LMFDB - Embedded newform 1028.6.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$1$
Dimension:	$49$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.5283 q^{3} +5.59326 q^{5} +131.433 q^{7} +358.638 q^{9} +O(q^{10})$	$q-24.5283 q^{3} +5.59326 q^{5} +131.433 q^{7} +358.638 q^{9} +794.852 q^{11} +405.436 q^{13} -137.193 q^{15} -1242.91 q^{17} +75.6100 q^{19} -3223.82 q^{21} -3826.19 q^{23} -3093.72 q^{25} -2836.41 q^{27} +8542.63 q^{29} -10257.0 q^{31} -19496.4 q^{33} +735.138 q^{35} +7081.26 q^{37} -9944.67 q^{39} -4692.12 q^{41} -15654.4 q^{43} +2005.96 q^{45} +15872.3 q^{47} +467.574 q^{49} +30486.6 q^{51} -37031.1 q^{53} +4445.82 q^{55} -1854.59 q^{57} -7081.64 q^{59} +41598.1 q^{61} +47136.8 q^{63} +2267.71 q^{65} +43598.6 q^{67} +93849.9 q^{69} -82324.1 q^{71} -32288.5 q^{73} +75883.6 q^{75} +104470. q^{77} +62099.5 q^{79} -17576.7 q^{81} +25189.8 q^{83} -6951.94 q^{85} -209536. q^{87} +122559. q^{89} +53287.6 q^{91} +251588. q^{93} +422.907 q^{95} -33723.5 q^{97} +285064. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	$ 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) $ 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * 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
$3$	−24.5283	−1.57349	−0.786746	−	0.617277i	$-0.788236\pi$
$3$	−24.5283	−1.57349	−0.786746	+	0.617277i	$0.788236\pi$
$4$	0	0
$5$	5.59326	0.100055	0.0500277	−	0.998748i	$-0.484069\pi$
$5$	5.59326	0.100055	0.0500277	+	0.998748i	$0.484069\pi$
$6$	0	0
$7$	131.433	1.01381	0.506907	−	0.862001i	$-0.330789\pi$
$7$	131.433	1.01381	0.506907	+	0.862001i	$0.330789\pi$
$8$	0	0
$9$	358.638	1.47588
$10$	0	0
$11$	794.852	1.98064	0.990318	−	0.138820i	$-0.0443310\pi$
$11$	794.852	1.98064	0.990318	+	0.138820i	$0.0443310\pi$
$12$	0	0
$13$	405.436	0.665371	0.332686	−	0.943038i	$-0.392045\pi$
$13$	405.436	0.665371	0.332686	+	0.943038i	$0.392045\pi$
$14$	0	0
$15$	−137.193	−0.157436
$16$	0	0
$17$	−1242.91	−1.04308	−0.521541	−	0.853226i	$-0.674643\pi$
$17$	−1242.91	−1.04308	−0.521541	+	0.853226i	$0.674643\pi$
$18$	0	0
$19$	75.6100	0.0480502	0.0240251	−	0.999711i	$-0.492352\pi$
$19$	75.6100	0.0480502	0.0240251	+	0.999711i	$0.492352\pi$
$20$	0	0
$21$	−3223.82	−1.59523
$22$	0	0
$23$	−3826.19	−1.50816	−0.754079	−	0.656784i	$-0.771916\pi$
$23$	−3826.19	−1.50816	−0.754079	+	0.656784i	$0.771916\pi$
$24$	0	0
$25$	−3093.72	−0.989989
$26$	0	0
$27$	−2836.41	−0.748789
$28$	0	0
$29$	8542.63	1.88624	0.943119	−	0.332455i	$-0.107877\pi$
$29$	8542.63	1.88624	0.943119	+	0.332455i	$0.107877\pi$
$30$	0	0
$31$	−10257.0	−1.91698	−0.958490	−	0.285127i	$-0.907964\pi$
$31$	−10257.0	−1.91698	−0.958490	+	0.285127i	$0.907964\pi$
$32$	0	0
$33$	−19496.4	−3.11651
$34$	0	0
$35$	735.138	0.101438
$36$	0	0
$37$	7081.26	0.850367	0.425183	−	0.905107i	$-0.360210\pi$
$37$	7081.26	0.850367	0.425183	+	0.905107i	$0.360210\pi$
$38$	0	0
$39$	−9944.67	−1.04696
$40$	0	0
$41$	−4692.12	−0.435922	−0.217961	−	0.975957i	$-0.569941\pi$
$41$	−4692.12	−0.435922	−0.217961	+	0.975957i	$0.569941\pi$
$42$	0	0
$43$	−15654.4	−1.29112	−0.645558	−	0.763712i	$-0.723375\pi$
$43$	−15654.4	−1.29112	−0.645558	+	0.763712i	$0.723375\pi$
$44$	0	0
$45$	2005.96	0.147669
$46$	0	0
$47$	15872.3	1.04808	0.524041	−	0.851693i	$-0.324424\pi$
$47$	15872.3	1.04808	0.524041	+	0.851693i	$0.324424\pi$
$48$	0	0
$49$	467.574	0.0278202
$50$	0	0
$51$	30486.6	1.64128
$52$	0	0
$53$	−37031.1	−1.81083	−0.905413	−	0.424532i	$-0.860439\pi$
$53$	−37031.1	−1.81083	−0.905413	+	0.424532i	$0.860439\pi$
$54$	0	0
$55$	4445.82	0.198173
$56$	0	0
$57$	−1854.59	−0.0756066
$58$	0	0
$59$	−7081.64	−0.264852	−0.132426	−	0.991193i	$-0.542277\pi$
$59$	−7081.64	−0.264852	−0.132426	+	0.991193i	$0.542277\pi$
$60$	0	0
$61$	41598.1	1.43136	0.715681	−	0.698428i	$-0.246117\pi$
$61$	41598.1	1.43136	0.715681	+	0.698428i	$0.246117\pi$
$62$	0	0
$63$	47136.8	1.49627
$64$	0	0
$65$	2267.71	0.0665739
$66$	0	0
$67$	43598.6	1.18655	0.593275	−	0.805000i	$-0.297835\pi$
$67$	43598.6	1.18655	0.593275	+	0.805000i	$0.297835\pi$
$68$	0	0
$69$	93849.9	2.37307
$70$	0	0
$71$	−82324.1	−1.93812	−0.969061	−	0.246822i	$-0.920614\pi$
$71$	−82324.1	−1.93812	−0.969061	+	0.246822i	$0.920614\pi$
$72$	0	0
$73$	−32288.5	−0.709153	−0.354577	−	0.935027i	$-0.615375\pi$
$73$	−32288.5	−0.709153	−0.354577	+	0.935027i	$0.615375\pi$
$74$	0	0
$75$	75883.6	1.55774
$76$	0	0
$77$	104470.	2.00800
$78$	0	0
$79$	62099.5	1.11949	0.559745	−	0.828665i	$-0.310899\pi$
$79$	62099.5	1.11949	0.559745	+	0.828665i	$0.310899\pi$
$80$	0	0
$81$	−17576.7	−0.297663
$82$	0	0
$83$	25189.8	0.401356	0.200678	−	0.979657i	$-0.435686\pi$
$83$	25189.8	0.401356	0.200678	+	0.979657i	$0.435686\pi$
$84$	0	0
$85$	−6951.94	−0.104366
$86$	0	0
$87$	−209536.	−2.96798
$88$	0	0
