Properties

Label 735.2.b.d
Level 735
Weight 2
Character orbit 735.b
Analytic conductor 5.869
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 735.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.856615824.2
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} - q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( -\beta_{3} - \beta_{5} ) q^{8} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -\beta_{3} q^{3} + ( -1 + \beta_{2} ) q^{4} - q^{5} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} ) q^{6} + ( -\beta_{3} - \beta_{5} ) q^{8} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + \beta_{1} q^{10} + ( -2 \beta_{1} + \beta_{7} ) q^{11} + ( -1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{12} + ( -\beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{13} + \beta_{3} q^{15} + ( \beta_{3} - \beta_{5} ) q^{16} + ( 3 + \beta_{6} ) q^{17} + ( -1 - \beta_{1} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{18} + ( \beta_{1} + \beta_{4} - \beta_{7} ) q^{19} + ( 1 - \beta_{2} ) q^{20} + ( -5 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{22} + ( -\beta_{1} - 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{23} + ( -3 - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{24} + q^{25} + ( 2 - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} ) q^{26} + ( 1 - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{27} + ( -3 \beta_{3} - 3 \beta_{5} - \beta_{7} ) q^{29} + ( -1 + \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} ) q^{30} + ( -\beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{31} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{32} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{33} + ( -3 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{34} + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{36} + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} ) q^{37} + ( 3 - \beta_{6} ) q^{38} + ( -2 + \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{39} + ( \beta_{3} + \beta_{5} ) q^{40} + ( -4 + \beta_{2} + \beta_{6} ) q^{41} + ( 2 + \beta_{6} ) q^{43} + ( 4 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} ) q^{44} + ( \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{45} + ( 3 - \beta_{2} + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{46} + ( 2 - 2 \beta_{2} + 2 \beta_{6} ) q^{47} + ( -3 + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{48} -\beta_{1} q^{50} + ( -1 - \beta_{2} - 3 \beta_{3} + \beta_{5} - \beta_{7} ) q^{51} + ( -\beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{52} -4 \beta_{7} q^{53} + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{54} + ( 2 \beta_{1} - \beta_{7} ) q^{55} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{5} - \beta_{6} ) q^{57} + ( 5 - 5 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{58} + ( 2 + 3 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{59} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{60} + ( -\beta_{1} - 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} - 3 \beta_{7} ) q^{61} + ( 5 - 4 \beta_{2} + 4 \beta_{3} - 4 \beta_{5} - \beta_{6} ) q^{62} + ( 6 - 4 \beta_{2} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{64} + ( \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{65} + ( -4 - \beta_{1} + 5 \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{66} + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{67} + ( 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{68} + ( -3 - 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{4} + \beta_{7} ) q^{71} + ( 1 - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} ) q^{72} + ( 5 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{73} + ( 6 \beta_{1} + 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} + 3 \beta_{7} ) q^{74} -\beta_{3} q^{75} + ( -\beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{76} + ( -1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{78} + ( 6 + 3 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{79} + ( -\beta_{3} + \beta_{5} ) q^{80} + ( 2 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{81} + ( 5 