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Properties

Label 242.3.d.c
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + (\beta_{6} - 4 \beta_{5} + 4 \beta_{3} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - 25 \beta_{7} - 48 \beta_{6} + \cdots + 20) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} + 20 q^{6} + 30 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} + 20 q^{6} + 30 q^{7} - 4 q^{9} + 24 q^{12} - 30 q^{13} + 16 q^{14} + 42 q^{15} - 8 q^{16} - 30 q^{17} - 40 q^{18} + 30 q^{19} - 4 q^{20} - 104 q^{23} + 40 q^{24} - 12 q^{25} - 96 q^{26} - 26 q^{27} + 40 q^{28} + 10 q^{29} + 60 q^{30} + 46 q^{31} + 112 q^{34} - 70 q^{35} - 12 q^{36} + 6 q^{37} + 108 q^{38} - 130 q^{39} - 80 q^{40} - 250 q^{41} + 56 q^{42} - 136 q^{45} + 160 q^{46} - 54 q^{47} - 8 q^{48} - 144 q^{49} + 80 q^{50} + 30 q^{51} + 40 q^{52} - 274 q^{53} + 48 q^{56} + 130 q^{57} + 64 q^{58} + 50 q^{59} + 116 q^{60} - 50 q^{61} - 20 q^{62} + 20 q^{63} + 16 q^{64} + 112 q^{67} - 60 q^{68} + 76 q^{69} + 4 q^{70} + 54 q^{71} + 80 q^{72} + 70 q^{73} + 40 q^{74} + 318 q^{75} + 104 q^{78} - 370 q^{79} + 48 q^{80} + 180 q^{81} - 96 q^{82} + 150 q^{83} + 120 q^{84} + 330 q^{85} - 72 q^{86} + 24 q^{89} - 160 q^{90} - 294 q^{91} - 112 q^{92} - 134 q^{93} + 20 q^{94} + 330 q^{95} - 18 q^{97}+O(q^{100}) \) 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 −0.295274 + 0.908759i −0.618034 − 1.90211i −2.42545 + 1.76219i −0.794285 − 1.09324i 3.70473 − 1.20374i 2.68999 + 0.874032i 6.54250 + 4.75340i − 4.23984i
161.2 0.831254 − 1.14412i −1.32276 + 4.07104i −0.618034 − 1.90211i 6.27955 − 4.56236i 3.55822 + 4.89747i 2.67724 − 0.869888i −2.68999 − 0.874032i −7.54250 − 5.47994i − 10.9771i
215.1 −1.34500 + 0.437016i −1.86723 − 1.35662i 1.61803 − 1.17557i −2.45454 + 7.55429i 3.10429 + 1.00865i 2.13277 + 2.93550i −1.66251 + 2.28825i −1.13502 − 3.49324i − 11.2332i
215.2 1.34500 − 0.437016i 2.48527 + 1.80565i 1.61803 − 1.17557i −0.399565 + 1.22973i 4.13178 + 1.34250i 6.48527 + 8.92621i 1.66251 − 2.28825i 0.135021 + 0.415553i 1.82860i
233.1 −1.34500 − 0.437016i −1.86723 + 1.35662i 1.61803 + 1.17557i −2.45454 − 7.55429i 3.10429 − 1.00865i 2.13277 − 2.93550i −1.66251 − 2.28825i −1.13502 + 3.49324i 11.2332i
233.2 1.34500 + 0.437016i 2.48527 − 1.80565i 1.61803 + 1.17557i −0.399565 − 1.22973i 4.13178 − 1.34250i 6.48527 − 8.92621i 1.66251 + 2.28825i 0.135021 − 0.415553i − 1.82860i
239.1 −0.831254 − 1.14412i −0.295274 − 0.908759i −0.618034 + 1.90211i −2.42545 − 1.76219i −0.794285 + 1.09324i 3.70473 + 1.20374i 2.68999 − 0.874032i 6.54250 − 4.75340i 4.23984i
239.2 0.831254 + 1.14412i −1.32276 − 4.07104i −0.618034 + 1.90211i 6.27955 + 4.56236i 3.55822 − 4.89747i 2.67724 + 0.869888i −2.68999 + 0.874032i −7.54250 + 5.47994i 10.9771i
\(n\): 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.c 8
11.b odd 2 1 22.3.d.a ✓ 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.d 8
11.c even 5 1 242.3.d.e 8
11.d odd 10 1 242.3.b.d 8
11.d odd 10 1 inner 242.3.d.c 8
11.d odd 10 1 242.3.d.d 8
11.d odd 10 1 242.3.d.e 8
33.d even 2 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.c even 2 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.b odd 2 1
22.3.d.a ✓ 8 11.c even 5 1
176.3.n.b 8 44.c even 2 1
176.3.n.b 8 44.h odd 10 1
198.3.j.a 8 33.d even 2 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 1.a even 1 1 trivial
242.3.d.c 8 11.d odd 10 1 inner
242.3.d.d 8 11.c even 5 1
242.3.d.d 8 11.d odd 10 1
242.3.d.e 8 11.c even 5 1
242.3.d.e 8 11.d odd 10 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 \(S_{3}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{8} + 2T_{3}^{7} + 13T_{3}^{6} - 16T_{3}^{5} + 274T_{3}^{3} + 1098T_{3}^{2} + 812T_{3} + 841 \) Copy content Toggle raw display
\( T_{7}^{8} - 30T_{7}^{7} + 473T_{7}^{6} - 4340T_{7}^{5} + 24884T_{7}^{4} - 92470T_{7}^{3} + 220062T_{7}^{2} - 307300T_{7} + 192721 \) 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
$3$ \( T^{8} + 2 T^{7} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$7$ \( T^{8} - 30 T^{7} + \cdots + 192721 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 2268521641 \) Copy content Toggle raw display
$17$ \( T^{8} + 30 T^{7} + \cdots + 22934521 \) 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
$31$ \( T^{8} - 46 T^{7} + \cdots + 996728041 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots + 20079361 \) Copy content Toggle raw display
$41$ \( T^{8} + 250 T^{7} + \cdots + 405257161 \) 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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