LMFDB - Embedded newform 1088.4.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,4,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1088.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:	$64.1940780862$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 383x^{4} - 964x^{3} - 1476x^{2} + 1978x + 1028 $ x^8 - 4x^7 - 34x^6 + 116x^5 + 383x^4 - 964x^3 - 1476x^2 + 1978*x + 1028
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.81367$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.96818 q^{3} -18.7155 q^{5} +19.9877 q^{7} +72.3646 q^{9} +O(q^{10})$	$q-9.96818 q^{3} -18.7155 q^{5} +19.9877 q^{7} +72.3646 q^{9} +67.7002 q^{11} +30.4613 q^{13} +186.560 q^{15} +17.0000 q^{17} -30.6630 q^{19} -199.241 q^{21} +30.3597 q^{23} +225.271 q^{25} -452.202 q^{27} +190.367 q^{29} -205.736 q^{31} -674.847 q^{33} -374.081 q^{35} +331.258 q^{37} -303.644 q^{39} +215.978 q^{41} +44.5779 q^{43} -1354.34 q^{45} -228.159 q^{47} +56.5089 q^{49} -169.459 q^{51} -207.572 q^{53} -1267.05 q^{55} +305.655 q^{57} +218.922 q^{59} +72.9869 q^{61} +1446.40 q^{63} -570.099 q^{65} +774.974 q^{67} -302.631 q^{69} +46.2918 q^{71} +1045.44 q^{73} -2245.54 q^{75} +1353.17 q^{77} -866.790 q^{79} +2553.79 q^{81} -361.544 q^{83} -318.164 q^{85} -1897.61 q^{87} -396.028 q^{89} +608.852 q^{91} +2050.81 q^{93} +573.875 q^{95} +931.215 q^{97} +4899.09 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{5} + 136 q^{9}+O(q^{10}) $ 8 * q - 16 * q^5 + 136 * q^9	$ 8 q - 16 q^{5} + 136 q^{9} + 56 q^{13} + 136 q^{17} - 312 q^{21} + 600 q^{25} + 128 q^{29} - 304 q^{33} + 112 q^{37} + 1328 q^{41} - 2592 q^{45} + 1624 q^{49} - 1216 q^{53} + 2560 q^{57} - 752 q^{61} + 1824 q^{65} - 1448 q^{69} + 3952 q^{73} + 1480 q^{77} + 6824 q^{81} - 272 q^{85} + 2304 q^{89} + 2680 q^{93} + 4560 q^{97}+O(q^{100}) $ 8 * q - 16 * q^5 + 136 * q^9 + 56 * q^13 + 136 * q^17 - 312 * q^21 + 600 * q^25 + 128 * q^29 - 304 * q^33 + 112 * q^37 + 1328 * q^41 - 2592 * q^45 + 1624 * q^49 - 1216 * q^53 + 2560 * q^57 - 752 * q^61 + 1824 * q^65 - 1448 * q^69 + 3952 * q^73 + 1480 * q^77 + 6824 * q^81 - 272 * q^85 + 2304 * q^89 + 2680 * q^93 + 4560 * q^97

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
$3$	−9.96818	−1.91838	−0.959188	−	0.282768i	$-0.908747\pi$
$3$	−9.96818	−1.91838	−0.959188	+	0.282768i	$0.908747\pi$
$4$	0	0
$5$	−18.7155	−1.67397	−0.836984	−	0.547227i	$-0.815683\pi$
$5$	−18.7155	−1.67397	−0.836984	+	0.547227i	$0.815683\pi$
$6$	0	0
$7$	19.9877	1.07924	0.539618	−	0.841910i	$-0.318569\pi$
$7$	19.9877	1.07924	0.539618	+	0.841910i	$0.318569\pi$
$8$	0	0
$9$	72.3646	2.68017
$10$	0	0
$11$	67.7002	1.85567	0.927835	−	0.372991i	$-0.121668\pi$
$11$	67.7002	1.85567	0.927835	+	0.372991i	$0.121668\pi$
$12$	0	0
$13$	30.4613	0.649880	0.324940	−	0.945735i	$-0.394656\pi$
$13$	30.4613	0.649880	0.324940	+	0.945735i	$0.394656\pi$
$14$	0	0
$15$	186.560	3.21130
$16$	0	0
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−30.6630	−0.370241	−0.185121	−	0.982716i	$-0.559268\pi$
$19$	−30.6630	−0.370241	−0.185121	+	0.982716i	$0.559268\pi$
$20$	0	0
$21$	−199.241	−2.07038
$22$	0	0
$23$	30.3597	0.275237	0.137618	−	0.990485i	$-0.456055\pi$
$23$	30.3597	0.275237	0.137618	+	0.990485i	$0.456055\pi$
$24$	0	0
$25$	225.271	1.80217
$26$	0	0
$27$	−452.202	−3.22320
$28$	0	0
$29$	190.367	1.21898	0.609488	−	0.792795i	$-0.291375\pi$
$29$	190.367	1.21898	0.609488	+	0.792795i	$0.291375\pi$
$30$	0	0
$31$	−205.736	−1.19197	−0.595987	−	0.802994i	$-0.703239\pi$
$31$	−205.736	−1.19197	−0.595987	+	0.802994i	$0.703239\pi$
$32$	0	0
$33$	−674.847	−3.55987
$34$	0	0
$35$	−374.081	−1.80661
$36$	0	0
$37$	331.258	1.47185	0.735926	−	0.677062i	$-0.236747\pi$
$37$	331.258	1.47185	0.735926	+	0.677062i	$0.236747\pi$
$38$	0	0
$39$	−303.644	−1.24671
$40$	0	0
$41$	215.978	0.822686	0.411343	−	0.911481i	$-0.365060\pi$
$41$	215.978	0.822686	0.411343	+	0.911481i	$0.365060\pi$
$42$	0	0
$43$	44.5779	0.158095	0.0790473	−	0.996871i	$-0.474812\pi$
$43$	44.5779	0.158095	0.0790473	+	0.996871i	$0.474812\pi$
$44$	0	0
$45$	−1354.34	−4.48652
$46$	0	0
$47$	−228.159	−0.708095	−0.354047	−	0.935227i	$-0.615195\pi$
$47$	−228.159	−0.708095	−0.354047	+	0.935227i	$0.615195\pi$
$48$	0	0
$49$	56.5089	0.164749
$50$	0	0
$51$	−169.459	−0.465275
$52$	0	0
$53$	−207.572	−0.537967	−0.268983	−	0.963145i	$-0.586688\pi$
$53$	−207.572	−0.537967	−0.268983	+	0.963145i	$0.586688\pi$
$54$	0	0
$55$	−1267.05	−3.10633
$56$	0	0
$57$	305.655	0.710262
$58$	0	0
$59$	218.922	0.483072	0.241536	−	0.970392i	$-0.422349\pi$
$59$	218.922	0.483072	0.241536	+	0.970392i	$0.422349\pi$
$60$	0	0
$61$	72.9869	0.153197	0.0765985	−	0.997062i	$-0.475594\pi$
