LMFDB - Embedded newform 6021.2.a.t.1.8

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:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
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)

$f(q)$	$=$	$q-2.00467 q^{2} +2.01871 q^{4} +4.10087 q^{5} +1.86508 q^{7} -0.0375024 q^{8} +O(q^{10})$	\(q-2.00467 q^{2} +2.01871 q^{4} +4.10087 q^{5} +1.86508 q^{7} -0.0375024 q^{8} -8.22090 q^{10} +1.63503 q^{11} +3.48558 q^{13} -3.73888 q^{14} -3.96224 q^{16} -0.0762526 q^{17} -1.54519 q^{19} +8.27846 q^{20} -3.27769 q^{22} +5.94244 q^{23} +11.8171 q^{25} -6.98744 q^{26} +3.76506 q^{28} -4.65086 q^{29} -4.20473 q^{31} +8.01798 q^{32} +0.152861 q^{34} +7.64847 q^{35} +3.70799 q^{37} +3.09760 q^{38} -0.153792 q^{40} -10.6522 q^{41} +4.08029 q^{43} +3.30064 q^{44} -11.9126 q^{46} -4.22184 q^{47} -3.52146 q^{49} -23.6895 q^{50} +7.03637 q^{52} -0.248254 q^{53} +6.70503 q^{55} -0.0699451 q^{56} +9.32345 q^{58} +1.05862 q^{59} +10.6849 q^{61} +8.42910 q^{62} -8.14895 q^{64} +14.2939 q^{65} +7.06557 q^{67} -0.153932 q^{68} -15.3327 q^{70} -7.40845 q^{71} -11.9188 q^{73} -7.43330 q^{74} -3.11929 q^{76} +3.04946 q^{77} +13.8246 q^{79} -16.2486 q^{80} +21.3541 q^{82} +4.44941 q^{83} -0.312702 q^{85} -8.17964 q^{86} -0.0613174 q^{88} -4.62291 q^{89} +6.50090 q^{91} +11.9960 q^{92} +8.46340 q^{94} -6.33662 q^{95} +18.4985 q^{97} +7.05937 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * 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 $

$2$	−2.00467	−1.41752	−0.708758	−	0.705451i	$-0.750744\pi$
$2$	−2.00467	−1.41752	−0.708758	+	0.705451i	$0.750744\pi$
$3$	0	0
$3$	0	0
$4$	2.01871	1.00935
$5$	4.10087	1.83397	0.916983	−	0.398927i	$-0.130617\pi$
$5$	4.10087	1.83397	0.916983	+	0.398927i	$0.130617\pi$
$6$	0	0
$7$	1.86508	0.704936	0.352468	−	0.935824i	$-0.385343\pi$
$7$	1.86508	0.704936	0.352468	+	0.935824i	$0.385343\pi$
$8$	−0.0375024	−0.0132591
$9$	0	0
$10$	−8.22090	−2.59968
$11$	1.63503	0.492979	0.246490	−	0.969145i	$-0.420723\pi$
$11$	1.63503	0.492979	0.246490	+	0.969145i	$0.420723\pi$
$12$	0	0
$13$	3.48558	0.966726	0.483363	−	0.875420i	$-0.339415\pi$
$13$	3.48558	0.966726	0.483363	+	0.875420i	$0.339415\pi$
$14$	−3.73888	−0.999258
$15$	0	0
$16$	−3.96224	−0.990559
$17$	−0.0762526	−0.0184940	−0.00924698	−	0.999957i	$-0.502943\pi$
$17$	−0.0762526	−0.0184940	−0.00924698	+	0.999957i	$0.502943\pi$
$18$	0	0
$19$	−1.54519	−0.354491	−0.177245	−	0.984167i	$-0.556719\pi$
$19$	−1.54519	−0.354491	−0.177245	+	0.984167i	$0.556719\pi$
$20$	8.27846	1.85112
$21$	0	0
$22$	−3.27769	−0.698806
$23$	5.94244	1.23908	0.619542	−	0.784963i	$-0.287318\pi$
$23$	5.94244	1.23908	0.619542	+	0.784963i	$0.287318\pi$
$24$	0	0
$25$	11.8171	2.36343
$26$	−6.98744	−1.37035
$27$	0	0
$28$	3.76506	0.711530
$29$	−4.65086	−0.863643	−0.431821	−	0.901959i	$-0.642129\pi$
$29$	−4.65086	−0.863643	−0.431821	+	0.901959i	$0.642129\pi$
$30$	0	0
$31$	−4.20473	−0.755192	−0.377596	−	0.925970i	$-0.623249\pi$
$31$	−4.20473	−0.755192	−0.377596	+	0.925970i	$0.623249\pi$
$32$	8.01798	1.41739
$33$	0	0
$34$	0.152861	0.0262155
$35$	7.64847	1.29283
$36$	0	0
$37$	3.70799	0.609590	0.304795	−	0.952418i	$-0.401412\pi$
$37$	3.70799	0.609590	0.304795	+	0.952418i	$0.401412\pi$
$38$	3.09760	0.502497
$39$	0	0
$40$	−0.153792	−0.0243167
$41$	−10.6522	−1.66359	−0.831795	−	0.555084i	$-0.812686\pi$
$41$	−10.6522	−1.66359	−0.831795	+	0.555084i	$0.812686\pi$
$42$	0	0
$43$	4.08029	0.622239	0.311119	−	0.950371i	$-0.399296\pi$
$43$	4.08029	0.622239	0.311119	+	0.950371i	$0.399296\pi$
$44$	3.30064	0.497590
$45$	0	0
$46$	−11.9126	−1.75642
$47$	−4.22184	−0.615819	−0.307909	−	0.951416i	$-0.599629\pi$
$47$	−4.22184	−0.615819	−0.307909	+	0.951416i	$0.599629\pi$
$48$	0	0
$49$	−3.52146	−0.503066
$50$	−23.6895	−3.35020
$51$	0	0
$52$	7.03637	0.975768
$53$	−0.248254	−0.0341003	−0.0170502	−	0.999855i	$-0.505427\pi$
$53$	−0.248254	−0.0341003	−0.0170502	+	0.999855i	$0.505427\pi$
$54$	0	0
$55$	6.70503	0.904107
$56$	−0.0699451	−0.00934681
$57$	0	0
$58$	9.32345	1.22423
$59$	1.05862	0.137820	0.0689102	−	0.997623i	$-0.478048\pi$
$59$	1.05862	0.137820	0.0689102	+	0.997623i	$0.478048\pi$
$60$	0	0
$61$	10.6849	1.36806	0.684028	−	0.729456i	$-0.260227\pi$
$61$	10.6849	1.36806	0.684028	+	0.729456i	$0.260227\pi$
$62$	8.42910	1.07050
$63$	0	0
$64$	−8.14895	−1.01862
$65$	14.2939	1.77294
$66$	0	0
$67$	7.06557	0.863197	0.431598	−	0.902066i	$-0.357950\pi$
$67$	7.06557	0.863197	0.431598	+	0.902066i	$0.357950\pi$
$68$	−0.153932	−0.0186670
$69$	0	0
$70$	−15.3327	−1.83261
$71$	−7.40845	−0.879221	−0.439610	−	0.898189i	$-0.644883\pi$
$71$	−7.40845	−0.879221	−0.439610	+	0.898189i	$0.644883\pi$
$72$	0	0
$73$	−11.9188	−1.39498	−0.697492	−	0.716592i	$-0.745701\pi$
$73$	−11.9188	−1.39498	−0.697492	+	0.716592i	$0.745701\pi$
$74$	−7.43330	−0.864103
$75$	0	0
$76$	−3.11929	−0.357807
$77$	3.04946	0.347519
