LMFDB - Embedded newform 160.6.a.d.1.1

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{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.16228$ 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)

$f(q)$	$=$	$q-6.32456 q^{3} +25.0000 q^{5} -44.2719 q^{7} -203.000 q^{9} +O(q^{10})$	$q-6.32456 q^{3} +25.0000 q^{5} -44.2719 q^{7} -203.000 q^{9} +720.999 q^{11} -146.000 q^{13} -158.114 q^{15} -702.000 q^{17} -2732.21 q^{19} +280.000 q^{21} +4091.99 q^{23} +625.000 q^{25} +2820.75 q^{27} -4010.00 q^{29} -4566.33 q^{31} -4560.00 q^{33} -1106.80 q^{35} -14778.0 q^{37} +923.385 q^{39} -4350.00 q^{41} -12427.8 q^{43} -5075.00 q^{45} -6014.65 q^{47} -14847.0 q^{49} +4439.84 q^{51} -18154.0 q^{53} +18025.0 q^{55} +17280.0 q^{57} +19707.3 q^{59} -42130.0 q^{61} +8987.19 q^{63} -3650.00 q^{65} -16184.5 q^{67} -25880.0 q^{69} -45448.3 q^{71} +26266.0 q^{73} -3952.85 q^{75} -31920.0 q^{77} +8677.29 q^{79} +31489.0 q^{81} +98757.9 q^{83} -17550.0 q^{85} +25361.5 q^{87} +30570.0 q^{89} +6463.70 q^{91} +28880.0 q^{93} -68305.2 q^{95} +66882.0 q^{97} -146363. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 50 q^{5} - 406 q^{9}+O(q^{10}) $ 2 * q + 50 * q^5 - 406 * q^9	$ 2 q + 50 q^{5} - 406 q^{9} - 292 q^{13} - 1404 q^{17} + 560 q^{21} + 1250 q^{25} - 8020 q^{29} - 9120 q^{33} - 29556 q^{37} - 8700 q^{41} - 10150 q^{45} - 29694 q^{49} - 36308 q^{53} + 34560 q^{57} - 84260 q^{61} - 7300 q^{65} - 51760 q^{69} + 52532 q^{73} - 63840 q^{77} + 62978 q^{81} - 35100 q^{85} + 61140 q^{89} + 57760 q^{93} + 133764 q^{97}+O(q^{100}) $ 2 * q + 50 * q^5 - 406 * q^9 - 292 * q^13 - 1404 * q^17 + 560 * q^21 + 1250 * q^25 - 8020 * q^29 - 9120 * q^33 - 29556 * q^37 - 8700 * q^41 - 10150 * q^45 - 29694 * q^49 - 36308 * q^53 + 34560 * q^57 - 84260 * q^61 - 7300 * q^65 - 51760 * q^69 + 52532 * q^73 - 63840 * q^77 + 62978 * q^81 - 35100 * q^85 + 61140 * q^89 + 57760 * q^93 + 133764 * 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
$3$	−6.32456	−0.405720	−0.202860	−	0.979208i	$-0.565024\pi$
$3$	−6.32456	−0.405720	−0.202860	+	0.979208i	$0.565024\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	−44.2719	−0.341494	−0.170747	−	0.985315i	$-0.554618\pi$
$7$	−44.2719	−0.341494	−0.170747	+	0.985315i	$0.554618\pi$
$8$	0	0
$9$	−203.000	−0.835391
$10$	0	0
$11$	720.999	1.79661	0.898304	−	0.439375i	$-0.144800\pi$
$11$	720.999	1.79661	0.898304	+	0.439375i	$0.144800\pi$
$12$	0	0
$13$	−146.000	−0.239604	−0.119802	−	0.992798i	$-0.538226\pi$
$13$	−146.000	−0.239604	−0.119802	+	0.992798i	$0.538226\pi$
$14$	0	0
$15$	−158.114	−0.181444
$16$	0	0
$17$	−702.000	−0.589135	−0.294567	−	0.955631i	$-0.595176\pi$
$17$	−702.000	−0.589135	−0.294567	+	0.955631i	$0.595176\pi$
$18$	0	0
$19$	−2732.21	−1.73632	−0.868160	−	0.496285i	$-0.834697\pi$
$19$	−2732.21	−1.73632	−0.868160	+	0.496285i	$0.834697\pi$
$20$	0	0
$21$	280.000	0.138551
$22$	0	0
$23$	4091.99	1.61293	0.806463	−	0.591284i	$-0.201379\pi$
$23$	4091.99	1.61293	0.806463	+	0.591284i	$0.201379\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2820.75	0.744656
$28$	0	0
$29$	−4010.00	−0.885420	−0.442710	−	0.896665i	$-0.645983\pi$
$29$	−4010.00	−0.885420	−0.442710	+	0.896665i	$0.645983\pi$
$30$	0	0
$31$	−4566.33	−0.853420	−0.426710	−	0.904388i	$-0.640328\pi$
$31$	−4566.33	−0.853420	−0.426710	+	0.904388i	$0.640328\pi$
$32$	0	0
$33$	−4560.00	−0.728920
$34$	0	0
$35$	−1106.80	−0.152721
$36$	0	0
$37$	−14778.0	−1.77464	−0.887322	−	0.461150i	$-0.847437\pi$
$37$	−14778.0	−1.77464	−0.887322	+	0.461150i	$0.847437\pi$
$38$	0	0
$39$	923.385	0.0972123
$40$	0	0
$41$	−4350.00	−0.404138	−0.202069	−	0.979371i	$-0.564767\pi$
$41$	−4350.00	−0.404138	−0.202069	+	0.979371i	$0.564767\pi$
$42$	0	0
$43$	−12427.8	−1.02499	−0.512497	−	0.858689i	$-0.671279\pi$
$43$	−12427.8	−1.02499	−0.512497	+	0.858689i	$0.671279\pi$
$44$	0	0
$45$	−5075.00	−0.373598
$46$	0	0
$47$	−6014.65	−0.397160	−0.198580	−	0.980085i	$-0.563633\pi$
$47$	−6014.65	−0.397160	−0.198580	+	0.980085i	$0.563633\pi$
$48$	0	0
$49$	−14847.0	−0.883382
$50$	0	0
$51$	4439.84	0.239024
$52$	0	0
$53$	−18154.0	−0.887734	−0.443867	−	0.896093i	$-0.646394\pi$
$53$	−18154.0	−0.887734	−0.443867	+	0.896093i	$0.646394\pi$
$54$	0	0
$55$	18025.0	0.803467
$56$	0	0
$57$	17280.0	0.704460
$58$	0	0
$59$	19707.3	0.737051	0.368525	−	0.929618i	$-0.379863\pi$
$59$	19707.3	0.737051	0.368525	+	0.929618i	$0.379863\pi$
$60$	0	0
$61$	−42130.0	−1.44966	−0.724831	−	0.688927i	$-0.758082\pi$
$61$	−42130.0	−1.44966	−0.724831	+	0.688927i	$0.758082\pi$
$62$	0	0
$63$	8987.19	0.285281
$64$	0	0
$65$	−3650.00	−0.107154
$66$	0	0
$67$	−16184.5	−0.440467	−0.220233	−	0.975447i	$-0.570682\pi$
$67$	−16184.5	−0.440467	−0.220233	+	0.975447i	$0.570682\pi$
$68$	0	0
$69$	−25880.0	−0.654397
