LMFDB - Embedded newform 441.6.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.44622$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +O(q^{10})$	$q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +304.064 q^{10} -295.570 q^{11} -1148.13 q^{13} -735.307 q^{16} -1032.38 q^{17} +2108.51 q^{19} +1416.19 q^{20} -2496.45 q^{22} +640.988 q^{23} -1829.00 q^{25} -9697.34 q^{26} -7631.58 q^{29} -966.976 q^{31} -8194.05 q^{32} -8719.74 q^{34} -1773.21 q^{37} +17809.0 q^{38} +2231.42 q^{40} +11976.4 q^{41} -19802.9 q^{43} -11627.3 q^{44} +5413.93 q^{46} +27966.1 q^{47} -15448.1 q^{50} -45165.8 q^{52} +7114.33 q^{53} -10640.5 q^{55} -64458.0 q^{58} +20869.5 q^{59} -23868.3 q^{61} -8167.30 q^{62} -45679.0 q^{64} -41332.6 q^{65} +34671.5 q^{67} -40612.6 q^{68} +28413.2 q^{71} -15292.7 q^{73} -14976.9 q^{74} +82946.0 q^{76} -73059.5 q^{79} -26471.0 q^{80} +101155. q^{82} +30340.9 q^{83} -37165.8 q^{85} -167260. q^{86} -18320.6 q^{88} +36089.5 q^{89} +25215.6 q^{92} +236208. q^{94} +75906.4 q^{95} -153963. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8	$ 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100}) $ 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * 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 $

$2$	8.44622	1.49310	0.746548	−	0.665332i	$-0.231710\pi$
$2$	8.44622	1.49310	0.746548	+	0.665332i	$0.231710\pi$
$3$	0	0
$3$	0	0
$4$	39.3387	1.22933
$5$	36.0000	0.643988	0.321994	−	0.946742i	$-0.395647\pi$
$5$	36.0000	0.643988	0.321994	+	0.946742i	$0.395647\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	61.9840	0.342416
$9$	0	0
$10$	304.064	0.961535
$11$	−295.570	−0.736509	−0.368255	−	0.929725i	$-0.620045\pi$
$11$	−295.570	−0.736509	−0.368255	+	0.929725i	$0.620045\pi$
$12$	0	0
$13$	−1148.13	−1.88422	−0.942111	−	0.335302i	$-0.891162\pi$
$13$	−1148.13	−1.88422	−0.942111	+	0.335302i	$0.891162\pi$
$14$	0	0
$15$	0	0
$16$	−735.307	−0.718073
$17$	−1032.38	−0.866401	−0.433200	−	0.901298i	$-0.642616\pi$
$17$	−1032.38	−0.866401	−0.433200	+	0.901298i	$0.642616\pi$
$18$	0	0
$19$	2108.51	1.33996	0.669980	−	0.742379i	$-0.266303\pi$
$19$	2108.51	1.33996	0.669980	+	0.742379i	$0.266303\pi$
$20$	1416.19	0.791675
$21$	0	0
$22$	−2496.45	−1.09968
$23$	640.988	0.252657	0.126328	−	0.991988i	$-0.459681\pi$
$23$	640.988	0.252657	0.126328	+	0.991988i	$0.459681\pi$
$24$	0	0
$25$	−1829.00	−0.585280
$26$	−9697.34	−2.81332
$27$	0	0
$28$	0	0
$29$	−7631.58	−1.68508	−0.842538	−	0.538637i	$-0.818940\pi$
$29$	−7631.58	−1.68508	−0.842538	+	0.538637i	$0.818940\pi$
$30$	0	0
$31$	−966.976	−0.180722	−0.0903611	−	0.995909i	$-0.528802\pi$
$31$	−966.976	−0.180722	−0.0903611	+	0.995909i	$0.528802\pi$
$32$	−8194.05	−1.41457
$33$	0	0
$34$	−8719.74	−1.29362
$35$	0	0
$36$	0	0
$37$	−1773.21	−0.212939	−0.106470	−	0.994316i	$-0.533955\pi$
$37$	−1773.21	−0.212939	−0.106470	+	0.994316i	$0.533955\pi$
$38$	17809.0	2.00069
$39$	0	0
$40$	2231.42	0.220512
$41$	11976.4	1.11267	0.556335	−	0.830958i	$-0.312207\pi$
$41$	11976.4	1.11267	0.556335	+	0.830958i	$0.312207\pi$
$42$	0	0
$43$	−19802.9	−1.63327	−0.816636	−	0.577153i	$-0.804163\pi$
$43$	−19802.9	−1.63327	−0.816636	+	0.577153i	$0.804163\pi$
$44$	−11627.3	−0.905416
$45$	0	0
$46$	5413.93	0.377240
$47$	27966.1	1.84666	0.923332	−	0.384002i	$-0.125455\pi$
$47$	27966.1	1.84666	0.923332	+	0.384002i	$0.125455\pi$
$48$	0	0
$49$	0	0
$50$	−15448.1	−0.873879
$51$	0	0
$52$	−45165.8	−2.31634
$53$	7114.33	0.347892	0.173946	−	0.984755i	$-0.444348\pi$
$53$	7114.33	0.347892	0.173946	+	0.984755i	$0.444348\pi$
$54$	0	0
$55$	−10640.5	−0.474303
$56$	0	0
$57$	0	0
$58$	−64458.0	−2.51598
$59$	20869.5	0.780518	0.390259	−	0.920705i	$-0.372386\pi$
$59$	20869.5	0.780518	0.390259	+	0.920705i	$0.372386\pi$
$60$	0	0
$61$	−23868.3	−0.821291	−0.410646	−	0.911795i	$-0.634697\pi$
$61$	−23868.3	−0.821291	−0.410646	+	0.911795i	$0.634697\pi$
$62$	−8167.30	−0.269835
$63$	0	0
$64$	−45679.0	−1.39401
$65$	−41332.6	−1.21342
$66$	0	0
$67$	34671.5	0.943595	0.471798	−	0.881707i	$-0.343605\pi$
$67$	34671.5	0.943595	0.471798	+	0.881707i	$0.343605\pi$
$68$	−40612.6	−1.06510
$69$	0	0
$70$	0	0
$71$	28413.2	0.668921	0.334461	−	0.942410i	$-0.391446\pi$
$71$	28413.2	0.668921	0.334461	+	0.942410i	$0.391446\pi$
$72$	0	0
$73$	−15292.7	−0.335874	−0.167937	−	0.985798i	$-0.553711\pi$
$73$	−15292.7	−0.335874	−0.167937	+	0.985798i	$0.553711\pi$
$74$	−14976.9	−0.317939
$75$	0	0
$76$	82946.0	1.64726
$77$	0	0
$78$	0	0
$79$	−73059.5	−1.31707	−0.658535	−	0.752550i	$-0.728824\pi$
$79$	−73059.5	−1.31707	−0.658535	+	0.752550i	$0.728824\pi$
$80$	−26471.0	−0.462430
$81$	0	0
$82$	101155.	1.66132
$83$	30340.9	0.483429	0.241715	−	0.970347i	$-0.422290\pi$
$83$	30340.9	0.483429	0.241715	+	0.970347i	$0.422290\pi$
$84$	0	0
$85$	−37165.8	−0.557951
$86$	−167260.	−2.43863
$87$	0	0
$88$	−18320.6	−0.252193
$89$	36089.5	0.482954	0.241477	−	0.970407i	$-0.422368\pi$
$89$	36089.5	0.482954	0.241477	+	0.970407i	$0.422368\pi$
$90$	0	0
$91$	0	0
