LMFDB - Embedded newform 3750.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3750,2,Mod(1,3750)] 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("3750.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3750, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-4,4,4,0,-4,4,-4,4,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.9439007580$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.72654 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.44246 q^{11} +1.00000 q^{12} +2.96917 q^{13} -1.72654 q^{14} +1.00000 q^{16} -3.34458 q^{17} -1.00000 q^{18} +2.24669 q^{19} +1.72654 q^{21} +2.44246 q^{22} -9.11798 q^{23} -1.00000 q^{24} -2.96917 q^{26} +1.00000 q^{27} +1.72654 q^{28} -9.37895 q^{29} -6.12569 q^{31} -1.00000 q^{32} -2.44246 q^{33} +3.34458 q^{34} +1.00000 q^{36} -3.30127 q^{37} -2.24669 q^{38} +2.96917 q^{39} -6.12756 q^{41} -1.72654 q^{42} -7.06997 q^{43} -2.44246 q^{44} +9.11798 q^{46} +4.99228 q^{47} +1.00000 q^{48} -4.01905 q^{49} -3.34458 q^{51} +2.96917 q^{52} +6.85108 q^{53} -1.00000 q^{54} -1.72654 q^{56} +2.24669 q^{57} +9.37895 q^{58} -0.182645 q^{59} +5.15131 q^{61} +6.12569 q^{62} +1.72654 q^{63} +1.00000 q^{64} +2.44246 q^{66} +11.8903 q^{67} -3.34458 q^{68} -9.11798 q^{69} -11.3618 q^{71} -1.00000 q^{72} -6.72539 q^{73} +3.30127 q^{74} +2.24669 q^{76} -4.21702 q^{77} -2.96917 q^{78} -11.3571 q^{79} +1.00000 q^{81} +6.12756 q^{82} +9.13922 q^{83} +1.72654 q^{84} +7.06997 q^{86} -9.37895 q^{87} +2.44246 q^{88} -8.11694 q^{89} +5.12641 q^{91} -9.11798 q^{92} -6.12569 q^{93} -4.99228 q^{94} -1.00000 q^{96} +9.88597 q^{97} +4.01905 q^{98} -2.44246 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} - 10 q^{11} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 4 q^{18} - 6 q^{19} + 4 q^{21} + 10 q^{22} - 4 q^{24} + 2 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 - 10 * q^11 + 4 * q^12 - 2 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 4 * q^18 - 6 * q^19 + 4 * q^21 + 10 * q^22 - 4 * q^24 + 2 * q^26 + 4 * q^27 + 4 * q^28 - 14 * q^29 - 18 * q^31 - 4 * q^32 - 10 * q^33 + 6 * q^34 + 4 * q^36 - 2 * q^37 + 6 * q^38 - 2 * q^39 - 14 * q^41 - 4 * q^42 + 14 * q^43 - 10 * q^44 - 10 * q^47 + 4 * q^48 - 4 * q^49 - 6 * q^51 - 2 * q^52 - 14 * q^53 - 4 * q^54 - 4 * q^56 - 6 * q^57 + 14 * q^58 - 20 * q^59 + 18 * q^62 + 4 * q^63 + 4 * q^64 + 10 * q^66 + 14 * q^67 - 6 * q^68 - 30 * q^71 - 4 * q^72 + 8 * q^73 + 2 * q^74 - 6 * q^76 - 20 * q^77 + 2 * q^78 - 28 * q^79 + 4 * q^81 + 14 * q^82 - 12 * q^83 + 4 * q^84 - 14 * q^86 - 14 * q^87 + 10 * q^88 - 28 * q^89 - 22 * q^91 - 18 * q^93 + 10 * q^94 - 4 * q^96 + 8 * q^97 + 4 * q^98 - 10 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	1.72654	0.652572	0.326286	−	0.945271i	$-0.394203\pi$
$7$	1.72654	0.652572	0.326286	+	0.945271i	$0.394203\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.44246	−0.736430	−0.368215	−	0.929741i	$-0.620031\pi$
$11$	−2.44246	−0.736430	−0.368215	+	0.929741i	$0.620031\pi$
$12$	1.00000	0.288675
$13$	2.96917	0.823501	0.411750	−	0.911297i	$-0.364918\pi$
$13$	2.96917	0.823501	0.411750	+	0.911297i	$0.364918\pi$
$14$	−1.72654	−0.461438
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.34458	−0.811179	−0.405589	−	0.914055i	$-0.632934\pi$
$17$	−3.34458	−0.811179	−0.405589	+	0.914055i	$0.632934\pi$
$18$	−1.00000	−0.235702
$19$	2.24669	0.515426	0.257713	−	0.966222i	$-0.417031\pi$
$19$	2.24669	0.515426	0.257713	+	0.966222i	$0.417031\pi$
$20$	0	0
$21$	1.72654	0.376762
$22$	2.44246	0.520735
$23$	−9.11798	−1.90123	−0.950615	−	0.310373i	$-0.899546\pi$
$23$	−9.11798	−1.90123	−0.950615	+	0.310373i	$0.899546\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.96917	−0.582303
$27$	1.00000	0.192450
$28$	1.72654	0.326286
$29$	−9.37895	−1.74163	−0.870814	−	0.491613i	$-0.836407\pi$
$29$	−9.37895	−1.74163	−0.870814	+	0.491613i	$0.836407\pi$
$30$	0	0
$31$	−6.12569	−1.10021	−0.550104	−	0.835096i	$-0.685412\pi$
$31$	−6.12569	−1.10021	−0.550104	+	0.835096i	$0.685412\pi$
$32$	−1.00000	−0.176777
$33$	−2.44246	−0.425178
$34$	3.34458	0.573590
$35$	0	0
$36$	1.00000	0.166667
$37$	−3.30127	−0.542725	−0.271362	−	0.962477i	$-0.587474\pi$
$37$	−3.30127	−0.542725	−0.271362	+	0.962477i	$0.587474\pi$
$38$	−2.24669	−0.364461
$39$	2.96917	0.475449
$40$	0	0
$41$	−6.12756	−0.956964	−0.478482	−	0.878097i	$-0.658813\pi$
$41$	−6.12756	−0.956964	−0.478482	+	0.878097i	$0.658813\pi$
$42$	−1.72654	−0.266411
$43$	−7.06997	−1.07816	−0.539080	−	0.842255i	$-0.681228\pi$
$43$	−7.06997	−1.07816	−0.539080	+	0.842255i	$0.681228\pi$
$44$	−2.44246	−0.368215
$45$	0	0
$46$	9.11798	1.34437
$47$	4.99228	0.728199	0.364100	−	0.931360i	$-0.381377\pi$
$47$	4.99228	0.728199	0.364100	+	0.931360i	$0.381377\pi$
$48$	1.00000	0.144338
$49$	−4.01905	−0.574150
$50$	0	0
$51$	−3.34458	−0.468334
$52$	2.96917	0.411750
$53$	6.85108	0.941069	0.470534	−	0.882382i	$-0.344061\pi$
$53$	6.85108	0.941069	0.470534	+	0.882382i	$0.344061\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−1.72654	−0.230719
$57$	2.24669	0.297581
$58$	9.37895	1.23152
$59$	−0.182645	−0.0237783	−0.0118892	−	0.999929i	$-0.503785\pi$
$59$	−0.182645	−0.0237783	−0.0118892	+	0.999929i	$0.503785\pi$
$60$	0	0
$61$	5.15131	0.659558	0.329779	−	0.944058i	$-0.393026\pi$
$61$	5.15131	0.659558	0.329779	+	0.944058i	$0.393026\pi$
$62$	6.12569	0.777964
$63$	1.72654	0.217524
$64$	1.00000	0.125000
$65$	0	0
$66$	2.44246	0.300646
$67$	11.8903	1.45264	0.726318	−	0.687359i	$-0.241230\pi$
