LMFDB - Embedded newform 280.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,4,Mod(1,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, 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("280.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	280.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:	$16.5205348016$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.11045.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 31x - 54 $ x^3 - x^2 - 31*x - 54
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.76369$ of defining polynomial
Character	$\chi$	$=$	280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.45643 q^{3} -5.00000 q^{5} +7.00000 q^{7} -15.0531 q^{9} -51.8821 q^{11} -5.91120 q^{13} -17.2821 q^{15} -103.850 q^{17} +118.470 q^{19} +24.1950 q^{21} -80.7029 q^{23} +25.0000 q^{25} -145.354 q^{27} -218.344 q^{29} +39.2738 q^{31} -179.327 q^{33} -35.0000 q^{35} +159.803 q^{37} -20.4316 q^{39} +233.322 q^{41} -524.515 q^{43} +75.2655 q^{45} -210.441 q^{47} +49.0000 q^{49} -358.949 q^{51} +348.028 q^{53} +259.411 q^{55} +409.485 q^{57} -85.1752 q^{59} -801.720 q^{61} -105.372 q^{63} +29.5560 q^{65} +716.049 q^{67} -278.944 q^{69} -520.412 q^{71} +230.005 q^{73} +86.4107 q^{75} -363.175 q^{77} -970.534 q^{79} -95.9705 q^{81} -1109.38 q^{83} +519.249 q^{85} -754.689 q^{87} +1127.45 q^{89} -41.3784 q^{91} +135.747 q^{93} -592.352 q^{95} +1136.37 q^{97} +780.987 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{3} - 15 q^{5} + 21 q^{7} + 25 q^{9} - 6 q^{11} + 8 q^{13} + 30 q^{15} - 52 q^{17} - 152 q^{19} - 42 q^{21} - 48 q^{23} + 75 q^{25} - 546 q^{27} - 468 q^{29} - 268 q^{31} - 82 q^{33} - 105 q^{35}+ \cdots - 1272 q^{99}+O(q^{100}) $ 3 * q - 6 * q^3 - 15 * q^5 + 21 * q^7 + 25 * q^9 - 6 * q^11 + 8 * q^13 + 30 * q^15 - 52 * q^17 - 152 * q^19 - 42 * q^21 - 48 * q^23 + 75 * q^25 - 546 * q^27 - 468 * q^29 - 268 * q^31 - 82 * q^33 - 105 * q^35 + 262 * q^37 - 674 * q^39 - 74 * q^41 - 692 * q^43 - 125 * q^45 - 774 * q^47 + 147 * q^49 - 366 * q^51 + 62 * q^53 + 30 * q^55 + 1432 * q^57 - 1088 * q^59 + 174 * q^61 + 175 * q^63 - 40 * q^65 + 212 * q^67 - 1784 * q^69 - 1056 * q^71 - 194 * q^73 - 150 * q^75 - 42 * q^77 + 362 * q^79 + 2579 * q^81 - 1696 * q^83 + 260 * q^85 + 2158 * q^87 + 158 * q^89 + 56 * q^91 + 1476 * q^93 + 760 * q^95 + 1620 * q^97 - 1272 * q^99

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
$3$	3.45643	0.665190	0.332595	−	0.943070i	$-0.392076\pi$
$3$	3.45643	0.665190	0.332595	+	0.943070i	$0.392076\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−15.0531	−0.557522
$10$	0	0
$11$	−51.8821	−1.42210	−0.711048	−	0.703144i	$-0.751779\pi$
$11$	−51.8821	−1.42210	−0.711048	+	0.703144i	$0.751779\pi$
$12$	0	0
$13$	−5.91120	−0.126113	−0.0630566	−	0.998010i	$-0.520085\pi$
$13$	−5.91120	−0.126113	−0.0630566	+	0.998010i	$0.520085\pi$
$14$	0	0
$15$	−17.2821	−0.297482
$16$	0	0
$17$	−103.850	−1.48160	−0.740802	−	0.671724i	$-0.765554\pi$
$17$	−103.850	−1.48160	−0.740802	+	0.671724i	$0.765554\pi$
$18$	0	0
$19$	118.470	1.43047	0.715236	−	0.698883i	$-0.246319\pi$
$19$	118.470	1.43047	0.715236	+	0.698883i	$0.246319\pi$
$20$	0	0
$21$	24.1950	0.251418
$22$	0	0
$23$	−80.7029	−0.731640	−0.365820	−	0.930686i	$-0.619211\pi$
$23$	−80.7029	−0.731640	−0.365820	+	0.930686i	$0.619211\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−145.354	−1.03605
$28$	0	0
$29$	−218.344	−1.39812	−0.699058	−	0.715065i	$-0.746397\pi$
$29$	−218.344	−1.39812	−0.699058	+	0.715065i	$0.746397\pi$
$30$	0	0
$31$	39.2738	0.227541	0.113771	−	0.993507i	$-0.463707\pi$
$31$	39.2738	0.227541	0.113771	+	0.993507i	$0.463707\pi$
$32$	0	0
$33$	−179.327	−0.945964
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	159.803	0.710040	0.355020	−	0.934859i	$-0.384474\pi$
$37$	159.803	0.710040	0.355020	+	0.934859i	$0.384474\pi$
$38$	0	0
$39$	−20.4316	−0.0838892
$40$	0	0
$41$	233.322	0.888750	0.444375	−	0.895841i	$-0.353426\pi$
$41$	233.322	0.888750	0.444375	+	0.895841i	$0.353426\pi$
$42$	0	0
$43$	−524.515	−1.86018	−0.930091	−	0.367329i	$-0.880272\pi$
$43$	−524.515	−1.86018	−0.930091	+	0.367329i	$0.880272\pi$
$44$	0	0
$45$	75.2655	0.249332
$46$	0	0
$47$	−210.441	−0.653105	−0.326552	−	0.945179i	$-0.605887\pi$
$47$	−210.441	−0.653105	−0.326552	+	0.945179i	$0.605887\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−358.949	−0.985548
$52$	0	0
$53$	348.028	0.901987	0.450994	−	0.892527i	$-0.351070\pi$
$53$	348.028	0.901987	0.450994	+	0.892527i	$0.351070\pi$
$54$	0	0
$55$	259.411	0.635980
$56$	0	0
$57$	409.485	0.951536
$58$	0	0
$59$	−85.1752	−0.187947	−0.0939735	−	0.995575i	$-0.529957\pi$
$59$	−85.1752	−0.187947	−0.0939735	+	0.995575i	$0.529957\pi$
$60$	0	0
$61$	−801.720	−1.68278	−0.841391	−	0.540427i	$-0.818263\pi$
