Properties

Label 825.4.c.q
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 574 x^{6} + 121601 x^{4} + 11262916 x^{2} + 384787456\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{2} + \beta_{3} ) q^{2} + 3 \beta_{3} q^{3} + ( 3 + \beta_{1} ) q^{4} + ( -3 + 3 \beta_{1} ) q^{6} + ( -\beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{7} + ( 11 \beta_{2} + 7 \beta_{3} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( \beta_{2} + \beta_{3} ) q^{2} + 3 \beta_{3} q^{3} + ( 3 + \beta_{1} ) q^{4} + ( -3 + 3 \beta_{1} ) q^{6} + ( -\beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{7} + ( 11 \beta_{2} + 7 \beta_{3} ) q^{8} -9 q^{9} -11 q^{11} + ( -3 \beta_{2} + 9 \beta_{3} ) q^{12} + ( 9 \beta_{2} + 11 \beta_{3} - \beta_{5} - \beta_{6} ) q^{13} + ( 2 \beta_{1} - \beta_{4} - \beta_{7} ) q^{14} + ( -27 + 15 \beta_{1} ) q^{16} + ( 19 \beta_{2} - 12 \beta_{3} + \beta_{6} ) q^{17} + ( -9 \beta_{2} - 9 \beta_{3} ) q^{18} + ( -29 - 2 \beta_{4} + \beta_{7} ) q^{19} + ( -6 - 3 \beta_{1} - 3 \beta_{4} ) q^{21} + ( -11 \beta_{2} - 11 \beta_{3} ) q^{22} + ( 26 \beta_{2} + 11 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{23} + ( -21 + 33 \beta_{1} ) q^{24} + ( -43 + 9 \beta_{1} - 3 \beta_{4} + \beta_{7} ) q^{26} -27 \beta_{3} q^{27} + ( -6 \beta_{2} + 8 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{28} + ( -55 - 17 \beta_{1} + 3 \beta_{4} + 3 \beta_{7} ) q^{29} + ( -27 - 15 \beta_{1} + 3 \beta_{4} - 3 \beta_{7} ) q^{31} + ( 61 \beta_{2} - 31 \beta_{3} ) q^{32} -33 \beta_{3} q^{33} + ( -66 - 12 \beta_{1} - \beta_{4} - \beta_{7} ) q^{34} + ( -27 - 9 \beta_{1} ) q^{36} + ( 75 \beta_{2} - 68 \beta_{3} - \beta_{5} + 3 \beta_{6} ) q^{37} + ( -31 \beta_{2} - 35 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{38} + ( -33 + 27 \beta_{1} + 3 \beta_{4} - 3 \beta_{7} ) q^{39} + ( -26 - \beta_{1} + 3 \beta_{4} + 4 \beta_{7} ) q^{41} + ( -6 \beta_{2} + 3 \beta_{5} - 3 \beta_{6} ) q^{42} + ( -109 \beta_{2} - 51 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{43} + ( -33 - 11 \beta_{1} ) q^{44} + ( -115 + 15 \beta_{1} + 10 \beta_{4} + 2 \beta_{7} ) q^{46} + ( 151 \beta_{2} - 90 \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{47} + ( -45 \beta_{2} - 81 \beta_{3} ) q^{48} + ( -287 + 75 \beta_{1} - 5 \beta_{4} + \beta_{7} ) q^{49} + ( 36 + 57 \beta_{1} - 3 \beta_{4} ) q^{51} + ( 27 \beta_{2} + \beta_{3} - 5 \beta_{5} - 7 \beta_{6} ) q^{52} + ( -18 \beta_{2} - 36 \beta_{3} - 8 \beta_{5} - 8 \beta_{6} ) q^{53} + ( 27 - 27 \beta_{1} ) q^{54} + ( 8 + 26 \beta_{1} - 7 \beta_{4} - 11 \beta_{7} ) q^{56} + ( -87 \beta_{3} - 3 \beta_{5} - 6 \beta_{6} ) q^{57} + ( -61 \beta_{2} + 13 \beta_{3} - 3 \beta_{5} + 15 \beta_{6} ) q^{58} + ( -284 + 85 \beta_{1} + 15 \beta_{4} - 5 \beta_{7} ) q^{59} + ( -110 + 36 \beta_{1} + 18 \beta_{4} + 4 \beta_{7} ) q^{61} + ( -21 \beta_{2} + 45 \beta_{3} - 3 \beta_{5} - 9 \beta_{6} ) q^{62} + ( 9 \beta_{2} - 18 \beta_{3} - 9 \beta_{6} ) q^{63} + ( -429 + 89 \beta_{1} ) q^{64} + ( 33 - 33 \beta_{1} ) q^{66} + ( 76 \beta_{2} - 30 \beta_{3} + 14 \beta_{5} + 6 \beta_{6} ) q^{67} + ( 88 \beta_{2} - 114 \beta_{3} + \beta_{5} + 3 \beta_{6} ) q^{68} + ( -33 + 78 \beta_{1} + 6 \beta_{4} + 6 \beta_{7} ) q^{69} + ( 339 + 86 \beta_{1} - 2 \beta_{4} + 4 \beta_{7} ) q^{71} + ( -99 \beta_{2} - 63 \beta_{3} ) q^{72} + ( -96 \beta_{2} - 190 \beta_{3} + 20 \beta_{5} + 2 \beta_{6} ) q^{73} + ( -236 - 70 \beta_{1} - 7 \beta_{4} - 3 \beta_{7} ) q^{74} + ( -81 - 31 \beta_{1} - 10 \beta_{4} + 6 \beta_{7} ) q^{76} + ( 11 \beta_{2} - 22 \beta_{3} - 11 \beta_{6} ) q^{77} + ( -27 \beta_{2} - 129 \beta_{3} - 3 \beta_{5} - 9 \beta_{6} ) q^{78} + ( -744 + 199 \beta_{1} + 11 \beta_{4} - 2 \beta_{7} ) q^{79} + 81 q^{81} + ( -34 \beta_{2} - 24 \beta_{3} - 3 \beta_{5} + 19 \beta_{6} ) q^{82} + ( 257 \beta_{2} - 113 \beta_{3} - 7 \beta_{5} + 5 \beta_{6} ) q^{83} + ( -24 - 18 \beta_{1} - 9 \beta_{4} + 3 \beta_{7} ) q^{84} + ( 491 - 57 \beta_{1} - 13 \beta_{4} - \beta_{7} ) q^{86} + ( 51 \beta_{2} - 165 \beta_{3} - 9 \beta_{5} + 9 \beta_{6} ) q^{87} + ( -121 \beta_{2} - 77 \beta_{3} ) q^{88} + ( -363 + 207 \beta_{1} - 15 \beta_{4} + 5 \beta_{7} ) q^{89} + ( 298 + 482 \beta_{1} - 6 \beta_{4} - 12 \beta_{7} ) q^{91} + ( 89 \beta_{2} - 71 \beta_{3} + 6 \beta_{5} + 2 \beta_{6} ) q^{92} + ( 45 \beta_{2} - 81 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} ) q^{93} + ( -510 - 88 \beta_{1} + 7 \beta_{4} + 3 \beta_{7} ) q^{94} + ( 93 + 183 \beta_{1} ) q^{96} + ( -12 \beta_{2} + 29 \beta_{3} - 10 \beta_{5} + 32 \beta_{6} ) q^{97} + ( -289 \beta_{2} - 599 \beta_{3} + 5 \beta_{5} - \beta_{6} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 28q^{4} - 12q^{6} - 72q^{9} + O(q^{10}) \) \( 8q + 28q^{4} - 12q^{6} - 72q^{9} - 88q^{11} + 6q^{14} - 156q^{16} - 236q^{19} - 66q^{21} - 36q^{24} - 314q^{26} - 502q^{29} - 270q^{31} - 578q^{34} - 252q^{36} - 150q^{39} - 206q^{41} - 308q^{44} - 840q^{46} - 2006q^{49} + 510q^{51} + 108q^{54} + 154q^{56} - 1902q^{59} - 700q^{61} - 3076q^{64} + 132q^{66} + 60q^{69} + 3052q^{71} - 2182q^{74} - 792q^{76} - 5134q^{79} + 648q^{81} - 282q^{84} + 3674q^{86} - 2106q^{89} + 4300q^{91} - 4418q^{94} + 1476q^{96} + 792q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 574 x^{6} + 121601 x^{4} + 11262916 x^{2} + 384787456\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 431 \nu^{4} + 59278 \nu^{2} + 2597792 \)\()/19320\)
\(\beta_{2}\)\(=\)\((\)\( 19 \nu^{7} + 2324 \nu^{5} - 542483 \nu^{3} - 63786772 \nu \)\()/23686320\)
\(\beta_{3}\)\(=\)\((\)\( -71 \nu^{7} - 30946 \nu^{5} - 4406423 \nu^{3} - 204733372 \nu \)\()/13535040\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} + 546 \nu^{4} + 93893 \nu^{2} + 5075812 \)\()/3220\)
\(\beta_{5}\)\(=\)\((\)\( 71 \nu^{7} + 30946 \nu^{5} + 4406423 \nu^{3} + 231803452 \nu \)\()/6767520\)
\(\beta_{6}\)\(=\)\((\)\( 4483 \nu^{7} + 1906298 \nu^{5} + 257684419 \nu^{3} + 10956545996 \nu \)\()/94745280\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 431 \nu^{4} + 60658 \nu^{2} + 2786162 \)\()/345\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + 2 \beta_{3}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - 56 \beta_{1} - 546\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-56 \beta_{6} - 151 \beta_{5} - 790 \beta_{3} + 112 \beta_{2}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-301 \beta_{7} + 112 \beta_{4} + 16184 \beta_{1} + 78154\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(16184 \beta_{6} + 23735 \beta_{5} + 186070 \beta_{3} - 48272 \beta_{2}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(70453 \beta_{7} - 48272 \beta_{4} - 3578456 \beta_{1} - 11709754\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-3578456 \beta_{6} - 3857279 \beta_{5} - 38600774 \beta_{3} + 14088816 \beta_{2}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
