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Properties

Label 770.2.bg.f
Level $770$
Weight $2$
Character orbit 770.bg
Analytic conductor $6.148$
Analytic rank $0$
Dimension $72$
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] = [770,2,Mod(81,770)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(770, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 20, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("770.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.bg (of order \(15\), degree \(8\), 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: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} - 4 q^{6} - 2 q^{7} - 18 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 9 q^{2} - 3 q^{3} + 9 q^{4} - 9 q^{5} - 4 q^{6} - 2 q^{7} - 18 q^{8} + 10 q^{9} + 36 q^{10} - 2 q^{11} + 2 q^{12} + 4 q^{13} - 2 q^{14} - 6 q^{15} + 9 q^{16} - 18 q^{17} + 15 q^{18} - 7 q^{19} + 18 q^{20} + 24 q^{21} - 16 q^{22} - 38 q^{23} + 2 q^{24} + 9 q^{25} - 7 q^{26} + 18 q^{27} - 6 q^{28} - 2 q^{29} + 3 q^{30} - 8 q^{31} - 36 q^{32} + 10 q^{33} - 24 q^{34} - 3 q^{35} - 30 q^{36} + 8 q^{37} + 8 q^{38} + 27 q^{39} - 9 q^{40} - 6 q^{41} + 38 q^{42} - 12 q^{43} - 2 q^{44} + 50 q^{45} + 12 q^{46} + 35 q^{47} + 6 q^{48} + 34 q^{49} - 18 q^{50} + 17 q^{51} - 7 q^{52} + 22 q^{53} + 26 q^{54} + 6 q^{55} - 2 q^{56} - 118 q^{57} + q^{58} + 7 q^{59} - 2 q^{60} + 11 q^{61} - 4 q^{62} + 49 q^{63} - 18 q^{64} - 18 q^{65} + 45 q^{66} - 44 q^{67} - 18 q^{68} - 56 q^{69} - 4 q^{70} + 38 q^{71} + 10 q^{72} - 10 q^{73} + 8 q^{74} + 2 q^{75} + 4 q^{76} - 61 q^{77} - 4 q^{78} + 2 q^{79} - 9 q^{80} - 52 q^{81} - 12 q^{82} - 2 q^{84} + 24 q^{85} + 11 q^{86} - 28 q^{87} - 7 q^{88} + 12 q^{89} + 20 q^{90} + 32 q^{91} - 14 q^{92} - 13 q^{93} - 30 q^{94} + 7 q^{95} - 3 q^{96} + 2 q^{97} - 28 q^{98} - 182 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 −0.104528 + 0.994522i −2.08179 − 2.31206i −0.978148 − 0.207912i −0.913545 − 0.406737i 2.51700 − 1.82871i 2.63803 + 0.201940i 0.309017 − 0.951057i −0.698197 + 6.64290i 0.500000 − 0.866025i
81.2 −0.104528 + 0.994522i −1.76906 − 1.96474i −0.978148 − 0.207912i −0.913545 − 0.406737i 2.13890 − 1.55400i −1.92084 + 1.81944i 0.309017 − 0.951057i −0.417047 + 3.96794i 0.500000 − 0.866025i
81.3 −0.104528 + 0.994522i −1.21034 − 1.34422i −0.978148 − 0.207912i −0.913545 − 0.406737i 1.46337 − 1.06320i −1.26771 − 2.32226i 0.309017 − 0.951057i −0.0284169 + 0.270368i 0.500000 − 0.866025i
81.4 −0.104528 + 0.994522i 0.151692 + 0.168472i −0.978148 − 0.207912i −0.913545 − 0.406737i −0.183405 + 0.133251i 2.49520 + 0.879768i 0.309017 − 0.951057i 0.308213 − 2.93245i 0.500000 − 0.866025i
81.5 −0.104528 + 0.994522i 0.158560 + 0.176098i −0.978148 − 0.207912i −0.913545 − 0.406737i −0.191708 + 0.139284i −1.28857 + 2.31076i 0.309017 − 0.951057i 0.307716 − 2.92772i 0.500000 − 0.866025i
81.6 −0.104528 + 0.994522i 0.570903 + 0.634052i −0.978148 − 0.207912i −0.913545 − 0.406737i −0.690254 + 0.501499i 2.53964 + 0.741773i 0.309017 − 0.951057i 0.237494 − 2.25960i 0.500000 − 0.866025i
81.7 −0.104528 + 0.994522i 1.07750 + 1.19669i −0.978148 − 0.207912i −0.913545 − 0.406737i −1.30276 + 0.946511i −0.639673 − 2.56726i 0.309017 − 0.951057i 0.0425361 − 0.404704i 0.500000 − 0.866025i
