LMFDB - Embedded newform 1288.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1288,4,Mod(1,1288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1288, 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("1288.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1288 = 2^{3} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1288.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:	$75.9944600874$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 197 x^{10} + 551 x^{9} + 13776 x^{8} - 35332 x^{7} - 433468 x^{6} + 942840 x^{5} + \cdots + 79691136 $ x^12 - 3x^11 - 197x^10 + 551x^9 + 13776x^8 - 35332x^7 - 433468x^6 + 942840x^5 + 6378956x^4 - 9860516x^3 - 40320768x^2 + 25815472*x + 79691136
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.50318$ of defining polynomial
Character	$\chi$	$=$	1288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.50318 q^{3} -17.0723 q^{5} -7.00000 q^{7} +15.2913 q^{9} +O(q^{10})$	$q-6.50318 q^{3} -17.0723 q^{5} -7.00000 q^{7} +15.2913 q^{9} -45.9151 q^{11} +40.2901 q^{13} +111.024 q^{15} -89.8566 q^{17} +31.5164 q^{19} +45.5223 q^{21} +23.0000 q^{23} +166.464 q^{25} +76.1435 q^{27} +175.700 q^{29} +279.265 q^{31} +298.594 q^{33} +119.506 q^{35} -290.401 q^{37} -262.014 q^{39} -55.0655 q^{41} -87.5806 q^{43} -261.058 q^{45} -541.475 q^{47} +49.0000 q^{49} +584.354 q^{51} +583.446 q^{53} +783.876 q^{55} -204.957 q^{57} +495.842 q^{59} +540.343 q^{61} -107.039 q^{63} -687.846 q^{65} +926.971 q^{67} -149.573 q^{69} +1078.47 q^{71} -265.881 q^{73} -1082.54 q^{75} +321.405 q^{77} -510.279 q^{79} -908.041 q^{81} -713.836 q^{83} +1534.06 q^{85} -1142.61 q^{87} -1228.77 q^{89} -282.031 q^{91} -1816.11 q^{93} -538.058 q^{95} +927.405 q^{97} -702.103 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} - 14 q^{5} - 84 q^{7} + 79 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 - 14 * q^5 - 84 * q^7 + 79 * q^9	$ 12 q - 3 q^{3} - 14 q^{5} - 84 q^{7} + 79 q^{9} + 8 q^{11} - 57 q^{13} + 10 q^{15} - 86 q^{17} + 160 q^{19} + 21 q^{21} + 276 q^{23} + 256 q^{25} + 15 q^{27} + 153 q^{29} + 253 q^{31} - 238 q^{33} + 98 q^{35} - 530 q^{37} + 123 q^{39} - 917 q^{41} - 174 q^{43} - 776 q^{45} - 571 q^{47} + 588 q^{49} + 792 q^{51} - 582 q^{53} - 388 q^{55} - 1214 q^{57} + 288 q^{59} - 1274 q^{61} - 553 q^{63} - 1446 q^{65} + 892 q^{67} - 69 q^{69} + 353 q^{71} - 1907 q^{73} - 217 q^{75} - 56 q^{77} + 102 q^{79} - 820 q^{81} - 2342 q^{83} - 2792 q^{85} - 2071 q^{87} - 1658 q^{89} + 399 q^{91} - 4729 q^{93} - 780 q^{95} - 3998 q^{97} - 1106 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 - 14 * q^5 - 84 * q^7 + 79 * q^9 + 8 * q^11 - 57 * q^13 + 10 * q^15 - 86 * q^17 + 160 * q^19 + 21 * q^21 + 276 * q^23 + 256 * q^25 + 15 * q^27 + 153 * q^29 + 253 * q^31 - 238 * q^33 + 98 * q^35 - 530 * q^37 + 123 * q^39 - 917 * q^41 - 174 * q^43 - 776 * q^45 - 571 * q^47 + 588 * q^49 + 792 * q^51 - 582 * q^53 - 388 * q^55 - 1214 * q^57 + 288 * q^59 - 1274 * q^61 - 553 * q^63 - 1446 * q^65 + 892 * q^67 - 69 * q^69 + 353 * q^71 - 1907 * q^73 - 217 * q^75 - 56 * q^77 + 102 * q^79 - 820 * q^81 - 2342 * q^83 - 2792 * q^85 - 2071 * q^87 - 1658 * q^89 + 399 * q^91 - 4729 * q^93 - 780 * q^95 - 3998 * q^97 - 1106 * 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
$3$	−6.50318	−1.25154	−0.625769	−	0.780009i	$-0.715215\pi$
$3$	−6.50318	−1.25154	−0.625769	+	0.780009i	$0.715215\pi$
$4$	0	0
$5$	−17.0723	−1.52699	−0.763497	−	0.645811i	$-0.776519\pi$
$5$	−17.0723	−1.52699	−0.763497	+	0.645811i	$0.776519\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	15.2913	0.566346
$10$	0	0
$11$	−45.9151	−1.25854	−0.629269	−	0.777188i	$-0.716645\pi$
$11$	−45.9151	−1.25854	−0.629269	+	0.777188i	$0.716645\pi$
$12$	0	0
$13$	40.2901	0.859575	0.429787	−	0.902930i	$-0.358589\pi$
$13$	40.2901	0.859575	0.429787	+	0.902930i	$0.358589\pi$
$14$	0	0
$15$	111.024	1.91109
$16$	0	0
$17$	−89.8566	−1.28197	−0.640983	−	0.767555i	$-0.721473\pi$
$17$	−89.8566	−1.28197	−0.640983	+	0.767555i	$0.721473\pi$
$18$	0	0
$19$	31.5164	0.380545	0.190273	−	0.981731i	$-0.439063\pi$
$19$	31.5164	0.380545	0.190273	+	0.981731i	$0.439063\pi$
$20$	0	0
$21$	45.5223	0.473037
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	166.464	1.33171
$26$	0	0
$27$	76.1435	0.542734
$28$	0	0
$29$	175.700	1.12506	0.562528	−	0.826778i	$-0.309829\pi$
$29$	175.700	1.12506	0.562528	+	0.826778i	$0.309829\pi$
$30$	0	0
$31$	279.265	1.61799	0.808993	−	0.587819i	$-0.200013\pi$
$31$	279.265	1.61799	0.808993	+	0.587819i	$0.200013\pi$
$32$	0	0
$33$	298.594	1.57511
$34$	0	0
$35$	119.506	0.577149
$36$	0	0
$37$	−290.401	−1.29031	−0.645156	−	0.764051i	$-0.723208\pi$
$37$	−290.401	−1.29031	−0.645156	+	0.764051i	$0.723208\pi$
$38$	0	0
$39$	−262.014	−1.07579
$40$	0	0
$41$	−55.0655	−0.209751	−0.104875	−	0.994485i	$-0.533444\pi$
$41$	−55.0655	−0.209751	−0.104875	+	0.994485i	$0.533444\pi$
