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Properties

Label 847.2.m.b
Level $847$
Weight $2$
Character orbit 847.m
Analytic conductor $6.763$
Analytic rank $0$
Dimension $340$
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] = [847,2,Mod(78,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.78");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.m (of order \(11\), degree \(10\), 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: \(6.76332905120\)
Analytic rank: \(0\)
Dimension: \(340\)
Relative dimension: \(34\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 340 q - 3 q^{2} + 16 q^{3} - 41 q^{4} - 4 q^{5} - 2 q^{6} - 34 q^{7} - 15 q^{8} + 372 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 340 q - 3 q^{2} + 16 q^{3} - 41 q^{4} - 4 q^{5} - 2 q^{6} - 34 q^{7} - 15 q^{8} + 372 q^{9} - 12 q^{10} - 8 q^{11} + 5 q^{12} + 54 q^{13} - 3 q^{14} - 7 q^{15} - 63 q^{16} - 10 q^{17} - 15 q^{18} - 26 q^{19} - 20 q^{20} - 6 q^{21} - 11 q^{22} + 35 q^{23} + 44 q^{24} - 60 q^{25} - 50 q^{26} + 40 q^{27} - 41 q^{28} + 16 q^{29} - 92 q^{30} - 63 q^{32} - 19 q^{33} + 44 q^{34} - 4 q^{35} - 139 q^{36} + 6 q^{37} - 12 q^{38} - 40 q^{39} - 78 q^{40} - 30 q^{41} - 2 q^{42} - 20 q^{43} + 15 q^{44} - 46 q^{45} - 92 q^{46} - 52 q^{47} - 116 q^{48} - 34 q^{49} + 43 q^{50} + 59 q^{51} - 18 q^{52} - 24 q^{53} - 18 q^{54} + 21 q^{55} + 29 q^{56} + 81 q^{57} + 52 q^{58} - 52 q^{59} + 46 q^{60} - 30 q^{61} - 47 q^{62} - 13 q^{63} - 51 q^{64} + 57 q^{65} - 109 q^{66} - 57 q^{67} - 116 q^{68} - 60 q^{69} + 10 q^{70} + 48 q^{71} - 163 q^{72} - 44 q^{73} - 2 q^{74} - 116 q^{75} + 30 q^{76} - 8 q^{77} - 48 q^{78} - 22 q^{79} - 17 q^{80} + 388 q^{81} - 62 q^{82} + 14 q^{83} + 5 q^{84} - 31 q^{85} - 28 q^{86} + 146 q^{87} + 308 q^{88} - 14 q^{89} - 162 q^{90} + 54 q^{91} - 130 q^{92} - 54 q^{93} + 294 q^{94} - 62 q^{95} - 254 q^{96} - 63 q^{97} + 8 q^{98} - 139 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
78.1 −1.82632 + 2.10769i −0.943531 −0.822265 − 5.71898i −1.12069 + 0.720226i 1.72319 − 1.98867i −0.959493 − 0.281733i 8.86325 + 5.69607i −2.10975 0.528735 − 3.67743i
78.2 −1.74820 + 2.01753i 2.59034 −0.729594 − 5.07444i 3.45296 − 2.21908i −4.52844 + 5.22609i −0.959493 − 0.281733i 7.02172 + 4.51259i 3.70988 −1.55940 + 10.8458i
78.3 −1.64070 + 1.89347i 1.35435 −0.608699 − 4.23360i −2.54045 + 1.63265i −2.22209 + 2.56443i −0.959493 − 0.281733i 4.79949 + 3.08445i −1.16572 1.07675 − 7.48896i
78.4 −1.58387 + 1.82788i −0.816529 −0.547879 − 3.81058i 1.93289 − 1.24220i 1.29327 − 1.49251i −0.959493 − 0.281733i 3.76368 + 2.41877i −2.33328 −0.790862 + 5.50056i
78.5 −1.42162 + 1.64064i 2.46870 −0.386058 − 2.68509i 0.0744161 − 0.0478243i −3.50956 + 4.05025i −0.959493 − 0.281733i 1.30159 + 0.836478i 3.09450 −0.0273291 + 0.190078i
78.6 −1.31368 + 1.51607i −2.82864 −0.288075 − 2.00361i −3.33220 + 2.14148i 3.71593 − 4.28841i −0.959493 − 0.281733i 0.0408604 + 0.0262594i 5.00121 1.13082 − 7.86506i
