Properties

Label 1380.2.f.b
Level $1380$
Weight $2$
Character orbit 1380.f
Analytic conductor $11.019$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Defining polynomial: \(x^{14} + 27 x^{12} + 283 x^{10} + 1441 x^{8} + 3596 x^{6} + 3740 x^{4} + 772 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9} +O(q^{10})\) \( q -\beta_{6} q^{3} + \beta_{10} q^{5} + \beta_{13} q^{7} - q^{9} + ( 1 - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{11} + ( -\beta_{1} + \beta_{7} + \beta_{10} + \beta_{12} ) q^{13} + \beta_{8} q^{15} + ( \beta_{1} + \beta_{8} + \beta_{9} - \beta_{12} - \beta_{13} ) q^{17} + ( \beta_{3} - \beta_{4} ) q^{19} -\beta_{2} q^{21} + \beta_{6} q^{23} + ( -1 + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{13} ) q^{25} + \beta_{6} q^{27} + ( -2 + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{29} + ( 1 - \beta_{3} + \beta_{5} + 2 \beta_{7} - 2 \beta_{10} ) q^{31} + ( -\beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{33} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} + \beta_{12} ) q^{35} + ( \beta_{1} + 3 \beta_{6} + \beta_{11} ) q^{37} + ( -\beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} ) q^{39} + ( 3 - \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{41} + ( \beta_{1} + \beta_{6} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{43} -\beta_{10} q^{45} + ( \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{47} + ( -1 - \beta_{2} - 2 \beta_{3} + \beta_{7} - \beta_{10} ) q^{49} + ( \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{10} ) q^{51} + ( -\beta_{6} + 2 \beta_{8} + 2 \beta_{9} + \beta_{11} - \beta_{12} ) q^{53} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{12} ) q^{55} + ( -\beta_{1} - \beta_{12} ) q^{57} + ( -2 + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{59} + ( 4 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{8} - \beta_{9} ) q^{61} -\beta_{13} q^{63} + ( -3 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{65} + ( -\beta_{1} + \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{67} + q^{69} + ( 3 + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{9} ) q^{71} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{12} - 2 \beta_{13} ) q^{73} + ( -1 - \beta_{2} + \beta_{6} + \beta_{11} + \beta_{13} ) q^{75} + ( -2 \beta_{1} + 3 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{77} + ( -2 - 2 \beta_{2} + 2 \beta_{4} + \beta_{8} - \beta_{9} ) q^{79} + q^{81} + ( \beta_{1} - 4 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} - \beta_{13} ) q^{83} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - \beta_{7} - \beta_{10} + 2 \beta_{11} - \beta_{12} + \beta_{13} ) q^{85} + ( -\beta_{1} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{87} + ( -2 + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{89} + ( -\beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 3 \beta_{9} ) q^{91} + ( \beta_{1} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} - \beta_{11} ) q^{93} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} - 