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Properties

Label 819.2.z.b
Level $819$
Weight $2$
Character orbit 819.z
Analytic conductor $6.540$
Analytic rank $0$
Dimension $214$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [819,2,Mod(394,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.394");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.z (of order \(6\), degree \(2\), 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: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(214\)
Relative dimension: \(107\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 214 q - 2 q^{3} - 204 q^{4} - 12 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 214 q - 2 q^{3} - 204 q^{4} - 12 q^{6} - 4 q^{7} + 4 q^{10} - 4 q^{12} + q^{13} + 8 q^{14} + 3 q^{15} + 200 q^{16} + 21 q^{17} - 6 q^{21} + 7 q^{22} - 8 q^{23} + 6 q^{24} - 198 q^{25} + 10 q^{26} - 14 q^{27} - 2 q^{28} - 11 q^{29} - 43 q^{30} + 3 q^{31} - 9 q^{33} + 15 q^{34} - 23 q^{35} + 14 q^{36} - 18 q^{37} - 12 q^{38} - 31 q^{39} + 8 q^{40} + 21 q^{42} - 4 q^{43} - 21 q^{44} + 27 q^{45} - 15 q^{46} + 3 q^{47} - 40 q^{48} - 8 q^{49} - 11 q^{51} - 12 q^{52} - 15 q^{53} + 39 q^{54} + q^{55} - 6 q^{56} + 24 q^{57} + 24 q^{58} + 3 q^{60} - q^{61} + 33 q^{62} - 21 q^{63} - 170 q^{64} - 4 q^{65} + 3 q^{66} + 9 q^{67} - 72 q^{68} + 14 q^{69} - 21 q^{70} - 39 q^{71} + 12 q^{72} - 45 q^{73} + 12 q^{74} - 37 q^{75} - 31 q^{77} - 40 q^{78} - 10 q^{79} - 60 q^{81} - 20 q^{82} + 45 q^{83} - 63 q^{84} - 39 q^{85} - 14 q^{87} - 16 q^{88} - 36 q^{89} - 68 q^{90} + 37 q^{91} + 59 q^{92} + 48 q^{93} + 28 q^{94} + 6 q^{95} + 12 q^{96} - 96 q^{98} + 42 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
394.1 − 2.76182i −0.992888 + 1.41922i −5.62766 3.80435i 3.91962 + 2.74218i −2.16953 + 1.51431i 10.0189i −1.02835 − 2.81824i 10.5069
394.2 − 2.70221i 0.859815 + 1.50357i −5.30195 − 2.59810i 4.06296 − 2.32340i 2.18755 − 1.48816i 8.92256i −1.52144 + 2.58558i −7.02062
394.3 − 2.69673i 0.196376 − 1.72088i −5.27238 − 1.06611i −4.64076 − 0.529574i −1.74299 − 1.99047i 8.82473i −2.92287 − 0.675880i −2.87502
394.4 − 2.66497i 1.68711 + 0.392011i −5.10208 0.680349i 1.04470 − 4.49609i 0.450689 + 2.60708i 8.26695i 2.69265 + 1.32273i 1.81311
394.5 − 2.66253i −1.24977 − 1.19920i −5.08904 2.74050i −3.19290 + 3.32754i 2.58315 − 0.572127i 8.22466i 0.123841 + 2.99744i 7.29664
394.6 − 2.59517i −1.42990 + 0.977431i −4.73488 − 0.836479i 2.53660 + 3.71084i 1.85385 − 1.88765i 7.09747i 1.08926 − 2.79527i −2.17080
394.7 − 2.58933i −1.21124 − 1.23811i −4.70461 − 1.17887i −3.20586 + 3.13628i −0.766577 + 2.53226i 7.00311i −0.0658180 + 2.99928i −3.05248
394.8 − 2.46182i 1.66579 − 0.474506i −4.06054 − 4.18797i −1.16815 − 4.10086i −2.52911 + 0.776921i 5.07267i 2.54969 − 1.58085i −10.3100
