Properties

Label 1638.2.bj.h
Level $1638$
Weight $2$
Character orbit 1638.bj
Analytic conductor $13.079$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + 6655 x^{8} + 478 x^{7} + 1106 x^{6} + 11266 x^{5} + 55249 x^{4} + 8856 x^{3} + \cdots + 97344 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{5} + 1) q^{4} - \beta_{9} q^{5} - \beta_{3} q^{7} + (\beta_{4} - \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{5} + 1) q^{4} - \beta_{9} q^{5} - \beta_{3} q^{7} + (\beta_{4} - \beta_{3}) q^{8} + \beta_{10} q^{10} + ( - \beta_{15} - \beta_{4} + \beta_{2} + \beta_1 - 1) q^{11} + (\beta_{15} - \beta_{7} - \beta_1 + 1) q^{13} - q^{14} - \beta_{5} q^{16} + ( - \beta_{15} - \beta_{14} + \beta_{13} - 2 \beta_{4} + \beta_{3}) q^{17} + (\beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} - \beta_1) q^{19} - \beta_{7} q^{20} + (\beta_{12} + \beta_{11} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{22} + ( - \beta_{12} - \beta_{10} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{23} + ( - 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \cdots - 2) q^{25}+ \cdots + \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} + 2 q^{10} - 12 q^{11} + 10 q^{13} - 16 q^{14} - 8 q^{16} - 6 q^{17} - 4 q^{22} - 12 q^{23} - 20 q^{25} - 2 q^{26} + 16 q^{29} + 2 q^{35} - 6 q^{37} + 4 q^{40} + 12 q^{41} - 6 q^{43} + 6 q^{46} + 8 q^{49} - 24 q^{50} - 4 q^{52} - 40 q^{53} + 20 q^{55} - 8 q^{56} + 6 q^{58} + 6 q^{59} - 2 q^{61} - 14 q^{62} - 16 q^{64} - 52 q^{65} - 30 q^{67} + 6 q^{68} + 12 q^{71} + 24 q^{74} + 8 q^{77} - 16 q^{79} + 2 q^{82} + 6 q^{85} + 4 q^{88} + 30 q^{89} + 4 q^{91} - 24 q^{92} - 8 q^{94} - 40 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 18 x^{13} + 143 x^{12} - 148 x^{11} + 172 x^{10} + 1612 x^{9} + 6655 x^{8} + 478 x^{7} + 1106 x^{6} + 11266 x^{5} + 55249 x^{4} + 8856 x^{3} + \cdots + 97344 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 282646183413977 \nu^{15} + \cdots - 56\!\cdots\!92 ) / 11\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 30\!\cdots\!11 \nu^{15} + \cdots - 67\!\cdots\!08 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 39\!\cdots\!57 \nu^{15} + \cdots - 66\!\cdots\!28 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 209664617 \nu^{15} - 2025754990 \nu^{14} + 4591800238 \nu^{13} - 476218986 \nu^{12} - 6913891517 \nu^{11} - 201926523452 \nu^{10} + \cdots + 19272755774688 ) / 35896513665792 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!30 \nu^{15} + \cdots - 73\!\cdots\!04 ) / 11\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 40\!\cdots\!40 \nu^{15} + \cdots + 11\!\cdots\!92 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10297601 \nu^{15} - 26746609 \nu^{14} + 27244757 \nu^{13} + 236512383 \nu^{12} + 1095488206 \nu^{11} - 1973529760 \nu^{10} + \cdots + 130830721008 ) / 230105856832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 66\!\cdots\!76 \nu^{15} + \cdots - 77\!\cdots\!28 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 88\!\cdots\!08 \nu^{15} + \cdots - 18\!\cdots\!00 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40\!\cdots\!63 \nu^{15} + \cdots - 57\!\cdots\!04 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 46\!\cdots\!27 \nu^{15} + \cdots + 17\!\cdots\!64 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 13\!\cdots\!15 \nu^{15} + \cdots + 42\!\cdots\!00 ) / 14\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 59\!\cdots\!81 \nu^{15} + \cdots + 94\!\cdots\!84 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 80\!\cdots\!01 \nu^{15} + \cdots + 35\!\cdots\!68 ) / 57\!