LMFDB - Embedded newform 160.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,6,Mod(1,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("160.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	160.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:	$25.6614111701$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{85}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 21 $ x^2 - x - 21
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.10977$ of defining polynomial
Character	$\chi$	$=$	160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.4391 q^{3} -25.0000 q^{5} -55.3173 q^{7} +97.0000 q^{9} +O(q^{10})$	$q+18.4391 q^{3} -25.0000 q^{5} -55.3173 q^{7} +97.0000 q^{9} -479.416 q^{11} +506.000 q^{13} -460.977 q^{15} -1838.00 q^{17} -2065.18 q^{19} -1020.00 q^{21} +1899.23 q^{23} +625.000 q^{25} -2692.11 q^{27} -4530.00 q^{29} +3650.94 q^{31} -8840.00 q^{33} +1382.93 q^{35} +338.000 q^{37} +9330.18 q^{39} -6330.00 q^{41} +18162.5 q^{43} -2425.00 q^{45} -4038.16 q^{47} -13747.0 q^{49} -33891.0 q^{51} -15486.0 q^{53} +11985.4 q^{55} -38080.0 q^{57} +7301.88 q^{59} -16750.0 q^{61} -5365.77 q^{63} -12650.0 q^{65} -13663.4 q^{67} +35020.0 q^{69} -43258.1 q^{71} -20806.0 q^{73} +11524.4 q^{75} +26520.0 q^{77} -69847.3 q^{79} -73211.0 q^{81} +106301. q^{83} +45950.0 q^{85} -83529.1 q^{87} -18310.0 q^{89} -27990.5 q^{91} +67320.0 q^{93} +51629.4 q^{95} +49978.0 q^{97} -46503.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 50 q^{5} + 194 q^{9}+O(q^{10}) $ 2 * q - 50 * q^5 + 194 * q^9	$ 2 q - 50 q^{5} + 194 q^{9} + 1012 q^{13} - 3676 q^{17} - 2040 q^{21} + 1250 q^{25} - 9060 q^{29} - 17680 q^{33} + 676 q^{37} - 12660 q^{41} - 4850 q^{45} - 27494 q^{49} - 30972 q^{53} - 76160 q^{57} - 33500 q^{61} - 25300 q^{65} + 70040 q^{69} - 41612 q^{73} + 53040 q^{77} - 146422 q^{81} + 91900 q^{85} - 36620 q^{89} + 134640 q^{93} + 99956 q^{97}+O(q^{100}) $ 2 * q - 50 * q^5 + 194 * q^9 + 1012 * q^13 - 3676 * q^17 - 2040 * q^21 + 1250 * q^25 - 9060 * q^29 - 17680 * q^33 + 676 * q^37 - 12660 * q^41 - 4850 * q^45 - 27494 * q^49 - 30972 * q^53 - 76160 * q^57 - 33500 * q^61 - 25300 * q^65 + 70040 * q^69 - 41612 * q^73 + 53040 * q^77 - 146422 * q^81 + 91900 * q^85 - 36620 * q^89 + 134640 * q^93 + 99956 * q^97

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
$3$	18.4391	1.18287	0.591434	−	0.806353i	$-0.298562\pi$
$3$	18.4391	1.18287	0.591434	+	0.806353i	$0.298562\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−55.3173	−0.426693	−0.213347	−	0.976977i	$-0.568436\pi$
$7$	−55.3173	−0.426693	−0.213347	+	0.976977i	$0.568436\pi$
$8$	0	0
$9$	97.0000	0.399177
$10$	0	0
$11$	−479.416	−1.19462	−0.597312	−	0.802009i	$-0.703765\pi$
$11$	−479.416	−1.19462	−0.597312	+	0.802009i	$0.703765\pi$
$12$	0	0
$13$	506.000	0.830409	0.415205	−	0.909728i	$-0.363710\pi$
$13$	506.000	0.830409	0.415205	+	0.909728i	$0.363710\pi$
$14$	0	0
$15$	−460.977	−0.528995
$16$	0	0
$17$	−1838.00	−1.54249	−0.771247	−	0.636537i	$-0.780366\pi$
$17$	−1838.00	−1.54249	−0.771247	+	0.636537i	$0.780366\pi$
$18$	0	0
$19$	−2065.18	−1.31242	−0.656211	−	0.754577i	$-0.727842\pi$
$19$	−2065.18	−1.31242	−0.656211	+	0.754577i	$0.727842\pi$
$20$	0	0
$21$	−1020.00	−0.504722
$22$	0	0
$23$	1899.23	0.748613	0.374306	−	0.927305i	$-0.377881\pi$
$23$	1899.23	0.748613	0.374306	+	0.927305i	$0.377881\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2692.11	−0.710694
$28$	0	0
$29$	−4530.00	−1.00024	−0.500119	−	0.865957i	$-0.666710\pi$
$29$	−4530.00	−1.00024	−0.500119	+	0.865957i	$0.666710\pi$
$30$	0	0
$31$	3650.94	0.682339	0.341170	−	0.940002i	$-0.389177\pi$
$31$	3650.94	0.682339	0.341170	+	0.940002i	$0.389177\pi$
$32$	0	0
$33$	−8840.00	−1.41308
$34$	0	0
$35$	1382.93	0.190823
$36$	0	0
$37$	338.000	0.0405894	0.0202947	−	0.999794i	$-0.493540\pi$
$37$	338.000	0.0405894	0.0202947	+	0.999794i	$0.493540\pi$
$38$	0	0
$39$	9330.18	0.982265
$40$	0	0
$41$	−6330.00	−0.588090	−0.294045	−	0.955792i	$-0.595002\pi$
$41$	−6330.00	−0.588090	−0.294045	+	0.955792i	$0.595002\pi$
$42$	0	0
$43$	18162.5	1.49797	0.748987	−	0.662584i	$-0.230540\pi$
$43$	18162.5	1.49797	0.748987	+	0.662584i	$0.230540\pi$
$44$	0	0
$45$	−2425.00	−0.178517
$46$	0	0
$47$	−4038.16	−0.266648	−0.133324	−	0.991072i	$-0.542565\pi$
$47$	−4038.16	−0.266648	−0.133324	+	0.991072i	$0.542565\pi$
$48$	0	0
$49$	−13747.0	−0.817933
$50$	0	0
$51$	−33891.0	−1.82457
$52$	0	0
$53$	−15486.0	−0.757268	−0.378634	−	0.925546i	$-0.623606\pi$
$53$	−15486.0	−0.757268	−0.378634	+	0.925546i	$0.623606\pi$
$54$	0	0
$55$	11985.4	0.534252
$56$	0	0
$57$	−38080.0	−1.55242
$58$	0	0
$59$	7301.88	0.273089	0.136545	−	0.990634i	$-0.456400\pi$
$59$	7301.88	0.273089	0.136545	+	0.990634i	$0.456400\pi$
$60$	0	0
$61$	−16750.0	−0.576355	−0.288178	−	0.957577i	$-0.593049\pi$
$61$	−16750.0	−0.576355	−0.288178	+	0.957577i	$0.593049\pi$
$62$	0	0
$63$	−5365.77	−0.170326
