LMFDB - Embedded newform 1840.4.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 $ x^10 - x^9 - 204x^8 + 42x^7 + 12958x^6 + 5872x^5 - 259871x^4 - 149461x^3 + 1222472x^2 + 627136x - 43712
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-9.58471$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.58471 q^{3} -5.00000 q^{5} -21.8969 q^{7} +64.8667 q^{9} +O(q^{10})$	$q-9.58471 q^{3} -5.00000 q^{5} -21.8969 q^{7} +64.8667 q^{9} -25.8615 q^{11} +17.7973 q^{13} +47.9236 q^{15} +71.2464 q^{17} +84.1980 q^{19} +209.876 q^{21} -23.0000 q^{23} +25.0000 q^{25} -362.941 q^{27} -257.106 q^{29} -205.205 q^{31} +247.875 q^{33} +109.485 q^{35} +19.8271 q^{37} -170.582 q^{39} -150.715 q^{41} +355.813 q^{43} -324.333 q^{45} +50.2544 q^{47} +136.475 q^{49} -682.876 q^{51} +46.6775 q^{53} +129.307 q^{55} -807.014 q^{57} -618.898 q^{59} -900.252 q^{61} -1420.38 q^{63} -88.9863 q^{65} -836.144 q^{67} +220.448 q^{69} +327.109 q^{71} +290.601 q^{73} -239.618 q^{75} +566.286 q^{77} -1277.03 q^{79} +1727.28 q^{81} +571.159 q^{83} -356.232 q^{85} +2464.29 q^{87} +1416.67 q^{89} -389.705 q^{91} +1966.83 q^{93} -420.990 q^{95} +457.158 q^{97} -1677.55 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * 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
$3$	−9.58471	−1.84458	−0.922289	−	0.386501i	$-0.873684\pi$
$3$	−9.58471	−1.84458	−0.922289	+	0.386501i	$0.873684\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−21.8969	−1.18232	−0.591161	−	0.806553i	$-0.701330\pi$
$7$	−21.8969	−1.18232	−0.591161	+	0.806553i	$0.701330\pi$
$8$	0	0
$9$	64.8667	2.40247
$10$	0	0
$11$	−25.8615	−0.708866	−0.354433	−	0.935082i	$-0.615326\pi$
$11$	−25.8615	−0.708866	−0.354433	+	0.935082i	$0.615326\pi$
$12$	0	0
$13$	17.7973	0.379698	0.189849	−	0.981813i	$-0.439200\pi$
$13$	17.7973	0.379698	0.189849	+	0.981813i	$0.439200\pi$
$14$	0	0
$15$	47.9236	0.824921
$16$	0	0
$17$	71.2464	1.01646	0.508229	−	0.861222i	$-0.330300\pi$
$17$	71.2464	1.01646	0.508229	+	0.861222i	$0.330300\pi$
$18$	0	0
$19$	84.1980	1.01665	0.508325	−	0.861165i	$-0.330265\pi$
$19$	84.1980	1.01665	0.508325	+	0.861165i	$0.330265\pi$
$20$	0	0
$21$	209.876	2.18089
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−362.941	−2.58696
$28$	0	0
$29$	−257.106	−1.64633	−0.823163	−	0.567806i	$-0.807792\pi$
$29$	−257.106	−1.64633	−0.823163	+	0.567806i	$0.807792\pi$
$30$	0	0
$31$	−205.205	−1.18890	−0.594449	−	0.804133i	$-0.702630\pi$
$31$	−205.205	−1.18890	−0.594449	+	0.804133i	$0.702630\pi$
$32$	0	0
$33$	247.875	1.30756
$34$	0	0
$35$	109.485	0.528751
$36$	0	0
$37$	19.8271	0.0880961	0.0440480	−	0.999029i	$-0.485975\pi$
$37$	19.8271	0.0880961	0.0440480	+	0.999029i	$0.485975\pi$
$38$	0	0
$39$	−170.582	−0.700383
$40$	0	0
$41$	−150.715	−0.574092	−0.287046	−	0.957917i	$-0.592673\pi$
$41$	−150.715	−0.574092	−0.287046	+	0.957917i	$0.592673\pi$
$42$	0	0
$43$	355.813	1.26188	0.630942	−	0.775830i	$-0.282669\pi$
$43$	355.813	1.26188	0.630942	+	0.775830i	$0.282669\pi$
$44$	0	0
$45$	−324.333	−1.07442
$46$	0	0
$47$	50.2544	0.155965	0.0779826	−	0.996955i	$-0.475152\pi$
$47$	50.2544	0.155965	0.0779826	+	0.996955i	$0.475152\pi$
$48$	0	0
$49$	136.475	0.397886
$50$	0	0
$51$	−682.876	−1.87494
$52$	0	0
$53$	46.6775	0.120974	0.0604872	−	0.998169i	$-0.480735\pi$
$53$	46.6775	0.120974	0.0604872	+	0.998169i	$0.480735\pi$
$54$	0	0
$55$	129.307	0.317014
$56$	0	0
$57$	−807.014	−1.87529
$58$	0	0
$59$	−618.898	−1.36565	−0.682827	−	0.730580i	$-0.739250\pi$
$59$	−618.898	−1.36565	−0.682827	+	0.730580i	$0.739250\pi$
$60$	0	0
$61$	−900.252	−1.88960	−0.944799	−	0.327651i	$-0.893743\pi$
$61$	−900.252	−1.88960	−0.944799	+	0.327651i	$0.893743\pi$
$62$	0	0
$63$	−1420.38	−2.84049
$64$	0	0
$65$	−88.9863	−0.169806
$66$	0	0
$67$	−836.144	−1.52464	−0.762322	−	0.647197i	$-0.775941\pi$
$67$	−836.144	−1.52464	−0.762322	+	0.647197i	$0.775941\pi$
$68$	0	0
$69$	220.448	0.384621
$70$	0	0
$71$	327.109	0.546770	0.273385	−	0.961905i	$-0.411857\pi$
$71$	327.109	0.546770	0.273385	+	0.961905i	$0.411857\pi$
$72$	0	0
$73$	290.601	0.465921	0.232961	−	0.972486i	$-0.425159\pi$
$73$	290.601	0.465921	0.232961	+	0.972486i	$0.425159\pi$
$74$	0	0
$75$	−239.618	−0.368916
$76$	0	0
$77$	566.286	0.838108
$78$	0	0
$79$	−1277.03	−1.81870	−0.909350	−	0.416032i	$-0.863420\pi$
$79$	−1277.03	−1.81870	−0.909350	+	0.416032i	$0.863420\pi$
$80$	0	0
$81$	1727.28	2.36939
$82$	0	0
$83$	571.159	0.755335	0.377668	−	0.925941i	$-0.376726\pi$
