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Properties

Label 770.2.bu.a
Level $770$
Weight $2$
Character orbit 770.bu
Analytic conductor $6.148$
Analytic rank $0$
Dimension $768$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [770,2,Mod(107,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([15, 20, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bu (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: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(768\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 8 q^{5} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 768 q - 8 q^{5} + 20 q^{7} - 4 q^{11} + 24 q^{15} - 96 q^{16} + 40 q^{17} - 24 q^{22} + 8 q^{23} + 16 q^{25} + 8 q^{26} + 20 q^{28} - 84 q^{33} + 160 q^{36} + 40 q^{37} + 40 q^{41} + 20 q^{42} - 32 q^{45} - 80 q^{47} + 120 q^{51} - 40 q^{53} - 8 q^{55} - 16 q^{56} + 160 q^{57} + 48 q^{58} - 160 q^{61} - 100 q^{63} - 32 q^{67} - 40 q^{68} - 68 q^{70} - 32 q^{71} + 40 q^{73} - 112 q^{75} + 152 q^{77} - 12 q^{80} - 24 q^{81} - 16 q^{82} - 160 q^{85} + 12 q^{88} - 320 q^{90} + 92 q^{91} + 16 q^{92} - 40 q^{93} - 20 q^{95} - 56 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −0.777146 − 0.629320i −0.177437 − 3.38569i 0.207912 + 0.978148i −0.473005 − 2.18547i −1.99279 + 2.74284i 0.564668 − 2.58479i 0.453990 − 0.891007i −8.44785 + 0.887905i −1.00777 + 1.99610i
107.2 −0.777146 − 0.629320i −0.146350 − 2.79252i 0.207912 + 0.978148i 0.557302 + 2.16551i −1.64366 + 2.26230i 0.386007 + 2.61744i 0.453990 − 0.891007i −4.79321 + 0.503786i 0.929691 − 2.03364i
107.3 −0.777146 − 0.629320i −0.126242 − 2.40884i 0.207912 + 0.978148i −0.380700 − 2.20342i −1.41782 + 1.95147i −1.63654 + 2.07888i 0.453990 − 0.891007i −2.80300 + 0.294607i −1.09080 + 1.95196i
107.4 −0.777146 − 0.629320i −0.126065 − 2.40546i 0.207912 + 0.978148i −2.08116 + 0.817793i −1.41583 + 1.94873i 2.64076 + 0.162395i 0.453990 − 0.891007i −2.78676 + 0.292900i 2.13202 + 0.674170i
107.5 −0.777146 − 0.629320i −0.125789 − 2.40020i 0.207912 + 0.978148i 1.98688 − 1.02582i −1.41274 + 1.94447i 0.756191 + 2.53538i 0.453990 − 0.891007i −2.76158 + 0.290254i −2.18966 − 0.453178i
107.6 −0.777146 − 0.629320i −0.111954 − 2.13621i 0.207912 + 0.978148i 2.21104 + 0.333610i −1.25736 + 1.73060i −2.58182 − 0.578094i 0.453990 − 0.891007i −1.56730 + 0.164730i −1.50835 − 1.65072i
107.7 −0.777146 − 0.629320i −0.0981813 − 1.87341i 0.207912 + 0.978148i 0.576944 + 2.16036i −1.10267 + 1.51770i −1.07789 − 2.41623i 0.453990 − 0.891007i −0.516464 + 0.0542826i 0.911186 − 2.04199i
107.8 −0.777146 − 0.629320i −0.0949368 − 1.81150i 0.207912 + 0.978148i −1.70636 + 1.44511i −1.06624 + 1.46755i 2.46408 − 0.963484i 0.453990 − 0.891007i −0.288960 + 0.0303709i 2.23553 − 0.0492189i
107.9 −0.777146 − 0.629320i −0.0602150 − 1.14897i 0.207912 + 0.978148i −2.18314 − 0.483623i −0.676275 + 0.930813i −2.30704 + 1.29522i 0.453990 − 0.891007i 1.66706 − 0.175215i 1.39227 + 1.74974i
