LMFDB - Embedded newform 1072.6.a.k.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1072,6,Mod(1,1072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1072, 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("1072.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1072 = 2^{4} \cdot 67 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1072.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:	$171.931454840$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 536)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	1072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.0663 q^{3} -22.6189 q^{5} +84.1376 q^{7} +200.789 q^{9} +O(q^{10})$	$q+21.0663 q^{3} -22.6189 q^{5} +84.1376 q^{7} +200.789 q^{9} -121.343 q^{11} -253.820 q^{13} -476.496 q^{15} -1017.53 q^{17} +1212.33 q^{19} +1772.47 q^{21} +98.9621 q^{23} -2613.39 q^{25} -889.226 q^{27} +1043.26 q^{29} +6653.60 q^{31} -2556.25 q^{33} -1903.10 q^{35} -12573.2 q^{37} -5347.04 q^{39} -2483.81 q^{41} +22905.0 q^{43} -4541.63 q^{45} -2943.77 q^{47} -9727.87 q^{49} -21435.7 q^{51} -22561.0 q^{53} +2744.65 q^{55} +25539.3 q^{57} -28530.5 q^{59} -51621.4 q^{61} +16893.9 q^{63} +5741.12 q^{65} -4489.00 q^{67} +2084.77 q^{69} -16023.2 q^{71} +57154.5 q^{73} -55054.4 q^{75} -10209.5 q^{77} -74842.6 q^{79} -67524.5 q^{81} +101508. q^{83} +23015.5 q^{85} +21977.6 q^{87} -99191.5 q^{89} -21355.8 q^{91} +140167. q^{93} -27421.6 q^{95} +88199.7 q^{97} -24364.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 36 q^{3} + 25 q^{5} - 171 q^{7} + 1740 q^{9}+O(q^{10}) $ 22 * q - 36 * q^3 + 25 * q^5 - 171 * q^7 + 1740 * q^9	$ 22 q - 36 q^{3} + 25 q^{5} - 171 q^{7} + 1740 q^{9} - 581 q^{11} + 874 q^{13} - 883 q^{15} - 2407 q^{17} - 3187 q^{19} + 2616 q^{21} - 5587 q^{23} + 14095 q^{25} - 15390 q^{27} + 4190 q^{29} - 15482 q^{31} - 11223 q^{33} + 5741 q^{35} - 7673 q^{37} + 9694 q^{39} - 36347 q^{41} - 16710 q^{43} - 5324 q^{45} - 20064 q^{47} + 62665 q^{49} - 31094 q^{51} + 26795 q^{53} - 94740 q^{55} + 46573 q^{57} - 54226 q^{59} + 82744 q^{61} - 146336 q^{63} + 37575 q^{65} - 98758 q^{67} + 192484 q^{69} - 124063 q^{71} + 94583 q^{73} - 114498 q^{75} + 215979 q^{77} - 5851 q^{79} + 397938 q^{81} - 212613 q^{83} + 302469 q^{85} - 330727 q^{87} + 181565 q^{89} - 202361 q^{91} + 246191 q^{93} - 224203 q^{95} + 172896 q^{97} - 532665 q^{99}+O(q^{100}) $ 22 * q - 36 * q^3 + 25 * q^5 - 171 * q^7 + 1740 * q^9 - 581 * q^11 + 874 * q^13 - 883 * q^15 - 2407 * q^17 - 3187 * q^19 + 2616 * q^21 - 5587 * q^23 + 14095 * q^25 - 15390 * q^27 + 4190 * q^29 - 15482 * q^31 - 11223 * q^33 + 5741 * q^35 - 7673 * q^37 + 9694 * q^39 - 36347 * q^41 - 16710 * q^43 - 5324 * q^45 - 20064 * q^47 + 62665 * q^49 - 31094 * q^51 + 26795 * q^53 - 94740 * q^55 + 46573 * q^57 - 54226 * q^59 + 82744 * q^61 - 146336 * q^63 + 37575 * q^65 - 98758 * q^67 + 192484 * q^69 - 124063 * q^71 + 94583 * q^73 - 114498 * q^75 + 215979 * q^77 - 5851 * q^79 + 397938 * q^81 - 212613 * q^83 + 302469 * q^85 - 330727 * q^87 + 181565 * q^89 - 202361 * q^91 + 246191 * q^93 - 224203 * q^95 + 172896 * q^97 - 532665 * 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
$3$	21.0663	1.35140	0.675702	−	0.737175i	$-0.263841\pi$
$3$	21.0663	1.35140	0.675702	+	0.737175i	$0.263841\pi$
$4$	0	0
$5$	−22.6189	−0.404619	−0.202309	−	0.979322i	$-0.564845\pi$
$5$	−22.6189	−0.404619	−0.202309	+	0.979322i	$0.564845\pi$
$6$	0	0
$7$	84.1376	0.649000	0.324500	−	0.945886i	$-0.394804\pi$
$7$	84.1376	0.649000	0.324500	+	0.945886i	$0.394804\pi$
$8$	0	0
$9$	200.789	0.826293
$10$	0	0
$11$	−121.343	−0.302367	−0.151183	−	0.988506i	$-0.548308\pi$
$11$	−121.343	−0.302367	−0.151183	+	0.988506i	$0.548308\pi$
$12$	0	0
$13$	−253.820	−0.416550	−0.208275	−	0.978070i	$-0.566785\pi$
$13$	−253.820	−0.416550	−0.208275	+	0.978070i	$0.566785\pi$
$14$	0	0
$15$	−476.496	−0.546804
$16$	0	0
$17$	−1017.53	−0.853938	−0.426969	−	0.904266i	$-0.640419\pi$
$17$	−1017.53	−0.853938	−0.426969	+	0.904266i	$0.640419\pi$
$18$	0	0
$19$	1212.33	0.770437	0.385218	−	0.922825i	$-0.374126\pi$
$19$	1212.33	0.770437	0.385218	+	0.922825i	$0.374126\pi$
$20$	0	0
$21$	1772.47	0.877061
$22$	0	0
$23$	98.9621	0.0390076	0.0195038	−	0.999810i	$-0.493791\pi$
$23$	98.9621	0.0390076	0.0195038	+	0.999810i	$0.493791\pi$
$24$	0	0
$25$	−2613.39	−0.836284
$26$	0	0
$27$	−889.226	−0.234748
$28$	0	0
$29$	1043.26	0.230355	0.115177	−	0.993345i	$-0.463256\pi$
$29$	1043.26	0.230355	0.115177	+	0.993345i	$0.463256\pi$
$30$	0	0
$31$	6653.60	1.24352	0.621759	−	0.783208i	$-0.286418\pi$
$31$	6653.60	1.24352	0.621759	+	0.783208i	$0.286418\pi$
$32$	0	0
$33$	−2556.25	−0.408620
$34$	0	0
$35$	−1903.10	−0.262598
$36$	0	0
$37$	−12573.2	−1.50987	−0.754936	−	0.655799i	$-0.772332\pi$
$37$	−12573.2	−1.50987	−0.754936	+	0.655799i	$0.772332\pi$
$38$	0	0
$39$	−5347.04	−0.562927
$40$	0	0
$41$	−2483.81	−0.230759	−0.115380	−	0.993321i	$-0.536808\pi$
$41$	−2483.81	−0.230759	−0.115380	+	0.993321i	$0.536808\pi$
$42$	0	0
$43$	22905.0	1.88912	0.944559	−	0.328342i	$-0.106490\pi$
