LMFDB - Embedded newform 891.6.a.f.1.1

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			1.1
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)

$f(q)$	$=$	$q-10.2329 q^{2} +72.7122 q^{4} -40.3523 q^{5} -223.015 q^{7} -416.604 q^{8} +O(q^{10})$	\(q-10.2329 q^{2} +72.7122 q^{4} -40.3523 q^{5} -223.015 q^{7} -416.604 q^{8} +412.921 q^{10} -121.000 q^{11} -462.257 q^{13} +2282.09 q^{14} +1936.28 q^{16} -734.739 q^{17} -1937.31 q^{19} -2934.10 q^{20} +1238.18 q^{22} +2026.57 q^{23} -1496.69 q^{25} +4730.23 q^{26} -16216.0 q^{28} +5760.55 q^{29} -5261.06 q^{31} -6482.38 q^{32} +7518.51 q^{34} +8999.18 q^{35} +15066.0 q^{37} +19824.3 q^{38} +16810.9 q^{40} -1278.33 q^{41} +7369.43 q^{43} -8798.18 q^{44} -20737.7 q^{46} +16726.0 q^{47} +32928.9 q^{49} +15315.5 q^{50} -33611.8 q^{52} +10034.6 q^{53} +4882.63 q^{55} +92909.1 q^{56} -58947.2 q^{58} -38862.4 q^{59} -37729.5 q^{61} +53835.9 q^{62} +4372.76 q^{64} +18653.1 q^{65} -17756.9 q^{67} -53424.5 q^{68} -92087.7 q^{70} +57646.0 q^{71} -19775.3 q^{73} -154169. q^{74} -140866. q^{76} +26984.9 q^{77} +78382.8 q^{79} -78133.1 q^{80} +13081.0 q^{82} +29192.4 q^{83} +29648.4 q^{85} -75410.7 q^{86} +50409.1 q^{88} +18044.1 q^{89} +103091. q^{91} +147356. q^{92} -171155. q^{94} +78175.1 q^{95} -119300. q^{97} -336958. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.2329	−1.80894	−0.904469	−	0.426539i	$-0.859733\pi$
$2$	−10.2329	−1.80894	−0.904469	+	0.426539i	$0.859733\pi$
$3$	0	0
$3$	0	0
$4$	72.7122	2.27226
$5$	−40.3523	−0.721844	−0.360922	−	0.932596i	$-0.617538\pi$
$5$	−40.3523	−0.721844	−0.360922	+	0.932596i	$0.617538\pi$
$6$	0	0
$7$	−223.015	−1.72024	−0.860122	−	0.510089i	$-0.829613\pi$
$7$	−223.015	−1.72024	−0.860122	+	0.510089i	$0.829613\pi$
$8$	−416.604	−2.30143
$9$	0	0
$10$	412.921	1.30577
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−462.257	−0.758622	−0.379311	−	0.925269i	$-0.623839\pi$
$13$	−462.257	−0.758622	−0.379311	+	0.925269i	$0.623839\pi$
$14$	2282.09	3.11181
$15$	0	0
$16$	1936.28	1.89089
$17$	−734.739	−0.616611	−0.308305	−	0.951287i	$-0.599762\pi$
$17$	−734.739	−0.616611	−0.308305	+	0.951287i	$0.599762\pi$
$18$	0	0
$19$	−1937.31	−1.23116	−0.615582	−	0.788072i	$-0.711079\pi$
$19$	−1937.31	−1.23116	−0.615582	+	0.788072i	$0.711079\pi$
$20$	−2934.10	−1.64021
$21$	0	0
$22$	1238.18	0.545415
$23$	2026.57	0.798808	0.399404	−	0.916775i	$-0.369217\pi$
$23$	2026.57	0.798808	0.399404	+	0.916775i	$0.369217\pi$
$24$	0	0
$25$	−1496.69	−0.478942
$26$	4730.23	1.37230
$27$	0	0
$28$	−16216.0	−3.90883
$29$	5760.55	1.27195	0.635974	−	0.771711i	$-0.280599\pi$
$29$	5760.55	1.27195	0.635974	+	0.771711i	$0.280599\pi$
$30$	0	0
$31$	−5261.06	−0.983261	−0.491631	−	0.870804i	$-0.663599\pi$
$31$	−5261.06	−0.983261	−0.491631	+	0.870804i	$0.663599\pi$
$32$	−6482.38	−1.11908
$33$	0	0
$34$	7518.51	1.11541
$35$	8999.18	1.24175
$36$	0	0
$37$	15066.0	1.80923	0.904614	−	0.426231i	$-0.140159\pi$
$37$	15066.0	1.80923	0.904614	+	0.426231i	$0.140159\pi$
$38$	19824.3	2.22710
$39$	0	0
$40$	16810.9	1.66128
$41$	−1278.33	−0.118763	−0.0593817	−	0.998235i	$-0.518913\pi$
$41$	−1278.33	−0.118763	−0.0593817	+	0.998235i	$0.518913\pi$
$42$	0	0
$43$	7369.43	0.607803	0.303902	−	0.952703i	$-0.401711\pi$
$43$	7369.43	0.607803	0.303902	+	0.952703i	$0.401711\pi$
$44$	−8798.18	−0.685111
$45$	0	0
$46$	−20737.7	−1.44499
$47$	16726.0	1.10445	0.552226	−	0.833694i	$-0.313779\pi$
$47$	16726.0	1.10445	0.552226	+	0.833694i	$0.313779\pi$
$48$	0	0
$49$	32928.9	1.95924
$50$	15315.5	0.866376
$51$	0	0
$52$	−33611.8	−1.72378
$53$	10034.6	0.490695	0.245348	−	0.969435i	$-0.421098\pi$
$53$	10034.6	0.490695	0.245348	+	0.969435i	$0.421098\pi$
$54$	0	0
$55$	4882.63	0.217644
$56$	92909.1	3.95903
$57$	0	0
$58$	−58947.2	−2.30087
$59$	−38862.4	−1.45345	−0.726723	−	0.686930i	$-0.758958\pi$
$59$	−38862.4	−1.45345	−0.726723	+	0.686930i	$0.758958\pi$
$60$	0	0
$61$	−37729.5	−1.29825	−0.649123	−	0.760684i	$-0.724864\pi$
$61$	−37729.5	−1.29825	−0.649123	+	0.760684i	$0.724864\pi$
$62$	53835.9	1.77866
$63$	0	0
$64$	4372.76	0.133446
$65$	18653.1	0.547607
$66$	0	0
$67$	−17756.9	−0.483260	−0.241630	−	0.970368i	$-0.577682\pi$
$67$	−17756.9	−0.483260	−0.241630	+	0.970368i	$0.577682\pi$
$68$	−53424.5	−1.40110
$69$	0	0
$70$	−92087.7	−2.24624
$71$	57646.0	1.35714	0.678568	−	0.734537i	$-0.262601\pi$
$71$	57646.0	1.35714	0.678568	+	0.734537i	$0.262601\pi$
$72$	0	0
$73$	−19775.3	−0.434326	−0.217163	−	0.976135i	$-0.569680\pi$
$73$	−19775.3	−0.434326	−0.217163	+	0.976135i	$0.569680\pi$
$74$	−154169.	−3.27278
$75$	0	0
$76$	−140866.	−2.79752
$77$	26984.9	0.518673
$78$	0	0
$79$	78382.8	1.41303	0.706517	−	0.707696i	$-0.250265\pi$
$79$	78382.8	1.41303	0.706517	+	0.707696i	$0.250265\pi$
$80$	−78133.1	−1.36493
$81$	0	0
$82$	13081.0	0.214836
$83$	29192.4	0.465131	0.232565	−	0.972581i	$-0.425288\pi$
$83$	29192.4	0.465131	0.232565	+	0.972581i	$0.425288\pi$
$84$	0	0
$85$	29648.4	0.445097
$86$	−75410.7	−1.09948
$87$	0	0
