LMFDB - Embedded newform 1078.2.c.a.1077.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1078,2,Mod(1077,1078)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1078, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("1078.1077");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1078 = 2 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1078.c (of order $2$, degree $1$, 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:	$8.60787333789$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1077.5
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1078.1077
Dual form			1078.2.c.a.1077.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -2.61313i q^{3} -1.00000 q^{4} +2.61313i q^{5} +2.61313 q^{6} -1.00000i q^{8} -3.82843 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -2.61313i q^{3} -1.00000 q^{4} +2.61313i q^{5} +2.61313 q^{6} -1.00000i q^{8} -3.82843 q^{9} -2.61313 q^{10} +(1.41421 - 3.00000i) q^{11} +2.61313i q^{12} -5.54328 q^{13} +6.82843 q^{15} +1.00000 q^{16} -3.82843i q^{18} -2.29610 q^{19} -2.61313i q^{20} +(3.00000 + 1.41421i) q^{22} -2.24264 q^{23} -2.61313 q^{24} -1.82843 q^{25} -5.54328i q^{26} +2.16478i q^{27} -10.2426i q^{29} +6.82843i q^{30} -7.07401i q^{31} +1.00000i q^{32} +(-7.83938 - 3.69552i) q^{33} +3.82843 q^{36} +6.00000 q^{37} -2.29610i q^{38} +14.4853i q^{39} +2.61313 q^{40} -11.0866 q^{41} +4.24264i q^{43} +(-1.41421 + 3.00000i) q^{44} -10.0042i q^{45} -2.24264i q^{46} -6.62567i q^{47} -2.61313i q^{48} -1.82843i q^{50} +5.54328 q^{52} -12.4853 q^{53} -2.16478 q^{54} +(7.83938 + 3.69552i) q^{55} +6.00000i q^{57} +10.2426 q^{58} +12.1689i q^{59} -6.82843 q^{60} -2.29610 q^{61} +7.07401 q^{62} -1.00000 q^{64} -14.4853i q^{65} +(3.69552 - 7.83938i) q^{66} +0.343146 q^{67} +5.86030i q^{69} +2.00000 q^{71} +3.82843i q^{72} -6.49435 q^{73} +6.00000i q^{74} +4.77791i q^{75} +2.29610 q^{76} -14.4853 q^{78} -8.48528i q^{79} +2.61313i q^{80} -5.82843 q^{81} -11.0866i q^{82} +0.951076 q^{83} -4.24264 q^{86} -26.7653 q^{87} +(-3.00000 - 1.41421i) q^{88} -12.3003i q^{89} +10.0042 q^{90} +2.24264 q^{92} -18.4853 q^{93} +6.62567 q^{94} -6.00000i q^{95} +2.61313 q^{96} +5.09494i q^{97} +(-5.41421 + 11.4853i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^4 - 8 * q^9	$ 8 q - 8 q^{4} - 8 q^{9} + 32 q^{15} + 8 q^{16} + 24 q^{22} + 16 q^{23} + 8 q^{25} + 8 q^{36} + 48 q^{37} - 32 q^{53} + 48 q^{58} - 32 q^{60} - 8 q^{64} + 48 q^{67} + 16 q^{71} - 48 q^{78} - 24 q^{81} - 24 q^{88} - 16 q^{92} - 80 q^{93} - 32 q^{99}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^9 + 32 * q^15 + 8 * q^16 + 24 * q^22 + 16 * q^23 + 8 * q^25 + 8 * q^36 + 48 * q^37 - 32 * q^53 + 48 * q^58 - 32 * q^60 - 8 * q^64 + 48 * q^67 + 16 * q^71 - 48 * q^78 - 24 * q^81 - 24 * q^88 - 16 * q^92 - 80 * q^93 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1078\mathbb{Z}\right)^\times$.

$n$	$199$	$981$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	2.61313i		−	1.50869i	−0.656479	−	0.754344i	$-0.727955\pi$
$3$		−	2.61313i		−	1.50869i	0.656479	−	0.754344i	$-0.272045\pi$
$4$	−1.00000			−0.500000
$5$			2.61313i			1.16863i	0.811529	+	0.584313i	$0.198636\pi$
$5$			2.61313i			1.16863i	−0.811529	+	0.584313i	$0.801364\pi$
$6$	2.61313			1.06680
$7$		0			0
$7$		0			0
$8$		−	1.00000i		−	0.353553i
$9$	−3.82843			−1.27614
$10$	−2.61313			−0.826343
$11$	1.41421	−	3.00000i	0.426401	−	0.904534i
$11$	1.41421	−	3.00000i	0.426401	−	0.904534i
$12$			2.61313i			0.754344i
$13$	−5.54328			−1.53743			−0.768714	−	0.639592i	$-0.779103\pi$
$13$	−5.54328			−1.53743			−0.768714	+	0.639592i	$0.779103\pi$
$14$		0			0
$15$	6.82843			1.76309
$16$	1.00000			0.250000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		−	3.82843i		−	0.902369i
$19$	−2.29610			−0.526762			−0.263381	−	0.964692i	$-0.584838\pi$
$19$	−2.29610			−0.526762			−0.263381	+	0.964692i	$0.584838\pi$
$20$		−	2.61313i		−	0.584313i
$21$		0			0
$22$	3.00000	+	1.41421i	0.639602	+	0.301511i
$23$	−2.24264			−0.467623			−0.233811	−	0.972282i	$-0.575120\pi$
$23$	−2.24264			−0.467623			−0.233811	+	0.972282i	$0.575120\pi$
$24$	−2.61313			−0.533402
$25$	−1.82843			−0.365685
$26$		−	5.54328i		−	1.08713i
$27$			2.16478i			0.416613i
$28$		0			0
$29$		−	10.2426i		−	1.90201i	−0.309175	−	0.951005i	$-0.600053\pi$
$29$		−	10.2426i		−	1.90201i	0.309175	−	0.951005i	$-0.399947\pi$
$30$			6.82843i			1.24669i
$31$		−	7.07401i		−	1.27053i	−0.772294	−	0.635265i	$-0.780891\pi$
$31$		−	7.07401i		−	1.27053i	0.772294	−	0.635265i	$-0.219109\pi$
$32$			1.00000i			0.176777i
$33$	−7.83938	−	3.69552i	−1.36466	−	0.643307i
$34$		0			0
$35$		0			0
$36$	3.82843			0.638071
$37$	6.00000			0.986394			0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000			0.986394			0.493197	+	0.869918i	$0.335828\pi$
$38$		−	2.29610i		−	0.372477i
$39$			14.4853i			2.31950i
$40$	2.61313			0.413171
$41$	−11.0866			−1.73143			−0.865714	−	0.500538i	$-0.833135\pi$
$41$	−11.0866			−1.73143			−0.865714	+	0.500538i	$0.833135\pi$
$42$		0			0
$43$			4.24264i			0.646997i	0.946229	+	0.323498i	$0.104859\pi$
$43$			4.24264i			0.646997i	−0.946229	+	0.323498i	$0.895141\pi$
$44$	−1.41421	+	3.00000i	−0.213201	+	0.452267i
$45$		−	10.0042i		−	1.49133i
$46$		−	2.24264i		−	0.330659i
$47$		−	6.62567i		−	0.966453i	−0.875495	−	0.483227i	$-0.839465\pi$
$47$		−	6.62567i		−	0.966453i	0.875495	−	0.483227i	$-0.160535\pi$
$48$		−	2.61313i		−	0.377172i
$49$		0			0
$50$		−	1.82843i		−	0.258579i
$51$		0			0
$52$	5.54328			0.768714
$53$	−12.4853			−1.71499			−0.857493	−	0.514496i	$-0.827979\pi$
$53$	−12.4853			−1.71499			−0.857493	+	0.514496i	$0.827979\pi$
$54$	−2.16478			−0.294590
$55$	7.83938	+	3.69552i	1.05706	+	0.498304i
$56$		0			0
$57$			6.00000i			0.794719i
$58$	10.2426			1.34492
$59$			12.1689i			1.58426i	0.610351	+	0.792131i	$0.291028\pi$
$59$			12.1689i			1.58426i	−0.610351	+	0.792131i	$0.708972\pi$
$60$	−6.82843			−0.881546
$61$	−2.29610			−0.293986			−0.146993	−	0.989138i	$-0.546959\pi$
$61$	−2.29610			−0.293986			−0.146993	+	0.989138i	$0.546959\pi$
$62$	7.07401			0.898400
$63$		0			0
$64$	−1.00000			−0.125000
$65$		−	14.4853i		−	1.79668i
$66$	3.69552	−	7.83938i	0.454887	−	0.964961i
$67$	0.343146			0.0419219			0.0209610	−	0.999780i	$-0.493327\pi$