$61$	−801.720	−1.68278	−0.841391	+	0.540427i	$0.818263\pi$
$62$	0	0
$63$	−105.372	−0.210724
$64$	0	0
$65$	29.5560	0.0563995
$66$	0	0
$67$	716.049	1.30566	0.652831	−	0.757504i	$-0.273581\pi$
$67$	716.049	1.30566	0.652831	+	0.757504i	$0.273581\pi$
$68$	0	0
$69$	−278.944	−0.486679
$70$	0	0
$71$	−520.412	−0.869880	−0.434940	−	0.900459i	$-0.643230\pi$
$71$	−520.412	−0.869880	−0.434940	+	0.900459i	$0.643230\pi$
$72$	0	0
$73$	230.005	0.368768	0.184384	−	0.982854i	$-0.440971\pi$
$73$	230.005	0.368768	0.184384	+	0.982854i	$0.440971\pi$
$74$	0	0
$75$	86.4107	0.133038
$76$	0	0
$77$	−363.175	−0.537502
$78$	0	0
$79$	−970.534	−1.38220	−0.691099	−	0.722760i	$-0.742873\pi$
$79$	−970.534	−1.38220	−0.691099	+	0.722760i	$0.742873\pi$
$80$	0	0
$81$	−95.9705	−0.131647
$82$	0	0
$83$	−1109.38	−1.46711	−0.733555	−	0.679630i	$-0.762140\pi$
$83$	−1109.38	−1.46711	−0.733555	+	0.679630i	$0.762140\pi$
$84$	0	0
$85$	519.249	0.662593
$86$	0	0
$87$	−754.689	−0.930013
$88$	0	0
$89$	1127.45	1.34280	0.671401	−	0.741094i	$-0.265693\pi$
$89$	1127.45	1.34280	0.671401	+	0.741094i	$0.265693\pi$
$90$	0	0
$91$	−41.3784	−0.0476663
$92$	0	0
$93$	135.747	0.151358
$94$	0	0
$95$	−592.352	−0.639727
$96$	0	0
$97$	1136.37	1.18950	0.594748	−	0.803912i	$-0.297252\pi$
$97$	1136.37	1.18950	0.594748	+	0.803912i	$0.297252\pi$
$98$	0	0
$99$	780.987	0.792850
$100$	0	0
$101$	858.838	0.846115	0.423057	−	0.906103i	$-0.360957\pi$
$101$	858.838	0.846115	0.423057	+	0.906103i	$0.360957\pi$
$102$	0	0
$103$	−348.379	−0.333270	−0.166635	−	0.986019i	$-0.553290\pi$
$103$	−348.379	−0.333270	−0.166635	+	0.986019i	$0.553290\pi$
$104$	0	0
$105$	−120.975	−0.112438
$106$	0	0
$107$	1860.54	1.68098	0.840490	−	0.541828i	$-0.182267\pi$
$107$	1860.54	1.68098	0.840490	+	0.541828i	$0.182267\pi$
$108$	0	0
$109$	1213.61	1.06645	0.533224	−	0.845974i	$-0.320980\pi$
$109$	1213.61	1.06645	0.533224	+	0.845974i	$0.320980\pi$
$110$	0	0
$111$	552.349	0.472312
$112$	0	0
$113$	−267.531	−0.222718	−0.111359	−	0.993780i	$-0.535520\pi$
$113$	−267.531	−0.222718	−0.111359	+	0.993780i	$0.535520\pi$
$114$	0	0
$115$	403.514	0.327199
$116$	0	0
$117$	88.9819	0.0703109
$118$	0	0
$119$	−726.948	−0.559994
$120$	0	0
$121$	1360.76	1.02236
$122$	0	0
$123$	806.460	0.591188
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−712.388	−0.497750	−0.248875	−	0.968536i	$-0.580061\pi$
$127$	−712.388	−0.497750	−0.248875	+	0.968536i	$0.580061\pi$
$128$	0	0
$129$	−1812.95	−1.23737
$130$	0	0
$131$	746.364	0.497787	0.248894	−	0.968531i	$-0.419933\pi$
$131$	746.364	0.497787	0.248894	+	0.968531i	$0.419933\pi$
$132$	0	0
$133$	829.293	0.540668
$134$	0	0
$135$	726.768	0.463335
$136$	0	0
$137$	2612.55	1.62924	0.814619	−	0.579996i	$-0.196946\pi$
$137$	2612.55	1.62924	0.814619	+	0.579996i	$0.196946\pi$
$138$	0	0
$139$	637.320	0.388898	0.194449	−	0.980913i	$-0.437708\pi$
$139$	637.320	0.388898	0.194449	+	0.980913i	$0.437708\pi$
$140$	0	0
$141$	−727.373	−0.434439
$142$	0	0
$143$	306.686	0.179345
$144$	0	0
$145$	1091.72	0.625257
$146$	0	0
$147$	169.365	0.0950271
$148$	0	0
$149$	−2151.92	−1.18317	−0.591585	−	0.806242i	$-0.701498\pi$
$149$	−2151.92	−1.18317	−0.591585	+	0.806242i	$0.701498\pi$
$150$	0	0
$151$	1685.18	0.908201	0.454101	−	0.890950i	$-0.349961\pi$
$151$	1685.18	0.908201	0.454101	+	0.890950i	$0.349961\pi$
$152$	0	0
$153$	1563.26	0.826027
$154$	0	0
$155$	−196.369	−0.101760
$156$	0	0
$157$	1139.31	0.579150	0.289575	−	0.957155i	$-0.406486\pi$
$157$	1139.31	0.579150	0.289575	+	0.957155i	$0.406486\pi$
$158$	0	0
$159$	1202.93	0.599993
$160$	0	0
$161$	−564.920	−0.276534
$162$	0	0
$163$	−1758.81	−0.845156	−0.422578	−	0.906327i	$-0.638875\pi$
$163$	−1758.81	−0.845156	−0.422578	+	0.906327i	$0.638875\pi$
$164$	0	0
$165$	896.634	0.423048
$166$	0	0
$167$	2350.50	1.08915	0.544573	−	0.838713i	$-0.316692\pi$
$167$	2350.50	1.08915	0.544573	+	0.838713i	$0.316692\pi$
$168$	0	0
$169$	−2162.06	−0.984095
$170$	0	0
$171$	−1783.35	−0.797520
$172$	0	0
$173$	−2403.94	−1.05646	−0.528231	−	0.849100i	$-0.677145\pi$
$173$	−2403.94	−1.05646	−0.528231	+	0.849100i	$0.677145\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−294.402	−0.125020
$178$	0	0
$179$	−3361.51	−1.40364	−0.701819	−	0.712355i	$-0.747628\pi$
$179$	−3361.51	−1.40364	−0.701819	+	0.712355i	$0.747628\pi$
$180$	0	0
$181$	−4681.04	−1.92231	−0.961156	−	0.276004i	$-0.910990\pi$
$181$	−4681.04	−1.92231	−0.961156	+	0.276004i	$0.910990\pi$
$182$	0	0
$183$	−2771.09	−1.11937
$184$	0	0
$185$	−799.016	−0.317540
$186$	0	0
$187$	5387.95	2.10698
$188$	0	0
$189$	−1017.47	−0.391589
$190$	0	0
$191$	2375.90	0.900075	0.450037	−	0.893010i	$-0.351411\pi$
$191$	2375.90	0.900075	0.450037	+	0.893010i	$0.351411\pi$
$192$	0	0
$193$	4940.87	1.84275	0.921377	−	0.388670i	$-0.127065\pi$
$193$	4940.87	1.84275	0.921377	+	0.388670i	$0.127065\pi$
