LMFDB - Embedded newform 891.6.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,6,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, 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("891.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	891.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:	$142.901983453$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.204234 q^{2} -31.9583 q^{4} -18.4559 q^{5} -133.899 q^{7} +13.0625 q^{8} +O(q^{10})$	\(q-0.204234 q^{2} -31.9583 q^{4} -18.4559 q^{5} -133.899 q^{7} +13.0625 q^{8} +3.76934 q^{10} -121.000 q^{11} -485.312 q^{13} +27.3468 q^{14} +1020.00 q^{16} +266.753 q^{17} +1642.09 q^{19} +589.821 q^{20} +24.7123 q^{22} -744.883 q^{23} -2784.38 q^{25} +99.1173 q^{26} +4279.19 q^{28} +1049.49 q^{29} +3237.77 q^{31} -626.317 q^{32} -54.4800 q^{34} +2471.24 q^{35} +13088.8 q^{37} -335.370 q^{38} -241.080 q^{40} +3398.57 q^{41} +11678.2 q^{43} +3866.95 q^{44} +152.131 q^{46} +640.836 q^{47} +1122.04 q^{49} +568.665 q^{50} +15509.7 q^{52} -19381.1 q^{53} +2233.17 q^{55} -1749.06 q^{56} -214.342 q^{58} +39338.6 q^{59} +19326.8 q^{61} -661.263 q^{62} -32512.0 q^{64} +8956.89 q^{65} -11867.3 q^{67} -8524.96 q^{68} -504.712 q^{70} -73798.6 q^{71} +10415.6 q^{73} -2673.18 q^{74} -52478.3 q^{76} +16201.8 q^{77} +31511.5 q^{79} -18825.0 q^{80} -694.103 q^{82} +104931. q^{83} -4923.17 q^{85} -2385.08 q^{86} -1580.56 q^{88} +22330.8 q^{89} +64983.0 q^{91} +23805.2 q^{92} -130.881 q^{94} -30306.3 q^{95} +33323.3 q^{97} -229.159 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

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 $

$2$	−0.204234	−0.0361038	−0.0180519	−	0.999837i	$-0.505746\pi$
$2$	−0.204234	−0.0361038	−0.0180519	+	0.999837i	$0.505746\pi$
$3$	0	0
$3$	0	0
$4$	−31.9583	−0.998697
$5$	−18.4559	−0.330150	−0.165075	−	0.986281i	$-0.552787\pi$
$5$	−18.4559	−0.330150	−0.165075	+	0.986281i	$0.552787\pi$
$6$	0	0
$7$	−133.899	−1.03284	−0.516420	−	0.856335i	$-0.672736\pi$
$7$	−133.899	−1.03284	−0.516420	+	0.856335i	$0.672736\pi$
$8$	13.0625	0.0721606
$9$	0	0
$10$	3.76934	0.0119197
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−485.312	−0.796458	−0.398229	−	0.917286i	$-0.630375\pi$
$13$	−485.312	−0.796458	−0.398229	+	0.917286i	$0.630375\pi$
$14$	27.3468	0.0372895
$15$	0	0
$16$	1020.00	0.996091
$17$	266.753	0.223865	0.111933	−	0.993716i	$-0.464296\pi$
$17$	266.753	0.223865	0.111933	+	0.993716i	$0.464296\pi$
$18$	0	0
$19$	1642.09	1.04355	0.521774	−	0.853084i	$-0.325271\pi$
$19$	1642.09	1.04355	0.521774	+	0.853084i	$0.325271\pi$
$20$	589.821	0.329720
$21$	0	0
$22$	24.7123	0.0108857
$23$	−744.883	−0.293609	−0.146804	−	0.989166i	$-0.546899\pi$
$23$	−744.883	−0.293609	−0.146804	+	0.989166i	$0.546899\pi$
$24$	0	0
$25$	−2784.38	−0.891001
$26$	99.1173	0.0287552
$27$	0	0
$28$	4279.19	1.03149
$29$	1049.49	0.231731	0.115866	−	0.993265i	$-0.463036\pi$
$29$	1049.49	0.231731	0.115866	+	0.993265i	$0.463036\pi$
$30$	0	0
$31$	3237.77	0.605121	0.302560	−	0.953130i	$-0.402159\pi$
$31$	3237.77	0.605121	0.302560	+	0.953130i	$0.402159\pi$
$32$	−626.317	−0.108123
$33$	0	0
$34$	−54.4800	−0.00808239
$35$	2471.24	0.340992
$36$	0	0
$37$	13088.8	1.57179	0.785897	−	0.618357i	$-0.212201\pi$
$37$	13088.8	1.57179	0.785897	+	0.618357i	$0.212201\pi$
$38$	−335.370	−0.0376761
$39$	0	0
$40$	−241.080	−0.0238238
$41$	3398.57	0.315745	0.157872	−	0.987460i	$-0.449537\pi$
$41$	3398.57	0.315745	0.157872	+	0.987460i	$0.449537\pi$
$42$	0	0
$43$	11678.2	0.963172	0.481586	−	0.876399i	$-0.340061\pi$
$43$	11678.2	0.963172	0.481586	+	0.876399i	$0.340061\pi$
$44$	3866.95	0.301118
$45$	0	0
$46$	152.131	0.0106004
$47$	640.836	0.0423158	0.0211579	−	0.999776i	$-0.493265\pi$
$47$	640.836	0.0423158	0.0211579	+	0.999776i	$0.493265\pi$
$48$	0	0
$49$	1122.04	0.0667602
$50$	568.665	0.0321686
$51$	0	0
$52$	15509.7	0.795419
$53$	−19381.1	−0.947739	−0.473869	−	0.880595i	$-0.657143\pi$
$53$	−19381.1	−0.947739	−0.473869	+	0.880595i	$0.657143\pi$
$54$	0	0
$55$	2233.17	0.0995440
$56$	−1749.06	−0.0745304
$57$	0	0
$58$	−214.342	−0.00836639
$59$	39338.6	1.47126	0.735629	−	0.677385i	$-0.236887\pi$
$59$	39338.6	1.47126	0.735629	+	0.677385i	$0.236887\pi$
$60$	0	0
$61$	19326.8	0.665020	0.332510	−	0.943100i	$-0.392104\pi$
$61$	19326.8	0.665020	0.332510	+	0.943100i	$0.392104\pi$
$62$	−661.263	−0.0218472
$63$	0	0
$64$	−32512.0	−0.992188
$65$	8956.89	0.262951
$66$	0	0
$67$	−11867.3	−0.322972	−0.161486	−	0.986875i	$-0.551629\pi$
$67$	−11867.3	−0.322972	−0.161486	+	0.986875i	$0.551629\pi$
$68$	−8524.96	−0.223573
$69$	0	0
$70$	−504.712	−0.0123111
$71$	−73798.6	−1.73741	−0.868705	−	0.495330i	$-0.835047\pi$
$71$	−73798.6	−1.73741	−0.868705	+	0.495330i	$0.835047\pi$
$72$	0	0
$73$	10415.6	0.228758	0.114379	−	0.993437i	$-0.463512\pi$
$73$	10415.6	0.228758	0.114379	+	0.993437i	$0.463512\pi$
$74$	−2673.18	−0.0567478
$75$	0	0
$76$	−52478.3	−1.04219
$77$	16201.8	0.311413
$78$	0	0
$79$	31511.5	0.568068	0.284034	−	0.958814i	$-0.408327\pi$
$79$	31511.5	0.568068	0.284034	+	0.958814i	$0.408327\pi$