$67$	0.343146			0.0419219			0.0209610	+	0.999780i	$0.493327\pi$
$68$		0			0
$69$			5.86030i			0.705498i
$70$		0			0
$71$	2.00000			0.237356			0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000			0.237356			0.118678	+	0.992933i	$0.462134\pi$
$72$			3.82843i			0.451184i
$73$	−6.49435			−0.760107			−0.380053	−	0.924965i	$-0.624094\pi$
$73$	−6.49435			−0.760107			−0.380053	+	0.924965i	$0.624094\pi$
$74$			6.00000i			0.697486i
$75$			4.77791i			0.551706i
$76$	2.29610			0.263381
$77$		0			0
$78$	−14.4853			−1.64014
$79$		−	8.48528i		−	0.954669i	−0.878722	−	0.477334i	$-0.841603\pi$
$79$		−	8.48528i		−	0.954669i	0.878722	−	0.477334i	$-0.158397\pi$
$80$			2.61313i			0.292156i
$81$	−5.82843			−0.647603
$82$		−	11.0866i		−	1.22431i
$83$	0.951076			0.104394			0.0521971	−	0.998637i	$-0.483378\pi$
$83$	0.951076			0.104394			0.0521971	+	0.998637i	$0.483378\pi$
$84$		0			0
$85$		0			0
$86$	−4.24264			−0.457496
$87$	−26.7653			−2.86954
$88$	−3.00000	−	1.41421i	−0.319801	−	0.150756i
$89$		−	12.3003i		−	1.30383i	−0.758294	−	0.651913i	$-0.773967\pi$
$89$		−	12.3003i		−	1.30383i	0.758294	−	0.651913i	$-0.226033\pi$
$90$	10.0042			1.05453
$91$		0			0
$92$	2.24264			0.233811
$93$	−18.4853			−1.91683
$94$	6.62567			0.683386
$95$		−	6.00000i		−	0.615587i
$96$	2.61313			0.266701
$97$			5.09494i			0.517312i	0.965969	+	0.258656i	$0.0832797\pi$
$97$			5.09494i			0.517312i	−0.965969	+	0.258656i	$0.916720\pi$
$98$		0			0
$99$	−5.41421	+	11.4853i	−0.544149	+	1.15431i
$100$	1.82843			0.182843
$101$	6.88830			0.685412			0.342706	−	0.939443i	$-0.388657\pi$
$101$	6.88830			0.685412			0.342706	+	0.939443i	$0.388657\pi$
$102$		0			0
$103$		−	10.3212i		−	1.01698i	−0.861069	−	0.508488i	$-0.830204\pi$
$103$		−	10.3212i		−	1.01698i	0.861069	−	0.508488i	$-0.169796\pi$
$104$			5.54328i			0.543563i
$105$		0			0
$106$		−	12.4853i		−	1.21268i
$107$			12.7279i			1.23045i	0.788350	+	0.615227i	$0.210936\pi$
$107$			12.7279i			1.23045i	−0.788350	+	0.615227i	$0.789064\pi$
$108$		−	2.16478i		−	0.208306i
$109$		−	20.4853i		−	1.96213i	−0.193668	−	0.981067i	$-0.562039\pi$
$109$		−	20.4853i		−	1.96213i	0.193668	−	0.981067i	$-0.437961\pi$
$110$	−3.69552	+	7.83938i	−0.352354	+	0.747455i
$111$		−	15.6788i		−	1.48816i
$112$		0			0
$113$	1.41421			0.133038			0.0665190	−	0.997785i	$-0.478811\pi$
$113$	1.41421			0.133038			0.0665190	+	0.997785i	$0.478811\pi$
$114$	−6.00000			−0.561951
$115$		−	5.86030i		−	0.546476i
$116$			10.2426i			0.951005i
$117$	21.2220			1.96198
$118$	−12.1689			−1.12024
$119$		0			0
$120$		−	6.82843i		−	0.623347i
$121$	−7.00000	−	8.48528i	−0.636364	−	0.771389i
$122$		−	2.29610i		−	0.207879i
$123$			28.9706i			2.61219i
$124$			7.07401i			0.635265i
$125$			8.28772i			0.741276i
$126$		0			0
$127$			6.00000i			0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$			6.00000i			0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	11.0866			0.976117
$130$	14.4853			1.27044
$131$	−16.6298			−1.45296			−0.726478	−	0.687190i	$-0.758844\pi$
$131$	−16.6298			−1.45296			−0.726478	+	0.687190i	$0.758844\pi$
$132$	7.83938	+	3.69552i	0.682330	+	0.321654i
$133$		0			0
$134$			0.343146i			0.0296433i
$135$	−5.65685			−0.486864
$136$		0			0
$137$	16.9706			1.44989			0.724947	−	0.688805i	$-0.241864\pi$
$137$	16.9706			1.44989			0.724947	+	0.688805i	$0.241864\pi$
$138$	−5.86030			−0.498862
$139$	0.951076			0.0806692			0.0403346	−	0.999186i	$-0.487158\pi$
$139$	0.951076			0.0806692			0.0403346	+	0.999186i	$0.487158\pi$
$140$		0			0
$141$	−17.3137			−1.45808
$142$			2.00000i			0.167836i
$143$	−7.83938	+	16.6298i	−0.655562	+	1.39066i
$144$	−3.82843			−0.319036
$145$	26.7653			2.22274
$146$		−	6.49435i		−	0.537476i
$147$		0			0
$148$	−6.00000			−0.493197
$149$			18.7279i			1.53425i	0.641497	+	0.767126i	$0.278314\pi$
$149$			18.7279i			1.53425i	−0.641497	+	0.767126i	$0.721686\pi$
$150$	−4.77791			−0.390115
$151$			6.00000i			0.488273i	0.969741	+	0.244137i	$0.0785045\pi$
$151$			6.00000i			0.488273i	−0.969741	+	0.244137i	$0.921495\pi$
$152$			2.29610i			0.186238i
$153$		0			0
$154$		0			0
$155$	18.4853			1.48477
$156$		−	14.4853i		−	1.15975i
$157$		−	1.71644i		−	0.136987i	−0.997652	−	0.0684935i	$-0.978181\pi$
$157$		−	1.71644i		−	0.136987i	0.997652	−	0.0684935i	$-0.0218192\pi$
$158$	8.48528			0.675053
$159$			32.6256i			2.58738i
$160$	−2.61313			−0.206586
$161$		0			0
$162$		−	5.82843i		−	0.457924i
$163$	20.1421			1.57765			0.788827	−	0.614615i	$-0.210689\pi$
$163$	20.1421			1.57765			0.788827	+	0.614615i	$0.210689\pi$
$164$	11.0866			0.865714
$165$	9.65685	−	20.4853i	0.751785	−	1.59478i
$166$			0.951076i			0.0738178i
$167$	17.5809			1.36045			0.680226	−	0.733003i	$-0.261882\pi$
$167$	17.5809			1.36045			0.680226	+	0.733003i	$0.261882\pi$
$168$		0			0
$169$	17.7279			1.36369
$170$		0			0
$171$	8.79045			0.672223
$172$		−	4.24264i		−	0.323498i
$173$	15.2848			1.16208			0.581041	−	0.813874i	$-0.302646\pi$
$173$	15.2848			1.16208			0.581041	+	0.813874i	$0.302646\pi$
$174$		−	26.7653i		−	2.02907i
$175$		0			0
$176$	1.41421	−	3.00000i	0.106600	−	0.226134i
$177$	31.7990			2.39016
$178$	12.3003			0.921944
$179$	5.65685			0.422813			0.211407	−	0.977398i	$-0.432196\pi$
$179$	5.65685			0.422813			0.211407	+	0.977398i	$0.432196\pi$
$180$			10.0042i			0.745666i
$181$		−	2.35049i		−	0.174711i	−0.996177	−	0.0873554i	$-0.972158\pi$
$181$		−	2.35049i		−	0.174711i	0.996177	−	0.0873554i	$-0.0278416\pi$
$182$		0			0
$183$			6.00000i			0.443533i
$184$			2.24264i			0.165330i
$185$			15.6788i			1.15273i
$186$		−	18.4853i		−	1.35541i
$187$		0			0
$188$			6.62567i			0.483227i
$189$		0			0
$190$	6.00000			0.435286
$191$	−10.2426			−0.741131			−0.370566	−	0.928806i	$-0.620836\pi$
$191$	−10.2426			−0.741131			−0.370566	+	0.928806i	$0.620836\pi$
$192$			2.61313i			0.188586i
$193$			14.4853i			1.04267i	0.853351	+	0.521337i	$0.174566\pi$
$193$			14.4853i			1.04267i	−0.853351	+	0.521337i	$0.825434\pi$
$194$	−5.09494			−0.365795
$195$	−37.8519			−2.71063
$196$		0			0
$197$		−	3.51472i		−	0.250413i	−0.992131	−	0.125207i	$-0.960041\pi$
$197$		−	3.51472i		−	0.250413i	0.992131	−	0.125207i	$-0.0399594\pi$
$198$	−11.4853	−	5.41421i	−0.816223	−	0.384771i
$199$			18.8715i			1.33777i	0.743367	+	0.668884i	$0.233228\pi$
