Properties

Label 825.4.a.u
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.91035289.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 3 q^{3} + ( - \beta_{3} - 4) q^{4} + 3 \beta_{3} q^{6} + ( - \beta_{3} + \beta_{2} - 3) q^{7} + ( - 11 \beta_{3} - 4) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 3 q^{3} + ( - \beta_{3} - 4) q^{4} + 3 \beta_{3} q^{6} + ( - \beta_{3} + \beta_{2} - 3) q^{7} + ( - 11 \beta_{3} - 4) q^{8} + 9 q^{9} - 11 q^{11} + ( - 3 \beta_{3} - 12) q^{12} + ( - 9 \beta_{3} + \beta_{2} + \beta_1 + 2) q^{13} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{14} + (15 \beta_{3} - 12) q^{16} + (19 \beta_{3} + \beta_{2} + 31) q^{17} + 9 \beta_{3} q^{18} + ( - 2 \beta_{2} - \beta_1 + 29) q^{19} + ( - 3 \beta_{3} + 3 \beta_{2} - 9) q^{21} - 11 \beta_{3} q^{22} + ( - 26 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 15) q^{23} + ( - 33 \beta_{3} - 12) q^{24} + (9 \beta_{3} + 3 \beta_{2} + \beta_1 - 34) q^{26} + 27 q^{27} + (6 \beta_{3} - 3 \beta_{2} - \beta_1 + 14) q^{28} + (17 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 72) q^{29} + ( - 15 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 42) q^{31} + (61 \beta_{3} + 92) q^{32} - 33 q^{33} + (12 \beta_{3} - \beta_{2} + \beta_1 + 78) q^{34} + ( - 9 \beta_{3} - 36) q^{36} + (75 \beta_{3} + 3 \beta_{2} - \beta_1 + 143) q^{37} + (31 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{38} + ( - 27 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{39} + ( - \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 27) q^{41} + ( - 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{42} + (109 \beta_{3} - \beta_{2} + 3 \beta_1 + 58) q^{43} + (11 \beta_{3} + 44) q^{44} + (15 \beta_{3} - 10 \beta_{2} + 2 \beta_1 - 100) q^{46} + (151 \beta_{3} - 3 \beta_{2} + \beta_1 + 241) q^{47} + (45 \beta_{3} - 36) q^{48} + ( - 75 \beta_{3} - 5 \beta_{2} - \beta_1 + 212) q^{49} + (57 \beta_{3} + 3 \beta_{2} + 93) q^{51} + (27 \beta_{3} - 7 \beta_{2} - 5 \beta_1 + 26) q^{52} + (18 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 18) q^{53} + 27 \beta_{3} q^{54} + (26 \beta_{3} + 7 \beta_{2} - 11 \beta_1 + 34) q^{56} + ( - 6 \beta_{2} - 3 \beta_1 + 87) q^{57} + (61 \beta_{3} - 15 \beta_{2} + 3 \beta_1 + 74) q^{58} + ( - 85 \beta_{3} + 15 \beta_{2} + 5 \beta_1 + 199) q^{59} + (36 \beta_{3} - 18 \beta_{2} + 4 \beta_1 - 74) q^{61} + ( - 21 \beta_{3} - 9 \beta_{2} - 3 \beta_1 - 66) q^{62} + ( - 9 \beta_{3} + 9 \beta_{2} - 27) q^{63} + ( - 89 \beta_{3} + 340) q^{64} - 33 \beta_{3} q^{66} + (76 \beta_{3} + 6 \beta_{2} + 14 \beta_1 + 106) q^{67} + ( - 88 \beta_{3} - 3 \beta_{2} - \beta_1 - 202) q^{68} + ( - 78 \beta_{3} + 6 \beta_{2} - 6 \beta_1 - 45) q^{69} + (86 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 425) q^{71} + ( - 99 \beta_{3} - 36) q^{72} + (96 \beta_{3} - 2 \beta_{2} - 20 \beta_1 - 94) q^{73} + (70 \beta_{3} - 7 \beta_{2} + 3 \beta_1 + 306) q^{74} + ( - 31 \beta_{3} + 10 \beta_{2} + 6 \beta_1 - 112) q^{76} + (11 \beta_{3} - 11 \beta_{2} + 33) q^{77} + (27 \beta_{3} + 9 \beta_{2} + 3 \beta_1 - 102) q^{78} + ( - 199 \beta_{3} + 11 \beta_{2} + 2 \beta_1 + 545) q^{79} + 81 q^{81} + ( - 34 \beta_{3} + 19 \beta_{2} - 3 \beta_1 - 10) q^{82} + ( - 257 \beta_{3} - 5 \beta_{2} + 7 \beta_1 - 370) q^{83} + (18 \beta_{3} - 9 \beta_{2} - 3 \beta_1 + 42) q^{84} + ( - 57 \beta_{3} + 13 \beta_{2} - \beta_1 + 434) q^{86} + (51 \beta_{3} + 9 \beta_{2} - 9 \beta_1 + 216) q^{87} + (121 \beta_{3} + 44) q^{88} + ( - 207 \beta_{3} - 15 \beta_{2} - 5 \beta_1 + 156) q^{89} + (482 \beta_{3} + 6 \beta_{2} - 12 \beta_1 + 780) q^{91} + (89 \beta_{3} + 2 \beta_{2} + 6 \beta_1 + 160) q^{92} + ( - 45 \beta_{3} - 9 \beta_{2} - 9 \beta_1 - 126) q^{93} + (88 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 598) q^{94} + (183 \beta_{3} + 276) q^{96} + ( - 12 \beta_{3} + 32 \beta_{2} - 10 \beta_1 - 41) q^{97} + (289 \beta_{3} + \beta_{2} - 5 \beta_1 - 310) q^{98} - 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 12 q^{3} - 14 q^{4} - 6 q^{6} - 11 q^{7} + 6 q^{8} + 36 q^{9} - 44 q^{11} - 42 q^{12} + 