LMFDB - Embedded newform 6021.2.a.l.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 20x^{8} + 139x^{6} - 384x^{4} + 331x^{2} - 63 $ x^10 - 20x^8 + 139x^6 - 384x^4 + 331x^2 - 63
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.02020$ of defining polynomial
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.02020 q^{2} -0.959195 q^{4} -2.96046 q^{5} +0.816945 q^{7} -3.01897 q^{8} +O(q^{10})$	\(q+1.02020 q^{2} -0.959195 q^{4} -2.96046 q^{5} +0.816945 q^{7} -3.01897 q^{8} -3.02025 q^{10} -4.92179 q^{11} +1.06465 q^{13} +0.833446 q^{14} -1.16155 q^{16} +0.905065 q^{17} -0.776141 q^{19} +2.83966 q^{20} -5.02120 q^{22} -6.93545 q^{23} +3.76431 q^{25} +1.08615 q^{26} -0.783610 q^{28} +1.72077 q^{29} -5.90638 q^{31} +4.85292 q^{32} +0.923346 q^{34} -2.41853 q^{35} +5.58855 q^{37} -0.791817 q^{38} +8.93752 q^{40} -5.11599 q^{41} -0.255947 q^{43} +4.72096 q^{44} -7.07553 q^{46} -9.75266 q^{47} -6.33260 q^{49} +3.84034 q^{50} -1.02120 q^{52} -13.7259 q^{53} +14.5708 q^{55} -2.46633 q^{56} +1.75553 q^{58} -9.97782 q^{59} -3.23898 q^{61} -6.02568 q^{62} +7.27405 q^{64} -3.15184 q^{65} +6.41020 q^{67} -0.868134 q^{68} -2.46738 q^{70} +7.97672 q^{71} +14.0235 q^{73} +5.70143 q^{74} +0.744470 q^{76} -4.02083 q^{77} -11.1936 q^{79} +3.43873 q^{80} -5.21932 q^{82} +10.5277 q^{83} -2.67941 q^{85} -0.261116 q^{86} +14.8587 q^{88} +18.2503 q^{89} +0.869757 q^{91} +6.65245 q^{92} -9.94965 q^{94} +2.29773 q^{95} +7.82848 q^{97} -6.46051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) $ 10 * q + 20 * q^4 + 2 * q^7	$ 10 q + 20 q^{4} + 2 q^{7} - 10 q^{10} + 2 q^{13} + 44 q^{16} + 28 q^{19} - 42 q^{22} + 22 q^{25} + 40 q^{28} - 18 q^{31} + 36 q^{34} + 20 q^{37} - 4 q^{40} + 2 q^{43} - 30 q^{46} - 32 q^{49} - 2 q^{52} + 52 q^{55} + 84 q^{58} + 40 q^{61} + 64 q^{64} + 18 q^{67} + 18 q^{70} + 32 q^{73} + 104 q^{76} - 16 q^{79} - 94 q^{82} - 40 q^{85} - 32 q^{88} + 14 q^{91} - 56 q^{94} - 18 q^{97}+O(q^{100}) $ 10 * q + 20 * q^4 + 2 * q^7 - 10 * q^10 + 2 * q^13 + 44 * q^16 + 28 * q^19 - 42 * q^22 + 22 * q^25 + 40 * q^28 - 18 * q^31 + 36 * q^34 + 20 * q^37 - 4 * q^40 + 2 * q^43 - 30 * q^46 - 32 * q^49 - 2 * q^52 + 52 * q^55 + 84 * q^58 + 40 * q^61 + 64 * q^64 + 18 * q^67 + 18 * q^70 + 32 * q^73 + 104 * q^76 - 16 * q^79 - 94 * q^82 - 40 * q^85 - 32 * q^88 + 14 * q^91 - 56 * q^94 - 18 * q^97

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 $

$2$	1.02020	0.721389	0.360695	−	0.932684i	$-0.382540\pi$
$2$	1.02020	0.721389	0.360695	+	0.932684i	$0.382540\pi$
$3$	0	0
$3$	0	0
$4$	−0.959195	−0.479598
$5$	−2.96046	−1.32396	−0.661978	−	0.749523i	$-0.730283\pi$
$5$	−2.96046	−1.32396	−0.661978	+	0.749523i	$0.730283\pi$
$6$	0	0
$7$	0.816945	0.308776	0.154388	−	0.988010i	$-0.450659\pi$
$7$	0.816945	0.308776	0.154388	+	0.988010i	$0.450659\pi$
$8$	−3.01897	−1.06737
$9$	0	0
$10$	−3.02025	−0.955088
$11$	−4.92179	−1.48398	−0.741988	−	0.670413i	$-0.766117\pi$
$11$	−4.92179	−1.48398	−0.741988	+	0.670413i	$0.766117\pi$
$12$	0	0
$13$	1.06465	0.295280	0.147640	−	0.989041i	$-0.452832\pi$
$13$	1.06465	0.295280	0.147640	+	0.989041i	$0.452832\pi$
$14$	0.833446	0.222748
$15$	0	0
$16$	−1.16155	−0.290388
$17$	0.905065	0.219511	0.109755	−	0.993959i	$-0.464993\pi$
$17$	0.905065	0.219511	0.109755	+	0.993959i	$0.464993\pi$
$18$	0	0
$19$	−0.776141	−0.178059	−0.0890294	−	0.996029i	$-0.528377\pi$
$19$	−0.776141	−0.178059	−0.0890294	+	0.996029i	$0.528377\pi$
$20$	2.83966	0.634967
$21$	0	0
$22$	−5.02120	−1.07052
$23$	−6.93545	−1.44614	−0.723070	−	0.690775i	$-0.757270\pi$
$23$	−6.93545	−1.44614	−0.723070	+	0.690775i	$0.757270\pi$
$24$	0	0
$25$	3.76431	0.752861
$26$	1.08615	0.213012
$27$	0	0
$28$	−0.783610	−0.148088
$29$	1.72077	0.319539	0.159770	−	0.987154i	$-0.448925\pi$
$29$	1.72077	0.319539	0.159770	+	0.987154i	$0.448925\pi$
$30$	0	0
$31$	−5.90638	−1.06082	−0.530409	−	0.847742i	$-0.677962\pi$
$31$	−5.90638	−1.06082	−0.530409	+	0.847742i	$0.677962\pi$
$32$	4.85292	0.857883
$33$	0	0
$34$	0.923346	0.158353
$35$	−2.41853	−0.408806
$36$	0	0
$37$	5.58855	0.918752	0.459376	−	0.888242i	$-0.348073\pi$
$37$	5.58855	0.918752	0.459376	+	0.888242i	$0.348073\pi$
$38$	−0.791817	−0.128450
$39$	0	0
$40$	8.93752	1.41315
$41$	−5.11599	−0.798983	−0.399491	−	0.916737i	$-0.630813\pi$
$41$	−5.11599	−0.798983	−0.399491	+	0.916737i	$0.630813\pi$
$42$	0	0
$43$	−0.255947	−0.0390315	−0.0195158	−	0.999810i	$-0.506212\pi$
$43$	−0.255947	−0.0390315	−0.0195158	+	0.999810i	$0.506212\pi$
$44$	4.72096	0.711711
$45$	0	0
$46$	−7.07553	−1.04323
$47$	−9.75266	−1.42257	−0.711286	−	0.702903i	$-0.751887\pi$
$47$	−9.75266	−1.42257	−0.711286	+	0.702903i	$0.751887\pi$
$48$	0	0
$49$	−6.33260	−0.904657
$50$	3.84034	0.543106
$51$	0	0
$52$	−1.02120	−0.141615
$53$	−13.7259	−1.88539	−0.942697	−	0.333650i	$-0.891720\pi$
$53$	−13.7259	−1.88539	−0.942697	+	0.333650i	$0.891720\pi$
$54$	0	0
$55$	14.5708	1.96472
$56$	−2.46633	−0.329577
$57$	0	0
$58$	1.75553	0.230512
$59$	−9.97782	−1.29900	−0.649501	−	0.760361i	$-0.725022\pi$
$59$	−9.97782	−1.29900	−0.649501	+	0.760361i	$0.725022\pi$
$60$	0	0
