Properties

Label 825.4.a.o
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $3$
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] = [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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 3 q^{3} + (2 \beta_{2} + 3) q^{4} - 3 \beta_1 q^{6} + ( - \beta_{2} + 4 \beta_1 - 7) q^{7} + ( - 2 \beta_{2} + \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 3 q^{3} + (2 \beta_{2} + 3) q^{4} - 3 \beta_1 q^{6} + ( - \beta_{2} + 4 \beta_1 - 7) q^{7} + ( - 2 \beta_{2} + \beta_1) q^{8} + 9 q^{9} + 11 q^{11} + (6 \beta_{2} + 9) q^{12} + (\beta_{2} - 5 \beta_1 - 13) q^{13} + ( - 7 \beta_{2} + 9 \beta_1 - 44) q^{14} + ( - 16 \beta_{2} + 4 \beta_1 - 35) q^{16} + (3 \beta_{2} + 4 \beta_1 + 19) q^{17} - 9 \beta_1 q^{18} + (8 \beta_{2} + 7 \beta_1 - 56) q^{19} + ( - 3 \beta_{2} + 12 \beta_1 - 21) q^{21} - 11 \beta_1 q^{22} + ( - 4 \beta_{2} - 12 \beta_1 - 49) q^{23} + ( - 6 \beta_{2} + 3 \beta_1) q^{24} + (9 \beta_{2} + 11 \beta_1 + 55) q^{26} + 27 q^{27} + ( - 3 \beta_{2} + 26 \beta_1 - 43) q^{28} + ( - 35 \beta_{2} - 21 \beta_1 - 97) q^{29} + ( - 47 \beta_{2} + 35 \beta_1 - 85) q^{31} + (24 \beta_{2} + 59 \beta_1 - 44) q^{32} + 33 q^{33} + ( - 11 \beta_{2} - 25 \beta_1 - 44) q^{34} + (18 \beta_{2} + 27) q^{36} + (73 \beta_{2} + 55 \beta_1 - 14) q^{37} + ( - 22 \beta_{2} + 40 \beta_1 - 77) q^{38} + (3 \beta_{2} - 15 \beta_1 - 39) q^{39} + ( - 55 \beta_{2} - 32 \beta_1 + 7) q^{41} + ( - 21 \beta_{2} + 27 \beta_1 - 132) q^{42} + ( - 23 \beta_{2} + 17 \beta_1 + 23) q^{43} + (22 \beta_{2} + 33) q^{44} + (28 \beta_{2} + 57 \beta_1 + 132) q^{46} + ( - 61 \beta_{2} + 65 \beta_1 - 132) q^{47} + ( - 48 \beta_{2} + 12 \beta_1 - 105) q^{48} + (35 \beta_{2} - 71 \beta_1 - 107) q^{49} + (9 \beta_{2} + 12 \beta_1 + 57) q^{51} + ( - 39 \beta_{2} - 33 \beta_1 - 17) q^{52} + ( - 34 \beta_{2} - 54 \beta_1 + 56) q^{53} - 27 \beta_1 q^{54} + (7 \beta_{2} - 23 \beta_1 + 66) q^{56} + (24 \beta_{2} + 21 \beta_1 - 168) q^{57} + (77 \beta_{2} + 167 \beta_1 + 231) q^{58} + (37 \beta_{2} - 119 \beta_1 - 176) q^{59} + (32 \beta_{2} + 158 \beta_1 - 388) q^{61} + ( - 23 \beta_{2} + 179 \beta_1 - 385) q^{62} + ( - 9 \beta_{2} + 36 \beta_1 - 63) q^{63} + ( - 14 \beta_{2} - 36 \beta_1 - 369) q^{64} - 33 \beta_1 q^{66} + ( - 44 \beta_{2} + 56 \beta_1 - 138) q^{67} + (37 \beta_{2} + 34 \beta_1 + 123) q^{68} + ( - 12 \beta_{2} - 36 \beta_1 - 147) q^{69} + (118 \beta_{2} + 60 \beta_1 - 467) q^{71} + ( - 18 \beta_{2} + 9 \beta_1) q^{72} + (112 \beta_{2} - 2 \beta_1 + 428) q^{73} + ( - 183 \beta_{2} - 132 \beta_1 - 605) q^{74} + ( - 122 \beta_{2} + 65 \beta_1 + 8) q^{76} + ( - 11 \beta_{2} + 44 \beta_1 - 77) q^{77} + (27 \beta_{2} + 33 \beta_1 + 165) q^{78} + (121 \beta_{2} + 2 \beta_1 - 129) q^{79} + 81 q^{81} + (119 \beta_{2} + 103 \beta_1 + 352) q^{82} + (5 \beta_{2} - 101 \beta_1 + 151) q^{83} + ( - 9 \beta_{2} + 78 \beta_1 - 129) q^{84} + ( - 11 \beta_{2} + 23 \beta_1 - 187) q^{86} + ( - 105 \beta_{2} - 63 \beta_1 - 291) q^{87} + ( - 22 \beta_{2} + 11 \beta_1) q^{88} + (165 \beta_{2} - 163 \beta_1 - 93) q^{89} + ( - 22 \beta_{2} - 140) q^{91} + ( - 110 \beta_{2} - 92 \beta_1 - 235) q^{92} + ( - 141 \beta_{2} + 105 \beta_1 - 255) q^{93} + ( - 69 \beta_{2} + 254 \beta_1 - 715) q^{94} + (72 \beta_{2} + 177 \beta_1 - 132) q^{96} + (62 \beta_{2} + 56 \beta_1 - 419) q^{97} + (107 \beta_{2} + 37 \beta_1 + 781) q^{98} + 99 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 9 q^{3} + 7 q^{4} - 3 q^{6} - 16 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 9 q^{3} + 7 q^{4} - 3 q^{6} - 16 q^{7} + 3 q^{8} + 27 q^{9} + 33 q^{11} + 21 q^{12} - 45 q^{13} - 116 q^{14} - 85 q^{16} + 58 