Properties

Label 1575.4.a.p
Level $1575$
Weight $4$
Character orbit 1575.a
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (3 \beta + 7) q^{4} - 7 q^{7} + ( - 5 \beta - 41) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (3 \beta + 7) q^{4} - 7 q^{7} + ( - 5 \beta - 41) q^{8} + ( - 10 \beta + 8) q^{11} + (12 \beta - 14) q^{13} + (7 \beta + 7) q^{14} + (27 \beta + 55) q^{16} + ( - 2 \beta - 2) q^{17} + ( - 24 \beta + 44) q^{19} + (12 \beta + 132) q^{22} + ( - 34 \beta + 20) q^{23} + ( - 10 \beta - 154) q^{26} + ( - 21 \beta - 49) q^{28} + ( - 24 \beta + 138) q^{29} + (72 \beta - 16) q^{31} + ( - 69 \beta - 105) q^{32} + (6 \beta + 30) q^{34} + (36 \beta + 106) q^{37} + (4 \beta + 292) q^{38} + (30 \beta + 210) q^{41} + (48 \beta - 212) q^{43} + ( - 76 \beta - 364) q^{44} + (48 \beta + 456) q^{46} + (68 \beta - 40) q^{47} + 49 q^{49} + (78 \beta + 406) q^{52} + (4 \beta - 554) q^{53} + (35 \beta + 287) q^{56} + ( - 90 \beta + 198) q^{58} + (116 \beta - 460) q^{59} + (72 \beta - 250) q^{61} + ( - 128 \beta - 992) q^{62} + (27 \beta + 631) q^{64} + ( - 108 \beta - 20) q^{67} + ( - 26 \beta - 98) q^{68} + (30 \beta - 492) q^{71} + ( - 12 \beta - 530) q^{73} + ( - 178 \beta - 610) q^{74} + ( - 108 \beta - 700) q^{76} + (70 \beta - 56) q^{77} + ( - 108 \beta - 232) q^{79} + ( - 270 \beta - 630) q^{82} + (96 \beta + 924) q^{83} + (116 \beta - 460) q^{86} + (420 \beta + 372) q^{88} + (142 \beta - 254) q^{89} + ( - 84 \beta + 98) q^{91} + ( - 280 \beta - 1288) q^{92} + ( - 96 \beta - 912) q^{94} + ( - 276 \beta - 266) q^{97} + ( - 49 \beta - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 17 q^{4} - 14 q^{7} - 87 q^{8} + 6 q^{11} - 16 q^{13} + 21 q^{14} + 137 q^{16} - 6 q^{17} + 64 q^{19} + 276 q^{22} + 6 q^{23} - 318 q^{26} - 119 q^{28} + 252 q^{29} + 40 q^{31} - 279 q^{32} + 66 q^{34} + 248 q^{37} + 588 q^{38} + 450 q^{41} - 376 q^{43} - 804 q^{44} + 960 q^{46} - 12 q^{47} + 98 q^{49} + 890 q^{52} - 1104 q^{53} + 609 q^{56} + 306 q^{58} - 804 q^{59} - 428 q^{61} - 2112 q^{62} + 1289 q^{64} - 148 q^{67} - 222 q^{68} - 954 q^{71} - 1072 q^{73} - 1398 q^{74} - 1508 q^{76} - 42 q^{77} - 572 q^{79} - 1530 q^{82} + 1944 q^{83} - 804 q^{86} + 1164 q^{88} - 366 q^{89} + 112 q^{91} - 2856 q^{92} - 1920 q^{94} - 808 q^{97} - 147 q^{98}+O(q^{100}) \) 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
4.27492
−3.27492
−5.27492 0 19.8248 0 0 −7.00000 −62.3746 0 0
1.2 2.27492 0 −2.82475 0 0 −7.00000 −24.6254 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.p 2
3.b odd 2 1 525.4.a.n 2
5.b even 2 1 63.4.a.e 2
15.d odd 2 1 21.4.a.c 2
15.e even 4 2 525.4.d.g 4
20.d odd 2 1 1008.4.a.ba 2
35.c odd 2 1 441.4.a.r 2
35.i odd 6 2 441.4.e.p 4
35.j even 6 2 441.4.e.q 4
60.h even 2 1 336.4.a.m 2
105.g even 2 1 147.4.a.i 2
105.o odd 6 2 147.4.e.l 4
105.p even 6 2 147.4.e.m 4
120.i odd 2 1 1344.4.a.bg 2
120.m even 2 1 1344.4.a.bo 2
420.o odd 2 1 2352.4.a.bz 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.c 2 15.d odd 2 1
63.4.a.e 2 5.b even 2 1
147.4.a.i 2 105.g even 2 1
147.4.e.l 4 105.o odd 6 2
147.4.e.m 4 105.p even 6 2
336.4.a.m 2 60.h even 2 1
441.4.a.r 2 35.c odd 2 1
441.4.e.p 4 35.i odd 6 2
441.4.e.q 4 35.j even 6 2
525.4.a.n 2 3.b odd 2 1
525.4.d.g 4 15.e even 4 2
1008.4.a.ba 2 20.d odd 2 1
1344.4.a.bg 2 120.i odd 2 1
1344.4.a.bo 2 120.m even 2 1
1575.4.a.p 2 1.a even 1 1 trivial
2352.4.a.bz 2 420.o odd 2 1

Hecke kernels

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

\( T_{2}^{2} + 3T_{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} - 1416 \) Copy content Toggle raw display
\( T_{13}^{2} + 16T_{13} - 1988 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 12 \) 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
$11$ \( T^{2} - 6T - 1416 \) Copy content Toggle raw display
$13$ \( T^{2} + 16T - 1988 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T - 48 \) Copy content Toggle raw display
$19$ \( T^{2} - 64T - 7184 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T - 16464 \) Copy content Toggle raw display
$29$ \( T^{2} - 252T + 7668 \) Copy content Toggle raw display
$31$ \( T^{2} - 40T - 73472 \) Copy content Toggle raw display
$37$ \( T^{2} - 248T - 3092 \) Copy content Toggle raw display
$41$ \( T^{2} - 450T + 37800 \) Copy content Toggle raw display
$43$ \( T^{2} + 376T + 2512 \) Copy content Toggle raw display
$47$ \( T^{2} + 12T - 65856 \) Copy content Toggle raw display
$53$ \( T^{2} + 1104 T + 304476 \) Copy content Toggle raw display
$59$ \( T^{2} + 804T - 30144 \) Copy content Toggle raw display
$61$ \( T^{2} + 428T - 28076 \) Copy content Toggle raw display
$67$ \( T^{2} + 148T - 160736 \) Copy content Toggle raw display
$71$ \( T^{2} + 954T + 214704 \) Copy content Toggle raw display
$73$ \( T^{2} + 1072 T + 285244 \) Copy content Toggle raw display
$79$ \( T^{2} + 572T - 84416 \) Copy content Toggle raw display
$83$ \( T^{2} - 1944 T + 813456 \) Copy content Toggle raw display
$89$ \( T^{2} + 366T - 253848 \) Copy content Toggle raw display
$97$ \( T^{2} + 808T - 922292 \) Copy content Toggle raw display
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