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Properties

Label 325.2.l.b
Level $325$
Weight $2$
Character orbit 325.l
Analytic conductor $2.595$
Analytic rank $0$
Dimension $56$
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] = [325,2,Mod(66,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.66");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.l (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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{3} - 24 q^{4} + 9 q^{5} - 5 q^{6} - 20 q^{7} + 13 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{3} - 24 q^{4} + 9 q^{5} - 5 q^{6} - 20 q^{7} + 13 q^{8} - 16 q^{9} - 39 q^{10} - 4 q^{11} + 13 q^{12} + 14 q^{13} - 36 q^{14} + 8 q^{15} - 10 q^{16} + 7 q^{17} - 64 q^{18} + q^{19} + 2 q^{20} + 25 q^{21} - 16 q^{22} + 40 q^{23} + 34 q^{24} + 17 q^{25} + 10 q^{26} + 25 q^{27} + 7 q^{28} - 9 q^{29} + 48 q^{30} + 6 q^{31} - 134 q^{32} - 24 q^{33} + 30 q^{34} - 48 q^{35} + 56 q^{36} - 4 q^{37} + 37 q^{38} - 8 q^{39} + 34 q^{40} + 4 q^{41} + 35 q^{42} - 56 q^{43} + 26 q^{44} + 73 q^{45} + 43 q^{46} + 2 q^{47} - 16 q^{48} + 36 q^{49} + 48 q^{50} - 118 q^{51} + 9 q^{52} + 14 q^{53} - 98 q^{54} - 17 q^{55} - 6 q^{56} - 36 q^{57} - q^{58} + 6 q^{59} - 70 q^{60} + 42 q^{61} - 36 q^{62} - 54 q^{63} - 49 q^{64} + q^{65} + 15 q^{66} + 78 q^{67} - 40 q^{68} + 30 q^{69} + 26 q^{70} + 43 q^{71} + 54 q^{72} - 13 q^{73} + 12 q^{74} + 27 q^{75} - 132 q^{76} + 3 q^{77} + 5 q^{78} + 2 q^{79} - 67 q^{80} - 26 q^{81} - 64 q^{82} + 80 q^{83} + 82 q^{84} - 144 q^{85} + 73 q^{86} + 8 q^{87} + 62 q^{88} - 11 q^{89} + 167 q^{90} - 10 q^{91} + 9 q^{92} - 22 q^{93} + 7 q^{94} - 90 q^{95} + 85 q^{96} + 49 q^{97} - 78 q^{98} + 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
66.1 −1.95307 − 1.41899i 0.403626 + 1.24223i 1.18291 + 3.64063i 2.08955 + 0.796101i 0.974403 − 2.99890i −2.22005 1.36369 − 4.19700i 1.04682 − 0.760561i −2.95138 − 4.51988i
66.2 −1.93262 − 1.40413i −0.232105 − 0.714347i 1.14541 + 3.52521i 0.356940 − 2.20740i −0.554466 + 1.70647i 1.52598 1.25982 − 3.87734i 1.97063 − 1.43175i −3.78931 + 3.76487i
66.3 −1.12971 − 0.820785i −0.966751 − 2.97535i −0.0154693 − 0.0476095i −2.22974 + 0.168118i −1.34997 + 4.15479i 2.59805 −0.884626 + 2.72260i −5.49108 + 3.98950i 2.65696 + 1.64021i
66.4 −1.11426 − 0.809555i 0.644694 + 1.98416i −0.0318443 − 0.0980066i −1.95882 − 1.07843i 0.887936 − 2.73279i 2.26479 −0.895076 + 2.75476i −1.09423 + 0.795001i 1.30959 + 2.78742i
66.5 −1.09745 − 0.797346i −0.778790 − 2.39687i −0.0493926 − 0.152015i 2.06181 − 0.865415i −1.05645 + 3.25141i −5.00125 −0.905381 + 2.78648i −2.71141 + 1.96995i −2.95277 − 0.694224i
66.6 −0.907449 − 0.659300i −0.433931 − 1.33550i −0.229247 − 0.705550i 0.939037 + 2.02934i −0.486727 + 1.49799i 4.07172 −0.950369 + 2.92493i 0.831781 − 0.604324i 0.485815 − 2.46063i
66.7 0.247295 + 0.179670i 0.407162 + 1.25312i −0.589161 − 1.81325i −0.383051 − 2.20301i −0.124459 + 0.383044i −3.93645 0.369007 − 1.13569i 1.02253 − 0.742914i 0.301090 − 0.613618i
