LMFDB - Embedded newform 4225.2.a.ca.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.242854$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.242854 q^{2} -1.19348 q^{3} -1.94102 q^{4} -0.289840 q^{6} -4.78046 q^{7} -0.957093 q^{8} -1.57562 q^{9} +O(q^{10})$	\(q+0.242854 q^{2} -1.19348 q^{3} -1.94102 q^{4} -0.289840 q^{6} -4.78046 q^{7} -0.957093 q^{8} -1.57562 q^{9} +1.30170 q^{11} +2.31656 q^{12} -1.16095 q^{14} +3.64961 q^{16} +5.83916 q^{17} -0.382645 q^{18} -4.27490 q^{19} +5.70536 q^{21} +0.316124 q^{22} +4.13523 q^{23} +1.14227 q^{24} +5.46089 q^{27} +9.27898 q^{28} -3.08654 q^{29} +5.43011 q^{31} +2.80051 q^{32} -1.55355 q^{33} +1.41806 q^{34} +3.05831 q^{36} +4.84188 q^{37} -1.03818 q^{38} -8.00178 q^{41} +1.38557 q^{42} +2.70433 q^{43} -2.52664 q^{44} +1.00426 q^{46} +6.31797 q^{47} -4.35572 q^{48} +15.8528 q^{49} -6.96890 q^{51} -0.506686 q^{53} +1.32620 q^{54} +4.57534 q^{56} +5.10199 q^{57} -0.749578 q^{58} -3.12373 q^{59} -2.79253 q^{61} +1.31872 q^{62} +7.53217 q^{63} -6.61911 q^{64} -0.377287 q^{66} -4.07096 q^{67} -11.3339 q^{68} -4.93529 q^{69} -7.18916 q^{71} +1.50801 q^{72} -5.23396 q^{73} +1.17587 q^{74} +8.29767 q^{76} -6.22275 q^{77} -7.18371 q^{79} -1.79058 q^{81} -1.94327 q^{82} +9.99869 q^{83} -11.0742 q^{84} +0.656757 q^{86} +3.68371 q^{87} -1.24585 q^{88} +12.3735 q^{89} -8.02657 q^{92} -6.48071 q^{93} +1.53434 q^{94} -3.34234 q^{96} -2.82773 q^{97} +3.84991 q^{98} -2.05099 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	$ 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

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 $

$2$	0.242854	0.171724	0.0858619	−	0.996307i	$-0.472636\pi$
$2$	0.242854	0.171724	0.0858619	+	0.996307i	$0.472636\pi$
$3$	−1.19348	−0.689053	−0.344527	−	0.938777i	$-0.611961\pi$
$3$	−1.19348	−0.689053	−0.344527	+	0.938777i	$0.611961\pi$
$4$	−1.94102	−0.970511
$5$	0	0
$5$	0	0
$6$	−0.289840	−0.118327
$7$	−4.78046	−1.80684	−0.903422	−	0.428753i	$-0.858953\pi$
$7$	−4.78046	−1.80684	−0.903422	+	0.428753i	$0.858953\pi$
$8$	−0.957093	−0.338383
$9$	−1.57562	−0.525205
$10$	0	0
$11$	1.30170	0.392479	0.196239	−	0.980556i	$-0.437127\pi$
$11$	1.30170	0.392479	0.196239	+	0.980556i	$0.437127\pi$
$12$	2.31656	0.668734
$13$	0	0
$13$	0	0
$14$	−1.16095	−0.310278
$15$	0	0
$16$	3.64961	0.912402
$17$	5.83916	1.41620	0.708102	−	0.706110i	$-0.249552\pi$
$17$	5.83916	1.41620	0.708102	+	0.706110i	$0.249552\pi$
$18$	−0.382645	−0.0901902
$19$	−4.27490	−0.980729	−0.490365	−	0.871517i	$-0.663136\pi$
$19$	−4.27490	−0.980729	−0.490365	+	0.871517i	$0.663136\pi$
$20$	0	0
$21$	5.70536	1.24501
$22$	0.316124	0.0673979
$23$	4.13523	0.862255	0.431127	−	0.902291i	$-0.358116\pi$
$23$	4.13523	0.862255	0.431127	+	0.902291i	$0.358116\pi$
$24$	1.14227	0.233164
$25$	0	0
$26$	0	0
$27$	5.46089	1.05095
$28$	9.27898	1.75356
$29$	−3.08654	−0.573156	−0.286578	−	0.958057i	$-0.592518\pi$
$29$	−3.08654	−0.573156	−0.286578	+	0.958057i	$0.592518\pi$
$30$	0	0
$31$	5.43011	0.975277	0.487639	−	0.873046i	$-0.337858\pi$
$31$	5.43011	0.975277	0.487639	+	0.873046i	$0.337858\pi$
$32$	2.80051	0.495065
$33$	−1.55355	−0.270439
$34$	1.41806	0.243196
$35$	0	0
$36$	3.05831	0.509718
$37$	4.84188	0.796000	0.398000	−	0.917385i	$-0.369704\pi$
$37$	4.84188	0.796000	0.398000	+	0.917385i	$0.369704\pi$
$38$	−1.03818	−0.168414
$39$	0	0
$40$	0	0
$41$	−8.00178	−1.24967	−0.624834	−	0.780757i	$-0.714833\pi$
$41$	−8.00178	−1.24967	−0.624834	+	0.780757i	$0.714833\pi$
$42$	1.38557	0.213798
$43$	2.70433	0.412406	0.206203	−	0.978509i	$-0.433889\pi$
$43$	2.70433	0.412406	0.206203	+	0.978509i	$0.433889\pi$
$44$	−2.52664	−0.380905
$45$	0	0
$46$	1.00426	0.148070
$47$	6.31797	0.921570	0.460785	−	0.887512i	$-0.347568\pi$
$47$	6.31797	0.921570	0.460785	+	0.887512i	$0.347568\pi$
$48$	−4.35572	−0.628694
$49$	15.8528	2.26468
$50$	0	0
$51$	−6.96890	−0.975841
$52$	0	0
$53$	−0.506686	−0.0695986	−0.0347993	−	0.999394i	$-0.511079\pi$
$53$	−0.506686	−0.0695986	−0.0347993	+	0.999394i	$0.511079\pi$
$54$	1.32620	0.180473
$55$	0	0
$56$	4.57534	0.611406
$57$	5.10199	0.675775
$58$	−0.749578	−0.0984244
$59$	−3.12373	−0.406675	−0.203337	−	0.979109i	$-0.565179\pi$
$59$	−3.12373	−0.406675	−0.203337	+	0.979109i	$0.565179\pi$
$60$	0	0
$61$	−2.79253	−0.357547	−0.178773	−	0.983890i	$-0.557213\pi$
$61$	−2.79253	−0.357547	−0.178773	+	0.983890i	$0.557213\pi$
$62$	1.31872	0.167478
$63$	7.53217	0.948964
$64$	−6.61911	−0.827388
$65$	0	0
$66$	−0.377287	−0.0464408
$67$	−4.07096	−0.497347	−0.248673	−	0.968587i	$-0.579995\pi$
$67$	−4.07096	−0.497347	−0.248673	+	0.968587i	$0.579995\pi$
$68$	−11.3339	−1.37444
$69$	−4.93529	−0.594140
$70$	0	0
$71$	−7.18916	−0.853196	−0.426598	−	0.904441i	$-0.640288\pi$
$71$	−7.18916	−0.853196	−0.426598	+	0.904441i	$0.640288\pi$
$72$	1.50801	0.177721
$73$	−5.23396	−0.612589	−0.306294	−	0.951937i	$-0.599089\pi$
$73$	−5.23396	−0.612589	−0.306294	+	0.951937i	$0.599089\pi$
$74$	1.17587	0.136692
$75$	0	0
$76$	8.29767	0.951808
$77$	−6.22275	−0.709148
$78$	0	0
$79$	−7.18371	−0.808231	−0.404116	−	0.914708i	$-0.632421\pi$
$79$	−7.18371	−0.808231	−0.404116	+	0.914708i	$0.632421\pi$
$80$	0	0
$81$	−1.79058	−0.198954
$82$	−1.94327	−0.214598
$83$	9.99869	1.09750	0.548749	−	0.835987i	$-0.315104\pi$
$83$	9.99869	1.09750	0.548749	+	0.835987i	$0.315104\pi$
$84$	−11.0742	−1.20830
$85$	0	0
$86$	0.656757	0.0708199