$80$	−18825.0	−0.328860
$81$	0	0
$82$	−694.103	−0.0113996
$83$	104931.	1.67190	0.835948	−	0.548808i	$-0.184918\pi$
$83$	104931.	1.67190	0.835948	+	0.548808i	$0.184918\pi$
$84$	0	0
$85$	−4923.17	−0.0739091
$86$	−2385.08	−0.0347742
$87$	0	0
$88$	−1580.56	−0.0217572
$89$	22330.8	0.298833	0.149416	−	0.988774i	$-0.452260\pi$
$89$	22330.8	0.298833	0.149416	+	0.988774i	$0.452260\pi$
$90$	0	0
$91$	64983.0	0.822614
$92$	23805.2	0.293226
$93$	0	0
$94$	−130.881	−0.00152776
$95$	−30306.3	−0.344527
$96$	0	0
$97$	33323.3	0.359599	0.179800	−	0.983703i	$-0.442455\pi$
$97$	33323.3	0.359599	0.179800	+	0.983703i	$0.442455\pi$
$98$	−229.159	−0.00241030
$99$	0	0
$100$	88984.0	0.889840
$101$	26453.6	0.258037	0.129018	−	0.991642i	$-0.458817\pi$
$101$	26453.6	0.258037	0.129018	+	0.991642i	$0.458817\pi$
$102$	0	0
$103$	−125171.	−1.16255	−0.581274	−	0.813708i	$-0.697446\pi$
$103$	−125171.	−1.16255	−0.581274	+	0.813708i	$0.697446\pi$
$104$	−6339.37	−0.0574729
$105$	0	0
$106$	3958.28	0.0342170
$107$	−205945.	−1.73897	−0.869486	−	0.493957i	$-0.835550\pi$
$107$	−205945.	−1.73897	−0.869486	+	0.493957i	$0.835550\pi$
$108$	0	0
$109$	65337.4	0.526739	0.263369	−	0.964695i	$-0.415166\pi$
$109$	65337.4	0.526739	0.263369	+	0.964695i	$0.415166\pi$
$110$	−456.090	−0.00359392
$111$	0	0
$112$	−136577.	−1.02880
$113$	185212.	1.36450	0.682251	−	0.731118i	$-0.261001\pi$
$113$	185212.	1.36450	0.682251	+	0.731118i	$0.261001\pi$
$114$	0	0
$115$	13747.5	0.0969349
$116$	−33540.0	−0.231429
$117$	0	0
$118$	−8034.28	−0.0531180
$119$	−35718.0	−0.231217
$120$	0	0
$121$	14641.0	0.0909091
$122$	−3947.19	−0.0240098
$123$	0	0
$124$	−103474.	−0.604332
$125$	109063.	0.624314
$126$	0	0
$127$	−280059.	−1.54078	−0.770388	−	0.637575i	$-0.779938\pi$
$127$	−280059.	−1.54078	−0.770388	+	0.637575i	$0.779938\pi$
$128$	26682.2	0.143945
$129$	0	0
$130$	−1829.30	−0.00949352
$131$	114848.	0.584718	0.292359	−	0.956309i	$-0.405560\pi$
$131$	114848.	0.584718	0.292359	+	0.956309i	$0.405560\pi$
$132$	0	0
$133$	−219874.	−1.07782
$134$	2423.71	0.0116605
$135$	0	0
$136$	3484.45	0.0161542
$137$	−273184.	−1.24352	−0.621761	−	0.783207i	$-0.713582\pi$
$137$	−273184.	−1.24352	−0.621761	+	0.783207i	$0.713582\pi$
$138$	0	0
$139$	−73813.1	−0.324038	−0.162019	−	0.986788i	$-0.551801\pi$
$139$	−73813.1	−0.324038	−0.162019	+	0.986788i	$0.551801\pi$
$140$	−78976.6	−0.340548
$141$	0	0
$142$	15072.2	0.0627272
$143$	58722.7	0.240141
$144$	0	0
$145$	−19369.4	−0.0765061
$146$	−2127.22	−0.00825906
$147$	0	0
$148$	−418296.	−1.56975
$149$	−452183.	−1.66859	−0.834294	−	0.551320i	$-0.814124\pi$
$149$	−452183.	−1.66859	−0.834294	+	0.551320i	$0.814124\pi$
$150$	0	0
$151$	−132408.	−0.472575	−0.236287	−	0.971683i	$-0.575931\pi$
$151$	−132408.	−0.472575	−0.236287	+	0.971683i	$0.575931\pi$
$152$	21449.7	0.0753030
$153$	0	0
$154$	−3308.97	−0.0112432
$155$	−59756.1	−0.199781
$156$	0	0
$157$	72251.5	0.233936	0.116968	−	0.993136i	$-0.462682\pi$
$157$	72251.5	0.233936	0.116968	+	0.993136i	$0.462682\pi$
$158$	−6435.71	−0.0205095
$159$	0	0
$160$	11559.3	0.0356969
$161$	99739.4	0.303251
$162$	0	0
$163$	−177247.	−0.522530	−0.261265	−	0.965267i	$-0.584140\pi$
$163$	−177247.	−0.522530	−0.261265	+	0.965267i	$0.584140\pi$
$164$	−108612.	−0.315333
$165$	0	0
$166$	−21430.5	−0.0603619
$167$	431796.	1.19808	0.599042	−	0.800718i	$-0.295548\pi$
$167$	431796.	1.19808	0.599042	+	0.800718i	$0.295548\pi$
$168$	0	0
$169$	−135765.	−0.365655
$170$	1005.48	0.00266840
$171$	0	0
$172$	−373214.	−0.961916
$173$	232191.	0.589835	0.294918	−	0.955523i	$-0.404708\pi$
$173$	232191.	0.589835	0.294918	+	0.955523i	$0.404708\pi$
$174$	0	0
$175$	372826.	0.920262
$176$	−123420.	−0.300333
$177$	0	0
$178$	−4560.70	−0.0107890
$179$	309541.	0.722080	0.361040	−	0.932550i	$-0.382422\pi$
$179$	309541.	0.722080	0.361040	+	0.932550i	$0.382422\pi$
$180$	0	0
$181$	167882.	0.380898	0.190449	−	0.981697i	$-0.439006\pi$
$181$	167882.	0.380898	0.190449	+	0.981697i	$0.439006\pi$
$182$	−13271.7	−0.0296995
$183$	0	0
$184$	−9730.01	−0.0211870
$185$	−241566.	−0.518928
$186$	0	0
$187$	−32277.1	−0.0674979
$188$	−20480.0	−0.0422606
$189$	0	0
$190$	6189.58	0.0124388
$191$	247391.	0.490683	0.245341	−	0.969437i	$-0.421100\pi$
$191$	247391.	0.490683	0.245341	+	0.969437i	$0.421100\pi$
$192$	0	0
$193$	545313.	1.05379	0.526893	−	0.849932i	$-0.323357\pi$
$193$	545313.	1.05379	0.526893	+	0.849932i	$0.323357\pi$
$194$	−6805.76	−0.0129829
$195$	0	0
$196$	−35858.5	−0.0666732
$197$	−434544.	−0.797753	−0.398876	−	0.917005i	$-0.630600\pi$
$197$	−434544.	−0.797753	−0.398876	+	0.917005i	$0.630600\pi$
$198$	0	0
$199$	−633381.	−1.13379	−0.566895	−	0.823790i	$-0.691855\pi$
$199$	−633381.	−1.13379	−0.566895	+	0.823790i	$0.691855\pi$
$200$	−36370.8	−0.0642952
$201$	0	0
$202$	−5402.73	−0.00931611
$203$	−140527.	−0.239342
$204$	0	0
$205$	−62723.8	−0.104243
$206$	25564.2	0.0419725
$207$	0	0
$208$	−495017.	−0.793344
$209$	−198693.	−0.314641
$210$	0	0
$211$	−741755.	−1.14698	−0.573488	−	0.819214i	$-0.694410\pi$
