LMFDB - Embedded newform 7605.2.a.bw.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7605,2,Mod(1,7605)] 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("7605.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,6,3,0,-3,6,0,0,0] 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:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.756.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 2 $ x^3 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.339877$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.339877 q^{2} -1.88448 q^{4} +1.00000 q^{5} -0.660123 q^{7} +1.32025 q^{8} -0.339877 q^{10} +0.679754 q^{11} +0.224361 q^{14} +3.32025 q^{16} -7.42909 q^{17} -0.115516 q^{19} -1.88448 q^{20} -0.231033 q^{22} +7.76897 q^{23} +1.00000 q^{25} +1.24399 q^{28} -5.54461 q^{29} +9.97370 q^{31} -3.76897 q^{32} +2.52498 q^{34} -0.660123 q^{35} -9.76897 q^{37} +0.0392613 q^{38} +1.32025 q^{40} +4.22436 q^{41} -0.544607 q^{43} -1.28098 q^{44} -2.64049 q^{46} -5.01963 q^{47} -6.56424 q^{49} -0.339877 q^{50} -0.679754 q^{53} +0.679754 q^{55} -0.871525 q^{56} +1.88448 q^{58} +2.22436 q^{59} -4.20473 q^{61} -3.38983 q^{62} -5.35951 q^{64} +7.63382 q^{67} +14.0000 q^{68} +0.224361 q^{70} +7.31357 q^{71} -8.01963 q^{73} +3.32025 q^{74} +0.217689 q^{76} -0.448721 q^{77} +9.97370 q^{79} +3.32025 q^{80} -1.43576 q^{82} +1.76897 q^{83} -7.42909 q^{85} +0.185099 q^{86} +0.897442 q^{88} +13.5446 q^{89} -14.6405 q^{92} +1.70606 q^{94} -0.115516 q^{95} -9.90411 q^{97} +2.23103 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40}+ \cdots + 30 q^{98}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 12 * q^14 + 12 * q^16 - 12 * q^19 + 6 * q^20 - 24 * q^22 + 3 * q^25 - 12 * q^28 - 6 * q^29 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 6 * q^37 - 6 * q^38 + 6 * q^40 + 9 * q^43 - 12 * q^44 - 12 * q^46 - 12 * q^47 - 6 * q^49 - 30 * q^56 - 6 * q^58 - 6 * q^59 - 3 * q^61 + 6 * q^62 - 12 * q^64 - 9 * q^67 + 42 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 12 * q^74 - 48 * q^76 + 24 * q^77 - 3 * q^79 + 12 * q^80 - 18 * q^82 - 18 * q^83 - 6 * q^86 - 48 * q^88 + 30 * q^89 - 48 * q^92 + 36 * q^94 - 12 * q^95 - 15 * q^97 + 30 * q^98

Coefficient data

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

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

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

$2$	−0.339877	−0.240329	−0.120165	−	0.992754i	$-0.538342\pi$
$2$	−0.339877	−0.240329	−0.120165	+	0.992754i	$0.538342\pi$
$3$	0	0
$3$	0	0
$4$	−1.88448	−0.942242
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.660123	−0.249503	−0.124752	−	0.992188i	$-0.539813\pi$
$7$	−0.660123	−0.249503	−0.124752	+	0.992188i	$0.539813\pi$
$8$	1.32025	0.466778
$9$	0	0
$10$	−0.339877	−0.107479
$11$	0.679754	0.204953	0.102477	−	0.994735i	$-0.467323\pi$
$11$	0.679754	0.204953	0.102477	+	0.994735i	$0.467323\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0.224361	0.0599629
$15$	0	0
$16$	3.32025	0.830062
$17$	−7.42909	−1.80182	−0.900910	−	0.434007i	$-0.857099\pi$
$17$	−7.42909	−1.80182	−0.900910	+	0.434007i	$0.857099\pi$
$18$	0	0
$19$	−0.115516	−0.0265013	−0.0132506	−	0.999912i	$-0.504218\pi$
$19$	−0.115516	−0.0265013	−0.0132506	+	0.999912i	$0.504218\pi$
$20$	−1.88448	−0.421383
$21$	0	0
$22$	−0.231033	−0.0492563
$23$	7.76897	1.61994	0.809971	−	0.586470i	$-0.199483\pi$
$23$	7.76897	1.61994	0.809971	+	0.586470i	$0.199483\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	1.24399	0.235092
$29$	−5.54461	−1.02961	−0.514804	−	0.857308i	$-0.672135\pi$
$29$	−5.54461	−1.02961	−0.514804	+	0.857308i	$0.672135\pi$
$30$	0	0
$31$	9.97370	1.79133	0.895664	−	0.444730i	$-0.146701\pi$
$31$	9.97370	1.79133	0.895664	+	0.444730i	$0.146701\pi$
$32$	−3.76897	−0.666266
$33$	0	0
$34$	2.52498	0.433030
$35$	−0.660123	−0.111581
$36$	0	0
$37$	−9.76897	−1.60601	−0.803004	−	0.595973i	$-0.796766\pi$
$37$	−9.76897	−1.60601	−0.803004	+	0.595973i	$0.796766\pi$
$38$	0.0392613	0.00636903
$39$	0	0
$40$	1.32025	0.208749
$41$	4.22436	0.659734	0.329867	−	0.944027i	$-0.392996\pi$
$41$	4.22436	0.659734	0.329867	+	0.944027i	$0.392996\pi$
$42$	0	0
$43$	−0.544607	−0.0830518	−0.0415259	−	0.999137i	$-0.513222\pi$
$43$	−0.544607	−0.0830518	−0.0415259	+	0.999137i	$0.513222\pi$
$44$	−1.28098	−0.193116
$45$	0	0
$46$	−2.64049	−0.389319
$47$	−5.01963	−0.732188	−0.366094	−	0.930578i	$-0.619305\pi$
$47$	−5.01963	−0.732188	−0.366094	+	0.930578i	$0.619305\pi$
$48$	0	0
$49$	−6.56424	−0.937748
$50$	−0.339877	−0.0480659
$51$	0	0
$52$	0	0
$53$	−0.679754	−0.0933714	−0.0466857	−	0.998910i	$-0.514866\pi$
$53$	−0.679754	−0.0933714	−0.0466857	+	0.998910i	$0.514866\pi$
$54$	0	0
$55$	0.679754	0.0916580
$56$	−0.871525	−0.116462
$57$	0	0
$58$	1.88448	0.247445
$59$	2.22436	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$59$	2.22436	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$60$	0	0
$61$	−4.20473	−0.538361	−0.269180	−	0.963090i	$-0.586753\pi$
$61$	−4.20473	−0.538361	−0.269180	+	0.963090i	$0.586753\pi$
$62$	−3.38983	−0.430509
$63$	0	0
