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Properties

Label 103.7.b.b
Level $103$
Weight $7$
Character orbit 103.b
Analytic conductor $23.696$
Analytic rank $0$
Dimension $46$
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] = [103,7,Mod(102,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.102");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 103.b (of order \(2\), degree \(1\), 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: \(23.6955706128\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 2 q^{2} + 1278 q^{4} - 810 q^{7} - 1096 q^{8} - 17938 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 2 q^{2} + 1278 q^{4} - 810 q^{7} - 1096 q^{8} - 17938 q^{9} + 2822 q^{13} - 6690 q^{14} + 4488 q^{15} + 36822 q^{16} + 478 q^{17} - 21436 q^{18} - 4650 q^{19} + 14246 q^{23} - 178082 q^{25} + 107000 q^{26} - 107086 q^{28} - 16666 q^{29} + 128416 q^{30} - 264566 q^{32} + 2280 q^{33} + 2840 q^{34} - 852564 q^{36} - 127366 q^{38} - 272394 q^{41} + 31656 q^{46} - 214532 q^{49} + 536876 q^{50} + 334070 q^{52} + 307608 q^{55} - 1074240 q^{56} + 31640 q^{58} + 183558 q^{59} + 215010 q^{60} + 118358 q^{61} + 132158 q^{63} + 1507580 q^{64} - 3093882 q^{66} - 350686 q^{68} - 1888056 q^{72} - 553676 q^{76} - 1485538 q^{79} + 3422742 q^{81} + 4547424 q^{82} + 3191126 q^{83} + 2101612 q^{91} + 2758828 q^{92} - 9030976 q^{93} - 1466602 q^{97} + 11212176 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
102.1 −15.4764 40.9460i 175.520 − 131.999i − 633.698i 19.5425 −1725.94 −947.573 2042.88i
102.2 −15.4764 − 40.9460i 175.520 131.999i 633.698i 19.5425 −1725.94 −947.573 − 2042.88i
102.3 −14.6146 17.1371i 149.586 209.903i − 250.452i −63.1388 −1250.81 435.319 − 3067.65i
102.4 −14.6146 − 17.1371i 149.586 − 209.903i 250.452i −63.1388 −1250.81 435.319 3067.65i
102.5 −13.3808 − 25.6432i 115.046 − 44.5050i 343.127i −160.733 −683.040 71.4270 595.514i
102.6 −13.3808 25.6432i 115.046 44.5050i − 343.127i −160.733 −683.040 71.4270 − 595.514i
102.7 −10.9861 51.3329i 56.6933 131.081i − 563.945i 471.141 80.2715 −1906.06 − 1440.06i
102.8 −10.9861 − 51.3329i 56.6933 − 131.081i 563.945i 471.141 80.2715 −1906.06 1440.06i
102.9 −9.88161 20.5090i 33.6462 − 133.068i − 202.662i 279.663 299.944 308.379 1314.93i
102.10 −9.88161 − 20.5090i 33.6462 133.068i 202.662i 279.663 299.944 308.379 − 1314.93i
102.11 −9.21030 − 42.8069i 20.8296 138.380i 394.264i −385.751 397.612 −1103.43 − 1274.52i
102.12 −9.21030 42.8069i 20.8296 − 138.380i − 394.264i −385.751 397.612 −1103.43 1274.52i
102.13 −7.79948 − 38.2654i −3.16808 − 153.041i 298.450i −439.752 523.876 −735.240 1193.64i
102.14 −7.79948 38.2654i −3.16808 153.041i − 298.450i −439.752 523.876 −735.240 − 1193.64i
102.15 −6.56259 − 14.2361i −20.9324 − 180.445i 93.4254i 81.8632 557.377 526.335 1184.19i
102.16 −6.56259 14.2361i −20.9324 180.445i − 93.4254i 81.8632 557.377 526.335 − 1184.19i
102.17 −4.40757 − 5.70633i −44.5733 − 65.2383i 25.1510i −200.996 478.544 696.438 287.542i
102.18 −4.40757 5.70633i −44.5733 65.2383i − 25.1510i −200.996 478.544 696.438 − 287.542i
102.19 −1.90279 28.0709i −60.3794 34.0768i − 53.4129i 476.704 236.668 −58.9733 − 64.8408i
102.20 −1.90279 − 28.0709i −60.3794 − 34.0768i 53.4129i 476.704 236.668 −58.9733 64.8408i
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 102.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.7.b.b ✓ 46
103.b odd 2 1 inner 103.7.b.b ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.7.b.b ✓ 46 1.a even 1 1 trivial
103.7.b.b ✓ 46 103.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{23} + T_{2}^{22} - 1055 T_{2}^{21} - 830 T_{2}^{20} + 474359 T_{2}^{19} + 295369 T_{2}^{18} + \cdots - 14\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(103, [\chi])\). Copy content Toggle raw display