$61$	72.9869	0.153197	0.0765985	+	0.997062i	$0.475594\pi$
$62$	0	0
$63$	1446.40	2.89253
$64$	0	0
$65$	−570.099	−1.08788
$66$	0	0
$67$	774.974	1.41311	0.706554	−	0.707659i	$-0.250249\pi$
$67$	774.974	1.41311	0.706554	+	0.707659i	$0.250249\pi$
$68$	0	0
$69$	−302.631	−0.528008
$70$	0	0
$71$	46.2918	0.0773779	0.0386890	−	0.999251i	$-0.487682\pi$
$71$	46.2918	0.0773779	0.0386890	+	0.999251i	$0.487682\pi$
$72$	0	0
$73$	1045.44	1.67616	0.838082	−	0.545544i	$-0.183677\pi$
$73$	1045.44	1.67616	0.838082	+	0.545544i	$0.183677\pi$
$74$	0	0
$75$	−2245.54	−3.45724
$76$	0	0
$77$	1353.17	2.00270
$78$	0	0
$79$	−866.790	−1.23445	−0.617225	−	0.786787i	$-0.711743\pi$
$79$	−866.790	−1.23445	−0.617225	+	0.786787i	$0.711743\pi$
$80$	0	0
$81$	2553.79	3.50314
$82$	0	0
$83$	−361.544	−0.478128	−0.239064	−	0.971004i	$-0.576841\pi$
$83$	−361.544	−0.478128	−0.239064	+	0.971004i	$0.576841\pi$
$84$	0	0
$85$	−318.164	−0.405997
$86$	0	0
$87$	−1897.61	−2.33846
$88$	0	0
$89$	−396.028	−0.471673	−0.235837	−	0.971793i	$-0.575783\pi$
$89$	−396.028	−0.471673	−0.235837	+	0.971793i	$0.575783\pi$
$90$	0	0
$91$	608.852	0.701373
$92$	0	0
$93$	2050.81	2.28665
$94$	0	0
$95$	573.875	0.619772
$96$	0	0
$97$	931.215	0.974748	0.487374	−	0.873193i	$-0.337955\pi$
$97$	931.215	0.974748	0.487374	+	0.873193i	$0.337955\pi$
$98$	0	0
$99$	4899.09	4.97351
$100$	0	0
$101$	−255.785	−0.251995	−0.125998	−	0.992031i	$-0.540213\pi$
$101$	−255.785	−0.251995	−0.125998	+	0.992031i	$0.540213\pi$
$102$	0	0
$103$	54.2222	0.0518706	0.0259353	−	0.999664i	$-0.491744\pi$
$103$	54.2222	0.0518706	0.0259353	+	0.999664i	$0.491744\pi$
$104$	0	0
$105$	3728.90	3.46575
$106$	0	0
$107$	−158.252	−0.142979	−0.0714895	−	0.997441i	$-0.522775\pi$
$107$	−158.252	−0.142979	−0.0714895	+	0.997441i	$0.522775\pi$
$108$	0	0
$109$	−1524.46	−1.33961	−0.669803	−	0.742539i	$-0.733621\pi$
$109$	−1524.46	−1.33961	−0.669803	+	0.742539i	$0.733621\pi$
$110$	0	0
$111$	−3302.04	−2.82357
$112$	0	0
$113$	−340.919	−0.283814	−0.141907	−	0.989880i	$-0.545323\pi$
$113$	−340.919	−0.283814	−0.141907	+	0.989880i	$0.545323\pi$
$114$	0	0
$115$	−568.199	−0.460737
$116$	0	0
$117$	2204.32	1.74179
$118$	0	0
$119$	339.791	0.261753
$120$	0	0
$121$	3252.31	2.44351
$122$	0	0
$123$	−2152.91	−1.57822
$124$	0	0
$125$	−1876.63	−1.34281
$126$	0	0
$127$	518.596	0.362346	0.181173	−	0.983451i	$-0.442011\pi$
$127$	518.596	0.362346	0.181173	+	0.983451i	$0.442011\pi$
$128$	0	0
$129$	−444.361	−0.303285
$130$	0	0
$131$	470.767	0.313978	0.156989	−	0.987600i	$-0.449821\pi$
$131$	470.767	0.313978	0.156989	+	0.987600i	$0.449821\pi$
$132$	0	0
$133$	−612.884	−0.399577
$134$	0	0
$135$	8463.21	5.39553
$136$	0	0
$137$	22.0500	0.0137508	0.00687538	−	0.999976i	$-0.497811\pi$
$137$	22.0500	0.0137508	0.00687538	+	0.999976i	$0.497811\pi$
$138$	0	0
$139$	−2001.82	−1.22153	−0.610764	−	0.791812i	$-0.709138\pi$
$139$	−2001.82	−1.22153	−0.610764	+	0.791812i	$0.709138\pi$
$140$	0	0
$141$	2274.33	1.35839
$142$	0	0
$143$	2062.23	1.20596
$144$	0	0
$145$	−3562.83	−2.04053
$146$	0	0
$147$	−563.290	−0.316050
$148$	0	0
$149$	−2611.90	−1.43608	−0.718038	−	0.696004i	$-0.754960\pi$
$149$	−2611.90	−1.43608	−0.718038	+	0.696004i	$0.754960\pi$
$150$	0	0
$151$	−168.464	−0.0907906	−0.0453953	−	0.998969i	$-0.514455\pi$
$151$	−168.464	−0.0907906	−0.0453953	+	0.998969i	$0.514455\pi$
$152$	0	0
$153$	1230.20	0.650037
$154$	0	0
$155$	3850.45	1.99533
$156$	0	0
$157$	800.895	0.407124	0.203562	−	0.979062i	$-0.434748\pi$
$157$	800.895	0.407124	0.203562	+	0.979062i	$0.434748\pi$
$158$	0	0
$159$	2069.12	1.03202
$160$	0	0
$161$	606.822	0.297045
$162$	0	0
$163$	1075.32	0.516722	0.258361	−	0.966048i	$-0.416818\pi$
$163$	1075.32	0.516722	0.258361	+	0.966048i	$0.416818\pi$
$164$	0	0
$165$	12630.1	5.95912
$166$	0	0
$167$	361.814	0.167653	0.0838263	−	0.996480i	$-0.473286\pi$
$167$	361.814	0.167653	0.0838263	+	0.996480i	$0.473286\pi$
$168$	0	0
$169$	−1269.11	−0.577656
$170$	0	0
$171$	−2218.92	−0.992309
$172$	0	0
$173$	−1364.98	−0.599871	−0.299936	−	0.953959i	$-0.596965\pi$
$173$	−1364.98	−0.599871	−0.299936	+	0.953959i	$0.596965\pi$
$174$	0	0
$175$	4502.66	1.94497
$176$	0	0
$177$	−2182.25	−0.926714
$178$	0	0
$179$	−2487.58	−1.03872	−0.519359	−	0.854556i	$-0.673829\pi$
$179$	−2487.58	−1.03872	−0.519359	+	0.854556i	$0.673829\pi$
$180$	0	0
$181$	2417.14	0.992621	0.496311	−	0.868145i	$-0.334688\pi$
$181$	2417.14	0.992621	0.496311	+	0.868145i	$0.334688\pi$
$182$	0	0
$183$	−727.546	−0.293889
$184$	0	0
$185$	−6199.68	−2.46383
$186$	0	0
$187$	1150.90	0.450066
$188$	0	0
$189$	−9038.49	−3.47859
$190$	0	0
$191$	−2109.39	−0.799109	−0.399554	−	0.916709i	$-0.630835\pi$
$191$	−2109.39	−0.799109	−0.399554	+	0.916709i	$0.630835\pi$
$192$	0	0
