LMFDB - Embedded newform 1089.4.a.bl.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,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, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$64.2530799963$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 77x^{10} + 2138x^{8} - 25937x^{6} + 133491x^{4} - 221760x^{2} + 9680 $ x^12 - 77x^10 + 2138x^8 - 25937x^6 + 133491x^4 - 221760*x^2 + 9680
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.60104$ 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)

$f(q)$	$=$	$q+3.60104 q^{2} +4.96749 q^{4} +15.9301 q^{5} -31.7130 q^{7} -10.9202 q^{8} +O(q^{10})$	$q+3.60104 q^{2} +4.96749 q^{4} +15.9301 q^{5} -31.7130 q^{7} -10.9202 q^{8} +57.3650 q^{10} +41.2318 q^{13} -114.200 q^{14} -79.0640 q^{16} +21.4777 q^{17} -135.247 q^{19} +79.1326 q^{20} -4.92310 q^{23} +128.769 q^{25} +148.477 q^{26} -157.534 q^{28} -59.4852 q^{29} +40.2789 q^{31} -197.351 q^{32} +77.3419 q^{34} -505.191 q^{35} +184.296 q^{37} -487.028 q^{38} -173.960 q^{40} -368.847 q^{41} -407.910 q^{43} -17.7283 q^{46} -524.061 q^{47} +662.712 q^{49} +463.701 q^{50} +204.818 q^{52} -427.268 q^{53} +346.312 q^{56} -214.209 q^{58} -434.633 q^{59} +141.970 q^{61} +145.046 q^{62} -78.1566 q^{64} +656.827 q^{65} -5.94006 q^{67} +106.690 q^{68} -1819.21 q^{70} +132.884 q^{71} -420.357 q^{73} +663.659 q^{74} -671.836 q^{76} -366.521 q^{79} -1259.50 q^{80} -1328.23 q^{82} +773.629 q^{83} +342.142 q^{85} -1468.90 q^{86} -14.6027 q^{89} -1307.58 q^{91} -24.4554 q^{92} -1887.16 q^{94} -2154.49 q^{95} +1194.84 q^{97} +2386.45 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 58 q^{4} - 66 q^{7}+O(q^{10}) $ 12 * q + 58 * q^4 - 66 * q^7	$ 12 q + 58 q^{4} - 66 q^{7} - 104 q^{10} - 120 q^{13} + 378 q^{16} - 360 q^{19} + 526 q^{25} - 560 q^{28} + 90 q^{31} - 1260 q^{34} - 428 q^{37} - 2376 q^{40} - 2004 q^{43} - 1282 q^{46} + 1186 q^{49} - 3198 q^{52} - 914 q^{58} - 2344 q^{61} + 1684 q^{64} - 968 q^{67} - 5922 q^{70} - 322 q^{73} - 7668 q^{76} - 3482 q^{79} + 4314 q^{82} - 636 q^{85} - 380 q^{91} - 5904 q^{94} + 3258 q^{97}+O(q^{100}) $ 12 * q + 58 * q^4 - 66 * q^7 - 104 * q^10 - 120 * q^13 + 378 * q^16 - 360 * q^19 + 526 * q^25 - 560 * q^28 + 90 * q^31 - 1260 * q^34 - 428 * q^37 - 2376 * q^40 - 2004 * q^43 - 1282 * q^46 + 1186 * q^49 - 3198 * q^52 - 914 * q^58 - 2344 * q^61 + 1684 * q^64 - 968 * q^67 - 5922 * q^70 - 322 * q^73 - 7668 * q^76 - 3482 * q^79 + 4314 * q^82 - 636 * q^85 - 380 * q^91 - 5904 * q^94 + 3258 * 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 $

$2$	3.60104	1.27316	0.636580	−	0.771211i	$-0.280349\pi$
$2$	3.60104	1.27316	0.636580	+	0.771211i	$0.280349\pi$
$3$	0	0
$3$	0	0
$4$	4.96749	0.620936
$5$	15.9301	1.42483	0.712417	−	0.701757i	$-0.247601\pi$
$5$	15.9301	1.42483	0.712417	+	0.701757i	$0.247601\pi$
$6$	0	0
$7$	−31.7130	−1.71234	−0.856170	−	0.516695i	$-0.827162\pi$
$7$	−31.7130	−1.71234	−0.856170	+	0.516695i	$0.827162\pi$
$8$	−10.9202	−0.482609
$9$	0	0
$10$	57.3650	1.81404
$11$	0	0
$11$	0	0
$12$	0	0
$13$	41.2318	0.879665	0.439832	−	0.898080i	$-0.355038\pi$
$13$	41.2318	0.879665	0.439832	+	0.898080i	$0.355038\pi$
$14$	−114.200	−2.18008
$15$	0	0
$16$	−79.0640	−1.23537
$17$	21.4777	0.306418	0.153209	−	0.988194i	$-0.451039\pi$
$17$	21.4777	0.306418	0.153209	+	0.988194i	$0.451039\pi$
$18$	0	0
$19$	−135.247	−1.63304	−0.816518	−	0.577319i	$-0.804099\pi$
$19$	−135.247	−1.63304	−0.816518	+	0.577319i	$0.804099\pi$
$20$	79.1326	0.884730
$21$	0	0
$22$	0	0
$23$	−4.92310	−0.0446320	−0.0223160	−	0.999751i	$-0.507104\pi$
$23$	−4.92310	−0.0446320	−0.0223160	+	0.999751i	$0.507104\pi$
$24$	0	0
$25$	128.769	1.03015
$26$	148.477	1.11995
$27$	0	0
$28$	−157.534	−1.06325
$29$	−59.4852	−0.380901	−0.190451	−	0.981697i	$-0.560995\pi$
$29$	−59.4852	−0.380901	−0.190451	+	0.981697i	$0.560995\pi$
$30$	0	0
$31$	40.2789	0.233365	0.116682	−	0.993169i	$-0.462774\pi$
$31$	40.2789	0.233365	0.116682	+	0.993169i	$0.462774\pi$
$32$	−197.351	−1.09022
$33$	0	0
$34$	77.3419	0.390118
$35$	−505.191	−2.43980
$36$	0	0
$37$	184.296	0.818869	0.409434	−	0.912340i	$-0.365726\pi$
$37$	184.296	0.818869	0.409434	+	0.912340i	$0.365726\pi$
$38$	−487.028	−2.07912
$39$	0	0
$40$	−173.960	−0.687638
$41$	−368.847	−1.40498	−0.702491	−	0.711692i	$-0.747929\pi$
$41$	−368.847	−1.40498	−0.702491	+	0.711692i	$0.747929\pi$
$42$	0	0
$43$	−407.910	−1.44664	−0.723322	−	0.690511i	$-0.757386\pi$
$43$	−407.910	−1.44664	−0.723322	+	0.690511i	$0.757386\pi$
$44$	0	0
$45$	0	0
$46$	−17.7283	−0.0568237
$47$	−524.061	−1.62643	−0.813214	−	0.581964i	$-0.802284\pi$
$47$	−524.061	−1.62643	−0.813214	+	0.581964i	$0.802284\pi$
$48$	0	0
$49$	662.712	1.93211
$50$	463.701	1.31154
$51$	0	0
$52$	204.818	0.546215
$53$	−427.268	−1.10735	−0.553677	−	0.832732i	$-0.686776\pi$
$53$	−427.268	−1.10735	−0.553677	+	0.832732i	$0.686776\pi$
$54$	0	0
$55$	0	0
$56$	346.312	0.826391
$57$	0	0
$58$	−214.209	−0.484948
$59$	−434.633	−0.959058	−0.479529	−	0.877526i	$-0.659192\pi$
$59$	−434.633	−0.959058	−0.479529	+	0.877526i	$0.659192\pi$