$87$	3.68371	0.394935
$88$	−1.24585	−0.132808
$89$	12.3735	1.31159	0.655794	−	0.754940i	$-0.272334\pi$
$89$	12.3735	1.31159	0.655794	+	0.754940i	$0.272334\pi$
$90$	0	0
$91$	0	0
$92$	−8.02657	−0.836828
$93$	−6.48071	−0.672018
$94$	1.53434	0.158256
$95$	0	0
$96$	−3.34234	−0.341126
$97$	−2.82773	−0.287112	−0.143556	−	0.989642i	$-0.545854\pi$
$97$	−2.82773	−0.287112	−0.143556	+	0.989642i	$0.545854\pi$
$98$	3.84991	0.388900
$99$	−2.05099	−0.206132
$100$	0	0
$101$	7.36950	0.733293	0.366646	−	0.930360i	$-0.380506\pi$
$101$	7.36950	0.733293	0.366646	+	0.930360i	$0.380506\pi$
$102$	−1.69242	−0.167575
$103$	−9.50730	−0.936783	−0.468391	−	0.883521i	$-0.655166\pi$
$103$	−9.50730	−0.936783	−0.468391	+	0.883521i	$0.655166\pi$
$104$	0	0
$105$	0	0
$106$	−0.123051	−0.0119517
$107$	−5.15118	−0.497984	−0.248992	−	0.968506i	$-0.580099\pi$
$107$	−5.15118	−0.497984	−0.248992	+	0.968506i	$0.580099\pi$
$108$	−10.5997	−1.01996
$109$	−8.78837	−0.841773	−0.420887	−	0.907113i	$-0.638281\pi$
$109$	−8.78837	−0.841773	−0.420887	+	0.907113i	$0.638281\pi$
$110$	0	0
$111$	−5.77866	−0.548486
$112$	−17.4468	−1.64857
$113$	12.2753	1.15477	0.577383	−	0.816473i	$-0.304074\pi$
$113$	12.2753	1.15477	0.577383	+	0.816473i	$0.304074\pi$
$114$	1.23904	0.116047
$115$	0	0
$116$	5.99104	0.556254
$117$	0	0
$118$	−0.758610	−0.0698357
$119$	−27.9139	−2.55886
$120$	0	0
$121$	−9.30556	−0.845960
$122$	−0.678177	−0.0613992
$123$	9.54993	0.861088
$124$	−10.5400	−0.946517
$125$	0	0
$126$	1.82922	0.162960
$127$	−12.5953	−1.11766	−0.558828	−	0.829284i	$-0.688749\pi$
$127$	−12.5953	−1.11766	−0.558828	+	0.829284i	$0.688749\pi$
$128$	−7.20849	−0.637147
$129$	−3.22755	−0.284170
$130$	0	0
$131$	3.91530	0.342082	0.171041	−	0.985264i	$-0.445287\pi$
$131$	3.91530	0.342082	0.171041	+	0.985264i	$0.445287\pi$
$132$	3.01548	0.262464
$133$	20.4360	1.77202
$134$	−0.988649	−0.0854062
$135$	0	0
$136$	−5.58862	−0.479220
$137$	14.4959	1.23847	0.619233	−	0.785207i	$-0.287444\pi$
$137$	14.4959	1.23847	0.619233	+	0.785207i	$0.287444\pi$
$138$	−1.19856	−0.102028
$139$	13.6934	1.16146	0.580730	−	0.814096i	$-0.302767\pi$
$139$	13.6934	1.16146	0.580730	+	0.814096i	$0.302767\pi$
$140$	0	0
$141$	−7.54034	−0.635011
$142$	−1.74592	−0.146514
$143$	0	0
$144$	−5.75038	−0.479199
$145$	0	0
$146$	−1.27109	−0.105196
$147$	−18.9199	−1.56049
$148$	−9.39819	−0.772527
$149$	−9.45832	−0.774856	−0.387428	−	0.921900i	$-0.626636\pi$
$149$	−9.45832	−0.774856	−0.387428	+	0.921900i	$0.626636\pi$
$150$	0	0
$151$	−9.59276	−0.780648	−0.390324	−	0.920678i	$-0.627637\pi$
$151$	−9.59276	−0.780648	−0.390324	+	0.920678i	$0.627637\pi$
$152$	4.09148	0.331863
$153$	−9.20028	−0.743798
$154$	−1.51122	−0.121778
$155$	0	0
$156$	0	0
$157$	−14.8447	−1.18473	−0.592366	−	0.805669i	$-0.701806\pi$
$157$	−14.8447	−1.18473	−0.592366	+	0.805669i	$0.701806\pi$
$158$	−1.74459	−0.138792
$159$	0.604717	0.0479572
$160$	0	0
$161$	−19.7683	−1.55796
$162$	−0.434851	−0.0341651
$163$	−1.68754	−0.132178	−0.0660891	−	0.997814i	$-0.521052\pi$
$163$	−1.68754	−0.132178	−0.0660891	+	0.997814i	$0.521052\pi$
$164$	15.5316	1.21282
$165$	0	0
$166$	2.42822	0.188467
$167$	−10.4585	−0.809305	−0.404652	−	0.914471i	$-0.632607\pi$
$167$	−10.4585	−0.809305	−0.404652	+	0.914471i	$0.632607\pi$
$168$	−5.46056	−0.421291
$169$	0	0
$170$	0	0
$171$	6.73560	0.515084
$172$	−5.24916	−0.400244
$173$	20.5699	1.56390	0.781950	−	0.623342i	$-0.214225\pi$
$173$	20.5699	1.56390	0.781950	+	0.623342i	$0.214225\pi$
$174$	0.894603	0.0678197
$175$	0	0
$176$	4.75072	0.358099
$177$	3.72809	0.280221
$178$	3.00495	0.225231
$179$	19.8999	1.48739	0.743694	−	0.668520i	$-0.233072\pi$
$179$	19.8999	1.48739	0.743694	+	0.668520i	$0.233072\pi$
$180$	0	0
$181$	−5.55507	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$181$	−5.55507	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$182$	0	0
$183$	3.33281	0.246369
$184$	−3.95780	−0.291773
$185$	0	0
$186$	−1.57387	−0.115401
$187$	7.60087	0.555830
$188$	−12.2633	−0.894394
$189$	−26.1055	−1.89890
$190$	0	0
$191$	7.20413	0.521272	0.260636	−	0.965437i	$-0.416068\pi$
$191$	7.20413	0.521272	0.260636	+	0.965437i	$0.416068\pi$
$192$	7.89974	0.570115
$193$	4.46771	0.321593	0.160796	−	0.986988i	$-0.448594\pi$
$193$	4.46771	0.321593	0.160796	+	0.986988i	$0.448594\pi$
$194$	−0.686725	−0.0493040
$195$	0	0
$196$	−30.7706	−2.19790
$197$	6.17342	0.439838	0.219919	−	0.975518i	$-0.429421\pi$
$197$	6.17342	0.439838	0.219919	+	0.975518i	$0.429421\pi$
$198$	−0.498091	−0.0353978
$199$	−0.734601	−0.0520745	−0.0260372	−	0.999661i	$-0.508289\pi$
$199$	−0.734601	−0.0520745	−0.0260372	+	0.999661i	$0.508289\pi$
$200$	0	0
$201$	4.85859	0.342698
$202$	1.78971	0.125924
$203$	14.7551	1.03560
$204$	13.5268	0.947064
$205$	0	0
$206$	−2.30889	−0.160868
$207$	−6.51553	−0.452861
$208$	0	0
$209$	−5.56466	−0.384915
$210$	0	0
$211$	−23.6319	−1.62689	−0.813443	−	0.581644i	$-0.802410\pi$
$211$	−23.6319	−1.62689	−0.813443	+	0.581644i	$0.802410\pi$
$212$	0.983488	0.0675462
$213$	8.58008	0.587898
$214$	−1.25099	−0.0855156
$215$	0	0
$216$	−5.22658	−0.355623
$217$	−25.9584	−1.76217
$218$	−2.13429	−0.144552
$219$	6.24660	0.422106
$220$	0	0
$221$	0	0
$222$	−1.40337	−0.0941881
$223$	−18.7656	−1.25664	−0.628320	−	0.777955i	$-0.716257\pi$
$223$	−18.7656	−1.25664	−0.628320	+	0.777955i	$0.716257\pi$
$224$	−13.3877	−0.894504
$225$	0	0
$226$	2.98111	0.198301