$194$	0	0
$195$	102.158	0.0375164
$196$	0	0
$197$	1335.30	0.482926	0.241463	−	0.970410i	$-0.422373\pi$
$197$	1335.30	0.482926	0.241463	+	0.970410i	$0.422373\pi$
$198$	0	0
$199$	−2235.61	−0.796374	−0.398187	−	0.917304i	$-0.630360\pi$
$199$	−2235.61	−0.796374	−0.398187	+	0.917304i	$0.630360\pi$
$200$	0	0
$201$	2474.97	0.868513
$202$	0	0
$203$	−1528.40	−0.528438
$204$	0	0
$205$	−1166.61	−0.397461
$206$	0	0
$207$	1214.83	0.407905
$208$	0	0
$209$	−6146.50	−2.03427
$210$	0	0
$211$	−875.868	−0.285769	−0.142884	−	0.989739i	$-0.545638\pi$
$211$	−875.868	−0.285769	−0.142884	+	0.989739i	$0.545638\pi$
$212$	0	0
$213$	−1798.77	−0.578636
$214$	0	0
$215$	2622.58	0.831899
$216$	0	0
$217$	274.917	0.0860025
$218$	0	0
$219$	794.995	0.245300
$220$	0	0
$221$	613.877	0.186850
$222$	0	0
$223$	−3950.23	−1.18622	−0.593110	−	0.805122i	$-0.702100\pi$
$223$	−3950.23	−1.18622	−0.593110	+	0.805122i	$0.702100\pi$
$224$	0	0
$225$	−376.328	−0.111504
$226$	0	0
$227$	−2254.32	−0.659138	−0.329569	−	0.944132i	$-0.606903\pi$
$227$	−2254.32	−0.659138	−0.329569	+	0.944132i	$0.606903\pi$
$228$	0	0
$229$	−3680.17	−1.06198	−0.530988	−	0.847379i	$-0.678179\pi$
$229$	−3680.17	−1.06198	−0.530988	+	0.847379i	$0.678179\pi$
$230$	0	0
$231$	−1255.29	−0.357541
$232$	0	0
$233$	−1109.74	−0.312024	−0.156012	−	0.987755i	$-0.549864\pi$
$233$	−1109.74	−0.312024	−0.156012	+	0.987755i	$0.549864\pi$
$234$	0	0
$235$	1052.20	0.292077
$236$	0	0
$237$	−3354.58	−0.919424
$238$	0	0
$239$	1079.71	0.292220	0.146110	−	0.989268i	$-0.453325\pi$
$239$	1079.71	0.292220	0.146110	+	0.989268i	$0.453325\pi$
$240$	0	0
$241$	81.1127	0.0216802	0.0108401	−	0.999941i	$-0.496549\pi$
$241$	81.1127	0.0216802	0.0108401	+	0.999941i	$0.496549\pi$
$242$	0	0
$243$	3592.83	0.948478
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−700.302	−0.180402
$248$	0	0
$249$	−3834.49	−0.975907
$250$	0	0
$251$	−2193.90	−0.551704	−0.275852	−	0.961200i	$-0.588960\pi$
$251$	−2193.90	−0.551704	−0.275852	+	0.961200i	$0.588960\pi$
$252$	0	0
$253$	4187.04	1.04046
$254$	0	0
$255$	1794.75	0.440750
$256$	0	0
$257$	−3933.07	−0.954623	−0.477312	−	0.878734i	$-0.658389\pi$
$257$	−3933.07	−0.954623	−0.477312	+	0.878734i	$0.658389\pi$
$258$	0	0
$259$	1118.62	0.268370
$260$	0	0
$261$	3286.75	0.779481
$262$	0	0
$263$	497.852	0.116726	0.0583629	−	0.998295i	$-0.481412\pi$
$263$	497.852	0.116726	0.0583629	+	0.998295i	$0.481412\pi$
$264$	0	0
$265$	−1740.14	−0.403381
$266$	0	0
$267$	3896.95	0.893219
$268$	0	0
$269$	−7672.17	−1.73896	−0.869481	−	0.493967i	$-0.835546\pi$
$269$	−7672.17	−1.73896	−0.869481	+	0.493967i	$0.835546\pi$
$270$	0	0
$271$	−3392.88	−0.760526	−0.380263	−	0.924878i	$-0.624167\pi$
$271$	−3392.88	−0.760526	−0.380263	+	0.924878i	$0.624167\pi$
$272$	0	0
$273$	−143.021	−0.0317072
$274$	0	0
$275$	−1297.05	−0.284419
$276$	0	0
$277$	2314.21	0.501975	0.250988	−	0.967990i	$-0.419245\pi$
$277$	2314.21	0.501975	0.250988	+	0.967990i	$0.419245\pi$
$278$	0	0
$279$	−591.192	−0.126859
$280$	0	0
$281$	−3604.79	−0.765280	−0.382640	−	0.923897i	$-0.624985\pi$
$281$	−3604.79	−0.765280	−0.382640	+	0.923897i	$0.624985\pi$
$282$	0	0
$283$	−2304.38	−0.484032	−0.242016	−	0.970272i	$-0.577809\pi$
$283$	−2304.38	−0.484032	−0.242016	+	0.970272i	$0.577809\pi$
$284$	0	0
$285$	−2047.42	−0.425540
$286$	0	0
$287$	1633.25	0.335916
$288$	0	0
$289$	5871.77	1.19515
$290$	0	0
$291$	3927.79	0.791241
$292$	0	0
$293$	−6680.29	−1.33197	−0.665984	−	0.745966i	$-0.731988\pi$
$293$	−6680.29	−1.33197	−0.665984	+	0.745966i	$0.731988\pi$
$294$	0	0
$295$	425.876	0.0840524
$296$	0	0
$297$	7541.25	1.47336
$298$	0	0
$299$	477.051	0.0922694
$300$	0	0
$301$	−3671.61	−0.703083
$302$	0	0
$303$	2968.51	0.562827
$304$	0	0
$305$	4008.60	0.752563
$306$	0	0
$307$	−3532.96	−0.656798	−0.328399	−	0.944539i	$-0.606509\pi$
$307$	−3532.96	−0.656798	−0.328399	+	0.944539i	$0.606509\pi$
$308$	0	0
$309$	−1204.15	−0.221688
$310$	0	0
$311$	−7713.50	−1.40641	−0.703204	−	0.710989i	$-0.748248\pi$
$311$	−7713.50	−1.40641	−0.703204	+	0.710989i	$0.748248\pi$
$312$	0	0
$313$	5776.97	1.04324	0.521619	−	0.853178i	$-0.325328\pi$
$313$	5776.97	1.04324	0.521619	+	0.853178i	$0.325328\pi$
$314$	0	0
$315$	526.859	0.0942385
$316$	0	0
$317$	2879.80	0.510239	0.255120	−	0.966910i	$-0.417885\pi$
$317$	2879.80	0.510239	0.255120	+	0.966910i	$0.417885\pi$
$318$	0	0
$319$	11328.1	1.98826
$320$	0	0
$321$	6430.81	1.11817
$322$	0	0
$323$	−12303.1	−2.11939
$324$	0	0
$325$	−147.780	−0.0252226
$326$	0	0
$327$	4194.76	0.709391
$328$	0	0
$329$	−1473.08	−0.246850
$330$	0	0
$331$	−2784.80	−0.462437	−0.231218	−	0.972902i	$-0.574271\pi$
$331$	−2784.80	−0.462437	−0.231218	+	0.972902i	$0.574271\pi$
$332$	0	0
$333$	−2405.53	−0.395863
$334$	0	0
$335$	−3580.25	−0.583910
$336$	0	0
