LMFDB - Embedded newform 1521.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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, 0]))

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

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.a (trivial)

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:	yes
Analytic conductor:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24698 q^{2} +3.04892 q^{4} +1.44504 q^{5} +2.04892 q^{7} +2.35690 q^{8} +O(q^{10})$	$q+2.24698 q^{2} +3.04892 q^{4} +1.44504 q^{5} +2.04892 q^{7} +2.35690 q^{8} +3.24698 q^{10} +2.55496 q^{11} +4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} -5.85086 q^{19} +4.40581 q^{20} +5.74094 q^{22} +1.89008 q^{23} -2.91185 q^{25} +6.24698 q^{28} -2.26875 q^{29} -4.26875 q^{31} -6.51573 q^{32} +11.8998 q^{34} +2.96077 q^{35} +5.35690 q^{37} -13.1468 q^{38} +3.40581 q^{40} -1.27413 q^{41} +6.13706 q^{43} +7.78986 q^{44} +4.24698 q^{46} +2.95108 q^{47} -2.80194 q^{49} -6.54288 q^{50} -5.52111 q^{53} +3.69202 q^{55} +4.82908 q^{56} -5.09783 q^{58} +12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -13.0368 q^{64} +0.576728 q^{67} +16.1468 q^{68} +6.65279 q^{70} +4.59419 q^{71} -10.5526 q^{73} +12.0368 q^{74} -17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} -1.15883 q^{80} -2.86294 q^{82} -7.72348 q^{83} +7.65279 q^{85} +13.7899 q^{86} +6.02177 q^{88} -6.61356 q^{89} +5.76271 q^{92} +6.63102 q^{94} -8.45473 q^{95} +11.9269 q^{97} -6.29590 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35} + 12 q^{37} - 12 q^{38} - 3 q^{40} + 7 q^{41} + 13 q^{43} + 8 q^{46} + 18 q^{47} - 4 q^{49} - q^{50} - q^{53} + 6 q^{55} + 4 q^{56} + 3 q^{58} + 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} - q^{67} + 21 q^{68} + 2 q^{70} + 27 q^{71} + 9 q^{73} + 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} + 5 q^{80} - 14 q^{82} + 7 q^{83} + 5 q^{85} + 18 q^{86} + 15 q^{88} + 11 q^{89} + 5 q^{94} - 3 q^{95} + 7 q^{97} - 5 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	2.24698	1.58885	0.794427	−	0.607359i	$-0.207771\pi$
$2$	2.24698	1.58885	0.794427	+	0.607359i	$0.207771\pi$
$3$	0	0
$3$	0	0
$4$	3.04892	1.52446
$5$	1.44504	0.646242	0.323121	−	0.946358i	$-0.395268\pi$
$5$	1.44504	0.646242	0.323121	+	0.946358i	$0.395268\pi$
$6$	0	0
$7$	2.04892	0.774418	0.387209	−	0.921992i	$-0.373439\pi$
$7$	2.04892	0.774418	0.387209	+	0.921992i	$0.373439\pi$
$8$	2.35690	0.833289
$9$	0	0
$10$	3.24698	1.02679
$11$	2.55496	0.770349	0.385174	−	0.922844i	$-0.374141\pi$
$11$	2.55496	0.770349	0.385174	+	0.922844i	$0.374141\pi$
$12$	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	$-0.278013\pi$
$17$	5.29590	1.28444	0.642222	+	0.766519i	$0.278013\pi$
$18$	0	0
$19$	−5.85086	−1.34228	−0.671139	−	0.741331i	$-0.734195\pi$
$19$	−5.85086	−1.34228	−0.671139	+	0.741331i	$0.734195\pi$
$20$	4.40581	0.985170
$21$	0	0
$22$	5.74094	1.22397
$23$	1.89008	0.394110	0.197055	−	0.980392i	$-0.436862\pi$
$23$	1.89008	0.394110	0.197055	+	0.980392i	$0.436862\pi$
$24$	0	0
$25$	−2.91185	−0.582371
$26$	0	0
$27$	0	0
$28$	6.24698	1.18057
$29$	−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
$31$	−4.26875	−0.766690	−0.383345	−	0.923605i	$-0.625228\pi$
$31$	−4.26875	−0.766690	−0.383345	+	0.923605i	$0.625228\pi$
$32$	−6.51573	−1.15183
$33$	0	0
$34$	11.8998	2.04079
$35$	2.96077	0.500462
$36$	0	0
$37$	5.35690	0.880668	0.440334	−	0.897834i	$-0.354860\pi$
$37$	5.35690	0.880668	0.440334	+	0.897834i	$0.354860\pi$
$38$	−13.1468	−2.13268
$39$	0	0
$40$	3.40581	0.538506
$41$	−1.27413	−0.198985	−0.0994926	−	0.995038i	$-0.531722\pi$
$41$	−1.27413	−0.198985	−0.0994926	+	0.995038i	$0.531722\pi$
$42$	0	0
$43$	6.13706	0.935893	0.467947	−	0.883757i	$-0.344994\pi$
$43$	6.13706	0.935893	0.467947	+	0.883757i	$0.344994\pi$
$44$	7.78986	1.17437
$45$	0	0
$46$	4.24698	0.626183
$47$	2.95108	0.430460	0.215230	−	0.976563i	$-0.430950\pi$
$47$	2.95108	0.430460	0.215230	+	0.976563i	$0.430950\pi$
$48$	0	0
$49$	−2.80194	−0.400277
$50$	−6.54288	−0.925302
$51$	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
$58$	−5.09783	−0.669378
$59$	12.2078	1.58931	0.794657	−	0.607059i	$-0.207651\pi$
$59$	12.2078	1.58931	0.794657	+	0.607059i	$0.207651\pi$
$60$	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
$67$	0.576728	0.0704586	0.0352293	−	0.999379i	$-0.488784\pi$
