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.
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 |
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 \(S_{2}^{\mathrm{new}}(325, [\chi])\).