LMFDB - Embedded newform 1560.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 41x^{2} - 30x - 4 $ x^4 - x^3 - 41x^2 - 30x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.49345$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -14.8428 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} -14.8428 q^{7} +9.00000 q^{9} -49.1048 q^{11} +13.0000 q^{13} +15.0000 q^{15} -27.0701 q^{17} +13.6333 q^{19} -44.5285 q^{21} +172.997 q^{23} +25.0000 q^{25} +27.0000 q^{27} +199.830 q^{29} -5.65451 q^{31} -147.314 q^{33} -74.2142 q^{35} +356.525 q^{37} +39.0000 q^{39} -219.843 q^{41} -542.329 q^{43} +45.0000 q^{45} -342.939 q^{47} -122.690 q^{49} -81.2102 q^{51} -355.526 q^{53} -245.524 q^{55} +40.8999 q^{57} -134.114 q^{59} +188.286 q^{61} -133.585 q^{63} +65.0000 q^{65} -309.396 q^{67} +518.990 q^{69} -73.1729 q^{71} -696.873 q^{73} +75.0000 q^{75} +728.854 q^{77} -877.332 q^{79} +81.0000 q^{81} -1469.74 q^{83} -135.350 q^{85} +599.489 q^{87} +984.177 q^{89} -192.957 q^{91} -16.9635 q^{93} +68.1665 q^{95} -606.420 q^{97} -441.943 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−14.8428	−0.801437	−0.400719	−	0.916201i	$-0.631240\pi$
$7$	−14.8428	−0.801437	−0.400719	+	0.916201i	$0.631240\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−49.1048	−1.34597	−0.672984	−	0.739657i	$-0.734988\pi$
$11$	−49.1048	−1.34597	−0.672984	+	0.739657i	$0.734988\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−27.0701	−0.386203	−0.193102	−	0.981179i	$-0.561855\pi$
$17$	−27.0701	−0.386203	−0.193102	+	0.981179i	$0.561855\pi$
$18$	0	0
$19$	13.6333	0.164615	0.0823077	−	0.996607i	$-0.473771\pi$
$19$	13.6333	0.164615	0.0823077	+	0.996607i	$0.473771\pi$
$20$	0	0
$21$	−44.5285	−0.462710
$22$	0	0
$23$	172.997	1.56836	0.784180	−	0.620534i	$-0.213084\pi$
$23$	172.997	1.56836	0.784180	+	0.620534i	$0.213084\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	199.830	1.27957	0.639783	−	0.768556i	$-0.279024\pi$
$29$	199.830	1.27957	0.639783	+	0.768556i	$0.279024\pi$
$30$	0	0
$31$	−5.65451	−0.0327606	−0.0163803	−	0.999866i	$-0.505214\pi$
$31$	−5.65451	−0.0327606	−0.0163803	+	0.999866i	$0.505214\pi$
$32$	0	0
$33$	−147.314	−0.777095
$34$	0	0
$35$	−74.2142	−0.358414
$36$	0	0
$37$	356.525	1.58412	0.792059	−	0.610445i	$-0.209009\pi$
$37$	356.525	1.58412	0.792059	+	0.610445i	$0.209009\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−219.843	−0.837406	−0.418703	−	0.908123i	$-0.637515\pi$
$41$	−219.843	−0.837406	−0.418703	+	0.908123i	$0.637515\pi$
$42$	0	0
$43$	−542.329	−1.92336	−0.961680	−	0.274176i	$-0.911595\pi$
$43$	−542.329	−1.92336	−0.961680	+	0.274176i	$0.911595\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−342.939	−1.06431	−0.532157	−	0.846646i	$-0.678618\pi$
$47$	−342.939	−1.06431	−0.532157	+	0.846646i	$0.678618\pi$
$48$	0	0
$49$	−122.690	−0.357698
$50$	0	0
$51$	−81.2102	−0.222975
$52$	0	0
$53$	−355.526	−0.921420	−0.460710	−	0.887551i	$-0.652405\pi$
$53$	−355.526	−0.921420	−0.460710	+	0.887551i	$0.652405\pi$
$54$	0	0
$55$	−245.524	−0.601936
$56$	0	0
$57$	40.8999	0.0950408
$58$	0	0
$59$	−134.114	−0.295935	−0.147967	−	0.988992i	$-0.547273\pi$
$59$	−134.114	−0.295935	−0.147967	+	0.988992i	$0.547273\pi$
$60$	0	0
$61$	188.286	0.395205	0.197602	−	0.980282i	$-0.436685\pi$
$61$	188.286	0.395205	0.197602	+	0.980282i	$0.436685\pi$
$62$	0	0
$63$	−133.585	−0.267146
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	−309.396	−0.564161	−0.282080	−	0.959391i	$-0.591024\pi$
$67$	−309.396	−0.564161	−0.282080	+	0.959391i	$0.591024\pi$
$68$	0	0
$69$	518.990	0.905493
$70$	0	0
$71$	−73.1729	−0.122310	−0.0611551	−	0.998128i	$-0.519478\pi$
$71$	−73.1729	−0.122310	−0.0611551	+	0.998128i	$0.519478\pi$
$72$	0	0
$73$	−696.873	−1.11730	−0.558650	−	0.829404i	$-0.688680\pi$
$73$	−696.873	−1.11730	−0.558650	+	0.829404i	$0.688680\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	728.854	1.07871
$78$	0	0
$79$	−877.332	−1.24946	−0.624731	−	0.780840i	$-0.714792\pi$
$79$	−877.332	−1.24946	−0.624731	+	0.780840i	$0.714792\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1469.74	−1.94368	−0.971839	−	0.235646i	$-0.924279\pi$
$83$	−1469.74	−1.94368	−0.971839	+	0.235646i	$0.924279\pi$
$84$	0	0
$85$	−135.350	−0.172715
$86$	0	0
$87$	599.489	0.738758
$88$	0	0
$89$	984.177	1.17216	0.586082	−	0.810252i	$-0.300670\pi$
$89$	984.177	1.17216	0.586082	+	0.810252i	$0.300670\pi$
$90$	0	0
$91$	−192.957	−0.222279
