LMFDB - Embedded newform 560.6.a.l.1.2

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			1.2
Root			$-3.53113$ 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)

$f(q)$	$=$	$q+10.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -130.780 q^{9} +O(q^{10})$	$q+10.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -130.780 q^{9} -90.5195 q^{11} -74.8502 q^{13} -264.835 q^{15} +1032.31 q^{17} +31.6771 q^{19} +519.076 q^{21} +3857.08 q^{23} +625.000 q^{25} -3959.60 q^{27} -866.917 q^{29} -3526.76 q^{31} -958.908 q^{33} -1225.00 q^{35} -9531.22 q^{37} -792.917 q^{39} -14503.9 q^{41} +9844.30 q^{43} +3269.50 q^{45} +16993.5 q^{47} +2401.00 q^{49} +10935.7 q^{51} -29621.1 q^{53} +2262.99 q^{55} +335.568 q^{57} -50697.4 q^{59} -2921.38 q^{61} -6408.23 q^{63} +1871.25 q^{65} +41086.2 q^{67} +40859.5 q^{69} -61753.1 q^{71} -23664.5 q^{73} +6620.87 q^{75} -4435.46 q^{77} -45191.5 q^{79} -10166.0 q^{81} -39095.9 q^{83} -25807.8 q^{85} -9183.58 q^{87} -41891.1 q^{89} -3667.66 q^{91} -37360.4 q^{93} -791.927 q^{95} +8036.53 q^{97} +11838.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	10.5934	0.679566	0.339783	−	0.940504i	$-0.389646\pi$
$3$	10.5934	0.679566	0.339783	+	0.940504i	$0.389646\pi$
$4$	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
$9$	−130.780	−0.538190
$10$	0	0
$11$	−90.5195	−0.225559	−0.112780	−	0.993620i	$-0.535975\pi$
$11$	−90.5195	−0.225559	−0.112780	+	0.993620i	$0.535975\pi$
$12$	0	0
$13$	−74.8502	−0.122838	−0.0614192	−	0.998112i	$-0.519563\pi$
$13$	−74.8502	−0.122838	−0.0614192	+	0.998112i	$0.519563\pi$
$14$	0	0
$15$	−264.835	−0.303911
$16$	0	0
$17$	1032.31	0.866342	0.433171	−	0.901312i	$-0.357395\pi$
$17$	1032.31	0.866342	0.433171	+	0.901312i	$0.357395\pi$
$18$	0	0
$19$	31.6771	0.0201308	0.0100654	−	0.999949i	$-0.496796\pi$
$19$	31.6771	0.0201308	0.0100654	+	0.999949i	$0.496796\pi$
$20$	0	0
$21$	519.076	0.256852
$22$	0	0
$23$	3857.08	1.52033	0.760167	−	0.649728i	$-0.225117\pi$
$23$	3857.08	1.52033	0.760167	+	0.649728i	$0.225117\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3959.60	−1.04530
$28$	0	0
$29$	−866.917	−0.191418	−0.0957089	−	0.995409i	$-0.530512\pi$
$29$	−866.917	−0.191418	−0.0957089	+	0.995409i	$0.530512\pi$
$30$	0	0
$31$	−3526.76	−0.659131	−0.329566	−	0.944133i	$-0.606902\pi$
$31$	−3526.76	−0.659131	−0.329566	+	0.944133i	$0.606902\pi$
$32$	0	0
$33$	−958.908	−0.153282
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−9531.22	−1.14458	−0.572288	−	0.820053i	$-0.693944\pi$
$37$	−9531.22	−1.14458	−0.572288	+	0.820053i	$0.693944\pi$
$38$	0	0
$39$	−792.917	−0.0834769
$40$	0	0
$41$	−14503.9	−1.34748	−0.673742	−	0.738967i	$-0.735314\pi$
$41$	−14503.9	−1.34748	−0.673742	+	0.738967i	$0.735314\pi$
$42$	0	0
$43$	9844.30	0.811921	0.405960	−	0.913891i	$-0.366937\pi$
$43$	9844.30	0.811921	0.405960	+	0.913891i	$0.366937\pi$
$44$	0	0
$45$	3269.50	0.240686
$46$	0	0
$47$	16993.5	1.12212	0.561059	−	0.827775i	$-0.310394\pi$
$47$	16993.5	1.12212	0.561059	+	0.827775i	$0.310394\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	10935.7	0.588736
$52$	0	0
$53$	−29621.1	−1.44848	−0.724239	−	0.689549i	$-0.757809\pi$
$53$	−29621.1	−1.44848	−0.724239	+	0.689549i	$0.757809\pi$
$54$	0	0
$55$	2262.99	0.100873
$56$	0	0
$57$	335.568	0.0136802
$58$	0	0
$59$	−50697.4	−1.89607	−0.948037	−	0.318159i	$-0.896935\pi$
$59$	−50697.4	−1.89607	−0.948037	+	0.318159i	$0.896935\pi$
$60$	0	0
$61$	−2921.38	−0.100522	−0.0502612	−	0.998736i	$-0.516005\pi$
$61$	−2921.38	−0.100522	−0.0502612	+	0.998736i	$0.516005\pi$
$62$	0	0
$63$	−6408.23	−0.203417
$64$	0	0
$65$	1871.25	0.0549350
$66$	0	0
$67$	41086.2	1.11817	0.559086	−	0.829109i	$-0.311152\pi$
$67$	41086.2	1.11817	0.559086	+	0.829109i	$0.311152\pi$
$68$	0	0
$69$	40859.5	1.03317
$70$	0	0
$71$	−61753.1	−1.45383	−0.726914	−	0.686729i	$-0.759046\pi$
$71$	−61753.1	−1.45383	−0.726914	+	0.686729i	$0.759046\pi$
$72$	0	0
$73$	−23664.5	−0.519745	−0.259872	−	0.965643i	$-0.583680\pi$
$73$	−23664.5	−0.519745	−0.259872	+	0.965643i	$0.583680\pi$
$74$	0	0
$75$	6620.87	0.135913
$76$	0	0
$77$	−4435.46	−0.0852533
$78$	0	0
$79$	−45191.5	−0.814683	−0.407341	−	0.913276i	$-0.633544\pi$
$79$	−45191.5	−0.814683	−0.407341	+	0.913276i	$0.633544\pi$
$80$	0	0
$81$	−10166.0	−0.172162
$82$	0	0
$83$	−39095.9	−0.622925	−0.311462	−	0.950259i	$-0.600819\pi$
$83$	−39095.9	−0.622925	−0.311462	+	0.950259i	$0.600819\pi$
$84$	0	0
$85$	−25807.8	−0.387440
$86$	0	0
$87$	−9183.58	−0.130081
$88$	0	0
$89$	−41891.1	−0.560592	−0.280296	−	0.959914i	$-0.590433\pi$
$89$	−41891.1	−0.560592	−0.280296	+	0.959914i	$0.590433\pi$
$90$	0	0
