Properties

Label 605.2.m.a
Level $605$
Weight $2$
Character orbit 605.m
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM discriminant -11
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: 16.0.3429742096000000000000.1
Defining polynomial: \(x^{16} + 5 x^{14} + 16 x^{12} + 35 x^{10} + 31 x^{8} + 315 x^{6} + 1296 x^{4} + 3645 x^{2} + 6561\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{3} + ( -2 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{14} ) q^{4} + ( 2 \beta_{10} - \beta_{12} ) q^{5} + ( \beta_{3} + 2 \beta_{8} + 3 \beta_{14} ) q^{9} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{3} + ( -2 \beta_{5} - 2 \beta_{7} + 2 \beta_{10} + 2 \beta_{14} ) q^{4} + ( 2 \beta_{10} - \beta_{12} ) q^{5} + ( \beta_{3} + 2 \beta_{8} + 3 \beta_{14} ) q^{9} + ( 2 \beta_{6} + 2 \beta_{15} ) q^{12} + ( 2 \beta_{2} - 4 \beta_{5} - 4 \beta_{7} + 4 \beta_{10} + \beta_{11} - 3 \beta_{13} + 4 \beta_{14} ) q^{15} + 4 \beta_{9} q^{16} + ( 2 \beta_{3} - 2 \beta_{8} ) q^{20} + ( 4 - \beta_{6} - 4 \beta_{7} + \beta_{15} ) q^{23} + ( 1 + \beta_{3} - 3 \beta_{4} - \beta_{9} + \beta_{13} ) q^{25} + ( 1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{8} - 6 \beta_{9} - 5 \beta_{10} - \beta_{12} + \beta_{13} ) q^{27} + ( -6 \beta_{1} - 6 \beta_{5} - 6 \beta_{7} + 6 \beta_{10} + 6 \beta_{11} - 6 \beta_{12} + 3 \beta_{14} - 6 \beta_{15} ) q^{31} + ( -6 - 4 \beta_{1} - 6 \beta_{3} - 2 \beta_{5} + 6 \beta_{9} - 6 \beta_{13} ) q^{36} + ( 3 \beta_{2} + 2 \beta_{5} + 2 \beta_{7} - 2 \beta_{10} + 3 \beta_{11} - 5 \beta_{13} - 2 \beta_{14} ) q^{37} + ( -3 + 3 \beta_{6} - 4 \beta_{7} + 3 \beta_{15} ) q^{45} + ( 7 + 2 \beta_{1} + 7 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 7 \beta_{9} + 7 \beta_{13} ) q^{47} + ( -4 \beta_{2} + 4 \beta_{11} + 4 \beta_{13} ) q^{48} -7 \beta_{10} q^{49} + ( 4 \beta_{1} - \beta_{3} + 4 \beta_{5} + 4 \beta_{7} + 4 \beta_{8} - 4 \beta_{10} - 4 \beta_{11} + 4 \beta_{12} - 5 \beta_{14} + 4 \beta_{15} ) q^{53} + ( -2 \beta_{2} + \beta_{13} ) q^{59} + ( 2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{8} + 6 \beta_{9} + 2 \beta_{10} + 4 \beta_{12} + 2 \beta_{13} ) q^{60} + 8 \beta_{14} q^{64} + ( 5 - 3 \beta_{6} + 5 \beta_{7} - 3 \beta_{15} ) q^{67} + ( -4 - 4 \beta_{3} + 8 \beta_{4} - \beta_{5} + 4 \beta_{9} - 4 \beta_{13} ) q^{69} + 3 \beta_{9} q^{71} + ( \beta_{1} + 7 \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - 9 \beta_{14} + \beta_{15} ) q^{75} + ( 4 \beta_{1} + 8 \beta_{5} ) q^{80} + ( 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{10} - 6 \beta_{11} - 2 \beta_{13} - 3 \beta_{14} ) q^{81} + 9 \beta_{7} q^{89} + ( -2 \beta_{2} - 8 \beta_{5} - 8 \beta_{7} + 8 \beta_{10} - 2 \beta_{11} + 10 \beta_{13} + 8 \beta_{14} ) q^{92} + ( -3 + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{8} - 15 \beta_{9} + 18 \beta_{10} - 3 \beta_{12} - 3 \beta_{13} ) q^{93} + ( 