Properties

Label 3744.2.m.h
Level $3744$
Weight $2$
Character orbit 3744.m
Analytic conductor $29.896$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3744.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(29.8959905168\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 936)
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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{5} + \beta_{5} q^{7} +O(q^{10})\) \( q + \beta_{14} q^{5} + \beta_{5} q^{7} + ( -\beta_{11} + \beta_{14} ) q^{11} + ( -\beta_{6} - \beta_{15} ) q^{13} -\beta_{3} q^{17} + ( -\beta_{6} + \beta_{12} - \beta_{15} ) q^{19} -\beta_{2} q^{23} + \beta_{1} q^{25} + \beta_{7} q^{29} + \beta_{10} q^{31} + ( -2 \beta_{7} + \beta_{9} ) q^{35} + ( -2 \beta_{13} + 2 \beta_{15} ) q^{37} + ( -3 \beta_{4} - 6 \beta_{8} ) q^{41} + ( \beta_{12} + \beta_{13} ) q^{43} + ( -4 \beta_{4} - \beta_{8} ) q^{47} + ( -8 - \beta_{1} ) q^{49} + ( -2 \beta_{7} - \beta_{9} ) q^{53} + ( 5 + 3 \beta_{1} ) q^{55} + ( -3 \beta_{11} - \beta_{14} ) q^{59} -3 \beta_{6} q^{61} + ( \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{8} ) q^{65} + ( 2 \beta_{6} - 2 \beta_{12} + \beta_{13} + \beta_{15} ) q^{67} + ( -4 \beta_{4} + \beta_{8} ) q^{71} + ( -\beta_{5} - \beta_{10} ) q^{73} + ( -\beta_{7} + 3 \beta_{9} ) q^{77} + ( -2 - 4 \beta_{1} ) q^{79} + ( \beta_{11} - 3 \beta_{14} ) q^{83} + ( -3 \beta_{6} + 3 \beta_{12} - \beta_{13} - 2 \beta_{15} ) q^{85} + ( \beta_{4} + 8 \beta_{8} ) q^{89} + ( 4 \beta_{6} - 3 \beta_{12} - \beta_{15} ) q^{91} + ( \beta_{2} - 2 \beta_{3} ) q^{95} + ( \beta_{5} - \beta_{10} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + O(q^{10}) \) \( 16 q - 128 q^{49} + 80 q^{55} - 32 q^{79} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 2 x^{14} - 16 x^{12} - 72 x^{10} + 26 x^{8} + 360 x^{6} + 725 x^{4} + 1000 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 396 \nu^{14} + 1812 \nu^{12} - 7496 \nu^{10} - 25482 \nu^{8} - 34944 \nu^{6} - 37770 \nu^{4} - 5400 \nu^{2} + 1991875 \)\()/881375\)
\(\beta_{2}\)\(=\)\((\)\( -7468 \nu^{14} - 15136 \nu^{12} + 146738 \nu^{10} + 531446 \nu^{8} - 573668 \nu^{6} - 3703230 \nu^{4} - 4157900 \nu^{2} - 4522500 \)\()/881375\)
\(\beta_{3}\)\(=\)\((\)\( 2458 \nu^{14} + 5148 \nu^{12} - 42734 \nu^{10} - 183828 \nu^{8} + 123374 \nu^{6} + 1088952 \nu^{4} + 1198950 \nu^{2} + 1368050 \)\()/176275\)
\(\beta_{4}\)\(=\)\((\)\( -7797 \nu^{15} - 1879 \nu^{13} + 194307 \nu^{11} + 471569 \nu^{9} - 1255952 \nu^{7} - 4205580 \nu^{5} - 3942175 \nu^{3} - 4720750 \nu \)\()/4406875\)
