LMFDB - Embedded newform 7220.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7220,2,Mod(1,7220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7220 = 2^{2} \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7220.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:	$57.6519902594$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	7220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.364448 q^{3} -1.00000 q^{5} +0.635552 q^{7} -2.86718 q^{9} +O(q^{10})$	$q+0.364448 q^{3} -1.00000 q^{5} +0.635552 q^{7} -2.86718 q^{9} +1.63555 q^{11} +1.00000 q^{13} -0.364448 q^{15} -6.59607 q^{17} +0.231626 q^{21} +0.867178 q^{23} +1.00000 q^{25} -2.13828 q^{27} +9.09880 q^{29} +1.86718 q^{31} +0.596074 q^{33} -0.635552 q^{35} -0.635552 q^{37} +0.364448 q^{39} -1.90666 q^{41} +3.96052 q^{43} +2.86718 q^{45} +9.09880 q^{47} -6.59607 q^{49} -2.40393 q^{51} +9.86718 q^{53} -1.63555 q^{55} -9.09880 q^{59} +2.27110 q^{61} -1.82224 q^{63} -1.00000 q^{65} -12.4633 q^{67} +0.316041 q^{69} -2.50273 q^{71} -0.403926 q^{73} +0.364448 q^{75} +1.03948 q^{77} -10.7289 q^{79} +7.82224 q^{81} -15.8672 q^{83} +6.59607 q^{85} +3.31604 q^{87} -0.542208 q^{89} +0.635552 q^{91} +0.680489 q^{93} +7.36991 q^{97} -4.68942 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) $ 3 * q + q^3 - 3 * q^5 + 2 * q^7 + 8 * q^9	$ 3 q + q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9} + 5 q^{11} + 3 q^{13} - q^{15} - 3 q^{17} - 16 q^{21} - 14 q^{23} + 3 q^{25} + 10 q^{27} - 6 q^{29} - 11 q^{31} - 15 q^{33} - 2 q^{35} - 2 q^{37} + q^{39} - 6 q^{41} - 5 q^{43} - 8 q^{45} - 6 q^{47} - 3 q^{49} - 24 q^{51} + 13 q^{53} - 5 q^{55} + 6 q^{59} + 7 q^{61} - 5 q^{63} - 3 q^{65} - 4 q^{67} - 15 q^{69} + 9 q^{71} - 18 q^{73} + q^{75} + 20 q^{77} - 32 q^{79} + 23 q^{81} - 31 q^{83} + 3 q^{85} - 6 q^{87} - 2 q^{89} + 2 q^{91} - 14 q^{93} - 11 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 + 2 * q^7 + 8 * q^9 + 5 * q^11 + 3 * q^13 - q^15 - 3 * q^17 - 16 * q^21 - 14 * q^23 + 3 * q^25 + 10 * q^27 - 6 * q^29 - 11 * q^31 - 15 * q^33 - 2 * q^35 - 2 * q^37 + q^39 - 6 * q^41 - 5 * q^43 - 8 * q^45 - 6 * q^47 - 3 * q^49 - 24 * q^51 + 13 * q^53 - 5 * q^55 + 6 * q^59 + 7 * q^61 - 5 * q^63 - 3 * q^65 - 4 * q^67 - 15 * q^69 + 9 * q^71 - 18 * q^73 + q^75 + 20 * q^77 - 32 * q^79 + 23 * q^81 - 31 * q^83 + 3 * q^85 - 6 * q^87 - 2 * q^89 + 2 * q^91 - 14 * q^93 - 11 * q^97 + 3 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	0.364448	0.210414	0.105207	−	0.994450i	$-0.466449\pi$
$3$	0.364448	0.210414	0.105207	+	0.994450i	$0.466449\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.635552	0.240216	0.120108	−	0.992761i	$-0.461676\pi$
$7$	0.635552	0.240216	0.120108	+	0.992761i	$0.461676\pi$
$8$	0	0
$9$	−2.86718	−0.955726
$10$	0	0
$11$	1.63555	0.493137	0.246569	−	0.969125i	$-0.420697\pi$
$11$	1.63555	0.493137	0.246569	+	0.969125i	$0.420697\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	−0.364448	−0.0941001
$16$	0	0
$17$	−6.59607	−1.59978	−0.799891	−	0.600145i	$-0.795110\pi$
$17$	−6.59607	−1.59978	−0.799891	+	0.600145i	$0.795110\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.231626	0.0505449
$22$	0	0
$23$	0.867178	0.180819	0.0904095	−	0.995905i	$-0.471182\pi$
$23$	0.867178	0.180819	0.0904095	+	0.995905i	$0.471182\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.13828	−0.411512
$28$	0	0
$29$	9.09880	1.68961	0.844803	−	0.535078i	$-0.179718\pi$
$29$	9.09880	1.68961	0.844803	+	0.535078i	$0.179718\pi$
$30$	0	0
$31$	1.86718	0.335355	0.167677	−	0.985842i	$-0.446373\pi$
$31$	1.86718	0.335355	0.167677	+	0.985842i	$0.446373\pi$
$32$	0	0
$33$	0.596074	0.103763
$34$	0	0
$35$	−0.635552	−0.107428
$36$	0	0
$37$	−0.635552	−0.104484	−0.0522420	−	0.998634i	$-0.516637\pi$
$37$	−0.635552	−0.104484	−0.0522420	+	0.998634i	$0.516637\pi$
$38$	0	0
$39$	0.364448	0.0583584
$40$	0	0
$41$	−1.90666	−0.297770	−0.148885	−	0.988855i	$-0.547568\pi$
