Properties

Label 605.2.b.g
Level $605$
Weight $2$
Character orbit 605.b
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1480160000.1
Defining polynomial: \(x^{8} + 9 x^{6} + 27 x^{4} + 31 x^{2} + 11\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{6} - \beta_{7} ) q^{3} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{4} + \beta_{6} ) q^{5} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{6} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( -\beta_{1} + \beta_{6} - \beta_{7} ) q^{3} + \beta_{2} q^{4} + ( -\beta_{3} + \beta_{4} + \beta_{6} ) q^{5} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{6} + \beta_{4} q^{7} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{8} + ( 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{9} + ( -2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{10} + ( \beta_{1} - \beta_{4} - 2 \beta_{6} ) q^{12} + ( -\beta_{1} + 2 \beta_{6} ) q^{13} + ( -1 - 2 \beta_{2} + 2 \beta_{5} ) q^{14} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} ) q^{15} + ( -2 + \beta_{2} + \beta_{3} ) q^{16} + ( 2 \beta_{1} - \beta_{6} + 3 \beta_{7} ) q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{18} + ( 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{19} + ( 2 \beta_{1} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{20} + ( -1 - \beta_{3} + 2 \beta_{5} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{4} + \beta_{6} ) q^{23} + ( 1 + \beta_{2} + 2 \beta_{3} ) q^{24} + ( -3 - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{25} + ( 4 - \beta_{2} + 2 \beta_{3} - 4 \beta_{5} ) q^{26} + ( -\beta_{4} - 2 \beta_{7} ) q^{27} + \beta_{1} q^{28} + ( -4 - \beta_{3} + \beta_{5} ) q^{29} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} ) q^{30} + ( 2 + \beta_{2} + \beta_{3} + \beta_{5} ) q^{31} + ( -\beta_{1} + 3 \beta_{4} + 2 \beta_{6} + \beta_{7} ) q^{32} + ( -2 + 2 \beta_{2} - 4 \beta_{3} - \beta_{5} ) q^{34} + ( -3 - \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{35} + ( 3 - 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{36} + ( 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{4} + \beta_{7} ) q^{38} + ( -2 + 5 \beta_{2} - \beta_{5} ) q^{39} + ( -4 - 2 \beta_{2} + \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{40} + ( 2 - 3 \beta_{2} + 2 \beta_{3} + 5 \beta_{5} ) q^{41} + ( -\beta_{1} + 3 \beta_{6} - \beta_{7} ) q^{42} + ( -3 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 + 3 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} ) q^{45} + ( 7 + 2 \beta_{2} + \beta_{3} - 6 \beta_{5} ) q^{46} + ( -\beta_{1} + 3 \beta_{4} - \beta_{6} + \beta_{7} ) q^{47} + ( 2 \beta_{1} - \beta_{4} - 5 \beta_{6} + 2 \beta_{7} ) q^{48} + ( 4 - \beta_{2} + 2 \beta_{3} + \beta_{5} ) q^{49} + ( -2 - 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{50} + ( 7 - \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{51} + ( 3 \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{52} + ( -3 \beta_{1} - \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -1 + 2 \beta_{2} + 2 \beta_{3} ) q^{54} + ( -4 - 3 \beta_{2} + 4 \beta_{5} ) q^{56} + ( \beta_{1} - \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{57} + ( -4 \beta_{1} + 2 \beta_{6} - \beta_{7} ) q^{58} + ( -5 \beta_{2} + 3 \beta_{3} + 2 \beta_{5} ) q^{59} + ( 5 - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + \beta_{7} ) q^{60} + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 5 \beta_{5} ) q^{61} + ( \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{62} + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{63} + ( -2 - 5 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{64} + ( -2 + 4 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{65} + ( 4 \beta_{1} - \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{67} + ( 