\(\beta_{5}\)\(=\)\((\)\( 10661 \nu^{15} + 14207 \nu^{13} - 225956 \nu^{11} - 888177 \nu^{9} + 521116 \nu^{7} + 5727895 \nu^{5} + 13225300 \nu^{3} + 26244000 \nu \)\()/4406875\)
\(\beta_{6}\)\(=\)\((\)\( 23588 \nu^{14} + 17416 \nu^{12} - 373728 \nu^{10} - 1083526 \nu^{8} + 2038808 \nu^{6} + 4471770 \nu^{4} + 6997400 \nu^{2} + 8274750 \)\()/881375\)
\(\beta_{7}\)\(=\)\((\)\( 28806 \nu^{14} + 29832 \nu^{12} - 448156 \nu^{10} - 1496702 \nu^{8} + 2047066 \nu^{6} + 6175530 \nu^{4} + 12354350 \nu^{2} + 13465750 \)\()/881375\)
\(\beta_{8}\)\(=\)\((\)\( -1496 \nu^{15} - 2572 \nu^{13} + 24401 \nu^{11} + 91992 \nu^{9} - 100886 \nu^{7} - 478015 \nu^{5} - 412275 \nu^{3} - 598500 \nu \)\()/400625\)
\(\beta_{9}\)\(=\)\((\)\( -44196 \nu^{14} - 27412 \nu^{12} + 754046 \nu^{10} + 2232082 \nu^{8} - 3966856 \nu^{6} - 11081230 \nu^{4} - 21249600 \nu^{2} - 22476000 \)\()/881375\)
\(\beta_{10}\)\(=\)\((\)\( 8289 \nu^{15} + 23943 \nu^{13} - 103294 \nu^{11} - 572573 \nu^{9} - 243116 \nu^{7} + 1825705 \nu^{5} + 5160650 \nu^{3} + 11379000 \nu \)\()/881375\)
\(\beta_{11}\)\(=\)\((\)\( -68641 \nu^{15} - 53217 \nu^{13} + 1141536 \nu^{11} + 3533287 \nu^{9} - 5705496 \nu^{7} - 17001745 \nu^{5} - 30732300 \nu^{3} - 30934000 \nu \)\()/4406875\)
\(\beta_{12}\)\(=\)\((\)\(-78923 \nu^{15} - 26760 \nu^{14} - 34611 \nu^{13} + 11580 \nu^{12} + 1339113 \nu^{11} + 580360 \nu^{10} + 3553821 \nu^{9} + 1252870 \nu^{8} - 8128668 \nu^{7} - 4374960 \nu^{6} - 15701920 \nu^{5} - 8822500 \nu^{4} - 26400825 \nu^{3} - 12863000 \nu^{2} - 32586750 \nu - 13844375\)\()/4406875\)
\(\beta_{13}\)\(=\)\((\)\(78923 \nu^{15} - 144700 \nu^{14} + 34611 \nu^{13} - 75500 \nu^{12} - 1339113 \nu^{11} + 2449000 \nu^{10} - 3553821 \nu^{9} + 6670500 \nu^{8} + 8128668 \nu^{7} - 14569000 \nu^{6} + 15701920 \nu^{5} - 31181350 \nu^{4} + 26400825 \nu^{3} - 47850000 \nu^{2} + 32586750 \nu - 55218125\)\()/4406875\)
\(\beta_{14}\)\(=\)\((\)\( 108813 \nu^{15} + 85331 \nu^{13} - 1829673 \nu^{11} - 5646091 \nu^{9} + 9386828 \nu^{7} + 27713410 \nu^{5} + 49911275 \nu^{3} + 50144500 \nu \)\()/4406875\)
