LMFDB - Embedded newform 1521.2.b.l.1351.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1351,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1521.1351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.6
Root			$0.445042i$ of defining polynomial
Character	$\chi$	$=$	1521.1351
Dual form			1521.2.b.l.1351.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698i q^{2} -3.04892 q^{4} +1.44504i q^{5} -2.04892i q^{7} -2.35690i q^{8} -3.24698 q^{10} -2.55496i q^{11} +4.60388 q^{14} -0.801938 q^{16} -5.29590 q^{17} -5.85086i q^{19} -4.40581i q^{20} +5.74094 q^{22} -1.89008 q^{23} +2.91185 q^{25} +6.24698i q^{28} -2.26875 q^{29} -4.26875i q^{31} -6.51573i q^{32} -11.8998i q^{34} +2.96077 q^{35} -5.35690i q^{37} +13.1468 q^{38} +3.40581 q^{40} -1.27413i q^{41} -6.13706 q^{43} +7.78986i q^{44} -4.24698i q^{46} -2.95108i q^{47} +2.80194 q^{49} +6.54288i q^{50} -5.52111 q^{53} +3.69202 q^{55} -4.82908 q^{56} -5.09783i q^{58} -12.2078i q^{59} +8.56465 q^{61} +9.59179 q^{62} +13.0368 q^{64} +0.576728i q^{67} +16.1468 q^{68} +6.65279i q^{70} +4.59419i q^{71} +10.5526i q^{73} +12.0368 q^{74} +17.8388i q^{76} -5.23490 q^{77} -15.7778 q^{79} -1.15883i q^{80} +2.86294 q^{82} -7.72348i q^{83} -7.65279i q^{85} -13.7899i q^{86} -6.02177 q^{88} +6.61356i q^{89} +5.76271 q^{92} +6.63102 q^{94} +8.45473 q^{95} +11.9269i q^{97} +6.29590i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62}+ \cdots + 6 q^{95}+O(q^{100}) $ 6 * q - 10 * q^10 + 10 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^22 - 10 * q^23 + 10 * q^25 + 2 * q^29 - 8 * q^35 + 24 * q^38 - 6 * q^40 - 26 * q^43 + 8 * q^49 - 2 * q^53 + 12 * q^55 - 8 * q^56 + 8 * q^61 + 2 * q^62 + 22 * q^64 + 42 * q^68 + 16 * q^74 + 16 * q^77 - 10 * q^79 + 28 * q^82 - 30 * q^88 + 10 * q^94 + 6 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.24698i	1.58885i	0.607359	+	0.794427i	$0.292229\pi$
$2$	2.24698i	1.58885i	−0.607359	+	0.794427i	$0.707771\pi$
$3$	0	0
$3$	0	0
$4$	−3.04892	−1.52446
$5$	1.44504i	0.646242i	0.946358	+	0.323121i	$0.104732\pi$
$5$	1.44504i	0.646242i	−0.946358	+	0.323121i	$0.895268\pi$
$6$	0	0
$7$	− 2.04892i	− 0.774418i	−0.921992	−	0.387209i	$-0.873439\pi$
$7$	− 2.04892i	− 0.774418i	0.921992	−	0.387209i	$-0.126561\pi$
$8$	− 2.35690i	− 0.833289i
$9$	0	0
$10$	−3.24698	−1.02679
$11$	− 2.55496i	− 0.770349i	−0.922844	−	0.385174i	$-0.874141\pi$
$11$	− 2.55496i	− 0.770349i	0.922844	−	0.385174i	$-0.125859\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	4.60388	1.23044
$15$	0	0
$16$	−0.801938	−0.200484
$17$	−5.29590	−1.28444	−0.642222	−	0.766519i	$-0.721987\pi$
$17$	−5.29590	−1.28444	−0.642222	+	0.766519i	$0.721987\pi$
$18$	0	0
$19$	− 5.85086i	− 1.34228i	−0.741331	−	0.671139i	$-0.765805\pi$
$19$	− 5.85086i	− 1.34228i	0.741331	−	0.671139i	$-0.234195\pi$
$20$	− 4.40581i	− 0.985170i
$21$	0	0
$22$	5.74094	1.22397
$23$	−1.89008	−0.394110	−0.197055	−	0.980392i	$-0.563138\pi$
$23$	−1.89008	−0.394110	−0.197055	+	0.980392i	$0.563138\pi$
$24$	0	0
$25$	2.91185	0.582371
$26$	0	0
$27$	0	0
$28$	6.24698i	1.18057i
$29$	−2.26875	−0.421296	−0.210648	−	0.977562i	$-0.567557\pi$
$29$	−2.26875	−0.421296	−0.210648	+	0.977562i	$0.567557\pi$
$30$	0	0
$31$	− 4.26875i	− 0.766690i	−0.923605	−	0.383345i	$-0.874772\pi$
$31$	− 4.26875i	− 0.766690i	0.923605	−	0.383345i	$-0.125228\pi$
$32$	− 6.51573i	− 1.15183i
$33$	0	0
$34$	− 11.8998i	− 2.04079i
$35$	2.96077	0.500462
$36$	0	0
$37$	− 5.35690i	− 0.880668i	−0.897834	−	0.440334i	$-0.854860\pi$
$37$	− 5.35690i	− 0.880668i	0.897834	−	0.440334i	$-0.145140\pi$
$38$	13.1468	2.13268
$39$	0	0
$40$	3.40581	0.538506
$41$	− 1.27413i	− 0.198985i	−0.995038	−	0.0994926i	$-0.968278\pi$
$41$	− 1.27413i	− 0.198985i	0.995038	−	0.0994926i	$-0.0317220\pi$
$42$	0	0
$43$	−6.13706	−0.935893	−0.467947	−	0.883757i	$-0.655006\pi$
$43$	−6.13706	−0.935893	−0.467947	+	0.883757i	$0.655006\pi$
$44$	7.78986i	1.17437i
$45$	0	0
$46$	− 4.24698i	− 0.626183i
$47$	− 2.95108i	− 0.430460i	−0.976563	−	0.215230i	$-0.930950\pi$
$47$	− 2.95108i	− 0.430460i	0.976563	−	0.215230i	$-0.0690501\pi$
$48$	0	0
$49$	2.80194	0.400277
$50$	6.54288i	0.925302i
$51$	0	0
$52$	0	0
$53$	−5.52111	−0.758382	−0.379191	−	0.925318i	$-0.623798\pi$
$53$	−5.52111	−0.758382	−0.379191	+	0.925318i	$0.623798\pi$
$54$	0	0
$55$	3.69202	0.497832
$56$	−4.82908	−0.645314
$57$	0	0
$58$	− 5.09783i	− 0.669378i
$59$	− 12.2078i	− 1.58931i	−0.607059	−	0.794657i	$-0.707651\pi$
$59$	− 12.2078i	− 1.58931i	0.607059	−	0.794657i	$-0.292349\pi$
$60$	0	0
$61$	8.56465	1.09659	0.548295	−	0.836285i	$-0.315277\pi$
$61$	8.56465	1.09659	0.548295	+	0.836285i	$0.315277\pi$
$62$	9.59179	1.21816
$63$	0	0
$64$	13.0368	1.62960
$65$	0	0
$66$	0	0
$67$	0.576728i	0.0704586i	0.999379	+	0.0352293i	$0.0112162\pi$
$67$	0.576728i	0.0704586i	−0.999379	+	0.0352293i	$0.988784\pi$
$68$	16.1468	1.95808
$69$	0	0
$70$	6.65279i	0.795161i
$71$	4.59419i	0.545230i	0.962123	+	0.272615i	$0.0878885\pi$
$71$	4.59419i	0.545230i	−0.962123	+	0.272615i	$0.912112\pi$
$72$	0	0
$73$	10.5526i	1.23508i	0.786538	+	0.617542i	$0.211872\pi$
$73$	10.5526i	1.23508i	−0.786538	+	0.617542i	$0.788128\pi$
$74$	12.0368	1.39925
$75$	0	0
