LMFDB - Embedded newform 1028.6.a.a.1.10

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			1.10
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)

$f(q)$	$=$	$q-19.7180 q^{3} +9.18763 q^{5} -4.07607 q^{7} +145.799 q^{9} +O(q^{10})$	$q-19.7180 q^{3} +9.18763 q^{5} -4.07607 q^{7} +145.799 q^{9} +345.607 q^{11} +759.335 q^{13} -181.161 q^{15} +90.7955 q^{17} -2313.13 q^{19} +80.3719 q^{21} -2942.17 q^{23} -3040.59 q^{25} +1916.62 q^{27} -2750.12 q^{29} +7031.77 q^{31} -6814.67 q^{33} -37.4494 q^{35} -2965.62 q^{37} -14972.5 q^{39} +3028.05 q^{41} +9839.21 q^{43} +1339.54 q^{45} -11972.0 q^{47} -16790.4 q^{49} -1790.30 q^{51} +16491.9 q^{53} +3175.31 q^{55} +45610.2 q^{57} +43288.0 q^{59} +10131.4 q^{61} -594.285 q^{63} +6976.49 q^{65} -8261.09 q^{67} +58013.7 q^{69} -23180.5 q^{71} -584.558 q^{73} +59954.2 q^{75} -1408.72 q^{77} +98385.3 q^{79} -73220.8 q^{81} +58501.6 q^{83} +834.195 q^{85} +54226.9 q^{87} -51116.6 q^{89} -3095.10 q^{91} -138652. q^{93} -21252.2 q^{95} -169271. q^{97} +50389.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−19.7180	−1.26491	−0.632454	−	0.774598i	$-0.717952\pi$
$3$	−19.7180	−1.26491	−0.632454	+	0.774598i	$0.717952\pi$
$4$	0	0
$5$	9.18763	0.164353	0.0821767	−	0.996618i	$-0.473813\pi$
$5$	9.18763	0.164353	0.0821767	+	0.996618i	$0.473813\pi$
$6$	0	0
$7$	−4.07607	−0.0314410	−0.0157205	−	0.999876i	$-0.505004\pi$
$7$	−4.07607	−0.0314410	−0.0157205	+	0.999876i	$0.505004\pi$
$8$	0	0
$9$	145.799	0.599994
$10$	0	0
$11$	345.607	0.861194	0.430597	−	0.902544i	$-0.358303\pi$
$11$	345.607	0.861194	0.430597	+	0.902544i	$0.358303\pi$
$12$	0	0
$13$	759.335	1.24616	0.623082	−	0.782157i	$-0.285880\pi$
$13$	759.335	1.24616	0.623082	+	0.782157i	$0.285880\pi$
$14$	0	0
$15$	−181.161	−0.207892
$16$	0	0
$17$	90.7955	0.0761977	0.0380989	−	0.999274i	$-0.487870\pi$
$17$	90.7955	0.0761977	0.0380989	+	0.999274i	$0.487870\pi$
$18$	0	0
$19$	−2313.13	−1.46999	−0.734997	−	0.678070i	$-0.762817\pi$
$19$	−2313.13	−1.46999	−0.734997	+	0.678070i	$0.762817\pi$
$20$	0	0
$21$	80.3719	0.0397700
$22$	0	0
$23$	−2942.17	−1.15971	−0.579854	−	0.814720i	$-0.696890\pi$
$23$	−2942.17	−1.15971	−0.579854	+	0.814720i	$0.696890\pi$
$24$	0	0
$25$	−3040.59	−0.972988
$26$	0	0
$27$	1916.62	0.505971
$28$	0	0
$29$	−2750.12	−0.607236	−0.303618	−	0.952794i	$-0.598195\pi$
$29$	−2750.12	−0.607236	−0.303618	+	0.952794i	$0.598195\pi$
$30$	0	0
$31$	7031.77	1.31420	0.657099	−	0.753804i	$-0.271783\pi$
$31$	7031.77	1.31420	0.657099	+	0.753804i	$0.271783\pi$
$32$	0	0
$33$	−6814.67	−1.08933
$34$	0	0
$35$	−37.4494	−0.00516744
$36$	0	0
$37$	−2965.62	−0.356132	−0.178066	−	0.984019i	$-0.556984\pi$
$37$	−2965.62	−0.356132	−0.178066	+	0.984019i	$0.556984\pi$
$38$	0	0
$39$	−14972.5	−1.57628
$40$	0	0
$41$	3028.05	0.281322	0.140661	−	0.990058i	$-0.455077\pi$
$41$	3028.05	0.281322	0.140661	+	0.990058i	$0.455077\pi$
$42$	0	0
$43$	9839.21	0.811501	0.405751	−	0.913984i	$-0.367010\pi$
$43$	9839.21	0.811501	0.405751	+	0.913984i	$0.367010\pi$
$44$	0	0
$45$	1339.54	0.0986110
$46$	0	0
$47$	−11972.0	−0.790535	−0.395268	−	0.918566i	$-0.629348\pi$
$47$	−11972.0	−0.790535	−0.395268	+	0.918566i	$0.629348\pi$
$48$	0	0
$49$	−16790.4	−0.999011
$50$	0	0
$51$	−1790.30	−0.0963831
$52$	0	0
$53$	16491.9	0.806454	0.403227	−	0.915100i	$-0.367888\pi$
$53$	16491.9	0.806454	0.403227	+	0.915100i	$0.367888\pi$
$54$	0	0
$55$	3175.31	0.141540
$56$	0	0
$57$	45610.2	1.85941
$58$	0	0
$59$	43288.0	1.61896	0.809482	−	0.587144i	$-0.199748\pi$
$59$	43288.0	1.61896	0.809482	+	0.587144i	$0.199748\pi$
$60$	0	0
$61$	10131.4	0.348613	0.174307	−	0.984691i	$-0.444232\pi$
$61$	10131.4	0.348613	0.174307	+	0.984691i	$0.444232\pi$
$62$	0	0
$63$	−594.285	−0.0188644
$64$	0	0
$65$	6976.49	0.204811
$66$	0	0
$67$	−8261.09	−0.224828	−0.112414	−	0.993661i	$-0.535858\pi$
$67$	−8261.09	−0.224828	−0.112414	+	0.993661i	$0.535858\pi$
$68$	0	0
$69$	58013.7	1.46692
$70$	0	0
$71$	−23180.5	−0.545728	−0.272864	−	0.962053i	$-0.587971\pi$
$71$	−23180.5	−0.545728	−0.272864	+	0.962053i	$0.587971\pi$
$72$	0	0
$73$	−584.558	−0.0128387	−0.00641935	−	0.999979i	$-0.502043\pi$
$73$	−584.558	−0.0128387	−0.00641935	+	0.999979i	$0.502043\pi$
$74$	0	0
$75$	59954.2	1.23074
$76$	0	0
$77$	−1408.72	−0.0270768
$78$	0	0
$79$	98385.3	1.77363	0.886814	−	0.462127i	$-0.152914\pi$
$79$	98385.3	1.77363	0.886814	+	0.462127i	$0.152914\pi$
$80$	0	0
$81$	−73220.8	−1.24000
$82$	0	0
$83$	58501.6	0.932122	0.466061	−	0.884753i	$-0.345673\pi$
$83$	58501.6	0.932122	0.466061	+	0.884753i	$0.345673\pi$
$84$	0	0
$85$	834.195	0.0125233