$43$	22905.0	1.88912	0.944559	+	0.328342i	$0.106490\pi$
$44$	0	0
$45$	−4541.63	−0.334334
$46$	0	0
$47$	−2943.77	−0.194383	−0.0971916	−	0.995266i	$-0.530986\pi$
$47$	−2943.77	−0.194383	−0.0971916	+	0.995266i	$0.530986\pi$
$48$	0	0
$49$	−9727.87	−0.578799
$50$	0	0
$51$	−21435.7	−1.15401
$52$	0	0
$53$	−22561.0	−1.10324	−0.551618	−	0.834097i	$-0.685989\pi$
$53$	−22561.0	−1.10324	−0.551618	+	0.834097i	$0.685989\pi$
$54$	0	0
$55$	2744.65	0.122343
$56$	0	0
$57$	25539.3	1.04117
$58$	0	0
$59$	−28530.5	−1.06704	−0.533518	−	0.845788i	$-0.679131\pi$
$59$	−28530.5	−1.06704	−0.533518	+	0.845788i	$0.679131\pi$
$60$	0	0
$61$	−51621.4	−1.77625	−0.888127	−	0.459599i	$-0.847993\pi$
$61$	−51621.4	−1.77625	−0.888127	+	0.459599i	$0.847993\pi$
$62$	0	0
$63$	16893.9	0.536264
$64$	0	0
$65$	5741.12	0.168544
$66$	0	0
$67$	−4489.00	−0.122169
$67$	−4489.00	−0.122169
$68$	0	0
$69$	2084.77	0.0527150
$70$	0	0
$71$	−16023.2	−0.377228	−0.188614	−	0.982051i	$-0.560400\pi$
$71$	−16023.2	−0.377228	−0.188614	+	0.982051i	$0.560400\pi$
$72$	0	0
$73$	57154.5	1.25529	0.627643	−	0.778501i	$-0.284020\pi$
$73$	57154.5	1.25529	0.627643	+	0.778501i	$0.284020\pi$
$74$	0	0
$75$	−55054.4	−1.13016
$76$	0	0
$77$	−10209.5	−0.196236
$78$	0	0
$79$	−74842.6	−1.34921	−0.674607	−	0.738177i	$-0.735687\pi$
$79$	−74842.6	−1.34921	−0.674607	+	0.738177i	$0.735687\pi$
$80$	0	0
$81$	−67524.5	−1.14353
$82$	0	0
$83$	101508.	1.61735	0.808676	−	0.588254i	$-0.200185\pi$
$83$	101508.	1.61735	0.808676	+	0.588254i	$0.200185\pi$
$84$	0	0
$85$	23015.5	0.345519
$86$	0	0
$87$	21977.6	0.311303
$88$	0	0
$89$	−99191.5	−1.32739	−0.663697	−	0.748002i	$-0.731013\pi$
$89$	−99191.5	−1.32739	−0.663697	+	0.748002i	$0.731013\pi$
$90$	0	0
$91$	−21355.8	−0.270341
$92$	0	0
$93$	140167.	1.68050
$94$	0	0
$95$	−27421.6	−0.311733
$96$	0	0
$97$	88199.7	0.951783	0.475891	−	0.879504i	$-0.342126\pi$
$97$	88199.7	0.951783	0.475891	+	0.879504i	$0.342126\pi$
$98$	0	0
$99$	−24364.4	−0.249843
$100$	0	0
$101$	115531.	1.12692	0.563461	−	0.826143i	$-0.309470\pi$
$101$	115531.	1.12692	0.563461	+	0.826143i	$0.309470\pi$
$102$	0	0
$103$	130247.	1.20969	0.604847	−	0.796341i	$-0.293234\pi$
$103$	130247.	1.20969	0.604847	+	0.796341i	$0.293234\pi$
$104$	0	0
$105$	−40091.2	−0.354876
$106$	0	0
$107$	−130043.	−1.09806	−0.549031	−	0.835802i	$-0.685003\pi$
$107$	−130043.	−1.09806	−0.549031	+	0.835802i	$0.685003\pi$
$108$	0	0
$109$	−64482.3	−0.519845	−0.259923	−	0.965629i	$-0.583697\pi$
$109$	−64482.3	−0.519845	−0.259923	+	0.965629i	$0.583697\pi$
$110$	0	0
$111$	−264870.	−2.04045
$112$	0	0
$113$	−74805.7	−0.551111	−0.275555	−	0.961285i	$-0.588862\pi$
$113$	−74805.7	−0.551111	−0.275555	+	0.961285i	$0.588862\pi$
$114$	0	0
$115$	−2238.41	−0.0157832
$116$	0	0
$117$	−50964.2	−0.344192
$118$	0	0
$119$	−85612.7	−0.554206
$120$	0	0
$121$	−146327.	−0.908574
$122$	0	0
$123$	−52324.7	−0.311849
$124$	0	0
$125$	129796.	0.742995
$126$	0	0
$127$	68579.6	0.377299	0.188649	−	0.982045i	$-0.439589\pi$
$127$	68579.6	0.377299	0.188649	+	0.982045i	$0.439589\pi$
$128$	0	0
$129$	482524.	2.55296
$130$	0	0
$131$	−303940.	−1.54743	−0.773714	−	0.633535i	$-0.781603\pi$
$131$	−303940.	−1.54743	−0.773714	+	0.633535i	$0.781603\pi$
$132$	0	0
$133$	102003.	0.500014
$134$	0	0
$135$	20113.3	0.0949836
$136$	0	0
$137$	−318441.	−1.44953	−0.724766	−	0.688995i	$-0.758052\pi$
$137$	−318441.	−1.44953	−0.724766	+	0.688995i	$0.758052\pi$
$138$	0	0
$139$	−200720.	−0.881156	−0.440578	−	0.897714i	$-0.645226\pi$
$139$	−200720.	−0.881156	−0.440578	+	0.897714i	$0.645226\pi$
$140$	0	0
$141$	−62014.3	−0.262690
$142$	0	0
$143$	30799.3	0.125951
$144$	0	0
$145$	−23597.4	−0.0932060
$146$	0	0
$147$	−204930.	−0.782191
$148$	0	0
$149$	71084.0	0.262305	0.131152	−	0.991362i	$-0.458132\pi$
$149$	71084.0	0.262305	0.131152	+	0.991362i	$0.458132\pi$
$150$	0	0
$151$	−520965.	−1.85937	−0.929687	−	0.368351i	$-0.879922\pi$
$151$	−520965.	−1.85937	−0.929687	+	0.368351i	$0.879922\pi$
$152$	0	0
$153$	−204310.	−0.705603
$154$	0	0
$155$	−150497.	−0.503151
$156$	0	0
$157$	222100.	0.719116	0.359558	−	0.933123i	$-0.382927\pi$
$157$	222100.	0.719116	0.359558	+	0.933123i	$0.382927\pi$
$158$	0	0
$159$	−475277.	−1.49092
$160$	0	0
$161$	8326.43	0.0253159
$162$	0	0
$163$	−394858.	−1.16405	−0.582025	−	0.813171i	$-0.697739\pi$
$163$	−394858.	−1.16405	−0.582025	+	0.813171i	$0.697739\pi$
$164$	0	0
$165$	57819.6	0.165335
$166$	0	0
$167$	−52099.8	−0.144559	−0.0722795	−	0.997384i	$-0.523027\pi$
$167$	−52099.8	−0.144559	−0.0722795	+	0.997384i	$0.523027\pi$
$168$	0	0
$169$	−306869.	−0.826486
$170$	0	0
$171$	243423.	0.636607
$172$	0	0
$173$	361455.	0.918204	0.459102	−	0.888384i	$-0.348171\pi$
$173$	361455.	0.918204	0.459102	+	0.888384i	$0.348171\pi$
$174$	0	0
$175$	−219884.	−0.542748
$176$	0	0
$177$	−601032.	−1.44200
$178$	0	0
$179$	−211023.	−0.492262	−0.246131	−	0.969237i	$-0.579159\pi$
$179$	−211023.	−0.492262	−0.246131	+	0.969237i	$0.579159\pi$
$180$	0	0
$181$	196197.	0.445140	0.222570	−	0.974917i	$-0.428555\pi$
$181$	196197.	0.445140	0.222570	+	0.974917i	$0.428555\pi$
