LMFDB - Embedded newform 3020.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3020,2,Mod(1,3020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3020, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3020 = 2^{2} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3020.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:	$24.1148214104$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.40310669.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 10x^{3} + 11x^{2} - 11x + 2 $ x^6 - 2x^5 - 7x^4 + 10x^3 + 11x^2 - 11*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.76047$ of defining polynomial
Character	$\chi$	$=$	3020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.66122 q^{3} +1.00000 q^{5} +2.25909 q^{7} -0.240356 q^{9} +O(q^{10})$	$q+1.66122 q^{3} +1.00000 q^{5} +2.25909 q^{7} -0.240356 q^{9} -2.37642 q^{11} +2.12579 q^{13} +1.66122 q^{15} +2.89376 q^{17} +6.37220 q^{19} +3.75283 q^{21} -2.59697 q^{23} +1.00000 q^{25} -5.38294 q^{27} +0.897983 q^{29} +4.03922 q^{31} -3.94775 q^{33} +2.25909 q^{35} +6.10404 q^{37} +3.53140 q^{39} -6.79295 q^{41} +8.48745 q^{43} -0.240356 q^{45} -2.53625 q^{47} -1.89653 q^{49} +4.80717 q^{51} -7.19062 q^{53} -2.37642 q^{55} +10.5856 q^{57} -4.26330 q^{59} +8.35347 q^{61} -0.542985 q^{63} +2.12579 q^{65} +11.1055 q^{67} -4.31413 q^{69} +14.5041 q^{71} +12.2221 q^{73} +1.66122 q^{75} -5.36853 q^{77} -8.95769 q^{79} -8.22116 q^{81} -2.16490 q^{83} +2.89376 q^{85} +1.49174 q^{87} +15.7445 q^{89} +4.80235 q^{91} +6.71002 q^{93} +6.37220 q^{95} -19.2585 q^{97} +0.571186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9	$ 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9} + 3 q^{11} + 2 q^{13} + 2 q^{15} + 9 q^{17} + 16 q^{19} - 12 q^{21} + 6 q^{23} + 6 q^{25} + 23 q^{27} + 2 q^{29} + 15 q^{31} - 14 q^{33} + 3 q^{35} + 9 q^{37} - 17 q^{39} - 16 q^{41} + 15 q^{43} + 20 q^{45} + 12 q^{47} - 7 q^{49} + 11 q^{51} - 14 q^{53} + 3 q^{55} + 27 q^{57} - 20 q^{59} + 17 q^{61} + 17 q^{63} + 2 q^{65} + 29 q^{67} - 40 q^{69} + 13 q^{71} - 6 q^{73} + 2 q^{75} + 13 q^{77} - 8 q^{79} + 30 q^{81} + 36 q^{83} + 9 q^{85} + 2 q^{87} - 15 q^{89} - 14 q^{91} - 31 q^{93} + 16 q^{95} - 23 q^{97} + 38 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9 + 3 * q^11 + 2 * q^13 + 2 * q^15 + 9 * q^17 + 16 * q^19 - 12 * q^21 + 6 * q^23 + 6 * q^25 + 23 * q^27 + 2 * q^29 + 15 * q^31 - 14 * q^33 + 3 * q^35 + 9 * q^37 - 17 * q^39 - 16 * q^41 + 15 * q^43 + 20 * q^45 + 12 * q^47 - 7 * q^49 + 11 * q^51 - 14 * q^53 + 3 * q^55 + 27 * q^57 - 20 * q^59 + 17 * q^61 + 17 * q^63 + 2 * q^65 + 29 * q^67 - 40 * q^69 + 13 * q^71 - 6 * q^73 + 2 * q^75 + 13 * q^77 - 8 * q^79 + 30 * q^81 + 36 * q^83 + 9 * q^85 + 2 * q^87 - 15 * q^89 - 14 * q^91 - 31 * q^93 + 16 * q^95 - 23 * q^97 + 38 * 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
$3$	1.66122	0.959104	0.479552	−	0.877513i	$-0.340799\pi$
$3$	1.66122	0.959104	0.479552	+	0.877513i	$0.340799\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.25909	0.853854	0.426927	−	0.904286i	$-0.359596\pi$
$7$	2.25909	0.853854	0.426927	+	0.904286i	$0.359596\pi$
$8$	0	0
$9$	−0.240356	−0.0801187
$10$	0	0
$11$	−2.37642	−0.716517	−0.358258	−	0.933623i	$-0.616629\pi$
$11$	−2.37642	−0.716517	−0.358258	+	0.933623i	$0.616629\pi$
$12$	0	0
$13$	2.12579	0.589588	0.294794	−	0.955561i	$-0.404749\pi$
$13$	2.12579	0.589588	0.294794	+	0.955561i	$0.404749\pi$
$14$	0	0
$15$	1.66122	0.428925
$16$	0	0
$17$	2.89376	0.701841	0.350920	−	0.936405i	$-0.385869\pi$
$17$	2.89376	0.701841	0.350920	+	0.936405i	$0.385869\pi$
$18$	0	0
$19$	6.37220	1.46188	0.730941	−	0.682440i	$-0.239081\pi$
$19$	6.37220	1.46188	0.730941	+	0.682440i	$0.239081\pi$
$20$	0	0
$21$	3.75283	0.818935
$22$	0	0
$23$	−2.59697	−0.541505	−0.270752	−	0.962649i	$-0.587272\pi$
$23$	−2.59697	−0.541505	−0.270752	+	0.962649i	$0.587272\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.38294	−1.03595
$28$	0	0
$29$	0.897983	0.166751	0.0833756	−	0.996518i	$-0.473430\pi$
$29$	0.897983	0.166751	0.0833756	+	0.996518i	$0.473430\pi$
$30$	0	0
$31$	4.03922	0.725465	0.362733	−	0.931893i	$-0.381844\pi$
$31$	4.03922	0.725465	0.362733	+	0.931893i	$0.381844\pi$
$32$	0	0
$33$	−3.94775	−0.687214
$34$	0	0
$35$	2.25909	0.381855
$36$	0	0
$37$	6.10404	1.00350	0.501749	−	0.865013i	$-0.332690\pi$
$37$	6.10404	1.00350	0.501749	+	0.865013i	$0.332690\pi$
$38$	0	0
$39$	3.53140	0.565477
$40$	0	0
$41$	−6.79295	−1.06088	−0.530441	−	0.847722i	$-0.677974\pi$
$41$	−6.79295	−1.06088	−0.530441	+	0.847722i	$0.677974\pi$
$42$	0	0
$43$	8.48745	1.29432	0.647162	−	0.762353i	$-0.275956\pi$
$43$	8.48745	1.29432	0.647162	+	0.762353i	$0.275956\pi$
$44$	0	0
$45$	−0.240356	−0.0358302
$46$	0	0
$47$	−2.53625	−0.369950	−0.184975	−	0.982743i	$-0.559220\pi$
$47$	−2.53625	−0.369950	−0.184975	+	0.982743i	$0.559220\pi$
$48$	0	0
