LMFDB - Embedded newform 1840.4.a.ba.1.4

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			1.4
Root			$4.57895$ 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)

$f(q)$	$=$	$q-4.57895 q^{3} +5.00000 q^{5} -22.7390 q^{7} -6.03324 q^{9} +O(q^{10})$	$q-4.57895 q^{3} +5.00000 q^{5} -22.7390 q^{7} -6.03324 q^{9} -45.6949 q^{11} +77.0979 q^{13} -22.8947 q^{15} -5.11820 q^{17} -53.0006 q^{19} +104.121 q^{21} +23.0000 q^{23} +25.0000 q^{25} +151.257 q^{27} -233.217 q^{29} -151.856 q^{31} +209.234 q^{33} -113.695 q^{35} -115.301 q^{37} -353.027 q^{39} -488.534 q^{41} -405.992 q^{43} -30.1662 q^{45} +43.9178 q^{47} +174.062 q^{49} +23.4360 q^{51} +130.889 q^{53} -228.474 q^{55} +242.687 q^{57} -262.986 q^{59} +821.666 q^{61} +137.190 q^{63} +385.490 q^{65} +505.879 q^{67} -105.316 q^{69} +561.760 q^{71} -885.893 q^{73} -114.474 q^{75} +1039.06 q^{77} -884.671 q^{79} -529.702 q^{81} +160.156 q^{83} -25.5910 q^{85} +1067.89 q^{87} +208.301 q^{89} -1753.13 q^{91} +695.340 q^{93} -265.003 q^{95} -819.333 q^{97} +275.688 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−4.57895	−0.881219	−0.440609	−	0.897699i	$-0.645238\pi$
$3$	−4.57895	−0.881219	−0.440609	+	0.897699i	$0.645238\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−22.7390	−1.22779	−0.613895	−	0.789387i	$-0.710398\pi$
$7$	−22.7390	−1.22779	−0.613895	+	0.789387i	$0.710398\pi$
$8$	0	0
$9$	−6.03324	−0.223453
$10$	0	0
$11$	−45.6949	−1.25250	−0.626251	−	0.779621i	$-0.715412\pi$
$11$	−45.6949	−1.25250	−0.626251	+	0.779621i	$0.715412\pi$
$12$	0	0
$13$	77.0979	1.64485	0.822427	−	0.568870i	$-0.192619\pi$
$13$	77.0979	1.64485	0.822427	+	0.568870i	$0.192619\pi$
$14$	0	0
$15$	−22.8947	−0.394093
$16$	0	0
$17$	−5.11820	−0.0730203	−0.0365101	−	0.999333i	$-0.511624\pi$
$17$	−5.11820	−0.0730203	−0.0365101	+	0.999333i	$0.511624\pi$
$18$	0	0
$19$	−53.0006	−0.639957	−0.319978	−	0.947425i	$-0.603676\pi$
$19$	−53.0006	−0.639957	−0.319978	+	0.947425i	$0.603676\pi$
$20$	0	0
$21$	104.121	1.08195
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	151.257	1.07813
$28$	0	0
$29$	−233.217	−1.49336	−0.746679	−	0.665184i	$-0.768353\pi$
$29$	−233.217	−1.49336	−0.746679	+	0.665184i	$0.768353\pi$
$30$	0	0
$31$	−151.856	−0.879810	−0.439905	−	0.898044i	$-0.644988\pi$
$31$	−151.856	−0.879810	−0.439905	+	0.898044i	$0.644988\pi$
$32$	0	0
$33$	209.234	1.10373
$34$	0	0
$35$	−113.695	−0.549085
$36$	0	0
$37$	−115.301	−0.512309	−0.256154	−	0.966636i	$-0.582456\pi$
$37$	−115.301	−0.512309	−0.256154	+	0.966636i	$0.582456\pi$
$38$	0	0
$39$	−353.027	−1.44948
$40$	0	0
$41$	−488.534	−1.86088	−0.930441	−	0.366441i	$-0.880576\pi$
$41$	−488.534	−1.86088	−0.930441	+	0.366441i	$0.880576\pi$
$42$	0	0
$43$	−405.992	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$43$	−405.992	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$44$	0	0
$45$	−30.1662	−0.0999314
$46$	0	0
$47$	43.9178	0.136299	0.0681497	−	0.997675i	$-0.478290\pi$
$47$	43.9178	0.136299	0.0681497	+	0.997675i	$0.478290\pi$
$48$	0	0
$49$	174.062	0.507470
$50$	0	0
$51$	23.4360	0.0643469
$52$	0	0
$53$	130.889	0.339225	0.169613	−	0.985511i	$-0.445748\pi$
$53$	130.889	0.339225	0.169613	+	0.985511i	$0.445748\pi$
$54$	0	0
$55$	−228.474	−0.560136
$56$	0	0
$57$	242.687	0.563942
$58$	0	0
$59$	−262.986	−0.580303	−0.290152	−	0.956981i	$-0.593706\pi$
$59$	−262.986	−0.580303	−0.290152	+	0.956981i	$0.593706\pi$
$60$	0	0
$61$	821.666	1.72465	0.862324	−	0.506356i	$-0.169008\pi$
$61$	821.666	1.72465	0.862324	+	0.506356i	$0.169008\pi$
$62$	0	0
$63$	137.190	0.274354
$64$	0	0
$65$	385.490	0.735601
$66$	0	0
$67$	505.879	0.922432	0.461216	−	0.887288i	$-0.347413\pi$
$67$	505.879	0.922432	0.461216	+	0.887288i	$0.347413\pi$
$68$	0	0
$69$	−105.316	−0.183747
$70$	0	0
$71$	561.760	0.938996	0.469498	−	0.882934i	$-0.344435\pi$
$71$	561.760	0.938996	0.469498	+	0.882934i	$0.344435\pi$
$72$	0	0
$73$	−885.893	−1.42036	−0.710178	−	0.704022i	$-0.751386\pi$
$73$	−885.893	−1.42036	−0.710178	+	0.704022i	$0.751386\pi$
$74$	0	0
$75$	−114.474	−0.176244
$76$	0	0
$77$	1039.06	1.53781
$78$	0	0
$79$	−884.671	−1.25992	−0.629958	−	0.776630i	$-0.716928\pi$
$79$	−884.671	−1.25992	−0.629958	+	0.776630i	$0.716928\pi$
$80$	0	0
$81$	−529.702	−0.726615
$82$	0	0
$83$	160.156	0.211800	0.105900	−	0.994377i	$-0.466228\pi$
$83$	160.156	0.211800	0.105900	+	0.994377i	$0.466228\pi$
$84$	0	0
$85$	−25.5910	−0.0326557
$86$	0	0
$87$	1067.89	1.31598
$88$	0	0