$64$	0	0
$65$	−12650.0	−0.371370
$66$	0	0
$67$	−13663.4	−0.371852	−0.185926	−	0.982564i	$-0.559529\pi$
$67$	−13663.4	−0.371852	−0.185926	+	0.982564i	$0.559529\pi$
$68$	0	0
$69$	35020.0	0.885510
$70$	0	0
$71$	−43258.1	−1.01841	−0.509204	−	0.860646i	$-0.670060\pi$
$71$	−43258.1	−1.01841	−0.509204	+	0.860646i	$0.670060\pi$
$72$	0	0
$73$	−20806.0	−0.456963	−0.228482	−	0.973548i	$-0.573376\pi$
$73$	−20806.0	−0.456963	−0.228482	+	0.973548i	$0.573376\pi$
$74$	0	0
$75$	11524.4	0.236574
$76$	0	0
$77$	26520.0	0.509738
$78$	0	0
$79$	−69847.3	−1.25916	−0.629581	−	0.776935i	$-0.716773\pi$
$79$	−69847.3	−1.25916	−0.629581	+	0.776935i	$0.716773\pi$
$80$	0	0
$81$	−73211.0	−1.23983
$82$	0	0
$83$	106301.	1.69373	0.846864	−	0.531810i	$-0.178488\pi$
$83$	106301.	1.69373	0.846864	+	0.531810i	$0.178488\pi$
$84$	0	0
$85$	45950.0	0.689824
$86$	0	0
$87$	−83529.1	−1.18315
$88$	0	0
$89$	−18310.0	−0.245027	−0.122513	−	0.992467i	$-0.539095\pi$
$89$	−18310.0	−0.245027	−0.122513	+	0.992467i	$0.539095\pi$
$90$	0	0
$91$	−27990.5	−0.354330
$92$	0	0
$93$	67320.0	0.807117
$94$	0	0
$95$	51629.4	0.586933
$96$	0	0
$97$	49978.0	0.539324	0.269662	−	0.962955i	$-0.413088\pi$
$97$	49978.0	0.539324	0.269662	+	0.962955i	$0.413088\pi$
$98$	0	0
$99$	−46503.4	−0.476866
$100$	0	0
$101$	153598.	1.49824	0.749121	−	0.662433i	$-0.230476\pi$
$101$	153598.	1.49824	0.749121	+	0.662433i	$0.230476\pi$
$102$	0	0
$103$	214834.	1.99531	0.997653	−	0.0684695i	$-0.0218116\pi$
$103$	214834.	1.99531	0.997653	+	0.0684695i	$0.0218116\pi$
$104$	0	0
$105$	25500.0	0.225718
$106$	0	0
$107$	204508.	1.72683	0.863417	−	0.504490i	$-0.168320\pi$
$107$	204508.	1.72683	0.863417	+	0.504490i	$0.168320\pi$
$108$	0	0
$109$	156386.	1.26076	0.630379	−	0.776288i	$-0.282900\pi$
$109$	156386.	1.26076	0.630379	+	0.776288i	$0.282900\pi$
$110$	0	0
$111$	6232.41	0.0480119
$112$	0	0
$113$	−179814.	−1.32473	−0.662365	−	0.749181i	$-0.730447\pi$
$113$	−179814.	−1.32473	−0.662365	+	0.749181i	$0.730447\pi$
$114$	0	0
$115$	−47480.7	−0.334790
$116$	0	0
$117$	49082.0	0.331480
$118$	0	0
$119$	101673.	0.658171
$120$	0	0
$121$	68789.0	0.427126
$122$	0	0
$123$	−116719.	−0.695633
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−167851.	−0.923453	−0.461726	−	0.887022i	$-0.652770\pi$
$127$	−167851.	−0.923453	−0.461726	+	0.887022i	$0.652770\pi$
$128$	0	0
$129$	334900.	1.77191
$130$	0	0
$131$	−310477.	−1.58071	−0.790354	−	0.612650i	$-0.790103\pi$
$131$	−310477.	−1.58071	−0.790354	+	0.612650i	$0.790103\pi$
$132$	0	0
$133$	114240.	0.560001
$134$	0	0
$135$	67302.7	0.317832
$136$	0	0
$137$	−109982.	−0.500634	−0.250317	−	0.968164i	$-0.580535\pi$
$137$	−109982.	−0.500634	−0.250317	+	0.968164i	$0.580535\pi$
$138$	0	0
$139$	181514.	0.796845	0.398423	−	0.917202i	$-0.369558\pi$
$139$	181514.	0.796845	0.398423	+	0.917202i	$0.369558\pi$
$140$	0	0
$141$	−74460.0	−0.315410
$142$	0	0
$143$	−242585.	−0.992026
$144$	0	0
$145$	113250.	0.447320
$146$	0	0
$147$	−253482.	−0.967507
$148$	0	0
$149$	394074.	1.45416	0.727080	−	0.686553i	$-0.240877\pi$
$149$	394074.	1.45416	0.727080	+	0.686553i	$0.240877\pi$
$150$	0	0
$151$	537057.	1.91681	0.958403	−	0.285420i	$-0.0921331\pi$
$151$	537057.	1.91681	0.958403	+	0.285420i	$0.0921331\pi$
$152$	0	0
$153$	−178286.	−0.615728
$154$	0	0
$155$	−91273.5	−0.305151
$156$	0	0
$157$	121402.	0.393076	0.196538	−	0.980496i	$-0.437030\pi$
$157$	121402.	0.393076	0.196538	+	0.980496i	$0.437030\pi$
$158$	0	0
$159$	−285548.	−0.895748
$160$	0	0
$161$	−105060.	−0.319428
$162$	0	0
$163$	−183856.	−0.542012	−0.271006	−	0.962578i	$-0.587356\pi$
$163$	−183856.	−0.542012	−0.271006	+	0.962578i	$0.587356\pi$
$164$	0	0
$165$	221000.	0.631950
$166$	0	0
$167$	−25280.0	−0.0701432	−0.0350716	−	0.999385i	$-0.511166\pi$
$167$	−25280.0	−0.0701432	−0.0350716	+	0.999385i	$0.511166\pi$
$168$	0	0
$169$	−115257.	−0.310421
$170$	0	0
$171$	−200322.	−0.523889
$172$	0	0
$173$	−554966.	−1.40978	−0.704890	−	0.709317i	$-0.749003\pi$
$173$	−554966.	−1.40978	−0.704890	+	0.709317i	$0.749003\pi$
$174$	0	0
$175$	−34573.3	−0.0853386
$176$	0	0
$177$	134640.	0.323029
$178$	0	0
$179$	426385.	0.994649	0.497324	−	0.867565i	$-0.334316\pi$
$179$	426385.	0.994649	0.497324	+	0.867565i	$0.334316\pi$
$180$	0	0
$181$	−803138.	−1.82219	−0.911095	−	0.412196i	$-0.864762\pi$
$181$	−803138.	−1.82219	−0.911095	+	0.412196i	$0.864762\pi$
$182$	0	0
$183$	−308855.	−0.681752
$184$	0	0
$185$	−8450.00	−0.0181521
$186$	0	0
$187$	881167.	1.84270
$188$	0	0
$189$	148920.	0.303248
$190$	0	0
$191$	688774.	1.36613	0.683067	−	0.730356i	$-0.260646\pi$
$191$	688774.	1.36613	0.683067	+	0.730356i	$0.260646\pi$
$192$	0	0
$193$	−548974.	−1.06086	−0.530431	−	0.847728i	$-0.677970\pi$
$193$	−548974.	−1.06086	−0.530431	+	0.847728i	$0.677970\pi$
$194$	0	0
$195$	−233254.	−0.439282
$196$	0	0
$197$	−705622.	−1.29541	−0.647704	−	0.761892i	$-0.724271\pi$
$197$	−705622.	−1.29541	−0.647704	+	0.761892i	$0.724271\pi$
