LMFDB - Embedded newform 624.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(1,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("624.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.69042$ of defining polynomial
Character	$\chi$	$=$	624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +9.38083 q^{5} +24.1425 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +9.38083 q^{5} +24.1425 q^{7} +9.00000 q^{9} +11.2383 q^{11} -13.0000 q^{13} +28.1425 q^{15} +2.28499 q^{17} +79.8575 q^{19} +72.4275 q^{21} +183.047 q^{23} -37.0000 q^{25} +27.0000 q^{27} -241.617 q^{29} +100.712 q^{31} +33.7150 q^{33} +226.477 q^{35} +222.855 q^{37} -39.0000 q^{39} -164.712 q^{41} -510.565 q^{43} +84.4275 q^{45} -298.757 q^{47} +239.860 q^{49} +6.85497 q^{51} +157.430 q^{53} +105.425 q^{55} +239.573 q^{57} +50.7667 q^{59} +41.4300 q^{61} +217.282 q^{63} -121.951 q^{65} +517.852 q^{67} +549.140 q^{69} +883.228 q^{71} +653.715 q^{73} -111.000 q^{75} +271.321 q^{77} -810.850 q^{79} +81.0000 q^{81} -505.430 q^{83} +21.4351 q^{85} -724.850 q^{87} -117.008 q^{89} -313.852 q^{91} +302.137 q^{93} +749.130 q^{95} +583.425 q^{97} +101.145 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 8 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9} + 60 q^{11} - 26 q^{13} - 108 q^{17} + 216 q^{19} - 24 q^{21} + 216 q^{23} - 74 q^{25} + 54 q^{27} - 108 q^{29} - 80 q^{31} + 180 q^{33} + 528 q^{35} + 108 q^{37} - 78 q^{39} - 48 q^{41} - 8 q^{43} + 228 q^{47} + 930 q^{49} - 324 q^{51} + 540 q^{53} - 352 q^{55} + 648 q^{57} + 852 q^{59} + 308 q^{61} - 72 q^{63} + 304 q^{67} + 648 q^{69} + 228 q^{71} + 1420 q^{73} - 222 q^{75} - 1296 q^{77} - 496 q^{79} + 162 q^{81} - 1236 q^{83} + 1056 q^{85} - 324 q^{87} - 1416 q^{89} + 104 q^{91} - 240 q^{93} - 528 q^{95} + 604 q^{97} + 540 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 8 * q^7 + 18 * q^9 + 60 * q^11 - 26 * q^13 - 108 * q^17 + 216 * q^19 - 24 * q^21 + 216 * q^23 - 74 * q^25 + 54 * q^27 - 108 * q^29 - 80 * q^31 + 180 * q^33 + 528 * q^35 + 108 * q^37 - 78 * q^39 - 48 * q^41 - 8 * q^43 + 228 * q^47 + 930 * q^49 - 324 * q^51 + 540 * q^53 - 352 * q^55 + 648 * q^57 + 852 * q^59 + 308 * q^61 - 72 * q^63 + 304 * q^67 + 648 * q^69 + 228 * q^71 + 1420 * q^73 - 222 * q^75 - 1296 * q^77 - 496 * q^79 + 162 * q^81 - 1236 * q^83 + 1056 * q^85 - 324 * q^87 - 1416 * q^89 + 104 * q^91 - 240 * q^93 - 528 * q^95 + 604 * q^97 + 540 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	9.38083	0.839047	0.419524	−	0.907744i	$-0.362197\pi$
$5$	9.38083	0.839047	0.419524	+	0.907744i	$0.362197\pi$
$6$	0	0
$7$	24.1425	1.30357	0.651786	−	0.758403i	$-0.274020\pi$
$7$	24.1425	1.30357	0.651786	+	0.758403i	$0.274020\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	11.2383	0.308044	0.154022	−	0.988067i	$-0.450777\pi$
$11$	11.2383	0.308044	0.154022	+	0.988067i	$0.450777\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	28.1425	0.484424
$16$	0	0
$17$	2.28499	0.0325995	0.0162997	−	0.999867i	$-0.494811\pi$
$17$	2.28499	0.0325995	0.0162997	+	0.999867i	$0.494811\pi$
$18$	0	0
$19$	79.8575	0.964240	0.482120	−	0.876105i	$-0.339867\pi$
$19$	79.8575	0.964240	0.482120	+	0.876105i	$0.339867\pi$
$20$	0	0
$21$	72.4275	0.752618
$22$	0	0
$23$	183.047	1.65947	0.829736	−	0.558156i	$-0.188491\pi$
$23$	183.047	1.65947	0.829736	+	0.558156i	$0.188491\pi$
$24$	0	0
$25$	−37.0000	−0.296000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−241.617	−1.54714	−0.773570	−	0.633710i	$-0.781531\pi$
$29$	−241.617	−1.54714	−0.773570	+	0.633710i	$0.781531\pi$
$30$	0	0
$31$	100.712	0.583500	0.291750	−	0.956495i	$-0.405763\pi$
$31$	100.712	0.583500	0.291750	+	0.956495i	$0.405763\pi$
$32$	0	0
$33$	33.7150	0.177849
$34$	0	0
$35$	226.477	1.09376
$36$	0	0
$37$	222.855	0.990193	0.495096	−	0.868838i	$-0.335133\pi$
$37$	222.855	0.990193	0.495096	+	0.868838i	$0.335133\pi$
$38$	0	0
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−164.712	−0.627409	−0.313704	−	0.949521i	$-0.601570\pi$
$41$	−164.712	−0.627409	−0.313704	+	0.949521i	$0.601570\pi$
$42$	0	0
$43$	−510.565	−1.81071	−0.905354	−	0.424658i	$-0.860394\pi$
$43$	−510.565	−1.81071	−0.905354	+	0.424658i	$0.860394\pi$
$44$	0	0
$45$	84.4275	0.279682
$46$	0	0
$47$	−298.757	−0.927194	−0.463597	−	0.886046i	$-0.653441\pi$
$47$	−298.757	−0.927194	−0.463597	+	0.886046i	$0.653441\pi$
$48$	0	0
$49$	239.860	0.699300
$50$	0	0
$51$	6.85497	0.0188213
$52$	0	0
$53$	157.430	0.408013	0.204006	−	0.978970i	$-0.434604\pi$
$53$	157.430	0.408013	0.204006	+	0.978970i	$0.434604\pi$
$54$	0	0
$55$	105.425	0.258464
$56$	0	0
$57$	239.573	0.556704
$58$	0	0
$59$	50.7667	0.112021	0.0560107	−	0.998430i	$-0.482162\pi$
$59$	50.7667	0.112021	0.0560107	+	0.998430i	$0.482162\pi$
$60$	0	0
$61$	41.4300	0.0869602	0.0434801	−	0.999054i	$-0.486155\pi$