$89$	208.301	0.248089	0.124044	−	0.992277i	$-0.460413\pi$
$89$	208.301	0.248089	0.124044	+	0.992277i	$0.460413\pi$
$90$	0	0
$91$	−1753.13	−2.01954
$92$	0	0
$93$	695.340	0.775305
$94$	0	0
$95$	−265.003	−0.286197
$96$	0	0
$97$	−819.333	−0.857636	−0.428818	−	0.903391i	$-0.641070\pi$
$97$	−819.333	−0.857636	−0.428818	+	0.903391i	$0.641070\pi$
$98$	0	0
$99$	275.688	0.279876
$100$	0	0
$101$	1457.19	1.43561	0.717803	−	0.696247i	$-0.245148\pi$
$101$	1457.19	1.43561	0.717803	+	0.696247i	$0.245148\pi$
$102$	0	0
$103$	−455.585	−0.435826	−0.217913	−	0.975968i	$-0.569925\pi$
$103$	−455.585	−0.435826	−0.217913	+	0.975968i	$0.569925\pi$
$104$	0	0
$105$	520.603	0.483864
$106$	0	0
$107$	−767.874	−0.693768	−0.346884	−	0.937908i	$-0.612760\pi$
$107$	−767.874	−0.693768	−0.346884	+	0.937908i	$0.612760\pi$
$108$	0	0
$109$	17.6019	0.0154675	0.00773375	−	0.999970i	$-0.497538\pi$
$109$	17.6019	0.0154675	0.00773375	+	0.999970i	$0.497538\pi$
$110$	0	0
$111$	527.959	0.451456
$112$	0	0
$113$	−1288.32	−1.07252	−0.536262	−	0.844052i	$-0.680164\pi$
$113$	−1288.32	−1.07252	−0.536262	+	0.844052i	$0.680164\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−465.151	−0.367549
$118$	0	0
$119$	116.383	0.0896536
$120$	0	0
$121$	757.022	0.568762
$122$	0	0
$123$	2236.97	1.63984
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	645.613	0.451094	0.225547	−	0.974232i	$-0.427583\pi$
$127$	645.613	0.451094	0.225547	+	0.974232i	$0.427583\pi$
$128$	0	0
$129$	1859.02	1.26882
$130$	0	0
$131$	754.190	0.503007	0.251504	−	0.967856i	$-0.419075\pi$
$131$	754.190	0.503007	0.251504	+	0.967856i	$0.419075\pi$
$132$	0	0
$133$	1205.18	0.785733
$134$	0	0
$135$	756.287	0.482154
$136$	0	0
$137$	965.453	0.602075	0.301037	−	0.953612i	$-0.402667\pi$
$137$	965.453	0.602075	0.301037	+	0.953612i	$0.402667\pi$
$138$	0	0
$139$	−1057.72	−0.645427	−0.322714	−	0.946497i	$-0.604595\pi$
$139$	−1057.72	−0.645427	−0.322714	+	0.946497i	$0.604595\pi$
$140$	0	0
$141$	−201.097	−0.120110
$142$	0	0
$143$	−3522.98	−2.06018
$144$	0	0
$145$	−1166.09	−0.667850
$146$	0	0
$147$	−797.021	−0.447192
$148$	0	0
$149$	885.023	0.486604	0.243302	−	0.969951i	$-0.421769\pi$
$149$	885.023	0.486604	0.243302	+	0.969951i	$0.421769\pi$
$150$	0	0
$151$	1597.03	0.860693	0.430347	−	0.902664i	$-0.358391\pi$
$151$	1597.03	0.860693	0.430347	+	0.902664i	$0.358391\pi$
$152$	0	0
$153$	30.8793	0.0163166
$154$	0	0
$155$	−759.279	−0.393463
$156$	0	0
$157$	1064.00	0.540867	0.270433	−	0.962739i	$-0.412833\pi$
$157$	1064.00	0.540867	0.270433	+	0.962739i	$0.412833\pi$
$158$	0	0
$159$	−599.332	−0.298931
$160$	0	0
$161$	−522.997	−0.256012
$162$	0	0
$163$	−3186.12	−1.53102	−0.765510	−	0.643424i	$-0.777513\pi$
$163$	−3186.12	−1.53102	−0.765510	+	0.643424i	$0.777513\pi$
$164$	0	0
$165$	1046.17	0.493602
$166$	0	0
$167$	2693.71	1.24818	0.624089	−	0.781354i	$-0.285470\pi$
$167$	2693.71	1.24818	0.624089	+	0.781354i	$0.285470\pi$
$168$	0	0
$169$	3747.09	1.70555
$170$	0	0
$171$	319.766	0.143001
$172$	0	0
$173$	−317.572	−0.139564	−0.0697819	−	0.997562i	$-0.522230\pi$
$173$	−317.572	−0.139564	−0.0697819	+	0.997562i	$0.522230\pi$
$174$	0	0
$175$	−568.475	−0.245558
$176$	0	0
$177$	1204.20	0.511374
$178$	0	0
$179$	1728.64	0.721812	0.360906	−	0.932602i	$-0.382467\pi$
$179$	1728.64	0.721812	0.360906	+	0.932602i	$0.382467\pi$
$180$	0	0
$181$	2268.32	0.931506	0.465753	−	0.884915i	$-0.345784\pi$
$181$	2268.32	0.931506	0.465753	+	0.884915i	$0.345784\pi$
$182$	0	0
$183$	−3762.37	−1.51979
$184$	0	0
$185$	−576.507	−0.229111
$186$	0	0
$187$	233.875	0.0914581
$188$	0	0
$189$	−3439.44	−1.32372
$190$	0	0
$191$	−1954.14	−0.740297	−0.370149	−	0.928973i	$-0.620693\pi$
$191$	−1954.14	−0.740297	−0.370149	+	0.928973i	$0.620693\pi$
$192$	0	0
$193$	−1598.53	−0.596191	−0.298095	−	0.954536i	$-0.596351\pi$
$193$	−1598.53	−0.596191	−0.298095	+	0.954536i	$0.596351\pi$
$194$	0	0
$195$	−1765.14	−0.648226
$196$	0	0
$197$	−1083.86	−0.391989	−0.195994	−	0.980605i	$-0.562793\pi$
$197$	−1083.86	−0.391989	−0.195994	+	0.980605i	$0.562793\pi$
$198$	0	0
$199$	3989.36	1.42110	0.710549	−	0.703648i	$-0.248447\pi$
$199$	3989.36	1.42110	0.710549	+	0.703648i	$0.248447\pi$
$200$	0	0
$201$	−2316.39	−0.812865
$202$	0	0
$203$	5303.13	1.83353
$204$	0	0
$205$	−2442.67	−0.832212
$206$	0	0
$207$	−138.765	−0.0465933
$208$	0	0
$209$	2421.86	0.801547
$210$	0	0
$211$	741.046	0.241781	0.120890	−	0.992666i	$-0.461425\pi$
$211$	741.046	0.241781	0.120890	+	0.992666i	$0.461425\pi$
$212$	0	0
$213$	−2572.27	−0.827461