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{68} + ( -2 + 4 \beta_{2} - \beta_{3} - 3 \beta_{5} ) q^{69} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 4 \beta_{5} + \beta_{6} ) q^{70} + ( -9 - 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} ) q^{71} + ( \beta_{1} + 2 \beta_{4} - \beta_{7} ) q^{72} + ( -2 \beta_{1} - 5 \beta_{4} - \beta_{7} ) q^{73} + ( 2 - 4 \beta_{2} + 2 \beta_{3} - 3 \beta_{5} ) q^{74} + ( -2 + 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{75} + ( 3 - 2 \beta_{2} + \beta_{3} + \beta_{5} ) q^{76} + ( -7 \beta_{1} + 5 \beta_{4} + 4 \beta_{6} ) q^{78} + ( -6 - 4 \beta_{2} - \beta_{3} + 5 \beta_{5} ) q^{79} + ( -1 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{80} + ( -1 + 4 \beta_{2} - 2 \beta_{3} - 7 \beta_{5} ) q^{81} + ( 5 \beta_{1} - 3 \beta_{4} + 2 \beta_{7} ) q^{82} + ( \beta_{1} - \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{83} + ( 2 - \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{84} + ( 1 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 5 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( 2 - 3 \beta_{2} + 6 \beta_{5} ) q^{86} + ( 5 \beta_{1} - \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{87} + ( 5 - 2 \beta_{2} + 6 \beta_{3} - 2 \beta_{5} ) q^{89} + ( -8 + \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} - \beta_{6} ) q^{90} + ( 1 + 2 \beta_{2} - 4 \beta_{3} ) q^{91} + ( \beta_{1} - 2 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{92} + ( -2 \beta_{1} - 2 \beta_{4} - \beta_{6} - 3 \beta_{7} ) q^{93} + ( -1 - 7 \beta_{2} - 2 \beta_{3} + 7 \beta_{5} ) q^{94} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{95} + ( -4 + 6 \beta_{2} - 3 \beta_{3} + 6 \beta_{5} ) q^{96} + ( \beta_{1} + \beta_{4} - 3 \beta_{7} ) q^{97} + ( 5 \beta_{1} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{4} + 4q^{5} + 6q^{6} - 4q^{9} + O(q^{10}) \) \( 8q - 2q^{4} + 4q^{5} + 6q^{6} - 4q^{9} + 4q^{14} - 8q^{15} - 22q^{16} - 12q^{19} - 4q^{20} + 4q^{21} - 2q^{24} - 8q^{25} + 10q^{26} - 24q^{29} - 22q^{30} + 14q^{31} - 8q^{34} - 14q^{35} + 20q^{36} - 30q^{39} - 24q^{40} + 34q^{41} + 6q^{45} + 24q^{46} + 30q^{49} - 16q^{50} + 54q^{51} - 20q^{54} - 10q^{56} + 6q^{59} + 34q^{60} + 20q^{61} - 14q^{64} - 20q^{65} - 32q^{69} + 8q^{70} - 42q^{71} + 4q^{74} - 20q^{75} + 28q^{76} - 16q^{79} - 28q^{80} - 36q^{81} + 6q^{84} + 4q^{85} + 46q^{86} + 12q^{89} - 46q^{90} + 20q^{91} + 42q^{94} - 26q^{95} - 8q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 9 x^{6} + 27 x^{4} + 31 x^{2} + 11\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} + 5 \nu^{2} + 4 \)
\(\beta_{4}\)\(=\)\( -\nu^{7} - 7 \nu^{5} - 13 \nu^{3} - 5 \nu \)
\(\beta_{5}\)\(=\)\( \nu^{6} + 7 \nu^{4} + 14 \nu^{2} + 8 \)
\(\beta_{6}\)\(=\)\( \nu^{7} + 7 \nu^{5} + 14 \nu^{3} + 8 \nu \)
\(\beta_{7}\)\(=\)\( \nu^{7} + 8 \nu^{5} + 19 \nu^{3} + 12 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} - 2\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{4} - 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{3} - 5 \beta_{2} + 6\)
\(\nu^{5}\)\(=\)\(\beta_{7} - 6 \beta_{6} - 5 \beta_{4} + 11 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{5} - 7 \beta_{3} + 21 \beta_{2} - 22\)
\(\nu^{7}\)\(=\)\(-7 \beta_{7} + 29 \beta_{6} + 21 \beta_{4} - 43 \beta_{1}\)

Character values

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

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
2.02368i
1.65458i
1.23399i
0.802699i
0.802699i
1.23399i
1.65458i
2.02368i
2.02368i 2.62059i −2.09529 −0.294963 + 2.21653i 5.30325 0.965823i 0.192845i −3.86752 4.48555 + 0.596911i
364.2 1.65458i 1.97479i −0.737640 2.19353 0.434096i −3.26745 2.24307i 2.08868i −0.899788 −0.718246 3.62937i