$41$	−1.90666	−0.297770	−0.148885	+	0.988855i	$0.547568\pi$
$42$	0	0
$43$	3.96052	0.603974	0.301987	−	0.953312i	$-0.402350\pi$
$43$	3.96052	0.603974	0.301987	+	0.953312i	$0.402350\pi$
$44$	0	0
$45$	2.86718	0.427414
$46$	0	0
$47$	9.09880	1.32720	0.663598	−	0.748089i	$-0.269028\pi$
$47$	9.09880	1.32720	0.663598	+	0.748089i	$0.269028\pi$
$48$	0	0
$49$	−6.59607	−0.942296
$50$	0	0
$51$	−2.40393	−0.336617
$52$	0	0
$53$	9.86718	1.35536	0.677681	−	0.735356i	$-0.262985\pi$
$53$	9.86718	1.35536	0.677681	+	0.735356i	$0.262985\pi$
$54$	0	0
$55$	−1.63555	−0.220538
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.09880	−1.18456	−0.592282	−	0.805731i	$-0.701773\pi$
$59$	−9.09880	−1.18456	−0.592282	+	0.805731i	$0.701773\pi$
$60$	0	0
$61$	2.27110	0.290785	0.145393	−	0.989374i	$-0.453555\pi$
$61$	2.27110	0.290785	0.145393	+	0.989374i	$0.453555\pi$
$62$	0	0
$63$	−1.82224	−0.229581
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−12.4633	−1.52263	−0.761314	−	0.648383i	$-0.775446\pi$
$67$	−12.4633	−1.52263	−0.761314	+	0.648383i	$0.775446\pi$
$68$	0	0
$69$	0.316041	0.0380469
$70$	0	0
$71$	−2.50273	−0.297019	−0.148510	−	0.988911i	$-0.547448\pi$
$71$	−2.50273	−0.297019	−0.148510	+	0.988911i	$0.547448\pi$
$72$	0	0
$73$	−0.403926	−0.0472760	−0.0236380	−	0.999721i	$-0.507525\pi$
$73$	−0.403926	−0.0472760	−0.0236380	+	0.999721i	$0.507525\pi$
$74$	0	0
$75$	0.364448	0.0420828
$76$	0	0
$77$	1.03948	0.118460
$78$	0	0
$79$	−10.7289	−1.20710	−0.603548	−	0.797327i	$-0.706247\pi$
$79$	−10.7289	−1.20710	−0.603548	+	0.797327i	$0.706247\pi$
$80$	0	0
$81$	7.82224	0.869138
$82$	0	0
$83$	−15.8672	−1.74165	−0.870825	−	0.491594i	$-0.836414\pi$
$83$	−15.8672	−1.74165	−0.870825	+	0.491594i	$0.836414\pi$
$84$	0	0
$85$	6.59607	0.715445
$86$	0	0
$87$	3.31604	0.355517
$88$	0	0
$89$	−0.542208	−0.0574739	−0.0287370	−	0.999587i	$-0.509149\pi$
$89$	−0.542208	−0.0574739	−0.0287370	+	0.999587i	$0.509149\pi$
$90$	0	0
$91$	0.635552	0.0666239
$92$	0	0
$93$	0.680489	0.0705634
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.36991	0.748301	0.374150	−	0.927368i	$-0.377934\pi$
$97$	7.36991	0.748301	0.374150	+	0.927368i	$0.377934\pi$
$98$	0	0
$99$	−4.68942	−0.471304
$100$	0	0
$101$	−9.32497	−0.927869	−0.463935	−	0.885869i	$-0.653563\pi$
$101$	−9.32497	−0.927869	−0.463935	+	0.885869i	$0.653563\pi$
$102$	0	0
$103$	10.8672	1.07077	0.535387	−	0.844607i	$-0.320166\pi$
$103$	10.8672	1.07077	0.535387	+	0.844607i	$0.320166\pi$
$104$	0	0
$105$	−0.231626	−0.0226044
$106$	0	0
$107$	−11.3304	−1.09535	−0.547677	−	0.836690i	$-0.684488\pi$
$107$	−11.3304	−1.09535	−0.547677	+	0.836690i	$0.684488\pi$
$108$	0	0
$109$	−13.2316	−1.26736	−0.633680	−	0.773595i	$-0.718456\pi$
$109$	−13.2316	−1.26736	−0.633680	+	0.773595i	$0.718456\pi$
$110$	0	0
$111$	−0.231626	−0.0219849
$112$	0	0
$113$	−7.09334	−0.667286	−0.333643	−	0.942700i	$-0.608278\pi$
$113$	−7.09334	−0.667286	−0.333643	+	0.942700i	$0.608278\pi$
$114$	0	0
$115$	−0.867178	−0.0808647
$116$	0	0
$117$	−2.86718	−0.265071
$118$	0	0
$119$	−4.19215	−0.384294
$120$	0	0
$121$	−8.32497	−0.756815
$122$	0	0
$123$	−0.694877	−0.0626550
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.2766	−1.44431	−0.722156	−	0.691731i	$-0.756849\pi$
$127$	−16.2766	−1.44431	−0.722156	+	0.691731i	$0.756849\pi$
$128$	0	0
$129$	1.44340	0.127085
$130$	0	0
$131$	−10.6356	−0.929232	−0.464616	−	0.885512i	$-0.653808\pi$
$131$	−10.6356	−0.929232	−0.464616	+	0.885512i	$0.653808\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.13828	0.184034
$136$	0	0
$137$	3.49727	0.298792	0.149396	−	0.988777i	$-0.452267\pi$
$137$	3.49727	0.298792	0.149396	+	0.988777i	$0.452267\pi$
$138$	0	0
$139$	0.734355	0.0622872	0.0311436	−	0.999515i	$-0.490085\pi$
$139$	0.734355	0.0622872	0.0311436	+	0.999515i	$0.490085\pi$
$140$	0	0
$141$	3.31604	0.279261
$142$	0	0
$143$	1.63555	0.136772
$144$	0	0
$145$	−9.09880	−0.755614
$146$	0	0
$147$	−2.40393	−0.198272
$148$	0	0
$149$	10.1383	0.830560	0.415280	−	0.909694i	$-0.363684\pi$
