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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [522,2,Mod(175,522)] 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("522.175"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(522, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 522.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,9,1,-9,4] 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: \(4.16819098551\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} + 4 x^{15} + 10 x^{14} - 36 x^{13} + 42 x^{12} - 18 x^{11} + 84 x^{10} - 216 x^{9} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{7} q^{3} + (\beta_{5} - 1) q^{4} + \beta_{3} q^{5} + \beta_1 q^{6} - \beta_{12} q^{7} - q^{8} + \beta_{8} q^{9} + ( - \beta_{15} + \beta_{3}) q^{10} + (\beta_{12} - \beta_{8} - \beta_{4}) q^{11}+ \cdots + ( - 2 \beta_{15} - 4 \beta_{14} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 9 q^{2} + q^{3} - 9 q^{4} + 4 q^{5} + 2 q^{6} - 3 q^{7} - 18 q^{8} + q^{9} + 8 q^{10} + 7 q^{11} + q^{12} - 4 q^{13} + 3 q^{14} + 4 q^{15} - 9 q^{16} - 28 q^{17} + 5 q^{18} + 6 q^{19} + 4 q^{20}+ \cdots + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 2 x^{17} + 4 x^{15} + 10 x^{14} - 36 x^{13} + 42 x^{12} - 18 x^{11} + 84 x^{10} - 216 x^{9} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 77 \nu^{17} + 724 \nu^{16} - 1464 \nu^{15} + 3148 \nu^{14} - 9290 \nu^{13} + 10983 \nu^{12} + \cdots + 3779136 ) / 4310577 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 455 \nu^{17} + 25 \nu^{16} - 3513 \nu^{15} + 21031 \nu^{14} - 21887 \nu^{13} + 24387 \nu^{12} + \cdots + 8548983 ) / 8621154 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 587 \nu^{17} - 2539 \nu^{16} + 1533 \nu^{15} - 1873 \nu^{14} + 9671 \nu^{13} - 29931 \nu^{12} + \cdots - 1935495 ) / 8621154 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1075 \nu^{17} + 1715 \nu^{16} - 2613 \nu^{15} - 7189 \nu^{14} + 11675 \nu^{13} + \cdots - 16002279 ) / 8621154 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 455 \nu^{17} + 511 \nu^{16} - 1407 \nu^{15} + 1267 \nu^{14} - 2933 \nu^{13} + 5109 \nu^{12} + \cdots - 3851307 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 133 \nu^{17} + 469 \nu^{16} - 1029 \nu^{15} - 539 \nu^{14} + 3757 \nu^{13} + 33 \nu^{12} + \cdots - 2985255 ) / 957906 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 952 \nu^{17} + 143 \nu^{16} + 7617 \nu^{15} - 8407 \nu^{14} - 3901 \nu^{13} + 5259 \nu^{12} + \cdots + 11606409 ) / 4310577 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2399 \nu^{17} - 889 \nu^{16} - 11769 \nu^{15} + 22709 \nu^{14} + 19889 \nu^{13} - 55869 \nu^{12} + \cdots - 17406333 ) / 8621154 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1160 \nu^{17} + 3856 \nu^{16} + 2256 \nu^{15} - 8519 \nu^{14} + 6586 \nu^{13} + 34287 \nu^{12} + \cdots + 931662 ) / 4310577 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3035 \nu^{17} - 3227 \nu^{16} + 12969 \nu^{15} - 9341 \nu^{14} - 15401 \nu^{13} + \cdots - 2565351 ) / 8621154 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1181 \nu^{17} - 1743 \nu^{16} - 2135 \nu^{15} + 5897 \nu^{14} + 6591 \nu^{13} - 15317 \nu^{12} + \cdots - 811377 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 1255 \nu^{17} + 695 \nu^{16} - 53 \nu^{15} - 6749 \nu^{14} + 15677 \nu^{13} - 6683 \nu^{12} + \cdots - 1388745 ) / 2873718 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2156 \nu^{17} + 2251 \nu^{16} - 585 \nu^{15} + 4813 \nu^{14} - 35420 \nu^{13} + 48870 \nu^{12} + \cdots + 17294796 ) / 4310577 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 2117 \nu^{17} + 4633 \nu^{16} + 5466 \nu^{15} - 8306 \nu^{14} - 21464 \nu^{13} + \cdots + 22871646 ) / 4310577 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 2296 \nu^{17} - 3521 \nu^{16} + 792 \nu^{15} - 7646 \nu^{14} + 31915 \nu^{13} - 61182 \nu^{12} + \cdots - 23429331 ) / 4310577 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} - \beta_{14} - \beta_{12} - \beta_{10} + \beta_{8} - \beta_{7} + \beta_{6} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{17} - \beta_{16} - 2 \beta_{15} - 