81.8 −0.104528 + 0.994522i 1.23838 + 1.37536i −0.978148 − 0.207912i −0.913545 − 0.406737i −1.49727 + 1.08783i −2.64362 + 0.106297i 0.309017 − 0.951057i −0.0444448 + 0.422864i 0.500000 − 0.866025i
81.9 −0.104528 + 0.994522i 2.27771 + 2.52965i −0.978148 − 0.207912i −0.913545 − 0.406737i −2.75388 + 2.00081i 1.17021 − 2.37289i 0.309017 − 0.951057i −0.897593 + 8.54003i 0.500000 − 0.866025i
191.1 0.913545 − 0.406737i −3.32959 + 0.707727i 0.669131 − 0.743145i 0.104528 + 0.994522i −2.75388 + 2.00081i −2.34147 − 1.23188i 0.309017 − 0.951057i 7.84468 − 3.49267i 0.500000 + 0.866025i
191.2 0.913545 − 0.406737i −1.81028 + 0.384788i 0.669131 − 0.743145i 0.104528 + 0.994522i −1.49727 + 1.08783i 2.20121 − 1.46788i 0.309017 − 0.951057i 0.388433 − 0.172942i 0.500000 + 0.866025i
191.3 0.913545 − 0.406737i −1.57511 + 0.334800i 0.669131 − 0.743145i 0.104528 + 0.994522i −1.30276 + 0.946511i −0.991491 − 2.45295i 0.309017 − 0.951057i −0.371752 + 0.165514i 0.500000 + 0.866025i
191.4 0.913545 − 0.406737i −0.834556 + 0.177390i 0.669131 − 0.743145i 0.104528 + 0.994522i −0.690254 + 0.501499i −1.61861 + 2.09287i 0.309017 − 0.951057i −2.07562 + 0.924125i 0.500000 + 0.866025i
191.5 0.913545 − 0.406737i −0.231786 + 0.0492675i 0.669131 − 0.743145i 0.104528 + 0.994522i −0.191708 + 0.139284i 2.40070 + 1.11204i 0.309017 − 0.951057i −2.68934 + 1.19737i 0.500000 + 0.866025i
191.6 0.913545 − 0.406737i −0.221747 + 0.0471337i 0.669131 − 0.743145i 0.104528 + 0.994522i −0.183405 + 0.133251i −1.50154 + 2.17839i 0.309017 − 0.951057i −2.69369 + 1.19931i 0.500000 + 0.866025i
191.7 0.913545 − 0.406737i 1.76930 − 0.376076i 0.669131 − 0.743145i 0.104528 + 0.994522i 1.46337 − 1.06320i −0.339391 − 2.62389i 0.309017 − 0.951057i 0.248354 − 0.110574i 0.500000 + 0.866025i
191.8 0.913545 − 0.406737i 2.58605 − 0.549681i 0.669131 − 0.743145i 0.104528 + 0.994522i 2.13890 − 1.55400i 2.62343 + 0.342920i 0.309017 − 0.951057i 3.64486 − 1.62280i 0.500000 + 0.866025i
191.9 0.913545 − 0.406737i 3.04320 − 0.646852i 0.669131 − 0.743145i 0.104528 + 0.994522i 2.51700 − 1.82871i −2.01552 + 1.71397i 0.309017 − 0.951057i 6.10202 − 2.71679i 0.500000 + 0.866025i
291.1 0.669131 − 0.743145i −2.96256 + 1.31902i −0.104528 − 0.994522i 0.978148 − 0.207912i −1.00212 + 3.08421i 0.527706 + 2.59259i −0.809017 − 0.587785i 5.02959 − 5.58592i 0.500000 − 0.866025i
291.2 0.669131 − 0.743145i −2.07433 + 0.923553i −0.104528 − 0.994522i 0.978148 − 0.207912i −0.701667 + 2.15951i 0.216838 − 2.63685i −0.809017 − 0.587785i 1.44252 − 1.60208i 0.500000 − 0.866025i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.bg.f ✓ 72
7.c even 3 1 inner 770.2.bg.f ✓ 72
11.c even 5 1 inner 770.2.bg.f ✓ 72
77.m even 15 1 inner 770.2.bg.f ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bg.f ✓ 72 1.a even 1 1 trivial
770.2.bg.f ✓ 72 7.c even 3 1 inner
770.2.bg.f ✓ 72 11.c even 5 1 inner
770.2.bg.f ✓ 72 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} + 3 T_{3}^{71} - 14 T_{3}^{70} - 49 T_{3}^{69} + 36 T_{3}^{68} + 162 T_{3}^{67} + \cdots + 907401035776 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display