$49$	−1.89653	−0.270933
$50$	0	0
$51$	4.80717	0.673139
$52$	0	0
$53$	−7.19062	−0.987707	−0.493854	−	0.869545i	$-0.664412\pi$
$53$	−7.19062	−0.987707	−0.493854	+	0.869545i	$0.664412\pi$
$54$	0	0
$55$	−2.37642	−0.320436
$56$	0	0
$57$	10.5856	1.40210
$58$	0	0
$59$	−4.26330	−0.555035	−0.277517	−	0.960721i	$-0.589512\pi$
$59$	−4.26330	−0.555035	−0.277517	+	0.960721i	$0.589512\pi$
$60$	0	0
$61$	8.35347	1.06955	0.534776	−	0.844994i	$-0.320396\pi$
$61$	8.35347	1.06955	0.534776	+	0.844994i	$0.320396\pi$
$62$	0	0
$63$	−0.542985	−0.0684097
$64$	0	0
$65$	2.12579	0.263672
$66$	0	0
$67$	11.1055	1.35675	0.678375	−	0.734716i	$-0.262685\pi$
$67$	11.1055	1.35675	0.678375	+	0.734716i	$0.262685\pi$
$68$	0	0
$69$	−4.31413	−0.519360
$70$	0	0
$71$	14.5041	1.72132	0.860660	−	0.509180i	$-0.170051\pi$
$71$	14.5041	1.72132	0.860660	+	0.509180i	$0.170051\pi$
$72$	0	0
$73$	12.2221	1.43048	0.715242	−	0.698876i	$-0.246316\pi$
$73$	12.2221	1.43048	0.715242	+	0.698876i	$0.246316\pi$
$74$	0	0
$75$	1.66122	0.191821
$76$	0	0
$77$	−5.36853	−0.611801
$78$	0	0
$79$	−8.95769	−1.00782	−0.503909	−	0.863757i	$-0.668105\pi$
$79$	−8.95769	−1.00782	−0.503909	+	0.863757i	$0.668105\pi$
$80$	0	0
$81$	−8.22116	−0.913462
$82$	0	0
$83$	−2.16490	−0.237629	−0.118814	−	0.992916i	$-0.537909\pi$
$83$	−2.16490	−0.237629	−0.118814	+	0.992916i	$0.537909\pi$
$84$	0	0
$85$	2.89376	0.313873
$86$	0	0
$87$	1.49174	0.159932
$88$	0	0
$89$	15.7445	1.66891	0.834456	−	0.551075i	$-0.185782\pi$
$89$	15.7445	1.66891	0.834456	+	0.551075i	$0.185782\pi$
$90$	0	0
$91$	4.80235	0.503423
$92$	0	0
$93$	6.71002	0.695797
$94$	0	0
$95$	6.37220	0.653774
$96$	0	0
$97$	−19.2585	−1.95541	−0.977703	−	0.209991i	$-0.932656\pi$
$97$	−19.2585	−1.95541	−0.977703	+	0.209991i	$0.932656\pi$
$98$	0	0
$99$	0.571186	0.0574064
$100$	0	0
$101$	−0.0374602	−0.00372743	−0.00186372	−	0.999998i	$-0.500593\pi$
$101$	−0.0374602	−0.00372743	−0.00186372	+	0.999998i	$0.500593\pi$
$102$	0	0
$103$	−7.34786	−0.724006	−0.362003	−	0.932177i	$-0.617907\pi$
$103$	−7.34786	−0.724006	−0.362003	+	0.932177i	$0.617907\pi$
$104$	0	0
$105$	3.75283	0.366239
$106$	0	0
$107$	−8.29333	−0.801746	−0.400873	−	0.916134i	$-0.631293\pi$
$107$	−8.29333	−0.801746	−0.400873	+	0.916134i	$0.631293\pi$
$108$	0	0
$109$	17.6026	1.68602	0.843011	−	0.537896i	$-0.180781\pi$
$109$	17.6026	1.68602	0.843011	+	0.537896i	$0.180781\pi$
$110$	0	0
$111$	10.1401	0.962460
$112$	0	0
$113$	4.61597	0.434234	0.217117	−	0.976146i	$-0.430335\pi$
$113$	4.61597	0.434234	0.217117	+	0.976146i	$0.430335\pi$
$114$	0	0
$115$	−2.59697	−0.242168
$116$	0	0
$117$	−0.510947	−0.0472370
$118$	0	0
$119$	6.53726	0.599270
$120$	0	0
$121$	−5.35264	−0.486604
$122$	0	0
$123$	−11.2846	−1.01750
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.80087	−0.869687	−0.434843	−	0.900506i	$-0.643196\pi$
$127$	−9.80087	−0.869687	−0.434843	+	0.900506i	$0.643196\pi$
$128$	0	0
$129$	14.0995	1.24139
$130$	0	0
$131$	1.53228	0.133876	0.0669381	−	0.997757i	$-0.478677\pi$
$131$	1.53228	0.133876	0.0669381	+	0.997757i	$0.478677\pi$
$132$	0	0
$133$	14.3953	1.24823
$134$	0	0
$135$	−5.38294	−0.463289
$136$	0	0
$137$	−19.1840	−1.63900	−0.819499	−	0.573081i	$-0.805748\pi$
$137$	−19.1840	−1.63900	−0.819499	+	0.573081i	$0.805748\pi$
$138$	0	0
$139$	6.18547	0.524644	0.262322	−	0.964980i	$-0.415512\pi$
$139$	6.18547	0.524644	0.262322	+	0.964980i	$0.415512\pi$
$140$	0	0
$141$	−4.21327	−0.354821
$142$	0	0
$143$	−5.05177	−0.422450
$144$	0	0
$145$	0.897983	0.0745734
$146$	0	0
$147$	−3.15055	−0.259853
$148$	0	0
$149$	11.8501	0.970800	0.485400	−	0.874292i	$-0.338674\pi$
$149$	11.8501	0.970800	0.485400	+	0.874292i	$0.338674\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−0.695534	−0.0562306
$154$	0	0
$155$	4.03922	0.324438
$156$	0	0
$157$	−10.1036	−0.806352	−0.403176	−	0.915122i	$-0.632094\pi$
$157$	−10.1036	−0.806352	−0.403176	+	0.915122i	$0.632094\pi$
$158$	0	0
$159$	−11.9452	−0.947314
$160$	0	0
$161$	−5.86677	−0.462366
$162$	0	0
$163$	−1.51946	−0.119013	−0.0595067	−	0.998228i	$-0.518953\pi$
$163$	−1.51946	−0.119013	−0.0595067	+	0.998228i	$0.518953\pi$
$164$	0	0
$165$	−3.94775	−0.307332
$166$	0	0
$167$	12.6997	0.982732	0.491366	−	0.870953i	$-0.336498\pi$
$167$	12.6997	0.982732	0.491366	+	0.870953i	$0.336498\pi$
$168$	0	0
$169$	−8.48101	−0.652385
$170$	0	0
$171$	−1.53160	−0.117124
$172$	0	0
$173$	−23.6829	−1.80057	−0.900287	−	0.435296i	$-0.856644\pi$
$173$	−23.6829	−1.80057	−0.900287	+	0.435296i	$0.856644\pi$
$174$	0	0
$175$	2.25909	0.170771
$176$	0	0
$177$	−7.08228	−0.532337
$178$	0	0
$179$	−25.0064	−1.86907	−0.934534	−	0.355873i	$-0.884183\pi$
$179$	−25.0064	−1.86907	−0.934534	+	0.355873i	$0.884183\pi$
$180$	0	0
$181$	−3.89423	−0.289456	−0.144728	−	0.989471i	$-0.546231\pi$
$181$	−3.89423	−0.289456	−0.144728	+	0.989471i	$0.546231\pi$
$182$	0	0
$183$	13.8769	1.02581
$184$	0	0