2 \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{16} - 2 \beta_{15} - 2 \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{11} - 2 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{17} - 3 \beta_{16} - \beta_{15} - 2 \beta_{12} - 2 \beta_{10} + 4 \beta_{9} + \beta_{8} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{16} - 11 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} + 7 \beta_{12} - \beta_{11} - 10 \beta_{9} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{17} + 8 \beta_{16} + 2 \beta_{15} - 4 \beta_{14} - 2 \beta_{13} + 8 \beta_{11} + 5 \beta_{10} + \cdots - 32 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 27 \beta_{17} - 3 \beta_{16} + \beta_{15} + 27 \beta_{14} + 6 \beta_{13} + 8 \beta_{12} + 14 \beta_{10} + \cdots - 61 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 54 \beta_{17} + 37 \beta_{16} - 16 \beta_{15} + 13 \beta_{14} - 10 \beta_{13} + 8 \beta_{12} + \cdots + 136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 27 \beta_{17} - 29 \beta_{16} + 91 \beta_{15} + 61 \beta_{14} + 53 \beta_{13} + 12 \beta_{12} + \cdots - 172 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 27 \beta_{17} - 42 \beta_{16} + 233 \beta_{15} + 117 \beta_{14} + 228 \beta_{13} + 172 \beta_{12} + \cdots - 38 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 36 \beta_{17} - 64 \beta_{16} + 232 \beta_{15} - 22 \beta_{14} + 199 \beta_{13} + 55 \beta_{12} + \cdots - 82 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 270 \beta_{17} + 200 \beta_{16} - 478 \beta_{15} - 349 \beta_{14} + 442 \beta_{13} + 258 \beta_{12} + \cdots - 215 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 540 \beta_{17} + 348 \beta_{16} - 1034 \beta_{15} - 540 \beta_{14} - 561 \beta_{13} + 206 \beta_{12} + \cdots + 542 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 504 \beta_{17} + 1342 \beta_{16} - 979 \beta_{15} - 878 \beta_{14} - 892 \beta_{13} - 406 \beta_{12} + \cdots - 1916 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 2772 \beta_{17} + 3670 \beta_{16} - 3068 \beta_{15} - 56 \beta_{14} - 550 \beta_{13} + 678 \beta_{12} + \cdots + 3590 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\)
\(\chi(n)\) \(1\) \(-1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
−0.563250 − 1.63791i
−1.54142 − 0.789949i
0.456799 − 1.67073i
1.46322 − 0.926821i
−1.45239 + 0.943691i
−1.35332 + 1.08098i
1.67283 + 0.449061i
1.28587 + 1.16041i
1.03168 + 1.39127i
−0.563250 + 1.63791i
−1.54142 + 0.789949i
0.456799 + 1.67073i
1.46322 + 0.926821i
−1.45239 − 0.943691i
−1.35332 − 1.08098i
1.67283 − 0.449061i
1.28587 − 1.16041i
1.03168 − 1.39127i
0.500000 + 0.866025i −1.70010 − 0.331166i −0.500000 + 0.866025i −0.808803 + 1.40089i −0.563250 − 1.63791i −1.48294 − 2.56852i −1.00000 2.78066 + 1.12603i −1.61761
175.2 0.500000 + 0.866025i −1.45483 + 0.939936i −0.500000 + 0.866025i 1.43905 − 2.49251i −1.54142 − 0.789949i −0.0642108 − 0.111216i −1.00000 1.23304 − 2.73489i 2.87810
175.3 0.500000 + 0.866025i −1.21849 − 1.23096i −0.500000 + 0.866025i 1.89583 − 3.28367i 0.456799 − 1.67073i −1.03953 − 1.80052i −1.00000 −0.0305428 + 2.99984i 3.79166
175.4 0.500000 + 0.866025i −0.0710418 − 1.73059i −0.500000 + 0.866025i −1.74060 + 3.01481i 1.46322 − 0.926821i −1.42445 − 2.46721i −1.00000 −2.98991 + 0.245889i −3.48121
175.5 0.500000 + 0.866025i 0.0910635 + 1.72966i −0.500000 + 0.866025i 0.300149 − 0.519873i −1.45239 + 0.943691i −0.568600 − 0.984845i −1.00000 −2.98341 + 0.315017i 0.600298
175.6 0.500000 + 0.866025i 0.259493 + 1.71250i −0.500000 + 0.866025i −1.05547 + 1.82814i −1.35332 + 1.08098i 1.81043 + 3.13576i −1.00000 −2.86533 + 0.888764i −2.11095
175.7 0.500000 + 0.866025i 1.22531 − 1.22418i −0.500000 + 0.866025i −0.765698 + 1.32623i 1.67283 + 0.449061i 1.35382 + 2.34489i −1.00000 0.00277391 − 3.00000i −1.53140
175.8 0.500000 + 0.866025i 1.64788 − 0.533390i −0.500000 + 0.866025i 1.61178 − 2.79168i 1.28587 + 1.16041i 2.24899 + 3.89537i −1.00000 2.43099 − 1.75792i 3.22355
175.9 0.500000 + 0.866025i 1.72072 − 0.197828i −0.500000 + 0.866025i 1.12377 − 1.94643i 1.03168 + 1.39127i −2.33353 − 4.04178i −1.00000 2.92173 − 0.680811i 2.24755
