LMFDB - Embedded newform 3645.2.a.c.1.5

Error: no document with id 279330657 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3645,2,Mod(1,3645)] 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("3645.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3645, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3645.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-3,0,9,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.1054715368$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3916917.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 12x^{3} + 18x^{2} - 3 $ x^6 - 3x^5 - 6x^4 + 12x^3 + 18x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.75528$ of defining polynomial
Character	$\chi$	$=$	3645.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.75528 q^{2} +1.08100 q^{4} +1.00000 q^{5} -0.223190 q^{7} -1.61309 q^{8} +1.75528 q^{10} -0.190142 q^{11} -6.78155 q^{13} -0.391761 q^{14} -4.99344 q^{16} +1.48457 q^{17} -0.985031 q^{19} +1.08100 q^{20} -0.333752 q^{22} +7.07520 q^{23} +1.00000 q^{25} -11.9035 q^{26} -0.241270 q^{28} -2.57705 q^{29} -2.51422 q^{31} -5.53869 q^{32} +2.60583 q^{34} -0.223190 q^{35} -9.93302 q^{37} -1.72900 q^{38} -1.61309 q^{40} -5.70334 q^{41} +6.55808 q^{43} -0.205544 q^{44} +12.4190 q^{46} -8.85966 q^{47} -6.95019 q^{49} +1.75528 q^{50} -7.33089 q^{52} -1.90890 q^{53} -0.190142 q^{55} +0.360027 q^{56} -4.52344 q^{58} -9.38596 q^{59} +0.0161816 q^{61} -4.41316 q^{62} +0.264929 q^{64} -6.78155 q^{65} -9.87354 q^{67} +1.60482 q^{68} -0.391761 q^{70} +5.03559 q^{71} +7.91456 q^{73} -17.4352 q^{74} -1.06482 q^{76} +0.0424378 q^{77} -9.91847 q^{79} -4.99344 q^{80} -10.0109 q^{82} +13.2221 q^{83} +1.48457 q^{85} +11.5113 q^{86} +0.306716 q^{88} -6.08753 q^{89} +1.51358 q^{91} +7.64832 q^{92} -15.5512 q^{94} -0.985031 q^{95} +3.75293 q^{97} -12.1995 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{2} + 9 q^{4} + 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 9 q^{11} - 6 q^{13} - 24 q^{14} - 9 q^{16} + 15 q^{17} - 9 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 6 q^{25} - 3 q^{26} + 12 q^{28} - 30 q^{29}+ \cdots - 15 q^{97}+O(q^{100}) $ 6 * q - 3 * q^2 + 9 * q^4 + 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 9 * q^11 - 6 * q^13 - 24 * q^14 - 9 * q^16 + 15 * q^17 - 9 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 6 * q^25 - 3 * q^26 + 12 * q^28 - 30 * q^29 - 3 * q^31 - 9 * q^32 + 15 * q^34 + 3 * q^35 - 18 * q^37 - 18 * q^38 - 3 * q^40 - 15 * q^41 - 3 * q^43 - 36 * q^44 + 36 * q^46 - 15 * q^47 - 3 * q^49 - 3 * q^50 + 18 * q^52 - 33 * q^53 - 9 * q^55 - 24 * q^56 + 6 * q^58 - 27 * q^59 + 3 * q^61 - 6 * q^62 - 27 * q^64 - 6 * q^65 - 33 * q^67 + 12 * q^68 - 24 * q^70 - 27 * q^71 - 9 * q^73 + 3 * q^74 - 6 * q^76 - 9 * q^77 + 18 * q^79 - 9 * q^80 - 27 * q^82 - 15 * q^83 + 15 * q^85 + 15 * q^86 + 33 * q^88 + 15 * q^89 - 21 * q^91 - 9 * q^92 + 18 * q^94 - 9 * q^95 - 15 * q^97

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 $

$2$	1.75528	1.24117	0.620585	−	0.784139i	$-0.286895\pi$
$2$	1.75528	1.24117	0.620585	+	0.784139i	$0.286895\pi$
$3$	0	0
$3$	0	0
$4$	1.08100	0.540502
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.223190	−0.0843580	−0.0421790	−	0.999110i	$-0.513430\pi$
$7$	−0.223190	−0.0843580	−0.0421790	+	0.999110i	$0.513430\pi$
$8$	−1.61309	−0.570315
$9$	0	0
$10$	1.75528	0.555068
$11$	−0.190142	−0.0573299	−0.0286649	−	0.999589i	$-0.509126\pi$
$11$	−0.190142	−0.0573299	−0.0286649	+	0.999589i	$0.509126\pi$
$12$	0	0
$13$	−6.78155	−1.88086	−0.940432	−	0.339981i	$-0.889579\pi$
$13$	−6.78155	−1.88086	−0.940432	+	0.339981i	$0.889579\pi$
$14$	−0.391761	−0.104703
$15$	0	0
$16$	−4.99344	−1.24836
$17$	1.48457	0.360060	0.180030	−	0.983661i	$-0.442380\pi$
$17$	1.48457	0.360060	0.180030	+	0.983661i	$0.442380\pi$
$18$	0	0
$19$	−0.985031	−0.225982	−0.112991	−	0.993596i	$-0.536043\pi$
$19$	−0.985031	−0.225982	−0.112991	+	0.993596i	$0.536043\pi$
$20$	1.08100	0.241720
$21$	0	0
$22$	−0.333752	−0.0711561
$23$	7.07520	1.47528	0.737641	−	0.675193i	$-0.235940\pi$
$23$	7.07520	1.47528	0.737641	+	0.675193i	$0.235940\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−11.9035	−2.33447
$27$	0	0
$28$	−0.241270	−0.0455957
$29$	−2.57705	−0.478546	−0.239273	−	0.970952i	$-0.576909\pi$
$29$	−2.57705	−0.478546	−0.239273	+	0.970952i	$0.576909\pi$
$30$	0	0
$31$	−2.51422	−0.451567	−0.225784	−	0.974177i	$-0.572494\pi$
$31$	−2.51422	−0.451567	−0.225784	+	0.974177i	$0.572494\pi$
$32$	−5.53869	−0.979111
$33$	0	0
$34$	2.60583	0.446896
$35$	−0.223190	−0.0377260
$36$	0	0
$37$	−9.93302	−1.63298	−0.816489	−	0.577361i	$-0.804082\pi$
$37$	−9.93302	−1.63298	−0.816489	+	0.577361i	$0.804082\pi$
$38$	−1.72900	−0.280482
$39$	0	0
$40$	−1.61309	−0.255052
$41$	−5.70334	−0.890712	−0.445356	−	0.895354i	$-0.646923\pi$
$41$	−5.70334	−0.890712	−0.445356	+	0.895354i	$0.646923\pi$
$42$	0	0
$43$	6.55808	1.00010	0.500049	−	0.865997i	$-0.333315\pi$
$43$	6.55808	1.00010	0.500049	+	0.865997i	$0.333315\pi$
$44$	−0.205544	−0.0309869
$45$	0	0
$46$	12.4190	1.83107
$47$	−8.85966	−1.29231	−0.646157	−	0.763204i	$-0.723625\pi$
$47$	−8.85966	−1.29231	−0.646157	+	0.763204i	$0.723625\pi$