$199$			18.8715i			1.33777i	−0.743367	+	0.668884i	$0.766772\pi$
$200$			1.82843i			0.129289i
$201$		−	0.896683i		−	0.0632471i
$202$			6.88830i			0.484659i
$203$		0			0
$204$		0			0
$205$		−	28.9706i		−	2.02339i
$206$	10.3212			0.719111
$207$	8.58579			0.596753
$208$	−5.54328			−0.384357
$209$	−3.24718	+	6.88830i	−0.224612	+	0.476474i
$210$		0			0
$211$		−	0.727922i		−	0.0501122i	−0.999686	−	0.0250561i	$-0.992024\pi$
$211$		−	0.727922i		−	0.0501122i	0.999686	−	0.0250561i	$-0.00797644\pi$
$212$	12.4853			0.857493
$213$		−	5.22625i		−	0.358097i
$214$	−12.7279			−0.870063
$215$	−11.0866			−0.756097
$216$	2.16478			0.147295
$217$		0			0
$218$	20.4853			1.38744
$219$			16.9706i			1.14676i
$220$	−7.83938	−	3.69552i	−0.528531	−	0.249152i
$221$		0			0
$222$	15.6788			1.05229
$223$		−	22.7528i		−	1.52364i	−0.647790	−	0.761819i	$-0.724307\pi$
$223$		−	22.7528i		−	1.52364i	0.647790	−	0.761819i	$-0.275693\pi$
$224$		0			0
$225$	7.00000			0.466667
$226$			1.41421i			0.0940721i
$227$	−13.3827			−0.888238			−0.444119	−	0.895968i	$-0.646483\pi$
$227$	−13.3827			−0.888238			−0.444119	+	0.895968i	$0.646483\pi$
$228$		−	6.00000i		−	0.397360i
$229$		−	9.63274i		−	0.636550i	−0.947998	−	0.318275i	$-0.896897\pi$
$229$		−	9.63274i		−	0.636550i	0.947998	−	0.318275i	$-0.103103\pi$
$230$	5.86030			0.386417
$231$		0			0
$232$	−10.2426			−0.672462
$233$		−	8.48528i		−	0.555889i	−0.960597	−	0.277945i	$-0.910347\pi$
$233$		−	8.48528i		−	0.555889i	0.960597	−	0.277945i	$-0.0896532\pi$
$234$			21.2220i			1.38733i
$235$	17.3137			1.12942
$236$		−	12.1689i		−	0.792131i
$237$	−22.1731			−1.44030
$238$		0			0
$239$		−	9.51472i		−	0.615456i	−0.951474	−	0.307728i	$-0.900431\pi$
$239$		−	9.51472i		−	0.615456i	0.951474	−	0.307728i	$-0.0995687\pi$
$240$	6.82843			0.440773
$241$	11.0866			0.714148			0.357074	−	0.934076i	$-0.383774\pi$
$241$	11.0866			0.714148			0.357074	+	0.934076i	$0.383774\pi$
$242$	8.48528	−	7.00000i	0.545455	−	0.449977i
$243$			21.7248i			1.39364i
$244$	2.29610			0.146993
$245$		0			0
$246$	−28.9706			−1.84710
$247$	12.7279			0.809858
$248$	−7.07401			−0.449200
$249$		−	2.48528i		−	0.157498i
$250$	−8.28772			−0.524161
$251$			7.20533i			0.454796i	0.973802	+	0.227398i	$0.0730219\pi$
$251$			7.20533i			0.454796i	−0.973802	+	0.227398i	$0.926978\pi$
$252$		0			0
$253$	−3.17157	+	6.72792i	−0.199395	+	0.422981i
$254$	−6.00000			−0.376473
$255$		0			0
$256$	1.00000			0.0625000
$257$			19.9539i			1.24469i	0.782742	+	0.622346i	$0.213820\pi$
$257$			19.9539i			1.24469i	−0.782742	+	0.622346i	$0.786180\pi$
$258$			11.0866i			0.690219i
$259$		0			0
$260$			14.4853i			0.898339i
$261$			39.2132i			2.42724i
$262$		−	16.6298i		−	1.02739i
$263$		−	24.0000i		−	1.47990i	−0.672660	−	0.739952i	$-0.734848\pi$
$263$		−	24.0000i		−	1.47990i	0.672660	−	0.739952i	$-0.265152\pi$
$264$	−3.69552	+	7.83938i	−0.227443	+	0.482480i
$265$		−	32.6256i		−	2.00418i
$266$		0			0
$267$	−32.1421			−1.96707
$268$	−0.343146			−0.0209610
$269$			13.6997i			0.835284i	0.908612	+	0.417642i	$0.137143\pi$
$269$			13.6997i			0.835284i	−0.908612	+	0.417642i	$0.862857\pi$
$270$		−	5.65685i		−	0.344265i
$271$	−1.90215			−0.115548			−0.0577738	−	0.998330i	$-0.518400\pi$
$271$	−1.90215			−0.115548			−0.0577738	+	0.998330i	$0.518400\pi$
$272$		0			0
$273$		0			0
$274$			16.9706i			1.02523i
$275$	−2.58579	+	5.48528i	−0.155929	+	0.330775i
$276$		−	5.86030i		−	0.352749i
$277$		−	6.72792i		−	0.404242i	−0.979361	−	0.202121i	$-0.935217\pi$
$277$		−	6.72792i		−	0.404242i	0.979361	−	0.202121i	$-0.0647834\pi$
$278$			0.951076i			0.0570417i
$279$			27.0823i			1.62138i
$280$		0			0
$281$			12.0000i			0.715860i	0.933748	+	0.357930i	$0.116517\pi$
$281$			12.0000i			0.715860i	−0.933748	+	0.357930i	$0.883483\pi$
$282$		−	17.3137i		−	1.03102i
$283$	23.1242			1.37459			0.687295	−	0.726378i	$-0.258798\pi$
$283$	23.1242			1.37459			0.687295	+	0.726378i	$0.258798\pi$
$284$	−2.00000			−0.118678
$285$	−15.6788			−0.928729
$286$	−16.6298	−	7.83938i	−0.983343	−	0.463552i
$287$		0			0
$288$		−	3.82843i		−	0.225592i
$289$	−17.0000			−1.00000
$290$			26.7653i			1.57171i
$291$	13.3137			0.780463
$292$	6.49435			0.380053
$293$	13.3827			0.781823			0.390912	−	0.920428i	$-0.372160\pi$
$293$	13.3827			0.781823			0.390912	+	0.920428i	$0.372160\pi$
$294$		0			0
$295$	−31.7990			−1.85141
$296$		−	6.00000i		−	0.348743i
$297$	6.49435	+	3.06147i	0.376841	+	0.177644i
$298$	−18.7279			−1.08488
$299$	12.4316			0.718937
$300$		−	4.77791i		−	0.275853i
$301$		0			0
$302$	−6.00000			−0.345261
$303$		−	18.0000i		−	1.03407i
$304$	−2.29610			−0.131690
$305$		−	6.00000i		−	0.343559i
$306$		0			0
$307$	29.0614			1.65862			0.829311	−	0.558787i	$-0.188733\pi$
$307$	29.0614			1.65862			0.829311	+	0.558787i	$0.188733\pi$
$308$		0			0
$309$	−26.9706			−1.53430
$310$			18.4853i			1.04989i
$311$		−	3.00707i		−	0.170516i	−0.996359	−	0.0852578i	$-0.972829\pi$
$311$		−	3.00707i		−	0.170516i	0.996359	−	0.0852578i	$-0.0271714\pi$
$312$	14.4853			0.820068
$313$		−	6.43996i		−	0.364008i	−0.983298	−	0.182004i	$-0.941742\pi$
$313$		−	6.43996i		−	0.364008i	0.983298	−	0.182004i	$-0.0582584\pi$
$314$	1.71644			0.0968645
$315$		0			0
$316$			8.48528i			0.477334i
$317$	10.9706			0.616168			0.308084	−	0.951359i	$-0.400312\pi$
$317$	10.9706			0.616168			0.308084	+	0.951359i	$0.400312\pi$
$318$	−32.6256			−1.82955
$319$	−30.7279	−	14.4853i	−1.72043	−	0.811020i
$320$		−	2.61313i		−	0.146078i
$321$	33.2597			1.85637
$322$		0			0
$323$		0			0
$324$	5.82843			0.323802
$325$	10.1355			0.562215
$326$			20.1421i			1.11557i
$327$	−53.5306			−2.96025
$328$			11.0866i			0.612153i
$329$		0			0
$330$	20.4853	+	9.65685i	1.12768	+	0.531592i
$331$	3.51472			0.193186			0.0965932	−	0.995324i	$-0.469205\pi$
$331$	3.51472			0.193186			0.0965932	+	0.995324i	$0.469205\pi$
$332$	−0.951076			−0.0521971
$333$	−22.9706			−1.25878
$334$			17.5809i			0.961984i
$335$			0.896683i			0.0489910i
$336$		0			0
$337$			9.51472i			0.518300i	0.965837	+	0.259150i	$0.0834424\pi$
$337$			9.51472i			0.518300i	−0.965837	+	0.259150i	$0.916558\pi$
$338$			17.7279i			0.964272i
$339$		−	3.69552i		−	0.200713i
$340$		0			0
$341$	−21.2220	−	10.0042i	−1.14924	−	0.541756i
$342$			8.79045i			0.475333i
$343$		0			0
$344$	4.24264			0.228748
$345$	−15.3137			−0.824462