$67$	11.8903	1.45264	0.726318	+	0.687359i	$0.241230\pi$
$68$	−3.34458	−0.405589
$69$	−9.11798	−1.09768
$70$	0	0
$71$	−11.3618	−1.34839	−0.674197	−	0.738552i	$-0.735510\pi$
$71$	−11.3618	−1.34839	−0.674197	+	0.738552i	$0.735510\pi$
$72$	−1.00000	−0.117851
$73$	−6.72539	−0.787147	−0.393574	−	0.919293i	$-0.628761\pi$
$73$	−6.72539	−0.787147	−0.393574	+	0.919293i	$0.628761\pi$
$74$	3.30127	0.383764
$75$	0	0
$76$	2.24669	0.257713
$77$	−4.21702	−0.480574
$78$	−2.96917	−0.336193
$79$	−11.3571	−1.27777	−0.638885	−	0.769302i	$-0.720604\pi$
$79$	−11.3571	−1.27777	−0.638885	+	0.769302i	$0.720604\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.12756	0.676676
$83$	9.13922	1.00316	0.501580	−	0.865111i	$-0.332752\pi$
$83$	9.13922	1.00316	0.501580	+	0.865111i	$0.332752\pi$
$84$	1.72654	0.188381
$85$	0	0
$86$	7.06997	0.762374
$87$	−9.37895	−1.00553
$88$	2.44246	0.260367
$89$	−8.11694	−0.860394	−0.430197	−	0.902735i	$-0.641556\pi$
$89$	−8.11694	−0.860394	−0.430197	+	0.902735i	$0.641556\pi$
$90$	0	0
$91$	5.12641	0.537393
$92$	−9.11798	−0.950615
$93$	−6.12569	−0.635205
$94$	−4.99228	−0.514915
$95$	0	0
$96$	−1.00000	−0.102062
$97$	9.88597	1.00377	0.501884	−	0.864935i	$-0.332640\pi$
$97$	9.88597	1.00377	0.501884	+	0.864935i	$0.332640\pi$
$98$	4.01905	0.405985
$99$	−2.44246	−0.245477
$100$	0	0
$101$	6.28885	0.625764	0.312882	−	0.949792i	$-0.398706\pi$
$101$	6.28885	0.625764	0.312882	+	0.949792i	$0.398706\pi$
$102$	3.34458	0.331162
$103$	−8.57949	−0.845362	−0.422681	−	0.906278i	$-0.638911\pi$
$103$	−8.57949	−0.845362	−0.422681	+	0.906278i	$0.638911\pi$
$104$	−2.96917	−0.291152
$105$	0	0
$106$	−6.85108	−0.665436
$107$	9.09673	0.879415	0.439707	−	0.898141i	$-0.355082\pi$
$107$	9.09673	0.879415	0.439707	+	0.898141i	$0.355082\pi$
$108$	1.00000	0.0962250
$109$	−13.6103	−1.30363	−0.651816	−	0.758377i	$-0.725993\pi$
$109$	−13.6103	−1.30363	−0.651816	+	0.758377i	$0.725993\pi$
$110$	0	0
$111$	−3.30127	−0.313342
$112$	1.72654	0.163143
$113$	−8.26761	−0.777751	−0.388875	−	0.921290i	$-0.627136\pi$
$113$	−8.26761	−0.777751	−0.388875	+	0.921290i	$0.627136\pi$
$114$	−2.24669	−0.210422
$115$	0	0
$116$	−9.37895	−0.870814
$117$	2.96917	0.274500
$118$	0.182645	0.0168138
$119$	−5.77455	−0.529352
$120$	0	0
$121$	−5.03437	−0.457670
$122$	−5.15131	−0.466378
$123$	−6.12756	−0.552503
$124$	−6.12569	−0.550104
$125$	0	0
$126$	−1.72654	−0.153813
$127$	17.8127	1.58062	0.790309	−	0.612709i	$-0.209920\pi$
$127$	17.8127	1.58062	0.790309	+	0.612709i	$0.209920\pi$
$128$	−1.00000	−0.0883883
$129$	−7.06997	−0.622476
$130$	0	0
$131$	10.3375	0.903192	0.451596	−	0.892223i	$-0.350855\pi$
$131$	10.3375	0.903192	0.451596	+	0.892223i	$0.350855\pi$
$132$	−2.44246	−0.212589
$133$	3.87901	0.336352
$134$	−11.8903	−1.02717
$135$	0	0
$136$	3.34458	0.286795
$137$	16.0816	1.37395	0.686973	−	0.726683i	$-0.258939\pi$
$137$	16.0816	1.37395	0.686973	+	0.726683i	$0.258939\pi$
$138$	9.11798	0.776174
$139$	−16.5094	−1.40031	−0.700155	−	0.713991i	$-0.746886\pi$
$139$	−16.5094	−1.40031	−0.700155	+	0.713991i	$0.746886\pi$
$140$	0	0
$141$	4.99228	0.420426
$142$	11.3618	0.953458
$143$	−7.25210	−0.606451
$144$	1.00000	0.0833333
$145$	0	0
$146$	6.72539	0.556597
$147$	−4.01905	−0.331486
$148$	−3.30127	−0.271362
$149$	20.5325	1.68209	0.841045	−	0.540965i	$-0.181941\pi$
$149$	20.5325	1.68209	0.841045	+	0.540965i	$0.181941\pi$
$150$	0	0
$151$	2.78808	0.226891	0.113445	−	0.993544i	$-0.463811\pi$
$151$	2.78808	0.226891	0.113445	+	0.993544i	$0.463811\pi$
$152$	−2.24669	−0.182231
$153$	−3.34458	−0.270393
$154$	4.21702	0.339817
$155$	0	0
$156$	2.96917	0.237724
$157$	−14.3416	−1.14458	−0.572291	−	0.820051i	$-0.693945\pi$
$157$	−14.3416	−1.14458	−0.572291	+	0.820051i	$0.693945\pi$
$158$	11.3571	0.903519
$159$	6.85108	0.543326
$160$	0	0
$161$	−15.7426	−1.24069
$162$	−1.00000	−0.0785674
$163$	5.90315	0.462371	0.231185	−	0.972910i	$-0.425740\pi$
$163$	5.90315	0.462371	0.231185	+	0.972910i	$0.425740\pi$
$164$	−6.12756	−0.478482
$165$	0	0
$166$	−9.13922	−0.709341
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	−1.72654	−0.133206
$169$	−4.18400	−0.321846
$170$	0	0
$171$	2.24669	0.171809
$172$	−7.06997	−0.539080
$173$	−25.9151	−1.97029	−0.985145	−	0.171722i	$-0.945067\pi$
$173$	−25.9151	−1.97029	−0.985145	+	0.171722i	$0.945067\pi$
$174$	9.37895	0.711016
$175$	0	0
$176$	−2.44246	−0.184108
$177$	−0.182645	−0.0137284
$178$	8.11694	0.608390
$179$	0.0480111	0.00358852	0.00179426	−	0.999998i	$-0.499429\pi$
$179$	0.0480111	0.00358852	0.00179426	+	0.999998i	$0.499429\pi$
$180$	0	0
$181$	20.5347	1.52633	0.763167	−	0.646202i	$-0.223644\pi$
$181$	20.5347	1.52633	0.763167	+	0.646202i	$0.223644\pi$
$182$	−5.12641	−0.379995
$183$	5.15131	0.380796
$184$	9.11798	0.672186
$185$	0	0
$186$	6.12569	0.449158
$187$	8.16901	0.597377
$188$	4.99228	0.364100
$189$	1.72654	0.125587
$190$	0	0
$191$	2.42040	0.175134	0.0875668	−	0.996159i	$-0.472091\pi$
$191$	2.42040	0.175134	0.0875668	+	0.996159i	$0.472091\pi$
$192$	1.00000	0.0721688
$193$	−13.3764	−0.962857	−0.481429	−	0.876485i	$-0.659882\pi$
$193$	−13.3764	−0.962857	−0.481429	+	0.876485i	$0.659882\pi$
$194$	−9.88597	−0.709771
$195$	0	0
$196$	−4.01905	−0.287075
$197$	−18.9236	−1.34825	−0.674124	−	0.738618i	$-0.735479\pi$
$197$	−18.9236	−1.34825	−0.674124	+	0.738618i	$0.735479\pi$
$198$	2.44246	0.173578
$199$	−22.6784	−1.60763	−0.803815	−	0.594879i	$-0.797200\pi$
$199$	−22.6784	−1.60763	−0.803815	+	0.594879i	$0.797200\pi$
