Properties

Label 507.2.s.a
Level $507$
Weight $2$
Character orbit 507.s
Analytic conductor $4.048$
Analytic rank $0$
Dimension $1392$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(1392\)
Relative dimension: \(58\) over \(\Q(\zeta_{52})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{52}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1392q - 22q^{3} - 52q^{4} - 22q^{6} - 56q^{7} - 22q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 1392q - 22q^{3} - 52q^{4} - 22q^{6} - 56q^{7} - 22q^{9} - 52q^{10} - 26q^{12} - 44q^{13} - 34q^{15} + 48q^{16} - 34q^{18} - 56q^{19} - 22q^{21} - 96q^{22} - 194q^{24} - 52q^{25} - 34q^{27} - 56q^{28} + 78q^{30} - 32q^{31} - 10q^{33} - 52q^{34} - 26q^{36} - 56q^{37} + 70q^{39} - 36q^{40} - 10q^{42} - 52q^{43} - 140q^{45} - 180q^{46} - 2q^{48} - 52q^{49} - 26q^{51} - 64q^{52} - 22q^{54} + 132q^{55} - 22q^{57} - 60q^{58} - 18q^{60} - 92q^{61} + 108q^{63} - 52q^{64} - 124q^{66} - 240q^{67} - 104q^{69} - 268q^{70} - 2q^{72} - 56q^{73} + 104q^{75} - 48q^{76} - 22q^{78} - 12q^{79} + 2q^{81} + 208q^{82} - 22q^{84} - 52q^{85} - 26q^{87} - 260q^{88} - 26q^{90} - 32q^{91} - 176q^{93} - 240q^{94} + 292q^{96} - 80q^{97} - 58q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.504808 + 2.75465i 1.69376 0.362196i −5.46324 2.07193i 2.58865 + 0.156585i 0.142700 + 4.84855i 0.0295242 + 0.0178480i 5.56770 9.21011i 2.73763 1.22694i −1.73811 + 7.05179i
5.2 −0.493960 + 2.69545i −1.66195 0.487770i −5.15143 1.95368i −2.69150 0.162806i 2.13570 4.23877i −3.53808 2.13884i 4.97528 8.23013i 2.52416 + 1.62130i 1.76833 7.17439i
5.3 −0.465956 + 2.54264i −0.218600 + 1.71820i −4.37786 1.66030i 1.00192 + 0.0606047i −4.26690 1.35643i −1.49633 0.904565i 3.58682 5.93332i −2.90443 0.751197i −0.620944 + 2.51927i
5.4 −0.450509 + 2.45835i 0.339251 1.69850i −3.97049 1.50581i −1.53833 0.0930519i 4.02267 + 1.59919i 0.917465 + 0.554627i 2.90459 4.80477i −2.76982 1.15244i 0.921786 3.73983i
5.5 −0.436480 + 2.38179i 1.15819 + 1.28787i −3.61239 1.37000i −0.777816 0.0470492i −3.57296 + 2.19643i 3.90248 + 2.35913i 2.33436 3.86150i −0.317208 + 2.98318i 0.451562 1.83206i
5.6 −0.422557 + 2.30582i 1.72797 + 0.118863i −3.26821 1.23947i −4.34895 0.263063i −1.00424 + 3.93415i −1.68324 1.01756i 1.81349 2.99989i 2.97174 + 0.410783i 2.44426 9.91674i
5.7 −0.407325 + 2.22270i −1.46444 + 0.924888i −2.90444 1.10151i 3.11001 + 0.188121i −1.45925 3.63174i 1.33442 + 0.806684i 1.29330 2.13939i 1.28916 2.70888i −1.68492 + 6.83599i
5.8 −0.400794 + 2.18706i 0.580417 1.63191i −2.75258 1.04392i 0.356016 + 0.0215350i 3.33645 + 1.92347i −1.89154 1.14347i 1.08575 1.79605i −2.32623 1.89437i −0.189788 + 0.769999i