$64$	−5.35951	−0.669938
$65$	0	0
$66$	0	0
$67$	7.63382	0.932620	0.466310	−	0.884621i	$-0.345583\pi$
$67$	7.63382	0.932620	0.466310	+	0.884621i	$0.345583\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	0.224361	0.0268162
$71$	7.31357	0.867962	0.433981	−	0.900922i	$-0.357109\pi$
$71$	7.31357	0.867962	0.433981	+	0.900922i	$0.357109\pi$
$72$	0	0
$73$	−8.01963	−0.938627	−0.469313	−	0.883032i	$-0.655499\pi$
$73$	−8.01963	−0.938627	−0.469313	+	0.883032i	$0.655499\pi$
$74$	3.32025	0.385971
$75$	0	0
$76$	0.217689	0.0249706
$77$	−0.448721	−0.0511365
$78$	0	0
$79$	9.97370	1.12213	0.561064	−	0.827772i	$-0.310392\pi$
$79$	9.97370	1.12213	0.561064	+	0.827772i	$0.310392\pi$
$80$	3.32025	0.371215
$81$	0	0
$82$	−1.43576	−0.158553
$83$	1.76897	0.194169	0.0970847	−	0.995276i	$-0.469048\pi$
$83$	1.76897	0.194169	0.0970847	+	0.995276i	$0.469048\pi$
$84$	0	0
$85$	−7.42909	−0.805798
$86$	0.185099	0.0199598
$87$	0	0
$88$	0.897442	0.0956677
$89$	13.5446	1.43573	0.717863	−	0.696185i	$-0.245121\pi$
$89$	13.5446	1.43573	0.717863	+	0.696185i	$0.245121\pi$
$90$	0	0
$91$	0	0
$92$	−14.6405	−1.52638
$93$	0	0
$94$	1.70606	0.175966
$95$	−0.115516	−0.0118517
$96$	0	0
$97$	−9.90411	−1.00561	−0.502805	−	0.864400i	$-0.667699\pi$
$97$	−9.90411	−1.00561	−0.502805	+	0.864400i	$0.667699\pi$
$98$	2.23103	0.225368
$99$	0	0
$100$	−1.88448	−0.188448
$101$	9.35951	0.931306	0.465653	−	0.884967i	$-0.345820\pi$
$101$	9.35951	0.931306	0.465653	+	0.884967i	$0.345820\pi$
$102$	0	0
$103$	4.50535	0.443925	0.221962	−	0.975055i	$-0.428754\pi$
$103$	4.50535	0.443925	0.221962	+	0.975055i	$0.428754\pi$
$104$	0	0
$105$	0	0
$106$	0.231033	0.0224399
$107$	−0.570909	−0.0551919	−0.0275960	−	0.999619i	$-0.508785\pi$
$107$	−0.570909	−0.0551919	−0.0275960	+	0.999619i	$0.508785\pi$
$108$	0	0
$109$	−15.4095	−1.47596	−0.737979	−	0.674823i	$-0.764220\pi$
$109$	−15.4095	−1.47596	−0.737979	+	0.674823i	$0.764220\pi$
$110$	−0.231033	−0.0220281
$111$	0	0
$112$	−2.19177	−0.207103
$113$	5.32025	0.500487	0.250243	−	0.968183i	$-0.419489\pi$
$113$	5.32025	0.500487	0.250243	+	0.968183i	$0.419489\pi$
$114$	0	0
$115$	7.76897	0.724460
$116$	10.4487	0.970139
$117$	0	0
$118$	−0.756009	−0.0695962
$119$	4.90411	0.449559
$120$	0	0
$121$	−10.5379	−0.957994
$122$	1.42909	0.129384
$123$	0	0
$124$	−18.7953	−1.68787
$125$	1.00000	0.0894427
$126$	0	0
$127$	−12.0196	−1.06657	−0.533285	−	0.845936i	$-0.679043\pi$
$127$	−12.0196	−1.06657	−0.533285	+	0.845936i	$0.679043\pi$
$128$	9.35951	0.827271
$129$	0	0
$130$	0	0
$131$	−10.9041	−0.952697	−0.476348	−	0.879257i	$-0.658040\pi$
$131$	−10.9041	−0.952697	−0.476348	+	0.879257i	$0.658040\pi$
$132$	0	0
$133$	0.0762550	0.00661215
$134$	−2.59456	−0.224136
$135$	0	0
$136$	−9.80823	−0.841049
$137$	3.38983	0.289613	0.144806	−	0.989460i	$-0.453744\pi$
$137$	3.38983	0.289613	0.144806	+	0.989460i	$0.453744\pi$
$138$	0	0
$139$	14.8082	1.25602	0.628009	−	0.778206i	$-0.283870\pi$
$139$	14.8082	1.25602	0.628009	+	0.778206i	$0.283870\pi$
$140$	1.24399	0.105136
$141$	0	0
$142$	−2.48571	−0.208597
$143$	0	0
$144$	0	0
$145$	−5.54461	−0.460455
$146$	2.72569	0.225579
$147$	0	0
$148$	18.4095	1.51325
$149$	−17.0892	−1.40000	−0.700001	−	0.714141i	$-0.746817\pi$
$149$	−17.0892	−1.40000	−0.700001	+	0.714141i	$0.746817\pi$
$150$	0	0
$151$	−13.0130	−1.05898	−0.529490	−	0.848316i	$-0.677617\pi$
$151$	−13.0130	−1.05898	−0.529490	+	0.848316i	$0.677617\pi$
$152$	−0.152510	−0.0123702
$153$	0	0
$154$	0.152510	0.0122896
$155$	9.97370	0.801107
$156$	0	0
$157$	−0.775639	−0.0619028	−0.0309514	−	0.999521i	$-0.509854\pi$
$157$	−0.775639	−0.0619028	−0.0309514	+	0.999521i	$0.509854\pi$
$158$	−3.38983	−0.269680
$159$	0	0
$160$	−3.76897	−0.297963
$161$	−5.12847	−0.404180
$162$	0	0
$163$	−12.1981	−0.955426	−0.477713	−	0.878516i	$-0.658534\pi$
$163$	−12.1981	−0.955426	−0.477713	+	0.878516i	$0.658534\pi$
$164$	−7.96074	−0.621629
$165$	0	0
$166$	−0.601231	−0.0466646
$167$	−1.59054	−0.123080	−0.0615398	−	0.998105i	$-0.519601\pi$
$167$	−1.59054	−0.123080	−0.0615398	+	0.998105i	$0.519601\pi$
$168$	0	0
$169$	0	0
$170$	2.52498	0.193657
$171$	0	0
$172$	1.02630	0.0782548
$173$	−12.7493	−0.969314	−0.484657	−	0.874704i	$-0.661056\pi$
$173$	−12.7493	−0.969314	−0.484657	+	0.874704i	$0.661056\pi$
$174$	0	0
$175$	−0.660123	−0.0499006
$176$	2.25695	0.170124
$177$	0	0
$178$	−4.60350	−0.345047
$179$	−17.7623	−1.32762	−0.663808	−	0.747903i	$-0.731061\pi$
$179$	−17.7623	−1.32762	−0.663808	+	0.747903i	$0.731061\pi$
$180$	0	0
$181$	7.02630	0.522261	0.261130	−	0.965304i	$-0.415905\pi$
$181$	7.02630	0.522261	0.261130	+	0.965304i	$0.415905\pi$
$182$	0	0
$183$	0	0
$184$	10.2569	0.756152
$185$	−9.76897	−0.718229
$186$	0	0
$187$	−5.04995	−0.369289
$188$	9.45941	0.689899
$189$	0	0
$190$	0.0392613	0.00284832
$191$	13.9541	1.00968	0.504840	−	0.863213i	$-0.331551\pi$
$191$	13.9541	1.00968	0.504840	+	0.863213i	$0.331551\pi$
$192$	0	0
$193$	−23.7493	−1.70951	−0.854757	−	0.519028i	$-0.826294\pi$
$193$	−23.7493	−1.70951	−0.854757	+	0.519028i	$0.826294\pi$
$194$	3.36618	0.241678
$195$	0	0
$196$	12.3702	0.883586
