LMFDB - Embedded newform 1680.4.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +19.0623 q^{11} -2.93774 q^{13} -15.0000 q^{15} -6.49806 q^{17} +5.43580 q^{19} -21.0000 q^{21} -49.3774 q^{23} +25.0000 q^{25} -27.0000 q^{27} -291.494 q^{29} -244.307 q^{31} -57.1868 q^{33} +35.0000 q^{35} -193.121 q^{37} +8.81323 q^{39} +315.113 q^{41} +300.996 q^{43} +45.0000 q^{45} -86.5058 q^{47} +49.0000 q^{49} +19.4942 q^{51} +509.677 q^{53} +95.3113 q^{55} -16.3074 q^{57} +83.3852 q^{59} -5.25291 q^{61} +63.0000 q^{63} -14.6887 q^{65} -205.992 q^{67} +148.132 q^{69} -1004.31 q^{71} -1007.29 q^{73} -75.0000 q^{75} +133.436 q^{77} +863.237 q^{79} +81.0000 q^{81} -1334.72 q^{83} -32.4903 q^{85} +874.483 q^{87} +326.249 q^{89} -20.5642 q^{91} +732.922 q^{93} +27.1790 q^{95} +1526.77 q^{97} +171.560 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	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
$9$	9.00000	0.333333
$10$	0	0
$11$	19.0623	0.522499	0.261249	−	0.965271i	$-0.415865\pi$
$11$	19.0623	0.522499	0.261249	+	0.965271i	$0.415865\pi$
$12$	0	0
$13$	−2.93774	−0.0626756	−0.0313378	−	0.999509i	$-0.509977\pi$
$13$	−2.93774	−0.0626756	−0.0313378	+	0.999509i	$0.509977\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−6.49806	−0.0927066	−0.0463533	−	0.998925i	$-0.514760\pi$
$17$	−6.49806	−0.0927066	−0.0463533	+	0.998925i	$0.514760\pi$
$18$	0	0
$19$	5.43580	0.0656347	0.0328173	−	0.999461i	$-0.489552\pi$
$19$	5.43580	0.0656347	0.0328173	+	0.999461i	$0.489552\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−49.3774	−0.447648	−0.223824	−	0.974630i	$-0.571854\pi$
$23$	−49.3774	−0.447648	−0.223824	+	0.974630i	$0.571854\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−291.494	−1.86652	−0.933261	−	0.359200i	$-0.883050\pi$
$29$	−291.494	−1.86652	−0.933261	+	0.359200i	$0.883050\pi$
$30$	0	0
$31$	−244.307	−1.41545	−0.707724	−	0.706489i	$-0.750278\pi$
$31$	−244.307	−1.41545	−0.707724	+	0.706489i	$0.750278\pi$
$32$	0	0
$33$	−57.1868	−0.301665
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−193.121	−0.858077	−0.429038	−	0.903286i	$-0.641147\pi$
$37$	−193.121	−0.858077	−0.429038	+	0.903286i	$0.641147\pi$
$38$	0	0
$39$	8.81323	0.0361858
$40$	0	0
$41$	315.113	1.20030	0.600151	−	0.799887i	$-0.295107\pi$
$41$	315.113	1.20030	0.600151	+	0.799887i	$0.295107\pi$
$42$	0	0
$43$	300.996	1.06748	0.533738	−	0.845650i	$-0.320787\pi$
$43$	300.996	1.06748	0.533738	+	0.845650i	$0.320787\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−86.5058	−0.268472	−0.134236	−	0.990949i	$-0.542858\pi$
$47$	−86.5058	−0.268472	−0.134236	+	0.990949i	$0.542858\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	19.4942	0.0535242
$52$	0	0
$53$	509.677	1.32093	0.660467	−	0.750855i	$-0.270358\pi$
$53$	509.677	1.32093	0.660467	+	0.750855i	$0.270358\pi$
$54$	0	0
$55$	95.3113	0.233669
$56$	0	0
$57$	−16.3074	−0.0378942
$58$	0	0
$59$	83.3852	0.183997	0.0919985	−	0.995759i	$-0.470674\pi$
$59$	83.3852	0.183997	0.0919985	+	0.995759i	$0.470674\pi$
$60$	0	0
$61$	−5.25291	−0.0110257	−0.00551283	−	0.999985i	$-0.501755\pi$
$61$	−5.25291	−0.0110257	−0.00551283	+	0.999985i	$0.501755\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−14.6887	−0.0280294
$66$	0	0
$67$	−205.992	−0.375611	−0.187806	−	0.982206i	$-0.560138\pi$
$67$	−205.992	−0.375611	−0.187806	+	0.982206i	$0.560138\pi$
$68$	0	0
$69$	148.132	0.258450
$70$	0	0
$71$	−1004.31	−1.67872	−0.839362	−	0.543573i	$-0.817071\pi$
$71$	−1004.31	−1.67872	−0.839362	+	0.543573i	$0.817071\pi$
$72$	0	0
$73$	−1007.29	−1.61499	−0.807494	−	0.589876i	$-0.799177\pi$
$73$	−1007.29	−1.61499	−0.807494	+	0.589876i	$0.799177\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	133.436	0.197486
$78$	0	0
$79$	863.237	1.22939	0.614695	−	0.788765i	$-0.289279\pi$
$79$	863.237	1.22939	0.614695	+	0.788765i	$0.289279\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1334.72	−1.76512	−0.882560	−	0.470200i	$-0.844182\pi$
$83$	−1334.72	−1.76512	−0.882560	+	0.470200i	$0.844182\pi$
$84$	0	0
$85$	−32.4903	−0.0414596
$86$	0	0
$87$	874.483	1.07764
$88$	0	0
$89$	326.249	0.388565	0.194283	−	0.980946i	$-0.437762\pi$
$89$	326.249	0.388565	0.194283	+	0.980946i	$0.437762\pi$
$90$	0	0
$91$	−20.5642	−0.0236892
$92$	0	0
$93$	732.922	0.817210
$94$	0	0
$95$	27.1790	0.0293527
$96$	0	0
$97$	1526.77	1.59815	0.799075	−	0.601232i	$-0.205323\pi$
$97$	1526.77	1.59815	0.799075	+	0.601232i	$0.205323\pi$
$98$	0	0
$99$	171.560	0.174166
$100$	0	0
$101$	96.8716	0.0954365	0.0477182	−	0.998861i	$-0.484805\pi$
$101$	96.8716	0.0954365	0.0477182	+	0.998861i	$0.484805\pi$
$102$	0	0
