LMFDB - Embedded newform 3528.2.s.w.3313.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, not 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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3528.3313
Dual form			3528.2.s.w.361.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{5} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{5} +6.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +10.0000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-3.00000 + 5.19615i) q^{37} +2.00000 q^{41} -4.00000 q^{43} +(4.00000 - 6.92820i) q^{47} +(-5.00000 - 8.66025i) q^{53} +(6.00000 + 10.3923i) q^{59} +(1.00000 - 1.73205i) q^{61} +(6.00000 - 10.3923i) q^{65} +(-6.00000 - 10.3923i) q^{67} +12.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(4.00000 - 6.92820i) q^{79} -12.0000 q^{83} -4.00000 q^{85} +(-1.00000 + 1.73205i) q^{89} +(4.00000 + 6.92820i) q^{95} +10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 12 q^{13} - 2 q^{17} - 4 q^{19} - 4 q^{23} + q^{25} + 20 q^{29} + 8 q^{31} - 6 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{53} + 12 q^{59} + 2 q^{61} + 12 q^{65} - 12 q^{67} + 24 q^{71} + 14 q^{73} + 8 q^{79} - 24 q^{83} - 8 q^{85} - 2 q^{89} + 8 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 12 * q^13 - 2 * q^17 - 4 * q^19 - 4 * q^23 + q^25 + 20 * q^29 + 8 * q^31 - 6 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^53 + 12 * q^59 + 2 * q^61 + 12 * q^65 - 12 * q^67 + 24 * q^71 + 14 * q^73 + 8 * q^79 - 24 * q^83 - 8 * q^85 - 2 * q^89 + 8 * q^95 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	−0.550990	−	0.834512i	$-0.685750\pi$
$5$	1.00000	−	1.73205i	0.447214	−	0.774597i	0.998203	+	0.0599153i	$0.0190830\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		0			0
$13$	6.00000			1.66410			0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000			1.66410			0.832050	+	0.554700i	$0.187167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	0.718900	−	0.695113i	$-0.244646\pi$
$17$	−1.00000	−	1.73205i	−0.242536	−	0.420084i	−0.961436	+	0.275029i	$0.911312\pi$
$18$		0			0
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	0.540068	+	0.841621i	$0.318398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	−0.995639	−	0.0932891i	$-0.970262\pi$
$23$	−2.00000	+	3.46410i	−0.417029	+	0.722315i	0.578610	+	0.815604i	$0.303595\pi$
$24$		0			0
$25$	0.500000	+	0.866025i	0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	10.0000			1.85695			0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000			1.85695			0.928477	+	0.371391i	$0.121119\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	$0.0884683\pi$
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	−0.243204	+	0.969975i	$0.578198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.00000	+	5.19615i	−0.493197	+	0.854242i	−0.999969	−	0.00783774i	$-0.997505\pi$
$37$	−3.00000	+	5.19615i	−0.493197	+	0.854242i	0.506772	+	0.862080i	$0.330838\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.00000			0.312348			0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000			0.312348			0.156174	+	0.987730i	$0.450084\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	−0.972867	−	0.231367i	$-0.925680\pi$
$53$	−5.00000	−	8.66025i	−0.686803	−	1.18958i	0.286064	−	0.958211i	$-0.407653\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	0.931282	+	0.364299i	$0.118692\pi$
$59$	6.00000	+	10.3923i	0.781133	+	1.35296i	−0.150148	+	0.988663i	$0.547975\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	−	10.3923i	0.744208	−	1.28901i
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	0.906208	+	0.422833i	$0.138964\pi$
$73$	7.00000	+	12.1244i	0.819288	+	1.41905i	−0.0869195	+	0.996215i	$0.527702\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	−0.548352	−	0.836247i	$-0.684745\pi$
$79$	4.00000	−	6.92820i	0.450035	−	0.779484i	0.998388	+	0.0567635i	$0.0180781\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	−4.00000			−0.433861
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.00000	+	1.73205i	−0.106000	+	0.183597i	−0.914146	−	0.405385i	$-0.867138\pi$
$89$	−1.00000	+	1.73205i	−0.106000	+	0.183597i	0.808146	+	0.588982i	$0.200471\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.00000	+	6.92820i	0.410391	+	0.710819i