$78$	0	0
$79$	13.8246	1.55539	0.777695	−	0.628642i	$-0.216389\pi$
$79$	13.8246	1.55539	0.777695	+	0.628642i	$0.216389\pi$
$80$	−16.2486	−1.81665
$81$	0	0
$82$	21.3541	2.35817
$83$	4.44941	0.488386	0.244193	−	0.969727i	$-0.421477\pi$
$83$	4.44941	0.488386	0.244193	+	0.969727i	$0.421477\pi$
$84$	0	0
$85$	−0.312702	−0.0339173
$86$	−8.17964	−0.882034
$87$	0	0
$88$	−0.0613174	−0.00653646
$89$	−4.62291	−0.490027	−0.245014	−	0.969520i	$-0.578792\pi$
$89$	−4.62291	−0.490027	−0.245014	+	0.969520i	$0.578792\pi$
$90$	0	0
$91$	6.50090	0.681480
$92$	11.9960	1.25067
$93$	0	0
$94$	8.46340	0.872933
$95$	−6.33662	−0.650124
$96$	0	0
$97$	18.4985	1.87824	0.939118	−	0.343595i	$-0.111645\pi$
$97$	18.4985	1.87824	0.939118	+	0.343595i	$0.111645\pi$
$98$	7.05937	0.713104
$99$	0	0
$100$	23.8554	2.38554
$101$	8.91588	0.887163	0.443582	−	0.896234i	$-0.353708\pi$
$101$	8.91588	0.887163	0.443582	+	0.896234i	$0.353708\pi$
$102$	0	0
$103$	7.14749	0.704264	0.352132	−	0.935950i	$-0.385457\pi$
$103$	7.14749	0.704264	0.352132	+	0.935950i	$0.385457\pi$
$104$	−0.130718	−0.0128179
$105$	0	0
$106$	0.497668	0.0483378
$107$	5.15735	0.498580	0.249290	−	0.968429i	$-0.419803\pi$
$107$	5.15735	0.498580	0.249290	+	0.968429i	$0.419803\pi$
$108$	0	0
$109$	6.34348	0.607595	0.303797	−	0.952737i	$-0.401745\pi$
$109$	6.34348	0.607595	0.303797	+	0.952737i	$0.401745\pi$
$110$	−13.4414	−1.28159
$111$	0	0
$112$	−7.38990	−0.698280
$113$	10.6194	0.998992	0.499496	−	0.866316i	$-0.333519\pi$
$113$	10.6194	0.998992	0.499496	+	0.866316i	$0.333519\pi$
$114$	0	0
$115$	24.3692	2.27244
$116$	−9.38873	−0.871721
$117$	0	0
$118$	−2.12218	−0.195363
$119$	−0.142218	−0.0130371
$120$	0	0
$121$	−8.32669	−0.756972
$122$	−21.4196	−1.93924
$123$	0	0
$124$	−8.48812	−0.762256
$125$	27.9562	2.50048
$126$	0	0
$127$	13.3158	1.18158	0.590792	−	0.806824i	$-0.298815\pi$
$127$	13.3158	1.18158	0.590792	+	0.806824i	$0.298815\pi$
$128$	0.300006	0.0265170
$129$	0	0
$130$	−28.6546	−2.51317
$131$	−13.6948	−1.19652	−0.598260	−	0.801302i	$-0.704141\pi$
$131$	−13.6948	−1.19652	−0.598260	+	0.801302i	$0.704141\pi$
$132$	0	0
$133$	−2.88191	−0.249893
$134$	−14.1641	−1.22360
$135$	0	0
$136$	0.00285965	0.000245213 0
$137$	2.02460	0.172973	0.0864866	−	0.996253i	$-0.472436\pi$
$137$	2.02460	0.172973	0.0864866	+	0.996253i	$0.472436\pi$
$138$	0	0
$139$	−4.83279	−0.409912	−0.204956	−	0.978771i	$-0.565705\pi$
$139$	−4.83279	−0.409912	−0.204956	+	0.978771i	$0.565705\pi$
$140$	15.4400	1.30492
$141$	0	0
$142$	14.8515	1.24631
$143$	5.69902	0.476576
$144$	0	0
$145$	−19.0726	−1.58389
$146$	23.8932	1.97741
$147$	0	0
$148$	7.48534	0.615291
$149$	−3.21603	−0.263467	−0.131734	−	0.991285i	$-0.542054\pi$
$149$	−3.21603	−0.263467	−0.131734	+	0.991285i	$0.542054\pi$
$150$	0	0
$151$	14.3148	1.16492	0.582462	−	0.812858i	$-0.302090\pi$
$151$	14.3148	1.16492	0.582462	+	0.812858i	$0.302090\pi$
$152$	0.0579483	0.00470023
$153$	0	0
$154$	−6.11317	−0.492613
$155$	−17.2431	−1.38500
$156$	0	0
$157$	3.88115	0.309749	0.154875	−	0.987934i	$-0.450503\pi$
$157$	3.88115	0.309749	0.154875	+	0.987934i	$0.450503\pi$
$158$	−27.7138	−2.20479
$159$	0	0
$160$	32.8807	2.59945
$161$	11.0832	0.873475
$162$	0	0
$163$	−5.63252	−0.441173	−0.220586	−	0.975367i	$-0.570797\pi$
$163$	−5.63252	−0.441173	−0.220586	+	0.975367i	$0.570797\pi$
$164$	−21.5036	−1.67915
$165$	0	0
$166$	−8.91961	−0.692296
$167$	8.40709	0.650560	0.325280	−	0.945618i	$-0.394541\pi$
$167$	8.40709	0.650560	0.325280	+	0.945618i	$0.394541\pi$
$168$	0	0
$169$	−0.850734	−0.0654411
$170$	0.626865	0.0480783
$171$	0	0
$172$	8.23692	0.628059
$173$	−24.4066	−1.85560	−0.927798	−	0.373082i	$-0.878301\pi$
$173$	−24.4066	−1.85560	−0.927798	+	0.373082i	$0.878301\pi$
$174$	0	0
$175$	22.0400	1.66607
$176$	−6.47836	−0.488325
$177$	0	0
$178$	9.26741	0.694622
$179$	−8.22417	−0.614704	−0.307352	−	0.951596i	$-0.599443\pi$
$179$	−8.22417	−0.614704	−0.307352	+	0.951596i	$0.599443\pi$
$180$	0	0
$181$	18.1140	1.34640	0.673200	−	0.739461i	$-0.264919\pi$
$181$	18.1140	1.34640	0.673200	+	0.739461i	$0.264919\pi$
$182$	−13.0322	−0.966009
$183$	0	0
$184$	−0.222856	−0.0164291
$185$	15.2060	1.11797
$186$	0	0
$187$	−0.124675	−0.00911714
$188$	−8.52266	−0.621579
$189$	0	0
$190$	12.7028	0.921561
$191$	−5.75631	−0.416512	−0.208256	−	0.978074i	$-0.566779\pi$
$191$	−5.75631	−0.416512	−0.208256	+	0.978074i	$0.566779\pi$
$192$	0	0
$193$	18.2015	1.31017	0.655085	−	0.755556i	$-0.272633\pi$
$193$	18.2015	1.31017	0.655085	+	0.755556i	$0.272633\pi$
$194$	−37.0834	−2.66243
$195$	0	0
$196$	−7.10880	−0.507771
$197$	21.7714	1.55115	0.775574	−	0.631257i	$-0.217461\pi$
$197$	21.7714	1.55115	0.775574	+	0.631257i	$0.217461\pi$
$198$	0	0
$199$	−9.47840	−0.671906	−0.335953	−	0.941879i	$-0.609058\pi$
$199$	−9.47840	−0.671906	−0.335953	+	0.941879i	$0.609058\pi$
$200$	−0.443171	−0.0313369
$201$	0	0
$202$	−17.8734	−1.25757