$42$	0	0
$43$	−87.5806	−0.310603	−0.155301	−	0.987867i	$-0.549635\pi$
$43$	−87.5806	−0.310603	−0.155301	+	0.987867i	$0.549635\pi$
$44$	0	0
$45$	−261.058	−0.864807
$46$	0	0
$47$	−541.475	−1.68047	−0.840236	−	0.542220i	$-0.817584\pi$
$47$	−541.475	−1.68047	−0.840236	+	0.542220i	$0.817584\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	584.354	1.60443
$52$	0	0
$53$	583.446	1.51212	0.756061	−	0.654501i	$-0.227121\pi$
$53$	583.446	1.51212	0.756061	+	0.654501i	$0.227121\pi$
$54$	0	0
$55$	783.876	1.92178
$56$	0	0
$57$	−204.957	−0.476266
$58$	0	0
$59$	495.842	1.09412	0.547060	−	0.837093i	$-0.315747\pi$
$59$	495.842	1.09412	0.547060	+	0.837093i	$0.315747\pi$
$60$	0	0
$61$	540.343	1.13416	0.567080	−	0.823662i	$-0.308073\pi$
$61$	540.343	1.13416	0.567080	+	0.823662i	$0.308073\pi$
$62$	0	0
$63$	−107.039	−0.214059
$64$	0	0
$65$	−687.846	−1.31257
$66$	0	0
$67$	926.971	1.69026	0.845131	−	0.534560i	$-0.179523\pi$
$67$	926.971	1.69026	0.845131	+	0.534560i	$0.179523\pi$
$68$	0	0
$69$	−149.573	−0.260964
$70$	0	0
$71$	1078.47	1.80269	0.901344	−	0.433104i	$-0.142582\pi$
$71$	1078.47	1.80269	0.901344	+	0.433104i	$0.142582\pi$
$72$	0	0
$73$	−265.881	−0.426287	−0.213144	−	0.977021i	$-0.568370\pi$
$73$	−265.881	−0.426287	−0.213144	+	0.977021i	$0.568370\pi$
$74$	0	0
$75$	−1082.54	−1.66669
$76$	0	0
$77$	321.405	0.475682
$78$	0	0
$79$	−510.279	−0.726720	−0.363360	−	0.931649i	$-0.618370\pi$
$79$	−510.279	−0.726720	−0.363360	+	0.931649i	$0.618370\pi$
$80$	0	0
$81$	−908.041	−1.24560
$82$	0	0
$83$	−713.836	−0.944020	−0.472010	−	0.881593i	$-0.656471\pi$
$83$	−713.836	−0.944020	−0.472010	+	0.881593i	$0.656471\pi$
$84$	0	0
$85$	1534.06	1.95755
$86$	0	0
$87$	−1142.61	−1.40805
$88$	0	0
$89$	−1228.77	−1.46347	−0.731736	−	0.681589i	$-0.761289\pi$
$89$	−1228.77	−1.46347	−0.731736	+	0.681589i	$0.761289\pi$
$90$	0	0
$91$	−282.031	−0.324889
$92$	0	0
$93$	−1816.11	−2.02497
$94$	0	0
$95$	−538.058	−0.581090
$96$	0	0
$97$	927.405	0.970760	0.485380	−	0.874303i	$-0.338681\pi$
$97$	927.405	0.970760	0.485380	+	0.874303i	$0.338681\pi$
$98$	0	0
$99$	−702.103	−0.712767
$100$	0	0
$101$	−757.746	−0.746520	−0.373260	−	0.927727i	$-0.621760\pi$
$101$	−757.746	−0.746520	−0.373260	+	0.927727i	$0.621760\pi$
$102$	0	0
$103$	−1278.87	−1.22340	−0.611702	−	0.791088i	$-0.709515\pi$
$103$	−1278.87	−1.22340	−0.611702	+	0.791088i	$0.709515\pi$
$104$	0	0
$105$	−777.170	−0.722324
$106$	0	0
$107$	1302.26	1.17658	0.588289	−	0.808651i	$-0.299802\pi$
$107$	1302.26	1.17658	0.588289	+	0.808651i	$0.299802\pi$
$108$	0	0
$109$	1195.72	1.05073	0.525365	−	0.850877i	$-0.323929\pi$
$109$	1195.72	1.05073	0.525365	+	0.850877i	$0.323929\pi$
$110$	0	0
$111$	1888.53	1.61487
$112$	0	0
$113$	−1393.19	−1.15983	−0.579913	−	0.814679i	$-0.696913\pi$
$113$	−1393.19	−1.15983	−0.579913	+	0.814679i	$0.696913\pi$
$114$	0	0
$115$	−392.663	−0.318400
$116$	0	0
$117$	616.090	0.486817
$118$	0	0
$119$	628.996	0.484538
$120$	0	0
$121$	777.192	0.583916
$122$	0	0
$123$	358.101	0.262511
$124$	0	0
$125$	−707.883	−0.506520
$126$	0	0
$127$	1633.26	1.14117	0.570585	−	0.821238i	$-0.306716\pi$
$127$	1633.26	1.14117	0.570585	+	0.821238i	$0.306716\pi$
$128$	0	0
$129$	569.553	0.388731
$130$	0	0
$131$	−2794.51	−1.86380	−0.931898	−	0.362722i	$-0.881848\pi$
$131$	−2794.51	−1.86380	−0.931898	+	0.362722i	$0.881848\pi$
$132$	0	0
$133$	−220.615	−0.143833
$134$	0	0
$135$	−1299.95	−0.828752
$136$	0	0
$137$	−2276.20	−1.41948	−0.709740	−	0.704464i	$-0.751187\pi$
$137$	−2276.20	−1.41948	−0.709740	+	0.704464i	$0.751187\pi$
$138$	0	0
$139$	−2801.40	−1.70943	−0.854717	−	0.519094i	$-0.826270\pi$
$139$	−2801.40	−1.70943	−0.854717	+	0.519094i	$0.826270\pi$
$140$	0	0
$141$	3521.31	2.10317
$142$	0	0
$143$	−1849.92	−1.08181
$144$	0	0
$145$	−2999.60	−1.71795
$146$	0	0
$147$	−318.656	−0.178791
$148$	0	0
$149$	1042.74	0.573317	0.286658	−	0.958033i	$-0.407456\pi$
$149$	1042.74	0.573317	0.286658	+	0.958033i	$0.407456\pi$
$150$	0	0
$151$	2441.83	1.31598	0.657992	−	0.753025i	$-0.271406\pi$
$151$	2441.83	1.31598	0.657992	+	0.753025i	$0.271406\pi$
$152$	0	0
$153$	−1374.03	−0.726036
$154$	0	0
$155$	−4767.71	−2.47065
$156$	0	0
$157$	71.6294	0.0364118	0.0182059	−	0.999834i	$-0.494205\pi$
$157$	71.6294	0.0364118	0.0182059	+	0.999834i	$0.494205\pi$
$158$	0	0
$159$	−3794.26	−1.89248
$160$	0	0
$161$	−161.000	−0.0788110
$162$	0	0
$163$	3362.14	1.61560	0.807801	−	0.589456i	$-0.200658\pi$
$163$	3362.14	1.61560	0.807801	+	0.589456i	$0.200658\pi$
$164$	0	0
$165$	−5097.69	−2.40518
$166$	0	0
$167$	3582.33	1.65993	0.829966	−	0.557814i	$-0.188360\pi$
$167$	3582.33	1.65993	0.829966	+	0.557814i	$0.188360\pi$
$168$	0	0
$169$	−573.705	−0.261131
$170$	0	0
$171$	481.928	0.215520
$172$	0	0
$173$	−2719.58	−1.19518	−0.597589	−	0.801803i	$-0.703875\pi$
$173$	−2719.58	−1.19518	−0.597589	+	0.801803i	$0.703875\pi$
$174$	0	0
$175$	−1165.25	−0.503339
$176$	0	0
$177$	−3224.55	−1.36933