$89$	122559.	1.64009	0.820047	−	0.572296i	$-0.193947\pi$
$89$	122559.	1.64009	0.820047	+	0.572296i	$0.193947\pi$
$90$	0	0
$91$	53287.6	0.674563
$92$	0	0
$93$	251588.	3.01635
$94$	0	0
$95$	422.907	0.00480768
$96$	0	0
$97$	−33723.5	−0.363918	−0.181959	−	0.983306i	$-0.558244\pi$
$97$	−33723.5	−0.363918	−0.181959	+	0.983306i	$0.558244\pi$
$98$	0	0
$99$	285064.	2.92317
$100$	0	0
$101$	−153142.	−1.49379	−0.746897	−	0.664940i	$-0.768457\pi$
$101$	−153142.	−1.49379	−0.746897	+	0.664940i	$0.768457\pi$
$102$	0	0
$103$	−86262.2	−0.801175	−0.400588	−	0.916258i	$-0.631194\pi$
$103$	−86262.2	−0.801175	−0.400588	+	0.916258i	$0.631194\pi$
$104$	0	0
$105$	−18031.7	−0.159611
$106$	0	0
$107$	−84672.3	−0.714960	−0.357480	−	0.933921i	$-0.616364\pi$
$107$	−84672.3	−0.714960	−0.357480	+	0.933921i	$0.616364\pi$
$108$	0	0
$109$	−117017.	−0.943375	−0.471688	−	0.881766i	$-0.656355\pi$
$109$	−117017.	−0.943375	−0.471688	+	0.881766i	$0.656355\pi$
$110$	0	0
$111$	−173691.	−1.33805
$112$	0	0
$113$	−161445.	−1.18940	−0.594702	−	0.803946i	$-0.702730\pi$
$113$	−161445.	−1.18940	−0.594702	+	0.803946i	$0.702730\pi$
$114$	0	0
$115$	−21400.9	−0.150899
$116$	0	0
$117$	145405.	0.982006
$118$	0	0
$119$	−163360.	−1.05749
$120$	0	0
$121$	470739.	2.92292
$122$	0	0
$123$	115090.	0.685920
$124$	0	0
$125$	−34782.9	−0.199109
$126$	0	0
$127$	−108348.	−0.596088	−0.298044	−	0.954552i	$-0.596334\pi$
$127$	−108348.	−0.596088	−0.298044	+	0.954552i	$0.596334\pi$
$128$	0	0
$129$	383976.	2.03156
$130$	0	0
$131$	70650.7	0.359698	0.179849	−	0.983694i	$-0.442439\pi$
$131$	70650.7	0.359698	0.179849	+	0.983694i	$0.442439\pi$
$132$	0	0
$133$	9937.64	0.0487140
$134$	0	0
$135$	−15864.8	−0.0749203
$136$	0	0
$137$	−150387.	−0.684558	−0.342279	−	0.939598i	$-0.611199\pi$
$137$	−150387.	−0.684558	−0.342279	+	0.939598i	$0.611199\pi$
$138$	0	0
$139$	−29146.8	−0.127954	−0.0639769	−	0.997951i	$-0.520378\pi$
$139$	−29146.8	−0.127954	−0.0639769	+	0.997951i	$0.520378\pi$
$140$	0	0
$141$	−389321.	−1.64915
$142$	0	0
$143$	322262.	1.31786
$144$	0	0
$145$	47781.2	0.188728
$146$	0	0
$147$	−11468.8	−0.0437748
$148$	0	0
$149$	−184909.	−0.682326	−0.341163	−	0.940004i	$-0.610821\pi$
$149$	−184909.	−0.682326	−0.341163	+	0.940004i	$0.610821\pi$
$150$	0	0
$151$	111119.	0.396594	0.198297	−	0.980142i	$-0.436459\pi$
$151$	111119.	0.396594	0.198297	+	0.980142i	$0.436459\pi$
$152$	0	0
$153$	−445756.	−1.53946
$154$	0	0
$155$	−57370.3	−0.191804
$156$	0	0
$157$	−435702.	−1.41072	−0.705359	−	0.708851i	$-0.749214\pi$
$157$	−435702.	−1.41072	−0.705359	+	0.708851i	$0.749214\pi$
$158$	0	0
$159$	908310.	2.84932
$160$	0	0
$161$	−502887.	−1.52899
$162$	0	0
$163$	309371.	0.912032	0.456016	−	0.889972i	$-0.349276\pi$
$163$	309371.	0.912032	0.456016	+	0.889972i	$0.349276\pi$
$164$	0	0
$165$	−109048.	−0.311824
$166$	0	0
$167$	−116680.	−0.323746	−0.161873	−	0.986812i	$-0.551753\pi$
$167$	−116680.	−0.323746	−0.161873	+	0.986812i	$0.551753\pi$
$168$	0	0
$169$	−206915.	−0.557281
$170$	0	0
$171$	27116.6	0.0709162
$172$	0	0
$173$	632998.	1.60800	0.804001	−	0.594627i	$-0.202700\pi$
$173$	632998.	1.60800	0.804001	+	0.594627i	$0.202700\pi$
$174$	0	0
$175$	−406616.	−1.00367
$176$	0	0
$177$	173701.	0.416743
$178$	0	0
$179$	−62850.9	−0.146615	−0.0733076	−	0.997309i	$-0.523355\pi$
$179$	−62850.9	−0.146615	−0.0733076	+	0.997309i	$0.523355\pi$
$180$	0	0
$181$	−25004.4	−0.0567310	−0.0283655	−	0.999598i	$-0.509030\pi$
$181$	−25004.4	−0.0567310	−0.0283655	+	0.999598i	$0.509030\pi$
$182$	0	0
$183$	−1.02033e6	−2.25224
$184$	0	0
$185$	39607.4	0.0850837
$186$	0	0
$187$	−987932.	−2.06597
$188$	0	0
$189$	−372797.	−0.759133
$190$	0	0
$191$	597658.	1.18541	0.592706	−	0.805419i	$-0.298059\pi$
$191$	597658.	1.18541	0.592706	+	0.805419i	$0.298059\pi$
$192$	0	0
$193$	−320799.	−0.619926	−0.309963	−	0.950749i	$-0.600317\pi$
$193$	−320799.	−0.619926	−0.309963	+	0.950749i	$0.600317\pi$
$194$	0	0
$195$	−55623.1	−0.104754
$196$	0	0
$197$	832575.	1.52847	0.764236	−	0.644936i	$-0.223116\pi$
$197$	832575.	1.52847	0.764236	+	0.644936i	$0.223116\pi$
$198$	0	0
$199$	553619.	0.991011	0.495506	−	0.868605i	$-0.334983\pi$
$199$	553619.	0.991011	0.495506	+	0.868605i	$0.334983\pi$
$200$	0	0
$201$	−1.06940e6	−1.86703
$202$	0	0
$203$	1.12278e6	1.91230
$204$	0	0
$205$	−26244.2	−0.0436163
$206$	0	0
$207$	−1.37222e6	−2.22586
$208$	0	0
$209$	60098.8	0.0951700
$210$	0	0
$211$	−463862.	−0.717270	−0.358635	−	0.933478i	$-0.616758\pi$
$211$	−463862.	−0.717270	−0.358635	+	0.933478i	$0.616758\pi$
$212$	0	0
$213$	2.01927e6	3.04962
$214$	0	0
$215$	−87559.1	−0.129183
$216$	0	0
$217$	−1.34811e6	−1.94346
$218$	0	0
$219$	791981.	1.11585
$220$	0	0
$221$	−503922.	−0.694037
$222$	0	0
$223$	385933.	0.519697	0.259848	−	0.965649i	$-0.416327\pi$
$223$	385933.	0.519697	0.259848	+	0.965649i	$0.416327\pi$