\(\beta_{15}\)\(=\)\((\)\(128244 \nu^{15} - 144700 \nu^{14} + 47208 \nu^{13} - 75500 \nu^{12} - 2180589 \nu^{11} + 2449000 \nu^{10} - 5658588 \nu^{9} + 6670500 \nu^{8} + 13459254 \nu^{7} - 14569000 \nu^{6} + 25923185 \nu^{5} - 31181350 \nu^{4} + 43314975 \nu^{3} - 47850000 \nu^{2} + 53041500 \nu - 55218125\)\()/4406875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{13} + \beta_{11} - \beta_{8} + \beta_{5} + \beta_{4}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - \beta_{1} - 1\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(10 \beta_{14} - \beta_{13} + \beta_{12} + 12 \beta_{11} + \beta_{10} + 8 \beta_{8} - \beta_{6} - 3 \beta_{5} - 2 \beta_{4}\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(3 \beta_{13} + 3 \beta_{12} - 2 \beta_{9} - 4 \beta_{7} + 5 \beta_{6} - 2 \beta_{2} - 8 \beta_{1} + 18\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(14 \beta_{15} - 19 \beta_{13} + 5 \beta_{12} - 6 \beta_{8} - 5 \beta_{6}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-11 \beta_{13} - 11 \beta_{12} + 11 \beta_{9} + 17 \beta_{7} - 15 \beta_{6} - 5 \beta_{3} - 6 \beta_{2} - 26 \beta_{1} + 56\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(10 \beta_{15} + 110 \beta_{14} - 9 \beta_{13} - \beta_{12} + 182 \beta_{11} - 9 \beta_{10} - 38 \beta_{8} + \beta_{6} + 17 \beta_{5} + 72 \beta_{4}\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(35 \beta_{13} + 35 \beta_{12} - 20 \beta_{9} - 32 \beta_{7} + 53 \beta_{6} - 32 \beta_{3} - 52 \beta_{2} - 18 \beta_{1} + 52\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(274 \beta_{15} + 114 \beta_{14} - 361 \beta_{13} + 87 \beta_{12} + 190 \beta_{11} + 87 \beta_{10} + 190 \beta_{8} - 87 \beta_{6} - 187 \beta_{5} - 304 \beta_{4}\)\()/8\)
\(\nu^{10}\)\(=\)\((\)\(21 \beta_{9} + 36 \beta_{7} - 275 \beta_{1} + 607\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(768 \beta_{15} + 682 \beta_{14} - 1007 \beta_{13} + 239 \beta_{12} + 1080 \beta_{11} - 239 \beta_{10} - 1080 \beta_{8} - 239 \beta_{6} + 529 \beta_{5} + 1762 \beta_{4}\)\()/8\)
\(\nu^{12}\)\(=\)\((\)\(-110 \beta_{13} - 110 \beta_{12} + 290 \beta_{9} + 462 \beta_{7} - 177 \beta_{6} - 462 \beta_{3} - 752 \beta_{2} - 67 \beta_{1} + 153\)\()/4\)
\(\nu^{13}\)\(=\)\((\)\(790 \beta_{15} + 4630 \beta_{14} - 1037 \beta_{13} + 247 \beta_{12} + 7514 \beta_{11} + 1037 \beta_{10} + 1746 \beta_{8} - 247 \beta_{6} - 2321 \beta_{5} - 2884 \beta_{4}\)\()/8\)
\(\nu^{14}\)\(=\)\((\)\(2071 \beta_{13} + 2071 \beta_{12} - 1039 \beta_{9} - 1678 \beta_{7} + 3360 \beta_{6} - 400 \beta_{3} - 639 \beta_{2} - 5431 \beta_{1} + 12151\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(4399 \beta_{15} - 5754 \beta_{13} + 1355 \beta_{12} - 1521 \beta_{8} - 1355 \beta_{6} + 2475 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times\).

