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Properties

Label 352.2.bc.a
Level $352$
Weight $2$
Character orbit 352.bc
Analytic conductor $2.811$
Analytic rank $0$
Dimension $736$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(19,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 35, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.bc (of order \(40\), degree \(16\), 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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(736\)
Relative dimension: \(46\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 20 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 736 q - 20 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 20 q^{6} - 20 q^{7} - 20 q^{8} - 12 q^{9} - 16 q^{11} - 20 q^{13} - 28 q^{14} - 24 q^{15} - 52 q^{16} - 120 q^{18} - 20 q^{19} + 4 q^{20} - 4 q^{22} - 48 q^{23} - 20 q^{24} - 12 q^{25} + 48 q^{26} - 12 q^{27} - 20 q^{28} - 20 q^{29} - 20 q^{30} - 32 q^{33} + 24 q^{34} - 20 q^{35} - 12 q^{36} - 12 q^{37} - 12 q^{38} - 20 q^{39} - 20 q^{40} - 20 q^{41} - 52 q^{42} + 24 q^{44} - 8 q^{45} - 20 q^{46} - 24 q^{47} - 12 q^{48} - 20 q^{50} - 20 q^{51} - 20 q^{52} - 60 q^{53} + 16 q^{55} - 248 q^{56} - 20 q^{57} - 12 q^{58} + 20 q^{59} + 68 q^{60} - 20 q^{61} - 20 q^{62} + 132 q^{64} - 8 q^{66} - 112 q^{67} - 20 q^{68} - 12 q^{69} + 36 q^{70} + 20 q^{71} - 320 q^{72} - 20 q^{73} - 20 q^{74} + 124 q^{75} - 32 q^{77} - 80 q^{78} - 40 q^{79} + 124 q^{80} + 28 q^{82} - 20 q^{83} - 20 q^{84} - 20 q^{85} - 116 q^{86} - 20 q^{88} - 32 q^{89} - 20 q^{90} - 156 q^{91} + 148 q^{92} - 36 q^{93} - 160 q^{94} - 20 q^{96} - 24 q^{97} - 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41245 + 0.0705991i 0.150757 − 1.91555i 1.99003 − 0.199436i −2.63276 − 0.632071i −0.0777007 + 2.71626i 2.34464 + 4.60161i −2.79674 + 0.422187i −0.683545 − 0.108263i 3.76327 + 0.706898i
19.2 −1.39865 − 0.209265i −0.134573 + 1.70991i 1.91242 + 0.585374i 0.794943 + 0.190849i 0.546042 − 2.36339i 1.17216 + 2.30049i −2.55229 − 1.21893i 0.0573940 + 0.00909032i −1.07190 − 0.433283i
19.3 −1.39496 − 0.232560i 0.00366593 − 0.0465800i 1.89183 + 0.648825i −0.0528412 − 0.0126860i −0.0159465 + 0.0641247i −1.30784 − 2.56679i −2.48814 − 1.34505i 2.96091 + 0.468962i 0.0707611 + 0.0299853i
19.4 −1.31946 + 0.508948i −0.158585 + 2.01502i 1.48194 − 1.34307i −3.93727 − 0.945256i −0.816293 − 2.73944i −0.0456497 − 0.0895925i −1.27181 + 2.52636i −1.07208 − 0.169801i 5.67616 − 0.756643i
19.5 −1.30861 + 0.536227i 0.141964 − 1.80382i 1.42492 − 1.40342i −1.17636 − 0.282420i 0.781484 + 2.43663i −0.850228 − 1.66867i −1.11211 + 2.60062i −0.270561 − 0.0428526i 1.69084 − 0.261221i
19.6 −1.29988 − 0.557061i 0.118949 − 1.51139i 1.37937 + 1.44822i 3.60772 + 0.866136i −0.996557 + 1.89836i −0.662879 − 1.30097i −0.986257 − 2.65090i 0.692913 + 0.109747i −4.20710 − 3.13559i
19.7 −1.27293 + 0.616165i 0.0557342 − 0.708171i 1.24068 − 1.56867i 2.69247 + 0.646406i 0.365405 + 0.935790i 0.404991 + 0.794840i −0.612737 + 2.76126i 2.46467 + 0.390365i −3.82561 + 0.836183i