$211$	−741755.	−1.14698	−0.573488	+	0.819214i	$0.694410\pi$
$212$	619387.	0.946503
$213$	0	0
$214$	42061.1	0.0627836
$215$	−215532.	−0.317991
$216$	0	0
$217$	−433535.	−0.624993
$218$	−13344.1	−0.0190173
$219$	0	0
$220$	−71368.3	−0.0994142
$221$	−129458.	−0.178299
$222$	0	0
$223$	−1.23824e6	−1.66741	−0.833703	−	0.552214i	$-0.813783\pi$
$223$	−1.23824e6	−1.66741	−0.833703	+	0.552214i	$0.813783\pi$
$224$	83863.5	0.111674
$225$	0	0
$226$	−37826.7	−0.0492637
$227$	1.49159e6	1.92125	0.960626	−	0.277846i	$-0.0896204\pi$
$227$	1.49159e6	1.92125	0.960626	+	0.277846i	$0.0896204\pi$
$228$	0	0
$229$	−187173.	−0.235860	−0.117930	−	0.993022i	$-0.537626\pi$
$229$	−187173.	−0.235860	−0.117930	+	0.993022i	$0.537626\pi$
$230$	−2807.72	−0.00349972
$231$	0	0
$232$	13709.0	0.0167219
$233$	1.30093e6	1.56988	0.784938	−	0.619574i	$-0.212695\pi$
$233$	1.30093e6	1.56988	0.784938	+	0.619574i	$0.212695\pi$
$234$	0	0
$235$	−11827.2	−0.0139706
$236$	−1.25719e6	−1.46934
$237$	0	0
$238$	7294.84	0.00834782
$239$	−1.49959e6	−1.69815	−0.849077	−	0.528269i	$-0.822841\pi$
$239$	−1.49959e6	−1.69815	−0.849077	+	0.528269i	$0.822841\pi$
$240$	0	0
$241$	−1.57999e6	−1.75231	−0.876157	−	0.482026i	$-0.839901\pi$
$241$	−1.57999e6	−1.75231	−0.876157	+	0.482026i	$0.839901\pi$
$242$	−2990.19	−0.00328217
$243$	0	0
$244$	−617651.	−0.664153
$245$	−20708.3	−0.0220409
$246$	0	0
$247$	−796925.	−0.831141
$248$	42293.3	0.0436659
$249$	0	0
$250$	−22274.4	−0.0225401
$251$	1.64669e6	1.64979	0.824893	−	0.565289i	$-0.191235\pi$
$251$	1.64669e6	1.64979	0.824893	+	0.565289i	$0.191235\pi$
$252$	0	0
$253$	90130.9	0.0885263
$254$	57197.5	0.0556279
$255$	0	0
$256$	1.03493e6	0.986991
$257$	−153285.	−0.144766	−0.0723830	−	0.997377i	$-0.523060\pi$
$257$	−153285.	−0.144766	−0.0723830	+	0.997377i	$0.523060\pi$
$258$	0	0
$259$	−1.75258e6	−1.62341
$260$	−286247.	−0.262608
$261$	0	0
$262$	−23455.9	−0.0211106
$263$	−2.12004e6	−1.88997	−0.944983	−	0.327119i	$-0.893922\pi$
$263$	−2.12004e6	−1.88997	−0.944983	+	0.327119i	$0.893922\pi$
$264$	0	0
$265$	357697.	0.312896
$266$	44905.9	0.0389134
$267$	0	0
$268$	379259.	0.322551
$269$	944003.	0.795413	0.397706	−	0.917513i	$-0.369806\pi$
$269$	944003.	0.795413	0.397706	+	0.917513i	$0.369806\pi$
$270$	0	0
$271$	−933742.	−0.772332	−0.386166	−	0.922429i	$-0.626201\pi$
$271$	−933742.	−0.772332	−0.386166	+	0.922429i	$0.626201\pi$
$272$	272087.	0.222990
$273$	0	0
$274$	55793.4	0.0448959
$275$	336910.	0.268647
$276$	0	0
$277$	1.14132e6	0.893733	0.446867	−	0.894601i	$-0.352540\pi$
$277$	1.14132e6	0.893733	0.446867	+	0.894601i	$0.352540\pi$
$278$	15075.2	0.0116990
$279$	0	0
$280$	32280.5	0.0246062
$281$	−373073.	−0.281856	−0.140928	−	0.990020i	$-0.545009\pi$
$281$	−373073.	−0.281856	−0.140928	+	0.990020i	$0.545009\pi$
$282$	0	0
$283$	188597.	0.139981	0.0699904	−	0.997548i	$-0.477703\pi$
$283$	188597.	0.139981	0.0699904	+	0.997548i	$0.477703\pi$
$284$	2.35848e6	1.73515
$285$	0	0
$286$	−11993.2	−0.00867001
$287$	−455066.	−0.326114
$288$	0	0
$289$	−1.34870e6	−0.949884
$290$	3955.89	0.00276216
$291$	0	0
$292$	−332865.	−0.228460
$293$	−1.82401e6	−1.24125	−0.620624	−	0.784109i	$-0.713121\pi$
$293$	−1.82401e6	−1.24125	−0.620624	+	0.784109i	$0.713121\pi$
$294$	0	0
$295$	−726031.	−0.485736
$296$	170972.	0.113422
$297$	0	0
$298$	92351.3	0.0602424
$299$	361501.	0.233847
$300$	0	0
$301$	−1.56370e6	−0.994803
$302$	27042.1	0.0170618
$303$	0	0
$304$	1.67492e6	1.03947
$305$	−356694.	−0.219556
$306$	0	0
$307$	1.43224e6	0.867304	0.433652	−	0.901080i	$-0.357225\pi$
$307$	1.43224e6	0.867304	0.433652	+	0.901080i	$0.357225\pi$
$308$	−517783.	−0.311007
$309$	0	0
$310$	12204.2	0.00721285
$311$	−552579.	−0.323961	−0.161981	−	0.986794i	$-0.551788\pi$
$311$	−552579.	−0.323961	−0.161981	+	0.986794i	$0.551788\pi$
$312$	0	0
$313$	62103.3	0.0358306	0.0179153	−	0.999840i	$-0.494297\pi$
$313$	62103.3	0.0358306	0.0179153	+	0.999840i	$0.494297\pi$
$314$	−14756.2	−0.00844600
$315$	0	0
$316$	−1.00705e6	−0.567328
$317$	3202.74	0.00179009	0.000895043	−	1.00000i	$-0.499715\pi$
$317$	3202.74	0.00179009	0.000895043		1.00000i	$0.499715\pi$
$318$	0	0
$319$	−126989.	−0.0698696
$320$	600040.	0.327571
$321$	0	0
$322$	−20370.2	−0.0109485
$323$	438031.	0.233614
$324$	0	0
$325$	1.35129e6	0.709644
$326$	36200.0	0.0188653
$327$	0	0
$328$	44393.7	0.0227843
$329$	−85807.6	−0.0437055
$330$	0	0
$331$	−1.88065e6	−0.943494	−0.471747	−	0.881734i	$-0.656376\pi$
$331$	−1.88065e6	−0.943494	−0.471747	+	0.881734i	$0.656376\pi$
$332$	−3.35342e6	−1.66972
$333$	0	0
$334$	−88187.4	−0.0432554
$335$	219022.	0.106629
$336$	0	0
$337$	−2.41239e6	−1.15711	−0.578553	−	0.815644i	$-0.696383\pi$
$337$	−2.41239e6	−1.15711	−0.578553	+	0.815644i	$0.696383\pi$
$338$	27727.9	0.0132016
$339$	0	0
$340$	157336.	0.0738127
$341$	−391770.	−0.182451
$342$	0	0
$343$	2.10021e6	0.963888
$344$	152546.	0.0695030
$345$	0	0
$346$	−47421.4	−0.0212953
$347$	398842.	0.177818	0.0889092	−	0.996040i	$-0.471662\pi$
$347$	398842.	0.177818	0.0889092	+	0.996040i	$0.471662\pi$
$348$	0	0