$149$	10.1383	0.830560	0.415280	+	0.909694i	$0.363684\pi$
$150$	0	0
$151$	−2.81331	−0.228944	−0.114472	−	0.993426i	$-0.536518\pi$
$151$	−2.81331	−0.228944	−0.114472	+	0.993426i	$0.536518\pi$
$152$	0	0
$153$	18.9121	1.52895
$154$	0	0
$155$	−1.86718	−0.149975
$156$	0	0
$157$	0.960522	0.0766580	0.0383290	−	0.999265i	$-0.487797\pi$
$157$	0.960522	0.0766580	0.0383290	+	0.999265i	$0.487797\pi$
$158$	0	0
$159$	3.59607	0.285187
$160$	0	0
$161$	0.551136	0.0434356
$162$	0	0
$163$	3.13828	0.245809	0.122905	−	0.992418i	$-0.460779\pi$
$163$	3.13828	0.245809	0.122905	+	0.992418i	$0.460779\pi$
$164$	0	0
$165$	−0.596074	−0.0464043
$166$	0	0
$167$	10.6356	0.823004	0.411502	−	0.911409i	$-0.365004\pi$
$167$	10.6356	0.823004	0.411502	+	0.911409i	$0.365004\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.3699	−0.940467	−0.470233	−	0.882542i	$-0.655830\pi$
$173$	−12.3699	−0.940467	−0.470233	+	0.882542i	$0.655830\pi$
$174$	0	0
$175$	0.635552	0.0480432
$176$	0	0
$177$	−3.31604	−0.249249
$178$	0	0
$179$	0.324970	0.0242894	0.0121447	−	0.999926i	$-0.496134\pi$
$179$	0.324970	0.0242894	0.0121447	+	0.999926i	$0.496134\pi$
$180$	0	0
$181$	−8.81331	−0.655088	−0.327544	−	0.944836i	$-0.606221\pi$
$181$	−8.81331	−0.655088	−0.327544	+	0.944836i	$0.606221\pi$
$182$	0	0
$183$	0.827699	0.0611853
$184$	0	0
$185$	0.635552	0.0467267
$186$	0	0
$187$	−10.7882	−0.788913
$188$	0	0
$189$	−1.35899	−0.0988519
$190$	0	0
$191$	25.1921	1.82284	0.911420	−	0.411478i	$-0.134987\pi$
$191$	25.1921	1.82284	0.911420	+	0.411478i	$0.134987\pi$
$192$	0	0
$193$	18.8726	1.35848	0.679241	−	0.733915i	$-0.262309\pi$
$193$	18.8726	1.35848	0.679241	+	0.733915i	$0.262309\pi$
$194$	0	0
$195$	−0.364448	−0.0260987
$196$	0	0
$197$	−7.36445	−0.524695	−0.262348	−	0.964973i	$-0.584497\pi$
$197$	−7.36445	−0.524695	−0.262348	+	0.964973i	$0.584497\pi$
$198$	0	0
$199$	2.76837	0.196245	0.0981224	−	0.995174i	$-0.468716\pi$
$199$	2.76837	0.196245	0.0981224	+	0.995174i	$0.468716\pi$
$200$	0	0
$201$	−4.54221	−0.320383
$202$	0	0
$203$	5.78276	0.405870
$204$	0	0
$205$	1.90666	0.133167
$206$	0	0
$207$	−2.48635	−0.172813
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−16.9265	−1.16527	−0.582634	−	0.812734i	$-0.697978\pi$
$211$	−16.9265	−1.16527	−0.582634	+	0.812734i	$0.697978\pi$
$212$	0	0
$213$	−0.912115	−0.0624971
$214$	0	0
$215$	−3.96052	−0.270105
$216$	0	0
$217$	1.18669	0.0805577
$218$	0	0
$219$	−0.147210	−0.00994754
$220$	0	0
$221$	−6.59607	−0.443700
$222$	0	0
$223$	−8.64101	−0.578645	−0.289322	−	0.957232i	$-0.593430\pi$
$223$	−8.64101	−0.578645	−0.289322	+	0.957232i	$0.593430\pi$
$224$	0	0
$225$	−2.86718	−0.191145
$226$	0	0
$227$	−20.9265	−1.38894	−0.694470	−	0.719521i	$-0.744361\pi$
$227$	−20.9265	−1.38894	−0.694470	+	0.719521i	$0.744361\pi$
$228$	0	0
$229$	22.3754	1.47861	0.739303	−	0.673373i	$-0.235155\pi$
$229$	22.3754	1.47861	0.739303	+	0.673373i	$0.235155\pi$
$230$	0	0
$231$	0.378836	0.0249256
$232$	0	0
$233$	−19.2371	−1.26026	−0.630132	−	0.776488i	$-0.716999\pi$
$233$	−19.2371	−1.26026	−0.630132	+	0.776488i	$0.716999\pi$
$234$	0	0
$235$	−9.09880	−0.593540
$236$	0	0
$237$	−3.91013	−0.253990
$238$	0	0
$239$	−2.13282	−0.137961	−0.0689804	−	0.997618i	$-0.521975\pi$
$239$	−2.13282	−0.137961	−0.0689804	+	0.997618i	$0.521975\pi$
$240$	0	0
$241$	−24.4633	−1.57582	−0.787908	−	0.615793i	$-0.788836\pi$
$241$	−24.4633	−1.57582	−0.787908	+	0.615793i	$0.788836\pi$
$242$	0	0
$243$	9.26564	0.594391
$244$	0	0
$245$	6.59607	0.421408
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.78276	−0.366468
$250$	0	0
$251$	3.28003	0.207034	0.103517	−	0.994628i	$-0.466990\pi$
$251$	3.28003	0.207034	0.103517	+	0.994628i	$0.466990\pi$
$252$	0	0
$253$	1.41831	0.0891686
$254$	0	0
$255$	2.40393	0.150540
$256$	0	0
$257$	7.63555	0.476293	0.238146	−	0.971229i	$-0.423460\pi$
$257$	7.63555	0.476293	0.238146	+	0.971229i	$0.423460\pi$
$258$	0	0
$259$	−0.403926	−0.0250988
$260$	0	0