$76$	17.8388i	2.04625i
$77$	−5.23490	−0.596572
$78$	0	0
$79$	−15.7778	−1.77514	−0.887569	−	0.460674i	$-0.847608\pi$
$79$	−15.7778	−1.77514	−0.887569	+	0.460674i	$0.847608\pi$
$80$	− 1.15883i	− 0.129562i
$81$	0	0
$82$	2.86294	0.316158
$83$	− 7.72348i	− 0.847762i	−0.905718	−	0.423881i	$-0.860667\pi$
$83$	− 7.72348i	− 0.847762i	0.905718	−	0.423881i	$-0.139333\pi$
$84$	0	0
$85$	− 7.65279i	− 0.830062i
$86$	− 13.7899i	− 1.48700i
$87$	0	0
$88$	−6.02177	−0.641923
$89$	6.61356i	0.701036i	0.936556	+	0.350518i	$0.113995\pi$
$89$	6.61356i	0.701036i	−0.936556	+	0.350518i	$0.886005\pi$
$90$	0	0
$91$	0	0
$92$	5.76271	0.600804
$93$	0	0
$94$	6.63102	0.683938
$95$	8.45473	0.867437
$96$	0	0
$97$	11.9269i	1.21100i	0.795847	+	0.605498i	$0.207026\pi$
$97$	11.9269i	1.21100i	−0.795847	+	0.605498i	$0.792974\pi$
$98$	6.29590i	0.635982i
$99$	0	0
$100$	−8.87800	−0.887800
$101$	13.0640	1.29991	0.649957	−	0.759971i	$-0.274787\pi$
$101$	13.0640	1.29991	0.649957	+	0.759971i	$0.274787\pi$
$102$	0	0
$103$	−9.16852	−0.903401	−0.451701	−	0.892170i	$-0.649182\pi$
$103$	−9.16852	−0.903401	−0.451701	+	0.892170i	$0.649182\pi$
$104$	0	0
$105$	0	0
$106$	− 12.4058i	− 1.20496i
$107$	6.89977	0.667026	0.333513	−	0.942745i	$-0.391766\pi$
$107$	6.89977	0.667026	0.333513	+	0.942745i	$0.391766\pi$
$108$	0	0
$109$	0.121998i	0.0116853i	0.999983	+	0.00584264i	$0.00185978\pi$
$109$	0.121998i	0.0116853i	−0.999983	+	0.00584264i	$0.998140\pi$
$110$	8.29590i	0.790983i
$111$	0	0
$112$	1.64310i	0.155259i
$113$	−7.30798	−0.687477	−0.343738	−	0.939065i	$-0.611693\pi$
$113$	−7.30798	−0.687477	−0.343738	+	0.939065i	$0.611693\pi$
$114$	0	0
$115$	− 2.73125i	− 0.254690i
$116$	6.91723	0.642249
$117$	0	0
$118$	27.4306	2.52519
$119$	10.8509i	0.994696i
$120$	0	0
$121$	4.47219	0.406563
$122$	19.2446i	1.74232i
$123$	0	0
$124$	13.0151i	1.16879i
$125$	11.4330i	1.02260i
$126$	0	0
$127$	18.9705	1.68336	0.841678	−	0.539980i	$-0.181568\pi$
$127$	18.9705	1.68336	0.841678	+	0.539980i	$0.181568\pi$
$128$	16.2620i	1.43738i
$129$	0	0
$130$	0	0
$131$	−3.25667	−0.284536	−0.142268	−	0.989828i	$-0.545440\pi$
$131$	−3.25667	−0.284536	−0.142268	+	0.989828i	$0.545440\pi$
$132$	0	0
$133$	−11.9879	−1.03948
$134$	−1.29590	−0.111948
$135$	0	0
$136$	12.4819i	1.07031i
$137$	0.792249i	0.0676864i	0.999427	+	0.0338432i	$0.0107747\pi$
$137$	0.792249i	0.0676864i	−0.999427	+	0.0338432i	$0.989225\pi$
$138$	0	0
$139$	−11.3394	−0.961799	−0.480899	−	0.876776i	$-0.659690\pi$
$139$	−11.3394	−0.961799	−0.480899	+	0.876776i	$0.659690\pi$
$140$	−9.02715	−0.762933
$141$	0	0
$142$	−10.3230	−0.866291
$143$	0	0
$144$	0	0
$145$	− 3.27844i	− 0.272260i
$146$	−23.7114	−1.96237
$147$	0	0
$148$	16.3327i	1.34254i
$149$	− 8.40581i	− 0.688631i	−0.938854	−	0.344316i	$-0.888111\pi$
$149$	− 8.40581i	− 0.688631i	0.938854	−	0.344316i	$-0.111889\pi$
$150$	0	0
$151$	− 14.1293i	− 1.14983i	−0.818215	−	0.574913i	$-0.805036\pi$
$151$	− 14.1293i	− 1.14983i	0.818215	−	0.574913i	$-0.194964\pi$
$152$	−13.7899	−1.11851
$153$	0	0
$154$	− 11.7627i	− 0.947866i
$155$	6.16852	0.495468
$156$	0	0
$157$	−9.43296	−0.752832	−0.376416	−	0.926451i	$-0.622844\pi$
$157$	−9.43296	−0.752832	−0.376416	+	0.926451i	$0.622844\pi$
$158$	− 35.4523i	− 2.82044i
$159$	0	0
$160$	9.41550	0.744361
$161$	3.87263i	0.305206i
$162$	0	0
$163$	− 8.70410i	− 0.681758i	−0.940107	−	0.340879i	$-0.889275\pi$
$163$	− 8.70410i	− 0.681758i	0.940107	−	0.340879i	$-0.110725\pi$
$164$	3.88471i	0.303345i
$165$	0	0
$166$	17.3545	1.34697
$167$	− 23.8538i	− 1.84587i	−0.384961	−	0.922933i	$-0.625785\pi$
$167$	− 23.8538i	− 1.84587i	0.384961	−	0.922933i	$-0.374215\pi$
$168$	0	0
$169$	0	0
$170$	17.1957	1.31885
$171$	0	0
$172$	18.7114	1.42673
$173$	−18.8552	−1.43353	−0.716766	−	0.697314i	$-0.754378\pi$
$173$	−18.8552	−1.43353	−0.716766	+	0.697314i	$0.754378\pi$
$174$	0	0
$175$	− 5.96615i	− 0.450998i
$176$	2.04892i	0.154443i
$177$	0	0
$178$	−14.8605	−1.11384
$179$	6.02177	0.450088	0.225044	−	0.974349i	$-0.427747\pi$
$179$	6.02177	0.450088	0.225044	+	0.974349i	$0.427747\pi$
$180$	0	0
$181$	4.77777	0.355129	0.177565	−	0.984109i	$-0.443178\pi$
$181$	4.77777	0.355129	0.177565	+	0.984109i	$0.443178\pi$
$182$	0	0
$183$	0	0
$184$	4.45473i	0.328407i
$185$	7.74094	0.569125
$186$	0	0
$187$	13.5308i	0.989470i
$188$	8.99761i	0.656218i
$189$	0	0
$190$	18.9976i	1.37823i
$191$	−18.4306	−1.33359	−0.666795	−	0.745242i	$-0.732334\pi$
$191$	−18.4306	−1.33359	−0.666795	+	0.745242i	$0.732334\pi$
$192$	0	0
$193$	− 6.05429i	− 0.435798i	−0.975971	−	0.217899i	$-0.930080\pi$
$193$	− 6.05429i	− 0.435798i	0.975971	−	0.217899i	$-0.0699203\pi$
$194$	−26.7995	−1.92410
$195$	0	0
$196$	−8.54288	−0.610205
$197$	11.4155i	0.813321i	0.913579	+	0.406660i	$0.133307\pi$
$197$	11.4155i	0.813321i	−0.913579	+	0.406660i	$0.866693\pi$
$198$	0	0
$199$	13.9051	0.985710	0.492855	−	0.870111i	$-0.335953\pi$
$199$	13.9051	0.985710	0.492855	+	0.870111i	$0.335953\pi$
$200$	− 6.86294i	− 0.485283i
$201$	0	0
$202$	29.3545i	2.06538i
$203$	4.64848i	0.326259i
$204$	0	0
$205$	1.84117	0.128593
$206$	− 20.6015i	− 1.43537i
$207$	0	0
$208$	0	0
$209$	−14.9487	−1.03402
$210$	0	0
$211$	−13.2446	−0.911795	−0.455897	−	0.890032i	$-0.650682\pi$