$67$	0.576728	0.0704586	0.0352293	+	0.999379i	$0.488784\pi$
$68$	16.1468	1.95808
$69$	0	0
$70$	6.65279	0.795161
$71$	4.59419	0.545230	0.272615	−	0.962123i	$-0.412112\pi$
$71$	4.59419	0.545230	0.272615	+	0.962123i	$0.412112\pi$
$72$	0	0
$73$	−10.5526	−1.23508	−0.617542	−	0.786538i	$-0.711872\pi$
$73$	−10.5526	−1.23508	−0.617542	+	0.786538i	$0.711872\pi$
$74$	12.0368	1.39925
$75$	0	0
$76$	−17.8388	−2.04625
$77$	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	$0.847608\pi$
$80$	−1.15883	−0.129562
$81$	0	0
$82$	−2.86294	−0.316158
$83$	−7.72348	−0.847762	−0.423881	−	0.905718i	$-0.639333\pi$
$83$	−7.72348	−0.847762	−0.423881	+	0.905718i	$0.639333\pi$
$84$	0	0
$85$	7.65279	0.830062
$86$	13.7899	1.48700
$87$	0	0
$88$	6.02177	0.641923
$89$	−6.61356	−0.701036	−0.350518	−	0.936556i	$-0.613995\pi$
$89$	−6.61356	−0.701036	−0.350518	+	0.936556i	$0.613995\pi$
$90$	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
$97$	11.9269	1.21100	0.605498	−	0.795847i	$-0.292974\pi$
$97$	11.9269	1.21100	0.605498	+	0.795847i	$0.292974\pi$
$98$	−6.29590	−0.635982
$99$	0	0
$100$	−8.87800	−0.887800
$101$	−13.0640	−1.29991	−0.649957	−	0.759971i	$-0.725213\pi$
$101$	−13.0640	−1.29991	−0.649957	+	0.759971i	$0.725213\pi$
$102$	0	0
$103$	9.16852	0.903401	0.451701	−	0.892170i	$-0.350818\pi$
$103$	9.16852	0.903401	0.451701	+	0.892170i	$0.350818\pi$
$104$	0	0
$105$	0	0
$106$	−12.4058	−1.20496
$107$	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
$109$	0.121998	0.0116853	0.00584264	−	0.999983i	$-0.498140\pi$
$109$	0.121998	0.0116853	0.00584264	+	0.999983i	$0.498140\pi$
$110$	8.29590	0.790983
$111$	0	0
$112$	−1.64310	−0.155259
$113$	−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
$115$	2.73125	0.254690
$116$	−6.91723	−0.642249
$117$	0	0
$118$	27.4306	2.52519
$119$	10.8509	0.994696
$120$	0	0
$121$	−4.47219	−0.406563
$122$	19.2446	1.74232
$123$	0	0
$124$	−13.0151	−1.16879
$125$	−11.4330	−1.02260
$126$	0	0
$127$	−18.9705	−1.68336	−0.841678	−	0.539980i	$-0.818432\pi$
$127$	−18.9705	−1.68336	−0.841678	+	0.539980i	$0.818432\pi$
$128$	−16.2620	−1.43738
$129$	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
$136$	12.4819	1.07031
$137$	−0.792249	−0.0676864	−0.0338432	−	0.999427i	$-0.510775\pi$
$137$	−0.792249	−0.0676864	−0.0338432	+	0.999427i	$0.510775\pi$
$138$	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
$145$	−3.27844	−0.272260
$146$	−23.7114	−1.96237
$147$	0	0
$148$	16.3327	1.34254
$149$	−8.40581	−0.688631	−0.344316	−	0.938854i	$-0.611889\pi$
$149$	−8.40581	−0.688631	−0.344316	+	0.938854i	$0.611889\pi$
$150$	0	0
$151$	14.1293	1.14983	0.574913	−	0.818215i	$-0.305036\pi$
$151$	14.1293	1.14983	0.574913	+	0.818215i	$0.305036\pi$
$152$	−13.7899	−1.11851
$153$	0	0
$154$	11.7627	0.947866
$155$	−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	$0.622844\pi$
$158$	−35.4523	−2.82044
$159$	0	0
$160$	−9.41550	−0.744361
$161$	3.87263	0.305206
$162$	0	0
$163$	8.70410	0.681758	0.340879	−	0.940107i	$-0.389275\pi$
$163$	8.70410	0.681758	0.340879	+	0.940107i	$0.389275\pi$
$164$	−3.88471	−0.303345
$165$	0	0
$166$	−17.3545	−1.34697
$167$	23.8538	1.84587	0.922933	−	0.384961i	$-0.125785\pi$
$167$	23.8538	1.84587	0.922933	+	0.384961i	$0.125785\pi$
$168$	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	$-0.245622\pi$
$173$	18.8552	1.43353	0.716766	+	0.697314i	$0.245622\pi$
$174$	0	0
$175$	−5.96615	−0.450998
$176$	−2.04892	−0.154443
$177$	0	0
$178$	−14.8605	−1.11384
$179$	−6.02177	−0.450088	−0.225044	−	0.974349i	$-0.572253\pi$
$179$	−6.02177	−0.450088	−0.225044	+	0.974349i	$0.572253\pi$
$180$	0	0
$181$	−4.77777	−0.355129	−0.177565	−	0.984109i	$-0.556822\pi$
$181$	−4.77777	−0.355129	−0.177565	+	0.984109i	$0.556822\pi$
$182$	0	0
$183$	0	0
$184$	4.45473	0.328407
$185$	7.74094	0.569125
$186$	0	0
$187$	13.5308	0.989470
$188$	8.99761	0.656218
$189$	0	0
$190$	−18.9976	−1.37823
$191$	−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
$193$	6.05429	0.435798	0.217899	−	0.975971i	$-0.430080\pi$
$193$	6.05429	0.435798	0.217899	+	0.975971i	$0.430080\pi$
$194$	26.7995	1.92410
$195$	0	0
$196$	−8.54288	−0.610205
$197$	11.4155	0.813321	0.406660	−	0.913579i	$-0.366693\pi$
$197$	11.4155	0.813321	0.406660	+	0.913579i	$0.366693\pi$
$198$	0	0
$199$	−13.9051	−0.985710	−0.492855	−	0.870111i	$-0.664047\pi$
$199$	−13.9051	−0.985710	−0.492855	+	0.870111i	$0.664047\pi$