$182$	0	0
$183$	−1.08747e6	−2.40044
$184$	0	0
$185$	284391.	0.610922
$186$	0	0
$187$	123471.	0.258202
$188$	0	0
$189$	−74817.3	−0.152352
$190$	0	0
$191$	−183266.	−0.363494	−0.181747	−	0.983345i	$-0.558175\pi$
$191$	−183266.	−0.363494	−0.181747	+	0.983345i	$0.558175\pi$
$192$	0	0
$193$	124564.	0.240714	0.120357	−	0.992731i	$-0.461596\pi$
$193$	124564.	0.240714	0.120357	+	0.992731i	$0.461596\pi$
$194$	0	0
$195$	120944.	0.227771
$196$	0	0
$197$	429361.	0.788237	0.394119	−	0.919060i	$-0.371050\pi$
$197$	429361.	0.788237	0.394119	+	0.919060i	$0.371050\pi$
$198$	0	0
$199$	831855.	1.48907	0.744534	−	0.667584i	$-0.232672\pi$
$199$	831855.	1.48907	0.744534	+	0.667584i	$0.232672\pi$
$200$	0	0
$201$	−94566.6	−0.165100
$202$	0	0
$203$	87777.3	0.149500
$204$	0	0
$205$	56181.0	0.0933695
$206$	0	0
$207$	19870.5	0.0322317
$208$	0	0
$209$	−147108.	−0.232954
$210$	0	0
$211$	1.10338e6	1.70616	0.853079	−	0.521782i	$-0.174732\pi$
$211$	1.10338e6	1.70616	0.853079	+	0.521782i	$0.174732\pi$
$212$	0	0
$213$	−337550.	−0.509788
$214$	0	0
$215$	−518085.	−0.764373
$216$	0	0
$217$	559818.	0.807044
$218$	0	0
$219$	1.20403e6	1.69640
$220$	0	0
$221$	258270.	0.355708
$222$	0	0
$223$	280096.	0.377177	0.188588	−	0.982056i	$-0.439609\pi$
$223$	280096.	0.377177	0.188588	+	0.982056i	$0.439609\pi$
$224$	0	0
$225$	−524740.	−0.691015
$226$	0	0
$227$	118150.	0.152184	0.0760920	−	0.997101i	$-0.475756\pi$
$227$	118150.	0.152184	0.0760920	+	0.997101i	$0.475756\pi$
$228$	0	0
$229$	−512364.	−0.645640	−0.322820	−	0.946460i	$-0.604631\pi$
$229$	−512364.	−0.645640	−0.322820	+	0.946460i	$0.604631\pi$
$230$	0	0
$231$	−215077.	−0.265194
$232$	0	0
$233$	−587692.	−0.709186	−0.354593	−	0.935021i	$-0.615381\pi$
$233$	−587692.	−0.709186	−0.354593	+	0.935021i	$0.615381\pi$
$234$	0	0
$235$	66584.7	0.0786511
$236$	0	0
$237$	−1.57666e6	−1.82333
$238$	0	0
$239$	236241.	0.267523	0.133761	−	0.991014i	$-0.457294\pi$
$239$	236241.	0.267523	0.133761	+	0.991014i	$0.457294\pi$
$240$	0	0
$241$	481270.	0.533760	0.266880	−	0.963730i	$-0.414007\pi$
$241$	481270.	0.533760	0.266880	+	0.963730i	$0.414007\pi$
$242$	0	0
$243$	−1.20641e6	−1.31063
$244$	0	0
$245$	220034.	0.234193
$246$	0	0
$247$	−307713.	−0.320925
$248$	0	0
$249$	2.13840e6	2.18570
$250$	0	0
$251$	−1.80604e6	−1.80943	−0.904716	−	0.426015i	$-0.859917\pi$
$251$	−1.80604e6	−1.80943	−0.904716	+	0.426015i	$0.859917\pi$
$252$	0	0
$253$	−12008.4	−0.0117946
$254$	0	0
$255$	484851.	0.466936
$256$	0	0
$257$	−1.81520e6	−1.71432	−0.857159	−	0.515051i	$-0.827773\pi$
$257$	−1.81520e6	−1.71432	−0.857159	+	0.515051i	$0.827773\pi$
$258$	0	0
$259$	−1.05787e6	−0.979907
$260$	0	0
$261$	209475.	0.190341
$262$	0	0
$263$	1.91874e6	1.71052	0.855259	−	0.518201i	$-0.173398\pi$
$263$	1.91874e6	1.71052	0.855259	+	0.518201i	$0.173398\pi$
$264$	0	0
$265$	510305.	0.446390
$266$	0	0
$267$	−2.08960e6	−1.79384
$268$	0	0
$269$	−538207.	−0.453491	−0.226745	−	0.973954i	$-0.572808\pi$
$269$	−538207.	−0.453491	−0.226745	+	0.973954i	$0.572808\pi$
$270$	0	0
$271$	514882.	0.425877	0.212939	−	0.977066i	$-0.431697\pi$
$271$	514882.	0.425877	0.212939	+	0.977066i	$0.431697\pi$
$272$	0	0
$273$	−449887.	−0.365340
$274$	0	0
$275$	317117.	0.252864
$276$	0	0
$277$	433983.	0.339839	0.169919	−	0.985458i	$-0.445649\pi$
$277$	433983.	0.339839	0.169919	+	0.985458i	$0.445649\pi$
$278$	0	0
$279$	1.33597e6	1.02751
$280$	0	0
$281$	1.24842e6	0.943182	0.471591	−	0.881817i	$-0.343680\pi$
$281$	1.24842e6	0.943182	0.471591	+	0.881817i	$0.343680\pi$
$282$	0	0
$283$	−1.29237e6	−0.959225	−0.479612	−	0.877480i	$-0.659223\pi$
$283$	−1.29237e6	−0.959225	−0.479612	+	0.877480i	$0.659223\pi$
$284$	0	0
$285$	−577671.	−0.421278
$286$	0	0
$287$	−208982.	−0.149763
$288$	0	0
$289$	−384484.	−0.270790
$290$	0	0
$291$	1.85804e6	1.28624
$292$	0	0
$293$	−609616.	−0.414846	−0.207423	−	0.978251i	$-0.566508\pi$
$293$	−609616.	−0.414846	−0.207423	+	0.978251i	$0.566508\pi$
$294$	0	0
$295$	645328.	0.431743
$296$	0	0
$297$	107902.	0.0709801
$298$	0	0
$299$	−25118.5	−0.0162486
$300$	0	0
$301$	1.92717e6	1.22604
$302$	0	0
$303$	2.43380e6	1.52293
$304$	0	0
$305$	1.16762e6	0.718705
$306$	0	0
$307$	1.92511e6	1.16576	0.582881	−	0.812558i	$-0.301925\pi$
$307$	1.92511e6	1.16576	0.582881	+	0.812558i	$0.301925\pi$
$308$	0	0
$309$	2.74383e6	1.63479
$310$	0	0
$311$	1.36702e6	0.801446	0.400723	−	0.916199i	$-0.368759\pi$
$311$	1.36702e6	0.801446	0.400723	+	0.916199i	$0.368759\pi$
$312$	0	0
$313$	419587.	0.242081	0.121041	−	0.992648i	$-0.461377\pi$
$313$	419587.	0.242081	0.121041	+	0.992648i	$0.461377\pi$
$314$	0	0
$315$	−382121.	−0.216983
$316$	0	0
$317$	−2.04029e6	−1.14037	−0.570184	−	0.821517i	$-0.693128\pi$
$317$	−2.04029e6	−1.14037	−0.570184	+	0.821517i	$0.693128\pi$
$318$	0	0
$319$	−126593.	−0.0696517
$320$	0	0
$321$	−2.73952e6	−1.48392
$322$	0	0
$323$	−1.23359e6	−0.657905
$324$	0	0
$325$	663329.	0.348354
$326$	0	0
$327$	−1.35840e6	−0.702521
$328$	0	0
$329$	−247681.	−0.126155
$330$	0	0
$331$	−761705.	−0.382135	−0.191068	−	0.981577i	$-0.561195\pi$
