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Properties

Label 234.8.h.b
Level $234$
Weight $8$
Character orbit 234.h
Analytic conductor $73.098$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [234,8,Mod(55,234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(234, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("234.55");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 234.h (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(73.0980959633\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 \beta_1 q^{2} + ( - 64 \beta_1 - 64) q^{4} + (\beta_{3} - 69) q^{5} + ( - 2 \beta_{6} - \beta_{4} + \cdots - 138) q^{7}+ \cdots + 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_1 q^{2} + ( - 64 \beta_1 - 64) q^{4} + (\beta_{3} - 69) q^{5} + ( - 2 \beta_{6} - \beta_{4} + \cdots - 138) q^{7}+ \cdots + ( - 3696 \beta_{7} - 1848 \beta_{6} + \cdots - 1434672) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} - 256 q^{4} - 556 q^{5} - 548 q^{7} + 4096 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} - 256 q^{4} - 556 q^{5} - 548 q^{7} + 4096 q^{8} + 2224 q^{10} + 7392 q^{11} - 25818 q^{13} + 8768 q^{14} - 16384 q^{16} - 28316 q^{17} - 99888 q^{19} + 17792 q^{20} + 59136 q^{22} + 33388 q^{23} + 173756 q^{25} + 156000 q^{26} - 35072 q^{28} - 93140 q^{29} + 622320 q^{31} - 131072 q^{32} + 453056 q^{34} - 141544 q^{35} - 9636 q^{37} + 1598208 q^{38} - 284672 q^{40} - 82892 q^{41} - 569264 q^{43} - 946176 q^{44} + 267104 q^{46} + 1148400 q^{47} - 717798 q^{49} - 695024 q^{50} + 404352 q^{52} - 2470700 q^{53} - 1092512 q^{55} - 280576 q^{56} - 745120 q^{58} - 231504 q^{59} + 685684 q^{61} - 2489280 q^{62} + 2097152 q^{64} + 6216678 q^{65} + 3271056 q^{67} - 1812224 q^{68} + 2264704 q^{70} + 175012 q^{71} + 14275780 q^{73} - 77088 q^{74} - 6392832 q^{76} + 27830412 q^{77} - 14107904 q^{79} + 1138688 q^{80} - 663136 q^{82} - 1314576 q^{83} + 11814998 q^{85} + 9108224 q^{86} + 3784704 q^{88} + 11452234 q^{89} + 16457168 q^{91} - 4273664 q^{92} - 4593600 q^{94} + 23334088 q^{95} - 428002 q^{97} - 5742384 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 4654x^{6} + 7012369x^{4} + 3763719168x^{2} + 637953638400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 4654\nu^{5} + 6213649\nu^{3} + 1905097728\nu - 3143761920 ) / 6287523840 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{4} - 6981\nu^{2} + 35424\nu - 2396160 ) / 2624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 3598\nu^{4} - 3472945\nu^{2} - 720634368 ) / 661248 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{4} + 6981\nu^{2} + 2396160 ) / 1312 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 4102\nu^{4} + 4976377\nu^{2} + 1507704960 ) / 330624 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 381 \nu^{7} - 13312 \nu^{6} - 1613430 \nu^{5} - 54605824 \nu^{4} - 2152864077 \nu^{3} + \cdots - 20070568427520 ) / 8802533376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 625 \nu^{7} - 6656 \nu^{6} - 2429518 \nu^{5} - 23948288 \nu^{4} - 2689763713 \nu^{3} + \cdots - 4792141086720 ) / 8802533376 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{2} ) / 27 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} - 2\beta_{4} + 6\beta_{3} - 3489 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 432 \beta_{7} - 1296 \beta_{6} - 648 \beta_{5} - 1703 \beta_{4} - 216 \beta_{3} - 3406 \beta_{2} + \cdots - 80136 ) / 27 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6981\beta_{5} + 5966\beta_{4} - 13962\beta_{3} + 5722743 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 580176 \beta_{7} + 3228336 \beta_{6} + 1614168 \beta_{5} + 3235009 \beta_{4} + 290088 \beta_{3} + \cdots + 310070808 ) / 27 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4899601\beta_{5} - 4839926\beta_{4} + 9137954\beta_{3} - 3545075771 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 15842736 \beta_{7} - 6971786640 \beta_{6} - 3485893320 \beta_{5} - 6378985367 \beta_{4} + \cdots - 860250992328 ) / 27 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(145\) \(209\)
\(\chi(n)\) \(-1 - \beta_{1}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
41.0833i
− 45.7672i
− 18.4011i
23.0850i
− 41.0833i
45.7672i
18.4011i
− 23.0850i
−4.00000 − 6.92820i 0 −32.0000 + 55.4256i −523.489 0 208.401 − 360.960i 512.000 0 2093.96 + 3626.84i
55.2 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i −132.638 0 −754.459 + 1306.76i 512.000 0 530.551 + 918.941i
55.3 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 54.4265 0 556.354 − 963.633i 512.000 0 −217.706 − 377.078i
55.4 −4.00000 − 6.92820i 0 −32.0000 + 55.4256i 323.700 0 −284.296 + 492.415i 512.000 0 −1294.80 − 2242.66i
217.1 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −523.489 0 208.401 + 360.960i 512.000 0 2093.96 − 3626.84i
217.2 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i −132.638 0 −754.459 − 1306.76i 512.000 0 530.551 − 918.941i
217.3 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 54.4265 0 556.354 + 963.633i 512.000 0 −217.706 + 377.078i
217.4 −4.00000 + 6.92820i 0 −32.0000 − 55.4256i 323.700 0 −284.296 − 492.415i 512.000 0 −1294.80 + 2242.66i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 234.8.h.b 8
3.b odd 2 1 26.8.c.b ✓ 8
12.b even 2 1 208.8.i.b 8
13.c even 3 1 inner 234.8.h.b 8
39.h odd 6 1 338.8.a.j 4
39.i odd 6 1 26.8.c.b ✓ 8
39.i odd 6 1 338.8.a.i 4
39.k even 12 2 338.8.b.h 8
156.p even 6 1 208.8.i.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.8.c.b ✓ 8 3.b odd 2 1
26.8.c.b ✓ 8 39.i odd 6 1
208.8.i.b 8 12.b even 2 1
208.8.i.b 8 156.p even 6 1
234.8.h.b 8 1.a even 1 1 trivial
234.8.h.b 8 13.c even 3 1 inner
338.8.a.i 4 39.i odd 6 1
338.8.a.j 4 39.h odd 6 1
338.8.b.h 8 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 278T_{5}^{3} - 161047T_{5}^{2} - 14695460T_{5} + 1223288100 \) acting on \(S_{8}^{\mathrm{new}}(234, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 278 T^{3} + \cdots + 1223288100)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 15\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 35\!\cdots\!41 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 64\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 65\!\cdots\!69 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 10\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 75\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 87\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 87\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 58\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 21\!\cdots\!40)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 46\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
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