$214$	0	0
$215$	−2029.96	−0.643917
$216$	0	0
$217$	3453.05	1.08022
$218$	0	0
$219$	4056.46	1.25164
$220$	0	0
$221$	−394.602	−0.120108
$222$	0	0
$223$	4993.00	1.49935	0.749677	−	0.661804i	$-0.230209\pi$
$223$	4993.00	1.49935	0.749677	+	0.661804i	$0.230209\pi$
$224$	0	0
$225$	−150.831	−0.0446907
$226$	0	0
$227$	1068.11	0.312303	0.156151	−	0.987733i	$-0.450091\pi$
$227$	1068.11	0.312303	0.156151	+	0.987733i	$0.450091\pi$
$228$	0	0
$229$	−1467.66	−0.423518	−0.211759	−	0.977322i	$-0.567919\pi$
$229$	−1467.66	−0.423518	−0.211759	+	0.977322i	$0.567919\pi$
$230$	0	0
$231$	−4757.78	−1.35515
$232$	0	0
$233$	4537.72	1.27586	0.637931	−	0.770094i	$-0.279790\pi$
$233$	4537.72	1.27586	0.637931	+	0.770094i	$0.279790\pi$
$234$	0	0
$235$	219.589	0.0609549
$236$	0	0
$237$	4050.86	1.11026
$238$	0	0
$239$	−1629.48	−0.441015	−0.220507	−	0.975385i	$-0.570771\pi$
$239$	−1629.48	−0.441015	−0.220507	+	0.975385i	$0.570771\pi$
$240$	0	0
$241$	5974.99	1.59702	0.798512	−	0.601978i	$-0.205621\pi$
$241$	5974.99	1.59702	0.798512	+	0.601978i	$0.205621\pi$
$242$	0	0
$243$	−1658.47	−0.437823
$244$	0	0
$245$	870.310	0.226947
$246$	0	0
$247$	−4086.24	−1.05264
$248$	0	0
$249$	−733.345	−0.186642
$250$	0	0
$251$	−572.090	−0.143865	−0.0719323	−	0.997410i	$-0.522917\pi$
$251$	−572.090	−0.143865	−0.0719323	+	0.997410i	$0.522917\pi$
$252$	0	0
$253$	−1050.98	−0.261165
$254$	0	0
$255$	117.180	0.0287768
$256$	0	0
$257$	−1021.88	−0.248027	−0.124013	−	0.992281i	$-0.539577\pi$
$257$	−1021.88	−0.248027	−0.124013	+	0.992281i	$0.539577\pi$
$258$	0	0
$259$	2621.84	0.629008
$260$	0	0
$261$	1407.06	0.333696
$262$	0	0
$263$	4589.03	1.07594	0.537969	−	0.842964i	$-0.319192\pi$
$263$	4589.03	1.07594	0.537969	+	0.842964i	$0.319192\pi$
$264$	0	0
$265$	654.443	0.151706
$266$	0	0
$267$	−953.801	−0.218621
$268$	0	0
$269$	2495.08	0.565532	0.282766	−	0.959189i	$-0.408748\pi$
$269$	2495.08	0.565532	0.282766	+	0.959189i	$0.408748\pi$
$270$	0	0
$271$	95.0343	0.0213023	0.0106511	−	0.999943i	$-0.496610\pi$
$271$	95.0343	0.0213023	0.0106511	+	0.999943i	$0.496610\pi$
$272$	0	0
$273$	8027.49	1.77965
$274$	0	0
$275$	−1142.37	−0.250500
$276$	0	0
$277$	−5398.24	−1.17094	−0.585468	−	0.810696i	$-0.699089\pi$
$277$	−5398.24	−1.17094	−0.585468	+	0.810696i	$0.699089\pi$
$278$	0	0
$279$	916.183	0.196597
$280$	0	0
$281$	6616.33	1.40462	0.702308	−	0.711873i	$-0.252153\pi$
$281$	6616.33	1.40462	0.702308	+	0.711873i	$0.252153\pi$
$282$	0	0
$283$	−5279.21	−1.10889	−0.554447	−	0.832219i	$-0.687070\pi$
$283$	−5279.21	−1.10889	−0.554447	+	0.832219i	$0.687070\pi$
$284$	0	0
$285$	1213.44	0.252203
$286$	0	0
$287$	11108.8	2.28477
$288$	0	0
$289$	−4886.80	−0.994668
$290$	0	0
$291$	3751.68	0.755765
$292$	0	0
$293$	8960.32	1.78658	0.893289	−	0.449483i	$-0.148392\pi$
$293$	8960.32	1.78658	0.893289	+	0.449483i	$0.148392\pi$
$294$	0	0
$295$	−1314.93	−0.259520
$296$	0	0
$297$	−6911.69	−1.35036
$298$	0	0
$299$	1773.25	0.342976
$300$	0	0
$301$	9231.85	1.76782
$302$	0	0
$303$	−6672.41	−1.26508
$304$	0	0
$305$	4108.33	0.771286
$306$	0	0
$307$	−2401.34	−0.446423	−0.223211	−	0.974770i	$-0.571654\pi$
$307$	−2401.34	−0.446423	−0.223211	+	0.974770i	$0.571654\pi$
$308$	0	0
$309$	2086.10	0.384058
$310$	0	0
$311$	9371.99	1.70880	0.854400	−	0.519616i	$-0.173925\pi$
$311$	9371.99	1.70880	0.854400	+	0.519616i	$0.173925\pi$
$312$	0	0
$313$	8616.91	1.55609	0.778045	−	0.628208i	$-0.216211\pi$
$313$	8616.91	1.55609	0.778045	+	0.628208i	$0.216211\pi$
$314$	0	0
$315$	685.950	0.122695
$316$	0	0
$317$	−9896.73	−1.75349	−0.876744	−	0.480957i	$-0.840289\pi$
$317$	−9896.73	−1.75349	−0.876744	+	0.480957i	$0.840289\pi$
$318$	0	0
$319$	10656.8	1.87043
$320$	0	0
$321$	3516.05	0.611361
$322$	0	0
$323$	271.268	0.0467298
$324$	0	0
$325$	1927.45	0.328971
$326$	0	0
$327$	−80.5983	−0.0136303
$328$	0	0
$329$	−998.647	−0.167347
$330$	0	0
$331$	7608.38	1.26343	0.631713	−	0.775202i	$-0.282352\pi$
$331$	7608.38	1.26343	0.631713	+	0.775202i	$0.282352\pi$
$332$	0	0
$333$	695.641	0.114477
$334$	0	0
$335$	2529.40	0.412524
$336$	0	0
$337$	−3979.18	−0.643204	−0.321602	−	0.946875i	$-0.604221\pi$
$337$	−3979.18	−0.643204	−0.321602	+	0.946875i	$0.604221\pi$
$338$	0	0
$339$	5899.16	0.945128
$340$	0	0
$341$	6939.03	1.10196
$342$	0	0
$343$	3841.48	0.604724
$344$	0	0
$345$	−526.579	−0.0821741
$346$	0	0
$347$	−4089.77	−0.632710	−0.316355	−	0.948641i	$-0.602459\pi$
$347$	−4089.77	−0.632710	−0.316355	+	0.948641i	$0.602459\pi$
$348$	0	0