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{82} + ( 3 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{83} + ( -3 - \beta_{6} ) q^{85} + ( -2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{86} + ( -8 + \beta_{1} + \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{87} + ( 2 - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{88} + ( -\beta_{2} - 4 \beta_{3} + 4 \beta_{5} - \beta_{6} ) q^{89} + ( 1 + \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{90} + ( 2 \beta_{1} + 5 \beta_{3} - 8 \beta_{4} + 5 \beta_{5} ) q^{92} + ( -5 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -4 \beta_{1} + 4 \beta_{4} - 2 \beta_{7} ) q^{94} + ( -\beta_{1} - \beta_{4} + \beta_{7} ) q^{95} + ( -5 + 4 \beta_{1} + 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{96} + ( 5 \beta_{1} + 5 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} + 3 \beta_{7} ) q^{97} + ( -4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{3} - 6q^{4} - 8q^{5} + 5q^{6} + q^{9} + O(q^{10}) \) \( 8q + q^{3} - 6q^{4} - 8q^{5} + 5q^{6} + q^{9} - 9q^{12} - q^{15} - 2q^{16} + 24q^{17} - 7q^{18} + 6q^{20} - 40q^{22} - 23q^{24} + 8q^{25} + 12q^{26} + 4q^{27} - 5q^{30} + 2q^{33} + 9q^{36} - 14q^{37} + 24q^{38} - 12q^{39} - 30q^{41} + 16q^{43} - q^{45} + 14q^{46} + 12q^{47} - 25q^{48} - 6q^{51} - 10q^{54} + 6q^{57} + 26q^{58} + 24q^{59} + 9q^{60} + 24q^{62} + 38q^{64} - 38q^{66} - 8q^{67} - 13q^{69} + q^{72} + q^{75} - 6q^{78} + 58q^{79} + 2q^{80} + 13q^{81} + 30q^{83} - 24q^{85} - 61q^{87} + 4q^{88} + 6q^{89} + 7q^{90} - 36q^{93} - 39q^{96} - 34q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 11 x^{6} + 36 x^{4} + 32 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + \nu^{3} + 6 \nu^{2} + 4 \nu + 4 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} + 10 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} - 6 \nu^{2} + 4 \nu - 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} + 9 \nu^{4} + 22 \nu^{2} + 10 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} + 11 \nu^{5} + 34 \nu^{3} + 22 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{3} - 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{5} + \beta_{3} - 6 \beta_{2} + 14\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 2 \beta_{4} - 7 \beta_{3} + 18 \beta_{1}\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} + 9 \beta_{5} - 9 \beta_{3} + 32 \beta_{2} - 70\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} + 43 \beta_{5} - 22 \beta_{4} + 43 \beta_{3} - 84 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
146.1
2.33086i
2.06288i
1.07834i
0.385731i
0.385731i
1.07834i
2.06288i
2.33086i
2.33086i −0.459555 + 1.66997i −3.43292 −1.00000 3.89248 + 1.07116i 0 3.33995i −2.57762 1.53489i 2.33086i
146.2 2.06288i 1.71189 + 0.263509i −2.25548 −1.00000 0.543588 3.53142i 0 0.527019i 2.86113 + 0.902197i 2.06288i
146.3 1.07834i 0.812371 1.52972i 0.837188 −1.00000 −1.64956 0.876010i 0 3.05945i −1.68011 2.48541i 1.07834i
146.4 0.385731i −1.56470 0.742765i 1.85121 −1.00000 −0.286507 + 0.603555i 0 1.48553i 1.89660 + 2.32442i 0.385731i
146.5 0.385731i −1.56470 + 0.742765i 1.85121 −1.00000 −0.286507 0.603555i 0 1.48553i 1.89660 2.32442i 0.385731i
146.6 1.07834i 0.812371 + 1.52972i 0.837188 −1.00000 −1.64956 + 0.876010i 0 3.05945i −1.68011 + 2.48541i 1.07834i
146.7 2.06288i 1.71189 0.263509i −2.25548 −1.00000 0.543588 + 3.53142i 0 0.527019i 2.86113 0.902197i 2.06288i
146.8 2.33086i −0.459555 1.66997i −3.43292 −1.00000 3.89248 1.07116i 0 3.33995i −2.57762 + 1.53489i 2.33086i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 146.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.b.d 8
3.b odd 2 1 735.2.b.c 8
7.b odd 2 1 735.2.b.c 8
7.c even 3 1 105.2.s.c 8
7.c even 3 1 735.2.s.l 8
7.d odd 6 1 105.2.s.d yes 8
7.d odd 6 1 735.2.s.k 8
21.c even 2 1 inner 735.2.b.d 8
21.g even 6 1 105.2.s.c 8
21.g even 6 1 735.2.s.l 8
21.h odd 6 1 105.2.s.d yes 8
21.h odd 6 1 735.2.s.k 8
35.i odd 6 1 525.2.t.f 8
35.j even 6 1 525.2.t.g 8
35.k even 12 2 525.2.q.e 16
35.l odd 12 2 525.2.q.f 16
105.o odd 6 1 525.2.t.f 8
105.p even 6 1 525.2.t.g 8
105.w odd 12 2 525.2.q.f 16
