LMFDB - Embedded newform 6021.2.a.l.1.1

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			1.1
Root			$-2.81643$ 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)

$f(q)$	$=$	$q-2.81643 q^{2} +5.93226 q^{4} -0.124866 q^{5} +3.22829 q^{7} -11.0749 q^{8} +O(q^{10})$	\(q-2.81643 q^{2} +5.93226 q^{4} -0.124866 q^{5} +3.22829 q^{7} -11.0749 q^{8} +0.351677 q^{10} +3.36004 q^{11} -0.920950 q^{13} -9.09225 q^{14} +19.3272 q^{16} -5.70034 q^{17} +3.70397 q^{19} -0.740741 q^{20} -9.46332 q^{22} +0.548387 q^{23} -4.98441 q^{25} +2.59379 q^{26} +19.1511 q^{28} -0.718746 q^{29} -5.26912 q^{31} -32.2839 q^{32} +16.0546 q^{34} -0.403105 q^{35} +1.21086 q^{37} -10.4320 q^{38} +1.38289 q^{40} +2.62407 q^{41} -5.63273 q^{43} +19.9327 q^{44} -1.54449 q^{46} +7.20734 q^{47} +3.42187 q^{49} +14.0382 q^{50} -5.46332 q^{52} +4.35904 q^{53} -0.419557 q^{55} -35.7531 q^{56} +2.02430 q^{58} +5.56560 q^{59} +7.15275 q^{61} +14.8401 q^{62} +52.2707 q^{64} +0.114996 q^{65} -10.6615 q^{67} -33.8159 q^{68} +1.13532 q^{70} -9.79102 q^{71} -8.48883 q^{73} -3.41031 q^{74} +21.9729 q^{76} +10.8472 q^{77} +8.39847 q^{79} -2.41332 q^{80} -7.39051 q^{82} -4.71189 q^{83} +0.711781 q^{85} +15.8642 q^{86} -37.2123 q^{88} +11.6874 q^{89} -2.97309 q^{91} +3.25318 q^{92} -20.2989 q^{94} -0.462502 q^{95} +0.873110 q^{97} -9.63744 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.81643	−1.99151	−0.995757	−	0.0920164i	$-0.970669\pi$
$2$	−2.81643	−1.99151	−0.995757	+	0.0920164i	$0.970669\pi$
$3$	0	0
$3$	0	0
$4$	5.93226	2.96613
$5$	−0.124866	−0.0558420	−0.0279210	−	0.999610i	$-0.508889\pi$
$5$	−0.124866	−0.0558420	−0.0279210	+	0.999610i	$0.508889\pi$
$6$	0	0
$7$	3.22829	1.22018	0.610090	−	0.792332i	$-0.291133\pi$
$7$	3.22829	1.22018	0.610090	+	0.792332i	$0.291133\pi$
$8$	−11.0749	−3.91558
$9$	0	0
$10$	0.351677	0.111210
$11$	3.36004	1.01309	0.506545	−	0.862213i	$-0.330922\pi$
$11$	3.36004	1.01309	0.506545	+	0.862213i	$0.330922\pi$
$12$	0	0
$13$	−0.920950	−0.255425	−0.127713	−	0.991811i	$-0.540764\pi$
$13$	−0.920950	−0.255425	−0.127713	+	0.991811i	$0.540764\pi$
$14$	−9.09225	−2.43001
$15$	0	0
$16$	19.3272	4.83181
$17$	−5.70034	−1.38254	−0.691268	−	0.722598i	$-0.742948\pi$
$17$	−5.70034	−1.38254	−0.691268	+	0.722598i	$0.742948\pi$
$18$	0	0
$19$	3.70397	0.849749	0.424875	−	0.905252i	$-0.360318\pi$
$19$	3.70397	0.849749	0.424875	+	0.905252i	$0.360318\pi$
$20$	−0.740741	−0.165635
$21$	0	0
$22$	−9.46332	−2.01759
$23$	0.548387	0.114347	0.0571733	−	0.998364i	$-0.481791\pi$
$23$	0.548387	0.114347	0.0571733	+	0.998364i	$0.481791\pi$
$24$	0	0
$25$	−4.98441	−0.996882
$26$	2.59379	0.508684
$27$	0	0
$28$	19.1511	3.61921
$29$	−0.718746	−0.133468	−0.0667339	−	0.997771i	$-0.521258\pi$
$29$	−0.718746	−0.133468	−0.0667339	+	0.997771i	$0.521258\pi$
$30$	0	0
$31$	−5.26912	−0.946362	−0.473181	−	0.880965i	$-0.656894\pi$
$31$	−5.26912	−0.946362	−0.473181	+	0.880965i	$0.656894\pi$
$32$	−32.2839	−5.70703
$33$	0	0
$34$	16.0546	2.75334
$35$	−0.403105	−0.0681372
$36$	0	0
$37$	1.21086	0.199065	0.0995323	−	0.995034i	$-0.468265\pi$
$37$	1.21086	0.199065	0.0995323	+	0.995034i	$0.468265\pi$
$38$	−10.4320	−1.69229
$39$	0	0
$40$	1.38289	0.218654
$41$	2.62407	0.409812	0.204906	−	0.978782i	$-0.434311\pi$
$41$	2.62407	0.409812	0.204906	+	0.978782i	$0.434311\pi$
$42$	0	0
$43$	−5.63273	−0.858983	−0.429492	−	0.903071i	$-0.641307\pi$
$43$	−5.63273	−0.858983	−0.429492	+	0.903071i	$0.641307\pi$
$44$	19.9327	3.00496
$45$	0	0
$46$	−1.54449	−0.227723
$47$	7.20734	1.05130	0.525649	−	0.850701i	$-0.323822\pi$
$47$	7.20734	1.05130	0.525649	+	0.850701i	$0.323822\pi$
$48$	0	0
$49$	3.42187	0.488838
$50$	14.0382	1.98530
$51$	0	0
$52$	−5.46332	−0.757626
$53$	4.35904	0.598761	0.299380	−	0.954134i	$-0.403220\pi$
$53$	4.35904	0.598761	0.299380	+	0.954134i	$0.403220\pi$
$54$	0	0
$55$	−0.419557	−0.0565730
$56$	−35.7531	−4.77771
$57$	0	0
$58$	2.02430	0.265803
$59$	5.56560	0.724580	0.362290	−	0.932065i	$-0.381995\pi$
$59$	5.56560	0.724580	0.362290	+	0.932065i	$0.381995\pi$
$60$	0	0
$61$	7.15275	0.915815	0.457908	−	0.889000i	$-0.348599\pi$
$61$	7.15275	0.915815	0.457908	+	0.889000i	$0.348599\pi$
$62$	14.8401	1.88469
$63$	0	0
$64$	52.2707	6.53384
$65$	0.114996	0.0142635
$66$	0	0
$67$	−10.6615	−1.30251	−0.651253	−	0.758861i	$-0.725756\pi$
$67$	−10.6615	−1.30251	−0.651253	+	0.758861i	$0.725756\pi$
$68$	−33.8159	−4.10078
$69$	0	0
$70$	1.13532	0.135696
$71$	−9.79102	−1.16198	−0.580990	−	0.813911i	$-0.697334\pi$
$71$	−9.79102	−1.16198	−0.580990	+	0.813911i	$0.697334\pi$
$72$	0	0
$73$	−8.48883	−0.993543	−0.496771	−	0.867881i	$-0.665481\pi$
$73$	−8.48883	−0.993543	−0.496771	+	0.867881i	$0.665481\pi$
$74$	−3.41031	−0.396440
$75$	0	0
$76$	21.9729	2.52047
$77$	10.8472	1.23615
$78$	0	0
$79$	8.39847	0.944901	0.472451	−	0.881357i	$-0.343369\pi$