$61$	−3.23898	−0.414709	−0.207355	−	0.978266i	$-0.566485\pi$
$61$	−3.23898	−0.414709	−0.207355	+	0.978266i	$0.566485\pi$
$62$	−6.02568	−0.765262
$63$	0	0
$64$	7.27405	0.909256
$65$	−3.15184	−0.390938
$66$	0	0
$67$	6.41020	0.783131	0.391566	−	0.920150i	$-0.371934\pi$
$67$	6.41020	0.783131	0.391566	+	0.920150i	$0.371934\pi$
$68$	−0.868134	−0.105277
$69$	0	0
$70$	−2.46738	−0.294908
$71$	7.97672	0.946662	0.473331	−	0.880885i	$-0.343051\pi$
$71$	7.97672	0.946662	0.473331	+	0.880885i	$0.343051\pi$
$72$	0	0
$73$	14.0235	1.64133	0.820666	−	0.571408i	$-0.193603\pi$
$73$	14.0235	1.64133	0.820666	+	0.571408i	$0.193603\pi$
$74$	5.70143	0.662777
$75$	0	0
$76$	0.744470	0.0853966
$77$	−4.02083	−0.458217
$78$	0	0
$79$	−11.1936	−1.25938	−0.629691	−	0.776845i	$-0.716819\pi$
$79$	−11.1936	−1.25938	−0.629691	+	0.776845i	$0.716819\pi$
$80$	3.43873	0.384462
$81$	0	0
$82$	−5.21932	−0.576378
$83$	10.5277	1.15557	0.577783	−	0.816190i	$-0.303918\pi$
$83$	10.5277	1.15557	0.577783	+	0.816190i	$0.303918\pi$
$84$	0	0
$85$	−2.67941	−0.290623
$86$	−0.261116	−0.0281569
$87$	0	0
$88$	14.8587	1.58395
$89$	18.2503	1.93453	0.967266	−	0.253764i	$-0.0816685\pi$
$89$	18.2503	1.93453	0.967266	+	0.253764i	$0.0816685\pi$
$90$	0	0
$91$	0.869757	0.0911754
$92$	6.65245	0.693566
$93$	0	0
$94$	−9.94965	−1.02623
$95$	2.29773	0.235742
$96$	0	0
$97$	7.82848	0.794862	0.397431	−	0.917632i	$-0.369902\pi$
$97$	7.82848	0.794862	0.397431	+	0.917632i	$0.369902\pi$
$98$	−6.46051	−0.652610
$99$	0	0
$100$	−3.61071	−0.361071
$101$	−4.14633	−0.412575	−0.206288	−	0.978491i	$-0.566138\pi$
$101$	−4.14633	−0.412575	−0.206288	+	0.978491i	$0.566138\pi$
$102$	0	0
$103$	6.55353	0.645738	0.322869	−	0.946444i	$-0.395353\pi$
$103$	6.55353	0.645738	0.322869	+	0.946444i	$0.395353\pi$
$104$	−3.21413	−0.315171
$105$	0	0
$106$	−14.0031	−1.36010
$107$	16.2385	1.56983	0.784916	−	0.619602i	$-0.212706\pi$
$107$	16.2385	1.56983	0.784916	+	0.619602i	$0.212706\pi$
$108$	0	0
$109$	−9.65373	−0.924659	−0.462330	−	0.886708i	$-0.652986\pi$
$109$	−9.65373	−0.924659	−0.462330	+	0.886708i	$0.652986\pi$
$110$	14.8651	1.41733
$111$	0	0
$112$	−0.948925	−0.0896650
$113$	14.5103	1.36501	0.682505	−	0.730881i	$-0.260891\pi$
$113$	14.5103	1.36501	0.682505	+	0.730881i	$0.260891\pi$
$114$	0	0
$115$	20.5321	1.91463
$116$	−1.65056	−0.153250
$117$	0	0
$118$	−10.1794	−0.937086
$119$	0.739389	0.0677797
$120$	0	0
$121$	13.2240	1.20218
$122$	−3.30441	−0.299167
$123$	0	0
$124$	5.66538	0.508766
$125$	3.65822	0.327201
$126$	0	0
$127$	21.5274	1.91025	0.955123	−	0.296210i	$-0.0957228\pi$
$127$	21.5274	1.91025	0.955123	+	0.296210i	$0.0957228\pi$
$128$	−2.28487	−0.201956
$129$	0	0
$130$	−3.21550	−0.282018
$131$	12.8466	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$131$	12.8466	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$132$	0	0
$133$	−0.634064	−0.0549803
$134$	6.53968	0.564942
$135$	0	0
$136$	−2.73236	−0.234298
$137$	−11.8104	−1.00903	−0.504515	−	0.863403i	$-0.668329\pi$
$137$	−11.8104	−1.00903	−0.504515	+	0.863403i	$0.668329\pi$
$138$	0	0
$139$	4.12665	0.350018	0.175009	−	0.984567i	$-0.444005\pi$
$139$	4.12665	0.350018	0.175009	+	0.984567i	$0.444005\pi$
$140$	2.31984	0.196063
$141$	0	0
$142$	8.13784	0.682912
$143$	−5.23997	−0.438188
$144$	0	0
$145$	−5.09427	−0.423056
$146$	14.3068	1.18404
$147$	0	0
$148$	−5.36051	−0.440631
$149$	−18.9392	−1.55156	−0.775779	−	0.631004i	$-0.782643\pi$
$149$	−18.9392	−1.55156	−0.775779	+	0.631004i	$0.782643\pi$
$150$	0	0
$151$	12.4638	1.01429	0.507145	−	0.861861i	$-0.330701\pi$
$151$	12.4638	1.01429	0.507145	+	0.861861i	$0.330701\pi$
$152$	2.34314	0.190054
$153$	0	0
$154$	−4.10205	−0.330552
$155$	17.4856	1.40448
$156$	0	0
$157$	10.4703	0.835622	0.417811	−	0.908534i	$-0.362797\pi$
$157$	10.4703	0.835622	0.417811	+	0.908534i	$0.362797\pi$
$158$	−11.4197	−0.908505
$159$	0	0
$160$	−14.3669	−1.13580
$161$	−5.66588	−0.446534
$162$	0	0
$163$	−3.62547	−0.283969	−0.141984	−	0.989869i	$-0.545348\pi$
$163$	−3.62547	−0.283969	−0.141984	+	0.989869i	$0.545348\pi$
$164$	4.90723	0.383190
$165$	0	0
$166$	10.7404	0.833613
$167$	22.9128	1.77305	0.886523	−	0.462685i	$-0.153114\pi$
$167$	22.9128	1.77305	0.886523	+	0.462685i	$0.153114\pi$
$168$	0	0
$169$	−11.8665	−0.912810
$170$	−2.73353	−0.209652
$171$	0	0
$172$	0.245503	0.0187194
$173$	−17.4304	−1.32521	−0.662605	−	0.748969i	$-0.730549\pi$
$173$	−17.4304	−1.32521	−0.662605	+	0.748969i	$0.730549\pi$
$174$	0	0
$175$	3.07523	0.232466
$176$	5.71692	0.430929
$177$	0	0
$178$	18.6190	1.39555
$179$	18.2971	1.36759	0.683796	−	0.729674i	$-0.260328\pi$
$179$	18.2971	1.36759	0.683796	+	0.729674i	$0.260328\pi$
$180$	0	0
$181$	−14.9203	−1.10901	−0.554507	−	0.832179i	$-0.687093\pi$
$181$	−14.9203	−1.10901	−0.554507	+	0.832179i	$0.687093\pi$
$182$	0.887325	0.0657729
$183$	0	0
$184$	20.9379	1.54356
$185$	−16.5447	−1.21639
$186$	0	0
$187$	−4.45454	−0.325748
$188$	9.35471	0.682262
$189$	0	0
$190$	2.34414	0.170062
$191$	−0.639251	−0.0462546	−0.0231273	−	0.999733i	$-0.507362\pi$
$191$	−0.639251	−0.0462546	−0.0231273	+	0.999733i	$0.507362\pi$
