LMFDB - Embedded newform 945.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,2,Mod(1,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("945.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	945.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:	$7.54586299101$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.144344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 5x + 9 $ x^4 - x^3 - 8x^2 + 5x + 9
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			1.2
Root			$1.51533$ of defining polynomial
Character	$\chi$	$=$	945.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.51533 q^{2} +0.296215 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.58179 q^{8} +O(q^{10})$	\(q-1.51533 q^{2} +0.296215 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.58179 q^{8} +1.51533 q^{10} -2.21911 q^{11} +1.18846 q^{13} -1.51533 q^{14} -4.50469 q^{16} -3.39333 q^{17} +7.61244 q^{19} -0.296215 q^{20} +3.36268 q^{22} -3.39333 q^{23} +1.00000 q^{25} -1.80090 q^{26} +0.296215 q^{28} -1.40757 q^{29} -4.42399 q^{31} +1.66249 q^{32} +5.14201 q^{34} -1.00000 q^{35} +5.03065 q^{37} -11.5353 q^{38} -2.58179 q^{40} +7.80090 q^{41} -1.42399 q^{43} -0.657335 q^{44} +5.14201 q^{46} +7.86221 q^{47} +1.00000 q^{49} -1.51533 q^{50} +0.352039 q^{52} +1.17422 q^{53} +2.21911 q^{55} +2.58179 q^{56} +2.13293 q^{58} +3.03065 q^{59} +5.39333 q^{61} +6.70378 q^{62} +6.49016 q^{64} -1.18846 q^{65} +3.01424 q^{67} -1.00516 q^{68} +1.51533 q^{70} +14.8764 q^{71} -3.18846 q^{73} -7.62308 q^{74} +2.25492 q^{76} -2.21911 q^{77} +11.4546 q^{79} +4.50469 q^{80} -11.8209 q^{82} -7.37692 q^{83} +3.39333 q^{85} +2.15780 q^{86} -5.72928 q^{88} +16.8929 q^{89} +1.18846 q^{91} -1.00516 q^{92} -11.9138 q^{94} -7.61244 q^{95} -1.75601 q^{97} -1.51533 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	$ 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.51533	−1.07150	−0.535749	−	0.844377i	$-0.679971\pi$
$2$	−1.51533	−1.07150	−0.535749	+	0.844377i	$0.679971\pi$
$3$	0	0
$3$	0	0
$4$	0.296215	0.148108
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.58179	0.912801
$9$	0	0
$10$	1.51533	0.479188
$11$	−2.21911	−0.669087	−0.334544	−	0.942380i	$-0.608582\pi$
$11$	−2.21911	−0.669087	−0.334544	+	0.942380i	$0.608582\pi$
$12$	0	0
$13$	1.18846	0.329619	0.164809	−	0.986325i	$-0.447299\pi$
$13$	1.18846	0.329619	0.164809	+	0.986325i	$0.447299\pi$
$14$	−1.51533	−0.404988
$15$	0	0
$16$	−4.50469	−1.12617
$17$	−3.39333	−0.823004	−0.411502	−	0.911409i	$-0.634996\pi$
$17$	−3.39333	−0.823004	−0.411502	+	0.911409i	$0.634996\pi$
$18$	0	0
$19$	7.61244	1.74641	0.873207	−	0.487349i	$-0.162036\pi$
$19$	7.61244	1.74641	0.873207	+	0.487349i	$0.162036\pi$
$20$	−0.296215	−0.0662357
$21$	0	0
$22$	3.36268	0.716926
$23$	−3.39333	−0.707559	−0.353779	−	0.935329i	$-0.615104\pi$
$23$	−3.39333	−0.707559	−0.353779	+	0.935329i	$0.615104\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.80090	−0.353186
$27$	0	0
$28$	0.296215	0.0559794
$29$	−1.40757	−0.261379	−0.130690	−	0.991423i	$-0.541719\pi$
$29$	−1.40757	−0.261379	−0.130690	+	0.991423i	$0.541719\pi$
$30$	0	0
$31$	−4.42399	−0.794571	−0.397286	−	0.917695i	$-0.630048\pi$
$31$	−4.42399	−0.794571	−0.397286	+	0.917695i	$0.630048\pi$
$32$	1.66249	0.293890
$33$	0	0
$34$	5.14201	0.881847
$35$	−1.00000	−0.169031
$36$	0	0
$37$	5.03065	0.827034	0.413517	−	0.910496i	$-0.364300\pi$
$37$	5.03065	0.827034	0.413517	+	0.910496i	$0.364300\pi$
$38$	−11.5353	−1.87128
$39$	0	0
$40$	−2.58179	−0.408217
$41$	7.80090	1.21830	0.609148	−	0.793056i	$-0.291511\pi$
$41$	7.80090	1.21830	0.609148	+	0.793056i	$0.291511\pi$
$42$	0	0
$43$	−1.42399	−0.217156	−0.108578	−	0.994088i	$-0.534630\pi$
$43$	−1.42399	−0.217156	−0.108578	+	0.994088i	$0.534630\pi$
$44$	−0.657335	−0.0990969
$45$	0	0
$46$	5.14201	0.758148
$47$	7.86221	1.14682	0.573411	−	0.819268i	$-0.305620\pi$
$47$	7.86221	1.14682	0.573411	+	0.819268i	$0.305620\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.51533	−0.214300
$51$	0	0
$52$	0.352039	0.0488191
$53$	1.17422	0.161292	0.0806458	−	0.996743i	$-0.474302\pi$
$53$	1.17422	0.161292	0.0806458	+	0.996743i	$0.474302\pi$
$54$	0	0
$55$	2.21911	0.299225
$56$	2.58179	0.345006
$57$	0	0
$58$	2.13293	0.280067
$59$	3.03065	0.394557	0.197279	−	0.980347i	$-0.436790\pi$
$59$	3.03065	0.394557	0.197279	+	0.980347i	$0.436790\pi$
$60$	0	0
$61$	5.39333	0.690545	0.345273	−	0.938502i	$-0.387786\pi$
$61$	5.39333	0.690545	0.345273	+	0.938502i	$0.387786\pi$
$62$	6.70378	0.851382
$63$	0	0
$64$	6.49016	0.811270
$65$	−1.18846	−0.147410
$66$	0	0
$67$	3.01424	0.368248	0.184124	−	0.982903i	$-0.441055\pi$
$67$	3.01424	0.368248	0.184124	+	0.982903i	$0.441055\pi$
$68$	−1.00516	−0.121893
$69$	0	0
$70$	1.51533	0.181116
$71$	14.8764	1.76551	0.882755	−	0.469834i	$-0.155686\pi$
$71$	14.8764	1.76551	0.882755	+	0.469834i	$0.155686\pi$
$72$	0	0
$73$	−3.18846	−0.373181	−0.186590	−	0.982438i	$-0.559744\pi$
$73$	−3.18846	−0.373181	−0.186590	+	0.982438i	$0.559744\pi$
$74$	−7.62308	−0.886166
$75$	0	0
$76$	2.25492	0.258657
$77$	−2.21911	−0.252891
$78$	0	0
$79$	11.4546	1.28875	0.644374	−	0.764711i	$-0.277118\pi$