\cdots\!32 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{9} - \beta_{8} - \beta_{5} - 5\beta_{4} + 5\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - 8 \beta_{6} + 2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 8 \beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - \beta_{14} - 14 \beta_{13} - 12 \beta_{11} + 14 \beta_{10} + \beta_{9} - 13 \beta_{8} - 2 \beta_{7} - 16 \beta_{6} + \beta_{5} - \beta_{4} + 11 \beta_{3} - 3 \beta_{2} - 3 \beta _1 - 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{15} - 30 \beta_{14} - 2 \beta_{13} + 4 \beta_{12} - 26 \beta_{11} + 30 \beta_{10} - 2 \beta_{9} - 28 \beta_{7} - 18 \beta_{6} - 22 \beta_{5} + 8 \beta_{4} - 34 \beta_{3} + 18 \beta_{2} - 81 \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 135 \beta_{14} + 24 \beta_{13} + 40 \beta_{12} + 20 \beta_{11} + 24 \beta_{10} - 163 \beta_{9} + 155 \beta_{8} + 56 \beta_{6} + 63 \beta_{5} + 355 \beta_{4} - 375 \beta_{3} + 211 \beta_{2} - 211 \beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 88 \beta_{15} - 290 \beta_{14} + 390 \beta_{13} + 88 \beta_{12} + 378 \beta_{11} - 56 \beta_{10} - 390 \beta_{9} + 534 \beta_{8} + 334 \beta_{7} + 896 \beta_{6} - 322 \beta_{5} + 760 \beta_{4} - 526 \beta_{3} + 896 \beta_{2} + \cdots + 760 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 622 \beta_{15} + 311 \beta_{14} + 1842 \beta_{13} + 1544 \beta_{11} - 1842 \beta_{10} - 411 \beta_{9} + 1863 \beta_{8} + 822 \beta_{7} + 2656 \beta_{6} - 311 \beta_{5} + 711 \beta_{4} - 833 \beta_{3} + 793 \beta_{2} + \cdots + 3614 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1452 \beta_{15} + 4598 \beta_{14} + 1006 \beta_{13} - 1452 \beta_{12} + 3146 \beta_{11} - 4922 \beta_{10} + 1006 \beta_{9} + 3916 \beta_{7} + 3678 \beta_{6} + 2662 \beta_{5} - 2004 \beta_{4} + 6118 \beta_{3} + \cdots + 2004 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 17921 \beta_{14} - 6136 \beta_{13} - 8896 \beta_{12} - 4448 \beta_{11} - 6136 \beta_{10} + 20953 \beta_{9} - 22689 \beta_{8} - 10280 \beta_{6} + 4735 \beta_{5} - 38365 \beta_{4} + 42813 \beta_{3} + \cdots - 11328 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 21488 \beta_{15} + 34122 \beta_{14} - 61474 \beta_{13} - 21488 \beta_{12} - 55610 \beta_{11} + 15072 \beta_{10} + 61474 \beta_{9} - 97490 \beta_{8} - 46402 \beta_{7} - 120840 \beta_{6} + 53578 \beta_{5} + \cdots - 111820 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 122050 \beta_{15} - 61025 \beta_{14} - 241902 \beta_{13} - 210020 \beta_{11} + 241902 \beta_{10} + 85305 \beta_{9} - 278773 \beta_{8} - 170610 \beta_{7} - 407248 \beta_{6} + 61025 \beta_{5} + \cdots - 418238 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 300388 \beta_{15} - 673430 \beta_{14} - 206178 \beta_{13} + 300388 \beta_{12} - 373042 \beta_{11} + 763598 \beta_{10} - 206178 \beta_{9} - 557420 \beta_{7} - 631474 \beta_{6} - 400830 \beta_{5} + \cdots - 273360 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 2478759 \beta_{14} + 1138040 \beta_{13} + 1631704 \beta_{12} + 815852 \beta_{11} + 1138040 \beta_{10} - 2834043 \beta_{9} + 3442451 \beta_{8} + 1578264 \beta_{6} - 1975057 \beta_{5} + \cdots + 2226908 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 4055624 \beta_{15} - 4121322 \beta_{14} + 9453366 \beta_{13} + 4055624 \beta_{12} + 8176946 \beta_{11} - 2683208 \beta_{10} - 9453366 \beta_{9} + 16126950 \beta_{8} + \cdots + 16260016 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(1 - \beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.41701 + 1.41701i
0.830471 0.830471i
2.24849 2.24849i
−2.02798 + 2.02798i
−0.977855 0.977855i
2.48294 + 2.48294i
−1.29491 1.29491i
1.15585 + 1.15585i
−2.02798 2.02798i
2.24849 + 2.24849i
0.830471 + 0.830471i
−1.41701 1.41701i
1.15585 1.15585i
−1.29491 + 1.29491i
2.48294 2.48294i
−0.977855 + 0.977855i
−0.866025 + 0.500000i 0 0.500000 0.866025i 3.18381i 0 0.866025 + 0.500000i 1.00000i 0 1.59191 + 2.75726i
127.2 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.25534i 0 0.866025 + 0.500000i 1.00000i 0 0.627670 + 1.08716i
127.3 −0.866025 + 0.500000i 0 0.500000 0.866025i 1.24768i 0 0.866025 + 0.500000i 1.00000i 0 −0.623842 1.08053i
127.4 −0.866025 + 0.500000i 0 0.500000 0.866025i 2.19147i 0 0.866025 + 0.500000i 1.00000i 0 −1.09573 1.89787i
127.5 0.866025 0.500000i 0 0.500000 0.866025i 4.11786i 0 −0.866025 0.500000i 1.00000i 0 −2.05893 3.56617i
127.6 0.866025 0.500000i 0 0.500000 0.866025i 0.145508i 0 −0.866025 0.500000i 1.00000i 0 0.0727538 + 0.126013i