$192$	0	0
$193$	−22.5496	−1.62316	−0.811578	−	0.584244i	$-0.801391\pi$
$193$	−22.5496	−1.62316	−0.811578	+	0.584244i	$0.801391\pi$
$194$	7.98660	0.573405
$195$	0	0
$196$	6.07420	0.433872
$197$	21.6184	1.54025	0.770124	−	0.637894i	$-0.220194\pi$
$197$	21.6184	1.54025	0.770124	+	0.637894i	$0.220194\pi$
$198$	0	0
$199$	−14.8123	−1.05002	−0.525008	−	0.851097i	$-0.675938\pi$
$199$	−14.8123	−1.05002	−0.525008	+	0.851097i	$0.675938\pi$
$200$	−11.3643	−0.803578
$201$	0	0
$202$	−4.23008	−0.297627
$203$	1.40578	0.0986661
$204$	0	0
$205$	15.1457	1.05782
$206$	6.68590	0.465829
$207$	0	0
$208$	−1.23664	−0.0857458
$209$	3.82000	0.264235
$210$	0	0
$211$	24.3958	1.67948	0.839739	−	0.542990i	$-0.182708\pi$
$211$	24.3958	1.67948	0.839739	+	0.542990i	$0.182708\pi$
$212$	13.1658	0.904231
$213$	0	0
$214$	16.5665	1.13246
$215$	0.757719	0.0516760
$216$	0	0
$217$	−4.82519	−0.327555
$218$	−9.84872	−0.667039
$219$	0	0
$220$	−13.9762	−0.942275
$221$	0.963574	0.0648170
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	3.96457	0.264894
$225$	0	0
$226$	14.8033	0.984703
$227$	0.738976	0.0490475	0.0245238	−	0.999699i	$-0.492193\pi$
$227$	0.738976	0.0490475	0.0245238	+	0.999699i	$0.492193\pi$
$228$	0	0
$229$	−20.4498	−1.35136	−0.675679	−	0.737196i	$-0.736150\pi$
$229$	−20.4498	−1.35136	−0.675679	+	0.737196i	$0.736150\pi$
$230$	20.9468	1.38119
$231$	0	0
$232$	−5.19495	−0.341065
$233$	−12.1591	−0.796566	−0.398283	−	0.917263i	$-0.630394\pi$
$233$	−12.1591	−0.796566	−0.398283	+	0.917263i	$0.630394\pi$
$234$	0	0
$235$	28.8723	1.88342
$236$	9.57068	0.622998
$237$	0	0
$238$	0.754323	0.0488955
$239$	−18.2519	−1.18062	−0.590309	−	0.807178i	$-0.700994\pi$
$239$	−18.2519	−1.18062	−0.590309	+	0.807178i	$0.700994\pi$
$240$	0	0
$241$	17.5927	1.13324	0.566622	−	0.823978i	$-0.308250\pi$
$241$	17.5927	1.13324	0.566622	+	0.823978i	$0.308250\pi$
$242$	13.4911	0.867243
$243$	0	0
$244$	3.10682	0.198894
$245$	18.7474	1.19773
$246$	0	0
$247$	−0.826315	−0.0525772
$248$	17.8312	1.13228
$249$	0	0
$250$	3.73211	0.236039
$251$	−3.70362	−0.233770	−0.116885	−	0.993145i	$-0.537291\pi$
$251$	−3.70362	−0.233770	−0.116885	+	0.993145i	$0.537291\pi$
$252$	0	0
$253$	34.1348	2.14604
$254$	21.9622	1.37803
$255$	0	0
$256$	−16.8791	−1.05494
$257$	−7.80033	−0.486571	−0.243286	−	0.969955i	$-0.578225\pi$
$257$	−7.80033	−0.486571	−0.243286	+	0.969955i	$0.578225\pi$
$258$	0	0
$259$	4.56554	0.283689
$260$	3.02323	0.187493
$261$	0	0
$262$	13.1061	0.809695
$263$	−1.31405	−0.0810278	−0.0405139	−	0.999179i	$-0.512900\pi$
$263$	−1.31405	−0.0810278	−0.0405139	+	0.999179i	$0.512900\pi$
$264$	0	0
$265$	40.6349	2.49618
$266$	−0.646871	−0.0396622
$267$	0	0
$268$	−6.14864	−0.375588
$269$	2.31324	0.141041	0.0705204	−	0.997510i	$-0.477534\pi$
$269$	2.31324	0.141041	0.0705204	+	0.997510i	$0.477534\pi$
$270$	0	0
$271$	−24.1417	−1.46651	−0.733253	−	0.679956i	$-0.761999\pi$
$271$	−24.1417	−1.46651	−0.733253	+	0.679956i	$0.761999\pi$
$272$	−1.05128	−0.0637433
$273$	0	0
$274$	−12.0489	−0.727903
$275$	−18.5271	−1.11723
$276$	0	0
$277$	13.9344	0.837237	0.418619	−	0.908162i	$-0.362514\pi$
$277$	13.9344	0.837237	0.418619	+	0.908162i	$0.362514\pi$
$278$	4.21001	0.252499
$279$	0	0
$280$	7.30146	0.436346
$281$	−1.54895	−0.0924024	−0.0462012	−	0.998932i	$-0.514712\pi$
$281$	−1.54895	−0.0924024	−0.0462012	+	0.998932i	$0.514712\pi$
$282$	0	0
$283$	−1.26359	−0.0751129	−0.0375564	−	0.999295i	$-0.511957\pi$
$283$	−1.26359	−0.0751129	−0.0375564	+	0.999295i	$0.511957\pi$
$284$	−7.65123	−0.454017
$285$	0	0
$286$	−5.34581	−0.316104
$287$	−4.17948	−0.246707
$288$	0	0
$289$	−16.1809	−0.951815
$290$	−5.19717	−0.305188
$291$	0	0
$292$	−13.4513	−0.787179
$293$	−27.9605	−1.63347	−0.816734	−	0.577015i	$-0.804218\pi$
$293$	−27.9605	−1.63347	−0.816734	+	0.577015i	$0.804218\pi$
$294$	0	0
$295$	29.5389	1.71982
$296$	−16.8716	−0.980644
$297$	0	0
$298$	−19.3217	−1.11928
$299$	−7.38380	−0.427016
$300$	0	0
$301$	−0.209094	−0.0120520
$302$	12.7155	0.731697
$303$	0	0
$304$	0.901529	0.0517062
$305$	9.58887	0.549057
$306$	0	0
$307$	12.8721	0.734649	0.367325	−	0.930093i	$-0.380274\pi$
$307$	12.8721	0.734649	0.367325	+	0.930093i	$0.380274\pi$
$308$	3.85677	0.219760
$309$	0	0
$310$	17.8388	1.01317
$311$	−21.4191	−1.21457	−0.607283	−	0.794485i	$-0.707741\pi$
$311$	−21.4191	−1.21457	−0.607283	+	0.794485i	$0.707741\pi$
$312$	0	0
$313$	−24.4709	−1.38318	−0.691588	−	0.722292i	$-0.743089\pi$
$313$	−24.4709	−1.38318	−0.691588	+	0.722292i	$0.743089\pi$
$314$	10.6818	0.602809
$315$	0	0
$316$	10.7369	0.603997
$317$	16.6996	0.937940	0.468970	−	0.883214i	$-0.344625\pi$
$317$	16.6996	0.937940	0.468970	+	0.883214i	$0.344625\pi$
$318$	0	0
$319$	−8.46928	−0.474189
$320$	−21.5345	−1.20382
$321$	0	0
$322$	−5.78032	−0.322125
$323$	−0.702458	−0.0390858
$324$	0	0
$325$	4.00765	0.222305
$326$	−3.69870	−0.204852
$327$	0	0
$328$	15.4450	0.852807
$329$	−7.96739	−0.439257
$330$	0	0
$331$	21.8132	1.19896	0.599481	−	0.800389i	$-0.295374\pi$
$331$	21.8132	1.19896	0.599481	+	0.800389i	$0.295374\pi$
$332$	−10.0981	−0.554207
$333$	0	0
$334$	23.3756	1.27906
