Properties

Label 2352.2.h.p
Level $2352$
Weight $2$
Character orbit 2352.h
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 12 x^{14} + 97 x^{12} - 432 x^{10} + 1392 x^{8} - 2502 x^{6} + 3181 x^{4} - 1650 x^{2} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{9} q^{3} + \beta_{13} q^{5} -\beta_{3} q^{9} +O(q^{10})\) \( q -\beta_{9} q^{3} + \beta_{13} q^{5} -\beta_{3} q^{9} -\beta_{7} q^{13} + \beta_{1} q^{15} + ( -\beta_{13} + \beta_{15} ) q^{17} + ( 2 \beta_{9} - \beta_{11} - \beta_{14} ) q^{19} + ( \beta_{1} + \beta_{8} ) q^{23} + ( -3 - \beta_{3} + \beta_{4} ) q^{25} + \beta_{12} q^{27} + \beta_{10} q^{29} + ( \beta_{9} + \beta_{14} ) q^{31} + ( -4 - \beta_{3} + \beta_{4} ) q^{37} + ( -\beta_{2} + \beta_{5} + \beta_{8} ) q^{39} + ( \beta_{13} + \beta_{15} ) q^{41} + ( \beta_{1} + 2 \beta_{2} - \beta_{8} ) q^{43} + ( -3 \beta_{6} - \beta_{7} - \beta_{13} + \beta_{15} ) q^{45} + ( -\beta_{9} - 2 \beta_{12} + \beta_{14} ) q^{47} + ( -\beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{8} ) q^{51} + ( -\beta_{3} - \beta_{4} - \beta_{10} ) q^{53} + ( 5 + 2 \beta_{3} + \beta_{4} + \beta_{10} ) q^{57} + ( -3 \beta_{9} - \beta_{11} ) q^{59} + ( -4 \beta_{6} + \beta_{7} ) q^{61} + ( 2 \beta_{3} + 2 \beta_{4} - \beta_{10} ) q^{65} + ( \beta_{2} - \beta_{5} ) q^{67} + ( \beta_{6} - 2 \beta_{7} - 3 \beta_{13} + \beta_{15} ) q^{69} + ( \beta_{1} - 3 \beta_{2} - \beta_{5} - \beta_{8} ) q^{71} + ( \beta_{6} - 4 \beta_{7} ) q^{73} + ( 3 \beta_{9} - \beta_{11} + \beta_{12} ) q^{75} + ( 2 \beta_{2} - 2 \beta_{5} ) q^{79} + ( 1 + 2 \beta_{4} - \beta_{10} ) q^{81} + ( -4 \beta_{9} - \beta_{11} - 2 \beta_{12} + \beta_{14} ) q^{83} + ( 10 + 3 \beta_{3} - 3 \beta_{4} ) q^{85} + ( \beta_{9} - \beta_{11} + \beta_{12} - 3 \beta_{14} ) q^{87} + ( -\beta_{13} + \beta_{15} ) q^{89} + ( 4 + \beta_{3} - \beta_{4} - \beta_{10} ) q^{93} + ( -2 \beta_{1} + 3 \beta_{2} + \beta_{5} ) q^{95} -7 \beta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9} + O(q^{10}) \) \( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 12 x^{14} + 97 x^{12} - 432 x^{10} + 1392 x^{8} - 2502 x^{6} + 3181 x^{4} - 1650 x^{2} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 1983 \nu^{14} - 260575 \nu^{12} + 3547200 \nu^{10} - 27979150 \nu^{8} + 125308500 \nu^{6} - 341211350 \nu^{4} + 429906927 \nu^{2} - 140609075 \)\()/45226950\)
\(\beta_{2}\)\(=\)\((\)\( 20867 \nu^{14} - 344205 \nu^{12} + 2915280 \nu^{10} - 16261730 \nu^{8} + 53541630 \nu^{6} - 131508430 \nu^{4} + 115210403 \nu^{2} - 156096575 \)\()/45226950\)