$61$	41.4300	0.0869602	0.0434801	+	0.999054i	$0.486155\pi$
$62$	0	0
$63$	217.282	0.434524
$64$	0	0
$65$	−121.951	−0.232710
$66$	0	0
$67$	517.852	0.944265	0.472132	−	0.881528i	$-0.343484\pi$
$67$	517.852	0.944265	0.472132	+	0.881528i	$0.343484\pi$
$68$	0	0
$69$	549.140	0.958097
$70$	0	0
$71$	883.228	1.47634	0.738168	−	0.674617i	$-0.235691\pi$
$71$	883.228	1.47634	0.738168	+	0.674617i	$0.235691\pi$
$72$	0	0
$73$	653.715	1.04810	0.524052	−	0.851686i	$-0.324420\pi$
$73$	653.715	1.04810	0.524052	+	0.851686i	$0.324420\pi$
$74$	0	0
$75$	−111.000	−0.170896
$76$	0	0
$77$	271.321	0.401558
$78$	0	0
$79$	−810.850	−1.15478	−0.577391	−	0.816468i	$-0.695929\pi$
$79$	−810.850	−1.15478	−0.577391	+	0.816468i	$0.695929\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−505.430	−0.668411	−0.334206	−	0.942500i	$-0.608468\pi$
$83$	−505.430	−0.668411	−0.334206	+	0.942500i	$0.608468\pi$
$84$	0	0
$85$	21.4351	0.0273525
$86$	0	0
$87$	−724.850	−0.893242
$88$	0	0
$89$	−117.008	−0.139357	−0.0696786	−	0.997569i	$-0.522197\pi$
$89$	−117.008	−0.139357	−0.0696786	+	0.997569i	$0.522197\pi$
$90$	0	0
$91$	−313.852	−0.361546
$92$	0	0
$93$	302.137	0.336884
$94$	0	0
$95$	749.130	0.809043
$96$	0	0
$97$	583.425	0.610699	0.305350	−	0.952240i	$-0.401227\pi$
$97$	583.425	0.610699	0.305350	+	0.952240i	$0.401227\pi$
$98$	0	0
$99$	101.145	0.102681
$100$	0	0
$101$	213.902	0.210733	0.105366	−	0.994433i	$-0.466398\pi$
$101$	213.902	0.210733	0.105366	+	0.994433i	$0.466398\pi$
$102$	0	0
$103$	647.715	0.619624	0.309812	−	0.950798i	$-0.399734\pi$
$103$	647.715	0.619624	0.309812	+	0.950798i	$0.399734\pi$
$104$	0	0
$105$	679.430	0.631482
$106$	0	0
$107$	1448.18	1.30842	0.654208	−	0.756315i	$-0.273002\pi$
$107$	1448.18	1.30842	0.654208	+	0.756315i	$0.273002\pi$
$108$	0	0
$109$	833.430	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$109$	833.430	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$110$	0	0
$111$	668.565	0.571688
$112$	0	0
$113$	−881.036	−0.733460	−0.366730	−	0.930327i	$-0.619523\pi$
$113$	−881.036	−0.733460	−0.366730	+	0.930327i	$0.619523\pi$
$114$	0	0
$115$	1717.13	1.39238
$116$	0	0
$117$	−117.000	−0.0924500
$118$	0	0
$119$	55.1653	0.0424958
$120$	0	0
$121$	−1204.70	−0.905109
$122$	0	0
$123$	−494.137	−0.362235
$124$	0	0
$125$	−1519.69	−1.08741
$126$	0	0
$127$	−1967.69	−1.37484	−0.687420	−	0.726260i	$-0.741257\pi$
$127$	−1967.69	−1.37484	−0.687420	+	0.726260i	$0.741257\pi$
$128$	0	0
$129$	−1531.69	−1.04541
$130$	0	0
$131$	−873.513	−0.582589	−0.291295	−	0.956633i	$-0.594086\pi$
$131$	−873.513	−0.582589	−0.291295	+	0.956633i	$0.594086\pi$
$132$	0	0
$133$	1927.96	1.25696
$134$	0	0
$135$	253.282	0.161475
$136$	0	0
$137$	2662.69	1.66050	0.830251	−	0.557390i	$-0.188197\pi$
$137$	2662.69	1.66050	0.830251	+	0.557390i	$0.188197\pi$
$138$	0	0
$139$	3189.71	1.94639	0.973193	−	0.229990i	$-0.0738693\pi$
$139$	3189.71	1.94639	0.973193	+	0.229990i	$0.0738693\pi$
$140$	0	0
$141$	−896.270	−0.535316
$142$	0	0
$143$	−146.098	−0.0854361
$144$	0	0
$145$	−2266.56	−1.29812
$146$	0	0
$147$	719.580	0.403741
$148$	0	0
$149$	−2691.85	−1.48003	−0.740016	−	0.672589i	$-0.765182\pi$
$149$	−2691.85	−1.48003	−0.740016	+	0.672589i	$0.765182\pi$
$150$	0	0
$151$	398.137	0.214569	0.107285	−	0.994228i	$-0.465784\pi$
$151$	398.137	0.214569	0.107285	+	0.994228i	$0.465784\pi$
$152$	0	0
$153$	20.5649	0.0108665
$154$	0	0
$155$	944.767	0.489584
$156$	0	0
$157$	457.440	0.232533	0.116267	−	0.993218i	$-0.462907\pi$
$157$	457.440	0.232533	0.116267	+	0.993218i	$0.462907\pi$
$158$	0	0
$159$	472.290	0.235566
$160$	0	0
$161$	4419.20	2.16324
$162$	0	0
$163$	1818.40	0.873793	0.436896	−	0.899512i	$-0.356078\pi$
$163$	1818.40	0.873793	0.436896	+	0.899512i	$0.356078\pi$
$164$	0	0
$165$	316.275	0.149224
$166$	0	0
$167$	−1296.87	−0.600927	−0.300464	−	0.953793i	$-0.597141\pi$
$167$	−1296.87	−0.600927	−0.300464	+	0.953793i	$0.597141\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	718.718	0.321413
$172$	0	0
$173$	−4199.58	−1.84560	−0.922798	−	0.385283i	$-0.874104\pi$
$173$	−4199.58	−1.84560	−0.922798	+	0.385283i	$0.874104\pi$
$174$	0	0
$175$	−893.272	−0.385857
$176$	0	0
$177$	152.300	0.0646756
$178$	0	0
$179$	210.840	0.0880386	0.0440193	−	0.999031i	$-0.485984\pi$
$179$	210.840	0.0880386	0.0440193	+	0.999031i	$0.485984\pi$
$180$	0	0
$181$	−1227.14	−0.503937	−0.251968	−	0.967735i	$-0.581078\pi$
$181$	−1227.14	−0.503937	−0.251968	+	0.967735i	$0.581078\pi$
$182$	0	0
$183$	124.290	0.0502065
$184$	0	0
$185$	2090.56	0.830818
$186$	0	0
$187$	25.6795	0.0100421
$188$	0	0
$189$	651.847	0.250873
$190$	0	0
$191$	−3019.03	−1.14371	−0.571856	−	0.820354i	$-0.693776\pi$
$191$	−3019.03	−1.14371	−0.571856	+	0.820354i	$0.693776\pi$
$192$	0	0
$193$	3431.11	1.27967	0.639836	−	0.768512i	$-0.279002\pi$
$193$	3431.11	1.27967	0.639836	+	0.768512i	$0.279002\pi$