$331$	−761705.	−0.382135	−0.191068	+	0.981577i	$0.561195\pi$
$332$	0	0
$333$	−2.52455e6	−1.24760
$334$	0	0
$335$	101536.	0.0494321
$336$	0	0
$337$	−3.13443e6	−1.50343	−0.751717	−	0.659486i	$-0.770774\pi$
$337$	−3.13443e6	−1.50343	−0.751717	+	0.659486i	$0.770774\pi$
$338$	0	0
$339$	−1.57588e6	−0.744773
$340$	0	0
$341$	−807369.	−0.375999
$342$	0	0
$343$	−2.23258e6	−1.02464
$344$	0	0
$345$	−47155.1	−0.0213295
$346$	0	0
$347$	−2.67845e6	−1.19415	−0.597076	−	0.802184i	$-0.703671\pi$
$347$	−2.67845e6	−1.19415	−0.597076	+	0.802184i	$0.703671\pi$
$348$	0	0
$349$	1.72350e6	0.757437	0.378719	−	0.925512i	$-0.376365\pi$
$349$	1.72350e6	0.757437	0.378719	+	0.925512i	$0.376365\pi$
$350$	0	0
$351$	225703.	0.0977844
$352$	0	0
$353$	2.03320e6	0.868446	0.434223	−	0.900806i	$-0.357023\pi$
$353$	2.03320e6	0.868446	0.434223	+	0.900806i	$0.357023\pi$
$354$	0	0
$355$	362427.	0.152634
$356$	0	0
$357$	−1.80354e6	−0.748956
$358$	0	0
$359$	−1.80314e6	−0.738403	−0.369202	−	0.929349i	$-0.620369\pi$
$359$	−1.80314e6	−0.738403	−0.369202	+	0.929349i	$0.620369\pi$
$360$	0	0
$361$	−1.00635e6	−0.406427
$362$	0	0
$363$	−3.08257e6	−1.22785
$364$	0	0
$365$	−1.29277e6	−0.507913
$366$	0	0
$367$	−1.10561e6	−0.428487	−0.214243	−	0.976780i	$-0.568729\pi$
$367$	−1.10561e6	−0.428487	−0.214243	+	0.976780i	$0.568729\pi$
$368$	0	0
$369$	−498722.	−0.190675
$370$	0	0
$371$	−1.89823e6	−0.716001
$372$	0	0
$373$	−2.25929e6	−0.840814	−0.420407	−	0.907336i	$-0.638113\pi$
$373$	−2.25929e6	−0.840814	−0.420407	+	0.907336i	$0.638113\pi$
$374$	0	0
$375$	2.73432e6	1.00409
$376$	0	0
$377$	−264800.	−0.0959543
$378$	0	0
$379$	−793938.	−0.283915	−0.141958	−	0.989873i	$-0.545340\pi$
$379$	−793938.	−0.283915	−0.141958	+	0.989873i	$0.545340\pi$
$380$	0	0
$381$	1.44472e6	0.509883
$382$	0	0
$383$	1.38096e6	0.481043	0.240522	−	0.970644i	$-0.422682\pi$
$383$	1.38096e6	0.481043	0.240522	+	0.970644i	$0.422682\pi$
$384$	0	0
$385$	230928.	0.0794008
$386$	0	0
$387$	4.59908e6	1.56096
$388$	0	0
$389$	538248.	0.180347	0.0901735	−	0.995926i	$-0.471258\pi$
$389$	538248.	0.180347	0.0901735	+	0.995926i	$0.471258\pi$
$390$	0	0
$391$	−100697.	−0.0333101
$392$	0	0
$393$	−6.40290e6	−2.09120
$394$	0	0
$395$	1.69285e6	0.545917
$396$	0	0
$397$	−2.34487e6	−0.746693	−0.373346	−	0.927692i	$-0.621790\pi$
$397$	−2.34487e6	−0.746693	−0.373346	+	0.927692i	$0.621790\pi$
$398$	0	0
$399$	2.14882e6	0.675720
$400$	0	0
$401$	−4.13951e6	−1.28555	−0.642774	−	0.766056i	$-0.722217\pi$
$401$	−4.13951e6	−1.28555	−0.642774	+	0.766056i	$0.722217\pi$
$402$	0	0
$403$	−1.68881e6	−0.517987
$404$	0	0
$405$	1.52733e6	0.462695
$406$	0	0
$407$	1.52567e6	0.456535
$408$	0	0
$409$	4.58658e6	1.35575	0.677877	−	0.735175i	$-0.262900\pi$
$409$	4.58658e6	1.35575	0.677877	+	0.735175i	$0.262900\pi$
$410$	0	0
$411$	−6.70838e6	−1.95890
$412$	0	0
$413$	−2.40049e6	−0.692507
$414$	0	0
$415$	−2.29600e6	−0.654411
$416$	0	0
$417$	−4.22842e6	−1.19080
$418$	0	0
$419$	−494462.	−0.137593	−0.0687967	−	0.997631i	$-0.521916\pi$
$419$	−494462.	−0.137593	−0.0687967	+	0.997631i	$0.521916\pi$
$420$	0	0
$421$	−5.44076e6	−1.49608	−0.748038	−	0.663655i	$-0.769004\pi$
$421$	−5.44076e6	−1.49608	−0.748038	+	0.663655i	$0.769004\pi$
$422$	0	0
$423$	−591077.	−0.160617
$424$	0	0
$425$	2.65921e6	0.714134
$426$	0	0
$427$	−4.34329e6	−1.15279
$428$	0	0
$429$	648827.	0.170210
$430$	0	0
$431$	−3.23735e6	−0.839452	−0.419726	−	0.907651i	$-0.637874\pi$
$431$	−3.23735e6	−0.839452	−0.419726	+	0.907651i	$0.637874\pi$
$432$	0	0
$433$	2.79950e6	0.717564	0.358782	−	0.933421i	$-0.383192\pi$
$433$	2.79950e6	0.717564	0.358782	+	0.933421i	$0.383192\pi$
$434$	0	0
$435$	−497109.	−0.125959
$436$	0	0
$437$	119975.	0.0300529
$438$	0	0
$439$	20034.5	0.00496156	0.00248078	−	0.999997i	$-0.499210\pi$
$439$	20034.5	0.00496156	0.00248078	+	0.999997i	$0.499210\pi$
$440$	0	0
$441$	−1.95325e6	−0.478257
$442$	0	0
$443$	5.05660e6	1.22419	0.612095	−	0.790784i	$-0.290327\pi$
$443$	5.05660e6	1.22419	0.612095	+	0.790784i	$0.290327\pi$
$444$	0	0
$445$	2.24360e6	0.537088
$446$	0	0
$447$	1.49748e6	0.354480
$448$	0	0
$449$	−7.87479e6	−1.84341	−0.921707	−	0.387886i	$-0.873205\pi$
$449$	−7.87479e6	−1.84341	−0.921707	+	0.387886i	$0.873205\pi$
$450$	0	0
$451$	301394.	0.0697739
$452$	0	0
$453$	−1.09748e7	−2.51276
$454$	0	0
$455$	483043.	0.109385
$456$	0	0
$457$	4.35558e6	0.975564	0.487782	−	0.872966i	$-0.337806\pi$
$457$	4.35558e6	0.975564	0.487782	+	0.872966i	$0.337806\pi$
$458$	0	0
$459$	904816.	0.200460
$460$	0	0
$461$	8.29988e6	1.81894	0.909472	−	0.415765i	$-0.136486\pi$
$461$	8.29988e6	1.81894	0.909472	+	0.415765i	$0.136486\pi$
$462$	0	0
$463$	1.65237e6	0.358224	0.179112	−	0.983829i	$-0.442678\pi$
$463$	1.65237e6	0.358224	0.179112	+	0.983829i	$0.442678\pi$
$464$	0	0
$465$	−3.17041e6	−0.679960
$466$	0	0
$467$	−430269.	−0.0912953	−0.0456476	−	0.998958i	$-0.514535\pi$
$467$	−430269.	−0.0912953	−0.0456476	+	0.998958i	$0.514535\pi$
$468$	0	0
$469$	−377693.	−0.0792880
$470$	0	0
$471$	4.67882e6	0.971816
$472$	0	0
$473$	−2.77937e6	−0.571206
$474$	0	0
$475$	−3.16829e6	−0.644304
$476$	0	0
