LMFDB - Embedded newform 731.2.d.b.560.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [731,2,Mod(560,731)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(731, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("731.560");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 731 = 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	731.d (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:	$5.83706438776$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 20x^{6} + 129x^{4} + 323x^{2} + 256 $ x^8 + 20x^6 + 129x^4 + 323*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			560.1
Root			$-3.20222i$ of defining polynomial
Character	$\chi$	$=$	731.560
Dual form			731.2.d.b.560.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20222i q^{3} -2.00000 q^{4} -0.837930i q^{5} -2.87630i q^{7} -7.25421 q^{9} +O(q^{10})$	\(q-3.20222i q^{3} -2.00000 q^{4} -0.837930i q^{5} -2.87630i q^{7} -7.25421 q^{9} +6.13648i q^{11} +6.40444i q^{12} -4.57097 q^{13} -2.68324 q^{15} +4.00000 q^{16} +(-1.31978 - 3.90617i) q^{17} +2.00000 q^{19} +1.67586i q^{20} -9.21054 q^{21} -5.48464i q^{23} +4.29787 q^{25} +13.6229i q^{27} +5.75260i q^{28} +1.67586i q^{29} +1.40790i q^{31} +19.6504 q^{33} -2.41014 q^{35} +14.5084 q^{36} +3.85406i q^{37} +14.6373i q^{39} -5.11246i q^{41} +1.00000 q^{43} -12.2730i q^{44} +6.07852i q^{45} -11.4647 q^{47} -12.8089i q^{48} -1.27310 q^{49} +(-12.5084 + 4.22624i) q^{51} +9.14194 q^{52} -2.82518 q^{53} +5.14194 q^{55} -6.40444i q^{57} -7.29787 q^{59} +5.36647 q^{60} +9.90867i q^{61} +20.8653i q^{63} -8.00000 q^{64} +3.83015i q^{65} -8.73785 q^{67} +(2.63957 + 7.81234i) q^{68} -17.5630 q^{69} +8.76873i q^{71} -15.6613i q^{73} -13.7627i q^{75} -4.00000 q^{76} +17.6504 q^{77} -8.56840i q^{79} -3.35172i q^{80} +21.8609 q^{81} -1.52730 q^{83} +18.4211 q^{84} +(-3.27310 + 1.10589i) q^{85} +5.36647 q^{87} +3.05461 q^{89} +13.1475i q^{91} +10.9693i q^{92} +4.50841 q^{93} -1.67586i q^{95} +7.16050i q^{97} -44.5153i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 16 * q^9	$ 8 q - 16 q^{4} - 16 q^{9} - 10 q^{13} - 6 q^{15} + 32 q^{16} - 3 q^{17} + 16 q^{19} - 32 q^{21} - 8 q^{25} + 20 q^{33} + 12 q^{35} + 32 q^{36} + 8 q^{43} - 8 q^{47} - 26 q^{49} - 16 q^{51} + 20 q^{52} + 46 q^{53} - 12 q^{55} - 16 q^{59} + 12 q^{60} - 64 q^{64} - 2 q^{67} + 6 q^{68} - 4 q^{69} - 32 q^{76} + 4 q^{77} - 4 q^{81} + 14 q^{83} + 64 q^{84} - 42 q^{85} + 12 q^{87} - 28 q^{89} - 48 q^{93}+O(q^{100}) $ 8 * q - 16 * q^4 - 16 * q^9 - 10 * q^13 - 6 * q^15 + 32 * q^16 - 3 * q^17 + 16 * q^19 - 32 * q^21 - 8 * q^25 + 20 * q^33 + 12 * q^35 + 32 * q^36 + 8 * q^43 - 8 * q^47 - 26 * q^49 - 16 * q^51 + 20 * q^52 + 46 * q^53 - 12 * q^55 - 16 * q^59 + 12 * q^60 - 64 * q^64 - 2 * q^67 + 6 * q^68 - 4 * q^69 - 32 * q^76 + 4 * q^77 - 4 * q^81 + 14 * q^83 + 64 * q^84 - 42 * q^85 + 12 * q^87 - 28 * q^89 - 48 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/731\mathbb{Z}\right)^\times$.

$n$	$173$	$562$
$\chi(n)$	$-1$	$1$

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 $

$2$		0			0			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		−	3.20222i		−	1.84880i	−0.381422	−	0.924401i	$-0.624566\pi$
$3$		−	3.20222i		−	1.84880i	0.381422	−	0.924401i	$-0.375434\pi$
$4$	−2.00000			−1.00000
$5$		−	0.837930i		−	0.374734i	−0.982290	−	0.187367i	$-0.940005\pi$
$5$		−	0.837930i		−	0.374734i	0.982290	−	0.187367i	$-0.0599953\pi$
$6$		0			0
$7$		−	2.87630i		−	1.08714i	−0.839364	−	0.543569i	$-0.817072\pi$
$7$		−	2.87630i		−	1.08714i	0.839364	−	0.543569i	$-0.182928\pi$
$8$		0			0
$9$	−7.25421			−2.41807
$10$		0			0
$11$			6.13648i			1.85022i	0.379701	+	0.925109i	$0.376027\pi$
$11$			6.13648i			1.85022i	−0.379701	+	0.925109i	$0.623973\pi$
$12$			6.40444i			1.84880i
$13$	−4.57097			−1.26776			−0.633880	−	0.773432i	$-0.718539\pi$
$13$	−4.57097			−1.26776			−0.633880	+	0.773432i	$0.718539\pi$
$14$		0			0
$15$	−2.68324			−0.692809
$16$	4.00000			1.00000
$17$	−1.31978	−	3.90617i	−0.320095	−	0.947386i
$17$	−1.31978	−	3.90617i	−0.320095	−	0.947386i
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$			1.67586i			0.374734i
$21$	−9.21054			−2.00990
$22$		0			0
$23$		−	5.48464i		−	1.14363i	−0.820384	−	0.571813i	$-0.806240\pi$
$23$		−	5.48464i		−	1.14363i	0.820384	−	0.571813i	$-0.193760\pi$
$24$		0			0
$25$	4.29787			0.859575
$26$		0			0
$27$			13.6229i			2.62173i
$28$			5.75260i			1.08714i
$29$			1.67586i			0.311199i	0.987820	+	0.155600i	$0.0497310\pi$
$29$			1.67586i			0.311199i	−0.987820	+	0.155600i	$0.950269\pi$
$30$		0			0
$31$			1.40790i			0.252867i	0.991975	+	0.126433i	$0.0403530\pi$
$31$			1.40790i			0.252867i	−0.991975	+	0.126433i	$0.959647\pi$
$32$		0			0
$33$	19.6504			3.42069
$34$		0			0
$35$	−2.41014			−0.407388
$36$	14.5084			2.41807
$37$			3.85406i			0.633603i	0.948492	+	0.316802i	$0.102609\pi$
$37$			3.85406i			0.633603i	−0.948492	+	0.316802i	$0.897391\pi$
$38$		0			0
$39$			14.6373i			2.34384i
$40$		0			0
$41$		−	5.11246i		−	0.798432i	−0.916857	−	0.399216i	$-0.869282\pi$
$41$		−	5.11246i		−	0.798432i	0.916857	−	0.399216i	$-0.130718\pi$
$42$		0			0
$43$	1.00000			0.152499
$43$	1.00000			0.152499
$44$		−	12.2730i		−	1.85022i
$45$			6.07852i			0.906132i
$46$		0			0
$47$	−11.4647			−1.67231			−0.836153	−	0.548497i	$-0.815200\pi$
$47$	−11.4647			−1.67231			−0.836153	+	0.548497i	$0.815200\pi$
$48$		−	12.8089i		−	1.84880i
$49$	−1.27310			−0.181871
$50$		0			0
$51$	−12.5084	+	4.22624i	−1.75153	+	0.591792i
$52$	9.14194			1.26776
$53$	−2.82518			−0.388068			−0.194034	−	0.980995i	$-0.562157\pi$
$53$	−2.82518			−0.388068			−0.194034	+	0.980995i	$0.562157\pi$
$54$		0			0
$55$	5.14194			0.693339
$56$		0			0
$57$		−	6.40444i		−	0.848289i
$58$		0			0
$59$	−7.29787			−0.950102			−0.475051	−	0.879958i	$-0.657570\pi$
