LMFDB - Embedded newform 6021.2.a.l.1.9

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			1.9
Root			$2.32314$ 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)

$f(q)$	$=$	$q+2.32314 q^{2} +3.39697 q^{4} -3.53749 q^{5} -1.40218 q^{7} +3.24535 q^{8} +O(q^{10})$	\(q+2.32314 q^{2} +3.39697 q^{4} -3.53749 q^{5} -1.40218 q^{7} +3.24535 q^{8} -8.21807 q^{10} -6.53720 q^{11} -3.29317 q^{13} -3.25745 q^{14} +0.745457 q^{16} +1.57991 q^{17} +5.79915 q^{19} -12.0167 q^{20} -15.1868 q^{22} +5.41288 q^{23} +7.51382 q^{25} -7.65049 q^{26} -4.76315 q^{28} +7.46990 q^{29} +4.41676 q^{31} -4.75890 q^{32} +3.67035 q^{34} +4.96018 q^{35} +5.73815 q^{37} +13.4722 q^{38} -11.4804 q^{40} -8.92699 q^{41} -1.70425 q^{43} -22.2066 q^{44} +12.5749 q^{46} +11.5501 q^{47} -5.03390 q^{49} +17.4556 q^{50} -11.1868 q^{52} +2.49870 q^{53} +23.1252 q^{55} -4.55055 q^{56} +17.3536 q^{58} +2.99467 q^{59} +8.38286 q^{61} +10.2607 q^{62} -12.5465 q^{64} +11.6496 q^{65} +7.33228 q^{67} +5.36691 q^{68} +11.5232 q^{70} -1.30439 q^{71} +11.3490 q^{73} +13.3305 q^{74} +19.6995 q^{76} +9.16630 q^{77} -4.15965 q^{79} -2.63705 q^{80} -20.7386 q^{82} -4.33184 q^{83} -5.58892 q^{85} -3.95922 q^{86} -21.2155 q^{88} -2.44930 q^{89} +4.61761 q^{91} +18.3874 q^{92} +26.8325 q^{94} -20.5144 q^{95} +1.32403 q^{97} -11.6944 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.32314	1.64271	0.821353	−	0.570420i	$-0.193220\pi$
$2$	2.32314	1.64271	0.821353	+	0.570420i	$0.193220\pi$
$3$	0	0
$3$	0	0
$4$	3.39697	1.69848
$5$	−3.53749	−1.58201	−0.791006	−	0.611808i	$-0.790442\pi$
$5$	−3.53749	−1.58201	−0.791006	+	0.611808i	$0.790442\pi$
$6$	0	0
$7$	−1.40218	−0.529973	−0.264987	−	0.964252i	$-0.585368\pi$
$7$	−1.40218	−0.529973	−0.264987	+	0.964252i	$0.585368\pi$
$8$	3.24535	1.14740
$9$	0	0
$10$	−8.21807	−2.59878
$11$	−6.53720	−1.97104	−0.985519	−	0.169563i	$-0.945764\pi$
$11$	−6.53720	−1.97104	−0.985519	+	0.169563i	$0.945764\pi$
$12$	0	0
$13$	−3.29317	−0.913362	−0.456681	−	0.889631i	$-0.650962\pi$
$13$	−3.29317	−0.913362	−0.456681	+	0.889631i	$0.650962\pi$
$14$	−3.25745	−0.870590
$15$	0	0
$16$	0.745457	0.186364
$17$	1.57991	0.383185	0.191593	−	0.981475i	$-0.438635\pi$
$17$	1.57991	0.383185	0.191593	+	0.981475i	$0.438635\pi$
$18$	0	0
$19$	5.79915	1.33042	0.665208	−	0.746658i	$-0.268343\pi$
$19$	5.79915	1.33042	0.665208	+	0.746658i	$0.268343\pi$
$20$	−12.0167	−2.68702
$21$	0	0
$22$	−15.1868	−3.23784
$23$	5.41288	1.12866	0.564332	−	0.825548i	$-0.309134\pi$
$23$	5.41288	1.12866	0.564332	+	0.825548i	$0.309134\pi$
$24$	0	0
$25$	7.51382	1.50276
$26$	−7.65049	−1.50039
$27$	0	0
$28$	−4.76315	−0.900151
$29$	7.46990	1.38712	0.693562	−	0.720397i	$-0.256040\pi$
$29$	7.46990	1.38712	0.693562	+	0.720397i	$0.256040\pi$
$30$	0	0
$31$	4.41676	0.793273	0.396636	−	0.917976i	$-0.370177\pi$
$31$	4.41676	0.793273	0.396636	+	0.917976i	$0.370177\pi$
$32$	−4.75890	−0.841263
$33$	0	0
$34$	3.67035	0.629460
$35$	4.96018	0.838424
$36$	0	0
$37$	5.73815	0.943347	0.471673	−	0.881773i	$-0.343650\pi$
$37$	5.73815	0.943347	0.471673	+	0.881773i	$0.343650\pi$
$38$	13.4722	2.18548
$39$	0	0
$40$	−11.4804	−1.81521
$41$	−8.92699	−1.39416	−0.697081	−	0.716993i	$-0.745518\pi$
$41$	−8.92699	−1.39416	−0.697081	+	0.716993i	$0.745518\pi$
$42$	0	0
$43$	−1.70425	−0.259896	−0.129948	−	0.991521i	$-0.541481\pi$
$43$	−1.70425	−0.259896	−0.129948	+	0.991521i	$0.541481\pi$
$44$	−22.2066	−3.34778
$45$	0	0
$46$	12.5749	1.85406
$47$	11.5501	1.68476	0.842379	−	0.538886i	$-0.181154\pi$
$47$	11.5501	1.68476	0.842379	+	0.538886i	$0.181154\pi$
$48$	0	0
$49$	−5.03390	−0.719129
$50$	17.4556	2.46860
$51$	0	0
$52$	−11.1868	−1.55133
$53$	2.49870	0.343223	0.171611	−	0.985165i	$-0.445103\pi$
$53$	2.49870	0.343223	0.171611	+	0.985165i	$0.445103\pi$
$54$	0	0
$55$	23.1252	3.11821
$56$	−4.55055	−0.608093
$57$	0	0
$58$	17.3536	2.27864
$59$	2.99467	0.389873	0.194936	−	0.980816i	$-0.437550\pi$
$59$	2.99467	0.389873	0.194936	+	0.980816i	$0.437550\pi$
$60$	0	0
$61$	8.38286	1.07331	0.536657	−	0.843800i	$-0.319687\pi$
$61$	8.38286	1.07331	0.536657	+	0.843800i	$0.319687\pi$
$62$	10.2607	1.30311
$63$	0	0
$64$	−12.5465	−1.56831
$65$	11.6496	1.44495
$66$	0	0
$67$	7.33228	0.895781	0.447890	−	0.894088i	$-0.352175\pi$
$67$	7.33228	0.895781	0.447890	+	0.894088i	$0.352175\pi$
$68$	5.36691	0.650834
$69$	0	0
$70$	11.5232	1.37728
$71$	−1.30439	−0.154802	−0.0774010	−	0.997000i	$-0.524662\pi$
$71$	−1.30439	−0.154802	−0.0774010	+	0.997000i	$0.524662\pi$
$72$	0	0
$73$	11.3490	1.32830	0.664152	−	0.747598i	$-0.268793\pi$
$73$	11.3490	1.32830	0.664152	+	0.747598i	$0.268793\pi$
$74$	13.3305	1.54964
$75$	0	0
$76$	19.6995	2.25969
$77$	9.16630	1.04460
$78$	0	0
$79$	−4.15965	−0.467997	−0.233999	−	0.972237i	$-0.575181\pi$
$79$	−4.15965	−0.467997	−0.233999	+	0.972237i	$0.575181\pi$