$60$	0	0
$61$	141.970	0.297990	0.148995	−	0.988838i	$-0.452396\pi$
$61$	141.970	0.297990	0.148995	+	0.988838i	$0.452396\pi$
$62$	145.046	0.297110
$63$	0	0
$64$	−78.1566	−0.152650
$65$	656.827	1.25338
$66$	0	0
$67$	−5.94006	−0.0108312	−0.00541562	−	0.999985i	$-0.501724\pi$
$67$	−5.94006	−0.0108312	−0.00541562	+	0.999985i	$0.501724\pi$
$68$	106.690	0.190266
$69$	0	0
$70$	−1819.21	−3.10625
$71$	132.884	0.222118	0.111059	−	0.993814i	$-0.464576\pi$
$71$	132.884	0.222118	0.111059	+	0.993814i	$0.464576\pi$
$72$	0	0
$73$	−420.357	−0.673960	−0.336980	−	0.941512i	$-0.609405\pi$
$73$	−420.357	−0.673960	−0.336980	+	0.941512i	$0.609405\pi$
$74$	663.659	1.04255
$75$	0	0
$76$	−671.836	−1.01401
$77$	0	0
$78$	0	0
$79$	−366.521	−0.521985	−0.260993	−	0.965341i	$-0.584050\pi$
$79$	−366.521	−0.521985	−0.260993	+	0.965341i	$0.584050\pi$
$80$	−1259.50	−1.76020
$81$	0	0
$82$	−1328.23	−1.78877
$83$	773.629	1.02309	0.511547	−	0.859255i	$-0.329073\pi$
$83$	773.629	1.02309	0.511547	+	0.859255i	$0.329073\pi$
$84$	0	0
$85$	342.142	0.436594
$86$	−1468.90	−1.84181
$87$	0	0
$88$	0	0
$89$	−14.6027	−0.0173919	−0.00869595	−	0.999962i	$-0.502768\pi$
$89$	−14.6027	−0.0173919	−0.00869595	+	0.999962i	$0.502768\pi$
$90$	0	0
$91$	−1307.58	−1.50628
$92$	−24.4554	−0.0277136
$93$	0	0
$94$	−1887.16	−2.07070
$95$	−2154.49	−2.32680
$96$	0	0
$97$	1194.84	1.25070	0.625351	−	0.780344i	$-0.284956\pi$
$97$	1194.84	1.25070	0.625351	+	0.780344i	$0.284956\pi$
$98$	2386.45	2.45988
$99$	0	0
$100$	639.656	0.639656
$101$	−1422.97	−1.40189	−0.700945	−	0.713215i	$-0.747238\pi$
$101$	−1422.97	−1.40189	−0.700945	+	0.713215i	$0.747238\pi$
$102$	0	0
$103$	61.1024	0.0584524	0.0292262	−	0.999573i	$-0.490696\pi$
$103$	61.1024	0.0584524	0.0292262	+	0.999573i	$0.490696\pi$
$104$	−450.259	−0.424534
$105$	0	0
$106$	−1538.61	−1.40984
$107$	890.989	0.805001	0.402501	−	0.915420i	$-0.368141\pi$
$107$	890.989	0.805001	0.402501	+	0.915420i	$0.368141\pi$
$108$	0	0
$109$	363.454	0.319382	0.159691	−	0.987167i	$-0.448950\pi$
$109$	363.454	0.319382	0.159691	+	0.987167i	$0.448950\pi$
$110$	0	0
$111$	0	0
$112$	2507.35	2.11538
$113$	1025.57	0.853786	0.426893	−	0.904302i	$-0.359608\pi$
$113$	1025.57	0.853786	0.426893	+	0.904302i	$0.359608\pi$
$114$	0	0
$115$	−78.4255	−0.0635932
$116$	−295.492	−0.236515
$117$	0	0
$118$	−1565.13	−1.22103
$119$	−681.120	−0.524691
$120$	0	0
$121$	0	0
$122$	511.240	0.379389
$123$	0	0
$124$	200.085	0.144904
$125$	60.0348	0.0429574
$126$	0	0
$127$	−2389.94	−1.66986	−0.834932	−	0.550353i	$-0.814493\pi$
$127$	−2389.94	−1.66986	−0.834932	+	0.550353i	$0.814493\pi$
$128$	1297.36	0.895873
$129$	0	0
$130$	2365.26	1.59575
$131$	−878.775	−0.586099	−0.293050	−	0.956097i	$-0.594670\pi$
$131$	−878.775	−0.586099	−0.293050	+	0.956097i	$0.594670\pi$
$132$	0	0
$133$	4289.07	2.79631
$134$	−21.3904	−0.0137899
$135$	0	0
$136$	−234.540	−0.147880
$137$	−984.886	−0.614193	−0.307097	−	0.951678i	$-0.599357\pi$
$137$	−984.886	−0.614193	−0.307097	+	0.951678i	$0.599357\pi$
$138$	0	0
$139$	1822.41	1.11205	0.556024	−	0.831166i	$-0.312326\pi$
$139$	1822.41	1.11205	0.556024	+	0.831166i	$0.312326\pi$
$140$	−2509.53	−1.51496
$141$	0	0
$142$	478.520	0.282792
$143$	0	0
$144$	0	0
$145$	−947.607	−0.542720
$146$	−1513.72	−0.858058
$147$	0	0
$148$	915.490	0.508465
$149$	3402.12	1.87055	0.935276	−	0.353919i	$-0.115151\pi$
$149$	3402.12	1.87055	0.935276	+	0.353919i	$0.115151\pi$
$150$	0	0
$151$	1059.37	0.570929	0.285464	−	0.958389i	$-0.407852\pi$
$151$	1059.37	0.570929	0.285464	+	0.958389i	$0.407852\pi$
$152$	1476.92	0.788119
$153$	0	0
$154$	0	0
$155$	641.648	0.332506
$156$	0	0
$157$	1516.60	0.770943	0.385471	−	0.922720i	$-0.374039\pi$
$157$	1516.60	0.770943	0.385471	+	0.922720i	$0.374039\pi$
$158$	−1319.86	−0.664571
$159$	0	0
$160$	−3143.82	−1.55338
$161$	156.126	0.0764252
$162$	0	0
$163$	−2395.86	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$163$	−2395.86	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$164$	−1832.24	−0.872404
$165$	0	0
$166$	2785.87	1.30256
$167$	2635.11	1.22102	0.610512	−	0.792007i	$-0.290964\pi$
$167$	2635.11	1.22102	0.610512	+	0.792007i	$0.290964\pi$
$168$	0	0
$169$	−496.940	−0.226190
$170$	1232.07	0.555854
$171$	0	0
$172$	−2026.29	−0.898273
$173$	433.778	0.190633	0.0953166	−	0.995447i	$-0.469614\pi$
$173$	433.778	0.190633	0.0953166	+	0.995447i	$0.469614\pi$
$174$	0	0
$175$	−4083.64	−1.76396
$176$	0	0
$177$	0	0
$178$	−52.5848	−0.0221427
$179$	1620.46	0.676643	0.338321	−	0.941031i	$-0.390141\pi$
$179$	1620.46	0.676643	0.338321	+	0.941031i	$0.390141\pi$
$180$	0	0
$181$	−1774.78	−0.728829	−0.364415	−	0.931237i	$-0.618731\pi$
$181$	−1774.78	−0.728829	−0.364415	+	0.931237i	$0.618731\pi$
$182$	−4708.66	−1.91774
$183$	0	0
$184$	53.7612	0.0215398
$185$	2935.86	1.16675
$186$	0	0
$187$	0	0
$188$	−2603.27	−1.00991
$189$	0	0
$190$	−7758.42	−2.96239
$191$	1987.39	0.752894	0.376447	−	0.926438i	$-0.377146\pi$
$191$	1987.39	0.752894	0.376447	+	0.926438i	$0.377146\pi$
$192$	0	0