$70$	0	0
$71$	−45448.3	−1.06997	−0.534985	−	0.844862i	$-0.679683\pi$
$71$	−45448.3	−1.06997	−0.534985	+	0.844862i	$0.679683\pi$
$72$	0	0
$73$	26266.0	0.576882	0.288441	−	0.957498i	$-0.406863\pi$
$73$	26266.0	0.576882	0.288441	+	0.957498i	$0.406863\pi$
$74$	0	0
$75$	−3952.85	−0.0811441
$76$	0	0
$77$	−31920.0	−0.613530
$78$	0	0
$79$	8677.29	0.156429	0.0782143	−	0.996937i	$-0.475078\pi$
$79$	8677.29	0.156429	0.0782143	+	0.996937i	$0.475078\pi$
$80$	0	0
$81$	31489.0	0.533269
$82$	0	0
$83$	98757.9	1.57354	0.786768	−	0.617249i	$-0.211753\pi$
$83$	98757.9	1.57354	0.786768	+	0.617249i	$0.211753\pi$
$84$	0	0
$85$	−17550.0	−0.263469
$86$	0	0
$87$	25361.5	0.359233
$88$	0	0
$89$	30570.0	0.409091	0.204546	−	0.978857i	$-0.434428\pi$
$89$	30570.0	0.409091	0.204546	+	0.978857i	$0.434428\pi$
$90$	0	0
$91$	6463.70	0.0818234
$92$	0	0
$93$	28880.0	0.346250
$94$	0	0
$95$	−68305.2	−0.776506
$96$	0	0
$97$	66882.0	0.721739	0.360869	−	0.932616i	$-0.382480\pi$
$97$	66882.0	0.721739	0.360869	+	0.932616i	$0.382480\pi$
$98$	0	0
$99$	−146363.	−1.50087
$100$	0	0
$101$	42798.0	0.417465	0.208732	−	0.977973i	$-0.433066\pi$
$101$	42798.0	0.417465	0.208732	+	0.977973i	$0.433066\pi$
$102$	0	0
$103$	−10505.1	−0.0975678	−0.0487839	−	0.998809i	$-0.515535\pi$
$103$	−10505.1	−0.0975678	−0.0487839	+	0.998809i	$0.515535\pi$
$104$	0	0
$105$	7000.00	0.0619619
$106$	0	0
$107$	−66856.9	−0.564529	−0.282265	−	0.959337i	$-0.591086\pi$
$107$	−66856.9	−0.564529	−0.282265	+	0.959337i	$0.591086\pi$
$108$	0	0
$109$	−111714.	−0.900620	−0.450310	−	0.892872i	$-0.648686\pi$
$109$	−111714.	−0.900620	−0.450310	+	0.892872i	$0.648686\pi$
$110$	0	0
$111$	93464.3	0.720009
$112$	0	0
$113$	216834.	1.59746	0.798732	−	0.601686i	$-0.205504\pi$
$113$	216834.	1.59746	0.798732	+	0.601686i	$0.205504\pi$
$114$	0	0
$115$	102300.	0.721323
$116$	0	0
$117$	29638.0	0.200163
$118$	0	0
$119$	31078.9	0.201186
$120$	0	0
$121$	358789.	2.22780
$122$	0	0
$123$	27511.8	0.163967
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−62277.9	−0.342629	−0.171315	−	0.985216i	$-0.554801\pi$
$127$	−62277.9	−0.342629	−0.171315	+	0.985216i	$0.554801\pi$
$128$	0	0
$129$	78600.0	0.415861
$130$	0	0
$131$	−176189.	−0.897019	−0.448510	−	0.893778i	$-0.648045\pi$
$131$	−176189.	−0.897019	−0.448510	+	0.893778i	$0.648045\pi$
$132$	0	0
$133$	120960.	0.592943
$134$	0	0
$135$	70518.8	0.333020
$136$	0	0
$137$	99802.0	0.454295	0.227147	−	0.973860i	$-0.427060\pi$
$137$	99802.0	0.454295	0.227147	+	0.973860i	$0.427060\pi$
$138$	0	0
$139$	271475.	1.19177	0.595886	−	0.803069i	$-0.296801\pi$
$139$	271475.	1.19177	0.595886	+	0.803069i	$0.296801\pi$
$140$	0	0
$141$	38040.0	0.161136
$142$	0	0
$143$	−105266.	−0.430475
$144$	0	0
$145$	−100250.	−0.395972
$146$	0	0
$147$	93900.7	0.358406
$148$	0	0
$149$	−413626.	−1.52631	−0.763154	−	0.646217i	$-0.776350\pi$
$149$	−413626.	−1.52631	−0.763154	+	0.646217i	$0.776350\pi$
$150$	0	0
$151$	−172496.	−0.615654	−0.307827	−	0.951442i	$-0.599602\pi$
$151$	−172496.	−0.615654	−0.307827	+	0.951442i	$0.599602\pi$
$152$	0	0
$153$	142506.	0.492158
$154$	0	0
$155$	−114158.	−0.381661
$156$	0	0
$157$	179358.	0.580726	0.290363	−	0.956917i	$-0.406224\pi$
$157$	179358.	0.580726	0.290363	+	0.956917i	$0.406224\pi$
$158$	0	0
$159$	114816.	0.360172
$160$	0	0
$161$	−181160.	−0.550805
$162$	0	0
$163$	−465924.	−1.37355	−0.686777	−	0.726868i	$-0.740975\pi$
$163$	−465924.	−1.37355	−0.686777	+	0.726868i	$0.740975\pi$
$164$	0	0
$165$	−114000.	−0.325983
$166$	0	0
$167$	609567.	1.69134	0.845669	−	0.533708i	$-0.179202\pi$
$167$	609567.	1.69134	0.845669	+	0.533708i	$0.179202\pi$
$168$	0	0
$169$	−349977.	−0.942590
$170$	0	0
$171$	554638.	1.45051
$172$	0	0
$173$	591086.	1.50153	0.750767	−	0.660567i	$-0.229684\pi$
$173$	591086.	1.50153	0.750767	+	0.660567i	$0.229684\pi$
$174$	0	0
$175$	−27669.9	−0.0682988
$176$	0	0
$177$	−124640.	−0.299037
$178$	0	0
$179$	−211215.	−0.492711	−0.246355	−	0.969180i	$-0.579233\pi$
$179$	−211215.	−0.492711	−0.246355	+	0.969180i	$0.579233\pi$
$180$	0	0
$181$	−97538.0	−0.221298	−0.110649	−	0.993860i	$-0.535293\pi$
$181$	−97538.0	−0.221298	−0.110649	+	0.993860i	$0.535293\pi$
$182$	0	0
$183$	266454.	0.588158
$184$	0	0
$185$	−369450.	−0.793645
$186$	0	0
$187$	−506142.	−1.05844
$188$	0	0
$189$	−124880.	−0.254295
$190$	0	0
$191$	−307158.	−0.609227	−0.304613	−	0.952476i	$-0.598527\pi$
$191$	−307158.	−0.609227	−0.304613	+	0.952476i	$0.598527\pi$
$192$	0	0
$193$	12434.0	0.0240280	0.0120140	−	0.999928i	$-0.496176\pi$
$193$	12434.0	0.0240280	0.0120140	+	0.999928i	$0.496176\pi$
$194$	0	0
$195$	23084.6	0.0434747
$196$	0	0
$197$	−378858.	−0.695522	−0.347761	−	0.937583i	$-0.613058\pi$
$197$	−378858.	−0.695522	−0.347761	+	0.937583i	$0.613058\pi$
$198$	0	0
