LMFDB - Embedded newform 3120.2.a.bj.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.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:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -4.91638 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -4.91638 q^{7} +1.00000 q^{9} -4.91638 q^{11} +1.00000 q^{13} -1.00000 q^{15} -4.33804 q^{17} -2.57834 q^{19} -4.91638 q^{21} +6.33804 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -1.42166 q^{31} -4.91638 q^{33} +4.91638 q^{35} +9.49472 q^{37} +1.00000 q^{39} +4.33804 q^{41} +1.15667 q^{43} -1.00000 q^{45} +5.42166 q^{47} +17.1708 q^{49} -4.33804 q^{51} -0.338044 q^{53} +4.91638 q^{55} -2.57834 q^{57} +11.2544 q^{59} -10.1708 q^{61} -4.91638 q^{63} -1.00000 q^{65} +7.25443 q^{67} +6.33804 q^{69} -0.916382 q^{71} -3.15667 q^{73} +1.00000 q^{75} +24.1708 q^{77} +3.49472 q^{79} +1.00000 q^{81} -11.2544 q^{83} +4.33804 q^{85} +6.00000 q^{87} -0.338044 q^{89} -4.91638 q^{91} -1.42166 q^{93} +2.57834 q^{95} -12.3380 q^{97} -4.91638 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - 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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.91638	−1.85822	−0.929109	−	0.369807i	$-0.879424\pi$
$7$	−4.91638	−1.85822	−0.929109	+	0.369807i	$0.879424\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.91638	−1.48234	−0.741172	−	0.671315i	$-0.765730\pi$
$11$	−4.91638	−1.48234	−0.741172	+	0.671315i	$0.765730\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−4.33804	−1.05213	−0.526065	−	0.850444i	$-0.676333\pi$
$17$	−4.33804	−1.05213	−0.526065	+	0.850444i	$0.676333\pi$
$18$	0	0
$19$	−2.57834	−0.591511	−0.295756	−	0.955264i	$-0.595571\pi$
$19$	−2.57834	−0.591511	−0.295756	+	0.955264i	$0.595571\pi$
$20$	0	0
$21$	−4.91638	−1.07284
$22$	0	0
$23$	6.33804	1.32157	0.660787	−	0.750574i	$-0.270223\pi$
$23$	6.33804	1.32157	0.660787	+	0.750574i	$0.270223\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−1.42166	−0.255338	−0.127669	−	0.991817i	$-0.540750\pi$
$31$	−1.42166	−0.255338	−0.127669	+	0.991817i	$0.540750\pi$
$32$	0	0
$33$	−4.91638	−0.855832
$34$	0	0
$35$	4.91638	0.831020
$36$	0	0
$37$	9.49472	1.56092	0.780461	−	0.625204i	$-0.214984\pi$
$37$	9.49472	1.56092	0.780461	+	0.625204i	$0.214984\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	4.33804	0.677489	0.338744	−	0.940878i	$-0.389998\pi$
$41$	4.33804	0.677489	0.338744	+	0.940878i	$0.389998\pi$
$42$	0	0
$43$	1.15667	0.176391	0.0881956	−	0.996103i	$-0.471890\pi$
$43$	1.15667	0.176391	0.0881956	+	0.996103i	$0.471890\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	5.42166	0.790831	0.395415	−	0.918502i	$-0.370601\pi$
$47$	5.42166	0.790831	0.395415	+	0.918502i	$0.370601\pi$
$48$	0	0
$49$	17.1708	2.45297
$50$	0	0
$51$	−4.33804	−0.607448
$52$	0	0
$53$	−0.338044	−0.0464340	−0.0232170	−	0.999730i	$-0.507391\pi$
$53$	−0.338044	−0.0464340	−0.0232170	+	0.999730i	$0.507391\pi$
$54$	0	0
$55$	4.91638	0.662925
$56$	0	0
$57$	−2.57834	−0.341509
$58$	0	0
$59$	11.2544	1.46520	0.732601	−	0.680659i	$-0.238306\pi$
$59$	11.2544	1.46520	0.732601	+	0.680659i	$0.238306\pi$
$60$	0	0
$61$	−10.1708	−1.30224	−0.651119	−	0.758975i	$-0.725700\pi$
$61$	−10.1708	−1.30224	−0.651119	+	0.758975i	$0.725700\pi$
$62$	0	0
$63$	−4.91638	−0.619406
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	7.25443	0.886269	0.443135	−	0.896455i	$-0.353866\pi$
$67$	7.25443	0.886269	0.443135	+	0.896455i	$0.353866\pi$
$68$	0	0
$69$	6.33804	0.763011
$70$	0	0
$71$	−0.916382	−0.108754	−0.0543772	−	0.998520i	$-0.517317\pi$
$71$	−0.916382	−0.108754	−0.0543772	+	0.998520i	$0.517317\pi$
$72$	0	0
$73$	−3.15667	−0.369461	−0.184730	−	0.982789i	$-0.559141\pi$
$73$	−3.15667	−0.369461	−0.184730	+	0.982789i	$0.559141\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	24.1708	2.75452
$78$	0	0
$79$	3.49472	0.393187	0.196593	−	0.980485i	$-0.437012\pi$
$79$	3.49472	0.393187	0.196593	+	0.980485i	$0.437012\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2544	−1.23533	−0.617667	−	0.786440i	$-0.711922\pi$
$83$	−11.2544	−1.23533	−0.617667	+	0.786440i	$0.711922\pi$
$84$	0	0
$85$	4.33804	0.470527
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	−0.338044	−0.0358326	−0.0179163	−	0.999839i	$-0.505703\pi$
$89$	−0.338044	−0.0358326	−0.0179163	+	0.999839i	$0.505703\pi$
$90$	0	0
$91$	−4.91638	−0.515377
$92$	0	0
$93$	−1.42166	−0.147420
$94$	0	0
$95$	2.57834	0.264532
$96$	0	0
