Properties

Label 1175.2.a.h
Level $1175$
Weight $2$
Character orbit 1175.a
Self dual yes
Analytic conductor $9.382$
Analytic rank $0$
Dimension $7$
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] = [1175,2,Mod(1,1175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1175.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1175 = 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1175.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: \(9.38242223750\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 28x^{3} - 17x^{2} - 19x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 235)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{6} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{8} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{4} q^{6} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{8} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{9} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{11} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{12} + (\beta_{6} + \beta_1) q^{13} + ( - \beta_{6} - 2 \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1) q^{14} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{16} + (\beta_{4} - \beta_{3} - 2) q^{17} + ( - \beta_{6} + \beta_{2} + 2) q^{18} + (2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{19} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{21} + (2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{22} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_1) q^{23} + ( - \beta_{6} - 2 \beta_1 + 1) q^{24} + (\beta_{5} + \beta_{4} - \beta_{3} - 2) q^{26} + (2 \beta_{6} + 2 \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 + 1) q^{27} + (\beta_{5} + 2 \beta_{3} + 3 \beta_{2} - \beta_1 + 5) q^{28} + ( - 2 \beta_1 + 4) q^{29} + (\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2}) q^{31} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{32} + ( - \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{33} + (2 \beta_{6} - \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_1 + 2) q^{34} + ( - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_1 - 6) q^{36} + ( - 2 \beta_{6} - \beta_{4} - 3 \beta_{3} - 2 \beta_1 - 2) q^{37} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{38} + ( - 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{39} + (2 \beta_{4} + 2) q^{41} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{42} + (\beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} + \beta_1 + 6) q^{43} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{44} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} + 2 \beta_1 - 6) q^{46} + q^{47} + (2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 5) q^{48} + ( - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 1) q^{49} + ( - 3 \beta_{5} - \beta_{4} - \beta_{2} + 3 \beta_1 - 3) q^{51} + ( - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{52} + ( - 3 \beta_{5} - \beta_{4} + \beta_{2} - \beta_1 - 1) q^{53} + (\beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{54} + ( - 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 4) q^{56} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{57} + (2 \beta_{2} - 4 \beta_1 + 6) q^{58} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + 1) q^{59} + (\beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{2} + 2 \beta_1 + 3) q^{61} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} - 4) q^{62} + ( - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{63} + ( - 2 \beta_{5} + 3 \beta_1 - 2) q^{64} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - 6) q^{66} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{67} + (\beta_{6} + 2 \beta_{4} - \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{68} + (4 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 4 \beta_1 + 4) q^{69} + (\beta_{6} + \beta_{5} + 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{71} + (2 \beta_{6} + 2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 3) q^{72} + ( - \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - \beta_1) q^{73} + (2 \beta_{6} - \beta_{5} + 5 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 10) q^{74} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{76} + (2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} - 2) q^{77} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{78} + ( - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{81} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{82} + (\beta_{5} - \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 4) q^{83} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 7) q^{84} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 4) q^{86} + (4 \beta_{5} - 2 \beta_{4}) q^{87} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - 2) q^{88} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - \beta_{2} + 4 \beta_1 - 1) q^{89} + (\beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{91} + (2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 2) q^{92} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + 4) q^{93} - \beta_1 q^{94} + (2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 4) q^{96} + ( - \beta_{5} - 3 \beta_{4} - 5 \beta_{2} + \beta_1 + 1) q^{97} + ( - 5 \beta_{4} + 2 \beta_{3} - 5 \beta_1) q^{98} + (2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - q^{2} - q^{3} + 7 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - q^{2} - q^{3} + 7 q^{4} + q^{6} + 3 q^{7} - 3 q^{8} + 10 q^{9} + q^{11} + 4 q^{12} - 2 q^{13} + 10 q^{14} - q^{16} - 12 q^{17} + 17 q^{18} + 3 q^{19} + 7 q^{21} + 5 q^{22} - q^{23} + 8 q^{24} - 13 q^{26} - q^{27} + 27 q^{28} + 26 q^{29} - 5 q^{31} - 12 q^{32} + 15 q^{33} + 6 q^{34} - 34 q^{36} + 7 q^{38} + q^{39} + 12 q^{41} + 18 q^{42} + 33 q^{43} + 3 q^{44} - 36 q^{46} + 7 q^{47} + 25 q^{48} + 6 q^{49} - 14 q^{51} + 11 q^{52} - 4 q^{53} - 2 q^{54} - 27 q^{56} + 21 q^{57} + 38 q^{58} + 13 q^{59} + 20 q^{61} - 15 q^{62} + 10 q^{63} - 9 q^{64} - 41 q^{66} + 32 q^{67} - 15 q^{68} + 16 q^{69} + 11 q^{71} + 8 q^{72} + 8 q^{73} + 48 q^{74} + 11 q^{76} - 29 q^{77} + 17 q^{78} + 17 q^{79} + 15 q^{81} - 6 q^{82} - 19 q^{83} + 46 q^{84} - 30 q^{86} - 2 q^{87} - 7 q^{88} + 13 q^{89} - 11 q^{91} - 16 q^{92} + 29 q^{93} - q^{94} - 29 q^{96} + 12 q^{97} - 6 q^{98} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 28x^{3} - 17x^{2} - 19x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 8\nu^{4} - 2\nu^{3} + 12\nu^{2} + 5\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 8\nu^{4} + 4\nu^{3} - 14\nu^{2} - 13\nu + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 10\nu^{4} - 24\nu^{2} + 3\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 2\nu^{5} - 10\nu^{4} - 18\nu^{3} + 22\nu^{2} + 31\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + 2\beta_{3} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} + \beta_{5} + 9\beta_{4} + 9\beta_{3} + 10\beta_{2} + 19\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{5} + 10\beta_{4} + 20\beta_{3} + 46\beta_{2} + 3\beta _1 + 84 \) 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
2.59170
1.68601
1.49903
0.0980806
−0.730051
−1.89749
−2.24729
−2.59170 0.345837 4.71692 0 −0.896307 3.55345 −7.04146 −2.88040 0
1.2 −1.68601 0.335153 0.842624 0 −0.565071 −4.20242 1.95135 −2.88767 0
1.3 −1.49903 −2.14267 0.247101 0 3.21194 −0.914748 2.62765 1.59105 0
1.4 −0.0980806 3.03215 −1.99038 0 −0.297395 −0.786854 0.391379 6.19391 0
1.5 0.730051 −3.14616 −1.46703 0 −2.29686 0.964297 −2.53111 6.89835 0
1.6 1.89749 −1.57213 1.60046 0 −2.98310 −0.327488 −0.758120 −0.528410 0
1.7 2.24729 2.14783 3.05029 0 4.82679 4.71377 2.36031 1.61318 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(47\) \(-1\)