$91$	−3667.66	−0.0464286
$92$	0	0
$93$	−37360.4	−0.447923
$94$	0	0
$95$	−791.927	−0.00900277
$96$	0	0
$97$	8036.53	0.0867240	0.0433620	−	0.999059i	$-0.486193\pi$
$97$	8036.53	0.0867240	0.0433620	+	0.999059i	$0.486193\pi$
$98$	0	0
$99$	11838.2	0.121394
$100$	0	0
$101$	124979.	1.21908	0.609542	−	0.792754i	$-0.291353\pi$
$101$	124979.	1.21908	0.609542	+	0.792754i	$0.291353\pi$
$102$	0	0
$103$	76635.1	0.711762	0.355881	−	0.934531i	$-0.384181\pi$
$103$	76635.1	0.711762	0.355881	+	0.934531i	$0.384181\pi$
$104$	0	0
$105$	−12976.9	−0.114868
$106$	0	0
$107$	−135296.	−1.14242	−0.571209	−	0.820804i	$-0.693526\pi$
$107$	−135296.	−1.14242	−0.571209	+	0.820804i	$0.693526\pi$
$108$	0	0
$109$	−199508.	−1.60840	−0.804202	−	0.594356i	$-0.797407\pi$
$109$	−199508.	−1.60840	−0.804202	+	0.594356i	$0.797407\pi$
$110$	0	0
$111$	−100968.	−0.777814
$112$	0	0
$113$	122569.	0.902991	0.451496	−	0.892273i	$-0.350891\pi$
$113$	122569.	0.902991	0.451496	+	0.892273i	$0.350891\pi$
$114$	0	0
$115$	−96427.0	−0.679914
$116$	0	0
$117$	9788.92	0.0661104
$118$	0	0
$119$	50583.4	0.327446
$120$	0	0
$121$	−152857.	−0.949123
$122$	0	0
$123$	−153645.	−0.915704
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−40046.2	−0.220319	−0.110160	−	0.993914i	$-0.535136\pi$
$127$	−40046.2	−0.220319	−0.110160	+	0.993914i	$0.535136\pi$
$128$	0	0
$129$	104284.	0.551754
$130$	0	0
$131$	−211963.	−1.07915	−0.539576	−	0.841937i	$-0.681415\pi$
$131$	−211963.	−1.07915	−0.539576	+	0.841937i	$0.681415\pi$
$132$	0	0
$133$	1552.18	0.00760873
$134$	0	0
$135$	98989.9	0.467473
$136$	0	0
$137$	40980.5	0.186542	0.0932709	−	0.995641i	$-0.470268\pi$
$137$	40980.5	0.186542	0.0932709	+	0.995641i	$0.470268\pi$
$138$	0	0
$139$	418948.	1.83917	0.919587	−	0.392886i	$-0.128523\pi$
$139$	418948.	1.83917	0.919587	+	0.392886i	$0.128523\pi$
$140$	0	0
$141$	180019.	0.762554
$142$	0	0
$143$	6775.40	0.0277073
$144$	0	0
$145$	21672.9	0.0856047
$146$	0	0
$147$	25434.7	0.0970809
$148$	0	0
$149$	−64962.7	−0.239717	−0.119858	−	0.992791i	$-0.538244\pi$
$149$	−64962.7	−0.239717	−0.119858	+	0.992791i	$0.538244\pi$
$150$	0	0
$151$	−379801.	−1.35554	−0.677772	−	0.735272i	$-0.737054\pi$
$151$	−379801.	−1.35554	−0.677772	+	0.735272i	$0.737054\pi$
$152$	0	0
$153$	−135006.	−0.466256
$154$	0	0
$155$	88169.1	0.294773
$156$	0	0
$157$	−281546.	−0.911590	−0.455795	−	0.890085i	$-0.650645\pi$
$157$	−281546.	−0.911590	−0.455795	+	0.890085i	$0.650645\pi$
$158$	0	0
$159$	−313788.	−0.984337
$160$	0	0
$161$	188997.	0.574632
$162$	0	0
$163$	382587.	1.12788	0.563938	−	0.825817i	$-0.309286\pi$
$163$	382587.	1.12788	0.563938	+	0.825817i	$0.309286\pi$
$164$	0	0
$165$	23972.7	0.0685499
$166$	0	0
$167$	388635.	1.07833	0.539164	−	0.842201i	$-0.318740\pi$
$167$	388635.	1.07833	0.539164	+	0.842201i	$0.318740\pi$
$168$	0	0
$169$	−365690.	−0.984911
$170$	0	0
$171$	−4142.73	−0.0108342
$172$	0	0
$173$	−389693.	−0.989936	−0.494968	−	0.868911i	$-0.664820\pi$
$173$	−389693.	−0.989936	−0.494968	+	0.868911i	$0.664820\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−537057.	−1.28851
$178$	0	0
$179$	149812.	0.349473	0.174737	−	0.984615i	$-0.444093\pi$
$179$	149812.	0.349473	0.174737	+	0.984615i	$0.444093\pi$
$180$	0	0
$181$	−821019.	−1.86276	−0.931380	−	0.364049i	$-0.881394\pi$
$181$	−821019.	−1.86276	−0.931380	+	0.364049i	$0.881394\pi$
$182$	0	0
$183$	−30947.3	−0.0683117
$184$	0	0
$185$	238281.	0.511870
$186$	0	0
$187$	−93444.5	−0.195411
$188$	0	0
$189$	−194020.	−0.395087
$190$	0	0
$191$	−286619.	−0.568487	−0.284244	−	0.958752i	$-0.591742\pi$
$191$	−286619.	−0.568487	−0.284244	+	0.958752i	$0.591742\pi$
$192$	0	0
$193$	993573.	1.92002	0.960012	−	0.279959i	$-0.0903209\pi$
$193$	993573.	1.92002	0.960012	+	0.279959i	$0.0903209\pi$
$194$	0	0
$195$	19822.9	0.0373320
$196$	0	0
$197$	−38209.0	−0.0701456	−0.0350728	−	0.999385i	$-0.511166\pi$
$197$	−38209.0	−0.0701456	−0.0350728	+	0.999385i	$0.511166\pi$
$198$	0	0
$199$	−730487.	−1.30761	−0.653807	−	0.756661i	$-0.726829\pi$
$199$	−730487.	−1.30761	−0.653807	+	0.756661i	$0.726829\pi$
$200$	0	0
$201$	435242.	0.759872
$202$	0	0
$203$	−42478.9	−0.0723491
$204$	0	0
$205$	362596.	0.602613
$206$	0	0
$207$	−504429.	−0.818228
$208$	0	0
$209$	−2867.39	−0.00454069
$210$	0	0
$211$	−194286.	−0.300424	−0.150212	−	0.988654i	$-0.547996\pi$
$211$	−194286.	−0.300424	−0.150212	+	0.988654i	$0.547996\pi$
$212$	0	0
$213$	−654175.	−0.987972
$214$	0	0
$215$	−246107.	−0.363102
$216$	0	0
$217$	−172811.	−0.249128
$218$	0	0
$219$	−250687.	−0.353201
$220$	0	0
$221$	−77268.8	−0.106420
$222$	0	0
$223$	−1.13569e6	−1.52931	−0.764656	−	0.644438i	$-0.777091\pi$
