LMFDB - Embedded newform 1840.4.a.ba.1.2

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} - 5 x^{9} - 192 x^{8} + 762 x^{7} + 12246 x^{6} - 33828 x^{5} - 298243 x^{4} + 383603 x^{3} + \cdots + 57408 $ x^10 - 5x^9 - 192x^8 + 762x^7 + 12246x^6 - 33828x^5 - 298243x^4 + 383603x^3 + 2423016x^2 + 864576*x + 57408
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.50847$ 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)

$f(q)$	$=$	$q-8.50847 q^{3} +5.00000 q^{5} -14.9751 q^{7} +45.3940 q^{9} +O(q^{10})$	$q-8.50847 q^{3} +5.00000 q^{5} -14.9751 q^{7} +45.3940 q^{9} +45.9901 q^{11} +76.1939 q^{13} -42.5423 q^{15} +105.202 q^{17} +93.8975 q^{19} +127.415 q^{21} +23.0000 q^{23} +25.0000 q^{25} -156.505 q^{27} +218.177 q^{29} +178.941 q^{31} -391.305 q^{33} -74.8757 q^{35} -85.2175 q^{37} -648.293 q^{39} +452.430 q^{41} +60.6941 q^{43} +226.970 q^{45} -232.963 q^{47} -118.745 q^{49} -895.109 q^{51} -352.776 q^{53} +229.950 q^{55} -798.924 q^{57} +490.342 q^{59} +445.773 q^{61} -679.782 q^{63} +380.970 q^{65} +171.205 q^{67} -195.695 q^{69} -297.395 q^{71} +501.109 q^{73} -212.712 q^{75} -688.708 q^{77} -494.725 q^{79} +105.978 q^{81} +220.907 q^{83} +526.011 q^{85} -1856.35 q^{87} +156.585 q^{89} -1141.01 q^{91} -1522.51 q^{93} +469.488 q^{95} +1347.98 q^{97} +2087.68 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) $ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9	$ 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) $ 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 + 56 * q^11 + 49 * q^13 - 25 * q^15 + 240 * q^17 - 88 * q^19 + 346 * q^21 + 230 * q^23 + 250 * q^25 - 449 * q^27 + 319 * q^29 + 109 * q^31 + 504 * q^33 - 70 * q^35 + 580 * q^37 - 107 * q^39 + 259 * q^41 - 330 * q^43 + 695 * q^45 - 227 * q^47 + 630 * q^49 + 192 * q^51 - 186 * q^53 + 280 * q^55 + 1708 * q^57 - 262 * q^59 + 1000 * q^61 - 722 * q^63 + 245 * q^65 - 354 * q^67 - 115 * q^69 - 599 * q^71 + 355 * q^73 - 125 * q^75 + 1776 * q^77 + 1068 * q^79 + 3490 * q^81 - 754 * q^83 + 1200 * q^85 - 2675 * q^87 + 1740 * q^89 - 690 * q^91 + 1669 * q^93 - 440 * q^95 + 2592 * q^97 - 916 * 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
$3$	−8.50847	−1.63746	−0.818728	−	0.574182i	$-0.805320\pi$
$3$	−8.50847	−1.63746	−0.818728	+	0.574182i	$0.805320\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−14.9751	−0.808582	−0.404291	−	0.914630i	$-0.632482\pi$
$7$	−14.9751	−0.808582	−0.404291	+	0.914630i	$0.632482\pi$
$8$	0	0
$9$	45.3940	1.68126
$10$	0	0
$11$	45.9901	1.26059	0.630297	−	0.776354i	$-0.282933\pi$
$11$	45.9901	1.26059	0.630297	+	0.776354i	$0.282933\pi$
$12$	0	0
$13$	76.1939	1.62557	0.812784	−	0.582565i	$-0.197951\pi$
$13$	76.1939	1.62557	0.812784	+	0.582565i	$0.197951\pi$
$14$	0	0
$15$	−42.5423	−0.732292
$16$	0	0
$17$	105.202	1.50090	0.750449	−	0.660929i	$-0.229837\pi$
$17$	105.202	1.50090	0.750449	+	0.660929i	$0.229837\pi$
$18$	0	0
$19$	93.8975	1.13377	0.566883	−	0.823798i	$-0.308149\pi$
$19$	93.8975	1.13377	0.566883	+	0.823798i	$0.308149\pi$
$20$	0	0
$21$	127.415	1.32402
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−156.505	−1.11553
$28$	0	0
$29$	218.177	1.39705	0.698524	−	0.715587i	$-0.253841\pi$
$29$	218.177	1.39705	0.698524	+	0.715587i	$0.253841\pi$
$30$	0	0
$31$	178.941	1.03673	0.518367	−	0.855158i	$-0.326540\pi$
$31$	178.941	1.03673	0.518367	+	0.855158i	$0.326540\pi$
$32$	0	0
$33$	−391.305	−2.06417
$34$	0	0
$35$	−74.8757	−0.361609
$36$	0	0
$37$	−85.2175	−0.378640	−0.189320	−	0.981915i	$-0.560628\pi$
$37$	−85.2175	−0.378640	−0.189320	+	0.981915i	$0.560628\pi$
$38$	0	0
$39$	−648.293	−2.66180
$40$	0	0
$41$	452.430	1.72336	0.861679	−	0.507454i	$-0.169413\pi$
$41$	452.430	1.72336	0.861679	+	0.507454i	$0.169413\pi$
$42$	0	0
$43$	60.6941	0.215250	0.107625	−	0.994192i	$-0.465675\pi$
$43$	60.6941	0.215250	0.107625	+	0.994192i	$0.465675\pi$
$44$	0	0
$45$	226.970	0.751882
$46$	0	0
$47$	−232.963	−0.723004	−0.361502	−	0.932371i	$-0.617736\pi$
$47$	−232.963	−0.723004	−0.361502	+	0.932371i	$0.617736\pi$
$48$	0	0
$49$	−118.745	−0.346196
$50$	0	0
$51$	−895.109	−2.45765
$52$	0	0
$53$	−352.776	−0.914294	−0.457147	−	0.889391i	$-0.651129\pi$
$53$	−352.776	−0.914294	−0.457147	+	0.889391i	$0.651129\pi$
$54$	0	0
$55$	229.950	0.563755
$56$	0	0
$57$	−798.924	−1.85649
$58$	0	0
$59$	490.342	1.08199	0.540993	−	0.841027i	$-0.318049\pi$
$59$	490.342	1.08199	0.540993	+	0.841027i	$0.318049\pi$
$60$	0	0
$61$	445.773	0.935662	0.467831	−	0.883818i	$-0.345036\pi$
$61$	445.773	0.935662	0.467831	+	0.883818i	$0.345036\pi$
$62$	0	0
$63$	−679.782	−1.35944
$64$	0	0
$65$	380.970	0.726976
$66$	0	0
$67$	171.205	0.312178	0.156089	−	0.987743i	$-0.450111\pi$
$67$	171.205	0.312178	0.156089	+	0.987743i	$0.450111\pi$