$335$	−18.9771	−1.03683
$336$	0	0
$337$	−9.50836	−0.517953	−0.258977	−	0.965884i	$-0.583385\pi$
$337$	−9.50836	−0.517953	−0.258977	+	0.965884i	$0.583385\pi$
$338$	−12.1062	−0.658491
$339$	0	0
$340$	2.57007	0.139382
$341$	29.0700	1.57423
$342$	0	0
$343$	−10.8920	−0.588113
$344$	0.772695	0.0416609
$345$	0	0
$346$	−17.7825	−0.955992
$347$	0.636113	0.0341483	0.0170742	−	0.999854i	$-0.494565\pi$
$347$	0.636113	0.0341483	0.0170742	+	0.999854i	$0.494565\pi$
$348$	0	0
$349$	27.9544	1.49637	0.748183	−	0.663492i	$-0.230926\pi$
$349$	27.9544	1.49637	0.748183	+	0.663492i	$0.230926\pi$
$350$	3.13735	0.167698
$351$	0	0
$352$	−23.8850	−1.27308
$353$	31.4811	1.67557	0.837784	−	0.546002i	$-0.183851\pi$
$353$	31.4811	1.67557	0.837784	+	0.546002i	$0.183851\pi$
$354$	0	0
$355$	−23.6147	−1.25334
$356$	−17.5056	−0.927797
$357$	0	0
$358$	18.6667	0.986566
$359$	15.5956	0.823102	0.411551	−	0.911387i	$-0.364987\pi$
$359$	15.5956	0.823102	0.411551	+	0.911387i	$0.364987\pi$
$360$	0	0
$361$	−18.3976	−0.968295
$362$	−15.2216	−0.800031
$363$	0	0
$364$	−0.834267	−0.0437275
$365$	−41.5161	−2.17305
$366$	0	0
$367$	5.18456	0.270632	0.135316	−	0.990802i	$-0.456795\pi$
$367$	5.18456	0.270632	0.135316	+	0.990802i	$0.456795\pi$
$368$	8.05589	0.419942
$369$	0	0
$370$	−16.8788	−0.877489
$371$	−11.2133	−0.582165
$372$	0	0
$373$	10.1724	0.526709	0.263354	−	0.964699i	$-0.415171\pi$
$373$	10.1724	0.526709	0.263354	+	0.964699i	$0.415171\pi$
$374$	−4.54452	−0.234991
$375$	0	0
$376$	29.4430	1.51840
$377$	1.83201	0.0943534
$378$	0	0
$379$	−10.3566	−0.531981	−0.265991	−	0.963976i	$-0.585699\pi$
$379$	−10.3566	−0.531981	−0.265991	+	0.963976i	$0.585699\pi$
$380$	−2.20397	−0.113061
$381$	0	0
$382$	−0.652163	−0.0333675
$383$	−23.1200	−1.18138	−0.590689	−	0.806899i	$-0.701144\pi$
$383$	−23.1200	−1.18138	−0.590689	+	0.806899i	$0.701144\pi$
$384$	0	0
$385$	11.9035	0.606659
$386$	−23.0051	−1.17093
$387$	0	0
$388$	−7.50904	−0.381214
$389$	38.1213	1.93282	0.966412	−	0.256997i	$-0.0827332\pi$
$389$	38.1213	1.93282	0.966412	+	0.256997i	$0.0827332\pi$
$390$	0	0
$391$	−6.27703	−0.317443
$392$	19.1179	0.965600
$393$	0	0
$394$	22.0551	1.11112
$395$	33.1383	1.66737
$396$	0	0
$397$	−20.7978	−1.04381	−0.521906	−	0.853003i	$-0.674779\pi$
$397$	−20.7978	−1.04381	−0.521906	+	0.853003i	$0.674779\pi$
$398$	−15.1115	−0.757470
$399$	0	0
$400$	−4.37244	−0.218622
$401$	−32.2088	−1.60843	−0.804215	−	0.594339i	$-0.797414\pi$
$401$	−32.2088	−1.60843	−0.804215	+	0.594339i	$0.797414\pi$
$402$	0	0
$403$	−6.28821	−0.313238
$404$	3.97714	0.197870
$405$	0	0
$406$	1.43417	0.0711767
$407$	−27.5057	−1.36341
$408$	0	0
$409$	2.32116	0.114774	0.0573869	−	0.998352i	$-0.481723\pi$
$409$	2.32116	0.114774	0.0573869	+	0.998352i	$0.481723\pi$
$410$	15.4516	0.763099
$411$	0	0
$412$	−6.28611	−0.309695
$413$	−8.15133	−0.401101
$414$	0	0
$415$	−31.1668	−1.52992
$416$	5.16664	0.253315
$417$	0	0
$418$	3.89716	0.190616
$419$	30.6786	1.49875	0.749374	−	0.662147i	$-0.230354\pi$
$419$	30.6786	1.49875	0.749374	+	0.662147i	$0.230354\pi$
$420$	0	0
$421$	10.2516	0.499632	0.249816	−	0.968293i	$-0.419630\pi$
$421$	10.2516	0.499632	0.249816	+	0.968293i	$0.419630\pi$
$422$	24.8886	1.21156
$423$	0	0
$424$	41.4380	2.01241
$425$	3.40694	0.165261
$426$	0	0
$427$	−2.64607	−0.128052
$428$	−15.5759	−0.752888
$429$	0	0
$430$	0.773024	0.0372785
$431$	5.19079	0.250031	0.125016	−	0.992155i	$-0.460102\pi$
$431$	5.19079	0.250031	0.125016	+	0.992155i	$0.460102\pi$
$432$	0	0
$433$	2.31455	0.111230	0.0556151	−	0.998452i	$-0.482288\pi$
$433$	2.31455	0.111230	0.0556151	+	0.998452i	$0.482288\pi$
$434$	−4.92265	−0.236295
$435$	0	0
$436$	9.25981	0.443464
$437$	5.38288	0.257498
$438$	0	0
$439$	37.7106	1.79983	0.899914	−	0.436068i	$-0.143629\pi$
$439$	37.7106	1.79983	0.899914	+	0.436068i	$0.143629\pi$
$440$	−43.9886	−2.09707
$441$	0	0
$442$	0.983037	0.0467583
$443$	−7.70764	−0.366201	−0.183100	−	0.983094i	$-0.558613\pi$
$443$	−7.70764	−0.366201	−0.183100	+	0.983094i	$0.558613\pi$
$444$	0	0
$445$	−54.0294	−2.56124
$446$	−1.02020	−0.0483078
$447$	0	0
$448$	5.94250	0.280757
$449$	−31.4408	−1.48378	−0.741892	−	0.670519i	$-0.766071\pi$
$449$	−31.4408	−1.48378	−0.741892	+	0.670519i	$0.766071\pi$
$450$	0	0
$451$	25.1798	1.18567
$452$	−13.9182	−0.654655
$453$	0	0
$454$	0.753902	0.0353824
$455$	−2.57488	−0.120712
$456$	0	0
$457$	22.5065	1.05281	0.526405	−	0.850234i	$-0.323539\pi$
$457$	22.5065	1.05281	0.526405	+	0.850234i	$0.323539\pi$
$458$	−20.8628	−0.974855
$459$	0	0
$460$	−19.6943	−0.918251
$461$	−31.3088	−1.45820	−0.729098	−	0.684410i	$-0.760060\pi$
$461$	−31.3088	−1.45820	−0.729098	+	0.684410i	$0.760060\pi$
$462$	0	0
$463$	35.9142	1.66907	0.834537	−	0.550952i	$-0.185735\pi$
$463$	35.9142	1.66907	0.834537	+	0.550952i	$0.185735\pi$
$464$	−1.99877	−0.0927905
$465$	0	0
$466$	−12.4046	−0.574634
$467$	33.8566	1.56670	0.783348	−	0.621584i	$-0.213511\pi$
$467$	33.8566	1.56670	0.783348	+	0.621584i	$0.213511\pi$
$468$	0	0
$469$	5.23679	0.241812
$470$	29.4555	1.35868
$471$	0	0
$472$	30.1227	1.38651
$473$	1.25972	0.0579218
$474$	0	0
