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Properties

Label 103.3.h.a
Level $103$
Weight $3$
Character orbit 103.h
Analytic conductor $2.807$
Analytic rank $0$
Dimension $512$
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] = [103,3,Mod(5,103)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(103, base_ring=CyclotomicField(102))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("103.5");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 103 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 103.h (of order \(102\), degree \(32\), 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: \(2.80654672291\)
Analytic rank: \(0\)
Dimension: \(512\)
Relative dimension: \(16\) over \(\Q(\zeta_{102})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{102}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 512 q - 33 q^{2} - 34 q^{3} - 5 q^{4} - 31 q^{5} - 31 q^{6} - 39 q^{7} - 44 q^{8} + 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 512 q - 33 q^{2} - 34 q^{3} - 5 q^{4} - 31 q^{5} - 31 q^{6} - 39 q^{7} - 44 q^{8} + 82 q^{9} + 34 q^{10} - 58 q^{11} - 262 q^{12} - 50 q^{13} + 216 q^{14} - 67 q^{15} - 29 q^{16} - 30 q^{17} - 56 q^{18} + 35 q^{19} - 43 q^{20} - 213 q^{21} - 34 q^{22} - 185 q^{23} - 34 q^{24} + 249 q^{25} - 50 q^{26} - 34 q^{27} + 418 q^{28} - 103 q^{29} - 594 q^{30} - 34 q^{31} + 67 q^{32} - 55 q^{33} - 638 q^{34} - 133 q^{35} + 148 q^{36} + 323 q^{37} + 265 q^{38} - 578 q^{39} + 113 q^{40} + 4 q^{41} - 34 q^{42} - 58 q^{43} + 143 q^{44} - 601 q^{45} + 732 q^{46} - 30 q^{47} - 242 q^{48} + 227 q^{49} - 583 q^{50} - 76 q^{51} - 7 q^{52} - 259 q^{53} - 139 q^{54} + 727 q^{55} - 944 q^{56} + 15 q^{57} - 50 q^{58} - 217 q^{59} + 1291 q^{60} - 38 q^{61} - 565 q^{62} + 190 q^{63} + 1478 q^{64} + 17 q^{65} + 286 q^{66} - 424 q^{67} - 2034 q^{68} - 578 q^{69} - 118 q^{70} + 432 q^{71} + 326 q^{72} + 1649 q^{73} - 121 q^{74} + 1146 q^{75} + 48 q^{76} - 399 q^{77} - 460 q^{78} + 198 q^{79} - 1462 q^{80} - 658 q^{81} + 1485 q^{82} - 466 q^{83} + 2423 q^{84} + 1272 q^{85} + 105 q^{86} + 773 q^{87} + 1695 q^{88} + 561 q^{89} + 3230 q^{90} + 853 q^{91} + 1423 q^{92} + 2074 q^{93} + 986 q^{94} + 952 q^{95} + 2045 q^{96} - 292 q^{97} + 730 q^{98} + 1208 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.815546 − 3.72389i −1.11864 − 2.88753i −9.56842 + 4.40217i 5.38194 + 0.165816i −9.84056 + 6.52059i −12.8404 − 1.58999i 15.0074 + 19.8730i −0.435409 + 0.396928i −3.77174 − 20.1770i
5.2 −0.749227 − 3.42107i 1.93063 + 4.98353i −7.50855 + 3.45448i −6.23571 − 0.192120i 15.6026 − 10.3386i −8.52281 − 1.05535i 9.00156 + 11.9200i −14.4572 + 13.1795i 4.01470 + 21.4768i
5.3 −0.603498 − 2.75566i 0.779442 + 2.01197i −3.59556 + 1.65422i 4.16820 + 0.128421i 5.07391 − 3.36209i 4.70695 + 0.582847i −0.0716647 − 0.0948994i 3.21058 − 2.92683i −2.16161 − 11.5636i
5.4 −0.581945 − 2.65724i −1.42643 − 3.68204i −3.08842 + 1.42090i −4.18943 − 0.129075i −8.95397 + 5.93312i 4.82223 + 0.597121i −0.984236 − 1.30334i −4.87164 + 4.44109i 2.09503 + 11.2074i
5.5 −0.382551 − 1.74678i 0.00968913 + 0.0250105i 0.728966 − 0.335378i −7.25782 − 0.223611i 0.0399813 − 0.0264926i −7.98241 − 0.988437i −5.17517 − 6.85304i 6.65055 − 6.06278i 2.38588 + 12.7634i
