LMFDB - Embedded newform 560.6.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.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:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 1151x^{3} - 5642x^{2} + 193596x + 1258056 $ x^5 - 1151x^3 - 5642x^2 + 193596*x + 1258056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-27.3320$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-30.3320 q^{3} +25.0000 q^{5} -49.0000 q^{7} +677.031 q^{9} +O(q^{10})$	$q-30.3320 q^{3} +25.0000 q^{5} -49.0000 q^{7} +677.031 q^{9} -604.346 q^{11} +383.964 q^{13} -758.300 q^{15} -791.640 q^{17} +2078.78 q^{19} +1486.27 q^{21} +2523.45 q^{23} +625.000 q^{25} -13165.0 q^{27} +372.040 q^{29} +2484.87 q^{31} +18331.0 q^{33} -1225.00 q^{35} -6552.46 q^{37} -11646.4 q^{39} -16194.7 q^{41} -15433.1 q^{43} +16925.8 q^{45} -22278.8 q^{47} +2401.00 q^{49} +24012.0 q^{51} -26272.4 q^{53} -15108.6 q^{55} -63053.6 q^{57} -39869.1 q^{59} +5893.86 q^{61} -33174.5 q^{63} +9599.11 q^{65} +32074.7 q^{67} -76541.3 q^{69} +57992.0 q^{71} +39480.1 q^{73} -18957.5 q^{75} +29612.9 q^{77} +8987.90 q^{79} +234804. q^{81} +52523.4 q^{83} -19791.0 q^{85} -11284.7 q^{87} +52068.7 q^{89} -18814.3 q^{91} -75371.1 q^{93} +51969.5 q^{95} +14928.3 q^{97} -409161. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} + 125 q^{5} - 245 q^{7} + 1132 q^{9}+O(q^{10}) $ 5 * q - 15 * q^3 + 125 * q^5 - 245 * q^7 + 1132 * q^9	$ 5 q - 15 q^{3} + 125 q^{5} - 245 q^{7} + 1132 q^{9} - 281 q^{11} + 909 q^{13} - 375 q^{15} + 1495 q^{17} + 422 q^{19} + 735 q^{21} + 62 q^{23} + 3125 q^{25} + 3363 q^{27} - 2047 q^{29} - 1636 q^{31} + 19181 q^{33} - 6125 q^{35} - 10358 q^{37} - 15685 q^{39} + 6424 q^{41} - 28306 q^{43} + 28300 q^{45} - 20955 q^{47} + 12005 q^{49} + 23577 q^{51} + 43748 q^{53} - 7025 q^{55} + 13690 q^{57} - 45788 q^{59} + 50432 q^{61} - 55468 q^{63} + 22725 q^{65} - 40712 q^{67} + 35050 q^{69} + 3096 q^{71} + 135438 q^{73} - 9375 q^{75} + 13769 q^{77} - 13191 q^{79} + 381101 q^{81} - 35108 q^{83} + 37375 q^{85} - 297289 q^{87} + 213772 q^{89} - 44541 q^{91} + 134244 q^{93} + 10550 q^{95} + 10659 q^{97} - 39462 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 + 125 * q^5 - 245 * q^7 + 1132 * q^9 - 281 * q^11 + 909 * q^13 - 375 * q^15 + 1495 * q^17 + 422 * q^19 + 735 * q^21 + 62 * q^23 + 3125 * q^25 + 3363 * q^27 - 2047 * q^29 - 1636 * q^31 + 19181 * q^33 - 6125 * q^35 - 10358 * q^37 - 15685 * q^39 + 6424 * q^41 - 28306 * q^43 + 28300 * q^45 - 20955 * q^47 + 12005 * q^49 + 23577 * q^51 + 43748 * q^53 - 7025 * q^55 + 13690 * q^57 - 45788 * q^59 + 50432 * q^61 - 55468 * q^63 + 22725 * q^65 - 40712 * q^67 + 35050 * q^69 + 3096 * q^71 + 135438 * q^73 - 9375 * q^75 + 13769 * q^77 - 13191 * q^79 + 381101 * q^81 - 35108 * q^83 + 37375 * q^85 - 297289 * q^87 + 213772 * q^89 - 44541 * q^91 + 134244 * q^93 + 10550 * q^95 + 10659 * q^97 - 39462 * 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
$3$	−30.3320	−1.94580	−0.972900	−	0.231227i	$-0.925726\pi$
$3$	−30.3320	−1.94580	−0.972900	+	0.231227i	$0.925726\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	677.031	2.78614
$10$	0	0
$11$	−604.346	−1.50593	−0.752963	−	0.658063i	$-0.771376\pi$
$11$	−604.346	−1.50593	−0.752963	+	0.658063i	$0.771376\pi$
$12$	0	0
$13$	383.964	0.630133	0.315067	−	0.949070i	$-0.397973\pi$
$13$	383.964	0.630133	0.315067	+	0.949070i	$0.397973\pi$
$14$	0	0
$15$	−758.300	−0.870188
$16$	0	0
$17$	−791.640	−0.664363	−0.332181	−	0.943216i	$-0.607785\pi$
$17$	−791.640	−0.664363	−0.332181	+	0.943216i	$0.607785\pi$
$18$	0	0
$19$	2078.78	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$19$	2078.78	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$20$	0	0
$21$	1486.27	0.735443
$22$	0	0
$23$	2523.45	0.994660	0.497330	−	0.867561i	$-0.334314\pi$
$23$	2523.45	0.994660	0.497330	+	0.867561i	$0.334314\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−13165.0	−3.47546
$28$	0	0
$29$	372.040	0.0821475	0.0410737	−	0.999156i	$-0.486922\pi$
$29$	372.040	0.0821475	0.0410737	+	0.999156i	$0.486922\pi$
$30$	0	0
$31$	2484.87	0.464408	0.232204	−	0.972667i	$-0.425406\pi$
$31$	2484.87	0.464408	0.232204	+	0.972667i	$0.425406\pi$
$32$	0	0
$33$	18331.0	2.93023
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−6552.46	−0.786865	−0.393433	−	0.919354i	$-0.628712\pi$
$37$	−6552.46	−0.786865	−0.393433	+	0.919354i	$0.628712\pi$
$38$	0	0
$39$	−11646.4	−1.22611
$40$	0	0
$41$	−16194.7	−1.50458	−0.752289	−	0.658834i	$-0.771050\pi$
$41$	−16194.7	−1.50458	−0.752289	+	0.658834i	$0.771050\pi$
$42$	0	0
$43$	−15433.1	−1.27287	−0.636433	−	0.771332i	$-0.719591\pi$
$43$	−15433.1	−1.27287	−0.636433	+	0.771332i	$0.719591\pi$
$44$	0	0
$45$	16925.8	1.24600
$46$	0	0
$47$	−22278.8	−1.47112	−0.735559	−	0.677460i	$-0.763081\pi$
$47$	−22278.8	−1.47112	−0.735559	+	0.677460i	$0.763081\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	24012.0	1.29272
$52$	0	0
$53$	−26272.4	−1.28472	−0.642362	−	0.766401i	$-0.722046\pi$
