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Properties

Label 775.2.bn.a
Level $775$
Weight $2$
Character orbit 775.bn
Analytic conductor $6.188$
Analytic rank $0$
Dimension $624$
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] = [775,2,Mod(41,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([6, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bn (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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(624\)
Relative dimension: \(78\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 624 q - 2 q^{2} - 7 q^{3} - 154 q^{4} - 3 q^{5} + 7 q^{6} - 14 q^{7} + 26 q^{8} + 67 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 624 q - 2 q^{2} - 7 q^{3} - 154 q^{4} - 3 q^{5} + 7 q^{6} - 14 q^{7} + 26 q^{8} + 67 q^{9} - 4 q^{10} - 16 q^{11} - 9 q^{12} - 8 q^{13} - 10 q^{14} - 34 q^{15} - 130 q^{16} - 8 q^{17} - 46 q^{18} - 8 q^{19} - 27 q^{20} - 28 q^{21} - 2 q^{22} - 9 q^{23} - 92 q^{24} - 13 q^{25} + 19 q^{26} + 23 q^{27} - 50 q^{28} + 34 q^{29} - 60 q^{30} - 16 q^{31} + 10 q^{32} + 41 q^{33} - 12 q^{34} + 13 q^{35} + 34 q^{36} - 17 q^{37} + 99 q^{38} - 14 q^{39} - 9 q^{40} + 3 q^{41} + 81 q^{42} + 4 q^{43} - 34 q^{44} + 14 q^{45} + 34 q^{46} - 2 q^{47} + 52 q^{48} + 60 q^{49} + 28 q^{50} - 43 q^{51} + 77 q^{52} - 21 q^{53} - 184 q^{54} - 33 q^{55} + 124 q^{56} - 52 q^{57} - 56 q^{58} + 12 q^{59} - 79 q^{60} + 12 q^{61} + 17 q^{62} - 72 q^{63} - 90 q^{64} - 9 q^{65} + 8 q^{66} - q^{67} - 32 q^{68} + 32 q^{69} - 60 q^{70} + 19 q^{71} - 101 q^{72} - 4 q^{73} - 43 q^{74} - 10 q^{75} - 86 q^{76} - 52 q^{77} - 59 q^{78} + 127 q^{79} + 18 q^{80} + 83 q^{81} + 29 q^{82} - 50 q^{83} - 39 q^{84} + 59 q^{85} - 52 q^{86} + 3 q^{87} + 22 q^{88} + 64 q^{89} - 84 q^{90} - 204 q^{91} - 78 q^{92} + 34 q^{93} - 29 q^{94} - 80 q^{95} + 30 q^{96} - 31 q^{97} + 27 q^{98} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −0.849848 − 2.61556i 2.04548 − 0.910707i −4.50089 + 3.27009i −2.12980 + 0.681130i −4.12036 − 4.57612i 3.45459 + 0.734296i 7.92833 + 5.76027i 1.34722 − 1.49623i 3.59155 + 4.99178i
41.2 −0.837596 − 2.57785i −1.75403 + 0.780942i −4.32573 + 3.14283i 0.762409 + 2.10208i 3.48232 + 3.86751i 2.13622 + 0.454068i 7.33926 + 5.33228i 0.459342 − 0.510151i 4.78026 − 3.72607i
41.3 −0.835719 − 2.57208i 0.754250 − 0.335814i −4.29913 + 3.12350i 2.22042 − 0.264035i −1.49408 − 1.65935i −3.63831 − 0.773347i 7.25088 + 5.26807i −1.55127 + 1.72286i −2.53477 − 5.49045i
41.4 −0.819464 − 2.52205i −1.35082 + 0.601423i −4.07119 + 2.95789i −0.583862 − 2.15850i 2.62377 + 2.91399i 0.294928 + 0.0626889i 6.50538 + 4.72644i −0.544393 + 0.604610i −4.96538 + 3.24134i
41.5 −0.801125 − 2.46561i 2.36154 − 1.05143i −3.81939 + 2.77495i 1.43252 + 1.71694i −4.48429 − 4.98031i 0.296723 + 0.0630705i 5.70701 + 4.14638i 2.46398 − 2.73653i 3.08568 − 4.90751i
41.6 −0.786465 − 2.42049i −1.25769 + 0.559960i −3.62221 + 2.63169i −2.19423 + 0.430551i 2.34451 + 2.60384i −2.24089 − 0.476317i 5.10074 + 3.70590i −0.739161 + 0.820922i 2.76783 + 4.97249i