$477$	−4.53000e6	−0.911597
$478$	0	0
$479$	−320965.	−0.0639173	−0.0319586	−	0.999489i	$-0.510174\pi$
$479$	−320965.	−0.0639173	−0.0319586	+	0.999489i	$0.510174\pi$
$480$	0	0
$481$	3.19131e6	0.628936
$482$	0	0
$483$	175407.	0.0342121
$484$	0	0
$485$	−1.99498e6	−0.385109
$486$	0	0
$487$	4.59916e6	0.878731	0.439365	−	0.898308i	$-0.355203\pi$
$487$	4.59916e6	0.878731	0.439365	+	0.898308i	$0.355203\pi$
$488$	0	0
$489$	−8.31819e6	−1.57310
$490$	0	0
$491$	−5.55766e6	−1.04037	−0.520186	−	0.854053i	$-0.674137\pi$
$491$	−5.55766e6	−1.04037	−0.520186	+	0.854053i	$0.674137\pi$
$492$	0	0
$493$	−1.06155e6	−0.196709
$494$	0	0
$495$	551096.	0.101091
$496$	0	0
$497$	−1.34816e6	−0.244821
$498$	0	0
$499$	−3.31191e6	−0.595426	−0.297713	−	0.954655i	$-0.596224\pi$
$499$	−3.31191e6	−0.595426	−0.297713	+	0.954655i	$0.596224\pi$
$500$	0	0
$501$	−1.09755e6	−0.195358
$502$	0	0
$503$	7.26717e6	1.28069	0.640347	−	0.768086i	$-0.278791\pi$
$503$	7.26717e6	1.28069	0.640347	+	0.768086i	$0.278791\pi$
$504$	0	0
$505$	−2.61317e6	−0.455974
$506$	0	0
$507$	−6.46459e6	−1.11692
$508$	0	0
$509$	492318.	0.0842270	0.0421135	−	0.999113i	$-0.486591\pi$
$509$	492318.	0.0842270	0.0421135	+	0.999113i	$0.486591\pi$
$510$	0	0
$511$	4.80884e6	0.814681
$512$	0	0
$513$	−1.07804e6	−0.180859
$514$	0	0
$515$	−2.94605e6	−0.489465
$516$	0	0
$517$	357206.	0.0587750
$518$	0	0
$519$	7.61453e6	1.24086
$520$	0	0
$521$	1.95420e6	0.315409	0.157704	−	0.987486i	$-0.449591\pi$
$521$	1.95420e6	0.315409	0.157704	+	0.987486i	$0.449591\pi$
$522$	0	0
$523$	4.47296e6	0.715057	0.357529	−	0.933902i	$-0.383619\pi$
$523$	4.47296e6	0.715057	0.357529	+	0.933902i	$0.383619\pi$
$524$	0	0
$525$	−4.63214e6	−0.733472
$526$	0	0
$527$	−6.77026e6	−1.06189
$528$	0	0
$529$	−6.42655e6	−0.998478
$530$	0	0
$531$	−5.72862e6	−0.881685
$532$	0	0
$533$	630440.	0.0961226
$534$	0	0
$535$	2.94142e6	0.444296
$536$	0	0
$537$	−4.44546e6	−0.665245
$538$	0	0
$539$	1.18041e6	0.175009
$540$	0	0
$541$	1.21568e7	1.78577	0.892883	−	0.450289i	$-0.148679\pi$
$541$	1.21568e7	1.78577	0.892883	+	0.450289i	$0.148679\pi$
$542$	0	0
$543$	4.13315e6	0.601564
$544$	0	0
$545$	1.45852e6	0.210339
$546$	0	0
$547$	765020.	0.109321	0.0546606	−	0.998505i	$-0.482592\pi$
$547$	765020.	0.109321	0.0546606	+	0.998505i	$0.482592\pi$
$548$	0	0
$549$	−1.03650e7	−1.46771
$550$	0	0
$551$	1.26478e6	0.177474
$552$	0	0
$553$	−6.29707e6	−0.875640
$554$	0	0
$555$	5.99106e6	0.825603
$556$	0	0
$557$	60208.0	0.00822273	0.00411137	−	0.999992i	$-0.498691\pi$
$557$	60208.0	0.00822273	0.00411137	+	0.999992i	$0.498691\pi$
$558$	0	0
$559$	−5.81374e6	−0.786911
$560$	0	0
$561$	2.60107e6	0.348936
$562$	0	0
$563$	−3.65083e6	−0.485424	−0.242712	−	0.970098i	$-0.578037\pi$
$563$	−3.65083e6	−0.485424	−0.242712	+	0.970098i	$0.578037\pi$
$564$	0	0
$565$	1.69202e6	0.222990
$566$	0	0
$567$	−5.68134e6	−0.742153
$568$	0	0
$569$	−8.57514e6	−1.11035	−0.555176	−	0.831733i	$-0.687349\pi$
$569$	−8.57514e6	−1.11035	−0.555176	+	0.831733i	$0.687349\pi$
$570$	0	0
$571$	1.72013e6	0.220786	0.110393	−	0.993888i	$-0.464789\pi$
$571$	1.72013e6	0.220786	0.110393	+	0.993888i	$0.464789\pi$
$572$	0	0
$573$	−3.86073e6	−0.491228
$574$	0	0
$575$	−258626.	−0.0326214
$576$	0	0
$577$	−2.18974e6	−0.273812	−0.136906	−	0.990584i	$-0.543716\pi$
$577$	−2.18974e6	−0.273812	−0.136906	+	0.990584i	$0.543716\pi$
$578$	0	0
$579$	2.62411e6	0.325302
$580$	0	0
$581$	8.54063e6	1.04966
$582$	0	0
$583$	2.73763e6	0.333582
$584$	0	0
$585$	1.15275e6	0.139267
$586$	0	0
$587$	1.16200e7	1.39191	0.695956	−	0.718084i	$-0.254981\pi$
$587$	1.16200e7	1.39191	0.695956	+	0.718084i	$0.254981\pi$
$588$	0	0
$589$	8.06636e6	0.958053
$590$	0	0
$591$	9.04505e6	1.06523
$592$	0	0
$593$	−999163.	−0.116681	−0.0583404	−	0.998297i	$-0.518581\pi$
$593$	−999163.	−0.116681	−0.0583404	+	0.998297i	$0.518581\pi$
$594$	0	0
$595$	1.93646e6	0.224242
$596$	0	0
$597$	1.75241e7	2.01233
$598$	0	0
$599$	−1.34261e7	−1.52892	−0.764459	−	0.644672i	$-0.776994\pi$
$599$	−1.34261e7	−1.52892	−0.764459	+	0.644672i	$0.776994\pi$
$600$	0	0
$601$	9.64322e6	1.08902	0.544510	−	0.838754i	$-0.316716\pi$
$601$	9.64322e6	1.08902	0.544510	+	0.838754i	$0.316716\pi$
$602$	0	0
$603$	−901343.	−0.100948
$604$	0	0
$605$	3.30975e6	0.367626
$606$	0	0
$607$	−1.30336e7	−1.43579	−0.717896	−	0.696150i	$-0.754895\pi$
$607$	−1.30336e7	−1.43579	−0.717896	+	0.696150i	$0.754895\pi$
$608$	0	0
$609$	1.84914e6	0.202035
$610$	0	0
$611$	747186.	0.0809703
$612$	0	0
$613$	3.21481e6	0.345545	0.172772	−	0.984962i	$-0.444727\pi$
$613$	3.21481e6	0.345545	0.172772	+	0.984962i	$0.444727\pi$
$614$	0	0
$615$	1.18353e6	0.126180
$616$	0	0
$617$	1.13090e7	1.19594	0.597971	−	0.801518i	$-0.295974\pi$
$617$	1.13090e7	1.19594	0.597971	+	0.801518i	$0.295974\pi$
$618$	0	0
$619$	−1.58167e7	−1.65916	−0.829580	−	0.558387i	$-0.811420\pi$
$619$	−1.58167e7	−1.65916	−0.829580	+	0.558387i	$0.811420\pi$
$620$	0	0
$621$	−87999.6	−0.00915697
$622$	0	0
$623$	−8.34573e6	−0.861478
$624$	0	0
$625$	5.23099e6	0.535654
$626$	0	0
$627$	−3.09902e6	−0.314816
$628$	0	0
$629$	1.27936e7	1.28934