$475$	−2.92163	−0.134054
$476$	−0.709218	−0.0325070
$477$	0	0
$478$	−18.6206	−0.851685
$479$	5.54372	0.253299	0.126649	−	0.991948i	$-0.459578\pi$
$479$	5.54372	0.253299	0.126649	+	0.991948i	$0.459578\pi$
$480$	0	0
$481$	5.94983	0.271289
$482$	17.9480	0.817510
$483$	0	0
$484$	−12.6844	−0.576565
$485$	−23.1759	−1.05236
$486$	0	0
$487$	1.87574	0.0849981	0.0424990	−	0.999097i	$-0.486468\pi$
$487$	1.87574	0.0849981	0.0424990	+	0.999097i	$0.486468\pi$
$488$	9.77838	0.442647
$489$	0	0
$490$	19.1261	0.864027
$491$	−28.8651	−1.30266	−0.651331	−	0.758794i	$-0.725789\pi$
$491$	−28.8651	−1.30266	−0.651331	+	0.758794i	$0.725789\pi$
$492$	0	0
$493$	1.55741	0.0701422
$494$	−0.843005	−0.0379286
$495$	0	0
$496$	6.86058	0.308049
$497$	6.51654	0.292307
$498$	0	0
$499$	26.4083	1.18220	0.591099	−	0.806599i	$-0.298694\pi$
$499$	26.4083	1.18220	0.591099	+	0.806599i	$0.298694\pi$
$500$	−3.50894	−0.156925
$501$	0	0
$502$	−3.77842	−0.168639
$503$	2.02512	0.0902955	0.0451478	−	0.998980i	$-0.485624\pi$
$503$	2.02512	0.0902955	0.0451478	+	0.998980i	$0.485624\pi$
$504$	0	0
$505$	12.2750	0.546232
$506$	34.8243	1.54813
$507$	0	0
$508$	−20.6490	−0.916149
$509$	−16.2338	−0.719551	−0.359775	−	0.933039i	$-0.617147\pi$
$509$	−16.2338	−0.719551	−0.359775	+	0.933039i	$0.617147\pi$
$510$	0	0
$511$	11.4565	0.506804
$512$	−12.6503	−0.559070
$513$	0	0
$514$	−7.95789	−0.351007
$515$	−19.4014	−0.854930
$516$	0	0
$517$	48.0006	2.11106
$518$	4.65775	0.204650
$519$	0	0
$520$	9.51530	0.417273
$521$	−8.99609	−0.394126	−0.197063	−	0.980391i	$-0.563140\pi$
$521$	−8.99609	−0.394126	−0.197063	+	0.980391i	$0.563140\pi$
$522$	0	0
$523$	27.4308	1.19947	0.599733	−	0.800200i	$-0.295273\pi$
$523$	27.4308	1.19947	0.599733	+	0.800200i	$0.295273\pi$
$524$	−12.3224	−0.538306
$525$	0	0
$526$	−1.34059	−0.0584526
$527$	−5.34566	−0.232861
$528$	0	0
$529$	25.1004	1.09132
$530$	41.4556	1.80072
$531$	0	0
$532$	0.608191	0.0263684
$533$	−5.44672	−0.235923
$534$	0	0
$535$	−48.0733	−2.07839
$536$	−19.3522	−0.835887
$537$	0	0
$538$	2.35996	0.101745
$539$	31.1677	1.34249
$540$	0	0
$541$	4.77733	0.205393	0.102697	−	0.994713i	$-0.467253\pi$
$541$	4.77733	0.205393	0.102697	+	0.994713i	$0.467253\pi$
$542$	−24.6294	−1.05792
$543$	0	0
$544$	4.39221	0.188314
$545$	28.5794	1.22421
$546$	0	0
$547$	−15.0689	−0.644299	−0.322150	−	0.946689i	$-0.604405\pi$
$547$	−15.0689	−0.644299	−0.322150	+	0.946689i	$0.604405\pi$
$548$	11.3285	0.483928
$549$	0	0
$550$	−18.9014	−0.805956
$551$	−1.33556	−0.0568968
$552$	0	0
$553$	−9.14459	−0.388868
$554$	14.2159	0.603974
$555$	0	0
$556$	−3.95827	−0.167868
$557$	39.7055	1.68238	0.841188	−	0.540743i	$-0.181857\pi$
$557$	39.7055	1.68238	0.841188	+	0.540743i	$0.181857\pi$
$558$	0	0
$559$	−0.272493	−0.0115252
$560$	2.80925	0.118713
$561$	0	0
$562$	−1.58023	−0.0666581
$563$	0.324778	0.0136878	0.00684388	−	0.999977i	$-0.497822\pi$
$563$	0.324778	0.0136878	0.00684388	+	0.999977i	$0.497822\pi$
$564$	0	0
$565$	−42.9570	−1.80721
$566$	−1.28912	−0.0541856
$567$	0	0
$568$	−24.0814	−1.01044
$569$	5.25116	0.220140	0.110070	−	0.993924i	$-0.464892\pi$
$569$	5.25116	0.220140	0.110070	+	0.993924i	$0.464892\pi$
$570$	0	0
$571$	−28.8442	−1.20709	−0.603545	−	0.797329i	$-0.706246\pi$
$571$	−28.8442	−1.20709	−0.603545	+	0.797329i	$0.706246\pi$
$572$	5.02615	0.210154
$573$	0	0
$574$	−4.26390	−0.177972
$575$	−26.1071	−1.08874
$576$	0	0
$577$	−36.9821	−1.53959	−0.769793	−	0.638294i	$-0.779640\pi$
$577$	−36.9821	−1.53959	−0.769793	+	0.638294i	$0.779640\pi$
$578$	−16.5077	−0.686629
$579$	0	0
$580$	4.88640	0.202897
$581$	8.60056	0.356811
$582$	0	0
$583$	67.5559	2.79788
$584$	−42.3366	−1.75190
$585$	0	0
$586$	−28.5252	−1.17837
$587$	−3.21656	−0.132762	−0.0663809	−	0.997794i	$-0.521145\pi$
$587$	−3.21656	−0.132762	−0.0663809	+	0.997794i	$0.521145\pi$
$588$	0	0
$589$	4.58418	0.188888
$590$	30.1355	1.24066
$591$	0	0
$592$	−6.49140	−0.266795
$593$	−23.4257	−0.961980	−0.480990	−	0.876726i	$-0.659723\pi$
$593$	−23.4257	−0.961980	−0.480990	+	0.876726i	$0.659723\pi$
$594$	0	0
$595$	−2.18893	−0.0897373
$596$	18.1664	0.744124
$597$	0	0
$598$	−7.53294	−0.308045
$599$	29.6434	1.21120	0.605598	−	0.795771i	$-0.292934\pi$
$599$	29.6434	1.21120	0.605598	+	0.795771i	$0.292934\pi$
$600$	0	0
$601$	6.47930	0.264296	0.132148	−	0.991230i	$-0.457813\pi$
$601$	6.47930	0.264296	0.132148	+	0.991230i	$0.457813\pi$
$602$	−0.213318	−0.00869419
$603$	0	0
$604$	−11.9552	−0.486451
$605$	−39.1492	−1.59164
$606$	0	0
$607$	19.9262	0.808778	0.404389	−	0.914587i	$-0.367484\pi$
$607$	19.9262	0.808778	0.404389	+	0.914587i	$0.367484\pi$
$608$	−3.76655	−0.152754
$609$	0	0
$610$	9.78255	0.396084
$611$	−10.3831	−0.420057
$612$	0	0
$613$	1.19872	0.0484157	0.0242078	−	0.999707i	$-0.492294\pi$
$613$	1.19872	0.0484157	0.0242078	+	0.999707i	$0.492294\pi$
$614$	13.1321	0.529968
$615$	0	0
$616$	12.1388	0.489085
$617$	−12.0721	−0.486003	−0.243002	−	0.970026i	$-0.578132\pi$
$617$	−12.0721	−0.486003	−0.243002	+	0.970026i	$0.578132\pi$
$618$	0	0
$619$	−5.08810	−0.204508	−0.102254	−	0.994758i	$-0.532605\pi$