$199$	−767194.	−1.37332	−0.686661	−	0.726978i	$-0.740924\pi$
$199$	−767194.	−1.37332	−0.686661	+	0.726978i	$0.740924\pi$
$200$	0	0
$201$	102360.	0.178706
$202$	0	0
$203$	177530.	0.302366
$204$	0	0
$205$	−108750.	−0.180736
$206$	0	0
$207$	−830673.	−1.34742
$208$	0	0
$209$	−1.96992e6	−3.11948
$210$	0	0
$211$	864529.	1.33682	0.668411	−	0.743793i	$-0.266975\pi$
$211$	864529.	1.33682	0.668411	+	0.743793i	$0.266975\pi$
$212$	0	0
$213$	287440.	0.434108
$214$	0	0
$215$	−310694.	−0.458391
$216$	0	0
$217$	202160.	0.291438
$218$	0	0
$219$	−166121.	−0.234053
$220$	0	0
$221$	102492.	0.141159
$222$	0	0
$223$	639710.	0.861432	0.430716	−	0.902488i	$-0.358261\pi$
$223$	639710.	0.861432	0.430716	+	0.902488i	$0.358261\pi$
$224$	0	0
$225$	−126875.	−0.167078
$226$	0	0
$227$	980515.	1.26296	0.631480	−	0.775392i	$-0.282448\pi$
$227$	980515.	1.26296	0.631480	+	0.775392i	$0.282448\pi$
$228$	0	0
$229$	−1.01261e6	−1.27601	−0.638004	−	0.770033i	$-0.720240\pi$
$229$	−1.01261e6	−1.27601	−0.638004	+	0.770033i	$0.720240\pi$
$230$	0	0
$231$	201880.	0.248922
$232$	0	0
$233$	−706326.	−0.852345	−0.426172	−	0.904642i	$-0.640138\pi$
$233$	−706326.	−0.852345	−0.426172	+	0.904642i	$0.640138\pi$
$234$	0	0
$235$	−150366.	−0.177616
$236$	0	0
$237$	−54880.0	−0.0634663
$238$	0	0
$239$	1.19390e6	1.35199	0.675994	−	0.736907i	$-0.263715\pi$
$239$	1.19390e6	1.35199	0.675994	+	0.736907i	$0.263715\pi$
$240$	0	0
$241$	404410.	0.448517	0.224259	−	0.974530i	$-0.428004\pi$
$241$	404410.	0.448517	0.224259	+	0.974530i	$0.428004\pi$
$242$	0	0
$243$	−884597.	−0.961014
$244$	0	0
$245$	−371175.	−0.395060
$246$	0	0
$247$	398902.	0.416030
$248$	0	0
$249$	−624600.	−0.638416
$250$	0	0
$251$	−1.43258e6	−1.43527	−0.717634	−	0.696420i	$-0.754775\pi$
$251$	−1.43258e6	−1.43527	−0.717634	+	0.696420i	$0.754775\pi$
$252$	0	0
$253$	2.95032e6	2.89780
$254$	0	0
$255$	110996.	0.106895
$256$	0	0
$257$	−987982.	−0.933074	−0.466537	−	0.884502i	$-0.654499\pi$
$257$	−987982.	−0.933074	−0.466537	+	0.884502i	$0.654499\pi$
$258$	0	0
$259$	654250.	0.606030
$260$	0	0
$261$	814030.	0.739672
$262$	0	0
$263$	−2.06222e6	−1.83842	−0.919210	−	0.393767i	$-0.871172\pi$
$263$	−2.06222e6	−1.83842	−0.919210	+	0.393767i	$0.871172\pi$
$264$	0	0
$265$	−453850.	−0.397007
$266$	0	0
$267$	−193342.	−0.165977
$268$	0	0
$269$	−780386.	−0.657550	−0.328775	−	0.944408i	$-0.606636\pi$
$269$	−780386.	−0.657550	−0.328775	+	0.944408i	$0.606636\pi$
$270$	0	0
$271$	562291.	0.465091	0.232545	−	0.972586i	$-0.425295\pi$
$271$	562291.	0.465091	0.232545	+	0.972586i	$0.425295\pi$
$272$	0	0
$273$	−40880.0	−0.0331974
$274$	0	0
$275$	450625.	0.359321
$276$	0	0
$277$	386758.	0.302859	0.151429	−	0.988468i	$-0.451612\pi$
$277$	386758.	0.302859	0.151429	+	0.988468i	$0.451612\pi$
$278$	0	0
$279$	926965.	0.712940
$280$	0	0
$281$	1.55485e6	1.17469	0.587344	−	0.809337i	$-0.300173\pi$
$281$	1.55485e6	1.17469	0.587344	+	0.809337i	$0.300173\pi$
$282$	0	0
$283$	1.75157e6	1.30005	0.650026	−	0.759912i	$-0.274758\pi$
$283$	1.75157e6	1.30005	0.650026	+	0.759912i	$0.274758\pi$
$284$	0	0
$285$	432000.	0.315044
$286$	0	0
$287$	192583.	0.138011
$288$	0	0
$289$	−927053.	−0.652920
$290$	0	0
$291$	−422999.	−0.292824
$292$	0	0
$293$	1.55309e6	1.05689	0.528444	−	0.848968i	$-0.322776\pi$
$293$	1.55309e6	1.05689	0.528444	+	0.848968i	$0.322776\pi$
$294$	0	0
$295$	492683.	0.329619
$296$	0	0
$297$	2.03376e6	1.33785
$298$	0	0
$299$	−597430.	−0.386464
$300$	0	0
$301$	550200.	0.350029
$302$	0	0
$303$	−270678.	−0.169374
$304$	0	0
$305$	−1.05325e6	−0.648309
$306$	0	0
$307$	−2.60409e6	−1.57692	−0.788461	−	0.615085i	$-0.789122\pi$
$307$	−2.60409e6	−1.57692	−0.788461	+	0.615085i	$0.789122\pi$
$308$	0	0
$309$	66440.0	0.0395853
$310$	0	0
$311$	302124.	0.177127	0.0885634	−	0.996071i	$-0.471772\pi$
$311$	302124.	0.177127	0.0885634	+	0.996071i	$0.471772\pi$
$312$	0	0
$313$	2.09455e6	1.20846	0.604228	−	0.796812i	$-0.293482\pi$
$313$	2.09455e6	1.20846	0.604228	+	0.796812i	$0.293482\pi$
$314$	0	0
$315$	224680.	0.127581
$316$	0	0
$317$	1.07624e6	0.601534	0.300767	−	0.953698i	$-0.402757\pi$
$317$	1.07624e6	0.601534	0.300767	+	0.953698i	$0.402757\pi$
$318$	0	0
$319$	−2.89121e6	−1.59075
$320$	0	0
$321$	422840.	0.229041
$322$	0	0
$323$	1.91801e6	1.02293
$324$	0	0
$325$	−91250.0	−0.0479208
$326$	0	0
$327$	706541.	0.365400
$328$	0	0
$329$	266280.	0.135628
$330$	0	0
$331$	−313002.	−0.157028	−0.0785141	−	0.996913i	$-0.525018\pi$
$331$	−313002.	−0.157028	−0.0785141	+	0.996913i	$0.525018\pi$
$332$	0	0
$333$	2.99993e6	1.48252
$334$	0	0
$335$	−404613.	−0.196983
$336$	0	0
$337$	1.55400e6	0.745378	0.372689	−	0.927956i	$-0.378436\pi$
$337$	1.55400e6	0.745378	0.372689	+	0.927956i	$0.378436\pi$
$338$	0	0
$339$	−1.37138e6	−0.648124
$340$	0	0
$341$	−3.29232e6	−1.53326
$342$	0	0