$97$	−12.3380	−1.25274	−0.626369	−	0.779526i	$-0.715460\pi$
$97$	−12.3380	−1.25274	−0.626369	+	0.779526i	$0.715460\pi$
$98$	0	0
$99$	−4.91638	−0.494115
$100$	0	0
$101$	10.6761	1.06231	0.531155	−	0.847274i	$-0.321758\pi$
$101$	10.6761	1.06231	0.531155	+	0.847274i	$0.321758\pi$
$102$	0	0
$103$	14.5089	1.42960	0.714800	−	0.699329i	$-0.246518\pi$
$103$	14.5089	1.42960	0.714800	+	0.699329i	$0.246518\pi$
$104$	0	0
$105$	4.91638	0.479790
$106$	0	0
$107$	−4.17081	−0.403207	−0.201604	−	0.979467i	$-0.564615\pi$
$107$	−4.17081	−0.403207	−0.201604	+	0.979467i	$0.564615\pi$
$108$	0	0
$109$	−3.83276	−0.367112	−0.183556	−	0.983009i	$-0.558761\pi$
$109$	−3.83276	−0.367112	−0.183556	+	0.983009i	$0.558761\pi$
$110$	0	0
$111$	9.49472	0.901199
$112$	0	0
$113$	−0.843326	−0.0793334	−0.0396667	−	0.999213i	$-0.512630\pi$
$113$	−0.843326	−0.0793334	−0.0396667	+	0.999213i	$0.512630\pi$
$114$	0	0
$115$	−6.33804	−0.591026
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	21.3275	1.95509
$120$	0	0
$121$	13.1708	1.19735
$122$	0	0
$123$	4.33804	0.391148
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.83276	−0.162631	−0.0813157	−	0.996688i	$-0.525912\pi$
$127$	−1.83276	−0.162631	−0.0813157	+	0.996688i	$0.525912\pi$
$128$	0	0
$129$	1.15667	0.101839
$130$	0	0
$131$	−5.83276	−0.509611	−0.254805	−	0.966992i	$-0.582011\pi$
$131$	−5.83276	−0.509611	−0.254805	+	0.966992i	$0.582011\pi$
$132$	0	0
$133$	12.6761	1.09916
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−16.5089	−1.41045	−0.705223	−	0.708985i	$-0.749153\pi$
$137$	−16.5089	−1.41045	−0.705223	+	0.708985i	$0.749153\pi$
$138$	0	0
$139$	−7.49472	−0.635694	−0.317847	−	0.948142i	$-0.602960\pi$
$139$	−7.49472	−0.635694	−0.317847	+	0.948142i	$0.602960\pi$
$140$	0	0
$141$	5.42166	0.456586
$142$	0	0
$143$	−4.91638	−0.411128
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	17.1708	1.41622
$148$	0	0
$149$	20.4842	1.67813	0.839064	−	0.544033i	$-0.183103\pi$
$149$	20.4842	1.67813	0.839064	+	0.544033i	$0.183103\pi$
$150$	0	0
$151$	16.4111	1.33552	0.667758	−	0.744378i	$-0.267254\pi$
$151$	16.4111	1.33552	0.667758	+	0.744378i	$0.267254\pi$
$152$	0	0
$153$	−4.33804	−0.350710
$154$	0	0
$155$	1.42166	0.114191
$156$	0	0
$157$	−21.6655	−1.72910	−0.864549	−	0.502549i	$-0.832396\pi$
$157$	−21.6655	−1.72910	−0.864549	+	0.502549i	$0.832396\pi$
$158$	0	0
$159$	−0.338044	−0.0268087
$160$	0	0
$161$	−31.1602	−2.45577
$162$	0	0
$163$	−6.07306	−0.475678	−0.237839	−	0.971305i	$-0.576439\pi$
$163$	−6.07306	−0.475678	−0.237839	+	0.971305i	$0.576439\pi$
$164$	0	0
$165$	4.91638	0.382740
$166$	0	0
$167$	0.745574	0.0576942	0.0288471	−	0.999584i	$-0.490816\pi$
$167$	0.745574	0.0576942	0.0288471	+	0.999584i	$0.490816\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.57834	−0.197170
$172$	0	0
$173$	0.843326	0.0641169	0.0320584	−	0.999486i	$-0.489794\pi$
$173$	0.843326	0.0641169	0.0320584	+	0.999486i	$0.489794\pi$
$174$	0	0
$175$	−4.91638	−0.371644
$176$	0	0
$177$	11.2544	0.845934
$178$	0	0
$179$	18.9894	1.41934	0.709669	−	0.704536i	$-0.248845\pi$
$179$	18.9894	1.41934	0.709669	+	0.704536i	$0.248845\pi$
$180$	0	0
$181$	17.4947	1.30037	0.650186	−	0.759775i	$-0.274691\pi$
$181$	17.4947	1.30037	0.650186	+	0.759775i	$0.274691\pi$
$182$	0	0
$183$	−10.1708	−0.751848
$184$	0	0
$185$	−9.49472	−0.698066
$186$	0	0
$187$	21.3275	1.55962
$188$	0	0
$189$	−4.91638	−0.357614
$190$	0	0
$191$	22.5089	1.62868	0.814342	−	0.580386i	$-0.197098\pi$
$191$	22.5089	1.62868	0.814342	+	0.580386i	$0.197098\pi$
$192$	0	0
$193$	2.65139	0.190851	0.0954257	−	0.995437i	$-0.469579\pi$
$193$	2.65139	0.190851	0.0954257	+	0.995437i	$0.469579\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−12.9894	−0.925459	−0.462730	−	0.886500i	$-0.653130\pi$
$197$	−12.9894	−0.925459	−0.462730	+	0.886500i	$0.653130\pi$
$198$	0	0
$199$	2.84333	0.201558	0.100779	−	0.994909i	$-0.467866\pi$
$199$	2.84333	0.201558	0.100779	+	0.994909i	$0.467866\pi$
$200$	0	0
$201$	7.25443	0.511688
$202$	0	0
$203$	−29.4983	−2.07037
$204$	0	0
$205$	−4.33804	−0.302982
$206$	0	0
$207$	6.33804	0.440525
$208$	0	0
$209$	12.6761	0.876823
$210$	0	0
$211$	−6.31335	−0.434629	−0.217314	−	0.976102i	$-0.569730\pi$
$211$	−6.31335	−0.434629	−0.217314	+	0.976102i	$0.569730\pi$
$212$	0	0
$213$	−0.916382	−0.0627894
$214$	0	0
$215$	−1.15667	−0.0788845
$216$	0	0
$217$	6.98944	0.474474
$218$	0	0
$219$	−3.15667	−0.213308
$220$	0	0
$221$	−4.33804	−0.291808
$222$	0	0