$223$	−1.13569e6	−1.52931	−0.764656	+	0.644438i	$0.777091\pi$
$224$	0	0
$225$	−81737.6	−0.107638
$226$	0	0
$227$	143806.	0.185231	0.0926155	−	0.995702i	$-0.470477\pi$
$227$	143806.	0.185231	0.0926155	+	0.995702i	$0.470477\pi$
$228$	0	0
$229$	−3832.50	−0.00482940	−0.00241470	−	0.999997i	$-0.500769\pi$
$229$	−3832.50	−0.00482940	−0.00241470	+	0.999997i	$0.500769\pi$
$230$	0	0
$231$	−46986.5	−0.0579353
$232$	0	0
$233$	−1.35599e6	−1.63631	−0.818157	−	0.574995i	$-0.805004\pi$
$233$	−1.35599e6	−1.63631	−0.818157	+	0.574995i	$0.805004\pi$
$234$	0	0
$235$	−424838.	−0.501827
$236$	0	0
$237$	−478731.	−0.553631
$238$	0	0
$239$	478372.	0.541716	0.270858	−	0.962619i	$-0.412693\pi$
$239$	478372.	0.541716	0.270858	+	0.962619i	$0.412693\pi$
$240$	0	0
$241$	−1.31082e6	−1.45379	−0.726894	−	0.686750i	$-0.759037\pi$
$241$	−1.31082e6	−1.45379	−0.726894	+	0.686750i	$0.759037\pi$
$242$	0	0
$243$	854490.	0.928307
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−2371.04	−0.00247284
$248$	0	0
$249$	−414157.	−0.423318
$250$	0	0
$251$	−340113.	−0.340753	−0.170376	−	0.985379i	$-0.554498\pi$
$251$	−340113.	−0.340753	−0.170376	+	0.985379i	$0.554498\pi$
$252$	0	0
$253$	−349141.	−0.342925
$254$	0	0
$255$	−273392.	−0.263291
$256$	0	0
$257$	−58839.6	−0.0555696	−0.0277848	−	0.999614i	$-0.508845\pi$
$257$	−58839.6	−0.0555696	−0.0277848	+	0.999614i	$0.508845\pi$
$258$	0	0
$259$	−467030.	−0.432609
$260$	0	0
$261$	113376.	0.103019
$262$	0	0
$263$	1380.79	0.00123095	0.000615473	−	1.00000i	$-0.499804\pi$
$263$	1380.79	0.00123095	0.000615473		1.00000i	$0.499804\pi$
$264$	0	0
$265$	740528.	0.647779
$266$	0	0
$267$	−443769.	−0.380959
$268$	0	0
$269$	886839.	0.747247	0.373624	−	0.927580i	$-0.378115\pi$
$269$	886839.	0.747247	0.373624	+	0.927580i	$0.378115\pi$
$270$	0	0
$271$	376955.	0.311793	0.155897	−	0.987773i	$-0.450173\pi$
$271$	376955.	0.311793	0.155897	+	0.987773i	$0.450173\pi$
$272$	0	0
$273$	−38852.9	−0.0315513
$274$	0	0
$275$	−56574.7	−0.0451118
$276$	0	0
$277$	−880363.	−0.689386	−0.344693	−	0.938716i	$-0.612017\pi$
$277$	−880363.	−0.689386	−0.344693	+	0.938716i	$0.612017\pi$
$278$	0	0
$279$	461231.	0.354738
$280$	0	0
$281$	2.08492e6	1.57515	0.787577	−	0.616217i	$-0.211335\pi$
$281$	2.08492e6	1.57515	0.787577	+	0.616217i	$0.211335\pi$
$282$	0	0
$283$	−692401.	−0.513915	−0.256958	−	0.966423i	$-0.582720\pi$
$283$	−692401.	−0.513915	−0.256958	+	0.966423i	$0.582720\pi$
$284$	0	0
$285$	−8389.19	−0.00611798
$286$	0	0
$287$	−710689.	−0.509301
$288$	0	0
$289$	−354186.	−0.249452
$290$	0	0
$291$	85134.1	0.0589347
$292$	0	0
$293$	2.27624e6	1.54899	0.774495	−	0.632580i	$-0.218004\pi$
$293$	2.27624e6	1.54899	0.774495	+	0.632580i	$0.218004\pi$
$294$	0	0
$295$	1.26743e6	0.847950
$296$	0	0
$297$	358421.	0.235777
$298$	0	0
$299$	−288703.	−0.186755
$300$	0	0
$301$	482371.	0.306877
$302$	0	0
$303$	1.32395e6	0.828448
$304$	0	0
$305$	73034.5	0.0449550
$306$	0	0
$307$	2.65187e6	1.60586	0.802928	−	0.596076i	$-0.203274\pi$
$307$	2.65187e6	1.60586	0.802928	+	0.596076i	$0.203274\pi$
$308$	0	0
$309$	811825.	0.483689
$310$	0	0
$311$	−2.57626e6	−1.51039	−0.755195	−	0.655501i	$-0.772458\pi$
$311$	−2.57626e6	−1.51039	−0.755195	+	0.655501i	$0.772458\pi$
$312$	0	0
$313$	2.51549e6	1.45131	0.725657	−	0.688057i	$-0.241536\pi$
$313$	2.51549e6	1.45131	0.725657	+	0.688057i	$0.241536\pi$
$314$	0	0
$315$	160206.	0.0909707
$316$	0	0
$317$	−2.30397e6	−1.28774	−0.643870	−	0.765135i	$-0.722672\pi$
$317$	−2.30397e6	−1.28774	−0.643870	+	0.765135i	$0.722672\pi$
$318$	0	0
$319$	78472.9	0.0431760
$320$	0	0
$321$	−1.43324e6	−0.776349
$322$	0	0
$323$	32700.7	0.0174402
$324$	0	0
$325$	−46781.4	−0.0245677
$326$	0	0
$327$	−2.11347e6	−1.09302
$328$	0	0
$329$	832683.	0.424121
$330$	0	0
$331$	−697305.	−0.349827	−0.174913	−	0.984584i	$-0.555965\pi$
$331$	−697305.	−0.349827	−0.174913	+	0.984584i	$0.555965\pi$
$332$	0	0
$333$	1.24649e6	0.615999
$334$	0	0
$335$	−1.02715e6	−0.500062
$336$	0	0
$337$	−1.14848e6	−0.550868	−0.275434	−	0.961320i	$-0.588822\pi$
$337$	−1.14848e6	−0.550868	−0.275434	+	0.961320i	$0.588822\pi$
$338$	0	0
$339$	1.29842e6	0.613642
$340$	0	0
$341$	319241.	0.148673
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.02149e6	−0.462046
$346$	0	0
$347$	2.43716e6	1.08658	0.543289	−	0.839546i	$-0.317179\pi$
$347$	2.43716e6	1.08658	0.543289	+	0.839546i	$0.317179\pi$
$348$	0	0
$349$	896801.	0.394124	0.197062	−	0.980391i	$-0.436860\pi$
$349$	896801.	0.394124	0.197062	+	0.980391i	$0.436860\pi$
$350$	0	0
$351$	296377.	0.128403
$352$	0	0
$353$	4.33422e6	1.85129	0.925644	−	0.378396i	$-0.123524\pi$
$353$	4.33422e6	1.85129	0.925644	+	0.378396i	$0.123524\pi$