$619$	−5.08810	−0.204508	−0.102254	+	0.994758i	$0.532605\pi$
$620$	−16.7721	−0.673584
$621$	0	0
$622$	−21.8517	−0.876175
$623$	14.9095	0.597338
$624$	0	0
$625$	−29.6515	−1.18606
$626$	−24.9652	−0.997808
$627$	0	0
$628$	−10.0431	−0.400762
$629$	5.05800	0.201676
$630$	0	0
$631$	14.2470	0.567166	0.283583	−	0.958948i	$-0.408477\pi$
$631$	14.2470	0.567166	0.283583	+	0.958948i	$0.408477\pi$
$632$	33.7932	1.34422
$633$	0	0
$634$	17.0369	0.676620
$635$	−63.7309	−2.52908
$636$	0	0
$637$	−6.74198	−0.267127
$638$	−8.64034	−0.342074
$639$	0	0
$640$	6.76425	0.267380
$641$	17.7071	0.699387	0.349693	−	0.936864i	$-0.386286\pi$
$641$	17.7071	0.699387	0.349693	+	0.936864i	$0.386286\pi$
$642$	0	0
$643$	−0.749927	−0.0295742	−0.0147871	−	0.999891i	$-0.504707\pi$
$643$	−0.749927	−0.0295742	−0.0147871	+	0.999891i	$0.504707\pi$
$644$	5.43468	0.214157
$645$	0	0
$646$	−0.716646	−0.0281961
$647$	14.2689	0.560968	0.280484	−	0.959859i	$-0.409505\pi$
$647$	14.2689	0.560968	0.280484	+	0.959859i	$0.409505\pi$
$648$	0	0
$649$	49.1087	1.92769
$650$	4.08860	0.160368
$651$	0	0
$652$	3.47753	0.136191
$653$	45.3896	1.77623	0.888116	−	0.459619i	$-0.152014\pi$
$653$	45.3896	1.77623	0.888116	+	0.459619i	$0.152014\pi$
$654$	0	0
$655$	−38.0318	−1.48602
$656$	5.94249	0.232015
$657$	0	0
$658$	−8.12832	−0.316875
$659$	−10.0092	−0.389901	−0.194951	−	0.980813i	$-0.562455\pi$
$659$	−10.0092	−0.389901	−0.194951	+	0.980813i	$0.562455\pi$
$660$	0	0
$661$	32.1175	1.24922	0.624612	−	0.780935i	$-0.285257\pi$
$661$	32.1175	1.24922	0.624612	+	0.780935i	$0.285257\pi$
$662$	22.2538	0.864918
$663$	0	0
$664$	−31.7828	−1.23341
$665$	1.87712	0.0727916
$666$	0	0
$667$	−11.9343	−0.462099
$668$	−21.9778	−0.850348
$669$	0	0
$670$	−19.3604	−0.747959
$671$	15.9416	0.615419
$672$	0	0
$673$	−15.4790	−0.596673	−0.298337	−	0.954461i	$-0.596432\pi$
$673$	−15.4790	−0.596673	−0.298337	+	0.954461i	$0.596432\pi$
$674$	−9.70041	−0.373646
$675$	0	0
$676$	11.3823	0.437782
$677$	−23.8787	−0.917732	−0.458866	−	0.888505i	$-0.651744\pi$
$677$	−23.8787	−0.917732	−0.458866	+	0.888505i	$0.651744\pi$
$678$	0	0
$679$	6.39544	0.245434
$680$	8.08904	0.310200
$681$	0	0
$682$	29.6572	1.13563
$683$	−40.9625	−1.56739	−0.783694	−	0.621147i	$-0.786667\pi$
$683$	−40.9625	−1.56739	−0.783694	+	0.621147i	$0.786667\pi$
$684$	0	0
$685$	34.9641	1.33591
$686$	−11.1120	−0.424258
$687$	0	0
$688$	0.297296	0.0113343
$689$	−14.6132	−0.556719
$690$	0	0
$691$	−2.07760	−0.0790356	−0.0395178	−	0.999219i	$-0.512582\pi$
$691$	−2.07760	−0.0790356	−0.0395178	+	0.999219i	$0.512582\pi$
$692$	16.7192	0.635568
$693$	0	0
$694$	0.648961	0.0246342
$695$	−12.2168	−0.463409
$696$	0	0
$697$	−4.63030	−0.175385
$698$	28.5191	1.07946
$699$	0	0
$700$	−2.94975	−0.111490
$701$	−27.6115	−1.04287	−0.521437	−	0.853290i	$-0.674604\pi$
$701$	−27.6115	−1.04287	−0.521437	+	0.853290i	$0.674604\pi$
$702$	0	0
$703$	−4.33750	−0.163592
$704$	−35.8013	−1.34931
$705$	0	0
$706$	32.1169	1.20874
$707$	−3.38733	−0.127394
$708$	0	0
$709$	−25.0503	−0.940783	−0.470391	−	0.882458i	$-0.655887\pi$
$709$	−25.0503	−0.940783	−0.470391	+	0.882458i	$0.655887\pi$
$710$	−24.0917	−0.904146
$711$	0	0
$712$	−55.0972	−2.06485
$713$	40.9634	1.53409
$714$	0	0
$715$	15.5127	0.580142
$716$	−17.5505	−0.655894
$717$	0	0
$718$	15.9106	0.593777
$719$	−47.3100	−1.76436	−0.882182	−	0.470909i	$-0.843926\pi$
$719$	−47.3100	−1.76436	−0.882182	+	0.470909i	$0.843926\pi$
$720$	0	0
$721$	5.35387	0.199389
$722$	−18.7692	−0.698518
$723$	0	0
$724$	14.3114	0.531881
$725$	6.47751	0.240569
$726$	0	0
$727$	45.1603	1.67490	0.837452	−	0.546511i	$-0.184044\pi$
$727$	45.1603	1.67490	0.837452	+	0.546511i	$0.184044\pi$
$728$	−2.62577	−0.0973175
$729$	0	0
$730$	−42.3547	−1.56762
$731$	−0.231649	−0.00856783
$732$	0	0
$733$	22.3570	0.825775	0.412887	−	0.910782i	$-0.364520\pi$
$733$	22.3570	0.825775	0.412887	+	0.910782i	$0.364520\pi$
$734$	5.28928	0.195231
$735$	0	0
$736$	−33.6571	−1.24062
$737$	−31.5497	−1.16215
$738$	0	0
$739$	8.83459	0.324986	0.162493	−	0.986710i	$-0.448047\pi$
$739$	8.83459	0.324986	0.162493	+	0.986710i	$0.448047\pi$
$740$	15.8696	0.583377
$741$	0	0
$742$	−11.4398	−0.419967
$743$	0.748826	0.0274718	0.0137359	−	0.999906i	$-0.495628\pi$
$743$	0.748826	0.0274718	0.0137359	+	0.999906i	$0.495628\pi$
$744$	0	0
$745$	56.0687	2.05420
$746$	10.3779	0.379962
$747$	0	0
$748$	4.27278	0.156228
$749$	13.2659	0.484727
$750$	0	0
$751$	−20.1417	−0.734980	−0.367490	−	0.930027i	$-0.619783\pi$
$751$	−20.1417	−0.734980	−0.367490	+	0.930027i	$0.619783\pi$
$752$	11.3282	0.413098
$753$	0	0
$754$	1.86902	0.0680655
$755$	−36.8985	−1.34287
$756$	0	0
$757$	−47.5005	−1.72644	−0.863218	−	0.504831i	$-0.831555\pi$
$757$	−47.5005	−1.72644	−0.863218	+	0.504831i	$0.831555\pi$
$758$	−10.5658	−0.383766
$759$	0	0
$760$	−6.93677	−0.251623
$761$	20.3548	0.737861	0.368931	−	0.929457i	$-0.379724\pi$
$761$	20.3548	0.737861	0.368931	+	0.929457i	$0.379724\pi$
$762$	0	0
$763$	−7.88657	−0.285513
$764$	0.613166	0.0221836
$765$	0	0
$766$	−23.5870	−0.852234
$767$	−10.6228	−0.383569
$768$	0	0
$769$	2.98985	0.107817	0.0539084	−	0.998546i	$-0.482832\pi$