$92$	25215.6	0.310599
$93$	0	0
$94$	236208.	2.75725
$95$	75906.4	0.862918
$96$	0	0
$97$	−153963.	−1.66145	−0.830724	−	0.556685i	$-0.812073\pi$
$97$	−153963.	−1.66145	−0.830724	+	0.556685i	$0.812073\pi$
$98$	0	0
$99$	0	0
$100$	−71950.4	−0.719504
$101$	−139809.	−1.36374	−0.681869	−	0.731474i	$-0.738833\pi$
$101$	−139809.	−1.36374	−0.681869	+	0.731474i	$0.738833\pi$
$102$	0	0
$103$	−115925.	−1.07668	−0.538339	−	0.842728i	$-0.680948\pi$
$103$	−115925.	−1.07668	−0.538339	+	0.842728i	$0.680948\pi$
$104$	−71165.6	−0.645188
$105$	0	0
$106$	60089.2	0.519436
$107$	−83061.8	−0.701361	−0.350681	−	0.936495i	$-0.614050\pi$
$107$	−83061.8	−0.701361	−0.350681	+	0.936495i	$0.614050\pi$
$108$	0	0
$109$	45356.2	0.365654	0.182827	−	0.983145i	$-0.441475\pi$
$109$	45356.2	0.365654	0.182827	+	0.983145i	$0.441475\pi$
$110$	−89872.1	−0.708179
$111$	0	0
$112$	0	0
$113$	355.533	0.00261929	0.00130965	−	0.999999i	$-0.499583\pi$
$113$	355.533	0.00261929	0.00130965	+	0.999999i	$0.499583\pi$
$114$	0	0
$115$	23075.6	0.162708
$116$	−300216.	−2.07152
$117$	0	0
$118$	176269.	1.16539
$119$	0	0
$120$	0	0
$121$	−73689.5	−0.457554
$122$	−201597.	−1.22627
$123$	0	0
$124$	−38039.6	−0.222168
$125$	−178344.	−1.02090
$126$	0	0
$127$	168967.	0.929593	0.464797	−	0.885417i	$-0.346127\pi$
$127$	168967.	0.929593	0.464797	+	0.885417i	$0.346127\pi$
$128$	−123605.	−0.666824
$129$	0	0
$130$	−349104.	−1.81174
$131$	173969.	0.885715	0.442858	−	0.896592i	$-0.353965\pi$
$131$	173969.	0.885715	0.442858	+	0.896592i	$0.353965\pi$
$132$	0	0
$133$	0	0
$134$	292843.	1.40888
$135$	0	0
$136$	−63991.3	−0.296670
$137$	−367723.	−1.67386	−0.836931	−	0.547308i	$-0.815653\pi$
$137$	−367723.	−1.67386	−0.836931	+	0.547308i	$0.815653\pi$
$138$	0	0
$139$	217967.	0.956870	0.478435	−	0.878123i	$-0.341204\pi$
$139$	217967.	0.956870	0.478435	+	0.878123i	$0.341204\pi$
$140$	0	0
$141$	0	0
$142$	239985.	0.998763
$143$	339352.	1.38775
$144$	0	0
$145$	−274737.	−1.08517
$146$	−129165.	−0.501492
$147$	0	0
$148$	−69755.7	−0.261773
$149$	64906.1	0.239508	0.119754	−	0.992804i	$-0.461789\pi$
$149$	64906.1	0.239508	0.119754	+	0.992804i	$0.461789\pi$
$150$	0	0
$151$	−223777.	−0.798681	−0.399341	−	0.916803i	$-0.630761\pi$
$151$	−223777.	−0.798681	−0.399341	+	0.916803i	$0.630761\pi$
$152$	130694.	0.458825
$153$	0	0
$154$	0	0
$155$	−34811.1	−0.116383
$156$	0	0
$157$	−459973.	−1.48930	−0.744652	−	0.667453i	$-0.767384\pi$
$157$	−459973.	−1.48930	−0.744652	+	0.667453i	$0.767384\pi$
$158$	−617077.	−1.96651
$159$	0	0
$160$	−294986.	−0.910964
$161$	0	0
$162$	0	0
$163$	91068.6	0.268472	0.134236	−	0.990949i	$-0.457142\pi$
$163$	91068.6	0.268472	0.134236	+	0.990949i	$0.457142\pi$
$164$	471135.	1.36784
$165$	0	0
$166$	256266.	0.721806
$167$	314772.	0.873384	0.436692	−	0.899611i	$-0.356150\pi$
$167$	314772.	0.873384	0.436692	+	0.899611i	$0.356150\pi$
$168$	0	0
$169$	946905.	2.55029
$170$	−313911.	−0.833075
$171$	0	0
$172$	−779021.	−2.00784
$173$	362143.	0.919951	0.459975	−	0.887932i	$-0.347858\pi$
$173$	362143.	0.919951	0.459975	+	0.887932i	$0.347858\pi$
$174$	0	0
$175$	0	0
$176$	217334.	0.528867
$177$	0	0
$178$	304820.	0.721096
$179$	173896.	0.405656	0.202828	−	0.979214i	$-0.434987\pi$
$179$	173896.	0.405656	0.202828	+	0.979214i	$0.434987\pi$
$180$	0	0
$181$	134973.	0.306233	0.153116	−	0.988208i	$-0.451069\pi$
$181$	134973.	0.306233	0.153116	+	0.988208i	$0.451069\pi$
$182$	0	0
$183$	0	0
$184$	39731.0	0.0865138
$185$	−63835.6	−0.137130
$186$	0	0
$187$	305141.	0.638112
$188$	1.10015e6	2.27017
$189$	0	0
$190$	641123.	1.28842
$191$	−181413.	−0.359821	−0.179910	−	0.983683i	$-0.557581\pi$
$191$	−181413.	−0.359821	−0.179910	+	0.983683i	$0.557581\pi$
$192$	0	0
$193$	965999.	1.86674	0.933369	−	0.358919i	$-0.116855\pi$
$193$	965999.	1.86674	0.933369	+	0.358919i	$0.116855\pi$
$194$	−1.30040e6	−2.48070
$195$	0	0
$196$	0	0
$197$	699058.	1.28336	0.641679	−	0.766974i	$-0.278238\pi$
$197$	699058.	1.28336	0.641679	+	0.766974i	$0.278238\pi$
$198$	0	0
$199$	−416191.	−0.745006	−0.372503	−	0.928031i	$-0.621500\pi$
$199$	−416191.	−0.745006	−0.372503	+	0.928031i	$0.621500\pi$
$200$	−113369.	−0.200410
$201$	0	0
$202$	−1.18086e6	−2.03619
$203$	0	0
$204$	0	0
$205$	431150.	0.716545
$206$	−979132.	−1.60758
$207$	0	0
$208$	844226.	1.35301
$209$	−623212.	−0.986894
$210$	0	0
$211$	−407152.	−0.629580	−0.314790	−	0.949161i	$-0.601934\pi$
$211$	−407152.	−0.629580	−0.314790	+	0.949161i	$0.601934\pi$
$212$	279868.	0.427675
$213$	0	0
$214$	−701558.	−1.04720
$215$	−712906.	−1.05181
$216$	0	0
$217$	0	0
$218$	383089.	0.545957
$219$	0	0
$220$	−418584.	−0.583076
$221$	1.18531e6	1.63249
$222$	0	0
$223$	882022.	1.18773	0.593865	−	0.804565i	$-0.297602\pi$
$223$	882022.	1.18773	0.593865	+	0.804565i	$0.297602\pi$
$224$	0	0
$225$	0	0
$226$	3002.91	0.00391085
$227$	1.12650e6	1.45100	0.725499	−	0.688223i	$-0.241609\pi$
$227$	1.12650e6	1.45100	0.725499	+	0.688223i	$0.241609\pi$
$228$	0	0
$229$	310084.	0.390743	0.195371	−	0.980729i	$-0.437409\pi$
