LMFDB - Embedded newform 9801.2.a.cg.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9801,2,Mod(1,9801)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9801.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9801, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,-1,0,11,2,0,-9,0,0,-8,0,0,-9,-21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$78.2613790211$
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} - x^{11} - 17 x^{10} + 16 x^{9} + 100 x^{8} - 84 x^{7} - 248 x^{6} + 152 x^{5} + 272 x^{4} + \cdots - 1 $ x^12 - x^11 - 17x^10 + 16x^9 + 100x^8 - 84x^7 - 248x^6 + 152x^5 + 272x^4 - 66x^3 - 110x^2 - 25x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 891)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.62930$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.62930 q^{2} +4.91321 q^{4} -0.239560 q^{5} -1.18583 q^{7} +7.65971 q^{8} -0.629874 q^{10} -6.03699 q^{13} -3.11791 q^{14} +10.3132 q^{16} -1.83436 q^{17} -3.25989 q^{19} -1.17701 q^{20} -5.55040 q^{23} -4.94261 q^{25} -15.8730 q^{26} -5.82625 q^{28} +6.22511 q^{29} +3.95750 q^{31} +11.7972 q^{32} -4.82307 q^{34} +0.284077 q^{35} -5.43325 q^{37} -8.57122 q^{38} -1.83496 q^{40} -10.6655 q^{41} +12.1439 q^{43} -14.5937 q^{46} +1.65475 q^{47} -5.59380 q^{49} -12.9956 q^{50} -29.6610 q^{52} -8.64187 q^{53} -9.08313 q^{56} +16.3677 q^{58} -3.92783 q^{59} -10.7259 q^{61} +10.4054 q^{62} +10.3918 q^{64} +1.44622 q^{65} +9.22102 q^{67} -9.01259 q^{68} +0.746925 q^{70} +7.14085 q^{71} -0.558900 q^{73} -14.2856 q^{74} -16.0165 q^{76} +4.42297 q^{79} -2.47063 q^{80} -28.0429 q^{82} -15.4189 q^{83} +0.439438 q^{85} +31.9300 q^{86} -0.0625956 q^{89} +7.15886 q^{91} -27.2703 q^{92} +4.35083 q^{94} +0.780938 q^{95} +6.09309 q^{97} -14.7078 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{2} + 11 q^{4} + 2 q^{5} - 9 q^{7} - 8 q^{10} - 9 q^{13} - 21 q^{14} + 21 q^{16} + 5 q^{17} - 18 q^{19} + 11 q^{20} + 3 q^{23} + 14 q^{25} + 8 q^{26} - 17 q^{28} + 6 q^{29} + 6 q^{31} + 9 q^{32}+ \cdots + 34 q^{98}+O(q^{100}) $ 12 * q - q^2 + 11 * q^4 + 2 * q^5 - 9 * q^7 - 8 * q^10 - 9 * q^13 - 21 * q^14 + 21 * q^16 + 5 * q^17 - 18 * q^19 + 11 * q^20 + 3 * q^23 + 14 * q^25 + 8 * q^26 - 17 * q^28 + 6 * q^29 + 6 * q^31 + 9 * q^32 - 21 * q^34 - 5 * q^35 - 7 * q^37 - 10 * q^38 - 21 * q^40 - 5 * q^41 - 15 * q^43 - 37 * q^46 + 17 * q^47 + 11 * q^49 + 2 * q^50 - 18 * q^52 - 19 * q^53 - 24 * q^56 + 14 * q^58 + 23 * q^59 - 31 * q^61 + 24 * q^62 + 44 * q^64 - 34 * q^65 - 7 * q^67 + 66 * q^68 - 15 * q^70 + 7 * q^71 - 25 * q^73 - 61 * q^74 - 51 * q^76 - 47 * q^79 + 49 * q^80 + 12 * q^82 - 24 * q^83 - 15 * q^85 + 25 * q^86 + 17 * q^89 + 29 * q^91 - 48 * q^92 - 45 * q^94 + 23 * q^95 + 15 * q^97 + 34 * q^98

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 $

$2$	2.62930	1.85920	0.929598	−	0.368576i	$-0.120155\pi$
$2$	2.62930	1.85920	0.929598	+	0.368576i	$0.120155\pi$
$3$	0	0
$3$	0	0
$4$	4.91321	2.45661
$5$	−0.239560	−0.107134	−0.0535671	−	0.998564i	$-0.517059\pi$
$5$	−0.239560	−0.107134	−0.0535671	+	0.998564i	$0.517059\pi$
$6$	0	0
$7$	−1.18583	−0.448203	−0.224101	−	0.974566i	$-0.571945\pi$
$7$	−1.18583	−0.448203	−0.224101	+	0.974566i	$0.571945\pi$
$8$	7.65971	2.70812
$9$	0	0
$10$	−0.629874	−0.199184
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.03699	−1.67436	−0.837180	−	0.546928i	$-0.815797\pi$
$13$	−6.03699	−1.67436	−0.837180	+	0.546928i	$0.815797\pi$
$14$	−3.11791	−0.833296
$15$	0	0
$16$	10.3132	2.57831
$17$	−1.83436	−0.444897	−0.222449	−	0.974944i	$-0.571405\pi$
$17$	−1.83436	−0.444897	−0.222449	+	0.974944i	$0.571405\pi$
$18$	0	0
$19$	−3.25989	−0.747870	−0.373935	−	0.927455i	$-0.621992\pi$
$19$	−3.25989	−0.747870	−0.373935	+	0.927455i	$0.621992\pi$
$20$	−1.17701	−0.263187
$21$	0	0
$22$	0	0
$23$	−5.55040	−1.15734	−0.578670	−	0.815562i	$-0.696428\pi$
$23$	−5.55040	−1.15734	−0.578670	+	0.815562i	$0.696428\pi$
$24$	0	0
$25$	−4.94261	−0.988522
$26$	−15.8730	−3.11296
$27$	0	0
$28$	−5.82625	−1.10106
$29$	6.22511	1.15597	0.577987	−	0.816046i	$-0.303838\pi$
$29$	6.22511	1.15597	0.577987	+	0.816046i	$0.303838\pi$
$30$	0	0
$31$	3.95750	0.710787	0.355394	−	0.934717i	$-0.384347\pi$
$31$	3.95750	0.710787	0.355394	+	0.934717i	$0.384347\pi$
$32$	11.7972	2.08546
$33$	0	0
$34$	−4.82307	−0.827150
$35$	0.284077	0.0480179
$36$	0	0
$37$	−5.43325	−0.893221	−0.446610	−	0.894729i	$-0.647369\pi$
$37$	−5.43325	−0.893221	−0.446610	+	0.894729i	$0.647369\pi$
$38$	−8.57122	−1.39044
$39$	0	0
$40$	−1.83496	−0.290132
$41$	−10.6655	−1.66568	−0.832838	−	0.553517i	$-0.813285\pi$
$41$	−10.6655	−1.66568	−0.832838	+	0.553517i	$0.813285\pi$
$42$	0	0
$43$	12.1439	1.85193	0.925964	−	0.377612i	$-0.123255\pi$
$43$	12.1439	1.85193	0.925964	+	0.377612i	$0.123255\pi$
$44$	0	0
$45$	0	0
$46$	−14.5937	−2.15172
$47$	1.65475	0.241370	0.120685	−	0.992691i	$-0.461491\pi$
$47$	1.65475	0.241370	0.120685	+	0.992691i	$0.461491\pi$
$48$	0	0
$49$	−5.59380	−0.799115
$50$	−12.9956	−1.83786
$51$	0	0
$52$	−29.6610	−4.11324
$53$	−8.64187	−1.18705	−0.593526	−	0.804815i	$-0.702265\pi$
$53$	−8.64187	−1.18705	−0.593526	+	0.804815i	$0.702265\pi$