$200$	−6.86294	−0.485283
$201$	0	0
$202$	−29.3545	−2.06538
$203$	−4.64848	−0.326259
$204$	0	0
$205$	−1.84117	−0.128593
$206$	20.6015	1.43537
$207$	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
$214$	15.5036	1.05981
$215$	8.86831	0.604814
$216$	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
$223$	−7.33513	−0.491196	−0.245598	−	0.969372i	$-0.578984\pi$
$223$	−7.33513	−0.491196	−0.245598	+	0.969372i	$0.578984\pi$
$224$	−13.3502	−0.891997
$225$	0	0
$226$	−16.4209	−1.09230
$227$	8.67456	0.575751	0.287875	−	0.957668i	$-0.407051\pi$
$227$	8.67456	0.575751	0.287875	+	0.957668i	$0.407051\pi$
$228$	0	0
$229$	−13.6866	−0.904439	−0.452219	−	0.891907i	$-0.649368\pi$
$229$	−13.6866	−0.904439	−0.452219	+	0.891907i	$0.649368\pi$
$230$	6.13706	0.404666
$231$	0	0
$232$	−5.34721	−0.351061
$233$	5.08815	0.333336	0.166668	−	0.986013i	$-0.446699\pi$
$233$	5.08815	0.333336	0.166668	+	0.986013i	$0.446699\pi$
$234$	0	0
$235$	4.26444	0.278181
$236$	37.2204	2.42284
$237$	0	0
$238$	24.3817	1.58043
$239$	10.9239	0.706611	0.353305	−	0.935508i	$-0.385058\pi$
$239$	10.9239	0.706611	0.353305	+	0.935508i	$0.385058\pi$
$240$	0	0
$241$	11.9148	0.767502	0.383751	−	0.923437i	$-0.374632\pi$
$241$	11.9148	0.767502	0.383751	+	0.923437i	$0.374632\pi$
$242$	−10.0489	−0.645969
$243$	0	0
$244$	26.1129	1.67171
$245$	−4.04892	−0.258676
$246$	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	$-0.749183\pi$
$251$	−22.3478	−1.41058	−0.705290	+	0.708919i	$0.749183\pi$
$252$	0	0
$253$	4.82908	0.303602
$254$	−42.6262	−2.67461
$255$	0	0
$256$	−10.4668	−0.654176
$257$	18.6601	1.16398	0.581992	−	0.813194i	$-0.302273\pi$
$257$	18.6601	1.16398	0.581992	+	0.813194i	$0.302273\pi$
$258$	0	0
$259$	10.9758	0.682005
$260$	0	0
$261$	0	0
$262$	−7.31767	−0.452087
$263$	−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
$265$	−7.97823	−0.490099
$266$	−26.9366	−1.65159
$267$	0	0
$268$	1.75840	0.107411
$269$	−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
$271$	−1.99569	−0.121229	−0.0606147	−	0.998161i	$-0.519306\pi$
$271$	−1.99569	−0.121229	−0.0606147	+	0.998161i	$0.519306\pi$
$272$	−4.24698	−0.257511
$273$	0	0
$274$	−1.78017	−0.107544
$275$	−7.43967	−0.448629
$276$	0	0
$277$	11.7845	0.708061	0.354030	−	0.935234i	$-0.384811\pi$
$277$	11.7845	0.708061	0.354030	+	0.935234i	$0.384811\pi$
$278$	−25.4795	−1.52816
$279$	0	0
$280$	6.97823	0.417029
$281$	6.47219	0.386098	0.193049	−	0.981189i	$-0.438162\pi$
$281$	6.47219	0.386098	0.193049	+	0.981189i	$0.438162\pi$
$282$	0	0
$283$	6.58104	0.391202	0.195601	−	0.980684i	$-0.437334\pi$
$283$	6.58104	0.391202	0.195601	+	0.980684i	$0.437334\pi$
$284$	14.0073	0.831180
$285$	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
$292$	−32.1739	−1.88284
$293$	24.3381	1.42185	0.710924	−	0.703269i	$-0.248277\pi$
$293$	24.3381	1.42185	0.710924	+	0.703269i	$0.248277\pi$
$294$	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
$301$	12.5743	0.724773
$302$	31.7482	1.82691
$303$	0	0
$304$	4.69202	0.269106
$305$	12.3763	0.708663
$306$	0	0
$307$	−14.0737	−0.803227	−0.401613	−	0.915809i	$-0.631550\pi$
$307$	−14.0737	−0.803227	−0.401613	+	0.915809i	$0.631550\pi$
$308$	15.9608	0.909449
$309$	0	0
$310$	−13.8605	−0.787226
$311$	29.7700	1.68810	0.844051	−	0.536263i	$-0.180164\pi$
$311$	29.7700	1.68810	0.844051	+	0.536263i	$0.180164\pi$
$312$	0	0
$313$	−7.47889	−0.422732	−0.211366	−	0.977407i	$-0.567791\pi$
$313$	−7.47889	−0.422732	−0.211366	+	0.977407i	$0.567791\pi$
$314$	−21.1957	−1.19614
$315$	0	0
$316$	−48.1051	−2.70613
$317$	−30.0301	−1.68666	−0.843330	−	0.537396i	$-0.819408\pi$
$317$	−30.0301	−1.68666	−0.843330	+	0.537396i	$0.819408\pi$
$318$	0	0
$319$	−5.79656	−0.324545
$320$	−18.8388	−1.05312
$321$	0	0
$322$	8.70171	0.484927
$323$	−30.9855	−1.72408
$324$	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
$331$	−15.7168	−0.863872	−0.431936	−	0.901904i	$-0.642169\pi$
$331$	−15.7168	−0.863872	−0.431936	+	0.901904i	$0.642169\pi$
$332$	−23.5483	−1.29238
$333$	0	0
$334$	53.5991	2.93281
$335$	0.833397	0.0455333
$336$	0	0
$337$	1.95407	0.106445	0.0532224	−	0.998583i	$-0.483051\pi$
$337$	1.95407	0.106445	0.0532224	+	0.998583i	$0.483051\pi$
$338$	0	0
$339$	0	0
$340$	23.3327	1.26540
$341$	−10.9065	−0.590619
$342$	0	0
$343$	−20.0834	−1.08440
$344$	14.4644	0.779869
$345$	0	0
$346$	42.3672	2.27767