127.7 0.866025 0.500000i 0 0.500000 0.866025i 1.34861i 0 −0.866025 0.500000i 1.00000i 0 0.674306 + 1.16793i
127.8 0.866025 0.500000i 0 0.500000 0.866025i 3.62374i 0 −0.866025 0.500000i 1.00000i 0 1.81187 + 3.13825i
1135.1 −0.866025 0.500000i 0 0.500000 + 0.866025i 2.19147i 0 0.866025 0.500000i 1.00000i 0 −1.09573 + 1.89787i
1135.2 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.24768i 0 0.866025 0.500000i 1.00000i 0 −0.623842 + 1.08053i
1135.3 −0.866025 0.500000i 0 0.500000 + 0.866025i 1.25534i 0 0.866025 0.500000i 1.00000i 0 0.627670 1.08716i
1135.4 −0.866025 0.500000i 0 0.500000 + 0.866025i 3.18381i 0 0.866025 0.500000i 1.00000i 0 1.59191 2.75726i
1135.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i 3.62374i 0 −0.866025 + 0.500000i 1.00000i 0 1.81187 3.13825i
1135.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i 1.34861i 0 −0.866025 + 0.500000i 1.00000i 0 0.674306 1.16793i
1135.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.145508i 0 −0.866025 + 0.500000i 1.00000i 0 0.0727538 0.126013i
1135.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 4.11786i 0 −0.866025 + 0.500000i 1.00000i 0 −2.05893 + 3.56617i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1135.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bj.h 16
3.b odd 2 1 1638.2.bj.i yes 16
13.e even 6 1 inner 1638.2.bj.h 16
39.h odd 6 1 1638.2.bj.i yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.bj.h 16 1.a even 1 1 trivial
1638.2.bj.h 16 13.e even 6 1 inner
1638.2.bj.i yes 16 3.b odd 2 1
1638.2.bj.i yes 16 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1638, [\chi])\):

\( T_{5}^{16} + 50 T_{5}^{14} + 953 T_{5}^{12} + 8752 T_{5}^{10} + 40824 T_{5}^{8} + 96800 T_{5}^{6} + 111760 T_{5}^{4} + 50688 T_{5}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{16} + 12 T_{11}^{15} + 18 T_{11}^{14} - 360 T_{11}^{13} - 881 T_{11}^{12} + 9276 T_{11}^{11} + 41282 T_{11}^{10} - 41736 T_{11}^{9} - 365343 T_{11}^{8} + 211620 T_{11}^{7} + 2108476 T_{11}^{6} - 1784016 T_{11}^{5} + \cdots + 11075584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 50 T^{14} + 953 T^{12} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + 12 T^{15} + 18 T^{14} + \cdots + 11075584 \) Copy content Toggle raw display
$13$ \( T^{16} - 10 T^{15} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 6 T^{15} + 108 T^{14} + \cdots + 184226329 \) Copy content Toggle raw display
$19$ \( T^{16} - 62 T^{14} + 3055 T^{12} + \cdots + 11075584 \) Copy content Toggle raw display
$23$ \( T^{16} + 12 T^{15} + \cdots + 112123183104 \) Copy content Toggle raw display
$29$ \( T^{16} - 16 T^{15} + \cdots + 37903417344 \) Copy content Toggle raw display
$31$ \( T^{16} + 330 T^{14} + \cdots + 645566464 \) Copy content Toggle raw display
$37$ \( T^{16} + 6 T^{15} - 137 T^{14} + \cdots + 897122304 \) Copy content Toggle raw display
$41$ \( T^{16} - 12 T^{15} + \cdots + 887876521984 \) Copy content Toggle raw display
$43$ \( T^{16} + 6 T^{15} + \cdots + 4269838336 \) Copy content Toggle raw display
$47$ \( T^{16} + 580 T^{14} + \cdots + 7653323063296 \) Copy content Toggle raw display
$53$ \( (T^{8} + 20 T^{7} + 64 T^{6} - 876 T^{5} + \cdots - 4239)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} - 6 T^{15} - 81 T^{14} + \cdots + 37748736 \) Copy content Toggle raw display
$61$ \( T^{16} + 2 T^{15} + \cdots + 3444868249 \) Copy content Toggle raw display
$67$ \( T^{16} + 30 T^{15} + \cdots + 102072582144 \) Copy content Toggle raw display
$71$ \( T^{16} - 12 T^{15} + \cdots + 72\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{16} + 678 T^{14} + \cdots + 5464074551296 \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} - 214 T^{6} + \cdots + 975168)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 374 T^{14} + 51953 T^{12} + \cdots + 331776 \) Copy content Toggle raw display
$89$ \( T^{16} - 30 T^{15} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{16} + 24 T^{15} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
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