$83$	571.159	0.755335	0.377668	+	0.925941i	$0.376726\pi$
$84$	0	0
$85$	−356.232	−0.454574
$86$	0	0
$87$	2464.29	3.03678
$88$	0	0
$89$	1416.67	1.68727	0.843635	−	0.536917i	$-0.180411\pi$
$89$	1416.67	1.68727	0.843635	+	0.536917i	$0.180411\pi$
$90$	0	0
$91$	−389.705	−0.448925
$92$	0	0
$93$	1966.83	2.19302
$94$	0	0
$95$	−420.990	−0.454660
$96$	0	0
$97$	457.158	0.478530	0.239265	−	0.970954i	$-0.423094\pi$
$97$	457.158	0.478530	0.239265	+	0.970954i	$0.423094\pi$
$98$	0	0
$99$	−1677.55	−1.70303
$100$	0	0
$101$	−1441.01	−1.41966	−0.709831	−	0.704372i	$-0.751229\pi$
$101$	−1441.01	−1.41966	−0.709831	+	0.704372i	$0.751229\pi$
$102$	0	0
$103$	−689.175	−0.659286	−0.329643	−	0.944106i	$-0.606928\pi$
$103$	−689.175	−0.659286	−0.329643	+	0.944106i	$0.606928\pi$
$104$	0	0
$105$	−1049.38	−0.975322
$106$	0	0
$107$	−432.925	−0.391144	−0.195572	−	0.980689i	$-0.562656\pi$
$107$	−432.925	−0.391144	−0.195572	+	0.980689i	$0.562656\pi$
$108$	0	0
$109$	−1291.66	−1.13503	−0.567515	−	0.823363i	$-0.692095\pi$
$109$	−1291.66	−1.13503	−0.567515	+	0.823363i	$0.692095\pi$
$110$	0	0
$111$	−190.037	−0.162500
$112$	0	0
$113$	963.150	0.801818	0.400909	−	0.916118i	$-0.368694\pi$
$113$	963.150	0.801818	0.400909	+	0.916118i	$0.368694\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	1154.45	0.912213
$118$	0	0
$119$	−1560.08	−1.20178
$120$	0	0
$121$	−662.185	−0.497510
$122$	0	0
$123$	1444.56	1.05896
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	360.608	0.251959	0.125980	−	0.992033i	$-0.459793\pi$
$127$	360.608	0.251959	0.125980	+	0.992033i	$0.459793\pi$
$128$	0	0
$129$	−3410.37	−2.32764
$130$	0	0
$131$	861.556	0.574615	0.287307	−	0.957838i	$-0.407240\pi$
$131$	861.556	0.574615	0.287307	+	0.957838i	$0.407240\pi$
$132$	0	0
$133$	−1843.68	−1.20201
$134$	0	0
$135$	1814.71	1.15693
$136$	0	0
$137$	418.161	0.260773	0.130386	−	0.991463i	$-0.458378\pi$
$137$	418.161	0.260773	0.130386	+	0.991463i	$0.458378\pi$
$138$	0	0
$139$	−94.2638	−0.0575205	−0.0287603	−	0.999586i	$-0.509156\pi$
$139$	−94.2638	−0.0575205	−0.0287603	+	0.999586i	$0.509156\pi$
$140$	0	0
$141$	−481.674	−0.287690
$142$	0	0
$143$	−460.263	−0.269155
$144$	0	0
$145$	1285.53	0.736259
$146$	0	0
$147$	−1308.07	−0.733932
$148$	0	0
$149$	−3456.74	−1.90059	−0.950294	−	0.311355i	$-0.899217\pi$
$149$	−3456.74	−1.90059	−0.950294	+	0.311355i	$0.899217\pi$
$150$	0	0
$151$	2658.81	1.43292	0.716459	−	0.697630i	$-0.245762\pi$
$151$	2658.81	1.43292	0.716459	+	0.697630i	$0.245762\pi$
$152$	0	0
$153$	4621.52	2.44201
$154$	0	0
$155$	1026.02	0.531692
$156$	0	0
$157$	1130.25	0.574548	0.287274	−	0.957848i	$-0.407251\pi$
$157$	1130.25	0.574548	0.287274	+	0.957848i	$0.407251\pi$
$158$	0	0
$159$	−447.390	−0.223147
$160$	0	0
$161$	503.629	0.246531
$162$	0	0
$163$	−1800.25	−0.865069	−0.432535	−	0.901617i	$-0.642381\pi$
$163$	−1800.25	−0.865069	−0.432535	+	0.901617i	$0.642381\pi$
$164$	0	0
$165$	−1239.37	−0.584758
$166$	0	0
$167$	−3479.14	−1.61212	−0.806060	−	0.591834i	$-0.798404\pi$
$167$	−3479.14	−1.61212	−0.806060	+	0.591834i	$0.798404\pi$
$168$	0	0
$169$	−1880.26	−0.855829
$170$	0	0
$171$	5461.65	2.44247
$172$	0	0
$173$	−2913.55	−1.28042	−0.640211	−	0.768199i	$-0.721153\pi$
$173$	−2913.55	−1.28042	−0.640211	+	0.768199i	$0.721153\pi$
$174$	0	0
$175$	−547.423	−0.236464
$176$	0	0
$177$	5931.95	2.51906
$178$	0	0
$179$	746.457	0.311692	0.155846	−	0.987781i	$-0.450190\pi$
$179$	746.457	0.311692	0.155846	+	0.987781i	$0.450190\pi$
$180$	0	0
$181$	675.750	0.277503	0.138752	−	0.990327i	$-0.455691\pi$
$181$	675.750	0.277503	0.138752	+	0.990327i	$0.455691\pi$
$182$	0	0
$183$	8628.66	3.48551
$184$	0	0
$185$	−99.1355	−0.0393978
$186$	0	0
$187$	−1842.54	−0.720532
$188$	0	0
$189$	7947.29	3.05863
$190$	0	0
$191$	2443.03	0.925505	0.462753	−	0.886487i	$-0.346862\pi$
$191$	2443.03	0.925505	0.462753	+	0.886487i	$0.346862\pi$
$192$	0	0
$193$	4352.30	1.62324	0.811621	−	0.584185i	$-0.198586\pi$
$193$	4352.30	1.62324	0.811621	+	0.584185i	$0.198586\pi$
$194$	0	0
$195$	852.908	0.313221
$196$	0	0
$197$	−3703.03	−1.33924	−0.669619	−	0.742705i	$-0.733542\pi$
$197$	−3703.03	−1.33924	−0.669619	+	0.742705i	$0.733542\pi$
$198$	0	0
$199$	634.780	0.226122	0.113061	−	0.993588i	$-0.463934\pi$
$199$	634.780	0.226122	0.113061	+	0.993588i	$0.463934\pi$
$200$	0	0
$201$	8014.19	2.81233
$202$	0	0
$203$	5629.83	1.94649
$204$	0	0
$205$	753.576	0.256742
$206$	0	0
$207$	−1491.93	−0.500949
$208$	0	0
$209$	−2177.48	−0.720668
$210$	0	0