$79$	8.39847	0.944901	0.472451	+	0.881357i	$0.343369\pi$
$80$	−2.41332	−0.269818
$81$	0	0
$82$	−7.39051	−0.816146
$83$	−4.71189	−0.517197	−0.258598	−	0.965985i	$-0.583261\pi$
$83$	−4.71189	−0.517197	−0.258598	+	0.965985i	$0.583261\pi$
$84$	0	0
$85$	0.711781	0.0772035
$86$	15.8642	1.71068
$87$	0	0
$88$	−37.2123	−3.96684
$89$	11.6874	1.23886	0.619430	−	0.785052i	$-0.287364\pi$
$89$	11.6874	1.23886	0.619430	+	0.785052i	$0.287364\pi$
$90$	0	0
$91$	−2.97309	−0.311665
$92$	3.25318	0.339167
$93$	0	0
$94$	−20.2989	−2.09368
$95$	−0.462502	−0.0474517
$96$	0	0
$97$	0.873110	0.0886509	0.0443254	−	0.999017i	$-0.485886\pi$
$97$	0.873110	0.0886509	0.0443254	+	0.999017i	$0.485886\pi$
$98$	−9.63744	−0.973529
$99$	0	0
$100$	−29.5688	−2.95688
$101$	−10.0105	−0.996078	−0.498039	−	0.867155i	$-0.665946\pi$
$101$	−10.0105	−0.996078	−0.498039	+	0.867155i	$0.665946\pi$
$102$	0	0
$103$	19.0427	1.87633	0.938165	−	0.346189i	$-0.112524\pi$
$103$	19.0427	1.87633	0.938165	+	0.346189i	$0.112524\pi$
$104$	10.1995	1.00014
$105$	0	0
$106$	−12.2769	−1.19244
$107$	5.35418	0.517608	0.258804	−	0.965930i	$-0.416672\pi$
$107$	5.35418	0.517608	0.258804	+	0.965930i	$0.416672\pi$
$108$	0	0
$109$	15.5008	1.48470	0.742352	−	0.670010i	$-0.233710\pi$
$109$	15.5008	1.48470	0.742352	+	0.670010i	$0.233710\pi$
$110$	1.18165	0.112666
$111$	0	0
$112$	62.3939	5.89567
$113$	11.6128	1.09244	0.546219	−	0.837642i	$-0.316067\pi$
$113$	11.6128	1.09244	0.546219	+	0.837642i	$0.316067\pi$
$114$	0	0
$115$	−0.0684752	−0.00638534
$116$	−4.26379	−0.395883
$117$	0	0
$118$	−15.6751	−1.44301
$119$	−18.4024	−1.68694
$120$	0	0
$121$	0.289886	0.0263532
$122$	−20.1452	−1.82386
$123$	0	0
$124$	−31.2578	−2.80704
$125$	1.24672	0.111510
$126$	0	0
$127$	16.0322	1.42262	0.711312	−	0.702877i	$-0.248101\pi$
$127$	16.0322	1.42262	0.711312	+	0.702877i	$0.248101\pi$
$128$	−82.6489	−7.30520
$129$	0	0
$130$	−0.323877	−0.0284059
$131$	14.9071	1.30244	0.651222	−	0.758887i	$-0.274257\pi$
$131$	14.9071	1.30244	0.651222	+	0.758887i	$0.274257\pi$
$132$	0	0
$133$	11.9575	1.03685
$134$	30.0273	2.59396
$135$	0	0
$136$	63.1309	5.41343
$137$	11.0481	0.943901	0.471951	−	0.881625i	$-0.343550\pi$
$137$	11.0481	0.943901	0.471951	+	0.881625i	$0.343550\pi$
$138$	0	0
$139$	13.4746	1.14290	0.571452	−	0.820636i	$-0.306380\pi$
$139$	13.4746	1.14290	0.571452	+	0.820636i	$0.306380\pi$
$140$	−2.39133	−0.202104
$141$	0	0
$142$	27.5757	2.31410
$143$	−3.09443	−0.258769
$144$	0	0
$145$	0.0897472	0.00745310
$146$	23.9082	1.97866
$147$	0	0
$148$	7.18316	0.590452
$149$	16.5331	1.35444	0.677222	−	0.735779i	$-0.263184\pi$
$149$	16.5331	1.35444	0.677222	+	0.735779i	$0.263184\pi$
$150$	0	0
$151$	14.3662	1.16911	0.584553	−	0.811355i	$-0.301270\pi$
$151$	14.3662	1.16911	0.584553	+	0.811355i	$0.301270\pi$
$152$	−41.0213	−3.32726
$153$	0	0
$154$	−30.5503	−2.46182
$155$	0.657937	0.0528467
$156$	0	0
$157$	−10.8959	−0.869586	−0.434793	−	0.900530i	$-0.643178\pi$
$157$	−10.8959	−0.869586	−0.434793	+	0.900530i	$0.643178\pi$
$158$	−23.6537	−1.88179
$159$	0	0
$160$	4.03117	0.318692
$161$	1.77035	0.139523
$162$	0	0
$163$	−16.0887	−1.26016	−0.630082	−	0.776529i	$-0.716979\pi$
$163$	−16.0887	−1.26016	−0.630082	+	0.776529i	$0.716979\pi$
$164$	15.5667	1.21556
$165$	0	0
$166$	13.2707	1.03000
$167$	20.6700	1.59949	0.799745	−	0.600340i	$-0.204968\pi$
$167$	20.6700	1.59949	0.799745	+	0.600340i	$0.204968\pi$
$168$	0	0
$169$	−12.1519	−0.934758
$170$	−2.00468	−0.153752
$171$	0	0
$172$	−33.4148	−2.54786
$173$	2.64449	0.201057	0.100528	−	0.994934i	$-0.467947\pi$
$173$	2.64449	0.201057	0.100528	+	0.994934i	$0.467947\pi$
$174$	0	0
$175$	−16.0911	−1.21637
$176$	64.9403	4.89506
$177$	0	0
$178$	−32.9167	−2.46721
$179$	−7.68071	−0.574083	−0.287042	−	0.957918i	$-0.592672\pi$
$179$	−7.68071	−0.574083	−0.287042	+	0.957918i	$0.592672\pi$
$180$	0	0
$181$	22.0382	1.63809	0.819044	−	0.573731i	$-0.194505\pi$
$181$	22.0382	1.63809	0.819044	+	0.573731i	$0.194505\pi$
$182$	8.37350	0.620685
$183$	0	0
$184$	−6.07336	−0.447734
$185$	−0.151196	−0.0111162
$186$	0	0
$187$	−19.1534	−1.40063
$188$	42.7558	3.11829
$189$	0	0
$190$	1.30260	0.0945007
$191$	9.82822	0.711145	0.355572	−	0.934649i	$-0.384286\pi$
$191$	9.82822	0.711145	0.355572	+	0.934649i	$0.384286\pi$
$192$	0	0
$193$	11.8951	0.856227	0.428113	−	0.903725i	$-0.359178\pi$
$193$	11.8951	0.856227	0.428113	+	0.903725i	$0.359178\pi$
$194$	−2.45905	−0.176550
$195$	0	0
$196$	20.2994	1.44996
$197$	−16.1282	−1.14909	−0.574544	−	0.818474i	$-0.694820\pi$
$197$	−16.1282	−1.14909	−0.574544	+	0.818474i	$0.694820\pi$
$198$	0	0
$199$	−17.3167	−1.22755	−0.613775	−	0.789481i	$-0.710350\pi$
$199$	−17.3167	−1.22755	−0.613775	+	0.789481i	$0.710350\pi$
$200$	55.2020	3.90337
$201$	0	0
$202$	28.1937	1.98371
$203$	−2.32032	−0.162855
$204$	0	0
$205$	−0.327659	−0.0228847
