Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{3} + 2 q^{4} -i q^{7} + 2 q^{9} +O(q^{10})\) \( q + i q^{3} + 2 q^{4} -i q^{7} + 2 q^{9} -3 q^{11} + 2 i q^{12} + 5 i q^{13} + 4 q^{16} -3 i q^{17} -2 q^{19} + q^{21} -6 i q^{23} + 5 i q^{27} -2 i q^{28} -3 q^{29} -4 q^{31} -3 i q^{33} + 4 q^{36} -2 i q^{37} -5 q^{39} -12 q^{41} -10 i q^{43} -6 q^{44} -9 i q^{47} + 4 i q^{48} - q^{49} + 3 q^{51} + 10 i q^{52} + 12 i q^{53} -2 i q^{57} + 8 q^{61} -2 i q^{63} + 8 q^{64} + 4 i q^{67} -6 i q^{68} + 6 q^{69} + 2 i q^{73} -4 q^{76} + 3 i q^{77} + q^{79} + q^{81} + 12 i q^{83} + 2 q^{84} -3 i q^{87} + 12 q^{89} + 5 q^{91} -12 i q^{92} -4 i q^{93} + i q^{97} -6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{4} + 4q^{9} + O(q^{10}) \) \( 2q + 4q^{4} + 4q^{9} - 6q^{11} + 8q^{16} - 4q^{19} + 2q^{21} - 6q^{29} - 8q^{31} + 8q^{36} - 10q^{39} - 24q^{41} - 12q^{44} - 2q^{49} + 6q^{51} + 16q^{61} + 16q^{64} + 12q^{69} - 8q^{76} + 2q^{79} + 2q^{81} + 4q^{84} + 24q^{89} + 10q^{91} - 12q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
0 1.00000i 2.00000 0 0 1.00000i 0 2.00000 0
99.2 0 1.00000i 2.00000 0 0 1.00000i 0 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.b.a 2
3.b odd 2 1 1575.2.d.c 2
4.b odd 2 1 2800.2.g.l 2
5.b even 2 1 inner 175.2.b.a 2
5.c odd 4 1 35.2.a.a 1
5.c odd 4 1 175.2.a.b 1
7.b odd 2 1 1225.2.b.d 2
15.d odd 2 1 1575.2.d.c 2
15.e even 4 1 315.2.a.b 1
15.e even 4 1 1575.2.a.f 1
20.d odd 2 1 2800.2.g.l 2
20.e even 4 1 560.2.a.b 1
20.e even 4 1 2800.2.a.z 1
35.c odd 2 1 1225.2.b.d 2
35.f even 4 1 245.2.a.c 1
35.f even 4 1 1225.2.a.e 1
35.k even 12 2 245.2.e.b 2
35.l odd 12 2 245.2.e.a 2
40.i odd 4 1 2240.2.a.k 1
40.k even 4 1 2240.2.a.u 1
55.e even 4 1 4235.2.a.c 1
60.l odd 4 1 5040.2.a.v 1
65.h odd 4 1 5915.2.a.f 1
105.k odd 4 1 2205.2.a.e 1
140.j odd 4 1 3920.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.a 1 5.c odd 4 1
175.2.a.b 1 5.c odd 4 1
175.2.b.a 2 1.a even 1 1 trivial
175.2.b.a 2 5.b even 2 1 inner
245.2.a.c 1 35.f even 4 1
245.2.e.a 2 35.l odd 12 2
245.2.e.b 2 35.k even 12 2
315.2.a.b 1 15.e even 4 1
560.2.a.b 1 20.e even 4 1
1225.2.a.e 1 35.f even 4 1
1225.2.b.d 2 7.b odd 2 1
1225.2.b.d 2 35.c odd 2 1
1575.2.a.f 1 15.e even 4 1
1575.2.d.c 2 3.b odd 2 1
1575.2.d.c 2 15.d odd 2 1
2205.2.a.e 1 105.k odd 4 1
2240.2.a.k 1 40.i odd 4 1
2240.2.a.u 1 40.k even 4 1
2800.2.a.z 1 20.e even 4 1
2800.2.g.l 2 4.b odd 2 1
2800.2.g.l 2 20.d odd 2 1
3920.2.a.ba 1 140.j odd 4 1
4235.2.a.c 1 55.e even 4 1
5040.2.a.v 1 60.l odd 4 1
5915.2.a.f 1 65.h odd 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 3 + T )^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 9 + T^{2} \)
$19$ \( ( 2 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 3 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( ( 12 + T )^{2} \)
$43$ \( 100 + T^{2} \)
$47$ \( 81 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -8 + T )^{2} \)
$67$ \( 16 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -1 + T )^{2} \)
$83$ \( 144 + T^{2} \)
$89$ \( ( -12 + T )^{2} \)
$97$ \( 1 + T^{2} \)
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