$224$	0	0
$225$	−1.10952e6	−1.46110
$226$	0	0
$227$	−83856.8	−0.108012	−0.0540062	−	0.998541i	$-0.517199\pi$
$227$	−83856.8	−0.108012	−0.0540062	+	0.998541i	$0.517199\pi$
$228$	0	0
$229$	710867.	0.895777	0.447888	−	0.894089i	$-0.352176\pi$
$229$	710867.	0.895777	0.447888	+	0.894089i	$0.352176\pi$
$230$	0	0
$231$	−2.56246e6	−3.15957
$232$	0	0
$233$	−1.05504e6	−1.27314	−0.636571	−	0.771218i	$-0.719648\pi$
$233$	−1.05504e6	−1.27314	−0.636571	+	0.771218i	$0.719648\pi$
$234$	0	0
$235$	88777.9	0.104866
$236$	0	0
$237$	−1.52320e6	−1.76151
$238$	0	0
$239$	67430.5	0.0763593	0.0381796	−	0.999271i	$-0.487844\pi$
$239$	67430.5	0.0763593	0.0381796	+	0.999271i	$0.487844\pi$
$240$	0	0
$241$	−26288.9	−0.0291561	−0.0145781	−	0.999894i	$-0.504641\pi$
$241$	−26288.9	−0.0291561	−0.0145781	+	0.999894i	$0.504641\pi$
$242$	0	0
$243$	1.12038e6	1.21716
$244$	0	0
$245$	2615.26	0.00278356
$246$	0	0
$247$	30655.0	0.0319712
$248$	0	0
$249$	−617863.	−0.631530
$250$	0	0
$251$	1.41726e6	1.41993	0.709963	−	0.704239i	$-0.248712\pi$
$251$	1.41726e6	1.41993	0.709963	+	0.704239i	$0.248712\pi$
$252$	0	0
$253$	−3.04125e6	−2.98711
$254$	0	0
$255$	170519.	0.164219
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	930710.	0.862114
$260$	0	0
$261$	3.06371e6	2.78386
$262$	0	0
$263$	−990673.	−0.883164	−0.441582	−	0.897221i	$-0.645582\pi$
$263$	−990673.	−0.883164	−0.441582	+	0.897221i	$0.645582\pi$
$264$	0	0
$265$	−207125.	−0.181183
$266$	0	0
$267$	−3.00616e6	−2.58068
$268$	0	0
$269$	−1.39516e6	−1.17556	−0.587780	−	0.809021i	$-0.699998\pi$
$269$	−1.39516e6	−1.17556	−0.587780	+	0.809021i	$0.699998\pi$
$270$	0	0
$271$	−215430.	−0.178190	−0.0890951	−	0.996023i	$-0.528397\pi$
$271$	−215430.	−0.178190	−0.0890951	+	0.996023i	$0.528397\pi$
$272$	0	0
$273$	−1.30705e6	−1.06142
$274$	0	0
$275$	−2.45905e6	−1.96081
$276$	0	0
$277$	−68889.1	−0.0539450	−0.0269725	−	0.999636i	$-0.508587\pi$
$277$	−68889.1	−0.0539450	−0.0269725	+	0.999636i	$0.508587\pi$
$278$	0	0
$279$	−3.67856e6	−2.82923
$280$	0	0
$281$	−269399.	−0.203530	−0.101765	−	0.994808i	$-0.532449\pi$
$281$	−269399.	−0.203530	−0.101765	+	0.994808i	$0.532449\pi$
$282$	0	0
$283$	−1.06206e6	−0.788286	−0.394143	−	0.919049i	$-0.628959\pi$
$283$	−1.06206e6	−0.788286	−0.394143	+	0.919049i	$0.628959\pi$
$284$	0	0
$285$	−10373.2	−0.00756485
$286$	0	0
$287$	−616698.	−0.441944
$288$	0	0
$289$	124977.	0.0880208
$290$	0	0
$291$	827182.	0.572622
$292$	0	0
$293$	−1.39496e6	−0.949274	−0.474637	−	0.880182i	$-0.657421\pi$
$293$	−1.39496e6	−0.949274	−0.474637	+	0.880182i	$0.657421\pi$
$294$	0	0
$295$	−39609.5	−0.0264999
$296$	0	0
$297$	−2.25453e6	−1.48308
$298$	0	0
$299$	−1.55127e6	−1.00348
$300$	0	0
$301$	−2.05750e6	−1.30895
$302$	0	0
$303$	3.75631e6	2.35047
$304$	0	0
$305$	232669.	0.143215
$306$	0	0
$307$	866256.	0.524566	0.262283	−	0.964991i	$-0.415525\pi$
$307$	866256.	0.524566	0.262283	+	0.964991i	$0.415525\pi$
$308$	0	0
$309$	2.11587e6	1.26064
$310$	0	0
$311$	−2.67321e6	−1.56723	−0.783615	−	0.621247i	$-0.786627\pi$
$311$	−2.67321e6	−1.56723	−0.783615	+	0.621247i	$0.786627\pi$
$312$	0	0
$313$	−1.15308e6	−0.665269	−0.332634	−	0.943056i	$-0.607937\pi$
$313$	−1.15308e6	−0.665269	−0.332634	+	0.943056i	$0.607937\pi$
$314$	0	0
$315$	263649.	0.149709
$316$	0	0
$317$	571215.	0.319265	0.159632	−	0.987177i	$-0.448969\pi$
$317$	571215.	0.319265	0.159632	+	0.987177i	$0.448969\pi$
$318$	0	0
$319$	6.79013e6	3.73595
$320$	0	0
$321$	2.07687e6	1.12498
$322$	0	0
$323$	−93976.7	−0.0501203
$324$	0	0
$325$	−1.25430e6	−0.658710
$326$	0	0
$327$	2.87024e6	1.48439
$328$	0	0
$329$	2.08614e6	1.06256
$330$	0	0
$331$	282357.	0.141654	0.0708270	−	0.997489i	$-0.477436\pi$
$331$	282357.	0.141654	0.0708270	+	0.997489i	$0.477436\pi$
$332$	0	0
$333$	2.53961e6	1.25504
$334$	0	0
$335$	243859.	0.118721
$336$	0	0
$337$	−1.52975e6	−0.733747	−0.366874	−	0.930271i	$-0.619572\pi$
$337$	−1.52975e6	−0.733747	−0.366874	+	0.930271i	$0.619572\pi$
$338$	0	0
$339$	3.95998e6	1.87152
$340$	0	0
$341$	−8.15282e6	−3.79684
$342$	0	0
$343$	−2.14754e6	−0.985610
$344$	0	0
$345$	524927.	0.237439
$346$	0	0
$347$	2.74649e6	1.22449	0.612243	−	0.790670i	$-0.290268\pi$
$347$	2.74649e6	1.22449	0.612243	+	0.790670i	$0.290268\pi$
$348$	0	0
$349$	125974.	0.0553625	0.0276813	−	0.999617i	$-0.491188\pi$
$349$	125974.	0.0553625	0.0276813	+	0.999617i	$0.491188\pi$
$350$	0	0
$351$	−1.14998e6	−0.498223
$352$	0	0
$353$	−3.44847e6	−1.47296	−0.736478	−	0.676462i	$-0.763512\pi$
$353$	−3.44847e6	−1.47296	−0.736478	+	0.676462i	$0.763512\pi$
$354$	0	0
$355$	−460460.	−0.193919
$356$	0	0
$357$	4.00693e6	1.66396
$358$	0	0
$359$	−2.75057e6	−1.12639	−0.563193	−	0.826325i	$-0.690427\pi$
$359$	−2.75057e6	−1.12639	−0.563193	+	0.826325i	$0.690427\pi$
$360$	0	0
$361$	−2.47038e6	−0.997691
