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Properties

Label 234.6.b.e
Level $234$
Weight $6$
Character orbit 234.b
Analytic conductor $37.530$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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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] = [234,6,Mod(181,234)] 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("234.181"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(234, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 234.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-192,0,0,0,0,0,-224] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.5298138362\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1616511x^{8} + 609939232149x^{4} + 125481622652164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - 16 q^{4} + (\beta_{4} - \beta_{3}) q^{5} - \beta_{7} q^{7} + 16 \beta_{3} q^{8} + ( - 2 \beta_1 - 18) q^{10} + (\beta_{5} + 3 \beta_{4} - 22 \beta_{3}) q^{11} + ( - \beta_{9} - \beta_{2} - \beta_1 - 37) q^{13}+ \cdots + ( - 232 \beta_{5} + \cdots + 749 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 192 q^{4} - 224 q^{10} - 452 q^{13} + 3072 q^{16} - 4320 q^{22} - 6772 q^{25} + 3584 q^{40} + 40160 q^{43} - 9532 q^{49} + 7232 q^{52} - 129120 q^{55} + 60136 q^{61} - 49152 q^{64} + 133520 q^{79}+ \cdots - 200160 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1616511x^{8} + 609939232149x^{4} + 125481622652164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 4432\nu^{8} + 4986886340\nu^{4} + 630446369467633 ) / 4774958026695 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2956\nu^{8} - 1410744980\nu^{4} + 612633725585336 ) / 1591652675565 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3594\nu^{10} + 5832144250\nu^{6} + 2246402667691266\nu^{2} ) / 8053056574977055 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -215151856\nu^{10} - 340173743335370\nu^{6} - 123860603519620408774\nu^{2} ) / 26744200885498799655 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -36806026\nu^{10} - 61519337273576\nu^{6} - 23940387725169549598\nu^{2} ) / 1782946725699919977 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10782\nu^{11} + 17496432750\nu^{7} + 6739208003073798\nu^{3} - 96636678899724660\nu ) / 8053056574977055 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12347502607 \nu^{11} + 223353846662 \nu^{9} + \cdots + 10\!\cdots\!18 \nu ) / 31\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7767645982 \nu^{11} + 24890528476 \nu^{9} + \cdots - 69\!\cdots\!16 \nu ) / 15\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24340377185 \nu^{11} - 198463318186 \nu^{9} - 1464575722352 \nu^{8} + \cdots - 20\!\cdots\!18 ) / 31\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9294264857 \nu^{11} - 57857596570 \nu^{9} + \cdots - 82\!\cdots\!50 \nu ) / 10\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 24695005214 \nu^{11} + 446707693324 \nu^{9} + \cdots + 21\!\cdots\!36 \nu ) / 15\!\cdots\!45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 2\beta_{9} - \beta_{7} - 2\beta_{6} - \beta _1 + 1 ) / 48 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + 9\beta_{4} + 301\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 72\beta_{11} + 1085\beta_{10} - 1702\beta_{9} - 234\beta_{8} - 905\beta_{7} + 1702\beta_{6} - 851\beta _1 + 851 ) / 48 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3324\beta_{2} + 6651\beta _1 - 2157565 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9999 \beta_{11} - 564677 \beta_{10} + 770452 \beta_{9} - 179451 \beta_{8} + 245771 \beta_{7} + \cdots - 385226 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3554181\beta_{5} - 4694706\beta_{4} - 249026899\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22120722 \beta_{11} - 1165762331 \beta_{10} + 1419002206 \beta_{9} + 456261228 \beta_{8} + \cdots - 709501103 ) / 48 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -3740164755\beta_{2} - 3174176205\beta _1 + 1858697190499 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 55069770960 \beta_{11} + 1196375456191 \beta_{10} - 1326506127002 \beta_{9} + 533122392690 \beta_{8} + \cdots + 663253063501 ) / 48 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3892400745558\beta_{5} + 1992926709549\beta_{4} + 220450669803301\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 40449670797429 \beta_{11} + 611263134037720 \beta_{10} - 628390744108166 \beta_{9} + \cdots + 314195372054083 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/234\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.67835 + 2.67835i
−2.67835 − 2.67835i
22.4536 + 22.4536i
−22.4536 − 22.4536i
−19.6764 − 19.6764i
19.6764 + 19.6764i
19.6764 − 19.6764i
−19.6764 + 19.6764i
−22.4536 + 22.4536i
22.4536 − 22.4536i
−2.67835 + 2.67835i
2.67835 − 2.67835i
− 4.00000i 0 −16.0000 − 70.4084i 0 − 182.316i 64.0000i 0 −281.634
181.2 − 4.00000i 0 −16.0000 − 70.4084i 0 182.316i 64.0000i 0 −281.634
181.3 − 4.00000i 0 −16.0000 − 19.3336i 0 − 14.8060i 64.0000i 0 −77.3346
181.4 − 4.00000i 0 −16.0000 − 19.3336i 0 14.8060i 64.0000i 0 −77.3346
181.5 − 4.00000i 0 −16.0000 75.7421i 0 − 139.088i 64.0000i 0 302.968
181.6 − 4.00000i 0 −16.0000 75.7421i 0 139.088i 64.0000i 0 302.968
181.7 4.00000i 0 −16.0000 − 75.7421i 0 − 139.088i − 64.0000i 0 302.968
181.8 4.00000i 0 −16.0000 − 75.7421i 0 139.088i − 64.0000i 0 302.968
181.9 4.00000i 0 −16.0000 19.3336i 0 − 14.8060i − 64.0000i 0 −77.3346
181.10 4.00000i 0 −16.0000 19.3336i 0 14.8060i − 64.0000i 0 −77.3346
181.11 4.00000i 0 −16.0000 70.4084i 0 − 182.316i − 64.0000i 0 −281.634
181.12 4.00000i 0 −16.0000 70.4084i 0 182.316i − 64.0000i 0 −281.634
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.6.b.e ✓ 12
3.b odd 2 1 inner 234.6.b.e ✓ 12
13.b even 2 1 inner 234.6.b.e ✓ 12
39.d odd 2 1 inner 234.6.b.e ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.6.b.e ✓ 12 1.a even 1 1 trivial
234.6.b.e ✓ 12 3.b odd 2 1 inner
234.6.b.e ✓ 12 13.b even 2 1 inner
234.6.b.e ✓ 12 39.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 11068T_{5}^{4} + 32437008T_{5}^{2} + 10630434816 \) acting on \(S_{6}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 11068 T^{4} + \cdots + 10630434816)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 52804 T^{4} + \cdots + 140963846400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 5563560038400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 51\!\cdots\!57)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 3416496537600)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots - 98\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 44\!\cdots\!44)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 10040 T^{2} + \cdots + 196255210112)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 77\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 1133585028152)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 86\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 60751859232960)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 28\!\cdots\!64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 95\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
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