$68$	0	0
$69$	−195.695	−0.341433
$70$	0	0
$71$	−297.395	−0.497102	−0.248551	−	0.968619i	$-0.579954\pi$
$71$	−297.395	−0.497102	−0.248551	+	0.968619i	$0.579954\pi$
$72$	0	0
$73$	501.109	0.803430	0.401715	−	0.915765i	$-0.368414\pi$
$73$	501.109	0.803430	0.401715	+	0.915765i	$0.368414\pi$
$74$	0	0
$75$	−212.712	−0.327491
$76$	0	0
$77$	−688.708	−1.01929
$78$	0	0
$79$	−494.725	−0.704569	−0.352284	−	0.935893i	$-0.614595\pi$
$79$	−494.725	−0.704569	−0.352284	+	0.935893i	$0.614595\pi$
$80$	0	0
$81$	105.978	0.145375
$82$	0	0
$83$	220.907	0.292141	0.146071	−	0.989274i	$-0.453337\pi$
$83$	220.907	0.292141	0.146071	+	0.989274i	$0.453337\pi$
$84$	0	0
$85$	526.011	0.671222
$86$	0	0
$87$	−1856.35	−2.28760
$88$	0	0
$89$	156.585	0.186495	0.0932473	−	0.995643i	$-0.470275\pi$
$89$	156.585	0.186495	0.0932473	+	0.995643i	$0.470275\pi$
$90$	0	0
$91$	−1141.01	−1.31440
$92$	0	0
$93$	−1522.51	−1.69761
$94$	0	0
$95$	469.488	0.507036
$96$	0	0
$97$	1347.98	1.41100	0.705501	−	0.708709i	$-0.250722\pi$
$97$	1347.98	1.41100	0.705501	+	0.708709i	$0.250722\pi$
$98$	0	0
$99$	2087.68	2.11939
$100$	0	0
$101$	−893.571	−0.880333	−0.440167	−	0.897916i	$-0.645081\pi$
$101$	−893.571	−0.880333	−0.440167	+	0.897916i	$0.645081\pi$
$102$	0	0
$103$	97.5123	0.0932832	0.0466416	−	0.998912i	$-0.485148\pi$
$103$	97.5123	0.0932832	0.0466416	+	0.998912i	$0.485148\pi$
$104$	0	0
$105$	637.077	0.592118
$106$	0	0
$107$	895.508	0.809085	0.404542	−	0.914519i	$-0.367431\pi$
$107$	895.508	0.809085	0.404542	+	0.914519i	$0.367431\pi$
$108$	0	0
$109$	−443.066	−0.389340	−0.194670	−	0.980869i	$-0.562364\pi$
$109$	−443.066	−0.389340	−0.194670	+	0.980869i	$0.562364\pi$
$110$	0	0
$111$	725.070	0.620006
$112$	0	0
$113$	−1966.85	−1.63740	−0.818699	−	0.574223i	$-0.805304\pi$
$113$	−1966.85	−1.63740	−0.818699	+	0.574223i	$0.805304\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	3458.75	2.73300
$118$	0	0
$119$	−1575.42	−1.21360
$120$	0	0
$121$	784.089	0.589098
$122$	0	0
$123$	−3849.48	−2.82192
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1137.03	−0.794451	−0.397226	−	0.917721i	$-0.630027\pi$
$127$	−1137.03	−0.794451	−0.397226	+	0.917721i	$0.630027\pi$
$128$	0	0
$129$	−516.414	−0.352463
$130$	0	0
$131$	−671.327	−0.447741	−0.223871	−	0.974619i	$-0.571869\pi$
$131$	−671.327	−0.447741	−0.223871	+	0.974619i	$0.571869\pi$
$132$	0	0
$133$	−1406.13	−0.916743
$134$	0	0
$135$	−782.524	−0.498881
$136$	0	0
$137$	−694.421	−0.433054	−0.216527	−	0.976277i	$-0.569473\pi$
$137$	−694.421	−0.433054	−0.216527	+	0.976277i	$0.569473\pi$
$138$	0	0
$139$	1406.56	0.858292	0.429146	−	0.903235i	$-0.358815\pi$
$139$	1406.56	0.858292	0.429146	+	0.903235i	$0.358815\pi$
$140$	0	0
$141$	1982.16	1.18389
$142$	0	0
$143$	3504.17	2.04918
$144$	0	0
$145$	1090.88	0.624779
$146$	0	0
$147$	1010.34	0.566880
$148$	0	0
$149$	−53.3959	−0.0293582	−0.0146791	−	0.999892i	$-0.504673\pi$
$149$	−53.3959	−0.0293582	−0.0146791	+	0.999892i	$0.504673\pi$
$150$	0	0
$151$	−1813.95	−0.977597	−0.488799	−	0.872397i	$-0.662565\pi$
$151$	−1813.95	−0.977597	−0.488799	+	0.872397i	$0.662565\pi$
$152$	0	0
$153$	4775.55	2.52340
$154$	0	0
$155$	894.705	0.463642
$156$	0	0
$157$	−2766.14	−1.40613	−0.703065	−	0.711126i	$-0.748186\pi$
$157$	−2766.14	−1.40613	−0.703065	+	0.711126i	$0.748186\pi$
$158$	0	0
$159$	3001.59	1.49711
$160$	0	0
$161$	−344.428	−0.168601
$162$	0	0
$163$	1804.39	0.867062	0.433531	−	0.901139i	$-0.357267\pi$
$163$	1804.39	0.867062	0.433531	+	0.901139i	$0.357267\pi$
$164$	0	0
$165$	−1956.53	−0.923123
$166$	0	0
$167$	2351.70	1.08970	0.544850	−	0.838534i	$-0.316587\pi$
$167$	2351.70	1.08970	0.544850	+	0.838534i	$0.316587\pi$
$168$	0	0
$169$	3608.51	1.64247
$170$	0	0
$171$	4262.39	1.90616
$172$	0	0
$173$	614.871	0.270218	0.135109	−	0.990831i	$-0.456862\pi$
$173$	614.871	0.270218	0.135109	+	0.990831i	$0.456862\pi$
$174$	0	0
$175$	−374.379	−0.161716
$176$	0	0
$177$	−4172.06	−1.77170
$178$	0	0
$179$	−3083.55	−1.28757	−0.643786	−	0.765206i	$-0.722637\pi$
$179$	−3083.55	−1.28757	−0.643786	+	0.765206i	$0.722637\pi$
$180$	0	0
$181$	−418.253	−0.171760	−0.0858798	−	0.996306i	$-0.527370\pi$
$181$	−418.253	−0.171760	−0.0858798	+	0.996306i	$0.527370\pi$
$182$	0	0
$183$	−3792.84	−1.53210
$184$	0	0
$185$	−426.088	−0.169333
$186$	0	0
$187$	4838.26	1.89202
$188$	0	0
$189$	2343.68	0.901999
$190$	0	0
$191$	4563.03	1.72863	0.864317	−	0.502947i	$-0.167751\pi$
$191$	4563.03	1.72863	0.864317	+	0.502947i	$0.167751\pi$
$192$	0	0
$193$	4611.12	1.71977	0.859884	−	0.510489i	$-0.170536\pi$
$193$	4611.12	1.71977	0.859884	+	0.510489i	$0.170536\pi$
$194$	0	0
$195$	−3241.47	−1.19039
$196$	0	0
$197$	−2043.76	−0.739148	−0.369574	−	0.929201i	$-0.620496\pi$
