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Properties

Label 403.2.cc.a
Level $403$
Weight $2$
Character orbit 403.cc
Analytic conductor $3.218$
Analytic rank $0$
Dimension $560$
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] = [403,2,Mod(24,403)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(403, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([35, 26]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("403.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 403 = 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 403.cc (of order \(60\), 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: \(3.21797120146\)
Analytic rank: \(0\)
Dimension: \(560\)
Relative dimension: \(35\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 560 q - 12 q^{2} - 4 q^{3} - 18 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 40 q^{8} - 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 560 q - 12 q^{2} - 4 q^{3} - 18 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 40 q^{8} - 72 q^{9} - 30 q^{10} - 18 q^{11} + 26 q^{12} - 24 q^{13} + 4 q^{14} - 62 q^{15} - 58 q^{16} - 30 q^{17} - 70 q^{18} - 24 q^{19} - 8 q^{20} + 114 q^{21} + 78 q^{22} - 30 q^{23} - 82 q^{24} - 24 q^{26} - 100 q^{27} - 62 q^{28} - 10 q^{29} - 4 q^{31} + 20 q^{32} - 110 q^{33} + 70 q^{34} - 2 q^{35} - 40 q^{37} - 108 q^{38} + 48 q^{39} - 28 q^{40} - 22 q^{41} - 10 q^{42} + 78 q^{43} - 24 q^{44} + 36 q^{45} + 44 q^{46} - 32 q^{47} - 10 q^{48} - 30 q^{49} - 30 q^{50} + 36 q^{51} - 252 q^{52} - 84 q^{53} + 82 q^{54} - 4 q^{55} + 164 q^{57} + 28 q^{58} - 2 q^{59} - 8 q^{60} + 36 q^{61} - 12 q^{62} + 78 q^{63} + 270 q^{64} - 72 q^{65} - 56 q^{66} - 46 q^{67} - 12 q^{68} + 150 q^{69} + 90 q^{70} - 74 q^{71} + 72 q^{72} + 30 q^{73} - 10 q^{74} - 16 q^{75} - 228 q^{76} + 72 q^{77} + 96 q^{78} - 32 q^{79} + 108 q^{80} - 104 q^{81} - 84 q^{82} + 4 q^{83} + 26 q^{84} + 12 q^{85} + 34 q^{86} + 112 q^{87} - 108 q^{88} - 154 q^{89} - 90 q^{90} - 4 q^{91} + 64 q^{93} - 24 q^{94} - 78 q^{95} - 4 q^{96} - 196 q^{97} + 50 q^{98} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −2.75171 + 0.144211i 2.34874 − 2.11482i 5.56208 − 0.584598i 0.937749 − 0.251269i −6.15808 + 6.15808i −0.0941873 + 0.594675i −9.77782 + 1.54865i 0.730552 − 6.95074i −2.54418 + 0.826654i
24.2 −2.75032 + 0.144138i −1.37473 + 1.23781i 5.55445 − 0.583796i −0.372115 + 0.0997080i 3.60254 − 3.60254i 0.787181 − 4.97006i −9.75199 + 1.54456i 0.0441185 − 0.419760i 1.00907 − 0.327865i
24.3 −2.48692 + 0.130334i −0.0686740 + 0.0618344i 4.17875 − 0.439204i −3.44926 + 0.924226i 0.162728 − 0.162728i −0.267734 + 1.69041i −5.41563 + 0.857752i −0.312693 + 2.97507i 8.45758 − 2.74803i
24.4 −2.30755 + 0.120933i 0.810915 − 0.730151i 3.32110 − 0.349061i 1.72963 − 0.463452i −1.78292 + 1.78292i 0.108867 − 0.687358i −3.05684 + 0.484156i −0.189123 + 1.79939i −3.93514 + 1.27861i
24.5 −2.17694 + 0.114088i 0.222847 − 0.200652i 2.73700 − 0.287670i 1.57540 − 0.422127i −0.462232 + 0.462232i −0.429485 + 2.71166i −1.61928 + 0.256469i −0.304186 + 2.89414i −3.38139 + 1.09868i
24.6 −2.04287 + 0.107062i −1.78599 + 1.60811i 2.17282 − 0.228372i 3.27471 − 0.877455i 3.47637 − 3.47637i 0.168500 − 1.06387i −0.373359 + 0.0591343i 0.290147 − 2.76056i −6.59586 + 2.14313i
