LMFDB - Embedded newform 560.6.a.l.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:	$1$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.53113$ 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)

$f(q)$	$=$	$q-13.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -58.2198 q^{9} +O(q^{10})$	$q-13.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -58.2198 q^{9} +691.520 q^{11} -502.150 q^{13} +339.835 q^{15} -991.313 q^{17} -661.677 q^{19} -666.076 q^{21} -3415.08 q^{23} +625.000 q^{25} +4094.60 q^{27} +6751.92 q^{29} +3922.76 q^{31} -9400.09 q^{33} -1225.00 q^{35} +627.222 q^{37} +6825.92 q^{39} +16277.9 q^{41} +17277.7 q^{43} +1455.50 q^{45} +4295.47 q^{47} +2401.00 q^{49} +13475.3 q^{51} -25960.9 q^{53} -17288.0 q^{55} +8994.43 q^{57} -8902.63 q^{59} -48924.6 q^{61} -2852.77 q^{63} +12553.7 q^{65} +4257.80 q^{67} +46422.5 q^{69} -18990.9 q^{71} +10132.5 q^{73} -8495.87 q^{75} +33884.5 q^{77} +96986.5 q^{79} -41512.0 q^{81} -70732.1 q^{83} +24782.8 q^{85} -91781.4 q^{87} +4241.12 q^{89} -24605.3 q^{91} -53323.6 q^{93} +16541.9 q^{95} -104376. q^{97} -40260.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9	$ 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9} + 601 q^{11} - 577 q^{13} + 75 q^{15} + 41 q^{17} - 630 q^{19} - 147 q^{21} + 442 q^{23} + 1250 q^{25} + 135 q^{27} + 5885 q^{29} + 396 q^{31} - 10359 q^{33} - 2450 q^{35} - 8904 q^{37} + 6033 q^{39} + 1774 q^{41} + 27122 q^{43} + 4725 q^{45} + 21289 q^{47} + 4802 q^{49} + 24411 q^{51} - 55582 q^{53} - 15025 q^{55} + 9330 q^{57} - 59600 q^{59} - 51846 q^{61} - 9261 q^{63} + 14425 q^{65} + 45344 q^{67} + 87282 q^{69} - 80744 q^{71} - 13532 q^{73} - 1875 q^{75} + 29449 q^{77} + 51795 q^{79} - 51678 q^{81} - 109828 q^{83} - 1025 q^{85} - 100965 q^{87} - 37650 q^{89} - 28273 q^{91} - 90684 q^{93} + 15750 q^{95} - 96339 q^{97} - 28422 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9 + 601 * q^11 - 577 * q^13 + 75 * q^15 + 41 * q^17 - 630 * q^19 - 147 * q^21 + 442 * q^23 + 1250 * q^25 + 135 * q^27 + 5885 * q^29 + 396 * q^31 - 10359 * q^33 - 2450 * q^35 - 8904 * q^37 + 6033 * q^39 + 1774 * q^41 + 27122 * q^43 + 4725 * q^45 + 21289 * q^47 + 4802 * q^49 + 24411 * q^51 - 55582 * q^53 - 15025 * q^55 + 9330 * q^57 - 59600 * q^59 - 51846 * q^61 - 9261 * q^63 + 14425 * q^65 + 45344 * q^67 + 87282 * q^69 - 80744 * q^71 - 13532 * q^73 - 1875 * q^75 + 29449 * q^77 + 51795 * q^79 - 51678 * q^81 - 109828 * q^83 - 1025 * q^85 - 100965 * q^87 - 37650 * q^89 - 28273 * q^91 - 90684 * q^93 + 15750 * q^95 - 96339 * q^97 - 28422 * 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
$3$	−13.5934	−0.872016	−0.436008	−	0.899943i	$-0.643608\pi$
$3$	−13.5934	−0.872016	−0.436008	+	0.899943i	$0.643608\pi$
$4$	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
$9$	−58.2198	−0.239588
$10$	0	0
$11$	691.520	1.72315	0.861574	−	0.507632i	$-0.169479\pi$
$11$	691.520	1.72315	0.861574	+	0.507632i	$0.169479\pi$
$12$	0	0
$13$	−502.150	−0.824091	−0.412045	−	0.911163i	$-0.635185\pi$
$13$	−502.150	−0.824091	−0.412045	+	0.911163i	$0.635185\pi$
$14$	0	0
$15$	339.835	0.389977
$16$	0	0
$17$	−991.313	−0.831934	−0.415967	−	0.909380i	$-0.636557\pi$
$17$	−991.313	−0.831934	−0.415967	+	0.909380i	$0.636557\pi$
$18$	0	0
$19$	−661.677	−0.420496	−0.210248	−	0.977648i	$-0.567427\pi$
$19$	−661.677	−0.420496	−0.210248	+	0.977648i	$0.567427\pi$
$20$	0	0
$21$	−666.076	−0.329591
$22$	0	0
$23$	−3415.08	−1.34611	−0.673056	−	0.739592i	$-0.735019\pi$
$23$	−3415.08	−1.34611	−0.673056	+	0.739592i	$0.735019\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	4094.60	1.08094
$28$	0	0
$29$	6751.92	1.49084	0.745422	−	0.666593i	$-0.232248\pi$
$29$	6751.92	1.49084	0.745422	+	0.666593i	$0.232248\pi$
$30$	0	0
$31$	3922.76	0.733142	0.366571	−	0.930390i	$-0.380532\pi$
$31$	3922.76	0.733142	0.366571	+	0.930390i	$0.380532\pi$
$32$	0	0
$33$	−9400.09	−1.50261
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	627.222	0.0753212	0.0376606	−	0.999291i	$-0.488009\pi$
$37$	627.222	0.0753212	0.0376606	+	0.999291i	$0.488009\pi$
$38$	0	0
$39$	6825.92	0.718620
$40$	0	0
$41$	16277.9	1.51230	0.756149	−	0.654399i	$-0.227078\pi$
$41$	16277.9	1.51230	0.756149	+	0.654399i	$0.227078\pi$
$42$	0	0
$43$	17277.7	1.42500	0.712500	−	0.701672i	$-0.247563\pi$
$43$	17277.7	1.42500	0.712500	+	0.701672i	$0.247563\pi$
$44$	0	0
$45$	1455.50	0.107147
$46$	0	0
$47$	4295.47	0.283639	0.141820	−	0.989893i	$-0.454705\pi$
$47$	4295.47	0.283639	0.141820	+	0.989893i	$0.454705\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	13475.3	0.725460
$52$	0	0
$53$	−25960.9	−1.26949	−0.634745	−	0.772721i	$-0.718895\pi$
$53$	−25960.9	−1.26949	−0.634745	+	0.772721i	$0.718895\pi$
$54$	0	0
$55$	−17288.0	−0.770615
$56$	0	0
$57$	8994.43	0.366679
$58$	0	0
$59$	−8902.63	−0.332957	−0.166479	−	0.986045i	$-0.553240\pi$
$59$	−8902.63	−0.332957	−0.166479	+	0.986045i	$0.553240\pi$
$60$	0	0
$61$	−48924.6	−1.68346	−0.841730	−	0.539898i	$-0.818463\pi$