$92$	0	0
$93$	−16.9635	−0.0189144
$94$	0	0
$95$	68.1665	0.0736183
$96$	0	0
$97$	−606.420	−0.634769	−0.317385	−	0.948297i	$-0.602805\pi$
$97$	−606.420	−0.634769	−0.317385	+	0.948297i	$0.602805\pi$
$98$	0	0
$99$	−441.943	−0.448656
$100$	0	0
$101$	−1941.86	−1.91309	−0.956546	−	0.291583i	$-0.905818\pi$
$101$	−1941.86	−1.91309	−0.956546	+	0.291583i	$0.905818\pi$
$102$	0	0
$103$	357.045	0.341560	0.170780	−	0.985309i	$-0.445371\pi$
$103$	357.045	0.341560	0.170780	+	0.985309i	$0.445371\pi$
$104$	0	0
$105$	−222.642	−0.206930
$106$	0	0
$107$	−4.83373	−0.00436724	−0.00218362	−	0.999998i	$-0.500695\pi$
$107$	−4.83373	−0.00436724	−0.00218362	+	0.999998i	$0.500695\pi$
$108$	0	0
$109$	−1874.41	−1.64712	−0.823561	−	0.567227i	$-0.808016\pi$
$109$	−1874.41	−1.64712	−0.823561	+	0.567227i	$0.808016\pi$
$110$	0	0
$111$	1069.58	0.914591
$112$	0	0
$113$	−480.245	−0.399802	−0.199901	−	0.979816i	$-0.564062\pi$
$113$	−480.245	−0.399802	−0.199901	+	0.979816i	$0.564062\pi$
$114$	0	0
$115$	864.983	0.701392
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	401.797	0.309518
$120$	0	0
$121$	1080.28	0.811632
$122$	0	0
$123$	−659.528	−0.483477
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	357.354	0.249685	0.124843	−	0.992177i	$-0.460157\pi$
$127$	357.354	0.249685	0.124843	+	0.992177i	$0.460157\pi$
$128$	0	0
$129$	−1626.99	−1.11045
$130$	0	0
$131$	−528.735	−0.352639	−0.176320	−	0.984333i	$-0.556419\pi$
$131$	−528.735	−0.352639	−0.176320	+	0.984333i	$0.556419\pi$
$132$	0	0
$133$	−202.357	−0.131929
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−221.540	−0.138156	−0.0690782	−	0.997611i	$-0.522006\pi$
$137$	−221.540	−0.138156	−0.0690782	+	0.997611i	$0.522006\pi$
$138$	0	0
$139$	1172.08	0.715211	0.357606	−	0.933873i	$-0.383593\pi$
$139$	1172.08	0.715211	0.357606	+	0.933873i	$0.383593\pi$
$140$	0	0
$141$	−1028.82	−0.614482
$142$	0	0
$143$	−638.363	−0.373305
$144$	0	0
$145$	999.148	0.572239
$146$	0	0
$147$	−368.071	−0.206517
$148$	0	0
$149$	−1099.97	−0.604786	−0.302393	−	0.953183i	$-0.597785\pi$
$149$	−1099.97	−0.604786	−0.302393	+	0.953183i	$0.597785\pi$
$150$	0	0
$151$	494.054	0.266262	0.133131	−	0.991098i	$-0.457497\pi$
$151$	494.054	0.266262	0.133131	+	0.991098i	$0.457497\pi$
$152$	0	0
$153$	−243.631	−0.128734
$154$	0	0
$155$	−28.2725	−0.0146510
$156$	0	0
$157$	1772.96	0.901257	0.450629	−	0.892711i	$-0.351200\pi$
$157$	1772.96	0.901257	0.450629	+	0.892711i	$0.351200\pi$
$158$	0	0
$159$	−1066.58	−0.531982
$160$	0	0
$161$	−2567.76	−1.25694
$162$	0	0
$163$	150.912	0.0725174	0.0362587	−	0.999342i	$-0.488456\pi$
$163$	150.912	0.0725174	0.0362587	+	0.999342i	$0.488456\pi$
$164$	0	0
$165$	−736.572	−0.347528
$166$	0	0
$167$	−2939.41	−1.36203	−0.681013	−	0.732272i	$-0.738460\pi$
$167$	−2939.41	−1.36203	−0.681013	+	0.732272i	$0.738460\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	122.700	0.0548718
$172$	0	0
$173$	1170.19	0.514267	0.257134	−	0.966376i	$-0.417222\pi$
$173$	1170.19	0.514267	0.257134	+	0.966376i	$0.417222\pi$
$174$	0	0
$175$	−371.071	−0.160287
$176$	0	0
$177$	−402.342	−0.170858
$178$	0	0
$179$	4304.43	1.79736	0.898682	−	0.438601i	$-0.144526\pi$
$179$	4304.43	1.79736	0.898682	+	0.438601i	$0.144526\pi$
$180$	0	0
$181$	−561.691	−0.230664	−0.115332	−	0.993327i	$-0.536793\pi$
$181$	−561.691	−0.230664	−0.115332	+	0.993327i	$0.536793\pi$
$182$	0	0
$183$	564.857	0.228172
$184$	0	0
$185$	1782.63	0.708439
$186$	0	0
$187$	1329.27	0.519818
$188$	0	0
$189$	−400.756	−0.154237
$190$	0	0
$191$	561.503	0.212717	0.106358	−	0.994328i	$-0.466081\pi$
$191$	561.503	0.212717	0.106358	+	0.994328i	$0.466081\pi$
$192$	0	0
$193$	1685.44	0.628603	0.314302	−	0.949323i	$-0.398230\pi$
$193$	1685.44	0.628603	0.314302	+	0.949323i	$0.398230\pi$
$194$	0	0
$195$	195.000	0.0716115
$196$	0	0
$197$	−2116.65	−0.765508	−0.382754	−	0.923850i	$-0.625024\pi$
$197$	−2116.65	−0.765508	−0.382754	+	0.923850i	$0.625024\pi$
$198$	0	0
$199$	−1824.47	−0.649915	−0.324957	−	0.945729i	$-0.605350\pi$
$199$	−1824.47	−0.649915	−0.324957	+	0.945729i	$0.605350\pi$
$200$	0	0
$201$	−928.189	−0.325718
$202$	0	0
$203$	−2966.04	−1.02549
$204$	0	0
$205$	−1099.21	−0.374499
$206$	0	0
$207$	1556.97	0.522786
$208$	0	0
$209$	−669.461	−0.221567
$210$	0	0
$211$	−3777.52	−1.23249	−0.616245	−	0.787554i	$-0.711347\pi$
$211$	−3777.52	−1.23249	−0.616245	+	0.787554i	$0.711347\pi$
$212$	0	0
$213$	−219.519	−0.0706158
$214$	0	0
$215$	−2711.65	−0.860152
$216$	0	0
