Properties

Label 242.4.c.l
Level $242$
Weight $4$
Character orbit 242.c
Analytic conductor $14.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [242,4,Mod(3,242)]
 
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([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.3");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.c (of order \(5\), 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: \(14.2784622214\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - 7 \zeta_{10}^{2} q^{3} - 4 \zeta_{10}^{3} q^{4} + 19 \zeta_{10} q^{5} - 14 \zeta_{10} q^{6} - 14 \zeta_{10}^{3} q^{7} - 8 \zeta_{10}^{2} q^{8} + (22 \zeta_{10}^{3} - 22 \zeta_{10}^{2} + 22 \zeta_{10} - 22) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{2} - 7 \zeta_{10}^{2} q^{3} - 4 \zeta_{10}^{3} q^{4} + 19 \zeta_{10} q^{5} - 14 \zeta_{10} q^{6} - 14 \zeta_{10}^{3} q^{7} - 8 \zeta_{10}^{2} q^{8} + (22 \zeta_{10}^{3} - 22 \zeta_{10}^{2} + 22 \zeta_{10} - 22) q^{9} + 38 q^{10} - 28 q^{12} + ( - 72 \zeta_{10}^{3} + 72 \zeta_{10}^{2} - 72 \zeta_{10} + 72) q^{13} - 28 \zeta_{10}^{2} q^{14} - 133 \zeta_{10}^{3} q^{15} - 16 \zeta_{10} q^{16} + 46 \zeta_{10} q^{17} + 44 \zeta_{10}^{3} q^{18} - 20 \zeta_{10}^{2} q^{19} + ( - 76 \zeta_{10}^{3} + 76 \zeta_{10}^{2} - 76 \zeta_{10} + 76) q^{20} - 98 q^{21} - 107 q^{23} + (56 \zeta_{10}^{3} - 56 \zeta_{10}^{2} + 56 \zeta_{10} - 56) q^{24} + 236 \zeta_{10}^{2} q^{25} - 144 \zeta_{10}^{3} q^{26} - 35 \zeta_{10} q^{27} - 56 \zeta_{10} q^{28} - 120 \zeta_{10}^{3} q^{29} - 266 \zeta_{10}^{2} q^{30} + (117 \zeta_{10}^{3} - 117 \zeta_{10}^{2} + 117 \zeta_{10} - 117) q^{31} - 32 q^{32} + 92 q^{34} + ( - 266 \zeta_{10}^{3} + 266 \zeta_{10}^{2} - 266 \zeta_{10} + 266) q^{35} + 88 \zeta_{10}^{2} q^{36} + 201 \zeta_{10}^{3} q^{37} - 40 \zeta_{10} q^{38} - 504 \zeta_{10} q^{39} - 152 \zeta_{10}^{3} q^{40} - 228 \zeta_{10}^{2} q^{41} + (196 \zeta_{10}^{3} - 196 \zeta_{10}^{2} + 196 \zeta_{10} - 196) q^{42} - 242 q^{43} - 418 q^{45} + (214 \zeta_{10}^{3} - 214 \zeta_{10}^{2} + 214 \zeta_{10} - 214) q^{46} - 96 \zeta_{10}^{2} q^{47} + 112 \zeta_{10}^{3} q^{48} + 147 \zeta_{10} q^{49} + 472 \zeta_{10} q^{50} - 322 \zeta_{10}^{3} q^{51} - 288 \zeta_{10}^{2} q^{52} + (458 \zeta_{10}^{3} - 458 \zeta_{10}^{2} + 458 \zeta_{10} - 458) q^{53} - 70 q^{54} - 112 q^{56} + (140 \zeta_{10}^{3} - 140 \zeta_{10}^{2} + 140 \zeta_{10} - 140) q^{57} - 240 \zeta_{10}^{2} q^{58} - 435 \zeta_{10}^{3} q^{59} - 532 \zeta_{10} q^{60} + 668 \zeta_{10} q^{61} + 234 \zeta_{10}^{3} q^{62} + 308 \zeta_{10}^{2} q^{63} + (64 \zeta_{10}^{3} - 64 \zeta_{10}^{2} + 64 \zeta_{10} - 64) q^{64} + 1368 q^{65} + 439 q^{67} + ( - 184 \zeta_{10}^{3} + 184 \zeta_{10}^{2} - 184 \zeta_{10} + 184) q^{68} + 749 \zeta_{10}^{2} q^{69} - 532 \zeta_{10}^{3} q^{70} + 1113 \zeta_{10} q^{71} + 176 \zeta_{10} q^{72} + 72 \zeta_{10}^{3} q^{73} + 402 \zeta_{10}^{2} q^{74} + ( - 1652 \zeta_{10}^{3} + 1652 \zeta_{10}^{2} - 1652 \zeta_{10} + 1652) q^{75} - 80 q^{76} - 1008 q^{78} + ( - 70 \zeta_{10}^{3} + 