$354$	0	0
$355$	1.54383e6	0.650172
$356$	0	0
$357$	535849.	0.222521
$358$	0	0
$359$	3.59167e6	1.47082	0.735411	−	0.677621i	$-0.236989\pi$
$359$	3.59167e6	1.47082	0.735411	+	0.677621i	$0.236989\pi$
$360$	0	0
$361$	−2.47510e6	−0.999595
$362$	0	0
$363$	−1.61928e6	−0.644992
$364$	0	0
$365$	591612.	0.232437
$366$	0	0
$367$	−361791.	−0.140215	−0.0701073	−	0.997539i	$-0.522334\pi$
$367$	−361791.	−0.140215	−0.0701073	+	0.997539i	$0.522334\pi$
$368$	0	0
$369$	1.89682e6	0.725202
$370$	0	0
$371$	−1.45144e6	−0.547473
$372$	0	0
$373$	−3.08325e6	−1.14746	−0.573729	−	0.819045i	$-0.694504\pi$
$373$	−3.08325e6	−1.14746	−0.573729	+	0.819045i	$0.694504\pi$
$374$	0	0
$375$	−165522.	−0.0607822
$376$	0	0
$377$	64888.9	0.0235135
$378$	0	0
$379$	5.04227e6	1.80313	0.901567	−	0.432639i	$-0.142418\pi$
$379$	5.04227e6	1.80313	0.901567	+	0.432639i	$0.142418\pi$
$380$	0	0
$381$	−424225.	−0.149721
$382$	0	0
$383$	3.11928e6	1.08657	0.543285	−	0.839548i	$-0.317180\pi$
$383$	3.11928e6	1.08657	0.543285	+	0.839548i	$0.317180\pi$
$384$	0	0
$385$	110886.	0.0381265
$386$	0	0
$387$	−1.28744e6	−0.436968
$388$	0	0
$389$	−1.64560e6	−0.551378	−0.275689	−	0.961247i	$-0.588906\pi$
$389$	−1.64560e6	−0.551378	−0.275689	+	0.961247i	$0.588906\pi$
$390$	0	0
$391$	3.98171e6	1.31713
$392$	0	0
$393$	−2.24541e6	−0.733355
$394$	0	0
$395$	1.12979e6	0.364337
$396$	0	0
$397$	3.72727e6	1.18690	0.593451	−	0.804870i	$-0.297765\pi$
$397$	3.72727e6	1.18690	0.593451	+	0.804870i	$0.297765\pi$
$398$	0	0
$399$	16442.8	0.00517063
$400$	0	0
$401$	4.84820e6	1.50564	0.752818	−	0.658229i	$-0.228694\pi$
$401$	4.84820e6	1.50564	0.752818	+	0.658229i	$0.228694\pi$
$402$	0	0
$403$	263979.	0.0809667
$404$	0	0
$405$	254149.	0.0769930
$406$	0	0
$407$	862761.	0.258169
$408$	0	0
$409$	−3.03126e6	−0.896014	−0.448007	−	0.894030i	$-0.647866\pi$
$409$	−3.03126e6	−0.896014	−0.448007	+	0.894030i	$0.647866\pi$
$410$	0	0
$411$	434123.	0.126768
$412$	0	0
$413$	−2.48417e6	−0.716649
$414$	0	0
$415$	977396.	0.278580
$416$	0	0
$417$	4.43808e6	1.24984
$418$	0	0
$419$	−1.66905e6	−0.464444	−0.232222	−	0.972663i	$-0.574600\pi$
$419$	−1.66905e6	−0.464444	−0.232222	+	0.972663i	$0.574600\pi$
$420$	0	0
$421$	−1.76031e6	−0.484043	−0.242022	−	0.970271i	$-0.577810\pi$
$421$	−1.76031e6	−0.484043	−0.242022	+	0.970271i	$0.577810\pi$
$422$	0	0
$423$	−2.22242e6	−0.603913
$424$	0	0
$425$	645196.	0.173268
$426$	0	0
$427$	−143148.	−0.0379939
$428$	0	0
$429$	71774.4	0.0188290
$430$	0	0
$431$	648832.	0.168244	0.0841219	−	0.996455i	$-0.473191\pi$
$431$	648832.	0.168244	0.0841219	+	0.996455i	$0.473191\pi$
$432$	0	0
$433$	−360993.	−0.0925293	−0.0462646	−	0.998929i	$-0.514732\pi$
$433$	−360993.	−0.0925293	−0.0462646	+	0.998929i	$0.514732\pi$
$434$	0	0
$435$	229590.	0.0581740
$436$	0	0
$437$	122181.	0.0306055
$438$	0	0
$439$	−1.07021e6	−0.265038	−0.132519	−	0.991180i	$-0.542307\pi$
$439$	−1.07021e6	−0.265038	−0.132519	+	0.991180i	$0.542307\pi$
$440$	0	0
$441$	−314003.	−0.0768843
$442$	0	0
$443$	797145.	0.192987	0.0964935	−	0.995334i	$-0.469237\pi$
$443$	797145.	0.192987	0.0964935	+	0.995334i	$0.469237\pi$
$444$	0	0
$445$	1.04728e6	0.250704
$446$	0	0
$447$	−688175.	−0.162903
$448$	0	0
$449$	6.40103e6	1.49842	0.749211	−	0.662331i	$-0.230433\pi$
$449$	6.40103e6	1.49842	0.749211	+	0.662331i	$0.230433\pi$
$450$	0	0
$451$	1.31288e6	0.303937
$452$	0	0
$453$	−4.02338e6	−0.921181
$454$	0	0
$455$	91691.5	0.0207635
$456$	0	0
$457$	5.91843e6	1.32561	0.662806	−	0.748791i	$-0.269366\pi$
$457$	5.91843e6	1.32561	0.662806	+	0.748791i	$0.269366\pi$
$458$	0	0
$459$	−4.08755e6	−0.905588
$460$	0	0
$461$	−8.84337e6	−1.93805	−0.969026	−	0.246960i	$-0.920569\pi$
$461$	−8.84337e6	−1.93805	−0.969026	+	0.246960i	$0.920569\pi$
$462$	0	0
$463$	6.76858e6	1.46739	0.733695	−	0.679479i	$-0.237794\pi$
$463$	6.76858e6	1.46739	0.733695	+	0.679479i	$0.237794\pi$
$464$	0	0
$465$	934009.	0.200317
$466$	0	0
$467$	1.04740e6	0.222239	0.111120	−	0.993807i	$-0.464556\pi$
$467$	1.04740e6	0.222239	0.111120	+	0.993807i	$0.464556\pi$
$468$	0	0
$469$	2.01322e6	0.422630
$470$	0	0
$471$	−2.98252e6	−0.619486
$472$	0	0
$473$	−891101.	−0.183136
$474$	0	0
$475$	19798.2	0.00402616
$476$	0	0
$477$	3.87386e6	0.779556
$478$	0	0
$479$	9.37725e6	1.86740	0.933699	−	0.358060i	$-0.116562\pi$
$479$	9.37725e6	1.86740	0.933699	+	0.358060i	$0.116562\pi$
$480$	0	0
$481$	713414.	0.140598
$482$	0	0
$483$	2.00212e6	0.390500
$484$	0	0
$485$	−200913.	−0.0387841
$486$	0	0
$487$	2.88960e6	0.552098	0.276049	−	0.961144i	$-0.410975\pi$
$487$	2.88960e6	0.552098	0.276049	+	0.961144i	$0.410975\pi$
$488$	0	0