$59$	−7.29787			−0.950102			−0.475051	+	0.879958i	$0.657570\pi$
$60$	5.36647			0.692809
$61$			9.90867i			1.26868i	0.773056	+	0.634338i	$0.218727\pi$
$61$			9.90867i			1.26868i	−0.773056	+	0.634338i	$0.781273\pi$
$62$		0			0
$63$			20.8653i			2.62878i
$64$	−8.00000			−1.00000
$65$			3.83015i			0.475072i
$66$		0			0
$67$	−8.73785			−1.06750			−0.533749	−	0.845643i	$-0.679217\pi$
$67$	−8.73785			−1.06750			−0.533749	+	0.845643i	$0.679217\pi$
$68$	2.63957	+	7.81234i	0.320095	+	0.947386i
$69$	−17.5630			−2.11434
$70$		0			0
$71$			8.76873i			1.04066i	0.853966	+	0.520328i	$0.174190\pi$
$71$			8.76873i			1.04066i	−0.853966	+	0.520328i	$0.825810\pi$
$72$		0			0
$73$		−	15.6613i		−	1.83301i	−0.400020	−	0.916507i	$-0.630997\pi$
$73$		−	15.6613i		−	1.83301i	0.400020	−	0.916507i	$-0.369003\pi$
$74$		0			0
$75$		−	13.7627i		−	1.58918i
$76$	−4.00000			−0.458831
$77$	17.6504			2.01144
$78$		0			0
$79$		−	8.56840i		−	0.964021i	−0.876165	−	0.482010i	$-0.839907\pi$
$79$		−	8.56840i		−	0.964021i	0.876165	−	0.482010i	$-0.160093\pi$
$80$		−	3.35172i		−	0.374734i
$81$	21.8609			2.42899
$82$		0			0
$83$	−1.52730			−0.167643			−0.0838217	−	0.996481i	$-0.526713\pi$
$83$	−1.52730			−0.167643			−0.0838217	+	0.996481i	$0.526713\pi$
$84$	18.4211			2.00990
$85$	−3.27310	+	1.10589i	−0.355017	+	0.119950i
$86$		0			0
$87$	5.36647			0.575346
$88$		0			0
$89$	3.05461			0.323788			0.161894	−	0.986808i	$-0.448240\pi$
$89$	3.05461			0.323788			0.161894	+	0.986808i	$0.448240\pi$
$90$		0			0
$91$			13.1475i			1.37823i
$92$			10.9693i			1.14363i
$93$	4.50841			0.467501
$94$		0			0
$95$		−	1.67586i		−	0.171940i
$96$		0			0
$97$			7.16050i			0.727039i	0.931587	+	0.363519i	$0.118425\pi$
$97$			7.16050i			0.727039i	−0.931587	+	0.363519i	$0.881575\pi$
$98$		0			0
$99$		−	44.5153i		−	4.47396i
$100$	−8.59575			−0.859575
$101$	−14.6504			−1.45776			−0.728882	−	0.684639i	$-0.759960\pi$
$101$	−14.6504			−1.45776			−0.728882	+	0.684639i	$0.759960\pi$
$102$		0			0
$103$	−0.248166			−0.0244525			−0.0122262	−	0.999925i	$-0.503892\pi$
$103$	−0.248166			−0.0244525			−0.0122262	+	0.999925i	$0.503892\pi$
$104$		0			0
$105$			7.71779i			0.753179i
$106$		0			0
$107$			12.9248i			1.24949i	0.780830	+	0.624744i	$0.214797\pi$
$107$			12.9248i			1.24949i	−0.780830	+	0.624744i	$0.785203\pi$
$108$		−	27.2458i		−	2.62173i
$109$		−	3.61968i		−	0.346702i	−0.984860	−	0.173351i	$-0.944540\pi$
$109$		−	3.61968i		−	0.346702i	0.984860	−	0.173351i	$-0.0554595\pi$
$110$		0			0
$111$	12.3415			1.17141
$112$		−	11.5052i		−	1.08714i
$113$		−	5.18009i		−	0.487302i	−0.969863	−	0.243651i	$-0.921655\pi$
$113$		−	5.18009i		−	0.487302i	0.969863	−	0.243651i	$-0.0783452\pi$
$114$		0			0
$115$	−4.59575			−0.428556
$116$		−	3.35172i		−	0.311199i
$117$	33.1588			3.06553
$118$		0			0
$119$	−11.2353	+	3.79610i	−1.02994	+	0.347988i
$120$		0			0
$121$	−26.6564			−2.42331
$122$		0			0
$123$	−16.3712			−1.47614
$124$		−	2.81581i		−	0.252867i
$125$		−	7.79097i		−	0.696845i
$126$		0			0
$127$	8.03572			0.713055			0.356527	−	0.934285i	$-0.383961\pi$
$127$	8.03572			0.713055			0.356527	+	0.934285i	$0.383961\pi$
$128$		0			0
$129$		−	3.20222i		−	0.281940i
$130$		0			0
$131$		−	10.1314i		−	0.885180i	−0.896724	−	0.442590i	$-0.854060\pi$
$131$		−	10.1314i		−	0.885180i	0.896724	−	0.442590i	$-0.145940\pi$
$132$	−39.3007			−3.42069
$133$		−	5.75260i		−	0.498814i
$134$		0			0
$135$	11.4150			0.982450
$136$		0			0
$137$	8.59575			0.734384			0.367192	−	0.930145i	$-0.380319\pi$
$137$	8.59575			0.734384			0.367192	+	0.930145i	$0.380319\pi$
$138$		0			0
$139$		−	12.9131i		−	1.09527i	−0.836716	−	0.547637i	$-0.815527\pi$
$139$		−	12.9131i		−	1.09527i	0.836716	−	0.547637i	$-0.184473\pi$
$140$	4.82028			0.407388
$141$			36.7126i			3.09176i
$142$		0			0
$143$		−	28.0497i		−	2.34563i
$144$	−29.0168			−2.41807
$145$	1.40425			0.116617
$146$		0			0
$147$			4.07674i			0.336244i
$148$		−	7.70812i		−	0.633603i
$149$	0.720861			0.0590552			0.0295276	−	0.999564i	$-0.490600\pi$
$149$	0.720861			0.0590552			0.0295276	+	0.999564i	$0.490600\pi$
$150$		0			0
$151$	−12.5957			−1.02503			−0.512514	−	0.858679i	$-0.671286\pi$
$151$	−12.5957			−1.02503			−0.512514	+	0.858679i	$0.671286\pi$
$152$		0			0
$153$	9.57399	+	28.3362i	0.774011	+	2.29084i
$154$		0			0
$155$	1.17972			0.0947577
$156$		−	29.2745i		−	2.34384i
$157$	−5.45380			−0.435261			−0.217630	−	0.976031i	$-0.569833\pi$
$157$	−5.45380			−0.435261			−0.217630	+	0.976031i	$0.569833\pi$
$158$		0			0
$159$			9.04684i			0.717461i
$160$		0			0
$161$	−15.7755			−1.24328
$162$		0			0
$163$		−	17.6168i		−	1.37985i	−0.723879	−	0.689927i	$-0.757643\pi$
$163$		−	17.6168i		−	1.37985i	0.723879	−	0.689927i	$-0.242357\pi$
$164$			10.2249i			0.798432i
$165$		−	16.4656i		−	1.28185i
$166$		0			0
$167$		−	5.48464i		−	0.424414i	−0.977225	−	0.212207i	$-0.931935\pi$
$167$		−	5.48464i		−	0.424414i	0.977225	−	0.212207i	$-0.0680652\pi$
$168$		0			0
$169$	7.89378			0.607214
$170$		0			0
$171$	−14.5084			−1.10949
$172$	−2.00000			−0.152499
$173$			18.9337i			1.43950i	0.694234	+	0.719750i	$0.255743\pi$
$173$			18.9337i			1.43950i	−0.694234	+	0.719750i	$0.744257\pi$
$174$		0			0
$175$		−	12.3620i		−	0.934477i
$176$			24.5459i			1.85022i
$177$			23.3694i			1.75655i
$178$		0			0
$179$	−17.6504			−1.31925			−0.659625	−	0.751595i	$-0.729285\pi$
$179$	−17.6504			−1.31925			−0.659625	+	0.751595i	$0.729285\pi$
$180$		−	12.1570i		−	0.906132i
$181$		−	8.43313i		−	0.626830i	−0.949616	−	0.313415i	$-0.898527\pi$
$181$		−	8.43313i		−	0.626830i	0.949616	−	0.313415i	$-0.101473\pi$
$182$		0			0
$183$	31.7297			2.34553
$184$		0			0
$185$	3.22943			0.237433
$186$		0			0
$187$	23.9701	−	8.09883i	1.75287	−	0.592245i
$188$	22.9295			1.67231
$189$	39.1835			2.85018
$190$		0			0
$191$	3.31661			0.239981			0.119991	−	0.992775i	$-0.461714\pi$