$197$	15.8082	1.12629	0.563145	−	0.826358i	$-0.309591\pi$
$197$	15.8082	1.12629	0.563145	+	0.826358i	$0.309591\pi$
$198$	0	0
$199$	8.76897	0.621616	0.310808	−	0.950473i	$-0.399400\pi$
$199$	8.76897	0.621616	0.310808	+	0.950473i	$0.399400\pi$
$200$	1.32025	0.0933555
$201$	0	0
$202$	−3.18108	−0.223820
$203$	3.66012	0.256890
$204$	0	0
$205$	4.22436	0.295042
$206$	−1.53126	−0.106688
$207$	0	0
$208$	0	0
$209$	−0.0785226	−0.00543152
$210$	0	0
$211$	8.80823	0.606383	0.303192	−	0.952930i	$-0.401948\pi$
$211$	8.80823	0.606383	0.303192	+	0.952930i	$0.401948\pi$
$212$	1.28098	0.0879784
$213$	0	0
$214$	0.194039	0.0132642
$215$	−0.544607	−0.0371419
$216$	0	0
$217$	−6.58387	−0.446942
$218$	5.23732	0.354716
$219$	0	0
$220$	−1.28098	−0.0863640
$221$	0	0
$222$	0	0
$223$	−10.0393	−0.672279	−0.336139	−	0.941812i	$-0.609121\pi$
$223$	−10.0393	−0.672279	−0.336139	+	0.941812i	$0.609121\pi$
$224$	2.48798	0.166235
$225$	0	0
$226$	−1.80823	−0.120282
$227$	1.21140	0.0804036	0.0402018	−	0.999192i	$-0.487200\pi$
$227$	1.21140	0.0804036	0.0402018	+	0.999192i	$0.487200\pi$
$228$	0	0
$229$	−19.2440	−1.27168	−0.635839	−	0.771821i	$-0.719346\pi$
$229$	−19.2440	−1.27168	−0.635839	+	0.771821i	$0.719346\pi$
$230$	−2.64049	−0.174109
$231$	0	0
$232$	−7.32025	−0.480598
$233$	23.9081	1.56627	0.783137	−	0.621849i	$-0.213618\pi$
$233$	23.9081	1.56627	0.783137	+	0.621849i	$0.213618\pi$
$234$	0	0
$235$	−5.01963	−0.327445
$236$	−4.19177	−0.272861
$237$	0	0
$238$	−1.66680	−0.108042
$239$	18.6798	1.20829	0.604146	−	0.796873i	$-0.293514\pi$
$239$	18.6798	1.20829	0.604146	+	0.796873i	$0.293514\pi$
$240$	0	0
$241$	−6.11552	−0.393935	−0.196968	−	0.980410i	$-0.563109\pi$
$241$	−6.11552	−0.393935	−0.196968	+	0.980410i	$0.563109\pi$
$242$	3.58160	0.230234
$243$	0	0
$244$	7.92375	0.507266
$245$	−6.56424	−0.419374
$246$	0	0
$247$	0	0
$248$	13.1677	0.836152
$249$	0	0
$250$	−0.339877	−0.0214957
$251$	−3.35951	−0.212050	−0.106025	−	0.994363i	$-0.533812\pi$
$251$	−3.35951	−0.212050	−0.106025	+	0.994363i	$0.533812\pi$
$252$	0	0
$253$	5.28098	0.332013
$254$	4.08519	0.256328
$255$	0	0
$256$	7.53793	0.471121
$257$	10.1088	0.630572	0.315286	−	0.948997i	$-0.397899\pi$
$257$	10.1088	0.630572	0.315286	+	0.948997i	$0.397899\pi$
$258$	0	0
$259$	6.44872	0.400704
$260$	0	0
$261$	0	0
$262$	3.70606	0.228961
$263$	22.5183	1.38854	0.694269	−	0.719716i	$-0.255728\pi$
$263$	22.5183	1.38854	0.694269	+	0.719716i	$0.255728\pi$
$264$	0	0
$265$	−0.679754	−0.0417569
$266$	−0.0259173	−0.00158909
$267$	0	0
$268$	−14.3858	−0.878753
$269$	−8.49465	−0.517928	−0.258964	−	0.965887i	$-0.583381\pi$
$269$	−8.49465	−0.517928	−0.258964	+	0.965887i	$0.583381\pi$
$270$	0	0
$271$	−9.37020	−0.569199	−0.284600	−	0.958647i	$-0.591861\pi$
$271$	−9.37020	−0.569199	−0.284600	+	0.958647i	$0.591861\pi$
$272$	−24.6664	−1.49562
$273$	0	0
$274$	−1.15212	−0.0696024
$275$	0.679754	0.0409907
$276$	0	0
$277$	0.719015	0.0432014	0.0216007	−	0.999767i	$-0.493124\pi$
$277$	0.719015	0.0432014	0.0216007	+	0.999767i	$0.493124\pi$
$278$	−5.03297	−0.301858
$279$	0	0
$280$	−0.871525	−0.0520836
$281$	1.54461	0.0921435	0.0460718	−	0.998938i	$-0.485330\pi$
$281$	1.54461	0.0921435	0.0460718	+	0.998938i	$0.485330\pi$
$282$	0	0
$283$	−18.1195	−1.07709	−0.538547	−	0.842595i	$-0.681027\pi$
$283$	−18.1195	−1.07709	−0.538547	+	0.842595i	$0.681027\pi$
$284$	−13.7823	−0.817830
$285$	0	0
$286$	0	0
$287$	−2.78860	−0.164606
$288$	0	0
$289$	38.1914	2.24655
$290$	1.88448	0.110661
$291$	0	0
$292$	15.1129	0.884413
$293$	−30.5050	−1.78212	−0.891059	−	0.453887i	$-0.850037\pi$
$293$	−30.5050	−1.78212	−0.891059	+	0.453887i	$0.850037\pi$
$294$	0	0
$295$	2.22436	0.129507
$296$	−12.8974	−0.749649
$297$	0	0
$298$	5.80823	0.336462
$299$	0	0
$300$	0	0
$301$	0.359508	0.0207217
$302$	4.42280	0.254504
$303$	0	0
$304$	−0.383543	−0.0219977
$305$	−4.20473	−0.240762
$306$	0	0
$307$	4.77564	0.272560	0.136280	−	0.990670i	$-0.456485\pi$
$307$	4.77564	0.272560	0.136280	+	0.990670i	$0.456485\pi$
$308$	0.845608	0.0481830
$309$	0	0
$310$	−3.38983	−0.192529
$311$	−30.5812	−1.73410	−0.867051	−	0.498220i	$-0.833987\pi$
$311$	−30.5812	−1.73410	−0.867051	+	0.498220i	$0.833987\pi$
$312$	0	0
$313$	−26.7230	−1.51048	−0.755238	−	0.655451i	$-0.772479\pi$
$313$	−26.7230	−1.51048	−0.755238	+	0.655451i	$0.772479\pi$
$314$	0.263622	0.0148770
$315$	0	0
$316$	−18.7953	−1.05732
$317$	−20.6271	−1.15854	−0.579268	−	0.815137i	$-0.696662\pi$
$317$	−20.6271	−1.15854	−0.579268	+	0.815137i	$0.696662\pi$
$318$	0	0
$319$	−3.76897	−0.211022
$320$	−5.35951	−0.299606
$321$	0	0
$322$	1.74305	0.0971364
$323$	0.858181	0.0477505
$324$	0	0
$325$	0	0
$326$	4.14584	0.229617
$327$	0	0
$328$	5.57720	0.307949
$329$	3.31357	0.182683
$330$	0	0
$331$	−5.37020	−0.295173	−0.147586	−	0.989049i	$-0.547150\pi$
$331$	−5.37020	−0.295173	−0.147586	+	0.989049i	$0.547150\pi$
$332$	−3.33359	−0.182955
$333$	0	0
$334$	0.540588	0.0295797
$335$	7.63382	0.417080
$336$	0	0
$337$	28.7623	1.56678	0.783391	−	0.621529i	$-0.213488\pi$
$337$	28.7623	1.56678	0.783391	+	0.621529i	$0.213488\pi$
$338$	0	0
$339$	0	0
$340$	14.0000	0.759257