$489$	4.05290e6	0.766467
$490$	0	0
$491$	1.04845e6	0.196265	0.0981324	−	0.995173i	$-0.468713\pi$
$491$	1.04845e6	0.196265	0.0981324	+	0.995173i	$0.468713\pi$
$492$	0	0
$493$	−894930.	−0.165833
$494$	0	0
$495$	−295954.	−0.0542889
$496$	0	0
$497$	−3.02590e6	−0.549495
$498$	0	0
$499$	−6.38524e6	−1.14796	−0.573979	−	0.818870i	$-0.694601\pi$
$499$	−6.38524e6	−1.14796	−0.573979	+	0.818870i	$0.694601\pi$
$500$	0	0
$501$	4.11696e6	0.732795
$502$	0	0
$503$	−1.35497e6	−0.238787	−0.119394	−	0.992847i	$-0.538095\pi$
$503$	−1.35497e6	−0.238787	−0.119394	+	0.992847i	$0.538095\pi$
$504$	0	0
$505$	−3.12448e6	−0.545191
$506$	0	0
$507$	−3.87390e6	−0.669312
$508$	0	0
$509$	6.43410e6	1.10076	0.550381	−	0.834914i	$-0.314482\pi$
$509$	6.43410e6	1.10076	0.550381	+	0.834914i	$0.314482\pi$
$510$	0	0
$511$	−1.15956e6	−0.196445
$512$	0	0
$513$	−125429.	−0.0210428
$514$	0	0
$515$	−1.91588e6	−0.318310
$516$	0	0
$517$	−1.53825e6	−0.253104
$518$	0	0
$519$	−4.12817e6	−0.672727
$520$	0	0
$521$	6.16552e6	0.995120	0.497560	−	0.867430i	$-0.334229\pi$
$521$	6.16552e6	0.995120	0.497560	+	0.867430i	$0.334229\pi$
$522$	0	0
$523$	−7.12400e6	−1.13886	−0.569429	−	0.822040i	$-0.692836\pi$
$523$	−7.12400e6	−1.13886	−0.569429	+	0.822040i	$0.692836\pi$
$524$	0	0
$525$	324422.	0.0513704
$526$	0	0
$527$	−3.64072e6	−0.571033
$528$	0	0
$529$	8.44071e6	1.31141
$530$	0	0
$531$	6.63021e6	1.02045
$532$	0	0
$533$	1.08562e6	0.165523
$534$	0	0
$535$	3.38240e6	0.510905
$536$	0	0
$537$	1.58702e6	0.237490
$538$	0	0
$539$	−217337.	−0.0322227
$540$	0	0
$541$	−9.51129e6	−1.39716	−0.698580	−	0.715532i	$-0.746184\pi$
$541$	−9.51129e6	−1.39716	−0.698580	+	0.715532i	$0.746184\pi$
$542$	0	0
$543$	−8.69737e6	−1.26587
$544$	0	0
$545$	4.98771e6	0.719300
$546$	0	0
$547$	−1.48406e6	−0.212071	−0.106036	−	0.994362i	$-0.533816\pi$
$547$	−1.48406e6	−0.212071	−0.106036	+	0.994362i	$0.533816\pi$
$548$	0	0
$549$	382058.	0.0541002
$550$	0	0
$551$	−27461.4	−0.00385340
$552$	0	0
$553$	−2.21438e6	−0.307921
$554$	0	0
$555$	2.52420e6	0.347849
$556$	0	0
$557$	−7.72192e6	−1.05460	−0.527299	−	0.849680i	$-0.676795\pi$
$557$	−7.72192e6	−1.05460	−0.527299	+	0.849680i	$0.676795\pi$
$558$	0	0
$559$	−736847.	−0.0997351
$560$	0	0
$561$	−989894.	−0.132795
$562$	0	0
$563$	−3.41747e6	−0.454395	−0.227198	−	0.973849i	$-0.572956\pi$
$563$	−3.41747e6	−0.454395	−0.227198	+	0.973849i	$0.572956\pi$
$564$	0	0
$565$	−3.06422e6	−0.403830
$566$	0	0
$567$	−498133.	−0.0650710
$568$	0	0
$569$	−2.42117e6	−0.313505	−0.156752	−	0.987638i	$-0.550102\pi$
$569$	−2.42117e6	−0.313505	−0.156752	+	0.987638i	$0.550102\pi$
$570$	0	0
$571$	−2.31941e6	−0.297705	−0.148853	−	0.988859i	$-0.547558\pi$
$571$	−2.31941e6	−0.297705	−0.148853	+	0.988859i	$0.547558\pi$
$572$	0	0
$573$	−3.03626e6	−0.386325
$574$	0	0
$575$	2.41067e6	0.304067
$576$	0	0
$577$	2.56645e6	0.320917	0.160459	−	0.987043i	$-0.448703\pi$
$577$	2.56645e6	0.320917	0.160459	+	0.987043i	$0.448703\pi$
$578$	0	0
$579$	1.05253e7	1.30478
$580$	0	0
$581$	−1.91570e6	−0.235443
$582$	0	0
$583$	2.68129e6	0.326718
$584$	0	0
$585$	−244723.	−0.0295655
$586$	0	0
$587$	−2.09851e6	−0.251371	−0.125685	−	0.992070i	$-0.540113\pi$
$587$	−2.09851e6	−0.251371	−0.125685	+	0.992070i	$0.540113\pi$
$588$	0	0
$589$	−111718.	−0.0132688
$590$	0	0
$591$	−404763.	−0.0476686
$592$	0	0
$593$	−5.11831e6	−0.597709	−0.298854	−	0.954299i	$-0.596604\pi$
$593$	−5.11831e6	−0.597709	−0.298854	+	0.954299i	$0.596604\pi$
$594$	0	0
$595$	−1.26458e6	−0.146438
$596$	0	0
$597$	−7.73833e6	−0.888610
$598$	0	0
$599$	2.61775e6	0.298099	0.149049	−	0.988830i	$-0.452379\pi$
$599$	2.61775e6	0.298099	0.149049	+	0.988830i	$0.452379\pi$
$600$	0	0
$601$	9.49925e6	1.07276	0.536381	−	0.843976i	$-0.319791\pi$
$601$	9.49925e6	1.07276	0.536381	+	0.843976i	$0.319791\pi$
$602$	0	0
$603$	−5.37326e6	−0.601789
$604$	0	0
$605$	3.82143e6	0.424461
$606$	0	0
$607$	−6.07366e6	−0.669081	−0.334541	−	0.942381i	$-0.608581\pi$
$607$	−6.07366e6	−0.669081	−0.334541	+	0.942381i	$0.608581\pi$
$608$	0	0
$609$	−449996.	−0.0491660
$610$	0	0
$611$	−1.27197e6	−0.137839
$612$	0	0
$613$	−9.20907e6	−0.989839	−0.494920	−	0.868939i	$-0.664802\pi$
$613$	−9.20907e6	−0.989839	−0.494920	+	0.868939i	$0.664802\pi$
$614$	0	0
$615$	3.84112e6	0.409515
$616$	0	0
$617$	1.37224e7	1.45117	0.725585	−	0.688133i	$-0.241569\pi$
$617$	1.37224e7	1.45117	0.725585	+	0.688133i	$0.241569\pi$
$618$	0	0
$619$	6.80356e6	0.713690	0.356845	−	0.934164i	$-0.383852\pi$
$619$	6.80356e6	0.713690	0.356845	+	0.934164i	$0.383852\pi$
$620$	0	0
$621$	−1.52725e7	−1.58921
$622$	0	0
$623$	−2.05266e6	−0.211884
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−30375.4	−0.00308570