$53$	−26272.4	−1.28472	−0.642362	+	0.766401i	$0.722046\pi$
$54$	0	0
$55$	−15108.6	−0.673471
$56$	0	0
$57$	−63053.6	−2.57053
$58$	0	0
$59$	−39869.1	−1.49110	−0.745549	−	0.666451i	$-0.767813\pi$
$59$	−39869.1	−1.49110	−0.745549	+	0.666451i	$0.767813\pi$
$60$	0	0
$61$	5893.86	0.202804	0.101402	−	0.994846i	$-0.467667\pi$
$61$	5893.86	0.202804	0.101402	+	0.994846i	$0.467667\pi$
$62$	0	0
$63$	−33174.5	−1.05306
$64$	0	0
$65$	9599.11	0.281804
$66$	0	0
$67$	32074.7	0.872922	0.436461	−	0.899723i	$-0.356232\pi$
$67$	32074.7	0.872922	0.436461	+	0.899723i	$0.356232\pi$
$68$	0	0
$69$	−76541.3	−1.93541
$70$	0	0
$71$	57992.0	1.36528	0.682641	−	0.730754i	$-0.260831\pi$
$71$	57992.0	1.36528	0.682641	+	0.730754i	$0.260831\pi$
$72$	0	0
$73$	39480.1	0.867103	0.433552	−	0.901129i	$-0.357260\pi$
$73$	39480.1	0.867103	0.433552	+	0.901129i	$0.357260\pi$
$74$	0	0
$75$	−18957.5	−0.389160
$76$	0	0
$77$	29612.9	0.569187
$78$	0	0
$79$	8987.90	0.162028	0.0810141	−	0.996713i	$-0.474184\pi$
$79$	8987.90	0.162028	0.0810141	+	0.996713i	$0.474184\pi$
$80$	0	0
$81$	234804.	3.97642
$82$	0	0
$83$	52523.4	0.836870	0.418435	−	0.908247i	$-0.362579\pi$
$83$	52523.4	0.836870	0.418435	+	0.908247i	$0.362579\pi$
$84$	0	0
$85$	−19791.0	−0.297112
$86$	0	0
$87$	−11284.7	−0.159843
$88$	0	0
$89$	52068.7	0.696790	0.348395	−	0.937348i	$-0.386727\pi$
$89$	52068.7	0.696790	0.348395	+	0.937348i	$0.386727\pi$
$90$	0	0
$91$	−18814.3	−0.238168
$92$	0	0
$93$	−75371.1	−0.903645
$94$	0	0
$95$	51969.5	0.590799
$96$	0	0
$97$	14928.3	0.161094	0.0805472	−	0.996751i	$-0.474333\pi$
$97$	14928.3	0.161094	0.0805472	+	0.996751i	$0.474333\pi$
$98$	0	0
$99$	−409161.	−4.19572
$100$	0	0
$101$	−137155.	−1.33785	−0.668927	−	0.743328i	$-0.733246\pi$
$101$	−137155.	−1.33785	−0.668927	+	0.743328i	$0.733246\pi$
$102$	0	0
$103$	112201.	1.04208	0.521041	−	0.853532i	$-0.325544\pi$
$103$	112201.	1.04208	0.521041	+	0.853532i	$0.325544\pi$
$104$	0	0
$105$	37156.7	0.328900
$106$	0	0
$107$	−195834.	−1.65360	−0.826798	−	0.562499i	$-0.809840\pi$
$107$	−195834.	−1.65360	−0.826798	+	0.562499i	$0.809840\pi$
$108$	0	0
$109$	220762.	1.77974	0.889872	−	0.456210i	$-0.150793\pi$
$109$	220762.	1.77974	0.889872	+	0.456210i	$0.150793\pi$
$110$	0	0
$111$	198749.	1.53108
$112$	0	0
$113$	−51394.7	−0.378636	−0.189318	−	0.981916i	$-0.560628\pi$
$113$	−51394.7	−0.378636	−0.189318	+	0.981916i	$0.560628\pi$
$114$	0	0
$115$	63086.2	0.444826
$116$	0	0
$117$	259956.	1.75564
$118$	0	0
$119$	38790.4	0.251106
$120$	0	0
$121$	204183.	1.26781
$122$	0	0
$123$	491219.	2.92761
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	35496.6	0.195289	0.0976445	−	0.995221i	$-0.468869\pi$
$127$	35496.6	0.195289	0.0976445	+	0.995221i	$0.468869\pi$
$128$	0	0
$129$	468118.	2.47674
$130$	0	0
$131$	−130196.	−0.662857	−0.331428	−	0.943480i	$-0.607531\pi$
$131$	−130196.	−0.662857	−0.331428	+	0.943480i	$0.607531\pi$
$132$	0	0
$133$	−101860.	−0.499316
$134$	0	0
$135$	−329126.	−1.55427
$136$	0	0
$137$	−260457.	−1.18559	−0.592795	−	0.805353i	$-0.701976\pi$
$137$	−260457.	−1.18559	−0.592795	+	0.805353i	$0.701976\pi$
$138$	0	0
$139$	−91256.3	−0.400614	−0.200307	−	0.979733i	$-0.564194\pi$
$139$	−91256.3	−0.400614	−0.200307	+	0.979733i	$0.564194\pi$
$140$	0	0
$141$	675762.	2.86250
$142$	0	0
$143$	−232047.	−0.948935
$144$	0	0
$145$	9300.99	0.0367375
$146$	0	0
$147$	−72827.2	−0.277971
$148$	0	0
$149$	450638.	1.66289	0.831443	−	0.555610i	$-0.187515\pi$
$149$	450638.	1.66289	0.831443	+	0.555610i	$0.187515\pi$
$150$	0	0
$151$	322713.	1.15179	0.575896	−	0.817523i	$-0.304653\pi$
$151$	322713.	1.15179	0.575896	+	0.817523i	$0.304653\pi$
$152$	0	0
$153$	−535965.	−1.85101
$154$	0	0
$155$	62121.8	0.207690
$156$	0	0
$157$	−106359.	−0.344371	−0.172186	−	0.985065i	$-0.555083\pi$
$157$	−106359.	−0.344371	−0.172186	+	0.985065i	$0.555083\pi$
$158$	0	0
$159$	796895.	2.49982
$160$	0	0
$161$	−123649.	−0.375946
$162$	0	0
$163$	134910.	0.397719	0.198860	−	0.980028i	$-0.436276\pi$
$163$	134910.	0.397719	0.198860	+	0.980028i	$0.436276\pi$
$164$	0	0
$165$	458276.	1.31044
$166$	0	0
$167$	232478.	0.645047	0.322523	−	0.946561i	$-0.395469\pi$
$167$	232478.	0.645047	0.322523	+	0.946561i	$0.395469\pi$
$168$	0	0
$169$	−223864.	−0.602932
$170$	0	0
$171$	1.40740e6	3.68067
$172$	0	0
$173$	−150150.	−0.381427	−0.190713	−	0.981646i	$-0.561080\pi$
$173$	−150150.	−0.381427	−0.190713	+	0.981646i	$0.561080\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	1.20931e6	2.90138
$178$	0	0
$179$	537663.	1.25423	0.627115	−	0.778926i	$-0.284235\pi$
$179$	537663.	1.25423	0.627115	+	0.778926i	$0.284235\pi$
$180$	0	0
$181$	−474105.	−1.07567	−0.537834	−	0.843051i	$-0.680757\pi$
$181$	−474105.	−1.07567	−0.537834	+	0.843051i	$0.680757\pi$
$182$	0	0
$183$	−178773.	−0.394615
$184$	0	0
$185$	−163812.	−0.351897
$186$	0	0
$187$	478424.	1.00048
$188$	0	0
$189$	645087.	1.31360
$190$	0	0
$191$	991261.	1.96610	0.983048	−	0.183348i	$-0.0586937\pi$
