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Properties

Label 435.2.bm.a
Level $435$
Weight $2$
Character orbit 435.bm
Analytic conductor $3.473$
Analytic rank $0$
Dimension $180$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,2,Mod(37,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 7, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.37");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.bm (of order \(28\), degree \(12\), 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: \(3.47349248793\)
Analytic rank: \(0\)
Dimension: \(180\)
Relative dimension: \(15\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 180 q - 30 q^{3} + 30 q^{4} + 10 q^{5} - 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 180 q - 30 q^{3} + 30 q^{4} + 10 q^{5} - 30 q^{9} - 4 q^{10} - 30 q^{11} - 180 q^{12} - 20 q^{13} - 10 q^{14} - 4 q^{15} - 14 q^{16} + 8 q^{19} + 2 q^{20} + 36 q^{22} + 64 q^{25} - 36 q^{26} - 30 q^{27} + 72 q^{28} + 12 q^{29} - 4 q^{30} - 20 q^{31} + 12 q^{33} + 40 q^{34} - 6 q^{35} + 30 q^{36} + 42 q^{37} + 16 q^{38} + 22 q^{39} + 18 q^{40} - 10 q^{41} + 4 q^{42} + 26 q^{43} + 4 q^{44} - 4 q^{45} + 12 q^{46} - 20 q^{47} - 70 q^{48} + 8 q^{50} + 12 q^{52} - 82 q^{53} + 48 q^{55} + 6 q^{56} + 8 q^{57} - 70 q^{58} - 40 q^{60} + 14 q^{61} + 110 q^{62} - 14 q^{63} - 74 q^{64} + 42 q^{65} + 22 q^{66} - 20 q^{67} - 98 q^{68} + 28 q^{69} + 8 q^{70} + 140 q^{71} + 98 q^{73} + 22 q^{75} - 4 q^{76} - 42 q^{77} + 34 q^{78} - 24 q^{79} - 62 q^{80} - 30 q^{81} + 6 q^{82} - 60 q^{83} - 68 q^{84} - 178 q^{85} - 44 q^{87} - 156 q^{88} - 12 q^{89} - 4 q^{90} - 56 q^{91} - 8 q^{92} + 8 q^{93} + 4 q^{95} - 42 q^{97} + 194 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −1.13407 − 2.35492i 0.623490 + 0.781831i −3.01255 + 3.77762i −2.06879 + 0.848599i 1.13407 − 2.35492i 0.499368 − 4.43202i 7.21599 + 1.64700i −0.222521 + 0.974928i 4.34453 + 3.90946i
37.2 −1.08732 − 2.25785i 0.623490 + 0.781831i −2.66862 + 3.34635i 0.923461 − 2.03647i 1.08732 − 2.25785i −0.0657335 + 0.583401i 5.57082 + 1.27150i −0.222521 + 0.974928i −5.60214 + 0.129268i
37.3 −0.828336 − 1.72006i 0.623490 + 0.781831i −1.02548 + 1.28591i 1.86055 + 1.24031i 0.828336 − 1.72006i −0.298117 + 2.64586i −0.661238 − 0.150923i −0.222521 + 0.974928i 0.592250 − 4.22764i
37.4 −0.781835 − 1.62350i 0.623490 + 0.781831i −0.777496 + 0.974949i −1.76781 − 1.36925i 0.781835 − 1.62350i −0.238154 + 2.11367i −1.32283 − 0.301928i −0.222521 + 0.974928i −0.840829 + 3.94056i
37.5 −0.616058 − 1.27926i 0.623490 + 0.781831i −0.00999478 + 0.0125331i 2.23587 + 0.0294969i 0.616058 − 1.27926i 0.433423 − 3.84674i −2.74635 − 0.626837i −0.222521 + 0.974928i −1.33969 − 2.87843i
37.6 −0.537625 − 1.11639i 0.623490 + 0.781831i 0.289694 − 0.363265i −1.26866 + 1.84133i 0.537625 − 1.11639i −0.377119 + 3.34702i −2.97736 − 0.679562i −0.222521 + 0.974928i 2.73771 + 0.426372i
37.7 −0.164366 − 0.341310i 0.623490 + 0.781831i 1.15750 − 1.45146i −2.12222 − 0.704390i 0.164366 − 0.341310i 0.0298844 − 0.265232i −1.42431 − 0.325089i −0.222521 + 0.974928i 0.108407 + 0.840114i