$217$	83.9289	0.0262556
$218$	0	0
$219$	−2090.62	−0.645073
$220$	0	0
$221$	−351.911	−0.107114
$222$	0	0
$223$	160.743	0.0482698	0.0241349	−	0.999709i	$-0.492317\pi$
$223$	160.743	0.0482698	0.0241349	+	0.999709i	$0.492317\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−1369.29	−0.400365	−0.200183	−	0.979759i	$-0.564154\pi$
$227$	−1369.29	−0.400365	−0.200183	+	0.979759i	$0.564154\pi$
$228$	0	0
$229$	6339.44	1.82935	0.914676	−	0.404187i	$-0.132445\pi$
$229$	6339.44	1.82935	0.914676	+	0.404187i	$0.132445\pi$
$230$	0	0
$231$	2186.56	0.622793
$232$	0	0
$233$	6997.06	1.96735	0.983675	−	0.179955i	$-0.0575954\pi$
$233$	6997.06	1.96735	0.983675	+	0.179955i	$0.0575954\pi$
$234$	0	0
$235$	−1714.69	−0.475976
$236$	0	0
$237$	−2632.00	−0.721378
$238$	0	0
$239$	−6642.89	−1.79788	−0.898939	−	0.438074i	$-0.855661\pi$
$239$	−6642.89	−1.79788	−0.898939	+	0.438074i	$0.855661\pi$
$240$	0	0
$241$	−3046.42	−0.814261	−0.407131	−	0.913370i	$-0.633471\pi$
$241$	−3046.42	−0.814261	−0.407131	+	0.913370i	$0.633471\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−613.452	−0.159967
$246$	0	0
$247$	177.233	0.0456561
$248$	0	0
$249$	−4409.23	−1.12218
$250$	0	0
$251$	6459.26	1.62432	0.812161	−	0.583434i	$-0.198291\pi$
$251$	6459.26	1.62432	0.812161	+	0.583434i	$0.198291\pi$
$252$	0	0
$253$	−8494.96	−2.11096
$254$	0	0
$255$	−406.051	−0.0997173
$256$	0	0
$257$	1822.65	0.442389	0.221194	−	0.975230i	$-0.429004\pi$
$257$	1822.65	0.442389	0.221194	+	0.975230i	$0.429004\pi$
$258$	0	0
$259$	−5291.84	−1.26957
$260$	0	0
$261$	1798.47	0.426522
$262$	0	0
$263$	−2532.04	−0.593658	−0.296829	−	0.954931i	$-0.595929\pi$
$263$	−2532.04	−0.593658	−0.296829	+	0.954931i	$0.595929\pi$
$264$	0	0
$265$	−1777.63	−0.412071
$266$	0	0
$267$	2952.53	0.676749
$268$	0	0
$269$	−961.992	−0.218044	−0.109022	−	0.994039i	$-0.534772\pi$
$269$	−961.992	−0.218044	−0.109022	+	0.994039i	$0.534772\pi$
$270$	0	0
$271$	−4063.41	−0.910829	−0.455415	−	0.890279i	$-0.650509\pi$
$271$	−4063.41	−0.910829	−0.455415	+	0.890279i	$0.650509\pi$
$272$	0	0
$273$	−578.870	−0.128333
$274$	0	0
$275$	−1227.62	−0.269194
$276$	0	0
$277$	2910.67	0.631354	0.315677	−	0.948867i	$-0.397768\pi$
$277$	2910.67	0.631354	0.315677	+	0.948867i	$0.397768\pi$
$278$	0	0
$279$	−50.8906	−0.0109202
$280$	0	0
$281$	5751.60	1.22104	0.610519	−	0.792001i	$-0.290961\pi$
$281$	5751.60	1.22104	0.610519	+	0.792001i	$0.290961\pi$
$282$	0	0
$283$	−5712.92	−1.19999	−0.599996	−	0.800003i	$-0.704831\pi$
$283$	−5712.92	−1.19999	−0.599996	+	0.800003i	$0.704831\pi$
$284$	0	0
$285$	204.500	0.0425035
$286$	0	0
$287$	3263.09	0.671129
$288$	0	0
$289$	−4180.21	−0.850847
$290$	0	0
$291$	−1819.26	−0.366484
$292$	0	0
$293$	3485.07	0.694880	0.347440	−	0.937702i	$-0.387051\pi$
$293$	3485.07	0.694880	0.347440	+	0.937702i	$0.387051\pi$
$294$	0	0
$295$	−670.570	−0.132346
$296$	0	0
$297$	−1325.83	−0.259032
$298$	0	0
$299$	2248.95	0.434985
$300$	0	0
$301$	8049.70	1.54145
$302$	0	0
$303$	−5825.58	−1.10452
$304$	0	0
$305$	941.428	0.176741
$306$	0	0
$307$	9657.78	1.79544	0.897718	−	0.440572i	$-0.145224\pi$
$307$	9657.78	1.79544	0.897718	+	0.440572i	$0.145224\pi$
$308$	0	0
$309$	1071.13	0.197200
$310$	0	0
$311$	8652.98	1.57770	0.788851	−	0.614585i	$-0.210676\pi$
$311$	8652.98	1.57770	0.788851	+	0.614585i	$0.210676\pi$
$312$	0	0
$313$	−3329.01	−0.601172	−0.300586	−	0.953755i	$-0.597182\pi$
$313$	−3329.01	−0.601172	−0.300586	+	0.953755i	$0.597182\pi$
$314$	0	0
$315$	−667.927	−0.119471
$316$	0	0
$317$	559.183	0.0990752	0.0495376	−	0.998772i	$-0.484225\pi$
$317$	559.183	0.0990752	0.0495376	+	0.998772i	$0.484225\pi$
$318$	0	0
$319$	−9812.59	−1.72226
$320$	0	0
$321$	−14.5012	−0.00252143
$322$	0	0
$323$	−369.054	−0.0635750
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	−5623.24	−0.950967
$328$	0	0
$329$	5090.18	0.852982
$330$	0	0
$331$	−8983.48	−1.49177	−0.745887	−	0.666073i	$-0.767974\pi$
$331$	−8983.48	−1.49177	−0.745887	+	0.666073i	$0.767974\pi$
$332$	0	0
$333$	3208.73	0.528039
$334$	0	0
$335$	−1546.98	−0.252300
$336$	0	0
$337$	−9302.39	−1.50366	−0.751830	−	0.659357i	$-0.770828\pi$
$337$	−9302.39	−1.50366	−0.751830	+	0.659357i	$0.770828\pi$
$338$	0	0
$339$	−1440.74	−0.230826
$340$	0	0
$341$	277.664	0.0440948
$342$	0	0
$343$	6912.16	1.08811
$344$	0	0
$345$	2594.95	0.404949
$346$	0	0
$347$	912.221	0.141126	0.0705628	−	0.997507i	$-0.477520\pi$
$347$	912.221	0.141126	0.0705628	+	0.997507i	$0.477520\pi$
