Properties

Label 1575.4.a.s
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
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] = [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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta + 2) q^{4} + 7 q^{7} + (5 \beta - 10) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta + 2) q^{4} + 7 q^{7} + (5 \beta - 10) q^{8} + (10 \beta + 23) q^{11} + ( - 8 \beta + 30) q^{13} - 7 \beta q^{14} + ( - 3 \beta - 66) q^{16} + ( - 5 \beta + 54) q^{17} + (7 \beta - 32) q^{19} + ( - 33 \beta - 100) q^{22} + (5 \beta + 13) q^{23} + ( - 22 \beta + 80) q^{26} + (7 \beta + 14) q^{28} + (27 \beta + 193) q^{29} + (66 \beta - 114) q^{31} + (29 \beta + 110) q^{32} + ( - 49 \beta + 50) q^{34} + (57 \beta + 9) q^{37} + (25 \beta - 70) q^{38} + (61 \beta + 222) q^{41} + (77 \beta - 75) q^{43} + (53 \beta + 146) q^{44} + ( - 18 \beta - 50) q^{46} + ( - 108 \beta - 58) q^{47} + 49 q^{49} + (6 \beta - 20) q^{52} + ( - 86 \beta + 174) q^{53} + (35 \beta - 70) q^{56} + ( - 220 \beta - 270) q^{58} + ( - 210 \beta + 10) q^{59} + (54 \beta + 468) q^{61} + (48 \beta - 660) q^{62} + ( - 115 \beta + 238) q^{64} + ( - 166 \beta + 537) q^{67} + (39 \beta + 58) q^{68} + (303 \beta - 215) q^{71} + ( - 269 \beta - 34) q^{73} + ( - 66 \beta - 570) q^{74} + ( - 11 \beta + 6) q^{76} + (70 \beta + 161) q^{77} + ( - 41 \beta - 539) q^{79} + ( - 283 \beta - 610) q^{82} + (69 \beta + 724) q^{83} + ( - 2 \beta - 770) q^{86} + (65 \beta + 270) q^{88} + (17 \beta + 848) q^{89} + ( - 56 \beta + 210) q^{91} + (28 \beta + 76) q^{92} + (166 \beta + 1080) q^{94} + ( - 272 \beta + 1018) q^{97} - 49 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + 5 q^{4} + 14 q^{7} - 15 q^{8} + 56 q^{11} + 52 q^{13} - 7 q^{14} - 135 q^{16} + 103 q^{17} - 57 q^{19} - 233 q^{22} + 31 q^{23} + 138 q^{26} + 35 q^{28} + 413 q^{29} - 162 q^{31} + 249 q^{32} + 51 q^{34} + 75 q^{37} - 115 q^{38} + 505 q^{41} - 73 q^{43} + 345 q^{44} - 118 q^{46} - 224 q^{47} + 98 q^{49} - 34 q^{52} + 262 q^{53} - 105 q^{56} - 760 q^{58} - 190 q^{59} + 990 q^{61} - 1272 q^{62} + 361 q^{64} + 908 q^{67} + 155 q^{68} - 127 q^{71} - 337 q^{73} - 1206 q^{74} + q^{76} + 392 q^{77} - 1119 q^{79} - 1503 q^{82} + 1517 q^{83} - 1542 q^{86} + 605 q^{88} + 1713 q^{89} + 364 q^{91} + 180 q^{92} + 2326 q^{94} + 1764 q^{97} - 49 q^{98}+O(q^{100}) \) 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
3.70156
−2.70156
−3.70156 0 5.70156 0 0 7.00000 8.50781 0 0
1.2 2.70156 0 −0.701562 0 0 7.00000 −23.5078 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.s 2
3.b odd 2 1 175.4.a.e yes 2
5.b even 2 1 1575.4.a.v 2
15.d odd 2 1 175.4.a.d 2
15.e even 4 2 175.4.b.d 4
21.c even 2 1 1225.4.a.t 2
105.g even 2 1 1225.4.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.4.a.d 2 15.d odd 2 1
175.4.a.e yes 2 3.b odd 2 1
175.4.b.d 4 15.e even 4 2
1225.4.a.r 2 105.g even 2 1
1225.4.a.t 2 21.c even 2 1
1575.4.a.s 2 1.a even 1 1 trivial
1575.4.a.v 2 5.b even 2 1

Hecke kernels

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

\( T_{2}^{2} + T_{2} - 10 \) Copy content Toggle raw display
\( T_{11}^{2} - 56T_{11} - 241 \) Copy content Toggle raw display
\( T_{13}^{2} - 52T_{13} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 10 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 56T - 241 \) Copy content Toggle raw display
$13$ \( T^{2} - 52T + 20 \) Copy content Toggle raw display
$17$ \( T^{2} - 103T + 2396 \) Copy content Toggle raw display
$19$ \( T^{2} + 57T + 310 \) Copy content Toggle raw display
$23$ \( T^{2} - 31T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 413T + 35170 \) Copy content Toggle raw display
$31$ \( T^{2} + 162T - 38088 \) Copy content Toggle raw display
$37$ \( T^{2} - 75T - 31896 \) Copy content Toggle raw display
$41$ \( T^{2} - 505T + 25616 \) Copy content Toggle raw display
$43$ \( T^{2} + 73T - 59440 \) Copy content Toggle raw display
$47$ \( T^{2} + 224T - 107012 \) Copy content Toggle raw display
$53$ \( T^{2} - 262T - 58648 \) Copy content Toggle raw display
$59$ \( T^{2} + 190T - 443000 \) Copy content Toggle raw display
$61$ \( T^{2} - 990T + 215136 \) Copy content Toggle raw display
$67$ \( T^{2} - 908T - 76333 \) Copy content Toggle raw display
$71$ \( T^{2} + 127T - 937010 \) Copy content Toggle raw display
$73$ \( T^{2} + 337T - 713308 \) Copy content Toggle raw display
$79$ \( T^{2} + 1119 T + 295810 \) Copy content Toggle raw display
$83$ \( T^{2} - 1517 T + 526522 \) Copy content Toggle raw display
$89$ \( T^{2} - 1713 T + 730630 \) Copy content Toggle raw display
$97$ \( T^{2} - 1764T + 19588 \) Copy content Toggle raw display
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