LMFDB - Embedded newform 6021.2.a.l.1.2

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			1.2
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	$-0.806780\pi$
$2$	−2.32314	−1.64271	−0.821353	+	0.570420i	$0.806780\pi$
$3$	0	0
$3$	0	0
$4$	3.39697	1.69848
$5$	3.53749	1.58201	0.791006	−	0.611808i	$-0.209558\pi$
$5$	3.53749	1.58201	0.791006	+	0.611808i	$0.209558\pi$
$6$	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	$-0.0542357\pi$
$11$	6.53720	1.97104	0.985519	+	0.169563i	$0.0542357\pi$
$12$	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	$-0.561365\pi$
$17$	−1.57991	−0.383185	−0.191593	+	0.981475i	$0.561365\pi$
$18$	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	$-0.690866\pi$
$23$	−5.41288	−1.12866	−0.564332	+	0.825548i	$0.690866\pi$
$24$	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	$-0.743960\pi$
$29$	−7.46990	−1.38712	−0.693562	+	0.720397i	$0.743960\pi$
$30$	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	$-0.254482\pi$
$41$	8.92699	1.39416	0.697081	+	0.716993i	$0.254482\pi$
$42$	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	$-0.818846\pi$
$47$	−11.5501	−1.68476	−0.842379	+	0.538886i	$0.818846\pi$
$48$	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	$-0.554897\pi$
$53$	−2.49870	−0.343223	−0.171611	+	0.985165i	$0.554897\pi$
$54$	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	$-0.562450\pi$
$59$	−2.99467	−0.389873	−0.194936	+	0.980816i	$0.562450\pi$
$60$	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	$-0.475338\pi$
$71$	1.30439	0.154802	0.0774010	+	0.997000i	$0.475338\pi$
$72$	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	$-0.423593\pi$
$83$	4.33184	0.475481	0.237741	+	0.971329i	$0.423593\pi$
$84$	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	$-0.458562\pi$
$89$	2.44930	0.259626	0.129813	+	0.991539i	$0.458562\pi$
$90$	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	$-0.340493\pi$
$101$	9.65585	0.960793	0.480397	+	0.877051i	$0.340493\pi$
$102$	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	$-0.365210\pi$
$107$	8.50101	0.821824	0.410912	+	0.911675i	$0.365210\pi$
$108$	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	$-0.294719\pi$
$113$	12.7801	1.20225	0.601126	+	0.799154i	$0.294719\pi$
$114$	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	$-0.513558\pi$
$131$	−0.974686	−0.0851587	−0.0425794	+	0.999093i	$0.513558\pi$
$132$	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	$-0.578494\pi$
$137$	−5.71434	−0.488209	−0.244105	+	0.969749i	$0.578494\pi$
$138$	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	$-0.602369\pi$
$149$	−7.71662	−0.632170	−0.316085	+	0.948731i	$0.602369\pi$
$150$	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	$-0.256936\pi$
$167$	17.8731	1.38306	0.691531	+	0.722346i	$0.256936\pi$
$168$	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	$-0.474667\pi$
$173$	2.09141	0.159007	0.0795035	+	0.996835i	$0.474667\pi$
$174$	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	$-0.237551\pi$
$179$	19.6462	1.46843	0.734214	+	0.678919i	$0.237551\pi$
$180$	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	$-0.565495\pi$
$191$	−5.64724	−0.408620	−0.204310	+	0.978906i	$0.565495\pi$
$192$	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	$-0.518372\pi$
$197$	−1.61935	−0.115374	−0.0576868	+	0.998335i	$0.518372\pi$
$198$	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	$-0.586867\pi$
$227$	−8.12162	−0.539051	−0.269525	+	0.962993i	$0.586867\pi$
$228$	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	$-0.682840\pi$
$233$	−16.5874	−1.08668	−0.543340	+	0.839513i	$0.682840\pi$
$234$	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	$-0.172576\pi$
$239$	26.4853	1.71319	0.856595	+	0.515989i	$0.172576\pi$
$240$	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	$-0.830438\pi$
$251$	−27.2956	−1.72288	−0.861442	+	0.507855i	$0.830438\pi$
$252$	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	$-0.386477\pi$
$257$	11.1940	0.698260	0.349130	+	0.937074i	$0.386477\pi$
