Properties

Label 77.4.n.a
Level $77$
Weight $4$
Character orbit 77.n
Analytic conductor $4.543$
Analytic rank $0$
Dimension $176$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [77,4,Mod(17,77)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(77, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([5, 27]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.n (of order \(30\), degree \(8\), 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: \(4.54314707044\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(22\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 5 q^{2} - 9 q^{3} - 93 q^{4} + 15 q^{5} - 100 q^{8} - 117 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 5 q^{2} - 9 q^{3} - 93 q^{4} + 15 q^{5} - 100 q^{8} - 117 q^{9} - 18 q^{11} + 24 q^{12} + 166 q^{14} - 36 q^{15} + 295 q^{16} - 45 q^{17} - 230 q^{18} - 15 q^{19} + 1128 q^{22} + 16 q^{23} - 1365 q^{24} - 97 q^{25} - 57 q^{26} + 350 q^{28} - 720 q^{29} + 265 q^{30} + 1233 q^{31} - 1614 q^{33} + 1800 q^{35} - 2486 q^{36} - 147 q^{37} - 135 q^{38} + 915 q^{39} + 2145 q^{40} - 2760 q^{42} - 646 q^{44} + 2610 q^{45} + 1370 q^{46} + 447 q^{47} - 1634 q^{49} + 1230 q^{50} - 325 q^{51} - 15 q^{52} - 287 q^{53} - 2536 q^{56} + 540 q^{57} + 68 q^{58} - 477 q^{59} - 1785 q^{60} - 2475 q^{61} - 245 q^{63} - 840 q^{64} - 1695 q^{66} - 3620 q^{67} + 465 q^{68} + 4727 q^{70} - 1656 q^{71} + 610 q^{72} + 2865 q^{73} + 1885 q^{74} + 2313 q^{75} - 597 q^{77} + 11060 q^{78} + 1795 q^{79} + 9495 q^{80} + 5465 q^{81} + 7959 q^{82} - 4145 q^{84} - 13780 q^{85} + 2844 q^{86} - 4849 q^{88} + 1536 q^{89} - 3624 q^{91} - 10386 q^{92} + 2601 q^{93} - 18615 q^{94} - 4775 q^{95} + 7635 q^{96} + 17560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −3.87348 3.48769i 0.846259 + 1.90073i 2.00358 + 19.0628i 0.657021 + 3.09104i 3.35119 10.3139i −3.03387 18.2701i 34.2149 47.0927i 15.1699 16.8479i 8.23565 14.2646i
17.2 −3.63700 3.27477i −3.23523 7.26645i 1.66743 + 15.8645i 1.97468 + 9.29016i −12.0294 + 37.0227i 11.9536 + 14.1461i 22.8750 31.4847i −24.2680 + 26.9523i 23.2412 40.2550i
17.3 −3.33798 3.00553i −1.94365 4.36551i 1.27267 + 12.1086i −3.69623 17.3894i −6.63281 + 20.4137i −17.7850 5.16648i 11.0235 15.1726i 2.78664 3.09487i −39.9264 + 69.1546i
17.4 −3.25272 2.92876i 3.30357 + 7.41994i 1.16631 + 11.0967i −3.29794 15.5156i 10.9857 33.8104i 3.14028 + 18.2521i 8.12421 11.1820i −26.0755 + 28.9597i −34.7141 + 60.1267i
17.5 −2.74310 2.46990i 0.806254 + 1.81088i 0.587969 + 5.59415i 0.434945 + 2.04626i 2.26104 6.95877i 18.4556 + 1.54614i −5.15296 + 7.09244i 15.4373 17.1449i 3.86094 6.68734i
17.6 −2.29364 2.06520i 4.04421 + 9.08345i 0.159492 + 1.51746i 3.09625 + 14.5667i 9.48319 29.1863i −8.23970 16.5864i −11.7450 + 16.1657i −48.0869 + 53.4059i 22.9815 39.8051i