$341$	6.77966	0.367139
$342$	0	0
$343$	8.95407	0.483474
$344$	−0.719015	−0.0387667
$345$	0	0
$346$	4.33320	0.232955
$347$	−22.3961	−1.20229	−0.601143	−	0.799141i	$-0.705288\pi$
$347$	−22.3961	−1.20229	−0.601143	+	0.799141i	$0.705288\pi$
$348$	0	0
$349$	11.9737	0.640937	0.320469	−	0.947259i	$-0.396160\pi$
$349$	11.9737	0.640937	0.320469	+	0.947259i	$0.396160\pi$
$350$	0.224361	0.0119926
$351$	0	0
$352$	−2.56197	−0.136553
$353$	−17.0366	−0.906767	−0.453384	−	0.891316i	$-0.649783\pi$
$353$	−17.0366	−0.906767	−0.453384	+	0.891316i	$0.649783\pi$
$354$	0	0
$355$	7.31357	0.388164
$356$	−25.5246	−1.35280
$357$	0	0
$358$	6.03699	0.319065
$359$	−35.0825	−1.85159	−0.925793	−	0.378031i	$-0.876601\pi$
$359$	−35.0825	−1.85159	−0.925793	+	0.378031i	$0.876601\pi$
$360$	0	0
$361$	−18.9867	−0.999298
$362$	−2.38808	−0.125515
$363$	0	0
$364$	0	0
$365$	−8.01963	−0.419767
$366$	0	0
$367$	20.0825	1.04830	0.524150	−	0.851626i	$-0.324383\pi$
$367$	20.0825	1.04830	0.524150	+	0.851626i	$0.324383\pi$
$368$	25.7949	1.34465
$369$	0	0
$370$	3.32025	0.172611
$371$	0.448721	0.0232964
$372$	0	0
$373$	−23.4790	−1.21570	−0.607849	−	0.794052i	$-0.707968\pi$
$373$	−23.4790	−1.21570	−0.607849	+	0.794052i	$0.707968\pi$
$374$	1.71636	0.0887510
$375$	0	0
$376$	−6.62715	−0.341769
$377$	0	0
$378$	0	0
$379$	−21.1414	−1.08596	−0.542981	−	0.839745i	$-0.682705\pi$
$379$	−21.1414	−1.08596	−0.542981	+	0.839745i	$0.682705\pi$
$380$	0.217689	0.0111672
$381$	0	0
$382$	−4.74266	−0.242656
$383$	1.73865	0.0888406	0.0444203	−	0.999013i	$-0.485856\pi$
$383$	1.73865	0.0888406	0.0444203	+	0.999013i	$0.485856\pi$
$384$	0	0
$385$	−0.448721	−0.0228689
$386$	8.07185	0.410846
$387$	0	0
$388$	18.6641	0.947528
$389$	26.6664	1.35204	0.676020	−	0.736883i	$-0.263703\pi$
$389$	26.6664	1.35204	0.676020	+	0.736883i	$0.263703\pi$
$390$	0	0
$391$	−57.7164	−2.91884
$392$	−8.66641	−0.437720
$393$	0	0
$394$	−5.37285	−0.270680
$395$	9.97370	0.501831
$396$	0	0
$397$	−14.5446	−0.729973	−0.364986	−	0.931013i	$-0.618926\pi$
$397$	−14.5446	−0.729973	−0.364986	+	0.931013i	$0.618926\pi$
$398$	−2.98037	−0.149392
$399$	0	0
$400$	3.32025	0.166012
$401$	−8.37020	−0.417988	−0.208994	−	0.977917i	$-0.567019\pi$
$401$	−8.37020	−0.417988	−0.208994	+	0.977917i	$0.567019\pi$
$402$	0	0
$403$	0	0
$404$	−17.6378	−0.877515
$405$	0	0
$406$	−1.24399	−0.0617382
$407$	−6.64049	−0.329157
$408$	0	0
$409$	−17.2547	−0.853189	−0.426595	−	0.904443i	$-0.640287\pi$
$409$	−17.2547	−0.853189	−0.426595	+	0.904443i	$0.640287\pi$
$410$	−1.43576	−0.0709073
$411$	0	0
$412$	−8.49025	−0.418285
$413$	−1.46835	−0.0722529
$414$	0	0
$415$	1.76897	0.0868352
$416$	0	0
$417$	0	0
$418$	0.0266880	0.00130535
$419$	−12.8649	−0.628489	−0.314245	−	0.949342i	$-0.601751\pi$
$419$	−12.8649	−0.628489	−0.314245	+	0.949342i	$0.601751\pi$
$420$	0	0
$421$	−24.5616	−1.19706	−0.598529	−	0.801101i	$-0.704248\pi$
$421$	−24.5616	−1.19706	−0.598529	+	0.801101i	$0.704248\pi$
$422$	−2.99371	−0.145732
$423$	0	0
$424$	−0.897442	−0.0435837
$425$	−7.42909	−0.360364
$426$	0	0
$427$	2.77564	0.134323
$428$	1.07587	0.0520041
$429$	0	0
$430$	0.185099	0.00892628
$431$	24.7797	1.19359	0.596797	−	0.802392i	$-0.296440\pi$
$431$	24.7797	1.19359	0.596797	+	0.802392i	$0.296440\pi$
$432$	0	0
$433$	−14.0589	−0.675627	−0.337814	−	0.941213i	$-0.609687\pi$
$433$	−14.0589	−0.675627	−0.337814	+	0.941213i	$0.609687\pi$
$434$	2.23770	0.107413
$435$	0	0
$436$	29.0389	1.39071
$437$	−0.897442	−0.0429305
$438$	0	0
$439$	20.8452	0.994888	0.497444	−	0.867496i	$-0.334272\pi$
$439$	20.8452	0.994888	0.497444	+	0.867496i	$0.334272\pi$
$440$	0.897442	0.0427839
$441$	0	0
$442$	0	0
$443$	11.4291	0.543012	0.271506	−	0.962437i	$-0.412478\pi$
$443$	11.4291	0.543012	0.271506	+	0.962437i	$0.412478\pi$
$444$	0	0
$445$	13.5446	0.642076
$446$	3.41211	0.161568
$447$	0	0
$448$	3.53793	0.167152
$449$	25.9081	1.22268	0.611340	−	0.791368i	$-0.290631\pi$
$449$	25.9081	1.22268	0.611340	+	0.791368i	$0.290631\pi$
$450$	0	0
$451$	2.87153	0.135215
$452$	−10.0259	−0.471579
$453$	0	0
$454$	−0.411728	−0.0193233
$455$	0	0
$456$	0	0
$457$	−5.90411	−0.276183	−0.138091	−	0.990419i	$-0.544097\pi$
$457$	−5.90411	−0.276183	−0.138091	+	0.990419i	$0.544097\pi$
$458$	6.54059	0.305622
$459$	0	0
$460$	−14.6405	−0.682616
$461$	−15.4920	−0.721534	−0.360767	−	0.932656i	$-0.617485\pi$
$461$	−15.4920	−0.721534	−0.360767	+	0.932656i	$0.617485\pi$
$462$	0	0
$463$	−17.4420	−0.810601	−0.405300	−	0.914184i	$-0.632833\pi$
$463$	−17.4420	−0.810601	−0.405300	+	0.914184i	$0.632833\pi$
$464$	−18.4095	−0.854638
$465$	0	0
$466$	−8.12582	−0.376421
$467$	20.4790	0.947657	0.473829	−	0.880617i	$-0.342872\pi$
$467$	20.4790	0.947657	0.473829	+	0.880617i	$0.342872\pi$
$468$	0	0
$469$	−5.03926	−0.232691
$470$	1.70606	0.0786945
$471$	0	0
$472$	2.93670	0.135173
$473$	−0.370199	−0.0170217
$474$	0	0
$475$	−0.115516	−0.00530025
$476$	−9.24172	−0.423594
$477$	0	0
$478$	−6.34882	−0.290388
$479$	−40.9907	−1.87291	−0.936456	−	0.350785i	$-0.885915\pi$
$479$	−40.9907	−1.87291	−0.936456	+	0.350785i	$0.885915\pi$
$480$	0	0
$481$	0	0
$482$	2.07852	0.0946741
