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Properties

Label 555.2.i.f
Level $555$
Weight $2$
Character orbit 555.i
Analytic conductor $4.432$
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] = [555,2,Mod(121,555)] 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("555.121"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(555, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-1,7,-9,7] 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: \(4.43169731218\)
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: \( x^{14} - x^{13} + 12 x^{12} - 7 x^{11} + 97 x^{10} - 56 x^{9} + 387 x^{8} - 181 x^{7} + 1087 x^{6} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 12 x^{12} - 7 x^{11} + 97 x^{10} - 56 x^{9} + 387 x^{8} - 181 x^{7} + 1087 x^{6} + \cdots + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 75317242994 \nu^{13} - 205892465950 \nu^{12} + 1114491784668 \nu^{11} + \cdots - 49499893316456 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 130575222956 \nu^{13} + 210684868740 \nu^{12} - 1527402784656 \nu^{11} + \cdots + 262811887335163 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 246010992014 \nu^{13} + 818777461010 \nu^{12} - 4045056138684 \nu^{11} + \cdots - 15312420752883 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4499990301496 \nu^{13} + 3671500628562 \nu^{12} - 51735066492502 \nu^{11} + \cdots + 414277997530804 ) / 997059337037269 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 805038744485 \nu^{13} + 1130393476534 \nu^{12} - 8998233995766 \nu^{11} + \cdots + 331331701345387 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10872828441396 \nu^{13} + 15333364964766 \nu^{12} + 114514851699928 \nu^{11} + \cdots + 31\!\cdots\!01 ) / 997059337037269 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11182437348348 \nu^{13} - 9131467128522 \nu^{12} + 130143807796807 \nu^{11} + \cdots + 14\!\cdots\!45 ) / 997059337037269 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11411900802457 \nu^{13} - 2184876062106 \nu^{12} + 134250073037686 \nu^{11} + \cdots + 20\!\cdots\!85 ) / 997059337037269 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12063643451972 \nu^{13} + 8696968329546 \nu^{12} - 138403768846290 \nu^{11} + \cdots - 16\!\cdots\!81 ) / 997059337037269 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1512367339492 \nu^{13} - 1696233019536 \nu^{12} - 16106493494511 \nu^{11} + \cdots - 406057884377068 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1907029001519 \nu^{13} + 2885225197241 \nu^{12} - 22490964726834 \nu^{11} + \cdots + 531040577297614 ) / 90641757912479 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2872139774631 \nu^{13} - 4762494074026 \nu^{12} + 34735284317346 \nu^{11} + \cdots - 936591838747367 ) / 90641757912479 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 3\beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 7\beta_{10} - \beta_{8} + \beta_{7} + 15\beta_{5} - \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 9 \beta_{10} + \beta_{9} - 8 \beta_{8} + 2 \beta_{7} + \cdots - 28 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -11\beta_{13} - 12\beta_{12} - \beta_{6} - 12\beta_{4} - 46\beta_{3} - 14\beta_{2} + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{11} + 72\beta_{10} - 12\beta_{9} + 58\beta_{8} - 25\beta_{7} - 51\beta_{5} + 165\beta _1 + 51 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 94 \beta_{13} + 106 \beta_{12} - 94 \beta_{11} + 306 \beta_{10} - 14 \beta_{9} + 108 \beta_{8} + \cdots + 138 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 216\beta_{13} + 230\beta_{12} + 104\beta_{6} + 414\beta_{4} + 554\beta_{3} + 1011\beta_{2} - 480 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 734\beta_{11} - 2069\beta_{10} + 140\beta_{9} - 874\beta_{8} + 838\beta_{7} + 3041\beta_{5} - 1188\beta _1 - 3041 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1748 \beta_{13} - 1888 \beta_{12} + 1748 \beta_{11} - 4167 \beta_{10} + 802 \beta_{9} + \cdots - 6409 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -5493\beta_{13} - 6295\beta_{12} - 1226\beta_{6} - 6717\beta_{4} - 14181\beta_{3} - 9549\beta_{2} + 19371 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 13436 \beta_{11} + 30871 \beta_{10} - 5871 \beta_{9} + 20900 \beta_{8} - 14662 \beta_{7} + \cdots + 31688 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(-\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.33616 − 2.31429i
1.04660 − 1.81277i
0.427269 − 0.740051i
0.296561 − 0.513660i
−0.397854 + 0.689103i
−1.07931 + 1.86942i
−1.12942 + 1.95622i
1.33616 + 2.31429i