$191$	991261.	1.96610	0.983048	+	0.183348i	$0.0586937\pi$
$192$	0	0
$193$	−398252.	−0.769598	−0.384799	−	0.923000i	$-0.625729\pi$
$193$	−398252.	−0.769598	−0.384799	+	0.923000i	$0.625729\pi$
$194$	0	0
$195$	−291160.	−0.548335
$196$	0	0
$197$	424086.	0.778553	0.389277	−	0.921121i	$-0.372725\pi$
$197$	424086.	0.778553	0.389277	+	0.921121i	$0.372725\pi$
$198$	0	0
$199$	178201.	0.318991	0.159495	−	0.987199i	$-0.449013\pi$
$199$	178201.	0.318991	0.159495	+	0.987199i	$0.449013\pi$
$200$	0	0
$201$	−972890.	−1.69853
$202$	0	0
$203$	−18229.9	−0.0310488
$204$	0	0
$205$	−404869.	−0.672867
$206$	0	0
$207$	1.70845e6	2.77126
$208$	0	0
$209$	−1.25630e6	−1.98943
$210$	0	0
$211$	−837166.	−1.29451	−0.647255	−	0.762273i	$-0.724083\pi$
$211$	−837166.	−1.29451	−0.647255	+	0.762273i	$0.724083\pi$
$212$	0	0
$213$	−1.75901e6	−2.65656
$214$	0	0
$215$	−385828.	−0.569243
$216$	0	0
$217$	−121759.	−0.175530
$218$	0	0
$219$	−1.19751e6	−1.68721
$220$	0	0
$221$	−303961.	−0.418637
$222$	0	0
$223$	545816.	0.734995	0.367497	−	0.930025i	$-0.380215\pi$
$223$	545816.	0.734995	0.367497	+	0.930025i	$0.380215\pi$
$224$	0	0
$225$	423145.	0.557227
$226$	0	0
$227$	−1.12689e6	−1.45151	−0.725753	−	0.687956i	$-0.758508\pi$
$227$	−1.12689e6	−1.45151	−0.725753	+	0.687956i	$0.758508\pi$
$228$	0	0
$229$	1.35322e6	1.70522	0.852610	−	0.522547i	$-0.175018\pi$
$229$	1.35322e6	1.70522	0.852610	+	0.522547i	$0.175018\pi$
$230$	0	0
$231$	−898220.	−1.10752
$232$	0	0
$233$	1.17172e6	1.41395	0.706975	−	0.707238i	$-0.250059\pi$
$233$	1.17172e6	1.41395	0.706975	+	0.707238i	$0.250059\pi$
$234$	0	0
$235$	−556971.	−0.657904
$236$	0	0
$237$	−272621.	−0.315274
$238$	0	0
$239$	1.63077e6	1.84671	0.923355	−	0.383946i	$-0.125435\pi$
$239$	1.63077e6	1.84671	0.923355	+	0.383946i	$0.125435\pi$
$240$	0	0
$241$	−459984.	−0.510153	−0.255076	−	0.966921i	$-0.582101\pi$
$241$	−459984.	−0.510153	−0.255076	+	0.966921i	$0.582101\pi$
$242$	0	0
$243$	−3.92296e6	−4.26185
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	798178.	0.832448
$248$	0	0
$249$	−1.59314e6	−1.62838
$250$	0	0
$251$	26214.6	0.0262639	0.0131319	−	0.999914i	$-0.495820\pi$
$251$	26214.6	0.0262639	0.0131319	+	0.999914i	$0.495820\pi$
$252$	0	0
$253$	−1.52504e6	−1.49789
$254$	0	0
$255$	600301.	0.578121
$256$	0	0
$257$	−1.10412e6	−1.04275	−0.521377	−	0.853326i	$-0.674582\pi$
$257$	−1.10412e6	−1.04275	−0.521377	+	0.853326i	$0.674582\pi$
$258$	0	0
$259$	321071.	0.297407
$260$	0	0
$261$	251882.	0.228874
$262$	0	0
$263$	−219876.	−0.196015	−0.0980074	−	0.995186i	$-0.531247\pi$
$263$	−219876.	−0.196015	−0.0980074	+	0.995186i	$0.531247\pi$
$264$	0	0
$265$	−656810.	−0.574546
$266$	0	0
$267$	−1.57935e6	−1.35581
$268$	0	0
$269$	1.85329e6	1.56157	0.780786	−	0.624799i	$-0.214819\pi$
$269$	1.85329e6	1.56157	0.780786	+	0.624799i	$0.214819\pi$
$270$	0	0
$271$	−1.70884e6	−1.41344	−0.706722	−	0.707492i	$-0.749827\pi$
$271$	−1.70884e6	−1.41344	−0.706722	+	0.707492i	$0.749827\pi$
$272$	0	0
$273$	570674.	0.463427
$274$	0	0
$275$	−377716.	−0.301185
$276$	0	0
$277$	188853.	0.147885	0.0739424	−	0.997263i	$-0.476442\pi$
$277$	188853.	0.147885	0.0739424	+	0.997263i	$0.476442\pi$
$278$	0	0
$279$	1.68233e6	1.29390
$280$	0	0
$281$	1.77467e6	1.34076	0.670380	−	0.742018i	$-0.266131\pi$
$281$	1.77467e6	1.34076	0.670380	+	0.742018i	$0.266131\pi$
$282$	0	0
$283$	−472156.	−0.350445	−0.175222	−	0.984529i	$-0.556064\pi$
$283$	−472156.	−0.350445	−0.175222	+	0.984529i	$0.556064\pi$
$284$	0	0
$285$	−1.57634e6	−1.14958
$286$	0	0
$287$	793543.	0.568677
$288$	0	0
$289$	−793163.	−0.558622
$290$	0	0
$291$	−452805.	−0.313458
$292$	0	0
$293$	−642119.	−0.436965	−0.218482	−	0.975841i	$-0.570111\pi$
$293$	−642119.	−0.436965	−0.218482	+	0.975841i	$0.570111\pi$
$294$	0	0
$295$	−996727.	−0.666839
$296$	0	0
$297$	7.95624e6	5.23379
$298$	0	0
$299$	968914.	0.626769
$300$	0	0
$301$	756223.	0.481098
$302$	0	0
$303$	4.16019e6	2.60320
$304$	0	0
$305$	147347.	0.0906965
$306$	0	0
$307$	2.54290e6	1.53987	0.769935	−	0.638123i	$-0.220289\pi$
$307$	2.54290e6	1.53987	0.769935	+	0.638123i	$0.220289\pi$
$308$	0	0
$309$	−3.40327e6	−2.02768
$310$	0	0
$311$	2.97181e6	1.74229	0.871143	−	0.491029i	$-0.163379\pi$
$311$	2.97181e6	1.74229	0.871143	+	0.491029i	$0.163379\pi$
$312$	0	0
$313$	41506.6	0.0239473	0.0119736	−	0.999928i	$-0.496189\pi$
$313$	41506.6	0.0239473	0.0119736	+	0.999928i	$0.496189\pi$
$314$	0	0
$315$	−829363.	−0.470943
$316$	0	0
$317$	762636.	0.426254	0.213127	−	0.977024i	$-0.431635\pi$
$317$	762636.	0.426254	0.213127	+	0.977024i	$0.431635\pi$
$318$	0	0
$319$	−224841.	−0.123708
$320$	0	0
$321$	5.94005e6	3.21757
$322$	0	0
$323$	−1.64565e6	−0.877668
$324$	0	0
$325$	239978.	0.126027
$326$	0	0
$327$	−6.69615e6	−3.46303
$328$	0	0
$329$	1.09166e6	0.556031
$330$	0	0
$331$	3.07851e6	1.54444	0.772219	−	0.635356i	$-0.219147\pi$
$331$	3.07851e6	1.54444	0.772219	+	0.635356i	$0.219147\pi$
$332$	0	0
$333$	−4.43622e6	−2.19231
$334$	0	0
$335$	801867.	0.390383
$336$	0	0