$178$	0	0
$179$	900.732	0.376111	0.188056	−	0.982158i	$-0.439782\pi$
$179$	900.732	0.376111	0.188056	+	0.982158i	$0.439782\pi$
$180$	0	0
$181$	−629.761	−0.258617	−0.129309	−	0.991604i	$-0.541276\pi$
$181$	−629.761	−0.258617	−0.129309	+	0.991604i	$0.541276\pi$
$182$	0	0
$183$	−3513.95	−1.41944
$184$	0	0
$185$	4957.81	1.97030
$186$	0	0
$187$	4125.77	1.61340
$188$	0	0
$189$	−533.005	−0.205134
$190$	0	0
$191$	−1469.56	−0.556720	−0.278360	−	0.960477i	$-0.589791\pi$
$191$	−1469.56	−0.556720	−0.278360	+	0.960477i	$0.589791\pi$
$192$	0	0
$193$	−3292.34	−1.22791	−0.613957	−	0.789339i	$-0.710423\pi$
$193$	−3292.34	−1.22791	−0.613957	+	0.789339i	$0.710423\pi$
$194$	0	0
$195$	4473.18	1.64272
$196$	0	0
$197$	−3686.76	−1.33336	−0.666678	−	0.745346i	$-0.732284\pi$
$197$	−3686.76	−1.33336	−0.666678	+	0.745346i	$0.732284\pi$
$198$	0	0
$199$	941.696	0.335453	0.167726	−	0.985834i	$-0.446358\pi$
$199$	941.696	0.335453	0.167726	+	0.985834i	$0.446358\pi$
$200$	0	0
$201$	−6028.26	−2.11543
$202$	0	0
$203$	−1229.90	−0.425231
$204$	0	0
$205$	940.095	0.320288
$206$	0	0
$207$	351.701	0.118091
$208$	0	0
$209$	−1447.08	−0.478930
$210$	0	0
$211$	5858.83	1.91156	0.955779	−	0.294085i	$-0.0950149\pi$
$211$	5858.83	1.91156	0.955779	+	0.294085i	$0.0950149\pi$
$212$	0	0
$213$	−7013.48	−2.25613
$214$	0	0
$215$	1495.20	0.474289
$216$	0	0
$217$	−1954.86	−0.611541
$218$	0	0
$219$	1729.07	0.533515
$220$	0	0
$221$	−3620.33	−1.10195
$222$	0	0
$223$	−1765.82	−0.530260	−0.265130	−	0.964213i	$-0.585415\pi$
$223$	−1765.82	−0.530260	−0.265130	+	0.964213i	$0.585415\pi$
$224$	0	0
$225$	2545.45	0.754209
$226$	0	0
$227$	−755.329	−0.220850	−0.110425	−	0.993884i	$-0.535221\pi$
$227$	−755.329	−0.220850	−0.110425	+	0.993884i	$0.535221\pi$
$228$	0	0
$229$	−3740.46	−1.07937	−0.539686	−	0.841866i	$-0.681457\pi$
$229$	−3740.46	−1.07937	−0.539686	+	0.841866i	$0.681457\pi$
$230$	0	0
$231$	−2090.16	−0.595334
$232$	0	0
$233$	834.229	0.234558	0.117279	−	0.993099i	$-0.462583\pi$
$233$	834.229	0.234558	0.117279	+	0.993099i	$0.462583\pi$
$234$	0	0
$235$	9244.23	2.56607
$236$	0	0
$237$	3318.43	0.909517
$238$	0	0
$239$	−2626.78	−0.710930	−0.355465	−	0.934690i	$-0.615677\pi$
$239$	−2626.78	−0.710930	−0.355465	+	0.934690i	$0.615677\pi$
$240$	0	0
$241$	−6345.17	−1.69597	−0.847984	−	0.530021i	$-0.822184\pi$
$241$	−6345.17	−1.69597	−0.847984	+	0.530021i	$0.822184\pi$
$242$	0	0
$243$	3849.28	1.01618
$244$	0	0
$245$	−836.543	−0.218142
$246$	0	0
$247$	1269.80	0.327107
$248$	0	0
$249$	4642.20	1.18148
$250$	0	0
$251$	4390.18	1.10401	0.552003	−	0.833842i	$-0.313864\pi$
$251$	4390.18	1.10401	0.552003	+	0.833842i	$0.313864\pi$
$252$	0	0
$253$	−1056.05	−0.262423
$254$	0	0
$255$	−9976.27	−2.44995
$256$	0	0
$257$	4393.84	1.06646	0.533230	−	0.845970i	$-0.320978\pi$
$257$	4393.84	1.06646	0.533230	+	0.845970i	$0.320978\pi$
$258$	0	0
$259$	2032.80	0.487692
$260$	0	0
$261$	2686.68	0.637171
$262$	0	0
$263$	3796.69	0.890168	0.445084	−	0.895489i	$-0.353174\pi$
$263$	3796.69	0.890168	0.445084	+	0.895489i	$0.353174\pi$
$264$	0	0
$265$	−9960.78	−2.30900
$266$	0	0
$267$	7990.89	1.83159
$268$	0	0
$269$	3715.83	0.842224	0.421112	−	0.907009i	$-0.361640\pi$
$269$	3715.83	0.842224	0.421112	+	0.907009i	$0.361640\pi$
$270$	0	0
$271$	1572.03	0.352376	0.176188	−	0.984357i	$-0.443623\pi$
$271$	1572.03	0.352376	0.176188	+	0.984357i	$0.443623\pi$
$272$	0	0
$273$	1834.10	0.406610
$274$	0	0
$275$	−7643.20	−1.67601
$276$	0	0
$277$	−5801.93	−1.25850	−0.629249	−	0.777204i	$-0.716638\pi$
$277$	−5801.93	−1.25850	−0.629249	+	0.777204i	$0.716638\pi$
$278$	0	0
$279$	4270.34	0.916339
$280$	0	0
$281$	−3118.77	−0.662101	−0.331051	−	0.943613i	$-0.607403\pi$
$281$	−3118.77	−0.662101	−0.331051	+	0.943613i	$0.607403\pi$
$282$	0	0
$283$	5470.36	1.14904	0.574522	−	0.818489i	$-0.305188\pi$
$283$	5470.36	1.14904	0.574522	+	0.818489i	$0.305188\pi$
$284$	0	0
$285$	3499.09	0.727256
$286$	0	0
$287$	385.458	0.0792784
$288$	0	0
$289$	3161.21	0.643438
$290$	0	0
$291$	−6031.08	−1.21494
$292$	0	0
$293$	−120.744	−0.0240748	−0.0120374	−	0.999928i	$-0.503832\pi$
$293$	−120.744	−0.0240748	−0.0120374	+	0.999928i	$0.503832\pi$
$294$	0	0
$295$	−8465.17	−1.67072
$296$	0	0
$297$	−3496.13	−0.683052
$298$	0	0
$299$	926.673	0.179234
$300$	0	0
$301$	613.064	0.117397
$302$	0	0
$303$	4927.76	0.934297
$304$	0	0
$305$	−9224.90	−1.73186
$306$	0	0
$307$	−8101.89	−1.50619	−0.753093	−	0.657914i	$-0.771439\pi$
$307$	−8101.89	−1.50619	−0.753093	+	0.657914i	$0.771439\pi$
$308$	0	0
$309$	8316.71	1.53114
$310$	0	0
$311$	1559.19	0.284288	0.142144	−	0.989846i	$-0.454600\pi$
$311$	1559.19	0.284288	0.142144	+	0.989846i	$0.454600\pi$
$312$	0	0
$313$	7720.64	1.39424	0.697119	−	0.716955i	$-0.254465\pi$
$313$	7720.64	1.39424	0.697119	+	0.716955i	$0.254465\pi$
$314$	0	0
$315$	1827.41	0.326866
$316$	0	0
$317$	7327.09	1.29820	0.649101	−	0.760702i	$-0.275145\pi$