3 \beta_{1} - 10 \beta_{3} + 3 \beta_{5} + 3 \beta_{7} - 3 \beta_{8} - 3 \beta_{10} - 3 \beta_{11} + 3 \beta_{12} + 7 \beta_{14} + 3 \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 2q^{3} + O(q^{10}) \) \( 16q + 2q^{3} - 16q^{12} + 8q^{15} + 16q^{16} - 12q^{20} + 72q^{23} - 2q^{25} - 22q^{27} - 24q^{36} + 14q^{37} - 72q^{45} + 24q^{47} - 8q^{48} + 12q^{53} + 28q^{60} + 104q^{67} + 12q^{71} - 32q^{75} + 8q^{81} - 36q^{92} - 66q^{93} + 34q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} + 5 x^{14} + 16 x^{12} + 35 x^{10} + 31 x^{8} + 315 x^{6} + 1296 x^{4} + 3645 x^{2} + 6561\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{14} + 718 \nu^{4} \)\()/2511\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{12} - 160 \nu^{2} \)\()/279\)
\(\beta_{4}\)\(=\)\((\)\( -5 \nu^{14} - 16 \nu^{12} - 35 \nu^{10} - 31 \nu^{8} + 403 \nu^{6} - 1296 \nu^{4} - 3645 \nu^{2} - 6561 \)\()/7533\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{11} - 253 \nu \)\()/93\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{10} - 253 \)\()/31\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{15} - 718 \nu^{5} \)\()/7533\)
\(\beta_{8}\)\(=\)\((\)\( -2 \nu^{12} - 599 \nu^{2} \)\()/279\)
\(\beta_{9}\)\(=\)\((\)\( 35 \nu^{14} + 175 \nu^{12} + 560 \nu^{10} + 496 \nu^{8} + 1085 \nu^{6} + 11025 \nu^{4} + 45360 \nu^{2} + 127575 \)\()/22599\)
\(\beta_{10}\)\(=\)\((\)\( 2 \nu^{13} + 599 \nu^{3} \)\()/837\)
\(\beta_{11}\)\(=\)\((\)\( \nu^{15} + 5 \nu^{13} + 16 \nu^{11} + 35 \nu^{9} + 31 \nu^{7} + 315 \nu^{5} + 1296 \nu^{3} + 3645 \nu \)\()/2187\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{13} - 160 \nu^{3} \)\()/279\)
\(\beta_{13}\)\(=\)\((\)\( -5 \nu^{14} - 1079 \nu^{4} \)\()/2511\)
\(\beta_{14}\)\(=\)\((\)\( 65 \nu^{15} + 208 \nu^{13} + 455 \nu^{11} + 403 \nu^{9} + 2294 \nu^{7} + 16848 \nu^{5} + 47385 \nu^{3} + 85293 \nu \)\()/67797\)
\(\beta_{15}\)\(=\)\((\)\( -5 \nu^{15} - 1079 \nu^{5} \)\()/2511\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{8} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\(2 \beta_{12} + 3 \beta_{10}\)
\(\nu^{4}\)\(=\)\(\beta_{13} + 5 \beta_{2}\)
\(\nu^{5}\)\(=\)\(\beta_{15} - 15 \beta_{7}\)
\(\nu^{6}\)\(=\)\(-3 \beta_{13} + 3 \beta_{9} + 16 \beta_{4} - 3 \beta_{3} - 3\)
\(\nu^{7}\)\(=\)\(13 \beta_{15} + 35 \beta_{14} + 13 \beta_{12} - 13 \beta_{11} - 13 \beta_{10} + 13 \beta_{7} + 13 \beta_{5} + 13 \beta_{1}\)
\(\nu^{8}\)\(=\)\(35 \beta_{13} + 39 \beta_{9} + 35 \beta_{8} + 35 \beta_{6} - 35 \beta_{4} + 35 \beta_{3} - 35 \beta_{2} + 35\)
\(\nu^{9}\)\(=\)\(-31 \beta_{14} + 74 \beta_{11} - 31 \beta_{10} + 31 \beta_{7} + 31 \beta_{5}\)
\(\nu^{10}\)\(=\)\(-31 \beta_{6} - 253\)
\(\nu^{11}\)\(=\)\(-93 \beta_{5} - 253 \beta_{1}\)
\(\nu^{12}\)\(=\)\(160 \beta_{8} - 599 \beta_{3}\)
\(\nu^{13}\)\(=\)\(-599 \beta_{12} - 480 \beta_{10}\)
\(\nu^{14}\)\(=\)\(-718 \beta_{13} - 1079 \beta_{2}\)