24.7 −1.77883 + 0.0932245i 1.52098 − 1.36950i 1.16650 − 0.122604i −1.15023 + 0.308204i −2.57789 + 2.57789i 0.653142 − 4.12378i 1.45511 − 0.230467i 0.124273 − 1.18238i 2.01733 − 0.655471i
24.8 −1.62065 + 0.0849346i 2.22876 − 2.00678i 0.630243 − 0.0662412i −2.61289 + 0.700121i −3.44158 + 3.44158i −0.735803 + 4.64568i 2.19001 − 0.346863i 0.626598 − 5.96168i 4.17511 − 1.35657i
24.9 −1.59013 + 0.0833353i −1.52733 + 1.37521i 0.532532 − 0.0559714i −0.213994 + 0.0573396i 2.31405 − 2.31405i 0.126040 − 0.795786i 2.30329 − 0.364805i 0.127936 − 1.21723i 0.335501 − 0.109011i
24.10 −1.27699 + 0.0669244i −0.436672 + 0.393181i −0.362811 + 0.0381330i −3.05329 + 0.818127i 0.531314 − 0.531314i −0.0822901 + 0.519559i 2.98676 − 0.473056i −0.277494 + 2.64018i 3.84428 − 1.24908i
24.11 −1.22640 + 0.0642731i −0.451855 + 0.406852i −0.489109 + 0.0514074i −1.63845 + 0.439022i 0.528007 − 0.528007i 0.600453 − 3.79111i 3.02247 − 0.478713i −0.274941 + 2.61589i 1.98119 − 0.643727i
24.12 −1.22597 + 0.0642504i 2.01198 − 1.81160i −0.490166 + 0.0515185i 3.36984 − 0.902946i −2.35024 + 2.35024i 0.394798 − 2.49266i 3.02270 − 0.478748i 0.452603 − 4.30623i −4.07331 + 1.32350i
24.13 −0.715710 + 0.0375088i −1.58346 + 1.42575i −1.47821 + 0.155366i 1.89550 − 0.507897i 1.07982 − 1.07982i −0.656203 + 4.14310i 2.46788 − 0.390874i 0.160984 − 1.53166i −1.33758 + 0.434604i
24.14 −0.690768 + 0.0362016i 0.0986013 − 0.0887810i −1.51319 + 0.159043i 3.88062 − 1.03981i −0.0648966 + 0.0648966i −0.213808 + 1.34993i 2.40591 − 0.381058i −0.311745 + 2.96606i −2.64297 + 0.858752i
24.15 −0.652040 + 0.0341720i 1.57276 − 1.41612i −1.56505 + 0.164494i 0.675785 − 0.181076i −0.977108 + 0.977108i −0.0634857 + 0.400833i 2.30465 − 0.365021i 0.154592 − 1.47084i −0.434452 + 0.141162i
24.16 −0.320349 + 0.0167888i −1.75222 + 1.57771i −1.88670 + 0.198300i −2.60602 + 0.698280i 0.534834 − 0.534834i −0.645048 + 4.07267i 1.23475 − 0.195566i 0.267533 − 2.54541i 0.823112 − 0.267445i
24.17 −0.235412 + 0.0123374i 0.964102 − 0.868082i −1.93378 + 0.203248i −1.50977 + 0.404542i −0.216251 + 0.216251i −0.125388 + 0.791668i 0.918393 − 0.145459i −0.137658 + 1.30973i 0.350428 − 0.113861i
24.18 −0.216740 + 0.0113589i −2.41844 + 2.17758i −1.94220 + 0.204133i 1.43655 − 0.384922i 0.499439 − 0.499439i 0.462411 − 2.91955i 0.847363 − 0.134209i 0.793446 − 7.54913i −0.306985 + 0.0997454i
24.19 0.298096 − 0.0156226i −0.463276 + 0.417135i −1.90043 + 0.199743i 0.387663 − 0.103874i −0.131584 + 0.131584i 0.399618 − 2.52309i −1.15305 + 0.182625i −0.272963 + 2.59707i 0.113938 − 0.0370207i
24.20 0.547115 − 0.0286731i −1.75246 + 1.57792i −1.69053 + 0.177682i −2.48381 + 0.665536i −0.913555 + 0.913555i 0.442675 − 2.79494i −2.00206 + 0.317096i 0.267694 − 2.54694i −1.33985 + 0.435343i
See next 80 embeddings (of 560 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.35
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
403.cc even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 403.2.cc.a ✓ 560
13.f odd 12 1 403.2.ch.a yes 560
31.h odd 30 1 403.2.ch.a yes 560
403.cc even 60 1 inner 403.2.cc.a ✓ 560
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.cc.a ✓ 560 1.a even 1 1 trivial
403.2.cc.a ✓ 560 403.cc even 60 1 inner
403.2.ch.a yes 560 13.f odd 12 1
403.2.ch.a yes 560 31.h odd 30 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(403, [\chi])\).