$349$	1.00215e6	0.440422	0.220211	−	0.975452i	$-0.429325\pi$
$349$	1.00215e6	0.440422	0.220211	+	0.975452i	$0.429325\pi$
$350$	−76143.9	−0.0332250
$351$	0	0
$352$	75784.4	0.0326004
$353$	2.71948e6	1.16158	0.580791	−	0.814053i	$-0.302743\pi$
$353$	2.71948e6	1.16158	0.580791	+	0.814053i	$0.302743\pi$
$354$	0	0
$355$	1.36202e6	0.573606
$356$	−713653.	−0.298443
$357$	0	0
$358$	−63218.8	−0.0260699
$359$	−3.37239e6	−1.38103	−0.690514	−	0.723319i	$-0.742615\pi$
$359$	−3.37239e6	−1.38103	−0.690514	+	0.723319i	$0.742615\pi$
$360$	0	0
$361$	220351.	0.0889913
$362$	−34287.3	−0.0137519
$363$	0	0
$364$	−2.07674e6	−0.821542
$365$	−192230.	−0.0755246
$366$	0	0
$367$	1.89673e6	0.735091	0.367545	−	0.930006i	$-0.380198\pi$
$367$	1.89673e6	0.735091	0.367545	+	0.930006i	$0.380198\pi$
$368$	−759779.	−0.292461
$369$	0	0
$370$	49336.1	0.0187353
$371$	2.59512e6	0.978863
$372$	0	0
$373$	284845.	0.106007	0.0530037	−	0.998594i	$-0.483120\pi$
$373$	284845.	0.106007	0.0530037	+	0.998594i	$0.483120\pi$
$374$	6592.08	0.00243693
$375$	0	0
$376$	8370.90	0.00305353
$377$	−509332.	−0.184564
$378$	0	0
$379$	−1.36001e6	−0.486346	−0.243173	−	0.969983i	$-0.578188\pi$
$379$	−1.36001e6	−0.486346	−0.243173	+	0.969983i	$0.578188\pi$
$380$	968537.	0.344078
$381$	0	0
$382$	−50525.7	−0.0177155
$383$	2.77594e6	0.966971	0.483485	−	0.875352i	$-0.339371\pi$
$383$	2.77594e6	0.966971	0.483485	+	0.875352i	$0.339371\pi$
$384$	0	0
$385$	−299020.	−0.102813
$386$	−111371.	−0.0380457
$387$	0	0
$388$	−1.06496e6	−0.359131
$389$	2.45004e6	0.820917	0.410459	−	0.911879i	$-0.365369\pi$
$389$	2.45004e6	0.820917	0.410459	+	0.911879i	$0.365369\pi$
$390$	0	0
$391$	−198700.	−0.0657287
$392$	14656.6	0.00481746
$393$	0	0
$394$	88748.8	0.0288019
$395$	−581574.	−0.187548
$396$	0	0
$397$	−853965.	−0.271934	−0.135967	−	0.990713i	$-0.543414\pi$
$397$	−853965.	−0.271934	−0.135967	+	0.990713i	$0.543414\pi$
$398$	129358.	0.0409342
$399$	0	0
$400$	−2.84006e6	−0.887518
$401$	−1.60787e6	−0.499334	−0.249667	−	0.968332i	$-0.580321\pi$
$401$	−1.60787e6	−0.499334	−0.249667	+	0.968332i	$0.580321\pi$
$402$	0	0
$403$	−1.57133e6	−0.481953
$404$	−845412.	−0.257700
$405$	0	0
$406$	28700.3	0.00864115
$407$	−1.58375e6	−0.473914
$408$	0	0
$409$	−5.12405e6	−1.51463	−0.757313	−	0.653053i	$-0.773488\pi$
$409$	−5.12405e6	−1.51463	−0.757313	+	0.653053i	$0.773488\pi$
$410$	12810.3	0.00376358
$411$	0	0
$412$	4.00025e6	1.16103
$413$	−5.26741e6	−1.51957
$414$	0	0
$415$	−1.93660e6	−0.551977
$416$	303959.	0.0861156
$417$	0	0
$418$	40579.8	0.0113598
$419$	986803.	0.274597	0.137298	−	0.990530i	$-0.456158\pi$
$419$	986803.	0.274597	0.137298	+	0.990530i	$0.456158\pi$
$420$	0	0
$421$	−4.27858e6	−1.17651	−0.588254	−	0.808676i	$-0.700184\pi$
$421$	−4.27858e6	−1.17651	−0.588254	+	0.808676i	$0.700184\pi$
$422$	151492.	0.0414103
$423$	0	0
$424$	−253165.	−0.0683894
$425$	−742740.	−0.199464
$426$	0	0
$427$	−2.58784e6	−0.686860
$428$	6.58166e6	1.73671
$429$	0	0
$430$	44018.9	0.0114807
$431$	4.72829e6	1.22606	0.613029	−	0.790061i	$-0.289951\pi$
$431$	4.72829e6	1.22606	0.613029	+	0.790061i	$0.289951\pi$
$432$	0	0
$433$	−3.99084e6	−1.02293	−0.511463	−	0.859305i	$-0.670896\pi$
$433$	−3.99084e6	−1.02293	−0.511463	+	0.859305i	$0.670896\pi$
$434$	88542.7	0.0225647
$435$	0	0
$436$	−2.08807e6	−0.526052
$437$	−1.22316e6	−0.306394
$438$	0	0
$439$	−5.85827e6	−1.45080	−0.725401	−	0.688327i	$-0.758345\pi$
$439$	−5.85827e6	−1.45080	−0.725401	+	0.688327i	$0.758345\pi$
$440$	29170.7	0.00718316
$441$	0	0
$442$	26439.8	0.00643728
$443$	−7.14607e6	−1.73005	−0.865024	−	0.501731i	$-0.832697\pi$
$443$	−7.14607e6	−1.73005	−0.865024	+	0.501731i	$0.832697\pi$
$444$	0	0
$445$	−412135.	−0.0986597
$446$	252890.	0.0601997
$447$	0	0
$448$	4.35334e6	1.02477
$449$	4.92038e6	1.15182	0.575908	−	0.817514i	$-0.304649\pi$
$449$	4.92038e6	1.15182	0.575908	+	0.817514i	$0.304649\pi$
$450$	0	0
$451$	−411226.	−0.0952006
$452$	−5.91907e6	−1.36272
$453$	0	0
$454$	−304633.	−0.0693645
$455$	−1.19932e6	−0.271586
$456$	0	0
$457$	2.27908e6	0.510468	0.255234	−	0.966879i	$-0.417847\pi$
$457$	2.27908e6	0.510468	0.255234	+	0.966879i	$0.417847\pi$
$458$	38227.2	0.00851546
$459$	0	0
$460$	−439348.	−0.0968085
$461$	6.48001e6	1.42012	0.710058	−	0.704144i	$-0.248669\pi$
$461$	6.48001e6	1.42012	0.710058	+	0.704144i	$0.248669\pi$
$462$	0	0
$463$	2.36408e6	0.512519	0.256259	−	0.966608i	$-0.417510\pi$
$463$	2.36408e6	0.512519	0.256259	+	0.966608i	$0.417510\pi$
$464$	1.07048e6	0.230826
$465$	0	0
$466$	−265695.	−0.0566786
$467$	−1.97130e6	−0.418274	−0.209137	−	0.977886i	$-0.567065\pi$
$467$	−1.97130e6	−0.418274	−0.209137	+	0.977886i	$0.567065\pi$
$468$	0	0
$469$	1.58902e6	0.333579
$470$	2415.53	0.000504391 0
$471$	0	0
$472$	513859.	0.106167
$473$	−1.41306e6	−0.290407
$474$	0	0
$475$	−4.57219e6	−0.929802
$476$	1.14149e6	0.230916
$477$	0	0
$478$	306267.	0.0613099
$479$	−4.20561e6	−0.837511	−0.418756	−	0.908099i	$-0.637534\pi$
$479$	−4.20561e6	−0.837511	−0.418756	+	0.908099i	$0.637534\pi$
$480$	0	0
$481$	−6.35216e6	−1.25187
$482$	322688.	0.0632653