$88$	50409.1	0.693908
$89$	18044.1	0.241468	0.120734	−	0.992685i	$-0.461475\pi$
$89$	18044.1	0.241468	0.120734	+	0.992685i	$0.461475\pi$
$90$	0	0
$91$	103091.	1.30501
$92$	147356.	1.81510
$93$	0	0
$94$	−171155.	−1.99789
$95$	78175.1	0.888709
$96$	0	0
$97$	−119300.	−1.28740	−0.643698	−	0.765280i	$-0.722601\pi$
$97$	−119300.	−1.28740	−0.643698	+	0.765280i	$0.722601\pi$
$98$	−336958.	−3.54414
$99$	0	0
$100$	−108828.	−1.08828
$101$	14192.9	0.138442	0.0692210	−	0.997601i	$-0.477949\pi$
$101$	14192.9	0.138442	0.0692210	+	0.997601i	$0.477949\pi$
$102$	0	0
$103$	98843.6	0.918027	0.459014	−	0.888429i	$-0.348203\pi$
$103$	98843.6	0.918027	0.459014	+	0.888429i	$0.348203\pi$
$104$	192578.	1.74592
$105$	0	0
$106$	−102683.	−0.887638
$107$	−210430.	−1.77684	−0.888418	−	0.459035i	$-0.848195\pi$
$107$	−210430.	−1.77684	−0.888418	+	0.459035i	$0.848195\pi$
$108$	0	0
$109$	−164498.	−1.32615	−0.663076	−	0.748552i	$-0.730749\pi$
$109$	−164498.	−1.32615	−0.663076	+	0.748552i	$0.730749\pi$
$110$	−49963.4	−0.393705
$111$	0	0
$112$	−431819.	−3.25280
$113$	154193.	1.13597	0.567986	−	0.823038i	$-0.307723\pi$
$113$	154193.	1.13597	0.567986	+	0.823038i	$0.307723\pi$
$114$	0	0
$115$	−81776.8	−0.576614
$116$	418863.	2.89019
$117$	0	0
$118$	397675.	2.62919
$119$	163858.	1.06072
$120$	0	0
$121$	14641.0	0.0909091
$122$	386082.	2.34845
$123$	0	0
$124$	−382543.	−2.23422
$125$	186496.	1.06756
$126$	0	0
$127$	81155.5	0.446487	0.223243	−	0.974763i	$-0.428336\pi$
$127$	81155.5	0.446487	0.223243	+	0.974763i	$0.428336\pi$
$128$	162690.	0.877681
$129$	0	0
$130$	−190876.	−0.990586
$131$	−97146.1	−0.494592	−0.247296	−	0.968940i	$-0.579542\pi$
$131$	−97146.1	−0.494592	−0.247296	+	0.968940i	$0.579542\pi$
$132$	0	0
$133$	432051.	2.11790
$134$	181705.	0.874188
$135$	0	0
$136$	306095.	1.41909
$137$	289825.	1.31927	0.659637	−	0.751584i	$-0.270710\pi$
$137$	289825.	1.31927	0.659637	+	0.751584i	$0.270710\pi$
$138$	0	0
$139$	236645.	1.03887	0.519435	−	0.854510i	$-0.326143\pi$
$139$	236645.	1.03887	0.519435	+	0.854510i	$0.326143\pi$
$140$	654351.	2.82157
$141$	0	0
$142$	−589886.	−2.45498
$143$	55933.1	0.228733
$144$	0	0
$145$	−232452.	−0.918147
$146$	202359.	0.785669
$147$	0	0
$148$	1.09548e6	4.11103
$149$	−84404.8	−0.311459	−0.155730	−	0.987800i	$-0.549773\pi$
$149$	−84404.8	−0.311459	−0.155730	+	0.987800i	$0.549773\pi$
$150$	0	0
$151$	462996.	1.65248	0.826238	−	0.563321i	$-0.190477\pi$
$151$	462996.	1.65248	0.826238	+	0.563321i	$0.190477\pi$
$152$	807093.	2.83344
$153$	0	0
$154$	−276133.	−0.938247
$155$	212296.	0.709761
$156$	0	0
$157$	197248.	0.638652	0.319326	−	0.947645i	$-0.396543\pi$
$157$	197248.	0.638652	0.319326	+	0.947645i	$0.396543\pi$
$158$	−802083.	−2.55609
$159$	0	0
$160$	261579.	0.807798
$161$	−451957.	−1.37414
$162$	0	0
$163$	454159.	1.33887	0.669436	−	0.742870i	$-0.266536\pi$
$163$	454159.	1.33887	0.669436	+	0.742870i	$0.266536\pi$
$164$	−92950.1	−0.269861
$165$	0	0
$166$	−298723.	−0.841393
$167$	42299.2	0.117366	0.0586828	−	0.998277i	$-0.481310\pi$
$167$	42299.2	0.117366	0.0586828	+	0.998277i	$0.481310\pi$
$168$	0	0
$169$	−157611.	−0.424492
$170$	−303389.	−0.805152
$171$	0	0
$172$	535848.	1.38108
$173$	−152704.	−0.387915	−0.193957	−	0.981010i	$-0.562132\pi$
$173$	−152704.	−0.387915	−0.193957	+	0.981010i	$0.562132\pi$
$174$	0	0
$175$	333786.	0.823897
$176$	−234289.	−0.570126
$177$	0	0
$178$	−184643.	−0.436801
$179$	−210986.	−0.492178	−0.246089	−	0.969247i	$-0.579146\pi$
$179$	−210986.	−0.492178	−0.246089	+	0.969247i	$0.579146\pi$
$180$	0	0
$181$	−573821.	−1.30191	−0.650954	−	0.759117i	$-0.725631\pi$
$181$	−573821.	−1.30191	−0.650954	+	0.759117i	$0.725631\pi$
$182$	−1.05492e6	−2.36069
$183$	0	0
$184$	−844278.	−1.83840
$185$	−607947.	−1.30598
$186$	0	0
$187$	88903.5	0.185915
$188$	1.21618e6	2.50960
$189$	0	0
$190$	−799958.	−1.60762
$191$	526350.	1.04398	0.521988	−	0.852953i	$-0.325190\pi$
$191$	526350.	1.04398	0.521988	+	0.852953i	$0.325190\pi$
$192$	0	0
$193$	295678.	0.571382	0.285691	−	0.958322i	$-0.407777\pi$
$193$	295678.	0.571382	0.285691	+	0.958322i	$0.407777\pi$
$194$	1.22079e6	2.32882
$195$	0	0
$196$	2.39433e6	4.45189
$197$	495889.	0.910372	0.455186	−	0.890396i	$-0.349573\pi$
$197$	495889.	0.910372	0.455186	+	0.890396i	$0.349573\pi$
$198$	0	0
$199$	−429236.	−0.768357	−0.384178	−	0.923259i	$-0.625515\pi$
$199$	−429236.	−0.768357	−0.384178	+	0.923259i	$0.625515\pi$
$200$	623528.	1.10225
$201$	0	0
$202$	−145235.	−0.250433
$203$	−1.28469e6	−2.18806
$204$	0	0
$205$	51583.5	0.0857286
$206$	−1.01146e6	−1.66065
$207$	0	0
$208$	−895058.	−1.43447
$209$	234415.	0.371210
$210$	0	0
$211$	231134.	0.357403	0.178702	−	0.983903i	$-0.442810\pi$
$211$	231134.	0.357403	0.178702	+	0.983903i	$0.442810\pi$
$212$	729641.	1.11499
$213$	0	0
$214$	2.15331e6	3.21419
$215$	−297373.	−0.438739
$216$	0	0
$217$	1.17330e6	1.69145
$218$	1.68329e6	2.39893
$219$	0	0
$220$	355027.	0.494543
$221$	339639.	0.467775
$222$	0	0
$223$	−997509.	−1.34324	−0.671621	−	0.740895i	$-0.734402\pi$
