Properties

Label 1575.4.a.o
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: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (3 \beta - 3) q^{4} - 7 q^{7} + (5 \beta - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (3 \beta - 3) q^{4} - 7 q^{7} + (5 \beta - 1) q^{8} + ( - 5 \beta + 18) q^{11} + (17 \beta + 11) q^{13} + (7 \beta + 7) q^{14} + ( - 33 \beta + 5) q^{16} + ( - 27 \beta + 53) q^{17} + (56 \beta - 56) q^{19} + ( - 8 \beta + 2) q^{22} + ( - 24 \beta - 115) q^{23} + ( - 45 \beta - 79) q^{26} + ( - 21 \beta + 21) q^{28} + ( - 84 \beta + 73) q^{29} + ( - 13 \beta - 61) q^{31} + (21 \beta + 135) q^{32} + (\beta + 55) q^{34} + ( - 19 \beta + 66) q^{37} + ( - 56 \beta - 168) q^{38} + ( - 45 \beta - 95) q^{41} + (58 \beta + 373) q^{43} + (54 \beta - 114) q^{44} + (163 \beta + 211) q^{46} + (88 \beta - 120) q^{47} + 49 q^{49} + (33 \beta + 171) q^{52} + (89 \beta - 119) q^{53} + ( - 35 \beta + 7) q^{56} + (95 \beta + 263) q^{58} + ( - 209 \beta + 325) q^{59} + ( - 273 \beta + 25) q^{61} + (87 \beta + 113) q^{62} + (87 \beta - 259) q^{64} + ( - 123 \beta + 640) q^{67} + (159 \beta - 483) q^{68} + ( - 235 \beta - 192) q^{71} + (88 \beta + 90) q^{73} + ( - 28 \beta + 10) q^{74} + ( - 168 \beta + 840) q^{76} + (35 \beta - 126) q^{77} + ( - 103 \beta - 162) q^{79} + (185 \beta + 275) q^{82} + (431 \beta - 821) q^{83} + ( - 489 \beta - 605) q^{86} + (70 \beta - 118) q^{88} + (522 \beta - 494) q^{89} + ( - 119 \beta - 77) q^{91} + ( - 345 \beta + 57) q^{92} + ( - 56 \beta - 232) q^{94} + (704 \beta - 266) q^{97} + ( - 49 \beta - 49) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 3 q^{4} - 14 q^{7} + 3 q^{8} + 31 q^{11} + 39 q^{13} + 21 q^{14} - 23 q^{16} + 79 q^{17} - 56 q^{19} - 4 q^{22} - 254 q^{23} - 203 q^{26} + 21 q^{28} + 62 q^{29} - 135 q^{31} + 291 q^{32} + 111 q^{34} + 113 q^{37} - 392 q^{38} - 235 q^{41} + 804 q^{43} - 174 q^{44} + 585 q^{46} - 152 q^{47} + 98 q^{49} + 375 q^{52} - 149 q^{53} - 21 q^{56} + 621 q^{58} + 441 q^{59} - 223 q^{61} + 313 q^{62} - 431 q^{64} + 1157 q^{67} - 807 q^{68} - 619 q^{71} + 268 q^{73} - 8 q^{74} + 1512 q^{76} - 217 q^{77} - 427 q^{79} + 735 q^{82} - 1211 q^{83} - 1699 q^{86} - 166 q^{88} - 466 q^{89} - 273 q^{91} - 231 q^{92} - 520 q^{94} + 172 q^{97} - 147 q^{98}+O(q^{100}) \) 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
2.56155
−1.56155
−3.56155 0 4.68466 0 0 −7.00000 11.8078 0 0
1.2 0.561553 0 −7.68466 0 0 −7.00000 −8.80776 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.o 2
3.b odd 2 1 525.4.a.m yes 2
5.b even 2 1 1575.4.a.x 2
15.d odd 2 1 525.4.a.j 2
15.e even 4 2 525.4.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.4.a.j 2 15.d odd 2 1
525.4.a.m yes 2 3.b odd 2 1
525.4.d.m 4 15.e even 4 2
1575.4.a.o 2 1.a even 1 1 trivial
1575.4.a.x 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1575))\):

\( T_{2}^{2} + 3T_{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 31T_{11} + 134 \) Copy content Toggle raw display
\( T_{13}^{2} - 39T_{13} - 848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T - 2 \) 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
$11$ \( T^{2} - 31T + 134 \) Copy content Toggle raw display
$13$ \( T^{2} - 39T - 848 \) Copy content Toggle raw display
$17$ \( T^{2} - 79T - 1538 \) Copy content Toggle raw display
$19$ \( T^{2} + 56T - 12544 \) Copy content Toggle raw display
$23$ \( T^{2} + 254T + 13681 \) Copy content Toggle raw display
$29$ \( T^{2} - 62T - 29027 \) Copy content Toggle raw display
$31$ \( T^{2} + 135T + 3838 \) Copy content Toggle raw display
$37$ \( T^{2} - 113T + 1658 \) Copy content Toggle raw display
$41$ \( T^{2} + 235T + 5200 \) Copy content Toggle raw display
$43$ \( T^{2} - 804T + 147307 \) Copy content Toggle raw display
$47$ \( T^{2} + 152T - 27136 \) Copy content Toggle raw display
$53$ \( T^{2} + 149T - 28114 \) Copy content Toggle raw display
$59$ \( T^{2} - 441T - 137024 \) Copy content Toggle raw display
$61$ \( T^{2} + 223T - 304316 \) Copy content Toggle raw display
$67$ \( T^{2} - 1157 T + 270364 \) Copy content Toggle raw display
$71$ \( T^{2} + 619T - 138916 \) Copy content Toggle raw display
$73$ \( T^{2} - 268T - 14956 \) Copy content Toggle raw display
$79$ \( T^{2} + 427T + 494 \) Copy content Toggle raw display
$83$ \( T^{2} + 1211 T - 422854 \) Copy content Toggle raw display
$89$ \( T^{2} + 466 T - 1103768 \) Copy content Toggle raw display
$97$ \( T^{2} - 172 T - 2098972 \) Copy content Toggle raw display
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