$346$			15.2848i			0.821716i
$347$			16.2426i			0.871951i	0.899959	+	0.435975i	$0.143597\pi$
$347$			16.2426i			0.871951i	−0.899959	+	0.435975i	$0.856403\pi$
$348$	26.7653			1.43477
$349$	−7.44543			−0.398545			−0.199272	−	0.979944i	$-0.563858\pi$
$349$	−7.44543			−0.398545			−0.199272	+	0.979944i	$0.563858\pi$
$350$		0			0
$351$		−	12.0000i		−	0.640513i
$352$	3.00000	+	1.41421i	0.159901	+	0.0753778i
$353$		−	8.15640i		−	0.434122i	−0.976158	−	0.217061i	$-0.930353\pi$
$353$		−	8.15640i		−	0.434122i	0.976158	−	0.217061i	$-0.0696470\pi$
$354$			31.7990i			1.69010i
$355$			5.22625i			0.277381i
$356$			12.3003i			0.651913i
$357$		0			0
$358$			5.65685i			0.298974i
$359$		−	14.4853i		−	0.764504i	−0.924058	−	0.382252i	$-0.875149\pi$
$359$		−	14.4853i		−	0.764504i	0.924058	−	0.382252i	$-0.124851\pi$
$360$	−10.0042			−0.527266
$361$	−13.7279			−0.722522
$362$	2.35049			0.123539
$363$	−22.1731	+	18.2919i	−1.16379	+	0.960075i
$364$		0			0
$365$		−	16.9706i		−	0.888280i
$366$	−6.00000			−0.313625
$367$		−	30.5921i		−	1.59690i	−0.602063	−	0.798448i	$-0.705654\pi$
$367$		−	30.5921i		−	1.59690i	0.602063	−	0.798448i	$-0.294346\pi$
$368$	−2.24264			−0.116906
$369$	42.4441			2.20955
$370$	−15.6788			−0.815100
$371$		0			0
$372$	18.4853			0.958417
$373$			3.51472i			0.181985i	0.995852	+	0.0909926i	$0.0290040\pi$
$373$			3.51472i			0.181985i	−0.995852	+	0.0909926i	$0.970996\pi$
$374$		0			0
$375$	21.6569			1.11836
$376$	−6.62567			−0.341693
$377$			56.7778i			2.92421i
$378$		0			0
$379$	−0.343146			−0.0176262			−0.00881311	−	0.999961i	$-0.502805\pi$
$379$	−0.343146			−0.0176262			−0.00881311	+	0.999961i	$0.502805\pi$
$380$			6.00000i			0.307794i
$381$	15.6788			0.803247
$382$		−	10.2426i		−	0.524059i
$383$			1.66205i			0.0849268i	0.999098	+	0.0424634i	$0.0135206\pi$
$383$			1.66205i			0.0849268i	−0.999098	+	0.0424634i	$0.986479\pi$
$384$	−2.61313			−0.133351
$385$		0			0
$386$	−14.4853			−0.737281
$387$		−	16.2426i		−	0.825660i
$388$		−	5.09494i		−	0.258656i
$389$	−32.0000			−1.62246			−0.811232	−	0.584724i	$-0.801203\pi$
$389$	−32.0000			−1.62246			−0.811232	+	0.584724i	$0.801203\pi$
$390$		−	37.8519i		−	1.91670i
$391$		0			0
$392$		0			0
$393$			43.4558i			2.19206i
$394$	3.51472			0.177069
$395$	22.1731			1.11565
$396$	5.41421	−	11.4853i	0.272074	−	0.577157i
$397$			11.5349i			0.578920i	0.957190	+	0.289460i	$0.0934757\pi$
$397$			11.5349i			0.578920i	−0.957190	+	0.289460i	$0.906524\pi$
$398$	−18.8715			−0.945945
$399$		0			0
$400$	−1.82843			−0.0914214
$401$	−16.4853			−0.823236			−0.411618	−	0.911357i	$-0.635036\pi$
$401$	−16.4853			−0.823236			−0.411618	+	0.911357i	$0.635036\pi$
$402$	0.896683			0.0447225
$403$			39.2132i			1.95335i
$404$	−6.88830			−0.342706
$405$		−	15.2304i		−	0.756805i
$406$		0			0
$407$	8.48528	−	18.0000i	0.420600	−	0.892227i
$408$		0			0
$409$	2.69005			0.133014			0.0665072	−	0.997786i	$-0.478814\pi$
$409$	2.69005			0.133014			0.0665072	+	0.997786i	$0.478814\pi$
$410$	28.9706			1.43075
$411$		−	44.3462i		−	2.18744i
$412$			10.3212i			0.508488i
$413$		0			0
$414$			8.58579i			0.421968i
$415$			2.48528i			0.121998i
$416$		−	5.54328i		−	0.271782i
$417$		−	2.48528i		−	0.121705i
$418$	−6.88830	−	3.24718i	−0.336918	−	0.158825i
$419$		−	5.67459i		−	0.277222i	−0.990347	−	0.138611i	$-0.955736\pi$
$419$		−	5.67459i		−	0.277222i	0.990347	−	0.138611i	$-0.0442638\pi$
$420$		0			0
$421$	6.34315			0.309146			0.154573	−	0.987981i	$-0.450600\pi$
$421$	6.34315			0.309146			0.154573	+	0.987981i	$0.450600\pi$
$422$	0.727922			0.0354347
$423$			25.3659i			1.23333i
$424$			12.4853i			0.606339i
$425$		0			0
$426$	5.22625			0.253213
$427$		0			0
$428$		−	12.7279i		−	0.615227i
$429$	43.4558	+	20.4853i	2.09807	+	0.989039i
$430$		−	11.0866i		−	0.534641i
$431$		−	14.4853i		−	0.697731i	−0.937173	−	0.348866i	$-0.886567\pi$
$431$		−	14.4853i		−	0.697731i	0.937173	−	0.348866i	$-0.113433\pi$
$432$			2.16478i			0.104153i
$433$		−	28.8757i		−	1.38768i	−0.720130	−	0.693839i	$-0.755918\pi$
$433$		−	28.8757i		−	1.38768i	0.720130	−	0.693839i	$-0.244082\pi$
$434$		0			0
$435$		−	69.9411i		−	3.35342i
$436$			20.4853i			0.981067i
$437$	5.14933			0.246326
$438$	−16.9706			−0.810885
$439$	15.6788			0.748306			0.374153	−	0.927367i	$-0.377933\pi$
$439$	15.6788			0.748306			0.374153	+	0.927367i	$0.377933\pi$
$440$	3.69552	−	7.83938i	0.176177	−	0.373728i
$441$		0			0
$442$		0			0
$443$	−22.6274			−1.07506			−0.537531	−	0.843244i	$-0.680643\pi$
$443$	−22.6274			−1.07506			−0.537531	+	0.843244i	$0.680643\pi$
$444$			15.6788i			0.744081i
$445$	32.1421			1.52368
$446$	22.7528			1.07737
$447$	48.9384			2.31471
$448$		0			0
$449$	−15.2721			−0.720734			−0.360367	−	0.932811i	$-0.617349\pi$
$449$	−15.2721			−0.720734			−0.360367	+	0.932811i	$0.617349\pi$
$450$			7.00000i			0.329983i
$451$	−15.6788	+	33.2597i	−0.738284	+	1.56614i
$452$	−1.41421			−0.0665190
$453$	15.6788			0.736652
$454$		−	13.3827i		−	0.628079i
$455$		0			0
$456$	6.00000			0.280976
$457$		−	19.4558i		−	0.910106i	−0.890464	−	0.455053i	$-0.849620\pi$
$457$		−	19.4558i		−	0.910106i	0.890464	−	0.455053i	$-0.150380\pi$
$458$	9.63274			0.450109
$459$		0			0
$460$			5.86030i			0.273238i
$461$	−7.44543			−0.346768			−0.173384	−	0.984854i	$-0.555470\pi$
$461$	−7.44543			−0.346768			−0.173384	+	0.984854i	$0.555470\pi$
$462$		0			0
$463$	17.3137			0.804636			0.402318	−	0.915500i	$-0.368205\pi$
$463$	17.3137			0.804636			0.402318	+	0.915500i	$0.368205\pi$
$464$		−	10.2426i		−	0.475503i
$465$		−	48.3044i		−	2.24006i
$466$	8.48528			0.393073
$467$			11.1635i			0.516584i	0.966067	+	0.258292i	$0.0831597\pi$
$467$			11.1635i			0.516584i	−0.966067	+	0.258292i	$0.916840\pi$
$468$	−21.2220			−0.980989
$469$		0			0
$470$			17.3137i			0.798622i
$471$	−4.48528			−0.206671
$472$	12.1689			0.560121
$473$	12.7279	+	6.00000i	0.585230	+	0.275880i
$474$		−	22.1731i		−	1.01844i
$475$	4.19825			0.192629
$476$		0			0
$477$	47.7990			2.18857
$478$	9.51472			0.435193
$479$	−22.1731			−1.01312			−0.506558	−	0.862206i	$-0.669082\pi$
$479$	−22.1731			−1.01312			−0.506558	+	0.862206i	$0.669082\pi$
$480$			6.82843i			0.311674i
$481$	−33.2597			−1.51651
$482$			11.0866i			0.504979i
$483$		0			0
$484$	7.00000	+	8.48528i	0.318182	+	0.385695i
$485$	−13.3137			−0.604544