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	0.909935	−	0.414751i	$-0.136131\pi$
$109$	1.00000	+	1.73205i	0.0957826	+	0.165900i	−0.814152	+	0.580651i	$0.802798\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$	4.00000	+	6.92820i	0.373002	+	0.646058i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	−	9.52628i	0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	12.0000			1.07331
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	0.905577	−	0.424182i	$-0.139438\pi$
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	−0.820141	+	0.572161i	$0.806105\pi$
$138$		0			0
$139$	−12.0000			−1.01783			−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000			−1.01783			−0.508913	+	0.860818i	$0.669953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	10.0000	−	17.3205i	0.830455	−	1.43839i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.0000	−	19.0526i	0.901155	−	1.56085i	0.0751583	−	0.997172i	$-0.476054\pi$
$149$	11.0000	−	19.0526i	0.901155	−	1.56085i	0.825997	−	0.563675i	$-0.190613\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0000			1.28515
$156$		0			0
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	0.961681	+	0.274169i	$0.0884028\pi$
$157$	9.00000	+	15.5885i	0.718278	+	1.24409i	−0.243403	+	0.969925i	$0.578264\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	−0.146772	−	0.989170i	$-0.546888\pi$
$163$	10.0000	−	17.3205i	0.783260	−	1.35665i	0.930033	−	0.367477i	$-0.119778\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	23.0000			1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	−0.825505	−	0.564396i	$-0.809109\pi$
$173$	1.00000	−	1.73205i	0.0760286	−	0.131685i	0.901533	+	0.432710i	$0.142443\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	−0.993124	−	0.117071i	$-0.962650\pi$
$179$	−8.00000	−	13.8564i	−0.597948	−	1.03568i	0.395175	−	0.918606i	$-0.370684\pi$
$180$		0			0
$181$	14.0000			1.04061			0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000			1.04061			0.520306	+	0.853980i	$0.325818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	6.00000	+	10.3923i	0.441129	+	0.764057i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	0.235983	+	0.971757i	$0.424169\pi$
$191$	−10.0000	+	17.3205i	−0.723575	+	1.25327i	−0.959558	+	0.281511i	$0.909164\pi$
$192$		0			0
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	−0.983639	−	0.180150i	$-0.942342\pi$
$193$	−9.00000	−	15.5885i	−0.647834	−	1.12208i	0.335805	−	0.941932i	$-0.390992\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.00000			−0.427482			−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000			−0.427482			−0.213741	+	0.976890i	$0.568565\pi$
$198$		0			0
$199$	−4.00000	−	6.92820i	−0.283552	−	0.491127i	0.688705	−	0.725042i	$-0.258180\pi$
$199$	−4.00000	−	6.92820i	−0.283552	−	0.491127i	−0.972257	+	0.233915i	$0.924846\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000	−	3.46410i	0.139686	−	0.241943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	+	6.92820i	−0.272798	+	0.472500i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6.00000	−	10.3923i	−0.403604	−	0.699062i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.0000	+	17.3205i	0.663723	+	1.14960i	0.979630	+	0.200812i	$0.0643581\pi$
$227$	10.0000	+	17.3205i	0.663723	+	1.14960i	−0.315906	+	0.948790i	$0.602309\pi$
$228$		0			0
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	−0.999088	−	0.0426906i	$-0.986407\pi$
$229$	−7.00000	+	12.1244i	−0.462573	+	0.801200i	0.536515	+	0.843891i	$0.319740\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00000	−	1.73205i	0.0655122	−	0.113470i	−0.831409	−	0.555661i	$-0.812465\pi$
$233$	1.00000	−	1.73205i	0.0655122	−	0.113470i	0.896921	+	0.442191i	$0.145799\pi$
$234$		0			0
$235$	−8.00000	−	13.8564i	−0.521862	−	0.903892i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	0.658838	−	0.752285i	$-0.271048\pi$
$241$	−5.00000	−	8.66025i	−0.322078	−	0.557856i	−0.980917	+	0.194429i	$0.937715\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−12.0000	+	20.7846i	−0.763542	+	1.32249i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	20.0000			1.26239			0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000			1.26239			0.631194	+	0.775625i	$0.282565\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	−0.757159	−	0.653231i	$-0.773413\pi$