$86$	0	0
$87$	54226.9	0.768098
$88$	0	0
$89$	−51116.6	−0.684049	−0.342024	−	0.939691i	$-0.611113\pi$
$89$	−51116.6	−0.684049	−0.342024	+	0.939691i	$0.611113\pi$
$90$	0	0
$91$	−3095.10	−0.0391807
$92$	0	0
$93$	−138652.	−1.66234
$94$	0	0
$95$	−21252.2	−0.241599
$96$	0	0
$97$	−169271.	−1.82664	−0.913320	−	0.407243i	$-0.866490\pi$
$97$	−169271.	−1.82664	−0.913320	+	0.407243i	$0.866490\pi$
$98$	0	0
$99$	50389.0	0.516711
$100$	0	0
$101$	−28454.0	−0.277549	−0.138775	−	0.990324i	$-0.544316\pi$
$101$	−28454.0	−0.277549	−0.138775	+	0.990324i	$0.544316\pi$
$102$	0	0
$103$	80300.5	0.745805	0.372903	−	0.927870i	$-0.378363\pi$
$103$	80300.5	0.745805	0.372903	+	0.927870i	$0.378363\pi$
$104$	0	0
$105$	738.427	0.00653633
$106$	0	0
$107$	71078.4	0.600176	0.300088	−	0.953912i	$-0.402984\pi$
$107$	71078.4	0.600176	0.300088	+	0.953912i	$0.402984\pi$
$108$	0	0
$109$	185564.	1.49598	0.747992	−	0.663707i	$-0.231018\pi$
$109$	185564.	1.49598	0.747992	+	0.663707i	$0.231018\pi$
$110$	0	0
$111$	58476.0	0.450474
$112$	0	0
$113$	−22668.1	−0.167001	−0.0835005	−	0.996508i	$-0.526610\pi$
$113$	−22668.1	−0.167001	−0.0835005	+	0.996508i	$0.526610\pi$
$114$	0	0
$115$	−27031.6	−0.190602
$116$	0	0
$117$	110710.	0.747691
$118$	0	0
$119$	−370.089	−0.00239573
$120$	0	0
$121$	−41606.8	−0.258346
$122$	0	0
$123$	−59707.0	−0.355846
$124$	0	0
$125$	−56647.1	−0.324267
$126$	0	0
$127$	−280058.	−1.54077	−0.770386	−	0.637578i	$-0.779936\pi$
$127$	−280058.	−1.54077	−0.770386	+	0.637578i	$0.779936\pi$
$128$	0	0
$129$	−194009.	−1.02648
$130$	0	0
$131$	−50896.8	−0.259127	−0.129563	−	0.991571i	$-0.541358\pi$
$131$	−50896.8	−0.259127	−0.129563	+	0.991571i	$0.541358\pi$
$132$	0	0
$133$	9428.48	0.0462181
$134$	0	0
$135$	17609.2	0.0831580
$136$	0	0
$137$	424029.	1.93017	0.965083	−	0.261946i	$-0.0843641\pi$
$137$	424029.	1.93017	0.965083	+	0.261946i	$0.0843641\pi$
$138$	0	0
$139$	−135970.	−0.596908	−0.298454	−	0.954424i	$-0.596471\pi$
$139$	−135970.	−0.596908	−0.298454	+	0.954424i	$0.596471\pi$
$140$	0	0
$141$	236063.	0.999955
$142$	0	0
$143$	262432.	1.07319
$144$	0	0
$145$	−25267.1	−0.0998012
$146$	0	0
$147$	331072.	1.26366
$148$	0	0
$149$	−60704.1	−0.224002	−0.112001	−	0.993708i	$-0.535726\pi$
$149$	−60704.1	−0.224002	−0.112001	+	0.993708i	$0.535726\pi$
$150$	0	0
$151$	−21582.8	−0.0770309	−0.0385154	−	0.999258i	$-0.512263\pi$
$151$	−21582.8	−0.0770309	−0.0385154	+	0.999258i	$0.512263\pi$
$152$	0	0
$153$	13237.8	0.0457182
$154$	0	0
$155$	64605.3	0.215993
$156$	0	0
$157$	−1158.56	−0.00375119	−0.00187559	−	0.999998i	$-0.500597\pi$
$157$	−1158.56	−0.00375119	−0.00187559	+	0.999998i	$0.500597\pi$
$158$	0	0
$159$	−325186.	−1.02009
$160$	0	0
$161$	11992.5	0.0364624
$162$	0	0
$163$	50278.9	0.148223	0.0741117	−	0.997250i	$-0.476388\pi$
$163$	50278.9	0.148223	0.0741117	+	0.997250i	$0.476388\pi$
$164$	0	0
$165$	−62610.7	−0.179035
$166$	0	0
$167$	72836.7	0.202097	0.101048	−	0.994882i	$-0.467780\pi$
$167$	72836.7	0.202097	0.101048	+	0.994882i	$0.467780\pi$
$168$	0	0
$169$	205297.	0.552924
$170$	0	0
$171$	−337251.	−0.881988
$172$	0	0
$173$	−696633.	−1.76966	−0.884828	−	0.465917i	$-0.845725\pi$
$173$	−696633.	−1.76966	−0.884828	+	0.465917i	$0.845725\pi$
$174$	0	0
$175$	12393.7	0.0305917
$176$	0	0
$177$	−853552.	−2.04784
$178$	0	0
$179$	−288799.	−0.673694	−0.336847	−	0.941559i	$-0.609361\pi$
$179$	−288799.	−0.673694	−0.336847	+	0.941559i	$0.609361\pi$
$180$	0	0
$181$	815406.	1.85002	0.925012	−	0.379938i	$-0.124055\pi$
$181$	815406.	1.85002	0.925012	+	0.379938i	$0.124055\pi$
$182$	0	0
$183$	−199770.	−0.440964
$184$	0	0
$185$	−27247.0	−0.0585315
$186$	0	0
$187$	31379.5	0.0656210
$188$	0	0
$189$	−7812.26	−0.0159083
$190$	0	0
$191$	276983.	0.549377	0.274688	−	0.961533i	$-0.411425\pi$
$191$	276983.	0.549377	0.274688	+	0.961533i	$0.411425\pi$
$192$	0	0
$193$	941208.	1.81883	0.909416	−	0.415888i	$-0.136529\pi$
$193$	941208.	1.81883	0.909416	+	0.415888i	$0.136529\pi$
$194$	0	0
$195$	−137562.	−0.259067
$196$	0	0
$197$	−561712.	−1.03121	−0.515606	−	0.856826i	$-0.672433\pi$
$197$	−561712.	−1.03121	−0.515606	+	0.856826i	$0.672433\pi$
$198$	0	0
$199$	−836555.	−1.49748	−0.748742	−	0.662862i	$-0.769342\pi$
$199$	−836555.	−1.49748	−0.748742	+	0.662862i	$0.769342\pi$
$200$	0	0
$201$	162892.	0.284387
$202$	0	0
$203$	11209.7	0.0190921
$204$	0	0
$205$	27820.6	0.0462362
$206$	0	0
$207$	−428964.	−0.695818
$208$	0	0
$209$	−799434.	−1.26595
$210$	0	0
$211$	718534.	1.11107	0.555535	−	0.831493i	$-0.312514\pi$
$211$	718534.	1.11107	0.555535	+	0.831493i	$0.312514\pi$
$212$	0	0
$213$	457072.	0.690296