$103$	−1321.99	−1.26466	−0.632329	−	0.774700i	$-0.717901\pi$
$103$	−1321.99	−1.26466	−0.632329	+	0.774700i	$0.717901\pi$
$104$	0	0
$105$	−105.000	−0.0975900
$106$	0	0
$107$	−1745.71	−1.57724	−0.788619	−	0.614883i	$-0.789203\pi$
$107$	−1745.71	−1.57724	−0.788619	+	0.614883i	$0.789203\pi$
$108$	0	0
$109$	476.856	0.419032	0.209516	−	0.977805i	$-0.432811\pi$
$109$	476.856	0.419032	0.209516	+	0.977805i	$0.432811\pi$
$110$	0	0
$111$	579.362	0.495411
$112$	0	0
$113$	1641.65	1.36666	0.683332	−	0.730108i	$-0.260530\pi$
$113$	1641.65	1.36666	0.683332	+	0.730108i	$0.260530\pi$
$114$	0	0
$115$	−246.887	−0.200194
$116$	0	0
$117$	−26.4397	−0.0208919
$118$	0	0
$119$	−45.4864	−0.0350398
$120$	0	0
$121$	−967.630	−0.726995
$122$	0	0
$123$	−945.339	−0.692994
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	844.016	0.589719	0.294859	−	0.955541i	$-0.404727\pi$
$127$	844.016	0.589719	0.294859	+	0.955541i	$0.404727\pi$
$128$	0	0
$129$	−902.988	−0.616308
$130$	0	0
$131$	2796.20	1.86493	0.932463	−	0.361265i	$-0.117655\pi$
$131$	2796.20	1.86493	0.932463	+	0.361265i	$0.117655\pi$
$132$	0	0
$133$	38.0506	0.0248076
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−2057.13	−1.28287	−0.641433	−	0.767179i	$-0.721660\pi$
$137$	−2057.13	−1.28287	−0.641433	+	0.767179i	$0.721660\pi$
$138$	0	0
$139$	1745.12	1.06489	0.532444	−	0.846465i	$-0.321274\pi$
$139$	1745.12	1.06489	0.532444	+	0.846465i	$0.321274\pi$
$140$	0	0
$141$	259.517	0.155002
$142$	0	0
$143$	−56.0000	−0.0327479
$144$	0	0
$145$	−1457.47	−0.834734
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1173.57	−0.645254	−0.322627	−	0.946526i	$-0.604566\pi$
$149$	−1173.57	−0.645254	−0.322627	+	0.946526i	$0.604566\pi$
$150$	0	0
$151$	−1540.07	−0.829994	−0.414997	−	0.909823i	$-0.636217\pi$
$151$	−1540.07	−0.829994	−0.414997	+	0.909823i	$0.636217\pi$
$152$	0	0
$153$	−58.4826	−0.0309022
$154$	0	0
$155$	−1221.54	−0.633008
$156$	0	0
$157$	−2544.53	−1.29348	−0.646738	−	0.762712i	$-0.723867\pi$
$157$	−2544.53	−1.29348	−0.646738	+	0.762712i	$0.723867\pi$
$158$	0	0
$159$	−1529.03	−0.762642
$160$	0	0
$161$	−345.642	−0.169195
$162$	0	0
$163$	594.708	0.285774	0.142887	−	0.989739i	$-0.454361\pi$
$163$	594.708	0.285774	0.142887	+	0.989739i	$0.454361\pi$
$164$	0	0
$165$	−285.934	−0.134909
$166$	0	0
$167$	−928.498	−0.430236	−0.215118	−	0.976588i	$-0.569014\pi$
$167$	−928.498	−0.430236	−0.215118	+	0.976588i	$0.569014\pi$
$168$	0	0
$169$	−2188.37	−0.996072
$170$	0	0
$171$	48.9222	0.0218782
$172$	0	0
$173$	−315.642	−0.138716	−0.0693578	−	0.997592i	$-0.522095\pi$
$173$	−315.642	−0.138716	−0.0693578	+	0.997592i	$0.522095\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−250.156	−0.106231
$178$	0	0
$179$	1445.49	0.603581	0.301791	−	0.953374i	$-0.402416\pi$
$179$	1445.49	0.603581	0.301791	+	0.953374i	$0.402416\pi$
$180$	0	0
$181$	−1843.81	−0.757180	−0.378590	−	0.925564i	$-0.623591\pi$
$181$	−1843.81	−0.757180	−0.378590	+	0.925564i	$0.623591\pi$
$182$	0	0
$183$	15.7587	0.00636567
$184$	0	0
$185$	−965.603	−0.383744
$186$	0	0
$187$	−123.868	−0.0484391
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	244.074	0.0924637	0.0462318	−	0.998931i	$-0.485279\pi$
$191$	244.074	0.0924637	0.0462318	+	0.998931i	$0.485279\pi$
$192$	0	0
$193$	−1733.03	−0.646355	−0.323178	−	0.946338i	$-0.604751\pi$
$193$	−1733.03	−0.646355	−0.323178	+	0.946338i	$0.604751\pi$
$194$	0	0
$195$	44.0661	0.0161828
$196$	0	0
$197$	358.230	0.129557	0.0647787	−	0.997900i	$-0.479366\pi$
$197$	358.230	0.129557	0.0647787	+	0.997900i	$0.479366\pi$
$198$	0	0
$199$	3203.63	1.14120	0.570601	−	0.821227i	$-0.306710\pi$
$199$	3203.63	1.14120	0.570601	+	0.821227i	$0.306710\pi$
$200$	0	0
$201$	617.977	0.216859
$202$	0	0
$203$	−2040.46	−0.705479
$204$	0	0
$205$	1575.56	0.536791
$206$	0	0
$207$	−444.397	−0.149216
$208$	0	0
$209$	103.619	0.0342940
$210$	0	0
$211$	−4943.16	−1.61280	−0.806401	−	0.591369i	$-0.798588\pi$
$211$	−4943.16	−1.61280	−0.806401	+	0.591369i	$0.798588\pi$
$212$	0	0
$213$	3012.92	0.969211
$214$	0	0
$215$	1504.98	0.477390
$216$	0	0
$217$	−1710.15	−0.534989
$218$	0	0
$219$	3021.86	0.932414
$220$	0	0
$221$	19.0896	0.00581044
$222$	0	0
$223$	−3160.15	−0.948965	−0.474482	−	0.880265i	$-0.657365\pi$
$223$	−3160.15	−0.948965	−0.474482	+	0.880265i	$0.657365\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3651.11	1.06755	0.533773	−	0.845628i	$-0.320774\pi$
$227$	3651.11	1.06755	0.533773	+	0.845628i	$0.320774\pi$
$228$	0	0
$229$	−4083.70	−1.17842	−0.589210	−	0.807980i	$-0.700561\pi$
$229$	−4083.70	−1.17842	−0.589210	+	0.807980i	$0.700561\pi$
$230$	0	0
$231$	−400.307	−0.114019