$257$	3.00000	−	5.19615i	0.187135	−	0.324127i	0.944294	+	0.329104i	$0.106747\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	0.989561	−	0.144112i	$-0.0460326\pi$
$263$	6.00000	+	10.3923i	0.369976	+	0.640817i	−0.619586	+	0.784929i	$0.712699\pi$
$264$		0			0
$265$	−20.0000			−1.22859
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.00000	−	5.19615i	−0.182913	−	0.316815i	0.759958	−	0.649972i	$-0.225219\pi$
$269$	−3.00000	−	5.19615i	−0.182913	−	0.316815i	−0.942871	+	0.333157i	$0.891886\pi$
$270$		0			0
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	−0.961563	−	0.274586i	$-0.911459\pi$
$271$	−4.00000	+	6.92820i	−0.242983	+	0.420858i	0.718580	+	0.695444i	$0.244792\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	−0.980373	−	0.197153i	$-0.936830\pi$
$277$	−11.0000	−	19.0526i	−0.660926	−	1.14476i	0.319447	−	0.947604i	$-0.396503\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.50000	−	11.2583i	0.382353	−	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$	24.0000			1.39733
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000	+	20.7846i	−0.693978	+	1.20201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.00000	−	3.46410i	−0.114520	−	0.198354i
$306$		0			0
$307$	4.00000			0.228292			0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000			0.228292			0.114146	+	0.993464i	$0.463587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	11.0000	−	19.0526i	0.621757	−	1.07691i	−0.367402	−	0.930062i	$-0.619753\pi$
$313$	11.0000	−	19.0526i	0.621757	−	1.07691i	0.989158	−	0.146852i	$-0.0469141\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000			0.445132
$324$		0			0
$325$	3.00000	+	5.19615i	0.166410	+	0.288231i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	−0.805812	−	0.592172i	$-0.798271\pi$
$331$	2.00000	−	3.46410i	0.109930	−	0.190404i	0.915742	+	0.401768i	$0.131604\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−24.0000			−1.31126
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	0.984487	+	0.175457i	$0.0561403\pi$
$347$	12.0000	+	20.7846i	0.644194	+	1.11578i	−0.340293	+	0.940319i	$0.610526\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	0.920677	+	0.390324i	$0.127637\pi$
$353$	15.0000	+	25.9808i	0.798369	+	1.38282i	−0.122308	+	0.992492i	$0.539030\pi$
$354$		0			0
$355$	12.0000	−	20.7846i	0.636894	−	1.10313i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−6.00000	+	10.3923i	−0.316668	+	0.548485i	−0.979791	−	0.200026i	$-0.935897\pi$
$359$	−6.00000	+	10.3923i	−0.316668	+	0.548485i	0.663123	+	0.748511i	$0.269231\pi$
$360$		0			0
$361$	1.50000	+	2.59808i	0.0789474	+	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	28.0000			1.46559
$366$		0			0
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	0.995697	−	0.0926670i	$-0.0295392\pi$
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	−0.578101	+	0.815966i	$0.696206\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	−0.707055	−	0.707159i	$-0.749977\pi$
$373$	5.00000	−	8.66025i	0.258890	−	0.448411i	0.965945	+	0.258748i	$0.0833099\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	60.0000			3.09016
$378$		0			0
$379$	12.0000			0.616399			0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000			0.616399			0.308199	+	0.951322i	$0.400274\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.00000	−	1.73205i	−0.0507020	−	0.0878185i	0.839561	−	0.543266i	$-0.182813\pi$
$389$	−1.00000	−	1.73205i	−0.0507020	−	0.0878185i	−0.890263	+	0.455448i	$0.849479\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−8.00000	−	13.8564i	−0.402524	−	0.697191i
$396$		0			0
$397$	1.00000	−	1.73205i	0.0501886	−	0.0869291i	−0.839840	−	0.542834i	$-0.817351\pi$
$397$	1.00000	−	1.73205i	0.0501886	−	0.0869291i	0.890028	+	0.455905i	$0.150684\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−3.00000	+	5.19615i	−0.149813	+	0.259483i	0.781345	+	0.624099i	$0.214534\pi$
$402$		0			0
$403$	24.0000	+	41.5692i	1.19553	+	2.07071i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−9.00000	−	15.5885i	−0.445021	−	0.770800i	0.553032	−	0.833160i	$-0.313471\pi$