$483$	0	0
$484$	−467901.	−0.0907906
$485$	−615013.	−0.118722
$486$	0	0
$487$	4.59957e6	0.878809	0.439404	−	0.898289i	$-0.355190\pi$
$487$	4.59957e6	0.878809	0.439404	+	0.898289i	$0.355190\pi$
$488$	252455.	0.0479883
$489$	0	0
$490$	4229.34	0.000795761 0
$491$	−6.05076e6	−1.13268	−0.566339	−	0.824173i	$-0.691641\pi$
$491$	−6.05076e6	−1.13268	−0.566339	+	0.824173i	$0.691641\pi$
$492$	0	0
$493$	279955.	0.0518766
$494$	162759.	0.0300074
$495$	0	0
$496$	3.30252e6	0.602755
$497$	9.88159e6	1.79447
$498$	0	0
$499$	6.19572e6	1.11389	0.556943	−	0.830551i	$-0.311974\pi$
$499$	6.19572e6	1.11389	0.556943	+	0.830551i	$0.311974\pi$
$500$	−3.48547e6	−0.623500
$501$	0	0
$502$	−336310.	−0.0595636
$503$	1.00722e7	1.77502	0.887508	−	0.460792i	$-0.152435\pi$
$503$	1.00722e7	1.77502	0.887508	+	0.460792i	$0.152435\pi$
$504$	0	0
$505$	−488226.	−0.0851908
$506$	−18407.8	−0.00319614
$507$	0	0
$508$	8.95019e6	1.53877
$509$	2.71440e6	0.464386	0.232193	−	0.972670i	$-0.425410\pi$
$509$	2.71440e6	0.464386	0.232193	+	0.972670i	$0.425410\pi$
$510$	0	0
$511$	−1.39464e6	−0.236271
$512$	−1.06520e6	−0.179579
$513$	0	0
$514$	31306.0	0.00522661
$515$	2.31015e6	0.383815
$516$	0	0
$517$	−77541.2	−0.0127587
$518$	357937.	0.0586115
$519$	0	0
$520$	116999.	0.0189747
$521$	−1.05219e7	−1.69824	−0.849121	−	0.528199i	$-0.822867\pi$
$521$	−1.05219e7	−1.69824	−0.849121	+	0.528199i	$0.822867\pi$
$522$	0	0
$523$	643830.	0.102924	0.0514620	−	0.998675i	$-0.483612\pi$
$523$	643830.	0.102924	0.0514620	+	0.998675i	$0.483612\pi$
$524$	−3.67035e6	−0.583956
$525$	0	0
$526$	432984.	0.0682350
$527$	863684.	0.135465
$528$	0	0
$529$	−5.88149e6	−0.913794
$530$	−73053.8	−0.0112967
$531$	0	0
$532$	7.02681e6	1.07641
$533$	−1.64936e6	−0.251477
$534$	0	0
$535$	3.80092e6	0.574122
$536$	−155016.	−0.0233059
$537$	0	0
$538$	−192798.	−0.0287175
$539$	−135767.	−0.0201290
$540$	0	0
$541$	−3.12700e6	−0.459341	−0.229670	−	0.973268i	$-0.573765\pi$
$541$	−3.12700e6	−0.459341	−0.229670	+	0.973268i	$0.573765\pi$
$542$	190702.	0.0278841
$543$	0	0
$544$	−167072.	−0.0242050
$545$	−1.20586e6	−0.173903
$546$	0	0
$547$	7.19824e6	1.02863	0.514314	−	0.857602i	$-0.328047\pi$
$547$	7.19824e6	1.02863	0.514314	+	0.857602i	$0.328047\pi$
$548$	8.73048e6	1.24190
$549$	0	0
$550$	−68808.5	−0.00969918
$551$	1.72336e6	0.241823
$552$	0	0
$553$	−4.21936e6	−0.586724
$554$	−233097.	−0.0322672
$555$	0	0
$556$	2.35894e6	0.323616
$557$	3.78257e6	0.516593	0.258297	−	0.966066i	$-0.416839\pi$
$557$	3.78257e6	0.516593	0.258297	+	0.966066i	$0.416839\pi$
$558$	0	0
$559$	−5.66756e6	−0.767125
$560$	2.52066e6	0.339660
$561$	0	0
$562$	76194.2	0.0101761
$563$	1.04802e7	1.39348	0.696738	−	0.717326i	$-0.254634\pi$
$563$	1.04802e7	1.39348	0.696738	+	0.717326i	$0.254634\pi$
$564$	0	0
$565$	−3.41827e6	−0.450490
$566$	−38517.9	−0.00505384
$567$	0	0
$568$	−963992.	−0.125373
$569$	−5.53639e6	−0.716879	−0.358440	−	0.933553i	$-0.616691\pi$
$569$	−5.53639e6	−0.716879	−0.358440	+	0.933553i	$0.616691\pi$
$570$	0	0
$571$	1.37354e7	1.76299	0.881495	−	0.472194i	$-0.156538\pi$
$571$	1.37354e7	1.76299	0.881495	+	0.472194i	$0.156538\pi$
$572$	−1.87668e6	−0.239828
$573$	0	0
$574$	92940.0	0.0117740
$575$	2.07404e6	0.261606
$576$	0	0
$577$	−2.03704e6	−0.254718	−0.127359	−	0.991857i	$-0.540650\pi$
$577$	−2.03704e6	−0.254718	−0.127359	+	0.991857i	$0.540650\pi$
$578$	275451.	0.0342945
$579$	0	0
$580$	619013.	0.0764064
$581$	−1.40502e7	−1.72680
$582$	0	0
$583$	2.34511e6	0.285754
$584$	136053.	0.0165074
$585$	0	0
$586$	372525.	0.0448138
$587$	−1.24125e7	−1.48684	−0.743422	−	0.668822i	$-0.766799\pi$
$587$	−1.24125e7	−1.48684	−0.743422	+	0.668822i	$0.766799\pi$
$588$	0	0
$589$	5.31670e6	0.631472
$590$	148280.	0.0175369
$591$	0	0
$592$	1.33506e7	1.56565
$593$	−5.38467e6	−0.628815	−0.314407	−	0.949288i	$-0.601806\pi$
$593$	−5.38467e6	−0.628815	−0.314407	+	0.949288i	$0.601806\pi$
$594$	0	0
$595$	659210.	0.0763363
$596$	1.44510e7	1.66641
$597$	0	0
$598$	−73830.8	−0.00844277
$599$	−3.96508e6	−0.451528	−0.225764	−	0.974182i	$-0.572488\pi$
$599$	−3.96508e6	−0.451528	−0.225764	+	0.974182i	$0.572488\pi$
$600$	0	0
$601$	7.89653e6	0.891764	0.445882	−	0.895092i	$-0.352890\pi$
$601$	7.89653e6	0.891764	0.445882	+	0.895092i	$0.352890\pi$
$602$	319361.	0.0359162
$603$	0	0
$604$	4.23152e6	0.471959
$605$	−270214.	−0.0300136
$606$	0	0
$607$	1.43363e7	1.57930	0.789651	−	0.613556i	$-0.210262\pi$
$607$	1.43363e7	1.57930	0.789651	+	0.613556i	$0.210262\pi$
$608$	−1.02847e6	−0.112832
$609$	0	0
$610$	72849.1	0.00792683
$611$	−311006.	−0.0337027
$612$	0	0
$613$	1.07870e7	1.15945	0.579724	−	0.814813i	$-0.303160\pi$
$613$	1.07870e7	1.15945	0.579724	+	0.814813i	$0.303160\pi$
$614$	−292513.	−0.0313130
$615$	0	0
$616$	211636.	0.0224718
$617$	6.10601e6	0.645721	0.322860	−	0.946447i	$-0.395356\pi$
$617$	6.10601e6	0.645721	0.322860	+	0.946447i	$0.395356\pi$
$618$	0	0
$619$	1.38454e7	1.45238	0.726190	−	0.687494i	$-0.241289\pi$
$619$	1.38454e7	1.45238	0.726190	+	0.687494i	$0.241289\pi$
$620$	1.90970e6	0.199520
$621$	0	0