$203$	−8.67425	−0.608813
$204$	0	0
$205$	−43.6832	−3.05096
$206$	−14.3284	−0.998305
$207$	0	0
$208$	−13.8107	−0.957599
$209$	−2.52643	−0.174757
$210$	0	0
$211$	−1.55253	−0.106881	−0.0534404	−	0.998571i	$-0.517019\pi$
$211$	−1.55253	−0.106881	−0.0534404	+	0.998571i	$0.517019\pi$
$212$	−0.501152	−0.0344193
$213$	0	0
$214$	−10.3388	−0.706746
$215$	16.7328	1.14116
$216$	0	0
$217$	−7.84218	−0.532362
$218$	−12.7166	−0.861276
$219$	0	0
$220$	13.5355	0.912564
$221$	−0.265784	−0.0178786
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	14.9542	0.999171
$225$	0	0
$226$	−21.2885	−1.41609
$227$	−12.5805	−0.834995	−0.417498	−	0.908678i	$-0.637093\pi$
$227$	−12.5805	−0.834995	−0.417498	+	0.908678i	$0.637093\pi$
$228$	0	0
$229$	−14.2597	−0.942311	−0.471155	−	0.882050i	$-0.656163\pi$
$229$	−14.2597	−0.942311	−0.471155	+	0.882050i	$0.656163\pi$
$230$	−48.8522	−3.22122
$231$	0	0
$232$	0.174418	0.0114511
$233$	4.50312	0.295009	0.147505	−	0.989061i	$-0.452876\pi$
$233$	4.50312	0.295009	0.147505	+	0.989061i	$0.452876\pi$
$234$	0	0
$235$	−17.3132	−1.12939
$236$	2.13704	0.139110
$237$	0	0
$238$	0.285099	0.0184802
$239$	−11.8868	−0.768891	−0.384446	−	0.923148i	$-0.625607\pi$
$239$	−11.8868	−0.768891	−0.384446	+	0.923148i	$0.625607\pi$
$240$	0	0
$241$	−6.89968	−0.444448	−0.222224	−	0.974996i	$-0.571332\pi$
$241$	−6.89968	−0.444448	−0.222224	+	0.974996i	$0.571332\pi$
$242$	16.6923	1.07302
$243$	0	0
$244$	21.5696	1.38085
$245$	−14.4411	−0.922605
$246$	0	0
$247$	−5.38588	−0.342695
$248$	0.157687	0.0100132
$249$	0	0
$250$	−56.0431	−3.54448
$251$	−13.3362	−0.841775	−0.420888	−	0.907113i	$-0.638281\pi$
$251$	−13.3362	−0.841775	−0.420888	+	0.907113i	$0.638281\pi$
$252$	0	0
$253$	9.71605	0.610843
$254$	−26.6938	−1.67492
$255$	0	0
$256$	15.6965	0.981031
$257$	−5.63537	−0.351525	−0.175762	−	0.984433i	$-0.556239\pi$
$257$	−5.63537	−0.351525	−0.175762	+	0.984433i	$0.556239\pi$
$258$	0	0
$259$	6.91571	0.429721
$260$	28.8552	1.78953
$261$	0	0
$262$	27.4536	1.69609
$263$	26.8911	1.65818	0.829089	−	0.559117i	$-0.188860\pi$
$263$	26.8911	1.65818	0.829089	+	0.559117i	$0.188860\pi$
$264$	0	0
$265$	−1.01806	−0.0625388
$266$	5.77728	0.354228
$267$	0	0
$268$	14.2633	0.871271
$269$	−15.7341	−0.959322	−0.479661	−	0.877454i	$-0.659240\pi$
$269$	−15.7341	−0.959322	−0.479661	+	0.877454i	$0.659240\pi$
$270$	0	0
$271$	22.3191	1.35579	0.677895	−	0.735159i	$-0.262892\pi$
$271$	22.3191	1.35579	0.677895	+	0.735159i	$0.262892\pi$
$272$	0.302131	0.0183194
$273$	0	0
$274$	−4.05865	−0.245192
$275$	19.3214	1.16512
$276$	0	0
$277$	−31.7213	−1.90595	−0.952975	−	0.303048i	$-0.901996\pi$
$277$	−31.7213	−1.90595	−0.952975	+	0.303048i	$0.901996\pi$
$278$	9.68816	0.581057
$279$	0	0
$280$	−0.286836	−0.0171417
$281$	−13.1157	−0.782415	−0.391208	−	0.920302i	$-0.627943\pi$
$281$	−13.1157	−0.782415	−0.391208	+	0.920302i	$0.627943\pi$
$282$	0	0
$283$	23.9529	1.42385	0.711926	−	0.702255i	$-0.247823\pi$
$283$	23.9529	1.42385	0.711926	+	0.702255i	$0.247823\pi$
$284$	−14.9555	−0.887445
$285$	0	0
$286$	−11.4247	−0.675554
$287$	−19.8672	−1.17272
$288$	0	0
$289$	−16.9942	−0.999658
$290$	38.2343	2.24519
$291$	0	0
$292$	−24.0605	−1.40803
$293$	−13.0623	−0.763110	−0.381555	−	0.924346i	$-0.624611\pi$
$293$	−13.0623	−0.763110	−0.381555	+	0.924346i	$0.624611\pi$
$294$	0	0
$295$	4.34126	0.252758
$296$	−0.139058	−0.00808261
$297$	0	0
$298$	6.44708	0.373470
$299$	20.7128	1.19785
$300$	0	0
$301$	7.61009	0.438638
$302$	−28.6965	−1.65130
$303$	0	0
$304$	6.12240	0.351144
$305$	43.8172	2.50897
$306$	0	0
$307$	24.1571	1.37872	0.689360	−	0.724418i	$-0.257892\pi$
$307$	24.1571	1.37872	0.689360	+	0.724418i	$0.257892\pi$
$308$	6.15597	0.350769
$309$	0	0
$310$	34.5667	1.96325
$311$	−21.6447	−1.22736	−0.613679	−	0.789556i	$-0.710311\pi$
$311$	−21.6447	−1.22736	−0.613679	+	0.789556i	$0.710311\pi$
$312$	0	0
$313$	−24.6968	−1.39594	−0.697971	−	0.716126i	$-0.745914\pi$
$313$	−24.6968	−1.39594	−0.697971	+	0.716126i	$0.745914\pi$
$314$	−7.78043	−0.439075
$315$	0	0
$316$	27.9078	1.56994
$317$	20.7180	1.16364	0.581821	−	0.813317i	$-0.302340\pi$
$317$	20.7180	1.16364	0.581821	+	0.813317i	$0.302340\pi$
$318$	0	0
$319$	−7.60428	−0.425758
$320$	−33.4178	−1.86811
$321$	0	0
$322$	−22.2181	−1.23817
$323$	0.117825	0.00655594
$324$	0	0
$325$	41.1896	2.28479
$326$	11.2913	0.625370
$327$	0	0
$328$	0.399482	0.0220577
$329$	−7.87409	−0.434112
$330$	0	0
$331$	18.0548	0.992379	0.496190	−	0.868214i	$-0.334732\pi$
$331$	18.0548	0.992379	0.496190	+	0.868214i	$0.334732\pi$
$332$	8.98206	0.492955
$333$	0	0
$334$	−16.8534	−0.922180
$335$	28.9750	1.58307
$336$	0	0
$337$	22.7574	1.23967	0.619837	−	0.784731i	$-0.287199\pi$
$337$	22.7574	1.23967	0.619837	+	0.784731i	$0.287199\pi$
$338$	1.70544	0.0927639
$339$	0	0
$340$	−0.631254	−0.0342346
$341$	−6.87485	−0.372294
$342$	0	0
$343$	−19.6234	−1.05956
