LMFDB - Embedded newform 1089.6.a.bo.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 469 x^{18} + 90890 x^{16} - 9428033 x^{14} + 567050771 x^{12} - 20037943016 x^{10} + \cdots + 14645220356096 $ x^20 - 469x^18 + 90890x^16 - 9428033x^14 + 567050771x^12 - 20037943016x^10 + 403500355680x^8 - 4333943247744x^6 + 23211804081664x^4 - 50981206538240*x^2 + 14645220356096
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8}\cdot 5^{2}\cdot 11^{9}\cdot 31^{2} $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-6.22592$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.22592 q^{2} +6.76211 q^{4} -7.30063 q^{5} +136.582 q^{7} +157.129 q^{8} +O(q^{10})$	$q-6.22592 q^{2} +6.76211 q^{4} -7.30063 q^{5} +136.582 q^{7} +157.129 q^{8} +45.4532 q^{10} -972.775 q^{13} -850.346 q^{14} -1194.66 q^{16} -1125.28 q^{17} -1252.80 q^{19} -49.3677 q^{20} -1709.16 q^{23} -3071.70 q^{25} +6056.42 q^{26} +923.579 q^{28} -7453.38 q^{29} -2156.31 q^{31} +2409.74 q^{32} +7005.91 q^{34} -997.131 q^{35} +11213.0 q^{37} +7799.83 q^{38} -1147.14 q^{40} +3238.53 q^{41} +15428.5 q^{43} +10641.1 q^{46} -15477.9 q^{47} +1847.52 q^{49} +19124.2 q^{50} -6578.01 q^{52} +17687.9 q^{53} +21460.9 q^{56} +46404.2 q^{58} -18841.0 q^{59} -31118.8 q^{61} +13425.0 q^{62} +23226.3 q^{64} +7101.87 q^{65} -1709.15 q^{67} -7609.27 q^{68} +6208.06 q^{70} -36063.3 q^{71} +55152.9 q^{73} -69811.0 q^{74} -8471.57 q^{76} +9603.01 q^{79} +8721.78 q^{80} -20162.9 q^{82} +37648.4 q^{83} +8215.26 q^{85} -96056.9 q^{86} -53878.6 q^{89} -132863. q^{91} -11557.6 q^{92} +96364.0 q^{94} +9146.22 q^{95} -64217.8 q^{97} -11502.5 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 298 q^{4} + 472 q^{7}+O(q^{10}) $ 20 * q + 298 * q^4 + 472 * q^7	$ 20 q + 298 q^{4} + 472 q^{7} + 836 q^{10} + 1816 q^{13} + 6794 q^{16} + 6478 q^{19} + 15250 q^{25} + 12264 q^{28} + 1226 q^{31} + 27548 q^{34} - 1032 q^{37} + 15224 q^{40} + 6128 q^{43} + 6710 q^{46} + 10268 q^{49} + 180082 q^{52} + 27898 q^{58} + 103962 q^{61} + 16948 q^{64} + 182882 q^{67} - 160154 q^{70} + 132276 q^{73} + 448700 q^{76} + 365012 q^{79} - 492270 q^{82} + 451616 q^{85} + 312092 q^{91} + 830808 q^{94} - 233530 q^{97}+O(q^{100}) $ 20 * q + 298 * q^4 + 472 * q^7 + 836 * q^10 + 1816 * q^13 + 6794 * q^16 + 6478 * q^19 + 15250 * q^25 + 12264 * q^28 + 1226 * q^31 + 27548 * q^34 - 1032 * q^37 + 15224 * q^40 + 6128 * q^43 + 6710 * q^46 + 10268 * q^49 + 180082 * q^52 + 27898 * q^58 + 103962 * q^61 + 16948 * q^64 + 182882 * q^67 - 160154 * q^70 + 132276 * q^73 + 448700 * q^76 + 365012 * q^79 - 492270 * q^82 + 451616 * q^85 + 312092 * q^91 + 830808 * q^94 - 233530 * 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 $

$2$	−6.22592	−1.10060	−0.550299	−	0.834968i	$-0.685486\pi$
$2$	−6.22592	−1.10060	−0.550299	+	0.834968i	$0.685486\pi$
$3$	0	0
$3$	0	0
$4$	6.76211	0.211316
$5$	−7.30063	−0.130598	−0.0652988	−	0.997866i	$-0.520800\pi$
$5$	−7.30063	−0.130598	−0.0652988	+	0.997866i	$0.520800\pi$
$6$	0	0
$7$	136.582	1.05353	0.526765	−	0.850011i	$-0.323405\pi$
$7$	136.582	1.05353	0.526765	+	0.850011i	$0.323405\pi$
$8$	157.129	0.868024
$9$	0	0
$10$	45.4532	0.143735
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−972.775	−1.59645	−0.798223	−	0.602363i	$-0.794226\pi$
$13$	−972.775	−1.59645	−0.798223	+	0.602363i	$0.794226\pi$
$14$	−850.346	−1.15951
$15$	0	0
$16$	−1194.66	−1.16666
$17$	−1125.28	−0.944362	−0.472181	−	0.881502i	$-0.656533\pi$
$17$	−1125.28	−0.944362	−0.472181	+	0.881502i	$0.656533\pi$
$18$	0	0
$19$	−1252.80	−0.796155	−0.398077	−	0.917352i	$-0.630322\pi$
$19$	−1252.80	−0.796155	−0.398077	+	0.917352i	$0.630322\pi$
$20$	−49.3677	−0.0275974
$21$	0	0
$22$	0	0
$23$	−1709.16	−0.673697	−0.336848	−	0.941559i	$-0.609361\pi$
$23$	−1709.16	−0.673697	−0.336848	+	0.941559i	$0.609361\pi$
$24$	0	0
$25$	−3071.70	−0.982944
$26$	6056.42	1.75704
$27$	0	0
$28$	923.579	0.222628
$29$	−7453.38	−1.64573	−0.822864	−	0.568238i	$-0.807625\pi$
$29$	−7453.38	−1.64573	−0.822864	+	0.568238i	$0.807625\pi$
$30$	0	0
$31$	−2156.31	−0.403002	−0.201501	−	0.979488i	$-0.564582\pi$
$31$	−2156.31	−0.403002	−0.201501	+	0.979488i	$0.564582\pi$
$32$	2409.74	0.416001
$33$	0	0
$34$	7005.91	1.03936
$35$	−997.131	−0.137589
$36$	0	0
$37$	11213.0	1.34653	0.673265	−	0.739401i	$-0.264891\pi$
$37$	11213.0	1.34653	0.673265	+	0.739401i	$0.264891\pi$
$38$	7799.83	0.876246
$39$	0	0
$40$	−1147.14	−0.113362
$41$	3238.53	0.300877	0.150438	−	0.988619i	$-0.451931\pi$
$41$	3238.53	0.300877	0.150438	+	0.988619i	$0.451931\pi$
$42$	0	0
$43$	15428.5	1.27249	0.636244	−	0.771488i	$-0.280487\pi$
$43$	15428.5	1.27249	0.636244	+	0.771488i	$0.280487\pi$
$44$	0	0
$45$	0	0
$46$	10641.1	0.741469
$47$	−15477.9	−1.02204	−0.511018	−	0.859570i	$-0.670732\pi$
$47$	−15477.9	−1.02204	−0.511018	+	0.859570i	$0.670732\pi$
$48$	0	0
$49$	1847.52	0.109925
$50$	19124.2	1.08183
$51$	0	0
$52$	−6578.01	−0.337354
$53$	17687.9	0.864942	0.432471	−	0.901648i	$-0.357642\pi$
$53$	17687.9	0.864942	0.432471	+	0.901648i	$0.357642\pi$
$54$	0	0
$55$	0	0
$56$	21460.9	0.914489
$57$	0	0
$58$	46404.2	1.81129
$59$	−18841.0	−0.704650	−0.352325	−	0.935878i	$-0.614609\pi$