\(\beta_{3}\)\(=\)\((\)\( -739 \nu^{14} + 3643 \nu^{12} - 18358 \nu^{10} - 76302 \nu^{8} + 441787 \nu^{6} - 2227272 \nu^{4} + 2610991 \nu^{2} - 2784875 \)\()/611175\)
\(\beta_{4}\)\(=\)\((\)\( 2786 \nu^{14} - 33307 \nu^{12} + 266042 \nu^{10} - 1141452 \nu^{8} + 3419887 \nu^{6} - 4966422 \nu^{4} + 4049116 \nu^{2} + 265475 \)\()/611175\)
\(\beta_{5}\)\(=\)\((\)\( 41947 \nu^{14} - 498361 \nu^{12} + 4000176 \nu^{10} - 17653306 \nu^{8} + 56108646 \nu^{6} - 99287606 \nu^{4} + 124734379 \nu^{2} - 65760095 \)\()/9045390\)
\(\beta_{6}\)\(=\)\((\)\( 203 \nu^{15} - 2381 \nu^{13} + 18686 \nu^{11} - 81166 \nu^{9} + 254896 \nu^{7} - 473576 \nu^{5} + 642763 \nu^{3} - 561525 \nu \)\()/210750\)
\(\beta_{7}\)\(=\)\((\)\( 1797 \nu^{15} - 13115 \nu^{13} + 74590 \nu^{11} - 7940 \nu^{9} - 684460 \nu^{7} + 4437260 \nu^{5} - 6536117 \nu^{3} + 6056225 \nu \)\()/1222350\)
\(\beta_{8}\)\(=\)\((\)\( -448577 \nu^{14} + 5115739 \nu^{12} - 39578784 \nu^{10} + 162926554 \nu^{8} - 477286824 \nu^{6} + 688243394 \nu^{4} - 612107777 \nu^{2} + 19459775 \)\()/45226950\)
\(\beta_{9}\)\(=\)\((\)\( 396239 \nu^{15} - 5143443 \nu^{13} + 42079208 \nu^{11} - 196431398 \nu^{9} + 625998838 \nu^{7} - 1165072378 \nu^{5} + 1301286409 \nu^{3} - 859155425 \nu \)\()/ 226134750 \)
\(\beta_{10}\)\(=\)\((\)\( 6854 \nu^{14} - 78723 \nu^{12} + 627888 \nu^{10} - 2676528 \nu^{8} + 8475618 \nu^{6} - 14170608 \nu^{4} + 17841074 \nu^{2} - 6203925 \)\()/611175\)
\(\beta_{11}\)\(=\)\((\)\( -148127 \nu^{15} + 1702799 \nu^{13} - 12387244 \nu^{11} + 45773164 \nu^{9} - 97649834 \nu^{7} - 1728946 \nu^{5} + 335203713 \nu^{3} - 596954175 \nu \)\()/75378250\)
\(\beta_{12}\)\(=\)\((\)\( -934049 \nu^{15} + 12587563 \nu^{13} - 106255328 \nu^{11} + 526954868 \nu^{9} - 1802844658 \nu^{7} + 3811230298 \nu^{5} - 4791091069 \nu^{3} + 2462381325 \nu \)\()/ 226134750 \)
\(\beta_{13}\)\(=\)\((\)\( 26459 \nu^{15} - 313703 \nu^{13} + 2495718 \nu^{11} - 10811208 \nu^{9} + 33158448 \nu^{7} - 53305788 \nu^{5} + 50468549 \nu^{3} - 12014375 \nu \)\()/6111750\)
\(\beta_{14}\)\(=\)\((\)\( -2209813 \nu^{15} + 25402981 \nu^{13} - 203257436 \nu^{11} + 872075666 \nu^{9} - 2767434646 \nu^{7} + 4607661826 \nu^{5} - 5646101103 \nu^{3} + 1635066675 \nu \)\()/ 226134750 \)
\(\beta_{15}\)\(=\)\((\)\( 32879 \nu^{15} - 382733 \nu^{13} + 3044898 \nu^{11} - 13003413 \nu^{9} + 40454928 \nu^{7} - 65035668 \nu^{5} + 75654434 \nu^{3} - 14658125 \nu \)\()/3055875\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{13} - 3 \beta_{12} - \beta_{11} + 3 \beta_{7} - 6 \beta_{6}\)\()/12\)
\(\nu^{2}\)\(=\)\((\)\(-3 \beta_{10} + \beta_{8} + 10 \beta_{5} - 6 \beta_{2} - \beta_{1} + 18\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{15} - 6 \beta_{14} - 12 \beta_{13} - 5 \beta_{11} + 9 \beta_{9}\)\()/6\)