$261$	−26.0879	−1.61480
$262$	0	0
$263$	14.7289	0.908223	0.454111	−	0.890945i	$-0.349957\pi$
$263$	14.7289	0.908223	0.454111	+	0.890945i	$0.349957\pi$
$264$	0	0
$265$	−9.86718	−0.606136
$266$	0	0
$267$	−0.197607	−0.0120933
$268$	0	0
$269$	13.4633	0.820869	0.410434	−	0.911890i	$-0.365377\pi$
$269$	13.4633	0.820869	0.410434	+	0.911890i	$0.365377\pi$
$270$	0	0
$271$	3.86172	0.234583	0.117291	−	0.993098i	$-0.462579\pi$
$271$	3.86172	0.234583	0.117291	+	0.993098i	$0.462579\pi$
$272$	0	0
$273$	0.231626	0.0140186
$274$	0	0
$275$	1.63555	0.0986275
$276$	0	0
$277$	−24.2766	−1.45864	−0.729319	−	0.684174i	$-0.760163\pi$
$277$	−24.2766	−1.45864	−0.729319	+	0.684174i	$0.760163\pi$
$278$	0	0
$279$	−5.35353	−0.320507
$280$	0	0
$281$	−0.497270	−0.0296647	−0.0148323	−	0.999890i	$-0.504721\pi$
$281$	−0.497270	−0.0296647	−0.0148323	+	0.999890i	$0.504721\pi$
$282$	0	0
$283$	−15.1581	−0.901057	−0.450529	−	0.892762i	$-0.648764\pi$
$283$	−15.1581	−0.901057	−0.450529	+	0.892762i	$0.648764\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.21178	−0.0715290
$288$	0	0
$289$	26.5082	1.55931
$290$	0	0
$291$	2.68595	0.157453
$292$	0	0
$293$	13.4094	0.783385	0.391692	−	0.920096i	$-0.371890\pi$
$293$	13.4094	0.783385	0.391692	+	0.920096i	$0.371890\pi$
$294$	0	0
$295$	9.09880	0.529753
$296$	0	0
$297$	−3.49727	−0.202932
$298$	0	0
$299$	0.867178	0.0501502
$300$	0	0
$301$	2.51712	0.145084
$302$	0	0
$303$	−3.39847	−0.195237
$304$	0	0
$305$	−2.27110	−0.130043
$306$	0	0
$307$	−32.2065	−1.83812	−0.919062	−	0.394113i	$-0.871052\pi$
$307$	−32.2065	−1.83812	−0.919062	+	0.394113i	$0.871052\pi$
$308$	0	0
$309$	3.96052	0.225306
$310$	0	0
$311$	12.3699	0.701433	0.350717	−	0.936482i	$-0.385938\pi$
$311$	12.3699	0.701433	0.350717	+	0.936482i	$0.385938\pi$
$312$	0	0
$313$	24.7882	1.40111	0.700557	−	0.713597i	$-0.252935\pi$
$313$	24.7882	1.40111	0.700557	+	0.713597i	$0.252935\pi$
$314$	0	0
$315$	1.82224	0.102672
$316$	0	0
$317$	−7.31058	−0.410603	−0.205302	−	0.978699i	$-0.565818\pi$
$317$	−7.31058	−0.410603	−0.205302	+	0.978699i	$0.565818\pi$
$318$	0	0
$319$	14.8816	0.833208
$320$	0	0
$321$	−4.12935	−0.230478
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−4.82224	−0.266670
$328$	0	0
$329$	5.78276	0.318814
$330$	0	0
$331$	−18.4722	−1.01532	−0.507661	−	0.861557i	$-0.669490\pi$
$331$	−18.4722	−1.01532	−0.507661	+	0.861557i	$0.669490\pi$
$332$	0	0
$333$	1.82224	0.0998582
$334$	0	0
$335$	12.4633	0.680940
$336$	0	0
$337$	−30.4633	−1.65944	−0.829720	−	0.558181i	$-0.811500\pi$
$337$	−30.4633	−1.65944	−0.829720	+	0.558181i	$0.811500\pi$
$338$	0	0
$339$	−2.58516	−0.140406
$340$	0	0
$341$	3.05387	0.165376
$342$	0	0
$343$	−8.64101	−0.466571
$344$	0	0
$345$	−0.316041	−0.0170151
$346$	0	0
$347$	23.3215	1.25196	0.625982	−	0.779838i	$-0.284698\pi$
$347$	23.3215	1.25196	0.625982	+	0.779838i	$0.284698\pi$
$348$	0	0
$349$	−17.6894	−0.946893	−0.473446	−	0.880823i	$-0.656990\pi$
$349$	−17.6894	−0.946893	−0.473446	+	0.880823i	$0.656990\pi$
$350$	0	0
$351$	−2.13828	−0.114133
$352$	0	0
$353$	−7.13828	−0.379932	−0.189966	−	0.981791i	$-0.560838\pi$
$353$	−7.13828	−0.379932	−0.189966	+	0.981791i	$0.560838\pi$
$354$	0	0
$355$	2.50273	0.132831
$356$	0	0
$357$	−1.52782	−0.0808608
$358$	0	0
$359$	−30.1437	−1.59093	−0.795463	−	0.606002i	$-0.792772\pi$
$359$	−30.1437	−1.59093	−0.795463	+	0.606002i	$0.792772\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−3.03402	−0.159245
$364$	0	0
$365$	0.403926	0.0211425
$366$	0	0
$367$	−30.2031	−1.57659	−0.788294	−	0.615299i	$-0.789035\pi$
$367$	−30.2031	−1.57659	−0.788294	+	0.615299i	$0.789035\pi$
$368$	0	0
$369$	5.46672	0.284586
$370$	0	0
$371$	6.27110	0.325579
$372$	0	0
$373$	−29.0449	−1.50389	−0.751945	−	0.659226i	$-0.770884\pi$
$373$	−29.0449	−1.50389	−0.751945	+	0.659226i	$0.770884\pi$
$374$	0	0
$375$	−0.364448	−0.0188200
$376$	0	0
$377$	9.09880	0.468612
$378$	0	0
$379$	−2.76837	−0.142202	−0.0711009	−	0.997469i	$-0.522651\pi$