$483$	0	0
$484$	19.8586	0.902662
$485$	−9.90411	−0.449723
$486$	0	0
$487$	−24.0629	−1.09039	−0.545197	−	0.838308i	$-0.683545\pi$
$487$	−24.0629	−1.09039	−0.545197	+	0.838308i	$0.683545\pi$
$488$	−5.55128	−0.251295
$489$	0	0
$490$	2.23103	0.100788
$491$	−36.7730	−1.65954	−0.829771	−	0.558104i	$-0.811529\pi$
$491$	−36.7730	−1.65954	−0.829771	+	0.558104i	$0.811529\pi$
$492$	0	0
$493$	41.1914	1.85517
$494$	0	0
$495$	0	0
$496$	33.1151	1.48691
$497$	−4.82786	−0.216559
$498$	0	0
$499$	−34.8212	−1.55881	−0.779405	−	0.626520i	$-0.784479\pi$
$499$	−34.8212	−1.55881	−0.779405	+	0.626520i	$0.784479\pi$
$500$	−1.88448	−0.0842767
$501$	0	0
$502$	1.14182	0.0509619
$503$	32.0259	1.42797	0.713983	−	0.700164i	$-0.246890\pi$
$503$	32.0259	1.42797	0.713983	+	0.700164i	$0.246890\pi$
$504$	0	0
$505$	9.35951	0.416493
$506$	−1.79488	−0.0797924
$507$	0	0
$508$	22.6508	1.00497
$509$	−41.3462	−1.83264	−0.916318	−	0.400451i	$-0.868854\pi$
$509$	−41.3462	−1.83264	−0.916318	+	0.400451i	$0.868854\pi$
$510$	0	0
$511$	5.29394	0.234190
$512$	−21.2810	−0.940496
$513$	0	0
$514$	−3.43576	−0.151545
$515$	4.50535	0.198529
$516$	0	0
$517$	−3.41211	−0.150065
$518$	−2.19177	−0.0963009
$519$	0	0
$520$	0	0
$521$	−4.40279	−0.192890	−0.0964448	−	0.995338i	$-0.530747\pi$
$521$	−4.40279	−0.192890	−0.0964448	+	0.995338i	$0.530747\pi$
$522$	0	0
$523$	−32.4487	−1.41888	−0.709442	−	0.704764i	$-0.751053\pi$
$523$	−32.4487	−1.41888	−0.709442	+	0.704764i	$0.751053\pi$
$524$	20.5486	0.897671
$525$	0	0
$526$	−7.65345	−0.333706
$527$	−74.0955	−3.22765
$528$	0	0
$529$	37.3569	1.62421
$530$	0.231033	0.0100354
$531$	0	0
$532$	−0.143701	−0.00623024
$533$	0	0
$534$	0	0
$535$	−0.570909	−0.0246826
$536$	10.0785	0.435326
$537$	0	0
$538$	2.88714	0.124473
$539$	−4.46207	−0.192195
$540$	0	0
$541$	8.57720	0.368762	0.184381	−	0.982855i	$-0.440972\pi$
$541$	8.57720	0.368762	0.184381	+	0.982855i	$0.440972\pi$
$542$	3.18471	0.136795
$543$	0	0
$544$	28.0000	1.20049
$545$	−15.4095	−0.660069
$546$	0	0
$547$	−32.4920	−1.38926	−0.694629	−	0.719368i	$-0.744431\pi$
$547$	−32.4920	−1.38926	−0.694629	+	0.719368i	$0.744431\pi$
$548$	−6.38808	−0.272885
$549$	0	0
$550$	−0.231033	−0.00985126
$551$	0.640492	0.0272859
$552$	0	0
$553$	−6.58387	−0.279975
$554$	−0.244377	−0.0103826
$555$	0	0
$556$	−27.9059	−1.18347
$557$	−41.2150	−1.74634	−0.873169	−	0.487418i	$-0.837939\pi$
$557$	−41.2150	−1.74634	−0.873169	+	0.487418i	$0.837939\pi$
$558$	0	0
$559$	0	0
$560$	−2.19177	−0.0926192
$561$	0	0
$562$	−0.524976	−0.0221448
$563$	12.1784	0.513260	0.256630	−	0.966510i	$-0.417388\pi$
$563$	12.1784	0.513260	0.256630	+	0.966510i	$0.417388\pi$
$564$	0	0
$565$	5.32025	0.223824
$566$	6.15841	0.258857
$567$	0	0
$568$	9.65572	0.405145
$569$	−6.94338	−0.291081	−0.145541	−	0.989352i	$-0.546492\pi$
$569$	−6.94338	−0.291081	−0.145541	+	0.989352i	$0.546492\pi$
$570$	0	0
$571$	3.51429	0.147068	0.0735341	−	0.997293i	$-0.476572\pi$
$571$	3.51429	0.147068	0.0735341	+	0.997293i	$0.476572\pi$
$572$	0	0
$573$	0	0
$574$	0.947780	0.0395596
$575$	7.76897	0.323988
$576$	0	0
$577$	−3.14182	−0.130796	−0.0653978	−	0.997859i	$-0.520832\pi$
$577$	−3.14182	−0.130796	−0.0653978	+	0.997859i	$0.520832\pi$
$578$	−12.9804	−0.539912
$579$	0	0
$580$	10.4487	0.433860
$581$	−1.16774	−0.0484459
$582$	0	0
$583$	−0.462065	−0.0191368
$584$	−10.5879	−0.438130
$585$	0	0
$586$	10.3679	0.428295
$587$	37.9777	1.56751	0.783754	−	0.621071i	$-0.213302\pi$
$587$	37.9777	1.56751	0.783754	+	0.621071i	$0.213302\pi$
$588$	0	0
$589$	−1.15212	−0.0474725
$590$	−0.756009	−0.0311244
$591$	0	0
$592$	−32.4354	−1.33309
$593$	10.4487	0.429078	0.214539	−	0.976715i	$-0.431175\pi$
$593$	10.4487	0.429078	0.214539	+	0.976715i	$0.431175\pi$
$594$	0	0
$595$	4.90411	0.201049
$596$	32.2043	1.31914
$597$	0	0
$598$	0	0
$599$	45.5705	1.86196	0.930981	−	0.365069i	$-0.118955\pi$
$599$	45.5705	1.86196	0.930981	+	0.365069i	$0.118955\pi$
$600$	0	0
$601$	−14.7819	−0.602967	−0.301484	−	0.953471i	$-0.597482\pi$
$601$	−14.7819	−0.602967	−0.301484	+	0.953471i	$0.597482\pi$
$602$	−0.122188	−0.00498002
$603$	0	0
$604$	24.5227	0.997815
$605$	−10.5379	−0.428428
$606$	0	0
$607$	−18.9344	−0.768525	−0.384263	−	0.923224i	$-0.625544\pi$
$607$	−18.9344	−0.768525	−0.384263	+	0.923224i	$0.625544\pi$
$608$	0.435377	0.0176569
$609$	0	0
$610$	1.42909	0.0578622
$611$	0	0
$612$	0	0
$613$	2.58387	0.104361	0.0521807	−	0.998638i	$-0.483383\pi$
$613$	2.58387	0.104361	0.0521807	+	0.998638i	$0.483383\pi$
$614$	−1.62313	−0.0655042
$615$	0	0
$616$	−0.592422	−0.0238694
$617$	14.0393	0.565199	0.282600	−	0.959238i	$-0.408803\pi$
$617$	14.0393	0.565199	0.282600	+	0.959238i	$0.408803\pi$
$618$	0	0
$619$	17.8582	0.717781	0.358890	−	0.933380i	$-0.383155\pi$
$619$	17.8582	0.717781	0.358890	+	0.933380i	$0.383155\pi$
$620$	−18.7953	−0.754836
$621$	0	0
$622$	10.3938	0.416755
$623$	−8.94111	−0.358218
$624$	0	0
$625$	1.00000	0.0400000
$626$	9.08254	0.363011
$627$	0	0
$628$	1.46168	0.0583274
$629$	72.5745	2.89374
$630$	0	0
$631$	24.9630	0.993762	0.496881	−	0.867819i	$-0.334479\pi$
