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Properties

Label 103.4.c.a
Level $103$
Weight $4$
Character orbit 103.c
Analytic conductor $6.077$
Analytic rank $0$
Dimension $50$
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] = [103,4,Mod(46,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 103.c (of order \(3\), degree \(2\), 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: \(6.07719673059\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q - q^{2} + 4 q^{3} - 97 q^{4} - 3 q^{5} - 17 q^{6} + 13 q^{7} - 30 q^{8} + 534 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - q^{2} + 4 q^{3} - 97 q^{4} - 3 q^{5} - 17 q^{6} + 13 q^{7} - 30 q^{8} + 534 q^{9} - 56 q^{10} - 17 q^{11} - 186 q^{12} - 16 q^{13} + 274 q^{14} + 70 q^{15} - 289 q^{16} - 41 q^{17} + 48 q^{18} + 133 q^{19} + 117 q^{20} - 416 q^{21} + 88 q^{22} - 184 q^{23} + 1002 q^{24} - 934 q^{25} - 522 q^{26} - 380 q^{27} + 578 q^{28} - 51 q^{29} + 1076 q^{30} - 216 q^{31} - 471 q^{32} - 182 q^{33} - 58 q^{34} - 27 q^{35} - 1486 q^{36} + 276 q^{37} + 1800 q^{38} + 220 q^{39} + 7 q^{40} - 133 q^{41} - 372 q^{42} + 549 q^{43} - 117 q^{44} - 251 q^{45} + 506 q^{46} + 767 q^{47} - 2082 q^{48} - 440 q^{49} - 1106 q^{50} - 456 q^{51} + 2053 q^{52} - 75 q^{53} - 2473 q^{54} + 201 q^{55} - 1278 q^{56} + 1176 q^{57} - 294 q^{58} + 871 q^{59} + 762 q^{60} - 2732 q^{61} - 603 q^{62} + 1165 q^{63} + 3518 q^{64} - 1392 q^{65} + 7624 q^{66} + 2635 q^{67} + 3025 q^{68} - 2468 q^{69} - 4724 q^{70} - 397 q^{71} - 6040 q^{72} + 3240 q^{73} - 1039 q^{74} - 3356 q^{75} - 2990 q^{76} - 319 q^{77} - 2654 q^{78} - 2340 q^{79} + 5284 q^{80} + 538 q^{81} - 687 q^{82} + 1243 q^{83} - 4271 q^{84} + 1397 q^{85} - 829 q^{86} - 1850 q^{87} + 991 q^{88} + 4068 q^{89} + 7258 q^{90} - 4778 q^{91} + 891 q^{92} - 3704 q^{93} + 5564 q^{94} - 3910 q^{95} - 3443 q^{96} + 2197 q^{97} + 994 q^{98} - 6033 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1 −2.64861 − 4.58754i 2.55193 −10.0303 + 17.3730i 9.53139 − 16.5088i −6.75908 − 11.7071i 4.69486 − 8.13173i 63.8880 −20.4877 −100.980
46.2 −2.64403 − 4.57959i 8.92200 −9.98176 + 17.2889i −7.77657 + 13.4694i −23.5900 − 40.8591i −5.52858 + 9.57578i 63.2637 52.6021 82.2458
46.3 −2.50384 − 4.33679i −5.03280 −8.53847 + 14.7891i −3.87381 + 6.70963i 12.6014 + 21.8262i 3.75920 − 6.51113i 45.4545 −1.67089 38.7976
46.4 −1.95104 − 3.37931i 1.84944 −3.61314 + 6.25814i −0.660396 + 1.14384i −3.60833 − 6.24981i −6.74939 + 11.6903i −3.01911 −23.5796 5.15385
46.5 −1.79965 − 3.11709i −4.88838 −2.47749 + 4.29115i 4.47004 − 7.74234i 8.79738 + 15.2375i −14.4709 + 25.0643i −10.9599 −3.10378 −32.1781
46.6 −1.60256 − 2.77571i 6.20268 −1.13639 + 1.96828i −2.29758 + 3.97952i −9.94015 − 17.2169i 8.45717 − 14.6483i −18.3564 11.4732 14.7280
46.7 −1.50190 − 2.60136i −7.36895 −0.511387 + 0.885748i 8.91504 − 15.4413i 11.0674 + 19.1693i 17.2527 − 29.8826i −20.9581 27.3014 −53.5579
46.8 −1.43400 − 2.48376i 9.01661 −0.112715 + 0.195229i 9.85197 − 17.0641i −12.9298 − 22.3951i −4.88992 + 8.46959i −22.2975 54.2992 −56.5109
46.9 −1.29037 − 2.23498i −9.58079 0.669895 − 1.16029i −7.09400 + 12.2872i 12.3628 + 21.4129i −3.66177 + 6.34237i −24.1036 64.7915 36.6155
46.10 −1.05755 − 1.83173i −1.54456 1.76317 − 3.05389i −6.86406 + 11.8889i 1.63345 + 2.82923i 12.4027 − 21.4820i −24.3794 −24.6143 29.0364
46.11 −0.454311 − 0.786890i 2.82553 3.58720 − 6.21322i −10.1544 + 17.5880i −1.28367 − 2.22338i −17.6352 + 30.5451i −13.7878 −19.0164 18.4531
46.12 −0.249872 − 0.432792i 2.03485 3.87513 − 6.71192i 5.16989 − 8.95451i −0.508454 − 0.880668i 1.82665 − 3.16386i −7.87111 −22.8594 −5.16725
46.13 −0.189714 − 0.328594i −4.93781 3.92802 − 6.80353i 3.76703 − 6.52469i 0.936770 + 1.62253i −6.95955 + 12.0543i −6.01622 −2.61807 −2.85863
46.14 0.209057 + 0.362098i 8.73562 3.91259 − 6.77681i −3.78825 + 6.56143i 1.82624 + 3.16315i 6.07830 − 10.5279i 6.61673 49.3111 −3.16784
46.15 0.523725 + 0.907118i −8.95690 3.45142 − 5.97804i −0.115841 + 0.200643i −4.69095 − 8.12497i −3.84563 + 6.66083i 15.6100 53.2261 −0.242676
46.16 0.654013 + 1.13278i −4.39112 3.14453 − 5.44649i −5.58120 + 9.66692i −2.87185 − 4.97419i 9.80040 − 16.9748i 18.6905 −7.71809 −14.6007
46.17 1.06645 + 1.84715i 6.25306 1.72535 − 2.98840i 4.69133 − 8.12561i 6.66859 + 11.5503i −13.8749 + 24.0320i 24.4233 12.1007 20.0123
46.18 1.48146 + 2.56597i −2.50003 −0.389451 + 0.674550i −5.63613 + 9.76206i −3.70369 − 6.41498i −6.53960 + 11.3269i 21.3955 −20.7499 −33.3988
46.19 1.51411 + 2.62252i 1.58526 −0.585088 + 1.01340i 5.43505 − 9.41379i 2.40027 + 4.15739i 15.2385 − 26.3938i 20.6823 −24.4869 32.9172
46.20 1.67715 + 2.90491i 6.12747 −1.62567 + 2.81574i −2.46278 + 4.26567i 10.2767 + 17.7998i −4.16214 + 7.20904i 15.9285 10.5459 −16.5218
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.4.c.a ✓ 50
103.c even 3 1 inner 103.4.c.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.4.c.a ✓ 50 1.a even 1 1 trivial
103.4.c.a ✓ 50 103.c even 3 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(103, [\chi])\).