Properties

Label 175.2.k.a
Level $175$
Weight $2$
Character orbit 175.k
Analytic conductor $1.397$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{3} + ( 1 - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} - q^{6} + ( -2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{8} + ( 2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{3} + ( 1 - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} - q^{6} + ( -2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{8} + ( 2 \zeta_{24} + 2 \zeta_{24}^{7} ) q^{9} + ( -2 - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{11} + ( \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{12} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{13} + ( -1 - 2 \zeta_{24} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{14} -3 \zeta_{24}^{4} q^{16} + ( 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{18} + ( -2 \zeta_{24} - 2 \zeta_{24}^{7} ) q^{19} + ( 4 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{21} + 2 \zeta_{24}^{6} q^{22} + ( -\zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{23} + ( 1 + 2 \zeta_{24}^{3} - \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{24} + ( -4 \zeta_{24} + 6 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{26} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{27} + ( \zeta_{24} - 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{28} + q^{29} + ( 6 - 6 \zeta_{24}^{4} ) q^{31} + ( -3 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{32} + ( 4 \zeta_{24} + 6 \zeta_{24}^{2} - 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{33} + ( -6 + 4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} ) q^{34} + ( -8 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{36} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{38} + ( 2 - 2 \zeta_{24}^{4} ) q^{39} + ( -5 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{41} + ( 2 \zeta_{24} + \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{42} + ( -\zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{43} + ( -2 \zeta_{24} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{44} + ( -1 + \zeta_{24}^{4} ) q^{46} + ( -2 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{47} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{6} ) q^{48} + ( 2 \zeta_{24} + 4 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{49} + 2 \zeta_{24}^{4} q^{51} + ( -10 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{52} + ( 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{53} + ( -2 \zeta_{24} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{54} + ( 5 - 6 \zeta_{24} - 4 \zeta_{24}^{3} - \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{56} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{57} + ( \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{58} + ( -4 - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{59} + ( 6 \zeta_{24} + 3 \zeta_{24}^{4} + 6 \zeta_{24}^{7} ) q^{61} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{62} + ( 2 \zeta_{24} - 8 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{63} + ( 7 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{64} + ( 2 + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{66} + ( 11 \zeta_{24}^{2} + \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{67} + ( -6 \zeta_{24} + 10 \zeta_{24}^{2} - 10 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{68} + ( 3 + 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{69} + ( -4 - 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{71} + ( -6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{72} + ( -2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{73} + ( 8 - 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{76} + ( 4 \zeta_{24} - 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{77} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{78} + ( -2 \zeta_{24} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{7} ) q^{79} + ( 1 + 6 \zeta_{24}^{3} - \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{81} + ( -7 \zeta_{24} + 9 \zeta_{24}^{2} - 9 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{82} + ( 9 \zeta_{24} - 9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + \zeta_{24}^{6} ) q^{83} + ( -5 + 2 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{84} + ( 4 \zeta_{24} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{86} + ( -\zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{87} + ( -2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{88} + ( 4 \zeta_{24} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{89} + ( -6 + 4 \zeta_{24} + 4 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 4 \zeta_{24}^{5} ) q^{91} + ( \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{92} + ( -6 \zeta_{24} - 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} + 6 \zeta_{24}^{7} ) q^{93} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{94} + ( 4 \zeta_{24} + 5 \zeta_{24}^{4} + 4 \zeta_{24}^{7} ) q^{96} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} ) q^{97} + ( -4 \zeta_{24} + 9 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 8 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{98} + ( -8 - 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} - 8q^{6} + O(q^{10}) \) \( 8q + 4q^{4} - 8q^{6} - 8q^{11} + 8q^{14} - 12q^{16} + 28q^{21} + 4q^{24} + 24q^{26} + 8q^{29} + 24q^{31} - 48q^{34} - 64q^{36} + 8q^{39} - 40q^{41} - 24q^{44} - 4q^{46} - 20q^{49} + 8q^{51} + 12q^{54} + 36q^{56} - 16q^{59} + 12q^{61} + 56q^{64} + 8q^{66} + 24q^{69} - 32q^{71} + 64q^{76} + 48q^{79} + 4q^{81} - 24q^{84} - 12q^{86} - 12q^{89} - 56q^{91} + 8q^{94} + 20q^{96} - 64q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{24}\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−2.09077 + 1.20711i 0.358719 + 0.207107i 1.91421 3.31552i 0 −1.00000 0.358719 2.62132i 4.41421i −1.41421 2.44949i 0
74.2 −0.358719 + 0.207107i 2.09077 + 1.20711i −0.914214 + 1.58346i 0 −1.00000 2.09077 1.62132i 1.58579i 1.41421 + 2.44949i 0
74.3 0.358719 0.207107i −2.09077 1.20711i −0.914214 + 1.58346i 0 −1.00000 −2.09077 + 1.62132i 1.58579i 1.41421 + 2.44949i 0
74.4 2.09077 1.20711i −0.358719 0.207107i 1.91421 3.31552i 0 −1.00000 −0.358719 + 2.62132i 4.41421i −1.41421 2.44949i 0
149.1 −2.09077 1.20711i 0.358719 0.207107i 1.91421 + 3.31552i 0 −1.00000 0.358719 + 2.62132i 4.41421i −1.41421 + 2.44949i 0
149.2 −0.358719 0.207107i 2.09077 1.20711i −0.914214 1.58346i 0 −1.00000 2.09077 + 1.62132i 1.58579i 1.41421 2.44949i 0
149.3 0.358719 + 0.207107i −2.09077 + 1.20711i −0.914214 1.58346i 0 −1.00000 −2.09077 1.62132i 1.58579i 1.41421 2.44949i 0
149.4 2.09077 + 1.20711i −0.358719 + 0.207107i 1.91421 + 3.31552i 0 −1.00000 −0.358719 2.62132i 4.41421i −1.41421 + 2.44949i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.k.a 8
5.b even 2 1 inner 175.2.k.a 8
5.c odd 4 1 35.2.e.a 4
5.c odd 4 1 175.2.e.c 4
7.c even 3 1 inner 175.2.k.a 8
7.c even 3 1 1225.2.b.g 4
7.d odd 6 1 1225.2.b.h 4
15.e even 4 1 315.2.j.e 4
20.e even 4 1 560.2.q.k 4
35.f even 4 1 245.2.e.e 4
35.i odd 6 1 1225.2.b.h 4
35.j even 6 1 inner 175.2.k.a 8
35.j even 6 1 1225.2.b.g 4
35.k even 12 1 245.2.a.g 2
35.k even 12 1 245.2.e.e 4
35.k even 12 1 1225.2.a.m 2
35.l odd 12 1 35.2.e.a 4
35.l odd 12 1 175.2.e.c 4
35.l odd 12 1 245.2.a.h 2
35.l odd 12 1 1225.2.a.k 2
105.w odd 12 1 2205.2.a.q 2
105.x even 12 1 315.2.j.e 4
105.x even 12 1 2205.2.a.n 2
140.w even 12 1 560.2.q.k 4
140.w even 12 1 3920.2.a.bq 2
140.x odd 12 1 3920.2.a.bv 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 5.c odd 4 1
35.2.e.a 4 35.l odd 12 1
175.2.e.c 4 5.c odd 4 1
175.2.e.c 4 35.l odd 12 1
175.2.k.a 8 1.a even 1 1 trivial
175.2.k.a 8 5.b even 2 1 inner
175.2.k.a 8 7.c even 3 1 inner
175.2.k.a 8 35.j even 6 1 inner
245.2.a.g 2 35.k even 12 1
245.2.a.h 2 35.l odd 12 1
245.2.e.e 4 35.f even 4 1
245.2.e.e 4 35.k even 12 1
315.2.j.e 4 15.e even 4 1
315.2.j.e 4 105.x even 12 1
560.2.q.k 4 20.e even 4 1
560.2.q.k 4 140.w even 12 1
1225.2.a.k 2 35.l odd 12 1
1225.2.a.m 2 35.k even 12 1
1225.2.b.g 4 7.c even 3 1
1225.2.b.g 4 35.j even 6 1
1225.2.b.h 4 7.d odd 6 1
1225.2.b.h 4 35.i odd 6 1
2205.2.a.n 2 105.x even 12 1
2205.2.a.q 2 105.w odd 12 1
3920.2.a.bq 2 140.w even 12 1
3920.2.a.bv 2 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 6 T_{2}^{6} + 35 T_{2}^{4} - 6 T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(175, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8} \)
$3$ \( 1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( 2401 + 490 T^{2} + 51 T^{4} + 10 T^{6} + T^{8} \)
$11$ \( ( 16 - 16 T + 20 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$13$ \( ( 16 + 24 T^{2} + T^{4} )^{2} \)
$17$ \( 256 - 384 T^{2} + 560 T^{4} - 24 T^{6} + T^{8} \)
$19$ \( ( 64 + 8 T^{2} + T^{4} )^{2} \)
$23$ \( 1 - 6 T^{2} + 35 T^{4} - 6 T^{6} + T^{8} \)
$29$ \( ( -1 + T )^{8} \)
$31$ \( ( 36 - 6 T + T^{2} )^{4} \)
$37$ \( T^{8} \)
$41$ \( ( 17 + 10 T + T^{2} )^{4} \)
$43$ \( ( 529 + 54 T^{2} + T^{4} )^{2} \)
$47$ \( ( 16 - 4 T^{2} + T^{4} )^{2} \)
$53$ \( 4096 - 3072 T^{2} + 2240 T^{4} - 48 T^{6} + T^{8} \)
$59$ \( ( 3136 - 448 T + 120 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$61$ \( ( 3969 + 378 T + 99 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$67$ \( 200533921 - 3483606 T^{2} + 46355 T^{4} - 246 T^{6} + T^{8} \)
$71$ \( ( -56 + 8 T + T^{2} )^{4} \)
$73$ \( 256 - 384 T^{2} + 560 T^{4} - 24 T^{6} + T^{8} \)
$79$ \( ( 18496 - 3264 T + 440 T^{2} - 24 T^{3} + T^{4} )^{2} \)
$83$ \( ( 25921 + 326 T^{2} + T^{4} )^{2} \)
$89$ \( ( 529 - 138 T + 59 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$97$ \( ( 16 + 136 T^{2} + T^{4} )^{2} \)
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