Properties

Label 1575.4.a.ba
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
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] = [1575,4,Mod(1,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.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: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.14360.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 17x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 7 q^{7} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 4) q^{4} - 7 q^{7} + ( - 3 \beta_{2} + \beta_1 - 4) q^{8} + ( - \beta_{2} - 3 \beta_1 + 25) q^{11} + ( - 5 \beta_{2} + 13 \beta_1 - 13) q^{13} + ( - 7 \beta_1 + 7) q^{14} + ( - \beta_{2} - 11 \beta_1 - 26) q^{16} + (11 \beta_{2} + 13 \beta_1 - 21) q^{17} + (6 \beta_{2} - 10 \beta_1 + 54) q^{19} + ( - \beta_{2} + 20 \beta_1 - 61) q^{22} + (2 \beta_{2} - 14 \beta_1 - 42) q^{23} + (23 \beta_{2} - 38 \beta_1 + 141) q^{26} + ( - 7 \beta_{2} + 7 \beta_1 - 28) q^{28} + ( - 17 \beta_{2} - 19 \beta_1 - 105) q^{29} + ( - 4 \beta_{2} + 24 \beta_1 + 108) q^{31} + (15 \beta_{2} - 39 \beta_1 - 66) q^{32} + ( - 9 \beta_{2} + 34 \beta_1 + 197) q^{34} + (12 \beta_{2} - 16 \beta_1 + 14) q^{37} + ( - 22 \beta_{2} + 84 \beta_1 - 146) q^{38} + ( - 2 \beta_{2} - 10 \beta_1 - 120) q^{41} + (34 \beta_{2} - 30 \beta_1 - 6) q^{43} + (30 \beta_{2} - 42 \beta_1 + 78) q^{44} + ( - 18 \beta_{2} - 32 \beta_1 - 106) q^{46} + ( - 13 \beta_{2} - 51 \beta_1 - 239) q^{47} + 49 q^{49} + ( - 44 \beta_{2} + 152 \beta_1 - 386) q^{52} + ( - 22 \beta_{2} - 130 \beta_1 + 44) q^{53} + (21 \beta_{2} - 7 \beta_1 + 28) q^{56} + (15 \beta_{2} - 190 \beta_1 - 155) q^{58} + ( - 48 \beta_{2} + 176 \beta_1 + 76) q^{59} + ( - 26 \beta_{2} - 34 \beta_1 + 416) q^{61} + (32 \beta_{2} + 88 \beta_1 + 144) q^{62} + ( - 61 \beta_{2} + 97 \beta_1 - 110) q^{64} + ( - 108 \beta_{2} + 12 \beta_1 - 32) q^{67} + ( - 36 \beta_{2} + 48 \beta_1 + 318) q^{68} + (40 \beta_{2} - 72 \beta_1 + 32) q^{71} + ( - 76 \beta_{2} - 124 \beta_1 - 78) q^{73} + ( - 40 \beta_{2} + 74 \beta_1 - 154) q^{74} + (80 \beta_{2} - 176 \beta_1 + 572) q^{76} + (7 \beta_{2} + 21 \beta_1 - 175) q^{77} + ( - 89 \beta_{2} - 83 \beta_1 - 315) q^{79} + ( - 6 \beta_{2} - 130 \beta_1 + 4) q^{82} + (8 \beta_{2} - 160 \beta_1 - 556) q^{83} + ( - 98 \beta_{2} + 164 \beta_1 - 222) q^{86} + ( - 94 \beta_{2} + 68 \beta_1 + 38) q^{88} + (82 \beta_{2} + 98 \beta_1 - 108) q^{89} + (35 \beta_{2} - 91 \beta_1 + 91) q^{91} + ( - 12 \beta_{2} - 84 \beta_1 + 36) q^{92} + ( - 25 \beta_{2} - 304 \beta_1 - 361) q^{94} + (65 \beta_{2} + 87 \beta_1 - 55) q^{97} + (49 \beta_1 - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 13 q^{4} - 21 q^{7} - 15 q^{8} + 74 q^{11} - 44 q^{13} + 21 q^{14} - 79 q^{16} - 52 q^{17} + 168 q^{19} - 184 q^{22} - 124 q^{23} + 446 q^{26} - 91 q^{28} - 332 q^{29} + 320 q^{31} - 183 q^{32} + 582 q^{34} + 54 q^{37} - 460 q^{38} - 362 q^{41} + 16 q^{43} + 264 q^{44} - 336 q^{46} - 730 q^{47} + 147 q^{49} - 1202 q^{52} + 110 q^{53} + 105 q^{56} - 450 q^{58} + 180 q^{59} + 1222 q^{61} + 464 q^{62} - 391 q^{64} - 204 q^{67} + 918 q^{68} + 136 q^{71} - 310 q^{73} - 502 q^{74} + 1796 q^{76} - 518 q^{77} - 1034 q^{79} + 6 q^{82} - 1660 q^{83} - 764 q^{86} + 20 q^{88} - 242 q^{89} + 308 q^{91} + 96 q^{92} - 1108 q^{94} - 100 q^{97} - 147 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 17x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 11 \) 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