$194$	0	0
$195$	−365.852	−0.134355
$196$	0	0
$197$	−1528.70	−0.552871	−0.276435	−	0.961033i	$-0.589153\pi$
$197$	−1528.70	−0.552871	−0.276435	+	0.961033i	$0.589153\pi$
$198$	0	0
$199$	−85.3945	−0.0304194	−0.0152097	−	0.999884i	$-0.504842\pi$
$199$	−85.3945	−0.0304194	−0.0152097	+	0.999884i	$0.504842\pi$
$200$	0	0
$201$	1553.56	0.545172
$202$	0	0
$203$	−5833.23	−2.01681
$204$	0	0
$205$	−1545.14	−0.526426
$206$	0	0
$207$	1647.42	0.553157
$208$	0	0
$209$	897.466	0.297029
$210$	0	0
$211$	612.020	0.199683	0.0998417	−	0.995003i	$-0.468166\pi$
$211$	612.020	0.199683	0.0998417	+	0.995003i	$0.468166\pi$
$212$	0	0
$213$	2649.68	0.852363
$214$	0	0
$215$	−4789.52	−1.51927
$216$	0	0
$217$	2431.45	0.760634
$218$	0	0
$219$	1961.15	0.605123
$220$	0	0
$221$	−29.7049	−0.00904147
$222$	0	0
$223$	5460.10	1.63962	0.819810	−	0.572635i	$-0.194079\pi$
$223$	5460.10	1.63962	0.819810	+	0.572635i	$0.194079\pi$
$224$	0	0
$225$	−333.000	−0.0986667
$226$	0	0
$227$	−4882.27	−1.42752	−0.713762	−	0.700388i	$-0.753010\pi$
$227$	−4882.27	−1.42752	−0.713762	+	0.700388i	$0.753010\pi$
$228$	0	0
$229$	−1486.57	−0.428975	−0.214488	−	0.976727i	$-0.568808\pi$
$229$	−1486.57	−0.428975	−0.214488	+	0.976727i	$0.568808\pi$
$230$	0	0
$231$	813.964	0.231840
$232$	0	0
$233$	−3638.10	−1.02292	−0.511459	−	0.859308i	$-0.670895\pi$
$233$	−3638.10	−1.02292	−0.511459	+	0.859308i	$0.670895\pi$
$234$	0	0
$235$	−2802.59	−0.777960
$236$	0	0
$237$	−2432.55	−0.666713
$238$	0	0
$239$	−4745.70	−1.28441	−0.642205	−	0.766533i	$-0.721980\pi$
$239$	−4745.70	−1.28441	−0.642205	+	0.766533i	$0.721980\pi$
$240$	0	0
$241$	3907.41	1.04439	0.522197	−	0.852825i	$-0.325113\pi$
$241$	3907.41	1.04439	0.522197	+	0.852825i	$0.325113\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	2250.09	0.586746
$246$	0	0
$247$	−1038.15	−0.267432
$248$	0	0
$249$	−1516.29	−0.385908
$250$	0	0
$251$	−4033.65	−1.01435	−0.507175	−	0.861843i	$-0.669310\pi$
$251$	−4033.65	−1.01435	−0.507175	+	0.861843i	$0.669310\pi$
$252$	0	0
$253$	2057.14	0.511191
$254$	0	0
$255$	64.3053	0.0157920
$256$	0	0
$257$	−3751.41	−0.910532	−0.455266	−	0.890355i	$-0.650456\pi$
$257$	−3751.41	−0.910532	−0.455266	+	0.890355i	$0.650456\pi$
$258$	0	0
$259$	5380.27	1.29079
$260$	0	0
$261$	−2174.55	−0.515714
$262$	0	0
$263$	2859.39	0.670409	0.335204	−	0.942145i	$-0.391195\pi$
$263$	2859.39	0.670409	0.335204	+	0.942145i	$0.391195\pi$
$264$	0	0
$265$	1476.82	0.342342
$266$	0	0
$267$	−351.023	−0.0804579
$268$	0	0
$269$	−6512.83	−1.47619	−0.738093	−	0.674699i	$-0.764274\pi$
$269$	−6512.83	−1.47619	−0.738093	+	0.674699i	$0.764274\pi$
$270$	0	0
$271$	−5396.38	−1.20962	−0.604809	−	0.796370i	$-0.706751\pi$
$271$	−5396.38	−1.20962	−0.604809	+	0.796370i	$0.706751\pi$
$272$	0	0
$273$	−941.557	−0.208739
$274$	0	0
$275$	−415.818	−0.0911811
$276$	0	0
$277$	−5601.98	−1.21513	−0.607564	−	0.794271i	$-0.707853\pi$
$277$	−5601.98	−1.21513	−0.607564	+	0.794271i	$0.707853\pi$
$278$	0	0
$279$	906.412	0.194500
$280$	0	0
$281$	5920.59	1.25691	0.628457	−	0.777844i	$-0.283687\pi$
$281$	5920.59	1.25691	0.628457	+	0.777844i	$0.283687\pi$
$282$	0	0
$283$	−1318.54	−0.276959	−0.138479	−	0.990365i	$-0.544221\pi$
$283$	−1318.54	−0.276959	−0.138479	+	0.990365i	$0.544221\pi$
$284$	0	0
$285$	2247.39	0.467101
$286$	0	0
$287$	−3976.57	−0.817873
$288$	0	0
$289$	−4907.78	−0.998937
$290$	0	0
$291$	1750.27	0.352587
$292$	0	0
$293$	−3009.35	−0.600028	−0.300014	−	0.953935i	$-0.596991\pi$
$293$	−3009.35	−0.600028	−0.300014	+	0.953935i	$0.596991\pi$
$294$	0	0
$295$	476.234	0.0939913
$296$	0	0
$297$	303.435	0.0592831
$298$	0	0
$299$	−2379.61	−0.460255
$300$	0	0
$301$	−12326.3	−2.36039
$302$	0	0
$303$	641.705	0.121667
$304$	0	0
$305$	388.648	0.0729637
$306$	0	0
$307$	−6763.51	−1.25737	−0.628687	−	0.777658i	$-0.716407\pi$
$307$	−6763.51	−1.25737	−0.628687	+	0.777658i	$0.716407\pi$
$308$	0	0
$309$	1943.15	0.357740
$310$	0	0
$311$	9902.24	1.80548	0.902740	−	0.430186i	$-0.141552\pi$
$311$	9902.24	1.80548	0.902740	+	0.430186i	$0.141552\pi$
$312$	0	0
$313$	5065.38	0.914735	0.457368	−	0.889278i	$-0.348792\pi$
$313$	5065.38	0.914735	0.457368	+	0.889278i	$0.348792\pi$
$314$	0	0
$315$	2038.29	0.364586
$316$	0	0
$317$	−8972.06	−1.58966	−0.794828	−	0.606834i	$-0.792439\pi$
$317$	−8972.06	−1.58966	−0.794828	+	0.606834i	$0.792439\pi$
$318$	0	0
$319$	−2715.37	−0.476588
$320$	0	0
$321$	4344.53	0.755414
$322$	0	0
$323$	182.474	0.0314337
$324$	0	0
$325$	481.000	0.0820956
$326$	0	0
$327$	2500.29	0.422833
$328$	0	0
$329$	−7212.73	−1.20866
$330$	0	0
$331$	−2891.88	−0.480217	−0.240109	−	0.970746i	$-0.577183\pi$
$331$	−2891.88	−0.480217	−0.240109	+	0.970746i	$0.577183\pi$
$332$	0	0
$333$	2005.69	0.330064
$334$	0	0
$335$	4857.89	0.792283
$336$	0	0
$337$	4973.39	0.803910	0.401955	−	0.915659i	$-0.368331\pi$
$337$	4973.39	0.803910	0.401955	+	0.915659i	$0.368331\pi$
