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Properties

Label 399.2.ci.b
Level $399$
Weight $2$
Character orbit 399.ci
Analytic conductor $3.186$
Analytic rank $0$
Dimension $288$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [399,2,Mod(17,399)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(399, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 3, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("399.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 399 = 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 399.ci (of order \(18\), 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: \(3.18603104065\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(48\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 9 q^{3} - 6 q^{4} - 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 9 q^{3} - 6 q^{4} - 6 q^{7} - 3 q^{9} - 18 q^{10} - 27 q^{12} - 36 q^{13} - 15 q^{15} + 6 q^{16} - 18 q^{18} - 72 q^{19} + 12 q^{21} - 9 q^{24} + 18 q^{25} + 27 q^{27} - 108 q^{28} - 6 q^{30} - 9 q^{33} + 36 q^{34} - 84 q^{36} - 24 q^{39} - 144 q^{40} + 39 q^{42} + 12 q^{43} - 6 q^{46} + 63 q^{48} - 30 q^{49} - 3 q^{51} + 90 q^{52} + 9 q^{54} - 90 q^{55} - 66 q^{57} + 24 q^{58} + 42 q^{60} + 54 q^{61} + 60 q^{63} + 54 q^{64} - 27 q^{66} + 102 q^{67} + 60 q^{70} - 39 q^{72} + 36 q^{73} - 63 q^{75} - 87 q^{78} + 6 q^{79} + 45 q^{81} - 18 q^{82} - 117 q^{84} + 36 q^{85} - 9 q^{87} - 204 q^{88} + 144 q^{90} - 198 q^{91} - 60 q^{93} - 180 q^{94} - 243 q^{96} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.944541 − 2.59511i −0.0712555 + 1.73058i −4.31033 + 3.61679i −0.697332 − 0.585131i 4.55835 − 1.44969i 2.08636 − 1.62699i 8.67392 + 5.00789i −2.98985 − 0.246627i −0.859818 + 2.36233i
17.2 −0.893748 − 2.45555i −0.925847 − 1.46383i −3.69886 + 3.10371i 2.22061 + 1.86331i −2.76705 + 3.58176i −0.00711626 + 2.64574i 6.40108 + 3.69566i −1.28562 + 2.71057i 2.59080 − 7.11815i
17.3 −0.886472 − 2.43556i 1.59092 + 0.684809i −3.61404 + 3.03254i 0.848702 + 0.712146i 0.257585 − 4.48185i −1.72566 + 2.00552i 6.10042 + 3.52208i 2.06207 + 2.17896i 0.982124 − 2.69836i
17.4 −0.858572 − 2.35891i 0.919907 − 1.46757i −3.29521 + 2.76501i −3.13043 − 2.62674i −4.25167 − 0.909959i 1.76041 + 1.97509i 5.00361 + 2.88883i −1.30754 − 2.70006i −3.50854 + 9.63963i
17.5 −0.832364 − 2.28690i 1.37101 − 1.05845i −3.00500 + 2.52149i 1.42292 + 1.19397i −3.56176 − 2.25435i −0.688785 − 2.55452i 4.05242 + 2.33967i 0.759356 − 2.90231i 1.54610 − 4.24789i
17.6 −0.801571 − 2.20230i −1.43427 + 0.971012i −2.67551 + 2.24502i −1.45856 − 1.22387i 3.28813 + 2.38036i −2.60326 + 0.472262i 3.02951 + 1.74909i 1.11427 − 2.78539i −1.52620 + 4.19320i
17.7 −0.790453 − 2.17175i −1.73099 + 0.0605998i −2.55960 + 2.14776i 0.227510 + 0.190904i 1.49987 + 3.71138i 2.62191 − 0.354408i 2.68464 + 1.54998i 2.99266 − 0.209795i 0.234759 − 0.644995i
17.8 −0.683230 − 1.87716i 1.56993 + 0.731654i −1.52483 + 1.27949i −0.743891 − 0.624199i 0.300807 − 3.44690i 2.28326 + 1.33668i −0.0163842 − 0.00945940i 1.92936 + 2.29729i −0.663472 + 1.82287i
17.9 −0.670554 − 1.84233i 0.104808 − 1.72888i −1.41246 + 1.18519i −1.01401 − 0.850853i −3.25545 + 0.966215i −2.63497 − 0.238651i −0.265159 − 0.153089i −2.97803 − 0.362399i −0.887608 + 2.43868i