$197$	−2043.76	−0.739148	−0.369574	+	0.929201i	$0.620496\pi$
$198$	0	0
$199$	2127.14	0.757733	0.378866	−	0.925451i	$-0.376314\pi$
$199$	2127.14	0.757733	0.378866	+	0.925451i	$0.376314\pi$
$200$	0	0
$201$	−1456.69	−0.511178
$202$	0	0
$203$	−3267.23	−1.12963
$204$	0	0
$205$	2262.15	0.770709
$206$	0	0
$207$	1044.06	0.350567
$208$	0	0
$209$	4318.36	1.42922
$210$	0	0
$211$	−5565.71	−1.81592	−0.907961	−	0.419056i	$-0.862361\pi$
$211$	−5565.71	−1.81592	−0.907961	+	0.419056i	$0.862361\pi$
$212$	0	0
$213$	2530.37	0.813983
$214$	0	0
$215$	303.470	0.0962628
$216$	0	0
$217$	−2679.67	−0.838284
$218$	0	0
$219$	−4263.67	−1.31558
$220$	0	0
$221$	8015.76	2.43981
$222$	0	0
$223$	−4705.09	−1.41290	−0.706449	−	0.707764i	$-0.749704\pi$
$223$	−4705.09	−1.41290	−0.706449	+	0.707764i	$0.749704\pi$
$224$	0	0
$225$	1134.85	0.336252
$226$	0	0
$227$	−1906.66	−0.557486	−0.278743	−	0.960366i	$-0.589918\pi$
$227$	−1906.66	−0.557486	−0.278743	+	0.960366i	$0.589918\pi$
$228$	0	0
$229$	−1510.97	−0.436017	−0.218008	−	0.975947i	$-0.569956\pi$
$229$	−1510.97	−0.436017	−0.218008	+	0.975947i	$0.569956\pi$
$230$	0	0
$231$	5859.85	1.66905
$232$	0	0
$233$	−2447.34	−0.688114	−0.344057	−	0.938949i	$-0.611801\pi$
$233$	−2447.34	−0.688114	−0.344057	+	0.938949i	$0.611801\pi$
$234$	0	0
$235$	−1164.82	−0.323337
$236$	0	0
$237$	4209.35	1.15370
$238$	0	0
$239$	4174.32	1.12977	0.564883	−	0.825171i	$-0.308921\pi$
$239$	4174.32	1.12977	0.564883	+	0.825171i	$0.308921\pi$
$240$	0	0
$241$	4167.94	1.11403	0.557014	−	0.830503i	$-0.311947\pi$
$241$	4167.94	1.11403	0.557014	+	0.830503i	$0.311947\pi$
$242$	0	0
$243$	3323.92	0.877488
$244$	0	0
$245$	−593.726	−0.154823
$246$	0	0
$247$	7154.42	1.84302
$248$	0	0
$249$	−1879.58	−0.478368
$250$	0	0
$251$	−4082.10	−1.02653	−0.513267	−	0.858229i	$-0.671565\pi$
$251$	−4082.10	−1.02653	−0.513267	+	0.858229i	$0.671565\pi$
$252$	0	0
$253$	1057.77	0.262852
$254$	0	0
$255$	−4475.54	−1.09910
$256$	0	0
$257$	−7124.37	−1.72921	−0.864603	−	0.502456i	$-0.832430\pi$
$257$	−7124.37	−1.72921	−0.864603	+	0.502456i	$0.832430\pi$
$258$	0	0
$259$	1276.14	0.306161
$260$	0	0
$261$	9903.91	2.34880
$262$	0	0
$263$	−4006.01	−0.939243	−0.469622	−	0.882868i	$-0.655610\pi$
$263$	−4006.01	−0.939243	−0.469622	+	0.882868i	$0.655610\pi$
$264$	0	0
$265$	−1763.88	−0.408885
$266$	0	0
$267$	−1332.30	−0.305377
$268$	0	0
$269$	6415.22	1.45406	0.727032	−	0.686604i	$-0.240899\pi$
$269$	6415.22	1.45406	0.727032	+	0.686604i	$0.240899\pi$
$270$	0	0
$271$	5513.26	1.23582	0.617909	−	0.786250i	$-0.287980\pi$
$271$	5513.26	1.23582	0.617909	+	0.786250i	$0.287980\pi$
$272$	0	0
$273$	9708.28	2.15228
$274$	0	0
$275$	1149.75	0.252119
$276$	0	0
$277$	−2986.58	−0.647821	−0.323910	−	0.946088i	$-0.604998\pi$
$277$	−2986.58	−0.647821	−0.323910	+	0.946088i	$0.604998\pi$
$278$	0	0
$279$	8122.85	1.74302
$280$	0	0
$281$	6974.00	1.48055	0.740274	−	0.672305i	$-0.234696\pi$
$281$	6974.00	1.48055	0.740274	+	0.672305i	$0.234696\pi$
$282$	0	0
$283$	−3419.16	−0.718191	−0.359096	−	0.933301i	$-0.616915\pi$
$283$	−3419.16	−0.718191	−0.359096	+	0.933301i	$0.616915\pi$
$284$	0	0
$285$	−3994.62	−0.830249
$286$	0	0
$287$	−6775.20	−1.39348
$288$	0	0
$289$	6154.48	1.25269
$290$	0	0
$291$	−11469.3	−2.31045
$292$	0	0
$293$	−7372.94	−1.47007	−0.735037	−	0.678027i	$-0.762835\pi$
$293$	−7372.94	−1.47007	−0.735037	+	0.678027i	$0.762835\pi$
$294$	0	0
$295$	2451.71	0.483878
$296$	0	0
$297$	−7197.67	−1.40623
$298$	0	0
$299$	1752.46	0.338954
$300$	0	0
$301$	−908.902	−0.174047
$302$	0	0
$303$	7602.92	1.44151
$304$	0	0
$305$	2228.86	0.418441
$306$	0	0
$307$	−7309.62	−1.35890	−0.679450	−	0.733722i	$-0.737781\pi$
$307$	−7309.62	−1.35890	−0.679450	+	0.733722i	$0.737781\pi$
$308$	0	0
$309$	−829.680	−0.152747
$310$	0	0
$311$	6229.49	1.13583	0.567913	−	0.823088i	$-0.307751\pi$
$311$	6229.49	1.13583	0.567913	+	0.823088i	$0.307751\pi$
$312$	0	0
$313$	3686.22	0.665678	0.332839	−	0.942984i	$-0.391993\pi$
$313$	3686.22	0.665678	0.332839	+	0.942984i	$0.391993\pi$
$314$	0	0
$315$	−3398.91	−0.607958
$316$	0	0
$317$	−8232.97	−1.45871	−0.729353	−	0.684137i	$-0.760179\pi$
$317$	−8232.97	−1.45871	−0.729353	+	0.684137i	$0.760179\pi$
$318$	0	0
$319$	10034.0	1.76111
$320$	0	0
$321$	−7619.40	−1.32484
$322$	0	0
$323$	9878.22	1.70167
$324$	0	0
$325$	1904.85	0.325114
$326$	0	0
$327$	3769.81	0.637526
$328$	0	0
$329$	3488.66	0.584608
$330$	0	0
$331$	7809.47	1.29682	0.648410	−	0.761292i	$-0.275434\pi$
$331$	7809.47	1.29682	0.648410	+	0.761292i	$0.275434\pi$
$332$	0	0
$333$	−3868.36	−0.636592
$334$	0	0
$335$	856.023	0.139610
$336$	0	0
$337$	−2776.22	−0.448754	−0.224377	−	0.974502i	$-0.572035\pi$
$337$	−2776.22	−0.448754	−0.224377	+	0.974502i	$0.572035\pi$
$338$	0	0
$339$	16734.9	2.68116
$340$	0	0
$341$	8229.51	1.30690