$59$	−18841.0	−0.704650	−0.352325	+	0.935878i	$0.614609\pi$
$60$	0	0
$61$	−31118.8	−1.07078	−0.535388	−	0.844606i	$-0.679835\pi$
$61$	−31118.8	−1.07078	−0.535388	+	0.844606i	$0.679835\pi$
$62$	13425.0	0.443543
$63$	0	0
$64$	23226.3	0.708811
$65$	7101.87	0.208492
$66$	0	0
$67$	−1709.15	−0.0465149	−0.0232575	−	0.999730i	$-0.507404\pi$
$67$	−1709.15	−0.0465149	−0.0232575	+	0.999730i	$0.507404\pi$
$68$	−7609.27	−0.199559
$69$	0	0
$70$	6208.06	0.151430
$71$	−36063.3	−0.849022	−0.424511	−	0.905423i	$-0.639554\pi$
$71$	−36063.3	−0.849022	−0.424511	+	0.905423i	$0.639554\pi$
$72$	0	0
$73$	55152.9	1.21133	0.605663	−	0.795721i	$-0.292908\pi$
$73$	55152.9	1.21133	0.605663	+	0.795721i	$0.292908\pi$
$74$	−69811.0	−1.48199
$75$	0	0
$76$	−8471.57	−0.168240
$77$	0	0
$78$	0	0
$79$	9603.01	0.173117	0.0865584	−	0.996247i	$-0.472413\pi$
$79$	9603.01	0.173117	0.0865584	+	0.996247i	$0.472413\pi$
$80$	8721.78	0.152363
$81$	0	0
$82$	−20162.9	−0.331144
$83$	37648.4	0.599863	0.299931	−	0.953961i	$-0.403036\pi$
$83$	37648.4	0.599863	0.299931	+	0.953961i	$0.403036\pi$
$84$	0	0
$85$	8215.26	0.123331
$86$	−96056.9	−1.40050
$87$	0	0
$88$	0	0
$89$	−53878.6	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$89$	−53878.6	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$90$	0	0
$91$	−132863.	−1.68190
$92$	−11557.6	−0.142363
$93$	0	0
$94$	96364.0	1.12485
$95$	9146.22	0.103976
$96$	0	0
$97$	−64217.8	−0.692989	−0.346495	−	0.938052i	$-0.612628\pi$
$97$	−64217.8	−0.692989	−0.346495	+	0.938052i	$0.612628\pi$
$98$	−11502.5	−0.120984
$99$	0	0
$100$	−20771.2	−0.207712
$101$	−184452.	−1.79920	−0.899600	−	0.436714i	$-0.856142\pi$
$101$	−184452.	−1.79920	−0.899600	+	0.436714i	$0.856142\pi$
$102$	0	0
$103$	4976.75	0.0462224	0.0231112	−	0.999733i	$-0.492643\pi$
$103$	4976.75	0.0462224	0.0231112	+	0.999733i	$0.492643\pi$
$104$	−152851.	−1.38575
$105$	0	0
$106$	−110124.	−0.951953
$107$	169567.	1.43180	0.715899	−	0.698204i	$-0.246017\pi$
$107$	169567.	1.43180	0.715899	+	0.698204i	$0.246017\pi$
$108$	0	0
$109$	−13274.8	−0.107019	−0.0535094	−	0.998567i	$-0.517041\pi$
$109$	−13274.8	−0.107019	−0.0535094	+	0.998567i	$0.517041\pi$
$110$	0	0
$111$	0	0
$112$	−163169.	−1.22911
$113$	−251251.	−1.85102	−0.925512	−	0.378719i	$-0.876365\pi$
$113$	−251251.	−1.85102	−0.925512	+	0.378719i	$0.876365\pi$
$114$	0	0
$115$	12478.0	0.0879832
$116$	−50400.6	−0.347769
$117$	0	0
$118$	117303.	0.775537
$119$	−153693.	−0.994914
$120$	0	0
$121$	0	0
$122$	193743.	1.17849
$123$	0	0
$124$	−14581.2	−0.0851607
$125$	45239.8	0.258968
$126$	0	0
$127$	208394.	1.14651	0.573253	−	0.819378i	$-0.305681\pi$
$127$	208394.	1.14651	0.573253	+	0.819378i	$0.305681\pi$
$128$	−221717.	−1.19612
$129$	0	0
$130$	−44215.7	−0.229466
$131$	291922.	1.48624	0.743119	−	0.669159i	$-0.233346\pi$
$131$	291922.	1.48624	0.743119	+	0.669159i	$0.233346\pi$
$132$	0	0
$133$	−171109.	−0.838773
$134$	10641.0	0.0511942
$135$	0	0
$136$	−176814.	−0.819729
$137$	76038.5	0.346124	0.173062	−	0.984911i	$-0.444634\pi$
$137$	76038.5	0.346124	0.173062	+	0.984911i	$0.444634\pi$
$138$	0	0
$139$	322549.	1.41598	0.707992	−	0.706220i	$-0.249601\pi$
$139$	322549.	1.41598	0.707992	+	0.706220i	$0.249601\pi$
$140$	−6742.71	−0.0290746
$141$	0	0
$142$	224527.	0.934432
$143$	0	0
$144$	0	0
$145$	54414.4	0.214928
$146$	−343377.	−1.33318
$147$	0	0
$148$	75823.3	0.284543
$149$	413294.	1.52508	0.762540	−	0.646940i	$-0.223952\pi$
$149$	413294.	1.52508	0.762540	+	0.646940i	$0.223952\pi$
$150$	0	0
$151$	34735.4	0.123974	0.0619869	−	0.998077i	$-0.480256\pi$
$151$	34735.4	0.123974	0.0619869	+	0.998077i	$0.480256\pi$
$152$	−196851.	−0.691081
$153$	0	0
$154$	0	0
$155$	15742.4	0.0526311
$156$	0	0
$157$	505640.	1.63716	0.818582	−	0.574390i	$-0.194761\pi$
$157$	505640.	1.63716	0.818582	+	0.574390i	$0.194761\pi$
$158$	−59787.6	−0.190532
$159$	0	0
$160$	−17592.6	−0.0543288
$161$	−233440.	−0.709759
$162$	0	0
$163$	−134283.	−0.395870	−0.197935	−	0.980215i	$-0.563423\pi$
$163$	−134283.	−0.395870	−0.197935	+	0.980215i	$0.563423\pi$
$164$	21899.3	0.0635800
$165$	0	0
$166$	−234396.	−0.660208
$167$	−323393.	−0.897304	−0.448652	−	0.893706i	$-0.648096\pi$
$167$	−323393.	−0.897304	−0.448652	+	0.893706i	$0.648096\pi$
$168$	0	0
$169$	574998.	1.54864
$170$	−51147.6	−0.135738
$171$	0	0
$172$	104329.	0.268897
$173$	−462607.	−1.17516	−0.587580	−	0.809166i	$-0.699919\pi$
$173$	−462607.	−1.17516	−0.587580	+	0.809166i	$0.699919\pi$
$174$	0	0
$175$	−419538.	−1.03556
$176$	0	0
$177$	0	0
$178$	335444.	0.793543
$179$	−595841.	−1.38995	−0.694973	−	0.719036i	$-0.744584\pi$
$179$	−595841.	−1.38995	−0.694973	+	0.719036i	$0.744584\pi$
$180$	0	0
$181$	321126.	0.728584	0.364292	−	0.931285i	$-0.381311\pi$
$181$	321126.	0.728584	0.364292	+	0.931285i	$0.381311\pi$
$182$	827195.	1.85110
$183$	0	0
$184$	−268560.	−0.584785
$185$	−81861.7	−0.175854
$186$	0	0
$187$	0	0
$188$	−104663.	−0.215973
$189$	0	0
$190$	−56943.7	−0.114436
$191$	−236413.	−0.468909	−0.234454	−	0.972127i	$-0.575330\pi$
$191$	−236413.	−0.468909	−0.234454	+	0.972127i	$0.575330\pi$
$192$	0	0
$193$	204411.	0.395012	0.197506	−	0.980302i	$-0.436716\pi$
$193$	204411.	0.395012	0.197506	+	0.980302i	$0.436716\pi$