$61$	−48924.6	−1.68346	−0.841730	+	0.539898i	$0.818463\pi$
$62$	0	0
$63$	−2852.77	−0.0905557
$64$	0	0
$65$	12553.7	0.368545
$66$	0	0
$67$	4257.80	0.115877	0.0579387	−	0.998320i	$-0.481547\pi$
$67$	4257.80	0.115877	0.0579387	+	0.998320i	$0.481547\pi$
$68$	0	0
$69$	46422.5	1.17383
$70$	0	0
$71$	−18990.9	−0.447095	−0.223547	−	0.974693i	$-0.571764\pi$
$71$	−18990.9	−0.447095	−0.223547	+	0.974693i	$0.571764\pi$
$72$	0	0
$73$	10132.5	0.222541	0.111270	−	0.993790i	$-0.464508\pi$
$73$	10132.5	0.222541	0.111270	+	0.993790i	$0.464508\pi$
$74$	0	0
$75$	−8495.87	−0.174403
$76$	0	0
$77$	33884.5	0.651289
$78$	0	0
$79$	96986.5	1.74841	0.874205	−	0.485557i	$-0.161383\pi$
$79$	96986.5	1.74841	0.874205	+	0.485557i	$0.161383\pi$
$80$	0	0
$81$	−41512.0	−0.703010
$82$	0	0
$83$	−70732.1	−1.12699	−0.563497	−	0.826118i	$-0.690544\pi$
$83$	−70732.1	−1.12699	−0.563497	+	0.826118i	$0.690544\pi$
$84$	0	0
$85$	24782.8	0.372052
$86$	0	0
$87$	−91781.4	−1.30004
$88$	0	0
$89$	4241.12	0.0567552	0.0283776	−	0.999597i	$-0.490966\pi$
$89$	4241.12	0.0567552	0.0283776	+	0.999597i	$0.490966\pi$
$90$	0	0
$91$	−24605.3	−0.311477
$92$	0	0
$93$	−53323.6	−0.639311
$94$	0	0
$95$	16541.9	0.188052
$96$	0	0
$97$	−104376.	−1.12634	−0.563170	−	0.826341i	$-0.690418\pi$
$97$	−104376.	−1.12634	−0.563170	+	0.826341i	$0.690418\pi$
$98$	0	0
$99$	−40260.2	−0.412845
$100$	0	0
$101$	−45715.1	−0.445919	−0.222959	−	0.974828i	$-0.571572\pi$
$101$	−45715.1	−0.445919	−0.222959	+	0.974828i	$0.571572\pi$
$102$	0	0
$103$	−89278.1	−0.829186	−0.414593	−	0.910007i	$-0.636076\pi$
$103$	−89278.1	−0.829186	−0.414593	+	0.910007i	$0.636076\pi$
$104$	0	0
$105$	16651.9	0.147398
$106$	0	0
$107$	106330.	0.897834	0.448917	−	0.893573i	$-0.351810\pi$
$107$	106330.	0.897834	0.448917	+	0.893573i	$0.351810\pi$
$108$	0	0
$109$	−49816.5	−0.401613	−0.200806	−	0.979631i	$-0.564356\pi$
$109$	−49816.5	−0.401613	−0.200806	+	0.979631i	$0.564356\pi$
$110$	0	0
$111$	−8526.08	−0.0656813
$112$	0	0
$113$	−37160.7	−0.273771	−0.136886	−	0.990587i	$-0.543709\pi$
$113$	−37160.7	−0.273771	−0.136886	+	0.990587i	$0.543709\pi$
$114$	0	0
$115$	85377.0	0.601999
$116$	0	0
$117$	29235.1	0.197442
$118$	0	0
$119$	−48574.4	−0.314441
$120$	0	0
$121$	317148.	1.96924
$122$	0	0
$123$	−221271.	−1.31875
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	46510.2	0.255882	0.127941	−	0.991782i	$-0.459163\pi$
$127$	46510.2	0.255882	0.127941	+	0.991782i	$0.459163\pi$
$128$	0	0
$129$	−234862.	−1.24262
$130$	0	0
$131$	−381771.	−1.94368	−0.971839	−	0.235646i	$-0.924279\pi$
$131$	−381771.	−1.94368	−0.971839	+	0.235646i	$0.924279\pi$
$132$	0	0
$133$	−32422.2	−0.158933
$134$	0	0
$135$	−102365.	−0.483411
$136$	0	0
$137$	−1894.54	−0.00862389	−0.00431194	−	0.999991i	$-0.501373\pi$
$137$	−1894.54	−0.00862389	−0.00431194	+	0.999991i	$0.501373\pi$
$138$	0	0
$139$	−201798.	−0.885889	−0.442944	−	0.896549i	$-0.646066\pi$
$139$	−201798.	−0.885889	−0.442944	+	0.896549i	$0.646066\pi$
$140$	0	0
$141$	−58390.0	−0.247338
$142$	0	0
$143$	−347246.	−1.42003
$144$	0	0
$145$	−168798.	−0.666726
$146$	0	0
$147$	−32637.7	−0.124574
$148$	0	0
$149$	−466237.	−1.72045	−0.860224	−	0.509917i	$-0.829676\pi$
$149$	−466237.	−1.72045	−0.860224	+	0.509917i	$0.829676\pi$
$150$	0	0
$151$	122212.	0.436185	0.218093	−	0.975928i	$-0.430017\pi$
$151$	122212.	0.436185	0.218093	+	0.975928i	$0.430017\pi$
$152$	0	0
$153$	57714.1	0.199321
$154$	0	0
$155$	−98069.1	−0.327871
$156$	0	0
$157$	−410638.	−1.32957	−0.664784	−	0.747036i	$-0.731476\pi$
$157$	−410638.	−1.32957	−0.664784	+	0.747036i	$0.731476\pi$
$158$	0	0
$159$	352896.	1.10702
$160$	0	0
$161$	−167339.	−0.508782
$162$	0	0
$163$	−78525.4	−0.231495	−0.115747	−	0.993279i	$-0.536926\pi$
$163$	−78525.4	−0.231495	−0.115747	+	0.993279i	$0.536926\pi$
$164$	0	0
$165$	235002.	0.671989
$166$	0	0
$167$	597714.	1.65845	0.829224	−	0.558916i	$-0.188783\pi$
$167$	597714.	1.65845	0.829224	+	0.558916i	$0.188783\pi$
$168$	0	0
$169$	−119139.	−0.320875
$170$	0	0
$171$	38522.7	0.100746
$172$	0	0
$173$	−59874.0	−0.152098	−0.0760490	−	0.997104i	$-0.524231\pi$
$173$	−59874.0	−0.152098	−0.0760490	+	0.997104i	$0.524231\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	121017.	0.290344
$178$	0	0
$179$	−616812.	−1.43887	−0.719433	−	0.694562i	$-0.755598\pi$
$179$	−616812.	−1.43887	−0.719433	+	0.694562i	$0.755598\pi$
$180$	0	0
$181$	−37287.0	−0.0845981	−0.0422990	−	0.999105i	$-0.513468\pi$
$181$	−37287.0	−0.0845981	−0.0422990	+	0.999105i	$0.513468\pi$
$182$	0	0
$183$	665051.	1.46800
$184$	0	0
$185$	−15680.6	−0.0336847
$186$	0	0
$187$	−685513.	−1.43355
$188$	0	0
$189$	200635.	0.408557
$190$	0	0
$191$	−326760.	−0.648106	−0.324053	−	0.946039i	$-0.605046\pi$
$191$	−326760.	−0.648106	−0.324053	+	0.946039i	$0.605046\pi$
$192$	0	0
$193$	−265735.	−0.513518	−0.256759	−	0.966475i	$-0.582655\pi$
$193$	−265735.	−0.513518	−0.256759	+	0.966475i	$0.582655\pi$
$194$	0	0
$195$	−170648.	−0.321377
$196$	0	0