$229$	310084.	0.390743	0.195371	+	0.980729i	$0.437409\pi$
$230$	194902.	0.242938
$231$	0	0
$232$	−473036.	−0.576998
$233$	−1.13654e6	−1.37149	−0.685746	−	0.727841i	$-0.740524\pi$
$233$	−1.13654e6	−1.37149	−0.685746	+	0.727841i	$0.740524\pi$
$234$	0	0
$235$	1.00678e6	1.18923
$236$	820980.	0.959516
$237$	0	0
$238$	0	0
$239$	−87506.8	−0.0990940	−0.0495470	−	0.998772i	$-0.515778\pi$
$239$	−87506.8	−0.0990940	−0.0495470	+	0.998772i	$0.515778\pi$
$240$	0	0
$241$	537768.	0.596421	0.298210	−	0.954500i	$-0.403610\pi$
$241$	537768.	0.596421	0.298210	+	0.954500i	$0.403610\pi$
$242$	−622398.	−0.683171
$243$	0	0
$244$	−938948.	−1.00964
$245$	0	0
$246$	0	0
$247$	−2.42084e6	−2.52478
$248$	−59937.1	−0.0618823
$249$	0	0
$250$	−1.50633e6	−1.52430
$251$	−1.35353e6	−1.35607	−0.678036	−	0.735028i	$-0.737169\pi$
$251$	−1.35353e6	−1.35607	−0.678036	+	0.735028i	$0.737169\pi$
$252$	0	0
$253$	−189457.	−0.186084
$254$	1.42713e6	1.38797
$255$	0	0
$256$	417731.	0.398380
$257$	976900.	0.922608	0.461304	−	0.887242i	$-0.347382\pi$
$257$	976900.	0.922608	0.461304	+	0.887242i	$0.347382\pi$
$258$	0	0
$259$	0	0
$260$	−1.62597e6	−1.49169
$261$	0	0
$262$	1.46938e6	1.32246
$263$	1.24375e6	1.10877	0.554387	−	0.832259i	$-0.312953\pi$
$263$	1.24375e6	1.10877	0.554387	+	0.832259i	$0.312953\pi$
$264$	0	0
$265$	256116.	0.224038
$266$	0	0
$267$	0	0
$268$	1.36393e6	1.15999
$269$	1.08408e6	0.913445	0.456722	−	0.889609i	$-0.349023\pi$
$269$	1.08408e6	0.913445	0.456722	+	0.889609i	$0.349023\pi$
$270$	0	0
$271$	−2.16627e6	−1.79180	−0.895900	−	0.444256i	$-0.853468\pi$
$271$	−2.16627e6	−1.79180	−0.895900	+	0.444256i	$0.853468\pi$
$272$	759119.	0.622139
$273$	0	0
$274$	−3.10587e6	−2.49924
$275$	540597.	0.431064
$276$	0	0
$277$	253859.	0.198789	0.0993946	−	0.995048i	$-0.468309\pi$
$277$	253859.	0.198789	0.0993946	+	0.995048i	$0.468309\pi$
$278$	1.84099e6	1.42870
$279$	0	0
$280$	0	0
$281$	−1.14116e6	−0.862143	−0.431072	−	0.902318i	$-0.641864\pi$
$281$	−1.14116e6	−0.862143	−0.431072	+	0.902318i	$0.641864\pi$
$282$	0	0
$283$	−609918.	−0.452694	−0.226347	−	0.974047i	$-0.572678\pi$
$283$	−609918.	−0.452694	−0.226347	+	0.974047i	$0.572678\pi$
$284$	1.11774e6	0.822327
$285$	0	0
$286$	2.86624e6	2.07204
$287$	0	0
$288$	0	0
$289$	−354040.	−0.249349
$290$	−2.32049e6	−1.62026
$291$	0	0
$292$	−601593.	−0.412901
$293$	−156438.	−0.106457	−0.0532283	−	0.998582i	$-0.516951\pi$
$293$	−156438.	−0.106457	−0.0532283	+	0.998582i	$0.516951\pi$
$294$	0	0
$295$	751303.	0.502644
$296$	−109911.	−0.0729139
$297$	0	0
$298$	548211.	0.357608
$299$	−735937.	−0.476061
$300$	0	0
$301$	0	0
$302$	−1.89007e6	−1.19251
$303$	0	0
$304$	−1.55040e6	−0.962189
$305$	−859259.	−0.528901
$306$	0	0
$307$	293229.	0.177566	0.0887831	−	0.996051i	$-0.471702\pi$
$307$	293229.	0.177566	0.0887831	+	0.996051i	$0.471702\pi$
$308$	0	0
$309$	0	0
$310$	−294023.	−0.173771
$311$	−2.45216e6	−1.43763	−0.718816	−	0.695200i	$-0.755316\pi$
$311$	−2.45216e6	−1.43763	−0.718816	+	0.695200i	$0.755316\pi$
$312$	0	0
$313$	1.83541e6	1.05894	0.529471	−	0.848328i	$-0.322390\pi$
$313$	1.83541e6	1.05894	0.529471	+	0.848328i	$0.322390\pi$
$314$	−3.88503e6	−2.22367
$315$	0	0
$316$	−2.87406e6	−1.61912
$317$	−589960.	−0.329742	−0.164871	−	0.986315i	$-0.552721\pi$
$317$	−589960.	−0.329742	−0.164871	+	0.986315i	$0.552721\pi$
$318$	0	0
$319$	2.25567e6	1.24107
$320$	−1.64444e6	−0.897726
$321$	0	0
$322$	0	0
$323$	−2.17679e6	−1.16094
$324$	0	0
$325$	2.09993e6	1.10280
$326$	769186.	0.400855
$327$	0	0
$328$	742344.	0.380996
$329$	0	0
$330$	0	0
$331$	177318.	0.0889577	0.0444789	−	0.999010i	$-0.485837\pi$
$331$	177318.	0.0889577	0.0444789	+	0.999010i	$0.485837\pi$
$332$	1.19357e6	0.594296
$333$	0	0
$334$	2.65864e6	1.30405
$335$	1.24817e6	0.607664
$336$	0	0
$337$	−3.04781e6	−1.46189	−0.730943	−	0.682438i	$-0.760920\pi$
$337$	−3.04781e6	−1.46189	−0.730943	+	0.682438i	$0.760920\pi$
$338$	7.99777e6	3.80783
$339$	0	0
$340$	−1.46205e6	−0.685908
$341$	285809.	0.133104
$342$	0	0
$343$	0	0
$344$	−1.22747e6	−0.559259
$345$	0	0
$346$	3.05874e6	1.37357
$347$	2.42361e6	1.08054	0.540268	−	0.841493i	$-0.318323\pi$
$347$	2.42361e6	1.08054	0.540268	+	0.841493i	$0.318323\pi$
$348$	0	0
$349$	−2.67690e6	−1.17644	−0.588218	−	0.808702i	$-0.700170\pi$
$349$	−2.67690e6	−1.17644	−0.588218	+	0.808702i	$0.700170\pi$
$350$	0	0
$351$	0	0
$352$	2.42191e6	1.04184
$353$	−950412.	−0.405953	−0.202976	−	0.979184i	$-0.565061\pi$
$353$	−950412.	−0.405953	−0.202976	+	0.979184i	$0.565061\pi$
$354$	0	0
$355$	1.02288e6	0.430777
$356$	1.41971e6	0.593711
$357$	0	0
$358$	1.46877e6	0.605683
$359$	−2.78881e6	−1.14204	−0.571022	−	0.820935i	$-0.693453\pi$
$359$	−2.78881e6	−1.14204	−0.571022	+	0.820935i	$0.693453\pi$
$360$	0	0
$361$	1.96972e6	0.795495
$362$	1.14001e6	0.457235
$363$	0	0
$364$	0	0
$365$	−550536.	−0.216299
$366$	0	0
$367$	153881.	0.0596377	0.0298189	−	0.999555i	$-0.490507\pi$
$367$	153881.	0.0596377	0.0298189	+	0.999555i	$0.490507\pi$
$368$	−471323.	−0.181426
$369$	0	0