$54$	0	0
$55$	0	0
$56$	−9.08313	−1.21378
$57$	0	0
$58$	16.3677	2.14918
$59$	−3.92783	−0.511359	−0.255680	−	0.966762i	$-0.582299\pi$
$59$	−3.92783	−0.511359	−0.255680	+	0.966762i	$0.582299\pi$
$60$	0	0
$61$	−10.7259	−1.37331	−0.686654	−	0.726984i	$-0.740921\pi$
$61$	−10.7259	−1.37331	−0.686654	+	0.726984i	$0.740921\pi$
$62$	10.4054	1.32149
$63$	0	0
$64$	10.3918	1.29898
$65$	1.44622	0.179381
$66$	0	0
$67$	9.22102	1.12653	0.563263	−	0.826277i	$-0.309546\pi$
$67$	9.22102	1.12653	0.563263	+	0.826277i	$0.309546\pi$
$68$	−9.01259	−1.09294
$69$	0	0
$70$	0.746925	0.0892746
$71$	7.14085	0.847462	0.423731	−	0.905788i	$-0.360720\pi$
$71$	7.14085	0.847462	0.423731	+	0.905788i	$0.360720\pi$
$72$	0	0
$73$	−0.558900	−0.0654144	−0.0327072	−	0.999465i	$-0.510413\pi$
$73$	−0.558900	−0.0654144	−0.0327072	+	0.999465i	$0.510413\pi$
$74$	−14.2856	−1.66067
$75$	0	0
$76$	−16.0165	−1.83722
$77$	0	0
$78$	0	0
$79$	4.42297	0.497623	0.248811	−	0.968552i	$-0.419960\pi$
$79$	4.42297	0.497623	0.248811	+	0.968552i	$0.419960\pi$
$80$	−2.47063	−0.276225
$81$	0	0
$82$	−28.0429	−3.09682
$83$	−15.4189	−1.69245	−0.846224	−	0.532828i	$-0.821129\pi$
$83$	−15.4189	−1.69245	−0.846224	+	0.532828i	$0.821129\pi$
$84$	0	0
$85$	0.439438	0.0476637
$86$	31.9300	3.44310
$87$	0	0
$88$	0	0
$89$	−0.0625956	−0.00663512	−0.00331756	−	0.999994i	$-0.501056\pi$
$89$	−0.0625956	−0.00663512	−0.00331756	+	0.999994i	$0.501056\pi$
$90$	0	0
$91$	7.15886	0.750452
$92$	−27.2703	−2.84313
$93$	0	0
$94$	4.35083	0.448754
$95$	0.780938	0.0801225
$96$	0	0
$97$	6.09309	0.618659	0.309330	−	0.950955i	$-0.399895\pi$
$97$	6.09309	0.618659	0.309330	+	0.950955i	$0.399895\pi$
$98$	−14.7078	−1.48571
$99$	0	0
$100$	−24.2841	−2.42841
$101$	−15.2575	−1.51818	−0.759091	−	0.650985i	$-0.774356\pi$
$101$	−15.2575	−1.51818	−0.759091	+	0.650985i	$0.774356\pi$
$102$	0	0
$103$	12.4965	1.23131	0.615657	−	0.788014i	$-0.288891\pi$
$103$	12.4965	1.23131	0.615657	+	0.788014i	$0.288891\pi$
$104$	−46.2416	−4.53436
$105$	0	0
$106$	−22.7221	−2.20696
$107$	10.7905	1.04316	0.521580	−	0.853202i	$-0.325343\pi$
$107$	10.7905	1.04316	0.521580	+	0.853202i	$0.325343\pi$
$108$	0	0
$109$	−5.16947	−0.495146	−0.247573	−	0.968869i	$-0.579633\pi$
$109$	−5.16947	−0.495146	−0.247573	+	0.968869i	$0.579633\pi$
$110$	0	0
$111$	0	0
$112$	−12.2298	−1.15560
$113$	2.03794	0.191713	0.0958566	−	0.995395i	$-0.469441\pi$
$113$	2.03794	0.191713	0.0958566	+	0.995395i	$0.469441\pi$
$114$	0	0
$115$	1.32965	0.123991
$116$	30.5853	2.83977
$117$	0	0
$118$	−10.3274	−0.950717
$119$	2.17524	0.199404
$120$	0	0
$121$	0	0
$122$	−28.2015	−2.55325
$123$	0	0
$124$	19.4440	1.74612
$125$	2.38185	0.213039
$126$	0	0
$127$	−1.19792	−0.106299	−0.0531493	−	0.998587i	$-0.516926\pi$
$127$	−1.19792	−0.106299	−0.0531493	+	0.998587i	$0.516926\pi$
$128$	3.72883	0.329585
$129$	0	0
$130$	3.80254	0.333505
$131$	−5.76699	−0.503864	−0.251932	−	0.967745i	$-0.581066\pi$
$131$	−5.76699	−0.503864	−0.251932	+	0.967745i	$0.581066\pi$
$132$	0	0
$133$	3.86568	0.335197
$134$	24.2448	2.09443
$135$	0	0
$136$	−14.0506	−1.20483
$137$	0.133756	0.0114275	0.00571376	−	0.999984i	$-0.498181\pi$
$137$	0.133756	0.0114275	0.00571376	+	0.999984i	$0.498181\pi$
$138$	0	0
$139$	11.1452	0.945320	0.472660	−	0.881245i	$-0.343294\pi$
$139$	11.1452	0.945320	0.472660	+	0.881245i	$0.343294\pi$
$140$	1.39573	0.117961
$141$	0	0
$142$	18.7754	1.57560
$143$	0	0
$144$	0	0
$145$	−1.49128	−0.123844
$146$	−1.46952	−0.121618
$147$	0	0
$148$	−26.6947	−2.19429
$149$	3.94488	0.323178	0.161589	−	0.986858i	$-0.448338\pi$
$149$	3.94488	0.323178	0.161589	+	0.986858i	$0.448338\pi$
$150$	0	0
$151$	−2.67715	−0.217863	−0.108932	−	0.994049i	$-0.534743\pi$
$151$	−2.67715	−0.217863	−0.108932	+	0.994049i	$0.534743\pi$
$152$	−24.9698	−2.02532
$153$	0	0
$154$	0	0
$155$	−0.948056	−0.0761497
$156$	0	0
$157$	−10.6750	−0.851954	−0.425977	−	0.904734i	$-0.640070\pi$
$157$	−10.6750	−0.851954	−0.425977	+	0.904734i	$0.640070\pi$
$158$	11.6293	0.925178
$159$	0	0
$160$	−2.82612	−0.223425
$161$	6.58185	0.518722
$162$	0	0
$163$	7.38361	0.578329	0.289165	−	0.957279i	$-0.406622\pi$
$163$	7.38361	0.578329	0.289165	+	0.957279i	$0.406622\pi$
$164$	−52.4020	−4.09191
$165$	0	0
$166$	−40.5410	−3.14659
$167$	−10.3219	−0.798730	−0.399365	−	0.916792i	$-0.630769\pi$
$167$	−10.3219	−0.798730	−0.399365	+	0.916792i	$0.630769\pi$
$168$	0	0
$169$	23.4452	1.80348
$170$	1.15541	0.0886162
$171$	0	0
$172$	59.6656	4.54946
$173$	16.0711	1.22186	0.610932	−	0.791683i	$-0.290795\pi$
$173$	16.0711	1.22186	0.610932	+	0.791683i	$0.290795\pi$
$174$	0	0
$175$	5.86111	0.443058
$176$	0	0
$177$	0	0
$178$	−0.164583	−0.0123360
$179$	−3.71559	−0.277716	−0.138858	−	0.990312i	$-0.544343\pi$
$179$	−3.71559	−0.277716	−0.138858	+	0.990312i	$0.544343\pi$
$180$	0	0
$181$	−15.1105	−1.12316	−0.561578	−	0.827424i	$-0.689805\pi$
$181$	−15.1105	−1.12316	−0.561578	+	0.827424i	$0.689805\pi$
$182$	18.8228	1.39524
$183$	0	0
$184$	−42.5145	−3.13421
$185$	1.30159	0.0956946
$186$	0	0
$187$	0	0
$188$	8.13014	0.592951
$189$	0	0
$190$	2.05332	0.148963
$191$	11.9839	0.867125	0.433562	−	0.901124i	$-0.357256\pi$