$197$	−517865.	−0.950716	−0.475358	−	0.879792i	$-0.657682\pi$
$197$	−517865.	−0.950716	−0.475358	+	0.879792i	$0.657682\pi$
$198$	0	0
$199$	148687.	0.266158	0.133079	−	0.991105i	$-0.457514\pi$
$199$	148687.	0.266158	0.133079	+	0.991105i	$0.457514\pi$
$200$	0	0
$201$	−57878.0	−0.101047
$202$	0	0
$203$	330844.	0.563486
$204$	0	0
$205$	−406946.	−0.676320
$206$	0	0
$207$	198825.	0.322512
$208$	0	0
$209$	−457563.	−0.724577
$210$	0	0
$211$	−7443.09	−0.0115093	−0.00575463	−	0.999983i	$-0.501832\pi$
$211$	−7443.09	−0.0115093	−0.00575463	+	0.999983i	$0.501832\pi$
$212$	0	0
$213$	258151.	0.389874
$214$	0	0
$215$	−431943.	−0.637279
$216$	0	0
$217$	192215.	0.277101
$218$	0	0
$219$	−137735.	−0.194059
$220$	0	0
$221$	497788.	0.685589
$222$	0	0
$223$	−119157.	−0.160456	−0.0802280	−	0.996777i	$-0.525565\pi$
$223$	−119157.	−0.160456	−0.0802280	+	0.996777i	$0.525565\pi$
$224$	0	0
$225$	−36387.4	−0.0479176
$226$	0	0
$227$	388843.	0.500852	0.250426	−	0.968136i	$-0.419429\pi$
$227$	388843.	0.500852	0.250426	+	0.968136i	$0.419429\pi$
$228$	0	0
$229$	732622.	0.923191	0.461595	−	0.887091i	$-0.347277\pi$
$229$	732622.	0.923191	0.461595	+	0.887091i	$0.347277\pi$
$230$	0	0
$231$	−460605.	−0.567934
$232$	0	0
$233$	−1.12639e6	−1.35925	−0.679626	−	0.733559i	$-0.737858\pi$
$233$	−1.12639e6	−1.35925	−0.679626	+	0.733559i	$0.737858\pi$
$234$	0	0
$235$	−107387.	−0.126847
$236$	0	0
$237$	−1.31837e6	−1.52464
$238$	0	0
$239$	−772317.	−0.874583	−0.437292	−	0.899320i	$-0.644062\pi$
$239$	−772317.	−0.874583	−0.437292	+	0.899320i	$0.644062\pi$
$240$	0	0
$241$	1.40297e6	1.55598	0.777991	−	0.628275i	$-0.216239\pi$
$241$	1.40297e6	1.55598	0.777991	+	0.628275i	$0.216239\pi$
$242$	0	0
$243$	−430698.	−0.467905
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	332261.	0.346527
$248$	0	0
$249$	961489.	0.982757
$250$	0	0
$251$	1.63922e6	1.64230	0.821151	−	0.570712i	$-0.193333\pi$
$251$	1.63922e6	1.64230	0.821151	+	0.570712i	$0.193333\pi$
$252$	0	0
$253$	−2.36159e6	−2.31955
$254$	0	0
$255$	−336883.	−0.324435
$256$	0	0
$257$	−223664.	−0.211234	−0.105617	−	0.994407i	$-0.533682\pi$
$257$	−223664.	−0.211234	−0.105617	+	0.994407i	$0.533682\pi$
$258$	0	0
$259$	30733.9	0.0284687
$260$	0	0
$261$	−393096.	−0.357188
$262$	0	0
$263$	−299519.	−0.267014	−0.133507	−	0.991048i	$-0.542624\pi$
$263$	−299519.	−0.267014	−0.133507	+	0.991048i	$0.542624\pi$
$264$	0	0
$265$	649022.	0.567734
$266$	0	0
$267$	−57651.2	−0.0494914
$268$	0	0
$269$	134341.	0.113195	0.0565974	−	0.998397i	$-0.481975\pi$
$269$	134341.	0.113195	0.0565974	+	0.998397i	$0.481975\pi$
$270$	0	0
$271$	−1.93414e6	−1.59980	−0.799898	−	0.600136i	$-0.795113\pi$
$271$	−1.93414e6	−1.59980	−0.799898	+	0.600136i	$0.795113\pi$
$272$	0	0
$273$	334470.	0.271613
$274$	0	0
$275$	432200.	0.344630
$276$	0	0
$277$	177599.	0.139072	0.0695362	−	0.997579i	$-0.477848\pi$
$277$	177599.	0.139072	0.0695362	+	0.997579i	$0.477848\pi$
$278$	0	0
$279$	−228383.	−0.175652
$280$	0	0
$281$	1.85131e6	1.39867	0.699333	−	0.714796i	$-0.253481\pi$
$281$	1.85131e6	1.39867	0.699333	+	0.714796i	$0.253481\pi$
$282$	0	0
$283$	−2.39851e6	−1.78023	−0.890114	−	0.455737i	$-0.849376\pi$
$283$	−2.39851e6	−1.78023	−0.890114	+	0.455737i	$0.849376\pi$
$284$	0	0
$285$	−224861.	−0.163984
$286$	0	0
$287$	797615.	0.571595
$288$	0	0
$289$	−437155.	−0.307887
$290$	0	0
$291$	1.41882e6	0.982186
$292$	0	0
$293$	2.49922e6	1.70073	0.850364	−	0.526196i	$-0.176382\pi$
$293$	2.49922e6	1.70073	0.850364	+	0.526196i	$0.176382\pi$
$294$	0	0
$295$	222566.	0.148903
$296$	0	0
$297$	2.83149e6	1.86262
$298$	0	0
$299$	1.71488e6	1.10932
$300$	0	0
$301$	846607.	0.538599
$302$	0	0
$303$	621422.	0.388848
$304$	0	0
$305$	1.22312e6	0.752866
$306$	0	0
$307$	−3.07195e6	−1.86024	−0.930119	−	0.367258i	$-0.880297\pi$
$307$	−3.07195e6	−1.86024	−0.930119	+	0.367258i	$0.880297\pi$
$308$	0	0
$309$	1.21359e6	0.723063
$310$	0	0
$311$	−661233.	−0.387662	−0.193831	−	0.981035i	$-0.562091\pi$
$311$	−661233.	−0.387662	−0.193831	+	0.981035i	$0.562091\pi$
$312$	0	0
$313$	−3.29393e6	−1.90043	−0.950217	−	0.311588i	$-0.899139\pi$
$313$	−3.29393e6	−1.90043	−0.950217	+	0.311588i	$0.899139\pi$
$314$	0	0
$315$	71319.3	0.0404977
$316$	0	0
$317$	639724.	0.357556	0.178778	−	0.983889i	$-0.442786\pi$
$317$	639724.	0.357556	0.178778	+	0.983889i	$0.442786\pi$
$318$	0	0
$319$	4.66908e6	2.56894
$320$	0	0
$321$	−1.44538e6	−0.782926
$322$	0	0
$323$	655929.	0.349825
$324$	0	0
$325$	−313844.	−0.164818
$326$	0	0
$327$	677175.	0.350213
$328$	0	0
$329$	210478.	0.107206
$330$	0	0
$331$	1.13876e6	0.571298	0.285649	−	0.958334i	$-0.407791\pi$
$331$	1.13876e6	0.571298	0.285649	+	0.958334i	$0.407791\pi$
$332$	0	0
$333$	−36516.8	−0.0180460
$334$	0	0
$335$	−106445.	−0.0518219
$336$	0	0
$337$	685493.	0.328797	0.164399	−	0.986394i	$-0.447432\pi$
$337$	685493.	0.328797	0.164399	+	0.986394i	$0.447432\pi$
$338$	0	0
$339$	505140.	0.238733
$340$	0	0
$341$	2.71267e6	1.26331
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.16056e6	−0.524953
$346$	0	0