$79$	11.4546	1.28875	0.644374	+	0.764711i	$0.277118\pi$
$80$	4.50469	0.503639
$81$	0	0
$82$	−11.8209	−1.30540
$83$	−7.37692	−0.809722	−0.404861	−	0.914378i	$-0.632680\pi$
$83$	−7.37692	−0.809722	−0.404861	+	0.914378i	$0.632680\pi$
$84$	0	0
$85$	3.39333	0.368059
$86$	2.15780	0.232682
$87$	0	0
$88$	−5.72928	−0.610743
$89$	16.8929	1.79064	0.895320	−	0.445424i	$-0.146947\pi$
$89$	16.8929	1.79064	0.895320	+	0.445424i	$0.146947\pi$
$90$	0	0
$91$	1.18846	0.124584
$92$	−1.00516	−0.104795
$93$	0	0
$94$	−11.9138	−1.22882
$95$	−7.61244	−0.781020
$96$	0	0
$97$	−1.75601	−0.178296	−0.0891480	−	0.996018i	$-0.528414\pi$
$97$	−1.75601	−0.178296	−0.0891480	+	0.996018i	$0.528414\pi$
$98$	−1.51533	−0.153071
$99$	0	0
$100$	0.296215	0.0296215
$101$	−0.393333	−0.0391381	−0.0195690	−	0.999809i	$-0.506229\pi$
$101$	−0.393333	−0.0391381	−0.0195690	+	0.999809i	$0.506229\pi$
$102$	0	0
$103$	−16.6938	−1.64489	−0.822443	−	0.568848i	$-0.807389\pi$
$103$	−16.6938	−1.64489	−0.822443	+	0.568848i	$0.807389\pi$
$104$	3.06835	0.300876
$105$	0	0
$106$	−1.77933	−0.172824
$107$	4.43822	0.429059	0.214530	−	0.976717i	$-0.431178\pi$
$107$	4.43822	0.429059	0.214530	+	0.976717i	$0.431178\pi$
$108$	0	0
$109$	15.4995	1.48459	0.742293	−	0.670076i	$-0.233738\pi$
$109$	15.4995	1.48459	0.742293	+	0.670076i	$0.233738\pi$
$110$	−3.36268	−0.320619
$111$	0	0
$112$	−4.50469	−0.425653
$113$	6.87285	0.646543	0.323272	−	0.946306i	$-0.395217\pi$
$113$	6.87285	0.646543	0.323272	+	0.946306i	$0.395217\pi$
$114$	0	0
$115$	3.39333	0.316430
$116$	−0.416944	−0.0387122
$117$	0	0
$118$	−4.59243	−0.422767
$119$	−3.39333	−0.311066
$120$	0	0
$121$	−6.07554	−0.552322
$122$	−8.17266	−0.739918
$123$	0	0
$124$	−1.31045	−0.117682
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	19.6182	1.74084	0.870418	−	0.492314i	$-0.163849\pi$
$127$	19.6182	1.74084	0.870418	+	0.492314i	$0.163849\pi$
$128$	−13.1597	−1.16316
$129$	0	0
$130$	1.80090	0.157950
$131$	15.8786	1.38732	0.693661	−	0.720302i	$-0.255997\pi$
$131$	15.8786	1.38732	0.693661	+	0.720302i	$0.255997\pi$
$132$	0	0
$133$	7.61244	0.660083
$134$	−4.56755	−0.394577
$135$	0	0
$136$	−8.76087	−0.751239
$137$	−1.17422	−0.100320	−0.0501602	−	0.998741i	$-0.515973\pi$
$137$	−1.17422	−0.100320	−0.0501602	+	0.998741i	$0.515973\pi$
$138$	0	0
$139$	8.78667	0.745275	0.372638	−	0.927977i	$-0.378453\pi$
$139$	8.78667	0.745275	0.372638	+	0.927977i	$0.378453\pi$
$140$	−0.296215	−0.0250348
$141$	0	0
$142$	−22.5427	−1.89174
$143$	−2.63732	−0.220544
$144$	0	0
$145$	1.40757	0.116892
$146$	4.83156	0.399862
$147$	0	0
$148$	1.49016	0.122490
$149$	23.2249	1.90266	0.951328	−	0.308179i	$-0.0997195\pi$
$149$	23.2249	1.90266	0.951328	+	0.308179i	$0.0997195\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	19.6537	1.59413
$153$	0	0
$154$	3.36268	0.270972
$155$	4.42399	0.355343
$156$	0	0
$157$	−15.8458	−1.26463	−0.632316	−	0.774711i	$-0.717896\pi$
$157$	−15.8458	−1.26463	−0.632316	+	0.774711i	$0.717896\pi$
$158$	−17.3575	−1.38089
$159$	0	0
$160$	−1.66249	−0.131431
$161$	−3.39333	−0.267432
$162$	0	0
$163$	−21.0707	−1.65038	−0.825192	−	0.564853i	$-0.808933\pi$
$163$	−21.0707	−1.65038	−0.825192	+	0.564853i	$0.808933\pi$
$164$	2.31075	0.180439
$165$	0	0
$166$	11.1784	0.867615
$167$	−0.485293	−0.0375531	−0.0187766	−	0.999824i	$-0.505977\pi$
$167$	−0.485293	−0.0375531	−0.0187766	+	0.999824i	$0.505977\pi$
$168$	0	0
$169$	−11.5876	−0.891351
$170$	−5.14201	−0.394374
$171$	0	0
$172$	−0.421806	−0.0321624
$173$	5.01642	0.381391	0.190696	−	0.981649i	$-0.438926\pi$
$173$	5.01642	0.381391	0.190696	+	0.981649i	$0.438926\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	9.99640	0.753507
$177$	0	0
$178$	−25.5982	−1.91867
$179$	−5.31107	−0.396968	−0.198484	−	0.980104i	$-0.563602\pi$
$179$	−5.31107	−0.396968	−0.198484	+	0.980104i	$0.563602\pi$
$180$	0	0
$181$	−23.9542	−1.78050	−0.890250	−	0.455473i	$-0.849470\pi$
$181$	−23.9542	−1.78050	−0.890250	+	0.455473i	$0.849470\pi$
$182$	−1.80090	−0.133492
$183$	0	0
$184$	−8.76087	−0.645860
$185$	−5.03065	−0.369861
$186$	0	0
$187$	7.53018	0.550662
$188$	2.32891	0.169853
$189$	0	0
$190$	11.5353	0.836862
$191$	−19.4440	−1.40692	−0.703459	−	0.710736i	$-0.748362\pi$
$191$	−19.4440	−1.40692	−0.703459	+	0.710736i	$0.748362\pi$
$192$	0	0
$193$	0.315609	0.0227180	0.0113590	−	0.999935i	$-0.496384\pi$
$193$	0.315609	0.0227180	0.0113590	+	0.999935i	$0.496384\pi$
$194$	2.66093	0.191044
$195$	0	0
$196$	0.296215	0.0211582
$197$	−3.20269	−0.228183	−0.114091	−	0.993470i	$-0.536396\pi$
$197$	−3.20269	−0.228183	−0.114091	+	0.993470i	$0.536396\pi$
$198$	0	0
$199$	4.43463	0.314362	0.157181	−	0.987570i	$-0.449759\pi$
$199$	4.43463	0.314362	0.157181	+	0.987570i	$0.449759\pi$
$200$	2.58179	0.182560
$201$	0	0
$202$	0.596028	0.0419364
$203$	−1.40757	−0.0987920
$204$	0	0
$205$	−7.80090	−0.544839
$206$	25.2965	1.76249
$207$	0	0
$208$	−5.35363	−0.371208
$209$	−16.8929	−1.16850
$210$	0	0
$211$	1.36050	0.0936606	0.0468303	−	0.998903i	$-0.485088\pi$
$211$	1.36050	0.0936606	0.0468303	+	0.998903i	$0.485088\pi$
$212$	0.347822	0.0238885
$213$	0	0
$214$	−6.72536	−0.459736