$409$	−9.00000	−	15.5885i	−0.445021	−	0.770800i	−0.998054	+	0.0623602i	$0.980137\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−12.0000	+	20.7846i	−0.589057	+	1.02028i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	−26.0000			−1.26716			−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000			−1.26716			−0.633581	+	0.773676i	$0.718416\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.00000	−	1.73205i	0.0485071	−	0.0840168i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	0.973574	−	0.228373i	$-0.0733406\pi$
$431$	6.00000	+	10.3923i	0.289010	+	0.500580i	−0.684564	+	0.728953i	$0.740007\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.00000	−	13.8564i	−0.382692	−	0.662842i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	4.00000	−	6.92820i	0.190046	−	0.329169i	−0.755219	−	0.655472i	$-0.772470\pi$
$443$	4.00000	−	6.92820i	0.190046	−	0.329169i	0.945265	+	0.326303i	$0.105803\pi$
$444$		0			0
$445$	2.00000	+	3.46410i	0.0948091	+	0.164214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.0000			−1.60456			−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000			−1.60456			−0.802280	+	0.596948i	$0.796380\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.0000			−0.838344			−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000			−0.838344			−0.419172	+	0.907907i	$0.637680\pi$
$462$		0			0
$463$	−40.0000			−1.85896			−0.929479	−	0.368875i	$-0.879743\pi$
$463$	−40.0000			−1.85896			−0.929479	+	0.368875i	$0.879743\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.0000	−	31.1769i	0.832941	−	1.44270i	−0.0627555	−	0.998029i	$-0.519989\pi$
$467$	18.0000	−	31.1769i	0.832941	−	1.44270i	0.895696	−	0.444667i	$-0.146678\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−4.00000			−0.183533
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.00000	−	6.92820i	−0.182765	−	0.316558i	0.760056	−	0.649857i	$-0.225171\pi$
$479$	−4.00000	−	6.92820i	−0.182765	−	0.316558i	−0.942821	+	0.333300i	$0.891838\pi$
$480$		0			0
$481$	−18.0000	+	31.1769i	−0.820729	+	1.42154i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	10.0000	−	17.3205i	0.454077	−	0.786484i
$486$		0			0
$487$	4.00000	+	6.92820i	0.181257	+	0.313947i	0.942309	−	0.334744i	$-0.108650\pi$
$487$	4.00000	+	6.92820i	0.181257	+	0.313947i	−0.761052	+	0.648691i	$0.775317\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$	−10.0000	−	17.3205i	−0.450377	−	0.780076i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	−0.907314	−	0.420455i	$-0.861871\pi$
$499$	−2.00000	+	3.46410i	−0.0895323	+	0.155074i	0.817781	+	0.575529i	$0.195204\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−40.0000			−1.78351			−0.891756	−	0.452517i	$-0.850526\pi$
$503$	−40.0000			−1.78351			−0.891756	+	0.452517i	$0.850526\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$		0			0
$508$		0			0
$509$	21.0000	−	36.3731i	0.930809	−	1.61221i	0.148866	−	0.988857i	$-0.452438\pi$
$509$	21.0000	−	36.3731i	0.930809	−	1.61221i	0.781943	−	0.623350i	$-0.214229\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	0.924229	−	0.381839i	$-0.124709\pi$
$521$	3.00000	+	5.19615i	0.131432	+	0.227648i	−0.792797	+	0.609486i	$0.791376\pi$
$522$		0			0
$523$	−10.0000	+	17.3205i	−0.437269	+	0.757373i	−0.997478	−	0.0709788i	$-0.977388\pi$
$523$	−10.0000	+	17.3205i	−0.437269	+	0.757373i	0.560208	+	0.828352i	$0.310721\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.00000	−	13.8564i	0.348485	−	0.603595i
$528$		0			0
$529$	3.50000	+	6.06218i	0.152174	+	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	12.0000			0.519778
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	−0.843728	−	0.536771i	$-0.819644\pi$
$541$	1.00000	−	1.73205i	0.0429934	−	0.0744667i	0.886721	+	0.462304i	$0.152977\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	4.00000			0.171341
$546$		0			0
$547$	−20.0000			−0.855138			−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000			−0.855138			−0.427569	+	0.903983i	$0.640630\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−20.0000	+	34.6410i	−0.852029	+	1.47576i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.00000	−	1.73205i	−0.0423714	−	0.0733893i	0.844062	−	0.536246i	$-0.180158\pi$
$557$	−1.00000	−	1.73205i	−0.0423714	−	0.0733893i	−0.886433	+	0.462856i	$0.846825\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	0.905088	−	0.425223i	$-0.139804\pi$
$563$	2.00000	+	3.46410i	0.0842900	+	0.145994i	−0.820798	+	0.571218i	$0.806471\pi$
$564$		0			0
