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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [342,2,Mod(293,342)] 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("342.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 342.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,-18,-1] 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: \(2.73088374913\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{6})\)
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: \( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{14} q^{3} + q^{4} - \beta_{10} q^{5} - \beta_{14} q^{6} - \beta_{5} q^{7} - q^{8} + \beta_{17} q^{9} + \beta_{10} q^{10} + ( - \beta_{16} - \beta_{14} + \cdots - \beta_1) q^{11}+ \cdots + (2 \beta_{17} - 2 \beta_{15} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{2} - q^{3} + 18 q^{4} + 3 q^{5} + q^{6} - 18 q^{8} + 5 q^{9} - 3 q^{10} + 15 q^{11} - q^{12} - 17 q^{15} + 18 q^{16} - 15 q^{17} - 5 q^{18} + 6 q^{19} + 3 q^{20} + q^{21} - 15 q^{22} + q^{24}+ \cdots - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} + 13 x^{16} - 30 x^{15} + 54 x^{14} - 69 x^{13} + 66 x^{12} + 36 x^{11} - 243 x^{10} + \cdots + 19683 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{17} + 5 \nu^{16} - 13 \nu^{15} + 30 \nu^{14} - 54 \nu^{13} + 69 \nu^{12} - 66 \nu^{11} + \cdots + 32805 ) / 6561 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10 \nu^{17} + 67 \nu^{16} - 140 \nu^{15} + 159 \nu^{14} - 792 \nu^{13} + 498 \nu^{12} + \cdots + 275562 ) / 32805 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53 \nu^{17} + 74 \nu^{16} - 457 \nu^{15} + 531 \nu^{14} - 1737 \nu^{13} + 3336 \nu^{12} + \cdots + 1135053 ) / 65610 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23 \nu^{17} - 28 \nu^{16} + 197 \nu^{15} - 414 \nu^{14} + 369 \nu^{13} - 1128 \nu^{12} + \cdots - 472392 ) / 32805 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28 \nu^{17} - 119 \nu^{16} + 124 \nu^{15} - 459 \nu^{14} + 747 \nu^{13} - 150 \nu^{12} + \cdots - 111537 ) / 32805 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47 \nu^{17} + 364 \nu^{16} - 707 \nu^{15} + 1071 \nu^{14} - 2997 \nu^{13} + 2406 \nu^{12} + \cdots + 610173 ) / 65610 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37 \nu^{17} + 194 \nu^{16} - 553 \nu^{15} + 1065 \nu^{14} - 1863 \nu^{13} + 2580 \nu^{12} + \cdots + 1148175 ) / 32805 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16 \nu^{17} - 71 \nu^{16} + 154 \nu^{15} - 318 \nu^{14} + 558 \nu^{13} - 591 \nu^{12} + \cdots - 223074 ) / 10935 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 145 \nu^{17} - 488 \nu^{16} + 1141 \nu^{15} - 2745 \nu^{14} + 3699 \nu^{13} - 4686 \nu^{12} + \cdots - 2066715 ) / 65610 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 85 \nu^{17} - 167 \nu^{16} + 328 \nu^{15} - 951 \nu^{14} + 765 \nu^{13} - 1005 \nu^{12} + \cdots - 249318 ) / 32805 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 76 \nu^{17} + 257 \nu^{16} - 571 \nu^{15} + 1428 \nu^{14} - 2016 \nu^{13} + 2193 \nu^{12} + \cdots + 977589 ) / 32805 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 79 \nu^{17} + 248 \nu^{16} - 535 \nu^{15} + 1377 \nu^{14} - 1773 \nu^{13} + 1914 \nu^{12} + \cdots + 929475 ) / 21870 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 79 \nu^{17} - 248 \nu^{16} + 535 \nu^{15} - 1377 \nu^{14} + 1773 \nu^{13} - 1914 \nu^{12} + \cdots - 951345 ) / 21870 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 105 \nu^{17} - 380 \nu^{16} + 877 \nu^{15} - 2009 \nu^{14} + 2925 \nu^{13} - 3546 \nu^{12} + \cdots - 1362501 ) / 21870 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 157 \nu^{17} + 665 \nu^{16} - 1378 \nu^{15} + 3051 \nu^{14} - 4896 \nu^{13} + 4866 \nu^{12} + \cdots + 1830519 ) / 32805 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 377 \nu^{17} - 1384 \nu^{16} + 2837 \nu^{15} - 6867 \nu^{14} + 10305 \nu^{13} - 9624 \nu^{12} + \cdots - 4625505 ) / 65610 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 191 \nu^{17} + 650 \nu^{16} - 1393 \nu^{15} + 3229 \nu^{14} - 4587 \nu^{13} + 4944 \nu^{12} + \cdots + 1826145 ) / 21870 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{14} + \beta_{13} - 2\beta_{12} + 