$215$	1.42399	0.0971151
$216$	0	0
$217$	−4.42399	−0.300320
$218$	−23.4869	−1.59073
$219$	0	0
$220$	0.657335	0.0443175
$221$	−4.03283	−0.271278
$222$	0	0
$223$	28.3168	1.89624	0.948118	−	0.317918i	$-0.102984\pi$
$223$	28.3168	1.89624	0.948118	+	0.317918i	$0.102984\pi$
$224$	1.66249	0.111080
$225$	0	0
$226$	−10.4146	−0.692770
$227$	14.9502	0.992283	0.496141	−	0.868242i	$-0.334750\pi$
$227$	14.9502	0.992283	0.496141	+	0.868242i	$0.334750\pi$
$228$	0	0
$229$	−8.16844	−0.539786	−0.269893	−	0.962890i	$-0.586988\pi$
$229$	−8.16844	−0.539786	−0.269893	+	0.962890i	$0.586988\pi$
$230$	−5.14201	−0.339054
$231$	0	0
$232$	−3.63405	−0.238587
$233$	−18.0364	−1.18161	−0.590803	−	0.806816i	$-0.701189\pi$
$233$	−18.0364	−1.18161	−0.590803	+	0.806816i	$0.701189\pi$
$234$	0	0
$235$	−7.86221	−0.512874
$236$	0.897726	0.0584370
$237$	0	0
$238$	5.14201	0.333307
$239$	12.0613	0.780181	0.390091	−	0.920777i	$-0.372444\pi$
$239$	12.0613	0.780181	0.390091	+	0.920777i	$0.372444\pi$
$240$	0	0
$241$	−22.2698	−1.43452	−0.717261	−	0.696804i	$-0.754605\pi$
$241$	−22.2698	−1.43452	−0.717261	+	0.696804i	$0.754605\pi$
$242$	9.20643	0.591812
$243$	0	0
$244$	1.59759	0.102275
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	9.04707	0.575651
$248$	−11.4218	−0.725285
$249$	0	0
$250$	1.51533	0.0958377
$251$	−15.8173	−0.998380	−0.499190	−	0.866492i	$-0.666369\pi$
$251$	−15.8173	−0.998380	−0.499190	+	0.866492i	$0.666369\pi$
$252$	0	0
$253$	7.53018	0.473419
$254$	−29.7280	−1.86530
$255$	0	0
$256$	6.96092	0.435057
$257$	−14.6182	−0.911860	−0.455930	−	0.890016i	$-0.650693\pi$
$257$	−14.6182	−0.911860	−0.455930	+	0.890016i	$0.650693\pi$
$258$	0	0
$259$	5.03065	0.312590
$260$	−0.352039	−0.0218326
$261$	0	0
$262$	−24.0613	−1.48651
$263$	−21.6631	−1.33580	−0.667902	−	0.744249i	$-0.732808\pi$
$263$	−21.6631	−1.33580	−0.667902	+	0.744249i	$0.732808\pi$
$264$	0	0
$265$	−1.17422	−0.0721318
$266$	−11.5353	−0.707277
$267$	0	0
$268$	0.892863	0.0545403
$269$	−24.2271	−1.47715	−0.738575	−	0.674171i	$-0.764501\pi$
$269$	−24.2271	−1.47715	−0.738575	+	0.674171i	$0.764501\pi$
$270$	0	0
$271$	−25.2142	−1.53166	−0.765828	−	0.643045i	$-0.777671\pi$
$271$	−25.2142	−1.53166	−0.765828	+	0.643045i	$0.777671\pi$
$272$	15.2859	0.926844
$273$	0	0
$274$	1.77933	0.107493
$275$	−2.21911	−0.133817
$276$	0	0
$277$	21.6346	1.29990	0.649950	−	0.759977i	$-0.274790\pi$
$277$	21.6346	1.29990	0.649950	+	0.759977i	$0.274790\pi$
$278$	−13.3147	−0.798561
$279$	0	0
$280$	−2.58179	−0.154292
$281$	22.9481	1.36897	0.684483	−	0.729028i	$-0.260028\pi$
$281$	22.9481	1.36897	0.684483	+	0.729028i	$0.260028\pi$
$282$	0	0
$283$	26.0729	1.54987	0.774935	−	0.632041i	$-0.217782\pi$
$283$	26.0729	1.54987	0.774935	+	0.632041i	$0.217782\pi$
$284$	4.40663	0.261485
$285$	0	0
$286$	3.99640	0.236312
$287$	7.80090	0.460473
$288$	0	0
$289$	−5.48529	−0.322664
$290$	−2.13293	−0.125250
$291$	0	0
$292$	−0.944470	−0.0552709
$293$	−2.66797	−0.155865	−0.0779324	−	0.996959i	$-0.524832\pi$
$293$	−2.66797	−0.155865	−0.0779324	+	0.996959i	$0.524832\pi$
$294$	0	0
$295$	−3.03065	−0.176451
$296$	12.9881	0.754918
$297$	0	0
$298$	−35.1933	−2.03869
$299$	−4.03283	−0.233225
$300$	0	0
$301$	−1.42399	−0.0820772
$302$	−12.1226	−0.697578
$303$	0	0
$304$	−34.2917	−1.96676
$305$	−5.39333	−0.308821
$306$	0	0
$307$	−4.63246	−0.264388	−0.132194	−	0.991224i	$-0.542202\pi$
$307$	−4.63246	−0.264388	−0.132194	+	0.991224i	$0.542202\pi$
$308$	−0.657335	−0.0374551
$309$	0	0
$310$	−6.70378	−0.380749
$311$	3.12355	0.177120	0.0885602	−	0.996071i	$-0.471773\pi$
$311$	3.12355	0.177120	0.0885602	+	0.996071i	$0.471773\pi$
$312$	0	0
$313$	−7.56537	−0.427620	−0.213810	−	0.976875i	$-0.568587\pi$
$313$	−7.56537	−0.427620	−0.213810	+	0.976875i	$0.568587\pi$
$314$	24.0116	1.35505
$315$	0	0
$316$	3.39304	0.190873
$317$	1.29684	0.0728375	0.0364188	−	0.999337i	$-0.488405\pi$
$317$	1.29684	0.0728375	0.0364188	+	0.999337i	$0.488405\pi$
$318$	0	0
$319$	3.12355	0.174885
$320$	−6.49016	−0.362811
$321$	0	0
$322$	5.14201	0.286553
$323$	−25.8316	−1.43731
$324$	0	0
$325$	1.18846	0.0659238
$326$	31.9290	1.76838
$327$	0	0
$328$	20.1403	1.11206
$329$	7.86221	0.433458
$330$	0	0
$331$	17.6631	0.970852	0.485426	−	0.874278i	$-0.338664\pi$
$331$	17.6631	0.970852	0.485426	+	0.874278i	$0.338664\pi$
$332$	−2.18515	−0.119926
$333$	0	0
$334$	0.735378	0.0402381
$335$	−3.01424	−0.164685
$336$	0	0
$337$	14.8480	0.808821	0.404410	−	0.914578i	$-0.367477\pi$
$337$	14.8480	0.808821	0.404410	+	0.914578i	$0.367477\pi$
$338$	17.5590	0.955081
$339$	0	0
$340$	1.00516	0.0545123
$341$	9.81732	0.531638
$342$	0	0
$343$	1.00000	0.0539949
$344$	−3.67643	−0.198220
$345$	0	0
$346$	−7.60151	−0.408660
$347$	−22.4995	−1.20784	−0.603919	−	0.797046i	$-0.706395\pi$
$347$	−22.4995	−1.20784	−0.603919	+	0.797046i	$0.706395\pi$
$348$	0	0
$349$	15.0564	0.805953	0.402976	−	0.915210i	$-0.367976\pi$
$349$	15.0564	0.805953	0.402976	+	0.915210i	$0.367976\pi$
$350$	−1.51533	−0.0809976
$351$	0	0
$352$	−3.68925	−0.196638
$353$	22.4382	1.19427	0.597133	−	0.802142i	$-0.296306\pi$
$353$	22.4382	1.19427	0.597133	+	0.802142i	$0.296306\pi$
