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Properties

Label 76.7.j.a
Level $76$
Weight $7$
Character orbit 76.j
Analytic conductor $17.484$
Analytic rank $0$
Dimension $60$
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] = [76,7,Mod(13,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.13");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.j (of order \(18\), degree \(6\), 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: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(10\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 30 q^{3} - 216 q^{7} + 690 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 30 q^{3} - 216 q^{7} + 690 q^{9} + 1680 q^{11} - 2940 q^{13} + 2496 q^{15} - 5112 q^{17} - 15792 q^{19} - 32076 q^{21} - 19272 q^{23} + 58896 q^{25} - 124830 q^{27} + 126840 q^{29} + 30780 q^{31} - 274470 q^{33} - 226284 q^{35} + 178968 q^{39} + 83394 q^{41} + 418848 q^{43} - 95472 q^{45} - 498696 q^{47} - 744330 q^{49} + 1334538 q^{51} + 458004 q^{53} - 260136 q^{55} - 981984 q^{57} - 523362 q^{59} - 644172 q^{61} + 926832 q^{63} + 1337220 q^{65} + 1719114 q^{67} + 1333800 q^{69} - 1895220 q^{71} - 1189704 q^{73} + 1337256 q^{77} + 147432 q^{79} + 272130 q^{81} + 442800 q^{83} + 1479096 q^{85} + 1300572 q^{87} - 301596 q^{89} + 661008 q^{91} + 3576 q^{93} - 5709984 q^{95} + 1386630 q^{97} + 3822798 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 0 −16.6973 + 45.8755i 0 9.11718 + 51.7061i 0 −320.368 + 554.894i 0 −1267.32 − 1063.40i 0
13.2 0 −13.8876 + 38.1558i 0 −5.33633 − 30.2638i 0 187.863 − 325.388i 0 −704.551 − 591.189i 0
13.3 0 −7.21536 + 19.8240i 0 −40.4206 − 229.237i 0 −48.6163 + 84.2059i 0 217.516 + 182.517i 0
13.4 0 −6.46108 + 17.7517i 0 14.8495 + 84.2156i 0 158.113 − 273.859i 0 285.070 + 239.202i 0
13.5 0 −3.50553 + 9.63135i 0 35.8010 + 203.037i 0 −107.477 + 186.155i 0 477.972 + 401.066i 0
13.6 0 −1.48422 + 4.07785i 0 −10.2967 − 58.3955i 0 −54.2532 + 93.9692i 0 544.020 + 456.487i 0
13.7 0 6.80225 − 18.6890i 0 −6.07240 − 34.4383i 0 −223.008 + 386.261i 0 255.437 + 214.337i 0
13.8 0 9.61067 − 26.4051i 0 26.4812 + 150.182i 0 197.423 − 341.946i 0 −46.4181 − 38.9494i 0
13.9 0 10.8061 − 29.6895i 0 −26.2824 − 149.055i 0 184.362 − 319.324i 0 −206.249 − 173.063i 0
13.10 0 17.6351 − 48.4521i 0 11.8840 + 67.3972i 0 −186.943 + 323.795i 0 −1478.16 − 1240.32i 0
21.1 0 −31.3285 + 37.3358i 0 37.0498 − 13.4850i 0 −148.463 − 257.146i 0 −285.900 − 1621.42i 0
21.2 0 −18.1708 + 21.6552i 0 40.6936 − 14.8113i 0 179.977 + 311.730i 0 −12.1770 − 69.0593i 0
21.3 0 −17.9376 + 21.3772i 0 −219.363 + 79.8415i 0 237.710 + 411.726i 0 −8.63778 − 48.9873i 0
21.4 0 −5.74881 + 6.85117i 0 55.2802 − 20.1204i 0 −190.315 − 329.635i 0 112.700 + 639.153i 0
21.5 0 −5.40108 + 6.43676i 0 −171.496 + 62.4193i 0 −231.676 − 401.275i 0 114.329 + 648.394i 0
21.6 0 1.99278 − 2.37490i 0 187.566 − 68.2686i 0 84.5123 + 146.380i 0 124.921 + 708.460i 0
21.7 0 15.2868 − 18.2181i 0 −90.7408 + 33.0269i 0 −57.6521 − 99.8564i 0 28.3767 + 160.932i 0
21.8 0 18.3534 − 21.8727i 0 −6.18195 + 2.25005i 0 237.753 + 411.801i 0 −14.9788 − 84.9490i 0
21.9 0 24.5774 − 29.2903i 0 −71.7580 + 26.1178i 0 38.3425 + 66.4112i 0 −127.279 − 721.834i 0
21.10 0 31.0369 − 36.9883i 0 186.326 − 67.8172i 0 −265.452 − 459.776i 0 −278.258 − 1578.08i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.7.j.a ✓ 60
19.f odd 18 1 inner 76.7.j.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.7.j.a ✓ 60 1.a even 1 1 trivial
76.7.j.a ✓ 60 19.f odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(76, [\chi])\).