$214$	0	0
$215$	90399.1	0.133373
$216$	0	0
$217$	−28662.0	−0.0413197
$218$	0	0
$219$	11526.3	0.0162398
$220$	0	0
$221$	68944.2	0.0949548
$222$	0	0
$223$	−351581.	−0.473438	−0.236719	−	0.971578i	$-0.576072\pi$
$223$	−351581.	−0.473438	−0.236719	+	0.971578i	$0.576072\pi$
$224$	0	0
$225$	−443313.	−0.583787
$226$	0	0
$227$	−646299.	−0.832470	−0.416235	−	0.909257i	$-0.636651\pi$
$227$	−646299.	−0.832470	−0.416235	+	0.909257i	$0.636651\pi$
$228$	0	0
$229$	−749600.	−0.944585	−0.472292	−	0.881442i	$-0.656573\pi$
$229$	−749600.	−0.944585	−0.472292	+	0.881442i	$0.656573\pi$
$230$	0	0
$231$	27777.1	0.0342497
$232$	0	0
$233$	744054.	0.897872	0.448936	−	0.893564i	$-0.351803\pi$
$233$	744054.	0.897872	0.448936	+	0.893564i	$0.351803\pi$
$234$	0	0
$235$	−109994.	−0.129927
$236$	0	0
$237$	−1.93996e6	−2.24348
$238$	0	0
$239$	−1.08671e6	−1.23061	−0.615305	−	0.788289i	$-0.710967\pi$
$239$	−1.08671e6	−1.23061	−0.615305	+	0.788289i	$0.710967\pi$
$240$	0	0
$241$	−1.04595e6	−1.16002	−0.580011	−	0.814609i	$-0.696952\pi$
$241$	−1.04595e6	−1.16002	−0.580011	+	0.814609i	$0.696952\pi$
$242$	0	0
$243$	978029.	1.06252
$244$	0	0
$245$	−154264.	−0.164191
$246$	0	0
$247$	−1.75644e6	−1.83185
$248$	0	0
$249$	−1.15353e6	−1.17905
$250$	0	0
$251$	−1.12208e6	−1.12419	−0.562097	−	0.827072i	$-0.690005\pi$
$251$	−1.12208e6	−1.12419	−0.562097	+	0.827072i	$0.690005\pi$
$252$	0	0
$253$	−1.01684e6	−0.998733
$254$	0	0
$255$	−16448.6	−0.0158409
$256$	0	0
$257$	66049.0	0.0623783
$257$	66049.0	0.0623783
$258$	0	0
$259$	12088.1	0.0111972
$260$	0	0
$261$	−400964.	−0.364338
$262$	0	0
$263$	1.72731e6	1.53986	0.769928	−	0.638131i	$-0.220292\pi$
$263$	1.72731e6	1.53986	0.769928	+	0.638131i	$0.220292\pi$
$264$	0	0
$265$	151521.	0.132543
$266$	0	0
$267$	1.00792e6	0.865259
$268$	0	0
$269$	−2.36117e6	−1.98951	−0.994756	−	0.102276i	$-0.967387\pi$
$269$	−2.36117e6	−1.98951	−0.994756	+	0.102276i	$0.967387\pi$
$270$	0	0
$271$	−121396.	−0.100411	−0.0502056	−	0.998739i	$-0.515988\pi$
$271$	−121396.	−0.100411	−0.0502056	+	0.998739i	$0.515988\pi$
$272$	0	0
$273$	61029.2	0.0495600
$274$	0	0
$275$	−1.05085e6	−0.837931
$276$	0	0
$277$	6685.77	0.00523542	0.00261771	−	0.999997i	$-0.499167\pi$
$277$	6685.77	0.00523542	0.00261771	+	0.999997i	$0.499167\pi$
$278$	0	0
$279$	1.02522e6	0.788511
$280$	0	0
$281$	−1.46554e6	−1.10721	−0.553606	−	0.832779i	$-0.686749\pi$
$281$	−1.46554e6	−1.10721	−0.553606	+	0.832779i	$0.686749\pi$
$282$	0	0
$283$	−25079.7	−0.0186147	−0.00930735	−	0.999957i	$-0.502963\pi$
$283$	−25079.7	−0.0186147	−0.00930735	+	0.999957i	$0.502963\pi$
$284$	0	0
$285$	419050.	0.305600
$286$	0	0
$287$	−12342.5	−0.00884504
$288$	0	0
$289$	−1.41161e6	−0.994194
$290$	0	0
$291$	3.33768e6	2.31053
$292$	0	0
$293$	−1.34695e6	−0.916607	−0.458304	−	0.888796i	$-0.651543\pi$
$293$	−1.34695e6	−0.916607	−0.458304	+	0.888796i	$0.651543\pi$
$294$	0	0
$295$	397714.	0.266082
$296$	0	0
$297$	662396.	0.435739
$298$	0	0
$299$	−2.23410e6	−1.44519
$300$	0	0
$301$	−40105.3	−0.0255144
$302$	0	0
$303$	561055.	0.351074
$304$	0	0
$305$	93083.3	0.0572957
$306$	0	0
$307$	367582.	0.222591	0.111296	−	0.993787i	$-0.464500\pi$
$307$	367582.	0.222591	0.111296	+	0.993787i	$0.464500\pi$
$308$	0	0
$309$	−1.58336e6	−0.943376
$310$	0	0
$311$	1.29187e6	0.757385	0.378693	−	0.925522i	$-0.376374\pi$
$311$	1.29187e6	0.757385	0.378693	+	0.925522i	$0.376374\pi$
$312$	0	0
$313$	−1.81446e6	−1.04685	−0.523427	−	0.852070i	$-0.675347\pi$
$313$	−1.81446e6	−1.04685	−0.523427	+	0.852070i	$0.675347\pi$
$314$	0	0
$315$	−5460.07	−0.00310043
$316$	0	0
$317$	−662693.	−0.370394	−0.185197	−	0.982701i	$-0.559292\pi$
$317$	−662693.	−0.370394	−0.185197	+	0.982701i	$0.559292\pi$
$318$	0	0
$319$	−950462.	−0.522947
$320$	0	0
$321$	−1.40152e6	−0.759167
$322$	0	0
$323$	−210022.	−0.112010
$324$	0	0
$325$	−2.30882e6	−1.21250
$326$	0	0
$327$	−3.65894e6	−1.89228
$328$	0	0
$329$	48798.6	0.0248552
$330$	0	0
$331$	−2.12155e6	−1.06435	−0.532174	−	0.846635i	$-0.678625\pi$
$331$	−2.12155e6	−1.06435	−0.532174	+	0.846635i	$0.678625\pi$
$332$	0	0
$333$	−432383.	−0.213677
$334$	0	0
$335$	−75899.8	−0.0369512
$336$	0	0
$337$	−1.72049e6	−0.825233	−0.412616	−	0.910905i	$-0.635385\pi$
$337$	−1.72049e6	−0.825233	−0.412616	+	0.910905i	$0.635385\pi$
$338$	0	0
$339$	446969.	0.211241
$340$	0	0
$341$	2.43023e6	1.13178
$342$	0	0
$343$	136945.	0.0628510
$344$	0	0
$345$	533008.	0.241094
$346$	0	0
$347$	682188.	0.304145	0.152072	−	0.988369i	$-0.451405\pi$
$347$	682188.	0.304145	0.152072	+	0.988369i	$0.451405\pi$
$348$	0	0
$349$	2.73987e6	1.20411	0.602054	−	0.798455i	$-0.294349\pi$