$379$	−2.76837	−0.142202	−0.0711009	+	0.997469i	$0.522651\pi$
$380$	0	0
$381$	−5.93196	−0.303904
$382$	0	0
$383$	−26.6105	−1.35973	−0.679866	−	0.733337i	$-0.737962\pi$
$383$	−26.6105	−1.35973	−0.679866	+	0.733337i	$0.737962\pi$
$384$	0	0
$385$	−1.03948	−0.0529767
$386$	0	0
$387$	−11.3555	−0.577233
$388$	0	0
$389$	−5.55660	−0.281731	−0.140865	−	0.990029i	$-0.544988\pi$
$389$	−5.55660	−0.281731	−0.140865	+	0.990029i	$0.544988\pi$
$390$	0	0
$391$	−5.71997	−0.289271
$392$	0	0
$393$	−3.87611	−0.195524
$394$	0	0
$395$	10.7289	0.539829
$396$	0	0
$397$	2.71451	0.136237	0.0681186	−	0.997677i	$-0.478300\pi$
$397$	2.71451	0.136237	0.0681186	+	0.997677i	$0.478300\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.8816	0.892963	0.446481	−	0.894793i	$-0.352677\pi$
$401$	17.8816	0.892963	0.446481	+	0.894793i	$0.352677\pi$
$402$	0	0
$403$	1.86718	0.0930107
$404$	0	0
$405$	−7.82224	−0.388690
$406$	0	0
$407$	−1.03948	−0.0515250
$408$	0	0
$409$	−26.3250	−1.30169	−0.650843	−	0.759212i	$-0.725584\pi$
$409$	−26.3250	−1.30169	−0.650843	+	0.759212i	$0.725584\pi$
$410$	0	0
$411$	1.27457	0.0628701
$412$	0	0
$413$	−5.78276	−0.284551
$414$	0	0
$415$	15.8672	0.778889
$416$	0	0
$417$	0.267634	0.0131061
$418$	0	0
$419$	15.3250	0.748674	0.374337	−	0.927293i	$-0.377870\pi$
$419$	15.3250	0.748674	0.374337	+	0.927293i	$0.377870\pi$
$420$	0	0
$421$	−21.8332	−1.06408	−0.532042	−	0.846718i	$-0.678575\pi$
$421$	−21.8332	−1.06408	−0.532042	+	0.846718i	$0.678575\pi$
$422$	0	0
$423$	−26.0879	−1.26844
$424$	0	0
$425$	−6.59607	−0.319957
$426$	0	0
$427$	1.44340	0.0698512
$428$	0	0
$429$	0.596074	0.0287787
$430$	0	0
$431$	−23.2855	−1.12162	−0.560811	−	0.827944i	$-0.689511\pi$
$431$	−23.2855	−1.12162	−0.560811	+	0.827944i	$0.689511\pi$
$432$	0	0
$433$	3.90120	0.187480	0.0937398	−	0.995597i	$-0.470118\pi$
$433$	3.90120	0.187480	0.0937398	+	0.995597i	$0.470118\pi$
$434$	0	0
$435$	−3.31604	−0.158992
$436$	0	0
$437$	0	0
$438$	0	0
$439$	19.4776	0.929617	0.464808	−	0.885411i	$-0.346123\pi$
$439$	19.4776	0.929617	0.464808	+	0.885411i	$0.346123\pi$
$440$	0	0
$441$	18.9121	0.900577
$442$	0	0
$443$	1.90666	0.0905880	0.0452940	−	0.998974i	$-0.485578\pi$
$443$	1.90666	0.0905880	0.0452940	+	0.998974i	$0.485578\pi$
$444$	0	0
$445$	0.542208	0.0257031
$446$	0	0
$447$	3.69488	0.174762
$448$	0	0
$449$	8.05933	0.380343	0.190172	−	0.981751i	$-0.439096\pi$
$449$	8.05933	0.380343	0.190172	+	0.981751i	$0.439096\pi$
$450$	0	0
$451$	−3.11843	−0.146841
$452$	0	0
$453$	−1.02531	−0.0481731
$454$	0	0
$455$	−0.635552	−0.0297951
$456$	0	0
$457$	11.3250	0.529760	0.264880	−	0.964281i	$-0.414668\pi$
$457$	11.3250	0.529760	0.264880	+	0.964281i	$0.414668\pi$
$458$	0	0
$459$	14.1043	0.658331
$460$	0	0
$461$	20.1043	0.936349	0.468174	−	0.883636i	$-0.344912\pi$
$461$	20.1043	0.936349	0.468174	+	0.883636i	$0.344912\pi$
$462$	0	0
$463$	1.33043	0.0618303	0.0309151	−	0.999522i	$-0.490158\pi$
$463$	1.33043	0.0618303	0.0309151	+	0.999522i	$0.490158\pi$
$464$	0	0
$465$	−0.680489	−0.0315569
$466$	0	0
$467$	10.0844	0.466651	0.233326	−	0.972399i	$-0.425039\pi$
$467$	10.0844	0.466651	0.233326	+	0.972399i	$0.425039\pi$
$468$	0	0
$469$	−7.92104	−0.365760
$470$	0	0
$471$	0.350060	0.0161299
$472$	0	0
$473$	6.47764	0.297842
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−28.2910	−1.29535
$478$	0	0
$479$	8.88157	0.405809	0.202905	−	0.979199i	$-0.434962\pi$
$479$	8.88157	0.405809	0.202905	+	0.979199i	$0.434962\pi$
$480$	0	0
$481$	−0.635552	−0.0289787
$482$	0	0
$483$	0.200861	0.00913947
$484$	0	0
$485$	−7.36991	−0.334650
$486$	0	0
$487$	−5.32497	−0.241297	−0.120649	−	0.992695i	$-0.538497\pi$
$487$	−5.32497	−0.241297	−0.120649	+	0.992695i	$0.538497\pi$
$488$	0	0
$489$	1.14374	0.0517217
$490$	0	0
$491$	−19.4094	−0.875933	−0.437967	−	0.898991i	$-0.644301\pi$
$491$	−19.4094	−0.875933	−0.437967	+	0.898991i	$0.644301\pi$
$492$	0	0
$493$	−60.0164	−2.70300
$494$	0	0
$495$	4.68942	0.210774
$496$	0	0