$185$	6.10404	0.448778
$186$	0	0
$187$	−6.87679	−0.502881
$188$	0	0
$189$	−12.1605	−0.884547
$190$	0	0
$191$	6.25397	0.452521	0.226261	−	0.974067i	$-0.427350\pi$
$191$	6.25397	0.452521	0.226261	+	0.974067i	$0.427350\pi$
$192$	0	0
$193$	3.74144	0.269315	0.134657	−	0.990892i	$-0.457007\pi$
$193$	3.74144	0.269315	0.134657	+	0.990892i	$0.457007\pi$
$194$	0	0
$195$	3.53140	0.252889
$196$	0	0
$197$	24.5005	1.74559	0.872793	−	0.488091i	$-0.162307\pi$
$197$	24.5005	1.74559	0.872793	+	0.488091i	$0.162307\pi$
$198$	0	0
$199$	−11.9373	−0.846214	−0.423107	−	0.906080i	$-0.639061\pi$
$199$	−11.9373	−0.846214	−0.423107	+	0.906080i	$0.639061\pi$
$200$	0	0
$201$	18.4486	1.30126
$202$	0	0
$203$	2.02862	0.142381
$204$	0	0
$205$	−6.79295	−0.474441
$206$	0	0
$207$	0.624196	0.0433847
$208$	0	0
$209$	−15.1430	−1.04746
$210$	0	0
$211$	−10.3122	−0.709920	−0.354960	−	0.934881i	$-0.615506\pi$
$211$	−10.3122	−0.709920	−0.354960	+	0.934881i	$0.615506\pi$
$212$	0	0
$213$	24.0945	1.65093
$214$	0	0
$215$	8.48745	0.578839
$216$	0	0
$217$	9.12494	0.619442
$218$	0	0
$219$	20.3035	1.37198
$220$	0	0
$221$	6.15154	0.413797
$222$	0	0
$223$	−9.54825	−0.639398	−0.319699	−	0.947519i	$-0.603582\pi$
$223$	−9.54825	−0.639398	−0.319699	+	0.947519i	$0.603582\pi$
$224$	0	0
$225$	−0.240356	−0.0160237
$226$	0	0
$227$	0.583799	0.0387481	0.0193741	−	0.999812i	$-0.493833\pi$
$227$	0.583799	0.0387481	0.0193741	+	0.999812i	$0.493833\pi$
$228$	0	0
$229$	26.6887	1.76364	0.881818	−	0.471589i	$-0.156319\pi$
$229$	26.6887	1.76364	0.881818	+	0.471589i	$0.156319\pi$
$230$	0	0
$231$	−8.91830	−0.586781
$232$	0	0
$233$	22.7355	1.48945	0.744727	−	0.667369i	$-0.232580\pi$
$233$	22.7355	1.48945	0.744727	+	0.667369i	$0.232580\pi$
$234$	0	0
$235$	−2.53625	−0.165447
$236$	0	0
$237$	−14.8807	−0.966603
$238$	0	0
$239$	7.62496	0.493218	0.246609	−	0.969115i	$-0.420684\pi$
$239$	7.62496	0.493218	0.246609	+	0.969115i	$0.420684\pi$
$240$	0	0
$241$	9.42005	0.606798	0.303399	−	0.952864i	$-0.401878\pi$
$241$	9.42005	0.606798	0.303399	+	0.952864i	$0.401878\pi$
$242$	0	0
$243$	2.49167	0.159841
$244$	0	0
$245$	−1.89653	−0.121165
$246$	0	0
$247$	13.5460	0.861909
$248$	0	0
$249$	−3.59637	−0.227911
$250$	0	0
$251$	1.03149	0.0651074	0.0325537	−	0.999470i	$-0.489636\pi$
$251$	1.03149	0.0651074	0.0325537	+	0.999470i	$0.489636\pi$
$252$	0	0
$253$	6.17147	0.387997
$254$	0	0
$255$	4.80717	0.301037
$256$	0	0
$257$	−30.8773	−1.92607	−0.963037	−	0.269369i	$-0.913185\pi$
$257$	−30.8773	−1.92607	−0.963037	+	0.269369i	$0.913185\pi$
$258$	0	0
$259$	13.7896	0.856842
$260$	0	0
$261$	−0.215835	−0.0133599
$262$	0	0
$263$	24.7517	1.52625	0.763127	−	0.646248i	$-0.223663\pi$
$263$	24.7517	1.52625	0.763127	+	0.646248i	$0.223663\pi$
$264$	0	0
$265$	−7.19062	−0.441716
$266$	0	0
$267$	26.1550	1.60066
$268$	0	0
$269$	−31.3577	−1.91192	−0.955958	−	0.293504i	$-0.905179\pi$
$269$	−31.3577	−1.91192	−0.955958	+	0.293504i	$0.905179\pi$
$270$	0	0
$271$	24.4505	1.48526	0.742630	−	0.669702i	$-0.233578\pi$
$271$	24.4505	1.48526	0.742630	+	0.669702i	$0.233578\pi$
$272$	0	0
$273$	7.97774	0.482835
$274$	0	0
$275$	−2.37642	−0.143303
$276$	0	0
$277$	−13.3707	−0.803366	−0.401683	−	0.915779i	$-0.631575\pi$
$277$	−13.3707	−0.803366	−0.401683	+	0.915779i	$0.631575\pi$
$278$	0	0
$279$	−0.970851	−0.0581233
$280$	0	0
$281$	−10.7007	−0.638351	−0.319175	−	0.947696i	$-0.603406\pi$
$281$	−10.7007	−0.638351	−0.319175	+	0.947696i	$0.603406\pi$
$282$	0	0
$283$	30.6584	1.82245	0.911225	−	0.411909i	$-0.135138\pi$
$283$	30.6584	1.82245	0.911225	+	0.411909i	$0.135138\pi$
$284$	0	0
$285$	10.5856	0.627037
$286$	0	0
$287$	−15.3459	−0.905838
$288$	0	0
$289$	−8.62613	−0.507419
$290$	0	0
$291$	−31.9926	−1.87544
$292$	0	0
$293$	−25.2174	−1.47322	−0.736608	−	0.676320i	$-0.763574\pi$
$293$	−25.2174	−1.47322	−0.736608	+	0.676320i	$0.763574\pi$
$294$	0	0
$295$	−4.26330	−0.248219
$296$	0	0
$297$	12.7921	0.742273
$298$	0	0
$299$	−5.52061	−0.319265
$300$	0	0
$301$	19.1739	1.10516
$302$	0	0
$303$	−0.0622296	−0.00357500
$304$	0	0
$305$	8.35347	0.478318
$306$	0	0
$307$	4.37149	0.249494	0.124747	−	0.992189i	$-0.460188\pi$
$307$	4.37149	0.249494	0.124747	+	0.992189i	$0.460188\pi$
$308$	0	0
$309$	−12.2064	−0.694397
$310$	0	0
$311$	−17.8839	−1.01410	−0.507051	−	0.861916i	$-0.669264\pi$
$311$	−17.8839	−1.01410	−0.507051	+	0.861916i	$0.669264\pi$
$312$	0	0
$313$	19.0300	1.07564	0.537820	−	0.843060i	$-0.319248\pi$
$313$	19.0300	1.07564	0.537820	+	0.843060i	$0.319248\pi$
$314$	0	0
$315$	−0.542985	−0.0305937
$316$	0	0
$317$	−27.7412	−1.55810	−0.779050	−	0.626962i	$-0.784298\pi$
$317$	−27.7412	−1.55810	−0.779050	+	0.626962i	$0.784298\pi$
$318$	0	0
$319$	−2.13398	−0.119480
$320$	0	0
$321$	−13.7770	−0.768958
$322$	0	0
$323$	18.4396	1.02601
$324$	0	0
$325$	2.12579	0.117918
$326$	0	0
$327$	29.2417	1.61707
$328$	0	0
$329$	−5.72961	−0.315884