$338$	0	0
$339$	−2643.11	−0.423463
$340$	0	0
$341$	1131.84	0.179744
$342$	0	0
$343$	−2490.06	−0.391984
$344$	0	0
$345$	5151.39	0.803888
$346$	0	0
$347$	−9022.73	−1.39587	−0.697933	−	0.716163i	$-0.745897\pi$
$347$	−9022.73	−1.39587	−0.697933	+	0.716163i	$0.745897\pi$
$348$	0	0
$349$	−10793.7	−1.65550	−0.827752	−	0.561094i	$-0.810381\pi$
$349$	−10793.7	−1.65550	−0.827752	+	0.561094i	$0.810381\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	0	0
$353$	−8322.79	−1.25489	−0.627446	−	0.778660i	$-0.715900\pi$
$353$	−8322.79	−1.25489	−0.627446	+	0.778660i	$0.715900\pi$
$354$	0	0
$355$	8285.41	1.23872
$356$	0	0
$357$	165.496	0.0245350
$358$	0	0
$359$	5156.83	0.758126	0.379063	−	0.925371i	$-0.376246\pi$
$359$	5156.83	0.758126	0.379063	+	0.925371i	$0.376246\pi$
$360$	0	0
$361$	−481.779	−0.0702404
$362$	0	0
$363$	−3614.10	−0.522565
$364$	0	0
$365$	6132.39	0.879408
$366$	0	0
$367$	8532.58	1.21362	0.606808	−	0.794848i	$-0.292450\pi$
$367$	8532.58	1.21362	0.606808	+	0.794848i	$0.292450\pi$
$368$	0	0
$369$	−1482.41	−0.209136
$370$	0	0
$371$	3800.75	0.531874
$372$	0	0
$373$	−1830.02	−0.254034	−0.127017	−	0.991901i	$-0.540540\pi$
$373$	−1830.02	−0.254034	−0.127017	+	0.991901i	$0.540540\pi$
$374$	0	0
$375$	−4559.08	−0.627814
$376$	0	0
$377$	3141.02	0.429100
$378$	0	0
$379$	−2376.71	−0.322120	−0.161060	−	0.986945i	$-0.551491\pi$
$379$	−2376.71	−0.322120	−0.161060	+	0.986945i	$0.551491\pi$
$380$	0	0
$381$	−5903.08	−0.793764
$382$	0	0
$383$	8960.30	1.19543	0.597715	−	0.801708i	$-0.296075\pi$
$383$	8960.30	1.19543	0.597715	+	0.801708i	$0.296075\pi$
$384$	0	0
$385$	2545.22	0.336926
$386$	0	0
$387$	−4595.08	−0.603569
$388$	0	0
$389$	−10962.3	−1.42881	−0.714407	−	0.699730i	$-0.753304\pi$
$389$	−10962.3	−1.42881	−0.714407	+	0.699730i	$0.753304\pi$
$390$	0	0
$391$	418.260	0.0540979
$392$	0	0
$393$	−2620.54	−0.336358
$394$	0	0
$395$	−7606.45	−0.968916
$396$	0	0
$397$	11379.4	1.43858	0.719290	−	0.694710i	$-0.244467\pi$
$397$	11379.4	1.43858	0.719290	+	0.694710i	$0.244467\pi$
$398$	0	0
$399$	5783.88	0.725704
$400$	0	0
$401$	−5236.32	−0.652093	−0.326046	−	0.945354i	$-0.605717\pi$
$401$	−5236.32	−0.652093	−0.326046	+	0.945354i	$0.605717\pi$
$402$	0	0
$403$	−1309.26	−0.161834
$404$	0	0
$405$	759.847	0.0932275
$406$	0	0
$407$	2504.52	0.305023
$408$	0	0
$409$	−4296.49	−0.519433	−0.259716	−	0.965685i	$-0.583629\pi$
$409$	−4296.49	−0.519433	−0.259716	+	0.965685i	$0.583629\pi$
$410$	0	0
$411$	7988.06	0.958691
$412$	0	0
$413$	1225.64	0.146028
$414$	0	0
$415$	−4741.35	−0.560829
$416$	0	0
$417$	9569.13	1.12375
$418$	0	0
$419$	−3753.08	−0.437589	−0.218795	−	0.975771i	$-0.570212\pi$
$419$	−3753.08	−0.437589	−0.218795	+	0.975771i	$0.570212\pi$
$420$	0	0
$421$	9944.23	1.15119	0.575596	−	0.817734i	$-0.304770\pi$
$421$	9944.23	1.15119	0.575596	+	0.817734i	$0.304770\pi$
$422$	0	0
$423$	−2688.81	−0.309065
$424$	0	0
$425$	−84.5446	−0.00964945
$426$	0	0
$427$	1000.22	0.113359
$428$	0	0
$429$	−438.295	−0.0493265
$430$	0	0
$431$	−8165.32	−0.912551	−0.456276	−	0.889839i	$-0.650817\pi$
$431$	−8165.32	−0.912551	−0.456276	+	0.889839i	$0.650817\pi$
$432$	0	0
$433$	5177.99	0.574684	0.287342	−	0.957828i	$-0.407228\pi$
$433$	5177.99	0.574684	0.287342	+	0.957828i	$0.407228\pi$
$434$	0	0
$435$	−6799.69	−0.749472
$436$	0	0
$437$	14617.6	1.60013
$438$	0	0
$439$	−11140.7	−1.21120	−0.605601	−	0.795768i	$-0.707067\pi$
$439$	−11140.7	−1.21120	−0.605601	+	0.795768i	$0.707067\pi$
$440$	0	0
$441$	2158.74	0.233100
$442$	0	0
$443$	−182.270	−0.0195483	−0.00977415	−	0.999952i	$-0.503111\pi$
$443$	−182.270	−0.0195483	−0.00977415	+	0.999952i	$0.503111\pi$
$444$	0	0
$445$	−1097.63	−0.116927
$446$	0	0
$447$	−8075.54	−0.854497
$448$	0	0
$449$	2978.68	0.313079	0.156540	−	0.987672i	$-0.449966\pi$
$449$	2978.68	0.313079	0.156540	+	0.987672i	$0.449966\pi$
$450$	0	0
$451$	−1851.09	−0.193270
$452$	0	0
$453$	1194.41	0.123882
$454$	0	0
$455$	−2944.20	−0.303354
$456$	0	0
$457$	14782.7	1.51314	0.756569	−	0.653914i	$-0.226874\pi$
$457$	14782.7	1.51314	0.756569	+	0.653914i	$0.226874\pi$
$458$	0	0
$459$	61.6947	0.00627377
$460$	0	0
$461$	−9223.03	−0.931799	−0.465900	−	0.884838i	$-0.654269\pi$
$461$	−9223.03	−0.931799	−0.465900	+	0.884838i	$0.654269\pi$
$462$	0	0
$463$	5109.25	0.512845	0.256422	−	0.966565i	$-0.417456\pi$
$463$	5109.25	0.512845	0.256422	+	0.966565i	$0.417456\pi$
$464$	0	0
$465$	2834.30	0.282661
$466$	0	0
$467$	19580.6	1.94021	0.970107	−	0.242677i	$-0.0780257\pi$
$467$	19580.6	1.94021	0.970107	+	0.242677i	$0.0780257\pi$
$468$	0	0
$469$	12502.2	1.23092
$470$	0	0
$471$	1372.32	0.134253
$472$	0	0
$473$	−5737.90	−0.557778
$474$	0	0
$475$	−2954.73	−0.285415
$476$	0	0
$477$	1416.87	0.136004
$478$	0	0
$479$	1739.46	0.165924	0.0829622	−	0.996553i	$-0.473562\pi$
$479$	1739.46	0.165924	0.0829622	+	0.996553i	$0.473562\pi$