$354$	0	0
$355$	−14.8764	−0.789560
$356$	5.00392	0.265207
$357$	0	0
$358$	8.04801	0.425351
$359$	−6.44182	−0.339986	−0.169993	−	0.985445i	$-0.554375\pi$
$359$	−6.44182	−0.339986	−0.169993	+	0.985445i	$0.554375\pi$
$360$	0	0
$361$	38.9493	2.04996
$362$	36.2984	1.90780
$363$	0	0
$364$	0.352039	0.0184519
$365$	3.18846	0.166892
$366$	0	0
$367$	−1.66311	−0.0868137	−0.0434069	−	0.999057i	$-0.513821\pi$
$367$	−1.66311	−0.0868137	−0.0434069	+	0.999057i	$0.513821\pi$
$368$	15.2859	0.796833
$369$	0	0
$370$	7.62308	0.396305
$371$	1.17422	0.0609625
$372$	0	0
$373$	0.843613	0.0436806	0.0218403	−	0.999761i	$-0.493047\pi$
$373$	0.843613	0.0436806	0.0218403	+	0.999761i	$0.493047\pi$
$374$	−11.4107	−0.590033
$375$	0	0
$376$	20.2986	1.04682
$377$	−1.67284	−0.0861555
$378$	0	0
$379$	18.5854	0.954667	0.477334	−	0.878722i	$-0.341603\pi$
$379$	18.5854	0.954667	0.477334	+	0.878722i	$0.341603\pi$
$380$	−2.25492	−0.115675
$381$	0	0
$382$	29.4640	1.50751
$383$	22.0871	1.12860	0.564299	−	0.825571i	$-0.309147\pi$
$383$	22.0871	1.12860	0.564299	+	0.825571i	$0.309147\pi$
$384$	0	0
$385$	2.21911	0.113096
$386$	−0.478251	−0.0243423
$387$	0	0
$388$	−0.520157	−0.0264070
$389$	13.3463	0.676682	0.338341	−	0.941024i	$-0.390134\pi$
$389$	13.3463	0.676682	0.338341	+	0.941024i	$0.390134\pi$
$390$	0	0
$391$	11.5147	0.582324
$392$	2.58179	0.130400
$393$	0	0
$394$	4.85313	0.244497
$395$	−11.4546	−0.576345
$396$	0	0
$397$	−28.6938	−1.44010	−0.720049	−	0.693923i	$-0.755881\pi$
$397$	−28.6938	−1.44010	−0.720049	+	0.693923i	$0.755881\pi$
$398$	−6.71991	−0.336839
$399$	0	0
$400$	−4.50469	−0.225234
$401$	3.71286	0.185412	0.0927058	−	0.995694i	$-0.470448\pi$
$401$	3.71286	0.185412	0.0927058	+	0.995694i	$0.470448\pi$
$402$	0	0
$403$	−5.25772	−0.261906
$404$	−0.116511	−0.00579665
$405$	0	0
$406$	2.13293	0.105855
$407$	−11.1636	−0.553358
$408$	0	0
$409$	−20.5854	−1.01788	−0.508941	−	0.860801i	$-0.669963\pi$
$409$	−20.5854	−1.01788	−0.508941	+	0.860801i	$0.669963\pi$
$410$	11.8209	0.583793
$411$	0	0
$412$	−4.94495	−0.243620
$413$	3.03065	0.149129
$414$	0	0
$415$	7.37692	0.362119
$416$	1.97580	0.0968716
$417$	0	0
$418$	25.5982	1.25205
$419$	33.3863	1.63103	0.815513	−	0.578738i	$-0.196455\pi$
$419$	33.3863	1.63103	0.815513	+	0.578738i	$0.196455\pi$
$420$	0	0
$421$	−31.9706	−1.55815	−0.779076	−	0.626930i	$-0.784311\pi$
$421$	−31.9706	−1.55815	−0.779076	+	0.626930i	$0.784311\pi$
$422$	−2.06160	−0.100357
$423$	0	0
$424$	3.03159	0.147227
$425$	−3.39333	−0.164601
$426$	0	0
$427$	5.39333	0.261002
$428$	1.31467	0.0635469
$429$	0	0
$430$	−2.15780	−0.104059
$431$	10.4133	0.501593	0.250797	−	0.968040i	$-0.419307\pi$
$431$	10.4133	0.501593	0.250797	+	0.968040i	$0.419307\pi$
$432$	0	0
$433$	1.35422	0.0650796	0.0325398	−	0.999470i	$-0.489640\pi$
$433$	1.35422	0.0650796	0.0325398	+	0.999470i	$0.489640\pi$
$434$	6.70378	0.321792
$435$	0	0
$436$	4.59120	0.219878
$437$	−25.8316	−1.23569
$438$	0	0
$439$	5.32531	0.254163	0.127082	−	0.991892i	$-0.459439\pi$
$439$	5.32531	0.254163	0.127082	+	0.991892i	$0.459439\pi$
$440$	5.72928	0.273133
$441$	0	0
$442$	6.11106	0.290673
$443$	1.89286	0.0899326	0.0449663	−	0.998989i	$-0.485682\pi$
$443$	1.89286	0.0899326	0.0449663	+	0.998989i	$0.485682\pi$
$444$	0	0
$445$	−16.8929	−0.800798
$446$	−42.9093	−2.03181
$447$	0	0
$448$	6.49016	0.306631
$449$	12.0729	0.569754	0.284877	−	0.958564i	$-0.408047\pi$
$449$	12.0729	0.569754	0.284877	+	0.958564i	$0.408047\pi$
$450$	0	0
$451$	−17.3111	−0.815147
$452$	2.03584	0.0957580
$453$	0	0
$454$	−22.6545	−1.06323
$455$	−1.18846	−0.0557158
$456$	0	0
$457$	26.7551	1.25155	0.625775	−	0.780004i	$-0.284783\pi$
$457$	26.7551	1.25155	0.625775	+	0.780004i	$0.284783\pi$
$458$	12.3779	0.578380
$459$	0	0
$460$	1.00516	0.0468657
$461$	−23.5751	−1.09800	−0.549000	−	0.835822i	$-0.684991\pi$
$461$	−23.5751	−1.09800	−0.549000	+	0.835822i	$0.684991\pi$
$462$	0	0
$463$	35.1769	1.63481	0.817404	−	0.576065i	$-0.195412\pi$
$463$	35.1769	1.63481	0.817404	+	0.576065i	$0.195412\pi$
$464$	6.34066	0.294358
$465$	0	0
$466$	27.3311	1.26609
$467$	−19.1493	−0.886126	−0.443063	−	0.896490i	$-0.646108\pi$
$467$	−19.1493	−0.886126	−0.443063	+	0.896490i	$0.646108\pi$
$468$	0	0
$469$	3.01424	0.139185
$470$	11.9138	0.549543
$471$	0	0
$472$	7.82451	0.360152
$473$	3.15998	0.145296
$474$	0	0
$475$	7.61244	0.349283
$476$	−1.00516	−0.0460713
$477$	0	0
$478$	−18.2768	−0.835962
$479$	2.96935	0.135673	0.0678365	−	0.997696i	$-0.478390\pi$
$479$	2.96935	0.135673	0.0678365	+	0.997696i	$0.478390\pi$
$480$	0	0
$481$	5.97872	0.272606
$482$	33.7460	1.53709
$483$	0	0
$484$	−1.79967	−0.0818031
$485$	1.75601	0.0797364
$486$	0	0
$487$	39.0440	1.76925	0.884625	−	0.466303i	$-0.154414\pi$
$487$	39.0440	1.76925	0.884625	+	0.466303i	$0.154414\pi$
$488$	13.9245	0.630331
$489$	0	0
$490$	1.51533	0.0684555
$491$	38.8764	1.75447	0.877235	−	0.480062i	$-0.159386\pi$
$491$	38.8764	1.75447	0.877235	+	0.480062i	$0.159386\pi$
$492$	0	0
$493$	4.77635	0.215116
$494$	−13.7093	−0.616809
$495$	0	0
$496$	19.9287	0.894824
$497$	14.8764	0.667300
$498$	0	0
$499$	−39.0893	−1.74988	−0.874938	−	0.484235i	$-0.839098\pi$