$211$	−13.2446	−0.911795	−0.455897	+	0.890032i	$0.650682\pi$
$212$	16.8334	1.15612
$213$	0	0
$214$	15.5036i	1.05981i
$215$	− 8.86831i	− 0.604814i
$216$	0	0
$217$	−8.74632	−0.593739
$218$	−0.274127	−0.0185662
$219$	0	0
$220$	−11.2567	−0.758924
$221$	0	0
$222$	0	0
$223$	− 7.33513i	− 0.491196i	−0.969372	−	0.245598i	$-0.921016\pi$
$223$	− 7.33513i	− 0.491196i	0.969372	−	0.245598i	$-0.0789844\pi$
$224$	−13.3502	−0.891997
$225$	0	0
$226$	− 16.4209i	− 1.09230i
$227$	8.67456i	0.575751i	0.957668	+	0.287875i	$0.0929489\pi$
$227$	8.67456i	0.575751i	−0.957668	+	0.287875i	$0.907051\pi$
$228$	0	0
$229$	13.6866i	0.904439i	0.891907	+	0.452219i	$0.149368\pi$
$229$	13.6866i	0.904439i	−0.891907	+	0.452219i	$0.850632\pi$
$230$	6.13706	0.404666
$231$	0	0
$232$	5.34721i	0.351061i
$233$	−5.08815	−0.333336	−0.166668	−	0.986013i	$-0.553301\pi$
$233$	−5.08815	−0.333336	−0.166668	+	0.986013i	$0.553301\pi$
$234$	0	0
$235$	4.26444	0.278181
$236$	37.2204i	2.42284i
$237$	0	0
$238$	−24.3817	−1.58043
$239$	10.9239i	0.706611i	0.935508	+	0.353305i	$0.114942\pi$
$239$	10.9239i	0.706611i	−0.935508	+	0.353305i	$0.885058\pi$
$240$	0	0
$241$	− 11.9148i	− 0.767502i	−0.923437	−	0.383751i	$-0.874632\pi$
$241$	− 11.9148i	− 0.767502i	0.923437	−	0.383751i	$-0.125368\pi$
$242$	10.0489i	0.645969i
$243$	0	0
$244$	−26.1129	−1.67171
$245$	4.04892i	0.258676i
$246$	0	0
$247$	0	0
$248$	−10.0610	−0.638874
$249$	0	0
$250$	−25.6896	−1.62475
$251$	22.3478	1.41058	0.705290	−	0.708919i	$-0.250817\pi$
$251$	22.3478	1.41058	0.705290	+	0.708919i	$0.250817\pi$
$252$	0	0
$253$	4.82908i	0.303602i
$254$	42.6262i	2.67461i
$255$	0	0
$256$	−10.4668	−0.654176
$257$	−18.6601	−1.16398	−0.581992	−	0.813194i	$-0.697727\pi$
$257$	−18.6601	−1.16398	−0.581992	+	0.813194i	$0.697727\pi$
$258$	0	0
$259$	−10.9758	−0.682005
$260$	0	0
$261$	0	0
$262$	− 7.31767i	− 0.452087i
$263$	−14.3991	−0.887887	−0.443944	−	0.896055i	$-0.646421\pi$
$263$	−14.3991	−0.887887	−0.443944	+	0.896055i	$0.646421\pi$
$264$	0	0
$265$	− 7.97823i	− 0.490099i
$266$	− 26.9366i	− 1.65159i
$267$	0	0
$268$	− 1.75840i	− 0.107411i
$269$	−0.652793	−0.0398015	−0.0199007	−	0.999802i	$-0.506335\pi$
$269$	−0.652793	−0.0398015	−0.0199007	+	0.999802i	$0.506335\pi$
$270$	0	0
$271$	1.99569i	0.121229i	0.998161	+	0.0606147i	$0.0193061\pi$
$271$	1.99569i	0.121229i	−0.998161	+	0.0606147i	$0.980694\pi$
$272$	4.24698	0.257511
$273$	0	0
$274$	−1.78017	−0.107544
$275$	− 7.43967i	− 0.448629i
$276$	0	0
$277$	−11.7845	−0.708061	−0.354030	−	0.935234i	$-0.615189\pi$
$277$	−11.7845	−0.708061	−0.354030	+	0.935234i	$0.615189\pi$
$278$	− 25.4795i	− 1.52816i
$279$	0	0
$280$	− 6.97823i	− 0.417029i
$281$	− 6.47219i	− 0.386098i	−0.981189	−	0.193049i	$-0.938162\pi$
$281$	− 6.47219i	− 0.386098i	0.981189	−	0.193049i	$-0.0618377\pi$
$282$	0	0
$283$	−6.58104	−0.391202	−0.195601	−	0.980684i	$-0.562666\pi$
$283$	−6.58104	−0.391202	−0.195601	+	0.980684i	$0.562666\pi$
$284$	− 14.0073i	− 0.831180i
$285$	0	0
$286$	0	0
$287$	−2.61058	−0.154098
$288$	0	0
$289$	11.0465	0.649796
$290$	7.36658	0.432581
$291$	0	0
$292$	− 32.1739i	− 1.88284i
$293$	− 24.3381i	− 1.42185i	−0.703269	−	0.710924i	$-0.748277\pi$
$293$	− 24.3381i	− 1.42185i	0.703269	−	0.710924i	$-0.251723\pi$
$294$	0	0
$295$	17.6407	1.02708
$296$	−12.6256	−0.733851
$297$	0	0
$298$	18.8877	1.09413
$299$	0	0
$300$	0	0
$301$	12.5743i	0.724773i
$302$	31.7482	1.82691
$303$	0	0
$304$	4.69202i	0.269106i
$305$	12.3763i	0.708663i
$306$	0	0
$307$	14.0737i	0.803227i	0.915809	+	0.401613i	$0.131550\pi$
$307$	14.0737i	0.803227i	−0.915809	+	0.401613i	$0.868450\pi$
$308$	15.9608	0.909449
$309$	0	0
$310$	13.8605i	0.787226i
$311$	−29.7700	−1.68810	−0.844051	−	0.536263i	$-0.819836\pi$
$311$	−29.7700	−1.68810	−0.844051	+	0.536263i	$0.819836\pi$
$312$	0	0
$313$	−7.47889	−0.422732	−0.211366	−	0.977407i	$-0.567791\pi$
$313$	−7.47889	−0.422732	−0.211366	+	0.977407i	$0.567791\pi$
$314$	− 21.1957i	− 1.19614i
$315$	0	0
$316$	48.1051	2.70613
$317$	− 30.0301i	− 1.68666i	−0.537396	−	0.843330i	$-0.680592\pi$
$317$	− 30.0301i	− 1.68666i	0.537396	−	0.843330i	$-0.319408\pi$
$318$	0	0
$319$	5.79656i	0.324545i
$320$	18.8388i	1.05312i
$321$	0	0
$322$	−8.70171	−0.484927
$323$	30.9855i	1.72408i
$324$	0	0
$325$	0	0
$326$	19.5579	1.08321
$327$	0	0
$328$	−3.00298	−0.165812
$329$	−6.04652	−0.333356
$330$	0	0
$331$	− 15.7168i	− 0.863872i	−0.901904	−	0.431936i	$-0.857831\pi$
$331$	− 15.7168i	− 0.863872i	0.901904	−	0.431936i	$-0.142169\pi$
$332$	23.5483i	1.29238i
$333$	0	0
$334$	53.5991	2.93281
$335$	−0.833397	−0.0455333
$336$	0	0
$337$	−1.95407	−0.106445	−0.0532224	−	0.998583i	$-0.516949\pi$
$337$	−1.95407	−0.106445	−0.0532224	+	0.998583i	$0.516949\pi$
$338$	0	0
$339$	0	0
$340$	23.3327i	1.26540i
$341$	−10.9065	−0.590619
$342$	0	0
$343$	− 20.0834i	− 1.08440i
$344$	14.4644i	0.779869i
$345$	0	0
$346$	− 42.3672i	− 2.27767i
$347$	17.1250	0.919317	0.459659	−	0.888096i	$-0.347972\pi$
$347$	17.1250	0.919317	0.459659	+	0.888096i	$0.347972\pi$
$348$	0	0
$349$	10.4668i	0.560276i	0.959960	+	0.280138i	$0.0903802\pi$
$349$	10.4668i	0.560276i	−0.959960	+	0.280138i	$0.909620\pi$
$350$	13.4058	0.716571
$351$	0	0
$352$	−16.6474	−0.887310