$80$	−2.63705	−0.294831
$81$	0	0
$82$	−20.7386	−2.29020
$83$	−4.33184	−0.475481	−0.237741	−	0.971329i	$-0.576407\pi$
$83$	−4.33184	−0.475481	−0.237741	+	0.971329i	$0.576407\pi$
$84$	0	0
$85$	−5.58892	−0.606203
$86$	−3.95922	−0.426933
$87$	0	0
$88$	−21.2155	−2.26158
$89$	−2.44930	−0.259626	−0.129813	−	0.991539i	$-0.541438\pi$
$89$	−2.44930	−0.259626	−0.129813	+	0.991539i	$0.541438\pi$
$90$	0	0
$91$	4.61761	0.484057
$92$	18.3874	1.91702
$93$	0	0
$94$	26.8325	2.76756
$95$	−20.5144	−2.10473
$96$	0	0
$97$	1.32403	0.134435	0.0672176	−	0.997738i	$-0.478588\pi$
$97$	1.32403	0.134435	0.0672176	+	0.997738i	$0.478588\pi$
$98$	−11.6944	−1.18132
$99$	0	0
$100$	25.5242	2.55242
$101$	−9.65585	−0.960793	−0.480397	−	0.877051i	$-0.659507\pi$
$101$	−9.65585	−0.960793	−0.480397	+	0.877051i	$0.659507\pi$
$102$	0	0
$103$	−7.66873	−0.755622	−0.377811	−	0.925883i	$-0.623323\pi$
$103$	−7.66873	−0.755622	−0.377811	+	0.925883i	$0.623323\pi$
$104$	−10.6875	−1.04800
$105$	0	0
$106$	5.80482	0.563814
$107$	−8.50101	−0.821824	−0.410912	−	0.911675i	$-0.634790\pi$
$107$	−8.50101	−0.821824	−0.410912	+	0.911675i	$0.634790\pi$
$108$	0	0
$109$	9.99005	0.956873	0.478437	−	0.878122i	$-0.341204\pi$
$109$	9.99005	0.956873	0.478437	+	0.878122i	$0.341204\pi$
$110$	53.7231	5.12230
$111$	0	0
$112$	−1.04526	−0.0987680
$113$	−12.7801	−1.20225	−0.601126	−	0.799154i	$-0.705281\pi$
$113$	−12.7801	−1.20225	−0.601126	+	0.799154i	$0.705281\pi$
$114$	0	0
$115$	−19.1480	−1.78556
$116$	25.3750	2.35601
$117$	0	0
$118$	6.95703	0.640447
$119$	−2.21532	−0.203078
$120$	0	0
$121$	31.7349	2.88499
$122$	19.4745	1.76314
$123$	0	0
$124$	15.0036	1.34736
$125$	−8.89259	−0.795378
$126$	0	0
$127$	−21.1443	−1.87625	−0.938127	−	0.346292i	$-0.887441\pi$
$127$	−21.1443	−1.87625	−0.938127	+	0.346292i	$0.887441\pi$
$128$	−19.6294	−1.73501
$129$	0	0
$130$	27.0635	2.37363
$131$	0.974686	0.0851587	0.0425794	−	0.999093i	$-0.486442\pi$
$131$	0.974686	0.0851587	0.0425794	+	0.999093i	$0.486442\pi$
$132$	0	0
$133$	−8.13143	−0.705084
$134$	17.0339	1.47150
$135$	0	0
$136$	5.12737	0.439668
$137$	5.71434	0.488209	0.244105	−	0.969749i	$-0.421506\pi$
$137$	5.71434	0.488209	0.244105	+	0.969749i	$0.421506\pi$
$138$	0	0
$139$	14.2906	1.21211	0.606056	−	0.795422i	$-0.292751\pi$
$139$	14.2906	1.21211	0.606056	+	0.795422i	$0.292751\pi$
$140$	16.8496	1.42405
$141$	0	0
$142$	−3.03027	−0.254294
$143$	21.5281	1.80027
$144$	0	0
$145$	−26.4247	−2.19445
$146$	26.3654	2.18201
$147$	0	0
$148$	19.4923	1.60226
$149$	7.71662	0.632170	0.316085	−	0.948731i	$-0.397631\pi$
$149$	7.71662	0.632170	0.316085	+	0.948731i	$0.397631\pi$
$150$	0	0
$151$	12.3850	1.00788	0.503939	−	0.863739i	$-0.331883\pi$
$151$	12.3850	1.00788	0.503939	+	0.863739i	$0.331883\pi$
$152$	18.8203	1.52652
$153$	0	0
$154$	21.2946	1.71597
$155$	−15.6242	−1.25497
$156$	0	0
$157$	15.9786	1.27523	0.637615	−	0.770355i	$-0.279921\pi$
$157$	15.9786	1.27523	0.637615	+	0.770355i	$0.279921\pi$
$158$	−9.66344	−0.768782
$159$	0	0
$160$	16.8345	1.33089
$161$	−7.58982	−0.598161
$162$	0	0
$163$	16.3354	1.27949	0.639743	−	0.768589i	$-0.279041\pi$
$163$	16.3354	1.27949	0.639743	+	0.768589i	$0.279041\pi$
$164$	−30.3247	−2.36796
$165$	0	0
$166$	−10.0635	−0.781076
$167$	−17.8731	−1.38306	−0.691531	−	0.722346i	$-0.743064\pi$
$167$	−17.8731	−1.38306	−0.691531	+	0.722346i	$0.743064\pi$
$168$	0	0
$169$	−2.15501	−0.165770
$170$	−12.9838	−0.995814
$171$	0	0
$172$	−5.78930	−0.441430
$173$	−2.09141	−0.159007	−0.0795035	−	0.996835i	$-0.525333\pi$
$173$	−2.09141	−0.159007	−0.0795035	+	0.996835i	$0.525333\pi$
$174$	0	0
$175$	−10.5357	−0.796424
$176$	−4.87320	−0.367331
$177$	0	0
$178$	−5.69007	−0.426489
$179$	−19.6462	−1.46843	−0.734214	−	0.678919i	$-0.762449\pi$
$179$	−19.6462	−1.46843	−0.734214	+	0.678919i	$0.762449\pi$
$180$	0	0
$181$	12.5283	0.931223	0.465611	−	0.884989i	$-0.345834\pi$
$181$	12.5283	0.931223	0.465611	+	0.884989i	$0.345834\pi$
$182$	10.7273	0.795164
$183$	0	0
$184$	17.5667	1.29503
$185$	−20.2986	−1.49239
$186$	0	0
$187$	−10.3282	−0.755272
$188$	39.2354	2.86153
$189$	0	0
$190$	−47.6578	−3.45746
$191$	5.64724	0.408620	0.204310	−	0.978906i	$-0.434505\pi$
$191$	5.64724	0.408620	0.204310	+	0.978906i	$0.434505\pi$
$192$	0	0
$193$	0.393200	0.0283031	0.0141516	−	0.999900i	$-0.495495\pi$
$193$	0.393200	0.0283031	0.0141516	+	0.999900i	$0.495495\pi$
$194$	3.07591	0.220838
$195$	0	0
$196$	−17.1000	−1.22143
$197$	1.61935	0.115374	0.0576868	−	0.998335i	$-0.481628\pi$
$197$	1.61935	0.115374	0.0576868	+	0.998335i	$0.481628\pi$
$198$	0	0
$199$	11.7301	0.831526	0.415763	−	0.909473i	$-0.363515\pi$
$199$	11.7301	0.831526	0.415763	+	0.909473i	$0.363515\pi$
$200$	24.3850	1.72428
$201$	0	0
$202$	−22.4319	−1.57830
$203$	−10.4741	−0.735139
$204$	0	0
$205$	31.5791	2.20558