$342$	0	0
$343$	6914.70	1.08851
$344$	0	0
$345$	−978.474	−0.152693
$346$	0	0
$347$	4671.68	0.722734	0.361367	−	0.932424i	$-0.382310\pi$
$347$	4671.68	0.722734	0.361367	+	0.932424i	$0.382310\pi$
$348$	0	0
$349$	−4872.47	−0.747327	−0.373664	−	0.927564i	$-0.621899\pi$
$349$	−4872.47	−0.747327	−0.373664	+	0.927564i	$0.621899\pi$
$350$	0	0
$351$	−11924.7	−1.81337
$352$	0	0
$353$	−1882.27	−0.283806	−0.141903	−	0.989881i	$-0.545322\pi$
$353$	−1882.27	−0.283806	−0.141903	+	0.989881i	$0.545322\pi$
$354$	0	0
$355$	−1486.97	−0.222311
$356$	0	0
$357$	13404.4	1.98721
$358$	0	0
$359$	8290.72	1.21885	0.609425	−	0.792843i	$-0.291400\pi$
$359$	8290.72	1.21885	0.609425	+	0.792843i	$0.291400\pi$
$360$	0	0
$361$	1957.75	0.285428
$362$	0	0
$363$	−6671.40	−0.964621
$364$	0	0
$365$	2505.55	0.359305
$366$	0	0
$367$	13217.6	1.87998	0.939991	−	0.341199i	$-0.110833\pi$
$367$	13217.6	1.87998	0.939991	+	0.341199i	$0.110833\pi$
$368$	0	0
$369$	20537.6	2.89741
$370$	0	0
$371$	5282.88	0.739281
$372$	0	0
$373$	9192.32	1.27603	0.638017	−	0.770023i	$-0.279755\pi$
$373$	9192.32	1.27603	0.638017	+	0.770023i	$0.279755\pi$
$374$	0	0
$375$	−1063.56	−0.146458
$376$	0	0
$377$	16623.7	2.27100
$378$	0	0
$379$	5741.12	0.778104	0.389052	−	0.921216i	$-0.372803\pi$
$379$	5741.12	0.778104	0.389052	+	0.921216i	$0.372803\pi$
$380$	0	0
$381$	9674.40	1.30088
$382$	0	0
$383$	−3473.84	−0.463460	−0.231730	−	0.972780i	$-0.574439\pi$
$383$	−3473.84	−0.463460	−0.231730	+	0.972780i	$0.574439\pi$
$384$	0	0
$385$	−3443.54	−0.455842
$386$	0	0
$387$	2755.15	0.361892
$388$	0	0
$389$	−7283.20	−0.949288	−0.474644	−	0.880178i	$-0.657423\pi$
$389$	−7283.20	−0.949288	−0.474644	+	0.880178i	$0.657423\pi$
$390$	0	0
$391$	2419.65	0.312959
$392$	0	0
$393$	5711.96	0.733156
$394$	0	0
$395$	−2473.63	−0.315093
$396$	0	0
$397$	−11743.2	−1.48457	−0.742287	−	0.670082i	$-0.766259\pi$
$397$	−11743.2	−1.48457	−0.742287	+	0.670082i	$0.766259\pi$
$398$	0	0
$399$	11964.0	1.50113
$400$	0	0
$401$	−15287.6	−1.90381	−0.951905	−	0.306393i	$-0.900878\pi$
$401$	−15287.6	−1.90381	−0.951905	+	0.306393i	$0.900878\pi$
$402$	0	0
$403$	13634.2	1.68528
$404$	0	0
$405$	529.890	0.0650135
$406$	0	0
$407$	−3919.16	−0.477311
$408$	0	0
$409$	−188.953	−0.0228438	−0.0114219	−	0.999935i	$-0.503636\pi$
$409$	−188.953	−0.0228438	−0.0114219	+	0.999935i	$0.503636\pi$
$410$	0	0
$411$	5908.46	0.709106
$412$	0	0
$413$	−7342.94	−0.874873
$414$	0	0
$415$	1104.54	0.130650
$416$	0	0
$417$	−11967.6	−1.40541
$418$	0	0
$419$	−10577.6	−1.23329	−0.616644	−	0.787242i	$-0.711508\pi$
$419$	−10577.6	−1.23329	−0.616644	+	0.787242i	$0.711508\pi$
$420$	0	0
$421$	−3111.05	−0.360150	−0.180075	−	0.983653i	$-0.557634\pi$
$421$	−3111.05	−0.360150	−0.180075	+	0.983653i	$0.557634\pi$
$422$	0	0
$423$	−10575.1	−1.21556
$424$	0	0
$425$	2630.05	0.300180
$426$	0	0
$427$	−6675.51	−0.756559
$428$	0	0
$429$	−29815.1	−3.35544
$430$	0	0
$431$	−7701.18	−0.860679	−0.430340	−	0.902667i	$-0.641606\pi$
$431$	−7701.18	−0.860679	−0.430340	+	0.902667i	$0.641606\pi$
$432$	0	0
$433$	10597.5	1.17617	0.588087	−	0.808798i	$-0.299881\pi$
$433$	10597.5	1.17617	0.588087	+	0.808798i	$0.299881\pi$
$434$	0	0
$435$	−9281.74	−1.02305
$436$	0	0
$437$	2159.64	0.236407
$438$	0	0
$439$	−2873.84	−0.312439	−0.156220	−	0.987722i	$-0.549931\pi$
$439$	−2873.84	−0.312439	−0.156220	+	0.987722i	$0.549931\pi$
$440$	0	0
$441$	−5390.32	−0.582045
$442$	0	0
$443$	12506.2	1.34128	0.670641	−	0.741782i	$-0.266019\pi$
$443$	12506.2	1.34128	0.670641	+	0.741782i	$0.266019\pi$
$444$	0	0
$445$	782.927	0.0834030
$446$	0	0
$447$	454.318	0.0480727
$448$	0	0
$449$	−12884.8	−1.35428	−0.677138	−	0.735856i	$-0.736780\pi$
$449$	−12884.8	−1.35428	−0.677138	+	0.735856i	$0.736780\pi$
$450$	0	0
$451$	20807.3	2.17245
$452$	0	0
$453$	15433.9	1.60077
$454$	0	0
$455$	−5705.07	−0.587820
$456$	0	0
$457$	−17642.4	−1.80586	−0.902930	−	0.429788i	$-0.858588\pi$
$457$	−17642.4	−1.80586	−0.902930	+	0.429788i	$0.858588\pi$
$458$	0	0
$459$	−16464.6	−1.67430
$460$	0	0
$461$	−12243.0	−1.23691	−0.618453	−	0.785822i	$-0.712240\pi$
$461$	−12243.0	−1.23691	−0.618453	+	0.785822i	$0.712240\pi$
$462$	0	0
$463$	10929.2	1.09702	0.548512	−	0.836142i	$-0.315194\pi$
$463$	10929.2	1.09702	0.548512	+	0.836142i	$0.315194\pi$
$464$	0	0
$465$	−7612.57	−0.759192
$466$	0	0
$467$	−12952.1	−1.28341	−0.641703	−	0.766953i	$-0.721772\pi$
$467$	−12952.1	−1.28341	−0.641703	+	0.766953i	$0.721772\pi$
$468$	0	0
$469$	−2563.81	−0.252422
$470$	0	0
$471$	23535.6	2.30247
$472$	0	0
$473$	2791.33	0.271343
$474$	0	0
$475$	2347.44	0.226753
$476$	0	0
$477$	−16013.9	−1.53717
$478$	0	0
$479$	6746.07	0.643499	0.321749	−	0.946825i	$-0.395729\pi$
$479$	6746.07	0.643499	0.321749	+	0.946825i	$0.395729\pi$
$480$	0	0
