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Properties

Label 806.2.cb.a
Level $806$
Weight $2$
Character orbit 806.cb
Analytic conductor $6.436$
Analytic rank $0$
Dimension $608$
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] = [806,2,Mod(137,806)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(806, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([55, 22]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("806.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 806 = 2 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 806.cb (of order \(60\), degree \(16\), 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: \(6.43594240292\)
Analytic rank: \(0\)
Dimension: \(608\)
Relative dimension: \(38\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 608 q - 12 q^{7} - 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 608 q - 12 q^{7} - 80 q^{9} + 8 q^{11} - 4 q^{12} + 20 q^{13} - 24 q^{14} - 76 q^{16} + 24 q^{17} + 32 q^{18} + 4 q^{19} - 112 q^{21} - 12 q^{22} + 60 q^{27} + 32 q^{28} - 12 q^{29} - 16 q^{31} + 92 q^{33} - 56 q^{34} + 4 q^{35} - 4 q^{37} + 108 q^{38} - 120 q^{39} + 8 q^{41} - 48 q^{43} - 48 q^{45} - 64 q^{46} + 8 q^{47} + 24 q^{51} + 4 q^{52} + 52 q^{53} - 148 q^{57} - 12 q^{58} - 4 q^{59} - 12 q^{61} + 36 q^{62} - 204 q^{63} + 128 q^{65} - 32 q^{66} + 52 q^{67} - 180 q^{69} - 48 q^{70} - 8 q^{71} - 32 q^{72} - 4 q^{73} + 116 q^{75} + 52 q^{76} - 144 q^{77} - 92 q^{78} + 28 q^{79} + 220 q^{81} - 56 q^{83} - 48 q^{84} - 104 q^{85} - 12 q^{86} - 32 q^{87} + 128 q^{89} - 104 q^{91} - 172 q^{93} + 8 q^{94} + 84 q^{97} - 72 q^{98} - 240 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.629320 + 0.777146i −3.30036 − 0.346881i −0.207912 − 0.978148i −0.646292 + 2.41199i 2.34656 − 2.34656i 1.50269 + 2.94920i 0.891007 + 0.453990i 7.83758 + 1.66593i −1.46775 − 2.02018i
137.2 −0.629320 + 0.777146i −2.94355 − 0.309380i −0.207912 − 0.978148i 0.135117 − 0.504262i 2.09287 − 2.09287i −2.18822 − 4.29462i 0.891007 + 0.453990i 5.63435 + 1.19762i 0.306853 + 0.422348i
137.3 −0.629320 + 0.777146i −2.29365 − 0.241072i −0.207912 − 0.978148i 0.754881 − 2.81726i 1.63079 − 1.63079i 0.805710 + 1.58129i 0.891007 + 0.453990i 2.26825 + 0.482132i 1.71436 + 2.35961i
137.4 −0.629320 + 0.777146i −2.20059 − 0.231291i −0.207912 − 0.978148i 0.866751 − 3.23476i 1.56462 − 1.56462i 1.28510 + 2.52216i 0.891007 + 0.453990i 1.85464 + 0.394217i 1.96842 + 2.70929i
137.5 −0.629320 + 0.777146i −1.92959 − 0.202808i −0.207912 − 0.978148i −0.669171 + 2.49738i 1.37194 − 1.37194i 0.601728 + 1.18096i 0.891007 + 0.453990i 0.747733 + 0.158935i −1.51971 − 2.09170i
137.6 −0.629320 + 0.777146i −1.86859 − 0.196397i −0.207912 − 0.978148i −0.523279 + 1.95291i 1.32857 − 1.32857i −0.353693 − 0.694162i 0.891007 + 0.453990i 0.518623 + 0.110237i −1.18838 − 1.63567i
137.7 −0.629320 + 0.777146i −1.20751 − 0.126914i −0.207912 − 0.978148i 0.453772 − 1.69350i 0.858539 − 0.858539i −0.767483 − 1.50627i 0.891007 + 0.453990i −1.49248 − 0.317236i 1.03053 + 1.41840i
