LMFDB - Embedded newform 882.6.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 7 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} +O(q^{10})$	$q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} -413.990 q^{10} -240.191 q^{11} +805.477 q^{13} +256.000 q^{16} +1293.27 q^{17} -275.377 q^{19} -1655.96 q^{20} -960.764 q^{22} -3796.57 q^{23} +7586.73 q^{25} +3221.91 q^{26} -1227.52 q^{29} +5624.86 q^{31} +1024.00 q^{32} +5173.06 q^{34} -9078.49 q^{37} -1101.51 q^{38} -6623.84 q^{40} +18207.4 q^{41} -11708.2 q^{43} -3843.05 q^{44} -15186.3 q^{46} +23048.6 q^{47} +30346.9 q^{50} +12887.6 q^{52} -17662.8 q^{53} +24859.2 q^{55} -4910.07 q^{58} -18376.3 q^{59} +11324.1 q^{61} +22499.5 q^{62} +4096.00 q^{64} -83364.9 q^{65} +36079.3 q^{67} +20692.3 q^{68} +63434.2 q^{71} -52982.4 q^{73} -36314.0 q^{74} -4406.03 q^{76} -48564.0 q^{79} -26495.4 q^{80} +72829.7 q^{82} -113161. q^{83} -133850. q^{85} -46832.9 q^{86} -15372.2 q^{88} -108366. q^{89} -60745.2 q^{92} +92194.4 q^{94} +28500.8 q^{95} +99641.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8} - 432 q^{10} - 124 q^{11} + 720 q^{13} + 512 q^{16} + 1260 q^{17} + 360 q^{19} - 1728 q^{20} - 496 q^{22} - 6524 q^{23} + 4482 q^{25} + 2880 q^{26} - 7088 q^{29} + 5904 q^{31} + 2048 q^{32} + 5040 q^{34} - 6040 q^{37} + 1440 q^{38} - 6912 q^{40} + 17388 q^{41} - 608 q^{43} - 1984 q^{44} - 26096 q^{46} + 30456 q^{47} + 17928 q^{50} + 11520 q^{52} - 3964 q^{53} + 24336 q^{55} - 28352 q^{58} - 40752 q^{59} - 1368 q^{61} + 23616 q^{62} + 8192 q^{64} - 82980 q^{65} - 16224 q^{67} + 20160 q^{68} + 3204 q^{71} + 23976 q^{73} - 24160 q^{74} + 5760 q^{76} - 82160 q^{79} - 27648 q^{80} + 69552 q^{82} - 173736 q^{83} - 133700 q^{85} - 2432 q^{86} - 7936 q^{88} - 200556 q^{89} - 104384 q^{92} + 121824 q^{94} + 25640 q^{95} + 251928 q^{97}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8 - 432 * q^10 - 124 * q^11 + 720 * q^13 + 512 * q^16 + 1260 * q^17 + 360 * q^19 - 1728 * q^20 - 496 * q^22 - 6524 * q^23 + 4482 * q^25 + 2880 * q^26 - 7088 * q^29 + 5904 * q^31 + 2048 * q^32 + 5040 * q^34 - 6040 * q^37 + 1440 * q^38 - 6912 * q^40 + 17388 * q^41 - 608 * q^43 - 1984 * q^44 - 26096 * q^46 + 30456 * q^47 + 17928 * q^50 + 11520 * q^52 - 3964 * q^53 + 24336 * q^55 - 28352 * q^58 - 40752 * q^59 - 1368 * q^61 + 23616 * q^62 + 8192 * q^64 - 82980 * q^65 - 16224 * q^67 + 20160 * q^68 + 3204 * q^71 + 23976 * q^73 - 24160 * q^74 + 5760 * q^76 - 82160 * q^79 - 27648 * q^80 + 69552 * q^82 - 173736 * q^83 - 133700 * q^85 - 2432 * q^86 - 7936 * q^88 - 200556 * q^89 - 104384 * q^92 + 121824 * q^94 + 25640 * q^95 + 251928 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−103.497	−1.85142	−0.925710	−	0.378235i	$-0.876531\pi$
$5$	−103.497	−1.85142	−0.925710	+	0.378235i	$0.876531\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	64.0000	0.353553
$9$	0	0
$10$	−413.990	−1.30915
$11$	−240.191	−0.598515	−0.299257	−	0.954172i	$-0.596739\pi$
$11$	−240.191	−0.598515	−0.299257	+	0.954172i	$0.596739\pi$
$12$	0	0
$13$	805.477	1.32189	0.660944	−	0.750435i	$-0.270156\pi$
$13$	805.477	1.32189	0.660944	+	0.750435i	$0.270156\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	1293.27	1.08534	0.542670	−	0.839946i	$-0.317414\pi$
$17$	1293.27	1.08534	0.542670	+	0.839946i	$0.317414\pi$
$18$	0	0
$19$	−275.377	−0.175002	−0.0875011	−	0.996164i	$-0.527888\pi$
$19$	−275.377	−0.175002	−0.0875011	+	0.996164i	$0.527888\pi$
$20$	−1655.96	−0.925710
$21$	0	0
$22$	−960.764	−0.423214
$23$	−3796.57	−1.49648	−0.748242	−	0.663426i	$-0.769102\pi$
$23$	−3796.57	−1.49648	−0.748242	+	0.663426i	$0.769102\pi$
$24$	0	0
$25$	7586.73	2.42775
$26$	3221.91	0.934717
$27$	0	0
$28$	0	0
$29$	−1227.52	−0.271040	−0.135520	−	0.990775i	$-0.543270\pi$
$29$	−1227.52	−0.271040	−0.135520	+	0.990775i	$0.543270\pi$
$30$	0	0
$31$	5624.86	1.05125	0.525627	−	0.850715i	$-0.323831\pi$
$31$	5624.86	1.05125	0.525627	+	0.850715i	$0.323831\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	5173.06	0.767451
$35$	0	0
$36$	0	0
$37$	−9078.49	−1.09021	−0.545104	−	0.838368i	$-0.683510\pi$
$37$	−9078.49	−1.09021	−0.545104	+	0.838368i	$0.683510\pi$
$38$	−1101.51	−0.123745
$39$	0	0
$40$	−6623.84	−0.654576
$41$	18207.4	1.69156	0.845782	−	0.533528i	$-0.179134\pi$
$41$	18207.4	1.69156	0.845782	+	0.533528i	$0.179134\pi$
$42$	0	0
$43$	−11708.2	−0.965650	−0.482825	−	0.875717i	$-0.660389\pi$
$43$	−11708.2	−0.965650	−0.482825	+	0.875717i	$0.660389\pi$
$44$	−3843.05	−0.299257
$45$	0	0
$46$	−15186.3	−1.05817
$47$	23048.6	1.52195	0.760974	−	0.648782i	$-0.224721\pi$
$47$	23048.6	1.52195	0.760974	+	0.648782i	$0.224721\pi$
$48$	0	0
$49$	0	0
$50$	30346.9	1.71668
$51$	0	0
$52$	12887.6	0.660944
$53$	−17662.8	−0.863714	−0.431857	−	0.901942i	$-0.642141\pi$
$53$	−17662.8	−0.863714	−0.431857	+	0.901942i	$0.642141\pi$
$54$	0	0
$55$	24859.2	1.10810
$56$	0	0
$57$	0	0
$58$	−4910.07	−0.191654
$59$	−18376.3	−0.687271	−0.343636	−	0.939103i	$-0.611658\pi$
$59$	−18376.3	−0.687271	−0.343636	+	0.939103i	$0.611658\pi$
$60$	0	0
$61$	11324.1	0.389654	0.194827	−	0.980838i	$-0.437586\pi$
$61$	11324.1	0.389654	0.194827	+	0.980838i	$0.437586\pi$
