Properties

Label 4140.2.i.b
Level $4140$
Weight $2$
Character orbit 4140.i
Analytic conductor $33.058$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{5} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - \beta_{9} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - \beta_{9} q^{7} - \beta_1 q^{11} - \beta_{5} q^{13} - \beta_{7} q^{17} + (\beta_{13} - \beta_{11}) q^{19} + ( - \beta_{10} - \beta_{4} + 1) q^{23} + q^{25} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_{9}) q^{29} + (\beta_{7} - \beta_{6} + \beta_{3} - 1) q^{31} - \beta_{9} q^{35} + ( - \beta_{13} + \beta_{10} + \beta_{9}) q^{37} + (\beta_{13} - \beta_{11} - \beta_{9} - \beta_{8}) q^{41} + ( - \beta_{14} - \beta_{12} + 2 \beta_{11}) q^{43} + (\beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} + 2 \beta_{11} + \beta_{10}) q^{47} + (\beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{49} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + \beta_1 - 1) q^{53} - \beta_1 q^{55} + (\beta_{15} - \beta_{11} + \beta_{9}) q^{59} + (\beta_{15} + \beta_{12} + \beta_{10} - \beta_{8}) q^{61} - \beta_{5} q^{65} + ( - \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8}) q^{67} + ( - \beta_{15} - \beta_{14} + \beta_{13} - \beta_{11} + \beta_{9} - \beta_{8}) q^{71} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - 2 \beta_{4} + 2) q^{73} + ( - \beta_{15} + \beta_{14} + 2 \beta_{12} - \beta_{11} - \beta_{10} - 2 \beta_{8}) q^{77} + 2 \beta_{12} q^{79} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{2} + 2 \beta_1 - 1) q^{83} - \beta_{7} q^{85} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} - 3) q^{89} + ( - \beta_{15} + 2 \beta_{14} + \beta_{12} - 4 \beta_{11} + \beta_{10} - \beta_{8}) q^{91} + (\beta_{13} - \beta_{11}) q^{95} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{11} + \beta_{10}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{5} + 8 q^{23} + 16 q^{25} - 8 q^{31} - 40 q^{49} - 4 q^{53} + 8 q^{73} - 20 q^{83} - 32 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 62x^{14} + 1303x^{12} + 12842x^{10} + 65359x^{8} + 170834x^{6} + 207293x^{4} + 91366x^{2} + 9604 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 124491880 \nu^{14} + 5531647309 \nu^{12} + 37688713508 \nu^{10} - 618906405792 \nu^{8} - 8576884945810 \nu^{6} + \cdots + 5081910751692 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1099532275 \nu^{14} - 65729686391 \nu^{12} - 1281439225700 \nu^{10} - 10996984617426 \nu^{8} - 43285810004227 \nu^{6} + \cdots - 1200536614236 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1705812243 \nu^{14} - 95894237026 \nu^{12} - 1666130880928 \nu^{10} - 12170363351342 \nu^{8} - 39670880077709 \nu^{6} + \cdots - 14661309337056 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1953768963 \nu^{14} + 112819382586 \nu^{12} + 2068772711892 \nu^{10} + 16453894067170 \nu^{8} + 60303704828537 \nu^{6} + \cdots - 1780511035288 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2166758938 \nu^{14} + 125161237541 \nu^{12} + 2294781870908 \nu^{10} + 18185274766792 \nu^{8} + 65524796371428 \nu^{6} + \cdots - 5607240199924 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2498489531 \nu^{14} + 151124599275 \nu^{12} + 3033785819160 \nu^{10} + 27878689600934 \nu^{8} + 127544892361671 \nu^{6} + \cdots + 48526335472260 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3048577016 \nu^{14} + 180777077505 \nu^{12} + 3498253609372 \nu^{10} + 30510475697496 \nu^{8} + 131178809583478 \nu^{6} + \cdots + 52453166710812 ) / 3306361853624 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14724018943 \nu^{15} - 1093972349025 \nu^{13} - 29349714587204 \nu^{11} - 364715172446186 \nu^{9} + \cdots - 19\!