$370$	−539169.	−0.204749
$371$	0	0
$372$	0	0
$373$	−2.38381e6	−0.887156	−0.443578	−	0.896236i	$-0.646291\pi$
$373$	−2.38381e6	−0.887156	−0.443578	+	0.896236i	$0.646291\pi$
$374$	2.57729e6	0.952763
$375$	0	0
$376$	1.73345e6	0.632328
$377$	8.76203e6	3.17506
$378$	0	0
$379$	3.65191e6	1.30594	0.652969	−	0.757385i	$-0.273523\pi$
$379$	3.65191e6	1.30594	0.652969	+	0.757385i	$0.273523\pi$
$380$	2.98606e6	1.06081
$381$	0	0
$382$	−1.53226e6	−0.537246
$383$	−2.15730e6	−0.751472	−0.375736	−	0.926727i	$-0.622610\pi$
$383$	−2.15730e6	−0.751472	−0.375736	+	0.926727i	$0.622610\pi$
$384$	0	0
$385$	0	0
$386$	8.15904e6	2.78722
$387$	0	0
$388$	−6.05669e6	−2.04247
$389$	3.66471e6	1.22791	0.613954	−	0.789342i	$-0.289578\pi$
$389$	3.66471e6	1.22791	0.613954	+	0.789342i	$0.289578\pi$
$390$	0	0
$391$	−661746.	−0.218902
$392$	0	0
$393$	0	0
$394$	5.90440e6	1.91617
$395$	−2.63014e6	−0.848177
$396$	0	0
$397$	−3.94648e6	−1.25671	−0.628353	−	0.777928i	$-0.716271\pi$
$397$	−3.94648e6	−1.25671	−0.628353	+	0.777928i	$0.716271\pi$
$398$	−3.51524e6	−1.11236
$399$	0	0
$400$	1.34488e6	0.420274
$401$	−25016.1	−0.00776887	−0.00388444	−	0.999992i	$-0.501236\pi$
$401$	−25016.1	−0.00776887	−0.00388444	+	0.999992i	$0.501236\pi$
$402$	0	0
$403$	1.11021e6	0.340521
$404$	−5.49989e6	−1.67649
$405$	0	0
$406$	0	0
$407$	524107.	0.156832
$408$	0	0
$409$	832700.	0.246139	0.123069	−	0.992398i	$-0.460726\pi$
$409$	832700.	0.246139	0.123069	+	0.992398i	$0.460726\pi$
$410$	3.64159e6	1.06987
$411$	0	0
$412$	−4.56035e6	−1.32360
$413$	0	0
$414$	0	0
$415$	1.09227e6	0.311323
$416$	9.40782e6	2.66536
$417$	0	0
$418$	−5.26379e6	−1.47353
$419$	−3.95178e6	−1.09966	−0.549828	−	0.835278i	$-0.685307\pi$
$419$	−3.95178e6	−1.09966	−0.549828	+	0.835278i	$0.685307\pi$
$420$	0	0
$421$	4.72285e6	1.29867	0.649336	−	0.760502i	$-0.275047\pi$
$421$	4.72285e6	1.29867	0.649336	+	0.760502i	$0.275047\pi$
$422$	−3.43890e6	−0.940023
$423$	0	0
$424$	440974.	0.119124
$425$	1.88823e6	0.507087
$426$	0	0
$427$	0	0
$428$	−3.26754e6	−0.862207
$429$	0	0
$430$	−6.02136e6	−1.57045
$431$	−4.07810e6	−1.05746	−0.528731	−	0.848790i	$-0.677332\pi$
$431$	−4.07810e6	−1.05746	−0.528731	+	0.848790i	$0.677332\pi$
$432$	0	0
$433$	−1.79927e6	−0.461186	−0.230593	−	0.973050i	$-0.574067\pi$
$433$	−1.79927e6	−0.461186	−0.230593	+	0.973050i	$0.574067\pi$
$434$	0	0
$435$	0	0
$436$	1.78425e6	0.449511
$437$	1.35153e6	0.338550
$438$	0	0
$439$	4.51827e6	1.11895	0.559475	−	0.828847i	$-0.311003\pi$
$439$	4.51827e6	1.11895	0.559475	+	0.828847i	$0.311003\pi$
$440$	−659542.	−0.162409
$441$	0	0
$442$	1.00114e7	2.43746
$443$	2.85256e6	0.690597	0.345299	−	0.938493i	$-0.387778\pi$
$443$	2.85256e6	0.690597	0.345299	+	0.938493i	$0.387778\pi$
$444$	0	0
$445$	1.29922e6	0.311016
$446$	7.44976e6	1.77339
$447$	0	0
$448$	0	0
$449$	1.90246e6	0.445348	0.222674	−	0.974893i	$-0.428522\pi$
$449$	1.90246e6	0.445348	0.222674	+	0.974893i	$0.428522\pi$
$450$	0	0
$451$	−3.53986e6	−0.819491
$452$	13986.2	0.00321998
$453$	0	0
$454$	9.51468e6	2.16648
$455$	0	0
$456$	0	0
$457$	2.64834e6	0.593176	0.296588	−	0.955006i	$-0.404151\pi$
$457$	2.64834e6	0.593176	0.296588	+	0.955006i	$0.404151\pi$
$458$	2.61904e6	0.583416
$459$	0	0
$460$	907763.	0.200022
$461$	−1.09031e6	−0.238944	−0.119472	−	0.992838i	$-0.538120\pi$
$461$	−1.09031e6	−0.238944	−0.119472	+	0.992838i	$0.538120\pi$
$462$	0	0
$463$	−2.50851e6	−0.543831	−0.271916	−	0.962321i	$-0.587657\pi$
$463$	−2.50851e6	−0.543831	−0.271916	+	0.962321i	$0.587657\pi$
$464$	5.61155e6	1.21001
$465$	0	0
$466$	−9.59943e6	−2.04777
$467$	3.20935e6	0.680966	0.340483	−	0.940251i	$-0.389409\pi$
$467$	3.20935e6	0.680966	0.340483	+	0.940251i	$0.389409\pi$
$468$	0	0
$469$	0	0
$470$	8.50350e6	1.77563
$471$	0	0
$472$	1.29358e6	0.267262
$473$	5.85315e6	1.20292
$474$	0	0
$475$	−3.85647e6	−0.784252
$476$	0	0
$477$	0	0
$478$	−739102.	−0.147957
$479$	−2.31462e6	−0.460936	−0.230468	−	0.973080i	$-0.574026\pi$
$479$	−2.31462e6	−0.460936	−0.230468	+	0.973080i	$0.574026\pi$
$480$	0	0
$481$	2.03587e6	0.401225
$482$	4.54211e6	0.890513
$483$	0	0
$484$	−2.89885e6	−0.562486
$485$	−5.54266e6	−1.06995
$486$	0	0
$487$	−4.63735e6	−0.886028	−0.443014	−	0.896515i	$-0.646091\pi$
$487$	−4.63735e6	−0.886028	−0.443014	+	0.896515i	$0.646091\pi$
$488$	−1.47945e6	−0.281224
$489$	0	0
$490$	0	0
$491$	−5.02151e6	−0.940007	−0.470003	−	0.882665i	$-0.655747\pi$
$491$	−5.02151e6	−0.940007	−0.470003	+	0.882665i	$0.655747\pi$
$492$	0	0
$493$	7.87872e6	1.45995
$494$	−2.04470e7	−3.76974
$495$	0	0
$496$	711024.	0.129772
$497$	0	0
$498$	0	0
$499$	3.37822e6	0.607347	0.303673	−	0.952776i	$-0.401787\pi$
$499$	3.37822e6	0.607347	0.303673	+	0.952776i	$0.401787\pi$
$500$	−7.01582e6	−1.25503
$501$	0	0
$502$	−1.14322e7	−2.02475
$503$	−5.03743e6	−0.887747	−0.443873	−	0.896090i	$-0.646396\pi$
$503$	−5.03743e6	−0.887747	−0.443873	+	0.896090i	$0.646396\pi$
$504$	0	0
$505$	−5.03311e6	−0.878230
$506$	−1.60019e6	−0.277841
$507$	0	0
$508$	6.64694e6	1.14278