$227$	−18.2444	−1.21092	−0.605461	−	0.795875i	$-0.707011\pi$
$227$	−18.2444	−1.21092	−0.605461	+	0.795875i	$0.707011\pi$
$228$	−9.90307	−0.655847
$229$	−5.07143	−0.335130	−0.167565	−	0.985861i	$-0.553590\pi$
$229$	−5.07143	−0.335130	−0.167565	+	0.985861i	$0.553590\pi$
$230$	0	0
$231$	7.42670	0.488641
$232$	2.95410	0.193946
$233$	30.1285	1.97379	0.986893	−	0.161374i	$-0.0515924\pi$
$233$	30.1285	1.97379	0.986893	+	0.161374i	$0.0515924\pi$
$234$	0	0
$235$	0	0
$236$	6.06323	0.394682
$237$	8.57359	0.556914
$238$	−6.77900	−0.439417
$239$	−7.85492	−0.508092	−0.254046	−	0.967192i	$-0.581762\pi$
$239$	−7.85492	−0.508092	−0.254046	+	0.967192i	$0.581762\pi$
$240$	0	0
$241$	−28.1490	−1.81324	−0.906619	−	0.421949i	$-0.861346\pi$
$241$	−28.1490	−1.81324	−0.906619	+	0.421949i	$0.861346\pi$
$242$	−2.25989	−0.145271
$243$	−14.2456	−0.913858
$244$	5.42036	0.347003
$245$	0	0
$246$	2.31924	0.147869
$247$	0	0
$248$	−5.19712	−0.330018
$249$	−11.9332	−0.756235
$250$	0	0
$251$	−22.9594	−1.44918	−0.724592	−	0.689178i	$-0.757972\pi$
$251$	−22.9594	−1.44918	−0.724592	+	0.689178i	$0.757972\pi$
$252$	−14.6201	−0.920980
$253$	5.38285	0.338417
$254$	−3.05883	−0.191928
$255$	0	0
$256$	11.4876	0.717975
$257$	18.1164	1.13007	0.565034	−	0.825068i	$-0.308863\pi$
$257$	18.1164	1.13007	0.565034	+	0.825068i	$0.308863\pi$
$258$	−0.783823	−0.0487987
$259$	−23.1464	−1.43825
$260$	0	0
$261$	4.86320	0.301024
$262$	0.950847	0.0587435
$263$	−6.49275	−0.400360	−0.200180	−	0.979759i	$-0.564153\pi$
$263$	−6.49275	−0.400360	−0.200180	+	0.979759i	$0.564153\pi$
$264$	1.48689	0.0915120
$265$	0	0
$266$	4.96296	0.304299
$267$	−14.7675	−0.903754
$268$	7.90182	0.482680
$269$	13.5473	0.825995	0.412998	−	0.910732i	$-0.364482\pi$
$269$	13.5473	0.825995	0.412998	+	0.910732i	$0.364482\pi$
$270$	0	0
$271$	0.147908	0.00898478	0.00449239	−	0.999990i	$-0.498570\pi$
$271$	0.147908	0.00898478	0.00449239	+	0.999990i	$0.498570\pi$
$272$	21.3107	1.29215
$273$	0	0
$274$	3.52038	0.212674
$275$	0	0
$276$	9.57952	0.576619
$277$	21.1703	1.27200	0.636000	−	0.771689i	$-0.280588\pi$
$277$	21.1703	1.27200	0.636000	+	0.771689i	$0.280588\pi$
$278$	3.32550	0.199450
$279$	−8.55577	−0.512221
$280$	0	0
$281$	−0.918162	−0.0547729	−0.0273865	−	0.999625i	$-0.508718\pi$
$281$	−0.918162	−0.0547729	−0.0273865	+	0.999625i	$0.508718\pi$
$282$	−1.83120	−0.109046
$283$	7.38223	0.438828	0.219414	−	0.975632i	$-0.429585\pi$
$283$	7.38223	0.438828	0.219414	+	0.975632i	$0.429585\pi$
$284$	13.9543	0.828036
$285$	0	0
$286$	0	0
$287$	38.2522	2.25796
$288$	−4.41253	−0.260011
$289$	17.0958	1.00564
$290$	0	0
$291$	3.37482	0.197836
$292$	10.1592	0.594524
$293$	8.29377	0.484527	0.242264	−	0.970210i	$-0.422110\pi$
$293$	8.29377	0.484527	0.242264	+	0.970210i	$0.422110\pi$
$294$	−4.59478	−0.267973
$295$	0	0
$296$	−4.63413	−0.269353
$297$	7.10846	0.412475
$298$	−2.29699	−0.133061
$299$	0	0
$300$	0	0
$301$	−12.9279	−0.745153
$302$	−2.32964	−0.134056
$303$	−8.79532	−0.505278
$304$	−15.6017	−0.894820
$305$	0	0
$306$	−2.23432	−0.127728
$307$	21.1166	1.20519	0.602595	−	0.798047i	$-0.294133\pi$
$307$	21.1166	1.20519	0.602595	+	0.798047i	$0.294133\pi$
$308$	12.0785	0.688236
$309$	11.3467	0.645493
$310$	0	0
$311$	13.7664	0.780623	0.390311	−	0.920683i	$-0.372367\pi$
$311$	13.7664	0.780623	0.390311	+	0.920683i	$0.372367\pi$
$312$	0	0
$313$	13.5602	0.766469	0.383234	−	0.923651i	$-0.374810\pi$
$313$	13.5602	0.766469	0.383234	+	0.923651i	$0.374810\pi$
$314$	−3.60509	−0.203447
$315$	0	0
$316$	13.9437	0.784397
$317$	−18.9348	−1.06348	−0.531741	−	0.846907i	$-0.678462\pi$
$317$	−18.9348	−1.06348	−0.531741	+	0.846907i	$0.678462\pi$
$318$	0.146858	0.00823538
$319$	−4.01776	−0.224951
$320$	0	0
$321$	6.14781	0.343137
$322$	−4.80081	−0.267539
$323$	−24.9618	−1.38891
$324$	3.47556	0.193087
$325$	0	0
$326$	−0.409825	−0.0226981
$327$	10.4887	0.580027
$328$	7.65845	0.422867
$329$	−30.2028	−1.66513
$330$	0	0
$331$	−16.6678	−0.916144	−0.458072	−	0.888915i	$-0.651460\pi$
$331$	−16.6678	−0.916144	−0.458072	+	0.888915i	$0.651460\pi$
$332$	−19.4077	−1.06513
$333$	−7.62894	−0.418063
$334$	−2.53989	−0.138977
$335$	0	0
$336$	20.8223	1.13595
$337$	26.1775	1.42598	0.712990	−	0.701174i	$-0.247341\pi$
$337$	26.1775	1.42598	0.712990	+	0.701174i	$0.247341\pi$
$338$	0	0
$339$	−14.6503	−0.795696
$340$	0	0
$341$	7.06840	0.382776
$342$	1.63577	0.0884522
$343$	−42.3204	−2.28509
$344$	−2.58829	−0.139551
$345$	0	0
$346$	4.99548	0.268559
$347$	12.5193	0.672074	0.336037	−	0.941849i	$-0.390913\pi$
$347$	12.5193	0.672074	0.336037	+	0.941849i	$0.390913\pi$
$348$	−7.15015	−0.383289
$349$	−4.30934	−0.230674	−0.115337	−	0.993326i	$-0.536795\pi$
$349$	−4.30934	−0.230674	−0.115337	+	0.993326i	$0.536795\pi$
$350$	0	0
$351$	0	0
$352$	3.64544	0.194302
$353$	20.5555	1.09406	0.547031	−	0.837113i	$-0.315758\pi$
$353$	20.5555	1.09406	0.547031	+	0.837113i	$0.315758\pi$
$354$	0.905383	0.0481205
$355$	0	0
$356$	−24.0172	−1.27291
$357$	33.3145	1.76319
$358$	4.83277	0.255420
$359$	−4.73134	−0.249710	−0.124855	−	0.992175i	$-0.539847\pi$
$359$	−4.73134	−0.249710	−0.124855	+	0.992175i	$0.539847\pi$
$360$	0	0
$361$	−0.725234	−0.0381702
$362$	−1.34907	−0.0709056
$363$	11.1060	0.582912
$364$	0	0
$365$	0	0
$366$	0.809387	0.0423073
$367$	−13.7595	−0.718240	−0.359120	−	0.933291i	$-0.616923\pi$
$367$	−13.7595	−0.718240	−0.359120	+	0.933291i	$0.616923\pi$
$368$	15.0920	0.786724