$206$	−17.8155	−1.24127
$207$	0	0
$208$	−2.45492	−0.170218
$209$	−37.9101	−2.62230
$210$	0	0
$211$	7.87538	0.542164	0.271082	−	0.962556i	$-0.412619\pi$
$211$	7.87538	0.542164	0.271082	+	0.962556i	$0.412619\pi$
$212$	8.48800	0.582958
$213$	0	0
$214$	−19.7490	−1.35002
$215$	6.02878	0.411159
$216$	0	0
$217$	−6.19307	−0.420413
$218$	23.2083	1.57186
$219$	0	0
$220$	78.5557	5.29623
$221$	−5.20292	−0.349987
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	6.67282	0.445846
$225$	0	0
$226$	−29.6900	−1.97495
$227$	8.12162	0.539051	0.269525	−	0.962993i	$-0.413133\pi$
$227$	8.12162	0.539051	0.269525	+	0.962993i	$0.413133\pi$
$228$	0	0
$229$	−26.7997	−1.77097	−0.885487	−	0.464665i	$-0.846175\pi$
$229$	−26.7997	−1.77097	−0.885487	+	0.464665i	$0.846175\pi$
$230$	−44.4834	−2.93315
$231$	0	0
$232$	24.2424	1.59159
$233$	16.5874	1.08668	0.543340	−	0.839513i	$-0.317160\pi$
$233$	16.5874	1.08668	0.543340	+	0.839513i	$0.317160\pi$
$234$	0	0
$235$	−40.8584	−2.66531
$236$	10.1728	0.662193
$237$	0	0
$238$	−5.14648	−0.333597
$239$	−26.4853	−1.71319	−0.856595	−	0.515989i	$-0.827424\pi$
$239$	−26.4853	−1.71319	−0.856595	+	0.515989i	$0.827424\pi$
$240$	0	0
$241$	−11.6051	−0.747549	−0.373775	−	0.927520i	$-0.621937\pi$
$241$	−11.6051	−0.747549	−0.373775	+	0.927520i	$0.621937\pi$
$242$	73.7246	4.73920
$243$	0	0
$244$	28.4763	1.82301
$245$	17.8074	1.13767
$246$	0	0
$247$	−19.0976	−1.21515
$248$	14.3339	0.910205
$249$	0	0
$250$	−20.6587	−1.30657
$251$	27.2956	1.72288	0.861442	−	0.507855i	$-0.169562\pi$
$251$	27.2956	1.72288	0.861442	+	0.507855i	$0.169562\pi$
$252$	0	0
$253$	−35.3851	−2.22464
$254$	−49.1211	−3.08213
$255$	0	0
$256$	−20.5089	−1.28181
$257$	−11.1940	−0.698260	−0.349130	−	0.937074i	$-0.613523\pi$
$257$	−11.1940	−0.698260	−0.349130	+	0.937074i	$0.613523\pi$
$258$	0	0
$259$	−8.04591	−0.499948
$260$	39.5732	2.45422
$261$	0	0
$262$	2.26433	0.139891
$263$	5.55378	0.342461	0.171230	−	0.985231i	$-0.445226\pi$
$263$	5.55378	0.342461	0.171230	+	0.985231i	$0.445226\pi$
$264$	0	0
$265$	−8.83911	−0.542982
$266$	−18.8904	−1.15825
$267$	0	0
$268$	24.9075	1.52147
$269$	19.0196	1.15964	0.579822	−	0.814743i	$-0.303122\pi$
$269$	19.0196	1.15964	0.579822	+	0.814743i	$0.303122\pi$
$270$	0	0
$271$	12.5633	0.763163	0.381582	−	0.924335i	$-0.375379\pi$
$271$	12.5633	0.763163	0.381582	+	0.924335i	$0.375379\pi$
$272$	1.17776	0.0714120
$273$	0	0
$274$	13.2752	0.801984
$275$	−49.1193	−2.96200
$276$	0	0
$277$	13.3244	0.800588	0.400294	−	0.916387i	$-0.368908\pi$
$277$	13.3244	0.800588	0.400294	+	0.916387i	$0.368908\pi$
$278$	33.1990	1.99115
$279$	0	0
$280$	16.0975	0.962011
$281$	24.8350	1.48153	0.740765	−	0.671764i	$-0.234463\pi$
$281$	24.8350	1.48153	0.740765	+	0.671764i	$0.234463\pi$
$282$	0	0
$283$	19.2156	1.14225	0.571123	−	0.820864i	$-0.306508\pi$
$283$	19.2156	1.14225	0.571123	+	0.820864i	$0.306508\pi$
$284$	−4.43096	−0.262929
$285$	0	0
$286$	50.0128	2.95732
$287$	12.5172	0.738868
$288$	0	0
$289$	−14.5039	−0.853169
$290$	−61.3881	−3.60483
$291$	0	0
$292$	38.5523	2.25610
$293$	−1.63777	−0.0956796	−0.0478398	−	0.998855i	$-0.515234\pi$
$293$	−1.63777	−0.0956796	−0.0478398	+	0.998855i	$0.515234\pi$
$294$	0	0
$295$	−10.5936	−0.616784
$296$	18.6223	1.08240
$297$	0	0
$298$	17.9268	1.03847
$299$	−17.8256	−1.03088
$300$	0	0
$301$	2.38967	0.137738
$302$	28.7721	1.65565
$303$	0	0
$304$	4.32301	0.247942
$305$	−29.6542	−1.69800
$306$	0	0
$307$	3.77984	0.215727	0.107863	−	0.994166i	$-0.465599\pi$
$307$	3.77984	0.215727	0.107863	+	0.994166i	$0.465599\pi$
$308$	31.1376	1.77423
$309$	0	0
$310$	−36.2972	−2.06154
$311$	−22.1528	−1.25617	−0.628084	−	0.778145i	$-0.716161\pi$
$311$	−22.1528	−1.25617	−0.628084	+	0.778145i	$0.716161\pi$
$312$	0	0
$313$	17.5961	0.994590	0.497295	−	0.867581i	$-0.334327\pi$
$313$	17.5961	0.994590	0.497295	+	0.867581i	$0.334327\pi$
$314$	37.1205	2.09483
$315$	0	0
$316$	−14.1302	−0.794886
$317$	16.3519	0.918413	0.459206	−	0.888330i	$-0.348134\pi$
$317$	16.3519	0.918413	0.459206	+	0.888330i	$0.348134\pi$
$318$	0	0
$319$	−48.8322	−2.73408
$320$	44.3831	2.48109
$321$	0	0
$322$	−17.6322	−0.982604
$323$	9.16214	0.509795
$324$	0	0
$325$	−24.7443	−1.37257
$326$	37.9493	2.10182
$327$	0	0
$328$	−28.9712	−1.59967
$329$	−16.1953	−0.892876
$330$	0	0
$331$	−21.6088	−1.18772	−0.593862	−	0.804567i	$-0.702398\pi$
$331$	−21.6088	−1.18772	−0.593862	+	0.804567i	$0.702398\pi$
$332$	−14.7151	−0.807598
$333$	0	0
$334$	−41.5217	−2.27197
$335$	−25.9379	−1.41714
$336$	0	0
$337$	−11.8096	−0.643308	−0.321654	−	0.946857i	$-0.604239\pi$
$337$	−11.8096	−0.643308	−0.321654	+	0.946857i	$0.604239\pi$
$338$	−5.00639	−0.272312
$339$	0	0
$340$	−18.9854	−1.02963
$341$	−28.8732	−1.56357
$342$	0	0
$343$	16.8737	0.911092
$344$	−5.53090	−0.298206
$345$	0	0
$346$	−4.85863	−0.261202