$481$	−6493.06	−0.615505
$482$	0	0
$483$	2930.56	0.276076
$484$	0	0
$485$	6739.92	0.631019
$486$	0	0
$487$	−9921.61	−0.923185	−0.461592	−	0.887092i	$-0.652722\pi$
$487$	−9921.61	−0.923185	−0.461592	+	0.887092i	$0.652722\pi$
$488$	0	0
$489$	−15352.6	−1.41978
$490$	0	0
$491$	1869.69	0.171849	0.0859247	−	0.996302i	$-0.472616\pi$
$491$	1869.69	0.171849	0.0859247	+	0.996302i	$0.472616\pi$
$492$	0	0
$493$	22952.6	2.09683
$494$	0	0
$495$	10438.4	0.947818
$496$	0	0
$497$	4453.53	0.401948
$498$	0	0
$499$	−10754.2	−0.964779	−0.482389	−	0.875957i	$-0.660231\pi$
$499$	−10754.2	−0.964779	−0.482389	+	0.875957i	$0.660231\pi$
$500$	0	0
$501$	−20009.3	−1.78433
$502$	0	0
$503$	5700.89	0.505348	0.252674	−	0.967551i	$-0.418690\pi$
$503$	5700.89	0.505348	0.252674	+	0.967551i	$0.418690\pi$
$504$	0	0
$505$	−4467.86	−0.393697
$506$	0	0
$507$	−30702.9	−2.68947
$508$	0	0
$509$	−11658.5	−1.01523	−0.507616	−	0.861583i	$-0.669473\pi$
$509$	−11658.5	−1.01523	−0.507616	+	0.861583i	$0.669473\pi$
$510$	0	0
$511$	−7504.18	−0.649639
$512$	0	0
$513$	−14695.4	−1.26475
$514$	0	0
$515$	487.561	0.0417175
$516$	0	0
$517$	−10714.0	−0.911415
$518$	0	0
$519$	−5231.61	−0.442470
$520$	0	0
$521$	−22047.6	−1.85398	−0.926990	−	0.375086i	$-0.877613\pi$
$521$	−22047.6	−1.85398	−0.926990	+	0.375086i	$0.877613\pi$
$522$	0	0
$523$	−7714.99	−0.645035	−0.322517	−	0.946564i	$-0.604529\pi$
$523$	−7714.99	−0.645035	−0.322517	+	0.946564i	$0.604529\pi$
$524$	0	0
$525$	3185.39	0.264803
$526$	0	0
$527$	18825.0	1.55603
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	22258.6	1.81910
$532$	0	0
$533$	34472.4	2.80144
$534$	0	0
$535$	4477.54	0.361834
$536$	0	0
$537$	26236.3	2.10834
$538$	0	0
$539$	−5461.10	−0.436412
$540$	0	0
$541$	7674.27	0.609875	0.304938	−	0.952372i	$-0.401364\pi$
$541$	7674.27	0.609875	0.304938	+	0.952372i	$0.401364\pi$
$542$	0	0
$543$	3558.69	0.281249
$544$	0	0
$545$	−2215.33	−0.174118
$546$	0	0
$547$	−20662.9	−1.61514	−0.807571	−	0.589770i	$-0.799218\pi$
$547$	−20662.9	−1.61514	−0.807571	+	0.589770i	$0.799218\pi$
$548$	0	0
$549$	20235.4	1.57309
$550$	0	0
$551$	20486.3	1.58393
$552$	0	0
$553$	7408.58	0.569701
$554$	0	0
$555$	3625.35	0.277275
$556$	0	0
$557$	−22210.5	−1.68956	−0.844782	−	0.535111i	$-0.820270\pi$
$557$	−22210.5	−1.68956	−0.844782	+	0.535111i	$0.820270\pi$
$558$	0	0
$559$	4624.52	0.349904
$560$	0	0
$561$	−41166.1	−3.09810
$562$	0	0
$563$	−5028.57	−0.376428	−0.188214	−	0.982128i	$-0.560270\pi$
$563$	−5028.57	−0.376428	−0.188214	+	0.982128i	$0.560270\pi$
$564$	0	0
$565$	−9834.26	−0.732266
$566$	0	0
$567$	−1587.04	−0.117547
$568$	0	0
$569$	12212.2	0.899756	0.449878	−	0.893090i	$-0.351467\pi$
$569$	12212.2	0.899756	0.449878	+	0.893090i	$0.351467\pi$
$570$	0	0
$571$	−19802.7	−1.45134	−0.725672	−	0.688041i	$-0.758471\pi$
$571$	−19802.7	−1.45134	−0.725672	+	0.688041i	$0.758471\pi$
$572$	0	0
$573$	−38824.4	−2.83056
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−25858.0	−1.86565	−0.932826	−	0.360328i	$-0.882665\pi$
$577$	−25858.0	−1.86565	−0.932826	+	0.360328i	$0.882665\pi$
$578$	0	0
$579$	−39233.5	−2.81604
$580$	0	0
$581$	−3308.12	−0.236220
$582$	0	0
$583$	−16224.2	−1.15255
$584$	0	0
$585$	17293.7	1.22224
$586$	0	0
$587$	8963.09	0.630232	0.315116	−	0.949053i	$-0.397957\pi$
$587$	8963.09	0.630232	0.315116	+	0.949053i	$0.397957\pi$
$588$	0	0
$589$	16802.1	1.17541
$590$	0	0
$591$	17389.3	1.21032
$592$	0	0
$593$	−9110.32	−0.630887	−0.315444	−	0.948944i	$-0.602153\pi$
$593$	−9110.32	−0.630887	−0.315444	+	0.948944i	$0.602153\pi$
$594$	0	0
$595$	−7877.08	−0.542738
$596$	0	0
$597$	−18098.7	−1.24075
$598$	0	0
$599$	−9764.41	−0.666048	−0.333024	−	0.942918i	$-0.608069\pi$
$599$	−9764.41	−0.666048	−0.333024	+	0.942918i	$0.608069\pi$
$600$	0	0
$601$	22126.4	1.50176	0.750878	−	0.660441i	$-0.229630\pi$
$601$	22126.4	1.50176	0.750878	+	0.660441i	$0.229630\pi$
$602$	0	0
$603$	7771.66	0.524853
$604$	0	0
$605$	3920.45	0.263453
$606$	0	0
$607$	−24374.9	−1.62990	−0.814948	−	0.579534i	$-0.803235\pi$
$607$	−24374.9	−1.62990	−0.814948	+	0.579534i	$0.803235\pi$
$608$	0	0
$609$	27799.1	1.84971
$610$	0	0
$611$	−17750.4	−1.17529
$612$	0	0
$613$	−2948.97	−0.194303	−0.0971514	−	0.995270i	$-0.530973\pi$
$613$	−2948.97	−0.194303	−0.0971514	+	0.995270i	$0.530973\pi$
$614$	0	0
$615$	−19247.4	−1.26200
$616$	0	0
$617$	12250.8	0.799349	0.399674	−	0.916657i	$-0.369123\pi$
$617$	12250.8	0.799349	0.399674	+	0.916657i	$0.369123\pi$
$618$	0	0
$619$	3519.41	0.228525	0.114263	−	0.993451i	$-0.463549\pi$
$619$	3519.41	0.228525	0.114263	+	0.993451i	$0.463549\pi$
$620$	0	0
$621$	−3599.61	−0.232605
$622$	0	0
$623$	−2344.89	−0.150796
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−36742.6	−2.34028
$628$	0	0
$629$	−8965.06	−0.568300
