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: \( x^{16} - 12x^{14} + 97x^{12} - 432x^{10} + 1392x^{8} - 2502x^{6} + 3181x^{4} - 1650x^{2} + 625 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + \beta_{13} q^{5} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{9} q^{3} + \beta_{13} q^{5} - \beta_{3} q^{9} - \beta_{7} q^{13} + \beta_1 q^{15} + (\beta_{15} - \beta_{13}) q^{17} + ( - \beta_{14} - \beta_{11} + 2 \beta_{9}) q^{19} + (\beta_{8} + \beta_1) q^{23} + (\beta_{4} - \beta_{3} - 3) q^{25} + \beta_{12} q^{27} + \beta_{10} q^{29} + (\beta_{14} + \beta_{9}) q^{31} + (\beta_{4} - \beta_{3} - 4) q^{37} + (\beta_{8} + \beta_{5} - \beta_{2}) q^{39} + (\beta_{15} + \beta_{13}) q^{41} + ( - \beta_{8} + 2 \beta_{2} + \beta_1) q^{43} + (\beta_{15} - \beta_{13} - \beta_{7} - 3 \beta_{6}) q^{45} + (\beta_{14} - 2 \beta_{12} - \beta_{9}) q^{47} + ( - \beta_{8} + 2 \beta_{5} - \beta_{2} - \beta_1) q^{51} + ( - \beta_{10} - \beta_{4} - \beta_{3}) q^{53} + (\beta_{10} + \beta_{4} + 2 \beta_{3} + 5) q^{57} + ( - \beta_{11} - 3 \beta_{9}) q^{59} + (\beta_{7} - 4 \beta_{6}) q^{61} + ( - \beta_{10} + 2 \beta_{4} + 2 \beta_{3}) q^{65} + ( - \beta_{5} + \beta_{2}) q^{67} + (\beta_{15} - 3 \beta_{13} - 2 \beta_{7} + \beta_{6}) q^{69} + ( - \beta_{8} - \beta_{5} - 3 \beta_{2} + \beta_1) q^{71} + ( - 4 \beta_{7} + \beta_{6}) q^{73} + (\beta_{12} - \beta_{11} + 3 \beta_{9}) q^{75} + ( - 2 \beta_{5} + 2 \beta_{2}) q^{79} + ( - \beta_{10} + 2 \beta_{4} + 1) q^{81} + (\beta_{14} - 2 \beta_{12} - \beta_{11} - 4 \beta_{9}) q^{83} + ( - 3 \beta_{4} + 3 \beta_{3} + 10) q^{85} + ( - 3 \beta_{14} + \beta_{12} - \beta_{11} + \beta_{9}) q^{87} + (\beta_{15} - \beta_{13}) q^{89} + ( - \beta_{10} - \beta_{4} + \beta_{3} + 4) q^{93} + (\beta_{5} + 3 \beta_{2} - 2 \beta_1) q^{95} - 7 \beta_{6} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{9} - 32 q^{25} - 48 q^{37} + 72 q^{57} + 32 q^{81} + 112 q^{85} + 48 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 1983 \nu^{14} - 260575 \nu^{12} + 3547200 \nu^{10} - 27979150 \nu^{8} + 125308500 \nu^{6} - 341211350 \nu^{4} + 429906927 \nu^{2} - 140609075 ) / 45226950 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20867 \nu^{14} - 344205 \nu^{12} + 2915280 \nu^{10} - 16261730 \nu^{8} + 53541630 \nu^{6} - 131508430 \nu^{4} + 115210403 \nu^{2} - 156096575 ) / 45226950 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 739 \nu^{14} + 3643 \nu^{12} - 18358 \nu^{10} - 76302 \nu^{8} + 441787 \nu^{6} - 2227272 \nu^{4} + 2610991 \nu^{2} - 2784875 ) / 611175 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2786 \nu^{14} - 33307 \nu^{12} + 266042 \nu^{10} - 1141452 \nu^{8} + 3419887 \nu^{6} - 4966422 \nu^{4} + 4049116 \nu^{2} + 265475 ) / 611175 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41947 \nu^{14} - 498361 \nu^{12} + 4000176 \nu^{10} - 17653306 \nu^{8} + 56108646 \nu^{6} - 99287606 \nu^{4} + 124734379 \nu^{2} - 65760095 ) / 9045390 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 203 \nu^{15} - 2381 \nu^{13} + 18686 \nu^{11} - 81166 \nu^{9} + 254896 \nu^{7} - 473576 \nu^{5} + 642763 \nu^{3} - 561525 \nu ) / 210750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1797 \nu^{15} - 13115 \nu^{13} + 74590 \nu^{11} - 7940 \nu^{9} - 684460 \nu^{7} + 4437260 \nu^{5} - 6536117 \nu^{3} + 6056225 \nu ) / 1222350 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 448577 \nu^{14} + 5115739 \nu^{12} - 39578784 \nu^{10} + 162926554 \nu^{8} - 477286824 \nu^{6} + 688243394 \nu^{4} - 612107777 \nu^{2} + \cdots + 19459775 ) / 45226950 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 396239 \nu^{15} - 5143443 \nu^{13} + 42079208 \nu^{11} - 196431398 \nu^{9} + 625998838 \nu^{7} - 1165072378 \nu^{5} + 1301286409 \nu^{3} + \cdots - 859155425 \nu ) / 226134750 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6854 \nu^{14} - 78723 \nu^{12} + 627888 \nu^{10} - 2676528 \nu^{8} + 8475618 \nu^{6} - 14170608 \nu^{4} + 17841074 \nu^{2} - 6203925 ) / 611175 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 148127 \nu^{15} + 1702799 \nu^{13} - 12387244 \nu^{11} + 45773164 \nu^{9} - 97649834 \nu^{7} - 1728946 \nu^{5} + 335203713 \nu^{3} + \cdots - 596954175 \nu ) / 75378250 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 934049 \nu^{15} + 12587563 \nu^{13} - 106255328 \nu^{11} + 526954868 \nu^{9} - 1802844658 \nu^{7} + 3811230298 \nu^{5} + \cdots + 2462381325 \nu ) / 226134750 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26459 \nu^{15} - 313703 \nu^{13} + 2495718 \nu^{11} - 10811208 \nu^{9} + 33158448 \nu^{7} - 53305788 \nu^{5} + 50468549 \nu^{3} - 12014375 \nu ) / 6111750 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2209813 \nu^{15} + 25402981 \nu^{13} - 203257436 \nu^{11} + 872075666 \nu^{9} - 2767434646 \nu^{7} + 4607661826 \nu^{5} + \cdots + 1635066675 \nu ) / 226134750 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 32879 \nu^{15} - 382733 \nu^{13} + 3044898 \nu^{11} - 13003413 \nu^{9} + 40454928 \nu^{7} - 65035668 \nu^{5} + 75654434 \nu^{3} - 14658125 \nu ) / 3055875 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -3\beta_{13} - 3\beta_{12} - \beta_{11} + 3\beta_{7} - 6\beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{10} + \beta_{8} + 10\beta_{5} - 6\beta_{2} - \beta _1 + 18 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{15} - 6\beta_{14} - 12\beta_{13} - 5\beta_{11} + 9\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{10} - 3\beta_{8} + 11\beta_{5} - \beta_{4} + \beta_{3} - 17\beta_{2} + 3\beta _1 - 24 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21 \beta_{15} - 57 \beta_{14} - 51 \beta_{13} + 51 \beta_{12} - 47 \beta_{11} + 9 \beta_{9} - 78 \beta_{7} + 111 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -35\beta_{8} - 32\beta_{5} - 27\beta_{4} + 27\beta_{3} - 96\beta_{2} + 29\beta _1 - 324 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 126 \beta_{15} + 57 \beta_{14} + 231 \beta_{13} + 231 \beta_{12} + 7 \beta_{11} - 489 \beta_{9} - 399 \beta_{7} + 636 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 168\beta_{10} - 7\beta_{8} - 374\beta_{5} - 25\beta_{4} + 97\beta_{3} + 28\beta_{2} - 17\beta _1 - 514 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 723\beta_{15} + 948\beta_{14} + 1092\beta_{13} + 679\beta_{11} - 1089\beta_{9} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2649 \beta_{10} + 2299 \beta_{8} - 4031 \beta_{5} + 1998 \beta_{4} - 252 \beta_{3} + 5313 \beta_{2} - 1717 \beta _1 + 7578 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4065 \beta_{15} + 7854 \beta_{14} + 5313 \beta_{13} - 5313 \beta_{12} + 7553 \beta_{11} + 4272 \beta_{9} + 10464 \beta_{7} - 20721 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1110\beta_{8} + 777\beta_{5} + 1101\beta_{4} - 1101\beta_{3} + 2331\beta_{2} - 444\beta _1 + 6341 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22590 \beta_{15} - 15210 \beta_{14} - 26403 \beta_{13} - 26403 \beta_{12} + 4519 \beta_{11} + 80370 \beta_{9} + 54213 \beta_{7} - 116256 \beta_{6} ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -74193\beta_{10} - 469\beta_{8} + 159560\beta_{5} - 75600\beta_{3} + 14964\beta_{2} + 25669\beta _1 + 194058 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( -124593\beta_{15} - 154911\beta_{14} - 133422\beta_{13} - 94405\beta_{11} + 128304\beta_{9} ) / 6 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

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

Hecke characteristic polynomials

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