Properties

Label 2475.4.a.p
Level $2475$
Weight $4$
Character orbit 2475.a
Self dual yes
Analytic conductor $146.030$
Analytic rank $0$
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] = [2475,4,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(146.029727264\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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 \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 16) q^{4} + (4 \beta - 14) q^{7} + (9 \beta + 24) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 16) q^{4} + (4 \beta - 14) q^{7} + (9 \beta + 24) q^{8} + 11 q^{11} + ( - 2 \beta - 14) q^{13} + ( - 10 \beta + 96) q^{14} + (25 \beta + 88) q^{16} + ( - 14 \beta + 60) q^{17} + ( - 2 \beta + 26) q^{19} + 11 \beta q^{22} + ( - 10 \beta + 72) q^{23} + ( - 16 \beta - 48) q^{26} + (54 \beta - 128) q^{28} + (6 \beta + 96) q^{29} + (8 \beta + 176) q^{31} + (41 \beta + 408) q^{32} + (46 \beta - 336) q^{34} + ( - 52 \beta + 190) q^{37} + (24 \beta - 48) q^{38} + (14 \beta + 384) q^{41} + (26 \beta - 206) q^{43} + (11 \beta + 176) q^{44} + (62 \beta - 240) q^{46} + (74 \beta + 96) q^{47} + ( - 96 \beta + 237) q^{49} + ( - 48 \beta - 272) q^{52} + ( - 54 \beta - 234) q^{53} + (6 \beta + 528) q^{56} + (102 \beta + 144) q^{58} + (100 \beta + 36) q^{59} + (34 \beta - 406) q^{61} + (184 \beta + 192) q^{62} + (249 \beta + 280) q^{64} + (96 \beta + 340) q^{67} + ( - 178 \beta + 624) q^{68} + ( - 54 \beta - 288) q^{71} + (28 \beta - 662) q^{73} + (138 \beta - 1248) q^{74} + ( - 8 \beta + 368) q^{76} + (44 \beta - 154) q^{77} + (144 \beta + 254) q^{79} + (398 \beta + 336) q^{82} + (156 \beta - 240) q^{83} + ( - 180 \beta + 624) q^{86} + (99 \beta + 264) q^{88} + ( - 72 \beta + 414) q^{89} + ( - 36 \beta + 4) q^{91} + ( - 98 \beta + 912) q^{92} + (170 \beta + 1776) q^{94} + ( - 192 \beta + 322) q^{97} + (141 \beta - 2304) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 33 q^{4} - 24 q^{7} + 57 q^{8} + 22 q^{11} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} + 50 q^{19} + 11 q^{22} + 134 q^{23} - 112 q^{26} - 202 q^{28} + 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 328 q^{37} - 72 q^{38} + 782 q^{41} - 386 q^{43} + 363 q^{44} - 418 q^{46} + 266 q^{47} + 378 q^{49} - 592 q^{52} - 522 q^{53} + 1062 q^{56} + 390 q^{58} + 172 q^{59} - 778 q^{61} + 568 q^{62} + 809 q^{64} + 776 q^{67} + 1070 q^{68} - 630 q^{71} - 1296 q^{73} - 2358 q^{74} + 728 q^{76} - 264 q^{77} + 652 q^{79} + 1070 q^{82} - 324 q^{83} + 1068 q^{86} + 627 q^{88} + 756 q^{89} - 28 q^{91} + 1726 q^{92} + 3722 q^{94} + 452 q^{97} - 4467 q^{98}+O(q^{100}) \) 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
−4.42443
5.42443
−4.42443 0 11.5756 0 0 −31.6977 −15.8199 0 0
1.2 5.42443 0 21.4244 0 0 7.69772 72.8199 0 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 2475.4.a.p 2
3.b odd 2 1 825.4.a.l 2
5.b even 2 1 99.4.a.f 2
15.d odd 2 1 33.4.a.c 2
15.e even 4 2 825.4.c.h 4
20.d odd 2 1 1584.4.a.bj 2
55.d odd 2 1 1089.4.a.u 2
60.h even 2 1 528.4.a.p 2
105.g even 2 1 1617.4.a.k 2
120.i odd 2 1 2112.4.a.bn 2
120.m even 2 1 2112.4.a.bg 2
165.d even 2 1 363.4.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.c 2 15.d odd 2 1
99.4.a.f 2 5.b even 2 1
363.4.a.i 2 165.d even 2 1
528.4.a.p 2 60.h even 2 1
825.4.a.l 2 3.b odd 2 1
825.4.c.h 4 15.e even 4 2
1089.4.a.u 2 55.d odd 2 1
1584.4.a.bj 2 20.d odd 2 1
1617.4.a.k 2 105.g even 2 1
2112.4.a.bg 2 120.m even 2 1
2112.4.a.bn 2 120.i odd 2 1
2475.4.a.p 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2475))\):

\( T_{2}^{2} - T_{2} - 24 \) Copy content Toggle raw display
\( T_{7}^{2} + 24T_{7} - 244 \) Copy content Toggle raw display
\( T_{29}^{2} - 198T_{29} + 8928 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 24 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 24T - 244 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 30T + 128 \) Copy content Toggle raw display
$17$ \( T^{2} - 106T - 1944 \) Copy content Toggle raw display
$19$ \( T^{2} - 50T + 528 \) Copy content Toggle raw display
$23$ \( T^{2} - 134T + 2064 \) Copy content Toggle raw display
$29$ \( T^{2} - 198T + 8928 \) Copy content Toggle raw display
$31$ \( T^{2} - 360T + 30848 \) Copy content Toggle raw display
$37$ \( T^{2} - 328T - 38676 \) Copy content Toggle raw display
$41$ \( T^{2} - 782T + 148128 \) Copy content Toggle raw display
$43$ \( T^{2} + 386T + 20856 \) Copy content Toggle raw display
$47$ \( T^{2} - 266T - 115104 \) Copy content Toggle raw display
$53$ \( T^{2} + 522T - 2592 \) Copy content Toggle raw display
$59$ \( T^{2} - 172T - 235104 \) Copy content Toggle raw display
$61$ \( T^{2} + 778T + 123288 \) Copy content Toggle raw display
$67$ \( T^{2} - 776T - 72944 \) Copy content Toggle raw display
$71$ \( T^{2} + 630T + 28512 \) Copy content Toggle raw display
$73$ \( T^{2} + 1296 T + 400892 \) Copy content Toggle raw display
$79$ \( T^{2} - 652T - 396572 \) Copy content Toggle raw display
$83$ \( T^{2} + 324T - 563904 \) Copy content Toggle raw display
$89$ \( T^{2} - 756T + 17172 \) Copy content Toggle raw display
$97$ \( T^{2} - 452T - 842876 \) Copy content Toggle raw display
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