$200$	0	0
$201$	11.8903	0.838680
$202$	−6.28885	−0.442482
$203$	−16.1932	−1.13654
$204$	−3.34458	−0.234167
$205$	0	0
$206$	8.57949	0.597762
$207$	−9.11798	−0.633743
$208$	2.96917	0.205875
$209$	−5.48746	−0.379575
$210$	0	0
$211$	2.55876	0.176152	0.0880761	−	0.996114i	$-0.471928\pi$
$211$	2.55876	0.176152	0.0880761	+	0.996114i	$0.471928\pi$
$212$	6.85108	0.470534
$213$	−11.3618	−0.778495
$214$	−9.09673	−0.621840
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−10.5763	−0.717964
$218$	13.6103	0.921807
$219$	−6.72539	−0.454460
$220$	0	0
$221$	−9.93063	−0.668007
$222$	3.30127	0.221566
$223$	18.6543	1.24918	0.624591	−	0.780952i	$-0.285266\pi$
$223$	18.6543	1.24918	0.624591	+	0.780952i	$0.285266\pi$
$224$	−1.72654	−0.115359
$225$	0	0
$226$	8.26761	0.549953
$227$	−11.0615	−0.734180	−0.367090	−	0.930185i	$-0.619646\pi$
$227$	−11.0615	−0.734180	−0.367090	+	0.930185i	$0.619646\pi$
$228$	2.24669	0.148791
$229$	27.0493	1.78746	0.893732	−	0.448600i	$-0.148077\pi$
$229$	27.0493	1.78746	0.893732	+	0.448600i	$0.148077\pi$
$230$	0	0
$231$	−4.21702	−0.277459
$232$	9.37895	0.615758
$233$	18.2774	1.19739	0.598696	−	0.800976i	$-0.295686\pi$
$233$	18.2774	1.19739	0.598696	+	0.800976i	$0.295686\pi$
$234$	−2.96917	−0.194101
$235$	0	0
$236$	−0.182645	−0.0118892
$237$	−11.3571	−0.737721
$238$	5.77455	0.374309
$239$	−2.56816	−0.166120	−0.0830602	−	0.996545i	$-0.526469\pi$
$239$	−2.56816	−0.166120	−0.0830602	+	0.996545i	$0.526469\pi$
$240$	0	0
$241$	13.3394	0.859264	0.429632	−	0.903004i	$-0.358643\pi$
$241$	13.3394	0.859264	0.429632	+	0.903004i	$0.358643\pi$
$242$	5.03437	0.323622
$243$	1.00000	0.0641500
$244$	5.15131	0.329779
$245$	0	0
$246$	6.12756	0.390679
$247$	6.67081	0.424454
$248$	6.12569	0.388982
$249$	9.13922	0.579175
$250$	0	0
$251$	−14.3409	−0.905191	−0.452595	−	0.891716i	$-0.649502\pi$
$251$	−14.3409	−0.905191	−0.452595	+	0.891716i	$0.649502\pi$
$252$	1.72654	0.108762
$253$	22.2703	1.40012
$254$	−17.8127	−1.11767
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.0010	−0.998117	−0.499059	−	0.866568i	$-0.666321\pi$
$257$	−16.0010	−0.998117	−0.499059	+	0.866568i	$0.666321\pi$
$258$	7.06997	0.440157
$259$	−5.69977	−0.354167
$260$	0	0
$261$	−9.37895	−0.580542
$262$	−10.3375	−0.638653
$263$	19.8584	1.22452	0.612260	−	0.790657i	$-0.290261\pi$
$263$	19.8584	1.22452	0.612260	+	0.790657i	$0.290261\pi$
$264$	2.44246	0.150323
$265$	0	0
$266$	−3.87901	−0.237837
$267$	−8.11694	−0.496749
$268$	11.8903	0.726318
$269$	−2.66084	−0.162234	−0.0811170	−	0.996705i	$-0.525849\pi$
$269$	−2.66084	−0.162234	−0.0811170	+	0.996705i	$0.525849\pi$
$270$	0	0
$271$	−13.4607	−0.817679	−0.408839	−	0.912606i	$-0.634066\pi$
$271$	−13.4607	−0.817679	−0.408839	+	0.912606i	$0.634066\pi$
$272$	−3.34458	−0.202795
$273$	5.12641	0.310264
$274$	−16.0816	−0.971527
$275$	0	0
$276$	−9.11798	−0.548838
$277$	5.34748	0.321299	0.160649	−	0.987012i	$-0.448641\pi$
$277$	5.34748	0.321299	0.160649	+	0.987012i	$0.448641\pi$
$278$	16.5094	0.990169
$279$	−6.12569	−0.366736
$280$	0	0
$281$	−16.7170	−0.997250	−0.498625	−	0.866818i	$-0.666162\pi$
$281$	−16.7170	−0.997250	−0.498625	+	0.866818i	$0.666162\pi$
$282$	−4.99228	−0.297286
$283$	14.1630	0.841901	0.420951	−	0.907084i	$-0.361697\pi$
$283$	14.1630	0.841901	0.420951	+	0.907084i	$0.361697\pi$
$284$	−11.3618	−0.674197
$285$	0	0
$286$	7.25210	0.428826
$287$	−10.5795	−0.624488
$288$	−1.00000	−0.0589256
$289$	−5.81381	−0.341989
$290$	0	0
$291$	9.88597	0.579526
$292$	−6.72539	−0.393574
$293$	−12.4643	−0.728175	−0.364088	−	0.931365i	$-0.618619\pi$
$293$	−12.4643	−0.728175	−0.364088	+	0.931365i	$0.618619\pi$
$294$	4.01905	0.234396
$295$	0	0
$296$	3.30127	0.191882
$297$	−2.44246	−0.141726
$298$	−20.5325	−1.18942
$299$	−27.0729	−1.56566
$300$	0	0
$301$	−12.2066	−0.703577
$302$	−2.78808	−0.160436
$303$	6.28885	0.361285
$304$	2.24669	0.128856
$305$	0	0
$306$	3.34458	0.191197
$307$	23.9974	1.36960	0.684801	−	0.728730i	$-0.259889\pi$
$307$	23.9974	1.36960	0.684801	+	0.728730i	$0.259889\pi$
$308$	−4.21702	−0.240287
$309$	−8.57949	−0.488070
$310$	0	0
$311$	−13.9591	−0.791547	−0.395773	−	0.918348i	$-0.629523\pi$
$311$	−13.9591	−0.791547	−0.395773	+	0.918348i	$0.629523\pi$
$312$	−2.96917	−0.168096
$313$	−9.62739	−0.544172	−0.272086	−	0.962273i	$-0.587714\pi$
$313$	−9.62739	−0.544172	−0.272086	+	0.962273i	$0.587714\pi$
$314$	14.3416	0.809341
$315$	0	0
$316$	−11.3571	−0.638885
$317$	0.556314	0.0312457	0.0156229	−	0.999878i	$-0.495027\pi$
$317$	0.556314	0.0312457	0.0156229	+	0.999878i	$0.495027\pi$
$318$	−6.85108	−0.384190
$319$	22.9077	1.28259
$320$	0	0
$321$	9.09673	0.507730
$322$	15.7426	0.877299
$323$	−7.51423	−0.418103
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.90315	−0.326945
$327$	−13.6103	−0.752652
$328$	6.12756	0.338338
$329$	8.61939	0.475202
$330$	0	0
$331$	−11.5651	−0.635678	−0.317839	−	0.948145i	$-0.602957\pi$
$331$	−11.5651	−0.635678	−0.317839	+	0.948145i	$0.602957\pi$
$332$	9.13922	0.501580
$333$	−3.30127	−0.180908
$334$	−8.94427	−0.489409
$335$	0	0
$336$	1.72654	0.0941906
$337$	29.8767	1.62749	0.813744	−	0.581224i	$-0.197426\pi$
$337$	29.8767	1.62749	0.813744	+	0.581224i	$0.197426\pi$
$338$	4.18400	0.227580
$339$	−8.26761	−0.449035
$340$	0	0
$341$	14.9618	0.810226
$342$	−2.24669	−0.121487
$343$	−19.0249	−1.02725
$344$	7.06997	0.381187
$345$	0	0
$346$	25.9151	1.39321
$347$	−4.24669	−0.227974	−0.113987	−	0.993482i	$-0.536362\pi$