$509$	−6.72466e6	−1.15047	−0.575236	−	0.817988i	$-0.695090\pi$
$509$	−6.72466e6	−1.15047	−0.575236	+	0.817988i	$0.695090\pi$
$510$	0	0
$511$	0	0
$512$	7.48361e6	1.26164
$513$	0	0
$514$	8.25112e6	1.37754
$515$	−4.17332e6	−0.693367
$516$	0	0
$517$	−8.26595e6	−1.36009
$518$	0	0
$519$	0	0
$520$	−2.56196e6	−0.415493
$521$	−4.42770e6	−0.714635	−0.357317	−	0.933983i	$-0.616309\pi$
$521$	−4.42770e6	−0.714635	−0.357317	+	0.933983i	$0.616309\pi$
$522$	0	0
$523$	8.95911e6	1.43222	0.716111	−	0.697986i	$-0.245920\pi$
$523$	8.95911e6	1.43222	0.716111	+	0.697986i	$0.245920\pi$
$524$	6.84372e6	1.08884
$525$	0	0
$526$	1.05050e7	1.65551
$527$	998291.	0.156578
$528$	0	0
$529$	−6.02548e6	−0.936165
$530$	2.16321e6	0.334510
$531$	0	0
$532$	0	0
$533$	−1.37504e7	−2.09652
$534$	0	0
$535$	−2.99022e6	−0.451668
$536$	2.14908e6	0.323103
$537$	0	0
$538$	9.15641e6	1.36386
$539$	0	0
$540$	0	0
$541$	−1.00467e7	−1.47581	−0.737907	−	0.674902i	$-0.764186\pi$
$541$	−1.00467e7	−1.47581	−0.737907	+	0.674902i	$0.764186\pi$
$542$	−1.82968e7	−2.67533
$543$	0	0
$544$	8.45941e6	1.22558
$545$	1.63282e6	0.235477
$546$	0	0
$547$	−1.31426e7	−1.87808	−0.939039	−	0.343811i	$-0.888282\pi$
$547$	−1.31426e7	−1.87808	−0.939039	+	0.343811i	$0.888282\pi$
$548$	−1.44657e7	−2.05774
$549$	0	0
$550$	4.56600e6	0.643620
$551$	−1.60913e7	−2.25794
$552$	0	0
$553$	0	0
$554$	2.14415e6	0.296811
$555$	0	0
$556$	8.57452e6	1.17631
$557$	−9.06752e6	−1.23837	−0.619185	−	0.785245i	$-0.712537\pi$
$557$	−9.06752e6	−1.23837	−0.619185	+	0.785245i	$0.712537\pi$
$558$	0	0
$559$	2.27363e7	3.07744
$560$	0	0
$561$	0	0
$562$	−9.63846e6	−1.28726
$563$	1.05180e7	1.39849	0.699247	−	0.714880i	$-0.253519\pi$
$563$	1.05180e7	1.39849	0.699247	+	0.714880i	$0.253519\pi$
$564$	0	0
$565$	12799.2	0.00168679
$566$	−5.15150e6	−0.675916
$567$	0	0
$568$	1.76117e6	0.229050
$569$	−7.32307e6	−0.948227	−0.474114	−	0.880464i	$-0.657231\pi$
$569$	−7.32307e6	−0.948227	−0.474114	+	0.880464i	$0.657231\pi$
$570$	0	0
$571$	−6.97981e6	−0.895887	−0.447943	−	0.894062i	$-0.647843\pi$
$571$	−6.97981e6	−0.895887	−0.447943	+	0.894062i	$0.647843\pi$
$572$	1.33497e7	1.70600
$573$	0	0
$574$	0	0
$575$	−1.17237e6	−0.147875
$576$	0	0
$577$	5.81210e6	0.726765	0.363382	−	0.931640i	$-0.381622\pi$
$577$	5.81210e6	0.726765	0.363382	+	0.931640i	$0.381622\pi$
$578$	−2.99030e6	−0.372302
$579$	0	0
$580$	−1.08078e7	−1.33403
$581$	0	0
$582$	0	0
$583$	−2.10278e6	−0.256226
$584$	−947901.	−0.115009
$585$	0	0
$586$	−1.32131e6	−0.158950
$587$	−7.37446e6	−0.883355	−0.441677	−	0.897174i	$-0.645616\pi$
$587$	−7.37446e6	−0.883355	−0.441677	+	0.897174i	$0.645616\pi$
$588$	0	0
$589$	−2.03888e6	−0.242161
$590$	6.34567e6	0.750495
$591$	0	0
$592$	1.30385e6	0.152906
$593$	9.46528e6	1.10534	0.552671	−	0.833399i	$-0.313609\pi$
$593$	9.46528e6	1.10534	0.552671	+	0.833399i	$0.313609\pi$
$594$	0	0
$595$	0	0
$596$	2.55332e6	0.294435
$597$	0	0
$598$	−6.21589e6	−0.710804
$599$	8.52195e6	0.970448	0.485224	−	0.874390i	$-0.338738\pi$
$599$	8.52195e6	0.970448	0.485224	+	0.874390i	$0.338738\pi$
$600$	0	0
$601$	657065.	0.0742031	0.0371016	−	0.999311i	$-0.488187\pi$
$601$	657065.	0.0742031	0.0371016	+	0.999311i	$0.488187\pi$
$602$	0	0
$603$	0	0
$604$	−8.80310e6	−0.981846
$605$	−2.65282e6	−0.294659
$606$	0	0
$607$	−5.86885e6	−0.646519	−0.323260	−	0.946310i	$-0.604779\pi$
$607$	−5.86885e6	−0.646519	−0.323260	+	0.946310i	$0.604779\pi$
$608$	−1.72773e7	−1.89547
$609$	0	0
$610$	−7.25750e6	−0.789700
$611$	−3.21087e7	−3.47952
$612$	0	0
$613$	3.84402e6	0.413175	0.206588	−	0.978428i	$-0.433764\pi$
$613$	3.84402e6	0.413175	0.206588	+	0.978428i	$0.433764\pi$
$614$	2.47667e6	0.265123
$615$	0	0
$616$	0	0
$617$	−6.44660e6	−0.681739	−0.340869	−	0.940111i	$-0.610721\pi$
$617$	−6.44660e6	−0.681739	−0.340869	+	0.940111i	$0.610721\pi$
$618$	0	0
$619$	−6.73740e6	−0.706749	−0.353375	−	0.935482i	$-0.614966\pi$
$619$	−6.73740e6	−0.706749	−0.353375	+	0.935482i	$0.614966\pi$
$620$	−1.36942e6	−0.143073
$621$	0	0
$622$	−2.07115e7	−2.14652
$623$	0	0
$624$	0	0
$625$	−704759.	−0.0721673
$626$	1.55023e7	1.58110
$627$	0	0
$628$	−1.80947e7	−1.83085
$629$	1.83063e6	0.184491
$630$	0	0
$631$	−9.14514e6	−0.914360	−0.457180	−	0.889374i	$-0.651140\pi$
$631$	−9.14514e6	−0.914360	−0.457180	+	0.889374i	$0.651140\pi$
$632$	−4.52852e6	−0.450987
$633$	0	0
$634$	−4.98293e6	−0.492336
$635$	6.08282e6	0.598647
$636$	0	0
$637$	0	0
$638$	1.90518e7	1.85304
$639$	0	0
$640$	−4.44978e6	−0.429426
$641$	1.04088e6	0.100059	0.0500296	−	0.998748i	$-0.484068\pi$
$641$	1.04088e6	0.100059	0.0500296	+	0.998748i	$0.484068\pi$
$642$	0	0
$643$	9.08713e6	0.866761	0.433381	−	0.901211i	$-0.357321\pi$
$643$	9.08713e6	0.866761	0.433381	+	0.901211i	$0.357321\pi$
$644$	0	0
$645$	0	0
$646$	−1.83857e7	−1.73340
$647$	2.20711e6	0.207283	0.103641	−	0.994615i	$-0.466951\pi$
$647$	2.20711e6	0.207283	0.103641	+	0.994615i	$0.466951\pi$
$648$	0	0
$649$	−6.16840e6	−0.574859
$650$	1.77364e7	1.64658
$651$	0	0
$652$	3.58252e6	0.330042
