Properties

Label 572.2.bn.a
Level $572$
Weight $2$
Character orbit 572.bn
Analytic conductor $4.567$
Analytic rank $0$
Dimension $640$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(80\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 640q - 15q^{2} - 3q^{4} - 15q^{6} - 78q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 640q - 15q^{2} - 3q^{4} - 15q^{6} - 78q^{9} - 4q^{12} - 20q^{13} - 12q^{14} - 19q^{16} - 10q^{17} - 45q^{20} - 15q^{22} - 15q^{24} + 104q^{25} - 20q^{26} - 15q^{28} - 10q^{29} + 45q^{30} - 24q^{33} + 4q^{36} - 18q^{37} - 14q^{38} - 20q^{40} - 30q^{41} - 13q^{42} - 108q^{45} - 15q^{46} + 35q^{48} - 74q^{49} - 90q^{50} + 15q^{52} - 8q^{53} + 50q^{56} + 27q^{58} - 10q^{61} - 5q^{62} - 84q^{64} - 146q^{66} - 70q^{68} - 26q^{69} - 15q^{72} - 55q^{74} + 44q^{77} + 24q^{78} - 39q^{80} + 30q^{81} - 29q^{82} + 30q^{84} - 30q^{85} - 75q^{88} - 24q^{89} + 30q^{90} + 14q^{92} + 66q^{93} - 5q^{94} - 30q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1 −1.41416 + 0.0119482i −1.52612 + 1.37412i 1.99971 0.0337935i −1.42290 1.95846i 2.14176 1.96147i −3.62797 3.26664i −2.82752 + 0.0716826i 0.127238 1.21059i 2.03562 + 2.75258i
95.2 −1.41338 0.0484849i 1.01203 0.911240i 1.99530 + 0.137055i 0.329599 + 0.453654i −1.47457 + 1.23886i 0.622444 + 0.560451i −2.81347 0.290453i −0.119730 + 1.13915i −0.443854 0.657168i
95.3 −1.41125 0.0914616i 2.42998 2.18797i 1.98327 + 0.258151i −2.46392 3.39130i −3.62944 + 2.86552i −1.52478 1.37292i −2.77528 0.545709i 0.804035 7.64988i 3.16704 + 5.01133i
95.4 −1.40667 + 0.145875i −1.99167 + 1.79330i 1.95744 0.410395i −0.674236 0.928006i 2.54002 2.81312i 0.302590 + 0.272453i −2.69361 + 0.862832i 0.437208 4.15976i 1.08380 + 1.20704i
95.5 −1.39373 0.239824i 0.309391 0.278577i 1.88497 + 0.668501i −2.28550 3.14573i −0.498018 + 0.314062i 2.11116 + 1.90090i −2.46682 1.38377i −0.295468 + 2.81119i 2.43096 + 4.93242i
95.6 −1.38627 0.279745i −0.527942 + 0.475361i 1.84349 + 0.775605i 1.38627 + 1.90804i 0.864849 0.511289i 3.14791 + 2.83439i −2.33860 1.59090i −0.260831 + 2.48164i −1.38798 3.03286i
95.7 −1.38137 0.302995i −0.868428 + 0.781936i 1.81639 + 0.837098i 1.42384 + 1.95975i 1.43655 0.817017i −2.34374 2.11032i −2.25548 1.70670i −0.170842 + 1.62546i −1.37306 3.13856i
95.8 −1.35854 + 0.392895i 2.09762 1.88870i 1.69127 1.06753i 2.14578 + 2.95342i −2.10764 + 3.39002i 0.819536 + 0.737914i −1.87823 + 2.11477i 0.519214 4.93999i −4.07552 3.16927i
95.9 −1.33917 + 0.454570i 0.706415 0.636059i 1.58673 1.21749i −0.839770 1.15584i −0.656874 + 1.17290i 0.861552 + 0.775745i −1.57146 + 2.35170i −0.219134 + 2.08492i 1.65000 + 1.16613i
95.10 −1.32607 + 0.491461i 1.23162 1.10895i 1.51693 1.30343i 0.467022 + 0.642801i −1.08820 + 2.07584i −3.46433 3.11929i −1.37098 + 2.47395i −0.0264818 + 0.251958i −0.935216 0.622876i