$258$	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	$-0.554774\pi$
$263$	−5.55378	−0.342461	−0.171230	+	0.985231i	$0.554774\pi$
$264$	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	$-0.696878\pi$
$269$	−19.0196	−1.15964	−0.579822	+	0.814743i	$0.696878\pi$
$270$	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	$-0.765537\pi$
$281$	−24.8350	−1.48153	−0.740765	+	0.671764i	$0.765537\pi$
$282$	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	$-0.484766\pi$
$293$	1.63777	0.0956796	0.0478398	+	0.998855i	$0.484766\pi$
$294$	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	$-0.283839\pi$
$311$	22.1528	1.25617	0.628084	+	0.778145i	$0.283839\pi$
$312$	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	$-0.651866\pi$
$317$	−16.3519	−0.918413	−0.459206	+	0.888330i	$0.651866\pi$
$318$	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	$-0.680574\pi$
$347$	−20.0194	−1.07470	−0.537349	+	0.843360i	$0.680574\pi$
$348$	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	$-0.661175\pi$
$353$	−18.2241	−0.969973	−0.484986	+	0.874522i	$0.661175\pi$
$354$	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	$-0.783431\pi$
$359$	−29.4569	−1.55468	−0.777338	+	0.629083i	$0.783431\pi$
$360$	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	$-0.752563\pi$
$383$	−27.8987	−1.42556	−0.712778	+	0.701389i	$0.752563\pi$
$384$	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	$-0.282586\pi$
$389$	24.8962	1.26229	0.631144	+	0.775666i	$0.282586\pi$
$390$	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	$-0.747624\pi$
$401$	−28.1074	−1.40362	−0.701808	+	0.712366i	$0.747624\pi$
$402$	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	$-0.691390\pi$
$419$	−23.1588	−1.13138	−0.565690	+	0.824618i	$0.691390\pi$
$420$	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	$-0.216946\pi$
$431$	32.2450	1.55319	0.776593	+	0.630003i	$0.216946\pi$
$432$	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	$-0.465365\pi$
$443$	4.57124	0.217186	0.108593	+	0.994086i	$0.465365\pi$
$444$	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	$-0.430225\pi$
$449$	9.21549	0.434906	0.217453	+	0.976071i	$0.430225\pi$
$450$	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	$-0.907433\pi$
$461$	−41.1388	−1.91603	−0.958013	+	0.286726i	$0.907433\pi$
$462$	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	$-0.755202\pi$
$467$	−31.0567	−1.43713	−0.718567	+	0.695458i	$0.755202\pi$
$468$	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	$-0.497822\pi$
$479$	0.299548	0.0136867	0.00684334	+	0.999977i	$0.497822\pi$
$480$	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	$-0.395841\pi$
$491$	14.2442	0.642833	0.321417	+	0.946938i	$0.395841\pi$
$492$	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	$-0.735457\pi$
$503$	−30.2358	−1.34815	−0.674073	+	0.738664i	$0.735457\pi$
$504$	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	$-0.617325\pi$
$509$	−16.2574	−0.720597	−0.360299	+	0.932837i	$0.617325\pi$
$510$	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	$-0.342381\pi$
$521$	21.6926	0.950371	0.475186	+	0.879886i	$0.342381\pi$
$522$	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	$-0.317709\pi$
$557$	25.5781	1.08378	0.541890	+	0.840449i	$0.317709\pi$
$558$	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	$-0.280120\pi$
$563$	30.2353	1.27427	0.637133	+	0.770754i	$0.280120\pi$
$564$	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	$-0.331254\pi$
$569$	24.1231	1.01129	0.505646	+	0.862741i	$0.331254\pi$
$570$	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	$-0.251399\pi$
$587$	34.1128	1.40799	0.703993	+	0.710207i	$0.251399\pi$
$588$	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	$-0.584636\pi$
$593$	−12.7977	−0.525537	−0.262769	+	0.964859i	$0.584636\pi$
$594$	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	$-0.292267\pi$
$599$	29.7249	1.21453	0.607263	+	0.794501i	$0.292267\pi$
$600$	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	$-0.141510\pi$
$617$	44.8501	1.80560	0.902798	+	0.430066i	$0.141510\pi$