$769$	2.98985	0.107817	0.0539084	+	0.998546i	$0.482832\pi$
$770$	12.1439	0.437637
$771$	0	0
$772$	21.6295	0.778462
$773$	19.3516	0.696030	0.348015	−	0.937489i	$-0.386856\pi$
$773$	19.3516	0.696030	0.348015	+	0.937489i	$0.386856\pi$
$774$	0	0
$775$	−22.2334	−0.798649
$776$	−23.6339	−0.848408
$777$	0	0
$778$	38.8912	1.39432
$779$	3.97072	0.142266
$780$	0	0
$781$	−39.2598	−1.40482
$782$	−6.40382	−0.229000
$783$	0	0
$784$	7.35565	0.262702
$785$	−30.9969	−1.10633
$786$	0	0
$787$	1.02888	0.0366758	0.0183379	−	0.999832i	$-0.494163\pi$
$787$	1.02888	0.0366758	0.0183379	+	0.999832i	$0.494163\pi$
$788$	−20.7363	−0.738700
$789$	0	0
$790$	33.8076	1.20282
$791$	11.8541	0.421483
$792$	0	0
$793$	−3.44837	−0.122455
$794$	−21.2179	−0.752994
$795$	0	0
$796$	14.2079	0.503585
$797$	8.72513	0.309060	0.154530	−	0.987988i	$-0.450614\pi$
$797$	8.72513	0.309060	0.154530	+	0.987988i	$0.450614\pi$
$798$	0	0
$799$	−8.82680	−0.312270
$800$	18.2679	0.645867
$801$	0	0
$802$	−32.8593	−1.16030
$803$	−69.0210	−2.43570
$804$	0	0
$805$	16.7736	0.591191
$806$	−6.41522	−0.225966
$807$	0	0
$808$	12.5176	0.440369
$809$	−24.2853	−0.853827	−0.426913	−	0.904293i	$-0.640399\pi$
$809$	−24.2853	−0.853827	−0.426913	+	0.904293i	$0.640399\pi$
$810$	0	0
$811$	−31.8476	−1.11832	−0.559161	−	0.829059i	$-0.688877\pi$
$811$	−31.8476	−1.11832	−0.559161	+	0.829059i	$0.688877\pi$
$812$	−1.34841	−0.0473200
$813$	0	0
$814$	−28.0612	−0.983546
$815$	10.7331	0.375962
$816$	0	0
$817$	0.198651	0.00694991
$818$	2.36804	0.0827966
$819$	0	0
$820$	−14.5276	−0.507327
$821$	5.30462	0.185133	0.0925663	−	0.995707i	$-0.470493\pi$
$821$	5.30462	0.185133	0.0925663	+	0.995707i	$0.470493\pi$
$822$	0	0
$823$	49.8141	1.73641	0.868206	−	0.496204i	$-0.165273\pi$
$823$	49.8141	1.73641	0.868206	+	0.496204i	$0.165273\pi$
$824$	−19.7849	−0.689239
$825$	0	0
$826$	−8.31597	−0.289350
$827$	18.2579	0.634889	0.317445	−	0.948277i	$-0.397175\pi$
$827$	18.2579	0.634889	0.317445	+	0.948277i	$0.397175\pi$
$828$	0	0
$829$	−19.7466	−0.685829	−0.342915	−	0.939367i	$-0.611414\pi$
$829$	−19.7466	−0.685829	−0.342915	+	0.939367i	$0.611414\pi$
$830$	−31.7964	−1.10367
$831$	0	0
$832$	7.74428	0.268485
$833$	−5.73142	−0.198582
$834$	0	0
$835$	−67.8323	−2.34744
$836$	−3.66413	−0.126727
$837$	0	0
$838$	31.2983	1.08118
$839$	−30.3683	−1.04843	−0.524215	−	0.851586i	$-0.675641\pi$
$839$	−30.3683	−1.04843	−0.524215	+	0.851586i	$0.675641\pi$
$840$	0	0
$841$	−26.0389	−0.897895
$842$	10.4587	0.360429
$843$	0	0
$844$	−23.4004	−0.805474
$845$	35.1303	1.20852
$846$	0	0
$847$	10.8033	0.371206
$848$	15.9433	0.547497
$849$	0	0
$850$	3.47576	0.119218
$851$	−38.7591	−1.32864
$852$	0	0
$853$	−17.6944	−0.605843	−0.302922	−	0.953015i	$-0.597962\pi$
$853$	−17.6944	−0.605843	−0.302922	+	0.953015i	$0.597962\pi$
$854$	−2.69952	−0.0923756
$855$	0	0
$856$	−49.0234	−1.67558
$857$	−33.5304	−1.14538	−0.572689	−	0.819773i	$-0.694100\pi$
$857$	−33.5304	−1.14538	−0.572689	+	0.819773i	$0.694100\pi$
$858$	0	0
$859$	−41.4206	−1.41325	−0.706627	−	0.707586i	$-0.749784\pi$
$859$	−41.4206	−1.41325	−0.706627	+	0.707586i	$0.749784\pi$
$860$	−0.726801	−0.0247837
$861$	0	0
$862$	5.29563	0.180370
$863$	−13.3151	−0.453251	−0.226626	−	0.973982i	$-0.572769\pi$
$863$	−13.3151	−0.453251	−0.226626	+	0.973982i	$0.572769\pi$
$864$	0	0
$865$	51.6020	1.75452
$866$	2.36130	0.0802403
$867$	0	0
$868$	4.62830	0.157095
$869$	55.0928	1.86889
$870$	0	0
$871$	6.82460	0.231243
$872$	29.1443	0.986950
$873$	0	0
$874$	5.49161	0.185756
$875$	2.98856	0.101032
$876$	0	0
$877$	33.4302	1.12886	0.564428	−	0.825482i	$-0.309097\pi$
$877$	33.4302	1.12886	0.564428	+	0.825482i	$0.309097\pi$
$878$	38.4723	1.29838
$879$	0	0
$880$	−16.9247	−0.570532
$881$	−35.1818	−1.18531	−0.592653	−	0.805458i	$-0.701919\pi$
$881$	−35.1818	−1.18531	−0.592653	+	0.805458i	$0.701919\pi$
$882$	0	0
$883$	37.5651	1.26417	0.632084	−	0.774900i	$-0.282200\pi$
$883$	37.5651	1.26417	0.632084	+	0.774900i	$0.282200\pi$
$884$	−0.924256	−0.0310861
$885$	0	0
$886$	−7.86332	−0.264173
$887$	−25.4376	−0.854113	−0.427056	−	0.904225i	$-0.640449\pi$
$887$	−25.4376	−0.854113	−0.427056	+	0.904225i	$0.640449\pi$
$888$	0	0
$889$	17.5867	0.589839
$890$	−55.1207	−1.84765
$891$	0	0
$892$	0.959195	0.0321162
$893$	7.56944	0.253302
$894$	0	0
$895$	−54.1679	−1.81063
$896$	−1.86661	−0.0623591
$897$	0	0
$898$	−32.0759	−1.07039
$899$	−10.1635	−0.338973
$900$	0	0
$901$	−12.4228	−0.413864
$902$	25.6884	0.855331
$903$	0	0
$904$	−43.8060	−1.45696
$905$	44.1708	1.46829
$906$	0	0
$907$	−9.19856	−0.305433	−0.152717	−	0.988270i	$-0.548802\pi$
$907$	−9.19856	−0.305433	−0.152717	+	0.988270i	$0.548802\pi$
$908$	−0.708822	−0.0235231
$909$	0	0
$910$	−2.62689	−0.0870805
$911$	10.4334	0.345675	0.172838	−	0.984950i	$-0.444706\pi$
$911$	10.4334	0.345675	0.172838	+	0.984950i	$0.444706\pi$
$912$	0	0
$913$	−51.8152	−1.71483
$914$	22.9611	0.759486
$915$	0	0
$916$	19.6153	0.648108
$917$	10.4950	0.346574
$918$	0	0
$919$	−40.9215	−1.34988	−0.674938	−	0.737874i	$-0.735830\pi$
$919$	−40.9215	−1.34988	−0.674938	+	0.737874i	$0.735830\pi$