$630$	0	0
$631$	8448.08	0.532984	0.266492	−	0.963837i	$-0.414135\pi$
$631$	8448.08	0.532984	0.266492	+	0.963837i	$0.414135\pi$
$632$	0	0
$633$	47355.7	2.97349
$634$	0	0
$635$	−5685.16	−0.355289
$636$	0	0
$637$	−9047.66	−0.562765
$638$	0	0
$639$	−13499.9	−0.835758
$640$	0	0
$641$	−17138.6	−1.05606	−0.528030	−	0.849226i	$-0.677069\pi$
$641$	−17138.6	−1.05606	−0.528030	+	0.849226i	$0.677069\pi$
$642$	0	0
$643$	−4278.61	−0.262414	−0.131207	−	0.991355i	$-0.541885\pi$
$643$	−4278.61	−0.262414	−0.131207	+	0.991355i	$0.541885\pi$
$644$	0	0
$645$	−2582.07	−0.157626
$646$	0	0
$647$	−25373.6	−1.54179	−0.770897	−	0.636960i	$-0.780191\pi$
$647$	−25373.6	−1.54179	−0.770897	+	0.636960i	$0.780191\pi$
$648$	0	0
$649$	22550.9	1.36394
$650$	0	0
$651$	22799.9	1.37265
$652$	0	0
$653$	−21915.2	−1.31334	−0.656668	−	0.754180i	$-0.728035\pi$
$653$	−21915.2	−1.31334	−0.656668	+	0.754180i	$0.728035\pi$
$654$	0	0
$655$	−3356.63	−0.200236
$656$	0	0
$657$	22747.4	1.35078
$658$	0	0
$659$	29764.0	1.75939	0.879697	−	0.475534i	$-0.157745\pi$
$659$	29764.0	1.75939	0.879697	+	0.475534i	$0.157745\pi$
$660$	0	0
$661$	10684.7	0.628723	0.314362	−	0.949303i	$-0.398210\pi$
$661$	10684.7	0.628723	0.314362	+	0.949303i	$0.398210\pi$
$662$	0	0
$663$	−68201.8	−3.99508
$664$	0	0
$665$	−7030.64	−0.409980
$666$	0	0
$667$	5018.06	0.291305
$668$	0	0
$669$	40033.1	2.31356
$670$	0	0
$671$	20501.1	1.17949
$672$	0	0
$673$	17962.6	1.02884	0.514420	−	0.857538i	$-0.328007\pi$
$673$	17962.6	1.02884	0.514420	+	0.857538i	$0.328007\pi$
$674$	0	0
$675$	−3912.62	−0.223106
$676$	0	0
$677$	−17452.7	−0.990786	−0.495393	−	0.868669i	$-0.664976\pi$
$677$	−17452.7	−0.990786	−0.495393	+	0.868669i	$0.664976\pi$
$678$	0	0
$679$	−20186.3	−1.14091
$680$	0	0
$681$	16222.7	0.912859
$682$	0	0
$683$	6304.38	0.353193	0.176596	−	0.984283i	$-0.443491\pi$
$683$	6304.38	0.353193	0.176596	+	0.984283i	$0.443491\pi$
$684$	0	0
$685$	−3472.10	−0.193668
$686$	0	0
$687$	12856.1	0.713958
$688$	0	0
$689$	−26879.4	−1.48625
$690$	0	0
$691$	−25618.6	−1.41039	−0.705193	−	0.709015i	$-0.749140\pi$
$691$	−25618.6	−1.41039	−0.705193	+	0.709015i	$0.749140\pi$
$692$	0	0
$693$	−31263.2	−1.71370
$694$	0	0
$695$	7032.78	0.383840
$696$	0	0
$697$	47596.6	2.58658
$698$	0	0
$699$	20823.1	1.12676
$700$	0	0
$701$	27137.4	1.46215	0.731073	−	0.682299i	$-0.239020\pi$
$701$	27137.4	1.46215	0.731073	+	0.682299i	$0.239020\pi$
$702$	0	0
$703$	−8001.71	−0.429289
$704$	0	0
$705$	9910.81	0.529450
$706$	0	0
$707$	13381.4	0.711821
$708$	0	0
$709$	−27259.4	−1.44393	−0.721966	−	0.691928i	$-0.756761\pi$
$709$	−27259.4	−1.44393	−0.721966	+	0.691928i	$0.756761\pi$
$710$	0	0
$711$	−22457.6	−1.18456
$712$	0	0
$713$	4115.64	0.216174
$714$	0	0
$715$	17520.8	0.916422
$716$	0	0
$717$	−35517.0	−1.84994
$718$	0	0
$719$	−28548.3	−1.48077	−0.740383	−	0.672186i	$-0.765356\pi$
$719$	−28548.3	−1.48077	−0.740383	+	0.672186i	$0.765356\pi$
$720$	0	0
$721$	−1460.26	−0.0754271
$722$	0	0
$723$	−35462.8	−1.82417
$724$	0	0
$725$	5454.42	0.279410
$726$	0	0
$727$	29743.4	1.51736	0.758681	−	0.651462i	$-0.225844\pi$
$727$	29743.4	1.51736	0.758681	+	0.651462i	$0.225844\pi$
$728$	0	0
$729$	−31142.9	−1.58222
$730$	0	0
$731$	6385.14	0.323069
$732$	0	0
$733$	15606.3	0.786399	0.393200	−	0.919453i	$-0.371368\pi$
$733$	15606.3	0.786399	0.393200	+	0.919453i	$0.371368\pi$
$734$	0	0
$735$	5051.70	0.253516
$736$	0	0
$737$	7873.71	0.393530
$738$	0	0
$739$	16178.6	0.805329	0.402665	−	0.915348i	$-0.368084\pi$
$739$	16178.6	0.805329	0.402665	+	0.915348i	$0.368084\pi$
$740$	0	0
$741$	−60873.2	−3.01786
$742$	0	0
$743$	−2658.96	−0.131289	−0.0656445	−	0.997843i	$-0.520910\pi$
$743$	−2658.96	−0.131289	−0.0656445	+	0.997843i	$0.520910\pi$
$744$	0	0
$745$	−266.980	−0.0131294
$746$	0	0
$747$	10027.9	0.491165
$748$	0	0
$749$	−13410.4	−0.654211
$750$	0	0
$751$	−245.799	−0.0119432	−0.00597160	−	0.999982i	$-0.501901\pi$
$751$	−245.799	−0.0119432	−0.00597160	+	0.999982i	$0.501901\pi$
$752$	0	0
$753$	34732.4	1.68090
$754$	0	0
$755$	−9069.75	−0.437195
$756$	0	0
$757$	−11042.7	−0.530190	−0.265095	−	0.964222i	$-0.585403\pi$
$757$	−11042.7	−0.530190	−0.265095	+	0.964222i	$0.585403\pi$
$758$	0	0
$759$	−9000.02	−0.430408
$760$	0	0
$761$	16639.8	0.792631	0.396315	−	0.918114i	$-0.370289\pi$
$761$	16639.8	0.792631	0.396315	+	0.918114i	$0.370289\pi$
$762$	0	0
$763$	6634.97	0.314813
$764$	0	0
$765$	23877.7	1.12850
$766$	0	0
$767$	37361.1	1.75884
$768$	0	0
$769$	−19893.8	−0.932886	−0.466443	−	0.884551i	$-0.654465\pi$
$769$	−19893.8	−0.932886	−0.466443	+	0.884551i	$0.654465\pi$
$770$	0	0
$771$	60617.4	2.83150
$772$	0	0
$773$	14775.4	0.687494	0.343747	−	0.939062i	$-0.388304\pi$
$773$	14775.4	0.687494	0.343747	+	0.939062i	$0.388304\pi$