$198$	0	0
$199$	−207698.	−0.371791	−0.185896	−	0.982569i	$-0.559519\pi$
$199$	−207698.	−0.371791	−0.185896	+	0.982569i	$0.559519\pi$
$200$	0	0
$201$	−251940.	−0.439852
$202$	0	0
$203$	250587.	0.426795
$204$	0	0
$205$	158250.	0.263002
$206$	0	0
$207$	184225.	0.298829
$208$	0	0
$209$	990080.	1.56785
$210$	0	0
$211$	644520.	0.996621	0.498311	−	0.866999i	$-0.333954\pi$
$211$	644520.	0.996621	0.498311	+	0.866999i	$0.333954\pi$
$212$	0	0
$213$	−797640.	−1.20464
$214$	0	0
$215$	−454063.	−0.669915
$216$	0	0
$217$	−201960.	−0.291149
$218$	0	0
$219$	−383644.	−0.540527
$220$	0	0
$221$	−930028.	−1.28090
$222$	0	0
$223$	−900510.	−1.21262	−0.606312	−	0.795227i	$-0.707352\pi$
$223$	−900510.	−1.21262	−0.606312	+	0.795227i	$0.707352\pi$
$224$	0	0
$225$	60625.0	0.0798354
$226$	0	0
$227$	562890.	0.725035	0.362517	−	0.931977i	$-0.381917\pi$
$227$	562890.	0.725035	0.362517	+	0.931977i	$0.381917\pi$
$228$	0	0
$229$	486150.	0.612606	0.306303	−	0.951934i	$-0.400908\pi$
$229$	486150.	0.612606	0.306303	+	0.951934i	$0.400908\pi$
$230$	0	0
$231$	489005.	0.602952
$232$	0	0
$233$	−1.43441e6	−1.73095	−0.865475	−	0.500951i	$-0.832984\pi$
$233$	−1.43441e6	−1.73095	−0.865475	+	0.500951i	$0.832984\pi$
$234$	0	0
$235$	100954.	0.119249
$236$	0	0
$237$	−1.28792e6	−1.48942
$238$	0	0
$239$	−16742.7	−0.0189597	−0.00947983	−	0.999955i	$-0.503018\pi$
$239$	−16742.7	−0.0189597	−0.00947983	+	0.999955i	$0.503018\pi$
$240$	0	0
$241$	1.16675e6	1.29400	0.647001	−	0.762489i	$-0.276023\pi$
$241$	1.16675e6	1.29400	0.647001	+	0.762489i	$0.276023\pi$
$242$	0	0
$243$	−695762.	−0.755867
$244$	0	0
$245$	343675.	0.365791
$246$	0	0
$247$	−1.04498e6	−1.08985
$248$	0	0
$249$	1.96010e6	2.00346
$250$	0	0
$251$	−1.04384e6	−1.04580	−0.522900	−	0.852394i	$-0.675150\pi$
$251$	−1.04384e6	−1.04580	−0.522900	+	0.852394i	$0.675150\pi$
$252$	0	0
$253$	−910520.	−0.894310
$254$	0	0
$255$	847276.	0.815971
$256$	0	0
$257$	532602.	0.503002	0.251501	−	0.967857i	$-0.419076\pi$
$257$	532602.	0.503002	0.251501	+	0.967857i	$0.419076\pi$
$258$	0	0
$259$	−18697.2	−0.0173192
$260$	0	0
$261$	−439410.	−0.399272
$262$	0	0
$263$	31623.0	0.0281912	0.0140956	−	0.999901i	$-0.495513\pi$
$263$	31623.0	0.0281912	0.0140956	+	0.999901i	$0.495513\pi$
$264$	0	0
$265$	387150.	0.338661
$266$	0	0
$267$	−337620.	−0.289834
$268$	0	0
$269$	−1.28829e6	−1.08550	−0.542752	−	0.839893i	$-0.682618\pi$
$269$	−1.28829e6	−1.08550	−0.542752	+	0.839893i	$0.682618\pi$
$270$	0	0
$271$	−557119.	−0.460813	−0.230406	−	0.973095i	$-0.574006\pi$
$271$	−557119.	−0.460813	−0.230406	+	0.973095i	$0.574006\pi$
$272$	0	0
$273$	−516120.	−0.419125
$274$	0	0
$275$	−299635.	−0.238925
$276$	0	0
$277$	1.15180e6	0.901942	0.450971	−	0.892539i	$-0.351078\pi$
$277$	1.15180e6	0.901942	0.450971	+	0.892539i	$0.351078\pi$
$278$	0	0
$279$	354141.	0.272374
$280$	0	0
$281$	−1.68073e6	−1.26979	−0.634895	−	0.772598i	$-0.718957\pi$
$281$	−1.68073e6	−1.26979	−0.634895	+	0.772598i	$0.718957\pi$
$282$	0	0
$283$	−1.33106e6	−0.987944	−0.493972	−	0.869478i	$-0.664455\pi$
$283$	−1.33106e6	−0.987944	−0.493972	+	0.869478i	$0.664455\pi$
$284$	0	0
$285$	952000.	0.694264
$286$	0	0
$287$	350158.	0.250934
$288$	0	0
$289$	1.95839e6	1.37928
$290$	0	0
$291$	921549.	0.637949
$292$	0	0
$293$	1.65543e6	1.12652	0.563262	−	0.826278i	$-0.309546\pi$
$293$	1.65543e6	1.12652	0.563262	+	0.826278i	$0.309546\pi$
$294$	0	0
$295$	−182547.	−0.122129
$296$	0	0
$297$	1.29064e6	0.849012
$298$	0	0
$299$	961008.	0.621655
$300$	0	0
$301$	−1.00470e6	−0.639176
$302$	0	0
$303$	2.83221e6	1.77222
$304$	0	0
$305$	418750.	0.257754
$306$	0	0
$307$	−939601.	−0.568980	−0.284490	−	0.958679i	$-0.591824\pi$
$307$	−939601.	−0.568980	−0.284490	+	0.958679i	$0.591824\pi$
$308$	0	0
$309$	3.96134e6	2.36018
$310$	0	0
$311$	−2.34158e6	−1.37280	−0.686401	−	0.727223i	$-0.740811\pi$
$311$	−2.34158e6	−1.37280	−0.686401	+	0.727223i	$0.740811\pi$
$312$	0	0
$313$	−1.36045e6	−0.784916	−0.392458	−	0.919770i	$-0.628375\pi$
$313$	−1.36045e6	−0.784916	−0.392458	+	0.919770i	$0.628375\pi$
$314$	0	0
$315$	134144.	0.0761721
$316$	0	0
$317$	−1.79744e6	−1.00463	−0.502315	−	0.864685i	$-0.667518\pi$
$317$	−1.79744e6	−1.00463	−0.502315	+	0.864685i	$0.667518\pi$
$318$	0	0
$319$	2.17176e6	1.19491
$320$	0	0
$321$	3.77094e6	2.04262
$322$	0	0
$323$	3.79580e6	2.02440
$324$	0	0
$325$	316250.	0.166082
$326$	0	0
$327$	2.88362e6	1.49131
$328$	0	0
$329$	223380.	0.113777
$330$	0	0
$331$	273820.	0.137371	0.0686856	−	0.997638i	$-0.478119\pi$
$331$	273820.	0.137371	0.0686856	+	0.997638i	$0.478119\pi$
$332$	0	0
$333$	32786.0	0.0162023
$334$	0	0
$335$	341584.	0.166297
$336$	0	0
$337$	956498.	0.458785	0.229393	−	0.973334i	$-0.426326\pi$
$337$	956498.	0.458785	0.229393	+	0.973334i	$0.426326\pi$
$338$	0	0
$339$	−3.31561e6	−1.56698
$340$	0	0
$341$	−1.75032e6	−0.815139
$342$	0	0
$343$	1.69016e6	0.775699
$344$	0	0
$345$	−875500.	−0.396012
$346$	0	0
