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Properties

Label 735.2.bf.a
Level $735$
Weight $2$
Character orbit 735.bf
Analytic conductor $5.869$
Analytic rank $0$
Dimension $228$
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] = [735,2,Mod(41,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 735.bf (of order \(14\), 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: \(5.86900454856\)
Analytic rank: \(0\)
Dimension: \(228\)
Relative dimension: \(38\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 228 q + 40 q^{4} - 38 q^{5} - 2 q^{6} + 4 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 228 q + 40 q^{4} - 38 q^{5} - 2 q^{6} + 4 q^{7} - 26 q^{9} + 49 q^{12} - 44 q^{16} + 2 q^{18} + 40 q^{20} - q^{21} - 12 q^{22} + 6 q^{24} - 38 q^{25} - 32 q^{26} + 51 q^{27} - 50 q^{28} - 42 q^{29} - 2 q^{30} - 25 q^{33} + 4 q^{35} - 14 q^{36} - 48 q^{37} - 16 q^{38} - 12 q^{39} + 46 q^{41} + 109 q^{42} + 24 q^{43} + 2 q^{45} - 30 q^{46} + 20 q^{47} - 10 q^{48} - 20 q^{49} + 14 q^{51} - 126 q^{52} + 50 q^{54} + 36 q^{56} + 10 q^{57} - 20 q^{58} - 16 q^{59} - 35 q^{60} - 84 q^{61} - 4 q^{62} + 42 q^{63} + 52 q^{64} - 42 q^{65} + 41 q^{66} - 24 q^{67} - 28 q^{68} + 10 q^{69} - 14 q^{71} - 18 q^{72} + 42 q^{74} - 6 q^{77} + 41 q^{78} - 8 q^{79} + 264 q^{80} + 22 q^{81} - 207 q^{84} + 98 q^{86} - 27 q^{87} - 16 q^{89} - 33 q^{90} - 22 q^{91} - 53 q^{93} - 28 q^{94} - 190 q^{96} - 8 q^{98} - 82 q^{99}+O(q^{100}) \) 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 −2.17091 + 1.73124i −0.629653 + 1.61355i 1.27061 − 5.56689i −0.900969 + 0.433884i −1.42652 − 4.59295i −2.11333 + 1.59180i 4.46973 + 9.28150i −2.20707 − 2.03195i 1.20476 − 2.50172i
41.2 −2.10945 + 1.68223i 0.816922 − 1.52730i 1.17484 − 5.14730i −0.900969 + 0.433884i 0.846009 + 4.59601i 0.0582761 − 2.64511i 3.83937 + 7.97254i −1.66528 − 2.49537i 1.17066 − 2.43089i
41.3 −2.05936 + 1.64228i −1.33601 − 1.10231i 1.09882 − 4.81422i −0.900969 + 0.433884i 4.56161 + 0.0759348i 1.78345 + 1.95431i 3.35775 + 6.97243i 0.569845 + 2.94538i 1.14286 − 2.37316i
41.4 −1.79280 + 1.42971i 1.00745 + 1.40891i 0.725023 − 3.17653i −0.900969 + 0.433884i −3.82051 − 1.08553i 0.192638 − 2.63873i 1.25184 + 2.59948i −0.970069 + 2.83883i 0.994931 − 2.06599i
41.5 −1.69813 + 1.35421i −1.72656 − 0.137845i 0.604706 − 2.64939i −0.900969 + 0.433884i 3.11858 − 2.10404i −2.51221 − 0.829938i 0.676187 + 1.40412i 2.96200 + 0.475993i 0.942389 − 1.95689i
41.6 −1.66589 + 1.32851i 1.26627 + 1.18177i 0.565233 − 2.47645i −0.900969 + 0.433884i −3.67945 − 0.286453i 1.35198 + 2.27424i 0.499356 + 1.03692i 0.206861 + 2.99286i 0.924502 − 1.91975i
41.7 −1.55299 + 1.23847i −0.467707 + 1.66771i 0.432935 − 1.89681i −0.900969 + 0.433884i −1.33906 − 3.16918i 1.08004 − 2.41527i −0.0468900 − 0.0973681i −2.56250 − 1.56000i 0.861846 − 1.78964i
41.8 −1.46520 + 1.16846i 1.28653 − 1.15967i 0.336476 − 1.47420i −0.900969 + 0.433884i −0.530002 + 3.20241i −2.35993 + 1.19613i −0.396715 − 0.823787i 0.310332 − 2.98391i 0.813125 − 1.68847i