$223$	−997509.	−1.34324	−0.671621	+	0.740895i	$0.734402\pi$
$224$	1.44567e6	1.92508
$225$	0	0
$226$	−1.57784e6	−2.05490
$227$	−961838.	−1.23890	−0.619451	−	0.785035i	$-0.712645\pi$
$227$	−961838.	−1.23890	−0.619451	+	0.785035i	$0.712645\pi$
$228$	0	0
$229$	954018.	1.20218	0.601088	−	0.799183i	$-0.294734\pi$
$229$	954018.	1.20218	0.601088	+	0.799183i	$0.294734\pi$
$230$	836813.	1.04306
$231$	0	0
$232$	−2.39987e6	−2.92730
$233$	142292.	0.171708	0.0858541	−	0.996308i	$-0.472638\pi$
$233$	142292.	0.171708	0.0858541	+	0.996308i	$0.472638\pi$
$234$	0	0
$235$	−674931.	−0.797242
$236$	−2.82577e6	−3.30260
$237$	0	0
$238$	−1.67675e6	−1.91878
$239$	−1.38513e6	−1.56854	−0.784269	−	0.620421i	$-0.786962\pi$
$239$	−1.38513e6	−1.56854	−0.784269	+	0.620421i	$0.786962\pi$
$240$	0	0
$241$	−174858.	−0.193929	−0.0969645	−	0.995288i	$-0.530913\pi$
$241$	−174858.	−0.193929	−0.0969645	+	0.995288i	$0.530913\pi$
$242$	−149820.	−0.164449
$243$	0	0
$244$	−2.74340e6	−2.94995
$245$	−1.32876e6	−1.41426
$246$	0	0
$247$	895538.	0.933989
$248$	2.19178e6	2.26291
$249$	0	0
$250$	−1.90839e6	−1.93116
$251$	264717.	0.265215	0.132607	−	0.991169i	$-0.457665\pi$
$251$	264717.	0.265215	0.132607	+	0.991169i	$0.457665\pi$
$252$	0	0
$253$	−245215.	−0.240850
$254$	−830456.	−0.807667
$255$	0	0
$256$	−1.80472e6	−1.72112
$257$	951943.	0.899038	0.449519	−	0.893271i	$-0.351595\pi$
$257$	951943.	0.899038	0.449519	+	0.893271i	$0.351595\pi$
$258$	0	0
$259$	−3.35995e6	−3.11231
$260$	1.35631e6	1.24430
$261$	0	0
$262$	994086.	0.894686
$263$	1.76764e6	1.57581	0.787906	−	0.615796i	$-0.211165\pi$
$263$	1.76764e6	1.57581	0.787906	+	0.615796i	$0.211165\pi$
$264$	0	0
$265$	−404921.	−0.354205
$266$	−4.42114e6	−3.83116
$267$	0	0
$268$	−1.29115e6	−1.09809
$269$	−67658.5	−0.0570087	−0.0285044	−	0.999594i	$-0.509074\pi$
$269$	−67658.5	−0.0570087	−0.0285044	+	0.999594i	$0.509074\pi$
$270$	0	0
$271$	210878.	0.174425	0.0872123	−	0.996190i	$-0.472204\pi$
$271$	210878.	0.174425	0.0872123	+	0.996190i	$0.472204\pi$
$272$	−1.42266e6	−1.16595
$273$	0	0
$274$	−2.96575e6	−2.38648
$275$	181100.	0.144406
$276$	0	0
$277$	−1.73484e6	−1.35850	−0.679249	−	0.733908i	$-0.737694\pi$
$277$	−1.73484e6	−1.35850	−0.679249	+	0.733908i	$0.737694\pi$
$278$	−2.42157e6	−1.87925
$279$	0	0
$280$	−3.74910e6	−2.85780
$281$	−202471.	−0.152967	−0.0764833	−	0.997071i	$-0.524369\pi$
$281$	−202471.	−0.152967	−0.0764833	+	0.997071i	$0.524369\pi$
$282$	0	0
$283$	719774.	0.534233	0.267116	−	0.963664i	$-0.413929\pi$
$283$	719774.	0.534233	0.267116	+	0.963664i	$0.413929\pi$
$284$	4.19157e6	3.08376
$285$	0	0
$286$	−572358.	−0.413764
$287$	285087.	0.204302
$288$	0	0
$289$	−880015.	−0.619791
$290$	2.37865e6	1.66087
$291$	0	0
$292$	−1.43791e6	−0.986901
$293$	1.20062e6	0.817027	0.408513	−	0.912752i	$-0.366047\pi$
$293$	1.20062e6	0.817027	0.408513	+	0.912752i	$0.366047\pi$
$294$	0	0
$295$	1.56818e6	1.04916
$296$	−6.27655e6	−4.16382
$297$	0	0
$298$	863706.	0.563411
$299$	−936798.	−0.605993
$300$	0	0
$301$	−1.64350e6	−1.04557
$302$	−4.73779e6	−2.98923
$303$	0	0
$304$	−3.75118e6	−2.32800
$305$	1.52247e6	0.937130
$306$	0	0
$307$	1.21082e6	0.733222	0.366611	−	0.930374i	$-0.380518\pi$
$307$	1.21082e6	0.733222	0.366611	+	0.930374i	$0.380518\pi$
$308$	1.96213e6	1.17856
$309$	0	0
$310$	−2.17240e6	−1.28391
$311$	1.65038e6	0.967572	0.483786	−	0.875186i	$-0.339261\pi$
$311$	1.65038e6	0.967572	0.483786	+	0.875186i	$0.339261\pi$
$312$	0	0
$313$	−389733.	−0.224857	−0.112428	−	0.993660i	$-0.535863\pi$
$313$	−389733.	−0.224857	−0.112428	+	0.993660i	$0.535863\pi$
$314$	−2.01842e6	−1.15528
$315$	0	0
$316$	5.69938e6	3.21078
$317$	−2.42797e6	−1.35705	−0.678524	−	0.734578i	$-0.737380\pi$
$317$	−2.42797e6	−1.35705	−0.678524	+	0.734578i	$0.737380\pi$
$318$	0	0
$319$	−697027.	−0.383507
$320$	−176451.	−0.0963271
$321$	0	0
$322$	4.62483e6	2.48574
$323$	1.42342e6	0.759149
$324$	0	0
$325$	691858.	0.363336
$326$	−4.64736e6	−2.42193
$327$	0	0
$328$	532557.	0.273326
$329$	−3.73015e6	−1.89993
$330$	0	0
$331$	3.38699e6	1.69920	0.849598	−	0.527431i	$-0.176845\pi$
$331$	3.38699e6	1.69920	0.849598	+	0.527431i	$0.176845\pi$
$332$	2.12265e6	1.05690
$333$	0	0
$334$	−432843.	−0.212307
$335$	716533.	0.348838
$336$	0	0
$337$	943731.	0.452661	0.226331	−	0.974051i	$-0.427327\pi$
$337$	943731.	0.452661	0.226331	+	0.974051i	$0.427327\pi$
$338$	1.61282e6	0.767881
$339$	0	0
$340$	2.15580e6	1.01137
$341$	636588.	0.296464
$342$	0	0
$343$	−3.59544e6	−1.65012
$344$	−3.07013e6	−1.39882
$345$	0	0
$346$	1.56261e6	0.701714
$347$	1.01945e6	0.454507	0.227254	−	0.973836i	$-0.427025\pi$
$347$	1.01945e6	0.454507	0.227254	+	0.973836i	$0.427025\pi$
$348$	0	0
$349$	−3.24750e6	−1.42720	−0.713601	−	0.700553i	$-0.752937\pi$
$349$	−3.24750e6	−1.42720	−0.713601	+	0.700553i	$0.752937\pi$
$350$	−3.41560e6	−1.49038
$351$	0	0
$352$	784368.	0.337414
$353$	−2.77228e6	−1.18413	−0.592067	−	0.805889i	$-0.701688\pi$
$353$	−2.77228e6	−1.18413	−0.592067	+	0.805889i	$0.701688\pi$