$48$	0	0
$49$	−6.95019	−0.992884
$50$	1.75528	0.248234
$51$	0	0
$52$	−7.33089	−1.01661
$53$	−1.90890	−0.262208	−0.131104	−	0.991369i	$-0.541852\pi$
$53$	−1.90890	−0.262208	−0.131104	+	0.991369i	$0.541852\pi$
$54$	0	0
$55$	−0.190142	−0.0256387
$56$	0.360027	0.0481106
$57$	0	0
$58$	−4.52344	−0.593957
$59$	−9.38596	−1.22195	−0.610974	−	0.791651i	$-0.709222\pi$
$59$	−9.38596	−1.22195	−0.610974	+	0.791651i	$0.709222\pi$
$60$	0	0
$61$	0.0161816	0.00207184	0.00103592	−	0.999999i	$-0.499670\pi$
$61$	0.0161816	0.00207184	0.00103592	+	0.999999i	$0.499670\pi$
$62$	−4.41316	−0.560472
$63$	0	0
$64$	0.264929	0.0331161
$65$	−6.78155	−0.841148
$66$	0	0
$67$	−9.87354	−1.20625	−0.603123	−	0.797649i	$-0.706077\pi$
$67$	−9.87354	−1.20625	−0.603123	+	0.797649i	$0.706077\pi$
$68$	1.60482	0.194613
$69$	0	0
$70$	−0.391761	−0.0468244
$71$	5.03559	0.597615	0.298807	−	0.954313i	$-0.403411\pi$
$71$	5.03559	0.597615	0.298807	+	0.954313i	$0.403411\pi$
$72$	0	0
$73$	7.91456	0.926329	0.463165	−	0.886272i	$-0.346714\pi$
$73$	7.91456	0.926329	0.463165	+	0.886272i	$0.346714\pi$
$74$	−17.4352	−2.02680
$75$	0	0
$76$	−1.06482	−0.122144
$77$	0.0424378	0.00483623
$78$	0	0
$79$	−9.91847	−1.11591	−0.557957	−	0.829870i	$-0.688415\pi$
$79$	−9.91847	−1.11591	−0.557957	+	0.829870i	$0.688415\pi$
$80$	−4.99344	−0.558283
$81$	0	0
$82$	−10.0109	−1.10552
$83$	13.2221	1.45132	0.725659	−	0.688055i	$-0.241535\pi$
$83$	13.2221	1.45132	0.725659	+	0.688055i	$0.241535\pi$
$84$	0	0
$85$	1.48457	0.161024
$86$	11.5113	1.24129
$87$	0	0
$88$	0.306716	0.0326961
$89$	−6.08753	−0.645277	−0.322639	−	0.946522i	$-0.604570\pi$
$89$	−6.08753	−0.645277	−0.322639	+	0.946522i	$0.604570\pi$
$90$	0	0
$91$	1.51358	0.158666
$92$	7.64832	0.797393
$93$	0	0
$94$	−15.5512	−1.60398
$95$	−0.985031	−0.101062
$96$	0	0
$97$	3.75293	0.381052	0.190526	−	0.981682i	$-0.438981\pi$
$97$	3.75293	0.381052	0.190526	+	0.981682i	$0.438981\pi$
$98$	−12.1995	−1.23234
$99$	0	0
$100$	1.08100	0.108100
$101$	−0.511790	−0.0509250	−0.0254625	−	0.999676i	$-0.508106\pi$
$101$	−0.511790	−0.0509250	−0.0254625	+	0.999676i	$0.508106\pi$
$102$	0	0
$103$	−1.19413	−0.117661	−0.0588305	−	0.998268i	$-0.518737\pi$
$103$	−1.19413	−0.117661	−0.0588305	+	0.998268i	$0.518737\pi$
$104$	10.9393	1.07268
$105$	0	0
$106$	−3.35066	−0.325444
$107$	−10.6681	−1.03133	−0.515664	−	0.856791i	$-0.672455\pi$
$107$	−10.6681	−1.03133	−0.515664	+	0.856791i	$0.672455\pi$
$108$	0	0
$109$	0.263959	0.0252827	0.0126413	−	0.999920i	$-0.495976\pi$
$109$	0.263959	0.0252827	0.0126413	+	0.999920i	$0.495976\pi$
$110$	−0.333752	−0.0318220
$111$	0	0
$112$	1.11449	0.105309
$113$	−12.8150	−1.20553	−0.602766	−	0.797918i	$-0.705935\pi$
$113$	−12.8150	−1.20553	−0.602766	+	0.797918i	$0.705935\pi$
$114$	0	0
$115$	7.07520	0.659766
$116$	−2.78580	−0.258655
$117$	0	0
$118$	−16.4750	−1.51664
$119$	−0.331341	−0.0303739
$120$	0	0
$121$	−10.9638	−0.996713
$122$	0.0284032	0.00257151
$123$	0	0
$124$	−2.71788	−0.244073
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.55770	−0.848108	−0.424054	−	0.905637i	$-0.639393\pi$
$127$	−9.55770	−0.848108	−0.424054	+	0.905637i	$0.639393\pi$
$128$	11.5424	1.02021
$129$	0	0
$130$	−11.9035	−1.04401
$131$	18.1182	1.58300	0.791498	−	0.611171i	$-0.209301\pi$
$131$	18.1182	1.58300	0.791498	+	0.611171i	$0.209301\pi$
$132$	0	0
$133$	0.219849	0.0190633
$134$	−17.3308	−1.49715
$135$	0	0
$136$	−2.39474	−0.205348
$137$	11.4313	0.976645	0.488323	−	0.872663i	$-0.337609\pi$
$137$	11.4313	0.976645	0.488323	+	0.872663i	$0.337609\pi$
$138$	0	0
$139$	11.9723	1.01548	0.507740	−	0.861510i	$-0.330481\pi$
$139$	11.9723	1.01548	0.507740	+	0.861510i	$0.330481\pi$
$140$	−0.241270	−0.0203910
$141$	0	0
$142$	8.83887	0.741741
$143$	1.28946	0.107830
$144$	0	0
$145$	−2.57705	−0.214012
$146$	13.8923	1.14973
$147$	0	0
$148$	−10.7376	−0.882628
$149$	−5.53546	−0.453483	−0.226741	−	0.973955i	$-0.572807\pi$
$149$	−5.53546	−0.453483	−0.226741	+	0.973955i	$0.572807\pi$
$150$	0	0
$151$	17.3970	1.41575	0.707873	−	0.706340i	$-0.249655\pi$
$151$	17.3970	1.41575	0.707873	+	0.706340i	$0.249655\pi$
$152$	1.58895	0.128881
$153$	0	0
$154$	0.0744901	0.00600258
$155$	−2.51422	−0.201947
$156$	0	0
$157$	8.87697	0.708460	0.354230	−	0.935158i	$-0.384743\pi$
$157$	8.87697	0.708460	0.354230	+	0.935158i	$0.384743\pi$
$158$	−17.4097	−1.38504
$159$	0	0
$160$	−5.53869	−0.437872
$161$	−1.57912	−0.124452
$162$	0	0
$163$	−2.76730	−0.216751	−0.108376	−	0.994110i	$-0.534565\pi$
$163$	−2.76730	−0.216751	−0.108376	+	0.994110i	$0.534565\pi$
$164$	−6.16533	−0.481432
$165$	0	0
$166$	23.2085	1.80133
$167$	7.69467	0.595431	0.297716	−	0.954655i	$-0.403775\pi$
$167$	7.69467	0.595431	0.297716	+	0.954655i	$0.403775\pi$
$168$	0	0
$169$	32.9895	2.53765
$170$	2.60583	0.199858
$171$	0	0
$172$	7.08932	0.540555
$173$	11.8826	0.903416	0.451708	−	0.892166i	$-0.350815\pi$
$173$	11.8826	0.903416	0.451708	+	0.892166i	$0.350815\pi$
$174$	0	0
$175$	−0.223190	−0.0168716
$176$	0.949461	0.0715683
$177$	0	0
$178$	−10.6853	−0.800899
$179$	−20.7864	−1.55365	−0.776826	−	0.629716i	$-0.783171\pi$
$179$	−20.7864	−1.55365	−0.776826	+	0.629716i	$0.783171\pi$
$180$	0	0
$181$	11.2246	0.834318	0.417159	−	0.908834i	$-0.363026\pi$