$347$	1.25151e6	0.557970	0.278985	−	0.960295i	$-0.410002\pi$
$347$	1.25151e6	0.557970	0.278985	+	0.960295i	$0.410002\pi$
$348$	0	0
$349$	−3.16606e6	−1.39141	−0.695706	−	0.718327i	$-0.744908\pi$
$349$	−3.16606e6	−1.39141	−0.695706	+	0.718327i	$0.744908\pi$
$350$	0	0
$351$	−2.05610e6	−0.890793
$352$	0	0
$353$	−2.43368e6	−1.03951	−0.519754	−	0.854316i	$-0.673976\pi$
$353$	−2.43368e6	−1.03951	−0.519754	+	0.854316i	$0.673976\pi$
$354$	0	0
$355$	474772.	0.199947
$356$	0	0
$357$	660290.	0.274198
$358$	0	0
$359$	2.13021e6	0.872341	0.436170	−	0.899864i	$-0.356335\pi$
$359$	2.13021e6	0.872341	0.436170	+	0.899864i	$0.356335\pi$
$360$	0	0
$361$	−2.03828e6	−0.823183
$362$	0	0
$363$	−4.31112e6	−1.71721
$364$	0	0
$365$	−253312.	−0.0995232
$366$	0	0
$367$	3.10976e6	1.20521	0.602604	−	0.798041i	$-0.294130\pi$
$367$	3.10976e6	1.20521	0.602604	+	0.798041i	$0.294130\pi$
$368$	0	0
$369$	−947694.	−0.362328
$370$	0	0
$371$	−1.27208e6	−0.479822
$372$	0	0
$373$	−3.15189e6	−1.17300	−0.586502	−	0.809948i	$-0.699495\pi$
$373$	−3.15189e6	−1.17300	−0.586502	+	0.809948i	$0.699495\pi$
$374$	0	0
$375$	212397.	0.0779955
$376$	0	0
$377$	−3.39047e6	−1.22859
$378$	0	0
$379$	−342350.	−0.122426	−0.0612129	−	0.998125i	$-0.519497\pi$
$379$	−342350.	−0.122426	−0.0612129	+	0.998125i	$0.519497\pi$
$380$	0	0
$381$	−632231.	−0.223133
$382$	0	0
$383$	−3.69387e6	−1.28672	−0.643361	−	0.765563i	$-0.722461\pi$
$383$	−3.69387e6	−1.28672	−0.643361	+	0.765563i	$0.722461\pi$
$384$	0	0
$385$	−847111.	−0.291265
$386$	0	0
$387$	−1.00590e6	−0.341413
$388$	0	0
$389$	2.05313e6	0.687928	0.343964	−	0.938983i	$-0.388230\pi$
$389$	2.05313e6	0.687928	0.343964	+	0.938983i	$0.388230\pi$
$390$	0	0
$391$	3.38541e6	1.11988
$392$	0	0
$393$	5.18956e6	1.69492
$394$	0	0
$395$	−2.42466e6	−0.781913
$396$	0	0
$397$	−2.28107e6	−0.726377	−0.363189	−	0.931716i	$-0.618312\pi$
$397$	−2.28107e6	−0.726377	−0.363189	+	0.931716i	$0.618312\pi$
$398$	0	0
$399$	440727.	0.138592
$400$	0	0
$401$	3.32082e6	1.03130	0.515649	−	0.856800i	$-0.327551\pi$
$401$	3.32082e6	1.03130	0.515649	+	0.856800i	$0.327551\pi$
$402$	0	0
$403$	−1.96981e6	−0.604175
$404$	0	0
$405$	1.03780e6	0.314396
$406$	0	0
$407$	433737.	0.129790
$408$	0	0
$409$	5.15938e6	1.52507	0.762534	−	0.646948i	$-0.223955\pi$
$409$	5.15938e6	1.52507	0.762534	+	0.646948i	$0.223955\pi$
$410$	0	0
$411$	25753.3	0.00752017
$412$	0	0
$413$	−436229.	−0.125846
$414$	0	0
$415$	1.76830e6	0.504007
$416$	0	0
$417$	2.74311e6	0.772509
$418$	0	0
$419$	4.85187e6	1.35012	0.675062	−	0.737761i	$-0.264117\pi$
$419$	4.85187e6	1.35012	0.675062	+	0.737761i	$0.264117\pi$
$420$	0	0
$421$	−6.14767e6	−1.69046	−0.845231	−	0.534401i	$-0.820537\pi$
$421$	−6.14767e6	−1.69046	−0.845231	+	0.534401i	$0.820537\pi$
$422$	0	0
$423$	−250082.	−0.0679565
$424$	0	0
$425$	−619571.	−0.166387
$426$	0	0
$427$	−2.39731e6	−0.636288
$428$	0	0
$429$	4.72025e6	1.23829
$430$	0	0
$431$	−3.55411e6	−0.921590	−0.460795	−	0.887507i	$-0.652436\pi$
$431$	−3.55411e6	−0.921590	−0.460795	+	0.887507i	$0.652436\pi$
$432$	0	0
$433$	2.82650e6	0.724485	0.362243	−	0.932084i	$-0.382011\pi$
$433$	2.82650e6	0.724485	0.362243	+	0.932084i	$0.382011\pi$
$434$	0	0
$435$	2.29454e6	0.581395
$436$	0	0
$437$	2.25968e6	0.566035
$438$	0	0
$439$	−4.64410e6	−1.15011	−0.575056	−	0.818114i	$-0.695020\pi$
$439$	−4.64410e6	−1.15011	−0.575056	+	0.818114i	$0.695020\pi$
$440$	0	0
$441$	−139786.	−0.0342268
$442$	0	0
$443$	6.15534e6	1.49019	0.745097	−	0.666957i	$-0.232403\pi$
$443$	6.15534e6	1.49019	0.745097	+	0.666957i	$0.232403\pi$
$444$	0	0
$445$	−106028.	−0.0253817
$446$	0	0
$447$	6.33774e6	1.50026
$448$	0	0
$449$	3.67035e6	0.859196	0.429598	−	0.903020i	$-0.358655\pi$
$449$	3.67035e6	0.859196	0.429598	+	0.903020i	$0.358655\pi$
$450$	0	0
$451$	1.12565e7	2.60591
$452$	0	0
$453$	−1.66127e6	−0.380360
$454$	0	0
$455$	615134.	0.139297
$456$	0	0
$457$	−866327.	−0.194040	−0.0970201	−	0.995282i	$-0.530931\pi$
$457$	−866327.	−0.194040	−0.0970201	+	0.995282i	$0.530931\pi$
$458$	0	0
$459$	−4.05903e6	−0.899271
$460$	0	0
$461$	−1.88572e6	−0.413261	−0.206631	−	0.978419i	$-0.566250\pi$
$461$	−1.88572e6	−0.413261	−0.206631	+	0.978419i	$0.566250\pi$
$462$	0	0
$463$	8.49017e6	1.84062	0.920309	−	0.391192i	$-0.127937\pi$
$463$	8.49017e6	1.84062	0.920309	+	0.391192i	$0.127937\pi$
$464$	0	0
$465$	1.33309e6	0.285909
$466$	0	0
$467$	−1.82738e6	−0.387737	−0.193868	−	0.981028i	$-0.562104\pi$
$467$	−1.82738e6	−0.387737	−0.193868	+	0.981028i	$0.562104\pi$
$468$	0	0
$469$	208632.	0.0437975
$470$	0	0
$471$	5.58197e6	1.15940
$472$	0	0
$473$	1.19479e7	2.45549
$474$	0	0
$475$	−413548.	−0.0840992
$476$	0	0
$477$	1.51144e6	0.304155
$478$	0	0
$479$	−4.68744e6	−0.933463	−0.466731	−	0.884399i	$-0.654568\pi$
$479$	−4.68744e6	−0.933463	−0.466731	+	0.884399i	$0.654568\pi$
$480$	0	0
$481$	−314960.	−0.0620715
$482$	0	0
$483$	2.27470e6	0.443666
$484$	0	0
$485$	2.60939e6	0.503714
$486$	0	0
$487$	−4.59651e6	−0.878225	−0.439112	−	0.898432i	$-0.644707\pi$