$353$	− 15.5308i	− 0.826621i	−0.910590	−	0.413310i	$-0.864372\pi$
$353$	− 15.5308i	− 0.826621i	0.910590	−	0.413310i	$-0.135628\pi$
$354$	0	0
$355$	−6.63879	−0.352351
$356$	− 20.1642i	− 1.06870i
$357$	0	0
$358$	13.5308i	0.715125i
$359$	− 21.4263i	− 1.13083i	−0.824805	−	0.565417i	$-0.808715\pi$
$359$	− 21.4263i	− 1.13083i	0.824805	−	0.565417i	$-0.191285\pi$
$360$	0	0
$361$	−15.2325	−0.801711
$362$	10.7356i	0.564249i
$363$	0	0
$364$	0	0
$365$	−15.2489	−0.798164
$366$	0	0
$367$	34.3032	1.79061	0.895306	−	0.445452i	$-0.146957\pi$
$367$	34.3032	1.79061	0.895306	+	0.445452i	$0.146957\pi$
$368$	1.51573	0.0790129
$369$	0	0
$370$	17.3937i	0.904257i
$371$	11.3123i	0.587305i
$372$	0	0
$373$	−12.5961	−0.652202	−0.326101	−	0.945335i	$-0.605735\pi$
$373$	−12.5961	−0.652202	−0.326101	+	0.945335i	$0.605735\pi$
$374$	−30.4034	−1.57212
$375$	0	0
$376$	−6.95539	−0.358697
$377$	0	0
$378$	0	0
$379$	16.5386i	0.849529i	0.905304	+	0.424765i	$0.139643\pi$
$379$	16.5386i	0.849529i	−0.905304	+	0.424765i	$0.860357\pi$
$380$	−25.7778	−1.32237
$381$	0	0
$382$	− 41.4131i	− 2.11888i
$383$	− 7.53617i	− 0.385080i	−0.981289	−	0.192540i	$-0.938327\pi$
$383$	− 7.53617i	− 0.385080i	0.981289	−	0.192540i	$-0.0616726\pi$
$384$	0	0
$385$	− 7.56465i	− 0.385530i
$386$	13.6039	0.692419
$387$	0	0
$388$	− 36.3642i	− 1.84611i
$389$	35.5555	1.80274	0.901369	−	0.433052i	$-0.142563\pi$
$389$	35.5555	1.80274	0.901369	+	0.433052i	$0.142563\pi$
$390$	0	0
$391$	10.0097	0.506212
$392$	− 6.60388i	− 0.333546i
$393$	0	0
$394$	−25.6504	−1.29225
$395$	− 22.7995i	− 1.14717i
$396$	0	0
$397$	− 1.35152i	− 0.0678308i	−0.999425	−	0.0339154i	$-0.989202\pi$
$397$	− 1.35152i	− 0.0678308i	0.999425	−	0.0339154i	$-0.0107977\pi$
$398$	31.2446i	1.56615i
$399$	0	0
$400$	−2.33513	−0.116756
$401$	0.579121i	0.0289199i	0.999895	+	0.0144600i	$0.00460291\pi$
$401$	0.579121i	0.0289199i	−0.999895	+	0.0144600i	$0.995397\pi$
$402$	0	0
$403$	0	0
$404$	−39.8310	−1.98167
$405$	0	0
$406$	−10.4450	−0.518379
$407$	−13.6866	−0.678422
$408$	0	0
$409$	− 15.1575i	− 0.749490i	−0.927128	−	0.374745i	$-0.877730\pi$
$409$	− 15.1575i	− 0.749490i	0.927128	−	0.374745i	$-0.122270\pi$
$410$	4.13706i	0.204315i
$411$	0	0
$412$	27.9541	1.37720
$413$	−25.0127	−1.23079
$414$	0	0
$415$	11.1608	0.547860
$416$	0	0
$417$	0	0
$418$	− 33.5894i	− 1.64291i
$419$	35.7235	1.74521	0.872603	−	0.488430i	$-0.162430\pi$
$419$	35.7235	1.74521	0.872603	+	0.488430i	$0.162430\pi$
$420$	0	0
$421$	− 35.0465i	− 1.70806i	−0.520221	−	0.854032i	$-0.674151\pi$
$421$	− 35.0465i	− 1.70806i	0.520221	−	0.854032i	$-0.325849\pi$
$422$	− 29.7603i	− 1.44871i
$423$	0	0
$424$	13.0127i	0.631951i
$425$	−15.4209	−0.748022
$426$	0	0
$427$	− 17.5483i	− 0.849220i
$428$	−21.0368	−1.01685
$429$	0	0
$430$	19.9269	0.960961
$431$	34.2814i	1.65128i	0.564199	+	0.825639i	$0.309185\pi$
$431$	34.2814i	1.65128i	−0.564199	+	0.825639i	$0.690815\pi$
$432$	0	0
$433$	−13.7385	−0.660232	−0.330116	−	0.943940i	$-0.607088\pi$
$433$	−13.7385	−0.660232	−0.330116	+	0.943940i	$0.607088\pi$
$434$	− 19.6528i	− 0.943364i
$435$	0	0
$436$	− 0.371961i	− 0.0178137i
$437$	11.0586i	0.529005i
$438$	0	0
$439$	−10.2403	−0.488742	−0.244371	−	0.969682i	$-0.578581\pi$
$439$	−10.2403	−0.488742	−0.244371	+	0.969682i	$0.578581\pi$
$440$	− 8.70171i	− 0.414838i
$441$	0	0
$442$	0	0
$443$	−12.1763	−0.578513	−0.289257	−	0.957252i	$-0.593408\pi$
$443$	−12.1763	−0.578513	−0.289257	+	0.957252i	$0.593408\pi$
$444$	0	0
$445$	−9.55688	−0.453039
$446$	16.4819	0.780440
$447$	0	0
$448$	− 26.7114i	− 1.26199i
$449$	− 12.9051i	− 0.609032i	−0.952507	−	0.304516i	$-0.901505\pi$
$449$	− 12.9051i	− 0.609032i	0.952507	−	0.304516i	$-0.0984947\pi$
$450$	0	0
$451$	−3.25534	−0.153288
$452$	22.2814	1.04803
$453$	0	0
$454$	−19.4916	−0.914785
$455$	0	0
$456$	0	0
$457$	4.65710i	0.217850i	0.994050	+	0.108925i	$0.0347409\pi$
$457$	4.65710i	0.217850i	−0.994050	+	0.108925i	$0.965259\pi$
$458$	−30.7536	−1.43702
$459$	0	0
$460$	8.32736i	0.388265i
$461$	31.5405i	1.46899i	0.678616	+	0.734493i	$0.262580\pi$
$461$	31.5405i	1.46899i	−0.678616	+	0.734493i	$0.737420\pi$
$462$	0	0
$463$	− 17.6504i	− 0.820284i	−0.912022	−	0.410142i	$-0.865479\pi$
$463$	− 17.6504i	− 0.820284i	0.912022	−	0.410142i	$-0.134521\pi$
$464$	1.81940	0.0844633
$465$	0	0
$466$	− 11.4330i	− 0.529622i
$467$	−32.1726	−1.48877	−0.744385	−	0.667751i	$-0.767257\pi$
$467$	−32.1726	−1.48877	−0.744385	+	0.667751i	$0.767257\pi$
$468$	0	0
$469$	1.18167	0.0545644
$470$	9.58211i	0.441990i
$471$	0	0
$472$	−28.7724	−1.32436
$473$	15.6799i	0.720964i
$474$	0	0
$475$	− 17.0368i	− 0.781704i
$476$	− 33.0834i	− 1.51637i
$477$	0	0
$478$	−24.5459	−1.12270
$479$	− 34.8998i	− 1.59461i	−0.603576	−	0.797306i	$-0.706258\pi$
$479$	− 34.8998i	− 1.59461i	0.603576	−	0.797306i	$-0.293742\pi$
$480$	0	0
$481$	0	0
$482$	26.7724	1.21945
$483$	0	0
$484$	−13.6353	−0.619788
$485$	−17.2349	−0.782596
$486$	0	0
$487$	41.8351i	1.89573i	0.318676	+	0.947864i	$0.396762\pi$
$487$	41.8351i	1.89573i	−0.318676	+	0.947864i	$0.603238\pi$
$488$	− 20.1860i	− 0.913776i
$489$	0	0
$490$	−9.09783	−0.410998
$491$	21.8455	0.985873	0.492936	−	0.870065i	$-0.335924\pi$