$362$	0	0
$363$	−1.15464e7	−4.59918
$364$	0	0
$365$	−180598.	−0.0709546
$366$	0	0
$367$	1.34972e6	0.523093	0.261546	−	0.965191i	$-0.415768\pi$
$367$	1.34972e6	0.523093	0.261546	+	0.965191i	$0.415768\pi$
$368$	0	0
$369$	−1.68277e6	−0.643368
$370$	0	0
$371$	−4.86710e6	−1.83584
$372$	0	0
$373$	−3.56633e6	−1.32724	−0.663619	−	0.748070i	$-0.730980\pi$
$373$	−3.56633e6	−1.32724	−0.663619	+	0.748070i	$0.730980\pi$
$374$	0	0
$375$	853166.	0.313296
$376$	0	0
$377$	3.46349e6	1.25505
$378$	0	0
$379$	1.43643e6	0.513673	0.256837	−	0.966455i	$-0.417320\pi$
$379$	1.43643e6	0.513673	0.256837	+	0.966455i	$0.417320\pi$
$380$	0	0
$381$	2.65759e6	0.937941
$382$	0	0
$383$	2.38366e6	0.830322	0.415161	−	0.909748i	$-0.363725\pi$
$383$	2.38366e6	0.830322	0.415161	+	0.909748i	$0.363725\pi$
$384$	0	0
$385$	584326.	0.200911
$386$	0	0
$387$	−5.61426e6	−1.90553
$388$	0	0
$389$	−996771.	−0.333981	−0.166990	−	0.985959i	$-0.553405\pi$
$389$	−996771.	−0.333981	−0.166990	+	0.985959i	$0.553405\pi$
$390$	0	0
$391$	4.75562e6	1.57313
$392$	0	0
$393$	−1.73294e6	−0.565982
$394$	0	0
$395$	347339.	0.112011
$396$	0	0
$397$	−488274.	−0.155485	−0.0777424	−	0.996973i	$-0.524771\pi$
$397$	−488274.	−0.155485	−0.0777424	+	0.996973i	$0.524771\pi$
$398$	0	0
$399$	−243753.	−0.0766511
$400$	0	0
$401$	−1.08770e6	−0.337790	−0.168895	−	0.985634i	$-0.554020\pi$
$401$	−1.08770e6	−0.337790	−0.168895	+	0.985634i	$0.554020\pi$
$402$	0	0
$403$	−4.15857e6	−1.27550
$404$	0	0
$405$	−98311.2	−0.0297828
$406$	0	0
$407$	5.62855e6	1.68427
$408$	0	0
$409$	−1.61498e6	−0.477373	−0.238687	−	0.971097i	$-0.576717\pi$
$409$	−1.61498e6	−0.477373	−0.238687	+	0.971097i	$0.576717\pi$
$410$	0	0
$411$	3.68875e6	1.07715
$412$	0	0
$413$	−930759.	−0.268511
$414$	0	0
$415$	140893.	0.0401578
$416$	0	0
$417$	714921.	0.201334
$418$	0	0
$419$	6.75825e6	1.88061	0.940305	−	0.340332i	$-0.110539\pi$
$419$	6.75825e6	1.88061	0.940305	+	0.340332i	$0.110539\pi$
$420$	0	0
$421$	−2.24385e6	−0.617005	−0.308502	−	0.951224i	$-0.599828\pi$
$421$	−2.24385e6	−0.617005	−0.308502	+	0.951224i	$0.599828\pi$
$422$	0	0
$423$	5.69241e6	1.54684
$424$	0	0
$425$	3.84522e6	1.03264
$426$	0	0
$427$	5.46736e6	1.45113
$428$	0	0
$429$	−7.90454e6	−2.07364
$430$	0	0
$431$	−2.69517e6	−0.698864	−0.349432	−	0.936962i	$-0.613625\pi$
$431$	−2.69517e6	−0.698864	−0.349432	+	0.936962i	$0.613625\pi$
$432$	0	0
$433$	−2.21937e6	−0.568867	−0.284433	−	0.958696i	$-0.591805\pi$
$433$	−2.21937e6	−0.568867	−0.284433	+	0.958696i	$0.591805\pi$
$434$	0	0
$435$	−1.17199e6	−0.296962
$436$	0	0
$437$	−289298.	−0.0724673
$438$	0	0
$439$	5.56091e6	1.37716	0.688580	−	0.725160i	$-0.258234\pi$
$439$	5.56091e6	1.37716	0.688580	+	0.725160i	$0.258234\pi$
$440$	0	0
$441$	167690.	0.0410592
$442$	0	0
$443$	−3.03054e6	−0.733686	−0.366843	−	0.930283i	$-0.619561\pi$
$443$	−3.03054e6	−0.733686	−0.366843	+	0.930283i	$0.619561\pi$
$444$	0	0
$445$	685503.	0.164100
$446$	0	0
$447$	4.53550e6	1.07363
$448$	0	0
$449$	6.81985e6	1.59646	0.798232	−	0.602350i	$-0.205769\pi$
$449$	6.81985e6	1.59646	0.798232	+	0.602350i	$0.205769\pi$
$450$	0	0
$451$	−3.72954e6	−0.863403
$452$	0	0
$453$	−2.72556e6	−0.624037
$454$	0	0
$455$	298052.	0.0674936
$456$	0	0
$457$	−2.53418e6	−0.567605	−0.283802	−	0.958883i	$-0.591596\pi$
$457$	−2.53418e6	−0.567605	−0.283802	+	0.958883i	$0.591596\pi$
$458$	0	0
$459$	3.52541e6	0.781049
$460$	0	0
$461$	6.49957e6	1.42440	0.712201	−	0.701976i	$-0.247699\pi$
$461$	6.49957e6	1.42440	0.712201	+	0.701976i	$0.247699\pi$
$462$	0	0
$463$	7.28190e6	1.57867	0.789336	−	0.613961i	$-0.210425\pi$
$463$	7.28190e6	1.57867	0.789336	+	0.613961i	$0.210425\pi$
$464$	0	0
$465$	1.40720e6	0.301802
$466$	0	0
$467$	−2.40030e6	−0.509301	−0.254650	−	0.967033i	$-0.581960\pi$
$467$	−2.40030e6	−0.509301	−0.254650	+	0.967033i	$0.581960\pi$
$468$	0	0
$469$	5.73029e6	1.20294
$470$	0	0
$471$	1.06870e7	2.21975
$472$	0	0
$473$	−1.24429e7	−2.55723
$474$	0	0
$475$	−233916.	−0.0475692
$476$	0	0
$477$	−1.32808e7	−2.67256
$478$	0	0
$479$	−701374.	−0.139672	−0.0698362	−	0.997558i	$-0.522248\pi$
$479$	−701374.	−0.139672	−0.0698362	+	0.997558i	$0.522248\pi$
$480$	0	0
$481$	2.87100e6	0.565810
$482$	0	0
$483$	1.23350e7	2.40586
$484$	0	0
$485$	−188625.	−0.0364120
$486$	0	0
$487$	−2.74404e6	−0.524285	−0.262143	−	0.965029i	$-0.584429\pi$
$487$	−2.74404e6	−0.524285	−0.262143	+	0.965029i	$0.584429\pi$
$488$	0	0
$489$	−7.58834e6	−1.43507
$490$	0	0
$491$	−7.05231e6	−1.32016	−0.660081	−	0.751194i	$-0.729478\pi$
$491$	−7.05231e6	−1.32016	−0.660081	+	0.751194i	$0.729478\pi$
$492$	0	0
$493$	−1.06178e7	−1.96750
$494$	0	0
$495$	1.59444e6	0.292479
$496$	0	0
$497$	−1.08201e7	−1.96490
$498$	0	0
$499$	−3.92055e6	−0.704848	−0.352424	−	0.935840i	$-0.614643\pi$
$499$	−3.92055e6	−0.704848	−0.352424	+	0.935840i	$0.614643\pi$