$223$	−19.2544	−1.28937	−0.644686	−	0.764448i	$-0.723012\pi$
$223$	−19.2544	−1.28937	−0.644686	+	0.764448i	$0.723012\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	13.0872	0.868627	0.434314	−	0.900762i	$-0.356991\pi$
$227$	13.0872	0.868627	0.434314	+	0.900762i	$0.356991\pi$
$228$	0	0
$229$	−24.5089	−1.61959	−0.809795	−	0.586713i	$-0.800422\pi$
$229$	−24.5089	−1.61959	−0.809795	+	0.586713i	$0.800422\pi$
$230$	0	0
$231$	24.1708	1.59032
$232$	0	0
$233$	8.33804	0.546243	0.273122	−	0.961979i	$-0.411944\pi$
$233$	8.33804	0.546243	0.273122	+	0.961979i	$0.411944\pi$
$234$	0	0
$235$	−5.42166	−0.353670
$236$	0	0
$237$	3.49472	0.227006
$238$	0	0
$239$	−8.91638	−0.576753	−0.288376	−	0.957517i	$-0.593115\pi$
$239$	−8.91638	−0.576753	−0.288376	+	0.957517i	$0.593115\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−17.1708	−1.09700
$246$	0	0
$247$	−2.57834	−0.164056
$248$	0	0
$249$	−11.2544	−0.713220
$250$	0	0
$251$	6.31335	0.398495	0.199248	−	0.979949i	$-0.436150\pi$
$251$	6.31335	0.398495	0.199248	+	0.979949i	$0.436150\pi$
$252$	0	0
$253$	−31.1602	−1.95903
$254$	0	0
$255$	4.33804	0.271659
$256$	0	0
$257$	−11.1567	−0.695934	−0.347967	−	0.937507i	$-0.613128\pi$
$257$	−11.1567	−0.695934	−0.347967	+	0.937507i	$0.613128\pi$
$258$	0	0
$259$	−46.6797	−2.90053
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0.338044	0.0207659
$266$	0	0
$267$	−0.338044	−0.0206880
$268$	0	0
$269$	18.6761	1.13870	0.569351	−	0.822095i	$-0.307195\pi$
$269$	18.6761	1.13870	0.569351	+	0.822095i	$0.307195\pi$
$270$	0	0
$271$	−6.57834	−0.399606	−0.199803	−	0.979836i	$-0.564030\pi$
$271$	−6.57834	−0.399606	−0.199803	+	0.979836i	$0.564030\pi$
$272$	0	0
$273$	−4.91638	−0.297553
$274$	0	0
$275$	−4.91638	−0.296469
$276$	0	0
$277$	25.6655	1.54209	0.771046	−	0.636779i	$-0.219734\pi$
$277$	25.6655	1.54209	0.771046	+	0.636779i	$0.219734\pi$
$278$	0	0
$279$	−1.42166	−0.0851127
$280$	0	0
$281$	3.15667	0.188311	0.0941557	−	0.995557i	$-0.469985\pi$
$281$	3.15667	0.188311	0.0941557	+	0.995557i	$0.469985\pi$
$282$	0	0
$283$	3.47002	0.206271	0.103136	−	0.994667i	$-0.467112\pi$
$283$	3.47002	0.206271	0.103136	+	0.994667i	$0.467112\pi$
$284$	0	0
$285$	2.57834	0.152728
$286$	0	0
$287$	−21.3275	−1.25892
$288$	0	0
$289$	1.81863	0.106978
$290$	0	0
$291$	−12.3380	−0.723269
$292$	0	0
$293$	28.6550	1.67404	0.837020	−	0.547172i	$-0.184296\pi$
$293$	28.6550	1.67404	0.837020	+	0.547172i	$0.184296\pi$
$294$	0	0
$295$	−11.2544	−0.655258
$296$	0	0
$297$	−4.91638	−0.285277
$298$	0	0
$299$	6.33804	0.366539
$300$	0	0
$301$	−5.68665	−0.327773
$302$	0	0
$303$	10.6761	0.613325
$304$	0	0
$305$	10.1708	0.582379
$306$	0	0
$307$	1.92694	0.109977	0.0549883	−	0.998487i	$-0.482488\pi$
$307$	1.92694	0.109977	0.0549883	+	0.998487i	$0.482488\pi$
$308$	0	0
$309$	14.5089	0.825380
$310$	0	0
$311$	29.9789	1.69995	0.849973	−	0.526826i	$-0.176618\pi$
$311$	29.9789	1.69995	0.849973	+	0.526826i	$0.176618\pi$
$312$	0	0
$313$	−16.3133	−0.922085	−0.461042	−	0.887378i	$-0.652524\pi$
$313$	−16.3133	−0.922085	−0.461042	+	0.887378i	$0.652524\pi$
$314$	0	0
$315$	4.91638	0.277007
$316$	0	0
$317$	30.6761	1.72294	0.861470	−	0.507808i	$-0.169544\pi$
$317$	30.6761	1.72294	0.861470	+	0.507808i	$0.169544\pi$
$318$	0	0
$319$	−29.4983	−1.65159
$320$	0	0
$321$	−4.17081	−0.232792
$322$	0	0
$323$	11.1849	0.622347
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−3.83276	−0.211952
$328$	0	0
$329$	−26.6550	−1.46954
$330$	0	0
$331$	10.0978	0.555023	0.277511	−	0.960722i	$-0.410490\pi$
$331$	10.0978	0.555023	0.277511	+	0.960722i	$0.410490\pi$
$332$	0	0
$333$	9.49472	0.520307
$334$	0	0
$335$	−7.25443	−0.396352
$336$	0	0
$337$	−1.32391	−0.0721180	−0.0360590	−	0.999350i	$-0.511480\pi$
$337$	−1.32391	−0.0721180	−0.0360590	+	0.999350i	$0.511480\pi$
$338$	0	0
$339$	−0.843326	−0.0458032
$340$	0	0
$341$	6.98944	0.378499
$342$	0	0
$343$	−50.0036	−2.69994
$344$	0	0
$345$	−6.33804	−0.341229
$346$	0	0
$347$	7.49472	0.402338	0.201169	−	0.979557i	$-0.435526\pi$
$347$	7.49472	0.402338	0.201169	+	0.979557i	$0.435526\pi$
$348$	0	0
$349$	−22.1461	−1.18545	−0.592727	−	0.805403i	$-0.701949\pi$
$349$	−22.1461	−1.18545	−0.592727	+	0.805403i	$0.701949\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	4.50885	0.239982	0.119991	−	0.992775i	$-0.461713\pi$
$353$	4.50885	0.239982	0.119991	+	0.992775i	$0.461713\pi$
$354$	0	0
$355$	0.916382	0.0486365
$356$	0	0
$357$	21.3275	1.12877
$358$	0	0
$359$	20.4111	1.07726	0.538628	−	0.842543i	$-0.318943\pi$