$630$	0	0
$631$	1.26244e7	1.26223	0.631114	−	0.775690i	$-0.282598\pi$
$631$	1.26244e7	1.26223	0.631114	+	0.775690i	$0.282598\pi$
$632$	0	0
$633$	2.32442e7	2.30571
$634$	0	0
$635$	−1.55119e6	−0.152662
$636$	0	0
$637$	2.46912e6	0.241098
$638$	0	0
$639$	−3.21729e6	−0.311701
$640$	0	0
$641$	−5.20187e6	−0.500052	−0.250026	−	0.968239i	$-0.580439\pi$
$641$	−5.20187e6	−0.500052	−0.250026	+	0.968239i	$0.580439\pi$
$642$	0	0
$643$	−8.93277e6	−0.852038	−0.426019	−	0.904714i	$-0.640084\pi$
$643$	−8.93277e6	−0.852038	−0.426019	+	0.904714i	$0.640084\pi$
$644$	0	0
$645$	−1.09141e7	−1.03298
$646$	0	0
$647$	−630565.	−0.0592201	−0.0296100	−	0.999562i	$-0.509427\pi$
$647$	−630565.	−0.0592201	−0.0296100	+	0.999562i	$0.509427\pi$
$648$	0	0
$649$	3.46198e6	0.322636
$650$	0	0
$651$	1.17933e7	1.09064
$652$	0	0
$653$	−8.42449e6	−0.773144	−0.386572	−	0.922259i	$-0.626341\pi$
$653$	−8.42449e6	−0.773144	−0.386572	+	0.922259i	$0.626341\pi$
$654$	0	0
$655$	6.87479e6	0.626118
$656$	0	0
$657$	1.14760e7	1.03723
$658$	0	0
$659$	−1.67708e7	−1.50432	−0.752160	−	0.658980i	$-0.770988\pi$
$659$	−1.67708e7	−1.50432	−0.752160	+	0.658980i	$0.770988\pi$
$660$	0	0
$661$	8.07158e6	0.718546	0.359273	−	0.933232i	$-0.383025\pi$
$661$	8.07158e6	0.718546	0.359273	+	0.933232i	$0.383025\pi$
$662$	0	0
$663$	5.44079e6	0.480705
$664$	0	0
$665$	−2.30718e6	−0.202315
$666$	0	0
$667$	103243.	0.00898560
$668$	0	0
$669$	5.90059e6	0.509718
$670$	0	0
$671$	6.26390e6	0.537080
$672$	0	0
$673$	2.04304e7	1.73876	0.869381	−	0.494143i	$-0.164518\pi$
$673$	2.04304e7	1.73876	0.869381	+	0.494143i	$0.164518\pi$
$674$	0	0
$675$	2.32389e6	0.196316
$676$	0	0
$677$	1.52916e7	1.28228	0.641138	−	0.767426i	$-0.278463\pi$
$677$	1.52916e7	1.28228	0.641138	+	0.767426i	$0.278463\pi$
$678$	0	0
$679$	7.42091e6	0.617707
$680$	0	0
$681$	2.48898e6	0.205662
$682$	0	0
$683$	1.29727e6	0.106409	0.0532044	−	0.998584i	$-0.483057\pi$
$683$	1.29727e6	0.106409	0.0532044	+	0.998584i	$0.483057\pi$
$684$	0	0
$685$	7.20279e6	0.586508
$686$	0	0
$687$	−1.07936e7	−0.872520
$688$	0	0
$689$	5.72642e6	0.459553
$690$	0	0
$691$	1.06428e7	0.847932	0.423966	−	0.905678i	$-0.360638\pi$
$691$	1.06428e7	0.847932	0.423966	+	0.905678i	$0.360638\pi$
$692$	0	0
$693$	−2.04996e6	−0.162148
$694$	0	0
$695$	4.54005e6	0.356532
$696$	0	0
$697$	2.52736e6	0.197054
$698$	0	0
$699$	−1.23805e7	−0.958397
$700$	0	0
$701$	2.98094e6	0.229117	0.114559	−	0.993416i	$-0.463455\pi$
$701$	2.98094e6	0.229117	0.114559	+	0.993416i	$0.463455\pi$
$702$	0	0
$703$	−1.52428e7	−1.16326
$704$	0	0
$705$	1.40269e6	0.106289
$706$	0	0
$707$	9.72047e6	0.731372
$708$	0	0
$709$	1.95929e7	1.46380	0.731902	−	0.681410i	$-0.238633\pi$
$709$	1.95929e7	1.46380	0.731902	+	0.681410i	$0.238633\pi$
$710$	0	0
$711$	−1.50276e7	−1.11485
$712$	0	0
$713$	658454.	0.0485067
$714$	0	0
$715$	−696646.	−0.0509620
$716$	0	0
$717$	4.97672e6	0.361531
$718$	0	0
$719$	4.43756e6	0.320127	0.160063	−	0.987107i	$-0.448830\pi$
$719$	4.43756e6	0.320127	0.160063	+	0.987107i	$0.448830\pi$
$720$	0	0
$721$	1.09587e7	0.785092
$722$	0	0
$723$	1.01386e7	0.721325
$724$	0	0
$725$	−2.72644e6	−0.192642
$726$	0	0
$727$	6.25395e6	0.438852	0.219426	−	0.975629i	$-0.429581\pi$
$727$	6.25395e6	0.438852	0.219426	+	0.975629i	$0.429581\pi$
$728$	0	0
$729$	−9.00614e6	−0.627653
$730$	0	0
$731$	−2.33066e7	−1.61319
$732$	0	0
$733$	1.36558e7	0.938764	0.469382	−	0.882995i	$-0.344477\pi$
$733$	1.36558e7	0.938764	0.469382	+	0.882995i	$0.344477\pi$
$734$	0	0
$735$	4.63529e6	0.316489
$736$	0	0
$737$	544710.	0.0369400
$738$	0	0
$739$	1.44368e7	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$739$	1.44368e7	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$740$	0	0
$741$	−6.48238e6	−0.433700
$742$	0	0
$743$	2.91893e7	1.93977	0.969887	−	0.243556i	$-0.0783140\pi$
$743$	2.91893e7	1.93977	0.969887	+	0.243556i	$0.0783140\pi$
$744$	0	0
$745$	−1.60784e6	−0.106133
$746$	0	0
$747$	2.03817e7	1.33641
$748$	0	0
$749$	−1.09415e7	−0.712642
$750$	0	0
$751$	1.20216e7	0.777792	0.388896	−	0.921282i	$-0.372857\pi$
$751$	1.20216e7	0.777792	0.388896	+	0.921282i	$0.372857\pi$
$752$	0	0
$753$	−3.80465e7	−2.44527
$754$	0	0
$755$	1.17837e7	0.752337
$756$	0	0
$757$	−9.66783e6	−0.613182	−0.306591	−	0.951841i	$-0.599188\pi$
$757$	−9.66783e6	−0.613182	−0.306591	+	0.951841i	$0.599188\pi$
$758$	0	0
$759$	−252972.	−0.0159393
$760$	0	0
$761$	2.06026e7	1.28962	0.644808	−	0.764344i	$-0.276937\pi$
$761$	2.06026e7	1.28962	0.644808	+	0.764344i	$0.276937\pi$
$762$	0	0
$763$	−5.42538e6	−0.337380
$764$	0	0
$765$	4.62126e6	0.285500
$766$	0	0
$767$	7.24160e6	0.444474
$768$	0	0
$769$	−253912.	−0.0154834	−0.00774171	−	0.999970i	$-0.502464\pi$
$769$	−253912.	−0.0154834	−0.00774171	+	0.999970i	$0.502464\pi$
$770$	0	0
$771$	−3.82396e7	−2.31674
$772$	0	0
$773$	3.02305e7	1.81969	0.909843	−	0.414953i	$-0.136202\pi$
$773$	3.02305e7	1.81969	0.909843	+	0.414953i	$0.136202\pi$
$774$	0	0
$775$	−1.73884e7	−1.03993
$776$	0	0
$777$	−2.22855e7	−1.32425
$778$	0	0
$779$	−3.01120e6	−0.177785
$780$	0	0
$781$	1.94431e6	0.114061