$348$	0	0
$349$	742.492	0.113882	0.0569408	−	0.998378i	$-0.481865\pi$
$349$	742.492	0.113882	0.0569408	+	0.998378i	$0.481865\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	−11896.2	−1.79368	−0.896840	−	0.442355i	$-0.854143\pi$
$353$	−11896.2	−1.79368	−0.896840	+	0.442355i	$0.854143\pi$
$354$	0	0
$355$	−365.864	−0.0546988
$356$	0	0
$357$	1205.39	0.178700
$358$	0	0
$359$	−6310.80	−0.927775	−0.463888	−	0.885894i	$-0.653546\pi$
$359$	−6310.80	−0.927775	−0.463888	+	0.885894i	$0.653546\pi$
$360$	0	0
$361$	−6673.13	−0.972902
$362$	0	0
$363$	3240.85	0.468596
$364$	0	0
$365$	−3484.37	−0.499672
$366$	0	0
$367$	−8989.29	−1.27858	−0.639288	−	0.768968i	$-0.720771\pi$
$367$	−8989.29	−1.27858	−0.639288	+	0.768968i	$0.720771\pi$
$368$	0	0
$369$	−1978.58	−0.279135
$370$	0	0
$371$	5277.01	0.738460
$372$	0	0
$373$	4302.03	0.597186	0.298593	−	0.954381i	$-0.403483\pi$
$373$	4302.03	0.597186	0.298593	+	0.954381i	$0.403483\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	2597.78	0.354888
$378$	0	0
$379$	−7255.31	−0.983325	−0.491663	−	0.870786i	$-0.663611\pi$
$379$	−7255.31	−0.983325	−0.491663	+	0.870786i	$0.663611\pi$
$380$	0	0
$381$	1072.06	0.144156
$382$	0	0
$383$	6848.87	0.913736	0.456868	−	0.889535i	$-0.348971\pi$
$383$	6848.87	0.913736	0.456868	+	0.889535i	$0.348971\pi$
$384$	0	0
$385$	3644.27	0.482414
$386$	0	0
$387$	−4880.96	−0.641120
$388$	0	0
$389$	−9865.40	−1.28585	−0.642925	−	0.765929i	$-0.722279\pi$
$389$	−9865.40	−1.28585	−0.642925	+	0.765929i	$0.722279\pi$
$390$	0	0
$391$	−4683.03	−0.605706
$392$	0	0
$393$	−1586.20	−0.203596
$394$	0	0
$395$	−4386.66	−0.558777
$396$	0	0
$397$	7695.91	0.972913	0.486457	−	0.873705i	$-0.338289\pi$
$397$	7695.91	0.972913	0.486457	+	0.873705i	$0.338289\pi$
$398$	0	0
$399$	−607.070	−0.0761692
$400$	0	0
$401$	12508.7	1.55774	0.778872	−	0.627183i	$-0.215792\pi$
$401$	12508.7	1.55774	0.778872	+	0.627183i	$0.215792\pi$
$402$	0	0
$403$	−73.5086	−0.00908617
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−17507.1	−2.13217
$408$	0	0
$409$	1287.64	0.155672	0.0778359	−	0.996966i	$-0.475199\pi$
$409$	1287.64	0.155672	0.0778359	+	0.996966i	$0.475199\pi$
$410$	0	0
$411$	−664.619	−0.0797646
$412$	0	0
$413$	1990.63	0.237173
$414$	0	0
$415$	−7348.72	−0.869239
$416$	0	0
$417$	3516.24	0.412928
$418$	0	0
$419$	5933.66	0.691833	0.345917	−	0.938265i	$-0.387568\pi$
$419$	5933.66	0.691833	0.345917	+	0.938265i	$0.387568\pi$
$420$	0	0
$421$	−1607.17	−0.186054	−0.0930271	−	0.995664i	$-0.529654\pi$
$421$	−1607.17	−0.186054	−0.0930271	+	0.995664i	$0.529654\pi$
$422$	0	0
$423$	−3086.45	−0.354772
$424$	0	0
$425$	−676.752	−0.0772407
$426$	0	0
$427$	−2794.69	−0.316732
$428$	0	0
$429$	−1915.09	−0.215528
$430$	0	0
$431$	259.297	0.0289789	0.0144895	−	0.999895i	$-0.495388\pi$
$431$	259.297	0.0289789	0.0144895	+	0.999895i	$0.495388\pi$
$432$	0	0
$433$	2929.17	0.325097	0.162549	−	0.986701i	$-0.448029\pi$
$433$	2929.17	0.325097	0.162549	+	0.986701i	$0.448029\pi$
$434$	0	0
$435$	2997.44	0.330383
$436$	0	0
$437$	2358.51	0.258176
$438$	0	0
$439$	−4748.59	−0.516259	−0.258130	−	0.966110i	$-0.583106\pi$
$439$	−4748.59	−0.516259	−0.258130	+	0.966110i	$0.583106\pi$
$440$	0	0
$441$	−1104.21	−0.119233
$442$	0	0
$443$	6595.03	0.707313	0.353656	−	0.935375i	$-0.384938\pi$
$443$	6595.03	0.707313	0.353656	+	0.935375i	$0.384938\pi$
$444$	0	0
$445$	4920.89	0.524208
$446$	0	0
$447$	−3299.91	−0.349173
$448$	0	0
$449$	15825.9	1.66341	0.831707	−	0.555215i	$-0.187364\pi$
$449$	15825.9	1.66341	0.831707	+	0.555215i	$0.187364\pi$
$450$	0	0
$451$	10795.3	1.12712
$452$	0	0
$453$	1482.16	0.153726
$454$	0	0
$455$	−964.784	−0.0994061
$456$	0	0
$457$	−900.430	−0.0921671	−0.0460835	−	0.998938i	$-0.514674\pi$
$457$	−900.430	−0.0921671	−0.0460835	+	0.998938i	$0.514674\pi$
$458$	0	0
$459$	−730.892	−0.0743249
$460$	0	0
$461$	−7091.10	−0.716411	−0.358205	−	0.933643i	$-0.616611\pi$
$461$	−7091.10	−0.716411	−0.358205	+	0.933643i	$0.616611\pi$
$462$	0	0
$463$	−19349.6	−1.94223	−0.971114	−	0.238617i	$-0.923306\pi$
$463$	−19349.6	−1.94223	−0.971114	+	0.238617i	$0.923306\pi$
$464$	0	0
$465$	−84.8176	−0.00845876
$466$	0	0
$467$	3591.33	0.355860	0.177930	−	0.984043i	$-0.443060\pi$
$467$	3591.33	0.355860	0.177930	+	0.984043i	$0.443060\pi$
$468$	0	0
$469$	4592.32	0.452140
$470$	0	0
$471$	5318.87	0.520341
$472$	0	0
$473$	26631.0	2.58878
$474$	0	0
$475$	340.833	0.0329231
$476$	0	0
$477$	−3199.73	−0.307140
$478$	0	0
