Properties

Label 175.4.a.c
Level $175$
Weight $4$
Character orbit 175.a
Self dual yes
Analytic conductor $10.325$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.3253342510\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 4) q^{2} + ( - 4 \beta - 1) q^{3} + ( - 8 \beta + 10) q^{4} + (15 \beta - 4) q^{6} + 7 q^{7} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 4) q^{2} + ( - 4 \beta - 1) q^{3} + ( - 8 \beta + 10) q^{4} + (15 \beta - 4) q^{6} + 7 q^{7} + (34 \beta - 24) q^{8} + (8 \beta + 6) q^{9} + (32 \beta - 7) q^{11} + ( - 32 \beta + 54) q^{12} + (4 \beta - 25) q^{13} + (7 \beta - 28) q^{14} + ( - 96 \beta + 84) q^{16} + (44 \beta + 25) q^{17} + ( - 26 \beta - 8) q^{18} + ( - 44 \beta + 18) q^{19} + ( - 28 \beta - 7) q^{21} + ( - 135 \beta + 92) q^{22} + ( - 68 \beta - 122) q^{23} + (62 \beta - 248) q^{24} + ( - 41 \beta + 108) q^{26} + (76 \beta - 43) q^{27} + ( - 56 \beta + 70) q^{28} + ( - 24 \beta - 13) q^{29} + (180 \beta - 60) q^{31} + (196 \beta - 336) q^{32} + ( - 4 \beta - 249) q^{33} + ( - 151 \beta - 12) q^{34} + (32 \beta - 68) q^{36} + ( - 60 \beta - 282) q^{37} + (194 \beta - 160) q^{38} + (96 \beta - 7) q^{39} + ( - 124 \beta - 164) q^{41} + (105 \beta - 28) q^{42} + (68 \beta + 130) q^{43} + (376 \beta - 582) q^{44} + (150 \beta + 352) q^{46} + ( - 132 \beta + 175) q^{47} + ( - 240 \beta + 684) q^{48} + 49 q^{49} + ( - 144 \beta - 377) q^{51} + (240 \beta - 314) q^{52} + (128 \beta + 28) q^{53} + ( - 347 \beta + 324) q^{54} + (238 \beta - 168) q^{56} + ( - 28 \beta + 334) q^{57} + (83 \beta + 4) q^{58} - 616 q^{59} + (108 \beta + 168) q^{61} + ( - 780 \beta + 600) q^{62} + (56 \beta + 42) q^{63} + ( - 352 \beta + 1064) q^{64} + ( - 233 \beta + 988) q^{66} + ( - 64 \beta + 76) q^{67} + (240 \beta - 454) q^{68} + (556 \beta + 666) q^{69} - 952 q^{71} + (12 \beta + 400) q^{72} + ( - 344 \beta - 338) q^{73} + ( - 42 \beta + 1008) q^{74} + ( - 584 \beta + 884) q^{76} + (224 \beta - 49) q^{77} + ( - 391 \beta + 220) q^{78} + ( - 248 \beta + 507) q^{79} + ( - 120 \beta - 727) q^{81} + (332 \beta + 408) q^{82} + (600 \beta + 188) q^{83} + ( - 224 \beta + 378) q^{84} + ( - 142 \beta - 384) q^{86} + (76 \beta + 205) q^{87} + ( - 1006 \beta + 2344) q^{88} + ( - 44 \beta - 108) q^{89} + (28 \beta - 175) q^{91} + (296 \beta - 132) q^{92} + (60 \beta - 1380) q^{93} + (703 \beta - 964) q^{94} + (1148 \beta - 1232) q^{96} + (220 \beta - 1371) q^{97} + (49 \beta - 196) q^{98} + (136 \beta + 470) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 2 q^{3} + 20 q^{4} - 8 q^{6} + 14 q^{7} - 48 q^{8} + 12 q^{9} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 56 q^{14} + 168 q^{16} + 50 q^{17} - 16 q^{18} + 36 q^{19} - 14 q^{21} + 184 q^{22} - 244 q^{23} - 496 q^{24} + 216 q^{26} - 86 q^{27} + 140 q^{28} - 26 q^{29} - 120 q^{31} - 672 q^{32} - 498 q^{33} - 24 q^{34} - 136 q^{36} - 564 q^{37} - 320 q^{38} - 14 q^{39} - 328 q^{41} - 56 q^{42} + 260 q^{43} - 1164 q^{44} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 98 q^{49} - 754 q^{51} - 628 q^{52} + 