37.8 −0.102759 − 0.213381i 0.623490 + 0.781831i 1.21201 − 1.51981i 1.24201 − 1.85941i 0.102759 − 0.213381i 0.0307732 − 0.273119i −0.910639 − 0.207847i −0.222521 + 0.974928i −0.524392 − 0.0739495i
37.9 0.110667 + 0.229802i 0.623490 + 0.781831i 1.20642 − 1.51280i 2.10538 + 0.753254i −0.110667 + 0.229802i −0.315615 + 2.80116i 0.978486 + 0.223333i −0.222521 + 0.974928i 0.0598960 + 0.567180i
37.10 0.537697 + 1.11654i 0.623490 + 0.781831i 0.289438 − 0.362944i −1.20234 − 1.88531i −0.537697 + 1.11654i 0.397131 − 3.52463i 2.97726 + 0.679540i −0.222521 + 0.974928i 1.45852 − 2.35618i
37.11 0.571502 + 1.18674i 0.623490 + 0.781831i 0.165250 − 0.207217i −0.713695 + 2.11911i −0.571502 + 1.18674i 0.00550243 − 0.0488354i 2.90866 + 0.663883i −0.222521 + 0.974928i −2.92271 + 0.364110i
37.12 0.654914 + 1.35994i 0.623490 + 0.781831i −0.173552 + 0.217628i 0.0447950 − 2.23562i −0.654914 + 1.35994i −0.524983 + 4.65935i 2.53353 + 0.578262i −0.222521 + 0.974928i 3.06965 − 1.40322i
37.13 0.792339 + 1.64531i 0.623490 + 0.781831i −0.832258 + 1.04362i 2.09574 − 0.779653i −0.792339 + 1.64531i 0.472013 − 4.18923i 1.18423 + 0.270293i −0.222521 + 0.974928i 2.94331 + 2.83039i
37.14 1.05750 + 2.19593i 0.623490 + 0.781831i −2.45682 + 3.08075i −2.23369 − 0.102994i −1.05750 + 2.19593i −0.0423490 + 0.375858i −4.61082 − 1.05239i −0.222521 + 0.974928i −2.13597 − 5.01396i
37.15 1.09386 + 2.27143i 0.623490 + 0.781831i −2.71589 + 3.40561i 1.75028 + 1.39159i −1.09386 + 2.27143i −0.143430 + 1.27298i −5.79065 − 1.32168i −0.222521 + 0.974928i −1.24635 + 5.49785i
43.1 −2.04531 − 1.63108i −0.222521 + 0.974928i 1.07783 + 4.72226i −1.64083 − 1.51910i 2.04531 − 1.63108i 0.0136741 + 0.0217622i 3.22777 − 6.70254i −0.900969 − 0.433884i 0.878234 + 5.78336i
43.2 −2.03116 − 1.61980i −0.222521 + 0.974928i 1.05683 + 4.63028i 1.16991 + 1.90560i 2.03116 − 1.61980i 1.52677 + 2.42985i 3.09909 − 6.43533i −0.900969 − 0.433884i 0.710399 − 5.76560i
43.3 −1.70467 − 1.35943i −0.222521 + 0.974928i 0.612805 + 2.68487i 2.19664 − 0.418062i 1.70467 − 1.35943i −1.82909 − 2.91097i 0.713223 − 1.48102i −0.900969 − 0.433884i −4.31286 − 2.27351i
43.4 −1.25592 − 1.00156i −0.222521 + 0.974928i 0.129161 + 0.565891i −2.00721 + 0.985438i 1.25592 − 1.00156i 0.245610 + 0.390886i −0.989403 + 2.05452i −0.900969 − 0.433884i 3.50787 + 0.772717i
43.5 −1.14581 − 0.913753i −0.222521 + 0.974928i 0.0328938 + 0.144117i 1.79018 − 1.33987i 1.14581 − 0.913753i 2.18772 + 3.48174i −1.17776 + 2.44563i −0.900969 − 0.433884i −3.27552 − 0.100541i
See next 80 embeddings (of 180 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.15
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.t even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 435.2.bm.a yes 180
5.c odd 4 1 435.2.bd.b ✓ 180
29.f odd 28 1 435.2.bd.b ✓ 180
145.t even 28 1 inner 435.2.bm.a yes 180
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.bd.b ✓ 180 5.c odd 4 1
435.2.bd.b ✓ 180 29.f odd 28 1
435.2.bm.a yes 180 1.a even 1 1 trivial
435.2.bm.a yes 180 145.t even 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{180} - 45 T_{2}^{178} + 1121 T_{2}^{176} - 20461 T_{2}^{174} + 305881 T_{2}^{172} + 210 T_{2}^{171} + \cdots + 12769 \) acting on \(S_{2}^{\mathrm{new}}(435, [\chi])\). Copy content Toggle raw display