$211$	−1120.29	−0.365515	−0.182757	−	0.983158i	$-0.558502\pi$
$211$	−1120.29	−0.365515	−0.182757	+	0.983158i	$0.558502\pi$
$212$	0	0
$213$	−3135.24	−1.00856
$214$	0	0
$215$	−1779.07	−0.564332
$216$	0	0
$217$	4493.35	1.40566
$218$	0	0
$219$	−2785.33	−0.859429
$220$	0	0
$221$	1267.99	0.385947
$222$	0	0
$223$	−989.498	−0.297138	−0.148569	−	0.988902i	$-0.547467\pi$
$223$	−989.498	−0.297138	−0.148569	+	0.988902i	$0.547467\pi$
$224$	0	0
$225$	1621.67	0.480494
$226$	0	0
$227$	1232.97	0.360507	0.180254	−	0.983620i	$-0.442308\pi$
$227$	1232.97	0.360507	0.180254	+	0.983620i	$0.442308\pi$
$228$	0	0
$229$	5202.87	1.50138	0.750689	−	0.660656i	$-0.229722\pi$
$229$	5202.87	1.50138	0.750689	+	0.660656i	$0.229722\pi$
$230$	0	0
$231$	−5427.69	−1.54596
$232$	0	0
$233$	1795.52	0.504844	0.252422	−	0.967617i	$-0.418773\pi$
$233$	1795.52	0.504844	0.252422	+	0.967617i	$0.418773\pi$
$234$	0	0
$235$	−251.272	−0.0697497
$236$	0	0
$237$	12240.0	3.35473
$238$	0	0
$239$	5044.34	1.36523	0.682617	−	0.730776i	$-0.260842\pi$
$239$	5044.34	1.36523	0.682617	+	0.730776i	$0.260842\pi$
$240$	0	0
$241$	−56.8541	−0.0151962	−0.00759812	−	0.999971i	$-0.502419\pi$
$241$	−56.8541	−0.0151962	−0.00759812	+	0.999971i	$0.502419\pi$
$242$	0	0
$243$	−6756.12	−1.78356
$244$	0	0
$245$	−682.375	−0.177940
$246$	0	0
$247$	1498.49	0.386020
$248$	0	0
$249$	−5474.39	−1.39327
$250$	0	0
$251$	4248.17	1.06829	0.534147	−	0.845391i	$-0.320633\pi$
$251$	4248.17	1.06829	0.534147	+	0.845391i	$0.320633\pi$
$252$	0	0
$253$	594.813	0.147809
$254$	0	0
$255$	3414.38	0.838497
$256$	0	0
$257$	−1894.91	−0.459927	−0.229963	−	0.973199i	$-0.573861\pi$
$257$	−1894.91	−0.459927	−0.229963	+	0.973199i	$0.573861\pi$
$258$	0	0
$259$	−434.152	−0.104158
$260$	0	0
$261$	−16677.6	−3.95525
$262$	0	0
$263$	2694.17	0.631671	0.315836	−	0.948814i	$-0.397715\pi$
$263$	2694.17	0.631671	0.315836	+	0.948814i	$0.397715\pi$
$264$	0	0
$265$	−233.387	−0.0541014
$266$	0	0
$267$	−13578.4	−3.11230
$268$	0	0
$269$	5170.60	1.17196	0.585980	−	0.810326i	$-0.300710\pi$
$269$	5170.60	1.17196	0.585980	+	0.810326i	$0.300710\pi$
$270$	0	0
$271$	1467.03	0.328840	0.164420	−	0.986390i	$-0.447425\pi$
$271$	1467.03	0.328840	0.164420	+	0.986390i	$0.447425\pi$
$272$	0	0
$273$	3735.21	0.828078
$274$	0	0
$275$	−646.536	−0.141773
$276$	0	0
$277$	7019.95	1.52270	0.761350	−	0.648341i	$-0.224537\pi$
$277$	7019.95	1.52270	0.761350	+	0.648341i	$0.224537\pi$
$278$	0	0
$279$	−13311.0	−2.85629
$280$	0	0
$281$	−7676.83	−1.62976	−0.814878	−	0.579633i	$-0.803196\pi$
$281$	−7676.83	−1.62976	−0.814878	+	0.579633i	$0.803196\pi$
$282$	0	0
$283$	−3201.28	−0.672426	−0.336213	−	0.941786i	$-0.609146\pi$
$283$	−3201.28	−0.672426	−0.336213	+	0.941786i	$0.609146\pi$
$284$	0	0
$285$	4035.07	0.838656
$286$	0	0
$287$	3300.20	0.678762
$288$	0	0
$289$	163.050	0.0331874
$290$	0	0
$291$	−4381.73	−0.882685
$292$	0	0
$293$	5872.87	1.17098	0.585489	−	0.810680i	$-0.300902\pi$
$293$	5872.87	1.17098	0.585489	+	0.810680i	$0.300902\pi$
$294$	0	0
$295$	3094.49	0.610739
$296$	0	0
$297$	9386.18	1.83381
$298$	0	0
$299$	−409.337	−0.0791725
$300$	0	0
$301$	−7791.21	−1.49195
$302$	0	0
$303$	13811.7	2.61868
$304$	0	0
$305$	4501.26	0.845054
$306$	0	0
$307$	3509.62	0.652457	0.326229	−	0.945291i	$-0.394222\pi$
$307$	3509.62	0.652457	0.326229	+	0.945291i	$0.394222\pi$
$308$	0	0
$309$	6605.55	1.21610
$310$	0	0
$311$	2150.62	0.392124	0.196062	−	0.980592i	$-0.437185\pi$
$311$	2150.62	0.392124	0.196062	+	0.980592i	$0.437185\pi$
$312$	0	0
$313$	−9646.52	−1.74202	−0.871012	−	0.491262i	$-0.836536\pi$
$313$	−9646.52	−1.74202	−0.871012	+	0.491262i	$0.836536\pi$
$314$	0	0
$315$	7101.90	1.27031
$316$	0	0
$317$	775.434	0.137390	0.0686951	−	0.997638i	$-0.478116\pi$
$317$	775.434	0.137390	0.0686951	+	0.997638i	$0.478116\pi$
$318$	0	0
$319$	6649.14	1.16702
$320$	0	0
$321$	4149.46	0.721496
$322$	0	0
$323$	5998.81	1.03338
$324$	0	0
$325$	444.932	0.0759396
$326$	0	0
$327$	12380.1	2.09365
$328$	0	0
$329$	−1100.42	−0.184401
$330$	0	0
$331$	−329.550	−0.0547242	−0.0273621	−	0.999626i	$-0.508711\pi$
$331$	−329.550	−0.0547242	−0.0273621	+	0.999626i	$0.508711\pi$
$332$	0	0
$333$	1286.12	0.211648
$334$	0	0
$335$	4180.72	0.681842
$336$	0	0
$337$	4930.84	0.797032	0.398516	−	0.917161i	$-0.369525\pi$
$337$	4930.84	0.797032	0.398516	+	0.917161i	$0.369525\pi$
$338$	0	0
$339$	−9231.51	−1.47902
$340$	0	0
$341$	5306.89	0.842769
$342$	0	0
$343$	4522.26	0.711893
$344$	0	0
$345$	−1102.24	−0.172008