$480$	0	0
$481$	−2897.11	−0.274630
$482$	0	0
$483$	13257.6	1.24895
$484$	0	0
$485$	5473.01	0.512405
$486$	0	0
$487$	−15150.7	−1.40975	−0.704873	−	0.709334i	$-0.748996\pi$
$487$	−15150.7	−1.40975	−0.704873	+	0.709334i	$0.748996\pi$
$488$	0	0
$489$	5455.21	0.504485
$490$	0	0
$491$	2627.16	0.241471	0.120735	−	0.992685i	$-0.461475\pi$
$491$	2627.16	0.241471	0.120735	+	0.992685i	$0.461475\pi$
$492$	0	0
$493$	−552.091	−0.0504360
$494$	0	0
$495$	948.825	0.0861545
$496$	0	0
$497$	21323.3	1.92451
$498$	0	0
$499$	7307.08	0.655531	0.327766	−	0.944759i	$-0.393704\pi$
$499$	7307.08	0.655531	0.327766	+	0.944759i	$0.393704\pi$
$500$	0	0
$501$	−3890.61	−0.346945
$502$	0	0
$503$	18250.0	1.61775	0.808875	−	0.587981i	$-0.200077\pi$
$503$	18250.0	1.61775	0.808875	+	0.587981i	$0.200077\pi$
$504$	0	0
$505$	2006.58	0.176815
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−12702.1	−1.10611	−0.553056	−	0.833144i	$-0.686538\pi$
$509$	−12702.1	−1.10611	−0.553056	+	0.833144i	$0.686538\pi$
$510$	0	0
$511$	15782.3	1.36628
$512$	0	0
$513$	2156.15	0.185568
$514$	0	0
$515$	6076.11	0.519894
$516$	0	0
$517$	−3357.53	−0.285617
$518$	0	0
$519$	−12598.7	−1.06556
$520$	0	0
$521$	−7027.65	−0.590954	−0.295477	−	0.955350i	$-0.595479\pi$
$521$	−7027.65	−0.590954	−0.295477	+	0.955350i	$0.595479\pi$
$522$	0	0
$523$	2774.95	0.232008	0.116004	−	0.993249i	$-0.462991\pi$
$523$	2774.95	0.232008	0.116004	+	0.993249i	$0.462991\pi$
$524$	0	0
$525$	−2679.82	−0.222775
$526$	0	0
$527$	230.127	0.0190218
$528$	0	0
$529$	21339.1	1.75385
$530$	0	0
$531$	456.901	0.0373405
$532$	0	0
$533$	2141.26	0.174012
$534$	0	0
$535$	13585.1	1.09782
$536$	0	0
$537$	632.519	0.0508291
$538$	0	0
$539$	2695.63	0.215415
$540$	0	0
$541$	2462.50	0.195695	0.0978476	−	0.995201i	$-0.468804\pi$
$541$	2462.50	0.195695	0.0978476	+	0.995201i	$0.468804\pi$
$542$	0	0
$543$	−3681.42	−0.290948
$544$	0	0
$545$	7818.27	0.614491
$546$	0	0
$547$	8329.07	0.651052	0.325526	−	0.945533i	$-0.394459\pi$
$547$	8329.07	0.651052	0.325526	+	0.945533i	$0.394459\pi$
$548$	0	0
$549$	372.870	0.0289867
$550$	0	0
$551$	−19294.9	−1.49182
$552$	0	0
$553$	−19575.9	−1.50534
$554$	0	0
$555$	6271.69	0.479673
$556$	0	0
$557$	17466.8	1.32871	0.664355	−	0.747417i	$-0.268706\pi$
$557$	17466.8	1.32871	0.664355	+	0.747417i	$0.268706\pi$
$558$	0	0
$559$	6637.34	0.502200
$560$	0	0
$561$	77.0384	0.00579780
$562$	0	0
$563$	−9209.44	−0.689399	−0.344700	−	0.938713i	$-0.612019\pi$
$563$	−9209.44	−0.689399	−0.344700	+	0.938713i	$0.612019\pi$
$564$	0	0
$565$	−8264.85	−0.615407
$566$	0	0
$567$	1955.54	0.144841
$568$	0	0
$569$	−14004.4	−1.03180	−0.515900	−	0.856649i	$-0.672542\pi$
$569$	−14004.4	−1.03180	−0.515900	+	0.856649i	$0.672542\pi$
$570$	0	0
$571$	24578.9	1.80139	0.900695	−	0.434451i	$-0.143058\pi$
$571$	24578.9	1.80139	0.900695	+	0.434451i	$0.143058\pi$
$572$	0	0
$573$	−9057.08	−0.660323
$574$	0	0
$575$	−6772.73	−0.491204
$576$	0	0
$577$	19354.4	1.39642	0.698209	−	0.715894i	$-0.253980\pi$
$577$	19354.4	1.39642	0.698209	+	0.715894i	$0.253980\pi$
$578$	0	0
$579$	10293.3	0.738819
$580$	0	0
$581$	−12202.3	−0.871323
$582$	0	0
$583$	1769.25	0.125686
$584$	0	0
$585$	−1097.56	−0.0775699
$586$	0	0
$587$	−21468.1	−1.50951	−0.754757	−	0.656005i	$-0.772245\pi$
$587$	−21468.1	−1.50951	−0.754757	+	0.656005i	$0.772245\pi$
$588$	0	0
$589$	8042.65	0.562634
$590$	0	0
$591$	−4586.11	−0.319200
$592$	0	0
$593$	10724.0	0.742634	0.371317	−	0.928506i	$-0.378906\pi$
$593$	10724.0	0.742634	0.371317	+	0.928506i	$0.378906\pi$
$594$	0	0
$595$	517.497	0.0356560
$596$	0	0
$597$	−256.183	−0.0175626
$598$	0	0
$599$	6008.83	0.409873	0.204937	−	0.978775i	$-0.434301\pi$
$599$	6008.83	0.409873	0.204937	+	0.978775i	$0.434301\pi$
$600$	0	0
$601$	−14460.2	−0.981437	−0.490719	−	0.871318i	$-0.663266\pi$
$601$	−14460.2	−0.981437	−0.490719	+	0.871318i	$0.663266\pi$
$602$	0	0
$603$	4660.67	0.314755
$604$	0	0
$605$	−11301.1	−0.759429
$606$	0	0
$607$	16674.9	1.11501	0.557506	−	0.830173i	$-0.311758\pi$
$607$	16674.9	1.11501	0.557506	+	0.830173i	$0.311758\pi$
$608$	0	0
$609$	−17499.7	−1.16441
$610$	0	0
$611$	3883.84	0.257157
$612$	0	0
$613$	−22692.3	−1.49516	−0.747579	−	0.664173i	$-0.768784\pi$
$613$	−22692.3	−1.49516	−0.747579	+	0.664173i	$0.768784\pi$
$614$	0	0
$615$	−4635.42	−0.303932
$616$	0	0
$617$	−9950.12	−0.649233	−0.324617	−	0.945846i	$-0.605235\pi$
$617$	−9950.12	−0.649233	−0.324617	+	0.945846i	$0.605235\pi$
$618$	0	0
$619$	21630.7	1.40454	0.702270	−	0.711910i	$-0.252170\pi$
$619$	21630.7	1.40454	0.702270	+	0.711910i	$0.252170\pi$
$620$	0	0
$621$	4942.26	0.319366
$622$	0	0
$623$	−2824.86	−0.181662
$624$	0	0
$625$	−9631.00	−0.616384
$626$	0	0
$627$	2692.40	0.171490
$628$	0	0
$629$	509.221	0.0322798
$630$	0	0
$631$	−22978.8	−1.44972	−0.724859	−	0.688897i	$-0.758095\pi$
