Properties

Label 175.6.b.a
Level $175$
Weight $6$
Character orbit 175.b
Analytic conductor $28.067$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 10 i q^{2} - 14 i q^{3} - 68 q^{4} + 140 q^{6} + 49 i q^{7} - 360 i q^{8} + 47 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 10 i q^{2} - 14 i q^{3} - 68 q^{4} + 140 q^{6} + 49 i q^{7} - 360 i q^{8} + 47 q^{9} + 232 q^{11} + 952 i q^{12} - 140 i q^{13} - 490 q^{14} + 1424 q^{16} + 1722 i q^{17} + 470 i q^{18} + 98 q^{19} + 686 q^{21} + 2320 i q^{22} + 1824 i q^{23} - 5040 q^{24} + 1400 q^{26} - 4060 i q^{27} - 3332 i q^{28} - 3418 q^{29} - 7644 q^{31} + 2720 i q^{32} - 3248 i q^{33} - 17220 q^{34} - 3196 q^{36} + 10398 i q^{37} + 980 i q^{38} - 1960 q^{39} - 17962 q^{41} + 6860 i q^{42} + 10880 i q^{43} - 15776 q^{44} - 18240 q^{46} - 9324 i q^{47} - 19936 i q^{48} - 2401 q^{49} + 24108 q^{51} + 9520 i q^{52} + 2262 i q^{53} + 40600 q^{54} + 17640 q^{56} - 1372 i q^{57} - 34180 i q^{58} + 2730 q^{59} + 25648 q^{61} - 76440 i q^{62} + 2303 i q^{63} + 18368 q^{64} + 32480 q^{66} + 48404 i q^{67} - 117096 i q^{68} + 25536 q^{69} - 58560 q^{71} - 16920 i q^{72} + 68082 i q^{73} - 103980 q^{74} - 6664 q^{76} + 11368 i q^{77} - 19600 i q^{78} - 31784 q^{79} - 45419 q^{81} - 179620 i q^{82} - 20538 i q^{83} - 46648 q^{84} - 108800 q^{86} + 47852 i q^{87} - 83520 i q^{88} + 50582 q^{89} + 6860 q^{91} - 124032 i q^{92} + 107016 i q^{93} + 93240 q^{94} + 38080 q^{96} + 58506 i q^{97} - 24010 i q^{98} + 10904 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 136 q^{4} + 280 q^{6} + 94 q^{9} + 464 q^{11} - 980 q^{14} + 2848 q^{16} + 196 q^{19} + 1372 q^{21} - 10080 q^{24} + 2800 q^{26} - 6836 q^{29} - 15288 q^{31} - 34440 q^{34} - 6392 q^{36} - 3920 q^{39} - 35924 q^{41} - 31552 q^{44} - 36480 q^{46} - 4802 q^{49} + 48216 q^{51} + 81200 q^{54} + 35280 q^{56} + 5460 q^{59} + 51296 q^{61} + 36736 q^{64} + 64960 q^{66} + 51072 q^{69} - 117120 q^{71} - 207960 q^{74} - 13328 q^{76} - 63568 q^{79} - 90838 q^{81} - 93296 q^{84} - 217600 q^{86} + 101164 q^{89} + 13720 q^{91} + 186480 q^{94} + 76160 q^{96} + 21808 q^{99}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
10.0000i 14.0000i −68.0000 0 140.000 49.0000i 360.000i 47.0000 0
99.2 10.0000i 14.0000i −68.0000 0 140.000 49.0000i 360.000i 47.0000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.6.b.a 2
5.b even 2 1 inner 175.6.b.a 2
5.c odd 4 1 7.6.a.a 1
5.c odd 4 1 175.6.a.b 1
15.e even 4 1 63.6.a.e 1
20.e even 4 1 112.6.a.g 1
35.f even 4 1 49.6.a.a 1
35.k even 12 2 49.6.c.b 2
35.l odd 12 2 49.6.c.c 2
40.i odd 4 1 448.6.a.m 1
40.k even 4 1 448.6.a.c 1
55.e even 4 1 847.6.a.b 1
60.l odd 4 1 1008.6.a.y 1
105.k odd 4 1 441.6.a.k 1
140.j odd 4 1 784.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 5.c odd 4 1
49.6.a.a 1 35.f even 4 1
49.6.c.b 2 35.k even 12 2
49.6.c.c 2 35.l odd 12 2
63.6.a.e 1 15.e even 4 1
112.6.a.g 1 20.e even 4 1
175.6.a.b 1 5.c odd 4 1
175.6.b.a 2 1.a even 1 1 trivial
175.6.b.a 2 5.b even 2 1 inner
441.6.a.k 1 105.k odd 4 1
448.6.a.c 1 40.k even 4 1
448.6.a.m 1 40.i odd 4 1
784.6.a.c 1 140.j odd 4 1
847.6.a.b 1 55.e even 4 1
1008.6.a.y 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 100 \) acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 100 \) Copy content Toggle raw display
$3$ \( T^{2} + 196 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 232)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 19600 \) Copy content Toggle raw display
$17$ \( T^{2} + 2965284 \) Copy content Toggle raw display
$19$ \( (T - 98)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3326976 \) Copy content Toggle raw display
$29$ \( (T + 3418)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7644)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 108118404 \) Copy content Toggle raw display
$41$ \( (T + 17962)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 118374400 \) Copy content Toggle raw display
$47$ \( T^{2} + 86936976 \) Copy content Toggle raw display
$53$ \( T^{2} + 5116644 \) Copy content Toggle raw display
$59$ \( (T - 2730)^{2} \) Copy content Toggle raw display
$61$ \( (T - 25648)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 2342947216 \) Copy content Toggle raw display
$71$ \( (T + 58560)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4635158724 \) Copy content Toggle raw display
$79$ \( (T + 31784)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 421809444 \) Copy content Toggle raw display
$89$ \( (T - 50582)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 3422952036 \) Copy content Toggle raw display
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