Properties

Label 175.10.b.d
Level $175$
Weight $10$
Character orbit 175.b
Analytic conductor $90.131$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(90.1312713287\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Defining polynomial: \( x^{6} + 853x^{4} + 185508x^{2} + 4064256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 7 \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + 28 \beta_1) q^{3} + ( - 7 \beta_{3} + 8 \beta_{2} - 519) q^{4} + ( - 6 \beta_{3} + 36 \beta_{2} + 1638) q^{6} - 2401 \beta_1 q^{7} + (147 \beta_{5} + 470 \beta_{4} + 4685 \beta_1) q^{8} + (126 \beta_{3} - 90 \beta_{2} + 8667) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 7 \beta_1) q^{2} + ( - \beta_{5} + \beta_{4} + 28 \beta_1) q^{3} + ( - 7 \beta_{3} + 8 \beta_{2} - 519) q^{4} + ( - 6 \beta_{3} + 36 \beta_{2} + 1638) q^{6} - 2401 \beta_1 q^{7} + (147 \beta_{5} + 470 \beta_{4} + 4685 \beta_1) q^{8} + (126 \beta_{3} - 90 \beta_{2} + 8667) q^{9} + (658 \beta_{3} + 650 \beta_{2} - 1148) q^{11} + ( - 182 \beta_{5} - 700 \beta_{4} + 35462 \beta_1) q^{12} + ( - 175 \beta_{5} - 3017 \beta_{4} - 6594 \beta_1) q^{13} + (2401 \beta_{2} - 16807) q^{14} + (1617 \beta_{3} - 10614 \beta_{2} + 160987) q^{16} + ( - 1574 \beta_{5} + 3030 \beta_{4} - 338898 \beta_1) q^{17} + ( - 2268 \beta_{5} - 16947 \beta_{4} - 91089 \beta_1) q^{18} + ( - 2437 \beta_{3} + 15371 \beta_{2} - 74284) q^{19} + ( - 2401 \beta_{3} - 2401 \beta_{2} + 67228) q^{21} + ( - 4004 \beta_{5} - 40972 \beta_{4} + 949016 \beta_1) q^{22} + ( - 3808 \beta_{5} + 24200 \beta_{4} + 628544 \beta_1) q^{23} + ( - 10338 \beta_{3} - 4500 \beta_{2} + 483210) q^{24} + ( - 23394 \beta_{3} + 20986 \beta_{2} - 2928352) q^{26} + ( - 15786 \beta_{5} + 33930 \beta_{4} - 183960 \beta_1) q^{27} + (16807 \beta_{5} + 19208 \beta_{4} + 1246119 \beta_1) q^{28} + ( - 18914 \beta_{3} - 54866 \beta_{2} - 1360606) q^{29} + ( - 70302 \beta_{3} + 55698 \beta_{2} + 956480) q^{31} + ( - 20055 \beta_{5} - 36066 \beta_{4} - 8407317 \beta_1) q^{32} + (65640 \beta_{5} + 76152 \beta_{4} - 6753264 \beta_1) q^{33} + (748 \beta_{3} + 438178 \beta_{2} + 1327214) q^{34} + ( - 83601 \beta_{3} + 209376 \beta_{2} - 11798793) q^{36} + ( - 60522 \beta_{5} + 209418 \beta_{4} - 465206 \beta_1) q^{37} + (139278 \beta_{5} + 248060 \beta_{4} + 14493290 \beta_1) q^{38} + (13748 \beta_{3} + 110012 \beta_{2} + 2996896) q^{39} + ( - 131894 \beta_{3} - 163478 \beta_{2} - 4806886) q^{41} + (14406 \beta_{5} + 86436 \beta_{4} - 3932838 \beta_1) q^{42} + (65366 \beta_{5} - 121982 \beta_{4} - 20543724 \beta_1) q^{43} + ( - 1960 \beta_{3} - 314984 \beta_{2} - 32337328) q^{44} + (119896 \beta_{3} - 405224 \beta_{2} + 29915888) q^{46} + ( - 83238 \beta_{5} - 534778 \beta_{4} + 3456320 \beta_1) q^{47} + (9710 \beta_{5} - 174140 \beta_{4} + 5599594 \beta_1) q^{48} - 5764801 q^{49} + ( - 186102 \beta_{3} - 587238 \beta_{2} - 8715576) q^{51} + (361424 \beta_{5} + 2925244 \beta_{4} + 26969348 \beta_1) q^{52} + ( - 450352 \beta_{5} - 1553376 \beta_{4} + \cdots + 22500870 \beta_1) q^{53}+ \cdots + ( - 376362 \beta_{3} + 7689150 \beta_{2} + \cdots + 633659724) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3114 q^{4} + 9828 q^{6} + 52002 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3114 q^{4} + 9828 q^{6} + 52002 q^{9} - 6888 q^{11} - 100842 q^{14} + 965922 q^{16} - 445704 q^{19} + 403368 q^{21} + 2899260 q^{24} - 17570112 q^{26} - 8163636 q^{29} + 5738880 q^{31} + 7963284 q^{34} - 70792758 q^{36} + 17981376 q^{39} - 28841316 q^{41} - 194023968 q^{44} + 179495328 q^{46} - 34588806 q^{49} - 52293456 q^{51} + 235758600 q^{54} + 67492110 q^{56} + 85180200 q^{59} + 383493684 q^{61} - 15704322 q^{64} - 16115904 q^{66} - 515807712 q^{69} + 593029008 q^{71} + 1381392924 q^{74} + 1457679972 q^{76} + 1920825312 q^{79} - 71655354 q^{81} + 510865572 q^{84} - 1761964512 q^{86} - 1013632956 q^{89} - 94993164 q^{91} - 2776009656 q^{94} + 666771588 q^{96} + 3801958344 