$628$	0	0
$629$	−9.83921e6	−0.991593
$630$	0	0
$631$	2.80897e6	0.280850	0.140425	−	0.990091i	$-0.455153\pi$
$631$	2.80897e6	0.280850	0.140425	+	0.990091i	$0.455153\pi$
$632$	0	0
$633$	−2.05815e6	−0.204158
$634$	0	0
$635$	1.00115e6	0.0985297
$636$	0	0
$637$	−179715.	−0.0175484
$638$	0	0
$639$	8.07608e6	0.782435
$640$	0	0
$641$	−3.83494e6	−0.368649	−0.184325	−	0.982865i	$-0.559010\pi$
$641$	−3.83494e6	−0.368649	−0.184325	+	0.982865i	$0.559010\pi$
$642$	0	0
$643$	−1.59562e7	−1.52195	−0.760977	−	0.648779i	$-0.775280\pi$
$643$	−1.59562e7	−1.52195	−0.760977	+	0.648779i	$0.775280\pi$
$644$	0	0
$645$	−2.60711e6	−0.246752
$646$	0	0
$647$	7.48025e6	0.702514	0.351257	−	0.936279i	$-0.385754\pi$
$647$	7.48025e6	0.702514	0.351257	+	0.936279i	$0.385754\pi$
$648$	0	0
$649$	4.58910e6	0.427677
$650$	0	0
$651$	−1.83066e6	−0.169299
$652$	0	0
$653$	2.62102e6	0.240540	0.120270	−	0.992741i	$-0.461624\pi$
$653$	2.62102e6	0.240540	0.120270	+	0.992741i	$0.461624\pi$
$654$	0	0
$655$	5.29908e6	0.482611
$656$	0	0
$657$	3.09485e6	0.279721
$658$	0	0
$659$	8.01276e6	0.718734	0.359367	−	0.933196i	$-0.382993\pi$
$659$	8.01276e6	0.718734	0.359367	+	0.933196i	$0.382993\pi$
$660$	0	0
$661$	−7.21439e6	−0.642238	−0.321119	−	0.947039i	$-0.604059\pi$
$661$	−7.21439e6	−0.642238	−0.321119	+	0.947039i	$0.604059\pi$
$662$	0	0
$663$	−818539.	−0.0723195
$664$	0	0
$665$	−38804.4	−0.00340273
$666$	0	0
$667$	−3.34377e6	−0.291019
$668$	0	0
$669$	−1.20308e7	−1.03927
$670$	0	0
$671$	264442.	0.0226738
$672$	0	0
$673$	1.43323e7	1.21977	0.609887	−	0.792488i	$-0.291215\pi$
$673$	1.43323e7	1.21977	0.609887	+	0.792488i	$0.291215\pi$
$674$	0	0
$675$	−2.47475e6	−0.209060
$676$	0	0
$677$	−4.94781e6	−0.414898	−0.207449	−	0.978246i	$-0.566516\pi$
$677$	−4.94781e6	−0.414898	−0.207449	+	0.978246i	$0.566516\pi$
$678$	0	0
$679$	393790.	0.0327786
$680$	0	0
$681$	1.52340e6	0.125877
$682$	0	0
$683$	−7.77331e6	−0.637608	−0.318804	−	0.947821i	$-0.603281\pi$
$683$	−7.77331e6	−0.637608	−0.318804	+	0.947821i	$0.603281\pi$
$684$	0	0
$685$	−1.02451e6	−0.0834241
$686$	0	0
$687$	−40599.1	−0.00328190
$688$	0	0
$689$	2.21715e6	0.177929
$690$	0	0
$691$	7.69151e6	0.612797	0.306398	−	0.951903i	$-0.400876\pi$
$691$	7.69151e6	0.612797	0.306398	+	0.951903i	$0.400876\pi$
$692$	0	0
$693$	580070.	0.0458825
$694$	0	0
$695$	−1.04737e7	−0.822504
$696$	0	0
$697$	−1.49725e7	−1.16738
$698$	0	0
$699$	−1.43645e7	−1.11198
$700$	0	0
$701$	1.53139e7	1.17704	0.588521	−	0.808482i	$-0.299711\pi$
$701$	1.53139e7	1.17704	0.588521	+	0.808482i	$0.299711\pi$
$702$	0	0
$703$	−301921.	−0.0230412
$704$	0	0
$705$	−4.50048e6	−0.341025
$706$	0	0
$707$	6.12397e6	0.460771
$708$	0	0
$709$	1.84381e7	1.37753	0.688764	−	0.724986i	$-0.258154\pi$
$709$	1.84381e7	1.37753	0.688764	+	0.724986i	$0.258154\pi$
$710$	0	0
$711$	5.91015e6	0.438454
$712$	0	0
$713$	−1.36030e7	−1.00210
$714$	0	0
$715$	−169385.	−0.0123911
$716$	0	0
$717$	5.06758e6	0.368132
$718$	0	0
$719$	99951.2	0.00721051	0.00360525	−	0.999994i	$-0.498852\pi$
$719$	99951.2	0.00721051	0.00360525	+	0.999994i	$0.498852\pi$
$720$	0	0
$721$	3.75512e6	0.269021
$722$	0	0
$723$	−1.38860e7	−0.987945
$724$	0	0
$725$	−541823.	−0.0382836
$726$	0	0
$727$	−1.25198e7	−0.878538	−0.439269	−	0.898356i	$-0.644762\pi$
$727$	−1.25198e7	−0.878538	−0.439269	+	0.898356i	$0.644762\pi$
$728$	0	0
$729$	1.15223e7	0.803007
$730$	0	0
$731$	1.01624e7	0.703401
$732$	0	0
$733$	−1.72341e7	−1.18476	−0.592378	−	0.805660i	$-0.701811\pi$
$733$	−1.72341e7	−1.18476	−0.592378	+	0.805660i	$0.701811\pi$
$734$	0	0
$735$	−635868.	−0.0434159
$736$	0	0
$737$	−3.71910e6	−0.252214
$738$	0	0
$739$	1.83842e7	1.23832	0.619160	−	0.785265i	$-0.287473\pi$
$739$	1.83842e7	1.23832	0.619160	+	0.785265i	$0.287473\pi$
$740$	0	0
$741$	−25117.3	−0.00168046
$742$	0	0
$743$	−2.79507e7	−1.85746	−0.928732	−	0.370752i	$-0.879100\pi$
$743$	−2.79507e7	−1.85746	−0.928732	+	0.370752i	$0.879100\pi$
$744$	0	0
$745$	1.62407e6	0.107205
$746$	0	0
$747$	5.11296e6	0.335252
$748$	0	0
$749$	−6.62950e6	−0.431794
$750$	0	0
$751$	−1.70136e7	−1.10077	−0.550383	−	0.834912i	$-0.685518\pi$
$751$	−1.70136e7	−1.10077	−0.550383	+	0.834912i	$0.685518\pi$
$752$	0	0
$753$	−3.60295e6	−0.231564
$754$	0	0
$755$	9.49502e6	0.606218
$756$	0	0
$757$	−1.91600e7	−1.21522	−0.607611	−	0.794234i	$-0.707872\pi$
$757$	−1.91600e7	−1.21522	−0.607611	+	0.794234i	$0.707872\pi$
$758$	0	0
$759$	−3.69858e6	−0.233040
$760$	0	0
$761$	−2.06404e7	−1.29198	−0.645992	−	0.763344i	$-0.723556\pi$
$761$	−2.06404e7	−1.29198	−0.645992	+	0.763344i	$0.723556\pi$
$762$	0	0
$763$	−9.77592e6	−0.607919