19.8 −1.19190 + 0.761162i −0.217455 + 2.76302i 0.841265 − 1.81446i 1.21827 + 0.292481i −1.84392 − 3.45877i 0.587203 + 1.15245i 0.378394 + 2.80300i −4.62394 − 0.732361i −1.67469 + 0.578692i
19.9 −1.17122 − 0.792623i −0.0864834 + 1.09888i 0.743499 + 1.85667i −1.85905 − 0.446319i 0.972285 − 1.21847i 0.571540 + 1.12171i 0.600838 − 2.76387i 1.76302 + 0.279234i 1.82359 + 1.99626i
19.10 −1.11604 − 0.868590i −0.259149 + 3.29279i 0.491101 + 1.93877i −1.78730 − 0.429093i 3.14931 − 3.44980i −2.10638 − 4.13400i 1.13591 − 2.59031i −7.81227 − 1.23734i 1.62200 + 2.03132i
19.11 −0.938875 − 1.05760i 0.233178 − 2.96281i −0.237028 + 1.98590i −1.45205 − 0.348606i −3.35239 + 2.53510i −0.603345 − 1.18413i 2.32283 − 1.61384i −5.76080 − 0.912421i 0.994606 + 1.86298i
19.12 −0.892958 − 1.09664i 0.121306 − 1.54134i −0.405253 + 1.95851i 0.448720 + 0.107728i −1.79862 + 1.24332i 1.16515 + 2.28673i 2.50966 − 1.30445i 0.602046 + 0.0953547i −0.282549 − 0.588282i
19.13 −0.873487 + 1.11221i 0.221517 − 2.81464i −0.474042 − 1.94301i −1.54483 − 0.370880i 2.93699 + 2.70493i −1.34669 − 2.64302i 2.57511 + 1.16996i −4.91008 − 0.777680i 1.76188 − 1.39422i
19.14 −0.823127 − 1.14998i −0.176573 + 2.24358i −0.644925 + 1.89316i 3.36360 + 0.807530i 2.72542 − 1.64369i 0.830538 + 1.63002i 2.70796 − 0.816660i −2.03939 − 0.323008i −1.84003 − 4.53279i
19.15 −0.821434 + 1.15119i −0.0829010 + 1.05336i −0.650493 − 1.89126i 2.88799 + 0.693345i −1.14452 − 0.960699i −1.79651 − 3.52584i 2.71154 + 0.804701i 1.86038 + 0.294654i −3.17047 + 2.75510i
19.16 −0.730601 + 1.21088i 0.222334 − 2.82502i −0.932444 − 1.76934i 2.89506 + 0.695043i 3.25831 + 2.33318i 2.11218 + 4.14539i 2.82369 + 0.163605i −4.96824 − 0.786892i −2.95675 + 2.99776i
19.17 −0.618838 + 1.27163i 0.00409177 − 0.0519908i −1.23408 − 1.57387i −2.91327 − 0.699414i 0.0635809 + 0.0373771i −0.210704 − 0.413530i 2.76507 − 0.595320i 2.96038 + 0.468878i 2.69224 − 3.27177i
19.18 −0.528312 − 1.31183i −0.00496317 + 0.0630631i −1.44177 + 1.38611i 0.438953 + 0.105383i 0.0853499 − 0.0268061i −1.74049 − 3.41591i 2.58004 + 1.15906i 2.95911 + 0.468677i −0.0936596 − 0.631506i
19.19 −0.367801 − 1.36555i −0.0569089 + 0.723097i −1.72945 + 1.00450i −2.99198 − 0.718310i 1.00835 − 0.188244i −0.251406 − 0.493413i 2.00778 + 1.99219i 2.44343 + 0.387002i 0.119564 + 4.34988i
19.20 −0.362021 + 1.36709i −0.202587 + 2.57412i −1.73788 − 0.989833i −1.06398 − 0.255438i −3.44571 − 1.20884i −1.31451 − 2.57988i 1.98234 − 2.01750i −3.62197 − 0.573664i 0.734389 − 1.36208i
See next 80 embeddings (of 736 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
32.h odd 8 1 inner
352.bc even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.bc.a ✓ 736
11.d odd 10 1 inner 352.2.bc.a ✓ 736
32.h odd 8 1 inner 352.2.bc.a ✓ 736
352.bc even 40 1 inner 352.2.bc.a ✓ 736
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.bc.a ✓ 736 1.a even 1 1 trivial
352.2.bc.a ✓ 736 11.d odd 10 1 inner
352.2.bc.a ✓ 736 32.h odd 8 1 inner
352.2.bc.a ✓ 736 352.bc even 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).