$359$	20.4111	1.07726	0.538628	+	0.842543i	$0.318943\pi$
$360$	0	0
$361$	−12.3522	−0.650115
$362$	0	0
$363$	13.1708	0.691288
$364$	0	0
$365$	3.15667	0.165228
$366$	0	0
$367$	10.3133	0.538352	0.269176	−	0.963091i	$-0.413249\pi$
$367$	10.3133	0.538352	0.269176	+	0.963091i	$0.413249\pi$
$368$	0	0
$369$	4.33804	0.225830
$370$	0	0
$371$	1.66196	0.0862844
$372$	0	0
$373$	18.6761	0.967011	0.483506	−	0.875341i	$-0.339363\pi$
$373$	18.6761	0.967011	0.483506	+	0.875341i	$0.339363\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	28.7527	1.47693	0.738464	−	0.674293i	$-0.235552\pi$
$379$	28.7527	1.47693	0.738464	+	0.674293i	$0.235552\pi$
$380$	0	0
$381$	−1.83276	−0.0938953
$382$	0	0
$383$	14.2439	0.727827	0.363914	−	0.931433i	$-0.381440\pi$
$383$	14.2439	0.727827	0.363914	+	0.931433i	$0.381440\pi$
$384$	0	0
$385$	−24.1708	−1.23186
$386$	0	0
$387$	1.15667	0.0587971
$388$	0	0
$389$	34.6761	1.75815	0.879074	−	0.476686i	$-0.158162\pi$
$389$	34.6761	1.75815	0.879074	+	0.476686i	$0.158162\pi$
$390$	0	0
$391$	−27.4947	−1.39047
$392$	0	0
$393$	−5.83276	−0.294224
$394$	0	0
$395$	−3.49472	−0.175838
$396$	0	0
$397$	7.18137	0.360423	0.180211	−	0.983628i	$-0.442322\pi$
$397$	7.18137	0.360423	0.180211	+	0.983628i	$0.442322\pi$
$398$	0	0
$399$	12.6761	0.634598
$400$	0	0
$401$	37.8610	1.89069	0.945345	−	0.326072i	$-0.105725\pi$
$401$	37.8610	1.89069	0.945345	+	0.326072i	$0.105725\pi$
$402$	0	0
$403$	−1.42166	−0.0708181
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−46.6797	−2.31382
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	−16.5089	−0.814322
$412$	0	0
$413$	−55.3311	−2.72266
$414$	0	0
$415$	11.2544	0.552458
$416$	0	0
$417$	−7.49472	−0.367018
$418$	0	0
$419$	33.4983	1.63650	0.818249	−	0.574864i	$-0.194945\pi$
$419$	33.4983	1.63650	0.818249	+	0.574864i	$0.194945\pi$
$420$	0	0
$421$	13.5194	0.658896	0.329448	−	0.944174i	$-0.393137\pi$
$421$	13.5194	0.658896	0.329448	+	0.944174i	$0.393137\pi$
$422$	0	0
$423$	5.42166	0.263610
$424$	0	0
$425$	−4.33804	−0.210426
$426$	0	0
$427$	50.0036	2.41984
$428$	0	0
$429$	−4.91638	−0.237365
$430$	0	0
$431$	−12.4111	−0.597822	−0.298911	−	0.954281i	$-0.596623\pi$
$431$	−12.4111	−0.597822	−0.298911	+	0.954281i	$0.596623\pi$
$432$	0	0
$433$	−17.3239	−0.832534	−0.416267	−	0.909242i	$-0.636662\pi$
$433$	−17.3239	−0.832534	−0.416267	+	0.909242i	$0.636662\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	−16.3416	−0.781725
$438$	0	0
$439$	−0.651393	−0.0310893	−0.0155446	−	0.999879i	$-0.504948\pi$
$439$	−0.651393	−0.0310893	−0.0155446	+	0.999879i	$0.504948\pi$
$440$	0	0
$441$	17.1708	0.817658
$442$	0	0
$443$	8.84690	0.420329	0.210164	−	0.977666i	$-0.432600\pi$
$443$	8.84690	0.420329	0.210164	+	0.977666i	$0.432600\pi$
$444$	0	0
$445$	0.338044	0.0160248
$446$	0	0
$447$	20.4842	0.968867
$448$	0	0
$449$	4.33804	0.204725	0.102362	−	0.994747i	$-0.467360\pi$
$449$	4.33804	0.204725	0.102362	+	0.994747i	$0.467360\pi$
$450$	0	0
$451$	−21.3275	−1.00427
$452$	0	0
$453$	16.4111	0.771061
$454$	0	0
$455$	4.91638	0.230484
$456$	0	0
$457$	15.3275	0.716989	0.358495	−	0.933532i	$-0.383290\pi$
$457$	15.3275	0.716989	0.358495	+	0.933532i	$0.383290\pi$
$458$	0	0
$459$	−4.33804	−0.202483
$460$	0	0
$461$	−11.8575	−0.552257	−0.276128	−	0.961121i	$-0.589052\pi$
$461$	−11.8575	−0.552257	−0.276128	+	0.961121i	$0.589052\pi$
$462$	0	0
$463$	26.4147	1.22759	0.613797	−	0.789464i	$-0.289641\pi$
$463$	26.4147	1.22759	0.613797	+	0.789464i	$0.289641\pi$
$464$	0	0
$465$	1.42166	0.0659280
$466$	0	0
$467$	33.6691	1.55802	0.779010	−	0.627012i	$-0.215722\pi$
$467$	33.6691	1.55802	0.779010	+	0.627012i	$0.215722\pi$
$468$	0	0
$469$	−35.6655	−1.64688
$470$	0	0
$471$	−21.6655	−0.998295
$472$	0	0
$473$	−5.68665	−0.261473
$474$	0	0
$475$	−2.57834	−0.118302
$476$	0	0
$477$	−0.338044	−0.0154780
$478$	0	0
$479$	−10.7491	−0.491141	−0.245570	−	0.969379i	$-0.578975\pi$
$479$	−10.7491	−0.491141	−0.245570	+	0.969379i	$0.578975\pi$
$480$	0	0
$481$	9.49472	0.432922
$482$	0	0
$483$	−31.1602	−1.41784
$484$	0	0
$485$	12.3380	0.560242
$486$	0	0
$487$	−22.7491	−1.03086	−0.515431	−	0.856931i	$-0.672368\pi$
$487$	−22.7491	−1.03086	−0.515431	+	0.856931i	$0.672368\pi$
$488$	0	0
$489$	−6.07306	−0.274633
$490$	0	0
$491$	17.6867	0.798187	0.399094	−	0.916910i	$-0.369325\pi$
$491$	17.6867	0.798187	0.399094	+	0.916910i	$0.369325\pi$
$492$	0	0
$493$	−26.0283	−1.17225
$494$	0	0
$495$	4.91638	0.220975
$496$	0	0
$497$	4.50528	0.202089