$343$	1.40138e6	0.643163
$344$	0	0
$345$	−647000.	−0.292655
$346$	0	0
$347$	2.22208e6	0.990684	0.495342	−	0.868698i	$-0.335043\pi$
$347$	2.22208e6	0.990684	0.495342	+	0.868698i	$0.335043\pi$
$348$	0	0
$349$	−774570.	−0.340406	−0.170203	−	0.985409i	$-0.554442\pi$
$349$	−774570.	−0.340406	−0.170203	+	0.985409i	$0.554442\pi$
$350$	0	0
$351$	−411830.	−0.178423
$352$	0	0
$353$	−2.26217e6	−0.966249	−0.483125	−	0.875552i	$-0.660498\pi$
$353$	−2.26217e6	−0.966249	−0.483125	+	0.875552i	$0.660498\pi$
$354$	0	0
$355$	−1.13621e6	−0.478505
$356$	0	0
$357$	−196560.	−0.0816253
$358$	0	0
$359$	869601.	0.356110	0.178055	−	0.984021i	$-0.443020\pi$
$359$	869601.	0.356110	0.178055	+	0.984021i	$0.443020\pi$
$360$	0	0
$361$	4.98886e6	2.01481
$362$	0	0
$363$	−2.26918e6	−0.903863
$364$	0	0
$365$	656650.	0.257989
$366$	0	0
$367$	1.25361e6	0.485844	0.242922	−	0.970046i	$-0.421894\pi$
$367$	1.25361e6	0.485844	0.242922	+	0.970046i	$0.421894\pi$
$368$	0	0
$369$	883050.	0.337613
$370$	0	0
$371$	803712.	0.303156
$372$	0	0
$373$	175206.	0.0652044	0.0326022	−	0.999468i	$-0.489621\pi$
$373$	175206.	0.0652044	0.0326022	+	0.999468i	$0.489621\pi$
$374$	0	0
$375$	−98821.2	−0.0362887
$376$	0	0
$377$	585460.	0.212150
$378$	0	0
$379$	796945.	0.284990	0.142495	−	0.989795i	$-0.454487\pi$
$379$	796945.	0.284990	0.142495	+	0.989795i	$0.454487\pi$
$380$	0	0
$381$	393880.	0.139012
$382$	0	0
$383$	3.36129e6	1.17087	0.585436	−	0.810719i	$-0.300923\pi$
$383$	3.36129e6	1.17087	0.585436	+	0.810719i	$0.300923\pi$
$384$	0	0
$385$	−798000.	−0.274379
$386$	0	0
$387$	2.52283e6	0.856271
$388$	0	0
$389$	−5.51959e6	−1.84941	−0.924705	−	0.380685i	$-0.875688\pi$
$389$	−5.51959e6	−1.84941	−0.924705	+	0.380685i	$0.875688\pi$
$390$	0	0
$391$	−2.87258e6	−0.950232
$392$	0	0
$393$	1.11432e6	0.363939
$394$	0	0
$395$	216932.	0.0699570
$396$	0	0
$397$	−1.74738e6	−0.556430	−0.278215	−	0.960519i	$-0.589743\pi$
$397$	−1.74738e6	−0.556430	−0.278215	+	0.960519i	$0.589743\pi$
$398$	0	0
$399$	−765018.	−0.240569
$400$	0	0
$401$	541122.	0.168048	0.0840242	−	0.996464i	$-0.473223\pi$
$401$	541122.	0.168048	0.0840242	+	0.996464i	$0.473223\pi$
$402$	0	0
$403$	666684.	0.204483
$404$	0	0
$405$	787225.	0.238485
$406$	0	0
$407$	−1.06549e7	−3.18834
$408$	0	0
$409$	3.79699e6	1.12236	0.561179	−	0.827694i	$-0.310348\pi$
$409$	3.79699e6	1.12236	0.561179	+	0.827694i	$0.310348\pi$
$410$	0	0
$411$	−631203.	−0.184317
$412$	0	0
$413$	−872480.	−0.251698
$414$	0	0
$415$	2.46895e6	0.703707
$416$	0	0
$417$	−1.71696e6	−0.483526
$418$	0	0
$419$	3.88108e6	1.07998	0.539992	−	0.841670i	$-0.318427\pi$
$419$	3.88108e6	1.07998	0.539992	+	0.841670i	$0.318427\pi$
$420$	0	0
$421$	−6.66081e6	−1.83156	−0.915781	−	0.401677i	$-0.868427\pi$
$421$	−6.66081e6	−1.83156	−0.915781	+	0.401677i	$0.868427\pi$
$422$	0	0
$423$	1.22097e6	0.331784
$424$	0	0
$425$	−438750.	−0.117827
$426$	0	0
$427$	1.86517e6	0.495051
$428$	0	0
$429$	665760.	0.174652
$430$	0	0
$431$	4.65337e6	1.20663	0.603315	−	0.797503i	$-0.293846\pi$
$431$	4.65337e6	1.20663	0.603315	+	0.797503i	$0.293846\pi$
$432$	0	0
$433$	−6.23873e6	−1.59910	−0.799552	−	0.600597i	$-0.794930\pi$
$433$	−6.23873e6	−1.59910	−0.799552	+	0.600597i	$0.794930\pi$
$434$	0	0
$435$	634037.	0.160654
$436$	0	0
$437$	−1.11802e7	−2.80056
$438$	0	0
$439$	4.75743e6	1.17818	0.589089	−	0.808068i	$-0.299487\pi$
$439$	4.75743e6	1.17818	0.589089	+	0.808068i	$0.299487\pi$
$440$	0	0
$441$	3.01394e6	0.737969
$442$	0	0
$443$	−5.77073e6	−1.39708	−0.698541	−	0.715570i	$-0.746167\pi$
$443$	−5.77073e6	−1.39708	−0.698541	+	0.715570i	$0.746167\pi$
$444$	0	0
$445$	764250.	0.182951
$446$	0	0
$447$	2.61600e6	0.619254
$448$	0	0
$449$	−4.22765e6	−0.989654	−0.494827	−	0.868991i	$-0.664769\pi$
$449$	−4.22765e6	−0.989654	−0.494827	+	0.868991i	$0.664769\pi$
$450$	0	0
$451$	−3.13635e6	−0.726077
$452$	0	0
$453$	1.09096e6	0.249783
$454$	0	0
$455$	161592.	0.0365925
$456$	0	0
$457$	4.31374e6	0.966192	0.483096	−	0.875567i	$-0.339512\pi$
$457$	4.31374e6	0.966192	0.483096	+	0.875567i	$0.339512\pi$
$458$	0	0
$459$	−1.98017e6	−0.438703
$460$	0	0
$461$	−5.94820e6	−1.30357	−0.651784	−	0.758405i	$-0.725979\pi$
$461$	−5.94820e6	−1.30357	−0.651784	+	0.758405i	$0.725979\pi$
$462$	0	0
$463$	−8.39722e6	−1.82047	−0.910234	−	0.414094i	$-0.864098\pi$
$463$	−8.39722e6	−1.82047	−0.910234	+	0.414094i	$0.864098\pi$
$464$	0	0
$465$	722000.	0.154848
$466$	0	0
$467$	−5.30976e6	−1.12663	−0.563317	−	0.826241i	$-0.690475\pi$
$467$	−5.30976e6	−1.12663	−0.563317	+	0.826241i	$0.690475\pi$
$468$	0	0
$469$	716520.	0.150417
$470$	0	0
$471$	−1.13436e6	−0.235613
$472$	0	0
$473$	−8.96040e6	−1.84151
$474$	0	0
$475$	−1.70763e6	−0.347264
$476$	0	0
$477$	3.68526e6	0.741605
$478$	0	0
$479$	−5.83653e6	−1.16229	−0.581147	−	0.813799i	$-0.697396\pi$