$191$	3.31661			0.239981			0.119991	+	0.992775i	$0.461714\pi$
$192$			25.6178i			1.84880i
$193$		−	23.4370i		−	1.68703i	−0.537103	−	0.843517i	$-0.680481\pi$
$193$		−	23.4370i		−	1.68703i	0.537103	−	0.843517i	$-0.319519\pi$
$194$		0			0
$195$	12.2650			0.878314
$196$	2.54620			0.181871
$197$			4.37574i			0.311759i	0.987776	+	0.155879i	$0.0498211\pi$
$197$			4.37574i			0.311759i	−0.987776	+	0.155879i	$0.950179\pi$
$198$		0			0
$199$			1.64228i			0.116418i	0.998304	+	0.0582092i	$0.0185390\pi$
$199$			1.64228i			0.116418i	−0.998304	+	0.0582092i	$0.981461\pi$
$200$		0			0
$201$			27.9805i			1.97359i
$202$		0			0
$203$	4.82028			0.338317
$204$	25.0168	−	8.45248i	1.75153	−	0.591792i
$205$	−4.28388			−0.299199
$206$		0			0
$207$			39.7867i			2.76537i
$208$	−18.2839			−1.26776
$209$			12.2730i			0.848939i
$210$		0			0
$211$		−	21.7271i		−	1.49576i	−0.663836	−	0.747878i	$-0.731073\pi$
$211$		−	21.7271i		−	1.49576i	0.663836	−	0.747878i	$-0.268927\pi$
$212$	5.65036			0.388068
$213$	28.0794			1.92397
$214$		0			0
$215$		−	0.837930i		−	0.0571464i
$216$		0			0
$217$	4.04955			0.274901
$218$		0			0
$219$	−50.1508			−3.38888
$220$	−10.2839			−0.693339
$221$	6.03270	+	17.8550i	0.405803	+	1.20106i
$222$		0			0
$223$	17.7876			1.19114			0.595571	−	0.803302i	$-0.296926\pi$
$223$	17.7876			1.19114			0.595571	+	0.803302i	$0.296926\pi$
$224$		0			0
$225$	−31.1777			−2.07851
$226$		0			0
$227$		−	14.8376i		−	0.984804i	−0.870368	−	0.492402i	$-0.836119\pi$
$227$		−	14.8376i		−	0.984804i	0.870368	−	0.492402i	$-0.163881\pi$
$228$			12.8089i			0.848289i
$229$	−1.54604			−0.102165			−0.0510826	−	0.998694i	$-0.516267\pi$
$229$	−1.54604			−0.102165			−0.0510826	+	0.998694i	$0.516267\pi$
$230$		0			0
$231$		−	56.5203i		−	3.71876i
$232$		0			0
$233$		−	11.2363i		−	0.736112i	−0.929804	−	0.368056i	$-0.880024\pi$
$233$		−	11.2363i		−	0.736112i	0.929804	−	0.368056i	$-0.119976\pi$
$234$		0			0
$235$			9.60666i			0.626669i
$236$	14.5957			0.950102
$237$	−27.4379			−1.78228
$238$		0			0
$239$	−6.36043			−0.411422			−0.205711	−	0.978613i	$-0.565951\pi$
$239$	−6.36043			−0.411422			−0.205711	+	0.978613i	$0.565951\pi$
$240$	−10.7329			−0.692809
$241$			27.5860i			1.77697i	0.458908	+	0.888484i	$0.348241\pi$
$241$			27.5860i			1.77697i	−0.458908	+	0.888484i	$0.651759\pi$
$242$		0			0
$243$		−	29.1347i		−	1.86899i
$244$		−	19.8173i		−	1.26868i
$245$			1.06677i			0.0681533i
$246$		0			0
$247$	−9.14194			−0.581688
$248$		0			0
$249$			4.89076i			0.309940i
$250$		0			0
$251$	23.1589			1.46178			0.730889	−	0.682496i	$-0.239106\pi$
$251$	23.1589			1.46178			0.730889	+	0.682496i	$0.239106\pi$
$252$		−	41.7305i		−	2.62878i
$253$	33.6564			2.11596
$254$		0			0
$255$	3.54129	+	10.4812i	0.221764	+	0.656357i
$256$	16.0000			1.00000
$257$	20.5957			1.28473			0.642364	−	0.766400i	$-0.277954\pi$
$257$	20.5957			1.28473			0.642364	+	0.766400i	$0.277954\pi$
$258$		0			0
$259$	11.0854			0.688815
$260$		−	7.66031i		−	0.475072i
$261$		−	12.1570i		−	0.752502i
$262$		0			0
$263$	−31.8749			−1.96549			−0.982745	−	0.184967i	$-0.940782\pi$
$263$	−31.8749			−1.96549			−0.982745	+	0.184967i	$0.940782\pi$
$264$		0			0
$265$			2.36730i			0.145422i
$266$		0			0
$267$		−	9.78153i		−	0.598620i
$268$	17.4757			1.06750
$269$		−	7.44016i		−	0.453635i	−0.973937	−	0.226817i	$-0.927168\pi$
$269$		−	7.44016i		−	0.453635i	0.973937	−	0.226817i	$-0.0728320\pi$
$270$		0			0
$271$	1.11717			0.0678631			0.0339315	−	0.999424i	$-0.489197\pi$
$271$	1.11717			0.0678631			0.0339315	+	0.999424i	$0.489197\pi$
$272$	−5.27914	−	15.6247i	−0.320095	−	0.947386i
$273$	42.1011			2.54808
$274$		0			0
$275$			26.3738i			1.59040i
$276$	35.1260			2.11434
$277$		−	7.39488i		−	0.444315i	−0.975011	−	0.222158i	$-0.928690\pi$
$277$		−	7.39488i		−	0.444315i	0.975011	−	0.222158i	$-0.0713100\pi$
$278$		0			0
$279$		−	10.2132i		−	0.611449i
$280$		0			0
$281$	−18.9921			−1.13297			−0.566485	−	0.824072i	$-0.691697\pi$
$281$	−18.9921			−1.13297			−0.566485	+	0.824072i	$0.691697\pi$
$282$		0			0
$283$			17.9407i			1.06646i	0.845969	+	0.533231i	$0.179022\pi$
$283$			17.9407i			1.06646i	−0.845969	+	0.533231i	$0.820978\pi$
$284$		−	17.5375i		−	1.04066i
$285$	−5.36647			−0.317882
$286$		0			0
$287$	−14.7050			−0.868007
$288$		0			0
$289$	−13.5163	+	10.3106i	−0.795079	+	0.606506i
$290$		0			0
$291$	22.9295			1.34415
$292$			31.3225i			1.83301i
$293$	−10.9187			−0.637878			−0.318939	−	0.947775i	$-0.603326\pi$
$293$	−10.9187			−0.637878			−0.318939	+	0.947775i	$0.603326\pi$
$294$		0			0
$295$			6.11511i			0.356035i
$296$		0			0
$297$	−83.5967			−4.85077
$298$		0			0
$299$			25.0701i			1.44984i
$300$			27.5255i			1.58918i
$301$		−	2.87630i		−	0.165787i
$302$		0			0
$303$			46.9136i			2.69512i
$304$	8.00000			0.458831
$305$	8.30278			0.475416
$306$		0			0
$307$	21.9295			1.25158			0.625791	−	0.779991i	$-0.284776\pi$
$307$	21.9295			1.25158			0.625791	+	0.779991i	$0.284776\pi$
$308$	−35.3007			−2.01144
$309$			0.794681i			0.0452078i
$310$		0			0
$311$		−	14.1690i		−	0.803449i	−0.915761	−	0.401724i	$-0.868411\pi$
$311$		−	14.1690i		−	0.803449i	0.915761	−	0.401724i	$-0.131589\pi$
$312$		0			0
$313$		−	19.5153i		−	1.10307i	−0.834151	−	0.551536i	$-0.814042\pi$
$313$		−	19.5153i		−	1.10307i	0.834151	−	0.551536i	$-0.185958\pi$
$314$		0			0
$315$	17.4836			0.985091
$316$			17.1368i			0.964021i
$317$			31.7064i			1.78081i	0.455169	+	0.890405i	$0.349579\pi$
$317$			31.7064i			1.78081i	−0.455169	+	0.890405i	$0.650421\pi$
$318$		0			0
$319$	−10.2839			−0.575787
$320$			6.70344i			0.374734i
$321$	41.3880			2.31006
$322$		0			0
$323$	−2.63957	−	7.81234i	−0.146870	−	0.434690i
$324$	−43.7218			−2.42899
$325$	−19.6455			−1.08973
$326$		0			0
$327$	−11.5910			−0.640984
$328$		0			0
$329$			32.9760i			1.81803i
$330$		0			0
$331$	−19.7876			−1.08762			−0.543811	−	0.839208i	$-0.683019\pi$