$499$	−39.0893	−1.74988	−0.874938	+	0.484235i	$0.839098\pi$
$500$	−0.296215	−0.0132471
$501$	0	0
$502$	23.9684	1.06976
$503$	−21.9706	−0.979620	−0.489810	−	0.871829i	$-0.662934\pi$
$503$	−21.9706	−0.979620	−0.489810	+	0.871829i	$0.662934\pi$
$504$	0	0
$505$	0.393333	0.0175031
$506$	−11.4107	−0.507267
$507$	0	0
$508$	5.81122	0.257831
$509$	−30.5396	−1.35364	−0.676821	−	0.736148i	$-0.736643\pi$
$509$	−30.5396	−1.35364	−0.676821	+	0.736148i	$0.736643\pi$
$510$	0	0
$511$	−3.18846	−0.141049
$512$	15.7713	0.697000
$513$	0	0
$514$	22.1514	0.977056
$515$	16.6938	0.735615
$516$	0	0
$517$	−17.4471	−0.767323
$518$	−7.62308	−0.334939
$519$	0	0
$520$	−3.06835	−0.134556
$521$	14.0280	0.614577	0.307288	−	0.951616i	$-0.400578\pi$
$521$	14.0280	0.614577	0.307288	+	0.951616i	$0.400578\pi$
$522$	0	0
$523$	−0.182681	−0.00798809	−0.00399404	−	0.999992i	$-0.501271\pi$
$523$	−0.182681	−0.00798809	−0.00399404	+	0.999992i	$0.501271\pi$
$524$	4.70349	0.205473
$525$	0	0
$526$	32.8267	1.43131
$527$	15.0121	0.653935
$528$	0	0
$529$	−11.4853	−0.499361
$530$	1.77933	0.0772891
$531$	0	0
$532$	2.25492	0.0977633
$533$	9.27104	0.401574
$534$	0	0
$535$	−4.43822	−0.191881
$536$	7.78213	0.336137
$537$	0	0
$538$	36.7119	1.58276
$539$	−2.21911	−0.0955839
$540$	0	0
$541$	−3.45246	−0.148433	−0.0742164	−	0.997242i	$-0.523646\pi$
$541$	−3.45246	−0.148433	−0.0742164	+	0.997242i	$0.523646\pi$
$542$	38.2078	1.64117
$543$	0	0
$544$	−5.64139	−0.241872
$545$	−15.4995	−0.663927
$546$	0	0
$547$	−31.8289	−1.36090	−0.680452	−	0.732793i	$-0.738217\pi$
$547$	−31.8289	−1.36090	−0.680452	+	0.732793i	$0.738217\pi$
$548$	−0.347822	−0.0148582
$549$	0	0
$550$	3.36268	0.143385
$551$	−10.7150	−0.456476
$552$	0	0
$553$	11.4546	0.487101
$554$	−32.7835	−1.39284
$555$	0	0
$556$	2.60274	0.110381
$557$	−16.3769	−0.693912	−0.346956	−	0.937881i	$-0.612785\pi$
$557$	−16.3769	−0.693912	−0.346956	+	0.937881i	$0.612785\pi$
$558$	0	0
$559$	−1.69235	−0.0715787
$560$	4.50469	0.190358
$561$	0	0
$562$	−34.7738	−1.46685
$563$	−21.1769	−0.892499	−0.446250	−	0.894909i	$-0.647241\pi$
$563$	−21.1769	−0.892499	−0.446250	+	0.894909i	$0.647241\pi$
$564$	0	0
$565$	−6.87285	−0.289143
$566$	−39.5089	−1.66068
$567$	0	0
$568$	38.4079	1.61156
$569$	31.6346	1.32619	0.663097	−	0.748534i	$-0.269242\pi$
$569$	31.6346	1.32619	0.663097	+	0.748534i	$0.269242\pi$
$570$	0	0
$571$	40.5685	1.69774	0.848869	−	0.528604i	$-0.177284\pi$
$571$	40.5685	1.69774	0.848869	+	0.528604i	$0.177284\pi$
$572$	−0.781215	−0.0326642
$573$	0	0
$574$	−11.8209	−0.493396
$575$	−3.39333	−0.141512
$576$	0	0
$577$	9.50531	0.395711	0.197856	−	0.980231i	$-0.436602\pi$
$577$	9.50531	0.395711	0.197856	+	0.980231i	$0.436602\pi$
$578$	8.31201	0.345734
$579$	0	0
$580$	0.416944	0.0173126
$581$	−7.37692	−0.306046
$582$	0	0
$583$	−2.60573	−0.107918
$584$	−8.23193	−0.340640
$585$	0	0
$586$	4.04285	0.167009
$587$	−21.6702	−0.894423	−0.447211	−	0.894428i	$-0.647583\pi$
$587$	−21.6702	−0.894423	−0.447211	+	0.894428i	$0.647583\pi$
$588$	0	0
$589$	−33.6773	−1.38765
$590$	4.59243	0.189067
$591$	0	0
$592$	−22.6615	−0.931383
$593$	−40.8422	−1.67719	−0.838593	−	0.544758i	$-0.816622\pi$
$593$	−40.8422	−1.67719	−0.838593	+	0.544758i	$0.816622\pi$
$594$	0	0
$595$	3.39333	0.139113
$596$	6.87957	0.281798
$597$	0	0
$598$	6.11106	0.249900
$599$	30.0977	1.22976	0.614880	−	0.788621i	$-0.289204\pi$
$599$	30.0977	1.22976	0.614880	+	0.788621i	$0.289204\pi$
$600$	0	0
$601$	−20.2298	−0.825189	−0.412594	−	0.910915i	$-0.635377\pi$
$601$	−20.2298	−0.825189	−0.412594	+	0.910915i	$0.635377\pi$
$602$	2.15780	0.0879455
$603$	0	0
$604$	2.36972	0.0964226
$605$	6.07554	0.247006
$606$	0	0
$607$	7.88894	0.320202	0.160101	−	0.987101i	$-0.448818\pi$
$607$	7.88894	0.320202	0.160101	+	0.987101i	$0.448818\pi$
$608$	12.6556	0.513253
$609$	0	0
$610$	8.17266	0.330901
$611$	9.34391	0.378014
$612$	0	0
$613$	−42.6725	−1.72353	−0.861763	−	0.507312i	$-0.830639\pi$
$613$	−42.6725	−1.72353	−0.861763	+	0.507312i	$0.830639\pi$
$614$	7.01969	0.283292
$615$	0	0
$616$	−5.72928	−0.230839
$617$	−34.0755	−1.37183	−0.685915	−	0.727682i	$-0.740598\pi$
$617$	−34.0755	−1.37183	−0.685915	+	0.727682i	$0.740598\pi$
$618$	0	0
$619$	−6.85157	−0.275388	−0.137694	−	0.990475i	$-0.543969\pi$
$619$	−6.85157	−0.275388	−0.137694	+	0.990475i	$0.543969\pi$
$620$	1.31045	0.0526290
$621$	0	0
$622$	−4.73320	−0.189784
$623$	16.8929	0.676798
$624$	0	0
$625$	1.00000	0.0400000
$626$	11.4640	0.458194
$627$	0	0
$628$	−4.69377	−0.187302
$629$	−17.0707	−0.680653
$630$	0	0
$631$	16.6795	0.664002	0.332001	−	0.943279i	$-0.392276\pi$
$631$	16.6795	0.664002	0.332001	+	0.943279i	$0.392276\pi$
$632$	29.5735	1.17637
$633$	0	0
$634$	−1.96513	−0.0780453
$635$	−19.6182	−0.778525
$636$	0	0
$637$	1.18846	0.0470884
$638$	−4.73320	−0.187389
$639$	0	0
$640$	13.1597	0.520182
$641$	18.5098	0.731095	0.365547	−	0.930793i	$-0.380882\pi$
$641$	18.5098	0.731095	0.365547	+	0.930793i	$0.380882\pi$
$642$	0	0
$643$	23.9399	0.944099	0.472049	−	0.881572i	$-0.343514\pi$
$643$	23.9399	0.944099	0.472049	+	0.881572i	$0.343514\pi$
$644$	−1.00516	−0.0396087
$645$	0	0