$479$	3635.32	0.346769	0.173384	−	0.984854i	$-0.444530\pi$
$479$	3635.32	0.346769	0.173384	+	0.984854i	$0.444530\pi$
$480$	0	0
$481$	4634.83	0.439355
$482$	0	0
$483$	−7703.27	−0.725696
$484$	0	0
$485$	−3032.10	−0.283877
$486$	0	0
$487$	−3397.71	−0.316150	−0.158075	−	0.987427i	$-0.550529\pi$
$487$	−3397.71	−0.316150	−0.158075	+	0.987427i	$0.550529\pi$
$488$	0	0
$489$	452.736	0.0418679
$490$	0	0
$491$	2170.44	0.199492	0.0997462	−	0.995013i	$-0.468197\pi$
$491$	2170.44	0.199492	0.0997462	+	0.995013i	$0.468197\pi$
$492$	0	0
$493$	−5409.40	−0.494173
$494$	0	0
$495$	−2209.72	−0.200645
$496$	0	0
$497$	1086.09	0.0980240
$498$	0	0
$499$	−14177.8	−1.27192	−0.635958	−	0.771724i	$-0.719395\pi$
$499$	−14177.8	−1.27192	−0.635958	+	0.771724i	$0.719395\pi$
$500$	0	0
$501$	−8818.22	−0.786366
$502$	0	0
$503$	11667.4	1.03424	0.517119	−	0.855913i	$-0.327004\pi$
$503$	11667.4	1.03424	0.517119	+	0.855913i	$0.327004\pi$
$504$	0	0
$505$	−9709.30	−0.855560
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−3926.43	−0.341917	−0.170959	−	0.985278i	$-0.554686\pi$
$509$	−3926.43	−0.341917	−0.170959	+	0.985278i	$0.554686\pi$
$510$	0	0
$511$	10343.6	0.895446
$512$	0	0
$513$	368.099	0.0316803
$514$	0	0
$515$	1785.22	0.152750
$516$	0	0
$517$	16840.0	1.43253
$518$	0	0
$519$	3510.58	0.296912
$520$	0	0
$521$	3917.77	0.329444	0.164722	−	0.986340i	$-0.447327\pi$
$521$	3917.77	0.329444	0.164722	+	0.986340i	$0.447327\pi$
$522$	0	0
$523$	4851.98	0.405664	0.202832	−	0.979214i	$-0.434985\pi$
$523$	4851.98	0.405664	0.202832	+	0.979214i	$0.434985\pi$
$524$	0	0
$525$	−1113.21	−0.0925420
$526$	0	0
$527$	153.068	0.0126523
$528$	0	0
$529$	17760.8	1.45975
$530$	0	0
$531$	−1207.03	−0.0986449
$532$	0	0
$533$	−2857.95	−0.232255
$534$	0	0
$535$	−24.1687	−0.00195309
$536$	0	0
$537$	12913.3	1.03771
$538$	0	0
$539$	6024.69	0.481450
$540$	0	0
$541$	−12685.3	−1.00810	−0.504052	−	0.863673i	$-0.668158\pi$
$541$	−12685.3	−1.00810	−0.504052	+	0.863673i	$0.668158\pi$
$542$	0	0
$543$	−1685.07	−0.133174
$544$	0	0
$545$	−9372.07	−0.736616
$546$	0	0
$547$	−11441.3	−0.894321	−0.447161	−	0.894454i	$-0.647565\pi$
$547$	−11441.3	−0.894321	−0.447161	+	0.894454i	$0.647565\pi$
$548$	0	0
$549$	1694.57	0.131735
$550$	0	0
$551$	2724.34	0.210636
$552$	0	0
$553$	13022.1	1.00137
$554$	0	0
$555$	5347.88	0.409017
$556$	0	0
$557$	14812.0	1.12676	0.563378	−	0.826199i	$-0.309501\pi$
$557$	14812.0	1.12676	0.563378	+	0.826199i	$0.309501\pi$
$558$	0	0
$559$	−7050.28	−0.533444
$560$	0	0
$561$	3987.81	0.300117
$562$	0	0
$563$	12101.8	0.905914	0.452957	−	0.891532i	$-0.350369\pi$
$563$	12101.8	0.905914	0.452957	+	0.891532i	$0.350369\pi$
$564$	0	0
$565$	−2401.23	−0.178797
$566$	0	0
$567$	−1202.27	−0.0890486
$568$	0	0
$569$	11602.5	0.854837	0.427418	−	0.904054i	$-0.359423\pi$
$569$	11602.5	0.854837	0.427418	+	0.904054i	$0.359423\pi$
$570$	0	0
$571$	14990.3	1.09864	0.549321	−	0.835611i	$-0.314886\pi$
$571$	14990.3	1.09864	0.549321	+	0.835611i	$0.314886\pi$
$572$	0	0
$573$	1684.51	0.122812
$574$	0	0
$575$	4324.91	0.313672
$576$	0	0
$577$	15998.2	1.15427	0.577135	−	0.816649i	$-0.304171\pi$
$577$	15998.2	1.15427	0.577135	+	0.816649i	$0.304171\pi$
$578$	0	0
$579$	5056.31	0.362924
$580$	0	0
$581$	21815.1	1.55774
$582$	0	0
$583$	17458.0	1.24020
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	−16085.0	−1.13100	−0.565500	−	0.824748i	$-0.691317\pi$
$587$	−16085.0	−1.13100	−0.565500	+	0.824748i	$0.691317\pi$
$588$	0	0
$589$	−77.0896	−0.00539291
$590$	0	0
$591$	−6349.95	−0.441966
$592$	0	0
$593$	792.054	0.0548495	0.0274248	−	0.999624i	$-0.491269\pi$
$593$	792.054	0.0548495	0.0274248	+	0.999624i	$0.491269\pi$
$594$	0	0
$595$	2008.98	0.138421
$596$	0	0
$597$	−5473.40	−0.375228
$598$	0	0
$599$	−10201.9	−0.695888	−0.347944	−	0.937515i	$-0.613120\pi$
$599$	−10201.9	−0.695888	−0.347944	+	0.937515i	$0.613120\pi$
$600$	0	0
$601$	−4870.97	−0.330601	−0.165300	−	0.986243i	$-0.552859\pi$
$601$	−4870.97	−0.330601	−0.165300	+	0.986243i	$0.552859\pi$
$602$	0	0
$603$	−2784.57	−0.188054
$604$	0	0
$605$	5401.41	0.362973
$606$	0	0
$607$	−13689.0	−0.915351	−0.457675	−	0.889119i	$-0.651318\pi$
$607$	−13689.0	−0.915351	−0.457675	+	0.889119i	$0.651318\pi$
$608$	0	0
$609$	−8898.11	−0.592068
$610$	0	0
$611$	−4458.21	−0.295188
$612$	0	0
$613$	16515.8	1.08820	0.544099	−	0.839021i	$-0.316872\pi$
$613$	16515.8	1.08820	0.544099	+	0.839021i	$0.316872\pi$
$614$	0	0
$615$	−3297.64	−0.216217