$618$	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	$-0.479353\pi$
$641$	3.28212	0.129636	0.0648179	+	0.997897i	$0.479353\pi$
$642$	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	$-0.637064\pi$
$647$	−21.2349	−0.834831	−0.417415	+	0.908716i	$0.637064\pi$
$648$	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	$-0.538810\pi$
$653$	−6.21595	−0.243249	−0.121624	+	0.992576i	$0.538810\pi$
$654$	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	$-0.417382\pi$
$659$	13.1767	0.513293	0.256646	+	0.966505i	$0.417382\pi$
$660$	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	$-0.438350\pi$
$677$	10.0158	0.384939	0.192470	+	0.981303i	$0.438350\pi$
$678$	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	$-0.400606\pi$
$683$	16.0572	0.614412	0.307206	+	0.951643i	$0.400606\pi$
$684$	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	$-0.301496\pi$
$701$	30.9232	1.16795	0.583977	+	0.811770i	$0.301496\pi$
$702$	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	$-0.872774\pi$
$719$	−49.4014	−1.84236	−0.921181	+	0.389134i	$0.872774\pi$
$720$	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	$-0.405048\pi$
$743$	16.0221	0.587795	0.293897	+	0.955837i	$0.405048\pi$
$744$	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	$-0.537775\pi$
$761$	−6.53224	−0.236793	−0.118397	+	0.992966i	$0.537775\pi$
$762$	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	$-0.792030\pi$
$773$	−44.1536	−1.58810	−0.794048	+	0.607856i	$0.792030\pi$
$774$	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	$-0.460495\pi$
$797$	6.98957	0.247583	0.123792	+	0.992308i	$0.460495\pi$
$798$	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	$-0.505574\pi$
$809$	−0.996074	−0.0350201	−0.0175101	+	0.999847i	$0.505574\pi$
$810$	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	$-0.501104\pi$
$821$	−0.198676	−0.00693384	−0.00346692	+	0.999994i	$0.501104\pi$
$822$	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	$-0.489221\pi$
$827$	1.94721	0.0677111	0.0338555	+	0.999427i	$0.489221\pi$
$828$	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	$-0.712716\pi$
$839$	−35.8956	−1.23925	−0.619626	+	0.784897i	$0.712716\pi$
$840$	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	$-0.722314\pi$
$857$	−37.6475	−1.28602	−0.643008	+	0.765860i	$0.722314\pi$
$858$	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	$-0.155856\pi$
$863$	51.8504	1.76501	0.882505	+	0.470304i	$0.155856\pi$
$864$	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	$-0.879677\pi$
$881$	−55.1724	−1.85880	−0.929402	+	0.369069i	$0.879677\pi$
$882$	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	$-0.282323\pi$
$887$	37.6323	1.26357	0.631784	+	0.775144i	$0.282323\pi$
$888$	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	$-0.443410\pi$
$911$	10.6755	0.353695	0.176847	+	0.984238i	$0.443410\pi$
$912$	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	$-0.379199\pi$
$929$	22.5830	0.740925	0.370462	+	0.928847i	$0.379199\pi$
$930$	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	$-0.560381\pi$
$941$	−11.5682	−0.377114	−0.188557	+	0.982062i	$0.560381\pi$
$942$	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	$-0.616035\pi$
$947$	−21.9424	−0.713031	−0.356515	+	0.934289i	$0.616035\pi$
$948$	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	$-0.269731\pi$
$953$	40.8694	1.32389	0.661945	+	0.749553i	$0.269731\pi$
$954$	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	$-0.566376\pi$
$971$	−12.9017	−0.414036	−0.207018	+	0.978337i	$0.566376\pi$
$972$	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	$-0.387841\pi$
$977$	21.5744	0.690225	0.345112	+	0.938561i	$0.387841\pi$
$978$	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	$-0.479106\pi$
$983$	4.11303	0.131185	0.0655927	+	0.997846i	$0.479106\pi$
$984$	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.1.2	✓	10
3.2	odd	2	inner	6021.2.a.l.1.9	yes	10

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

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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