$486$	−21.7248			−0.985455
$487$	−33.5563			−1.52058			−0.760292	−	0.649582i	$-0.774944\pi$
$487$	−33.5563			−1.52058			−0.760292	+	0.649582i	$0.774944\pi$
$488$			2.29610i			0.103940i
$489$		−	52.6339i		−	2.38019i
$490$		0			0
$491$		−	27.9411i		−	1.26097i	−0.776203	−	0.630483i	$-0.782857\pi$
$491$		−	27.9411i		−	1.26097i	0.776203	−	0.630483i	$-0.217143\pi$
$492$		−	28.9706i		−	1.30609i
$493$		0			0
$494$			12.7279i			0.572656i
$495$	−30.0125	−	14.1480i	−1.34896	−	0.635906i
$496$		−	7.07401i		−	0.317632i
$497$		0			0
$498$	2.48528			0.111368
$499$	−16.6274			−0.744345			−0.372173	−	0.928163i	$-0.621387\pi$
$499$	−16.6274			−0.744345			−0.372173	+	0.928163i	$0.621387\pi$
$500$		−	8.28772i		−	0.370638i
$501$		−	45.9411i		−	2.05250i
$502$	−7.20533			−0.321589
$503$	15.6788			0.699081			0.349541	−	0.936921i	$-0.386338\pi$
$503$	15.6788			0.699081			0.349541	+	0.936921i	$0.386338\pi$
$504$		0			0
$505$			18.0000i			0.800989i
$506$	−6.72792	−	3.17157i	−0.299093	−	0.140994i
$507$		−	46.3253i		−	2.05738i
$508$		−	6.00000i		−	0.266207i
$509$		−	10.3756i		−	0.459890i	−0.973204	−	0.229945i	$-0.926145\pi$
$509$		−	10.3756i		−	0.459890i	0.973204	−	0.229945i	$-0.0738546\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			0.0441942i
$513$		−	4.97056i		−	0.219456i
$514$	−19.9539			−0.880130
$515$	26.9706			1.18846
$516$	−11.0866			−0.488058
$517$	−19.8770	−	9.37011i	−0.874190	−	0.412097i
$518$		0			0
$519$		−	39.9411i		−	1.75322i
$520$	−14.4853			−0.635222
$521$		−	20.4023i		−	0.893840i	−0.894574	−	0.446920i	$-0.852521\pi$
$521$		−	20.4023i		−	0.893840i	0.894574	−	0.446920i	$-0.147479\pi$
$522$	−39.2132			−1.71632
$523$	13.3827			0.585183			0.292591	−	0.956238i	$-0.405482\pi$
$523$	13.3827			0.585183			0.292591	+	0.956238i	$0.405482\pi$
$524$	16.6298			0.726478
$525$		0			0
$526$	24.0000			1.04645
$527$		0			0
$528$	−7.83938	−	3.69552i	−0.341165	−	0.160827i
$529$	−17.9706			−0.781329
$530$	32.6256			1.41717
$531$		−	46.5879i		−	2.02174i
$532$		0			0
$533$	61.4558			2.66195
$534$		−	32.1421i		−	1.39093i
$535$	−33.2597			−1.43794
$536$		−	0.343146i		−	0.0148216i
$537$		−	14.7821i		−	0.637894i
$538$	−13.6997			−0.590635
$539$		0			0
$540$	5.65685			0.243432
$541$		−	22.9706i		−	0.987582i	−0.869581	−	0.493791i	$-0.835611\pi$
$541$		−	22.9706i		−	0.987582i	0.869581	−	0.493791i	$-0.164389\pi$
$542$		−	1.90215i		−	0.0817044i
$543$	−6.14214			−0.263584
$544$		0			0
$545$	53.5306			2.29300
$546$		0			0
$547$		−	24.0000i		−	1.02617i	−0.858339	−	0.513083i	$-0.828503\pi$
$547$		−	24.0000i		−	1.02617i	0.858339	−	0.513083i	$-0.171497\pi$
$548$	−16.9706			−0.724947
$549$	8.79045			0.375167
$550$	−5.48528	−	2.58579i	−0.233893	−	0.110258i
$551$			23.5181i			1.00191i
$552$	5.86030			0.249431
$553$		0			0
$554$	6.72792			0.285842
$555$	40.9706			1.73910
$556$	−0.951076			−0.0403346
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	−27.0823			−1.14649
$559$		−	23.5181i		−	0.994711i
$560$		0			0
$561$		0			0
$562$	−12.0000			−0.506189
$563$	−34.2107			−1.44181			−0.720905	−	0.693034i	$-0.756274\pi$
$563$	−34.2107			−1.44181			−0.720905	+	0.693034i	$0.756274\pi$
$564$	17.3137			0.729039
$565$			3.69552i			0.155472i
$566$			23.1242i			0.971982i
$567$		0			0
$568$		−	2.00000i		−	0.0839181i
$569$			30.0000i			1.25767i	0.777541	+	0.628833i	$0.216467\pi$
$569$			30.0000i			1.25767i	−0.777541	+	0.628833i	$0.783533\pi$
$570$		−	15.6788i		−	0.656711i
$571$			36.0000i			1.50655i	0.657704	+	0.753277i	$0.271528\pi$
$571$			36.0000i			1.50655i	−0.657704	+	0.753277i	$0.728472\pi$
$572$	7.83938	−	16.6298i	0.327781	−	0.695328i
$573$			26.7653i			1.11814i
$574$		0			0
$575$	4.10051			0.171003
$576$	3.82843			0.159518
$577$		−	12.2233i		−	0.508864i	−0.967091	−	0.254432i	$-0.918111\pi$
$577$		−	12.2233i		−	0.508864i	0.967091	−	0.254432i	$-0.0818886\pi$
$578$		−	17.0000i		−	0.707107i
$579$	37.8519			1.57307
$580$	−26.7653			−1.11137
$581$		0			0
$582$			13.3137i			0.551871i
$583$	−17.6569	+	37.4558i	−0.731272	+	1.55126i
$584$			6.49435i			0.268738i
$585$			55.4558i			2.29282i
$586$			13.3827i			0.552832i
$587$			21.4621i			0.885837i	0.896562	+	0.442919i	$0.146057\pi$
$587$			21.4621i			0.885837i	−0.896562	+	0.442919i	$0.853943\pi$
$588$		0			0
$589$			16.2426i			0.669266i
$590$		−	31.7990i		−	1.30914i
$591$	−9.18440			−0.377796
$592$	6.00000			0.246598
$593$	−24.8632			−1.02101			−0.510504	−	0.859876i	$-0.670541\pi$
$593$	−24.8632			−1.02101			−0.510504	+	0.859876i	$0.670541\pi$
$594$	−3.06147	+	6.49435i	−0.125614	+	0.266467i
$595$		0			0
$596$		−	18.7279i		−	0.767126i
$597$	49.3137			2.01828
$598$			12.4316i			0.508365i
$599$	−26.0000			−1.06233			−0.531166	−	0.847268i	$-0.678246\pi$
$599$	−26.0000			−1.06233			−0.531166	+	0.847268i	$0.678246\pi$
$600$	4.77791			0.195057
$601$	24.0753			0.982050			0.491025	−	0.871145i	$-0.336622\pi$
$601$	24.0753			0.982050			0.491025	+	0.871145i	$0.336622\pi$
$602$		0			0
$603$	−1.31371			−0.0534983
$604$		−	6.00000i		−	0.244137i
$605$	22.1731	−	18.2919i	0.901465	−	0.743671i
$606$	18.0000			0.731200
$607$	45.1341			1.83194			0.915969	−	0.401250i	$-0.131424\pi$
$607$	45.1341			1.83194			0.915969	+	0.401250i	$0.131424\pi$
$608$		−	2.29610i		−	0.0931192i
$609$		0			0
$610$	6.00000			0.242933
$611$			36.7279i			1.48585i
$612$		0			0
$613$		−	22.2426i		−	0.898372i	−0.893438	−	0.449186i	$-0.851714\pi$
$613$		−	22.2426i		−	0.898372i	0.893438	−	0.449186i	$-0.148286\pi$
$614$			29.0614i			1.17282i
$615$	−75.7037			−3.05267
$616$		0			0
$617$	−1.41421			−0.0569341			−0.0284670	−	0.999595i	$-0.509063\pi$
$617$	−1.41421			−0.0569341			−0.0284670	+	0.999595i	$0.509063\pi$
$618$		−	26.9706i		−	1.08492i
$619$		−	11.5349i		−	0.463627i	−0.972760	−	0.231813i	$-0.925534\pi$
$619$		−	11.5349i		−	0.463627i	0.972760	−	0.231813i	$-0.0744658\pi$
$620$	−18.4853			−0.742387
$621$		−	4.85483i		−	0.194818i
$622$	3.00707			0.120573
$623$		0			0
$624$			14.4853i			0.579875i
$625$	−30.7990			−1.23196
$626$	6.43996			0.257393
$627$	18.0000	+	8.48528i	0.718851	+	0.338869i
$628$			1.71644i			0.0684935i
$629$		0			0
$630$		0			0
$631$	−27.2132			−1.08334			−0.541670	−	0.840591i	$-0.682208\pi$
$631$	−27.2132			−1.08334			−0.541670	+	0.840591i	$0.682208\pi$
$632$	−8.48528			−0.337526