\(\nu^{4}\)\(=\)\((\)\(-6 \beta_{10} - 3 \beta_{8} + 11 \beta_{5} - \beta_{4} + \beta_{3} - 17 \beta_{2} + 3 \beta_{1} - 24\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-21 \beta_{15} - 57 \beta_{14} - 51 \beta_{13} + 51 \beta_{12} - 47 \beta_{11} + 9 \beta_{9} - 78 \beta_{7} + 111 \beta_{6}\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(-35 \beta_{8} - 32 \beta_{5} - 27 \beta_{4} + 27 \beta_{3} - 96 \beta_{2} + 29 \beta_{1} - 324\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(126 \beta_{15} + 57 \beta_{14} + 231 \beta_{13} + 231 \beta_{12} + 7 \beta_{11} - 489 \beta_{9} - 399 \beta_{7} + 636 \beta_{6}\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(168 \beta_{10} - 7 \beta_{8} - 374 \beta_{5} - 25 \beta_{4} + 97 \beta_{3} + 28 \beta_{2} - 17 \beta_{1} - 514\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(723 \beta_{15} + 948 \beta_{14} + 1092 \beta_{13} + 679 \beta_{11} - 1089 \beta_{9}\)\()/6\)
\(\nu^{10}\)\(=\)\((\)\(2649 \beta_{10} + 2299 \beta_{8} - 4031 \beta_{5} + 1998 \beta_{4} - 252 \beta_{3} + 5313 \beta_{2} - 1717 \beta_{1} + 7578\)\()/12\)
\(\nu^{11}\)\(=\)\((\)\(4065 \beta_{15} + 7854 \beta_{14} + 5313 \beta_{13} - 5313 \beta_{12} + 7553 \beta_{11} + 4272 \beta_{9} + 10464 \beta_{7} - 20721 \beta_{6}\)\()/12\)
\(\nu^{12}\)\(=\)\(1110 \beta_{8} + 777 \beta_{5} + 1101 \beta_{4} - 1101 \beta_{3} + 2331 \beta_{2} - 444 \beta_{1} + 6341\)
\(\nu^{13}\)\(=\)\((\)\(-22590 \beta_{15} - 15210 \beta_{14} - 26403 \beta_{13} - 26403 \beta_{12} + 4519 \beta_{11} + 80370 \beta_{9} + 54213 \beta_{7} - 116256 \beta_{6}\)\()/12\)
\(\nu^{14}\)\(=\)\((\)\(-74193 \beta_{10} - 469 \beta_{8} + 159560 \beta_{5} - 75600 \beta_{3} + 14964 \beta_{2} + 25669 \beta_{1} + 194058\)\()/12\)
\(\nu^{15}\)\(=\)\((\)\(-124593 \beta_{15} - 154911 \beta_{14} - 133422 \beta_{13} - 94405 \beta_{11} + 128304 \beta_{9}\)\()/6\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2255.1
2.01367 + 1.16260i
−0.667172 + 0.385192i
−0.667172 0.385192i
2.01367 1.16260i
−1.75344 + 1.01235i
−1.19392 0.689309i
−1.19392 + 0.689309i
−1.75344 1.01235i
1.75344 + 1.01235i
1.19392 0.689309i
1.19392 + 0.689309i
1.75344 1.01235i
−2.01367 + 1.16260i
0.667172 + 0.385192i
0.667172 0.385192i
−2.01367 1.16260i