$194$	399815.	0.762702
$195$	0	0
$196$	12493.1	0.0232290
$197$	691384.	1.26927	0.634635	−	0.772812i	$-0.281150\pi$
$197$	691384.	1.26927	0.634635	+	0.772812i	$0.281150\pi$
$198$	0	0
$199$	85831.0	0.153643	0.0768213	−	0.997045i	$-0.475523\pi$
$199$	85831.0	0.153643	0.0768213	+	0.997045i	$0.475523\pi$
$200$	−482654.	−0.853219
$201$	0	0
$202$	1.14838e6	1.98020
$203$	−1.01799e6	−1.73382
$204$	0	0
$205$	−23643.3	−0.0392938
$206$	−30984.8	−0.0508723
$207$	0	0
$208$	1.16214e6	1.86251
$209$	0	0
$210$	0	0
$211$	−861229.	−1.33172	−0.665859	−	0.746078i	$-0.731935\pi$
$211$	−861229.	−1.33172	−0.665859	+	0.746078i	$0.731935\pi$
$212$	119608.	0.182776
$213$	0	0
$214$	−1.05571e6	−1.57583
$215$	−112638.	−0.166184
$216$	0	0
$217$	−294512.	−0.424575
$218$	82647.6	0.117785
$219$	0	0
$220$	0	0
$221$	1.09464e6	1.50762
$222$	0	0
$223$	376435.	0.506906	0.253453	−	0.967348i	$-0.418434\pi$
$223$	376435.	0.506906	0.253453	+	0.967348i	$0.418434\pi$
$224$	329126.	0.438270
$225$	0	0
$226$	1.56427e6	2.03723
$227$	−1.02387e6	−1.31881	−0.659404	−	0.751788i	$-0.729191\pi$
$227$	−1.02387e6	−1.31881	−0.659404	+	0.751788i	$0.729191\pi$
$228$	0	0
$229$	−643453.	−0.810826	−0.405413	−	0.914134i	$-0.632872\pi$
$229$	−643453.	−0.810826	−0.405413	+	0.914134i	$0.632872\pi$
$230$	−77686.9	−0.0968341
$231$	0	0
$232$	−1.17114e6	−1.42853
$233$	−1.15815e6	−1.39757	−0.698786	−	0.715331i	$-0.746276\pi$
$233$	−1.15815e6	−1.39757	−0.698786	+	0.715331i	$0.746276\pi$
$234$	0	0
$235$	112998.	0.133476
$236$	−127405.	−0.148904
$237$	0	0
$238$	956878.	1.09500
$239$	74991.4	0.0849214	0.0424607	−	0.999098i	$-0.486480\pi$
$239$	74991.4	0.0849214	0.0424607	+	0.999098i	$0.486480\pi$
$240$	0	0
$241$	1.34455e6	1.49119	0.745597	−	0.666397i	$-0.232164\pi$
$241$	1.34455e6	1.49119	0.745597	+	0.666397i	$0.232164\pi$
$242$	0	0
$243$	0	0
$244$	−210429.	−0.226272
$245$	−13488.0	−0.0143560
$246$	0	0
$247$	1.21869e6	1.27102
$248$	−338819.	−0.349815
$249$	0	0
$250$	−281660.	−0.285019
$251$	−1.26165e6	−1.26402	−0.632009	−	0.774961i	$-0.717769\pi$
$251$	−1.26165e6	−1.26402	−0.632009	+	0.774961i	$0.717769\pi$
$252$	0	0
$253$	0	0
$254$	−1.29745e6	−1.26184
$255$	0	0
$256$	637150.	0.607633
$257$	−266506.	−0.251695	−0.125848	−	0.992050i	$-0.540165\pi$
$257$	−266506.	−0.251695	−0.125848	+	0.992050i	$0.540165\pi$
$258$	0	0
$259$	1.53148e6	1.41861
$260$	48023.6	0.0440577
$261$	0	0
$262$	−1.81748e6	−1.63575
$263$	−1.10324e6	−0.983514	−0.491757	−	0.870732i	$-0.663645\pi$
$263$	−1.10324e6	−0.983514	−0.491757	+	0.870732i	$0.663645\pi$
$264$	0	0
$265$	−129133.	−0.112959
$266$	1.06531e6	0.923152
$267$	0	0
$268$	−11557.4	−0.00982934
$269$	1.09528e6	0.922875	0.461437	−	0.887173i	$-0.347334\pi$
$269$	1.09528e6	0.922875	0.461437	+	0.887173i	$0.347334\pi$
$270$	0	0
$271$	463110.	0.383055	0.191527	−	0.981487i	$-0.438656\pi$
$271$	463110.	0.383055	0.191527	+	0.981487i	$0.438656\pi$
$272$	1.34433e6	1.10175
$273$	0	0
$274$	−473410.	−0.380944
$275$	0	0
$276$	0	0
$277$	−61863.8	−0.0484437	−0.0242218	−	0.999707i	$-0.507711\pi$
$277$	−61863.8	−0.0484437	−0.0242218	+	0.999707i	$0.507711\pi$
$278$	−2.00816e6	−1.55843
$279$	0	0
$280$	−156678.	−0.119430
$281$	464704.	0.351084	0.175542	−	0.984472i	$-0.443832\pi$
$281$	464704.	0.351084	0.175542	+	0.984472i	$0.443832\pi$
$282$	0	0
$283$	2.41351e6	1.79136	0.895678	−	0.444702i	$-0.146690\pi$
$283$	2.41351e6	1.79136	0.895678	+	0.444702i	$0.146690\pi$
$284$	−243864.	−0.179412
$285$	0	0
$286$	0	0
$287$	442324.	0.316983
$288$	0	0
$289$	−153600.	−0.108180
$290$	−338780.	−0.236550
$291$	0	0
$292$	372950.	0.255972
$293$	1.44771e6	0.985176	0.492588	−	0.870263i	$-0.336051\pi$
$293$	1.44771e6	0.985176	0.492588	+	0.870263i	$0.336051\pi$
$294$	0	0
$295$	137551.	0.0920257
$296$	1.76188e6	1.16882
$297$	0	0
$298$	−2.57313e6	−1.67850
$299$	1.66263e6	1.07552
$300$	0	0
$301$	2.10725e6	1.34060
$302$	−216260.	−0.136445
$303$	0	0
$304$	1.49667e6	0.928843
$305$	227187.	0.139841
$306$	0	0
$307$	−546831.	−0.331137	−0.165568	−	0.986198i	$-0.552946\pi$
$307$	−546831.	−0.331137	−0.165568	+	0.986198i	$0.552946\pi$
$308$	0	0
$309$	0	0
$310$	−98011.1	−0.0579257
$311$	1.36252e6	0.798806	0.399403	−	0.916775i	$-0.369217\pi$
$311$	1.36252e6	0.798806	0.399403	+	0.916775i	$0.369217\pi$
$312$	0	0
$313$	64631.0	0.0372889	0.0186445	−	0.999826i	$-0.494065\pi$
$313$	64631.0	0.0372889	0.0186445	+	0.999826i	$0.494065\pi$
$314$	−3.14807e6	−1.80186
$315$	0	0
$316$	64936.6	0.0365824
$317$	813429.	0.454644	0.227322	−	0.973820i	$-0.427003\pi$
$317$	813429.	0.454644	0.227322	+	0.973820i	$0.427003\pi$
$318$	0	0
$319$	0	0
$320$	−169567.	−0.0925691
$321$	0	0
$322$	1.45338e6	0.781160
$323$	1.40975e6	0.751858
$324$	0	0
$325$	2.98807e6	1.56922
$326$	836036.	0.435694
$327$	0	0
$328$	508868.	0.261168
$329$	−2.11399e6	−1.07675
$330$	0	0
$331$	−2.45104e6	−1.22965	−0.614824	−	0.788664i	$-0.710773\pi$
$331$	−2.45104e6	−1.22965	−0.614824	+	0.788664i	$0.710773\pi$
$332$	254583.	0.126761
$333$	0	0
$334$	2.01342e6	0.987571
$335$	12477.8	0.00607474
$336$	0	0
$337$	3.02049e6	1.44878	0.724391	−	0.689390i	$-0.242121\pi$
$337$	3.02049e6	1.44878	0.724391	+	0.689390i	$0.242121\pi$
$338$	−3.57989e6	−1.70443