$191$	11.9839	0.867125	0.433562	+	0.901124i	$0.357256\pi$
$192$	0	0
$193$	−22.5333	−1.62199	−0.810993	−	0.585056i	$-0.801073\pi$
$193$	−22.5333	−1.62199	−0.810993	+	0.585056i	$0.801073\pi$
$194$	16.0205	1.15021
$195$	0	0
$196$	−27.4835	−1.96311
$197$	−9.95308	−0.709127	−0.354564	−	0.935032i	$-0.615371\pi$
$197$	−9.95308	−0.709127	−0.354564	+	0.935032i	$0.615371\pi$
$198$	0	0
$199$	−15.5617	−1.10314	−0.551570	−	0.834128i	$-0.685971\pi$
$199$	−15.5617	−1.10314	−0.551570	+	0.834128i	$0.685971\pi$
$200$	−37.8590	−2.67703
$201$	0	0
$202$	−40.1166	−2.82260
$203$	−7.38194	−0.518110
$204$	0	0
$205$	2.55503	0.178451
$206$	32.8570	2.28925
$207$	0	0
$208$	−62.2609	−4.31702
$209$	0	0
$210$	0	0
$211$	−2.39488	−0.164870	−0.0824352	−	0.996596i	$-0.526270\pi$
$211$	−2.39488	−0.164870	−0.0824352	+	0.996596i	$0.526270\pi$
$212$	−42.4593	−2.91612
$213$	0	0
$214$	28.3716	1.93944
$215$	−2.90919	−0.198405
$216$	0	0
$217$	−4.69293	−0.318577
$218$	−13.5921	−0.920572
$219$	0	0
$220$	0	0
$221$	11.0740	0.744918
$222$	0	0
$223$	12.9839	0.869465	0.434732	−	0.900560i	$-0.356843\pi$
$223$	12.9839	0.869465	0.434732	+	0.900560i	$0.356843\pi$
$224$	−13.9895	−0.934710
$225$	0	0
$226$	5.35835	0.356432
$227$	18.4506	1.22461	0.612306	−	0.790621i	$-0.290242\pi$
$227$	18.4506	1.22461	0.612306	+	0.790621i	$0.290242\pi$
$228$	0	0
$229$	20.1398	1.33087	0.665436	−	0.746454i	$-0.268245\pi$
$229$	20.1398	1.33087	0.665436	+	0.746454i	$0.268245\pi$
$230$	3.49605	0.230523
$231$	0	0
$232$	47.6825	3.13051
$233$	−7.35914	−0.482114	−0.241057	−	0.970511i	$-0.577494\pi$
$233$	−7.35914	−0.482114	−0.241057	+	0.970511i	$0.577494\pi$
$234$	0	0
$235$	−0.396411	−0.0258590
$236$	−19.2982	−1.25621
$237$	0	0
$238$	5.71936	0.370731
$239$	9.62582	0.622643	0.311321	−	0.950305i	$-0.399228\pi$
$239$	9.62582	0.622643	0.311321	+	0.950305i	$0.399228\pi$
$240$	0	0
$241$	−26.5631	−1.71108	−0.855540	−	0.517736i	$-0.826775\pi$
$241$	−26.5631	−1.71108	−0.855540	+	0.517736i	$0.826775\pi$
$242$	0	0
$243$	0	0
$244$	−52.6985	−3.37368
$245$	1.34005	0.0856126
$246$	0	0
$247$	19.6799	1.25220
$248$	30.3133	1.92489
$249$	0	0
$250$	6.26259	0.396081
$251$	24.8619	1.56927	0.784635	−	0.619957i	$-0.212850\pi$
$251$	24.8619	1.56927	0.784635	+	0.619957i	$0.212850\pi$
$252$	0	0
$253$	0	0
$254$	−3.14970	−0.197630
$255$	0	0
$256$	−10.9794	−0.686213
$257$	13.4101	0.836497	0.418249	−	0.908333i	$-0.362644\pi$
$257$	13.4101	0.836497	0.418249	+	0.908333i	$0.362644\pi$
$258$	0	0
$259$	6.44292	0.400344
$260$	7.10558	0.440669
$261$	0	0
$262$	−15.1631	−0.936782
$263$	−0.781901	−0.0482141	−0.0241071	−	0.999709i	$-0.507674\pi$
$263$	−0.781901	−0.0482141	−0.0241071	+	0.999709i	$0.507674\pi$
$264$	0	0
$265$	2.07024	0.127174
$266$	10.1640	0.623197
$267$	0	0
$268$	45.3048	2.76743
$269$	−10.6650	−0.650254	−0.325127	−	0.945670i	$-0.605407\pi$
$269$	−10.6650	−0.650254	−0.325127	+	0.945670i	$0.605407\pi$
$270$	0	0
$271$	−24.5987	−1.49426	−0.747132	−	0.664676i	$-0.768570\pi$
$271$	−24.5987	−1.49426	−0.747132	+	0.664676i	$0.768570\pi$
$272$	−18.9182	−1.14708
$273$	0	0
$274$	0.351684	0.0212460
$275$	0	0
$276$	0	0
$277$	−8.97223	−0.539089	−0.269544	−	0.962988i	$-0.586873\pi$
$277$	−8.97223	−0.539089	−0.269544	+	0.962988i	$0.586873\pi$
$278$	29.3040	1.75753
$279$	0	0
$280$	2.17595	0.130038
$281$	16.7261	0.997796	0.498898	−	0.866661i	$-0.333738\pi$
$281$	16.7261	0.997796	0.498898	+	0.866661i	$0.333738\pi$
$282$	0	0
$283$	−5.85197	−0.347864	−0.173932	−	0.984758i	$-0.555647\pi$
$283$	−5.85197	−0.347864	−0.173932	+	0.984758i	$0.555647\pi$
$284$	35.0845	2.08188
$285$	0	0
$286$	0	0
$287$	12.6475	0.746560
$288$	0	0
$289$	−13.6351	−0.802067
$290$	−3.92103	−0.230251
$291$	0	0
$292$	−2.74600	−0.160697
$293$	14.9829	0.875308	0.437654	−	0.899144i	$-0.355810\pi$
$293$	14.9829	0.875308	0.437654	+	0.899144i	$0.355810\pi$
$294$	0	0
$295$	0.940948	0.0547841
$296$	−41.6171	−2.41895
$297$	0	0
$298$	10.3723	0.600850
$299$	33.5077	1.93780
$300$	0	0
$301$	−14.4006	−0.830039
$302$	−7.03902	−0.405050
$303$	0	0
$304$	−33.6200	−1.92824
$305$	2.56949	0.147128
$306$	0	0
$307$	6.52424	0.372358	0.186179	−	0.982516i	$-0.440390\pi$
$307$	6.52424	0.372358	0.186179	+	0.982516i	$0.440390\pi$
$308$	0	0
$309$	0	0
$310$	−2.49272	−0.141577
$311$	14.4149	0.817396	0.408698	−	0.912670i	$-0.365983\pi$
$311$	14.4149	0.817396	0.408698	+	0.912670i	$0.365983\pi$
$312$	0	0
$313$	15.7384	0.889588	0.444794	−	0.895633i	$-0.353277\pi$
$313$	15.7384	0.889588	0.444794	+	0.895633i	$0.353277\pi$
$314$	−28.0676	−1.58395
$315$	0	0
$316$	21.7310	1.22246
$317$	29.3176	1.64664	0.823320	−	0.567577i	$-0.192119\pi$
$317$	29.3176	1.64664	0.823320	+	0.567577i	$0.192119\pi$
$318$	0	0
$319$	0	0
$320$	−2.48946	−0.139165
$321$	0	0
$322$	17.3056	0.964406
$323$	5.97980	0.332725
$324$	0	0
$325$	29.8385	1.65514
$326$	19.4137	1.07523
$327$	0	0
$328$	−81.6948	−4.51084
$329$	−1.96226	−0.108183
$330$	0	0
$331$	13.4116	0.737169	0.368584	−	0.929594i	$-0.379843\pi$
$331$	13.4116	0.737169	0.368584	+	0.929594i	$0.379843\pi$
$332$	−75.7565	−4.15768
$333$	0	0
$334$	−27.1393	−1.48499
