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Properties

Label 952.2.q.e
Level $952$
Weight $2$
Character orbit 952.q
Analytic conductor $7.602$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [952,2,Mod(137,952)] 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("952.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(952, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 952 = 2^{3} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 952.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,3] 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: no
Analytic conductor: \(7.60175827243\)
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: \( x^{14} - 3 x^{13} + 17 x^{12} - 18 x^{11} + 102 x^{10} - 59 x^{9} + 462 x^{8} - 28 x^{7} + 1148 x^{6} + \cdots + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{13} q^{5} + (\beta_{10} - \beta_{7} + \beta_{4}) q^{7} + ( - \beta_{10} - \beta_{5} + \cdots + \beta_1) q^{9} + ( - \beta_{9} - \beta_{6} + \beta_{2} + \cdots + 1) q^{11} + ( - \beta_{11} - \beta_{8} - 2 \beta_{2} + 1) q^{13}+ \cdots + ( - \beta_{12} + \beta_{10} - \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 3 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} + 12 q^{11} + 4 q^{13} - 12 q^{15} - 7 q^{17} + 3 q^{19} + 18 q^{21} + 18 q^{23} - 15 q^{25} - 36 q^{27} + 10 q^{29} + 10 q^{31} - 21 q^{33} + 19 q^{35} - 11 q^{37}+ \cdots - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 3 x^{13} + 17 x^{12} - 18 x^{11} + 102 x^{10} - 59 x^{9} + 462 x^{8} - 28 x^{7} + 1148 x^{6} + \cdots + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13383848435 \nu^{13} + 372134565145 \nu^{12} - 1410959577022 \nu^{11} + \cdots - 134661973506033 ) / 128077844913159 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 68249653197 \nu^{13} + 300431515169 \nu^{12} - 1424824094530 \nu^{11} + \cdots + 66765103875070 ) / 128077844913159 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4956982022081 \nu^{13} - 38967737412845 \nu^{12} + 142273759506660 \nu^{11} + \cdots - 17\!\cdots\!64 ) / 73\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 163932208775 \nu^{13} - 565011505350 \nu^{12} + 2856580447814 \nu^{11} + \cdots - 426360513810430 ) / 128077844913159 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3513952835530 \nu^{13} + 9245115095847 \nu^{12} - 54028999415799 \nu^{11} + \cdots + 10\!\cdots\!38 ) / 24\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13380138014665 \nu^{13} + 29778268983605 \nu^{12} - 141080000860020 \nu^{11} + \cdots - 30\!\cdots\!00 ) / 73\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 242299645231 \nu^{13} - 2943336831557 \nu^{12} + 13214585819510 \nu^{11} + \cdots - 37382559803388 ) / 128077844913159 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7078514128868 \nu^{13} - 14312843662206 \nu^{12} + 93209533722057 \nu^{11} + \cdots - 7634131528826 ) / 24\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8723889950608 \nu^{13} - 22708325474102 \nu^{12} + 134883627535001 \nu^{11} + \cdots + 36\!\cdots\!26 ) / 24\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 686167420275 \nu^{13} - 4149823599977 \nu^{12} + 19607112866878 \nu^{11} + \cdots - 271344537108367 ) / 128077844913159 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17657632823322 \nu^{13} - 38077501399189 \nu^{12} + 199325376388787 \nu^{11} + \cdots + 81\!\cdots\!41 ) / 24\!\cdots\!21 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 26700864673484 \nu^{13} + 64574228981655 \nu^{12} - 348222941195961 \nu^{11} + \cdots - 50\!\cdots\!74 ) / 24\!\cdots\!21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} - 3\beta_{6} - \beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{5} - 5\beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} + 8\beta_{10} - 3\beta_{9} + \beta_{7} + 16\beta_{6} - \beta_{5} + \beta_{4} + 3\beta_{2} - 9\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{13} - \beta_{12} - 3 \beta_{11} + 24 \beta_{10} - 13 \beta_{9} - 3 \beta_{7} + 33 \beta_{6} + \cdots - 31 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 15 \beta_{12} - 10 \beta_{11} + 15 \beta_{10} - 4 \beta_{8} - 30 \beta_{7} + 83 \beta_{5} + \cdots + 114 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20 \beta_{13} - 25 \beta_{12} + 23 \beta_{11} - 195 \beta_{10} + 133 \beta_{9} - 20 \beta_{8} + \cdots + 299 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 71 \beta_{13} + 88 \beta_{12} + 176 \beta_{11} - 774 \beta_{10} + 404 \beta_{9} + 176 \beta_{7} + \cdots + 588 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 563 \beta_{12} + 245 \beta_{11} - 563 \beta_{10} + 264 \beta_{8} + 1126 \beta_{7} - 2329 \beta_{5} + \cdots - 2625 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 881 \beta_{13} + 774 \beta_{12} - 1091 \beta_{11} + 5334 \beta_{10} - 3911 \beta_{9} + 881 \beta_{8} + \cdots - 7874 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2956 \beta_{13} - 3627 \beta_{12} - 5883 \beta_{11} + 21804 \beta_{10} - 12094 \beta_{9} + \cdots - 14001 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18677 \beta_{12} - 6882 \beta_{11} + 18677 \beta_{10} - 9510 \beta_{8} - 37354 \beta_{7} + \cdots + 68790 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 30472 \beta_{13} - 20433 \beta_{12} + 37697 \beta_{11} - 143587 \beta_{10} + 112568 \beta_{9} + \cdots + 203904 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/952\mathbb{Z}\right)^\times\).