$344$	−0.153021	−0.00825032
$345$	0	0
$346$	48.9271	2.63034
$347$	−36.5157	−1.96027	−0.980133	−	0.198344i	$-0.936444\pi$
$347$	−36.5157	−1.96027	−0.980133	+	0.198344i	$0.936444\pi$
$348$	0	0
$349$	−24.2947	−1.30046	−0.650232	−	0.759736i	$-0.725328\pi$
$349$	−24.2947	−1.30046	−0.650232	+	0.759736i	$0.725328\pi$
$350$	−44.1829	−2.36168
$351$	0	0
$352$	13.1096	0.698745
$353$	14.7292	0.783958	0.391979	−	0.919974i	$-0.371791\pi$
$353$	14.7292	0.783958	0.391979	+	0.919974i	$0.371791\pi$
$354$	0	0
$355$	−30.3811	−1.61246
$356$	−9.33230	−0.494611
$357$	0	0
$358$	16.4868	0.871353
$359$	31.6493	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$359$	31.6493	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$360$	0	0
$361$	−16.6124	−0.874336
$362$	−36.3125	−1.90854
$363$	0	0
$364$	13.1234	0.687854
$365$	−48.8773	−2.55835
$366$	0	0
$367$	−17.9798	−0.938536	−0.469268	−	0.883056i	$-0.655482\pi$
$367$	−17.9798	−0.938536	−0.469268	+	0.883056i	$0.655482\pi$
$368$	−23.5453	−1.22739
$369$	0	0
$370$	−30.4830	−1.58474
$371$	−0.463015	−0.0240385
$372$	0	0
$373$	−17.8202	−0.922693	−0.461347	−	0.887220i	$-0.652634\pi$
$373$	−17.8202	−0.922693	−0.461347	+	0.887220i	$0.652634\pi$
$374$	0.249932	0.0129237
$375$	0	0
$376$	0.158329	0.00816520
$377$	−16.2109	−0.834906
$378$	0	0
$379$	−8.91882	−0.458129	−0.229064	−	0.973411i	$-0.573567\pi$
$379$	−8.91882	−0.458129	−0.229064	+	0.973411i	$0.573567\pi$
$380$	−12.7918	−0.656205
$381$	0	0
$382$	11.5395	0.590413
$383$	−13.3664	−0.682989	−0.341495	−	0.939884i	$-0.610933\pi$
$383$	−13.3664	−0.682989	−0.341495	+	0.939884i	$0.610933\pi$
$384$	0	0
$385$	12.5055	0.637337
$386$	−36.4879	−1.85719
$387$	0	0
$388$	37.3430	1.89580
$389$	−9.71759	−0.492701	−0.246351	−	0.969181i	$-0.579232\pi$
$389$	−9.71759	−0.492701	−0.246351	+	0.969181i	$0.579232\pi$
$390$	0	0
$391$	−0.453126	−0.0229156
$392$	0.132063	0.00667020
$393$	0	0
$394$	−43.6445	−2.19878
$395$	56.6929	2.85253
$396$	0	0
$397$	4.61343	0.231542	0.115771	−	0.993276i	$-0.463066\pi$
$397$	4.61343	0.231542	0.115771	+	0.993276i	$0.463066\pi$
$398$	19.0011	0.952438
$399$	0	0
$400$	−46.8223	−2.34112
$401$	6.62434	0.330804	0.165402	−	0.986226i	$-0.447108\pi$
$401$	6.62434	0.330804	0.165402	+	0.986226i	$0.447108\pi$
$402$	0	0
$403$	−14.6559	−0.730063
$404$	17.9986	0.895462
$405$	0	0
$406$	17.3890	0.863002
$407$	6.06266	0.300515
$408$	0	0
$409$	−16.2700	−0.804500	−0.402250	−	0.915530i	$-0.631772\pi$
$409$	−16.2700	−0.804500	−0.402250	+	0.915530i	$0.631772\pi$
$410$	87.5704	4.32479
$411$	0	0
$412$	14.4287	0.710851
$413$	1.97441	0.0971546
$414$	0	0
$415$	18.2465	0.895684
$416$	27.9473	1.37023
$417$	0	0
$418$	5.06465	0.247720
$419$	6.88186	0.336201	0.168100	−	0.985770i	$-0.446237\pi$
$419$	6.88186	0.336201	0.168100	+	0.985770i	$0.446237\pi$
$420$	0	0
$421$	16.4401	0.801242	0.400621	−	0.916244i	$-0.368794\pi$
$421$	16.4401	0.801242	0.400621	+	0.916244i	$0.368794\pi$
$422$	3.11232	0.151505
$423$	0	0
$424$	0.00931012	0.000452139 0
$425$	−0.901088	−0.0437092
$426$	0	0
$427$	19.9282	0.964391
$428$	10.4112	0.503244
$429$	0	0
$430$	−33.5437	−1.61762
$431$	−0.0473787	−0.00228215	−0.00114108	−	0.999999i	$-0.500363\pi$
$431$	−0.0473787	−0.00228215	−0.00114108	+	0.999999i	$0.500363\pi$
$432$	0	0
$433$	−24.4507	−1.17503	−0.587513	−	0.809215i	$-0.699893\pi$
$433$	−24.4507	−1.17503	−0.587513	+	0.809215i	$0.699893\pi$
$434$	15.7210	0.754632
$435$	0	0
$436$	12.8056	0.613278
$437$	−9.18219	−0.439244
$438$	0	0
$439$	8.11798	0.387450	0.193725	−	0.981056i	$-0.437943\pi$
$439$	8.11798	0.387450	0.193725	+	0.981056i	$0.437943\pi$
$440$	−0.251455	−0.0119876
$441$	0	0
$442$	0.532810	0.0253432
$443$	−19.1680	−0.910700	−0.455350	−	0.890313i	$-0.650486\pi$
$443$	−19.1680	−0.910700	−0.455350	+	0.890313i	$0.650486\pi$
$444$	0	0
$445$	−18.9580	−0.898693
$446$	−2.00467	−0.0949239
$447$	0	0
$448$	−15.1985	−0.718061
$449$	−15.9129	−0.750976	−0.375488	−	0.926827i	$-0.622525\pi$
$449$	−15.9129	−0.750976	−0.375488	+	0.926827i	$0.622525\pi$
$450$	0	0
$451$	−17.4166	−0.820115
$452$	21.4375	1.00834
$453$	0	0
$454$	25.2197	1.18362
$455$	26.6594	1.24981
$456$	0	0
$457$	0.0552170	0.00258294	0.00129147	−	0.999999i	$-0.499589\pi$
$457$	0.0552170	0.00258294	0.00129147	+	0.999999i	$0.499589\pi$
$458$	28.5861	1.33574
$459$	0	0
$460$	49.1943	2.29369
$461$	19.6181	0.913705	0.456852	−	0.889543i	$-0.348977\pi$
$461$	19.6181	0.913705	0.456852	+	0.889543i	$0.348977\pi$
$462$	0	0
$463$	−15.4472	−0.717892	−0.358946	−	0.933358i	$-0.616864\pi$
$463$	−15.4472	−0.717892	−0.358946	+	0.933358i	$0.616864\pi$
$464$	18.4278	0.855489
$465$	0	0
$466$	−9.02727	−0.418180
$467$	36.5279	1.69031	0.845154	−	0.534523i	$-0.179509\pi$
$467$	36.5279	1.69031	0.845154	+	0.534523i	$0.179509\pi$
$468$	0	0
$469$	13.1779	0.608498
$470$	34.7073	1.60093
$471$	0	0
$472$	−0.0397007	−0.00182737
$473$	6.67139	0.306751