3 \beta_{8} - \beta_{9} - 2 \beta_{11} + \beta_{12} - 2 \beta_{13} ) q^{95} + ( 3 \beta_{1} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{97} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{9} + O(q^{10}) \) \( 14q - 14q^{9} + 2q^{15} + 4q^{19} - 2q^{21} - 6q^{25} - 30q^{29} + 6q^{31} - 14q^{35} + 4q^{39} + 46q^{41} - 20q^{49} + 2q^{51} - 16q^{55} - 10q^{59} + 64q^{61} - 36q^{65} + 14q^{69} + 42q^{71} - 16q^{75} - 32q^{79} + 14q^{81} - 42q^{85} - 52q^{89} + 28q^{91} - 44q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} + 27 x^{12} + 283 x^{10} + 1441 x^{8} + 3596 x^{6} + 3740 x^{4} + 772 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 6 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{12} + 6 \nu^{10} - 153 \nu^{8} - 1732 \nu^{6} - 5726 \nu^{4} - 5036 \nu^{2} + 136 \)\()/372\)
\(\beta_{4}\)\(=\)\((\)\( 15 \nu^{12} + 338 \nu^{10} + 2851 \nu^{8} + 11282 \nu^{6} + 21122 \nu^{4} + 15228 \nu^{2} + 800 \)\()/372\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{12} - 86 \nu^{10} - 659 \nu^{8} - 2124 \nu^{6} - 2423 \nu^{4} - 6 \nu^{2} + 200 \)\()/62\)
\(\beta_{6}\)\(=\)\((\)\( 6 \nu^{13} + 160 \nu^{11} + 1655 \nu^{9} + 8332 \nu^{7} + 20731 \nu^{5} + 22050 \nu^{3} + 5218 \nu \)\()/372\)
\(\beta_{7}\)\(=\)\((\)\( -21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 12 \nu^{10} - 6459 \nu^{9} - 306 \nu^{8} - 34215 \nu^{7} - 3464 \nu^{6} - 88260 \nu^{5} - 11824 \nu^{4} - 93078 \nu^{3} - 12676 \nu^{2} - 16992 \nu - 1216 \)\()/744\)
\(\beta_{8}\)\(=\)\((\)\( -21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 74 \nu^{10} - 6459 \nu^{9} - 934 \nu^{8} - 34215 \nu^{7} - 5154 \nu^{6} - 88260 \nu^{5} - 12604 \nu^{4} - 93078 \nu^{3} - 11504 \nu^{2} - 16248 \nu - 1264 \)\()/744\)
\(\beta_{9}\)\(=\)\((\)\( -21 \nu^{13} + 2 \nu^{12} - 591 \nu^{11} + 74 \nu^{10} - 6459 \nu^{9} + 934 \nu^{8} - 34215 \nu^{7} + 5154 \nu^{6} - 88260 \nu^{5} + 12604 \nu^{4} - 93078 \nu^{3} + 11504 \nu^{2} - 16248 \nu + 1264 \)\()/744\)
\(\beta_{10}\)\(=\)\((\)\( -21 \nu^{13} - 2 \nu^{12} - 591 \nu^{11} - 12 \nu^{10} - 6459 \nu^{9} + 306 \nu^{8} - 34215 \nu^{7} + 3464 \nu^{6} - 88260 \nu^{5} + 11824 \nu^{4} - 93078 \nu^{3} + 12676 \nu^{2} - 16992 \nu + 1216 \)\()/744\)
\(\beta_{11}\)\(=\)\((\)\( 16 \nu^{13} + 468 \nu^{11} + 5271 \nu^{9} + 28460 \nu^{7} + 74017 \nu^{5} + 78826 \nu^{3} + 16870 \nu \)\()/372\)
\(\beta_{12}\)\(=\)\((\)\( -27 \nu^{13} - 689 \nu^{11} - 6781 \nu^{9} - 32255 \nu^{7} - 74798 \nu^{5} - 71046 \nu^{3} - 10244 \nu \)\()/372\)
\(\beta_{13}\)\(=\)\((\)\( -22 \nu^{13} - 597 \nu^{11} - 6306 \nu^{9} - 32483 \nu^{7} - 82534 \nu^{5} - 88786 \nu^{3} - 20848 \nu \)\()/372\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{10} - 3 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} - \beta_{7} + 2 \beta_{3} - 7 \beta_{2} + 24\)