$349$	8988.18	1.37858	0.689292	−	0.724483i	$-0.257922\pi$
$349$	8988.18	1.37858	0.689292	+	0.724483i	$0.257922\pi$
$350$	0	0
$351$	11661.6	1.77337
$352$	0	0
$353$	565.369	0.0852452	0.0426226	−	0.999091i	$-0.486429\pi$
$353$	565.369	0.0852452	0.0426226	+	0.999091i	$0.486429\pi$
$354$	0	0
$355$	2808.80	0.419932
$356$	0	0
$357$	−532.910	−0.0790045
$358$	0	0
$359$	−7315.90	−1.07554	−0.537770	−	0.843092i	$-0.680733\pi$
$359$	−7315.90	−1.07554	−0.537770	+	0.843092i	$0.680733\pi$
$360$	0	0
$361$	−4049.93	−0.590455
$362$	0	0
$363$	−3466.36	−0.501204
$364$	0	0
$365$	−4429.47	−0.635202
$366$	0	0
$367$	3607.99	0.513176	0.256588	−	0.966521i	$-0.417402\pi$
$367$	3607.99	0.513176	0.256588	+	0.966521i	$0.417402\pi$
$368$	0	0
$369$	2947.45	0.415821
$370$	0	0
$371$	−2976.27	−0.416497
$372$	0	0
$373$	−5382.09	−0.747115	−0.373557	−	0.927607i	$-0.621862\pi$
$373$	−5382.09	−0.747115	−0.373557	+	0.927607i	$0.621862\pi$
$374$	0	0
$375$	−572.368	−0.0788186
$376$	0	0
$377$	−17980.6	−2.45636
$378$	0	0
$379$	10362.8	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$379$	10362.8	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$380$	0	0
$381$	−2956.23	−0.397512
$382$	0	0
$383$	9102.19	1.21436	0.607181	−	0.794564i	$-0.292300\pi$
$383$	9102.19	1.21436	0.607181	+	0.794564i	$0.292300\pi$
$384$	0	0
$385$	5195.28	0.687730
$386$	0	0
$387$	2449.45	0.321738
$388$	0	0
$389$	−13298.3	−1.73329	−0.866643	−	0.498928i	$-0.833727\pi$
$389$	−13298.3	−1.73329	−0.866643	+	0.498928i	$0.833727\pi$
$390$	0	0
$391$	−117.719	−0.0152258
$392$	0	0
$393$	−3453.40	−0.443259
$394$	0	0
$395$	−4423.36	−0.563451
$396$	0	0
$397$	−11177.1	−1.41300	−0.706500	−	0.707713i	$-0.749727\pi$
$397$	−11177.1	−1.41300	−0.706500	+	0.707713i	$0.749727\pi$
$398$	0	0
$399$	−5518.46	−0.692403
$400$	0	0
$401$	−15640.8	−1.94779	−0.973894	−	0.227002i	$-0.927108\pi$
$401$	−15640.8	−1.94779	−0.973894	+	0.227002i	$0.927108\pi$
$402$	0	0
$403$	−11707.8	−1.44716
$404$	0	0
$405$	−2648.51	−0.324952
$406$	0	0
$407$	5268.68	0.641668
$408$	0	0
$409$	−7724.78	−0.933902	−0.466951	−	0.884283i	$-0.654648\pi$
$409$	−7724.78	−0.933902	−0.466951	+	0.884283i	$0.654648\pi$
$410$	0	0
$411$	−4420.76	−0.530559
$412$	0	0
$413$	5980.05	0.712491
$414$	0	0
$415$	800.779	0.0947198
$416$	0	0
$417$	4843.23	0.568763
$418$	0	0
$419$	10258.9	1.19613	0.598064	−	0.801448i	$-0.295937\pi$
$419$	10258.9	1.19613	0.598064	+	0.801448i	$0.295937\pi$
$420$	0	0
$421$	2228.68	0.258003	0.129002	−	0.991644i	$-0.458823\pi$
$421$	2228.68	0.258003	0.129002	+	0.991644i	$0.458823\pi$
$422$	0	0
$423$	−264.967	−0.0304566
$424$	0	0
$425$	−127.955	−0.0146041
$426$	0	0
$427$	−18683.9	−2.11751
$428$	0	0
$429$	16131.5	1.81547
$430$	0	0
$431$	13457.0	1.50395	0.751975	−	0.659191i	$-0.229101\pi$
$431$	13457.0	1.50395	0.751975	+	0.659191i	$0.229101\pi$
$432$	0	0
$433$	10782.8	1.19674	0.598371	−	0.801219i	$-0.295815\pi$
$433$	10782.8	1.19674	0.598371	+	0.801219i	$0.295815\pi$
$434$	0	0
$435$	5339.45	0.588522
$436$	0	0
$437$	−1219.01	−0.133440
$438$	0	0
$439$	17043.1	1.85290	0.926448	−	0.376424i	$-0.122846\pi$
$439$	17043.1	1.85290	0.926448	+	0.376424i	$0.122846\pi$
$440$	0	0
$441$	−1050.16	−0.113396
$442$	0	0
$443$	−4874.29	−0.522764	−0.261382	−	0.965235i	$-0.584178\pi$
$443$	−4874.29	−0.522764	−0.261382	+	0.965235i	$0.584178\pi$
$444$	0	0
$445$	1041.51	0.110949
$446$	0	0
$447$	−4052.48	−0.428804
$448$	0	0
$449$	−3846.88	−0.404333	−0.202167	−	0.979351i	$-0.564798\pi$
$449$	−3846.88	−0.404333	−0.202167	+	0.979351i	$0.564798\pi$
$450$	0	0
$451$	22323.5	2.33076
$452$	0	0
$453$	−7312.73	−0.758459
$454$	0	0
$455$	−8765.65	−0.903164
$456$	0	0
$457$	2729.02	0.279339	0.139670	−	0.990198i	$-0.455396\pi$
$457$	2729.02	0.279339	0.139670	+	0.990198i	$0.455396\pi$
$458$	0	0
$459$	−774.166	−0.0787254
$460$	0	0
$461$	−9619.57	−0.971861	−0.485931	−	0.873997i	$-0.661519\pi$
$461$	−9619.57	−0.971861	−0.485931	+	0.873997i	$0.661519\pi$
$462$	0	0
$463$	13868.4	1.39205	0.696026	−	0.718016i	$-0.254950\pi$
$463$	13868.4	1.39205	0.696026	+	0.718016i	$0.254950\pi$
$464$	0	0
$465$	3476.70	0.346727
$466$	0	0
$467$	−1775.39	−0.175921	−0.0879606	−	0.996124i	$-0.528035\pi$
$467$	−1775.39	−0.175921	−0.0879606	+	0.996124i	$0.528035\pi$
$468$	0	0
$469$	−11503.2	−1.13255
$470$	0	0
$471$	−4871.98	−0.476622
$472$	0	0
$473$	18551.8	1.80341
$474$	0	0
$475$	−1325.02	−0.127991
$476$	0	0
$477$	−789.683	−0.0758010
$478$	0	0
$479$	−9515.52	−0.907673	−0.453837	−	0.891085i	$-0.649945\pi$
