LMFDB - Embedded newform 1521.4.a.v.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,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, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 25x^{2} + 24x + 78 $ x^4 - 2x^3 - 25x^2 + 24*x + 78
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.22605$ 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)

$f(q)$	$=$	$q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +O(q^{10})$	$q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +24.7627 q^{10} +33.8892 q^{11} -101.955 q^{14} -45.6659 q^{16} +49.3956 q^{17} -76.8548 q^{19} +57.7723 q^{20} +143.218 q^{22} -6.29163 q^{23} -90.6659 q^{25} -237.866 q^{28} -100.995 q^{29} -307.580 q^{31} -255.854 q^{32} +208.749 q^{34} -141.364 q^{35} -76.0189 q^{37} -324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -268.184 q^{43} +334.132 q^{44} -26.5888 q^{46} +460.912 q^{47} +239.037 q^{49} -383.159 q^{50} -67.8057 q^{53} +198.575 q^{55} -189.589 q^{56} -426.812 q^{58} -25.2021 q^{59} -588.832 q^{61} -1299.85 q^{62} -715.927 q^{64} -1004.46 q^{67} +487.018 q^{68} -597.411 q^{70} -895.481 q^{71} +968.599 q^{73} -321.260 q^{74} -757.753 q^{76} -817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +2173.96 q^{82} -480.784 q^{83} +289.435 q^{85} -1133.36 q^{86} +266.318 q^{88} -1085.91 q^{89} -62.0325 q^{92} +1947.84 q^{94} -450.333 q^{95} -16.6552 q^{97} +1010.18 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8	$ 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * 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 $

$2$	4.22605	1.49414	0.747068	−	0.664748i	$-0.231461\pi$
$2$	4.22605	1.49414	0.747068	+	0.664748i	$0.231461\pi$
$3$	0	0
$3$	0	0
$4$	9.85953	1.23244
$5$	5.85953	0.524093	0.262046	−	0.965055i	$-0.415603\pi$
$5$	5.85953	0.524093	0.262046	+	0.965055i	$0.415603\pi$
$6$	0	0
$7$	−24.1254	−1.30265	−0.651326	−	0.758798i	$-0.725787\pi$
$7$	−24.1254	−1.30265	−0.651326	+	0.758798i	$0.725787\pi$
$8$	7.85849	0.347299
$9$	0	0
$10$	24.7627	0.783065
$11$	33.8892	0.928907	0.464453	−	0.885598i	$-0.346251\pi$
$11$	33.8892	0.928907	0.464453	+	0.885598i	$0.346251\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−101.955	−1.94634
$15$	0	0
$16$	−45.6659	−0.713529
$17$	49.3956	0.704718	0.352359	−	0.935865i	$-0.385380\pi$
$17$	49.3956	0.704718	0.352359	+	0.935865i	$0.385380\pi$
$18$	0	0
$19$	−76.8548	−0.927985	−0.463992	−	0.885839i	$-0.653583\pi$
$19$	−76.8548	−0.927985	−0.463992	+	0.885839i	$0.653583\pi$
$20$	57.7723	0.645913
$21$	0	0
$22$	143.218	1.38791
$23$	−6.29163	−0.0570389	−0.0285195	−	0.999593i	$-0.509079\pi$
$23$	−6.29163	−0.0570389	−0.0285195	+	0.999593i	$0.509079\pi$
$24$	0	0
$25$	−90.6659	−0.725327
$26$	0	0
$27$	0	0
$28$	−237.866	−1.60544
$29$	−100.995	−0.646702	−0.323351	−	0.946279i	$-0.604809\pi$
$29$	−100.995	−0.646702	−0.323351	+	0.946279i	$0.604809\pi$
$30$	0	0
$31$	−307.580	−1.78203	−0.891016	−	0.453972i	$-0.850007\pi$
$31$	−307.580	−1.78203	−0.891016	+	0.453972i	$0.850007\pi$
$32$	−255.854	−1.41341
$33$	0	0
$34$	208.749	1.05294
$35$	−141.364	−0.682710
$36$	0	0
$37$	−76.0189	−0.337768	−0.168884	−	0.985636i	$-0.554016\pi$
$37$	−76.0189	−0.337768	−0.168884	+	0.985636i	$0.554016\pi$
$38$	−324.793	−1.38653
$39$	0	0
$40$	46.0471	0.182017
$41$	514.418	1.95948	0.979740	−	0.200274i	$-0.0641833\pi$
$41$	514.418	1.95948	0.979740	+	0.200274i	$0.0641833\pi$
$42$	0	0
$43$	−268.184	−0.951108	−0.475554	−	0.879686i	$-0.657752\pi$
$43$	−268.184	−0.951108	−0.475554	+	0.879686i	$0.657752\pi$
$44$	334.132	1.14482
$45$	0	0
$46$	−26.5888	−0.0852239
$47$	460.912	1.43045	0.715223	−	0.698896i	$-0.246325\pi$
$47$	460.912	1.43045	0.715223	+	0.698896i	$0.246325\pi$
$48$	0	0
$49$	239.037	0.696901
$50$	−383.159	−1.08374
$51$	0	0
$52$	0	0
$53$	−67.8057	−0.175733	−0.0878663	−	0.996132i	$-0.528005\pi$
$53$	−67.8057	−0.175733	−0.0878663	+	0.996132i	$0.528005\pi$
$54$	0	0
$55$	198.575	0.486833
$56$	−189.589	−0.452410
$57$	0	0
$58$	−426.812	−0.966261
$59$	−25.2021	−0.0556107	−0.0278053	−	0.999613i	$-0.508852\pi$
$59$	−25.2021	−0.0556107	−0.0278053	+	0.999613i	$0.508852\pi$
$60$	0	0
$61$	−588.832	−1.23594	−0.617969	−	0.786202i	$-0.712044\pi$
$61$	−588.832	−1.23594	−0.617969	+	0.786202i	$0.712044\pi$
$62$	−1299.85	−2.66260
$63$	0	0
$64$	−715.927	−1.39830
$65$	0	0
$66$	0	0
$67$	−1004.46	−1.83156	−0.915778	−	0.401684i	$-0.868425\pi$
$67$	−1004.46	−1.83156	−0.915778	+	0.401684i	$0.868425\pi$
$68$	487.018	0.868523
$69$	0	0
$70$	−597.411	−1.02006
$71$	−895.481	−1.49682	−0.748408	−	0.663238i	$-0.769182\pi$
$71$	−895.481	−1.49682	−0.748408	+	0.663238i	$0.769182\pi$
$72$	0	0
$73$	968.599	1.55296	0.776479	−	0.630143i	$-0.217004\pi$
$73$	968.599	1.55296	0.776479	+	0.630143i	$0.217004\pi$
$74$	−321.260	−0.504672
$75$	0	0
$76$	−757.753	−1.14369
$77$	−817.592	−1.21004
$78$	0	0
$79$	−119.053	−0.169551	−0.0847755	−	0.996400i	$-0.527017\pi$
$79$	−119.053	−0.169551	−0.0847755	+	0.996400i	$0.527017\pi$