$764$	0	0
$765$	3.37515e6	0.208516
$766$	0	0
$767$	3.79471e6	0.232911
$768$	0	0
$769$	1.11748e7	0.681432	0.340716	−	0.940166i	$-0.389331\pi$
$769$	1.11748e7	0.681432	0.340716	+	0.940166i	$0.389331\pi$
$770$	0	0
$771$	−623311.	−0.0377632
$772$	0	0
$773$	−3.07555e7	−1.85129	−0.925645	−	0.378394i	$-0.876477\pi$
$773$	−3.07555e7	−1.85129	−0.925645	+	0.378394i	$0.876477\pi$
$774$	0	0
$775$	−2.20423e6	−0.131826
$776$	0	0
$777$	−4.94743e6	−0.293986
$778$	0	0
$779$	−459440.	−0.0271259
$780$	0	0
$781$	5.58986e6	0.327924
$782$	0	0
$783$	3.43264e6	0.200089
$784$	0	0
$785$	7.03864e6	0.407675
$786$	0	0
$787$	−3.52357e6	−0.202790	−0.101395	−	0.994846i	$-0.532331\pi$
$787$	−3.52357e6	−0.202790	−0.101395	+	0.994846i	$0.532331\pi$
$788$	0	0
$789$	14627.3	0.000836509 0
$790$	0	0
$791$	6.00587e6	0.341299
$792$	0	0
$793$	218666.	0.0123480
$794$	0	0
$795$	7.84470e6	0.440209
$796$	0	0
$797$	−3.57469e6	−0.199339	−0.0996695	−	0.995021i	$-0.531779\pi$
$797$	−3.57469e6	−0.199339	−0.0996695	+	0.995021i	$0.531779\pi$
$798$	0	0
$799$	1.75426e7	0.972139
$800$	0	0
$801$	5.47853e6	0.301705
$802$	0	0
$803$	2.14210e6	0.117233
$804$	0	0
$805$	−4.72492e6	−0.256983
$806$	0	0
$807$	9.39463e6	0.507804
$808$	0	0
$809$	3.38466e7	1.81821	0.909104	−	0.416569i	$-0.136767\pi$
$809$	3.38466e7	1.81821	0.909104	+	0.416569i	$0.136767\pi$
$810$	0	0
$811$	−3.21058e7	−1.71408	−0.857042	−	0.515247i	$-0.827700\pi$
$811$	−3.21058e7	−1.71408	−0.857042	+	0.515247i	$0.827700\pi$
$812$	0	0
$813$	3.99323e6	0.211884
$814$	0	0
$815$	−9.56468e6	−0.504402
$816$	0	0
$817$	311839.	0.0163446
$818$	0	0
$819$	479657.	0.0249874
$820$	0	0
$821$	3.04076e6	0.157443	0.0787216	−	0.996897i	$-0.474916\pi$
$821$	3.04076e6	0.157443	0.0787216	+	0.996897i	$0.474916\pi$
$822$	0	0
$823$	−3.72147e6	−0.191520	−0.0957601	−	0.995404i	$-0.530528\pi$
$823$	−3.72147e6	−0.191520	−0.0957601	+	0.995404i	$0.530528\pi$
$824$	0	0
$825$	−599318.	−0.0306565
$826$	0	0
$827$	8.77895e6	0.446354	0.223177	−	0.974778i	$-0.428357\pi$
$827$	8.77895e6	0.446354	0.223177	+	0.974778i	$0.428357\pi$
$828$	0	0
$829$	6.61553e6	0.334332	0.167166	−	0.985929i	$-0.446538\pi$
$829$	6.61553e6	0.334332	0.167166	+	0.985929i	$0.446538\pi$
$830$	0	0
$831$	−9.32602e6	−0.468483
$832$	0	0
$833$	2.47858e6	0.123763
$834$	0	0
$835$	−9.71588e6	−0.482243
$836$	0	0
$837$	1.39646e7	0.688991
$838$	0	0
$839$	4.67110e6	0.229094	0.114547	−	0.993418i	$-0.463458\pi$
$839$	4.67110e6	0.229094	0.114547	+	0.993418i	$0.463458\pi$
$840$	0	0
$841$	−1.97596e7	−0.963359
$842$	0	0
$843$	2.20863e7	1.07042
$844$	0	0
$845$	9.14226e6	0.440465
$846$	0	0
$847$	−7.49000e6	−0.358735
$848$	0	0
$849$	−7.33487e6	−0.349239
$850$	0	0
$851$	−3.67627e7	−1.74014
$852$	0	0
$853$	−3.66146e7	−1.72299	−0.861494	−	0.507768i	$-0.830471\pi$
$853$	−3.66146e7	−1.72299	−0.861494	+	0.507768i	$0.830471\pi$
$854$	0	0
$855$	103568.	0.00484520
$856$	0	0
$857$	1.15520e7	0.537288	0.268644	−	0.963240i	$-0.413425\pi$
$857$	1.15520e7	0.537288	0.268644	+	0.963240i	$0.413425\pi$
$858$	0	0
$859$	−3.53878e7	−1.63633	−0.818165	−	0.574983i	$-0.805009\pi$
$859$	−3.53878e7	−1.63633	−0.818165	+	0.574983i	$0.805009\pi$
$860$	0	0
$861$	−7.52860e6	−0.346104
$862$	0	0
$863$	−1.07433e7	−0.491034	−0.245517	−	0.969392i	$-0.578958\pi$
$863$	−1.07433e7	−0.491034	−0.245517	+	0.969392i	$0.578958\pi$
$864$	0	0
$865$	9.74232e6	0.442713
$866$	0	0
$867$	−3.75203e6	−0.169519
$868$	0	0
$869$	4.09071e6	0.183759
$870$	0	0
$871$	−3.07531e6	−0.137355
$872$	0	0
$873$	−1.05102e6	−0.0466740
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	4.58293e6	0.201208	0.100604	−	0.994927i	$-0.467923\pi$
$877$	4.58293e6	0.201208	0.100604	+	0.994927i	$0.467923\pi$
$878$	0	0
$879$	2.41131e7	1.05264
$880$	0	0
$881$	1.68070e7	0.729543	0.364772	−	0.931097i	$-0.381147\pi$
$881$	1.68070e7	0.729543	0.364772	+	0.931097i	$0.381147\pi$
$882$	0	0
$883$	5.28260e6	0.228006	0.114003	−	0.993480i	$-0.463633\pi$
$883$	5.28260e6	0.228006	0.114003	+	0.993480i	$0.463633\pi$
$884$	0	0
$885$	1.34264e7	0.576238
$886$	0	0
$887$	−4.43107e7	−1.89104	−0.945518	−	0.325569i	$-0.894444\pi$
$887$	−4.43107e7	−1.89104	−0.945518	+	0.325569i	$0.894444\pi$
$888$	0	0
$889$	−1.96226e6	−0.0832728
$890$	0	0
$891$	920219.	0.0388326
$892$	0	0
$893$	538305.	0.0225892
$894$	0	0
$895$	−3.74530e6	−0.156289
$896$	0	0
$897$	−3.05834e6	−0.126913
$898$	0	0
$899$	3.05741e6	0.126170
$900$	0	0
$901$	−3.05783e7	−1.25488
$902$	0	0
$903$	5.10994e6	0.208543
$904$	0	0
$905$	2.05255e7	0.833051
$906$	0	0
$907$	3.95608e7	1.59679	0.798394	−	0.602135i	$-0.205683\pi$
$907$	3.95608e7	1.59679	0.798394	+	0.602135i	$0.205683\pi$