$565$	−18.0000	+	31.1769i	−0.757266	+	1.31162i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	0.358954	+	0.933355i	$0.383134\pi$
$569$	−15.0000	+	25.9808i	−0.628833	+	1.08917i	−0.987786	+	0.155815i	$0.950200\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.00000			−0.166812
$576$		0			0
$577$	−9.00000	−	15.5885i	−0.374675	−	0.648956i	0.615603	−	0.788056i	$-0.288912\pi$
$577$	−9.00000	−	15.5885i	−0.374675	−	0.648956i	−0.990278	+	0.139100i	$0.955579\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−28.0000			−1.15568			−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000			−1.15568			−0.577842	+	0.816149i	$0.696105\pi$
$588$		0			0
$589$	−32.0000			−1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.00000	+	15.5885i	−0.369586	+	0.640141i	−0.989501	−	0.144528i	$-0.953834\pi$
$593$	−9.00000	+	15.5885i	−0.369586	+	0.640141i	0.619915	+	0.784669i	$0.287167\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.0000	−	38.1051i	−0.898896	−	1.55693i	−0.828908	−	0.559385i	$-0.811037\pi$
$599$	−22.0000	−	38.1051i	−0.898896	−	1.55693i	−0.0699877	−	0.997548i	$-0.522296\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.0000	−	19.0526i	−0.447214	−	0.774597i
$606$		0			0
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	−0.981454	−	0.191700i	$-0.938600\pi$
$607$	−8.00000	+	13.8564i	−0.324710	+	0.562414i	0.656744	+	0.754114i	$0.271933\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24.0000	−	41.5692i	0.970936	−	1.68171i
$612$		0			0
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	0.999574	+	0.0291886i	$0.00929235\pi$
$613$	13.0000	+	22.5167i	0.525065	+	0.909439i	−0.474509	+	0.880251i	$0.657374\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	30.0000			1.20775			0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000			1.20775			0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	6.00000	+	10.3923i	0.241160	+	0.417702i	0.961045	−	0.276392i	$-0.0891387\pi$
$619$	6.00000	+	10.3923i	0.241160	+	0.417702i	−0.719885	+	0.694094i	$0.755805\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.50000	−	16.4545i	0.380000	−	0.658179i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	12.0000			0.478471
$630$		0			0
$631$	−16.0000			−0.636950			−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000			−0.636950			−0.318475	+	0.947931i	$0.603171\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	−	13.8564i	0.317470	−	0.549875i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.00000	+	8.66025i	0.197488	+	0.342059i	0.947713	−	0.319123i	$-0.103388\pi$
$641$	5.00000	+	8.66025i	0.197488	+	0.342059i	−0.750225	+	0.661182i	$0.770055\pi$
$642$		0			0
$643$	4.00000			0.157745			0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000			0.157745			0.0788723	+	0.996885i	$0.474868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000	−	41.5692i	−0.943537	−	1.63425i	−0.758654	−	0.651494i	$-0.774142\pi$
$647$	−24.0000	−	41.5692i	−0.943537	−	1.63425i	−0.184884	−	0.982760i	$-0.559191\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	−0.947118	−	0.320884i	$-0.896020\pi$
$653$	−5.00000	+	8.66025i	−0.195665	+	0.338902i	0.751453	+	0.659786i	$0.229353\pi$
$654$		0			0
$655$	4.00000	+	6.92820i	0.156293	+	0.270707i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−8.00000			−0.311636			−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000			−0.311636			−0.155818	+	0.987786i	$0.549801\pi$
$660$		0			0
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	0.946729	−	0.322031i	$-0.104366\pi$
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	−0.752252	+	0.658876i	$0.771032\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−20.0000	+	34.6410i	−0.774403	+	1.34131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	−0.917899	−	0.396813i	$-0.870116\pi$
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	0.802600	+	0.596518i	$0.203449\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	0.539750	−	0.841825i	$-0.318519\pi$
$683$	−12.0000	−	20.7846i	−0.459167	−	0.795301i	−0.998917	+	0.0465244i	$0.985185\pi$
$684$		0			0
$685$	4.00000			0.152832
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.0000	−	51.9615i	−1.14291	−	1.97958i
$690$		0			0
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	0.0555386	+	0.998457i	$0.482312\pi$
