Properties

Label 175.4.b.b
Level $175$
Weight $4$
Character orbit 175.b
Analytic conductor $10.325$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + i q^{2} -2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} +O(q^{10})\) \( q + i q^{2} -2 i q^{3} + 7 q^{4} + 2 q^{6} + 7 i q^{7} + 15 i q^{8} + 23 q^{9} -8 q^{11} -14 i q^{12} + 28 i q^{13} -7 q^{14} + 41 q^{16} -54 i q^{17} + 23 i q^{18} + 110 q^{19} + 14 q^{21} -8 i q^{22} + 48 i q^{23} + 30 q^{24} -28 q^{26} -100 i q^{27} + 49 i q^{28} + 110 q^{29} + 12 q^{31} + 161 i q^{32} + 16 i q^{33} + 54 q^{34} + 161 q^{36} + 246 i q^{37} + 110 i q^{38} + 56 q^{39} + 182 q^{41} + 14 i q^{42} + 128 i q^{43} -56 q^{44} -48 q^{46} -324 i q^{47} -82 i q^{48} -49 q^{49} -108 q^{51} + 196 i q^{52} -162 i q^{53} + 100 q^{54} -105 q^{56} -220 i q^{57} + 110 i q^{58} -810 q^{59} -488 q^{61} + 12 i q^{62} + 161 i q^{63} + 167 q^{64} -16 q^{66} -244 i q^{67} -378 i q^{68} + 96 q^{69} -768 q^{71} + 345 i q^{72} -702 i q^{73} -246 q^{74} + 770 q^{76} -56 i q^{77} + 56 i q^{78} -440 q^{79} + 421 q^{81} + 182 i q^{82} -1302 i q^{83} + 98 q^{84} -128 q^{86} -220 i q^{87} -120 i q^{88} -730 q^{89} -196 q^{91} + 336 i q^{92} -24 i q^{93} + 324 q^{94} + 322 q^{96} -294 i q^{97} -49 i q^{98} -184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 14q^{4} + 4q^{6} + 46q^{9} + O(q^{10}) \) \( 2q + 14q^{4} + 4q^{6} + 46q^{9} - 16q^{11} - 14q^{14} + 82q^{16} + 220q^{19} + 28q^{21} + 60q^{24} - 56q^{26} + 220q^{29} + 24q^{31} + 108q^{34} + 322q^{36} + 112q^{39} + 364q^{41} - 112q^{44} - 96q^{46} - 98q^{49} - 216q^{51} + 200q^{54} - 210q^{56} - 1620q^{59} - 976q^{61} + 334q^{64} - 32q^{66} + 192q^{69} - 1536q^{71} - 492q^{74} + 1540q^{76} - 880q^{79} + 842q^{81} + 196q^{84} - 256q^{86} - 1460q^{89} - 392q^{91} + 648q^{94} + 644q^{96} - 368q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
1.00000i 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 0
99.2 1.00000i 2.00000i 7.00000 0 2.00000 7.00000i 15.0000i 23.0000 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 175.4.b.b 2
5.b even 2 1 inner 175.4.b.b 2
5.c odd 4 1 7.4.a.a 1
5.c odd 4 1 175.4.a.b 1
15.e even 4 1 63.4.a.b 1
15.e even 4 1 1575.4.a.e 1
20.e even 4 1 112.4.a.f 1
35.f even 4 1 49.4.a.b 1
35.f even 4 1 1225.4.a.j 1
35.k even 12 2 49.4.c.b 2
35.l odd 12 2 49.4.c.c 2
40.i odd 4 1 448.4.a.i 1
40.k even 4 1 448.4.a.e 1
55.e even 4 1 847.4.a.b 1
60.l odd 4 1 1008.4.a.c 1
65.h odd 4 1 1183.4.a.b 1
85.g odd 4 1 2023.4.a.a 1
105.k odd 4 1 441.4.a.i 1
105.w odd 12 2 441.4.e.e 2
105.x even 12 2 441.4.e.h 2
140.j odd 4 1 784.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 5.c odd 4 1
49.4.a.b 1 35.f even 4 1
49.4.c.b 2 35.k even 12 2
49.4.c.c 2 35.l odd 12 2
63.4.a.b 1 15.e even 4 1
112.4.a.f 1 20.e even 4 1
175.4.a.b 1 5.c odd 4 1
175.4.b.b 2 1.a even 1 1 trivial
175.4.b.b 2 5.b even 2 1 inner
441.4.a.i 1 105.k odd 4 1
441.4.e.e 2 105.w odd 12 2
441.4.e.h 2 105.x even 12 2
448.4.a.e 1 40.k even 4 1
448.4.a.i 1 40.i odd 4 1
784.4.a.g 1 140.j odd 4 1
847.4.a.b 1 55.e even 4 1
1008.4.a.c 1 60.l odd 4 1
1183.4.a.b 1 65.h odd 4 1
1225.4.a.j 1 35.f even 4 1
1575.4.a.e 1 15.e even 4 1
2023.4.a.a 1 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(175, [\chi])\):

\( T_{2}^{2} + 1 \)
\( T_{3}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 49 + T^{2} \)
$11$ \( ( 8 + T )^{2} \)
$13$ \( 784 + T^{2} \)
$17$ \( 2916 + T^{2} \)
$19$ \( ( -110 + T )^{2} \)
$23$ \( 2304 + T^{2} \)
$29$ \( ( -110 + T )^{2} \)
$31$ \( ( -12 + T )^{2} \)
$37$ \( 60516 + T^{2} \)
$41$ \( ( -182 + T )^{2} \)
$43$ \( 16384 + T^{2} \)
$47$ \( 104976 + T^{2} \)
$53$ \( 26244 + T^{2} \)
$59$ \( ( 810 + T )^{2} \)
$61$ \( ( 488 + T )^{2} \)
$67$ \( 59536 + T^{2} \)
$71$ \( ( 768 + T )^{2} \)
$73$ \( 492804 + T^{2} \)
$79$ \( ( 440 + T )^{2} \)
$83$ \( 1695204 + T^{2} \)
$89$ \( ( 730 + T )^{2} \)
$97$ \( 86436 + T^{2} \)
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