25 q^{13} - 3 q^{14} - 78 q^{16} + 85 q^{17} - 18 q^{18} + 118 q^{19} - 33 q^{21} + 22 q^{22} - 10 q^{23} + 18 q^{24} - 157 q^{26} + 108 q^{27} + 47 q^{28} + 251 q^{29} - 135 q^{31} + 246 q^{32} - 132 q^{33} + 289 q^{34} - 126 q^{36} + 419 q^{37} - 76 q^{38} + 75 q^{39} - 103 q^{41} - 9 q^{42} + 15 q^{43} + 154 q^{44} - 420 q^{46} + 665 q^{47} - 234 q^{48} + 1003 q^{49} + 255 q^{51} + 57 q^{52} - 116 q^{53} - 54 q^{54} + 77 q^{56} + 354 q^{57} + 189 q^{58} + 951 q^{59} - 350 q^{61} - 213 q^{62} - 99 q^{63} + 1538 q^{64} + 66 q^{66} + 266 q^{67} - 629 q^{68} - 30 q^{69} + 1526 q^{71} + 54 q^{72} - 566 q^{73} + 1091 q^{74} - 396 q^{76} + 121 q^{77} - 471 q^{78} + 2567 q^{79} + 324 q^{81} + 9 q^{82} - 961 q^{83} + 141 q^{84} + 1837 q^{86} + 753 q^{87} - 66 q^{88} + 1053 q^{89} + 2150 q^{91} + 460 q^{92} - 405 q^{93} + 2209 q^{94} + 738 q^{96} - 172 q^{97} - 1819 q^{98} - 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 285x^{2} + 286x + 19616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 150\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} - \nu - 150 ) / 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 56\beta_{3} + \beta _1 + 602 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 56\beta_{3} + 56\beta_{2} + 151\beta _1 + 902 ) / 4 \) 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
11.1952
−10.1952
−12.6191
13.6191
−2.56155 3.00000 −1.43845 0 −7.68466 −29.1157 24.1771 9.00000 0
1.2 −2.56155 3.00000 −1.43845 0 −7.68466 25.6772 24.1771 9.00000 0
1.3 1.56155 3.00000 −5.56155 0 4.68466 −24.2670 −21.1771 9.00000 0
1.4 1.56155 3.00000 −5.56155 0 4.68466 16.7054 −21.1771 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 825.4.a.u 4
3.b odd 2 1 2475.4.a.bd 4
5.b even 2 1 825.4.a.v yes 4
5.c odd 4 2 825.4.c.q 8
15.d odd 2 1 2475.4.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.u 4 1.a even 1 1 trivial
825.4.a.v yes 4 5.b even 2 1
825.4.c.q 8 5.c odd 4 2
2475.4.a.bb 4 15.d odd 2 1
2475.4.a.bd 4 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{2} + T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{4} + 11T_{7}^{3} - 1127T_{7}^{2} - 7047T_{7} + 303074 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$3$ \( (T - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 11 T^{3} - 1127 T^{2} + \cdots + 303074 \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 25 T^{3} - 5940 T^{2} + \cdots - 1706944 \) Copy content Toggle raw display
$17$ \( T^{4} - 85 T^{3} - 1693 T^{2} + \cdots - 1313888 \) Copy content Toggle raw display
$19$ \( T^{4} - 118 T^{3} - 3664 T^{2} + \cdots - 6797601 \) Copy content Toggle raw display
$23$ \( T^{4} + 10 T^{3} - 29068 T^{2} + \cdots + 20292331 \) Copy content Toggle raw display
$29$ \( T^{4} - 251 T^{3} + \cdots - 42323912 \) Copy content Toggle raw display
$31$ \( T^{4} + 135 T^{3} + \cdots + 92275200 \) Copy content Toggle raw display
$37$ \( T^{4} - 419 T^{3} + \cdots - 881580586 \) Copy content Toggle raw display
$41$ \( T^{4} + 103 T^{3} + \cdots + 1343813418 \) Copy content Toggle raw display
$43$ \( T^{4} - 15 T^{3} + \cdots + 795036672 \) Copy content Toggle raw display
$47$ \( T^{4} - 665 T^{3} + \cdots + 1019998564 \) Copy content Toggle raw display
$53$ \( T^{4} + 116 T^{3} + \cdots + 3274784192 \) Copy content Toggle raw display
$59$ \( T^{4} - 951 T^{3} + \cdots - 37958294854 \) Copy content Toggle raw display
$61$ \( T^{4} + 350 T^{3} + \cdots + 10189762592 \) Copy content Toggle raw display
$67$ \( T^{4} - 266 T^{3} + \cdots + 122607629152 \) Copy content Toggle raw display
$71$ \( T^{4} - 1526 T^{3} + \cdots + 6876641459 \) Copy content Toggle raw display
$73$ \( T^{4} + 566 T^{3} + \cdots + 625767485216 \) Copy content Toggle raw display
$79$ \( T^{4} - 2567 T^{3} + \cdots - 51952998816 \) Copy content Toggle raw display
$83$ \( T^{4} + 961 T^{3} + \cdots - 55004349600 \) Copy content Toggle raw display
$89$ \( T^{4} - 1053 T^{3} + \cdots - 134213353376 \) Copy content Toggle raw display
$97$ \( T^{4} + 172 T^{3} + \cdots + 27014010329 \) Copy content Toggle raw display
show more
show less