$497$	−1.59061	−0.0713488
$498$	0	0
$499$	11.5871	0.518712	0.259356	−	0.965782i	$-0.416490\pi$
$499$	11.5871	0.518712	0.259356	+	0.965782i	$0.416490\pi$
$500$	0	0
$501$	3.87611	0.173172
$502$	0	0
$503$	35.1043	1.56522	0.782611	−	0.622511i	$-0.213888\pi$
$503$	35.1043	1.56522	0.782611	+	0.622511i	$0.213888\pi$
$504$	0	0
$505$	9.32497	0.414956
$506$	0	0
$507$	−4.37338	−0.194228
$508$	0	0
$509$	15.8672	0.703300	0.351650	−	0.936131i	$-0.385621\pi$
$509$	15.8672	0.703300	0.351650	+	0.936131i	$0.385621\pi$
$510$	0	0
$511$	−0.256716	−0.0113565
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.8672	−0.478865
$516$	0	0
$517$	14.8816	0.654490
$518$	0	0
$519$	−4.50819	−0.197888
$520$	0	0
$521$	−36.6949	−1.60763	−0.803816	−	0.594878i	$-0.797200\pi$
$521$	−36.6949	−1.60763	−0.803816	+	0.594878i	$0.797200\pi$
$522$	0	0
$523$	31.0253	1.35664	0.678321	−	0.734766i	$-0.262708\pi$
$523$	31.0253	1.35664	0.678321	+	0.734766i	$0.262708\pi$
$524$	0	0
$525$	0.231626	0.0101090
$526$	0	0
$527$	−12.3160	−0.536495
$528$	0	0
$529$	−22.2480	−0.967304
$530$	0	0
$531$	26.0879	1.13212
$532$	0	0
$533$	−1.90666	−0.0825864
$534$	0	0
$535$	11.3304	0.489857
$536$	0	0
$537$	0.118435	0.00511083
$538$	0	0
$539$	−10.7882	−0.464682
$540$	0	0
$541$	−7.58715	−0.326197	−0.163098	−	0.986610i	$-0.552149\pi$
$541$	−7.58715	−0.326197	−0.163098	+	0.986610i	$0.552149\pi$
$542$	0	0
$543$	−3.21199	−0.137840
$544$	0	0
$545$	13.2316	0.566781
$546$	0	0
$547$	30.7828	1.31618	0.658088	−	0.752941i	$-0.271365\pi$
$547$	30.7828	1.31618	0.658088	+	0.752941i	$0.271365\pi$
$548$	0	0
$549$	−6.51166	−0.277911
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−6.81877	−0.289964
$554$	0	0
$555$	0.231626	0.00983196
$556$	0	0
$557$	31.0737	1.31664	0.658318	−	0.752740i	$-0.271268\pi$
$557$	31.0737	1.31664	0.658318	+	0.752740i	$0.271268\pi$
$558$	0	0
$559$	3.96052	0.167512
$560$	0	0
$561$	−3.93175	−0.165998
$562$	0	0
$563$	23.2766	0.980990	0.490495	−	0.871444i	$-0.336816\pi$
$563$	23.2766	0.980990	0.490495	+	0.871444i	$0.336816\pi$
$564$	0	0
$565$	7.09334	0.298419
$566$	0	0
$567$	4.97144	0.208781
$568$	0	0
$569$	−24.1976	−1.01442	−0.507208	−	0.861824i	$-0.669322\pi$
$569$	−24.1976	−1.01442	−0.507208	+	0.861824i	$0.669322\pi$
$570$	0	0
$571$	29.8475	1.24908	0.624540	−	0.780992i	$-0.285286\pi$
$571$	29.8475	1.24908	0.624540	+	0.780992i	$0.285286\pi$
$572$	0	0
$573$	9.18123	0.383551
$574$	0	0
$575$	0.867178	0.0361638
$576$	0	0
$577$	33.7882	1.40662	0.703311	−	0.710882i	$-0.251704\pi$
$577$	33.7882	1.40662	0.703311	+	0.710882i	$0.251704\pi$
$578$	0	0
$579$	6.87810	0.285844
$580$	0	0
$581$	−10.0844	−0.418372
$582$	0	0
$583$	16.1383	0.668379
$584$	0	0
$585$	2.86718	0.118543
$586$	0	0
$587$	−30.6949	−1.26691	−0.633457	−	0.773778i	$-0.718364\pi$
$587$	−30.6949	−1.26691	−0.633457	+	0.773778i	$0.718364\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.68396	−0.110403
$592$	0	0
$593$	6.99454	0.287231	0.143616	−	0.989634i	$-0.454127\pi$
$593$	6.99454	0.287231	0.143616	+	0.989634i	$0.454127\pi$
$594$	0	0
$595$	4.19215	0.171861
$596$	0	0
$597$	1.00893	0.0412927
$598$	0	0
$599$	4.53675	0.185367	0.0926833	−	0.995696i	$-0.470456\pi$
$599$	4.53675	0.185367	0.0926833	+	0.995696i	$0.470456\pi$
$600$	0	0
$601$	−16.0790	−0.655874	−0.327937	−	0.944700i	$-0.606353\pi$
$601$	−16.0790	−0.655874	−0.327937	+	0.944700i	$0.606353\pi$
$602$	0	0
$603$	35.7344	1.45522
$604$	0	0
$605$	8.32497	0.338458
$606$	0	0
$607$	34.0702	1.38287	0.691434	−	0.722439i	$-0.256979\pi$
$607$	34.0702	1.38287	0.691434	+	0.722439i	$0.256979\pi$
$608$	0	0
$609$	2.10752	0.0854009
$610$	0	0
$611$	9.09880	0.368098
$612$	0	0
$613$	−39.4382	−1.59289	−0.796446	−	0.604709i	$-0.793289\pi$
$613$	−39.4382	−1.59289	−0.796446	+	0.604709i	$0.793289\pi$
$614$	0	0
$615$	0.694877	0.0280201
$616$	0	0
$617$	−39.2515	−1.58020	−0.790102	−	0.612975i	$-0.789973\pi$
$617$	−39.2515	−1.58020	−0.790102	+	0.612975i	$0.789973\pi$
$618$	0	0