$232$	0	0
$233$	−3682.51	−1.03540	−0.517702	−	0.855561i	$-0.673212\pi$
$233$	−3682.51	−1.03540	−0.517702	+	0.855561i	$0.673212\pi$
$234$	0	0
$235$	−432.529	−0.120064
$236$	0	0
$237$	−2589.71	−0.709789
$238$	0	0
$239$	−2658.78	−0.719591	−0.359796	−	0.933031i	$-0.617154\pi$
$239$	−2658.78	−0.719591	−0.359796	+	0.933031i	$0.617154\pi$
$240$	0	0
$241$	−4820.39	−1.28842	−0.644209	−	0.764850i	$-0.722813\pi$
$241$	−4820.39	−1.28842	−0.644209	+	0.764850i	$0.722813\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−15.9690	−0.00411369
$248$	0	0
$249$	4004.17	1.01909
$250$	0	0
$251$	−1672.27	−0.420530	−0.210265	−	0.977644i	$-0.567433\pi$
$251$	−1672.27	−0.420530	−0.210265	+	0.977644i	$0.567433\pi$
$252$	0	0
$253$	−941.245	−0.233896
$254$	0	0
$255$	97.4709	0.0239367
$256$	0	0
$257$	−3697.74	−0.897506	−0.448753	−	0.893656i	$-0.648132\pi$
$257$	−3697.74	−0.897506	−0.448753	+	0.893656i	$0.648132\pi$
$258$	0	0
$259$	−1351.84	−0.324323
$260$	0	0
$261$	−2623.45	−0.622174
$262$	0	0
$263$	−7319.00	−1.71600	−0.858002	−	0.513646i	$-0.828294\pi$
$263$	−7319.00	−1.71600	−0.858002	+	0.513646i	$0.828294\pi$
$264$	0	0
$265$	2548.39	0.590740
$266$	0	0
$267$	−978.747	−0.224338
$268$	0	0
$269$	−815.097	−0.184749	−0.0923743	−	0.995724i	$-0.529446\pi$
$269$	−815.097	−0.184749	−0.0923743	+	0.995724i	$0.529446\pi$
$270$	0	0
$271$	5106.02	1.14453	0.572267	−	0.820068i	$-0.306064\pi$
$271$	5106.02	1.14453	0.572267	+	0.820068i	$0.306064\pi$
$272$	0	0
$273$	61.6926	0.0136769
$274$	0	0
$275$	476.556	0.104500
$276$	0	0
$277$	1398.72	0.303398	0.151699	−	0.988427i	$-0.451526\pi$
$277$	1398.72	0.303398	0.151699	+	0.988427i	$0.451526\pi$
$278$	0	0
$279$	−2198.77	−0.471816
$280$	0	0
$281$	−7102.38	−1.50780	−0.753901	−	0.656988i	$-0.771830\pi$
$281$	−7102.38	−1.50780	−0.753901	+	0.656988i	$0.771830\pi$
$282$	0	0
$283$	−4465.18	−0.937907	−0.468953	−	0.883223i	$-0.655369\pi$
$283$	−4465.18	−0.937907	−0.468953	+	0.883223i	$0.655369\pi$
$284$	0	0
$285$	−81.5371	−0.0169468
$286$	0	0
$287$	2205.79	0.453671
$288$	0	0
$289$	−4870.78	−0.991405
$290$	0	0
$291$	−4580.32	−0.922692
$292$	0	0
$293$	7590.61	1.51348	0.756738	−	0.653718i	$-0.226792\pi$
$293$	7590.61	1.51348	0.756738	+	0.653718i	$0.226792\pi$
$294$	0	0
$295$	416.926	0.0822860
$296$	0	0
$297$	−514.681	−0.100555
$298$	0	0
$299$	145.058	0.0280566
$300$	0	0
$301$	2106.97	0.403468
$302$	0	0
$303$	−290.615	−0.0551003
$304$	0	0
$305$	−26.2645	−0.00493083
$306$	0	0
$307$	−9480.12	−1.76241	−0.881203	−	0.472737i	$-0.843266\pi$
$307$	−9480.12	−1.76241	−0.881203	+	0.472737i	$0.843266\pi$
$308$	0	0
$309$	3965.98	0.730151
$310$	0	0
$311$	−7078.01	−1.29054	−0.645268	−	0.763956i	$-0.723254\pi$
$311$	−7078.01	−1.29054	−0.645268	+	0.763956i	$0.723254\pi$
$312$	0	0
$313$	5593.84	1.01017	0.505084	−	0.863070i	$-0.331461\pi$
$313$	5593.84	1.01017	0.505084	+	0.863070i	$0.331461\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	3567.81	0.632139	0.316070	−	0.948736i	$-0.397637\pi$
$317$	3567.81	0.632139	0.316070	+	0.948736i	$0.397637\pi$
$318$	0	0
$319$	−5556.54	−0.975255
$320$	0	0
$321$	5237.14	0.910618
$322$	0	0
$323$	−35.3222	−0.00608477
$324$	0	0
$325$	−73.4436	−0.0125351
$326$	0	0
$327$	−1430.57	−0.241928
$328$	0	0
$329$	−605.541	−0.101473
$330$	0	0
$331$	4389.67	0.728936	0.364468	−	0.931216i	$-0.381251\pi$
$331$	4389.67	0.728936	0.364468	+	0.931216i	$0.381251\pi$
$332$	0	0
$333$	−1738.09	−0.286026
$334$	0	0
$335$	−1029.96	−0.167978
$336$	0	0
$337$	2348.83	0.379671	0.189835	−	0.981816i	$-0.439205\pi$
$337$	2348.83	0.379671	0.189835	+	0.981816i	$0.439205\pi$
$338$	0	0
$339$	−4924.94	−0.789044
$340$	0	0
$341$	−4657.05	−0.739570
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	740.661	0.115582
$346$	0	0
$347$	−558.436	−0.0863931	−0.0431965	−	0.999067i	$-0.513754\pi$
$347$	−558.436	−0.0863931	−0.0431965	+	0.999067i	$0.513754\pi$
$348$	0	0
$349$	3233.89	0.496006	0.248003	−	0.968759i	$-0.420226\pi$
$349$	3233.89	0.496006	0.248003	+	0.968759i	$0.420226\pi$
$350$	0	0
$351$	79.3190	0.0120619
$352$	0	0
$353$	−7516.35	−1.13330	−0.566650	−	0.823959i	$-0.691761\pi$
$353$	−7516.35	−1.13330	−0.566650	+	0.823959i	$0.691761\pi$
$354$	0	0
$355$	−5021.54	−0.750748
$356$	0	0
$357$	136.459	0.0202302
$358$	0	0
$359$	6577.76	0.967021	0.483511	−	0.875338i	$-0.339361\pi$
$359$	6577.76	0.967021	0.483511	+	0.875338i	$0.339361\pi$
$360$	0	0
$361$	−6829.45	−0.995692
$362$	0	0
$363$	2902.89	0.419731
$364$	0	0
$365$	−5036.44	−0.722245
$366$	0	0
$367$	−8307.17	−1.18155	−0.590777	−	0.806835i	$-0.701179\pi$
$367$	−8307.17	−1.18155	−0.590777	+	0.806835i	$0.701179\pi$
