Properties

Label 175.3.c.d
Level $175$
Weight $3$
Character orbit 175.c
Analytic conductor $4.768$
Analytic rank $0$
Dimension $4$
CM no
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} - 7 \beta_1 q^{7} + 7 \beta_1 q^{8} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + 3 q^{4} + \beta_{2} q^{6} - 7 \beta_1 q^{7} + 7 \beta_1 q^{8} + 11 q^{9} + 2 q^{11} + 3 \beta_{3} q^{12} - 3 \beta_{3} q^{13} + 7 q^{14} + 5 q^{16} - 6 \beta_{3} q^{17} + 11 \beta_1 q^{18} - 3 \beta_{2} q^{19} - 7 \beta_{2} q^{21} + 2 \beta_1 q^{22} + 26 \beta_1 q^{23} + 7 \beta_{2} q^{24} - 3 \beta_{2} q^{26} + 2 \beta_{3} q^{27} - 21 \beta_1 q^{28} + 22 q^{29} + 12 \beta_{2} q^{31} + 33 \beta_1 q^{32} + 2 \beta_{3} q^{33} - 6 \beta_{2} q^{34} + 33 q^{36} - 14 \beta_1 q^{37} + 3 \beta_{3} q^{38} - 60 q^{39} + 6 \beta_{2} q^{41} + 7 \beta_{3} q^{42} - 34 \beta_1 q^{43} + 6 q^{44} - 26 q^{46} + 6 \beta_{3} q^{47} + 5 \beta_{3} q^{48} - 49 q^{49} - 120 q^{51} - 9 \beta_{3} q^{52} - 34 \beta_1 q^{53} + 2 \beta_{2} q^{54} + 49 q^{56} - 60 \beta_1 q^{57} + 22 \beta_1 q^{58} - 9 \beta_{2} q^{59} - 21 \beta_{2} q^{61} - 12 \beta_{3} q^{62} - 77 \beta_1 q^{63} - 13 q^{64} + 2 \beta_{2} q^{66} - 14 \beta_1 q^{67} - 18 \beta_{3} q^{68} + 26 \beta_{2} q^{69} + 62 q^{71} + 77 \beta_1 q^{72} - 12 \beta_{3} q^{73} + 14 q^{74} - 9 \beta_{2} q^{76} - 14 \beta_1 q^{77} - 60 \beta_1 q^{78} - 38 q^{79} - 59 q^{81} - 6 \beta_{3} q^{82} + 9 \beta_{3} q^{83} - 21 \beta_{2} q^{84} + 34 q^{86} + 22 \beta_{3} q^{87} + 14 \beta_1 q^{88} - 6 \beta_{2} q^{89} + 21 \beta_{2} q^{91} + 78 \beta_1 q^{92} + 240 \beta_1 q^{93} + 6 \beta_{2} q^{94} + 33 \beta_{2} q^{96} + 6 \beta_{3} q^{97} - 49 \beta_1 q^{98} + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{4} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{4} + 44 q^{9} + 8 q^{11} + 28 q^{14} + 20 q^{16} + 88 q^{29} + 132 q^{36} - 240 q^{39} + 24 q^{44} - 104 q^{46} - 196 q^{49} - 480 q^{51} + 196 q^{56} - 52 q^{64} + 248 q^{71} + 56 q^{74} - 152 q^{79} - 236 q^{81} + 136 q^{86} + 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 8\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display

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
1.61803i
0.618034i
1.61803i
0.618034i
1.00000i −4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.2 1.00000i 4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.3 1.00000i −4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 0
174.4 1.00000i 4.47214 3.00000 0 4.47214i 7.00000i 7.00000i 11.0000 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
7.b odd 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.d 4
5.b even 2 1 inner 175.3.c.d 4
5.c odd 4 1 35.3.d.a 2
5.c odd 4 1 175.3.d.f 2
7.b odd 2 1 inner 175.3.c.d 4
15.e even 4 1 315.3.h.b 2
20.e even 4 1 560.3.f.a 2
35.c odd 2 1 inner 175.3.c.d 4
35.f even 4 1 35.3.d.a 2
35.f even 4 1 175.3.d.f 2
35.k even 12 2 245.3.h.b 4
35.l odd 12 2 245.3.h.b 4
105.k odd 4 1 315.3.h.b 2
140.j odd 4 1 560.3.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 5.c odd 4 1
35.3.d.a 2 35.f even 4 1
175.3.c.d 4 1.a even 1 1 trivial
175.3.c.d 4 5.b even 2 1 inner
175.3.c.d 4 7.b odd 2 1 inner
175.3.c.d 4 35.c odd 2 1 inner
175.3.d.f 2 5.c odd 4 1
175.3.d.f 2 35.f even 4 1
245.3.h.b 4 35.k even 12 2
245.3.h.b 4 35.l odd 12 2
315.3.h.b 2 15.e even 4 1
315.3.h.b 2 105.k odd 4 1
560.3.f.a 2 20.e even 4 1
560.3.f.a 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 720)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 676)^{2} \) Copy content Toggle raw display
$29$ \( (T - 22)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 720)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 1620)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 8820)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$71$ \( (T - 62)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2880)^{2} \) Copy content Toggle raw display
$79$ \( (T + 38)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 1620)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 720)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 720)^{2} \) Copy content Toggle raw display
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