Properties

Label 175.5.d.e
Level $175$
Weight $5$
Character orbit 175.d
Analytic conductor $18.090$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.0897435397\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 10 q^{4} + \beta_{3} q^{6} + ( - 14 \beta_{2} + 7 \beta_1) q^{7} + 26 \beta_{2} q^{8} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - \beta_1 q^{3} - 10 q^{4} + \beta_{3} q^{6} + ( - 14 \beta_{2} + 7 \beta_1) q^{7} + 26 \beta_{2} q^{8} + 56 q^{9} + 89 q^{11} + 10 \beta_1 q^{12} + \beta_1 q^{13} + ( - 7 \beta_{3} + 84) q^{14} + 4 q^{16} - 97 \beta_1 q^{17} - 56 \beta_{2} q^{18} + 18 \beta_{3} q^{19} + (14 \beta_{3} + 175) q^{21} - 89 \beta_{2} q^{22} - 286 \beta_{2} q^{23} - 26 \beta_{3} q^{24} - \beta_{3} q^{26} - 137 \beta_1 q^{27} + (140 \beta_{2} - 70 \beta_1) q^{28} - 191 q^{29} + 86 \beta_{3} q^{31} - 420 \beta_{2} q^{32} - 89 \beta_1 q^{33} + 97 \beta_{3} q^{34} - 560 q^{36} - 666 \beta_{2} q^{37} - 108 \beta_1 q^{38} + 25 q^{39} - 238 \beta_{3} q^{41} + ( - 175 \beta_{2} - 84 \beta_1) q^{42} + 154 \beta_{2} q^{43} - 890 q^{44} + 1716 q^{46} - 439 \beta_1 q^{47} - 4 \beta_1 q^{48} + ( - 196 \beta_{3} - 49) q^{49} - 2425 q^{51} - 10 \beta_1 q^{52} - 648 \beta_{2} q^{53} + 137 \beta_{3} q^{54} + (182 \beta_{3} - 2184) q^{56} + 450 \beta_{2} q^{57} + 191 \beta_{2} q^{58} - 296 \beta_{3} q^{59} - 158 \beta_{3} q^{61} - 516 \beta_1 q^{62} + ( - 784 \beta_{2} + 392 \beta_1) q^{63} + 2456 q^{64} + 89 \beta_{3} q^{66} - 836 \beta_{2} q^{67} + 970 \beta_1 q^{68} + 286 \beta_{3} q^{69} + 4454 q^{71} + 1456 \beta_{2} q^{72} - 1730 \beta_1 q^{73} + 3996 q^{74} - 180 \beta_{3} q^{76} + ( - 1246 \beta_{2} + 623 \beta_1) q^{77} - 25 \beta_{2} q^{78} - 5561 q^{79} + 1111 q^{81} + 1428 \beta_1 q^{82} - 398 \beta_1 q^{83} + ( - 140 \beta_{3} - 1750) q^{84} - 924 q^{86} + 191 \beta_1 q^{87} + 2314 \beta_{2} q^{88} - 66 \beta_{3} q^{89} + ( - 14 \beta_{3} - 175) q^{91} + 2860 \beta_{2} q^{92} + 2150 \beta_{2} q^{93} + 439 \beta_{3} q^{94} + 420 \beta_{3} q^{96} + 1847 \beta_1 q^{97} + (49 \beta_{2} + 1176 \beta_1) q^{98} + 4984 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{4} + 224 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{4} + 224 q^{9} + 356 q^{11} + 336 q^{14} + 16 q^{16} + 700 q^{21} - 764 q^{29} - 2240 q^{36} + 100 q^{39} - 3560 q^{44} + 6864 q^{46} - 196 q^{49} - 9700 q^{51} - 8736 q^{56} + 9824 q^{64} + 17816 q^{71} + 15984 q^{74} - 22244 q^{79} + 4444 q^{81} - 7000 q^{84} - 3696 q^{86} - 700 q^{91} + 19936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{3} + 15\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 5\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 15\beta_{2} ) / 10 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
1.22474 + 1.22474i
1.22474 1.22474i
−1.22474 1.22474i
−1.22474 + 1.22474i
−2.44949 5.00000i −10.0000 0 12.2474i −34.2929 + 35.0000i 63.6867 56.0000 0
76.2 −2.44949 5.00000i −10.0000 0 12.2474i −34.2929 35.0000i 63.6867 56.0000 0
76.3 2.44949 5.00000i −10.0000 0 12.2474i 34.2929 + 35.0000i −63.6867 56.0000 0
76.4 2.44949 5.00000i −10.0000 0 12.2474i 34.2929 35.0000i −63.6867 56.0000 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 175.5.d.e 4
5.b even 2 1 inner 175.5.d.e 4
5.c odd 4 1 35.5.c.c 2
5.c odd 4 1 35.5.c.d yes 2
7.b odd 2 1 inner 175.5.d.e 4
15.e even 4 1 315.5.e.c 2
15.e even 4 1 315.5.e.d 2
20.e even 4 1 560.5.p.d 2
20.e even 4 1 560.5.p.e 2
35.c odd 2 1 inner 175.5.d.e 4
35.f even 4 1 35.5.c.c 2
35.f even 4 1 35.5.c.d yes 2
105.k odd 4 1 315.5.e.c 2
105.k odd 4 1 315.5.e.d 2
140.j odd 4 1 560.5.p.d 2
140.j odd 4 1 560.5.p.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 5.c odd 4 1
35.5.c.c 2 35.f even 4 1
35.5.c.d yes 2 5.c odd 4 1
35.5.c.d yes 2 35.f even 4 1
175.5.d.e 4 1.a even 1 1 trivial
175.5.d.e 4 5.b even 2 1 inner
175.5.d.e 4 7.b odd 2 1 inner
175.5.d.e 4 35.c odd 2 1 inner
315.5.e.c 2 15.e even 4 1
315.5.e.c 2 105.k odd 4 1
315.5.e.d 2 15.e even 4 1
315.5.e.d 2 105.k odd 4 1
560.5.p.d 2 20.e even 4 1
560.5.p.d 2 140.j odd 4 1
560.5.p.e 2 20.e even 4 1
560.5.p.e 2 140.j odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6 \) acting on \(S_{5}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 98 T^{2} + 5764801 \) Copy content Toggle raw display
$11$ \( (T - 89)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 235225)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 48600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 490776)^{2} \) Copy content Toggle raw display
$29$ \( (T + 191)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1109400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2661336)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8496600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 142296)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4818025)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 2519424)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 13142400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3744600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 4193376)^{2} \) Copy content Toggle raw display
$71$ \( (T - 4454)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 74822500)^{2} \) Copy content Toggle raw display
$79$ \( (T + 5561)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 3960100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 653400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 85285225)^{2} \) Copy content Toggle raw display
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