Properties

Label 275.2.i.b
Level $275$
Weight $2$
Character orbit 275.i
Analytic conductor $2.196$
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] = [275,2,Mod(56,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.i (of order \(5\), degree \(4\), 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{2} - 2 q^{3} - 19 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{2} - 2 q^{3} - 19 q^{4} + 6 q^{5} - 4 q^{6} - 6 q^{8} - 27 q^{9} - 11 q^{10} - 15 q^{11} + 12 q^{12} - 14 q^{13} - 14 q^{14} + q^{15} - 27 q^{16} - 2 q^{17} + 118 q^{18} - 10 q^{19} - 10 q^{20} - 18 q^{21} - q^{22} - 12 q^{23} + 34 q^{24} - 8 q^{25} + 42 q^{26} - 20 q^{27} - 44 q^{28} - 12 q^{29} + 65 q^{30} - 28 q^{31} - 38 q^{32} - 2 q^{33} - 29 q^{34} + 18 q^{35} - 29 q^{36} + 30 q^{37} - 10 q^{38} + 4 q^{39} - 69 q^{40} - 24 q^{41} + 76 q^{42} + 60 q^{43} - 19 q^{44} - 53 q^{45} - 34 q^{46} - 39 q^{47} - 17 q^{48} + 140 q^{49} + 25 q^{50} + 10 q^{51} + 68 q^{52} - 14 q^{53} + 23 q^{54} - 4 q^{55} - 34 q^{56} - 80 q^{57} - 28 q^{58} + 14 q^{59} + 31 q^{60} - 50 q^{61} + 2 q^{62} - 72 q^{63} - 46 q^{64} - 17 q^{65} - 4 q^{66} + 73 q^{67} + 228 q^{68} - 32 q^{69} - 108 q^{70} - 23 q^{71} - 80 q^{72} + 36 q^{73} - 16 q^{74} + 36 q^{75} + 80 q^{76} - 135 q^{78} + q^{79} + 120 q^{80} + 10 q^{81} - 130 q^{82} + 30 q^{83} - 32 q^{84} + 10 q^{85} + 13 q^{86} + 73 q^{87} + 9 q^{88} - 18 q^{89} - 24 q^{90} - 6 q^{91} + 182 q^{92} - 4 q^{93} - 15 q^{94} - 50 q^{95} - 4 q^{96} - 65 q^{97} - 97 q^{98} + 78 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −0.847393 2.60801i 0.282837 0.205493i −4.46560 + 3.24445i 1.18952 + 1.89342i −0.775602 0.563508i 4.12997 7.80866 + 5.67332i −0.889282 + 2.73693i 3.93007 4.70675i
56.2 −0.749343 2.30624i 1.41740 1.02980i −3.13919 + 2.28076i −2.10136 + 0.764385i −3.43709 2.49719i −4.46925 3.68870 + 2.68000i 0.0214834 0.0661192i 3.33749 + 4.27346i
56.3 −0.614887 1.89243i −1.67850 + 1.21950i −1.58516 + 1.15169i 2.23365 0.103885i 3.33991 + 2.42659i −3.97514 −0.0654098 0.0475230i 0.403133 1.24071i −1.57004 4.16315i
56.4 −0.593152 1.82553i −1.65284 + 1.20086i −1.36271 + 0.990068i 0.437546 2.19284i 3.17259 + 2.30502i 3.96949 −0.490086 0.356068i 0.362767 1.11648i −4.26264 + 0.501933i
56.5 −0.383688 1.18087i 1.38363 1.00527i 0.370797 0.269400i 0.160480 2.23030i −1.71797 1.24818i −0.528113 −2.46941 1.79413i −0.0231746 + 0.0713241i −2.69527 + 0.666234i
56.6 −0.253584 0.780452i 2.37547 1.72588i 1.07323 0.779750i 0.790188 + 2.09179i −1.94935 1.41628i −1.59619 −2.20850 1.60457i 1.73714 5.34636i 1.43217 1.14715i
56.7 −0.0516459 0.158950i −0.303083 + 0.220202i 1.59544 1.15915i 1.93201 + 1.12576i 0.0506541 + 0.0368023i 0.678243 −0.537066 0.390201i −0.883681 + 2.71969i 0.0791593 0.365233i