$349$	2.73987e6	1.20411	0.602054	+	0.798455i	$0.294349\pi$
$350$	0	0
$351$	1.45535e6	0.630523
$352$	0	0
$353$	−2.97047e6	−1.26879	−0.634393	−	0.773010i	$-0.718750\pi$
$353$	−2.97047e6	−1.26879	−0.634393	+	0.773010i	$0.718750\pi$
$354$	0	0
$355$	−212974.	−0.0896922
$356$	0	0
$357$	7297.40	0.00303038
$358$	0	0
$359$	1.75885e6	0.720267	0.360133	−	0.932901i	$-0.382731\pi$
$359$	1.75885e6	0.720267	0.360133	+	0.932901i	$0.382731\pi$
$360$	0	0
$361$	2.87447e6	1.16089
$362$	0	0
$363$	820402.	0.326784
$364$	0	0
$365$	−5370.71	−0.00211008
$366$	0	0
$367$	−4.19905e6	−1.62737	−0.813684	−	0.581307i	$-0.802541\pi$
$367$	−4.19905e6	−1.62737	−0.813684	+	0.581307i	$0.802541\pi$
$368$	0	0
$369$	441485.	0.168791
$370$	0	0
$371$	−67222.0	−0.0253557
$372$	0	0
$373$	−4.60150e6	−1.71249	−0.856244	−	0.516571i	$-0.827208\pi$
$373$	−4.60150e6	−1.71249	−0.856244	+	0.516571i	$0.827208\pi$
$374$	0	0
$375$	1.11697e6	0.410168
$376$	0	0
$377$	−2.08827e6	−0.756715
$378$	0	0
$379$	−3.18271e6	−1.13815	−0.569074	−	0.822287i	$-0.692698\pi$
$379$	−3.18271e6	−1.13815	−0.569074	+	0.822287i	$0.692698\pi$
$380$	0	0
$381$	5.52217e6	1.94894
$382$	0	0
$383$	−589297.	−0.205276	−0.102638	−	0.994719i	$-0.532728\pi$
$383$	−589297.	−0.205276	−0.102638	+	0.994719i	$0.532728\pi$
$384$	0	0
$385$	−12942.8	−0.00445016
$386$	0	0
$387$	1.43454e6	0.486896
$388$	0	0
$389$	2.71593e6	0.910005	0.455003	−	0.890490i	$-0.349638\pi$
$389$	2.71593e6	0.910005	0.455003	+	0.890490i	$0.349638\pi$
$390$	0	0
$391$	−267136.	−0.0883671
$392$	0	0
$393$	1.00358e6	0.327772
$394$	0	0
$395$	903927.	0.291501
$396$	0	0
$397$	−3.04186e6	−0.968643	−0.484321	−	0.874890i	$-0.660933\pi$
$397$	−3.04186e6	−0.968643	−0.484321	+	0.874890i	$0.660933\pi$
$398$	0	0
$399$	−185910.	−0.0584617
$400$	0	0
$401$	−4.81204e6	−1.49440	−0.747202	−	0.664597i	$-0.768603\pi$
$401$	−4.81204e6	−1.49440	−0.747202	+	0.664597i	$0.768603\pi$
$402$	0	0
$403$	5.33947e6	1.63771
$404$	0	0
$405$	−672726.	−0.203798
$406$	0	0
$407$	−1.02494e6	−0.306699
$408$	0	0
$409$	2.98699e6	0.882928	0.441464	−	0.897279i	$-0.354459\pi$
$409$	2.98699e6	0.882928	0.441464	+	0.897279i	$0.354459\pi$
$410$	0	0
$411$	−8.36100e6	−2.44148
$412$	0	0
$413$	−176445.	−0.0509019
$414$	0	0
$415$	537491.	0.153197
$416$	0	0
$417$	2.68106e6	0.755034
$418$	0	0
$419$	5.38001e6	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$419$	5.38001e6	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$420$	0	0
$421$	−98921.6	−0.0272011	−0.0136005	−	0.999908i	$-0.504329\pi$
$421$	−98921.6	−0.0272011	−0.0136005	+	0.999908i	$0.504329\pi$
$422$	0	0
$423$	−1.74550e6	−0.474316
$424$	0	0
$425$	−276072.	−0.0741395
$426$	0	0
$427$	−41296.2	−0.0109608
$428$	0	0
$429$	−5.17462e6	−1.35748
$430$	0	0
$431$	3.25179e6	0.843197	0.421598	−	0.906783i	$-0.361469\pi$
$431$	3.25179e6	0.843197	0.421598	+	0.906783i	$0.361469\pi$
$432$	0	0
$433$	3.76480e6	0.964988	0.482494	−	0.875899i	$-0.339731\pi$
$433$	3.76480e6	0.964988	0.482494	+	0.875899i	$0.339731\pi$
$434$	0	0
$435$	498216.	0.126239
$436$	0	0
$437$	6.80563e6	1.70476
$438$	0	0
$439$	1.06673e6	0.264175	0.132087	−	0.991238i	$-0.457832\pi$
$439$	1.06673e6	0.264175	0.132087	+	0.991238i	$0.457832\pi$
$440$	0	0
$441$	−2.44801e6	−0.599401
$442$	0	0
$443$	839643.	0.203276	0.101638	−	0.994821i	$-0.467592\pi$
$443$	839643.	0.203276	0.101638	+	0.994821i	$0.467592\pi$
$444$	0	0
$445$	−469641.	−0.112426
$446$	0	0
$447$	1.19696e6	0.283342
$448$	0	0
$449$	−7.55492e6	−1.76854	−0.884268	−	0.466980i	$-0.845342\pi$
$449$	−7.55492e6	−1.76854	−0.884268	+	0.466980i	$0.845342\pi$
$450$	0	0
$451$	1.04652e6	0.242273
$452$	0	0
$453$	425568.	0.0974370
$454$	0	0
$455$	−28436.7	−0.00643947
$456$	0	0
$457$	444098.	0.0994692	0.0497346	−	0.998762i	$-0.484162\pi$
$457$	444098.	0.0994692	0.0497346	+	0.998762i	$0.484162\pi$
$458$	0	0
$459$	174020.	0.0385538
$460$	0	0
$461$	−1.55870e6	−0.341594	−0.170797	−	0.985306i	$-0.554634\pi$
$461$	−1.55870e6	−0.341594	−0.170797	+	0.985306i	$0.554634\pi$
$462$	0	0
$463$	6.25381e6	1.35579	0.677894	−	0.735159i	$-0.262893\pi$
$463$	6.25381e6	1.35579	0.677894	+	0.735159i	$0.262893\pi$
$464$	0	0
$465$	−1.27389e6	−0.273211
$466$	0	0
$467$	−404774.	−0.0858855	−0.0429428	−	0.999078i	$-0.513673\pi$
$467$	−404774.	−0.0858855	−0.0429428	+	0.999078i	$0.513673\pi$
$468$	0	0
$469$	33672.8	0.00706882
$470$	0	0
$471$	22844.4	0.00474491
$472$	0	0
$473$	3.40050e6	0.698860
$474$	0	0
$475$	7.03327e6	1.43029
$476$	0	0
$477$	2.40449e6	0.483868
$478$	0	0
$479$	2.85151e6	0.567854	0.283927	−	0.958846i	$-0.408363\pi$