$347$	20.0194	1.07470	0.537349	−	0.843360i	$-0.319426\pi$
$347$	20.0194	1.07470	0.537349	+	0.843360i	$0.319426\pi$
$348$	0	0
$349$	−27.1679	−1.45426	−0.727131	−	0.686499i	$-0.759147\pi$
$349$	−27.1679	−1.45426	−0.727131	+	0.686499i	$0.759147\pi$
$350$	−24.4759	−1.30829
$351$	0	0
$352$	31.1099	1.65816
$353$	18.2241	0.969973	0.484986	−	0.874522i	$-0.338825\pi$
$353$	18.2241	0.969973	0.484986	+	0.874522i	$0.338825\pi$
$354$	0	0
$355$	4.61425	0.244899
$356$	−8.32020	−0.440970
$357$	0	0
$358$	−45.6409	−2.41219
$359$	29.4569	1.55468	0.777338	−	0.629083i	$-0.216569\pi$
$359$	29.4569	1.55468	0.777338	+	0.629083i	$0.216569\pi$
$360$	0	0
$361$	14.6301	0.770004
$362$	29.1050	1.52973
$363$	0	0
$364$	15.6859	0.822163
$365$	−40.1470	−2.10139
$366$	0	0
$367$	16.0386	0.837208	0.418604	−	0.908169i	$-0.362519\pi$
$367$	16.0386	0.837208	0.418604	+	0.908169i	$0.362519\pi$
$368$	4.03507	0.210343
$369$	0	0
$370$	−47.1565	−2.45155
$371$	−3.50362	−0.181899
$372$	0	0
$373$	−7.02715	−0.363852	−0.181926	−	0.983312i	$-0.558233\pi$
$373$	−7.02715	−0.363852	−0.181926	+	0.983312i	$0.558233\pi$
$374$	−23.9938	−1.24069
$375$	0	0
$376$	37.4842	1.93310
$377$	−24.5997	−1.26695
$378$	0	0
$379$	24.4557	1.25620	0.628102	−	0.778131i	$-0.283832\pi$
$379$	24.4557	1.25620	0.628102	+	0.778131i	$0.283832\pi$
$380$	−69.6868	−3.57486
$381$	0	0
$382$	13.1193	0.671243
$383$	27.8987	1.42556	0.712778	−	0.701389i	$-0.247437\pi$
$383$	27.8987	1.42556	0.712778	+	0.701389i	$0.247437\pi$
$384$	0	0
$385$	−32.4257	−1.65257
$386$	0.913457	0.0464937
$387$	0	0
$388$	4.49770	0.228336
$389$	−24.8962	−1.26229	−0.631144	−	0.775666i	$-0.717414\pi$
$389$	−24.8962	−1.26229	−0.631144	+	0.775666i	$0.717414\pi$
$390$	0	0
$391$	8.55188	0.432487
$392$	−16.3368	−0.825131
$393$	0	0
$394$	3.76196	0.189525
$395$	14.7147	0.740378
$396$	0	0
$397$	5.21207	0.261586	0.130793	−	0.991410i	$-0.458248\pi$
$397$	5.21207	0.261586	0.130793	+	0.991410i	$0.458248\pi$
$398$	27.2507	1.36595
$399$	0	0
$400$	5.60123	0.280061
$401$	28.1074	1.40362	0.701808	−	0.712366i	$-0.252376\pi$
$401$	28.1074	1.40362	0.701808	+	0.712366i	$0.252376\pi$
$402$	0	0
$403$	−14.5451	−0.724545
$404$	−32.8006	−1.63189
$405$	0	0
$406$	−24.3328	−1.20762
$407$	−37.5114	−1.85937
$408$	0	0
$409$	12.6479	0.625399	0.312699	−	0.949852i	$-0.398767\pi$
$409$	12.6479	0.625399	0.312699	+	0.949852i	$0.398767\pi$
$410$	73.3626	3.62312
$411$	0	0
$412$	−26.0504	−1.28341
$413$	−4.19906	−0.206622
$414$	0	0
$415$	15.3238	0.752217
$416$	15.6719	0.768377
$417$	0	0
$418$	−88.0705	−4.30767
$419$	23.1588	1.13138	0.565690	−	0.824618i	$-0.308610\pi$
$419$	23.1588	1.13138	0.565690	+	0.824618i	$0.308610\pi$
$420$	0	0
$421$	25.7550	1.25522	0.627612	−	0.778527i	$-0.284033\pi$
$421$	25.7550	1.25522	0.627612	+	0.778527i	$0.284033\pi$
$422$	18.2956	0.890616
$423$	0	0
$424$	8.10915	0.393815
$425$	11.8712	0.575836
$426$	0	0
$427$	−11.7542	−0.568828
$428$	−28.8777	−1.39586
$429$	0	0
$430$	14.0057	0.675414
$431$	−32.2450	−1.55319	−0.776593	−	0.630003i	$-0.783054\pi$
$431$	−32.2450	−1.55319	−0.776593	+	0.630003i	$0.783054\pi$
$432$	0	0
$433$	−7.11035	−0.341702	−0.170851	−	0.985297i	$-0.554652\pi$
$433$	−7.11035	−0.341702	−0.170851	+	0.985297i	$0.554652\pi$
$434$	−14.3874	−0.690615
$435$	0	0
$436$	33.9359	1.62523
$437$	31.3901	1.50159
$438$	0	0
$439$	−19.4782	−0.929644	−0.464822	−	0.885404i	$-0.653882\pi$
$439$	−19.4782	−0.929644	−0.464822	+	0.885404i	$0.653882\pi$
$440$	75.0495	3.57784
$441$	0	0
$442$	−12.0871	−0.574925
$443$	−4.57124	−0.217186	−0.108593	−	0.994086i	$-0.534635\pi$
$443$	−4.57124	−0.217186	−0.108593	+	0.994086i	$0.534635\pi$
$444$	0	0
$445$	8.66438	0.410731
$446$	−2.32314	−0.110004
$447$	0	0
$448$	17.5924	0.831163
$449$	−9.21549	−0.434906	−0.217453	−	0.976071i	$-0.569775\pi$
$449$	−9.21549	−0.434906	−0.217453	+	0.976071i	$0.569775\pi$
$450$	0	0
$451$	58.3575	2.74795
$452$	−43.4136	−2.04201
$453$	0	0
$454$	18.8676	0.885502
$455$	−16.3347	−0.765784
$456$	0	0
$457$	−6.45955	−0.302165	−0.151082	−	0.988521i	$-0.548276\pi$
$457$	−6.45955	−0.302165	−0.151082	+	0.988521i	$0.548276\pi$
$458$	−62.2594	−2.90919
$459$	0	0
$460$	−65.0452	−3.03275
$461$	41.1388	1.91603	0.958013	−	0.286726i	$-0.0925670\pi$
$461$	41.1388	1.91603	0.958013	+	0.286726i	$0.0925670\pi$
$462$	0	0
$463$	−19.8971	−0.924698	−0.462349	−	0.886698i	$-0.652993\pi$
$463$	−19.8971	−0.924698	−0.462349	+	0.886698i	$0.652993\pi$
$464$	5.56849	0.258511
$465$	0	0
$466$	38.5349	1.78510
$467$	31.0567	1.43713	0.718567	−	0.695458i	$-0.244798\pi$
$467$	31.0567	1.43713	0.718567	+	0.695458i	$0.244798\pi$
$468$	0	0
$469$	−10.2812	−0.474740
$470$	−94.9197	−4.37832
$471$	0	0
$472$	9.71876	0.447342
$473$	11.1410	0.512266
$474$	0	0
$475$	43.5737	1.99930
$476$	−7.52536	−0.344924
$477$	0	0
$478$	−61.5289	−2.81427