$330$	0	0
$331$	−19.4520	−1.06918	−0.534590	−	0.845112i	$-0.679534\pi$
$331$	−19.4520	−1.06918	−0.534590	+	0.845112i	$0.679534\pi$
$332$	0	0
$333$	−1.46714	−0.0803990
$334$	0	0
$335$	11.1055	0.606757
$336$	0	0
$337$	−22.3242	−1.21608	−0.608038	−	0.793908i	$-0.708043\pi$
$337$	−22.3242	−1.21608	−0.608038	+	0.793908i	$0.708043\pi$
$338$	0	0
$339$	7.66813	0.416476
$340$	0	0
$341$	−9.59887	−0.519808
$342$	0	0
$343$	−20.0980	−1.08519
$344$	0	0
$345$	−4.31413	−0.232265
$346$	0	0
$347$	5.72778	0.307483	0.153742	−	0.988111i	$-0.450868\pi$
$347$	5.72778	0.307483	0.153742	+	0.988111i	$0.450868\pi$
$348$	0	0
$349$	−6.03118	−0.322842	−0.161421	−	0.986886i	$-0.551608\pi$
$349$	−6.03118	−0.322842	−0.161421	+	0.986886i	$0.551608\pi$
$350$	0	0
$351$	−11.4430	−0.610782
$352$	0	0
$353$	36.1958	1.92651	0.963255	−	0.268588i	$-0.0865570\pi$
$353$	36.1958	1.92651	0.963255	+	0.268588i	$0.0865570\pi$
$354$	0	0
$355$	14.5041	0.769798
$356$	0	0
$357$	10.8598	0.574762
$358$	0	0
$359$	−3.86917	−0.204207	−0.102104	−	0.994774i	$-0.532557\pi$
$359$	−3.86917	−0.204207	−0.102104	+	0.994774i	$0.532557\pi$
$360$	0	0
$361$	21.6049	1.13710
$362$	0	0
$363$	−8.89190	−0.466704
$364$	0	0
$365$	12.2221	0.639732
$366$	0	0
$367$	−25.1739	−1.31407	−0.657033	−	0.753862i	$-0.728189\pi$
$367$	−25.1739	−1.31407	−0.657033	+	0.753862i	$0.728189\pi$
$368$	0	0
$369$	1.63273	0.0849964
$370$	0	0
$371$	−16.2442	−0.843358
$372$	0	0
$373$	−14.7741	−0.764974	−0.382487	−	0.923961i	$-0.624932\pi$
$373$	−14.7741	−0.764974	−0.382487	+	0.923961i	$0.624932\pi$
$374$	0	0
$375$	1.66122	0.0857849
$376$	0	0
$377$	1.90892	0.0983146
$378$	0	0
$379$	−25.2569	−1.29736	−0.648681	−	0.761061i	$-0.724679\pi$
$379$	−25.2569	−1.29736	−0.648681	+	0.761061i	$0.724679\pi$
$380$	0	0
$381$	−16.2814	−0.834120
$382$	0	0
$383$	28.9525	1.47940	0.739701	−	0.672936i	$-0.234967\pi$
$383$	28.9525	1.47940	0.739701	+	0.672936i	$0.234967\pi$
$384$	0	0
$385$	−5.36853	−0.273606
$386$	0	0
$387$	−2.04001	−0.103699
$388$	0	0
$389$	−4.71275	−0.238946	−0.119473	−	0.992837i	$-0.538120\pi$
$389$	−4.71275	−0.238946	−0.119473	+	0.992837i	$0.538120\pi$
$390$	0	0
$391$	−7.51501	−0.380050
$392$	0	0
$393$	2.54546	0.128401
$394$	0	0
$395$	−8.95769	−0.450710
$396$	0	0
$397$	−34.0662	−1.70973	−0.854867	−	0.518847i	$-0.826361\pi$
$397$	−34.0662	−1.70973	−0.854867	+	0.518847i	$0.826361\pi$
$398$	0	0
$399$	23.9138	1.19719
$400$	0	0
$401$	−15.4480	−0.771437	−0.385719	−	0.922616i	$-0.626046\pi$
$401$	−15.4480	−0.771437	−0.385719	+	0.922616i	$0.626046\pi$
$402$	0	0
$403$	8.58654	0.427726
$404$	0	0
$405$	−8.22116	−0.408513
$406$	0	0
$407$	−14.5058	−0.719023
$408$	0	0
$409$	−7.37891	−0.364863	−0.182432	−	0.983219i	$-0.558397\pi$
$409$	−7.37891	−0.364863	−0.182432	+	0.983219i	$0.558397\pi$
$410$	0	0
$411$	−31.8688	−1.57197
$412$	0	0
$413$	−9.63117	−0.473919
$414$	0	0
$415$	−2.16490	−0.106271
$416$	0	0
$417$	10.2754	0.503189
$418$	0	0
$419$	26.0723	1.27371	0.636857	−	0.770982i	$-0.280234\pi$
$419$	26.0723	1.27371	0.636857	+	0.770982i	$0.280234\pi$
$420$	0	0
$421$	−7.60746	−0.370765	−0.185383	−	0.982666i	$-0.559352\pi$
$421$	−7.60746	−0.370765	−0.185383	+	0.982666i	$0.559352\pi$
$422$	0	0
$423$	0.609603	0.0296399
$424$	0	0
$425$	2.89376	0.140368
$426$	0	0
$427$	18.8712	0.913242
$428$	0	0
$429$	−8.39208	−0.405174
$430$	0	0
$431$	5.98489	0.288282	0.144141	−	0.989557i	$-0.453958\pi$
$431$	5.98489	0.288282	0.144141	+	0.989557i	$0.453958\pi$
$432$	0	0
$433$	24.6307	1.18368	0.591838	−	0.806057i	$-0.298403\pi$
$433$	24.6307	1.18368	0.591838	+	0.806057i	$0.298403\pi$
$434$	0	0
$435$	1.49174	0.0715237
$436$	0	0
$437$	−16.5484	−0.791617
$438$	0	0
$439$	−34.6304	−1.65282	−0.826410	−	0.563069i	$-0.809621\pi$
$439$	−34.6304	−1.65282	−0.826410	+	0.563069i	$0.809621\pi$
$440$	0	0
$441$	0.455842	0.0217068
$442$	0	0
$443$	16.2207	0.770668	0.385334	−	0.922777i	$-0.374086\pi$
$443$	16.2207	0.770668	0.385334	+	0.922777i	$0.374086\pi$
$444$	0	0
$445$	15.7445	0.746360
$446$	0	0
$447$	19.6856	0.931098
$448$	0	0
$449$	1.01449	0.0478770	0.0239385	−	0.999713i	$-0.492379\pi$
$449$	1.01449	0.0478770	0.0239385	+	0.999713i	$0.492379\pi$
$450$	0	0
$451$	16.1429	0.760139
$452$	0	0
$453$	1.66122	0.0780508
$454$	0	0
$455$	4.80235	0.225137
$456$	0	0
$457$	−7.46204	−0.349060	−0.174530	−	0.984652i	$-0.555841\pi$
$457$	−7.46204	−0.349060	−0.174530	+	0.984652i	$0.555841\pi$
$458$	0	0
$459$	−15.5769	−0.727070
$460$	0	0
$461$	1.78047	0.0829247	0.0414624	−	0.999140i	$-0.486798\pi$
$461$	1.78047	0.0829247	0.0414624	+	0.999140i	$0.486798\pi$
$462$	0	0
$463$	2.06503	0.0959703	0.0479851	−	0.998848i	$-0.484720\pi$
$463$	2.06503	0.0959703	0.0479851	+	0.998848i	$0.484720\pi$
$464$	0	0
$465$	6.71002	0.311170
$466$	0	0
$467$	19.4453	0.899821	0.449910	−	0.893074i	$-0.351456\pi$
$467$	19.4453	0.899821	0.449910	+	0.893074i	$0.351456\pi$
