Properties

Label 825.4.c.p
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: \(x^{8} + 54 x^{6} + 913 x^{4} + 5292 x^{2} + 8464\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -7 + \beta_{3} - \beta_{5} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 9 \beta_{2} - \beta_{4} + \beta_{6} ) q^{7} + ( -5 \beta_{1} + 11 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{8} -9 q^{9} +O(q^{10})\) \( q + ( \beta_{1} - \beta_{2} ) q^{2} + 3 \beta_{2} q^{3} + ( -7 + \beta_{3} - \beta_{5} ) q^{4} + ( 3 - 3 \beta_{3} ) q^{6} + ( -2 \beta_{1} - 9 \beta_{2} - \beta_{4} + \beta_{6} ) q^{7} + ( -5 \beta_{1} + 11 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{8} -9 q^{9} -11 q^{11} + ( 3 \beta_{1} - 18 \beta_{2} - 3 \beta_{6} ) q^{12} + ( 8 \beta_{1} + 3 \beta_{2} + 5 \beta_{4} - 5 \beta_{6} ) q^{13} + ( 16 + 4 \beta_{3} + 6 \beta_{5} + 4 \beta_{7} ) q^{14} + ( 19 - 13 \beta_{3} + 5 \beta_{5} + 8 \beta_{7} ) q^{16} + ( 6 \beta_{1} - 19 \beta_{2} + 7 \beta_{4} + \beta_{6} ) q^{17} + ( -9 \beta_{1} + 9 \beta_{2} ) q^{18} + ( -36 - 10 \beta_{3} - 4 \beta_{5} ) q^{19} + ( 24 + 6 \beta_{3} - 3 \beta_{5} - 3 \beta_{7} ) q^{21} + ( -11 \beta_{1} + 11 \beta_{2} ) q^{22} + ( -8 \beta_{1} - 8 \beta_{2} - 16 \beta_{6} ) q^{23} + ( -39 + 15 \beta_{3} - 6 \beta_{5} - 6 \beta_{7} ) q^{24} + ( -94 + 22 \beta_{3} - 28 \beta_{5} - 20 \beta_{7} ) q^{26} -27 \beta_{2} q^{27} + ( 28 \beta_{1} - 2 \beta_{2} + 12 \beta_{4} - 6 \beta_{6} ) q^{28} + ( -22 - 10 \beta_{3} - 18 \beta_{5} + 2 \beta_{7} ) q^{29} + ( 128 + 28 \beta_{3} + 10 \beta_{5} + 10 \beta_{7} ) q^{31} + ( -7 \beta_{1} - 39 \beta_{2} + 10 \beta_{4} - 26 \beta_{6} ) q^{32} -33 \beta_{2} q^{33} + ( -74 - 2 \beta_{3} - 26 \beta_{5} - 12 \beta_{7} ) q^{34} + ( 63 - 9 \beta_{3} + 9 \beta_{5} ) q^{36} + ( -52 \beta_{1} - 8 \beta_{2} + 2 \beta_{4} + 18 \beta_{6} ) q^{37} + ( -60 \beta_{1} - 98 \beta_{2} - 8 \beta_{4} - 6 \beta_{6} ) q^{38} + ( 6 - 24 \beta_{3} + 15 \beta_{5} + 15 \beta_{7} ) q^{39} + ( 82 + 30 \beta_{3} + 30 \beta_{5} + 26 \beta_{7} ) q^{41} + ( 12 \beta_{1} + 30 \beta_{2} - 12 \beta_{4} + 18 \beta_{6} ) q^{42} + ( -2 \beta_{1} + 133 \beta_{2} - 19 \beta_{4} + 7 \beta_{6} ) q^{43} + ( 77 - 11 \beta_{3} + 11 \beta_{5} ) q^{44} + ( 88 + 120 \beta_{3} - 8 \beta_{5} - 32 \beta_{7} ) q^{46} + ( -32 \beta_{1} + 74 \beta_{2} - 26 \beta_{4} - 10 \beta_{6} ) q^{47} + ( -39 \beta_{1} + 42 \beta_{2} - 24 \beta_{4} + 15 \beta_{6} ) q^{48} + ( 127 + 6 \beta_{5} + 14 \beta_{7} ) q^{49} + ( 54 - 18 \beta_{3} - 3 \beta_{5} + 