$337$	2009.52	0.324824	0.162412	−	0.986723i	$-0.448073\pi$
$337$	2009.52	0.324824	0.162412	+	0.986723i	$0.448073\pi$
$338$	0	0
$339$	−924.701	−0.148150
$340$	0	0
$341$	−2037.61	−0.323586
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	1394.72	0.217650
$346$	0	0
$347$	5117.20	0.791659	0.395829	−	0.918324i	$-0.370457\pi$
$347$	5117.20	0.791659	0.395829	+	0.918324i	$0.370457\pi$
$348$	0	0
$349$	−9330.98	−1.43116	−0.715581	−	0.698530i	$-0.753838\pi$
$349$	−9330.98	−1.43116	−0.715581	+	0.698530i	$0.753838\pi$
$350$	0	0
$351$	859.214	0.130659
$352$	0	0
$353$	−1549.68	−0.233657	−0.116829	−	0.993152i	$-0.537273\pi$
$353$	−1549.68	−0.233657	−0.116829	+	0.993152i	$0.537273\pi$
$354$	0	0
$355$	2602.06	0.389022
$356$	0	0
$357$	−2512.64	−0.372502
$358$	0	0
$359$	9513.09	1.39856	0.699278	−	0.714850i	$-0.253505\pi$
$359$	9513.09	1.39856	0.699278	+	0.714850i	$0.253505\pi$
$360$	0	0
$361$	7176.25	1.04625
$362$	0	0
$363$	4703.35	0.680061
$364$	0	0
$365$	−1150.02	−0.164918
$366$	0	0
$367$	−6441.14	−0.916144	−0.458072	−	0.888915i	$-0.651460\pi$
$367$	−6441.14	−0.916144	−0.458072	+	0.888915i	$0.651460\pi$
$368$	0	0
$369$	−3512.22	−0.495498
$370$	0	0
$371$	2436.20	0.340919
$372$	0	0
$373$	−7428.14	−1.03114	−0.515569	−	0.856848i	$-0.672419\pi$
$373$	−7428.14	−1.03114	−0.515569	+	0.856848i	$0.672419\pi$
$374$	0	0
$375$	−432.054	−0.0594964
$376$	0	0
$377$	1290.67	0.176321
$378$	0	0
$379$	−10369.5	−1.40539	−0.702696	−	0.711491i	$-0.748020\pi$
$379$	−10369.5	−1.40539	−0.702696	+	0.711491i	$0.748020\pi$
$380$	0	0
$381$	−2462.32	−0.331098
$382$	0	0
$383$	5924.21	0.790374	0.395187	−	0.918601i	$-0.370680\pi$
$383$	5924.21	0.790374	0.395187	+	0.918601i	$0.370680\pi$
$384$	0	0
$385$	1815.87	0.240378
$386$	0	0
$387$	7895.58	1.03709
$388$	0	0
$389$	1330.48	0.173413	0.0867067	−	0.996234i	$-0.472366\pi$
$389$	1330.48	0.173413	0.0867067	+	0.996234i	$0.472366\pi$
$390$	0	0
$391$	8380.97	1.08400
$392$	0	0
$393$	2579.75	0.331123
$394$	0	0
$395$	4852.67	0.618137
$396$	0	0
$397$	10355.1	1.30909	0.654544	−	0.756023i	$-0.272860\pi$
$397$	10355.1	1.30909	0.654544	+	0.756023i	$0.272860\pi$
$398$	0	0
$399$	2866.39	0.359647
$400$	0	0
$401$	3844.38	0.478751	0.239376	−	0.970927i	$-0.423057\pi$
$401$	3844.38	0.478751	0.239376	+	0.970927i	$0.423057\pi$
$402$	0	0
$403$	−232.155	−0.0286960
$404$	0	0
$405$	479.852	0.0588742
$406$	0	0
$407$	−8290.93	−1.00975
$408$	0	0
$409$	8351.01	1.00961	0.504805	−	0.863233i	$-0.331564\pi$
$409$	8351.01	1.00961	0.504805	+	0.863233i	$0.331564\pi$
$410$	0	0
$411$	9030.11	1.08375
$412$	0	0
$413$	−596.227	−0.0710373
$414$	0	0
$415$	5546.89	0.656112
$416$	0	0
$417$	2202.85	0.258691
$418$	0	0
$419$	−1966.13	−0.229240	−0.114620	−	0.993409i	$-0.536565\pi$
$419$	−1966.13	−0.229240	−0.114620	+	0.993409i	$0.536565\pi$
$420$	0	0
$421$	12154.0	1.40700	0.703501	−	0.710694i	$-0.251619\pi$
$421$	12154.0	1.40700	0.703501	+	0.710694i	$0.251619\pi$
$422$	0	0
$423$	3167.78	0.364120
$424$	0	0
$425$	−2596.24	−0.296321
$426$	0	0
$427$	−5612.04	−0.636032
$428$	0	0
$429$	1060.04	0.119299
$430$	0	0
$431$	−15045.0	−1.68143	−0.840713	−	0.541481i	$-0.817864\pi$
$431$	−15045.0	−1.68143	−0.840713	+	0.541481i	$0.817864\pi$
$432$	0	0
$433$	1945.75	0.215950	0.107975	−	0.994154i	$-0.465563\pi$
$433$	1945.75	0.215950	0.107975	+	0.994154i	$0.465563\pi$
$434$	0	0
$435$	3773.44	0.415915
$436$	0	0
$437$	−9560.91	−1.04659
$438$	0	0
$439$	8484.35	0.922405	0.461203	−	0.887295i	$-0.347418\pi$
$439$	8484.35	0.922405	0.461203	+	0.887295i	$0.347418\pi$
$440$	0	0
$441$	−737.602	−0.0796460
$442$	0	0
$443$	−14875.3	−1.59537	−0.797683	−	0.603078i	$-0.793941\pi$
$443$	−14875.3	−1.59537	−0.797683	+	0.603078i	$0.793941\pi$
$444$	0	0
$445$	−5637.25	−0.600519
$446$	0	0
$447$	−7437.97	−0.787033
$448$	0	0
$449$	17995.8	1.89148	0.945742	−	0.324918i	$-0.105337\pi$
$449$	17995.8	1.89148	0.945742	+	0.324918i	$0.105337\pi$
$450$	0	0
$451$	−12105.2	−1.26389
$452$	0	0
$453$	5824.72	0.604126
$454$	0	0
$455$	206.892	0.0213170
$456$	0	0
$457$	4243.31	0.434340	0.217170	−	0.976134i	$-0.430317\pi$
$457$	4243.31	0.434340	0.217170	+	0.976134i	$0.430317\pi$
$458$	0	0
$459$	15094.9	1.53501
$460$	0	0
$461$	−11234.8	−1.13504	−0.567522	−	0.823358i	$-0.692098\pi$
$461$	−11234.8	−1.13504	−0.567522	+	0.823358i	$0.692098\pi$
$462$	0	0
$463$	−226.775	−0.0227627	−0.0113814	−	0.999935i	$-0.503623\pi$
$463$	−226.775	−0.0227627	−0.0113814	+	0.999935i	$0.503623\pi$
$464$	0	0
$465$	−678.735	−0.0676895
$466$	0	0
$467$	−11886.0	−1.17777	−0.588885	−	0.808217i	$-0.700433\pi$
$467$	−11886.0	−1.17777	−0.588885	+	0.808217i	$0.700433\pi$
$468$	0	0
$469$	5012.35	0.493494
$470$	0	0
$471$	3937.93	0.385245
$472$	0	0
$473$	27213.0	2.64536
$474$	0	0
$475$	2961.76	0.286095
$476$	0	0
$477$	−5238.90	−0.502878
$478$	0	0