$468$	0	0
$469$	25.0882	1.15847
$470$	0	0
$471$	−16.7842	−0.773376
$472$	0	0
$473$	−20.1697	−0.927404
$474$	0	0
$475$	6.37220	0.292377
$476$	0	0
$477$	1.72831	0.0791338
$478$	0	0
$479$	−35.0630	−1.60207	−0.801034	−	0.598619i	$-0.795716\pi$
$479$	−35.0630	−1.60207	−0.801034	+	0.598619i	$0.795716\pi$
$480$	0	0
$481$	12.9759	0.591651
$482$	0	0
$483$	−9.74598	−0.443458
$484$	0	0
$485$	−19.2585	−0.874484
$486$	0	0
$487$	8.69432	0.393977	0.196989	−	0.980406i	$-0.436884\pi$
$487$	8.69432	0.393977	0.196989	+	0.980406i	$0.436884\pi$
$488$	0	0
$489$	−2.52416	−0.114146
$490$	0	0
$491$	12.6643	0.571531	0.285766	−	0.958300i	$-0.407752\pi$
$491$	12.6643	0.571531	0.285766	+	0.958300i	$0.407752\pi$
$492$	0	0
$493$	2.59855	0.117033
$494$	0	0
$495$	0.571186	0.0256729
$496$	0	0
$497$	32.7660	1.46976
$498$	0	0
$499$	−1.01019	−0.0452223	−0.0226111	−	0.999744i	$-0.507198\pi$
$499$	−1.01019	−0.0452223	−0.0226111	+	0.999744i	$0.507198\pi$
$500$	0	0
$501$	21.0970	0.942543
$502$	0	0
$503$	−5.86293	−0.261415	−0.130708	−	0.991421i	$-0.541725\pi$
$503$	−5.86293	−0.261415	−0.130708	+	0.991421i	$0.541725\pi$
$504$	0	0
$505$	−0.0374602	−0.00166696
$506$	0	0
$507$	−14.0888	−0.625706
$508$	0	0
$509$	−1.80778	−0.0801284	−0.0400642	−	0.999197i	$-0.512756\pi$
$509$	−1.80778	−0.0801284	−0.0400642	+	0.999197i	$0.512756\pi$
$510$	0	0
$511$	27.6107	1.22143
$512$	0	0
$513$	−34.3011	−1.51443
$514$	0	0
$515$	−7.34786	−0.323785
$516$	0	0
$517$	6.02719	0.265076
$518$	0	0
$519$	−39.3424	−1.72694
$520$	0	0
$521$	19.8441	0.869384	0.434692	−	0.900579i	$-0.356857\pi$
$521$	19.8441	0.869384	0.434692	+	0.900579i	$0.356857\pi$
$522$	0	0
$523$	11.9493	0.522507	0.261254	−	0.965270i	$-0.415864\pi$
$523$	11.9493	0.522507	0.261254	+	0.965270i	$0.415864\pi$
$524$	0	0
$525$	3.75283	0.163787
$526$	0	0
$527$	11.6885	0.509161
$528$	0	0
$529$	−16.2558	−0.706772
$530$	0	0
$531$	1.02471	0.0444687
$532$	0	0
$533$	−14.4404	−0.625483
$534$	0	0
$535$	−8.29333	−0.358552
$536$	0	0
$537$	−41.5411	−1.79263
$538$	0	0
$539$	4.50695	0.194128
$540$	0	0
$541$	−40.9130	−1.75899	−0.879494	−	0.475910i	$-0.842119\pi$
$541$	−40.9130	−1.75899	−0.879494	+	0.475910i	$0.842119\pi$
$542$	0	0
$543$	−6.46917	−0.277619
$544$	0	0
$545$	17.6026	0.754012
$546$	0	0
$547$	−25.7630	−1.10155	−0.550773	−	0.834655i	$-0.685667\pi$
$547$	−25.7630	−1.10155	−0.550773	+	0.834655i	$0.685667\pi$
$548$	0	0
$549$	−2.00781	−0.0856911
$550$	0	0
$551$	5.72212	0.243771
$552$	0	0
$553$	−20.2362	−0.860530
$554$	0	0
$555$	10.1401	0.430425
$556$	0	0
$557$	−15.1783	−0.643126	−0.321563	−	0.946888i	$-0.604208\pi$
$557$	−15.1783	−0.643126	−0.321563	+	0.946888i	$0.604208\pi$
$558$	0	0
$559$	18.0425	0.763118
$560$	0	0
$561$	−11.4238	−0.482315
$562$	0	0
$563$	45.5973	1.92170	0.960849	−	0.277073i	$-0.0893643\pi$
$563$	45.5973	1.92170	0.960849	+	0.277073i	$0.0893643\pi$
$564$	0	0
$565$	4.61597	0.194195
$566$	0	0
$567$	−18.5723	−0.779964
$568$	0	0
$569$	4.01811	0.168448	0.0842240	−	0.996447i	$-0.473159\pi$
$569$	4.01811	0.168448	0.0842240	+	0.996447i	$0.473159\pi$
$570$	0	0
$571$	−0.193734	−0.00810751	−0.00405375	−	0.999992i	$-0.501290\pi$
$571$	−0.193734	−0.00810751	−0.00405375	+	0.999992i	$0.501290\pi$
$572$	0	0
$573$	10.3892	0.434015
$574$	0	0
$575$	−2.59697	−0.108301
$576$	0	0
$577$	−21.9005	−0.911729	−0.455865	−	0.890049i	$-0.650670\pi$
$577$	−21.9005	−0.911729	−0.455865	+	0.890049i	$0.650670\pi$
$578$	0	0
$579$	6.21535	0.258301
$580$	0	0
$581$	−4.89070	−0.202900
$582$	0	0
$583$	17.0879	0.707709
$584$	0	0
$585$	−0.510947	−0.0211250
$586$	0	0
$587$	28.7183	1.18533	0.592666	−	0.805448i	$-0.298075\pi$
$587$	28.7183	1.18533	0.592666	+	0.805448i	$0.298075\pi$
$588$	0	0
$589$	25.7387	1.06054
$590$	0	0
$591$	40.7006	1.67420
$592$	0	0
$593$	−38.0504	−1.56254	−0.781271	−	0.624192i	$-0.785428\pi$
$593$	−38.0504	−1.56254	−0.781271	+	0.624192i	$0.785428\pi$
$594$	0	0
$595$	6.53726	0.268002
$596$	0	0
$597$	−19.8305	−0.811607
$598$	0	0
$599$	−25.9029	−1.05836	−0.529182	−	0.848509i	$-0.677501\pi$
$599$	−25.9029	−1.05836	−0.529182	+	0.848509i	$0.677501\pi$
$600$	0	0
$601$	29.4146	1.19985	0.599923	−	0.800058i	$-0.295198\pi$
$601$	29.4146	1.19985	0.599923	+	0.800058i	$0.295198\pi$
$602$	0	0
$603$	−2.66927	−0.108701
$604$	0	0
$605$	−5.35264	−0.217616
$606$	0	0
$607$	35.4041	1.43701	0.718505	−	0.695522i	$-0.244827\pi$
$607$	35.4041	1.43701	0.718505	+	0.695522i	$0.244827\pi$
$608$	0	0
$609$	3.36998	0.136558
$610$	0	0
$611$	−5.39154	−0.218118
$612$	0	0
$613$	6.83612	0.276108	0.138054	−	0.990425i	$-0.455915\pi$
$613$	6.83612	0.276108	0.138054	+	0.990425i	$0.455915\pi$
$614$	0	0
$615$	−11.2846	−0.455038
$616$	0	0
$617$	−23.1563	−0.932238	−0.466119	−	0.884722i	$-0.654348\pi$
$617$	−23.1563	−0.932238	−0.466119	+	0.884722i	$0.654348\pi$
$618$	0	0
$619$	35.3277	1.41994	0.709971	−	0.704231i	$-0.248708\pi$