$491$	21.8455	0.985873	0.492936	+	0.870065i	$0.335924\pi$
$492$	0	0
$493$	12.0151	0.541131
$494$	0	0
$495$	0	0
$496$	3.42327i	0.153709i
$497$	9.41311	0.422236
$498$	0	0
$499$	23.5472i	1.05412i	0.849829	+	0.527058i	$0.176705\pi$
$499$	23.5472i	1.05412i	−0.849829	+	0.527058i	$0.823295\pi$
$500$	− 34.8582i	− 1.55890i
$501$	0	0
$502$	50.2150i	2.24121i
$503$	7.08682	0.315986	0.157993	−	0.987440i	$-0.449498\pi$
$503$	7.08682	0.315986	0.157993	+	0.987440i	$0.449498\pi$
$504$	0	0
$505$	18.8780i	0.840060i
$506$	−10.8509	−0.482379
$507$	0	0
$508$	−57.8394	−2.56621
$509$	− 7.61894i	− 0.337704i	−0.985641	−	0.168852i	$-0.945994\pi$
$509$	− 7.61894i	− 0.337704i	0.985641	−	0.168852i	$-0.0540059\pi$
$510$	0	0
$511$	21.6213	0.956471
$512$	9.00538i	0.397985i
$513$	0	0
$514$	− 41.9288i	− 1.84940i
$515$	− 13.2489i	− 0.583816i
$516$	0	0
$517$	−7.53989	−0.331604
$518$	− 24.6625i	− 1.08361i
$519$	0	0
$520$	0	0
$521$	39.5133	1.73111	0.865555	−	0.500813i	$-0.166966\pi$
$521$	39.5133	1.73111	0.865555	+	0.500813i	$0.166966\pi$
$522$	0	0
$523$	−15.8194	−0.691734	−0.345867	−	0.938284i	$-0.612415\pi$
$523$	−15.8194	−0.691734	−0.345867	+	0.938284i	$0.612415\pi$
$524$	9.92931	0.433764
$525$	0	0
$526$	− 32.3545i	− 1.41072i
$527$	22.6069i	0.984770i
$528$	0	0
$529$	−19.4276	−0.844678
$530$	17.9269	0.778696
$531$	0	0
$532$	36.5502	1.58465
$533$	0	0
$534$	0	0
$535$	9.97046i	0.431061i
$536$	1.35929	0.0587123
$537$	0	0
$538$	− 1.46681i	− 0.0632388i
$539$	− 7.15883i	− 0.308353i
$540$	0	0
$541$	− 34.4819i	− 1.48249i	−0.671234	−	0.741246i	$-0.734235\pi$
$541$	− 34.4819i	− 1.48249i	0.671234	−	0.741246i	$-0.265765\pi$
$542$	−4.48427	−0.192616
$543$	0	0
$544$	34.5066i	1.47946i
$545$	−0.176292	−0.00755152
$546$	0	0
$547$	36.8582	1.57594	0.787970	−	0.615713i	$-0.211132\pi$
$547$	36.8582	1.57594	0.787970	+	0.615713i	$0.211132\pi$
$548$	− 2.41550i	− 0.103185i
$549$	0	0
$550$	16.7168	0.712806
$551$	13.2741i	0.565497i
$552$	0	0
$553$	32.3274i	1.37470i
$554$	− 26.4795i	− 1.12501i
$555$	0	0
$556$	34.5730	1.46622
$557$	1.27652i	0.0540879i	0.999634	+	0.0270439i	$0.00860940\pi$
$557$	1.27652i	0.0540879i	−0.999634	+	0.0270439i	$0.991391\pi$
$558$	0	0
$559$	0	0
$560$	−2.37435	−0.100335
$561$	0	0
$562$	14.5429	0.613454
$563$	−9.12737	−0.384673	−0.192336	−	0.981329i	$-0.561607\pi$
$563$	−9.12737	−0.384673	−0.192336	+	0.981329i	$0.561607\pi$
$564$	0	0
$565$	− 10.5603i	− 0.444277i
$566$	− 14.7875i	− 0.621563i
$567$	0	0
$568$	10.8280	0.454334
$569$	−5.72156	−0.239860	−0.119930	−	0.992782i	$-0.538267\pi$
$569$	−5.72156	−0.239860	−0.119930	+	0.992782i	$0.538267\pi$
$570$	0	0
$571$	−7.60148	−0.318112	−0.159056	−	0.987270i	$-0.550845\pi$
$571$	−7.60148	−0.318112	−0.159056	+	0.987270i	$0.550845\pi$
$572$	0	0
$573$	0	0
$574$	− 5.86592i	− 0.244839i
$575$	−5.50365	−0.229518
$576$	0	0
$577$	45.1564i	1.87989i	0.341330	+	0.939944i	$0.389123\pi$
$577$	45.1564i	1.87989i	−0.341330	+	0.939944i	$0.610877\pi$
$578$	24.8213i	1.03243i
$579$	0	0
$580$	9.99569i	0.415048i
$581$	−15.8248	−0.656522
$582$	0	0
$583$	14.1062i	0.584219i
$584$	24.8713	1.02918
$585$	0	0
$586$	54.6872	2.25911
$587$	32.4040i	1.33746i	0.743507	+	0.668728i	$0.233161\pi$
$587$	32.4040i	1.33746i	−0.743507	+	0.668728i	$0.766839\pi$
$588$	0	0
$589$	−24.9758	−1.02911
$590$	39.6383i	1.63188i
$591$	0	0
$592$	4.29590i	0.176560i
$593$	36.6848i	1.50647i	0.657754	+	0.753233i	$0.271507\pi$
$593$	36.6848i	1.50647i	−0.657754	+	0.753233i	$0.728493\pi$
$594$	0	0
$595$	−15.6799	−0.642815
$596$	25.6286i	1.04979i
$597$	0	0
$598$	0	0
$599$	9.99223	0.408271	0.204136	−	0.978943i	$-0.434562\pi$
$599$	9.99223	0.408271	0.204136	+	0.978943i	$0.434562\pi$
$600$	0	0
$601$	−1.81163	−0.0738978	−0.0369489	−	0.999317i	$-0.511764\pi$
$601$	−1.81163	−0.0738978	−0.0369489	+	0.999317i	$0.511764\pi$
$602$	−28.2543	−1.15156
$603$	0	0
$604$	43.0790i	1.75286i
$605$	6.46250i	0.262738i
$606$	0	0
$607$	11.2161	0.455248	0.227624	−	0.973749i	$-0.426904\pi$
$607$	11.2161	0.455248	0.227624	+	0.973749i	$0.426904\pi$
$608$	−38.1226	−1.54608
$609$	0	0
$610$	−27.8092	−1.12596
$611$	0	0
$612$	0	0
$613$	− 20.8944i	− 0.843917i	−0.906615	−	0.421958i	$-0.861343\pi$
$613$	− 20.8944i	− 0.843917i	0.906615	−	0.421958i	$-0.138657\pi$
$614$	−31.6233	−1.27621
$615$	0	0
$616$	12.3381i	0.497117i
$617$	− 12.0992i	− 0.487094i	−0.969889	−	0.243547i	$-0.921689\pi$
$617$	− 12.0992i	− 0.487094i	0.969889	−	0.243547i	$-0.0783110\pi$
$618$	0	0
$619$	10.5526i	0.424143i	0.977254	+	0.212072i	$0.0680210\pi$
$619$	10.5526i	0.424143i	−0.977254	+	0.212072i	$0.931979\pi$
$620$	−18.8073	−0.755320
$621$	0	0
$622$	− 66.8926i	− 2.68215i
$623$	13.5506	0.542895
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	− 16.8049i	− 0.671660i
$627$	0	0
$628$	28.7603	1.14766
$629$	28.3696i	1.13117i
$630$	0	0
$631$	13.8514i	0.551417i	0.961241	+	0.275709i	$0.0889125\pi$
$631$	13.8514i	0.551417i	−0.961241	+	0.275709i	$0.911087\pi$
$632$	37.1866i	1.47920i
$633$	0	0
$634$	67.4771	2.67986
$635$	27.4131i	1.08786i
$636$	0	0
$637$	0	0
$638$	−13.0248	−0.515655
$639$	0	0
$640$	−23.4993	−0.928893
$641$	34.9608	1.38087	0.690434	−	0.723396i	$-0.257420\pi$
$641$	34.9608	1.38087	0.690434	+	0.723396i	$0.257420\pi$