$631$	−22978.8	−1.44972	−0.724859	+	0.688897i	$0.758095\pi$
$632$	0	0
$633$	1836.06	0.115287
$634$	0	0
$635$	−18458.6	−1.15356
$636$	0	0
$637$	−3118.18	−0.193951
$638$	0	0
$639$	7949.05	0.492112
$640$	0	0
$641$	16673.9	1.02742	0.513712	−	0.857963i	$-0.328270\pi$
$641$	16673.9	1.02742	0.513712	+	0.857963i	$0.328270\pi$
$642$	0	0
$643$	−9486.50	−0.581821	−0.290911	−	0.956750i	$-0.593958\pi$
$643$	−9486.50	−0.581821	−0.290911	+	0.956750i	$0.593958\pi$
$644$	0	0
$645$	−14368.6	−0.877150
$646$	0	0
$647$	23910.5	1.45289	0.726444	−	0.687225i	$-0.241172\pi$
$647$	23910.5	1.45289	0.726444	+	0.687225i	$0.241172\pi$
$648$	0	0
$649$	570.534	0.0345076
$650$	0	0
$651$	7294.35	0.439152
$652$	0	0
$653$	14726.5	0.882528	0.441264	−	0.897377i	$-0.354530\pi$
$653$	14726.5	0.882528	0.441264	+	0.897377i	$0.354530\pi$
$654$	0	0
$655$	−8194.28	−0.488820
$656$	0	0
$657$	5883.44	0.349368
$658$	0	0
$659$	6100.69	0.360621	0.180310	−	0.983610i	$-0.442290\pi$
$659$	6100.69	0.360621	0.180310	+	0.983610i	$0.442290\pi$
$660$	0	0
$661$	−25017.6	−1.47212	−0.736061	−	0.676915i	$-0.763316\pi$
$661$	−25017.6	−1.47212	−0.736061	+	0.676915i	$0.763316\pi$
$662$	0	0
$663$	−89.1146	−0.00522010
$664$	0	0
$665$	18085.9	1.05465
$666$	0	0
$667$	−44227.1	−2.56744
$668$	0	0
$669$	16380.3	0.946636
$670$	0	0
$671$	465.605	0.0267876
$672$	0	0
$673$	11593.8	0.664054	0.332027	−	0.943270i	$-0.392268\pi$
$673$	11593.8	0.664054	0.332027	+	0.943270i	$0.392268\pi$
$674$	0	0
$675$	−999.000	−0.0569652
$676$	0	0
$677$	2529.48	0.143598	0.0717990	−	0.997419i	$-0.477126\pi$
$677$	2529.48	0.143598	0.0717990	+	0.997419i	$0.477126\pi$
$678$	0	0
$679$	14085.3	0.796091
$680$	0	0
$681$	−14646.8	−0.824181
$682$	0	0
$683$	−10313.9	−0.577819	−0.288909	−	0.957356i	$-0.593293\pi$
$683$	−10313.9	−0.577819	−0.288909	+	0.957356i	$0.593293\pi$
$684$	0	0
$685$	24978.2	1.39324
$686$	0	0
$687$	−4459.71	−0.247669
$688$	0	0
$689$	−2046.59	−0.113162
$690$	0	0
$691$	−20461.8	−1.12649	−0.563244	−	0.826291i	$-0.690447\pi$
$691$	−20461.8	−1.12649	−0.563244	+	0.826291i	$0.690447\pi$
$692$	0	0
$693$	2441.89	0.133853
$694$	0	0
$695$	29922.1	1.63311
$696$	0	0
$697$	−376.366	−0.0204532
$698$	0	0
$699$	−10914.3	−0.590582
$700$	0	0
$701$	17137.2	0.923342	0.461671	−	0.887051i	$-0.347250\pi$
$701$	17137.2	0.923342	0.461671	+	0.887051i	$0.347250\pi$
$702$	0	0
$703$	17796.6	0.954784
$704$	0	0
$705$	−8407.76	−0.449155
$706$	0	0
$707$	5164.12	0.274705
$708$	0	0
$709$	−28396.7	−1.50418	−0.752088	−	0.659062i	$-0.770953\pi$
$709$	−28396.7	−1.50418	−0.752088	+	0.659062i	$0.770953\pi$
$710$	0	0
$711$	−7297.65	−0.384927
$712$	0	0
$713$	18435.1	0.968302
$714$	0	0
$715$	−1370.52	−0.0716849
$716$	0	0
$717$	−14237.1	−0.741555
$718$	0	0
$719$	29552.4	1.53285	0.766425	−	0.642333i	$-0.222034\pi$
$719$	29552.4	1.53285	0.766425	+	0.642333i	$0.222034\pi$
$720$	0	0
$721$	15637.5	0.807724
$722$	0	0
$723$	11722.2	0.602981
$724$	0	0
$725$	8939.82	0.457954
$726$	0	0
$727$	−34674.0	−1.76889	−0.884447	−	0.466640i	$-0.845464\pi$
$727$	−34674.0	−1.76889	−0.884447	+	0.466640i	$0.845464\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1166.64	−0.0590281
$732$	0	0
$733$	−31865.2	−1.60569	−0.802844	−	0.596190i	$-0.796681\pi$
$733$	−31865.2	−1.60569	−0.802844	+	0.596190i	$0.796681\pi$
$734$	0	0
$735$	6750.26	0.338758
$736$	0	0
$737$	5819.80	0.290875
$738$	0	0
$739$	207.318	0.0103198	0.00515989	−	0.999987i	$-0.498358\pi$
$739$	207.318	0.0103198	0.00515989	+	0.999987i	$0.498358\pi$
$740$	0	0
$741$	−3114.44	−0.154402
$742$	0	0
$743$	−36713.3	−1.81276	−0.906379	−	0.422465i	$-0.861165\pi$
$743$	−36713.3	−1.81276	−0.906379	+	0.422465i	$0.861165\pi$
$744$	0	0
$745$	−25251.8	−1.24182
$746$	0	0
$747$	−4548.87	−0.222804
$748$	0	0
$749$	34962.6	1.70561
$750$	0	0
$751$	−7912.71	−0.384473	−0.192236	−	0.981349i	$-0.561574\pi$
$751$	−7912.71	−0.384473	−0.192236	+	0.981349i	$0.561574\pi$
$752$	0	0
$753$	−12100.9	−0.585635
$754$	0	0
$755$	3734.86	0.180034
$756$	0	0
$757$	20599.2	0.989021	0.494510	−	0.869172i	$-0.335347\pi$
$757$	20599.2	0.989021	0.494510	+	0.869172i	$0.335347\pi$
$758$	0	0
$759$	6171.42	0.295136
$760$	0	0
$761$	6505.46	0.309885	0.154943	−	0.987923i	$-0.450481\pi$
$761$	6505.46	0.309885	0.154943	+	0.987923i	$0.450481\pi$
$762$	0	0
$763$	20121.1	0.954695
$764$	0	0
$765$	192.916	0.00911750
$766$	0	0
$767$	−659.968	−0.0310692
$768$	0	0
$769$	10566.4	0.495492	0.247746	−	0.968825i	$-0.420310\pi$
$769$	10566.4	0.495492	0.247746	+	0.968825i	$0.420310\pi$
$770$	0	0
$771$	−11254.2	−0.525696
$772$	0	0
$773$	−24200.7	−1.12605	−0.563025	−	0.826440i	$-0.690363\pi$
$773$	−24200.7	−1.12605	−0.563025	+	0.826440i	$0.690363\pi$
$774$	0	0
$775$	−3726.36	−0.172716
$776$	0	0
$777$	16140.8	0.745237
$778$	0	0
$779$	−13153.5	−0.604973
$780$	0	0
$781$	9926.02	0.454777
$782$	0	0
$783$	−6523.65	−0.297747
$784$	0	0