105.x even 12 2 525.2.q.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.s.c 8 7.c even 3 1
105.2.s.c 8 21.g even 6 1
105.2.s.d yes 8 7.d odd 6 1
105.2.s.d yes 8 21.h odd 6 1
525.2.q.e 16 35.k even 12 2
525.2.q.e 16 105.x even 12 2
525.2.q.f 16 35.l odd 12 2
525.2.q.f 16 105.w odd 12 2
525.2.t.f 8 35.i odd 6 1
525.2.t.f 8 105.o odd 6 1
525.2.t.g 8 35.j even 6 1
525.2.t.g 8 105.p even 6 1
735.2.b.c 8 3.b odd 2 1
735.2.b.c 8 7.b odd 2 1
735.2.b.d 8 1.a even 1 1 trivial
735.2.b.d 8 21.c even 2 1 inner
735.2.s.k 8 7.d odd 6 1
735.2.s.k 8 21.h odd 6 1
735.2.s.l 8 7.c even 3 1
735.2.s.l 8 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{8} + 11 T_{2}^{6} + 36 T_{2}^{4} + 32 T_{2}^{2} + 4 \)
\( T_{17}^{4} - 12 T_{17}^{3} + 42 T_{17}^{2} - 42 T_{17} + 12 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T^{2} + 16 T^{4} - 44 T^{6} + 100 T^{8} - 176 T^{10} + 256 T^{12} - 320 T^{14} + 256 T^{16} \)
$3$ \( 1 - T - T^{3} - 2 T^{4} - 3 T^{5} - 27 T^{7} + 81 T^{8} \)
$5$ \( ( 1 + T )^{8} \)
$7$ 1
$11$ \( 1 - 32 T^{2} + 772 T^{4} - 12116 T^{6} + 157534 T^{8} - 1466036 T^{10} + 11302852 T^{12} - 56689952 T^{14} + 214358881 T^{16} \)
$13$ \( 1 - 83 T^{2} + 3217 T^{4} - 76058 T^{6} + 1197778 T^{8} - 12853802 T^{10} + 91880737 T^{12} - 400625147 T^{14} + 815730721 T^{16} \)
$17$ \( ( 1 - 12 T + 110 T^{2} - 654 T^{3} + 3174 T^{4} - 11118 T^{5} + 31790 T^{6} - 58956 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( 1 - 119 T^{2} + 6541 T^{4} - 219914 T^{6} + 5006074 T^{8} - 79388954 T^{10} + 852429661 T^{12} - 5598459839 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 - 53 T^{2} + 2410 T^{4} - 73019 T^{6} + 1993390 T^{8} - 38627051 T^{10} + 674416810 T^{12} - 7845902117 T^{14} + 78310985281 T^{16} \)
$29$ \( 1 - 53 T^{2} + 3250 T^{4} - 128951 T^{6} + 4063174 T^{8} - 108447791 T^{10} + 2298663250 T^{12} - 31525636013 T^{14} + 500246412961 T^{16} \)
$31$ \( 1 - 83 T^{2} + 5089 T^{4} - 218834 T^{6} + 7580470 T^{8} - 210299474 T^{10} + 4699798369 T^{12} - 73662805523 T^{14} + 852891037441 T^{16} \)
$37$ \( ( 1 + 7 T + 73 T^{2} + 502 T^{3} + 2788 T^{4} + 18574 T^{5} + 99937 T^{6} + 354571 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( ( 1 + 15 T + 218 T^{2} + 1791 T^{3} + 14136 T^{4} + 73431 T^{5} + 366458 T^{6} + 1033815 T^{7} + 2825761 T^{8} )^{2} \)
$43$ \( ( 1 - 8 T + 184 T^{2} - 1022 T^{3} + 12127 T^{4} - 43946 T^{5} + 340216 T^{6} - 636056 T^{7} + 3418801 T^{8} )^{2} \)
$47$ \( ( 1 - 6 T + 152 T^{2} - 582 T^{3} + 9486 T^{4} - 27354 T^{5} + 335768 T^{6} - 622938 T^{7} + 4879681 T^{8} )^{2} \)
$53$ \( 1 - 104 T^{2} + 7612 T^{4} - 399896 T^{6} + 19004518 T^{8} - 1123307864 T^{10} + 60062341372 T^{12} - 2305093557416 T^{14} + 62259690411361 T^{16} \)
$59$ \( ( 1 - 12 T + 224 T^{2} - 1926 T^{3} + 19374 T^{4} - 113634 T^{5} + 779744 T^{6} - 2464548 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( 1 - 221 T^{2} + 26290 T^{4} - 2283227 T^{6} + 155613610 T^{8} - 8495887667 T^{10} + 364007159890 T^{12} - 11386002733781 T^{14} + 191707312997281 T^{16} \)
$67$ \( ( 1 + 4 T + 250 T^{2} + 706 T^{3} + 24421 T^{4} + 47302 T^{5} + 1122250 T^{6} + 1203052 T^{7} + 20151121 T^{8} )^{2} \)
$71$ \( 1 - 464 T^{2} + 99532 T^{4} - 12936548 T^{6} + 1114829374 T^{8} - 65213138468 T^{10} + 2529275433292 T^{12} - 59438531739344 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 335 T^{2} + 48517 T^{4} - 4323158 T^{6} + 317931826 T^{8} - 23038108982 T^{10} + 1377797458597 T^{12} - 50696965806815 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 - 29 T + 547 T^{2} - 6944 T^{3} + 70456 T^{4} - 548576 T^{5} + 3413827 T^{6} - 14298131 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( ( 1 - 15 T + 380 T^{2} - 3759 T^{3} + 49260 T^{4} - 311997 T^{5} + 2617820 T^{6} - 8576805 T^{7} + 47458321 T^{8} )^{2} \)
$89$ \( ( 1 - 3 T + 62 T^{2} + 1317 T^{3} - 6432 T^{4} + 117213 T^{5} + 491102 T^{6} - 2114907 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 - 368 T^{2} + 81676 T^{4} - 12257504 T^{6} + 1385094598 T^{8} - 115330855136 T^{10} + 7230717554956 T^{12} - 306533697813872 T^{14} + 7837433594376961 T^{16} \)
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