Properties

Label 175.2.a.f
Level $175$
Weight $2$
Character orbit 175.a
Self dual yes
Analytic conductor $1.397$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.39738203537\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 2) q^{4} - 4 q^{6} + q^{7} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 1) q^{3} + (\beta + 2) q^{4} - 4 q^{6} + q^{7} + (\beta + 4) q^{8} + ( - \beta + 2) q^{9} + ( - \beta + 1) q^{11} + ( - 2 \beta - 2) q^{12} + (\beta - 3) q^{13} + \beta q^{14} + 3 \beta q^{16} + ( - \beta + 3) q^{17} + (\beta - 4) q^{18} + ( - 2 \beta - 2) q^{19} + ( - \beta + 1) q^{21} - 4 q^{22} + ( - 2 \beta + 2) q^{23} - 4 \beta q^{24} + ( - 2 \beta + 4) q^{26} + (\beta + 3) q^{27} + (\beta + 2) q^{28} + (3 \beta - 1) q^{29} + (\beta + 4) q^{32} + ( - \beta + 5) q^{33} + (2 \beta - 4) q^{34} - \beta q^{36} - 6 q^{37} + ( - 4 \beta - 8) q^{38} + (3 \beta - 7) q^{39} + 2 \beta q^{41} - 4 q^{42} + (2 \beta - 6) q^{43} + ( - 2 \beta - 2) q^{44} - 8 q^{46} + (3 \beta + 1) q^{47} - 12 q^{48} + q^{49} + ( - 3 \beta + 7) q^{51} - 2 q^{52} + 2 \beta q^{53} + (4 \beta + 4) q^{54} + (\beta + 4) q^{56} + (2 \beta + 6) q^{57} + (2 \beta + 12) q^{58} - 4 q^{59} + 6 \beta q^{61} + ( - \beta + 2) q^{63} + ( - \beta + 4) q^{64} + (4 \beta - 4) q^{66} - 4 \beta q^{67} + 2 q^{68} + ( - 2 \beta + 10) q^{69} + 8 q^{71} + ( - 3 \beta + 4) q^{72} + (4 \beta + 2) q^{73} - 6 \beta q^{74} + ( - 8 \beta - 12) q^{76} + ( - \beta + 1) q^{77} + ( - 4 \beta + 12) q^{78} + (\beta - 5) q^{79} - 7 q^{81} + (2 \beta + 8) q^{82} - 4 q^{83} + ( - 2 \beta - 2) q^{84} + ( - 4 \beta + 8) q^{86} + (\beta - 13) q^{87} - 4 \beta q^{88} + ( - 2 \beta + 4) q^{89} + (\beta - 3) q^{91} + ( - 4 \beta - 4) q^{92} + (4 \beta + 12) q^{94} - 4 \beta q^{96} + ( - 5 \beta + 7) q^{97} + \beta q^{98} + ( - 2 \beta + 6) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} + 5 q^{4} - 8 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + q^{11} - 6 q^{12} - 5 q^{13} + q^{14} + 3 q^{16} + 5 q^{17} - 7 q^{18} - 6 q^{19} + q^{21} - 8 q^{22} + 2 q^{23} - 4 q^{24} + 6 q^{26} + 7 q^{27} + 5 q^{28} + q^{29} + 9 q^{32} + 9 q^{33} - 6 q^{34} - q^{36} - 12 q^{37} - 20 q^{38} - 11 q^{39} + 2 q^{41} - 8 q^{42} - 10 q^{43} - 6 q^{44} - 16 q^{46} + 5 q^{47} - 24 q^{48} + 2 q^{49} + 11 q^{51} - 4 q^{52} + 2 q^{53} + 12 q^{54} + 9 q^{56} + 14 q^{57} + 26 q^{58} - 8 q^{59} + 6 q^{61} + 3 q^{63} + 7 q^{64} - 4 q^{66} - 4 q^{67} + 4 q^{68} + 18 q^{69} + 16 q^{71} + 5 q^{72} + 8 q^{73} - 6 q^{74} - 32 q^{76} + q^{77} + 20 q^{78} - 9 q^{79} - 14 q^{81} + 18 q^{82} - 8 q^{83} - 6 q^{84} + 12 q^{86} - 25 q^{87} - 4 q^{88} + 6 q^{89} - 5 q^{91} - 12 q^{92} + 28 q^{94} - 4 q^{96} + 9 q^{97} + q^{98} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.56155
2.56155
−1.56155 2.56155 0.438447 0 −4.00000 1.00000 2.43845 3.56155 0
1.2 2.56155 −1.56155 4.56155 0 −4.00000 1.00000 6.56155 −0.561553 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.2.a.f 2
3.b odd 2 1 1575.2.a.p 2
4.b odd 2 1 2800.2.a.bi 2
5.b even 2 1 35.2.a.b 2
5.c odd 4 2 175.2.b.b 4
7.b odd 2 1 1225.2.a.s 2
15.d odd 2 1 315.2.a.e 2
15.e even 4 2 1575.2.d.e 4
20.d odd 2 1 560.2.a.i 2
20.e even 4 2 2800.2.g.t 4
35.c odd 2 1 245.2.a.d 2
35.f even 4 2 1225.2.b.f 4
35.i odd 6 2 245.2.e.h 4
35.j even 6 2 245.2.e.i 4
40.e odd 2 1 2240.2.a.bd 2
40.f even 2 1 2240.2.a.bh 2
55.d odd 2 1 4235.2.a.m 2
60.h even 2 1 5040.2.a.bt 2
65.d even 2 1 5915.2.a.l 2
105.g even 2 1 2205.2.a.x 2
140.c even 2 1 3920.2.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.a.b 2 5.b even 2 1
175.2.a.f 2 1.a even 1 1 trivial
175.2.b.b 4 5.c odd 4 2
245.2.a.d 2 35.c odd 2 1
245.2.e.h 4 35.i odd 6 2
245.2.e.i 4 35.j even 6 2
315.2.a.e 2 15.d odd 2 1
560.2.a.i 2 20.d odd 2 1
1225.2.a.s 2 7.b odd 2 1
1225.2.b.f 4 35.f even 4 2
1575.2.a.p 2 3.b odd 2 1
1575.2.d.e 4 15.e even 4 2
2205.2.a.x 2 105.g even 2 1
2240.2.a.bd 2 40.e odd 2 1
2240.2.a.bh 2 40.f even 2 1
2800.2.a.bi 2 4.b odd 2 1
2800.2.g.t 4 20.e even 4 2
3920.2.a.bs 2 140.c even 2 1
4235.2.a.m 2 55.d odd 2 1
5040.2.a.bt 2 60.h even 2 1
5915.2.a.l 2 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 4 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$29$ \( T^{2} - T - 38 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$47$ \( T^{2} - 5T - 32 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$67$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 52 \) Copy content Toggle raw display
$79$ \( T^{2} + 9T + 16 \) Copy content Toggle raw display
$83$ \( (T + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T - 86 \) Copy content Toggle raw display
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