$619$	24.3644	0.979290	0.489645	−	0.871922i	$-0.337126\pi$
$619$	24.3644	0.979290	0.489645	+	0.871922i	$0.337126\pi$
$620$	0	0
$621$	−1.85427	−0.0744093
$622$	0	0
$623$	−0.344601	−0.0138062
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.19215	0.167152
$630$	0	0
$631$	−35.3898	−1.40884	−0.704422	−	0.709781i	$-0.748794\pi$
$631$	−35.3898	−1.40884	−0.704422	+	0.709781i	$0.748794\pi$
$632$	0	0
$633$	−6.16883	−0.245189
$634$	0	0
$635$	16.2766	0.645916
$636$	0	0
$637$	−6.59607	−0.261346
$638$	0	0
$639$	7.17577	0.283869
$640$	0	0
$641$	−35.7093	−1.41043	−0.705216	−	0.708993i	$-0.749150\pi$
$641$	−35.7093	−1.41043	−0.705216	+	0.708993i	$0.749150\pi$
$642$	0	0
$643$	21.1097	0.832486	0.416243	−	0.909253i	$-0.363347\pi$
$643$	21.1097	0.832486	0.416243	+	0.909253i	$0.363347\pi$
$644$	0	0
$645$	−1.44340	−0.0568340
$646$	0	0
$647$	43.7058	1.71825	0.859126	−	0.511764i	$-0.171008\pi$
$647$	43.7058	1.71825	0.859126	+	0.511764i	$0.171008\pi$
$648$	0	0
$649$	−14.8816	−0.584153
$650$	0	0
$651$	0.432486	0.0169505
$652$	0	0
$653$	−24.1887	−0.946576	−0.473288	−	0.880908i	$-0.656933\pi$
$653$	−24.1887	−0.946576	−0.473288	+	0.880908i	$0.656933\pi$
$654$	0	0
$655$	10.6356	0.415565
$656$	0	0
$657$	1.15813	0.0451829
$658$	0	0
$659$	−33.9913	−1.32411	−0.662056	−	0.749454i	$-0.730316\pi$
$659$	−33.9913	−1.32411	−0.662056	+	0.749454i	$0.730316\pi$
$660$	0	0
$661$	−4.55660	−0.177231	−0.0886155	−	0.996066i	$-0.528244\pi$
$661$	−4.55660	−0.177231	−0.0886155	+	0.996066i	$0.528244\pi$
$662$	0	0
$663$	−2.40393	−0.0933608
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.89028	0.305513
$668$	0	0
$669$	−3.14920	−0.121755
$670$	0	0
$671$	3.71451	0.143397
$672$	0	0
$673$	22.7487	0.876900	0.438450	−	0.898756i	$-0.355528\pi$
$673$	22.7487	0.876900	0.438450	+	0.898756i	$0.355528\pi$
$674$	0	0
$675$	−2.13828	−0.0823025
$676$	0	0
$677$	15.6949	0.603203	0.301602	−	0.953434i	$-0.402479\pi$
$677$	15.6949	0.603203	0.301602	+	0.953434i	$0.402479\pi$
$678$	0	0
$679$	4.68396	0.179754
$680$	0	0
$681$	−7.62662	−0.292253
$682$	0	0
$683$	8.82770	0.337783	0.168891	−	0.985635i	$-0.445981\pi$
$683$	8.82770	0.337783	0.168891	+	0.985635i	$0.445981\pi$
$684$	0	0
$685$	−3.49727	−0.133624
$686$	0	0
$687$	8.15466	0.311120
$688$	0	0
$689$	9.86718	0.375910
$690$	0	0
$691$	−23.0198	−0.875716	−0.437858	−	0.899044i	$-0.644263\pi$
$691$	−23.0198	−0.875716	−0.437858	+	0.899044i	$0.644263\pi$
$692$	0	0
$693$	−2.98037	−0.113215
$694$	0	0
$695$	−0.734355	−0.0278557
$696$	0	0
$697$	12.5764	0.476367
$698$	0	0
$699$	−7.01092	−0.265177
$700$	0	0
$701$	−9.91211	−0.374375	−0.187188	−	0.982324i	$-0.559937\pi$
$701$	−9.91211	−0.374375	−0.187188	+	0.982324i	$0.559937\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−3.31604	−0.124889
$706$	0	0
$707$	−5.92650	−0.222889
$708$	0	0
$709$	−36.1043	−1.35592	−0.677962	−	0.735097i	$-0.737137\pi$
$709$	−36.1043	−1.35592	−0.677962	+	0.735097i	$0.737137\pi$
$710$	0	0
$711$	30.7617	1.15365
$712$	0	0
$713$	1.61917	0.0606386
$714$	0	0
$715$	−1.63555	−0.0611662
$716$	0	0
$717$	−0.777303	−0.0290289
$718$	0	0
$719$	−19.0109	−0.708988	−0.354494	−	0.935058i	$-0.615347\pi$
$719$	−19.0109	−0.708988	−0.354494	+	0.935058i	$0.615347\pi$
$720$	0	0
$721$	6.90666	0.257217
$722$	0	0
$723$	−8.91558	−0.331574
$724$	0	0
$725$	9.09880	0.337921
$726$	0	0
$727$	−2.88702	−0.107074	−0.0535369	−	0.998566i	$-0.517049\pi$
$727$	−2.88702	−0.107074	−0.0535369	+	0.998566i	$0.517049\pi$
$728$	0	0
$729$	−20.0899	−0.744069
$730$	0	0
$731$	−26.1239	−0.966227
$732$	0	0
$733$	44.9463	1.66013	0.830066	−	0.557666i	$-0.188303\pi$
$733$	44.9463	1.66013	0.830066	+	0.557666i	$0.188303\pi$
$734$	0	0
$735$	2.40393	0.0886702
$736$	0	0
$737$	−20.3843	−0.750865
$738$	0	0
$739$	50.9913	1.87574	0.937872	−	0.346980i	$-0.112793\pi$
$739$	50.9913	1.87574	0.937872	+	0.346980i	$0.112793\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.8761	−0.692497	−0.346249	−	0.938143i	$-0.612545\pi$