$335$	−2.20898	−0.120690
$336$	0	0
$337$	12.9222	0.703917	0.351959	−	0.936016i	$-0.385516\pi$
$337$	12.9222	0.703917	0.351959	+	0.936016i	$0.385516\pi$
$338$	61.6445	3.35302
$339$	0	0
$340$	2.15905	0.117091
$341$	0	0
$342$	0	0
$343$	14.9341	0.806368
$344$	93.0188	5.01524
$345$	0	0
$346$	42.2558	2.27168
$347$	−34.8960	−1.87331	−0.936656	−	0.350250i	$-0.886097\pi$
$347$	−34.8960	−1.87331	−0.936656	+	0.350250i	$0.886097\pi$
$348$	0	0
$349$	−3.72027	−0.199142	−0.0995709	−	0.995030i	$-0.531747\pi$
$349$	−3.72027	−0.199142	−0.0995709	+	0.995030i	$0.531747\pi$
$350$	15.4106	0.823732
$351$	0	0
$352$	0	0
$353$	−3.95252	−0.210371	−0.105186	−	0.994453i	$-0.533544\pi$
$353$	−3.95252	−0.210371	−0.105186	+	0.994453i	$0.533544\pi$
$354$	0	0
$355$	−1.71066	−0.0907923
$356$	−0.307546	−0.0162999
$357$	0	0
$358$	−9.76940	−0.516329
$359$	−7.40400	−0.390768	−0.195384	−	0.980727i	$-0.562595\pi$
$359$	−7.40400	−0.390768	−0.195384	+	0.980727i	$0.562595\pi$
$360$	0	0
$361$	−8.37312	−0.440691
$362$	−39.7301	−2.08817
$363$	0	0
$364$	35.1730	1.84357
$365$	0.133890	0.00700812
$366$	0	0
$367$	−25.4159	−1.32670	−0.663350	−	0.748310i	$-0.730866\pi$
$367$	−25.4159	−1.32670	−0.663350	+	0.748310i	$0.730866\pi$
$368$	−57.2426	−2.98398
$369$	0	0
$370$	3.42226	0.177915
$371$	10.2478	0.532040
$372$	0	0
$373$	−4.77475	−0.247227	−0.123614	−	0.992330i	$-0.539448\pi$
$373$	−4.77475	−0.247227	−0.123614	+	0.992330i	$0.539448\pi$
$374$	0	0
$375$	0	0
$376$	12.6749	0.653658
$377$	−37.5809	−1.93552
$378$	0	0
$379$	−17.4486	−0.896272	−0.448136	−	0.893965i	$-0.647912\pi$
$379$	−17.4486	−0.896272	−0.448136	+	0.893965i	$0.647912\pi$
$380$	3.83691	0.196829
$381$	0	0
$382$	31.5093	1.61215
$383$	3.38813	0.173125	0.0865626	−	0.996246i	$-0.472412\pi$
$383$	3.38813	0.173125	0.0865626	+	0.996246i	$0.472412\pi$
$384$	0	0
$385$	0	0
$386$	−59.2469	−3.01559
$387$	0	0
$388$	29.9366	1.51980
$389$	18.7542	0.950878	0.475439	−	0.879749i	$-0.342289\pi$
$389$	18.7542	0.950878	0.475439	+	0.879749i	$0.342289\pi$
$390$	0	0
$391$	10.1814	0.514897
$392$	−42.8469	−2.16409
$393$	0	0
$394$	−26.1696	−1.31841
$395$	−1.05956	−0.0533125
$396$	0	0
$397$	10.7157	0.537806	0.268903	−	0.963167i	$-0.413339\pi$
$397$	10.7157	0.537806	0.268903	+	0.963167i	$0.413339\pi$
$398$	−40.9164	−2.05095
$399$	0	0
$400$	−50.9743	−2.54872
$401$	2.83265	0.141456	0.0707279	−	0.997496i	$-0.477468\pi$
$401$	2.83265	0.141456	0.0707279	+	0.997496i	$0.477468\pi$
$402$	0	0
$403$	−23.8914	−1.19011
$404$	−74.9635	−3.72957
$405$	0	0
$406$	−19.4093	−0.963268
$407$	0	0
$408$	0	0
$409$	23.5437	1.16416	0.582081	−	0.813131i	$-0.302239\pi$
$409$	23.5437	1.16416	0.582081	+	0.813131i	$0.302239\pi$
$410$	6.71794	0.331775
$411$	0	0
$412$	61.3978	3.02485
$413$	4.65774	0.229193
$414$	0	0
$415$	3.69375	0.181319
$416$	−71.2194	−3.49182
$417$	0	0
$418$	0	0
$419$	21.1174	1.03165	0.515826	−	0.856693i	$-0.327485\pi$
$419$	21.1174	1.03165	0.515826	+	0.856693i	$0.327485\pi$
$420$	0	0
$421$	−23.5172	−1.14616	−0.573080	−	0.819500i	$-0.694252\pi$
$421$	−23.5172	−1.14616	−0.573080	+	0.819500i	$0.694252\pi$
$422$	−6.29686	−0.306526
$423$	0	0
$424$	−66.1942	−3.21467
$425$	9.06652	0.439791
$426$	0	0
$427$	12.7191	0.615520
$428$	53.0162	2.56264
$429$	0	0
$430$	−7.64913	−0.368874
$431$	−4.97989	−0.239873	−0.119936	−	0.992782i	$-0.538269\pi$
$431$	−4.97989	−0.239873	−0.119936	+	0.992782i	$0.538269\pi$
$432$	0	0
$433$	−25.5853	−1.22955	−0.614776	−	0.788702i	$-0.710753\pi$
$433$	−25.5853	−1.22955	−0.614776	+	0.788702i	$0.710753\pi$
$434$	−12.3391	−0.592296
$435$	0	0
$436$	−25.3987	−1.21638
$437$	18.0937	0.865539
$438$	0	0
$439$	−4.66005	−0.222412	−0.111206	−	0.993797i	$-0.535471\pi$
$439$	−4.66005	−0.222412	−0.111206	+	0.993797i	$0.535471\pi$
$440$	0	0
$441$	0	0
$442$	29.1168	1.38495
$443$	−23.1237	−1.09864	−0.549321	−	0.835612i	$-0.685114\pi$
$443$	−23.1237	−1.09864	−0.549321	+	0.835612i	$0.685114\pi$
$444$	0	0
$445$	0.0149954	0.000710849 0
$446$	34.1385	1.61650
$447$	0	0
$448$	−12.3229	−0.582204
$449$	13.7641	0.649569	0.324784	−	0.945788i	$-0.394708\pi$
$449$	13.7641	0.649569	0.324784	+	0.945788i	$0.394708\pi$
$450$	0	0
$451$	0	0
$452$	10.0128	0.470964
$453$	0	0
$454$	48.5122	2.27679
$455$	−1.71497	−0.0803992
$456$	0	0
$457$	10.8395	0.507051	0.253526	−	0.967329i	$-0.418410\pi$
$457$	10.8395	0.507051	0.253526	+	0.967329i	$0.418410\pi$
$458$	52.9535	2.47435
$459$	0	0
$460$	6.53287	0.304596
$461$	2.92792	0.136367	0.0681833	−	0.997673i	$-0.478280\pi$
$461$	2.92792	0.136367	0.0681833	+	0.997673i	$0.478280\pi$
$462$	0	0
$463$	−27.0835	−1.25867	−0.629337	−	0.777132i	$-0.716674\pi$
$463$	−27.0835	−1.25867	−0.629337	+	0.777132i	$0.716674\pi$
$464$	64.2010	2.98046
$465$	0	0
$466$	−19.3494	−0.896343
$467$	−11.3066	−0.523206	−0.261603	−	0.965176i	$-0.584251\pi$
$467$	−11.3066	−0.523206	−0.261603	+	0.965176i	$0.584251\pi$
$468$	0	0
$469$	−10.9346	−0.504912
$470$	−1.04228	−0.0480769
$471$	0	0
$472$	−30.0860	−1.38482
$473$	0	0
$474$	0	0
$475$	16.1124	0.739286
$476$	10.6874	0.489857
$477$	0	0
$478$	25.3092	1.15761
$479$	−30.3858	−1.38836	−0.694181	−	0.719801i	$-0.744233\pi$