$331$	−19.7876			−1.08762			−0.543811	+	0.839208i	$0.683019\pi$
$332$	3.05461			0.167643
$333$		−	27.9581i		−	1.53210i
$334$		0			0
$335$			7.32170i			0.400027i
$336$	−36.8422			−2.00990
$337$			1.66416i			0.0906525i	0.998972	+	0.0453263i	$0.0144327\pi$
$337$			1.66416i			0.0906525i	−0.998972	+	0.0453263i	$0.985567\pi$
$338$		0			0
$339$	−16.5878			−0.900926
$340$	6.54620	−	2.21177i	0.355017	−	0.119950i
$341$	−8.63957			−0.467859
$342$		0			0
$343$		−	16.4723i		−	0.889420i
$344$		0			0
$345$			14.7166i			0.792314i
$346$		0			0
$347$		−	4.08796i		−	0.219453i	−0.993962	−	0.109727i	$-0.965002\pi$
$347$		−	4.08796i		−	0.219453i	0.993962	−	0.109727i	$-0.0349975\pi$
$348$	−10.7329			−0.575346
$349$	29.4258			1.57513			0.787564	−	0.616233i	$-0.211342\pi$
$349$	29.4258			1.57513			0.787564	+	0.616233i	$0.211342\pi$
$350$		0			0
$351$		−	62.2699i		−	3.32372i
$352$		0			0
$353$	22.1231			1.17749			0.588746	−	0.808318i	$-0.299622\pi$
$353$	22.1231			1.17749			0.588746	+	0.808318i	$0.299622\pi$
$354$		0			0
$355$	7.34758			0.389969
$356$	−6.10922			−0.323788
$357$	12.1559	+	35.9779i	0.643360	+	1.90415i
$358$		0			0
$359$	0.697225			0.0367981			0.0183991	−	0.999831i	$-0.494143\pi$
$359$	0.697225			0.0367981			0.0183991	+	0.999831i	$0.494143\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$			85.3596i			4.48022i
$364$		−	26.2950i		−	1.37823i
$365$	−13.1231			−0.686892
$366$		0			0
$367$		−	15.0460i		−	0.785396i	−0.919668	−	0.392698i	$-0.871542\pi$
$367$		−	15.0460i		−	0.785396i	0.919668	−	0.392698i	$-0.128458\pi$
$368$		−	21.9386i		−	1.14363i
$369$			37.0868i			1.93066i
$370$		0			0
$371$			8.12606i			0.421884i
$372$	−9.01683			−0.467501
$373$	25.7596			1.33378			0.666890	−	0.745156i	$-0.267625\pi$
$373$	25.7596			1.33378			0.666890	+	0.745156i	$0.267625\pi$
$374$		0			0
$375$	−24.9484			−1.28833
$376$		0			0
$377$		−	7.66031i		−	0.394526i
$378$		0			0
$379$		−	10.0973i		−	0.518663i	−0.965788	−	0.259332i	$-0.916498\pi$
$379$		−	10.0973i		−	0.518663i	0.965788	−	0.259332i	$-0.0835022\pi$
$380$			3.35172i			0.171940i
$381$		−	25.7321i		−	1.31830i
$382$		0			0
$383$	−20.3337			−1.03901			−0.519503	−	0.854469i	$-0.673883\pi$
$383$	−20.3337			−1.03901			−0.519503	+	0.854469i	$0.673883\pi$
$384$		0			0
$385$		−	14.7898i		−	0.753756i
$386$		0			0
$387$	−7.25421			−0.368752
$388$		−	14.3210i		−	0.727039i
$389$	12.8301			0.650511			0.325255	−	0.945626i	$-0.394550\pi$
$389$	12.8301			0.650511			0.325255	+	0.945626i	$0.394550\pi$
$390$		0			0
$391$	−21.4239	+	7.23855i	−1.08346	+	0.366069i
$392$		0			0
$393$	−32.4428			−1.63652
$394$		0			0
$395$	−7.17972			−0.361251
$396$			89.0306i			4.47396i
$397$		−	26.9214i		−	1.35115i	−0.737292	−	0.675574i	$-0.763896\pi$
$397$		−	26.9214i		−	1.35115i	0.737292	−	0.675574i	$-0.236104\pi$
$398$		0			0
$399$	−18.4211			−0.922208
$400$	17.1915			0.859575
$401$			19.9500i			0.996257i	0.867103	+	0.498129i	$0.165979\pi$
$401$			19.9500i			0.996257i	−0.867103	+	0.498129i	$0.834021\pi$
$402$		0			0
$403$		−	6.43548i		−	0.320574i
$404$	29.3007			1.45776
$405$		−	18.3179i		−	0.910224i
$406$		0			0
$407$	−23.6504			−1.17230
$408$		0			0
$409$	24.6176			1.21726			0.608632	−	0.793453i	$-0.291719\pi$
$409$	24.6176			1.21726			0.608632	+	0.793453i	$0.291719\pi$
$410$		0			0
$411$		−	27.5255i		−	1.35773i
$412$	0.496332			0.0244525
$413$			20.9909i			1.03289i
$414$		0			0
$415$			1.27977i			0.0628217i
$416$		0			0
$417$	−41.3506			−2.02495
$418$		0			0
$419$		−	16.6360i		−	0.812723i	−0.913712	−	0.406361i	$-0.866797\pi$
$419$		−	16.6360i		−	0.812723i	0.913712	−	0.406361i	$-0.133203\pi$
$420$		−	15.4356i		−	0.753179i
$421$	−31.4258			−1.53160			−0.765801	−	0.643078i	$-0.777657\pi$
$421$	−31.4258			−1.53160			−0.765801	+	0.643078i	$0.777657\pi$
$422$		0			0
$423$	83.1676			4.04375
$424$		0			0
$425$	−5.67227	−	16.7882i	−0.275145	−	0.814349i
$426$		0			0
$427$	28.5003			1.37923
$428$		−	25.8496i		−	1.24949i
$429$	−89.8212			−4.33661
$430$		0			0
$431$			1.10890i			0.0534138i	0.999643	+	0.0267069i	$0.00850209\pi$
$431$			1.10890i			0.0534138i	−0.999643	+	0.0267069i	$0.991498\pi$
$432$			54.4916i			2.62173i
$433$	13.2414			0.636339			0.318169	−	0.948034i	$-0.396932\pi$
$433$	13.2414			0.636339			0.318169	+	0.948034i	$0.396932\pi$
$434$		0			0
$435$		−	4.49673i		−	0.215602i
$436$			7.23936i			0.346702i
$437$		−	10.9693i		−	0.524732i
$438$		0			0
$439$		−	23.9058i		−	1.14096i	−0.821311	−	0.570480i	$-0.806757\pi$
$439$		−	23.9058i		−	1.14096i	0.821311	−	0.570480i	$-0.193243\pi$
$440$		0			0
$441$	9.23532			0.439777
$442$		0			0
$443$	−36.6188			−1.73981			−0.869905	−	0.493219i	$-0.835820\pi$
$443$	−36.6188			−1.73981			−0.869905	+	0.493219i	$0.835820\pi$
$444$	−24.6831			−1.17141
$445$		−	2.55955i		−	0.121334i
$446$		0			0
$447$		−	2.30835i		−	0.109181i
$448$			23.0104i			1.08714i
$449$			17.6391i			0.832443i	0.909263	+	0.416221i	$0.136646\pi$
$449$			17.6391i			0.832443i	−0.909263	+	0.416221i	$0.863354\pi$
$450$		0			0
$451$	31.3725			1.47727
$452$			10.3602i			0.487302i
$453$			40.3343i			1.89507i
$454$		0			0
$455$	11.0167			0.516469
$456$		0			0
$457$	−2.92949			−0.137036			−0.0685180	−	0.997650i	$-0.521827\pi$
$457$	−2.92949			−0.137036			−0.0685180	+	0.997650i	$0.521827\pi$
$458$		0			0
$459$	53.2134	−	17.9793i	2.48379	−	0.839202i
$460$	9.19149			0.428556
$461$	32.3136			1.50499			0.752496	−	0.658596i	$-0.228850\pi$
$461$	32.3136			1.50499			0.752496	+	0.658596i	$0.228850\pi$
$462$		0			0
$463$	3.50367			0.162829			0.0814146	−	0.996680i	$-0.474056\pi$
$463$	3.50367			0.162829			0.0814146	+	0.996680i	$0.474056\pi$
$464$			6.70344i			0.311199i
$465$		−	3.77774i		−	0.175188i
$466$		0			0
$467$	33.7498			1.56175			0.780877	−	0.624685i	$-0.214773\pi$
$467$	33.7498			1.56175			0.780877	+	0.624685i	$0.214773\pi$
