Properties

Label 175.4.k.a
Level $175$
Weight $4$
Character orbit 175.k
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \zeta_{12} q^{2} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{3} -4 \zeta_{12}^{2} q^{4} + 14 q^{6} + ( -14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{7} -24 \zeta_{12}^{3} q^{8} + ( 22 - 22 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + 2 \zeta_{12} q^{2} + ( 7 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{3} -4 \zeta_{12}^{2} q^{4} + 14 q^{6} + ( -14 \zeta_{12} - 7 \zeta_{12}^{3} ) q^{7} -24 \zeta_{12}^{3} q^{8} + ( 22 - 22 \zeta_{12}^{2} ) q^{9} + 5 \zeta_{12}^{2} q^{11} -28 \zeta_{12} q^{12} -14 \zeta_{12}^{3} q^{13} + ( 14 - 42 \zeta_{12}^{2} ) q^{14} + ( 16 - 16 \zeta_{12}^{2} ) q^{16} + ( 21 \zeta_{12} - 21 \zeta_{12}^{3} ) q^{17} + ( 44 \zeta_{12} - 44 \zeta_{12}^{3} ) q^{18} + ( 49 - 49 \zeta_{12}^{2} ) q^{19} + ( -98 - 49 \zeta_{12}^{2} ) q^{21} + 10 \zeta_{12}^{3} q^{22} + 159 \zeta_{12} q^{23} -168 \zeta_{12}^{2} q^{24} + ( 28 - 28 \zeta_{12}^{2} ) q^{26} + 35 \zeta_{12}^{3} q^{27} + ( -28 \zeta_{12} + 84 \zeta_{12}^{3} ) q^{28} -58 q^{29} -147 \zeta_{12}^{2} q^{31} + ( -160 \zeta_{12} + 160 \zeta_{12}^{3} ) q^{32} + 35 \zeta_{12} q^{33} + 42 q^{34} -88 q^{36} + 219 \zeta_{12} q^{37} + ( 98 \zeta_{12} - 98 \zeta_{12}^{3} ) q^{38} -98 \zeta_{12}^{2} q^{39} + 350 q^{41} + ( -196 \zeta_{12} - 98 \zeta_{12}^{3} ) q^{42} -124 \zeta_{12}^{3} q^{43} + ( 20 - 20 \zeta_{12}^{2} ) q^{44} + 318 \zeta_{12}^{2} q^{46} + 525 \zeta_{12} q^{47} -112 \zeta_{12}^{3} q^{48} + ( -245 + 392 \zeta_{12}^{2} ) q^{49} + ( 147 - 147 \zeta_{12}^{2} ) q^{51} + ( -56 \zeta_{12} + 56 \zeta_{12}^{3} ) q^{52} + ( 303 \zeta_{12} - 303 \zeta_{12}^{3} ) q^{53} + ( -70 + 70 \zeta_{12}^{2} ) q^{54} + ( -504 + 336 \zeta_{12}^{2} ) q^{56} -343 \zeta_{12}^{3} q^{57} -116 \zeta_{12} q^{58} -105 \zeta_{12}^{2} q^{59} + ( 413 - 413 \zeta_{12}^{2} ) q^{61} -294 \zeta_{12}^{3} q^{62} + ( -462 \zeta_{12} + 308 \zeta_{12}^{3} ) q^{63} -448 q^{64} + 70 \zeta_{12}^{2} q^{66} + ( -415 \zeta_{12} + 415 \zeta_{12}^{3} ) q^{67} -84 \zeta_{12} q^{68} + 1113 q^{69} -432 q^{71} -528 \zeta_{12} q^{72} + ( -1113 \zeta_{12} + 1113 \zeta_{12}^{3} ) q^{73} + 438 \zeta_{12}^{2} q^{74} -196 q^{76} + ( 35 \zeta_{12} - 105 \zeta_{12}^{3} ) q^{77} -196 \zeta_{12}^{3} q^{78} + ( -103 + 103 \zeta_{12}^{2} ) q^{79} + 839 \zeta_{12}^{2} q^{81} + 700 \zeta_{12} q^{82} + 1092 \zeta_{12}^{3} q^{83} + ( -196 + 588 \zeta_{12}^{2} ) q^{84} + ( 248 - 248 \zeta_{12}^{2} ) q^{86} + ( -406 \zeta_{12} + 406 \zeta_{12}^{3} ) q^{87} + ( 120 \zeta_{12} - 120 \zeta_{12}^{3} ) q^{88} + ( -329 + 329 \zeta_{12}^{2} ) q^{89} + ( -294 + 196 \zeta_{12}^{2} ) q^{91} -636 \zeta_{12}^{3} q^{92} -1029 \zeta_{12} q^{93} + 1050 \zeta_{12}^{2} q^{94} + ( -1120 + 1120 \zeta_{12}^{2} ) q^{96} + 882 \zeta_{12}^{3} q^{97} + ( -490 \zeta_{12} + 784 \zeta_{12}^{3} ) q^{98} + 110 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{4} + 56q^{6} + 44q^{9} + O(q^{10}) \) \( 4q - 8q^{4} + 56q^{6} + 44q^{9} + 10q^{11} - 28q^{14} + 32q^{16} + 98q^{19} - 490q^{21} - 336q^{24} + 56q^{26} - 232q^{29} - 294q^{31} + 168q^{34} - 352q^{36} - 196q^{39} + 1400q^{41} + 40q^{44} + 636q^{46} - 196q^{49} + 294q^{51} - 140q^{54} - 1344q^{56} - 210q^{59} + 826q^{61} - 1792q^{64} + 140q^{66} + 4452q^{69} - 1728q^{71} + 876q^{74} - 784q^{76} - 206q^{79} + 1678q^{81} + 392q^{84} + 496q^{86} - 658q^{89} - 784q^{91} + 2100q^{94} - 2240q^{96} + 440q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.73205 + 1.00000i −6.06218 3.50000i −2.00000 + 3.46410i 0 14.0000 12.1244 14.0000i 24.0000i 11.0000 + 19.0526i 0