$479$	−30.3858	−1.38836	−0.694181	+	0.719801i	$0.744233\pi$
$480$	0	0
$481$	32.8005	1.49557
$482$	−69.8424	−3.18123
$483$	0	0
$484$	0	0
$485$	−1.45966	−0.0662796
$486$	0	0
$487$	−3.19964	−0.144989	−0.0724947	−	0.997369i	$-0.523096\pi$
$487$	−3.19964	−0.144989	−0.0724947	+	0.997369i	$0.523096\pi$
$488$	−82.1571	−3.71908
$489$	0	0
$490$	3.52339	0.159170
$491$	−5.98163	−0.269947	−0.134974	−	0.990849i	$-0.543095\pi$
$491$	−5.98163	−0.269947	−0.134974	+	0.990849i	$0.543095\pi$
$492$	0	0
$493$	−11.4191	−0.514289
$494$	51.7444	2.32809
$495$	0	0
$496$	40.8146	1.83263
$497$	−8.46785	−0.379835
$498$	0	0
$499$	29.3773	1.31511	0.657554	−	0.753407i	$-0.271591\pi$
$499$	29.3773	1.31511	0.657554	+	0.753407i	$0.271591\pi$
$500$	11.7025	0.523353
$501$	0	0
$502$	65.3694	2.91758
$503$	−14.4066	−0.642358	−0.321179	−	0.947019i	$-0.604079\pi$
$503$	−14.4066	−0.642358	−0.321179	+	0.947019i	$0.604079\pi$
$504$	0	0
$505$	3.65509	0.162649
$506$	0	0
$507$	0	0
$508$	−5.88566	−0.261134
$509$	−43.8413	−1.94323	−0.971616	−	0.236562i	$-0.923979\pi$
$509$	−43.8413	−1.94323	−0.971616	+	0.236562i	$0.923979\pi$
$510$	0	0
$511$	0.662762	0.0293189
$512$	−36.3258	−1.60539
$513$	0	0
$514$	35.2591	1.55521
$515$	−2.99365	−0.131916
$516$	0	0
$517$	0	0
$518$	16.9404	0.744317
$519$	0	0
$520$	11.0776	0.485785
$521$	20.1646	0.883429	0.441715	−	0.897156i	$-0.354370\pi$
$521$	20.1646	0.883429	0.441715	+	0.897156i	$0.354370\pi$
$522$	0	0
$523$	39.1717	1.71286	0.856429	−	0.516265i	$-0.172678\pi$
$523$	39.1717	1.71286	0.856429	+	0.516265i	$0.172678\pi$
$524$	−28.3345	−1.23780
$525$	0	0
$526$	−2.05585	−0.0896394
$527$	−7.25946	−0.316227
$528$	0	0
$529$	7.80699	0.339434
$530$	5.44329	0.236441
$531$	0	0
$532$	18.9929	0.823448
$533$	64.3877	2.78894
$534$	0	0
$535$	−2.58498	−0.111758
$536$	70.6303	3.05077
$537$	0	0
$538$	−28.0414	−1.20895
$539$	0	0
$540$	0	0
$541$	18.0264	0.775017	0.387509	−	0.921866i	$-0.373336\pi$
$541$	18.0264	0.775017	0.387509	+	0.921866i	$0.373336\pi$
$542$	−64.6773	−2.77813
$543$	0	0
$544$	−21.6402	−0.927817
$545$	1.23840	0.0530471
$546$	0	0
$547$	−0.0895697	−0.00382972	−0.00191486	−	0.999998i	$-0.500610\pi$
$547$	−0.0895697	−0.00382972	−0.00191486	+	0.999998i	$0.500610\pi$
$548$	0.657170	0.0280729
$549$	0	0
$550$	0	0
$551$	−20.2932	−0.864518
$552$	0	0
$553$	−5.24490	−0.223036
$554$	−23.5907	−1.00227
$555$	0	0
$556$	54.7585	2.32228
$557$	−3.07514	−0.130298	−0.0651489	−	0.997876i	$-0.520752\pi$
$557$	−3.07514	−0.130298	−0.0651489	+	0.997876i	$0.520752\pi$
$558$	0	0
$559$	−73.3126	−3.10079
$560$	2.92976	0.123805
$561$	0	0
$562$	43.9779	1.85510
$563$	17.8141	0.750775	0.375387	−	0.926868i	$-0.377510\pi$
$563$	17.8141	0.750775	0.375387	+	0.926868i	$0.377510\pi$
$564$	0	0
$565$	−0.488208	−0.0205391
$566$	−15.3866	−0.646746
$567$	0	0
$568$	54.6968	2.29503
$569$	−4.24899	−0.178127	−0.0890634	−	0.996026i	$-0.528387\pi$
$569$	−4.24899	−0.178127	−0.0890634	+	0.996026i	$0.528387\pi$
$570$	0	0
$571$	−45.0573	−1.88559	−0.942795	−	0.333373i	$-0.891813\pi$
$571$	−45.0573	−1.88559	−0.942795	+	0.333373i	$0.891813\pi$
$572$	0	0
$573$	0	0
$574$	33.2541	1.38800
$575$	27.4335	1.14406
$576$	0	0
$577$	−15.2751	−0.635909	−0.317954	−	0.948106i	$-0.602996\pi$
$577$	−15.2751	−0.635909	−0.317954	+	0.948106i	$0.602996\pi$
$578$	−35.8508	−1.49120
$579$	0	0
$580$	−7.32700	−0.304237
$581$	18.2843	0.758559
$582$	0	0
$583$	0	0
$584$	−4.28101	−0.177150
$585$	0	0
$586$	39.3944	1.62737
$587$	−42.5860	−1.75771	−0.878856	−	0.477086i	$-0.841693\pi$
$587$	−42.5860	−1.75771	−0.878856	+	0.477086i	$0.841693\pi$
$588$	0	0
$589$	−12.9010	−0.531576
$590$	2.47403	0.101854
$591$	0	0
$592$	−56.0344	−2.30300
$593$	−10.6230	−0.436236	−0.218118	−	0.975922i	$-0.569992\pi$
$593$	−10.6230	−0.436236	−0.218118	+	0.975922i	$0.569992\pi$
$594$	0	0
$595$	−0.521100	−0.0213630
$596$	19.3821	0.793920
$597$	0	0
$598$	88.1018	3.60275
$599$	34.7679	1.42058	0.710289	−	0.703911i	$-0.248564\pi$
$599$	34.7679	1.42058	0.710289	+	0.703911i	$0.248564\pi$
$600$	0	0
$601$	9.08763	0.370692	0.185346	−	0.982673i	$-0.440659\pi$
$601$	9.08763	0.370692	0.185346	+	0.982673i	$0.440659\pi$
$602$	−37.8636	−1.54320
$603$	0	0
$604$	−13.1534	−0.535204
$605$	0	0
$606$	0	0
$607$	−15.5962	−0.633029	−0.316515	−	0.948588i	$-0.602513\pi$
$607$	−15.5962	−0.633029	−0.316515	+	0.948588i	$0.602513\pi$
$608$	−38.4575	−1.55966
$609$	0	0
$610$	6.75595	0.273540
$611$	−9.98970	−0.404140
$612$	0	0
$613$	−27.8024	−1.12293	−0.561464	−	0.827501i	$-0.689762\pi$
$613$	−27.8024	−1.12293	−0.561464	+	0.827501i	$0.689762\pi$
$614$	17.1542	0.692286
$615$	0	0
$616$	0	0
$617$	−8.03380	−0.323428	−0.161714	−	0.986838i	$-0.551702\pi$
$617$	−8.03380	−0.323428	−0.161714	+	0.986838i	$0.551702\pi$
$618$	0	0
$619$	3.60011	0.144701	0.0723503	−	0.997379i	$-0.476950\pi$
$619$	3.60011	0.144701	0.0723503	+	0.997379i	$0.476950\pi$
$620$	−4.65800	−0.187070
$621$	0	0
$622$	37.9012	1.51970
$623$	0.0742279	0.00297388
$624$	0	0
$625$	24.1425	0.965698
$626$	41.3810	1.65392
$627$	0	0
$628$	−52.4483	−2.09292
$629$	9.96652	0.397391
$630$	0	0
$631$	−20.2394	−0.805718	−0.402859	−	0.915262i	$-0.631984\pi$