$80$	−267.581	−0.373955
$81$	0	0
$82$	2173.96	2.92773
$83$	−480.784	−0.635818	−0.317909	−	0.948121i	$-0.602981\pi$
$83$	−480.784	−0.635818	−0.317909	+	0.948121i	$0.602981\pi$
$84$	0	0
$85$	289.435	0.369337
$86$	−1133.36	−1.42109
$87$	0	0
$88$	266.318	0.322609
$89$	−1085.91	−1.29333	−0.646663	−	0.762776i	$-0.723836\pi$
$89$	−1085.91	−1.29333	−0.646663	+	0.762776i	$0.723836\pi$
$90$	0	0
$91$	0	0
$92$	−62.0325	−0.0702972
$93$	0	0
$94$	1947.84	2.13728
$95$	−450.333	−0.486350
$96$	0	0
$97$	−16.6552	−0.0174338	−0.00871692	−	0.999962i	$-0.502775\pi$
$97$	−16.6552	−0.0174338	−0.00871692	+	0.999962i	$0.502775\pi$
$98$	1010.18	1.04126
$99$	0	0
$100$	−893.923	−0.893923
$101$	958.004	0.943811	0.471906	−	0.881649i	$-0.343566\pi$
$101$	958.004	0.943811	0.471906	+	0.881649i	$0.343566\pi$
$102$	0	0
$103$	−2.70560	−0.00258826	−0.00129413	−	0.999999i	$-0.500412\pi$
$103$	−2.70560	−0.00258826	−0.00129413	+	0.999999i	$0.500412\pi$
$104$	0	0
$105$	0	0
$106$	−286.551	−0.262568
$107$	−1351.28	−1.22087	−0.610437	−	0.792065i	$-0.709006\pi$
$107$	−1351.28	−1.22087	−0.610437	+	0.792065i	$0.709006\pi$
$108$	0	0
$109$	448.455	0.394075	0.197037	−	0.980396i	$-0.436868\pi$
$109$	448.455	0.394075	0.197037	+	0.980396i	$0.436868\pi$
$110$	839.188	0.727395
$111$	0	0
$112$	1101.71	0.929480
$113$	−1398.85	−1.16453	−0.582267	−	0.812997i	$-0.697834\pi$
$113$	−1398.85	−1.16453	−0.582267	+	0.812997i	$0.697834\pi$
$114$	0	0
$115$	−36.8660	−0.0298937
$116$	−995.767	−0.797023
$117$	0	0
$118$	−106.505	−0.0830899
$119$	−1191.69	−0.918002
$120$	0	0
$121$	−182.523	−0.137132
$122$	−2488.44	−1.84666
$123$	0	0
$124$	−3032.59	−2.19625
$125$	−1263.70	−0.904231
$126$	0	0
$127$	−119.504	−0.0834985	−0.0417492	−	0.999128i	$-0.513293\pi$
$127$	−119.504	−0.0834985	−0.0417492	+	0.999128i	$0.513293\pi$
$128$	−978.713	−0.675834
$129$	0	0
$130$	0	0
$131$	2251.70	1.50177	0.750886	−	0.660432i	$-0.229627\pi$
$131$	2251.70	1.50177	0.750886	+	0.660432i	$0.229627\pi$
$132$	0	0
$133$	1854.16	1.20884
$134$	−4244.90	−2.73659
$135$	0	0
$136$	388.175	0.244748
$137$	−1130.62	−0.705076	−0.352538	−	0.935797i	$-0.614681\pi$
$137$	−1130.62	−0.705076	−0.352538	+	0.935797i	$0.614681\pi$
$138$	0	0
$139$	−595.287	−0.363249	−0.181624	−	0.983368i	$-0.558136\pi$
$139$	−595.287	−0.363249	−0.181624	+	0.983368i	$0.558136\pi$
$140$	−1393.78	−0.841400
$141$	0	0
$142$	−3784.35	−2.23645
$143$	0	0
$144$	0	0
$145$	−591.786	−0.338932
$146$	4093.35	2.32033
$147$	0	0
$148$	−749.511	−0.416280
$149$	793.175	0.436103	0.218052	−	0.975937i	$-0.430030\pi$
$149$	793.175	0.436103	0.218052	+	0.975937i	$0.430030\pi$
$150$	0	0
$151$	−134.213	−0.0723317	−0.0361659	−	0.999346i	$-0.511514\pi$
$151$	−134.213	−0.0723317	−0.0361659	+	0.999346i	$0.511514\pi$
$152$	−603.963	−0.322288
$153$	0	0
$154$	−3455.19	−1.80797
$155$	−1802.28	−0.933950
$156$	0	0
$157$	1509.07	0.767114	0.383557	−	0.923517i	$-0.374699\pi$
$157$	1509.07	0.767114	0.383557	+	0.923517i	$0.374699\pi$
$158$	−503.125	−0.253332
$159$	0	0
$160$	−1499.19	−0.740757
$161$	151.788	0.0743019
$162$	0	0
$163$	1175.08	0.564658	0.282329	−	0.959318i	$-0.408893\pi$
$163$	1175.08	0.564658	0.282329	+	0.959318i	$0.408893\pi$
$164$	5071.93	2.41494
$165$	0	0
$166$	−2031.82	−0.949999
$167$	−1474.01	−0.683010	−0.341505	−	0.939880i	$-0.610937\pi$
$167$	−1474.01	−0.683010	−0.341505	+	0.939880i	$0.610937\pi$
$168$	0	0
$169$	0	0
$170$	1223.17	0.551840
$171$	0	0
$172$	−2644.17	−1.17219
$173$	2328.31	1.02322	0.511612	−	0.859216i	$-0.329048\pi$
$173$	2328.31	1.02322	0.511612	+	0.859216i	$0.329048\pi$
$174$	0	0
$175$	2187.35	0.944848
$176$	−1547.58	−0.662802
$177$	0	0
$178$	−4589.10	−1.93240
$179$	2133.85	0.891015	0.445508	−	0.895278i	$-0.353023\pi$
$179$	2133.85	0.891015	0.445508	+	0.895278i	$0.353023\pi$
$180$	0	0
$181$	−2485.41	−1.02066	−0.510329	−	0.859979i	$-0.670476\pi$
$181$	−2485.41	−1.02066	−0.510329	+	0.859979i	$0.670476\pi$
$182$	0	0
$183$	0	0
$184$	−49.4427	−0.0198096
$185$	−445.435	−0.177022
$186$	0	0
$187$	1673.98	0.654617
$188$	4544.38	1.76294
$189$	0	0
$190$	−1903.13	−0.726673
$191$	2325.07	0.880816	0.440408	−	0.897798i	$-0.354834\pi$
$191$	2325.07	0.880816	0.440408	+	0.897798i	$0.354834\pi$
$192$	0	0
$193$	3350.12	1.24946	0.624732	−	0.780839i	$-0.285208\pi$
$193$	3350.12	1.24946	0.624732	+	0.780839i	$0.285208\pi$
$194$	−70.3859	−0.0260485
$195$	0	0
$196$	2356.79	0.858890
$197$	3859.30	1.39576	0.697878	−	0.716217i	$-0.254128\pi$
$197$	3859.30	1.39576	0.697878	+	0.716217i	$0.254128\pi$
$198$	0	0
$199$	−4083.60	−1.45467	−0.727333	−	0.686284i	$-0.759241\pi$
$199$	−4083.60	−1.45467	−0.727333	+	0.686284i	$0.759241\pi$
$200$	−712.497	−0.251906
$201$	0	0
$202$	4048.58	1.41018
$203$	2436.56	0.842428
$204$	0	0
$205$	3014.25	1.02695
$206$	−11.4340	−0.00386721
$207$	0	0
$208$	0	0
$209$	−2604.55	−0.862011
$210$	0	0
$211$	3027.30	0.987714	0.493857	−	0.869543i	$-0.335587\pi$