78.7 −1.26849 + 1.46391i 1.39476 −0.249348 − 1.73425i 0.453698 − 0.291574i −1.76923 + 2.04180i −0.959493 − 0.281733i −0.403980 − 0.259622i −1.05465 −0.148671 + 1.03403i
78.8 −1.24361 + 1.43520i −1.82277 −0.228612 − 1.59003i −1.49325 + 0.959653i 2.26682 − 2.61605i −0.959493 − 0.281733i −0.628833 − 0.404127i 0.322485 0.479724 − 3.33656i
78.9 −0.991759 + 1.14455i −2.40742 −0.0417814 − 0.290596i 2.39066 − 1.53639i 2.38758 − 2.75541i −0.959493 − 0.281733i −2.17405 − 1.39718i 2.79567 −0.612490 + 4.25996i
78.10 −0.976435 + 1.12687i −0.133482 −0.0317722 − 0.220980i 1.81196 − 1.16448i 0.130337 − 0.150417i −0.959493 − 0.281733i −2.22868 − 1.43228i −2.98218 −0.457053 + 3.17887i
78.11 −0.911048 + 1.05140i 2.17341 0.00918498 + 0.0638829i −1.15437 + 0.741867i −1.98008 + 2.28513i −0.959493 − 0.281733i −2.41625 − 1.55283i 1.72370 0.271681 − 1.88959i
78.12 −0.764706 + 0.882518i −0.916792 0.0905672 + 0.629909i −1.77473 + 1.14055i 0.701077 − 0.809086i −0.959493 − 0.281733i −2.58989 − 1.66442i −2.15949 0.350591 − 2.43842i
78.13 −0.524678 + 0.605510i 2.86031 0.193274 + 1.34425i 3.51183 − 2.25692i −1.50074 + 1.73195i −0.959493 − 0.281733i −2.26340 − 1.45460i 5.18138 −0.475993 + 3.31060i
78.14 −0.431479 + 0.497954i −3.42175 0.222846 + 1.54993i 0.152943 − 0.0982904i 1.47641 − 1.70387i −0.959493 − 0.281733i −1.97653 − 1.27024i 8.70834 −0.0170476 + 0.118569i
78.15 −0.314465 + 0.362912i 0.708785 0.251813 + 1.75140i −2.53911 + 1.63179i −0.222888 + 0.257226i −0.959493 − 0.281733i −1.52273 − 0.978599i −2.49762 0.206266 − 1.43461i
78.16 −0.234491 + 0.270617i −1.93807 0.266382 + 1.85273i 2.28735 − 1.46999i 0.454460 − 0.524475i −0.959493 − 0.281733i −1.16631 − 0.749542i 0.756131 −0.138558 + 0.963694i
78.17 −0.0595554 + 0.0687305i −0.261084 0.283453 + 1.97146i 1.63947 − 1.05362i 0.0155490 − 0.0179445i −0.959493 − 0.281733i −0.305394 − 0.196265i −2.93183 −0.0252231 + 0.175430i
78.18 0.0789979 − 0.0911684i 3.10250 0.282559 + 1.96524i −2.61465 + 1.68034i 0.245091 − 0.282850i −0.959493 − 0.281733i 0.404455 + 0.259928i 6.62548 −0.0533586 + 0.371117i
78.19 0.193213 − 0.222980i 0.105122 0.272241 + 1.89348i −0.174930 + 0.112421i 0.0203110 − 0.0234402i −0.959493 − 0.281733i 0.971223 + 0.624167i −2.98895 −0.00873124 + 0.0607271i
78.20 0.402854 − 0.464918i 2.97088 0.230772 + 1.60506i 0.308803 − 0.198456i 1.19683 − 1.38122i −0.959493 − 0.281733i 1.87422 + 1.20449i 5.82613 0.0321368 − 0.223516i
See next 80 embeddings (of 340 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 78.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
121.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.m.b ✓ 340
121.e even 11 1 inner 847.2.m.b ✓ 340
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
847.2.m.b ✓ 340 1.a even 1 1 trivial
847.2.m.b ✓ 340 121.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{340} + 3 T_{2}^{339} + 59 T_{2}^{338} + 177 T_{2}^{337} + 1906 T_{2}^{336} + \cdots + 10\!\cdots\!89 \) acting on \(S_{2}^{\mathrm{new}}(847, [\chi])\). Copy content Toggle raw display