394.9 − 2.44831i 0.682008 + 1.59213i −3.99422 − 0.408972i 3.89802 − 1.66977i −2.62366 − 0.341205i 4.88247i −2.06973 + 2.17169i −1.00129
394.10 − 2.37060i −1.73197 + 0.0168448i −3.61977 − 0.487580i 0.0399323 + 4.10581i −1.60500 − 2.10332i 3.83983i 2.99943 − 0.0583492i −1.15586
394.11 − 2.35048i 1.17794 − 1.26982i −3.52477 2.58894i −2.98470 − 2.76873i 2.25978 + 1.37601i 3.58395i −0.224901 − 2.99156i 6.08526
394.12 − 2.30537i 0.223929 − 1.71751i −3.31474 − 2.56056i −3.95951 − 0.516239i 2.05906 + 1.66141i 3.03096i −2.89971 − 0.769202i −5.90304
394.13 − 2.24730i 0.648956 + 1.60588i −3.05037 3.72791i 3.60890 − 1.45840i 2.15243 − 1.53852i 2.36049i −2.15771 + 2.08429i 8.37773
394.14 − 2.23456i 0.184149 − 1.72223i −2.99328 3.61610i −3.84844 − 0.411493i −2.57385 − 0.612611i 2.21954i −2.93218 − 0.634296i 8.08042
394.15 − 2.13152i 1.45460 − 0.940293i −2.54339 − 1.57543i −2.00426 − 3.10051i 0.889392 − 2.49178i 1.15825i 1.23170 − 2.73549i −3.35807
394.16 − 2.11108i −1.46005 + 0.931800i −2.45664 1.69779i 1.96710 + 3.08228i 2.26000 + 1.37564i 0.964007i 1.26350 − 2.72095i 3.58416
394.17 − 2.09393i 1.37411 − 1.05443i −2.38455 1.40312i −2.20791 − 2.87729i −2.28777 + 1.32896i 0.805228i 0.776354 − 2.89781i 2.93804
394.18 − 2.00990i 0.0696116 + 1.73065i −2.03970 − 0.824644i 3.47844 − 0.139912i −0.243919 + 2.63448i 0.0797908i −2.99031 + 0.240947i −1.65745
394.19 − 2.00952i −1.72468 + 0.159638i −2.03819 − 1.10872i 0.320796 + 3.46578i −1.19498 + 2.36051i 0.0767405i 2.94903 − 0.550647i −2.22799
394.20 − 1.96652i −1.66992 − 0.459757i −1.86721 − 3.73480i −0.904122 + 3.28393i 2.62978 + 0.290271i − 0.261140i 2.57725 + 1.53551i −7.34456
See next 80 embeddings (of 214 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 394.107
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
819.z even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.z.b ✓ 214
7.c even 3 1 819.2.dj.b yes 214
9.c even 3 1 819.2.dr.b yes 214
13.e even 6 1 819.2.cv.b yes 214
63.g even 3 1 819.2.cv.b yes 214
91.u even 6 1 819.2.dr.b yes 214
117.r even 6 1 819.2.dj.b yes 214
819.z even 6 1 inner 819.2.z.b ✓ 214
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
819.2.z.b ✓ 214 1.a even 1 1 trivial
819.2.z.b ✓ 214 819.z even 6 1 inner
819.2.cv.b yes 214 13.e even 6 1
819.2.cv.b yes 214 63.g even 3 1
819.2.dj.b yes 214 7.c even 3 1
819.2.dj.b yes 214 117.r even 6 1
819.2.dr.b yes 214 9.c even 3 1
819.2.dr.b yes 214 91.u even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{214} + 316 T_{2}^{212} + 49144 T_{2}^{210} + 5014351 T_{2}^{208} + 377569192 T_{2}^{206} + \cdots + 216397886119947 \) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\). Copy content Toggle raw display