$691$	−22.0000	+	38.1051i	−0.836919	+	1.44959i	−0.892458	+	0.451130i	$0.851021\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000	+	20.7846i	−0.455186	+	0.788405i
$696$		0			0
$697$	−2.00000	−	3.46410i	−0.0757554	−	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−22.0000			−0.830929			−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		0			0
$703$	−12.0000	−	20.7846i	−0.452589	−	0.783906i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	0.582115	+	0.813107i	$0.302225\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−32.0000			−1.19841
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.00000	+	8.66025i	0.185695	+	0.321634i
$726$		0			0
$727$	−16.0000			−0.593407			−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000			−0.593407			−0.296704	+	0.954970i	$0.595887\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.00000	+	6.92820i	0.147945	+	0.256249i
$732$		0			0
$733$	13.0000	−	22.5167i	0.480166	−	0.831672i	−0.519575	−	0.854425i	$-0.673910\pi$
$733$	13.0000	−	22.5167i	0.480166	−	0.831672i	0.999741	+	0.0227529i	$0.00724310\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	0.989230	−	0.146369i	$-0.0467586\pi$
$739$	10.0000	+	17.3205i	0.367856	+	0.637145i	−0.621374	+	0.783514i	$0.713425\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−20.0000			−0.733729			−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000			−0.733729			−0.366864	+	0.930274i	$0.619569\pi$
$744$		0			0
$745$	−22.0000	−	38.1051i	−0.806018	−	1.39606i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	0.411170	+	0.911559i	$0.365120\pi$
$751$	−16.0000	+	27.7128i	−0.583848	+	1.01125i	−0.995018	+	0.0996961i	$0.968213\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	16.0000			0.582300
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.00000	+	15.5885i	−0.326250	+	0.565081i	−0.981764	−	0.190101i	$-0.939118\pi$
$761$	−9.00000	+	15.5885i	−0.326250	+	0.565081i	0.655515	+	0.755182i	$0.272452\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	36.0000	+	62.3538i	1.29988	+	2.25147i
$768$		0			0
$769$	34.0000			1.22607			0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000			1.22607			0.613036	+	0.790055i	$0.289948\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.00000	+	8.66025i	0.179838	+	0.311488i	0.941825	−	0.336104i	$-0.109109\pi$
$773$	5.00000	+	8.66025i	0.179838	+	0.311488i	−0.761987	+	0.647592i	$0.775776\pi$
$774$		0			0
$775$	−4.00000	+	6.92820i	−0.143684	+	0.248868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.00000	+	6.92820i	−0.143315	+	0.248229i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	36.0000			1.28490
$786$		0			0
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	0.500953	−	0.865474i	$-0.332983\pi$
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	−0.999999	+	0.00110111i	$0.999650\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.00000	−	10.3923i	0.213066	−	0.369042i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$	−16.0000			−0.566039
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.00000	−	5.19615i	−0.105474	−	0.182687i	0.808458	−	0.588555i	$-0.200303\pi$
$809$	−3.00000	−	5.19615i	−0.105474	−	0.182687i	−0.913932	+	0.405868i	$0.866969\pi$
$810$		0			0
$811$	36.0000			1.26413			0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000			1.26413			0.632065	+	0.774915i	$0.282207\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−20.0000	−	34.6410i	−0.700569	−	1.21342i
$816$		0			0
$817$	8.00000	−	13.8564i	0.279885	−	0.484774i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	−0.808915	−	0.587925i	$-0.799945\pi$
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	0.913616	+	0.406578i	$0.133278\pi$
$822$		0			0
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	0.971102	−	0.238664i	$-0.0767093\pi$
$823$	8.00000	+	13.8564i	0.278862	+	0.483004i	−0.692240	+	0.721668i	$0.743376\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−19.0000	−	32.9090i	−0.659897	−	1.14298i	−0.980642	−	0.195810i	$-0.937266\pi$
$829$	−19.0000	−	32.9090i	−0.659897	−	1.14298i	0.320745	−	0.947166i	$-0.396067\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	71.0000			2.44828
$842$		0			0
$843$		0			0
$844$		0			0