$479$	−0.299548	−0.0136867	−0.00684334	−	0.999977i	$-0.502178\pi$
$479$	−0.299548	−0.0136867	−0.00684334	+	0.999977i	$0.502178\pi$
$480$	0	0
$481$	−18.8967	−0.861617
$482$	−26.9602	−1.22800
$483$	0	0
$484$	107.803	4.90012
$485$	−4.68375	−0.212678
$486$	0	0
$487$	26.6462	1.20745	0.603727	−	0.797191i	$-0.293682\pi$
$487$	26.6462	1.20745	0.603727	+	0.797191i	$0.293682\pi$
$488$	27.2053	1.23153
$489$	0	0
$490$	41.3689	1.86886
$491$	−14.2442	−0.642833	−0.321417	−	0.946938i	$-0.604159\pi$
$491$	−14.2442	−0.642833	−0.321417	+	0.946938i	$0.604159\pi$
$492$	0	0
$493$	11.8018	0.531525
$494$	−44.3663	−1.99614
$495$	0	0
$496$	3.29250	0.147838
$497$	1.82898	0.0820409
$498$	0	0
$499$	10.5703	0.473193	0.236596	−	0.971608i	$-0.423968\pi$
$499$	10.5703	0.473193	0.236596	+	0.971608i	$0.423968\pi$
$500$	−30.2079	−1.35094
$501$	0	0
$502$	63.4115	2.83019
$503$	30.2358	1.34815	0.674073	−	0.738664i	$-0.264543\pi$
$503$	30.2358	1.34815	0.674073	+	0.738664i	$0.264543\pi$
$504$	0	0
$505$	34.1575	1.51999
$506$	−82.2044	−3.65443
$507$	0	0
$508$	−71.8265	−3.18679
$509$	16.2574	0.720597	0.360299	−	0.932837i	$-0.382675\pi$
$509$	16.2574	0.720597	0.360299	+	0.932837i	$0.382675\pi$
$510$	0	0
$511$	−15.9133	−0.703965
$512$	−8.38609	−0.370616
$513$	0	0
$514$	−26.0051	−1.14704
$515$	27.1280	1.19540
$516$	0	0
$517$	−75.5054	−3.32072
$518$	−18.6917	−0.821268
$519$	0	0
$520$	37.8069	1.65794
$521$	−21.6926	−0.950371	−0.475186	−	0.879886i	$-0.657619\pi$
$521$	−21.6926	−0.950371	−0.475186	+	0.879886i	$0.657619\pi$
$522$	0	0
$523$	6.17954	0.270212	0.135106	−	0.990831i	$-0.456862\pi$
$523$	6.17954	0.270212	0.135106	+	0.990831i	$0.456862\pi$
$524$	3.31098	0.144641
$525$	0	0
$526$	12.9022	0.562562
$527$	6.97809	0.303970
$528$	0	0
$529$	6.29929	0.273882
$530$	−20.5345	−0.891961
$531$	0	0
$532$	−27.6222	−1.19757
$533$	29.3981	1.27337
$534$	0	0
$535$	30.0722	1.30014
$536$	23.7958	1.02782
$537$	0	0
$538$	44.1851	1.90495
$539$	32.9076	1.41743
$540$	0	0
$541$	18.4621	0.793746	0.396873	−	0.917874i	$-0.370095\pi$
$541$	18.4621	0.793746	0.396873	+	0.917874i	$0.370095\pi$
$542$	29.1862	1.25365
$543$	0	0
$544$	−7.51864	−0.322359
$545$	−35.3397	−1.51379
$546$	0	0
$547$	36.6114	1.56539	0.782694	−	0.622406i	$-0.213845\pi$
$547$	36.6114	1.56539	0.782694	+	0.622406i	$0.213845\pi$
$548$	19.4114	0.829216
$549$	0	0
$550$	−114.111	−4.86570
$551$	43.3190	1.84545
$552$	0	0
$553$	5.83257	0.248026
$554$	30.9545	1.31513
$555$	0	0
$556$	48.5447	2.05875
$557$	−25.5781	−1.08378	−0.541890	−	0.840449i	$-0.682291\pi$
$557$	−25.5781	−1.08378	−0.541890	+	0.840449i	$0.682291\pi$
$558$	0	0
$559$	5.61240	0.237379
$560$	3.69760	0.156252
$561$	0	0
$562$	57.6950	2.43372
$563$	−30.2353	−1.27427	−0.637133	−	0.770754i	$-0.719880\pi$
$563$	−30.2353	−1.27427	−0.637133	+	0.770754i	$0.719880\pi$
$564$	0	0
$565$	45.2095	1.90198
$566$	44.6404	1.87638
$567$	0	0
$568$	−4.23319	−0.177621
$569$	−24.1231	−1.01129	−0.505646	−	0.862741i	$-0.668746\pi$
$569$	−24.1231	−1.01129	−0.505646	+	0.862741i	$0.668746\pi$
$570$	0	0
$571$	36.6082	1.53200	0.766002	−	0.642838i	$-0.222243\pi$
$571$	36.6082	1.53200	0.766002	+	0.642838i	$0.222243\pi$
$572$	73.1303	3.05773
$573$	0	0
$574$	29.0792	1.21374
$575$	40.6714	1.69611
$576$	0	0
$577$	31.0593	1.29302	0.646509	−	0.762907i	$-0.276228\pi$
$577$	31.0593	1.29302	0.646509	+	0.762907i	$0.276228\pi$
$578$	−33.6945	−1.40151
$579$	0	0
$580$	−89.7637	−3.72724
$581$	6.07401	0.251992
$582$	0	0
$583$	−16.3345	−0.676505
$584$	36.8316	1.52410
$585$	0	0
$586$	−3.80477	−0.157173
$587$	−34.1128	−1.40799	−0.703993	−	0.710207i	$-0.748601\pi$
$587$	−34.1128	−1.40799	−0.703993	+	0.710207i	$0.748601\pi$
$588$	0	0
$589$	25.6134	1.05538
$590$	−24.6104	−1.01319
$591$	0	0
$592$	4.27755	0.175806
$593$	12.7977	0.525537	0.262769	−	0.964859i	$-0.415364\pi$
$593$	12.7977	0.525537	0.262769	+	0.964859i	$0.415364\pi$
$594$	0	0
$595$	7.83665	0.321271
$596$	26.2131	1.07373
$597$	0	0
$598$	−41.4112	−1.69343
$599$	−29.7249	−1.21453	−0.607263	−	0.794501i	$-0.707733\pi$
$599$	−29.7249	−1.21453	−0.607263	+	0.794501i	$0.707733\pi$
$600$	0	0
$601$	40.7464	1.66208	0.831041	−	0.556211i	$-0.187745\pi$
$601$	40.7464	1.66208	0.831041	+	0.556211i	$0.187745\pi$
$602$	5.55152	0.226263
$603$	0	0
$604$	42.0715	1.71187
$605$	−112.262	−4.56410
$606$	0	0
$607$	8.90470	0.361431	0.180715	−	0.983535i	$-0.442159\pi$
$607$	8.90470	0.361431	0.180715	+	0.983535i	$0.442159\pi$
$608$	−27.5976	−1.11923
$609$	0	0
$610$	−68.8909	−2.78931
$611$	−38.0365	−1.53879
$612$	0	0
$613$	32.0669	1.29517	0.647585	−	0.761994i	$-0.275779\pi$
$613$	32.0669	1.29517	0.647585	+	0.761994i	$0.275779\pi$
$614$	8.78108	0.354375
$615$	0	0
$616$	29.7479	1.19858
$617$	−44.8501	−1.80560	−0.902798	−	0.430066i	$-0.858490\pi$
$617$	−44.8501	−1.80560	−0.902798	+	0.430066i	$0.858490\pi$
$618$	0	0