$479$	−5.83653e6	−1.16229	−0.581147	+	0.813799i	$0.697396\pi$
$480$	0	0
$481$	2.15759e6	0.425212
$482$	0	0
$483$	1.14576e6	0.223473
$484$	0	0
$485$	1.67205e6	0.322771
$486$	0	0
$487$	8.75327e6	1.67243	0.836215	−	0.548402i	$-0.184764\pi$
$487$	8.75327e6	1.67243	0.836215	+	0.548402i	$0.184764\pi$
$488$	0	0
$489$	2.94676e6	0.557279
$490$	0	0
$491$	5.16834e6	0.967492	0.483746	−	0.875209i	$-0.339276\pi$
$491$	5.16834e6	0.967492	0.483746	+	0.875209i	$0.339276\pi$
$492$	0	0
$493$	2.81502e6	0.521632
$494$	0	0
$495$	−3.65907e6	−0.671209
$496$	0	0
$497$	2.01208e6	0.365388
$498$	0	0
$499$	5.36171e6	0.963943	0.481972	−	0.876187i	$-0.339921\pi$
$499$	5.36171e6	0.963943	0.481972	+	0.876187i	$0.339921\pi$
$500$	0	0
$501$	−3.85524e6	−0.686210
$502$	0	0
$503$	5.26635e6	0.928089	0.464045	−	0.885812i	$-0.346398\pi$
$503$	5.26635e6	0.928089	0.464045	+	0.885812i	$0.346398\pi$
$504$	0	0
$505$	1.06995e6	0.186696
$506$	0	0
$507$	2.21345e6	0.382428
$508$	0	0
$509$	5.46847e6	0.935559	0.467780	−	0.883845i	$-0.345054\pi$
$509$	5.46847e6	0.935559	0.467780	+	0.883845i	$0.345054\pi$
$510$	0	0
$511$	−1.16285e6	−0.197002
$512$	0	0
$513$	−7.70688e6	−1.29296
$514$	0	0
$515$	−262627.	−0.0436337
$516$	0	0
$517$	−4.33656e6	−0.713541
$518$	0	0
$519$	−3.73836e6	−0.609203
$520$	0	0
$521$	−3.05762e6	−0.493503	−0.246751	−	0.969079i	$-0.579363\pi$
$521$	−3.05762e6	−0.493503	−0.246751	+	0.969079i	$0.579363\pi$
$522$	0	0
$523$	9.64117e6	1.54126	0.770629	−	0.637284i	$-0.219942\pi$
$523$	9.64117e6	1.54126	0.770629	+	0.637284i	$0.219942\pi$
$524$	0	0
$525$	175000.	0.0277102
$526$	0	0
$527$	3.20556e6	0.502780
$528$	0	0
$529$	1.03080e7	1.60153
$530$	0	0
$531$	−4.00058e6	−0.615726
$532$	0	0
$533$	635100.	0.0968332
$534$	0	0
$535$	−1.67142e6	−0.252465
$536$	0	0
$537$	1.33584e6	0.199903
$538$	0	0
$539$	−1.07047e7	−1.58709
$540$	0	0
$541$	−4.15820e6	−0.610819	−0.305409	−	0.952221i	$-0.598793\pi$
$541$	−4.15820e6	−0.610819	−0.305409	+	0.952221i	$0.598793\pi$
$542$	0	0
$543$	616884.	0.0897851
$544$	0	0
$545$	−2.79285e6	−0.402769
$546$	0	0
$547$	−5.32628e6	−0.761125	−0.380562	−	0.924755i	$-0.624270\pi$
$547$	−5.32628e6	−0.761125	−0.380562	+	0.924755i	$0.624270\pi$
$548$	0	0
$549$	8.55239e6	1.21103
$550$	0	0
$551$	1.09562e7	1.53737
$552$	0	0
$553$	−384160.	−0.0534194
$554$	0	0
$555$	2.33661e6	0.321998
$556$	0	0
$557$	999102.	0.136449	0.0682247	−	0.997670i	$-0.478267\pi$
$557$	999102.	0.136449	0.0682247	+	0.997670i	$0.478267\pi$
$558$	0	0
$559$	1.81445e6	0.245593
$560$	0	0
$561$	3.20112e6	0.429432
$562$	0	0
$563$	−5.31886e6	−0.707208	−0.353604	−	0.935395i	$-0.615044\pi$
$563$	−5.31886e6	−0.707208	−0.353604	+	0.935395i	$0.615044\pi$
$564$	0	0
$565$	5.42085e6	0.714408
$566$	0	0
$567$	−1.39408e6	−0.182108
$568$	0	0
$569$	6.56759e6	0.850404	0.425202	−	0.905099i	$-0.360203\pi$
$569$	6.56759e6	0.850404	0.425202	+	0.905099i	$0.360203\pi$
$570$	0	0
$571$	164173.	0.0210723	0.0105361	−	0.999944i	$-0.496646\pi$
$571$	164173.	0.0210723	0.0105361	+	0.999944i	$0.496646\pi$
$572$	0	0
$573$	1.94264e6	0.247176
$574$	0	0
$575$	2.55749e6	0.322585
$576$	0	0
$577$	9.27762e6	1.16010	0.580052	−	0.814579i	$-0.303032\pi$
$577$	9.27762e6	1.16010	0.580052	+	0.814579i	$0.303032\pi$
$578$	0	0
$579$	−78639.5	−0.00974865
$580$	0	0
$581$	−4.37220e6	−0.537353
$582$	0	0
$583$	−1.30890e7	−1.59491
$584$	0	0
$585$	740950.	0.0895157
$586$	0	0
$587$	5.44474e6	0.652202	0.326101	−	0.945335i	$-0.394265\pi$
$587$	5.44474e6	0.652202	0.326101	+	0.945335i	$0.394265\pi$
$588$	0	0
$589$	1.24762e7	1.48181
$590$	0	0
$591$	2.39611e6	0.282187
$592$	0	0
$593$	−1.50978e7	−1.76310	−0.881548	−	0.472094i	$-0.843498\pi$
$593$	−1.50978e7	−1.76310	−0.881548	+	0.472094i	$0.843498\pi$
$594$	0	0
$595$	776972.	0.0899731
$596$	0	0
$597$	4.85216e6	0.557185
$598$	0	0
$599$	6.98593e6	0.795531	0.397765	−	0.917487i	$-0.369786\pi$
$599$	6.98593e6	0.795531	0.397765	+	0.917487i	$0.369786\pi$
$600$	0	0
$601$	6.16941e6	0.696719	0.348359	−	0.937361i	$-0.386739\pi$
$601$	6.16941e6	0.696719	0.348359	+	0.937361i	$0.386739\pi$
$602$	0	0
$603$	3.28546e6	0.367962
$604$	0	0
$605$	8.96972e6	0.996301
$606$	0	0
$607$	−1.00769e7	−1.11009	−0.555043	−	0.831822i	$-0.687298\pi$
$607$	−1.00769e7	−1.11009	−0.555043	+	0.831822i	$0.687298\pi$
$608$	0	0
$609$	−1.12280e6	−0.122676
$610$	0	0
$611$	878139.	0.0951613
$612$	0	0
$613$	5.26885e6	0.566324	0.283162	−	0.959072i	$-0.408617\pi$
$613$	5.26885e6	0.566324	0.283162	+	0.959072i	$0.408617\pi$
$614$	0	0
$615$	687795.	0.0733283
$616$	0	0
$617$	−1.38112e7	−1.46056	−0.730280	−	0.683148i	$-0.760610\pi$
$617$	−1.38112e7	−1.46056	−0.730280	+	0.683148i	$0.760610\pi$
$618$	0	0
$619$	3.20903e6	0.336625	0.168313	−	0.985734i	$-0.446168\pi$
$619$	3.20903e6	0.336625	0.168313	+	0.985734i	$0.446168\pi$
