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Properties

Label 215.2.x.a
Level $215$
Weight $2$
Character orbit 215.x
Analytic conductor $1.717$
Analytic rank $0$
Dimension $480$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [215,2,Mod(3,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(84))
 
chi = DirichletCharacter(H, H._module([63, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.x (of order \(84\), degree \(24\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(20\) over \(\Q(\zeta_{84})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 28 q^{2} - 22 q^{3} - 22 q^{5} - 20 q^{6} - 30 q^{7} - 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q - 28 q^{2} - 22 q^{3} - 22 q^{5} - 20 q^{6} - 30 q^{7} - 28 q^{8} - 18 q^{10} - 40 q^{11} - 48 q^{12} - 46 q^{13} - 26 q^{15} - 24 q^{16} - 24 q^{17} - 10 q^{18} - 46 q^{20} - 24 q^{21} - 28 q^{22} - 14 q^{23} - 30 q^{25} - 44 q^{26} - 28 q^{27} + 46 q^{28} + 116 q^{30} - 12 q^{31} - 168 q^{32} + 76 q^{33} - 64 q^{35} - 148 q^{36} - 36 q^{37} - 20 q^{38} - 38 q^{40} - 72 q^{41} + 68 q^{43} - 28 q^{45} + 52 q^{46} + 24 q^{47} + 166 q^{48} - 12 q^{50} - 280 q^{51} - 80 q^{52} + 40 q^{53} + 32 q^{55} - 136 q^{56} - 28 q^{57} - 36 q^{58} - 98 q^{60} - 8 q^{61} - 28 q^{62} + 94 q^{63} + 56 q^{65} + 184 q^{66} + 54 q^{67} + 114 q^{68} - 168 q^{70} - 80 q^{71} + 406 q^{72} - 94 q^{73} - 28 q^{75} - 236 q^{76} + 8 q^{77} + 56 q^{78} + 30 q^{80} - 288 q^{81} + 196 q^{82} + 40 q^{83} - 144 q^{86} + 304 q^{87} + 28 q^{88} + 68 q^{90} - 68 q^{91} + 96 q^{92} - 144 q^{93} - 110 q^{95} + 32 q^{96} + 100 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −0.887646 − 2.53675i 0.198401 + 0.230546i −4.08351 + 3.25649i −0.338813 + 2.21025i 0.408727 − 0.707936i −0.680398 + 2.53928i 7.33435 + 4.60848i 0.433338 − 2.87501i 5.90759 − 1.10244i
3.2 −0.759523 − 2.17059i −1.39537 − 1.62144i −2.57093 + 2.05025i −0.925647 − 2.03548i −2.45968 + 4.26029i 0.274047 − 1.02276i 2.50862 + 1.57627i −0.234909 + 1.55852i −3.71515 + 3.55520i
3.3 −0.746624 − 2.13373i 0.202390 + 0.235181i −2.43168 + 1.93920i 2.12678 − 0.690517i 0.350703 − 0.607436i −0.0201265 + 0.0751130i 2.12510 + 1.33529i 0.432778 − 2.87130i −3.06128 − 4.02241i
3.4 −0.535193 − 1.52949i 1.99539 + 2.31868i −0.489256 + 0.390169i 0.115705 + 2.23307i 2.47849 − 4.29287i −0.274715 + 1.02525i −1.88550 − 1.18474i −0.947587 + 6.28683i 3.35355 − 1.37210i
3.5 −0.514048 − 1.46906i −1.67912 − 1.95117i −0.330241 + 0.263358i −1.60561 + 1.55629i −2.00325 + 3.46973i −1.14826 + 4.28535i −2.07904 − 1.30635i −0.540507 + 3.58603i 3.11164 + 1.55873i
3.6 −0.443258 − 1.26676i −1.05297 − 1.22357i 0.155463 − 0.123977i 1.26114 + 1.84649i −1.08323 + 1.87621i 1.01958 − 3.80513i −2.49869 − 1.57003i 0.0587443 − 0.389743i 1.78005 − 2.41603i
3.7 −0.405368 − 1.15848i 1.45993 + 1.69647i 0.385919 − 0.307760i 1.03230 − 1.98352i 1.37351 − 2.37900i 0.279825 − 1.04432i −2.59143 − 1.62830i −0.299491 + 1.98699i −2.71633 − 0.391832i