$616$	0	0
$617$	−9597.25	−0.626208	−0.313104	−	0.949719i	$-0.601369\pi$
$617$	−9597.25	−0.626208	−0.313104	+	0.949719i	$0.601369\pi$
$618$	0	0
$619$	−8104.49	−0.526247	−0.263124	−	0.964762i	$-0.584753\pi$
$619$	−8104.49	−0.526247	−0.263124	+	0.964762i	$0.584753\pi$
$620$	0	0
$621$	4670.91	0.301831
$622$	0	0
$623$	−14608.0	−0.939416
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−2008.38	−0.127922
$628$	0	0
$629$	−9651.16	−0.611792
$630$	0	0
$631$	−5340.31	−0.336916	−0.168458	−	0.985709i	$-0.553879\pi$
$631$	−5340.31	−0.336916	−0.168458	+	0.985709i	$0.553879\pi$
$632$	0	0
$633$	−11332.6	−0.711578
$634$	0	0
$635$	1786.77	0.111663
$636$	0	0
$637$	−1594.98	−0.0992076
$638$	0	0
$639$	−658.556	−0.0407701
$640$	0	0
$641$	11209.8	0.690736	0.345368	−	0.938467i	$-0.387754\pi$
$641$	11209.8	0.690736	0.345368	+	0.938467i	$0.387754\pi$
$642$	0	0
$643$	−18292.7	−1.12192	−0.560958	−	0.827844i	$-0.689567\pi$
$643$	−18292.7	−1.12192	−0.560958	+	0.827844i	$0.689567\pi$
$644$	0	0
$645$	−8134.94	−0.496609
$646$	0	0
$647$	−12985.7	−0.789057	−0.394528	−	0.918884i	$-0.629092\pi$
$647$	−12985.7	−0.789057	−0.394528	+	0.918884i	$0.629092\pi$
$648$	0	0
$649$	6585.64	0.398319
$650$	0	0
$651$	251.787	0.0151587
$652$	0	0
$653$	15746.2	0.943642	0.471821	−	0.881694i	$-0.343597\pi$
$653$	15746.2	0.943642	0.471821	+	0.881694i	$0.343597\pi$
$654$	0	0
$655$	−2643.67	−0.157705
$656$	0	0
$657$	−6271.86	−0.372433
$658$	0	0
$659$	−15165.9	−0.896480	−0.448240	−	0.893913i	$-0.647949\pi$
$659$	−15165.9	−0.896480	−0.448240	+	0.893913i	$0.647949\pi$
$660$	0	0
$661$	5124.93	0.301568	0.150784	−	0.988567i	$-0.451820\pi$
$661$	5124.93	0.301568	0.150784	+	0.988567i	$0.451820\pi$
$662$	0	0
$663$	−1055.73	−0.0618420
$664$	0	0
$665$	−1011.78	−0.0590004
$666$	0	0
$667$	34569.8	2.00682
$668$	0	0
$669$	482.230	0.0278686
$670$	0	0
$671$	−9245.73	−0.531933
$672$	0	0
$673$	−28225.6	−1.61667	−0.808333	−	0.588726i	$-0.799630\pi$
$673$	−28225.6	−1.61667	−0.808333	+	0.588726i	$0.799630\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	28869.8	1.63893	0.819466	−	0.573128i	$-0.194270\pi$
$677$	28869.8	1.63893	0.819466	+	0.573128i	$0.194270\pi$
$678$	0	0
$679$	9000.99	0.508728
$680$	0	0
$681$	−4107.87	−0.231151
$682$	0	0
$683$	−31997.3	−1.79260	−0.896298	−	0.443453i	$-0.853753\pi$
$683$	−31997.3	−1.79260	−0.896298	+	0.443453i	$0.853753\pi$
$684$	0	0
$685$	−1107.70	−0.0617854
$686$	0	0
$687$	19018.3	1.05618
$688$	0	0
$689$	−4621.84	−0.255556
$690$	0	0
$691$	−1853.01	−0.102014	−0.0510070	−	0.998698i	$-0.516243\pi$
$691$	−1853.01	−0.102014	−0.0510070	+	0.998698i	$0.516243\pi$
$692$	0	0
$693$	6559.69	0.359570
$694$	0	0
$695$	5860.39	0.319852
$696$	0	0
$697$	5951.16	0.323409
$698$	0	0
$699$	20991.2	1.13585
$700$	0	0
$701$	75.8612	0.00408736	0.00204368	−	0.999998i	$-0.499349\pi$
$701$	75.8612	0.00408736	0.00204368	+	0.999998i	$0.499349\pi$
$702$	0	0
$703$	4860.61	0.260770
$704$	0	0
$705$	−5144.08	−0.274805
$706$	0	0
$707$	28822.7	1.53322
$708$	0	0
$709$	8215.20	0.435160	0.217580	−	0.976043i	$-0.430184\pi$
$709$	8215.20	0.435160	0.217580	+	0.976043i	$0.430184\pi$
$710$	0	0
$711$	−7895.99	−0.416488
$712$	0	0
$713$	−978.210	−0.0513805
$714$	0	0
$715$	−3191.81	−0.166947
$716$	0	0
$717$	−19928.7	−1.03800
$718$	0	0
$719$	35510.4	1.84188	0.920942	−	0.389699i	$-0.127421\pi$
$719$	35510.4	1.84188	0.920942	+	0.389699i	$0.127421\pi$
$720$	0	0
$721$	−5299.56	−0.273739
$722$	0	0
$723$	−9139.25	−0.470114
$724$	0	0
$725$	4995.74	0.255913
$726$	0	0
$727$	−10471.2	−0.534189	−0.267094	−	0.963670i	$-0.586064\pi$
$727$	−10471.2	−0.534189	−0.267094	+	0.963670i	$0.586064\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	14680.9	0.742808
$732$	0	0
$733$	1411.16	0.0711082	0.0355541	−	0.999368i	$-0.488680\pi$
$733$	1411.16	0.0711082	0.0355541	+	0.999368i	$0.488680\pi$
$734$	0	0
$735$	−1840.36	−0.0923572
$736$	0	0
$737$	15192.8	0.759343
$738$	0	0
$739$	−23784.0	−1.18391	−0.591955	−	0.805971i	$-0.701644\pi$
$739$	−23784.0	−1.18391	−0.591955	+	0.805971i	$0.701644\pi$
$740$	0	0
$741$	531.699	0.0263596
$742$	0	0
$743$	24478.8	1.20867	0.604334	−	0.796731i	$-0.293439\pi$
$743$	24478.8	1.20867	0.604334	+	0.796731i	$0.293439\pi$
$744$	0	0
$745$	−5499.85	−0.270468
$746$	0	0
$747$	−13227.7	−0.647893
$748$	0	0
$749$	71.7462	0.00350007
$750$	0	0
$751$	24470.6	1.18901	0.594505	−	0.804092i	$-0.297348\pi$
$751$	24470.6	1.18901	0.594505	+	0.804092i	$0.297348\pi$
$752$	0	0