$368$	0	0
$369$	2836.02	0.400101
$370$	0	0
$371$	3567.74	0.499266
$372$	0	0
$373$	−4551.09	−0.631760	−0.315880	−	0.948799i	$-0.602300\pi$
$373$	−4551.09	−0.631760	−0.315880	+	0.948799i	$0.602300\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	856.335	0.116985
$378$	0	0
$379$	1788.29	0.242370	0.121185	−	0.992630i	$-0.461331\pi$
$379$	1788.29	0.242370	0.121185	+	0.992630i	$0.461331\pi$
$380$	0	0
$381$	−2532.05	−0.340474
$382$	0	0
$383$	1358.47	0.181240	0.0906199	−	0.995886i	$-0.471115\pi$
$383$	1358.47	0.181240	0.0906199	+	0.995886i	$0.471115\pi$
$384$	0	0
$385$	667.179	0.0883184
$386$	0	0
$387$	2708.97	0.355825
$388$	0	0
$389$	9722.54	1.26723	0.633615	−	0.773649i	$-0.281570\pi$
$389$	9722.54	1.26723	0.633615	+	0.773649i	$0.281570\pi$
$390$	0	0
$391$	320.858	0.0414999
$392$	0	0
$393$	−8388.61	−1.07672
$394$	0	0
$395$	4316.19	0.549800
$396$	0	0
$397$	−4788.04	−0.605302	−0.302651	−	0.953101i	$-0.597872\pi$
$397$	−4788.04	−0.605302	−0.302651	+	0.953101i	$0.597872\pi$
$398$	0	0
$399$	−114.152	−0.0143227
$400$	0	0
$401$	9681.41	1.20565	0.602826	−	0.797873i	$-0.294041\pi$
$401$	9681.41	1.20565	0.602826	+	0.797873i	$0.294041\pi$
$402$	0	0
$403$	717.712	0.0887141
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−3681.32	−0.448344
$408$	0	0
$409$	11113.1	1.34353	0.671767	−	0.740763i	$-0.265536\pi$
$409$	11113.1	1.34353	0.671767	+	0.740763i	$0.265536\pi$
$410$	0	0
$411$	6171.40	0.740663
$412$	0	0
$413$	583.696	0.0695443
$414$	0	0
$415$	−6673.62	−0.789386
$416$	0	0
$417$	−5235.37	−0.614814
$418$	0	0
$419$	1230.09	0.143421	0.0717107	−	0.997425i	$-0.477154\pi$
$419$	1230.09	0.143421	0.0717107	+	0.997425i	$0.477154\pi$
$420$	0	0
$421$	−12356.5	−1.43044	−0.715222	−	0.698897i	$-0.753674\pi$
$421$	−12356.5	−1.43044	−0.715222	+	0.698897i	$0.753674\pi$
$422$	0	0
$423$	−778.552	−0.0894906
$424$	0	0
$425$	−162.452	−0.0185413
$426$	0	0
$427$	−36.7703	−0.00416731
$428$	0	0
$429$	168.000	0.0189070
$430$	0	0
$431$	7375.27	0.824256	0.412128	−	0.911126i	$-0.364786\pi$
$431$	7375.27	0.824256	0.412128	+	0.911126i	$0.364786\pi$
$432$	0	0
$433$	−690.067	−0.0765877	−0.0382939	−	0.999267i	$-0.512192\pi$
$433$	−690.067	−0.0765877	−0.0382939	+	0.999267i	$0.512192\pi$
$434$	0	0
$435$	4372.41	0.481934
$436$	0	0
$437$	−268.406	−0.0293812
$438$	0	0
$439$	−8408.79	−0.914191	−0.457095	−	0.889418i	$-0.651110\pi$
$439$	−8408.79	−0.914191	−0.457095	+	0.889418i	$0.651110\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−6568.55	−0.704473	−0.352236	−	0.935911i	$-0.614579\pi$
$443$	−6568.55	−0.704473	−0.352236	+	0.935911i	$0.614579\pi$
$444$	0	0
$445$	1631.25	0.173772
$446$	0	0
$447$	3520.72	0.372537
$448$	0	0
$449$	2954.55	0.310543	0.155271	−	0.987872i	$-0.450375\pi$
$449$	2954.55	0.310543	0.155271	+	0.987872i	$0.450375\pi$
$450$	0	0
$451$	6006.76	0.627156
$452$	0	0
$453$	4620.21	0.479197
$454$	0	0
$455$	−102.821	−0.0105941
$456$	0	0
$457$	−8144.84	−0.833697	−0.416849	−	0.908976i	$-0.636865\pi$
$457$	−8144.84	−0.833697	−0.416849	+	0.908976i	$0.636865\pi$
$458$	0	0
$459$	175.448	0.0178414
$460$	0	0
$461$	2495.26	0.252095	0.126048	−	0.992024i	$-0.459771\pi$
$461$	2495.26	0.252095	0.126048	+	0.992024i	$0.459771\pi$
$462$	0	0
$463$	5755.66	0.577728	0.288864	−	0.957370i	$-0.406722\pi$
$463$	5755.66	0.577728	0.288864	+	0.957370i	$0.406722\pi$
$464$	0	0
$465$	3664.61	0.365467
$466$	0	0
$467$	4143.73	0.410598	0.205299	−	0.978699i	$-0.434183\pi$
$467$	4143.73	0.410598	0.205299	+	0.978699i	$0.434183\pi$
$468$	0	0
$469$	−1441.95	−0.141968
$470$	0	0
$471$	7633.60	0.746789
$472$	0	0
$473$	5737.67	0.557755
$474$	0	0
$475$	135.895	0.0131269
$476$	0	0
$477$	4587.09	0.440312
$478$	0	0
$479$	6765.96	0.645396	0.322698	−	0.946502i	$-0.395410\pi$
$479$	6765.96	0.645396	0.322698	+	0.946502i	$0.395410\pi$
$480$	0	0
$481$	567.339	0.0537805
$482$	0	0
$483$	1036.93	0.0976848
$484$	0	0
$485$	7633.87	0.714714
$486$	0	0
$487$	−6360.42	−0.591824	−0.295912	−	0.955215i	$-0.595623\pi$
$487$	−6360.42	−0.591824	−0.295912	+	0.955215i	$0.595623\pi$
$488$	0	0
$489$	−1784.12	−0.164992
$490$	0	0
$491$	7072.54	0.650060	0.325030	−	0.945704i	$-0.394626\pi$
$491$	7072.54	0.650060	0.325030	+	0.945704i	$0.394626\pi$
$492$	0	0
$493$	1894.15	0.173039
$494$	0	0
$495$	857.802	0.0778895
$496$	0	0
$497$	−7030.15	−0.634498
$498$	0	0
$499$	18473.9	1.65732	0.828661	−	0.559751i	$-0.189103\pi$
$499$	18473.9	1.65732	0.828661	+	0.559751i	$0.189103\pi$
$500$	0	0
$501$	2785.49	0.248397
$502$	0	0
$503$	−11379.2	−1.00869	−0.504347	−	0.863501i	$-0.668267\pi$
$503$	−11379.2	−1.00869	−0.504347	+	0.863501i	$0.668267\pi$
$504$	0	0