$642$	0	0
$643$	33.3980i	1.31709i	0.752541	+	0.658545i	$0.228828\pi$
$643$	33.3980i	1.31709i	−0.752541	+	0.658545i	$0.771172\pi$
$644$	− 11.8073i	− 0.465273i
$645$	0	0
$646$	−69.6238	−2.73931
$647$	2.32842	0.0915397	0.0457698	−	0.998952i	$-0.485426\pi$
$647$	2.32842	0.0915397	0.0457698	+	0.998952i	$0.485426\pi$
$648$	0	0
$649$	−31.1903	−1.22433
$650$	0	0
$651$	0	0
$652$	26.5381i	1.03931i
$653$	−14.5714	−0.570221	−0.285111	−	0.958495i	$-0.592030\pi$
$653$	−14.5714	−0.570221	−0.285111	+	0.958495i	$0.592030\pi$
$654$	0	0
$655$	− 4.70602i	− 0.183879i
$656$	1.02177i	0.0398934i
$657$	0	0
$658$	− 13.5864i	− 0.529654i
$659$	−11.1395	−0.433932	−0.216966	−	0.976179i	$-0.569616\pi$
$659$	−11.1395	−0.433932	−0.216966	+	0.976179i	$0.569616\pi$
$660$	0	0
$661$	13.8498i	0.538694i	0.963043	+	0.269347i	$0.0868079\pi$
$661$	13.8498i	0.538694i	−0.963043	+	0.269347i	$0.913192\pi$
$662$	35.3153	1.37257
$663$	0	0
$664$	−18.2034	−0.706430
$665$	− 17.3230i	− 0.671759i
$666$	0	0
$667$	4.28813	0.166037
$668$	72.7284i	2.81395i
$669$	0	0
$670$	− 1.87263i	− 0.0723458i
$671$	− 21.8823i	− 0.844757i
$672$	0	0
$673$	6.52973	0.251703	0.125851	−	0.992049i	$-0.459834\pi$
$673$	6.52973	0.251703	0.125851	+	0.992049i	$0.459834\pi$
$674$	− 4.39075i	− 0.169125i
$675$	0	0
$676$	0	0
$677$	11.3104	0.434693	0.217346	−	0.976095i	$-0.430260\pi$
$677$	11.3104	0.434693	0.217346	+	0.976095i	$0.430260\pi$
$678$	0	0
$679$	24.4373	0.937816
$680$	−18.0368	−0.691681
$681$	0	0
$682$	− 24.5066i	− 0.938407i
$683$	14.1793i	0.542555i	0.962501	+	0.271277i	$0.0874461\pi$
$683$	14.1793i	0.542555i	−0.962501	+	0.271277i	$0.912554\pi$
$684$	0	0
$685$	−1.14483	−0.0437418
$686$	45.1269	1.72295
$687$	0	0
$688$	4.92154	0.187632
$689$	0	0
$690$	0	0
$691$	30.7952i	1.17151i	0.810490	+	0.585753i	$0.199201\pi$
$691$	30.7952i	1.17151i	−0.810490	+	0.585753i	$0.800799\pi$
$692$	57.4878	2.18536
$693$	0	0
$694$	38.4795i	1.46066i
$695$	− 16.3860i	− 0.621555i
$696$	0	0
$697$	6.74764i	0.255585i
$698$	−23.5187	−0.890196
$699$	0	0
$700$	18.1903i	0.687528i
$701$	6.73184	0.254258	0.127129	−	0.991886i	$-0.459424\pi$
$701$	6.73184	0.254258	0.127129	+	0.991886i	$0.459424\pi$
$702$	0	0
$703$	−31.3424	−1.18210
$704$	− 33.3086i	− 1.25536i
$705$	0	0
$706$	34.8974	1.31338
$707$	− 26.7670i	− 1.00668i
$708$	0	0
$709$	47.6252i	1.78860i	0.447467	+	0.894300i	$0.352326\pi$
$709$	47.6252i	1.78860i	−0.447467	+	0.894300i	$0.647674\pi$
$710$	− 14.9172i	− 0.559834i
$711$	0	0
$712$	15.5875	0.584166
$713$	8.06829i	0.302160i
$714$	0	0
$715$	0	0
$716$	−18.3599	−0.686141
$717$	0	0
$718$	48.1444	1.79673
$719$	−5.99330	−0.223512	−0.111756	−	0.993736i	$-0.535648\pi$
$719$	−5.99330	−0.223512	−0.111756	+	0.993736i	$0.535648\pi$
$720$	0	0
$721$	18.7855i	0.699610i
$722$	− 34.2271i	− 1.27380i
$723$	0	0
$724$	−14.5670	−0.541380
$725$	−6.60627	−0.245351
$726$	0	0
$727$	24.1226	0.894657	0.447329	−	0.894370i	$-0.352375\pi$
$727$	24.1226	0.894657	0.447329	+	0.894370i	$0.352375\pi$
$728$	0	0
$729$	0	0
$730$	− 34.2640i	− 1.26817i
$731$	32.5013	1.20210
$732$	0	0
$733$	36.0646i	1.33208i	0.745918	+	0.666038i	$0.232011\pi$
$733$	36.0646i	1.33208i	−0.745918	+	0.666038i	$0.767989\pi$
$734$	77.0786i	2.84502i
$735$	0	0
$736$	12.3153i	0.453947i
$737$	1.47352	0.0542777
$738$	0	0
$739$	− 27.5254i	− 1.01254i	−0.862375	−	0.506269i	$-0.831024\pi$
$739$	− 27.5254i	− 1.01254i	0.862375	−	0.506269i	$-0.168976\pi$
$740$	−23.6015	−0.867608
$741$	0	0
$742$	−25.4185	−0.933142
$743$	10.4692i	0.384078i	0.981387	+	0.192039i	$0.0615100\pi$
$743$	10.4692i	0.384078i	−0.981387	+	0.192039i	$0.938490\pi$
$744$	0	0
$745$	12.1468	0.445023
$746$	− 28.3032i	− 1.03625i
$747$	0	0
$748$	− 41.2543i	− 1.50841i
$749$	− 14.1371i	− 0.516557i
$750$	0	0
$751$	−4.06770	−0.148433	−0.0742163	−	0.997242i	$-0.523646\pi$
$751$	−4.06770	−0.148433	−0.0742163	+	0.997242i	$0.523646\pi$
$752$	2.36658i	0.0863005i
$753$	0	0
$754$	0	0
$755$	20.4174	0.743066
$756$	0	0
$757$	20.4336	0.742670	0.371335	−	0.928499i	$-0.378900\pi$
$757$	20.4336	0.742670	0.371335	+	0.928499i	$0.378900\pi$
$758$	−37.1618	−1.34978
$759$	0	0
$760$	− 19.9269i	− 0.722825i
$761$	− 27.0237i	− 0.979608i	−0.871833	−	0.489804i	$-0.837068\pi$
$761$	− 27.0237i	− 0.979608i	0.871833	−	0.489804i	$-0.162932\pi$
$762$	0	0
$763$	0.249964	0.00904929
$764$	56.1933	2.03300
$765$	0	0
$766$	16.9336	0.611837
$767$	0	0
$768$	0	0
$769$	− 37.9407i	− 1.36818i	−0.729400	−	0.684088i	$-0.760201\pi$
$769$	− 37.9407i	− 1.36818i	0.729400	−	0.684088i	$-0.239799\pi$
$770$	16.9976	0.612551
$771$	0	0
$772$	18.4590i	0.664355i
$773$	16.3375i	0.587620i	0.955864	+	0.293810i	$0.0949232\pi$
$773$	16.3375i	0.587620i	−0.955864	+	0.293810i	$0.905077\pi$
$774$	0	0
$775$	− 12.4300i	− 0.446498i
$776$	28.1105	1.00911
$777$	0	0
$778$	79.8926i	2.86429i
$779$	−7.45473	−0.267093
$780$	0	0
$781$	11.7380	0.420017
$782$	22.4916i	0.804297i
$783$	0	0
$784$	−2.24698	−0.0802493
$785$	− 13.6310i	− 0.486512i
$786$	0	0
$787$	18.6907i	0.666251i	0.942882	+	0.333126i	$0.108103\pi$
$787$	18.6907i	0.666251i	−0.942882	+	0.333126i	$0.891897\pi$
$788$	− 34.8049i	− 1.23987i
$789$	0	0
$790$	51.2301	1.82269
$791$	14.9734i	0.532394i
$792$	0	0