$347$	−4.10899e6	−1.83194	−0.915969	−	0.401249i	$-0.868576\pi$
$347$	−4.10899e6	−1.83194	−0.915969	+	0.401249i	$0.868576\pi$
$348$	0	0
$349$	4.43371e6	1.94851	0.974257	−	0.225438i	$-0.0723814\pi$
$349$	4.43371e6	1.94851	0.974257	+	0.225438i	$0.0723814\pi$
$350$	0	0
$351$	−1.36221e6	−0.590167
$352$	0	0
$353$	−2.23853e6	−0.956149	−0.478074	−	0.878319i	$-0.658665\pi$
$353$	−2.23853e6	−0.956149	−0.478074	+	0.878319i	$0.658665\pi$
$354$	0	0
$355$	1.08145e6	0.455446
$356$	0	0
$357$	1.87476e6	0.778530
$358$	0	0
$359$	−3.36447e6	−1.37778	−0.688891	−	0.724865i	$-0.741902\pi$
$359$	−3.36447e6	−1.37778	−0.688891	+	0.724865i	$0.741902\pi$
$360$	0	0
$361$	1.78886e6	0.722451
$362$	0	0
$363$	1.26841e6	0.505233
$364$	0	0
$365$	520150.	0.204360
$366$	0	0
$367$	−3.05357e6	−1.18343	−0.591715	−	0.806147i	$-0.701549\pi$
$367$	−3.05357e6	−1.18343	−0.591715	+	0.806147i	$0.701549\pi$
$368$	0	0
$369$	−614010.	−0.234752
$370$	0	0
$371$	856643.	0.323121
$372$	0	0
$373$	3.87919e6	1.44367	0.721837	−	0.692063i	$-0.243298\pi$
$373$	3.87919e6	1.44367	0.721837	+	0.692063i	$0.243298\pi$
$374$	0	0
$375$	−288111.	−0.105799
$376$	0	0
$377$	−2.29218e6	−0.830607
$378$	0	0
$379$	285585.	0.102126	0.0510631	−	0.998695i	$-0.483739\pi$
$379$	285585.	0.102126	0.0510631	+	0.998695i	$0.483739\pi$
$380$	0	0
$381$	−3.09502e6	−1.09232
$382$	0	0
$383$	−2.29819e6	−0.800552	−0.400276	−	0.916395i	$-0.631086\pi$
$383$	−2.29819e6	−0.800552	−0.400276	+	0.916395i	$0.631086\pi$
$384$	0	0
$385$	−663000.	−0.227962
$386$	0	0
$387$	1.76176e6	0.597957
$388$	0	0
$389$	−3.49829e6	−1.17215	−0.586074	−	0.810258i	$-0.699327\pi$
$389$	−3.49829e6	−1.17215	−0.586074	+	0.810258i	$0.699327\pi$
$390$	0	0
$391$	−3.49078e6	−1.15473
$392$	0	0
$393$	−5.72492e6	−1.86977
$394$	0	0
$395$	1.74618e6	0.563114
$396$	0	0
$397$	−510822.	−0.162665	−0.0813324	−	0.996687i	$-0.525918\pi$
$397$	−510822.	−0.162665	−0.0813324	+	0.996687i	$0.525918\pi$
$398$	0	0
$399$	2.10648e6	0.662408
$400$	0	0
$401$	−1.75768e6	−0.545856	−0.272928	−	0.962034i	$-0.587992\pi$
$401$	−1.75768e6	−0.545856	−0.272928	+	0.962034i	$0.587992\pi$
$402$	0	0
$403$	1.84738e6	0.566621
$404$	0	0
$405$	1.83028e6	0.554471
$406$	0	0
$407$	−162043.	−0.0484890
$408$	0	0
$409$	−2.55911e6	−0.756450	−0.378225	−	0.925714i	$-0.623465\pi$
$409$	−2.55911e6	−0.756450	−0.378225	+	0.925714i	$0.623465\pi$
$410$	0	0
$411$	−2.02797e6	−0.592184
$412$	0	0
$413$	−403920.	−0.116525
$414$	0	0
$415$	−2.65753e6	−0.757458
$416$	0	0
$417$	3.34696e6	0.942563
$418$	0	0
$419$	−201134.	−0.0559693	−0.0279846	−	0.999608i	$-0.508909\pi$
$419$	−201134.	−0.0559693	−0.0279846	+	0.999608i	$0.508909\pi$
$420$	0	0
$421$	4.75945e6	1.30873	0.654367	−	0.756177i	$-0.272935\pi$
$421$	4.75945e6	1.30873	0.654367	+	0.756177i	$0.272935\pi$
$422$	0	0
$423$	−391702.	−0.106440
$424$	0	0
$425$	−1.14875e6	−0.308499
$426$	0	0
$427$	926564.	0.245927
$428$	0	0
$429$	−4.47304e6	−1.17344
$430$	0	0
$431$	−547235.	−0.141900	−0.0709498	−	0.997480i	$-0.522603\pi$
$431$	−547235.	−0.141900	−0.0709498	+	0.997480i	$0.522603\pi$
$432$	0	0
$433$	−1.72241e6	−0.441485	−0.220742	−	0.975332i	$-0.570848\pi$
$433$	−1.72241e6	−0.441485	−0.220742	+	0.975332i	$0.570848\pi$
$434$	0	0
$435$	2.08823e6	0.529120
$436$	0	0
$437$	−3.92224e6	−0.982496
$438$	0	0
$439$	526915.	0.130491	0.0652454	−	0.997869i	$-0.479217\pi$
$439$	526915.	0.130491	0.0652454	+	0.997869i	$0.479217\pi$
$440$	0	0
$441$	−1.33346e6	−0.326500
$442$	0	0
$443$	−5.19331e6	−1.25729	−0.628645	−	0.777693i	$-0.716390\pi$
$443$	−5.19331e6	−1.25729	−0.628645	+	0.777693i	$0.716390\pi$
$444$	0	0
$445$	457750.	0.109579
$446$	0	0
$447$	7.26637e6	1.72008
$448$	0	0
$449$	−829754.	−0.194238	−0.0971188	−	0.995273i	$-0.530963\pi$
$449$	−829754.	−0.194238	−0.0971188	+	0.995273i	$0.530963\pi$
$450$	0	0
$451$	3.03471e6	0.702547
$452$	0	0
$453$	9.90284e6	2.26733
$454$	0	0
$455$	699763.	0.158461
$456$	0	0
$457$	−6.95844e6	−1.55855	−0.779276	−	0.626681i	$-0.784413\pi$
$457$	−6.95844e6	−1.55855	−0.779276	+	0.626681i	$0.784413\pi$
$458$	0	0
$459$	4.94809e6	1.09624
$460$	0	0
$461$	−6.58020e6	−1.44207	−0.721036	−	0.692898i	$-0.756334\pi$
$461$	−6.58020e6	−1.44207	−0.721036	+	0.692898i	$0.756334\pi$
$462$	0	0
$463$	−1.59965e6	−0.346794	−0.173397	−	0.984852i	$-0.555474\pi$
$463$	−1.59965e6	−0.346794	−0.173397	+	0.984852i	$0.555474\pi$
$464$	0	0
$465$	−1.68300e6	−0.360954
$466$	0	0
$467$	−4.46471e6	−0.947330	−0.473665	−	0.880705i	$-0.657069\pi$
$467$	−4.46471e6	−0.947330	−0.473665	+	0.880705i	$0.657069\pi$
$468$	0	0
$469$	755820.	0.158667
$470$	0	0
$471$	2.23854e6	0.464957
$472$	0	0
$473$	−8.70740e6	−1.78952
$474$	0	0
$475$	−1.29074e6	−0.262484
$476$	0	0
$477$	−1.50214e6	−0.302284
$478$	0	0
$479$	4.01242e6	0.799038	0.399519	−	0.916725i	$-0.369177\pi$
$479$	4.01242e6	0.799038	0.399519	+	0.916725i	$0.369177\pi$
$480$	0	0
$481$	171028.	0.0337058
$482$	0	0
$483$	−1.93721e6	−0.377841
$484$	0	0