$181$	11.2246	0.834318	0.417159	+	0.908834i	$0.363026\pi$
$182$	2.65675	0.196931
$183$	0	0
$184$	−11.4130	−0.841375
$185$	−9.93302	−0.730290
$186$	0	0
$187$	−0.282278	−0.0206422
$188$	−9.57733	−0.698499
$189$	0	0
$190$	−1.72900	−0.125435
$191$	14.7637	1.06826	0.534131	−	0.845402i	$-0.320639\pi$
$191$	14.7637	1.06826	0.534131	+	0.845402i	$0.320639\pi$
$192$	0	0
$193$	−25.8760	−1.86260	−0.931298	−	0.364257i	$-0.881323\pi$
$193$	−25.8760	−1.86260	−0.931298	+	0.364257i	$0.881323\pi$
$194$	6.58743	0.472950
$195$	0	0
$196$	−7.51318	−0.536656
$197$	−23.5478	−1.67771	−0.838857	−	0.544352i	$-0.816776\pi$
$197$	−23.5478	−1.67771	−0.838857	+	0.544352i	$0.816776\pi$
$198$	0	0
$199$	24.9325	1.76741	0.883707	−	0.468040i	$-0.155040\pi$
$199$	24.9325	1.76741	0.883707	+	0.468040i	$0.155040\pi$
$200$	−1.61309	−0.114063
$201$	0	0
$202$	−0.898335	−0.0632066
$203$	0.575172	0.0403691
$204$	0	0
$205$	−5.70334	−0.398338
$206$	−2.09603	−0.146037
$207$	0	0
$208$	33.8633	2.34800
$209$	0.187295	0.0129555
$210$	0	0
$211$	16.2681	1.11994	0.559970	−	0.828513i	$-0.310813\pi$
$211$	16.2681	1.11994	0.559970	+	0.828513i	$0.310813\pi$
$212$	−2.06353	−0.141724
$213$	0	0
$214$	−18.7256	−1.28005
$215$	6.55808	0.447257
$216$	0	0
$217$	0.561149	0.0380933
$218$	0.463322	0.0313801
$219$	0	0
$220$	−0.205544	−0.0138578
$221$	−10.0677	−0.677224
$222$	0	0
$223$	−10.4632	−0.700665	−0.350333	−	0.936625i	$-0.613931\pi$
$223$	−10.4632	−0.700665	−0.350333	+	0.936625i	$0.613931\pi$
$224$	1.23618	0.0825958
$225$	0	0
$226$	−22.4939	−1.49627
$227$	−21.6774	−1.43878	−0.719388	−	0.694608i	$-0.755578\pi$
$227$	−21.6774	−1.43878	−0.719388	+	0.694608i	$0.755578\pi$
$228$	0	0
$229$	24.9218	1.64688	0.823438	−	0.567406i	$-0.192053\pi$
$229$	24.9218	1.64688	0.823438	+	0.567406i	$0.192053\pi$
$230$	12.4190	0.818881
$231$	0	0
$232$	4.15702	0.272922
$233$	−14.9463	−0.979167	−0.489584	−	0.871956i	$-0.662851\pi$
$233$	−14.9463	−0.979167	−0.489584	+	0.871956i	$0.662851\pi$
$234$	0	0
$235$	−8.85966	−0.577940
$236$	−10.1463	−0.660465
$237$	0	0
$238$	−0.581595	−0.0376992
$239$	−5.53501	−0.358030	−0.179015	−	0.983846i	$-0.557291\pi$
$239$	−5.53501	−0.358030	−0.179015	+	0.983846i	$0.557291\pi$
$240$	0	0
$241$	−1.88415	−0.121369	−0.0606843	−	0.998157i	$-0.519328\pi$
$241$	−1.88415	−0.121369	−0.0606843	+	0.998157i	$0.519328\pi$
$242$	−19.2446	−1.23709
$243$	0	0
$244$	0.0174924	0.00111983
$245$	−6.95019	−0.444031
$246$	0	0
$247$	6.68004	0.425041
$248$	4.05567	0.257536
$249$	0	0
$250$	1.75528	0.111014
$251$	−22.4584	−1.41756	−0.708779	−	0.705430i	$-0.750754\pi$
$251$	−22.4584	−1.41756	−0.708779	+	0.705430i	$0.750754\pi$
$252$	0	0
$253$	−1.34529	−0.0845777
$254$	−16.7764	−1.05265
$255$	0	0
$256$	19.7303	1.23314
$257$	−25.8465	−1.61226	−0.806131	−	0.591737i	$-0.798443\pi$
$257$	−25.8465	−1.61226	−0.806131	+	0.591737i	$0.798443\pi$
$258$	0	0
$259$	2.21695	0.137755
$260$	−7.33089	−0.454643
$261$	0	0
$262$	31.8025	1.96477
$263$	22.3794	1.37997	0.689987	−	0.723822i	$-0.257616\pi$
$263$	22.3794	1.37997	0.689987	+	0.723822i	$0.257616\pi$
$264$	0	0
$265$	−1.90890	−0.117263
$266$	0.385897	0.0236608
$267$	0	0
$268$	−10.6733	−0.651978
$269$	18.6149	1.13497	0.567486	−	0.823383i	$-0.307916\pi$
$269$	18.6149	1.13497	0.567486	+	0.823383i	$0.307916\pi$
$270$	0	0
$271$	−19.1389	−1.16261	−0.581304	−	0.813686i	$-0.697457\pi$
$271$	−19.1389	−1.16261	−0.581304	+	0.813686i	$0.697457\pi$
$272$	−7.41309	−0.449485
$273$	0	0
$274$	20.0652	1.21218
$275$	−0.190142	−0.0114660
$276$	0	0
$277$	−4.01820	−0.241430	−0.120715	−	0.992687i	$-0.538519\pi$
$277$	−4.01820	−0.241430	−0.120715	+	0.992687i	$0.538519\pi$
$278$	21.0148	1.26038
$279$	0	0
$280$	0.360027	0.0215157
$281$	15.3460	0.915466	0.457733	−	0.889090i	$-0.348662\pi$
$281$	15.3460	0.915466	0.457733	+	0.889090i	$0.348662\pi$
$282$	0	0
$283$	−31.2708	−1.85886	−0.929429	−	0.369001i	$-0.879700\pi$
$283$	−31.2708	−1.85886	−0.929429	+	0.369001i	$0.879700\pi$
$284$	5.44350	0.323012
$285$	0	0
$286$	2.26336	0.133835
$287$	1.27293	0.0751386
$288$	0	0
$289$	−14.7961	−0.870357
$290$	−4.52344	−0.265625
$291$	0	0
$292$	8.55568	0.500683
$293$	−3.57602	−0.208913	−0.104457	−	0.994529i	$-0.533310\pi$
$293$	−3.57602	−0.208913	−0.104457	+	0.994529i	$0.533310\pi$
$294$	0	0
$295$	−9.38596	−0.546472
$296$	16.0229	0.931311
$297$	0	0
$298$	−9.71628	−0.562849
$299$	−47.9809	−2.77480
$300$	0	0
$301$	−1.46370	−0.0843662
$302$	30.5366	1.75718
$303$	0	0
$304$	4.91869	0.282106
$305$	0.0161816	0.000926556 0
$306$	0	0
$307$	−11.1729	−0.637671	−0.318835	−	0.947810i	$-0.603292\pi$
$307$	−11.1729	−0.637671	−0.318835	+	0.947810i	$0.603292\pi$
$308$	0.0458754	0.00261399
$309$	0	0
$310$	−4.41316	−0.250651
$311$	−15.5457	−0.881517	−0.440758	−	0.897626i	$-0.645290\pi$
$311$	−15.5457	−0.881517	−0.440758	+	0.897626i	$0.645290\pi$
$312$	0	0
$313$	29.2641	1.65410	0.827052	−	0.562125i	$-0.190016\pi$
$313$	29.2641	1.65410	0.827052	+	0.562125i	$0.190016\pi$
$314$	15.5816	0.879319
$315$	0	0
$316$	−10.7219	−0.603154
$317$	32.9848	1.85261	0.926306	−	0.376773i	$-0.122966\pi$
$317$	32.9848	1.85261	0.926306	+	0.376773i	$0.122966\pi$
$318$	0	0
$319$	0.490004	0.0274350
$320$	0.264929	0.0148100
$321$	0	0