$211$	3027.30	0.987714	0.493857	+	0.869543i	$0.335587\pi$
$212$	−668.533	−0.216580
$213$	0	0
$214$	−5710.60	−1.82415
$215$	−1571.43	−0.498469
$216$	0	0
$217$	7420.50	2.32137
$218$	1895.19	0.588801
$219$	0	0
$220$	1957.86	0.599994
$221$	0	0
$222$	0	0
$223$	−1724.76	−0.517930	−0.258965	−	0.965887i	$-0.583381\pi$
$223$	−1724.76	−0.517930	−0.258965	+	0.965887i	$0.583381\pi$
$224$	6172.60	1.84118
$225$	0	0
$226$	−5911.60	−1.73997
$227$	1923.27	0.562344	0.281172	−	0.959657i	$-0.409277\pi$
$227$	1923.27	0.562344	0.281172	+	0.959657i	$0.409277\pi$
$228$	0	0
$229$	373.993	0.107922	0.0539610	−	0.998543i	$-0.482815\pi$
$229$	373.993	0.107922	0.0539610	+	0.998543i	$0.482815\pi$
$230$	−155.798	−0.0446652
$231$	0	0
$232$	−793.671	−0.224599
$233$	−3094.49	−0.870073	−0.435036	−	0.900413i	$-0.643265\pi$
$233$	−3094.49	−0.870073	−0.435036	+	0.900413i	$0.643265\pi$
$234$	0	0
$235$	2700.73	0.749686
$236$	−248.481	−0.0685369
$237$	0	0
$238$	−5036.15	−1.37162
$239$	1221.18	0.330510	0.165255	−	0.986251i	$-0.447155\pi$
$239$	1221.18	0.330510	0.165255	+	0.986251i	$0.447155\pi$
$240$	0	0
$241$	145.401	0.0388634	0.0194317	−	0.999811i	$-0.493814\pi$
$241$	145.401	0.0388634	0.0194317	+	0.999811i	$0.493814\pi$
$242$	−771.350	−0.204894
$243$	0	0
$244$	−5805.61	−1.52322
$245$	1400.65	0.365241
$246$	0	0
$247$	0	0
$248$	−2417.11	−0.618899
$249$	0	0
$250$	−5340.47	−1.35104
$251$	985.670	0.247868	0.123934	−	0.992290i	$-0.460449\pi$
$251$	985.670	0.247868	0.123934	+	0.992290i	$0.460449\pi$
$252$	0	0
$253$	−213.218	−0.0529839
$254$	−505.032	−0.124758
$255$	0	0
$256$	1591.33	0.388507
$257$	−2929.32	−0.710995	−0.355498	−	0.934677i	$-0.615689\pi$
$257$	−2929.32	−0.710995	−0.355498	+	0.934677i	$0.615689\pi$
$258$	0	0
$259$	1833.99	0.439995
$260$	0	0
$261$	0	0
$262$	9515.82	2.24385
$263$	−2238.00	−0.524719	−0.262360	−	0.964970i	$-0.584501\pi$
$263$	−2238.00	−0.524719	−0.262360	+	0.964970i	$0.584501\pi$
$264$	0	0
$265$	−397.310	−0.0921002
$266$	7835.77	1.80617
$267$	0	0
$268$	−9903.50	−2.25729
$269$	−1925.98	−0.436540	−0.218270	−	0.975888i	$-0.570041\pi$
$269$	−1925.98	−0.436540	−0.218270	+	0.975888i	$0.570041\pi$
$270$	0	0
$271$	−3562.29	−0.798500	−0.399250	−	0.916842i	$-0.630729\pi$
$271$	−3562.29	−0.798500	−0.399250	+	0.916842i	$0.630729\pi$
$272$	−2255.69	−0.502837
$273$	0	0
$274$	−4778.06	−1.05348
$275$	−3072.59	−0.673761
$276$	0	0
$277$	−1437.42	−0.311792	−0.155896	−	0.987773i	$-0.549827\pi$
$277$	−1437.42	−0.311792	−0.155896	+	0.987773i	$0.549827\pi$
$278$	−2515.72	−0.542743
$279$	0	0
$280$	−1110.91	−0.237105
$281$	3913.51	0.830820	0.415410	−	0.909634i	$-0.363638\pi$
$281$	3913.51	0.830820	0.415410	+	0.909634i	$0.363638\pi$
$282$	0	0
$283$	−3212.31	−0.674743	−0.337371	−	0.941372i	$-0.609538\pi$
$283$	−3212.31	−0.674743	−0.337371	+	0.941372i	$0.609538\pi$
$284$	−8829.02	−1.84474
$285$	0	0
$286$	0	0
$287$	−12410.6	−2.55252
$288$	0	0
$289$	−2473.07	−0.503373
$290$	−2500.92	−0.506410
$291$	0	0
$292$	9549.94	1.91393
$293$	4901.77	0.977353	0.488677	−	0.872465i	$-0.337480\pi$
$293$	4901.77	0.977353	0.488677	+	0.872465i	$0.337480\pi$
$294$	0	0
$295$	−147.672	−0.0291452
$296$	−597.394	−0.117307
$297$	0	0
$298$	3352.00	0.651598
$299$	0	0
$300$	0	0
$301$	6470.06	1.23896
$302$	−567.191	−0.108073
$303$	0	0
$304$	3509.64	0.662144
$305$	−3450.28	−0.647746
$306$	0	0
$307$	5800.63	1.07837	0.539185	−	0.842188i	$-0.318733\pi$
$307$	5800.63	1.07837	0.539185	+	0.842188i	$0.318733\pi$
$308$	−8061.07	−1.49131
$309$	0	0
$310$	−7616.51	−1.39545
$311$	4913.51	0.895884	0.447942	−	0.894063i	$-0.352157\pi$
$311$	4913.51	0.895884	0.447942	+	0.894063i	$0.352157\pi$
$312$	0	0
$313$	−8104.97	−1.46364	−0.731822	−	0.681496i	$-0.761330\pi$
$313$	−8104.97	−1.46364	−0.731822	+	0.681496i	$0.761330\pi$
$314$	6377.41	1.14617
$315$	0	0
$316$	−1173.81	−0.208962
$317$	−5149.92	−0.912455	−0.456227	−	0.889863i	$-0.650800\pi$
$317$	−5149.92	−0.912455	−0.456227	+	0.889863i	$0.650800\pi$
$318$	0	0
$319$	−3422.65	−0.600726
$320$	−4195.00	−0.732836
$321$	0	0
$322$	641.466	0.111017
$323$	−3796.29	−0.653967
$324$	0	0
$325$	0	0
$326$	4965.95	0.843676
$327$	0	0
$328$	4042.55	0.680526
$329$	−11119.7	−1.86337
$330$	0	0
$331$	6061.98	1.00664	0.503318	−	0.864101i	$-0.332112\pi$
$331$	6061.98	1.00664	0.503318	+	0.864101i	$0.332112\pi$
$332$	−4740.31	−0.783609
$333$	0	0
$334$	−6229.26	−1.02051
$335$	−5885.67	−0.959905
$336$	0	0
$337$	3743.50	0.605108	0.302554	−	0.953132i	$-0.402161\pi$
$337$	3743.50	0.605108	0.302554	+	0.953132i	$0.402161\pi$
$338$	0	0
$339$	0	0
$340$	2853.70	0.455187
$341$	−10423.6	−1.65534
$342$	0	0
$343$	2508.15	0.394832
$344$	−2107.52	−0.330319
$345$	0	0
$346$	9839.55	1.52884
$347$	2520.41	0.389921	0.194961	−	0.980811i	$-0.437542\pi$
$347$	2520.41	0.389921	0.194961	+	0.980811i	$0.437542\pi$
$348$	0	0
$349$	10650.7	1.63359	0.816793	−	0.576931i	$-0.195750\pi$
$349$	10650.7	1.63359	0.816793	+	0.576931i	$0.195750\pi$
$350$	9243.88	1.41173
$351$	0	0