$631$	24.9630	0.993762	0.496881	+	0.867819i	$0.334479\pi$
$632$	13.1677	0.523784
$633$	0	0
$634$	7.01069	0.278430
$635$	−12.0196	−0.476984
$636$	0	0
$637$	0	0
$638$	1.28098	0.0507147
$639$	0	0
$640$	9.35951	0.369967
$641$	−23.3069	−0.920567	−0.460284	−	0.887772i	$-0.652252\pi$
$641$	−23.3069	−0.920567	−0.460284	+	0.887772i	$0.652252\pi$
$642$	0	0
$643$	−45.4790	−1.79352	−0.896759	−	0.442519i	$-0.854085\pi$
$643$	−45.4790	−1.79352	−0.896759	+	0.442519i	$0.854085\pi$
$644$	9.66453	0.380836
$645$	0	0
$646$	−0.291676	−0.0114758
$647$	−6.40946	−0.251982	−0.125991	−	0.992031i	$-0.540211\pi$
$647$	−6.40946	−0.251982	−0.125991	+	0.992031i	$0.540211\pi$
$648$	0	0
$649$	1.51202	0.0593519
$650$	0	0
$651$	0	0
$652$	22.9870	0.900242
$653$	−1.48170	−0.0579832	−0.0289916	−	0.999580i	$-0.509230\pi$
$653$	−1.48170	−0.0579832	−0.0289916	+	0.999580i	$0.509230\pi$
$654$	0	0
$655$	−10.9041	−0.426059
$656$	14.0259	0.547620
$657$	0	0
$658$	−1.12621	−0.0439041
$659$	7.08921	0.276157	0.138078	−	0.990421i	$-0.455907\pi$
$659$	7.08921	0.276157	0.138078	+	0.990421i	$0.455907\pi$
$660$	0	0
$661$	−1.17843	−0.0458355	−0.0229178	−	0.999737i	$-0.507296\pi$
$661$	−1.17843	−0.0458355	−0.0229178	+	0.999737i	$0.507296\pi$
$662$	1.82521	0.0709387
$663$	0	0
$664$	2.33547	0.0906339
$665$	0.0762550	0.00295704
$666$	0	0
$667$	−43.0759	−1.66790
$668$	2.99735	0.115971
$669$	0	0
$670$	−2.59456	−0.100237
$671$	−2.85818	−0.110339
$672$	0	0
$673$	13.5576	0.522606	0.261303	−	0.965257i	$-0.415848\pi$
$673$	13.5576	0.522606	0.261303	+	0.965257i	$0.415848\pi$
$674$	−9.77564	−0.376544
$675$	0	0
$676$	0	0
$677$	−9.53793	−0.366573	−0.183286	−	0.983060i	$-0.558674\pi$
$677$	−9.53793	−0.366573	−0.183286	+	0.983060i	$0.558674\pi$
$678$	0	0
$679$	6.53793	0.250903
$680$	−9.80823	−0.376128
$681$	0	0
$682$	−2.30425	−0.0882343
$683$	−44.5183	−1.70345	−0.851723	−	0.523993i	$-0.824442\pi$
$683$	−44.5183	−1.70345	−0.851723	+	0.523993i	$0.824442\pi$
$684$	0	0
$685$	3.38983	0.129519
$686$	−3.04328	−0.116193
$687$	0	0
$688$	−1.80823	−0.0689381
$689$	0	0
$690$	0	0
$691$	5.28060	0.200883	0.100442	−	0.994943i	$-0.467974\pi$
$691$	5.28060	0.200883	0.100442	+	0.994943i	$0.467974\pi$
$692$	24.0259	0.913328
$693$	0	0
$694$	7.61192	0.288945
$695$	14.8082	0.561708
$696$	0	0
$697$	−31.3832	−1.18872
$698$	−4.06958	−0.154036
$699$	0	0
$700$	1.24399	0.0470184
$701$	−20.1392	−0.760646	−0.380323	−	0.924854i	$-0.624187\pi$
$701$	−20.1392	−0.760646	−0.380323	+	0.924854i	$0.624187\pi$
$702$	0	0
$703$	1.12847	0.0425612
$704$	−3.64315	−0.137306
$705$	0	0
$706$	5.79035	0.217923
$707$	−6.17843	−0.232364
$708$	0	0
$709$	−1.54059	−0.0578580	−0.0289290	−	0.999581i	$-0.509210\pi$
$709$	−1.54059	−0.0578580	−0.0289290	+	0.999581i	$0.509210\pi$
$710$	−2.48571	−0.0932872
$711$	0	0
$712$	17.8822	0.670164
$713$	77.4853	2.90185
$714$	0	0
$715$	0	0
$716$	33.4728	1.25094
$717$	0	0
$718$	11.9237	0.444990
$719$	−37.0040	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$719$	−37.0040	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$720$	0	0
$721$	−2.97408	−0.110761
$722$	6.45313	0.240160
$723$	0	0
$724$	−13.2410	−0.492096
$725$	−5.54461	−0.205922
$726$	0	0
$727$	−44.9015	−1.66530	−0.832652	−	0.553797i	$-0.813178\pi$
$727$	−44.9015	−1.66530	−0.832652	+	0.553797i	$0.813178\pi$
$728$	0	0
$729$	0	0
$730$	2.72569	0.100882
$731$	4.04593	0.149644
$732$	0	0
$733$	4.94072	0.182490	0.0912449	−	0.995828i	$-0.470915\pi$
$733$	4.94072	0.182490	0.0912449	+	0.995828i	$0.470915\pi$
$734$	−6.82559	−0.251937
$735$	0	0
$736$	−29.2810	−1.07931
$737$	5.18912	0.191144
$738$	0	0
$739$	−6.03926	−0.222158	−0.111079	−	0.993812i	$-0.535431\pi$
$739$	−6.03926	−0.222158	−0.111079	+	0.993812i	$0.535431\pi$
$740$	18.4095	0.676745
$741$	0	0
$742$	−0.152510	−0.00559882
$743$	−13.0589	−0.479084	−0.239542	−	0.970886i	$-0.576997\pi$
$743$	−13.0589	−0.479084	−0.239542	+	0.970886i	$0.576997\pi$
$744$	0	0
$745$	−17.0892	−0.626100
$746$	7.97998	0.292168
$747$	0	0
$748$	9.51655	0.347960
$749$	0.376871	0.0137706
$750$	0	0
$751$	48.0236	1.75241	0.876204	−	0.481941i	$-0.160068\pi$
$751$	48.0236	1.75241	0.876204	+	0.481941i	$0.160068\pi$
$752$	−16.6664	−0.607761
$753$	0	0
$754$	0	0
$755$	−13.0130	−0.473590
$756$	0	0
$757$	−4.56424	−0.165890	−0.0829450	−	0.996554i	$-0.526433\pi$
$757$	−4.56424	−0.165890	−0.0829450	+	0.996554i	$0.526433\pi$
$758$	7.18548	0.260989
$759$	0	0
$760$	−0.152510	−0.00553212
$761$	21.0892	0.764483	0.382242	−	0.924062i	$-0.375152\pi$
$761$	21.0892	0.764483	0.382242	+	0.924062i	$0.375152\pi$
$762$	0	0
$763$	10.1721	0.368256
$764$	−26.2962	−0.951364
$765$	0	0
$766$	−0.590926	−0.0213510
$767$	0	0
$768$	0	0
$769$	−42.3332	−1.52657	−0.763287	−	0.646059i	$-0.776416\pi$
$769$	−42.3332	−1.52657	−0.763287	+	0.646059i	$0.776416\pi$
$770$	0.152510	0.00549608
$771$	0	0
$772$	44.7552	1.61078
$773$	50.2436	1.80714	0.903568	−	0.428444i	$-0.140938\pi$
$773$	50.2436	1.80714	0.903568	+	0.428444i	$0.140938\pi$
$774$	0	0
$775$	9.97370	0.358266
$776$	−13.0759	−0.469396
$777$	0	0
$778$	−9.06330	−0.324935
$779$	−0.487982	−0.0174838
$780$	0	0