$620$	0	0
$621$	1.15425e7	1.20108
$622$	0	0
$623$	−1.35339e6	−0.139702
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	1.24589e7	1.26564
$628$	0	0
$629$	1.03742e7	1.04551
$630$	0	0
$631$	−6.07204e6	−0.607102	−0.303551	−	0.952815i	$-0.598172\pi$
$631$	−6.07204e6	−0.607102	−0.303551	+	0.952815i	$0.598172\pi$
$632$	0	0
$633$	−5.46776e6	−0.542376
$634$	0	0
$635$	−1.55695e6	−0.153229
$636$	0	0
$637$	2.16766e6	0.211662
$638$	0	0
$639$	9.22600e6	0.893843
$640$	0	0
$641$	−2.02767e6	−0.194918	−0.0974591	−	0.995240i	$-0.531072\pi$
$641$	−2.02767e6	−0.194918	−0.0974591	+	0.995240i	$0.531072\pi$
$642$	0	0
$643$	−1.19769e7	−1.14240	−0.571199	−	0.820811i	$-0.693522\pi$
$643$	−1.19769e7	−1.14240	−0.571199	+	0.820811i	$0.693522\pi$
$644$	0	0
$645$	1.96500e6	0.185979
$646$	0	0
$647$	−1.44768e6	−0.135961	−0.0679803	−	0.997687i	$-0.521656\pi$
$647$	−1.44768e6	−0.135961	−0.0679803	+	0.997687i	$0.521656\pi$
$648$	0	0
$649$	1.42090e7	1.32419
$650$	0	0
$651$	−1.27857e6	−0.118242
$652$	0	0
$653$	4.82477e6	0.442785	0.221393	−	0.975185i	$-0.428940\pi$
$653$	4.82477e6	0.442785	0.221393	+	0.975185i	$0.428940\pi$
$654$	0	0
$655$	−4.40474e6	−0.401159
$656$	0	0
$657$	−5.33200e6	−0.481922
$658$	0	0
$659$	4.37616e6	0.392536	0.196268	−	0.980550i	$-0.437118\pi$
$659$	4.37616e6	0.392536	0.196268	+	0.980550i	$0.437118\pi$
$660$	0	0
$661$	−7.34953e6	−0.654268	−0.327134	−	0.944978i	$-0.606083\pi$
$661$	−7.34953e6	−0.654268	−0.327134	+	0.944978i	$0.606083\pi$
$662$	0	0
$663$	−648216.	−0.0572712
$664$	0	0
$665$	3.02400e6	0.265172
$666$	0	0
$667$	−1.64089e7	−1.42812
$668$	0	0
$669$	−4.04588e6	−0.349500
$670$	0	0
$671$	−3.03757e7	−2.60447
$672$	0	0
$673$	−1.75692e7	−1.49525	−0.747626	−	0.664119i	$-0.768807\pi$
$673$	−1.75692e7	−1.49525	−0.747626	+	0.664119i	$0.768807\pi$
$674$	0	0
$675$	1.76297e6	0.148931
$676$	0	0
$677$	5.60338e6	0.469871	0.234935	−	0.972011i	$-0.424512\pi$
$677$	5.60338e6	0.469871	0.234935	+	0.972011i	$0.424512\pi$
$678$	0	0
$679$	−2.96099e6	−0.246469
$680$	0	0
$681$	−6.20132e6	−0.512409
$682$	0	0
$683$	5.05809e6	0.414892	0.207446	−	0.978246i	$-0.433485\pi$
$683$	5.05809e6	0.414892	0.207446	+	0.978246i	$0.433485\pi$
$684$	0	0
$685$	2.49505e6	0.203167
$686$	0	0
$687$	6.40431e6	0.517703
$688$	0	0
$689$	2.65048e6	0.212705
$690$	0	0
$691$	1.47421e7	1.17453	0.587267	−	0.809393i	$-0.300204\pi$
$691$	1.47421e7	1.17453	0.587267	+	0.809393i	$0.300204\pi$
$692$	0	0
$693$	6.47976e6	0.512538
$694$	0	0
$695$	6.78688e6	0.532977
$696$	0	0
$697$	3.05370e6	0.238092
$698$	0	0
$699$	4.46720e6	0.345814
$700$	0	0
$701$	−8.83317e6	−0.678925	−0.339462	−	0.940620i	$-0.610245\pi$
$701$	−8.83317e6	−0.678925	−0.339462	+	0.940620i	$0.610245\pi$
$702$	0	0
$703$	4.03766e7	3.08135
$704$	0	0
$705$	951000.	0.0720622
$706$	0	0
$707$	−1.89475e6	−0.142562
$708$	0	0
$709$	−1.93101e6	−0.144268	−0.0721338	−	0.997395i	$-0.522981\pi$
$709$	−1.93101e6	−0.144268	−0.0721338	+	0.997395i	$0.522981\pi$
$710$	0	0
$711$	−1.76149e6	−0.130679
$712$	0	0
$713$	−1.86854e7	−1.37650
$714$	0	0
$715$	−2.63165e6	−0.192514
$716$	0	0
$717$	−7.55088e6	−0.548529
$718$	0	0
$719$	−7.22373e6	−0.521122	−0.260561	−	0.965457i	$-0.583907\pi$
$719$	−7.22373e6	−0.521122	−0.260561	+	0.965457i	$0.583907\pi$
$720$	0	0
$721$	465080.	0.0333188
$722$	0	0
$723$	−2.55771e6	−0.181973
$724$	0	0
$725$	−2.50625e6	−0.177084
$726$	0	0
$727$	9.81106e6	0.688462	0.344231	−	0.938885i	$-0.388140\pi$
$727$	9.81106e6	0.688462	0.344231	+	0.938885i	$0.388140\pi$
$728$	0	0
$729$	−2.05715e6	−0.143366
$730$	0	0
$731$	8.72428e6	0.603860
$732$	0	0
$733$	−2.40813e7	−1.65547	−0.827733	−	0.561122i	$-0.810370\pi$
$733$	−2.40813e7	−1.65547	−0.827733	+	0.561122i	$0.810370\pi$
$734$	0	0
$735$	2.34752e6	0.160284
$736$	0	0
$737$	−1.16690e7	−0.791346
$738$	0	0
$739$	−2.40518e7	−1.62008	−0.810040	−	0.586374i	$-0.800555\pi$
$739$	−2.40518e7	−1.62008	−0.810040	+	0.586374i	$0.800555\pi$
$740$	0	0
$741$	−2.52288e6	−0.168792
$742$	0	0
$743$	−4.70160e6	−0.312445	−0.156223	−	0.987722i	$-0.549932\pi$
$743$	−4.70160e6	−0.312445	−0.156223	+	0.987722i	$0.549932\pi$
$744$	0	0
$745$	−1.03406e7	−0.682586
$746$	0	0
$747$	−2.00479e7	−1.31452
$748$	0	0
$749$	2.95988e6	0.192783
$750$	0	0
$751$	2.31282e7	1.49638	0.748189	−	0.663485i	$-0.230923\pi$
$751$	2.31282e7	1.49638	0.748189	+	0.663485i	$0.230923\pi$
$752$	0	0
$753$	9.06040e6	0.582318
$754$	0	0
$755$	−4.31240e6	−0.275329
$756$	0	0
$757$	−1.26635e7	−0.803181	−0.401591	−	0.915819i	$-0.631542\pi$
$757$	−1.26635e7	−0.803181	−0.401591	+	0.915819i	$0.631542\pi$
$758$	0	0
$759$	−1.86595e7	−1.17570
$760$	0	0
$761$	−4.33524e6	−0.271363	−0.135682	−	0.990752i	$-0.543322\pi$
$761$	−4.33524e6	−0.271363	−0.135682	+	0.990752i	$0.543322\pi$
$762$	0	0
$763$	4.94579e6	0.307556
$764$	0	0