$619$	9.28436	0.373170	0.186585	−	0.982439i	$-0.440258\pi$
$619$	9.28436	0.373170	0.186585	+	0.982439i	$0.440258\pi$
$620$	−53.0750	−2.13154
$621$	0	0
$622$	−51.4639	−2.06352
$623$	3.43436	0.137595
$624$	0	0
$625$	−6.11165	−0.244466
$626$	40.8782	1.63382
$627$	0	0
$628$	54.2787	2.16596
$629$	9.06578	0.361476
$630$	0	0
$631$	1.10110	0.0438340	0.0219170	−	0.999760i	$-0.493023\pi$
$631$	1.10110	0.0438340	0.0219170	+	0.999760i	$0.493023\pi$
$632$	−13.4995	−0.536982
$633$	0	0
$634$	37.9877	1.50868
$635$	74.7977	2.96826
$636$	0	0
$637$	16.5775	0.656825
$638$	−113.444	−4.49129
$639$	0	0
$640$	69.4389	2.74481
$641$	−3.28212	−0.129636	−0.0648179	−	0.997897i	$-0.520647\pi$
$641$	−3.28212	−0.129636	−0.0648179	+	0.997897i	$0.520647\pi$
$642$	0	0
$643$	12.9431	0.510425	0.255213	−	0.966885i	$-0.417855\pi$
$643$	12.9431	0.510425	0.255213	+	0.966885i	$0.417855\pi$
$644$	−25.7824	−1.01597
$645$	0	0
$646$	21.2849	0.837444
$647$	21.2349	0.834831	0.417415	−	0.908716i	$-0.362936\pi$
$647$	21.2349	0.834831	0.417415	+	0.908716i	$0.362936\pi$
$648$	0	0
$649$	−19.5768	−0.768455
$650$	−57.4844	−2.25472
$651$	0	0
$652$	55.4908	2.17319
$653$	6.21595	0.243249	0.121624	−	0.992576i	$-0.461190\pi$
$653$	6.21595	0.243249	0.121624	+	0.992576i	$0.461190\pi$
$654$	0	0
$655$	−3.44794	−0.134722
$656$	−6.65469	−0.259822
$657$	0	0
$658$	−37.6239	−1.46673
$659$	−13.1767	−0.513293	−0.256646	−	0.966505i	$-0.582618\pi$
$659$	−13.1767	−0.513293	−0.256646	+	0.966505i	$0.582618\pi$
$660$	0	0
$661$	26.9021	1.04637	0.523186	−	0.852219i	$-0.324743\pi$
$661$	26.9021	1.04637	0.523186	+	0.852219i	$0.324743\pi$
$662$	−50.2001	−1.95108
$663$	0	0
$664$	−14.0583	−0.545569
$665$	28.7648	1.11545
$666$	0	0
$667$	40.4337	1.56560
$668$	−60.7144	−2.34911
$669$	0	0
$670$	−60.2572	−2.32794
$671$	−54.8004	−2.11554
$672$	0	0
$673$	−23.6460	−0.911485	−0.455742	−	0.890112i	$-0.650626\pi$
$673$	−23.6460	−0.911485	−0.455742	+	0.890112i	$0.650626\pi$
$674$	−27.4352	−1.05677
$675$	0	0
$676$	−7.32050	−0.281558
$677$	−10.0158	−0.384939	−0.192470	−	0.981303i	$-0.561650\pi$
$677$	−10.0158	−0.384939	−0.192470	+	0.981303i	$0.561650\pi$
$678$	0	0
$679$	−1.85653	−0.0712470
$680$	−18.1380	−0.695560
$681$	0	0
$682$	−67.0764	−2.56849
$683$	−16.0572	−0.614412	−0.307206	−	0.951643i	$-0.599394\pi$
$683$	−16.0572	−0.614412	−0.307206	+	0.951643i	$0.599394\pi$
$684$	0	0
$685$	−20.2144	−0.772353
$686$	39.1998	1.49666
$687$	0	0
$688$	−1.27045	−0.0484354
$689$	−8.22865	−0.313486
$690$	0	0
$691$	−48.7163	−1.85325	−0.926627	−	0.375982i	$-0.877306\pi$
$691$	−48.7163	−1.85325	−0.926627	+	0.375982i	$0.877306\pi$
$692$	−7.10446	−0.270071
$693$	0	0
$694$	46.5078	1.76541
$695$	−50.5528	−1.91758
$696$	0	0
$697$	−14.1039	−0.534222
$698$	−63.1147	−2.38892
$699$	0	0
$700$	−35.7894	−1.35271
$701$	−30.9232	−1.16795	−0.583977	−	0.811770i	$-0.698504\pi$
$701$	−30.9232	−1.16795	−0.583977	+	0.811770i	$0.698504\pi$
$702$	0	0
$703$	33.2764	1.25504
$704$	82.0189	3.09120
$705$	0	0
$706$	42.3372	1.59338
$707$	13.5392	0.509195
$708$	0	0
$709$	3.57456	0.134245	0.0671227	−	0.997745i	$-0.478618\pi$
$709$	3.57456	0.134245	0.0671227	+	0.997745i	$0.478618\pi$
$710$	10.7195	0.402297
$711$	0	0
$712$	−7.94884	−0.297896
$713$	23.9074	0.895338
$714$	0	0
$715$	−76.1554	−2.84805
$716$	−66.7376	−2.49410
$717$	0	0
$718$	68.4325	2.55388
$719$	49.4014	1.84236	0.921181	−	0.389134i	$-0.127226\pi$
$719$	49.4014	1.84236	0.921181	+	0.389134i	$0.127226\pi$
$720$	0	0
$721$	10.7529	0.400459
$722$	33.9877	1.26489
$723$	0	0
$724$	42.5583	1.58167
$725$	56.1274	2.08452
$726$	0	0
$727$	12.1853	0.451929	0.225964	−	0.974136i	$-0.427447\pi$
$727$	12.1853	0.451929	0.225964	+	0.974136i	$0.427447\pi$
$728$	14.9858	0.555409
$729$	0	0
$730$	−93.2671	−3.45197
$731$	−2.69257	−0.0995883
$732$	0	0
$733$	16.0114	0.591396	0.295698	−	0.955281i	$-0.404448\pi$
$733$	16.0114	0.591396	0.295698	+	0.955281i	$0.404448\pi$
$734$	37.2599	1.37529
$735$	0	0
$736$	−25.7594	−0.949503
$737$	−47.9326	−1.76562
$738$	0	0
$739$	5.37767	0.197821	0.0989104	−	0.995096i	$-0.468464\pi$
$739$	5.37767	0.197821	0.0989104	+	0.995096i	$0.468464\pi$
$740$	−68.9539	−2.53479
$741$	0	0
$742$	−8.13939	−0.298806
$743$	−16.0221	−0.587795	−0.293897	−	0.955837i	$-0.594952\pi$
$743$	−16.0221	−0.587795	−0.293897	+	0.955837i	$0.594952\pi$
$744$	0	0
$745$	−27.2975	−1.00010
$746$	−16.3250	−0.597702
$747$	0	0
$748$	−35.0846	−1.28282
$749$	11.9199	0.435545
$750$	0	0
$751$	−49.1863	−1.79483	−0.897416	−	0.441185i	$-0.854558\pi$
$751$	−49.1863	−1.79483	−0.897416	+	0.441185i	$0.854558\pi$
$752$	8.61012	0.313979
$753$	0	0
$754$	−57.1484	−2.08122
$755$	−43.8119	−1.59448
$756$	0	0
$757$	−41.9297	−1.52396	−0.761981	−	0.647599i	$-0.775773\pi$
$757$	−41.9297	−1.52396	−0.761981	+	0.647599i	$0.775773\pi$
$758$	56.8139	2.06357
$759$	0	0