$653$	1.83610e7	1.68505	0.842524	−	0.538658i	$-0.181069\pi$
$653$	1.83610e7	1.68505	0.842524	+	0.538658i	$0.181069\pi$
$654$	0	0
$655$	6.26289e6	0.570390
$656$	−8.80632e6	−0.798978
$657$	0	0
$658$	0	0
$659$	−6.21208e6	−0.557216	−0.278608	−	0.960405i	$-0.589873\pi$
$659$	−6.21208e6	−0.557216	−0.278608	+	0.960405i	$0.589873\pi$
$660$	0	0
$661$	−1.54230e7	−1.37298	−0.686491	−	0.727138i	$-0.740850\pi$
$661$	−1.54230e7	−1.37298	−0.686491	+	0.727138i	$0.740850\pi$
$662$	1.49767e6	0.132822
$663$	0	0
$664$	1.88065e6	0.165534
$665$	0	0
$666$	0	0
$667$	−4.89176e6	−0.425746
$668$	1.23827e7	1.07368
$669$	0	0
$670$	1.05424e7	0.907300
$671$	7.05475e6	0.604889
$672$	0	0
$673$	−2.27201e7	−1.93362	−0.966811	−	0.255491i	$-0.917763\pi$
$673$	−2.27201e7	−1.93362	−0.966811	+	0.255491i	$0.917763\pi$
$674$	−2.57425e7	−2.18274
$675$	0	0
$676$	3.72500e7	3.13516
$677$	1.36173e7	1.14188	0.570940	−	0.820992i	$-0.306579\pi$
$677$	1.36173e7	1.14188	0.570940	+	0.820992i	$0.306579\pi$
$678$	0	0
$679$	0	0
$680$	−2.30369e6	−0.191052
$681$	0	0
$682$	2.41401e6	0.198736
$683$	−2.39985e6	−0.196849	−0.0984245	−	0.995145i	$-0.531380\pi$
$683$	−2.39985e6	−0.196849	−0.0984245	+	0.995145i	$0.531380\pi$
$684$	0	0
$685$	−1.32380e7	−1.07795
$686$	0	0
$687$	0	0
$688$	1.45612e7	1.17281
$689$	−8.16816e6	−0.655505
$690$	0	0
$691$	1.40365e6	0.111831	0.0559156	−	0.998435i	$-0.482192\pi$
$691$	1.40365e6	0.111831	0.0559156	+	0.998435i	$0.482192\pi$
$692$	1.42462e7	1.13093
$693$	0	0
$694$	2.04703e7	1.61334
$695$	7.84680e6	0.616212
$696$	0	0
$697$	−1.23642e7	−0.964018
$698$	−2.26097e7	−1.75653
$699$	0	0
$700$	0	0
$701$	5.78991e6	0.445017	0.222509	−	0.974931i	$-0.428575\pi$
$701$	5.78991e6	0.445017	0.222509	+	0.974931i	$0.428575\pi$
$702$	0	0
$703$	−3.73884e6	−0.285330
$704$	1.35013e7	1.02670
$705$	0	0
$706$	−8.02739e6	−0.606126
$707$	0	0
$708$	0	0
$709$	−1.13143e7	−0.845304	−0.422652	−	0.906292i	$-0.638901\pi$
$709$	−1.13143e7	−0.845304	−0.422652	+	0.906292i	$0.638901\pi$
$710$	8.63944e6	0.643191
$711$	0	0
$712$	2.23697e6	0.165371
$713$	−619821.	−0.0456607
$714$	0	0
$715$	1.22167e7	0.893692
$716$	6.84085e6	0.498686
$717$	0	0
$718$	−2.35549e7	−1.70518
$719$	−2.73780e7	−1.97506	−0.987529	−	0.157437i	$-0.949677\pi$
$719$	−2.73780e7	−1.97506	−0.987529	+	0.157437i	$0.949677\pi$
$720$	0	0
$721$	0	0
$722$	1.66367e7	1.18775
$723$	0	0
$724$	5.30967e6	0.376462
$725$	1.39582e7	0.986241
$726$	0	0
$727$	9.86471e6	0.692226	0.346113	−	0.938193i	$-0.387501\pi$
$727$	9.86471e6	0.692226	0.346113	+	0.938193i	$0.387501\pi$
$728$	0	0
$729$	0	0
$730$	−4.64995e6	−0.322954
$731$	2.04442e7	1.41507
$732$	0	0
$733$	−3.87876e6	−0.266645	−0.133322	−	0.991073i	$-0.542565\pi$
$733$	−3.87876e6	−0.266645	−0.133322	+	0.991073i	$0.542565\pi$
$734$	1.29972e6	0.0890448
$735$	0	0
$736$	−5.25229e6	−0.357400
$737$	−1.02479e7	−0.694967
$738$	0	0
$739$	−7.95498e6	−0.535831	−0.267916	−	0.963442i	$-0.586335\pi$
$739$	−7.95498e6	−0.535831	−0.267916	+	0.963442i	$0.586335\pi$
$740$	−2.51121e6	−0.168579
$741$	0	0
$742$	0	0
$743$	−1.65977e7	−1.10300	−0.551500	−	0.834175i	$-0.685944\pi$
$743$	−1.65977e7	−1.10300	−0.551500	+	0.834175i	$0.685944\pi$
$744$	0	0
$745$	2.33662e6	0.154240
$746$	−2.01342e7	−1.32461
$747$	0	0
$748$	1.20039e7	0.784453
$749$	0	0
$750$	0	0
$751$	1.51072e7	0.977426	0.488713	−	0.872445i	$-0.337467\pi$
$751$	1.51072e7	0.977426	0.488713	+	0.872445i	$0.337467\pi$
$752$	−2.05637e7	−1.32604
$753$	0	0
$754$	7.40061e7	4.74066
$755$	−8.05598e6	−0.514341
$756$	0	0
$757$	5.80923e6	0.368450	0.184225	−	0.982884i	$-0.441022\pi$
$757$	5.80923e6	0.368450	0.184225	+	0.982884i	$0.441022\pi$
$758$	3.08449e7	1.94989
$759$	0	0
$760$	4.70498e6	0.295477
$761$	2.54270e7	1.59160	0.795799	−	0.605561i	$-0.207051\pi$
$761$	2.54270e7	1.59160	0.795799	+	0.605561i	$0.207051\pi$
$762$	0	0
$763$	0	0
$764$	−7.13656e6	−0.442340
$765$	0	0
$766$	−1.82210e7	−1.12202
$767$	−2.39609e7	−1.47067
$768$	0	0
$769$	−1.53909e7	−0.938532	−0.469266	−	0.883057i	$-0.655481\pi$
$769$	−1.53909e7	−0.938532	−0.469266	+	0.883057i	$0.655481\pi$
$770$	0	0
$771$	0	0
$772$	3.80011e7	2.29484
$773$	905393.	0.0544990	0.0272495	−	0.999629i	$-0.491325\pi$
$773$	905393.	0.0544990	0.0272495	+	0.999629i	$0.491325\pi$
$774$	0	0
$775$	1.76860e6	0.105773
$776$	−9.54323e6	−0.568907
$777$	0	0
$778$	3.09530e7	1.83338
$779$	2.52523e7	1.49093
$780$	0	0
$781$	−8.39810e6	−0.492667
$782$	−5.58926e6	−0.326841
$783$	0	0
$784$	0	0
$785$	−1.65590e7	−0.959093
$786$	0	0
$787$	2.79334e7	1.60763	0.803817	−	0.594877i	$-0.202799\pi$
$787$	2.79334e7	1.60763	0.803817	+	0.594877i	$0.202799\pi$
$788$	2.75000e7	1.57767
$789$	0	0
$790$	−2.22148e7	−1.26641
$791$	0	0
$792$	0	0
$793$	2.74039e7	1.54749
$794$	−3.33328e7	−1.87638
$795$	0	0
$796$	−1.63724e7	−0.915860
$797$	2.18824e7	1.22025	0.610126	−	0.792304i	$-0.291119\pi$
$797$	2.18824e7	1.22025	0.610126	+	0.792304i	$0.291119\pi$
$798$	0	0
$799$	−2.88718e7	−1.59995
$800$	1.49869e7	0.827918
$801$	0	0
$802$	−211291.	−0.0115997