$619$	35.3277	1.41994	0.709971	+	0.704231i	$0.248708\pi$
$620$	0	0
$621$	13.9793	0.560970
$622$	0	0
$623$	35.5681	1.42501
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−25.1558	−1.00463
$628$	0	0
$629$	17.6637	0.704296
$630$	0	0
$631$	−15.7275	−0.626103	−0.313052	−	0.949736i	$-0.601351\pi$
$631$	−15.7275	−0.626103	−0.313052	+	0.949736i	$0.601351\pi$
$632$	0	0
$633$	−17.1308	−0.680888
$634$	0	0
$635$	−9.80087	−0.388936
$636$	0	0
$637$	−4.03163	−0.159739
$638$	0	0
$639$	−3.48615	−0.137910
$640$	0	0
$641$	−45.6336	−1.80242	−0.901210	−	0.433383i	$-0.857320\pi$
$641$	−45.6336	−1.80242	−0.901210	+	0.433383i	$0.857320\pi$
$642$	0	0
$643$	−12.8284	−0.505904	−0.252952	−	0.967479i	$-0.581401\pi$
$643$	−12.8284	−0.505904	−0.252952	+	0.967479i	$0.581401\pi$
$644$	0	0
$645$	14.0995	0.555167
$646$	0	0
$647$	−25.3936	−0.998327	−0.499164	−	0.866508i	$-0.666359\pi$
$647$	−25.3936	−0.998327	−0.499164	+	0.866508i	$0.666359\pi$
$648$	0	0
$649$	10.1314	0.397692
$650$	0	0
$651$	15.1585	0.594109
$652$	0	0
$653$	3.02753	0.118476	0.0592381	−	0.998244i	$-0.481133\pi$
$653$	3.02753	0.118476	0.0592381	+	0.998244i	$0.481133\pi$
$654$	0	0
$655$	1.53228	0.0598713
$656$	0	0
$657$	−2.93765	−0.114609
$658$	0	0
$659$	−10.9431	−0.426283	−0.213142	−	0.977021i	$-0.568370\pi$
$659$	−10.9431	−0.426283	−0.213142	+	0.977021i	$0.568370\pi$
$660$	0	0
$661$	−39.6182	−1.54097	−0.770484	−	0.637460i	$-0.779985\pi$
$661$	−39.6182	−1.54097	−0.770484	+	0.637460i	$0.779985\pi$
$662$	0	0
$663$	10.2190	0.396875
$664$	0	0
$665$	14.3953	0.558228
$666$	0	0
$667$	−2.33203	−0.0902966
$668$	0	0
$669$	−15.8617	−0.613249
$670$	0	0
$671$	−19.8513	−0.766352
$672$	0	0
$673$	−17.2693	−0.665683	−0.332841	−	0.942983i	$-0.608007\pi$
$673$	−17.2693	−0.665683	−0.332841	+	0.942983i	$0.608007\pi$
$674$	0	0
$675$	−5.38294	−0.207189
$676$	0	0
$677$	15.0001	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$677$	15.0001	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$678$	0	0
$679$	−43.5067	−1.66963
$680$	0	0
$681$	0.969817	0.0371635
$682$	0	0
$683$	−13.9444	−0.533566	−0.266783	−	0.963757i	$-0.585961\pi$
$683$	−13.9444	−0.533566	−0.266783	+	0.963757i	$0.585961\pi$
$684$	0	0
$685$	−19.1840	−0.732982
$686$	0	0
$687$	44.3357	1.69151
$688$	0	0
$689$	−15.2857	−0.582341
$690$	0	0
$691$	−24.5840	−0.935218	−0.467609	−	0.883936i	$-0.654884\pi$
$691$	−24.5840	−0.935218	−0.467609	+	0.883936i	$0.654884\pi$
$692$	0	0
$693$	1.29036	0.0490167
$694$	0	0
$695$	6.18547	0.234628
$696$	0	0
$697$	−19.6572	−0.744570
$698$	0	0
$699$	37.7687	1.42854
$700$	0	0
$701$	−47.7857	−1.80484	−0.902421	−	0.430855i	$-0.858212\pi$
$701$	−47.7857	−1.80484	−0.902421	+	0.430855i	$0.858212\pi$
$702$	0	0
$703$	38.8962	1.46700
$704$	0	0
$705$	−4.21327	−0.158681
$706$	0	0
$707$	−0.0846259	−0.00318268
$708$	0	0
$709$	20.6579	0.775824	0.387912	−	0.921696i	$-0.373196\pi$
$709$	20.6579	0.775824	0.387912	+	0.921696i	$0.373196\pi$
$710$	0	0
$711$	2.15303	0.0807451
$712$	0	0
$713$	−10.4897	−0.392843
$714$	0	0
$715$	−5.05177	−0.188925
$716$	0	0
$717$	12.6667	0.473047
$718$	0	0
$719$	−0.593212	−0.0221231	−0.0110615	−	0.999939i	$-0.503521\pi$
$719$	−0.593212	−0.0221231	−0.0110615	+	0.999939i	$0.503521\pi$
$720$	0	0
$721$	−16.5994	−0.618196
$722$	0	0
$723$	15.6487	0.581983
$724$	0	0
$725$	0.897983	0.0333502
$726$	0	0
$727$	2.12677	0.0788775	0.0394388	−	0.999222i	$-0.487443\pi$
$727$	2.12677	0.0788775	0.0394388	+	0.999222i	$0.487443\pi$
$728$	0	0
$729$	28.8027	1.06677
$730$	0	0
$731$	24.5607	0.908409
$732$	0	0
$733$	−6.00535	−0.221813	−0.110906	−	0.993831i	$-0.535375\pi$
$733$	−6.00535	−0.221813	−0.110906	+	0.993831i	$0.535375\pi$
$734$	0	0
$735$	−3.15055	−0.116210
$736$	0	0
$737$	−26.3912	−0.972134
$738$	0	0
$739$	48.7104	1.79184	0.895921	−	0.444213i	$-0.146517\pi$
$739$	48.7104	1.79184	0.895921	+	0.444213i	$0.146517\pi$
$740$	0	0
$741$	22.5028	0.826661
$742$	0	0
$743$	20.7005	0.759427	0.379713	−	0.925104i	$-0.376023\pi$
$743$	20.7005	0.759427	0.379713	+	0.925104i	$0.376023\pi$
$744$	0	0
$745$	11.8501	0.434155
$746$	0	0
$747$	0.520347	0.0190385
$748$	0	0
$749$	−18.7353	−0.684574
$750$	0	0
$751$	−2.12159	−0.0774180	−0.0387090	−	0.999251i	$-0.512325\pi$
$751$	−2.12159	−0.0774180	−0.0387090	+	0.999251i	$0.512325\pi$
$752$	0	0
$753$	1.71354	0.0624448
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	−1.07287	−0.0389943	−0.0194971	−	0.999810i	$-0.506207\pi$
$757$	−1.07287	−0.0389943	−0.0194971	+	0.999810i	$0.506207\pi$
$758$	0	0
$759$	10.2522	0.372130
$760$	0	0
$761$	14.1688	0.513620	0.256810	−	0.966462i	$-0.417329\pi$
$761$	14.1688	0.513620	0.256810	+	0.966462i	$0.417329\pi$
$762$	0	0
$763$	39.7658	1.43962
$764$	0	0
$765$	−0.695534	−0.0251471
$766$	0	0
$767$	−9.06290	−0.327242
$768$	0	0
$769$	−40.4822	−1.45982	−0.729912	−	0.683541i	$-0.760439\pi$
$769$	−40.4822	−1.45982	−0.729912	+	0.683541i	$0.760439\pi$