$206$	−53.6323	−3.73674
$207$	0	0
$208$	−17.7994	−1.23417
$209$	12.4455	0.860873
$210$	0	0
$211$	8.62066	0.593470	0.296735	−	0.954960i	$-0.404102\pi$
$211$	8.62066	0.593470	0.296735	+	0.954960i	$0.404102\pi$
$212$	25.8590	1.77600
$213$	0	0
$214$	−15.0797	−1.03082
$215$	0.703339	0.0479673
$216$	0	0
$217$	−17.0103	−1.15473
$218$	−43.6568	−2.95681
$219$	0	0
$220$	−2.48892	−0.167803
$221$	5.24973	0.353135
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	−104.222	−6.96361
$225$	0	0
$226$	−32.7065	−2.17561
$227$	−2.53641	−0.168347	−0.0841737	−	0.996451i	$-0.526825\pi$
$227$	−2.53641	−0.168347	−0.0841737	+	0.996451i	$0.526825\pi$
$228$	0	0
$229$	11.1798	0.738783	0.369392	−	0.929274i	$-0.379566\pi$
$229$	11.1798	0.738783	0.369392	+	0.929274i	$0.379566\pi$
$230$	0.192855	0.0127165
$231$	0	0
$232$	7.96006	0.522604
$233$	21.2564	1.39256	0.696278	−	0.717773i	$-0.254838\pi$
$233$	21.2564	1.39256	0.696278	+	0.717773i	$0.254838\pi$
$234$	0	0
$235$	−0.899955	−0.0587066
$236$	33.0166	2.14920
$237$	0	0
$238$	51.8289	3.35957
$239$	−22.8566	−1.47847	−0.739234	−	0.673448i	$-0.764812\pi$
$239$	−22.8566	−1.47847	−0.739234	+	0.673448i	$0.764812\pi$
$240$	0	0
$241$	−11.0813	−0.713812	−0.356906	−	0.934140i	$-0.616168\pi$
$241$	−11.0813	−0.713812	−0.356906	+	0.934140i	$0.616168\pi$
$242$	−0.816442	−0.0524829
$243$	0	0
$244$	42.4320	2.71643
$245$	−0.427277	−0.0272977
$246$	0	0
$247$	−3.41117	−0.217048
$248$	58.3552	3.70556
$249$	0	0
$250$	−3.51129	−0.222073
$251$	−19.7549	−1.24692	−0.623461	−	0.781855i	$-0.714274\pi$
$251$	−19.7549	−1.24692	−0.623461	+	0.781855i	$0.714274\pi$
$252$	0	0
$253$	1.84261	0.115844
$254$	−45.1534	−2.83318
$255$	0	0
$256$	128.233	8.01458
$257$	−1.90691	−0.118950	−0.0594748	−	0.998230i	$-0.518943\pi$
$257$	−1.90691	−0.118950	−0.0594748	+	0.998230i	$0.518943\pi$
$258$	0	0
$259$	3.90902	0.242895
$260$	0.682185	0.0423073
$261$	0	0
$262$	−41.9849	−2.59384
$263$	21.2964	1.31319	0.656596	−	0.754243i	$-0.271996\pi$
$263$	21.2964	1.31319	0.656596	+	0.754243i	$0.271996\pi$
$264$	0	0
$265$	−0.544298	−0.0334360
$266$	−33.6774	−2.06490
$267$	0	0
$268$	−63.2467	−3.86341
$269$	8.82113	0.537834	0.268917	−	0.963163i	$-0.413334\pi$
$269$	8.82113	0.537834	0.268917	+	0.963163i	$0.413334\pi$
$270$	0	0
$271$	20.1858	1.22620	0.613099	−	0.790006i	$-0.289923\pi$
$271$	20.1858	1.22620	0.613099	+	0.790006i	$0.289923\pi$
$272$	−110.172	−6.68015
$273$	0	0
$274$	−31.1161	−1.87979
$275$	−16.7478	−1.00993
$276$	0	0
$277$	8.10596	0.487040	0.243520	−	0.969896i	$-0.421698\pi$
$277$	8.10596	0.487040	0.243520	+	0.969896i	$0.421698\pi$
$278$	−37.9503	−2.27611
$279$	0	0
$280$	4.46437	0.266797
$281$	22.0442	1.31505	0.657525	−	0.753433i	$-0.271604\pi$
$281$	22.0442	1.31505	0.657525	+	0.753433i	$0.271604\pi$
$282$	0	0
$283$	8.14881	0.484396	0.242198	−	0.970227i	$-0.422132\pi$
$283$	8.14881	0.484396	0.242198	+	0.970227i	$0.422132\pi$
$284$	−58.0829	−3.44659
$285$	0	0
$286$	8.71524	0.515343
$287$	8.47128	0.500044
$288$	0	0
$289$	15.4939	0.911406
$290$	−0.252767	−0.0148430
$291$	0	0
$292$	−50.3580	−2.94698
$293$	17.0223	0.994455	0.497227	−	0.867620i	$-0.334351\pi$
$293$	17.0223	0.994455	0.497227	+	0.867620i	$0.334351\pi$
$294$	0	0
$295$	−0.694957	−0.0404620
$296$	−13.4102	−0.779454
$297$	0	0
$298$	−46.5643	−2.69740
$299$	−0.505037	−0.0292071
$300$	0	0
$301$	−18.1841	−1.04811
$302$	−40.4614	−2.32829
$303$	0	0
$304$	71.5875	4.10582
$305$	−0.893138	−0.0511409
$306$	0	0
$307$	−17.9252	−1.02305	−0.511524	−	0.859269i	$-0.670919\pi$
$307$	−17.9252	−1.02305	−0.511524	+	0.859269i	$0.670919\pi$
$308$	64.3484	3.66659
$309$	0	0
$310$	−1.85303	−0.105245
$311$	−22.3036	−1.26472	−0.632362	−	0.774673i	$-0.717914\pi$
$311$	−22.3036	−1.26472	−0.632362	+	0.774673i	$0.717914\pi$
$312$	0	0
$313$	−6.94362	−0.392476	−0.196238	−	0.980556i	$-0.562873\pi$
$313$	−6.94362	−0.392476	−0.196238	+	0.980556i	$0.562873\pi$
$314$	30.6875	1.73179
$315$	0	0
$316$	49.8219	2.80270
$317$	10.7454	0.603524	0.301762	−	0.953383i	$-0.402425\pi$
$317$	10.7454	0.603524	0.301762	+	0.953383i	$0.402425\pi$
$318$	0	0
$319$	−2.41502	−0.135215
$320$	−6.52686	−0.364862
$321$	0	0
$322$	−4.98608	−0.277863
$323$	−21.1139	−1.17481
$324$	0	0
$325$	4.59039	0.254629
$326$	45.3126	2.50964
$327$	0	0
$328$	−29.0615	−1.60465
$329$	23.2674	1.28277
$330$	0	0
$331$	−22.0439	−1.21164	−0.605821	−	0.795601i	$-0.707155\pi$
$331$	−22.0439	−1.21164	−0.605821	+	0.795601i	$0.707155\pi$
$332$	−27.9521	−1.53407
$333$	0	0
$334$	−58.2155	−3.18541
$335$	1.33126	0.0727345
$336$	0	0
$337$	4.61714	0.251512	0.125756	−	0.992061i	$-0.459864\pi$
$337$	4.61714	0.251512	0.125756	+	0.992061i	$0.459864\pi$
$338$	34.2248	1.86158
$339$	0	0
$340$	4.22248	0.228996
$341$	−17.7045	−0.958751
$342$	0	0
$343$	−11.5513	−0.623709
$344$	62.3821	3.36342
$345$	0	0