$322$	−2.77179	−0.154466
$323$	−1.46234	−0.0813670
$324$	0	0
$325$	−6.78155	−0.376173
$326$	−4.85738	−0.269025
$327$	0	0
$328$	9.20002	0.507986
$329$	1.97739	0.109017
$330$	0	0
$331$	21.6263	1.18869	0.594345	−	0.804210i	$-0.297412\pi$
$331$	21.6263	1.18869	0.594345	+	0.804210i	$0.297412\pi$
$332$	14.2932	0.784440
$333$	0	0
$334$	13.5063	0.739031
$335$	−9.87354	−0.539449
$336$	0	0
$337$	19.5789	1.06653	0.533264	−	0.845949i	$-0.320965\pi$
$337$	19.5789	1.06653	0.533264	+	0.845949i	$0.320965\pi$
$338$	57.9057	3.14966
$339$	0	0
$340$	1.60482	0.0870337
$341$	0.478058	0.0258883
$342$	0	0
$343$	3.11354	0.168116
$344$	−10.5788	−0.570371
$345$	0	0
$346$	20.8572	1.12129
$347$	−9.66065	−0.518611	−0.259305	−	0.965795i	$-0.583494\pi$
$347$	−9.66065	−0.518611	−0.259305	+	0.965795i	$0.583494\pi$
$348$	0	0
$349$	−0.438994	−0.0234988	−0.0117494	−	0.999931i	$-0.503740\pi$
$349$	−0.438994	−0.0234988	−0.0117494	+	0.999931i	$0.503740\pi$
$350$	−0.391761	−0.0209405
$351$	0	0
$352$	1.05314	0.0561323
$353$	14.0546	0.748051	0.374025	−	0.927418i	$-0.377977\pi$
$353$	14.0546	0.748051	0.374025	+	0.927418i	$0.377977\pi$
$354$	0	0
$355$	5.03559	0.267261
$356$	−6.58065	−0.348774
$357$	0	0
$358$	−36.4860	−1.92834
$359$	−5.28300	−0.278826	−0.139413	−	0.990234i	$-0.544522\pi$
$359$	−5.28300	−0.278826	−0.139413	+	0.990234i	$0.544522\pi$
$360$	0	0
$361$	−18.0297	−0.948932
$362$	19.7023	1.03553
$363$	0	0
$364$	1.63618	0.0857593
$365$	7.91456	0.414267
$366$	0	0
$367$	5.35973	0.279775	0.139888	−	0.990167i	$-0.455326\pi$
$367$	5.35973	0.279775	0.139888	+	0.990167i	$0.455326\pi$
$368$	−35.3296	−1.84168
$369$	0	0
$370$	−17.4352	−0.906414
$371$	0.426048	0.0221193
$372$	0	0
$373$	24.5532	1.27132	0.635659	−	0.771970i	$-0.280729\pi$
$373$	24.5532	1.27132	0.635659	+	0.771970i	$0.280729\pi$
$374$	−0.495477	−0.0256205
$375$	0	0
$376$	14.2915	0.737026
$377$	17.4764	0.900080
$378$	0	0
$379$	37.9306	1.94836	0.974182	−	0.225764i	$-0.0724878\pi$
$379$	37.9306	1.94836	0.974182	+	0.225764i	$0.0724878\pi$
$380$	−1.06482	−0.0546243
$381$	0	0
$382$	25.9144	1.32589
$383$	−4.45900	−0.227844	−0.113922	−	0.993490i	$-0.536341\pi$
$383$	−4.45900	−0.227844	−0.113922	+	0.993490i	$0.536341\pi$
$384$	0	0
$385$	0.0424378	0.00216283
$386$	−45.4196	−2.31180
$387$	0	0
$388$	4.05693	0.205959
$389$	−30.7009	−1.55660	−0.778298	−	0.627895i	$-0.783917\pi$
$389$	−30.7009	−1.55660	−0.778298	+	0.627895i	$0.783917\pi$
$390$	0	0
$391$	10.5036	0.531190
$392$	11.2113	0.566256
$393$	0	0
$394$	−41.3330	−2.08233
$395$	−9.91847	−0.499052
$396$	0	0
$397$	−6.41781	−0.322101	−0.161050	−	0.986946i	$-0.551488\pi$
$397$	−6.41781	−0.322101	−0.161050	+	0.986946i	$0.551488\pi$
$398$	43.7634	2.19366
$399$	0	0
$400$	−4.99344	−0.249672
$401$	−22.8020	−1.13868	−0.569338	−	0.822104i	$-0.692800\pi$
$401$	−22.8020	−1.13868	−0.569338	+	0.822104i	$0.692800\pi$
$402$	0	0
$403$	17.0503	0.849337
$404$	−0.553248	−0.0275251
$405$	0	0
$406$	1.00959	0.0501050
$407$	1.88868	0.0936184
$408$	0	0
$409$	35.8450	1.77242	0.886211	−	0.463281i	$-0.153328\pi$
$409$	35.8450	1.77242	0.886211	+	0.463281i	$0.153328\pi$
$410$	−10.0109	−0.494405
$411$	0	0
$412$	−1.29086	−0.0635960
$413$	2.09485	0.103081
$414$	0	0
$415$	13.2221	0.649049
$416$	37.5609	1.84158
$417$	0	0
$418$	0.328756	0.0160800
$419$	−4.75964	−0.232524	−0.116262	−	0.993219i	$-0.537091\pi$
$419$	−4.75964	−0.232524	−0.116262	+	0.993219i	$0.537091\pi$
$420$	0	0
$421$	−22.7101	−1.10682	−0.553412	−	0.832908i	$-0.686674\pi$
$421$	−22.7101	−1.10682	−0.553412	+	0.832908i	$0.686674\pi$
$422$	28.5550	1.39004
$423$	0	0
$424$	3.07924	0.149541
$425$	1.48457	0.0720120
$426$	0	0
$427$	−0.00361157	−0.000174776 0
$428$	−11.5323	−0.557436
$429$	0	0
$430$	11.5113	0.555122
$431$	26.1473	1.25947	0.629736	−	0.776809i	$-0.283163\pi$
$431$	26.1473	1.25947	0.629736	+	0.776809i	$0.283163\pi$
$432$	0	0
$433$	16.2090	0.778953	0.389476	−	0.921036i	$-0.372656\pi$
$433$	16.2090	0.778953	0.389476	+	0.921036i	$0.372656\pi$
$434$	0.984974	0.0472803
$435$	0	0
$436$	0.285341	0.0136653
$437$	−6.96929	−0.333386
$438$	0	0
$439$	21.9212	1.04624	0.523121	−	0.852258i	$-0.324767\pi$
$439$	21.9212	1.04624	0.523121	+	0.852258i	$0.324767\pi$
$440$	0.306716	0.0146221
$441$	0	0
$442$	−17.6716	−0.840550
$443$	−15.2120	−0.722746	−0.361373	−	0.932421i	$-0.617692\pi$
$443$	−15.2120	−0.722746	−0.361373	+	0.932421i	$0.617692\pi$
$444$	0	0
$445$	−6.08753	−0.288577
$446$	−18.3658	−0.869644
$447$	0	0
$448$	−0.0591295	−0.00279361
$449$	−5.37836	−0.253821	−0.126910	−	0.991914i	$-0.540506\pi$
$449$	−5.37836	−0.253821	−0.126910	+	0.991914i	$0.540506\pi$
$450$	0	0
$451$	1.08444	0.0510644
$452$	−13.8531	−0.651593
$453$	0	0
$454$	−38.0498	−1.78577
$455$	1.51358	0.0709575
$456$	0	0
$457$	1.23577	0.0578068	0.0289034	−	0.999582i	$-0.490798\pi$
$457$	1.23577	0.0578068	0.0289034	+	0.999582i	$0.490798\pi$
$458$	43.7446	2.04405
$459$	0	0
$460$	7.64832	0.356605
$461$	−10.0263	−0.466970	−0.233485	−	0.972360i	$-0.575013\pi$
$461$	−10.0263	−0.466970	−0.233485	+	0.972360i	$0.575013\pi$
$462$	0	0
$463$	−21.3391	−0.991712	−0.495856	−	0.868405i	$-0.665146\pi$
$463$	−21.3391	−0.991712	−0.495856	+	0.868405i	$0.665146\pi$
$464$	12.8683	0.597397