$193$	−2192.87	−0.817855	−0.408927	−	0.912567i	$-0.634097\pi$
$193$	−2192.87	−0.817855	−0.408927	+	0.912567i	$0.634097\pi$
$194$	4302.68	1.59234
$195$	0	0
$196$	3292.02	1.19971
$197$	−4825.72	−1.74527	−0.872636	−	0.488372i	$-0.837591\pi$
$197$	−4825.72	−1.74527	−0.872636	+	0.488372i	$0.837591\pi$
$198$	0	0
$199$	−3488.25	−1.24259	−0.621296	−	0.783576i	$-0.713393\pi$
$199$	−3488.25	−1.24259	−0.621296	+	0.783576i	$0.713393\pi$
$200$	−1406.18	−0.497159
$201$	0	0
$202$	−5124.17	−1.78483
$203$	1886.45	0.652232
$204$	0	0
$205$	−5875.78	−2.00187
$206$	220.032	0.0744193
$207$	0	0
$208$	−3259.95	−1.08672
$209$	0	0
$210$	0	0
$211$	1525.30	0.497657	0.248829	−	0.968548i	$-0.419954\pi$
$211$	1525.30	0.497657	0.248829	+	0.968548i	$0.419954\pi$
$212$	−2122.45	−0.687595
$213$	0	0
$214$	3208.48	1.02489
$215$	−6498.05	−2.06123
$216$	0	0
$217$	−1277.36	−0.399599
$218$	1308.81	0.406624
$219$	0	0
$220$	0	0
$221$	885.562	0.269545
$222$	0	0
$223$	2755.17	0.827352	0.413676	−	0.910424i	$-0.364245\pi$
$223$	2755.17	0.827352	0.413676	+	0.910424i	$0.364245\pi$
$224$	6258.58	1.86683
$225$	0	0
$226$	3693.13	1.08701
$227$	−4711.54	−1.37760	−0.688802	−	0.724949i	$-0.741863\pi$
$227$	−4711.54	−1.37760	−0.688802	+	0.724949i	$0.741863\pi$
$228$	0	0
$229$	4066.26	1.17339	0.586695	−	0.809808i	$-0.300429\pi$
$229$	4066.26	1.17339	0.586695	+	0.809808i	$0.300429\pi$
$230$	−282.413	−0.0809643
$231$	0	0
$232$	649.591	0.183826
$233$	−2784.09	−0.782797	−0.391399	−	0.920221i	$-0.628009\pi$
$233$	−2784.09	−0.782797	−0.391399	+	0.920221i	$0.628009\pi$
$234$	0	0
$235$	−8348.35	−2.31739
$236$	−2159.03	−0.595513
$237$	0	0
$238$	−2452.74	−0.668015
$239$	1554.86	0.420817	0.210409	−	0.977614i	$-0.432521\pi$
$239$	1554.86	0.420817	0.210409	+	0.977614i	$0.432521\pi$
$240$	0	0
$241$	924.309	0.247054	0.123527	−	0.992341i	$-0.460579\pi$
$241$	924.309	0.247054	0.123527	+	0.992341i	$0.460579\pi$
$242$	0	0
$243$	0	0
$244$	705.235	0.185033
$245$	10557.1	2.75293
$246$	0	0
$247$	−5576.46	−1.43652
$248$	−439.854	−0.112624
$249$	0	0
$250$	216.188	0.0546916
$251$	3410.93	0.857754	0.428877	−	0.903363i	$-0.358909\pi$
$251$	3410.93	0.857754	0.428877	+	0.903363i	$0.358909\pi$
$252$	0	0
$253$	0	0
$254$	−8606.27	−2.12600
$255$	0	0
$256$	5297.11	1.29324
$257$	−5845.43	−1.41879	−0.709393	−	0.704813i	$-0.751031\pi$
$257$	−5845.43	−1.41879	−0.709393	+	0.704813i	$0.751031\pi$
$258$	0	0
$259$	−5844.59	−1.40218
$260$	3262.78	0.778266
$261$	0	0
$262$	−3164.51	−0.746198
$263$	−426.069	−0.0998955	−0.0499477	−	0.998752i	$-0.515905\pi$
$263$	−426.069	−0.0998955	−0.0499477	+	0.998752i	$0.515905\pi$
$264$	0	0
$265$	−6806.42	−1.57779
$266$	15445.1	3.56015
$267$	0	0
$268$	−29.5071	−0.00672551
$269$	6563.67	1.48771	0.743855	−	0.668341i	$-0.232995\pi$
$269$	6563.67	1.48771	0.743855	+	0.668341i	$0.232995\pi$
$270$	0	0
$271$	−2980.25	−0.668034	−0.334017	−	0.942567i	$-0.608404\pi$
$271$	−2980.25	−0.668034	−0.334017	+	0.942567i	$0.608404\pi$
$272$	−1698.11	−0.378540
$273$	0	0
$274$	−3546.61	−0.781966
$275$	0	0
$276$	0	0
$277$	−431.890	−0.0936814	−0.0468407	−	0.998902i	$-0.514915\pi$
$277$	−431.890	−0.0936814	−0.0468407	+	0.998902i	$0.514915\pi$
$278$	6562.57	1.41582
$279$	0	0
$280$	5516.79	1.17747
$281$	2831.10	0.601030	0.300515	−	0.953777i	$-0.402841\pi$
$281$	2831.10	0.601030	0.300515	+	0.953777i	$0.402841\pi$
$282$	0	0
$283$	2058.70	0.432428	0.216214	−	0.976346i	$-0.430629\pi$
$283$	2058.70	0.432428	0.216214	+	0.976346i	$0.430629\pi$
$284$	660.099	0.137921
$285$	0	0
$286$	0	0
$287$	11697.2	2.40581
$288$	0	0
$289$	−4451.71	−0.906108
$290$	−3412.37	−0.690970
$291$	0	0
$292$	−2088.12	−0.418486
$293$	7638.46	1.52302	0.761508	−	0.648156i	$-0.224460\pi$
$293$	7638.46	1.52302	0.761508	+	0.648156i	$0.224460\pi$
$294$	0	0
$295$	−6923.75	−1.36650
$296$	−2012.55	−0.395194
$297$	0	0
$298$	12251.2	2.38151
$299$	−202.988	−0.0392612
$300$	0	0
$301$	12936.0	2.47715
$302$	3814.83	0.726884
$303$	0	0
$304$	10693.1	2.01741
$305$	2261.60	0.424587
$306$	0	0
$307$	4284.72	0.796553	0.398277	−	0.917265i	$-0.369608\pi$
$307$	4284.72	0.796553	0.398277	+	0.917265i	$0.369608\pi$
$308$	0	0
$309$	0	0
$310$	2310.60	0.423333
$311$	−743.786	−0.135615	−0.0678074	−	0.997698i	$-0.521600\pi$
$311$	−743.786	−0.135615	−0.0678074	+	0.997698i	$0.521600\pi$
$312$	0	0
$313$	1182.19	0.213487	0.106743	−	0.994287i	$-0.465958\pi$
$313$	1182.19	0.213487	0.106743	+	0.994287i	$0.465958\pi$
$314$	5461.34	0.981533
$315$	0	0
$316$	−1820.69	−0.324119
$317$	1022.21	0.181114	0.0905571	−	0.995891i	$-0.471135\pi$
$317$	1022.21	0.181114	0.0905571	+	0.995891i	$0.471135\pi$
$318$	0	0
$319$	0	0
$320$	−1245.04	−0.217500
$321$	0	0
$322$	562.216	0.0973014
$323$	−2904.78	−0.500391
$324$	0	0
$325$	5309.36	0.906186
$326$	−8627.60	−1.46576
$327$	0	0
$328$	4027.89	0.678058
$329$	16619.5	2.78500
$330$	0	0
$331$	4137.81	0.687113	0.343557	−	0.939132i	$-0.388368\pi$
$331$	4137.81	0.687113	0.343557	+	0.939132i	$0.388368\pi$
$332$	3842.99	0.635276
$333$	0	0
$334$	9489.14	1.55456
$335$	−94.6258	−0.0154327