$505$	484.358	0.0426805
$506$	0	0
$507$	6565.11	0.575082
$508$	0	0
$509$	6064.48	0.528101	0.264051	−	0.964509i	$-0.414941\pi$
$509$	6064.48	0.528101	0.264051	+	0.964509i	$0.414941\pi$
$510$	0	0
$511$	−7051.02	−0.610408
$512$	0	0
$513$	−146.767	−0.0126314
$514$	0	0
$515$	−6609.96	−0.565572
$516$	0	0
$517$	−1649.00	−0.140276
$518$	0	0
$519$	946.926	0.0800875
$520$	0	0
$521$	2682.88	0.225603	0.112801	−	0.993618i	$-0.464018\pi$
$521$	2682.88	0.225603	0.112801	+	0.993618i	$0.464018\pi$
$522$	0	0
$523$	4309.02	0.360268	0.180134	−	0.983642i	$-0.442347\pi$
$523$	4309.02	0.360268	0.180134	+	0.983642i	$0.442347\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	1587.52	0.131221
$528$	0	0
$529$	−9728.87	−0.799611
$530$	0	0
$531$	750.467	0.0613323
$532$	0	0
$533$	−925.720	−0.0752296
$534$	0	0
$535$	−8728.56	−0.705362
$536$	0	0
$537$	−4336.47	−0.348478
$538$	0	0
$539$	934.051	0.0746427
$540$	0	0
$541$	4081.47	0.324355	0.162178	−	0.986762i	$-0.448148\pi$
$541$	4081.47	0.324355	0.162178	+	0.986762i	$0.448148\pi$
$542$	0	0
$543$	5531.44	0.437158
$544$	0	0
$545$	2384.28	0.187397
$546$	0	0
$547$	−8844.82	−0.691366	−0.345683	−	0.938351i	$-0.612353\pi$
$547$	−8844.82	−0.691366	−0.345683	+	0.938351i	$0.612353\pi$
$548$	0	0
$549$	−47.2762	−0.00367522
$550$	0	0
$551$	−1584.51	−0.122509
$552$	0	0
$553$	6042.66	0.464666
$554$	0	0
$555$	2896.81	0.221554
$556$	0	0
$557$	−11144.7	−0.847787	−0.423894	−	0.905712i	$-0.639337\pi$
$557$	−11144.7	−0.847787	−0.423894	+	0.905712i	$0.639337\pi$
$558$	0	0
$559$	−884.249	−0.0669047
$560$	0	0
$561$	371.603	0.0279663
$562$	0	0
$563$	21857.5	1.63621	0.818104	−	0.575071i	$-0.195025\pi$
$563$	21857.5	1.63621	0.818104	+	0.575071i	$0.195025\pi$
$564$	0	0
$565$	8208.23	0.611191
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	23496.4	1.73115	0.865573	−	0.500783i	$-0.166954\pi$
$569$	23496.4	1.73115	0.865573	+	0.500783i	$0.166954\pi$
$570$	0	0
$571$	−11067.8	−0.811164	−0.405582	−	0.914059i	$-0.632931\pi$
$571$	−11067.8	−0.811164	−0.405582	+	0.914059i	$0.632931\pi$
$572$	0	0
$573$	−732.222	−0.0533839
$574$	0	0
$575$	−1234.44	−0.0895296
$576$	0	0
$577$	20482.9	1.47784	0.738922	−	0.673791i	$-0.235335\pi$
$577$	20482.9	1.47784	0.738922	+	0.673791i	$0.235335\pi$
$578$	0	0
$579$	5199.10	0.373173
$580$	0	0
$581$	−9343.07	−0.667153
$582$	0	0
$583$	9715.60	0.690187
$584$	0	0
$585$	−132.198	−0.00934313
$586$	0	0
$587$	23444.3	1.64847	0.824235	−	0.566248i	$-0.191606\pi$
$587$	23444.3	1.64847	0.824235	+	0.566248i	$0.191606\pi$
$588$	0	0
$589$	−1328.01	−0.0929025
$590$	0	0
$591$	−1074.69	−0.0748000
$592$	0	0
$593$	4404.69	0.305024	0.152512	−	0.988302i	$-0.451264\pi$
$593$	4404.69	0.305024	0.152512	+	0.988302i	$0.451264\pi$
$594$	0	0
$595$	−227.432	−0.0156703
$596$	0	0
$597$	−9610.89	−0.658874
$598$	0	0
$599$	−3327.05	−0.226945	−0.113472	−	0.993541i	$-0.536197\pi$
$599$	−3327.05	−0.226945	−0.113472	+	0.993541i	$0.536197\pi$
$600$	0	0
$601$	−14244.8	−0.966818	−0.483409	−	0.875395i	$-0.660602\pi$
$601$	−14244.8	−0.966818	−0.483409	+	0.875395i	$0.660602\pi$
$602$	0	0
$603$	−1853.93	−0.125204
$604$	0	0
$605$	−4838.15	−0.325122
$606$	0	0
$607$	−11446.5	−0.765402	−0.382701	−	0.923872i	$-0.625006\pi$
$607$	−11446.5	−0.765402	−0.382701	+	0.923872i	$0.625006\pi$
$608$	0	0
$609$	6121.38	0.407308
$610$	0	0
$611$	254.132	0.0168266
$612$	0	0
$613$	−19436.4	−1.28063	−0.640316	−	0.768111i	$-0.721197\pi$
$613$	−19436.4	−1.28063	−0.640316	+	0.768111i	$0.721197\pi$
$614$	0	0
$615$	−4726.69	−0.309917
$616$	0	0
$617$	−20530.1	−1.33956	−0.669781	−	0.742558i	$-0.733612\pi$
$617$	−20530.1	−1.33956	−0.669781	+	0.742558i	$0.733612\pi$
$618$	0	0
$619$	−5833.35	−0.378776	−0.189388	−	0.981902i	$-0.560650\pi$
$619$	−5833.35	−0.378776	−0.189388	+	0.981902i	$0.560650\pi$
$620$	0	0
$621$	1333.19	0.0861499
$622$	0	0
$623$	2283.74	0.146864
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−310.856	−0.0197997
$628$	0	0
$629$	1254.91	0.0795493
$630$	0	0
$631$	−24776.6	−1.56314	−0.781568	−	0.623820i	$-0.785580\pi$
$631$	−24776.6	−1.56314	−0.781568	+	0.623820i	$0.785580\pi$
$632$	0	0
$633$	14829.5	0.931152
$634$	0	0
$635$	4220.08	0.263730
$636$	0	0
$637$	−143.949	−0.00895366
$638$	0	0
$639$	−9038.77	−0.559574
$640$	0	0
$641$	27219.4	1.67723	0.838613	−	0.544728i	$-0.183367\pi$
$641$	27219.4	1.67723	0.838613	+	0.544728i	$0.183367\pi$
$642$	0	0
$643$	−7091.79	−0.434950	−0.217475	−	0.976066i	$-0.569782\pi$
$643$	−7091.79	−0.434950	−0.217475	+	0.976066i	$0.569782\pi$
$644$	0	0
$645$	−4514.94	−0.275621
$646$	0	0
$647$	−27773.0	−1.68758	−0.843792	−	0.536670i	$-0.819682\pi$