$346$	0	0
$347$	859.081	0.132905	0.0664523	−	0.997790i	$-0.478832\pi$
$347$	859.081	0.132905	0.0664523	+	0.997790i	$0.478832\pi$
$348$	0	0
$349$	−1893.98	−0.290494	−0.145247	−	0.989395i	$-0.546398\pi$
$349$	−1893.98	−0.290494	−0.145247	+	0.989395i	$0.546398\pi$
$350$	0	0
$351$	−6459.36	−0.982265
$352$	0	0
$353$	−11460.2	−1.72795	−0.863975	−	0.503534i	$-0.832033\pi$
$353$	−11460.2	−1.72795	−0.863975	+	0.503534i	$0.832033\pi$
$354$	0	0
$355$	−1635.54	−0.244523
$356$	0	0
$357$	14952.9	2.21678
$358$	0	0
$359$	−3857.60	−0.567121	−0.283560	−	0.958954i	$-0.591516\pi$
$359$	−3857.60	−0.567121	−0.283560	+	0.958954i	$0.591516\pi$
$360$	0	0
$361$	230.308	0.0335774
$362$	0	0
$363$	6346.85	0.917696
$364$	0	0
$365$	−1453.00	−0.208366
$366$	0	0
$367$	1135.37	0.161487	0.0807434	−	0.996735i	$-0.474271\pi$
$367$	1135.37	0.161487	0.0807434	+	0.996735i	$0.474271\pi$
$368$	0	0
$369$	−9776.40	−1.37924
$370$	0	0
$371$	−1022.09	−0.143031
$372$	0	0
$373$	−2864.98	−0.397703	−0.198851	−	0.980030i	$-0.563721\pi$
$373$	−2864.98	−0.397703	−0.198851	+	0.980030i	$0.563721\pi$
$374$	0	0
$375$	1198.09	0.164984
$376$	0	0
$377$	−4575.79	−0.625106
$378$	0	0
$379$	6412.64	0.869116	0.434558	−	0.900644i	$-0.356905\pi$
$379$	6412.64	0.869116	0.434558	+	0.900644i	$0.356905\pi$
$380$	0	0
$381$	−3456.33	−0.464758
$382$	0	0
$383$	−7868.43	−1.04976	−0.524880	−	0.851176i	$-0.675890\pi$
$383$	−7868.43	−1.04976	−0.524880	+	0.851176i	$0.675890\pi$
$384$	0	0
$385$	−2831.43	−0.374813
$386$	0	0
$387$	23080.4	3.03164
$388$	0	0
$389$	−1615.39	−0.210549	−0.105275	−	0.994443i	$-0.533572\pi$
$389$	−1615.39	−0.210549	−0.105275	+	0.994443i	$0.533572\pi$
$390$	0	0
$391$	−1638.67	−0.211946
$392$	0	0
$393$	−8257.77	−1.05992
$394$	0	0
$395$	6385.16	0.813347
$396$	0	0
$397$	−7204.15	−0.910746	−0.455373	−	0.890301i	$-0.650494\pi$
$397$	−7204.15	−0.910746	−0.455373	+	0.890301i	$0.650494\pi$
$398$	0	0
$399$	17671.1	2.21720
$400$	0	0
$401$	4260.02	0.530512	0.265256	−	0.964178i	$-0.414543\pi$
$401$	4260.02	0.530512	0.265256	+	0.964178i	$0.414543\pi$
$402$	0	0
$403$	−3652.08	−0.451423
$404$	0	0
$405$	−8636.42	−1.05962
$406$	0	0
$407$	−512.758	−0.0624483
$408$	0	0
$409$	10497.9	1.26917	0.634584	−	0.772853i	$-0.281171\pi$
$409$	10497.9	1.26917	0.634584	+	0.772853i	$0.281171\pi$
$410$	0	0
$411$	−4007.95	−0.481016
$412$	0	0
$413$	13551.9	1.61464
$414$	0	0
$415$	−2855.79	−0.337796
$416$	0	0
$417$	903.491	0.106101
$418$	0	0
$419$	3442.58	0.401387	0.200693	−	0.979654i	$-0.435680\pi$
$419$	3442.58	0.401387	0.200693	+	0.979654i	$0.435680\pi$
$420$	0	0
$421$	−9439.30	−1.09274	−0.546370	−	0.837544i	$-0.683991\pi$
$421$	−9439.30	−1.09274	−0.546370	+	0.837544i	$0.683991\pi$
$422$	0	0
$423$	3259.84	0.374701
$424$	0	0
$425$	1781.16	0.203292
$426$	0	0
$427$	19712.7	2.23411
$428$	0	0
$429$	4411.49	0.496477
$430$	0	0
$431$	7837.69	0.875935	0.437968	−	0.898991i	$-0.355698\pi$
$431$	7837.69	0.875935	0.437968	+	0.898991i	$0.355698\pi$
$432$	0	0
$433$	−12240.3	−1.35850	−0.679252	−	0.733905i	$-0.737696\pi$
$433$	−12240.3	−1.35850	−0.679252	+	0.733905i	$0.737696\pi$
$434$	0	0
$435$	−12321.4	−1.35809
$436$	0	0
$437$	−1936.55	−0.211986
$438$	0	0
$439$	−5147.60	−0.559639	−0.279820	−	0.960053i	$-0.590275\pi$
$439$	−5147.60	−0.559639	−0.279820	+	0.960053i	$0.590275\pi$
$440$	0	0
$441$	8852.68	0.955910
$442$	0	0
$443$	−10609.3	−1.13784	−0.568918	−	0.822394i	$-0.692638\pi$
$443$	−10609.3	−1.13784	−0.568918	+	0.822394i	$0.692638\pi$
$444$	0	0
$445$	−7083.36	−0.754570
$446$	0	0
$447$	33131.9	3.50578
$448$	0	0
$449$	−7160.78	−0.752646	−0.376323	−	0.926488i	$-0.622812\pi$
$449$	−7160.78	−0.752646	−0.376323	+	0.926488i	$0.622812\pi$
$450$	0	0
$451$	3897.72	0.406954
$452$	0	0
$453$	−25483.9	−2.64313
$454$	0	0
$455$	1948.53	0.200766
$456$	0	0
$457$	18416.8	1.88513	0.942563	−	0.334029i	$-0.108408\pi$
$457$	18416.8	1.88513	0.942563	+	0.334029i	$0.108408\pi$
$458$	0	0
$459$	−25858.2	−2.62954
$460$	0	0
$461$	−13758.0	−1.38997	−0.694984	−	0.719025i	$-0.744589\pi$
$461$	−13758.0	−1.38997	−0.694984	+	0.719025i	$0.744589\pi$
$462$	0	0
$463$	15476.1	1.55343	0.776714	−	0.629854i	$-0.216885\pi$
$463$	15476.1	1.55343	0.776714	+	0.629854i	$0.216885\pi$
$464$	0	0
$465$	−9834.14	−0.980747
$466$	0	0
$467$	−7980.97	−0.790825	−0.395412	−	0.918504i	$-0.629398\pi$
$467$	−7980.97	−0.790825	−0.395412	+	0.918504i	$0.629398\pi$
$468$	0	0
$469$	18309.0	1.80262
$470$	0	0
$471$	−10833.1	−1.05980
$472$	0	0
