Properties

Label 825.4.c.s
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 43 x^{8} + 631 x^{6} + 3625 x^{4} + 7104 x^{2} + 900\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} -3 \beta_{4} q^{6} + ( 4 \beta_{2} - \beta_{7} - \beta_{9} ) q^{7} + ( 2 \beta_{1} + \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + 3 \beta_{2} q^{3} + ( -1 + \beta_{3} ) q^{4} -3 \beta_{4} q^{6} + ( 4 \beta_{2} - \beta_{7} - \beta_{9} ) q^{7} + ( 2 \beta_{1} + \beta_{5} ) q^{8} -9 q^{9} -11 q^{11} + ( -3 \beta_{2} + 3 \beta_{7} ) q^{12} + ( 4 \beta_{1} - 6 \beta_{2} - 3 \beta_{5} - \beta_{7} - \beta_{9} ) q^{13} + ( 1 - \beta_{3} - 11 \beta_{4} + \beta_{6} ) q^{14} + ( -27 + 5 \beta_{3} + 2 \beta_{6} ) q^{16} + ( 10 \beta_{2} - 6 \beta_{7} + 7 \beta_{9} ) q^{17} -9 \beta_{1} q^{18} + ( 18 - 9 \beta_{3} - 16 \beta_{4} + 2 \beta_{6} - \beta_{8} ) q^{19} + ( -12 + 3 \beta_{3} + 3 \beta_{6} ) q^{21} -11 \beta_{1} q^{22} + ( 12 \beta_{1} - 29 \beta_{2} + 2 \beta_{5} - 15 \beta_{7} + 4 \beta_{9} ) q^{23} + ( -6 \beta_{4} + 3 \beta_{8} ) q^{24} + ( -32 + 18 \beta_{3} - \beta_{4} - 5 \beta_{6} ) q^{26} -27 \beta_{2} q^{27} + ( 4 \beta_{1} - 66 \beta_{2} - 2 \beta_{5} + 2 \beta_{7} - 7 \beta_{9} ) q^{28} + ( 22 + 6 \beta_{3} - 4 \beta_{4} + \beta_{6} + 11 \beta_{8} ) q^{29} + ( -58 + 3 \beta_{3} + 3 \beta_{6} + 11 \beta_{8} ) q^{31} + ( -40 \beta_{1} + 2 \beta_{2} + 11 \beta_{5} - 2 \beta_{7} + 2 \beta_{9} ) q^{32} -33 \beta_{2} q^{33} + ( -7 + 7 \beta_{3} - 26 \beta_{4} - 7 \beta_{6} - 13 \beta_{8} ) q^{34} + ( 9 - 9 \beta_{3} ) q^{36} + ( -24 \beta_{1} - 39 \beta_{2} - 11 \beta_{5} - 15 \beta_{7} - 5 \beta_{9} ) q^{37} + ( 59 \beta_{1} - 141 \beta_{2} - 11 \beta_{5} + 19 \beta_{7} ) q^{38} + ( 18 + 3 \beta_{3} - 12 \beta_{4} + 3 \beta_{6} - 9 \beta_{8} ) q^{39} + ( -16 + 4 \beta_{3} - 68 \beta_{4} - 7 \beta_{6} - 12 \beta_{8} ) q^{41} + ( -33 \beta_{1} + 3 \beta_{2} - 3 \beta_{7} + 3 \beta_{9} ) q^{42} + ( 12 \beta_{1} - 62 \beta_{2} + 11 \beta_{5} - 45 \beta_{7} + 3 \beta_{9} ) q^{43} + ( 11 - 11 \beta_{3} ) q^{44} + ( -114 + 6 \beta_{3} - 38 \beta_{4} - 19 \beta_{8} ) q^{46} + ( -4 \beta_{1} - 33 \beta_{2} - 11 \beta_{5} + 33 \beta_{7} - 33 \beta_{9} ) q^{47} + ( -81 \beta_{2} + 15 \beta_{7} + 6 \beta_{9} ) q^{48} + ( 94 - 18 \beta_{3} - 8 \beta_{4} - 11 \beta_{6} + 11 \beta_{8} ) q^{49} + ( -30 + 18 \beta_{3} - 21 \beta_{6} ) q^{51} + ( -80 \beta_{1} - 62 \beta_{2} - \beta_{5} - 2 \beta_{7} - 13 \beta_{9} ) q^{52} + ( 24 \beta_{1} - 182 \beta_{2} + 26 \beta_{5} + 24 \beta_{7} - 14 \beta_{9} ) q^{53} + 27 \beta_{4} q^{54} + ( -19 - \beta_{3} - 26 \beta_{4} + 11 \beta_{6} + 9 \beta_{8} ) q^{56} + ( -48 \beta_{1} + 54 \beta_{2} + 3 \beta_{5} - 27 \beta_{7} + 6 \beta_{9} ) q^{57} + ( -10 \beta_{1} - 46 \beta_{2} + 5 \beta_{5} - 52 \beta_{7} + 23 \beta_{9} ) q^{58} + ( 235 + 5 \beta_{3} - 44 \beta_{4} - 5 \beta_{6} - 11 \beta_{8} ) q^{59} + ( -24 - 53 \beta_{3} - 116 \beta_{4} + 22 \beta_{6} + 2 \beta_{8} ) q^{61} + ( -79 \beta_{1} - 8 \beta_{2} - 58 \beta_{7} + 25 \beta_{9} ) q^{62} + ( -36 \beta_{2} + 9 \beta_{7} + 9 \beta_{9} ) q^{63} + ( 131 - 53 \beta_{3} - 8 \beta_{4} + 36 \beta_{6} - 4 \beta_{8} ) q^{64} + 33 \beta_{4} q^{66} + ( -160 \beta_{1} - 150 \beta_{2} + 20 \beta_{5} - 19 \beta_{7} + 22 \beta_{9} ) q^{67} + ( -28 \beta_{1} - 148 \beta_{2} + 14 \beta_{5} + 50 \beta_{7} + 23 \beta_{9} ) q^{68} + ( 87 + 45 \beta_{3} - 36 \beta_{4} - 12 \beta_{6} + 6 \beta_{8} ) q^{69} + ( -95 - 39 \beta_{3} - 132 \beta_{4} - 2 \beta_{6} + 22 \beta_{8} ) q^{71} + ( -18 \beta_{1} - 9 \beta_{5} ) q^{72} + ( -8 \beta_{1} - 64 \beta_{2} - 38 \beta_{5} + 22 \beta_{7} - 44 \beta_{9} ) q^{73} + ( 232 + 26 \beta_{3} - 46 \beta_{4} - 17 \beta_{6} - 10 \beta_{8} ) q^{74} + ( -376 + 42 \beta_{3} + 108 \beta_{4} - 6 \beta_{6} + 11 \beta_{8} ) q^{76} + ( -44 \beta_{2} + 11 \beta_{7} + 11 \beta_{9} ) q^{77} + ( -3 \beta_{1} - 96 \beta_{2} + 54 \beta_{7} - 15 \beta_{9} ) q^{78} + ( 520 + 2 \beta_{3} - 116 \beta_{4} - 39 \beta_{6} - 6 \beta_{8} ) q^{79} + 81 q^{81} + ( -22 \beta_{1} - 607 \beta_{2} + 11 \beta_{5} + 135 \beta_{7} - 31 \beta_{9} ) q^{82} + ( 32 \beta_{1} - 330 \beta_{2} - 35 \beta_{5} - 2 \beta_{7} - 15 \beta_{9} ) q^{83} + ( 198 - 6 \beta_{3} - 12 \beta_{4} + 21 \beta_{6} - 6 \beta_{8} ) q^{84} + ( -122 - 40 \beta_{3} - 157 \beta_{4} + 19 \beta_{6} - 48 \beta_{8} ) q^{86} + ( -12 \beta_{1} + 66 \beta_{2} - 33 \beta_{5} + 18 \beta_{7} + 3 \beta_{9} ) q^{87} + ( -22 \beta_{1} - 11 \beta_{5} ) q^{88} + ( 494 + 14 \beta_{3} - 28 \beta_{4} + 47 \beta_{6} + 35 \beta_{8} ) q^{89} + ( -142 - 35 \beta_{3} - 40 \beta_{4} - 44 \beta_{6} - 16 \beta_{8} ) q^{91} + ( -48 \beta_{1} - 555 \beta_{2} + 22 \beta_{5} + 13 \beta_{7} - 6 \beta_{9} ) q^{92} + ( -174 \beta_{2} - 33 \beta_{5} + 9 \beta_{7} + 9 \beta_{9} ) q^{93} + ( 80 + 18 \beta_{3} + 132 \beta_{4} + 11 \beta_{6} + 66 \beta_{8} ) q^{94} + ( -6 + 6 \beta_{3} + 120 \beta_{4} - 6 \beta_{6} + 33 \beta_{8} ) q^{96} + ( -112 \beta_{1} - 393 \beta_{2} - 4 \beta_{5} + 90 \beta_{7} + 58 \beta_{9} ) q^{97} + ( 206 \beta_{1} - 94 \beta_{2} - 7 \beta_{5} - 36 \beta_{7} + 11 \beta_{9} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 6q^{4} - 6q^{6} - 90q^{9} + O(q^{10}) \) \( 10q - 6q^{4} - 6q^{6} - 90q^{9} - 110q^{11} - 16q^{14} - 250q^{16} + 114q^{19} - 108q^{21} - 18q^{24} - 250q^{26} + 214q^{29} - 590q^{31} - 68q^{34} + 54q^{36} + 186q^{39} - 256q^{41} + 66q^{44} - 1154q^{46} + 830q^{49} - 228q^{51} + 54q^{54} - 264q^{56} + 2304q^{59} - 688q^{61} + 1090q^{64} + 66q^{66} + 966q^{69} - 1414q^{71} + 2352q^{74} - 3398q^{76} + 4988q^{79} + 810q^{81} + 1944q^{84} - 1598q^{86} + 4870q^{89} - 1608q^{91} + 1004q^{94} + 138q^{96} + 990q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 43 x^{8} + 631 x^{6} + 3625 x^{4} + 7104 x^{2} + 900\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{9} + 37 \nu^{7} + 445 \nu^{5} + 1891 \nu^{3} + 2202 \nu \)\()/720\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 9 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{8} + 31 \nu^{6} + 289 \nu^{4} + 817 \nu^{2} + 150 \)\()/120\)