$336$	0	0
$337$	−2023.80	−0.327132	−0.163566	−	0.986532i	$-0.552300\pi$
$337$	−2023.80	−0.327132	−0.163566	+	0.986532i	$0.552300\pi$
$338$	−1789.50	−0.287976
$339$	0	0
$340$	1699.58	0.271097
$341$	0	0
$342$	0	0
$343$	−10139.0	−1.59608
$344$	4454.46	0.698164
$345$	0	0
$346$	1562.05	0.242706
$347$	−7311.66	−1.13115	−0.565577	−	0.824695i	$-0.691347\pi$
$347$	−7311.66	−1.13115	−0.565577	+	0.824695i	$0.691347\pi$
$348$	0	0
$349$	−10218.9	−1.56735	−0.783673	−	0.621173i	$-0.786656\pi$
$349$	−10218.9	−1.56735	−0.783673	+	0.621173i	$0.786656\pi$
$350$	−14705.3	−2.24581
$351$	0	0
$352$	0	0
$353$	−4008.76	−0.604433	−0.302216	−	0.953239i	$-0.597726\pi$
$353$	−4008.76	−0.604433	−0.302216	+	0.953239i	$0.597726\pi$
$354$	0	0
$355$	2116.86	0.316482
$356$	−72.5385	−0.0107993
$357$	0	0
$358$	5835.35	0.861474
$359$	−798.351	−0.117369	−0.0586843	−	0.998277i	$-0.518691\pi$
$359$	−798.351	−0.117369	−0.0586843	+	0.998277i	$0.518691\pi$
$360$	0	0
$361$	11432.6	1.66681
$362$	−6391.04	−0.927916
$363$	0	0
$364$	−6495.40	−0.935306
$365$	−6696.33	−0.960280
$366$	0	0
$367$	2504.05	0.356159	0.178080	−	0.984016i	$-0.443012\pi$
$367$	2504.05	0.356159	0.178080	+	0.984016i	$0.443012\pi$
$368$	389.239	0.0551373
$369$	0	0
$370$	10572.2	1.48546
$371$	13549.9	1.89616
$372$	0	0
$373$	9807.89	1.36148	0.680742	−	0.732524i	$-0.261658\pi$
$373$	9807.89	1.36148	0.680742	+	0.732524i	$0.261658\pi$
$374$	0	0
$375$	0	0
$376$	5722.85	0.784929
$377$	−2452.68	−0.335065
$378$	0	0
$379$	−10895.6	−1.47670	−0.738348	−	0.674420i	$-0.764394\pi$
$379$	−10895.6	−1.47670	−0.738348	+	0.674420i	$0.764394\pi$
$380$	−10702.4	−1.44480
$381$	0	0
$382$	7156.68	0.958554
$383$	9693.76	1.29328	0.646642	−	0.762793i	$-0.276173\pi$
$383$	9693.76	1.29328	0.646642	+	0.762793i	$0.276173\pi$
$384$	0	0
$385$	0	0
$386$	−7896.60	−1.04126
$387$	0	0
$388$	5935.38	0.776606
$389$	12288.6	1.60169	0.800843	−	0.598874i	$-0.204385\pi$
$389$	12288.6	1.60169	0.800843	+	0.598874i	$0.204385\pi$
$390$	0	0
$391$	−105.737	−0.0136760
$392$	−7236.95	−0.932452
$393$	0	0
$394$	−17377.6	−2.22201
$395$	−5838.72	−0.743742
$396$	0	0
$397$	−2988.47	−0.377801	−0.188901	−	0.981996i	$-0.560492\pi$
$397$	−2988.47	−0.377801	−0.188901	+	0.981996i	$0.560492\pi$
$398$	−12561.3	−1.58202
$399$	0	0
$400$	−10181.0	−1.27262
$401$	−13527.7	−1.68465	−0.842323	−	0.538973i	$-0.818813\pi$
$401$	−13527.7	−1.68465	−0.842323	+	0.538973i	$0.818813\pi$
$402$	0	0
$403$	1660.77	0.205283
$404$	−7068.59	−0.870484
$405$	0	0
$406$	6793.19	0.830396
$407$	0	0
$408$	0	0
$409$	4030.65	0.487293	0.243647	−	0.969864i	$-0.421656\pi$
$409$	4030.65	0.487293	0.243647	+	0.969864i	$0.421656\pi$
$410$	−21158.9	−2.54870
$411$	0	0
$412$	303.525	0.0362952
$413$	13783.5	1.64223
$414$	0	0
$415$	12324.0	1.45774
$416$	−8137.13	−0.959028
$417$	0	0
$418$	0	0
$419$	8327.13	0.970900	0.485450	−	0.874264i	$-0.338656\pi$
$419$	8327.13	0.970900	0.485450	+	0.874264i	$0.338656\pi$
$420$	0	0
$421$	635.158	0.0735290	0.0367645	−	0.999324i	$-0.488295\pi$
$421$	635.158	0.0735290	0.0367645	+	0.999324i	$0.488295\pi$
$422$	5492.65	0.633597
$423$	0	0
$424$	4665.85	0.534419
$425$	2765.65	0.315656
$426$	0	0
$427$	−4502.29	−0.510261
$428$	4425.97	0.499854
$429$	0	0
$430$	−23399.7	−2.62427
$431$	−9689.22	−1.08286	−0.541431	−	0.840745i	$-0.682117\pi$
$431$	−9689.22	−1.08286	−0.541431	+	0.840745i	$0.682117\pi$
$432$	0	0
$433$	3904.36	0.433330	0.216665	−	0.976246i	$-0.430482\pi$
$433$	3904.36	0.433330	0.216665	+	0.976246i	$0.430482\pi$
$434$	−4599.84	−0.508754
$435$	0	0
$436$	1805.45	0.198316
$437$	665.832	0.0728857
$438$	0	0
$439$	−1221.08	−0.132754	−0.0663771	−	0.997795i	$-0.521144\pi$
$439$	−1221.08	−0.132754	−0.0663771	+	0.997795i	$0.521144\pi$
$440$	0	0
$441$	0	0
$442$	3188.95	0.343173
$443$	−15261.1	−1.63674	−0.818370	−	0.574691i	$-0.805122\pi$
$443$	−15261.1	−1.63674	−0.818370	+	0.574691i	$0.805122\pi$
$444$	0	0
$445$	−232.622	−0.0247805
$446$	9921.46	1.05335
$447$	0	0
$448$	2478.58	0.261388
$449$	−801.923	−0.0842875	−0.0421438	−	0.999112i	$-0.513419\pi$
$449$	−801.923	−0.0842875	−0.0421438	+	0.999112i	$0.513419\pi$
$450$	0	0
$451$	0	0
$452$	5094.52	0.530146
$453$	0	0
$454$	−16966.5	−1.75391
$455$	−20829.9	−2.14620
$456$	0	0
$457$	12383.2	1.26753	0.633766	−	0.773525i	$-0.281508\pi$
$457$	12383.2	1.26753	0.633766	+	0.773525i	$0.281508\pi$
$458$	14642.8	1.49391
$459$	0	0
$460$	−389.578	−0.0394873
$461$	4583.73	0.463092	0.231546	−	0.972824i	$-0.425622\pi$
$461$	4583.73	0.463092	0.231546	+	0.972824i	$0.425622\pi$
$462$	0	0
$463$	−144.386	−0.0144928	−0.00724642	−	0.999974i	$-0.502307\pi$
$463$	−144.386	−0.0144928	−0.00724642	+	0.999974i	$0.502307\pi$
$464$	4703.14	0.470556
$465$	0	0
$466$	−10025.6	−0.996626
$467$	−3444.78	−0.341339	−0.170670	−	0.985328i	$-0.554593\pi$
$467$	−3444.78	−0.341339	−0.170670	+	0.985328i	$0.554593\pi$
$468$	0	0
$469$	188.377	0.0185468
$470$	−30062.7	−2.95041
$471$	0	0
$472$	4746.28	0.462850
$473$	0	0
$474$	0	0
$475$	−17415.5	−1.68227
$476$	−3383.46	−0.325799
$477$	0	0