$346$	−7.44801	−0.400408
$347$	−18.6511	−1.00124	−0.500621	−	0.865666i	$-0.666895\pi$
$347$	−18.6511	−1.00124	−0.500621	+	0.865666i	$0.666895\pi$
$348$	0	0
$349$	8.18611	0.438192	0.219096	−	0.975703i	$-0.429689\pi$
$349$	8.18611	0.438192	0.219096	+	0.975703i	$0.429689\pi$
$350$	45.3195	2.42243
$351$	0	0
$352$	−108.475	−5.78174
$353$	−20.6078	−1.09684	−0.548420	−	0.836203i	$-0.684771\pi$
$353$	−20.6078	−1.09684	−0.548420	+	0.836203i	$0.684771\pi$
$354$	0	0
$355$	1.22257	0.0648873
$356$	69.3326	3.67462
$357$	0	0
$358$	21.6322	1.14330
$359$	−0.312281	−0.0164816	−0.00824078	−	0.999966i	$-0.502623\pi$
$359$	−0.312281	−0.0164816	−0.00824078	+	0.999966i	$0.502623\pi$
$360$	0	0
$361$	−5.28059	−0.277926
$362$	−62.0690	−3.26227
$363$	0	0
$364$	−17.6372	−0.924439
$365$	1.05997	0.0554814
$366$	0	0
$367$	−19.0254	−0.993117	−0.496558	−	0.868003i	$-0.665403\pi$
$367$	−19.0254	−0.993117	−0.496558	+	0.868003i	$0.665403\pi$
$368$	10.5988	0.552501
$369$	0	0
$370$	0.425833	0.0221380
$371$	14.0723	0.730596
$372$	0	0
$373$	−13.8618	−0.717736	−0.358868	−	0.933388i	$-0.616837\pi$
$373$	−13.8618	−0.717736	−0.358868	+	0.933388i	$0.616837\pi$
$374$	53.9441	2.78938
$375$	0	0
$376$	−79.8208	−4.11644
$377$	0.661929	0.0340911
$378$	0	0
$379$	−38.1466	−1.95946	−0.979728	−	0.200330i	$-0.935798\pi$
$379$	−38.1466	−1.95946	−0.979728	+	0.200330i	$0.935798\pi$
$380$	−2.74368	−0.140748
$381$	0	0
$382$	−27.6805	−1.41626
$383$	12.6600	0.646894	0.323447	−	0.946246i	$-0.395158\pi$
$383$	12.6600	0.646894	0.323447	+	0.946246i	$0.395158\pi$
$384$	0	0
$385$	−1.35445	−0.0690292
$386$	−33.5016	−1.70519
$387$	0	0
$388$	5.17952	0.262950
$389$	4.72242	0.239436	0.119718	−	0.992808i	$-0.461801\pi$
$389$	4.72242	0.239436	0.119718	+	0.992808i	$0.461801\pi$
$390$	0	0
$391$	−3.12600	−0.158088
$392$	−37.8970	−1.91409
$393$	0	0
$394$	45.4239	2.28843
$395$	−1.04869	−0.0527652
$396$	0	0
$397$	6.45086	0.323759	0.161880	−	0.986810i	$-0.448244\pi$
$397$	6.45086	0.323759	0.161880	+	0.986810i	$0.448244\pi$
$398$	48.7713	2.44468
$399$	0	0
$400$	−96.3348	−4.81674
$401$	−16.2464	−0.811308	−0.405654	−	0.914027i	$-0.632956\pi$
$401$	−16.2464	−0.811308	−0.405654	+	0.914027i	$0.632956\pi$
$402$	0	0
$403$	4.85260	0.241725
$404$	−59.3847	−2.95450
$405$	0	0
$406$	6.53502	0.324327
$407$	4.06855	0.201671
$408$	0	0
$409$	25.7248	1.27201	0.636004	−	0.771686i	$-0.280586\pi$
$409$	25.7248	1.27201	0.636004	+	0.771686i	$0.280586\pi$
$410$	0.922827	0.0455752
$411$	0	0
$412$	112.966	5.56544
$413$	17.9674	0.884118
$414$	0	0
$415$	0.588356	0.0288813
$416$	29.7318	1.45772
$417$	0	0
$418$	−35.0519	−1.71444
$419$	28.7142	1.40278	0.701390	−	0.712777i	$-0.252563\pi$
$419$	28.7142	1.40278	0.701390	+	0.712777i	$0.252563\pi$
$420$	0	0
$421$	11.2771	0.549613	0.274807	−	0.961500i	$-0.411386\pi$
$421$	11.2771	0.549613	0.274807	+	0.961500i	$0.411386\pi$
$422$	−24.2795	−1.18191
$423$	0	0
$424$	−48.2761	−2.34450
$425$	28.4128	1.37822
$426$	0	0
$427$	23.0912	1.11746
$428$	31.7624	1.53529
$429$	0	0
$430$	−1.98090	−0.0955277
$431$	5.62073	0.270741	0.135371	−	0.990795i	$-0.456778\pi$
$431$	5.62073	0.270741	0.135371	+	0.990795i	$0.456778\pi$
$432$	0	0
$433$	40.5673	1.94954	0.974771	−	0.223207i	$-0.0716526\pi$
$433$	40.5673	1.94954	0.974771	+	0.223207i	$0.0716526\pi$
$434$	47.9082	2.29967
$435$	0	0
$436$	91.9546	4.40383
$437$	2.03121	0.0971660
$438$	0	0
$439$	−26.4139	−1.26067	−0.630333	−	0.776325i	$-0.717082\pi$
$439$	−26.4139	−1.26067	−0.630333	+	0.776325i	$0.717082\pi$
$440$	4.64656	0.221516
$441$	0	0
$442$	−14.7855	−0.703273
$443$	−4.94120	−0.234764	−0.117382	−	0.993087i	$-0.537450\pi$
$443$	−4.94120	−0.234764	−0.117382	+	0.993087i	$0.537450\pi$
$444$	0	0
$445$	−1.45936	−0.0691804
$446$	2.81643	0.133362
$447$	0	0
$448$	168.745	7.97246
$449$	38.9599	1.83863	0.919315	−	0.393523i	$-0.128744\pi$
$449$	38.9599	1.83863	0.919315	+	0.393523i	$0.128744\pi$
$450$	0	0
$451$	8.81700	0.415176
$452$	68.8901	3.24032
$453$	0	0
$454$	7.14361	0.335266
$455$	0.371240	0.0174040
$456$	0	0
$457$	−22.2593	−1.04125	−0.520624	−	0.853786i	$-0.674301\pi$
$457$	−22.2593	−1.04125	−0.520624	+	0.853786i	$0.674301\pi$
$458$	−31.4872	−1.47130
$459$	0	0
$460$	−0.406213	−0.0189398
$461$	−36.9696	−1.72185	−0.860924	−	0.508734i	$-0.830114\pi$
$461$	−36.9696	−1.72185	−0.860924	+	0.508734i	$0.830114\pi$
$462$	0	0
$463$	−20.3582	−0.946128	−0.473064	−	0.881028i	$-0.656852\pi$
$463$	−20.3582	−0.946128	−0.473064	+	0.881028i	$0.656852\pi$
$464$	−13.8914	−0.644890
$465$	0	0
$466$	−59.8672	−2.77329
$467$	−9.05488	−0.419010	−0.209505	−	0.977808i	$-0.567185\pi$
$467$	−9.05488	−0.419010	−0.209505	+	0.977808i	$0.567185\pi$
$468$	0	0
$469$	−34.4184	−1.58929
$470$	2.53466	0.116915
$471$	0	0
$472$	−61.6387	−2.83715
$473$	−18.9262	−0.870228
$474$	0	0
$475$	−18.4621	−0.847100