$479$	2.85151e6	0.567854	0.283927	+	0.958846i	$0.408363\pi$
$480$	0	0
$481$	−2.25190e6	−0.443799
$482$	0	0
$483$	−236468.	−0.0461216
$484$	0	0
$485$	−1.55520e6	−0.300214
$486$	0	0
$487$	−799293.	−0.152716	−0.0763578	−	0.997080i	$-0.524329\pi$
$487$	−799293.	−0.152716	−0.0763578	+	0.997080i	$0.524329\pi$
$488$	0	0
$489$	−991398.	−0.187489
$490$	0	0
$491$	−3.75114e6	−0.702198	−0.351099	−	0.936338i	$-0.614192\pi$
$491$	−3.75114e6	−0.702198	−0.351099	+	0.936338i	$0.614192\pi$
$492$	0	0
$493$	−249699.	−0.0462700
$494$	0	0
$495$	462955.	0.0849232
$496$	0	0
$497$	94485.2	0.0171583
$498$	0	0
$499$	7.67383e6	1.37962	0.689812	−	0.723988i	$-0.257693\pi$
$499$	7.67383e6	1.37962	0.689812	+	0.723988i	$0.257693\pi$
$500$	0	0
$501$	−1.43619e6	−0.255634
$502$	0	0
$503$	−5.38187e6	−0.948448	−0.474224	−	0.880404i	$-0.657271\pi$
$503$	−5.38187e6	−0.948448	−0.474224	+	0.880404i	$0.657271\pi$
$504$	0	0
$505$	−261425.	−0.0456161
$506$	0	0
$507$	−4.04804e6	−0.699398
$508$	0	0
$509$	503974.	0.0862212	0.0431106	−	0.999070i	$-0.486273\pi$
$509$	503974.	0.0862212	0.0431106	+	0.999070i	$0.486273\pi$
$510$	0	0
$511$	2382.70	0.000403662 0
$512$	0	0
$513$	−4.43338e6	−0.743775
$514$	0	0
$515$	737772.	0.122576
$516$	0	0
$517$	−4.13760e6	−0.680804
$518$	0	0
$519$	1.37362e7	2.23845
$520$	0	0
$521$	−9.57725e6	−1.54578	−0.772888	−	0.634543i	$-0.781189\pi$
$521$	−9.57725e6	−1.54578	−0.772888	+	0.634543i	$0.781189\pi$
$522$	0	0
$523$	−5.45583e6	−0.872181	−0.436091	−	0.899903i	$-0.643637\pi$
$523$	−5.45583e6	−0.872181	−0.436091	+	0.899903i	$0.643637\pi$
$524$	0	0
$525$	−244378.	−0.0386958
$526$	0	0
$527$	638453.	0.100139
$528$	0	0
$529$	2.22004e6	0.344922
$530$	0	0
$531$	6.31132e6	0.971369
$532$	0	0
$533$	2.29930e6	0.350573
$534$	0	0
$535$	653042.	0.0986408
$536$	0	0
$537$	5.69453e6	0.852161
$538$	0	0
$539$	−5.80287e6	−0.860342
$540$	0	0
$541$	−3.72669e6	−0.547432	−0.273716	−	0.961811i	$-0.588253\pi$
$541$	−3.72669e6	−0.547432	−0.273716	+	0.961811i	$0.588253\pi$
$542$	0	0
$543$	−1.60782e7	−2.34011
$544$	0	0
$545$	1.70489e6	0.245870
$546$	0	0
$547$	6.94475e6	0.992403	0.496202	−	0.868207i	$-0.334728\pi$
$547$	6.94475e6	0.992403	0.496202	+	0.868207i	$0.334728\pi$
$548$	0	0
$549$	1.47714e6	0.209166
$550$	0	0
$551$	6.36139e6	0.892633
$552$	0	0
$553$	−401025.	−0.0557646
$554$	0	0
$555$	537256.	0.0740370
$556$	0	0
$557$	9.94529e6	1.35825	0.679124	−	0.734023i	$-0.262360\pi$
$557$	9.94529e6	1.35825	0.679124	+	0.734023i	$0.262360\pi$
$558$	0	0
$559$	7.47126e6	1.01126
$560$	0	0
$561$	−618741.	−0.0830045
$562$	0	0
$563$	1.01195e7	1.34552	0.672758	−	0.739863i	$-0.265110\pi$
$563$	1.01195e7	1.34552	0.672758	+	0.739863i	$0.265110\pi$
$564$	0	0
$565$	−208266.	−0.0274472
$566$	0	0
$567$	298453.	0.0389869
$568$	0	0
$569$	−1.35542e7	−1.75507	−0.877533	−	0.479516i	$-0.840812\pi$
$569$	−1.35542e7	−1.75507	−0.877533	+	0.479516i	$0.840812\pi$
$570$	0	0
$571$	696832.	0.0894413	0.0447206	−	0.999000i	$-0.485760\pi$
$571$	696832.	0.0894413	0.0447206	+	0.999000i	$0.485760\pi$
$572$	0	0
$573$	−5.46155e6	−0.694911
$574$	0	0
$575$	8.94593e6	1.12838
$576$	0	0
$577$	−1.04161e7	−1.30246	−0.651232	−	0.758879i	$-0.725747\pi$
$577$	−1.04161e7	−1.30246	−0.651232	+	0.758879i	$0.725747\pi$
$578$	0	0
$579$	−1.85587e7	−2.30066
$580$	0	0
$581$	−238457.	−0.0293069
$582$	0	0
$583$	5.69970e6	0.694513
$584$	0	0
$585$	1.01716e6	0.122885
$586$	0	0
$587$	7.49480e6	0.897769	0.448885	−	0.893590i	$-0.351821\pi$
$587$	7.49480e6	0.897769	0.448885	+	0.893590i	$0.351821\pi$
$588$	0	0
$589$	−1.62654e7	−1.93186
$590$	0	0
$591$	1.10758e7	1.30439
$592$	0	0
$593$	1.55582e7	1.81686	0.908432	−	0.418033i	$-0.137280\pi$
$593$	1.55582e7	1.81686	0.908432	+	0.418033i	$0.137280\pi$
$594$	0	0
$595$	−3400.24	−0.000393747 0
$596$	0	0
$597$	1.64952e7	1.89418
$598$	0	0
$599$	−1.31160e7	−1.49360	−0.746798	−	0.665051i	$-0.768410\pi$
$599$	−1.31160e7	−1.49360	−0.746798	+	0.665051i	$0.768410\pi$
$600$	0	0
$601$	3.99621e6	0.451296	0.225648	−	0.974209i	$-0.427550\pi$
$601$	3.99621e6	0.451296	0.225648	+	0.974209i	$0.427550\pi$
$602$	0	0
$603$	−1.20445e6	−0.134895
$604$	0	0
$605$	−382268.	−0.0424599
$606$	0	0
$607$	−1.54856e7	−1.70591	−0.852956	−	0.521983i	$-0.825193\pi$
$607$	−1.54856e7	−1.70591	−0.852956	+	0.521983i	$0.825193\pi$
$608$	0	0
$609$	−221033.	−0.0241498
$610$	0	0
$611$	−9.09074e6	−0.985136
$612$	0	0
$613$	1.64396e6	0.176702	0.0883508	−	0.996089i	$-0.471840\pi$
$613$	1.64396e6	0.176702	0.0883508	+	0.996089i	$0.471840\pi$
$614$	0	0
$615$	−548566.	−0.0584845
$616$	0	0
$617$	−5.96503e6	−0.630812	−0.315406	−	0.948957i	$-0.602141\pi$