41.9 −1.41029 + 1.12467i 0.195488 − 1.72098i 0.279001 − 1.22238i −0.900969 + 0.433884i 1.65985 + 2.64695i 2.33363 + 1.24666i −0.584002 − 1.21269i −2.92357 − 0.672864i 0.782654 − 1.62520i
41.10 −1.33545 + 1.06499i −0.891497 + 1.48500i 0.204194 − 0.894634i −0.900969 + 0.433884i −0.390959 − 2.93259i −0.634287 + 2.56859i −0.802159 − 1.66570i −1.41047 − 2.64775i 0.741121 − 1.53895i
41.11 −1.31640 + 1.04980i 1.69921 − 0.335702i 0.185804 − 0.814061i −0.900969 + 0.433884i −1.88443 + 2.22574i −1.46748 − 2.20148i −0.851093 − 1.76731i 2.77461 − 1.14085i 0.730549 − 1.51700i
41.12 −1.10925 + 0.884601i −1.73021 − 0.0798930i 0.00288467 − 0.0126386i −0.900969 + 0.433884i 1.98991 − 1.44192i 2.56766 − 0.638055i −1.22320 − 2.54000i 2.98723 + 0.276463i 0.615590 − 1.27829i
41.13 −0.805152 + 0.642087i 1.70059 − 0.328606i −0.209048 + 0.915900i −0.900969 + 0.433884i −1.15824 + 1.35651i 2.64445 + 0.0828173i −1.31342 − 2.72735i 2.78404 − 1.11765i 0.446826 − 0.927843i
41.14 −0.755513 + 0.602501i −0.628123 − 1.61414i −0.237250 + 1.03946i −0.900969 + 0.433884i 1.44708 + 0.841062i −1.83427 − 1.90669i −1.28559 − 2.66955i −2.21092 + 2.02776i 0.419278 − 0.870640i
41.15 −0.654189 + 0.521698i −1.10900 − 1.33046i −0.289248 + 1.26728i −0.900969 + 0.433884i 1.41959 + 0.291813i −1.29669 + 2.30621i −1.19801 − 2.48769i −0.540259 + 2.95095i 0.363048 − 0.753876i
41.16 −0.546191 + 0.435573i 0.505817 + 1.65655i −0.336441 + 1.47404i −0.900969 + 0.433884i −0.997820 − 0.684472i −2.50085 − 0.863579i −1.06452 − 2.21050i −2.48830 + 1.67582i 0.303113 − 0.629421i
41.17 −0.130915 + 0.104401i −0.0587352 + 1.73105i −0.438803 + 1.92252i −0.900969 + 0.433884i −0.173035 − 0.232753i 1.88627 + 1.85526i −0.288572 − 0.599227i −2.99310 − 0.203348i 0.0726524 − 0.150864i
41.18 −0.120472 + 0.0960733i −1.39461 + 1.02716i −0.439758 + 1.92671i −0.900969 + 0.433884i 0.0693285 − 0.257729i 1.03731 − 2.43392i −0.265840 − 0.552023i 0.889869 − 2.86498i 0.0668570 − 0.138830i
41.19 −0.00450508 + 0.00359268i 0.896820 − 1.48179i −0.445034 + 1.94982i −0.900969 + 0.433884i 0.00128337 + 0.00989758i −0.343815 − 2.62332i −0.0100004 − 0.0207661i −1.39143 − 2.65780i 0.00250013 − 0.00519157i
41.20 0.0565473 − 0.0450950i −0.237340 − 1.71571i −0.443878 + 1.94476i −0.900969 + 0.433884i −0.0907910 − 0.0863161i 2.50735 − 0.844500i 0.125361 + 0.260316i −2.88734 + 0.814416i −0.0313814 + 0.0651641i
See next 80 embeddings (of 228 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
147.k even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 735.2.bf.a ✓ 228
3.b odd 2 1 735.2.bf.b yes 228
49.f odd 14 1 735.2.bf.b yes 228
147.k even 14 1 inner 735.2.bf.a ✓ 228
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
735.2.bf.a ✓ 228 1.a even 1 1 trivial
735.2.bf.a ✓ 228 147.k even 14 1 inner
735.2.bf.b yes 228 3.b odd 2 1
735.2.bf.b yes 228 49.f odd 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{228} - 58 T_{2}^{226} + 1829 T_{2}^{224} - 41620 T_{2}^{222} + 765664 T_{2}^{220} + \cdots + 133905496761 \) acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\). Copy content Toggle raw display