$487$	−4.59651e6	−0.878225	−0.439112	+	0.898432i	$0.644707\pi$
$488$	0	0
$489$	1.06743e6	0.201867
$490$	0	0
$491$	−6.62099e6	−1.23942	−0.619711	−	0.784830i	$-0.712750\pi$
$491$	−6.62099e6	−1.23942	−0.619711	+	0.784830i	$0.712750\pi$
$492$	0	0
$493$	−6.69327e6	−1.24028
$494$	0	0
$495$	1.00650e6	0.184630
$496$	0	0
$497$	−930554.	−0.168986
$498$	0	0
$499$	4.52632e6	0.813756	0.406878	−	0.913482i	$-0.366617\pi$
$499$	4.52632e6	0.813756	0.406878	+	0.913482i	$0.366617\pi$
$500$	0	0
$501$	−8.12495e6	−1.44619
$502$	0	0
$503$	−3.83316e6	−0.675518	−0.337759	−	0.941233i	$-0.609669\pi$
$503$	−3.83316e6	−0.675518	−0.337759	+	0.941233i	$0.609669\pi$
$504$	0	0
$505$	1.14288e6	0.199421
$506$	0	0
$507$	1.61950e6	0.279808
$508$	0	0
$509$	−3.25460e6	−0.556806	−0.278403	−	0.960464i	$-0.589805\pi$
$509$	−3.25460e6	−0.556806	−0.278403	+	0.960464i	$0.589805\pi$
$510$	0	0
$511$	496492.	0.0841124
$512$	0	0
$513$	−2.70930e6	−0.454531
$514$	0	0
$515$	2.23195e6	0.370823
$516$	0	0
$517$	2.97040e6	0.488752
$518$	0	0
$519$	813891.	0.132632
$520$	0	0
$521$	−1.07842e6	−0.174057	−0.0870287	−	0.996206i	$-0.527737\pi$
$521$	−1.07842e6	−0.174057	−0.0870287	+	0.996206i	$0.527737\pi$
$522$	0	0
$523$	−408626.	−0.0653238	−0.0326619	−	0.999466i	$-0.510398\pi$
$523$	−408626.	−0.0653238	−0.0326619	+	0.999466i	$0.510398\pi$
$524$	0	0
$525$	−416297.	−0.0659182
$526$	0	0
$527$	−3.88869e6	−0.609925
$528$	0	0
$529$	5.22642e6	0.812016
$530$	0	0
$531$	518310.	0.0797724
$532$	0	0
$533$	−8.17392e6	−1.24627
$534$	0	0
$535$	−2.65825e6	−0.401524
$536$	0	0
$537$	8.38456e6	1.25471
$538$	0	0
$539$	1.66034e6	0.246164
$540$	0	0
$541$	1.13918e7	1.67340	0.836701	−	0.547659i	$-0.184481\pi$
$541$	1.13918e7	1.67340	0.836701	+	0.547659i	$0.184481\pi$
$542$	0	0
$543$	506856.	0.0737709
$544$	0	0
$545$	1.24541e6	0.179607
$546$	0	0
$547$	3.44866e6	0.492813	0.246406	−	0.969167i	$-0.420750\pi$
$547$	3.44866e6	0.492813	0.246406	+	0.969167i	$0.420750\pi$
$548$	0	0
$549$	2.84838e6	0.403337
$550$	0	0
$551$	−4.46759e6	−0.626894
$552$	0	0
$553$	4.75234e6	0.660837
$554$	0	0
$555$	213152.	0.0293736
$556$	0	0
$557$	−4.69772e6	−0.641577	−0.320789	−	0.947151i	$-0.603948\pi$
$557$	−4.69772e6	−0.641577	−0.320789	+	0.947151i	$0.603948\pi$
$558$	0	0
$559$	−8.67599e6	−1.17433
$560$	0	0
$561$	9.31844e6	1.25007
$562$	0	0
$563$	561642.	0.0746772	0.0373386	−	0.999303i	$-0.488112\pi$
$563$	561642.	0.0746772	0.0373386	+	0.999303i	$0.488112\pi$
$564$	0	0
$565$	929018.	0.122434
$566$	0	0
$567$	−2.03409e6	−0.265713
$568$	0	0
$569$	5.19019e6	0.672051	0.336026	−	0.941853i	$-0.390917\pi$
$569$	5.19019e6	0.672051	0.336026	+	0.941853i	$0.390917\pi$
$570$	0	0
$571$	5.20274e6	0.667793	0.333897	−	0.942610i	$-0.391636\pi$
$571$	5.20274e6	0.667793	0.333897	+	0.942610i	$0.391636\pi$
$572$	0	0
$573$	4.44178e6	0.565159
$574$	0	0
$575$	−2.13442e6	−0.269222
$576$	0	0
$577$	8.70662e6	1.08870	0.544352	−	0.838857i	$-0.316775\pi$
$577$	8.70662e6	1.08870	0.544352	+	0.838857i	$0.316775\pi$
$578$	0	0
$579$	3.61224e6	0.447796
$580$	0	0
$581$	−3.46588e6	−0.425964
$582$	0	0
$583$	−1.79524e7	−2.18752
$584$	0	0
$585$	−730877.	−0.0882988
$586$	0	0
$587$	4.35717e6	0.521926	0.260963	−	0.965349i	$-0.415960\pi$
$587$	4.35717e6	0.521926	0.260963	+	0.965349i	$0.415960\pi$
$588$	0	0
$589$	−2.59560e6	−0.308283
$590$	0	0
$591$	7.03954e6	0.829040
$592$	0	0
$593$	−3.22991e6	−0.377184	−0.188592	−	0.982055i	$-0.560392\pi$
$593$	−3.22991e6	−0.377184	−0.188592	+	0.982055i	$0.560392\pi$
$594$	0	0
$595$	1.21436e6	0.140622
$596$	0	0
$597$	−2.02115e6	−0.232094
$598$	0	0
$599$	7.23988e6	0.824450	0.412225	−	0.911082i	$-0.364752\pi$
$599$	7.23988e6	0.824450	0.412225	+	0.911082i	$0.364752\pi$
$600$	0	0
$601$	−1.06837e7	−1.20652	−0.603262	−	0.797543i	$-0.706133\pi$
$601$	−1.06837e7	−1.20652	−0.603262	+	0.797543i	$0.706133\pi$
$602$	0	0
$603$	−247889.	−0.0277628
$604$	0	0
$605$	−7.92871e6	−0.880671
$606$	0	0
$607$	−2.51528e6	−0.277086	−0.138543	−	0.990356i	$-0.544242\pi$
$607$	−2.51528e6	−0.277086	−0.138543	+	0.990356i	$0.544242\pi$
$608$	0	0
$609$	−4.49729e6	−0.491369
$610$	0	0
$611$	−2.15697e6	−0.233744
$612$	0	0
$613$	213999.	0.0230017	0.0115009	−	0.999934i	$-0.496339\pi$
$613$	213999.	0.0230017	0.0115009	+	0.999934i	$0.496339\pi$
$614$	0	0
$615$	5.53178e6	0.589762
$616$	0	0
$617$	−127951.	−0.0135310	−0.00676552	−	0.999977i	$-0.502154\pi$
$617$	−127951.	−0.0135310	−0.00676552	+	0.999977i	$0.502154\pi$
$618$	0	0
$619$	1.23980e7	1.30054	0.650272	−	0.759701i	$-0.274655\pi$
$619$	1.23980e7	1.30054	0.650272	+	0.759701i	$0.274655\pi$
$620$	0	0
$621$	−1.39834e7	−1.45507
$622$	0	0
$623$	207815.	0.0214514
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	6.21983e6	0.631843
$628$	0	0
$629$	−621774.	−0.0626622
$630$	0	0
$631$	5.87683e6	0.587584	0.293792	−	0.955869i	$-0.405083\pi$
$631$	5.87683e6	0.587584	0.293792	+	0.955869i	$0.405083\pi$
$632$	0	0
$633$	101177.	0.0100363
$634$	0	0
$635$	−1.16275e6	−0.114434
$636$	0	0
$637$	−1.20566e6	−0.117727
$638$	0	0