$803$	4.52005e6	0.247374
$804$	0	0
$805$	0	0
$806$	9.37710e6	0.508430
$807$	0	0
$808$	−8.66590e6	−0.466966
$809$	−2.44194e7	−1.31179	−0.655893	−	0.754854i	$-0.727708\pi$
$809$	−2.44194e7	−1.31179	−0.655893	+	0.754854i	$0.727708\pi$
$810$	0	0
$811$	−4.46711e6	−0.238492	−0.119246	−	0.992865i	$-0.538048\pi$
$811$	−4.46711e6	−0.238492	−0.119246	+	0.992865i	$0.538048\pi$
$812$	0	0
$813$	0	0
$814$	4.42673e6	0.234165
$815$	3.27847e6	0.172893
$816$	0	0
$817$	−4.17547e7	−2.18852
$818$	7.03317e6	0.367509
$819$	0	0
$820$	1.69609e7	0.880873
$821$	−1.34708e7	−0.697485	−0.348743	−	0.937219i	$-0.613391\pi$
$821$	−1.34708e7	−0.697485	−0.348743	+	0.937219i	$0.613391\pi$
$822$	0	0
$823$	−4.03958e6	−0.207891	−0.103946	−	0.994583i	$-0.533147\pi$
$823$	−4.03958e6	−0.207891	−0.103946	+	0.994583i	$0.533147\pi$
$824$	−7.18552e6	−0.368672
$825$	0	0
$826$	0	0
$827$	−1.24927e7	−0.635175	−0.317588	−	0.948229i	$-0.602873\pi$
$827$	−1.24927e7	−0.635175	−0.317588	+	0.948229i	$0.602873\pi$
$828$	0	0
$829$	1.45980e7	0.737749	0.368874	−	0.929479i	$-0.379743\pi$
$829$	1.45980e7	0.737749	0.368874	+	0.929479i	$0.379743\pi$
$830$	9.22557e6	0.464834
$831$	0	0
$832$	5.24453e7	2.62663
$833$	0	0
$834$	0	0
$835$	1.13318e7	0.562449
$836$	−2.45163e7	−1.21322
$837$	0	0
$838$	−3.33776e7	−1.64189
$839$	−9.15983e6	−0.449244	−0.224622	−	0.974446i	$-0.572115\pi$
$839$	−9.15983e6	−0.449244	−0.224622	+	0.974446i	$0.572115\pi$
$840$	0	0
$841$	3.77299e7	1.83948
$842$	3.98903e7	1.93904
$843$	0	0
$844$	−1.60168e7	−0.773964
$845$	3.40886e7	1.64236
$846$	0	0
$847$	0	0
$848$	−5.23121e6	−0.249812
$849$	0	0
$850$	1.59484e7	0.757129
$851$	−1.13661e6	−0.0538005
$852$	0	0
$853$	8.68253e6	0.408577	0.204289	−	0.978911i	$-0.434512\pi$
$853$	8.68253e6	0.408577	0.204289	+	0.978911i	$0.434512\pi$
$854$	0	0
$855$	0	0
$856$	−5.14850e6	−0.240158
$857$	1.04988e7	0.488302	0.244151	−	0.969737i	$-0.421491\pi$
$857$	1.04988e7	0.488302	0.244151	+	0.969737i	$0.421491\pi$
$858$	0	0
$859$	−9.03780e6	−0.417907	−0.208954	−	0.977926i	$-0.567006\pi$
$859$	−9.03780e6	−0.417907	−0.208954	+	0.977926i	$0.567006\pi$
$860$	−2.80448e7	−1.29302
$861$	0	0
$862$	−3.44445e7	−1.57889
$863$	1.59858e7	0.730645	0.365322	−	0.930881i	$-0.380959\pi$
$863$	1.59858e7	0.730645	0.365322	+	0.930881i	$0.380959\pi$
$864$	0	0
$865$	1.30371e7	0.592437
$866$	−1.51970e7	−0.688594
$867$	0	0
$868$	0	0
$869$	2.15942e7	0.970035
$870$	0	0
$871$	−3.98073e7	−1.77794
$872$	2.81136e6	0.125206
$873$	0	0
$874$	1.14153e7	0.505487
$875$	0	0
$876$	0	0
$877$	2.67453e7	1.17422	0.587109	−	0.809508i	$-0.300266\pi$
$877$	2.67453e7	1.17422	0.587109	+	0.809508i	$0.300266\pi$
$878$	3.81623e7	1.67070
$879$	0	0
$880$	7.82404e6	0.340584
$881$	−6.40715e6	−0.278115	−0.139058	−	0.990284i	$-0.544407\pi$
$881$	−6.40715e6	−0.278115	−0.139058	+	0.990284i	$0.544407\pi$
$882$	0	0
$883$	4.96462e6	0.214281	0.107141	−	0.994244i	$-0.465831\pi$
$883$	4.96462e6	0.214281	0.107141	+	0.994244i	$0.465831\pi$
$884$	4.66285e7	2.00688
$885$	0	0
$886$	2.40933e7	1.03113
$887$	8.71625e6	0.371981	0.185990	−	0.982552i	$-0.440451\pi$
$887$	8.71625e6	0.371981	0.185990	+	0.982552i	$0.440451\pi$
$888$	0	0
$889$	0	0
$890$	1.09735e7	0.464377
$891$	0	0
$892$	3.46976e7	1.46011
$893$	5.89669e7	2.47446
$894$	0	0
$895$	6.26026e6	0.261237
$896$	0	0
$897$	0	0
$898$	1.60686e7	0.664946
$899$	7.37956e6	0.304531
$900$	0	0
$901$	−7.34472e6	−0.301414
$902$	−2.98984e7	−1.22358
$903$	0	0
$904$	22037.4	0.000896888 0
$905$	4.85904e6	0.197210
$906$	0	0
$907$	−2.36255e7	−0.953594	−0.476797	−	0.879014i	$-0.658202\pi$
$907$	−2.36255e7	−0.953594	−0.476797	+	0.879014i	$0.658202\pi$
$908$	4.43151e7	1.78376
$909$	0	0
$910$	0	0
$911$	1.69360e7	0.676105	0.338053	−	0.941127i	$-0.390232\pi$
$911$	1.69360e7	0.676105	0.338053	+	0.941127i	$0.390232\pi$
$912$	0	0
$913$	−8.96785e6	−0.356050
$914$	2.23685e7	0.885668
$915$	0	0
$916$	1.21983e7	0.480353
$917$	0	0
$918$	0	0
$919$	−3.13804e7	−1.22566	−0.612829	−	0.790216i	$-0.709968\pi$
$919$	−3.13804e7	−1.22566	−0.612829	+	0.790216i	$0.709968\pi$
$920$	1.43032e6	0.0557138
$921$	0	0
$922$	−9.20896e6	−0.356766
$923$	−3.26220e7	−1.26040
$924$	0	0
$925$	3.24320e6	0.124629
$926$	−2.11875e7	−0.811992
$927$	0	0
$928$	6.25336e7	2.38365
$929$	−1.43089e7	−0.543959	−0.271980	−	0.962303i	$-0.587678\pi$
$929$	−1.43089e7	−0.543959	−0.271980	+	0.962303i	$0.587678\pi$
$930$	0	0
$931$	0	0
$932$	−4.47098e7	−1.68602
$933$	0	0
$934$	2.71069e7	1.01675
$935$	1.09851e7	0.410937
$936$	0	0
$937$	−2.81206e7	−1.04635	−0.523173	−	0.852227i	$-0.675252\pi$
$937$	−2.81206e7	−1.04635	−0.523173	+	0.852227i	$0.675252\pi$
$938$	0	0
$939$	0	0
$940$	3.96054e7	1.46196
$941$	−3.29569e7	−1.21331	−0.606656	−	0.794964i	$-0.707490\pi$
$941$	−3.29569e7	−1.21331	−0.606656	+	0.794964i	$0.707490\pi$
$942$	0	0
$943$	7.67672e6	0.281123
$944$	−1.53455e7	−0.560469
$945$	0	0
$946$	4.94370e7	1.79607
$947$	−1.62975e7	−0.590535	−0.295268	−	0.955415i	$-0.595409\pi$
