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Properties

Label 79.3.h.a
Level $79$
Weight $3$
Character orbit 79.h
Analytic conductor $2.153$
Analytic rank $0$
Dimension $288$
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] = [79,3,Mod(3,79)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(79, base_ring=CyclotomicField(78))
 
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("79.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 79 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 79.h (of order \(78\), degree \(24\), 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: \(2.15259408845\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(12\) over \(\Q(\zeta_{78})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{78}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 24 q^{2} - 26 q^{3} - 12 q^{4} - 25 q^{5} - 23 q^{6} - 20 q^{7} - 6 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 24 q^{2} - 26 q^{3} - 12 q^{4} - 25 q^{5} - 23 q^{6} - 20 q^{7} - 6 q^{8} - 54 q^{9} + 42 q^{10} - 123 q^{11} - 26 q^{12} - 31 q^{13} + 169 q^{14} + 104 q^{15} + 28 q^{16} - 26 q^{17} - 319 q^{18} - 42 q^{19} - 100 q^{20} + 6 q^{21} + 142 q^{22} - 43 q^{23} - 231 q^{24} - 175 q^{25} - 111 q^{26} + 247 q^{27} - 185 q^{28} - 47 q^{29} + 187 q^{30} - 353 q^{31} + 160 q^{32} - 26 q^{33} - 248 q^{34} - 317 q^{35} + 3 q^{36} + 14 q^{37} - 24 q^{38} + 411 q^{39} + 499 q^{40} - 26 q^{41} - 349 q^{42} + 232 q^{43} - 665 q^{44} + 29 q^{45} + 578 q^{46} - 36 q^{47} + 210 q^{48} - 178 q^{49} + 372 q^{50} - 108 q^{51} - 346 q^{52} + 584 q^{53} - 866 q^{54} + 82 q^{55} + 120 q^{56} - 338 q^{57} - 26 q^{58} - 237 q^{59} + 1332 q^{60} - 26 q^{61} - 36 q^{62} + 1447 q^{63} + 502 q^{64} + 388 q^{65} - 136 q^{66} + 345 q^{67} + 951 q^{68} + 1027 q^{69} + 1954 q^{70} + 338 q^{71} + 1334 q^{72} + 461 q^{73} - 831 q^{74} + 163 q^{75} + 58 q^{76} - 617 q^{77} - 336 q^{79} - 58 q^{80} - 402 q^{81} + 137 q^{82} - 119 q^{83} - 881 q^{84} - 826 q^{85} - 71 q^{86} - 291 q^{87} - 2310 q^{88} - 244 q^{89} - 3720 q^{90} - 793 q^{91} - 1192 q^{92} - 1547 q^{93} - 650 q^{94} - 769 q^{95} - 286 q^{96} - 2201 q^{97} + 595 q^{98} + 284 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −3.51774 − 0.571688i −1.36674 − 0.0550778i 8.25350 + 2.75542i 1.33953 − 1.00659i 4.77635 + 0.975098i 2.03732 − 3.22176i −14.8357 − 7.78639i −7.10587 − 0.573645i −5.28755 + 2.77512i
3.2 −3.08516 − 0.501388i 4.33497 + 0.174693i 5.47270 + 1.82706i 2.61772 − 1.96709i −13.2865 − 2.71246i −4.19384 + 6.63203i −4.89765 − 2.57048i 9.79066 + 0.790384i −9.06238 + 4.75630i
3.3 −2.18948 − 0.355826i 2.57208 + 0.103651i 0.873080 + 0.291477i −7.54859 + 5.67240i −5.59464 − 1.14215i 0.521531 − 0.824735i 6.04863 + 3.17456i −2.36598 − 0.191002i 18.5459 − 9.73364i
3.4 −2.15325 − 0.349936i −4.71643 − 0.190066i 0.719865 + 0.240326i 0.189824 − 0.142644i 10.0891 + 2.05971i −0.243267 + 0.384695i 6.26053 + 3.28578i 13.2378 + 1.06866i −0.458655 + 0.240721i
3.5 −1.13019 − 0.183673i 1.53769 + 0.0619668i −2.55056 − 0.851502i 2.06270 − 1.55002i −1.72650 − 0.352467i 5.88474 − 9.30597i 6.78165 + 3.55929i −6.61016 − 0.533628i −2.61593 + 1.37295i
