Properties

Label 175.3.c.a
Level $175$
Weight $3$
Character orbit 175.c
Analytic conductor $4.768$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 175.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.76840462631\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 i q^{2} -5 q^{4} + 7 i q^{7} -3 i q^{8} -9 q^{9} +O(q^{10})\) \( q + 3 i q^{2} -5 q^{4} + 7 i q^{7} -3 i q^{8} -9 q^{9} -6 q^{11} -21 q^{14} -11 q^{16} -27 i q^{18} -18 i q^{22} + 18 i q^{23} -35 i q^{28} + 54 q^{29} -45 i q^{32} + 45 q^{36} + 38 i q^{37} + 58 i q^{43} + 30 q^{44} -54 q^{46} -49 q^{49} -6 i q^{53} + 21 q^{56} + 162 i q^{58} -63 i q^{63} + 91 q^{64} + 118 i q^{67} + 114 q^{71} + 27 i q^{72} -114 q^{74} -42 i q^{77} + 94 q^{79} + 81 q^{81} -174 q^{86} + 18 i q^{88} -90 i q^{92} -147 i q^{98} + 54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 10q^{4} - 18q^{9} + O(q^{10}) \) \( 2q - 10q^{4} - 18q^{9} - 12q^{11} - 42q^{14} - 22q^{16} + 108q^{29} + 90q^{36} + 60q^{44} - 108q^{46} - 98q^{49} + 42q^{56} + 182q^{64} + 228q^{71} - 228q^{74} + 188q^{79} + 162q^{81} - 348q^{86} + 108q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
174.1
1.00000i
1.00000i
3.00000i 0 −5.00000 0 0 7.00000i 3.00000i −9.00000 0
174.2 3.00000i 0 −5.00000 0 0 7.00000i 3.00000i −9.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
5.b even 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.3.c.a 2
5.b even 2 1 inner 175.3.c.a 2
5.c odd 4 1 7.3.b.a 1
5.c odd 4 1 175.3.d.a 1
7.b odd 2 1 CM 175.3.c.a 2
15.e even 4 1 63.3.d.a 1
20.e even 4 1 112.3.c.a 1
35.c odd 2 1 inner 175.3.c.a 2
35.f even 4 1 7.3.b.a 1
35.f even 4 1 175.3.d.a 1
35.k even 12 2 49.3.d.a 2
35.l odd 12 2 49.3.d.a 2
40.i odd 4 1 448.3.c.a 1
40.k even 4 1 448.3.c.b 1
60.l odd 4 1 1008.3.f.a 1
105.k odd 4 1 63.3.d.a 1
105.w odd 12 2 441.3.m.a 2
105.x even 12 2 441.3.m.a 2
140.j odd 4 1 112.3.c.a 1
140.w even 12 2 784.3.s.a 2
140.x odd 12 2 784.3.s.a 2
280.s even 4 1 448.3.c.a 1
280.y odd 4 1 448.3.c.b 1
420.w even 4 1 1008.3.f.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.3.b.a 1 5.c odd 4 1
7.3.b.a 1 35.f even 4 1
49.3.d.a 2 35.k even 12 2
49.3.d.a 2 35.l odd 12 2
63.3.d.a 1 15.e even 4 1
63.3.d.a 1 105.k odd 4 1
112.3.c.a 1 20.e even 4 1
112.3.c.a 1 140.j odd 4 1
175.3.c.a 2 1.a even 1 1 trivial
175.3.c.a 2 5.b even 2 1 inner
175.3.c.a 2 7.b odd 2 1 CM
175.3.c.a 2 35.c odd 2 1 inner
175.3.d.a 1 5.c odd 4 1
175.3.d.a 1 35.f even 4 1
441.3.m.a 2 105.w odd 12 2
441.3.m.a 2 105.x even 12 2
448.3.c.a 1 40.i odd 4 1
448.3.c.a 1 280.s even 4 1
448.3.c.b 1 40.k even 4 1
448.3.c.b 1 280.y odd 4 1
784.3.s.a 2 140.w even 12 2
784.3.s.a 2 140.x odd 12 2
1008.3.f.a 1 60.l odd 4 1
1008.3.f.a 1 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 9 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 16 T^{4} \)
$3$ \( ( 1 + 9 T^{2} )^{2} \)
$5$ 1
$7$ \( 1 + 49 T^{2} \)
$11$ \( ( 1 + 6 T + 121 T^{2} )^{2} \)
$13$ \( ( 1 + 169 T^{2} )^{2} \)
$17$ \( ( 1 + 289 T^{2} )^{2} \)
$19$ \( ( 1 - 19 T )^{2}( 1 + 19 T )^{2} \)
$23$ \( 1 - 734 T^{2} + 279841 T^{4} \)
$29$ \( ( 1 - 54 T + 841 T^{2} )^{2} \)
$31$ \( ( 1 - 31 T )^{2}( 1 + 31 T )^{2} \)
$37$ \( 1 - 1294 T^{2} + 1874161 T^{4} \)
$41$ \( ( 1 - 41 T )^{2}( 1 + 41 T )^{2} \)
$43$ \( 1 - 334 T^{2} + 3418801 T^{4} \)
$47$ \( ( 1 + 2209 T^{2} )^{2} \)
$53$ \( 1 - 5582 T^{2} + 7890481 T^{4} \)
$59$ \( ( 1 - 59 T )^{2}( 1 + 59 T )^{2} \)
$61$ \( ( 1 - 61 T )^{2}( 1 + 61 T )^{2} \)
$67$ \( 1 + 4946 T^{2} + 20151121 T^{4} \)
$71$ \( ( 1 - 114 T + 5041 T^{2} )^{2} \)
$73$ \( ( 1 + 5329 T^{2} )^{2} \)
$79$ \( ( 1 - 94 T + 6241 T^{2} )^{2} \)
$83$ \( ( 1 + 6889 T^{2} )^{2} \)
$89$ \( ( 1 - 89 T )^{2}( 1 + 89 T )^{2} \)
$97$ \( ( 1 + 9409 T^{2} )^{2} \)
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