21 \beta_{7} ) q^{51} + ( -158 \beta_{1} + 236 \beta_{2} - 56 \beta_{4} + 70 \beta_{6} ) q^{52} + ( -68 \beta_{1} - 50 \beta_{2} + 24 \beta_{4} + 8 \beta_{6} ) q^{53} + ( -27 + 27 \beta_{3} ) q^{54} + ( -224 + 52 \beta_{3} - 22 \beta_{5} - 4 \beta_{7} ) q^{56} + ( -30 \beta_{1} - 96 \beta_{2} - 12 \beta_{6} ) q^{57} + ( -134 \beta_{1} - 112 \beta_{2} - 32 \beta_{4} + 2 \beta_{6} ) q^{58} + ( 244 + 84 \beta_{3} - 28 \beta_{5} - 48 \beta_{7} ) q^{59} + ( 6 + 52 \beta_{3} - 46 \beta_{5} + 14 \beta_{7} ) q^{61} + ( 168 \beta_{1} + 316 \beta_{2} + 40 \beta_{4} - 12 \beta_{6} ) q^{62} + ( 18 \beta_{1} + 81 \beta_{2} + 9 \beta_{4} - 9 \beta_{6} ) q^{63} + ( 225 + 97 \beta_{3} - 9 \beta_{5} - 8 \beta_{7} ) q^{64} + ( -33 + 33 \beta_{3} ) q^{66} + ( -32 \beta_{1} + 136 \beta_{2} + 12 \beta_{4} + 4 \beta_{6} ) q^{67} + ( -158 \beta_{1} - 214 \beta_{2} - 20 \beta_{4} + 68 \beta_{6} ) q^{68} + ( 72 + 24 \beta_{3} + 48 \beta_{5} ) q^{69} + ( -240 + 116 \beta_{3} - 20 \beta_{5} - 72 \beta_{7} ) q^{71} + ( 45 \beta_{1} - 99 \beta_{2} + 18 \beta_{4} - 18 \beta_{6} ) q^{72} + ( -124 \beta_{1} + 213 \beta_{2} + 47 \beta_{4} + 37 \beta_{6} ) q^{73} + ( 746 - 122 \beta_{3} + 64 \beta_{5} + 32 \beta_{7} ) q^{74} + ( 416 + 76 \beta_{3} + 46 \beta_{5} + 4 \beta_{7} ) q^{76} + ( 22 \beta_{1} + 99 \beta_{2} + 11 \beta_{4} - 11 \beta_{6} ) q^{77} + ( 66 \beta_{1} - 198 \beta_{2} + 60 \beta_{4} - 84 \beta_{6} ) q^{78} + ( -328 + 70 \beta_{3} - 80 \beta_{5} - 24 \beta_{7} ) q^{79} + 81 q^{81} + ( 210 \beta_{1} + 520 \beta_{2} + 112 \beta_{4} - 78 \beta_{6} ) q^{82} + ( -180 \beta_{1} - 47 \beta_{2} - 35 \beta_{4} + 27 \beta_{6} ) q^{83} + ( 24 - 84 \beta_{3} + 18 \beta_{5} + 36 \beta_{7} ) q^{84} + ( 92 - 144 \beta_{3} + 66 \beta_{5} + 52 \beta_{7} ) q^{86} + ( -30 \beta_{1} - 12 \beta_{2} - 6 \beta_{4} - 54 \beta_{6} ) q^{87} + ( 55 \beta_{1} - 121 \beta_{2} + 22 \beta_{4} - 22 \beta_{6} ) q^{88} + ( 254 - 332 \beta_{3} - 24 \beta_{5} - 12 \beta_{7} ) q^{89} + ( 696 - 96 \beta_{3} + 2 \beta_{5} - 34 \beta_{7} ) q^{91} + ( 40 \beta_{1} + 1176 \beta_{2} - 80 \beta_{4} + 96 \beta_{6} ) q^{92} + ( 84 \beta_{1} + 354 \beta_{2} - 30 \beta_{4} + 30 \beta_{6} ) q^{93} + ( 408 + 48 \beta_{3} + 100 \beta_{5} + 32 \beta_{7} ) q^{94} + ( 195 + 21 \beta_{3} + 78 \beta_{5} + 30 \beta_{7} ) q^{96} + ( 84 \beta_{1} + 426 \beta_{2} - 24 \beta_{4} - 28 \beta_{6} ) q^{97} + ( 135 \beta_{1} - 23 \beta_{2} + 40 \beta_{4} - 48 \beta_{6} ) q^{98} + 99 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 52q^{4} + 