$633$	−1.90215			−0.0756038
$634$			10.9706i			0.435697i
$635$	−15.6788			−0.622192
$636$		−	32.6256i		−	1.29369i
$637$		0			0
$638$	14.4853	−	30.7279i	0.573478	−	1.21653i
$639$	−7.65685			−0.302900
$640$	2.61313			0.103293
$641$	45.2132			1.78581			0.892907	−	0.450241i	$-0.148662\pi$
$641$	45.2132			1.78581			0.892907	+	0.450241i	$0.148662\pi$
$642$			33.2597i			1.31265i
$643$			17.3952i			0.686000i	0.939335	+	0.343000i	$0.111443\pi$
$643$			17.3952i			0.686000i	−0.939335	+	0.343000i	$0.888557\pi$
$644$		0			0
$645$			28.9706i			1.14071i
$646$		0			0
$647$			12.3003i			0.483573i	0.970329	+	0.241787i	$0.0777334\pi$
$647$			12.3003i			0.483573i	−0.970329	+	0.241787i	$0.922267\pi$
$648$			5.82843i			0.228962i
$649$	36.5068	+	17.2095i	1.43302	+	0.675532i
$650$			10.1355i			0.397546i
$651$		0			0
$652$	−20.1421			−0.788827
$653$	−29.4558			−1.15270			−0.576348	−	0.817204i	$-0.695523\pi$
$653$	−29.4558			−1.15270			−0.576348	+	0.817204i	$0.695523\pi$
$654$		−	53.5306i		−	2.09321i
$655$		−	43.4558i		−	1.69796i
$656$	−11.0866			−0.432857
$657$	24.8632			0.970004
$658$		0			0
$659$			33.2132i			1.29380i	0.762574	+	0.646901i	$0.223935\pi$
$659$			33.2132i			1.29380i	−0.762574	+	0.646901i	$0.776065\pi$
$660$	−9.65685	+	20.4853i	−0.375893	+	0.797388i
$661$		−	29.0070i		−	1.12824i	−0.825692	−	0.564121i	$-0.809215\pi$
$661$		−	29.0070i		−	1.12824i	0.825692	−	0.564121i	$-0.190785\pi$
$662$			3.51472i			0.136603i
$663$		0			0
$664$		−	0.951076i		−	0.0369089i
$665$		0			0
$666$		−	22.9706i		−	0.890091i
$667$			22.9706i			0.889424i
$668$	−17.5809			−0.680226
$669$	−59.4558			−2.29870
$670$	−0.896683			−0.0346419
$671$	−3.24718	+	6.88830i	−0.125356	+	0.265920i
$672$		0			0
$673$		−	21.9411i		−	0.845768i	−0.906184	−	0.422884i	$-0.861018\pi$
$673$		−	21.9411i		−	0.845768i	0.906184	−	0.422884i	$-0.138982\pi$
$674$	−9.51472			−0.366493
$675$		−	3.95815i		−	0.152349i
$676$	−17.7279			−0.681843
$677$	−14.7277			−0.566031			−0.283015	−	0.959115i	$-0.591335\pi$
$677$	−14.7277			−0.566031			−0.283015	+	0.959115i	$0.591335\pi$
$678$	3.69552			0.141926
$679$		0			0
$680$		0			0
$681$			34.9706i			1.34007i
$682$	10.0042	−	21.2220i	0.383079	−	0.812634i
$683$	−24.0000			−0.918334			−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000			−0.918334			−0.459167	+	0.888350i	$0.651852\pi$
$684$	−8.79045			−0.336111
$685$			44.3462i			1.69438i
$686$		0			0
$687$	−25.1716			−0.960355
$688$			4.24264i			0.161749i
$689$	69.2094			2.63667
$690$		−	15.3137i		−	0.582983i
$691$		−	47.2220i		−	1.79641i	−0.439578	−	0.898204i	$-0.644872\pi$
$691$		−	47.2220i		−	1.79641i	0.439578	−	0.898204i	$-0.355128\pi$
$692$	−15.2848			−0.581041
$693$		0			0
$694$	−16.2426			−0.616562
$695$			2.48528i			0.0942721i
$696$			26.7653i			1.01454i
$697$		0			0
$698$		−	7.44543i		−	0.281814i
$699$	−22.1731			−0.838664
$700$		0			0
$701$			27.9411i			1.05532i	0.849455	+	0.527661i	$0.176931\pi$
$701$			27.9411i			1.05532i	−0.849455	+	0.527661i	$0.823069\pi$
$702$	12.0000			0.452911
$703$	−13.7766			−0.519594
$704$	−1.41421	+	3.00000i	−0.0533002	+	0.113067i
$705$		−	45.2429i		−	1.70395i
$706$	8.15640			0.306970
$707$		0			0
$708$	−31.7990			−1.19508
$709$	−0.686292			−0.0257742			−0.0128871	−	0.999917i	$-0.504102\pi$
$709$	−0.686292			−0.0257742			−0.0128871	+	0.999917i	$0.504102\pi$
$710$	−5.22625			−0.196138
$711$			32.4853i			1.21829i
$712$	−12.3003			−0.460972
$713$			15.8645i			0.594129i
$714$		0			0
$715$	−43.4558	−	20.4853i	−1.62516	−	0.766106i
$716$	−5.65685			−0.211407
$717$	−24.8632			−0.928532
$718$	14.4853			0.540586
$719$		−	14.2024i		−	0.529661i	−0.964295	−	0.264830i	$-0.914684\pi$
$719$		−	14.2024i		−	0.529661i	0.964295	−	0.264830i	$-0.0853160\pi$
$720$		−	10.0042i		−	0.372833i
$721$		0			0
$722$		−	13.7279i		−	0.510900i
$723$		−	28.9706i		−	1.07743i
$724$			2.35049i			0.0873554i
$725$			18.7279i			0.695538i
$726$	−18.2919	−	22.1731i	−0.678875	−	0.822921i
$727$			0.0543929i			0.00201732i	0.999999	+	0.00100866i	$0.000321067\pi$
$727$			0.0543929i			0.00201732i	−0.999999	+	0.00100866i	$0.999679\pi$
$728$		0			0
$729$	39.2843			1.45497
$730$	16.9706			0.628109
$731$		0			0
$732$		−	6.00000i		−	0.221766i
$733$	−13.3827			−0.494300			−0.247150	−	0.968977i	$-0.579494\pi$
$733$	−13.3827			−0.494300			−0.247150	+	0.968977i	$0.579494\pi$
$734$	30.5921			1.12918
$735$		0			0
$736$		−	2.24264i		−	0.0826648i
$737$	0.485281	−	1.02944i	0.0178756	−	0.0379198i
$738$			42.4441i			1.56239i
$739$			24.0000i			0.882854i	0.897297	+	0.441427i	$0.145528\pi$
$739$			24.0000i			0.882854i	−0.897297	+	0.441427i	$0.854472\pi$
$740$		−	15.6788i		−	0.576363i
$741$		−	33.2597i		−	1.22182i
$742$		0			0
$743$			48.4264i			1.77659i	0.459271	+	0.888296i	$0.348111\pi$
$743$			48.4264i			1.77659i	−0.459271	+	0.888296i	$0.651889\pi$
$744$			18.4853i			0.677703i
$745$	−48.9384			−1.79296
$746$	−3.51472			−0.128683
$747$	−3.64113			−0.133222
$748$		0			0
$749$		0			0
$750$			21.6569i			0.790797i
$751$	16.5858			0.605224			0.302612	−	0.953114i	$-0.402141\pi$
$751$	16.5858			0.605224			0.302612	+	0.953114i	$0.402141\pi$
$752$		−	6.62567i		−	0.241613i
$753$	18.8284			0.686146
$754$	−56.7778			−2.06773
$755$	−15.6788			−0.570608
$756$		0			0
$757$	−9.85786			−0.358290			−0.179145	−	0.983823i	$-0.557333\pi$
$757$	−9.85786			−0.358290			−0.179145	+	0.983823i	$0.557333\pi$
$758$		−	0.343146i		−	0.0124636i
$759$	17.5809	+	8.28772i	0.638147	+	0.300825i
$760$	−6.00000			−0.217643
$761$	24.8632			0.901289			0.450644	−	0.892704i	$-0.351194\pi$
$761$	24.8632			0.901289			0.450644	+	0.892704i	$0.351194\pi$
$762$			15.6788i			0.567981i
$763$		0			0
$764$	10.2426			0.370566
$765$		0			0
$766$	−1.66205			−0.0600523
$767$		−	67.4558i		−	2.43569i
$768$		−	2.61313i		−	0.0942931i
$769$	−15.6788			−0.565390			−0.282695	−	0.959210i	$-0.591228\pi$
$769$	−15.6788			−0.565390			−0.282695	+	0.959210i	$0.591228\pi$
$770$		0			0
$771$	52.1421			1.87785
$772$		−	14.4853i		−	0.521337i
$773$			4.51528i			0.162403i	0.996698	+	0.0812016i	$0.0258758\pi$
$773$			4.51528i			0.162403i	−0.996698	+	0.0812016i	$0.974124\pi$
$774$	16.2426			0.583830
$775$			12.9343i			0.464614i
$776$	5.09494			0.182898
$777$		0			0
$778$		−	32.0000i		−	1.14726i