$473$	−9201.85	−0.894506
$474$	0	0
$475$	2104.95	0.203330
$476$	0	0
$477$	3027.81	0.290637
$478$	0	0
$479$	13404.3	1.27862	0.639309	−	0.768950i	$-0.279220\pi$
$479$	13404.3	1.27862	0.639309	+	0.768950i	$0.279220\pi$
$480$	0	0
$481$	352.868	0.0334499
$482$	0	0
$483$	−4827.14	−0.454746
$484$	0	0
$485$	−2285.79	−0.214005
$486$	0	0
$487$	20868.0	1.94172	0.970861	−	0.239645i	$-0.0770311\pi$
$487$	20868.0	1.94172	0.970861	+	0.239645i	$0.0770311\pi$
$488$	0	0
$489$	17254.9	1.59569
$490$	0	0
$491$	10428.4	0.958511	0.479255	−	0.877676i	$-0.340907\pi$
$491$	10428.4	0.958511	0.479255	+	0.877676i	$0.340907\pi$
$492$	0	0
$493$	−18317.9	−1.67342
$494$	0	0
$495$	8387.73	0.761617
$496$	0	0
$497$	−7162.67	−0.646458
$498$	0	0
$499$	−11272.3	−1.01126	−0.505630	−	0.862750i	$-0.668740\pi$
$499$	−11272.3	−1.01126	−0.505630	+	0.862750i	$0.668740\pi$
$500$	0	0
$501$	33346.6	2.97368
$502$	0	0
$503$	−964.052	−0.0854572	−0.0427286	−	0.999087i	$-0.513605\pi$
$503$	−964.052	−0.0854572	−0.0427286	+	0.999087i	$0.513605\pi$
$504$	0	0
$505$	7205.05	0.634892
$506$	0	0
$507$	18021.7	1.57864
$508$	0	0
$509$	−8969.58	−0.781080	−0.390540	−	0.920586i	$-0.627712\pi$
$509$	−8969.58	−0.781080	−0.390540	+	0.920586i	$0.627712\pi$
$510$	0	0
$511$	−6363.26	−0.550869
$512$	0	0
$513$	−30558.9	−2.63004
$514$	0	0
$515$	3445.88	0.294842
$516$	0	0
$517$	−1299.65	−0.110558
$518$	0	0
$519$	27925.5	2.36184
$520$	0	0
$521$	3908.39	0.328656	0.164328	−	0.986406i	$-0.447454\pi$
$521$	3908.39	0.328656	0.164328	+	0.986406i	$0.447454\pi$
$522$	0	0
$523$	−14063.0	−1.17577	−0.587887	−	0.808943i	$-0.700040\pi$
$523$	−14063.0	−1.17577	−0.587887	+	0.808943i	$0.700040\pi$
$524$	0	0
$525$	5246.89	0.436177
$526$	0	0
$527$	−14620.1	−1.20847
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−40145.8	−3.28094
$532$	0	0
$533$	−2682.32	−0.217982
$534$	0	0
$535$	2164.62	0.174925
$536$	0	0
$537$	−7154.58	−0.574940
$538$	0	0
$539$	−3529.44	−0.282048
$540$	0	0
$541$	−16091.6	−1.27880	−0.639400	−	0.768874i	$-0.720817\pi$
$541$	−16091.6	−1.27880	−0.639400	+	0.768874i	$0.720817\pi$
$542$	0	0
$543$	−6476.87	−0.511877
$544$	0	0
$545$	6458.28	0.507601
$546$	0	0
$547$	−10900.9	−0.852085	−0.426043	−	0.904703i	$-0.640093\pi$
$547$	−10900.9	−0.852085	−0.426043	+	0.904703i	$0.640093\pi$
$548$	0	0
$549$	−58396.4	−4.53970
$550$	0	0
$551$	−21647.8	−1.67374
$552$	0	0
$553$	27963.1	2.15029
$554$	0	0
$555$	950.185	0.0726723
$556$	0	0
$557$	20006.3	1.52189	0.760946	−	0.648816i	$-0.224735\pi$
$557$	20006.3	1.52189	0.760946	+	0.648816i	$0.224735\pi$
$558$	0	0
$559$	6332.50	0.479135
$560$	0	0
$561$	17660.2	1.32908
$562$	0	0
$563$	7983.16	0.597602	0.298801	−	0.954315i	$-0.403413\pi$
$563$	7983.16	0.597602	0.298801	+	0.954315i	$0.403413\pi$
$564$	0	0
$565$	−4815.75	−0.358584
$566$	0	0
$567$	−37822.2	−2.80138
$568$	0	0
$569$	19494.2	1.43627	0.718135	−	0.695904i	$-0.244996\pi$
$569$	19494.2	1.43627	0.718135	+	0.695904i	$0.244996\pi$
$570$	0	0
$571$	−24755.5	−1.81433	−0.907166	−	0.420773i	$-0.861759\pi$
$571$	−24755.5	−1.81433	−0.907166	+	0.420773i	$0.861759\pi$
$572$	0	0
$573$	−23415.7	−1.70717
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−18139.8	−1.30878	−0.654392	−	0.756155i	$-0.727075\pi$
$577$	−18139.8	−1.30878	−0.654392	+	0.756155i	$0.727075\pi$
$578$	0	0
$579$	−41715.6	−2.99420
$580$	0	0
$581$	−12506.6	−0.893050
$582$	0	0
$583$	−1207.15	−0.0857546
$584$	0	0
$585$	−5772.25	−0.407954
$586$	0	0
$587$	−8359.18	−0.587769	−0.293884	−	0.955841i	$-0.594948\pi$
$587$	−8359.18	−0.587769	−0.293884	+	0.955841i	$0.594948\pi$
$588$	0	0
$589$	−17277.8	−1.20869
$590$	0	0
$591$	35492.4	2.47033
$592$	0	0
$593$	−17587.0	−1.21789	−0.608947	−	0.793211i	$-0.708408\pi$
$593$	−17587.0	−1.21789	−0.608947	+	0.793211i	$0.708408\pi$
$594$	0	0
$595$	7800.38	0.537453
$596$	0	0
$597$	−6084.18	−0.417100
$598$	0	0
$599$	23626.0	1.61157	0.805785	−	0.592208i	$-0.201744\pi$
$599$	23626.0	1.61157	0.805785	+	0.592208i	$0.201744\pi$
$600$	0	0
$601$	22705.8	1.54108	0.770540	−	0.637392i	$-0.219987\pi$
$601$	22705.8	1.54108	0.770540	+	0.637392i	$0.219987\pi$
$602$	0	0
$603$	−54237.9	−3.66291
$604$	0	0
$605$	3310.93	0.222493
$606$	0	0
$607$	−6594.61	−0.440967	−0.220484	−	0.975391i	$-0.570764\pi$
$607$	−6594.61	−0.440967	−0.220484	+	0.975391i	$0.570764\pi$
$608$	0	0
$609$	−53960.3	−3.59045
$610$	0	0
$611$	894.391	0.0592197
$612$	0	0
$613$	−1240.87	−0.0817588	−0.0408794	−	0.999164i	$-0.513016\pi$