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 853x^{4} + 185508x^{2} + 4064256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 1163\nu^{3} + 675324\nu ) / 3205440 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{4} - 6287\nu^{2} - 480096 ) / 9540 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{4} - 1817\nu^{2} + 77688 ) / 1908 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 692\nu^{3} + 94501\nu ) / 11130 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\nu^{5} + 19697\nu^{3} + 4509828\nu ) / 320544 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11\beta_{3} - 25\beta_{2} - 1706 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -415\beta_{5} + 451\beta_{4} + 9538\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6287\beta_{3} + 9085\beta_{2} + 713186 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 192679\beta_{5} - 150811\beta_{4} - 6789298\beta_1 ) / 6 \) 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
4.96128i
18.2745i
22.2358i
22.2358i
18.2745i
4.96128i
41.8019i 0.232339i −1235.40 0 9.71222 2401.00i 30239.6i 19682.9 0
99.2 34.1627i 79.6469i −655.088 0 2720.95 2401.00i 4888.28i 13339.4 0
99.3 13.3607i 163.415i 333.491 0 2183.34 2401.00i 11296.4i −7021.32 0
99.4 13.3607i 163.415i 333.491 0 2183.34 2401.00i 11296.4i −7021.32 0
99.5 34.1627i 79.6469i −655.088 0 2720.95 2401.00i 4888.28i 13339.4 0
99.6 41.8019i 0.232339i −1235.40 0 9.71222 2401.00i 30239.6i 19682.9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
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 175.10.b.d 6
5.b even 2 1 inner 175.10.b.d 6
5.c odd 4 1 7.10.a.b 3
5.c odd 4 1 175.10.a.d 3
15.e even 4 1 63.10.a.e 3
20.e even 4 1 112.10.a.h 3
35.f even 4 1 49.10.a.c 3
35.k even 12 2 49.10.c.e 6
35.l odd 12 2 49.10.c.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.a.b 3 5.c odd 4 1
49.10.a.c 3 35.f even 4 1
49.10.c.d 6 35.l odd 12 2
49.10.c.e 6 35.k even 12 2
63.10.a.e 3 15.e even 4 1
112.10.a.h 3 20.e even 4 1
175.10.a.d 3 5.c odd 4 1
175.10.b.d 6 1.a even 1 1 trivial
175.10.b.d 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3093T_{2}^{4} + 2559636T_{2}^{2} + 364046400 \) acting on \(S_{10}^{\mathrm{new}}(175, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 3093 T^{4} + \cdots + 364046400 \) Copy content Toggle raw display
$3$ \( T^{6} + 33048 T^{4} + \cdots + 9144576 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{2} + 5764801)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} + 3444 T^{2} + \cdots + 108859759460352)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 26253893652 T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{6} + 446677667052 T^{4} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{3} + 222852 T^{2} + \cdots + 43\!\cdots\!60)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 3527218647168 T^{4} + \cdots + 94\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{3} + 4081818 T^{2} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 2869440 T^{2} + \cdots - 74\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 256366243984332 T^{4} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$41$ \( (T^{3} + 14420658 T^{2} + \cdots - 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{6} + 928644247418784 T^{4} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 57\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{3} - 42590100 T^{2} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 191746842 T^{2} + \cdots + 51\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} - 296514504 T^{2} + \cdots + 16\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 38\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{3} - 960412656 T^{2} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 36\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{3} + 506816478 T^{2} + \cdots - 19\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
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