$485$	−1.24945e6	−0.241193
$486$	0	0
$487$	5.03418e6	0.961849	0.480924	−	0.876762i	$-0.340301\pi$
$487$	5.03418e6	0.961849	0.480924	+	0.876762i	$0.340301\pi$
$488$	0	0
$489$	−3.39014e6	−0.641129
$490$	0	0
$491$	8.49020e6	1.58933	0.794666	−	0.607047i	$-0.207646\pi$
$491$	8.49020e6	1.58933	0.794666	+	0.607047i	$0.207646\pi$
$492$	0	0
$493$	8.32614e6	1.54286
$494$	0	0
$495$	1.16258e6	0.213261
$496$	0	0
$497$	2.39292e6	0.434547
$498$	0	0
$499$	−8.15229e6	−1.46564	−0.732822	−	0.680421i	$-0.761797\pi$
$499$	−8.15229e6	−1.46564	−0.732822	+	0.680421i	$0.761797\pi$
$500$	0	0
$501$	−466140.	−0.0829702
$502$	0	0
$503$	7.03390e6	1.23959	0.619793	−	0.784766i	$-0.287217\pi$
$503$	7.03390e6	1.23959	0.619793	+	0.784766i	$0.287217\pi$
$504$	0	0
$505$	−3.83995e6	−0.670034
$506$	0	0
$507$	−2.12523e6	−0.367187
$508$	0	0
$509$	−7.88765e6	−1.34944	−0.674719	−	0.738074i	$-0.735735\pi$
$509$	−7.88765e6	−1.34944	−0.674719	+	0.738074i	$0.735735\pi$
$510$	0	0
$511$	1.15093e6	0.194983
$512$	0	0
$513$	5.55968e6	0.932731
$514$	0	0
$515$	−5.37085e6	−0.892328
$516$	0	0
$517$	1.93596e6	0.318544
$518$	0	0
$519$	−1.02331e7	−1.66758
$520$	0	0
$521$	1.07500e7	1.73506	0.867528	−	0.497389i	$-0.165708\pi$
$521$	1.07500e7	1.73506	0.867528	+	0.497389i	$0.165708\pi$
$522$	0	0
$523$	−4.34873e6	−0.695198	−0.347599	−	0.937643i	$-0.613003\pi$
$523$	−4.34873e6	−0.695198	−0.347599	+	0.937643i	$0.613003\pi$
$524$	0	0
$525$	−637500.	−0.100944
$526$	0	0
$527$	−6.71043e6	−1.05250
$528$	0	0
$529$	−2.82928e6	−0.439579
$530$	0	0
$531$	708282.	0.109011
$532$	0	0
$533$	−3.20298e6	−0.488356
$534$	0	0
$535$	−5.11270e6	−0.772264
$536$	0	0
$537$	7.86216e6	1.17654
$538$	0	0
$539$	6.59054e6	0.977122
$540$	0	0
$541$	−1.57900e6	−0.231947	−0.115974	−	0.993252i	$-0.536999\pi$
$541$	−1.57900e6	−0.231947	−0.115974	+	0.993252i	$0.536999\pi$
$542$	0	0
$543$	−1.48091e7	−2.15541
$544$	0	0
$545$	−3.90965e6	−0.563828
$546$	0	0
$547$	1.50609e6	0.215219	0.107610	−	0.994193i	$-0.465680\pi$
$547$	1.50609e6	0.215219	0.107610	+	0.994193i	$0.465680\pi$
$548$	0	0
$549$	−1.62475e6	−0.230068
$550$	0	0
$551$	9.35526e6	1.31273
$552$	0	0
$553$	3.86376e6	0.537276
$554$	0	0
$555$	−155810.	−0.0214716
$556$	0	0
$557$	3.87866e6	0.529716	0.264858	−	0.964287i	$-0.414675\pi$
$557$	3.87866e6	0.529716	0.264858	+	0.964287i	$0.414675\pi$
$558$	0	0
$559$	9.19023e6	1.24393
$560$	0	0
$561$	1.62479e7	2.17967
$562$	0	0
$563$	−7.07646e6	−0.940904	−0.470452	−	0.882426i	$-0.655909\pi$
$563$	−7.07646e6	−0.940904	−0.470452	+	0.882426i	$0.655909\pi$
$564$	0	0
$565$	4.49535e6	0.592437
$566$	0	0
$567$	4.04983e6	0.529029
$568$	0	0
$569$	−6.25891e6	−0.810435	−0.405218	−	0.914220i	$-0.632804\pi$
$569$	−6.25891e6	−0.810435	−0.405218	+	0.914220i	$0.632804\pi$
$570$	0	0
$571$	−1.66214e6	−0.213342	−0.106671	−	0.994294i	$-0.534019\pi$
$571$	−1.66214e6	−0.213342	−0.106671	+	0.994294i	$0.534019\pi$
$572$	0	0
$573$	1.27004e7	1.61596
$574$	0	0
$575$	1.18702e6	0.149723
$576$	0	0
$577$	1.17539e7	1.46974	0.734872	−	0.678206i	$-0.237242\pi$
$577$	1.17539e7	1.46974	0.734872	+	0.678206i	$0.237242\pi$
$578$	0	0
$579$	−1.01226e7	−1.25486
$580$	0	0
$581$	−5.88030e6	−0.722702
$582$	0	0
$583$	7.42424e6	0.904650
$584$	0	0
$585$	−1.22705e6	−0.148242
$586$	0	0
$587$	8.11045e6	0.971516	0.485758	−	0.874093i	$-0.338544\pi$
$587$	8.11045e6	0.971516	0.485758	+	0.874093i	$0.338544\pi$
$588$	0	0
$589$	−7.53984e6	−0.895517
$590$	0	0
$591$	−1.30110e7	−1.53230
$592$	0	0
$593$	4.72379e6	0.551638	0.275819	−	0.961210i	$-0.411051\pi$
$593$	4.72379e6	0.551638	0.275819	+	0.961210i	$0.411051\pi$
$594$	0	0
$595$	−2.54183e6	−0.294343
$596$	0	0
$597$	−3.82976e6	−0.439780
$598$	0	0
$599$	1.08518e6	0.123576	0.0617879	−	0.998089i	$-0.480320\pi$
$599$	1.08518e6	0.123576	0.0617879	+	0.998089i	$0.480320\pi$
$600$	0	0
$601$	6.24943e6	0.705755	0.352878	−	0.935669i	$-0.385203\pi$
$601$	6.24943e6	0.705755	0.352878	+	0.935669i	$0.385203\pi$
$602$	0	0
$603$	−1.32535e6	−0.148435
$604$	0	0
$605$	−1.71972e6	−0.191016
$606$	0	0
$607$	−1.05137e7	−1.15820	−0.579102	−	0.815255i	$-0.696597\pi$
$607$	−1.05137e7	−1.15820	−0.579102	+	0.815255i	$0.696597\pi$
$608$	0	0
$609$	4.62060e6	0.504842
$610$	0	0
$611$	−2.04331e6	−0.221427
$612$	0	0
$613$	1.59932e7	1.71904	0.859518	−	0.511106i	$-0.170764\pi$
$613$	1.59932e7	1.71904	0.859518	+	0.511106i	$0.170764\pi$
$614$	0	0
$615$	2.91799e6	0.311097
$616$	0	0
$617$	−83598.0	−0.00884063	−0.00442031	−	0.999990i	$-0.501407\pi$
$617$	−83598.0	−0.00884063	−0.00442031	+	0.999990i	$0.501407\pi$
$618$	0	0
$619$	1.71954e7	1.80379	0.901895	−	0.431956i	$-0.142176\pi$
$619$	1.71954e7	1.80379	0.901895	+	0.431956i	$0.142176\pi$
$620$	0	0
$621$	−5.11292e6	−0.532035
$622$	0	0
$623$	1.01286e6	0.104551
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.82562e7	1.85456
$628$	0	0
$629$	−621244.	−0.0626088
$630$	0	0
$631$	1.60606e7	1.60578	0.802892	−	0.596124i	$-0.203293\pi$