$193$	2837.60	1.05831	0.529157	−	0.848524i	$-0.322508\pi$
$193$	2837.60	1.05831	0.529157	+	0.848524i	$0.322508\pi$
$194$	0	0
$195$	5682.85	2.08696
$196$	0	0
$197$	−1536.36	−0.555639	−0.277820	−	0.960633i	$-0.589612\pi$
$197$	−1536.36	−0.555639	−0.277820	+	0.960633i	$0.589612\pi$
$198$	0	0
$199$	−1998.95	−0.712071	−0.356036	−	0.934472i	$-0.615872\pi$
$199$	−1998.95	−0.712071	−0.356036	+	0.934472i	$0.615872\pi$
$200$	0	0
$201$	−7725.08	−2.71087
$202$	0	0
$203$	3805.01	1.31556
$204$	0	0
$205$	−4042.15	−1.37715
$206$	0	0
$207$	2196.97	0.737681
$208$	0	0
$209$	−2075.89	−0.687045
$210$	0	0
$211$	3577.01	1.16707	0.583535	−	0.812088i	$-0.301669\pi$
$211$	3577.01	1.16707	0.583535	+	0.812088i	$0.301669\pi$
$212$	0	0
$213$	−461.445	−0.148440
$214$	0	0
$215$	−834.300	−0.264646
$216$	0	0
$217$	−4112.18	−1.28642
$218$	0	0
$219$	−10421.2	−3.21551
$220$	0	0
$221$	517.842	0.157619
$222$	0	0
$223$	4120.79	1.23744	0.618719	−	0.785612i	$-0.287652\pi$
$223$	4120.79	1.23744	0.618719	+	0.785612i	$0.287652\pi$
$224$	0	0
$225$	16301.7	4.83012
$226$	0	0
$227$	−32.0875	−0.00938205	−0.00469102	−	0.999989i	$-0.501493\pi$
$227$	−32.0875	−0.00938205	−0.00469102	+	0.999989i	$0.501493\pi$
$228$	0	0
$229$	1025.47	0.295917	0.147958	−	0.988994i	$-0.452730\pi$
$229$	1025.47	0.295917	0.147958	+	0.988994i	$0.452730\pi$
$230$	0	0
$231$	−13488.7	−3.84194
$232$	0	0
$233$	6131.15	1.72388	0.861942	−	0.507007i	$-0.169248\pi$
$233$	6131.15	1.72388	0.861942	+	0.507007i	$0.169248\pi$
$234$	0	0
$235$	4270.12	1.18533
$236$	0	0
$237$	8640.32	2.36814
$238$	0	0
$239$	−1290.54	−0.349280	−0.174640	−	0.984632i	$-0.555876\pi$
$239$	−1290.54	−0.349280	−0.174640	+	0.984632i	$0.555876\pi$
$240$	0	0
$241$	5592.44	1.49477	0.747387	−	0.664389i	$-0.231308\pi$
$241$	5592.44	1.49477	0.747387	+	0.664389i	$0.231308\pi$
$242$	0	0
$243$	−13247.2	−3.49714
$244$	0	0
$245$	−1057.59	−0.275784
$246$	0	0
$247$	−934.035	−0.240612
$248$	0	0
$249$	3603.94	0.917230
$250$	0	0
$251$	−652.353	−0.164048	−0.0820242	−	0.996630i	$-0.526138\pi$
$251$	−652.353	−0.164048	−0.0820242	+	0.996630i	$0.526138\pi$
$252$	0	0
$253$	2055.36	0.510748
$254$	0	0
$255$	3171.52	0.778855
$256$	0	0
$257$	−5000.23	−1.21364	−0.606821	−	0.794839i	$-0.707555\pi$
$257$	−5000.23	−1.21364	−0.606821	+	0.794839i	$0.707555\pi$
$258$	0	0
$259$	6621.10	1.58847
$260$	0	0
$261$	13775.8	3.26706
$262$	0	0
$263$	−7341.18	−1.72120	−0.860602	−	0.509279i	$-0.829912\pi$
$263$	−7341.18	−1.72120	−0.860602	+	0.509279i	$0.829912\pi$
$264$	0	0
$265$	3884.83	0.900539
$266$	0	0
$267$	3947.68	0.904847
$268$	0	0
$269$	6905.65	1.56522	0.782612	−	0.622510i	$-0.213887\pi$
$269$	6905.65	1.56522	0.782612	+	0.622510i	$0.213887\pi$
$270$	0	0
$271$	3187.35	0.714457	0.357228	−	0.934017i	$-0.383722\pi$
$271$	3187.35	0.714457	0.357228	+	0.934017i	$0.383722\pi$
$272$	0	0
$273$	−6069.14	−1.34550
$274$	0	0
$275$	15250.9	3.34423
$276$	0	0
$277$	4996.23	1.08373	0.541867	−	0.840464i	$-0.317718\pi$
$277$	4996.23	1.08373	0.541867	+	0.840464i	$0.317718\pi$
$278$	0	0
$279$	−14888.0	−3.19469
$280$	0	0
$281$	3184.98	0.676157	0.338079	−	0.941118i	$-0.390223\pi$
$281$	3184.98	0.676157	0.338079	+	0.941118i	$0.390223\pi$
$282$	0	0
$283$	8878.85	1.86499	0.932496	−	0.361180i	$-0.117626\pi$
$283$	8878.85	1.86499	0.932496	+	0.361180i	$0.117626\pi$
$284$	0	0
$285$	−5720.49	−1.18896
$286$	0	0
$287$	4316.91	0.887871
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−9282.52	−1.86993
$292$	0	0
$293$	1328.09	0.264805	0.132403	−	0.991196i	$-0.457731\pi$
$293$	1328.09	0.264805	0.132403	+	0.991196i	$0.457731\pi$
$294$	0	0
$295$	−4097.25	−0.808647
$296$	0	0
$297$	−30614.2	−5.98119
$298$	0	0
$299$	924.796	0.178871
$300$	0	0
$301$	891.011	0.170621
$302$	0	0
$303$	2549.71	0.483422
$304$	0	0
$305$	−1365.99	−0.256447
$306$	0	0
$307$	−9131.42	−1.69758	−0.848791	−	0.528729i	$-0.822669\pi$
$307$	−9131.42	−1.69758	−0.848791	+	0.528729i	$0.822669\pi$
$308$	0	0
$309$	−540.497	−0.0995074
$310$	0	0
$311$	−1807.37	−0.329539	−0.164769	−	0.986332i	$-0.552688\pi$
$311$	−1807.37	−0.329539	−0.164769	+	0.986332i	$0.552688\pi$
$312$	0	0
$313$	8781.00	1.58572	0.792862	−	0.609402i	$-0.208590\pi$
$313$	8781.00	1.58572	0.792862	+	0.609402i	$0.208590\pi$
$314$	0	0
$315$	−27070.2	−4.84201
$316$	0	0
$317$	−8561.68	−1.51695	−0.758473	−	0.651705i	$-0.774054\pi$
$317$	−8561.68	−1.51695	−0.758473	+	0.651705i	$0.774054\pi$
$318$	0	0
$319$	12887.9	2.26202
$320$	0	0
$321$	1577.48	0.274288
$322$	0	0
$323$	−521.271	−0.0897967
$324$	0	0
$325$	6862.05	1.17119
$326$	0	0
$327$	15196.1	2.56987
$328$	0	0
$329$	−4560.38	−0.764201
$330$	0	0
$331$	6976.82	1.15855	0.579276	−	0.815132i	$-0.303335\pi$
$331$	6976.82	1.15855	0.579276	+	0.815132i	$0.303335\pi$