3\beta_{10} - 3\beta_{9} + 3\beta_{8} - 3\beta_{3} - 3\beta _1 - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{14} - 2\beta_{13} + \beta_{12} - 3\beta_{10} + 3\beta_{8} + 3\beta_{6} + 3\beta_{3} + 3\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 6 \beta_{17} + 6 \beta_{15} - 2 \beta_{13} + 4 \beta_{12} - 9 \beta_{10} + 9 \beta_{9} - 3 \beta_{8} + \cdots + 13 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{17} + 3 \beta_{16} + 6 \beta_{15} + 6 \beta_{14} + 10 \beta_{13} - 5 \beta_{12} + 6 \beta_{10} + \cdots - 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{17} + 6 \beta_{16} - 3 \beta_{15} - 33 \beta_{14} + \beta_{13} - 8 \beta_{12} - 18 \beta_{11} + \cdots + 28 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 15 \beta_{17} - 27 \beta_{16} + 15 \beta_{15} + 34 \beta_{13} - 5 \beta_{12} - 18 \beta_{11} + \cdots - 41 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 24 \beta_{17} + 21 \beta_{16} + 42 \beta_{15} + 69 \beta_{14} - 53 \beta_{13} + 4 \beta_{12} + \cdots - 113 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21 \beta_{17} + 42 \beta_{16} + 15 \beta_{15} + 48 \beta_{14} - 17 \beta_{13} - 62 \beta_{12} + \cdots - 71 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 24 \beta_{17} - 18 \beta_{16} + 105 \beta_{15} + 61 \beta_{13} + 58 \beta_{12} - 126 \beta_{11} + \cdots + 130 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 84 \beta_{17} - 33 \beta_{16} + 42 \beta_{15} + 69 \beta_{14} + 199 \beta_{13} - 203 \beta_{12} + \cdots + 85 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 102 \beta_{17} + 258 \beta_{16} - 255 \beta_{15} + 183 \beta_{14} - 170 \beta_{13} + 55 \beta_{12} + \cdots - 845 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 132 \beta_{17} - 18 \beta_{16} + 429 \beta_{15} + 459 \beta_{14} - 92 \beta_{13} - 284 \beta_{12} + \cdots - 968 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 462 \beta_{17} + 642 \beta_{16} + 555 \beta_{15} + 3255 \beta_{14} - 1385 \beta_{13} + 1345 \beta_{12} + \cdots + 1147 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 438 \beta_{17} + 1230 \beta_{16} - 606 \beta_{15} - 1437 \beta_{14} - 350 \beta_{13} + 10 \beta_{12} + \cdots + 6967 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 1515 \beta_{17} + 900 \beta_{16} + 726 \beta_{15} + 4320 \beta_{14} + 4048 \beta_{13} + 967 \beta_{12} + \cdots - 3200 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 1299 \beta_{17} + 3531 \beta_{16} + 1095 \beta_{15} - 1902 \beta_{14} - 4940 \beta_{13} + 3622 \beta_{12} + \cdots + 14845 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/342\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(325\)
\(\chi(n)\) \(-\beta_{9}\) \(1 + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.565455 − 1.63715i
1.61535 − 0.625018i
−0.290848 − 1.70746i
1.73155 + 0.0417686i
1.64560 + 0.540356i
−1.68196 − 0.413538i
0.539325 + 1.64594i
−0.599249 + 1.62508i
−1.02522 + 1.39604i
0.565455 + 1.63715i
1.61535 + 0.625018i
−0.290848 + 1.70746i
1.73155 − 0.0417686i
1.64560 − 0.540356i
−1.68196 + 0.413538i
0.539325 − 1.64594i
−0.599249 − 1.62508i
−1.02522 − 1.39604i
−1.00000 −1.70054 − 0.328876i 1.00000 −2.21060 + 1.27629i 1.70054 + 0.328876i 0.762878 + 1.32134i −1.00000 2.78368 + 1.11854i 2.21060 − 1.27629i
293.2 −1.00000 −1.34896 + 1.08642i 1.00000 3.41119 − 1.96945i 1.34896 − 1.08642i −1.35148 − 2.34083i −1.00000 0.639363 − 2.93108i −3.41119 + 1.96945i
293.3 −1.00000 −1.33328 − 1.10561i 1.00000 2.55682 − 1.47618i 1.33328 + 1.10561i 1.79740 + 3.11318i −1.00000 0.555252 + 2.94817i −2.55682 + 1.47618i
293.4 −1.00000 −0.829601 + 1.52045i 1.00000 0.820646 − 0.473800i 0.829601 − 1.52045i 1.09312 + 1.89335i −1.00000 −1.62352 − 2.52273i −0.820646 + 0.473800i
293.5 −1.00000 −0.354840 + 1.69531i 1.00000 −2.76146 + 1.59433i 0.354840 − 1.69531i −1.55443 − 2.69235i −1.00000 −2.74818 − 1.20313i 2.76146 − 1.59433i
293.6 −1.00000 0.482845 − 1.66339i 1.00000 1.62258 − 0.936797i −0.482845 + 1.66339i −0.813726 − 1.40941i −1.00000 −2.53372 − 1.60632i −1.62258 + 0.936797i
293.7 −1.00000 1.15577 + 1.29004i 1.00000 0.400415 − 0.231180i −1.15577 − 1.29004i 1.36198 + 2.35901i −1.00000 −0.328410 + 2.98197i −0.400415 + 0.231180i