$337$	1.44180e6	0.691560	0.345780	−	0.938316i	$-0.387614\pi$
$337$	1.44180e6	0.691560	0.345780	+	0.938316i	$0.387614\pi$
$338$	0	0
$339$	1.55891e6	0.736751
$340$	0	0
$341$	−1.50172e6	−0.699364
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.91353e6	−0.865542
$346$	0	0
$347$	1.48182e6	0.660649	0.330324	−	0.943868i	$-0.392842\pi$
$347$	1.48182e6	0.660649	0.330324	+	0.943868i	$0.392842\pi$
$348$	0	0
$349$	−1.90464e6	−0.837046	−0.418523	−	0.908206i	$-0.637452\pi$
$349$	−1.90464e6	−0.837046	−0.418523	+	0.908206i	$0.637452\pi$
$350$	0	0
$351$	−5.05491e6	−2.19001
$352$	0	0
$353$	−298729.	−0.127597	−0.0637986	−	0.997963i	$-0.520322\pi$
$353$	−298729.	−0.127597	−0.0637986	+	0.997963i	$0.520322\pi$
$354$	0	0
$355$	1.44980e6	0.610573
$356$	0	0
$357$	−1.17659e6	−0.488601
$358$	0	0
$359$	3.36818e6	1.37930	0.689651	−	0.724142i	$-0.257764\pi$
$359$	3.36818e6	1.37930	0.689651	+	0.724142i	$0.257764\pi$
$360$	0	0
$361$	1.84523e6	0.745217
$362$	0	0
$363$	−6.19327e6	−2.46691
$364$	0	0
$365$	987002.	0.387780
$366$	0	0
$367$	23037.2	0.00892820	0.00446410	−	0.999990i	$-0.498579\pi$
$367$	23037.2	0.00892820	0.00446410	+	0.999990i	$0.498579\pi$
$368$	0	0
$369$	−1.09643e7	−4.19196
$370$	0	0
$371$	1.28735e6	0.485580
$372$	0	0
$373$	−1.71485e6	−0.638195	−0.319097	−	0.947722i	$-0.603380\pi$
$373$	−1.71485e6	−0.638195	−0.319097	+	0.947722i	$0.603380\pi$
$374$	0	0
$375$	−473938.	−0.174038
$376$	0	0
$377$	142850.	0.0517639
$378$	0	0
$379$	1.90452e6	0.681064	0.340532	−	0.940233i	$-0.389393\pi$
$379$	1.90452e6	0.681064	0.340532	+	0.940233i	$0.389393\pi$
$380$	0	0
$381$	−1.07668e6	−0.379993
$382$	0	0
$383$	−1.77535e6	−0.618424	−0.309212	−	0.950993i	$-0.600065\pi$
$383$	−1.77535e6	−0.618424	−0.309212	+	0.950993i	$0.600065\pi$
$384$	0	0
$385$	740323.	0.254548
$386$	0	0
$387$	−1.04487e7	−3.54638
$388$	0	0
$389$	−2.47325e6	−0.828693	−0.414346	−	0.910119i	$-0.635990\pi$
$389$	−2.47325e6	−0.828693	−0.414346	+	0.910119i	$0.635990\pi$
$390$	0	0
$391$	−1.99766e6	−0.660815
$392$	0	0
$393$	3.94911e6	1.28979
$394$	0	0
$395$	224697.	0.0724612
$396$	0	0
$397$	298950.	0.0951968	0.0475984	−	0.998867i	$-0.484843\pi$
$397$	298950.	0.0951968	0.0475984	+	0.998867i	$0.484843\pi$
$398$	0	0
$399$	3.08963e6	0.971569
$400$	0	0
$401$	3.60863e6	1.12068	0.560340	−	0.828263i	$-0.310671\pi$
$401$	3.60863e6	1.12068	0.560340	+	0.828263i	$0.310671\pi$
$402$	0	0
$403$	954102.	0.292639
$404$	0	0
$405$	5.87009e6	1.77831
$406$	0	0
$407$	3.95995e6	1.18496
$408$	0	0
$409$	−431739.	−0.127618	−0.0638092	−	0.997962i	$-0.520325\pi$
$409$	−431739.	−0.127618	−0.0638092	+	0.997962i	$0.520325\pi$
$410$	0	0
$411$	7.90019e6	2.30692
$412$	0	0
$413$	1.95358e6	0.563582
$414$	0	0
$415$	1.31309e6	0.374260
$416$	0	0
$417$	2.76799e6	0.779514
$418$	0	0
$419$	−6.10143e6	−1.69784	−0.848920	−	0.528522i	$-0.822747\pi$
$419$	−6.10143e6	−1.69784	−0.848920	+	0.528522i	$0.822747\pi$
$420$	0	0
$421$	768074.	0.211202	0.105601	−	0.994409i	$-0.466323\pi$
$421$	768074.	0.211202	0.105601	+	0.994409i	$0.466323\pi$
$422$	0	0
$423$	−1.50835e7	−4.09874
$424$	0	0
$425$	−494775.	−0.132873
$426$	0	0
$427$	−288799.	−0.0766525
$428$	0	0
$429$	7.03846e6	1.84644
$430$	0	0
$431$	1.21891e6	0.316067	0.158034	−	0.987434i	$-0.449485\pi$
$431$	1.21891e6	0.316067	0.158034	+	0.987434i	$0.449485\pi$
$432$	0	0
$433$	2.44392e6	0.626423	0.313211	−	0.949683i	$-0.398595\pi$
$433$	2.44392e6	0.626423	0.313211	+	0.949683i	$0.398595\pi$
$434$	0	0
$435$	−282118.	−0.0714838
$436$	0	0
$437$	5.24570e6	1.31401
$438$	0	0
$439$	−495858.	−0.122799	−0.0613997	−	0.998113i	$-0.519556\pi$
$439$	−495858.	−0.122799	−0.0613997	+	0.998113i	$0.519556\pi$
$440$	0	0
$441$	1.62555e6	0.398020
$442$	0	0
$443$	−5.75805e6	−1.39401	−0.697006	−	0.717066i	$-0.745485\pi$
$443$	−5.75805e6	−1.39401	−0.697006	+	0.717066i	$0.745485\pi$
$444$	0	0
$445$	1.30172e6	0.311614
$446$	0	0
$447$	−1.36688e7	−3.23564
$448$	0	0
$449$	4.74045e6	1.10970	0.554848	−	0.831952i	$-0.312777\pi$
$449$	4.74045e6	1.10970	0.554848	+	0.831952i	$0.312777\pi$
$450$	0	0
$451$	9.78723e6	2.26578
$452$	0	0
$453$	−9.78853e6	−2.24116
$454$	0	0
$455$	−470356.	−0.106512
$456$	0	0
$457$	−2.61547e6	−0.585814	−0.292907	−	0.956141i	$-0.594623\pi$
$457$	−2.61547e6	−0.585814	−0.292907	+	0.956141i	$0.594623\pi$
$458$	0	0
$459$	1.04220e7	2.30897
$460$	0	0
$461$	−110237.	−0.0241588	−0.0120794	−	0.999927i	$-0.503845\pi$
$461$	−110237.	−0.0241588	−0.0120794	+	0.999927i	$0.503845\pi$
$462$	0	0
$463$	1.53603e6	0.333003	0.166501	−	0.986041i	$-0.446753\pi$
$463$	1.53603e6	0.333003	0.166501	+	0.986041i	$0.446753\pi$
$464$	0	0
$465$	−1.88428e6	−0.404122
$466$	0	0
$467$	−4.37245e6	−0.927753	−0.463877	−	0.885900i	$-0.653542\pi$
$467$	−4.37245e6	−0.927753	−0.463877	+	0.885900i	$0.653542\pi$
$468$	0	0
$469$	−1.57166e6	−0.329933
$470$	0	0
$471$	3.22610e6	0.670078
$472$	0	0
$473$	9.32694e6	1.91684
$474$	0	0
$475$	1.29924e6	0.264213
$476$	0	0
$477$	−1.77872e7	−3.57942
$478$	0	0