$317$	7327.09	1.29820	0.649101	+	0.760702i	$0.275145\pi$
$318$	0	0
$319$	−8067.26	−1.41592
$320$	0	0
$321$	−8468.81	−1.47253
$322$	0	0
$323$	−2831.96	−0.487846
$324$	0	0
$325$	6706.85	1.14470
$326$	0	0
$327$	−7776.01	−1.31503
$328$	0	0
$329$	3790.32	0.635159
$330$	0	0
$331$	9332.74	1.54977	0.774885	−	0.632102i	$-0.217808\pi$
$331$	9332.74	1.54977	0.774885	+	0.632102i	$0.217808\pi$
$332$	0	0
$333$	−4440.61	−0.730763
$334$	0	0
$335$	−15825.5	−2.58102
$336$	0	0
$337$	6848.27	1.10697	0.553485	−	0.832859i	$-0.313298\pi$
$337$	6848.27	1.10697	0.553485	+	0.832859i	$0.313298\pi$
$338$	0	0
$339$	9060.16	1.45156
$340$	0	0
$341$	−12822.5	−2.03629
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	2553.56	0.398490
$346$	0	0
$347$	5710.81	0.883494	0.441747	−	0.897140i	$-0.354359\pi$
$347$	5710.81	0.883494	0.441747	+	0.897140i	$0.354359\pi$
$348$	0	0
$349$	64.6322	0.00991312	0.00495656	−	0.999988i	$-0.498422\pi$
$349$	64.6322	0.00991312	0.00495656	+	0.999988i	$0.498422\pi$
$350$	0	0
$351$	3067.83	0.466521
$352$	0	0
$353$	54.9660	0.00828767	0.00414383	−	0.999991i	$-0.498681\pi$
$353$	54.9660	0.00828767	0.00414383	+	0.999991i	$0.498681\pi$
$354$	0	0
$355$	−18412.0	−2.75269
$356$	0	0
$357$	−4090.48	−0.606417
$358$	0	0
$359$	1793.26	0.263635	0.131817	−	0.991274i	$-0.457919\pi$
$359$	1793.26	0.263635	0.131817	+	0.991274i	$0.457919\pi$
$360$	0	0
$361$	−5865.72	−0.855185
$362$	0	0
$363$	−5054.22	−0.730793
$364$	0	0
$365$	4539.20	0.650938
$366$	0	0
$367$	−12141.1	−1.72687	−0.863433	−	0.504463i	$-0.831690\pi$
$367$	−12141.1	−1.72687	−0.863433	+	0.504463i	$0.831690\pi$
$368$	0	0
$369$	−842.025	−0.118792
$370$	0	0
$371$	−4084.12	−0.571529
$372$	0	0
$373$	−3958.76	−0.549536	−0.274768	−	0.961511i	$-0.588601\pi$
$373$	−3958.76	−0.549536	−0.274768	+	0.961511i	$0.588601\pi$
$374$	0	0
$375$	4603.49	0.633929
$376$	0	0
$377$	7078.96	0.967069
$378$	0	0
$379$	5709.86	0.773868	0.386934	−	0.922107i	$-0.373534\pi$
$379$	5709.86	0.773868	0.386934	+	0.922107i	$0.373534\pi$
$380$	0	0
$381$	−10621.4	−1.42822
$382$	0	0
$383$	−5187.12	−0.692036	−0.346018	−	0.938228i	$-0.612466\pi$
$383$	−5187.12	−0.692036	−0.346018	+	0.938228i	$0.612466\pi$
$384$	0	0
$385$	−5487.13	−0.726364
$386$	0	0
$387$	−1339.23	−0.175909
$388$	0	0
$389$	−2776.78	−0.361924	−0.180962	−	0.983490i	$-0.557921\pi$
$389$	−2776.78	−0.361924	−0.180962	+	0.983490i	$0.557921\pi$
$390$	0	0
$391$	−2066.70	−0.267308
$392$	0	0
$393$	18173.2	2.33261
$394$	0	0
$395$	8711.64	1.10970
$396$	0	0
$397$	−1596.36	−0.201811	−0.100906	−	0.994896i	$-0.532174\pi$
$397$	−1596.36	−0.201811	−0.100906	+	0.994896i	$0.532174\pi$
$398$	0	0
$399$	1434.70	0.180012
$400$	0	0
$401$	4235.70	0.527484	0.263742	−	0.964593i	$-0.415043\pi$
$401$	4235.70	0.527484	0.263742	+	0.964593i	$0.415043\pi$
$402$	0	0
$403$	11251.6	1.39078
$404$	0	0
$405$	15502.4	1.90202
$406$	0	0
$407$	13333.8	1.62391
$408$	0	0
$409$	−6723.81	−0.812888	−0.406444	−	0.913676i	$-0.633231\pi$
$409$	−6723.81	−0.812888	−0.406444	+	0.913676i	$0.633231\pi$
$410$	0	0
$411$	14802.5	1.77653
$412$	0	0
$413$	−3470.89	−0.413539
$414$	0	0
$415$	12186.8	1.44151
$416$	0	0
$417$	18218.0	2.13942
$418$	0	0
$419$	−12585.8	−1.46744	−0.733722	−	0.679450i	$-0.762219\pi$
$419$	−12585.8	−1.46744	−0.733722	+	0.679450i	$0.762219\pi$
$420$	0	0
$421$	8378.28	0.969911	0.484955	−	0.874539i	$-0.338836\pi$
$421$	8378.28	0.969911	0.484955	+	0.874539i	$0.338836\pi$
$422$	0	0
$423$	−8279.87	−0.951729
$424$	0	0
$425$	−14957.9	−1.70721
$426$	0	0
$427$	−3782.40	−0.428672
$428$	0	0
$429$	12030.4	1.35392
$430$	0	0
$431$	4355.54	0.486773	0.243387	−	0.969929i	$-0.421742\pi$
$431$	4355.54	0.486773	0.243387	+	0.969929i	$0.421742\pi$
$432$	0	0
$433$	8389.90	0.931161	0.465581	−	0.885005i	$-0.345846\pi$
$433$	8389.90	0.931161	0.465581	+	0.885005i	$0.345846\pi$
$434$	0	0
$435$	19506.9	2.15008
$436$	0	0
$437$	724.877	0.0793491
$438$	0	0
$439$	14952.0	1.62556	0.812782	−	0.582568i	$-0.197952\pi$
$439$	14952.0	1.62556	0.812782	+	0.582568i	$0.197952\pi$
$440$	0	0
$441$	749.276	0.0809065
$442$	0	0
$443$	−14877.5	−1.59560	−0.797801	−	0.602921i	$-0.794003\pi$
$443$	−14877.5	−1.59560	−0.797801	+	0.602921i	$0.794003\pi$
$444$	0	0
$445$	20977.9	2.23471
$446$	0	0
$447$	−6781.09	−0.717527
$448$	0	0
$449$	16.1499	0.00169747	0.000848733	−	1.00000i	$-0.499730\pi$
$449$	16.1499	0.00169747	0.000848733		1.00000i	$0.499730\pi$
$450$	0	0
$451$	2528.34	0.263979
$452$	0	0
$453$	−15879.7	−1.64700
$454$	0	0
$455$	4814.92	0.496103
$456$	0	0
$457$	−14761.6	−1.51098	−0.755490	−	0.655160i	$-0.772601\pi$
$457$	−14761.6	−1.51098	−0.755490	+	0.655160i	$0.772601\pi$
$458$	0	0
$459$	−6842.00	−0.695767
$460$	0	0
$461$	17598.7	1.77799	0.888996	−	0.457915i	$-0.151404\pi$
$461$	17598.7	1.77799	0.888996	+	0.457915i	$0.151404\pi$
$462$	0	0
$463$	9402.39	0.943771	0.471886	−	0.881660i	$-0.343574\pi$
$463$	9402.39	0.943771	0.471886	+	0.881660i	$0.343574\pi$
