \beta_{12} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{2} + 7 q^{3} + 26 q^{4} - 7 q^{5} + q^{6} + q^{7} + 12 q^{8} - 7 q^{9} - q^{10} - 6 q^{11} + 13 q^{12} - 2 q^{13} + 4 q^{14} + 7 q^{15} + 50 q^{16} + 3 q^{17} - q^{18} - 21 q^{19} - 13 q^{20}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 14 x^{12} - 7 x^{11} + 131 x^{10} - 59 x^{9} + 627 x^{8} - 130 x^{7} + 2078 x^{6} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 34152678019 \nu^{13} + 224276026287 \nu^{12} - 872569103874 \nu^{11} + \cdots + 46746819043760 ) / 423701950560204 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47530837067 \nu^{13} + 98607902902 \nu^{12} - 667658304764 \nu^{11} + \cdots + 423565339848128 ) / 105925487640051 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1803172448125 \nu^{13} + 4595878819803 \nu^{12} - 25915925978696 \nu^{11} + \cdots + 39\!\cdots\!72 ) / 25\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1648689210381 \nu^{13} - 5913156913841 \nu^{12} + 31273103615254 \nu^{11} + \cdots - 20\!\cdots\!88 ) / 847403901120408 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1687409126727 \nu^{13} - 4459259874187 \nu^{12} + 27888679014626 \nu^{11} + \cdots - 41\!\cdots\!04 ) / 847403901120408 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2515274326213 \nu^{13} - 4673858702667 \nu^{12} + 38462097282780 \nu^{11} + \cdots - 527173103063696 ) / 847403901120408 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1306224907935 \nu^{13} - 3495868306663 \nu^{12} + 20518990154450 \nu^{11} + \cdots - 23\!\cdots\!52 ) / 423701950560204 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2921676190235 \nu^{13} - 2887523512216 \nu^{12} + 40679190637003 \nu^{11} + \cdots + 30522575571452 ) / 423701950560204 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4853156790880 \nu^{13} - 4729975767348 \nu^{12} + 68482219866263 \nu^{11} + \cdots - 17\!\cdots\!84 ) / 635552925840306 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 27788600761243 \nu^{13} + 24436700454255 \nu^{12} - 387059109165110 \nu^{11} + \cdots - 23\!\cdots\!40 ) / 25\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2874145353168 \nu^{13} + 2788915609314 \nu^{12} - 40011532332239 \nu^{11} + \cdots - 30385964859376 ) / 105925487640051 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14788050684574 \nu^{13} + 15181370929791 \nu^{12} - 207694293386805 \nu^{11} + \cdots + 21\!\cdots\!32 ) / 423701950560204 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + 4\beta_{9} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{3} - 5\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{13} - 8\beta_{12} - \beta_{11} - 24\beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{13} - 11 \beta_{12} - 8 \beta_{11} + \beta_{10} - 12 \beta_{9} + 9 \beta_{7} + 8 \beta_{6} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{13} + 6 \beta_{11} - 3 \beta_{10} - 10 \beta_{8} - 3 \beta_{7} - 15 \beta_{6} + 14 \beta_{5} + \cdots + 152 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 100 \beta_{13} + 99 \beta_{12} + 89 \beta_{11} - 32 \beta_{10} + 116 \beta_{9} - 13 \beta_{8} + \cdots + 161 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 199 \beta_{13} + 433 \beta_{12} + 97 \beta_{11} - 45 \beta_{10} + 989 \beta_{9} - 114 \beta_{7} + \cdots - 989 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 175 \beta_{13} - 350 \beta_{11} + 175 \beta_{10} + 131 \beta_{8} + 175 \beta_{7} + 721 \beta_{6} + \cdots - 1021 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2167 \beta_{13} - 3200 \beta_{12} - 1762 \beta_{11} + 962 \beta_{10} - 6582 \beta_{9} + 687 \beta_{8} + \cdots - 151 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5367 \beta_{13} - 6799 \beta_{12} - 2918 \beta_{11} + 1649 \beta_{10} - 8533 \beta_{9} + 4169 \beta_{7} + \cdots + 8533 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 4496 \beta_{13} + 8992 \beta_{11} - 4496 \beta_{10} - 5420 \beta_{8} - 4496 \beta_{7} - 12891 \beta_{6} + \cdots + 44741 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 56553 \beta_{13} + 54338 \beta_{12} + 50152 \beta_{11} - 28824 \beta_{10} + 69135 \beta_{9} + \cdots + 31949 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/645\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(431\) \(517\)
\(\chi(n)\) \(-\beta_{9}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
436.1
−1.20038 − 2.07912i
−1.17297 − 2.03164i
−0.401020 − 0.694586i
0.0525125 + 0.0909543i
0.630659 + 1.09233i
1.20961 + 2.09511i
1.38158 + 2.39297i
−1.20038 + 2.07912i
−1.17297 + 2.03164i
−0.401020 + 0.694586i
0.0525125 − 0.0909543i
0.630659 − 1.09233i
1.20961 − 2.09511i
