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Properties

Label 77.4.m.a
Level $77$
Weight $4$
Character orbit 77.m
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(4,77)]
 
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([20, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("77.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 77.m (of order \(15\), 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 \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 176 q - q^{2} - 3 q^{3} + 67 q^{4} - 11 q^{5} - 92 q^{6} + 12 q^{7} + 72 q^{8} + 111 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - q^{2} - 3 q^{3} + 67 q^{4} - 11 q^{5} - 92 q^{6} + 12 q^{7} + 72 q^{8} + 111 q^{9} - 176 q^{10} - 10 q^{11} - 100 q^{12} - 116 q^{13} - 74 q^{14} + 12 q^{15} + 511 q^{16} + 331 q^{17} - 228 q^{18} + 217 q^{19} - 4 q^{20} - 270 q^{21} - 1260 q^{22} - 312 q^{23} + 785 q^{24} + 99 q^{25} - 139 q^{26} - 156 q^{27} - 202 q^{28} + 424 q^{29} + 377 q^{30} - 417 q^{31} + 656 q^{32} + 628 q^{33} + 1200 q^{34} - 2182 q^{35} - 426 q^{36} - 483 q^{37} - 77 q^{38} - 969 q^{39} + 695 q^{40} - 1744 q^{41} + 5104 q^{42} + 1616 q^{43} - 2918 q^{44} - 1358 q^{45} - 1522 q^{46} + 185 q^{47} + 482 q^{48} - 1458 q^{49} + 454 q^{50} + 653 q^{51} + 1545 q^{52} - 895 q^{53} - 4388 q^{54} + 4490 q^{55} - 1080 q^{56} + 2728 q^{57} + 1264 q^{58} - 1903 q^{59} + 2699 q^{60} - 351 q^{61} - 4688 q^{62} - 4729 q^{63} - 12632 q^{64} - 1398 q^{65} - 2513 q^{66} + 3380 q^{67} + 3019 q^{68} + 3678 q^{69} + 4299 q^{70} + 480 q^{71} + 6134 q^{72} + 6065 q^{73} - 927 q^{74} + 2543 q^{75} + 7616 q^{76} + 5553 q^{77} + 10204 q^{78} + 1785 q^{79} - 1909 q^{80} + 2413 q^{81} - 2435 q^{82} - 4844 q^{83} + 5445 q^{84} + 11192 q^{85} - 4054 q^{86} - 8998 q^{87} - 569 q^{88} - 1576 q^{89} - 9974 q^{90} - 1236 q^{91} - 2062 q^{92} - 2727 q^{93} + 2177 q^{94} - 5253 q^{95} - 3003 q^{96} + 6960 q^{97} + 900 q^{98} - 12140 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.568278 + 5.40681i −0.983216 − 1.09197i −21.0855 − 4.48185i −9.50411 − 4.23150i 6.46282 − 4.69552i 13.8137 + 12.3362i 22.7749 − 70.0940i 2.59658 − 24.7048i 28.2799 − 48.9822i
4.2 −0.551188 + 5.24420i 6.14548 + 6.82524i −19.3727 − 4.11779i 8.77277 + 3.90589i −39.1803 + 28.4661i −8.73125 − 16.3330i 19.2367 − 59.2045i −5.99480 + 57.0367i −25.3187 + 43.8533i
4.3 −0.448354 + 4.26581i −2.25201 − 2.50111i −10.1709 − 2.16189i −3.51595 − 1.56540i 11.6789 − 8.48525i −12.9938 − 13.1970i 3.17864 − 9.78285i 1.63826 − 15.5870i 8.25409 − 14.2965i
4.4 −0.430706 + 4.09789i −0.0241478 − 0.0268189i −8.78205 − 1.86668i 19.3803 + 8.62865i 0.120301 − 0.0874041i −5.11662 + 17.7994i 1.24560 − 3.83356i 2.82213 − 26.8508i −43.7065 + 75.7018i
4.5 −0.381286 + 3.62769i −5.27480 − 5.85826i −5.18958 − 1.10308i 5.26133 + 2.34250i 23.2632 − 16.9017i 17.6901 − 5.48266i −3.03720 + 9.34755i −3.67342 + 34.9503i −10.5039 + 18.1933i
4.6 −0.345571 + 3.28788i 3.70401 + 4.11372i −2.86559 − 0.609099i −17.3073 − 7.70570i −14.8054 + 10.7568i −16.8141 + 7.76446i −5.17996 + 15.9423i −0.380732 + 3.62243i 31.3163 − 54.2415i