$354$	0	0
$355$	−2.32615e6	−0.979640
$356$	1.31202e6	0.548677
$357$	0	0
$358$	2.15900e6	0.890319
$359$	−3.17234e6	−1.29910	−0.649551	−	0.760318i	$-0.725043\pi$
$359$	−3.17234e6	−1.29910	−0.649551	+	0.760318i	$0.725043\pi$
$360$	0	0
$361$	1.27709e6	0.515767
$362$	5.87185e6	2.35507
$363$	0	0
$364$	7.49594e6	2.96533
$365$	797979.	0.313516
$366$	0	0
$367$	517282.	0.200476	0.100238	−	0.994964i	$-0.468040\pi$
$367$	517282.	0.200476	0.100238	+	0.994964i	$0.468040\pi$
$368$	3.92400e6	1.51046
$369$	0	0
$370$	6.22106e6	2.36244
$371$	−2.23788e6	−0.844116
$372$	0	0
$373$	1.14512e6	0.426164	0.213082	−	0.977034i	$-0.431650\pi$
$373$	1.14512e6	0.426164	0.213082	+	0.977034i	$0.431650\pi$
$374$	−909740.	−0.336309
$375$	0	0
$376$	−6.96811e6	−2.54182
$377$	−2.66286e6	−0.964928
$378$	0	0
$379$	−3.96614e6	−1.41831	−0.709153	−	0.705054i	$-0.750923\pi$
$379$	−3.96614e6	−1.41831	−0.709153	+	0.705054i	$0.750923\pi$
$380$	5.68428e6	2.01937
$381$	0	0
$382$	−5.38608e6	−1.88849
$383$	1.00695e6	0.350761	0.175380	−	0.984501i	$-0.443884\pi$
$383$	1.00695e6	0.350761	0.175380	+	0.984501i	$0.443884\pi$
$384$	0	0
$385$	−1.08890e6	−0.374401
$386$	−3.02565e6	−1.03359
$387$	0	0
$388$	−8.67459e6	−2.92529
$389$	1.47053e6	0.492720	0.246360	−	0.969178i	$-0.420766\pi$
$389$	1.47053e6	0.492720	0.246360	+	0.969178i	$0.420766\pi$
$390$	0	0
$391$	−1.48900e6	−0.492553
$392$	−1.37183e7	−4.50906
$393$	0	0
$394$	−5.07438e6	−1.64681
$395$	−3.16292e6	−1.01999
$396$	0	0
$397$	−2.95456e6	−0.940842	−0.470421	−	0.882442i	$-0.655898\pi$
$397$	−2.95456e6	−0.940842	−0.470421	+	0.882442i	$0.655898\pi$
$398$	4.39232e6	1.38991
$399$	0	0
$400$	−2.89801e6	−0.905628
$401$	−158419.	−0.0491978	−0.0245989	−	0.999697i	$-0.507831\pi$
$401$	−158419.	−0.0491978	−0.0245989	+	0.999697i	$0.507831\pi$
$402$	0	0
$403$	2.43196e6	0.745924
$404$	1.03200e6	0.314576
$405$	0	0
$406$	1.31461e7	3.95806
$407$	−1.82299e6	−0.545503
$408$	0	0
$409$	−1.23065e6	−0.363770	−0.181885	−	0.983320i	$-0.558220\pi$
$409$	−1.23065e6	−0.363770	−0.181885	+	0.983320i	$0.558220\pi$
$410$	−527848.	−0.155078
$411$	0	0
$412$	7.18714e6	2.08599
$413$	8.66691e6	2.50028
$414$	0	0
$415$	−1.17798e6	−0.335752
$416$	2.99653e6	0.848956
$417$	0	0
$418$	−2.39875e6	−0.671496
$419$	−4.47988e6	−1.24661	−0.623306	−	0.781978i	$-0.714211\pi$
$419$	−4.47988e6	−1.24661	−0.623306	+	0.781978i	$0.714211\pi$
$420$	0	0
$421$	−64272.2	−0.0176733	−0.00883666	−	0.999961i	$-0.502813\pi$
$421$	−64272.2	−0.0176733	−0.00883666	+	0.999961i	$0.502813\pi$
$422$	−2.36517e6	−0.646520
$423$	0	0
$424$	−4.18047e6	−1.12930
$425$	1.09968e6	0.295321
$426$	0	0
$427$	8.41427e6	2.23330
$428$	−1.53008e7	−4.03743
$429$	0	0
$430$	3.04299e6	0.793651
$431$	−3.41553e6	−0.885655	−0.442827	−	0.896607i	$-0.646025\pi$
$431$	−3.41553e6	−0.885655	−0.442827	+	0.896607i	$0.646025\pi$
$432$	0	0
$433$	4.90253e6	1.25661	0.628305	−	0.777967i	$-0.283749\pi$
$433$	4.90253e6	1.25661	0.628305	+	0.777967i	$0.283749\pi$
$434$	−1.20062e7	−3.05973
$435$	0	0
$436$	−1.19610e7	−3.01336
$437$	−3.92611e6	−0.983464
$438$	0	0
$439$	813390.	0.201436	0.100718	−	0.994915i	$-0.467886\pi$
$439$	813390.	0.201436	0.100718	+	0.994915i	$0.467886\pi$
$440$	−2.03412e6	−0.500893
$441$	0	0
$442$	−3.47549e6	−0.846175
$443$	−2.90428e6	−0.703119	−0.351560	−	0.936165i	$-0.614349\pi$
$443$	−2.90428e6	−0.703119	−0.351560	+	0.936165i	$0.614349\pi$
$444$	0	0
$445$	−728120.	−0.174302
$446$	1.02074e7	2.42984
$447$	0	0
$448$	−975192.	−0.229560
$449$	−2.48960e6	−0.582791	−0.291396	−	0.956603i	$-0.594120\pi$
$449$	−2.48960e6	−0.582791	−0.291396	+	0.956603i	$0.594120\pi$
$450$	0	0
$451$	154678.	0.0358085
$452$	1.12117e7	2.58122
$453$	0	0
$454$	9.84239e6	2.24110
$455$	−4.15994e6	−0.942017
$456$	0	0
$457$	−777529.	−0.174151	−0.0870756	−	0.996202i	$-0.527752\pi$
$457$	−777529.	−0.174151	−0.0870756	+	0.996202i	$0.527752\pi$
$458$	−9.76237e6	−2.17466
$459$	0	0
$460$	−5.94617e6	−1.31022
$461$	−2.70427e6	−0.592648	−0.296324	−	0.955087i	$-0.595761\pi$
$461$	−2.70427e6	−0.592648	−0.296324	+	0.955087i	$0.595761\pi$
$462$	0	0
$463$	1.22317e6	0.265176	0.132588	−	0.991171i	$-0.457671\pi$
$463$	1.22317e6	0.265176	0.132588	+	0.991171i	$0.457671\pi$
$464$	1.11540e7	2.40512
$465$	0	0
$466$	−1.45606e6	−0.310610
$467$	8.18365e6	1.73642	0.868210	−	0.496197i	$-0.165271\pi$
$467$	8.18365e6	1.73642	0.868210	+	0.496197i	$0.165271\pi$
$468$	0	0
$469$	3.96007e6	0.831325
$470$	6.90651e6	1.44216
$471$	0	0
$472$	1.61902e7	3.34501
$473$	−891701.	−0.183260
$474$	0	0
$475$	2.89957e6	0.589656
$476$	1.19145e7	2.41023
$477$	0	0
$478$	1.41739e7	2.83739
$479$	−3.46407e6	−0.689839	−0.344919	−	0.938632i	$-0.612094\pi$
$479$	−3.46407e6	−0.689839	−0.344919	+	0.938632i	$0.612094\pi$
$480$	0	0
$481$	−6.96437e6	−1.37252
$482$	1.78930e6	0.350806
$483$	0	0
$484$	1.06458e6	0.206569
$485$	4.81404e6	0.929298
$486$	0	0
$487$	−2.78657e6	−0.532411	−0.266205	−	0.963916i	$-0.585770\pi$
$487$	−2.78657e6	−0.532411	−0.266205	+	0.963916i	$0.585770\pi$