q^{17} - 9 q^{18} - 169 q^{19} - 48 q^{21} - 11 q^{22} - 155 q^{23} + 9 q^{24} + 167 q^{26} + 81 q^{27} - 100 q^{28} - 277 q^{29} - 173 q^{31} - 97 q^{32} + 99 q^{33} - 146 q^{34} + 63 q^{36} - 60 q^{37} - 169 q^{38} - 135 q^{39} + 44 q^{41} - 348 q^{42} + 109 q^{43} + 77 q^{44} + 425 q^{46} - 270 q^{47} - 255 q^{48} - 427 q^{49} + 174 q^{51} - 45 q^{52} + 148 q^{53} - 27 q^{54} + 168 q^{56} - 507 q^{57} + 783 q^{58} - 684 q^{59} - 1038 q^{61} - 953 q^{62} - 144 q^{63} - 1129 q^{64} - 33 q^{66} - 314 q^{67} + 366 q^{68} - 465 q^{69} - 1459 q^{71} + 27 q^{72} + 1170 q^{73} - 1764 q^{74} + 211 q^{76} - 176 q^{77} + 501 q^{78} - 506 q^{79} + 243 q^{81} + 1040 q^{82} + 347 q^{83} - 300 q^{84} - 527 q^{86} - 831 q^{87} + 33 q^{88} - 607 q^{89} - 398 q^{91} - 687 q^{92} - 519 q^{93} - 1822 q^{94} - 291 q^{96} - 1263 q^{97} + 2273 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 15x + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 11 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 11 \) 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
4.03932
0.723686
−3.76300
−4.03932 3.00000 8.31608 0 −12.1180 6.49923 −1.27677 9.00000 0
1.2 −0.723686 3.00000 −7.47628 0 −2.17106 1.13288 11.2000 9.00000 0
1.3 3.76300 3.00000 6.16019 0 11.2890 −23.6321 −6.92320 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.o 3
3.b odd 2 1 2475.4.a.y 3
5.b even 2 1 825.4.a.q yes 3
5.c odd 4 2 825.4.c.n 6
15.d odd 2 1 2475.4.a.v 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.o 3 1.a even 1 1 trivial
825.4.a.q yes 3 5.b even 2 1
825.4.c.n 6 5.c odd 4 2
2475.4.a.v 3 15.d odd 2 1
2475.4.a.y 3 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{3} + T_{2}^{2} - 15T_{2} - 11 \) Copy content Toggle raw display
\( T_{7}^{3} + 16T_{7}^{2} - 173T_{7} + 174 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 15 T - 11 \) Copy content Toggle raw display
$3$ \( (T - 3)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 16 T^{2} - 173 T + 174 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 45 T^{2} + 280 T - 4936 \) Copy content Toggle raw display
$17$ \( T^{3} - 58 T^{2} + 695 T - 2316 \) Copy content Toggle raw display
$19$ \( T^{3} + 169 T^{2} + 7521 T + 41805 \) Copy content Toggle raw display
$23$ \( T^{3} + 155 T^{2} + 5443 T + 40361 \) Copy content Toggle raw display
$29$ \( T^{3} + 277 T^{2} - 4624 T - 1432824 \) Copy content Toggle raw display
$31$ \( T^{3} + 173 T^{2} - 47112 T - 3720456 \) Copy content Toggle raw display
$37$ \( T^{3} + 60 T^{2} - 148235 T - 15147422 \) Copy content Toggle raw display
$41$ \( T^{3} - 44 T^{2} - 72861 T + 2957382 \) Copy content Toggle raw display
$43$ \( T^{3} - 109 T^{2} - 9648 T + 367740 \) Copy content Toggle raw display
$47$ \( T^{3} + 270 T^{2} - 103415 T - 3504204 \) Copy content Toggle raw display
$53$ \( T^{3} - 148 T^{2} + \cdots + 10118048 \) Copy content Toggle raw display
$59$ \( T^{3} + 684 T^{2} + \cdots - 84071174 \) Copy content Toggle raw display
$61$ \( T^{3} + 1038 T^{2} + \cdots - 137885416 \) Copy content Toggle raw display
$67$ \( T^{3} + 314 T^{2} - 47428 T - 1599464 \) Copy content Toggle raw display
$71$ \( T^{3} + 1459 T^{2} + \cdots - 46945715 \) Copy content Toggle raw display
$73$ \( T^{3} - 1170 T^{2} + \cdots + 70378904 \) Copy content Toggle raw display
$79$ \( T^{3} + 506 T^{2} - 183457 T + 8353348 \) Copy content Toggle raw display
$83$ \( T^{3} - 347 T^{2} - 116064 T + 6792264 \) Copy content Toggle raw display
$89$ \( T^{3} + 607 T^{2} + \cdots - 262736468 \) Copy content Toggle raw display
$97$ \( T^{3} + 1263 T^{2} + \cdots + 10470601 \) Copy content Toggle raw display
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