$478$	5599.10	0.535768
$479$	13373.0	1.27563	0.637815	−	0.770190i	$-0.279838\pi$
$479$	13373.0	1.27563	0.637815	+	0.770190i	$0.279838\pi$
$480$	0	0
$481$	7598.87	0.720330
$482$	3328.48	0.314539
$483$	0	0
$484$	0	0
$485$	19034.0	1.78204
$486$	0	0
$487$	17998.9	1.67476	0.837379	−	0.546623i	$-0.184087\pi$
$487$	17998.9	1.67476	0.837379	+	0.546623i	$0.184087\pi$
$488$	−1550.34	−0.143813
$489$	0	0
$490$	38016.5	3.50492
$491$	−13308.5	−1.22323	−0.611614	−	0.791156i	$-0.709479\pi$
$491$	−13308.5	−1.22323	−0.611614	+	0.791156i	$0.709479\pi$
$492$	0	0
$493$	−1277.60	−0.116715
$494$	−20081.1	−1.82893
$495$	0	0
$496$	−3184.61	−0.288293
$497$	−4214.14	−0.380342
$498$	0	0
$499$	20341.2	1.82484	0.912421	−	0.409254i	$-0.134211\pi$
$499$	20341.2	1.82484	0.912421	+	0.409254i	$0.134211\pi$
$500$	298.222	0.0266738
$501$	0	0
$502$	12282.9	1.09206
$503$	2052.86	0.181973	0.0909866	−	0.995852i	$-0.470998\pi$
$503$	2052.86	0.181973	0.0909866	+	0.995852i	$0.470998\pi$
$504$	0	0
$505$	−22668.1	−1.99746
$506$	0	0
$507$	0	0
$508$	−11872.0	−1.03688
$509$	−7928.34	−0.690408	−0.345204	−	0.938528i	$-0.612190\pi$
$509$	−7928.34	−0.690408	−0.345204	+	0.938528i	$0.612190\pi$
$510$	0	0
$511$	13330.8	1.15405
$512$	8696.19	0.750627
$513$	0	0
$514$	−21049.6	−1.80634
$515$	973.369	0.0832849
$516$	0	0
$517$	0	0
$518$	−21046.6	−1.78520
$519$	0	0
$520$	−7172.69	−0.604890
$521$	17091.1	1.43718	0.718592	−	0.695432i	$-0.244787\pi$
$521$	17091.1	1.43718	0.718592	+	0.695432i	$0.244787\pi$
$522$	0	0
$523$	−10831.2	−0.905571	−0.452786	−	0.891619i	$-0.649570\pi$
$523$	−10831.2	−0.905571	−0.452786	+	0.891619i	$0.649570\pi$
$524$	−4365.31	−0.363930
$525$	0	0
$526$	−1534.29	−0.127183
$527$	865.097	0.0715070
$528$	0	0
$529$	−12142.8	−0.998008
$530$	−24510.2	−2.00878
$531$	0	0
$532$	21305.9	1.73633
$533$	−15208.2	−1.23591
$534$	0	0
$535$	14193.6	1.14699
$536$	64.8666	0.00522726
$537$	0	0
$538$	23636.0	1.89409
$539$	0	0
$540$	0	0
$541$	−20819.9	−1.65456	−0.827282	−	0.561787i	$-0.810114\pi$
$541$	−20819.9	−1.65456	−0.827282	+	0.561787i	$0.810114\pi$
$542$	−10732.0	−0.850514
$543$	0	0
$544$	−4238.64	−0.334062
$545$	5789.87	0.455066
$546$	0	0
$547$	4309.33	0.336844	0.168422	−	0.985715i	$-0.446133\pi$
$547$	4309.33	0.336844	0.168422	+	0.985715i	$0.446133\pi$
$548$	−4892.41	−0.381375
$549$	0	0
$550$	0	0
$551$	8045.18	0.622026
$552$	0	0
$553$	11623.5	0.893816
$554$	−1555.25	−0.119271
$555$	0	0
$556$	9052.80	0.690511
$557$	1871.85	0.142393	0.0711963	−	0.997462i	$-0.477318\pi$
$557$	1871.85	0.142393	0.0711963	+	0.997462i	$0.477318\pi$
$558$	0	0
$559$	−16818.9	−1.27256
$560$	39942.4	3.01406
$561$	0	0
$562$	10194.9	0.765207
$563$	−1073.87	−0.0803876	−0.0401938	−	0.999192i	$-0.512798\pi$
$563$	−1073.87	−0.0803876	−0.0401938	+	0.999192i	$0.512798\pi$
$564$	0	0
$565$	16337.5	1.21650
$566$	7413.47	0.550550
$567$	0	0
$568$	−1451.12	−0.107196
$569$	13847.6	1.02025	0.510126	−	0.860100i	$-0.329599\pi$
$569$	13847.6	1.02025	0.510126	+	0.860100i	$0.329599\pi$
$570$	0	0
$571$	781.386	0.0572680	0.0286340	−	0.999590i	$-0.490884\pi$
$571$	781.386	0.0572680	0.0286340	+	0.999590i	$0.490884\pi$
$572$	0	0
$573$	0	0
$574$	42122.3	3.06298
$575$	−633.940	−0.0459776
$576$	0	0
$577$	8759.01	0.631963	0.315981	−	0.948765i	$-0.397666\pi$
$577$	8759.01	0.631963	0.315981	+	0.948765i	$0.397666\pi$
$578$	−16030.8	−1.15362
$579$	0	0
$580$	−4707.22	−0.336995
$581$	−24534.1	−1.75188
$582$	0	0
$583$	0	0
$584$	4590.38	0.325259
$585$	0	0
$586$	27506.4	1.93904
$587$	961.648	0.0676175	0.0338087	−	0.999428i	$-0.489236\pi$
$587$	961.648	0.0676175	0.0338087	+	0.999428i	$0.489236\pi$
$588$	0	0
$589$	−5447.58	−0.381093
$590$	−24932.7	−1.73977
$591$	0	0
$592$	−14571.2	−1.01161
$593$	−1880.24	−0.130206	−0.0651030	−	0.997879i	$-0.520738\pi$
$593$	−1880.24	−0.130206	−0.0651030	+	0.997879i	$0.520738\pi$
$594$	0	0
$595$	−10850.3	−0.747597
$596$	16900.0	1.16149
$597$	0	0
$598$	−730.968	−0.0499858
$599$	−5062.67	−0.345334	−0.172667	−	0.984980i	$-0.555238\pi$
$599$	−5062.67	−0.345334	−0.172667	+	0.984980i	$0.555238\pi$
$600$	0	0
$601$	−14187.5	−0.962929	−0.481465	−	0.876466i	$-0.659895\pi$
$601$	−14187.5	−0.962929	−0.481465	+	0.876466i	$0.659895\pi$
$602$	46583.2	3.15380
$603$	0	0
$604$	5262.40	0.354510
$605$	0	0
$606$	0	0
$607$	−8470.75	−0.566420	−0.283210	−	0.959058i	$-0.591399\pi$
$607$	−8470.75	−0.566420	−0.283210	+	0.959058i	$0.591399\pi$
$608$	26691.0	1.78037
$609$	0	0
$610$	8144.11	0.540566
$611$	−21608.0	−1.43071
$612$	0	0
$613$	11660.0	0.768261	0.384130	−	0.923279i	$-0.374501\pi$
$613$	11660.0	0.768261	0.384130	+	0.923279i	$0.374501\pi$
$614$	15429.4	1.01414
$615$	0	0
$616$	0	0
$617$	−29514.0	−1.92575	−0.962877	−	0.269941i	$-0.912996\pi$
$617$	−29514.0	−1.92575	−0.962877	+	0.269941i	$0.912996\pi$
$618$	0	0
$619$	14933.4	0.969669	0.484834	−	0.874606i	$-0.338880\pi$
$619$	14933.4	0.969669	0.484834	+	0.874606i	$0.338880\pi$
$620$	3187.38	0.206465
$621$	0	0
$622$	−2678.40	−0.172659
$623$	463.094	0.0297808
$624$	0	0
$625$	−15139.7	−0.968942
$626$	4257.11	0.271802