\(\nu^{15}\)\(=\)\(-718 \beta_{15} + 3237 \beta_{7}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(\beta_{7}\) \(-\beta_{13}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1
−1.04771 + 1.37924i
1.63550 0.570223i
0.0369185 + 1.73166i
−0.987975 1.42264i
−1.63550 + 0.570223i
1.04771 1.37924i
0.0369185 1.73166i
−0.987975 + 1.42264i
−0.0369185 + 1.73166i
0.987975 1.42264i
−1.63550 0.570223i
1.04771 + 1.37924i
−1.04771 1.37924i
1.63550 + 0.570223i
−0.0369185 1.73166i
0.987975 + 1.42264i
0 −0.256255 + 1.61793i 1.17557 1.61803i 0.914138 2.04067i 0 0 0 0.301128 + 0.0978424i 0
112.2 0 0.477487 3.01474i 1.17557 1.61803i 1.93903 + 1.11362i 0 0 0 −6.00747 1.95194i 0
118.1 0 −0.743682 + 1.45956i −1.90211 0.618034i 2.22328 0.238794i 0 0 0 0.186107 + 0.256155i 0
118.2 0 1.38572 2.71963i −1.90211 0.618034i −0.459925 2.18826i 0 0 0 −3.71282 5.11026i 0
233.1 0 −1.61793 0.256255i −1.17557 + 1.61803i −1.93903 1.11362i 0 0 0 −0.301128 0.0978424i 0
233.2 0 3.01474 + 0.477487i −1.17557 + 1.61803i −0.914138 + 2.04067i 0 0 0 6.00747 + 1.95194i 0
282.1 0 −0.743682 1.45956i −1.90211 + 0.618034i 2.22328 + 0.238794i 0 0 0 0.186107 0.256155i 0
282.2 0 1.38572 + 2.71963i −1.90211 + 0.618034i −0.459925 + 2.18826i 0 0 0 −3.71282 + 5.11026i 0
403.1 0 −2.71963 + 1.38572i 1.90211 0.618034i −2.22328 0.238794i 0 0 0 3.71282 5.11026i 0
403.2 0 1.45956 0.743682i 1.90211 0.618034i 0.459925 2.18826i 0 0 0 −0.186107 + 0.256155i 0
457.1 0 −1.61793 + 0.256255i −1.17557 1.61803i −1.93903 + 1.11362i 0 0 0 −0.301128 + 0.0978424i 0
457.2 0 3.01474 0.477487i −1.17557 1.61803i −0.914138 2.04067i 0 0 0 6.00747 1.95194i 0
578.1 0 −0.256255 1.61793i 1.17557 + 1.61803i 0.914138 + 2.04067i 0 0 0 0.301128 0.0978424i 0
578.2 0 0.477487 + 3.01474i 1.17557 + 1.61803i 1.93903 1.11362i 0 0 0 −6.00747 + 1.95194i 0
602.1 0 −2.71963 1.38572i 1.90211 + 0.618034i −2.22328 + 0.238794i 0 0 0 3.71282 + 5.11026i 0
602.2 0 1.45956 + 0.743682i 1.90211 + 0.618034i 0.459925 + 2.18826i 0 0 0 −0.186107 0.256155i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 602.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.c odd 4 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.a 16
5.c odd 4 1 inner 605.2.m.a 16
11.b odd 2 1 CM 605.2.m.a 16
11.c even 5 1 55.2.e.b 4
11.c even 5 3 inner 605.2.m.a 16
11.d odd 10 1 55.2.e.b 4
11.d odd 10 3 inner 605.2.m.a 16
33.f even 10 1 495.2.k.a 4
33.h odd 10 1 495.2.k.a 4
44.g even 10 1 880.2.bd.d 4
44.h odd 10 1 880.2.bd.d 4
55.e even 4 1 inner 605.2.m.a 16
55.h odd 10 1 275.2.e.a 4
55.j even 10 1 275.2.e.a 4
55.k odd 20 1 55.2.e.b 4
55.k odd 20 1 275.2.e.a 4
55.k odd 20 3 inner 605.2.m.a 16
55.l even 20 1 55.2.e.b 4
55.l even 20 1 275.2.e.a 4
55.l even 20 3 inner 605.2.m.a 16
165.u odd 20 1 495.2.k.a 4
165.v even 20 1 495.2.k.a 4
220.v even 20 1 880.2.bd.d 4
220.w odd 20 1 880.2.bd.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.b 4 11.c even 5 1