1.04660 + 1.81277i
0.427269 + 0.740051i
0.296561 + 0.513660i
−0.397854 − 0.689103i
−1.07931 − 1.86942i
−1.12942 − 1.95622i
−1.33616 + 2.31429i 0.500000 + 0.866025i −2.57064 − 4.45248i 0.500000 + 0.866025i −2.67232 0.483850 + 0.838052i 8.39448 −0.500000 + 0.866025i −2.67232
121.2 −1.04660 + 1.81277i 0.500000 + 0.866025i −1.19075 − 2.06244i 0.500000 + 0.866025i −2.09320 −1.58311 − 2.74203i 0.798546 −0.500000 + 0.866025i −2.09320
121.3 −0.427269 + 0.740051i 0.500000 + 0.866025i 0.634883 + 1.09965i 0.500000 + 0.866025i −0.854538 1.94023 + 3.36057i −2.79414 −0.500000 + 0.866025i −0.854538
121.4 −0.296561 + 0.513660i 0.500000 + 0.866025i 0.824103 + 1.42739i 0.500000 + 0.866025i −0.593123 −0.949485 − 1.64456i −2.16383 −0.500000 + 0.866025i −0.593123
121.5 0.397854 − 0.689103i 0.500000 + 0.866025i 0.683424 + 1.18373i 0.500000 + 0.866025i 0.795708 0.666727 + 1.15480i 2.67903 −0.500000 + 0.866025i 0.795708
121.6 1.07931 − 1.86942i 0.500000 + 0.866025i −1.32983 − 2.30333i 0.500000 + 0.866025i 2.15862 −1.74628 − 3.02465i −1.42395 −0.500000 + 0.866025i 2.15862
121.7 1.12942 − 1.95622i 0.500000 + 0.866025i −1.55120 − 2.68675i 0.500000 + 0.866025i 2.25885 1.68808 + 2.92384i −2.49014 −0.500000 + 0.866025i 2.25885
211.1 −1.33616 − 2.31429i 0.500000 − 0.866025i −2.57064 + 4.45248i 0.500000 − 0.866025i −2.67232 0.483850 − 0.838052i 8.39448 −0.500000 − 0.866025i −2.67232
211.2 −1.04660 − 1.81277i 0.500000 − 0.866025i −1.19075 + 2.06244i 0.500000 − 0.866025i −2.09320 −1.58311 + 2.74203i 0.798546 −0.500000 − 0.866025i −2.09320
211.3 −0.427269 − 0.740051i 0.500000 − 0.866025i 0.634883 − 1.09965i 0.500000 − 0.866025i −0.854538 1.94023 − 3.36057i −2.79414 −0.500000 − 0.866025i −0.854538
211.4 −0.296561 − 0.513660i 0.500000 − 0.866025i 0.824103 − 1.42739i 0.500000 − 0.866025i −0.593123 −0.949485 + 1.64456i −2.16383 −0.500000 − 0.866025i −0.593123
211.5 0.397854 + 0.689103i 0.500000 − 0.866025i 0.683424 − 1.18373i 0.500000 − 0.866025i 0.795708 0.666727 − 1.15480i 2.67903 −0.500000 − 0.866025i 0.795708
211.6 1.07931 + 1.86942i 0.500000 − 0.866025i −1.32983 + 2.30333i 0.500000 − 0.866025i 2.15862 −1.74628 + 3.02465i −1.42395 −0.500000 − 0.866025i 2.15862
211.7 1.12942 + 1.95622i 0.500000 − 0.866025i −1.55120 + 2.68675i 0.500000 − 0.866025i 2.25885 1.68808 − 2.92384i −2.49014 −0.500000 − 0.866025i 2.25885
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 121.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.i.f ✓ 14
37.c even 3 1 inner 555.2.i.f ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.i.f ✓ 14 1.a even 1 1 trivial
555.2.i.f ✓ 14 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} + 12 T_{2}^{12} + 7 T_{2}^{11} + 97 T_{2}^{10} + 56 T_{2}^{9} + 387 T_{2}^{8} + \cdots + 121 \) acting on \(S_{2}^{\mathrm{new}}(555, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + T^{13} + \cdots + 121 \) 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
$7$ \( T^{14} - T^{13} + \cdots + 126025 \) Copy content Toggle raw display
$11$ \( (T^{7} - 6 T^{6} + \cdots - 956)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 1575025 \) Copy content Toggle raw display
$17$ \( T^{14} - 8 T^{13} + \cdots + 8294400 \) Copy content Toggle raw display
$19$ \( T^{14} + 12 T^{13} + \cdots + 15366400 \) Copy content Toggle raw display
$23$ \( (T^{7} + 6 T^{6} + \cdots - 17600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} - 6 T^{6} + \cdots + 53824)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 11 T^{6} + \cdots - 16339)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 94931877133 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10390132624 \) Copy content Toggle raw display
$43$ \( (T^{7} + 5 T^{6} + \cdots - 9601)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + 2 T^{6} + \cdots - 604)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + 12 T^{13} + \cdots + 262144 \) Copy content Toggle raw display
$59$ \( T^{14} - 6 T^{13} + \cdots + 85895824 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 230016160000 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 274222937569 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 349630959616 \) Copy content Toggle raw display
$73$ \( (T^{7} - 19 T^{6} + \cdots - 162293)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 596873761 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 412143456256 \) Copy content Toggle raw display
$89$ \( T^{14} - 18 T^{13} + \cdots + 17472400 \) Copy content Toggle raw display
$97$ \( (T^{7} + 39 T^{6} + \cdots - 302461)^{2} \) Copy content Toggle raw display
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