Properties

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

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 4 \beta_1) q^{2} + ( - 4 \beta_{2} - \beta_1) q^{3} + (8 \beta_{3} - 10) q^{4} + (15 \beta_{3} - 4) q^{6} - 7 \beta_1 q^{7} + (34 \beta_{2} - 24 \beta_1) q^{8} + ( - 8 \beta_{3} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 4 \beta_1) q^{2} + ( - 4 \beta_{2} - \beta_1) q^{3} + (8 \beta_{3} - 10) q^{4} + (15 \beta_{3} - 4) q^{6} - 7 \beta_1 q^{7} + (34 \beta_{2} - 24 \beta_1) q^{8} + ( - 8 \beta_{3} - 6) q^{9} + (32 \beta_{3} - 7) q^{11} + (32 \beta_{2} - 54 \beta_1) q^{12} + (4 \beta_{2} - 25 \beta_1) q^{13} + ( - 7 \beta_{3} + 28) q^{14} + ( - 96 \beta_{3} + 84) q^{16} + ( - 44 \beta_{2} - 25 \beta_1) q^{17} + ( - 26 \beta_{2} - 8 \beta_1) q^{18} + (44 \beta_{3} - 18) q^{19} + ( - 28 \beta_{3} - 7) q^{21} + (135 \beta_{2} - 92 \beta_1) q^{22} + ( - 68 \beta_{2} - 122 \beta_1) q^{23} + ( - 62 \beta_{3} + 248) q^{24} + ( - 41 \beta_{3} + 108) q^{26} + ( - 76 \beta_{2} + 43 \beta_1) q^{27} + ( - 56 \beta_{2} + 70 \beta_1) q^{28} + (24 \beta_{3} + 13) q^{29} + (180 \beta_{3} - 60) q^{31} + ( - 196 \beta_{2} + 336 \beta_1) q^{32} + ( - 4 \beta_{2} - 249 \beta_1) q^{33} + (151 \beta_{3} + 12) q^{34} + (32 \beta_{3} - 68) q^{36} + (60 \beta_{2} + 282 \beta_1) q^{37} + (194 \beta_{2} - 160 \beta_1) q^{38} + ( - 96 \beta_{3} + 7) q^{39} + ( - 124 \beta_{3} - 164) q^{41} + ( - 105 \beta_{2} + 28 \beta_1) q^{42} + (68 \beta_{2} + 130 \beta_1) q^{43} + ( - 376 \beta_{3} + 582) q^{44} + (150 \beta_{3} + 352) q^{46} + (132 \beta_{2} - 175 \beta_1) q^{47} + ( - 240 \beta_{2} + 684 \beta_1) q^{48} - 49 q^{49} + ( - 144 \beta_{3} - 377) q^{51} + ( - 240 \beta_{2} + 314 \beta_1) q^{52} + (128 \beta_{2} + 28 \beta_1) q^{53} + (347 \beta_{3} - 324) q^{54} + (238 \beta_{3} - 168) q^{56} + (28 \beta_{2} - 334 \beta_1) q^{57} + (83 \beta_{2} + 4 \beta_1) q^{58} + 616 q^{59} + (108 \beta_{3} + 168) q^{61} + (780 \beta_{2} - 600 \beta_1) q^{62} + (56 \beta_{2} + 42 \beta_1) q^{63} + (352 \beta_{3} - 1064) q^{64} + ( - 233 \beta_{3} + 988) q^{66} + (64 \beta_{2} - 76 \beta_1) q^{67} + (240 \beta_{2} - 454 \beta_1) q^{68} + ( - 556 \beta_{3} - 666) q^{69} - 952 q^{71} + ( - 12 \beta_{2} - 400 \beta_1) q^{72} + ( - 344 \beta_{2} - 338 \beta_1) q^{73} + (42 \beta_{3} - 1008) q^{74} + ( - 584 \beta_{3} + 884) q^{76} + ( - 224 \beta_{2} + 49 \beta_1) q^{77} + ( - 391 \beta_{2} + 220 \beta_1) q^{78} + (248 \beta_{3} - 507) q^{79} + ( - 120 \beta_{3} - 727) q^{81} + ( - 332 \beta_{2} - 408 \beta_1) q^{82} + (600 \beta_{2} + 188 \beta_1) q^{83} + (224 \beta_{3} - 378) q^{84} + ( - 142 \beta_{3} - 384) q^{86} + ( - 76 \beta_{2} - 205 \beta_1) q^{87} + ( - 1006 \beta_{2} + 2344 \beta_1) q^{88} + (44 \beta_{3} + 108) q^{89} + (28 \beta_{3} - 175) q^{91} + ( - 296 \beta_{2} + 132 \beta_1) q^{92} + (60 \beta_{2} - 1380 \beta_1) q^{93} + ( - 703 \beta_{3} + 964) q^{94} + (1148 \beta_{3} - 1232) q^{96} + ( - 220 \beta_{2} + 1371 \beta_1) q^{97} + (49 \beta_{2} - 196 \beta_1) q^{98} + ( - 136 \beta_{3} - 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{4} - 16 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{4} - 16 q^{6} - 24 q^{9} - 28 q^{11} + 112 q^{14} + 336 q^{16} - 72 q^{19} - 28 q^{21} + 992 q^{24} + 432 q^{26} + 52 q^{29} - 240 q^{31} + 48 q^{34} - 272 q^{36} + 28 q^{39} - 656 q^{41} + 2328 q^{44} + 1408 q^{46} - 196 q^{49} - 1508 q^{51} - 1296 q^{54} - 672 q^{56} + 2464 q^{59} + 672 q^{61} - 4256 q^{64} + 3952 q^{66} - 2664 q^{69} - 3808 q^{71} - 4032 q^{74} + 3536 q^{76} - 2028 q^{79} - 2908 q^{81} - 1512 q^{84} - 1536 q^{86} + 432 q^{89} - 700 q^{91} + 3856 q^{94} - 4928 q^{96} - 1880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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
−0.707107 + 0.707107i
0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i
5.41421i 4.65685i −21.3137 0 −25.2132 7.00000i 72.0833i 5.31371 0
99.2 2.58579i 6.65685i 1.31371 0 17.2132 7.00000i 24.0833i −17.3137 0
99.3 2.58579i 6.65685i 1.31371 0 17.2132 7.00000i 24.0833i −17.3137 0
99.4 5.41421i 4.65685i −21.3137 0 −25.2132 7.00000i 72.0833i 5.31371 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.c 4
5.b even 2 1 inner 175.4.b.c 4
5.c odd 4 1 35.4.a.b 2
5.c odd 4 1 175.4.a.c 2
15.e even 4 1 315.4.a.f 2
15.e even 4 1 1575.4.a.z 2
20.e even 4 1 560.4.a.r 2
35.f even 4 1 245.4.a.k 2
35.f even 4 1 1225.4.a.m 2
35.k even 12 2 245.4.e.i 4
35.l odd 12 2 245.4.e.h 4
40.i odd 4 1 2240.4.a.bn 2
40.k even 4 1 2240.4.a.bo 2
105.k odd 4 1 2205.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 5.c odd 4 1
175.4.a.c 2 5.c odd 4 1
175.4.b.c 4 1.a even 1 1 trivial
175.4.b.c 4 5.b even 2 1 inner
245.4.a.k 2 35.f even 4 1
245.4.e.h 4 35.l odd 12 2
245.4.e.i 4 35.k even 12 2
315.4.a.f 2 15.e even 4 1
560.4.a.r 2 20.e even 4 1
1225.4.a.m 2 35.f even 4 1
1575.4.a.z 2 15.e even 4 1
2205.4.a.u 2 105.k odd 4 1
2240.4.a.bn 2 40.i odd 4 1
2240.4.a.bo 2 40.k even 4 1