$347$	−4.24669	−0.227974	−0.113987	+	0.993482i	$0.536362\pi$
$348$	−9.37895	−0.502764
$349$	−19.3711	−1.03691	−0.518456	−	0.855104i	$-0.673493\pi$
$349$	−19.3711	−1.03691	−0.518456	+	0.855104i	$0.673493\pi$
$350$	0	0
$351$	2.96917	0.158483
$352$	2.44246	0.130184
$353$	−0.243275	−0.0129482	−0.00647411	−	0.999979i	$-0.502061\pi$
$353$	−0.243275	−0.0129482	−0.00647411	+	0.999979i	$0.502061\pi$
$354$	0.182645	0.00970747
$355$	0	0
$356$	−8.11694	−0.430197
$357$	−5.77455	−0.305622
$358$	−0.0480111	−0.00253746
$359$	21.0432	1.11062	0.555309	−	0.831644i	$-0.312600\pi$
$359$	21.0432	1.11062	0.555309	+	0.831644i	$0.312600\pi$
$360$	0	0
$361$	−13.9524	−0.734336
$362$	−20.5347	−1.07928
$363$	−5.03437	−0.264236
$364$	5.12641	0.268697
$365$	0	0
$366$	−5.15131	−0.269263
$367$	−7.83339	−0.408900	−0.204450	−	0.978877i	$-0.565541\pi$
$367$	−7.83339	−0.408900	−0.204450	+	0.978877i	$0.565541\pi$
$368$	−9.11798	−0.475307
$369$	−6.12756	−0.318988
$370$	0	0
$371$	11.8287	0.614115
$372$	−6.12569	−0.317602
$373$	−15.5256	−0.803883	−0.401941	−	0.915665i	$-0.631664\pi$
$373$	−15.5256	−0.803883	−0.401941	+	0.915665i	$0.631664\pi$
$374$	−8.16901	−0.422409
$375$	0	0
$376$	−4.99228	−0.257457
$377$	−27.8477	−1.43423
$378$	−1.72654	−0.0888038
$379$	−10.8637	−0.558030	−0.279015	−	0.960287i	$-0.590008\pi$
$379$	−10.8637	−0.558030	−0.279015	+	0.960287i	$0.590008\pi$
$380$	0	0
$381$	17.8127	0.912570
$382$	−2.42040	−0.123838
$383$	−9.59615	−0.490340	−0.245170	−	0.969480i	$-0.578844\pi$
$383$	−9.59615	−0.490340	−0.245170	+	0.969480i	$0.578844\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	13.3764	0.680843
$387$	−7.06997	−0.359387
$388$	9.88597	0.501884
$389$	34.8220	1.76555	0.882773	−	0.469800i	$-0.155674\pi$
$389$	34.8220	1.76555	0.882773	+	0.469800i	$0.155674\pi$
$390$	0	0
$391$	30.4958	1.54224
$392$	4.01905	0.202993
$393$	10.3375	0.521458
$394$	18.9236	0.953355
$395$	0	0
$396$	−2.44246	−0.122738
$397$	16.0058	0.803309	0.401654	−	0.915791i	$-0.368435\pi$
$397$	16.0058	0.803309	0.401654	+	0.915791i	$0.368435\pi$
$398$	22.6784	1.13677
$399$	3.87901	0.194193
$400$	0	0
$401$	11.5284	0.575701	0.287850	−	0.957675i	$-0.407059\pi$
$401$	11.5284	0.575701	0.287850	+	0.957675i	$0.407059\pi$
$402$	−11.8903	−0.593036
$403$	−18.1883	−0.906022
$404$	6.28885	0.312882
$405$	0	0
$406$	16.1932	0.803653
$407$	8.06322	0.399679
$408$	3.34458	0.165581
$409$	−14.5906	−0.721461	−0.360730	−	0.932670i	$-0.617473\pi$
$409$	−14.5906	−0.721461	−0.360730	+	0.932670i	$0.617473\pi$
$410$	0	0
$411$	16.0816	0.793248
$412$	−8.57949	−0.422681
$413$	−0.315344	−0.0155171
$414$	9.11798	0.448124
$415$	0	0
$416$	−2.96917	−0.145576
$417$	−16.5094	−0.808469
$418$	5.48746	0.268400
$419$	−14.1068	−0.689165	−0.344582	−	0.938756i	$-0.611979\pi$
$419$	−14.1068	−0.689165	−0.344582	+	0.938756i	$0.611979\pi$
$420$	0	0
$421$	−12.8882	−0.628130	−0.314065	−	0.949401i	$-0.601691\pi$
$421$	−12.8882	−0.628130	−0.314065	+	0.949401i	$0.601691\pi$
$422$	−2.55876	−0.124558
$423$	4.99228	0.242733
$424$	−6.85108	−0.332718
$425$	0	0
$426$	11.3618	0.550479
$427$	8.89396	0.430409
$428$	9.09673	0.439707
$429$	−7.25210	−0.350135
$430$	0	0
$431$	−23.1424	−1.11473	−0.557366	−	0.830267i	$-0.688188\pi$
$431$	−23.1424	−1.11473	−0.557366	+	0.830267i	$0.688188\pi$
$432$	1.00000	0.0481125
$433$	−23.5106	−1.12985	−0.564923	−	0.825144i	$-0.691094\pi$
$433$	−23.5106	−1.12985	−0.564923	+	0.825144i	$0.691094\pi$
$434$	10.5763	0.507677
$435$	0	0
$436$	−13.6103	−0.651816
$437$	−20.4853	−0.979943
$438$	6.72539	0.321352
$439$	1.99240	0.0950919	0.0475459	−	0.998869i	$-0.484860\pi$
$439$	1.99240	0.0950919	0.0475459	+	0.998869i	$0.484860\pi$
$440$	0	0
$441$	−4.01905	−0.191383
$442$	9.93063	0.472352
$443$	−13.0327	−0.619202	−0.309601	−	0.950867i	$-0.600195\pi$
$443$	−13.0327	−0.619202	−0.309601	+	0.950867i	$0.600195\pi$
$444$	−3.30127	−0.156671
$445$	0	0
$446$	−18.6543	−0.883305
$447$	20.5325	0.971155
$448$	1.72654	0.0815715
$449$	12.0846	0.570310	0.285155	−	0.958481i	$-0.407955\pi$
$449$	12.0846	0.570310	0.285155	+	0.958481i	$0.407955\pi$
$450$	0	0
$451$	14.9663	0.704737
$452$	−8.26761	−0.388875
$453$	2.78808	0.130995
$454$	11.0615	0.519144
$455$	0	0
$456$	−2.24669	−0.105211
$457$	5.62762	0.263249	0.131624	−	0.991300i	$-0.457981\pi$
$457$	5.62762	0.263249	0.131624	+	0.991300i	$0.457981\pi$
$458$	−27.0493	−1.26393
$459$	−3.34458	−0.156111
$460$	0	0
$461$	−4.62686	−0.215494	−0.107747	−	0.994178i	$-0.534364\pi$
$461$	−4.62686	−0.215494	−0.107747	+	0.994178i	$0.534364\pi$
$462$	4.21702	0.196193
$463$	−5.75912	−0.267649	−0.133824	−	0.991005i	$-0.542726\pi$
$463$	−5.75912	−0.267649	−0.133824	+	0.991005i	$0.542726\pi$
$464$	−9.37895	−0.435407
$465$	0	0
$466$	−18.2774	−0.846684
$467$	−24.5818	−1.13751	−0.568755	−	0.822507i	$-0.692575\pi$
$467$	−24.5818	−1.13751	−0.568755	+	0.822507i	$0.692575\pi$
$468$	2.96917	0.137250
$469$	20.5292	0.947949
$470$	0	0
$471$	−14.3416	−0.660824
$472$	0.182645	0.00840692
$473$	17.2681	0.793990
$474$	11.3571	0.521647
$475$	0	0
$476$	−5.77455	−0.264676
$477$	6.85108	0.313690
$478$	2.56816	0.117465
$479$	−5.84150	−0.266905	−0.133453	−	0.991055i	$-0.542606\pi$
$479$	−5.84150	−0.266905	−0.133453	+	0.991055i	$0.542606\pi$
$480$	0	0
$481$	−9.80203	−0.446934
$482$	−13.3394	−0.607592
$483$	−15.7426	−0.716312
$484$	−5.03437	−0.228835
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−5.10374	−0.231272	−0.115636	−	0.993292i	$-0.536891\pi$
$487$	−5.10374	−0.231272	−0.115636	+	0.993292i	$0.536891\pi$
