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Properties

Label 19.6.c.a
Level $19$
Weight $6$
Character orbit 19.c
Analytic conductor $3.047$
Analytic rank $0$
Dimension $16$
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] = [19,6,Mod(7,19)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(19, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("19.7"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 19.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.04729257645\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 210 x^{14} - 467 x^{13} + 30342 x^{12} - 66627 x^{11} + 2132053 x^{10} + \cdots + 629547033600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_1) q^{2} + (\beta_{5} + 3 \beta_{2} + 3) q^{3} + ( - \beta_{10} + 19 \beta_{2}) q^{4} + ( - \beta_{9} - \beta_{8} + \beta_{3} + \cdots + 2) q^{5} + ( - \beta_{12} + \beta_{10} + \cdots - 4 \beta_1) q^{6}+ \cdots + (201 \beta_{15} + 248 \beta_{14} + \cdots + 4707 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} + 28 q^{3} - 155 q^{4} + 10 q^{5} + 149 q^{6} + 208 q^{7} - 78 q^{8} - 616 q^{9} + 580 q^{10} + 632 q^{11} - 2238 q^{12} + 786 q^{13} + 2054 q^{14} + 796 q^{15} - 3779 q^{16} - 746 q^{17}+ \cdots + 163952 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 210 x^{14} - 467 x^{13} + 30342 x^{12} - 66627 x^{11} + 2132053 x^{10} + \cdots + 629547033600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 15\!\cdots\!49 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 68\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 44\!\cdots\!75 \nu^{15} + \cdots + 22\!\cdots\!40 ) / 15\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!97 \nu^{15} + \cdots + 13\!\cdots\!88 ) / 26\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!17 \nu^{15} + \cdots + 20\!\cdots\!40 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!97 \nu^{15} + \cdots - 64\!\cdots\!60 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!19 \nu^{15} + \cdots + 95\!\cdots\!00 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47\!\cdots\!33 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 26\!\cdots\!15 \nu^{15} + \cdots - 14\!\cdots\!52 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!97 \nu^{15} + \cdots - 34\!\cdots\!00 ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18\!\cdots\!03 \nu^{15} + \cdots + 13\!\cdots\!80 ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 59\!\cdots\!51 \nu^{15} + \cdots + 98\!\cdots\!92 ) / 31\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57\!\cdots\!65 \nu^{15} + \cdots + 48\!\cdots\!60 ) / 25\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 45\!\cdots\!55 \nu^{15} + \cdots - 14\!\cdots\!00 ) / 14\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 46\!\cdots\!13 \nu^{15} + \cdots - 93\!\cdots\!00 ) / 14\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{4} - 51\beta_{2} - 51 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{13} - \beta_{11} - 6\beta_{7} + 6\beta_{5} + \beta_{4} + 83\beta_{3} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} + 6 \beta_{14} + 8 \beta_{12} - 119 \beta_{10} - 12 \beta_{8} - 4 \beta_{7} + \cdots - 56 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 145 \beta_{14} + 145 \beta_{13} + 137 \beta_{11} - 265 \beta_{10} - 16 \beta_{9} - 16 \beta_{8} + \cdots + 2624 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 426 \beta_{15} + 958 \beta_{13} - 1480 \beta_{12} + 426 \beta_{11} - 1996 \beta_{9} + 1124 \beta_{7} + \cdots + 437891 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 16633 \beta_{15} - 18369 \beta_{14} - 64 \beta_{12} + 45145 \beta_{10} + 4464 \beta_{8} + \cdots + 887827 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 129182 \beta_{14} - 129182 \beta_{13} - 67978 \beta_{11} + 1589903 \beta_{10} + 266060 \beta_{9} + \cdots - 47974531 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1985145 \beta_{15} - 2256065 \beta_{13} + 66752 \beta_{12} - 1985145 \beta_{11} + 789872 \beta_{9} + \cdots - 126036928 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 9781226 \beta_{15} + 16817406 \beta_{14} + 25625288 \beta_{12} - 187097327 \beta_{10} + \cdots - 388871912 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 275250817 \beta_{14} + 275250817 \beta_{13} + 237026489 \beta_{11} - 945918937 \beta_{10} + \cdots + 19639719008 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1341805002 \beta_{15} + 2165740510 \beta_{13} - 3092247880 \beta_{12} + 1341805002 \beta_{11} + \cdots + 636529745795 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 28413826297 \beta_{15} - 33580450881 \beta_{14} - 4288890304 \beta_{12} + 128477265913 \beta_{10} + \cdots + 1379333669779 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 277341967550 \beta_{14} - 277341967550 \beta_{13} - 179339801002 \beta_{11} + 2661163792559 \beta_{10} + \cdots - 75126742995715 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3421391799609 \beta_{15} - 4103826821633 \beta_{13} + 757052626496 \beta_{12} + \cdots - 393160958698912 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−5.16020 + 8.93772i