$347$	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
$349$	−10.4668	−0.560276	−0.280138	−	0.959960i	$-0.590380\pi$
$349$	−10.4668	−0.560276	−0.280138	+	0.959960i	$0.590380\pi$
$350$	−13.4058	−0.716571
$351$	0	0
$352$	−16.6474	−0.887310
$353$	−15.5308	−0.826621	−0.413310	−	0.910590i	$-0.635628\pi$
$353$	−15.5308	−0.826621	−0.413310	+	0.910590i	$0.635628\pi$
$354$	0	0
$355$	6.63879	0.352351
$356$	−20.1642	−1.06870
$357$	0	0
$358$	−13.5308	−0.715125
$359$	21.4263	1.13083	0.565417	−	0.824805i	$-0.308715\pi$
$359$	21.4263	1.13083	0.565417	+	0.824805i	$0.308715\pi$
$360$	0	0
$361$	15.2325	0.801711
$362$	−10.7356	−0.564249
$363$	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
$370$	17.3937	0.904257
$371$	−11.3123	−0.587305
$372$	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
$379$	16.5386	0.849529	0.424765	−	0.905304i	$-0.360357\pi$
$379$	16.5386	0.849529	0.424765	+	0.905304i	$0.360357\pi$
$380$	−25.7778	−1.32237
$381$	0	0
$382$	−41.4131	−2.11888
$383$	−7.53617	−0.385080	−0.192540	−	0.981289i	$-0.561673\pi$
$383$	−7.53617	−0.385080	−0.192540	+	0.981289i	$0.561673\pi$
$384$	0	0
$385$	7.56465	0.385530
$386$	13.6039	0.692419
$387$	0	0
$388$	36.3642	1.84611
$389$	−35.5555	−1.80274	−0.901369	−	0.433052i	$-0.857437\pi$
$389$	−35.5555	−1.80274	−0.901369	+	0.433052i	$0.857437\pi$
$390$	0	0
$391$	10.0097	0.506212
$392$	−6.60388	−0.333546
$393$	0	0
$394$	25.6504	1.29225
$395$	−22.7995	−1.14717
$396$	0	0
$397$	1.35152	0.0678308	0.0339154	−	0.999425i	$-0.489202\pi$
$397$	1.35152	0.0678308	0.0339154	+	0.999425i	$0.489202\pi$
$398$	−31.2446	−1.56615
$399$	0	0
$400$	2.33513	0.116756
$401$	−0.579121	−0.0289199	−0.0144600	−	0.999895i	$-0.504603\pi$
$401$	−0.579121	−0.0289199	−0.0144600	+	0.999895i	$0.504603\pi$
$402$	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
$409$	−15.1575	−0.749490	−0.374745	−	0.927128i	$-0.622270\pi$
$409$	−15.1575	−0.749490	−0.374745	+	0.927128i	$0.622270\pi$
$410$	−4.13706	−0.204315
$411$	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
$418$	−33.5894	−1.64291
$419$	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
$421$	−35.0465	−1.70806	−0.854032	−	0.520221i	$-0.825849\pi$
$421$	−35.0465	−1.70806	−0.854032	+	0.520221i	$0.825849\pi$
$422$	−29.7603	−1.44871
$423$	0	0
$424$	−13.0127	−0.631951
$425$	−15.4209	−0.748022
$426$	0	0
$427$	17.5483	0.849220
$428$	21.0368	1.01685
$429$	0	0
$430$	19.9269	0.960961
$431$	34.2814	1.65128	0.825639	−	0.564199i	$-0.190815\pi$
$431$	34.2814	1.65128	0.825639	+	0.564199i	$0.190815\pi$
$432$	0	0
$433$	13.7385	0.660232	0.330116	−	0.943940i	$-0.392912\pi$
$433$	13.7385	0.660232	0.330116	+	0.943940i	$0.392912\pi$
$434$	−19.6528	−0.943364
$435$	0	0
$436$	0.371961	0.0178137
$437$	−11.0586	−0.529005
$438$	0	0
$439$	10.2403	0.488742	0.244371	−	0.969682i	$-0.421419\pi$
$439$	10.2403	0.488742	0.244371	+	0.969682i	$0.421419\pi$
$440$	8.70171	0.414838
$441$	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
$448$	−26.7114	−1.26199
$449$	12.9051	0.609032	0.304516	−	0.952507i	$-0.401505\pi$
$449$	12.9051	0.609032	0.304516	+	0.952507i	$0.401505\pi$
$450$	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
$457$	4.65710	0.217850	0.108925	−	0.994050i	$-0.465259\pi$
$457$	4.65710	0.217850	0.108925	+	0.994050i	$0.465259\pi$
$458$	−30.7536	−1.43702
$459$	0	0
$460$	8.32736	0.388265
$461$	31.5405	1.46899	0.734493	−	0.678616i	$-0.237420\pi$
$461$	31.5405	1.46899	0.734493	+	0.678616i	$0.237420\pi$
$462$	0	0
$463$	17.6504	0.820284	0.410142	−	0.912022i	$-0.365479\pi$
$463$	17.6504	0.820284	0.410142	+	0.912022i	$0.365479\pi$
$464$	1.81940	0.0844633
$465$	0	0
$466$	11.4330	0.529622
$467$	32.1726	1.48877	0.744385	−	0.667751i	$-0.232743\pi$
$467$	32.1726	1.48877	0.744385	+	0.667751i	$0.232743\pi$
$468$	0	0
$469$	1.18167	0.0545644
$470$	9.58211	0.441990
$471$	0	0
$472$	28.7724	1.32436
$473$	15.6799	0.720964
$474$	0	0
$475$	17.0368	0.781704
$476$	33.0834	1.51637
$477$	0	0
$478$	24.5459	1.12270
$479$	34.8998	1.59461	0.797306	−	0.603576i	$-0.206258\pi$
$479$	34.8998	1.59461	0.797306	+	0.603576i	$0.206258\pi$
$480$	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
$487$	41.8351	1.89573	0.947864	−	0.318676i	$-0.103238\pi$
$487$	41.8351	1.89573	0.947864	+	0.318676i	$0.103238\pi$
$488$	20.1860	0.913776
$489$	0	0
$490$	−9.09783	−0.410998
$491$	−21.8455	−0.985873	−0.492936	−	0.870065i	$-0.664076\pi$