5.6 −0.278814 − 1.27311i −1.60142 − 4.13373i 2.09080 − 0.961922i 7.40412 + 0.228119i −4.81618 + 3.19131i 3.69271 + 0.457256i −4.94917 − 6.55377i −7.87212 + 7.17638i −1.77396 − 9.48983i
5.7 −0.0981144 − 0.448004i 1.40155 + 3.61782i 3.44278 − 1.58393i −3.00938 − 0.0927180i 1.48328 − 0.982860i 7.98836 + 0.989173i −2.15292 − 2.85093i −4.47319 + 4.07785i 0.253725 + 1.35731i
5.8 −0.0701122 − 0.320142i 1.83407 + 4.73429i 3.53629 − 1.62695i 7.46186 + 0.229897i 1.38706 − 0.919096i −12.1672 − 1.50662i −1.55880 − 2.06418i −12.3986 + 11.3028i −0.449567 − 2.40497i
5.9 −0.0683083 − 0.311905i −0.202171 − 0.521863i 3.54124 − 1.62923i 1.35167 + 0.0416446i −0.148962 + 0.0987057i −2.06391 − 0.255568i −1.51974 − 2.01246i 6.41961 − 5.85225i −0.0793413 − 0.424438i
5.10 0.199090 + 0.909072i −1.97027 − 5.08585i 2.84709 − 1.30987i −2.78036 − 0.0856618i 4.23114 − 2.80365i −11.2905 − 1.39807i 4.00088 + 5.29802i −15.3328 + 13.9777i −0.475668 − 2.54460i
5.11 0.264478 + 1.20764i −0.695366 − 1.79495i 2.24541 − 1.03305i −5.43330 − 0.167398i 1.98375 − 1.31448i 11.6179 + 1.43861i 4.82148 + 6.38468i 3.91278 − 3.56697i −1.23483 − 6.60576i
5.12 0.417969 + 1.90850i −0.285822 − 0.737792i 0.166171 − 0.0764506i 5.00497 + 0.154202i 1.28861 − 0.853867i −2.98923 − 0.370147i 4.92492 + 6.52165i 6.18844 − 5.64151i 1.79763 + 9.61646i
5.13 0.491855 + 2.24588i 1.43163 + 3.69546i −1.16819 + 0.537453i −2.96019 − 0.0912023i −7.59540 + 5.03289i −0.823930 − 0.102025i 3.76045 + 4.97964i −4.95577 + 4.51778i −1.25115 − 6.69308i
5.14 0.688569 + 3.14410i −1.83623 − 4.73985i −5.77738 + 2.65802i 4.38796 + 0.135191i 13.6382 − 9.03699i 11.5643 + 1.43197i −4.57661 − 6.06040i −12.4433 + 11.3436i 2.59635 + 13.8893i
5.15 0.703367 + 3.21167i −0.721967 − 1.86361i −6.18624 + 2.84612i −9.82716 − 0.302771i 5.47750 − 3.62952i −3.15039 − 0.390103i −5.56667 − 7.37146i 3.69927 − 3.37233i −5.93969 − 31.7745i
5.16 0.791253 + 3.61297i 0.862467 + 2.22628i −8.79362 + 4.04571i 3.43253 + 0.105755i −7.36107 + 4.87762i 0.998760 + 0.123673i −12.6594 − 16.7637i 2.43859 − 2.22307i 2.33391 + 12.4853i
6.1 −3.03022 + 2.14505i −3.74103 + 4.10372i 3.25159 − 9.22735i 6.08407 − 4.03145i 2.53348 − 20.4599i −2.38778 − 2.46248i 5.87606 + 20.6522i −2.01476 − 21.7427i −9.78842 + 25.2668i
6.2 −2.94388 + 2.08393i 0.850104 − 0.932520i 2.99426 − 8.49709i −1.97277 + 1.30720i −0.559300 + 4.51679i 2.15597 + 2.22341i 4.94439 + 17.3777i 0.683499 + 7.37613i 3.08347 − 7.95936i
6.3 −2.18695 + 1.54811i 1.15545 − 1.26747i 1.05669 − 2.99866i 2.67437 − 1.77210i −0.564731 + 4.56065i −8.00798 − 8.25851i −0.601718 − 2.11482i 0.559004 + 6.03262i −3.10530 + 8.01571i
6.4 −2.07382 + 1.46803i 3.85519 − 4.22894i 0.816214 − 2.31625i −0.537226 + 0.355979i −1.78677 + 14.4296i 5.63538 + 5.81168i −1.07370 − 3.77365i −2.19105 − 23.6452i 0.591525 − 1.52690i
See next 80 embeddings (of 512 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
103.h odd 102 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 103.3.h.a ✓ 512
103.h odd 102 1 inner 103.3.h.a ✓ 512
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
103.3.h.a ✓ 512 1.a even 1 1 trivial
103.3.h.a ✓ 512 103.h odd 102 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(103, [\chi])\).