$498$	0	0
$499$	−19.9305	−0.892212	−0.446106	−	0.894980i	$-0.647190\pi$
$499$	−19.9305	−0.892212	−0.446106	+	0.894980i	$0.647190\pi$
$500$	0	0
$501$	0.745574	0.0333098
$502$	0	0
$503$	−33.3522	−1.48710	−0.743550	−	0.668680i	$-0.766860\pi$
$503$	−33.3522	−1.48710	−0.743550	+	0.668680i	$0.766860\pi$
$504$	0	0
$505$	−10.6761	−0.475080
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	13.8363	0.613285	0.306642	−	0.951825i	$-0.400794\pi$
$509$	13.8363	0.613285	0.306642	+	0.951825i	$0.400794\pi$
$510$	0	0
$511$	15.5194	0.686538
$512$	0	0
$513$	−2.57834	−0.113836
$514$	0	0
$515$	−14.5089	−0.639336
$516$	0	0
$517$	−26.6550	−1.17228
$518$	0	0
$519$	0.843326	0.0370179
$520$	0	0
$521$	−23.3522	−1.02308	−0.511539	−	0.859260i	$-0.670924\pi$
$521$	−23.3522	−1.02308	−0.511539	+	0.859260i	$0.670924\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	−4.91638	−0.214568
$526$	0	0
$527$	6.16724	0.268649
$528$	0	0
$529$	17.1708	0.746557
$530$	0	0
$531$	11.2544	0.488400
$532$	0	0
$533$	4.33804	0.187902
$534$	0	0
$535$	4.17081	0.180320
$536$	0	0
$537$	18.9894	0.819455
$538$	0	0
$539$	−84.4182	−3.63615
$540$	0	0
$541$	16.1744	0.695391	0.347695	−	0.937608i	$-0.386964\pi$
$541$	16.1744	0.695391	0.347695	+	0.937608i	$0.386964\pi$
$542$	0	0
$543$	17.4947	0.750770
$544$	0	0
$545$	3.83276	0.164178
$546$	0	0
$547$	−9.68665	−0.414171	−0.207086	−	0.978323i	$-0.566398\pi$
$547$	−9.68665	−0.414171	−0.207086	+	0.978323i	$0.566398\pi$
$548$	0	0
$549$	−10.1708	−0.434079
$550$	0	0
$551$	−15.4700	−0.659045
$552$	0	0
$553$	−17.1814	−0.730626
$554$	0	0
$555$	−9.49472	−0.403028
$556$	0	0
$557$	0.647822	0.0274491	0.0137246	−	0.999906i	$-0.495631\pi$
$557$	0.647822	0.0274491	0.0137246	+	0.999906i	$0.495631\pi$
$558$	0	0
$559$	1.15667	0.0489221
$560$	0	0
$561$	21.3275	0.900447
$562$	0	0
$563$	16.3169	0.687676	0.343838	−	0.939029i	$-0.388273\pi$
$563$	16.3169	0.687676	0.343838	+	0.939029i	$0.388273\pi$
$564$	0	0
$565$	0.843326	0.0354790
$566$	0	0
$567$	−4.91638	−0.206469
$568$	0	0
$569$	44.6550	1.87203	0.936017	−	0.351956i	$-0.114483\pi$
$569$	44.6550	1.87203	0.936017	+	0.351956i	$0.114483\pi$
$570$	0	0
$571$	6.67252	0.279236	0.139618	−	0.990205i	$-0.455413\pi$
$571$	6.67252	0.279236	0.139618	+	0.990205i	$0.455413\pi$
$572$	0	0
$573$	22.5089	0.940321
$574$	0	0
$575$	6.33804	0.264315
$576$	0	0
$577$	15.3275	0.638091	0.319046	−	0.947739i	$-0.396638\pi$
$577$	15.3275	0.638091	0.319046	+	0.947739i	$0.396638\pi$
$578$	0	0
$579$	2.65139	0.110188
$580$	0	0
$581$	55.3311	2.29552
$582$	0	0
$583$	1.66196	0.0688312
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	−12.2650	−0.506230	−0.253115	−	0.967436i	$-0.581455\pi$
$587$	−12.2650	−0.506230	−0.253115	+	0.967436i	$0.581455\pi$
$588$	0	0
$589$	3.66553	0.151035
$590$	0	0
$591$	−12.9894	−0.534314
$592$	0	0
$593$	−5.85389	−0.240390	−0.120195	−	0.992750i	$-0.538352\pi$
$593$	−5.85389	−0.240390	−0.120195	+	0.992750i	$0.538352\pi$
$594$	0	0
$595$	−21.3275	−0.874342
$596$	0	0
$597$	2.84333	0.116370
$598$	0	0
$599$	−27.1355	−1.10873	−0.554364	−	0.832274i	$-0.687039\pi$
$599$	−27.1355	−1.10873	−0.554364	+	0.832274i	$0.687039\pi$
$600$	0	0
$601$	34.1708	1.39386	0.696928	−	0.717141i	$-0.254550\pi$
$601$	34.1708	1.39386	0.696928	+	0.717141i	$0.254550\pi$
$602$	0	0
$603$	7.25443	0.295423
$604$	0	0
$605$	−13.1708	−0.535469
$606$	0	0
$607$	−10.3133	−0.418606	−0.209303	−	0.977851i	$-0.567119\pi$
$607$	−10.3133	−0.418606	−0.209303	+	0.977851i	$0.567119\pi$
$608$	0	0
$609$	−29.4983	−1.19533
$610$	0	0
$611$	5.42166	0.219337
$612$	0	0
$613$	0.484156	0.0195549	0.00977744	−	0.999952i	$-0.496888\pi$
$613$	0.484156	0.0195549	0.00977744	+	0.999952i	$0.496888\pi$
$614$	0	0
$615$	−4.33804	−0.174927
$616$	0	0
$617$	−7.15667	−0.288117	−0.144058	−	0.989569i	$-0.546015\pi$
$617$	−7.15667	−0.288117	−0.144058	+	0.989569i	$0.546015\pi$
$618$	0	0
$619$	−5.42166	−0.217915	−0.108958	−	0.994046i	$-0.534751\pi$
$619$	−5.42166	−0.217915	−0.108958	+	0.994046i	$0.534751\pi$
$620$	0	0
$621$	6.33804	0.254337
$622$	0	0
$623$	1.66196	0.0665848
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.6761	0.506234
$628$	0	0
$629$	−41.1885	−1.64229
$630$	0	0
$631$	10.7244	0.426934	0.213467	−	0.976950i	$-0.431524\pi$
$631$	10.7244	0.426934	0.213467	+	0.976950i	$0.431524\pi$
$632$	0	0
$633$	−6.31335	−0.250933
$634$	0	0
$635$	1.83276	0.0727310
$636$	0	0
$637$	17.1708	0.680332
$638$	0	0
$639$	−0.916382	−0.0362515
$640$	0	0