$622$	112855.	0.0116962
$623$	−2.99007e6	−0.308647
$624$	0	0
$625$	6.68832e6	0.684884
$626$	−12683.6	−0.00129362
$627$	0	0
$628$	−2.30903e6	−0.233631
$629$	3.49147e6	0.351870
$630$	0	0
$631$	−2.17332e6	−0.217296	−0.108648	−	0.994080i	$-0.534652\pi$
$631$	−2.17332e6	−0.217296	−0.108648	+	0.994080i	$0.534652\pi$
$632$	411617.	0.0409922
$633$	0	0
$634$	−654.109	−6.46290e−5 0
$635$	5.16875e6	0.508687
$636$	0	0
$637$	−544539.	−0.0531717
$638$	25935.4	0.00252256
$639$	0	0
$640$	−492446.	−0.0475235
$641$	−2.72173e6	−0.261637	−0.130819	−	0.991406i	$-0.541761\pi$
$641$	−2.72173e6	−0.261637	−0.130819	+	0.991406i	$0.541761\pi$
$642$	0	0
$643$	1.75669e7	1.67559	0.837794	−	0.545986i	$-0.183845\pi$
$643$	1.75669e7	1.67559	0.837794	+	0.545986i	$0.183845\pi$
$644$	−3.18750e6	−0.302856
$645$	0	0
$646$	−89460.9	−0.00843436
$647$	1.50300e7	1.41156	0.705779	−	0.708432i	$-0.250597\pi$
$647$	1.50300e7	1.41156	0.705779	+	0.708432i	$0.250597\pi$
$648$	0	0
$649$	−4.75997e6	−0.443601
$650$	−275980.	−0.0256209
$651$	0	0
$652$	5.66453e6	0.521849
$653$	−5.45622e6	−0.500736	−0.250368	−	0.968151i	$-0.580552\pi$
$653$	−5.45622e6	−0.500736	−0.250368	+	0.968151i	$0.580552\pi$
$654$	0	0
$655$	−2.11963e6	−0.193045
$656$	3.46653e6	0.314511
$657$	0	0
$658$	17524.8	0.00157794
$659$	−2.15403e6	−0.193214	−0.0966070	−	0.995323i	$-0.530799\pi$
$659$	−2.15403e6	−0.193214	−0.0966070	+	0.995323i	$0.530799\pi$
$660$	0	0
$661$	4.74952e6	0.422810	0.211405	−	0.977398i	$-0.432196\pi$
$661$	4.74952e6	0.422810	0.211405	+	0.977398i	$0.432196\pi$
$662$	384094.	0.0340637
$663$	0	0
$664$	1.37066e6	0.120645
$665$	4.05799e6	0.355842
$666$	0	0
$667$	−781750.	−0.0680383
$668$	−1.37994e7	−1.19652
$669$	0	0
$670$	−44731.8	−0.00384973
$671$	−2.33854e6	−0.200511
$672$	0	0
$673$	−8.84599e6	−0.752851	−0.376425	−	0.926447i	$-0.622847\pi$
$673$	−8.84599e6	−0.752851	−0.376425	+	0.926447i	$0.622847\pi$
$674$	492693.	0.0417760
$675$	0	0
$676$	4.33883e6	0.365179
$677$	−5.95663e6	−0.499492	−0.249746	−	0.968311i	$-0.580347\pi$
$677$	−5.95663e6	−0.499492	−0.249746	+	0.968311i	$0.580347\pi$
$678$	0	0
$679$	−4.46197e6	−0.371409
$680$	−64308.8	−0.00533332
$681$	0	0
$682$	80012.9	0.00658717
$683$	−1.71760e7	−1.40887	−0.704433	−	0.709771i	$-0.748798\pi$
$683$	−1.71760e7	−1.40887	−0.704433	+	0.709771i	$0.748798\pi$
$684$	0	0
$685$	5.04186e6	0.410549
$686$	−428934.	−0.0348001
$687$	0	0
$688$	1.19117e7	0.959407
$689$	9.40588e6	0.754834
$690$	0	0
$691$	1.94330e7	1.54826	0.774130	−	0.633026i	$-0.218187\pi$
$691$	1.94330e7	1.54826	0.774130	+	0.633026i	$0.218187\pi$
$692$	−7.42044e6	−0.589067
$693$	0	0
$694$	−81457.1	−0.00641993
$695$	1.36229e6	0.106981
$696$	0	0
$697$	906576.	0.0706842
$698$	−204673.	−0.0159009
$699$	0	0
$700$	−1.19149e7	−0.919063
$701$	1.83197e7	1.40807	0.704035	−	0.710165i	$-0.251380\pi$
$701$	1.83197e7	1.40807	0.704035	+	0.710165i	$0.251380\pi$
$702$	0	0
$703$	2.14930e7	1.64024
$704$	3.93395e6	0.299156
$705$	0	0
$706$	−555412.	−0.0419376
$707$	−3.54212e6	−0.266511
$708$	0	0
$709$	192174.	0.0143575	0.00717874	−	0.999974i	$-0.497715\pi$
$709$	192174.	0.0143575	0.00717874	+	0.999974i	$0.497715\pi$
$710$	−278172.	−0.0207094
$711$	0	0
$712$	291695.	0.0215640
$713$	−2.41176e6	−0.177669
$714$	0	0
$715$	−1.08378e6	−0.0792826
$716$	−9.89240e6	−0.721139
$717$	0	0
$718$	688758.	0.0498604
$719$	−1.15700e7	−0.834663	−0.417332	−	0.908754i	$-0.637035\pi$
$719$	−1.15700e7	−0.834663	−0.417332	+	0.908754i	$0.637035\pi$
$720$	0	0
$721$	1.67603e7	1.20073
$722$	−45003.2	−0.00321293
$723$	0	0
$724$	−5.36523e6	−0.380401
$725$	−2.92219e6	−0.206473
$726$	0	0
$727$	3.90368e6	0.273929	0.136965	−	0.990576i	$-0.456265\pi$
$727$	3.90368e6	0.273929	0.136965	+	0.990576i	$0.456265\pi$
$728$	848838.	0.0593603
$729$	0	0
$730$	39259.9	0.00272673
$731$	3.11518e6	0.215621
$732$	0	0
$733$	−6.77636e6	−0.465840	−0.232920	−	0.972496i	$-0.574828\pi$
$733$	−6.77636e6	−0.465840	−0.232920	+	0.972496i	$0.574828\pi$
$734$	−387378.	−0.0265396
$735$	0	0
$736$	466533.	0.0317459
$737$	1.43594e6	0.0973798
$738$	0	0
$739$	2.04716e7	1.37893	0.689463	−	0.724321i	$-0.257847\pi$
$739$	2.04716e7	1.37893	0.689463	+	0.724321i	$0.257847\pi$
$740$	7.72005e6	0.518252
$741$	0	0
$742$	−530011.	−0.0353407
$743$	−7.80246e6	−0.518513	−0.259256	−	0.965809i	$-0.583477\pi$
$743$	−7.80246e6	−0.518513	−0.259256	+	0.965809i	$0.583477\pi$
$744$	0	0
$745$	8.34547e6	0.550884
$746$	−58175.0	−0.00382727
$747$	0	0
$748$	1.03152e6	0.0674099
$749$	2.75760e7	1.79608
$750$	0	0
$751$	−2.42552e7	−1.56930	−0.784649	−	0.619941i	$-0.787157\pi$
$751$	−2.42552e7	−1.56930	−0.784649	+	0.619941i	$0.787157\pi$
$752$	653652.	0.0421504
$753$	0	0
$754$	104023.	0.00666347
$755$	2.44371e6	0.156021
$756$	0	0
$757$	−1.12699e7	−0.714792	−0.357396	−	0.933953i	$-0.616335\pi$
$757$	−1.12699e7	−0.714792	−0.357396	+	0.933953i	$0.616335\pi$
$758$	277761.	0.0175589
$759$	0	0
$760$	−395875.	−0.0248613
$761$	1.76585e7	1.10533	0.552666	−	0.833403i	$-0.313610\pi$
$761$	1.76585e7	1.10533	0.552666	+	0.833403i	$0.313610\pi$
$762$	0	0