41.7 −0.751484 − 2.31283i 2.26392 − 1.00796i −3.16642 + 2.30054i −1.65137 − 1.50764i −4.03254 − 4.47858i −3.82593 − 0.813227i 3.76543 + 2.73575i 2.10194 − 2.33444i −2.24594 + 4.95230i
41.8 −0.726425 − 2.23571i −3.06233 + 1.36344i −2.85265 + 2.07257i 0.500651 − 2.17930i 5.27279 + 5.85603i −2.43079 − 0.516680i 2.90229 + 2.10864i 5.51150 − 6.12114i −5.23596 + 0.463790i
41.9 −0.712159 − 2.19180i −0.549476 + 0.244643i −2.67878 + 1.94625i 1.90704 − 1.16757i 0.927522 + 1.03012i 2.53687 + 0.539229i 2.44459 + 1.77610i −1.76532 + 1.96058i −3.91719 − 3.34834i
41.10 −0.675126 − 2.07782i 3.04155 − 1.35418i −2.24352 + 1.63001i 1.52524 − 1.63512i −4.86718 − 5.40555i 0.552001 + 0.117331i 1.36653 + 0.992845i 5.40980 − 6.00820i −4.42723 − 2.06527i
41.11 −0.673135 − 2.07170i 0.237725 − 0.105842i −2.22079 + 1.61350i 1.19379 + 1.89073i −0.379294 − 0.421249i 1.45125 + 0.308472i 1.31298 + 0.953939i −1.96208 + 2.17911i 3.11344 − 3.74589i
41.12 −0.673088 − 2.07155i 0.755102 − 0.336193i −2.22024 + 1.61310i −2.23143 − 0.143953i −1.20469 − 1.33795i 2.85259 + 0.606337i 1.31171 + 0.953013i −1.55024 + 1.72171i 1.20374 + 4.71941i
41.13 −0.663462 − 2.04193i 0.232852 − 0.103672i −2.11124 + 1.53391i −1.05826 + 1.96979i −0.366180 − 0.406684i −3.64707 − 0.775209i 1.05892 + 0.769352i −1.96392 + 2.18115i 4.72428 + 0.854014i
41.14 −0.661059 − 2.03453i −2.79378 + 1.24387i −2.08428 + 1.51432i −1.58513 + 1.57714i 4.37755 + 4.86177i 0.688494 + 0.146344i 0.997409 + 0.724660i 4.25062 − 4.72079i 4.25660 + 2.18242i
41.15 −0.610183 − 1.87795i 1.67252 − 0.744652i −1.53634 + 1.11622i −0.803655 + 2.08666i −2.41896 − 2.68653i −0.900058 − 0.191313i −0.161307 − 0.117196i 0.235413 − 0.261452i 4.40902 + 0.235982i
41.16 −0.590276 − 1.81668i −1.43559 + 0.639166i −1.33388 + 0.969118i −1.45460 − 1.69827i 2.00856 + 2.23073i 4.17730 + 0.887913i −0.542790 − 0.394360i −0.355006 + 0.394275i −2.22661 + 3.64500i
41.17 −0.577506 − 1.77738i 1.77068 − 0.788356i −1.20753 + 0.877325i −0.0191727 − 2.23599i −2.42379 − 2.69189i 1.11616 + 0.237248i −0.767162 − 0.557376i 0.506401 − 0.562416i −3.96313 + 1.32537i
41.18 −0.569318 − 1.75218i −0.607528 + 0.270489i −1.12798 + 0.819528i 1.65765 − 1.50074i 0.819822 + 0.910504i −3.46097 − 0.735651i −0.902847 − 0.655956i −1.71147 + 1.90078i −3.57329 − 2.05010i
41.19 −0.545286 − 1.67822i −2.49542 + 1.11103i −0.901042 + 0.654645i 2.20409 + 0.376790i 3.22527 + 3.58203i 3.82361 + 0.812734i −1.26519 − 0.919216i 2.98535 − 3.31556i −0.569526 − 3.90441i
41.20 −0.485746 − 1.49497i 0.730624 − 0.325295i −0.380958 + 0.276782i −0.748388 − 2.10711i −0.841204 − 0.934252i −1.14026 − 0.242369i −1.94457 − 1.41281i −1.57940 + 1.75410i −2.78654 + 2.14234i
See next 80 embeddings (of 624 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
775.bn even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bn.a ✓ 624
25.d even 5 1 775.2.bp.a yes 624
31.g even 15 1 775.2.bp.a yes 624
775.bn even 15 1 inner 775.2.bn.a ✓ 624
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bn.a ✓ 624 1.a even 1 1 trivial
775.2.bn.a ✓ 624 775.bn even 15 1 inner
775.2.bp.a yes 624 25.d even 5 1
775.2.bp.a yes 624 31.g even 15 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(775, [\chi])\).