$369$	12.6077	0.656333
$370$	0	0
$371$	2.42219	0.125754
$372$	12.5792	0.652201
$373$	−27.5451	−1.42623	−0.713117	−	0.701045i	$-0.752717\pi$
$373$	−27.5451	−1.42623	−0.713117	+	0.701045i	$0.752717\pi$
$374$	1.84590	0.0954492
$375$	0	0
$376$	−6.04688	−0.311844
$377$	0	0
$378$	−6.33984	−0.326086
$379$	8.61694	0.442623	0.221311	−	0.975203i	$-0.428966\pi$
$379$	8.61694	0.442623	0.221311	+	0.975203i	$0.428966\pi$
$380$	0	0
$381$	15.0322	0.770124
$382$	1.74955	0.0895149
$383$	−26.2816	−1.34292	−0.671462	−	0.741039i	$-0.734334\pi$
$383$	−26.2816	−1.34292	−0.671462	+	0.741039i	$0.734334\pi$
$384$	8.60316	0.439028
$385$	0	0
$386$	1.08500	0.0552251
$387$	−4.26098	−0.216598
$388$	5.48868	0.278646
$389$	32.3511	1.64027	0.820133	−	0.572173i	$-0.193899\pi$
$389$	32.3511	1.64027	0.820133	+	0.572173i	$0.193899\pi$
$390$	0	0
$391$	24.1463	1.22113
$392$	−15.1726	−0.766332
$393$	−4.67282	−0.235713
$394$	1.49924	0.0755306
$395$	0	0
$396$	3.98101	0.200053
$397$	0.377739	0.0189582	0.00947909	−	0.999955i	$-0.496983\pi$
$397$	0.377739	0.0189582	0.00947909	+	0.999955i	$0.496983\pi$
$398$	−0.178401	−0.00894242
$399$	−24.3898	−1.22102
$400$	0	0
$401$	−22.2378	−1.11050	−0.555252	−	0.831682i	$-0.687378\pi$
$401$	−22.2378	−1.11050	−0.555252	+	0.831682i	$0.687378\pi$
$402$	1.17993	0.0588495
$403$	0	0
$404$	−14.3044	−0.711669
$405$	0	0
$406$	3.58333	0.177838
$407$	6.30270	0.312413
$408$	6.66988	0.330208
$409$	−22.6239	−1.11868	−0.559341	−	0.828938i	$-0.688946\pi$
$409$	−22.6239	−1.11868	−0.559341	+	0.828938i	$0.688946\pi$
$410$	0	0
$411$	−17.3005	−0.853369
$412$	18.4539	0.909158
$413$	14.9329	0.734798
$414$	−1.58232	−0.0777670
$415$	0	0
$416$	0	0
$417$	−16.3428	−0.800308
$418$	−1.35140	−0.0660991
$419$	−40.2612	−1.96689	−0.983444	−	0.181212i	$-0.941998\pi$
$419$	−40.2612	−1.96689	−0.983444	+	0.181212i	$0.941998\pi$
$420$	0	0
$421$	36.3027	1.76929	0.884644	−	0.466268i	$-0.154402\pi$
$421$	36.3027	1.76929	0.884644	+	0.466268i	$0.154402\pi$
$422$	−5.73910	−0.279375
$423$	−9.95469	−0.484014
$424$	0.484945	0.0235510
$425$	0	0
$426$	2.08371	0.100956
$427$	13.3496	0.646031
$428$	9.99856	0.483299
$429$	0	0
$430$	0	0
$431$	−37.0989	−1.78699	−0.893495	−	0.449073i	$-0.851754\pi$
$431$	−37.0989	−1.78699	−0.893495	+	0.449073i	$0.851754\pi$
$432$	19.9301	0.958888
$433$	−33.0641	−1.58896	−0.794480	−	0.607290i	$-0.792256\pi$
$433$	−33.0641	−1.58896	−0.794480	+	0.607290i	$0.792256\pi$
$434$	−6.30411	−0.302607
$435$	0	0
$436$	17.0584	0.816950
$437$	−17.6777	−0.845639
$438$	1.51701	0.0724857
$439$	9.86163	0.470670	0.235335	−	0.971914i	$-0.424381\pi$
$439$	9.86163	0.470670	0.235335	+	0.971914i	$0.424381\pi$
$440$	0	0
$441$	−24.9779	−1.18942
$442$	0	0
$443$	−23.9028	−1.13566	−0.567828	−	0.823147i	$-0.692216\pi$
$443$	−23.9028	−1.13566	−0.567828	+	0.823147i	$0.692216\pi$
$444$	11.2165	0.532312
$445$	0	0
$446$	−4.55731	−0.215795
$447$	11.2883	0.533917
$448$	31.6424	1.49496
$449$	15.5735	0.734957	0.367479	−	0.930032i	$-0.380221\pi$
$449$	15.5735	0.734957	0.367479	+	0.930032i	$0.380221\pi$
$450$	0	0
$451$	−10.4160	−0.490468
$452$	−23.8267	−1.12071
$453$	11.4487	0.537908
$454$	−4.43072	−0.207944
$455$	0	0
$456$	−4.88308	−0.228671
$457$	−32.5970	−1.52483	−0.762413	−	0.647091i	$-0.775985\pi$
$457$	−32.5970	−1.52483	−0.762413	+	0.647091i	$0.775985\pi$
$458$	−1.23162	−0.0575497
$459$	31.8870	1.48836
$460$	0	0
$461$	10.2984	0.479644	0.239822	−	0.970817i	$-0.422911\pi$
$461$	10.2984	0.479644	0.239822	+	0.970817i	$0.422911\pi$
$462$	1.80360	0.0839112
$463$	−2.97479	−0.138250	−0.0691251	−	0.997608i	$-0.522021\pi$
$463$	−2.97479	−0.138250	−0.0691251	+	0.997608i	$0.522021\pi$
$464$	−11.2647	−0.522949
$465$	0	0
$466$	7.31684	0.338946
$467$	−34.9628	−1.61789	−0.808943	−	0.587887i	$-0.799960\pi$
$467$	−34.9628	−1.61789	−0.808943	+	0.587887i	$0.799960\pi$
$468$	0	0
$469$	19.4611	0.898628
$470$	0	0
$471$	17.7167	0.816344
$472$	2.98970	0.137612
$473$	3.52024	0.161861
$474$	2.08213	0.0956354
$475$	0	0
$476$	54.1814	2.48340
$477$	0.798342	0.0365536
$478$	−1.90760	−0.0872515
$479$	−13.7338	−0.627511	−0.313756	−	0.949504i	$-0.601587\pi$
$479$	−13.7338	−0.627511	−0.313756	+	0.949504i	$0.601587\pi$
$480$	0	0
$481$	0	0
$482$	−6.83611	−0.311376
$483$	23.5930	1.07352
$484$	18.0623	0.821014
$485$	0	0
$486$	−3.45961	−0.156931
$487$	−39.4097	−1.78582	−0.892911	−	0.450233i	$-0.851341\pi$
$487$	−39.4097	−1.78582	−0.892911	+	0.450233i	$0.851341\pi$
$488$	2.67271	0.120988
$489$	2.01404	0.0910778
$490$	0	0
$491$	3.97696	0.179477	0.0897387	−	0.995965i	$-0.471397\pi$
$491$	3.97696	0.179477	0.0897387	+	0.995965i	$0.471397\pi$
$492$	−18.5366	−0.835696
$493$	−18.0228	−0.811706
$494$	0	0
$495$	0	0
$496$	19.8178	0.889845
$497$	34.3675	1.54159
$498$	−2.89802	−0.129864
$499$	−4.86965	−0.217996	−0.108998	−	0.994042i	$-0.534764\pi$
$499$	−4.86965	−0.217996	−0.108998	+	0.994042i	$0.534764\pi$
$500$	0	0
$501$	12.4820	0.557654
$502$	−5.57578	−0.248859
$503$	−12.6904	−0.565836	−0.282918	−	0.959144i	$-0.591302\pi$
$503$	−12.6904	−0.565836	−0.282918	+	0.959144i	$0.591302\pi$
$504$	−7.20899	−0.321114
$505$	0	0
$506$	1.30725	0.0581142
$507$	0	0
$508$	24.4478	1.08470
$509$	14.6745	0.650434	0.325217	−	0.945639i	$-0.394563\pi$
$509$	14.6745	0.650434	0.325217	+	0.945639i	$0.394563\pi$
$510$	0	0
$511$	25.0207	1.10685
$512$	17.2068	0.760440
$513$	−23.3447	−1.03070
$514$	4.39963	0.194059
$515$	0	0