$62$	22499.5	0.743349
$63$	0	0
$64$	4096.00	0.125000
$65$	−83364.9	−2.44737
$66$	0	0
$67$	36079.3	0.981910	0.490955	−	0.871185i	$-0.336648\pi$
$67$	36079.3	0.981910	0.490955	+	0.871185i	$0.336648\pi$
$68$	20692.3	0.542670
$69$	0	0
$70$	0	0
$71$	63434.2	1.49341	0.746703	−	0.665158i	$-0.231636\pi$
$71$	63434.2	1.49341	0.746703	+	0.665158i	$0.231636\pi$
$72$	0	0
$73$	−52982.4	−1.16366	−0.581828	−	0.813312i	$-0.697662\pi$
$73$	−52982.4	−1.16366	−0.581828	+	0.813312i	$0.697662\pi$
$74$	−36314.0	−0.770893
$75$	0	0
$76$	−4406.03	−0.0875011
$77$	0	0
$78$	0	0
$79$	−48564.0	−0.875481	−0.437741	−	0.899101i	$-0.644221\pi$
$79$	−48564.0	−0.875481	−0.437741	+	0.899101i	$0.644221\pi$
$80$	−26495.4	−0.462855
$81$	0	0
$82$	72829.7	1.19612
$83$	−113161.	−1.80303	−0.901513	−	0.432753i	$-0.857542\pi$
$83$	−113161.	−1.80303	−0.901513	+	0.432753i	$0.857542\pi$
$84$	0	0
$85$	−133850.	−2.00942
$86$	−46832.9	−0.682818
$87$	0	0
$88$	−15372.2	−0.211607
$89$	−108366.	−1.45017	−0.725083	−	0.688662i	$-0.758199\pi$
$89$	−108366.	−1.45017	−0.725083	+	0.688662i	$0.758199\pi$
$90$	0	0
$91$	0	0
$92$	−60745.2	−0.748242
$93$	0	0
$94$	92194.4	1.07618
$95$	28500.8	0.324002
$96$	0	0
$97$	99641.2	1.07525	0.537625	−	0.843184i	$-0.319321\pi$
$97$	99641.2	1.07525	0.537625	+	0.843184i	$0.319321\pi$
$98$	0	0
$99$	0	0
$100$	121388.	1.21388
$101$	−165200.	−1.61141	−0.805705	−	0.592317i	$-0.798213\pi$
$101$	−165200.	−1.61141	−0.805705	+	0.592317i	$0.798213\pi$
$102$	0	0
$103$	−107247.	−0.996077	−0.498038	−	0.867155i	$-0.665946\pi$
$103$	−107247.	−0.996077	−0.498038	+	0.867155i	$0.665946\pi$
$104$	51550.5	0.467358
$105$	0	0
$106$	−70651.2	−0.610738
$107$	97422.3	0.822620	0.411310	−	0.911496i	$-0.365071\pi$
$107$	97422.3	0.822620	0.411310	+	0.911496i	$0.365071\pi$
$108$	0	0
$109$	59307.2	0.478125	0.239062	−	0.971004i	$-0.423160\pi$
$109$	59307.2	0.478125	0.239062	+	0.971004i	$0.423160\pi$
$110$	99436.6	0.783546
$111$	0	0
$112$	0	0
$113$	−157268.	−1.15863	−0.579313	−	0.815105i	$-0.696679\pi$
$113$	−157268.	−1.15863	−0.579313	+	0.815105i	$0.696679\pi$
$114$	0	0
$115$	392936.	2.77062
$116$	−19640.3	−0.135520
$117$	0	0
$118$	−73505.2	−0.485974
$119$	0	0
$120$	0	0
$121$	−103359.	−0.641780
$122$	45296.3	0.275527
$123$	0	0
$124$	89997.8	0.525627
$125$	−461778.	−2.64337
$126$	0	0
$127$	−92477.3	−0.508775	−0.254387	−	0.967102i	$-0.581874\pi$
$127$	−92477.3	−0.508775	−0.254387	+	0.967102i	$0.581874\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	−333459.	−1.73055
$131$	72328.4	0.368240	0.184120	−	0.982904i	$-0.441057\pi$
$131$	72328.4	0.368240	0.184120	+	0.982904i	$0.441057\pi$
$132$	0	0
$133$	0	0
$134$	144317.	0.694315
$135$	0	0
$136$	82769.0	0.383725
$137$	−225662.	−1.02720	−0.513602	−	0.858028i	$-0.671689\pi$
$137$	−225662.	−1.02720	−0.513602	+	0.858028i	$0.671689\pi$
$138$	0	0
$139$	−327971.	−1.43979	−0.719894	−	0.694084i	$-0.755810\pi$
$139$	−327971.	−1.43979	−0.719894	+	0.694084i	$0.755810\pi$
$140$	0	0
$141$	0	0
$142$	253737.	1.05600
$143$	−193468.	−0.791170
$144$	0	0
$145$	127045.	0.501808
$146$	−211930.	−0.822829
$147$	0	0
$148$	−145256.	−0.545104
$149$	−164011.	−0.605211	−0.302606	−	0.953116i	$-0.597856\pi$
$149$	−164011.	−0.605211	−0.302606	+	0.953116i	$0.597856\pi$
$150$	0	0
$151$	62308.8	0.222386	0.111193	−	0.993799i	$-0.464533\pi$
$151$	62308.8	0.222386	0.111193	+	0.993799i	$0.464533\pi$
$152$	−17624.1	−0.0618726
$153$	0	0
$154$	0	0
$155$	−582159.	−1.94631
$156$	0	0
$157$	−239807.	−0.776447	−0.388224	−	0.921565i	$-0.626911\pi$
$157$	−239807.	−0.776447	−0.388224	+	0.921565i	$0.626911\pi$
$158$	−194256.	−0.619059
$159$	0	0
$160$	−105981.	−0.327288
$161$	0	0
$162$	0	0
$163$	−436795.	−1.28768	−0.643841	−	0.765160i	$-0.722660\pi$
$163$	−436795.	−1.28768	−0.643841	+	0.765160i	$0.722660\pi$
$164$	291319.	0.845782
$165$	0	0
$166$	−452644.	−1.27493
$167$	−38846.7	−0.107786	−0.0538931	−	0.998547i	$-0.517163\pi$
$167$	−38846.7	−0.107786	−0.0538931	+	0.998547i	$0.517163\pi$
$168$	0	0
$169$	277501.	0.747390
$170$	−535399.	−1.42087
$171$	0	0
$172$	−187331.	−0.482825
$173$	−396277.	−1.00666	−0.503331	−	0.864093i	$-0.667892\pi$
$173$	−396277.	−1.00666	−0.503331	+	0.864093i	$0.667892\pi$
$174$	0	0
$175$	0	0
$176$	−61488.9	−0.149629
$177$	0	0
$178$	−433464.	−1.02542
$179$	−243883.	−0.568918	−0.284459	−	0.958688i	$-0.591814\pi$
$179$	−243883.	−0.568918	−0.284459	+	0.958688i	$0.591814\pi$
$180$	0	0
$181$	−299273.	−0.679002	−0.339501	−	0.940606i	$-0.610258\pi$
$181$	−299273.	−0.679002	−0.339501	+	0.940606i	$0.610258\pi$
$182$	0	0
$183$	0	0
$184$	−242981.	−0.529087
$185$	939601.	2.01843
$186$	0	0
$187$	−310631.	−0.649592
$188$	368778.	0.760974
$189$	0	0
$190$	114003.	0.229104
$191$	776996.	1.54112	0.770558	−	0.637369i	$-0.219977\pi$
$191$	776996.	1.54112	0.770558	+	0.637369i	$0.219977\pi$
$192$	0	0
$193$	276421.	0.534168	0.267084	−	0.963673i	$-0.413940\pi$
$193$	276421.	0.534168	0.267084	+	0.963673i	$0.413940\pi$
$194$	398565.	0.760317
$195$	0	0
$196$	0	0
$197$	131615.	0.241624	0.120812	−	0.992675i	$-0.461450\pi$
$197$	131615.	0.241624	0.120812	+	0.992675i	$0.461450\pi$
$198$	0	0