$646$	39.1432	1.54007
$647$	−26.5690	−1.04453	−0.522267	−	0.852782i	$-0.674914\pi$
$647$	−26.5690	−1.04453	−0.522267	+	0.852782i	$0.674914\pi$
$648$	0	0
$649$	−6.72536	−0.263993
$650$	−1.80090	−0.0706372
$651$	0	0
$652$	−6.24146	−0.244434
$653$	40.4312	1.58219	0.791097	−	0.611690i	$-0.209510\pi$
$653$	40.4312	1.58219	0.791097	+	0.611690i	$0.209510\pi$
$654$	0	0
$655$	−15.8786	−0.620429
$656$	−35.1406	−1.37201
$657$	0	0
$658$	−11.9138	−0.464449
$659$	−23.2364	−0.905163	−0.452582	−	0.891723i	$-0.649497\pi$
$659$	−23.2364	−0.905163	−0.452582	+	0.891723i	$0.649497\pi$
$660$	0	0
$661$	−2.43822	−0.0948359	−0.0474179	−	0.998875i	$-0.515099\pi$
$661$	−2.43822	−0.0948359	−0.0474179	+	0.998875i	$0.515099\pi$
$662$	−26.7654	−1.04027
$663$	0	0
$664$	−19.0457	−0.739115
$665$	−7.61244	−0.295198
$666$	0	0
$667$	4.77635	0.184941
$668$	−0.143751	−0.00556190
$669$	0	0
$670$	4.56755	0.176460
$671$	−11.9684	−0.462035
$672$	0	0
$673$	24.2956	0.936525	0.468263	−	0.883589i	$-0.344880\pi$
$673$	24.2956	0.936525	0.468263	+	0.883589i	$0.344880\pi$
$674$	−22.4995	−0.866650
$675$	0	0
$676$	−3.43241	−0.132016
$677$	7.06944	0.271701	0.135850	−	0.990729i	$-0.456623\pi$
$677$	7.06944	0.271701	0.135850	+	0.990729i	$0.456623\pi$
$678$	0	0
$679$	−1.75601	−0.0673895
$680$	8.76087	0.335964
$681$	0	0
$682$	−14.8764	−0.569649
$683$	1.05644	0.0404237	0.0202119	−	0.999796i	$-0.493566\pi$
$683$	1.05644	0.0404237	0.0202119	+	0.999796i	$0.493566\pi$
$684$	0	0
$685$	1.17422	0.0448647
$686$	−1.51533	−0.0578554
$687$	0	0
$688$	6.41461	0.244555
$689$	1.39551	0.0531648
$690$	0	0
$691$	3.18577	0.121193	0.0605963	−	0.998162i	$-0.480700\pi$
$691$	3.18577	0.121193	0.0605963	+	0.998162i	$0.480700\pi$
$692$	1.48594	0.0564869
$693$	0	0
$694$	34.0941	1.29420
$695$	−8.78667	−0.333297
$696$	0	0
$697$	−26.4711	−1.00266
$698$	−22.8154	−0.863577
$699$	0	0
$700$	0.296215	0.0111959
$701$	−33.1467	−1.25193	−0.625966	−	0.779850i	$-0.715295\pi$
$701$	−33.1467	−1.25193	−0.625966	+	0.779850i	$0.715295\pi$
$702$	0	0
$703$	38.2956	1.44434
$704$	−14.4024	−0.542810
$705$	0	0
$706$	−34.0012	−1.27965
$707$	−0.393333	−0.0147928
$708$	0	0
$709$	−6.26760	−0.235385	−0.117692	−	0.993050i	$-0.537550\pi$
$709$	−6.26760	−0.235385	−0.117692	+	0.993050i	$0.537550\pi$
$710$	22.5427	0.846012
$711$	0	0
$712$	43.6138	1.63450
$713$	15.0121	0.562206
$714$	0	0
$715$	2.63732	0.0986302
$716$	−1.57322	−0.0587940
$717$	0	0
$718$	9.76146	0.364295
$719$	28.4995	1.06285	0.531427	−	0.847104i	$-0.321656\pi$
$719$	28.4995	1.06285	0.531427	+	0.847104i	$0.321656\pi$
$720$	0	0
$721$	−16.6938	−0.621708
$722$	−59.0209	−2.19653
$723$	0	0
$724$	−7.09559	−0.263706
$725$	−1.40757	−0.0522758
$726$	0	0
$727$	42.0444	1.55934	0.779670	−	0.626191i	$-0.215387\pi$
$727$	42.0444	1.55934	0.779670	+	0.626191i	$0.215387\pi$
$728$	3.06835	0.113721
$729$	0	0
$730$	−4.83156	−0.178824
$731$	4.83206	0.178720
$732$	0	0
$733$	−37.1284	−1.37137	−0.685684	−	0.727899i	$-0.740497\pi$
$733$	−37.1284	−1.37137	−0.685684	+	0.727899i	$0.740497\pi$
$734$	2.52016	0.0930207
$735$	0	0
$736$	−5.64139	−0.207944
$737$	−6.68893	−0.246390
$738$	0	0
$739$	45.2404	1.66419	0.832097	−	0.554630i	$-0.187140\pi$
$739$	45.2404	1.66419	0.832097	+	0.554630i	$0.187140\pi$
$740$	−1.49016	−0.0547792
$741$	0	0
$742$	−1.77933	−0.0653212
$743$	27.6018	1.01261	0.506306	−	0.862354i	$-0.331011\pi$
$743$	27.6018	1.01261	0.506306	+	0.862354i	$0.331011\pi$
$744$	0	0
$745$	−23.2249	−0.850894
$746$	−1.27835	−0.0468037
$747$	0	0
$748$	2.23056	0.0815572
$749$	4.43822	0.162169
$750$	0	0
$751$	10.5454	0.384806	0.192403	−	0.981316i	$-0.438372\pi$
$751$	10.5454	0.384806	0.192403	+	0.981316i	$0.438372\pi$
$752$	−35.4168	−1.29152
$753$	0	0
$754$	2.53490	0.0923154
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−21.1636	−0.769203	−0.384602	−	0.923083i	$-0.625661\pi$
$757$	−21.1636	−0.769203	−0.384602	+	0.923083i	$0.625661\pi$
$758$	−28.1629	−1.02292
$759$	0	0
$760$	−19.6537	−0.712916
$761$	−39.1708	−1.41994	−0.709970	−	0.704232i	$-0.751291\pi$
$761$	−39.1708	−1.41994	−0.709970	+	0.704232i	$0.751291\pi$
$762$	0	0
$763$	15.4995	0.561121
$764$	−5.75961	−0.208375
$765$	0	0
$766$	−33.4692	−1.20929
$767$	3.60180	0.130054
$768$	0	0
$769$	−37.0236	−1.33511	−0.667553	−	0.744562i	$-0.732658\pi$
$769$	−37.0236	−1.33511	−0.667553	+	0.744562i	$0.732658\pi$
$770$	−3.36268	−0.121183
$771$	0	0
$772$	0.0934882	0.00336471
$773$	33.7490	1.21387	0.606933	−	0.794753i	$-0.292400\pi$
$773$	33.7490	1.21387	0.606933	+	0.794753i	$0.292400\pi$
$774$	0	0
$775$	−4.42399	−0.158914
$776$	−4.53365	−0.162749
$777$	0	0
$778$	−20.2239	−0.725064
$779$	59.3839	2.12765
$780$	0	0
$781$	−33.0125	−1.18128
$782$	−17.4485	−0.623959
$783$	0	0
$784$	−4.50469	−0.160882
$785$	15.8458	0.565561
$786$	0	0
$787$	−17.7275	−0.631918	−0.315959	−	0.948773i	$-0.602326\pi$
$787$	−17.7275	−0.631918	−0.315959	+	0.948773i	$0.602326\pi$
$788$	−0.948687	−0.0337956
$789$	0	0
$790$	17.3575	0.617553
$791$	6.87285	0.244370
$792$	0	0
$793$	6.40975	0.227617
$794$	43.4804	1.54306
$795$	0	0
$796$	1.31360	0.0465595
$797$	−5.74177	−0.203384	−0.101692	−	0.994816i	$-0.532426\pi$