$781$	4.97143	0.177892
$782$	19.6165	0.701483
$783$	0	0
$784$	−21.7949	−0.778389
$785$	−0.775639	−0.0276838
$786$	0	0
$787$	10.5816	0.377193	0.188597	−	0.982055i	$-0.439606\pi$
$787$	10.5816	0.377193	0.188597	+	0.982055i	$0.439606\pi$
$788$	−29.7903	−1.06124
$789$	0	0
$790$	−3.38983	−0.120605
$791$	−3.51202	−0.124873
$792$	0	0
$793$	0	0
$794$	4.94338	0.175434
$795$	0	0
$796$	−16.5250	−0.585712
$797$	20.1481	0.713683	0.356841	−	0.934165i	$-0.383854\pi$
$797$	20.1481	0.713683	0.356841	+	0.934165i	$0.383854\pi$
$798$	0	0
$799$	37.2913	1.31927
$800$	−3.76897	−0.133253
$801$	0	0
$802$	2.84484	0.100455
$803$	−5.45137	−0.192375
$804$	0	0
$805$	−5.12847	−0.180755
$806$	0	0
$807$	0	0
$808$	12.3569	0.434713
$809$	−12.9108	−0.453919	−0.226960	−	0.973904i	$-0.572879\pi$
$809$	−12.9108	−0.453919	−0.226960	+	0.973904i	$0.572879\pi$
$810$	0	0
$811$	15.8974	0.558235	0.279117	−	0.960257i	$-0.409958\pi$
$811$	15.8974	0.558235	0.279117	+	0.960257i	$0.409958\pi$
$812$	−6.89744	−0.242053
$813$	0	0
$814$	2.25695	0.0791061
$815$	−12.1981	−0.427279
$816$	0	0
$817$	0.0629110	0.00220098
$818$	5.86447	0.205046
$819$	0	0
$820$	−7.96074	−0.278001
$821$	−37.2284	−1.29928	−0.649640	−	0.760242i	$-0.725080\pi$
$821$	−37.2284	−1.29928	−0.649640	+	0.760242i	$0.725080\pi$
$822$	0	0
$823$	−4.89744	−0.170714	−0.0853571	−	0.996350i	$-0.527203\pi$
$823$	−4.89744	−0.170714	−0.0853571	+	0.996350i	$0.527203\pi$
$824$	5.94817	0.207214
$825$	0	0
$826$	0.499059	0.0173645
$827$	37.6334	1.30864	0.654321	−	0.756217i	$-0.272954\pi$
$827$	37.6334	1.30864	0.654321	+	0.756217i	$0.272954\pi$
$828$	0	0
$829$	−48.9104	−1.69873	−0.849364	−	0.527807i	$-0.823014\pi$
$829$	−48.9104	−1.69873	−0.849364	+	0.527807i	$0.823014\pi$
$830$	−0.601231	−0.0208690
$831$	0	0
$832$	0	0
$833$	48.7663	1.68965
$834$	0	0
$835$	−1.59054	−0.0550429
$836$	0.147975	0.00511781
$837$	0	0
$838$	4.37247	0.151044
$839$	48.2783	1.66675	0.833377	−	0.552706i	$-0.186405\pi$
$839$	48.2783	1.66675	0.833377	+	0.552706i	$0.186405\pi$
$840$	0	0
$841$	1.74266	0.0600919
$842$	8.34791	0.287688
$843$	0	0
$844$	−16.5990	−0.571360
$845$	0	0
$846$	0	0
$847$	6.95633	0.239022
$848$	−2.25695	−0.0775040
$849$	0	0
$850$	2.52498	0.0866060
$851$	−75.8948	−2.60164
$852$	0	0
$853$	14.4291	0.494043	0.247021	−	0.969010i	$-0.420548\pi$
$853$	14.4291	0.494043	0.247021	+	0.969010i	$0.420548\pi$
$854$	−0.943376	−0.0322817
$855$	0	0
$856$	−0.753741	−0.0257623
$857$	−3.64678	−0.124572	−0.0622858	−	0.998058i	$-0.519839\pi$
$857$	−3.64678	−0.124572	−0.0622858	+	0.998058i	$0.519839\pi$
$858$	0	0
$859$	21.7427	0.741850	0.370925	−	0.928663i	$-0.379041\pi$
$859$	21.7427	0.741850	0.370925	+	0.928663i	$0.379041\pi$
$860$	1.02630	0.0349966
$861$	0	0
$862$	−8.42203	−0.286856
$863$	−17.0892	−0.581724	−0.290862	−	0.956765i	$-0.593942\pi$
$863$	−17.0892	−0.581724	−0.290862	+	0.956765i	$0.593942\pi$
$864$	0	0
$865$	−12.7493	−0.433490
$866$	4.77829	0.162373
$867$	0	0
$868$	12.4072	0.421128
$869$	6.77966	0.229984
$870$	0	0
$871$	0	0
$872$	−20.3443	−0.688944
$873$	0	0
$874$	0.305020	0.0103175
$875$	−0.660123	−0.0223162
$876$	0	0
$877$	4.85818	0.164049	0.0820246	−	0.996630i	$-0.473861\pi$
$877$	4.85818	0.164049	0.0820246	+	0.996630i	$0.473861\pi$
$878$	−7.08481	−0.239101
$879$	0	0
$880$	2.25695	0.0760818
$881$	23.3202	0.785679	0.392840	−	0.919607i	$-0.371493\pi$
$881$	23.3202	0.785679	0.392840	+	0.919607i	$0.371493\pi$
$882$	0	0
$883$	−12.8934	−0.433898	−0.216949	−	0.976183i	$-0.569611\pi$
$883$	−12.8934	−0.433898	−0.216949	+	0.976183i	$0.569611\pi$
$884$	0	0
$885$	0	0
$886$	−3.88448	−0.130502
$887$	−30.9278	−1.03845	−0.519226	−	0.854637i	$-0.673780\pi$
$887$	−30.9278	−1.03845	−0.519226	+	0.854637i	$0.673780\pi$
$888$	0	0
$889$	7.93444	0.266112
$890$	−4.60350	−0.154310
$891$	0	0
$892$	18.9188	0.633449
$893$	0.579849	0.0194039
$894$	0	0
$895$	−17.7623	−0.593728
$896$	−6.17843	−0.206407
$897$	0	0
$898$	−8.80558	−0.293846
$899$	−55.3002	−1.84437
$900$	0	0
$901$	5.04995	0.168238
$902$	−0.975965	−0.0324961
$903$	0	0
$904$	7.02404	0.233616
$905$	7.02630	0.233562
$906$	0	0
$907$	−16.9604	−0.563159	−0.281580	−	0.959538i	$-0.590858\pi$
$907$	−16.9604	−0.563159	−0.281580	+	0.959538i	$0.590858\pi$
$908$	−2.28287	−0.0757596
$909$	0	0
$910$	0	0
$911$	−37.7297	−1.25004	−0.625020	−	0.780608i	$-0.714909\pi$
$911$	−37.7297	−1.25004	−0.625020	+	0.780608i	$0.714909\pi$
$912$	0	0
$913$	1.20246	0.0397957
$914$	2.00667	0.0663748
$915$	0	0
$916$	36.2650	1.19823
$917$	7.19806	0.237701
$918$	0	0
$919$	40.1628	1.32485	0.662425	−	0.749129i	$-0.269528\pi$
$919$	40.1628	1.32485	0.662425	+	0.749129i	$0.269528\pi$
$920$	10.2569	0.338162
$921$	0	0
$922$	5.26537	0.173406
$923$	0	0
$924$	0	0
$925$	−9.76897	−0.321202
$926$	5.92815	0.194811
$927$	0	0
$928$	20.8974	0.685992
$929$	23.7690	0.779835	0.389917	−	0.920850i	$-0.372504\pi$
$929$	23.7690	0.779835	0.389917	+	0.920850i	$0.372504\pi$
$930$	0	0
$931$	0.758276	0.0248515
$932$	−45.0545	−1.47581
$933$	0	0
$934$	−6.96035	−0.227750
$935$	−5.04995	−0.165151
$936$	0	0
$937$	−7.43803	−0.242990	−0.121495	−	0.992592i	$-0.538769\pi$