$199$	564960.	1.01131	0.505656	−	0.862735i	$-0.331250\pi$
$199$	564960.	1.01131	0.505656	+	0.862735i	$0.331250\pi$
$200$	485551.	0.858340
$201$	0	0
$202$	−660799.	−1.13944
$203$	0	0
$204$	0	0
$205$	−1.88442e6	−3.13180
$206$	−428989.	−0.704333
$207$	0	0
$208$	206202.	0.330472
$209$	66143.0	0.104741
$210$	0	0
$211$	313637.	0.484977	0.242489	−	0.970154i	$-0.422036\pi$
$211$	313637.	0.484977	0.242489	+	0.970154i	$0.422036\pi$
$212$	−282605.	−0.431857
$213$	0	0
$214$	389689.	0.581680
$215$	1.21177e6	1.78782
$216$	0	0
$217$	0	0
$218$	237229.	0.338085
$219$	0	0
$220$	397746.	0.554051
$221$	1.04170e6	1.43470
$222$	0	0
$223$	−279483.	−0.376351	−0.188176	−	0.982135i	$-0.560257\pi$
$223$	−279483.	−0.376351	−0.188176	+	0.982135i	$0.560257\pi$
$224$	0	0
$225$	0	0
$226$	−629071.	−0.819273
$227$	1.22351e6	1.57595	0.787974	−	0.615709i	$-0.211130\pi$
$227$	1.22351e6	1.57595	0.787974	+	0.615709i	$0.211130\pi$
$228$	0	0
$229$	254775.	0.321046	0.160523	−	0.987032i	$-0.448682\pi$
$229$	254775.	0.321046	0.160523	+	0.987032i	$0.448682\pi$
$230$	1.57174e6	1.95912
$231$	0	0
$232$	−78561.2	−0.0958270
$233$	−746173.	−0.900429	−0.450215	−	0.892920i	$-0.648653\pi$
$233$	−746173.	−0.900429	−0.450215	+	0.892920i	$0.648653\pi$
$234$	0	0
$235$	−2.38547e6	−2.81776
$236$	−294021.	−0.343636
$237$	0	0
$238$	0	0
$239$	−321036.	−0.363546	−0.181773	−	0.983341i	$-0.558184\pi$
$239$	−321036.	−0.363546	−0.181773	+	0.983341i	$0.558184\pi$
$240$	0	0
$241$	1.24254e6	1.37805	0.689027	−	0.724735i	$-0.258038\pi$
$241$	1.24254e6	1.37805	0.689027	+	0.724735i	$0.258038\pi$
$242$	−413437.	−0.453807
$243$	0	0
$244$	181185.	0.194827
$245$	0	0
$246$	0	0
$247$	−221810.	−0.231333
$248$	359991.	0.371675
$249$	0	0
$250$	−1.84711e6	−1.86914
$251$	−690235.	−0.691532	−0.345766	−	0.938321i	$-0.612381\pi$
$251$	−690235.	−0.691532	−0.345766	+	0.938321i	$0.612381\pi$
$252$	0	0
$253$	911902.	0.895668
$254$	−369909.	−0.359758
$255$	0	0
$256$	65536.0	0.0625000
$257$	−13786.7	−0.0130205	−0.00651024	−	0.999979i	$-0.502072\pi$
$257$	−13786.7	−0.0130205	−0.00651024	+	0.999979i	$0.502072\pi$
$258$	0	0
$259$	0	0
$260$	−1.33384e6	−1.22369
$261$	0	0
$262$	289314.	0.260385
$263$	−574833.	−0.512451	−0.256226	−	0.966617i	$-0.582479\pi$
$263$	−574833.	−0.512451	−0.256226	+	0.966617i	$0.582479\pi$
$264$	0	0
$265$	1.82806e6	1.59910
$266$	0	0
$267$	0	0
$268$	577270.	0.490955
$269$	1.11893e6	0.942804	0.471402	−	0.881918i	$-0.343748\pi$
$269$	1.11893e6	0.942804	0.471402	+	0.881918i	$0.343748\pi$
$270$	0	0
$271$	694869.	0.574751	0.287376	−	0.957818i	$-0.407217\pi$
$271$	694869.	0.574751	0.287376	+	0.957818i	$0.407217\pi$
$272$	331076.	0.271335
$273$	0	0
$274$	−902648.	−0.726343
$275$	−1.82226e6	−1.45305
$276$	0	0
$277$	−837654.	−0.655941	−0.327971	−	0.944688i	$-0.606365\pi$
$277$	−837654.	−0.655941	−0.327971	+	0.944688i	$0.606365\pi$
$278$	−1.31188e6	−1.01808
$279$	0	0
$280$	0	0
$281$	917687.	0.693312	0.346656	−	0.937992i	$-0.387317\pi$
$281$	917687.	0.693312	0.346656	+	0.937992i	$0.387317\pi$
$282$	0	0
$283$	−415263.	−0.308218	−0.154109	−	0.988054i	$-0.549251\pi$
$283$	−415263.	−0.308218	−0.154109	+	0.988054i	$0.549251\pi$
$284$	1.01495e6	0.746703
$285$	0	0
$286$	−773873.	−0.559442
$287$	0	0
$288$	0	0
$289$	252680.	0.177962
$290$	508180.	0.354832
$291$	0	0
$292$	−847718.	−0.581828
$293$	1.21491e6	0.826752	0.413376	−	0.910561i	$-0.364350\pi$
$293$	1.21491e6	0.826752	0.413376	+	0.910561i	$0.364350\pi$
$294$	0	0
$295$	1.90190e6	1.27243
$296$	−581023.	−0.385447
$297$	0	0
$298$	−656043.	−0.427949
$299$	−3.05805e6	−1.97819
$300$	0	0
$301$	0	0
$302$	249235.	0.157250
$303$	0	0
$304$	−70496.5	−0.0437505
$305$	−1.17201e6	−0.721412
$306$	0	0
$307$	−2.60119e6	−1.57517	−0.787584	−	0.616207i	$-0.788668\pi$
$307$	−2.60119e6	−1.57517	−0.787584	+	0.616207i	$0.788668\pi$
$308$	0	0
$309$	0	0
$310$	−2.32864e6	−1.37625
$311$	819164.	0.480253	0.240126	−	0.970742i	$-0.422811\pi$
$311$	819164.	0.480253	0.240126	+	0.970742i	$0.422811\pi$
$312$	0	0
$313$	−881939.	−0.508836	−0.254418	−	0.967094i	$-0.581884\pi$
$313$	−881939.	−0.508836	−0.254418	+	0.967094i	$0.581884\pi$
$314$	−959226.	−0.549031
$315$	0	0
$316$	−777024.	−0.437741
$317$	1.60438e6	0.896726	0.448363	−	0.893852i	$-0.352007\pi$
$317$	1.60438e6	0.896726	0.448363	+	0.893852i	$0.352007\pi$
$318$	0	0
$319$	294839.	0.162221
$320$	−423926.	−0.231427
$321$	0	0
$322$	0	0
$323$	−356135.	−0.189937
$324$	0	0
$325$	6.11094e6	3.20922
$326$	−1.74718e6	−0.910528
$327$	0	0
$328$	1.16527e6	0.598058
$329$	0	0
$330$	0	0
$331$	1.00308e6	0.503229	0.251614	−	0.967828i	$-0.419039\pi$
$331$	1.00308e6	0.503229	0.251614	+	0.967828i	$0.419039\pi$
$332$	−1.81058e6	−0.901513
$333$	0	0
$334$	−155387.	−0.0762164
$335$	−3.73412e6	−1.81793
$336$	0	0
$337$	−2.62317e6	−1.25821	−0.629103	−	0.777322i	$-0.716578\pi$
$337$	−2.62317e6	−1.25821	−0.629103	+	0.777322i	$0.716578\pi$
$338$	1.11000e6	0.528484
$339$	0	0
$340$	−2.14160e6	−1.00471
$341$	−1.35104e6	−0.629191
$342$	0	0
$343$	0	0
$344$	−749326.	−0.341409
$345$	0	0
$346$	−1.58511e6	−0.711818
$347$	−226234.	−0.100863	−0.0504317	−	0.998728i	$-0.516060\pi$
$347$	−226234.	−0.100863	−0.0504317	+	0.998728i	$0.516060\pi$