$631$	1.60606e7	1.60578	0.802892	+	0.596124i	$0.203293\pi$
$632$	0	0
$633$	1.18844e7	1.17887
$634$	0	0
$635$	4.19628e6	0.412981
$636$	0	0
$637$	−6.95598e6	−0.679219
$638$	0	0
$639$	−4.19604e6	−0.406525
$640$	0	0
$641$	9.25283e6	0.889466	0.444733	−	0.895663i	$-0.353299\pi$
$641$	9.25283e6	0.889466	0.444733	+	0.895663i	$0.353299\pi$
$642$	0	0
$643$	1.41822e7	1.35274	0.676372	−	0.736560i	$-0.263551\pi$
$643$	1.41822e7	1.35274	0.676372	+	0.736560i	$0.263551\pi$
$644$	0	0
$645$	−8.37250e6	−0.792421
$646$	0	0
$647$	−1.31705e7	−1.23692	−0.618460	−	0.785816i	$-0.712243\pi$
$647$	−1.31705e7	−1.23692	−0.618460	+	0.785816i	$0.712243\pi$
$648$	0	0
$649$	−3.50064e6	−0.326239
$650$	0	0
$651$	−3.72396e6	−0.344391
$652$	0	0
$653$	8.35731e6	0.766980	0.383490	−	0.923545i	$-0.374722\pi$
$653$	8.35731e6	0.766980	0.383490	+	0.923545i	$0.374722\pi$
$654$	0	0
$655$	7.76193e6	0.706914
$656$	0	0
$657$	−2.01818e6	−0.182409
$658$	0	0
$659$	8.57809e6	0.769444	0.384722	−	0.923033i	$-0.374297\pi$
$659$	8.57809e6	0.769444	0.384722	+	0.923033i	$0.374297\pi$
$660$	0	0
$661$	8.77657e6	0.781306	0.390653	−	0.920538i	$-0.372249\pi$
$661$	8.77657e6	0.781306	0.390653	+	0.920538i	$0.372249\pi$
$662$	0	0
$663$	−1.71489e7	−1.51514
$664$	0	0
$665$	−2.85600e6	−0.250440
$666$	0	0
$667$	−8.60349e6	−0.748790
$668$	0	0
$669$	−1.66046e7	−1.43437
$670$	0	0
$671$	8.03022e6	0.688528
$672$	0	0
$673$	−1.17288e7	−0.998197	−0.499098	−	0.866545i	$-0.666335\pi$
$673$	−1.17288e7	−0.998197	−0.499098	+	0.866545i	$0.666335\pi$
$674$	0	0
$675$	−1.68257e6	−0.142139
$676$	0	0
$677$	−8.89498e6	−0.745888	−0.372944	−	0.927854i	$-0.621652\pi$
$677$	−8.89498e6	−0.745888	−0.372944	+	0.927854i	$0.621652\pi$
$678$	0	0
$679$	−2.76465e6	−0.230126
$680$	0	0
$681$	1.03792e7	0.857621
$682$	0	0
$683$	8.77304e6	0.719612	0.359806	−	0.933027i	$-0.382843\pi$
$683$	8.77304e6	0.719612	0.359806	+	0.933027i	$0.382843\pi$
$684$	0	0
$685$	2.74955e6	0.223890
$686$	0	0
$687$	8.96416e6	0.724633
$688$	0	0
$689$	−7.83592e6	−0.628842
$690$	0	0
$691$	−1.51415e7	−1.20635	−0.603175	−	0.797609i	$-0.706098\pi$
$691$	−1.51415e7	−1.20635	−0.603175	+	0.797609i	$0.706098\pi$
$692$	0	0
$693$	2.57244e6	0.203476
$694$	0	0
$695$	−4.53786e6	−0.356360
$696$	0	0
$697$	1.16345e7	0.907125
$698$	0	0
$699$	−2.64493e7	−2.04749
$700$	0	0
$701$	−1.30259e6	−0.100118	−0.0500591	−	0.998746i	$-0.515941\pi$
$701$	−1.30259e6	−0.100118	−0.0500591	+	0.998746i	$0.515941\pi$
$702$	0	0
$703$	−698030.	−0.0532704
$704$	0	0
$705$	1.86150e6	0.141056
$706$	0	0
$707$	−8.49662e6	−0.639290
$708$	0	0
$709$	−1.33291e7	−0.995826	−0.497913	−	0.867227i	$-0.665900\pi$
$709$	−1.33291e7	−0.995826	−0.497913	+	0.867227i	$0.665900\pi$
$710$	0	0
$711$	−6.77519e6	−0.502629
$712$	0	0
$713$	6.93396e6	0.510808
$714$	0	0
$715$	6.06462e6	0.443648
$716$	0	0
$717$	−308720.	−0.0224268
$718$	0	0
$719$	−3.51250e6	−0.253393	−0.126696	−	0.991942i	$-0.540437\pi$
$719$	−3.51250e6	−0.253393	−0.126696	+	0.991942i	$0.540437\pi$
$720$	0	0
$721$	−1.18840e7	−0.851383
$722$	0	0
$723$	2.15138e7	1.53063
$724$	0	0
$725$	−2.83125e6	−0.200048
$726$	0	0
$727$	−3.58078e6	−0.251270	−0.125635	−	0.992077i	$-0.540097\pi$
$727$	−3.58078e6	−0.251270	−0.125635	+	0.992077i	$0.540097\pi$
$728$	0	0
$729$	4.96105e6	0.345744
$730$	0	0
$731$	−3.33827e7	−2.31062
$732$	0	0
$733$	−5.42649e6	−0.373043	−0.186521	−	0.982451i	$-0.559721\pi$
$733$	−5.42649e6	−0.373043	−0.186521	+	0.982451i	$0.559721\pi$
$734$	0	0
$735$	6.33705e6	0.432682
$736$	0	0
$737$	6.55044e6	0.444224
$738$	0	0
$739$	740440.	0.0498745	0.0249373	−	0.999689i	$-0.492061\pi$
$739$	740440.	0.0498745	0.0249373	+	0.999689i	$0.492061\pi$
$740$	0	0
$741$	−1.92685e7	−1.28915
$742$	0	0
$743$	2.46833e7	1.64033	0.820164	−	0.572128i	$-0.193882\pi$
$743$	2.46833e7	1.64033	0.820164	+	0.572128i	$0.193882\pi$
$744$	0	0
$745$	−9.85185e6	−0.650320
$746$	0	0
$747$	1.03112e7	0.676097
$748$	0	0
$749$	−1.13128e7	−0.736828
$750$	0	0
$751$	−9.14568e6	−0.591720	−0.295860	−	0.955231i	$-0.595606\pi$
$751$	−9.14568e6	−0.591720	−0.295860	+	0.955231i	$0.595606\pi$
$752$	0	0
$753$	−1.92474e7	−1.23704
$754$	0	0
$755$	−1.34264e7	−0.857221
$756$	0	0
$757$	285562.	0.0181118	0.00905588	−	0.999959i	$-0.497117\pi$
$757$	285562.	0.0181118	0.00905588	+	0.999959i	$0.497117\pi$
$758$	0	0
$759$	−1.67892e7	−1.05785
$760$	0	0
$761$	−1.06300e7	−0.665385	−0.332693	−	0.943035i	$-0.607957\pi$
$761$	−1.06300e7	−0.665385	−0.332693	+	0.943035i	$0.607957\pi$
$762$	0	0
$763$	−8.65085e6	−0.537957
$764$	0	0
$765$	4.45715e6	0.275362
$766$	0	0
$767$	3.69475e6	0.226776
$768$	0	0
$769$	−1.86623e7	−1.13802	−0.569008	−	0.822332i	$-0.692673\pi$
$769$	−1.86623e7	−1.13802	−0.569008	+	0.822332i	$0.692673\pi$
$770$	0	0
$771$	9.82070e6	0.594985
$772$	0	0
$773$	−6.18973e6	−0.372583	−0.186292	−	0.982495i	$-0.559647\pi$
$773$	−6.18973e6	−0.372583	−0.186292	+	0.982495i	$0.559647\pi$
$774$	0	0
$775$	2.28184e6	0.136468