\(n\) \(703\) \(2017\) \(2081\) \(2341\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1585.1
0.752864 0.902863i
0.0783900 1.17295i
0.752864 + 0.902863i
0.0783900 + 1.17295i
0.556839 1.81878i
−1.90184 + 0.0324487i
0.556839 + 1.81878i
−1.90184 0.0324487i
1.90184 + 0.0324487i
−0.556839 1.81878i
1.90184 0.0324487i
−0.556839 + 1.81878i
−0.0783900 1.17295i
−0.752864 0.902863i
−0.0783900 + 1.17295i
−0.752864 + 0.902863i
0 0 0 −2.68999 0 4.15163i 0 0 0
1585.2 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.3 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.4 0 0 0 −2.68999 0 4.15163i 0 0 0
1585.5 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.6 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.7 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.8 0 0 0 −1.66251 0 3.57266i 0 0 0
1585.9 0 0 0 1.66251 0 3.57266i 0 0 0
1585.10 0 0 0 1.66251 0 3.57266i 0 0 0
1585.11 0 0 0 1.66251 0 3.57266i 0 0 0
1585.12 0 0 0 1.66251 0 3.57266i 0 0 0
1585.13 0 0 0 2.68999 0 4.15163i 0 0 0
1585.14 0 0 0 2.68999 0 4.15163i 0 0 0
1585.15 0 0 0 2.68999 0 4.15163i 0 0 0
1585.16 0 0 0 2.68999 0 4.15163i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1585.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
13.b even 2 1 inner
24.h odd 2 1 inner
39.d odd 2 1 inner
104.e even 2 1 inner
312.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3744.2.m.h 16
3.b odd 2 1 inner 3744.2.m.h 16
4.b odd 2 1 936.2.m.h 16
8.b even 2 1 inner 3744.2.m.h 16
8.d odd 2 1 936.2.m.h 16
12.b even 2 1 936.2.m.h 16
13.b even 2 1 inner 3744.2.m.h 16
24.f even 2 1 936.2.m.h 16
24.h odd 2 1 inner 3744.2.m.h 16
39.d odd 2 1 inner 3744.2.m.h 16
52.b odd 2 1 936.2.m.h 16
104.e even 2 1 inner 3744.2.m.h 16
104.h odd 2 1 936.2.m.h 16
156.h even 2 1 936.2.m.h 16
312.b odd 2 1 inner 3744.2.m.h 16
312.h even 2 1 936.2.m.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.m.h 16 4.b odd 2 1
936.2.m.h 16 8.d odd 2 1
936.2.m.h 16 12.b even 2 1
936.2.m.h 16 24.f even 2 1
936.2.m.h 16 52.b odd 2 1
936.2.m.h 16 104.h odd 2 1
936.2.m.h 16 156.h even 2 1
936.2.m.h 16 312.h even 2 1
3744.2.m.h 16 1.a even 1 1 trivial
3744.2.m.h 16 3.b odd 2 1 inner
3744.2.m.h 16 8.b even 2 1 inner
3744.2.m.h 16 13.b even 2 1 inner
3744.2.m.h 16 24.h odd 2 1 inner
3744.2.m.h 16 39.d odd 2 1 inner
3744.2.m.h 16 104.e even 2 1 inner
3744.2.m.h 16 312.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 10 T_{5}^{2} + 20 \) acting on \(S_{2}^{\mathrm{new}}(3744, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 20 - 10 T^{2} + T^{4} )^{4} \)
$7$ \( ( 220 + 30 T^{2} + T^{4} )^{4} \)
$11$ \( ( 20 - 20 T^{2} + T^{4} )^{4} \)
$13$ \( ( 28561 - 2028 T^{2} + 54 T^{4} - 12 T^{6} + T^{8} )^{2} \)
$17$ \( ( 880 - 60 T^{2} + T^{4} )^{4} \)
$19$ \( ( 44 - 34 T^{2} + T^{4} )^{4} \)
$23$ \( ( 880 - 80 T^{2} + T^{4} )^{4} \)
$29$ \( ( 176 + 28 T^{2} + T^{4} )^{4} \)
$31$ \( ( 5500 + 150 T^{2} + T^{4} )^{4} \)
$37$ \( ( 704 - 104 T^{2} + T^{4} )^{4} \)
$41$ \( ( 324 + 126 T^{2} + T^{4} )^{4} \)
$43$ \( ( 80 + 20 T^{2} + T^{4} )^{4} \)
$47$ \( ( 1444 + 84 T^{2} + T^{4} )^{4} \)
$53$ \( ( 176 + 128 T^{2} + T^{4} )^{4} \)
$59$ \( ( 500 - 100 T^{2} + T^{4} )^{4} \)
$61$ \( ( 6480 + 180 T^{2} + T^{4} )^{4} \)
$67$ \( ( 5324 - 154 T^{2} + T^{4} )^{4} \)
$71$ \( ( 484 + 116 T^{2} + T^{4} )^{4} \)
$73$ \( ( 3520 + 200 T^{2} + T^{4} )^{4} \)
$79$ \( ( -76 + 4 T + T^{2} )^{8} \)
$83$ \( ( 500 - 100 T^{2} + T^{4} )^{4} \)
$89$ \( ( 12100 + 230 T^{2} + T^{4} )^{4} \)
$97$ \( ( 3520 + 160 T^{2} + T^{4} )^{4} \)
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