$641$	−0.362741	−0.0143274	−0.00716370	−	0.999974i	$-0.502280\pi$
$641$	−0.362741	−0.0143274	−0.00716370	+	0.999974i	$0.502280\pi$
$642$	0	0
$643$	9.39697	0.370580	0.185290	−	0.982684i	$-0.440678\pi$
$643$	9.39697	0.370580	0.185290	+	0.982684i	$0.440678\pi$
$644$	0	0
$645$	−1.15667	−0.0455440
$646$	0	0
$647$	−18.0036	−0.707793	−0.353897	−	0.935285i	$-0.615144\pi$
$647$	−18.0036	−0.707793	−0.353897	+	0.935285i	$0.615144\pi$
$648$	0	0
$649$	−55.3311	−2.17193
$650$	0	0
$651$	6.98944	0.273938
$652$	0	0
$653$	−34.8222	−1.36270	−0.681349	−	0.731959i	$-0.738606\pi$
$653$	−34.8222	−1.36270	−0.681349	+	0.731959i	$0.738606\pi$
$654$	0	0
$655$	5.83276	0.227905
$656$	0	0
$657$	−3.15667	−0.123154
$658$	0	0
$659$	−11.4700	−0.446809	−0.223404	−	0.974726i	$-0.571717\pi$
$659$	−11.4700	−0.446809	−0.223404	+	0.974726i	$0.571717\pi$
$660$	0	0
$661$	12.1672	0.473251	0.236625	−	0.971601i	$-0.423959\pi$
$661$	12.1672	0.473251	0.236625	+	0.971601i	$0.423959\pi$
$662$	0	0
$663$	−4.33804	−0.168476
$664$	0	0
$665$	−12.6761	−0.491558
$666$	0	0
$667$	38.0283	1.47246
$668$	0	0
$669$	−19.2544	−0.744419
$670$	0	0
$671$	50.0036	1.93037
$672$	0	0
$673$	−27.9789	−1.07851	−0.539253	−	0.842144i	$-0.681293\pi$
$673$	−27.9789	−1.07851	−0.539253	+	0.842144i	$0.681293\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	22.9930	0.883693	0.441847	−	0.897091i	$-0.354324\pi$
$677$	22.9930	0.883693	0.441847	+	0.897091i	$0.354324\pi$
$678$	0	0
$679$	60.6585	2.32786
$680$	0	0
$681$	13.0872	0.501502
$682$	0	0
$683$	28.6066	1.09460	0.547301	−	0.836936i	$-0.315655\pi$
$683$	28.6066	1.09460	0.547301	+	0.836936i	$0.315655\pi$
$684$	0	0
$685$	16.5089	0.630771
$686$	0	0
$687$	−24.5089	−0.935071
$688$	0	0
$689$	−0.338044	−0.0128785
$690$	0	0
$691$	19.4005	0.738031	0.369016	−	0.929423i	$-0.379695\pi$
$691$	19.4005	0.738031	0.369016	+	0.929423i	$0.379695\pi$
$692$	0	0
$693$	24.1708	0.918173
$694$	0	0
$695$	7.49472	0.284291
$696$	0	0
$697$	−18.8186	−0.712806
$698$	0	0
$699$	8.33804	0.315374
$700$	0	0
$701$	−38.9683	−1.47181	−0.735906	−	0.677083i	$-0.763244\pi$
$701$	−38.9683	−1.47181	−0.735906	+	0.677083i	$0.763244\pi$
$702$	0	0
$703$	−24.4806	−0.923303
$704$	0	0
$705$	−5.42166	−0.204192
$706$	0	0
$707$	−52.4877	−1.97400
$708$	0	0
$709$	−17.5194	−0.657955	−0.328978	−	0.944338i	$-0.606704\pi$
$709$	−17.5194	−0.657955	−0.328978	+	0.944338i	$0.606704\pi$
$710$	0	0
$711$	3.49472	0.131062
$712$	0	0
$713$	−9.01056	−0.337448
$714$	0	0
$715$	4.91638	0.183862
$716$	0	0
$717$	−8.91638	−0.332988
$718$	0	0
$719$	4.33447	0.161649	0.0808243	−	0.996728i	$-0.474245\pi$
$719$	4.33447	0.161649	0.0808243	+	0.996728i	$0.474245\pi$
$720$	0	0
$721$	−71.3311	−2.65651
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−22.1672	−0.822137	−0.411069	−	0.911604i	$-0.634844\pi$
$727$	−22.1672	−0.822137	−0.411069	+	0.911604i	$0.634844\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.01770	−0.185586
$732$	0	0
$733$	−2.83976	−0.104889	−0.0524444	−	0.998624i	$-0.516701\pi$
$733$	−2.83976	−0.104889	−0.0524444	+	0.998624i	$0.516701\pi$
$734$	0	0
$735$	−17.1708	−0.633355
$736$	0	0
$737$	−35.6655	−1.31376
$738$	0	0
$739$	43.9305	1.61601	0.808005	−	0.589176i	$-0.200547\pi$
$739$	43.9305	1.61601	0.808005	+	0.589176i	$0.200547\pi$
$740$	0	0
$741$	−2.57834	−0.0947176
$742$	0	0
$743$	−41.0872	−1.50734	−0.753671	−	0.657251i	$-0.771719\pi$
$743$	−41.0872	−1.50734	−0.753671	+	0.657251i	$0.771719\pi$
$744$	0	0
$745$	−20.4842	−0.750481
$746$	0	0
$747$	−11.2544	−0.411778
$748$	0	0
$749$	20.5053	0.749247
$750$	0	0
$751$	−23.6902	−0.864468	−0.432234	−	0.901761i	$-0.642275\pi$
$751$	−23.6902	−0.864468	−0.432234	+	0.901761i	$0.642275\pi$
$752$	0	0
$753$	6.31335	0.230071
$754$	0	0
$755$	−16.4111	−0.597261
$756$	0	0
$757$	9.32391	0.338883	0.169442	−	0.985540i	$-0.445804\pi$
$757$	9.32391	0.338883	0.169442	+	0.985540i	$0.445804\pi$
$758$	0	0
$759$	−31.1602	−1.13105
$760$	0	0
$761$	−42.8222	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$761$	−42.8222	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$762$	0	0
$763$	18.8433	0.682174
$764$	0	0
$765$	4.33804	0.156842
$766$	0	0
$767$	11.2544	0.406374
$768$	0	0
$769$	−17.3239	−0.624716	−0.312358	−	0.949964i	$-0.601119\pi$
$769$	−17.3239	−0.624716	−0.312358	+	0.949964i	$0.601119\pi$
$770$	0	0
$771$	−11.1567	−0.401798
$772$	0	0
$773$	−11.6373	−0.418563	−0.209282	−	0.977855i	$-0.567113\pi$
$773$	−11.6373	−0.418563	−0.209282	+	0.977855i	$0.567113\pi$
