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Properties

Label 770.2.bg.e
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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} - 4 q^{6} + 2 q^{7} + 18 q^{8} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 9 q^{2} + 3 q^{3} + 9 q^{4} + 9 q^{5} - 4 q^{6} + 2 q^{7} + 18 q^{8} + 22 q^{9} + 36 q^{10} - 2 q^{12} + 28 q^{13} - 2 q^{14} - 6 q^{15} + 9 q^{16} - 4 q^{17} - 7 q^{18} - q^{19} - 18 q^{20} - 32 q^{21} - 20 q^{22} - 30 q^{23} + 2 q^{24} + 9 q^{25} - 11 q^{26} + 18 q^{27} + 6 q^{28} + 22 q^{29} - 3 q^{30} - 8 q^{31} + 36 q^{32} - 24 q^{33} + 32 q^{34} - 3 q^{35} - 14 q^{36} + 20 q^{37} - 4 q^{38} - 25 q^{39} - 9 q^{40} + 34 q^{41} + 22 q^{42} + 12 q^{43} - 58 q^{45} - 20 q^{46} - 13 q^{47} - 6 q^{48} + 6 q^{49} + 18 q^{50} - 37 q^{51} + 11 q^{52} + 2 q^{53} + 14 q^{54} - 10 q^{55} - 2 q^{56} - 18 q^{57} + 11 q^{58} + 29 q^{59} - 2 q^{60} + 21 q^{61} + 44 q^{62} + 31 q^{63} - 18 q^{64} + 6 q^{65} - q^{66} - 20 q^{67} - 4 q^{68} - 104 q^{69} + 4 q^{70} - 70 q^{71} - 22 q^{72} - 28 q^{73} - 20 q^{74} - 2 q^{75} + 12 q^{76} - 47 q^{77} + 20 q^{78} - 24 q^{79} + 9 q^{80} + 44 q^{81} - 28 q^{82} - 56 q^{83} - 46 q^{84} + 8 q^{85} + 11 q^{86} - 88 q^{87} - 5 q^{88} - 20 q^{89} + 44 q^{90} - 36 q^{91} + 10 q^{92} + 77 q^{93} - 22 q^{94} - q^{95} - 3 q^{96} - 178 q^{97} - 68 q^{98} + 90 q^{99}+O(q^{100}) \) 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 −2.21721 − 2.46246i −0.978148 − 0.207912i 0.913545 + 0.406737i −2.68073 + 1.94767i 0.483074 − 2.60128i −0.309017 + 0.951057i −0.834111 + 7.93603i 0.500000 − 0.866025i
81.2 0.104528 − 0.994522i −1.85952 − 2.06520i −0.978148 − 0.207912i 0.913545 + 0.406737i −2.24826 + 1.63346i −2.50940 + 0.838393i −0.309017 + 0.951057i −0.493674 + 4.69700i 0.500000 − 0.866025i
81.3 0.104528 − 0.994522i −1.07365 − 1.19241i −0.978148 − 0.207912i 0.913545 + 0.406737i −1.29811 + 0.943132i −0.138091 + 2.64215i −0.309017 + 0.951057i 0.0444679 − 0.423084i 0.500000 − 0.866025i
81.4 0.104528 − 0.994522i −0.957195 − 1.06307i −0.978148 − 0.207912i 0.913545 + 0.406737i −1.15730 + 0.840830i 1.99487 − 1.73796i −0.309017 + 0.951057i 0.0996839 − 0.948429i 0.500000 − 0.866025i
81.5 0.104528 − 0.994522i 0.00718702 + 0.00798200i −0.978148 − 0.207912i 0.913545 + 0.406737i 0.00868952 − 0.00631331i 1.85370 + 1.88780i −0.309017 + 0.951057i 0.313573 − 2.98345i 0.500000 − 0.866025i
81.6 0.104528 − 0.994522i 0.497999 + 0.553084i −0.978148 − 0.207912i 0.913545 + 0.406737i 0.602109 − 0.437458i −2.64472 + 0.0737512i −0.309017 + 0.951057i 0.255687 − 2.43270i 0.500000 − 0.866025i
81.7 0.104528 − 0.994522i 1.20541 + 1.33875i −0.978148 − 0.207912i 0.913545 + 0.406737i 1.45741 − 1.05887i −2.40187 + 1.10950i −0.309017 + 0.951057i −0.0256378 + 0.243928i 0.500000 − 0.866025i
81.8 0.104528 − 0.994522i 1.94146 + 2.15621i −0.978148 − 0.207912i 0.913545 + 0.406737i 2.34733 − 1.70544i 0.194710 − 2.63858i −0.309017 + 0.951057i −0.566387 + 5.38881i 0.500000 − 0.866025i