$479$	−9515.52	−0.907673	−0.453837	+	0.891085i	$0.649945\pi$
$480$	0	0
$481$	−8889.49	−0.842674
$482$	0	0
$483$	2394.78	0.225603
$484$	0	0
$485$	−4096.67	−0.383546
$486$	0	0
$487$	−147.076	−0.0136851	−0.00684255	−	0.999977i	$-0.502178\pi$
$487$	−147.076	−0.0136851	−0.00684255	+	0.999977i	$0.502178\pi$
$488$	0	0
$489$	14589.1	1.34916
$490$	0	0
$491$	7750.33	0.712358	0.356179	−	0.934418i	$-0.384079\pi$
$491$	7750.33	0.712358	0.356179	+	0.934418i	$0.384079\pi$
$492$	0	0
$493$	1193.65	0.109045
$494$	0	0
$495$	1378.44	0.125164
$496$	0	0
$497$	−12773.9	−1.15289
$498$	0	0
$499$	−6927.06	−0.621439	−0.310719	−	0.950502i	$-0.600570\pi$
$499$	−6927.06	−0.621439	−0.310719	+	0.950502i	$0.600570\pi$
$500$	0	0
$501$	−12334.4	−1.09992
$502$	0	0
$503$	−1903.44	−0.168728	−0.0843638	−	0.996435i	$-0.526886\pi$
$503$	−1903.44	−0.168728	−0.0843638	+	0.996435i	$0.526886\pi$
$504$	0	0
$505$	7285.97	0.642022
$506$	0	0
$507$	−17157.7	−1.50296
$508$	0	0
$509$	6672.83	0.581076	0.290538	−	0.956863i	$-0.406166\pi$
$509$	6672.83	0.581076	0.290538	+	0.956863i	$0.406166\pi$
$510$	0	0
$511$	20144.3	1.74390
$512$	0	0
$513$	−8016.74	−0.689957
$514$	0	0
$515$	−2277.92	−0.194907
$516$	0	0
$517$	−2006.82	−0.170715
$518$	0	0
$519$	1454.15	0.122986
$520$	0	0
$521$	12557.4	1.05595	0.527976	−	0.849260i	$-0.322951\pi$
$521$	12557.4	1.05595	0.527976	+	0.849260i	$0.322951\pi$
$522$	0	0
$523$	9783.91	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$523$	9783.91	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$524$	0	0
$525$	2603.02	0.216390
$526$	0	0
$527$	777.228	0.0642440
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	1586.66	0.129671
$532$	0	0
$533$	−37665.0	−3.06088
$534$	0	0
$535$	−3839.37	−0.310262
$536$	0	0
$537$	−7915.34	−0.636075
$538$	0	0
$539$	−7953.74	−0.635607
$540$	0	0
$541$	−17311.5	−1.37575	−0.687873	−	0.725831i	$-0.741456\pi$
$541$	−17311.5	−1.37575	−0.687873	+	0.725831i	$0.741456\pi$
$542$	0	0
$543$	−10386.5	−0.820860
$544$	0	0
$545$	88.0096	0.00691728
$546$	0	0
$547$	−16852.2	−1.31727	−0.658635	−	0.752462i	$-0.728866\pi$
$547$	−16852.2	−1.31727	−0.658635	+	0.752462i	$0.728866\pi$
$548$	0	0
$549$	−4957.31	−0.385379
$550$	0	0
$551$	12360.7	0.955685
$552$	0	0
$553$	20116.5	1.54691
$554$	0	0
$555$	2639.79	0.201897
$556$	0	0
$557$	13426.1	1.02133	0.510666	−	0.859779i	$-0.329399\pi$
$557$	13426.1	1.02133	0.510666	+	0.859779i	$0.329399\pi$
$558$	0	0
$559$	−31301.1	−2.36833
$560$	0	0
$561$	−1070.90	−0.0805946
$562$	0	0
$563$	−21437.2	−1.60475	−0.802373	−	0.596823i	$-0.796430\pi$
$563$	−21437.2	−1.60475	−0.802373	+	0.596823i	$0.796430\pi$
$564$	0	0
$565$	−6441.61	−0.479647
$566$	0	0
$567$	12044.9	0.892131
$568$	0	0
$569$	−3931.05	−0.289628	−0.144814	−	0.989459i	$-0.546258\pi$
$569$	−3931.05	−0.289628	−0.144814	+	0.989459i	$0.546258\pi$
$570$	0	0
$571$	−19378.6	−1.42026	−0.710131	−	0.704070i	$-0.751364\pi$
$571$	−19378.6	−1.42026	−0.710131	+	0.704070i	$0.751364\pi$
$572$	0	0
$573$	8947.91	0.652364
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	8639.15	0.623314	0.311657	−	0.950195i	$-0.399116\pi$
$577$	8639.15	0.623314	0.311657	+	0.950195i	$0.399116\pi$
$578$	0	0
$579$	7319.59	0.525374
$580$	0	0
$581$	−3641.78	−0.260046
$582$	0	0
$583$	−5980.94	−0.424880
$584$	0	0
$585$	−2325.75	−0.164373
$586$	0	0
$587$	17791.9	1.25102	0.625511	−	0.780215i	$-0.284891\pi$
$587$	17791.9	1.25102	0.625511	+	0.780215i	$0.284891\pi$
$588$	0	0
$589$	8048.45	0.563040
$590$	0	0
$591$	4962.93	0.345428
$592$	0	0
$593$	3992.65	0.276490	0.138245	−	0.990398i	$-0.455854\pi$
$593$	3992.65	0.276490	0.138245	+	0.990398i	$0.455854\pi$
$594$	0	0
$595$	581.913	0.0400943
$596$	0	0
$597$	−18267.1	−1.25230
$598$	0	0
$599$	−14419.2	−0.983562	−0.491781	−	0.870719i	$-0.663654\pi$
$599$	−14419.2	−0.983562	−0.491781	+	0.870719i	$0.663654\pi$
$600$	0	0
$601$	−16541.2	−1.12268	−0.561338	−	0.827586i	$-0.689713\pi$
$601$	−16541.2	−1.12268	−0.561338	+	0.827586i	$0.689713\pi$
$602$	0	0
$603$	−3052.09	−0.206121
$604$	0	0
$605$	3785.11	0.254358
$606$	0	0
$607$	9777.97	0.653832	0.326916	−	0.945053i	$-0.393991\pi$
$607$	9777.97	0.653832	0.326916	+	0.945053i	$0.393991\pi$
$608$	0	0
$609$	−24282.8	−1.61574
$610$	0	0
$611$	3385.97	0.224193
$612$	0	0
$613$	14266.6	0.940004	0.470002	−	0.882665i	$-0.344253\pi$
$613$	14266.6	0.940004	0.470002	+	0.882665i	$0.344253\pi$
$614$	0	0
$615$	11184.9	0.733361
$616$	0	0
$617$	−7353.06	−0.479778	−0.239889	−	0.970800i	$-0.577111\pi$