$352$	−8670.70	−1.31293
$353$	9002.82	1.35743	0.678714	−	0.734403i	$-0.262538\pi$
$353$	9002.82	1.35743	0.678714	+	0.734403i	$0.262538\pi$
$354$	0	0
$355$	−5247.10	−0.784471
$356$	−10706.5	−1.59395
$357$	0	0
$358$	9017.78	1.33130
$359$	11360.9	1.67021	0.835106	−	0.550089i	$-0.185406\pi$
$359$	11360.9	1.67021	0.835106	+	0.550089i	$0.185406\pi$
$360$	0	0
$361$	−952.336	−0.138845
$362$	−10503.5	−1.52500
$363$	0	0
$364$	0	0
$365$	5675.54	0.813894
$366$	0	0
$367$	−13938.8	−1.98257	−0.991283	−	0.131753i	$-0.957939\pi$
$367$	−13938.8	−1.98257	−0.991283	+	0.131753i	$0.957939\pi$
$368$	287.313	0.0406990
$369$	0	0
$370$	−1882.43	−0.264495
$371$	1635.84	0.228918
$372$	0	0
$373$	1593.07	0.221142	0.110571	−	0.993868i	$-0.464732\pi$
$373$	1593.07	0.221142	0.110571	+	0.993868i	$0.464732\pi$
$374$	7074.32	0.978087
$375$	0	0
$376$	3622.07	0.496793
$377$	0	0
$378$	0	0
$379$	9137.56	1.23843	0.619215	−	0.785221i	$-0.287451\pi$
$379$	9137.56	1.23843	0.619215	+	0.785221i	$0.287451\pi$
$380$	−4440.08	−0.599398
$381$	0	0
$382$	9825.86	1.31606
$383$	−9551.07	−1.27425	−0.637124	−	0.770761i	$-0.719876\pi$
$383$	−9551.07	−1.27425	−0.637124	+	0.770761i	$0.719876\pi$
$384$	0	0
$385$	−4790.71	−0.634174
$386$	14157.8	1.86687
$387$	0	0
$388$	−164.213	−0.0214862
$389$	−7366.50	−0.960145	−0.480072	−	0.877229i	$-0.659390\pi$
$389$	−7366.50	−0.960145	−0.480072	+	0.877229i	$0.659390\pi$
$390$	0	0
$391$	−310.779	−0.0401964
$392$	1878.47	0.242033
$393$	0	0
$394$	16309.6	2.08545
$395$	−697.596	−0.0888604
$396$	0	0
$397$	−11696.5	−1.47866	−0.739332	−	0.673342i	$-0.764858\pi$
$397$	−11696.5	−1.47866	−0.739332	+	0.673342i	$0.764858\pi$
$398$	−17257.5	−2.17347
$399$	0	0
$400$	4140.34	0.517542
$401$	14167.6	1.76433	0.882167	−	0.470937i	$-0.156084\pi$
$401$	14167.6	1.76433	0.882167	+	0.470937i	$0.156084\pi$
$402$	0	0
$403$	0	0
$404$	9445.47	1.16319
$405$	0	0
$406$	10297.0	1.25870
$407$	−2576.22	−0.313755
$408$	0	0
$409$	2703.69	0.326868	0.163434	−	0.986554i	$-0.447743\pi$
$409$	2703.69	0.326868	0.163434	+	0.986554i	$0.447743\pi$
$410$	12738.4	1.53440
$411$	0	0
$412$	−26.6759	−0.00318988
$413$	608.011	0.0724414
$414$	0	0
$415$	−2817.17	−0.333228
$416$	0	0
$417$	0	0
$418$	−11007.0	−1.28796
$419$	−7142.52	−0.832781	−0.416390	−	0.909186i	$-0.636705\pi$
$419$	−7142.52	−0.832781	−0.416390	+	0.909186i	$0.636705\pi$
$420$	0	0
$421$	−3406.45	−0.394347	−0.197174	−	0.980369i	$-0.563176\pi$
$421$	−3406.45	−0.394347	−0.197174	+	0.980369i	$0.563176\pi$
$422$	12793.5	1.47578
$423$	0	0
$424$	−532.850	−0.0610318
$425$	−4478.50	−0.511151
$426$	0	0
$427$	14205.8	1.61000
$428$	−13323.0	−1.50466
$429$	0	0
$430$	−6640.96	−0.744780
$431$	−5172.97	−0.578128	−0.289064	−	0.957310i	$-0.593344\pi$
$431$	−5172.97	−0.578128	−0.289064	+	0.957310i	$0.593344\pi$
$432$	0	0
$433$	10955.0	1.21585	0.607924	−	0.793995i	$-0.292002\pi$
$433$	10955.0	1.21585	0.607924	+	0.793995i	$0.292002\pi$
$434$	31359.4	3.46844
$435$	0	0
$436$	4421.55	0.485674
$437$	483.542	0.0529313
$438$	0	0
$439$	11832.4	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$439$	11832.4	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$440$	1560.50	0.169077
$441$	0	0
$442$	0	0
$443$	−13479.8	−1.44570	−0.722852	−	0.691003i	$-0.757169\pi$
$443$	−13479.8	−1.44570	−0.722852	+	0.691003i	$0.757169\pi$
$444$	0	0
$445$	−6362.91	−0.677822
$446$	−7288.92	−0.773857
$447$	0	0
$448$	17272.1	1.82149
$449$	6774.34	0.712028	0.356014	−	0.934481i	$-0.384135\pi$
$449$	6774.34	0.712028	0.356014	+	0.934481i	$0.384135\pi$
$450$	0	0
$451$	17433.2	1.82017
$452$	−13792.0	−1.43522
$453$	0	0
$454$	8127.86	0.840219
$455$	0	0
$456$	0	0
$457$	−4642.36	−0.475187	−0.237594	−	0.971365i	$-0.576359\pi$
$457$	−4642.36	−0.475187	−0.237594	+	0.971365i	$0.576359\pi$
$458$	1580.51	0.161250
$459$	0	0
$460$	−363.482	−0.0368422
$461$	−2460.37	−0.248570	−0.124285	−	0.992247i	$-0.539664\pi$
$461$	−2460.37	−0.248570	−0.124285	+	0.992247i	$0.539664\pi$
$462$	0	0
$463$	4290.01	0.430613	0.215306	−	0.976547i	$-0.430925\pi$
$463$	4290.01	0.430613	0.215306	+	0.976547i	$0.430925\pi$
$464$	4612.04	0.461441
$465$	0	0
$466$	−13077.5	−1.30001
$467$	8798.99	0.871882	0.435941	−	0.899975i	$-0.356416\pi$
$467$	8798.99	0.871882	0.435941	+	0.899975i	$0.356416\pi$
$468$	0	0
$469$	24233.0	2.38588
$470$	11413.4	1.12013
$471$	0	0
$472$	−198.050	−0.0193136
$473$	−9088.54	−0.883491
$474$	0	0
$475$	6968.11	0.673092
$476$	−11749.5	−1.13138
$477$	0	0
$478$	5160.79	0.493826
$479$	10973.4	1.04673	0.523367	−	0.852107i	$-0.324676\pi$
$479$	10973.4	1.04673	0.523367	+	0.852107i	$0.324676\pi$
$480$	0	0
$481$	0	0
$482$	614.470	0.0580671
$483$	0	0
$484$	−1799.59	−0.169007
$485$	−97.5919	−0.00913694
$486$	0	0
$487$	−5209.58	−0.484740	−0.242370	−	0.970184i	$-0.577925\pi$
$487$	−5209.58	−0.484740	−0.242370	+	0.970184i	$0.577925\pi$
$488$	−4627.33	−0.429241
$489$	0	0
$490$	5919.20	0.545719
$491$	8779.22	0.806926	0.403463	−	0.914996i	$-0.367806\pi$
