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Properties

Label 242.3.d.d
Level $242$
Weight $3$
Character orbit 242.d
Analytic conductor $6.594$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [242,3,Mod(161,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 242.d (of order \(10\), degree \(4\), not 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: \(6.59402239752\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 2) q^{3}+ \cdots + (4 \beta_{7} + 4 \beta_{3} - \beta_{2} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 2) q^{3}+ \cdots + (25 \beta_{7} + 8 \beta_{6} + \cdots - 28) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 4 q^{4} + 12 q^{5} - 20 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 4 q^{4} + 12 q^{5} - 20 q^{7} + 6 q^{9} + 24 q^{12} + 20 q^{13} - 4 q^{14} - 58 q^{15} - 8 q^{16} - 20 q^{17} - 40 q^{18} - 40 q^{19} - 24 q^{20} - 104 q^{23} - 22 q^{25} + 104 q^{26} + 14 q^{27} + 60 q^{28} + 180 q^{29} + 120 q^{30} + 16 q^{31} + 112 q^{34} - 70 q^{35} + 8 q^{36} - 24 q^{37} - 112 q^{38} - 130 q^{39} - 120 q^{41} - 104 q^{42} - 136 q^{45} - 80 q^{46} + 16 q^{47} + 32 q^{48} + 106 q^{49} + 240 q^{50} - 150 q^{51} + 60 q^{52} + 296 q^{53} + 48 q^{56} + 240 q^{57} - 56 q^{58} - 100 q^{59} - 84 q^{60} - 140 q^{61} - 60 q^{62} - 10 q^{63} + 16 q^{64} + 112 q^{67} - 40 q^{68} + 36 q^{69} + 64 q^{70} + 84 q^{71} - 80 q^{72} - 20 q^{73} + 80 q^{74} - 342 q^{75} + 104 q^{78} - 220 q^{79} + 8 q^{80} - 40 q^{81} + 64 q^{82} + 160 q^{83} - 60 q^{84} - 220 q^{85} + 8 q^{86} + 24 q^{89} + 80 q^{90} + 206 q^{91} + 8 q^{92} + 86 q^{93} - 200 q^{94} + 250 q^{95} - 80 q^{96} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(\beta_{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.831254 + 1.14412i
0.831254 − 1.14412i
−1.34500 + 0.437016i
1.34500 − 0.437016i
−1.34500 − 0.437016i
1.34500 + 0.437016i
−0.831254 − 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i 0.713220 − 2.19507i −0.618034 − 1.90211i 6.42606 − 4.66881i 1.91856 + 2.64067i −3.45089 + 1.12126i 2.68999 + 0.874032i 2.97152 + 2.15894i 11.2332i
161.2 0.831254 − 1.14412i −0.949288 + 2.92161i −0.618034 − 1.90211i 1.04607 − 0.760017i 2.55358 + 3.51470i −10.4934 + 3.40951i −2.68999 − 0.874032i −0.353491 − 0.256826i − 1.82860i
215.1 −1.34500 + 0.437016i 3.46303 + 2.51604i 1.61803 − 1.17557i −2.39858 + 7.38206i −5.75732 − 1.87067i 1.65463 + 2.27740i −1.66251 + 2.28825i 2.88098 + 8.86674i − 10.9771i
215.2 1.34500 − 0.437016i 0.773037 + 0.561644i 1.61803 − 1.17557i 0.926440 − 2.85129i 1.28518 + 0.417580i 2.28965 + 3.15143i 1.66251 − 2.28825i −2.49901 − 7.69117i − 4.23984i
233.1 −1.34500 − 0.437016i 3.46303 − 2.51604i 1.61803 + 1.17557i −2.39858 − 7.38206i −5.75732 + 1.87067i 1.65463 − 2.27740i −1.66251 − 2.28825i 2.88098 − 8.86674i 10.9771i
233.2 1.34500 + 0.437016i 0.773037 − 0.561644i 1.61803 + 1.17557i 0.926440 + 2.85129i 1.28518 − 0.417580i 2.28965 − 3.15143i 1.66251 + 2.28825i −2.49901 + 7.69117i 4.23984i
239.1 −0.831254 − 1.14412i 0.713220 + 2.19507i −0.618034 + 1.90211i 6.42606 + 4.66881i 1.91856 − 2.64067i −3.45089 − 1.12126i 2.68999 − 0.874032i 2.97152 − 2.15894i − 11.2332i