$631$	−20.2394	−0.805718	−0.402859	+	0.915262i	$0.631984\pi$
$632$	33.8786	1.34762
$633$	0	0
$634$	77.0847	3.06143
$635$	0.286974	0.0113882
$636$	0	0
$637$	33.7697	1.33800
$638$	0	0
$639$	0	0
$640$	−0.893277	−0.0353099
$641$	27.8787	1.10114	0.550572	−	0.834788i	$-0.314410\pi$
$641$	27.8787	1.10114	0.550572	+	0.834788i	$0.314410\pi$
$642$	0	0
$643$	0.0894662	0.00352820	0.00176410	−	0.999998i	$-0.499438\pi$
$643$	0.0894662	0.00352820	0.00176410	+	0.999998i	$0.499438\pi$
$644$	32.3380	1.27430
$645$	0	0
$646$	15.7227	0.618601
$647$	47.1805	1.85486	0.927428	−	0.374001i	$-0.122014\pi$
$647$	47.1805	1.85486	0.927428	+	0.374001i	$0.122014\pi$
$648$	0	0
$649$	0	0
$650$	78.4543	3.07723
$651$	0	0
$652$	36.2773	1.42073
$653$	−17.2808	−0.676249	−0.338124	−	0.941101i	$-0.609792\pi$
$653$	−17.2808	−0.676249	−0.338124	+	0.941101i	$0.609792\pi$
$654$	0	0
$655$	1.38154	0.0539812
$656$	−109.996	−4.29463
$657$	0	0
$658$	−5.15936	−0.201133
$659$	12.0864	0.470818	0.235409	−	0.971896i	$-0.424357\pi$
$659$	12.0864	0.470818	0.235409	+	0.971896i	$0.424357\pi$
$660$	0	0
$661$	38.5830	1.50071	0.750353	−	0.661037i	$-0.229884\pi$
$661$	38.5830	1.50071	0.750353	+	0.661037i	$0.229884\pi$
$662$	35.2631	1.37054
$663$	0	0
$664$	−118.104	−4.58334
$665$	−0.926061	−0.0359111
$666$	0	0
$667$	−34.5519	−1.33785
$668$	−50.7135	−1.96216
$669$	0	0
$670$	−5.80808	−0.224386
$671$	0	0
$672$	0	0
$673$	28.8426	1.11180	0.555900	−	0.831249i	$-0.312374\pi$
$673$	28.8426	1.11180	0.555900	+	0.831249i	$0.312374\pi$
$674$	33.9763	1.30872
$675$	0	0
$676$	115.191	4.43044
$677$	7.04711	0.270842	0.135421	−	0.990788i	$-0.456761\pi$
$677$	7.04711	0.270842	0.135421	+	0.990788i	$0.456761\pi$
$678$	0	0
$679$	−7.22538	−0.277285
$680$	3.36597	0.129079
$681$	0	0
$682$	0	0
$683$	−32.5733	−1.24638	−0.623191	−	0.782070i	$-0.714164\pi$
$683$	−32.5733	−1.24638	−0.623191	+	0.782070i	$0.714164\pi$
$684$	0	0
$685$	−0.0320424	−0.00122428
$686$	39.2663	1.49919
$687$	0	0
$688$	125.243	4.77484
$689$	52.1709	1.98755
$690$	0	0
$691$	43.1809	1.64268	0.821339	−	0.570441i	$-0.193228\pi$
$691$	43.1809	1.64268	0.821339	+	0.570441i	$0.193228\pi$
$692$	78.9608	3.00164
$693$	0	0
$694$	−91.7519	−3.48285
$695$	−2.66993	−0.101276
$696$	0	0
$697$	19.5644	0.741054
$698$	−9.78171	−0.370243
$699$	0	0
$700$	28.7969	1.08842
$701$	−27.1707	−1.02622	−0.513111	−	0.858322i	$-0.671507\pi$
$701$	−27.1707	−1.02622	−0.513111	+	0.858322i	$0.671507\pi$
$702$	0	0
$703$	17.7118	0.668013
$704$	0	0
$705$	0	0
$706$	−10.3924	−0.391121
$707$	18.0929	0.680453
$708$	0	0
$709$	−21.4612	−0.805994	−0.402997	−	0.915201i	$-0.632031\pi$
$709$	−21.4612	−0.805994	−0.402997	+	0.915201i	$0.632031\pi$
$710$	−4.49783	−0.168801
$711$	0	0
$712$	−0.479464	−0.0179687
$713$	−21.9657	−0.822622
$714$	0	0
$715$	0	0
$716$	−18.2555	−0.682240
$717$	0	0
$718$	−19.4673	−0.726514
$719$	−18.1638	−0.677396	−0.338698	−	0.940895i	$-0.609987\pi$
$719$	−18.1638	−0.677396	−0.338698	+	0.940895i	$0.609987\pi$
$720$	0	0
$721$	−14.8187	−0.551878
$722$	−22.0154	−0.819330
$723$	0	0
$724$	−74.2412	−2.75915
$725$	−30.7683	−1.14271
$726$	0	0
$727$	−25.5506	−0.947619	−0.473810	−	0.880627i	$-0.657122\pi$
$727$	−25.5506	−0.947619	−0.473810	+	0.880627i	$0.657122\pi$
$728$	54.8348	2.03231
$729$	0	0
$730$	0.352037	0.0130295
$731$	−22.2763	−0.823917
$732$	0	0
$733$	−40.7368	−1.50465	−0.752324	−	0.658793i	$-0.771067\pi$
$733$	−40.7368	−1.50465	−0.752324	+	0.658793i	$0.771067\pi$
$734$	−66.8260	−2.46659
$735$	0	0
$736$	−65.4790	−2.41359
$737$	0	0
$738$	0	0
$739$	−32.1099	−1.18118	−0.590591	−	0.806971i	$-0.701105\pi$
$739$	−32.1099	−1.18118	−0.590591	+	0.806971i	$0.701105\pi$
$740$	6.39497	0.235084
$741$	0	0
$742$	26.9446	0.989166
$743$	17.9623	0.658974	0.329487	−	0.944160i	$-0.393124\pi$
$743$	17.9623	0.658974	0.329487	+	0.944160i	$0.393124\pi$
$744$	0	0
$745$	−0.945035	−0.0346234
$746$	−12.5542	−0.459643
$747$	0	0
$748$	0	0
$749$	−12.7958	−0.467547
$750$	0	0
$751$	−24.8837	−0.908019	−0.454009	−	0.890997i	$-0.650007\pi$
$751$	−24.8837	−0.908019	−0.454009	+	0.890997i	$0.650007\pi$
$752$	17.0658	0.622327
$753$	0	0
$754$	−98.8115	−3.59850
$755$	0.641337	0.0233406
$756$	0	0
$757$	−32.5184	−1.18190	−0.590951	−	0.806708i	$-0.701247\pi$
$757$	−32.5184	−1.18190	−0.590951	+	0.806708i	$0.701247\pi$
$758$	−45.8775	−1.66635
$759$	0	0
$760$	5.98175	0.216981
$761$	−8.82315	−0.319839	−0.159919	−	0.987130i	$-0.551123\pi$
$761$	−8.82315	−0.319839	−0.159919	+	0.987130i	$0.551123\pi$
$762$	0	0
$763$	6.13013	0.221925
$764$	58.8795	2.13018
$765$	0	0
$766$	8.90840	0.321874
$767$	23.7122	0.856199
$768$	0	0
$769$	−26.9649	−0.972378	−0.486189	−	0.873854i	$-0.661613\pi$
$769$	−26.9649	−0.972378	−0.486189	+	0.873854i	$0.661613\pi$
$770$	0	0
$771$	0	0
$772$	−110.711	−3.98458
$773$	−26.6678	−0.959175	−0.479588	−	0.877494i	$-0.659214\pi$
$773$	−26.6678	−0.959175	−0.479588	+	0.877494i	$0.659214\pi$
$774$	0	0
$775$	−19.5604	−0.702629
$776$	46.6713	1.67540
$777$	0	0
$778$	49.3105	1.76787
$779$	34.7684	1.24571
$780$	0	0
$781$	0	0
$782$	26.7700	0.957294
$783$	0	0
$784$	−57.6902	−2.06036
$785$	2.55729	0.0912735
$786$	0	0