$500$	0	0
$501$	2.86195e6	0.509411
$502$	0	0
$503$	5.77337e6	1.01744	0.508720	−	0.860932i	$-0.330119\pi$
$503$	5.77337e6	1.01744	0.508720	+	0.860932i	$0.330119\pi$
$504$	0	0
$505$	−856563.	−0.149462
$506$	0	0
$507$	5.07526e6	0.876877
$508$	0	0
$509$	−6.85695e6	−1.17310	−0.586552	−	0.809911i	$-0.699515\pi$
$509$	−6.85695e6	−1.17310	−0.586552	+	0.809911i	$0.699515\pi$
$510$	0	0
$511$	−4.24376e6	−0.718950
$512$	0	0
$513$	−214461.	−0.0359795
$514$	0	0
$515$	−482487.	−0.0801618
$516$	0	0
$517$	1.26161e7	2.07587
$518$	0	0
$519$	−1.55264e7	−2.53018
$520$	0	0
$521$	6.25077e6	1.00888	0.504440	−	0.863447i	$-0.331699\pi$
$521$	6.25077e6	1.00888	0.504440	+	0.863447i	$0.331699\pi$
$522$	0	0
$523$	3.18237e6	0.508740	0.254370	−	0.967107i	$-0.418132\pi$
$523$	3.18237e6	0.508740	0.254370	+	0.967107i	$0.418132\pi$
$524$	0	0
$525$	9.97360e6	1.57926
$526$	0	0
$527$	1.27486e7	1.99957
$528$	0	0
$529$	8.20337e6	1.27454
$530$	0	0
$531$	−2.53975e6	−0.390890
$532$	0	0
$533$	−1.90235e6	−0.290050
$534$	0	0
$535$	−473595.	−0.0715356
$536$	0	0
$537$	1.54163e6	0.230698
$538$	0	0
$539$	371652.	0.0551016
$540$	0	0
$541$	−6.92939e6	−1.01789	−0.508946	−	0.860798i	$-0.669965\pi$
$541$	−6.92939e6	−1.01789	−0.508946	+	0.860798i	$0.669965\pi$
$542$	0	0
$543$	613316.	0.0892657
$544$	0	0
$545$	−654509.	−0.0943897
$546$	0	0
$547$	−8.32913e6	−1.19023	−0.595116	−	0.803640i	$-0.702894\pi$
$547$	−8.32913e6	−1.19023	−0.595116	+	0.803640i	$0.702894\pi$
$548$	0	0
$549$	1.49187e7	2.11251
$550$	0	0
$551$	645909.	0.0906342
$552$	0	0
$553$	8.16191e6	1.13496
$554$	0	0
$555$	−971502.	−0.133879
$556$	0	0
$557$	1.08882e7	1.48702	0.743510	−	0.668725i	$-0.233159\pi$
$557$	1.08882e7	1.48702	0.743510	+	0.668725i	$0.233159\pi$
$558$	0	0
$559$	−6.34686e6	−0.859071
$560$	0	0
$561$	2.42323e7	3.25078
$562$	0	0
$563$	7.14701e6	0.950284	0.475142	−	0.879909i	$-0.342397\pi$
$563$	7.14701e6	0.950284	0.475142	+	0.879909i	$0.342397\pi$
$564$	0	0
$565$	−903006.	−0.119006
$566$	0	0
$567$	−2.31016e6	−0.301776
$568$	0	0
$569$	−1.37579e6	−0.178144	−0.0890719	−	0.996025i	$-0.528390\pi$
$569$	−1.37579e6	−0.178144	−0.0890719	+	0.996025i	$0.528390\pi$
$570$	0	0
$571$	1.22271e6	0.156939	0.0784696	−	0.996917i	$-0.474997\pi$
$571$	1.22271e6	0.156939	0.0784696	+	0.996917i	$0.474997\pi$
$572$	0	0
$573$	−1.46596e7	−1.86524
$574$	0	0
$575$	1.18371e7	1.49306
$576$	0	0
$577$	−2.72270e6	−0.340455	−0.170228	−	0.985405i	$-0.554450\pi$
$577$	−2.72270e6	−0.340455	−0.170228	+	0.985405i	$0.554450\pi$
$578$	0	0
$579$	7.86866e6	0.975449
$580$	0	0
$581$	3.31077e6	0.406900
$582$	0	0
$583$	−2.94342e7	−3.58659
$584$	0	0
$585$	813288.	0.0982550
$586$	0	0
$587$	−1.15061e6	−0.137826	−0.0689131	−	0.997623i	$-0.521953\pi$
$587$	−1.15061e6	−0.137826	−0.0689131	+	0.997623i	$0.521953\pi$
$588$	0	0
$589$	−775535.	−0.0921113
$590$	0	0
$591$	−2.04217e7	−2.40504
$592$	0	0
$593$	−896439.	−0.104685	−0.0523425	−	0.998629i	$-0.516669\pi$
$593$	−896439.	−0.104685	−0.0523425	+	0.998629i	$0.516669\pi$
$594$	0	0
$595$	−913713.	−0.105808
$596$	0	0
$597$	−1.35794e7	−1.55935
$598$	0	0
$599$	5.13967e6	0.585286	0.292643	−	0.956222i	$-0.405465\pi$
$599$	5.13967e6	0.585286	0.292643	+	0.956222i	$0.405465\pi$
$600$	0	0
$601$	−4.02113e6	−0.454111	−0.227055	−	0.973882i	$-0.572910\pi$
$601$	−4.02113e6	−0.454111	−0.227055	+	0.973882i	$0.572910\pi$
$602$	0	0
$603$	1.56361e7	1.75120
$604$	0	0
$605$	2.63296e6	0.292453
$606$	0	0
$607$	1.26773e7	1.39654	0.698271	−	0.715834i	$-0.253953\pi$
$607$	1.26773e7	1.39654	0.698271	+	0.715834i	$0.253953\pi$
$608$	0	0
$609$	−2.75399e7	−3.00898
$610$	0	0
$611$	6.43520e6	0.697364
$612$	0	0
$613$	−6.33751e6	−0.681189	−0.340595	−	0.940210i	$-0.610628\pi$
$613$	−6.33751e6	−0.681189	−0.340595	+	0.940210i	$0.610628\pi$
$614$	0	0
$615$	643727.	0.0686300
$616$	0	0
$617$	−1.35210e7	−1.42987	−0.714936	−	0.699190i	$-0.753544\pi$
$617$	−1.35210e7	−1.42987	−0.714936	+	0.699190i	$0.753544\pi$
$618$	0	0
$619$	5.56568e6	0.583837	0.291919	−	0.956443i	$-0.405706\pi$
$619$	5.56568e6	0.583837	0.291919	+	0.956443i	$0.405706\pi$
$620$	0	0
$621$	1.08526e7	1.12929
$622$	0	0
$623$	1.61082e7	1.66275
$624$	0	0
$625$	9.47331e6	0.970067
$626$	0	0
$627$	−1.47412e6	−0.149749
$628$	0	0
$629$	−8.80140e6	−0.887003
$630$	0	0
$631$	−1.16750e7	−1.16730	−0.583650	−	0.812005i	$-0.698376\pi$
$631$	−1.16750e7	−1.16730	−0.583650	+	0.812005i	$0.698376\pi$
$632$	0	0
$633$	1.13778e7	1.12862
$634$	0	0
$635$	−606018.	−0.0596418
$636$	0	0
$637$	189571.	0.0185108
$638$	0	0
$639$	−2.95246e7	−2.86043
$640$	0	0
$641$	4.93204e6	0.474113	0.237056	−	0.971496i	$-0.423817\pi$
$641$	4.93204e6	0.474113	0.237056	+	0.971496i	$0.423817\pi$
$642$	0	0
$643$	−948839.	−0.0905034	−0.0452517	−	0.998976i	$-0.514409\pi$