$474$	0	0
$475$	−18.2597	−0.837814
$476$	−0.287096	−0.0131590
$477$	0	0
$478$	23.8291	1.08992
$479$	22.2720	1.01764	0.508818	−	0.860874i	$-0.330083\pi$
$479$	22.2720	1.01764	0.508818	+	0.860874i	$0.330083\pi$
$480$	0	0
$481$	12.9245	0.589306
$482$	13.8316	0.630012
$483$	0	0
$484$	−16.8091	−0.764052
$485$	75.8599	3.44462
$486$	0	0
$487$	−15.3495	−0.695552	−0.347776	−	0.937578i	$-0.613063\pi$
$487$	−15.3495	−0.695552	−0.347776	+	0.937578i	$0.613063\pi$
$488$	−0.400708	−0.0181392
$489$	0	0
$490$	28.9496	1.30781
$491$	0.459624	0.0207425	0.0103713	−	0.999946i	$-0.496699\pi$
$491$	0.459624	0.0207425	0.0103713	+	0.999946i	$0.496699\pi$
$492$	0	0
$493$	0.354640	0.0159722
$494$	10.7969	0.485776
$495$	0	0
$496$	16.6601	0.748062
$497$	−13.8174	−0.619794
$498$	0	0
$499$	−24.7041	−1.10591	−0.552953	−	0.833212i	$-0.686499\pi$
$499$	−24.7041	−1.10591	−0.552953	+	0.833212i	$0.686499\pi$
$500$	56.4355	2.52387
$501$	0	0
$502$	26.7348	1.19323
$503$	27.9158	1.24470	0.622351	−	0.782738i	$-0.286178\pi$
$503$	27.9158	1.24470	0.622351	+	0.782738i	$0.286178\pi$
$504$	0	0
$505$	36.5629	1.62703
$506$	−19.4775	−0.865880
$507$	0	0
$508$	26.8807	1.19264
$509$	−15.8695	−0.703404	−0.351702	−	0.936112i	$-0.614397\pi$
$509$	−15.8695	−0.703404	−0.351702	+	0.936112i	$0.614397\pi$
$510$	0	0
$511$	−22.2295	−0.983375
$512$	−32.0663	−1.41714
$513$	0	0
$514$	11.2971	0.498292
$515$	29.3110	1.29160
$516$	0	0
$517$	−6.90282	−0.303586
$518$	−13.8637	−0.609137
$519$	0	0
$520$	−0.536056	−0.0235076
$521$	26.2509	1.15007	0.575037	−	0.818128i	$-0.304988\pi$
$521$	26.2509	1.15007	0.575037	+	0.818128i	$0.304988\pi$
$522$	0	0
$523$	40.7836	1.78334	0.891671	−	0.452685i	$-0.149534\pi$
$523$	40.7836	1.78334	0.891671	+	0.452685i	$0.149534\pi$
$524$	−27.6458	−1.20771
$525$	0	0
$526$	−53.9079	−2.35049
$527$	0.320622	0.0139665
$528$	0	0
$529$	12.3126	0.535330
$530$	2.04087	0.0886498
$531$	0	0
$532$	−5.81773	−0.252231
$533$	−37.1290	−1.60823
$534$	0	0
$535$	21.1496	0.914379
$536$	−0.264976	−0.0114452
$537$	0	0
$538$	31.5416	1.35986
$539$	−5.75768	−0.248001
$540$	0	0
$541$	−6.33889	−0.272530	−0.136265	−	0.990672i	$-0.543510\pi$
$541$	−6.33889	−0.272530	−0.136265	+	0.990672i	$0.543510\pi$
$542$	−44.7425	−1.92186
$543$	0	0
$544$	−0.611392	−0.0262132
$545$	26.0138	1.11431
$546$	0	0
$547$	−26.1634	−1.11866	−0.559332	−	0.828943i	$-0.688942\pi$
$547$	−26.1634	−1.11866	−0.559332	+	0.828943i	$0.688942\pi$
$548$	4.08707	0.174591
$549$	0	0
$550$	−38.7330	−1.65158
$551$	7.18646	0.306153
$552$	0	0
$553$	25.7841	1.09645
$554$	63.5909	2.70172
$555$	0	0
$556$	−9.75599	−0.413746
$557$	−35.9310	−1.52244	−0.761222	−	0.648491i	$-0.775400\pi$
$557$	−35.9310	−1.52244	−0.761222	+	0.648491i	$0.775400\pi$
$558$	0	0
$559$	14.2222	0.601534
$560$	−30.3050	−1.28062
$561$	0	0
$562$	26.2926	1.10909
$563$	19.3537	0.815661	0.407831	−	0.913058i	$-0.366285\pi$
$563$	19.3537	0.815661	0.407831	+	0.913058i	$0.366285\pi$
$564$	0	0
$565$	43.5489	1.83212
$566$	−48.0177	−2.01833
$567$	0	0
$568$	0.277834	0.0116577
$569$	4.93197	0.206759	0.103379	−	0.994642i	$-0.467034\pi$
$569$	4.93197	0.206759	0.103379	+	0.994642i	$0.467034\pi$
$570$	0	0
$571$	14.4739	0.605712	0.302856	−	0.953036i	$-0.402060\pi$
$571$	14.4739	0.605712	0.302856	+	0.953036i	$0.402060\pi$
$572$	11.5046	0.481033
$573$	0	0
$574$	39.8272	1.66236
$575$	70.2227	2.92849
$576$	0	0
$577$	−6.88263	−0.286528	−0.143264	−	0.989685i	$-0.545760\pi$
$577$	−6.88263	−0.286528	−0.143264	+	0.989685i	$0.545760\pi$
$578$	34.0678	1.41703
$579$	0	0
$580$	−38.5020	−1.59871
$581$	8.29853	0.344281
$582$	0	0
$583$	−0.405902	−0.0168107
$584$	0.446982	0.0184962
$585$	0	0
$586$	26.1857	1.08172
$587$	−29.5621	−1.22016	−0.610078	−	0.792341i	$-0.708862\pi$
$587$	−29.5621	−1.22016	−0.610078	+	0.792341i	$0.708862\pi$
$588$	0	0
$589$	6.49710	0.267708
$590$	−8.70280	−0.358289
$591$	0	0
$592$	−14.6919	−0.603834
$593$	−10.7939	−0.443251	−0.221626	−	0.975132i	$-0.571136\pi$
$593$	−10.7939	−0.443251	−0.221626	+	0.975132i	$0.571136\pi$
$594$	0	0
$595$	−0.583216	−0.0239095
$596$	−6.49222	−0.265932
$597$	0	0
$598$	−41.5225	−1.69798
$599$	21.1844	0.865570	0.432785	−	0.901497i	$-0.357531\pi$
$599$	21.1844	0.865570	0.432785	+	0.901497i	$0.357531\pi$
$600$	0	0
$601$	34.7041	1.41561	0.707805	−	0.706407i	$-0.249685\pi$
$601$	34.7041	1.41561	0.707805	+	0.706407i	$0.249685\pi$
$602$	−15.2557	−0.621777
$603$	0	0
$604$	28.8974	1.17582
$605$	−34.1467	−1.38826
$606$	0	0
$607$	18.3424	0.744493	0.372247	−	0.928134i	$-0.378588\pi$
$607$	18.3424	0.744493	0.372247	+	0.928134i	$0.378588\pi$
$608$	−12.3893	−0.502453
$609$	0	0
$610$	−87.8391	−3.55650
$611$	−14.7156	−0.595328
$612$	0	0
$613$	−47.1344	−1.90374	−0.951870	−	0.306503i	$-0.900841\pi$
$613$	−47.1344	−1.90374	−0.951870	+	0.306503i	$0.900841\pi$
$614$	−48.4271	−1.95436
$615$	0	0
$616$	−0.114362	−0.00460778