$763$	−8.74863e6	−0.544038
$764$	−7.90620e6	−0.490043
$765$	0	0
$766$	−566942.	−0.0349113
$767$	−1.90915e7	−1.17179
$768$	0	0
$769$	−2.13086e7	−1.29939	−0.649695	−	0.760195i	$-0.725103\pi$
$769$	−2.13086e7	−1.29939	−0.649695	+	0.760195i	$0.725103\pi$
$770$	61070.1	0.00371195
$771$	0	0
$772$	−1.74273e7	−1.05241
$773$	−1.26033e6	−0.0758639	−0.0379320	−	0.999280i	$-0.512077\pi$
$773$	−1.26033e6	−0.0758639	−0.0379320	+	0.999280i	$0.512077\pi$
$774$	0	0
$775$	−9.01518e6	−0.539163
$776$	435285.	0.0259489
$777$	0	0
$778$	−500382.	−0.0296383
$779$	5.58074e6	0.329495
$780$	0	0
$781$	8.92963e6	0.523849
$782$	40581.2	0.00237306
$783$	0	0
$784$	1.14448e6	0.0664993
$785$	−1.33347e6	−0.0772341
$786$	0	0
$787$	−1.28175e7	−0.737676	−0.368838	−	0.929494i	$-0.620244\pi$
$787$	−1.28175e7	−0.737676	−0.368838	+	0.929494i	$0.620244\pi$
$788$	1.38873e7	0.796713
$789$	0	0
$790$	118777.	0.00677120
$791$	−2.47998e7	−1.40931
$792$	0	0
$793$	−9.37952e6	−0.529660
$794$	174409.	0.00981787
$795$	0	0
$796$	2.02418e7	1.13231
$797$	990679.	0.0552443	0.0276221	−	0.999618i	$-0.491206\pi$
$797$	990679.	0.0552443	0.0276221	+	0.999618i	$0.491206\pi$
$798$	0	0
$799$	170945.	0.00947303
$800$	1.74390e6	0.0963380
$801$	0	0
$802$	328383.	0.0180279
$803$	−1.26029e6	−0.0689733
$804$	0	0
$805$	−1.84079e6	−0.100118
$806$	320919.	0.0174003
$807$	0	0
$808$	345549.	0.0186201
$809$	−2.02544e7	−1.08805	−0.544024	−	0.839070i	$-0.683100\pi$
$809$	−2.02544e7	−1.08805	−0.544024	+	0.839070i	$0.683100\pi$
$810$	0	0
$811$	−2.21681e7	−1.18352	−0.591762	−	0.806113i	$-0.701567\pi$
$811$	−2.21681e7	−1.18352	−0.591762	+	0.806113i	$0.701567\pi$
$812$	4.49099e6	0.239030
$813$	0	0
$814$	323455.	0.0171101
$815$	3.27127e6	0.172513
$816$	0	0
$817$	1.91766e7	1.00512
$818$	1.04651e6	0.0546838
$819$	0	0
$820$	2.00454e6	0.104107
$821$	2.00313e6	0.103717	0.0518587	−	0.998654i	$-0.483485\pi$
$821$	2.00313e6	0.103717	0.0518587	+	0.998654i	$0.483485\pi$
$822$	0	0
$823$	−2.91552e7	−1.50043	−0.750217	−	0.661192i	$-0.770051\pi$
$823$	−2.91552e7	−1.50043	−0.750217	+	0.661192i	$0.770051\pi$
$824$	−1.63504e6	−0.0838902
$825$	0	0
$826$	1.07579e6	0.0548625
$827$	−1.92308e7	−0.977765	−0.488882	−	0.872350i	$-0.662595\pi$
$827$	−1.92308e7	−0.977765	−0.488882	+	0.872350i	$0.662595\pi$
$828$	0	0
$829$	1.56404e7	0.790427	0.395214	−	0.918589i	$-0.370671\pi$
$829$	1.56404e7	0.790427	0.395214	+	0.918589i	$0.370671\pi$
$830$	395521.	0.0199285
$831$	0	0
$832$	1.57785e7	0.790235
$833$	299307.	0.0149453
$834$	0	0
$835$	−7.96920e6	−0.395547
$836$	6.34987e6	0.314231
$837$	0	0
$838$	−201539.	−0.00991399
$839$	3.68277e7	1.80621	0.903107	−	0.429416i	$-0.141280\pi$
$839$	3.68277e7	1.80621	0.903107	+	0.429416i	$0.141280\pi$
$840$	0	0
$841$	−1.94097e7	−0.946301
$842$	873833.	0.0424765
$843$	0	0
$844$	2.37052e7	1.14548
$845$	2.50568e6	0.120721
$846$	0	0
$847$	−1.96042e6	−0.0938946
$848$	−1.97687e7	−0.944034
$849$	0	0
$850$	151693.	0.00720142
$851$	−9.74964e6	−0.461492
$852$	0	0
$853$	−3.15483e7	−1.48458	−0.742289	−	0.670080i	$-0.766260\pi$
$853$	−3.15483e7	−1.48458	−0.742289	+	0.670080i	$0.766260\pi$
$854$	528526.	0.0247983
$855$	0	0
$856$	−2.69016e6	−0.125485
$857$	3.68633e7	1.71452	0.857260	−	0.514883i	$-0.172165\pi$
$857$	3.68633e7	1.71452	0.857260	+	0.514883i	$0.172165\pi$
$858$	0	0
$859$	−2.94173e6	−0.136025	−0.0680126	−	0.997684i	$-0.521666\pi$
$859$	−2.94173e6	−0.136025	−0.0680126	+	0.997684i	$0.521666\pi$
$860$	6.88802e6	0.317577
$861$	0	0
$862$	−965678.	−0.0442654
$863$	3.87778e7	1.77238	0.886188	−	0.463325i	$-0.153344\pi$
$863$	3.87778e7	1.77238	0.886188	+	0.463325i	$0.153344\pi$
$864$	0	0
$865$	−4.28531e6	−0.194734
$866$	815065.	0.0369316
$867$	0	0
$868$	1.38551e7	0.624179
$869$	−3.81289e6	−0.171279
$870$	0	0
$871$	5.75934e6	0.257234
$872$	853467.	0.0380098
$873$	0	0
$874$	249812.	0.0110620
$875$	−1.46035e7	−0.644817
$876$	0	0
$877$	1.06223e6	0.0466360	0.0233180	−	0.999728i	$-0.492577\pi$
$877$	1.06223e6	0.0466360	0.0233180	+	0.999728i	$0.492577\pi$
$878$	1.19646e6	0.0523795
$879$	0	0
$880$	2.27783e6	0.0991549
$881$	−2.77645e7	−1.20518	−0.602588	−	0.798052i	$-0.705864\pi$
$881$	−2.77645e7	−1.20518	−0.602588	+	0.798052i	$0.705864\pi$
$882$	0	0
$883$	−3.87561e7	−1.67278	−0.836389	−	0.548136i	$-0.815338\pi$
$883$	−3.87561e7	−1.67278	−0.836389	+	0.548136i	$0.815338\pi$
$884$	4.13726e6	0.178067
$885$	0	0
$886$	1.45947e6	0.0624613
$887$	3.40511e7	1.45319	0.726595	−	0.687066i	$-0.241102\pi$
$887$	3.40511e7	1.45319	0.726595	+	0.687066i	$0.241102\pi$
$888$	0	0
$889$	3.74997e7	1.59138
$890$	84172.1	0.00356199
$891$	0	0
$892$	3.95719e7	1.66523
$893$	1.05231e6	0.0441585
$894$	0	0
$895$	−5.71287e6	−0.238395
$896$	−3.57273e6	−0.148672
$897$	0	0
$898$	−1.00491e6	−0.0415850
$899$	3.39802e6	0.140225
$900$	0	0
$901$	−5.16996e6	−0.212166
$902$	83986.5	0.00343711
$903$	0	0
$904$	2.41933e6	0.0984633
$905$	−3.09843e6	−0.125753
$906$	0	0
$907$	4.59307e7	1.85389	0.926947	−	0.375192i	$-0.122423\pi$
$907$	4.59307e7	1.85389	0.926947	+	0.375192i	$0.122423\pi$