81.9 0.104528 − 0.994522i 2.04197 + 2.26784i −0.978148 − 0.207912i 0.913545 + 0.406737i 2.46886 − 1.79373i 2.08506 + 1.62865i −0.309017 + 0.951057i −0.659864 + 6.27818i 0.500000 − 0.866025i
191.1 −0.913545 + 0.406737i −2.98500 + 0.634480i 0.669131 − 0.743145i −0.104528 − 0.994522i 2.46886 − 1.79373i −0.729555 + 2.54318i −0.309017 + 0.951057i 5.76700 − 2.56763i 0.500000 + 0.866025i
191.2 −0.913545 + 0.406737i −2.83806 + 0.603248i 0.669131 − 0.743145i −0.104528 − 0.994522i 2.34733 − 1.70544i −1.70844 − 2.02021i −0.309017 + 0.951057i 4.95004 − 2.20390i 0.500000 + 0.866025i
191.3 −0.913545 + 0.406737i −1.76210 + 0.374545i 0.669131 − 0.743145i −0.104528 − 0.994522i 1.45741 − 1.05887i 2.59531 − 0.514179i −0.309017 + 0.951057i 0.224066 − 0.0997608i 0.500000 + 0.866025i
191.4 −0.913545 + 0.406737i −0.727984 + 0.154738i 0.669131 − 0.743145i −0.104528 − 0.994522i 0.602109 − 0.437458i 2.18298 − 1.49486i −0.309017 + 0.951057i −2.23462 + 0.994917i 0.500000 + 0.866025i
191.5 −0.913545 + 0.406737i −0.0105061 + 0.00223315i 0.669131 − 0.743145i −0.104528 − 0.994522i 0.00868952 − 0.00631331i −0.390052 + 2.61684i −0.309017 + 0.951057i −2.74053 + 1.22016i 0.500000 + 0.866025i
191.6 −0.913545 + 0.406737i 1.39925 − 0.297419i 0.669131 − 0.743145i −0.104528 − 0.994522i −1.15730 + 0.840830i −2.63543 − 0.233487i −0.309017 + 0.951057i −0.871205 + 0.387886i 0.500000 + 0.866025i
191.7 −0.913545 + 0.406737i 1.56949 − 0.333605i 0.669131 − 0.743145i −0.104528 − 0.994522i −1.29811 + 0.943132i 1.66473 + 2.05637i −0.309017 + 0.951057i −0.388635 + 0.173032i 0.500000 + 0.866025i
191.8 −0.913545 + 0.406737i 2.71828 − 0.577788i 0.669131 − 0.743145i −0.104528 − 0.994522i −2.24826 + 1.63346i 2.52294 − 0.796716i −0.309017 + 0.951057i 4.31456 − 1.92096i 0.500000 + 0.866025i
191.9 −0.913545 + 0.406737i 3.24116 − 0.688930i 0.669131 − 0.743145i −0.104528 − 0.994522i −2.68073 + 1.94767i −1.91981 − 1.82053i −0.309017 + 0.951057i 7.28986 − 3.24565i 0.500000 + 0.866025i
291.1 −0.669131 + 0.743145i −2.93977 + 1.30887i −0.104528 − 0.994522i −0.978148 + 0.207912i 0.994412 − 3.06048i 1.31802 + 2.29409i 0.809017 + 0.587785i 4.92174 − 5.46614i 0.500000 − 0.866025i
291.2 −0.669131 + 0.743145i −1.49289 + 0.664675i −0.104528 − 0.994522i −0.978148 + 0.207912i 0.504985 − 1.55418i −1.80895 − 1.93072i 0.809017 + 0.587785i −0.220478 + 0.244866i 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:
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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.e ✓ 72
7.c even 3 1 inner 770.2.bg.e ✓ 72
11.c even 5 1 inner 770.2.bg.e ✓ 72
77.m even 15 1 inner 770.2.bg.e ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.bg.e ✓ 72 1.a even 1 1 trivial
770.2.bg.e ✓ 72 7.c even 3 1 inner
770.2.bg.e ✓ 72 11.c even 5 1 inner
770.2.bg.e ✓ 72 77.m even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 3 T_{3}^{71} - 20 T_{3}^{70} + 55 T_{3}^{69} + 168 T_{3}^{68} - 268 T_{3}^{67} + \cdots + 959512576 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display