\(\beta_{5}\)\(=\)\( \nu^{3} + 14 \nu \)
\(\beta_{6}\)\(=\)\((\)\( \nu^{4} + 19 \nu^{2} + 46 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{9} + 49 \nu^{7} + 757 \nu^{5} + 4039 \nu^{3} + 6306 \nu \)\()/240\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{8} - 46 \nu^{6} - 619 \nu^{4} - 2242 \nu^{2} - 600 \)\()/60\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{9} + 55 \nu^{7} + 1093 \nu^{5} + 8893 \nu^{3} + 22398 \nu \)\()/360\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 9\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 14 \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} - 19 \beta_{3} + 125\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 2 \beta_{7} - 21 \beta_{5} + 2 \beta_{2} + 216 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-4 \beta_{8} - 44 \beta_{6} - 8 \beta_{4} + 323 \beta_{3} - 1925\)
\(\nu^{7}\)\(=\)\(-52 \beta_{9} + 72 \beta_{7} + 367 \beta_{5} - 112 \beta_{2} - 3452 \beta_{1}\)
\(\nu^{8}\)\(=\)\(124 \beta_{8} + 786 \beta_{6} + 368 \beta_{4} - 5339 \beta_{3} + 30753\)
\(\nu^{9}\)\(=\)\(1034 \beta_{9} - 1774 \beta_{7} - 6125 \beta_{5} + 3974 \beta_{2} + 55876 \beta_{1}\)

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
4.08549i
3.98707i
2.51748i
1.98454i
0.368634i
0.368634i
1.98454i
2.51748i
3.98707i
4.08549i
4.08549i 3.00000i −8.69126 0 −12.2565 7.95915i 2.82414i −9.00000 0
199.2 3.98707i 3.00000i −7.89672 0 11.9612 12.5627i 0.411800i −9.00000 0
199.3 2.51748i 3.00000i 1.66228 0 −7.55245 18.4627i 24.3246i −9.00000 0
199.4 1.98454i 3.00000i 4.06159 0 5.95363 5.59777i 23.9367i −9.00000 0
199.5 0.368634i 3.00000i 7.86411 0 −1.10590 26.5824i 5.84806i −9.00000 0
199.6 0.368634i 3.00000i 7.86411 0 −1.10590 26.5824i 5.84806i −9.00000 0
199.7 1.98454i 3.00000i 4.06159 0 5.95363 5.59777i 23.9367i −9.00000 0
199.8 2.51748i 3.00000i 1.66228 0 −7.55245 18.4627i 24.3246i −9.00000 0
199.9 3.98707i 3.00000i −7.89672 0 11.9612 12.5627i 0.411800i −9.00000 0
199.10 4.08549i 3.00000i −8.69126 0 −12.2565 7.95915i 2.82414i −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.10
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.s 10
5.b even 2 1 inner 825.4.c.s 10
5.c odd 4 1 825.4.a.x 5
5.c odd 4 1 825.4.a.y yes 5
15.e even 4 1 2475.4.a.bi 5
15.e even 4 1 2475.4.a.bj 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.4.a.x 5 5.c odd 4 1
825.4.a.y yes 5 5.c odd 4 1
825.4.c.s 10 1.a even 1 1 trivial
825.4.c.s 10 5.b even 2 1 inner
2475.4.a.bi 5 15.e even 4 1
2475.4.a.bj 5 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{10} + 43 T_{2}^{8} + 631 T_{2}^{6} + 3625 T_{2}^{4} + 7104 T_{2}^{2} + 900 \)
