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Properties

Label 946.2.z.a
Level $946$
Weight $2$
Character orbit 946.z
Analytic conductor $7.554$
Analytic rank $0$
Dimension $528$
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(39,946)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(946, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([63, 55]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("946.39");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 946 = 2 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 946.z (of order \(70\), 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: \(7.55384803121\)
Analytic rank: \(0\)
Dimension: \(528\)
Relative dimension: \(22\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 528 q - 22 q^{2} + 22 q^{4} - 5 q^{6} - 22 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 528 q - 22 q^{2} + 22 q^{4} - 5 q^{6} - 22 q^{8} - 20 q^{9} + 7 q^{11} - 12 q^{13} + 35 q^{14} - 20 q^{15} + 22 q^{16} + 24 q^{17} + 25 q^{18} - 55 q^{19} + 12 q^{22} + 18 q^{23} - 2 q^{24} - 6 q^{25} + 43 q^{26} + 15 q^{27} - 8 q^{29} - 8 q^{30} + 4 q^{31} + 88 q^{32} + 115 q^{33} - 21 q^{34} - 60 q^{35} + 129 q^{36} + 20 q^{37} - 29 q^{38} + 62 q^{39} - 21 q^{40} - 28 q^{41} + 65 q^{43} - 6 q^{44} - 112 q^{45} - 5 q^{46} + 17 q^{47} - 166 q^{49} - 118 q^{50} - 14 q^{51} - 20 q^{52} - 21 q^{54} - 63 q^{55} - 51 q^{57} - 6 q^{58} + 55 q^{59} - 12 q^{60} - 87 q^{61} - 2 q^{62} - 36 q^{63} + 22 q^{64} - 32 q^{65} - 95 q^{66} + 16 q^{67} + 25 q^{68} - 54 q^{69} - 37 q^{71} + 20 q^{72} + 70 q^{73} + 34 q^{74} - 266 q^{75} + 24 q^{76} - 21 q^{77} - 4 q^{78} + 20 q^{79} - 87 q^{81} - 23 q^{82} + 65 q^{83} + 82 q^{85} - 23 q^{86} - 9 q^{88} - 42 q^{89} - 30 q^{91} - 14 q^{92} + 480 q^{93} + 4 q^{94} + 120 q^{95} - 77 q^{97} + 52 q^{98} + 230 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
39.1 −0.936235 − 0.351375i −3.28978 + 0.147744i 0.753071 + 0.657939i −2.79994 − 2.67702i 3.13192 + 1.01762i −0.223883 + 0.162661i −0.473869 − 0.880596i 7.81290 − 0.703174i 1.68077 + 3.49015i
39.2 −0.936235 − 0.351375i −2.77511 + 0.124630i 0.753071 + 0.657939i 0.579938 + 0.554477i 2.64194 + 0.858419i −0.988155 + 0.717937i −0.473869 − 0.880596i 4.69776 − 0.422806i −0.348129 − 0.722897i
39.3 −0.936235 − 0.351375i −2.36063 + 0.106016i 0.753071 + 0.657939i 0.714526 + 0.683157i 2.24735 + 0.730210i −0.762149 + 0.553733i −0.473869 − 0.880596i 2.57341 − 0.231611i −0.428920 − 0.890662i
39.4 −0.936235 − 0.351375i −2.31928 + 0.104159i 0.753071 + 0.657939i −1.57958 − 1.51023i 2.20799 + 0.717418i 4.11045 − 2.98642i −0.473869 − 0.880596i 2.38027 − 0.214228i 0.948200 + 1.96896i
39.5 −0.936235 − 0.351375i −2.26337 + 0.101648i 0.753071 + 0.657939i 2.88892 + 2.76209i 2.15476 + 0.700124i 1.83605 − 1.33397i −0.473869 − 0.880596i 2.12458 − 0.191216i −1.73418 − 3.60106i
39.6 −0.936235 − 0.351375i −2.17374 + 0.0976227i 0.753071 + 0.657939i 0.480962 + 0.459847i 2.06943 + 0.672399i 0.944533 − 0.686243i −0.473869 − 0.880596i 1.72769 − 0.155494i −0.288715 − 0.599523i
39.7 −0.936235 − 0.351375i −1.10745 + 0.0497358i 0.753071 + 0.657939i 2.36629 + 2.26240i 1.05431 + 0.342567i −2.86862 + 2.08417i −0.473869 − 0.880596i −1.76394 + 0.158758i −1.42045 − 2.94959i