$779$	25.4558			0.912050
$780$	37.8519			1.35531
$781$	2.82843	−	6.00000i	0.101209	−	0.214697i
$782$		0			0
$783$	22.1731			0.792402
$784$		0			0
$785$	4.48528			0.160087
$786$	−43.4558			−1.55002
$787$	10.1355			0.361291			0.180645	−	0.983548i	$-0.442181\pi$
$787$	10.1355			0.361291			0.180645	+	0.983548i	$0.442181\pi$
$788$			3.51472i			0.125207i
$789$	−62.7150			−2.23271
$790$			22.1731i			0.788884i
$791$		0			0
$792$	11.4853	+	5.41421i	0.408112	+	0.192386i
$793$	12.7279			0.451982
$794$	−11.5349			−0.409358
$795$	−85.2548			−3.02368
$796$		−	18.8715i		−	0.668884i
$797$			25.9456i			0.919039i	0.888168	+	0.459519i	$0.151978\pi$
$797$			25.9456i			0.919039i	−0.888168	+	0.459519i	$0.848022\pi$
$798$		0			0
$799$		0			0
$800$		−	1.82843i		−	0.0646447i
$801$			47.0907i			1.66387i
$802$		−	16.4853i		−	0.582116i
$803$	−9.18440	+	19.4831i	−0.324111	+	0.687542i
$804$			0.896683i			0.0316236i
$805$		0			0
$806$	−39.2132			−1.38123
$807$	35.7990			1.26018
$808$		−	6.88830i		−	0.242330i
$809$			2.48528i			0.0873778i	0.999045	+	0.0436889i	$0.0139110\pi$
$809$			2.48528i			0.0873778i	−0.999045	+	0.0436889i	$0.986089\pi$
$810$	15.2304			0.535142
$811$	16.6298			0.583952			0.291976	−	0.956426i	$-0.405687\pi$
$811$	16.6298			0.583952			0.291976	+	0.956426i	$0.405687\pi$
$812$		0			0
$813$			4.97056i			0.174325i
$814$	18.0000	+	8.48528i	0.630900	+	0.297409i
$815$			52.6339i			1.84369i
$816$		0			0
$817$		−	9.74153i		−	0.340813i
$818$			2.69005i			0.0940554i
$819$		0			0
$820$			28.9706i			1.01170i
$821$			22.2426i			0.776274i	0.921602	+	0.388137i	$0.126881\pi$
$821$			22.2426i			0.776274i	−0.921602	+	0.388137i	$0.873119\pi$
$822$	44.3462			1.54675
$823$	51.2132			1.78518			0.892590	−	0.450869i	$-0.148886\pi$
$823$	51.2132			1.78518			0.892590	+	0.450869i	$0.148886\pi$
$824$	−10.3212			−0.359556
$825$	14.3337	+	6.75699i	0.499036	+	0.235248i
$826$		0			0
$827$			16.9706i			0.590124i	0.955478	+	0.295062i	$0.0953404\pi$
$827$			16.9706i			0.590124i	−0.955478	+	0.295062i	$0.904660\pi$
$828$	−8.58579			−0.298377
$829$		−	6.04601i		−	0.209987i	−0.994473	−	0.104993i	$-0.966518\pi$
$829$		−	6.04601i		−	0.209987i	0.994473	−	0.104993i	$-0.0334821\pi$
$830$	−2.48528			−0.0862654
$831$	−17.5809			−0.609875
$832$	5.54328			0.192179
$833$		0			0
$834$	2.48528			0.0860583
$835$			45.9411i			1.58986i
$836$	3.24718	−	6.88830i	0.112306	−	0.238237i
$837$	15.3137			0.529319
$838$	5.67459			0.196026
$839$			2.55873i			0.0883373i	0.999024	+	0.0441686i	$0.0140639\pi$
$839$			2.55873i			0.0883373i	−0.999024	+	0.0441686i	$0.985936\pi$
$840$		0			0
$841$	−75.9117			−2.61764
$842$			6.34315i			0.218599i
$843$	31.3575			1.08001
$844$			0.727922i			0.0250561i
$845$			46.3253i			1.59364i
$846$	−25.3659			−0.872097
$847$		0			0
$848$	−12.4853			−0.428746
$849$		−	60.4264i		−	2.07383i
$850$		0			0
$851$	−13.4558			−0.461260
$852$			5.22625i			0.179048i
$853$	2.29610			0.0786170			0.0393085	−	0.999227i	$-0.487484\pi$
$853$	2.29610			0.0786170			0.0393085	+	0.999227i	$0.487484\pi$
$854$		0			0
$855$			22.9706i			0.785577i
$856$	12.7279			0.435031
$857$	26.7653			0.914286			0.457143	−	0.889393i	$-0.348873\pi$
$857$	26.7653			0.914286			0.457143	+	0.889393i	$0.348873\pi$
$858$	−20.4853	+	43.4558i	−0.699356	+	1.48356i
$859$		−	1.71644i		−	0.0585643i	−0.999571	−	0.0292821i	$-0.990678\pi$
$859$		−	1.71644i		−	0.0585643i	0.999571	−	0.0292821i	$-0.00932213\pi$
$860$	11.0866			0.378048
$861$		0			0
$862$	14.4853			0.493371
$863$	−5.27208			−0.179464			−0.0897318	−	0.995966i	$-0.528601\pi$
$863$	−5.27208			−0.179464			−0.0897318	+	0.995966i	$0.528601\pi$
$864$	−2.16478			−0.0736475
$865$			39.9411i			1.35804i
$866$	28.8757			0.981236
$867$			44.4231i			1.50869i
$868$		0			0
$869$	−25.4558	−	12.0000i	−0.863530	−	0.407072i
$870$	69.9411			2.37123
$871$	−1.90215			−0.0644520
$872$	−20.4853			−0.693719
$873$		−	19.5056i		−	0.660164i
$874$			5.14933i			0.174179i
$875$		0			0
$876$		−	16.9706i		−	0.573382i
$877$		−	54.0000i		−	1.82345i	−0.410801	−	0.911725i	$-0.634751\pi$
$877$		−	54.0000i		−	1.82345i	0.410801	−	0.911725i	$-0.365249\pi$
$878$			15.6788i			0.529132i
$879$		−	34.9706i		−	1.17953i
$880$	7.83938	+	3.69552i	0.264265	+	0.124576i
$881$		−	23.7582i		−	0.800435i	−0.916420	−	0.400218i	$-0.868935\pi$
$881$		−	23.7582i		−	0.800435i	0.916420	−	0.400218i	$-0.131065\pi$
$882$		0			0
$883$	42.4264			1.42776			0.713881	−	0.700267i	$-0.246936\pi$
$883$	42.4264			1.42776			0.713881	+	0.700267i	$0.246936\pi$
$884$		0			0
$885$			83.0948i			2.79320i
$886$		−	22.6274i		−	0.760183i
$887$	22.1731			0.744500			0.372250	−	0.928132i	$-0.378586\pi$
$887$	22.1731			0.744500			0.372250	+	0.928132i	$0.378586\pi$
$888$	−15.6788			−0.526145
$889$		0			0
$890$			32.1421i			1.07741i
$891$	−8.24264	+	17.4853i	−0.276139	+	0.585779i
$892$			22.7528i			0.761819i
$893$			15.2132i			0.509090i
$894$			48.9384i			1.63675i
$895$			14.7821i			0.494110i
$896$		0			0
$897$		−	32.4853i		−	1.08465i
$898$		−	15.2721i		−	0.509636i
$899$	−72.4566			−2.41656
$900$	−7.00000			−0.233333
$901$		0			0
$902$	−33.2597	−	15.6788i	−1.10743	−	0.522045i
$903$		0			0
$904$		−	1.41421i		−	0.0470360i
$905$	6.14214			0.204171
$906$			15.6788i			0.520892i
$907$	37.7990			1.25509			0.627547	−	0.778578i	$-0.284059\pi$
$907$	37.7990			1.25509			0.627547	+	0.778578i	$0.284059\pi$
$908$	13.3827			0.444119
$909$	−26.3714			−0.874683
$910$		0			0
$911$	29.7574			0.985905			0.492953	−	0.870056i	$-0.335918\pi$
$911$	29.7574			0.985905			0.492953	+	0.870056i	$0.335918\pi$
$912$			6.00000i			0.198680i
$913$	1.34502	−	2.85323i	0.0445138	−	0.0944281i
$914$	19.4558			0.643542
$915$	−15.6788			−0.518324
$916$			9.63274i			0.318275i
$917$		0			0
$918$		0			0
$919$		−	54.4264i		−	1.79536i	−0.440646	−	0.897681i	$-0.645251\pi$
$919$		−	54.4264i		−	1.79536i	0.440646	−	0.897681i	$-0.354749\pi$
$920$	−5.86030			−0.193208
$921$		−	75.9411i		−	2.50235i
$922$		−	7.44543i		−	0.245202i
$923$	−11.0866			−0.364918
$924$		0			0
$925$	−10.9706			−0.360710
$926$			17.3137i			0.568964i
$927$			39.5139i			1.29781i
$928$	10.2426			0.336231
$929$			49.6269i			1.62820i	0.580722	+	0.814102i	$0.302771\pi$
$929$			49.6269i			1.62820i	−0.580722	+	0.814102i	$0.697229\pi$
$930$	48.3044			1.58396