$339$	0	0
$340$	55552.5	0.0260619
$341$	0	0
$342$	0	0
$343$	−2.04319e6	−0.937720
$344$	2.42427e6	1.10455
$345$	0	0
$346$	2.88016e6	1.29338
$347$	1.67145e6	0.745195	0.372598	−	0.927993i	$-0.378467\pi$
$347$	1.67145e6	0.745195	0.372598	+	0.927993i	$0.378467\pi$
$348$	0	0
$349$	−1.62846e6	−0.715672	−0.357836	−	0.933784i	$-0.616485\pi$
$349$	−1.62846e6	−0.715672	−0.357836	+	0.933784i	$0.616485\pi$
$350$	2.61201e6	1.13974
$351$	0	0
$352$	0	0
$353$	−529652.	−0.226232	−0.113116	−	0.993582i	$-0.536083\pi$
$353$	−529652.	−0.226232	−0.113116	+	0.993582i	$0.536083\pi$
$354$	0	0
$355$	263284.	0.110880
$356$	−364333.	−0.152361
$357$	0	0
$358$	3.70966e6	1.52977
$359$	−1.67125e6	−0.684391	−0.342195	−	0.939629i	$-0.611170\pi$
$359$	−1.67125e6	−0.684391	−0.342195	+	0.939629i	$0.611170\pi$
$360$	0	0
$361$	−906593.	−0.366138
$362$	−1.99931e6	−0.801878
$363$	0	0
$364$	−898435.	−0.355413
$365$	−402651.	−0.158196
$366$	0	0
$367$	−1.24006e6	−0.480595	−0.240297	−	0.970699i	$-0.577245\pi$
$367$	−1.24006e6	−0.480595	−0.240297	+	0.970699i	$0.577245\pi$
$368$	2.04187e6	0.785976
$369$	0	0
$370$	509664.	0.193544
$371$	2.41584e6	0.911242
$372$	0	0
$373$	4.46056e6	1.66003	0.830017	−	0.557738i	$-0.188331\pi$
$373$	4.46056e6	1.66003	0.830017	+	0.557738i	$0.188331\pi$
$374$	0	0
$375$	0	0
$376$	−2.43202e6	−0.887152
$377$	7.25046e6	2.62732
$378$	0	0
$379$	−4.23483e6	−1.51439	−0.757196	−	0.653188i	$-0.773431\pi$
$379$	−4.23483e6	−1.51439	−0.757196	+	0.653188i	$0.773431\pi$
$380$	61847.8	0.0219718
$381$	0	0
$382$	1.47189e6	0.516080
$383$	5.03255e6	1.75304	0.876518	−	0.481368i	$-0.159860\pi$
$383$	5.03255e6	1.75304	0.876518	+	0.481368i	$0.159860\pi$
$384$	0	0
$385$	0	0
$386$	−1.27264e6	−0.434749
$387$	0	0
$388$	−434248.	−0.146440
$389$	−2.93874e6	−0.984661	−0.492330	−	0.870408i	$-0.663855\pi$
$389$	−2.93874e6	−0.984661	−0.492330	+	0.870408i	$0.663855\pi$
$390$	0	0
$391$	1.92329e6	0.636214
$392$	290299.	0.0954180
$393$	0	0
$394$	−4.30450e6	−1.39696
$395$	−70108.0	−0.0226087
$396$	0	0
$397$	751757.	0.239387	0.119694	−	0.992811i	$-0.461809\pi$
$397$	751757.	0.239387	0.119694	+	0.992811i	$0.461809\pi$
$398$	−534377.	−0.169099
$399$	0	0
$400$	3.66964e6	1.14676
$401$	415251.	0.128958	0.0644792	−	0.997919i	$-0.479461\pi$
$401$	415251.	0.128958	0.0644792	+	0.997919i	$0.479461\pi$
$402$	0	0
$403$	2.09760e6	0.643370
$404$	−1.24728e6	−0.380200
$405$	0	0
$406$	6.33795e6	1.90824
$407$	0	0
$408$	0	0
$409$	5.47596e6	1.61865	0.809323	−	0.587364i	$-0.199834\pi$
$409$	5.47596e6	1.61865	0.809323	+	0.587364i	$0.199834\pi$
$410$	147202.	0.0432467
$411$	0	0
$412$	33653.3	0.00976753
$413$	−2.57333e6	−0.742370
$414$	0	0
$415$	−274857.	−0.0783406
$416$	−2.34413e6	−0.664123
$417$	0	0
$418$	0	0
$419$	962820.	0.267923	0.133961	−	0.990987i	$-0.457230\pi$
$419$	962820.	0.267923	0.133961	+	0.990987i	$0.457230\pi$
$420$	0	0
$421$	284931.	0.0783491	0.0391745	−	0.999232i	$-0.487527\pi$
$421$	284931.	0.0783491	0.0391745	+	0.999232i	$0.487527\pi$
$422$	5.36194e6	1.46569
$423$	0	0
$424$	2.77929e6	0.750790
$425$	3.45653e6	0.928255
$426$	0	0
$427$	−4.25026e6	−1.12809
$428$	1.14663e6	0.302562
$429$	0	0
$430$	701276.	0.182902
$431$	6.90167e6	1.78962	0.894811	−	0.446446i	$-0.147310\pi$
$431$	6.90167e6	1.78962	0.894811	+	0.446446i	$0.147310\pi$
$432$	0	0
$433$	6.18278e6	1.58476	0.792381	−	0.610026i	$-0.208841\pi$
$433$	6.18278e6	1.58476	0.792381	+	0.610026i	$0.208841\pi$
$434$	1.83361e6	0.467286
$435$	0	0
$436$	−89765.4	−0.0226148
$437$	2.14124e6	0.536367
$438$	0	0
$439$	−105064.	−0.0260191	−0.0130096	−	0.999915i	$-0.504141\pi$
$439$	−105064.	−0.0260191	−0.0130096	+	0.999915i	$0.504141\pi$
$440$	0	0
$441$	0	0
$442$	−6.81517e6	−1.65929
$443$	−783890.	−0.189778	−0.0948890	−	0.995488i	$-0.530250\pi$
$443$	−783890.	−0.189778	−0.0948890	+	0.995488i	$0.530250\pi$
$444$	0	0
$445$	393348.	0.0941623
$446$	−2.34365e6	−0.557900
$447$	0	0
$448$	3.17229e6	0.746754
$449$	5.85382e6	1.37033	0.685163	−	0.728390i	$-0.259731\pi$
$449$	5.85382e6	1.37033	0.685163	+	0.728390i	$0.259731\pi$
$450$	0	0
$451$	0	0
$452$	−1.69899e6	−0.391151
$453$	0	0
$454$	6.37456e6	1.45148
$455$	969984.	0.219653
$456$	0	0
$457$	−845043.	−0.189273	−0.0946364	−	0.995512i	$-0.530169\pi$
$457$	−845043.	−0.189273	−0.0946364	+	0.995512i	$0.530169\pi$
$458$	4.00609e6	0.892394
$459$	0	0
$460$	84377.5	0.0185922
$461$	−718001.	−0.157352	−0.0786761	−	0.996900i	$-0.525069\pi$
$461$	−718001.	−0.157352	−0.0786761	+	0.996900i	$0.525069\pi$
$462$	0	0
$463$	−6.59154e6	−1.42901	−0.714504	−	0.699632i	$-0.753347\pi$
$463$	−6.59154e6	−1.42901	−0.714504	+	0.699632i	$0.753347\pi$
$464$	8.90427e6	1.92001
$465$	0	0
$466$	7.21053e6	1.53816
$467$	6.41662e6	1.36149	0.680745	−	0.732521i	$-0.261656\pi$
$467$	6.41662e6	1.36149	0.680745	+	0.732521i	$0.261656\pi$
$468$	0	0
$469$	−233438.	−0.0490049
$470$	−703518.	−0.146903
$471$	0	0
$472$	−2.96047e6	−0.611653
$473$	0	0
$474$	0	0
$475$	3.84822e6	0.782576
$476$	−1.03929e6	−0.210241
$477$	0	0
$478$	−466891.	−0.0934643
$479$	−8.52593e6	−1.69786	−0.848932	−	0.528502i	$-0.822754\pi$
$479$	−8.52593e6	−1.69786	−0.848932	+	0.528502i	$0.822754\pi$
$480$	0	0
$481$	−1.09077e7	−2.14966
