Properties

Label 390.2.bn.b
Level $390$
Weight $2$
Character orbit 390.bn
Analytic conductor $3.114$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bn (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{14} + 1) q^{2} - \beta_{15} q^{3} + \beta_{14} q^{4} + ( - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{13} + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} + 1) q^{2} - \beta_{15} q^{3} + \beta_{14} q^{4} + ( - \beta_{15} - \beta_{14} - \beta_{12} - \beta_{11} - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{5}+ \cdots + ( - 3 \beta_{15} - \beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + 2 \beta_{2} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 8 q^{4} + 24 q^{7} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 8 q^{4} + 24 q^{7} - 16 q^{8} + 12 q^{11} + 8 q^{13} - 8 q^{15} - 8 q^{16} - 4 q^{17} - 4 q^{19} + 4 q^{23} + 4 q^{25} + 4 q^{26} - 24 q^{28} - 48 q^{29} - 4 q^{30} + 16 q^{31} + 8 q^{32} + 16 q^{33} - 20 q^{34} - 12 q^{35} + 12 q^{37} + 4 q^{38} + 4 q^{39} - 12 q^{41} + 20 q^{43} - 12 q^{44} + 8 q^{45} - 4 q^{46} - 4 q^{49} - 16 q^{50} - 4 q^{52} + 32 q^{53} + 16 q^{55} - 24 q^{56} - 8 q^{57} - 48 q^{58} + 4 q^{59} + 4 q^{60} + 4 q^{61} - 4 q^{62} + 4 q^{63} + 16 q^{64} - 8 q^{65} + 32 q^{66} - 28 q^{67} - 16 q^{68} + 4 q^{69} - 36 q^{71} + 32 q^{73} + 12 q^{74} - 24 q^{75} + 8 q^{76} + 4 q^{77} + 20 q^{78} + 8 q^{81} - 24 q^{82} + 52 q^{85} + 16 q^{86} + 36 q^{87} - 12 q^{88} - 24 q^{89} + 4 q^{90} - 8 q^{92} + 36 q^{94} - 40 q^{95} - 4 q^{97} + 4 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1434138866353 \nu^{15} - 74224360591111 \nu^{14} + 217326371958939 \nu^{13} - 648847305484636 \nu^{12} + \cdots + 46\!\cdots\!94 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 123434729989149 \nu^{15} + 492304781090243 \nu^{14} + \cdots + 195044141171618 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 137809854283897 \nu^{15} + 666808683532056 \nu^{14} + \cdots + 69\!\cdots\!44 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3909951128714 \nu^{15} - 13126607646508 \nu^{14} + 38145193554947 \nu^{13} - 162143292202288 \nu^{12} + \cdots - 16575048540328 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4143762135082 \nu^{15} + 12665097411614 \nu^{14} - 36598537974476 \nu^{13} + 160755388928989 \nu^{12} + \cdots + 3631882693804 ) / 28585593772190 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!27 \nu^{15} + \cdots + 35\!\cdots\!44 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6100180981 \nu^{15} - 22259671750 \nu^{14} + 65510725326 \nu^{13} - 270219726184 \nu^{12} + 314950092835 \nu^{11} - 40514369326 \nu^{10} + \cdots - 77929949288 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6072104849199 \nu^{15} + 21780610045393 \nu^{14} - 63731636791632 \nu^{13} + 264606089742658 \nu^{12} + \cdots + 65633529574178 ) / 16530327389030 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20\!\cdots\!87 \nu^{15} + \cdots - 28\!\cdots\!14 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30320067 \nu^{15} - 111020724 \nu^{14} + 326080511 \nu^{13} - 1344589864 \nu^{12} + 1575363211 \nu^{11} - 188871116 \nu^{10} + 3515753069 \nu^{9} + \cdots - 330526984 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 567889621787826 \nu^{15} + \cdots - 77\!