$479$	−2.50712e6	−0.499271	−0.249635	−	0.968340i	$-0.580311\pi$
$479$	−2.50712e6	−0.499271	−0.249635	+	0.968340i	$0.580311\pi$
$480$	0	0
$481$	−2.51591e6	−0.495830
$482$	0	0
$483$	3.75052e6	0.731516
$484$	0	0
$485$	373207.	0.0720436
$486$	0	0
$487$	3.35121e6	0.640295	0.320147	−	0.947368i	$-0.396268\pi$
$487$	3.35121e6	0.640295	0.320147	+	0.947368i	$0.396268\pi$
$488$	0	0
$489$	−4.09211e6	−0.773882
$490$	0	0
$491$	−7.04673e6	−1.31912	−0.659559	−	0.751652i	$-0.729257\pi$
$491$	−7.04673e6	−1.31912	−0.659559	+	0.751652i	$0.729257\pi$
$492$	0	0
$493$	−294521.	−0.0545757
$494$	0	0
$495$	−1.02290e7	−1.87638
$496$	0	0
$497$	−2.84161e6	−0.516028
$498$	0	0
$499$	8.36969e6	1.50473	0.752364	−	0.658747i	$-0.228913\pi$
$499$	8.36969e6	1.50473	0.752364	+	0.658747i	$0.228913\pi$
$500$	0	0
$501$	−7.05153e6	−1.25513
$502$	0	0
$503$	−3.04525e6	−0.536665	−0.268333	−	0.963326i	$-0.586473\pi$
$503$	−3.04525e6	−0.536665	−0.268333	+	0.963326i	$0.586473\pi$
$504$	0	0
$505$	−3.42888e6	−0.598307
$506$	0	0
$507$	6.79026e6	1.17318
$508$	0	0
$509$	564309.	0.0965434	0.0482717	−	0.998834i	$-0.484629\pi$
$509$	564309.	0.0965434	0.0482717	+	0.998834i	$0.484629\pi$
$510$	0	0
$511$	−1.93452e6	−0.327734
$512$	0	0
$513$	−2.73672e7	−4.59132
$514$	0	0
$515$	2.80501e6	0.466033
$516$	0	0
$517$	1.34641e7	2.21540
$518$	0	0
$519$	4.55436e6	0.742180
$520$	0	0
$521$	−3.99644e6	−0.645029	−0.322515	−	0.946564i	$-0.604528\pi$
$521$	−3.99644e6	−0.645029	−0.322515	+	0.946564i	$0.604528\pi$
$522$	0	0
$523$	4.52509e6	0.723391	0.361695	−	0.932296i	$-0.382198\pi$
$523$	4.52509e6	0.723391	0.361695	+	0.932296i	$0.382198\pi$
$524$	0	0
$525$	928918.	0.147089
$526$	0	0
$527$	−1.96712e6	−0.308535
$528$	0	0
$529$	−68550.6	−0.0106505
$530$	0	0
$531$	−2.69926e7	−4.15440
$532$	0	0
$533$	−6.21821e6	−0.948085
$534$	0	0
$535$	−4.89586e6	−0.739511
$536$	0	0
$537$	−1.63084e7	−2.44048
$538$	0	0
$539$	−1.45103e6	−0.215132
$540$	0	0
$541$	−5.68445e6	−0.835018	−0.417509	−	0.908673i	$-0.637097\pi$
$541$	−5.68445e6	−0.835018	−0.417509	+	0.908673i	$0.637097\pi$
$542$	0	0
$543$	1.43806e7	2.09304
$544$	0	0
$545$	5.51904e6	0.795926
$546$	0	0
$547$	−7.82153e6	−1.11770	−0.558848	−	0.829270i	$-0.688756\pi$
$547$	−7.82153e6	−1.11770	−0.558848	+	0.829270i	$0.688756\pi$
$548$	0	0
$549$	3.99033e6	0.565038
$550$	0	0
$551$	773389.	0.108522
$552$	0	0
$553$	−440407.	−0.0612409
$554$	0	0
$555$	4.96874e6	0.684721
$556$	0	0
$557$	7.04122e6	0.961634	0.480817	−	0.876821i	$-0.340340\pi$
$557$	7.04122e6	0.961634	0.480817	+	0.876821i	$0.340340\pi$
$558$	0	0
$559$	−5.92577e6	−0.802076
$560$	0	0
$561$	−1.45116e7	−1.94674
$562$	0	0
$563$	7.83612e6	1.04191	0.520955	−	0.853584i	$-0.325576\pi$
$563$	7.83612e6	1.04191	0.520955	+	0.853584i	$0.325576\pi$
$564$	0	0
$565$	−1.28487e6	−0.169331
$566$	0	0
$567$	−1.15054e7	−1.50295
$568$	0	0
$569$	6.57277e6	0.851075	0.425537	−	0.904941i	$-0.360085\pi$
$569$	6.57277e6	0.851075	0.425537	+	0.904941i	$0.360085\pi$
$570$	0	0
$571$	881273.	0.113115	0.0565575	−	0.998399i	$-0.481988\pi$
$571$	881273.	0.113115	0.0565575	+	0.998399i	$0.481988\pi$
$572$	0	0
$573$	−3.00670e7	−3.82563
$574$	0	0
$575$	1.57716e6	0.198932
$576$	0	0
$577$	1.35985e7	1.70041	0.850203	−	0.526455i	$-0.176479\pi$
$577$	1.35985e7	1.70041	0.850203	+	0.526455i	$0.176479\pi$
$578$	0	0
$579$	1.20798e7	1.49748
$580$	0	0
$581$	−2.57365e6	−0.316307
$582$	0	0
$583$	1.58776e7	1.93470
$584$	0	0
$585$	6.49890e6	0.785145
$586$	0	0
$587$	5.56764e6	0.666923	0.333462	−	0.942764i	$-0.391783\pi$
$587$	5.56764e6	0.666923	0.333462	+	0.942764i	$0.391783\pi$
$588$	0	0
$589$	5.16550e6	0.613514
$590$	0	0
$591$	−1.28634e7	−1.51491
$592$	0	0
$593$	4.27729e6	0.499496	0.249748	−	0.968311i	$-0.419652\pi$
$593$	4.27729e6	0.499496	0.249748	+	0.968311i	$0.419652\pi$
$594$	0	0
$595$	969759.	0.112298
$596$	0	0
$597$	−5.40520e6	−0.620692
$598$	0	0
$599$	−6.53192e6	−0.743830	−0.371915	−	0.928267i	$-0.621299\pi$
$599$	−6.53192e6	−0.743830	−0.371915	+	0.928267i	$0.621299\pi$
$600$	0	0
$601$	−3.26670e6	−0.368913	−0.184456	−	0.982841i	$-0.559052\pi$
$601$	−3.26670e6	−0.368913	−0.184456	+	0.982841i	$0.559052\pi$
$602$	0	0
$603$	2.17156e7	2.43208
$604$	0	0
$605$	5.10457e6	0.566984
$606$	0	0
$607$	2.17041e6	0.239094	0.119547	−	0.992829i	$-0.461856\pi$
$607$	2.17041e6	0.239094	0.119547	+	0.992829i	$0.461856\pi$
$608$	0	0
$609$	552951.	0.0604148
$610$	0	0
$611$	−8.55427e6	−0.927001
$612$	0	0
$613$	9.89798e6	1.06389	0.531943	−	0.846780i	$-0.321462\pi$
$613$	9.89798e6	1.06389	0.531943	+	0.846780i	$0.321462\pi$
$614$	0	0
$615$	1.22805e7	1.30927
$616$	0	0
$617$	1.68570e7	1.78266	0.891329	−	0.453357i	$-0.149774\pi$
$617$	1.68570e7	1.78266	0.891329	+	0.453357i	$0.149774\pi$
$618$	0	0
$619$	−1.38025e7	−1.44787	−0.723935	−	0.689868i	$-0.757669\pi$
$619$	−1.38025e7	−1.44787	−0.723935	+	0.689868i	$0.757669\pi$
$620$	0	0
$621$	−3.32213e7	−3.45691
$622$	0	0
$623$	−2.55137e6	−0.263362
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	3.81062e7	3.87103
$628$	0	0