5.9 −0.392585 + 2.14227i −1.02029 1.39965i −2.56516 0.972837i −1.56541 0.0946898i 3.39897 1.63625i 2.97174 + 1.79648i 0.837658 1.38566i −0.918027 + 2.85609i 0.817408 3.31635i
5.10 −0.378713 + 2.06657i −1.55175 0.769460i −2.25726 0.856068i 2.99779 + 0.181333i 2.17781 2.91540i −2.77544 1.67781i 0.450143 0.744627i 1.81586 + 2.38802i −1.51004 + 6.12649i
5.11 −0.367556 + 2.00569i 1.27896 + 1.16801i −2.01765 0.765194i 1.15023 + 0.0695764i −2.81276 + 2.13589i −2.33236 1.40996i 0.166545 0.275500i 0.271498 + 2.98769i −0.562324 + 2.28144i
5.12 −0.320681 + 1.74990i −1.48788 + 0.886686i −1.08928 0.413111i −3.06762 0.185557i −1.07448 2.88799i 1.60183 + 0.968339i −0.768515 + 1.27128i 1.42758 2.63857i 1.30844 5.30853i
5.13 −0.315367 + 1.72090i 1.06756 1.36393i −0.992015 0.376222i 3.73990 + 0.226222i 2.01052 + 2.26731i 2.39720 + 1.44916i −0.849938 + 1.40597i −0.720612 2.91217i −1.56875 + 6.36465i
5.14 −0.308001 + 1.68071i −1.68938 0.382115i −0.859883 0.326111i −0.599116 0.0362399i 1.16255 2.72165i 0.705424 + 0.426444i −0.955005 + 1.57977i 2.70798 + 1.29107i 0.245437 0.995778i
5.15 −0.289269 + 1.57849i −1.18129 + 1.26671i −0.537915 0.204004i −1.19167 0.0720827i −1.65777 2.23107i −3.74112 2.26159i −1.18280 + 1.95659i −0.209095 2.99270i 0.458494 1.86018i
5.16 −0.287822 + 1.57060i 1.61828 0.617401i −0.513895 0.194895i −0.563686 0.0340967i 0.503912 + 2.71936i 2.05136 + 1.24009i −1.19811 + 1.98191i 2.23763 1.99825i 0.215793 0.875508i
5.17 −0.259021 + 1.41343i 0.863683 + 1.50135i −0.0606581 0.0230046i −2.17786 0.131736i −2.34576 + 0.831874i −1.46195 0.883779i −1.43857 + 2.37968i −1.50810 + 2.59338i 0.750309 3.04412i
5.18 −0.250240 + 1.36552i 0.0535762 + 1.73122i 0.0680148 + 0.0257946i 1.85720 + 0.112340i −2.37742 0.360062i 3.00163 + 1.81455i −1.48864 + 2.46251i −2.99426 + 0.185505i −0.618149 + 2.50793i
5.19 −0.230989 + 1.26047i −0.545148 1.64402i 0.334608 + 0.126900i 2.48769 + 0.150478i 2.19816 0.307389i 0.0931907 + 0.0563358i −1.56314 + 2.58575i −2.40563 + 1.79247i −0.764303 + 3.10090i
5.20 −0.205529 + 1.12154i 0.653740 1.60394i 0.654428 + 0.248192i −3.58890 0.217089i 1.66452 + 1.06285i −1.80817 1.09308i −1.59261 + 2.63450i −2.14525 2.09712i 0.981098 3.98047i
See next 80 embeddings (of 1392 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 476.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
169.j odd 52 1 inner
507.s even 52 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.s.a 1392
3.b odd 2 1 inner 507.2.s.a 1392
169.j odd 52 1 inner 507.2.s.a 1392
507.s even 52 1 inner 507.2.s.a 1392
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.s.a 1392 1.a even 1 1 trivial
507.2.s.a 1392 3.b odd 2 1 inner
507.2.s.a 1392 169.j odd 52 1 inner
507.2.s.a 1392 507.s even 52 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(507, [\chi])\).