$617$	−7353.06	−0.479778	−0.239889	+	0.970800i	$0.577111\pi$
$618$	0	0
$619$	−13600.4	−0.883109	−0.441554	−	0.897234i	$-0.645573\pi$
$619$	−13600.4	−0.883109	−0.441554	+	0.897234i	$0.645573\pi$
$620$	0	0
$621$	3478.92	0.224806
$622$	0	0
$623$	−4736.57	−0.304601
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−11089.6	−0.706339
$628$	0	0
$629$	590.135	0.0374089
$630$	0	0
$631$	−11179.5	−0.705308	−0.352654	−	0.935754i	$-0.614721\pi$
$631$	−11179.5	−0.705308	−0.352654	+	0.935754i	$0.614721\pi$
$632$	0	0
$633$	−3393.21	−0.213062
$634$	0	0
$635$	3228.06	0.201735
$636$	0	0
$637$	13419.8	0.834714
$638$	0	0
$639$	−3389.24	−0.209822
$640$	0	0
$641$	−8839.18	−0.544660	−0.272330	−	0.962204i	$-0.587794\pi$
$641$	−8839.18	−0.544660	−0.272330	+	0.962204i	$0.587794\pi$
$642$	0	0
$643$	12131.8	0.744059	0.372030	−	0.928221i	$-0.378662\pi$
$643$	12131.8	0.744059	0.372030	+	0.928221i	$0.378662\pi$
$644$	0	0
$645$	9295.08	0.567432
$646$	0	0
$647$	9300.88	0.565155	0.282577	−	0.959244i	$-0.408811\pi$
$647$	9300.88	0.565155	0.282577	+	0.959244i	$0.408811\pi$
$648$	0	0
$649$	12017.1	0.726831
$650$	0	0
$651$	−15811.3	−0.951912
$652$	0	0
$653$	−20143.8	−1.20718	−0.603591	−	0.797294i	$-0.706264\pi$
$653$	−20143.8	−1.20718	−0.603591	+	0.797294i	$0.706264\pi$
$654$	0	0
$655$	3770.95	0.224952
$656$	0	0
$657$	5344.81	0.317383
$658$	0	0
$659$	−8800.58	−0.520215	−0.260108	−	0.965580i	$-0.583758\pi$
$659$	−8800.58	−0.520215	−0.260108	+	0.965580i	$0.583758\pi$
$660$	0	0
$661$	−30793.5	−1.81199	−0.905997	−	0.423284i	$-0.860877\pi$
$661$	−30793.5	−1.81199	−0.905997	+	0.423284i	$0.860877\pi$
$662$	0	0
$663$	1806.86	0.105841
$664$	0	0
$665$	6025.91	0.351390
$666$	0	0
$667$	−5364.00	−0.311387
$668$	0	0
$669$	−22862.7	−1.32126
$670$	0	0
$671$	−37545.9	−2.16013
$672$	0	0
$673$	19786.4	1.13330	0.566648	−	0.823960i	$-0.308240\pi$
$673$	19786.4	1.13330	0.566648	+	0.823960i	$0.308240\pi$
$674$	0	0
$675$	3781.44	0.215626
$676$	0	0
$677$	10152.1	0.576330	0.288165	−	0.957581i	$-0.406955\pi$
$677$	10152.1	0.576330	0.288165	+	0.957581i	$0.406955\pi$
$678$	0	0
$679$	18630.8	1.05300
$680$	0	0
$681$	−4890.80	−0.275207
$682$	0	0
$683$	−1273.06	−0.0713208	−0.0356604	−	0.999364i	$-0.511353\pi$
$683$	−1273.06	−0.0713208	−0.0356604	+	0.999364i	$0.511353\pi$
$684$	0	0
$685$	4827.26	0.269256
$686$	0	0
$687$	6720.33	0.373212
$688$	0	0
$689$	10091.2	0.557976
$690$	0	0
$691$	−6383.65	−0.351440	−0.175720	−	0.984440i	$-0.556225\pi$
$691$	−6383.65	−0.351440	−0.175720	+	0.984440i	$0.556225\pi$
$692$	0	0
$693$	−6268.88	−0.343629
$694$	0	0
$695$	−5288.59	−0.288644
$696$	0	0
$697$	2500.41	0.135882
$698$	0	0
$699$	−20778.0	−1.12431
$700$	0	0
$701$	−2647.58	−0.142650	−0.0713250	−	0.997453i	$-0.522723\pi$
$701$	−2647.58	−0.142650	−0.0713250	+	0.997453i	$0.522723\pi$
$702$	0	0
$703$	6111.04	0.327856
$704$	0	0
$705$	−1005.49	−0.0537146
$706$	0	0
$707$	−33135.1	−1.76262
$708$	0	0
$709$	−4713.06	−0.249651	−0.124826	−	0.992179i	$-0.539837\pi$
$709$	−4713.06	−0.249651	−0.124826	+	0.992179i	$0.539837\pi$
$710$	0	0
$711$	5337.44	0.281532
$712$	0	0
$713$	−3492.68	−0.183453
$714$	0	0
$715$	−17614.9	−0.921343
$716$	0	0
$717$	7461.32	0.388631
$718$	0	0
$719$	−20538.5	−1.06531	−0.532655	−	0.846333i	$-0.678806\pi$
$719$	−20538.5	−1.06531	−0.532655	+	0.846333i	$0.678806\pi$
$720$	0	0
$721$	10359.5	0.535103
$722$	0	0
$723$	−27359.2	−1.40733
$724$	0	0
$725$	−5830.44	−0.298672
$726$	0	0
$727$	−35056.0	−1.78838	−0.894191	−	0.447686i	$-0.852248\pi$
$727$	−35056.0	−1.78838	−0.894191	+	0.447686i	$0.852248\pi$
$728$	0	0
$729$	21896.0	1.11243
$730$	0	0
$731$	2077.95	0.105138
$732$	0	0
$733$	22602.0	1.13891	0.569457	−	0.822021i	$-0.307153\pi$
$733$	22602.0	1.13891	0.569457	+	0.822021i	$0.307153\pi$
$734$	0	0
$735$	−3985.10	−0.199990
$736$	0	0
$737$	−23116.1	−1.15535
$738$	0	0
$739$	12123.3	0.603466	0.301733	−	0.953392i	$-0.402435\pi$
$739$	12123.3	0.603466	0.301733	+	0.953392i	$0.402435\pi$
$740$	0	0
$741$	18710.7	0.927603
$742$	0	0
$743$	8708.16	0.429975	0.214987	−	0.976617i	$-0.431029\pi$
$743$	8708.16	0.429975	0.214987	+	0.976617i	$0.431029\pi$
$744$	0	0
$745$	4425.12	0.217616
$746$	0	0
$747$	−966.260	−0.0473274
$748$	0	0
$749$	17460.7	0.851802
$750$	0	0
$751$	33105.8	1.60859	0.804294	−	0.594232i	$-0.202544\pi$
$751$	33105.8	1.60859	0.804294	+	0.594232i	$0.202544\pi$
$752$	0	0
$753$	2619.57	0.126776
$754$	0	0
$755$	7985.16	0.384914
$756$	0	0
$757$	7927.36	0.380614	0.190307	−	0.981725i	$-0.439052\pi$
