Properties

Label 242.4.a.h
Level $242$
Weight $4$
Character orbit 242.a
Self dual yes
Analytic conductor $14.278$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [242,4,Mod(1,242)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(242, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("242.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 242.a (trivial)

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: yes
Analytic conductor: \(14.2784622214\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 22)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - 3 \beta - 5) q^{3} + 4 q^{4} - \beta q^{5} + (6 \beta + 10) q^{6} + ( - 11 \beta + 3) q^{7} - 8 q^{8} + (39 \beta + 7) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + ( - 3 \beta - 5) q^{3} + 4 q^{4} - \beta q^{5} + (6 \beta + 10) q^{6} + ( - 11 \beta + 3) q^{7} - 8 q^{8} + (39 \beta + 7) q^{9} + 2 \beta q^{10} + ( - 12 \beta - 20) q^{12} + (29 \beta + 4) q^{13} + (22 \beta - 6) q^{14} + (8 \beta + 3) q^{15} + 16 q^{16} + ( - 27 \beta + 104) q^{17} + ( - 78 \beta - 14) q^{18} + ( - 9 \beta + 41) q^{19} - 4 \beta q^{20} + (79 \beta + 18) q^{21} + (24 \beta - 68) q^{23} + (24 \beta + 40) q^{24} + (\beta - 124) q^{25} + ( - 58 \beta - 8) q^{26} + ( - 252 \beta - 17) q^{27} + ( - 44 \beta + 12) q^{28} + (159 \beta - 107) q^{29} + ( - 16 \beta - 6) q^{30} + (143 \beta - 202) q^{31} - 32 q^{32} + (54 \beta - 208) q^{34} + (8 \beta + 11) q^{35} + (156 \beta + 28) q^{36} + ( - 79 \beta - 97) q^{37} + (18 \beta - 82) q^{38} + ( - 244 \beta - 107) q^{39} + 8 \beta q^{40} + ( - 129 \beta + 89) q^{41} + ( - 158 \beta - 36) q^{42} + (68 \beta - 324) q^{43} + ( - 46 \beta - 39) q^{45} + ( - 48 \beta + 136) q^{46} + (21 \beta - 209) q^{47} + ( - 48 \beta - 80) q^{48} + (55 \beta - 213) q^{49} + ( - 2 \beta + 248) q^{50} + ( - 96 \beta - 439) q^{51} + (116 \beta + 16) q^{52} + (223 \beta - 424) q^{53} + (504 \beta + 34) q^{54} + (88 \beta - 24) q^{56} + ( - 51 \beta - 178) q^{57} + ( - 318 \beta + 214) q^{58} + ( - 301 \beta + 225) q^{59} + (32 \beta + 12) q^{60} + (453 \beta - 24) q^{61} + ( - 286 \beta + 404) q^{62} + ( - 389 \beta - 408) q^{63} + 64 q^{64} + ( - 33 \beta - 29) q^{65} + ( - 116 \beta + 36) q^{67} + ( - 108 \beta + 416) q^{68} + (12 \beta + 268) q^{69} + ( - 16 \beta - 22) q^{70} + (441 \beta + 342) q^{71} + ( - 312 \beta - 56) q^{72} + (51 \beta - 51) q^{73} + (158 \beta + 194) q^{74} + (364 \beta + 617) q^{75} + ( - 36 \beta + 164) q^{76} + (488 \beta + 214) q^{78} + (537 \beta - 934) q^{79} - 16 \beta q^{80} + (1014 \beta + 652) q^{81} + (258 \beta - 178) q^{82} + (475 \beta - 538) q^{83} + (316 \beta + 72) q^{84} + ( - 77 \beta + 27) q^{85} + ( - 136 \beta + 648) q^{86} + ( - 951 \beta + 58) q^{87} + ( - 940 \beta + 902) q^{89} + (92 \beta + 78) q^{90} + ( - 276 \beta - 307) q^{91} + (96 \beta - 272) q^{92} + ( - 538 \beta + 581) q^{93} + ( - 42 \beta + 418) q^{94} + ( - 32 \beta + 9) q^{95} + (96 \beta + 160) q^{96} + ( - 665 \beta - 24) q^{97} + ( - 110 \beta + 426) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 13 q^{3} + 8 q^{4} - q^{5} + 26 q^{6} - 5 q^{7} - 16 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} - 13 q^{3} + 8 q^{4} - q^{5} + 26 q^{6} - 5 q^{7} - 16 q^{8} + 53 q^{9} + 2 q^{10} - 52 q^{12} + 37 q^{13} + 10 q^{14} + 14 q^{15} + 32 q^{16} + 181 q^{17} - 106 q^{18} + 73 q^{19} - 4 q^{20} + 115 q^{21} - 112 q^{23} + 104 q^{24} - 247 q^{25} - 74 q^{26} - 286 q^{27} - 20 q^{28} - 55 q^{29} - 28 q^{30} - 261 q^{31} - 64 q^{32} - 362 q^{34} + 30 q^{35} + 212 q^{36} - 273 q^{37} - 146 q^{38} - 458 q^{39} + 8 q^{40} + 49 q^{41} - 230 q^{42} - 580 q^{43} - 124 q^{45} + 224 q^{46} - 397 q^{47} - 208 q^{48} - 371 q^{49} + 494 q^{50} - 974 q^{51} + 148 q^{52} - 625 q^{53} + 572 q^{54} + 40 q^{56} - 407 q^{57} + 110 q^{58} + 149 q^{59} + 56 q^{60} + 405 q^{61} + 522 q^{62} - 1205 q^{63} + 128 q^{64} - 91 q^{65} - 44 q^{67} + 724 q^{68} + 548 q^{69} - 60 q^{70} + 1125 q^{71} - 424 q^{72} - 51 q^{73} + 546 q^{74} + 1598 q^{75} + 292 q^{76} + 916 q^{78} - 1331 q^{79} - 16 q^{80} + 2318 q^{81} - 98 q^{82} - 601 q^{83} + 460 q^{84} - 23 q^{85} + 1160 q^{86} - 835 q^{87} + 864 q^{89} + 248 q^{90} - 890 q^{91} - 448 q^{92} + 624 q^{93} + 794 q^{94} - 14 q^{95} + 416 q^{96} - 713 q^{97} + 742 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
−2.00000 −9.85410 4.00000 −1.61803 19.7082 −14.7984 −8.00000 70.1033 3.23607
1.2 −2.00000 −3.14590 4.00000 0.618034 6.29180 9.79837 −8.00000 −17.1033 −1.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.4.a.h 2
3.b odd 2 1 2178.4.a.bi 2
4.b odd 2 1 1936.4.a.bc 2
11.b odd 2 1 242.4.a.k 2
11.c even 5 2 242.4.c.j 4
11.c even 5 2 242.4.c.m 4
11.d odd 10 2 22.4.c.a 4
11.d odd 10 2 242.4.c.f 4
33.d even 2 1 2178.4.a.z 2
33.f even 10 2 198.4.f.b 4
44.c even 2 1 1936.4.a.bb 2
44.g even 10 2 176.4.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.a 4 11.d odd 10 2
176.4.m.a 4 44.g even 10 2
198.4.f.b 4 33.f even 10 2
242.4.a.h 2 1.a even 1 1 trivial
242.4.a.k 2 11.b odd 2 1
242.4.c.f 4 11.d odd 10 2
242.4.c.j 4 11.c even 5 2
242.4.c.m 4 11.c even 5 2
1936.4.a.bb 2 44.c even 2 1
1936.4.a.bc 2 4.b odd 2 1
2178.4.a.z 2 33.d even 2 1
2178.4.a.bi 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(242))\):