3.6 −0.191269 − 0.0310843i −1.16745 − 0.0470465i −3.75853 − 1.25478i −1.42288 + 1.06923i 0.221834 + 0.0452878i −5.23201 + 8.27376i 1.36622 + 0.717046i −7.61010 − 0.614351i 0.305390 − 0.160281i
3.7 0.542375 + 0.0881446i 4.89218 + 0.197148i −3.50774 − 1.17106i 3.24191 − 2.43614i 2.63602 + 0.538148i −0.890896 + 1.40884i −3.74549 − 1.96579i 14.9238 + 1.20477i 1.97306 − 1.03554i
3.8 1.18497 + 0.192577i −4.08877 − 0.164772i −2.42707 − 0.810275i 7.99070 − 6.00462i −4.81335 − 0.982653i 3.51865 − 5.56430i −6.97200 − 3.65919i 7.72010 + 0.623231i 10.6251 − 5.57649i
3.9 1.39994 + 0.227513i −3.87566 − 0.156184i −1.88607 − 0.629661i −6.04018 + 4.53890i −5.39017 − 1.10041i 2.79076 − 4.41323i −7.52054 − 3.94708i 6.02555 + 0.486433i −9.48857 + 4.97999i
3.10 2.41234 + 0.392043i 3.55695 + 0.143340i 1.87153 + 0.624808i −5.33794 + 4.01120i 8.52436 + 1.74026i 2.40238 − 3.79906i −4.38636 − 2.30214i 3.66051 + 0.295507i −14.4495 + 7.58367i
3.11 2.69538 + 0.438042i 0.668546 + 0.0269415i 3.27904 + 1.09471i 2.76927 − 2.08097i 1.79018 + 0.365469i −2.70969 + 4.28504i −1.31306 − 0.689149i −8.52459 − 0.688176i 8.37580 − 4.39596i
3.12 3.75203 + 0.609764i −2.61347 − 0.105319i 9.91177 + 3.30904i −1.14028 + 0.856865i −9.74160 − 1.98876i 3.06803 − 4.85170i 21.7081 + 11.3933i −2.15167 − 0.173701i −4.80085 + 2.51968i
6.1 −2.69202 − 2.80269i −1.50817 − 0.307896i −0.447035 + 11.0931i 0.542679 − 0.0438096i 3.19710 + 5.05581i 3.16570 + 9.48244i 20.6586 − 18.3019i −6.10003 − 2.59898i −1.58369 − 1.40303i
6.2 −2.16347 − 2.25241i 2.30741 + 0.471062i −0.231688 + 5.74928i −6.25117 + 0.504647i −3.93099 − 6.21637i −4.35825 − 13.0545i 4.10020 − 3.63246i −3.17756 − 1.35383i 14.6609 + 12.9884i
6.3 −1.83869 − 1.91428i 5.26186 + 1.07422i −0.122620 + 3.04278i 2.90211 − 0.234283i −7.61859 − 12.0478i 3.18204 + 9.53139i −1.89686 + 1.68047i 18.2534 + 7.77706i −5.78457 − 5.12468i
6.4 −1.63482 − 1.70203i −0.823959 − 0.168212i −0.0631997 + 1.56828i 9.08916 − 0.733753i 1.06072 + 1.67740i −2.30486 − 6.90391i −4.29333 + 3.80355i −7.62920 − 3.25050i −16.1080 − 14.2705i
6.5 −1.45674 − 1.51663i −4.41239 − 0.900795i −0.0170022 + 0.421906i −0.184490 + 0.0148936i 5.06153 + 8.00417i 0.611805 + 1.83258i −5.63157 + 4.98914i 10.3779 + 4.42161i 0.291343 + 0.258107i
6.6 −0.691313 − 0.719733i −0.946095 − 0.193147i 0.120962 − 3.00164i −7.06131 + 0.570048i 0.515034 + 0.814461i 1.71732 + 5.14399i −5.23194 + 4.63509i −7.42202 − 3.16223i 5.29186 + 4.68818i
6.7 −0.0770222 − 0.0801886i 3.09369 + 0.631582i 0.160566 − 3.98440i 1.41615 − 0.114324i −0.187637 − 0.296725i −0.578626 − 1.73320i −0.664771 + 0.588935i 0.892219 + 0.380139i −0.118243 − 0.104754i
6.8 0.711316 + 0.740558i −3.76185 − 0.767988i 0.118608 − 2.94321i 0.369010 − 0.0297896i −2.10713 − 3.33215i −2.86947 − 8.59511i 5.33839 − 4.72940i 5.28193 + 2.25042i 0.284544 + 0.252084i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.h odd 78 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 79.3.h.a ✓ 288
79.h odd 78 1 inner 79.3.h.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
79.3.h.a ✓ 288 1.a even 1 1 trivial
79.3.h.a ✓ 288 79.h odd 78 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(79, [\chi])\).