$464$	0	0
$465$	31005.2	3.09212
$466$	0	0
$467$	4583.78	0.454202	0.227101	−	0.973871i	$-0.427075\pi$
$467$	4583.78	0.454202	0.227101	+	0.973871i	$0.427075\pi$
$468$	0	0
$469$	−6488.80	−0.638859
$470$	0	0
$471$	−465.819	−0.0455707
$472$	0	0
$473$	4021.27	0.390905
$474$	0	0
$475$	5246.34	0.506776
$476$	0	0
$477$	8921.67	0.856384
$478$	0	0
$479$	−7361.95	−0.702246	−0.351123	−	0.936329i	$-0.614200\pi$
$479$	−7361.95	−0.702246	−0.351123	+	0.936329i	$0.614200\pi$
$480$	0	0
$481$	−11700.3	−1.10912
$482$	0	0
$483$	1047.01	0.0986350
$484$	0	0
$485$	−15833.0	−1.48235
$486$	0	0
$487$	6971.43	0.648677	0.324339	−	0.945941i	$-0.394858\pi$
$487$	6971.43	0.648677	0.324339	+	0.945941i	$0.394858\pi$
$488$	0	0
$489$	−21864.6	−2.02199
$490$	0	0
$491$	−5798.16	−0.532927	−0.266464	−	0.963845i	$-0.585855\pi$
$491$	−5798.16	−0.532927	−0.266464	+	0.963845i	$0.585855\pi$
$492$	0	0
$493$	−15787.8	−1.44228
$494$	0	0
$495$	11986.5	1.08839
$496$	0	0
$497$	−7549.29	−0.681352
$498$	0	0
$499$	9409.32	0.844126	0.422063	−	0.906567i	$-0.361306\pi$
$499$	9409.32	0.844126	0.422063	+	0.906567i	$0.361306\pi$
$500$	0	0
$501$	−23296.5	−2.07747
$502$	0	0
$503$	−3615.66	−0.320506	−0.160253	−	0.987076i	$-0.551231\pi$
$503$	−3615.66	−0.320506	−0.160253	+	0.987076i	$0.551231\pi$
$504$	0	0
$505$	12936.5	1.13993
$506$	0	0
$507$	3730.91	0.326816
$508$	0	0
$509$	3169.75	0.276025	0.138013	−	0.990430i	$-0.455929\pi$
$509$	3169.75	0.276025	0.138013	+	0.990430i	$0.455929\pi$
$510$	0	0
$511$	1861.16	0.161121
$512$	0	0
$513$	2399.77	0.206535
$514$	0	0
$515$	21833.2	1.86813
$516$	0	0
$517$	24861.8	2.11494
$518$	0	0
$519$	17685.9	1.49581
$520$	0	0
$521$	12254.6	1.03049	0.515244	−	0.857044i	$-0.327701\pi$
$521$	12254.6	1.03049	0.515244	+	0.857044i	$0.327701\pi$
$522$	0	0
$523$	−1024.05	−0.0856184	−0.0428092	−	0.999083i	$-0.513631\pi$
$523$	−1024.05	−0.0856184	−0.0428092	+	0.999083i	$0.513631\pi$
$524$	0	0
$525$	7577.81	0.629948
$526$	0	0
$527$	−25093.8	−2.07420
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	7582.09	0.619651
$532$	0	0
$533$	−2218.60	−0.180297
$534$	0	0
$535$	−22232.5	−1.79663
$536$	0	0
$537$	−5857.62	−0.470717
$538$	0	0
$539$	−2249.84	−0.179791
$540$	0	0
$541$	47.7937	0.00379817	0.00189909	−	0.999998i	$-0.499396\pi$
$541$	47.7937	0.00379817	0.00189909	+	0.999998i	$0.499396\pi$
$542$	0	0
$543$	4095.45	0.323669
$544$	0	0
$545$	−20413.8	−1.60446
$546$	0	0
$547$	12567.5	0.982356	0.491178	−	0.871059i	$-0.336567\pi$
$547$	12567.5	0.982356	0.491178	+	0.871059i	$0.336567\pi$
$548$	0	0
$549$	8262.56	0.642327
$550$	0	0
$551$	5537.42	0.428134
$552$	0	0
$553$	3571.95	0.274674
$554$	0	0
$555$	−32241.5	−2.46590
$556$	0	0
$557$	16277.8	1.23826	0.619132	−	0.785287i	$-0.287485\pi$
$557$	16277.8	1.23826	0.619132	+	0.785287i	$0.287485\pi$
$558$	0	0
$559$	−3528.64	−0.266986
$560$	0	0
$561$	−26830.6	−2.01923
$562$	0	0
$563$	−4904.79	−0.367162	−0.183581	−	0.983005i	$-0.558769\pi$
$563$	−4904.79	−0.367162	−0.183581	+	0.983005i	$0.558769\pi$
$564$	0	0
$565$	23785.0	1.77105
$566$	0	0
$567$	6356.29	0.470792
$568$	0	0
$569$	−10988.1	−0.809573	−0.404786	−	0.914411i	$-0.632654\pi$
$569$	−10988.1	−0.809573	−0.404786	+	0.914411i	$0.632654\pi$
$570$	0	0
$571$	−14364.5	−1.05277	−0.526387	−	0.850245i	$-0.676454\pi$
$571$	−14364.5	−1.05277	−0.526387	+	0.850245i	$0.676454\pi$
$572$	0	0
$573$	9556.80	0.696756
$574$	0	0
$575$	3828.67	0.277681
$576$	0	0
$577$	−15516.9	−1.11955	−0.559773	−	0.828646i	$-0.689112\pi$
$577$	−15516.9	−1.11955	−0.559773	+	0.828646i	$0.689112\pi$
$578$	0	0
$579$	21410.7	1.53678
$580$	0	0
$581$	4996.85	0.356806
$582$	0	0
$583$	−26789.0	−1.90306
$584$	0	0
$585$	−10518.1	−0.743366
$586$	0	0
$587$	−1014.59	−0.0713399	−0.0356699	−	0.999364i	$-0.511357\pi$
$587$	−1014.59	−0.0713399	−0.0356699	+	0.999364i	$0.511357\pi$
$588$	0	0
$589$	8801.44	0.615716
$590$	0	0
$591$	23975.7	1.66874
$592$	0	0
$593$	8997.19	0.623053	0.311526	−	0.950237i	$-0.399160\pi$
$593$	8997.19	0.623053	0.311526	+	0.950237i	$0.399160\pi$
$594$	0	0
$595$	−10738.4	−0.739886
$596$	0	0
$597$	−6124.02	−0.419831
$598$	0	0
$599$	6652.35	0.453769	0.226884	−	0.973922i	$-0.427146\pi$
$599$	6652.35	0.453769	0.226884	+	0.973922i	$0.427146\pi$
$600$	0	0
$601$	27260.1	1.85019	0.925094	−	0.379737i	$-0.123986\pi$
$601$	27260.1	1.85019	0.925094	+	0.379737i	$0.123986\pi$
$602$	0	0
$603$	14174.6	0.957272
$604$	0	0
$605$	−13268.5	−0.891636
$606$	0	0
$607$	−19632.4	−1.31278	−0.656389	−	0.754422i	$-0.727917\pi$
$607$	−19632.4	−1.31278	−0.656389	+	0.754422i	$0.727917\pi$
$608$	0	0
$609$	7998.25	0.532193
$610$	0	0
$611$	−21816.1	−1.44449
$612$	0	0
$613$	−25166.3	−1.65817	−0.829084	−	0.559124i	$-0.811138\pi$
$613$	−25166.3	−1.65817	−0.829084	+	0.559124i	$0.811138\pi$
$614$	0	0
$615$	−6113.61	−0.400853
$616$	0	0
$617$	4420.47	0.288430	0.144215	−	0.989546i	$-0.453934\pi$
$617$	4420.47	0.288430	0.144215	+	0.989546i	$0.453934\pi$