$476$	−109.168	−5.00369
$477$	0	0
$478$	64.3739	2.94439
$479$	−14.9804	−0.684471	−0.342235	−	0.939614i	$-0.611184\pi$
$479$	−14.9804	−0.684471	−0.342235	+	0.939614i	$0.611184\pi$
$480$	0	0
$481$	−1.11514	−0.0508462
$482$	31.2098	1.42157
$483$	0	0
$484$	1.71968	0.0781672
$485$	−0.109022	−0.00495044
$486$	0	0
$487$	−6.44452	−0.292029	−0.146014	−	0.989282i	$-0.546645\pi$
$487$	−6.44452	−0.292029	−0.146014	+	0.989282i	$0.546645\pi$
$488$	−79.2162	−3.58595
$489$	0	0
$490$	1.20339	0.0543638
$491$	40.2913	1.81832	0.909159	−	0.416448i	$-0.136725\pi$
$491$	40.2913	1.81832	0.909159	+	0.416448i	$0.136725\pi$
$492$	0	0
$493$	4.09710	0.184524
$494$	9.60732	0.432254
$495$	0	0
$496$	−101.838	−4.57264
$497$	−31.6083	−1.41782
$498$	0	0
$499$	9.08047	0.406498	0.203249	−	0.979127i	$-0.434850\pi$
$499$	9.08047	0.406498	0.203249	+	0.979127i	$0.434850\pi$
$500$	7.39586	0.330753
$501$	0	0
$502$	55.6384	2.48326
$503$	−28.9079	−1.28894	−0.644470	−	0.764629i	$-0.722922\pi$
$503$	−28.9079	−1.28894	−0.644470	+	0.764629i	$0.722922\pi$
$504$	0	0
$505$	1.24997	0.0556230
$506$	−5.18956	−0.230704
$507$	0	0
$508$	95.1070	4.21969
$509$	−24.6861	−1.09419	−0.547096	−	0.837070i	$-0.684267\pi$
$509$	−24.6861	−1.09419	−0.547096	+	0.837070i	$0.684267\pi$
$510$	0	0
$511$	−27.4044	−1.21230
$512$	−195.862	−8.65595
$513$	0	0
$514$	5.37067	0.236890
$515$	−2.37779	−0.104778
$516$	0	0
$517$	24.2170	1.06506
$518$	−11.0095	−0.483728
$519$	0	0
$520$	−1.27357	−0.0558497
$521$	−19.1245	−0.837858	−0.418929	−	0.908019i	$-0.637594\pi$
$521$	−19.1245	−0.837858	−0.418929	+	0.908019i	$0.637594\pi$
$522$	0	0
$523$	5.61026	0.245319	0.122660	−	0.992449i	$-0.460858\pi$
$523$	5.61026	0.245319	0.122660	+	0.992449i	$0.460858\pi$
$524$	88.4331	3.86322
$525$	0	0
$526$	−59.9797	−2.61524
$527$	30.0358	1.30838
$528$	0	0
$529$	−22.6993	−0.986925
$530$	1.53298	0.0665883
$531$	0	0
$532$	70.9351	3.07542
$533$	−2.41664	−0.104676
$534$	0	0
$535$	−0.668557	−0.0289043
$536$	118.075	5.10007
$537$	0	0
$538$	−24.8441	−1.07110
$539$	11.4976	0.495238
$540$	0	0
$541$	36.7588	1.58039	0.790193	−	0.612858i	$-0.209980\pi$
$541$	36.7588	1.58039	0.790193	+	0.612858i	$0.209980\pi$
$542$	−56.8517	−2.44199
$543$	0	0
$544$	184.029	7.89018
$545$	−1.93553	−0.0829088
$546$	0	0
$547$	−19.8365	−0.848149	−0.424075	−	0.905627i	$-0.639401\pi$
$547$	−19.8365	−0.848149	−0.424075	+	0.905627i	$0.639401\pi$
$548$	65.5401	2.79974
$549$	0	0
$550$	47.1690	2.01129
$551$	−2.66221	−0.113414
$552$	0	0
$553$	27.1127	1.15295
$554$	−22.8298	−0.969947
$555$	0	0
$556$	79.9351	3.39000
$557$	8.54518	0.362071	0.181035	−	0.983477i	$-0.442055\pi$
$557$	8.54518	0.362071	0.181035	+	0.983477i	$0.442055\pi$
$558$	0	0
$559$	5.18746	0.219406
$560$	−7.79091	−0.329226
$561$	0	0
$562$	−62.0860	−2.61894
$563$	−4.75798	−0.200525	−0.100262	−	0.994961i	$-0.531968\pi$
$563$	−4.75798	−0.200525	−0.100262	+	0.994961i	$0.531968\pi$
$564$	0	0
$565$	−1.45005	−0.0610039
$566$	−22.9505	−0.964683
$567$	0	0
$568$	108.435	4.54983
$569$	−10.6637	−0.447047	−0.223524	−	0.974699i	$-0.571756\pi$
$569$	−10.6637	−0.447047	−0.223524	+	0.974699i	$0.571756\pi$
$570$	0	0
$571$	4.51604	0.188990	0.0944952	−	0.995525i	$-0.469876\pi$
$571$	4.51604	0.188990	0.0944952	+	0.995525i	$0.469876\pi$
$572$	−18.3570	−0.767544
$573$	0	0
$574$	−23.8587	−0.995845
$575$	−2.73339	−0.113990
$576$	0	0
$577$	40.4163	1.68255	0.841277	−	0.540604i	$-0.181804\pi$
$577$	40.4163	1.68255	0.841277	+	0.540604i	$0.181804\pi$
$578$	−43.6374	−1.81508
$579$	0	0
$580$	0.532404	0.0221069
$581$	−15.2113	−0.631073
$582$	0	0
$583$	14.6466	0.606599
$584$	94.0133	3.89030
$585$	0	0
$586$	−47.9421	−1.98047
$587$	−45.9663	−1.89723	−0.948616	−	0.316430i	$-0.897516\pi$
$587$	−45.9663	−1.89723	−0.948616	+	0.316430i	$0.897516\pi$
$588$	0	0
$589$	−19.5167	−0.804171
$590$	1.95730	0.0805806
$591$	0	0
$592$	23.4026	0.961842
$593$	28.5047	1.17055	0.585274	−	0.810835i	$-0.300987\pi$
$593$	28.5047	1.17055	0.585274	+	0.810835i	$0.300987\pi$
$594$	0	0
$595$	2.29784	0.0942022
$596$	98.0787	4.01746
$597$	0	0
$598$	1.42240	0.0581663
$599$	39.8156	1.62682	0.813411	−	0.581690i	$-0.197608\pi$
$599$	39.8156	1.62682	0.813411	+	0.581690i	$0.197608\pi$
$600$	0	0
$601$	−2.51500	−0.102589	−0.0512946	−	0.998684i	$-0.516335\pi$
$601$	−2.51500	−0.102589	−0.0512946	+	0.998684i	$0.516335\pi$
$602$	51.2142	2.08733
$603$	0	0
$604$	85.2242	3.46772
$605$	−0.0361970	−0.00147162
$606$	0	0
$607$	19.6959	0.799432	0.399716	−	0.916639i	$-0.369109\pi$
$607$	19.6959	0.799432	0.399716	+	0.916639i	$0.369109\pi$
$608$	−119.579	−4.84955
$609$	0	0
$610$	2.51546	0.101848
$611$	−6.63759	−0.268528
$612$	0	0
$613$	23.4907	0.948781	0.474390	−	0.880315i	$-0.342669\pi$
$613$	23.4907	0.948781	0.474390	+	0.880315i	$0.342669\pi$
$614$	50.4851	2.03741
$615$	0	0
$616$	−120.132	−4.84026