24q^{6} - 72q^{9} + O(q^{10}) \) \( 8q - 52q^{4} + 24q^{6} - 72q^{9} - 88q^{11} + 104q^{14} + 132q^{16} - 272q^{19} + 204q^{21} - 288q^{24} - 640q^{26} - 104q^{29} + 984q^{31} - 488q^{34} + 468q^{36} - 12q^{39} + 536q^{41} + 572q^{44} + 736q^{46} + 992q^{49} + 444q^{51} - 216q^{54} - 1704q^{56} + 2064q^{59} + 232q^{61} + 1836q^{64} - 264q^{66} + 384q^{69} - 1840q^{71} + 5712q^{74} + 3144q^{76} - 2304q^{79} + 648q^{81} + 120q^{84} + 472q^{86} + 2128q^{89} + 5560q^{91} + 2864q^{94} + 1248q^{96} + 792q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 54 x^{6} + 913 x^{4} + 5292 x^{2} + 8464\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 54 \nu^{5} + 821 \nu^{3} + 2808 \nu \)\()/1656\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 27 \nu^{2} + 92 \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} - 45 \nu^{3} - 452 \nu \)\()/36\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{4} - 45 \nu^{2} - 344 \)\()/18\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} - 610 \nu^{5} - 8189 \nu^{3} - 29696 \nu \)\()/1656\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{6} + 47 \nu^{4} + 614 \nu^{2} + 1840 \)\()/36\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} - \beta_{3} - 14\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - 2 \beta_{4} - 13 \beta_{2} - 21 \beta_{1}\)
\(\nu^{4}\)\(=\)\(27 \beta_{5} + 45 \beta_{3} + 286\)
\(\nu^{5}\)\(=\)\(45 \beta_{6} + 54 \beta_{4} + 585 \beta_{2} + 493 \beta_{1}\)
\(\nu^{6}\)\(=\)\(36 \beta_{7} - 655 \beta_{5} - 1501 \beta_{3} - 6686\)
\(\nu^{7}\)\(=\)\(-1609 \beta_{6} - 1274 \beta_{4} - 19261 \beta_{2} - 12189 \beta_{1}\)

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
4.17080i
5.20196i
2.63835i
1.60719i
1.60719i
2.63835i
5.20196i
4.17080i
5.17080i 3.00000i −18.7372 0 15.5124 11.1745i 55.5199i −9.00000 0
199.2 4.20196i 3.00000i −9.65650 0 −12.6059 15.3793i 6.96057i −9.00000 0
199.3 3.63835i 3.00000i −5.23763 0 10.9151 20.8444i 10.0505i −9.00000 0
199.4 0.607192i 3.00000i 7.63132 0 −1.82158 8.95080i 9.49121i −9.00000 0
199.5 0.607192i 3.00000i 7.63132 0 −1.82158 8.95080i 9.49121i −9.00000 0
199.6 3.63835i 3.00000i −5.23763 0 10.9151 20.8444i 10.0505i −9.00000 0
199.7 4.20196i 3.00000i −9.65650 0 −12.6059 15.3793i 6.96057i −9.00000 0
199.8 5.17080i 3.00000i −18.7372 0 15.5124 11.1745i 55.5199i −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.p 8
5.b even 2 1 inner 825.4.c.p 8
5.c odd 4 1 165.4.a.h 4
5.c odd 4 1 825.4.a.t 4
15.e even 4 1 495.4.a.m 4
15.e even 4 1 2475.4.a.be 4