\(n\) \(239\) \(409\) \(477\) \(785\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1
−0.882673 + 1.52883i
−0.733007 + 1.26961i
−0.658657 + 1.14083i
0.237215 − 0.410869i
0.778178 − 1.34784i
1.25290 − 2.17009i
1.50604 − 2.60854i
−0.882673 − 1.52883i
−0.733007 − 1.26961i
−0.658657 − 1.14083i
0.237215 + 0.410869i
0.778178 + 1.34784i
1.25290 + 2.17009i
1.50604 + 2.60854i
0 −0.882673 + 1.52883i 0 −1.23147 − 2.13296i 0 0.310485 + 2.62747i 0 −0.0582232 − 0.100846i 0
137.2 0 −0.733007 + 1.26961i 0 1.99695 + 3.45882i 0 −2.63946 − 0.182280i 0 0.425400 + 0.736815i 0
137.3 0 −0.658657 + 1.14083i 0 −0.533302 − 0.923707i 0 −0.740531 − 2.54000i 0 0.632341 + 1.09525i 0
137.4 0 0.237215 − 0.410869i 0 −0.817607 − 1.41614i 0 2.62656 − 0.318109i 0 1.38746 + 2.40315i 0
137.5 0 0.778178 − 1.34784i 0 1.56126 + 2.70418i 0 2.52616 − 0.786442i 0 0.288879 + 0.500353i 0
137.6 0 1.25290 − 2.17009i 0 −1.89712 − 3.28590i 0 −2.54429 + 0.725680i 0 −1.63954 − 2.83977i 0
137.7 0 1.50604 − 2.60854i 0 −0.0787158 − 0.136340i 0 1.46107 + 2.20573i 0 −3.03631 − 5.25905i 0
681.1 0 −0.882673 − 1.52883i 0 −1.23147 + 2.13296i 0 0.310485 − 2.62747i 0 −0.0582232 + 0.100846i 0
681.2 0 −0.733007 − 1.26961i 0 1.99695 − 3.45882i 0 −2.63946 + 0.182280i 0 0.425400 − 0.736815i 0
681.3 0 −0.658657 − 1.14083i 0 −0.533302 + 0.923707i 0 −0.740531 + 2.54000i 0 0.632341 − 1.09525i 0
681.4 0 0.237215 + 0.410869i 0 −0.817607 + 1.41614i 0 2.62656 + 0.318109i 0 1.38746 − 2.40315i 0
681.5 0 0.778178 + 1.34784i 0 1.56126 − 2.70418i 0 2.52616 + 0.786442i 0 0.288879 − 0.500353i 0
681.6 0 1.25290 + 2.17009i 0 −1.89712 + 3.28590i 0 −2.54429 − 0.725680i 0 −1.63954 + 2.83977i 0
681.7 0 1.50604 + 2.60854i 0 −0.0787158 + 0.136340i 0 1.46107 − 2.20573i 0 −3.03631 + 5.25905i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 952.2.q.e ✓ 14
7.c even 3 1 inner 952.2.q.e ✓ 14
7.c even 3 1 6664.2.a.v 7
7.d odd 6 1 6664.2.a.y 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
952.2.q.e ✓ 14 1.a even 1 1 trivial
952.2.q.e ✓ 14 7.c even 3 1 inner
6664.2.a.v 7 7.c even 3 1
6664.2.a.y 7 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 3 T_{3}^{13} + 17 T_{3}^{12} - 18 T_{3}^{11} + 102 T_{3}^{10} - 59 T_{3}^{9} + 462 T_{3}^{8} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(952, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 3 T^{13} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{14} - 2 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} - 12 T^{13} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{7} - 2 T^{6} + \cdots - 5123)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 377913600 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 117679104 \) Copy content Toggle raw display
$29$ \( (T^{7} - 5 T^{6} + \cdots + 14368)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 1680672016 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 671846400 \) Copy content Toggle raw display
$41$ \( (T^{7} - 161 T^{5} + \cdots + 366240)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + 15 T^{6} + \cdots - 72752)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 898977044736 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 6726132169 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 149035264 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 25743560704 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 973205926144 \) Copy content Toggle raw display
$71$ \( (T^{7} + 27 T^{6} + \cdots + 2411)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 574465024 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 15587548883449 \) Copy content Toggle raw display
$83$ \( (T^{7} + 12 T^{6} + \cdots - 56768)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + 32 T^{13} + \cdots + 200704 \) Copy content Toggle raw display
$97$ \( (T^{7} + 39 T^{6} + \cdots + 14288)^{2} \) Copy content Toggle raw display
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