$468$	−66.3175			−3.06553
$469$			25.1327i			1.16052i
$470$		0			0
$471$			17.4643i			0.804711i
$472$		0			0
$473$			6.13648i			0.282156i
$474$		0			0
$475$	8.59575			0.394400
$476$	22.4706	−	7.59219i	1.02994	−	0.347988i
$477$	20.4944			0.938375
$478$		0			0
$479$		−	1.45571i		−	0.0665131i	−0.999447	−	0.0332566i	$-0.989412\pi$
$479$		−	1.45571i		−	0.0665131i	0.999447	−	0.0332566i	$-0.0105878\pi$
$480$		0			0
$481$		−	17.6168i		−	0.803256i
$482$		0			0
$483$			50.5165i			2.29858i
$484$	53.3128			2.42331
$485$	6.00000			0.272446
$486$		0			0
$487$		−	25.8262i		−	1.17030i	−0.810926	−	0.585148i	$-0.801036\pi$
$487$		−	25.8262i		−	1.17030i	0.810926	−	0.585148i	$-0.198964\pi$
$488$		0			0
$489$	−56.4128			−2.55108
$490$		0			0
$491$	23.2791			1.05057			0.525286	−	0.850925i	$-0.323958\pi$
$491$	23.2791			1.05057			0.525286	+	0.850925i	$0.323958\pi$
$492$	32.7424			1.47614
$493$	6.54620	−	2.21177i	0.294826	−	0.0996133i
$494$		0			0
$495$	−37.3007			−1.67654
$496$			5.63161i			0.252867i
$497$	25.2215			1.13134
$498$		0			0
$499$			16.0955i			0.720533i	0.932849	+	0.360267i	$0.117314\pi$
$499$			16.0955i			0.720533i	−0.932849	+	0.360267i	$0.882686\pi$
$500$			15.5819i			0.696845i
$501$	−17.5630			−0.784658
$502$		0			0
$503$			26.0265i			1.16047i	0.814451	+	0.580233i	$0.197038\pi$
$503$			26.0265i			1.16047i	−0.814451	+	0.580233i	$0.802962\pi$
$504$		0			0
$505$			12.2760i			0.546274i
$506$		0			0
$507$		−	25.2776i		−	1.12262i
$508$	−16.0714			−0.713055
$509$	−0.533189			−0.0236332			−0.0118166	−	0.999930i	$-0.503761\pi$
$509$	−0.533189			−0.0236332			−0.0118166	+	0.999930i	$0.503761\pi$
$510$		0			0
$511$	−45.0465			−1.99274
$512$		0			0
$513$			27.2458i			1.20293i
$514$		0			0
$515$			0.207946i			0.00916318i
$516$			6.40444i			0.281940i
$517$		−	70.3532i		−	3.09413i
$518$		0			0
$519$	60.6297			2.66135
$520$		0			0
$521$		−	20.1575i		−	0.883116i	−0.897233	−	0.441558i	$-0.854426\pi$
$521$		−	20.1575i		−	0.883116i	0.897233	−	0.441558i	$-0.145574\pi$
$522$		0			0
$523$	12.9295			0.565367			0.282684	−	0.959213i	$-0.408775\pi$
$523$	12.9295			0.565367			0.282684	+	0.959213i	$0.408775\pi$
$524$			20.2627i			0.885180i
$525$	−39.5857			−1.72766
$526$		0			0
$527$	5.49951	−	1.85813i	0.239562	−	0.0809414i
$528$	78.6014			3.42069
$529$	−7.08129			−0.307882
$530$		0			0
$531$	52.9403			2.29741
$532$			11.5052i			0.498814i
$533$			23.3689i			1.01222i
$534$		0			0
$535$	10.8301			0.468225
$536$		0			0
$537$			56.5203i			2.43903i
$538$		0			0
$539$		−	7.81234i		−	0.336501i
$540$	−22.8301			−0.982450
$541$			3.61968i			0.155622i	0.996968	+	0.0778111i	$0.0247931\pi$
$541$			3.61968i			0.155622i	−0.996968	+	0.0778111i	$0.975207\pi$
$542$		0			0
$543$	−27.0047			−1.15888
$544$		0			0
$545$	−3.03304			−0.129921
$546$		0			0
$547$			7.41676i			0.317118i	0.987350	+	0.158559i	$0.0506848\pi$
$547$			7.41676i			0.317118i	−0.987350	+	0.158559i	$0.949315\pi$
$548$	−17.1915			−0.734384
$549$		−	71.8796i		−	3.06774i
$550$		0			0
$551$			3.35172i			0.142788i
$552$		0			0
$553$	−24.6453			−1.04802
$554$		0			0
$555$		−	10.3413i		−	0.438966i
$556$			25.8262i			1.09527i
$557$	−17.0497			−0.722419			−0.361210	−	0.932485i	$-0.617636\pi$
$557$	−17.0497			−0.722419			−0.361210	+	0.932485i	$0.617636\pi$
$558$		0			0
$559$	−4.57097			−0.193331
$560$	−9.64055			−0.407388
$561$	−25.9342	−	76.7576i	−1.09494	−	3.24071i
$562$		0			0
$563$	−9.83008			−0.414288			−0.207144	−	0.978310i	$-0.566417\pi$
$563$	−9.83008			−0.414288			−0.207144	+	0.978310i	$0.566417\pi$
$564$		−	73.4253i		−	3.09176i
$565$	−4.34056			−0.182609
$566$		0			0
$567$		−	62.8785i		−	2.64065i
$568$		0			0
$569$	35.5301			1.48950			0.744751	−	0.667343i	$-0.232568\pi$
$569$	35.5301			1.48950			0.744751	+	0.667343i	$0.232568\pi$
$570$		0			0
$571$		−	17.8013i		−	0.744963i	−0.928040	−	0.372481i	$-0.878507\pi$
$571$		−	17.8013i		−	0.744963i	0.928040	−	0.372481i	$-0.121493\pi$
$572$			56.0994i			2.34563i
$573$		−	10.6205i		−	0.443678i
$574$		0			0
$575$		−	23.5723i		−	0.983033i
$576$	58.0337			2.41807
$577$	10.6456			0.443183			0.221591	−	0.975140i	$-0.428875\pi$
$577$	10.6456			0.443183			0.221591	+	0.975140i	$0.428875\pi$
$578$		0			0
$579$	−75.0505			−3.11899
$580$	−2.80851			−0.116617
$581$			4.39299i			0.182252i
$582$		0			0
$583$		−	17.3366i		−	0.718011i
$584$		0			0
$585$		−	27.7847i		−	1.14876i
$586$		0			0
$587$	−20.2461			−0.835646			−0.417823	−	0.908528i	$-0.637207\pi$
$587$	−20.2461			−0.835646			−0.417823	+	0.908528i	$0.637207\pi$
$588$		−	8.15348i		−	0.336244i
$589$			2.81581i			0.116023i
$590$		0			0
$591$	14.0121			0.576380
$592$			15.4162i			0.633603i
$593$	29.2791			1.20235			0.601175	−	0.799117i	$-0.294699\pi$
$593$	29.2791			1.20235			0.601175	+	0.799117i	$0.294699\pi$
$594$		0			0
$595$	3.18086	+	9.41441i	0.130403	+	0.385953i
$596$	−1.44172			−0.0590552
$597$	5.25895			0.215235
$598$		0			0
$599$	−1.25405			−0.0512391			−0.0256195	−	0.999672i	$-0.508156\pi$
$599$	−1.25405			−0.0512391			−0.0256195	+	0.999672i	$0.508156\pi$
$600$		0			0
$601$		−	11.8072i		−	0.481626i	−0.970571	−	0.240813i	$-0.922586\pi$
$601$		−	11.8072i		−	0.481626i	0.970571	−	0.240813i	$-0.0774141\pi$
$602$		0			0
$603$	63.3861			2.58128
$604$	25.1915			1.02503
$605$			22.3362i			0.908096i
$606$		0			0
$607$			18.9113i			0.767586i	0.923419	+	0.383793i	$0.125382\pi$
$607$			18.9113i			0.767586i	−0.923419	+	0.383793i	$0.874618\pi$
$608$		0			0
$609$		−	15.4356i		−	0.625481i
$610$		0			0
$611$	52.4050			2.12008
$612$	−19.1480	−	56.6723i	−0.774011	−	2.29084i
$613$	−8.55698			−0.345613			−0.172807	−	0.984956i	$-0.555284\pi$
$613$	−8.55698			−0.345613			−0.172807	+	0.984956i	$0.555284\pi$
$614$		0			0
$615$			13.7179i			0.553161i
$616$		0			0
$617$		−	5.34937i		−	0.215358i	−0.994186	−	0.107679i	$-0.965658\pi$