$793$	0	0
$794$	3.03684	0.107773
$795$	0	0
$796$	−42.3957	−1.50267
$797$	29.2519	1.03615	0.518077	−	0.855334i	$-0.326648\pi$
$797$	29.2519	1.03615	0.518077	+	0.855334i	$0.326648\pi$
$798$	0	0
$799$	15.6286i	0.552901i
$800$	− 18.9729i	− 0.670792i
$801$	0	0
$802$	−1.30127	−0.0459496
$803$	26.9614	0.951446
$804$	0	0
$805$	−5.59611	−0.197237
$806$	0	0
$807$	0	0
$808$	− 30.7904i	− 1.08320i
$809$	6.65087	0.233832	0.116916	−	0.993142i	$-0.462699\pi$
$809$	6.65087	0.233832	0.116916	+	0.993142i	$0.462699\pi$
$810$	0	0
$811$	3.89200i	0.136667i	0.997663	+	0.0683333i	$0.0217681\pi$
$811$	3.89200i	0.136667i	−0.997663	+	0.0683333i	$0.978232\pi$
$812$	− 14.1728i	− 0.497369i
$813$	0	0
$814$	− 30.7536i	− 1.07791i
$815$	12.5778	0.440581
$816$	0	0
$817$	35.9071i	1.25623i
$818$	34.0586	1.19083
$819$	0	0
$820$	−5.61356	−0.196034
$821$	− 45.9982i	− 1.60535i	−0.596418	−	0.802674i	$-0.703410\pi$
$821$	− 45.9982i	− 1.60535i	0.596418	−	0.802674i	$-0.296590\pi$
$822$	0	0
$823$	−7.95300	−0.277224	−0.138612	−	0.990347i	$-0.544264\pi$
$823$	−7.95300	−0.277224	−0.138612	+	0.990347i	$0.544264\pi$
$824$	21.6093i	0.752794i
$825$	0	0
$826$	− 56.2030i	− 1.95555i
$827$	27.9648i	0.972432i	0.873839	+	0.486216i	$0.161623\pi$
$827$	27.9648i	0.972432i	−0.873839	+	0.486216i	$0.838377\pi$
$828$	0	0
$829$	−27.6310	−0.959665	−0.479833	−	0.877360i	$-0.659303\pi$
$829$	−27.6310	−0.959665	−0.479833	+	0.877360i	$0.659303\pi$
$830$	25.0780i	0.870470i
$831$	0	0
$832$	0	0
$833$	−14.8388	−0.514133
$834$	0	0
$835$	34.4698	1.19288
$836$	45.5773	1.57632
$837$	0	0
$838$	80.2699i	2.77288i
$839$	28.6848i	0.990311i	0.868804	+	0.495155i	$0.164889\pi$
$839$	28.6848i	0.990311i	−0.868804	+	0.495155i	$0.835111\pi$
$840$	0	0
$841$	−23.8528	−0.822509
$842$	78.7488	2.71386
$843$	0	0
$844$	40.3817	1.38999
$845$	0	0
$846$	0	0
$847$	− 9.16315i	− 0.314849i
$848$	4.42758	0.152044
$849$	0	0
$850$	− 34.6504i	− 1.18850i
$851$	10.1250i	0.347080i
$852$	0	0
$853$	43.2078i	1.47941i	0.672934	+	0.739703i	$0.265034\pi$
$853$	43.2078i	1.47941i	−0.672934	+	0.739703i	$0.734966\pi$
$854$	39.4306	1.34929
$855$	0	0
$856$	− 16.2620i	− 0.555825i
$857$	35.1685	1.20133	0.600667	−	0.799499i	$-0.294902\pi$
$857$	35.1685	1.20133	0.600667	+	0.799499i	$0.294902\pi$
$858$	0	0
$859$	27.3793	0.934168	0.467084	−	0.884213i	$-0.345305\pi$
$859$	27.3793	0.934168	0.467084	+	0.884213i	$0.345305\pi$
$860$	27.0388i	0.922014i
$861$	0	0
$862$	−77.0297	−2.62364
$863$	41.3913i	1.40898i	0.709715	+	0.704489i	$0.248824\pi$
$863$	41.3913i	1.40898i	−0.709715	+	0.704489i	$0.751176\pi$
$864$	0	0
$865$	− 27.2465i	− 0.926409i
$866$	− 30.8702i	− 1.04901i
$867$	0	0
$868$	26.6668	0.905130
$869$	40.3116i	1.36748i
$870$	0	0
$871$	0	0
$872$	0.287536	0.00973721
$873$	0	0
$874$	−24.8485	−0.840512
$875$	23.4252	0.791916
$876$	0	0
$877$	− 24.7472i	− 0.835653i	−0.908527	−	0.417826i	$-0.862792\pi$
$877$	− 24.7472i	− 0.835653i	0.908527	−	0.417826i	$-0.137208\pi$
$878$	− 23.0097i	− 0.776539i
$879$	0	0
$880$	−2.96077	−0.0998076
$881$	−28.5875	−0.963137	−0.481568	−	0.876409i	$-0.659933\pi$
$881$	−28.5875	−0.963137	−0.481568	+	0.876409i	$0.659933\pi$
$882$	0	0
$883$	−9.61702	−0.323639	−0.161819	−	0.986820i	$-0.551736\pi$
$883$	−9.61702	−0.323639	−0.161819	+	0.986820i	$0.551736\pi$
$884$	0	0
$885$	0	0
$886$	− 27.3599i	− 0.919173i
$887$	−15.9661	−0.536091	−0.268045	−	0.963406i	$-0.586378\pi$
$887$	−15.9661	−0.536091	−0.268045	+	0.963406i	$0.586378\pi$
$888$	0	0
$889$	− 38.8689i	− 1.30362i
$890$	− 21.4741i	− 0.719814i
$891$	0	0
$892$	22.3642i	0.748809i
$893$	−17.2664	−0.577797
$894$	0	0
$895$	8.70171i	0.290866i
$896$	33.3196	1.11313
$897$	0	0
$898$	28.9976	0.967663
$899$	9.68473i	0.323004i
$900$	0	0
$901$	29.2392	0.974099
$902$	− 7.31468i	− 0.243552i
$903$	0	0
$904$	17.2241i	0.572867i
$905$	6.90408i	0.229500i
$906$	0	0
$907$	28.8364	0.957496	0.478748	−	0.877952i	$-0.341091\pi$
$907$	28.8364	0.957496	0.478748	+	0.877952i	$0.341091\pi$
$908$	− 26.4480i	− 0.877709i
$909$	0	0
$910$	0	0
$911$	−38.5633	−1.27766	−0.638830	−	0.769348i	$-0.720581\pi$
$911$	−38.5633	−1.27766	−0.638830	+	0.769348i	$0.720581\pi$
$912$	0	0
$913$	−19.7332	−0.653073
$914$	−10.4644	−0.346132
$915$	0	0
$916$	− 41.7294i	− 1.37878i
$917$	6.67264i	0.220350i
$918$	0	0
$919$	8.87502	0.292760	0.146380	−	0.989228i	$-0.453238\pi$
$919$	8.87502	0.292760	0.146380	+	0.989228i	$0.453238\pi$
$920$	−6.43727	−0.212231
$921$	0	0
$922$	−70.8708	−2.33401
$923$	0	0
$924$	0	0
$925$	− 15.5985i	− 0.512875i
$926$	39.6601	1.30331
$927$	0	0
$928$	14.7826i	0.485261i
$929$	− 24.2295i	− 0.794945i	−0.917614	−	0.397472i	$-0.869887\pi$
$929$	− 24.2295i	− 0.794945i	0.917614	−	0.397472i	$-0.130113\pi$
$930$	0	0
$931$	− 16.3937i	− 0.537283i
$932$	15.5133	0.508156
$933$	0	0
$934$	− 72.2911i	− 2.36544i
$935$	−19.5526	−0.639437
$936$	0	0
$937$	17.2644	0.564005	0.282002	−	0.959414i	$-0.409001\pi$
$937$	17.2644	0.564005	0.282002	+	0.959414i	$0.409001\pi$
$938$	2.65519i	0.0866949i
$939$	0	0
$940$	−13.0019	−0.424076
$941$	− 4.34050i	− 0.141496i	−0.997494	−	0.0707482i	$-0.977461\pi$
$941$	− 4.34050i	− 0.141496i	0.997494	−	0.0707482i	$-0.0225387\pi$
$942$	0	0
$943$	2.40821i	0.0784220i