$937$	−7.43803	−0.242990	−0.121495	+	0.992592i	$0.538769\pi$
$938$	1.71273	0.0559226
$939$	0	0
$940$	9.45941	0.308532
$941$	−19.3528	−0.630884	−0.315442	−	0.948945i	$-0.602153\pi$
$941$	−19.3528	−0.630884	−0.315442	+	0.948945i	$0.602153\pi$
$942$	0	0
$943$	32.8189	1.06873
$944$	7.38542	0.240375
$945$	0	0
$946$	0.125822	0.00409082
$947$	13.9171	0.452244	0.226122	−	0.974099i	$-0.427395\pi$
$947$	13.9171	0.452244	0.226122	+	0.974099i	$0.427395\pi$
$948$	0	0
$949$	0	0
$950$	0.0392613	0.00127381
$951$	0	0
$952$	6.47464	0.209844
$953$	10.6271	0.344247	0.172124	−	0.985075i	$-0.444937\pi$
$953$	10.6271	0.344247	0.172124	+	0.985075i	$0.444937\pi$
$954$	0	0
$955$	13.9541	0.451543
$956$	−35.2017	−1.13850
$957$	0	0
$958$	13.9318	0.450115
$959$	−2.23770	−0.0722593
$960$	0	0
$961$	68.4746	2.20886
$962$	0	0
$963$	0	0
$964$	11.5246	0.371182
$965$	−23.7493	−0.764518
$966$	0	0
$967$	−51.6771	−1.66182	−0.830912	−	0.556404i	$-0.812181\pi$
$967$	−51.6771	−1.66182	−0.830912	+	0.556404i	$0.812181\pi$
$968$	−13.9127	−0.447170
$969$	0	0
$970$	3.36618	0.108082
$971$	42.8974	1.37664	0.688322	−	0.725405i	$-0.258348\pi$
$971$	42.8974	1.37664	0.688322	+	0.725405i	$0.258348\pi$
$972$	0	0
$973$	−9.77525	−0.313380
$974$	8.17843	0.262054
$975$	0	0
$976$	−13.9607	−0.446872
$977$	18.1651	0.581153	0.290576	−	0.956852i	$-0.406153\pi$
$977$	18.1651	0.581153	0.290576	+	0.956852i	$0.406153\pi$
$978$	0	0
$979$	9.20700	0.294257
$980$	12.3702	0.395151
$981$	0	0
$982$	12.4983	0.398836
$983$	−10.8582	−0.346322	−0.173161	−	0.984894i	$-0.555398\pi$
$983$	−10.8582	−0.346322	−0.173161	+	0.984894i	$0.555398\pi$
$984$	0	0
$985$	15.8082	0.503692
$986$	−14.0000	−0.445851
$987$	0	0
$988$	0	0
$989$	−4.23103	−0.134539
$990$	0	0
$991$	−22.3725	−0.710685	−0.355342	−	0.934736i	$-0.615636\pi$
$991$	−22.3725	−0.710685	−0.355342	+	0.934736i	$0.615636\pi$
$992$	−37.5905	−1.19350
$993$	0	0
$994$	1.64088	0.0520455
$995$	8.76897	0.277995
$996$	0	0
$997$	−24.2373	−0.767604	−0.383802	−	0.923415i	$-0.625385\pi$
$997$	−24.2373	−0.767604	−0.383802	+	0.923415i	$0.625385\pi$
$998$	11.8349	0.374628
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.bw.1.2		3
3.2	odd	2		2535.2.a.ba.1.2		3
13.4	even	6		585.2.j.f.406.2		6
13.10	even	6		585.2.j.f.451.2		6
13.12	even	2		7605.2.a.bv.1.2		3
39.17	odd	6		195.2.i.d.16.2	✓	6
39.23	odd	6		195.2.i.d.61.2	yes	6
39.38	odd	2		2535.2.a.bb.1.2		3
195.17	even	12		975.2.bb.k.874.3		12
195.23	even	12		975.2.bb.k.724.3		12
195.62	even	12		975.2.bb.k.724.4		12
195.134	odd	6		975.2.i.l.601.2		6
195.173	even	12		975.2.bb.k.874.4		12
195.179	odd	6		975.2.i.l.451.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.i.d.16.2	✓	6	39.17	odd	6
195.2.i.d.61.2	yes	6	39.23	odd	6
585.2.j.f.406.2		6	13.4	even	6
585.2.j.f.451.2		6	13.10	even	6
975.2.i.l.451.2		6	195.179	odd	6
975.2.i.l.601.2		6	195.134	odd	6
975.2.bb.k.724.3		12	195.23	even	12
975.2.bb.k.724.4		12	195.62	even	12
975.2.bb.k.874.3		12	195.17	even	12
975.2.bb.k.874.4		12	195.173	even	12
2535.2.a.ba.1.2		3	3.2	odd	2
2535.2.a.bb.1.2		3	39.38	odd	2
7605.2.a.bv.1.2		3	13.12	even	2
7605.2.a.bw.1.2		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.339877 q^{2} -1.88448 q^{4} +1.00000 q^{5} -0.660123 q^{7} +1.32025 q^{8} -0.339877 q^{10} +0.679754 q^{11} +0.224361 q^{14} +3.32025 q^{16} -7.42909 q^{17} -0.115516 q^{19} -1.88448 q^{20} -0.231033 q^{22} +7.76897 q^{23} +1.00000 q^{25} +1.24399 q^{28} -5.54461 q^{29} +9.97370 q^{31} -3.76897 q^{32} +2.52498 q^{34} -0.660123 q^{35} -9.76897 q^{37} +0.0392613 q^{38} +1.32025 q^{40} +4.22436 q^{41} -0.544607 q^{43} -1.28098 q^{44} -2.64049 q^{46} -5.01963 q^{47} -6.56424 q^{49} -0.339877 q^{50} -0.679754 q^{53} +0.679754 q^{55} -0.871525 q^{56} +1.88448 q^{58} +2.22436 q^{59} -4.20473 q^{61} -3.38983 q^{62} -5.35951 q^{64} +7.63382 q^{67} +14.0000 q^{68} +0.224361 q^{70} +7.31357 q^{71} -8.01963 q^{73} +3.32025 q^{74} +0.217689 q^{76} -0.448721 q^{77} +9.97370 q^{79} +3.32025 q^{80} -1.43576 q^{82} +1.76897 q^{83} -7.42909 q^{85} +0.185099 q^{86} +0.897442 q^{88} +13.5446 q^{89} -14.6405 q^{92} +1.70606 q^{94} -0.115516 q^{95} -9.90411 q^{97} +2.23103 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} - 12 q^{14} + 12 q^{16} - 12 q^{19} + 6 q^{20} - 24 q^{22} + 3 q^{25} - 12 q^{28} - 6 q^{29} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 6 q^{37} - 6 q^{38} + 6 q^{40}+ \cdots + 30 q^{98}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 - 12 * q^14 + 12 * q^16 - 12 * q^19 + 6 * q^20 - 24 * q^22 + 3 * q^25 - 12 * q^28 - 6 * q^29 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 6 * q^37 - 6 * q^38 + 6 * q^40 + 9 * q^43 - 12 * q^44 - 12 * q^46 - 12 * q^47 - 6 * q^49 - 30 * q^56 - 6 * q^58 - 6 * q^59 - 3 * q^61 + 6 * q^62 - 12 * q^64 - 9 * q^67 + 42 * q^68 - 12 * q^70 - 12 * q^71 - 21 * q^73 + 12 * q^74 - 48 * q^76 + 24 * q^77 - 3 * q^79 + 12 * q^80 - 18 * q^82 - 18 * q^83 - 6 * q^86 - 48 * q^88 + 30 * q^89 - 48 * q^92 + 36 * q^94 - 12 * q^95 - 15 * q^97 + 30 * q^98

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