$491$	−21.8455	−0.985873	−0.492936	+	0.870065i	$0.664076\pi$
$492$	0	0
$493$	−12.0151	−0.541131
$494$	0	0
$495$	0	0
$496$	3.42327	0.153709
$497$	9.41311	0.422236
$498$	0	0
$499$	23.5472	1.05412	0.527058	−	0.849829i	$-0.323295\pi$
$499$	23.5472	1.05412	0.527058	+	0.849829i	$0.323295\pi$
$500$	−34.8582	−1.55890
$501$	0	0
$502$	−50.2150	−2.24121
$503$	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
$505$	−18.8780	−0.840060
$506$	10.8509	0.482379
$507$	0	0
$508$	−57.8394	−2.56621
$509$	−7.61894	−0.337704	−0.168852	−	0.985641i	$-0.554006\pi$
$509$	−7.61894	−0.337704	−0.168852	+	0.985641i	$0.554006\pi$
$510$	0	0
$511$	−21.6213	−0.956471
$512$	9.00538	0.397985
$513$	0	0
$514$	41.9288	1.84940
$515$	13.2489	0.583816
$516$	0	0
$517$	7.53989	0.331604
$518$	24.6625	1.08361
$519$	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
$526$	−32.3545	−1.41072
$527$	−22.6069	−0.984770
$528$	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
$535$	9.97046	0.431061
$536$	1.35929	0.0587123
$537$	0	0
$538$	−1.46681	−0.0632388
$539$	−7.15883	−0.308353
$540$	0	0
$541$	34.4819	1.48249	0.741246	−	0.671234i	$-0.234235\pi$
$541$	34.4819	1.48249	0.741246	+	0.671234i	$0.234235\pi$
$542$	−4.48427	−0.192616
$543$	0	0
$544$	−34.5066	−1.47946
$545$	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	$0.211132\pi$
$548$	−2.41550	−0.103185
$549$	0	0
$550$	−16.7168	−0.712806
$551$	13.2741	0.565497
$552$	0	0
$553$	−32.3274	−1.37470
$554$	26.4795	1.12501
$555$	0	0
$556$	−34.5730	−1.46622
$557$	−1.27652	−0.0540879	−0.0270439	−	0.999634i	$-0.508609\pi$
$557$	−1.27652	−0.0540879	−0.0270439	+	0.999634i	$0.508609\pi$
$558$	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	$-0.438393\pi$
$563$	9.12737	0.384673	0.192336	+	0.981329i	$0.438393\pi$
$564$	0	0
$565$	−10.5603	−0.444277
$566$	14.7875	0.621563
$567$	0	0
$568$	10.8280	0.454334
$569$	5.72156	0.239860	0.119930	−	0.992782i	$-0.461733\pi$
$569$	5.72156	0.239860	0.119930	+	0.992782i	$0.461733\pi$
$570$	0	0
$571$	7.60148	0.318112	0.159056	−	0.987270i	$-0.449155\pi$
$571$	7.60148	0.318112	0.159056	+	0.987270i	$0.449155\pi$
$572$	0	0
$573$	0	0
$574$	−5.86592	−0.244839
$575$	−5.50365	−0.229518
$576$	0	0
$577$	45.1564	1.87989	0.939944	−	0.341330i	$-0.110877\pi$
$577$	45.1564	1.87989	0.939944	+	0.341330i	$0.110877\pi$
$578$	24.8213	1.03243
$579$	0	0
$580$	−9.99569	−0.415048
$581$	−15.8248	−0.656522
$582$	0	0
$583$	−14.1062	−0.584219
$584$	−24.8713	−1.02918
$585$	0	0
$586$	54.6872	2.25911
$587$	32.4040	1.33746	0.668728	−	0.743507i	$-0.266839\pi$
$587$	32.4040	1.33746	0.668728	+	0.743507i	$0.266839\pi$
$588$	0	0
$589$	24.9758	1.02911
$590$	39.6383	1.63188
$591$	0	0
$592$	−4.29590	−0.176560
$593$	−36.6848	−1.50647	−0.753233	−	0.657754i	$-0.771507\pi$
$593$	−36.6848	−1.50647	−0.753233	+	0.657754i	$0.771507\pi$
$594$	0	0
$595$	15.6799	0.642815
$596$	−25.6286	−1.04979
$597$	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
$604$	43.0790	1.75286
$605$	−6.46250	−0.262738
$606$	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
$613$	−20.8944	−0.843917	−0.421958	−	0.906615i	$-0.638657\pi$
$613$	−20.8944	−0.843917	−0.421958	+	0.906615i	$0.638657\pi$
$614$	−31.6233	−1.27621
$615$	0	0
$616$	12.3381	0.497117
$617$	−12.0992	−0.487094	−0.243547	−	0.969889i	$-0.578311\pi$
$617$	−12.0992	−0.487094	−0.243547	+	0.969889i	$0.578311\pi$
$618$	0	0
$619$	−10.5526	−0.424143	−0.212072	−	0.977254i	$-0.568021\pi$
$619$	−10.5526	−0.424143	−0.212072	+	0.977254i	$0.568021\pi$
$620$	−18.8073	−0.755320
$621$	0	0
$622$	66.8926	2.68215
$623$	−13.5506	−0.542895
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	−16.8049	−0.671660
$627$	0	0
$628$	−28.7603	−1.14766
$629$	28.3696	1.13117
$630$	0	0
$631$	−13.8514	−0.551417	−0.275709	−	0.961241i	$-0.588913\pi$
$631$	−13.8514	−0.551417	−0.275709	+	0.961241i	$0.588913\pi$
$632$	−37.1866	−1.47920
$633$	0	0
$634$	−67.4771	−2.67986
$635$	−27.4131	−1.08786
$636$	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	$-0.742580\pi$
$641$	−34.9608	−1.38087	−0.690434	+	0.723396i	$0.742580\pi$
$642$	0	0
$643$	33.3980	1.31709	0.658545	−	0.752541i	$-0.271172\pi$
$643$	33.3980	1.31709	0.658545	+	0.752541i	$0.271172\pi$
$644$	11.8073	0.465273
$645$	0	0
$646$	−69.6238	−2.73931
$647$	−2.32842	−0.0915397	−0.0457698	−	0.998952i	$-0.514574\pi$