$491$	8779.22	0.806926	0.403463	+	0.914996i	$0.367806\pi$
$492$	0	0
$493$	−4988.73	−0.455742
$494$	0	0
$495$	0	0
$496$	14045.9	1.27153
$497$	21603.9	1.94983
$498$	0	0
$499$	−15590.1	−1.39861	−0.699305	−	0.714823i	$-0.746507\pi$
$499$	−15590.1	−1.39861	−0.699305	+	0.714823i	$0.746507\pi$
$500$	−12459.5	−1.11441
$501$	0	0
$502$	4165.49	0.370349
$503$	64.3909	0.00570785	0.00285393	−	0.999996i	$-0.499092\pi$
$503$	64.3909	0.00570785	0.00285393	+	0.999996i	$0.499092\pi$
$504$	0	0
$505$	5613.45	0.494644
$506$	−901.072	−0.0791651
$507$	0	0
$508$	−1178.26	−0.102907
$509$	−3214.15	−0.279891	−0.139946	−	0.990159i	$-0.544693\pi$
$509$	−3214.15	−0.279891	−0.139946	+	0.990159i	$0.544693\pi$
$510$	0	0
$511$	−23367.9	−2.02296
$512$	14554.7	1.25632
$513$	0	0
$514$	−12379.5	−1.06232
$515$	−15.8535	−0.00135649
$516$	0	0
$517$	15619.9	1.32875
$518$	7750.54	0.657412
$519$	0	0
$520$	0	0
$521$	3053.01	0.256727	0.128363	−	0.991727i	$-0.459028\pi$
$521$	3053.01	0.256727	0.128363	+	0.991727i	$0.459028\pi$
$522$	0	0
$523$	5096.02	0.426067	0.213034	−	0.977045i	$-0.431666\pi$
$523$	5096.02	0.426067	0.213034	+	0.977045i	$0.431666\pi$
$524$	22200.7	1.85085
$525$	0	0
$526$	−9457.92	−0.784002
$527$	−15193.1	−1.25583
$528$	0	0
$529$	−12127.4	−0.996747
$530$	−1679.05	−0.137610
$531$	0	0
$532$	18281.1	1.48983
$533$	0	0
$534$	0	0
$535$	−7917.89	−0.639851
$536$	−7893.53	−0.636098
$537$	0	0
$538$	−8139.31	−0.652250
$539$	8100.77	0.647356
$540$	0	0
$541$	7861.99	0.624793	0.312397	−	0.949952i	$-0.398868\pi$
$541$	7861.99	0.624793	0.312397	+	0.949952i	$0.398868\pi$
$542$	−15054.4	−1.19307
$543$	0	0
$544$	−12638.1	−0.996054
$545$	2627.73	0.206532
$546$	0	0
$547$	−6317.48	−0.493814	−0.246907	−	0.969039i	$-0.579414\pi$
$547$	−6317.48	−0.493814	−0.246907	+	0.969039i	$0.579414\pi$
$548$	−11147.4	−0.868965
$549$	0	0
$550$	−12984.9	−1.00669
$551$	7761.98	0.600130
$552$	0	0
$553$	2872.21	0.220866
$554$	−6074.63	−0.465860
$555$	0	0
$556$	−5869.25	−0.447683
$557$	971.234	0.0738824	0.0369412	−	0.999317i	$-0.488239\pi$
$557$	971.234	0.0738824	0.0369412	+	0.999317i	$0.488239\pi$
$558$	0	0
$559$	0	0
$560$	6455.50	0.487134
$561$	0	0
$562$	16538.7	1.24136
$563$	−9328.49	−0.698311	−0.349155	−	0.937065i	$-0.613531\pi$
$563$	−9328.49	−0.698311	−0.349155	+	0.937065i	$0.613531\pi$
$564$	0	0
$565$	−8196.59	−0.610324
$566$	−13575.4	−1.00816
$567$	0	0
$568$	−7037.12	−0.519843
$569$	−17452.2	−1.28582	−0.642911	−	0.765941i	$-0.722273\pi$
$569$	−17452.2	−1.28582	−0.642911	+	0.765941i	$0.722273\pi$
$570$	0	0
$571$	−20181.4	−1.47910	−0.739548	−	0.673103i	$-0.764961\pi$
$571$	−20181.4	−1.47910	−0.739548	+	0.673103i	$0.764961\pi$
$572$	0	0
$573$	0	0
$574$	−52447.8	−3.81381
$575$	570.436	0.0413719
$576$	0	0
$577$	6382.72	0.460513	0.230257	−	0.973130i	$-0.426043\pi$
$577$	6382.72	0.460513	0.230257	+	0.973130i	$0.426043\pi$
$578$	−10451.3	−0.752107
$579$	0	0
$580$	−5834.73	−0.417714
$581$	11599.1	0.828250
$582$	0	0
$583$	−2297.88	−0.163239
$584$	7611.72	0.539341
$585$	0	0
$586$	20715.2	1.46030
$587$	775.527	0.0545305	0.0272653	−	0.999628i	$-0.491320\pi$
$587$	775.527	0.0545305	0.0272653	+	0.999628i	$0.491320\pi$
$588$	0	0
$589$	23639.0	1.65370
$590$	−624.072	−0.0435468
$591$	0	0
$592$	3471.47	0.241008
$593$	17843.3	1.23564	0.617821	−	0.786319i	$-0.288016\pi$
$593$	17843.3	1.23564	0.617821	+	0.786319i	$0.288016\pi$
$594$	0	0
$595$	−6982.76	−0.481118
$596$	7820.33	0.537472
$597$	0	0
$598$	0	0
$599$	24373.3	1.66255	0.831274	−	0.555863i	$-0.187612\pi$
$599$	24373.3	1.66255	0.831274	+	0.555863i	$0.187612\pi$
$600$	0	0
$601$	−3526.99	−0.239383	−0.119691	−	0.992811i	$-0.538190\pi$
$601$	−3526.99	−0.239383	−0.119691	+	0.992811i	$0.538190\pi$
$602$	27342.8	1.85118
$603$	0	0
$604$	−1323.28	−0.0891446
$605$	−1069.50	−0.0718698
$606$	0	0
$607$	7991.55	0.534377	0.267189	−	0.963644i	$-0.413905\pi$
$607$	7991.55	0.534377	0.267189	+	0.963644i	$0.413905\pi$
$608$	19663.6	1.31162
$609$	0	0
$610$	−14581.1	−0.967821
$611$	0	0
$612$	0	0
$613$	−16332.2	−1.07610	−0.538051	−	0.842912i	$-0.680839\pi$
$613$	−16332.2	−1.07610	−0.538051	+	0.842912i	$0.680839\pi$
$614$	24513.8	1.61123
$615$	0	0
$616$	−6425.04	−0.420247
$617$	19353.6	1.26280	0.631401	−	0.775457i	$-0.282480\pi$
$617$	19353.6	1.26280	0.631401	+	0.775457i	$0.282480\pi$
$618$	0	0
$619$	−9982.52	−0.648193	−0.324096	−	0.946024i	$-0.605060\pi$
$619$	−9982.52	−0.648193	−0.324096	+	0.946024i	$0.605060\pi$
$620$	−17769.6	−1.15104
$621$	0	0
$622$	20764.8	1.33857
$623$	26198.0	1.68475
$624$	0	0
$625$	3928.53	0.251426
$626$	−34252.1	−2.18688
$627$	0	0
$628$	14878.7	0.945423
$629$	−3755.00	−0.238031
$630$	0	0
$631$	−575.775	−0.0363253	−0.0181626	−	0.999835i	$-0.505782\pi$
$631$	−575.775	−0.0363253	−0.0181626	+	0.999835i	$0.505782\pi$
$632$	−935.578	−0.0588850
$633$	0	0
$634$	−21763.8	−1.36333
$635$	−700.240	−0.0437609
$636$	0	0
$637$	0	0
$638$	−14464.3	−0.897566
$639$	0	0
$640$	−5734.80	−0.354200