$488$	1.57183e7	2.98783
$489$	0	0
$490$	1.35970e7	2.55831
$491$	−5.62204e6	−1.05242	−0.526211	−	0.850354i	$-0.676388\pi$
$491$	−5.62204e6	−1.05242	−0.526211	+	0.850354i	$0.676388\pi$
$492$	0	0
$493$	−4.23251e6	−0.784297
$494$	−9.16395e6	−1.68953
$495$	0	0
$496$	−1.01869e7	−1.85924
$497$	−1.28560e7	−2.33461
$498$	0	0
$499$	−832339.	−0.149640	−0.0748202	−	0.997197i	$-0.523838\pi$
$499$	−832339.	−0.149640	−0.0748202	+	0.997197i	$0.523838\pi$
$500$	1.35605e7	2.42578
$501$	0	0
$502$	−2.70882e6	−0.479757
$503$	−5.24218e6	−0.923830	−0.461915	−	0.886924i	$-0.652838\pi$
$503$	−5.24218e6	−0.923830	−0.461915	+	0.886924i	$0.652838\pi$
$504$	0	0
$505$	−572716.	−0.0999335
$506$	2.50926e6	0.435682
$507$	0	0
$508$	5.90100e6	1.01453
$509$	−1.01443e7	−1.73552	−0.867758	−	0.496986i	$-0.834440\pi$
$509$	−1.01443e7	−1.73552	−0.867758	+	0.496986i	$0.834440\pi$
$510$	0	0
$511$	4.41020e6	0.747147
$512$	1.32614e7	2.23571
$513$	0	0
$514$	−9.74113e6	−1.62630
$515$	−3.98856e6	−0.662672
$516$	0	0
$517$	−2.02384e6	−0.333005
$518$	3.43820e7	5.62998
$519$	0	0
$520$	−7.77097e6	−1.26028
$521$	−5.64231e6	−0.910673	−0.455337	−	0.890319i	$-0.650481\pi$
$521$	−5.64231e6	−0.910673	−0.455337	+	0.890319i	$0.650481\pi$
$522$	0	0
$523$	3.92857e6	0.628029	0.314015	−	0.949418i	$-0.398326\pi$
$523$	3.92857e6	0.628029	0.314015	+	0.949418i	$0.398326\pi$
$524$	−7.06370e6	−1.12384
$525$	0	0
$526$	−1.80881e7	−2.85054
$527$	3.86551e6	0.606289
$528$	0	0
$529$	−2.32935e6	−0.361906
$530$	4.14351e6	0.640736
$531$	0	0
$532$	3.14154e7	4.81242
$533$	590917.	0.0900966
$534$	0	0
$535$	8.49132e6	1.28260
$536$	7.39761e6	1.11219
$537$	0	0
$538$	692342.	0.103125
$539$	−3.98440e6	−0.590732
$540$	0	0
$541$	1.06829e7	1.56926	0.784631	−	0.619963i	$-0.212852\pi$
$541$	1.06829e7	1.56926	0.784631	+	0.619963i	$0.212852\pi$
$542$	−2.15789e6	−0.315523
$543$	0	0
$544$	4.76286e6	0.690035
$545$	6.63785e6	0.957274
$546$	0	0
$547$	1.23258e7	1.76136	0.880678	−	0.473716i	$-0.157088\pi$
$547$	1.23258e7	1.76136	0.880678	+	0.473716i	$0.157088\pi$
$548$	2.10738e7	2.99773
$549$	0	0
$550$	−1.85318e6	−0.261222
$551$	−1.11600e7	−1.56598
$552$	0	0
$553$	−1.74806e7	−2.43076
$554$	1.77524e7	2.45744
$555$	0	0
$556$	1.72070e7	2.36058
$557$	8.11938e6	1.10888	0.554440	−	0.832223i	$-0.312932\pi$
$557$	8.11938e6	1.10888	0.554440	+	0.832223i	$0.312932\pi$
$558$	0	0
$559$	−3.40657e6	−0.461093
$560$	1.74249e7	2.34801
$561$	0	0
$562$	2.07186e6	0.276707
$563$	1.85829e6	0.247082	0.123541	−	0.992339i	$-0.460575\pi$
$563$	1.85829e6	0.247082	0.123541	+	0.992339i	$0.460575\pi$
$564$	0	0
$565$	−6.22203e6	−0.819994
$566$	−7.36538e6	−0.966394
$567$	0	0
$568$	−2.40156e7	−3.12336
$569$	5.37098e6	0.695461	0.347731	−	0.937595i	$-0.386952\pi$
$569$	5.37098e6	0.695461	0.347731	+	0.937595i	$0.386952\pi$
$570$	0	0
$571$	−5.26186e6	−0.675382	−0.337691	−	0.941257i	$-0.609646\pi$
$571$	−5.26186e6	−0.675382	−0.337691	+	0.941257i	$0.609646\pi$
$572$	4.06702e6	0.519741
$573$	0	0
$574$	−2.91727e6	−0.369570
$575$	−3.03316e6	−0.382582
$576$	0	0
$577$	−7.27426e6	−0.909598	−0.454799	−	0.890594i	$-0.650289\pi$
$577$	−7.27426e6	−0.909598	−0.454799	+	0.890594i	$0.650289\pi$
$578$	9.00510e6	1.12116
$579$	0	0
$580$	−1.69021e7	−2.08627
$581$	−6.51036e6	−0.800138
$582$	0	0
$583$	−1.21419e6	−0.147950
$584$	8.23847e6	0.999573
$585$	0	0
$586$	−1.22858e7	−1.47795
$587$	1.49186e7	1.78703	0.893516	−	0.449032i	$-0.148231\pi$
$587$	1.49186e7	1.78703	0.893516	+	0.449032i	$0.148231\pi$
$588$	0	0
$589$	1.01923e7	1.21056
$590$	−1.60471e7	−1.89787
$591$	0	0
$592$	2.91719e7	3.42106
$593$	−1.31320e6	−0.153354	−0.0766771	−	0.997056i	$-0.524431\pi$
$593$	−1.31320e6	−0.153354	−0.0766771	+	0.997056i	$0.524431\pi$
$594$	0	0
$595$	−6.61206e6	−0.765674
$596$	−6.13726e6	−0.707716
$597$	0	0
$598$	9.58615e6	1.09620
$599$	1.38614e7	1.57849	0.789243	−	0.614081i	$-0.210473\pi$
$599$	1.38614e7	1.57849	0.789243	+	0.614081i	$0.210473\pi$
$600$	0	0
$601$	−382565.	−0.0432035	−0.0216018	−	0.999767i	$-0.506877\pi$
$601$	−382565.	−0.0432035	−0.0216018	+	0.999767i	$0.506877\pi$
$602$	1.68177e7	1.89137
$603$	0	0
$604$	3.36655e7	3.75485
$605$	−590798.	−0.0656221
$606$	0	0
$607$	−1.11340e7	−1.22653	−0.613267	−	0.789876i	$-0.710145\pi$
$607$	−1.11340e7	−1.22653	−0.613267	+	0.789876i	$0.710145\pi$
$608$	1.25584e7	1.37777
$609$	0	0
$610$	−1.55793e7	−1.69521
$611$	−7.73171e6	−0.837862
$612$	0	0
$613$	−5.20428e6	−0.559383	−0.279692	−	0.960090i	$-0.590232\pi$
$613$	−5.20428e6	−0.559383	−0.279692	+	0.960090i	$0.590232\pi$
$614$	−1.23902e7	−1.32635
$615$	0	0
$616$	−1.12420e7	−1.19369
$617$	−1.11765e7	−1.18194	−0.590969	−	0.806694i	$-0.701254\pi$
$617$	−1.11765e7	−1.18194	−0.590969	+	0.806694i	$0.701254\pi$
$618$	0	0
$619$	−7.14952e6	−0.749981	−0.374990	−	0.927029i	$-0.622354\pi$
$619$	−7.14952e6	−0.749981	−0.374990	+	0.927029i	$0.622354\pi$
$620$	1.54365e7	1.61276
$621$	0	0
$622$	−1.68882e7	−1.75028
$623$	−4.02411e6	−0.415384
$624$	0	0
$625$	−2.84837e6	−0.291673