293.8 −1.00000 1.70699 + 0.293577i 1.00000 1.10624 − 0.638690i −1.70699 − 0.293577i −1.74174 − 3.01679i −1.00000 2.82762 + 1.00227i −1.10624 + 0.638690i
293.9 −1.00000 1.72161 − 0.189853i 1.00000 −3.44583 + 1.98945i −1.72161 + 0.189853i 0.446005 + 0.772503i −1.00000 2.92791 − 0.653708i 3.44583 − 1.98945i
335.1 −1.00000 −1.70054 + 0.328876i 1.00000 −2.21060 − 1.27629i 1.70054 − 0.328876i 0.762878 − 1.32134i −1.00000 2.78368 − 1.11854i 2.21060 + 1.27629i
335.2 −1.00000 −1.34896 − 1.08642i 1.00000 3.41119 + 1.96945i 1.34896 + 1.08642i −1.35148 + 2.34083i −1.00000 0.639363 + 2.93108i −3.41119 − 1.96945i
335.3 −1.00000 −1.33328 + 1.10561i 1.00000 2.55682 + 1.47618i 1.33328 − 1.10561i 1.79740 − 3.11318i −1.00000 0.555252 − 2.94817i −2.55682 − 1.47618i
335.4 −1.00000 −0.829601 − 1.52045i 1.00000 0.820646 + 0.473800i 0.829601 + 1.52045i 1.09312 − 1.89335i −1.00000 −1.62352 + 2.52273i −0.820646 − 0.473800i
335.5 −1.00000 −0.354840 − 1.69531i 1.00000 −2.76146 − 1.59433i 0.354840 + 1.69531i −1.55443 + 2.69235i −1.00000 −2.74818 + 1.20313i 2.76146 + 1.59433i
335.6 −1.00000 0.482845 + 1.66339i 1.00000 1.62258 + 0.936797i −0.482845 − 1.66339i −0.813726 + 1.40941i −1.00000 −2.53372 + 1.60632i −1.62258 − 0.936797i
335.7 −1.00000 1.15577 − 1.29004i 1.00000 0.400415 + 0.231180i −1.15577 + 1.29004i 1.36198 − 2.35901i −1.00000 −0.328410 − 2.98197i −0.400415 − 0.231180i
335.8 −1.00000 1.70699 − 0.293577i 1.00000 1.10624 + 0.638690i −1.70699 + 0.293577i −1.74174 + 3.01679i −1.00000 2.82762 − 1.00227i −1.10624 − 0.638690i
335.9 −1.00000 1.72161 + 0.189853i 1.00000 −3.44583 − 1.98945i −1.72161 − 0.189853i 0.446005 − 0.772503i −1.00000 2.92791 + 0.653708i 3.44583 + 1.98945i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 293.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.t even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.2.n.f yes 18
3.b odd 2 1 1026.2.n.f 18
9.c even 3 1 1026.2.j.f 18
9.d odd 6 1 342.2.j.f ✓ 18
19.d odd 6 1 342.2.j.f ✓ 18
57.f even 6 1 1026.2.j.f 18
171.i odd 6 1 1026.2.n.f 18
171.t even 6 1 inner 342.2.n.f yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.2.j.f ✓ 18 9.d odd 6 1
342.2.j.f ✓ 18 19.d odd 6 1
342.2.n.f yes 18 1.a even 1 1 trivial
342.2.n.f yes 18 171.t even 6 1 inner
1026.2.j.f 18 9.c even 3 1
1026.2.j.f 18 57.f even 6 1
1026.2.n.f 18 3.b odd 2 1
1026.2.n.f 18 171.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} - 3 T_{5}^{17} - 27 T_{5}^{16} + 90 T_{5}^{15} + 540 T_{5}^{14} - 1977 T_{5}^{13} + \cdots + 155952 \) acting on \(S_{2}^{\mathrm{new}}(342, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + T^{17} + \cdots + 19683 \) Copy content Toggle raw display
$5$ \( T^{18} - 3 T^{17} + \cdots + 155952 \) Copy content Toggle raw display
$7$ \( T^{18} + 30 T^{16} + \cdots + 1926544 \) Copy content Toggle raw display
$11$ \( T^{18} - 15 T^{17} + \cdots + 87723 \) Copy content Toggle raw display
$13$ \( T^{18} + 147 T^{16} + \cdots + 45349632 \) Copy content Toggle raw display
$17$ \( T^{18} + 15 T^{17} + \cdots + 5015547 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{18} + 198 T^{16} + \cdots + 5419008 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 172186884 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 2293235712 \) Copy content Toggle raw display
$37$ \( T^{18} + 294 T^{16} + \cdots + 11337408 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 867167742034944 \) Copy content Toggle raw display
$43$ \( (T^{9} - 6 T^{8} + \cdots + 58276)^{2} \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 87121505633328 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 182079717264 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 79127439616 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 142397118885888 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 259843051643904 \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 13\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 862152512709843 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 713791570066884 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 53197952688 \) Copy content Toggle raw display
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