$753$	19377.8	0.937802
$754$	0	0
$755$	2470.27	0.119076
$756$	0	0
$757$	4692.57	0.225303	0.112651	−	0.993635i	$-0.464066\pi$
$757$	4692.57	0.225303	0.112651	+	0.993635i	$0.464066\pi$
$758$	0	0
$759$	−25484.9	−1.21876
$760$	0	0
$761$	29556.4	1.40791	0.703954	−	0.710245i	$-0.251416\pi$
$761$	29556.4	1.40791	0.703954	+	0.710245i	$0.251416\pi$
$762$	0	0
$763$	27821.6	1.32007
$764$	0	0
$765$	−1218.15	−0.0575718
$766$	0	0
$767$	−1743.48	−0.0820775
$768$	0	0
$769$	30530.8	1.43169	0.715844	−	0.698260i	$-0.246042\pi$
$769$	30530.8	1.43169	0.715844	+	0.698260i	$0.246042\pi$
$770$	0	0
$771$	5467.96	0.255413
$772$	0	0
$773$	24866.7	1.15704	0.578521	−	0.815667i	$-0.303630\pi$
$773$	24866.7	1.15704	0.578521	+	0.815667i	$0.303630\pi$
$774$	0	0
$775$	−141.363	−0.00655213
$776$	0	0
$777$	−15875.5	−0.732987
$778$	0	0
$779$	−2997.18	−0.137850
$780$	0	0
$781$	3593.14	0.164626
$782$	0	0
$783$	5395.40	0.246253
$784$	0	0
$785$	8864.79	0.403055
$786$	0	0
$787$	41283.1	1.86986	0.934931	−	0.354828i	$-0.115461\pi$
$787$	41283.1	1.86986	0.934931	+	0.354828i	$0.115461\pi$
$788$	0	0
$789$	−7596.11	−0.342748
$790$	0	0
$791$	7128.20	0.320417
$792$	0	0
$793$	2447.71	0.109610
$794$	0	0
$795$	−5332.89	−0.237910
$796$	0	0
$797$	−34421.6	−1.52983	−0.764915	−	0.644131i	$-0.777219\pi$
$797$	−34421.6	−1.52983	−0.764915	+	0.644131i	$0.777219\pi$
$798$	0	0
$799$	9283.38	0.411042
$800$	0	0
$801$	8857.60	0.390721
$802$	0	0
$803$	34219.8	1.50385
$804$	0	0
$805$	−12838.8	−0.562122
$806$	0	0
$807$	−2885.98	−0.125888
$808$	0	0
$809$	−14195.4	−0.616915	−0.308457	−	0.951238i	$-0.599813\pi$
$809$	−14195.4	−0.616915	−0.308457	+	0.951238i	$0.599813\pi$
$810$	0	0
$811$	−29109.5	−1.26039	−0.630193	−	0.776439i	$-0.717024\pi$
$811$	−29109.5	−1.26039	−0.630193	+	0.776439i	$0.717024\pi$
$812$	0	0
$813$	−12190.2	−0.525868
$814$	0	0
$815$	754.559	0.0324307
$816$	0	0
$817$	−7393.74	−0.316615
$818$	0	0
$819$	−1736.61	−0.0740929
$820$	0	0
$821$	−5954.16	−0.253108	−0.126554	−	0.991960i	$-0.540392\pi$
$821$	−5954.16	−0.253108	−0.126554	+	0.991960i	$0.540392\pi$
$822$	0	0
$823$	−6647.52	−0.281553	−0.140776	−	0.990041i	$-0.544960\pi$
$823$	−6647.52	−0.281553	−0.140776	+	0.990041i	$0.544960\pi$
$824$	0	0
$825$	−3682.86	−0.155419
$826$	0	0
$827$	20555.8	0.864322	0.432161	−	0.901797i	$-0.357751\pi$
$827$	20555.8	0.864322	0.432161	+	0.901797i	$0.357751\pi$
$828$	0	0
$829$	−26720.6	−1.11947	−0.559737	−	0.828671i	$-0.689098\pi$
$829$	−26720.6	−1.11947	−0.559737	+	0.828671i	$0.689098\pi$
$830$	0	0
$831$	8732.00	0.364512
$832$	0	0
$833$	3321.24	0.138144
$834$	0	0
$835$	−14697.0	−0.609116
$836$	0	0
$837$	−152.672	−0.00630479
$838$	0	0
$839$	4995.75	0.205569	0.102785	−	0.994704i	$-0.467225\pi$
$839$	4995.75	0.205569	0.102785	+	0.994704i	$0.467225\pi$
$840$	0	0
$841$	15542.9	0.637290
$842$	0	0
$843$	17254.8	0.704967
$844$	0	0
$845$	845.000	0.0344010
$846$	0	0
$847$	−16034.4	−0.650472
$848$	0	0
$849$	−17138.8	−0.692816
$850$	0	0
$851$	61677.6	2.48447
$852$	0	0
$853$	15051.3	0.604157	0.302079	−	0.953283i	$-0.402320\pi$
$853$	15051.3	0.604157	0.302079	+	0.953283i	$0.402320\pi$
$854$	0	0
$855$	613.499	0.0245394
$856$	0	0
$857$	12206.4	0.486538	0.243269	−	0.969959i	$-0.421780\pi$
$857$	12206.4	0.486538	0.243269	+	0.969959i	$0.421780\pi$
$858$	0	0
$859$	−26741.3	−1.06217	−0.531083	−	0.847320i	$-0.678215\pi$
$859$	−26741.3	−1.06217	−0.531083	+	0.847320i	$0.678215\pi$
$860$	0	0
$861$	9789.26	0.387476
$862$	0	0
$863$	27750.8	1.09461	0.547305	−	0.836933i	$-0.315654\pi$
$863$	27750.8	1.09461	0.547305	+	0.836933i	$0.315654\pi$
$864$	0	0
$865$	5850.97	0.229987
$866$	0	0
$867$	−12540.6	−0.491237
$868$	0	0
$869$	43081.2	1.68174
$870$	0	0
$871$	−4022.15	−0.156470
$872$	0	0
$873$	−5457.78	−0.211590
$874$	0	0
$875$	−1855.35	−0.0716827
$876$	0	0
$877$	−5774.59	−0.222342	−0.111171	−	0.993801i	$-0.535460\pi$
$877$	−5774.59	−0.222342	−0.111171	+	0.993801i	$0.535460\pi$
$878$	0	0
$879$	10455.2	0.401189
$880$	0	0
$881$	−44144.7	−1.68817	−0.844083	−	0.536213i	$-0.819854\pi$
$881$	−44144.7	−1.68817	−0.844083	+	0.536213i	$0.819854\pi$
$882$	0	0
$883$	32585.2	1.24188	0.620940	−	0.783858i	$-0.286751\pi$
$883$	32585.2	1.24188	0.620940	+	0.783858i	$0.286751\pi$
$884$	0	0
$885$	−2011.71	−0.0764100
$886$	0	0
$887$	45208.8	1.71135	0.855673	−	0.517517i	$-0.173144\pi$
$887$	45208.8	1.71135	0.855673	+	0.517517i	$0.173144\pi$
$888$	0	0
$889$	−5304.14	−0.200107
$890$	0	0