$785$	4291.17	0.195106
$786$	0	0
$787$	11588.9	0.524906	0.262453	−	0.964945i	$-0.415469\pi$
$787$	11588.9	0.524906	0.262453	+	0.964945i	$0.415469\pi$
$788$	0	0
$789$	8578.17	0.387061
$790$	0	0
$791$	−21270.4	−0.956118
$792$	0	0
$793$	−538.590	−0.0241184
$794$	0	0
$795$	4430.47	0.197651
$796$	0	0
$797$	4964.19	0.220628	0.110314	−	0.993897i	$-0.464814\pi$
$797$	4964.19	0.220628	0.110314	+	0.993897i	$0.464814\pi$
$798$	0	0
$799$	−682.656	−0.0302261
$800$	0	0
$801$	−1053.07	−0.0464524
$802$	0	0
$803$	7346.67	0.322862
$804$	0	0
$805$	41455.8	1.81506
$806$	0	0
$807$	−19538.5	−0.852277
$808$	0	0
$809$	42543.6	1.84889	0.924446	−	0.381313i	$-0.124528\pi$
$809$	42543.6	1.84889	0.924446	+	0.381313i	$0.124528\pi$
$810$	0	0
$811$	20021.9	0.866911	0.433456	−	0.901175i	$-0.357294\pi$
$811$	20021.9	0.866911	0.433456	+	0.901175i	$0.357294\pi$
$812$	0	0
$813$	−16189.1	−0.698373
$814$	0	0
$815$	17058.1	0.733153
$816$	0	0
$817$	−40772.4	−1.74596
$818$	0	0
$819$	−2824.67	−0.120515
$820$	0	0
$821$	45853.4	1.94920	0.974601	−	0.223951i	$-0.0718954\pi$
$821$	45853.4	1.94920	0.974601	+	0.223951i	$0.0718954\pi$
$822$	0	0
$823$	−20695.7	−0.876557	−0.438279	−	0.898839i	$-0.644412\pi$
$823$	−20695.7	−0.876557	−0.438279	+	0.898839i	$0.644412\pi$
$824$	0	0
$825$	−1247.46	−0.0526434
$826$	0	0
$827$	20730.7	0.871677	0.435839	−	0.900025i	$-0.356452\pi$
$827$	20730.7	0.871677	0.435839	+	0.900025i	$0.356452\pi$
$828$	0	0
$829$	−11772.2	−0.493202	−0.246601	−	0.969117i	$-0.579314\pi$
$829$	−11772.2	−0.493202	−0.246601	+	0.969117i	$0.579314\pi$
$830$	0	0
$831$	−16805.9	−0.701554
$832$	0	0
$833$	548.078	0.0227968
$834$	0	0
$835$	−12165.7	−0.504206
$836$	0	0
$837$	2719.24	0.112295
$838$	0	0
$839$	−20034.9	−0.824410	−0.412205	−	0.911091i	$-0.635241\pi$
$839$	−20034.9	−0.824410	−0.412205	+	0.911091i	$0.635241\pi$
$840$	0	0
$841$	33989.6	1.39364
$842$	0	0
$843$	17761.8	0.725680
$844$	0	0
$845$	1585.36	0.0645421
$846$	0	0
$847$	−29084.5	−1.17987
$848$	0	0
$849$	−3955.63	−0.159902
$850$	0	0
$851$	40792.9	1.64320
$852$	0	0
$853$	−36273.3	−1.45601	−0.728003	−	0.685574i	$-0.759551\pi$
$853$	−36273.3	−1.45601	−0.728003	+	0.685574i	$0.759551\pi$
$854$	0	0
$855$	6742.17	0.269681
$856$	0	0
$857$	−18222.9	−0.726349	−0.363174	−	0.931721i	$-0.618307\pi$
$857$	−18222.9	−0.726349	−0.363174	+	0.931721i	$0.618307\pi$
$858$	0	0
$859$	8658.51	0.343917	0.171958	−	0.985104i	$-0.444991\pi$
$859$	8658.51	0.343917	0.171958	+	0.985104i	$0.444991\pi$
$860$	0	0
$861$	−11929.7	−0.472199
$862$	0	0
$863$	29360.1	1.15809	0.579043	−	0.815297i	$-0.303426\pi$
$863$	29360.1	1.15809	0.579043	+	0.815297i	$0.303426\pi$
$864$	0	0
$865$	−39395.6	−1.54854
$866$	0	0
$867$	−14723.3	−0.576737
$868$	0	0
$869$	−9112.60	−0.355724
$870$	0	0
$871$	−6732.08	−0.261892
$872$	0	0
$873$	5250.82	0.203566
$874$	0	0
$875$	−36689.2	−1.41751
$876$	0	0
$877$	17742.3	0.683142	0.341571	−	0.939856i	$-0.389041\pi$
$877$	17742.3	0.683142	0.341571	+	0.939856i	$0.389041\pi$
$878$	0	0
$879$	−9028.05	−0.346426
$880$	0	0
$881$	−30449.5	−1.16444	−0.582219	−	0.813032i	$-0.697815\pi$
$881$	−30449.5	−1.16444	−0.582219	+	0.813032i	$0.697815\pi$
$882$	0	0
$883$	−29634.0	−1.12940	−0.564702	−	0.825295i	$-0.691009\pi$
$883$	−29634.0	−1.12940	−0.564702	+	0.825295i	$0.691009\pi$
$884$	0	0
$885$	1428.70	0.0542659
$886$	0	0
$887$	47303.3	1.79063	0.895315	−	0.445433i	$-0.146950\pi$
$887$	47303.3	1.79063	0.895315	+	0.445433i	$0.146950\pi$
$888$	0	0
$889$	−47505.1	−1.79220
$890$	0	0
$891$	910.305	0.0342271
$892$	0	0
$893$	−23858.0	−0.894038
$894$	0	0
$895$	1977.85	0.0738685
$896$	0	0
$897$	−7138.82	−0.265728
$898$	0	0
$899$	−24333.8	−0.902756
$900$	0	0
$901$	359.726	0.0133010
$902$	0	0
$903$	−36978.9	−1.36277
$904$	0	0
$905$	−11511.6	−0.422827
$906$	0	0
$907$	−38196.4	−1.39833	−0.699167	−	0.714958i	$-0.746446\pi$
$907$	−38196.4	−1.39833	−0.699167	+	0.714958i	$0.746446\pi$
$908$	0	0
$909$	1925.11	0.0702442
$910$	0	0
$911$	10106.0	0.367536	0.183768	−	0.982970i	$-0.441171\pi$
$911$	10106.0	0.367536	0.183768	+	0.982970i	$0.441171\pi$
$912$	0	0
$913$	−5680.19	−0.205900
$914$	0	0
$915$	1165.94	0.0421256
$916$	0	0
$917$	−21088.8	−0.759447
$918$	0	0
$919$	2314.67	0.0830836	0.0415418	−	0.999137i	$-0.486773\pi$
$919$	2314.67	0.0830836	0.0415418	+	0.999137i	$0.486773\pi$
$920$	0	0
$921$	−20290.5	−0.725946
$922$	0	0
$923$	−11482.0	−0.409462
$924$	0	0
$925$	−8245.63	−0.293097
$926$	0	0
$927$	5829.44	0.206541
$928$	0	0
$929$	−55806.9	−1.97090	−0.985449	−	0.169974i	$-0.945632\pi$
$929$	−55806.9	−1.97090	−0.985449	+	0.169974i	$0.945632\pi$
$930$	0	0
$931$	19154.6	0.674294
$932$	0	0
$933$	29706.7	1.04239
$934$	0	0
$935$	240.895	0.00842578
$936$	0	0
$937$	−29671.8	−1.03451	−0.517256	−	0.855831i	$-0.673046\pi$
$937$	−29671.8	−1.03451	−0.517256	+	0.855831i	$0.673046\pi$
$938$	0	0
$939$	15196.1	0.528123