$618$	0	0
$619$	9186.51	0.596506	0.298253	−	0.954487i	$-0.403596\pi$
$619$	9186.51	0.596506	0.298253	+	0.954487i	$0.403596\pi$
$620$	0	0
$621$	1751.30	0.113168
$622$	0	0
$623$	8601.36	0.553140
$624$	0	0
$625$	−8722.77	−0.558257
$626$	0	0
$627$	9410.60	0.599399
$628$	0	0
$629$	26094.4	1.65414
$630$	0	0
$631$	−14023.1	−0.884710	−0.442355	−	0.896840i	$-0.645857\pi$
$631$	−14023.1	−0.884710	−0.442355	+	0.896840i	$0.645857\pi$
$632$	0	0
$633$	−38101.1	−2.39239
$634$	0	0
$635$	−27883.6	−1.74256
$636$	0	0
$637$	1974.22	0.122796
$638$	0	0
$639$	16491.3	1.02094
$640$	0	0
$641$	10716.9	0.660364	0.330182	−	0.943917i	$-0.392890\pi$
$641$	10716.9	0.660364	0.330182	+	0.943917i	$0.392890\pi$
$642$	0	0
$643$	13307.2	0.816151	0.408075	−	0.912948i	$-0.366200\pi$
$643$	13307.2	0.816151	0.408075	+	0.912948i	$0.366200\pi$
$644$	0	0
$645$	−9723.58	−0.593590
$646$	0	0
$647$	7269.11	0.441697	0.220849	−	0.975308i	$-0.429117\pi$
$647$	7269.11	0.441697	0.220849	+	0.975308i	$0.429117\pi$
$648$	0	0
$649$	−22766.6	−1.37699
$650$	0	0
$651$	12712.8	0.765366
$652$	0	0
$653$	−2044.20	−0.122505	−0.0612525	−	0.998122i	$-0.519509\pi$
$653$	−2044.20	−0.122505	−0.0612525	+	0.998122i	$0.519509\pi$
$654$	0	0
$655$	47708.7	2.84600
$656$	0	0
$657$	−4065.67	−0.241426
$658$	0	0
$659$	−6048.29	−0.357524	−0.178762	−	0.983892i	$-0.557209\pi$
$659$	−6048.29	−0.357524	−0.178762	+	0.983892i	$0.557209\pi$
$660$	0	0
$661$	2006.18	0.118051	0.0590253	−	0.998256i	$-0.481201\pi$
$661$	2006.18	0.118051	0.0590253	+	0.998256i	$0.481201\pi$
$662$	0	0
$663$	23543.7	1.37913
$664$	0	0
$665$	3766.40	0.219631
$666$	0	0
$667$	4041.09	0.234590
$668$	0	0
$669$	11483.4	0.663640
$670$	0	0
$671$	−24809.9	−1.42738
$672$	0	0
$673$	5475.40	0.313612	0.156806	−	0.987629i	$-0.449880\pi$
$673$	5475.40	0.313612	0.156806	+	0.987629i	$0.449880\pi$
$674$	0	0
$675$	12675.1	0.722765
$676$	0	0
$677$	−10927.9	−0.620376	−0.310188	−	0.950675i	$-0.600392\pi$
$677$	−10927.9	−0.620376	−0.310188	+	0.950675i	$0.600392\pi$
$678$	0	0
$679$	−6491.84	−0.366913
$680$	0	0
$681$	4912.04	0.276402
$682$	0	0
$683$	−5688.45	−0.318686	−0.159343	−	0.987223i	$-0.550938\pi$
$683$	−5688.45	−0.318686	−0.159343	+	0.987223i	$0.550938\pi$
$684$	0	0
$685$	38859.9	2.16754
$686$	0	0
$687$	24324.9	1.35088
$688$	0	0
$689$	23507.1	1.29978
$690$	0	0
$691$	19952.1	1.09843	0.549215	−	0.835681i	$-0.314927\pi$
$691$	19952.1	1.09843	0.549215	+	0.835681i	$0.314927\pi$
$692$	0	0
$693$	4914.72	0.269401
$694$	0	0
$695$	47826.3	2.61030
$696$	0	0
$697$	4948.00	0.268894
$698$	0	0
$699$	−5425.14	−0.293559
$700$	0	0
$701$	−3260.89	−0.175695	−0.0878474	−	0.996134i	$-0.527999\pi$
$701$	−3260.89	−0.175695	−0.0878474	+	0.996134i	$0.527999\pi$
$702$	0	0
$703$	−9152.38	−0.491022
$704$	0	0
$705$	−60116.9	−3.21154
$706$	0	0
$707$	5304.22	0.282158
$708$	0	0
$709$	−7172.13	−0.379909	−0.189954	−	0.981793i	$-0.560834\pi$
$709$	−7172.13	−0.379909	−0.189954	+	0.981793i	$0.560834\pi$
$710$	0	0
$711$	−7802.85	−0.411575
$712$	0	0
$713$	6423.10	0.337373
$714$	0	0
$715$	31582.5	1.65191
$716$	0	0
$717$	17082.4	0.889755
$718$	0	0
$719$	−25764.5	−1.33638	−0.668188	−	0.743993i	$-0.732930\pi$
$719$	−25764.5	−1.33638	−0.668188	+	0.743993i	$0.732930\pi$
$720$	0	0
$721$	8952.08	0.462403
$722$	0	0
$723$	41263.8	2.12257
$724$	0	0
$725$	29247.6	1.49825
$726$	0	0
$727$	−24281.6	−1.23873	−0.619363	−	0.785105i	$-0.712609\pi$
$727$	−24281.6	−1.23873	−0.619363	+	0.785105i	$0.712609\pi$
$728$	0	0
$729$	−515.437	−0.0261869
$730$	0	0
$731$	7869.70	0.398182
$732$	0	0
$733$	2716.37	0.136878	0.0684390	−	0.997655i	$-0.478198\pi$
$733$	2716.37	0.136878	0.0684390	+	0.997655i	$0.478198\pi$
$734$	0	0
$735$	5440.19	0.273013
$736$	0	0
$737$	−42561.9	−2.12726
$738$	0	0
$739$	−25941.2	−1.29129	−0.645644	−	0.763638i	$-0.723411\pi$
$739$	−25941.2	−1.29129	−0.645644	+	0.763638i	$0.723411\pi$
$740$	0	0
$741$	−8257.74	−0.409387
$742$	0	0
$743$	−29601.3	−1.46160	−0.730799	−	0.682593i	$-0.760852\pi$
$743$	−29601.3	−1.46160	−0.730799	+	0.682593i	$0.760852\pi$
$744$	0	0
$745$	−17801.9	−0.875451
$746$	0	0
$747$	−10915.5	−0.534642
$748$	0	0
$749$	−9115.80	−0.444705
$750$	0	0
$751$	−28807.2	−1.39972	−0.699861	−	0.714279i	$-0.746755\pi$
$751$	−28807.2	−1.39972	−0.699861	+	0.714279i	$0.746755\pi$
$752$	0	0
$753$	−28550.1	−1.38171
$754$	0	0
$755$	−41687.7	−2.00950
$756$	0	0
$757$	2438.15	0.117062	0.0585310	−	0.998286i	$-0.481358\pi$
$757$	2438.15	0.117062	0.0585310	+	0.998286i	$0.481358\pi$
$758$	0	0
$759$	6867.66	0.328432
$760$	0	0
$761$	−7933.73	−0.377920	−0.188960	−	0.981985i	$-0.560512\pi$
$761$	−7933.73	−0.377920	−0.188960	+	0.981985i	$0.560512\pi$
$762$	0	0
$763$	−8370.07	−0.397139
$764$	0	0
$765$	23457.8	1.10865
$766$	0	0
$767$	19977.5	0.940478
$768$	0	0
$769$	−23161.4	−1.08611	−0.543057	−	0.839696i	$-0.682733\pi$
$769$	−23161.4	−1.08611	−0.543057	+	0.839696i	$0.682733\pi$
$770$	0	0