3.8 −0.313323 − 0.895425i 0.472112 + 0.548604i 0.860048 − 0.685865i −2.19173 + 0.443070i 0.343311 − 0.594632i 0.438889 − 1.63796i −2.49012 − 1.56465i 0.369050 − 2.44849i 1.08346 + 1.82371i
3.9 −0.0608299 − 0.173842i 0.165761 + 0.192618i 1.53714 − 1.22583i 2.06703 + 0.852862i 0.0234018 − 0.0405332i −1.02651 + 3.83098i −0.618499 − 0.388629i 0.437502 − 2.90264i 0.0225258 − 0.411216i
3.10 −0.0434215 − 0.124091i −2.14943 − 2.49768i 1.55015 − 1.23620i 1.59909 − 1.56298i −0.216610 + 0.375179i 0.0586673 − 0.218949i −0.443348 − 0.278574i −1.17125 + 7.77071i −0.263388 − 0.130566i
3.11 −0.0432148 − 0.123501i −0.624336 − 0.725491i 1.55028 − 1.23631i −1.73791 − 1.40701i −0.0626182 + 0.108458i −0.0414457 + 0.154677i −0.441256 − 0.277260i 0.310584 − 2.06059i −0.0986634 + 0.275437i
3.12 0.187365 + 0.535458i 1.75833 + 2.04321i 1.31205 − 1.04633i −0.692469 − 2.12614i −0.764606 + 1.32434i −0.340298 + 1.27001i 1.76678 + 1.11014i −0.635877 + 4.21877i 1.00872 − 0.769153i
3.13 0.270701 + 0.773620i 1.16584 + 1.35473i 1.03845 − 0.828139i −0.817213 + 2.08138i −0.732454 + 1.26865i 0.663021 − 2.47443i 2.30975 + 1.45131i −0.0289874 + 0.192319i −1.83142 − 0.0687786i
3.14 0.466045 + 1.33188i −0.679346 − 0.789414i 0.00695788 − 0.00554873i 0.262688 − 2.22058i 0.734799 − 1.27271i −0.588200 + 2.19519i 2.40020 + 1.50814i 0.285463 − 1.89392i 3.07997 − 0.685023i
3.15 0.529382 + 1.51289i −0.301191 − 0.349990i −0.444920 + 0.354812i 2.12745 − 0.688451i 0.370051 − 0.640947i 0.927283 − 3.46067i 1.94199 + 1.22023i 0.415350 − 2.75566i 2.16778 + 2.85414i
3.16 0.616290 + 1.76126i 0.0383317 + 0.0445423i −1.15855 + 0.923912i −1.72480 + 1.42304i −0.0548269 + 0.0949630i −1.01557 + 3.79015i 0.818672 + 0.514406i 0.446612 − 2.96308i −3.56933 − 2.16081i
3.17 0.683038 + 1.95201i −1.62979 − 1.89385i −1.78014 + 1.41962i −2.16306 − 0.566733i 2.58360 − 4.47493i 1.13320 − 4.22914i −0.484854 − 0.304654i −0.483323 + 3.20664i −0.371180 − 4.60941i
3.18 0.754785 + 2.15705i −1.97486 − 2.29483i −2.51951 + 2.00924i 1.55742 + 1.60450i 3.45947 − 5.99197i −0.646699 + 2.41352i −2.36570 − 1.48647i −0.919037 + 6.09741i −2.28548 + 4.57050i
3.19 0.779570 + 2.22788i 1.55695 + 1.80921i −2.79207 + 2.22661i −1.83714 − 1.27472i −2.81696 + 4.87912i 0.477255 − 1.78114i −3.14013 − 1.97307i −0.402020 + 2.66723i 1.40776 − 5.08667i
3.20 0.870593 + 2.48801i 0.633174 + 0.735762i −3.86861 + 3.08511i 1.45525 + 1.69772i −1.27935 + 2.21589i 0.288853 − 1.07801i −6.57997 − 4.13447i 0.306691 − 2.03476i −2.95702 + 5.09870i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
43.h odd 42 1 inner
215.x even 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.x.a ✓ 480
5.c odd 4 1 inner 215.2.x.a ✓ 480
43.h odd 42 1 inner 215.2.x.a ✓ 480
215.x even 84 1 inner 215.2.x.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.x.a ✓ 480 1.a even 1 1 trivial
215.2.x.a ✓ 480 5.c odd 4 1 inner
215.2.x.a ✓ 480 43.h odd 42 1 inner
215.2.x.a ✓ 480 215.x even 84 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(215, [\chi])\).