70 \zeta_{10}^{2} - 70 \zeta_{10} + 70) q^{79} - 304 \zeta_{10}^{2} q^{80} + 839 \zeta_{10}^{3} q^{81} - 456 \zeta_{10} q^{82} - 358 \zeta_{10} q^{83} + 392 \zeta_{10}^{3} q^{84} + 874 \zeta_{10}^{2} q^{85} + (484 \zeta_{10}^{3} - 484 \zeta_{10}^{2} + 484 \zeta_{10} - 484) q^{86} - 840 q^{87} + 895 q^{89} + (836 \zeta_{10}^{3} - 836 \zeta_{10}^{2} + 836 \zeta_{10} - 836) q^{90} - 1008 \zeta_{10}^{2} q^{91} + 428 \zeta_{10}^{3} q^{92} + 819 \zeta_{10} q^{93} - 192 \zeta_{10} q^{94} - 380 \zeta_{10}^{3} q^{95} + 224 \zeta_{10}^{2} q^{96} + (409 \zeta_{10}^{3} - 409 \zeta_{10}^{2} + 409 \zeta_{10} - 409) q^{97} + 294 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 7 q^{3} - 4 q^{4} + 19 q^{5} - 14 q^{6} - 14 q^{7} + 8 q^{8} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 7 q^{3} - 4 q^{4} + 19 q^{5} - 14 q^{6} - 14 q^{7} + 8 q^{8} - 22 q^{9} + 152 q^{10} - 112 q^{12} + 72 q^{13} + 28 q^{14} - 133 q^{15} - 16 q^{16} + 46 q^{17} + 44 q^{18} + 20 q^{19} + 76 q^{20} - 392 q^{21} - 428 q^{23} - 56 q^{24} - 236 q^{25} - 144 q^{26} - 35 q^{27} - 56 q^{28} - 120 q^{29} + 266 q^{30} - 117 q^{31} - 128 q^{32} + 368 q^{34} + 266 q^{35} - 88 q^{36} + 201 q^{37} - 40 q^{38} - 504 q^{39} - 152 q^{40} + 228 q^{41} - 196 q^{42} - 968 q^{43} - 1672 q^{45} - 214 q^{46} + 96 q^{47} + 112 q^{48} + 147 q^{49} + 472 q^{50} - 322 q^{51} + 288 q^{52} - 458 q^{53} - 280 q^{54} - 448 q^{56} - 140 q^{57} + 240 q^{58} - 435 q^{59} - 532 q^{60} + 668 q^{61} + 234 q^{62} - 308 q^{63} - 64 q^{64} + 5472 q^{65} + 1756 q^{67} + 184 q^{68} - 749 q^{69} - 532 q^{70} + 1113 q^{71} + 176 q^{72} + 72 q^{73} - 402 q^{74} + 1652 q^{75} - 320 q^{76} - 4032 q^{78} + 70 q^{79} + 304 q^{80} + 839 q^{81} - 456 q^{82} - 358 q^{83} + 392 q^{84} - 874 q^{85} - 484 q^{86} - 3360 q^{87} + 3580 q^{89} - 836 q^{90} + 1008 q^{91} + 428 q^{92} + 819 q^{93} - 192 q^{94} - 380 q^{95} - 224 q^{96} - 409 q^{97} + 1176 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(123\)
\(\chi(n)\) \(-\zeta_{10}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0.809017 + 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
1.61803 1.17557i −2.16312 6.65740i 1.23607 3.80423i 15.3713 + 11.1679i −11.3262 8.22899i 4.32624 13.3148i −2.47214 7.60845i −17.7984 + 12.9313i 38.0000
9.1 −0.618034 + 1.90211i 5.66312 4.11450i −3.23607 2.35114i −5.87132 18.0701i 4.32624 + 13.3148i −11.3262 8.22899i 6.47214 4.70228i 6.79837 20.9232i 38.0000
27.1 −0.618034 1.90211i 5.66312 + 4.11450i −3.23607 + 2.35114i −5.87132 + 18.0701i 4.32624 13.3148i −11.3262 + 8.22899i 6.47214 + 4.70228i 6.79837 + 20.9232i 38.0000
81.1 1.61803 + 1.17557i −2.16312 + 6.65740i 1.23607 + 3.80423i 15.3713 11.1679i −11.3262 + 8.22899i 4.32624 + 13.3148i −2.47214 + 7.60845i −17.7984 12.9313i 38.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.c.l 4
11.b odd 2 1 242.4.c.e 4
11.c even 5 1 22.4.a.a 1
11.c even 5 3 inner 242.4.c.l 4
11.d odd 10 1 242.4.a.d 1
11.d odd 10 3 242.4.c.e 4
33.f even 10 1 2178.4.a.l 1
33.h odd 10 1 198.4.a.g 1