$770$	0	0
$771$	−51.2939	−1.84731
$772$	0	0
$773$	43.1713	1.55276	0.776382	−	0.630263i	$-0.217053\pi$
$773$	43.1713	1.55276	0.776382	+	0.630263i	$0.217053\pi$
$774$	0	0
$775$	4.03922	0.145093
$776$	0	0
$777$	22.9075	0.821801
$778$	0	0
$779$	−43.2861	−1.55088
$780$	0	0
$781$	−34.4678	−1.23335
$782$	0	0
$783$	−4.83378	−0.172745
$784$	0	0
$785$	−10.1036	−0.360611
$786$	0	0
$787$	2.76943	0.0987195	0.0493597	−	0.998781i	$-0.484282\pi$
$787$	2.76943	0.0987195	0.0493597	+	0.998781i	$0.484282\pi$
$788$	0	0
$789$	41.1179	1.46384
$790$	0	0
$791$	10.4279	0.370773
$792$	0	0
$793$	17.7577	0.630595
$794$	0	0
$795$	−11.9452	−0.423652
$796$	0	0
$797$	−21.1007	−0.747423	−0.373712	−	0.927545i	$-0.621915\pi$
$797$	−21.1007	−0.747423	−0.373712	+	0.927545i	$0.621915\pi$
$798$	0	0
$799$	−7.33931	−0.259646
$800$	0	0
$801$	−3.78428	−0.133711
$802$	0	0
$803$	−29.0447	−1.02497
$804$	0	0
$805$	−5.86677	−0.206777
$806$	0	0
$807$	−52.0920	−1.83373
$808$	0	0
$809$	−18.1348	−0.637587	−0.318794	−	0.947824i	$-0.603278\pi$
$809$	−18.1348	−0.637587	−0.318794	+	0.947824i	$0.603278\pi$
$810$	0	0
$811$	3.37723	0.118591	0.0592953	−	0.998240i	$-0.481115\pi$
$811$	3.37723	0.118591	0.0592953	+	0.998240i	$0.481115\pi$
$812$	0	0
$813$	40.6175	1.42452
$814$	0	0
$815$	−1.51946	−0.0532244
$816$	0	0
$817$	54.0837	1.89215
$818$	0	0
$819$	−1.15427	−0.0403335
$820$	0	0
$821$	−55.3176	−1.93060	−0.965299	−	0.261146i	$-0.915900\pi$
$821$	−55.3176	−1.93060	−0.965299	+	0.261146i	$0.915900\pi$
$822$	0	0
$823$	−32.5310	−1.13396	−0.566981	−	0.823731i	$-0.691889\pi$
$823$	−32.5310	−1.13396	−0.566981	+	0.823731i	$0.691889\pi$
$824$	0	0
$825$	−3.94775	−0.137443
$826$	0	0
$827$	−10.4918	−0.364836	−0.182418	−	0.983221i	$-0.558392\pi$
$827$	−10.4918	−0.364836	−0.182418	+	0.983221i	$0.558392\pi$
$828$	0	0
$829$	−29.9411	−1.03990	−0.519948	−	0.854198i	$-0.674049\pi$
$829$	−29.9411	−1.03990	−0.519948	+	0.854198i	$0.674049\pi$
$830$	0	0
$831$	−22.2116	−0.770512
$832$	0	0
$833$	−5.48811	−0.190152
$834$	0	0
$835$	12.6997	0.439491
$836$	0	0
$837$	−21.7429	−0.751543
$838$	0	0
$839$	−18.5976	−0.642060	−0.321030	−	0.947069i	$-0.604029\pi$
$839$	−18.5976	−0.642060	−0.321030	+	0.947069i	$0.604029\pi$
$840$	0	0
$841$	−28.1936	−0.972194
$842$	0	0
$843$	−17.7762	−0.612245
$844$	0	0
$845$	−8.48101	−0.291756
$846$	0	0
$847$	−12.0921	−0.415489
$848$	0	0
$849$	50.9302	1.74792
$850$	0	0
$851$	−15.8520	−0.543399
$852$	0	0
$853$	−25.5739	−0.875635	−0.437817	−	0.899064i	$-0.644248\pi$
$853$	−25.5739	−0.875635	−0.437817	+	0.899064i	$0.644248\pi$
$854$	0	0
$855$	−1.53160	−0.0523795
$856$	0	0
$857$	7.11527	0.243053	0.121527	−	0.992588i	$-0.461221\pi$
$857$	7.11527	0.243053	0.121527	+	0.992588i	$0.461221\pi$
$858$	0	0
$859$	−56.9738	−1.94392	−0.971960	−	0.235146i	$-0.924443\pi$
$859$	−56.9738	−1.94392	−0.971960	+	0.235146i	$0.924443\pi$
$860$	0	0
$861$	−25.4928	−0.868793
$862$	0	0
$863$	12.5659	0.427750	0.213875	−	0.976861i	$-0.431392\pi$
$863$	12.5659	0.427750	0.213875	+	0.976861i	$0.431392\pi$
$864$	0	0
$865$	−23.6829	−0.805242
$866$	0	0
$867$	−14.3299	−0.486668
$868$	0	0
$869$	21.2872	0.722119
$870$	0	0
$871$	23.6079	0.799924
$872$	0	0
$873$	4.62890	0.156665
$874$	0	0
$875$	2.25909	0.0763710
$876$	0	0
$877$	−0.900600	−0.0304111	−0.0152055	−	0.999884i	$-0.504840\pi$
$877$	−0.900600	−0.0304111	−0.0152055	+	0.999884i	$0.504840\pi$
$878$	0	0
$879$	−41.8916	−1.41297
$880$	0	0
$881$	41.4025	1.39489	0.697443	−	0.716641i	$-0.254321\pi$
$881$	41.4025	1.39489	0.697443	+	0.716641i	$0.254321\pi$
$882$	0	0
$883$	17.2025	0.578911	0.289455	−	0.957192i	$-0.406526\pi$
$883$	17.2025	0.578911	0.289455	+	0.957192i	$0.406526\pi$
$884$	0	0
$885$	−7.08228	−0.238068
$886$	0	0
$887$	35.4977	1.19190	0.595948	−	0.803023i	$-0.296777\pi$
$887$	35.4977	1.19190	0.595948	+	0.803023i	$0.296777\pi$
$888$	0	0
$889$	−22.1410	−0.742586
$890$	0	0
$891$	19.5369	0.654511
$892$	0	0
$893$	−16.1615	−0.540824
$894$	0	0
$895$	−25.0064	−0.835873
$896$	0	0
$897$	−9.17093	−0.306209
$898$	0	0
$899$	3.62715	0.120972
$900$	0	0
$901$	−20.8079	−0.693213
$902$	0	0
$903$	31.8520	1.05997
$904$	0	0
$905$	−3.89423	−0.129449
$906$	0	0
$907$	26.4426	0.878012	0.439006	−	0.898484i	$-0.355331\pi$
$907$	26.4426	0.878012	0.439006	+	0.898484i	$0.355331\pi$
$908$	0	0
$909$	0.00900379	0.000298637 0
$910$	0	0
$911$	−26.6060	−0.881495	−0.440747	−	0.897631i	$-0.645287\pi$
$911$	−26.6060	−0.881495	−0.440747	+	0.897631i	$0.645287\pi$
$912$	0	0
$913$	5.14471	0.170265
$914$	0	0
$915$	13.8769	0.458757
$916$	0	0
$917$	3.46156	0.114311
$918$	0	0
$919$	46.4039	1.53072	0.765362	−	0.643600i	$-0.222560\pi$
$919$	46.4039	1.53072	0.765362	+	0.643600i	$0.222560\pi$
$920$	0	0
$921$	7.26200	0.239291
$922$	0	0
$923$	30.8327	1.01487
$924$	0	0
$925$	6.10404	0.200700
$926$	0	0
$927$	1.76610	0.0580064
$928$	0	0