56 q^{53} + 648 q^{54} - 336 q^{56} + 668 q^{57} + 8 q^{58} - 1232 q^{59} + 336 q^{61} + 1200 q^{62} + 84 q^{63} + 2128 q^{64} + 1976 q^{66} + 152 q^{67} - 908 q^{68} + 1332 q^{69} - 1904 q^{71} + 800 q^{72} - 676 q^{73} + 2016 q^{74} + 1768 q^{76} - 98 q^{77} + 440 q^{78} + 1014 q^{79} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 756 q^{84} - 768 q^{86} + 410 q^{87} + 4688 q^{88} - 216 q^{89} - 350 q^{91} - 264 q^{92} - 2760 q^{93} - 1928 q^{94} - 2464 q^{96} - 2742 q^{97} - 392 q^{98} + 940 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
−5.41421 4.65685 21.3137 0 −25.2132 7.00000 −72.0833 −5.31371 0
1.2 −2.58579 −6.65685 −1.31371 0 17.2132 7.00000 24.0833 17.3137 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 175.4.a.c 2
3.b odd 2 1 1575.4.a.z 2
5.b even 2 1 35.4.a.b 2
5.c odd 4 2 175.4.b.c 4
7.b odd 2 1 1225.4.a.m 2
15.d odd 2 1 315.4.a.f 2
20.d odd 2 1 560.4.a.r 2
35.c odd 2 1 245.4.a.k 2
35.i odd 6 2 245.4.e.i 4
35.j even 6 2 245.4.e.h 4
40.e odd 2 1 2240.4.a.bo 2
40.f even 2 1 2240.4.a.bn 2
105.g even 2 1 2205.4.a.u 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.a.b 2 5.b even 2 1
175.4.a.c 2 1.a even 1 1 trivial
175.4.b.c 4 5.c odd 4 2
245.4.a.k 2 35.c odd 2 1
245.4.e.h 4 35.j even 6 2
245.4.e.i 4 35.i odd 6 2
315.4.a.f 2 15.d odd 2 1
560.4.a.r 2 20.d odd 2 1
1225.4.a.m 2 7.b odd 2 1
1575.4.a.z 2 3.b odd 2 1
2205.4.a.u 2 105.g even 2 1
2240.4.a.bn 2 40.f even 2 1
2240.4.a.bo 2 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 8T_{2} + 14 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 14 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 31 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 14T - 1999 \) Copy content Toggle raw display
$13$ \( T^{2} + 50T + 593 \) Copy content Toggle raw display
$17$ \( T^{2} - 50T - 3247 \) Copy content Toggle raw display
$19$ \( T^{2} - 36T - 3548 \) Copy content Toggle raw display
$23$ \( T^{2} + 244T + 5636 \) Copy content Toggle raw display
$29$ \( T^{2} + 26T - 983 \) Copy content Toggle raw display
$31$ \( T^{2} + 120T - 61200 \) Copy content Toggle raw display
$37$ \( T^{2} + 564T + 72324 \) Copy content Toggle raw display
$41$ \( T^{2} + 328T - 3856 \) Copy content Toggle raw display
$43$ \( T^{2} - 260T + 7652 \) Copy content Toggle raw display
$47$ \( T^{2} - 350T - 4223 \) Copy content Toggle raw display
$53$ \( T^{2} - 56T - 31984 \) Copy content Toggle raw display
$59$ \( (T + 616)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 336T + 4896 \) Copy content Toggle raw display
$67$ \( T^{2} - 152T - 2416 \) Copy content Toggle raw display
$71$ \( (T + 952)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 676T - 122428 \) Copy content Toggle raw display
$79$ \( T^{2} - 1014 T + 134041 \) Copy content Toggle raw display
$83$ \( T^{2} - 376T - 684656 \) Copy content Toggle raw display
$89$ \( T^{2} + 216T + 7792 \) Copy content Toggle raw display
$97$ \( T^{2} + 2742 T + 1782841 \) Copy content Toggle raw display
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