44.g even 10 1 1936.4.a.n 1
44.h odd 10 1 176.4.a.f 1
55.j even 10 1 550.4.a.n 1
55.k odd 20 2 550.4.b.k 2
77.j odd 10 1 1078.4.a.d 1
88.l odd 10 1 704.4.a.b 1
88.o even 10 1 704.4.a.l 1
132.o even 10 1 1584.4.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.a.a 1 11.c even 5 1
176.4.a.f 1 44.h odd 10 1
198.4.a.g 1 33.h odd 10 1
242.4.a.d 1 11.d odd 10 1
242.4.c.e 4 11.b odd 2 1
242.4.c.e 4 11.d odd 10 3
242.4.c.l 4 1.a even 1 1 trivial
242.4.c.l 4 11.c even 5 3 inner
550.4.a.n 1 55.j even 10 1
550.4.b.k 2 55.k odd 20 2
704.4.a.b 1 88.l odd 10 1
704.4.a.l 1 88.o even 10 1
1078.4.a.d 1 77.j odd 10 1
1584.4.a.v 1 132.o even 10 1
1936.4.a.n 1 44.g even 10 1
2178.4.a.l 1 33.f even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{4} - 7T_{3}^{3} + 49T_{3}^{2} - 343T_{3} + 2401 \) Copy content Toggle raw display
\( T_{5}^{4} - 19T_{5}^{3} + 361T_{5}^{2} - 6859T_{5} + 130321 \) Copy content Toggle raw display
\( T_{7}^{4} + 14T_{7}^{3} + 196T_{7}^{2} + 2744T_{7} + 38416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 4 T^{2} - 8 T + 16 \) Copy content Toggle raw display
$3$ \( T^{4} - 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{4} - 19 T^{3} + 361 T^{2} + \cdots + 130321 \) Copy content Toggle raw display
$7$ \( T^{4} + 14 T^{3} + 196 T^{2} + \cdots + 38416 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 72 T^{3} + 5184 T^{2} + \cdots + 26873856 \) Copy content Toggle raw display
$17$ \( T^{4} - 46 T^{3} + 2116 T^{2} + \cdots + 4477456 \) Copy content Toggle raw display
$19$ \( T^{4} - 20 T^{3} + 400 T^{2} + \cdots + 160000 \) Copy content Toggle raw display
$23$ \( (T + 107)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 120 T^{3} + \cdots + 207360000 \) Copy content Toggle raw display
$31$ \( T^{4} + 117 T^{3} + \cdots + 187388721 \) Copy content Toggle raw display
$37$ \( T^{4} - 201 T^{3} + \cdots + 1632240801 \) Copy content Toggle raw display
$41$ \( T^{4} - 228 T^{3} + \cdots + 2702336256 \) Copy content Toggle raw display
$43$ \( (T + 242)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 96 T^{3} + 9216 T^{2} + \cdots + 84934656 \) Copy content Toggle raw display
$53$ \( T^{4} + 458 T^{3} + \cdots + 44000935696 \) Copy content Toggle raw display
$59$ \( T^{4} + 435 T^{3} + \cdots + 35806100625 \) Copy content Toggle raw display
$61$ \( T^{4} - 668 T^{3} + \cdots + 199115858176 \) Copy content Toggle raw display
$67$ \( (T - 439)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} - 1113 T^{3} + \cdots + 1534548635361 \) Copy content Toggle raw display
$73$ \( T^{4} - 72 T^{3} + 5184 T^{2} + \cdots + 26873856 \) Copy content Toggle raw display
$79$ \( T^{4} - 70 T^{3} + 4900 T^{2} + \cdots + 24010000 \) Copy content Toggle raw display
$83$ \( T^{4} + 358 T^{3} + \cdots + 16426010896 \) Copy content Toggle raw display
$89$ \( (T - 895)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 409 T^{3} + \cdots + 27982932961 \) Copy content Toggle raw display
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