$482$	−8.37106e6	−1.64121
$483$	0	0
$484$	0	0
$485$	468831.	0.0905027
$486$	0	0
$487$	−172775.	−0.0330110	−0.0165055	−	0.999864i	$-0.505254\pi$
$487$	−172775.	−0.0330110	−0.0165055	+	0.999864i	$0.505254\pi$
$488$	−4.88968e6	−0.929460
$489$	0	0
$490$	83975.5	0.0158002
$491$	−7.85916e6	−1.47120	−0.735601	−	0.677415i	$-0.763100\pi$
$491$	−7.85916e6	−1.47120	−0.735601	+	0.677415i	$0.763100\pi$
$492$	0	0
$493$	8.38715e6	1.55416
$494$	−7.58748e6	−1.39888
$495$	0	0
$496$	2.57606e6	0.470167
$497$	−4.92557e6	−0.894470
$498$	0	0
$499$	−8.69278e6	−1.56281	−0.781407	−	0.624021i	$-0.785498\pi$
$499$	−8.69278e6	−1.56281	−0.781407	+	0.624021i	$0.785498\pi$
$500$	305917.	0.0547240
$501$	0	0
$502$	7.85491e6	1.39118
$503$	5.68491e6	1.00185	0.500926	−	0.865490i	$-0.332993\pi$
$503$	5.68491e6	1.00185	0.500926	+	0.865490i	$0.332993\pi$
$504$	0	0
$505$	1.34661e6	0.234971
$506$	0	0
$507$	0	0
$508$	1.40918e6	0.242275
$509$	6.72641e6	1.15077	0.575385	−	0.817883i	$-0.304852\pi$
$509$	6.72641e6	1.15077	0.575385	+	0.817883i	$0.304852\pi$
$510$	0	0
$511$	7.53286e6	1.27617
$512$	3.12810e6	0.527358
$513$	0	0
$514$	1.65925e6	0.277015
$515$	−36333.4	−0.00603654
$516$	0	0
$517$	0	0
$518$	−9.53490e6	−1.56132
$519$	0	0
$520$	1.11591e6	0.180976
$521$	−488912.	−0.0789108	−0.0394554	−	0.999221i	$-0.512562\pi$
$521$	−488912.	−0.0789108	−0.0394554	+	0.999221i	$0.512562\pi$
$522$	0	0
$523$	2.18850e6	0.349859	0.174929	−	0.984581i	$-0.444030\pi$
$523$	2.18850e6	0.349859	0.174929	+	0.984581i	$0.444030\pi$
$524$	1.97401e6	0.314066
$525$	0	0
$526$	6.86869e6	1.08245
$527$	2.42645e6	0.380580
$528$	0	0
$529$	−3.51510e6	−0.546133
$530$	803972.	0.124323
$531$	0	0
$532$	−1.15706e6	−0.177246
$533$	−3.15036e6	−0.480333
$534$	0	0
$535$	−1.23795e6	−0.186989
$536$	−268557.	−0.0403761
$537$	0	0
$538$	−6.81910e6	−1.01571
$539$	0	0
$540$	0	0
$541$	−2.60674e6	−0.382916	−0.191458	−	0.981501i	$-0.561322\pi$
$541$	−2.60674e6	−0.382916	−0.191458	+	0.981501i	$0.561322\pi$
$542$	−2.88329e6	−0.421589
$543$	0	0
$544$	−2.71163e6	−0.392856
$545$	96914.1	0.0139764
$546$	0	0
$547$	9.91563e6	1.41694	0.708471	−	0.705740i	$-0.249385\pi$
$547$	9.91563e6	1.41694	0.708471	+	0.705740i	$0.249385\pi$
$548$	514181.	0.0731416
$549$	0	0
$550$	0	0
$551$	9.33759e6	1.31025
$552$	0	0
$553$	1.31159e6	0.182384
$554$	385159.	0.0533170
$555$	0	0
$556$	2.18111e6	0.299220
$557$	8.08079e6	1.10361	0.551805	−	0.833973i	$-0.313939\pi$
$557$	8.08079e6	1.10361	0.551805	+	0.833973i	$0.313939\pi$
$558$	0	0
$559$	−1.50085e7	−2.03146
$560$	1.19123e6	0.160519
$561$	0	0
$562$	−2.89321e6	−0.386402
$563$	−1.42992e7	−1.90126	−0.950630	−	0.310326i	$-0.899562\pi$
$563$	−1.42992e7	−1.90126	−0.950630	+	0.310326i	$0.899562\pi$
$564$	0	0
$565$	1.83429e6	0.241739
$566$	−1.50263e7	−1.97156
$567$	0	0
$568$	−5.66659e6	−0.736972
$569$	−26713.6	−0.00345901	−0.00172950	−	0.999999i	$-0.500551\pi$
$569$	−26713.6	−0.00345901	−0.00172950	+	0.999999i	$0.500551\pi$
$570$	0	0
$571$	444503.	0.0570538	0.0285269	−	0.999593i	$-0.490918\pi$
$571$	444503.	0.0570538	0.0285269	+	0.999593i	$0.490918\pi$
$572$	0	0
$573$	0	0
$574$	−2.75387e6	−0.348871
$575$	5.25004e6	0.662206
$576$	0	0
$577$	201538.	0.0252010	0.0126005	−	0.999921i	$-0.495989\pi$
$577$	201538.	0.0252010	0.0126005	+	0.999921i	$0.495989\pi$
$578$	956303.	0.119063
$579$	0	0
$580$	367956.	0.0454178
$581$	5.14208e6	0.631973
$582$	0	0
$583$	0	0
$584$	8.66612e6	1.05146
$585$	0	0
$586$	−9.01336e6	−1.08428
$587$	4.44425e6	0.532358	0.266179	−	0.963924i	$-0.414239\pi$
$587$	4.44425e6	0.532358	0.266179	+	0.963924i	$0.414239\pi$
$588$	0	0
$589$	2.70142e6	0.320852
$590$	−856382.	−0.101283
$591$	0	0
$592$	−1.33957e7	−1.57094
$593$	6.22055e6	0.726427	0.363214	−	0.931706i	$-0.381680\pi$
$593$	6.22055e6	0.726427	0.363214	+	0.931706i	$0.381680\pi$
$594$	0	0
$595$	1.12205e6	0.129933
$596$	2.79474e6	0.322274
$597$	0	0
$598$	−1.03514e7	−1.18371
$599$	2.95826e6	0.336876	0.168438	−	0.985712i	$-0.446128\pi$
$599$	2.95826e6	0.336876	0.168438	+	0.985712i	$0.446128\pi$
$600$	0	0
$601$	−1.28776e7	−1.45428	−0.727139	−	0.686490i	$-0.759151\pi$
$601$	−1.28776e7	−1.45428	−0.727139	+	0.686490i	$0.759151\pi$
$602$	−1.31196e7	−1.47547
$603$	0	0
$604$	234885.	0.0261976
$605$	0	0
$606$	0	0
$607$	−894221.	−0.0985083	−0.0492542	−	0.998786i	$-0.515684\pi$
$607$	−894221.	−0.0985083	−0.0492542	+	0.998786i	$0.515684\pi$
$608$	−3.01892e6	−0.331201
$609$	0	0
$610$	−1.41445e6	−0.153909
$611$	1.50565e7	1.63163
$612$	0	0
$613$	7.22854e6	0.776962	0.388481	−	0.921457i	$-0.373000\pi$
$613$	7.22854e6	0.776962	0.388481	+	0.921457i	$0.373000\pi$
$614$	3.40453e6	0.364449
$615$	0	0
$616$	0	0
$617$	−837999.	−0.0886198	−0.0443099	−	0.999018i	$-0.514109\pi$
$617$	−837999.	−0.0886198	−0.0443099	+	0.999018i	$0.514109\pi$
$618$	0	0
$619$	−8.29484e6	−0.870125	−0.435062	−	0.900400i	$-0.643274\pi$
$619$	−8.29484e6	−0.870125	−0.435062	+	0.900400i	$0.643274\pi$
$620$	106452.	0.0111218
$621$	0	0
$622$	−8.48293e6	−0.879164
$623$	−7.35883e6	−0.759606
$624$	0	0
$625$	9.26879e6	0.949124
$626$	−402387.	−0.0410401
$627$	0	0
$628$	3.41919e6	0.345959
$629$	−1.26177e7	−1.27161
$630$	0	0
$631$	−1.34362e7	−1.34339	−0.671696	−	0.740827i	$-0.734434\pi$