$771$	−28573.9	−1.33472
$772$	0	0
$773$	−9807.68	−0.456349	−0.228175	−	0.973620i	$-0.573276\pi$
$773$	−9807.68	−0.456349	−0.228175	+	0.973620i	$0.573276\pi$
$774$	0	0
$775$	46487.6	2.15469
$776$	0	0
$777$	−13219.7	−0.610365
$778$	0	0
$779$	−1735.47	−0.0798197
$780$	0	0
$781$	−49518.0	−2.26875
$782$	0	0
$783$	13378.4	0.610606
$784$	0	0
$785$	−1222.88	−0.0556006
$786$	0	0
$787$	−8020.60	−0.363283	−0.181641	−	0.983365i	$-0.558141\pi$
$787$	−8020.60	−0.363283	−0.181641	+	0.983365i	$0.558141\pi$
$788$	0	0
$789$	−24690.6	−1.11408
$790$	0	0
$791$	9752.33	0.438373
$792$	0	0
$793$	21770.5	0.974896
$794$	0	0
$795$	64776.7	2.88980
$796$	0	0
$797$	5282.79	0.234788	0.117394	−	0.993085i	$-0.462546\pi$
$797$	5282.79	0.234788	0.117394	+	0.993085i	$0.462546\pi$
$798$	0	0
$799$	48655.1	2.15431
$800$	0	0
$801$	−18789.5	−0.828831
$802$	0	0
$803$	12207.9	0.536499
$804$	0	0
$805$	2748.64	0.120344
$806$	0	0
$807$	−24164.7	−1.05407
$808$	0	0
$809$	−42539.6	−1.84872	−0.924360	−	0.381522i	$-0.875400\pi$
$809$	−42539.6	−1.84872	−0.924360	+	0.381522i	$0.875400\pi$
$810$	0	0
$811$	30280.3	1.31108	0.655540	−	0.755160i	$-0.272441\pi$
$811$	30280.3	1.31108	0.655540	+	0.755160i	$0.272441\pi$
$812$	0	0
$813$	−10223.2	−0.441012
$814$	0	0
$815$	−57399.5	−2.46701
$816$	0	0
$817$	−2760.23	−0.118198
$818$	0	0
$819$	−4312.63	−0.183999
$820$	0	0
$821$	17496.5	0.743765	0.371883	−	0.928280i	$-0.378712\pi$
$821$	17496.5	0.743765	0.371883	+	0.928280i	$0.378712\pi$
$822$	0	0
$823$	42199.7	1.78735	0.893675	−	0.448716i	$-0.148118\pi$
$823$	42199.7	1.78735	0.893675	+	0.448716i	$0.148118\pi$
$824$	0	0
$825$	49705.1	2.09759
$826$	0	0
$827$	−39217.3	−1.64900	−0.824498	−	0.565865i	$-0.808542\pi$
$827$	−39217.3	−1.64900	−0.824498	+	0.565865i	$0.808542\pi$
$828$	0	0
$829$	−8693.31	−0.364211	−0.182106	−	0.983279i	$-0.558291\pi$
$829$	−8693.31	−0.364211	−0.182106	+	0.983279i	$0.558291\pi$
$830$	0	0
$831$	37731.0	1.57506
$832$	0	0
$833$	−4402.97	−0.183138
$834$	0	0
$835$	−61158.6	−2.53471
$836$	0	0
$837$	21264.3	0.878136
$838$	0	0
$839$	6104.75	0.251203	0.125601	−	0.992081i	$-0.459914\pi$
$839$	6104.75	0.251203	0.125601	+	0.992081i	$0.459914\pi$
$840$	0	0
$841$	6481.38	0.265750
$842$	0	0
$843$	20281.9	0.828645
$844$	0	0
$845$	9794.48	0.398746
$846$	0	0
$847$	−5440.35	−0.220700
$848$	0	0
$849$	−35574.7	−1.43807
$850$	0	0
$851$	−6679.22	−0.269049
$852$	0	0
$853$	−27998.3	−1.12385	−0.561925	−	0.827188i	$-0.689939\pi$
$853$	−27998.3	−1.12385	−0.561925	+	0.827188i	$0.689939\pi$
$854$	0	0
$855$	−8227.62	−0.329098
$856$	0	0
$857$	−12915.7	−0.514808	−0.257404	−	0.966304i	$-0.582867\pi$
$857$	−12915.7	−0.514808	−0.257404	+	0.966304i	$0.582867\pi$
$858$	0	0
$859$	11937.1	0.474141	0.237071	−	0.971492i	$-0.423813\pi$
$859$	11937.1	0.474141	0.237071	+	0.971492i	$0.423813\pi$
$860$	0	0
$861$	−2506.71	−0.0992199
$862$	0	0
$863$	−45181.5	−1.78215	−0.891076	−	0.453855i	$-0.850048\pi$
$863$	−45181.5	−1.78215	−0.891076	+	0.453855i	$0.850048\pi$
$864$	0	0
$865$	46429.5	1.82503
$866$	0	0
$867$	−20557.9	−0.805286
$868$	0	0
$869$	23429.5	0.914604
$870$	0	0
$871$	37347.8	1.45291
$872$	0	0
$873$	14181.3	0.549786
$874$	0	0
$875$	4955.18	0.191447
$876$	0	0
$877$	−8157.55	−0.314094	−0.157047	−	0.987591i	$-0.550197\pi$
$877$	−8157.55	−0.314094	−0.157047	+	0.987591i	$0.550197\pi$
$878$	0	0
$879$	785.219	0.0301306
$880$	0	0
$881$	−26779.5	−1.02409	−0.512046	−	0.858958i	$-0.671112\pi$
$881$	−26779.5	−1.02409	−0.512046	+	0.858958i	$0.671112\pi$
$882$	0	0
$883$	3975.16	0.151500	0.0757502	−	0.997127i	$-0.475865\pi$
$883$	3975.16	0.151500	0.0757502	+	0.997127i	$0.475865\pi$
$884$	0	0
$885$	55050.5	2.09096
$886$	0	0
$887$	−6292.10	−0.238183	−0.119091	−	0.992883i	$-0.537998\pi$
$887$	−6292.10	−0.238183	−0.119091	+	0.992883i	$0.537998\pi$
$888$	0	0
$889$	−11432.8	−0.431322
$890$	0	0
$891$	41692.8	1.56763
$892$	0	0
$893$	−17065.3	−0.639496
$894$	0	0
$895$	−15377.6	−0.574319
$896$	0	0
$897$	−6026.32	−0.224318
$898$	0	0
$899$	49066.8	1.82032
$900$	0	0
$901$	−52426.5	−1.93849
$902$	0	0
$903$	−3986.87	−0.146927
$904$	0	0
$905$	10751.5	0.394907
$906$	0	0
$907$	−5656.75	−0.207089	−0.103544	−	0.994625i	$-0.533018\pi$
$907$	−5656.75	−0.207089	−0.103544	+	0.994625i	$0.533018\pi$
$908$	0	0
$909$	−11586.9	−0.422788
$910$	0	0
$911$	7442.11	0.270656	0.135328	−	0.990801i	$-0.456791\pi$
$911$	7442.11	0.270656	0.135328	+	0.990801i	$0.456791\pi$
$912$	0	0
$913$	32775.8	1.18808
$914$	0	0
$915$	59991.2	2.16748
$916$	0	0
$917$	19561.5	0.704448
$918$	0	0
$919$	−18067.2	−0.648510	−0.324255	−	0.945970i	$-0.605114\pi$
$919$	−18067.2	−0.648510	−0.324255	+	0.945970i	$0.605114\pi$
$920$	0	0
$921$	52688.0	1.88505
$922$	0	0
$923$	43451.7	1.54955
$924$	0	0
$925$	−48341.2	−1.71832
$926$	0	0
$927$	−19555.6	−0.692870
$928$	0	0
$929$	39344.3	1.38950	0.694749	−	0.719252i	$-0.255515\pi$
$929$	39344.3	1.38950	0.694749	+	0.719252i	$0.255515\pi$
$930$	0	0