$929$	−17.6802	−0.580068	−0.290034	−	0.957016i	$-0.593667\pi$
$929$	−17.6802	−0.580068	−0.290034	+	0.957016i	$0.593667\pi$
$930$	0	0
$931$	−12.0851	−0.396072
$932$	0	0
$933$	−29.7090	−0.972630
$934$	0	0
$935$	−6.87679	−0.224895
$936$	0	0
$937$	14.1323	0.461681	0.230840	−	0.972992i	$-0.425853\pi$
$937$	14.1323	0.461681	0.230840	+	0.972992i	$0.425853\pi$
$938$	0	0
$939$	31.6130	1.03165
$940$	0	0
$941$	2.31063	0.0753244	0.0376622	−	0.999291i	$-0.488009\pi$
$941$	2.31063	0.0753244	0.0376622	+	0.999291i	$0.488009\pi$
$942$	0	0
$943$	17.6411	0.574472
$944$	0	0
$945$	−12.1605	−0.395582
$946$	0	0
$947$	−22.2355	−0.722556	−0.361278	−	0.932458i	$-0.617659\pi$
$947$	−22.2355	−0.722556	−0.361278	+	0.932458i	$0.617659\pi$
$948$	0	0
$949$	25.9816	0.843397
$950$	0	0
$951$	−46.0841	−1.49438
$952$	0	0
$953$	−32.8977	−1.06566	−0.532830	−	0.846222i	$-0.678872\pi$
$953$	−32.8977	−1.06566	−0.532830	+	0.846222i	$0.678872\pi$
$954$	0	0
$955$	6.25397	0.202374
$956$	0	0
$957$	−3.54501	−0.114594
$958$	0	0
$959$	−43.3382	−1.39947
$960$	0	0
$961$	−14.6847	−0.473700
$962$	0	0
$963$	1.99335	0.0642348
$964$	0	0
$965$	3.74144	0.120441
$966$	0	0
$967$	13.3813	0.430312	0.215156	−	0.976580i	$-0.430974\pi$
$967$	13.3813	0.430312	0.215156	+	0.976580i	$0.430974\pi$
$968$	0	0
$969$	30.6323	0.984050
$970$	0	0
$971$	−3.06239	−0.0982768	−0.0491384	−	0.998792i	$-0.515648\pi$
$971$	−3.06239	−0.0982768	−0.0491384	+	0.998792i	$0.515648\pi$
$972$	0	0
$973$	13.9735	0.447970
$974$	0	0
$975$	3.53140	0.113095
$976$	0	0
$977$	−0.541611	−0.0173277	−0.00866384	−	0.999962i	$-0.502758\pi$
$977$	−0.541611	−0.0173277	−0.00866384	+	0.999962i	$0.502758\pi$
$978$	0	0
$979$	−37.4155	−1.19580
$980$	0	0
$981$	−4.23089	−0.135082
$982$	0	0
$983$	−13.9628	−0.445343	−0.222671	−	0.974894i	$-0.571478\pi$
$983$	−13.9628	−0.445343	−0.222671	+	0.974894i	$0.571478\pi$
$984$	0	0
$985$	24.5005	0.780650
$986$	0	0
$987$	−9.51813	−0.302965
$988$	0	0
$989$	−22.0416	−0.700882
$990$	0	0
$991$	−14.6278	−0.464667	−0.232334	−	0.972636i	$-0.574636\pi$
$991$	−14.6278	−0.464667	−0.232334	+	0.972636i	$0.574636\pi$
$992$	0	0
$993$	−32.3140	−1.02546
$994$	0	0
$995$	−11.9373	−0.378438
$996$	0	0
$997$	−27.6041	−0.874232	−0.437116	−	0.899405i	$-0.644000\pi$
$997$	−27.6041	−0.874232	−0.437116	+	0.899405i	$0.644000\pi$
$998$	0	0
$999$	−32.8577	−1.03957

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3020.2.a.f.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3020.2.a.f.1.4	✓	6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3020 = 2^{2} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3020.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66122 q^{3} +1.00000 q^{5} +2.25909 q^{7} -0.240356 q^{9} +O(q^{10})\)	\(q+1.66122 q^{3} +1.00000 q^{5} +2.25909 q^{7} -0.240356 q^{9} -2.37642 q^{11} +2.12579 q^{13} +1.66122 q^{15} +2.89376 q^{17} +6.37220 q^{19} +3.75283 q^{21} -2.59697 q^{23} +1.00000 q^{25} -5.38294 q^{27} +0.897983 q^{29} +4.03922 q^{31} -3.94775 q^{33} +2.25909 q^{35} +6.10404 q^{37} +3.53140 q^{39} -6.79295 q^{41} +8.48745 q^{43} -0.240356 q^{45} -2.53625 q^{47} -1.89653 q^{49} +4.80717 q^{51} -7.19062 q^{53} -2.37642 q^{55} +10.5856 q^{57} -4.26330 q^{59} +8.35347 q^{61} -0.542985 q^{63} +2.12579 q^{65} +11.1055 q^{67} -4.31413 q^{69} +14.5041 q^{71} +12.2221 q^{73} +1.66122 q^{75} -5.36853 q^{77} -8.95769 q^{79} -8.22116 q^{81} -2.16490 q^{83} +2.89376 q^{85} +1.49174 q^{87} +15.7445 q^{89} +4.80235 q^{91} +6.71002 q^{93} +6.37220 q^{95} -19.2585 q^{97} +0.571186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9	\( 6 q + 2 q^{3} + 6 q^{5} + 3 q^{7} + 20 q^{9} + 3 q^{11} + 2 q^{13} + 2 q^{15} + 9 q^{17} + 16 q^{19} - 12 q^{21} + 6 q^{23} + 6 q^{25} + 23 q^{27} + 2 q^{29} + 15 q^{31} - 14 q^{33} + 3 q^{35} + 9 q^{37} - 17 q^{39} - 16 q^{41} + 15 q^{43} + 20 q^{45} + 12 q^{47} - 7 q^{49} + 11 q^{51} - 14 q^{53} + 3 q^{55} + 27 q^{57} - 20 q^{59} + 17 q^{61} + 17 q^{63} + 2 q^{65} + 29 q^{67} - 40 q^{69} + 13 q^{71} - 6 q^{73} + 2 q^{75} + 13 q^{77} - 8 q^{79} + 30 q^{81} + 36 q^{83} + 9 q^{85} + 2 q^{87} - 15 q^{89} - 14 q^{91} - 31 q^{93} + 16 q^{95} - 23 q^{97} + 38 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 6 * q^5 + 3 * q^7 + 20 * q^9 + 3 * q^11 + 2 * q^13 + 2 * q^15 + 9 * q^17 + 16 * q^19 - 12 * q^21 + 6 * q^23 + 6 * q^25 + 23 * q^27 + 2 * q^29 + 15 * q^31 - 14 * q^33 + 3 * q^35 + 9 * q^37 - 17 * q^39 - 16 * q^41 + 15 * q^43 + 20 * q^45 + 12 * q^47 - 7 * q^49 + 11 * q^51 - 14 * q^53 + 3 * q^55 + 27 * q^57 - 20 * q^59 + 17 * q^61 + 17 * q^63 + 2 * q^65 + 29 * q^67 - 40 * q^69 + 13 * q^71 - 6 * q^73 + 2 * q^75 + 13 * q^77 - 8 * q^79 + 30 * q^81 + 36 * q^83 + 9 * q^85 + 2 * q^87 - 15 * q^89 - 14 * q^91 - 31 * q^93 + 16 * q^95 - 23 * q^97 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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