$908$	−4.76686e7	−1.91875
$909$	0	0
$910$	244943.	0.00980530
$911$	−1.91956e7	−0.766312	−0.383156	−	0.923684i	$-0.625163\pi$
$911$	−1.91956e7	−0.766312	−0.383156	+	0.923684i	$0.625163\pi$
$912$	0	0
$913$	−1.26967e7	−0.504096
$914$	−465465.	−0.0184299
$915$	0	0
$916$	5.98173e6	0.235553
$917$	−1.53781e7	−0.603920
$918$	0	0
$919$	1.51176e7	0.590463	0.295232	−	0.955426i	$-0.404603\pi$
$919$	1.51176e7	0.590463	0.295232	+	0.955426i	$0.404603\pi$
$920$	179577.	0.00699488
$921$	0	0
$922$	−1.32344e6	−0.0512716
$923$	3.58154e7	1.38377
$924$	0	0
$925$	−3.64442e7	−1.40047
$926$	−482826.	−0.0185039
$927$	0	0
$928$	−657316.	−0.0250556
$929$	−1.85558e7	−0.705406	−0.352703	−	0.935735i	$-0.614737\pi$
$929$	−1.85558e7	−0.705406	−0.352703	+	0.935735i	$0.614737\pi$
$930$	0	0
$931$	1.84249e6	0.0696675
$932$	−4.15756e7	−1.56783
$933$	0	0
$934$	402607.	0.0151013
$935$	595704.	0.0222844
$936$	0	0
$937$	−1.99378e7	−0.741870	−0.370935	−	0.928659i	$-0.620963\pi$
$937$	−1.99378e7	−0.741870	−0.370935	+	0.928659i	$0.620963\pi$
$938$	−324533.	−0.0120435
$939$	0	0
$940$	377979.	0.0139524
$941$	−1.91969e6	−0.0706737	−0.0353368	−	0.999375i	$-0.511250\pi$
$941$	−1.91969e6	−0.0706737	−0.0353368	+	0.999375i	$0.511250\pi$
$942$	0	0
$943$	−2.53154e6	−0.0927053
$944$	4.01253e7	1.46551
$945$	0	0
$946$	288595.	0.0104848
$947$	−1.50540e7	−0.545477	−0.272739	−	0.962088i	$-0.587929\pi$
$947$	−1.50540e7	−0.545477	−0.272739	+	0.962088i	$0.587929\pi$
$948$	0	0
$949$	−5.05482e6	−0.182196
$950$	933798.	0.0335694
$951$	0	0
$952$	−466565.	−0.0166848
$953$	1.12446e7	0.401063	0.200532	−	0.979687i	$-0.435733\pi$
$953$	1.12446e7	0.401063	0.200532	+	0.979687i	$0.435733\pi$
$954$	0	0
$955$	−4.56584e6	−0.161999
$956$	4.79243e7	1.69594
$957$	0	0
$958$	858930.	0.0302374
$959$	3.65791e7	1.28436
$960$	0	0
$961$	−1.81460e7	−0.633829
$962$	1.29733e6	0.0451972
$963$	0	0
$964$	5.04938e7	1.75003
$965$	−1.00643e7	−0.347908
$966$	0	0
$967$	3.09515e7	1.06443	0.532214	−	0.846610i	$-0.321360\pi$
$967$	3.09515e7	1.06443	0.532214	+	0.846610i	$0.321360\pi$
$968$	191248.	0.00656006
$969$	0	0
$970$	125607.	0.00428631
$971$	1.93456e7	0.658467	0.329233	−	0.944249i	$-0.393210\pi$
$971$	1.93456e7	0.658467	0.329233	+	0.944249i	$0.393210\pi$
$972$	0	0
$973$	9.88353e6	0.334680
$974$	−939388.	−0.0317284
$975$	0	0
$976$	1.97133e7	0.662421
$977$	−2.59609e7	−0.870129	−0.435064	−	0.900399i	$-0.643274\pi$
$977$	−2.59609e7	−0.870129	−0.435064	+	0.900399i	$0.643274\pi$
$978$	0	0
$979$	−2.70202e6	−0.0901015
$980$	661802.	0.0220122
$981$	0	0
$982$	1.23577e6	0.0408940
$983$	−4.98685e7	−1.64605	−0.823023	−	0.568008i	$-0.807714\pi$
$983$	−4.98685e7	−1.64605	−0.823023	+	0.568008i	$0.807714\pi$
$984$	0	0
$985$	8.01993e6	0.263378
$986$	−57176.4	−0.00187294
$987$	0	0
$988$	2.54683e7	0.830058
$989$	−8.69888e6	−0.282795
$990$	0	0
$991$	−1.32579e7	−0.428834	−0.214417	−	0.976742i	$-0.568785\pi$
$991$	−1.32579e7	−0.428834	−0.214417	+	0.976742i	$0.568785\pi$
$992$	−2.02787e6	−0.0654276
$993$	0	0
$994$	−2.01816e6	−0.0647872
$995$	1.16897e7	0.374321
$996$	0	0
$997$	−3.33264e7	−1.06182	−0.530909	−	0.847429i	$-0.678150\pi$
$997$	−3.33264e7	−1.06182	−0.530909	+	0.847429i	$0.678150\pi$
$998$	−1.26538e6	−0.0402155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.f.1.12		23
3.2	odd	2		891.6.a.e.1.12		23
9.2	odd	6		297.6.e.a.199.12		46
9.4	even	3		99.6.e.a.34.12	✓	46
9.5	odd	6		297.6.e.a.100.12		46
9.7	even	3		99.6.e.a.67.12	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.12	✓	46	9.4	even	3
99.6.e.a.67.12	yes	46	9.7	even	3
297.6.e.a.100.12		46	9.5	odd	6
297.6.e.a.199.12		46	9.2	odd	6
891.6.a.e.1.12		23	3.2	odd	2
891.6.a.f.1.12		23	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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q-0.204234 q^{2} -31.9583 q^{4} -18.4559 q^{5} -133.899 q^{7} +13.0625 q^{8} +O(q^{10})\)	\(q-0.204234 q^{2} -31.9583 q^{4} -18.4559 q^{5} -133.899 q^{7} +13.0625 q^{8} +3.76934 q^{10} -121.000 q^{11} -485.312 q^{13} +27.3468 q^{14} +1020.00 q^{16} +266.753 q^{17} +1642.09 q^{19} +589.821 q^{20} +24.7123 q^{22} -744.883 q^{23} -2784.38 q^{25} +99.1173 q^{26} +4279.19 q^{28} +1049.49 q^{29} +3237.77 q^{31} -626.317 q^{32} -54.4800 q^{34} +2471.24 q^{35} +13088.8 q^{37} -335.370 q^{38} -241.080 q^{40} +3398.57 q^{41} +11678.2 q^{43} +3866.95 q^{44} +152.131 q^{46} +640.836 q^{47} +1122.04 q^{49} +568.665 q^{50} +15509.7 q^{52} -19381.1 q^{53} +2233.17 q^{55} -1749.06 q^{56} -214.342 q^{58} +39338.6 q^{59} +19326.8 q^{61} -661.263 q^{62} -32512.0 q^{64} +8956.89 q^{65} -11867.3 q^{67} -8524.96 q^{68} -504.712 q^{70} -73798.6 q^{71} +10415.6 q^{73} -2673.18 q^{74} -52478.3 q^{76} +16201.8 q^{77} +31511.5 q^{79} -18825.0 q^{80} -694.103 q^{82} +104931. q^{83} -4923.17 q^{85} -2385.08 q^{86} -1580.56 q^{88} +22330.8 q^{89} +64983.0 q^{91} +23805.2 q^{92} -130.881 q^{94} -30306.3 q^{95} +33323.3 q^{97} -229.159 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

Introduction

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