$465$	0	0
$466$	−26.2350	−1.21531
$467$	−3.39845	−0.157261	−0.0786307	−	0.996904i	$-0.525055\pi$
$467$	−3.39845	−0.157261	−0.0786307	+	0.996904i	$0.525055\pi$
$468$	0	0
$469$	2.20368	0.101756
$470$	−15.5512	−0.717322
$471$	0	0
$472$	15.1404	0.696895
$473$	−1.24696	−0.0573355
$474$	0	0
$475$	−0.985031	−0.0451963
$476$	−0.358181	−0.0164172
$477$	0	0
$478$	−9.71548	−0.444376
$479$	−21.5971	−0.986798	−0.493399	−	0.869803i	$-0.664246\pi$
$479$	−21.5971	−0.986798	−0.493399	+	0.869803i	$0.664246\pi$
$480$	0	0
$481$	67.3613	3.07141
$482$	−3.30720	−0.150639
$483$	0	0
$484$	−11.8520	−0.538726
$485$	3.75293	0.170412
$486$	0	0
$487$	18.1927	0.824390	0.412195	−	0.911096i	$-0.364762\pi$
$487$	18.1927	0.824390	0.412195	+	0.911096i	$0.364762\pi$
$488$	−0.0261024	−0.00118160
$489$	0	0
$490$	−12.1995	−0.551118
$491$	0.913715	0.0412354	0.0206177	−	0.999787i	$-0.493437\pi$
$491$	0.913715	0.0412354	0.0206177	+	0.999787i	$0.493437\pi$
$492$	0	0
$493$	−3.82580	−0.172305
$494$	11.7253	0.527548
$495$	0	0
$496$	12.5546	0.563719
$497$	−1.12389	−0.0504135
$498$	0	0
$499$	−6.93641	−0.310516	−0.155258	−	0.987874i	$-0.549621\pi$
$499$	−6.93641	−0.310516	−0.155258	+	0.987874i	$0.549621\pi$
$500$	1.08100	0.0483440
$501$	0	0
$502$	−39.4207	−1.75943
$503$	−6.21600	−0.277158	−0.138579	−	0.990351i	$-0.544253\pi$
$503$	−6.21600	−0.277158	−0.138579	+	0.990351i	$0.544253\pi$
$504$	0	0
$505$	−0.511790	−0.0227744
$506$	−2.36136	−0.104975
$507$	0	0
$508$	−10.3319	−0.458405
$509$	−5.16768	−0.229053	−0.114527	−	0.993420i	$-0.536535\pi$
$509$	−5.16768	−0.229053	−0.114527	+	0.993420i	$0.536535\pi$
$510$	0	0
$511$	−1.76645	−0.0781433
$512$	11.5473	0.510325
$513$	0	0
$514$	−45.3679	−2.00109
$515$	−1.19413	−0.0526196
$516$	0	0
$517$	1.68459	0.0740882
$518$	3.89137	0.170977
$519$	0	0
$520$	10.9393	0.479719
$521$	34.4176	1.50786	0.753931	−	0.656954i	$-0.228155\pi$
$521$	34.4176	1.50786	0.753931	+	0.656954i	$0.228155\pi$
$522$	0	0
$523$	−42.1655	−1.84377	−0.921885	−	0.387464i	$-0.873351\pi$
$523$	−42.1655	−1.84377	−0.921885	+	0.387464i	$0.873351\pi$
$524$	19.5859	0.855613
$525$	0	0
$526$	39.2821	1.71278
$527$	−3.73253	−0.162591
$528$	0	0
$529$	27.0585	1.17646
$530$	−3.35066	−0.145543
$531$	0	0
$532$	0.237658	0.0103038
$533$	38.6775	1.67531
$534$	0	0
$535$	−10.6681	−0.461224
$536$	15.9269	0.687939
$537$	0	0
$538$	32.6744	1.40869
$539$	1.32152	0.0569219
$540$	0	0
$541$	20.0990	0.864122	0.432061	−	0.901844i	$-0.357787\pi$
$541$	20.0990	0.864122	0.432061	+	0.901844i	$0.357787\pi$
$542$	−33.5942	−1.44299
$543$	0	0
$544$	−8.22255	−0.352539
$545$	0.263959	0.0113068
$546$	0	0
$547$	−20.9334	−0.895047	−0.447523	−	0.894272i	$-0.647694\pi$
$547$	−20.9334	−0.895047	−0.447523	+	0.894272i	$0.647694\pi$
$548$	12.3573	0.527879
$549$	0	0
$550$	−0.333752	−0.0142312
$551$	2.53847	0.108143
$552$	0	0
$553$	2.21370	0.0941363
$554$	−7.05306	−0.299656
$555$	0	0
$556$	12.9422	0.548870
$557$	−9.92402	−0.420494	−0.210247	−	0.977648i	$-0.567427\pi$
$557$	−9.92402	−0.420494	−0.210247	+	0.977648i	$0.567427\pi$
$558$	0	0
$559$	−44.4740	−1.88105
$560$	1.11449	0.0470956
$561$	0	0
$562$	26.9365	1.13625
$563$	−20.9705	−0.883800	−0.441900	−	0.897064i	$-0.645695\pi$
$563$	−20.9705	−0.883800	−0.441900	+	0.897064i	$0.645695\pi$
$564$	0	0
$565$	−12.8150	−0.539130
$566$	−54.8890	−2.30716
$567$	0	0
$568$	−8.12288	−0.340828
$569$	−13.9928	−0.586610	−0.293305	−	0.956019i	$-0.594755\pi$
$569$	−13.9928	−0.586610	−0.293305	+	0.956019i	$0.594755\pi$
$570$	0	0
$571$	9.51846	0.398335	0.199168	−	0.979965i	$-0.436176\pi$
$571$	9.51846	0.398335	0.199168	+	0.979965i	$0.436176\pi$
$572$	1.39391	0.0582822
$573$	0	0
$574$	2.23435	0.0932598
$575$	7.07520	0.295056
$576$	0	0
$577$	25.4415	1.05914	0.529571	−	0.848266i	$-0.322353\pi$
$577$	25.4415	1.05914	0.529571	+	0.848266i	$0.322353\pi$
$578$	−25.9712	−1.08026
$579$	0	0
$580$	−2.78580	−0.115674
$581$	−2.95105	−0.122430
$582$	0	0
$583$	0.362962	0.0150323
$584$	−12.7669	−0.528299
$585$	0	0
$586$	−6.27691	−0.259297
$587$	−26.0782	−1.07636	−0.538181	−	0.842829i	$-0.680888\pi$
$587$	−26.0782	−1.07636	−0.538181	+	0.842829i	$0.680888\pi$
$588$	0	0
$589$	2.47659	0.102046
$590$	−16.4750	−0.678264
$591$	0	0
$592$	49.5999	2.03854
$593$	18.7253	0.768957	0.384478	−	0.923134i	$-0.374381\pi$
$593$	18.7253	0.768957	0.384478	+	0.923134i	$0.374381\pi$
$594$	0	0
$595$	−0.331341	−0.0135836
$596$	−5.98386	−0.245108
$597$	0	0
$598$	−84.2198	−3.44400
$599$	−9.75105	−0.398417	−0.199208	−	0.979957i	$-0.563837\pi$
$599$	−9.75105	−0.398417	−0.199208	+	0.979957i	$0.563837\pi$
$600$	0	0
$601$	−32.7671	−1.33660	−0.668300	−	0.743892i	$-0.732978\pi$
$601$	−32.7671	−1.33660	−0.668300	+	0.743892i	$0.732978\pi$
$602$	−2.56920	−0.104713
$603$	0	0
$604$	18.8062	0.765214
$605$	−10.9638	−0.445744
$606$	0	0
$607$	18.7538	0.761192	0.380596	−	0.924741i	$-0.375719\pi$
$607$	18.7538	0.761192	0.380596	+	0.924741i	$0.375719\pi$
$608$	5.45578	0.221261
$609$	0	0
$610$	0.0284032	0.00115001
$611$	60.0823	2.43067
$612$	0	0
$613$	24.1403	0.975018	0.487509	−	0.873118i	$-0.337906\pi$
$613$	24.1403	0.975018	0.487509	+	0.873118i	$0.337906\pi$
$614$	−19.6115	−0.791458
$615$	0	0
$616$	−0.0684561	−0.00275817