55.e even 4 1 1815.4.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.h 4 5.c odd 4 1
495.4.a.m 4 15.e even 4 1
825.4.a.t 4 5.c odd 4 1
825.4.c.p 8 1.a even 1 1 trivial
825.4.c.p 8 5.b even 2 1 inner
1815.4.a.t 4 55.e even 4 1
2475.4.a.be 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{8} + 58 T_{2}^{6} + 1081 T_{2}^{4} + 6640 T_{2}^{2} + 2304 \)
\( T_{7}^{8} + 876 T_{7}^{6} + 250320 T_{7}^{4} + 27778816 T_{7}^{2} + 1028100096 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 2304 + 6640 T^{2} + 1081 T^{4} + 58 T^{6} + T^{8} \)
$3$ \( ( 9 + T^{2} )^{4} \)
$5$ \( T^{8} \)
$7$ \( 1028100096 + 27778816 T^{2} + 250320 T^{4} + 876 T^{6} + T^{8} \)
$11$ \( ( 11 + T )^{8} \)
$13$ \( 4643604736 + 21917680448 T^{2} + 44065280 T^{4} + 13172 T^{6} + T^{8} \)
$17$ \( 565050932640000 + 556873934400 T^{2} + 192576256 T^{4} + 26244 T^{6} + T^{8} \)
$19$ \( ( 576 - 23472 T + 1780 T^{2} + 136 T^{3} + T^{4} )^{2} \)
$23$ \( 61270839508598784 + 22107249442816 T^{2} + 2259095552 T^{4} + 83072 T^{6} + T^{8} \)
$29$ \( ( 315474624 - 2737584 T - 57364 T^{2} + 52 T^{3} + T^{4} )^{2} \)
$31$ \( ( -903269376 + 8037504 T + 40496 T^{2} - 492 T^{3} + T^{4} )^{2} \)
$37$ \( 1587623022803466496 + 333393894033152 T^{2} + 19776098528 T^{4} + 262832 T^{6} + T^{8} \)
$41$ \( ( -6228069696 + 73712016 T - 188244 T^{2} - 268 T^{3} + T^{4} )^{2} \)
$43$ \( 1441324182680211456 + 268458404647680 T^{2} + 13485332176 T^{4} + 212300 T^{6} + T^{8} \)
$47$ \( 31683297217807908864 + 2360389329469440 T^{2} + 55533542656 T^{4} + 455728 T^{6} + T^{8} \)
$53$ \( \)\(16\!\cdots\!96\)\( + 8352515137212160 T^{2} + 130161534944 T^{4} + 670640 T^{6} + T^{8} \)
$59$ \( ( -2612441088 + 266215936 T - 143792 T^{2} - 1032 T^{3} + T^{4} )^{2} \)
$61$ \( ( -8546777488 - 133895280 T - 551360 T^{2} - 116 T^{3} + T^{4} )^{2} \)
$67$ \( 827336766023335936 + 264716973654016 T^{2} + 15059351040 T^{4} + 233536 T^{6} + T^{8} \)
$71$ \( ( 291456592896 - 487321600 T - 886640 T^{2} + 920 T^{3} + T^{4} )^{2} \)
$73$ \( \)\(17\!\cdots\!84\)\( + 1236054089219172416 T^{2} + 3483295672832 T^{4} + 3257348 T^{6} + T^{8} \)
$79$ \( ( 48148922944 - 252278064 T - 419564 T^{2} + 1152 T^{3} + T^{4} )^{2} \)
$83$ \( \)\(30\!\cdots\!04\)\( + 34275389012757504 T^{2} + 989822539152 T^{4} + 1972012 T^{6} + T^{8} \)
$89$ \( ( 479041129296 + 2802564256 T - 2716632 T^{2} - 1064 T^{3} + T^{4} )^{2} \)
$97$ \( 2583874927180960000 + 3293920387308800 T^{2} + 413304907616 T^{4} + 1856336 T^{6} + T^{8} \)
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