$643$	−948839.	−0.0905034	−0.0452517	+	0.998976i	$0.514409\pi$
$644$	0	0
$645$	2.14768e6	0.203268
$646$	0	0
$647$	1.53603e7	1.44258	0.721290	−	0.692633i	$-0.243549\pi$
$647$	1.53603e7	1.44258	0.721290	+	0.692633i	$0.243549\pi$
$648$	0	0
$649$	−5.62885e6	−0.524576
$650$	0	0
$651$	3.30669e7	3.05802
$652$	0	0
$653$	−1.41878e7	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$653$	−1.41878e7	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$654$	0	0
$655$	395168.	0.0359897
$656$	0	0
$657$	−1.15799e7	−1.04662
$658$	0	0
$659$	−1.80234e7	−1.61668	−0.808340	−	0.588716i	$-0.799634\pi$
$659$	−1.80234e7	−1.61668	−0.808340	+	0.588716i	$0.799634\pi$
$660$	0	0
$661$	4.89327e6	0.435607	0.217804	−	0.975993i	$-0.430111\pi$
$661$	4.89327e6	0.435607	0.217804	+	0.975993i	$0.430111\pi$
$662$	0	0
$663$	1.23604e7	1.09206
$664$	0	0
$665$	55583.8	0.00487410
$666$	0	0
$667$	−3.26857e7	−2.84474
$668$	0	0
$669$	−9.46629e6	−0.817738
$670$	0	0
$671$	3.30644e7	2.83500
$672$	0	0
$673$	−1.91986e7	−1.63392	−0.816961	−	0.576693i	$-0.804343\pi$
$673$	−1.91986e7	−1.63392	−0.816961	+	0.576693i	$0.804343\pi$
$674$	0	0
$675$	8.77505e6	0.741293
$676$	0	0
$677$	−2.31107e7	−1.93794	−0.968972	−	0.247171i	$-0.920499\pi$
$677$	−2.31107e7	−1.93794	−0.968972	+	0.247171i	$0.920499\pi$
$678$	0	0
$679$	−4.43238e6	−0.368946
$680$	0	0
$681$	2.05687e6	0.169957
$682$	0	0
$683$	1.23727e7	1.01487	0.507437	−	0.861689i	$-0.330593\pi$
$683$	1.23727e7	1.01487	0.507437	+	0.861689i	$0.330593\pi$
$684$	0	0
$685$	−841156.	−0.0684936
$686$	0	0
$687$	−1.74364e7	−1.40950
$688$	0	0
$689$	−1.50137e7	−1.20487
$690$	0	0
$691$	2.84755e6	0.226869	0.113435	−	0.993545i	$-0.463815\pi$
$691$	2.84755e6	0.226869	0.113435	+	0.993545i	$0.463815\pi$
$692$	0	0
$693$	3.74668e7	2.96356
$694$	0	0
$695$	−163026.	−0.0128025
$696$	0	0
$697$	5.83189e6	0.454703
$698$	0	0
$699$	2.58782e7	2.00328
$700$	0	0
$701$	2.22957e7	1.71366	0.856832	−	0.515596i	$-0.172429\pi$
$701$	2.22957e7	1.71366	0.856832	+	0.515596i	$0.172429\pi$
$702$	0	0
$703$	535414.	0.0408603
$704$	0	0
$705$	−2.17757e6	−0.165006
$706$	0	0
$707$	−2.01279e7	−1.51443
$708$	0	0
$709$	3.05522e6	0.228258	0.114129	−	0.993466i	$-0.463592\pi$
$709$	3.05522e6	0.228258	0.114129	+	0.993466i	$0.463592\pi$
$710$	0	0
$711$	2.22712e7	1.65223
$712$	0	0
$713$	3.92453e7	2.89111
$714$	0	0
$715$	1.80249e6	0.131859
$716$	0	0
$717$	−1.65396e6	−0.120151
$718$	0	0
$719$	−2.34831e7	−1.69408	−0.847040	−	0.531529i	$-0.821618\pi$
$719$	−2.34831e7	−1.69408	−0.847040	+	0.531529i	$0.821618\pi$
$720$	0	0
$721$	−1.13377e7	−0.812243
$722$	0	0
$723$	644823.	0.0458770
$724$	0	0
$725$	−2.64285e7	−1.86736
$726$	0	0
$727$	−1.73878e7	−1.22013	−0.610067	−	0.792350i	$-0.708858\pi$
$727$	−1.73878e7	−1.22013	−0.610067	+	0.792350i	$0.708858\pi$
$728$	0	0
$729$	−2.32098e7	−1.61753
$730$	0	0
$731$	1.94571e7	1.34674
$732$	0	0
$733$	−1.43687e7	−0.987771	−0.493886	−	0.869527i	$-0.664424\pi$
$733$	−1.43687e7	−0.987771	−0.493886	+	0.869527i	$0.664424\pi$
$734$	0	0
$735$	−64148.0	−0.00437991
$736$	0	0
$737$	3.46544e7	2.35012
$738$	0	0
$739$	2.70997e7	1.82538	0.912691	−	0.408651i	$-0.134001\pi$
$739$	2.70997e7	1.82538	0.912691	+	0.408651i	$0.134001\pi$
$740$	0	0
$741$	−751916.	−0.0503065
$742$	0	0
$743$	−2.81706e7	−1.87208	−0.936038	−	0.351900i	$-0.885536\pi$
$743$	−2.81706e7	−1.87208	−0.936038	+	0.351900i	$0.885536\pi$
$744$	0	0
$745$	−1.03424e6	−0.0682703
$746$	0	0
$747$	9.03402e6	0.592352
$748$	0	0
$749$	−1.11287e7	−0.724837
$750$	0	0
$751$	−2.54911e7	−1.64926	−0.824630	−	0.565672i	$-0.808617\pi$
$751$	−2.54911e7	−1.64926	−0.824630	+	0.565672i	$0.808617\pi$
$752$	0	0
$753$	−3.47630e7	−2.23424
$754$	0	0
$755$	621518.	0.0396813
$756$	0	0
$757$	−2.13145e7	−1.35187	−0.675936	−	0.736961i	$-0.736260\pi$
$757$	−2.13145e7	−1.35187	−0.675936	+	0.736961i	$0.736260\pi$
$758$	0	0
$759$	7.45968e7	4.70019
$760$	0	0
$761$	1.11656e7	0.698906	0.349453	−	0.936954i	$-0.386367\pi$
$761$	1.11656e7	0.698906	0.349453	+	0.936954i	$0.386367\pi$
$762$	0	0
$763$	−1.53799e7	−0.956408
$764$	0	0
$765$	−2.49323e6	−0.154031
$766$	0	0
$767$	−2.87115e6	−0.176225
$768$	0	0
$769$	−418129.	−0.0254973	−0.0127486	−	0.999919i	$-0.504058\pi$
$769$	−418129.	−0.0254973	−0.0127486	+	0.999919i	$0.504058\pi$
$770$	0	0
$771$	−1.62007e6	−0.0981517
$772$	0	0
$773$	2.27997e7	1.37240	0.686200	−	0.727413i	$-0.259278\pi$
$773$	2.27997e7	1.37240	0.686200	+	0.727413i	$0.259278\pi$
$774$	0	0
$775$	3.17323e7	1.89779
$776$	0	0
$777$	−2.28287e7	−1.35653
$778$	0	0
$779$	−354771.	−0.0209462
$780$	0	0
$781$	−6.54354e7	−3.83871
$782$	0	0
$783$	−2.42304e7	−1.41240
$784$	0	0
$785$	−2.43699e6	−0.141150
$786$	0	0
$787$	2.49776e7	1.43752	0.718759	−	0.695259i	$-0.244710\pi$
$787$	2.49776e7	1.43752	0.718759	+	0.695259i	$0.244710\pi$