$891$	−3977.49	−0.149552
$892$	0	0
$893$	−4675.39	−0.175203
$894$	0	0
$895$	21522.1	0.803806
$896$	0	0
$897$	6746.86	0.251138
$898$	0	0
$899$	−1129.94	−0.0419194
$900$	0	0
$901$	9624.11	0.355855
$902$	0	0
$903$	24149.1	0.889958
$904$	0	0
$905$	−2808.45	−0.103156
$906$	0	0
$907$	−9628.66	−0.352497	−0.176248	−	0.984346i	$-0.556396\pi$
$907$	−9628.66	−0.352497	−0.176248	+	0.984346i	$0.556396\pi$
$908$	0	0
$909$	−17476.7	−0.637697
$910$	0	0
$911$	29141.6	1.05983	0.529915	−	0.848051i	$-0.322224\pi$
$911$	29141.6	1.05983	0.529915	+	0.848051i	$0.322224\pi$
$912$	0	0
$913$	72171.5	2.61613
$914$	0	0
$915$	2824.28	0.102041
$916$	0	0
$917$	7847.92	0.282618
$918$	0	0
$919$	−13926.6	−0.499887	−0.249943	−	0.968260i	$-0.580412\pi$
$919$	−13926.6	−0.499887	−0.249943	+	0.968260i	$0.580412\pi$
$920$	0	0
$921$	28973.3	1.03659
$922$	0	0
$923$	−951.247	−0.0339227
$924$	0	0
$925$	8913.13	0.316824
$926$	0	0
$927$	3213.40	0.113853
$928$	0	0
$929$	−17594.5	−0.621373	−0.310687	−	0.950512i	$-0.600559\pi$
$929$	−17594.5	−0.621373	−0.310687	+	0.950512i	$0.600559\pi$
$930$	0	0
$931$	−1672.67	−0.0588826
$932$	0	0
$933$	25958.9	0.910887
$934$	0	0
$935$	6646.35	0.232470
$936$	0	0
$937$	36350.3	1.26736	0.633678	−	0.773597i	$-0.281544\pi$
$937$	36350.3	1.26736	0.633678	+	0.773597i	$0.281544\pi$
$938$	0	0
$939$	−9987.03	−0.347087
$940$	0	0
$941$	−22164.8	−0.767856	−0.383928	−	0.923363i	$-0.625429\pi$
$941$	−22164.8	−0.767856	−0.383928	+	0.923363i	$0.625429\pi$
$942$	0	0
$943$	−38032.0	−1.31335
$944$	0	0
$945$	−2003.78	−0.0689768
$946$	0	0
$947$	19401.2	0.665739	0.332869	−	0.942973i	$-0.391983\pi$
$947$	19401.2	0.665739	0.332869	+	0.942973i	$0.391983\pi$
$948$	0	0
$949$	−9059.35	−0.309883
$950$	0	0
$951$	1677.55	0.0572011
$952$	0	0
$953$	41700.1	1.41742	0.708709	−	0.705501i	$-0.249278\pi$
$953$	41700.1	1.41742	0.708709	+	0.705501i	$0.249278\pi$
$954$	0	0
$955$	2807.51	0.0951299
$956$	0	0
$957$	−29437.8	−0.994345
$958$	0	0
$959$	3288.28	0.110724
$960$	0	0
$961$	−29759.0	−0.998927
$962$	0	0
$963$	−43.5036	−0.00145575
$964$	0	0
$965$	8427.19	0.281120
$966$	0	0
$967$	−1413.18	−0.0469955	−0.0234978	−	0.999724i	$-0.507480\pi$
$967$	−1413.18	−0.0469955	−0.0234978	+	0.999724i	$0.507480\pi$
$968$	0	0
$969$	−1107.16	−0.0367051
$970$	0	0
$971$	2393.52	0.0791059	0.0395530	−	0.999217i	$-0.487407\pi$
$971$	2393.52	0.0791059	0.0395530	+	0.999217i	$0.487407\pi$
$972$	0	0
$973$	−17397.0	−0.573197
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	42069.7	1.37761	0.688807	−	0.724945i	$-0.258135\pi$
$977$	42069.7	1.37761	0.688807	+	0.724945i	$0.258135\pi$
$978$	0	0
$979$	−48327.8	−1.57770
$980$	0	0
$981$	−16869.7	−0.549041
$982$	0	0
$983$	−32578.1	−1.05705	−0.528525	−	0.848918i	$-0.677255\pi$
$983$	−32578.1	−1.05705	−0.528525	+	0.848918i	$0.677255\pi$
$984$	0	0
$985$	−10583.3	−0.342346
$986$	0	0
$987$	15270.6	0.492469
$988$	0	0
$989$	−93821.1	−3.01652
$990$	0	0
$991$	−13983.9	−0.448247	−0.224123	−	0.974561i	$-0.571952\pi$
$991$	−13983.9	−0.448247	−0.224123	+	0.974561i	$0.571952\pi$
$992$	0	0
$993$	−26950.5	−0.861276
$994$	0	0
$995$	−9122.33	−0.290651
$996$	0	0
$997$	29440.5	0.935197	0.467598	−	0.883941i	$-0.345119\pi$
$997$	29440.5	0.935197	0.467598	+	0.883941i	$0.345119\pi$
$998$	0	0
$999$	9626.18	0.304864

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.n.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.n.1.2	✓	4	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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -14.8428 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} -14.8428 q^{7} +9.00000 q^{9} -49.1048 q^{11} +13.0000 q^{13} +15.0000 q^{15} -27.0701 q^{17} +13.6333 q^{19} -44.5285 q^{21} +172.997 q^{23} +25.0000 q^{25} +27.0000 q^{27} +199.830 q^{29} -5.65451 q^{31} -147.314 q^{33} -74.2142 q^{35} +356.525 q^{37} +39.0000 q^{39} -219.843 q^{41} -542.329 q^{43} +45.0000 q^{45} -342.939 q^{47} -122.690 q^{49} -81.2102 q^{51} -355.526 q^{53} -245.524 q^{55} +40.8999 q^{57} -134.114 q^{59} +188.286 q^{61} -133.585 q^{63} +65.0000 q^{65} -309.396 q^{67} +518.990 q^{69} -73.1729 q^{71} -696.873 q^{73} +75.0000 q^{75} +728.854 q^{77} -877.332 q^{79} +81.0000 q^{81} -1469.74 q^{83} -135.350 q^{85} +599.489 q^{87} +984.177 q^{89} -192.957 q^{91} -16.9635 q^{93} +68.1665 q^{95} -606.420 q^{97} -441.943 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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