Properties

Label 975.2.bc.j
Level $975$
Weight $2$
Character orbit 975.bc
Analytic conductor $7.785$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(751,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.751");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bc (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.191102976.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 6x^{4} + 36x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 6x^{4} + 36x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 9\nu^{4} + 18\nu^{2} + 42 ) / 30 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 4\nu^{4} + 12\nu^{2} - 42 ) / 30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 13\nu^{4} + 36\nu^{2} + 24 ) / 30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 9\nu^{5} - 3\nu^{3} - 72\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 18\nu^{5} - 51\nu^{3} - 9\nu ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} + 21\nu^{5} - 12\nu^{3} - 198\nu ) / 90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 12\nu^{3} ) / 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} + 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{3} - 12\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{7} - 10\beta_{6} - 2\beta_{5} + 14\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36\beta_{3} - 18\beta_{2} - 60\beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18\beta_{7} - 36\beta_{6} + 12\beta_{5} + 48\beta_{4} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1 - \beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.291439 + 1.08766i
−2.10121 + 0.563016i
2.10121 0.563016i
−0.291439 1.08766i
0.291439 1.08766i
−2.10121 0.563016i
2.10121 + 0.563016i
−0.291439 + 1.08766i
−1.88389 + 1.08766i −0.500000 0.866025i 1.36603 2.36603i 0 1.88389 + 1.08766i 0.383889 + 0.221638i 1.59245i −0.500000 + 0.866025i 0
751.2 −0.975173 + 0.563016i −0.500000 0.866025i −0.366025 + 0.633975i 0 0.975173 + 0.563016i −0.524827 0.303009i 3.07638i −0.500000 + 0.866025i 0
751.3 0.975173 0.563016i −0.500000 0.866025i −0.366025 + 0.633975i 0 −0.975173 0.563016i −2.47517 1.42904i 3.07638i −0.500000 + 0.866025i 0
751.4 1.88389 1.08766i −0.500000 0.866025i 1.36603 2.36603i 0 −1.88389 1.08766i −3.38389 1.95369i 1.59245i −0.500000 + 0.866025i 0
901.1 −1.88389 1.08766i −0.500000 + 0.866025i 1.36603 + 2.36603i 0 1.88389 1.08766i 0.383889 0.221638i 1.59245i −0.500000 0.866025i 0
901.2 −0.975173 0.563016i −0.500000 + 0.866025i −0.366025 0.633975i 0 0.975173 0.563016i −0.524827 + 0.303009i 3.07638i −0.500000 0.866025i 0
901.3 0.975173 + 0.563016i −0.500000 + 0.866025i −0.366025 0.633975i 0 −0.975173 + 0.563016i −2.47517 + 1.42904i 3.07638i −0.500000 0.866025i 0
901.4 1.88389 + 1.08766i −0.500000 + 0.866025i 1.36603 + 2.36603i 0 −1.88389 + 1.08766i −3.38389 + 1.95369i 1.59245i −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bc.j 8
5.b even 2 1 195.2.bb.b 8
5.c odd 4 1 975.2.w.h 8
5.c odd 4 1 975.2.w.i 8
13.e even 6 1 inner 975.2.bc.j 8
15.d odd 2 1 585.2.bu.d 8
65.l even 6 1 195.2.bb.b 8
65.r odd 12 1 975.2.w.h 8
65.r odd 12 1 975.2.w.i 8
65.s odd 12 1 2535.2.a.bj 4
65.s odd 12 1 2535.2.a.bk 4
195.y odd 6 1 585.2.bu.d 8
195.bh even 12 1 7605.2.a.ch 4
195.bh even 12 1 7605.2.a.ci 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.b 8 5.b even 2 1
195.2.bb.b 8 65.l even 6 1
585.2.bu.d 8 15.d odd 2 1
585.2.bu.d 8 195.y odd 6 1
975.2.w.h 8 5.c odd 4 1
975.2.w.h 8 65.r odd 12 1
975.2.w.i 8 5.c odd 4 1
975.2.w.i 8 65.r odd 12 1
975.2.bc.j 8 1.a even 1 1 trivial
975.2.bc.j 8 13.e even 6 1 inner
2535.2.a.bj 4 65.s odd 12 1
2535.2.a.bk 4 65.s odd 12 1
7605.2.a.ch 4 195.bh even 12 1
7605.2.a.ci 4 195.bh even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{8} - 6T_{2}^{6} + 30T_{2}^{4} - 36T_{2}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{8} + 12T_{7}^{7} + 60T_{7}^{6} + 144T_{7}^{5} + 147T_{7}^{4} - 36T_{7}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 6 T^{6} + 30 T^{4} - 36 T^{2} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{7} + 60 T^{6} + 144 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{6} + 120 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + 16 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 18 T^{6} + 270 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} - 576 T^{5} + \cdots + 197136 \) Copy content Toggle raw display
$23$ \( T^{8} + 60 T^{6} + 288 T^{5} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{7} + 114 T^{6} + \cdots + 54756 \) Copy content Toggle raw display
$31$ \( T^{8} + 132 T^{6} + 4422 T^{4} + \cdots + 1089 \) Copy content Toggle raw display
$37$ \( T^{8} - 96 T^{6} + 8688 T^{4} + \cdots + 278784 \) Copy content Toggle raw display
$41$ \( T^{8} - 36 T^{7} + 570 T^{6} + \cdots + 49284 \) Copy content Toggle raw display
$43$ \( T^{8} + 16 T^{7} + 220 T^{6} + \cdots + 2319529 \) Copy content Toggle raw display
$47$ \( T^{8} + 180 T^{6} + 9408 T^{4} + \cdots + 427716 \) Copy content Toggle raw display
$53$ \( (T^{4} - 144 T^{2} + 3456)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 36 T^{7} + 522 T^{6} + \cdots + 4435236 \) Copy content Toggle raw display
$61$ \( (T^{4} - 16 T^{3} + 195 T^{2} - 976 T + 3721)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 48 T^{7} + 1020 T^{6} + \cdots + 962361 \) Copy content Toggle raw display
$71$ \( T^{8} - 36 T^{7} + 462 T^{6} + \cdots + 1313316 \) Copy content Toggle raw display
$73$ \( T^{8} + 264 T^{6} + 20250 T^{4} + \cdots + 1390041 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 102 T^{2} - 904 T - 1703)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 132 T^{2} + 4056)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 36 T^{7} + 462 T^{6} + \cdots + 4356 \) Copy content Toggle raw display
$97$ \( T^{8} - 180 T^{6} + \cdots + 26101881 \) Copy content Toggle raw display
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