$348$	0	0
$349$	−1.46280e6	−0.642866	−0.321433	−	0.946932i	$-0.604165\pi$
$349$	−1.46280e6	−0.642866	−0.321433	+	0.946932i	$0.604165\pi$
$350$	0	0
$351$	0	0
$352$	−245955.	−0.105803
$353$	−2.80957e6	−1.20006	−0.600031	−	0.799977i	$-0.704845\pi$
$353$	−2.80957e6	−1.20006	−0.600031	+	0.799977i	$0.704845\pi$
$354$	0	0
$355$	−6.56528e6	−2.76492
$356$	−1.73385e6	−0.725083
$357$	0	0
$358$	−975534.	−0.402286
$359$	3.69900e6	1.51478	0.757388	−	0.652965i	$-0.226475\pi$
$359$	3.69900e6	1.51478	0.757388	+	0.652965i	$0.226475\pi$
$360$	0	0
$361$	−2.40027e6	−0.969374
$362$	−1.19709e6	−0.480127
$363$	0	0
$364$	0	0
$365$	5.48354e6	2.15441
$366$	0	0
$367$	1.16925e6	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$367$	1.16925e6	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$368$	−971923.	−0.374121
$369$	0	0
$370$	3.75840e6	1.42725
$371$	0	0
$372$	0	0
$373$	−4.57690e6	−1.70333	−0.851666	−	0.524084i	$-0.824408\pi$
$373$	−4.57690e6	−1.70333	−0.851666	+	0.524084i	$0.824408\pi$
$374$	−1.24252e6	−0.459331
$375$	0	0
$376$	1.47511e6	0.538090
$377$	−988738.	−0.358284
$378$	0	0
$379$	−2.35520e6	−0.842229	−0.421114	−	0.907007i	$-0.638361\pi$
$379$	−2.35520e6	−0.842229	−0.421114	+	0.907007i	$0.638361\pi$
$380$	456013.	0.162001
$381$	0	0
$382$	3.10799e6	1.08973
$383$	−3.03946e6	−1.05876	−0.529382	−	0.848384i	$-0.677576\pi$
$383$	−3.03946e6	−1.05876	−0.529382	+	0.848384i	$0.677576\pi$
$384$	0	0
$385$	0	0
$386$	1.10568e6	0.377714
$387$	0	0
$388$	1.59426e6	0.537625
$389$	459273.	0.153885	0.0769426	−	0.997036i	$-0.475484\pi$
$389$	459273.	0.153885	0.0769426	+	0.997036i	$0.475484\pi$
$390$	0	0
$391$	−4.90998e6	−1.62419
$392$	0	0
$393$	0	0
$394$	526461.	0.170854
$395$	5.02625e6	1.62088
$396$	0	0
$397$	5.42500e6	1.72752	0.863761	−	0.503901i	$-0.168102\pi$
$397$	5.42500e6	1.72752	0.863761	+	0.503901i	$0.168102\pi$
$398$	2.25984e6	0.715106
$399$	0	0
$400$	1.94220e6	0.606938
$401$	−2.11127e6	−0.655667	−0.327833	−	0.944736i	$-0.606318\pi$
$401$	−2.11127e6	−0.655667	−0.327833	+	0.944736i	$0.606318\pi$
$402$	0	0
$403$	4.53070e6	1.38964
$404$	−2.64320e6	−0.805705
$405$	0	0
$406$	0	0
$407$	2.18057e6	0.652506
$408$	0	0
$409$	2.55745e6	0.755959	0.377980	−	0.925814i	$-0.376619\pi$
$409$	2.55745e6	0.755959	0.377980	+	0.925814i	$0.376619\pi$
$410$	−7.53769e6	−2.21451
$411$	0	0
$412$	−1.71595e6	−0.498038
$413$	0	0
$414$	0	0
$415$	1.17119e7	3.33816
$416$	824809.	0.233679
$417$	0	0
$418$	264572.	0.0740633
$419$	−2.17401e6	−0.604959	−0.302480	−	0.953156i	$-0.597814\pi$
$419$	−2.17401e6	−0.604959	−0.302480	+	0.953156i	$0.597814\pi$
$420$	0	0
$421$	−5.47930e6	−1.50668	−0.753338	−	0.657633i	$-0.771558\pi$
$421$	−5.47930e6	−1.50668	−0.753338	+	0.657633i	$0.771558\pi$
$422$	1.25455e6	0.342931
$423$	0	0
$424$	−1.13042e6	−0.305369
$425$	9.81166e6	2.63494
$426$	0	0
$427$	0	0
$428$	1.55876e6	0.411310
$429$	0	0
$430$	4.84708e6	1.26418
$431$	6.16874e6	1.59957	0.799785	−	0.600286i	$-0.204947\pi$
$431$	6.16874e6	1.59957	0.799785	+	0.600286i	$0.204947\pi$
$432$	0	0
$433$	4.89596e6	1.25493	0.627463	−	0.778647i	$-0.284093\pi$
$433$	4.89596e6	1.25493	0.627463	+	0.778647i	$0.284093\pi$
$434$	0	0
$435$	0	0
$436$	948915.	0.239062
$437$	1.04549e6	0.261888
$438$	0	0
$439$	−3.54588e6	−0.878137	−0.439069	−	0.898454i	$-0.644691\pi$
$439$	−3.54588e6	−0.878137	−0.439069	+	0.898454i	$0.644691\pi$
$440$	1.59099e6	0.391773
$441$	0	0
$442$	4.16679e6	1.01448
$443$	−6.05652e6	−1.46627	−0.733134	−	0.680084i	$-0.761943\pi$
$443$	−6.05652e6	−1.46627	−0.733134	+	0.680084i	$0.761943\pi$
$444$	0	0
$445$	1.12156e7	2.68486
$446$	−1.11793e6	−0.266121
$447$	0	0
$448$	0	0
$449$	1.20801e6	0.282783	0.141392	−	0.989954i	$-0.454842\pi$
$449$	1.20801e6	0.282783	0.141392	+	0.989954i	$0.454842\pi$
$450$	0	0
$451$	−4.37326e6	−1.01243
$452$	−2.51628e6	−0.579313
$453$	0	0
$454$	4.89403e6	1.11436
$455$	0	0
$456$	0	0
$457$	−5.46550e6	−1.22416	−0.612082	−	0.790794i	$-0.709668\pi$
$457$	−5.46550e6	−1.22416	−0.612082	+	0.790794i	$0.709668\pi$
$458$	1.01910e6	0.227014
$459$	0	0
$460$	6.28697e6	1.38531
$461$	−1.15956e6	−0.254121	−0.127061	−	0.991895i	$-0.540554\pi$
$461$	−1.15956e6	−0.254121	−0.127061	+	0.991895i	$0.540554\pi$
$462$	0	0
$463$	−4.31335e6	−0.935108	−0.467554	−	0.883964i	$-0.654865\pi$
$463$	−4.31335e6	−0.935108	−0.467554	+	0.883964i	$0.654865\pi$
$464$	−314245.	−0.0677599
$465$	0	0
$466$	−2.98469e6	−0.636700
$467$	5.98923e6	1.27081	0.635403	−	0.772181i	$-0.280834\pi$
$467$	5.98923e6	1.27081	0.635403	+	0.772181i	$0.280834\pi$
$468$	0	0
$469$	0	0
$470$	−9.54189e6	−1.99246
$471$	0	0
$472$	−1.17608e6	−0.242987
$473$	2.81221e6	0.577956
$474$	0	0
$475$	−2.08921e6	−0.424862
$476$	0	0
$477$	0	0
$478$	−1.28415e6	−0.257066
$479$	−3.54652e6	−0.706258	−0.353129	−	0.935575i	$-0.614882\pi$
$479$	−3.54652e6	−0.706258	−0.353129	+	0.935575i	$0.614882\pi$
$480$	0	0
$481$	−7.31252e6	−1.44113
$482$	4.97015e6	0.974432
$483$	0	0
$484$	−1.65375e6	−0.320890
$485$	−1.03126e7	−1.99074
$486$	0	0
$487$	−8.78320e6	−1.67815	−0.839074	−	0.544017i	$-0.816903\pi$
$487$	−8.78320e6	−1.67815	−0.839074	+	0.544017i	$0.816903\pi$
$488$	724742.	0.137763
$489$	0	0
$490$	0	0