$626$	3.98810e6	0.406752
$627$	0	0
$628$	1.43424e7	1.45118
$629$	−1.10696e7	−1.11559
$630$	0	0
$631$	6.48132e6	0.648022	0.324011	−	0.946053i	$-0.394968\pi$
$631$	6.48132e6	0.648022	0.324011	+	0.946053i	$0.394968\pi$
$632$	−3.26546e7	−3.25201
$633$	0	0
$634$	2.48452e7	2.45482
$635$	−3.27481e6	−0.322294
$636$	0	0
$637$	−1.52216e7	−1.48632
$638$	7.13261e6	0.693740
$639$	0	0
$640$	−6.56492e6	−0.633548
$641$	2.02563e7	1.94722	0.973609	−	0.228223i	$-0.0732914\pi$
$641$	2.02563e7	1.94722	0.973609	+	0.228223i	$0.0732914\pi$
$642$	0	0
$643$	−1.53474e7	−1.46389	−0.731943	−	0.681366i	$-0.761386\pi$
$643$	−1.53474e7	−1.46389	−0.731943	+	0.681366i	$0.761386\pi$
$644$	−3.28628e7	−3.12241
$645$	0	0
$646$	−1.45657e7	−1.37325
$647$	−3.21017e6	−0.301486	−0.150743	−	0.988573i	$-0.548167\pi$
$647$	−3.21017e6	−0.301486	−0.150743	+	0.988573i	$0.548167\pi$
$648$	0	0
$649$	4.70234e6	0.438231
$650$	−7.07971e6	−0.657252
$651$	0	0
$652$	3.30229e7	3.04226
$653$	790806.	0.0725750	0.0362875	−	0.999341i	$-0.488447\pi$
$653$	790806.	0.0725750	0.0362875	+	0.999341i	$0.488447\pi$
$654$	0	0
$655$	3.92007e6	0.357018
$656$	−2.47520e6	−0.224569
$657$	0	0
$658$	3.81703e7	3.43685
$659$	−16412.1	−0.00147215	−0.000736075	−	1.00000i	$-0.500234\pi$
$659$	−16412.1	−0.00147215	−0.000736075		1.00000i	$0.500234\pi$
$660$	0	0
$661$	8.16451e6	0.726819	0.363410	−	0.931629i	$-0.381613\pi$
$661$	8.16451e6	0.726819	0.363410	+	0.931629i	$0.381613\pi$
$662$	−3.46587e7	−3.07374
$663$	0	0
$664$	−1.21617e7	−1.07047
$665$	−1.74343e7	−1.52880
$666$	0	0
$667$	1.16742e7	1.01604
$668$	3.07567e6	0.266685
$669$	0	0
$670$	−7.33221e6	−0.631027
$671$	4.56527e6	0.391436
$672$	0	0
$673$	−6.42509e6	−0.546816	−0.273408	−	0.961898i	$-0.588151\pi$
$673$	−6.42509e6	−0.546816	−0.273408	+	0.961898i	$0.588151\pi$
$674$	−9.65711e6	−0.818837
$675$	0	0
$676$	−1.14603e7	−0.964556
$677$	−1.36660e7	−1.14596	−0.572980	−	0.819569i	$-0.694213\pi$
$677$	−1.36660e7	−1.14596	−0.572980	+	0.819569i	$0.694213\pi$
$678$	0	0
$679$	2.66058e7	2.21463
$680$	−1.23516e7	−1.02436
$681$	0	0
$682$	−6.51414e6	−0.536286
$683$	1.60884e7	1.31965	0.659827	−	0.751418i	$-0.270630\pi$
$683$	1.60884e7	1.31965	0.659827	+	0.751418i	$0.270630\pi$
$684$	0	0
$685$	−1.16951e7	−0.952309
$686$	3.67917e7	2.98497
$687$	0	0
$688$	1.42693e7	1.14929
$689$	−4.63859e6	−0.372252
$690$	0	0
$691$	−1.47580e7	−1.17580	−0.587900	−	0.808934i	$-0.700045\pi$
$691$	−1.47580e7	−1.17580	−0.587900	+	0.808934i	$0.700045\pi$
$692$	−1.11035e7	−0.881442
$693$	0	0
$694$	−1.04319e7	−0.822176
$695$	−9.54918e6	−0.749901
$696$	0	0
$697$	939238.	0.0732308
$698$	3.32313e7	2.58172
$699$	0	0
$700$	2.42703e7	1.87210
$701$	55146.6	0.00423861	0.00211931	−	0.999998i	$-0.499325\pi$
$701$	55146.6	0.00423861	0.00211931	+	0.999998i	$0.499325\pi$
$702$	0	0
$703$	−2.91876e7	−2.22746
$704$	−529104.	−0.0402355
$705$	0	0
$706$	2.83685e7	2.14203
$707$	−3.16524e6	−0.238154
$708$	0	0
$709$	1.67435e7	1.25092	0.625462	−	0.780255i	$-0.284911\pi$
$709$	1.67435e7	1.25092	0.625462	+	0.780255i	$0.284911\pi$
$710$	2.38032e7	1.77211
$711$	0	0
$712$	−7.51723e6	−0.555723
$713$	−1.06619e7	−0.785437
$714$	0	0
$715$	−2.25703e6	−0.165110
$716$	−1.53413e7	−1.11835
$717$	0	0
$718$	3.24622e7	2.34999
$719$	1.47240e7	1.06220	0.531098	−	0.847311i	$-0.321780\pi$
$719$	1.47240e7	1.06220	0.531098	+	0.847311i	$0.321780\pi$
$720$	0	0
$721$	−2.20437e7	−1.57923
$722$	−1.30683e7	−0.932991
$723$	0	0
$724$	−4.17238e7	−2.95827
$725$	−8.62178e6	−0.609189
$726$	0	0
$727$	−1.02730e7	−0.720876	−0.360438	−	0.932783i	$-0.617373\pi$
$727$	−1.02730e7	−0.720876	−0.360438	+	0.932783i	$0.617373\pi$
$728$	−4.29479e7	−3.00340
$729$	0	0
$730$	−8.16564e6	−0.567130
$731$	−5.41461e6	−0.374778
$732$	0	0
$733$	8.25057e6	0.567184	0.283592	−	0.958945i	$-0.408474\pi$
$733$	8.25057e6	0.567184	0.283592	+	0.958945i	$0.408474\pi$
$734$	−5.29329e6	−0.362648
$735$	0	0
$736$	−1.31370e7	−0.893927
$737$	2.14859e6	0.145708
$738$	0	0
$739$	8.85615e6	0.596532	0.298266	−	0.954483i	$-0.403592\pi$
$739$	8.85615e6	0.596532	0.298266	+	0.954483i	$0.403592\pi$
$740$	−4.42052e7	−2.96752
$741$	0	0
$742$	2.29000e7	1.52695
$743$	9.67155e6	0.642723	0.321362	−	0.946957i	$-0.395860\pi$
$743$	9.67155e6	0.642723	0.321362	+	0.946957i	$0.395860\pi$
$744$	0	0
$745$	3.40593e6	0.224825
$746$	−1.17178e7	−0.770905
$747$	0	0
$748$	6.46437e6	0.422447
$749$	4.69291e7	3.05659
$750$	0	0
$751$	3.58970e6	0.232251	0.116126	−	0.993235i	$-0.462952\pi$
$751$	3.58970e6	0.232251	0.116126	+	0.993235i	$0.462952\pi$
$752$	3.23861e7	2.08840
$753$	0	0
$754$	2.72488e7	1.74549
$755$	−1.86830e7	−1.19283
$756$	0	0
$757$	−2.21496e7	−1.40484	−0.702420	−	0.711763i	$-0.747897\pi$
$757$	−2.21496e7	−1.40484	−0.702420	+	0.711763i	$0.747897\pi$
$758$	4.05851e7	2.56563
$759$	0	0
$760$	−3.25681e7	−2.04530
$761$	4.73451e6	0.296356	0.148178	−	0.988961i	$-0.452659\pi$
$761$	4.73451e6	0.296356	0.148178	+	0.988961i	$0.452659\pi$
$762$	0	0
$763$	3.66855e7	2.28130
$764$	3.82720e7	2.37218