$647$	−2.32842	−0.0915397	−0.0457698	+	0.998952i	$0.514574\pi$
$648$	0	0
$649$	31.1903	1.22433
$650$	0	0
$651$	0	0
$652$	26.5381	1.03931
$653$	−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
$655$	−4.70602	−0.183879
$656$	1.02177	0.0398934
$657$	0	0
$658$	13.5864	0.529654
$659$	−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
$661$	−13.8498	−0.538694	−0.269347	−	0.963043i	$-0.586808\pi$
$661$	−13.8498	−0.538694	−0.269347	+	0.963043i	$0.586808\pi$
$662$	−35.3153	−1.37257
$663$	0	0
$664$	−18.2034	−0.706430
$665$	−17.3230	−0.671759
$666$	0	0
$667$	−4.28813	−0.166037
$668$	72.7284	2.81395
$669$	0	0
$670$	1.87263	0.0723458
$671$	21.8823	0.844757
$672$	0	0
$673$	−6.52973	−0.251703	−0.125851	−	0.992049i	$-0.540166\pi$
$673$	−6.52973	−0.251703	−0.125851	+	0.992049i	$0.540166\pi$
$674$	4.39075	0.169125
$675$	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
$682$	−24.5066	−0.938407
$683$	−14.1793	−0.542555	−0.271277	−	0.962501i	$-0.587446\pi$
$683$	−14.1793	−0.542555	−0.271277	+	0.962501i	$0.587446\pi$
$684$	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
$691$	30.7952	1.17151	0.585753	−	0.810490i	$-0.300799\pi$
$691$	30.7952	1.17151	0.585753	+	0.810490i	$0.300799\pi$
$692$	57.4878	2.18536
$693$	0	0
$694$	38.4795	1.46066
$695$	−16.3860	−0.621555
$696$	0	0
$697$	−6.74764	−0.255585
$698$	−23.5187	−0.890196
$699$	0	0
$700$	−18.1903	−0.687528
$701$	−6.73184	−0.254258	−0.127129	−	0.991886i	$-0.540576\pi$
$701$	−6.73184	−0.254258	−0.127129	+	0.991886i	$0.540576\pi$
$702$	0	0
$703$	−31.3424	−1.18210
$704$	−33.3086	−1.25536
$705$	0	0
$706$	−34.8974	−1.31338
$707$	−26.7670	−1.00668
$708$	0	0
$709$	−47.6252	−1.78860	−0.894300	−	0.447467i	$-0.852326\pi$
$709$	−47.6252	−1.78860	−0.894300	+	0.447467i	$0.852326\pi$
$710$	14.9172	0.559834
$711$	0	0
$712$	−15.5875	−0.584166
$713$	−8.06829	−0.302160
$714$	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	$-0.464352\pi$
$719$	5.99330	0.223512	0.111756	+	0.993736i	$0.464352\pi$
$720$	0	0
$721$	18.7855	0.699610
$722$	34.2271	1.27380
$723$	0	0
$724$	−14.5670	−0.541380
$725$	6.60627	0.245351
$726$	0	0
$727$	−24.1226	−0.894657	−0.447329	−	0.894370i	$-0.647625\pi$
$727$	−24.1226	−0.894657	−0.447329	+	0.894370i	$0.647625\pi$
$728$	0	0
$729$	0	0
$730$	−34.2640	−1.26817
$731$	32.5013	1.20210
$732$	0	0
$733$	36.0646	1.33208	0.666038	−	0.745918i	$-0.267989\pi$
$733$	36.0646	1.33208	0.666038	+	0.745918i	$0.267989\pi$
$734$	77.0786	2.84502
$735$	0	0
$736$	−12.3153	−0.453947
$737$	1.47352	0.0542777
$738$	0	0
$739$	27.5254	1.01254	0.506269	−	0.862375i	$-0.331024\pi$
$739$	27.5254	1.01254	0.506269	+	0.862375i	$0.331024\pi$
$740$	23.6015	0.867608
$741$	0	0
$742$	−25.4185	−0.933142
$743$	10.4692	0.384078	0.192039	−	0.981387i	$-0.438490\pi$
$743$	10.4692	0.384078	0.192039	+	0.981387i	$0.438490\pi$
$744$	0	0
$745$	−12.1468	−0.445023
$746$	−28.3032	−1.03625
$747$	0	0
$748$	41.2543	1.50841
$749$	14.1371	0.516557
$750$	0	0
$751$	4.06770	0.148433	0.0742163	−	0.997242i	$-0.476354\pi$
$751$	4.06770	0.148433	0.0742163	+	0.997242i	$0.476354\pi$
$752$	−2.36658	−0.0863005
$753$	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
$760$	−19.9269	−0.722825
$761$	27.0237	0.979608	0.489804	−	0.871833i	$-0.337068\pi$
$761$	27.0237	0.979608	0.489804	+	0.871833i	$0.337068\pi$
$762$	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
$769$	−37.9407	−1.36818	−0.684088	−	0.729400i	$-0.739799\pi$
$769$	−37.9407	−1.36818	−0.684088	+	0.729400i	$0.739799\pi$
$770$	16.9976	0.612551
$771$	0	0
$772$	18.4590	0.664355
$773$	16.3375	0.587620	0.293810	−	0.955864i	$-0.405077\pi$
$773$	16.3375	0.587620	0.293810	+	0.955864i	$0.405077\pi$
$774$	0	0
$775$	12.4300	0.446498
$776$	28.1105	1.00911
$777$	0	0
$778$	−79.8926	−2.86429
$779$	7.45473	0.267093
$780$	0	0
$781$	11.7380	0.420017
$782$	22.4916	0.804297
$783$	0	0
$784$	2.24698	0.0802493
$785$	−13.6310	−0.486512
$786$	0	0
$787$	−18.6907	−0.666251	−0.333126	−	0.942882i	$-0.608103\pi$
$787$	−18.6907	−0.666251	−0.333126	+	0.942882i	$0.608103\pi$
$788$	34.8049	1.23987
$789$	0	0
$790$	−51.2301	−1.82269
$791$	−14.9734	−0.532394
$792$	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	$-0.673352\pi$
$797$	−29.2519	−1.03615	−0.518077	+	0.855334i	$0.673352\pi$
$798$	0	0
$799$	15.6286	0.552901
$800$	18.9729	0.670792
$801$	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