$332$	0	0
$333$	23971.4	3.94481
$334$	0	0
$335$	−14504.1	−2.36550
$336$	0	0
$337$	2496.41	0.403525	0.201762	−	0.979434i	$-0.435333\pi$
$337$	2496.41	0.403525	0.201762	+	0.979434i	$0.435333\pi$
$338$	0	0
$339$	3398.34	0.544462
$340$	0	0
$341$	−13928.3	−2.21191
$342$	0	0
$343$	−5726.30	−0.901433
$344$	0	0
$345$	5663.91	0.883868
$346$	0	0
$347$	1801.40	0.278687	0.139344	−	0.990244i	$-0.455501\pi$
$347$	1801.40	0.278687	0.139344	+	0.990244i	$0.455501\pi$
$348$	0	0
$349$	900.161	0.138064	0.0690322	−	0.997614i	$-0.478009\pi$
$349$	900.161	0.138064	0.0690322	+	0.997614i	$0.478009\pi$
$350$	0	0
$351$	−13774.7	−2.09469
$352$	0	0
$353$	1787.54	0.269521	0.134761	−	0.990878i	$-0.456973\pi$
$353$	1787.54	0.269521	0.134761	+	0.990878i	$0.456973\pi$
$354$	0	0
$355$	−866.377	−0.129528
$356$	0	0
$357$	−3387.10	−0.502141
$358$	0	0
$359$	−148.066	−0.0217678	−0.0108839	−	0.999941i	$-0.503465\pi$
$359$	−148.066	−0.0217678	−0.0108839	+	0.999941i	$0.503465\pi$
$360$	0	0
$361$	−5918.78	−0.862922
$362$	0	0
$363$	−32419.6	−4.68757
$364$	0	0
$365$	−19566.1	−2.80585
$366$	0	0
$367$	−13059.0	−1.85743	−0.928713	−	0.370799i	$-0.879084\pi$
$367$	−13059.0	−1.85743	−0.928713	+	0.370799i	$0.879084\pi$
$368$	0	0
$369$	15629.2	2.20494
$370$	0	0
$371$	−4148.90	−0.580593
$372$	0	0
$373$	−5263.69	−0.730680	−0.365340	−	0.930874i	$-0.619047\pi$
$373$	−5263.69	−0.730680	−0.365340	+	0.930874i	$0.619047\pi$
$374$	0	0
$375$	18706.6	2.57601
$376$	0	0
$377$	5798.83	0.792188
$378$	0	0
$379$	9591.46	1.29995	0.649974	−	0.759956i	$-0.274780\pi$
$379$	9591.46	1.29995	0.649974	+	0.759956i	$0.274780\pi$
$380$	0	0
$381$	−5169.46	−0.695116
$382$	0	0
$383$	3930.98	0.524448	0.262224	−	0.965007i	$-0.415544\pi$
$383$	3930.98	0.524448	0.262224	+	0.965007i	$0.415544\pi$
$384$	0	0
$385$	−25325.3	−3.35246
$386$	0	0
$387$	3225.86	0.423721
$388$	0	0
$389$	−4535.48	−0.591152	−0.295576	−	0.955319i	$-0.595511\pi$
$389$	−4535.48	−0.591152	−0.295576	+	0.955319i	$0.595511\pi$
$390$	0	0
$391$	516.115	0.0667547
$392$	0	0
$393$	−4692.69	−0.602328
$394$	0	0
$395$	16222.4	2.06643
$396$	0	0
$397$	8688.83	1.09844	0.549219	−	0.835678i	$-0.314925\pi$
$397$	8688.83	1.09844	0.549219	+	0.835678i	$0.314925\pi$
$398$	0	0
$399$	6109.34	0.766540
$400$	0	0
$401$	4079.50	0.508031	0.254016	−	0.967200i	$-0.418249\pi$
$401$	4079.50	0.508031	0.254016	+	0.967200i	$0.418249\pi$
$402$	0	0
$403$	−6266.97	−0.774640
$404$	0	0
$405$	−47795.5	−5.86415
$406$	0	0
$407$	22426.2	2.73127
$408$	0	0
$409$	−11239.8	−1.35886	−0.679428	−	0.733742i	$-0.737772\pi$
$409$	−11239.8	−1.35886	−0.679428	+	0.733742i	$0.737772\pi$
$410$	0	0
$411$	−219.798	−0.0263792
$412$	0	0
$413$	4375.75	0.521348
$414$	0	0
$415$	6766.49	0.800371
$416$	0	0
$417$	19954.5	2.34335
$418$	0	0
$419$	−6658.88	−0.776390	−0.388195	−	0.921577i	$-0.626901\pi$
$419$	−6658.88	−0.776390	−0.388195	+	0.921577i	$0.626901\pi$
$420$	0	0
$421$	15017.4	1.73848	0.869242	−	0.494387i	$-0.164608\pi$
$421$	15017.4	1.73848	0.869242	+	0.494387i	$0.164608\pi$
$422$	0	0
$423$	−16510.7	−1.89781
$424$	0	0
$425$	3829.61	0.437091
$426$	0	0
$427$	1458.84	0.165336
$428$	0	0
$429$	−20556.7	−2.31349
$430$	0	0
$431$	10252.4	1.14580	0.572901	−	0.819625i	$-0.305818\pi$
$431$	10252.4	1.14580	0.572901	+	0.819625i	$0.305818\pi$
$432$	0	0
$433$	12528.6	1.39050	0.695248	−	0.718770i	$-0.255294\pi$
$433$	12528.6	1.39050	0.695248	+	0.718770i	$0.255294\pi$
$434$	0	0
$435$	35514.9	3.91450
$436$	0	0
$437$	−930.921	−0.101904
$438$	0	0
$439$	−10042.7	−1.09182	−0.545912	−	0.837842i	$-0.683817\pi$
$439$	−10042.7	−1.09182	−0.545912	+	0.837842i	$0.683817\pi$
$440$	0	0
$441$	4089.24	0.441555
$442$	0	0
$443$	−17018.2	−1.82519	−0.912595	−	0.408865i	$-0.865925\pi$
$443$	−17018.2	−1.82519	−0.912595	+	0.408865i	$0.865925\pi$
$444$	0	0
$445$	7411.88	0.789566
$446$	0	0
$447$	26035.9	2.75494
$448$	0	0
$449$	−1073.19	−0.112800	−0.0563998	−	0.998408i	$-0.517962\pi$
$449$	−1073.19	−0.112800	−0.0563998	+	0.998408i	$0.517962\pi$
$450$	0	0
$451$	14621.8	1.52663
$452$	0	0
$453$	1679.28	0.174171
$454$	0	0
$455$	−11395.0	−1.17408
$456$	0	0
$457$	−6058.44	−0.620135	−0.310068	−	0.950715i	$-0.600352\pi$
$457$	−6058.44	−0.620135	−0.310068	+	0.950715i	$0.600352\pi$
$458$	0	0
$459$	−7687.44	−0.781740
$460$	0	0
$461$	−2316.22	−0.234007	−0.117003	−	0.993132i	$-0.537329\pi$
$461$	−2316.22	−0.234007	−0.117003	+	0.993132i	$0.537329\pi$
$462$	0	0
$463$	9045.41	0.907939	0.453970	−	0.891017i	$-0.350007\pi$
$463$	9045.41	0.907939	0.453970	+	0.891017i	$0.350007\pi$
$464$	0	0
$465$	−38382.0	−3.82779
$466$	0	0
$467$	−13960.5	−1.38333	−0.691665	−	0.722218i	$-0.743123\pi$
$467$	−13960.5	−1.38333	−0.691665	+	0.722218i	$0.743123\pi$