$782$	0	0
$783$	−927694.	−0.0540754
$784$	0	0
$785$	−5.02365e6	−0.290968
$786$	0	0
$787$	1.10493e7	0.635911	0.317956	−	0.948106i	$-0.397004\pi$
$787$	1.10493e7	0.635911	0.317956	+	0.948106i	$0.397004\pi$
$788$	0	0
$789$	4.04208e7	2.31160
$790$	0	0
$791$	−6.29397e6	−0.357671
$792$	0	0
$793$	1.31025e7	0.739898
$794$	0	0
$795$	1.07502e7	0.603254
$796$	0	0
$797$	−7.24922e6	−0.404246	−0.202123	−	0.979360i	$-0.564784\pi$
$797$	−7.24922e6	−0.404246	−0.202123	+	0.979360i	$0.564784\pi$
$798$	0	0
$799$	2.99538e6	0.165991
$800$	0	0
$801$	−1.99166e7	−1.09682
$802$	0	0
$803$	−6.93531e6	−0.379557
$804$	0	0
$805$	−188335.	−0.0102433
$806$	0	0
$807$	−1.13380e7	−0.612849
$808$	0	0
$809$	1.31389e7	0.705809	0.352905	−	0.935659i	$-0.385194\pi$
$809$	1.31389e7	0.705809	0.352905	+	0.935659i	$0.385194\pi$
$810$	0	0
$811$	2.73502e7	1.46019	0.730094	−	0.683347i	$-0.239476\pi$
$811$	2.73502e7	1.46019	0.730094	+	0.683347i	$0.239476\pi$
$812$	0	0
$813$	1.08467e7	0.575532
$814$	0	0
$815$	8.93124e6	0.470996
$816$	0	0
$817$	2.77684e7	1.45545
$818$	0	0
$819$	−4.28801e6	−0.223381
$820$	0	0
$821$	−1.27538e7	−0.660360	−0.330180	−	0.943918i	$-0.607109\pi$
$821$	−1.27538e7	−0.660360	−0.330180	+	0.943918i	$0.607109\pi$
$822$	0	0
$823$	3.26420e6	0.167987	0.0839937	−	0.996466i	$-0.473232\pi$
$823$	3.26420e6	0.167987	0.0839937	+	0.996466i	$0.473232\pi$
$824$	0	0
$825$	6.68048e6	0.341722
$826$	0	0
$827$	2.99760e7	1.52409	0.762043	−	0.647527i	$-0.224197\pi$
$827$	2.99760e7	1.52409	0.762043	+	0.647527i	$0.224197\pi$
$828$	0	0
$829$	−5.59206e6	−0.282609	−0.141304	−	0.989966i	$-0.545130\pi$
$829$	−5.59206e6	−0.282609	−0.141304	+	0.989966i	$0.545130\pi$
$830$	0	0
$831$	9.14241e6	0.459260
$832$	0	0
$833$	9.89843e6	0.494258
$834$	0	0
$835$	1.17844e6	0.0584913
$836$	0	0
$837$	−5.91655e6	−0.291914
$838$	0	0
$839$	−2.77473e6	−0.136087	−0.0680433	−	0.997682i	$-0.521676\pi$
$839$	−2.77473e6	−0.136087	−0.0680433	+	0.997682i	$0.521676\pi$
$840$	0	0
$841$	−1.94228e7	−0.946937
$842$	0	0
$843$	2.62996e7	1.27462
$844$	0	0
$845$	6.94102e6	0.334412
$846$	0	0
$847$	−1.23116e7	−0.589665
$848$	0	0
$849$	−2.72254e7	−1.29630
$850$	0	0
$851$	−1.24427e6	−0.0588965
$852$	0	0
$853$	−1.48259e7	−0.697668	−0.348834	−	0.937184i	$-0.613422\pi$
$853$	−1.48259e7	−0.697668	−0.348834	+	0.937184i	$0.613422\pi$
$854$	0	0
$855$	−5.50595e6	−0.257583
$856$	0	0
$857$	−1.07150e7	−0.498356	−0.249178	−	0.968458i	$-0.580161\pi$
$857$	−1.07150e7	−0.498356	−0.249178	+	0.968458i	$0.580161\pi$
$858$	0	0
$859$	−2.35740e7	−1.09006	−0.545031	−	0.838416i	$-0.683482\pi$
$859$	−2.35740e7	−1.09006	−0.545031	+	0.838416i	$0.683482\pi$
$860$	0	0
$861$	−4.40247e6	−0.202390
$862$	0	0
$863$	−3.39176e7	−1.55024	−0.775118	−	0.631816i	$-0.782310\pi$
$863$	−3.39176e7	−1.55024	−0.775118	+	0.631816i	$0.782310\pi$
$864$	0	0
$865$	−8.17571e6	−0.371523
$866$	0	0
$867$	−8.09965e6	−0.365947
$868$	0	0
$869$	9.08164e6	0.407957
$870$	0	0
$871$	1.13940e6	0.0508896
$872$	0	0
$873$	1.77095e7	0.786451
$874$	0	0
$875$	1.09207e7	0.482204
$876$	0	0
$877$	−1.78515e7	−0.783746	−0.391873	−	0.920019i	$-0.628173\pi$
$877$	−1.78515e7	−0.783746	−0.391873	+	0.920019i	$0.628173\pi$
$878$	0	0
$879$	−1.28424e7	−0.560625
$880$	0	0
$881$	−3.72325e7	−1.61615	−0.808076	−	0.589078i	$-0.799491\pi$
$881$	−3.72325e7	−1.61615	−0.808076	+	0.589078i	$0.799491\pi$
$882$	0	0
$883$	3.35174e7	1.44667	0.723333	−	0.690500i	$-0.242609\pi$
$883$	3.35174e7	1.44667	0.723333	+	0.690500i	$0.242609\pi$
$884$	0	0
$885$	1.35947e7	0.583460
$886$	0	0
$887$	3.33089e6	0.142152	0.0710758	−	0.997471i	$-0.477357\pi$
$887$	3.33089e6	0.142152	0.0710758	+	0.997471i	$0.477357\pi$
$888$	0	0
$889$	5.77012e6	0.244867
$890$	0	0
$891$	8.19364e6	0.345766
$892$	0	0
$893$	−3.56882e6	−0.149760
$894$	0	0
$895$	4.77309e6	0.199178
$896$	0	0
$897$	−529154.	−0.0219584
$898$	0	0
$899$	6.94143e6	0.286451
$900$	0	0
$901$	2.29566e7	0.942095
$902$	0	0
$903$	4.05984e7	1.65687
$904$	0	0
$905$	−4.43776e6	−0.180112
$906$	0	0
$907$	61218.6	0.00247095	0.00123548	−	0.999999i	$-0.499607\pi$
$907$	61218.6	0.00247095	0.00123548	+	0.999999i	$0.499607\pi$
$908$	0	0
$909$	2.31973e7	0.931168
$910$	0	0
$911$	4.23940e7	1.69242	0.846211	−	0.532848i	$-0.178878\pi$
$911$	4.23940e7	1.69242	0.846211	+	0.532848i	$0.178878\pi$
$912$	0	0
$913$	−1.23173e7	−0.489033
$914$	0	0
$915$	2.45974e7	0.971261
$916$	0	0
$917$	−2.55728e7	−1.00428
$918$	0	0
$919$	−4.43190e7	−1.73102	−0.865508	−	0.500895i	$-0.833004\pi$
$919$	−4.43190e7	−1.73102	−0.865508	+	0.500895i	$0.833004\pi$
$920$	0	0
$921$	4.05550e7	1.57542
$922$	0	0
$923$	4.06701e6	0.157134
$924$	0	0
$925$	3.28585e7	1.26268
$926$	0	0
$927$	2.61523e7	0.999562
$928$	0	0
$929$	1.15781e7	0.440147	0.220074	−	0.975483i	$-0.429370\pi$
$929$	1.15781e7	0.440147	0.220074	+	0.975483i	$0.429370\pi$
$930$	0	0
$931$	−1.17934e7	−0.445928
$932$	0	0
$933$	2.87981e7	1.08308
$934$	0	0
$935$	−2.79277e6	−0.104474
$936$	0	0
$937$	−3.36163e7	−1.25084	−0.625420	−	0.780289i	$-0.715072\pi$
$937$	−3.36163e7	−1.25084	−0.625420	+	0.780289i	$0.715072\pi$
$938$	0	0