$641$	24521.8	1.51100	0.755500	−	0.655149i	$-0.227394\pi$
$641$	24521.8	1.51100	0.755500	+	0.655149i	$0.227394\pi$
$642$	0	0
$643$	−22667.0	−1.39020	−0.695099	−	0.718914i	$-0.744640\pi$
$643$	−22667.0	−1.39020	−0.695099	+	0.718914i	$0.744640\pi$
$644$	1496.56	0.0915727
$645$	0	0
$646$	−16043.3	−0.977116
$647$	2397.45	0.145678	0.0728389	−	0.997344i	$-0.476794\pi$
$647$	2397.45	0.145678	0.0728389	+	0.997344i	$0.476794\pi$
$648$	0	0
$649$	−854.078	−0.0516572
$650$	0	0
$651$	0	0
$652$	11585.7	0.695908
$653$	−20002.1	−1.19868	−0.599342	−	0.800493i	$-0.704571\pi$
$653$	−20002.1	−1.19868	−0.599342	+	0.800493i	$0.704571\pi$
$654$	0	0
$655$	13193.9	0.787068
$656$	−23491.4	−1.39815
$657$	0	0
$658$	−46992.5	−2.78413
$659$	3517.96	0.207952	0.103976	−	0.994580i	$-0.466844\pi$
$659$	3517.96	0.207952	0.103976	+	0.994580i	$0.466844\pi$
$660$	0	0
$661$	13583.4	0.799294	0.399647	−	0.916669i	$-0.369133\pi$
$661$	13583.4	0.799294	0.399647	+	0.916669i	$0.369133\pi$
$662$	25618.2	1.50405
$663$	0	0
$664$	−3778.24	−0.220819
$665$	10864.5	0.633544
$666$	0	0
$667$	635.425	0.0368872
$668$	−14533.1	−0.841770
$669$	0	0
$670$	−24873.1	−1.43423
$671$	−19955.1	−1.14807
$672$	0	0
$673$	−10895.8	−0.624077	−0.312038	−	0.950069i	$-0.601012\pi$
$673$	−10895.8	−0.624077	−0.312038	+	0.950069i	$0.601012\pi$
$674$	15820.2	0.904113
$675$	0	0
$676$	0	0
$677$	−1449.03	−0.0822609	−0.0411305	−	0.999154i	$-0.513096\pi$
$677$	−1449.03	−0.0822609	−0.0411305	+	0.999154i	$0.513096\pi$
$678$	0	0
$679$	401.815	0.0227102
$680$	2274.52	0.128271
$681$	0	0
$682$	−44050.9	−2.47331
$683$	15366.4	0.860878	0.430439	−	0.902620i	$-0.358359\pi$
$683$	15366.4	0.860878	0.430439	+	0.902620i	$0.358359\pi$
$684$	0	0
$685$	−6624.91	−0.369525
$686$	10599.6	0.589933
$687$	0	0
$688$	12246.9	0.678644
$689$	0	0
$690$	0	0
$691$	−2019.71	−0.111191	−0.0555957	−	0.998453i	$-0.517706\pi$
$691$	−2019.71	−0.111191	−0.0555957	+	0.998453i	$0.517706\pi$
$692$	22956.0	1.26106
$693$	0	0
$694$	10651.4	0.582595
$695$	−3488.10	−0.190376
$696$	0	0
$697$	25410.0	1.38088
$698$	45010.6	2.44080
$699$	0	0
$700$	21566.3	1.16447
$701$	−28031.6	−1.51033	−0.755164	−	0.655536i	$-0.772443\pi$
$701$	−28031.6	−1.51033	−0.755164	+	0.655536i	$0.772443\pi$
$702$	0	0
$703$	5842.42	0.313444
$704$	−24262.2	−1.29889
$705$	0	0
$706$	38046.4	2.02818
$707$	−23112.3	−1.22946
$708$	0	0
$709$	19604.4	1.03845	0.519224	−	0.854638i	$-0.326221\pi$
$709$	19604.4	1.03845	0.519224	+	0.854638i	$0.326221\pi$
$710$	−22174.5	−1.17211
$711$	0	0
$712$	−8533.59	−0.449171
$713$	1935.18	0.101645
$714$	0	0
$715$	0	0
$716$	21038.8	1.09812
$717$	0	0
$718$	48011.8	2.49552
$719$	14727.4	0.763894	0.381947	−	0.924184i	$-0.375254\pi$
$719$	14727.4	0.763894	0.381947	+	0.924184i	$0.375254\pi$
$720$	0	0
$721$	65.2737	0.00337160
$722$	−4024.62	−0.207453
$723$	0	0
$724$	−24505.0	−1.25790
$725$	9156.83	0.469071
$726$	0	0
$727$	−16890.5	−0.861668	−0.430834	−	0.902431i	$-0.641781\pi$
$727$	−16890.5	−0.861668	−0.430834	+	0.902431i	$0.641781\pi$
$728$	0	0
$729$	0	0
$730$	23985.1	1.21607
$731$	−13247.1	−0.670263
$732$	0	0
$733$	12553.6	0.632578	0.316289	−	0.948663i	$-0.397563\pi$
$733$	12553.6	0.632578	0.316289	+	0.948663i	$0.397563\pi$
$734$	−58906.3	−2.96222
$735$	0	0
$736$	1609.74	0.0806194
$737$	−34040.3	−1.70135
$738$	0	0
$739$	−37874.8	−1.88531	−0.942656	−	0.333766i	$-0.891680\pi$
$739$	−37874.8	−1.88531	−0.942656	+	0.333766i	$0.891680\pi$
$740$	−4391.78	−0.218169
$741$	0	0
$742$	6913.16	0.342035
$743$	−35882.6	−1.77174	−0.885872	−	0.463929i	$-0.846439\pi$
$743$	−35882.6	−1.77174	−0.885872	+	0.463929i	$0.846439\pi$
$744$	0	0
$745$	4647.63	0.228559
$746$	6732.40	0.330416
$747$	0	0
$748$	16504.6	0.806777
$749$	32600.3	1.59037
$750$	0	0
$751$	−181.689	−0.00882815	−0.00441407	−	0.999990i	$-0.501405\pi$
$751$	−181.689	−0.00882815	−0.00441407	+	0.999990i	$0.501405\pi$
$752$	−21048.0	−1.02067
$753$	0	0
$754$	0	0
$755$	−786.425	−0.0379085
$756$	0	0
$757$	−491.053	−0.0235768	−0.0117884	−	0.999931i	$-0.503752\pi$
$757$	−491.053	−0.0235768	−0.0117884	+	0.999931i	$0.503752\pi$
$758$	38615.8	1.85038
$759$	0	0
$760$	−3538.94	−0.168909
$761$	8113.01	0.386460	0.193230	−	0.981153i	$-0.438104\pi$
$761$	8113.01	0.386460	0.193230	+	0.981153i	$0.438104\pi$
$762$	0	0
$763$	−10819.2	−0.513342
$764$	22924.1	1.08555
$765$	0	0
$766$	−40363.3	−1.90390
$767$	0	0
$768$	0	0
$769$	19864.7	0.931519	0.465759	−	0.884911i	$-0.345781\pi$
$769$	19864.7	0.931519	0.465759	+	0.884911i	$0.345781\pi$
$770$	−20245.8	−0.947542
$771$	0	0
$772$	33030.6	1.53989
$773$	−9047.42	−0.420974	−0.210487	−	0.977597i	$-0.567505\pi$
$773$	−9047.42	−0.420974	−0.210487	+	0.977597i	$0.567505\pi$
$774$	0	0
$775$	27887.0	1.29256
$776$	−130.885	−0.00605476
$777$	0	0
$778$	−31131.2	−1.43459
$779$	−39535.5	−1.81837
$780$	0	0
$781$	−30347.1	−1.39040
$782$	−1313.37	−0.0600588
$783$	0	0
$784$	−10915.8	−0.497259
$785$	8842.45	0.402039
$786$	0	0