$845$	23.0000	−	39.8372i	0.791224	−	1.37044i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−12.0000	−	20.7846i	−0.411355	−	0.712487i
$852$		0			0
$853$	−26.0000			−0.890223			−0.445112	−	0.895475i	$-0.646836\pi$
$853$	−26.0000			−0.890223			−0.445112	+	0.895475i	$0.646836\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	0.670366	−	0.742030i	$-0.266137\pi$
$857$	−9.00000	−	15.5885i	−0.307434	−	0.532492i	−0.977800	+	0.209539i	$0.932804\pi$
$858$		0			0
$859$	10.0000	−	17.3205i	0.341196	−	0.590968i	−0.643459	−	0.765480i	$-0.722501\pi$
$859$	10.0000	−	17.3205i	0.341196	−	0.590968i	0.984655	+	0.174512i	$0.0558348\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	10.0000	−	17.3205i	0.340404	−	0.589597i	−0.644104	−	0.764938i	$-0.722770\pi$
$863$	10.0000	−	17.3205i	0.340404	−	0.589597i	0.984508	+	0.175341i	$0.0561028\pi$
$864$		0			0
$865$	−2.00000	−	3.46410i	−0.0680020	−	0.117783i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−36.0000	−	62.3538i	−1.21981	−	2.11278i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	−0.959671	−	0.281126i	$-0.909292\pi$
$877$	−7.00000	+	12.1244i	−0.236373	+	0.409410i	0.723298	+	0.690536i	$0.242625\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		0			0
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.0000	+	41.5692i	−0.805841	+	1.39576i	0.109881	+	0.993945i	$0.464953\pi$
$887$	−24.0000	+	41.5692i	−0.805841	+	1.39576i	−0.915722	+	0.401813i	$0.868380\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.0000	+	27.7128i	0.535420	+	0.927374i
$894$		0			0
$895$	−32.0000			−1.06964
$896$		0			0
$897$		0			0
$898$		0			0
$899$	40.0000	+	69.2820i	1.33407	+	2.31069i
$900$		0			0
$901$	−10.0000	+	17.3205i	−0.333148	+	0.577030i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	14.0000	−	24.2487i	0.465376	−	0.806054i
$906$		0			0
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	−0.956671	−	0.291172i	$-0.905955\pi$
$907$	−22.0000	−	38.1051i	−0.730498	−	1.26526i	0.226173	−	0.974087i	$-0.427379\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	20.0000			0.662630			0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000			0.662630			0.331315	+	0.943520i	$0.392508\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	4.00000	−	6.92820i	0.131948	−	0.228540i	−0.792480	−	0.609898i	$-0.791210\pi$
$919$	4.00000	−	6.92820i	0.131948	−	0.228540i	0.924427	+	0.381358i	$0.124544\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	72.0000			2.36991
$924$		0			0
$925$	−6.00000			−0.197279
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.0000	+	22.5167i	−0.426516	+	0.738748i	−0.996561	−	0.0828661i	$-0.973593\pi$
$929$	−13.0000	+	22.5167i	−0.426516	+	0.738748i	0.570045	+	0.821614i	$0.306926\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−54.0000			−1.76410			−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000			−1.76410			−0.882052	+	0.471153i	$0.843838\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	21.0000	+	36.3731i	0.684580	+	1.18573i	0.973568	+	0.228395i	$0.0733479\pi$
$941$	21.0000	+	36.3731i	0.684580	+	1.18573i	−0.288988	+	0.957333i	$0.593319\pi$
$942$		0			0
$943$	−4.00000	+	6.92820i	−0.130258	+	0.225613i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.0000	+	48.4974i	−0.909878	+	1.57595i	−0.0956453	+	0.995415i	$0.530491\pi$
$947$	−28.0000	+	48.4974i	−0.909878	+	1.57595i	−0.814232	+	0.580539i	$0.802842\pi$
$948$		0			0
$949$	42.0000	+	72.7461i	1.36338	+	2.36144i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−58.0000			−1.87880			−0.939402	−	0.342817i	$-0.888619\pi$
$953$	−58.0000			−1.87880			−0.939402	+	0.342817i	$0.888619\pi$
$954$		0			0
$955$	20.0000	+	34.6410i	0.647185	+	1.12096i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.0000			−1.15888
$966$		0			0
$967$	−48.0000			−1.54358			−0.771788	−	0.635880i	$-0.780637\pi$
$967$	−48.0000			−1.54358			−0.771788	+	0.635880i	$0.780637\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.00000	−	3.46410i	0.0641831	−	0.111168i	−0.832148	−	0.554553i	$-0.812889\pi$
$971$	2.00000	−	3.46410i	0.0641831	−	0.111168i	0.896331	+	0.443385i	$0.146223\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	0.814038	−	0.580812i	$-0.197265\pi$
$977$	−3.00000	−	5.19615i	−0.0959785	−	0.166240i	−0.910017	+	0.414572i	$0.863931\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	0.608712	−	0.793391i	$-0.291686\pi$