$617$	−12.3356	−0.496613	−0.248307	−	0.968681i	$-0.579874\pi$
$617$	−12.3356	−0.496613	−0.248307	+	0.968681i	$0.579874\pi$
$618$	0	0
$619$	27.7293	1.11453	0.557267	−	0.830333i	$-0.311850\pi$
$619$	27.7293	1.11453	0.557267	+	0.830333i	$0.311850\pi$
$620$	−34.8087	−1.39795
$621$	0	0
$622$	43.3905	1.73980
$623$	−8.62212	−0.345438
$624$	0	0
$625$	55.5592	2.22237
$626$	49.5089	1.97877
$627$	0	0
$628$	7.83491	0.312647
$629$	−0.282744	−0.0112737
$630$	0	0
$631$	35.3612	1.40771	0.703853	−	0.710346i	$-0.251461\pi$
$631$	35.3612	1.40771	0.703853	+	0.710346i	$0.251461\pi$
$632$	−0.518456	−0.0206231
$633$	0	0
$634$	−41.5329	−1.64948
$635$	54.6063	2.16699
$636$	0	0
$637$	−12.2743	−0.486327
$638$	15.2441	0.603519
$639$	0	0
$640$	1.23029	0.0486313
$641$	33.2576	1.31360	0.656798	−	0.754066i	$-0.271910\pi$
$641$	33.2576	1.31360	0.656798	+	0.754066i	$0.271910\pi$
$642$	0	0
$643$	5.17983	0.204273	0.102136	−	0.994770i	$-0.467432\pi$
$643$	5.17983	0.204273	0.102136	+	0.994770i	$0.467432\pi$
$644$	22.3736	0.881645
$645$	0	0
$646$	−0.236200	−0.00929315
$647$	7.33803	0.288488	0.144244	−	0.989542i	$-0.453925\pi$
$647$	7.33803	0.288488	0.144244	+	0.989542i	$0.453925\pi$
$648$	0	0
$649$	1.73087	0.0679426
$650$	−82.5716	−3.23873
$651$	0	0
$652$	−11.3704	−0.445299
$653$	−11.5620	−0.452458	−0.226229	−	0.974074i	$-0.572640\pi$
$653$	−11.5620	−0.452458	−0.226229	+	0.974074i	$0.572640\pi$
$654$	0	0
$655$	−56.1606	−2.19438
$656$	42.2064	1.64788
$657$	0	0
$658$	15.7850	0.615362
$659$	−6.23977	−0.243067	−0.121533	−	0.992587i	$-0.538781\pi$
$659$	−6.23977	−0.243067	−0.121533	+	0.992587i	$0.538781\pi$
$660$	0	0
$661$	−4.84695	−0.188525	−0.0942623	−	0.995547i	$-0.530049\pi$
$661$	−4.84695	−0.188525	−0.0942623	+	0.995547i	$0.530049\pi$
$662$	−36.1939	−1.40671
$663$	0	0
$664$	−0.166864	−0.00647556
$665$	−11.8183	−0.458295
$666$	0	0
$667$	−27.6375	−1.07013
$668$	16.9715	0.656645
$669$	0	0
$670$	−58.0854	−2.24403
$671$	17.4700	0.674423
$672$	0	0
$673$	−5.19634	−0.200304	−0.100152	−	0.994972i	$-0.531933\pi$
$673$	−5.19634	−0.200304	−0.100152	+	0.994972i	$0.531933\pi$
$674$	−45.6211	−1.75726
$675$	0	0
$676$	−1.71738	−0.0660532
$677$	25.7906	0.991213	0.495607	−	0.868547i	$-0.334946\pi$
$677$	25.7906	0.991213	0.495607	+	0.868547i	$0.334946\pi$
$678$	0	0
$679$	34.5012	1.32404
$680$	0.0117271	0.000449713 0
$681$	0	0
$682$	13.7818	0.527733
$683$	3.46291	0.132505	0.0662523	−	0.997803i	$-0.478896\pi$
$683$	3.46291	0.132505	0.0662523	+	0.997803i	$0.478896\pi$
$684$	0	0
$685$	8.30262	0.317227
$686$	39.3385	1.50195
$687$	0	0
$688$	−16.1671	−0.616364
$689$	−0.865309	−0.0329656
$690$	0	0
$691$	−13.4438	−0.511425	−0.255713	−	0.966753i	$-0.582310\pi$
$691$	−13.4438	−0.511425	−0.255713	+	0.966753i	$0.582310\pi$
$692$	−49.2697	−1.87295
$693$	0	0
$694$	73.2020	2.77871
$695$	−19.8187	−0.751764
$696$	0	0
$697$	0.812255	0.0307664
$698$	48.7028	1.84343
$699$	0	0
$700$	44.4923	1.68165
$701$	16.0052	0.604510	0.302255	−	0.953227i	$-0.402261\pi$
$701$	16.0052	0.604510	0.302255	+	0.953227i	$0.402261\pi$
$702$	0	0
$703$	−5.72954	−0.216094
$704$	−13.3238	−0.502158
$705$	0	0
$706$	−29.5273	−1.11127
$707$	16.6289	0.625393
$708$	0	0
$709$	25.5515	0.959605	0.479803	−	0.877376i	$-0.340708\pi$
$709$	25.5515	0.959605	0.479803	+	0.877376i	$0.340708\pi$
$710$	60.9041	2.28569
$711$	0	0
$712$	0.173370	0.00649732
$713$	−24.9864	−0.935746
$714$	0	0
$715$	23.3709	0.874023
$716$	−16.6022	−0.620453
$717$	0	0
$718$	−63.4464	−2.36780
$719$	3.01102	0.112292	0.0561461	−	0.998423i	$-0.482119\pi$
$719$	3.01102	0.112292	0.0561461	+	0.998423i	$0.482119\pi$
$720$	0	0
$721$	13.3307	0.496461
$722$	33.3024	1.23939
$723$	0	0
$724$	36.5668	1.35899
$725$	−54.9599	−2.04116
$726$	0	0
$727$	17.7783	0.659362	0.329681	−	0.944092i	$-0.393059\pi$
$727$	17.7783	0.659362	0.329681	+	0.944092i	$0.393059\pi$
$728$	−0.243799	−0.00903580
$729$	0	0
$730$	97.9829	3.62651
$731$	−0.311133	−0.0115077
$732$	0	0
$733$	27.6435	1.02104	0.510519	−	0.859867i	$-0.329453\pi$
$733$	27.6435	1.02104	0.510519	+	0.859867i	$0.329453\pi$
$734$	36.0435	1.33039
$735$	0	0
$736$	47.6464	1.75627
$737$	11.5524	0.425538
$738$	0	0
$739$	−7.37183	−0.271177	−0.135589	−	0.990765i	$-0.543293\pi$
$739$	−7.37183	−0.271177	−0.135589	+	0.990765i	$0.543293\pi$
$740$	30.6964	1.12842
$741$	0	0
$742$	0.928192	0.0340750
$743$	−17.1285	−0.628384	−0.314192	−	0.949360i	$-0.601734\pi$
$743$	−17.1285	−0.628384	−0.314192	+	0.949360i	$0.601734\pi$
$744$	0	0
$745$	−13.1885	−0.483190
$746$	35.7236	1.30793
$747$	0	0
$748$	−0.251682	−0.00920242
$749$	9.61890	0.351467
$750$	0	0
$751$	−14.0637	−0.513190	−0.256595	−	0.966519i	$-0.582601\pi$
$751$	−14.0637	−0.513190	−0.256595	+	0.966519i	$0.582601\pi$
$752$	16.7279	0.610004
$753$	0	0
$754$	32.4976	1.18349
$755$	58.7032	2.13643
$756$	0	0
$757$	15.8094	0.574604	0.287302	−	0.957840i	$-0.407242\pi$