$797$	−5.74177	−0.203384	−0.101692	+	0.994816i	$0.532426\pi$
$798$	0	0
$799$	−26.6791	−0.943838
$800$	1.66249	0.0587779
$801$	0	0
$802$	−5.62620	−0.198668
$803$	7.07554	0.249691
$804$	0	0
$805$	3.39333	0.119599
$806$	7.96717	0.280631
$807$	0	0
$808$	−1.01550	−0.0357253
$809$	−21.1964	−0.745226	−0.372613	−	0.927987i	$-0.621538\pi$
$809$	−21.1964	−0.745226	−0.372613	+	0.927987i	$0.621538\pi$
$810$	0	0
$811$	−39.3226	−1.38080	−0.690402	−	0.723426i	$-0.742566\pi$
$811$	−39.3226	−1.38080	−0.690402	+	0.723426i	$0.742566\pi$
$812$	−0.416944	−0.0146318
$813$	0	0
$814$	16.9165	0.592922
$815$	21.0707	0.738074
$816$	0	0
$817$	−10.8400	−0.379244
$818$	31.1936	1.09066
$819$	0	0
$820$	−2.31075	−0.0806948
$821$	−29.5690	−1.03196	−0.515982	−	0.856599i	$-0.672573\pi$
$821$	−29.5690	−1.03196	−0.515982	+	0.856599i	$0.672573\pi$
$822$	0	0
$823$	−26.0249	−0.907169	−0.453585	−	0.891213i	$-0.649855\pi$
$823$	−26.0249	−0.907169	−0.453585	+	0.891213i	$0.649855\pi$
$824$	−43.0998	−1.50145
$825$	0	0
$826$	−4.59243	−0.159791
$827$	−39.7490	−1.38221	−0.691104	−	0.722756i	$-0.742875\pi$
$827$	−39.7490	−1.38221	−0.691104	+	0.722756i	$0.742875\pi$
$828$	0	0
$829$	3.19206	0.110865	0.0554323	−	0.998462i	$-0.482346\pi$
$829$	3.19206	0.110865	0.0554323	+	0.998462i	$0.482346\pi$
$830$	−11.1784	−0.388009
$831$	0	0
$832$	7.71328	0.267410
$833$	−3.39333	−0.117572
$834$	0	0
$835$	0.485293	0.0167943
$836$	−5.00392	−0.173064
$837$	0	0
$838$	−50.5911	−1.74764
$839$	1.66623	0.0575247	0.0287623	−	0.999586i	$-0.490843\pi$
$839$	1.66623	0.0575247	0.0287623	+	0.999586i	$0.490843\pi$
$840$	0	0
$841$	−27.0187	−0.931681
$842$	48.4459	1.66956
$843$	0	0
$844$	0.403001	0.0138719
$845$	11.5876	0.398624
$846$	0	0
$847$	−6.07554	−0.208758
$848$	−5.28950	−0.181642
$849$	0	0
$850$	5.14201	0.176369
$851$	−17.0707	−0.585175
$852$	0	0
$853$	−30.4097	−1.04121	−0.520605	−	0.853798i	$-0.674294\pi$
$853$	−30.4097	−1.04121	−0.520605	+	0.853798i	$0.674294\pi$
$854$	−8.17266	−0.279663
$855$	0	0
$856$	11.4586	0.391646
$857$	−18.7365	−0.640026	−0.320013	−	0.947413i	$-0.603687\pi$
$857$	−18.7365	−0.640026	−0.320013	+	0.947413i	$0.603687\pi$
$858$	0	0
$859$	−35.5299	−1.21226	−0.606132	−	0.795364i	$-0.707280\pi$
$859$	−35.5299	−1.21226	−0.606132	+	0.795364i	$0.707280\pi$
$860$	0.421806	0.0143835
$861$	0	0
$862$	−15.7796	−0.537456
$863$	36.4039	1.23920	0.619602	−	0.784916i	$-0.287294\pi$
$863$	36.4039	1.23920	0.619602	+	0.784916i	$0.287294\pi$
$864$	0	0
$865$	−5.01642	−0.170563
$866$	−2.05208	−0.0697327
$867$	0	0
$868$	−1.31045	−0.0444796
$869$	−25.4191	−0.862285
$870$	0	0
$871$	3.58229	0.121381
$872$	40.0165	1.35513
$873$	0	0
$874$	39.1432	1.32404
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	21.8502	0.737827	0.368914	−	0.929464i	$-0.379730\pi$
$877$	21.8502	0.737827	0.368914	+	0.929464i	$0.379730\pi$
$878$	−8.06958	−0.272335
$879$	0	0
$880$	−9.99640	−0.336979
$881$	−10.9481	−0.368850	−0.184425	−	0.982847i	$-0.559042\pi$
$881$	−10.9481	−0.368850	−0.184425	+	0.982847i	$0.559042\pi$
$882$	0	0
$883$	−59.1605	−1.99091	−0.995454	−	0.0952432i	$-0.969637\pi$
$883$	−59.1605	−1.99091	−0.995454	+	0.0952432i	$0.969637\pi$
$884$	−1.19459	−0.0401783
$885$	0	0
$886$	−2.86831	−0.0963626
$887$	15.4426	0.518511	0.259256	−	0.965809i	$-0.416523\pi$
$887$	15.4426	0.518511	0.259256	+	0.965809i	$0.416523\pi$
$888$	0	0
$889$	19.6182	0.657974
$890$	25.5982	0.858054
$891$	0	0
$892$	8.38788	0.280847
$893$	59.8506	2.00282
$894$	0	0
$895$	5.31107	0.177530
$896$	−13.1597	−0.439634
$897$	0	0
$898$	−18.2943	−0.610490
$899$	6.22707	0.207684
$900$	0	0
$901$	−3.98452	−0.132744
$902$	26.2319	0.873428
$903$	0	0
$904$	17.7443	0.590165
$905$	23.9542	0.796264
$906$	0	0
$907$	−0.483113	−0.0160415	−0.00802076	−	0.999968i	$-0.502553\pi$
$907$	−0.483113	−0.0160415	−0.00802076	+	0.999968i	$0.502553\pi$
$908$	4.42849	0.146965
$909$	0	0
$910$	1.80090	0.0596993
$911$	39.0173	1.29270	0.646351	−	0.763040i	$-0.276294\pi$
$911$	39.0173	1.29270	0.646351	+	0.763040i	$0.276294\pi$
$912$	0	0
$913$	16.3702	0.541775
$914$	−40.5427	−1.34103
$915$	0	0
$916$	−2.41962	−0.0799464
$917$	15.8786	0.524358
$918$	0	0
$919$	24.7939	0.817874	0.408937	−	0.912563i	$-0.365900\pi$
$919$	24.7939	0.817874	0.408937	+	0.912563i	$0.365900\pi$
$920$	8.76087	0.288837
$921$	0	0
$922$	35.7239	1.17651
$923$	17.6800	0.581945
$924$	0	0
$925$	5.03065	0.165407
$926$	−53.3045	−1.75169
$927$	0	0
$928$	−2.34007	−0.0768166
$929$	19.2298	0.630908	0.315454	−	0.948941i	$-0.397843\pi$
$929$	19.2298	0.630908	0.315454	+	0.948941i	$0.397843\pi$
$930$	0	0
$931$	7.61244	0.249488
$932$	−5.34267	−0.175005
$933$	0	0
$934$	29.0175	0.949482
$935$	−7.53018	−0.246263
$936$	0	0
$937$	−17.5867	−0.574531	−0.287265	−	0.957851i	$-0.592746\pi$
$937$	−17.5867	−0.574531	−0.287265	+	0.957851i	$0.592746\pi$
$938$	−4.56755	−0.149136
$939$	0	0
$940$	−2.32891	−0.0759605
$941$	−3.50533	−0.114271	−0.0571353	−	0.998366i	$-0.518197\pi$
$941$	−3.50533	−0.114271	−0.0571353	+	0.998366i	$0.518197\pi$
$942$	0	0
$943$	−26.4711	−0.862016
$944$	−13.6521	−0.444339
$945$	0	0
$946$	−4.78841	−0.155685