$757$	7927.36	0.380614	0.190307	+	0.981725i	$0.439052\pi$
$758$	0	0
$759$	4812.39	0.230143
$760$	0	0
$761$	7726.91	0.368069	0.184034	−	0.982920i	$-0.441084\pi$
$761$	7726.91	0.368069	0.184034	+	0.982920i	$0.441084\pi$
$762$	0	0
$763$	−400.250	−0.0189909
$764$	0	0
$765$	154.397	0.00729702
$766$	0	0
$767$	−20275.7	−0.954515
$768$	0	0
$769$	24366.8	1.14264	0.571320	−	0.820727i	$-0.306431\pi$
$769$	24366.8	1.14264	0.571320	+	0.820727i	$0.306431\pi$
$770$	0	0
$771$	4679.12	0.218566
$772$	0	0
$773$	27017.2	1.25711	0.628553	−	0.777767i	$-0.283648\pi$
$773$	27017.2	1.25711	0.628553	+	0.777767i	$0.283648\pi$
$774$	0	0
$775$	−3796.39	−0.175962
$776$	0	0
$777$	−12005.3	−0.554294
$778$	0	0
$779$	25892.6	1.19088
$780$	0	0
$781$	−25669.6	−1.17609
$782$	0	0
$783$	−35275.9	−1.61003
$784$	0	0
$785$	5319.98	0.241883
$786$	0	0
$787$	−915.686	−0.0414748	−0.0207374	−	0.999785i	$-0.506601\pi$
$787$	−915.686	−0.0414748	−0.0207374	+	0.999785i	$0.506601\pi$
$788$	0	0
$789$	−21012.9	−0.948137
$790$	0	0
$791$	29295.2	1.31683
$792$	0	0
$793$	63348.8	2.83680
$794$	0	0
$795$	−2996.66	−0.133686
$796$	0	0
$797$	−33851.1	−1.50448	−0.752238	−	0.658891i	$-0.771026\pi$
$797$	−33851.1	−1.50448	−0.752238	+	0.658891i	$0.771026\pi$
$798$	0	0
$799$	−224.780	−0.00995262
$800$	0	0
$801$	−1256.73	−0.0554363
$802$	0	0
$803$	40480.8	1.77900
$804$	0	0
$805$	−2614.98	−0.114492
$806$	0	0
$807$	−11424.9	−0.498357
$808$	0	0
$809$	−25273.4	−1.09835	−0.549176	−	0.835707i	$-0.685058\pi$
$809$	−25273.4	−1.09835	−0.549176	+	0.835707i	$0.685058\pi$
$810$	0	0
$811$	−17503.8	−0.757881	−0.378940	−	0.925421i	$-0.623711\pi$
$811$	−17503.8	−0.757881	−0.378940	+	0.925421i	$0.623711\pi$
$812$	0	0
$813$	−435.157	−0.0187720
$814$	0	0
$815$	−15930.6	−0.684693
$816$	0	0
$817$	21517.8	0.921437
$818$	0	0
$819$	10577.1	0.451273
$820$	0	0
$821$	20455.5	0.869553	0.434777	−	0.900538i	$-0.356827\pi$
$821$	20455.5	0.869553	0.434777	+	0.900538i	$0.356827\pi$
$822$	0	0
$823$	−22655.6	−0.959569	−0.479785	−	0.877386i	$-0.659285\pi$
$823$	−22655.6	−0.959569	−0.479785	+	0.877386i	$0.659285\pi$
$824$	0	0
$825$	5230.86	0.220746
$826$	0	0
$827$	33765.8	1.41977	0.709887	−	0.704315i	$-0.248746\pi$
$827$	33765.8	1.41977	0.709887	+	0.704315i	$0.248746\pi$
$828$	0	0
$829$	−11322.2	−0.474351	−0.237175	−	0.971467i	$-0.576222\pi$
$829$	−11322.2	−0.474351	−0.237175	+	0.971467i	$0.576222\pi$
$830$	0	0
$831$	24718.3	1.03185
$832$	0	0
$833$	−890.884	−0.0370556
$834$	0	0
$835$	13468.6	0.558202
$836$	0	0
$837$	−22969.3	−0.948550
$838$	0	0
$839$	9158.98	0.376881	0.188441	−	0.982085i	$-0.439657\pi$
$839$	9158.98	0.376881	0.188441	+	0.982085i	$0.439657\pi$
$840$	0	0
$841$	30001.4	1.23012
$842$	0	0
$843$	−30295.8	−1.23777
$844$	0	0
$845$	18735.4	0.762744
$846$	0	0
$847$	−17213.9	−0.698321
$848$	0	0
$849$	24173.2	0.977177
$850$	0	0
$851$	−2651.93	−0.106824
$852$	0	0
$853$	32135.6	1.28992	0.644959	−	0.764217i	$-0.276874\pi$
$853$	32135.6	1.28992	0.644959	+	0.764217i	$0.276874\pi$
$854$	0	0
$855$	1598.83	0.0639518
$856$	0	0
$857$	41850.6	1.66813	0.834065	−	0.551666i	$-0.186008\pi$
$857$	41850.6	1.66813	0.834065	+	0.551666i	$0.186008\pi$
$858$	0	0
$859$	37768.0	1.50015	0.750074	−	0.661353i	$-0.230018\pi$
$859$	37768.0	1.50015	0.750074	+	0.661353i	$0.230018\pi$
$860$	0	0
$861$	−50866.5	−2.01339
$862$	0	0
$863$	−49709.6	−1.96076	−0.980379	−	0.197122i	$-0.936840\pi$
$863$	−49709.6	−1.96076	−0.980379	+	0.197122i	$0.936840\pi$
$864$	0	0
$865$	−1587.86	−0.0624149
$866$	0	0
$867$	22376.4	0.876520
$868$	0	0
$869$	40425.0	1.57805
$870$	0	0
$871$	39002.2	1.51727
$872$	0	0
$873$	4943.24	0.191642
$874$	0	0
$875$	−2842.37	−0.109817
$876$	0	0
$877$	−44943.1	−1.73047	−0.865234	−	0.501368i	$-0.832830\pi$
$877$	−44943.1	−1.73047	−0.865234	+	0.501368i	$0.832830\pi$
$878$	0	0
$879$	−41028.8	−1.57437
$880$	0	0
$881$	8717.67	0.333378	0.166689	−	0.986010i	$-0.446692\pi$
$881$	8717.67	0.333378	0.166689	+	0.986010i	$0.446692\pi$
$882$	0	0
$883$	−13935.8	−0.531117	−0.265559	−	0.964095i	$-0.585556\pi$
$883$	−13935.8	−0.531117	−0.265559	+	0.964095i	$0.585556\pi$
$884$	0	0
$885$	6021.00	0.228694
$886$	0	0
$887$	18238.5	0.690405	0.345202	−	0.938528i	$-0.387810\pi$
$887$	18238.5	0.690405	0.345202	+	0.938528i	$0.387810\pi$
$888$	0	0
$889$	−14680.6	−0.553848
$890$	0	0
$891$	24204.7	0.910087
$892$	0	0
$893$	−2327.67	−0.0872257
$894$	0	0
$895$	8643.19	0.322804
$896$	0	0
$897$	−8119.63	−0.302237
$898$	0	0
$899$	35415.4	1.31387