$617$	−5.96503e6	−0.630812	−0.315406	+	0.948957i	$0.602141\pi$
$618$	0	0
$619$	9.44065e6	0.990320	0.495160	−	0.868802i	$-0.335110\pi$
$619$	9.44065e6	0.990320	0.495160	+	0.868802i	$0.335110\pi$
$620$	0	0
$621$	−5.63902e6	−0.586779
$622$	0	0
$623$	208355.	0.0215072
$624$	0	0
$625$	8.98138e6	0.919694
$626$	0	0
$627$	1.57632e7	1.60131
$628$	0	0
$629$	−269265.	−0.0271364
$630$	0	0
$631$	9.10439e6	0.910285	0.455143	−	0.890419i	$-0.349588\pi$
$631$	9.10439e6	0.910285	0.455143	+	0.890419i	$0.349588\pi$
$632$	0	0
$633$	−1.41680e7	−1.40540
$634$	0	0
$635$	−2.57307e6	−0.253231
$636$	0	0
$637$	−1.27495e7	−1.24493
$638$	0	0
$639$	−3.37968e6	−0.327434
$640$	0	0
$641$	−7.34169e6	−0.705750	−0.352875	−	0.935670i	$-0.614796\pi$
$641$	−7.34169e6	−0.705750	−0.352875	+	0.935670i	$0.614796\pi$
$642$	0	0
$643$	−4.13298e6	−0.394217	−0.197109	−	0.980382i	$-0.563155\pi$
$643$	−4.13298e6	−0.394217	−0.197109	+	0.980382i	$0.563155\pi$
$644$	0	0
$645$	−1.78249e6	−0.168705
$646$	0	0
$647$	−1.60011e7	−1.50276	−0.751379	−	0.659871i	$-0.770611\pi$
$647$	−1.60011e7	−1.50276	−0.751379	+	0.659871i	$0.770611\pi$
$648$	0	0
$649$	1.49606e7	1.39424
$650$	0	0
$651$	565157.	0.0522657
$652$	0	0
$653$	3.56121e6	0.326825	0.163412	−	0.986558i	$-0.447750\pi$
$653$	3.56121e6	0.326825	0.163412	+	0.986558i	$0.447750\pi$
$654$	0	0
$655$	−467621.	−0.0425883
$656$	0	0
$657$	−85227.7	−0.00770314
$658$	0	0
$659$	4.03789e6	0.362194	0.181097	−	0.983465i	$-0.442035\pi$
$659$	4.03789e6	0.362194	0.181097	+	0.983465i	$0.442035\pi$
$660$	0	0
$661$	−1.19166e7	−1.06084	−0.530419	−	0.847735i	$-0.677965\pi$
$661$	−1.19166e7	−1.06084	−0.530419	+	0.847735i	$0.677965\pi$
$662$	0	0
$663$	−1.35944e6	−0.120109
$664$	0	0
$665$	86625.4	0.00759610
$666$	0	0
$667$	8.09134e6	0.704216
$668$	0	0
$669$	6.93246e6	0.598855
$670$	0	0
$671$	3.50147e6	0.300223
$672$	0	0
$673$	−6.40767e6	−0.545334	−0.272667	−	0.962108i	$-0.587906\pi$
$673$	−6.40767e6	−0.545334	−0.272667	+	0.962108i	$0.587906\pi$
$674$	0	0
$675$	−5.82764e6	−0.492304
$676$	0	0
$677$	−1.17790e6	−0.0987729	−0.0493865	−	0.998780i	$-0.515727\pi$
$677$	−1.17790e6	−0.0987729	−0.0493865	+	0.998780i	$0.515727\pi$
$678$	0	0
$679$	689960.	0.0574314
$680$	0	0
$681$	1.27437e7	1.05300
$682$	0	0
$683$	−1.93241e7	−1.58507	−0.792533	−	0.609829i	$-0.791238\pi$
$683$	−1.93241e7	−1.58507	−0.792533	+	0.609829i	$0.791238\pi$
$684$	0	0
$685$	3.89582e6	0.317229
$686$	0	0
$687$	1.47806e7	1.19481
$688$	0	0
$689$	1.25228e7	1.00497
$690$	0	0
$691$	4.27154e6	0.340321	0.170161	−	0.985416i	$-0.445571\pi$
$691$	4.27154e6	0.340321	0.170161	+	0.985416i	$0.445571\pi$
$692$	0	0
$693$	−205389.	−0.0162459
$694$	0	0
$695$	−1.24925e6	−0.0981038
$696$	0	0
$697$	274933.	0.0214361
$698$	0	0
$699$	−1.46712e7	−1.13573
$700$	0	0
$701$	−632675.	−0.0486279	−0.0243140	−	0.999704i	$-0.507740\pi$
$701$	−632675.	−0.0486279	−0.0243140	+	0.999704i	$0.507740\pi$
$702$	0	0
$703$	6.85986e6	0.523512
$704$	0	0
$705$	2.16886e6	0.164346
$706$	0	0
$707$	115981.	0.00872643
$708$	0	0
$709$	−7.96075e6	−0.594755	−0.297378	−	0.954760i	$-0.596112\pi$
$709$	−7.96075e6	−0.594755	−0.297378	+	0.954760i	$0.596112\pi$
$710$	0	0
$711$	1.43444e7	1.06417
$712$	0	0
$713$	−2.06887e7	−1.52409
$714$	0	0
$715$	2.41112e6	0.176382
$716$	0	0
$717$	2.14278e7	1.55661
$718$	0	0
$719$	332748.	0.0240045	0.0120023	−	0.999928i	$-0.496179\pi$
$719$	332748.	0.0240045	0.0120023	+	0.999928i	$0.496179\pi$
$720$	0	0
$721$	−327311.	−0.0234489
$722$	0	0
$723$	2.06239e7	1.46732
$724$	0	0
$725$	8.36199e6	0.590833
$726$	0	0
$727$	4.50315e6	0.315995	0.157998	−	0.987440i	$-0.449496\pi$
$727$	4.50315e6	0.315995	0.157998	+	0.987440i	$0.449496\pi$
$728$	0	0
$729$	−1.49208e6	−0.103986
$730$	0	0
$731$	893356.	0.0618345
$732$	0	0
$733$	−3.41191e6	−0.234551	−0.117276	−	0.993099i	$-0.537416\pi$
$733$	−3.41191e6	−0.234551	−0.117276	+	0.993099i	$0.537416\pi$
$734$	0	0
$735$	3.04177e6	0.207686
$736$	0	0
$737$	−2.85509e6	−0.193620
$738$	0	0
$739$	6.80909e6	0.458646	0.229323	−	0.973350i	$-0.426349\pi$
$739$	6.80909e6	0.458646	0.229323	+	0.973350i	$0.426349\pi$
$740$	0	0
$741$	3.46334e7	2.31713
$742$	0	0
$743$	−1.66946e7	−1.10944	−0.554721	−	0.832037i	$-0.687175\pi$
$743$	−1.66946e7	−1.10944	−0.554721	+	0.832037i	$0.687175\pi$
$744$	0	0
$745$	−557727.	−0.0368155
$746$	0	0
$747$	8.52945e6	0.559268
$748$	0	0
$749$	−289721.	−0.0188701
$750$	0	0
$751$	−2.02864e7	−1.31252	−0.656258	−	0.754537i	$-0.727862\pi$
$751$	−2.02864e7	−1.31252	−0.656258	+	0.754537i	$0.727862\pi$
$752$	0	0
$753$	2.21252e7	1.42200
$754$	0	0