$488$	−5.15131	−0.233189
$489$	5.90315	0.266950
$490$	0	0
$491$	39.0842	1.76385	0.881923	−	0.471394i	$-0.156249\pi$
$491$	39.0842	1.76385	0.881923	+	0.471394i	$0.156249\pi$
$492$	−6.12756	−0.276252
$493$	31.3686	1.41277
$494$	−6.67081	−0.300134
$495$	0	0
$496$	−6.12569	−0.275052
$497$	−19.6166	−0.879923
$498$	−9.13922	−0.409538
$499$	−12.0527	−0.539553	−0.269777	−	0.962923i	$-0.586950\pi$
$499$	−12.0527	−0.539553	−0.269777	+	0.962923i	$0.586950\pi$
$500$	0	0
$501$	8.94427	0.399601
$502$	14.3409	0.640066
$503$	−9.88108	−0.440576	−0.220288	−	0.975435i	$-0.570700\pi$
$503$	−9.88108	−0.440576	−0.220288	+	0.975435i	$0.570700\pi$
$504$	−1.72654	−0.0769063
$505$	0	0
$506$	−22.2703	−0.990037
$507$	−4.18400	−0.185818
$508$	17.8127	0.790309
$509$	20.0102	0.886936	0.443468	−	0.896290i	$-0.353748\pi$
$509$	20.0102	0.886936	0.443468	+	0.896290i	$0.353748\pi$
$510$	0	0
$511$	−11.6117	−0.513670
$512$	−1.00000	−0.0441942
$513$	2.24669	0.0991938
$514$	16.0010	0.705776
$515$	0	0
$516$	−7.06997	−0.311238
$517$	−12.1935	−0.536268
$518$	5.69977	0.250434
$519$	−25.9151	−1.13755
$520$	0	0
$521$	10.7786	0.472220	0.236110	−	0.971726i	$-0.424128\pi$
$521$	10.7786	0.472220	0.236110	+	0.971726i	$0.424128\pi$
$522$	9.37895	0.410505
$523$	−26.4549	−1.15679	−0.578396	−	0.815756i	$-0.696321\pi$
$523$	−26.4549	−1.15679	−0.578396	+	0.815756i	$0.696321\pi$
$524$	10.3375	0.451596
$525$	0	0
$526$	−19.8584	−0.865866
$527$	20.4879	0.892465
$528$	−2.44246	−0.106295
$529$	60.1375	2.61467
$530$	0	0
$531$	−0.182645	−0.00792612
$532$	3.87901	0.168176
$533$	−18.1938	−0.788061
$534$	8.11694	0.351254
$535$	0	0
$536$	−11.8903	−0.513584
$537$	0.0480111	0.00207183
$538$	2.66084	0.114717
$539$	9.81639	0.422822
$540$	0	0
$541$	−28.0936	−1.20784	−0.603919	−	0.797046i	$-0.706395\pi$
$541$	−28.0936	−1.20784	−0.603919	+	0.797046i	$0.706395\pi$
$542$	13.4607	0.578186
$543$	20.5347	0.881229
$544$	3.34458	0.143398
$545$	0	0
$546$	−5.12641	−0.219390
$547$	−7.37748	−0.315438	−0.157719	−	0.987484i	$-0.550414\pi$
$547$	−7.37748	−0.315438	−0.157719	+	0.987484i	$0.550414\pi$
$548$	16.0816	0.686973
$549$	5.15131	0.219853
$550$	0	0
$551$	−21.0716	−0.897680
$552$	9.11798	0.388087
$553$	−19.6085	−0.833836
$554$	−5.34748	−0.227193
$555$	0	0
$556$	−16.5094	−0.700155
$557$	−17.2189	−0.729587	−0.364794	−	0.931088i	$-0.618860\pi$
$557$	−17.2189	−0.729587	−0.364794	+	0.931088i	$0.618860\pi$
$558$	6.12569	0.259321
$559$	−20.9920	−0.887866
$560$	0	0
$561$	8.16901	0.344896
$562$	16.7170	0.705162
$563$	−0.199504	−0.00840807	−0.00420404	−	0.999991i	$-0.501338\pi$
$563$	−0.199504	−0.00840807	−0.00420404	+	0.999991i	$0.501338\pi$
$564$	4.99228	0.210213
$565$	0	0
$566$	−14.1630	−0.595314
$567$	1.72654	0.0725080
$568$	11.3618	0.476729
$569$	−18.8463	−0.790077	−0.395038	−	0.918665i	$-0.629269\pi$
$569$	−18.8463	−0.790077	−0.395038	+	0.918665i	$0.629269\pi$
$570$	0	0
$571$	18.1761	0.760646	0.380323	−	0.924854i	$-0.375813\pi$
$571$	18.1761	0.760646	0.380323	+	0.924854i	$0.375813\pi$
$572$	−7.25210	−0.303226
$573$	2.42040	0.101113
$574$	10.5795	0.441579
$575$	0	0
$576$	1.00000	0.0416667
$577$	0.379704	0.0158073	0.00790364	−	0.999969i	$-0.497484\pi$
$577$	0.379704	0.0158073	0.00790364	+	0.999969i	$0.497484\pi$
$578$	5.81381	0.241823
$579$	−13.3764	−0.555906
$580$	0	0
$581$	15.7793	0.654634
$582$	−9.88597	−0.409787
$583$	−16.7335	−0.693032
$584$	6.72539	0.278299
$585$	0	0
$586$	12.4643	0.514898
$587$	15.5322	0.641082	0.320541	−	0.947235i	$-0.396135\pi$
$587$	15.5322	0.641082	0.320541	+	0.947235i	$0.396135\pi$
$588$	−4.01905	−0.165743
$589$	−13.7625	−0.567075
$590$	0	0
$591$	−18.9236	−0.778411
$592$	−3.30127	−0.135681
$593$	4.62686	0.190002	0.0950012	−	0.995477i	$-0.469715\pi$
$593$	4.62686	0.190002	0.0950012	+	0.995477i	$0.469715\pi$
$594$	2.44246	0.100215
$595$	0	0
$596$	20.5325	0.841045
$597$	−22.6784	−0.928166
$598$	27.0729	1.10709
$599$	28.6517	1.17067	0.585337	−	0.810790i	$-0.300962\pi$
$599$	28.6517	1.17067	0.585337	+	0.810790i	$0.300962\pi$
$600$	0	0
$601$	7.54545	0.307785	0.153893	−	0.988088i	$-0.450819\pi$
$601$	7.54545	0.307785	0.153893	+	0.988088i	$0.450819\pi$
$602$	12.2066	0.497504
$603$	11.8903	0.484212
$604$	2.78808	0.113445
$605$	0	0
$606$	−6.28885	−0.255467
$607$	0.971865	0.0394468	0.0197234	−	0.999805i	$-0.493721\pi$
$607$	0.971865	0.0394468	0.0197234	+	0.999805i	$0.493721\pi$
$608$	−2.24669	−0.0911153
$609$	−16.1932	−0.656180
$610$	0	0
$611$	14.8230	0.599673
$612$	−3.34458	−0.135196
$613$	−41.0725	−1.65890	−0.829452	−	0.558578i	$-0.811347\pi$
$613$	−41.0725	−1.65890	−0.829452	+	0.558578i	$0.811347\pi$
$614$	−23.9974	−0.968455
$615$	0	0
$616$	4.21702	0.169908
$617$	8.20866	0.330468	0.165234	−	0.986254i	$-0.447162\pi$
$617$	8.20866	0.330468	0.165234	+	0.986254i	$0.447162\pi$
$618$	8.57949	0.345118
$619$	45.6024	1.83292	0.916458	−	0.400131i	$-0.131035\pi$
$619$	45.6024	1.83292	0.916458	+	0.400131i	$0.131035\pi$
$620$	0	0
$621$	−9.11798	−0.365892
$622$	13.9591	0.559708
$623$	−14.0142	−0.561469
$624$	2.96917	0.118862
$625$	0	0
$626$	9.62739	0.384788
$627$	−5.48746	−0.219148
$628$	−14.3416	−0.572291
$629$	11.0413	0.440247
$630$	0	0
$631$	34.1315	1.35875	0.679377	−	0.733790i	$-0.262250\pi$
$631$	34.1315	1.35875	0.679377	+	0.733790i	$0.262250\pi$
$632$	11.3571	0.451760
$633$	2.55876	0.101702
$634$	−0.556314	−0.0220941
$635$	0	0
$636$	6.85108	0.271663
$637$	−11.9333	−0.472813
$638$	−22.9077	−0.906926
$639$	−11.3618	−0.449464