$774$	0	0
$775$	4473.52	0.207347
$776$	0	0
$777$	−10858.0	−0.501325
$778$	0	0
$779$	42482.0	1.95389
$780$	0	0
$781$	−13677.2	−0.626644
$782$	0	0
$783$	−34145.7	−1.55845
$784$	0	0
$785$	−13830.7	−0.628840
$786$	0	0
$787$	−35343.4	−1.60083	−0.800416	−	0.599444i	$-0.795388\pi$
$787$	−35343.4	−1.60083	−0.800416	+	0.599444i	$0.795388\pi$
$788$	0	0
$789$	34085.0	1.53797
$790$	0	0
$791$	29453.9	1.32397
$792$	0	0
$793$	33965.2	1.52098
$794$	0	0
$795$	15007.9	0.669530
$796$	0	0
$797$	37024.5	1.64551	0.822757	−	0.568393i	$-0.192435\pi$
$797$	37024.5	1.64551	0.822757	+	0.568393i	$0.192435\pi$
$798$	0	0
$799$	−24508.2	−1.08516
$800$	0	0
$801$	7108.04	0.313546
$802$	0	0
$803$	23046.1	1.01280
$804$	0	0
$805$	−1722.14	−0.0754006
$806$	0	0
$807$	−54583.7	−2.38096
$808$	0	0
$809$	28859.7	1.25421	0.627103	−	0.778937i	$-0.284241\pi$
$809$	28859.7	1.25421	0.627103	+	0.778937i	$0.284241\pi$
$810$	0	0
$811$	9536.18	0.412898	0.206449	−	0.978457i	$-0.433809\pi$
$811$	9536.18	0.412898	0.206449	+	0.978457i	$0.433809\pi$
$812$	0	0
$813$	−46909.4	−2.02360
$814$	0	0
$815$	9021.97	0.387762
$816$	0	0
$817$	5699.02	0.244044
$818$	0	0
$819$	−51795.2	−2.20986
$820$	0	0
$821$	17969.1	0.763855	0.381927	−	0.924192i	$-0.375260\pi$
$821$	17969.1	0.763855	0.381927	+	0.924192i	$0.375260\pi$
$822$	0	0
$823$	−16028.7	−0.678887	−0.339443	−	0.940626i	$-0.610239\pi$
$823$	−16028.7	−0.678887	−0.339443	+	0.940626i	$0.610239\pi$
$824$	0	0
$825$	−9782.63	−0.412833
$826$	0	0
$827$	24895.1	1.04678	0.523391	−	0.852093i	$-0.324667\pi$
$827$	24895.1	1.04678	0.523391	+	0.852093i	$0.324667\pi$
$828$	0	0
$829$	5426.24	0.227336	0.113668	−	0.993519i	$-0.463740\pi$
$829$	5426.24	0.227336	0.113668	+	0.993519i	$0.463740\pi$
$830$	0	0
$831$	25411.2	1.06078
$832$	0	0
$833$	−12492.2	−0.519604
$834$	0	0
$835$	11758.5	0.487328
$836$	0	0
$837$	−28005.1	−1.15651
$838$	0	0
$839$	−18653.7	−0.767577	−0.383788	−	0.923421i	$-0.625381\pi$
$839$	−18653.7	−0.767577	−0.383788	+	0.923421i	$0.625381\pi$
$840$	0	0
$841$	23212.0	0.951743
$842$	0	0
$843$	−59338.0	−2.42433
$844$	0	0
$845$	18042.6	0.734536
$846$	0	0
$847$	−11741.8	−0.476334
$848$	0	0
$849$	29091.8	1.17601
$850$	0	0
$851$	−1960.00	−0.0789519
$852$	0	0
$853$	18449.2	0.740550	0.370275	−	0.928922i	$-0.379263\pi$
$853$	18449.2	0.740550	0.370275	+	0.928922i	$0.379263\pi$
$854$	0	0
$855$	21311.9	0.852459
$856$	0	0
$857$	−23872.2	−0.951529	−0.475765	−	0.879573i	$-0.657829\pi$
$857$	−23872.2	−0.951529	−0.475765	+	0.879573i	$0.657829\pi$
$858$	0	0
$859$	−32745.6	−1.30066	−0.650329	−	0.759653i	$-0.725369\pi$
$859$	−32745.6	−1.30066	−0.650329	+	0.759653i	$0.725369\pi$
$860$	0	0
$861$	57646.6	2.28175
$862$	0	0
$863$	−29732.1	−1.17276	−0.586380	−	0.810036i	$-0.699448\pi$
$863$	−29732.1	−1.17276	−0.586380	+	0.810036i	$0.699448\pi$
$864$	0	0
$865$	3074.35	0.120845
$866$	0	0
$867$	−52365.2	−2.05123
$868$	0	0
$869$	−22752.5	−0.888175
$870$	0	0
$871$	13044.7	0.507467
$872$	0	0
$873$	61190.4	2.37226
$874$	0	0
$875$	−1871.89	−0.0723217
$876$	0	0
$877$	−22342.8	−0.860275	−0.430138	−	0.902763i	$-0.641535\pi$
$877$	−22342.8	−0.860275	−0.430138	+	0.902763i	$0.641535\pi$
$878$	0	0
$879$	62732.4	2.40718
$880$	0	0
$881$	8618.90	0.329600	0.164800	−	0.986327i	$-0.447302\pi$
$881$	8618.90	0.329600	0.164800	+	0.986327i	$0.447302\pi$
$882$	0	0
$883$	46569.0	1.77483	0.887413	−	0.460976i	$-0.152501\pi$
$883$	46569.0	1.77483	0.887413	+	0.460976i	$0.152501\pi$
$884$	0	0
$885$	−20860.3	−0.792329
$886$	0	0
$887$	18720.6	0.708653	0.354327	−	0.935122i	$-0.384710\pi$
$887$	18720.6	0.708653	0.354327	+	0.935122i	$0.384710\pi$
$888$	0	0
$889$	17027.2	0.642379
$890$	0	0
$891$	4873.94	0.183258
$892$	0	0
$893$	−21874.7	−0.819718
$894$	0	0
$895$	−15417.7	−0.575819
$896$	0	0
$897$	−14910.7	−0.555023
$898$	0	0
$899$	39040.7	1.44837
$900$	0	0
$901$	−37112.8	−1.37226
$902$	0	0
$903$	7733.37	0.284995
$904$	0	0
$905$	−2091.26	−0.0768132
$906$	0	0
$907$	16330.1	0.597829	0.298915	−	0.954280i	$-0.403375\pi$
$907$	16330.1	0.597829	0.298915	+	0.954280i	$0.403375\pi$
$908$	0	0
$909$	−40562.8	−1.48007
$910$	0	0
$911$	−39566.4	−1.43896	−0.719481	−	0.694512i	$-0.755620\pi$
$911$	−39566.4	−1.43896	−0.719481	+	0.694512i	$0.755620\pi$
$912$	0	0
$913$	10159.5	0.368272
$914$	0	0
$915$	−18964.2	−0.685178
$916$	0	0
$917$	10053.2	0.362035
$918$	0	0
$919$	−1885.33	−0.0676728	−0.0338364	−	0.999427i	$-0.510773\pi$
$919$	−1885.33	−0.0676728	−0.0338364	+	0.999427i	$0.510773\pi$
$920$	0	0
$921$	62193.7	2.22514
$922$	0	0
$923$	−22659.7	−0.808074
$924$	0	0
$925$	−2130.44	−0.0757280
$926$	0	0
$927$	4426.47	0.156833
$928$	0	0
$929$	16038.9	0.566437	0.283218	−	0.959055i	$-0.408598\pi$
$929$	16038.9	0.566437	0.283218	+	0.959055i	$0.408598\pi$
$930$	0	0