39.8 −0.936235 − 0.351375i −1.02853 + 0.0461914i 0.753071 + 0.657939i −1.96515 − 1.87888i 0.979177 + 0.318154i −3.38348 + 2.45824i −0.473869 − 0.880596i −1.93218 + 0.173899i 1.17965 + 2.44957i
39.9 −0.936235 − 0.351375i −0.768509 + 0.0345138i 0.753071 + 0.657939i 0.692846 + 0.662429i 0.731632 + 0.237722i −3.73554 + 2.71403i −0.473869 − 0.880596i −2.39851 + 0.215870i −0.415906 − 0.863638i
39.10 −0.936235 − 0.351375i −0.503985 + 0.0226340i 0.753071 + 0.657939i −2.08529 − 1.99374i 0.479801 + 0.155897i 1.16686 − 0.847773i −0.473869 − 0.880596i −2.73443 + 0.246104i 1.25177 + 2.59932i
39.11 −0.936235 − 0.351375i −0.477519 + 0.0214454i 0.753071 + 0.657939i −2.30479 − 2.20361i 0.454606 + 0.147710i −0.833445 + 0.605533i −0.473869 − 0.880596i −2.76036 + 0.248437i 1.38354 + 2.87294i
39.12 −0.936235 − 0.351375i 0.0961812 − 0.00431950i 0.753071 + 0.657939i −0.0420406 − 0.0401949i −0.0915660 − 0.0297516i 3.18576 − 2.31459i −0.473869 − 0.880596i −2.97869 + 0.268087i 0.0252364 + 0.0524038i
39.13 −0.936235 − 0.351375i 0.602956 − 0.0270788i 0.753071 + 0.657939i 2.22745 + 2.12966i −0.574023 − 0.186511i 1.28898 − 0.936496i −0.473869 − 0.880596i −2.62510 + 0.236263i −1.33711 − 2.77654i
39.14 −0.936235 − 0.351375i 0.656645 − 0.0294900i 0.753071 + 0.657939i 0.796910 + 0.761924i −0.625136 − 0.203119i 1.80939 − 1.31460i −0.473869 − 0.880596i −2.55761 + 0.230189i −0.478374 − 0.993354i
39.15 −0.936235 − 0.351375i 0.846124 − 0.0379995i 0.753071 + 0.657939i −0.106606 − 0.101926i −0.805523 − 0.261730i −0.617453 + 0.448606i −0.473869 − 0.880596i −2.27344 + 0.204613i 0.0639940 + 0.132885i
39.16 −0.936235 − 0.351375i 1.51478 − 0.0680290i 0.753071 + 0.657939i 2.31369 + 2.21211i −1.44210 − 0.468566i −2.66874 + 1.93895i −0.473869 − 0.880596i −0.697981 + 0.0628194i −1.38888 − 2.88403i
39.17 −0.936235 − 0.351375i 2.02076 − 0.0907524i 0.753071 + 0.657939i −2.15464 − 2.06004i −1.92379 − 0.625078i −2.53957 + 1.84511i −0.473869 − 0.880596i 1.08731 − 0.0978592i 1.29340 + 2.68577i
39.18 −0.936235 − 0.351375i 2.03596 − 0.0914352i 0.753071 + 0.657939i −1.44680 − 1.38328i −1.93827 − 0.629781i 0.467575 − 0.339713i −0.473869 − 0.880596i 1.14886 − 0.103399i 0.868491 + 1.80344i
39.19 −0.936235 − 0.351375i 2.59471 − 0.116528i 0.753071 + 0.657939i 0.345031 + 0.329883i −2.47020 − 0.802617i 1.29458 − 0.940569i −0.473869 − 0.880596i 3.73100 − 0.335796i −0.207117 − 0.430083i
39.20 −0.936235 − 0.351375i 2.65647 − 0.119302i 0.753071 + 0.657939i −2.97416 − 2.84359i −2.52900 − 0.821722i 4.09666 − 2.97640i −0.473869 − 0.880596i 4.05469 − 0.364928i 1.78535 + 3.70731i
See next 80 embeddings (of 528 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
473.bb even 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 946.2.z.a ✓ 528
11.d odd 10 1 946.2.z.b yes 528
43.f odd 14 1 946.2.z.b yes 528
473.bb even 70 1 inner 946.2.z.a ✓ 528
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
946.2.z.a ✓ 528 1.a even 1 1 trivial
946.2.z.a ✓ 528 473.bb even 70 1 inner
946.2.z.b yes 528 11.d odd 10 1
946.2.z.b yes 528 43.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{528} + 43 T_{3}^{526} + 5 T_{3}^{525} + 854 T_{3}^{524} + 89 T_{3}^{523} + 12030 T_{3}^{522} + \cdots + 49\!\cdots\!81 \) acting on \(S_{2}^{\mathrm{new}}(946, [\chi])\). Copy content Toggle raw display