$479$	5052.68	0.481969	0.240984	−	0.970529i	$-0.422530\pi$
$479$	5052.68	0.481969	0.240984	+	0.970529i	$0.422530\pi$
$480$	0	0
$481$	−944.629	−0.0895455
$482$	0	0
$483$	−1952.61	−0.183948
$484$	0	0
$485$	−5681.86	−0.531959
$486$	0	0
$487$	3987.21	0.371001	0.185501	−	0.982644i	$-0.440609\pi$
$487$	3987.21	0.371001	0.185501	+	0.982644i	$0.440609\pi$
$488$	0	0
$489$	−6079.19	−0.562189
$490$	0	0
$491$	−4969.66	−0.456777	−0.228389	−	0.973570i	$-0.573346\pi$
$491$	−4969.66	−0.456777	−0.228389	+	0.973570i	$0.573346\pi$
$492$	0	0
$493$	22674.9	2.07145
$494$	0	0
$495$	−3904.93	−0.354573
$496$	0	0
$497$	−3642.88	−0.328784
$498$	0	0
$499$	11651.2	1.04525	0.522625	−	0.852562i	$-0.324953\pi$
$499$	11651.2	1.04525	0.522625	+	0.852562i	$0.324953\pi$
$500$	0	0
$501$	8124.35	0.724490
$502$	0	0
$503$	1024.77	0.0908398	0.0454199	−	0.998968i	$-0.485537\pi$
$503$	1024.77	0.0908398	0.0454199	+	0.998968i	$0.485537\pi$
$504$	0	0
$505$	−4294.19	−0.378394
$506$	0	0
$507$	−7473.00	−0.654610
$508$	0	0
$509$	−6350.40	−0.552999	−0.276500	−	0.961014i	$-0.589174\pi$
$509$	−6350.40	−0.552999	−0.276500	+	0.961014i	$0.589174\pi$
$510$	0	0
$511$	1610.03	0.139381
$512$	0	0
$513$	−17220.1	−1.48204
$514$	0	0
$515$	1741.89	0.149043
$516$	0	0
$517$	10918.1	0.928778
$518$	0	0
$519$	−8309.04	−0.702749
$520$	0	0
$521$	11664.5	0.980864	0.490432	−	0.871479i	$-0.336839\pi$
$521$	11664.5	0.980864	0.490432	+	0.871479i	$0.336839\pi$
$522$	0	0
$523$	2475.61	0.206980	0.103490	−	0.994630i	$-0.466999\pi$
$523$	2475.61	0.206980	0.103490	+	0.994630i	$0.466999\pi$
$524$	0	0
$525$	604.875	0.0502836
$526$	0	0
$527$	−4078.57	−0.337126
$528$	0	0
$529$	−5654.05	−0.464703
$530$	0	0
$531$	1282.15	0.104785
$532$	0	0
$533$	−1379.21	−0.112083
$534$	0	0
$535$	−9302.68	−0.751757
$536$	0	0
$537$	−11618.8	−0.933686
$538$	0	0
$539$	−2542.22	−0.203157
$540$	0	0
$541$	−5689.60	−0.452153	−0.226077	−	0.974110i	$-0.572590\pi$
$541$	−5689.60	−0.452153	−0.226077	+	0.974110i	$0.572590\pi$
$542$	0	0
$543$	−16179.7	−1.27870
$544$	0	0
$545$	−6068.06	−0.476930
$546$	0	0
$547$	−13796.6	−1.07843	−0.539213	−	0.842169i	$-0.681278\pi$
$547$	−13796.6	−1.07843	−0.539213	+	0.842169i	$0.681278\pi$
$548$	0	0
$549$	12068.4	0.938188
$550$	0	0
$551$	−25867.3	−1.99997
$552$	0	0
$553$	−6793.74	−0.522422
$554$	0	0
$555$	−2761.74	−0.211224
$556$	0	0
$557$	−14995.2	−1.14069	−0.570347	−	0.821404i	$-0.693191\pi$
$557$	−14995.2	−1.14069	−0.570347	+	0.821404i	$0.693191\pi$
$558$	0	0
$559$	3100.51	0.234593
$560$	0	0
$561$	18623.1	1.40154
$562$	0	0
$563$	−7110.77	−0.532297	−0.266149	−	0.963932i	$-0.585751\pi$
$563$	−7110.77	−0.532297	−0.266149	+	0.963932i	$0.585751\pi$
$564$	0	0
$565$	1337.65	0.0996027
$566$	0	0
$567$	−671.793	−0.0497578
$568$	0	0
$569$	−19858.3	−1.46310	−0.731549	−	0.681789i	$-0.761202\pi$
$569$	−19858.3	−1.46310	−0.731549	+	0.681789i	$0.761202\pi$
$570$	0	0
$571$	−8680.50	−0.636196	−0.318098	−	0.948058i	$-0.603044\pi$
$571$	−8680.50	−0.636196	−0.318098	+	0.948058i	$0.603044\pi$
$572$	0	0
$573$	8212.14	0.598721
$574$	0	0
$575$	−2017.57	−0.146328
$576$	0	0
$577$	−4639.54	−0.334743	−0.167372	−	0.985894i	$-0.553528\pi$
$577$	−4639.54	−0.334743	−0.167372	+	0.985894i	$0.553528\pi$
$578$	0	0
$579$	17077.8	1.22578
$580$	0	0
$581$	−7765.65	−0.554516
$582$	0	0
$583$	−18056.4	−1.28271
$584$	0	0
$585$	−444.909	−0.0314440
$586$	0	0
$587$	11286.4	0.793597	0.396799	−	0.917906i	$-0.370121\pi$
$587$	11286.4	0.793597	0.396799	+	0.917906i	$0.370121\pi$
$588$	0	0
$589$	4652.78	0.325492
$590$	0	0
$591$	4615.38	0.321237
$592$	0	0
$593$	−2434.90	−0.168616	−0.0843082	−	0.996440i	$-0.526868\pi$
$593$	−2434.90	−0.168616	−0.0843082	+	0.996440i	$0.526868\pi$
$594$	0	0
$595$	3634.74	0.250437
$596$	0	0
$597$	−7727.24	−0.529740
$598$	0	0
$599$	−5668.75	−0.386676	−0.193338	−	0.981132i	$-0.561931\pi$
$599$	−5668.75	−0.386676	−0.193338	+	0.981132i	$0.561931\pi$
$600$	0	0
$601$	22312.6	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$601$	22312.6	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$602$	0	0
$603$	−10778.8	−0.727936
$604$	0	0
$605$	−6803.78	−0.457211
$606$	0	0
$607$	−13583.7	−0.908314	−0.454157	−	0.890922i	$-0.650059\pi$
$607$	−13583.7	−0.908314	−0.454157	+	0.890922i	$0.650059\pi$
$608$	0	0
$609$	−5282.82	−0.351512
$610$	0	0
$611$	1243.96	0.0823651
$612$	0	0
$613$	6448.94	0.424911	0.212455	−	0.977171i	$-0.431854\pi$
$613$	6448.94	0.424911	0.212455	+	0.977171i	$0.431854\pi$
$614$	0	0
$615$	−4032.30	−0.264387
$616$	0	0
$617$	4075.91	0.265948	0.132974	−	0.991120i	$-0.457547\pi$
$617$	4075.91	0.265948	0.132974	+	0.991120i	$0.457547\pi$
$618$	0	0
$619$	−17730.9	−1.15131	−0.575657	−	0.817691i	$-0.695254\pi$
$619$	−17730.9	−1.15131	−0.575657	+	0.817691i	$0.695254\pi$
$620$	0	0
$621$	11730.4	0.758014
$622$	0	0
$623$	7892.14	0.507531