$639$	1.10565e6	0.107118
$640$	0	0
$641$	−2.60122e6	−0.250053	−0.125026	−	0.992153i	$-0.539902\pi$
$641$	−2.60122e6	−0.250053	−0.125026	+	0.992153i	$0.539902\pi$
$642$	0	0
$643$	−1.31345e7	−1.25282	−0.626408	−	0.779495i	$-0.715476\pi$
$643$	−1.31345e7	−1.25282	−0.626408	+	0.779495i	$0.715476\pi$
$644$	0	0
$645$	5.87156e6	0.555718
$646$	0	0
$647$	−5.54662e6	−0.520916	−0.260458	−	0.965485i	$-0.583874\pi$
$647$	−5.54662e6	−0.520916	−0.260458	+	0.965485i	$0.583874\pi$
$648$	0	0
$649$	−6.15634e6	−0.573734
$650$	0	0
$651$	−2.61286e6	−0.241637
$652$	0	0
$653$	−1.54993e7	−1.42242	−0.711212	−	0.702978i	$-0.751853\pi$
$653$	−1.54993e7	−1.42242	−0.711212	+	0.702978i	$0.751853\pi$
$654$	0	0
$655$	9.54427e6	0.869239
$656$	0	0
$657$	−589912.	−0.0533180
$658$	0	0
$659$	4.30145e6	0.385835	0.192917	−	0.981215i	$-0.438205\pi$
$659$	4.30145e6	0.385835	0.192917	+	0.981215i	$0.438205\pi$
$660$	0	0
$661$	861980.	0.0767350	0.0383675	−	0.999264i	$-0.487784\pi$
$661$	861980.	0.0767350	0.0383675	+	0.999264i	$0.487784\pi$
$662$	0	0
$663$	−6.76662e6	−0.597844
$664$	0	0
$665$	810554.	0.0710768
$666$	0	0
$667$	−2.30583e7	−2.00684
$668$	0	0
$669$	1.61974e6	0.139920
$670$	0	0
$671$	−3.38323e7	−2.90085
$672$	0	0
$673$	−953818.	−0.0811760	−0.0405880	−	0.999176i	$-0.512923\pi$
$673$	−953818.	−0.0811760	−0.0405880	+	0.999176i	$0.512923\pi$
$674$	0	0
$675$	2.55912e6	0.216188
$676$	0	0
$677$	−1.48606e7	−1.24613	−0.623065	−	0.782170i	$-0.714113\pi$
$677$	−1.48606e7	−1.24613	−0.623065	+	0.782170i	$0.714113\pi$
$678$	0	0
$679$	−5.11440e6	−0.425716
$680$	0	0
$681$	−5.28569e6	−0.436751
$682$	0	0
$683$	1.57605e7	1.29276	0.646381	−	0.763015i	$-0.276282\pi$
$683$	1.57605e7	1.29276	0.646381	+	0.763015i	$0.276282\pi$
$684$	0	0
$685$	47363.6	0.00385672
$686$	0	0
$687$	−9.95882e6	−0.805037
$688$	0	0
$689$	1.30362e7	1.04618
$690$	0	0
$691$	−1.59740e7	−1.27267	−0.636337	−	0.771411i	$-0.719551\pi$
$691$	−1.59740e7	−1.27267	−0.636337	+	0.771411i	$0.719551\pi$
$692$	0	0
$693$	−1.97275e6	−0.156041
$694$	0	0
$695$	5.04494e6	0.396182
$696$	0	0
$697$	−1.61365e7	−1.25813
$698$	0	0
$699$	1.53115e7	1.18529
$700$	0	0
$701$	−1.85736e7	−1.42758	−0.713790	−	0.700360i	$-0.753023\pi$
$701$	−1.85736e7	−1.42758	−0.713790	+	0.700360i	$0.753023\pi$
$702$	0	0
$703$	−415019.	−0.0316723
$704$	0	0
$705$	1.45975e6	0.110613
$706$	0	0
$707$	−2.24004e6	−0.168541
$708$	0	0
$709$	−1.71029e7	−1.27778	−0.638888	−	0.769300i	$-0.720605\pi$
$709$	−1.71029e7	−1.27778	−0.638888	+	0.769300i	$0.720605\pi$
$710$	0	0
$711$	−5.64654e6	−0.418898
$712$	0	0
$713$	−1.33965e7	−0.986890
$714$	0	0
$715$	8.68116e6	0.635057
$716$	0	0
$717$	1.04984e7	0.762651
$718$	0	0
$719$	1.40945e7	1.01678	0.508390	−	0.861127i	$-0.330241\pi$
$719$	1.40945e7	1.01678	0.508390	+	0.861127i	$0.330241\pi$
$720$	0	0
$721$	−4.37463e6	−0.313403
$722$	0	0
$723$	−1.90711e7	−1.35684
$724$	0	0
$725$	4.21995e6	0.298169
$726$	0	0
$727$	−2.24196e7	−1.57323	−0.786616	−	0.617443i	$-0.788169\pi$
$727$	−2.24196e7	−1.57323	−0.786616	+	0.617443i	$0.788169\pi$
$728$	0	0
$729$	1.59421e7	1.11103
$730$	0	0
$731$	−1.71276e7	−1.18551
$732$	0	0
$733$	−8.02464e6	−0.551653	−0.275826	−	0.961208i	$-0.588951\pi$
$733$	−8.02464e6	−0.551653	−0.275826	+	0.961208i	$0.588951\pi$
$734$	0	0
$735$	815943.	0.0557111
$736$	0	0
$737$	2.94435e6	0.199674
$738$	0	0
$739$	−1.15678e7	−0.779181	−0.389591	−	0.920988i	$-0.627384\pi$
$739$	−1.15678e7	−0.779181	−0.389591	+	0.920988i	$0.627384\pi$
$740$	0	0
$741$	−4.51655e6	−0.302177
$742$	0	0
$743$	−3.79387e6	−0.252122	−0.126061	−	0.992023i	$-0.540233\pi$
$743$	−3.79387e6	−0.252122	−0.126061	+	0.992023i	$0.540233\pi$
$744$	0	0
$745$	1.16559e7	0.769407
$746$	0	0
$747$	4.11801e6	0.270014
$748$	0	0
$749$	5.21017e6	0.339349
$750$	0	0
$751$	9.07884e6	0.587395	0.293698	−	0.955898i	$-0.405114\pi$
$751$	9.07884e6	0.587395	0.293698	+	0.955898i	$0.405114\pi$
$752$	0	0
$753$	−2.22825e7	−1.43211
$754$	0	0
$755$	−3.05530e6	−0.195068
$756$	0	0
$757$	−2.33210e7	−1.47913	−0.739565	−	0.673085i	$-0.764969\pi$
$757$	−2.33210e7	−1.47913	−0.739565	+	0.673085i	$0.764969\pi$
$758$	0	0
$759$	3.21020e7	2.02269
$760$	0	0
$761$	1.44859e7	0.906745	0.453373	−	0.891321i	$-0.350221\pi$
$761$	1.44859e7	0.906745	0.453373	+	0.891321i	$0.350221\pi$
$762$	0	0
$763$	−2.44101e6	−0.151795
$764$	0	0
$765$	−1.44285e6	−0.0891391
$766$	0	0
$767$	4.47045e6	0.274387
$768$	0	0
$769$	1.59424e7	0.972162	0.486081	−	0.873914i	$-0.338426\pi$
$769$	1.59424e7	0.972162	0.486081	+	0.873914i	$0.338426\pi$
$770$	0	0
$771$	3.04036e6	0.184200
$772$	0	0
$773$	−3.36066e6	−0.202290	−0.101145	−	0.994872i	$-0.532251\pi$
$773$	−3.36066e6	−0.202290	−0.101145	+	0.994872i	$0.532251\pi$
$774$	0	0
$775$	2.45173e6	0.146628
$776$	0	0
$777$	−417778.	−0.0248252
$778$	0	0
$779$	−1.07707e7	−0.635916
$780$	0	0
$781$	−1.31326e7	−0.770411
$782$	0	0
$783$	2.76464e7	1.61151
$784$	0	0
$785$	1.02660e7	0.594601
$786$	0	0
$787$	3.28918e6	0.189300	0.0946500	−	0.995511i	$-0.469827\pi$