$468$	0	0
$469$	15490.0	1.52508
$470$	0	0
$471$	−7983.47	−0.781016
$472$	0	0
$473$	3017.93	0.293372
$474$	0	0
$475$	−6907.50	−0.667238
$476$	0	0
$477$	−15020.9	−1.44184
$478$	0	0
$479$	−460.666	−0.0439423	−0.0219712	−	0.999759i	$-0.506994\pi$
$479$	−460.666	−0.0439423	−0.0219712	+	0.999759i	$0.506994\pi$
$480$	0	0
$481$	10090.6	0.956527
$482$	0	0
$483$	−6048.91	−0.569844
$484$	0	0
$485$	−17428.2	−1.63170
$486$	0	0
$487$	18516.7	1.72294	0.861470	−	0.507808i	$-0.169544\pi$
$487$	18516.7	1.72294	0.861470	+	0.507808i	$0.169544\pi$
$488$	0	0
$489$	−10719.0	−0.991267
$490$	0	0
$491$	5890.57	0.541421	0.270710	−	0.962661i	$-0.412741\pi$
$491$	5890.57	0.541421	0.270710	+	0.962661i	$0.412741\pi$
$492$	0	0
$493$	3236.24	0.295645
$494$	0	0
$495$	−91689.2	−8.32550
$496$	0	0
$497$	925.268	0.0835090
$498$	0	0
$499$	5912.17	0.530391	0.265196	−	0.964195i	$-0.414563\pi$
$499$	5912.17	0.530391	0.265196	+	0.964195i	$0.414563\pi$
$500$	0	0
$501$	−3606.62	−0.321621
$502$	0	0
$503$	14783.8	1.31049	0.655244	−	0.755417i	$-0.272566\pi$
$503$	14783.8	1.31049	0.655244	+	0.755417i	$0.272566\pi$
$504$	0	0
$505$	4787.14	0.421832
$506$	0	0
$507$	12650.7	1.10816
$508$	0	0
$509$	8460.64	0.736761	0.368380	−	0.929675i	$-0.379912\pi$
$509$	8460.64	0.736761	0.368380	+	0.929675i	$0.379912\pi$
$510$	0	0
$511$	20896.0	1.80898
$512$	0	0
$513$	13865.9	1.19336
$514$	0	0
$515$	−1014.80	−0.0868298
$516$	0	0
$517$	−15446.4	−1.31399
$518$	0	0
$519$	13606.4	1.15078
$520$	0	0
$521$	−6765.31	−0.568894	−0.284447	−	0.958692i	$-0.591810\pi$
$521$	−6765.31	−0.568894	−0.284447	+	0.958692i	$0.591810\pi$
$522$	0	0
$523$	3008.53	0.251537	0.125769	−	0.992060i	$-0.459860\pi$
$523$	3008.53	0.251537	0.125769	+	0.992060i	$0.459860\pi$
$524$	0	0
$525$	−44883.3	−3.73118
$526$	0	0
$527$	−3497.50	−0.289096
$528$	0	0
$529$	−11245.3	−0.924245
$530$	0	0
$531$	15842.2	1.29471
$532$	0	0
$533$	6578.97	0.534647
$534$	0	0
$535$	2961.76	0.239342
$536$	0	0
$537$	24796.7	1.99265
$538$	0	0
$539$	3825.66	0.305719
$540$	0	0
$541$	6852.16	0.544543	0.272271	−	0.962221i	$-0.412225\pi$
$541$	6852.16	0.544543	0.272271	+	0.962221i	$0.412225\pi$
$542$	0	0
$543$	−24094.5	−1.90422
$544$	0	0
$545$	28531.1	2.24246
$546$	0	0
$547$	−8591.54	−0.671568	−0.335784	−	0.941939i	$-0.609001\pi$
$547$	−8591.54	−0.671568	−0.335784	+	0.941939i	$0.609001\pi$
$548$	0	0
$549$	5281.67	0.410594
$550$	0	0
$551$	−5837.24	−0.451315
$552$	0	0
$553$	−17325.2	−1.33226
$554$	0	0
$555$	61799.5	4.72656
$556$	0	0
$557$	159.441	0.0121288	0.00606438	−	0.999982i	$-0.498070\pi$
$557$	159.441	0.0121288	0.00606438	+	0.999982i	$0.498070\pi$
$558$	0	0
$559$	1357.90	0.102743
$560$	0	0
$561$	−11472.4	−0.863396
$562$	0	0
$563$	2804.89	0.209968	0.104984	−	0.994474i	$-0.466521\pi$
$563$	2804.89	0.209968	0.104984	+	0.994474i	$0.466521\pi$
$564$	0	0
$565$	6380.48	0.475095
$566$	0	0
$567$	51044.4	3.78071
$568$	0	0
$569$	−8415.81	−0.620051	−0.310025	−	0.950728i	$-0.600338\pi$
$569$	−8415.81	−0.620051	−0.310025	+	0.950728i	$0.600338\pi$
$570$	0	0
$571$	−3502.18	−0.256676	−0.128338	−	0.991731i	$-0.540964\pi$
$571$	−3502.18	−0.256676	−0.128338	+	0.991731i	$0.540964\pi$
$572$	0	0
$573$	21026.7	1.53299
$574$	0	0
$575$	6839.18	0.496023
$576$	0	0
$577$	24364.5	1.75790	0.878950	−	0.476914i	$-0.158245\pi$
$577$	24364.5	1.75790	0.878950	+	0.476914i	$0.158245\pi$
$578$	0	0
$579$	−28285.7	−2.03025
$580$	0	0
$581$	−7226.44	−0.516013
$582$	0	0
$583$	−14052.7	−0.998289
$584$	0	0
$585$	−41255.0	−2.91570
$586$	0	0
$587$	15923.3	1.11963	0.559817	−	0.828616i	$-0.310871\pi$
$587$	15923.3	1.11963	0.559817	+	0.828616i	$0.310871\pi$
$588$	0	0
$589$	6308.47	0.441318
$590$	0	0
$591$	15314.7	1.06593
$592$	0	0
$593$	6122.01	0.423948	0.211974	−	0.977275i	$-0.432011\pi$
$593$	6122.01	0.423948	0.211974	+	0.977275i	$0.432011\pi$
$594$	0	0
$595$	−6359.37	−0.438166
$596$	0	0
$597$	19925.9	1.36602
$598$	0	0
$599$	−13172.4	−0.898512	−0.449256	−	0.893403i	$-0.648311\pi$
$599$	−13172.4	−0.898512	−0.449256	+	0.893403i	$0.648311\pi$
$600$	0	0
$601$	−20390.9	−1.38396	−0.691981	−	0.721916i	$-0.743262\pi$
$601$	−20390.9	−1.38396	−0.691981	+	0.721916i	$0.743262\pi$
$602$	0	0
$603$	56080.7	3.78737
$604$	0	0
$605$	−60868.8	−4.09036
$606$	0	0
$607$	−5823.69	−0.389417	−0.194709	−	0.980861i	$-0.562376\pi$
$607$	−5823.69	−0.389417	−0.194709	+	0.980861i	$0.562376\pi$
$608$	0	0
$609$	−37929.0	−2.52374
$610$	0	0
$611$	−6950.03	−0.460177
$612$	0	0
$613$	−7215.05	−0.475389	−0.237694	−	0.971340i	$-0.576392\pi$
$613$	−7215.05	−0.475389	−0.237694	+	0.971340i	$0.576392\pi$
$614$	0	0
$615$	40292.8	2.64189
$616$	0	0
$617$	−1128.91	−0.0736600	−0.0368300	−	0.999322i	$-0.511726\pi$