$931$	−11149.9	−0.392505
$932$	0	0
$933$	−53003.4	−1.85987
$934$	0	0
$935$	24191.3	0.846138
$936$	0	0
$937$	17041.5	0.594155	0.297077	−	0.954853i	$-0.403988\pi$
$937$	17041.5	0.594155	0.297077	+	0.954853i	$0.403988\pi$
$938$	0	0
$939$	−31364.0	−1.09002
$940$	0	0
$941$	−12535.5	−0.434269	−0.217134	−	0.976142i	$-0.569671\pi$
$941$	−12535.5	−0.434269	−0.217134	+	0.976142i	$0.569671\pi$
$942$	0	0
$943$	10405.9	0.359345
$944$	0	0
$945$	11718.4	0.403386
$946$	0	0
$947$	−12368.7	−0.424423	−0.212212	−	0.977224i	$-0.568067\pi$
$947$	−12368.7	−0.424423	−0.212212	+	0.977224i	$0.568067\pi$
$948$	0	0
$949$	38181.5	1.30603
$950$	0	0
$951$	70050.0	2.38857
$952$	0	0
$953$	14183.0	0.482091	0.241046	−	0.970514i	$-0.422510\pi$
$953$	14183.0	0.482091	0.241046	+	0.970514i	$0.422510\pi$
$954$	0	0
$955$	22815.1	0.773069
$956$	0	0
$957$	−85373.7	−2.88374
$958$	0	0
$959$	10399.0	0.350159
$960$	0	0
$961$	2228.88	0.0748173
$962$	0	0
$963$	40650.7	1.36028
$964$	0	0
$965$	23055.6	0.769104
$966$	0	0
$967$	−48306.2	−1.60644	−0.803218	−	0.595685i	$-0.796881\pi$
$967$	−48306.2	−1.60644	−0.803218	+	0.595685i	$0.796881\pi$
$968$	0	0
$969$	−84048.5	−2.78641
$970$	0	0
$971$	43219.1	1.42839	0.714196	−	0.699946i	$-0.246793\pi$
$971$	43219.1	1.42839	0.714196	+	0.699946i	$0.246793\pi$
$972$	0	0
$973$	−21063.4	−0.693999
$974$	0	0
$975$	−16207.3	−0.532359
$976$	0	0
$977$	34443.1	1.12787	0.563937	−	0.825818i	$-0.309286\pi$
$977$	34443.1	1.12787	0.563937	+	0.825818i	$0.309286\pi$
$978$	0	0
$979$	7201.38	0.235094
$980$	0	0
$981$	−20112.5	−0.654581
$982$	0	0
$983$	−23680.6	−0.768355	−0.384177	−	0.923259i	$-0.625515\pi$
$983$	−23680.6	−0.768355	−0.384177	+	0.923259i	$0.625515\pi$
$984$	0	0
$985$	−10218.8	−0.330557
$986$	0	0
$987$	−29683.1	−0.957270
$988$	0	0
$989$	1395.96	0.0448828
$990$	0	0
$991$	38712.7	1.24092	0.620458	−	0.784239i	$-0.286947\pi$
$991$	38712.7	1.24092	0.620458	+	0.784239i	$0.286947\pi$
$992$	0	0
$993$	−66446.6	−2.12348
$994$	0	0
$995$	10635.7	0.338868
$996$	0	0
$997$	47032.6	1.49402	0.747010	−	0.664813i	$-0.231489\pi$
$997$	47032.6	1.49402	0.747010	+	0.664813i	$0.231489\pi$
$998$	0	0
$999$	13337.0	0.422385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.ba.1.2		10
4.3	odd	2		920.4.a.h.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.h.1.9	✓	10	4.3	odd	2
1840.4.a.ba.1.2		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)

\(f(q)\)	\(=\)	\(q-8.50847 q^{3} +5.00000 q^{5} -14.9751 q^{7} +45.3940 q^{9} +O(q^{10})\)	\(q-8.50847 q^{3} +5.00000 q^{5} -14.9751 q^{7} +45.3940 q^{9} +45.9901 q^{11} +76.1939 q^{13} -42.5423 q^{15} +105.202 q^{17} +93.8975 q^{19} +127.415 q^{21} +23.0000 q^{23} +25.0000 q^{25} -156.505 q^{27} +218.177 q^{29} +178.941 q^{31} -391.305 q^{33} -74.8757 q^{35} -85.2175 q^{37} -648.293 q^{39} +452.430 q^{41} +60.6941 q^{43} +226.970 q^{45} -232.963 q^{47} -118.745 q^{49} -895.109 q^{51} -352.776 q^{53} +229.950 q^{55} -798.924 q^{57} +490.342 q^{59} +445.773 q^{61} -679.782 q^{63} +380.970 q^{65} +171.205 q^{67} -195.695 q^{69} -297.395 q^{71} +501.109 q^{73} -212.712 q^{75} -688.708 q^{77} -494.725 q^{79} +105.978 q^{81} +220.907 q^{83} +526.011 q^{85} -1856.35 q^{87} +156.585 q^{89} -1141.01 q^{91} -1522.51 q^{93} +469.488 q^{95} +1347.98 q^{97} +2087.68 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9	\( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9} + 56 q^{11} + 49 q^{13} - 25 q^{15} + 240 q^{17} - 88 q^{19} + 346 q^{21} + 230 q^{23} + 250 q^{25} - 449 q^{27} + 319 q^{29} + 109 q^{31} + 504 q^{33} - 70 q^{35} + 580 q^{37} - 107 q^{39} + 259 q^{41} - 330 q^{43} + 695 q^{45} - 227 q^{47} + 630 q^{49} + 192 q^{51} - 186 q^{53} + 280 q^{55} + 1708 q^{57} - 262 q^{59} + 1000 q^{61} - 722 q^{63} + 245 q^{65} - 354 q^{67} - 115 q^{69} - 599 q^{71} + 355 q^{73} - 125 q^{75} + 1776 q^{77} + 1068 q^{79} + 3490 q^{81} - 754 q^{83} + 1200 q^{85} - 2675 q^{87} + 1740 q^{89} - 690 q^{91} + 1669 q^{93} - 440 q^{95} + 2592 q^{97} - 916 q^{99}+O(q^{100}) \) 10 * q - 5 * q^3 + 50 * q^5 - 14 * q^7 + 139 * q^9 + 56 * q^11 + 49 * q^13 - 25 * q^15 + 240 * q^17 - 88 * q^19 + 346 * q^21 + 230 * q^23 + 250 * q^25 - 449 * q^27 + 319 * q^29 + 109 * q^31 + 504 * q^33 - 70 * q^35 + 580 * q^37 - 107 * q^39 + 259 * q^41 - 330 * q^43 + 695 * q^45 - 227 * q^47 + 630 * q^49 + 192 * q^51 - 186 * q^53 + 280 * q^55 + 1708 * q^57 - 262 * q^59 + 1000 * q^61 - 722 * q^63 + 245 * q^65 - 354 * q^67 - 115 * q^69 - 599 * q^71 + 355 * q^73 - 125 * q^75 + 1776 * q^77 + 1068 * q^79 + 3490 * q^81 - 754 * q^83 + 1200 * q^85 - 2675 * q^87 + 1740 * q^89 - 690 * q^91 + 1669 * q^93 - 440 * q^95 + 2592 * q^97 - 916 * q^99

Introduction

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