$765$	3.56265e6	0.220100
$766$	0	0
$767$	−2.87727e6	−0.176600
$768$	0	0
$769$	−364270.	−0.0222130	−0.0111065	−	0.999938i	$-0.503535\pi$
$769$	−364270.	−0.0222130	−0.0111065	+	0.999938i	$0.503535\pi$
$770$	0	0
$771$	6.24855e6	0.378567
$772$	0	0
$773$	−1.03051e6	−0.0620300	−0.0310150	−	0.999519i	$-0.509874\pi$
$773$	−1.03051e6	−0.0620300	−0.0310150	+	0.999519i	$0.509874\pi$
$774$	0	0
$775$	−2.85396e6	−0.170684
$776$	0	0
$777$	−4.13784e6	−0.245879
$778$	0	0
$779$	1.18851e7	0.701713
$780$	0	0
$781$	−3.27682e7	−1.92231
$782$	0	0
$783$	−1.13112e7	−0.659333
$784$	0	0
$785$	4.48395e6	0.259709
$786$	0	0
$787$	1.95300e7	1.12400	0.561999	−	0.827138i	$-0.310032\pi$
$787$	1.95300e7	1.12400	0.561999	+	0.827138i	$0.310032\pi$
$788$	0	0
$789$	1.30426e7	0.745885
$790$	0	0
$791$	−9.59965e6	−0.545524
$792$	0	0
$793$	6.15098e6	0.347345
$794$	0	0
$795$	2.87040e6	0.161074
$796$	0	0
$797$	7.86344e6	0.438497	0.219249	−	0.975669i	$-0.429639\pi$
$797$	7.86344e6	0.438497	0.219249	+	0.975669i	$0.429639\pi$
$798$	0	0
$799$	4.22229e6	0.233981
$800$	0	0
$801$	−6.20571e6	−0.341751
$802$	0	0
$803$	1.89378e7	1.03643
$804$	0	0
$805$	−4.52900e6	−0.246327
$806$	0	0
$807$	4.93559e6	0.266781
$808$	0	0
$809$	−1.05014e7	−0.564127	−0.282064	−	0.959396i	$-0.591019\pi$
$809$	−1.05014e7	−0.564127	−0.282064	+	0.959396i	$0.591019\pi$
$810$	0	0
$811$	−1.87803e7	−1.00265	−0.501325	−	0.865259i	$-0.667154\pi$
$811$	−1.87803e7	−1.00265	−0.501325	+	0.865259i	$0.667154\pi$
$812$	0	0
$813$	−3.55624e6	−0.188697
$814$	0	0
$815$	−1.16481e7	−0.614272
$816$	0	0
$817$	3.39552e7	1.77972
$818$	0	0
$819$	−1.31213e6	−0.0683545
$820$	0	0
$821$	2.00558e7	1.03844	0.519220	−	0.854641i	$-0.326223\pi$
$821$	2.00558e7	1.03844	0.519220	+	0.854641i	$0.326223\pi$
$822$	0	0
$823$	−1.02444e7	−0.527212	−0.263606	−	0.964630i	$-0.584912\pi$
$823$	−1.02444e7	−0.527212	−0.263606	+	0.964630i	$0.584912\pi$
$824$	0	0
$825$	−2.85000e6	−0.145784
$826$	0	0
$827$	−827524.	−0.0420743	−0.0210371	−	0.999779i	$-0.506697\pi$
$827$	−827524.	−0.0420743	−0.0210371	+	0.999779i	$0.506697\pi$
$828$	0	0
$829$	−3.80328e7	−1.92208	−0.961041	−	0.276407i	$-0.910856\pi$
$829$	−3.80328e7	−1.92208	−0.961041	+	0.276407i	$0.910856\pi$
$830$	0	0
$831$	−2.44607e6	−0.122876
$832$	0	0
$833$	1.04226e7	0.520431
$834$	0	0
$835$	1.52392e7	0.756389
$836$	0	0
$837$	−1.28805e7	−0.635504
$838$	0	0
$839$	3.57628e7	1.75399	0.876993	−	0.480503i	$-0.159546\pi$
$839$	3.57628e7	1.75399	0.876993	+	0.480503i	$0.159546\pi$
$840$	0	0
$841$	−4.43105e6	−0.216031
$842$	0	0
$843$	−9.83373e6	−0.476595
$844$	0	0
$845$	−8.74942e6	−0.421539
$846$	0	0
$847$	−1.58843e7	−0.760779
$848$	0	0
$849$	−1.10779e7	−0.527457
$850$	0	0
$851$	−6.04714e7	−2.86237
$852$	0	0
$853$	970214.	0.0456557	0.0228278	−	0.999739i	$-0.492733\pi$
$853$	970214.	0.0456557	0.0228278	+	0.999739i	$0.492733\pi$
$854$	0	0
$855$	1.38660e7	0.648686
$856$	0	0
$857$	1.86468e6	0.0867267	0.0433633	−	0.999059i	$-0.486193\pi$
$857$	1.86468e6	0.0867267	0.0433633	+	0.999059i	$0.486193\pi$
$858$	0	0
$859$	−1.05785e6	−0.0489147	−0.0244573	−	0.999701i	$-0.507786\pi$
$859$	−1.05785e6	−0.0489147	−0.0244573	+	0.999701i	$0.507786\pi$
$860$	0	0
$861$	−1.21800e6	−0.0559937
$862$	0	0
$863$	−8.24333e6	−0.376769	−0.188385	−	0.982095i	$-0.560325\pi$
$863$	−8.24333e6	−0.376769	−0.188385	+	0.982095i	$0.560325\pi$
$864$	0	0
$865$	1.47772e7	0.671507
$866$	0	0
$867$	5.86320e6	0.264903
$868$	0	0
$869$	6.25632e6	0.281041
$870$	0	0
$871$	2.36294e6	0.105538
$872$	0	0
$873$	−1.35770e7	−0.602934
$874$	0	0
$875$	−691748.	−0.0305441
$876$	0	0
$877$	−2.40362e7	−1.05528	−0.527640	−	0.849468i	$-0.676923\pi$
$877$	−2.40362e7	−1.05528	−0.527640	+	0.849468i	$0.676923\pi$
$878$	0	0
$879$	−9.82263e6	−0.428801
$880$	0	0
$881$	3.68605e7	1.60000	0.800002	−	0.599997i	$-0.204831\pi$
$881$	3.68605e7	1.60000	0.800002	+	0.599997i	$0.204831\pi$
$882$	0	0
$883$	−1.67403e7	−0.722541	−0.361271	−	0.932461i	$-0.617657\pi$
$883$	−1.67403e7	−0.722541	−0.361271	+	0.932461i	$0.617657\pi$
$884$	0	0
$885$	−3.11600e6	−0.133733
$886$	0	0
$887$	9.25364e6	0.394915	0.197457	−	0.980311i	$-0.436732\pi$
$887$	9.25364e6	0.394915	0.197457	+	0.980311i	$0.436732\pi$
$888$	0	0
$889$	2.75716e6	0.117006
$890$	0	0
$891$	2.27035e7	0.958075
$892$	0	0
$893$	1.64333e7	0.689597
$894$	0	0
$895$	−5.28037e6	−0.220347
$896$	0	0
$897$	3.77848e6	0.156796
$898$	0	0
$899$	1.83110e7	0.755635
$900$	0	0
$901$	1.27441e7	0.522995
$902$	0	0
$903$	−3.47977e6	−0.142014
$904$	0	0
$905$	−2.43845e6	−0.0989675
$906$	0	0
$907$	1.03030e7	0.415859	0.207929	−	0.978144i	$-0.433328\pi$
$907$	1.03030e7	0.415859	0.207929	+	0.978144i	$0.433328\pi$
$908$	0	0
$909$	−8.68799e6	−0.348746
$910$	0	0
$911$	3.82374e7	1.52649	0.763243	−	0.646111i	$-0.223606\pi$