$617$		−	5.34937i		−	0.215358i	0.994186	−	0.107679i	$-0.0343418\pi$
$618$		0			0
$619$		−	31.6928i		−	1.27384i	−0.770930	−	0.636920i	$-0.780208\pi$
$619$		−	31.6928i		−	1.27384i	0.770930	−	0.636920i	$-0.219792\pi$
$620$	−2.35945			−0.0947577
$621$	74.7167			2.99828
$622$		0			0
$623$		−	8.78597i		−	0.352003i
$624$			58.5490i			2.34384i
$625$	14.9611			0.598443
$626$		0			0
$627$	39.3007			1.56952
$628$	10.9076			0.435261
$629$	15.0546	−	5.08653i	0.600267	−	0.202813i
$630$		0			0
$631$	−16.8301			−0.669995			−0.334997	−	0.942219i	$-0.608735\pi$
$631$	−16.8301			−0.669995			−0.334997	+	0.942219i	$0.608735\pi$
$632$		0			0
$633$	−69.5749			−2.76536
$634$		0			0
$635$		−	6.73337i		−	0.267206i
$636$		−	18.0937i		−	0.717461i
$637$	5.81929			0.230569
$638$		0			0
$639$		−	63.6102i		−	2.51638i
$640$		0			0
$641$		−	36.7589i		−	1.45189i	−0.687753	−	0.725945i	$-0.741403\pi$
$641$		−	36.7589i		−	1.45189i	0.687753	−	0.725945i	$-0.258597\pi$
$642$		0			0
$643$			39.8909i			1.57315i	0.617498	+	0.786573i	$0.288146\pi$
$643$			39.8909i			1.57315i	−0.617498	+	0.786573i	$0.711854\pi$
$644$	31.5509			1.24328
$645$	−2.68324			−0.105652
$646$		0			0
$647$	−38.6173			−1.51820			−0.759102	−	0.650972i	$-0.774362\pi$
$647$	−38.6173			−1.51820			−0.759102	+	0.650972i	$0.774362\pi$
$648$		0			0
$649$		−	44.7833i		−	1.75790i
$650$		0			0
$651$		−	12.9675i		−	0.508238i
$652$			35.2336i			1.37985i
$653$			31.4736i			1.23166i	0.787880	+	0.615828i	$0.211179\pi$
$653$			31.4736i			1.23166i	−0.787880	+	0.615828i	$0.788821\pi$
$654$		0			0
$655$	−8.48937			−0.331707
$656$		−	20.4498i		−	0.798432i
$657$			113.610i			4.43235i
$658$		0			0
$659$	−50.6611			−1.97348			−0.986739	−	0.162315i	$-0.948104\pi$
$659$	−50.6611			−1.97348			−0.986739	+	0.162315i	$0.948104\pi$
$660$			32.9312i			1.28185i
$661$	−1.39011			−0.0540689			−0.0270345	−	0.999635i	$-0.508606\pi$
$661$	−1.39011			−0.0540689			−0.0270345	+	0.999635i	$0.508606\pi$
$662$		0			0
$663$	57.1756	−	19.3180i	2.22052	−	0.750250i
$664$		0			0
$665$	−4.82028			−0.186922
$666$		0			0
$667$	9.19149			0.355896
$668$			10.9693i			0.424414i
$669$		−	56.9596i		−	2.20219i
$670$		0			0
$671$	−60.8044			−2.34733
$672$		0			0
$673$		−	4.72557i		−	0.182157i	−0.995844	−	0.0910786i	$-0.970969\pi$
$673$		−	4.72557i		−	0.182157i	0.995844	−	0.0910786i	$-0.0290314\pi$
$674$		0			0
$675$			58.5495i			2.25357i
$676$	−15.7876			−0.607214
$677$			9.56040i			0.367436i	0.982979	+	0.183718i	$0.0588133\pi$
$677$			9.56040i			0.367436i	−0.982979	+	0.183718i	$0.941187\pi$
$678$		0			0
$679$	20.5957			0.790392
$680$		0			0
$681$	−47.5132			−1.82071
$682$		0			0
$683$			0.364537i			0.0139486i	0.999976	+	0.00697432i	$0.00222001\pi$
$683$			0.364537i			0.0139486i	−0.999976	+	0.00697432i	$0.997780\pi$
$684$	29.0168			1.10949
$685$		−	7.20263i		−	0.275199i
$686$		0			0
$687$			4.95076i			0.188883i
$688$	4.00000			0.152499
$689$	12.9138			0.491977
$690$		0			0
$691$		−	37.5439i		−	1.42824i	−0.700025	−	0.714119i	$-0.746828\pi$
$691$		−	37.5439i		−	1.42824i	0.700025	−	0.714119i	$-0.253172\pi$
$692$		−	37.8673i		−	1.43950i
$693$	−128.039			−4.86381
$694$		0			0
$695$	−10.8203			−0.410436
$696$		0			0
$697$	−19.9701	+	6.74735i	−0.756423	+	0.255574i
$698$		0			0
$699$	−35.9810			−1.36092
$700$			24.7239i			0.934477i
$701$	36.0418			1.36128			0.680639	−	0.732619i	$-0.261702\pi$
$701$	36.0418			1.36128			0.680639	+	0.732619i	$0.261702\pi$
$702$		0			0
$703$			7.70812i			0.290717i
$704$		−	49.0918i		−	1.85022i
$705$	30.7626			1.15859
$706$		0			0
$707$			42.1388i			1.58479i
$708$		−	46.7388i		−	1.75655i
$709$			13.5171i			0.507647i	0.967251	+	0.253823i	$0.0816882\pi$
$709$			13.5171i			0.507647i	−0.967251	+	0.253823i	$0.918312\pi$
$710$		0			0
$711$			62.1570i			2.33107i
$712$		0			0
$713$	7.72184			0.289185
$714$		0			0
$715$	−23.5037			−0.878987
$716$	35.3007			1.31925
$717$			20.3675i			0.760638i
$718$		0			0
$719$			12.9248i			0.482014i	0.970523	+	0.241007i	$0.0774776\pi$
$719$			12.9248i			0.482014i	−0.970523	+	0.241007i	$0.922522\pi$
$720$			24.3141i			0.906132i
$721$			0.713799i			0.0265833i
$722$		0			0
$723$	88.3363			3.28526
$724$			16.8663i			0.626830i
$725$			7.20263i			0.267499i
$726$		0			0
$727$	−41.8371			−1.55165			−0.775826	−	0.630947i	$-0.782667\pi$
$727$	−41.8371			−1.55165			−0.775826	+	0.630947i	$0.782667\pi$
$728$		0			0
$729$	−27.7129			−1.02640
$730$		0			0
$731$	−1.31978	−	3.90617i	−0.0488140	−	0.144475i
$732$	−63.4595			−2.34553
$733$	23.1638			0.855576			0.427788	−	0.903879i	$-0.359293\pi$
$733$	23.1638			0.855576			0.427788	+	0.903879i	$0.359293\pi$
$734$		0			0
$735$	3.41602			0.126002
$736$		0			0
$737$		−	53.6196i		−	1.97510i
$738$		0			0
$739$	−11.4039			−0.419501			−0.209750	−	0.977755i	$-0.567265\pi$
$739$	−11.4039			−0.419501			−0.209750	+	0.977755i	$0.567265\pi$
$740$	−6.45886			−0.237433
$741$			29.2745i			1.07543i
$742$		0			0
$743$		−	5.23092i		−	0.191904i	−0.995386	−	0.0959518i	$-0.969411\pi$
$743$		−	5.23092i		−	0.191904i	0.995386	−	0.0959518i	$-0.0305895\pi$
$744$		0			0
$745$		−	0.604031i		−	0.0221300i
$746$		0			0
$747$	11.0794			0.405373
$748$	−47.9403	+	16.1977i	−1.75287	+	0.592245i
$749$	37.1756			1.35837
$750$		0			0
$751$			16.3223i			0.595608i	0.954627	+	0.297804i	$0.0962543\pi$
$751$			16.3223i			0.595608i	−0.954627	+	0.297804i	$0.903746\pi$
$752$	−45.8590			−1.67231
$753$		−	74.1600i		−	2.70254i
$754$		0			0
$755$			10.5544i			0.384112i
$756$	−78.3671			−2.85018
$757$	−16.4332			−0.597273			−0.298637	−	0.954367i	$-0.596532\pi$
$757$	−16.4332			−0.597273			−0.298637	+	0.954367i	$0.596532\pi$
$758$		0			0
$759$		−	107.775i		−	3.91199i
$760$		0			0
$761$	32.7050			1.18555			0.592777	−	0.805367i	$-0.298032\pi$
$761$	32.7050			1.18555			0.592777	+	0.805367i	$0.298032\pi$
$762$		0			0