$516$	6.26474	0.275790
$517$	8.22413	0.361697
$518$	−5.62120	−0.246981
$519$	−24.5496	−1.07761
$520$	0	0
$521$	6.04993	0.265052	0.132526	−	0.991180i	$-0.457691\pi$
$521$	6.04993	0.265052	0.132526	+	0.991180i	$0.457691\pi$
$522$	1.18105	0.0516930
$523$	−28.2967	−1.23733	−0.618664	−	0.785655i	$-0.712326\pi$
$523$	−28.2967	−1.23733	−0.618664	+	0.785655i	$0.712326\pi$
$524$	−7.59969	−0.331994
$525$	0	0
$526$	−1.57679	−0.0687514
$527$	31.7073	1.38119
$528$	−5.66986	−0.246749
$529$	−5.89988	−0.256516
$530$	0	0
$531$	4.92180	0.213588
$532$	−39.6667	−1.71977
$533$	0	0
$534$	−3.58634	−0.155196
$535$	0	0
$536$	3.89629	0.168294
$537$	−23.7501	−1.02489
$538$	3.29002	0.141843
$539$	20.6357	0.888841
$540$	0	0
$541$	−6.32246	−0.271824	−0.135912	−	0.990721i	$-0.543396\pi$
$541$	−6.32246	−0.271824	−0.135912	+	0.990721i	$0.543396\pi$
$542$	0.0359201	0.00154290
$543$	6.62984	0.284514
$544$	16.3526	0.701113
$545$	0	0
$546$	0	0
$547$	−4.21492	−0.180217	−0.0901084	−	0.995932i	$-0.528721\pi$
$547$	−4.21492	−0.180217	−0.0901084	+	0.995932i	$0.528721\pi$
$548$	−28.1368	−1.20195
$549$	4.39995	0.187785
$550$	0	0
$551$	13.1946	0.562110
$552$	4.72354	0.201047
$553$	34.3415	1.46035
$554$	5.14129	0.218433
$555$	0	0
$556$	−26.5792	−1.12721
$557$	−18.3347	−0.776867	−0.388434	−	0.921477i	$-0.626984\pi$
$557$	−18.3347	−0.776867	−0.388434	+	0.921477i	$0.626984\pi$
$558$	−2.07780	−0.0879605
$559$	0	0
$560$	0	0
$561$	−9.07145	−0.382997
$562$	−0.222979	−0.00940581
$563$	−13.2135	−0.556884	−0.278442	−	0.960453i	$-0.589818\pi$
$563$	−13.2135	−0.556884	−0.278442	+	0.960453i	$0.589818\pi$
$564$	14.6360	0.616285
$565$	0	0
$566$	1.79280	0.0753571
$567$	8.55982	0.359478
$568$	6.88069	0.288707
$569$	−25.3262	−1.06173	−0.530865	−	0.847457i	$-0.678133\pi$
$569$	−25.3262	−1.06173	−0.530865	+	0.847457i	$0.678133\pi$
$570$	0	0
$571$	−3.47261	−0.145324	−0.0726621	−	0.997357i	$-0.523149\pi$
$571$	−3.47261	−0.145324	−0.0726621	+	0.997357i	$0.523149\pi$
$572$	0	0
$573$	−8.59795	−0.359185
$574$	9.28970	0.387745
$575$	0	0
$576$	10.4292	0.434549
$577$	9.33439	0.388596	0.194298	−	0.980943i	$-0.437757\pi$
$577$	9.33439	0.388596	0.194298	+	0.980943i	$0.437757\pi$
$578$	4.15178	0.172691
$579$	−5.33210	−0.221595
$580$	0	0
$581$	−47.7983	−1.98301
$582$	0.819590	0.0339731
$583$	−0.659555	−0.0273160
$584$	5.00939	0.207290
$585$	0	0
$586$	2.01418	0.0832048
$587$	14.4210	0.595218	0.297609	−	0.954688i	$-0.403811\pi$
$587$	14.4210	0.595218	0.297609	+	0.954688i	$0.403811\pi$
$588$	36.7240	1.51447
$589$	−23.2132	−0.956483
$590$	0	0
$591$	−7.36783	−0.303072
$592$	17.6710	0.726272
$593$	−19.2593	−0.790883	−0.395441	−	0.918491i	$-0.629408\pi$
$593$	−19.2593	−0.790883	−0.395441	+	0.918491i	$0.629408\pi$
$594$	1.72632	0.0708317
$595$	0	0
$596$	18.3588	0.752006
$597$	0.876728	0.0358821
$598$	0	0
$599$	35.3956	1.44623	0.723113	−	0.690730i	$-0.242711\pi$
$599$	35.3956	1.44623	0.723113	+	0.690730i	$0.242711\pi$
$600$	0	0
$601$	−12.2161	−0.498305	−0.249152	−	0.968464i	$-0.580152\pi$
$601$	−12.2161	−0.498305	−0.249152	+	0.968464i	$0.580152\pi$
$602$	−3.13960	−0.127960
$603$	6.41427	0.261209
$604$	18.6198	0.757628
$605$	0	0
$606$	−2.13598	−0.0867682
$607$	0.852469	0.0346007	0.0173003	−	0.999850i	$-0.494493\pi$
$607$	0.852469	0.0346007	0.0173003	+	0.999850i	$0.494493\pi$
$608$	−11.9719	−0.485524
$609$	−17.6098	−0.713585
$610$	0	0
$611$	0	0
$612$	17.8579	0.721864
$613$	23.2544	0.939234	0.469617	−	0.882870i	$-0.344392\pi$
$613$	23.2544	0.939234	0.469617	+	0.882870i	$0.344392\pi$
$614$	5.12826	0.206960
$615$	0	0
$616$	5.95575	0.239964
$617$	35.4404	1.42678	0.713388	−	0.700770i	$-0.247160\pi$
$617$	35.4404	1.42678	0.713388	+	0.700770i	$0.247160\pi$
$618$	2.75560	0.110846
$619$	−1.77450	−0.0713232	−0.0356616	−	0.999364i	$-0.511354\pi$
$619$	−1.77450	−0.0713232	−0.0356616	+	0.999364i	$0.511354\pi$
$620$	0	0
$621$	22.5820	0.906185
$622$	3.34323	0.134051
$623$	−59.1510	−2.36984
$624$	0	0
$625$	0	0
$626$	3.29315	0.131621
$627$	6.64128	0.265227
$628$	28.8138	1.14980
$629$	28.2725	1.12730
$630$	0	0
$631$	−0.678894	−0.0270263	−0.0135132	−	0.999909i	$-0.504302\pi$
$631$	−0.678894	−0.0270263	−0.0135132	+	0.999909i	$0.504302\pi$
$632$	6.87548	0.273492
$633$	28.2041	1.12101
$634$	−4.59839	−0.182625
$635$	0	0
$636$	−1.17377	−0.0465430
$637$	0	0
$638$	−0.975729	−0.0386295
$639$	11.3274	0.448103
$640$	0	0
$641$	−34.5481	−1.36457	−0.682283	−	0.731088i	$-0.739013\pi$
$641$	−34.5481	−1.36457	−0.682283	+	0.731088i	$0.739013\pi$
$642$	1.49302	0.0589248
$643$	11.3200	0.446418	0.223209	−	0.974771i	$-0.428347\pi$
$643$	11.3200	0.446418	0.223209	+	0.974771i	$0.428347\pi$
$644$	38.3707	1.51202
$645$	0	0
$646$	−6.06208	−0.238509
$647$	8.22475	0.323348	0.161674	−	0.986844i	$-0.448311\pi$
$647$	8.22475	0.323348	0.161674	+	0.986844i	$0.448311\pi$
$648$	1.71376	0.0673227
$649$	−4.06617	−0.159611
$650$	0	0
$651$	30.9808	1.21423
$652$	3.27555	0.128280
$653$	−5.22441	−0.204447	−0.102223	−	0.994761i	$-0.532596\pi$
$653$	−5.22441	−0.204447	−0.102223	+	0.994761i	$0.532596\pi$
$654$	2.54722	0.0996043
$655$	0	0
$656$	−29.2034	−1.14020
$657$	8.24671	0.321735
$658$	−7.33487	−0.285943
$659$	−21.1150	−0.822525	−0.411263	−	0.911517i	$-0.634912\pi$
$659$	−21.1150	−0.822525	−0.411263	+	0.911517i	$0.634912\pi$
$660$	0	0
$661$	−1.52692	−0.0593904	−0.0296952	−	0.999559i	$-0.509454\pi$
$661$	−1.52692	−0.0593904	−0.0296952	+	0.999559i	$0.509454\pi$