95.11 −1.26700 0.628253i 1.54872 1.39447i 1.21060 + 1.59200i 0.336497 + 0.463148i −2.83831 + 0.793815i −3.04914 2.74546i −0.533654 2.77763i 0.140390 1.33572i −0.135369 0.798216i
95.12 −1.26470 0.632879i 1.96015 1.76492i 1.19893 + 1.60080i 1.52195 + 2.09479i −3.59598 + 0.991561i 1.53627 + 1.38326i −0.503167 2.78331i 0.413632 3.93545i −0.599063 3.61249i
95.13 −1.25254 + 0.656612i −1.23162 + 1.10895i 1.13772 1.64487i 0.467022 + 0.642801i 0.814500 2.19770i 3.46433 + 3.11929i −0.345007 + 2.80731i −0.0264818 + 0.251958i −1.00703 0.498483i
95.14 −1.23389 + 0.691028i −0.706415 + 0.636059i 1.04496 1.70530i −0.839770 1.15584i 0.432103 1.27298i −0.861552 0.775745i −0.110954 + 2.82625i −0.219134 + 2.08492i 1.83490 + 0.845879i
95.15 −1.20947 0.732934i −2.33850 + 2.10559i 0.925615 + 1.77292i 1.48866 + 2.04896i 4.37160 0.832678i −0.625058 0.562805i 0.179932 2.82270i 0.721466 6.86429i −0.298727 3.56924i
95.16 −1.20102 + 0.746694i −2.09762 + 1.88870i 0.884895 1.79359i 2.14578 + 2.95342i 1.10900 3.83465i −0.819536 0.737914i 0.276486 + 2.81488i 0.519214 4.93999i −4.78243 1.94487i
95.17 −1.17222 0.791139i −1.67971 + 1.51242i 0.748199 + 1.85478i −0.747543 1.02890i 3.16552 0.444002i 0.696450 + 0.627087i 0.590331 2.76614i 0.220432 2.09727i 0.0622786 + 1.79751i
95.18 −1.10180 0.886582i 0.133436 0.120147i 0.427944 + 1.95368i −1.12761 1.55202i −0.253541 + 0.0140758i −0.344634 0.310310i 1.26059 2.53198i −0.310215 + 2.95150i −0.133590 + 2.70974i
95.19 −1.06851 0.926436i 0.0906924 0.0816598i 0.283431 + 1.97981i −1.37126 1.88738i −0.172559 + 0.00323364i −2.06611 1.86033i 1.53132 2.37803i −0.312029 + 2.96875i −0.283329 + 3.28707i
95.20 −1.04965 + 0.947750i 1.99167 1.79330i 0.203539 1.98962i −0.674236 0.928006i −0.390952 + 3.76995i −0.302590 0.272453i 1.67201 + 2.28131i 0.437208 4.15976i 1.58723 + 0.335076i
See next 80 embeddings (of 640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 563.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
13.e even 6 1 inner
44.g even 10 1 inner
52.i odd 6 1 inner
143.v odd 30 1 inner
572.bn even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.bn.a 640
4.b odd 2 1 inner 572.2.bn.a 640
11.d odd 10 1 inner 572.2.bn.a 640
13.e even 6 1 inner 572.2.bn.a 640
44.g even 10 1 inner 572.2.bn.a 640
52.i odd 6 1 inner 572.2.bn.a 640
143.v odd 30 1 inner 572.2.bn.a 640
572.bn even 30 1 inner 572.2.bn.a 640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.bn.a 640 1.a even 1 1 trivial
572.2.bn.a 640 4.b odd 2 1 inner
572.2.bn.a 640 11.d odd 10 1 inner
572.2.bn.a 640 13.e even 6 1 inner
572.2.bn.a 640 44.g even 10 1 inner
572.2.bn.a 640 52.i odd 6 1 inner
572.2.bn.a 640 143.v odd 30 1 inner
572.2.bn.a 640 572.bn even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(572, [\chi])\).