$617$	−1128.91	−0.0736600	−0.0368300	+	0.999322i	$0.511726\pi$
$618$	0	0
$619$	−19279.2	−1.25185	−0.625927	−	0.779882i	$-0.715279\pi$
$619$	−19279.2	−1.25185	−0.625927	+	0.779882i	$0.715279\pi$
$620$	0	0
$621$	−13728.7	−0.887142
$622$	0	0
$623$	−7915.70	−0.509046
$624$	0	0
$625$	6963.26	0.445649
$626$	0	0
$627$	20692.9	1.31801
$628$	0	0
$629$	5631.39	0.356976
$630$	0	0
$631$	−17300.5	−1.09148	−0.545738	−	0.837956i	$-0.683751\pi$
$631$	−17300.5	−1.09148	−0.545738	+	0.837956i	$0.683751\pi$
$632$	0	0
$633$	−35656.3	−2.23888
$634$	0	0
$635$	−9705.80	−0.606556
$636$	0	0
$637$	1721.33	0.107067
$638$	0	0
$639$	3349.89	0.207386
$640$	0	0
$641$	17606.5	1.08489	0.542444	−	0.840092i	$-0.317499\pi$
$641$	17606.5	1.08489	0.542444	+	0.840092i	$0.317499\pi$
$642$	0	0
$643$	1977.02	0.121254	0.0606268	−	0.998161i	$-0.480690\pi$
$643$	1977.02	0.121254	0.0606268	+	0.998161i	$0.480690\pi$
$644$	0	0
$645$	8316.45	0.507690
$646$	0	0
$647$	29850.5	1.81382	0.906911	−	0.421322i	$-0.138434\pi$
$647$	29850.5	1.81382	0.906911	+	0.421322i	$0.138434\pi$
$648$	0	0
$649$	14821.1	0.896422
$650$	0	0
$651$	40991.0	2.46784
$652$	0	0
$653$	16467.9	0.986888	0.493444	−	0.869778i	$-0.335738\pi$
$653$	16467.9	0.986888	0.493444	+	0.869778i	$0.335738\pi$
$654$	0	0
$655$	−8810.66	−0.525589
$656$	0	0
$657$	75653.1	4.49240
$658$	0	0
$659$	−17716.9	−1.04727	−0.523635	−	0.851943i	$-0.675424\pi$
$659$	−17716.9	−1.04727	−0.523635	+	0.851943i	$0.675424\pi$
$660$	0	0
$661$	22702.8	1.33591	0.667956	−	0.744201i	$-0.267169\pi$
$661$	22702.8	1.33591	0.667956	+	0.744201i	$0.267169\pi$
$662$	0	0
$663$	−5161.94	−0.302373
$664$	0	0
$665$	11470.5	0.668880
$666$	0	0
$667$	5779.50	0.335507
$668$	0	0
$669$	−41076.8	−2.37387
$670$	0	0
$671$	4941.22	0.284283
$672$	0	0
$673$	−18951.1	−1.08545	−0.542727	−	0.839909i	$-0.682608\pi$
$673$	−18951.1	−1.08545	−0.542727	+	0.839909i	$0.682608\pi$
$674$	0	0
$675$	−101868.	−5.80875
$676$	0	0
$677$	−8341.21	−0.473528	−0.236764	−	0.971567i	$-0.576087\pi$
$677$	−8341.21	−0.473528	−0.236764	+	0.971567i	$0.576087\pi$
$678$	0	0
$679$	18612.9	1.05198
$680$	0	0
$681$	319.854	0.0179983
$682$	0	0
$683$	24527.7	1.37412	0.687061	−	0.726600i	$-0.258901\pi$
$683$	24527.7	1.37412	0.687061	+	0.726600i	$0.258901\pi$
$684$	0	0
$685$	−412.677	−0.0230184
$686$	0	0
$687$	−10222.1	−0.567680
$688$	0	0
$689$	−6322.92	−0.349614
$690$	0	0
$691$	−33120.5	−1.82339	−0.911695	−	0.410867i	$-0.865226\pi$
$691$	−33120.5	−1.82339	−0.911695	+	0.410867i	$0.865226\pi$
$692$	0	0
$693$	97921.7	5.36759
$694$	0	0
$695$	37465.2	2.04480
$696$	0	0
$697$	3671.63	0.199531
$698$	0	0
$699$	−61116.4	−3.30706
$700$	0	0
$701$	−12331.8	−0.664433	−0.332216	−	0.943203i	$-0.607796\pi$
$701$	−12331.8	−0.664433	−0.332216	+	0.943203i	$0.607796\pi$
$702$	0	0
$703$	−10157.4	−0.544940
$704$	0	0
$705$	−42565.4	−2.27391
$706$	0	0
$707$	−5112.55	−0.271962
$708$	0	0
$709$	−1663.55	−0.0881182	−0.0440591	−	0.999029i	$-0.514029\pi$
$709$	−1663.55	−0.0881182	−0.0440591	+	0.999029i	$0.514029\pi$
$710$	0	0
$711$	−62724.9	−3.30854
$712$	0	0
$713$	−6246.08	−0.328075
$714$	0	0
$715$	−38595.8	−2.01874
$716$	0	0
$717$	12864.3	0.670050
$718$	0	0
$719$	35709.5	1.85221	0.926105	−	0.377267i	$-0.123136\pi$
$719$	35709.5	1.85221	0.926105	+	0.377267i	$0.123136\pi$
$720$	0	0
$721$	1083.78	0.0559806
$722$	0	0
$723$	−55746.4	−2.86754
$724$	0	0
$725$	42884.3	2.19680
$726$	0	0
$727$	21578.7	1.10084	0.550419	−	0.834888i	$-0.314468\pi$
$727$	21578.7	1.10084	0.550419	+	0.834888i	$0.314468\pi$
$728$	0	0
$729$	63097.8	3.20570
$730$	0	0
$731$	757.825	0.0383436
$732$	0	0
$733$	1262.81	0.0636331	0.0318165	−	0.999494i	$-0.489871\pi$
$733$	1262.81	0.0636331	0.0318165	+	0.999494i	$0.489871\pi$
$734$	0	0
$735$	10542.3	0.529058
$736$	0	0
$737$	52465.9	2.62226
$738$	0	0
$739$	30811.1	1.53370	0.766850	−	0.641827i	$-0.221823\pi$
$739$	30811.1	1.53370	0.766850	+	0.641827i	$0.221823\pi$
$740$	0	0
$741$	9310.63	0.461585
$742$	0	0
$743$	−26048.7	−1.28618	−0.643092	−	0.765789i	$-0.722349\pi$
$743$	−26048.7	−1.28618	−0.643092	+	0.765789i	$0.722349\pi$
$744$	0	0
$745$	48883.2	2.40395
$746$	0	0
$747$	−26163.0	−1.28146
$748$	0	0
$749$	−3163.09	−0.154308
$750$	0	0
$751$	−22341.9	−1.08558	−0.542788	−	0.839870i	$-0.682631\pi$
$751$	−22341.9	−1.08558	−0.542788	+	0.839870i	$0.682631\pi$
$752$	0	0
$753$	6502.77	0.314707
$754$	0	0
$755$	3152.89	0.151981
$756$	0	0
$757$	−3390.50	−0.162787	−0.0813935	−	0.996682i	$-0.525937\pi$
$757$	−3390.50	−0.162787	−0.0813935	+	0.996682i	$0.525937\pi$
$758$	0	0
$759$	−20488.2	−0.979808
$760$	0	0
$761$	11138.0	0.530553	0.265277	−	0.964172i	$-0.414537\pi$
$761$	11138.0	0.530553	0.265277	+	0.964172i	$0.414537\pi$
$762$	0	0
$763$	−30470.5	−1.44575