107.10 −0.777146 − 0.629320i −0.0433710 − 0.827568i 0.207912 + 0.978148i 1.56058 − 1.60144i −0.487100 + 0.670435i 0.947044 − 2.47045i 0.453990 − 0.891007i 2.30058 − 0.241801i −2.22061 + 0.262445i
107.11 −0.777146 − 0.629320i −0.00892362 − 0.170273i 0.207912 + 0.978148i −1.44824 − 1.70370i −0.100221 + 0.137943i 2.39388 + 1.12664i 0.453990 − 0.891007i 2.95465 − 0.310546i 0.0533237 + 2.23543i
107.12 −0.777146 − 0.629320i 0.00338324 + 0.0645561i 0.207912 + 0.978148i 0.912363 + 2.04147i 0.0379972 − 0.0522986i −0.671448 + 2.55913i 0.453990 − 0.891007i 2.97941 − 0.313149i 0.575698 − 2.16069i
107.13 −0.777146 − 0.629320i 0.0203177 + 0.387685i 0.207912 + 0.978148i 1.83542 + 1.27720i 0.228188 − 0.314074i 2.36026 + 1.19549i 0.453990 − 0.891007i 2.83368 − 0.297832i −0.622616 − 2.14764i
107.14 −0.777146 − 0.629320i 0.0298324 + 0.569236i 0.207912 + 0.978148i −0.963434 + 2.01787i 0.335048 − 0.461154i −0.508492 − 2.59643i 0.453990 − 0.891007i 2.66043 − 0.279622i 2.01862 − 0.961870i
107.15 −0.777146 − 0.629320i 0.0304960 + 0.581898i 0.207912 + 0.978148i −2.17858 − 0.503768i 0.342501 − 0.471412i −1.25399 − 2.32970i 0.453990 − 0.891007i 2.64589 − 0.278094i 1.37604 + 1.76253i
107.16 −0.777146 − 0.629320i 0.0578754 + 1.10433i 0.207912 + 0.978148i 1.31735 − 1.80681i 0.649999 − 0.894646i 2.57564 − 0.605038i 0.453990 − 0.891007i 1.76737 − 0.185759i −2.16084 + 0.575121i
107.17 −0.777146 − 0.629320i 0.0622478 + 1.18776i 0.207912 + 0.978148i 2.03094 − 0.935576i 0.699106 − 0.962236i −2.42806 + 1.05095i 0.453990 − 0.891007i 1.57667 − 0.165714i −2.16711 − 0.551030i
107.18 −0.777146 − 0.629320i 0.0854328 + 1.63015i 0.207912 + 0.978148i 0.0157199 − 2.23601i 0.959495 − 1.32063i −2.36707 − 1.18194i 0.453990 − 0.891007i 0.333462 − 0.0350483i −1.41939 + 1.72782i
107.19 −0.777146 − 0.629320i 0.109440 + 2.08823i 0.207912 + 0.978148i −0.664170 − 2.13515i 1.22912 − 1.69173i −0.371690 + 2.61951i 0.453990 − 0.891007i −1.36517 + 0.143486i −0.827538 + 2.07730i
107.20 −0.777146 − 0.629320i 0.118071 + 2.25292i 0.207912 + 0.978148i −1.26889 + 1.84117i 1.32605 − 1.82515i 1.90020 + 1.84099i 0.453990 − 0.891007i −2.07815 + 0.218422i 2.14480 − 0.632320i
See next 80 embeddings (of 768 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
11.d odd 10 1 inner
35.l odd 12 1 inner
55.l even 20 1 inner
77.o odd 30 1 inner
385.bv even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bu.a ✓ 768
5.c odd 4 1 inner 770.2.bu.a ✓ 768
7.c even 3 1 inner 770.2.bu.a ✓ 768
11.d odd 10 1 inner 770.2.bu.a ✓ 768
35.l odd 12 1 inner 770.2.bu.a ✓ 768
55.l even 20 1 inner 770.2.bu.a ✓ 768
77.o odd 30 1 inner 770.2.bu.a ✓ 768
385.bv even 60 1 inner 770.2.bu.a ✓ 768
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bu.a ✓ 768 1.a even 1 1 trivial
770.2.bu.a ✓ 768 5.c odd 4 1 inner
770.2.bu.a ✓ 768 7.c even 3 1 inner
770.2.bu.a ✓ 768 11.d odd 10 1 inner
770.2.bu.a ✓ 768 35.l odd 12 1 inner
770.2.bu.a ✓ 768 55.l even 20 1 inner
770.2.bu.a ✓ 768 77.o odd 30 1 inner
770.2.bu.a ✓ 768 385.bv even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(770, [\chi])\).