$662$	−4.04784	−0.157324
$663$	0	0
$664$	−9.56968	−0.371375
$665$	0	0
$666$	−1.85272	−0.0717914
$667$	−12.7635	−0.494206
$668$	20.3002	0.785439
$669$	22.3963	0.865892
$670$	0	0
$671$	−3.63505	−0.140329
$672$	15.9779	0.616361
$673$	−7.81728	−0.301334	−0.150667	−	0.988585i	$-0.548142\pi$
$673$	−7.81728	−0.301334	−0.150667	+	0.988585i	$0.548142\pi$
$674$	6.35731	0.244875
$675$	0	0
$676$	0	0
$677$	−22.8535	−0.878332	−0.439166	−	0.898406i	$-0.644726\pi$
$677$	−22.8535	−0.878332	−0.439166	+	0.898406i	$0.644726\pi$
$678$	−3.55789	−0.136640
$679$	13.5178	0.518767
$680$	0	0
$681$	21.7742	0.834391
$682$	1.71659	0.0657316
$683$	−30.7945	−1.17832	−0.589160	−	0.808016i	$-0.700541\pi$
$683$	−30.7945	−1.17832	−0.589160	+	0.808016i	$0.700541\pi$
$684$	−13.0740	−0.499895
$685$	0	0
$686$	−10.2777	−0.392404
$687$	6.05263	0.230922
$688$	9.86974	0.376280
$689$	0	0
$690$	0	0
$691$	31.1778	1.18606	0.593030	−	0.805180i	$-0.297931\pi$
$691$	31.1778	1.18606	0.593030	+	0.805180i	$0.297931\pi$
$692$	−39.9266	−1.51778
$693$	9.80466	0.372448
$694$	3.04037	0.115411
$695$	0	0
$696$	−3.52565	−0.133639
$697$	−46.7237	−1.76979
$698$	−1.04654	−0.0396121
$699$	−35.9577	−1.36004
$700$	0	0
$701$	42.7533	1.61477	0.807385	−	0.590025i	$-0.200882\pi$
$701$	42.7533	1.61477	0.807385	+	0.590025i	$0.200882\pi$
$702$	0	0
$703$	−20.6985	−0.780660
$704$	−8.61612	−0.324732
$705$	0	0
$706$	4.99200	0.187876
$707$	−35.2296	−1.32495
$708$	−7.23631	−0.271957
$709$	7.54090	0.283204	0.141602	−	0.989924i	$-0.454775\pi$
$709$	7.54090	0.283204	0.141602	+	0.989924i	$0.454775\pi$
$710$	0	0
$711$	11.3188	0.424487
$712$	−11.8426	−0.443820
$713$	22.4548	0.840937
$714$	8.09057	0.302782
$715$	0	0
$716$	−38.6262	−1.44353
$717$	9.37465	0.350103
$718$	−1.14902	−0.0428812
$719$	32.9984	1.23063	0.615316	−	0.788281i	$-0.289029\pi$
$719$	32.9984	1.23063	0.615316	+	0.788281i	$0.289029\pi$
$720$	0	0
$721$	45.4493	1.69262
$722$	−0.176126	−0.00655473
$723$	33.5952	1.24942
$724$	10.7825	0.400729
$725$	0	0
$726$	2.69713	0.100100
$727$	31.2676	1.15965	0.579826	−	0.814741i	$-0.303121\pi$
$727$	31.2676	1.15965	0.579826	+	0.814741i	$0.303121\pi$
$728$	0	0
$729$	22.3736	0.828651
$730$	0	0
$731$	15.7910	0.584051
$732$	−6.46906	−0.239103
$733$	28.2677	1.04409	0.522046	−	0.852917i	$-0.325169\pi$
$733$	28.2677	1.04409	0.522046	+	0.852917i	$0.325169\pi$
$734$	−3.34155	−0.123339
$735$	0	0
$736$	11.5807	0.426872
$737$	−5.29919	−0.195198
$738$	3.06184	0.112708
$739$	−38.1632	−1.40385	−0.701927	−	0.712249i	$-0.747677\pi$
$739$	−38.1632	−1.40385	−0.701927	+	0.712249i	$0.747677\pi$
$740$	0	0
$741$	0	0
$742$	0.588239	0.0215949
$743$	−5.93960	−0.217903	−0.108951	−	0.994047i	$-0.534749\pi$
$743$	−5.93960	−0.217903	−0.108951	+	0.994047i	$0.534749\pi$
$744$	6.20264	0.227400
$745$	0	0
$746$	−6.68945	−0.244918
$747$	−15.7541	−0.576412
$748$	−14.7534	−0.539439
$749$	24.6250	0.899779
$750$	0	0
$751$	22.6332	0.825897	0.412948	−	0.910754i	$-0.364499\pi$
$751$	22.6332	0.825897	0.412948	+	0.910754i	$0.364499\pi$
$752$	23.0581	0.840843
$753$	27.4015	0.998565
$754$	0	0
$755$	0	0
$756$	50.6714	1.84290
$757$	−40.2878	−1.46429	−0.732143	−	0.681151i	$-0.761480\pi$
$757$	−40.2878	−1.46429	−0.732143	+	0.681151i	$0.761480\pi$
$758$	2.09266	0.0760088
$759$	−6.42430	−0.233187
$760$	0	0
$761$	34.7739	1.26055	0.630277	−	0.776371i	$-0.282941\pi$
$761$	34.7739	1.26055	0.630277	+	0.776371i	$0.282941\pi$
$762$	3.65064	0.132249
$763$	42.0124	1.52095
$764$	−13.9834	−0.505901
$765$	0	0
$766$	−6.38258	−0.230612
$767$	0	0
$768$	−13.7102	−0.494723
$769$	10.2840	0.370851	0.185425	−	0.982658i	$-0.440634\pi$
$769$	10.2840	0.370851	0.185425	+	0.982658i	$0.440634\pi$
$770$	0	0
$771$	−21.6214	−0.778677
$772$	−8.67192	−0.312109
$773$	39.0773	1.40551	0.702757	−	0.711430i	$-0.251952\pi$
$773$	39.0773	1.40551	0.702757	+	0.711430i	$0.251952\pi$
$774$	−1.03480	−0.0371950
$775$	0	0
$776$	2.70640	0.0971541
$777$	27.6247	0.991029
$778$	7.85660	0.281673
$779$	34.2068	1.22559
$780$	0	0
$781$	−9.35816	−0.334861
$782$	5.86402	0.209697
$783$	−16.8552	−0.602357
$784$	57.8565	2.06630
$785$	0	0
$786$	−1.13481	−0.0404774
$787$	−31.9154	−1.13766	−0.568831	−	0.822454i	$-0.692604\pi$
$787$	−31.9154	−1.13766	−0.568831	+	0.822454i	$0.692604\pi$
$788$	−11.9827	−0.426868
$789$	7.74894	0.275870
$790$	0	0
$791$	−58.6817	−2.08648
$792$	1.96299	0.0697517
$793$	0	0
$794$	0.0917354	0.00325557
$795$	0	0
$796$	1.42588	0.0505388
$797$	23.7632	0.841737	0.420869	−	0.907122i	$-0.361725\pi$
$797$	23.7632	0.841737	0.420869	+	0.907122i	$0.361725\pi$
$798$	−5.92317	−0.209678
$799$	36.8916	1.30513
$800$	0	0
$801$	−19.4959	−0.688853
$802$	−5.40055	−0.190700
$803$	−6.81307	−0.240428
$804$	−9.43063	−0.332593
$805$	0	0
$806$	0	0
$807$	−16.1684	−0.569155
$808$	−7.05330	−0.248134
$809$	22.9840	0.808076	0.404038	−	0.914742i	$-0.367606\pi$
$809$	22.9840	0.808076	0.404038	+	0.914742i	$0.367606\pi$
$810$	0	0
$811$	−21.1171	−0.741520	−0.370760	−	0.928729i	$-0.620903\pi$
$811$	−21.1171	−0.741520	−0.370760	+	0.928729i	$0.620903\pi$
$812$	−28.6399	−1.00506
$813$	−0.176525	−0.00619099
$814$	1.53064	0.0536487
$815$	0	0
$816$	−25.4338	−0.890359
$817$	−11.5607	−0.404459
$818$	−5.49432	−0.192104
$819$	0	0
$820$	0	0
$821$	−23.7584	−0.829175	−0.414587	−	0.910010i	$-0.636074\pi$
$821$	−23.7584	−0.829175	−0.414587	+	0.910010i	$0.636074\pi$