\(\nu^{5}\)\(=\)\(-2 \beta_{13} + \beta_{11} - 19 \beta_{10} + 21 \beta_{9} + 21 \beta_{8} - 19 \beta_{7} - 3 \beta_{6} - 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-17 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} + 17 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 24 \beta_{3} + 49 \beta_{2} - 156\)
\(\nu^{7}\)\(=\)\(30 \beta_{13} + 2 \beta_{12} - 15 \beta_{11} + 128 \beta_{10} - 156 \beta_{9} - 156 \beta_{8} + 128 \beta_{7} + 61 \beta_{6} + 99 \beta_{1}\)
\(\nu^{8}\)\(=\)\(197 \beta_{10} - 68 \beta_{9} + 68 \beta_{8} - 197 \beta_{7} - 30 \beta_{5} - 28 \beta_{4} + 230 \beta_{3} - 353 \beta_{2} + 1072\)
\(\nu^{9}\)\(=\)\(-338 \beta_{13} - 32 \beta_{12} + 157 \beta_{11} - 904 \beta_{10} + 1194 \beta_{9} + 1194 \beta_{8} - 904 \beta_{7} - 787 \beta_{6} - 841 \beta_{1}\)
\(\nu^{10}\)\(=\)\(-1965 \beta_{10} + 810 \beta_{9} - 810 \beta_{8} + 1965 \beta_{7} + 322 \beta_{5} + 282 \beta_{4} - 2052 \beta_{3} + 2617 \beta_{2} - 7692\)
\(\nu^{11}\)\(=\)\(3390 \beta_{13} + 362 \beta_{12} - 1437 \beta_{11} + 6624 \beta_{10} - 9320 \beta_{9} - 9320 \beta_{8} + 6624 \beta_{7} + 8455 \beta_{6} + 7003 \beta_{1}\)
\(\nu^{12}\)\(=\)\(18213 \beta_{10} - 8336 \beta_{9} + 8336 \beta_{8} - 18213 \beta_{7} - 3058 \beta_{5} - 2512 \beta_{4} + 17758 \beta_{3} - 19889 \beta_{2} + 57120\)
\(\nu^{13}\)\(=\)\(-31918 \beta_{13} - 3604 \beta_{12} + 12389 \beta_{11} - 49978 \beta_{10} + 73852 \beta_{9} + 73852 \beta_{8} - 49978 \beta_{7} - 82667 \beta_{6} - 57917 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
829.1
2.39625i
0.511427i
2.47928i
2.04690i
2.88373i
0.0729221i
1.52925i
2.39625i
0.511427i
2.47928i
2.04690i
2.88373i
0.0729221i
1.52925i
0 1.00000i 0 −2.22987 0.166381i 0 1.74202i 0 −1.00000 0
829.2 0 1.00000i 0 −1.81604 + 1.30461i 0 3.73844i 0 −1.00000 0
829.3 0 1.00000i 0 −0.258163 2.22111i 0 2.14682i 0 −1.00000 0
829.4 0 1.00000i 0 −0.181772 + 2.22867i 0 0.189781i 0 −1.00000 0
829.5 0 1.00000i 0 0.792997 + 2.09073i 0 4.31589i 0 −1.00000 0
829.6 0 1.00000i 0 1.54426 1.61718i 0 3.99468i 0 −1.00000 0
829.7 0 1.00000i 0 2.14859 0.619333i 0 1.66138i 0 −1.00000 0
829.8 0 1.00000i 0 −2.22987 + 0.166381i 0 1.74202i 0 −1.00000 0
829.9 0 1.00000i 0 −1.81604 1.30461i 0 3.73844i 0 −1.00000 0
829.10 0 1.00000i 0 −0.258163 + 2.22111i 0 2.14682i 0 −1.00000 0
829.11 0 1.00000i 0 −0.181772 2.22867i 0 0.189781i 0 −1.00000 0
829.12 0 1.00000i 0 0.792997 2.09073i 0 4.31589i 0 −1.00000 0
829.13 0 1.00000i 0 1.54426 + 1.61718i 0 3.99468i 0 −1.00000 0
829.14 0 1.00000i 0 2.14859 + 0.619333i 0 1.66138i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 829.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.f.b 14
3.b odd 2 1 4140.2.f.c 14
5.b even 2 1 inner 1380.2.f.b 14
5.c odd 4 1 6900.2.a.bc 7
5.c odd 4 1 6900.2.a.bd 7
15.d odd 2 1 4140.2.f.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.f.b 14 1.a even 1 1 trivial
1380.2.f.b 14 5.b even 2 1 inner
4140.2.f.c 14 3.b odd 2 1
4140.2.f.c 14 15.d odd 2 1
6900.2.a.bc 7 5.c odd 4 1