$755$	−198294.	−0.0126603
$756$	0	0
$757$	−1.42527e7	−0.903975	−0.451987	−	0.892024i	$-0.649285\pi$
$757$	−1.42527e7	−0.903975	−0.451987	+	0.892024i	$0.649285\pi$
$758$	0	0
$759$	2.00499e7	1.26331
$760$	0	0
$761$	9.48757e6	0.593873	0.296936	−	0.954897i	$-0.404035\pi$
$761$	9.48757e6	0.593873	0.296936	+	0.954897i	$0.404035\pi$
$762$	0	0
$763$	−756372.	−0.0470353
$764$	0	0
$765$	121624.	0.00751393
$766$	0	0
$767$	3.28701e7	2.01750
$768$	0	0
$769$	3.09080e6	0.188476	0.0942379	−	0.995550i	$-0.469959\pi$
$769$	3.09080e6	0.188476	0.0942379	+	0.995550i	$0.469959\pi$
$770$	0	0
$771$	−1.30235e6	−0.0789028
$772$	0	0
$773$	−2.31671e7	−1.39451	−0.697257	−	0.716821i	$-0.745596\pi$
$773$	−2.31671e7	−1.39451	−0.697257	+	0.716821i	$0.745596\pi$
$774$	0	0
$775$	−2.13807e7	−1.27870
$776$	0	0
$777$	−238352.	−0.0141634
$778$	0	0
$779$	−7.00427e6	−0.413542
$780$	0	0
$781$	−8.01133e6	−0.469978
$782$	0	0
$783$	−5.27093e6	−0.307244
$784$	0	0
$785$	−10644.4	−0.000616520 0
$786$	0	0
$787$	6.11513e6	0.351940	0.175970	−	0.984396i	$-0.443694\pi$
$787$	6.11513e6	0.351940	0.175970	+	0.984396i	$0.443694\pi$
$788$	0	0
$789$	−3.40590e7	−1.94778
$790$	0	0
$791$	92396.9	0.00525069
$792$	0	0
$793$	7.69311e6	0.434429
$794$	0	0
$795$	−2.98769e6	−0.167655
$796$	0	0
$797$	−8.48212e6	−0.472997	−0.236499	−	0.971632i	$-0.576000\pi$
$797$	−8.48212e6	−0.472997	−0.236499	+	0.971632i	$0.576000\pi$
$798$	0	0
$799$	−1.08700e6	−0.0602370
$800$	0	0
$801$	−7.45273e6	−0.410425
$802$	0	0
$803$	−202027.	−0.0110566
$804$	0	0
$805$	110183.	0.00599272
$806$	0	0
$807$	4.65575e7	2.51655
$808$	0	0
$809$	−2.08461e7	−1.11983	−0.559917	−	0.828549i	$-0.689167\pi$
$809$	−2.08461e7	−1.11983	−0.559917	+	0.828549i	$0.689167\pi$
$810$	0	0
$811$	1.58201e7	0.844610	0.422305	−	0.906454i	$-0.361221\pi$
$811$	1.58201e7	0.844610	0.422305	+	0.906454i	$0.361221\pi$
$812$	0	0
$813$	2.39369e6	0.127011
$814$	0	0
$815$	461944.	0.0243610
$816$	0	0
$817$	−2.27594e7	−1.19290
$818$	0	0
$819$	−451262.	−0.0235082
$820$	0	0
$821$	2.01938e7	1.04558	0.522792	−	0.852460i	$-0.324890\pi$
$821$	2.01938e7	1.04558	0.522792	+	0.852460i	$0.324890\pi$
$822$	0	0
$823$	3.72294e7	1.91596	0.957981	−	0.286831i	$-0.0926019\pi$
$823$	3.72294e7	1.91596	0.957981	+	0.286831i	$0.0926019\pi$
$824$	0	0
$825$	2.07206e7	1.05991
$826$	0	0
$827$	−9.94610e6	−0.505695	−0.252848	−	0.967506i	$-0.581367\pi$
$827$	−9.94610e6	−0.505695	−0.252848	+	0.967506i	$0.581367\pi$
$828$	0	0
$829$	−1.29088e7	−0.652378	−0.326189	−	0.945305i	$-0.605765\pi$
$829$	−1.29088e7	−0.652378	−0.326189	+	0.945305i	$0.605765\pi$
$830$	0	0
$831$	−131830.	−0.00662233
$832$	0	0
$833$	−1.52449e6	−0.0761224
$834$	0	0
$835$	669197.	0.0332153
$836$	0	0
$837$	1.34772e7	0.664946
$838$	0	0
$839$	1.02381e6	0.0502130	0.0251065	−	0.999685i	$-0.492008\pi$
$839$	1.02381e6	0.0502130	0.0251065	+	0.999685i	$0.492008\pi$
$840$	0	0
$841$	−1.29480e7	−0.631265
$842$	0	0
$843$	2.88974e7	1.40052
$844$	0	0
$845$	1.88619e6	0.0908749
$846$	0	0
$847$	169592.	0.00812265
$848$	0	0
$849$	494521.	0.0235459
$850$	0	0
$851$	8.72536e6	0.413009
$852$	0	0
$853$	−1.49268e7	−0.702414	−0.351207	−	0.936298i	$-0.614229\pi$
$853$	−1.49268e7	−0.702414	−0.351207	+	0.936298i	$0.614229\pi$
$854$	0	0
$855$	−3.09854e6	−0.144958
$856$	0	0
$857$	2.06704e7	0.961386	0.480693	−	0.876889i	$-0.340385\pi$
$857$	2.06704e7	0.961386	0.480693	+	0.876889i	$0.340385\pi$
$858$	0	0
$859$	−1.26701e7	−0.585866	−0.292933	−	0.956133i	$-0.594631\pi$
$859$	−1.26701e7	−0.585866	−0.292933	+	0.956133i	$0.594631\pi$
$860$	0	0
$861$	243370.	0.0111882
$862$	0	0
$863$	9.96218e6	0.455331	0.227666	−	0.973739i	$-0.426891\pi$
$863$	9.96218e6	0.455331	0.227666	+	0.973739i	$0.426891\pi$
$864$	0	0
$865$	−6.40041e6	−0.290849
$866$	0	0
$867$	2.78342e7	1.25756
$868$	0	0
$869$	3.40026e7	1.52744
$870$	0	0
$871$	−6.27293e6	−0.280172
$872$	0	0
$873$	−2.46794e7	−1.09597
$874$	0	0
$875$	230898.	0.0101953
$876$	0	0
$877$	−2.90469e7	−1.27526	−0.637632	−	0.770341i	$-0.720086\pi$
$877$	−2.90469e7	−1.27526	−0.637632	+	0.770341i	$0.720086\pi$
$878$	0	0
$879$	2.65592e7	1.15942
$880$	0	0
$881$	4.21597e7	1.83003	0.915013	−	0.403423i	$-0.132180\pi$
$881$	4.21597e7	1.83003	0.915013	+	0.403423i	$0.132180\pi$
$882$	0	0
$883$	−2.82895e7	−1.22102	−0.610511	−	0.792008i	$-0.709036\pi$
$883$	−2.82895e7	−1.22102	−0.610511	+	0.792008i	$0.709036\pi$
$884$	0	0
$885$	−7.84211e6	−0.336570
$886$	0	0
$887$	−4.10232e7	−1.75073	−0.875367	−	0.483458i	$-0.839381\pi$
$887$	−4.10232e7	−1.75073	−0.875367	+	0.483458i	$0.839381\pi$
$888$	0	0
$889$	1.14154e6	0.0484434
$890$	0	0