$491$	586608.	0.109811	0.0549053	−	0.998492i	$-0.482514\pi$
$491$	586608.	0.109811	0.0549053	+	0.998492i	$0.482514\pi$
$492$	0	0
$493$	−1.58751e6	−0.294170
$494$	−887239.	−0.163577
$495$	0	0
$496$	1.43997e6	0.262814
$497$	0	0
$498$	0	0
$499$	−9.59772e6	−1.72551	−0.862753	−	0.505626i	$-0.831262\pi$
$499$	−9.59772e6	−1.72551	−0.862753	+	0.505626i	$0.831262\pi$
$500$	−7.38844e6	−1.32168
$501$	0	0
$502$	−2.76094e6	−0.488987
$503$	7.21481e6	1.27147	0.635733	−	0.771909i	$-0.280698\pi$
$503$	7.21481e6	1.27147	0.635733	+	0.771909i	$0.280698\pi$
$504$	0	0
$505$	1.70978e7	2.98340
$506$	3.64761e6	0.633333
$507$	0	0
$508$	−1.47964e6	−0.254387
$509$	−4.38419e6	−0.750058	−0.375029	−	0.927013i	$-0.622367\pi$
$509$	−4.38419e6	−0.750058	−0.375029	+	0.927013i	$0.622367\pi$
$510$	0	0
$511$	0	0
$512$	262144.	0.0441942
$513$	0	0
$514$	−55146.8	−0.00920688
$515$	1.10998e7	1.84416
$516$	0	0
$517$	−5.53606e6	−0.910909
$518$	0	0
$519$	0	0
$520$	−5.33535e6	−0.865276
$521$	−2.06581e6	−0.333424	−0.166712	−	0.986006i	$-0.553315\pi$
$521$	−2.06581e6	−0.333424	−0.166712	+	0.986006i	$0.553315\pi$
$522$	0	0
$523$	7.96570e6	1.27341	0.636707	−	0.771106i	$-0.280296\pi$
$523$	7.96570e6	1.27341	0.636707	+	0.771106i	$0.280296\pi$
$524$	1.15725e6	0.184120
$525$	0	0
$526$	−2.29933e6	−0.362358
$527$	7.27445e6	1.14097
$528$	0	0
$529$	7.97762e6	1.23946
$530$	7.31222e6	1.13073
$531$	0	0
$532$	0	0
$533$	1.46657e7	2.23606
$534$	0	0
$535$	−1.00830e7	−1.52301
$536$	2.30908e6	0.347158
$537$	0	0
$538$	4.47571e6	0.666663
$539$	0	0
$540$	0	0
$541$	8.23423e6	1.20957	0.604784	−	0.796390i	$-0.293260\pi$
$541$	8.23423e6	1.20957	0.604784	+	0.796390i	$0.293260\pi$
$542$	2.77948e6	0.406411
$543$	0	0
$544$	1.32430e6	0.191863
$545$	−6.13815e6	−0.885209
$546$	0	0
$547$	5.20365e6	0.743601	0.371800	−	0.928313i	$-0.378741\pi$
$547$	5.20365e6	0.743601	0.371800	+	0.928313i	$0.378741\pi$
$548$	−3.61059e6	−0.513602
$549$	0	0
$550$	−7.28905e6	−1.02746
$551$	338030.	0.0474325
$552$	0	0
$553$	0	0
$554$	−3.35061e6	−0.463821
$555$	0	0
$556$	−5.24754e6	−0.719894
$557$	4.09165e6	0.558806	0.279403	−	0.960174i	$-0.409864\pi$
$557$	4.09165e6	0.558806	0.279403	+	0.960174i	$0.409864\pi$
$558$	0	0
$559$	−9.43070e6	−1.27648
$560$	0	0
$561$	0	0
$562$	3.67075e6	0.490246
$563$	3.59052e6	0.477404	0.238702	−	0.971093i	$-0.423278\pi$
$563$	3.59052e6	0.477404	0.238702	+	0.971093i	$0.423278\pi$
$564$	0	0
$565$	1.62768e7	2.14510
$566$	−1.66105e6	−0.217943
$567$	0	0
$568$	4.05979e6	0.527999
$569$	−9.79717e6	−1.26859	−0.634293	−	0.773093i	$-0.718709\pi$
$569$	−9.79717e6	−1.26859	−0.634293	+	0.773093i	$0.718709\pi$
$570$	0	0
$571$	−2.21804e6	−0.284694	−0.142347	−	0.989817i	$-0.545465\pi$
$571$	−2.21804e6	−0.284694	−0.142347	+	0.989817i	$0.545465\pi$
$572$	−3.09549e6	−0.395585
$573$	0	0
$574$	0	0
$575$	−2.88036e7	−3.63309
$576$	0	0
$577$	−2.65534e6	−0.332032	−0.166016	−	0.986123i	$-0.553090\pi$
$577$	−2.65534e6	−0.332032	−0.166016	+	0.986123i	$0.553090\pi$
$578$	1.01072e6	0.125838
$579$	0	0
$580$	2.03272e6	0.250904
$581$	0	0
$582$	0	0
$583$	4.24244e6	0.516945
$584$	−3.39087e6	−0.411414
$585$	0	0
$586$	4.85964e6	0.584602
$587$	2.62690e6	0.314665	0.157332	−	0.987546i	$-0.449711\pi$
$587$	2.62690e6	0.314665	0.157332	+	0.987546i	$0.449711\pi$
$588$	0	0
$589$	−1.54896e6	−0.183972
$590$	7.60760e6	0.899742
$591$	0	0
$592$	−2.32409e6	−0.272552
$593$	1.15062e7	1.34368	0.671839	−	0.740697i	$-0.265505\pi$
$593$	1.15062e7	1.34368	0.671839	+	0.740697i	$0.265505\pi$
$594$	0	0
$595$	0	0
$596$	−2.62417e6	−0.302606
$597$	0	0
$598$	−1.22322e7	−1.39879
$599$	−6.46264e6	−0.735941	−0.367970	−	0.929838i	$-0.619947\pi$
$599$	−6.46264e6	−0.735941	−0.367970	+	0.929838i	$0.619947\pi$
$600$	0	0
$601$	8.55332e6	0.965937	0.482968	−	0.875638i	$-0.339559\pi$
$601$	8.55332e6	0.965937	0.482968	+	0.875638i	$0.339559\pi$
$602$	0	0
$603$	0	0
$604$	996941.	0.111193
$605$	1.06974e7	1.18820
$606$	0	0
$607$	2.99042e6	0.329428	0.164714	−	0.986341i	$-0.447330\pi$
$607$	2.99042e6	0.329428	0.164714	+	0.986341i	$0.447330\pi$
$608$	−281986.	−0.0309363
$609$	0	0
$610$	−4.68806e6	−0.510115
$611$	1.85651e7	2.01185
$612$	0	0
$613$	−1.77145e6	−0.190405	−0.0952023	−	0.995458i	$-0.530350\pi$
$613$	−1.77145e6	−0.190405	−0.0952023	+	0.995458i	$0.530350\pi$
$614$	−1.04048e7	−1.11381
$615$	0	0
$616$	0	0
$617$	−1.40871e7	−1.48974	−0.744869	−	0.667210i	$-0.767488\pi$
$617$	−1.40871e7	−1.48974	−0.744869	+	0.667210i	$0.767488\pi$
$618$	0	0
$619$	−1.13430e7	−1.18988	−0.594940	−	0.803770i	$-0.702824\pi$
$619$	−1.13430e7	−1.18988	−0.594940	+	0.803770i	$0.702824\pi$
$620$	−9.31455e6	−0.973156
$621$	0	0
$622$	3.27666e6	0.339590
$623$	0	0
$624$	0	0
$625$	2.40843e7	2.46623
$626$	−3.52776e6	−0.359801
$627$	0	0
$628$	−3.83690e6	−0.388224
$629$	−1.17409e7	−1.18325
$630$	0	0
$631$	−2.32628e6	−0.232588	−0.116294	−	0.993215i	$-0.537102\pi$
$631$	−2.32628e6	−0.232588	−0.116294	+	0.993215i	$0.537102\pi$
$632$	−3.10810e6	−0.309529
$633$	0	0
$634$	6.41753e6	0.634081
$635$	9.57116e6	0.941956
$636$	0	0
$637$	0	0
$638$	1.17935e6	0.114708
$639$	0	0
$640$	−1.69570e6	−0.163644
$641$	2.56385e6	0.246461	0.123230	−	0.992378i	$-0.460675\pi$