$627$	0	0
$628$	7533.70	0.478706
$629$	3958.26	0.250916
$630$	0	0
$631$	5684.96	0.358660	0.179330	−	0.983789i	$-0.442607\pi$
$631$	5684.96	0.358660	0.179330	+	0.983789i	$0.442607\pi$
$632$	4002.48	0.251915
$633$	0	0
$634$	3681.03	0.230587
$635$	−38072.0	−2.37928
$636$	0	0
$637$	27324.8	1.69961
$638$	0	0
$639$	0	0
$640$	20667.1	1.27647
$641$	9160.03	0.564429	0.282215	−	0.959351i	$-0.408931\pi$
$641$	9160.03	0.564429	0.282215	+	0.959351i	$0.408931\pi$
$642$	0	0
$643$	−29979.9	−1.83871	−0.919355	−	0.393430i	$-0.871288\pi$
$643$	−29979.9	−1.83871	−0.919355	+	0.393430i	$0.871288\pi$
$644$	775.554	0.0474551
$645$	0	0
$646$	−10460.2	−0.637078
$647$	−13103.7	−0.796230	−0.398115	−	0.917336i	$-0.630335\pi$
$647$	−13103.7	−0.796230	−0.398115	+	0.917336i	$0.630335\pi$
$648$	0	0
$649$	0	0
$650$	19119.2	1.15372
$651$	0	0
$652$	−11901.4	−0.714870
$653$	−11747.5	−0.704005	−0.352003	−	0.935999i	$-0.614499\pi$
$653$	−11747.5	−0.704005	−0.352003	+	0.935999i	$0.614499\pi$
$654$	0	0
$655$	−13999.0	−0.835093
$656$	29162.5	1.73568
$657$	0	0
$658$	59847.6	3.54575
$659$	−1585.28	−0.0937081	−0.0468541	−	0.998902i	$-0.514920\pi$
$659$	−1585.28	−0.0937081	−0.0468541	+	0.998902i	$0.514920\pi$
$660$	0	0
$661$	−9070.54	−0.533741	−0.266871	−	0.963732i	$-0.585990\pi$
$661$	−9070.54	−0.533741	−0.266871	+	0.963732i	$0.585990\pi$
$662$	14900.4	0.874805
$663$	0	0
$664$	−8448.18	−0.493755
$665$	68325.4	3.98428
$666$	0	0
$667$	292.852	0.0170004
$668$	13089.9	0.758177
$669$	0	0
$670$	−340.751	−0.0196483
$671$	0	0
$672$	0	0
$673$	−15746.6	−0.901913	−0.450957	−	0.892546i	$-0.648917\pi$
$673$	−15746.6	−0.901913	−0.450957	+	0.892546i	$0.648917\pi$
$674$	−7287.78	−0.416491
$675$	0	0
$676$	−2468.54	−0.140450
$677$	18047.8	1.02457	0.512285	−	0.858815i	$-0.328799\pi$
$677$	18047.8	1.02457	0.512285	+	0.858815i	$0.328799\pi$
$678$	0	0
$679$	−37892.1	−2.14163
$680$	−3736.26	−0.210704
$681$	0	0
$682$	0	0
$683$	10422.1	0.583881	0.291941	−	0.956436i	$-0.405699\pi$
$683$	10422.1	0.583881	0.291941	+	0.956436i	$0.405699\pi$
$684$	0	0
$685$	−15689.3	−0.875123
$686$	−36511.0	−2.03207
$687$	0	0
$688$	32251.0	1.78715
$689$	−17617.0	−0.974100
$690$	0	0
$691$	−8031.22	−0.442145	−0.221072	−	0.975257i	$-0.570956\pi$
$691$	−8031.22	−0.442145	−0.221072	+	0.975257i	$0.570956\pi$
$692$	2154.79	0.118371
$693$	0	0
$694$	−26329.6	−1.44014
$695$	29031.2	1.58448
$696$	0	0
$697$	−7921.98	−0.430511
$698$	−36798.6	−1.99548
$699$	0	0
$700$	−20285.4	−1.09531
$701$	−5193.62	−0.279829	−0.139915	−	0.990164i	$-0.544683\pi$
$701$	−5193.62	−0.279829	−0.139915	+	0.990164i	$0.544683\pi$
$702$	0	0
$703$	−24925.5	−1.33724
$704$	0	0
$705$	0	0
$706$	−14435.7	−0.769539
$707$	45126.6	2.40051
$708$	0	0
$709$	−32288.8	−1.71034	−0.855171	−	0.518347i	$-0.826548\pi$
$709$	−32288.8	−1.71034	−0.855171	+	0.518347i	$0.826548\pi$
$710$	7622.88	0.402932
$711$	0	0
$712$	159.464	0.00839349
$713$	−198.297	−0.0104155
$714$	0	0
$715$	0	0
$716$	8049.62	0.420152
$717$	0	0
$718$	−2874.89	−0.149429
$719$	−8844.76	−0.458768	−0.229384	−	0.973336i	$-0.573671\pi$
$719$	−8844.76	−0.458768	−0.229384	+	0.973336i	$0.573671\pi$
$720$	0	0
$721$	−1937.74	−0.100090
$722$	41169.4	2.12211
$723$	0	0
$724$	−8816.18	−0.452556
$725$	−7659.83	−0.392385
$726$	0	0
$727$	17537.4	0.894670	0.447335	−	0.894367i	$-0.352373\pi$
$727$	17537.4	0.894670	0.447335	+	0.894367i	$0.352373\pi$
$728$	14279.1	0.726947
$729$	0	0
$730$	−24113.8	−1.22259
$731$	−8760.95	−0.443277
$732$	0	0
$733$	882.175	0.0444527	0.0222264	−	0.999753i	$-0.492925\pi$
$733$	882.175	0.0444527	0.0222264	+	0.999753i	$0.492925\pi$
$734$	9017.19	0.453447
$735$	0	0
$736$	971.577	0.0486587
$737$	0	0
$738$	0	0
$739$	1441.71	0.0717650	0.0358825	−	0.999356i	$-0.488576\pi$
$739$	1441.71	0.0717650	0.0358825	+	0.999356i	$0.488576\pi$
$740$	14583.9	0.724478
$741$	0	0
$742$	48793.8	2.41412
$743$	−31943.3	−1.57723	−0.788617	−	0.614884i	$-0.789203\pi$
$743$	−31943.3	−1.57723	−0.788617	+	0.614884i	$0.789203\pi$
$744$	0	0
$745$	54196.1	2.66522
$746$	35318.6	1.73339
$747$	0	0
$748$	0	0
$749$	−28255.9	−1.37843
$750$	0	0
$751$	−32895.5	−1.59837	−0.799183	−	0.601088i	$-0.794734\pi$
$751$	−32895.5	−1.59837	−0.799183	+	0.601088i	$0.794734\pi$
$752$	41434.3	2.00925
$753$	0	0
$754$	−8832.21	−0.426592
$755$	16875.9	0.813478
$756$	0	0
$757$	1794.93	0.0861792	0.0430896	−	0.999071i	$-0.486280\pi$
$757$	1794.93	0.0861792	0.0430896	+	0.999071i	$0.486280\pi$
$758$	−39235.4	−1.88007
$759$	0	0
$760$	23527.5	1.12294
$761$	304.527	0.0145060	0.00725302	−	0.999974i	$-0.497691\pi$
$761$	304.527	0.0145060	0.00725302	+	0.999974i	$0.497691\pi$
$762$	0	0
$763$	−11526.2	−0.546890
$764$	9872.35	0.467499
$765$	0	0
$766$	34907.6	1.64656
$767$	−17920.7	−0.843649
$768$	0	0
$769$	−23116.6	−1.08402	−0.542008	−	0.840374i	$-0.682336\pi$
$769$	−23116.6	−1.08402	−0.542008	+	0.840374i	$0.682336\pi$
$770$	0	0
$771$	0	0
$772$	−10893.0	−0.507835
$773$	10896.9	0.507030	0.253515	−	0.967331i	$-0.418413\pi$
$773$	10896.9	0.507030	0.253515	+	0.967331i	$0.418413\pi$