$763$	−10.4113			−0.376914
$764$	−6.63321			−0.239981
$765$	23.7437	−	8.02234i	0.858456	−	0.290048i
$766$		0			0
$767$	33.3584			1.20450
$768$		−	51.2355i		−	1.84880i
$769$	−30.8906			−1.11394			−0.556971	−	0.830532i	$-0.688037\pi$
$769$	−30.8906			−1.11394			−0.556971	+	0.830532i	$0.688037\pi$
$770$		0			0
$771$		−	65.9521i		−	2.37521i
$772$			46.8740i			1.68703i
$773$	−43.5252			−1.56549			−0.782747	−	0.622340i	$-0.786182\pi$
$773$	−43.5252			−1.56549			−0.782747	+	0.622340i	$0.786182\pi$
$774$		0			0
$775$			6.05099i			0.217358i
$776$		0			0
$777$		−	35.4980i		−	1.27348i
$778$		0			0
$779$		−	10.2249i		−	0.366346i
$780$	−24.5300			−0.878314
$781$	−53.8091			−1.92544
$782$		0			0
$783$	−22.8301			−0.815881
$784$	−5.09239			−0.181871
$785$			4.56991i			0.163107i
$786$		0			0
$787$			21.4027i			0.762922i	0.924385	+	0.381461i	$0.124579\pi$
$787$			21.4027i			0.762922i	−0.924385	+	0.381461i	$0.875421\pi$
$788$		−	8.75148i		−	0.311759i
$789$			102.070i			3.63380i
$790$		0			0
$791$	−14.8995			−0.529765
$792$		0			0
$793$		−	45.2923i		−	1.60838i
$794$		0			0
$795$	7.58062			0.268857
$796$		−	3.28457i		−	0.116418i
$797$	13.4835			0.477609			0.238805	−	0.971068i	$-0.423244\pi$
$797$	13.4835			0.477609			0.238805	+	0.971068i	$0.423244\pi$
$798$		0			0
$799$	15.1310	+	44.7833i	0.535296	+	1.58432i
$800$		0			0
$801$	−22.1588			−0.782942
$802$		0			0
$803$	96.1051			3.39148
$804$		−	55.9610i		−	1.97359i
$805$			13.2187i			0.465899i
$806$		0			0
$807$	−23.8250			−0.838680
$808$		0			0
$809$		−	29.8785i		−	1.05047i	−0.850956	−	0.525237i	$-0.823977\pi$
$809$		−	29.8785i		−	1.05047i	0.850956	−	0.525237i	$-0.176023\pi$
$810$		0			0
$811$		−	6.41712i		−	0.225336i	−0.993633	−	0.112668i	$-0.964060\pi$
$811$		−	6.41712i		−	0.225336i	0.993633	−	0.112668i	$-0.0359396\pi$
$812$	−9.64055			−0.338317
$813$		−	3.57741i		−	0.125465i
$814$		0			0
$815$	−14.7616			−0.517078
$816$	−50.0337	+	16.9050i	−1.75153	+	0.591792i
$817$	2.00000			0.0699711
$818$		0			0
$819$		−	95.3745i		−	3.33266i
$820$	8.56777			0.299199
$821$		−	1.82789i		−	0.0637939i	−0.999491	−	0.0318969i	$-0.989845\pi$
$821$		−	1.82789i		−	0.0637939i	0.999491	−	0.0318969i	$-0.0101548\pi$
$822$		0			0
$823$		−	44.7044i		−	1.55830i	−0.626839	−	0.779149i	$-0.715652\pi$
$823$		−	44.7044i		−	1.55830i	0.626839	−	0.779149i	$-0.284348\pi$
$824$		0			0
$825$	84.4547			2.94034
$826$		0			0
$827$			30.9270i			1.07544i	0.843125	+	0.537718i	$0.180713\pi$
$827$			30.9270i			1.07544i	−0.843125	+	0.537718i	$0.819287\pi$
$828$		−	79.5734i		−	2.76537i
$829$	46.6173			1.61909			0.809543	−	0.587060i	$-0.199715\pi$
$829$	46.6173			1.61909			0.809543	+	0.587060i	$0.199715\pi$
$830$		0			0
$831$	−23.6800			−0.821451
$832$	36.5678			1.26776
$833$	1.68022	+	4.97294i	0.0582160	+	0.172302i
$834$		0			0
$835$	−4.59575			−0.159042
$836$		−	24.5459i		−	0.848939i
$837$	−19.1797			−0.662948
$838$		0			0
$839$			39.7384i			1.37192i	0.727638	+	0.685962i	$0.240618\pi$
$839$			39.7384i			1.37192i	−0.727638	+	0.685962i	$0.759382\pi$
$840$		0			0
$841$	26.1915			0.903155
$842$		0			0
$843$			60.8167i			2.09464i
$844$			43.4542i			1.49576i
$845$		−	6.61443i		−	0.227543i
$846$		0			0
$847$			76.6718i			2.63447i
$848$	−11.3007			−0.388068
$849$	57.4500			1.97168
$850$		0			0
$851$	21.1381			0.724606
$852$	−56.1588			−1.92397
$853$		−	34.8070i		−	1.19177i	−0.803071	−	0.595884i	$-0.796802\pi$
$853$		−	34.8070i		−	1.19177i	0.803071	−	0.595884i	$-0.203198\pi$
$854$		0			0
$855$			12.1570i			0.415762i
$856$		0			0
$857$		−	25.5953i		−	0.874319i	−0.899384	−	0.437160i	$-0.855984\pi$
$857$		−	25.5953i		−	0.874319i	0.899384	−	0.437160i	$-0.144016\pi$
$858$		0			0
$859$	2.36173			0.0805811			0.0402905	−	0.999188i	$-0.487172\pi$
$859$	2.36173			0.0805811			0.0402905	+	0.999188i	$0.487172\pi$
$860$			1.67586i			0.0571464i
$861$			47.0885i			1.60477i
$862$		0			0
$863$	−6.37122			−0.216879			−0.108439	−	0.994103i	$-0.534585\pi$
$863$	−6.37122			−0.216879			−0.108439	+	0.994103i	$0.534585\pi$
$864$		0			0
$865$	15.8651			0.539429
$866$		0			0
$867$	33.0168	+	43.2823i	1.12131	+	1.46994i
$868$	−8.09910			−0.274901
$869$	52.5799			1.78365
$870$		0			0
$871$	39.9404			1.35333
$872$		0			0
$873$		−	51.9438i		−	1.75803i
$874$		0			0
$875$	−22.4092			−0.757568
$876$	100.302			3.38888
$877$			26.2528i			0.886495i	0.896399	+	0.443248i	$0.146174\pi$
$877$			26.2528i			0.886495i	−0.896399	+	0.443248i	$0.853826\pi$
$878$		0			0
$879$			34.9641i			1.17931i
$880$	20.5678			0.693339
$881$		−	35.0835i		−	1.18199i	−0.806674	−	0.590997i	$-0.798735\pi$
$881$		−	35.0835i		−	1.18199i	0.806674	−	0.590997i	$-0.201265\pi$
$882$		0			0
$883$	−30.4081			−1.02331			−0.511657	−	0.859190i	$-0.670968\pi$
$883$	−30.4081			−1.02331			−0.511657	+	0.859190i	$0.670968\pi$
$884$	−12.0654	−	35.7100i	−0.405803	−	1.20106i
$885$	19.5819			0.658239
$886$		0			0
$887$		−	32.7773i		−	1.10055i	−0.834982	−	0.550277i	$-0.814522\pi$
$887$		−	32.7773i		−	1.10055i	0.834982	−	0.550277i	$-0.185478\pi$
$888$		0			0
$889$		−	23.1131i		−	0.775190i
$890$		0			0
$891$			134.149i			4.49416i
$892$	−35.5751			−1.19114
$893$	−22.9295			−0.767306
$894$		0			0
$895$			14.7898i			0.494367i
$896$		0			0
$897$	80.2801			2.68047
$898$		0			0
$899$	−2.35945			−0.0786920
$900$	62.3553			2.07851
$901$	3.72863	+	11.0356i	0.124219	+	0.367650i
$902$		0			0
$903$	−9.21054			−0.306508
$904$		0			0
$905$	−7.06638			−0.234894
$906$		0			0
$907$		−	40.6948i		−	1.35125i	−0.737246	−	0.675625i	$-0.763874\pi$
$907$		−	40.6948i		−	1.35125i	0.737246	−	0.675625i	$-0.236126\pi$
$908$			29.6751i			0.984804i
$909$	106.277			3.52498
$910$		0			0
$911$		−	15.1589i		−	0.502238i	−0.967956	−	0.251119i	$-0.919201\pi$
$911$		−	15.1589i		−	0.502238i	0.967956	−	0.251119i	$-0.0807985\pi$
$912$		−	25.6178i		−	0.848289i