\cdots\!96 ) / 10\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 52\!\cdots\!58 \nu^{15} + \cdots - 60\!\cdots\!06 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 19482487322 \nu^{15} + 71829768307 \nu^{14} - 211530176114 \nu^{13} + 869648666130 \nu^{12} - 1035106924390 \nu^{11} + \cdots + 258618081490 ) / 18265555126 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 82631746 \nu^{15} + 300206917 \nu^{14} - 880560228 \nu^{13} + 3640243297 \nu^{12} - 4191737118 \nu^{11} + 407798693 \nu^{10} + \cdots + 791533522 ) / 63923990 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 69\!\cdots\!34 \nu^{15} + \cdots - 83\!\cdots\!68 ) / 51\!\cdots\!90 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{15} + 4\beta_{14} + 2\beta_{13} + 3\beta_{12} + 2\beta_{11} - 2\beta_{8} + \beta_{6} - 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{15} - 6 \beta_{14} - 11 \beta_{12} - 4 \beta_{11} - 4 \beta_{10} + 9 \beta_{9} - 10 \beta_{8} - 9 \beta_{7} - 3 \beta_{6} - 15 \beta_{5} + 9 \beta_{4} - 10 \beta_{3} - 20 \beta_{2} - 9 \beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 23 \beta_{15} - 16 \beta_{14} - 22 \beta_{13} - 21 \beta_{12} - 2 \beta_{11} + 21 \beta_{10} - 2 \beta_{9} + 5 \beta_{8} + 22 \beta_{6} - 2 \beta_{5} + 39 \beta_{4} - 21 \beta_{3} + 14 \beta_{2} + 7 \beta _1 - 37 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 28 \beta_{15} + 154 \beta_{14} + 23 \beta_{13} + 140 \beta_{12} + 89 \beta_{11} + 95 \beta_{10} - 56 \beta_{9} + 12 \beta_{8} + 89 \beta_{7} + 84 \beta_{6} + 121 \beta_{5} - 12 \beta_{4} + 56 \beta_{3} + 173 \beta_{2} - 39 \beta _1 + 93 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 163 \beta_{15} + 184 \beta_{14} + 240 \beta_{13} + 50 \beta_{12} - 94 \beta_{10} + 200 \beta_{9} - 351 \beta_{8} - 35 \beta_{7} - 219 \beta_{6} - 128 \beta_{5} - 94 \beta_{4} + 35 \beta_{3} - 440 \beta_{2} - 400 \beta _1 + 985 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 645 \beta_{15} - 1488 \beta_{14} - 329 \beta_{13} - 1794 \beta_{12} - 914 \beta_{11} - 242 \beta_{10} + 914 \beta_{9} - 672 \beta_{8} - 663 \beta_{7} - 645 \beta_{6} - 1243 \beta_{5} + 914 \beta_{4} - 914 \beta_{3} + \cdots + 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1828 \beta_{15} - 186 \beta_{14} - 1819 \beta_{13} - 195 \beta_{12} + 477 \beta_{11} + 3325 \beta_{10} - 1299 \beta_{9} + 1776 \beta_{8} + 1915 \beta_{7} + 2296 \beta_{6} + 2491 \beta_{5} + 1915 \beta_{4} + \cdots - 5566 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6400 \beta_{15} + 16412 \beta_{14} + 6539 \beta_{13} + 14863 \beta_{12} + 7481 \beta_{11} + 6042 \beta_{10} - 3465 \beta_{9} - 1493 \beta_{8} + 8974 \beta_{7} + 1665 \beta_{6} + 11899 \beta_{5} - 7481 \beta_{4} + \cdots + 17408 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8014 \beta_{15} - 12665 \beta_{14} + 18481 \beta_{13} - 24749 \beta_{12} - 18707 \beta_{11} - 21849 \beta_{10} + 32416 \beta_{9} - 37414 \beta_{8} - 15992 \beta_{7} - 38812 \beta_{6} - 32352 \beta_{5} + \cdots + 77085 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 91679 \beta_{15} - 170862 \beta_{14} - 70257 \beta_{13} - 181429 \beta_{12} - 89750 \beta_{11} + 7240 \beta_{10} + 55995 \beta_{9} - 7240 \beta_{8} - 43472 \beta_{7} - 31804 \beta_{6} + \cdots - 166538 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 55995 \beta_{15} + 241852 \beta_{14} - 92227 \beta_{13} + 278084 \beta_{12} + 185857 \beta_{11} + 372602 \beta_{10} - 253893 \beta_{9} + 253893 \beta_{8} + 322709 \beta_{7} + 281000 \beta_{6} + \cdots - 469625 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 880388 \beta_{15} + 1459906 \beta_{14} + 949204 \beta_{13} + 1273161 \beta_{12} + 510702 \beta_{11} + 13265 \beta_{10} + 13265 \beta_{9} - 533645 \beta_{8} + 533645 \beta_{7} - 400341 \beta_{6} + \cdots + 2650257 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 533645 \beta_{15} - 4295841 \beta_{14} + 533645 \beta_{13} - 5220149 \beta_{12} - 3264759 \beta_{11} - 2855121 \beta_{10} + 3769764 \beta_{9} - 3264759 \beta_{8} - 2855121 \beta_{7} + \cdots + 3840272 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{13}\) \(\beta_{5} - \beta_{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.277956 + 0.213283i
−1.09227 0.838128i
0.792206 1.03242i
−0.709944 + 0.925217i
0.339278 + 0.0446668i
2.69978 + 0.355433i
0.117630 0.893490i
−0.424637 + 3.22544i
0.277956 0.213283i
−1.09227 + 0.838128i
0.792206 + 1.03242i
−0.709944 0.925217i
0.339278 0.0446668i
2.69978 0.355433i
0.117630 + 0.893490i
−0.424637 3.22544i
0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i −1.90502 1.17085i −0.258819 0.965926i 3.41596 + 1.97220i −1.00000 0.866025 + 0.500000i 0.0614757 2.23522i
67.2 0.500000 + 0.866025i −0.965926 0.258819i −0.500000 + 0.866025i 2.06394 + 0.860320i −0.258819 0.965926i 0.450069 + 0.259847i −1.00000 0.866025 + 0.500000i 0.286912 + 2.21758i
67.3 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i −0.0421887 2.23567i 0.258819 + 0.965926i 2.21194 + 1.27707i −1.00000 0.866025 + 0.500000i 1.91505 1.15437i
67.4 0.500000 + 0.866025i 0.965926 + 0.258819i −0.500000 + 0.866025i 1.61532 + 1.54620i 0.258819 + 0.965926i 1.65408 + 0.954985i −1.00000 0.866025 + 0.500000i −0.531389 + 2.17201i
97.1 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i −1.66815 1.48905i −0.965926 0.258819i −0.568824 + 0.328411i −1.00000 −0.866025 + 0.500000i −2.12363 + 0.700141i
97.2 0.500000 0.866025i −0.258819 0.965926i −0.500000 0.866025i 1.50924 1.64991i −0.965926 0.258819i 2.70280 1.56046i −1.00000 −0.866025 + 0.500000i −0.674247 2.13199i
97.3 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i −2.23607 + 0.00342112i 0.965926 + 0.258819i 3.25724 1.88057i −1.00000 −0.866025 + 0.500000i −1.11507 + 1.93820i
97.4 0.500000 0.866025i 0.258819 + 0.965926i −0.500000 0.866025i 0.662933 + 2.13554i 0.965926 + 0.258819i −1.12326 + 0.648516i −1.00000 −0.866025 + 0.500000i 2.18090 + 0.493652i
163.1 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i −1.90502 + 1.17085i −0.258819 + 0.965926i 3.41596 1.97220i −1.00000 0.866025 0.500000i 0.0614757 + 2.23522i
163.2 0.500000 0.866025i −0.965926 + 0.258819i −0.500000 0.866025i 2.06394 0.860320i −0.258819 + 0.965926i 0.450069 0.259847i −1.00000 0.866025 0.500000i 0.286912 2.21758i
163.3 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i −0.0421887 + 2.23567i 0.258819 0.965926i 2.21194 1.27707i −1.00000 0.866025 0.500000i 1.91505 + 1.15437i
163.4 0.500000 0.866025i 0.965926 0.258819i −0.500000 0.866025i 1.61532 1.54620i 0.258819 0.965926i 1.65408 0.954985i −1.00000 0.866025 0.500000i −0.531389 2.17201i