$939$	8.83914e6	0.327149
$940$	0	0
$941$	−4.31858e7	−1.58989	−0.794945	−	0.606682i	$-0.792500\pi$
$941$	−4.31858e7	−1.58989	−0.794945	+	0.606682i	$0.792500\pi$
$942$	0	0
$943$	−245803.	−0.00900136
$944$	0	0
$945$	1.69228e6	0.0616444
$946$	0	0
$947$	3.88643e7	1.40824	0.704119	−	0.710082i	$-0.251342\pi$
$947$	3.88643e7	1.40824	0.704119	+	0.710082i	$0.251342\pi$
$948$	0	0
$949$	−1.45069e7	−0.522889
$950$	0	0
$951$	−4.29815e7	−1.54110
$952$	0	0
$953$	−1.03960e7	−0.370794	−0.185397	−	0.982664i	$-0.559357\pi$
$953$	−1.03960e7	−0.370794	−0.185397	+	0.982664i	$0.559357\pi$
$954$	0	0
$955$	4.14526e6	0.147077
$956$	0	0
$957$	−2.66684e6	−0.0941275
$958$	0	0
$959$	−2.67929e7	−0.940747
$960$	0	0
$961$	1.56412e7	0.546339
$962$	0	0
$963$	−2.61112e7	−0.907321
$964$	0	0
$965$	−2.81751e6	−0.0973973
$966$	0	0
$967$	−1.78946e7	−0.615396	−0.307698	−	0.951484i	$-0.599559\pi$
$967$	−1.78946e7	−0.615396	−0.307698	+	0.951484i	$0.599559\pi$
$968$	0	0
$969$	−2.59871e7	−0.889096
$970$	0	0
$971$	1.67950e7	0.571651	0.285825	−	0.958282i	$-0.407732\pi$
$971$	1.67950e7	0.571651	0.285825	+	0.958282i	$0.407732\pi$
$972$	0	0
$973$	−1.68881e7	−0.571870
$974$	0	0
$975$	1.39739e7	0.470767
$976$	0	0
$977$	−3.02919e7	−1.01529	−0.507645	−	0.861567i	$-0.669484\pi$
$977$	−3.02919e7	−1.01529	−0.507645	+	0.861567i	$0.669484\pi$
$978$	0	0
$979$	1.20362e7	0.401359
$980$	0	0
$981$	−1.29473e7	−0.429545
$982$	0	0
$983$	−2.50887e7	−0.828123	−0.414062	−	0.910249i	$-0.635890\pi$
$983$	−2.50887e7	−0.828123	−0.414062	+	0.910249i	$0.635890\pi$
$984$	0	0
$985$	−9.71166e6	−0.318936
$986$	0	0
$987$	−5.21773e6	−0.170486
$988$	0	0
$989$	2.26673e6	0.0736900
$990$	0	0
$991$	−2.06448e7	−0.667768	−0.333884	−	0.942614i	$-0.608359\pi$
$991$	−2.06448e7	−0.667768	−0.333884	+	0.942614i	$0.608359\pi$
$992$	0	0
$993$	−1.60463e7	−0.516419
$994$	0	0
$995$	−1.88156e7	−0.602505
$996$	0	0
$997$	2.51393e7	0.800968	0.400484	−	0.916304i	$-0.368842\pi$
$997$	2.51393e7	0.800968	0.400484	+	0.916304i	$0.368842\pi$
$998$	0	0
$999$	1.11804e7	0.354440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1072.6.a.k.1.19		22
4.3	odd	2		536.6.a.c.1.4	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.6.a.c.1.4	✓	22	4.3	odd	2
1072.6.a.k.1.19		22	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:	\( N \)	\(=\)	\( 1072 = 2^{4} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1072.a (trivial)

\(f(q)\)	\(=\)	\(q+21.0663 q^{3} -22.6189 q^{5} +84.1376 q^{7} +200.789 q^{9} +O(q^{10})\)	\(q+21.0663 q^{3} -22.6189 q^{5} +84.1376 q^{7} +200.789 q^{9} -121.343 q^{11} -253.820 q^{13} -476.496 q^{15} -1017.53 q^{17} +1212.33 q^{19} +1772.47 q^{21} +98.9621 q^{23} -2613.39 q^{25} -889.226 q^{27} +1043.26 q^{29} +6653.60 q^{31} -2556.25 q^{33} -1903.10 q^{35} -12573.2 q^{37} -5347.04 q^{39} -2483.81 q^{41} +22905.0 q^{43} -4541.63 q^{45} -2943.77 q^{47} -9727.87 q^{49} -21435.7 q^{51} -22561.0 q^{53} +2744.65 q^{55} +25539.3 q^{57} -28530.5 q^{59} -51621.4 q^{61} +16893.9 q^{63} +5741.12 q^{65} -4489.00 q^{67} +2084.77 q^{69} -16023.2 q^{71} +57154.5 q^{73} -55054.4 q^{75} -10209.5 q^{77} -74842.6 q^{79} -67524.5 q^{81} +101508. q^{83} +23015.5 q^{85} +21977.6 q^{87} -99191.5 q^{89} -21355.8 q^{91} +140167. q^{93} -27421.6 q^{95} +88199.7 q^{97} -24364.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 36 q^{3} + 25 q^{5} - 171 q^{7} + 1740 q^{9}+O(q^{10}) \) 22 * q - 36 * q^3 + 25 * q^5 - 171 * q^7 + 1740 * q^9	\( 22 q - 36 q^{3} + 25 q^{5} - 171 q^{7} + 1740 q^{9} - 581 q^{11} + 874 q^{13} - 883 q^{15} - 2407 q^{17} - 3187 q^{19} + 2616 q^{21} - 5587 q^{23} + 14095 q^{25} - 15390 q^{27} + 4190 q^{29} - 15482 q^{31} - 11223 q^{33} + 5741 q^{35} - 7673 q^{37} + 9694 q^{39} - 36347 q^{41} - 16710 q^{43} - 5324 q^{45} - 20064 q^{47} + 62665 q^{49} - 31094 q^{51} + 26795 q^{53} - 94740 q^{55} + 46573 q^{57} - 54226 q^{59} + 82744 q^{61} - 146336 q^{63} + 37575 q^{65} - 98758 q^{67} + 192484 q^{69} - 124063 q^{71} + 94583 q^{73} - 114498 q^{75} + 215979 q^{77} - 5851 q^{79} + 397938 q^{81} - 212613 q^{83} + 302469 q^{85} - 330727 q^{87} + 181565 q^{89} - 202361 q^{91} + 246191 q^{93} - 224203 q^{95} + 172896 q^{97} - 532665 q^{99}+O(q^{100}) \) 22 * q - 36 * q^3 + 25 * q^5 - 171 * q^7 + 1740 * q^9 - 581 * q^11 + 874 * q^13 - 883 * q^15 - 2407 * q^17 - 3187 * q^19 + 2616 * q^21 - 5587 * q^23 + 14095 * q^25 - 15390 * q^27 + 4190 * q^29 - 15482 * q^31 - 11223 * q^33 + 5741 * q^35 - 7673 * q^37 + 9694 * q^39 - 36347 * q^41 - 16710 * q^43 - 5324 * q^45 - 20064 * q^47 + 62665 * q^49 - 31094 * q^51 + 26795 * q^53 - 94740 * q^55 + 46573 * q^57 - 54226 * q^59 + 82744 * q^61 - 146336 * q^63 + 37575 * q^65 - 98758 * q^67 + 192484 * q^69 - 124063 * q^71 + 94583 * q^73 - 114498 * q^75 + 215979 * q^77 - 5851 * q^79 + 397938 * q^81 - 212613 * q^83 + 302469 * q^85 - 330727 * q^87 + 181565 * q^89 - 202361 * q^91 + 246191 * q^93 - 224203 * q^95 + 172896 * q^97 - 532665 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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