349.1 0.500000 − 0.866025i −1.70010 + 0.331166i −0.500000 − 0.866025i −0.808803 − 1.40089i −0.563250 + 1.63791i −1.48294 + 2.56852i −1.00000 2.78066 − 1.12603i −1.61761
349.2 0.500000 − 0.866025i −1.45483 − 0.939936i −0.500000 − 0.866025i 1.43905 + 2.49251i −1.54142 + 0.789949i −0.0642108 + 0.111216i −1.00000 1.23304 + 2.73489i 2.87810
349.3 0.500000 − 0.866025i −1.21849 + 1.23096i −0.500000 − 0.866025i 1.89583 + 3.28367i 0.456799 + 1.67073i −1.03953 + 1.80052i −1.00000 −0.0305428 − 2.99984i 3.79166
349.4 0.500000 − 0.866025i −0.0710418 + 1.73059i −0.500000 − 0.866025i −1.74060 − 3.01481i 1.46322 + 0.926821i −1.42445 + 2.46721i −1.00000 −2.98991 − 0.245889i −3.48121
349.5 0.500000 − 0.866025i 0.0910635 − 1.72966i −0.500000 − 0.866025i 0.300149 + 0.519873i −1.45239 − 0.943691i −0.568600 + 0.984845i −1.00000 −2.98341 − 0.315017i 0.600298
349.6 0.500000 − 0.866025i 0.259493 − 1.71250i −0.500000 − 0.866025i −1.05547 − 1.82814i −1.35332 − 1.08098i 1.81043 − 3.13576i −1.00000 −2.86533 − 0.888764i −2.11095
349.7 0.500000 − 0.866025i 1.22531 + 1.22418i −0.500000 − 0.866025i −0.765698 − 1.32623i 1.67283 − 0.449061i 1.35382 − 2.34489i −1.00000 0.00277391 + 3.00000i −1.53140
349.8 0.500000 − 0.866025i 1.64788 + 0.533390i −0.500000 − 0.866025i 1.61178 + 2.79168i 1.28587 − 1.16041i 2.24899 − 3.89537i −1.00000 2.43099 + 1.75792i 3.22355
349.9 0.500000 − 0.866025i 1.72072 + 0.197828i −0.500000 − 0.866025i 1.12377 + 1.94643i 1.03168 − 1.39127i −2.33353 + 4.04178i −1.00000 2.92173 + 0.680811i 2.24755
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 175.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 522.2.e.i ✓ 18
3.b odd 2 1 1566.2.e.i 18
9.c even 3 1 inner 522.2.e.i ✓ 18
9.c even 3 1 4698.2.a.bk 9
9.d odd 6 1 1566.2.e.i 18
9.d odd 6 1 4698.2.a.bl 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
522.2.e.i ✓ 18 1.a even 1 1 trivial
522.2.e.i ✓ 18 9.c even 3 1 inner
1566.2.e.i 18 3.b odd 2 1
1566.2.e.i 18 9.d odd 6 1
4698.2.a.bk 9 9.c even 3 1
4698.2.a.bl 9 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\):

\( T_{5}^{18} - 4 T_{5}^{17} + 38 T_{5}^{16} - 88 T_{5}^{15} + 677 T_{5}^{14} - 1258 T_{5}^{13} + \cdots + 746496 \) Copy content Toggle raw display
\( T_{7}^{18} + 3 T_{7}^{17} + 47 T_{7}^{16} + 130 T_{7}^{15} + 1410 T_{7}^{14} + 3776 T_{7}^{13} + \cdots + 278784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{18} - T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} - 4 T^{17} + \cdots + 746496 \) Copy content Toggle raw display
$7$ \( T^{18} + 3 T^{17} + \cdots + 278784 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 615684969 \) Copy content Toggle raw display
$13$ \( T^{18} + 4 T^{17} + \cdots + 944784 \) Copy content Toggle raw display
$17$ \( (T^{9} + 14 T^{8} + \cdots - 2304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} - 3 T^{8} + \cdots + 10681)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 31066177536 \) Copy content Toggle raw display
$29$ \( (T^{2} - T + 1)^{9} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 7815468819456 \) Copy content Toggle raw display
$37$ \( (T^{9} - 8 T^{8} + \cdots - 18893664)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 25424829628416 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 1038705450561 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 11539915776 \) Copy content Toggle raw display
$53$ \( (T^{9} + 18 T^{8} + \cdots - 34956)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 25397729427456 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 18648716749056 \) Copy content Toggle raw display
$71$ \( (T^{9} + 19 T^{8} + \cdots - 1611888)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} - 4 T^{8} + \cdots + 62805888)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 168069133365904 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 57127475625984 \) Copy content Toggle raw display
$89$ \( (T^{9} + 26 T^{8} + \cdots - 148718592)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 149708341207296 \) Copy content Toggle raw display
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