$913$		−	9.37228i		−	0.310177i
$914$		0			0
$915$		−	26.5873i		−	0.878949i
$916$	3.09208			0.102165
$917$	−29.1408			−0.962314
$918$		0			0
$919$	7.20564			0.237692			0.118846	−	0.992913i	$-0.462081\pi$
$919$	7.20564			0.237692			0.118846	+	0.992913i	$0.462081\pi$
$920$		0			0
$921$		−	70.2230i		−	2.31393i
$922$		0			0
$923$		−	40.0816i		−	1.31930i
$924$			113.041i			3.71876i
$925$			16.5643i			0.544629i
$926$		0			0
$927$	1.80025			0.0591278
$928$		0			0
$929$			10.5971i			0.347680i	0.984774	+	0.173840i	$0.0556175\pi$
$929$			10.5971i			0.347680i	−0.984774	+	0.173840i	$0.944383\pi$
$930$		0			0
$931$	−2.54620			−0.0834482
$932$			22.4725i			0.736112i
$933$	−45.3721			−1.48542
$934$		0			0
$935$	−6.78626	−	20.0853i	−0.221934	−	0.656860i
$936$		0			0
$937$	−39.3128			−1.28429			−0.642146	−	0.766582i	$-0.721956\pi$
$937$	−39.3128			−1.28429			−0.642146	+	0.766582i	$0.721956\pi$
$938$		0			0
$939$	−62.4924			−2.03936
$940$		−	19.2133i		−	0.626669i
$941$			46.6365i			1.52031i	0.649743	+	0.760154i	$0.274876\pi$
$941$			46.6365i			1.52031i	−0.649743	+	0.760154i	$0.725124\pi$
$942$		0			0
$943$	−28.0400			−0.913108
$944$	−29.1915			−0.950102
$945$		−	32.8331i		−	1.06806i
$946$		0			0
$947$		−	10.0048i		−	0.325111i	−0.986699	−	0.162556i	$-0.948026\pi$
$947$		−	10.0048i		−	0.325111i	0.986699	−	0.162556i	$-0.0519737\pi$
$948$	54.8758			1.78228
$949$			71.5872i			2.32382i
$950$		0			0
$951$	101.531			3.29237
$952$		0			0
$953$	−38.2461			−1.23891			−0.619456	−	0.785031i	$-0.712647\pi$
$953$	−38.2461			−1.23891			−0.619456	+	0.785031i	$0.712647\pi$
$954$		0			0
$955$		−	2.77909i		−	0.0899291i
$956$	12.7209			0.411422
$957$			32.9312i			1.06452i
$958$		0			0
$959$		−	24.7239i		−	0.798378i
$960$	21.4659			0.692809
$961$	29.0178			0.936058
$962$		0			0
$963$		−	93.7592i		−	3.02135i
$964$		−	55.1719i		−	1.77697i
$965$	−19.6386			−0.632189
$966$		0			0
$967$	55.0435			1.77008			0.885039	−	0.465517i	$-0.154131\pi$
$967$	55.0435			1.77008			0.885039	+	0.465517i	$0.154131\pi$
$968$		0			0
$969$	−25.0168	+	8.45248i	−0.803656	+	0.271533i
$970$		0			0
$971$	−2.36631			−0.0759386			−0.0379693	−	0.999279i	$-0.512089\pi$
$971$	−2.36631			−0.0759386			−0.0379693	+	0.999279i	$0.512089\pi$
$972$			58.2693i			1.86899i
$973$	−37.1419			−1.19072
$974$		0			0
$975$			62.9090i			2.01470i
$976$			39.6347i			1.26868i
$977$	29.9231			0.957326			0.478663	−	0.877999i	$-0.341122\pi$
$977$	29.9231			0.957326			0.478663	+	0.877999i	$0.341122\pi$
$978$		0			0
$979$			18.7446i			0.599079i
$980$		−	2.13353i		−	0.0681533i
$981$			26.2579i			0.838350i
$982$		0			0
$983$		−	35.9637i		−	1.14706i	−0.819183	−	0.573532i	$-0.805573\pi$
$983$		−	35.9637i		−	1.14706i	0.819183	−	0.573532i	$-0.194427\pi$
$984$		0			0
$985$	3.66657			0.116826
$986$		0			0
$987$	105.597			3.36117
$988$	18.2839			0.581688
$989$		−	5.48464i		−	0.174401i
$990$		0			0
$991$		−	19.6618i		−	0.624578i	−0.949987	−	0.312289i	$-0.898904\pi$
$991$		−	19.6618i		−	0.624578i	0.949987	−	0.312289i	$-0.101096\pi$
$992$		0			0
$993$			63.3641i			2.01080i
$994$		0			0
$995$	1.37612			0.0436259
$996$		−	9.78153i		−	0.309940i
$997$		−	2.75531i		−	0.0872616i	−0.999048	−	0.0436308i	$-0.986107\pi$
$997$		−	2.75531i		−	0.0872616i	0.999048	−	0.0436308i	$-0.0138925\pi$
$998$		0			0
$999$	−52.5035			−1.66114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.d.b.560.1	✓	8
17.16	even	2	inner	731.2.d.b.560.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.d.b.560.1	✓	8	1.1	even	1	trivial
731.2.d.b.560.8	yes	8	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.20222i q^{3} -2.00000 q^{4} -0.837930i q^{5} -2.87630i q^{7} -7.25421 q^{9} +O(q^{10})\)	\(q-3.20222i q^{3} -2.00000 q^{4} -0.837930i q^{5} -2.87630i q^{7} -7.25421 q^{9} +6.13648i q^{11} +6.40444i q^{12} -4.57097 q^{13} -2.68324 q^{15} +4.00000 q^{16} +(-1.31978 - 3.90617i) q^{17} +2.00000 q^{19} +1.67586i q^{20} -9.21054 q^{21} -5.48464i q^{23} +4.29787 q^{25} +13.6229i q^{27} +5.75260i q^{28} +1.67586i q^{29} +1.40790i q^{31} +19.6504 q^{33} -2.41014 q^{35} +14.5084 q^{36} +3.85406i q^{37} +14.6373i q^{39} -5.11246i q^{41} +1.00000 q^{43} -12.2730i q^{44} +6.07852i q^{45} -11.4647 q^{47} -12.8089i q^{48} -1.27310 q^{49} +(-12.5084 + 4.22624i) q^{51} +9.14194 q^{52} -2.82518 q^{53} +5.14194 q^{55} -6.40444i q^{57} -7.29787 q^{59} +5.36647 q^{60} +9.90867i q^{61} +20.8653i q^{63} -8.00000 q^{64} +3.83015i q^{65} -8.73785 q^{67} +(2.63957 + 7.81234i) q^{68} -17.5630 q^{69} +8.76873i q^{71} -15.6613i q^{73} -13.7627i q^{75} -4.00000 q^{76} +17.6504 q^{77} -8.56840i q^{79} -3.35172i q^{80} +21.8609 q^{81} -1.52730 q^{83} +18.4211 q^{84} +(-3.27310 + 1.10589i) q^{85} +5.36647 q^{87} +3.05461 q^{89} +13.1475i q^{91} +10.9693i q^{92} +4.50841 q^{93} -1.67586i q^{95} +7.16050i q^{97} -44.5153i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 16 * q^9	\( 8 q - 16 q^{4} - 16 q^{9} - 10 q^{13} - 6 q^{15} + 32 q^{16} - 3 q^{17} + 16 q^{19} - 32 q^{21} - 8 q^{25} + 20 q^{33} + 12 q^{35} + 32 q^{36} + 8 q^{43} - 8 q^{47} - 26 q^{49} - 16 q^{51} + 20 q^{52} + 46 q^{53} - 12 q^{55} - 16 q^{59} + 12 q^{60} - 64 q^{64} - 2 q^{67} + 6 q^{68} - 4 q^{69} - 32 q^{76} + 4 q^{77} - 4 q^{81} + 14 q^{83} + 64 q^{84} - 42 q^{85} + 12 q^{87} - 28 q^{89} - 48 q^{93}+O(q^{100}) \) 8 * q - 16 * q^4 - 16 * q^9 - 10 * q^13 - 6 * q^15 + 32 * q^16 - 3 * q^17 + 16 * q^19 - 32 * q^21 - 8 * q^25 + 20 * q^33 + 12 * q^35 + 32 * q^36 + 8 * q^43 - 8 * q^47 - 26 * q^49 - 16 * q^51 + 20 * q^52 + 46 * q^53 - 12 * q^55 - 16 * q^59 + 12 * q^60 - 64 * q^64 - 2 * q^67 + 6 * q^68 - 4 * q^69 - 32 * q^76 + 4 * q^77 - 4 * q^81 + 14 * q^83 + 64 * q^84 - 42 * q^85 + 12 * q^87 - 28 * q^89 - 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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