−3.62456
−0.861086
4.48565
−4.62456 0 13.3866 0 0 −7.00000 −24.9107 0 0
1.2 −1.86109 0 −4.53636 0 0 −7.00000 23.3312 0 0
1.3 3.48565 0 4.14976 0 0 −7.00000 −13.4206 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( +1 \)

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 1575.4.a.ba 3
3.b odd 2 1 175.4.a.f 3
5.b even 2 1 315.4.a.p 3
15.d odd 2 1 35.4.a.c 3
15.e even 4 2 175.4.b.e 6
21.c even 2 1 1225.4.a.y 3
35.c odd 2 1 2205.4.a.bm 3
60.h even 2 1 560.4.a.u 3
105.g even 2 1 245.4.a.l 3
105.o odd 6 2 245.4.e.m 6
105.p even 6 2 245.4.e.n 6
120.i odd 2 1 2240.4.a.bt 3
120.m even 2 1 2240.4.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.c 3 15.d odd 2 1
175.4.a.f 3 3.b odd 2 1
175.4.b.e 6 15.e even 4 2
245.4.a.l 3 105.g even 2 1
245.4.e.m 6 105.o odd 6 2
245.4.e.n 6 105.p even 6 2
315.4.a.p 3 5.b even 2 1
560.4.a.u 3 60.h even 2 1
1225.4.a.y 3 21.c even 2 1
1575.4.a.ba 3 1.a even 1 1 trivial
2205.4.a.bm 3 35.c odd 2 1
2240.4.a.bt 3 120.i odd 2 1
2240.4.a.bv 3 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{3} + 3T_{2}^{2} - 14T_{2} - 30 \) Copy content Toggle raw display
\( T_{11}^{3} - 74T_{11}^{2} + 1577T_{11} - 7692 \) Copy content Toggle raw display
\( T_{13}^{3} + 44T_{13}^{2} - 3491T_{13} + 44870 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3 T^{2} + \cdots - 30 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( (T + 7)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 74 T^{2} + \cdots - 7692 \) Copy content Toggle raw display
$13$ \( T^{3} + 44 T^{2} + \cdots + 44870 \) Copy content Toggle raw display
$17$ \( T^{3} + 52 T^{2} + \cdots - 56706 \) Copy content Toggle raw display
$19$ \( T^{3} - 168 T^{2} + \cdots - 28720 \) Copy content Toggle raw display
$23$ \( T^{3} + 124 T^{2} + \cdots - 94368 \) Copy content Toggle raw display
$29$ \( T^{3} + 332 T^{2} + \cdots - 2565450 \) Copy content Toggle raw display
$31$ \( T^{3} - 320 T^{2} + \cdots + 50176 \) Copy content Toggle raw display
$37$ \( T^{3} - 54 T^{2} + \cdots - 25736 \) Copy content Toggle raw display
$41$ \( T^{3} + 362 T^{2} + \cdots + 1536192 \) Copy content Toggle raw display
$43$ \( T^{3} - 16 T^{2} + \cdots + 1524560 \) Copy content Toggle raw display
$47$ \( T^{3} + 730 T^{2} + \cdots + 4968912 \) Copy content Toggle raw display
$53$ \( T^{3} - 110 T^{2} + \cdots + 90318336 \) Copy content Toggle raw display
$59$ \( T^{3} - 180 T^{2} + \cdots + 202459200 \) Copy content Toggle raw display
$61$ \( T^{3} - 1222 T^{2} + \cdots - 38393792 \) Copy content Toggle raw display
$67$ \( T^{3} + 204 T^{2} + \cdots - 324944128 \) Copy content Toggle raw display
$71$ \( T^{3} - 136 T^{2} + \cdots - 15575040 \) Copy content Toggle raw display
$73$ \( T^{3} + 310 T^{2} + \cdots - 48718616 \) Copy content Toggle raw display
$79$ \( T^{3} + 1034 T^{2} + \cdots - 343615600 \) Copy content Toggle raw display
$83$ \( T^{3} + 1660 T^{2} + \cdots - 42727104 \) Copy content Toggle raw display
$89$ \( T^{3} + 242 T^{2} + \cdots + 6359520 \) Copy content Toggle raw display
$97$ \( T^{3} + 100 T^{2} + \cdots + 1978018 \) Copy content Toggle raw display
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