$776$	0	0
$777$	−344760.	−0.0204863
$778$	0	0
$779$	1.30726e7	0.771823
$780$	0	0
$781$	2.07386e7	1.21661
$782$	0	0
$783$	1.21952e7	0.710863
$784$	0	0
$785$	−3.03505e6	−0.175789
$786$	0	0
$787$	−2.06471e7	−1.18829	−0.594144	−	0.804359i	$-0.702509\pi$
$787$	−2.06471e7	−1.18829	−0.594144	+	0.804359i	$0.702509\pi$
$788$	0	0
$789$	583100.	0.0333465
$790$	0	0
$791$	9.94682e6	0.565253
$792$	0	0
$793$	−8.47550e6	−0.478611
$794$	0	0
$795$	7.13869e6	0.400591
$796$	0	0
$797$	−2.89792e6	−0.161600	−0.0807998	−	0.996730i	$-0.525747\pi$
$797$	−2.89792e6	−0.161600	−0.0807998	+	0.996730i	$0.525747\pi$
$798$	0	0
$799$	7.42214e6	0.411303
$800$	0	0
$801$	−1.77607e6	−0.0978090
$802$	0	0
$803$	9.97474e6	0.545899
$804$	0	0
$805$	2.62650e6	0.142852
$806$	0	0
$807$	−2.37548e7	−1.28401
$808$	0	0
$809$	1.11249e7	0.597619	0.298810	−	0.954313i	$-0.403410\pi$
$809$	1.11249e7	0.597619	0.298810	+	0.954313i	$0.403410\pi$
$810$	0	0
$811$	−1.37881e7	−0.736124	−0.368062	−	0.929801i	$-0.619979\pi$
$811$	−1.37881e7	−0.736124	−0.368062	+	0.929801i	$0.619979\pi$
$812$	0	0
$813$	−1.02728e7	−0.545081
$814$	0	0
$815$	4.59640e6	0.242395
$816$	0	0
$817$	−3.75088e7	−1.96598
$818$	0	0
$819$	−2.71508e6	−0.141440
$820$	0	0
$821$	−2.73122e7	−1.41416	−0.707079	−	0.707134i	$-0.749988\pi$
$821$	−2.73122e7	−1.41416	−0.707079	+	0.707134i	$0.749988\pi$
$822$	0	0
$823$	2.63577e7	1.35646	0.678231	−	0.734848i	$-0.262747\pi$
$823$	2.63577e7	1.35646	0.678231	+	0.734848i	$0.262747\pi$
$824$	0	0
$825$	−5.52500e6	−0.282616
$826$	0	0
$827$	7.94251e6	0.403826	0.201913	−	0.979403i	$-0.435284\pi$
$827$	7.94251e6	0.403826	0.201913	+	0.979403i	$0.435284\pi$
$828$	0	0
$829$	3.07561e7	1.55434	0.777168	−	0.629293i	$-0.216655\pi$
$829$	3.07561e7	1.55434	0.777168	+	0.629293i	$0.216655\pi$
$830$	0	0
$831$	2.12382e7	1.06688
$832$	0	0
$833$	2.52670e7	1.26166
$834$	0	0
$835$	632000.	0.0313690
$836$	0	0
$837$	−9.82872e6	−0.484935
$838$	0	0
$839$	6.30373e6	0.309167	0.154583	−	0.987980i	$-0.450596\pi$
$839$	6.30373e6	0.309167	0.154583	+	0.987980i	$0.450596\pi$
$840$	0	0
$841$	9751.00	0.000475400 0
$842$	0	0
$843$	−3.09911e7	−1.50199
$844$	0	0
$845$	2.88142e6	0.138824
$846$	0	0
$847$	−3.80522e6	−0.182252
$848$	0	0
$849$	−2.45436e7	−1.16861
$850$	0	0
$851$	641938.	0.0303857
$852$	0	0
$853$	1.52783e7	0.718958	0.359479	−	0.933153i	$-0.382954\pi$
$853$	1.52783e7	0.718958	0.359479	+	0.933153i	$0.382954\pi$
$854$	0	0
$855$	5.00806e6	0.234290
$856$	0	0
$857$	−1.39360e7	−0.648165	−0.324082	−	0.946029i	$-0.605056\pi$
$857$	−1.39360e7	−0.648165	−0.324082	+	0.946029i	$0.605056\pi$
$858$	0	0
$859$	3.51792e7	1.62668	0.813342	−	0.581786i	$-0.197646\pi$
$859$	3.51792e7	1.62668	0.813342	+	0.581786i	$0.197646\pi$
$860$	0	0
$861$	6.45660e6	0.296822
$862$	0	0
$863$	−1.90668e7	−0.871469	−0.435734	−	0.900075i	$-0.643511\pi$
$863$	−1.90668e7	−0.871469	−0.435734	+	0.900075i	$0.643511\pi$
$864$	0	0
$865$	1.38742e7	0.630472
$866$	0	0
$867$	3.61109e7	1.63151
$868$	0	0
$869$	3.34859e7	1.50422
$870$	0	0
$871$	−6.91366e6	−0.308790
$872$	0	0
$873$	4.84787e6	0.215286
$874$	0	0
$875$	864332.	0.0381646
$876$	0	0
$877$	3.99993e7	1.75612	0.878059	−	0.478553i	$-0.158838\pi$
$877$	3.99993e7	1.75612	0.878059	+	0.478553i	$0.158838\pi$
$878$	0	0
$879$	3.05245e7	1.33253
$880$	0	0
$881$	−7.29981e6	−0.316863	−0.158432	−	0.987370i	$-0.550644\pi$
$881$	−7.29981e6	−0.316863	−0.158432	+	0.987370i	$0.550644\pi$
$882$	0	0
$883$	−4.18787e6	−0.180755	−0.0903777	−	0.995908i	$-0.528807\pi$
$883$	−4.18787e6	−0.180755	−0.0903777	+	0.995908i	$0.528807\pi$
$884$	0	0
$885$	−3.36600e6	−0.144463
$886$	0	0
$887$	−5.29606e6	−0.226018	−0.113009	−	0.993594i	$-0.536049\pi$
$887$	−5.29606e6	−0.226018	−0.113009	+	0.993594i	$0.536049\pi$
$888$	0	0
$889$	9.28506e6	0.394031
$890$	0	0
$891$	3.50985e7	1.48114
$892$	0	0
$893$	8.33952e6	0.349955
$894$	0	0
$895$	−1.06596e7	−0.444821
$896$	0	0
$897$	1.77201e7	0.735336
$898$	0	0
$899$	−1.65388e7	−0.682502
$900$	0	0
$901$	2.84633e7	1.16808
$902$	0	0
$903$	−1.85258e7	−0.756060
$904$	0	0
$905$	2.00784e7	0.814908
$906$	0	0
$907$	−1.93678e7	−0.781741	−0.390871	−	0.920446i	$-0.627826\pi$
$907$	−1.93678e7	−0.781741	−0.390871	+	0.920446i	$0.627826\pi$
$908$	0	0
$909$	1.48990e7	0.598064
$910$	0	0
$911$	−3.88005e7	−1.54896	−0.774481	−	0.632597i	$-0.781989\pi$
$911$	−3.88005e7	−1.54896	−0.774481	+	0.632597i	$0.781989\pi$
$912$	0	0
$913$	−5.09626e7	−2.02337
$914$	0	0
$915$	7.72137e6	0.304889
$916$	0	0
$917$	1.71748e7	0.674477
$918$	0	0
$919$	−4.08440e7	−1.59529	−0.797644	−	0.603128i	$-0.793921\pi$
$919$	−4.08440e7	−1.59529	−0.797644	+	0.603128i	$0.793921\pi$
$920$	0	0
$921$	−1.73254e7	−0.673029
$922$	0	0
$923$	−2.18886e7	−0.845695
$924$	0	0
$925$	211250.	0.00811788
$926$	0	0
$927$	2.08389e7	0.796480
$928$	0	0
$929$	2.52903e7	0.961423	0.480712	−	0.876879i	$-0.340378\pi$
$929$	2.52903e7	0.961423	0.480712	+	0.876879i	$0.340378\pi$
