Properties

Label 5.12.b.a
Level $5$
Weight $12$
Character orbit 5.b
Analytic conductor $3.842$
Analytic rank $0$
Dimension $4$
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] = [5,12,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 5.b (of order \(2\), degree \(1\), 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: \(3.84171590280\)
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: \( x^{4} - x^{3} - 142x^{2} - 2144x + 28656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{2} - 18) q^{4} + (5 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 75) q^{5} + ( - 11 \beta_{2} - 438) q^{6} + ( - 119 \beta_{3} - 156 \beta_1) q^{7} + (112 \beta_{3} + 1196 \beta_1) q^{8} + (78 \beta_{2} + 6363) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{3} q^{3} + ( - \beta_{2} - 18) q^{4} + (5 \beta_{3} - 5 \beta_{2} - 10 \beta_1 - 75) q^{5} + ( - 11 \beta_{2} - 438) q^{6} + ( - 119 \beta_{3} - 156 \beta_1) q^{7} + (112 \beta_{3} + 1196 \beta_1) q^{8} + (78 \beta_{2} + 6363) q^{9} + (560 \beta_{3} + 65 \beta_{2} - 4245 \beta_1 + 22850) q^{10} + (220 \beta_{2} - 81588) q^{11} + ( - 816 \beta_{3} - 9612 \beta_1) q^{12} + (110 \beta_{3} + 39468 \beta_1) q^{13} + ( - 1153 \beta_{2} + 270174) q^{14} + ( - 4095 \beta_{3} - 280 \beta_{2} - 48060 \beta_1 + 858300) q^{15} + ( - 2012 \beta_{2} - 2458744) q^{16} + (12648 \beta_{3} - 29480 \beta_1) q^{17} + ( - 8736 \beta_{3} + 71415 \beta_1) q^{18} + (7436 \beta_{2} + 3865220) q^{19} + (2960 \beta_{3} + 165 \beta_{2} + 56580 \beta_1 + 8861850) q^{20} + (10998 \beta_{2} - 20254968) q^{21} + ( - 24640 \beta_{3} + 101892 \beta_1) q^{22} + (46641 \beta_{3} - 624404 \beta_1) q^{23} + ( - 21892 \beta_{2} + 18603960) q^{24} + (29750 \beta_{3} + 1500 \beta_{2} + 565500 \beta_1 + 39788125) q^{25} + ( - 38258 \beta_{2} - 81492708) q^{26} + ( - 118458 \beta_{3} + 749736 \beta_1) q^{27} + ( - 114576 \beta_{3} - 1010916 \beta_1) q^{28} + (4024 \beta_{2} + 54060630) q^{29} + (31360 \beta_{3} + 3015 \beta_{2} + 624780 \beta_1 + 97498350) q^{30} + (75560 \beta_{2} - 171010768) q^{31} + (454720 \beta_{3} - 1687344 \beta_1) q^{32} + (265068 \beta_{3} + 2114640 \beta_1) q^{33} + (168608 \beta_{2} + 66445504) q^{34} + ( - 574665 \beta_{3} - 43460 \beta_{2} - 5056920 \beta_1 + 98573100) q^{35} + ( - 7767 \beta_{2} - 138338334) q^{36} + ( - 35238 \beta_{3} + 661212 \beta_1) q^{37} + ( - 832832 \beta_{3} + 10066844 \beta_1) q^{38} + ( - 442728 \beta_{2} + 1499256) q^{39} + (1128400 \beta_{3} + 109100 \beta_{2} + 305700 \beta_1 - 68801000) q^{40} + ( - 426490 \beta_{2} + 253218342) q^{41} + ( - 1231776 \beta_{3} - 11082636 \beta_1) q^{42} + (1380883 \beta_{3} - 4432008 \beta_1) q^{43} + (77628 \beta_{2} - 388393416) q^{44} + ( - 206085 \beta_{3} - 37665 \beta_{2} + \cdots - 691596225) q^{45}+ \cdots + ( - 4964004 \beta_{2} + 29890091556) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 72 q^{4} - 300 q^{5} - 1752 q^{6} + 25452 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 72 q^{4} - 300 q^{5} - 1752 q^{6} + 25452 q^{9} + 91400 q^{10} - 326352 q^{11} + 1080696 q^{14} + 3433200 q^{15} - 9834976 q^{16} + 15460880 q^{19} + 35447400 q^{20} - 81019872 q^{21} + 74415840 q^{24} + 159152500 q^{25} - 325970832 q^{26} + 216242520 q^{29} + 389993400 q^{30} - 684043072 q^{31} + 265782016 q^{34} + 394292400 q^{35} - 553353336 q^{36} + 5997024 q^{39} - 275204000 q^{40} + 1012873368 q^{41} - 1553573664 q^{44} - 2766384900 q^{45} + 5241789688 q^{46} - 1900646372 q^{49} - 4621170000 q^{50} + 8691953088 q^{51} - 6403356720 q^{54} - 