$787$	3.28918e6	0.189300	0.0946500	+	0.995511i	$0.469827\pi$
$788$	0	0
$789$	4.07147e6	0.232841
$790$	0	0
$791$	−1.82088e6	−0.103476
$792$	0	0
$793$	2.45675e7	1.38732
$794$	0	0
$795$	−8.82240e6	−0.495073
$796$	0	0
$797$	−2.51939e6	−0.140492	−0.0702458	−	0.997530i	$-0.522378\pi$
$797$	−2.51939e6	−0.140492	−0.0702458	+	0.997530i	$0.522378\pi$
$798$	0	0
$799$	−4.25816e6	−0.235969
$800$	0	0
$801$	−246917.	−0.0135978
$802$	0	0
$803$	7.00682e6	0.383470
$804$	0	0
$805$	4.18347e6	0.227534
$806$	0	0
$807$	−1.82614e6	−0.0987077
$808$	0	0
$809$	8.93808e6	0.480146	0.240073	−	0.970755i	$-0.422829\pi$
$809$	8.93808e6	0.480146	0.240073	+	0.970755i	$0.422829\pi$
$810$	0	0
$811$	−3.01341e7	−1.60881	−0.804406	−	0.594080i	$-0.797516\pi$
$811$	−3.01341e7	−1.60881	−0.804406	+	0.594080i	$0.797516\pi$
$812$	0	0
$813$	2.62915e7	1.39505
$814$	0	0
$815$	1.96313e6	0.103528
$816$	0	0
$817$	−1.14323e7	−0.599207
$818$	0	0
$819$	1.43252e6	0.0746261
$820$	0	0
$821$	−3.25440e7	−1.68505	−0.842524	−	0.538658i	$-0.818931\pi$
$821$	−3.25440e7	−1.68505	−0.842524	+	0.538658i	$0.818931\pi$
$822$	0	0
$823$	499640.	0.0257133	0.0128566	−	0.999917i	$-0.495907\pi$
$823$	499640.	0.0257133	0.0128566	+	0.999917i	$0.495907\pi$
$824$	0	0
$825$	−5.87506e6	−0.300523
$826$	0	0
$827$	4.64084e6	0.235957	0.117978	−	0.993016i	$-0.462359\pi$
$827$	4.64084e6	0.235957	0.117978	+	0.993016i	$0.462359\pi$
$828$	0	0
$829$	−3.08794e7	−1.56057	−0.780283	−	0.625427i	$-0.784925\pi$
$829$	−3.08794e7	−1.56057	−0.780283	+	0.625427i	$0.784925\pi$
$830$	0	0
$831$	−2.41417e6	−0.121273
$832$	0	0
$833$	−2.38014e6	−0.118848
$834$	0	0
$835$	−1.49428e7	−0.741681
$836$	0	0
$837$	1.60621e7	0.792482
$838$	0	0
$839$	−1.93485e7	−0.948948	−0.474474	−	0.880269i	$-0.657362\pi$
$839$	−1.93485e7	−0.948948	−0.474474	+	0.880269i	$0.657362\pi$
$840$	0	0
$841$	2.50772e7	1.22261
$842$	0	0
$843$	−2.51656e7	−1.21966
$844$	0	0
$845$	2.97846e6	0.143500
$846$	0	0
$847$	1.55403e7	0.744303
$848$	0	0
$849$	3.26039e7	1.55239
$850$	0	0
$851$	−2.14201e6	−0.101391
$852$	0	0
$853$	−3.05998e7	−1.43994	−0.719972	−	0.694003i	$-0.755845\pi$
$853$	−3.05998e7	−1.43994	−0.719972	+	0.694003i	$0.755845\pi$
$854$	0	0
$855$	−963068.	−0.0450549
$856$	0	0
$857$	−1.33039e7	−0.618766	−0.309383	−	0.950937i	$-0.600123\pi$
$857$	−1.33039e7	−0.618766	−0.309383	+	0.950937i	$0.600123\pi$
$858$	0	0
$859$	1.27708e7	0.590520	0.295260	−	0.955417i	$-0.404594\pi$
$859$	1.27708e7	0.590520	0.295260	+	0.955417i	$0.404594\pi$
$860$	0	0
$861$	−1.08423e7	−0.498440
$862$	0	0
$863$	−1.37987e7	−0.630684	−0.315342	−	0.948978i	$-0.602119\pi$
$863$	−1.37987e7	−0.630684	−0.315342	+	0.948978i	$0.602119\pi$
$864$	0	0
$865$	1.49685e6	0.0680203
$866$	0	0
$867$	5.94241e6	0.268482
$868$	0	0
$869$	6.70680e7	3.01277
$870$	0	0
$871$	−2.13806e6	−0.0954934
$872$	0	0
$873$	6.07673e6	0.269857
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	6.68992e6	0.293712	0.146856	−	0.989158i	$-0.453085\pi$
$877$	6.68992e6	0.293712	0.146856	+	0.989158i	$0.453085\pi$
$878$	0	0
$879$	−3.39728e7	−1.48306
$880$	0	0
$881$	−7.25051e6	−0.314723	−0.157362	−	0.987541i	$-0.550299\pi$
$881$	−7.25051e6	−0.314723	−0.157362	+	0.987541i	$0.550299\pi$
$882$	0	0
$883$	2.55944e7	1.10470	0.552348	−	0.833613i	$-0.313732\pi$
$883$	2.55944e7	1.10470	0.552348	+	0.833613i	$0.313732\pi$
$884$	0	0
$885$	−3.02542e6	−0.129846
$886$	0	0
$887$	2.33204e7	0.995240	0.497620	−	0.867395i	$-0.334207\pi$
$887$	2.33204e7	0.995240	0.497620	+	0.867395i	$0.334207\pi$
$888$	0	0
$889$	2.27900e6	0.0967141
$890$	0	0
$891$	−2.87064e7	−1.21139
$892$	0	0
$893$	−2.84222e6	−0.119269
$894$	0	0
$895$	1.54203e7	0.643480
$896$	0	0
$897$	−2.33110e7	−0.967343
$898$	0	0
$899$	2.64862e7	1.09300
$900$	0	0
$901$	2.57354e7	1.05613
$902$	0	0
$903$	−1.15083e7	−0.469667
$904$	0	0
$905$	932174.	0.0378334
$906$	0	0
$907$	−3.71721e7	−1.50037	−0.750187	−	0.661226i	$-0.770036\pi$
$907$	−3.71721e7	−1.50037	−0.750187	+	0.661226i	$0.770036\pi$
$908$	0	0
$909$	2.66152e6	0.106837
$910$	0	0
$911$	2.89923e7	1.15741	0.578704	−	0.815537i	$-0.303559\pi$
$911$	2.89923e7	1.15741	0.578704	+	0.815537i	$0.303559\pi$
$912$	0	0
$913$	−4.89127e7	−1.94198
$914$	0	0
$915$	−1.66263e7	−0.656512
$916$	0	0
$917$	−1.87068e7	−0.734641
$918$	0	0
$919$	1.25020e6	0.0488306	0.0244153	−	0.999702i	$-0.492228\pi$
$919$	1.25020e6	0.0488306	0.0244153	+	0.999702i	$0.492228\pi$
$920$	0	0
$921$	4.17582e7	1.62216
$922$	0	0
$923$	9.53627e6	0.368447
$924$	0	0
$925$	392014.	0.0150642
$926$	0	0
$927$	5.19776e6	0.198663
$928$	0	0
$929$	6.87875e6	0.261499	0.130749	−	0.991415i	$-0.458262\pi$
$929$	6.87875e6	0.261499	0.130749	+	0.991415i	$0.458262\pi$
$930$	0	0
$931$	−1.58869e6	−0.0600709
$932$	0	0
$933$	8.98840e6	0.338048
$934$	0	0
$935$	1.71378e7	0.641101
$936$	0	0
$937$	2.18600e7	0.813396	0.406698	−	0.913563i	$-0.366680\pi$
$937$	2.18600e7	0.813396	0.406698	+	0.913563i	$0.366680\pi$
$938$	0	0
$939$	4.47756e7	1.65721
$940$	0	0
$941$	2.26450e6	0.0833679	0.0416840	−	0.999131i	$-0.486728\pi$