$787$	−15018.4	−0.680240	−0.340120	−	0.940382i	$-0.610468\pi$
$787$	−15018.4	−0.680240	−0.340120	+	0.940382i	$0.610468\pi$
$788$	38050.9	1.72019
$789$	0	0
$790$	−2948.08	−0.132770
$791$	33747.8	1.51698
$792$	0	0
$793$	0	0
$794$	−49429.9	−2.20932
$795$	0	0
$796$	−40262.4	−1.79279
$797$	−31941.1	−1.41959	−0.709794	−	0.704409i	$-0.751212\pi$
$797$	−31941.1	−1.41959	−0.709794	+	0.704409i	$0.751212\pi$
$798$	0	0
$799$	22767.1	1.00806
$800$	23197.3	1.02518
$801$	0	0
$802$	59873.2	2.63615
$803$	32825.0	1.44255
$804$	0	0
$805$	889.409	0.0389411
$806$	0	0
$807$	0	0
$808$	7528.46	0.327785
$809$	−27260.1	−1.18469	−0.592344	−	0.805685i	$-0.701797\pi$
$809$	−27260.1	−1.18469	−0.592344	+	0.805685i	$0.701797\pi$
$810$	0	0
$811$	−20707.8	−0.896607	−0.448303	−	0.893881i	$-0.647972\pi$
$811$	−20707.8	−0.896607	−0.448303	+	0.893881i	$0.647972\pi$
$812$	24023.3	1.03824
$813$	0	0
$814$	−10887.2	−0.468793
$815$	6885.42	0.295933
$816$	0	0
$817$	20611.2	0.882614
$818$	11426.0	0.488385
$819$	0	0
$820$	29719.1	1.26565
$821$	−15658.6	−0.665636	−0.332818	−	0.942991i	$-0.608000\pi$
$821$	−15658.6	−0.665636	−0.332818	+	0.942991i	$0.608000\pi$
$822$	0	0
$823$	−4106.58	−0.173932	−0.0869662	−	0.996211i	$-0.527717\pi$
$823$	−4106.58	−0.173932	−0.0869662	+	0.996211i	$0.527717\pi$
$824$	−21.2619	−0.000898900 0
$825$	0	0
$826$	2569.49	0.108237
$827$	−16747.3	−0.704184	−0.352092	−	0.935965i	$-0.614530\pi$
$827$	−16747.3	−0.704184	−0.352092	+	0.935965i	$0.614530\pi$
$828$	0	0
$829$	−29157.1	−1.22155	−0.610776	−	0.791803i	$-0.709142\pi$
$829$	−29157.1	−1.22155	−0.610776	+	0.791803i	$0.709142\pi$
$830$	−11905.5	−0.497887
$831$	0	0
$832$	0	0
$833$	11807.4	0.491119
$834$	0	0
$835$	−8637.03	−0.357960
$836$	−25679.6	−1.06238
$837$	0	0
$838$	−30184.7	−1.24429
$839$	−45819.4	−1.88541	−0.942707	−	0.333621i	$-0.891729\pi$
$839$	−45819.4	−1.88541	−0.942707	+	0.333621i	$0.891729\pi$
$840$	0	0
$841$	−14188.9	−0.581776
$842$	−14395.8	−0.589209
$843$	0	0
$844$	29847.7	1.21730
$845$	0	0
$846$	0	0
$847$	4403.44	0.178635
$848$	3096.41	0.125390
$849$	0	0
$850$	−18926.4	−0.763729
$851$	478.283	0.0192660
$852$	0	0
$853$	17351.1	0.696471	0.348235	−	0.937407i	$-0.386781\pi$
$853$	17351.1	0.696471	0.348235	+	0.937407i	$0.386781\pi$
$854$	60034.7	2.40556
$855$	0	0
$856$	−10619.0	−0.424009
$857$	21768.1	0.867659	0.433829	−	0.900995i	$-0.357162\pi$
$857$	21768.1	0.867659	0.433829	+	0.900995i	$0.357162\pi$
$858$	0	0
$859$	−29878.4	−1.18677	−0.593387	−	0.804918i	$-0.702209\pi$
$859$	−29878.4	−1.18677	−0.593387	+	0.804918i	$0.702209\pi$
$860$	−15493.6	−0.614334
$861$	0	0
$862$	−21861.2	−0.863801
$863$	15067.7	0.594335	0.297168	−	0.954825i	$-0.403958\pi$
$863$	15067.7	0.594335	0.297168	+	0.954825i	$0.403958\pi$
$864$	0	0
$865$	13642.8	0.536265
$866$	46296.3	1.81664
$867$	0	0
$868$	73162.7	2.86095
$869$	−4034.62	−0.157497
$870$	0	0
$871$	0	0
$872$	3524.17	0.136862
$873$	0	0
$874$	2043.48	0.0790865
$875$	30487.4	1.17790
$876$	0	0
$877$	−21119.5	−0.813177	−0.406588	−	0.913611i	$-0.633282\pi$
$877$	−21119.5	−0.813177	−0.406588	+	0.913611i	$0.633282\pi$
$878$	50004.5	1.92206
$879$	0	0
$880$	−9068.09	−0.347370
$881$	−31652.5	−1.21044	−0.605221	−	0.796057i	$-0.706915\pi$
$881$	−31652.5	−1.21044	−0.605221	+	0.796057i	$0.706915\pi$
$882$	0	0
$883$	11701.6	0.445969	0.222984	−	0.974822i	$-0.428420\pi$
$883$	11701.6	0.445969	0.222984	+	0.974822i	$0.428420\pi$
$884$	0	0
$885$	0	0
$886$	−56966.6	−2.16008
$887$	−22507.9	−0.852020	−0.426010	−	0.904719i	$-0.640081\pi$
$887$	−22507.9	−0.852020	−0.426010	+	0.904719i	$0.640081\pi$
$888$	0	0
$889$	2883.10	0.108769
$890$	−26890.0	−1.01276
$891$	0	0
$892$	−17005.3	−0.638318
$893$	−35423.3	−1.32743
$894$	0	0
$895$	12503.4	0.466974
$896$	23611.9	0.880377
$897$	0	0
$898$	28628.7	1.06387
$899$	31064.1	1.15244
$900$	0	0
$901$	−3349.31	−0.123842
$902$	73673.8	2.71959
$903$	0	0
$904$	−10992.8	−0.404442
$905$	−14563.3	−0.534919
$906$	0	0
$907$	18718.6	0.685273	0.342636	−	0.939468i	$-0.388680\pi$
$907$	18718.6	0.685273	0.342636	+	0.939468i	$0.388680\pi$
$908$	18962.6	0.693057
$909$	0	0
$910$	0	0
$911$	−18616.7	−0.677057	−0.338529	−	0.940956i	$-0.609929\pi$
$911$	−18616.7	−0.677057	−0.338529	+	0.940956i	$0.609929\pi$
$912$	0	0
$913$	−16293.4	−0.590616
$914$	−19618.9	−0.709994
$915$	0	0
$916$	3687.39	0.133008
$917$	−54323.3	−1.95629
$918$	0	0
$919$	−54764.4	−1.96573	−0.982867	−	0.184316i	$-0.940993\pi$
$919$	−54764.4	−1.96573	−0.982867	+	0.184316i	$0.940993\pi$
$920$	−289.711	−0.0103821
$921$	0	0
$922$	−10397.6	−0.371397
$923$	0	0
$924$	0	0
$925$	6892.32	0.244993
$926$	18129.8	0.643394
$927$	0	0
$928$	25840.1	0.914055
$929$	31832.1	1.12419	0.562097	−	0.827071i	$-0.309995\pi$
$929$	31832.1	1.12419	0.562097	+	0.827071i	$0.309995\pi$
$930$	0	0
$931$	−18371.2	−0.646713
$932$	−30510.3	−1.07231
$933$	0	0
$934$	37185.0	1.30271
$935$	9808.73	0.343080
$936$	0	0
$937$	−27408.7	−0.955607	−0.477803	−	0.878467i	$-0.658567\pi$