56.8 −0.00137816 0.00424154i −2.06095 + 1.49737i 1.61802 1.17556i −2.22854 0.183378i 0.00919146 + 0.00667799i 4.51694 −0.0144322 0.0104856i 1.07835 3.31882i 0.00229348 + 0.00970516i
56.9 0.126287 + 0.388670i 0.876499 0.636814i 1.48292 1.07740i −1.49442 1.66334i 0.358200 + 0.260248i 0.810085 1.26727 + 0.920727i −0.564333 + 1.73684i 0.457765 0.790895i
56.10 0.317055 + 0.975794i −0.474307 + 0.344604i 0.766384 0.556811i −1.56781 + 1.59435i −0.486644 0.353567i −5.27769 2.44644 + 1.77744i −0.820836 + 2.52627i −2.05284 1.02437i
56.11 0.532896 + 1.64009i −0.485785 + 0.352943i −0.787872 + 0.572422i 2.09251 0.788291i −0.837731 0.608647i −0.652091 1.43161 + 1.04012i −0.815633 + 2.51026i 2.40796 + 3.01182i
56.12 0.549271 + 1.69048i 2.64305 1.92029i −0.937997 + 0.681495i −2.13719 0.657589i 4.69797 + 3.41327i −0.0570090 1.20875 + 0.878209i 2.37116 7.29768i −0.0622530 3.97407i
56.13 0.643133 + 1.97936i −2.55218 + 1.85427i −1.88621 + 1.37041i −0.839175 2.07263i −5.31166 3.85915i −2.50218 −0.558133 0.405507i 2.14827 6.61170i 3.56277 2.99400i
56.14 0.783991 + 2.41288i −1.82681 + 1.32725i −3.58930 + 2.60778i −0.0917842 + 2.23418i −4.63470 3.36730i 2.83405 −5.00118 3.63357i 0.648573 1.99610i −5.46277 + 1.53012i
56.15 0.851456 + 2.62051i 1.55557 1.13018i −4.52407 + 3.28693i 2.00634 0.987214i 4.28616 + 3.11408i 2.11888 −8.00720 5.81757i 0.215417 0.662985i 4.29532 + 4.41707i
111.1 −2.10502 + 1.52938i 0.913514 2.81151i 1.47404 4.53664i 1.95157 + 1.09151i 2.37691 + 7.31538i 3.15600 2.22729 + 6.85489i −4.64302 3.37335i −5.77741 + 0.687058i
111.2 −1.96894 + 1.43052i −1.01785 + 3.13261i 1.21231 3.73111i 1.19246 + 1.89157i −2.47718 7.62397i −0.337264 1.44632 + 4.45132i −6.35016 4.61366i −5.05382 2.01855i
111.3 −1.88069 + 1.36640i −0.178254 + 0.548608i 1.05190 3.23742i 1.77953 1.35399i −0.414378 1.27533i −4.13856 1.00860 + 3.10414i 2.15785 + 1.56777i −1.49665 + 4.97797i
111.4 −1.48661 + 1.08008i 0.310838 0.956660i 0.425384 1.30920i −2.11835 + 0.715970i 0.571178 + 1.75791i 2.93566 −0.354003 1.08951i 1.60847 + 1.16862i 2.37584 3.35235i
111.5 −1.00729 + 0.731839i −0.570382 + 1.75545i −0.138989 + 0.427764i −0.988997 2.00546i −0.710170 2.18568i −1.68501 −0.942554 2.90088i −0.329235 0.239203i 2.46388 + 1.29630i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.i.b 60
25.d even 5 1 inner 275.2.i.b 60
25.d even 5 1 6875.2.a.h 30
25.e even 10 1 6875.2.a.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.i.b 60 1.a even 1 1 trivial
275.2.i.b 60 25.d even 5 1 inner
6875.2.a.g 30 25.e even 10 1
6875.2.a.h 30 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} + T_{2}^{59} + 25 T_{2}^{58} + 28 T_{2}^{57} + 381 T_{2}^{56} + 454 T_{2}^{55} + 4589 T_{2}^{54} + 5781 T_{2}^{53} + 48247 T_{2}^{52} + 62508 T_{2}^{51} + 428533 T_{2}^{50} + 563785 T_{2}^{49} + 3299463 T_{2}^{48} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display