$788$	0	0
$789$	2.42995e7	1.38965
$790$	0	0
$791$	−2.12192e7	−1.20584
$792$	0	0
$793$	1.68654e7	0.952387
$794$	0	0
$795$	5.08042e6	0.285090
$796$	0	0
$797$	−2.12032e7	−1.18238	−0.591189	−	0.806533i	$-0.701341\pi$
$797$	−2.12032e7	−1.18238	−0.591189	+	0.806533i	$0.701341\pi$
$798$	0	0
$799$	−1.97279e7	−1.09324
$800$	0	0
$801$	4.39542e7	2.42058
$802$	0	0
$803$	−2.56645e7	−1.40457
$804$	0	0
$805$	−2.81278e6	−0.152984
$806$	0	0
$807$	3.42210e7	1.84973
$808$	0	0
$809$	−1.79856e7	−0.966171	−0.483086	−	0.875573i	$-0.660484\pi$
$809$	−1.79856e7	−0.966171	−0.483086	+	0.875573i	$0.660484\pi$
$810$	0	0
$811$	−2.56180e7	−1.36771	−0.683853	−	0.729620i	$-0.739697\pi$
$811$	−2.56180e7	−1.36771	−0.683853	+	0.729620i	$0.739697\pi$
$812$	0	0
$813$	5.28414e6	0.280381
$814$	0	0
$815$	1.73039e6	0.0912536
$816$	0	0
$817$	−1.18363e6	−0.0620384
$818$	0	0
$819$	1.91110e7	0.995573
$820$	0	0
$821$	−2.32275e6	−0.120266	−0.0601331	−	0.998190i	$-0.519153\pi$
$821$	−2.32275e6	−0.120266	−0.0601331	+	0.998190i	$0.519153\pi$
$822$	0	0
$823$	1.43877e6	0.0740441	0.0370221	−	0.999314i	$-0.488213\pi$
$823$	1.43877e6	0.0740441	0.0370221	+	0.999314i	$0.488213\pi$
$824$	0	0
$825$	6.03162e7	3.08531
$826$	0	0
$827$	7.44847e6	0.378707	0.189353	−	0.981909i	$-0.439361\pi$
$827$	7.44847e6	0.378707	0.189353	+	0.981909i	$0.439361\pi$
$828$	0	0
$829$	3.00333e7	1.51781	0.758904	−	0.651202i	$-0.225735\pi$
$829$	3.00333e7	1.51781	0.758904	+	0.651202i	$0.225735\pi$
$830$	0	0
$831$	1.68973e6	0.0848821
$832$	0	0
$833$	−581154.	−0.0290187
$834$	0	0
$835$	−652620.	−0.0323925
$836$	0	0
$837$	2.90931e7	1.43541
$838$	0	0
$839$	−1.93064e7	−0.946883	−0.473442	−	0.880825i	$-0.656989\pi$
$839$	−1.93064e7	−0.946883	−0.473442	+	0.880825i	$0.656989\pi$
$840$	0	0
$841$	5.24654e7	2.55790
$842$	0	0
$843$	6.60789e6	0.320254
$844$	0	0
$845$	−1.15733e6	−0.0557589
$846$	0	0
$847$	6.18705e7	2.96329
$848$	0	0
$849$	2.60506e7	1.24036
$850$	0	0
$851$	−2.70942e7	−1.28249
$852$	0	0
$853$	−2.15308e7	−1.01318	−0.506590	−	0.862187i	$-0.669094\pi$
$853$	−2.15308e7	−1.01318	−0.506590	+	0.862187i	$0.669094\pi$
$854$	0	0
$855$	151671.	0.00709555
$856$	0	0
$857$	1.56530e7	0.728026	0.364013	−	0.931394i	$-0.381406\pi$
$857$	1.56530e7	0.728026	0.364013	+	0.931394i	$0.381406\pi$
$858$	0	0
$859$	−1.03331e7	−0.477803	−0.238902	−	0.971044i	$-0.576787\pi$
$859$	−1.03331e7	−0.477803	−0.238902	+	0.971044i	$0.576787\pi$
$860$	0	0
$861$	1.51266e7	0.695396
$862$	0	0
$863$	2.28173e7	1.04289	0.521444	−	0.853286i	$-0.325394\pi$
$863$	2.28173e7	1.04289	0.521444	+	0.853286i	$0.325394\pi$
$864$	0	0
$865$	3.54052e6	0.160889
$866$	0	0
$867$	−3.06547e6	−0.138500
$868$	0	0
$869$	4.93599e7	2.21730
$870$	0	0
$871$	1.76765e7	0.789496
$872$	0	0
$873$	−1.20945e7	−0.537099
$874$	0	0
$875$	−4.57161e6	−0.201860
$876$	0	0
$877$	4.24096e6	0.186194	0.0930969	−	0.995657i	$-0.470323\pi$
$877$	4.24096e6	0.186194	0.0930969	+	0.995657i	$0.470323\pi$
$878$	0	0
$879$	3.42159e7	1.49368
$880$	0	0
$881$	−2.47881e7	−1.07598	−0.537989	−	0.842952i	$-0.680816\pi$
$881$	−2.47881e7	−1.07598	−0.537989	+	0.842952i	$0.680816\pi$
$882$	0	0
$883$	1.58198e7	0.682809	0.341405	−	0.939916i	$-0.389097\pi$
$883$	1.58198e7	0.682809	0.341405	+	0.939916i	$0.389097\pi$
$884$	0	0
$885$	971553.	0.0416974
$886$	0	0
$887$	1.27597e7	0.544542	0.272271	−	0.962221i	$-0.412225\pi$
$887$	1.27597e7	0.544542	0.272271	+	0.962221i	$0.412225\pi$
$888$	0	0
$889$	−1.42404e7	−0.604323
$890$	0	0
$891$	−1.39709e7	−0.589563
$892$	0	0
$893$	1.20011e6	0.0503606
$894$	0	0
$895$	−351542.	−0.0146696
$896$	0	0
$897$	3.80502e7	1.57898
$898$	0	0
$899$	−8.76220e7	−3.61588
$900$	0	0
$901$	4.60264e7	1.88884
$902$	0	0
$903$	5.04670e7	2.05962
$904$	0	0
$905$	−139856.	−0.00567624
$906$	0	0
$907$	−2.95607e7	−1.19315	−0.596577	−	0.802556i	$-0.703473\pi$
$907$	−2.95607e7	−1.19315	−0.596577	+	0.802556i	$0.703473\pi$
$908$	0	0
$909$	−5.49225e7	−2.20466
$910$	0	0
$911$	−1.02319e7	−0.408469	−0.204235	−	0.978922i	$-0.565471\pi$
$911$	−1.02319e7	−0.408469	−0.204235	+	0.978922i	$0.565471\pi$
$912$	0	0
$913$	2.00222e7	0.794939
$914$	0	0
$915$	−5.70698e6	−0.225348
$916$	0	0
$917$	9.28581e6	0.364667
$918$	0	0
$919$	1.79030e7	0.699257	0.349629	−	0.936888i	$-0.386308\pi$
$919$	1.79030e7	0.699257	0.349629	+	0.936888i	$0.386308\pi$
$920$	0	0
$921$	−2.12478e7	−0.825401
$922$	0	0
$923$	−3.33772e7	−1.28957
$924$	0	0
$925$	−2.19074e7	−0.841854
$926$	0	0
$927$	−3.09369e7	−1.18244
$928$	0	0
$929$	3.32173e7	1.26277	0.631387	−	0.775468i	$-0.282486\pi$
$929$	3.32173e7	1.26277	0.631387	+	0.775468i	$0.282486\pi$
$930$	0	0
$931$	35353.3	0.00133677
$932$	0	0
$933$	6.55694e7	2.46602
$934$	0	0
$935$	−5.52576e6	−0.206711
$936$	0	0
$937$	1.33921e7	0.498311	0.249156	−	0.968463i	$-0.419847\pi$