$617$	21.7940	0.877393	0.438696	−	0.898635i	$-0.355440\pi$
$617$	21.7940	0.877393	0.438696	+	0.898635i	$0.355440\pi$
$618$	0	0
$619$	5.79745	0.233019	0.116510	−	0.993190i	$-0.462829\pi$
$619$	5.79745	0.233019	0.116510	+	0.993190i	$0.462829\pi$
$620$	−2.71788	−0.109153
$621$	0	0
$622$	−27.2871	−1.09411
$623$	1.35868	0.0544343
$624$	0	0
$625$	1.00000	0.0400000
$626$	51.3667	2.05302
$627$	0	0
$628$	9.59605	0.382924
$629$	−14.7462	−0.587970
$630$	0	0
$631$	39.1135	1.55709	0.778543	−	0.627591i	$-0.215959\pi$
$631$	39.1135	1.55709	0.778543	+	0.627591i	$0.215959\pi$
$632$	15.9994	0.636423
$633$	0	0
$634$	57.8975	2.29940
$635$	−9.55770	−0.379286
$636$	0	0
$637$	47.1331	1.86748
$638$	0.860094	0.0340515
$639$	0	0
$640$	11.5424	0.456254
$641$	−11.1146	−0.439000	−0.219500	−	0.975613i	$-0.570443\pi$
$641$	−11.1146	−0.439000	−0.219500	+	0.975613i	$0.570443\pi$
$642$	0	0
$643$	−15.8727	−0.625960	−0.312980	−	0.949760i	$-0.601327\pi$
$643$	−15.8727	−0.625960	−0.312980	+	0.949760i	$0.601327\pi$
$644$	−1.70703	−0.0672664
$645$	0	0
$646$	−2.56682	−0.100990
$647$	4.73752	0.186251	0.0931256	−	0.995654i	$-0.470314\pi$
$647$	4.73752	0.186251	0.0931256	+	0.995654i	$0.470314\pi$
$648$	0	0
$649$	1.78466	0.0700541
$650$	−11.9035	−0.466894
$651$	0	0
$652$	−2.99146	−0.117155
$653$	−27.0730	−1.05945	−0.529725	−	0.848170i	$-0.677705\pi$
$653$	−27.0730	−1.05945	−0.529725	+	0.848170i	$0.677705\pi$
$654$	0	0
$655$	18.1182	0.707938
$656$	28.4793	1.11193
$657$	0	0
$658$	3.47087	0.135309
$659$	−23.6004	−0.919341	−0.459670	−	0.888090i	$-0.652032\pi$
$659$	−23.6004	−0.919341	−0.459670	+	0.888090i	$0.652032\pi$
$660$	0	0
$661$	−18.6986	−0.727292	−0.363646	−	0.931537i	$-0.618468\pi$
$661$	−18.6986	−0.727292	−0.363646	+	0.931537i	$0.618468\pi$
$662$	37.9602	1.47537
$663$	0	0
$664$	−21.3285	−0.827708
$665$	0.219849	0.00852539
$666$	0	0
$667$	−18.2331	−0.705990
$668$	8.31797	0.321832
$669$	0	0
$670$	−17.3308	−0.669548
$671$	−0.00307680	−0.000118778 0
$672$	0	0
$673$	30.8041	1.18741	0.593706	−	0.804682i	$-0.297664\pi$
$673$	30.8041	1.18741	0.593706	+	0.804682i	$0.297664\pi$
$674$	34.3664	1.32374
$675$	0	0
$676$	35.6618	1.37161
$677$	41.4658	1.59366	0.796830	−	0.604204i	$-0.206509\pi$
$677$	41.4658	1.59366	0.796830	+	0.604204i	$0.206509\pi$
$678$	0	0
$679$	−0.837616	−0.0321448
$680$	−2.39474	−0.0918342
$681$	0	0
$682$	0.839126	0.0321318
$683$	16.4639	0.629973	0.314987	−	0.949096i	$-0.398000\pi$
$683$	16.4639	0.629973	0.314987	+	0.949096i	$0.398000\pi$
$684$	0	0
$685$	11.4313	0.436769
$686$	5.46514	0.208660
$687$	0	0
$688$	−32.7474	−1.24848
$689$	12.9453	0.493178
$690$	0	0
$691$	28.9840	1.10260	0.551301	−	0.834306i	$-0.314132\pi$
$691$	28.9840	1.10260	0.551301	+	0.834306i	$0.314132\pi$
$692$	12.8451	0.488298
$693$	0	0
$694$	−16.9571	−0.643684
$695$	11.9723	0.454137
$696$	0	0
$697$	−8.46698	−0.320710
$698$	−0.770557	−0.0291660
$699$	0	0
$700$	−0.241270	−0.00911913
$701$	−4.97591	−0.187937	−0.0939687	−	0.995575i	$-0.529955\pi$
$701$	−4.97591	−0.187937	−0.0939687	+	0.995575i	$0.529955\pi$
$702$	0	0
$703$	9.78433	0.369023
$704$	−0.0503740	−0.00189854
$705$	0	0
$706$	24.6697	0.928458
$707$	0.114227	0.00429593
$708$	0	0
$709$	−24.0390	−0.902805	−0.451402	−	0.892321i	$-0.649076\pi$
$709$	−24.0390	−0.902805	−0.451402	+	0.892321i	$0.649076\pi$
$710$	8.83887	0.331717
$711$	0	0
$712$	9.81976	0.368011
$713$	−17.7886	−0.666189
$714$	0	0
$715$	1.28946	0.0482229
$716$	−22.4702	−0.839752
$717$	0	0
$718$	−9.27313	−0.346070
$719$	50.8582	1.89669	0.948346	−	0.317238i	$-0.102755\pi$
$719$	50.8582	1.89669	0.948346	+	0.317238i	$0.102755\pi$
$720$	0	0
$721$	0.266518	0.00992563
$722$	−31.6472	−1.17779
$723$	0	0
$724$	12.1338	0.450951
$725$	−2.57705	−0.0957092
$726$	0	0
$727$	−42.0631	−1.56004	−0.780018	−	0.625757i	$-0.784790\pi$
$727$	−42.0631	−1.56004	−0.780018	+	0.625757i	$0.784790\pi$
$728$	−2.44154	−0.0904895
$729$	0	0
$730$	13.8923	0.514176
$731$	9.73590	0.360095
$732$	0	0
$733$	−28.5875	−1.05591	−0.527953	−	0.849274i	$-0.677040\pi$
$733$	−28.5875	−1.05591	−0.527953	+	0.849274i	$0.677040\pi$
$734$	9.40781	0.347249
$735$	0	0
$736$	−39.1873	−1.44446
$737$	1.87737	0.0691539
$738$	0	0
$739$	13.0317	0.479379	0.239690	−	0.970850i	$-0.422954\pi$
$739$	13.0317	0.479379	0.239690	+	0.970850i	$0.422954\pi$
$740$	−10.7376	−0.394723
$741$	0	0
$742$	0.747833	0.0274538
$743$	2.50040	0.0917307	0.0458654	−	0.998948i	$-0.485395\pi$
$743$	2.50040	0.0917307	0.0458654	+	0.998948i	$0.485395\pi$
$744$	0	0
$745$	−5.53546	−0.202804
$746$	43.0977	1.57792
$747$	0	0
$748$	−0.305144	−0.0111572
$749$	2.38103	0.0870008
$750$	0	0
$751$	−12.9896	−0.473997	−0.236999	−	0.971510i	$-0.576164\pi$
$751$	−12.9896	−0.473997	−0.236999	+	0.971510i	$0.576164\pi$
$752$	44.2402	1.61327
$753$	0	0
$754$	30.6759	1.11715
$755$	17.3970	0.633141
$756$	0	0
$757$	−0.526184	−0.0191245	−0.00956223	−	0.999954i	$-0.503044\pi$
$757$	−0.526184	−0.0191245	−0.00956223	+	0.999954i	$0.503044\pi$
$758$	66.5788	2.41825
$759$	0	0
$760$	1.58895	0.0576372
$761$	27.7858	1.00723	0.503617	−	0.863927i	$-0.332002\pi$
$761$	27.7858	1.00723	0.503617	+	0.863927i	$0.332002\pi$
$762$	0	0
$763$	−0.0589130	−0.00213280
$764$	15.9596	0.577398
$765$	0	0