$629$	5.18719e6	0.522764
$630$	0	0
$631$	9.76100e6	0.975935	0.487967	−	0.872862i	$-0.337738\pi$
$631$	9.76100e6	0.975935	0.487967	+	0.872862i	$0.337738\pi$
$632$	0	0
$633$	2.53929e7	2.51886
$634$	0	0
$635$	887415.	0.0873359
$636$	0	0
$637$	921898.	0.0900191
$638$	0	0
$639$	3.92624e7	3.80386
$640$	0	0
$641$	−1.72746e7	−1.66059	−0.830295	−	0.557324i	$-0.811828\pi$
$641$	−1.72746e7	−1.66059	−0.830295	+	0.557324i	$0.811828\pi$
$642$	0	0
$643$	3.88097e6	0.370180	0.185090	−	0.982722i	$-0.440742\pi$
$643$	3.88097e6	0.370180	0.185090	+	0.982722i	$0.440742\pi$
$644$	0	0
$645$	1.17029e7	1.10763
$646$	0	0
$647$	−1.53463e7	−1.44126	−0.720632	−	0.693318i	$-0.756148\pi$
$647$	−1.53463e7	−1.44126	−0.720632	+	0.693318i	$0.756148\pi$
$648$	0	0
$649$	2.40947e7	2.24548
$650$	0	0
$651$	3.69319e6	0.341546
$652$	0	0
$653$	−436146.	−0.0400266	−0.0200133	−	0.999800i	$-0.506371\pi$
$653$	−436146.	−0.0400266	−0.0200133	+	0.999800i	$0.506371\pi$
$654$	0	0
$655$	−3.25490e6	−0.296439
$656$	0	0
$657$	2.67292e7	2.41587
$658$	0	0
$659$	6.04128e6	0.541895	0.270948	−	0.962594i	$-0.412663\pi$
$659$	6.04128e6	0.541895	0.270948	+	0.962594i	$0.412663\pi$
$660$	0	0
$661$	5.67242e6	0.504969	0.252485	−	0.967601i	$-0.418752\pi$
$661$	5.67242e6	0.504969	0.252485	+	0.967601i	$0.418752\pi$
$662$	0	0
$663$	9.21977e6	0.814584
$664$	0	0
$665$	−2.54651e6	−0.223301
$666$	0	0
$667$	938823.	0.0817088
$668$	0	0
$669$	−1.65557e7	−1.43015
$670$	0	0
$671$	−3.56193e6	−0.305407
$672$	0	0
$673$	−585604.	−0.0498387	−0.0249193	−	0.999689i	$-0.507933\pi$
$673$	−585604.	−0.0498387	−0.0249193	+	0.999689i	$0.507933\pi$
$674$	0	0
$675$	−8.22815e6	−0.695093
$676$	0	0
$677$	3.94361e6	0.330691	0.165346	−	0.986236i	$-0.447126\pi$
$677$	3.94361e6	0.330691	0.165346	+	0.986236i	$0.447126\pi$
$678$	0	0
$679$	−731486.	−0.0608880
$680$	0	0
$681$	3.41810e7	2.82434
$682$	0	0
$683$	−1.30389e7	−1.06952	−0.534761	−	0.845003i	$-0.679598\pi$
$683$	−1.30389e7	−1.06952	−0.534761	+	0.845003i	$0.679598\pi$
$684$	0	0
$685$	−6.51143e6	−0.530212
$686$	0	0
$687$	−4.10460e7	−3.31802
$688$	0	0
$689$	−1.00877e7	−0.809548
$690$	0	0
$691$	−1.39341e7	−1.11016	−0.555079	−	0.831798i	$-0.687312\pi$
$691$	−1.39341e7	−1.11016	−0.555079	+	0.831798i	$0.687312\pi$
$692$	0	0
$693$	2.00489e7	1.58583
$694$	0	0
$695$	−2.28141e6	−0.179160
$696$	0	0
$697$	1.28204e7	0.999585
$698$	0	0
$699$	−3.55406e7	−2.75126
$700$	0	0
$701$	2.17868e7	1.67455	0.837277	−	0.546779i	$-0.184146\pi$
$701$	2.17868e7	1.67455	0.837277	+	0.546779i	$0.184146\pi$
$702$	0	0
$703$	−1.36211e7	−1.03950
$704$	0	0
$705$	1.68940e7	1.28015
$706$	0	0
$707$	6.72060e6	0.505661
$708$	0	0
$709$	9.10065e6	0.679919	0.339959	−	0.940440i	$-0.389587\pi$
$709$	9.10065e6	0.679919	0.339959	+	0.940440i	$0.389587\pi$
$710$	0	0
$711$	6.08509e6	0.451432
$712$	0	0
$713$	6.27044e6	0.461928
$714$	0	0
$715$	−5.80118e6	−0.424376
$716$	0	0
$717$	−4.94646e7	−3.59333
$718$	0	0
$719$	−2.49384e7	−1.79906	−0.899532	−	0.436854i	$-0.856093\pi$
$719$	−2.49384e7	−1.79906	−0.899532	+	0.436854i	$0.856093\pi$
$720$	0	0
$721$	−5.49783e6	−0.393870
$722$	0	0
$723$	1.39522e7	0.992655
$724$	0	0
$725$	232525.	0.0164295
$726$	0	0
$727$	2.51596e7	1.76550	0.882750	−	0.469843i	$-0.155690\pi$
$727$	2.51596e7	1.76550	0.882750	+	0.469843i	$0.155690\pi$
$728$	0	0
$729$	6.19341e7	4.31629
$730$	0	0
$731$	1.22175e7	0.845645
$732$	0	0
$733$	4.35013e6	0.299049	0.149524	−	0.988758i	$-0.452226\pi$
$733$	4.35013e6	0.299049	0.149524	+	0.988758i	$0.452226\pi$
$734$	0	0
$735$	−1.82068e6	−0.124313
$736$	0	0
$737$	−1.93842e7	−1.31456
$738$	0	0
$739$	2.16737e7	1.45990	0.729948	−	0.683502i	$-0.239544\pi$
$739$	2.16737e7	1.45990	0.729948	+	0.683502i	$0.239544\pi$
$740$	0	0
$741$	−2.42103e7	−1.61978
$742$	0	0
$743$	1.05226e7	0.699281	0.349641	−	0.936884i	$-0.386304\pi$
$743$	1.05226e7	0.699281	0.349641	+	0.936884i	$0.386304\pi$
$744$	0	0
$745$	1.12660e7	0.743665
$746$	0	0
$747$	3.55600e7	2.33163
$748$	0	0
$749$	9.59588e6	0.625001
$750$	0	0
$751$	−2.04846e7	−1.32534	−0.662670	−	0.748912i	$-0.730577\pi$
$751$	−2.04846e7	−1.32534	−0.662670	+	0.748912i	$0.730577\pi$
$752$	0	0
$753$	−795142.	−0.0511042
$754$	0	0
$755$	8.06782e6	0.515097
$756$	0	0
$757$	4.48469e6	0.284441	0.142221	−	0.989835i	$-0.454576\pi$
$757$	4.48469e6	0.284441	0.142221	+	0.989835i	$0.454576\pi$
$758$	0	0
$759$	4.62574e7	2.91458
$760$	0	0
$761$	−3.92897e6	−0.245933	−0.122967	−	0.992411i	$-0.539241\pi$
$761$	−3.92897e6	−0.245933	−0.122967	+	0.992411i	$0.539241\pi$
$762$	0	0
$763$	−1.08173e7	−0.672680
$764$	0	0
$765$	−1.33991e7	−0.827795
$766$	0	0
$767$	−1.53083e7	−0.939590
$768$	0	0
$769$	1.23798e6	0.0754915	0.0377458	−	0.999287i	$-0.487982\pi$
$769$	1.23798e6	0.0754915	0.0377458	+	0.999287i	$0.487982\pi$
$770$	0	0
$771$	3.34901e7	2.02899
$772$	0	0
$773$	−4.21618e6	−0.253788	−0.126894	−	0.991916i	$-0.540501\pi$
$773$	−4.21618e6	−0.253788	−0.126894	+	0.991916i	$0.540501\pi$
$774$	0	0
$775$	1.55304e6	0.0928816
$776$	0	0