6900.2.a.bd 7 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 59 T_{7}^{12} + 1323 T_{7}^{10} + 14065 T_{7}^{8} + 72984 T_{7}^{6} + 178488 T_{7}^{4} + 166704 T_{7}^{2} + 5776 \) acting on \(S_{2}^{\mathrm{new}}(1380, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \)
$3$ \( ( 1 + T^{2} )^{7} \)
$5$ \( 78125 + 9375 T^{2} + 5000 T^{3} - 1875 T^{4} + 800 T^{5} + 95 T^{6} - 144 T^{7} + 19 T^{8} + 32 T^{9} - 15 T^{10} + 8 T^{11} + 3 T^{12} + T^{14} \)
$7$ \( 5776 + 166704 T^{2} + 178488 T^{4} + 72984 T^{6} + 14065 T^{8} + 1323 T^{10} + 59 T^{12} + T^{14} \)
$11$ \( ( 9216 - 4608 T - 2112 T^{2} + 1200 T^{3} + 62 T^{4} - 64 T^{5} + T^{7} )^{2} \)
$13$ \( 6071296 + 12337152 T^{2} + 5286784 T^{4} + 958912 T^{6} + 88004 T^{8} + 4268 T^{10} + 104 T^{12} + T^{14} \)
$17$ \( 10000 + 6370864 T^{2} + 23376664 T^{4} + 4556280 T^{6} + 323441 T^{8} + 10683 T^{10} + 167 T^{12} + T^{14} \)
$19$ \( ( -36680 - 35928 T - 2216 T^{2} + 4108 T^{3} + 158 T^{4} - 118 T^{5} - 2 T^{6} + T^{7} )^{2} \)
$23$ \( ( 1 + T^{2} )^{7} \)
$29$ \( ( -12272 + 20976 T + 6504 T^{2} - 1792 T^{3} - 595 T^{4} + 17 T^{5} + 15 T^{6} + T^{7} )^{2} \)
$31$ \( ( 54720 - 23616 T - 13920 T^{2} + 4248 T^{3} + 499 T^{4} - 141 T^{5} - 3 T^{6} + T^{7} )^{2} \)
$37$ \( 16126968064 + 4925021696 T^{2} + 590900192 T^{4} + 36366992 T^{6} + 1247449 T^{8} + 24043 T^{10} + 243 T^{12} + T^{14} \)
$41$ \( ( 15248 - 45552 T + 41576 T^{2} - 12896 T^{3} + 1185 T^{4} + 103 T^{5} - 23 T^{6} + T^{7} )^{2} \)
$43$ \( 389193984 + 483577344 T^{2} + 194891904 T^{4} + 29281296 T^{6} + 1720720 T^{8} + 37928 T^{10} + 336 T^{12} + T^{14} \)
$47$ \( 1973013529600 + 304913816576 T^{2} + 18562826624 T^{4} + 577880384 T^{6} + 10016260 T^{8} + 97288 T^{10} + 492 T^{12} + T^{14} \)
$53$ \( 374345104 + 274681456 T^{2} + 76576024 T^{4} + 10169496 T^{6} + 662233 T^{8} + 19775 T^{10} + 251 T^{12} + T^{14} \)
$59$ \( ( -1483200 - 308160 T + 81312 T^{2} + 16872 T^{3} - 1217 T^{4} - 255 T^{5} + 5 T^{6} + T^{7} )^{2} \)
$61$ \( ( 377248 - 459232 T + 192464 T^{2} - 32672 T^{3} + 1226 T^{4} + 274 T^{5} - 32 T^{6} + T^{7} )^{2} \)
$67$ \( 300304 + 803120048 T^{2} + 345348536 T^{4} + 45589400 T^{6} + 1979401 T^{8} + 37075 T^{10} + 315 T^{12} + T^{14} \)
$71$ \( ( -784 + 5840 T - 3848 T^{2} - 3184 T^{3} + 875 T^{4} + 55 T^{5} - 21 T^{6} + T^{7} )^{2} \)
$73$ \( 47127199744 + 28443672576 T^{2} + 3597107200 T^{4} + 192441856 T^{6} + 5104292 T^{8} + 68876 T^{10} + 440 T^{12} + T^{14} \)
$79$ \( ( -4755584 - 697856 T + 264784 T^{2} + 23540 T^{3} - 3792 T^{4} - 264 T^{5} + 16 T^{6} + T^{7} )^{2} \)
$83$ \( 1638400 + 13500416 T^{2} + 25436672 T^{4} + 11084416 T^{6} + 1252593 T^{8} + 47443 T^{10} + 415 T^{12} + T^{14} \)
$89$ \( ( -526112 + 457456 T + 64264 T^{2} - 23724 T^{3} - 4000 T^{4} + 12 T^{5} + 26 T^{6} + T^{7} )^{2} \)
$97$ \( 56070144 + 16510058496 T^{2} + 34971683328 T^{4} + 1825932288 T^{6} + 31417360 T^{8} + 237264 T^{10} + 804 T^{12} + T^{14} \)
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