\cdots\!52 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15363299351 \nu^{15} - 937621248815 \nu^{13} - 19163955374068 \nu^{11} - 181475071162522 \nu^{9} + \cdots - 547253380343924 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16590392249 \nu^{15} - 991263848858 \nu^{13} - 19272328078224 \nu^{11} - 163184490956034 \nu^{9} + \cdots + 18\!\cdots\!88 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 101161287 \nu^{15} - 6326074969 \nu^{13} - 134946581276 \nu^{11} - 1356493811342 \nu^{9} - 7051373895279 \nu^{7} + \cdots - 5858744937772 \nu ) / 839439019832 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53520056427 \nu^{15} + 3044902710698 \nu^{13} + 54206097656324 \nu^{11} + 411579426630098 \nu^{9} + \cdots + 55390175420520 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12538314823 \nu^{15} - 765686789820 \nu^{13} - 15657214632828 \nu^{11} - 148299700666470 \nu^{9} + \cdots - 501097207085760 \nu ) / 23144532975368 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 92337525767 \nu^{15} - 5452625870940 \nu^{13} - 104585770514544 \nu^{11} - 897509336354710 \nu^{9} + \cdots - 31\!\cdots\!64 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 147338037348 \nu^{15} + 8962991653737 \nu^{13} + 181689302841464 \nu^{11} + \cdots + 16\!\cdots\!96 \nu ) / 162011730827576 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{13} - \beta_{12} + \beta_{11} + \beta_{8} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{15} - 3\beta_{14} + 15\beta_{13} + 12\beta_{12} - 25\beta_{11} - 5\beta_{10} - 5\beta_{9} - 13\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 31\beta_{7} - 28\beta_{6} + 75\beta_{5} - 56\beta_{4} + 25\beta_{3} + 29\beta_{2} - 25\beta _1 + 176 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 103 \beta_{15} + 110 \beta_{14} - 485 \beta_{13} - 348 \beta_{12} + 889 \beta_{11} + 188 \beta_{10} + 191 \beta_{9} + 377 \beta_{8} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -945\beta_{7} + 842\beta_{6} - 2462\beta_{5} + 1980\beta_{4} - 698\beta_{3} - 868\beta_{2} + 789\beta _1 - 5098 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3263 \beta_{15} - 3637 \beta_{14} + 15605 \beta_{13} + 10734 \beta_{12} - 29123 \beta_{11} - 6111 \beta_{10} - 6431 \beta_{9} - 11633 \beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 29697 \beta_{7} - 26434 \beta_{6} + 79015 \beta_{5} - 64918 \beta_{4} + 21133 \beta_{3} + 26957 \beta_{2} - 25473 \beta _1 + 158532 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 103411 \beta_{15} + 117348 \beta_{14} - 500033 \beta_{13} - 338856 \beta_{12} + 936583 \beta_{11} + 195286 \beta_{10} + 209433 \beta_{9} + 367341 \beta_{8} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 944115 \beta_{7} + 840704 \beta_{6} - 2526354 \beta_{5} + 2089968 \beta_{4} - 662368 \beta_{3} - 851820 \beta_{2} + 818717 \beta _1 - 5024736 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3292133 \beta_{15} - 3760505 \beta_{14} + 15994667 \beta_{13} + 10783900 \beta_{12} - 29982539 \beta_{11} - 6233921 \beta_{10} - 6739253 \beta_{9} - 11691485 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 30122223 \beta_{7} - 26830090 \beta_{6} + 80730559 \beta_{5} - 66939762 \beta_{4} + 21025123 \beta_{3} + 27106515 \beta_{2} - 26223785 \beta _1 + 160149968 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 105047815 \beta_{15} + 120270062 \beta_{14} - 511290731 \beta_{13} - 344125986 \beta_{12} + 958615535 \beta_{11} + 199094092 \beta_{10} + 215896467 \beta_{9} + 373095275 \beta_{8} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 962087017 \beta_{7} + 857039202 \beta_{6} - 2579652506 \beta_{5} + 2140659196 \beta_{4} - 670326578 \beta_{3} - 864880484 \beta_{2} + 838709617 \beta _1 - 5113220882 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 3355026899 \beta_{15} - 3844249297 \beta_{14} + 16340259777 \beta_{13} + 10991421558 \beta_{12} - 30637798611 \beta_{11} - 6360566579 \beta_{10} + \cdots - 11916729213 \beta_{8} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1241.1