$822$	−4.20149	−0.146544
$823$	41.8058	1.45726	0.728629	−	0.684908i	$-0.240158\pi$
$823$	41.8058	1.45726	0.728629	+	0.684908i	$0.240158\pi$
$824$	9.09937	0.316992
$825$	0	0
$826$	3.62651	0.126182
$827$	−4.92854	−0.171382	−0.0856910	−	0.996322i	$-0.527310\pi$
$827$	−4.92854	−0.171382	−0.0856910	+	0.996322i	$0.527310\pi$
$828$	12.6468	0.439507
$829$	−46.6538	−1.62035	−0.810177	−	0.586186i	$-0.800629\pi$
$829$	−46.6538	−1.62035	−0.810177	+	0.586186i	$0.800629\pi$
$830$	0	0
$831$	−25.2662	−0.876476
$832$	0	0
$833$	92.5670	3.20726
$834$	−3.96891	−0.137432
$835$	0	0
$836$	10.8011	0.373565
$837$	29.6532	1.02497
$838$	−9.77759	−0.337761
$839$	9.95523	0.343693	0.171846	−	0.985124i	$-0.445027\pi$
$839$	9.95523	0.343693	0.171846	+	0.985124i	$0.445027\pi$
$840$	0	0
$841$	−19.4733	−0.671493
$842$	8.81627	0.303829
$843$	1.09580	0.0377415
$844$	45.8700	1.57891
$845$	0	0
$846$	−2.41754	−0.0831167
$847$	44.4849	1.52852
$848$	−1.84921	−0.0635020
$849$	−8.81051	−0.302376
$850$	0	0
$851$	20.0223	0.686355
$852$	−16.6541	−0.570561
$853$	12.8126	0.438695	0.219348	−	0.975647i	$-0.429607\pi$
$853$	12.8126	0.438695	0.219348	+	0.975647i	$0.429607\pi$
$854$	3.24200	0.110939
$855$	0	0
$856$	4.93016	0.168509
$857$	13.1145	0.447984	0.223992	−	0.974591i	$-0.428091\pi$
$857$	13.1145	0.447984	0.223992	+	0.974591i	$0.428091\pi$
$858$	0	0
$859$	27.9112	0.952317	0.476158	−	0.879359i	$-0.342029\pi$
$859$	27.9112	0.952317	0.476158	+	0.879359i	$0.342029\pi$
$860$	0	0
$861$	−45.6531	−1.55585
$862$	−9.00961	−0.306869
$863$	−17.9152	−0.609842	−0.304921	−	0.952378i	$-0.598630\pi$
$863$	−17.9152	−0.609842	−0.304921	+	0.952378i	$0.598630\pi$
$864$	15.2933	0.520287
$865$	0	0
$866$	−8.02975	−0.272862
$867$	−20.4034	−0.692936
$868$	50.3859	1.71021
$869$	−9.35108	−0.317214
$870$	0	0
$871$	0	0
$872$	8.41129	0.284842
$873$	4.45541	0.150793
$874$	−4.29310	−0.145216
$875$	0	0
$876$	−12.1248	−0.409659
$877$	24.7957	0.837292	0.418646	−	0.908149i	$-0.362505\pi$
$877$	24.7957	0.837292	0.418646	+	0.908149i	$0.362505\pi$
$878$	2.39494	0.0808252
$879$	−9.89841	−0.333865
$880$	0	0
$881$	−26.5986	−0.896130	−0.448065	−	0.894001i	$-0.647887\pi$
$881$	−26.5986	−0.896130	−0.448065	+	0.894001i	$0.647887\pi$
$882$	−6.06599	−0.204252
$883$	−11.1078	−0.373808	−0.186904	−	0.982378i	$-0.559845\pi$
$883$	−11.1078	−0.373808	−0.186904	+	0.982378i	$0.559845\pi$
$884$	0	0
$885$	0	0
$886$	−5.80489	−0.195019
$887$	4.20024	0.141030	0.0705152	−	0.997511i	$-0.477536\pi$
$887$	4.20024	0.141030	0.0705152	+	0.997511i	$0.477536\pi$
$888$	5.53072	0.185599
$889$	60.2115	2.01943
$890$	0	0
$891$	−2.33081	−0.0780852
$892$	36.4245	1.21958
$893$	−27.0087	−0.903811
$894$	2.74140	0.0916862
$895$	0	0
$896$	34.4599	1.15122
$897$	0	0
$898$	3.78208	0.126210
$899$	−16.7602	−0.558985
$900$	0	0
$901$	−2.95862	−0.0985659
$902$	−2.52956	−0.0842251
$903$	15.4292	0.513450
$904$	−11.7486	−0.390754
$905$	0	0
$906$	2.78037	0.0923716
$907$	−2.73644	−0.0908621	−0.0454311	−	0.998967i	$-0.514466\pi$
$907$	−2.73644	−0.0908621	−0.0454311	+	0.998967i	$0.514466\pi$
$908$	35.4128	1.17521
$909$	−11.6115	−0.385129
$910$	0	0
$911$	33.0840	1.09612	0.548061	−	0.836438i	$-0.315366\pi$
$911$	33.0840	1.09612	0.548061	+	0.836438i	$0.315366\pi$
$912$	18.6203	0.616579
$913$	13.0153	0.430745
$914$	−7.91632	−0.261849
$915$	0	0
$916$	9.84376	0.325247
$917$	−18.7170	−0.618088
$918$	7.74388	0.255586
$919$	−25.5915	−0.844185	−0.422092	−	0.906553i	$-0.638704\pi$
$919$	−25.5915	−0.844185	−0.422092	+	0.906553i	$0.638704\pi$
$920$	0	0
$921$	−25.2022	−0.830441
$922$	2.50100	0.0823662
$923$	0	0
$924$	−14.4154	−0.474231
$925$	0	0
$926$	−0.722440	−0.0237409
$927$	14.9799	0.492003
$928$	−8.64387	−0.283749
$929$	54.8425	1.79932	0.899661	−	0.436589i	$-0.143813\pi$
$929$	54.8425	1.79932	0.899661	+	0.436589i	$0.143813\pi$
$930$	0	0
$931$	−67.7691	−2.22104
$932$	−58.4802	−1.91558
$933$	−16.4299	−0.537891
$934$	−8.49086	−0.277830
$935$	0	0
$936$	0	0
$937$	48.7703	1.59326	0.796629	−	0.604469i	$-0.206615\pi$
$937$	48.7703	1.59326	0.796629	+	0.604469i	$0.206615\pi$
$938$	4.72619	0.154316
$939$	−16.1838	−0.528138
$940$	0	0
$941$	−28.7281	−0.936510	−0.468255	−	0.883593i	$-0.655117\pi$
$941$	−28.7281	−0.936510	−0.468255	+	0.883593i	$0.655117\pi$
$942$	4.30258	0.140186
$943$	−33.0892	−1.07753
$944$	−11.4004	−0.371051
$945$	0	0
$946$	0.854903	0.0277953
$947$	44.2195	1.43694	0.718470	−	0.695558i	$-0.244843\pi$
$947$	44.2195	1.43694	0.718470	+	0.695558i	$0.244843\pi$
$948$	−16.6415	−0.540491
$949$	0	0
$950$	0	0
$951$	22.5982	0.732796
$952$	26.7162	0.865876
$953$	−10.9105	−0.353426	−0.176713	−	0.984262i	$-0.556546\pi$
$953$	−10.9105	−0.353426	−0.176713	+	0.984262i	$0.556546\pi$
$954$	0.193881	0.00627712
$955$	0	0
$956$	15.2466	0.493109
$957$	4.79510	0.155004
$958$	−3.33530	−0.107759
$959$	−69.2970	−2.23772
$960$	0	0
$961$	−1.51387	−0.0488347
$962$	0	0
$963$	8.11629	0.261544
$964$	54.6379	1.75977
$965$	0	0
$966$	5.72965	0.184348
$967$	−14.1899	−0.456317	−0.228159	−	0.973624i	$-0.573271\pi$
$967$	−14.1899	−0.456317	−0.228159	+	0.973624i	$0.573271\pi$
$968$	8.90629	0.286259
$969$	29.7913	0.957035
$970$	0	0
$971$	0.308213	0.00989102	0.00494551	−	0.999988i	$-0.498426\pi$
$971$	0.308213	0.00989102	0.00494551	+	0.999988i	$0.498426\pi$
$972$	27.6511	0.886909
$973$	−65.4608	−2.09858
$974$	−9.57079	−0.306668
$975$	0	0
$976$	−10.1916	−0.326226