$808$	−30.7904	−1.08320
$809$	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
$811$	3.89200	0.136667	0.0683333	−	0.997663i	$-0.478232\pi$
$811$	3.89200	0.136667	0.0683333	+	0.997663i	$0.478232\pi$
$812$	−14.1728	−0.497369
$813$	0	0
$814$	30.7536	1.07791
$815$	12.5778	0.440581
$816$	0	0
$817$	−35.9071	−1.25623
$818$	−34.0586	−1.19083
$819$	0	0
$820$	−5.61356	−0.196034
$821$	−45.9982	−1.60535	−0.802674	−	0.596418i	$-0.796590\pi$
$821$	−45.9982	−1.60535	−0.802674	+	0.596418i	$0.796590\pi$
$822$	0	0
$823$	7.95300	0.277224	0.138612	−	0.990347i	$-0.455736\pi$
$823$	7.95300	0.277224	0.138612	+	0.990347i	$0.455736\pi$
$824$	21.6093	0.752794
$825$	0	0
$826$	56.2030	1.95555
$827$	−27.9648	−0.972432	−0.486216	−	0.873839i	$-0.661623\pi$
$827$	−27.9648	−0.972432	−0.486216	+	0.873839i	$0.661623\pi$
$828$	0	0
$829$	27.6310	0.959665	0.479833	−	0.877360i	$-0.340697\pi$
$829$	27.6310	0.959665	0.479833	+	0.877360i	$0.340697\pi$
$830$	−25.0780	−0.870470
$831$	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
$838$	80.2699	2.77288
$839$	−28.6848	−0.990311	−0.495155	−	0.868804i	$-0.664889\pi$
$839$	−28.6848	−0.990311	−0.495155	+	0.868804i	$0.664889\pi$
$840$	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
$847$	−9.16315	−0.314849
$848$	4.42758	0.152044
$849$	0	0
$850$	−34.6504	−1.18850
$851$	10.1250	0.347080
$852$	0	0
$853$	−43.2078	−1.47941	−0.739703	−	0.672934i	$-0.765034\pi$
$853$	−43.2078	−1.47941	−0.739703	+	0.672934i	$0.765034\pi$
$854$	39.4306	1.34929
$855$	0	0
$856$	16.2620	0.555825
$857$	−35.1685	−1.20133	−0.600667	−	0.799499i	$-0.705098\pi$
$857$	−35.1685	−1.20133	−0.600667	+	0.799499i	$0.705098\pi$
$858$	0	0
$859$	27.3793	0.934168	0.467084	−	0.884213i	$-0.345305\pi$
$859$	27.3793	0.934168	0.467084	+	0.884213i	$0.345305\pi$
$860$	27.0388	0.922014
$861$	0	0
$862$	77.0297	2.62364
$863$	41.3913	1.40898	0.704489	−	0.709715i	$-0.251176\pi$
$863$	41.3913	1.40898	0.704489	+	0.709715i	$0.251176\pi$
$864$	0	0
$865$	27.2465	0.926409
$866$	30.8702	1.04901
$867$	0	0
$868$	−26.6668	−0.905130
$869$	−40.3116	−1.36748
$870$	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
$877$	−24.7472	−0.835653	−0.417826	−	0.908527i	$-0.637208\pi$
$877$	−24.7472	−0.835653	−0.417826	+	0.908527i	$0.637208\pi$
$878$	23.0097	0.776539
$879$	0	0
$880$	−2.96077	−0.0998076
$881$	28.5875	0.963137	0.481568	−	0.876409i	$-0.340067\pi$
$881$	28.5875	0.963137	0.481568	+	0.876409i	$0.340067\pi$
$882$	0	0
$883$	9.61702	0.323639	0.161819	−	0.986820i	$-0.448264\pi$
$883$	9.61702	0.323639	0.161819	+	0.986820i	$0.448264\pi$
$884$	0	0
$885$	0	0
$886$	−27.3599	−0.919173
$887$	−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
$889$	−38.8689	−1.30362
$890$	−21.4741	−0.719814
$891$	0	0
$892$	−22.3642	−0.748809
$893$	−17.2664	−0.577797
$894$	0	0
$895$	−8.70171	−0.290866
$896$	−33.3196	−1.11313
$897$	0	0
$898$	28.9976	0.967663
$899$	9.68473	0.323004
$900$	0	0
$901$	−29.2392	−0.974099
$902$	−7.31468	−0.243552
$903$	0	0
$904$	−17.2241	−0.572867
$905$	−6.90408	−0.229500
$906$	0	0
$907$	−28.8364	−0.957496	−0.478748	−	0.877952i	$-0.658909\pi$
$907$	−28.8364	−0.957496	−0.478748	+	0.877952i	$0.658909\pi$
$908$	26.4480	0.877709
$909$	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
$916$	−41.7294	−1.37878
$917$	−6.67264	−0.220350
$918$	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
$925$	−15.5985	−0.512875
$926$	39.6601	1.30331
$927$	0	0
$928$	14.7826	0.485261
$929$	−24.2295	−0.794945	−0.397472	−	0.917614i	$-0.630113\pi$
$929$	−24.2295	−0.794945	−0.397472	+	0.917614i	$0.630113\pi$
$930$	0	0
$931$	16.3937	0.537283
$932$	15.5133	0.508156
$933$	0	0
$934$	72.2911	2.36544
$935$	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	$0.409001\pi$
$938$	2.65519	0.0866949
$939$	0	0
$940$	13.0019	0.424076
$941$	−4.34050	−0.141496	−0.0707482	−	0.997494i	$-0.522539\pi$
$941$	−4.34050	−0.141496	−0.0707482	+	0.997494i	$0.522539\pi$
$942$	0	0
$943$	−2.40821	−0.0784220
$944$	−9.78986	−0.318633
$945$	0	0
$946$	35.2325	1.14551
$947$	45.0146	1.46278	0.731389	−	0.681961i	$-0.238872\pi$
$947$	45.0146	1.46278	0.731389	+	0.681961i	$0.238872\pi$
$948$	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	$-0.225497\pi$
$953$	46.8859	1.51878	0.759391	+	0.650634i	$0.225497\pi$
$954$	0	0
$955$	−26.6329	−0.861822
$956$	33.3062	1.07720