$631$	−1.34362e7	−1.34339	−0.671696	+	0.740827i	$0.734434\pi$
$632$	1.50891e6	0.150270
$633$	0	0
$634$	−5.06435e6	−0.500381
$635$	−1.52141e6	−0.149731
$636$	0	0
$637$	−1.79722e6	−0.175490
$638$	0	0
$639$	0	0
$640$	1.61867e6	0.156210
$641$	−1.42328e7	−1.36818	−0.684091	−	0.729397i	$-0.739801\pi$
$641$	−1.42328e7	−1.36818	−0.684091	+	0.729397i	$0.739801\pi$
$642$	0	0
$643$	−8.26148e6	−0.788007	−0.394004	−	0.919109i	$-0.628910\pi$
$643$	−8.26148e6	−0.788007	−0.394004	+	0.919109i	$0.628910\pi$
$644$	−1.57855e6	−0.149983
$645$	0	0
$646$	−8.77700e6	−0.827494
$647$	−1.61204e7	−1.51397	−0.756983	−	0.653435i	$-0.773327\pi$
$647$	−1.61204e7	−1.51397	−0.756983	+	0.653435i	$0.773327\pi$
$648$	0	0
$649$	0	0
$650$	−1.86035e7	−1.72708
$651$	0	0
$652$	−908037.	−0.0836536
$653$	−1.60056e7	−1.46889	−0.734443	−	0.678670i	$-0.762556\pi$
$653$	−1.60056e7	−1.46889	−0.734443	+	0.678670i	$0.762556\pi$
$654$	0	0
$655$	−2.13121e6	−0.194099
$656$	−3.86895e6	−0.351021
$657$	0	0
$658$	1.31615e7	1.18506
$659$	2.15185e6	0.193018	0.0965091	−	0.995332i	$-0.469232\pi$
$659$	2.15185e6	0.193018	0.0965091	+	0.995332i	$0.469232\pi$
$660$	0	0
$661$	1.75543e7	1.56272	0.781359	−	0.624082i	$-0.214527\pi$
$661$	1.75543e7	1.56272	0.781359	+	0.624082i	$0.214527\pi$
$662$	1.52600e7	1.35335
$663$	0	0
$664$	5.91567e6	0.520695
$665$	1.24921e6	0.109542
$666$	0	0
$667$	1.27391e7	1.10872
$668$	−2.18682e6	−0.189615
$669$	0	0
$670$	−77686.1	−0.00668585
$671$	0	0
$672$	0	0
$673$	563466.	0.0479546	0.0239773	−	0.999713i	$-0.492367\pi$
$673$	563466.	0.0479546	0.0239773	+	0.999713i	$0.492367\pi$
$674$	−1.88053e7	−1.59453
$675$	0	0
$676$	3.88820e6	0.327252
$677$	2.02172e7	1.69531	0.847656	−	0.530546i	$-0.178013\pi$
$677$	2.02172e7	1.69531	0.847656	+	0.530546i	$0.178013\pi$
$678$	0	0
$679$	−8.77097e6	−0.730085
$680$	1.29086e6	0.107055
$681$	0	0
$682$	0	0
$683$	1.91722e7	1.57261	0.786305	−	0.617839i	$-0.211991\pi$
$683$	1.91722e7	1.57261	0.786305	+	0.617839i	$0.211991\pi$
$684$	0	0
$685$	−555129.	−0.0452030
$686$	1.27207e7	1.03205
$687$	0	0
$688$	−1.84319e7	−1.48456
$689$	−1.72064e7	−1.38083
$690$	0	0
$691$	2.16046e7	1.72128	0.860640	−	0.509214i	$-0.170064\pi$
$691$	2.16046e7	1.72128	0.860640	+	0.509214i	$0.170064\pi$
$692$	−3.12820e6	−0.248330
$693$	0	0
$694$	−1.04063e7	−0.820161
$695$	−2.35481e6	−0.184924
$696$	0	0
$697$	−3.64426e6	−0.284137
$698$	1.01387e7	0.787668
$699$	0	0
$700$	−2.83696e6	−0.218831
$701$	4.24431e6	0.326221	0.163110	−	0.986608i	$-0.447847\pi$
$701$	4.24431e6	0.326221	0.163110	+	0.986608i	$0.447847\pi$
$702$	0	0
$703$	−1.40476e7	−1.07205
$704$	0	0
$705$	0	0
$706$	3.29757e6	0.248990
$707$	−2.51927e7	−1.89551
$708$	0	0
$709$	1.02466e7	0.765537	0.382769	−	0.923844i	$-0.374971\pi$
$709$	1.02466e7	0.765537	0.382769	+	0.923844i	$0.374971\pi$
$710$	−1.63919e6	−0.122035
$711$	0	0
$712$	−8.46590e6	−0.625854
$713$	3.68549e6	0.271501
$714$	0	0
$715$	0	0
$716$	−4.02914e6	−0.293718
$717$	0	0
$718$	1.04050e7	0.753239
$719$	748477.	0.0539954	0.0269977	−	0.999635i	$-0.491405\pi$
$719$	748477.	0.0539954	0.0269977	+	0.999635i	$0.491405\pi$
$720$	0	0
$721$	679732.	0.0486967
$722$	5.64438e6	0.402970
$723$	0	0
$724$	2.17149e6	0.153961
$725$	2.28946e7	1.61766
$726$	0	0
$727$	2.52466e7	1.77161	0.885803	−	0.464062i	$-0.153608\pi$
$727$	2.52466e7	1.77161	0.885803	+	0.464062i	$0.153608\pi$
$728$	−2.08767e7	−1.45993
$729$	0	0
$730$	2.50687e6	0.174110
$731$	−1.73614e7	−1.20169
$732$	0	0
$733$	−4.07191e6	−0.279923	−0.139961	−	0.990157i	$-0.544698\pi$
$733$	−4.07191e6	−0.279923	−0.139961	+	0.990157i	$0.544698\pi$
$734$	7.72054e6	0.528942
$735$	0	0
$736$	−4.11864e6	−0.280259
$737$	0	0
$738$	0	0
$739$	2.61295e7	1.76003	0.880015	−	0.474947i	$-0.157533\pi$
$739$	2.61295e7	1.76003	0.880015	+	0.474947i	$0.157533\pi$
$740$	−553558.	−0.0371607
$741$	0	0
$742$	−1.50408e7	−1.00291
$743$	−2.48602e6	−0.165208	−0.0826041	−	0.996582i	$-0.526324\pi$
$743$	−2.48602e6	−0.165208	−0.0826041	+	0.996582i	$0.526324\pi$
$744$	0	0
$745$	−3.01730e6	−0.199172
$746$	−2.77711e7	−1.82703
$747$	0	0
$748$	0	0
$749$	2.31597e7	1.50844
$750$	0	0
$751$	542606.	0.0351063	0.0175531	−	0.999846i	$-0.494412\pi$
$751$	542606.	0.0351063	0.0175531	+	0.999846i	$0.494412\pi$
$752$	1.84908e7	1.19237
$753$	0	0
$754$	−4.51408e7	−2.89162
$755$	−253590.	−0.0161907
$756$	0	0
$757$	4.07118e6	0.258214	0.129107	−	0.991631i	$-0.458789\pi$
$757$	4.07118e6	0.258214	0.129107	+	0.991631i	$0.458789\pi$
$758$	2.63657e7	1.66674
$759$	0	0
$760$	1.43714e6	0.0902536
$761$	−8.45477e6	−0.529225	−0.264612	−	0.964355i	$-0.585244\pi$
$761$	−8.45477e6	−0.529225	−0.264612	+	0.964355i	$0.585244\pi$
$762$	0	0
$763$	−1.81309e6	−0.112748
$764$	−1.59865e6	−0.0990879
$765$	0	0
$766$	−3.13323e7	−1.92939
$767$	1.83280e7	1.12494
$768$	0	0
$769$	1.31072e6	0.0799273	0.0399637	−	0.999201i	$-0.487276\pi$
$769$	1.31072e6	0.0799273	0.0399637	+	0.999201i	$0.487276\pi$
$770$	0	0
$771$	0	0
$772$	1.38225e6	0.0834723
$773$	1.30223e7	0.783864	0.391932	−	0.919994i	$-0.371807\pi$
$773$	1.30223e7	0.783864	0.391932	+	0.919994i	$0.371807\pi$
$774$	0	0
$775$	6.62354e6	0.396128
$776$	−1.00905e7	−0.601531
$777$	0	0