$787$	−16.5454	−0.589781	−0.294891	−	0.955531i	$-0.595283\pi$
$787$	−16.5454	−0.589781	−0.294891	+	0.955531i	$0.595283\pi$
$788$	−48.9016	−1.74205
$789$	0	0
$790$	−2.78591	−0.0991183
$791$	−2.41666	−0.0859264
$792$	0	0
$793$	64.7520	2.29941
$794$	28.1748	0.999886
$795$	0	0
$796$	−76.4580	−2.70998
$797$	44.5185	1.57693	0.788463	−	0.615082i	$-0.210877\pi$
$797$	44.5185	1.57693	0.788463	+	0.615082i	$0.210877\pi$
$798$	0	0
$799$	−3.03540	−0.107385
$800$	−58.3088	−2.06153
$801$	0	0
$802$	7.44789	0.262994
$803$	0	0
$804$	0	0
$805$	−1.57674	−0.0555730
$806$	−62.8175	−2.21265
$807$	0	0
$808$	−116.868	−4.11141
$809$	−49.0736	−1.72533	−0.862667	−	0.505772i	$-0.831208\pi$
$809$	−49.0736	−1.72533	−0.862667	+	0.505772i	$0.831208\pi$
$810$	0	0
$811$	−42.7314	−1.50050	−0.750251	−	0.661153i	$-0.770067\pi$
$811$	−42.7314	−1.50050	−0.750251	+	0.661153i	$0.770067\pi$
$812$	−36.2690	−1.27279
$813$	0	0
$814$	0	0
$815$	−1.76882	−0.0619589
$816$	0	0
$817$	−39.5878	−1.38500
$818$	61.9035	2.16441
$819$	0	0
$820$	12.5534	0.438384
$821$	28.9770	1.01130	0.505652	−	0.862737i	$-0.331252\pi$
$821$	28.9770	1.01130	0.505652	+	0.862737i	$0.331252\pi$
$822$	0	0
$823$	19.2295	0.670298	0.335149	−	0.942165i	$-0.391213\pi$
$823$	19.2295	0.670298	0.335149	+	0.942165i	$0.391213\pi$
$824$	95.7193	3.33454
$825$	0	0
$826$	12.2466	0.426114
$827$	−43.6052	−1.51630	−0.758150	−	0.652080i	$-0.773897\pi$
$827$	−43.6052	−1.51630	−0.758150	+	0.652080i	$0.773897\pi$
$828$	0	0
$829$	21.6237	0.751022	0.375511	−	0.926818i	$-0.377467\pi$
$829$	21.6237	0.751022	0.375511	+	0.926818i	$0.377467\pi$
$830$	9.71198	0.337108
$831$	0	0
$832$	−62.7352	−2.17495
$833$	10.2610	0.355524
$834$	0	0
$835$	2.47270	0.0855713
$836$	0	0
$837$	0	0
$838$	55.5239	1.91804
$839$	22.2036	0.766554	0.383277	−	0.923634i	$-0.374796\pi$
$839$	22.2036	0.766554	0.383277	+	0.923634i	$0.374796\pi$
$840$	0	0
$841$	9.75201	0.336276
$842$	−61.8338	−2.13093
$843$	0	0
$844$	−11.7666	−0.405022
$845$	−5.61653	−0.193214
$846$	0	0
$847$	0	0
$848$	−89.1256	−3.06059
$849$	0	0
$850$	23.8386	0.817657
$851$	30.1567	1.03376
$852$	0	0
$853$	26.3161	0.901047	0.450523	−	0.892765i	$-0.351237\pi$
$853$	26.3161	0.901047	0.450523	+	0.892765i	$0.351237\pi$
$854$	33.4423	1.14437
$855$	0	0
$856$	82.6524	2.82500
$857$	27.9702	0.955442	0.477721	−	0.878512i	$-0.341463\pi$
$857$	27.9702	0.955442	0.477721	+	0.878512i	$0.341463\pi$
$858$	0	0
$859$	15.3807	0.524784	0.262392	−	0.964961i	$-0.415489\pi$
$859$	15.3807	0.524784	0.262392	+	0.964961i	$0.415489\pi$
$860$	−14.2935	−0.487403
$861$	0	0
$862$	−13.0936	−0.445970
$863$	32.8422	1.11796	0.558980	−	0.829181i	$-0.311193\pi$
$863$	32.8422	1.11796	0.558980	+	0.829181i	$0.311193\pi$
$864$	0	0
$865$	−3.84999	−0.130904
$866$	−67.2714	−2.28598
$867$	0	0
$868$	−23.0573	−0.782617
$869$	0	0
$870$	0	0
$871$	−55.6672	−1.88621
$872$	−39.5966	−1.34091
$873$	0	0
$874$	47.5738	1.60921
$875$	−2.82447	−0.0954846
$876$	0	0
$877$	50.0173	1.68896	0.844482	−	0.535583i	$-0.179908\pi$
$877$	50.0173	1.68896	0.844482	+	0.535583i	$0.179908\pi$
$878$	−12.2527	−0.413507
$879$	0	0
$880$	0	0
$881$	32.0798	1.08080	0.540399	−	0.841409i	$-0.318273\pi$
$881$	32.0798	1.08080	0.540399	+	0.841409i	$0.318273\pi$
$882$	0	0
$883$	46.6107	1.56857	0.784287	−	0.620398i	$-0.213029\pi$
$883$	46.6107	1.56857	0.784287	+	0.620398i	$0.213029\pi$
$884$	54.4089	1.82997
$885$	0	0
$886$	−60.7992	−2.04259
$887$	−29.3501	−0.985478	−0.492739	−	0.870177i	$-0.664004\pi$
$887$	−29.3501	−0.985478	−0.492739	+	0.870177i	$0.664004\pi$
$888$	0	0
$889$	1.42054	0.0476433
$890$	0.0394273	0.00132161
$891$	0	0
$892$	63.7925	2.13593
$893$	−5.39430	−0.180513
$894$	0	0
$895$	0.890105	0.0297529
$896$	−4.42177	−0.147721
$897$	0	0
$898$	36.1900	1.20768
$899$	24.6358	0.821652
$900$	0	0
$901$	15.8523	0.528116
$902$	0	0
$903$	0	0
$904$	15.6100	0.519182
$905$	3.61987	0.120328
$906$	0	0
$907$	−25.1195	−0.834081	−0.417040	−	0.908888i	$-0.636933\pi$
$907$	−25.1195	−0.834081	−0.417040	+	0.908888i	$0.636933\pi$
$908$	90.6519	3.00839
$909$	0	0
$910$	−4.50918	−0.149478
$911$	−2.76894	−0.0917392	−0.0458696	−	0.998947i	$-0.514606\pi$
$911$	−2.76894	−0.0917392	−0.0458696	+	0.998947i	$0.514606\pi$
$912$	0	0
$913$	0	0
$914$	28.5003	0.942707
$915$	0	0
$916$	98.9509	3.26943
$917$	6.83869	0.225833
$918$	0	0
$919$	11.8435	0.390679	0.195340	−	0.980736i	$-0.437419\pi$
$919$	11.8435	0.390679	0.195340	+	0.980736i	$0.437419\pi$
$920$	10.1847	0.335781
$921$	0	0
$922$	7.69836	0.253532
$923$	−43.1092	−1.41896
$924$	0	0
$925$	26.8544	0.882969
$926$	−71.2105	−2.34012
$927$	0	0
$928$	73.4387	2.41074
$929$	−11.2585	−0.369380	−0.184690	−	0.982797i	$-0.559128\pi$
$929$	−11.2585	−0.369380	−0.184690	+	0.982797i	$0.559128\pi$
$930$	0	0
$931$	18.2352	0.597634
$932$	−36.1570	−1.18436
$933$	0	0
$934$	−29.7284	−0.972742
$935$	0	0
$936$	0	0
$937$	−31.7496	−1.03721	−0.518607	−	0.855013i	$-0.673549\pi$
$937$	−31.7496	−1.03721	−0.518607	+	0.855013i	$0.673549\pi$
$938$	−28.7503	−0.938730
$939$	0	0
$940$	−1.94765	−0.0635254
$941$	−12.6099	−0.411070	−0.205535	−	0.978650i	$-0.565894\pi$