$947$	−1.62975e7	−0.590535	−0.295268	+	0.955415i	$0.595409\pi$
$948$	0	0
$949$	1.75579e7	0.632861
$950$	−3.25726e7	−1.17096
$951$	0	0
$952$	0	0
$953$	3.03230e7	1.08153	0.540767	−	0.841172i	$-0.318134\pi$
$953$	3.03230e7	1.08153	0.540767	+	0.841172i	$0.318134\pi$
$954$	0	0
$955$	−6.53088e6	−0.231720
$956$	−3.44240e6	−0.121820
$957$	0	0
$958$	−1.95498e7	−0.688221
$959$	0	0
$960$	0	0
$961$	−2.76941e7	−0.967339
$962$	1.71954e7	0.599067
$963$	0	0
$964$	2.11551e7	0.733200
$965$	3.47759e7	1.20216
$966$	0	0
$967$	−1.49059e7	−0.512616	−0.256308	−	0.966595i	$-0.582506\pi$
$967$	−1.49059e7	−0.512616	−0.256308	+	0.966595i	$0.582506\pi$
$968$	−4.56757e6	−0.156674
$969$	0	0
$970$	−4.68145e7	−1.59754
$971$	1.65608e7	0.563679	0.281840	−	0.959462i	$-0.409055\pi$
$971$	1.65608e7	0.563679	0.281840	+	0.959462i	$0.409055\pi$
$972$	0	0
$973$	0	0
$974$	−3.91681e7	−1.32292
$975$	0	0
$976$	1.75505e7	0.589747
$977$	−7.56862e6	−0.253676	−0.126838	−	0.991923i	$-0.540483\pi$
$977$	−7.56862e6	−0.253676	−0.126838	+	0.991923i	$0.540483\pi$
$978$	0	0
$979$	−1.06670e7	−0.355700
$980$	0	0
$981$	0	0
$982$	−4.24128e7	−1.40352
$983$	−3.32042e7	−1.09600	−0.547998	−	0.836480i	$-0.684610\pi$
$983$	−3.32042e7	−1.09600	−0.547998	+	0.836480i	$0.684610\pi$
$984$	0	0
$985$	2.51661e7	0.826466
$986$	6.65454e7	2.17985
$987$	0	0
$988$	−9.52327e7	−3.10380
$989$	−1.26935e7	−0.412657
$990$	0	0
$991$	3.48003e7	1.12564	0.562819	−	0.826580i	$-0.309717\pi$
$991$	3.48003e7	1.12564	0.562819	+	0.826580i	$0.309717\pi$
$992$	7.92345e6	0.255644
$993$	0	0
$994$	0	0
$995$	−1.49829e7	−0.479774
$996$	0	0
$997$	−9.40852e6	−0.299767	−0.149883	−	0.988704i	$-0.547890\pi$
$997$	−9.40852e6	−0.299767	−0.149883	+	0.988704i	$0.547890\pi$
$998$	2.85332e7	0.906826
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.r.1.2		2
3.2	odd	2		147.6.a.j.1.1	yes	2
7.6	odd	2		441.6.a.q.1.2		2
21.2	odd	6		147.6.e.m.67.2		4
21.5	even	6		147.6.e.n.67.2		4
21.11	odd	6		147.6.e.m.79.2		4
21.17	even	6		147.6.e.n.79.2		4
21.20	even	2		147.6.a.h.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.h.1.1	✓	2	21.20	even	2
147.6.a.j.1.1	yes	2	3.2	odd	2
147.6.e.m.67.2		4	21.2	odd	6
147.6.e.m.79.2		4	21.11	odd	6
147.6.e.n.67.2		4	21.5	even	6
147.6.e.n.79.2		4	21.17	even	6
441.6.a.q.1.2		2	7.6	odd	2
441.6.a.r.1.2		2	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +O(q^{10})\)	\(q+8.44622 q^{2} +39.3387 q^{4} +36.0000 q^{5} +61.9840 q^{8} +304.064 q^{10} -295.570 q^{11} -1148.13 q^{13} -735.307 q^{16} -1032.38 q^{17} +2108.51 q^{19} +1416.19 q^{20} -2496.45 q^{22} +640.988 q^{23} -1829.00 q^{25} -9697.34 q^{26} -7631.58 q^{29} -966.976 q^{31} -8194.05 q^{32} -8719.74 q^{34} -1773.21 q^{37} +17809.0 q^{38} +2231.42 q^{40} +11976.4 q^{41} -19802.9 q^{43} -11627.3 q^{44} +5413.93 q^{46} +27966.1 q^{47} -15448.1 q^{50} -45165.8 q^{52} +7114.33 q^{53} -10640.5 q^{55} -64458.0 q^{58} +20869.5 q^{59} -23868.3 q^{61} -8167.30 q^{62} -45679.0 q^{64} -41332.6 q^{65} +34671.5 q^{67} -40612.6 q^{68} +28413.2 q^{71} -15292.7 q^{73} -14976.9 q^{74} +82946.0 q^{76} -73059.5 q^{79} -26471.0 q^{80} +101155. q^{82} +30340.9 q^{83} -37165.8 q^{85} -167260. q^{86} -18320.6 q^{88} +36089.5 q^{89} +25215.6 q^{92} +236208. q^{94} +75906.4 q^{95} -153963. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8	\( 2 q + 3 q^{2} + 37 q^{4} + 72 q^{5} + 249 q^{8} + 108 q^{10} - 480 q^{11} - 1296 q^{13} - 1679 q^{16} + 936 q^{17} + 216 q^{19} + 1332 q^{20} - 1492 q^{22} + 504 q^{23} - 3658 q^{25} - 8892 q^{26} - 6372 q^{29} - 9936 q^{31} - 9039 q^{32} - 19440 q^{34} + 11124 q^{37} + 28116 q^{38} + 8964 q^{40} + 20952 q^{41} - 6264 q^{43} - 11196 q^{44} + 6160 q^{46} + 7920 q^{47} - 5487 q^{50} - 44820 q^{52} - 2220 q^{53} - 17280 q^{55} - 71318 q^{58} + 29736 q^{59} + 17280 q^{61} + 40680 q^{62} - 10879 q^{64} - 46656 q^{65} - 20680 q^{67} - 45216 q^{68} + 92280 q^{71} - 56592 q^{73} - 85218 q^{74} + 87372 q^{76} - 56096 q^{79} - 60444 q^{80} + 52272 q^{82} - 71352 q^{83} + 33696 q^{85} - 240996 q^{86} - 52812 q^{88} + 123192 q^{89} + 25536 q^{92} + 345384 q^{94} + 7776 q^{95} - 35856 q^{97}+O(q^{100}) \) 2 * q + 3 * q^2 + 37 * q^4 + 72 * q^5 + 249 * q^8 + 108 * q^10 - 480 * q^11 - 1296 * q^13 - 1679 * q^16 + 936 * q^17 + 216 * q^19 + 1332 * q^20 - 1492 * q^22 + 504 * q^23 - 3658 * q^25 - 8892 * q^26 - 6372 * q^29 - 9936 * q^31 - 9039 * q^32 - 19440 * q^34 + 11124 * q^37 + 28116 * q^38 + 8964 * q^40 + 20952 * q^41 - 6264 * q^43 - 11196 * q^44 + 6160 * q^46 + 7920 * q^47 - 5487 * q^50 - 44820 * q^52 - 2220 * q^53 - 17280 * q^55 - 71318 * q^58 + 29736 * q^59 + 17280 * q^61 + 40680 * q^62 - 10879 * q^64 - 46656 * q^65 - 20680 * q^67 - 45216 * q^68 + 92280 * q^71 - 56592 * q^73 - 85218 * q^74 + 87372 * q^76 - 56096 * q^79 - 60444 * q^80 + 52272 * q^82 - 71352 * q^83 + 33696 * q^85 - 240996 * q^86 - 52812 * q^88 + 123192 * q^89 + 25536 * q^92 + 345384 * q^94 + 7776 * q^95 - 35856 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more