Properties

Label 175.5.d.c
Level $175$
Weight $5$
Character orbit 175.d
Analytic conductor $18.090$
Analytic rank $0$
Dimension $2$
CM discriminant -35
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 175.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.0897435397\)
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 35)
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.

\(f(q)\) \(=\) \( q + 17 i q^{3} -16 q^{4} + 49 i q^{7} -208 q^{9} +O(q^{10})\) \( q + 17 i q^{3} -16 q^{4} + 49 i q^{7} -208 q^{9} -73 q^{11} -272 i q^{12} -23 i q^{13} + 256 q^{16} + 263 i q^{17} -833 q^{21} -2159 i q^{27} -784 i q^{28} + 1153 q^{29} -1241 i q^{33} + 3328 q^{36} + 391 q^{39} + 1168 q^{44} -3457 i q^{47} + 4352 i q^{48} -2401 q^{49} -4471 q^{51} + 368 i q^{52} -10192 i q^{63} -4096 q^{64} -4208 i q^{68} -10078 q^{71} + 9502 i q^{73} -3577 i q^{77} -12167 q^{79} + 19855 q^{81} + 6382 i q^{83} + 13328 q^{84} + 19601 i q^{87} + 1127 q^{91} + 3383 i q^{97} + 15184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} - 416q^{9} + O(q^{10}) \) \( 2q - 32q^{4} - 416q^{9} - 146q^{11} + 512q^{16} - 1666q^{21} + 2306q^{29} + 6656q^{36} + 782q^{39} + 2336q^{44} - 4802q^{49} - 8942q^{51} - 8192q^{64} - 20156q^{71} - 24334q^{79} + 39710q^{81} + 26656q^{84} + 2254q^{91} + 30368q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.00000i
1.00000i
0 17.0000i −16.0000 0 0 49.0000i 0 −208.000 0
76.2 0 17.0000i −16.0000 0 0 49.0000i 0 −208.000 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
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
5.b even 2 1 inner
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.5.d.c 2
5.b even 2 1 inner 175.5.d.c 2
5.c odd 4 1 35.5.c.a 1
5.c odd 4 1 35.5.c.b yes 1
7.b odd 2 1 inner 175.5.d.c 2
15.e even 4 1 315.5.e.a 1
15.e even 4 1 315.5.e.b 1
20.e even 4 1 560.5.p.a 1
20.e even 4 1 560.5.p.b 1
35.c odd 2 1 CM 175.5.d.c 2
35.f even 4 1 35.5.c.a 1
35.f even 4 1 35.5.c.b yes 1
105.k odd 4 1 315.5.e.a 1
105.k odd 4 1 315.5.e.b 1
140.j odd 4 1 560.5.p.a 1
140.j odd 4 1 560.5.p.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 5.c odd 4 1
35.5.c.a 1 35.f even 4 1
35.5.c.b yes 1 5.c odd 4 1
35.5.c.b yes 1 35.f even 4 1
175.5.d.c 2 1.a even 1 1 trivial
175.5.d.c 2 5.b even 2 1 inner
175.5.d.c 2 7.b odd 2 1 inner
175.5.d.c 2 35.c odd 2 1 CM
315.5.e.a 1 15.e even 4 1
315.5.e.a 1 105.k odd 4 1
315.5.e.b 1 15.e even 4 1
315.5.e.b 1 105.k odd 4 1
560.5.p.a 1 20.e even 4 1
560.5.p.a 1 140.j odd 4 1
560.5.p.b 1 20.e even 4 1
560.5.p.b 1 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( 289 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 2401 + T^{2} \)
$11$ \( ( 73 + T )^{2} \)
$13$ \( 529 + T^{2} \)
$17$ \( 69169 + T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -1153 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( 11950849 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( ( 10078 + T )^{2} \)
$73$ \( 90288004 + T^{2} \)
$79$ \( ( 12167 + T )^{2} \)
$83$ \( 40729924 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 11444689 + T^{2} \)
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