−4.04451 + 7.00529i
−2.22796 + 3.85894i
−0.699534 + 1.21163i
1.70954 − 2.96100i
2.91004 − 5.04034i
3.43239 − 5.94507i
5.58024 − 9.66525i
−5.16020 − 8.93772i
−4.04451 − 7.00529i
−2.22796 − 3.85894i
−0.699534 − 1.21163i
1.70954 + 2.96100i
2.91004 + 5.04034i
3.43239 + 5.94507i
5.58024 + 9.66525i
−5.16020 − 8.93772i 14.4299 + 24.9933i −37.2553 + 64.5280i 18.6343 + 32.2756i 148.922 − 257.941i −30.1607 438.725 −294.945 + 510.859i 192.314 − 333.097i
7.2 −4.04451 − 7.00529i −8.84337 − 15.3172i −16.7161 + 28.9531i 15.3484 + 26.5843i −71.5341 + 123.901i −20.6961 11.5845 −34.9104 + 60.4666i 124.154 − 215.041i
7.3 −2.22796 − 3.85894i 2.01841 + 3.49598i 6.07237 − 10.5176i −34.5040 − 59.7627i 8.99386 − 15.5778i −107.943 −196.706 113.352 − 196.332i −153.747 + 266.298i
7.4 −0.699534 − 1.21163i 5.27989 + 9.14503i 15.0213 − 26.0177i 26.3538 + 45.6461i 7.38692 − 12.7945i 195.855 −86.8018 65.7456 − 113.875i 36.8707 − 63.8619i
7.5 1.70954 + 2.96100i −10.6018 − 18.3629i 10.1550 − 17.5889i −15.7296 − 27.2444i 36.2483 − 62.7839i 24.4401 178.851 −103.296 + 178.914i 53.7805 − 93.1505i
7.6 2.91004 + 5.04034i 3.53109 + 6.11602i −0.936658 + 1.62234i 36.3205 + 62.9090i −20.5512 + 35.5957i −226.435 175.340 96.5629 − 167.252i −211.388 + 366.135i
7.7 3.43239 + 5.94507i 13.1276 + 22.7377i −7.56257 + 13.0988i −49.8844 − 86.4023i −90.1180 + 156.089i 92.8461 115.842 −223.168 + 386.538i 342.445 − 593.132i
7.8 5.58024 + 9.66525i −4.94172 − 8.55931i −46.2781 + 80.1560i 8.46093 + 14.6548i 55.1519 − 95.5260i 176.093 −675.836 72.6588 − 125.849i −94.4280 + 163.554i
11.1 −5.16020 + 8.93772i 14.4299 − 24.9933i −37.2553 − 64.5280i 18.6343 − 32.2756i 148.922 + 257.941i −30.1607 438.725 −294.945 − 510.859i 192.314 + 333.097i
11.2 −4.04451 + 7.00529i −8.84337 + 15.3172i −16.7161 − 28.9531i 15.3484 − 26.5843i −71.5341 − 123.901i −20.6961 11.5845 −34.9104 − 60.4666i 124.154 + 215.041i
11.3 −2.22796 + 3.85894i 2.01841 − 3.49598i 6.07237 + 10.5176i −34.5040 + 59.7627i 8.99386 + 15.5778i −107.943 −196.706 113.352 + 196.332i −153.747 − 266.298i
11.4 −0.699534 + 1.21163i 5.27989 − 9.14503i 15.0213 + 26.0177i 26.3538 − 45.6461i 7.38692 + 12.7945i 195.855 −86.8018 65.7456 + 113.875i 36.8707 + 63.8619i
11.5 1.70954 − 2.96100i −10.6018 + 18.3629i 10.1550 + 17.5889i −15.7296 + 27.2444i 36.2483 + 62.7839i 24.4401 178.851 −103.296 − 178.914i 53.7805 + 93.1505i
11.6 2.91004 − 5.04034i 3.53109 − 6.11602i −0.936658 − 1.62234i 36.3205 − 62.9090i −20.5512 − 35.5957i −226.435 175.340 96.5629 + 167.252i −211.388 − 366.135i
11.7 3.43239 − 5.94507i 13.1276 − 22.7377i −7.56257 − 13.0988i −49.8844 + 86.4023i −90.1180 − 156.089i 92.8461 115.842 −223.168 − 386.538i 342.445 + 593.132i
11.8 5.58024 − 9.66525i −4.94172 + 8.55931i −46.2781 − 80.1560i 8.46093 − 14.6548i 55.1519 + 95.5260i 176.093 −675.836 72.6588 + 125.849i −94.4280 − 163.554i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.6.c.a ✓ 16
3.b odd 2 1 171.6.f.b 16
4.b odd 2 1 304.6.i.c 16
19.c even 3 1 inner 19.6.c.a ✓ 16
19.c even 3 1 361.6.a.g 8
19.d odd 6 1 361.6.a.h 8
57.h odd 6 1 171.6.f.b 16
76.g odd 6 1 304.6.i.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.c.a ✓ 16 1.a even 1 1 trivial
19.6.c.a ✓ 16 19.c even 3 1 inner
171.6.f.b 16 3.b odd 2 1
171.6.f.b 16 57.h odd 6 1
304.6.i.c 16 4.b odd 2 1
304.6.i.c 16 76.g odd 6 1
361.6.a.g 8 19.c even 3 1
361.6.a.h 8 19.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 629547033600 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 71\!\cdots\!09 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 11\!\cdots\!92)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 20\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 17\!\cdots\!44 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 27\!\cdots\!88)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 51\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 66\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 55\!\cdots\!81 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
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