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Properties

Label 946.2.t.a
Level $946$
Weight $2$
Character orbit 946.t
Analytic conductor $7.554$
Analytic rank $0$
Dimension $176$
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] = [946,2,Mod(7,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([21, 25]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.t (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: \(7.55384803121\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 44 q^{2} - 44 q^{4} - 5 q^{6} - 15 q^{7} - 44 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 44 q^{2} - 44 q^{4} - 5 q^{6} - 15 q^{7} - 44 q^{8} - 20 q^{9} - 5 q^{11} - 5 q^{13} + 15 q^{14} + 41 q^{15} - 44 q^{16} - 2 q^{17} - 25 q^{18} + 34 q^{19} + 5 q^{22} - 4 q^{23} + 5 q^{24} - 24 q^{25} + 20 q^{26} - 60 q^{27} - 3 q^{29} - 49 q^{30} + 40 q^{31} + 176 q^{32} - 47 q^{33} + 3 q^{34} - 60 q^{35} - 25 q^{36} - 8 q^{37} + 14 q^{38} + 12 q^{39} - 19 q^{43} - 10 q^{44} + 26 q^{46} - 4 q^{47} - 21 q^{49} + 6 q^{50} + 45 q^{51} - 20 q^{52} - 27 q^{53} + 68 q^{55} + 71 q^{57} - 3 q^{58} - 21 q^{59} + 6 q^{60} + 30 q^{61} + 71 q^{63} - 44 q^{64} - 16 q^{65} + 43 q^{66} - 3 q^{67} + 8 q^{68} - 42 q^{69} + 3 q^{71} - 20 q^{72} - 19 q^{73} - 28 q^{74} + 9 q^{76} + 38 q^{77} + 92 q^{78} + 32 q^{79} - 23 q^{81} - 5 q^{82} - 26 q^{83} + 2 q^{85} + 36 q^{86} + 5 q^{88} + 57 q^{89} + 30 q^{91} + 6 q^{92} - 10 q^{93} + 6 q^{94} - 58 q^{95} - 37 q^{97} - 106 q^{98} - 119 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −0.809017 − 0.587785i −0.636377 − 2.99392i 0.309017 + 0.951057i −0.725164 + 1.62875i −1.24494 + 2.79618i −0.238910 − 0.0507819i 0.309017 − 0.951057i −5.81792 + 2.59031i 1.54402 − 0.891442i
7.2 −0.809017 − 0.587785i −0.617735 − 2.90621i 0.309017 + 0.951057i 1.20517 − 2.70685i −1.20847 + 2.71427i −3.61575 − 0.768551i 0.309017 − 0.951057i −5.32385 + 2.37033i −2.56604 + 1.48151i
7.3 −0.809017 − 0.587785i −0.599378 − 2.81985i 0.309017 + 0.951057i 0.127711 − 0.286844i −1.17256 + 2.63361i 3.11044 + 0.661144i 0.309017 − 0.951057i −4.85168 + 2.16011i −0.271923 + 0.156995i
7.4 −0.809017 − 0.587785i −0.499118 − 2.34817i 0.309017 + 0.951057i −1.19686 + 2.68819i −0.976423 + 2.19308i −4.21054 − 0.894978i 0.309017 − 0.951057i −2.52413 + 1.12382i 2.54836 − 1.47129i
7.5 −0.809017 − 0.587785i −0.455192 − 2.14151i 0.309017 + 0.951057i 1.26725 − 2.84630i −0.890490 + 2.00007i 1.93358 + 0.410995i 0.309017 − 0.951057i −1.63823 + 0.729387i −2.69824 + 1.55783i
7.6 −0.809017 − 0.587785i −0.290867 − 1.36842i 0.309017 + 0.951057i −1.00355 + 2.25400i −0.569022 + 1.27804i −2.35239 − 0.500016i 0.309017 − 0.951057i 0.952659 − 0.424151i 2.13676 − 1.23366i
7.7 −0.809017 − 0.587785i −0.274655 − 1.29215i 0.309017 + 0.951057i −0.122824 + 0.275867i −0.537307 + 1.20681i 2.26550 + 0.481546i 0.309017 − 0.951057i 1.14642 − 0.510417i 0.261517 − 0.150987i
7.8 −0.809017 − 0.587785i −0.198366 − 0.933241i 0.309017 + 0.951057i 0.694375 − 1.55959i −0.388063 + 0.871604i −2.29842 − 0.488545i 0.309017 − 0.951057i 1.90905 − 0.849963i −1.47847 + 0.853593i