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−21244.9	−1.35318
$628$	0	0
$629$	−16595.5	−1.05200
$630$	0	0
$631$	−24687.2	−1.55750	−0.778748	−	0.627337i	$-0.784145\pi$
$631$	−24687.2	−1.55750	−0.778748	+	0.627337i	$0.784145\pi$
$632$	0	0
$633$	−3027.38	−0.190091
$634$	0	0
$635$	3561.94	0.222600
$636$	0	0
$637$	−289.649	−0.0180162
$638$	0	0
$639$	7833.81	0.484977
$640$	0	0
$641$	−3640.27	−0.224309	−0.112154	−	0.993691i	$-0.535775\pi$
$641$	−3640.27	−0.224309	−0.112154	+	0.993691i	$0.535775\pi$
$642$	0	0
$643$	18809.9	1.15364	0.576821	−	0.816871i	$-0.304293\pi$
$643$	18809.9	1.15364	0.576821	+	0.816871i	$0.304293\pi$
$644$	0	0
$645$	9064.75	0.553371
$646$	0	0
$647$	7914.91	0.480939	0.240469	−	0.970657i	$-0.422699\pi$
$647$	7914.91	0.480939	0.240469	+	0.970657i	$0.422699\pi$
$648$	0	0
$649$	4419.07	0.267279
$650$	0	0
$651$	950.229	0.0572080
$652$	0	0
$653$	20629.9	1.23631	0.618155	−	0.786056i	$-0.287880\pi$
$653$	20629.9	1.23631	0.618155	+	0.786056i	$0.287880\pi$
$654$	0	0
$655$	−3731.82	−0.222617
$656$	0	0
$657$	−3462.29	−0.205596
$658$	0	0
$659$	20033.4	1.18420	0.592101	−	0.805864i	$-0.298299\pi$
$659$	20033.4	1.18420	0.592101	+	0.805864i	$0.298299\pi$
$660$	0	0
$661$	−33181.9	−1.95254	−0.976268	−	0.216565i	$-0.930515\pi$
$661$	−33181.9	−1.95254	−0.976268	+	0.216565i	$0.930515\pi$
$662$	0	0
$663$	2121.82	0.124291
$664$	0	0
$665$	−4146.47	−0.241794
$666$	0	0
$667$	17621.0	1.02292
$668$	0	0
$669$	−13653.7	−0.789061
$670$	0	0
$671$	41594.9	2.39308
$672$	0	0
$673$	9881.92	0.566003	0.283001	−	0.959119i	$-0.408670\pi$
$673$	9881.92	0.566003	0.283001	+	0.959119i	$0.408670\pi$
$674$	0	0
$675$	−3633.84	−0.207210
$676$	0	0
$677$	−2059.42	−0.116913	−0.0584563	−	0.998290i	$-0.518618\pi$
$677$	−2059.42	−0.116913	−0.0584563	+	0.998290i	$0.518618\pi$
$678$	0	0
$679$	7954.61	0.449587
$680$	0	0
$681$	−7791.89	−0.438452
$682$	0	0
$683$	29350.2	1.64429	0.822147	−	0.569275i	$-0.192776\pi$
$683$	29350.2	1.64429	0.822147	+	0.569275i	$0.192776\pi$
$684$	0	0
$685$	−13062.8	−0.728617
$686$	0	0
$687$	−12720.3	−0.706416
$688$	0	0
$689$	−2057.26	−0.113753
$690$	0	0
$691$	−11343.4	−0.624491	−0.312246	−	0.950001i	$-0.601081\pi$
$691$	−11343.4	−0.624491	−0.312246	+	0.950001i	$0.601081\pi$
$692$	0	0
$693$	5466.91	0.299669
$694$	0	0
$695$	−3186.60	−0.173920
$696$	0	0
$697$	−24230.4	−1.31678
$698$	0	0
$699$	−3835.74	−0.207555
$700$	0	0
$701$	−16939.1	−0.912667	−0.456334	−	0.889809i	$-0.650838\pi$
$701$	−16939.1	−0.912667	−0.456334	+	0.889809i	$0.650838\pi$
$702$	0	0
$703$	18932.0	1.01569
$704$	0	0
$705$	3636.87	0.194287
$706$	0	0
$707$	6011.87	0.319801
$708$	0	0
$709$	20813.0	1.10247	0.551233	−	0.834351i	$-0.314157\pi$
$709$	20813.0	1.10247	0.551233	+	0.834351i	$0.314157\pi$
$710$	0	0
$711$	14609.5	0.770606
$712$	0	0
$713$	−3169.51	−0.166478
$714$	0	0
$715$	−1533.43	−0.0802055
$716$	0	0
$717$	3731.94	0.194382
$718$	0	0
$719$	15526.6	0.805348	0.402674	−	0.915344i	$-0.368081\pi$
$719$	15526.6	0.805348	0.402674	+	0.915344i	$0.368081\pi$
$720$	0	0
$721$	−2438.65	−0.125964
$722$	0	0
$723$	280.360	0.0144214
$724$	0	0
$725$	−5458.59	−0.279623
$726$	0	0
$727$	−18301.0	−0.933628	−0.466814	−	0.884356i	$-0.654598\pi$
$727$	−18301.0	−0.933628	−0.466814	+	0.884356i	$0.654598\pi$
$728$	0	0
$729$	15009.6	0.762565
$730$	0	0
$731$	54470.8	2.75605
$732$	0	0
$733$	−31175.7	−1.57094	−0.785472	−	0.618897i	$-0.787580\pi$
$733$	−31175.7	−1.57094	−0.785472	+	0.618897i	$0.787580\pi$
$734$	0	0
$735$	−846.825	−0.0424974
$736$	0	0
$737$	−37150.2	−1.85678
$738$	0	0
$739$	−38874.4	−1.93507	−0.967537	−	0.252730i	$-0.918672\pi$
$739$	−38874.4	−1.93507	−0.967537	+	0.252730i	$0.918672\pi$
$740$	0	0
$741$	−2420.55	−0.120001
$742$	0	0
$743$	−15899.8	−0.785071	−0.392536	−	0.919737i	$-0.628402\pi$
$743$	−15899.8	−0.785071	−0.392536	+	0.919737i	$0.628402\pi$
$744$	0	0
$745$	10759.6	0.529130
$746$	0	0
$747$	16699.6	0.817947
$748$	0	0
$749$	13023.7	0.635350
$750$	0	0
$751$	20419.5	0.992168	0.496084	−	0.868275i	$-0.334771\pi$
$751$	20419.5	0.992168	0.496084	+	0.868275i	$0.334771\pi$
$752$	0	0
$753$	−7583.05	−0.366988
$754$	0	0
$755$	−8425.92	−0.406160
$756$	0	0
$757$	33358.9	1.60165	0.800826	−	0.598897i	$-0.204394\pi$
$757$	33358.9	1.60165	0.800826	+	0.598897i	$0.204394\pi$
$758$	0	0
$759$	14472.2	0.692105
$760$	0	0
$761$	−7773.61	−0.370293	−0.185147	−	0.982711i	$-0.559276\pi$
$761$	−7773.61	−0.370293	−0.185147	+	0.982711i	$0.559276\pi$
$762$	0	0
$763$	8495.28	0.403080
$764$	0	0
$765$	−7816.30	−0.369411
$766$	0	0
$767$	503.488	0.0237026
$768$	0	0
$769$	30704.0	1.43981	0.719906	−	0.694072i	$-0.244185\pi$
$769$	30704.0	1.43981	0.719906	+	0.694072i	$0.244185\pi$
$770$	0	0
$771$	−13594.4	−0.635006
$772$	0	0
$773$	31749.5	1.47730	0.738648	−	0.674091i	$-0.235464\pi$
