Properties

Label 176.2.m.d
Level $176$
Weight $2$
Character orbit 176.m
Analytic conductor $1.405$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [176,2,Mod(49,176)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(176, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("176.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 176.m (of order \(5\), degree \(4\), not 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: \(1.40536707557\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.682515625.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + (2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3}) q^{5} + (2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 2) q^{7} + (\beta_{5} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 1) q^{11} + (\beta_{6} - \beta_{5} + \beta_{3} + 1) q^{13} + ( - \beta_{7} - \beta_{6} - 3 \beta_{3} + \beta_1 + 1) q^{15} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{17} + ( - 4 \beta_{7} + 4 \beta_{3} - 3 \beta_{2} + 3) q^{19} + ( - 3 \beta_{7} - \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{21} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{2}) q^{23} + (6 \beta_{2} + \beta_1 - 6) q^{25} + (5 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{27} + ( - 2 \beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2) q^{29} + (\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{31} + (5 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 5 \beta_{2} + \beta_1 - 1) q^{33} + (2 \beta_{6} + \beta_{5} - 5 \beta_{3} + 7 \beta_{2} - 5) q^{35} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + 5 \beta_{3} - \beta_{2} + 2 \beta_1 + 2) q^{37} + ( - 7 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 5 \beta_{3} - 7 \beta_{2} - 3 \beta_1) q^{39} + ( - 2 \beta_{7} - \beta_{6} + \beta_{4} + 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 4) q^{41} + ( - \beta_{5} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_1 + 6) q^{45} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{47} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + 2 \beta_1) q^{49} + (8 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + \cdots - 8) q^{51}+ \cdots + ( - 3 \beta_{7} + 4 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 3 q^{5} - 7 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 3 q^{5} - 7 q^{7} - 13 q^{9} - 7 q^{11} + 7 q^{13} + 13 q^{15} + q^{17} + 2 q^{19} - 2 q^{21} + 4 q^{23} - 33 q^{25} + 22 q^{27} + 17 q^{29} + 13 q^{31} + 16 q^{33} - 11 q^{35} + q^{37} - 39 q^{39} + 9 q^{41} - 6 q^{43} + 44 q^{45} + q^{47} + 3 q^{49} - 38 q^{51} - 33 q^{53} - 13 q^{55} - 6 q^{57} + 30 q^{59} - 9 q^{61} + 10 q^{63} - 10 q^{65} + 10 q^{67} - 38 q^{69} + 25 q^{71} - 7 q^{73} + 6 q^{75} - 7 q^{77} + q^{79} + 14 q^{81} + 39 q^{85} + 6 q^{87} + 22 q^{89} - 7 q^{91} - 5 q^{93} - 7 q^{95} + 8 q^{97} + 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} + 2x^{5} + 19x^{4} + 28x^{3} + 100x^{2} + 88x + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 528 \nu^{7} + 2098 \nu^{6} - 15725 \nu^{5} + 33439 \nu^{4} + 71401 \nu^{3} - 332708 \nu^{2} + 319181 \nu + 440220 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5794 \nu^{7} - 9973 \nu^{6} - 30517 \nu^{5} + 195125 \nu^{4} - 61888 \nu^{3} + 104068 \nu^{2} + 501961 \nu + 528473 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7409 \nu^{7} - 59487 \nu^{6} + 183537 \nu^{5} - 171974 \nu^{4} - 58164 \nu^{3} - 77439 \nu^{2} + 18601 \nu - 701074 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8817 \nu^{7} + 16927 \nu^{6} - 106264 \nu^{5} + 200474 \nu^{4} + 521745 \nu^{3} + 380907 \nu^{2} + 2179908 \nu + 2809884 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11971 \nu^{7} + 3536 \nu^{6} + 58156 \nu^{5} - 228404 \nu^{4} - 102852 \nu^{3} - 979996 \nu^{2} - 1085964 \nu - 2305776 ) / 1168519 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 13790 \nu^{7} + 57068 \nu^{6} - 113608 \nu^{5} + 65418 \nu^{4} - 266949 \nu^{3} + 6060 \nu^{2} - 742824 \nu + 665808 ) / 1168519 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{3} - 5\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 6\beta_{6} + 6\beta_{5} + 2\beta_{4} + 4\beta_{3} - 10\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} + 10\beta_{6} + 13\beta_{5} + 13\beta_{4} + 14\beta_{3} - 13\beta_{2} - 10\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 43\beta_{7} + 25\beta_{5} + 49\beta_{4} + 18\beta_{2} - 25\beta _1 - 62 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 97\beta_{7} - 92\beta_{6} + 92\beta_{4} - 97\beta_{3} + 221\beta_{2} - 44\beta _1 - 221 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -449\beta_{6} - 260\beta_{5} - 412\beta_{3} + 896\beta_{2} - 412 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/176\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(133\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.20316 0.874145i