$647$	−27773.0	−1.68758	−0.843792	+	0.536670i	$0.819682\pi$
$648$	0	0
$649$	1589.51	0.0961382
$650$	0	0
$651$	5130.46	0.308876
$652$	0	0
$653$	21380.4	1.28129	0.640643	−	0.767839i	$-0.278668\pi$
$653$	21380.4	1.28129	0.640643	+	0.767839i	$0.278668\pi$
$654$	0	0
$655$	13981.0	0.834020
$656$	0	0
$657$	−9065.59	−0.538329
$658$	0	0
$659$	17232.3	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$659$	17232.3	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$660$	0	0
$661$	26577.7	1.56392	0.781962	−	0.623326i	$-0.214219\pi$
$661$	26577.7	1.56392	0.781962	+	0.623326i	$0.214219\pi$
$662$	0	0
$663$	−57.2689	−0.00335466
$664$	0	0
$665$	190.253	0.0110943
$666$	0	0
$667$	14393.2	0.835544
$668$	0	0
$669$	9480.44	0.547885
$670$	0	0
$671$	−100.132	−0.00576090
$672$	0	0
$673$	−31695.2	−1.81540	−0.907698	−	0.419624i	$-0.862162\pi$
$673$	−31695.2	−1.81540	−0.907698	+	0.419624i	$0.862162\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	20440.3	1.16039	0.580195	−	0.814477i	$-0.302976\pi$
$677$	20440.3	1.16039	0.580195	+	0.814477i	$0.302976\pi$
$678$	0	0
$679$	10687.4	0.604044
$680$	0	0
$681$	−10953.3	−0.616348
$682$	0	0
$683$	22896.9	1.28276	0.641381	−	0.767223i	$-0.278362\pi$
$683$	22896.9	1.28276	0.641381	+	0.767223i	$0.278362\pi$
$684$	0	0
$685$	−10285.7	−0.573715
$686$	0	0
$687$	12251.1	0.680361
$688$	0	0
$689$	−1497.30	−0.0827904
$690$	0	0
$691$	23764.0	1.30829	0.654143	−	0.756371i	$-0.273029\pi$
$691$	23764.0	1.30829	0.654143	+	0.756371i	$0.273029\pi$
$692$	0	0
$693$	1200.92	0.0658287
$694$	0	0
$695$	8725.62	0.476233
$696$	0	0
$697$	−2047.62	−0.111276
$698$	0	0
$699$	11047.5	0.597791
$700$	0	0
$701$	26259.5	1.41485	0.707423	−	0.706791i	$-0.249858\pi$
$701$	26259.5	1.41485	0.707423	+	0.706791i	$0.249858\pi$
$702$	0	0
$703$	−1049.77	−0.0563196
$704$	0	0
$705$	1297.59	0.0693191
$706$	0	0
$707$	678.101	0.0360716
$708$	0	0
$709$	12783.0	0.677116	0.338558	−	0.940945i	$-0.390061\pi$
$709$	12783.0	0.677116	0.338558	+	0.940945i	$0.390061\pi$
$710$	0	0
$711$	7769.14	0.409797
$712$	0	0
$713$	12063.3	0.633623
$714$	0	0
$715$	−280.000	−0.0146453
$716$	0	0
$717$	7976.35	0.415456
$718$	0	0
$719$	27609.0	1.43205	0.716025	−	0.698075i	$-0.245960\pi$
$719$	27609.0	1.43205	0.716025	+	0.698075i	$0.245960\pi$
$720$	0	0
$721$	−9253.95	−0.477996
$722$	0	0
$723$	14461.2	0.743868
$724$	0	0
$725$	−7287.35	−0.373304
$726$	0	0
$727$	−31306.2	−1.59709	−0.798544	−	0.601937i	$-0.794396\pi$
$727$	−31306.2	−1.59709	−0.798544	+	0.601937i	$0.794396\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1955.89	−0.0989621
$732$	0	0
$733$	−15765.8	−0.794441	−0.397220	−	0.917723i	$-0.630025\pi$
$733$	−15765.8	−0.794441	−0.397220	+	0.917723i	$0.630025\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	−3926.68	−0.196256
$738$	0	0
$739$	−3966.51	−0.197443	−0.0987216	−	0.995115i	$-0.531475\pi$
$739$	−3966.51	−0.197443	−0.0987216	+	0.995115i	$0.531475\pi$
$740$	0	0
$741$	47.9070	0.00237504
$742$	0	0
$743$	−8224.50	−0.406094	−0.203047	−	0.979169i	$-0.565084\pi$
$743$	−8224.50	−0.406094	−0.203047	+	0.979169i	$0.565084\pi$
$744$	0	0
$745$	−5867.86	−0.288566
$746$	0	0
$747$	−12012.5	−0.588373
$748$	0	0
$749$	−12220.0	−0.596140
$750$	0	0
$751$	−18929.2	−0.919754	−0.459877	−	0.887983i	$-0.652106\pi$
$751$	−18929.2	−0.919754	−0.459877	+	0.887983i	$0.652106\pi$
$752$	0	0
$753$	5016.82	0.242793
$754$	0	0
$755$	−7700.35	−0.371185
$756$	0	0
$757$	−34906.8	−1.67597	−0.837984	−	0.545695i	$-0.816266\pi$
$757$	−34906.8	−1.67597	−0.837984	+	0.545695i	$0.816266\pi$
$758$	0	0
$759$	2823.74	0.135040
$760$	0	0
$761$	−13683.4	−0.651803	−0.325902	−	0.945404i	$-0.605668\pi$
$761$	−13683.4	−0.651803	−0.325902	+	0.945404i	$0.605668\pi$
$762$	0	0
$763$	3337.99	0.158379
$764$	0	0
$765$	−292.413	−0.0138199
$766$	0	0
$767$	−244.964	−0.0115321
$768$	0	0
$769$	41837.3	1.96189	0.980943	−	0.194294i	$-0.0622416\pi$
$769$	41837.3	1.96189	0.980943	+	0.194294i	$0.0622416\pi$
$770$	0	0
$771$	11093.2	0.518175
$772$	0	0
$773$	19640.0	0.913843	0.456921	−	0.889507i	$-0.348952\pi$
$773$	19640.0	0.913843	0.456921	+	0.889507i	$0.348952\pi$
$774$	0	0
$775$	−6107.69	−0.283090
$776$	0	0
$777$	4055.53	0.187248
$778$	0	0
$779$	1712.89	0.0787814
$780$	0	0
$781$	−19144.4	−0.877131
$782$	0	0
$783$	7870.34	0.359212
$784$	0	0
$785$	−12722.7	−0.578460
$786$	0	0
$787$	−24935.3	−1.12941	−0.564705	−	0.825293i	$-0.691010\pi$
$787$	−24935.3	−1.12941	−0.564705	+	0.825293i	$0.691010\pi$
$788$	0	0
$789$	21957.0	0.990736
$790$	0	0
$791$	11491.5	0.516551
$792$	0	0
$793$	15.4317	0.000691041 0
$794$	0	0
$795$	−7645.16	−0.341064
$796$	0	0