55.2.e.b 4 11.d odd 10 1
55.2.e.b 4 55.k odd 20 1
55.2.e.b 4 55.l even 20 1
275.2.e.a 4 55.h odd 10 1
275.2.e.a 4 55.j even 10 1
275.2.e.a 4 55.k odd 20 1
275.2.e.a 4 55.l even 20 1
495.2.k.a 4 33.f even 10 1
495.2.k.a 4 33.h odd 10 1
495.2.k.a 4 165.u odd 20 1
495.2.k.a 4 165.v even 20 1
605.2.m.a 16 1.a even 1 1 trivial
605.2.m.a 16 5.c odd 4 1 inner
605.2.m.a 16 11.b odd 2 1 CM
605.2.m.a 16 11.c even 5 3 inner
605.2.m.a 16 11.d odd 10 3 inner
605.2.m.a 16 55.e even 4 1 inner
605.2.m.a 16 55.k odd 20 3 inner
605.2.m.a 16 55.l even 20 3 inner
880.2.bd.d 4 44.g even 10 1
880.2.bd.d 4 44.h odd 10 1
880.2.bd.d 4 220.v even 20 1
880.2.bd.d 4 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 390625 + 156250 T + 31250 T^{2} - 31250 T^{3} - 43125 T^{4} + 21250 T^{5} + 13200 T^{6} + 8280 T^{7} + 2281 T^{8} - 1656 T^{9} + 528 T^{10} - 170 T^{11} - 69 T^{12} + 10 T^{13} + 2 T^{14} - 2 T^{15} + T^{16} \)
$5$ \( 390625 + 15625 T^{2} - 15000 T^{4} - 1225 T^{6} + 551 T^{8} - 49 T^{10} - 24 T^{12} + T^{14} + T^{16} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( T^{16} \)
$17$ \( T^{16} \)
$19$ \( T^{16} \)
$23$ \( ( 1225 - 630 T + 162 T^{2} - 18 T^{3} + T^{4} )^{4} \)
$29$ \( T^{16} \)
$31$ \( ( 96059601 + 970299 T^{2} + 9801 T^{4} + 99 T^{6} + T^{8} )^{2} \)
$37$ \( 152587890625 + 85449218750 T + 23925781250 T^{2} - 3417968750 T^{3} - 7823828125 T^{4} + 946093750 T^{5} + 1794870000 T^{6} + 1037502200 T^{7} + 290891241 T^{8} - 41500088 T^{9} + 2871792 T^{10} - 60550 T^{11} - 20029 T^{12} + 350 T^{13} + 98 T^{14} - 14 T^{15} + T^{16} \)
$41$ \( T^{16} \)
$43$ \( T^{16} \)
$47$ \( 39062500000000 - 18750000000000 T + 4500000000000 T^{2} - 375000000000 T^{3} - 174025000000 T^{4} + 20700000000 T^{5} + 11911680000 T^{6} - 6281606400 T^{7} + 1513835536 T^{8} - 125632128 T^{9} + 4764672 T^{10} + 165600 T^{11} - 27844 T^{12} - 1200 T^{13} + 288 T^{14} - 24 T^{15} + T^{16} \)
$53$ \( 576480100000000 + 98825160000000 T + 8470728000000 T^{2} - 1411788000000 T^{3} - 726158440000 T^{4} + 81249840000 T^{5} + 26327347200 T^{6} + 5385851520 T^{7} + 485654416 T^{8} - 76940736 T^{9} + 5372928 T^{10} - 236880 T^{11} - 30244 T^{12} + 840 T^{13} + 72 T^{14} - 12 T^{15} + T^{16} \)
$59$ \( ( 14641 - 1331 T^{2} + 121 T^{4} - 11 T^{6} + T^{8} )^{2} \)
$61$ \( T^{16} \)
$67$ \( ( 1225 - 910 T + 338 T^{2} - 26 T^{3} + T^{4} )^{4} \)
$71$ \( ( 81 - 27 T + 9 T^{2} - 3 T^{3} + T^{4} )^{4} \)
$73$ \( T^{16} \)
$79$ \( T^{16} \)
$83$ \( T^{16} \)
$89$ \( ( 81 + T^{2} )^{8} \)
$97$ \( 6634204312890625 - 2374346806718750 T + 424883112781250 T^{2} - 24993124281250 T^{3} - 10056627218125 T^{4} + 811411126250 T^{5} + 400748554800 T^{6} - 154736289560 T^{7} + 27771102441 T^{8} - 1628803048 T^{9} + 44404272 T^{10} + 946390 T^{11} - 123469 T^{12} - 3230 T^{13} + 578 T^{14} - 34 T^{15} + T^{16} \)
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