$760$	−66.5764	−2.41498
$761$	6.53224	0.236793	0.118397	−	0.992966i	$-0.462225\pi$
$761$	6.53224	0.236793	0.118397	+	0.992966i	$0.462225\pi$
$762$	0	0
$763$	−14.0078	−0.507117
$764$	19.1835	0.694035
$765$	0	0
$766$	64.8125	2.34177
$767$	−9.86197	−0.356095
$768$	0	0
$769$	−10.5327	−0.379820	−0.189910	−	0.981802i	$-0.560820\pi$
$769$	−10.5327	−0.379820	−0.189910	+	0.981802i	$0.560820\pi$
$770$	−75.3293	−2.71468
$771$	0	0
$772$	1.33569	0.0480724
$773$	44.1536	1.58810	0.794048	−	0.607856i	$-0.207970\pi$
$773$	44.1536	1.58810	0.794048	+	0.607856i	$0.207970\pi$
$774$	0	0
$775$	33.1867	1.19210
$776$	4.29695	0.154252
$777$	0	0
$778$	−57.8373	−2.07357
$779$	−51.7689	−1.85481
$780$	0	0
$781$	8.52702	0.305121
$782$	19.8672	0.710449
$783$	0	0
$784$	−3.75256	−0.134020
$785$	−56.5240	−2.01743
$786$	0	0
$787$	1.54523	0.0550815	0.0275408	−	0.999621i	$-0.491232\pi$
$787$	1.54523	0.0550815	0.0275408	+	0.999621i	$0.491232\pi$
$788$	5.50087	0.195960
$789$	0	0
$790$	34.1843	1.21622
$791$	17.9200	0.637161
$792$	0	0
$793$	−27.6062	−0.980325
$794$	12.1084	0.429710
$795$	0	0
$796$	39.8469	1.41233
$797$	−6.98957	−0.247583	−0.123792	−	0.992308i	$-0.539505\pi$
$797$	−6.98957	−0.247583	−0.123792	+	0.992308i	$0.539505\pi$
$798$	0	0
$799$	18.2482	0.645574
$800$	−35.7575	−1.26422
$801$	0	0
$802$	65.2973	2.30573
$803$	−74.1908	−2.61814
$804$	0	0
$805$	26.8489	0.946299
$806$	−33.7904	−1.19021
$807$	0	0
$808$	−31.3366	−1.10242
$809$	0.996074	0.0350201	0.0175101	−	0.999847i	$-0.494426\pi$
$809$	0.996074	0.0350201	0.0175101	+	0.999847i	$0.494426\pi$
$810$	0	0
$811$	−24.4456	−0.858401	−0.429201	−	0.903209i	$-0.641205\pi$
$811$	−24.4456	−0.858401	−0.429201	+	0.903209i	$0.641205\pi$
$812$	−35.5802	−1.24862
$813$	0	0
$814$	−87.1442	−3.05440
$815$	−57.7862	−2.02416
$816$	0	0
$817$	−9.88321	−0.345770
$818$	29.3828	1.02735
$819$	0	0
$820$	107.273	3.74614
$821$	0.198676	0.00693384	0.00346692	−	0.999994i	$-0.498896\pi$
$821$	0.198676	0.00693384	0.00346692	+	0.999994i	$0.498896\pi$
$822$	0	0
$823$	−14.5672	−0.507782	−0.253891	−	0.967233i	$-0.581710\pi$
$823$	−14.5672	−0.507782	−0.253891	+	0.967233i	$0.581710\pi$
$824$	−24.8877	−0.867004
$825$	0	0
$826$	−9.75499	−0.339420
$827$	−1.94721	−0.0677111	−0.0338555	−	0.999427i	$-0.510779\pi$
$827$	−1.94721	−0.0677111	−0.0338555	+	0.999427i	$0.510779\pi$
$828$	0	0
$829$	−13.9260	−0.483668	−0.241834	−	0.970318i	$-0.577749\pi$
$829$	−13.9260	−0.483668	−0.241834	+	0.970318i	$0.577749\pi$
$830$	35.5994	1.23567
$831$	0	0
$832$	41.3178	1.43244
$833$	−7.95312	−0.275559
$834$	0	0
$835$	63.2259	2.18802
$836$	−128.780	−4.45393
$837$	0	0
$838$	53.8010	1.85853
$839$	35.8956	1.23925	0.619626	−	0.784897i	$-0.287284\pi$
$839$	35.8956	1.23925	0.619626	+	0.784897i	$0.287284\pi$
$840$	0	0
$841$	26.7993	0.924115
$842$	59.8325	2.06196
$843$	0	0
$844$	26.7524	0.920856
$845$	7.62332	0.262250
$846$	0	0
$847$	−44.4980	−1.52897
$848$	1.86267	0.0639644
$849$	0	0
$850$	27.5784	0.945930
$851$	31.0599	1.06472
$852$	0	0
$853$	−34.8890	−1.19458	−0.597289	−	0.802026i	$-0.703755\pi$
$853$	−34.8890	−1.19458	−0.597289	+	0.802026i	$0.703755\pi$
$854$	−27.3067	−0.934417
$855$	0	0
$856$	−27.5888	−0.942965
$857$	37.6475	1.28602	0.643008	−	0.765860i	$-0.277686\pi$
$857$	37.6475	1.28602	0.643008	+	0.765860i	$0.277686\pi$
$858$	0	0
$859$	−7.50501	−0.256068	−0.128034	−	0.991770i	$-0.540867\pi$
$859$	−7.50501	−0.256068	−0.128034	+	0.991770i	$0.540867\pi$
$860$	20.4796	0.698347
$861$	0	0
$862$	−74.9095	−2.55143
$863$	−51.8504	−1.76501	−0.882505	−	0.470304i	$-0.844144\pi$
$863$	−51.8504	−1.76501	−0.882505	+	0.470304i	$0.844144\pi$
$864$	0	0
$865$	7.39834	0.251551
$866$	−16.5183	−0.561316
$867$	0	0
$868$	−21.0377	−0.714065
$869$	27.1925	0.922441
$870$	0	0
$871$	−24.1465	−0.818172
$872$	32.4212	1.09792
$873$	0	0
$874$	72.9235	2.46667
$875$	12.4690	0.421529
$876$	0	0
$877$	45.5895	1.53945	0.769724	−	0.638376i	$-0.220394\pi$
$877$	45.5895	1.53945	0.769724	+	0.638376i	$0.220394\pi$
$878$	−45.2505	−1.52713
$879$	0	0
$880$	17.2389	0.581122
$881$	55.1724	1.85880	0.929402	−	0.369069i	$-0.120323\pi$
$881$	55.1724	1.85880	0.929402	+	0.369069i	$0.120323\pi$
$882$	0	0
$883$	23.6530	0.795988	0.397994	−	0.917388i	$-0.369706\pi$
$883$	23.6530	0.795988	0.397994	+	0.917388i	$0.369706\pi$
$884$	−17.6742	−0.594447
$885$	0	0
$886$	−10.6196	−0.356773
$887$	−37.6323	−1.26357	−0.631784	−	0.775144i	$-0.717677\pi$
$887$	−37.6323	−1.26357	−0.631784	+	0.775144i	$0.717677\pi$
$888$	0	0
$889$	29.6481	0.994364
$890$	20.1285	0.674710
$891$	0	0
$892$	−3.39697	−0.113739
$893$	66.9808	2.24143
$894$	0	0
$895$	69.4982	2.32307
$896$	27.5239	0.919510
$897$	0	0
$898$	−21.4088	−0.714422
$899$	32.9927	1.10037
$900$	0	0
$901$	3.94772	0.131518
$902$	135.572	4.51407
$903$	0	0
$904$	−41.4759	−1.37947