$774$	0	0
$775$	5186.66	0.240400
$776$	−13047.9	−0.603600
$777$	0	0
$778$	44251.7	2.03920
$779$	49885.4	2.29439
$780$	0	0
$781$	0	0
$782$	−380.762	−0.0174118
$783$	0	0
$784$	−52396.7	−2.38687
$785$	24159.6	1.09846
$786$	0	0
$787$	−19310.9	−0.874662	−0.437331	−	0.899301i	$-0.644076\pi$
$787$	−19310.9	−0.874662	−0.437331	+	0.899301i	$0.644076\pi$
$788$	−23971.7	−1.08370
$789$	0	0
$790$	−21025.5	−0.946902
$791$	−32524.0	−1.46197
$792$	0	0
$793$	5853.68	0.262132
$794$	−10761.6	−0.481001
$795$	0	0
$796$	−17327.9	−0.771570
$797$	−33816.1	−1.50292	−0.751460	−	0.659778i	$-0.770650\pi$
$797$	−33816.1	−1.50292	−0.751460	+	0.659778i	$0.770650\pi$
$798$	0	0
$799$	−11255.6	−0.498366
$800$	−25412.6	−1.12309
$801$	0	0
$802$	−48713.9	−2.14482
$803$	0	0
$804$	0	0
$805$	2487.10	0.108893
$806$	5980.50	0.261358
$807$	0	0
$808$	15539.1	0.676565
$809$	−31725.9	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$809$	−31725.9	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$810$	0	0
$811$	−13034.9	−0.564387	−0.282194	−	0.959357i	$-0.591062\pi$
$811$	−13034.9	−0.564387	−0.282194	+	0.959357i	$0.591062\pi$
$812$	9370.93	0.404994
$813$	0	0
$814$	0	0
$815$	−38166.4	−1.64038
$816$	0	0
$817$	55168.4	2.36242
$818$	14514.5	0.620402
$819$	0	0
$820$	−29187.9	−1.24303
$821$	−1908.17	−0.0811154	−0.0405577	−	0.999177i	$-0.512913\pi$
$821$	−1908.17	−0.0811154	−0.0405577	+	0.999177i	$0.512913\pi$
$822$	0	0
$823$	17024.9	0.721084	0.360542	−	0.932743i	$-0.382592\pi$
$823$	17024.9	0.721084	0.360542	+	0.932743i	$0.382592\pi$
$824$	−667.251	−0.0282097
$825$	0	0
$826$	49634.9	2.09082
$827$	−20952.3	−0.880995	−0.440498	−	0.897754i	$-0.645198\pi$
$827$	−20952.3	−0.880995	−0.440498	+	0.897754i	$0.645198\pi$
$828$	0	0
$829$	6064.60	0.254080	0.127040	−	0.991898i	$-0.459452\pi$
$829$	6064.60	0.254080	0.127040	+	0.991898i	$0.459452\pi$
$830$	44379.2	1.85593
$831$	0	0
$832$	−3222.54	−0.134281
$833$	14233.5	0.592031
$834$	0	0
$835$	41977.6	1.73975
$836$	0	0
$837$	0	0
$838$	29986.3	1.23611
$839$	−14693.0	−0.604598	−0.302299	−	0.953213i	$-0.597754\pi$
$839$	−14693.0	−0.604598	−0.302299	+	0.953213i	$0.597754\pi$
$840$	0	0
$841$	−20850.5	−0.854914
$842$	2287.23	0.0936142
$843$	0	0
$844$	7576.89	0.309013
$845$	−7916.30	−0.322283
$846$	0	0
$847$	0	0
$848$	33781.5	1.36800
$849$	0	0
$850$	9959.21	0.401880
$851$	−907.309	−0.0365478
$852$	0	0
$853$	−1193.20	−0.0478949	−0.0239474	−	0.999713i	$-0.507623\pi$
$853$	−1193.20	−0.0478949	−0.0239474	+	0.999713i	$0.507623\pi$
$854$	−16212.9	−0.649643
$855$	0	0
$856$	−9729.77	−0.388501
$857$	−12579.5	−0.501410	−0.250705	−	0.968064i	$-0.580662\pi$
$857$	−12579.5	−0.501410	−0.250705	+	0.968064i	$0.580662\pi$
$858$	0	0
$859$	−23421.2	−0.930293	−0.465146	−	0.885234i	$-0.653998\pi$
$859$	−23421.2	−0.930293	−0.465146	+	0.885234i	$0.653998\pi$
$860$	−32279.0	−1.27989
$861$	0	0
$862$	−34891.3	−1.37866
$863$	40751.6	1.60742	0.803708	−	0.595024i	$-0.202857\pi$
$863$	40751.6	1.60742	0.803708	+	0.595024i	$0.202857\pi$
$864$	0	0
$865$	6910.14	0.271620
$866$	14059.8	0.551698
$867$	0	0
$868$	−6345.29	−0.248126
$869$	0	0
$870$	0	0
$871$	−244.919	−0.00952786
$872$	−3968.99	−0.154137
$873$	0	0
$874$	2397.69	0.0927952
$875$	−1903.88	−0.0735576
$876$	0	0
$877$	−22946.4	−0.883517	−0.441758	−	0.897134i	$-0.645645\pi$
$877$	−22946.4	−0.883517	−0.441758	+	0.897134i	$0.645645\pi$
$878$	−4397.17	−0.169017
$879$	0	0
$880$	0	0
$881$	−34636.1	−1.32454	−0.662269	−	0.749266i	$-0.730406\pi$
$881$	−34636.1	−1.32454	−0.662269	+	0.749266i	$0.730406\pi$
$882$	0	0
$883$	−21216.8	−0.808608	−0.404304	−	0.914625i	$-0.632486\pi$
$883$	−21216.8	−0.808608	−0.404304	+	0.914625i	$0.632486\pi$
$884$	4399.02	0.167370
$885$	0	0
$886$	−54955.8	−2.08383
$887$	19141.6	0.724590	0.362295	−	0.932064i	$-0.381993\pi$
$887$	19141.6	0.724590	0.362295	+	0.932064i	$0.381993\pi$
$888$	0	0
$889$	75792.1	2.85938
$890$	−837.681	−0.0315496
$891$	0	0
$892$	13686.2	0.513733
$893$	70877.5	2.65602
$894$	0	0
$895$	25814.1	0.964103
$896$	−41143.2	−1.53404
$897$	0	0
$898$	−2887.76	−0.107311
$899$	−2396.00	−0.0888889
$900$	0	0
$901$	−9176.71	−0.339312
$902$	0	0
$903$	0	0
$904$	−11199.5	−0.412045
$905$	−28272.4	−1.03846
$906$	0	0
$907$	6740.07	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$907$	6740.07	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$908$	−23404.5	−0.855404
$909$	0	0
$910$	−75009.4	−2.73246
$911$	21836.9	0.794169	0.397084	−	0.917782i	$-0.370022\pi$
$911$	21836.9	0.794169	0.397084	+	0.917782i	$0.370022\pi$
$912$	0	0
$913$	0	0
$914$	44592.4	1.61377
$915$	0	0
$916$	20199.1	0.728599
$917$	27868.6	1.00360
$918$	0	0
$919$	−11490.8	−0.412456	−0.206228	−	0.978504i	$-0.566119\pi$
$919$	−11490.8	−0.412456	−0.206228	+	0.978504i	$0.566119\pi$
$920$	856.422	0.0306906
$921$	0	0
$922$	16506.2	0.589590
$923$	5479.04	0.195390
$924$	0	0
$925$	23731.6	0.843557
$926$	−519.939	−0.0184517
$927$	0	0
$928$	11739.5	0.415266
$929$	22645.0	0.799740	0.399870	−	0.916572i	$-0.369055\pi$