7.9 −0.809017 − 0.587785i −0.154782 − 0.728191i 0.309017 + 0.951057i −0.915164 + 2.05549i −0.302799 + 0.680098i 0.349041 + 0.0741910i 0.309017 − 0.951057i 2.23433 − 0.994788i 1.94857 − 1.12501i
7.10 −0.809017 − 0.587785i −0.122515 − 0.576387i 0.309017 + 0.951057i −1.58246 + 3.55426i −0.239675 + 0.538319i 4.84535 + 1.02991i 0.309017 − 0.951057i 2.42342 − 1.07898i 3.36938 − 1.94531i
7.11 −0.809017 − 0.587785i −0.00732925 − 0.0344814i 0.309017 + 0.951057i 1.44488 − 3.24526i −0.0143382 + 0.0322041i 0.108803 + 0.0231267i 0.309017 − 0.951057i 2.73950 − 1.21970i −3.07645 + 1.77619i
7.12 −0.809017 − 0.587785i 0.0695351 + 0.327137i 0.309017 + 0.951057i 1.13525 − 2.54982i 0.136031 − 0.305531i 2.70230 + 0.574392i 0.309017 − 0.951057i 2.63845 − 1.17471i −2.41718 + 1.39556i
7.13 −0.809017 − 0.587785i 0.0791260 + 0.372258i 0.309017 + 0.951057i 0.505501 − 1.13537i 0.154794 − 0.347673i −3.40513 − 0.723782i 0.309017 − 0.951057i 2.60832 − 1.16130i −1.07631 + 0.621410i
7.14 −0.809017 − 0.587785i 0.124160 + 0.584127i 0.309017 + 0.951057i −0.795552 + 1.78684i 0.242894 − 0.545548i 1.84422 + 0.392001i 0.309017 − 0.951057i 2.41485 − 1.07516i 1.69389 − 0.977969i
7.15 −0.809017 − 0.587785i 0.348414 + 1.63916i 0.309017 + 0.951057i −0.0366065 + 0.0822195i 0.681600 − 1.53090i −4.09051 − 0.869465i 0.309017 − 0.951057i 0.175192 − 0.0780004i 0.0779427 − 0.0450002i
7.16 −0.809017 − 0.587785i 0.378939 + 1.78277i 0.309017 + 0.951057i −1.75643 + 3.94500i 0.741316 − 1.66502i −2.93928 − 0.624763i 0.309017 − 0.951057i −0.294028 + 0.130910i 3.73979 − 2.15917i
7.17 −0.809017 − 0.587785i 0.406414 + 1.91203i 0.309017 + 0.951057i 0.762919 − 1.71354i 0.795065 − 1.78575i −2.76446 − 0.587604i 0.309017 − 0.951057i −0.750038 + 0.333938i −1.62441 + 0.937854i
7.18 −0.809017 − 0.587785i 0.408426 + 1.92149i 0.309017 + 0.951057i −0.341490 + 0.766999i 0.799001 − 1.79459i 4.50849 + 0.958309i 0.309017 − 0.951057i −0.784684 + 0.349364i 0.727102 − 0.419793i
7.19 −0.809017 − 0.587785i 0.509879 + 2.39879i 0.309017 + 0.951057i −1.24348 + 2.79289i 0.997474 − 2.24036i −0.500265 − 0.106335i 0.309017 − 0.951057i −2.75359 + 1.22598i 2.64761 − 1.52860i
7.20 −0.809017 − 0.587785i 0.565191 + 2.65902i 0.309017 + 0.951057i 1.63374 − 3.66945i 1.10568 − 2.48340i −2.00185 − 0.425507i 0.309017 − 0.951057i −4.01029 + 1.78550i −3.47857 + 2.00835i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
473.s even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.t.a ✓ 176
11.d odd 10 1 946.2.t.b yes 176
43.d odd 6 1 946.2.t.b yes 176
473.s even 30 1 inner 946.2.t.a ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.t.a ✓ 176 1.a even 1 1 trivial
946.2.t.a ✓ 176 473.s even 30 1 inner
946.2.t.b yes 176 11.d odd 10 1
946.2.t.b yes 176 43.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{176} + 43 T_{3}^{174} + 40 T_{3}^{173} + 775 T_{3}^{172} + 1720 T_{3}^{171} + \cdots + 17\!\cdots\!41 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display