$957$	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
$964$	36.3274	1.17003
$965$	8.74871	0.281631
$966$	0	0
$967$	6.29457	0.202420	0.101210	−	0.994865i	$-0.467729\pi$
$967$	6.29457	0.202420	0.101210	+	0.994865i	$0.467729\pi$
$968$	−10.5405	−0.338784
$969$	0	0
$970$	38.7265	1.24343
$971$	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
$973$	−23.2336	−0.744834
$974$	94.0025	3.01203
$975$	0	0
$976$	−6.86831	−0.219849
$977$	−23.7530	−0.759926	−0.379963	−	0.925002i	$-0.624063\pi$
$977$	−23.7530	−0.759926	−0.379963	+	0.925002i	$0.624063\pi$
$978$	0	0
$979$	−16.8974	−0.540043
$980$	−12.3448	−0.394341
$981$	0	0
$982$	−49.0863	−1.56641
$983$	55.7251	1.77736	0.888678	−	0.458532i	$-0.151625\pi$
$983$	55.7251	1.77736	0.888678	+	0.458532i	$0.151625\pi$
$984$	0	0
$985$	16.4959	0.525602
$986$	−26.9976	−0.859779
$987$	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
$994$	21.1511	0.670871
$995$	−20.0935	−0.637007
$996$	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.a.r.1.3		3
3.2	odd	2		169.2.a.b.1.1	✓	3
12.11	even	2		2704.2.a.z.1.2		3
13.5	odd	4		1521.2.b.l.1351.1		6
13.8	odd	4		1521.2.b.l.1351.6		6
13.12	even	2		1521.2.a.o.1.1		3
15.14	odd	2		4225.2.a.bg.1.3		3
21.20	even	2		8281.2.a.bf.1.1		3
39.2	even	12		169.2.e.b.147.1		12
39.5	even	4		169.2.b.b.168.6		6
39.8	even	4		169.2.b.b.168.1		6
39.11	even	12		169.2.e.b.147.6		12
39.17	odd	6		169.2.c.b.146.1		6
39.20	even	12		169.2.e.b.23.1		12
39.23	odd	6		169.2.c.b.22.1		6
39.29	odd	6		169.2.c.c.22.3		6
39.32	even	12		169.2.e.b.23.6		12
39.35	odd	6		169.2.c.c.146.3		6
39.38	odd	2		169.2.a.c.1.3	yes	3
156.47	odd	4		2704.2.f.o.337.3		6
156.83	odd	4		2704.2.f.o.337.4		6
156.155	even	2		2704.2.a.ba.1.2		3
195.194	odd	2		4225.2.a.bb.1.1		3
273.272	even	2		8281.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.1	✓	3	3.2	odd	2
169.2.a.c.1.3	yes	3	39.38	odd	2
169.2.b.b.168.1		6	39.8	even	4
169.2.b.b.168.6		6	39.5	even	4
169.2.c.b.22.1		6	39.23	odd	6
169.2.c.b.146.1		6	39.17	odd	6
169.2.c.c.22.3		6	39.29	odd	6
169.2.c.c.146.3		6	39.35	odd	6
169.2.e.b.23.1		12	39.20	even	12
169.2.e.b.23.6		12	39.32	even	12
169.2.e.b.147.1		12	39.2	even	12
169.2.e.b.147.6		12	39.11	even	12
1521.2.a.o.1.1		3	13.12	even	2
1521.2.a.r.1.3		3	1.1	even	1	trivial
1521.2.b.l.1351.1		6	13.5	odd	4
1521.2.b.l.1351.6		6	13.8	odd	4
2704.2.a.z.1.2		3	12.11	even	2
2704.2.a.ba.1.2		3	156.155	even	2
2704.2.f.o.337.3		6	156.47	odd	4
2704.2.f.o.337.4		6	156.83	odd	4
4225.2.a.bb.1.1		3	195.194	odd	2
4225.2.a.bg.1.3		3	15.14	odd	2
8281.2.a.bf.1.1		3	21.20	even	2
8281.2.a.bj.1.3		3	273.272	even	2

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.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24698 q^{2} +3.04892 q^{4} +1.44504 q^{5} +2.04892 q^{7} +2.35690 q^{8} +O(q^{10})\)	\(q+2.24698 q^{2} +3.04892 q^{4} +1.44504 q^{5} +2.04892 q^{7} +2.35690 q^{8} +3.24698 q^{10} +2.55496 q^{11} +4.60388 q^{14} -0.801938 q^{16} +5.29590 q^{17} -5.85086 q^{19} +4.40581 q^{20} +5.74094 q^{22} +1.89008 q^{23} -2.91185 q^{25} +6.24698 q^{28} -2.26875 q^{29} -4.26875 q^{31} -6.51573 q^{32} +11.8998 q^{34} +2.96077 q^{35} +5.35690 q^{37} -13.1468 q^{38} +3.40581 q^{40} -1.27413 q^{41} +6.13706 q^{43} +7.78986 q^{44} +4.24698 q^{46} +2.95108 q^{47} -2.80194 q^{49} -6.54288 q^{50} -5.52111 q^{53} +3.69202 q^{55} +4.82908 q^{56} -5.09783 q^{58} +12.2078 q^{59} +8.56465 q^{61} -9.59179 q^{62} -13.0368 q^{64} +0.576728 q^{67} +16.1468 q^{68} +6.65279 q^{70} +4.59419 q^{71} -10.5526 q^{73} +12.0368 q^{74} -17.8388 q^{76} +5.23490 q^{77} -15.7778 q^{79} -1.15883 q^{80} -2.86294 q^{82} -7.72348 q^{83} +7.65279 q^{85} +13.7899 q^{86} +6.02177 q^{88} -6.61356 q^{89} +5.76271 q^{92} +6.63102 q^{94} -8.45473 q^{95} +11.9269 q^{97} -6.29590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35} + 12 q^{37} - 12 q^{38} - 3 q^{40} + 7 q^{41} + 13 q^{43} + 8 q^{46} + 18 q^{47} - 4 q^{49} - q^{50} - q^{53} + 6 q^{55} + 4 q^{56} + 3 q^{58} + 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} - q^{67} + 21 q^{68} + 2 q^{70} + 27 q^{71} + 9 q^{73} + 8 q^{74} - 21 q^{76} - 8 q^{77} - 5 q^{79} + 5 q^{80} - 14 q^{82} + 7 q^{83} + 5 q^{85} + 18 q^{86} + 15 q^{88} + 11 q^{89} + 5 q^{94} - 3 q^{95} + 7 q^{97} - 5 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

Introduction

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