2.51217 + 1.82520i
0.581882 1.79085i
−0.390899 + 1.20306i
−1.20316 + 0.874145i
2.51217 1.82520i
0.581882 + 1.79085i
−0.390899 1.20306i
0 −0.743592 0.540251i 0 −1.24359 + 3.82738i 0 −3.01217 + 2.18847i 0 −0.665993 2.04972i 0
49.2 0 1.55261 + 1.12804i 0 1.05261 3.23960i 0 0.703158 0.510874i 0 0.211078 + 0.649631i 0
81.1 0 −0.941506 + 2.89766i 0 −1.44151 + 1.04732i 0 −0.109101 0.335777i 0 −5.08293 3.69296i 0
81.2 0 0.632489 1.94660i 0 0.132489 0.0962586i 0 −1.08188 3.32969i 0 −0.962157 0.699048i 0
97.1 0 −0.743592 + 0.540251i 0 −1.24359 3.82738i 0 −3.01217 2.18847i 0 −0.665993 + 2.04972i 0
97.2 0 1.55261 1.12804i 0 1.05261 + 3.23960i 0 0.703158 + 0.510874i 0 0.211078 0.649631i 0
113.1 0 −0.941506 2.89766i 0 −1.44151 1.04732i 0 −0.109101 + 0.335777i 0 −5.08293 + 3.69296i 0
113.2 0 0.632489 + 1.94660i 0 0.132489 + 0.0962586i 0 −1.08188 + 3.32969i 0 −0.962157 + 0.699048i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.2.m.d 8
4.b odd 2 1 88.2.i.b 8
8.b even 2 1 704.2.m.i 8
8.d odd 2 1 704.2.m.l 8
11.c even 5 1 inner 176.2.m.d 8
11.c even 5 1 1936.2.a.bb 4
11.d odd 10 1 1936.2.a.bc 4
12.b even 2 1 792.2.r.g 8
44.c even 2 1 968.2.i.p 8
44.g even 10 1 968.2.a.m 4
44.g even 10 1 968.2.i.p 8
44.g even 10 2 968.2.i.t 8
44.h odd 10 1 88.2.i.b 8
44.h odd 10 1 968.2.a.n 4
44.h odd 10 2 968.2.i.s 8
88.k even 10 1 7744.2.a.dh 4
88.l odd 10 1 704.2.m.l 8
88.l odd 10 1 7744.2.a.di 4
88.o even 10 1 704.2.m.i 8
88.o even 10 1 7744.2.a.dr 4
88.p odd 10 1 7744.2.a.ds 4
132.n odd 10 1 8712.2.a.cd 4
132.o even 10 1 792.2.r.g 8
132.o even 10 1 8712.2.a.ce 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.i.b 8 4.b odd 2 1
88.2.i.b 8 44.h odd 10 1
176.2.m.d 8 1.a even 1 1 trivial
176.2.m.d 8 11.c even 5 1 inner
704.2.m.i 8 8.b even 2 1
704.2.m.i 8 88.o even 10 1
704.2.m.l 8 8.d odd 2 1
704.2.m.l 8 88.l odd 10 1
792.2.r.g 8 12.b even 2 1
792.2.r.g 8 132.o even 10 1
968.2.a.m 4 44.g even 10 1
968.2.a.n 4 44.h odd 10 1
968.2.i.p 8 44.c even 2 1
968.2.i.p 8 44.g even 10 1
968.2.i.s 8 44.h odd 10 2
968.2.i.t 8 44.g even 10 2
1936.2.a.bb 4 11.c even 5 1
1936.2.a.bc 4 11.d odd 10 1
7744.2.a.dh 4 88.k even 10 1
7744.2.a.di 4 88.l odd 10 1
7744.2.a.dr 4 88.o even 10 1
7744.2.a.ds 4 88.p odd 10 1
8712.2.a.cd 4 132.n odd 10 1
8712.2.a.ce 4 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{7} + 10T_{3}^{6} - 19T_{3}^{5} + 49T_{3}^{4} - 29T_{3}^{3} + 20T_{3}^{2} + 99T_{3} + 121 \) acting on \(S_{2}^{\mathrm{new}}(176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + 10 T^{6} - 19 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{8} + 3 T^{7} + 26 T^{6} + 54 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} + 7 T^{7} + 30 T^{6} + 62 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 7 T^{7} + 23 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 7 T^{7} + 20 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$17$ \( T^{8} - T^{7} + 22 T^{6} - 153 T^{5} + \cdots + 39601 \) Copy content Toggle raw display
$19$ \( (T^{4} - T^{3} + 16 T^{2} - 66 T + 121)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} - 52 T^{2} + 128 T - 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 17 T^{7} + 130 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$31$ \( T^{8} - 13 T^{7} + 96 T^{6} + \cdots + 55696 \) Copy content Toggle raw display
$37$ \( T^{8} - T^{7} + 110 T^{6} + 726 T^{5} + \cdots + 1936 \) Copy content Toggle raw display
$41$ \( T^{8} - 9 T^{7} + 14 T^{6} + 267 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{3} - 15 T^{2} - 44 T - 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + 12 T^{6} - 78 T^{5} + \cdots + 30976 \) Copy content Toggle raw display
$53$ \( T^{8} + 33 T^{7} + 536 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$59$ \( (T^{4} - 15 T^{3} + 100 T^{2} - 250 T + 625)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 9 T^{7} + 164 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$67$ \( (T^{4} - 5 T^{3} - 101 T^{2} - 260 T - 176)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 25 T^{7} + 402 T^{6} + \cdots + 512656 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + 130 T^{6} + \cdots + 609961 \) Copy content Toggle raw display
$79$ \( T^{8} - T^{7} + 204 T^{6} + \cdots + 712336 \) Copy content Toggle raw display
$83$ \( T^{8} + 167 T^{6} - 1680 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{4} - 11 T^{3} + 27 T^{2} + 36 T - 124)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 8 T^{7} + 11 T^{6} + \cdots + 241081 \) Copy content Toggle raw display
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