$773$	31749.5	1.47730	0.738648	+	0.674091i	$0.235464\pi$
$774$	0	0
$775$	981.845	0.0455083
$776$	0	0
$777$	3866.44	0.178517
$778$	0	0
$779$	27641.7	1.27133
$780$	0	0
$781$	27000.1	1.23705
$782$	0	0
$783$	31737.0	1.44852
$784$	0	0
$785$	−5696.54	−0.259004
$786$	0	0
$787$	4536.07	0.205455	0.102728	−	0.994710i	$-0.467243\pi$
$787$	4536.07	0.205455	0.102728	+	0.994710i	$0.467243\pi$
$788$	0	0
$789$	1720.79	0.0776448
$790$	0	0
$791$	−1872.71	−0.0841796
$792$	0	0
$793$	4739.13	0.212221
$794$	0	0
$795$	−6014.67	−0.268325
$796$	0	0
$797$	−237.980	−0.0105768	−0.00528840	−	0.999986i	$-0.501683\pi$
$797$	−237.980	−0.0105768	−0.00528840	+	0.999986i	$0.501683\pi$
$798$	0	0
$799$	21854.2	0.967643
$800$	0	0
$801$	−16971.6	−0.748642
$802$	0	0
$803$	−11933.1	−0.524423
$804$	0	0
$805$	2824.60	0.123670
$806$	0	0
$807$	−26518.3	−1.15674
$808$	0	0
$809$	−33008.4	−1.43450	−0.717251	−	0.696815i	$-0.754600\pi$
$809$	−33008.4	−1.43450	−0.717251	+	0.696815i	$0.754600\pi$
$810$	0	0
$811$	−16028.1	−0.693988	−0.346994	−	0.937867i	$-0.612798\pi$
$811$	−16028.1	−0.693988	−0.346994	+	0.937867i	$0.612798\pi$
$812$	0	0
$813$	−11727.2	−0.505894
$814$	0	0
$815$	8794.04	0.377965
$816$	0	0
$817$	−62139.6	−2.66094
$818$	0	0
$819$	622.873	0.0265750
$820$	0	0
$821$	6121.99	0.260242	0.130121	−	0.991498i	$-0.458463\pi$
$821$	6121.99	0.260242	0.130121	+	0.991498i	$0.458463\pi$
$822$	0	0
$823$	10372.7	0.439331	0.219666	−	0.975575i	$-0.429503\pi$
$823$	10372.7	0.439331	0.219666	+	0.975575i	$0.429503\pi$
$824$	0	0
$825$	−4483.17	−0.189193
$826$	0	0
$827$	−5018.82	−0.211030	−0.105515	−	0.994418i	$-0.533649\pi$
$827$	−5018.82	−0.211030	−0.105515	+	0.994418i	$0.533649\pi$
$828$	0	0
$829$	−13231.4	−0.554336	−0.277168	−	0.960822i	$-0.589396\pi$
$829$	−13231.4	−0.554336	−0.277168	+	0.960822i	$0.589396\pi$
$830$	0	0
$831$	7998.89	0.333909
$832$	0	0
$833$	−5088.64	−0.211658
$834$	0	0
$835$	−11752.5	−0.487081
$836$	0	0
$837$	−5708.59	−0.235744
$838$	0	0
$839$	−9522.56	−0.391842	−0.195921	−	0.980620i	$-0.562770\pi$
$839$	−9522.56	−0.391842	−0.195921	+	0.980620i	$0.562770\pi$
$840$	0	0
$841$	23284.9	0.954730
$842$	0	0
$843$	−12459.7	−0.509057
$844$	0	0
$845$	10810.3	0.440101
$846$	0	0
$847$	9525.29	0.386414
$848$	0	0
$849$	−7964.92	−0.321973
$850$	0	0
$851$	−12896.6	−0.519494
$852$	0	0
$853$	43594.1	1.74986	0.874932	−	0.484245i	$-0.160906\pi$
$853$	43594.1	1.74986	0.874932	+	0.484245i	$0.160906\pi$
$854$	0	0
$855$	8916.74	0.356662
$856$	0	0
$857$	31892.6	1.27122	0.635608	−	0.772012i	$-0.280749\pi$
$857$	31892.6	1.27122	0.635608	+	0.772012i	$0.280749\pi$
$858$	0	0
$859$	13058.6	0.518689	0.259345	−	0.965785i	$-0.416493\pi$
$859$	13058.6	0.518689	0.259345	+	0.965785i	$0.416493\pi$
$860$	0	0
$861$	5645.22	0.223448
$862$	0	0
$863$	−21487.2	−0.847548	−0.423774	−	0.905768i	$-0.639295\pi$
$863$	−21487.2	−0.847548	−0.423774	+	0.905768i	$0.639295\pi$
$864$	0	0
$865$	12019.7	0.472465
$866$	0	0
$867$	20295.4	0.795002
$868$	0	0
$869$	50353.4	1.96562
$870$	0	0
$871$	−4232.71	−0.164661
$872$	0	0
$873$	−17105.9	−0.663171
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−2743.60	−0.105638	−0.0528192	−	0.998604i	$-0.516821\pi$
$877$	−2743.60	−0.105638	−0.0528192	+	0.998604i	$0.516821\pi$
$878$	0	0
$879$	−23089.9	−0.886012
$880$	0	0
$881$	−12554.8	−0.480115	−0.240057	−	0.970759i	$-0.577166\pi$
$881$	−12554.8	−0.480115	−0.240057	+	0.970759i	$0.577166\pi$
$882$	0	0
$883$	26353.2	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$883$	26353.2	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$884$	0	0
$885$	1472.01	0.0559109
$886$	0	0
$887$	14016.6	0.530587	0.265293	−	0.964168i	$-0.414531\pi$
$887$	14016.6	0.530587	0.265293	+	0.964168i	$0.414531\pi$
$888$	0	0
$889$	−4986.72	−0.188132
$890$	0	0
$891$	4979.15	0.187214
$892$	0	0
$893$	−24931.0	−0.934249
$894$	0	0
$895$	16807.6	0.627726
$896$	0	0
$897$	1648.89	0.0613767
$898$	0	0
$899$	−8575.18	−0.318129
$900$	0	0
$901$	−36142.6	−1.33639
$902$	0	0
$903$	−12690.6	−0.467684
$904$	0	0
$905$	23405.2	0.859684
$906$	0	0
$907$	30710.0	1.12427	0.562134	−	0.827047i	$-0.309981\pi$
$907$	30710.0	1.12427	0.562134	+	0.827047i	$0.309981\pi$
$908$	0	0
$909$	−12928.2	−0.471728
$910$	0	0
$911$	47099.3	1.71292	0.856459	−	0.516215i	$-0.172659\pi$
$911$	47099.3	1.71292	0.856459	+	0.516215i	$0.172659\pi$
$912$	0	0
$913$	57556.9	2.08637
$914$	0	0
$915$	13855.4	0.500597
$916$	0	0
$917$	5224.55	0.188146
$918$	0	0
$919$	7922.89	0.284387	0.142194	−	0.989839i	$-0.454584\pi$
$919$	7922.89	0.284387	0.142194	+	0.989839i	$0.454584\pi$
$920$	0	0
$921$	−12211.4	−0.436895
$922$	0	0
$923$	3076.26	0.109703
$924$	0	0
$925$	3995.08	0.142008
$926$	0	0
$927$	5244.18	0.185805
$928$	0	0
$929$	22388.1	0.790668	0.395334	−	0.918537i	$-0.370629\pi$