$941$	−12.6099	−0.411070	−0.205535	+	0.978650i	$0.565894\pi$
$942$	0	0
$943$	59.1980	1.92775
$944$	−40.5086	−1.31844
$945$	0	0
$946$	0	0
$947$	0.369770	0.0120159	0.00600796	−	0.999982i	$-0.498088\pi$
$947$	0.369770	0.0120159	0.00600796	+	0.999982i	$0.498088\pi$
$948$	0	0
$949$	3.37408	0.109527
$950$	42.3642	1.37448
$951$	0	0
$952$	16.6617	0.540009
$953$	−16.9401	−0.548743	−0.274372	−	0.961624i	$-0.588470\pi$
$953$	−16.9401	−0.548743	−0.274372	+	0.961624i	$0.588470\pi$
$954$	0	0
$955$	−2.87086	−0.0928988
$956$	47.2937	1.52959
$957$	0	0
$958$	−79.8933	−2.58124
$959$	−0.158612	−0.00512184
$960$	0	0
$961$	−15.3382	−0.494782
$962$	86.2422	2.78056
$963$	0	0
$964$	−130.510	−4.20345
$965$	5.39808	0.173770
$966$	0	0
$967$	24.8161	0.798031	0.399016	−	0.916944i	$-0.369352\pi$
$967$	24.8161	0.798031	0.399016	+	0.916944i	$0.369352\pi$
$968$	0	0
$969$	0	0
$970$	−3.83788	−0.123227
$971$	−12.1354	−0.389445	−0.194722	−	0.980858i	$-0.562381\pi$
$971$	−12.1354	−0.389445	−0.194722	+	0.980858i	$0.562381\pi$
$972$	0	0
$973$	−13.2163	−0.423695
$974$	−8.41280	−0.269564
$975$	0	0
$976$	−110.619	−3.54081
$977$	35.3764	1.13179	0.565896	−	0.824476i	$-0.308530\pi$
$977$	35.3764	1.13179	0.565896	+	0.824476i	$0.308530\pi$
$978$	0	0
$979$	0	0
$980$	6.58394	0.210316
$981$	0	0
$982$	−15.7275	−0.501885
$983$	22.4859	0.717190	0.358595	−	0.933493i	$-0.383256\pi$
$983$	22.4859	0.717190	0.358595	+	0.933493i	$0.383256\pi$
$984$	0	0
$985$	2.38435	0.0759719
$986$	−30.0242	−0.956164
$987$	0	0
$988$	96.6916	3.07617
$989$	−67.4036	−2.14331
$990$	0	0
$991$	16.2761	0.517028	0.258514	−	0.966008i	$-0.416767\pi$
$991$	16.2761	0.517028	0.258514	+	0.966008i	$0.416767\pi$
$992$	46.6872	1.48232
$993$	0	0
$994$	−22.2645	−0.706187
$995$	3.72796	0.118184
$996$	0	0
$997$	4.62418	0.146449	0.0732246	−	0.997315i	$-0.476671\pi$
$997$	4.62418	0.146449	0.0732246	+	0.997315i	$0.476671\pi$
$998$	77.2417	2.44504
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.cg.1.12		12
3.2	odd	2		9801.2.a.ck.1.1		12
11.5	even	5		891.2.f.c.487.1	✓	24
11.9	even	5		891.2.f.c.730.1	yes	24
11.10	odd	2		9801.2.a.cl.1.1		12
33.5	odd	10		891.2.f.d.487.6	yes	24
33.20	odd	10		891.2.f.d.730.6	yes	24
33.32	even	2		9801.2.a.cf.1.12		12
99.5	odd	30		891.2.n.j.784.6		48
99.16	even	15		891.2.n.k.190.6		48
99.20	odd	30		891.2.n.j.433.6		48
99.31	even	15		891.2.n.k.136.6		48
99.38	odd	30		891.2.n.j.190.1		48
99.49	even	15		891.2.n.k.784.1		48
99.86	odd	30		891.2.n.j.136.1		48
99.97	even	15		891.2.n.k.433.1		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
891.2.f.c.487.1	✓	24	11.5	even	5
891.2.f.c.730.1	yes	24	11.9	even	5
891.2.f.d.487.6	yes	24	33.5	odd	10
891.2.f.d.730.6	yes	24	33.20	odd	10
891.2.n.j.136.1		48	99.86	odd	30
891.2.n.j.190.1		48	99.38	odd	30
891.2.n.j.433.6		48	99.20	odd	30
891.2.n.j.784.6		48	99.5	odd	30
891.2.n.k.136.6		48	99.31	even	15
891.2.n.k.190.6		48	99.16	even	15
891.2.n.k.433.1		48	99.97	even	15
891.2.n.k.784.1		48	99.49	even	15
9801.2.a.cf.1.12		12	33.32	even	2
9801.2.a.cg.1.12		12	1.1	even	1	trivial
9801.2.a.ck.1.1		12	3.2	odd	2
9801.2.a.cl.1.1		12	11.10	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q+2.62930 q^{2} +4.91321 q^{4} -0.239560 q^{5} -1.18583 q^{7} +7.65971 q^{8} -0.629874 q^{10} -6.03699 q^{13} -3.11791 q^{14} +10.3132 q^{16} -1.83436 q^{17} -3.25989 q^{19} -1.17701 q^{20} -5.55040 q^{23} -4.94261 q^{25} -15.8730 q^{26} -5.82625 q^{28} +6.22511 q^{29} +3.95750 q^{31} +11.7972 q^{32} -4.82307 q^{34} +0.284077 q^{35} -5.43325 q^{37} -8.57122 q^{38} -1.83496 q^{40} -10.6655 q^{41} +12.1439 q^{43} -14.5937 q^{46} +1.65475 q^{47} -5.59380 q^{49} -12.9956 q^{50} -29.6610 q^{52} -8.64187 q^{53} -9.08313 q^{56} +16.3677 q^{58} -3.92783 q^{59} -10.7259 q^{61} +10.4054 q^{62} +10.3918 q^{64} +1.44622 q^{65} +9.22102 q^{67} -9.01259 q^{68} +0.746925 q^{70} +7.14085 q^{71} -0.558900 q^{73} -14.2856 q^{74} -16.0165 q^{76} +4.42297 q^{79} -2.47063 q^{80} -28.0429 q^{82} -15.4189 q^{83} +0.439438 q^{85} +31.9300 q^{86} -0.0625956 q^{89} +7.15886 q^{91} -27.2703 q^{92} +4.35083 q^{94} +0.780938 q^{95} +6.09309 q^{97} -14.7078 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{2} + 11 q^{4} + 2 q^{5} - 9 q^{7} - 8 q^{10} - 9 q^{13} - 21 q^{14} + 21 q^{16} + 5 q^{17} - 18 q^{19} + 11 q^{20} + 3 q^{23} + 14 q^{25} + 8 q^{26} - 17 q^{28} + 6 q^{29} + 6 q^{31} + 9 q^{32}+ \cdots + 34 q^{98}+O(q^{100}) \) 12 * q - q^2 + 11 * q^4 + 2 * q^5 - 9 * q^7 - 8 * q^10 - 9 * q^13 - 21 * q^14 + 21 * q^16 + 5 * q^17 - 18 * q^19 + 11 * q^20 + 3 * q^23 + 14 * q^25 + 8 * q^26 - 17 * q^28 + 6 * q^29 + 6 * q^31 + 9 * q^32 - 21 * q^34 - 5 * q^35 - 7 * q^37 - 10 * q^38 - 21 * q^40 - 5 * q^41 - 15 * q^43 - 37 * q^46 + 17 * q^47 + 11 * q^49 + 2 * q^50 - 18 * q^52 - 19 * q^53 - 24 * q^56 + 14 * q^58 + 23 * q^59 - 31 * q^61 + 24 * q^62 + 44 * q^64 - 34 * q^65 - 7 * q^67 + 66 * q^68 - 15 * q^70 + 7 * q^71 - 25 * q^73 - 61 * q^74 - 51 * q^76 - 47 * q^79 + 49 * q^80 + 12 * q^82 - 24 * q^83 - 15 * q^85 + 25 * q^86 + 17 * q^89 + 29 * q^91 - 48 * q^92 - 45 * q^94 + 23 * q^95 + 15 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more