1.38158 − 2.39297i
−2.40076 0.500000 − 0.866025i 3.76365 −0.500000 + 0.866025i −1.20038 + 2.07912i −2.61068 − 4.52183i −4.23411 −0.500000 − 0.866025i 1.20038 − 2.07912i
436.2 −2.34593 0.500000 − 0.866025i 3.50340 −0.500000 + 0.866025i −1.17297 + 2.03164i 1.81242 + 3.13921i −3.52687 −0.500000 − 0.866025i 1.17297 − 2.03164i
436.3 −0.802039 0.500000 − 0.866025i −1.35673 −0.500000 + 0.866025i −0.401020 + 0.694586i −0.385006 − 0.666851i 2.69223 −0.500000 − 0.866025i 0.401020 − 0.694586i
436.4 0.105025 0.500000 − 0.866025i −1.98897 −0.500000 + 0.866025i 0.0525125 − 0.0909543i 0.504268 + 0.873418i −0.418941 −0.500000 − 0.866025i −0.0525125 + 0.0909543i
436.5 1.26132 0.500000 − 0.866025i −0.409078 −0.500000 + 0.866025i 0.630659 − 1.09233i 2.10034 + 3.63790i −3.03861 −0.500000 − 0.866025i −0.630659 + 1.09233i
436.6 2.41922 0.500000 − 0.866025i 3.85265 −0.500000 + 0.866025i 1.20961 − 2.09511i 1.39824 + 2.42182i 4.48197 −0.500000 − 0.866025i −1.20961 + 2.09511i
436.7 2.76317 0.500000 − 0.866025i 5.63508 −0.500000 + 0.866025i 1.38158 − 2.39297i −2.31958 − 4.01763i 10.0443 −0.500000 − 0.866025i −1.38158 + 2.39297i
466.1 −2.40076 0.500000 + 0.866025i 3.76365 −0.500000 − 0.866025i −1.20038 − 2.07912i −2.61068 + 4.52183i −4.23411 −0.500000 + 0.866025i 1.20038 + 2.07912i
466.2 −2.34593 0.500000 + 0.866025i 3.50340 −0.500000 − 0.866025i −1.17297 − 2.03164i 1.81242 − 3.13921i −3.52687 −0.500000 + 0.866025i 1.17297 + 2.03164i
466.3 −0.802039 0.500000 + 0.866025i −1.35673 −0.500000 − 0.866025i −0.401020 − 0.694586i −0.385006 + 0.666851i 2.69223 −0.500000 + 0.866025i 0.401020 + 0.694586i
466.4 0.105025 0.500000 + 0.866025i −1.98897 −0.500000 − 0.866025i 0.0525125 + 0.0909543i 0.504268 − 0.873418i −0.418941 −0.500000 + 0.866025i −0.0525125 − 0.0909543i
466.5 1.26132 0.500000 + 0.866025i −0.409078 −0.500000 − 0.866025i 0.630659 + 1.09233i 2.10034 − 3.63790i −3.03861 −0.500000 + 0.866025i −0.630659 − 1.09233i
466.6 2.41922 0.500000 + 0.866025i 3.85265 −0.500000 − 0.866025i 1.20961 + 2.09511i 1.39824 − 2.42182i 4.48197 −0.500000 + 0.866025i −1.20961 − 2.09511i
466.7 2.76317 0.500000 + 0.866025i 5.63508 −0.500000 − 0.866025i 1.38158 + 2.39297i −2.31958 + 4.01763i 10.0443 −0.500000 + 0.866025i −1.38158 − 2.39297i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 436.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 645.2.i.g ✓ 14
43.c even 3 1 inner 645.2.i.g ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
645.2.i.g ✓ 14 1.a even 1 1 trivial
645.2.i.g ✓ 14 43.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(645, [\chi])\):

\( T_{2}^{7} - T_{2}^{6} - 13T_{2}^{5} + 10T_{2}^{4} + 48T_{2}^{3} - 25T_{2}^{2} - 36T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{14} - T_{7}^{13} + 45 T_{7}^{12} - 110 T_{7}^{11} + 1516 T_{7}^{10} - 3591 T_{7}^{9} + \cdots + 641601 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} - T^{6} - 13 T^{5} + \cdots + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} - T^{13} + \cdots + 641601 \) Copy content Toggle raw display
$11$ \( (T^{7} + 3 T^{6} + \cdots + 394)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + 2 T^{13} + \cdots + 44568976 \) Copy content Toggle raw display
$17$ \( T^{14} - 3 T^{13} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{14} + 21 T^{13} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( T^{14} - 9 T^{13} + \cdots + 1871424 \) Copy content Toggle raw display
$29$ \( T^{14} + 8 T^{13} + \cdots + 1478656 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 111260304 \) Copy content Toggle raw display
$37$ \( T^{14} - 7 T^{13} + \cdots + 3444736 \) Copy content Toggle raw display
$41$ \( (T^{7} + 10 T^{6} + \cdots + 27472)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 271818611107 \) Copy content Toggle raw display
$47$ \( (T^{7} + 6 T^{6} + \cdots - 115968)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} - 18 T^{13} + \cdots + 41616 \) Copy content Toggle raw display
$59$ \( (T^{7} + 12 T^{6} + \cdots - 925872)^{2} \) Copy content Toggle raw display
$61$ \( T^{14} + 11 T^{13} + \cdots + 36048016 \) Copy content Toggle raw display
$67$ \( T^{14} + 15 T^{13} + \cdots + 21316689 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 973071765136 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 5960457616 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 956480813496576 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 675584064 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 69898899456 \) Copy content Toggle raw display
$97$ \( (T^{7} + 27 T^{6} + \cdots + 7177008)^{2} \) Copy content Toggle raw display
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