$765$	0	0
$766$	−1.03040e7	−0.634505
$767$	1.79644e7	1.10262
$768$	0	0
$769$	−2.46202e7	−1.50133	−0.750665	−	0.660683i	$-0.770267\pi$
$769$	−2.46202e7	−1.50133	−0.750665	+	0.660683i	$0.770267\pi$
$770$	1.11426e7	0.677268
$771$	0	0
$772$	2.14994e7	1.29833
$773$	8.15876e6	0.491106	0.245553	−	0.969383i	$-0.421030\pi$
$773$	8.15876e6	0.491106	0.245553	+	0.969383i	$0.421030\pi$
$774$	0	0
$775$	7.87419e6	0.470925
$776$	4.97010e7	2.96286
$777$	0	0
$778$	−1.50478e7	−0.891299
$779$	2.47652e6	0.146217
$780$	0	0
$781$	−6.97517e6	−0.409192
$782$	1.52368e7	0.890999
$783$	0	0
$784$	6.37594e7	3.70471
$785$	−7.95942e6	−0.461007
$786$	0	0
$787$	6.19045e6	0.356275	0.178137	−	0.984006i	$-0.442993\pi$
$787$	6.19045e6	0.356275	0.178137	+	0.984006i	$0.442993\pi$
$788$	3.60572e7	2.06860
$789$	0	0
$790$	3.23659e7	1.84510
$791$	−3.43873e7	−1.95415
$792$	0	0
$793$	1.74408e7	0.984878
$794$	3.02337e7	1.70192
$795$	0	0
$796$	−3.12107e7	−1.74590
$797$	8.16767e6	0.455462	0.227731	−	0.973724i	$-0.426869\pi$
$797$	8.16767e6	0.455462	0.227731	+	0.973724i	$0.426869\pi$
$798$	0	0
$799$	−1.22892e7	−0.681017
$800$	9.70214e6	0.535973
$801$	0	0
$802$	1.62108e6	0.0889957
$803$	2.39281e6	0.130954
$804$	0	0
$805$	1.82375e7	0.991917
$806$	−2.48860e7	−1.34933
$807$	0	0
$808$	−5.91282e6	−0.318615
$809$	−7.38740e6	−0.396844	−0.198422	−	0.980117i	$-0.563582\pi$
$809$	−7.38740e6	−0.396844	−0.198422	+	0.980117i	$0.563582\pi$
$810$	0	0
$811$	1.98716e7	1.06091	0.530457	−	0.847712i	$-0.322020\pi$
$811$	1.98716e7	1.06091	0.530457	+	0.847712i	$0.322020\pi$
$812$	−9.34129e7	−4.97183
$813$	0	0
$814$	1.86544e7	0.986781
$815$	−1.83263e7	−0.966455
$816$	0	0
$817$	−1.42769e7	−0.748306
$818$	1.25931e7	0.658038
$819$	0	0
$820$	3.75075e6	0.194797
$821$	1.30916e7	0.677850	0.338925	−	0.940813i	$-0.389937\pi$
$821$	1.30916e7	0.677850	0.338925	+	0.940813i	$0.389937\pi$
$822$	0	0
$823$	1.66634e7	0.857561	0.428781	−	0.903409i	$-0.358943\pi$
$823$	1.66634e7	0.857561	0.428781	+	0.903409i	$0.358943\pi$
$824$	−4.11786e7	−2.11278
$825$	0	0
$826$	−8.86876e7	−4.52285
$827$	−2.39612e7	−1.21827	−0.609137	−	0.793065i	$-0.708484\pi$
$827$	−2.39612e7	−1.21827	−0.609137	+	0.793065i	$0.708484\pi$
$828$	0	0
$829$	1.08421e7	0.547931	0.273965	−	0.961740i	$-0.411665\pi$
$829$	1.08421e7	0.547931	0.273965	+	0.961740i	$0.411665\pi$
$830$	1.20542e7	0.607354
$831$	0	0
$832$	−2.02134e6	−0.101235
$833$	−2.41942e7	−1.20809
$834$	0	0
$835$	−1.70687e6	−0.0847196
$836$	1.70448e7	0.843485
$837$	0	0
$838$	4.58421e7	2.25504
$839$	693864.	0.0340306	0.0170153	−	0.999855i	$-0.494584\pi$
$839$	693864.	0.0340306	0.0170153	+	0.999855i	$0.494584\pi$
$840$	0	0
$841$	1.26728e7	0.617851
$842$	657691.	0.0319699
$843$	0	0
$844$	1.68063e7	0.812111
$845$	6.35997e6	0.306417
$846$	0	0
$847$	−3.26517e6	−0.156386
$848$	1.94298e7	0.927853
$849$	0	0
$850$	−1.12529e7	−0.534217
$851$	3.05323e7	1.44523
$852$	0	0
$853$	−2.96408e7	−1.39482	−0.697409	−	0.716673i	$-0.745664\pi$
$853$	−2.96408e7	−1.39482	−0.697409	+	0.716673i	$0.745664\pi$
$854$	−8.61024e7	−4.03990
$855$	0	0
$856$	8.76658e7	4.08927
$857$	−1.12865e7	−0.524935	−0.262468	−	0.964941i	$-0.584536\pi$
$857$	−1.12865e7	−0.524935	−0.262468	+	0.964941i	$0.584536\pi$
$858$	0	0
$859$	−8.27254e6	−0.382522	−0.191261	−	0.981539i	$-0.561258\pi$
$859$	−8.27254e6	−0.382522	−0.191261	+	0.981539i	$0.561258\pi$
$860$	−2.16227e7	−0.996927
$861$	0	0
$862$	3.49507e7	1.60209
$863$	−1.81204e7	−0.828212	−0.414106	−	0.910229i	$-0.635906\pi$
$863$	−1.81204e7	−0.828212	−0.414106	+	0.910229i	$0.635906\pi$
$864$	0	0
$865$	6.16197e6	0.280014
$866$	−5.01671e7	−2.27313
$867$	0	0
$868$	8.53131e7	3.84341
$869$	−9.48431e6	−0.426046
$870$	0	0
$871$	8.20827e6	0.366612
$872$	6.85303e7	3.05205
$873$	0	0
$874$	4.01755e7	1.77903
$875$	−4.15915e7	−1.83647
$876$	0	0
$877$	−2.01754e7	−0.885774	−0.442887	−	0.896577i	$-0.646046\pi$
$877$	−2.01754e7	−0.885774	−0.442887	+	0.896577i	$0.646046\pi$
$878$	−8.32334e6	−0.364386
$879$	0	0
$880$	9.45411e6	0.411542
$881$	1.56531e7	0.679456	0.339728	−	0.940524i	$-0.389665\pi$
$881$	1.56531e7	0.679456	0.339728	+	0.940524i	$0.389665\pi$
$882$	0	0
$883$	1.94221e7	0.838289	0.419144	−	0.907920i	$-0.362330\pi$
$883$	1.94221e7	0.838289	0.419144	+	0.907920i	$0.362330\pi$
$884$	2.46959e7	1.06290
$885$	0	0
$886$	2.97192e7	1.27190
$887$	−1.71471e6	−0.0731781	−0.0365891	−	0.999330i	$-0.511649\pi$
$887$	−1.71471e6	−0.0731781	−0.0365891	+	0.999330i	$0.511649\pi$
$888$	0	0
$889$	−1.80989e7	−0.768066
$890$	7.45077e6	0.315302
$891$	0	0
$892$	−7.25311e7	−3.05219
$893$	−3.24035e7	−1.35976
$894$	0	0
$895$	8.51378e6	0.355275
$896$	−3.62825e7	−1.50983
$897$	0	0
$898$	2.54758e7	1.05423
$899$	−3.03066e7	−1.25066
$900$	0	0
$901$	−7.37284e6	−0.302568
$902$	−1.58280e6	−0.0647754
$903$	0	0
$904$	−6.42373e7	−2.61436
$905$	2.31550e7	0.939774
$906$	0	0
$907$	1.04320e7	0.421066	0.210533	−	0.977587i	$-0.432480\pi$
$907$	1.04320e7	0.421066	0.210533	+	0.977587i	$0.432480\pi$
$908$	−6.99374e7	−2.81510