$778$	1.82963e7	1.08372
$779$	−4.05723e6	−0.239544
$780$	0	0
$781$	0	0
$782$	−1.19743e7	−0.700215
$783$	0	0
$784$	−2.20716e6	−0.128246
$785$	−3.69149e6	−0.213810
$786$	0	0
$787$	−3.08159e7	−1.77353	−0.886763	−	0.462225i	$-0.847051\pi$
$787$	−3.08159e7	−1.77353	−0.886763	+	0.462225i	$0.847051\pi$
$788$	4.67522e6	0.268217
$789$	0	0
$790$	436487.	0.0248830
$791$	−3.43163e7	−1.95011
$792$	0	0
$793$	3.02716e7	1.70944
$794$	−4.68038e6	−0.263469
$795$	0	0
$796$	580399.	0.0324671
$797$	−2.94660e7	−1.64314	−0.821570	−	0.570107i	$-0.806902\pi$
$797$	−2.94660e7	−1.64314	−0.821570	+	0.570107i	$0.806902\pi$
$798$	0	0
$799$	1.74169e7	0.965173
$800$	−7.40199e6	−0.408906
$801$	0	0
$802$	−2.58532e6	−0.141931
$803$	0	0
$804$	0	0
$805$	1.70426e6	0.0926929
$806$	−1.30595e7	−0.708092
$807$	0	0
$808$	−2.89828e7	−1.56175
$809$	1.63214e7	0.876772	0.438386	−	0.898787i	$-0.355550\pi$
$809$	1.63214e7	0.876772	0.438386	+	0.898787i	$0.355550\pi$
$810$	0	0
$811$	596949.	0.0318702	0.0159351	−	0.999873i	$-0.494927\pi$
$811$	596949.	0.0318702	0.0159351	+	0.999873i	$0.494927\pi$
$812$	−6.88379e6	−0.366385
$813$	0	0
$814$	0	0
$815$	980351.	0.0516997
$816$	0	0
$817$	−1.93289e7	−1.01310
$818$	−3.40929e7	−1.78148
$819$	0	0
$820$	−159879.	−0.00830340
$821$	−3.18257e7	−1.64786	−0.823930	−	0.566692i	$-0.808223\pi$
$821$	−3.18257e7	−1.64786	−0.823930	+	0.566692i	$0.808223\pi$
$822$	0	0
$823$	−2.45106e7	−1.26140	−0.630701	−	0.776026i	$-0.717233\pi$
$823$	−2.45106e7	−1.26140	−0.630701	+	0.776026i	$0.717233\pi$
$824$	781992.	0.0401221
$825$	0	0
$826$	1.60214e7	0.817051
$827$	−1.83895e7	−0.934987	−0.467494	−	0.883996i	$-0.654843\pi$
$827$	−1.83895e7	−0.934987	−0.467494	+	0.883996i	$0.654843\pi$
$828$	0	0
$829$	−3.14936e7	−1.59161	−0.795804	−	0.605554i	$-0.792952\pi$
$829$	−3.14936e7	−1.59161	−0.795804	+	0.605554i	$0.792952\pi$
$830$	1.71124e6	0.0862215
$831$	0	0
$832$	−2.25940e7	−1.13158
$833$	−2.07898e6	−0.103809
$834$	0	0
$835$	2.36097e6	0.117186
$836$	0	0
$837$	0	0
$838$	−5.99444e6	−0.294875
$839$	2.33750e7	1.14643	0.573214	−	0.819406i	$-0.305696\pi$
$839$	2.33750e7	1.14643	0.573214	+	0.819406i	$0.305696\pi$
$840$	0	0
$841$	3.50417e7	1.70842
$842$	−1.77396e6	−0.0862308
$843$	0	0
$844$	−5.82372e6	−0.281413
$845$	−4.19785e6	−0.202248
$846$	0	0
$847$	0	0
$848$	−2.11311e7	−1.00909
$849$	0	0
$850$	−2.15201e7	−1.02164
$851$	−1.91648e7	−0.907152
$852$	0	0
$853$	3.07580e7	1.44739	0.723695	−	0.690120i	$-0.242442\pi$
$853$	3.07580e7	1.44739	0.723695	+	0.690120i	$0.242442\pi$
$854$	2.64618e7	1.24158
$855$	0	0
$856$	2.66439e7	1.24283
$857$	−1.04807e7	−0.487461	−0.243731	−	0.969843i	$-0.578371\pi$
$857$	−1.04807e7	−0.487461	−0.243731	+	0.969843i	$0.578371\pi$
$858$	0	0
$859$	−3.92265e6	−0.181383	−0.0906915	−	0.995879i	$-0.528908\pi$
$859$	−3.92265e6	−0.181383	−0.0906915	+	0.995879i	$0.528908\pi$
$860$	−761671.	−0.0351173
$861$	0	0
$862$	−4.29693e7	−1.96965
$863$	1.31389e7	0.600524	0.300262	−	0.953857i	$-0.402926\pi$
$863$	1.31389e7	0.600524	0.300262	+	0.953857i	$0.402926\pi$
$864$	0	0
$865$	3.37732e6	0.153473
$866$	−3.84935e7	−1.74419
$867$	0	0
$868$	−1.99152e6	−0.0897194
$869$	0	0
$870$	0	0
$871$	1.66261e6	0.0742585
$872$	−2.08585e6	−0.0928950
$873$	0	0
$874$	−1.33312e7	−0.590324
$875$	6.17892e6	0.272830
$876$	0	0
$877$	−5.70219e6	−0.250347	−0.125174	−	0.992135i	$-0.539949\pi$
$877$	−5.70219e6	−0.250347	−0.125174	+	0.992135i	$0.539949\pi$
$878$	654120.	0.0286366
$879$	0	0
$880$	0	0
$881$	1.01646e7	0.441213	0.220607	−	0.975363i	$-0.429196\pi$
$881$	1.01646e7	0.441213	0.220607	+	0.975363i	$0.429196\pi$
$882$	0	0
$883$	−2.47595e6	−0.106866	−0.0534331	−	0.998571i	$-0.517016\pi$
$883$	−2.47595e6	−0.106866	−0.0534331	+	0.998571i	$0.517016\pi$
$884$	7.40211e6	0.318585
$885$	0	0
$886$	4.88044e6	0.208869
$887$	3.23893e7	1.38227	0.691135	−	0.722726i	$-0.257111\pi$
$887$	3.23893e7	1.38227	0.691135	+	0.722726i	$0.257111\pi$
$888$	0	0
$889$	2.84628e7	1.20788
$890$	−2.44895e6	−0.103635
$891$	0	0
$892$	2.54549e6	0.107117
$893$	1.93907e7	0.813699
$894$	0	0
$895$	4.35002e6	0.181524
$896$	−3.02824e7	−1.26015
$897$	0	0
$898$	−3.64454e7	−1.50818
$899$	1.60718e7	0.663232
$900$	0	0
$901$	−1.99039e7	−0.816819
$902$	0	0
$903$	0	0
$904$	−3.94789e7	−1.60673
$905$	−2.34442e6	−0.0951513
$906$	0	0
$907$	−1.70320e7	−0.687461	−0.343731	−	0.939068i	$-0.611691\pi$
$907$	−1.70320e7	−0.687461	−0.343731	+	0.939068i	$0.611691\pi$
$908$	−6.92355e6	−0.278685
$909$	0	0
$910$	−6.03905e6	−0.241749
$911$	1.55492e7	0.620744	0.310372	−	0.950615i	$-0.399546\pi$
$911$	1.55492e7	0.620744	0.310372	+	0.950615i	$0.399546\pi$
$912$	0	0
$913$	0	0
$914$	5.26117e6	0.208313
$915$	0	0
$916$	−4.35110e6	−0.171341
$917$	3.98711e7	1.56580
$918$	0	0
$919$	−4.45440e7	−1.73980	−0.869902	−	0.493225i	$-0.835818\pi$
$919$	−4.45440e7	−1.73980	−0.869902	+	0.493225i	$0.835818\pi$
$920$	1.96065e6	0.0763715
$921$	0	0
$922$	4.47022e6	0.173181
$923$	3.50814e7	1.35542
$924$	0	0
$925$	−3.44429e7	−1.32356
$926$	4.10384e7	1.57276
$927$	0	0
$928$	−1.79607e7	−0.684625
$929$	2.07733e7	0.789709	0.394855	−	0.918744i	$-0.370795\pi$
$929$	2.07733e7	0.789709	0.394855	+	0.918744i	$0.370795\pi$
$930$	0	0