$931$		0			0
$932$			8.48528i			0.277945i
$933$	−7.85786			−0.257255
$934$	−11.1635			−0.365280
$935$		0			0
$936$		−	21.2220i		−	0.693664i
$937$	24.0753			0.786504			0.393252	−	0.919431i	$-0.371350\pi$
$937$	24.0753			0.786504			0.393252	+	0.919431i	$0.371350\pi$
$938$		0			0
$939$	−16.8284			−0.549175
$940$	−17.3137			−0.564711
$941$	2.85323			0.0930126			0.0465063	−	0.998918i	$-0.485191\pi$
$941$	2.85323			0.0930126			0.0465063	+	0.998918i	$0.485191\pi$
$942$		−	4.48528i		−	0.146138i
$943$	24.8632			0.809656
$944$			12.1689i			0.396065i
$945$		0			0
$946$	−6.00000	+	12.7279i	−0.195077	+	0.413820i
$947$	−20.4853			−0.665682			−0.332841	−	0.942983i	$-0.608007\pi$
$947$	−20.4853			−0.665682			−0.332841	+	0.942983i	$0.608007\pi$
$948$	22.1731			0.720149
$949$	36.0000			1.16861
$950$			4.19825i			0.136209i
$951$		−	28.6675i		−	0.929606i
$952$		0			0
$953$		−	21.9411i		−	0.710743i	−0.934725	−	0.355371i	$-0.884354\pi$
$953$		−	21.9411i		−	0.710743i	0.934725	−	0.355371i	$-0.115646\pi$
$954$			47.7990i			1.54755i
$955$		−	26.7653i		−	0.866105i
$956$			9.51472i			0.307728i
$957$	−37.8519	+	80.2959i	−1.22358	+	2.59560i
$958$		−	22.1731i		−	0.716381i
$959$		0			0
$960$	−6.82843			−0.220387
$961$	−19.0416			−0.614246
$962$		−	33.2597i		−	1.07233i
$963$		−	48.7279i		−	1.57024i
$964$	−11.0866			−0.357074
$965$	−37.8519			−1.21849
$966$		0			0
$967$		−	31.4558i		−	1.01155i	−0.862665	−	0.505776i	$-0.831206\pi$
$967$		−	31.4558i		−	1.01155i	0.862665	−	0.505776i	$-0.168794\pi$
$968$	−8.48528	+	7.00000i	−0.272727	+	0.224989i
$969$		0			0
$970$		−	13.3137i		−	0.427477i
$971$		−	43.1550i		−	1.38491i	−0.721461	−	0.692456i	$-0.756529\pi$
$971$		−	43.1550i		−	1.38491i	0.721461	−	0.692456i	$-0.243471\pi$
$972$		−	21.7248i		−	0.696822i
$973$		0			0
$974$		−	33.5563i		−	1.07521i
$975$		−	26.4853i		−	0.848208i
$976$	−2.29610			−0.0734964
$977$	−35.7574			−1.14398			−0.571990	−	0.820261i	$-0.693828\pi$
$977$	−35.7574			−1.14398			−0.571990	+	0.820261i	$0.693828\pi$
$978$	52.6339			1.68305
$979$	−36.9008	−	17.3952i	−1.17935	−	0.555953i
$980$		0			0
$981$			78.4264i			2.50396i
$982$	27.9411			0.891637
$983$		−	35.4788i		−	1.13160i	−0.824543	−	0.565800i	$-0.808568\pi$
$983$		−	35.4788i		−	1.13160i	0.824543	−	0.565800i	$-0.191432\pi$
$984$	28.9706			0.923548
$985$	9.18440			0.292639
$986$		0			0
$987$		0			0
$988$	−12.7279			−0.404929
$989$		−	9.51472i		−	0.302550i
$990$	14.1480	−	30.0125i	0.449654	−	0.953859i
$991$	21.5980			0.686082			0.343041	−	0.939320i	$-0.388543\pi$
$991$	21.5980			0.686082			0.343041	+	0.939320i	$0.388543\pi$
$992$	7.07401			0.224600
$993$		−	9.18440i		−	0.291458i
$994$		0			0
$995$	−49.3137			−1.56335
$996$			2.48528i			0.0787492i
$997$	−47.9873			−1.51977			−0.759887	−	0.650055i	$-0.774746\pi$
$997$	−47.9873			−1.51977			−0.759887	+	0.650055i	$0.774746\pi$
$998$		−	16.6274i		−	0.526332i
$999$			12.9887i			0.410944i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1078.2.c.a.1077.5	yes	8
7.2	even	3		1078.2.i.a.1011.8		16
7.3	odd	6		1078.2.i.a.901.4		16
7.4	even	3		1078.2.i.a.901.1		16
7.5	odd	6		1078.2.i.a.1011.5		16
7.6	odd	2	inner	1078.2.c.a.1077.8	yes	8
11.10	odd	2	inner	1078.2.c.a.1077.1	✓	8
77.10	even	6		1078.2.i.a.901.8		16
77.32	odd	6		1078.2.i.a.901.5		16
77.54	even	6		1078.2.i.a.1011.1		16
77.65	odd	6		1078.2.i.a.1011.4		16
77.76	even	2	inner	1078.2.c.a.1077.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1078.2.c.a.1077.1	✓	8	11.10	odd	2	inner
1078.2.c.a.1077.4	yes	8	77.76	even	2	inner
1078.2.c.a.1077.5	yes	8	1.1	even	1	trivial
1078.2.c.a.1077.8	yes	8	7.6	odd	2	inner
1078.2.i.a.901.1		16	7.4	even	3
1078.2.i.a.901.4		16	7.3	odd	6
1078.2.i.a.901.5		16	77.32	odd	6
1078.2.i.a.901.8		16	77.10	even	6
1078.2.i.a.1011.1		16	77.54	even	6
1078.2.i.a.1011.4		16	77.65	odd	6
1078.2.i.a.1011.5		16	7.5	odd	6
1078.2.i.a.1011.8		16	7.2	even	3

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:	\( N \)	\(=\)	\( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1078.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -2.61313i q^{3} -1.00000 q^{4} +2.61313i q^{5} +2.61313 q^{6} -1.00000i q^{8} -3.82843 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -2.61313i q^{3} -1.00000 q^{4} +2.61313i q^{5} +2.61313 q^{6} -1.00000i q^{8} -3.82843 q^{9} -2.61313 q^{10} +(1.41421 - 3.00000i) q^{11} +2.61313i q^{12} -5.54328 q^{13} +6.82843 q^{15} +1.00000 q^{16} -3.82843i q^{18} -2.29610 q^{19} -2.61313i q^{20} +(3.00000 + 1.41421i) q^{22} -2.24264 q^{23} -2.61313 q^{24} -1.82843 q^{25} -5.54328i q^{26} +2.16478i q^{27} -10.2426i q^{29} +6.82843i q^{30} -7.07401i q^{31} +1.00000i q^{32} +(-7.83938 - 3.69552i) q^{33} +3.82843 q^{36} +6.00000 q^{37} -2.29610i q^{38} +14.4853i q^{39} +2.61313 q^{40} -11.0866 q^{41} +4.24264i q^{43} +(-1.41421 + 3.00000i) q^{44} -10.0042i q^{45} -2.24264i q^{46} -6.62567i q^{47} -2.61313i q^{48} -1.82843i q^{50} +5.54328 q^{52} -12.4853 q^{53} -2.16478 q^{54} +(7.83938 + 3.69552i) q^{55} +6.00000i q^{57} +10.2426 q^{58} +12.1689i q^{59} -6.82843 q^{60} -2.29610 q^{61} +7.07401 q^{62} -1.00000 q^{64} -14.4853i q^{65} +(3.69552 - 7.83938i) q^{66} +0.343146 q^{67} +5.86030i q^{69} +2.00000 q^{71} +3.82843i q^{72} -6.49435 q^{73} +6.00000i q^{74} +4.77791i q^{75} +2.29610 q^{76} -14.4853 q^{78} -8.48528i q^{79} +2.61313i q^{80} -5.82843 q^{81} -11.0866i q^{82} +0.951076 q^{83} -4.24264 q^{86} -26.7653 q^{87} +(-3.00000 - 1.41421i) q^{88} -12.3003i q^{89} +10.0042 q^{90} +2.24264 q^{92} -18.4853 q^{93} +6.62567 q^{94} -6.00000i q^{95} +2.61313 q^{96} +5.09494i q^{97} +(-5.41421 + 11.4853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^4 - 8 * q^9	\( 8 q - 8 q^{4} - 8 q^{9} + 32 q^{15} + 8 q^{16} + 24 q^{22} + 16 q^{23} + 8 q^{25} + 8 q^{36} + 48 q^{37} - 32 q^{53} + 48 q^{58} - 32 q^{60} - 8 q^{64} + 48 q^{67} + 16 q^{71} - 48 q^{78} - 24 q^{81} - 24 q^{88} - 16 q^{92} - 80 q^{93} - 32 q^{99}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^9 + 32 * q^15 + 8 * q^16 + 24 * q^22 + 16 * q^23 + 8 * q^25 + 8 * q^36 + 48 * q^37 - 32 * q^53 + 48 * q^58 - 32 * q^60 - 8 * q^64 + 48 * q^67 + 16 * q^71 - 48 * q^78 - 24 * q^81 - 24 * q^88 - 16 * q^92 - 80 * q^93 - 32 * q^99

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