$983$	−12.0000	−	20.7846i	−0.382741	−	0.662926i	−0.991453	+	0.130465i	$0.958353\pi$
$984$		0			0
$985$	−6.00000	+	10.3923i	−0.191176	+	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.00000	−	13.8564i	0.254385	−	0.440608i
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.0000			−0.507234
$996$		0			0
$997$	13.0000	+	22.5167i	0.411714	+	0.713110i	0.995077	−	0.0991016i	$-0.0315969\pi$
$997$	13.0000	+	22.5167i	0.411714	+	0.713110i	−0.583363	+	0.812211i	$0.698264\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.w.3313.1		2
3.2	odd	2		1176.2.q.f.961.1		2
7.2	even	3		504.2.a.e.1.1		1
7.3	odd	6		3528.2.s.g.361.1		2
7.4	even	3	inner	3528.2.s.w.361.1		2
7.5	odd	6		3528.2.a.v.1.1		1
7.6	odd	2		3528.2.s.g.3313.1		2
12.11	even	2		2352.2.q.d.961.1		2
21.2	odd	6		168.2.a.a.1.1	✓	1
21.5	even	6		1176.2.a.f.1.1		1
21.11	odd	6		1176.2.q.f.361.1		2
21.17	even	6		1176.2.q.d.361.1		2
21.20	even	2		1176.2.q.d.961.1		2
28.19	even	6		7056.2.a.bq.1.1		1
28.23	odd	6		1008.2.a.b.1.1		1
56.37	even	6		4032.2.a.bh.1.1		1
56.51	odd	6		4032.2.a.bc.1.1		1
84.11	even	6		2352.2.q.d.1537.1		2
84.23	even	6		336.2.a.e.1.1		1
84.47	odd	6		2352.2.a.c.1.1		1
84.59	odd	6		2352.2.q.w.1537.1		2
84.83	odd	2		2352.2.q.w.961.1		2
105.2	even	12		4200.2.t.j.1849.2		2
105.23	even	12		4200.2.t.j.1849.1		2
105.44	odd	6		4200.2.a.t.1.1		1
168.5	even	6		9408.2.a.be.1.1		1
168.107	even	6		1344.2.a.b.1.1		1
168.131	odd	6		9408.2.a.da.1.1		1
168.149	odd	6		1344.2.a.m.1.1		1
336.107	even	12		5376.2.c.bb.2689.1		2
336.149	odd	12		5376.2.c.d.2689.2		2
336.275	even	12		5376.2.c.bb.2689.2		2
336.317	odd	12		5376.2.c.d.2689.1		2
420.359	even	6		8400.2.a.y.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.a.a.1.1	✓	1	21.2	odd	6
336.2.a.e.1.1		1	84.23	even	6
504.2.a.e.1.1		1	7.2	even	3
1008.2.a.b.1.1		1	28.23	odd	6
1176.2.a.f.1.1		1	21.5	even	6
1176.2.q.d.361.1		2	21.17	even	6
1176.2.q.d.961.1		2	21.20	even	2
1176.2.q.f.361.1		2	21.11	odd	6
1176.2.q.f.961.1		2	3.2	odd	2
1344.2.a.b.1.1		1	168.107	even	6
1344.2.a.m.1.1		1	168.149	odd	6
2352.2.a.c.1.1		1	84.47	odd	6
2352.2.q.d.961.1		2	12.11	even	2
2352.2.q.d.1537.1		2	84.11	even	6
2352.2.q.w.961.1		2	84.83	odd	2
2352.2.q.w.1537.1		2	84.59	odd	6
3528.2.a.v.1.1		1	7.5	odd	6
3528.2.s.g.361.1		2	7.3	odd	6
3528.2.s.g.3313.1		2	7.6	odd	2
3528.2.s.w.361.1		2	7.4	even	3	inner
3528.2.s.w.3313.1		2	1.1	even	1	trivial
4032.2.a.bc.1.1		1	56.51	odd	6
4032.2.a.bh.1.1		1	56.37	even	6
4200.2.a.t.1.1		1	105.44	odd	6
4200.2.t.j.1849.1		2	105.23	even	12
4200.2.t.j.1849.2		2	105.2	even	12
5376.2.c.d.2689.1		2	336.317	odd	12
5376.2.c.d.2689.2		2	336.149	odd	12
5376.2.c.bb.2689.1		2	336.107	even	12
5376.2.c.bb.2689.2		2	336.275	even	12
7056.2.a.bq.1.1		1	28.19	even	6
8400.2.a.y.1.1		1	420.359	even	6
9408.2.a.be.1.1		1	168.5	even	6
9408.2.a.da.1.1		1	168.131	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{5} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{5} +6.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(-2.00000 + 3.46410i) q^{23} +(0.500000 + 0.866025i) q^{25} +10.0000 q^{29} +(4.00000 + 6.92820i) q^{31} +(-3.00000 + 5.19615i) q^{37} +2.00000 q^{41} -4.00000 q^{43} +(4.00000 - 6.92820i) q^{47} +(-5.00000 - 8.66025i) q^{53} +(6.00000 + 10.3923i) q^{59} +(1.00000 - 1.73205i) q^{61} +(6.00000 - 10.3923i) q^{65} +(-6.00000 - 10.3923i) q^{67} +12.0000 q^{71} +(7.00000 + 12.1244i) q^{73} +(4.00000 - 6.92820i) q^{79} -12.0000 q^{83} -4.00000 q^{85} +(-1.00000 + 1.73205i) q^{89} +(4.00000 + 6.92820i) q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 12 q^{13} - 2 q^{17} - 4 q^{19} - 4 q^{23} + q^{25} + 20 q^{29} + 8 q^{31} - 6 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{53} + 12 q^{59} + 2 q^{61} + 12 q^{65} - 12 q^{67} + 24 q^{71} + 14 q^{73} + 8 q^{79} - 24 q^{83} - 8 q^{85} - 2 q^{89} + 8 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 12 * q^13 - 2 * q^17 - 4 * q^19 - 4 * q^23 + q^25 + 20 * q^29 + 8 * q^31 - 6 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^53 + 12 * q^59 + 2 * q^61 + 12 * q^65 - 12 * q^67 + 24 * q^71 + 14 * q^73 + 8 * q^79 - 24 * q^83 - 8 * q^85 - 2 * q^89 + 8 * q^95 + 20 * q^97

Introduction

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