$944$	9.78986i	0.318633i
$945$	0	0
$946$	−35.2325	−1.14551
$947$	− 45.0146i	− 1.46278i	−0.681961	−	0.731389i	$-0.738872\pi$
$947$	− 45.0146i	− 1.46278i	0.681961	−	0.731389i	$-0.261128\pi$
$948$	0	0
$949$	0	0
$950$	38.2814	1.24201
$951$	0	0
$952$	25.5743	0.828869
$953$	−46.8859	−1.51878	−0.759391	−	0.650634i	$-0.774503\pi$
$953$	−46.8859	−1.51878	−0.759391	+	0.650634i	$0.774503\pi$
$954$	0	0
$955$	− 26.6329i	− 0.861822i
$956$	− 33.3062i	− 1.07720i
$957$	0	0
$958$	78.4191	2.53361
$959$	1.62325	0.0524176
$960$	0	0
$961$	12.7778	0.412186
$962$	0	0
$963$	0	0
$964$	36.3274i	1.17003i
$965$	8.74871	0.281631
$966$	0	0
$967$	6.29457i	0.202420i	0.994865	+	0.101210i	$0.0322714\pi$
$967$	6.29457i	0.202420i	−0.994865	+	0.101210i	$0.967729\pi$
$968$	− 10.5405i	− 0.338784i
$969$	0	0
$970$	− 38.7265i	− 1.24343i
$971$	41.8068	1.34165	0.670823	−	0.741618i	$-0.265941\pi$
$971$	41.8068	1.34165	0.670823	+	0.741618i	$0.265941\pi$
$972$	0	0
$973$	23.2336i	0.744834i
$974$	−94.0025	−3.01203
$975$	0	0
$976$	−6.86831	−0.219849
$977$	− 23.7530i	− 0.759926i	−0.925002	−	0.379963i	$-0.875937\pi$
$977$	− 23.7530i	− 0.759926i	0.925002	−	0.379963i	$-0.124063\pi$
$978$	0	0
$979$	16.8974	0.540043
$980$	− 12.3448i	− 0.394341i
$981$	0	0
$982$	49.0863i	1.56641i
$983$	− 55.7251i	− 1.77736i	−0.458532	−	0.888678i	$-0.651625\pi$
$983$	− 55.7251i	− 1.77736i	0.458532	−	0.888678i	$-0.348375\pi$
$984$	0	0
$985$	−16.4959	−0.525602
$986$	26.9976i	0.859779i
$987$	0	0
$988$	0	0
$989$	11.5996	0.368845
$990$	0	0
$991$	−35.5512	−1.12932	−0.564661	−	0.825323i	$-0.690993\pi$
$991$	−35.5512	−1.12932	−0.564661	+	0.825323i	$0.690993\pi$
$992$	−27.8140	−0.883096
$993$	0	0
$994$	21.1511i	0.670871i
$995$	20.0935i	0.637007i
$996$	0	0
$997$	6.61058	0.209359	0.104680	−	0.994506i	$-0.466618\pi$
$997$	6.61058	0.209359	0.104680	+	0.994506i	$0.466618\pi$
$998$	−52.9101	−1.67484
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.b.l.1351.6		6
3.2	odd	2		169.2.b.b.168.1		6
12.11	even	2		2704.2.f.o.337.3		6
13.5	odd	4		1521.2.a.r.1.3		3
13.8	odd	4		1521.2.a.o.1.1		3
13.12	even	2	inner	1521.2.b.l.1351.1		6
39.2	even	12		169.2.c.c.22.3		6
39.5	even	4		169.2.a.b.1.1	✓	3
39.8	even	4		169.2.a.c.1.3	yes	3
39.11	even	12		169.2.c.b.22.1		6
39.17	odd	6		169.2.e.b.23.6		12
39.20	even	12		169.2.c.b.146.1		6
39.23	odd	6		169.2.e.b.147.1		12
39.29	odd	6		169.2.e.b.147.6		12
39.32	even	12		169.2.c.c.146.3		6
39.35	odd	6		169.2.e.b.23.1		12
39.38	odd	2		169.2.b.b.168.6		6
156.47	odd	4		2704.2.a.ba.1.2		3
156.83	odd	4		2704.2.a.z.1.2		3
156.155	even	2		2704.2.f.o.337.4		6
195.44	even	4		4225.2.a.bg.1.3		3
195.164	even	4		4225.2.a.bb.1.1		3
273.83	odd	4		8281.2.a.bf.1.1		3
273.125	odd	4		8281.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.1	✓	3	39.5	even	4
169.2.a.c.1.3	yes	3	39.8	even	4
169.2.b.b.168.1		6	3.2	odd	2
169.2.b.b.168.6		6	39.38	odd	2
169.2.c.b.22.1		6	39.11	even	12
169.2.c.b.146.1		6	39.20	even	12
169.2.c.c.22.3		6	39.2	even	12
169.2.c.c.146.3		6	39.32	even	12
169.2.e.b.23.1		12	39.35	odd	6
169.2.e.b.23.6		12	39.17	odd	6
169.2.e.b.147.1		12	39.23	odd	6
169.2.e.b.147.6		12	39.29	odd	6
1521.2.a.o.1.1		3	13.8	odd	4
1521.2.a.r.1.3		3	13.5	odd	4
1521.2.b.l.1351.1		6	13.12	even	2	inner
1521.2.b.l.1351.6		6	1.1	even	1	trivial
2704.2.a.z.1.2		3	156.83	odd	4
2704.2.a.ba.1.2		3	156.47	odd	4
2704.2.f.o.337.3		6	12.11	even	2
2704.2.f.o.337.4		6	156.155	even	2
4225.2.a.bb.1.1		3	195.164	even	4
4225.2.a.bg.1.3		3	195.44	even	4
8281.2.a.bf.1.1		3	273.83	odd	4
8281.2.a.bj.1.3		3	273.125	odd	4

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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.24698i q^{2} -3.04892 q^{4} +1.44504i q^{5} -2.04892i q^{7} -2.35690i q^{8} -3.24698 q^{10} -2.55496i q^{11} +4.60388 q^{14} -0.801938 q^{16} -5.29590 q^{17} -5.85086i q^{19} -4.40581i q^{20} +5.74094 q^{22} -1.89008 q^{23} +2.91185 q^{25} +6.24698i q^{28} -2.26875 q^{29} -4.26875i q^{31} -6.51573i q^{32} -11.8998i q^{34} +2.96077 q^{35} -5.35690i q^{37} +13.1468 q^{38} +3.40581 q^{40} -1.27413i q^{41} -6.13706 q^{43} +7.78986i q^{44} -4.24698i q^{46} -2.95108i q^{47} +2.80194 q^{49} +6.54288i q^{50} -5.52111 q^{53} +3.69202 q^{55} -4.82908 q^{56} -5.09783i q^{58} -12.2078i q^{59} +8.56465 q^{61} +9.59179 q^{62} +13.0368 q^{64} +0.576728i q^{67} +16.1468 q^{68} +6.65279i q^{70} +4.59419i q^{71} +10.5526i q^{73} +12.0368 q^{74} +17.8388i q^{76} -5.23490 q^{77} -15.7778 q^{79} -1.15883i q^{80} +2.86294 q^{82} -7.72348i q^{83} -7.65279i q^{85} -13.7899i q^{86} -6.02177 q^{88} +6.61356i q^{89} +5.76271 q^{92} +6.63102 q^{94} +8.45473 q^{95} +11.9269i q^{97} +6.29590i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62}+ \cdots + 6 q^{95}+O(q^{100}) \) 6 * q - 10 * q^10 + 10 * q^14 + 4 * q^16 - 4 * q^17 + 6 * q^22 - 10 * q^23 + 10 * q^25 + 2 * q^29 - 8 * q^35 + 24 * q^38 - 6 * q^40 - 26 * q^43 + 8 * q^49 - 2 * q^53 + 12 * q^55 - 8 * q^56 + 8 * q^61 + 2 * q^62 + 22 * q^64 + 42 * q^68 + 16 * q^74 + 16 * q^77 - 10 * q^79 + 28 * q^82 - 30 * q^88 + 10 * q^94 + 6 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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