7772763600 q^{55} + 10366738080 q^{56} + 3200971440 q^{59} + 1922954400 q^{60} - 2310471352 q^{61} - 5401150592 q^{64} + 3229723200 q^{65} - 17010985824 q^{66} + 32956101984 q^{69} + 40783573800 q^{70} - 60335466912 q^{71} - 5525992944 q^{74} + 19332540000 q^{75} - 52987638240 q^{76} + 74637768320 q^{79} + 72046927200 q^{80} - 77727716316 q^{81} - 76499865504 q^{84} - 46117585600 q^{85} + 39045421128 q^{86} + 118272499560 q^{89} + 36519766200 q^{90} + 51565095648 q^{91} - 266098749224 q^{94} - 264706278000 q^{95} + 313591828608 q^{96} + 119560366224 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 142x^{2} - 2144x + 28656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 12\nu^{2} + 31\nu + 2562 ) / 15 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 3\nu^{2} + 316\nu + 1422 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{3} + 57\nu^{2} + 34\nu - 22932 ) / 30 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 6\beta _1 + 30 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -38\beta_{3} + 5\beta_{2} - 234\beta _1 + 8550 ) / 120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 518\beta_{3} - 29\beta_{2} + 1194\beta _1 + 205770 ) / 120 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
11.3434 + 1.39818i
−10.8434 10.0894i
−10.8434 + 10.0894i
11.3434 1.39818i
58.2855i 258.747i −1349.20 −6731.01 + 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. + 392321.i
4.2 27.1071i 524.040i 1313.20 6581.01 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 178392.i
4.3 27.1071i 524.040i 1313.20 6581.01 + 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 + 178392.i
4.4 58.2855i 258.747i −1349.20 −6731.01 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. 392321.i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.12.b.a 4
3.b odd 2 1 45.12.b.b 4
4.b odd 2 1 80.12.c.a 4
5.b even 2 1 inner 5.12.b.a 4
5.c odd 4 2 25.12.a.e 4
15.d odd 2 1 45.12.b.b 4
15.e even 4 2 225.12.a.r 4
20.d odd 2 1 80.12.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.12.b.a 4 1.a even 1 1 trivial
5.12.b.a 4 5.b even 2 1 inner
25.12.a.e 4 5.c odd 4 2
45.12.b.b 4 3.b odd 2 1
45.12.b.b 4 15.d odd 2 1
80.12.c.a 4 4.b odd 2 1
80.12.c.a 4 20.d odd 2 1
225.12.a.r 4 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4132 T^{2} + \cdots + 2496256 \) Copy content Toggle raw display
$3$ \( T^{4} + 341568 T^{2} + \cdots + 18385718256 \) Copy content Toggle raw display
$5$ \( T^{4} + 300 T^{3} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{4} + 4904976672 T^{2} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( (T^{2} + 163176 T - 79113038256)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 6433038235008 T^{2} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{4} + 58885538244352 T^{2} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7730440 T - 83046741873200)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{2} - 108121260 T + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 342021536 T + 19\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{2} - 506436684 T - 25\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{2} - 1600485720 T - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 1155235676 T - 44\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} + 30167733456 T + 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{2} - 37318884160 T - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{2} - 59136249780 T - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
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