$908$	0	0
$909$	−1.63448e7	−0.656099
$910$	0	0
$911$	2.40982e7	0.962030	0.481015	−	0.876712i	$-0.340268\pi$
$911$	2.40982e7	0.962030	0.481015	+	0.876712i	$0.340268\pi$
$912$	0	0
$913$	3.53894e6	0.140506
$914$	0	0
$915$	773682.	0.0305499
$916$	0	0
$917$	−1.03862e7	−0.407881
$918$	0	0
$919$	2.24660e7	0.877480	0.438740	−	0.898614i	$-0.355425\pi$
$919$	2.24660e7	0.877480	0.438740	+	0.898614i	$0.355425\pi$
$920$	0	0
$921$	2.80923e7	1.09129
$922$	0	0
$923$	4.62223e6	0.178586
$924$	0	0
$925$	−5.95701e6	−0.228915
$926$	0	0
$927$	−1.00224e7	−0.383063
$928$	0	0
$929$	−1.16280e7	−0.442043	−0.221021	−	0.975269i	$-0.570939\pi$
$929$	−1.16280e7	−0.442043	−0.221021	+	0.975269i	$0.570939\pi$
$930$	0	0
$931$	76056.7	0.00287583
$932$	0	0
$933$	−2.72913e7	−1.02641
$934$	0	0
$935$	2.33611e6	0.0873906
$936$	0	0
$937$	−2.40956e7	−0.896580	−0.448290	−	0.893888i	$-0.647967\pi$
$937$	−2.40956e7	−0.896580	−0.448290	+	0.893888i	$0.647967\pi$
$938$	0	0
$939$	2.66475e7	0.986264
$940$	0	0
$941$	−567590.	−0.0208959	−0.0104479	−	0.999945i	$-0.503326\pi$
$941$	−567590.	−0.0208959	−0.0104479	+	0.999945i	$0.503326\pi$
$942$	0	0
$943$	−5.59425e7	−2.04863
$944$	0	0
$945$	4.85051e6	0.176688
$946$	0	0
$947$	−9.04501e6	−0.327743	−0.163872	−	0.986482i	$-0.552398\pi$
$947$	−9.04501e6	−0.327743	−0.163872	+	0.986482i	$0.552398\pi$
$948$	0	0
$949$	1.77129e6	0.0638446
$950$	0	0
$951$	−2.44068e7	−0.875105
$952$	0	0
$953$	−1.48053e7	−0.528063	−0.264032	−	0.964514i	$-0.585052\pi$
$953$	−1.48053e7	−0.528063	−0.264032	+	0.964514i	$0.585052\pi$
$954$	0	0
$955$	7.16546e6	0.254235
$956$	0	0
$957$	831294.	0.0293410
$958$	0	0
$959$	2.00805e6	0.0705062
$960$	0	0
$961$	−1.61911e7	−0.565546
$962$	0	0
$963$	1.76940e7	0.614838
$964$	0	0
$965$	−2.48393e7	−0.858661
$966$	0	0
$967$	2.02304e7	0.695726	0.347863	−	0.937545i	$-0.386907\pi$
$967$	2.02304e7	0.695726	0.347863	+	0.937545i	$0.386907\pi$
$968$	0	0
$969$	346411.	0.0118517
$970$	0	0
$971$	255216.	0.00868679	0.00434340	−	0.999991i	$-0.498617\pi$
$971$	255216.	0.00868679	0.00434340	+	0.999991i	$0.498617\pi$
$972$	0	0
$973$	2.05284e7	0.695142
$974$	0	0
$975$	−495573.	−0.0166954
$976$	0	0
$977$	3.15610e7	1.05783	0.528914	−	0.848675i	$-0.322599\pi$
$977$	3.15610e7	1.05783	0.528914	+	0.848675i	$0.322599\pi$
$978$	0	0
$979$	3.79196e6	0.126447
$980$	0	0
$981$	2.60917e7	0.865627
$982$	0	0
$983$	5.24054e7	1.72978	0.864892	−	0.501958i	$-0.167387\pi$
$983$	5.24054e7	1.72978	0.864892	+	0.501958i	$0.167387\pi$
$984$	0	0
$985$	955225.	0.0313701
$986$	0	0
$987$	8.82093e6	0.288218
$988$	0	0
$989$	3.79702e7	1.23439
$990$	0	0
$991$	−3.73103e7	−1.20683	−0.603413	−	0.797429i	$-0.706193\pi$
$991$	−3.73103e7	−1.20683	−0.603413	+	0.797429i	$0.706193\pi$
$992$	0	0
$993$	−7.38682e6	−0.237730
$994$	0	0
$995$	1.82622e7	0.584783
$996$	0	0
$997$	−2.27918e7	−0.726174	−0.363087	−	0.931755i	$-0.618277\pi$
$997$	−2.27918e7	−0.726174	−0.363087	+	0.931755i	$0.618277\pi$
$998$	0	0
$999$	3.77398e7	1.19643

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.l.1.2		2
4.3	odd	2		35.6.a.b.1.2	✓	2
12.11	even	2		315.6.a.c.1.1		2
20.3	even	4		175.6.b.d.99.1		4
20.7	even	4		175.6.b.d.99.4		4
20.19	odd	2		175.6.a.d.1.1		2
28.27	even	2		245.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.a.b.1.2	✓	2	4.3	odd	2
175.6.a.d.1.1		2	20.19	odd	2
175.6.b.d.99.1		4	20.3	even	4
175.6.b.d.99.4		4	20.7	even	4
245.6.a.c.1.2		2	28.27	even	2
315.6.a.c.1.1		2	12.11	even	2
560.6.a.l.1.2		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)

\(f(q)\)	\(=\)	\(q+10.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -130.780 q^{9} +O(q^{10})\)	\(q+10.5934 q^{3} -25.0000 q^{5} +49.0000 q^{7} -130.780 q^{9} -90.5195 q^{11} -74.8502 q^{13} -264.835 q^{15} +1032.31 q^{17} +31.6771 q^{19} +519.076 q^{21} +3857.08 q^{23} +625.000 q^{25} -3959.60 q^{27} -866.917 q^{29} -3526.76 q^{31} -958.908 q^{33} -1225.00 q^{35} -9531.22 q^{37} -792.917 q^{39} -14503.9 q^{41} +9844.30 q^{43} +3269.50 q^{45} +16993.5 q^{47} +2401.00 q^{49} +10935.7 q^{51} -29621.1 q^{53} +2262.99 q^{55} +335.568 q^{57} -50697.4 q^{59} -2921.38 q^{61} -6408.23 q^{63} +1871.25 q^{65} +41086.2 q^{67} +40859.5 q^{69} -61753.1 q^{71} -23664.5 q^{73} +6620.87 q^{75} -4435.46 q^{77} -45191.5 q^{79} -10166.0 q^{81} -39095.9 q^{83} -25807.8 q^{85} -9183.58 q^{87} -41891.1 q^{89} -3667.66 q^{91} -37360.4 q^{93} -791.927 q^{95} +8036.53 q^{97} +11838.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 50 q^{5} + 98 q^{7} - 189 q^{9}+O(q^{10}) \) 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

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