$617$	39.9835	1.60967	0.804837	−	0.593496i	$-0.202253\pi$
$617$	39.9835	1.60967	0.804837	+	0.593496i	$0.202253\pi$
$618$	0	0
$619$	43.2012	1.73640	0.868202	−	0.496211i	$-0.165276\pi$
$619$	43.2012	1.73640	0.868202	+	0.496211i	$0.165276\pi$
$620$	3.90305	0.156750
$621$	0	0
$622$	62.8166	2.51872
$623$	37.7303	1.51163
$624$	0	0
$625$	24.7664	0.990655
$626$	19.5562	0.781623
$627$	0	0
$628$	−64.6372	−2.57931
$629$	−6.90233	−0.275214
$630$	0	0
$631$	15.6714	0.623868	0.311934	−	0.950104i	$-0.399023\pi$
$631$	15.6714	0.623868	0.311934	+	0.950104i	$0.399023\pi$
$632$	−93.0125	−3.69984
$633$	0	0
$634$	−30.2637	−1.20193
$635$	−2.00188	−0.0794421
$636$	0	0
$637$	−3.15137	−0.124862
$638$	6.80172	0.269283
$639$	0	0
$640$	10.3201	0.407937
$641$	−0.378793	−0.0149614	−0.00748070	−	0.999972i	$-0.502381\pi$
$641$	−0.378793	−0.0149614	−0.00748070	+	0.999972i	$0.502381\pi$
$642$	0	0
$643$	6.70228	0.264312	0.132156	−	0.991229i	$-0.457810\pi$
$643$	6.70228	0.264312	0.132156	+	0.991229i	$0.457810\pi$
$644$	10.5022	0.413845
$645$	0	0
$646$	59.4658	2.33965
$647$	−4.10987	−0.161576	−0.0807879	−	0.996731i	$-0.525744\pi$
$647$	−4.10987	−0.161576	−0.0807879	+	0.996731i	$0.525744\pi$
$648$	0	0
$649$	18.7007	0.734065
$650$	−12.9285	−0.507097
$651$	0	0
$652$	−95.4424	−3.73781
$653$	40.7473	1.59456	0.797282	−	0.603607i	$-0.206270\pi$
$653$	40.7473	1.59456	0.797282	+	0.603607i	$0.206270\pi$
$654$	0	0
$655$	−1.86140	−0.0727310
$656$	50.7161	1.98013
$657$	0	0
$658$	−65.5309	−2.55466
$659$	16.5558	0.644924	0.322462	−	0.946582i	$-0.395490\pi$
$659$	16.5558	0.644924	0.322462	+	0.946582i	$0.395490\pi$
$660$	0	0
$661$	39.2496	1.52663	0.763316	−	0.646025i	$-0.223570\pi$
$661$	39.2496	1.52663	0.763316	+	0.646025i	$0.223570\pi$
$662$	62.0850	2.41300
$663$	0	0
$664$	52.1838	2.02513
$665$	−1.49309	−0.0578996
$666$	0	0
$667$	−0.394151	−0.0152616
$668$	122.620	4.74430
$669$	0	0
$670$	−3.74940	−0.144852
$671$	24.0335	0.927804
$672$	0	0
$673$	−15.4796	−0.596693	−0.298346	−	0.954458i	$-0.596435\pi$
$673$	−15.4796	−0.596693	−0.298346	+	0.954458i	$0.596435\pi$
$674$	−13.0038	−0.500889
$675$	0	0
$676$	−72.0880	−2.77262
$677$	29.9970	1.15288	0.576440	−	0.817140i	$-0.304442\pi$
$677$	29.9970	1.15288	0.576440	+	0.817140i	$0.304442\pi$
$678$	0	0
$679$	2.81865	0.108170
$680$	−7.88293	−0.302297
$681$	0	0
$682$	49.8634	1.90937
$683$	−41.6586	−1.59402	−0.797011	−	0.603964i	$-0.793587\pi$
$683$	−41.6586	−1.59402	−0.797011	+	0.603964i	$0.793587\pi$
$684$	0	0
$685$	−1.37953	−0.0527093
$686$	32.5333	1.24213
$687$	0	0
$688$	−108.865	−4.15044
$689$	−4.01446	−0.152939
$690$	0	0
$691$	23.3193	0.887106	0.443553	−	0.896248i	$-0.353718\pi$
$691$	23.3193	0.887106	0.443553	+	0.896248i	$0.353718\pi$
$692$	15.6878	0.596361
$693$	0	0
$694$	52.5294	1.99399
$695$	−1.68253	−0.0638220
$696$	0	0
$697$	−14.9581	−0.566579
$698$	−23.0556	−0.872666
$699$	0	0
$700$	−95.4568	−3.60793
$701$	−20.6930	−0.781563	−0.390782	−	0.920483i	$-0.627795\pi$
$701$	−20.6930	−0.781563	−0.390782	+	0.920483i	$0.627795\pi$
$702$	0	0
$703$	4.48500	0.169155
$704$	175.632	6.61937
$705$	0	0
$706$	58.0403	2.18437
$707$	−32.3167	−1.21539
$708$	0	0
$709$	−39.7331	−1.49221	−0.746104	−	0.665829i	$-0.768078\pi$
$709$	−39.7331	−1.49221	−0.746104	+	0.665829i	$0.768078\pi$
$710$	−3.44328	−0.129224
$711$	0	0
$712$	−129.437	−4.85086
$713$	−2.88952	−0.108213
$714$	0	0
$715$	0.386390	0.0144502
$716$	−45.5640	−1.70281
$717$	0	0
$718$	0.879516	0.0328233
$719$	23.9908	0.894707	0.447354	−	0.894357i	$-0.352367\pi$
$719$	23.9908	0.894707	0.447354	+	0.894357i	$0.352367\pi$
$720$	0	0
$721$	61.4753	2.28946
$722$	14.8724	0.553494
$723$	0	0
$724$	130.736	4.85878
$725$	3.58252	0.133052
$726$	0	0
$727$	25.8523	0.958810	0.479405	−	0.877594i	$-0.340852\pi$
$727$	25.8523	0.958810	0.479405	+	0.877594i	$0.340852\pi$
$728$	32.9268	1.22035
$729$	0	0
$730$	−2.98533	−0.110492
$731$	32.1085	1.18758
$732$	0	0
$733$	−25.8872	−0.956164	−0.478082	−	0.878315i	$-0.658668\pi$
$733$	−25.8872	−0.956164	−0.478082	+	0.878315i	$0.658668\pi$
$734$	53.5836	1.97781
$735$	0	0
$736$	−17.7041	−0.652580
$737$	−35.8230	−1.31956
$738$	0	0
$739$	28.4514	1.04660	0.523301	−	0.852148i	$-0.324700\pi$
$739$	28.4514	1.04660	0.523301	+	0.852148i	$0.324700\pi$
$740$	−0.896935	−0.0329720
$741$	0	0
$742$	−39.6335	−1.45499
$743$	17.6118	0.646113	0.323057	−	0.946380i	$-0.395290\pi$
$743$	17.6118	0.646113	0.323057	+	0.946380i	$0.395290\pi$
$744$	0	0
$745$	−2.06443	−0.0756348
$746$	39.0407	1.42938
$747$	0	0
$748$	−113.623	−4.15447
$749$	17.2849	0.631575
$750$	0	0
$751$	27.2532	0.994483	0.497242	−	0.867612i	$-0.334346\pi$
$751$	27.2532	0.994483	0.497242	+	0.867612i	$0.334346\pi$
$752$	139.298	5.07967
$753$	0	0
$754$	−1.86427	−0.0678929
$755$	−1.79386	−0.0652852
$756$	0	0
$757$	10.8676	0.394988	0.197494	−	0.980304i	$-0.436720\pi$