$930$	0	0
$931$	2.83900e7	1.07347
$932$	0	0
$933$	−4.31766e7	−1.62384
$934$	0	0
$935$	−2.20292e7	−0.824080
$936$	0	0
$937$	−3.20866e7	−1.19392	−0.596960	−	0.802271i	$-0.703625\pi$
$937$	−3.20866e7	−1.19392	−0.596960	+	0.802271i	$0.703625\pi$
$938$	0	0
$939$	−2.50855e7	−0.928452
$940$	0	0
$941$	2.92051e7	1.07519	0.537595	−	0.843203i	$-0.319333\pi$
$941$	2.92051e7	1.07519	0.537595	+	0.843203i	$0.319333\pi$
$942$	0	0
$943$	−1.20221e7	−0.440252
$944$	0	0
$945$	−3.72300e6	−0.135617
$946$	0	0
$947$	−5.98557e6	−0.216885	−0.108443	−	0.994103i	$-0.534586\pi$
$947$	−5.98557e6	−0.216885	−0.108443	+	0.994103i	$0.534586\pi$
$948$	0	0
$949$	−1.05278e7	−0.379467
$950$	0	0
$951$	−3.31431e7	−1.18834
$952$	0	0
$953$	3.73699e6	0.133287	0.0666437	−	0.997777i	$-0.478771\pi$
$953$	3.73699e6	0.133287	0.0666437	+	0.997777i	$0.478771\pi$
$954$	0	0
$955$	−1.72193e7	−0.610953
$956$	0	0
$957$	4.00452e7	1.41342
$958$	0	0
$959$	6.08390e6	0.213617
$960$	0	0
$961$	−1.52998e7	−0.534413
$962$	0	0
$963$	1.98373e7	0.689313
$964$	0	0
$965$	1.37243e7	0.474431
$966$	0	0
$967$	1.17760e7	0.404977	0.202488	−	0.979285i	$-0.435097\pi$
$967$	1.17760e7	0.404977	0.202488	+	0.979285i	$0.435097\pi$
$968$	0	0
$969$	6.99910e7	2.39460
$970$	0	0
$971$	8.49470e6	0.289135	0.144567	−	0.989495i	$-0.453821\pi$
$971$	8.49470e6	0.289135	0.144567	+	0.989495i	$0.453821\pi$
$972$	0	0
$973$	−1.00409e7	−0.340008
$974$	0	0
$975$	5.83136e6	0.196453
$976$	0	0
$977$	3.86010e7	1.29379	0.646893	−	0.762581i	$-0.276068\pi$
$977$	3.86010e7	1.29379	0.646893	+	0.762581i	$0.276068\pi$
$978$	0	0
$979$	8.77811e6	0.292715
$980$	0	0
$981$	1.51694e7	0.503265
$982$	0	0
$983$	−2.03309e7	−0.671077	−0.335539	−	0.942026i	$-0.608918\pi$
$983$	−2.03309e7	−0.671077	−0.335539	+	0.942026i	$0.608918\pi$
$984$	0	0
$985$	1.76406e7	0.579324
$986$	0	0
$987$	4.11892e6	0.134583
$988$	0	0
$989$	3.44947e7	1.12140
$990$	0	0
$991$	3.36656e7	1.08894	0.544468	−	0.838782i	$-0.316732\pi$
$991$	3.36656e7	1.08894	0.544468	+	0.838782i	$0.316732\pi$
$992$	0	0
$993$	5.04900e6	0.162492
$994$	0	0
$995$	5.19245e6	0.166270
$996$	0	0
$997$	−4.95466e7	−1.57862	−0.789308	−	0.613998i	$-0.789560\pi$
$997$	−4.95466e7	−1.57862	−0.789308	+	0.613998i	$0.789560\pi$
$998$	0	0
$999$	−909932.	−0.0288466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.6.a.c.1.2	yes	2
4.3	odd	2	inner	160.6.a.c.1.1	✓	2
5.2	odd	4		800.6.c.e.449.1		4
5.3	odd	4		800.6.c.e.449.3		4
5.4	even	2		800.6.a.j.1.1		2
8.3	odd	2		320.6.a.u.1.2		2
8.5	even	2		320.6.a.u.1.1		2
20.3	even	4		800.6.c.e.449.2		4
20.7	even	4		800.6.c.e.449.4		4
20.19	odd	2		800.6.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.c.1.1	✓	2	4.3	odd	2	inner
160.6.a.c.1.2	yes	2	1.1	even	1	trivial
320.6.a.u.1.1		2	8.5	even	2
320.6.a.u.1.2		2	8.3	odd	2
800.6.a.j.1.1		2	5.4	even	2
800.6.a.j.1.2		2	20.19	odd	2
800.6.c.e.449.1		4	5.2	odd	4
800.6.c.e.449.2		4	20.3	even	4
800.6.c.e.449.3		4	5.3	odd	4
800.6.c.e.449.4		4	20.7	even	4

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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)

\(f(q)\)	\(=\)	\(q+18.4391 q^{3} -25.0000 q^{5} -55.3173 q^{7} +97.0000 q^{9} +O(q^{10})\)	\(q+18.4391 q^{3} -25.0000 q^{5} -55.3173 q^{7} +97.0000 q^{9} -479.416 q^{11} +506.000 q^{13} -460.977 q^{15} -1838.00 q^{17} -2065.18 q^{19} -1020.00 q^{21} +1899.23 q^{23} +625.000 q^{25} -2692.11 q^{27} -4530.00 q^{29} +3650.94 q^{31} -8840.00 q^{33} +1382.93 q^{35} +338.000 q^{37} +9330.18 q^{39} -6330.00 q^{41} +18162.5 q^{43} -2425.00 q^{45} -4038.16 q^{47} -13747.0 q^{49} -33891.0 q^{51} -15486.0 q^{53} +11985.4 q^{55} -38080.0 q^{57} +7301.88 q^{59} -16750.0 q^{61} -5365.77 q^{63} -12650.0 q^{65} -13663.4 q^{67} +35020.0 q^{69} -43258.1 q^{71} -20806.0 q^{73} +11524.4 q^{75} +26520.0 q^{77} -69847.3 q^{79} -73211.0 q^{81} +106301. q^{83} +45950.0 q^{85} -83529.1 q^{87} -18310.0 q^{89} -27990.5 q^{91} +67320.0 q^{93} +51629.4 q^{95} +49978.0 q^{97} -46503.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 50 q^{5} + 194 q^{9}+O(q^{10}) \) 2 * q - 50 * q^5 + 194 * q^9	\( 2 q - 50 q^{5} + 194 q^{9} + 1012 q^{13} - 3676 q^{17} - 2040 q^{21} + 1250 q^{25} - 9060 q^{29} - 17680 q^{33} + 676 q^{37} - 12660 q^{41} - 4850 q^{45} - 27494 q^{49} - 30972 q^{53} - 76160 q^{57} - 33500 q^{61} - 25300 q^{65} + 70040 q^{69} - 41612 q^{73} + 53040 q^{77} - 146422 q^{81} + 91900 q^{85} - 36620 q^{89} + 134640 q^{93} + 99956 q^{97}+O(q^{100}) \) 2 * q - 50 * q^5 + 194 * q^9 + 1012 * q^13 - 3676 * q^17 - 2040 * q^21 + 1250 * q^25 - 9060 * q^29 - 17680 * q^33 + 676 * q^37 - 12660 * q^41 - 4850 * q^45 - 27494 * q^49 - 30972 * q^53 - 76160 * q^57 - 33500 * q^61 - 25300 * q^65 + 70040 * q^69 - 41612 * q^73 + 53040 * q^77 - 146422 * q^81 + 91900 * q^85 - 36620 * q^89 + 134640 * q^93 + 99956 * q^97

Introduction

L-functions

Modular forms

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