364.3 1.23399i 0.363982i 0.477260 1.29496 1.82293i −0.449152 2.58558i 3.05692i 2.86752 −2.24948 1.59798i
364.4 0.802699i 1.76074i 1.35567 −1.19353 1.89090i 1.41335 0.592103i 2.69360i −0.100212 −1.51782 + 0.958043i
364.5 0.802699i 1.76074i 1.35567 −1.19353 + 1.89090i 1.41335 0.592103i 2.69360i −0.100212 −1.51782 0.958043i
364.6 1.23399i 0.363982i 0.477260 1.29496 + 1.82293i −0.449152 2.58558i 3.05692i 2.86752 −2.24948 + 1.59798i
364.7 1.65458i 1.97479i −0.737640 2.19353 + 0.434096i −3.26745 2.24307i 2.08868i −0.899788 −0.718246 + 3.62937i
364.8 2.02368i 2.62059i −2.09529 −0.294963 2.21653i 5.30325 0.965823i 0.192845i −3.86752 4.48555 0.596911i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.8
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 605.2.b.g 8
5.b even 2 1 inner 605.2.b.g 8
5.c odd 4 2 3025.2.a.bl 8
11.b odd 2 1 605.2.b.f 8
11.c even 5 2 55.2.j.a 16
11.c even 5 2 605.2.j.h 16
11.d odd 10 2 605.2.j.d 16
11.d odd 10 2 605.2.j.g 16
33.h odd 10 2 495.2.ba.a 16
44.h odd 10 2 880.2.cd.c 16
55.d odd 2 1 605.2.b.f 8
55.e even 4 2 3025.2.a.bk 8
55.h odd 10 2 605.2.j.d 16
55.h odd 10 2 605.2.j.g 16
55.j even 10 2 55.2.j.a 16
55.j even 10 2 605.2.j.h 16
55.k odd 20 4 275.2.h.d 16
165.o odd 10 2 495.2.ba.a 16
220.n odd 10 2 880.2.cd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.j.a 16 11.c even 5 2
55.2.j.a 16 55.j even 10 2
275.2.h.d 16 55.k odd 20 4
495.2.ba.a 16 33.h odd 10 2
495.2.ba.a 16 165.o odd 10 2
605.2.b.f 8 11.b odd 2 1
605.2.b.f 8 55.d odd 2 1
605.2.b.g 8 1.a even 1 1 trivial
605.2.b.g 8 5.b even 2 1 inner
605.2.j.d 16 11.d odd 10 2
605.2.j.d 16 55.h odd 10 2
605.2.j.g 16 11.d odd 10 2
605.2.j.g 16 55.h odd 10 2
605.2.j.h 16 11.c even 5 2
605.2.j.h 16 55.j even 10 2
880.2.cd.c 16 44.h odd 10 2
880.2.cd.c 16 220.n odd 10 2
3025.2.a.bk 8 55.e even 4 2
3025.2.a.bl 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} + 9 T_{2}^{6} + 27 T_{2}^{4} + 31 T_{2}^{2} + 11 \)
\( T_{19}^{4} + 6 T_{19}^{3} - 4 T_{19}^{2} - 39 T_{19} + 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 + 31 T^{2} + 27 T^{4} + 9 T^{6} + T^{8} \)
$3$ \( 11 + 91 T^{2} + 62 T^{4} + 14 T^{6} + T^{8} \)
$5$ \( 625 - 500 T + 300 T^{2} - 180 T^{3} + 86 T^{4} - 36 T^{5} + 12 T^{6} - 4 T^{7} + T^{8} \)
$7$ \( 11 + 47 T^{2} + 49 T^{4} + 13 T^{6} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 6875 + 3875 T^{2} + 675 T^{4} + 45 T^{6} + T^{8} \)
$17$ \( 40931 + 15524 T^{2} + 1842 T^{4} + 81 T^{6} + T^{8} \)
$19$ \( ( 11 - 39 T - 4 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$23$ \( 26411 + 16366 T^{2} + 2232 T^{4} + 99 T^{6} + T^{8} \)
$29$ \( ( 11 + 67 T + 48 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$31$ \( ( -1 + T + 5 T^{2} - 7 T^{3} + T^{4} )^{2} \)
$37$ \( 3971 + 15304 T^{2} + 2822 T^{4} + 101 T^{6} + T^{8} \)
$41$ \( ( -1969 + 438 T + 48 T^{2} - 17 T^{3} + T^{4} )^{2} \)
$43$ \( 212531 + 79707 T^{2} + 8319 T^{4} + 173 T^{6} + T^{8} \)
$47$ \( 244211 + 90271 T^{2} + 7351 T^{4} + 191 T^{6} + T^{8} \)
$53$ \( 489731 + 164921 T^{2} + 15237 T^{4} + 249 T^{6} + T^{8} \)
$59$ \( ( -1 + 29 T - 75 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$61$ \( ( 209 + 10 T - 61 T^{2} - 10 T^{3} + T^{4} )^{2} \)
$67$ \( 18491 + 46104 T^{2} + 5736 T^{4} + 149 T^{6} + T^{8} \)
$71$ \( ( -2511 - 432 T + 90 T^{2} + 21 T^{3} + T^{4} )^{2} \)
$73$ \( 1279091 + 671988 T^{2} + 25504 T^{4} + 297 T^{6} + T^{8} \)
$79$ \( ( -319 - 377 T - 52 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$83$ \( 212531 + 50451 T^{2} + 3931 T^{4} + 111 T^{6} + T^{8} \)
$89$ \( ( 1871 + 486 T - 128 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$97$ \( 161051 + 55902 T^{2} + 5588 T^{4} + 162 T^{6} + T^{8} \)
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