$937$	1.33921e7	0.498311	0.249156	+	0.968463i	$0.419847\pi$
$938$	0	0
$939$	2.82830e7	1.04679
$940$	0	0
$941$	3.56650e7	1.31301	0.656506	−	0.754321i	$-0.272034\pi$
$941$	3.56650e7	1.31301	0.656506	+	0.754321i	$0.272034\pi$
$942$	0	0
$943$	1.79529e7	0.657439
$944$	0	0
$945$	−2.08515e6	−0.0759553
$946$	0	0
$947$	−1.58428e7	−0.574060	−0.287030	−	0.957922i	$-0.592668\pi$
$947$	−1.58428e7	−0.574060	−0.287030	+	0.957922i	$0.592668\pi$
$948$	0	0
$949$	−1.30909e7	−0.471850
$950$	0	0
$951$	−1.40109e7	−0.502361
$952$	0	0
$953$	3.22385e7	1.14985	0.574927	−	0.818204i	$-0.305030\pi$
$953$	3.22385e7	1.14985	0.574927	+	0.818204i	$0.305030\pi$
$954$	0	0
$955$	3.34286e6	0.118607
$956$	0	0
$957$	−1.66550e8	−5.87849
$958$	0	0
$959$	−1.97658e7	−0.694015
$960$	0	0
$961$	7.65776e7	2.67481
$962$	0	0
$963$	−3.03667e7	−1.05519
$964$	0	0
$965$	−1.79431e6	−0.0620269
$966$	0	0
$967$	3.76769e7	1.29571	0.647856	−	0.761763i	$-0.275666\pi$
$967$	3.76769e7	1.29571	0.647856	+	0.761763i	$0.275666\pi$
$968$	0	0
$969$	2.30509e6	0.0788640
$970$	0	0
$971$	−5.15932e7	−1.75608	−0.878040	−	0.478588i	$-0.841149\pi$
$971$	−5.15932e7	−1.75608	−0.878040	+	0.478588i	$0.841149\pi$
$972$	0	0
$973$	−3.83084e6	−0.129722
$974$	0	0
$975$	3.07660e7	1.03648
$976$	0	0
$977$	−5.45711e6	−0.182905	−0.0914527	−	0.995809i	$-0.529151\pi$
$977$	−5.45711e6	−0.182905	−0.0914527	+	0.995809i	$0.529151\pi$
$978$	0	0
$979$	9.74160e7	3.24843
$980$	0	0
$981$	−4.19669e7	−1.39231
$982$	0	0
$983$	4.80255e6	0.158521	0.0792607	−	0.996854i	$-0.474744\pi$
$983$	4.80255e6	0.158521	0.0792607	+	0.996854i	$0.474744\pi$
$984$	0	0
$985$	4.65681e6	0.152932
$986$	0	0
$987$	−5.11695e7	−1.67193
$988$	0	0
$989$	5.98966e7	1.94721
$990$	0	0
$991$	−2.74651e7	−0.888375	−0.444188	−	0.895934i	$-0.646508\pi$
$991$	−2.74651e7	−0.888375	−0.444188	+	0.895934i	$0.646508\pi$
$992$	0	0
$993$	−6.92574e6	−0.222891
$994$	0	0
$995$	3.09654e6	0.0991560
$996$	0	0
$997$	−3.63436e7	−1.15795	−0.578975	−	0.815345i	$-0.696547\pi$
$997$	−3.63436e7	−1.15795	−0.578975	+	0.815345i	$0.696547\pi$
$998$	0	0
$999$	−2.00854e7	−0.636746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.6	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.6	✓	49	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:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-24.5283 q^{3} +5.59326 q^{5} +131.433 q^{7} +358.638 q^{9} +O(q^{10})\)	\(q-24.5283 q^{3} +5.59326 q^{5} +131.433 q^{7} +358.638 q^{9} +794.852 q^{11} +405.436 q^{13} -137.193 q^{15} -1242.91 q^{17} +75.6100 q^{19} -3223.82 q^{21} -3826.19 q^{23} -3093.72 q^{25} -2836.41 q^{27} +8542.63 q^{29} -10257.0 q^{31} -19496.4 q^{33} +735.138 q^{35} +7081.26 q^{37} -9944.67 q^{39} -4692.12 q^{41} -15654.4 q^{43} +2005.96 q^{45} +15872.3 q^{47} +467.574 q^{49} +30486.6 q^{51} -37031.1 q^{53} +4445.82 q^{55} -1854.59 q^{57} -7081.64 q^{59} +41598.1 q^{61} +47136.8 q^{63} +2267.71 q^{65} +43598.6 q^{67} +93849.9 q^{69} -82324.1 q^{71} -32288.5 q^{73} +75883.6 q^{75} +104470. q^{77} +62099.5 q^{79} -17576.7 q^{81} +25189.8 q^{83} -6951.94 q^{85} -209536. q^{87} +122559. q^{89} +53287.6 q^{91} +251588. q^{93} +422.907 q^{95} -33723.5 q^{97} +285064. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9} - 484 q^{11} - 2922 q^{13} - 244 q^{15} - 3638 q^{17} - 3567 q^{19} - 3811 q^{21} + 3712 q^{23} + 13568 q^{25} - 8814 q^{27} - 5911 q^{29} - 3539 q^{31} - 17215 q^{33} - 10524 q^{35} - 31531 q^{37} - 14986 q^{39} - 35365 q^{41} - 44830 q^{43} - 34403 q^{45} + 12410 q^{47} + 68891 q^{49} - 48608 q^{51} - 79861 q^{53} - 34265 q^{55} - 109824 q^{57} - 51564 q^{59} - 66228 q^{61} + 24981 q^{63} - 31903 q^{65} - 80135 q^{67} + 50903 q^{69} + 162703 q^{71} - 58717 q^{73} + 207827 q^{75} + 89720 q^{77} + 125647 q^{79} + 420081 q^{81} - 28722 q^{83} - 125617 q^{85} + 65848 q^{87} + 93837 q^{89} - 66437 q^{91} - 278711 q^{93} - 83170 q^{95} - 347060 q^{97} + 7934 q^{99}+O(q^{100}) \) 49 * q - 27 * q^3 - 89 * q^5 + 88 * q^7 + 3430 * q^9 - 484 * q^11 - 2922 * q^13 - 244 * q^15 - 3638 * q^17 - 3567 * q^19 - 3811 * q^21 + 3712 * q^23 + 13568 * q^25 - 8814 * q^27 - 5911 * q^29 - 3539 * q^31 - 17215 * q^33 - 10524 * q^35 - 31531 * q^37 - 14986 * q^39 - 35365 * q^41 - 44830 * q^43 - 34403 * q^45 + 12410 * q^47 + 68891 * q^49 - 48608 * q^51 - 79861 * q^53 - 34265 * q^55 - 109824 * q^57 - 51564 * q^59 - 66228 * q^61 + 24981 * q^63 - 31903 * q^65 - 80135 * q^67 + 50903 * q^69 + 162703 * q^71 - 58717 * q^73 + 207827 * q^75 + 89720 * q^77 + 125647 * q^79 + 420081 * q^81 - 28722 * q^83 - 125617 * q^85 + 65848 * q^87 + 93837 * q^89 - 66437 * q^91 - 278711 * q^93 - 83170 * q^95 - 347060 * q^97 + 7934 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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