137.8 −0.629320 + 0.777146i −0.588695 − 0.0618744i −0.207912 − 0.978148i −0.658162 + 2.45629i 0.418563 − 0.418563i −1.40813 − 2.76362i 0.891007 + 0.453990i −2.59171 − 0.550885i −1.49470 − 2.05728i
137.9 −0.629320 + 0.777146i −0.501636 − 0.0527240i −0.207912 − 0.978148i 0.159150 − 0.593955i 0.356664 − 0.356664i 1.31445 + 2.57975i 0.891007 + 0.453990i −2.68558 − 0.570839i 0.361433 + 0.497470i
137.10 −0.629320 + 0.777146i 0.0603854 + 0.00634676i −0.207912 − 0.978148i 1.09517 − 4.08724i −0.0429341 + 0.0429341i −1.71456 − 3.36501i 0.891007 + 0.453990i −2.93084 − 0.622969i 2.48717 + 3.42329i
137.11 −0.629320 + 0.777146i 0.212569 + 0.0223419i −0.207912 − 0.978148i −0.785715 + 2.93233i −0.151137 + 0.151137i 0.558115 + 1.09536i 0.891007 + 0.453990i −2.88976 − 0.614237i −1.78438 − 2.45599i
137.12 −0.629320 + 0.777146i 0.928740 + 0.0976145i −0.207912 − 0.978148i 0.142749 − 0.532745i −0.660335 + 0.660335i −0.880423 − 1.72793i 0.891007 + 0.453990i −2.08141 − 0.442418i 0.324186 + 0.446204i
137.13 −0.629320 + 0.777146i 0.947124 + 0.0995467i −0.207912 − 0.978148i 0.225025 − 0.839805i −0.673406 + 0.673406i 2.38077 + 4.67252i 0.891007 + 0.453990i −2.04731 − 0.435169i 0.511038 + 0.703384i
137.14 −0.629320 + 0.777146i 1.11691 + 0.117392i −0.207912 − 0.978148i 0.543073 − 2.02678i −0.794127 + 0.794127i 0.187044 + 0.367094i 0.891007 + 0.453990i −1.70073 − 0.361501i 1.23333 + 1.69754i
137.15 −0.629320 + 0.777146i 1.47196 + 0.154709i −0.207912 − 0.978148i −1.02139 + 3.81189i −1.04657 + 1.04657i 1.56508 + 3.07165i 0.891007 + 0.453990i −0.791706 − 0.168282i −2.31961 − 3.19267i
137.16 −0.629320 + 0.777146i 2.23170 + 0.234561i −0.207912 − 0.978148i 0.0842571 − 0.314452i −1.58674 + 1.58674i 0.291461 + 0.572024i 0.891007 + 0.453990i 1.99100 + 0.423201i 0.191350 + 0.263371i
137.17 −0.629320 + 0.777146i 2.62609 + 0.276013i −0.207912 − 0.978148i −0.365925 + 1.36565i −1.86716 + 1.86716i −1.80947 − 3.55129i 0.891007 + 0.453990i 3.88573 + 0.825937i −0.831025 − 1.14381i
137.18 −0.629320 + 0.777146i 3.11222 + 0.327108i −0.207912 − 0.978148i −0.707798 + 2.64154i −2.21280 + 2.21280i −0.0448989 − 0.0881190i 0.891007 + 0.453990i 6.64449 + 1.41233i −1.60743 − 2.21244i
137.19 −0.629320 + 0.777146i 3.13194 + 0.329180i −0.207912 − 0.978148i 0.917788 − 3.42523i −2.22681 + 2.22681i 0.780743 + 1.53229i 0.891007 + 0.453990i 6.76622 + 1.43821i 2.08432 + 2.86882i
137.20 0.629320 − 0.777146i −3.13984 − 0.330010i −0.207912 − 0.978148i −0.0143719 + 0.0536367i −2.23243 + 2.23243i 0.292070 + 0.573220i −0.891007 − 0.453990i 6.81523 + 1.44862i 0.0326390 + 0.0449238i
See next 80 embeddings (of 608 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.cc even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 806.2.cb.a ✓ 608
13.f odd 12 1 806.2.ch.a yes 608
31.h odd 30 1 806.2.ch.a yes 608
403.cc even 60 1 inner 806.2.cb.a ✓ 608
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
806.2.cb.a ✓ 608 1.a even 1 1 trivial
806.2.cb.a ✓ 608 403.cc even 60 1 inner
806.2.ch.a yes 608 13.f odd 12 1
806.2.ch.a yes 608 31.h odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(806, [\chi])\).