3.31268i
1.41255i
2.05179i
2.59852i
2.31631i
5.65286i
0.387906i
0.773378i
0.773378i
0.387906i
5.65286i
2.31631i
2.59852i
2.05179i
1.41255i
3.31268i
0 0 0 1.00000 0 4.85908i 0 0 0
1241.2 0 0 0 1.00000 0 4.35549i 0 0 0
1241.3 0 0 0 1.00000 0 4.17184i 0 0 0
1241.4 0 0 0 1.00000 0 3.38743i 0 0 0
1241.5 0 0 0 1.00000 0 1.73994i 0 0 0
1241.6 0 0 0 1.00000 0 0.826970i 0 0 0
1241.7 0 0 0 1.00000 0 0.807745i 0 0 0
1241.8 0 0 0 1.00000 0 0.420015i 0 0 0
1241.9 0 0 0 1.00000 0 0.420015i 0 0 0
1241.10 0 0 0 1.00000 0 0.807745i 0 0 0
1241.11 0 0 0 1.00000 0 0.826970i 0 0 0
1241.12 0 0 0 1.00000 0 1.73994i 0 0 0
1241.13 0 0 0 1.00000 0 3.38743i 0 0 0
1241.14 0 0 0 1.00000 0 4.17184i 0 0 0
1241.15 0 0 0 1.00000 0 4.35549i 0 0 0
1241.16 0 0 0 1.00000 0 4.85908i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1241.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4140.2.i.b yes 16
3.b odd 2 1 4140.2.i.a 16
23.b odd 2 1 4140.2.i.a 16
69.c even 2 1 inner 4140.2.i.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4140.2.i.a 16 3.b odd 2 1
4140.2.i.a 16 23.b odd 2 1
4140.2.i.b yes 16 1.a even 1 1 trivial
4140.2.i.b yes 16 69.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{8} - 56T_{11}^{6} + 16T_{11}^{5} + 930T_{11}^{4} - 636T_{11}^{3} - 4164T_{11}^{2} + 5024T_{11} - 256 \) acting on \(S_{2}^{\mathrm{new}}(4140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 76 T^{14} + 2207 T^{12} + \cdots + 21316 \) Copy content Toggle raw display
$11$ \( (T^{8} - 56 T^{6} + 16 T^{5} + 930 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 52 T^{6} - 36 T^{5} + 662 T^{4} + \cdots + 208)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 73 T^{6} - 48 T^{5} + 1377 T^{4} + \cdots + 448)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 152 T^{14} + \cdots + 14500864 \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + \cdots + 78310985281 \) Copy content Toggle raw display
$29$ \( T^{16} + 188 T^{14} + \cdots + 11237272036 \) Copy content Toggle raw display
$31$ \( (T^{8} + 4 T^{7} - 113 T^{6} + \cdots - 372356)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 196 T^{14} + \cdots + 13410244 \) Copy content Toggle raw display
$41$ \( T^{16} + 324 T^{14} + \cdots + 14273284 \) Copy content Toggle raw display
$43$ \( T^{16} + 272 T^{14} + \cdots + 14546289664 \) Copy content Toggle raw display
$47$ \( T^{16} + 384 T^{14} + \cdots + 805651456 \) Copy content Toggle raw display
$53$ \( (T^{8} + 2 T^{7} - 185 T^{6} + \cdots - 110404)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + 420 T^{14} + \cdots + 62188996 \) Copy content Toggle raw display
$61$ \( T^{16} + 744 T^{14} + \cdots + 3487272395776 \) Copy content Toggle raw display
$67$ \( T^{16} + 412 T^{14} + \cdots + 201024896164 \) Copy content Toggle raw display
$71$ \( T^{16} + 548 T^{14} + \cdots + 69650599396 \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} - 396 T^{6} + \cdots + 33260432)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + 640 T^{14} + \cdots + 1073741824 \) Copy content Toggle raw display
$83$ \( (T^{8} + 10 T^{7} - 353 T^{6} + \cdots - 104384)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} - 336 T^{6} + \cdots - 541184)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 792 T^{14} + \cdots + 4196794737664 \) Copy content Toggle raw display
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