$766$	−7.82679	−0.282794
$767$	63.6514	2.29832
$768$	0	0
$769$	−28.3274	−1.02151	−0.510756	−	0.859726i	$-0.670635\pi$
$769$	−28.3274	−1.02151	−0.510756	+	0.859726i	$0.670635\pi$
$770$	0.0744901	0.00268444
$771$	0	0
$772$	−27.9721	−1.00674
$773$	−0.566959	−0.0203921	−0.0101960	−	0.999948i	$-0.503246\pi$
$773$	−0.566959	−0.0203921	−0.0101960	+	0.999948i	$0.503246\pi$
$774$	0	0
$775$	−2.51422	−0.0903135
$776$	−6.05382	−0.217320
$777$	0	0
$778$	−53.8886	−1.93200
$779$	5.61796	0.201284
$780$	0	0
$781$	−0.957476	−0.0342612
$782$	18.4368	0.659297
$783$	0	0
$784$	34.7053	1.23948
$785$	8.87697	0.316833
$786$	0	0
$787$	−31.5499	−1.12463	−0.562316	−	0.826923i	$-0.690089\pi$
$787$	−31.5499	−1.12463	−0.562316	+	0.826923i	$0.690089\pi$
$788$	−25.4553	−0.906808
$789$	0	0
$790$	−17.4097	−0.619409
$791$	2.86018	0.101696
$792$	0	0
$793$	−0.109736	−0.00389685
$794$	−11.2651	−0.399782
$795$	0	0
$796$	26.9521	0.955292
$797$	−51.6290	−1.82879	−0.914396	−	0.404821i	$-0.867334\pi$
$797$	−51.6290	−1.82879	−0.914396	+	0.404821i	$0.867334\pi$
$798$	0	0
$799$	−13.1528	−0.465311
$800$	−5.53869	−0.195822
$801$	0	0
$802$	−40.0238	−1.41329
$803$	−1.50489	−0.0531064
$804$	0	0
$805$	−1.57912	−0.0556565
$806$	29.9281	1.05417
$807$	0	0
$808$	0.825566	0.0290433
$809$	−38.2838	−1.34599	−0.672993	−	0.739649i	$-0.734992\pi$
$809$	−38.2838	−1.34599	−0.672993	+	0.739649i	$0.734992\pi$
$810$	0	0
$811$	4.03388	0.141649	0.0708243	−	0.997489i	$-0.477437\pi$
$811$	4.03388	0.141649	0.0708243	+	0.997489i	$0.477437\pi$
$812$	0.621763	0.0218196
$813$	0	0
$814$	3.31516	0.116196
$815$	−2.76730	−0.0969341
$816$	0	0
$817$	−6.45991	−0.226004
$818$	62.9180	2.19988
$819$	0	0
$820$	−6.16533	−0.215303
$821$	38.7589	1.35270	0.676348	−	0.736582i	$-0.263562\pi$
$821$	38.7589	1.35270	0.676348	+	0.736582i	$0.263562\pi$
$822$	0	0
$823$	16.6299	0.579680	0.289840	−	0.957075i	$-0.406398\pi$
$823$	16.6299	0.579680	0.289840	+	0.957075i	$0.406398\pi$
$824$	1.92624	0.0671037
$825$	0	0
$826$	3.67705	0.127941
$827$	−42.0718	−1.46298	−0.731490	−	0.681853i	$-0.761175\pi$
$827$	−42.0718	−1.46298	−0.731490	+	0.681853i	$0.761175\pi$
$828$	0	0
$829$	−41.4389	−1.43923	−0.719616	−	0.694372i	$-0.755682\pi$
$829$	−41.4389	−1.43923	−0.719616	+	0.694372i	$0.755682\pi$
$830$	23.2085	0.805580
$831$	0	0
$832$	−1.79663	−0.0622869
$833$	−10.3180	−0.357498
$834$	0	0
$835$	7.69467	0.266285
$836$	0.202467	0.00700248
$837$	0	0
$838$	−8.35450	−0.288601
$839$	9.54808	0.329636	0.164818	−	0.986324i	$-0.447296\pi$
$839$	9.54808	0.329636	0.164818	+	0.986324i	$0.447296\pi$
$840$	0	0
$841$	−22.3588	−0.770994
$842$	−39.8626	−1.37376
$843$	0	0
$844$	17.5859	0.605330
$845$	32.9895	1.13487
$846$	0	0
$847$	2.44702	0.0840807
$848$	9.53199	0.327330
$849$	0	0
$850$	2.60583	0.0893792
$851$	−70.2781	−2.40910
$852$	0	0
$853$	9.42489	0.322702	0.161351	−	0.986897i	$-0.448415\pi$
$853$	9.42489	0.322702	0.161351	+	0.986897i	$0.448415\pi$
$854$	−0.00633932	−0.000216927 0
$855$	0	0
$856$	17.2087	0.588182
$857$	−14.0842	−0.481106	−0.240553	−	0.970636i	$-0.577329\pi$
$857$	−14.0842	−0.481106	−0.240553	+	0.970636i	$0.577329\pi$
$858$	0	0
$859$	−38.5901	−1.31668	−0.658338	−	0.752722i	$-0.728740\pi$
$859$	−38.5901	−1.31668	−0.658338	+	0.752722i	$0.728740\pi$
$860$	7.08932	0.241744
$861$	0	0
$862$	45.8958	1.56322
$863$	−19.4106	−0.660745	−0.330372	−	0.943851i	$-0.607174\pi$
$863$	−19.4106	−0.660745	−0.330372	+	0.943851i	$0.607174\pi$
$864$	0	0
$865$	11.8826	0.404020
$866$	28.4512	0.966813
$867$	0	0
$868$	0.606605	0.0205895
$869$	1.88591	0.0639753
$870$	0	0
$871$	66.9580	2.26878
$872$	−0.425790	−0.0144191
$873$	0	0
$874$	−12.2331	−0.413789
$875$	−0.223190	−0.00754520
$876$	0	0
$877$	5.72736	0.193399	0.0966996	−	0.995314i	$-0.469171\pi$
$877$	5.72736	0.193399	0.0966996	+	0.995314i	$0.469171\pi$
$878$	38.4778	1.29856
$879$	0	0
$880$	0.949461	0.0320063
$881$	43.6561	1.47081	0.735406	−	0.677627i	$-0.236991\pi$
$881$	43.6561	1.47081	0.735406	+	0.677627i	$0.236991\pi$
$882$	0	0
$883$	−44.8290	−1.50861	−0.754307	−	0.656521i	$-0.772027\pi$
$883$	−44.8290	−1.50861	−0.754307	+	0.656521i	$0.772027\pi$
$884$	−10.8832	−0.366041
$885$	0	0
$886$	−26.7014	−0.897051
$887$	−7.02999	−0.236044	−0.118022	−	0.993011i	$-0.537655\pi$
$887$	−7.02999	−0.236044	−0.118022	+	0.993011i	$0.537655\pi$
$888$	0	0
$889$	2.13318	0.0715447
$890$	−10.6853	−0.358173
$891$	0	0
$892$	−11.3107	−0.378711
$893$	8.72704	0.292039
$894$	0	0
$895$	−20.7864	−0.694814
$896$	−2.57615	−0.0860632
$897$	0	0
$898$	−9.44053	−0.315034
$899$	6.47927	0.216096
$900$	0	0
$901$	−2.83389	−0.0944106
$902$	1.90350	0.0633796
$903$	0	0
$904$	20.6718	0.687533
$905$	11.2246	0.373118
$906$	0	0
$907$	−39.6162	−1.31544	−0.657718	−	0.753264i	$-0.728478\pi$
$907$	−39.6162	−1.31544	−0.657718	+	0.753264i	$0.728478\pi$
$908$	−23.4333	−0.777662
$909$	0	0
$910$	2.65675	0.0880704
$911$	23.3132	0.772400	0.386200	−	0.922415i	$-0.373787\pi$
$911$	23.3132	0.772400	0.386200	+	0.922415i	$0.373787\pi$
$912$	0	0
$913$	−2.51408	−0.0832039
$914$	2.16912	0.0717480
$915$	0	0
$916$	26.9405	0.890140
$917$	−4.04381	−0.133538
$918$	0	0
$919$	−35.5890	−1.17397	−0.586987	−	0.809596i	$-0.699686\pi$
$919$	−35.5890	−1.17397	−0.586987	+	0.809596i	$0.699686\pi$