$920$	−61.9857	−2.04361
$921$	0	0
$922$	−31.9412	−1.05193
$923$	8.49238	0.279530
$924$	0	0
$925$	21.0370	0.691693
$926$	36.6396	1.20405
$927$	0	0
$928$	8.35076	0.274127
$929$	−4.21200	−0.138191	−0.0690956	−	0.997610i	$-0.522011\pi$
$929$	−4.21200	−0.138191	−0.0690956	+	0.997610i	$0.522011\pi$
$930$	0	0
$931$	4.91499	0.161082
$932$	11.6629	0.382031
$933$	0	0
$934$	34.5404	1.13020
$935$	13.1875	0.431277
$936$	0	0
$937$	0.573206	0.0187258	0.00936292	−	0.999956i	$-0.497020\pi$
$937$	0.573206	0.0187258	0.00936292	+	0.999956i	$0.497020\pi$
$938$	5.34256	0.174441
$939$	0	0
$940$	−27.6942	−0.903286
$941$	−2.73202	−0.0890613	−0.0445307	−	0.999008i	$-0.514179\pi$
$941$	−2.73202	−0.0890613	−0.0445307	+	0.999008i	$0.514179\pi$
$942$	0	0
$943$	35.4816	1.15544
$944$	11.5898	0.377215
$945$	0	0
$946$	1.28516	0.0417842
$947$	37.7361	1.22626	0.613129	−	0.789982i	$-0.289910\pi$
$947$	37.7361	1.22626	0.613129	+	0.789982i	$0.289910\pi$
$948$	0	0
$949$	14.9301	0.484652
$950$	−2.98064	−0.0967048
$951$	0	0
$952$	−2.23219	−0.0723457
$953$	43.9311	1.42307	0.711533	−	0.702652i	$-0.248001\pi$
$953$	43.9311	1.42307	0.711533	+	0.702652i	$0.248001\pi$
$954$	0	0
$955$	1.89247	0.0612390
$956$	17.5071	0.566221
$957$	0	0
$958$	5.65569	0.182727
$959$	−9.64844	−0.311564
$960$	0	0
$961$	3.88536	0.125334
$962$	6.07000	0.195705
$963$	0	0
$964$	−16.8748	−0.543501
$965$	66.7572	2.14899
$966$	0	0
$967$	−42.4540	−1.36523	−0.682615	−	0.730779i	$-0.739157\pi$
$967$	−42.4540	−1.36523	−0.682615	+	0.730779i	$0.739157\pi$
$968$	−39.9229	−1.28317
$969$	0	0
$970$	−23.6440	−0.759163
$971$	15.2680	0.489973	0.244986	−	0.969526i	$-0.421216\pi$
$971$	15.2680	0.489973	0.244986	+	0.969526i	$0.421216\pi$
$972$	0	0
$973$	3.37125	0.108077
$974$	1.91363	0.0613167
$975$	0	0
$976$	3.76225	0.120427
$977$	37.5625	1.20173	0.600866	−	0.799350i	$-0.294823\pi$
$977$	37.5625	1.20173	0.600866	+	0.799350i	$0.294823\pi$
$978$	0	0
$979$	−89.8244	−2.87080
$980$	−17.9824	−0.574427
$981$	0	0
$982$	−29.4481	−0.939726
$983$	−21.5300	−0.686702	−0.343351	−	0.939207i	$-0.611562\pi$
$983$	−21.5300	−0.686702	−0.343351	+	0.939207i	$0.611562\pi$
$984$	0	0
$985$	−64.0004	−2.03922
$986$	1.58887	0.0505998
$987$	0	0
$988$	0.792598	0.0252159
$989$	1.77510	0.0564450
$990$	0	0
$991$	−17.0084	−0.540289	−0.270145	−	0.962820i	$-0.587071\pi$
$991$	−17.0084	−0.540289	−0.270145	+	0.962820i	$0.587071\pi$
$992$	−28.6632	−0.910057
$993$	0	0
$994$	6.64817	0.210867
$995$	43.8512	1.39018
$996$	0	0
$997$	−35.7124	−1.13102	−0.565512	−	0.824740i	$-0.691321\pi$
$997$	−35.7124	−1.13102	−0.565512	+	0.824740i	$0.691321\pi$
$998$	26.9417	0.852825
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.l.1.7	yes	10
3.2	odd	2	inner	6021.2.a.l.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.l.1.4	✓	10	3.2	odd	2	inner
6021.2.a.l.1.7	yes	10	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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q+1.02020 q^{2} -0.959195 q^{4} -2.96046 q^{5} +0.816945 q^{7} -3.01897 q^{8} +O(q^{10})\)	\(q+1.02020 q^{2} -0.959195 q^{4} -2.96046 q^{5} +0.816945 q^{7} -3.01897 q^{8} -3.02025 q^{10} -4.92179 q^{11} +1.06465 q^{13} +0.833446 q^{14} -1.16155 q^{16} +0.905065 q^{17} -0.776141 q^{19} +2.83966 q^{20} -5.02120 q^{22} -6.93545 q^{23} +3.76431 q^{25} +1.08615 q^{26} -0.783610 q^{28} +1.72077 q^{29} -5.90638 q^{31} +4.85292 q^{32} +0.923346 q^{34} -2.41853 q^{35} +5.58855 q^{37} -0.791817 q^{38} +8.93752 q^{40} -5.11599 q^{41} -0.255947 q^{43} +4.72096 q^{44} -7.07553 q^{46} -9.75266 q^{47} -6.33260 q^{49} +3.84034 q^{50} -1.02120 q^{52} -13.7259 q^{53} +14.5708 q^{55} -2.46633 q^{56} +1.75553 q^{58} -9.97782 q^{59} -3.23898 q^{61} -6.02568 q^{62} +7.27405 q^{64} -3.15184 q^{65} +6.41020 q^{67} -0.868134 q^{68} -2.46738 q^{70} +7.97672 q^{71} +14.0235 q^{73} +5.70143 q^{74} +0.744470 q^{76} -4.02083 q^{77} -11.1936 q^{79} +3.43873 q^{80} -5.21932 q^{82} +10.5277 q^{83} -2.67941 q^{85} -0.261116 q^{86} +14.8587 q^{88} +18.2503 q^{89} +0.869757 q^{91} +6.65245 q^{92} -9.94965 q^{94} +2.29773 q^{95} +7.82848 q^{97} -6.46051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) 10 * q + 20 * q^4 + 2 * q^7	\( 10 q + 20 q^{4} + 2 q^{7} - 10 q^{10} + 2 q^{13} + 44 q^{16} + 28 q^{19} - 42 q^{22} + 22 q^{25} + 40 q^{28} - 18 q^{31} + 36 q^{34} + 20 q^{37} - 4 q^{40} + 2 q^{43} - 30 q^{46} - 32 q^{49} - 2 q^{52} + 52 q^{55} + 84 q^{58} + 40 q^{61} + 64 q^{64} + 18 q^{67} + 18 q^{70} + 32 q^{73} + 104 q^{76} - 16 q^{79} - 94 q^{82} - 40 q^{85} - 32 q^{88} + 14 q^{91} - 56 q^{94} - 18 q^{97}+O(q^{100}) \) 10 * q + 20 * q^4 + 2 * q^7 - 10 * q^10 + 2 * q^13 + 44 * q^16 + 28 * q^19 - 42 * q^22 + 22 * q^25 + 40 * q^28 - 18 * q^31 + 36 * q^34 + 20 * q^37 - 4 * q^40 + 2 * q^43 - 30 * q^46 - 32 * q^49 - 2 * q^52 + 52 * q^55 + 84 * q^58 + 40 * q^61 + 64 * q^64 + 18 * q^67 + 18 * q^70 + 32 * q^73 + 104 * q^76 - 16 * q^79 - 94 * q^82 - 40 * q^85 - 32 * q^88 + 14 * q^91 - 56 * q^94 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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