Hecke kernels

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

\( T_{2}^{4} + 36T_{2}^{2} + 196 \) Copy content Toggle raw display
\( T_{3}^{4} + 66T_{3}^{2} + 961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 36T^{2} + 196 \) Copy content Toggle raw display
$3$ \( T^{4} + 66T^{2} + 961 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 14 T - 1999)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1314 T^{2} + 351649 \) Copy content Toggle raw display
$17$ \( T^{4} + 8994 T^{2} + \cdots + 10543009 \) Copy content Toggle raw display
$19$ \( (T^{2} + 36 T - 3548)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 48264 T^{2} + \cdots + 31764496 \) Copy content Toggle raw display
$29$ \( (T^{2} - 26 T - 983)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 120 T - 61200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 173448 T^{2} + \cdots + 5230760976 \) Copy content Toggle raw display
$41$ \( (T^{2} + 328 T - 3856)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 52296 T^{2} + \cdots + 58553104 \) Copy content Toggle raw display
$47$ \( T^{4} + 130946 T^{2} + \cdots + 17833729 \) Copy content Toggle raw display
$53$ \( T^{4} + 67104 T^{2} + \cdots + 1022976256 \) Copy content Toggle raw display
$59$ \( (T - 616)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 336 T + 4896)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 27936 T^{2} + \cdots + 5837056 \) Copy content Toggle raw display
$71$ \( (T + 952)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 701832 T^{2} + \cdots + 14988615184 \) Copy content Toggle raw display
$79$ \( (T^{2} + 1014 T + 134041)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 1510688 T^{2} + \cdots + 468753838336 \) Copy content Toggle raw display
$89$ \( (T^{2} - 216 T + 7792)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 3952882 T^{2} + \cdots + 3178522031281 \) Copy content Toggle raw display
show more
show less