$757$	15.8094	0.574604	0.287302	+	0.957840i	$0.407242\pi$
$758$	17.8793	0.649406
$759$	0	0
$760$	0.237638	0.00862005
$761$	−13.8556	−0.502266	−0.251133	−	0.967953i	$-0.580803\pi$
$761$	−13.8556	−0.502266	−0.251133	+	0.967953i	$0.580803\pi$
$762$	0	0
$763$	11.8311	0.428315
$764$	−11.6203	−0.420408
$765$	0	0
$766$	26.7952	0.968149
$767$	3.68990	0.133235
$768$	0	0
$769$	−38.1887	−1.37712	−0.688560	−	0.725180i	$-0.741757\pi$
$769$	−38.1887	−1.37712	−0.688560	+	0.725180i	$0.741757\pi$
$770$	−25.0693	−0.903436
$771$	0	0
$772$	36.7434	1.32242
$773$	31.6650	1.13891	0.569455	−	0.822023i	$-0.307154\pi$
$773$	31.6650	1.13891	0.569455	+	0.822023i	$0.307154\pi$
$774$	0	0
$775$	−49.6879	−1.78484
$776$	−0.693737	−0.0249037
$777$	0	0
$778$	19.4806	0.698413
$779$	16.4596	0.589727
$780$	0	0
$781$	−12.1130	−0.433437
$782$	0.908370	0.0324832
$783$	0	0
$784$	13.9528	0.498316
$785$	15.9161	0.568070
$786$	0	0
$787$	17.0655	0.608320	0.304160	−	0.952621i	$-0.401624\pi$
$787$	17.0655	0.608320	0.304160	+	0.952621i	$0.401624\pi$
$788$	43.9501	1.56566
$789$	0	0
$790$	−113.651	−4.04351
$791$	19.8061	0.704225
$792$	0	0
$793$	37.2429	1.32253
$794$	−9.24842	−0.328214
$795$	0	0
$796$	−19.1341	−0.678191
$797$	−10.6489	−0.377202	−0.188601	−	0.982054i	$-0.560395\pi$
$797$	−10.6489	−0.377202	−0.188601	+	0.982054i	$0.560395\pi$
$798$	0	0
$799$	0.321926	0.0113889
$800$	94.7497	3.34991
$801$	0	0
$802$	−13.2796	−0.468920
$803$	−19.4875	−0.687698
$804$	0	0
$805$	45.4506	1.60192
$806$	29.3803	1.03488
$807$	0	0
$808$	−0.334367	−0.0117630
$809$	−42.0544	−1.47855	−0.739276	−	0.673402i	$-0.764832\pi$
$809$	−42.0544	−1.47855	−0.739276	+	0.673402i	$0.764832\pi$
$810$	0	0
$811$	35.8566	1.25909	0.629547	−	0.776962i	$-0.283240\pi$
$811$	35.8566	1.25909	0.629547	+	0.776962i	$0.283240\pi$
$812$	−17.5108	−0.614507
$813$	0	0
$814$	−12.1536	−0.425985
$815$	−23.0982	−0.809095
$816$	0	0
$817$	−6.30482	−0.220578
$818$	32.6160	1.14039
$819$	0	0
$820$	−88.1835	−3.07950
$821$	−39.9135	−1.39299	−0.696495	−	0.717561i	$-0.745258\pi$
$821$	−39.9135	−1.39299	−0.696495	+	0.717561i	$0.745258\pi$
$822$	0	0
$823$	−24.3943	−0.850331	−0.425165	−	0.905116i	$-0.639784\pi$
$823$	−24.3943	−0.850331	−0.425165	+	0.905116i	$0.639784\pi$
$824$	−0.268048	−0.00933790
$825$	0	0
$826$	−3.95805	−0.137718
$827$	13.1774	0.458222	0.229111	−	0.973400i	$-0.426418\pi$
$827$	13.1774	0.458222	0.229111	+	0.973400i	$0.426418\pi$
$828$	0	0
$829$	−30.4666	−1.05815	−0.529075	−	0.848575i	$-0.677461\pi$
$829$	−30.4666	−1.05815	−0.529075	+	0.848575i	$0.677461\pi$
$830$	−36.5782	−1.26965
$831$	0	0
$832$	−28.4038	−0.984726
$833$	0.268520	0.00930368
$834$	0	0
$835$	34.4764	1.19310
$836$	−5.10012	−0.176391
$837$	0	0
$838$	−13.7959	−0.476570
$839$	5.59075	0.193014	0.0965071	−	0.995332i	$-0.469233\pi$
$839$	5.59075	0.193014	0.0965071	+	0.995332i	$0.469233\pi$
$840$	0	0
$841$	−7.36950	−0.254121
$842$	−32.9570	−1.13577
$843$	0	0
$844$	−3.13411	−0.107881
$845$	−3.48875	−0.120017
$846$	0	0
$847$	−15.5300	−0.533616
$848$	0.983641	0.0337784
$849$	0	0
$850$	1.80639	0.0619585
$851$	22.0345	0.755333
$852$	0	0
$853$	2.55958	0.0876382	0.0438191	−	0.999039i	$-0.486047\pi$
$853$	2.55958	0.0876382	0.0438191	+	0.999039i	$0.486047\pi$
$854$	−39.9494	−1.36704
$855$	0	0
$856$	−0.193413	−0.00661072
$857$	−8.01104	−0.273652	−0.136826	−	0.990595i	$-0.543690\pi$
$857$	−8.01104	−0.273652	−0.136826	+	0.990595i	$0.543690\pi$
$858$	0	0
$859$	−4.30603	−0.146920	−0.0734599	−	0.997298i	$-0.523404\pi$
$859$	−4.30603	−0.146920	−0.0734599	+	0.997298i	$0.523404\pi$
$860$	33.7785	1.15184
$861$	0	0
$862$	0.0949788	0.00323499
$863$	36.0083	1.22574	0.612869	−	0.790185i	$-0.290015\pi$
$863$	36.0083	1.22574	0.612869	+	0.790185i	$0.290015\pi$
$864$	0	0
$865$	−100.088	−3.40310
$866$	49.0156	1.66562
$867$	0	0
$868$	−15.8311	−0.537341
$869$	22.6036	0.766775
$870$	0	0
$871$	24.6276	0.834475
$872$	−0.237895	−0.00805616
$873$	0	0
$874$	18.4073	0.622636
$875$	52.1408	1.76268
$876$	0	0
$877$	−30.9984	−1.04674	−0.523371	−	0.852105i	$-0.675326\pi$
$877$	−30.9984	−1.04674	−0.523371	+	0.852105i	$0.675326\pi$
$878$	−16.2739	−0.549217
$879$	0	0
$880$	−26.5669	−0.895571
$881$	−20.7414	−0.698797	−0.349399	−	0.936974i	$-0.613614\pi$
$881$	−20.7414	−0.698797	−0.349399	+	0.936974i	$0.613614\pi$
$882$	0	0
$883$	−38.3070	−1.28913	−0.644567	−	0.764548i	$-0.722962\pi$
$883$	−38.3070	−1.28913	−0.644567	+	0.764548i	$0.722962\pi$
$884$	−0.536541	−0.0180458
$885$	0	0
$886$	38.4256	1.29093
$887$	−23.8662	−0.801347	−0.400674	−	0.916221i	$-0.631224\pi$
$887$	−23.8662	−0.801347	−0.400674	+	0.916221i	$0.631224\pi$
$888$	0	0
$889$	24.8351	0.832941
$890$	38.0045	1.27391
$891$	0	0
$892$	2.01871	0.0675913
$893$	6.52354	0.218302
$894$	0	0
$895$	−33.7263	−1.12735
$896$	0.559537	0.0186928
$897$	0	0
$898$	31.9001	1.06452
$899$	19.5556	0.652216
$900$	0	0
$901$	0.0189300	0.000630650 0