$774$	0	0
$775$	−1.42166	−0.0510676
$776$	0	0
$777$	−46.6797	−1.67462
$778$	0	0
$779$	−11.1849	−0.400742
$780$	0	0
$781$	4.50528	0.161212
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	21.6655	0.773276
$786$	0	0
$787$	−20.9411	−0.746469	−0.373234	−	0.927737i	$-0.621751\pi$
$787$	−20.9411	−0.746469	−0.373234	+	0.927737i	$0.621751\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	4.14611	0.147419
$792$	0	0
$793$	−10.1708	−0.361176
$794$	0	0
$795$	0.338044	0.0119892
$796$	0	0
$797$	20.3380	0.720410	0.360205	−	0.932873i	$-0.382707\pi$
$797$	20.3380	0.720410	0.360205	+	0.932873i	$0.382707\pi$
$798$	0	0
$799$	−23.5194	−0.832057
$800$	0	0
$801$	−0.338044	−0.0119442
$802$	0	0
$803$	15.5194	0.547668
$804$	0	0
$805$	31.1602	1.09825
$806$	0	0
$807$	18.6761	0.657429
$808$	0	0
$809$	7.68665	0.270248	0.135124	−	0.990829i	$-0.456857\pi$
$809$	7.68665	0.270248	0.135124	+	0.990829i	$0.456857\pi$
$810$	0	0
$811$	44.4111	1.55948	0.779742	−	0.626101i	$-0.215350\pi$
$811$	44.4111	1.55948	0.779742	+	0.626101i	$0.215350\pi$
$812$	0	0
$813$	−6.57834	−0.230712
$814$	0	0
$815$	6.07306	0.212730
$816$	0	0
$817$	−2.98230	−0.104337
$818$	0	0
$819$	−4.91638	−0.171792
$820$	0	0
$821$	−46.4630	−1.62157	−0.810785	−	0.585343i	$-0.800960\pi$
$821$	−46.4630	−1.62157	−0.810785	+	0.585343i	$0.800960\pi$
$822$	0	0
$823$	−46.5089	−1.62120	−0.810598	−	0.585603i	$-0.800858\pi$
$823$	−46.5089	−1.62120	−0.810598	+	0.585603i	$0.800858\pi$
$824$	0	0
$825$	−4.91638	−0.171166
$826$	0	0
$827$	−39.4005	−1.37009	−0.685045	−	0.728500i	$-0.740218\pi$
$827$	−39.4005	−1.37009	−0.685045	+	0.728500i	$0.740218\pi$
$828$	0	0
$829$	47.6444	1.65476	0.827379	−	0.561644i	$-0.189831\pi$
$829$	47.6444	1.65476	0.827379	+	0.561644i	$0.189831\pi$
$830$	0	0
$831$	25.6655	0.890327
$832$	0	0
$833$	−74.4877	−2.58085
$834$	0	0
$835$	−0.745574	−0.0258017
$836$	0	0
$837$	−1.42166	−0.0491399
$838$	0	0
$839$	39.9058	1.37770	0.688851	−	0.724903i	$-0.258115\pi$
$839$	39.9058	1.37770	0.688851	+	0.724903i	$0.258115\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	3.15667	0.108722
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−64.7527	−2.22493
$848$	0	0
$849$	3.47002	0.119091
$850$	0	0
$851$	60.1779	2.06287
$852$	0	0
$853$	29.5019	1.01012	0.505062	−	0.863083i	$-0.331470\pi$
$853$	29.5019	1.01012	0.505062	+	0.863083i	$0.331470\pi$
$854$	0	0
$855$	2.57834	0.0881773
$856$	0	0
$857$	8.33804	0.284822	0.142411	−	0.989808i	$-0.454515\pi$
$857$	8.33804	0.284822	0.142411	+	0.989808i	$0.454515\pi$
$858$	0	0
$859$	4.17081	0.142306	0.0711531	−	0.997465i	$-0.477332\pi$
$859$	4.17081	0.142306	0.0711531	+	0.997465i	$0.477332\pi$
$860$	0	0
$861$	−21.3275	−0.726839
$862$	0	0
$863$	3.93051	0.133796	0.0668981	−	0.997760i	$-0.478690\pi$
$863$	3.93051	0.133796	0.0668981	+	0.997760i	$0.478690\pi$
$864$	0	0
$865$	−0.843326	−0.0286739
$866$	0	0
$867$	1.81863	0.0617639
$868$	0	0
$869$	−17.1814	−0.582838
$870$	0	0
$871$	7.25443	0.245807
$872$	0	0
$873$	−12.3380	−0.417580
$874$	0	0
$875$	4.91638	0.166204
$876$	0	0
$877$	−23.0177	−0.777253	−0.388626	−	0.921395i	$-0.627050\pi$
$877$	−23.0177	−0.777253	−0.388626	+	0.921395i	$0.627050\pi$
$878$	0	0
$879$	28.6550	0.966508
$880$	0	0
$881$	−15.3522	−0.517228	−0.258614	−	0.965981i	$-0.583266\pi$
$881$	−15.3522	−0.517228	−0.258614	+	0.965981i	$0.583266\pi$
$882$	0	0
$883$	−42.8011	−1.44037	−0.720185	−	0.693782i	$-0.755943\pi$
$883$	−42.8011	−1.44037	−0.720185	+	0.693782i	$0.755943\pi$
$884$	0	0
$885$	−11.2544	−0.378313
$886$	0	0
$887$	53.1885	1.78590	0.892948	−	0.450160i	$-0.148633\pi$
$887$	53.1885	1.78590	0.892948	+	0.450160i	$0.148633\pi$
$888$	0	0
$889$	9.01056	0.302205
$890$	0	0
$891$	−4.91638	−0.164705
$892$	0	0
$893$	−13.9789	−0.467785
$894$	0	0
$895$	−18.9894	−0.634747
$896$	0	0
$897$	6.33804	0.211621
$898$	0	0
$899$	−8.52998	−0.284491
$900$	0	0
$901$	1.46645	0.0488546
$902$	0	0
$903$	−5.68665	−0.189240
$904$	0	0
$905$	−17.4947	−0.581544
$906$	0	0
$907$	−11.8116	−0.392199	−0.196099	−	0.980584i	$-0.562828\pi$
$907$	−11.8116	−0.392199	−0.196099	+	0.980584i	$0.562828\pi$
$908$	0	0
$909$	10.6761	0.354104
$910$	0	0
$911$	−44.1955	−1.46426	−0.732131	−	0.681164i	$-0.761474\pi$
$911$	−44.1955	−1.46426	−0.732131	+	0.681164i	$0.761474\pi$
$912$	0	0
$913$	55.3311	1.83119
$914$	0	0
$915$	10.1708	0.336237
$916$	0	0
$917$	28.6761	0.946968
$918$	0	0
$919$	−55.2096	−1.82120	−0.910599	−	0.413291i	$-0.864379\pi$
$919$	−55.2096	−1.82120	−0.910599	+	0.413291i	$0.864379\pi$