$641$	2.56385e6	0.246461	0.123230	+	0.992378i	$0.460675\pi$
$642$	0	0
$643$	−5.95343e6	−0.567858	−0.283929	−	0.958845i	$-0.591638\pi$
$643$	−5.95343e6	−0.567858	−0.283929	+	0.958845i	$0.591638\pi$
$644$	0	0
$645$	0	0
$646$	−1.42454e6	−0.134306
$647$	1.53522e6	0.144182	0.0720909	−	0.997398i	$-0.477033\pi$
$647$	1.53522e6	0.144182	0.0720909	+	0.997398i	$0.477033\pi$
$648$	0	0
$649$	4.41382e6	0.411342
$650$	2.44437e7	2.26926
$651$	0	0
$652$	−6.98871e6	−0.643841
$653$	1.99582e7	1.83163	0.915815	−	0.401600i	$-0.131546\pi$
$653$	1.99582e7	1.83163	0.915815	+	0.401600i	$0.131546\pi$
$654$	0	0
$655$	−7.48581e6	−0.681766
$656$	4.66110e6	0.422891
$657$	0	0
$658$	0	0
$659$	1.05477e7	0.946119	0.473060	−	0.881030i	$-0.343150\pi$
$659$	1.05477e7	0.946119	0.473060	+	0.881030i	$0.343150\pi$
$660$	0	0
$661$	−1.22044e7	−1.08646	−0.543229	−	0.839585i	$-0.682798\pi$
$661$	−1.22044e7	−1.08646	−0.543229	+	0.839585i	$0.682798\pi$
$662$	4.01232e6	0.355837
$663$	0	0
$664$	−7.24231e6	−0.637466
$665$	0	0
$666$	0	0
$667$	4.66036e6	0.405607
$668$	−621548.	−0.0538931
$669$	0	0
$670$	−1.49365e7	−1.28547
$671$	−2.71994e6	−0.233213
$672$	0	0
$673$	−7.97959e6	−0.679114	−0.339557	−	0.940585i	$-0.610277\pi$
$673$	−7.97959e6	−0.679114	−0.339557	+	0.940585i	$0.610277\pi$
$674$	−1.04927e7	−0.889686
$675$	0	0
$676$	4.44001e6	0.373695
$677$	−5.08381e6	−0.426302	−0.213151	−	0.977019i	$-0.568373\pi$
$677$	−5.08381e6	−0.426302	−0.213151	+	0.977019i	$0.568373\pi$
$678$	0	0
$679$	0	0
$680$	−8.56639e6	−0.710437
$681$	0	0
$682$	−5.40416e6	−0.444905
$683$	1.60331e7	1.31512	0.657559	−	0.753403i	$-0.271589\pi$
$683$	1.60331e7	1.31512	0.657559	+	0.753403i	$0.271589\pi$
$684$	0	0
$685$	2.33554e7	1.90179
$686$	0	0
$687$	0	0
$688$	−2.99730e6	−0.241412
$689$	−1.42270e7	−1.14173
$690$	0	0
$691$	−1.10131e7	−0.877438	−0.438719	−	0.898624i	$-0.644568\pi$
$691$	−1.10131e7	−0.877438	−0.438719	+	0.898624i	$0.644568\pi$
$692$	−6.34044e6	−0.503331
$693$	0	0
$694$	−904935.	−0.0713212
$695$	3.39442e7	2.66565
$696$	0	0
$697$	2.35470e7	1.83592
$698$	−5.85119e6	−0.454575
$699$	0	0
$700$	0	0
$701$	3.10660e6	0.238776	0.119388	−	0.992848i	$-0.461907\pi$
$701$	3.10660e6	0.238776	0.119388	+	0.992848i	$0.461907\pi$
$702$	0	0
$703$	2.50001e6	0.190789
$704$	−983822.	−0.0748143
$705$	0	0
$706$	−1.12383e7	−0.848571
$707$	0	0
$708$	0	0
$709$	−1.38036e7	−1.03128	−0.515640	−	0.856805i	$-0.672446\pi$
$709$	−1.38036e7	−1.03128	−0.515640	+	0.856805i	$0.672446\pi$
$710$	−2.62611e7	−1.95509
$711$	0	0
$712$	−6.93542e6	−0.512711
$713$	−2.13552e7	−1.57319
$714$	0	0
$715$	2.00235e7	1.46479
$716$	−3.90214e6	−0.284459
$717$	0	0
$718$	1.47960e7	1.07111
$719$	1.29855e7	0.936777	0.468389	−	0.883523i	$-0.344835\pi$
$719$	1.29855e7	0.936777	0.468389	+	0.883523i	$0.344835\pi$
$720$	0	0
$721$	0	0
$722$	−9.60107e6	−0.685451
$723$	0	0
$724$	−4.78837e6	−0.339501
$725$	−9.31285e6	−0.658017
$726$	0	0
$727$	−8.61224e6	−0.604338	−0.302169	−	0.953254i	$-0.597711\pi$
$727$	−8.61224e6	−0.604338	−0.302169	+	0.953254i	$0.597711\pi$
$728$	0	0
$729$	0	0
$730$	2.19342e7	1.52340
$731$	−1.51418e7	−1.04806
$732$	0	0
$733$	6.02001e6	0.413844	0.206922	−	0.978357i	$-0.433655\pi$
$733$	6.02001e6	0.413844	0.206922	+	0.978357i	$0.433655\pi$
$734$	4.67698e6	0.320424
$735$	0	0
$736$	−3.88769e6	−0.264544
$737$	−8.66593e6	−0.587688
$738$	0	0
$739$	−93692.5	−0.00631094	−0.00315547	−	0.999995i	$-0.501004\pi$
$739$	−93692.5	−0.00631094	−0.00315547	+	0.999995i	$0.501004\pi$
$740$	1.50336e7	1.00922
$741$	0	0
$742$	0	0
$743$	1.29874e7	0.863081	0.431541	−	0.902094i	$-0.357970\pi$
$743$	1.29874e7	0.863081	0.431541	+	0.902094i	$0.357970\pi$
$744$	0	0
$745$	1.69747e7	1.12050
$746$	−1.83076e7	−1.20444
$747$	0	0
$748$	−4.97009e6	−0.324796
$749$	0	0
$750$	0	0
$751$	−637441.	−0.0412420	−0.0206210	−	0.999787i	$-0.506564\pi$
$751$	−637441.	−0.0412420	−0.0206210	+	0.999787i	$0.506564\pi$
$752$	5.90044e6	0.380487
$753$	0	0
$754$	−3.95495e6	−0.253345
$755$	−6.44880e6	−0.411729
$756$	0	0
$757$	−4.86768e6	−0.308732	−0.154366	−	0.988014i	$-0.549334\pi$
$757$	−4.86768e6	−0.308732	−0.154366	+	0.988014i	$0.549334\pi$
$758$	−9.42081e6	−0.595546
$759$	0	0
$760$	1.82405e6	0.114552
$761$	−5.81787e6	−0.364169	−0.182084	−	0.983283i	$-0.558284\pi$
$761$	−5.81787e6	−0.364169	−0.182084	+	0.983283i	$0.558284\pi$
$762$	0	0
$763$	0	0
$764$	1.24319e7	0.770558
$765$	0	0
$766$	−1.21578e7	−0.748659
$767$	−1.48017e7	−0.908496
$768$	0	0
$769$	1.94310e7	1.18490	0.592448	−	0.805609i	$-0.298161\pi$
$769$	1.94310e7	1.18490	0.592448	+	0.805609i	$0.298161\pi$
$770$	0	0
$771$	0	0
$772$	4.42274e6	0.267084
$773$	−8.24179e6	−0.496104	−0.248052	−	0.968747i	$-0.579790\pi$
$773$	−8.24179e6	−0.496104	−0.248052	+	0.968747i	$0.579790\pi$
$774$	0	0
$775$	4.26743e7	2.55219
$776$	6.37704e6	0.380159
$777$	0	0
$778$	1.83709e6	0.108813
$779$	−5.01390e6	−0.296027
$780$	0	0
$781$	−1.52363e7	−0.893826
$782$	−1.96399e7	−1.14848
$783$	0	0
$784$	0	0
$785$	2.48194e7	1.43753
$786$	0	0
$787$	−1.47863e7	−0.850986	−0.425493	−	0.904962i	$-0.639899\pi$
$787$	−1.47863e7	−0.850986	−0.425493	+	0.904962i	$0.639899\pi$
$788$	2.10585e6	0.120812
$789$	0	0