$797$	−1168.33	−0.0519251	−0.0259625	−	0.999663i	$-0.508265\pi$
$797$	−1168.33	−0.0519251	−0.0259625	+	0.999663i	$0.508265\pi$
$798$	0	0
$799$	562.120	0.0248891
$800$	0	0
$801$	2936.24	0.129522
$802$	0	0
$803$	−19201.2	−0.843829
$804$	0	0
$805$	−1728.21	−0.0756663
$806$	0	0
$807$	2445.29	0.106665
$808$	0	0
$809$	−35175.7	−1.52869	−0.764345	−	0.644807i	$-0.776938\pi$
$809$	−35175.7	−1.52869	−0.764345	+	0.644807i	$0.776938\pi$
$810$	0	0
$811$	15256.5	0.660577	0.330288	−	0.943880i	$-0.392854\pi$
$811$	15256.5	0.660577	0.330288	+	0.943880i	$0.392854\pi$
$812$	0	0
$813$	−15318.1	−0.660797
$814$	0	0
$815$	2973.54	0.127802
$816$	0	0
$817$	1636.16	0.0700635
$818$	0	0
$819$	−185.078	−0.00789639
$820$	0	0
$821$	−15971.9	−0.678956	−0.339478	−	0.940614i	$-0.610250\pi$
$821$	−15971.9	−0.678956	−0.339478	+	0.940614i	$0.610250\pi$
$822$	0	0
$823$	−2312.41	−0.0979409	−0.0489705	−	0.998800i	$-0.515594\pi$
$823$	−2312.41	−0.0979409	−0.0489705	+	0.998800i	$0.515594\pi$
$824$	0	0
$825$	−1429.67	−0.0603330
$826$	0	0
$827$	−10422.4	−0.438238	−0.219119	−	0.975698i	$-0.570318\pi$
$827$	−10422.4	−0.438238	−0.219119	+	0.975698i	$0.570318\pi$
$828$	0	0
$829$	−13213.4	−0.553584	−0.276792	−	0.960930i	$-0.589271\pi$
$829$	−13213.4	−0.553584	−0.276792	+	0.960930i	$0.589271\pi$
$830$	0	0
$831$	−4196.17	−0.175167
$832$	0	0
$833$	−318.405	−0.0132438
$834$	0	0
$835$	−4642.49	−0.192407
$836$	0	0
$837$	6596.30	0.272403
$838$	0	0
$839$	10119.6	0.416409	0.208205	−	0.978085i	$-0.433238\pi$
$839$	10119.6	0.416409	0.208205	+	0.978085i	$0.433238\pi$
$840$	0	0
$841$	60579.9	2.48390
$842$	0	0
$843$	21307.1	0.870530
$844$	0	0
$845$	−10941.8	−0.445457
$846$	0	0
$847$	−6773.41	−0.274778
$848$	0	0
$849$	13395.5	0.541501
$850$	0	0
$851$	9535.80	0.384116
$852$	0	0
$853$	35378.1	1.42007	0.710037	−	0.704165i	$-0.248678\pi$
$853$	35378.1	1.42007	0.710037	+	0.704165i	$0.248678\pi$
$854$	0	0
$855$	244.611	0.00978424
$856$	0	0
$857$	6697.57	0.266960	0.133480	−	0.991052i	$-0.457385\pi$
$857$	6697.57	0.266960	0.133480	+	0.991052i	$0.457385\pi$
$858$	0	0
$859$	24298.4	0.965135	0.482568	−	0.875859i	$-0.339704\pi$
$859$	24298.4	0.965135	0.482568	+	0.875859i	$0.339704\pi$
$860$	0	0
$861$	−6617.37	−0.261927
$862$	0	0
$863$	−24942.9	−0.983853	−0.491926	−	0.870637i	$-0.663707\pi$
$863$	−24942.9	−0.983853	−0.491926	+	0.870637i	$0.663707\pi$
$864$	0	0
$865$	−1578.21	−0.0620355
$866$	0	0
$867$	14612.3	0.572388
$868$	0	0
$869$	16455.3	0.642355
$870$	0	0
$871$	605.152	0.0235417
$872$	0	0
$873$	13741.0	0.532716
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−16276.6	−0.626705	−0.313353	−	0.949637i	$-0.601452\pi$
$877$	−16276.6	−0.626705	−0.313353	+	0.949637i	$0.601452\pi$
$878$	0	0
$879$	−22771.8	−0.873806
$880$	0	0
$881$	26636.5	1.01862	0.509311	−	0.860582i	$-0.329900\pi$
$881$	26636.5	1.01862	0.509311	+	0.860582i	$0.329900\pi$
$882$	0	0
$883$	21788.3	0.830392	0.415196	−	0.909732i	$-0.363713\pi$
$883$	21788.3	0.830392	0.415196	+	0.909732i	$0.363713\pi$
$884$	0	0
$885$	−1250.78	−0.0475078
$886$	0	0
$887$	26813.2	1.01499	0.507496	−	0.861654i	$-0.330571\pi$
$887$	26813.2	1.01499	0.507496	+	0.861654i	$0.330571\pi$
$888$	0	0
$889$	5908.11	0.222893
$890$	0	0
$891$	1544.04	0.0580554
$892$	0	0
$893$	−470.229	−0.0176211
$894$	0	0
$895$	7227.45	0.269930
$896$	0	0
$897$	−435.174	−0.0161985
$898$	0	0
$899$	71214.2	2.64196
$900$	0	0
$901$	−3311.91	−0.122459
$902$	0	0
$903$	−6320.92	−0.232942
$904$	0	0
$905$	−9219.07	−0.338621
$906$	0	0
$907$	−15543.0	−0.569014	−0.284507	−	0.958674i	$-0.591830\pi$
$907$	−15543.0	−0.569014	−0.284507	+	0.958674i	$0.591830\pi$
$908$	0	0
$909$	871.844	0.0318122
$910$	0	0
$911$	−48711.1	−1.77154	−0.885768	−	0.464128i	$-0.846368\pi$
$911$	−48711.1	−1.77154	−0.885768	+	0.464128i	$0.846368\pi$
$912$	0	0
$913$	−25442.8	−0.922273
$914$	0	0
$915$	78.7936	0.00284682
$916$	0	0
$917$	19573.4	0.704876
$918$	0	0
$919$	−1030.47	−0.0369883	−0.0184941	−	0.999829i	$-0.505887\pi$
$919$	−1030.47	−0.0369883	−0.0184941	+	0.999829i	$0.505887\pi$
$920$	0	0
$921$	28440.4	1.01753
$922$	0	0
$923$	2950.40	0.105215
$924$	0	0
$925$	−4828.02	−0.171615
$926$	0	0
$927$	−11897.9	−0.421553
$928$	0	0
$929$	879.756	0.0310698	0.0155349	−	0.999879i	$-0.495055\pi$
$929$	879.756	0.0310698	0.0155349	+	0.999879i	$0.495055\pi$
$930$	0	0
$931$	266.354	0.00937638
$932$	0	0
$933$	21234.0	0.745092
$934$	0	0
$935$	−619.339	−0.0216626
$936$	0	0
$937$	−18668.1	−0.650864	−0.325432	−	0.945565i	$-0.605510\pi$
$937$	−18668.1	−0.650864	−0.325432	+	0.945565i	$0.605510\pi$
$938$	0	0
$939$	−16781.5	−0.583221
$940$	0	0