$764$	0	0
$765$	−23023.8	−1.08814
$766$	0	0
$767$	6668.65	0.313939
$768$	0	0
$769$	−35895.4	−1.68325	−0.841626	−	0.540061i	$-0.818401\pi$
$769$	−35895.4	−1.68325	−0.841626	+	0.540061i	$0.818401\pi$
$770$	0	0
$771$	49843.2	2.32822
$772$	0	0
$773$	−18690.6	−0.869671	−0.434836	−	0.900510i	$-0.643193\pi$
$773$	−18690.6	−0.869671	−0.434836	+	0.900510i	$0.643193\pi$
$774$	0	0
$775$	−46346.3	−2.14814
$776$	0	0
$777$	−66000.3	−3.04729
$778$	0	0
$779$	−6622.54	−0.304592
$780$	0	0
$781$	3133.97	0.143588
$782$	0	0
$783$	−86084.5	−3.92900
$784$	0	0
$785$	−14989.2	−0.681512
$786$	0	0
$787$	−9069.97	−0.410813	−0.205406	−	0.978677i	$-0.565852\pi$
$787$	−9069.97	−0.410813	−0.205406	+	0.978677i	$0.565852\pi$
$788$	0	0
$789$	73178.2	3.30192
$790$	0	0
$791$	−6814.19	−0.306302
$792$	0	0
$793$	2223.27	0.0995596
$794$	0	0
$795$	−38724.6	−1.72757
$796$	0	0
$797$	32361.1	1.43825	0.719127	−	0.694879i	$-0.244542\pi$
$797$	32361.1	1.43825	0.719127	+	0.694879i	$0.244542\pi$
$798$	0	0
$799$	−3878.71	−0.171738
$800$	0	0
$801$	−28658.4	−1.26416
$802$	0	0
$803$	70776.8	3.11041
$804$	0	0
$805$	−11357.0	−0.497244
$806$	0	0
$807$	−68836.8	−3.00269
$808$	0	0
$809$	−45554.9	−1.97976	−0.989878	−	0.141918i	$-0.954673\pi$
$809$	−45554.9	−1.97976	−0.989878	+	0.141918i	$0.954673\pi$
$810$	0	0
$811$	6458.07	0.279622	0.139811	−	0.990178i	$-0.455350\pi$
$811$	6458.07	0.279622	0.139811	+	0.990178i	$0.455350\pi$
$812$	0	0
$813$	−31772.1	−1.37060
$814$	0	0
$815$	−20125.2	−0.864976
$816$	0	0
$817$	−1366.89	−0.0585332
$818$	0	0
$819$	44059.3	1.87980
$820$	0	0
$821$	−15250.6	−0.648294	−0.324147	−	0.946007i	$-0.605077\pi$
$821$	−15250.6	−0.648294	−0.324147	+	0.946007i	$0.605077\pi$
$822$	0	0
$823$	20885.5	0.884595	0.442298	−	0.896868i	$-0.354163\pi$
$823$	20885.5	0.884595	0.442298	+	0.896868i	$0.354163\pi$
$824$	0	0
$825$	−152024.	−6.41550
$826$	0	0
$827$	−5270.11	−0.221596	−0.110798	−	0.993843i	$-0.535341\pi$
$827$	−5270.11	−0.221596	−0.110798	+	0.993843i	$0.535341\pi$
$828$	0	0
$829$	18930.4	0.793102	0.396551	−	0.918013i	$-0.370207\pi$
$829$	18930.4	0.793102	0.396551	+	0.918013i	$0.370207\pi$
$830$	0	0
$831$	−49803.3	−2.07901
$832$	0	0
$833$	960.650	0.0399575
$834$	0	0
$835$	−6771.54	−0.280645
$836$	0	0
$837$	93034.1	3.84197
$838$	0	0
$839$	23432.6	0.964222	0.482111	−	0.876110i	$-0.339870\pi$
$839$	23432.6	0.964222	0.482111	+	0.876110i	$0.339870\pi$
$840$	0	0
$841$	11850.7	0.485903
$842$	0	0
$843$	−31748.5	−1.29712
$844$	0	0
$845$	23752.1	0.966978
$846$	0	0
$847$	65006.3	2.63712
$848$	0	0
$849$	−88505.9	−3.57776
$850$	0	0
$851$	10056.9	0.405108
$852$	0	0
$853$	−33084.8	−1.32802	−0.664010	−	0.747724i	$-0.731147\pi$
$853$	−33084.8	−1.32802	−0.664010	+	0.747724i	$0.731147\pi$
$854$	0	0
$855$	41528.2	1.66109
$856$	0	0
$857$	29921.7	1.19266	0.596328	−	0.802741i	$-0.296626\pi$
$857$	29921.7	1.19266	0.596328	+	0.802741i	$0.296626\pi$
$858$	0	0
$859$	22775.4	0.904640	0.452320	−	0.891856i	$-0.350596\pi$
$859$	22775.4	0.904640	0.452320	+	0.891856i	$0.350596\pi$
$860$	0	0
$861$	−43031.7	−1.70327
$862$	0	0
$863$	42887.2	1.69165	0.845826	−	0.533459i	$-0.179108\pi$
$863$	42887.2	1.69165	0.845826	+	0.533459i	$0.179108\pi$
$864$	0	0
$865$	25546.4	1.00417
$866$	0	0
$867$	−2880.80	−0.112846
$868$	0	0
$869$	−58681.9	−2.29073
$870$	0	0
$871$	23606.7	0.918350
$872$	0	0
$873$	67387.0	2.61249
$874$	0	0
$875$	−37509.6	−1.44921
$876$	0	0
$877$	−33049.2	−1.27251	−0.636255	−	0.771479i	$-0.719517\pi$
$877$	−33049.2	−1.27251	−0.636255	+	0.771479i	$0.719517\pi$
$878$	0	0
$879$	−13238.6	−0.507996
$880$	0	0
$881$	5862.29	0.224183	0.112092	−	0.993698i	$-0.464245\pi$
$881$	5862.29	0.224183	0.112092	+	0.993698i	$0.464245\pi$
$882$	0	0
$883$	27542.6	1.04970	0.524848	−	0.851196i	$-0.324122\pi$
$883$	27542.6	1.04970	0.524848	+	0.851196i	$0.324122\pi$
$884$	0	0
$885$	40842.1	1.55129
$886$	0	0
$887$	51006.9	1.93083	0.965413	−	0.260725i	$-0.0839615\pi$
$887$	51006.9	1.93083	0.965413	+	0.260725i	$0.0839615\pi$
$888$	0	0
$889$	10365.5	0.391057
$890$	0	0
$891$	172892.	6.50067
$892$	0	0
$893$	6996.06	0.262166
$894$	0	0
$895$	46556.4	1.73878
$896$	0	0
$897$	−9218.54	−0.343142
$898$	0	0
$899$	−39165.3	−1.45299
$900$	0	0
$901$	−3528.73	−0.130476
$902$	0	0
$903$	−8881.76	−0.327316
$904$	0	0
$905$	−45238.0	−1.66162
$906$	0	0
$907$	26854.4	0.983117	0.491559	−	0.870845i	$-0.336427\pi$
$907$	26854.4	0.983117	0.491559	+	0.870845i	$0.336427\pi$
$908$	0	0
$909$	−18509.7	−0.675390
$910$	0	0
$911$	−40437.0	−1.47062	−0.735311	−	0.677730i	$-0.762964\pi$
$911$	−40437.0	−1.47062	−0.735311	+	0.677730i	$0.762964\pi$
$912$	0	0
$913$	−24476.6	−0.887248
$914$	0	0
$915$	13616.4	0.491962
$916$	0	0
$917$	9409.56	0.338856
$918$	0	0