$929$	22645.0	0.799740	0.399870	+	0.916572i	$0.369055\pi$
$930$	0	0
$931$	−89629.6	−3.15520
$932$	−13829.9	−0.486067
$933$	0	0
$934$	−12404.8	−0.434579
$935$	0	0
$936$	0	0
$937$	21984.6	0.766494	0.383247	−	0.923646i	$-0.374806\pi$
$937$	21984.6	0.766494	0.383247	+	0.923646i	$0.374806\pi$
$938$	678.352	0.0236130
$939$	0	0
$940$	−41470.3	−1.43895
$941$	−1475.41	−0.0511126	−0.0255563	−	0.999673i	$-0.508136\pi$
$941$	−1475.41	−0.0511126	−0.0255563	+	0.999673i	$0.508136\pi$
$942$	0	0
$943$	1815.87	0.0627072
$944$	34363.8	1.18480
$945$	0	0
$946$	0	0
$947$	−44129.0	−1.51426	−0.757128	−	0.653266i	$-0.773398\pi$
$947$	−44129.0	−1.51426	−0.757128	+	0.653266i	$0.773398\pi$
$948$	0	0
$949$	−17332.1	−0.592859
$950$	−62714.0	−2.14180
$951$	0	0
$952$	7437.97	0.253221
$953$	46294.9	1.57360	0.786799	−	0.617209i	$-0.211737\pi$
$953$	46294.9	1.57360	0.786799	+	0.617209i	$0.211737\pi$
$954$	0	0
$955$	31659.4	1.07275
$956$	7723.73	0.261300
$957$	0	0
$958$	48156.6	1.62408
$959$	31233.6	1.05171
$960$	0	0
$961$	−28168.6	−0.945541
$962$	27363.8	0.917095
$963$	0	0
$964$	4591.50	0.153405
$965$	−34932.6	−1.16531
$966$	0	0
$967$	41724.4	1.38755	0.693777	−	0.720189i	$-0.255945\pi$
$967$	41724.4	1.38755	0.693777	+	0.720189i	$0.255945\pi$
$968$	0	0
$969$	0	0
$970$	68542.2	2.26882
$971$	4172.09	0.137887	0.0689437	−	0.997621i	$-0.478037\pi$
$971$	4172.09	0.137887	0.0689437	+	0.997621i	$0.478037\pi$
$972$	0	0
$973$	−57794.0	−1.90420
$974$	64814.7	2.13223
$975$	0	0
$976$	−11224.7	−0.368130
$977$	−20138.9	−0.659470	−0.329735	−	0.944074i	$-0.606959\pi$
$977$	−20138.9	−0.659470	−0.329735	+	0.944074i	$0.606959\pi$
$978$	0	0
$979$	0	0
$980$	52442.2	1.70939
$981$	0	0
$982$	−47924.5	−1.55736
$983$	−38495.2	−1.24904	−0.624520	−	0.781008i	$-0.714706\pi$
$983$	−38495.2	−1.24904	−0.624520	+	0.781008i	$0.714706\pi$
$984$	0	0
$985$	−76874.3	−2.48672
$986$	−4600.70	−0.148597
$987$	0	0
$988$	−27701.0	−0.891990
$989$	2008.18	0.0645666
$990$	0	0
$991$	−920.782	−0.0295152	−0.0147576	−	0.999891i	$-0.504698\pi$
$991$	−920.782	−0.0295152	−0.0147576	+	0.999891i	$0.504698\pi$
$992$	−7949.08	−0.254419
$993$	0	0
$994$	−15175.3	−0.484236
$995$	−55568.3	−1.77049
$996$	0	0
$997$	−55986.4	−1.77844	−0.889222	−	0.457476i	$-0.848753\pi$
$997$	−55986.4	−1.77844	−0.889222	+	0.457476i	$0.848753\pi$
$998$	73249.4	2.32331
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.bl.1.10		12
3.2	odd	2	inner	1089.4.a.bl.1.3		12
11.2	odd	10		99.4.f.e.37.5	yes	24
11.6	odd	10		99.4.f.e.91.5	yes	24
11.10	odd	2		1089.4.a.bm.1.3		12
33.2	even	10		99.4.f.e.37.2	✓	24
33.17	even	10		99.4.f.e.91.2	yes	24
33.32	even	2		1089.4.a.bm.1.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.4.f.e.37.2	✓	24	33.2	even	10
99.4.f.e.37.5	yes	24	11.2	odd	10
99.4.f.e.91.2	yes	24	33.17	even	10
99.4.f.e.91.5	yes	24	11.6	odd	10
1089.4.a.bl.1.3		12	3.2	odd	2	inner
1089.4.a.bl.1.10		12	1.1	even	1	trivial
1089.4.a.bm.1.3		12	11.10	odd	2
1089.4.a.bm.1.10		12	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+3.60104 q^{2} +4.96749 q^{4} +15.9301 q^{5} -31.7130 q^{7} -10.9202 q^{8} +O(q^{10})\)	\(q+3.60104 q^{2} +4.96749 q^{4} +15.9301 q^{5} -31.7130 q^{7} -10.9202 q^{8} +57.3650 q^{10} +41.2318 q^{13} -114.200 q^{14} -79.0640 q^{16} +21.4777 q^{17} -135.247 q^{19} +79.1326 q^{20} -4.92310 q^{23} +128.769 q^{25} +148.477 q^{26} -157.534 q^{28} -59.4852 q^{29} +40.2789 q^{31} -197.351 q^{32} +77.3419 q^{34} -505.191 q^{35} +184.296 q^{37} -487.028 q^{38} -173.960 q^{40} -368.847 q^{41} -407.910 q^{43} -17.7283 q^{46} -524.061 q^{47} +662.712 q^{49} +463.701 q^{50} +204.818 q^{52} -427.268 q^{53} +346.312 q^{56} -214.209 q^{58} -434.633 q^{59} +141.970 q^{61} +145.046 q^{62} -78.1566 q^{64} +656.827 q^{65} -5.94006 q^{67} +106.690 q^{68} -1819.21 q^{70} +132.884 q^{71} -420.357 q^{73} +663.659 q^{74} -671.836 q^{76} -366.521 q^{79} -1259.50 q^{80} -1328.23 q^{82} +773.629 q^{83} +342.142 q^{85} -1468.90 q^{86} -14.6027 q^{89} -1307.58 q^{91} -24.4554 q^{92} -1887.16 q^{94} -2154.49 q^{95} +1194.84 q^{97} +2386.45 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 58 q^{4} - 66 q^{7}+O(q^{10}) \) 12 * q + 58 * q^4 - 66 * q^7	\( 12 q + 58 q^{4} - 66 q^{7} - 104 q^{10} - 120 q^{13} + 378 q^{16} - 360 q^{19} + 526 q^{25} - 560 q^{28} + 90 q^{31} - 1260 q^{34} - 428 q^{37} - 2376 q^{40} - 2004 q^{43} - 1282 q^{46} + 1186 q^{49} - 3198 q^{52} - 914 q^{58} - 2344 q^{61} + 1684 q^{64} - 968 q^{67} - 5922 q^{70} - 322 q^{73} - 7668 q^{76} - 3482 q^{79} + 4314 q^{82} - 636 q^{85} - 380 q^{91} - 5904 q^{94} + 3258 q^{97}+O(q^{100}) \) 12 * q + 58 * q^4 - 66 * q^7 - 104 * q^10 - 120 * q^13 + 378 * q^16 - 360 * q^19 + 526 * q^25 - 560 * q^28 + 90 * q^31 - 1260 * q^34 - 428 * q^37 - 2376 * q^40 - 2004 * q^43 - 1282 * q^46 + 1186 * q^49 - 3198 * q^52 - 914 * q^58 - 2344 * q^61 + 1684 * q^64 - 968 * q^67 - 5922 * q^70 - 322 * q^73 - 7668 * q^76 - 3482 * q^79 + 4314 * q^82 - 636 * q^85 - 380 * q^91 - 5904 * q^94 + 3258 * q^97

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