$941$	29613.4	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$941$	29613.4	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$942$	0	0
$943$	−15559.5	−0.537313
$944$	0	0
$945$	−945.000	−0.0325300
$946$	0	0
$947$	−20738.9	−0.711640	−0.355820	−	0.934554i	$-0.615798\pi$
$947$	−20738.9	−0.711640	−0.355820	+	0.934554i	$0.615798\pi$
$948$	0	0
$949$	2959.15	0.101220
$950$	0	0
$951$	−10703.4	−0.364966
$952$	0	0
$953$	−45776.5	−1.55598	−0.777988	−	0.628279i	$-0.783760\pi$
$953$	−45776.5	−1.55598	−0.777988	+	0.628279i	$0.783760\pi$
$954$	0	0
$955$	1220.37	0.0413510
$956$	0	0
$957$	16669.6	0.563064
$958$	0	0
$959$	−14399.9	−0.484878
$960$	0	0
$961$	29895.1	1.00349
$962$	0	0
$963$	−15711.4	−0.525746
$964$	0	0
$965$	−8665.17	−0.289059
$966$	0	0
$967$	−34461.0	−1.14601	−0.573005	−	0.819552i	$-0.694222\pi$
$967$	−34461.0	−1.14601	−0.573005	+	0.819552i	$0.694222\pi$
$968$	0	0
$969$	105.967	0.00351304
$970$	0	0
$971$	22762.8	0.752309	0.376154	−	0.926557i	$-0.377246\pi$
$971$	22762.8	0.752309	0.376154	+	0.926557i	$0.377246\pi$
$972$	0	0
$973$	12215.9	0.402490
$974$	0	0
$975$	220.331	0.00723716
$976$	0	0
$977$	4809.57	0.157494	0.0787470	−	0.996895i	$-0.474908\pi$
$977$	4809.57	0.157494	0.0787470	+	0.996895i	$0.474908\pi$
$978$	0	0
$979$	6219.04	0.203025
$980$	0	0
$981$	4291.70	0.139677
$982$	0	0
$983$	27591.6	0.895256	0.447628	−	0.894220i	$-0.352269\pi$
$983$	27591.6	0.895256	0.447628	+	0.894220i	$0.352269\pi$
$984$	0	0
$985$	1791.15	0.0579399
$986$	0	0
$987$	1816.62	0.0585853
$988$	0	0
$989$	−14862.4	−0.477854
$990$	0	0
$991$	22263.4	0.713643	0.356822	−	0.934173i	$-0.383860\pi$
$991$	22263.4	0.713643	0.356822	+	0.934173i	$0.383860\pi$
$992$	0	0
$993$	−13169.0	−0.420851
$994$	0	0
$995$	16018.2	0.510361
$996$	0	0
$997$	30378.2	0.964983	0.482491	−	0.875901i	$-0.339732\pi$
$997$	30378.2	0.964983	0.482491	+	0.875901i	$0.339732\pi$
$998$	0	0
$999$	5214.26	0.165137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.bg.1.2		2
4.3	odd	2		105.4.a.f.1.2	✓	2
12.11	even	2		315.4.a.i.1.1		2
20.3	even	4		525.4.d.h.274.1		4
20.7	even	4		525.4.d.h.274.4		4
20.19	odd	2		525.4.a.k.1.1		2
28.27	even	2		735.4.a.p.1.2		2
60.59	even	2		1575.4.a.w.1.2		2
84.83	odd	2		2205.4.a.z.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.2	✓	2	4.3	odd	2
315.4.a.i.1.1		2	12.11	even	2
525.4.a.k.1.1		2	20.19	odd	2
525.4.d.h.274.1		4	20.3	even	4
525.4.d.h.274.4		4	20.7	even	4
735.4.a.p.1.2		2	28.27	even	2
1575.4.a.w.1.2		2	60.59	even	2
1680.4.a.bg.1.2		2	1.1	even	1	trivial
2205.4.a.z.1.1		2	84.83	odd	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +19.0623 q^{11} -2.93774 q^{13} -15.0000 q^{15} -6.49806 q^{17} +5.43580 q^{19} -21.0000 q^{21} -49.3774 q^{23} +25.0000 q^{25} -27.0000 q^{27} -291.494 q^{29} -244.307 q^{31} -57.1868 q^{33} +35.0000 q^{35} -193.121 q^{37} +8.81323 q^{39} +315.113 q^{41} +300.996 q^{43} +45.0000 q^{45} -86.5058 q^{47} +49.0000 q^{49} +19.4942 q^{51} +509.677 q^{53} +95.3113 q^{55} -16.3074 q^{57} +83.3852 q^{59} -5.25291 q^{61} +63.0000 q^{63} -14.6887 q^{65} -205.992 q^{67} +148.132 q^{69} -1004.31 q^{71} -1007.29 q^{73} -75.0000 q^{75} +133.436 q^{77} +863.237 q^{79} +81.0000 q^{81} -1334.72 q^{83} -32.4903 q^{85} +874.483 q^{87} +326.249 q^{89} -20.5642 q^{91} +732.922 q^{93} +27.1790 q^{95} +1526.77 q^{97} +171.560 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} - 22 q^{13} - 30 q^{15} + 116 q^{17} - 102 q^{19} - 42 q^{21} - 260 q^{23} + 50 q^{25} - 54 q^{27} - 196 q^{29} - 150 q^{31} - 66 q^{33} + 70 q^{35} - 96 q^{37} + 66 q^{39} - 176 q^{41} + 344 q^{43} + 90 q^{45} - 560 q^{47} + 98 q^{49} - 348 q^{51} + 326 q^{53} + 110 q^{55} + 306 q^{57} + 844 q^{59} - 204 q^{61} + 126 q^{63} - 110 q^{65} + 104 q^{67} + 780 q^{69} - 1670 q^{71} - 386 q^{73} - 150 q^{75} + 154 q^{77} + 888 q^{79} + 162 q^{81} - 928 q^{83} + 580 q^{85} + 588 q^{87} + 588 q^{89} - 154 q^{91} + 450 q^{93} - 510 q^{95} + 522 q^{97} + 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 - 22 * q^13 - 30 * q^15 + 116 * q^17 - 102 * q^19 - 42 * q^21 - 260 * q^23 + 50 * q^25 - 54 * q^27 - 196 * q^29 - 150 * q^31 - 66 * q^33 + 70 * q^35 - 96 * q^37 + 66 * q^39 - 176 * q^41 + 344 * q^43 + 90 * q^45 - 560 * q^47 + 98 * q^49 - 348 * q^51 + 326 * q^53 + 110 * q^55 + 306 * q^57 + 844 * q^59 - 204 * q^61 + 126 * q^63 - 110 * q^65 + 104 * q^67 + 780 * q^69 - 1670 * q^71 - 386 * q^73 - 150 * q^75 + 154 * q^77 + 888 * q^79 + 162 * q^81 - 928 * q^83 + 580 * q^85 + 588 * q^87 + 588 * q^89 - 154 * q^91 + 450 * q^93 - 510 * q^95 + 522 * q^97 + 198 * q^99

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