$937$	−27408.7	−0.955607	−0.477803	+	0.878467i	$0.658567\pi$
$938$	102410.	3.56483
$939$	0	0
$940$	26627.9	0.923945
$941$	−54837.8	−1.89975	−0.949874	−	0.312634i	$-0.898789\pi$
$941$	−54837.8	−1.89975	−0.949874	+	0.312634i	$0.898789\pi$
$942$	0	0
$943$	−3236.53	−0.111767
$944$	1150.87	0.0396799
$945$	0	0
$946$	−38408.6	−1.32006
$947$	39707.8	1.36255	0.681273	−	0.732030i	$-0.261427\pi$
$947$	39707.8	1.36255	0.681273	+	0.732030i	$0.261427\pi$
$948$	0	0
$949$	0	0
$950$	29447.6	1.00569
$951$	0	0
$952$	−9364.89	−0.318821
$953$	17106.6	0.581468	0.290734	−	0.956804i	$-0.406101\pi$
$953$	17106.6	0.581468	0.290734	+	0.956804i	$0.406101\pi$
$954$	0	0
$955$	13623.8	0.461629
$956$	12040.3	0.407334
$957$	0	0
$958$	46374.0	1.56396
$959$	27276.7	0.918469
$960$	0	0
$961$	64814.4	2.17564
$962$	0	0
$963$	0	0
$964$	1433.58	0.0478968
$965$	19630.1	0.654835
$966$	0	0
$967$	−23417.5	−0.778756	−0.389378	−	0.921078i	$-0.627310\pi$
$967$	−23417.5	−0.778756	−0.389378	+	0.921078i	$0.627310\pi$
$968$	−1434.35	−0.0476258
$969$	0	0
$970$	−412.428	−0.0136518
$971$	−16430.6	−0.543032	−0.271516	−	0.962434i	$-0.587525\pi$
$971$	−16430.6	−0.543032	−0.271516	+	0.962434i	$0.587525\pi$
$972$	0	0
$973$	14361.6	0.473187
$974$	−22015.9	−0.724267
$975$	0	0
$976$	26889.5	0.881879
$977$	10554.6	0.345622	0.172811	−	0.984955i	$-0.444715\pi$
$977$	10554.6	0.345622	0.172811	+	0.984955i	$0.444715\pi$
$978$	0	0
$979$	−36800.5	−1.20138
$980$	13809.7	0.450138
$981$	0	0
$982$	37101.5	1.20566
$983$	−1534.33	−0.0497839	−0.0248919	−	0.999690i	$-0.507924\pi$
$983$	−1534.33	−0.0497839	−0.0248919	+	0.999690i	$0.507924\pi$
$984$	0	0
$985$	22613.7	0.731505
$986$	−21082.6	−0.680941
$987$	0	0
$988$	0	0
$989$	1687.31	0.0542502
$990$	0	0
$991$	−18018.2	−0.577563	−0.288782	−	0.957395i	$-0.593250\pi$
$991$	−18018.2	−0.577563	−0.288782	+	0.957395i	$0.593250\pi$
$992$	78695.7	2.51874
$993$	0	0
$994$	91299.1	2.91331
$995$	−23928.0	−0.762380
$996$	0	0
$997$	48287.7	1.53389	0.766944	−	0.641714i	$-0.221776\pi$
$997$	48287.7	1.53389	0.766944	+	0.641714i	$0.221776\pi$
$998$	−65884.4	−2.08971
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.v.1.4		4
3.2	odd	2		507.4.a.m.1.1		4
13.3	even	3		117.4.g.e.100.1		8
13.9	even	3		117.4.g.e.55.1		8
13.12	even	2		1521.4.a.bb.1.1		4
39.5	even	4		507.4.b.h.337.7		8
39.8	even	4		507.4.b.h.337.2		8
39.29	odd	6		39.4.e.c.22.4	yes	8
39.35	odd	6		39.4.e.c.16.4	✓	8
39.38	odd	2		507.4.a.i.1.4		4
156.35	even	6		624.4.q.i.289.2		8
156.107	even	6		624.4.q.i.529.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.e.c.16.4	✓	8	39.35	odd	6
39.4.e.c.22.4	yes	8	39.29	odd	6
117.4.g.e.55.1		8	13.9	even	3
117.4.g.e.100.1		8	13.3	even	3
507.4.a.i.1.4		4	39.38	odd	2
507.4.a.m.1.1		4	3.2	odd	2
507.4.b.h.337.2		8	39.8	even	4
507.4.b.h.337.7		8	39.5	even	4
624.4.q.i.289.2		8	156.35	even	6
624.4.q.i.529.2		8	156.107	even	6
1521.4.a.v.1.4		4	1.1	even	1	trivial
1521.4.a.bb.1.1		4	13.12	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +O(q^{10})\)	\(q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +24.7627 q^{10} +33.8892 q^{11} -101.955 q^{14} -45.6659 q^{16} +49.3956 q^{17} -76.8548 q^{19} +57.7723 q^{20} +143.218 q^{22} -6.29163 q^{23} -90.6659 q^{25} -237.866 q^{28} -100.995 q^{29} -307.580 q^{31} -255.854 q^{32} +208.749 q^{34} -141.364 q^{35} -76.0189 q^{37} -324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -268.184 q^{43} +334.132 q^{44} -26.5888 q^{46} +460.912 q^{47} +239.037 q^{49} -383.159 q^{50} -67.8057 q^{53} +198.575 q^{55} -189.589 q^{56} -426.812 q^{58} -25.2021 q^{59} -588.832 q^{61} -1299.85 q^{62} -715.927 q^{64} -1004.46 q^{67} +487.018 q^{68} -597.411 q^{70} -895.481 q^{71} +968.599 q^{73} -321.260 q^{74} -757.753 q^{76} -817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +2173.96 q^{82} -480.784 q^{83} +289.435 q^{85} -1133.36 q^{86} +266.318 q^{88} -1085.91 q^{89} -62.0325 q^{92} +1947.84 q^{94} -450.333 q^{95} -16.6552 q^{97} +1010.18 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8	\( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 22 * q^4 + 6 * q^5 - 14 * q^7 - 54 * q^8 - 62 * q^10 - 40 * q^11 - 40 * q^14 + 122 * q^16 - 98 * q^17 + 124 * q^19 + 466 * q^20 + 220 * q^22 - 104 * q^23 - 58 * q^25 - 144 * q^28 - 194 * q^29 + 26 * q^31 - 654 * q^32 + 1062 * q^34 - 88 * q^35 + 102 * q^37 - 332 * q^38 - 998 * q^40 + 1054 * q^41 + 450 * q^43 + 44 * q^44 - 172 * q^46 + 96 * q^47 + 1070 * q^49 - 996 * q^50 - 262 * q^53 + 204 * q^55 - 2164 * q^56 + 722 * q^58 - 308 * q^59 - 928 * q^61 - 2780 * q^62 + 1026 * q^64 - 1134 * q^67 - 1786 * q^68 - 2324 * q^70 - 1064 * q^71 + 952 * q^73 - 1158 * q^74 - 1708 * q^76 - 2508 * q^77 - 746 * q^79 + 2922 * q^80 + 1734 * q^82 + 404 * q^83 - 1394 * q^85 - 3168 * q^86 + 3060 * q^88 - 1620 * q^89 - 332 * q^92 - 772 * q^94 - 2204 * q^95 + 2166 * q^97 + 1906 * q^98

Introduction

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