$891$	−2.53056e7	−1.06788
$892$	0	0
$893$	2.76927e7	1.16208
$894$	0	0
$895$	−2.65338e6	−0.110724
$896$	0	0
$897$	4.40518e7	1.82803
$898$	0	0
$899$	−1.93382e7	−0.798028
$900$	0	0
$901$	1.49739e6	0.0614500
$902$	0	0
$903$	790796.	0.0322734
$904$	0	0
$905$	7.49165e6	0.304058
$906$	0	0
$907$	−4.11413e7	−1.66058	−0.830290	−	0.557331i	$-0.811825\pi$
$907$	−4.11413e7	−1.66058	−0.830290	+	0.557331i	$0.811825\pi$
$908$	0	0
$909$	−4.14855e6	−0.166528
$910$	0	0
$911$	−1.59487e7	−0.636694	−0.318347	−	0.947974i	$-0.603128\pi$
$911$	−1.59487e7	−0.636694	−0.318347	+	0.947974i	$0.603128\pi$
$912$	0	0
$913$	2.02186e7	0.802738
$914$	0	0
$915$	−1.83541e6	−0.0724739
$916$	0	0
$917$	207459.	0.00814721
$918$	0	0
$919$	2.46075e7	0.961124	0.480562	−	0.876961i	$-0.340433\pi$
$919$	2.46075e7	0.961124	0.480562	+	0.876961i	$0.340433\pi$
$920$	0	0
$921$	−7.24797e6	−0.281558
$922$	0	0
$923$	−1.76017e7	−0.680067
$924$	0	0
$925$	9.01722e6	0.346512
$926$	0	0
$927$	1.17077e7	0.447479
$928$	0	0
$929$	2.15011e7	0.817375	0.408687	−	0.912674i	$-0.365987\pi$
$929$	2.15011e7	0.817375	0.408687	+	0.912674i	$0.365987\pi$
$930$	0	0
$931$	3.88383e7	1.46854
$932$	0	0
$933$	−2.54730e7	−0.958023
$934$	0	0
$935$	288304.	0.0107850
$936$	0	0
$937$	7.36218e6	0.273941	0.136971	−	0.990575i	$-0.456263\pi$
$937$	7.36218e6	0.273941	0.136971	+	0.990575i	$0.456263\pi$
$938$	0	0
$939$	3.57775e7	1.32418
$940$	0	0
$941$	3.10438e7	1.14288	0.571440	−	0.820644i	$-0.306385\pi$
$941$	3.10438e7	1.14288	0.571440	+	0.820644i	$0.306385\pi$
$942$	0	0
$943$	−8.90905e6	−0.326251
$944$	0	0
$945$	−71776.2	−0.00261457
$946$	0	0
$947$	3.66902e7	1.32946	0.664730	−	0.747084i	$-0.268547\pi$
$947$	3.66902e7	1.32946	0.664730	+	0.747084i	$0.268547\pi$
$948$	0	0
$949$	−443876.	−0.0159991
$950$	0	0
$951$	1.30670e7	0.468515
$952$	0	0
$953$	2.21916e7	0.791510	0.395755	−	0.918356i	$-0.370483\pi$
$953$	2.21916e7	0.791510	0.395755	+	0.918356i	$0.370483\pi$
$954$	0	0
$955$	2.54482e6	0.0902919
$956$	0	0
$957$	1.87412e7	0.661481
$958$	0	0
$959$	−1.72837e6	−0.0606864
$960$	0	0
$961$	2.08167e7	0.727116
$962$	0	0
$963$	1.03631e7	0.360102
$964$	0	0
$965$	8.64747e6	0.298931
$966$	0	0
$967$	−1.63614e7	−0.562671	−0.281335	−	0.959609i	$-0.590777\pi$
$967$	−1.63614e7	−0.562671	−0.281335	+	0.959609i	$0.590777\pi$
$968$	0	0
$969$	4.14120e6	0.141683
$970$	0	0
$971$	1.11524e7	0.379596	0.189798	−	0.981823i	$-0.439217\pi$
$971$	1.11524e7	0.379596	0.189798	+	0.981823i	$0.439217\pi$
$972$	0	0
$973$	554225.	0.0187674
$974$	0	0
$975$	4.55253e7	1.53370
$976$	0	0
$977$	−9.42556e6	−0.315915	−0.157958	−	0.987446i	$-0.550491\pi$
$977$	−9.42556e6	−0.315915	−0.157958	+	0.987446i	$0.550491\pi$
$978$	0	0
$979$	−1.76663e7	−0.589099
$980$	0	0
$981$	2.70549e7	0.897582
$982$	0	0
$983$	2.77662e7	0.916500	0.458250	−	0.888823i	$-0.348476\pi$
$983$	2.77662e7	0.916500	0.458250	+	0.888823i	$0.348476\pi$
$984$	0	0
$985$	−5.16080e6	−0.169483
$986$	0	0
$987$	−962210.	−0.0314396
$988$	0	0
$989$	−2.89487e7	−0.941105
$990$	0	0
$991$	4.06398e7	1.31452	0.657261	−	0.753663i	$-0.271715\pi$
$991$	4.06398e7	1.31452	0.657261	+	0.753663i	$0.271715\pi$
$992$	0	0
$993$	4.18327e7	1.34630
$994$	0	0
$995$	−7.68596e6	−0.246116
$996$	0	0
$997$	−2.27311e7	−0.724241	−0.362121	−	0.932131i	$-0.617947\pi$
$997$	−2.27311e7	−0.724241	−0.362121	+	0.932131i	$0.617947\pi$
$998$	0	0
$999$	−5.68395e6	−0.180192

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.a.1.10	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.a.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-19.7180 q^{3} +9.18763 q^{5} -4.07607 q^{7} +145.799 q^{9} +O(q^{10})\)	\(q-19.7180 q^{3} +9.18763 q^{5} -4.07607 q^{7} +145.799 q^{9} +345.607 q^{11} +759.335 q^{13} -181.161 q^{15} +90.7955 q^{17} -2313.13 q^{19} +80.3719 q^{21} -2942.17 q^{23} -3040.59 q^{25} +1916.62 q^{27} -2750.12 q^{29} +7031.77 q^{31} -6814.67 q^{33} -37.4494 q^{35} -2965.62 q^{37} -14972.5 q^{39} +3028.05 q^{41} +9839.21 q^{43} +1339.54 q^{45} -11972.0 q^{47} -16790.4 q^{49} -1790.30 q^{51} +16491.9 q^{53} +3175.31 q^{55} +45610.2 q^{57} +43288.0 q^{59} +10131.4 q^{61} -594.285 q^{63} +6976.49 q^{65} -8261.09 q^{67} +58013.7 q^{69} -23180.5 q^{71} -584.558 q^{73} +59954.2 q^{75} -1408.72 q^{77} +98385.3 q^{79} -73220.8 q^{81} +58501.6 q^{83} +834.195 q^{85} +54226.9 q^{87} -51116.6 q^{89} -3095.10 q^{91} -138652. q^{93} -21252.2 q^{95} -169271. q^{97} +50389.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q - 27 q^{3} - 89 q^{5} + 88 q^{7} + 3430 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more