$931$	−2.31457e6	−0.0875177
$932$	−7.83152e6	−0.295329
$933$	0	0
$934$	−3.99494e7	−1.49845
$935$	0	0
$936$	0	0
$937$	3.94221e7	1.46687	0.733433	−	0.679762i	$-0.237917\pi$
$937$	3.94221e7	1.46687	0.733433	+	0.679762i	$0.237917\pi$
$938$	1.45337e6	0.0539347
$939$	0	0
$940$	764106.	0.0282055
$941$	1.74816e6	0.0643587	0.0321794	−	0.999482i	$-0.489755\pi$
$941$	1.74816e6	0.0643587	0.0321794	+	0.999482i	$0.489755\pi$
$942$	0	0
$943$	−5.53519e6	−0.202700
$944$	2.25086e7	0.822088
$945$	0	0
$946$	0	0
$947$	1.75885e7	0.637313	0.318656	−	0.947870i	$-0.396768\pi$
$947$	1.75885e7	0.637313	0.318656	+	0.947870i	$0.396768\pi$
$948$	0	0
$949$	−5.36513e7	−1.93381
$950$	−2.39587e7	−0.861301
$951$	0	0
$952$	−2.41496e7	−0.863609
$953$	1.92786e7	0.687612	0.343806	−	0.939041i	$-0.388284\pi$
$953$	1.92786e7	0.687612	0.343806	+	0.939041i	$0.388284\pi$
$954$	0	0
$955$	1.72597e6	0.0612384
$956$	507100.	0.0179452
$957$	0	0
$958$	5.30818e7	1.86867
$959$	1.03855e7	0.364652
$960$	0	0
$961$	−2.39795e7	−0.837590
$962$	6.79104e7	2.36591
$963$	0	0
$964$	9.09199e6	0.315113
$965$	−1.49233e6	−0.0515876
$966$	0	0
$967$	−3.74926e7	−1.28938	−0.644688	−	0.764445i	$-0.723013\pi$
$967$	−3.74926e7	−1.28938	−0.644688	+	0.764445i	$0.723013\pi$
$968$	0	0
$969$	0	0
$970$	−2.91890e6	−0.0996071
$971$	1.98769e7	0.676550	0.338275	−	0.941047i	$-0.390157\pi$
$971$	1.98769e7	0.676550	0.338275	+	0.941047i	$0.390157\pi$
$972$	0	0
$973$	4.40542e7	1.49178
$974$	1.07568e6	0.0363318
$975$	0	0
$976$	3.71765e7	1.24923
$977$	2.33856e7	0.783813	0.391907	−	0.920005i	$-0.371816\pi$
$977$	2.33856e7	0.783813	0.391907	+	0.920005i	$0.371816\pi$
$978$	0	0
$979$	0	0
$980$	−91207.6	−0.00303365
$981$	0	0
$982$	4.89305e7	1.61920
$983$	8.75560e6	0.289003	0.144501	−	0.989505i	$-0.453842\pi$
$983$	8.75560e6	0.289003	0.144501	+	0.989505i	$0.453842\pi$
$984$	0	0
$985$	−5.04754e6	−0.165764
$986$	−5.22177e7	−1.71051
$987$	0	0
$988$	8.24093e6	0.268586
$989$	−2.63699e7	−0.857271
$990$	0	0
$991$	602269.	0.0194808	0.00974040	−	0.999953i	$-0.496899\pi$
$991$	602269.	0.0194808	0.00974040	+	0.999953i	$0.496899\pi$
$992$	−5.19614e6	−0.167649
$993$	0	0
$994$	3.06662e7	0.984452
$995$	−626621.	−0.0200654
$996$	0	0
$997$	−2.83153e7	−0.902159	−0.451080	−	0.892484i	$-0.648961\pi$
$997$	−2.83153e7	−0.902159	−0.451080	+	0.892484i	$0.648961\pi$
$998$	5.41206e7	1.72003
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bo.1.6		20
3.2	odd	2	inner	1089.6.a.bo.1.15		20
11.2	odd	10		99.6.f.d.37.3	✓	40
11.6	odd	10		99.6.f.d.91.3	yes	40
11.10	odd	2		1089.6.a.bn.1.15		20
33.2	even	10		99.6.f.d.37.8	yes	40
33.17	even	10		99.6.f.d.91.8	yes	40
33.32	even	2		1089.6.a.bn.1.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.f.d.37.3	✓	40	11.2	odd	10
99.6.f.d.37.8	yes	40	33.2	even	10
99.6.f.d.91.3	yes	40	11.6	odd	10
99.6.f.d.91.8	yes	40	33.17	even	10
1089.6.a.bn.1.6		20	33.32	even	2
1089.6.a.bn.1.15		20	11.10	odd	2
1089.6.a.bo.1.6		20	1.1	even	1	trivial
1089.6.a.bo.1.15		20	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-6.22592 q^{2} +6.76211 q^{4} -7.30063 q^{5} +136.582 q^{7} +157.129 q^{8} +O(q^{10})\)	\(q-6.22592 q^{2} +6.76211 q^{4} -7.30063 q^{5} +136.582 q^{7} +157.129 q^{8} +45.4532 q^{10} -972.775 q^{13} -850.346 q^{14} -1194.66 q^{16} -1125.28 q^{17} -1252.80 q^{19} -49.3677 q^{20} -1709.16 q^{23} -3071.70 q^{25} +6056.42 q^{26} +923.579 q^{28} -7453.38 q^{29} -2156.31 q^{31} +2409.74 q^{32} +7005.91 q^{34} -997.131 q^{35} +11213.0 q^{37} +7799.83 q^{38} -1147.14 q^{40} +3238.53 q^{41} +15428.5 q^{43} +10641.1 q^{46} -15477.9 q^{47} +1847.52 q^{49} +19124.2 q^{50} -6578.01 q^{52} +17687.9 q^{53} +21460.9 q^{56} +46404.2 q^{58} -18841.0 q^{59} -31118.8 q^{61} +13425.0 q^{62} +23226.3 q^{64} +7101.87 q^{65} -1709.15 q^{67} -7609.27 q^{68} +6208.06 q^{70} -36063.3 q^{71} +55152.9 q^{73} -69811.0 q^{74} -8471.57 q^{76} +9603.01 q^{79} +8721.78 q^{80} -20162.9 q^{82} +37648.4 q^{83} +8215.26 q^{85} -96056.9 q^{86} -53878.6 q^{89} -132863. q^{91} -11557.6 q^{92} +96364.0 q^{94} +9146.22 q^{95} -64217.8 q^{97} -11502.5 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 298 q^{4} + 472 q^{7}+O(q^{10}) \) 20 * q + 298 * q^4 + 472 * q^7	\( 20 q + 298 q^{4} + 472 q^{7} + 836 q^{10} + 1816 q^{13} + 6794 q^{16} + 6478 q^{19} + 15250 q^{25} + 12264 q^{28} + 1226 q^{31} + 27548 q^{34} - 1032 q^{37} + 15224 q^{40} + 6128 q^{43} + 6710 q^{46} + 10268 q^{49} + 180082 q^{52} + 27898 q^{58} + 103962 q^{61} + 16948 q^{64} + 182882 q^{67} - 160154 q^{70} + 132276 q^{73} + 448700 q^{76} + 365012 q^{79} - 492270 q^{82} + 451616 q^{85} + 312092 q^{91} + 830808 q^{94} - 233530 q^{97}+O(q^{100}) \) 20 * q + 298 * q^4 + 472 * q^7 + 836 * q^10 + 1816 * q^13 + 6794 * q^16 + 6478 * q^19 + 15250 * q^25 + 12264 * q^28 + 1226 * q^31 + 27548 * q^34 - 1032 * q^37 + 15224 * q^40 + 6128 * q^43 + 6710 * q^46 + 10268 * q^49 + 180082 * q^52 + 27898 * q^58 + 103962 * q^61 + 16948 * q^64 + 182882 * q^67 - 160154 * q^70 + 132276 * q^73 + 448700 * q^76 + 365012 * q^79 - 492270 * q^82 + 451616 * q^85 + 312092 * q^91 + 830808 * q^94 - 233530 * q^97

Introduction

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