4.7 −0.309956 + 2.94904i 4.37350 + 4.85726i −0.775559 − 0.164850i −1.74642 − 0.777557i −15.6798 + 11.3921i 16.4645 + 8.48058i −6.60404 + 20.3251i −1.64324 + 15.6343i 2.83436 − 4.90925i
4.8 −0.172632 + 1.64248i −5.44538 − 6.04771i 5.15724 + 1.09621i −5.36561 − 2.38893i 10.8733 − 7.89991i −13.1299 + 13.0616i −6.77360 + 20.8470i −4.10034 + 39.0121i 4.85004 − 8.40051i
4.9 −0.164887 + 1.56880i 2.21270 + 2.45746i 5.39125 + 1.14595i 9.29228 + 4.13719i −4.22009 + 3.06608i 2.80174 − 18.3071i −6.58634 + 20.2707i 1.67923 − 15.9768i −8.02258 + 13.8955i
4.10 −0.0847812 + 0.806640i −1.84507 − 2.04916i 7.18170 + 1.52652i −15.7997 − 7.03449i 1.80936 − 1.31458i 11.1236 − 14.8076i −3.84533 + 11.8347i 2.02750 − 19.2904i 7.01382 − 12.1483i
4.11 −0.0109611 + 0.104288i −0.827916 − 0.919494i 7.81442 + 1.66101i 0.870759 + 0.387687i 0.104967 − 0.0762629i 2.97445 + 18.2798i −0.518111 + 1.59458i 2.66224 − 25.3296i −0.0499755 + 0.0865601i
4.12 0.0556486 − 0.529461i 5.01512 + 5.56986i 7.54795 + 1.60437i 5.64635 + 2.51392i 3.22810 − 2.34535i −18.3329 + 2.62769i 2.58559 − 7.95763i −3.04959 + 29.0149i 1.64523 − 2.84963i
4.13 0.0825489 − 0.785401i −5.02276 − 5.57834i 7.21514 + 1.53363i 18.9844 + 8.45241i −4.79586 + 3.48439i −12.4159 − 13.7421i 3.75242 − 11.5488i −3.06750 + 29.1853i 8.20567 − 14.2126i
4.14 0.210594 − 2.00367i 6.06944 + 6.74079i 3.85484 + 0.819371i −11.6544 − 5.18888i 14.7845 − 10.7416i 18.4839 − 1.16010i 7.43418 − 22.8801i −5.77795 + 54.9735i −12.8512 + 22.2589i
4.15 0.247686 − 2.35658i −1.53059 − 1.69989i 2.33307 + 0.495910i 9.11950 + 4.06026i −4.38502 + 3.18591i 18.1718 + 3.57592i 7.60439 − 23.4039i 2.27534 − 21.6484i 11.8271 − 20.4851i
4.16 0.253253 − 2.40954i 0.295132 + 0.327777i 2.08343 + 0.442846i −12.5228 − 5.57552i 0.864536 − 0.628122i −15.6943 − 9.83299i 7.58422 − 23.3418i 2.80193 − 26.6586i −16.6059 + 28.7623i
4.17 0.343592 − 3.26906i −6.44146 − 7.15397i −2.74352 − 0.583153i −11.0442 − 4.91719i −25.6000 + 18.5995i 17.6279 + 5.67963i 5.27706 − 16.2411i −6.86457 + 65.3120i −19.8693 + 34.4146i
4.18 0.401957 − 3.82437i −3.38886 − 3.76371i −6.63905 − 1.41117i −1.45017 − 0.645656i −15.7560 + 11.4474i −17.8979 + 4.76096i 1.44097 − 4.43485i 0.141123 − 1.34270i −3.05214 + 5.28645i
4.19 0.426331 − 4.05627i 3.03081 + 3.36605i −8.44640 − 1.79534i 3.74775 + 1.66861i 14.9457 − 10.8587i 4.97684 − 17.8390i −0.800457 + 2.46355i 0.677755 − 6.44841i 8.36611 − 14.4905i
4.20 0.437036 − 4.15812i 3.86364 + 4.29100i −9.27381 − 1.97121i 14.9930 + 6.67531i 19.5311 − 14.1901i −3.82327 + 18.1213i −1.91345 + 5.88899i −0.662750 + 6.30565i 34.3092 − 59.4253i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 77.4.m.a ✓ 176
7.c even 3 1 inner 77.4.m.a ✓ 176
11.c even 5 1 inner 77.4.m.a ✓ 176
77.m even 15 1 inner 77.4.m.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.4.m.a ✓ 176 1.a even 1 1 trivial
77.4.m.a ✓ 176 7.c even 3 1 inner
77.4.m.a ✓ 176 11.c even 5 1 inner
77.4.m.a ✓ 176 77.m even 15 1 inner

Hecke kernels

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