17.7 −2.16383 1.94832i −0.570022 1.28029i 0.0499798 + 0.475526i 3.23118 + 15.2015i −1.26099 + 3.88092i −15.1390 + 10.6683i −12.8734 + 17.7187i 16.7523 18.6053i 22.6257 39.1889i
17.8 −1.30063 1.17109i −2.30577 5.17884i −0.516047 4.90986i 0.329952 + 1.55230i −3.06595 + 9.43603i 1.71585 18.4406i −13.3085 + 18.3176i −3.43732 + 3.81753i 1.38875 2.40538i
17.9 −1.27371 1.14685i −2.99266 6.72163i −0.529166 5.03467i −3.10847 14.6242i −3.89693 + 11.9935i 16.0251 + 9.28422i −13.1594 + 18.1124i −18.1577 + 20.1662i −12.8125 + 22.1919i
17.10 −1.04448 0.940451i 1.04120 + 2.33857i −0.629744 5.99162i −1.55589 7.31990i 1.11180 3.42177i −15.9430 + 9.42442i −11.5860 + 15.9468i 13.6817 15.1951i −5.25891 + 9.10871i
17.11 −0.101647 0.0915238i 2.43582 + 5.47095i −0.834272 7.93757i −2.91889 13.7323i 0.253127 0.779044i 0.0206170 18.5202i −1.28485 + 1.76845i −5.93152 + 6.58762i −0.960133 + 1.66300i
17.12 0.105220 + 0.0947403i 2.36435 + 5.31043i −0.834132 7.93624i 1.95794 + 9.21137i −0.254335 + 0.782762i 12.6485 + 13.5283i 1.32990 1.83045i −4.54395 + 5.04656i −0.666675 + 1.15471i
17.13 0.631608 + 0.568703i −3.77378 8.47605i −0.760721 7.23778i 1.53376 + 7.21577i 2.43680 7.49970i −14.3667 + 11.6875i 7.63220 10.5048i −39.5355 + 43.9086i −3.13489 + 5.42980i
17.14 0.677422 + 0.609954i −0.922770 2.07258i −0.749370 7.12978i 3.21414 + 15.1213i 0.639070 1.96686i 12.8687 13.3190i 8.12762 11.1867i 14.6225 16.2399i −7.04599 + 12.2040i
17.15 1.76800 + 1.59191i −0.459138 1.03124i −0.244598 2.32720i −2.74908 12.9334i 0.829889 2.55413i 13.9884 + 12.1377i 14.4593 19.9015i 17.2139 19.1179i 15.7284 27.2425i
17.16 1.95261 + 1.75814i −1.42219 3.19430i −0.114589 1.09024i −1.58425 7.45329i 2.83903 8.73764i −16.4147 8.57654i 14.0483 19.3358i 9.88562 10.9791i 10.0105 17.3387i
17.17 2.24857 + 2.02462i 2.64431 + 5.93922i 0.120749 + 1.14885i 1.63909 + 7.71131i −6.07876 + 18.7085i −16.6915 + 8.02459i 12.1735 16.7553i −10.2154 + 11.3453i −11.9269 + 20.6580i
17.18 2.86713 + 2.58158i 3.38904 + 7.61192i 0.719679 + 6.84729i −0.771490 3.62958i −9.93392 + 30.5735i 14.6454 11.3363i 2.52852 3.48021i −28.3892 + 31.5293i 7.15807 12.3981i
17.19 3.11976 + 2.80905i −1.91488 4.30089i 1.00595 + 9.57096i 3.24497 + 15.2664i 6.10744 18.7968i 8.74619 + 16.3250i −4.00652 + 5.51451i 3.23561 3.59351i −32.7605 + 56.7428i
17.20 3.42618 + 3.08494i −3.94032 8.85010i 1.38558 + 13.1829i −2.35545 11.0815i 13.8018 42.4777i 11.0019 14.8983i −14.2421 + 19.6026i −44.7316 + 49.6795i 26.1156 45.2336i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
11.d odd 10 1 inner
77.n even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.n.a 176
7.d odd 6 1 inner 77.4.n.a 176
11.d odd 10 1 inner 77.4.n.a 176
77.n even 30 1 inner 77.4.n.a 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.n.a 176 1.a even 1 1 trivial
77.4.n.a 176 7.d odd 6 1 inner
77.4.n.a 176 11.d odd 10 1 inner
77.4.n.a 176 77.n even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(77, [\chi])\).