$640$	0	0
$641$	−34.3957	−1.35855	−0.679274	−	0.733884i	$-0.737705\pi$
$641$	−34.3957	−1.35855	−0.679274	+	0.733884i	$0.737705\pi$
$642$	−9.09673	−0.359019
$643$	−2.71413	−0.107035	−0.0535173	−	0.998567i	$-0.517043\pi$
$643$	−2.71413	−0.107035	−0.0535173	+	0.998567i	$0.517043\pi$
$644$	−15.7426	−0.620344
$645$	0	0
$646$	7.51423	0.295643
$647$	−28.6041	−1.12455	−0.562273	−	0.826952i	$-0.690073\pi$
$647$	−28.6041	−1.12455	−0.562273	+	0.826952i	$0.690073\pi$
$648$	−1.00000	−0.0392837
$649$	0.446104	0.0175111
$650$	0	0
$651$	−10.5763	−0.414517
$652$	5.90315	0.231185
$653$	−34.2818	−1.34155	−0.670776	−	0.741660i	$-0.734039\pi$
$653$	−34.2818	−1.34155	−0.670776	+	0.741660i	$0.734039\pi$
$654$	13.6103	0.532206
$655$	0	0
$656$	−6.12756	−0.239241
$657$	−6.72539	−0.262382
$658$	−8.61939	−0.336019
$659$	18.5057	0.720879	0.360440	−	0.932783i	$-0.382627\pi$
$659$	18.5057	0.720879	0.360440	+	0.932783i	$0.382627\pi$
$660$	0	0
$661$	13.3123	0.517787	0.258894	−	0.965906i	$-0.416642\pi$
$661$	13.3123	0.517787	0.258894	+	0.965906i	$0.416642\pi$
$662$	11.5651	0.449492
$663$	−9.93063	−0.385674
$664$	−9.13922	−0.354671
$665$	0	0
$666$	3.30127	0.127921
$667$	85.5170	3.31123
$668$	8.94427	0.346064
$669$	18.6543	0.721216
$670$	0	0
$671$	−12.5819	−0.485718
$672$	−1.72654	−0.0666028
$673$	35.9582	1.38609	0.693044	−	0.720896i	$-0.256269\pi$
$673$	35.9582	1.38609	0.693044	+	0.720896i	$0.256269\pi$
$674$	−29.8767	−1.15081
$675$	0	0
$676$	−4.18400	−0.160923
$677$	−6.11998	−0.235210	−0.117605	−	0.993060i	$-0.537522\pi$
$677$	−6.11998	−0.235210	−0.117605	+	0.993060i	$0.537522\pi$
$678$	8.26761	0.317515
$679$	17.0685	0.655031
$680$	0	0
$681$	−11.0615	−0.423879
$682$	−14.9618	−0.572916
$683$	−7.39955	−0.283136	−0.141568	−	0.989929i	$-0.545214\pi$
$683$	−7.39955	−0.283136	−0.141568	+	0.989929i	$0.545214\pi$
$684$	2.24669	0.0859043
$685$	0	0
$686$	19.0249	0.726373
$687$	27.0493	1.03199
$688$	−7.06997	−0.269540
$689$	20.3421	0.774971
$690$	0	0
$691$	−32.6446	−1.24186	−0.620930	−	0.783866i	$-0.713245\pi$
$691$	−32.6446	−1.24186	−0.620930	+	0.783866i	$0.713245\pi$
$692$	−25.9151	−0.985145
$693$	−4.21702	−0.160191
$694$	4.24669	0.161202
$695$	0	0
$696$	9.37895	0.355508
$697$	20.4941	0.776269
$698$	19.3711	0.733208
$699$	18.2774	0.691315
$700$	0	0
$701$	15.4733	0.584418	0.292209	−	0.956354i	$-0.405610\pi$
$701$	15.4733	0.584418	0.292209	+	0.956354i	$0.405610\pi$
$702$	−2.96917	−0.112064
$703$	−7.41692	−0.279734
$704$	−2.44246	−0.0920538
$705$	0	0
$706$	0.243275	0.00915578
$707$	10.8580	0.408356
$708$	−0.182645	−0.00686422
$709$	6.31260	0.237075	0.118537	−	0.992950i	$-0.462179\pi$
$709$	6.31260	0.237075	0.118537	+	0.992950i	$0.462179\pi$
$710$	0	0
$711$	−11.3571	−0.425923
$712$	8.11694	0.304195
$713$	55.8539	2.09175
$714$	5.77455	0.216107
$715$	0	0
$716$	0.0480111	0.00179426
$717$	−2.56816	−0.0959096
$718$	−21.0432	−0.785325
$719$	29.7954	1.11118	0.555591	−	0.831456i	$-0.312492\pi$
$719$	29.7954	1.11118	0.555591	+	0.831456i	$0.312492\pi$
$720$	0	0
$721$	−14.8129	−0.551660
$722$	13.9524	0.519254
$723$	13.3394	0.496096
$724$	20.5347	0.763167
$725$	0	0
$726$	5.03437	0.186843
$727$	9.68374	0.359150	0.179575	−	0.983744i	$-0.442528\pi$
$727$	9.68374	0.359150	0.179575	+	0.983744i	$0.442528\pi$
$728$	−5.12641	−0.189997
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.6460	0.874581
$732$	5.15131	0.190398
$733$	−34.7176	−1.28232	−0.641162	−	0.767405i	$-0.721548\pi$
$733$	−34.7176	−1.28232	−0.641162	+	0.767405i	$0.721548\pi$
$734$	7.83339	0.289136
$735$	0	0
$736$	9.11798	0.336093
$737$	−29.0417	−1.06977
$738$	6.12756	0.225559
$739$	−22.6966	−0.834907	−0.417453	−	0.908698i	$-0.637077\pi$
$739$	−22.6966	−0.834907	−0.417453	+	0.908698i	$0.637077\pi$
$740$	0	0
$741$	6.67081	0.245058
$742$	−11.8287	−0.434245
$743$	9.53920	0.349959	0.174980	−	0.984572i	$-0.444014\pi$
$743$	9.53920	0.349959	0.174980	+	0.984572i	$0.444014\pi$
$744$	6.12569	0.224579
$745$	0	0
$746$	15.5256	0.568431
$747$	9.13922	0.334387
$748$	8.16901	0.298688
$749$	15.7059	0.573881
$750$	0	0
$751$	22.5289	0.822090	0.411045	−	0.911615i	$-0.365164\pi$
$751$	22.5289	0.822090	0.411045	+	0.911615i	$0.365164\pi$
$752$	4.99228	0.182050
$753$	−14.3409	−0.522612
$754$	27.8477	1.01415
$755$	0	0
$756$	1.72654	0.0627937
$757$	38.3200	1.39276	0.696382	−	0.717672i	$-0.254792\pi$
$757$	38.3200	1.39276	0.696382	+	0.717672i	$0.254792\pi$
$758$	10.8637	0.394587
$759$	22.2703	0.808362
$760$	0	0
$761$	20.4089	0.739823	0.369912	−	0.929067i	$-0.379388\pi$
$761$	20.4089	0.739823	0.369912	+	0.929067i	$0.379388\pi$
$762$	−17.8127	−0.645284
$763$	−23.4988	−0.850713
$764$	2.42040	0.0875668
$765$	0	0
$766$	9.59615	0.346723
$767$	−0.542305	−0.0195815
$768$	1.00000	0.0360844
$769$	−30.0402	−1.08328	−0.541638	−	0.840612i	$-0.682196\pi$
$769$	−30.0402	−1.08328	−0.541638	+	0.840612i	$0.682196\pi$
$770$	0	0
$771$	−16.0010	−0.576263
$772$	−13.3764	−0.481429
$773$	21.7468	0.782179	0.391090	−	0.920353i	$-0.372098\pi$
$773$	21.7468	0.782179	0.391090	+	0.920353i	$0.372098\pi$
$774$	7.06997	0.254125
$775$	0	0
$776$	−9.88597	−0.354886
$777$	−5.69977	−0.204478
$778$	−34.8220	−1.24843
$779$	−13.7667	−0.493244
$780$	0	0
$781$	27.7507	0.992998
$782$	−30.4958	−1.09053
$783$	−9.37895	−0.335176
$784$	−4.01905	−0.143538
$785$	0	0
$786$	−10.3375	−0.368726
$787$	13.1966	0.470409	0.235205	−	0.971946i	$-0.424424\pi$
$787$	13.1966	0.470409	0.235205	+	0.971946i	$0.424424\pi$
$788$	−18.9236	−0.674124