$920$	0	0
$921$	1.92694	0.0634950
$922$	0	0
$923$	−0.916382	−0.0301631
$924$	0	0
$925$	9.49472	0.312184
$926$	0	0
$927$	14.5089	0.476533
$928$	0	0
$929$	22.9930	0.754376	0.377188	−	0.926137i	$-0.376891\pi$
$929$	22.9930	0.754376	0.377188	+	0.926137i	$0.376891\pi$
$930$	0	0
$931$	−44.2721	−1.45096
$932$	0	0
$933$	29.9789	0.981464
$934$	0	0
$935$	−21.3275	−0.697483
$936$	0	0
$937$	7.97887	0.260658	0.130329	−	0.991471i	$-0.458397\pi$
$937$	7.97887	0.260658	0.130329	+	0.991471i	$0.458397\pi$
$938$	0	0
$939$	−16.3133	−0.532366
$940$	0	0
$941$	41.5019	1.35292	0.676461	−	0.736478i	$-0.263513\pi$
$941$	41.5019	1.35292	0.676461	+	0.736478i	$0.263513\pi$
$942$	0	0
$943$	27.4947	0.895351
$944$	0	0
$945$	4.91638	0.159930
$946$	0	0
$947$	−47.4499	−1.54192	−0.770958	−	0.636886i	$-0.780222\pi$
$947$	−47.4499	−1.54192	−0.770958	+	0.636886i	$0.780222\pi$
$948$	0	0
$949$	−3.15667	−0.102470
$950$	0	0
$951$	30.6761	0.994740
$952$	0	0
$953$	30.3663	0.983661	0.491831	−	0.870691i	$-0.336328\pi$
$953$	30.3663	0.983661	0.491831	+	0.870691i	$0.336328\pi$
$954$	0	0
$955$	−22.5089	−0.728369
$956$	0	0
$957$	−29.4983	−0.953544
$958$	0	0
$959$	81.1638	2.62092
$960$	0	0
$961$	−28.9789	−0.934802
$962$	0	0
$963$	−4.17081	−0.134402
$964$	0	0
$965$	−2.65139	−0.0853514
$966$	0	0
$967$	41.0943	1.32150	0.660752	−	0.750604i	$-0.270237\pi$
$967$	41.0943	1.32150	0.660752	+	0.750604i	$0.270237\pi$
$968$	0	0
$969$	11.1849	0.359312
$970$	0	0
$971$	−38.1744	−1.22507	−0.612537	−	0.790442i	$-0.709851\pi$
$971$	−38.1744	−1.22507	−0.612537	+	0.790442i	$0.709851\pi$
$972$	0	0
$973$	36.8469	1.18126
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−10.4806	−0.335304	−0.167652	−	0.985846i	$-0.553618\pi$
$977$	−10.4806	−0.335304	−0.167652	+	0.985846i	$0.553618\pi$
$978$	0	0
$979$	1.66196	0.0531163
$980$	0	0
$981$	−3.83276	−0.122371
$982$	0	0
$983$	0.0766264	0.00244400	0.00122200	−	0.999999i	$-0.499611\pi$
$983$	0.0766264	0.00244400	0.00122200	+	0.999999i	$0.499611\pi$
$984$	0	0
$985$	12.9894	0.413878
$986$	0	0
$987$	−26.6550	−0.848437
$988$	0	0
$989$	7.33105	0.233114
$990$	0	0
$991$	13.8575	0.440197	0.220098	−	0.975478i	$-0.429362\pi$
$991$	13.8575	0.440197	0.220098	+	0.975478i	$0.429362\pi$
$992$	0	0
$993$	10.0978	0.320442
$994$	0	0
$995$	−2.84333	−0.0901395
$996$	0	0
$997$	−10.3416	−0.327522	−0.163761	−	0.986500i	$-0.552363\pi$
$997$	−10.3416	−0.327522	−0.163761	+	0.986500i	$0.552363\pi$
$998$	0	0
$999$	9.49472	0.300400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bj.1.1		3
3.2	odd	2		9360.2.a.dd.1.1		3
4.3	odd	2		195.2.a.e.1.2	✓	3
12.11	even	2		585.2.a.n.1.2		3
20.3	even	4		975.2.c.i.274.4		6
20.7	even	4		975.2.c.i.274.3		6
20.19	odd	2		975.2.a.o.1.2		3
28.27	even	2		9555.2.a.bq.1.2		3
52.51	odd	2		2535.2.a.bc.1.2		3
60.23	odd	4		2925.2.c.w.2224.3		6
60.47	odd	4		2925.2.c.w.2224.4		6
60.59	even	2		2925.2.a.bh.1.2		3
156.155	even	2		7605.2.a.bx.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.2	✓	3	4.3	odd	2
585.2.a.n.1.2		3	12.11	even	2
975.2.a.o.1.2		3	20.19	odd	2
975.2.c.i.274.3		6	20.7	even	4
975.2.c.i.274.4		6	20.3	even	4
2535.2.a.bc.1.2		3	52.51	odd	2
2925.2.a.bh.1.2		3	60.59	even	2
2925.2.c.w.2224.3		6	60.23	odd	4
2925.2.c.w.2224.4		6	60.47	odd	4
3120.2.a.bj.1.1		3	1.1	even	1	trivial
7605.2.a.bx.1.2		3	156.155	even	2
9360.2.a.dd.1.1		3	3.2	odd	2
9555.2.a.bq.1.2		3	28.27	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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.91638 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -4.91638 q^{7} +1.00000 q^{9} -4.91638 q^{11} +1.00000 q^{13} -1.00000 q^{15} -4.33804 q^{17} -2.57834 q^{19} -4.91638 q^{21} +6.33804 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} -1.42166 q^{31} -4.91638 q^{33} +4.91638 q^{35} +9.49472 q^{37} +1.00000 q^{39} +4.33804 q^{41} +1.15667 q^{43} -1.00000 q^{45} +5.42166 q^{47} +17.1708 q^{49} -4.33804 q^{51} -0.338044 q^{53} +4.91638 q^{55} -2.57834 q^{57} +11.2544 q^{59} -10.1708 q^{61} -4.91638 q^{63} -1.00000 q^{65} +7.25443 q^{67} +6.33804 q^{69} -0.916382 q^{71} -3.15667 q^{73} +1.00000 q^{75} +24.1708 q^{77} +3.49472 q^{79} +1.00000 q^{81} -11.2544 q^{83} +4.33804 q^{85} +6.00000 q^{87} -0.338044 q^{89} -4.91638 q^{91} -1.42166 q^{93} +2.57834 q^{95} -12.3380 q^{97} -4.91638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - q^99

Introduction

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