$947$	−28.3880	−0.922487	−0.461244	−	0.887274i	$-0.652597\pi$
$947$	−28.3880	−0.922487	−0.461244	+	0.887274i	$0.652597\pi$
$948$	0	0
$949$	−3.78935	−0.123007
$950$	−11.5353	−0.374256
$951$	0	0
$952$	−8.76087	−0.283942
$953$	−4.49953	−0.145754	−0.0728770	−	0.997341i	$-0.523218\pi$
$953$	−4.49953	−0.145754	−0.0728770	+	0.997341i	$0.523218\pi$
$954$	0	0
$955$	19.4440	0.629193
$956$	3.57274	0.115551
$957$	0	0
$958$	−4.49953	−0.145373
$959$	−1.17422	−0.0379176
$960$	0	0
$961$	−11.4283	−0.368656
$962$	−9.05972	−0.292097
$963$	0	0
$964$	−6.59665	−0.212464
$965$	−0.315609	−0.0101598
$966$	0	0
$967$	−11.0574	−0.355581	−0.177791	−	0.984068i	$-0.556895\pi$
$967$	−11.0574	−0.355581	−0.177791	+	0.984068i	$0.556895\pi$
$968$	−15.6858	−0.504160
$969$	0	0
$970$	−2.66093	−0.0854374
$971$	−44.4570	−1.42669	−0.713346	−	0.700812i	$-0.752821\pi$
$971$	−44.4570	−1.42669	−0.713346	+	0.700812i	$0.752821\pi$
$972$	0	0
$973$	8.78667	0.281688
$974$	−59.1643	−1.89575
$975$	0	0
$976$	−24.2953	−0.777673
$977$	27.2942	0.873217	0.436609	−	0.899652i	$-0.356179\pi$
$977$	27.2942	0.873217	0.436609	+	0.899652i	$0.356179\pi$
$978$	0	0
$979$	−37.4871	−1.19809
$980$	−0.296215	−0.00946225
$981$	0	0
$982$	−58.9105	−1.87991
$983$	26.1440	0.833866	0.416933	−	0.908937i	$-0.363105\pi$
$983$	26.1440	0.833866	0.416933	+	0.908937i	$0.363105\pi$
$984$	0	0
$985$	3.20269	0.102046
$986$	−7.23773	−0.230496
$987$	0	0
$988$	2.67988	0.0852583
$989$	4.83206	0.153651
$990$	0	0
$991$	36.2413	1.15124	0.575621	−	0.817716i	$-0.304760\pi$
$991$	36.2413	1.15124	0.575621	+	0.817716i	$0.304760\pi$
$992$	−7.35484	−0.233516
$993$	0	0
$994$	−22.5427	−0.715010
$995$	−4.43463	−0.140587
$996$	0	0
$997$	2.94683	0.0933269	0.0466635	−	0.998911i	$-0.485141\pi$
$997$	2.94683	0.0933269	0.0466635	+	0.998911i	$0.485141\pi$
$998$	59.2330	1.87499
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.2.a.m.1.2	✓	4
3.2	odd	2		945.2.a.n.1.3	yes	4
5.4	even	2		4725.2.a.bx.1.3		4
7.6	odd	2		6615.2.a.be.1.2		4
15.14	odd	2		4725.2.a.bo.1.2		4
21.20	even	2		6615.2.a.bh.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
945.2.a.m.1.2	✓	4	1.1	even	1	trivial
945.2.a.n.1.3	yes	4	3.2	odd	2
4725.2.a.bo.1.2		4	15.14	odd	2
4725.2.a.bx.1.3		4	5.4	even	2
6615.2.a.be.1.2		4	7.6	odd	2
6615.2.a.bh.1.3		4	21.20	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	945.a (trivial)

\(f(q)\)	\(=\)	\(q-1.51533 q^{2} +0.296215 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.58179 q^{8} +O(q^{10})\)	\(q-1.51533 q^{2} +0.296215 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.58179 q^{8} +1.51533 q^{10} -2.21911 q^{11} +1.18846 q^{13} -1.51533 q^{14} -4.50469 q^{16} -3.39333 q^{17} +7.61244 q^{19} -0.296215 q^{20} +3.36268 q^{22} -3.39333 q^{23} +1.00000 q^{25} -1.80090 q^{26} +0.296215 q^{28} -1.40757 q^{29} -4.42399 q^{31} +1.66249 q^{32} +5.14201 q^{34} -1.00000 q^{35} +5.03065 q^{37} -11.5353 q^{38} -2.58179 q^{40} +7.80090 q^{41} -1.42399 q^{43} -0.657335 q^{44} +5.14201 q^{46} +7.86221 q^{47} +1.00000 q^{49} -1.51533 q^{50} +0.352039 q^{52} +1.17422 q^{53} +2.21911 q^{55} +2.58179 q^{56} +2.13293 q^{58} +3.03065 q^{59} +5.39333 q^{61} +6.70378 q^{62} +6.49016 q^{64} -1.18846 q^{65} +3.01424 q^{67} -1.00516 q^{68} +1.51533 q^{70} +14.8764 q^{71} -3.18846 q^{73} -7.62308 q^{74} +2.25492 q^{76} -2.21911 q^{77} +11.4546 q^{79} +4.50469 q^{80} -11.8209 q^{82} -7.37692 q^{83} +3.39333 q^{85} +2.15780 q^{86} -5.72928 q^{88} +16.8929 q^{89} +1.18846 q^{91} -1.00516 q^{92} -11.9138 q^{94} -7.61244 q^{95} -1.75601 q^{97} -1.51533 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8	\( 4 q - q^{2} + 9 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + q^{10} + 4 q^{11} + 2 q^{13} - q^{14} + 19 q^{16} + 4 q^{19} - 9 q^{20} + 10 q^{22} + 4 q^{25} + 22 q^{26} + 9 q^{28} + 10 q^{29} + 6 q^{31} - 23 q^{32} - 13 q^{34} - 4 q^{35} + 10 q^{37} + q^{38} + 6 q^{40} + 2 q^{41} + 18 q^{43} + 36 q^{44} - 13 q^{46} - 18 q^{47} + 4 q^{49} - q^{50} - 34 q^{52} + 4 q^{53} - 4 q^{55} - 6 q^{56} - 14 q^{58} + 2 q^{59} + 8 q^{61} + 19 q^{62} + 54 q^{64} - 2 q^{65} + 10 q^{67} - 13 q^{68} + q^{70} + 8 q^{71} - 10 q^{73} - 36 q^{74} - 5 q^{76} + 4 q^{77} + 12 q^{79} - 19 q^{80} + 24 q^{82} - 24 q^{83} + 16 q^{86} + 4 q^{88} + 8 q^{89} + 2 q^{91} - 13 q^{92} - 38 q^{94} - 4 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 4 * q - q^2 + 9 * q^4 - 4 * q^5 + 4 * q^7 - 6 * q^8 + q^10 + 4 * q^11 + 2 * q^13 - q^14 + 19 * q^16 + 4 * q^19 - 9 * q^20 + 10 * q^22 + 4 * q^25 + 22 * q^26 + 9 * q^28 + 10 * q^29 + 6 * q^31 - 23 * q^32 - 13 * q^34 - 4 * q^35 + 10 * q^37 + q^38 + 6 * q^40 + 2 * q^41 + 18 * q^43 + 36 * q^44 - 13 * q^46 - 18 * q^47 + 4 * q^49 - q^50 - 34 * q^52 + 4 * q^53 - 4 * q^55 - 6 * q^56 - 14 * q^58 + 2 * q^59 + 8 * q^61 + 19 * q^62 + 54 * q^64 - 2 * q^65 + 10 * q^67 - 13 * q^68 + q^70 + 8 * q^71 - 10 * q^73 - 36 * q^74 - 5 * q^76 + 4 * q^77 + 12 * q^79 - 19 * q^80 + 24 * q^82 - 24 * q^83 + 16 * q^86 + 4 * q^88 + 8 * q^89 + 2 * q^91 - 13 * q^92 - 38 * q^94 - 4 * q^95 + 10 * q^97 - q^98

Introduction

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