$613$	−1240.87	−0.0817588	−0.0408794	+	0.999164i	$0.513016\pi$
$614$	0	0
$615$	−7222.81	−0.473580
$616$	0	0
$617$	−7488.30	−0.488602	−0.244301	−	0.969699i	$-0.578558\pi$
$617$	−7488.30	−0.488602	−0.244301	+	0.969699i	$0.578558\pi$
$618$	0	0
$619$	−10417.1	−0.676411	−0.338206	−	0.941072i	$-0.609820\pi$
$619$	−10417.1	−0.676411	−0.338206	+	0.941072i	$0.609820\pi$
$620$	0	0
$621$	8347.64	0.539419
$622$	0	0
$623$	−31020.8	−1.99490
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	20870.5	1.32933
$628$	0	0
$629$	1412.61	0.0895460
$630$	0	0
$631$	−19840.3	−1.25171	−0.625856	−	0.779939i	$-0.715250\pi$
$631$	−19840.3	−1.25171	−0.625856	+	0.779939i	$0.715250\pi$
$632$	0	0
$633$	10737.6	0.674221
$634$	0	0
$635$	−1803.04	−0.112680
$636$	0	0
$637$	2428.88	0.151077
$638$	0	0
$639$	21218.4	1.31360
$640$	0	0
$641$	20002.1	1.23251	0.616253	−	0.787548i	$-0.288650\pi$
$641$	20002.1	1.23251	0.616253	+	0.787548i	$0.288650\pi$
$642$	0	0
$643$	−3293.94	−0.202022	−0.101011	−	0.994885i	$-0.532208\pi$
$643$	−3293.94	−0.202022	−0.101011	+	0.994885i	$0.532208\pi$
$644$	0	0
$645$	17051.8	1.04095
$646$	0	0
$647$	−5662.22	−0.344057	−0.172028	−	0.985092i	$-0.555032\pi$
$647$	−5662.22	−0.344057	−0.172028	+	0.985092i	$0.555032\pi$
$648$	0	0
$649$	16005.6	0.968065
$650$	0	0
$651$	−43067.5	−2.59285
$652$	0	0
$653$	−14023.2	−0.840382	−0.420191	−	0.907436i	$-0.638037\pi$
$653$	−14023.2	−0.840382	−0.420191	+	0.907436i	$0.638037\pi$
$654$	0	0
$655$	−4307.78	−0.256975
$656$	0	0
$657$	18850.3	1.11936
$658$	0	0
$659$	26542.3	1.56896	0.784478	−	0.620156i	$-0.212931\pi$
$659$	26542.3	1.56896	0.784478	+	0.620156i	$0.212931\pi$
$660$	0	0
$661$	17080.8	1.00509	0.502545	−	0.864551i	$-0.332397\pi$
$661$	17080.8	1.00509	0.502545	+	0.864551i	$0.332397\pi$
$662$	0	0
$663$	−12153.3	−0.711910
$664$	0	0
$665$	9218.39	0.537554
$666$	0	0
$667$	5913.44	0.343283
$668$	0	0
$669$	9484.06	0.548094
$670$	0	0
$671$	23281.8	1.33947
$672$	0	0
$673$	−900.526	−0.0515791	−0.0257896	−	0.999667i	$-0.508210\pi$
$673$	−900.526	−0.0515791	−0.0257896	+	0.999667i	$0.508210\pi$
$674$	0	0
$675$	−9073.53	−0.517393
$676$	0	0
$677$	14833.6	0.842099	0.421050	−	0.907038i	$-0.361662\pi$
$677$	14833.6	0.842099	0.421050	+	0.907038i	$0.361662\pi$
$678$	0	0
$679$	−10010.4	−0.565776
$680$	0	0
$681$	−11817.7	−0.664984
$682$	0	0
$683$	26813.0	1.50216	0.751078	−	0.660213i	$-0.229534\pi$
$683$	26813.0	1.50216	0.751078	+	0.660213i	$0.229534\pi$
$684$	0	0
$685$	−2090.80	−0.116621
$686$	0	0
$687$	−49868.0	−2.76941
$688$	0	0
$689$	830.731	0.0459337
$690$	0	0
$691$	−11830.0	−0.651281	−0.325641	−	0.945494i	$-0.605580\pi$
$691$	−11830.0	−0.651281	−0.325641	+	0.945494i	$0.605580\pi$
$692$	0	0
$693$	36733.1	2.01353
$694$	0	0
$695$	471.319	0.0257240
$696$	0	0
$697$	−10737.9	−0.583541
$698$	0	0
$699$	−17209.6	−0.931225
$700$	0	0
$701$	6784.29	0.365534	0.182767	−	0.983156i	$-0.441495\pi$
$701$	6784.29	0.365534	0.182767	+	0.983156i	$0.441495\pi$
$702$	0	0
$703$	1669.40	0.0895629
$704$	0	0
$705$	2408.37	0.128659
$706$	0	0
$707$	31553.7	1.67850
$708$	0	0
$709$	−3055.22	−0.161835	−0.0809177	−	0.996721i	$-0.525785\pi$
$709$	−3055.22	−0.161835	−0.0809177	+	0.996721i	$0.525785\pi$
$710$	0	0
$711$	−82836.8	−4.36937
$712$	0	0
$713$	4719.71	0.247903
$714$	0	0
$715$	2301.32	0.120370
$716$	0	0
$717$	−48348.5	−2.51828
$718$	0	0
$719$	3484.92	0.180759	0.0903795	−	0.995907i	$-0.471192\pi$
$719$	3484.92	0.180759	0.0903795	+	0.995907i	$0.471192\pi$
$720$	0	0
$721$	15090.8	0.779489
$722$	0	0
$723$	544.930	0.0280307
$724$	0	0
$725$	−6427.66	−0.329265
$726$	0	0
$727$	6355.27	0.324214	0.162107	−	0.986773i	$-0.448171\pi$
$727$	6355.27	0.324214	0.162107	+	0.986773i	$0.448171\pi$
$728$	0	0
$729$	18118.7	0.920527
$730$	0	0
$731$	25350.4	1.28265
$732$	0	0
$733$	12048.0	0.607100	0.303550	−	0.952816i	$-0.401828\pi$
$733$	12048.0	0.607100	0.303550	+	0.952816i	$0.401828\pi$
$734$	0	0
$735$	6540.37	0.328225
$736$	0	0
$737$	21623.9	1.08077
$738$	0	0
$739$	22389.0	1.11447	0.557234	−	0.830355i	$-0.311863\pi$
$739$	22389.0	1.11447	0.557234	+	0.830355i	$0.311863\pi$
$740$	0	0
$741$	−14362.6	−0.712044
$742$	0	0
$743$	23891.4	1.17967	0.589833	−	0.807525i	$-0.299194\pi$
$743$	23891.4	1.17967	0.589833	+	0.807525i	$0.299194\pi$
$744$	0	0
$745$	17283.7	0.849969
$746$	0	0
$747$	37049.2	1.81467
$748$	0	0
$749$	9479.72	0.462458
$750$	0	0
$751$	−7623.41	−0.370416	−0.185208	−	0.982699i	$-0.559296\pi$
$751$	−7623.41	−0.370416	−0.185208	+	0.982699i	$0.559296\pi$