$902$	34.9145	1.16253
$903$	0	0
$904$	−0.398254	−0.0132457
$905$	74.2830	2.46925
$906$	0	0
$907$	−24.9984	−0.830060	−0.415030	−	0.909808i	$-0.636229\pi$
$907$	−24.9984	−0.830060	−0.415030	+	0.909808i	$0.636229\pi$
$908$	−25.3963	−0.842806
$909$	0	0
$910$	−53.4433	−1.77163
$911$	31.4025	1.04041	0.520205	−	0.854041i	$-0.325856\pi$
$911$	31.4025	1.04041	0.520205	+	0.854041i	$0.325856\pi$
$912$	0	0
$913$	7.27491	0.240764
$914$	−0.110692	−0.00366137
$915$	0	0
$916$	−28.7863	−0.951125
$917$	−25.5420	−0.843470
$918$	0	0
$919$	−39.1711	−1.29213	−0.646067	−	0.763281i	$-0.723587\pi$
$919$	−39.1711	−1.29213	−0.646067	+	0.763281i	$0.723587\pi$
$920$	−0.913903	−0.0301305
$921$	0	0
$922$	−39.3278	−1.29519
$923$	−25.8227	−0.849965
$924$	0	0
$925$	43.8178	1.44072
$926$	30.9665	1.01762
$927$	0	0
$928$	−37.2905	−1.22412
$929$	−26.4561	−0.867995	−0.433998	−	0.900914i	$-0.642897\pi$
$929$	−26.4561	−0.867995	−0.433998	+	0.900914i	$0.642897\pi$
$930$	0	0
$931$	5.44132	0.178332
$932$	9.09048	0.297768
$933$	0	0
$934$	−73.2264	−2.39604
$935$	−0.511276	−0.0167205
$936$	0	0
$937$	23.5017	0.767769	0.383884	−	0.923381i	$-0.374586\pi$
$937$	23.5017	0.767769	0.383884	+	0.923381i	$0.374586\pi$
$938$	−26.4173	−0.862557
$939$	0	0
$940$	−34.9503	−1.13995
$941$	−27.5968	−0.899631	−0.449816	−	0.893121i	$-0.648510\pi$
$941$	−27.5968	−0.899631	−0.449816	+	0.893121i	$0.648510\pi$
$942$	0	0
$943$	−63.2999	−2.06133
$944$	−4.19450	−0.136519
$945$	0	0
$946$	−13.3739	−0.434824
$947$	−5.12684	−0.166600	−0.0833000	−	0.996525i	$-0.526546\pi$
$947$	−5.12684	−0.166600	−0.0833000	+	0.996525i	$0.526546\pi$
$948$	0	0
$949$	−41.5438	−1.34857
$950$	36.6048	1.18762
$951$	0	0
$952$	0.00533350	0.000172860 0
$953$	−46.8403	−1.51731	−0.758653	−	0.651494i	$-0.774142\pi$
$953$	−46.8403	−1.51731	−0.758653	+	0.651494i	$0.774142\pi$
$954$	0	0
$955$	−23.6059	−0.763869
$956$	−23.9959	−0.776083
$957$	0	0
$958$	−44.6481	−1.44252
$959$	3.77605	0.121935
$960$	0	0
$961$	−13.3202	−0.429685
$962$	−25.9094	−0.835351
$963$	0	0
$964$	−13.9284	−0.448605
$965$	74.6418	2.40280
$966$	0	0
$967$	25.1230	0.807903	0.403951	−	0.914780i	$-0.367636\pi$
$967$	25.1230	0.807903	0.403951	+	0.914780i	$0.367636\pi$
$968$	0.312271	0.0100368
$969$	0	0
$970$	−152.074	−4.88281
$971$	−28.8876	−0.927046	−0.463523	−	0.886085i	$-0.653415\pi$
$971$	−28.8876	−0.927046	−0.463523	+	0.886085i	$0.653415\pi$
$972$	0	0
$973$	−9.01356	−0.288962
$974$	30.7707	0.985957
$975$	0	0
$976$	−42.3359	−1.35514
$977$	−8.27878	−0.264861	−0.132431	−	0.991192i	$-0.542278\pi$
$977$	−8.27878	−0.264861	−0.132431	+	0.991192i	$0.542278\pi$
$978$	0	0
$979$	−7.55858	−0.241573
$980$	−29.1523	−0.931235
$981$	0	0
$982$	−0.921394	−0.0294029
$983$	38.1383	1.21642	0.608212	−	0.793775i	$-0.291887\pi$
$983$	38.1383	1.21642	0.608212	+	0.793775i	$0.291887\pi$
$984$	0	0
$985$	89.2817	2.84475
$986$	−0.710937	−0.0226408
$987$	0	0
$988$	−10.8725	−0.345901
$989$	24.2469	0.771006
$990$	0	0
$991$	−14.3983	−0.457377	−0.228689	−	0.973500i	$-0.573444\pi$
$991$	−14.3983	−0.457377	−0.228689	+	0.973500i	$0.573444\pi$
$992$	−33.7135	−1.07040
$993$	0	0
$994$	27.6993	0.878568
$995$	−38.8697	−1.23225
$996$	0	0
$997$	62.2346	1.97099	0.985495	−	0.169706i	$-0.0542817\pi$
$997$	62.2346	1.97099	0.985495	+	0.169706i	$0.0542817\pi$
$998$	49.5235	1.56764
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.8	✓	40
3.2	odd	2	inner	6021.2.a.t.1.33	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.8	✓	40	1.1	even	1	trivial
6021.2.a.t.1.33	yes	40	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)

\(f(q)\)	\(=\)	\(q-2.00467 q^{2} +2.01871 q^{4} +4.10087 q^{5} +1.86508 q^{7} -0.0375024 q^{8} +O(q^{10})\)	\(q-2.00467 q^{2} +2.01871 q^{4} +4.10087 q^{5} +1.86508 q^{7} -0.0375024 q^{8} -8.22090 q^{10} +1.63503 q^{11} +3.48558 q^{13} -3.73888 q^{14} -3.96224 q^{16} -0.0762526 q^{17} -1.54519 q^{19} +8.27846 q^{20} -3.27769 q^{22} +5.94244 q^{23} +11.8171 q^{25} -6.98744 q^{26} +3.76506 q^{28} -4.65086 q^{29} -4.20473 q^{31} +8.01798 q^{32} +0.152861 q^{34} +7.64847 q^{35} +3.70799 q^{37} +3.09760 q^{38} -0.153792 q^{40} -10.6522 q^{41} +4.08029 q^{43} +3.30064 q^{44} -11.9126 q^{46} -4.22184 q^{47} -3.52146 q^{49} -23.6895 q^{50} +7.03637 q^{52} -0.248254 q^{53} +6.70503 q^{55} -0.0699451 q^{56} +9.32345 q^{58} +1.05862 q^{59} +10.6849 q^{61} +8.42910 q^{62} -8.14895 q^{64} +14.2939 q^{65} +7.06557 q^{67} -0.153932 q^{68} -15.3327 q^{70} -7.40845 q^{71} -11.9188 q^{73} -7.43330 q^{74} -3.11929 q^{76} +3.04946 q^{77} +13.8246 q^{79} -16.2486 q^{80} +21.3541 q^{82} +4.44941 q^{83} -0.312702 q^{85} -8.17964 q^{86} -0.0613174 q^{88} -4.62291 q^{89} +6.50090 q^{91} +11.9960 q^{92} +8.46340 q^{94} -6.33662 q^{95} +18.4985 q^{97} +7.05937 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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