$919$	−12761.1	−0.458052	−0.229026	−	0.973420i	$-0.573554\pi$
$919$	−12761.1	−0.458052	−0.229026	+	0.973420i	$0.573554\pi$
$920$	0	0
$921$	91023.6	3.25660
$922$	0	0
$923$	1410.11	0.0502864
$924$	0	0
$925$	74623.0	2.65253
$926$	0	0
$927$	3923.77	0.139022
$928$	0	0
$929$	46295.0	1.63497	0.817487	−	0.575947i	$-0.195367\pi$
$929$	46295.0	1.63497	0.817487	+	0.575947i	$0.195367\pi$
$930$	0	0
$931$	−1732.73	−0.0609968
$932$	0	0
$933$	18016.2	0.632179
$934$	0	0
$935$	−21539.8	−0.753396
$936$	0	0
$937$	39313.6	1.37067	0.685336	−	0.728227i	$-0.259656\pi$
$937$	39313.6	1.37067	0.685336	+	0.728227i	$0.259656\pi$
$938$	0	0
$939$	−87530.5	−3.04201
$940$	0	0
$941$	23082.1	0.799634	0.399817	−	0.916595i	$-0.369074\pi$
$941$	23082.1	0.799634	0.399817	+	0.916595i	$0.369074\pi$
$942$	0	0
$943$	6557.04	0.226433
$944$	0	0
$945$	169160.	5.82305
$946$	0	0
$947$	10979.5	0.376754	0.188377	−	0.982097i	$-0.439677\pi$
$947$	10979.5	0.376754	0.188377	+	0.982097i	$0.439677\pi$
$948$	0	0
$949$	31845.6	1.08931
$950$	0	0
$951$	85344.3	2.91007
$952$	0	0
$953$	−34231.0	−1.16354	−0.581768	−	0.813355i	$-0.697639\pi$
$953$	−34231.0	−1.16354	−0.581768	+	0.813355i	$0.697639\pi$
$954$	0	0
$955$	39478.3	1.33768
$956$	0	0
$957$	−128469.	−4.33940
$958$	0	0
$959$	440.728	0.0148403
$960$	0	0
$961$	12536.1	0.420802
$962$	0	0
$963$	−11451.8	−0.383208
$964$	0	0
$965$	−53107.2	−1.77159
$966$	0	0
$967$	−53802.5	−1.78922	−0.894608	−	0.446852i	$-0.852545\pi$
$967$	−53802.5	−1.78922	−0.894608	+	0.446852i	$0.852545\pi$
$968$	0	0
$969$	5196.13	0.172264
$970$	0	0
$971$	14560.0	0.481207	0.240603	−	0.970623i	$-0.422655\pi$
$971$	14560.0	0.481207	0.240603	+	0.970623i	$0.422655\pi$
$972$	0	0
$973$	−40011.9	−1.31832
$974$	0	0
$975$	−68402.2	−2.24679
$976$	0	0
$977$	−24105.6	−0.789363	−0.394682	−	0.918818i	$-0.629145\pi$
$977$	−24105.6	−0.789363	−0.394682	+	0.918818i	$0.629145\pi$
$978$	0	0
$979$	−26811.2	−0.875270
$980$	0	0
$981$	−110317.	−3.59037
$982$	0	0
$983$	41756.9	1.35487	0.677435	−	0.735583i	$-0.263091\pi$
$983$	41756.9	1.35487	0.677435	+	0.735583i	$0.263091\pi$
$984$	0	0
$985$	28753.8	0.930123
$986$	0	0
$987$	45458.7	1.46603
$988$	0	0
$989$	1353.37	0.0435134
$990$	0	0
$991$	7388.19	0.236825	0.118413	−	0.992964i	$-0.462219\pi$
$991$	7388.19	0.236825	0.118413	+	0.992964i	$0.462219\pi$
$992$	0	0
$993$	−69546.2	−2.22254
$994$	0	0
$995$	37411.5	1.19198
$996$	0	0
$997$	−12635.8	−0.401385	−0.200693	−	0.979654i	$-0.564319\pi$
$997$	−12635.8	−0.401385	−0.200693	+	0.979654i	$0.564319\pi$
$998$	0	0
$999$	−149796.	−4.74407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.4.a.bi.1.1		8
4.3	odd	2	inner	1088.4.a.bi.1.8		8
8.3	odd	2		544.4.a.m.1.1	✓	8
8.5	even	2		544.4.a.m.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.4.a.m.1.1	✓	8	8.3	odd	2
544.4.a.m.1.8	yes	8	8.5	even	2
1088.4.a.bi.1.1		8	1.1	even	1	trivial
1088.4.a.bi.1.8		8	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q-9.96818 q^{3} -18.7155 q^{5} +19.9877 q^{7} +72.3646 q^{9} +O(q^{10})\)	\(q-9.96818 q^{3} -18.7155 q^{5} +19.9877 q^{7} +72.3646 q^{9} +67.7002 q^{11} +30.4613 q^{13} +186.560 q^{15} +17.0000 q^{17} -30.6630 q^{19} -199.241 q^{21} +30.3597 q^{23} +225.271 q^{25} -452.202 q^{27} +190.367 q^{29} -205.736 q^{31} -674.847 q^{33} -374.081 q^{35} +331.258 q^{37} -303.644 q^{39} +215.978 q^{41} +44.5779 q^{43} -1354.34 q^{45} -228.159 q^{47} +56.5089 q^{49} -169.459 q^{51} -207.572 q^{53} -1267.05 q^{55} +305.655 q^{57} +218.922 q^{59} +72.9869 q^{61} +1446.40 q^{63} -570.099 q^{65} +774.974 q^{67} -302.631 q^{69} +46.2918 q^{71} +1045.44 q^{73} -2245.54 q^{75} +1353.17 q^{77} -866.790 q^{79} +2553.79 q^{81} -361.544 q^{83} -318.164 q^{85} -1897.61 q^{87} -396.028 q^{89} +608.852 q^{91} +2050.81 q^{93} +573.875 q^{95} +931.215 q^{97} +4899.09 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{5} + 136 q^{9}+O(q^{10}) \) 8 * q - 16 * q^5 + 136 * q^9	\( 8 q - 16 q^{5} + 136 q^{9} + 56 q^{13} + 136 q^{17} - 312 q^{21} + 600 q^{25} + 128 q^{29} - 304 q^{33} + 112 q^{37} + 1328 q^{41} - 2592 q^{45} + 1624 q^{49} - 1216 q^{53} + 2560 q^{57} - 752 q^{61} + 1824 q^{65} - 1448 q^{69} + 3952 q^{73} + 1480 q^{77} + 6824 q^{81} - 272 q^{85} + 2304 q^{89} + 2680 q^{93} + 4560 q^{97}+O(q^{100}) \) 8 * q - 16 * q^5 + 136 * q^9 + 56 * q^13 + 136 * q^17 - 312 * q^21 + 600 * q^25 + 128 * q^29 - 304 * q^33 + 112 * q^37 + 1328 * q^41 - 2592 * q^45 + 1624 * q^49 - 1216 * q^53 + 2560 * q^57 - 752 * q^61 + 1824 * q^65 - 1448 * q^69 + 3952 * q^73 + 1480 * q^77 + 6824 * q^81 - 272 * q^85 + 2304 * q^89 + 2680 * q^93 + 4560 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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