74.2 1.73205 1.00000i 6.06218 + 3.50000i −2.00000 + 3.46410i 0 14.0000 −12.1244 + 14.0000i 24.0000i 11.0000 + 19.0526i 0
149.1 −1.73205 1.00000i −6.06218 + 3.50000i −2.00000 3.46410i 0 14.0000 12.1244 + 14.0000i 24.0000i 11.0000 19.0526i 0
149.2 1.73205 + 1.00000i 6.06218 3.50000i −2.00000 3.46410i 0 14.0000 −12.1244 14.0000i 24.0000i 11.0000 19.0526i 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.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.k.a 4
5.b even 2 1 inner 175.4.k.a 4
5.c odd 4 1 7.4.c.a 2
5.c odd 4 1 175.4.e.a 2
7.c even 3 1 inner 175.4.k.a 4
15.e even 4 1 63.4.e.b 2
20.e even 4 1 112.4.i.c 2
35.f even 4 1 49.4.c.a 2
35.j even 6 1 inner 175.4.k.a 4
35.k even 12 1 49.4.a.c 1
35.k even 12 1 49.4.c.a 2
35.k even 12 1 1225.4.a.d 1
35.l odd 12 1 7.4.c.a 2
35.l odd 12 1 49.4.a.d 1
35.l odd 12 1 175.4.e.a 2
35.l odd 12 1 1225.4.a.c 1
40.i odd 4 1 448.4.i.f 2
40.k even 4 1 448.4.i.a 2
105.k odd 4 1 441.4.e.k 2
105.w odd 12 1 441.4.a.e 1
105.w odd 12 1 441.4.e.k 2
105.x even 12 1 63.4.e.b 2
105.x even 12 1 441.4.a.d 1
140.w even 12 1 112.4.i.c 2
140.w even 12 1 784.4.a.b 1
140.x odd 12 1 784.4.a.r 1
280.br even 12 1 448.4.i.a 2
280.bt odd 12 1 448.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.c.a 2 5.c odd 4 1
7.4.c.a 2 35.l odd 12 1
49.4.a.c 1 35.k even 12 1
49.4.a.d 1 35.l odd 12 1
49.4.c.a 2 35.f even 4 1
49.4.c.a 2 35.k even 12 1
63.4.e.b 2 15.e even 4 1
63.4.e.b 2 105.x even 12 1
112.4.i.c 2 20.e even 4 1
112.4.i.c 2 140.w even 12 1
175.4.e.a 2 5.c odd 4 1
175.4.e.a 2 35.l odd 12 1
175.4.k.a 4 1.a even 1 1 trivial
175.4.k.a 4 5.b even 2 1 inner
175.4.k.a 4 7.c even 3 1 inner
175.4.k.a 4 35.j even 6 1 inner
441.4.a.d 1 105.x even 12 1
441.4.a.e 1 105.w odd 12 1
441.4.e.k 2 105.k odd 4 1
441.4.e.k 2 105.w odd 12 1
448.4.i.a 2 40.k even 4 1
448.4.i.a 2 280.br even 12 1
448.4.i.f 2 40.i odd 4 1
448.4.i.f 2 280.bt odd 12 1
784.4.a.b 1 140.w even 12 1
784.4.a.r 1 140.x odd 12 1
1225.4.a.c 1 35.l odd 12 1
1225.4.a.d 1 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4 T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 - 4 T^{2} + T^{4} \)
$3$ \( 2401 - 49 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 117649 + 98 T^{2} + T^{4} \)
$11$ \( ( 25 - 5 T + T^{2} )^{2} \)
$13$ \( ( 196 + T^{2} )^{2} \)
$17$ \( 194481 - 441 T^{2} + T^{4} \)
$19$ \( ( 2401 - 49 T + T^{2} )^{2} \)
$23$ \( 639128961 - 25281 T^{2} + T^{4} \)
$29$ \( ( 58 + T )^{4} \)
$31$ \( ( 21609 + 147 T + T^{2} )^{2} \)
$37$ \( 2300257521 - 47961 T^{2} + T^{4} \)
$41$ \( ( -350 + T )^{4} \)
$43$ \( ( 15376 + T^{2} )^{2} \)
$47$ \( 75969140625 - 275625 T^{2} + T^{4} \)
$53$ \( 8428892481 - 91809 T^{2} + T^{4} \)
$59$ \( ( 11025 + 105 T + T^{2} )^{2} \)
$61$ \( ( 170569 - 413 T + T^{2} )^{2} \)
$67$ \( 29661450625 - 172225 T^{2} + T^{4} \)
$71$ \( ( 432 + T )^{4} \)
$73$ \( 1534548635361 - 1238769 T^{2} + T^{4} \)
$79$ \( ( 10609 + 103 T + T^{2} )^{2} \)
$83$ \( ( 1192464 + T^{2} )^{2} \)
$89$ \( ( 108241 + 329 T + T^{2} )^{2} \)
$97$ \( ( 777924 + T^{2} )^{2} \)
show more
show less