$929$	22388.1	0.790668	0.395334	+	0.918537i	$0.370629\pi$
$930$	0	0
$931$	5805.05	0.204353
$932$	0	0
$933$	−26661.2	−0.935528
$934$	0	0
$935$	−26939.7	−0.942271
$936$	0	0
$937$	−15161.4	−0.528603	−0.264302	−	0.964440i	$-0.585141\pi$
$937$	−15161.4	−0.528603	−0.264302	+	0.964440i	$0.585141\pi$
$938$	0	0
$939$	19967.7	0.693952
$940$	0	0
$941$	32547.3	1.12754	0.563768	−	0.825933i	$-0.309351\pi$
$941$	32547.3	1.12754	0.563768	+	0.825933i	$0.309351\pi$
$942$	0	0
$943$	−18829.7	−0.650245
$944$	0	0
$945$	5087.37	0.175124
$946$	0	0
$947$	16496.7	0.566072	0.283036	−	0.959109i	$-0.408658\pi$
$947$	16496.7	0.566072	0.283036	+	0.959109i	$0.408658\pi$
$948$	0	0
$949$	−1359.60	−0.0465065
$950$	0	0
$951$	9953.83	0.339406
$952$	0	0
$953$	−42608.4	−1.44829	−0.724146	−	0.689647i	$-0.757766\pi$
$953$	−42608.4	−1.44829	−0.724146	+	0.689647i	$0.757766\pi$
$954$	0	0
$955$	−11879.5	−0.402526
$956$	0	0
$957$	39154.9	1.32257
$958$	0	0
$959$	18287.9	0.615794
$960$	0	0
$961$	−28248.6	−0.948225
$962$	0	0
$963$	−28006.8	−0.937183
$964$	0	0
$965$	−24704.3	−0.824105
$966$	0	0
$967$	−25980.4	−0.863983	−0.431992	−	0.901878i	$-0.642189\pi$
$967$	−25980.4	−0.863983	−0.431992	+	0.901878i	$0.642189\pi$
$968$	0	0
$969$	−42524.9	−1.40980
$970$	0	0
$971$	475.566	0.0157174	0.00785872	−	0.999969i	$-0.497498\pi$
$971$	475.566	0.0157174	0.00785872	+	0.999969i	$0.497498\pi$
$972$	0	0
$973$	4461.24	0.146990
$974$	0	0
$975$	−510.791	−0.0167778
$976$	0	0
$977$	17562.9	0.575115	0.287557	−	0.957763i	$-0.407157\pi$
$977$	17562.9	0.575115	0.287557	+	0.957763i	$0.407157\pi$
$978$	0	0
$979$	−58494.5	−1.90959
$980$	0	0
$981$	−18268.6	−0.594569
$982$	0	0
$983$	−20943.8	−0.679557	−0.339779	−	0.940505i	$-0.610352\pi$
$983$	−20943.8	−0.679557	−0.339779	+	0.940505i	$0.610352\pi$
$984$	0	0
$985$	−6676.51	−0.215971
$986$	0	0
$987$	−5091.61	−0.164202
$988$	0	0
$989$	42329.9	1.36098
$990$	0	0
$991$	−55297.8	−1.77254	−0.886272	−	0.463164i	$-0.846714\pi$
$991$	−55297.8	−1.77254	−0.886272	+	0.463164i	$0.846714\pi$
$992$	0	0
$993$	−9625.47	−0.307608
$994$	0	0
$995$	11178.1	0.356149
$996$	0	0
$997$	−1485.25	−0.0471800	−0.0235900	−	0.999722i	$-0.507510\pi$
$997$	−1485.25	−0.0471800	−0.0235900	+	0.999722i	$0.507510\pi$
$998$	0	0
$999$	−23228.0	−0.735636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.4.a.f.1.3	✓	3
4.3	odd	2		560.4.a.w.1.1		3
5.2	odd	4		1400.4.g.l.449.2		6
5.3	odd	4		1400.4.g.l.449.5		6
5.4	even	2		1400.4.a.m.1.1		3
7.6	odd	2		1960.4.a.r.1.1		3
8.3	odd	2		2240.4.a.br.1.3		3
8.5	even	2		2240.4.a.bz.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.4.a.f.1.3	✓	3	1.1	even	1	trivial
560.4.a.w.1.1		3	4.3	odd	2
1400.4.a.m.1.1		3	5.4	even	2
1400.4.g.l.449.2		6	5.2	odd	4
1400.4.g.l.449.5		6	5.3	odd	4
1960.4.a.r.1.1		3	7.6	odd	2
2240.4.a.br.1.3		3	8.3	odd	2
2240.4.a.bz.1.1		3	8.5	even	2

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:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	280.a (trivial)

\(f(q)\)	\(=\)	\(q+3.45643 q^{3} -5.00000 q^{5} +7.00000 q^{7} -15.0531 q^{9} -51.8821 q^{11} -5.91120 q^{13} -17.2821 q^{15} -103.850 q^{17} +118.470 q^{19} +24.1950 q^{21} -80.7029 q^{23} +25.0000 q^{25} -145.354 q^{27} -218.344 q^{29} +39.2738 q^{31} -179.327 q^{33} -35.0000 q^{35} +159.803 q^{37} -20.4316 q^{39} +233.322 q^{41} -524.515 q^{43} +75.2655 q^{45} -210.441 q^{47} +49.0000 q^{49} -358.949 q^{51} +348.028 q^{53} +259.411 q^{55} +409.485 q^{57} -85.1752 q^{59} -801.720 q^{61} -105.372 q^{63} +29.5560 q^{65} +716.049 q^{67} -278.944 q^{69} -520.412 q^{71} +230.005 q^{73} +86.4107 q^{75} -363.175 q^{77} -970.534 q^{79} -95.9705 q^{81} -1109.38 q^{83} +519.249 q^{85} -754.689 q^{87} +1127.45 q^{89} -41.3784 q^{91} +135.747 q^{93} -592.352 q^{95} +1136.37 q^{97} +780.987 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} - 15 q^{5} + 21 q^{7} + 25 q^{9} - 6 q^{11} + 8 q^{13} + 30 q^{15} - 52 q^{17} - 152 q^{19} - 42 q^{21} - 48 q^{23} + 75 q^{25} - 546 q^{27} - 468 q^{29} - 268 q^{31} - 82 q^{33} - 105 q^{35}+ \cdots - 1272 q^{99}+O(q^{100}) \) 3 * q - 6 * q^3 - 15 * q^5 + 21 * q^7 + 25 * q^9 - 6 * q^11 + 8 * q^13 + 30 * q^15 - 52 * q^17 - 152 * q^19 - 42 * q^21 - 48 * q^23 + 75 * q^25 - 546 * q^27 - 468 * q^29 - 268 * q^31 - 82 * q^33 - 105 * q^35 + 262 * q^37 - 674 * q^39 - 74 * q^41 - 692 * q^43 - 125 * q^45 - 774 * q^47 + 147 * q^49 - 366 * q^51 + 62 * q^53 + 30 * q^55 + 1432 * q^57 - 1088 * q^59 + 174 * q^61 + 175 * q^63 - 40 * q^65 + 212 * q^67 - 1784 * q^69 - 1056 * q^71 - 194 * q^73 - 150 * q^75 - 42 * q^77 + 362 * q^79 + 2579 * q^81 - 1696 * q^83 + 260 * q^85 + 2158 * q^87 + 158 * q^89 + 56 * q^91 + 1476 * q^93 + 760 * q^95 + 1620 * q^97 - 1272 * q^99

Introduction

L-functions

Modular forms

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