$752$	0	0
$753$	−40717.5	−1.97055
$754$	0	0
$755$	−13294.0	−0.640820
$756$	0	0
$757$	12184.2	0.584999	0.292499	−	0.956266i	$-0.405513\pi$
$757$	12184.2	0.584999	0.292499	+	0.956266i	$0.405513\pi$
$758$	0	0
$759$	−5701.11	−0.272645
$760$	0	0
$761$	−25501.9	−1.21477	−0.607386	−	0.794407i	$-0.707782\pi$
$761$	−25501.9	−1.21477	−0.607386	+	0.794407i	$0.707782\pi$
$762$	0	0
$763$	28283.3	1.34197
$764$	0	0
$765$	−23107.6	−1.09210
$766$	0	0
$767$	−11014.7	−0.518536
$768$	0	0
$769$	−29984.2	−1.40606	−0.703029	−	0.711161i	$-0.748170\pi$
$769$	−29984.2	−1.40606	−0.703029	+	0.711161i	$0.748170\pi$
$770$	0	0
$771$	18162.2	0.848371
$772$	0	0
$773$	8060.15	0.375037	0.187518	−	0.982261i	$-0.439956\pi$
$773$	8060.15	0.375037	0.187518	+	0.982261i	$0.439956\pi$
$774$	0	0
$775$	−5130.12	−0.237780
$776$	0	0
$777$	4161.23	0.192128
$778$	0	0
$779$	−12689.9	−0.583651
$780$	0	0
$781$	−8459.50	−0.387586
$782$	0	0
$783$	93314.4	4.25898
$784$	0	0
$785$	−5651.26	−0.256946
$786$	0	0
$787$	24517.3	1.11048	0.555241	−	0.831690i	$-0.312626\pi$
$787$	24517.3	1.11048	0.555241	+	0.831690i	$0.312626\pi$
$788$	0	0
$789$	−25822.8	−1.16517
$790$	0	0
$791$	−21090.0	−0.948008
$792$	0	0
$793$	−16022.0	−0.717476
$794$	0	0
$795$	2236.95	0.0997943
$796$	0	0
$797$	18153.0	0.806789	0.403395	−	0.915026i	$-0.367830\pi$
$797$	18153.0	0.806789	0.403395	+	0.915026i	$0.367830\pi$
$798$	0	0
$799$	3580.45	0.158532
$800$	0	0
$801$	91894.8	4.05361
$802$	0	0
$803$	−7515.36	−0.330276
$804$	0	0
$805$	−2518.15	−0.110252
$806$	0	0
$807$	−49558.7	−2.16177
$808$	0	0
$809$	26131.6	1.13565	0.567823	−	0.823151i	$-0.307786\pi$
$809$	26131.6	1.13565	0.567823	+	0.823151i	$0.307786\pi$
$810$	0	0
$811$	27873.2	1.20685	0.603427	−	0.797418i	$-0.293801\pi$
$811$	27873.2	1.20685	0.603427	+	0.797418i	$0.293801\pi$
$812$	0	0
$813$	−14061.0	−0.606570
$814$	0	0
$815$	9001.24	0.386871
$816$	0	0
$817$	29958.8	1.28289
$818$	0	0
$819$	−25278.9	−1.07853
$820$	0	0
$821$	−22706.3	−0.965232	−0.482616	−	0.875832i	$-0.660313\pi$
$821$	−22706.3	−0.965232	−0.482616	+	0.875832i	$0.660313\pi$
$822$	0	0
$823$	15620.9	0.661616	0.330808	−	0.943698i	$-0.392679\pi$
$823$	15620.9	0.661616	0.330808	+	0.943698i	$0.392679\pi$
$824$	0	0
$825$	6196.86	0.261512
$826$	0	0
$827$	8881.39	0.373442	0.186721	−	0.982413i	$-0.440214\pi$
$827$	8881.39	0.373442	0.186721	+	0.982413i	$0.440214\pi$
$828$	0	0
$829$	5318.64	0.222827	0.111414	−	0.993774i	$-0.464462\pi$
$829$	5318.64	0.222827	0.111414	+	0.993774i	$0.464462\pi$
$830$	0	0
$831$	−67284.2	−2.80874
$832$	0	0
$833$	9723.35	0.404435
$834$	0	0
$835$	17395.7	0.720962
$836$	0	0
$837$	74477.2	3.07564
$838$	0	0
$839$	−35909.2	−1.47762	−0.738811	−	0.673913i	$-0.764612\pi$
$839$	−35909.2	−1.47762	−0.738811	+	0.673913i	$0.764612\pi$
$840$	0	0
$841$	41714.6	1.71039
$842$	0	0
$843$	73580.2	3.00621
$844$	0	0
$845$	9401.29	0.382739
$846$	0	0
$847$	14499.8	0.588217
$848$	0	0
$849$	30683.4	1.24034
$850$	0	0
$851$	−456.023	−0.0183693
$852$	0	0
$853$	−41861.6	−1.68032	−0.840161	−	0.542337i	$-0.817540\pi$
$853$	−41861.6	−1.68032	−0.840161	+	0.542337i	$0.817540\pi$
$854$	0	0
$855$	−27308.2	−1.09231
$856$	0	0
$857$	9758.74	0.388976	0.194488	−	0.980905i	$-0.437696\pi$
$857$	9758.74	0.388976	0.194488	+	0.980905i	$0.437696\pi$
$858$	0	0
$859$	−33052.9	−1.31286	−0.656432	−	0.754385i	$-0.727935\pi$
$859$	−33052.9	−1.31286	−0.656432	+	0.754385i	$0.727935\pi$
$860$	0	0
$861$	−31631.5	−1.25203
$862$	0	0
$863$	−11136.3	−0.439263	−0.219631	−	0.975583i	$-0.570485\pi$
$863$	−11136.3	−0.439263	−0.219631	+	0.975583i	$0.570485\pi$
$864$	0	0
$865$	14567.8	0.572623
$866$	0	0
$867$	−1562.79	−0.0612168
$868$	0	0
$869$	33025.9	1.28921
$870$	0	0
$871$	−14881.1	−0.578905
$872$	0	0
$873$	29654.3	1.14965
$874$	0	0
$875$	2737.11	0.105750
$876$	0	0
$877$	51513.0	1.98343	0.991717	−	0.128441i	$-0.0409972\pi$
$877$	51513.0	1.98343	0.991717	+	0.128441i	$0.0409972\pi$
$878$	0	0
$879$	−56289.8	−2.15996
$880$	0	0
$881$	−862.527	−0.0329844	−0.0164922	−	0.999864i	$-0.505250\pi$
$881$	−862.527	−0.0329844	−0.0164922	+	0.999864i	$0.505250\pi$
$882$	0	0
$883$	−27670.6	−1.05458	−0.527288	−	0.849686i	$-0.676791\pi$
$883$	−27670.6	−1.05458	−0.527288	+	0.849686i	$0.676791\pi$
$884$	0	0
$885$	−29659.8	−1.12656
$886$	0	0
$887$	27956.9	1.05829	0.529143	−	0.848533i	$-0.322514\pi$
$887$	27956.9	1.05829	0.529143	+	0.848533i	$0.322514\pi$
$888$	0	0
$889$	−7896.21	−0.297897
$890$	0	0
$891$	−44670.1	−1.67958