17.10 −0.643648 − 1.76841i −1.72242 − 0.182359i −1.18089 + 0.990888i 2.53819 + 2.12979i 0.786150 + 3.16332i −1.59960 − 2.10743i −0.747172 − 0.431380i 2.93349 + 0.628198i 2.13265 − 5.85939i
17.11 −0.573945 − 1.57690i 0.763523 + 1.55468i −0.625116 + 0.524535i −2.02899 − 1.70252i 2.01336 − 2.09630i −1.81327 − 1.92667i −1.72064 − 0.993411i −1.83407 + 2.37407i −1.52018 + 4.17667i
17.12 −0.564636 − 1.55133i −0.554355 + 1.64094i −0.555707 + 0.466294i 1.94099 + 1.62868i 2.85864 − 0.0665512i 2.16148 + 1.52579i −1.82227 − 1.05209i −2.38538 − 1.81933i 1.43066 − 3.93072i
17.13 −0.545272 − 1.49812i −1.37264 − 1.05634i −0.414961 + 0.348193i −2.91953 − 2.44978i −0.834058 + 2.63238i 1.45517 − 2.20963i −2.01345 − 1.16247i 0.768304 + 2.89995i −2.07813 + 5.70961i
17.14 −0.486094 − 1.33553i 1.30387 − 1.14014i −0.0152680 + 0.0128114i 3.09898 + 2.60035i −2.15650 − 1.18715i 0.690003 + 2.55419i −2.43713 − 1.40708i 0.400166 − 2.97319i 1.96646 − 5.40280i
17.15 −0.474788 − 1.30447i −1.17011 − 1.27705i 0.0558696 − 0.0468801i −0.769103 − 0.645354i −1.11032 + 2.13270i −0.646012 + 2.56567i −2.49209 − 1.43881i −0.261701 + 2.98856i −0.476684 + 1.30968i
17.16 −0.452293 − 1.24266i 1.69039 − 0.377593i 0.192445 − 0.161480i −0.715648 − 0.600500i −1.23377 − 1.92981i 1.73245 − 1.99965i −2.57820 − 1.48852i 2.71485 − 1.27656i −0.422537 + 1.16091i
17.17 −0.302600 − 0.831388i 1.58612 + 0.695859i 0.932451 − 0.782419i 2.27405 + 1.90816i 0.0985676 − 1.52925i −2.64547 − 0.0388495i −2.46508 − 1.42321i 2.03156 + 2.20743i 0.898290 − 2.46803i
17.18 −0.300160 − 0.824683i −0.896180 + 1.48218i 0.942083 − 0.790502i 0.767308 + 0.643848i 1.49133 + 0.294172i −0.609997 − 2.57447i −2.45475 − 1.41725i −1.39372 − 2.65660i 0.300655 − 0.826043i
17.19 −0.285944 − 0.785625i −1.56622 + 0.739569i 0.996646 − 0.836286i −0.861028 − 0.722488i 1.02887 + 1.01898i −0.200186 + 2.63817i −2.39006 − 1.37990i 1.90607 − 2.31665i −0.321399 + 0.883036i
17.20 −0.259349 − 0.712555i −0.812731 − 1.52953i 1.09162 − 0.915975i 2.36676 + 1.98594i −0.879095 + 0.975798i 1.09932 − 2.40655i −2.24918 − 1.29856i −1.67894 + 2.48620i 0.801279 − 2.20150i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
133.x odd 18 1 inner
399.ci even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 399.2.ci.b yes 288
3.b odd 2 1 inner 399.2.ci.b yes 288
7.d odd 6 1 399.2.cb.b ✓ 288
19.e even 9 1 399.2.cb.b ✓ 288
21.g even 6 1 399.2.cb.b ✓ 288
57.l odd 18 1 399.2.cb.b ✓ 288
133.x odd 18 1 inner 399.2.ci.b yes 288
399.ci even 18 1 inner 399.2.ci.b yes 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
399.2.cb.b ✓ 288 7.d odd 6 1
399.2.cb.b ✓ 288 19.e even 9 1
399.2.cb.b ✓ 288 21.g even 6 1
399.2.cb.b ✓ 288 57.l odd 18 1
399.2.ci.b yes 288 1.a even 1 1 trivial
399.2.ci.b yes 288 3.b odd 2 1 inner
399.2.ci.b yes 288 133.x odd 18 1 inner
399.2.ci.b yes 288 399.ci even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{288} + 3 T_{2}^{286} - 6 T_{2}^{284} - 1974 T_{2}^{282} - 5031 T_{2}^{280} + \cdots + 13\!\cdots\!29 \) acting on \(S_{2}^{\mathrm{new}}(399, [\chi])\). Copy content Toggle raw display