$905$	−44.3188	−1.47321
$906$	0	0
$907$	−29.3700	−0.975214	−0.487607	−	0.873063i	$-0.662130\pi$
$907$	−29.3700	−0.975214	−0.487607	+	0.873063i	$0.662130\pi$
$908$	27.5889	0.915569
$909$	0	0
$910$	−37.9478	−1.25796
$911$	−10.6755	−0.353695	−0.176847	−	0.984238i	$-0.556590\pi$
$911$	−10.6755	−0.353695	−0.176847	+	0.984238i	$0.556590\pi$
$912$	0	0
$913$	28.3181	0.937192
$914$	−15.0064	−0.496368
$915$	0	0
$916$	−91.0377	−3.00797
$917$	−1.36668	−0.0451318
$918$	0	0
$919$	−31.5040	−1.03922	−0.519610	−	0.854404i	$-0.673923\pi$
$919$	−31.5040	−1.03922	−0.519610	+	0.854404i	$0.673923\pi$
$920$	−62.1420	−2.04876
$921$	0	0
$922$	95.5711	3.14747
$923$	4.29557	0.141390
$924$	0	0
$925$	43.1154	1.41763
$926$	−46.2238	−1.51901
$927$	0	0
$928$	−35.5485	−1.16694
$929$	−22.5830	−0.740925	−0.370462	−	0.928847i	$-0.620801\pi$
$929$	−22.5830	−0.740925	−0.370462	+	0.928847i	$0.620801\pi$
$930$	0	0
$931$	−29.1923	−0.956740
$932$	56.3470	1.84571
$933$	0	0
$934$	72.1490	2.36079
$935$	36.5359	1.19485
$936$	0	0
$937$	29.5549	0.965516	0.482758	−	0.875754i	$-0.339635\pi$
$937$	29.5549	0.965516	0.482758	+	0.875754i	$0.339635\pi$
$938$	−23.8845	−0.779858
$939$	0	0
$940$	−138.795	−4.52698
$941$	11.5682	0.377114	0.188557	−	0.982062i	$-0.439619\pi$
$941$	11.5682	0.377114	0.188557	+	0.982062i	$0.439619\pi$
$942$	0	0
$943$	−48.3207	−1.57354
$944$	2.23240	0.0726584
$945$	0	0
$946$	25.8822	0.841502
$947$	21.9424	0.713031	0.356515	−	0.934289i	$-0.383965\pi$
$947$	21.9424	0.713031	0.356515	+	0.934289i	$0.383965\pi$
$948$	0	0
$949$	−37.3743	−1.21322
$950$	101.228	3.28426
$951$	0	0
$952$	−7.18948	−0.233012
$953$	−40.8694	−1.32389	−0.661945	−	0.749553i	$-0.730269\pi$
$953$	−40.8694	−1.32389	−0.661945	+	0.749553i	$0.730269\pi$
$954$	0	0
$955$	−19.9770	−0.646442
$956$	−89.9697	−2.90983
$957$	0	0
$958$	−0.695891	−0.0224832
$959$	−8.01252	−0.258738
$960$	0	0
$961$	−11.4923	−0.370718
$962$	−43.8997	−1.41538
$963$	0	0
$964$	−39.4221	−1.26970
$965$	−1.39094	−0.0447759
$966$	0	0
$967$	5.34299	0.171819	0.0859095	−	0.996303i	$-0.472620\pi$
$967$	5.34299	0.171819	0.0859095	+	0.996303i	$0.472620\pi$
$968$	102.991	3.31025
$969$	0	0
$970$	−10.8810	−0.349368
$971$	12.9017	0.414036	0.207018	−	0.978337i	$-0.433624\pi$
$971$	12.9017	0.414036	0.207018	+	0.978337i	$0.433624\pi$
$972$	0	0
$973$	−20.0379	−0.642387
$974$	61.9028	1.98349
$975$	0	0
$976$	6.24906	0.200028
$977$	−21.5744	−0.690225	−0.345112	−	0.938561i	$-0.612159\pi$
$977$	−21.5744	−0.690225	−0.345112	+	0.938561i	$0.612159\pi$
$978$	0	0
$979$	16.0116	0.511732
$980$	60.4910	1.93232
$981$	0	0
$982$	−33.0913	−1.05599
$983$	−4.11303	−0.131185	−0.0655927	−	0.997846i	$-0.520894\pi$
$983$	−4.11303	−0.131185	−0.0655927	+	0.997846i	$0.520894\pi$
$984$	0	0
$985$	−5.72841	−0.182522
$986$	27.4172	0.873140
$987$	0	0
$988$	−64.8739	−2.06391
$989$	−9.22492	−0.293336
$990$	0	0
$991$	−48.2771	−1.53357	−0.766786	−	0.641903i	$-0.778145\pi$
$991$	−48.2771	−1.53357	−0.766786	+	0.641903i	$0.778145\pi$
$992$	−21.0189	−0.667351
$993$	0	0
$994$	4.24897	0.134769
$995$	−41.4952	−1.31548
$996$	0	0
$997$	−47.3843	−1.50068	−0.750338	−	0.661054i	$-0.770109\pi$
$997$	−47.3843	−1.50068	−0.750338	+	0.661054i	$0.770109\pi$
$998$	24.5563	0.777317
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.l.1.9	yes	10
3.2	odd	2	inner	6021.2.a.l.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.l.1.2	✓	10	3.2	odd	2	inner
6021.2.a.l.1.9	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)

\(f(q)\)	\(=\)	\(q+2.32314 q^{2} +3.39697 q^{4} -3.53749 q^{5} -1.40218 q^{7} +3.24535 q^{8} +O(q^{10})\)	\(q+2.32314 q^{2} +3.39697 q^{4} -3.53749 q^{5} -1.40218 q^{7} +3.24535 q^{8} -8.21807 q^{10} -6.53720 q^{11} -3.29317 q^{13} -3.25745 q^{14} +0.745457 q^{16} +1.57991 q^{17} +5.79915 q^{19} -12.0167 q^{20} -15.1868 q^{22} +5.41288 q^{23} +7.51382 q^{25} -7.65049 q^{26} -4.76315 q^{28} +7.46990 q^{29} +4.41676 q^{31} -4.75890 q^{32} +3.67035 q^{34} +4.96018 q^{35} +5.73815 q^{37} +13.4722 q^{38} -11.4804 q^{40} -8.92699 q^{41} -1.70425 q^{43} -22.2066 q^{44} +12.5749 q^{46} +11.5501 q^{47} -5.03390 q^{49} +17.4556 q^{50} -11.1868 q^{52} +2.49870 q^{53} +23.1252 q^{55} -4.55055 q^{56} +17.3536 q^{58} +2.99467 q^{59} +8.38286 q^{61} +10.2607 q^{62} -12.5465 q^{64} +11.6496 q^{65} +7.33228 q^{67} +5.36691 q^{68} +11.5232 q^{70} -1.30439 q^{71} +11.3490 q^{73} +13.3305 q^{74} +19.6995 q^{76} +9.16630 q^{77} -4.15965 q^{79} -2.63705 q^{80} -20.7386 q^{82} -4.33184 q^{83} -5.58892 q^{85} -3.95922 q^{86} -21.2155 q^{88} -2.44930 q^{89} +4.61761 q^{91} +18.3874 q^{92} +26.8325 q^{94} -20.5144 q^{95} +1.32403 q^{97} -11.6944 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) 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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