$777$	−9.73872e6	−0.578695
$778$	0	0
$779$	−3.36653e7	−1.98765
$780$	0	0
$781$	−3.50472e7	−2.05601
$782$	0	0
$783$	−4.89792e6	−0.285501
$784$	0	0
$785$	−2.65899e6	−0.154008
$786$	0	0
$787$	2.70556e7	1.55711	0.778557	−	0.627574i	$-0.215952\pi$
$787$	2.70556e7	1.55711	0.778557	+	0.627574i	$0.215952\pi$
$788$	0	0
$789$	6.66929e6	0.381406
$790$	0	0
$791$	2.51834e6	0.143111
$792$	0	0
$793$	2.26303e6	0.127793
$794$	0	0
$795$	1.99224e7	1.11795
$796$	0	0
$797$	2.70075e7	1.50605	0.753024	−	0.657993i	$-0.228594\pi$
$797$	2.70075e7	1.50605	0.753024	+	0.657993i	$0.228594\pi$
$798$	0	0
$799$	1.76368e7	0.977356
$800$	0	0
$801$	3.52522e7	1.94135
$802$	0	0
$803$	−2.38596e7	−1.30579
$804$	0	0
$805$	−3.09122e6	−0.168128
$806$	0	0
$807$	−5.62139e7	−3.03850
$808$	0	0
$809$	−1.12425e7	−0.603937	−0.301969	−	0.953318i	$-0.597644\pi$
$809$	−1.12425e7	−0.603937	−0.301969	+	0.953318i	$0.597644\pi$
$810$	0	0
$811$	2.73965e7	1.46266	0.731331	−	0.682023i	$-0.238900\pi$
$811$	2.73965e7	1.46266	0.731331	+	0.682023i	$0.238900\pi$
$812$	0	0
$813$	5.18326e7	2.75028
$814$	0	0
$815$	3.37276e6	0.177865
$816$	0	0
$817$	−3.20821e7	−1.68154
$818$	0	0
$819$	−1.27378e7	−0.663569
$820$	0	0
$821$	2.83040e7	1.46551	0.732756	−	0.680491i	$-0.238233\pi$
$821$	2.83040e7	1.46551	0.732756	+	0.680491i	$0.238233\pi$
$822$	0	0
$823$	1.63941e7	0.843701	0.421850	−	0.906665i	$-0.361381\pi$
$823$	1.63941e7	0.843701	0.421850	+	0.906665i	$0.361381\pi$
$824$	0	0
$825$	1.14569e7	0.586046
$826$	0	0
$827$	−3.47958e7	−1.76914	−0.884570	−	0.466407i	$-0.845548\pi$
$827$	−3.47958e7	−1.76914	−0.884570	+	0.466407i	$0.845548\pi$
$828$	0	0
$829$	−1.57884e7	−0.797905	−0.398952	−	0.916972i	$-0.630626\pi$
$829$	−1.57884e7	−0.797905	−0.398952	+	0.916972i	$0.630626\pi$
$830$	0	0
$831$	−5.72828e6	−0.287754
$832$	0	0
$833$	−1.90073e6	−0.0949090
$834$	0	0
$835$	5.81196e6	0.288474
$836$	0	0
$837$	−3.27134e7	−1.61403
$838$	0	0
$839$	1.22890e7	0.602714	0.301357	−	0.953511i	$-0.402560\pi$
$839$	1.22890e7	0.602714	0.301357	+	0.953511i	$0.402560\pi$
$840$	0	0
$841$	−2.03727e7	−0.993252
$842$	0	0
$843$	−5.38292e7	−2.60885
$844$	0	0
$845$	−5.59661e6	−0.269639
$846$	0	0
$847$	−1.00050e7	−0.479189
$848$	0	0
$849$	1.43214e7	0.681895
$850$	0	0
$851$	−1.65348e7	−0.782664
$852$	0	0
$853$	3.39815e7	1.59908	0.799539	−	0.600615i	$-0.205077\pi$
$853$	3.39815e7	1.59908	0.799539	+	0.600615i	$0.205077\pi$
$854$	0	0
$855$	3.51850e7	1.64605
$856$	0	0
$857$	7.74121e6	0.360045	0.180022	−	0.983662i	$-0.442383\pi$
$857$	7.74121e6	0.360045	0.180022	+	0.983662i	$0.442383\pi$
$858$	0	0
$859$	3.03603e7	1.40386	0.701930	−	0.712246i	$-0.252322\pi$
$859$	3.03603e7	1.40386	0.701930	+	0.712246i	$0.252322\pi$
$860$	0	0
$861$	−2.40697e7	−1.10653
$862$	0	0
$863$	3.02445e7	1.38235	0.691177	−	0.722686i	$-0.257093\pi$
$863$	3.02445e7	1.38235	0.691177	+	0.722686i	$0.257093\pi$
$864$	0	0
$865$	−3.75376e6	−0.170579
$866$	0	0
$867$	2.40582e7	1.08697
$868$	0	0
$869$	−5.43180e6	−0.244002
$870$	0	0
$871$	1.23155e7	0.550057
$872$	0	0
$873$	1.01069e7	0.448831
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	2.48236e7	1.08985	0.544924	−	0.838485i	$-0.316558\pi$
$877$	2.48236e7	1.08985	0.544924	+	0.838485i	$0.316558\pi$
$878$	0	0
$879$	1.94768e7	0.850246
$880$	0	0
$881$	4.40667e7	1.91281	0.956403	−	0.292050i	$-0.0943374\pi$
$881$	4.40667e7	1.91281	0.956403	+	0.292050i	$0.0943374\pi$
$882$	0	0
$883$	3.46407e6	0.149515	0.0747574	−	0.997202i	$-0.476182\pi$
$883$	3.46407e6	0.149515	0.0747574	+	0.997202i	$0.476182\pi$
$884$	0	0
$885$	3.02327e7	1.29754
$886$	0	0
$887$	−1.67314e7	−0.714039	−0.357020	−	0.934097i	$-0.616207\pi$
$887$	−1.67314e7	−0.714039	−0.357020	+	0.934097i	$0.616207\pi$
$888$	0	0
$889$	−1.73933e6	−0.0738123
$890$	0	0
$891$	−1.41903e8	−5.98820
$892$	0	0
$893$	−4.63128e7	−1.94345
$894$	0	0
$895$	1.34416e7	0.560909
$896$	0	0
$897$	−2.93891e7	−1.21957
$898$	0	0
$899$	924470.	0.0381499
$900$	0	0
$901$	2.07983e7	0.853523
$902$	0	0
$903$	−2.29378e7	−0.936121
$904$	0	0
$905$	−1.18526e7	−0.481054
$906$	0	0
$907$	−1.76952e7	−0.714227	−0.357114	−	0.934061i	$-0.616239\pi$
$907$	−1.76952e7	−0.714227	−0.357114	+	0.934061i	$0.616239\pi$
$908$	0	0
$909$	−9.28583e7	−3.72744
$910$	0	0
$911$	−4.39235e6	−0.175348	−0.0876740	−	0.996149i	$-0.527943\pi$
$911$	−4.39235e6	−0.175348	−0.0876740	+	0.996149i	$0.527943\pi$
$912$	0	0
$913$	−3.17423e7	−1.26026
$914$	0	0
$915$	−4.46932e6	−0.176477
$916$	0	0
$917$	6.37961e6	0.250536
$918$	0	0
$919$	−2.35758e7	−0.920825	−0.460412	−	0.887705i	$-0.652298\pi$
$919$	−2.35758e7	−0.920825	−0.460412	+	0.887705i	$0.652298\pi$
$920$	0	0
$921$	−7.71314e7	−2.99628
$922$	0	0
$923$	2.22669e7	0.860310
$924$	0	0
$925$	−4.09529e6	−0.157373
$926$	0	0
$927$	7.59633e7	2.90338
$928$	0	0
$929$	2.38326e7	0.906007	0.453004	−	0.891509i	$-0.350352\pi$
$929$	2.38326e7	0.906007	0.453004	+	0.891509i	$0.350352\pi$
$930$	0	0
$931$	4.99115e6	0.188724
$932$	0	0
$933$	−9.01409e7	−3.39014
$934$	0	0
$935$	1.19606e7	0.447429