$900$	0	0
$901$	−669.913	−0.0247703
$902$	0	0
$903$	−42272.2	−1.55784
$904$	0	0
$905$	11341.6	0.416582
$906$	0	0
$907$	753.756	0.0275943	0.0137972	−	0.999905i	$-0.495608\pi$
$907$	753.756	0.0275943	0.0137972	+	0.999905i	$0.495608\pi$
$908$	0	0
$909$	−8791.60	−0.320791
$910$	0	0
$911$	11803.7	0.429281	0.214640	−	0.976693i	$-0.431142\pi$
$911$	11803.7	0.429281	0.214640	+	0.976693i	$0.431142\pi$
$912$	0	0
$913$	−7318.30	−0.265280
$914$	0	0
$915$	−18811.8	−0.679672
$916$	0	0
$917$	−17149.5	−0.617587
$918$	0	0
$919$	−17147.1	−0.615485	−0.307743	−	0.951470i	$-0.599574\pi$
$919$	−17147.1	−0.615485	−0.307743	+	0.951470i	$0.599574\pi$
$920$	0	0
$921$	10995.6	0.393396
$922$	0	0
$923$	43310.6	1.54451
$924$	0	0
$925$	−2882.53	−0.102462
$926$	0	0
$927$	2748.65	0.0973869
$928$	0	0
$929$	−38863.8	−1.37253	−0.686265	−	0.727352i	$-0.740751\pi$
$929$	−38863.8	−1.37253	−0.686265	+	0.727352i	$0.740751\pi$
$930$	0	0
$931$	−9225.40	−0.324759
$932$	0	0
$933$	−42913.8	−1.50583
$934$	0	0
$935$	1169.38	0.0409013
$936$	0	0
$937$	27006.0	0.941565	0.470782	−	0.882249i	$-0.343972\pi$
$937$	27006.0	0.941565	0.470782	+	0.882249i	$0.343972\pi$
$938$	0	0
$939$	−39456.4	−1.37126
$940$	0	0
$941$	−40150.3	−1.39093	−0.695463	−	0.718562i	$-0.744801\pi$
$941$	−40150.3	−1.39093	−0.695463	+	0.718562i	$0.744801\pi$
$942$	0	0
$943$	−11236.3	−0.388021
$944$	0	0
$945$	−17197.2	−0.591985
$946$	0	0
$947$	−24511.5	−0.841095	−0.420547	−	0.907271i	$-0.638162\pi$
$947$	−24511.5	−0.841095	−0.420547	+	0.907271i	$0.638162\pi$
$948$	0	0
$949$	−68300.5	−2.33628
$950$	0	0
$951$	45316.6	1.54521
$952$	0	0
$953$	39783.7	1.35228	0.676139	−	0.736774i	$-0.263652\pi$
$953$	39783.7	1.35228	0.676139	+	0.736774i	$0.263652\pi$
$954$	0	0
$955$	−9770.71	−0.331071
$956$	0	0
$957$	−48797.1	−1.64826
$958$	0	0
$959$	−21953.4	−0.739221
$960$	0	0
$961$	−6730.82	−0.225935
$962$	0	0
$963$	4632.77	0.155025
$964$	0	0
$965$	−7992.66	−0.266625
$966$	0	0
$967$	−21786.1	−0.724504	−0.362252	−	0.932080i	$-0.617992\pi$
$967$	−21786.1	−0.724504	−0.362252	+	0.932080i	$0.617992\pi$
$968$	0	0
$969$	−1242.12	−0.0411792
$970$	0	0
$971$	49798.9	1.64585	0.822926	−	0.568148i	$-0.192340\pi$
$971$	49798.9	1.64585	0.822926	+	0.568148i	$0.192340\pi$
$972$	0	0
$973$	24051.4	0.792450
$974$	0	0
$975$	−8825.68	−0.289895
$976$	0	0
$977$	−26331.4	−0.862247	−0.431123	−	0.902293i	$-0.641883\pi$
$977$	−26331.4	−0.862247	−0.431123	+	0.902293i	$0.641883\pi$
$978$	0	0
$979$	−9518.31	−0.310732
$980$	0	0
$981$	−106.197	−0.00345627
$982$	0	0
$983$	−30723.1	−0.996861	−0.498430	−	0.866930i	$-0.666090\pi$
$983$	−30723.1	−0.996861	−0.498430	+	0.866930i	$0.666090\pi$
$984$	0	0
$985$	−5419.30	−0.175303
$986$	0	0
$987$	4572.75	0.147469
$988$	0	0
$989$	−9337.81	−0.300228
$990$	0	0
$991$	−42304.9	−1.35606	−0.678032	−	0.735032i	$-0.737167\pi$
$991$	−42304.9	−1.35606	−0.678032	+	0.735032i	$0.737167\pi$
$992$	0	0
$993$	−34838.4	−1.11336
$994$	0	0
$995$	19946.8	0.635534
$996$	0	0
$997$	324.783	0.0103169	0.00515847	−	0.999987i	$-0.498358\pi$
$997$	324.783	0.0103169	0.00515847	+	0.999987i	$0.498358\pi$
$998$	0	0
$999$	−17440.2	−0.552336

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.ba.1.4		10
4.3	odd	2		920.4.a.h.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.h.1.7	✓	10	4.3	odd	2
1840.4.a.ba.1.4		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)

\(f(q)\)	\(=\)	\(q-4.57895 q^{3} +5.00000 q^{5} -22.7390 q^{7} -6.03324 q^{9} +O(q^{10})\)	\(q-4.57895 q^{3} +5.00000 q^{5} -22.7390 q^{7} -6.03324 q^{9} -45.6949 q^{11} +77.0979 q^{13} -22.8947 q^{15} -5.11820 q^{17} -53.0006 q^{19} +104.121 q^{21} +23.0000 q^{23} +25.0000 q^{25} +151.257 q^{27} -233.217 q^{29} -151.856 q^{31} +209.234 q^{33} -113.695 q^{35} -115.301 q^{37} -353.027 q^{39} -488.534 q^{41} -405.992 q^{43} -30.1662 q^{45} +43.9178 q^{47} +174.062 q^{49} +23.4360 q^{51} +130.889 q^{53} -228.474 q^{55} +242.687 q^{57} -262.986 q^{59} +821.666 q^{61} +137.190 q^{63} +385.490 q^{65} +505.879 q^{67} -105.316 q^{69} +561.760 q^{71} -885.893 q^{73} -114.474 q^{75} +1039.06 q^{77} -884.671 q^{79} -529.702 q^{81} +160.156 q^{83} -25.5910 q^{85} +1067.89 q^{87} +208.301 q^{89} -1753.13 q^{91} +695.340 q^{93} -265.003 q^{95} -819.333 q^{97} +275.688 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 5 q^{3} + 50 q^{5} - 14 q^{7} + 139 q^{9}+O(q^{10}) \) 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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