$911$	3.82374e7	1.52649	0.763243	+	0.646111i	$0.223606\pi$
$912$	0	0
$913$	7.12044e7	2.82703
$914$	0	0
$915$	6.66134e6	0.263032
$916$	0	0
$917$	7.80024e6	0.306327
$918$	0	0
$919$	−7.72150e6	−0.301587	−0.150794	−	0.988565i	$-0.548183\pi$
$919$	−7.72150e6	−0.301587	−0.150794	+	0.988565i	$0.548183\pi$
$920$	0	0
$921$	1.64697e7	0.639790
$922$	0	0
$923$	6.63545e6	0.256369
$924$	0	0
$925$	−9.23625e6	−0.354929
$926$	0	0
$927$	2.13253e6	0.0815073
$928$	0	0
$929$	3.39039e6	0.128888	0.0644438	−	0.997921i	$-0.479473\pi$
$929$	3.39039e6	0.128888	0.0644438	+	0.997921i	$0.479473\pi$
$930$	0	0
$931$	4.05651e7	1.53383
$932$	0	0
$933$	−1.91080e6	−0.0718640
$934$	0	0
$935$	−1.26535e7	−0.473351
$936$	0	0
$937$	−7.67490e6	−0.285577	−0.142789	−	0.989753i	$-0.545607\pi$
$937$	−7.67490e6	−0.285577	−0.142789	+	0.989753i	$0.545607\pi$
$938$	0	0
$939$	−1.32471e7	−0.490295
$940$	0	0
$941$	4.55115e7	1.67551	0.837756	−	0.546045i	$-0.183867\pi$
$941$	4.55115e7	1.67551	0.837756	+	0.546045i	$0.183867\pi$
$942$	0	0
$943$	−1.78001e7	−0.651845
$944$	0	0
$945$	−3.12200e6	−0.113724
$946$	0	0
$947$	−2.28294e7	−0.827218	−0.413609	−	0.910455i	$-0.635732\pi$
$947$	−2.28294e7	−0.827218	−0.413609	+	0.910455i	$0.635732\pi$
$948$	0	0
$949$	−3.83484e6	−0.138223
$950$	0	0
$951$	−6.80673e6	−0.244055
$952$	0	0
$953$	4.64478e7	1.65666	0.828330	−	0.560241i	$-0.189291\pi$
$953$	4.64478e7	1.65666	0.828330	+	0.560241i	$0.189291\pi$
$954$	0	0
$955$	−7.67896e6	−0.272454
$956$	0	0
$957$	1.82856e7	0.645401
$958$	0	0
$959$	−4.41842e6	−0.155139
$960$	0	0
$961$	−7.77779e6	−0.271674
$962$	0	0
$963$	1.35719e7	0.471603
$964$	0	0
$965$	310850.	0.0107456
$966$	0	0
$967$	−2.63000e7	−0.904460	−0.452230	−	0.891901i	$-0.649371\pi$
$967$	−2.63000e7	−0.904460	−0.452230	+	0.891901i	$0.649371\pi$
$968$	0	0
$969$	−1.21306e7	−0.415022
$970$	0	0
$971$	2.19967e7	0.748704	0.374352	−	0.927287i	$-0.377865\pi$
$971$	2.19967e7	0.748704	0.374352	+	0.927287i	$0.377865\pi$
$972$	0	0
$973$	−1.20187e7	−0.406983
$974$	0	0
$975$	577116.	0.0194425
$976$	0	0
$977$	4.46750e7	1.49737	0.748683	−	0.662929i	$-0.230687\pi$
$977$	4.46750e7	1.49737	0.748683	+	0.662929i	$0.230687\pi$
$978$	0	0
$979$	2.20409e7	0.734977
$980$	0	0
$981$	2.26779e7	0.752369
$982$	0	0
$983$	2.02740e7	0.669201	0.334600	−	0.942360i	$-0.391399\pi$
$983$	2.02740e7	0.669201	0.334600	+	0.942360i	$0.391399\pi$
$984$	0	0
$985$	−9.47145e6	−0.311047
$986$	0	0
$987$	−1.68410e6	−0.0550270
$988$	0	0
$989$	−5.08542e7	−1.65324
$990$	0	0
$991$	−1.49135e7	−0.482386	−0.241193	−	0.970477i	$-0.577539\pi$
$991$	−1.49135e7	−0.482386	−0.241193	+	0.970477i	$0.577539\pi$
$992$	0	0
$993$	1.97960e6	0.0637095
$994$	0	0
$995$	−1.91798e7	−0.614168
$996$	0	0
$997$	3.17032e7	1.01010	0.505052	−	0.863089i	$-0.331473\pi$
$997$	3.17032e7	1.01010	0.505052	+	0.863089i	$0.331473\pi$
$998$	0	0
$999$	−4.16851e7	−1.32150

Twists

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

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

\(f(q)\)	\(=\)	\(q-6.32456 q^{3} +25.0000 q^{5} -44.2719 q^{7} -203.000 q^{9} +O(q^{10})\)	\(q-6.32456 q^{3} +25.0000 q^{5} -44.2719 q^{7} -203.000 q^{9} +720.999 q^{11} -146.000 q^{13} -158.114 q^{15} -702.000 q^{17} -2732.21 q^{19} +280.000 q^{21} +4091.99 q^{23} +625.000 q^{25} +2820.75 q^{27} -4010.00 q^{29} -4566.33 q^{31} -4560.00 q^{33} -1106.80 q^{35} -14778.0 q^{37} +923.385 q^{39} -4350.00 q^{41} -12427.8 q^{43} -5075.00 q^{45} -6014.65 q^{47} -14847.0 q^{49} +4439.84 q^{51} -18154.0 q^{53} +18025.0 q^{55} +17280.0 q^{57} +19707.3 q^{59} -42130.0 q^{61} +8987.19 q^{63} -3650.00 q^{65} -16184.5 q^{67} -25880.0 q^{69} -45448.3 q^{71} +26266.0 q^{73} -3952.85 q^{75} -31920.0 q^{77} +8677.29 q^{79} +31489.0 q^{81} +98757.9 q^{83} -17550.0 q^{85} +25361.5 q^{87} +30570.0 q^{89} +6463.70 q^{91} +28880.0 q^{93} -68305.2 q^{95} +66882.0 q^{97} -146363. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{5} - 406 q^{9}+O(q^{10}) \) 2 * q + 50 * q^5 - 406 * q^9	\( 2 q + 50 q^{5} - 406 q^{9} - 292 q^{13} - 1404 q^{17} + 560 q^{21} + 1250 q^{25} - 8020 q^{29} - 9120 q^{33} - 29556 q^{37} - 8700 q^{41} - 10150 q^{45} - 29694 q^{49} - 36308 q^{53} + 34560 q^{57} - 84260 q^{61} - 7300 q^{65} - 51760 q^{69} + 52532 q^{73} - 63840 q^{77} + 62978 q^{81} - 35100 q^{85} + 61140 q^{89} + 57760 q^{93} + 133764 q^{97}+O(q^{100}) \) 2 * q + 50 * q^5 - 406 * q^9 - 292 * q^13 - 1404 * q^17 + 560 * q^21 + 1250 * q^25 - 8020 * q^29 - 9120 * q^33 - 29556 * q^37 - 8700 * q^41 - 10150 * q^45 - 29694 * q^49 - 36308 * q^53 + 34560 * q^57 - 84260 * q^61 - 7300 * q^65 - 51760 * q^69 + 52532 * q^73 - 63840 * q^77 + 62978 * q^81 - 35100 * q^85 + 61140 * q^89 + 57760 * q^93 + 133764 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more