\( T_{3}^{2} + 13T_{3} + 31 \) Copy content Toggle raw display
\( T_{5}^{2} + T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 5T_{7} - 145 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 13T + 31 \) Copy content Toggle raw display
$5$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 5T - 145 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 37T - 709 \) Copy content Toggle raw display
$17$ \( T^{2} - 181T + 7279 \) Copy content Toggle raw display
$19$ \( T^{2} - 73T + 1231 \) Copy content Toggle raw display
$23$ \( T^{2} + 112T + 2416 \) Copy content Toggle raw display
$29$ \( T^{2} + 55T - 30845 \) Copy content Toggle raw display
$31$ \( T^{2} + 261T - 8531 \) Copy content Toggle raw display
$37$ \( T^{2} + 273T + 10831 \) Copy content Toggle raw display
$41$ \( T^{2} - 49T - 20201 \) Copy content Toggle raw display
$43$ \( T^{2} + 580T + 78320 \) Copy content Toggle raw display
$47$ \( T^{2} + 397T + 38851 \) Copy content Toggle raw display
$53$ \( T^{2} + 625T + 35495 \) Copy content Toggle raw display
$59$ \( T^{2} - 149T - 107701 \) Copy content Toggle raw display
$61$ \( T^{2} - 405T - 215505 \) Copy content Toggle raw display
$67$ \( T^{2} + 44T - 16336 \) Copy content Toggle raw display
$71$ \( T^{2} - 1125T + 73305 \) Copy content Toggle raw display
$73$ \( T^{2} + 51T - 2601 \) Copy content Toggle raw display
$79$ \( T^{2} + 1331T + 82429 \) Copy content Toggle raw display
$83$ \( T^{2} + 601T - 191731 \) Copy content Toggle raw display
$89$ \( T^{2} - 864T - 917876 \) Copy content Toggle raw display
$97$ \( T^{2} + 713T - 425689 \) Copy content Toggle raw display
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