$757$	10.8676	0.394988	0.197494	+	0.980304i	$0.436720\pi$
$758$	107.437	3.90229
$759$	0	0
$760$	5.12218	0.185801
$761$	−7.55148	−0.273741	−0.136871	−	0.990589i	$-0.543704\pi$
$761$	−7.55148	−0.273741	−0.136871	+	0.990589i	$0.543704\pi$
$762$	0	0
$763$	50.0410	1.81161
$764$	58.3036	2.10935
$765$	0	0
$766$	−35.6559	−1.28830
$767$	−5.12564	−0.185076
$768$	0	0
$769$	−22.4120	−0.808196	−0.404098	−	0.914716i	$-0.632415\pi$
$769$	−22.4120	−0.808196	−0.404098	+	0.914716i	$0.632415\pi$
$770$	3.81471	0.137473
$771$	0	0
$772$	70.5648	2.53968
$773$	28.5340	1.02630	0.513148	−	0.858300i	$-0.328479\pi$
$773$	28.5340	1.02630	0.513148	+	0.858300i	$0.328479\pi$
$774$	0	0
$775$	26.2635	0.943411
$776$	−9.66964	−0.347120
$777$	0	0
$778$	−13.3004	−0.476841
$779$	9.71950	0.348237
$780$	0	0
$781$	−32.8982	−1.17719
$782$	8.80414	0.314835
$783$	0	0
$784$	66.1352	2.36197
$785$	1.36053	0.0485594
$786$	0	0
$787$	−4.76770	−0.169950	−0.0849751	−	0.996383i	$-0.527081\pi$
$787$	−4.76770	−0.169950	−0.0849751	+	0.996383i	$0.527081\pi$
$788$	−95.6768	−3.40835
$789$	0	0
$790$	2.95355	0.105083
$791$	37.4894	1.33297
$792$	0	0
$793$	−6.58732	−0.233923
$794$	−18.1684	−0.644772
$795$	0	0
$796$	−102.727	−3.64107
$797$	23.6867	0.839027	0.419513	−	0.907749i	$-0.362201\pi$
$797$	23.6867	0.839027	0.419513	+	0.907749i	$0.362201\pi$
$798$	0	0
$799$	−41.0843	−1.45346
$800$	160.916	5.68924
$801$	0	0
$802$	45.7569	1.61573
$803$	−28.5228	−1.00655
$804$	0	0
$805$	−0.221058	−0.00779127
$806$	−13.6670	−0.481399
$807$	0	0
$808$	110.865	3.90023
$809$	35.9271	1.26313	0.631565	−	0.775323i	$-0.282413\pi$
$809$	35.9271	1.26313	0.631565	+	0.775323i	$0.282413\pi$
$810$	0	0
$811$	6.52186	0.229013	0.114507	−	0.993422i	$-0.463471\pi$
$811$	6.52186	0.229013	0.114507	+	0.993422i	$0.463471\pi$
$812$	−13.7648	−0.483048
$813$	0	0
$814$	−11.4588	−0.401630
$815$	2.00894	0.0703700
$816$	0	0
$817$	−20.8635	−0.729921
$818$	−72.4520	−2.53322
$819$	0	0
$820$	−1.94376	−0.0678790
$821$	22.6680	0.791118	0.395559	−	0.918441i	$-0.370551\pi$
$821$	22.6680	0.791118	0.395559	+	0.918441i	$0.370551\pi$
$822$	0	0
$823$	39.9644	1.39307	0.696536	−	0.717521i	$-0.254723\pi$
$823$	39.9644	1.39307	0.696536	+	0.717521i	$0.254723\pi$
$824$	−210.896	−7.34692
$825$	0	0
$826$	−50.6039	−1.76073
$827$	28.6853	0.997486	0.498743	−	0.866750i	$-0.333795\pi$
$827$	28.6853	0.997486	0.498743	+	0.866750i	$0.333795\pi$
$828$	0	0
$829$	−31.4349	−1.09178	−0.545889	−	0.837857i	$-0.683808\pi$
$829$	−31.4349	−1.09178	−0.545889	+	0.837857i	$0.683808\pi$
$830$	−1.65706	−0.0575175
$831$	0	0
$832$	−48.1387	−1.66891
$833$	−19.5058	−0.675837
$834$	0	0
$835$	−2.58099	−0.0893187
$836$	73.8300	2.55346
$837$	0	0
$838$	−80.8715	−2.79366
$839$	−16.6270	−0.574027	−0.287014	−	0.957927i	$-0.592662\pi$
$839$	−16.6270	−0.574027	−0.287014	+	0.957927i	$0.592662\pi$
$840$	0	0
$841$	−28.4834	−0.982186
$842$	−31.7612	−1.09456
$843$	0	0
$844$	51.1400	1.76031
$845$	1.51736	0.0521987
$846$	0	0
$847$	0.935836	0.0321557
$848$	84.2482	2.89310
$849$	0	0
$850$	−80.0227	−2.74476
$851$	0.664022	0.0227624
$852$	0	0
$853$	1.23407	0.0422536	0.0211268	−	0.999777i	$-0.493275\pi$
$853$	1.23407	0.0422536	0.0211268	+	0.999777i	$0.493275\pi$
$854$	−65.0346	−2.22544
$855$	0	0
$856$	−59.2972	−2.02674
$857$	−51.2719	−1.75142	−0.875708	−	0.482842i	$-0.839605\pi$
$857$	−51.2719	−1.75142	−0.875708	+	0.482842i	$0.839605\pi$
$858$	0	0
$859$	−55.3263	−1.88771	−0.943854	−	0.330362i	$-0.892829\pi$
$859$	−55.3263	−1.88771	−0.943854	+	0.330362i	$0.892829\pi$
$860$	4.17239	0.142277
$861$	0	0
$862$	−15.8304	−0.539185
$863$	31.4922	1.07201	0.536004	−	0.844216i	$-0.319933\pi$
$863$	31.4922	1.07201	0.536004	+	0.844216i	$0.319933\pi$
$864$	0	0
$865$	−0.330208	−0.0112274
$866$	−114.255	−3.88254
$867$	0	0
$868$	−100.909	−3.42509
$869$	28.2192	0.957271
$870$	0	0
$871$	9.81868	0.332693
$872$	−171.670	−5.81348
$873$	0	0
$874$	−5.72076	−0.193508
$875$	4.02477	0.136062
$876$	0	0
$877$	−0.648770	−0.0219074	−0.0109537	−	0.999940i	$-0.503487\pi$
$877$	−0.648770	−0.0219074	−0.0109537	+	0.999940i	$0.503487\pi$
$878$	74.3928	2.51064
$879$	0	0
$880$	−8.10887	−0.273350
$881$	−10.9876	−0.370181	−0.185090	−	0.982722i	$-0.559258\pi$
$881$	−10.9876	−0.370181	−0.185090	+	0.982722i	$0.559258\pi$
$882$	0	0
$883$	−0.518059	−0.0174341	−0.00871703	−	0.999962i	$-0.502775\pi$
$883$	−0.518059	−0.0174341	−0.00871703	+	0.999962i	$0.502775\pi$
$884$	31.1428	1.04744
$885$	0	0
$886$	13.9165	0.467535
$887$	−8.30618	−0.278894	−0.139447	−	0.990230i	$-0.544533\pi$
$887$	−8.30618	−0.278894	−0.139447	+	0.990230i	$0.544533\pi$
$888$	0	0
$889$	51.7565	1.73586
$890$	4.11019	0.137774
$891$	0	0
$892$	−5.93226	−0.198627
$893$	26.6958	0.893340
$894$	0	0
$895$	0.959063	0.0320579
$896$	−266.815	−8.91366
$897$	0	0
$898$	−109.728	−3.66166