$909$	0	0
$910$	4.25682e7	1.70405
$911$	1.01553e7	0.405410	0.202705	−	0.979240i	$-0.435027\pi$
$911$	1.01553e7	0.405410	0.202705	+	0.979240i	$0.435027\pi$
$912$	0	0
$913$	−3.53228e6	−0.140242
$914$	7.95638e6	0.315029
$915$	0	0
$916$	6.93687e7	2.73165
$917$	2.16651e7	0.850818
$918$	0	0
$919$	4.61971e7	1.80437	0.902187	−	0.431346i	$-0.141961\pi$
$919$	4.61971e7	1.80437	0.902187	+	0.431346i	$0.141961\pi$
$920$	3.40685e7	1.32704
$921$	0	0
$922$	2.76725e7	1.07206
$923$	−2.66473e7	−1.02955
$924$	0	0
$925$	−2.25492e7	−0.866515
$926$	−1.25166e7	−0.479687
$927$	0	0
$928$	−3.73421e7	−1.42341
$929$	3.03402e7	1.15340	0.576699	−	0.816957i	$-0.304341\pi$
$929$	3.03402e7	1.15340	0.576699	+	0.816957i	$0.304341\pi$
$930$	0	0
$931$	−6.37937e7	−2.41214
$932$	1.03464e7	0.390165
$933$	0	0
$934$	−8.37424e7	−3.14108
$935$	−3.58746e6	−0.134202
$936$	0	0
$937$	−1.89578e7	−0.705404	−0.352702	−	0.935736i	$-0.614737\pi$
$937$	−1.89578e7	−0.705404	−0.352702	+	0.935736i	$0.614737\pi$
$938$	−4.05230e7	−1.50382
$939$	0	0
$940$	−4.90758e7	−1.81154
$941$	3.09489e7	1.13939	0.569694	−	0.821857i	$-0.307062\pi$
$941$	3.09489e7	1.13939	0.569694	+	0.821857i	$0.307062\pi$
$942$	0	0
$943$	−2.59062e6	−0.0948692
$944$	−7.52482e7	−2.74831
$945$	0	0
$946$	9.12469e6	0.331505
$947$	2.79318e7	1.01210	0.506050	−	0.862504i	$-0.331105\pi$
$947$	2.79318e7	1.01210	0.506050	+	0.862504i	$0.331105\pi$
$948$	0	0
$949$	9.14128e6	0.329490
$950$	−2.96710e7	−1.06665
$951$	0	0
$952$	−6.82640e7	−2.44118
$953$	1.13778e7	0.405813	0.202906	−	0.979198i	$-0.434961\pi$
$953$	1.13778e7	0.405813	0.202906	+	0.979198i	$0.434961\pi$
$954$	0	0
$955$	−2.12394e7	−0.753588
$956$	−1.00716e8	−3.56412
$957$	0	0
$958$	3.54474e7	1.24788
$959$	−6.46355e7	−2.26947
$960$	0	0
$961$	−950411.	−0.0331973
$962$	7.12657e7	2.48281
$963$	0	0
$964$	−1.27143e7	−0.440657
$965$	−1.19313e7	−0.412448
$966$	0	0
$967$	−4.40091e7	−1.51348	−0.756739	−	0.653717i	$-0.773209\pi$
$967$	−4.40091e7	−1.51348	−0.756739	+	0.653717i	$0.773209\pi$
$968$	−6.09950e6	−0.209221
$969$	0	0
$970$	−4.92616e7	−1.68104
$971$	9.40911e6	0.320258	0.160129	−	0.987096i	$-0.448809\pi$
$971$	9.40911e6	0.320258	0.160129	+	0.987096i	$0.448809\pi$
$972$	0	0
$973$	−5.27756e7	−1.78711
$974$	2.85146e7	0.963098
$975$	0	0
$976$	−7.30548e7	−2.45484
$977$	−6.69270e6	−0.224319	−0.112159	−	0.993690i	$-0.535777\pi$
$977$	−6.69270e6	−0.224319	−0.112159	+	0.993690i	$0.535777\pi$
$978$	0	0
$979$	−2.18333e6	−0.0728053
$980$	−9.66168e7	−3.21357
$981$	0	0
$982$	5.75298e7	1.90377
$983$	1.40612e7	0.464128	0.232064	−	0.972700i	$-0.425452\pi$
$983$	1.40612e7	0.464128	0.232064	+	0.972700i	$0.425452\pi$
$984$	0	0
$985$	−2.00102e7	−0.657146
$986$	4.33108e7	1.41874
$987$	0	0
$988$	6.51166e7	2.12226
$989$	1.49347e7	0.485518
$990$	0	0
$991$	−4.11327e7	−1.33046	−0.665231	−	0.746637i	$-0.731667\pi$
$991$	−4.11327e7	−1.33046	−0.665231	+	0.746637i	$0.731667\pi$
$992$	3.41042e7	1.10034
$993$	0	0
$994$	1.31554e8	4.22316
$995$	1.73206e7	0.554633
$996$	0	0
$997$	4.74444e7	1.51164	0.755818	−	0.654782i	$-0.227239\pi$
$997$	4.74444e7	1.51164	0.755818	+	0.654782i	$0.227239\pi$
$998$	8.51724e6	0.270690
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.f.1.1		23
3.2	odd	2		891.6.a.e.1.23		23
9.2	odd	6		297.6.e.a.199.1		46
9.4	even	3		99.6.e.a.34.23	✓	46
9.5	odd	6		297.6.e.a.100.1		46
9.7	even	3		99.6.e.a.67.23	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.23	✓	46	9.4	even	3
99.6.e.a.67.23	yes	46	9.7	even	3
297.6.e.a.100.1		46	9.5	odd	6
297.6.e.a.199.1		46	9.2	odd	6
891.6.a.e.1.23		23	3.2	odd	2
891.6.a.f.1.1		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)

\(f(q)\)	\(=\)	\(q-10.2329 q^{2} +72.7122 q^{4} -40.3523 q^{5} -223.015 q^{7} -416.604 q^{8} +O(q^{10})\)	\(q-10.2329 q^{2} +72.7122 q^{4} -40.3523 q^{5} -223.015 q^{7} -416.604 q^{8} +412.921 q^{10} -121.000 q^{11} -462.257 q^{13} +2282.09 q^{14} +1936.28 q^{16} -734.739 q^{17} -1937.31 q^{19} -2934.10 q^{20} +1238.18 q^{22} +2026.57 q^{23} -1496.69 q^{25} +4730.23 q^{26} -16216.0 q^{28} +5760.55 q^{29} -5261.06 q^{31} -6482.38 q^{32} +7518.51 q^{34} +8999.18 q^{35} +15066.0 q^{37} +19824.3 q^{38} +16810.9 q^{40} -1278.33 q^{41} +7369.43 q^{43} -8798.18 q^{44} -20737.7 q^{46} +16726.0 q^{47} +32928.9 q^{49} +15315.5 q^{50} -33611.8 q^{52} +10034.6 q^{53} +4882.63 q^{55} +92909.1 q^{56} -58947.2 q^{58} -38862.4 q^{59} -37729.5 q^{61} +53835.9 q^{62} +4372.76 q^{64} +18653.1 q^{65} -17756.9 q^{67} -53424.5 q^{68} -92087.7 q^{70} +57646.0 q^{71} -19775.3 q^{73} -154169. q^{74} -140866. q^{76} +26984.9 q^{77} +78382.8 q^{79} -78133.1 q^{80} +13081.0 q^{82} +29192.4 q^{83} +29648.4 q^{85} -75410.7 q^{86} +50409.1 q^{88} +18044.1 q^{89} +103091. q^{91} +147356. q^{92} -171155. q^{94} +78175.1 q^{95} -119300. q^{97} -336958. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) 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

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