193.1 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i −1.66815 + 1.48905i −0.965926 + 0.258819i −0.568824 0.328411i −1.00000 −0.866025 0.500000i −2.12363 0.700141i
193.2 0.500000 + 0.866025i −0.258819 + 0.965926i −0.500000 + 0.866025i 1.50924 + 1.64991i −0.965926 + 0.258819i 2.70280 + 1.56046i −1.00000 −0.866025 0.500000i −0.674247 + 2.13199i
193.3 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i −2.23607 0.00342112i 0.965926 0.258819i 3.25724 + 1.88057i −1.00000 −0.866025 0.500000i −1.11507 1.93820i
193.4 0.500000 + 0.866025i 0.258819 0.965926i −0.500000 + 0.866025i 0.662933 2.13554i 0.965926 0.258819i −1.12326 0.648516i −1.00000 −0.866025 0.500000i 2.18090 0.493652i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.o even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.bn.b yes 16
5.c odd 4 1 390.2.bd.b 16
13.f odd 12 1 390.2.bd.b 16
65.o even 12 1 inner 390.2.bn.b yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bd.b 16 5.c odd 4 1
390.2.bd.b 16 13.f odd 12 1
390.2.bn.b yes 16 1.a even 1 1 trivial
390.2.bn.b yes 16 65.o even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} - 24 T_{7}^{15} + 262 T_{7}^{14} - 1680 T_{7}^{13} + 6823 T_{7}^{12} - 17400 T_{7}^{11} + 24326 T_{7}^{10} - 5472 T_{7}^{9} - 33171 T_{7}^{8} + 17400 T_{7}^{7} + 87604 T_{7}^{6} - 136800 T_{7}^{5} + \cdots + 10000 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{14} + 16 T^{13} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 24 T^{15} + 262 T^{14} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( T^{16} - 12 T^{15} + 78 T^{14} + \cdots + 10000 \) Copy content Toggle raw display
$13$ \( T^{16} - 8 T^{15} + 20 T^{14} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 4 T^{15} - 58 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$19$ \( T^{16} + 4 T^{15} + 26 T^{14} + 68 T^{13} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{16} - 4 T^{15} - 28 T^{14} + 64 T^{13} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{16} + 48 T^{15} + 1030 T^{14} + \cdots + 24010000 \) Copy content Toggle raw display
$31$ \( T^{16} - 16 T^{15} + 128 T^{14} + \cdots + 21827584 \) Copy content Toggle raw display
$37$ \( T^{16} - 12 T^{15} + 576 T^{13} + \cdots + 418609 \) Copy content Toggle raw display
$41$ \( T^{16} + 12 T^{15} + 90 T^{14} + \cdots + 85264 \) Copy content Toggle raw display
$43$ \( T^{16} - 20 T^{15} + \cdots + 151388416 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 610001280825616 \) Copy content Toggle raw display
$53$ \( T^{16} - 32 T^{15} + \cdots + 14952153841 \) Copy content Toggle raw display
$59$ \( T^{16} - 4 T^{15} + \cdots + 2922403926016 \) Copy content Toggle raw display
$61$ \( T^{16} - 4 T^{15} + \cdots + 1536975103504 \) Copy content Toggle raw display
$67$ \( T^{16} + 28 T^{15} + \cdots + 2412728463616 \) Copy content Toggle raw display
$71$ \( T^{16} + 36 T^{15} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} - 16 T^{7} - 54 T^{6} + 864 T^{5} + \cdots - 86528)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 243571706822656 \) Copy content Toggle raw display
$83$ \( T^{16} + 232 T^{14} + \cdots + 64770304 \) Copy content Toggle raw display
$89$ \( T^{16} + 24 T^{15} + \cdots + 30073246952464 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 344958409179136 \) Copy content Toggle raw display
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