$940$	0	0
$941$	−40838.9	−1.41478	−0.707391	−	0.706823i	$-0.750128\pi$
$941$	−40838.9	−1.41478	−0.707391	+	0.706823i	$0.750128\pi$
$942$	0	0
$943$	−30150.1	−1.04117
$944$	0	0
$945$	6114.87	0.210494
$946$	0	0
$947$	20232.2	0.694253	0.347127	−	0.937818i	$-0.387157\pi$
$947$	20232.2	0.694253	0.347127	+	0.937818i	$0.387157\pi$
$948$	0	0
$949$	−8498.30	−0.290692
$950$	0	0
$951$	−26916.2	−0.917789
$952$	0	0
$953$	−49933.2	−1.69727	−0.848634	−	0.528981i	$-0.822574\pi$
$953$	−49933.2	−1.69727	−0.848634	+	0.528981i	$0.822574\pi$
$954$	0	0
$955$	−28321.0	−0.959629
$956$	0	0
$957$	−8146.11	−0.275158
$958$	0	0
$959$	64283.9	2.16458
$960$	0	0
$961$	−19648.0	−0.659528
$962$	0	0
$963$	13033.6	0.436139
$964$	0	0
$965$	32186.7	1.07370
$966$	0	0
$967$	18683.9	0.621339	0.310670	−	0.950518i	$-0.399447\pi$
$967$	18683.9	0.621339	0.310670	+	0.950518i	$0.399447\pi$
$968$	0	0
$969$	547.421	0.0181483
$970$	0	0
$971$	−34323.1	−1.13438	−0.567189	−	0.823588i	$-0.691969\pi$
$971$	−34323.1	−1.13438	−0.567189	+	0.823588i	$0.691969\pi$
$972$	0	0
$973$	77007.6	2.53725
$974$	0	0
$975$	1443.00	0.0473979
$976$	0	0
$977$	37636.5	1.23245	0.616223	−	0.787572i	$-0.288662\pi$
$977$	37636.5	1.23245	0.616223	+	0.787572i	$0.288662\pi$
$978$	0	0
$979$	−1314.97	−0.0429281
$980$	0	0
$981$	7500.87	0.244123
$982$	0	0
$983$	16477.5	0.534640	0.267320	−	0.963608i	$-0.413862\pi$
$983$	16477.5	0.534640	0.267320	+	0.963608i	$0.413862\pi$
$984$	0	0
$985$	−14340.5	−0.463885
$986$	0	0
$987$	−21638.2	−0.697823
$988$	0	0
$989$	−93457.2	−3.00482
$990$	0	0
$991$	37180.6	1.19181	0.595903	−	0.803056i	$-0.296794\pi$
$991$	37180.6	1.19181	0.595903	+	0.803056i	$0.296794\pi$
$992$	0	0
$993$	−8675.63	−0.277254
$994$	0	0
$995$	−801.071	−0.0255233
$996$	0	0
$997$	1949.92	0.0619404	0.0309702	−	0.999520i	$-0.490140\pi$
$997$	1949.92	0.0619404	0.0309702	+	0.999520i	$0.490140\pi$
$998$	0	0
$999$	6017.08	0.190563

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.p.1.2		2
3.2	odd	2		1872.4.a.y.1.1		2
4.3	odd	2		156.4.a.c.1.2	✓	2
8.3	odd	2		2496.4.a.bf.1.1		2
8.5	even	2		2496.4.a.w.1.1		2
12.11	even	2		468.4.a.g.1.1		2
52.31	even	4		2028.4.b.e.337.2		4
52.47	even	4		2028.4.b.e.337.3		4
52.51	odd	2		2028.4.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.4.a.c.1.2	✓	2	4.3	odd	2
468.4.a.g.1.1		2	12.11	even	2
624.4.a.p.1.2		2	1.1	even	1	trivial
1872.4.a.y.1.1		2	3.2	odd	2
2028.4.a.d.1.1		2	52.51	odd	2
2028.4.b.e.337.2		4	52.31	even	4
2028.4.b.e.337.3		4	52.47	even	4
2496.4.a.w.1.1		2	8.5	even	2
2496.4.a.bf.1.1		2	8.3	odd	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +9.38083 q^{5} +24.1425 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +9.38083 q^{5} +24.1425 q^{7} +9.00000 q^{9} +11.2383 q^{11} -13.0000 q^{13} +28.1425 q^{15} +2.28499 q^{17} +79.8575 q^{19} +72.4275 q^{21} +183.047 q^{23} -37.0000 q^{25} +27.0000 q^{27} -241.617 q^{29} +100.712 q^{31} +33.7150 q^{33} +226.477 q^{35} +222.855 q^{37} -39.0000 q^{39} -164.712 q^{41} -510.565 q^{43} +84.4275 q^{45} -298.757 q^{47} +239.860 q^{49} +6.85497 q^{51} +157.430 q^{53} +105.425 q^{55} +239.573 q^{57} +50.7667 q^{59} +41.4300 q^{61} +217.282 q^{63} -121.951 q^{65} +517.852 q^{67} +549.140 q^{69} +883.228 q^{71} +653.715 q^{73} -111.000 q^{75} +271.321 q^{77} -810.850 q^{79} +81.0000 q^{81} -505.430 q^{83} +21.4351 q^{85} -724.850 q^{87} -117.008 q^{89} -313.852 q^{91} +302.137 q^{93} +749.130 q^{95} +583.425 q^{97} +101.145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 8 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 8 q^{7} + 18 q^{9} + 60 q^{11} - 26 q^{13} - 108 q^{17} + 216 q^{19} - 24 q^{21} + 216 q^{23} - 74 q^{25} + 54 q^{27} - 108 q^{29} - 80 q^{31} + 180 q^{33} + 528 q^{35} + 108 q^{37} - 78 q^{39} - 48 q^{41} - 8 q^{43} + 228 q^{47} + 930 q^{49} - 324 q^{51} + 540 q^{53} - 352 q^{55} + 648 q^{57} + 852 q^{59} + 308 q^{61} - 72 q^{63} + 304 q^{67} + 648 q^{69} + 228 q^{71} + 1420 q^{73} - 222 q^{75} - 1296 q^{77} - 496 q^{79} + 162 q^{81} - 1236 q^{83} + 1056 q^{85} - 324 q^{87} - 1416 q^{89} + 104 q^{91} - 240 q^{93} - 528 q^{95} + 604 q^{97} + 540 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 8 * q^7 + 18 * q^9 + 60 * q^11 - 26 * q^13 - 108 * q^17 + 216 * q^19 - 24 * q^21 + 216 * q^23 - 74 * q^25 + 54 * q^27 - 108 * q^29 - 80 * q^31 + 180 * q^33 + 528 * q^35 + 108 * q^37 - 78 * q^39 - 48 * q^41 - 8 * q^43 + 228 * q^47 + 930 * q^49 - 324 * q^51 + 540 * q^53 - 352 * q^55 + 648 * q^57 + 852 * q^59 + 308 * q^61 - 72 * q^63 + 304 * q^67 + 648 * q^69 + 228 * q^71 + 1420 * q^73 - 222 * q^75 - 1296 * q^77 - 496 * q^79 + 162 * q^81 - 1236 * q^83 + 1056 * q^85 - 324 * q^87 - 1416 * q^89 + 104 * q^91 - 240 * q^93 - 528 * q^95 + 604 * q^97 + 540 * q^99

Introduction

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