$931$	1544.30	0.0543636
$932$	0	0
$933$	−10139.7	−0.355798
$934$	0	0
$935$	−70436.5	−2.46366
$936$	0	0
$937$	−6915.81	−0.241120	−0.120560	−	0.992706i	$-0.538469\pi$
$937$	−6915.81	−0.241120	−0.120560	+	0.992706i	$0.538469\pi$
$938$	0	0
$939$	−50208.7	−1.74494
$940$	0	0
$941$	−11839.5	−0.410157	−0.205078	−	0.978746i	$-0.565745\pi$
$941$	−11839.5	−0.410157	−0.205078	+	0.978746i	$0.565745\pi$
$942$	0	0
$943$	−1266.51	−0.0437361
$944$	0	0
$945$	9099.62	0.313239
$946$	0	0
$947$	−20081.7	−0.689091	−0.344545	−	0.938770i	$-0.611967\pi$
$947$	−20081.7	−0.689091	−0.344545	+	0.938770i	$0.611967\pi$
$948$	0	0
$949$	−10712.4	−0.366426
$950$	0	0
$951$	−47649.3	−1.62475
$952$	0	0
$953$	22134.0	0.752352	0.376176	−	0.926548i	$-0.377239\pi$
$953$	22134.0	0.752352	0.376176	+	0.926548i	$0.377239\pi$
$954$	0	0
$955$	25088.7	0.850108
$956$	0	0
$957$	52462.8	1.77208
$958$	0	0
$959$	15933.4	0.536513
$960$	0	0
$961$	48198.2	1.61788
$962$	0	0
$963$	19913.2	0.666350
$964$	0	0
$965$	56207.8	1.87502
$966$	0	0
$967$	−48237.2	−1.60414	−0.802070	−	0.597230i	$-0.796268\pi$
$967$	−48237.2	−1.60414	−0.802070	+	0.597230i	$0.796268\pi$
$968$	0	0
$969$	18416.7	0.610558
$970$	0	0
$971$	14566.3	0.481417	0.240709	−	0.970597i	$-0.422620\pi$
$971$	14566.3	0.481417	0.240709	+	0.970597i	$0.422620\pi$
$972$	0	0
$973$	19609.8	0.646106
$974$	0	0
$975$	−43615.8	−1.43264
$976$	0	0
$977$	13866.1	0.454059	0.227030	−	0.973888i	$-0.427099\pi$
$977$	13866.1	0.454059	0.227030	+	0.973888i	$0.427099\pi$
$978$	0	0
$979$	56418.9	1.84183
$980$	0	0
$981$	18284.2	0.595077
$982$	0	0
$983$	−6895.15	−0.223724	−0.111862	−	0.993724i	$-0.535682\pi$
$983$	−6895.15	−0.223724	−0.111862	+	0.993724i	$0.535682\pi$
$984$	0	0
$985$	62941.6	2.03603
$986$	0	0
$987$	−24649.2	−0.794925
$988$	0	0
$989$	−2014.35	−0.0647652
$990$	0	0
$991$	−84.3710	−0.00270447	−0.00135224	−	0.999999i	$-0.500430\pi$
$991$	−84.3710	−0.00270447	−0.00135224	+	0.999999i	$0.500430\pi$
$992$	0	0
$993$	−60692.5	−1.93960
$994$	0	0
$995$	−16076.9	−0.512234
$996$	0	0
$997$	−23704.9	−0.753002	−0.376501	−	0.926416i	$-0.622873\pi$
$997$	−23704.9	−0.753002	−0.376501	+	0.926416i	$0.622873\pi$
$998$	0	0
$999$	−22112.1	−0.700297

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1288.4.a.d.1.2	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.4.a.d.1.2	✓	12	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1288.a (trivial)

\(f(q)\)	\(=\)	\(q-6.50318 q^{3} -17.0723 q^{5} -7.00000 q^{7} +15.2913 q^{9} +O(q^{10})\)	\(q-6.50318 q^{3} -17.0723 q^{5} -7.00000 q^{7} +15.2913 q^{9} -45.9151 q^{11} +40.2901 q^{13} +111.024 q^{15} -89.8566 q^{17} +31.5164 q^{19} +45.5223 q^{21} +23.0000 q^{23} +166.464 q^{25} +76.1435 q^{27} +175.700 q^{29} +279.265 q^{31} +298.594 q^{33} +119.506 q^{35} -290.401 q^{37} -262.014 q^{39} -55.0655 q^{41} -87.5806 q^{43} -261.058 q^{45} -541.475 q^{47} +49.0000 q^{49} +584.354 q^{51} +583.446 q^{53} +783.876 q^{55} -204.957 q^{57} +495.842 q^{59} +540.343 q^{61} -107.039 q^{63} -687.846 q^{65} +926.971 q^{67} -149.573 q^{69} +1078.47 q^{71} -265.881 q^{73} -1082.54 q^{75} +321.405 q^{77} -510.279 q^{79} -908.041 q^{81} -713.836 q^{83} +1534.06 q^{85} -1142.61 q^{87} -1228.77 q^{89} -282.031 q^{91} -1816.11 q^{93} -538.058 q^{95} +927.405 q^{97} -702.103 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} - 14 q^{5} - 84 q^{7} + 79 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 - 14 * q^5 - 84 * q^7 + 79 * q^9	\( 12 q - 3 q^{3} - 14 q^{5} - 84 q^{7} + 79 q^{9} + 8 q^{11} - 57 q^{13} + 10 q^{15} - 86 q^{17} + 160 q^{19} + 21 q^{21} + 276 q^{23} + 256 q^{25} + 15 q^{27} + 153 q^{29} + 253 q^{31} - 238 q^{33} + 98 q^{35} - 530 q^{37} + 123 q^{39} - 917 q^{41} - 174 q^{43} - 776 q^{45} - 571 q^{47} + 588 q^{49} + 792 q^{51} - 582 q^{53} - 388 q^{55} - 1214 q^{57} + 288 q^{59} - 1274 q^{61} - 553 q^{63} - 1446 q^{65} + 892 q^{67} - 69 q^{69} + 353 q^{71} - 1907 q^{73} - 217 q^{75} - 56 q^{77} + 102 q^{79} - 820 q^{81} - 2342 q^{83} - 2792 q^{85} - 2071 q^{87} - 1658 q^{89} + 399 q^{91} - 4729 q^{93} - 780 q^{95} - 3998 q^{97} - 1106 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 - 14 * q^5 - 84 * q^7 + 79 * q^9 + 8 * q^11 - 57 * q^13 + 10 * q^15 - 86 * q^17 + 160 * q^19 + 21 * q^21 + 276 * q^23 + 256 * q^25 + 15 * q^27 + 153 * q^29 + 253 * q^31 - 238 * q^33 + 98 * q^35 - 530 * q^37 + 123 * q^39 - 917 * q^41 - 174 * q^43 - 776 * q^45 - 571 * q^47 + 588 * q^49 + 792 * q^51 - 582 * q^53 - 388 * q^55 - 1214 * q^57 + 288 * q^59 - 1274 * q^61 - 553 * q^63 - 1446 * q^65 + 892 * q^67 - 69 * q^69 + 353 * q^71 - 1907 * q^73 - 217 * q^75 - 56 * q^77 + 102 * q^79 - 820 * q^81 - 2342 * q^83 - 2792 * q^85 - 2071 * q^87 - 1658 * q^89 + 399 * q^91 - 4729 * q^93 - 780 * q^95 - 3998 * q^97 - 1106 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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