$790$	2.01050e7	1.14614
$791$	0	0
$792$	0	0
$793$	9.12129e6	0.515079
$794$	2.17000e7	1.22154
$795$	0	0
$796$	9.03937e6	0.505656
$797$	1.58158e7	0.881956	0.440978	−	0.897518i	$-0.354632\pi$
$797$	1.58158e7	0.881956	0.440978	+	0.897518i	$0.354632\pi$
$798$	0	0
$799$	2.98080e7	1.65183
$800$	7.76881e6	0.429170
$801$	0	0
$802$	−8.44508e6	−0.463626
$803$	1.27259e7	0.696465
$804$	0	0
$805$	0	0
$806$	1.81228e7	0.982625
$807$	0	0
$808$	−1.05728e7	−0.569720
$809$	−1.19304e7	−0.640888	−0.320444	−	0.947267i	$-0.603832\pi$
$809$	−1.19304e7	−0.640888	−0.320444	+	0.947267i	$0.603832\pi$
$810$	0	0
$811$	1.89393e7	1.01114	0.505571	−	0.862785i	$-0.331282\pi$
$811$	1.89393e7	1.01114	0.505571	+	0.862785i	$0.331282\pi$
$812$	0	0
$813$	0	0
$814$	8.72228e6	0.461391
$815$	4.52071e7	2.38404
$816$	0	0
$817$	3.22417e6	0.168991
$818$	1.02298e7	0.534544
$819$	0	0
$820$	−3.01507e7	−1.56590
$821$	−3.20087e7	−1.65733	−0.828667	−	0.559741i	$-0.810900\pi$
$821$	−3.20087e7	−1.65733	−0.828667	+	0.559741i	$0.810900\pi$
$822$	0	0
$823$	3.61430e7	1.86005	0.930025	−	0.367495i	$-0.119785\pi$
$823$	3.61430e7	1.86005	0.930025	+	0.367495i	$0.119785\pi$
$824$	−6.86382e6	−0.352166
$825$	0	0
$826$	0	0
$827$	2.45831e7	1.24989	0.624946	−	0.780668i	$-0.285121\pi$
$827$	2.45831e7	1.24989	0.624946	+	0.780668i	$0.285121\pi$
$828$	0	0
$829$	2.10424e7	1.06343	0.531715	−	0.846923i	$-0.321548\pi$
$829$	2.10424e7	1.06343	0.531715	+	0.846923i	$0.321548\pi$
$830$	4.68475e7	2.36043
$831$	0	0
$832$	3.29923e6	0.165236
$833$	0	0
$834$	0	0
$835$	4.02054e6	0.199557
$836$	1.05829e6	0.0523707
$837$	0	0
$838$	−8.69604e6	−0.427771
$839$	−1.12403e7	−0.551283	−0.275642	−	0.961260i	$-0.588890\pi$
$839$	−1.12403e7	−0.551283	−0.275642	+	0.961260i	$0.588890\pi$
$840$	0	0
$841$	−1.90043e7	−0.926537
$842$	−2.19172e7	−1.06538
$843$	0	0
$844$	5.01820e6	0.242489
$845$	−2.87206e7	−1.38373
$846$	0	0
$847$	0	0
$848$	−4.52168e6	−0.215928
$849$	0	0
$850$	3.92466e7	1.86318
$851$	3.44672e7	1.63148
$852$	0	0
$853$	−2.63779e7	−1.24127	−0.620636	−	0.784099i	$-0.713126\pi$
$853$	−2.63779e7	−1.24127	−0.620636	+	0.784099i	$0.713126\pi$
$854$	0	0
$855$	0	0
$856$	6.23503e6	0.290840
$857$	−4.26663e7	−1.98442	−0.992208	−	0.124595i	$-0.960237\pi$
$857$	−4.26663e7	−1.98442	−0.992208	+	0.124595i	$0.960237\pi$
$858$	0	0
$859$	2.92617e7	1.35306	0.676530	−	0.736415i	$-0.263483\pi$
$859$	2.92617e7	1.35306	0.676530	+	0.736415i	$0.263483\pi$
$860$	1.93883e7	0.893911
$861$	0	0
$862$	2.46750e7	1.13107
$863$	−1.39448e7	−0.637359	−0.318680	−	0.947862i	$-0.603239\pi$
$863$	−1.39448e7	−0.637359	−0.318680	+	0.947862i	$0.603239\pi$
$864$	0	0
$865$	4.10137e7	1.86375
$866$	1.95838e7	0.887366
$867$	0	0
$868$	0	0
$869$	1.16646e7	0.523988
$870$	0	0
$871$	2.90611e7	1.29798
$872$	3.79566e6	0.169043
$873$	0	0
$874$	4.18195e6	0.185183
$875$	0	0
$876$	0	0
$877$	3.40772e6	0.149611	0.0748057	−	0.997198i	$-0.476166\pi$
$877$	3.40772e6	0.149611	0.0748057	+	0.997198i	$0.476166\pi$
$878$	−1.41835e7	−0.620937
$879$	0	0
$880$	6.36394e6	0.277025
$881$	3.51391e6	0.152528	0.0762641	−	0.997088i	$-0.475701\pi$
$881$	3.51391e6	0.152528	0.0762641	+	0.997088i	$0.475701\pi$
$882$	0	0
$883$	2.14895e7	0.927524	0.463762	−	0.885960i	$-0.346499\pi$
$883$	2.14895e7	0.927524	0.463762	+	0.885960i	$0.346499\pi$
$884$	1.66671e7	0.717349
$885$	0	0
$886$	−2.42261e7	−1.03681
$887$	−2.45357e7	−1.04710	−0.523550	−	0.851995i	$-0.675393\pi$
$887$	−2.45357e7	−1.04710	−0.523550	+	0.851995i	$0.675393\pi$
$888$	0	0
$889$	0	0
$890$	4.48624e7	1.89849
$891$	0	0
$892$	−4.47173e6	−0.188176
$893$	−6.34705e6	−0.266344
$894$	0	0
$895$	2.52413e7	1.05331
$896$	0	0
$897$	0	0
$898$	4.83203e6	0.199958
$899$	−6.90462e6	−0.284932
$900$	0	0
$901$	−2.28427e7	−0.937423
$902$	−1.74930e7	−0.715894
$903$	0	0
$904$	−1.00651e7	−0.409636
$905$	3.09740e7	1.25712
$906$	0	0
$907$	−8.14146e6	−0.328613	−0.164306	−	0.986409i	$-0.552539\pi$
$907$	−8.14146e6	−0.328613	−0.164306	+	0.986409i	$0.552539\pi$
$908$	1.95761e7	0.787974
$909$	0	0
$910$	0	0
$911$	−1.01384e7	−0.404739	−0.202370	−	0.979309i	$-0.564864\pi$
$911$	−1.01384e7	−0.404739	−0.202370	+	0.979309i	$0.564864\pi$
$912$	0	0
$913$	2.71803e7	1.07914
$914$	−2.18620e7	−0.865615
$915$	0	0
$916$	4.07639e6	0.160523
$917$	0	0
$918$	0	0
$919$	−1.01176e7	−0.395175	−0.197587	−	0.980285i	$-0.563311\pi$
$919$	−1.01176e7	−0.395175	−0.197587	+	0.980285i	$0.563311\pi$
$920$	2.51479e7	0.979562
$921$	0	0
$922$	−4.63824e6	−0.179691
$923$	5.10948e7	1.97412
$924$	0	0
$925$	−6.88760e7	−2.64676
$926$	−1.72534e7	−0.661221
$927$	0	0
$928$	−1.25698e6	−0.0479135
$929$	−2.05111e7	−0.779739	−0.389870	−	0.920870i	$-0.627480\pi$
$929$	−2.05111e7	−0.779739	−0.389870	+	0.920870i	$0.627480\pi$
$930$	0	0
$931$	0	0
$932$	−1.19388e7	−0.450215
$933$	0	0
$934$	2.39569e7	0.898595
$935$	3.21495e7	1.20267
$936$	0	0
$937$	−1.46527e7	−0.545218	−0.272609	−	0.962125i	$-0.587886\pi$
$937$	−1.46527e7	−0.545218	−0.272609	+	0.962125i	$0.587886\pi$
$938$	0	0
$939$	0	0
$940$	−3.81676e7	−1.40888
$941$	−2.59341e7	−0.954765	−0.477382	−	0.878696i	$-0.658414\pi$
$941$	−2.59341e7	−0.954765	−0.477382	+	0.878696i	$0.658414\pi$
$942$	0	0