$899$	3.78716	0.126309
$900$	0	0
$901$	−24.8480	−0.827808
$902$	−24.8324	−0.826830
$903$	0	0
$904$	−128.611	−4.27753
$905$	−2.75183	−0.0914740
$906$	0	0
$907$	38.3627	1.27381	0.636906	−	0.770942i	$-0.280214\pi$
$907$	38.3627	1.27381	0.636906	+	0.770942i	$0.280214\pi$
$908$	−15.0466	−0.499340
$909$	0	0
$910$	−1.04557	−0.0346603
$911$	−59.0975	−1.95799	−0.978995	−	0.203887i	$-0.934643\pi$
$911$	−59.0975	−1.95799	−0.978995	+	0.203887i	$0.934643\pi$
$912$	0	0
$913$	−15.8321	−0.523967
$914$	62.6918	2.07366
$915$	0	0
$916$	66.3217	2.19133
$917$	48.1246	1.58922
$918$	0	0
$919$	−14.5910	−0.481313	−0.240656	−	0.970610i	$-0.577363\pi$
$919$	−14.5910	−0.481313	−0.240656	+	0.970610i	$0.577363\pi$
$920$	0.758358	0.0250023
$921$	0	0
$922$	104.122	3.42909
$923$	9.01703	0.296799
$924$	0	0
$925$	−6.03543	−0.198444
$926$	57.3375	1.88423
$927$	0	0
$928$	23.2039	0.761705
$929$	−3.60368	−0.118233	−0.0591164	−	0.998251i	$-0.518828\pi$
$929$	−3.60368	−0.118233	−0.0591164	+	0.998251i	$0.518828\pi$
$930$	0	0
$931$	12.6745	0.415390
$932$	126.099	4.13050
$933$	0	0
$934$	25.5024	0.834465
$935$	2.39162	0.0782142
$936$	0	0
$937$	−43.7736	−1.43002	−0.715010	−	0.699114i	$-0.753578\pi$
$937$	−43.7736	−1.43002	−0.715010	+	0.699114i	$0.753578\pi$
$938$	96.9368	3.16510
$939$	0	0
$940$	−5.33877	−0.174131
$941$	20.0613	0.653979	0.326990	−	0.945028i	$-0.393966\pi$
$941$	20.0613	0.653979	0.326990	+	0.945028i	$0.393966\pi$
$942$	0	0
$943$	1.43901	0.0468606
$944$	107.568	3.50103
$945$	0	0
$946$	53.3043	1.73307
$947$	18.8381	0.612155	0.306077	−	0.952007i	$-0.400983\pi$
$947$	18.8381	0.612155	0.306077	+	0.952007i	$0.400983\pi$
$948$	0	0
$949$	7.81779	0.253776
$950$	51.9972	1.68701
$951$	0	0
$952$	203.805	6.60536
$953$	5.68693	0.184218	0.0921089	−	0.995749i	$-0.470639\pi$
$953$	5.68693	0.184218	0.0921089	+	0.995749i	$0.470639\pi$
$954$	0	0
$955$	−1.22721	−0.0397117
$956$	−135.591	−4.38533
$957$	0	0
$958$	42.1911	1.36313
$959$	35.6664	1.15173
$960$	0	0
$961$	−3.23635	−0.104398
$962$	3.14072	0.101261
$963$	0	0
$964$	−65.7375	−2.11726
$965$	−1.48530	−0.0478134
$966$	0	0
$967$	−32.9496	−1.05959	−0.529794	−	0.848126i	$-0.677731\pi$
$967$	−32.9496	−1.05959	−0.529794	+	0.848126i	$0.677731\pi$
$968$	−3.21047	−0.103188
$969$	0	0
$970$	0.307053	0.00985888
$971$	49.7206	1.59561	0.797806	−	0.602915i	$-0.205994\pi$
$971$	49.7206	1.59561	0.797806	+	0.602915i	$0.205994\pi$
$972$	0	0
$973$	43.5000	1.39455
$974$	18.1505	0.581580
$975$	0	0
$976$	138.243	4.42504
$977$	47.2849	1.51278	0.756389	−	0.654122i	$-0.226962\pi$
$977$	47.2849	1.51278	0.756389	+	0.654122i	$0.226962\pi$
$978$	0	0
$979$	39.2701	1.25508
$980$	−2.53472	−0.0809686
$981$	0	0
$982$	−113.477	−3.62121
$983$	−62.0881	−1.98030	−0.990151	−	0.140006i	$-0.955288\pi$
$983$	−62.0881	−1.98030	−0.990151	+	0.140006i	$0.955288\pi$
$984$	0	0
$985$	2.01387	0.0641673
$986$	−11.5392	−0.367482
$987$	0	0
$988$	−20.2360	−0.643792
$989$	−3.08892	−0.0982219
$990$	0	0
$991$	47.5154	1.50938	0.754689	−	0.656083i	$-0.227788\pi$
$991$	47.5154	1.50938	0.754689	+	0.656083i	$0.227788\pi$
$992$	170.108	5.40092
$993$	0	0
$994$	89.0224	2.82362
$995$	2.16228	0.0685488
$996$	0	0
$997$	0.479420	0.0151834	0.00759169	−	0.999971i	$-0.497583\pi$
$997$	0.479420	0.0151834	0.00759169	+	0.999971i	$0.497583\pi$
$998$	−25.5745	−0.809546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.l.1.1	✓	10
3.2	odd	2	inner	6021.2.a.l.1.10	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.l.1.1	✓	10	1.1	even	1	trivial
6021.2.a.l.1.10	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)

\(f(q)\)	\(=\)	\(q-2.81643 q^{2} +5.93226 q^{4} -0.124866 q^{5} +3.22829 q^{7} -11.0749 q^{8} +O(q^{10})\)	\(q-2.81643 q^{2} +5.93226 q^{4} -0.124866 q^{5} +3.22829 q^{7} -11.0749 q^{8} +0.351677 q^{10} +3.36004 q^{11} -0.920950 q^{13} -9.09225 q^{14} +19.3272 q^{16} -5.70034 q^{17} +3.70397 q^{19} -0.740741 q^{20} -9.46332 q^{22} +0.548387 q^{23} -4.98441 q^{25} +2.59379 q^{26} +19.1511 q^{28} -0.718746 q^{29} -5.26912 q^{31} -32.2839 q^{32} +16.0546 q^{34} -0.403105 q^{35} +1.21086 q^{37} -10.4320 q^{38} +1.38289 q^{40} +2.62407 q^{41} -5.63273 q^{43} +19.9327 q^{44} -1.54449 q^{46} +7.20734 q^{47} +3.42187 q^{49} +14.0382 q^{50} -5.46332 q^{52} +4.35904 q^{53} -0.419557 q^{55} -35.7531 q^{56} +2.02430 q^{58} +5.56560 q^{59} +7.15275 q^{61} +14.8401 q^{62} +52.2707 q^{64} +0.114996 q^{65} -10.6615 q^{67} -33.8159 q^{68} +1.13532 q^{70} -9.79102 q^{71} -8.48883 q^{73} -3.41031 q^{74} +21.9729 q^{76} +10.8472 q^{77} +8.39847 q^{79} -2.41332 q^{80} -7.39051 q^{82} -4.71189 q^{83} +0.711781 q^{85} +15.8642 q^{86} -37.2123 q^{88} +11.6874 q^{89} -2.97309 q^{91} +3.25318 q^{92} -20.2989 q^{94} -0.462502 q^{95} +0.873110 q^{97} -9.63744 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more