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 1175.2.a.h 7
5.b even 2 1 235.2.a.e 7
5.c odd 4 2 1175.2.c.g 14
15.d odd 2 1 2115.2.a.v 7
20.d odd 2 1 3760.2.a.bi 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
235.2.a.e 7 5.b even 2 1
1175.2.a.h 7 1.a even 1 1 trivial
1175.2.c.g 14 5.c odd 4 2
2115.2.a.v 7 15.d odd 2 1
3760.2.a.bi 7 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1175))\):

\( T_{2}^{7} + T_{2}^{6} - 10T_{2}^{5} - 8T_{2}^{4} + 28T_{2}^{3} + 17T_{2}^{2} - 19T_{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{7} - T_{11}^{6} - 46T_{11}^{5} + 40T_{11}^{4} + 512T_{11}^{3} - 80T_{11}^{2} - 1408T_{11} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} + T^{6} - 10 T^{5} - 8 T^{4} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( T^{7} + T^{6} - 15 T^{5} - 13 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 3 T^{6} - 23 T^{5} + 53 T^{4} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{7} - T^{6} - 46 T^{5} + 40 T^{4} + \cdots - 256 \) Copy content Toggle raw display
$13$ \( T^{7} + 2 T^{6} - 35 T^{5} - 36 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$17$ \( T^{7} + 12 T^{6} + 15 T^{5} + \cdots + 3424 \) Copy content Toggle raw display
$19$ \( T^{7} - 3 T^{6} - 100 T^{5} + \cdots + 4352 \) Copy content Toggle raw display
$23$ \( T^{7} + T^{6} - 130 T^{5} + \cdots + 152576 \) Copy content Toggle raw display
$29$ \( T^{7} - 26 T^{6} + 248 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( T^{7} + 5 T^{6} - 74 T^{5} + \cdots + 1024 \) Copy content Toggle raw display
$37$ \( T^{7} - 201 T^{5} + 74 T^{4} + \cdots + 397984 \) Copy content Toggle raw display
$41$ \( T^{7} - 12 T^{6} - 36 T^{5} + \cdots - 54784 \) Copy content Toggle raw display
$43$ \( T^{7} - 33 T^{6} + 300 T^{5} + \cdots - 65536 \) Copy content Toggle raw display
$47$ \( (T - 1)^{7} \) Copy content Toggle raw display
$53$ \( T^{7} + 4 T^{6} - 131 T^{5} + \cdots + 16448 \) Copy content Toggle raw display
$59$ \( T^{7} - 13 T^{6} - 197 T^{5} + \cdots - 191656 \) Copy content Toggle raw display
$61$ \( T^{7} - 20 T^{6} - 18 T^{5} + \cdots - 6218 \) Copy content Toggle raw display
$67$ \( T^{7} - 32 T^{6} + 304 T^{5} + \cdots - 200704 \) Copy content Toggle raw display
$71$ \( T^{7} - 11 T^{6} - 227 T^{5} + \cdots + 550688 \) Copy content Toggle raw display
$73$ \( T^{7} - 8 T^{6} - 209 T^{5} + \cdots - 31648 \) Copy content Toggle raw display
$79$ \( T^{7} - 17 T^{6} - 157 T^{5} + \cdots + 1921952 \) Copy content Toggle raw display
$83$ \( T^{7} + 19 T^{6} - 22 T^{5} + \cdots + 2048 \) Copy content Toggle raw display
$89$ \( T^{7} - 13 T^{6} - 187 T^{5} + \cdots + 14252 \) Copy content Toggle raw display
$97$ \( T^{7} - 12 T^{6} - 431 T^{5} + \cdots - 473984 \) Copy content Toggle raw display
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