# 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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$403 = 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 403.cc (of order $$60$$, degree $$16$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$560q - 12q^{2} - 4q^{3} - 18q^{4} - 8q^{5} - 42q^{6} - 8q^{7} - 40q^{8} - 72q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$560q - 12q^{2} - 4q^{3} - 18q^{4} - 8q^{5} - 42q^{6} - 8q^{7} - 40q^{8} - 72q^{9} - 30q^{10} - 18q^{11} + 26q^{12} - 24q^{13} + 4q^{14} - 62q^{15} - 58q^{16} - 30q^{17} - 70q^{18} - 24q^{19} - 8q^{20} + 114q^{21} + 78q^{22} - 30q^{23} - 82q^{24} - 24q^{26} - 100q^{27} - 62q^{28} - 10q^{29} - 4q^{31} + 20q^{32} - 110q^{33} + 70q^{34} - 2q^{35} - 40q^{37} - 108q^{38} + 48q^{39} - 28q^{40} - 22q^{41} - 10q^{42} + 78q^{43} - 24q^{44} + 36q^{45} + 44q^{46} - 32q^{47} - 10q^{48} - 30q^{49} - 30q^{50} + 36q^{51} - 252q^{52} - 84q^{53} + 82q^{54} - 4q^{55} + 164q^{57} + 28q^{58} - 2q^{59} - 8q^{60} + 36q^{61} - 12q^{62} + 78q^{63} + 270q^{64} - 72q^{65} - 56q^{66} - 46q^{67} - 12q^{68} + 150q^{69} + 90q^{70} - 74q^{71} + 72q^{72} + 30q^{73} - 10q^{74} - 16q^{75} - 228q^{76} + 72q^{77} + 96q^{78} - 32q^{79} + 108q^{80} - 104q^{81} - 84q^{82} + 4q^{83} + 26q^{84} + 12q^{85} + 34q^{86} + 112q^{87} - 108q^{88} - 154q^{89} - 90q^{90} - 4q^{91} + 64q^{93} - 24q^{94} - 78q^{95} - 4q^{96} - 196q^{97} + 50q^{98} + 96q^{99} + O(q^{100})$$

## 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.

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 383.35 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(403, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database