66.8 0.412531 + 0.299721i 0.593396 + 1.82628i −0.537685 − 1.65482i 2.15220 + 0.606651i −0.302582 + 0.931252i 3.35489 0.589320 − 1.81374i −0.556143 + 0.404062i 0.706024 + 0.895323i
66.9 0.657973 + 0.478045i −0.0994647 − 0.306121i −0.413633 − 1.27303i 1.76444 + 1.37359i 0.0808946 − 0.248968i −2.18029 0.839054 − 2.58234i 2.34323 − 1.70246i 0.504315 + 1.74727i
66.10 0.693919 + 0.504162i −0.748985 − 2.30514i −0.390689 − 1.20242i −0.766393 + 2.10063i 0.642428 − 1.97719i −3.08835 0.865214 − 2.66285i −2.32563 + 1.68967i −1.59087 + 1.07128i
66.11 1.21398 + 0.882005i 0.955535 + 2.94083i 0.0777703 + 0.239352i −1.83202 + 1.28207i −1.43383 + 4.41289i 0.883175 0.810696 − 2.49507i −5.30840 + 3.85678i −3.35482 − 0.0594577i
66.12 1.75104 + 1.27221i −1.01142 − 3.11285i 0.829608 + 2.55327i −0.806156 − 2.08569i 2.18914 − 6.73747i −0.702622 −0.457933 + 1.40937i −6.23978 + 4.53346i 1.24182 − 4.67774i
66.13 2.07473 + 1.50738i 0.0369605 + 0.113753i 1.41427 + 4.35267i −1.93094 + 1.12760i −0.0947852 + 0.291719i −3.74246 −2.04195 + 6.28446i 2.41548 − 1.75495i −5.70588 − 0.571203i
66.14 2.20113 + 1.59922i −0.387955 − 1.19400i 1.66946 + 5.13806i 2.23413 − 0.0930839i 1.05553 − 3.24858i −1.06321 −2.86066 + 8.80420i 1.15192 − 0.836918i 5.06648 + 3.36797i
131.1 −0.839430 − 2.58350i −1.82945 + 1.32917i −4.35179 + 3.16176i 1.27957 − 1.83376i 4.96962 + 3.61064i 3.49199 7.42611 + 5.39538i 0.653138 − 2.01015i −5.81164 − 1.76646i
131.2 −0.789910 − 2.43109i 0.0823912 − 0.0598607i −3.66822 + 2.66512i 1.51845 + 1.64143i −0.210609 − 0.153016i −3.18998 5.24069 + 3.80758i −0.923846 + 2.84331i 2.79104 − 4.98808i
131.3 −0.776911 − 2.39109i 2.72175 − 1.97747i −3.49567 + 2.53976i −2.09046 + 0.793709i −6.84286 − 4.97163i −2.36912 4.72065 + 3.42975i 2.57050 − 7.91119i 3.52193 + 4.38183i
131.4 −0.597263 − 1.83819i 1.93910 − 1.40884i −1.40417 + 1.02019i 1.19756 − 1.88834i −3.74787 − 2.72298i 1.36128 −0.413347 − 0.300314i 0.848236 − 2.61060i −4.18639 − 1.07351i
131.5 −0.536973 − 1.65263i −2.03424 + 1.47796i −0.824818 + 0.599266i −1.84471 − 1.26374i 3.53486 + 2.56822i 0.233067 −1.37835 − 1.00143i 1.02671 − 3.15988i −1.09794 + 3.72722i
131.6 −0.340856 − 1.04905i 1.77913 − 1.29262i 0.633719 − 0.460424i 0.135337 + 2.23197i −1.96244 − 1.42580i 3.06266 −2.48376 − 1.80455i 0.567410 − 1.74631i 2.29531 − 0.902754i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 66.14
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 325.2.l.b ✓ 56
25.d even 5 1 inner 325.2.l.b ✓ 56
25.d even 5 1 8125.2.a.g 28
25.e even 10 1 8125.2.a.h 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.l.b ✓ 56 1.a even 1 1 trivial
325.2.l.b ✓ 56 25.d even 5 1 inner
8125.2.a.g 28 25.d even 5 1
8125.2.a.h 28 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} + 26 T_{2}^{54} - 11 T_{2}^{53} + 382 T_{2}^{52} - 152 T_{2}^{51} + 4306 T_{2}^{50} + \cdots + 8094025 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display