$789$	19.8584	0.706976
$790$	0	0
$791$	−14.2744	−0.507538
$792$	2.44246	0.0867892
$793$	15.2951	0.543146
$794$	−16.0058	−0.568025
$795$	0	0
$796$	−22.6784	−0.803815
$797$	−16.6732	−0.590595	−0.295297	−	0.955405i	$-0.595419\pi$
$797$	−16.6732	−0.590595	−0.295297	+	0.955405i	$0.595419\pi$
$798$	−3.87901	−0.137315
$799$	−16.6971	−0.590700
$800$	0	0
$801$	−8.11694	−0.286798
$802$	−11.5284	−0.407082
$803$	16.4265	0.579679
$804$	11.8903	0.419340
$805$	0	0
$806$	18.1883	0.640654
$807$	−2.66084	−0.0936658
$808$	−6.28885	−0.221241
$809$	−27.4913	−0.966542	−0.483271	−	0.875471i	$-0.660551\pi$
$809$	−27.4913	−0.966542	−0.483271	+	0.875471i	$0.660551\pi$
$810$	0	0
$811$	−25.8449	−0.907538	−0.453769	−	0.891119i	$-0.649921\pi$
$811$	−25.8449	−0.907538	−0.453769	+	0.891119i	$0.649921\pi$
$812$	−16.1932	−0.568268
$813$	−13.4607	−0.472087
$814$	−8.06322	−0.282616
$815$	0	0
$816$	−3.34458	−0.117084
$817$	−15.8840	−0.555712
$818$	14.5906	0.510150
$819$	5.12641	0.179131
$820$	0	0
$821$	−32.6620	−1.13991	−0.569956	−	0.821675i	$-0.693040\pi$
$821$	−32.6620	−1.13991	−0.569956	+	0.821675i	$0.693040\pi$
$822$	−16.0816	−0.560911
$823$	49.0183	1.70867	0.854336	−	0.519722i	$-0.173964\pi$
$823$	49.0183	1.70867	0.854336	+	0.519722i	$0.173964\pi$
$824$	8.57949	0.298881
$825$	0	0
$826$	0.315344	0.0109722
$827$	9.46963	0.329291	0.164646	−	0.986353i	$-0.447352\pi$
$827$	9.46963	0.329291	0.164646	+	0.986353i	$0.447352\pi$
$828$	−9.11798	−0.316872
$829$	−55.5976	−1.93098	−0.965492	−	0.260432i	$-0.916135\pi$
$829$	−55.5976	−1.93098	−0.965492	+	0.260432i	$0.916135\pi$
$830$	0	0
$831$	5.34748	0.185502
$832$	2.96917	0.102938
$833$	13.4420	0.465739
$834$	16.5094	0.571674
$835$	0	0
$836$	−5.48746	−0.189788
$837$	−6.12569	−0.211735
$838$	14.1068	0.487313
$839$	−6.13233	−0.211712	−0.105856	−	0.994381i	$-0.533758\pi$
$839$	−6.13233	−0.211712	−0.105856	+	0.994381i	$0.533758\pi$
$840$	0	0
$841$	58.9647	2.03326
$842$	12.8882	0.444155
$843$	−16.7170	−0.575763
$844$	2.55876	0.0880761
$845$	0	0
$846$	−4.99228	−0.171638
$847$	−8.69206	−0.298663
$848$	6.85108	0.235267
$849$	14.1630	0.486072
$850$	0	0
$851$	30.1009	1.03184
$852$	−11.3618	−0.389248
$853$	30.4772	1.04352	0.521760	−	0.853092i	$-0.325276\pi$
$853$	30.4772	1.04352	0.521760	+	0.853092i	$0.325276\pi$
$854$	−8.89396	−0.304345
$855$	0	0
$856$	−9.09673	−0.310920
$857$	−7.53137	−0.257267	−0.128633	−	0.991692i	$-0.541059\pi$
$857$	−7.53137	−0.257267	−0.128633	+	0.991692i	$0.541059\pi$
$858$	7.25210	0.247583
$859$	49.3400	1.68346	0.841729	−	0.539900i	$-0.181538\pi$
$859$	49.3400	1.68346	0.841729	+	0.539900i	$0.181538\pi$
$860$	0	0
$861$	−10.5795	−0.360548
$862$	23.1424	0.788235
$863$	−31.4871	−1.07183	−0.535916	−	0.844271i	$-0.680034\pi$
$863$	−31.4871	−1.07183	−0.535916	+	0.844271i	$0.680034\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	23.5106	0.798922
$867$	−5.81381	−0.197447
$868$	−10.5763	−0.358982
$869$	27.7392	0.940988
$870$	0	0
$871$	35.3045	1.19625
$872$	13.6103	0.460903
$873$	9.88597	0.334589
$874$	20.4853	0.692924
$875$	0	0
$876$	−6.72539	−0.227230
$877$	7.87907	0.266057	0.133029	−	0.991112i	$-0.457530\pi$
$877$	7.87907	0.266057	0.133029	+	0.991112i	$0.457530\pi$
$878$	−1.99240	−0.0672401
$879$	−12.4643	−0.420412
$880$	0	0
$881$	−33.0782	−1.11443	−0.557216	−	0.830368i	$-0.688130\pi$
$881$	−33.0782	−1.11443	−0.557216	+	0.830368i	$0.688130\pi$
$882$	4.01905	0.135328
$883$	−7.81473	−0.262987	−0.131493	−	0.991317i	$-0.541977\pi$
$883$	−7.81473	−0.262987	−0.131493	+	0.991317i	$0.541977\pi$
$884$	−9.93063	−0.334003
$885$	0	0
$886$	13.0327	0.437842
$887$	10.1119	0.339525	0.169762	−	0.985485i	$-0.445700\pi$
$887$	10.1119	0.339525	0.169762	+	0.985485i	$0.445700\pi$
$888$	3.30127	0.110783
$889$	30.7543	1.03147
$890$	0	0
$891$	−2.44246	−0.0818256
$892$	18.6543	0.624591
$893$	11.2161	0.375333
$894$	−20.5325	−0.686710
$895$	0	0
$896$	−1.72654	−0.0576797
$897$	−27.0729	−0.903937
$898$	−12.0846	−0.403270
$899$	57.4526	1.91615
$900$	0	0
$901$	−22.9140	−0.763375
$902$	−14.9663	−0.498325
$903$	−12.2066	−0.406210
$904$	8.26761	0.274976
$905$	0	0
$906$	−2.78808	−0.0926277
$907$	−10.2207	−0.339373	−0.169686	−	0.985498i	$-0.554275\pi$
$907$	−10.2207	−0.339373	−0.169686	+	0.985498i	$0.554275\pi$
$908$	−11.0615	−0.367090
$909$	6.28885	0.208588
$910$	0	0
$911$	42.4554	1.40661	0.703305	−	0.710889i	$-0.251707\pi$
$911$	42.4554	1.40661	0.703305	+	0.710889i	$0.251707\pi$
$912$	2.24669	0.0743953
$913$	−22.3222	−0.738757
$914$	−5.62762	−0.186145
$915$	0	0
$916$	27.0493	0.893732
$917$	17.8481	0.589397
$918$	3.34458	0.110387
$919$	52.0693	1.71761	0.858805	−	0.512303i	$-0.171208\pi$
$919$	52.0693	1.71761	0.858805	+	0.512303i	$0.171208\pi$
$920$	0	0
$921$	23.9974	0.790741
$922$	4.62686	0.152378
$923$	−33.7351	−1.11040
$924$	−4.21702	−0.138730
$925$	0	0
$926$	5.75912	0.189256
$927$	−8.57949	−0.281787
$928$	9.37895	0.307879
$929$	26.3223	0.863605	0.431803	−	0.901968i	$-0.357878\pi$
$929$	26.3223	0.863605	0.431803	+	0.901968i	$0.357878\pi$
$930$	0	0
$931$	−9.02956	−0.295932
$932$	18.2774	0.598696
$933$	−13.9591	−0.457000
$934$	24.5818	0.804341
$935$	0	0
$936$	−2.96917	−0.0970505
$937$	−10.0676	−0.328894	−0.164447	−	0.986386i	$-0.552584\pi$
$937$	−10.0676	−0.328894	−0.164447	+	0.986386i	$0.552584\pi$
$938$	−20.5292	−0.670301
$939$	−9.62739	−0.314178
$940$	0	0
$941$	9.99318	0.325768	0.162884	−	0.986645i	$-0.447920\pi$
$941$	9.99318	0.325768	0.162884	+	0.986645i	$0.447920\pi$