$977$	−5.50140	−0.176005	−0.0880027	−	0.996120i	$-0.528048\pi$
$977$	−5.50140	−0.176005	−0.0880027	+	0.996120i	$0.528048\pi$
$978$	0.489117	0.0156402
$979$	16.1066	0.514771
$980$	0	0
$981$	13.8471	0.442104
$982$	0.965820	0.0308205
$983$	−13.0306	−0.415611	−0.207805	−	0.978170i	$-0.566632\pi$
$983$	−13.0306	−0.415611	−0.207805	+	0.978170i	$0.566632\pi$
$984$	−9.14017	−0.291378
$985$	0	0
$986$	−4.37691	−0.139389
$987$	36.0463	1.14737
$988$	0	0
$989$	11.1830	0.355599
$990$	0	0
$991$	5.85779	0.186079	0.0930393	−	0.995662i	$-0.470342\pi$
$991$	5.85779	0.186079	0.0930393	+	0.995662i	$0.470342\pi$
$992$	15.2071	0.482825
$993$	19.8926	0.631272
$994$	8.34628	0.264728
$995$	0	0
$996$	23.1626	0.733935
$997$	−11.4851	−0.363738	−0.181869	−	0.983323i	$-0.558215\pi$
$997$	−11.4851	−0.363738	−0.181869	+	0.983323i	$0.558215\pi$
$998$	−1.18261	−0.0374350
$999$	26.4409	0.836554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.ca.1.10		18
5.2	odd	4		845.2.b.g.339.10	yes	18
5.3	odd	4		845.2.b.g.339.9	✓	18
5.4	even	2	inner	4225.2.a.ca.1.9		18
13.12	even	2		4225.2.a.cb.1.9		18
65.2	even	12		845.2.l.g.654.17		72
65.3	odd	12		845.2.n.i.529.10		36
65.7	even	12		845.2.l.g.699.20		72
65.8	even	4		845.2.d.e.844.19		36
65.12	odd	4		845.2.b.h.339.9	yes	18
65.17	odd	12		845.2.n.h.484.9		36
65.18	even	4		845.2.d.e.844.17		36
65.22	odd	12		845.2.n.i.484.10		36
65.23	odd	12		845.2.n.h.529.9		36
65.28	even	12		845.2.l.g.654.20		72
65.32	even	12		845.2.l.g.699.18		72
65.33	even	12		845.2.l.g.699.17		72
65.37	even	12		845.2.l.g.654.19		72
65.38	odd	4		845.2.b.h.339.10	yes	18
65.42	odd	12		845.2.n.i.529.9		36
65.43	odd	12		845.2.n.h.484.10		36
65.47	even	4		845.2.d.e.844.18		36
65.48	odd	12		845.2.n.i.484.9		36
65.57	even	4		845.2.d.e.844.20		36
65.58	even	12		845.2.l.g.699.19		72
65.62	odd	12		845.2.n.h.529.10		36
65.63	even	12		845.2.l.g.654.18		72
65.64	even	2		4225.2.a.cb.1.10		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.9	✓	18	5.3	odd	4
845.2.b.g.339.10	yes	18	5.2	odd	4
845.2.b.h.339.9	yes	18	65.12	odd	4
845.2.b.h.339.10	yes	18	65.38	odd	4
845.2.d.e.844.17		36	65.18	even	4
845.2.d.e.844.18		36	65.47	even	4
845.2.d.e.844.19		36	65.8	even	4
845.2.d.e.844.20		36	65.57	even	4
845.2.l.g.654.17		72	65.2	even	12
845.2.l.g.654.18		72	65.63	even	12
845.2.l.g.654.19		72	65.37	even	12
845.2.l.g.654.20		72	65.28	even	12
845.2.l.g.699.17		72	65.33	even	12
845.2.l.g.699.18		72	65.32	even	12
845.2.l.g.699.19		72	65.58	even	12
845.2.l.g.699.20		72	65.7	even	12
845.2.n.h.484.9		36	65.17	odd	12
845.2.n.h.484.10		36	65.43	odd	12
845.2.n.h.529.9		36	65.23	odd	12
845.2.n.h.529.10		36	65.62	odd	12
845.2.n.i.484.9		36	65.48	odd	12
845.2.n.i.484.10		36	65.22	odd	12
845.2.n.i.529.9		36	65.42	odd	12
845.2.n.i.529.10		36	65.3	odd	12
4225.2.a.ca.1.9		18	5.4	even	2	inner
4225.2.a.ca.1.10		18	1.1	even	1	trivial
4225.2.a.cb.1.9		18	13.12	even	2
4225.2.a.cb.1.10		18	65.64	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.242854 q^{2} -1.19348 q^{3} -1.94102 q^{4} -0.289840 q^{6} -4.78046 q^{7} -0.957093 q^{8} -1.57562 q^{9} +O(q^{10})\)	\(q+0.242854 q^{2} -1.19348 q^{3} -1.94102 q^{4} -0.289840 q^{6} -4.78046 q^{7} -0.957093 q^{8} -1.57562 q^{9} +1.30170 q^{11} +2.31656 q^{12} -1.16095 q^{14} +3.64961 q^{16} +5.83916 q^{17} -0.382645 q^{18} -4.27490 q^{19} +5.70536 q^{21} +0.316124 q^{22} +4.13523 q^{23} +1.14227 q^{24} +5.46089 q^{27} +9.27898 q^{28} -3.08654 q^{29} +5.43011 q^{31} +2.80051 q^{32} -1.55355 q^{33} +1.41806 q^{34} +3.05831 q^{36} +4.84188 q^{37} -1.03818 q^{38} -8.00178 q^{41} +1.38557 q^{42} +2.70433 q^{43} -2.52664 q^{44} +1.00426 q^{46} +6.31797 q^{47} -4.35572 q^{48} +15.8528 q^{49} -6.96890 q^{51} -0.506686 q^{53} +1.32620 q^{54} +4.57534 q^{56} +5.10199 q^{57} -0.749578 q^{58} -3.12373 q^{59} -2.79253 q^{61} +1.31872 q^{62} +7.53217 q^{63} -6.61911 q^{64} -0.377287 q^{66} -4.07096 q^{67} -11.3339 q^{68} -4.93529 q^{69} -7.18916 q^{71} +1.50801 q^{72} -5.23396 q^{73} +1.17587 q^{74} +8.29767 q^{76} -6.22275 q^{77} -7.18371 q^{79} -1.79058 q^{81} -1.94327 q^{82} +9.99869 q^{83} -11.0742 q^{84} +0.656757 q^{86} +3.68371 q^{87} -1.24585 q^{88} +12.3735 q^{89} -8.02657 q^{92} -6.48071 q^{93} +1.53434 q^{94} -3.34234 q^{96} -2.82773 q^{97} +3.84991 q^{98} -2.05099 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9}+O(q^{10}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9	\( 18 q + 16 q^{4} - 16 q^{6} + 18 q^{9} - 22 q^{11} + 4 q^{14} - 12 q^{16} - 28 q^{19} - 26 q^{21} - 34 q^{24} - 20 q^{29} - 32 q^{31} - 18 q^{34} + 32 q^{36} - 52 q^{41} - 50 q^{44} - 30 q^{46} + 44 q^{49} - 40 q^{51} - 90 q^{54} - 20 q^{56} - 76 q^{59} + 8 q^{61} - 68 q^{64} + 8 q^{66} + 30 q^{69} - 72 q^{71} + 30 q^{74} - 4 q^{76} + 16 q^{79} - 30 q^{81} - 78 q^{84} + 30 q^{86} - 94 q^{89} - 128 q^{94} + 18 q^{96} - 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 - 16 * q^6 + 18 * q^9 - 22 * q^11 + 4 * q^14 - 12 * q^16 - 28 * q^19 - 26 * q^21 - 34 * q^24 - 20 * q^29 - 32 * q^31 - 18 * q^34 + 32 * q^36 - 52 * q^41 - 50 * q^44 - 30 * q^46 + 44 * q^49 - 40 * q^51 - 90 * q^54 - 20 * q^56 - 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 - 72 * q^71 + 30 * q^74 - 4 * q^76 + 16 * q^79 - 30 * q^81 - 78 * q^84 + 30 * q^86 - 94 * q^89 - 128 * q^94 + 18 * q^96 - 76 * q^99

Introduction

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