$892$	0	0
$893$	4231.32	0.158562
$894$	0	0
$895$	−3732.29	−0.139393
$896$	0	0
$897$	3923.38	0.146040
$898$	0	0
$899$	52759.4	1.95731
$900$	0	0
$901$	3325.60	0.122965
$902$	0	0
$903$	74676.5	2.75203
$904$	0	0
$905$	−3378.75	−0.124103
$906$	0	0
$907$	5866.55	0.214769	0.107385	−	0.994218i	$-0.465752\pi$
$907$	5866.55	0.214769	0.107385	+	0.994218i	$0.465752\pi$
$908$	0	0
$909$	−93473.5	−3.41069
$910$	0	0
$911$	−9663.57	−0.351447	−0.175724	−	0.984440i	$-0.556227\pi$
$911$	−9663.57	−0.351447	−0.175724	+	0.984440i	$0.556227\pi$
$912$	0	0
$913$	−14771.0	−0.535431
$914$	0	0
$915$	−43143.3	−1.55877
$916$	0	0
$917$	−18865.4	−0.679380
$918$	0	0
$919$	−27983.5	−1.00445	−0.502226	−	0.864737i	$-0.667485\pi$
$919$	−27983.5	−1.00445	−0.502226	+	0.864737i	$0.667485\pi$
$920$	0	0
$921$	−33638.7	−1.20351
$922$	0	0
$923$	5821.64	0.207607
$924$	0	0
$925$	495.678	0.0176192
$926$	0	0
$927$	−44704.5	−1.58391
$928$	0	0
$929$	39607.1	1.39878	0.699390	−	0.714741i	$-0.253455\pi$
$929$	39607.1	1.39878	0.699390	+	0.714741i	$0.253455\pi$
$930$	0	0
$931$	11490.9	0.404511
$932$	0	0
$933$	−20613.1	−0.723303
$934$	0	0
$935$	9212.68	0.322232
$936$	0	0
$937$	−47936.1	−1.67129	−0.835647	−	0.549267i	$-0.814907\pi$
$937$	−47936.1	−1.67129	−0.835647	+	0.549267i	$0.814907\pi$
$938$	0	0
$939$	92459.1	3.21330
$940$	0	0
$941$	25913.7	0.897728	0.448864	−	0.893600i	$-0.351829\pi$
$941$	25913.7	0.897728	0.448864	+	0.893600i	$0.351829\pi$
$942$	0	0
$943$	3466.45	0.119706
$944$	0	0
$945$	−39736.5	−1.36786
$946$	0	0
$947$	37712.9	1.29409	0.647046	−	0.762451i	$-0.276004\pi$
$947$	37712.9	1.29409	0.647046	+	0.762451i	$0.276004\pi$
$948$	0	0
$949$	5171.90	0.176909
$950$	0	0
$951$	−7432.31	−0.253427
$952$	0	0
$953$	16785.4	0.570548	0.285274	−	0.958446i	$-0.407915\pi$
$953$	16785.4	0.570548	0.285274	+	0.958446i	$0.407915\pi$
$954$	0	0
$955$	−12215.2	−0.413899
$956$	0	0
$957$	−63730.1	−2.15267
$958$	0	0
$959$	−9156.44	−0.308318
$960$	0	0
$961$	12318.0	0.413481
$962$	0	0
$963$	−28082.4	−0.939712
$964$	0	0
$965$	−21761.5	−0.725936
$966$	0	0
$967$	50355.9	1.67460	0.837299	−	0.546745i	$-0.184133\pi$
$967$	50355.9	1.67460	0.837299	+	0.546745i	$0.184133\pi$
$968$	0	0
$969$	−57496.8	−1.90615
$970$	0	0
$971$	36370.3	1.20204	0.601018	−	0.799235i	$-0.294762\pi$
$971$	36370.3	1.20204	0.601018	+	0.799235i	$0.294762\pi$
$972$	0	0
$973$	2064.09	0.0680078
$974$	0	0
$975$	−4264.54	−0.140077
$976$	0	0
$977$	−25701.4	−0.841617	−0.420808	−	0.907150i	$-0.638254\pi$
$977$	−25701.4	−0.841617	−0.420808	+	0.907150i	$0.638254\pi$
$978$	0	0
$979$	−36637.2	−1.19605
$980$	0	0
$981$	−83785.4	−2.72687
$982$	0	0
$983$	29857.8	0.968787	0.484393	−	0.874850i	$-0.339040\pi$
$983$	29857.8	0.968787	0.484393	+	0.874850i	$0.339040\pi$
$984$	0	0
$985$	18515.1	0.598925
$986$	0	0
$987$	10547.2	0.340142
$988$	0	0
$989$	−8183.70	−0.263121
$990$	0	0
$991$	−21202.2	−0.679627	−0.339813	−	0.940493i	$-0.610364\pi$
$991$	−21202.2	−0.679627	−0.339813	+	0.940493i	$0.610364\pi$
$992$	0	0
$993$	3158.64	0.100943
$994$	0	0
$995$	−3173.90	−0.101125
$996$	0	0
$997$	43316.3	1.37597	0.687985	−	0.725725i	$-0.258496\pi$
$997$	43316.3	1.37597	0.687985	+	0.725725i	$0.258496\pi$
$998$	0	0
$999$	−7196.07	−0.227901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.bb.1.1		10
4.3	odd	2		920.4.a.g.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.g.1.10	✓	10	4.3	odd	2
1840.4.a.bb.1.1		10	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-9.58471 q^{3} -5.00000 q^{5} -21.8969 q^{7} +64.8667 q^{9} +O(q^{10})\)	\(q-9.58471 q^{3} -5.00000 q^{5} -21.8969 q^{7} +64.8667 q^{9} -25.8615 q^{11} +17.7973 q^{13} +47.9236 q^{15} +71.2464 q^{17} +84.1980 q^{19} +209.876 q^{21} -23.0000 q^{23} +25.0000 q^{25} -362.941 q^{27} -257.106 q^{29} -205.205 q^{31} +247.875 q^{33} +109.485 q^{35} +19.8271 q^{37} -170.582 q^{39} -150.715 q^{41} +355.813 q^{43} -324.333 q^{45} +50.2544 q^{47} +136.475 q^{49} -682.876 q^{51} +46.6775 q^{53} +129.307 q^{55} -807.014 q^{57} -618.898 q^{59} -900.252 q^{61} -1420.38 q^{63} -88.9863 q^{65} -836.144 q^{67} +220.448 q^{69} +327.109 q^{71} +290.601 q^{73} -239.618 q^{75} +566.286 q^{77} -1277.03 q^{79} +1727.28 q^{81} +571.159 q^{83} -356.232 q^{85} +2464.29 q^{87} +1416.67 q^{89} -389.705 q^{91} +1966.83 q^{93} -420.990 q^{95} +457.158 q^{97} -1677.55 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * q^99

Introduction

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