$942$	14.3416	0.467273
$943$	55.8709	1.81941
$944$	−0.182645	−0.00594459
$945$	0	0
$946$	−17.2681	−0.561435
$947$	55.4456	1.80174	0.900869	−	0.434090i	$-0.142930\pi$
$947$	55.4456	1.80174	0.900869	+	0.434090i	$0.142930\pi$
$948$	−11.3571	−0.368860
$949$	−19.9689	−0.648217
$950$	0	0
$951$	0.556314	0.0180397
$952$	5.77455	0.187154
$953$	12.1049	0.392117	0.196058	−	0.980592i	$-0.437186\pi$
$953$	12.1049	0.392117	0.196058	+	0.980592i	$0.437186\pi$
$954$	−6.85108	−0.221812
$955$	0	0
$956$	−2.56816	−0.0830602
$957$	22.9077	0.740502
$958$	5.84150	0.188730
$959$	27.7656	0.896598
$960$	0	0
$961$	6.52413	0.210456
$962$	9.80203	0.316030
$963$	9.09673	0.293138
$964$	13.3394	0.429632
$965$	0	0
$966$	15.7426	0.506509
$967$	2.91557	0.0937584	0.0468792	−	0.998901i	$-0.485072\pi$
$967$	2.91557	0.0937584	0.0468792	+	0.998901i	$0.485072\pi$
$968$	5.03437	0.161811
$969$	−7.51423	−0.241392
$970$	0	0
$971$	11.3329	0.363691	0.181845	−	0.983327i	$-0.441793\pi$
$971$	11.3329	0.363691	0.181845	+	0.983327i	$0.441793\pi$
$972$	1.00000	0.0320750
$973$	−28.5042	−0.913803
$974$	5.10374	0.163534
$975$	0	0
$976$	5.15131	0.164889
$977$	4.80307	0.153664	0.0768320	−	0.997044i	$-0.475519\pi$
$977$	4.80307	0.153664	0.0768320	+	0.997044i	$0.475519\pi$
$978$	−5.90315	−0.188762
$979$	19.8253	0.633620
$980$	0	0
$981$	−13.6103	−0.434544
$982$	−39.0842	−1.24723
$983$	−7.76040	−0.247518	−0.123759	−	0.992312i	$-0.539495\pi$
$983$	−7.76040	−0.247518	−0.123759	+	0.992312i	$0.539495\pi$
$984$	6.12756	0.195339
$985$	0	0
$986$	−31.3686	−0.998980
$987$	8.61939	0.274358
$988$	6.67081	0.212227
$989$	64.4638	2.04983
$990$	0	0
$991$	−44.8119	−1.42350	−0.711749	−	0.702434i	$-0.752097\pi$
$991$	−44.8119	−1.42350	−0.711749	+	0.702434i	$0.752097\pi$
$992$	6.12569	0.194491
$993$	−11.5651	−0.367009
$994$	19.6166	0.622200
$995$	0	0
$996$	9.13922	0.289587
$997$	3.99189	0.126424	0.0632121	−	0.998000i	$-0.479866\pi$
$997$	3.99189	0.126424	0.0632121	+	0.998000i	$0.479866\pi$
$998$	12.0527	0.381522
$999$	−3.30127	−0.104447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.m.1.3		4
5.2	odd	4		3750.2.c.e.1249.3		8
5.3	odd	4		3750.2.c.e.1249.6		8
5.4	even	2		3750.2.a.o.1.2		4
25.3	odd	20		150.2.h.a.109.2	✓	8
25.4	even	10		750.2.g.c.451.1		8
25.6	even	5		750.2.g.e.301.2		8
25.8	odd	20		750.2.h.c.199.1		8
25.17	odd	20		150.2.h.a.139.2	yes	8
25.19	even	10		750.2.g.c.301.1		8
25.21	even	5		750.2.g.e.451.2		8
25.22	odd	20		750.2.h.c.49.1		8
75.17	even	20		450.2.l.a.289.1		8
75.53	even	20		450.2.l.a.109.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.h.a.109.2	✓	8	25.3	odd	20
150.2.h.a.139.2	yes	8	25.17	odd	20
450.2.l.a.109.1		8	75.53	even	20
450.2.l.a.289.1		8	75.17	even	20
750.2.g.c.301.1		8	25.19	even	10
750.2.g.c.451.1		8	25.4	even	10
750.2.g.e.301.2		8	25.6	even	5
750.2.g.e.451.2		8	25.21	even	5
750.2.h.c.49.1		8	25.22	odd	20
750.2.h.c.199.1		8	25.8	odd	20
3750.2.a.m.1.3		4	1.1	even	1	trivial
3750.2.a.o.1.2		4	5.4	even	2
3750.2.c.e.1249.3		8	5.2	odd	4
3750.2.c.e.1249.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.72654 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.44246 q^{11} +1.00000 q^{12} +2.96917 q^{13} -1.72654 q^{14} +1.00000 q^{16} -3.34458 q^{17} -1.00000 q^{18} +2.24669 q^{19} +1.72654 q^{21} +2.44246 q^{22} -9.11798 q^{23} -1.00000 q^{24} -2.96917 q^{26} +1.00000 q^{27} +1.72654 q^{28} -9.37895 q^{29} -6.12569 q^{31} -1.00000 q^{32} -2.44246 q^{33} +3.34458 q^{34} +1.00000 q^{36} -3.30127 q^{37} -2.24669 q^{38} +2.96917 q^{39} -6.12756 q^{41} -1.72654 q^{42} -7.06997 q^{43} -2.44246 q^{44} +9.11798 q^{46} +4.99228 q^{47} +1.00000 q^{48} -4.01905 q^{49} -3.34458 q^{51} +2.96917 q^{52} +6.85108 q^{53} -1.00000 q^{54} -1.72654 q^{56} +2.24669 q^{57} +9.37895 q^{58} -0.182645 q^{59} +5.15131 q^{61} +6.12569 q^{62} +1.72654 q^{63} +1.00000 q^{64} +2.44246 q^{66} +11.8903 q^{67} -3.34458 q^{68} -9.11798 q^{69} -11.3618 q^{71} -1.00000 q^{72} -6.72539 q^{73} +3.30127 q^{74} +2.24669 q^{76} -4.21702 q^{77} -2.96917 q^{78} -11.3571 q^{79} +1.00000 q^{81} +6.12756 q^{82} +9.13922 q^{83} +1.72654 q^{84} +7.06997 q^{86} -9.37895 q^{87} +2.44246 q^{88} -8.11694 q^{89} +5.12641 q^{91} -9.11798 q^{92} -6.12569 q^{93} -4.99228 q^{94} -1.00000 q^{96} +9.88597 q^{97} +4.01905 q^{98} -2.44246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{6} + 4 q^{7} - 4 q^{8} + 4 q^{9} - 10 q^{11} + 4 q^{12} - 2 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 4 q^{18} - 6 q^{19} + 4 q^{21} + 10 q^{22} - 4 q^{24} + 2 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 - 4 * q^6 + 4 * q^7 - 4 * q^8 + 4 * q^9 - 10 * q^11 + 4 * q^12 - 2 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 4 * q^18 - 6 * q^19 + 4 * q^21 + 10 * q^22 - 4 * q^24 + 2 * q^26 + 4 * q^27 + 4 * q^28 - 14 * q^29 - 18 * q^31 - 4 * q^32 - 10 * q^33 + 6 * q^34 + 4 * q^36 - 2 * q^37 + 6 * q^38 - 2 * q^39 - 14 * q^41 - 4 * q^42 + 14 * q^43 - 10 * q^44 - 10 * q^47 + 4 * q^48 - 4 * q^49 - 6 * q^51 - 2 * q^52 - 14 * q^53 - 4 * q^54 - 4 * q^56 - 6 * q^57 + 14 * q^58 - 20 * q^59 + 18 * q^62 + 4 * q^63 + 4 * q^64 + 10 * q^66 + 14 * q^67 - 6 * q^68 - 30 * q^71 - 4 * q^72 + 8 * q^73 + 2 * q^74 - 6 * q^76 - 20 * q^77 + 2 * q^78 - 28 * q^79 + 4 * q^81 + 14 * q^82 - 12 * q^83 + 4 * q^84 - 14 * q^86 - 14 * q^87 + 10 * q^88 - 28 * q^89 - 22 * q^91 - 18 * q^93 + 10 * q^94 - 4 * q^96 + 8 * q^97 + 4 * q^98 - 10 * q^99

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