10.1952i
11.1952i
12.6191i
13.6191i
13.6191i
12.6191i
11.1952i
10.1952i
2.56155i 3.00000i 1.43845 0 −7.68466 29.1157i 24.1771i −9.00000 0
199.2 2.56155i 3.00000i 1.43845 0 −7.68466 25.6772i 24.1771i −9.00000 0
199.3 1.56155i 3.00000i 5.56155 0 4.68466 16.7054i 21.1771i −9.00000 0
199.4 1.56155i 3.00000i 5.56155 0 4.68466 24.2670i 21.1771i −9.00000 0
199.5 1.56155i 3.00000i 5.56155 0 4.68466 24.2670i 21.1771i −9.00000 0
199.6 1.56155i 3.00000i 5.56155 0 4.68466 16.7054i 21.1771i −9.00000 0
199.7 2.56155i 3.00000i 1.43845 0 −7.68466 25.6772i 24.1771i −9.00000 0
199.8 2.56155i 3.00000i 1.43845 0 −7.68466 29.1157i 24.1771i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.8
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 825.4.c.q 8
5.b even 2 1 inner 825.4.c.q 8
5.c odd 4 1 825.4.a.u 4
5.c odd 4 1 825.4.a.v yes 4
15.e even 4 1 2475.4.a.bb 4
15.e even 4 1 2475.4.a.bd 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.u 4 5.c odd 4 1
825.4.a.v yes 4 5.c odd 4 1
825.4.c.q 8 1.a even 1 1 trivial
825.4.c.q 8 5.b even 2 1 inner
2475.4.a.bb 4 15.e even 4 1
2475.4.a.bd 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 9 T_{2}^{2} + 16 \)
\( T_{7}^{8} + 2375 T_{7}^{6} + 2031311 T_{7}^{4} + 732789005 T_{7}^{2} + 91853849476 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 16 + 9 T^{2} + T^{4} )^{2} \)
$3$ \( ( 9 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 91853849476 + 732789005 T^{2} + 2031311 T^{4} + 2375 T^{6} + T^{8} \)
$11$ \( ( 11 + T )^{8} \)
$13$ \( 2913657819136 + 44481575680 T^{2} + 44593712 T^{4} + 12505 T^{6} + T^{8} \)
$17$ \( 1726301676544 + 26288628273 T^{2} + 30043043 T^{4} + 10611 T^{6} + T^{8} \)
$19$ \( ( -6797601 - 456966 T - 3664 T^{2} + 118 T^{3} + T^{4} )^{2} \)
$23$ \( 411778697413561 + 4256546683116 T^{2} + 920615086 T^{4} + 58236 T^{6} + T^{8} \)
$29$ \( ( -42323912 - 8773676 T - 32818 T^{2} + 251 T^{3} + T^{4} )^{2} \)
$31$ \( ( 92275200 - 265680 T - 45522 T^{2} + 135 T^{3} + T^{4} )^{2} \)
$37$ \( 777184329612103396 + 197065022778685 T^{2} + 9980124431 T^{4} + 174775 T^{6} + T^{8} \)
$41$ \( ( 1343813418 - 4808235 T - 82279 T^{2} + 103 T^{3} + T^{4} )^{2} \)
$43$ \( 632083309824835584 + 239849545540608 T^{2} + 22141657168 T^{4} + 286201 T^{6} + T^{8} \)
$47$ \( 1040397070562062096 + 3631539664209297 T^{2} + 82874146099 T^{4} + 521907 T^{6} + T^{8} \)
$53$ \( 10724211504173092864 + 4855870593444352 T^{2} + 144192938768 T^{4} + 723304 T^{6} + T^{8} \)
$59$ \( ( -37958294854 - 234590839 T - 75259 T^{2} + 951 T^{3} + T^{4} )^{2} \)
$61$ \( ( 10189762592 - 33725752 T - 426732 T^{2} + 350 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(15\!\cdots\!04\)\( + 235577950644501312 T^{2} + 1123064318704 T^{4} + 1972380 T^{6} + T^{8} \)
$71$ \( ( 6876641459 - 132382682 T + 732444 T^{2} - 1526 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(39\!\cdots\!56\)\( + 2788688056849048384 T^{2} + 5286353309936 T^{4} + 3894556 T^{6} + T^{8} \)
$79$ \( ( -51952998816 + 399027621 T + 1997131 T^{2} + 2567 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(30\!\cdots\!00\)\( + 170697354634569600 T^{2} + 1037049626596 T^{4} + 1896989 T^{6} + T^{8} \)
$89$ \( ( -134213353376 - 576920604 T - 338640 T^{2} + 1053 T^{3} + T^{4} )^{2} \)
$97$ \( \)\(72\!\cdots\!41\)\( + 106607143161643540 T^{2} + 3011993633366 T^{4} + 3445300 T^{6} + T^{8} \)
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