$936$	0	0
$937$	2.52434e6	0.0939288	0.0469644	−	0.998897i	$-0.485045\pi$
$937$	2.52434e6	0.0939288	0.0469644	+	0.998897i	$0.485045\pi$
$938$	0	0
$939$	−1.25898e6	−0.0465966
$940$	0	0
$941$	3.95629e7	1.45651	0.728257	−	0.685305i	$-0.240331\pi$
$941$	3.95629e7	1.45651	0.728257	+	0.685305i	$0.240331\pi$
$942$	0	0
$943$	−4.08666e7	−1.49654
$944$	0	0
$945$	1.61272e7	0.587461
$946$	0	0
$947$	1.96005e7	0.710218	0.355109	−	0.934825i	$-0.384444\pi$
$947$	1.96005e7	0.710218	0.355109	+	0.934825i	$0.384444\pi$
$948$	0	0
$949$	1.51589e7	0.546391
$950$	0	0
$951$	−2.31323e7	−0.829406
$952$	0	0
$953$	1.69856e7	0.605828	0.302914	−	0.953018i	$-0.402041\pi$
$953$	1.69856e7	0.605828	0.302914	+	0.953018i	$0.402041\pi$
$954$	0	0
$955$	2.47815e7	0.879265
$956$	0	0
$957$	6.81987e6	0.240711
$958$	0	0
$959$	1.27624e7	0.448111
$960$	0	0
$961$	−2.24546e7	−0.784325
$962$	0	0
$963$	−1.32586e8	−4.60715
$964$	0	0
$965$	−9.95629e6	−0.344175
$966$	0	0
$967$	−1.70160e7	−0.585184	−0.292592	−	0.956237i	$-0.594518\pi$
$967$	−1.70160e7	−0.585184	−0.292592	+	0.956237i	$0.594518\pi$
$968$	0	0
$969$	4.99158e7	1.70777
$970$	0	0
$971$	1.53215e7	0.521499	0.260750	−	0.965406i	$-0.416030\pi$
$971$	1.53215e7	0.521499	0.260750	+	0.965406i	$0.416030\pi$
$972$	0	0
$973$	4.47156e6	0.151418
$974$	0	0
$975$	−7.27901e6	−0.245223
$976$	0	0
$977$	1.60455e7	0.537795	0.268898	−	0.963169i	$-0.413341\pi$
$977$	1.60455e7	0.537795	0.268898	+	0.963169i	$0.413341\pi$
$978$	0	0
$979$	−3.14675e7	−1.04931
$980$	0	0
$981$	1.49463e8	4.95861
$982$	0	0
$983$	−4.45298e7	−1.46983	−0.734914	−	0.678160i	$-0.762778\pi$
$983$	−4.45298e7	−1.46983	−0.734914	+	0.678160i	$0.762778\pi$
$984$	0	0
$985$	1.06021e7	0.348180
$986$	0	0
$987$	−3.31123e7	−1.08192
$988$	0	0
$989$	−3.89447e7	−1.26607
$990$	0	0
$991$	9.31862e6	0.301417	0.150708	−	0.988578i	$-0.451845\pi$
$991$	9.31862e6	0.301417	0.150708	+	0.988578i	$0.451845\pi$
$992$	0	0
$993$	−9.33774e7	−3.00517
$994$	0	0
$995$	4.45503e6	0.142657
$996$	0	0
$997$	−4.78418e7	−1.52430	−0.762149	−	0.647402i	$-0.775856\pi$
$997$	−4.78418e7	−1.52430	−0.762149	+	0.647402i	$0.775856\pi$
$998$	0	0
$999$	8.62634e7	2.73472

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.y.1.1		5
4.3	odd	2		280.6.a.j.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.j.1.5	✓	5	4.3	odd	2
560.6.a.y.1.1		5	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-30.3320 q^{3} +25.0000 q^{5} -49.0000 q^{7} +677.031 q^{9} +O(q^{10})\)	\(q-30.3320 q^{3} +25.0000 q^{5} -49.0000 q^{7} +677.031 q^{9} -604.346 q^{11} +383.964 q^{13} -758.300 q^{15} -791.640 q^{17} +2078.78 q^{19} +1486.27 q^{21} +2523.45 q^{23} +625.000 q^{25} -13165.0 q^{27} +372.040 q^{29} +2484.87 q^{31} +18331.0 q^{33} -1225.00 q^{35} -6552.46 q^{37} -11646.4 q^{39} -16194.7 q^{41} -15433.1 q^{43} +16925.8 q^{45} -22278.8 q^{47} +2401.00 q^{49} +24012.0 q^{51} -26272.4 q^{53} -15108.6 q^{55} -63053.6 q^{57} -39869.1 q^{59} +5893.86 q^{61} -33174.5 q^{63} +9599.11 q^{65} +32074.7 q^{67} -76541.3 q^{69} +57992.0 q^{71} +39480.1 q^{73} -18957.5 q^{75} +29612.9 q^{77} +8987.90 q^{79} +234804. q^{81} +52523.4 q^{83} -19791.0 q^{85} -11284.7 q^{87} +52068.7 q^{89} -18814.3 q^{91} -75371.1 q^{93} +51969.5 q^{95} +14928.3 q^{97} -409161. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} + 125 q^{5} - 245 q^{7} + 1132 q^{9}+O(q^{10}) \) 5 * q - 15 * q^3 + 125 * q^5 - 245 * q^7 + 1132 * q^9	\( 5 q - 15 q^{3} + 125 q^{5} - 245 q^{7} + 1132 q^{9} - 281 q^{11} + 909 q^{13} - 375 q^{15} + 1495 q^{17} + 422 q^{19} + 735 q^{21} + 62 q^{23} + 3125 q^{25} + 3363 q^{27} - 2047 q^{29} - 1636 q^{31} + 19181 q^{33} - 6125 q^{35} - 10358 q^{37} - 15685 q^{39} + 6424 q^{41} - 28306 q^{43} + 28300 q^{45} - 20955 q^{47} + 12005 q^{49} + 23577 q^{51} + 43748 q^{53} - 7025 q^{55} + 13690 q^{57} - 45788 q^{59} + 50432 q^{61} - 55468 q^{63} + 22725 q^{65} - 40712 q^{67} + 35050 q^{69} + 3096 q^{71} + 135438 q^{73} - 9375 q^{75} + 13769 q^{77} - 13191 q^{79} + 381101 q^{81} - 35108 q^{83} + 37375 q^{85} - 297289 q^{87} + 213772 q^{89} - 44541 q^{91} + 134244 q^{93} + 10550 q^{95} + 10659 q^{97} - 39462 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 + 125 * q^5 - 245 * q^7 + 1132 * q^9 - 281 * q^11 + 909 * q^13 - 375 * q^15 + 1495 * q^17 + 422 * q^19 + 735 * q^21 + 62 * q^23 + 3125 * q^25 + 3363 * q^27 - 2047 * q^29 - 1636 * q^31 + 19181 * q^33 - 6125 * q^35 - 10358 * q^37 - 15685 * q^39 + 6424 * q^41 - 28306 * q^43 + 28300 * q^45 - 20955 * q^47 + 12005 * q^49 + 23577 * q^51 + 43748 * q^53 - 7025 * q^55 + 13690 * q^57 - 45788 * q^59 + 50432 * q^61 - 55468 * q^63 + 22725 * q^65 - 40712 * q^67 + 35050 * q^69 + 3096 * q^71 + 135438 * q^73 - 9375 * q^75 + 13769 * q^77 - 13191 * q^79 + 381101 * q^81 - 35108 * q^83 + 37375 * q^85 - 297289 * q^87 + 213772 * q^89 - 44541 * q^91 + 134244 * q^93 + 10550 * q^95 + 10659 * q^97 - 39462 * q^99

Introduction

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