\( T_{7}^{10} + 1300 T_{7}^{8} + 522294 T_{7}^{6} + 78865816 T_{7}^{4} + 4405598425 T_{7}^{2} + 75458991204 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 900 + 7104 T^{2} + 3625 T^{4} + 631 T^{6} + 43 T^{8} + T^{10} \)
$3$ \( ( 9 + T^{2} )^{5} \)
$5$ \( T^{10} \)
$7$ \( 75458991204 + 4405598425 T^{2} + 78865816 T^{4} + 522294 T^{6} + 1300 T^{8} + T^{10} \)
$11$ \( ( 11 + T )^{10} \)
$13$ \( 25911930697436224 + 69886893907728 T^{2} + 74058213313 T^{4} + 38577043 T^{6} + 9891 T^{8} + T^{10} \)
$17$ \( 8989649377695360000 + 10356565119872400 T^{2} + 3788205649704 T^{4} + 591042217 T^{6} + 40518 T^{8} + T^{10} \)
$19$ \( ( 92396817 + 3849987 T - 252712 T^{2} - 11988 T^{3} - 57 T^{4} + T^{5} )^{2} \)
$23$ \( 13299414298805129616 + 18434318630344248 T^{2} + 7551384503569 T^{4} + 1179799355 T^{6} + 64211 T^{8} + T^{10} \)
$29$ \( ( -16041054816 + 396997440 T + 2770092 T^{2} - 60112 T^{3} - 107 T^{4} + T^{5} )^{2} \)
$31$ \( ( 9419016168 + 261146472 T - 7920319 T^{2} - 33433 T^{3} + 295 T^{4} + T^{5} )^{2} \)
$37$ \( \)\(13\!\cdots\!16\)\( + 4717474279099681968 T^{2} + 508282630702540 T^{4} + 18140901625 T^{6} + 238398 T^{8} + T^{10} \)
$41$ \( ( 144243119304 + 3215304180 T - 188058 T^{2} - 184077 T^{3} + 128 T^{4} + T^{5} )^{2} \)
$43$ \( \)\(11\!\cdots\!04\)\( + \)\(13\!\cdots\!76\)\( T^{2} + 6012855295602169 T^{4} + 91651241931 T^{6} + 545979 T^{8} + T^{10} \)
$47$ \( \)\(98\!\cdots\!00\)\( + \)\(36\!\cdots\!44\)\( T^{2} + 50483210112941520 T^{4} + 319610068801 T^{6} + 928978 T^{8} + T^{10} \)
$53$ \( \)\(42\!\cdots\!24\)\( + \)\(36\!\cdots\!36\)\( T^{2} + 77578583189829376 T^{4} + 486382853984 T^{6} + 1187744 T^{8} + T^{10} \)
$59$ \( ( 849531990720 - 6150914688 T - 35969354 T^{2} + 416773 T^{3} - 1152 T^{4} + T^{5} )^{2} \)
$61$ \( ( 105472451720 - 5498810188 T - 146032310 T^{2} - 621055 T^{3} + 344 T^{4} + T^{5} )^{2} \)
$67$ \( \)\(11\!\cdots\!00\)\( + \)\(38\!\cdots\!56\)\( T^{2} + 335025092514202156 T^{4} + 1168843160425 T^{6} + 1790126 T^{8} + T^{10} \)
$71$ \( ( 13467667404912 + 113335765416 T - 283106455 T^{2} - 621005 T^{3} + 707 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(79\!\cdots\!56\)\( + 36338954132823652608 T^{2} + 205783209482291584 T^{4} + 1660481463904 T^{6} + 2445732 T^{8} + T^{10} \)
$79$ \( ( -57405986771584 - 322674693392 T + 336709760 T^{2} + 1537579 T^{3} - 2494 T^{4} + T^{5} )^{2} \)
$83$ \( \)\(49\!\cdots\!56\)\( + \)\(29\!\cdots\!40\)\( T^{2} + 307503627025814352 T^{4} + 1153179964200 T^{6} + 1803841 T^{8} + T^{10} \)
$89$ \( ( 205516733649744 - 1628635153200 T + 2330847928 T^{2} + 528072 T^{3} - 2435 T^{4} + T^{5} )^{2} \)
$97$ \( \)\(21\!\cdots\!25\)\( + \)\(46\!\cdots\!25\)\( T^{2} + 16220519649170023306 T^{4} + 16900938077482 T^{6} + 6962517 T^{8} + T^{10} \)
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