$941$	2.26450e6	0.0833679	0.0416840	+	0.999131i	$0.486728\pi$
$942$	0	0
$943$	−5.55901e7	−2.03572
$944$	0	0
$945$	−5.01588e6	−0.182712
$946$	0	0
$947$	2.45846e7	0.890815	0.445408	−	0.895328i	$-0.353059\pi$
$947$	2.45846e7	0.890815	0.445408	+	0.895328i	$0.353059\pi$
$948$	0	0
$949$	−5.08803e6	−0.183394
$950$	0	0
$951$	−8.69601e6	−0.311795
$952$	0	0
$953$	4.59681e7	1.63955	0.819774	−	0.572687i	$-0.194099\pi$
$953$	4.59681e7	1.63955	0.819774	+	0.572687i	$0.194099\pi$
$954$	0	0
$955$	8.16901e6	0.289842
$956$	0	0
$957$	−6.34686e7	−2.24016
$958$	0	0
$959$	−92832.6	−0.00325952
$960$	0	0
$961$	−1.32411e7	−0.462503
$962$	0	0
$963$	−6.19051e6	−0.215110
$964$	0	0
$965$	6.64338e6	0.229652
$966$	0	0
$967$	5.30202e7	1.82337	0.911686	−	0.410888i	$-0.134781\pi$
$967$	5.30202e7	1.82337	0.911686	+	0.410888i	$0.134781\pi$
$968$	0	0
$969$	−8.91630e6	−0.305053
$970$	0	0
$971$	5.43523e7	1.84999	0.924996	−	0.379976i	$-0.124068\pi$
$971$	5.43523e7	1.84999	0.924996	+	0.379976i	$0.124068\pi$
$972$	0	0
$973$	−9.88809e6	−0.334835
$974$	0	0
$975$	4.26620e6	0.143724
$976$	0	0
$977$	1.08929e7	0.365097	0.182549	−	0.983197i	$-0.441565\pi$
$977$	1.08929e7	0.365097	0.182549	+	0.983197i	$0.441565\pi$
$978$	0	0
$979$	2.93282e6	0.0977976
$980$	0	0
$981$	2.90031e6	0.0962215
$982$	0	0
$983$	3.42136e7	1.12932	0.564658	−	0.825325i	$-0.309008\pi$
$983$	3.42136e7	1.12932	0.564658	+	0.825325i	$0.309008\pi$
$984$	0	0
$985$	1.29466e7	0.425173
$986$	0	0
$987$	−2.86111e6	−0.0934850
$988$	0	0
$989$	−5.90047e7	−1.91821
$990$	0	0
$991$	−4.22117e6	−0.136537	−0.0682683	−	0.997667i	$-0.521747\pi$
$991$	−4.22117e6	−0.136537	−0.0682683	+	0.997667i	$0.521747\pi$
$992$	0	0
$993$	−1.54796e7	−0.498181
$994$	0	0
$995$	−3.71716e6	−0.119029
$996$	0	0
$997$	4.30323e7	1.37106	0.685530	−	0.728044i	$-0.259571\pi$
$997$	4.30323e7	1.37106	0.685530	+	0.728044i	$0.259571\pi$
$998$	0	0
$999$	2.56822e6	0.0814177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.l.1.1		2
4.3	odd	2		35.6.a.b.1.1	✓	2
12.11	even	2		315.6.a.c.1.2		2
20.3	even	4		175.6.b.d.99.3		4
20.7	even	4		175.6.b.d.99.2		4
20.19	odd	2		175.6.a.d.1.2		2
28.27	even	2		245.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.1	✓	2	4.3	odd	2
175.6.a.d.1.2		2	20.19	odd	2
175.6.b.d.99.2		4	20.7	even	4
175.6.b.d.99.3		4	20.3	even	4
245.6.a.c.1.1		2	28.27	even	2
315.6.a.c.1.2		2	12.11	even	2
560.6.a.l.1.1		2	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)

\(f(q)\)	\(=\)	\(q-13.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -58.2198 q^{9} +O(q^{10})\)	\(q-13.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -58.2198 q^{9} +691.520 q^{11} -502.150 q^{13} +339.835 q^{15} -991.313 q^{17} -661.677 q^{19} -666.076 q^{21} -3415.08 q^{23} +625.000 q^{25} +4094.60 q^{27} +6751.92 q^{29} +3922.76 q^{31} -9400.09 q^{33} -1225.00 q^{35} +627.222 q^{37} +6825.92 q^{39} +16277.9 q^{41} +17277.7 q^{43} +1455.50 q^{45} +4295.47 q^{47} +2401.00 q^{49} +13475.3 q^{51} -25960.9 q^{53} -17288.0 q^{55} +8994.43 q^{57} -8902.63 q^{59} -48924.6 q^{61} -2852.77 q^{63} +12553.7 q^{65} +4257.80 q^{67} +46422.5 q^{69} -18990.9 q^{71} +10132.5 q^{73} -8495.87 q^{75} +33884.5 q^{77} +96986.5 q^{79} -41512.0 q^{81} -70732.1 q^{83} +24782.8 q^{85} -91781.4 q^{87} +4241.12 q^{89} -24605.3 q^{91} -53323.6 q^{93} +16541.9 q^{95} -104376. q^{97} -40260.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9	\( 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9} + 601 q^{11} - 577 q^{13} + 75 q^{15} + 41 q^{17} - 630 q^{19} - 147 q^{21} + 442 q^{23} + 1250 q^{25} + 135 q^{27} + 5885 q^{29} + 396 q^{31} - 10359 q^{33} - 2450 q^{35} - 8904 q^{37} + 6033 q^{39} + 1774 q^{41} + 27122 q^{43} + 4725 q^{45} + 21289 q^{47} + 4802 q^{49} + 24411 q^{51} - 55582 q^{53} - 15025 q^{55} + 9330 q^{57} - 59600 q^{59} - 51846 q^{61} - 9261 q^{63} + 14425 q^{65} + 45344 q^{67} + 87282 q^{69} - 80744 q^{71} - 13532 q^{73} - 1875 q^{75} + 29449 q^{77} + 51795 q^{79} - 51678 q^{81} - 109828 q^{83} - 1025 q^{85} - 100965 q^{87} - 37650 q^{89} - 28273 q^{91} - 90684 q^{93} + 15750 q^{95} - 96339 q^{97} - 28422 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 50 * q^5 + 98 * q^7 - 189 * q^9 + 601 * q^11 - 577 * q^13 + 75 * q^15 + 41 * q^17 - 630 * q^19 - 147 * q^21 + 442 * q^23 + 1250 * q^25 + 135 * q^27 + 5885 * q^29 + 396 * q^31 - 10359 * q^33 - 2450 * q^35 - 8904 * q^37 + 6033 * q^39 + 1774 * q^41 + 27122 * q^43 + 4725 * q^45 + 21289 * q^47 + 4802 * q^49 + 24411 * q^51 - 55582 * q^53 - 15025 * q^55 + 9330 * q^57 - 59600 * q^59 - 51846 * q^61 - 9261 * q^63 + 14425 * q^65 + 45344 * q^67 + 87282 * q^69 - 80744 * q^71 - 13532 * q^73 - 1875 * q^75 + 29449 * q^77 + 51795 * q^79 - 51678 * q^81 - 109828 * q^83 - 1025 * q^85 - 100965 * q^87 - 37650 * q^89 - 28273 * q^91 - 90684 * q^93 + 15750 * q^95 - 96339 * q^97 - 28422 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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