$920$	−11.4130	−0.376274
$921$	0	0
$922$	−17.5989	−0.579589
$923$	−34.1491	−1.12403
$924$	0	0
$925$	−9.93302	−0.326596
$926$	−37.4561	−1.23088
$927$	0	0
$928$	14.2735	0.468550
$929$	23.5289	0.771959	0.385979	−	0.922507i	$-0.373864\pi$
$929$	23.5289	0.771959	0.385979	+	0.922507i	$0.373864\pi$
$930$	0	0
$931$	6.84615	0.224373
$932$	−16.1571	−0.529242
$933$	0	0
$934$	−5.96522	−0.195188
$935$	−0.282278	−0.00923148
$936$	0	0
$937$	2.42367	0.0791779	0.0395889	−	0.999216i	$-0.487395\pi$
$937$	2.42367	0.0791779	0.0395889	+	0.999216i	$0.487395\pi$
$938$	3.86807	0.126297
$939$	0	0
$940$	−9.57733	−0.312378
$941$	25.9943	0.847388	0.423694	−	0.905805i	$-0.360733\pi$
$941$	25.9943	0.847388	0.423694	+	0.905805i	$0.360733\pi$
$942$	0	0
$943$	−40.3523	−1.31405
$944$	46.8682	1.52543
$945$	0	0
$946$	−2.18877	−0.0711631
$947$	−28.4680	−0.925085	−0.462542	−	0.886597i	$-0.653063\pi$
$947$	−28.4680	−0.925085	−0.462542	+	0.886597i	$0.653063\pi$
$948$	0	0
$949$	−53.6730	−1.74230
$950$	−1.72900	−0.0560963
$951$	0	0
$952$	0.534483	0.0173227
$953$	−5.57136	−0.180474	−0.0902370	−	0.995920i	$-0.528762\pi$
$953$	−5.57136	−0.180474	−0.0902370	+	0.995920i	$0.528762\pi$
$954$	0	0
$955$	14.7637	0.477741
$956$	−5.98337	−0.193516
$957$	0	0
$958$	−37.9090	−1.22478
$959$	−2.55136	−0.0823878
$960$	0	0
$961$	−24.6787	−0.796087
$962$	118.238	3.81214
$963$	0	0
$964$	−2.03677	−0.0656000
$965$	−25.8760	−0.832979
$966$	0	0
$967$	−6.74985	−0.217060	−0.108530	−	0.994093i	$-0.534614\pi$
$967$	−6.74985	−0.217060	−0.108530	+	0.994093i	$0.534614\pi$
$968$	17.6857	0.568440
$969$	0	0
$970$	6.58743	0.211510
$971$	12.4963	0.401026	0.200513	−	0.979691i	$-0.435739\pi$
$971$	12.4963	0.401026	0.200513	+	0.979691i	$0.435739\pi$
$972$	0	0
$973$	−2.67211	−0.0856639
$974$	31.9333	1.02321
$975$	0	0
$976$	−0.0808018	−0.00258640
$977$	4.07191	0.130272	0.0651359	−	0.997876i	$-0.479252\pi$
$977$	4.07191	0.130272	0.0651359	+	0.997876i	$0.479252\pi$
$978$	0	0
$979$	1.15749	0.0369937
$980$	−7.51318	−0.240000
$981$	0	0
$982$	1.60382	0.0511801
$983$	1.37070	0.0437184	0.0218592	−	0.999761i	$-0.493041\pi$
$983$	1.37070	0.0437184	0.0218592	+	0.999761i	$0.493041\pi$
$984$	0	0
$985$	−23.5478	−0.750297
$986$	−6.71534	−0.213860
$987$	0	0
$988$	7.22115	0.229736
$989$	46.3997	1.47543
$990$	0	0
$991$	8.58059	0.272571	0.136286	−	0.990670i	$-0.456484\pi$
$991$	8.58059	0.272571	0.136286	+	0.990670i	$0.456484\pi$
$992$	13.9255	0.442135
$993$	0	0
$994$	−1.97275	−0.0625718
$995$	24.9325	0.790412
$996$	0	0
$997$	1.86063	0.0589266	0.0294633	−	0.999566i	$-0.490620\pi$
$997$	1.86063	0.0589266	0.0294633	+	0.999566i	$0.490620\pi$
$998$	−12.1753	−0.385404
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.2.a.c.1.5	✓	6
3.2	odd	2		3645.2.a.d.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3645.2.a.c.1.5	✓	6	1.1	even	1	trivial
3645.2.a.d.1.2	yes	6	3.2	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:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3645.a (trivial)

\(f(q)\)	\(=\)	\(q+1.75528 q^{2} +1.08100 q^{4} +1.00000 q^{5} -0.223190 q^{7} -1.61309 q^{8} +1.75528 q^{10} -0.190142 q^{11} -6.78155 q^{13} -0.391761 q^{14} -4.99344 q^{16} +1.48457 q^{17} -0.985031 q^{19} +1.08100 q^{20} -0.333752 q^{22} +7.07520 q^{23} +1.00000 q^{25} -11.9035 q^{26} -0.241270 q^{28} -2.57705 q^{29} -2.51422 q^{31} -5.53869 q^{32} +2.60583 q^{34} -0.223190 q^{35} -9.93302 q^{37} -1.72900 q^{38} -1.61309 q^{40} -5.70334 q^{41} +6.55808 q^{43} -0.205544 q^{44} +12.4190 q^{46} -8.85966 q^{47} -6.95019 q^{49} +1.75528 q^{50} -7.33089 q^{52} -1.90890 q^{53} -0.190142 q^{55} +0.360027 q^{56} -4.52344 q^{58} -9.38596 q^{59} +0.0161816 q^{61} -4.41316 q^{62} +0.264929 q^{64} -6.78155 q^{65} -9.87354 q^{67} +1.60482 q^{68} -0.391761 q^{70} +5.03559 q^{71} +7.91456 q^{73} -17.4352 q^{74} -1.06482 q^{76} +0.0424378 q^{77} -9.91847 q^{79} -4.99344 q^{80} -10.0109 q^{82} +13.2221 q^{83} +1.48457 q^{85} +11.5113 q^{86} +0.306716 q^{88} -6.08753 q^{89} +1.51358 q^{91} +7.64832 q^{92} -15.5512 q^{94} -0.985031 q^{95} +3.75293 q^{97} -12.1995 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{2} + 9 q^{4} + 6 q^{5} + 3 q^{7} - 3 q^{8} - 3 q^{10} - 9 q^{11} - 6 q^{13} - 24 q^{14} - 9 q^{16} + 15 q^{17} - 9 q^{19} + 9 q^{20} + 3 q^{22} - 6 q^{23} + 6 q^{25} - 3 q^{26} + 12 q^{28} - 30 q^{29}+ \cdots - 15 q^{97}+O(q^{100}) \) 6 * q - 3 * q^2 + 9 * q^4 + 6 * q^5 + 3 * q^7 - 3 * q^8 - 3 * q^10 - 9 * q^11 - 6 * q^13 - 24 * q^14 - 9 * q^16 + 15 * q^17 - 9 * q^19 + 9 * q^20 + 3 * q^22 - 6 * q^23 + 6 * q^25 - 3 * q^26 + 12 * q^28 - 30 * q^29 - 3 * q^31 - 9 * q^32 + 15 * q^34 + 3 * q^35 - 18 * q^37 - 18 * q^38 - 3 * q^40 - 15 * q^41 - 3 * q^43 - 36 * q^44 + 36 * q^46 - 15 * q^47 - 3 * q^49 - 3 * q^50 + 18 * q^52 - 33 * q^53 - 9 * q^55 - 24 * q^56 + 6 * q^58 - 27 * q^59 + 3 * q^61 - 6 * q^62 - 27 * q^64 - 6 * q^65 - 33 * q^67 + 12 * q^68 - 24 * q^70 - 27 * q^71 - 9 * q^73 + 3 * q^74 - 6 * q^76 - 9 * q^77 + 18 * q^79 - 9 * q^80 - 27 * q^82 - 15 * q^83 + 15 * q^85 + 15 * q^86 + 33 * q^88 + 15 * q^89 - 21 * q^91 - 9 * q^92 + 18 * q^94 - 9 * q^95 - 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more