0 −1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.2 0 −1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.3 0 −1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.4 0 −1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.5 0 −0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.6 0 −0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.7 0 −0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.8 0 −0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.9 0 0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.10 0 0.777403 1.54779i 0 3.40332i 0 0 0 −1.79129 2.40651i 0
2255.11 0 0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.12 0 0.777403 + 1.54779i 0 3.40332i 0 0 0 −1.79129 + 2.40651i 0
2255.13 0 1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.14 0 1.70166 0.323042i 0 1.55481i 0 0 0 2.79129 1.09941i 0
2255.15 0 1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
2255.16 0 1.70166 + 0.323042i 0 1.55481i 0 0 0 2.79129 + 1.09941i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2255.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.h.p 16
3.b odd 2 1 inner 2352.2.h.p 16
4.b odd 2 1 inner 2352.2.h.p 16
7.b odd 2 1 inner 2352.2.h.p 16
12.b even 2 1 inner 2352.2.h.p 16
21.c even 2 1 inner 2352.2.h.p 16
28.d even 2 1 inner 2352.2.h.p 16
84.h odd 2 1 inner 2352.2.h.p 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.h.p 16 1.a even 1 1 trivial
2352.2.h.p 16 3.b odd 2 1 inner
2352.2.h.p 16 4.b odd 2 1 inner
2352.2.h.p 16 7.b odd 2 1 inner
2352.2.h.p 16 12.b even 2 1 inner
2352.2.h.p 16 21.c even 2 1 inner
2352.2.h.p 16 28.d even 2 1 inner
2352.2.h.p 16 84.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{4} + 14 T_{5}^{2} + 28 \)
\( T_{11} \)
\( T_{13}^{4} - 22 T_{13}^{2} + 100 \)
\( T_{47}^{4} - 168 T_{47}^{2} + 4032 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( ( 81 - 18 T^{2} - 2 T^{4} - 2 T^{6} + T^{8} )^{2} \)
$5$ \( ( 28 + 14 T^{2} + T^{4} )^{4} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( ( 100 - 22 T^{2} + T^{4} )^{4} \)
$17$ \( ( 700 + 56 T^{2} + T^{4} )^{4} \)
$19$ \( ( 900 + 66 T^{2} + T^{4} )^{4} \)
$23$ \( ( 1008 - 84 T^{2} + T^{4} )^{4} \)
$29$ \( ( 1008 + 84 T^{2} + T^{4} )^{4} \)
$31$ \( ( 24 + T^{2} )^{8} \)
$37$ \( ( -12 + 6 T + T^{2} )^{8} \)
$41$ \( ( 28 + 56 T^{2} + T^{4} )^{4} \)
$43$ \( ( 3600 + 132 T^{2} + T^{4} )^{4} \)
$47$ \( ( 4032 - 168 T^{2} + T^{4} )^{4} \)
$53$ \( ( 112 + 112 T^{2} + T^{4} )^{4} \)
$59$ \( ( 2268 - 126 T^{2} + T^{4} )^{4} \)
$61$ \( ( 196 - 70 T^{2} + T^{4} )^{4} \)
$67$ \( ( 12 + T^{2} )^{8} \)
$71$ \( ( 9072 - 252 T^{2} + T^{4} )^{4} \)
$73$ \( ( 27556 - 340 T^{2} + T^{4} )^{4} \)
$79$ \( ( 48 + T^{2} )^{8} \)
$83$ \( ( 6300 - 294 T^{2} + T^{4} )^{4} \)
$89$ \( ( 700 + 56 T^{2} + T^{4} )^{4} \)
$97$ \( ( -98 + T^{2} )^{8} \)
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