$943$	−6.91258e7	−2.53140
$944$	−4.70433e6	−0.171818
$945$	0	0
$946$	1.12488e7	0.408676
$947$	−1.04490e7	−0.378615	−0.189307	−	0.981918i	$-0.560624\pi$
$947$	−1.04490e7	−0.378615	−0.189307	+	0.981918i	$0.560624\pi$
$948$	0	0
$949$	−4.26761e7	−1.53822
$950$	−8.35683e6	−0.300423
$951$	0	0
$952$	0	0
$953$	−1.15666e6	−0.0412548	−0.0206274	−	0.999787i	$-0.506566\pi$
$953$	−1.15666e6	−0.0412548	−0.0206274	+	0.999787i	$0.506566\pi$
$954$	0	0
$955$	−8.04172e7	−2.85325
$956$	−5.13658e6	−0.181773
$957$	0	0
$958$	−1.41861e7	−0.499400
$959$	0	0
$960$	0	0
$961$	3.00994e6	0.105135
$962$	−2.92501e7	−1.01904
$963$	0	0
$964$	1.98806e7	0.689027
$965$	−2.86089e7	−0.988969
$966$	0	0
$967$	2.00516e7	0.689577	0.344789	−	0.938680i	$-0.387951\pi$
$967$	2.00516e7	0.689577	0.344789	+	0.938680i	$0.387951\pi$
$968$	−6.61500e6	−0.226904
$969$	0	0
$970$	−4.12505e7	−1.40767
$971$	−6.10209e6	−0.207697	−0.103849	−	0.994593i	$-0.533116\pi$
$971$	−6.10209e6	−0.207697	−0.103849	+	0.994593i	$0.533116\pi$
$972$	0	0
$973$	0	0
$974$	−3.51328e7	−1.18663
$975$	0	0
$976$	2.89897e6	0.0974134
$977$	−1.36743e7	−0.458320	−0.229160	−	0.973389i	$-0.573598\pi$
$977$	−1.36743e7	−0.458320	−0.229160	+	0.973389i	$0.573598\pi$
$978$	0	0
$979$	2.60285e7	0.867945
$980$	0	0
$981$	0	0
$982$	2.34643e6	0.0776478
$983$	1.67033e7	0.551338	0.275669	−	0.961253i	$-0.411101\pi$
$983$	1.67033e7	0.551338	0.275669	+	0.961253i	$0.411101\pi$
$984$	0	0
$985$	−1.36219e7	−0.447348
$986$	−6.35003e6	−0.208010
$987$	0	0
$988$	−3.54896e6	−0.115667
$989$	4.44511e7	1.44508
$990$	0	0
$991$	5.74906e7	1.85957	0.929785	−	0.368102i	$-0.119992\pi$
$991$	5.74906e7	1.85957	0.929785	+	0.368102i	$0.119992\pi$
$992$	5.75986e6	0.185837
$993$	0	0
$994$	0	0
$995$	−5.84720e7	−1.87236
$996$	0	0
$997$	3.94489e7	1.25689	0.628444	−	0.777855i	$-0.283692\pi$
$997$	3.94489e7	1.25689	0.628444	+	0.777855i	$0.283692\pi$
$998$	−3.83909e7	−1.22012
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bk.1.1		2
3.2	odd	2		294.6.a.q.1.2	yes	2
7.6	odd	2		882.6.a.bu.1.2		2
21.2	odd	6		294.6.e.x.67.1		4
21.5	even	6		294.6.e.z.67.2		4
21.11	odd	6		294.6.e.x.79.1		4
21.17	even	6		294.6.e.z.79.2		4
21.20	even	2		294.6.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.n.1.1	✓	2	21.20	even	2
294.6.a.q.1.2	yes	2	3.2	odd	2
294.6.e.x.67.1		4	21.2	odd	6
294.6.e.x.79.1		4	21.11	odd	6
294.6.e.z.67.2		4	21.5	even	6
294.6.e.z.79.2		4	21.17	even	6
882.6.a.bk.1.1		2	1.1	even	1	trivial
882.6.a.bu.1.2		2	7.6	odd	2

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -103.497 q^{5} +64.0000 q^{8} -413.990 q^{10} -240.191 q^{11} +805.477 q^{13} +256.000 q^{16} +1293.27 q^{17} -275.377 q^{19} -1655.96 q^{20} -960.764 q^{22} -3796.57 q^{23} +7586.73 q^{25} +3221.91 q^{26} -1227.52 q^{29} +5624.86 q^{31} +1024.00 q^{32} +5173.06 q^{34} -9078.49 q^{37} -1101.51 q^{38} -6623.84 q^{40} +18207.4 q^{41} -11708.2 q^{43} -3843.05 q^{44} -15186.3 q^{46} +23048.6 q^{47} +30346.9 q^{50} +12887.6 q^{52} -17662.8 q^{53} +24859.2 q^{55} -4910.07 q^{58} -18376.3 q^{59} +11324.1 q^{61} +22499.5 q^{62} +4096.00 q^{64} -83364.9 q^{65} +36079.3 q^{67} +20692.3 q^{68} +63434.2 q^{71} -52982.4 q^{73} -36314.0 q^{74} -4406.03 q^{76} -48564.0 q^{79} -26495.4 q^{80} +72829.7 q^{82} -113161. q^{83} -133850. q^{85} -46832.9 q^{86} -15372.2 q^{88} -108366. q^{89} -60745.2 q^{92} +92194.4 q^{94} +28500.8 q^{95} +99641.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} - 108 q^{5} + 128 q^{8} - 432 q^{10} - 124 q^{11} + 720 q^{13} + 512 q^{16} + 1260 q^{17} + 360 q^{19} - 1728 q^{20} - 496 q^{22} - 6524 q^{23} + 4482 q^{25} + 2880 q^{26} - 7088 q^{29} + 5904 q^{31} + 2048 q^{32} + 5040 q^{34} - 6040 q^{37} + 1440 q^{38} - 6912 q^{40} + 17388 q^{41} - 608 q^{43} - 1984 q^{44} - 26096 q^{46} + 30456 q^{47} + 17928 q^{50} + 11520 q^{52} - 3964 q^{53} + 24336 q^{55} - 28352 q^{58} - 40752 q^{59} - 1368 q^{61} + 23616 q^{62} + 8192 q^{64} - 82980 q^{65} - 16224 q^{67} + 20160 q^{68} + 3204 q^{71} + 23976 q^{73} - 24160 q^{74} + 5760 q^{76} - 82160 q^{79} - 27648 q^{80} + 69552 q^{82} - 173736 q^{83} - 133700 q^{85} - 2432 q^{86} - 7936 q^{88} - 200556 q^{89} - 104384 q^{92} + 121824 q^{94} + 25640 q^{95} + 251928 q^{97}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 - 108 * q^5 + 128 * q^8 - 432 * q^10 - 124 * q^11 + 720 * q^13 + 512 * q^16 + 1260 * q^17 + 360 * q^19 - 1728 * q^20 - 496 * q^22 - 6524 * q^23 + 4482 * q^25 + 2880 * q^26 - 7088 * q^29 + 5904 * q^31 + 2048 * q^32 + 5040 * q^34 - 6040 * q^37 + 1440 * q^38 - 6912 * q^40 + 17388 * q^41 - 608 * q^43 - 1984 * q^44 - 26096 * q^46 + 30456 * q^47 + 17928 * q^50 + 11520 * q^52 - 3964 * q^53 + 24336 * q^55 - 28352 * q^58 - 40752 * q^59 - 1368 * q^61 + 23616 * q^62 + 8192 * q^64 - 82980 * q^65 - 16224 * q^67 + 20160 * q^68 + 3204 * q^71 + 23976 * q^73 - 24160 * q^74 + 5760 * q^76 - 82160 * q^79 - 27648 * q^80 + 69552 * q^82 - 173736 * q^83 - 133700 * q^85 - 2432 * q^86 - 7936 * q^88 - 200556 * q^89 - 104384 * q^92 + 121824 * q^94 + 25640 * q^95 + 251928 * q^97

Introduction

L-functions

Modular forms

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