Properties

Label 361.6.a.e
Level $361$
Weight $6$
Character orbit 361.a
Self dual yes
Analytic conductor $57.899$
Analytic rank $0$
Dimension $4$
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] = [361,6,Mod(1,361)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(361, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("361.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 361.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: \(57.8985589525\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 84x^{2} - 154x + 484 \) 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 19)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 3) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + ( - \beta_{3} - 3 \beta_{2} + \cdots + 23) q^{4}+ \cdots + ( - 27 \beta_{2} + 72 \beta_1 + 180) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 3) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + ( - \beta_{3} - 3 \beta_{2} + \cdots + 23) q^{4}+ \cdots + (3096 \beta_{3} - 17271 \beta_{2} + \cdots - 62406) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 9 q^{2} - 6 q^{3} + 97 q^{4} + 90 q^{5} + 369 q^{6} - 190 q^{7} + 225 q^{8} + 846 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 9 q^{2} - 6 q^{3} + 97 q^{4} + 90 q^{5} + 369 q^{6} - 190 q^{7} + 225 q^{8} + 846 q^{9} + 264 q^{10} - 162 q^{11} + 2091 q^{12} + 52 q^{13} - 1395 q^{14} + 1944 q^{15} - 551 q^{16} - 288 q^{17} + 7506 q^{18} + 900 q^{20} - 6096 q^{21} + 222 q^{22} + 9900 q^{23} + 3879 q^{24} + 6862 q^{25} - 711 q^{26} - 1350 q^{27} - 9433 q^{28} - 4176 q^{29} + 4536 q^{30} - 13580 q^{31} + 3321 q^{32} + 12564 q^{33} + 17445 q^{34} - 19314 q^{35} - 1854 q^{36} - 1172 q^{37} - 26484 q^{39} + 29520 q^{40} - 9540 q^{41} - 39285 q^{42} + 13370 q^{43} + 42822 q^{44} + 21438 q^{45} - 7287 q^{46} + 28098 q^{47} - 20409 q^{48} - 29568 q^{49} - 89379 q^{50} + 84726 q^{51} - 24281 q^{52} - 34740 q^{53} + 96795 q^{54} - 95046 q^{55} - 44289 q^{56} + 48525 q^{58} - 9702 q^{59} + 115848 q^{60} - 37978 q^{61} + 56340 q^{62} - 75690 q^{63} - 130079 q^{64} - 94356 q^{65} + 73458 q^{66} + 2974 q^{67} + 81819 q^{68} - 10980 q^{69} + 21000 q^{70} - 32220 q^{71} + 204606 q^{72} - 86908 q^{73} - 189414 q^{74} + 70770 q^{75} + 27270 q^{77} + 23103 q^{78} + 165736 q^{79} - 109368 q^{80} + 437724 q^{81} + 50256 q^{82} - 146448 q^{83} - 269139 q^{84} + 150186 q^{85} + 25200 q^{86} + 210996 q^{87} - 153366 q^{88} + 26604 q^{89} + 296784 q^{90} + 104882 q^{91} + 306279 q^{92} - 41532 q^{93} - 56304 q^{94} - 28017 q^{96} - 313820 q^{97} + 355590 q^{98} - 229338 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 84x^{2} - 154x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 62\nu - 132 ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 13\nu^{2} + 102\nu - 220 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + 2\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 26\beta_{2} + 64\beta _1 + 176 \) 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
−4.37522
−6.51487
10.2303
1.65978
−9.02832 −7.95930 49.5106 −73.8370 71.8591 41.7769 −158.091 −179.649 666.624
1.2 −7.37689 3.41393 22.4185 108.514 −25.1842 −80.6465 70.6815 −231.345 −800.499
1.3 −1.84953 −30.2395 −28.5793 10.2772 55.9287 19.2784 112.043 671.426 −19.0079
1.4 9.25474 28.7849 53.6502 45.0454 266.396 −170.409 200.367 585.568 416.883
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \( -1 \)

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 361.6.a.e 4
19.b odd 2 1 19.6.a.d 4
57.d even 2 1 171.6.a.i 4
76.d even 2 1 304.6.a.l 4
95.d odd 2 1 475.6.a.e 4
133.c even 2 1 931.6.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.6.a.d 4 19.b odd 2 1
171.6.a.i 4 57.d even 2 1
304.6.a.l 4 76.d even 2 1
361.6.a.e 4 1.a even 1 1 trivial
475.6.a.e 4 95.d odd 2 1
931.6.a.d 4 133.c even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 9T_{2}^{3} - 72T_{2}^{2} - 774T_{2} - 1140 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(361))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9 T^{3} + \cdots - 1140 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + \cdots + 23652 \) Copy content Toggle raw display
$5$ \( T^{4} - 90 T^{3} + \cdots - 3709248 \) Copy content Toggle raw display
$7$ \( T^{4} + 190 T^{3} + \cdots + 11068411 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 23023022796 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 19147784368 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 597258958449 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 6908113340160 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 56401950031212 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 5096621276672 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 63\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 813179406508032 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 942340690593024 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 10\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 12\!\cdots\!37 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 20\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 13\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 38\!\cdots\!72 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 42\!\cdots\!12 \) Copy content Toggle raw display
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