239.2 0.831254 + 1.14412i −0.949288 − 2.92161i −0.618034 + 1.90211i 1.04607 + 0.760017i 2.55358 − 3.51470i −10.4934 − 3.40951i −2.68999 + 0.874032i −0.353491 + 0.256826i 1.82860i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.3.d.d 8
11.b odd 2 1 242.3.d.e 8
11.c even 5 1 22.3.d.a ✓ 8
11.c even 5 1 242.3.b.d 8
11.c even 5 1 242.3.d.c 8
11.c even 5 1 242.3.d.e 8
11.d odd 10 1 22.3.d.a ✓ 8
11.d odd 10 1 242.3.b.d 8
11.d odd 10 1 242.3.d.c 8
11.d odd 10 1 inner 242.3.d.d 8
33.f even 10 1 198.3.j.a 8
33.f even 10 1 2178.3.d.l 8
33.h odd 10 1 198.3.j.a 8
33.h odd 10 1 2178.3.d.l 8
44.g even 10 1 176.3.n.b 8
44.h odd 10 1 176.3.n.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.d.a ✓ 8 11.c even 5 1
22.3.d.a ✓ 8 11.d odd 10 1
176.3.n.b 8 44.g even 10 1
176.3.n.b 8 44.h odd 10 1
198.3.j.a 8 33.f even 10 1
198.3.j.a 8 33.h odd 10 1
242.3.b.d 8 11.c even 5 1
242.3.b.d 8 11.d odd 10 1
242.3.d.c 8 11.c even 5 1
242.3.d.c 8 11.d odd 10 1
242.3.d.d 8 1.a even 1 1 trivial
242.3.d.d 8 11.d odd 10 1 inner
242.3.d.e 8 11.b odd 2 1
242.3.d.e 8 11.c even 5 1
2178.3.d.l 8 33.f even 10 1
2178.3.d.l 8 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{8} - 8T_{3}^{7} + 38T_{3}^{6} - 126T_{3}^{5} + 440T_{3}^{4} - 936T_{3}^{3} + 1823T_{3}^{2} - 1798T_{3} + 841 \) Copy content Toggle raw display
\( T_{7}^{8} + 20T_{7}^{7} + 98T_{7}^{6} - 110T_{7}^{5} + 1204T_{7}^{4} + 9220T_{7}^{3} - 1633T_{7}^{2} - 4390T_{7} + 192721 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} - 8 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( T^{8} - 12 T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$7$ \( T^{8} + 20 T^{7} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2268521641 \) Copy content Toggle raw display
$17$ \( T^{8} + 20 T^{7} + \cdots + 22934521 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 3189877441 \) Copy content Toggle raw display
$23$ \( (T^{4} + 52 T^{3} + \cdots - 90224)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 206760274681 \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + \cdots + 996728041 \) Copy content Toggle raw display
$37$ \( T^{8} + 24 T^{7} + \cdots + 20079361 \) Copy content Toggle raw display
$41$ \( T^{8} + 120 T^{7} + \cdots + 405257161 \) Copy content Toggle raw display
$43$ \( T^{8} + 3632 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 7428543721 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 189900554154481 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 47196831300025 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 4097365398025 \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{3} + \cdots + 112576)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 14204072031241 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 36635874025 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 33\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 4768279282321 \) Copy content Toggle raw display
$89$ \( (T^{4} - 12 T^{3} + \cdots - 22808304)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 185279454481 \) Copy content Toggle raw display
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