$743$	−18.8761	−0.692497	−0.346249	+	0.938143i	$0.612545\pi$
$744$	0	0
$745$	−10.1383	−0.371438
$746$	0	0
$747$	45.4940	1.66454
$748$	0	0
$749$	−7.20108	−0.263122
$750$	0	0
$751$	−26.0253	−0.949677	−0.474838	−	0.880073i	$-0.657494\pi$
$751$	−26.0253	−0.949677	−0.474838	+	0.880073i	$0.657494\pi$
$752$	0	0
$753$	1.19540	0.0435629
$754$	0	0
$755$	2.81331	0.102387
$756$	0	0
$757$	−11.9156	−0.433079	−0.216540	−	0.976274i	$-0.569477\pi$
$757$	−11.9156	−0.433079	−0.216540	+	0.976274i	$0.569477\pi$
$758$	0	0
$759$	0.516902	0.0187623
$760$	0	0
$761$	−2.39500	−0.0868186	−0.0434093	−	0.999057i	$-0.513822\pi$
$761$	−2.39500	−0.0868186	−0.0434093	+	0.999057i	$0.513822\pi$
$762$	0	0
$763$	−8.40939	−0.304440
$764$	0	0
$765$	−18.9121	−0.683769
$766$	0	0
$767$	−9.09880	−0.328539
$768$	0	0
$769$	5.32497	0.192023	0.0960117	−	0.995380i	$-0.469391\pi$
$769$	5.32497	0.192023	0.0960117	+	0.995380i	$0.469391\pi$
$770$	0	0
$771$	2.78276	0.100219
$772$	0	0
$773$	24.4973	0.881106	0.440553	−	0.897727i	$-0.354782\pi$
$773$	24.4973	0.881106	0.440553	+	0.897727i	$0.354782\pi$
$774$	0	0
$775$	1.86718	0.0670710
$776$	0	0
$777$	−0.147210	−0.00528113
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−4.09334	−0.146471
$782$	0	0
$783$	−19.4558	−0.695294
$784$	0	0
$785$	−0.960522	−0.0342825
$786$	0	0
$787$	20.5171	0.731356	0.365678	−	0.930741i	$-0.380837\pi$
$787$	20.5171	0.731356	0.365678	+	0.930741i	$0.380837\pi$
$788$	0	0
$789$	5.36792	0.191103
$790$	0	0
$791$	−4.50819	−0.160293
$792$	0	0
$793$	2.27110	0.0806493
$794$	0	0
$795$	−3.59607	−0.127540
$796$	0	0
$797$	23.0144	0.815211	0.407606	−	0.913158i	$-0.366364\pi$
$797$	23.0144	0.815211	0.407606	+	0.913158i	$0.366364\pi$
$798$	0	0
$799$	−60.0164	−2.12323
$800$	0	0
$801$	1.55461	0.0549293
$802$	0	0
$803$	−0.660642	−0.0233136
$804$	0	0
$805$	−0.551136	−0.0194250
$806$	0	0
$807$	4.90666	0.172722
$808$	0	0
$809$	−28.3195	−0.995661	−0.497830	−	0.867274i	$-0.665870\pi$
$809$	−28.3195	−0.995661	−0.497830	+	0.867274i	$0.665870\pi$
$810$	0	0
$811$	−39.5531	−1.38890	−0.694449	−	0.719542i	$-0.744352\pi$
$811$	−39.5531	−1.38890	−0.694449	+	0.719542i	$0.744352\pi$
$812$	0	0
$813$	1.40740	0.0493595
$814$	0	0
$815$	−3.13828	−0.109929
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−1.82224	−0.0636742
$820$	0	0
$821$	30.8026	1.07502	0.537509	−	0.843258i	$-0.319365\pi$
$821$	30.8026	1.07502	0.537509	+	0.843258i	$0.319365\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	0	0
$825$	0.596074	0.0207526
$826$	0	0
$827$	−20.2855	−0.705396	−0.352698	−	0.935737i	$-0.614736\pi$
$827$	−20.2855	−0.705396	−0.352698	+	0.935737i	$0.614736\pi$
$828$	0	0
$829$	−44.0792	−1.53093	−0.765466	−	0.643476i	$-0.777492\pi$
$829$	−44.0792	−1.53093	−0.765466	+	0.643476i	$0.777492\pi$
$830$	0	0
$831$	−8.84755	−0.306918
$832$	0	0
$833$	43.5082	1.50747
$834$	0	0
$835$	−10.6356	−0.368058
$836$	0	0
$837$	−3.99255	−0.138003
$838$	0	0
$839$	−53.9069	−1.86107	−0.930536	−	0.366201i	$-0.880658\pi$
$839$	−53.9069	−1.86107	−0.930536	+	0.366201i	$0.880658\pi$
$840$	0	0
$841$	53.7882	1.85477
$842$	0	0
$843$	−0.181229	−0.00624187
$844$	0	0
$845$	12.0000	0.412813
$846$	0	0
$847$	−5.29095	−0.181799
$848$	0	0
$849$	−5.52435	−0.189595
$850$	0	0
$851$	−0.551136	−0.0188927
$852$	0	0
$853$	27.0648	0.926681	0.463340	−	0.886180i	$-0.346651\pi$
$853$	27.0648	0.926681	0.463340	+	0.886180i	$0.346651\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.35353	0.114554	0.0572772	−	0.998358i	$-0.481758\pi$
$857$	3.35353	0.114554	0.0572772	+	0.998358i	$0.481758\pi$
$858$	0	0
$859$	−53.7058	−1.83242	−0.916209	−	0.400701i	$-0.868767\pi$
$859$	−53.7058	−1.83242	−0.916209	+	0.400701i	$0.868767\pi$
$860$	0	0
$861$	−0.441630	−0.0150507
$862$	0	0
$863$	39.1527	1.33277	0.666386	−	0.745607i	$-0.267840\pi$
$863$	39.1527	1.33277	0.666386	+	0.745607i	$0.267840\pi$
$864$	0	0
$865$	12.3699	0.420589
$866$	0	0
$867$	9.66086	0.328100
$868$	0	0
$869$	−17.5477	−0.595264
$870$	0	0
$871$	−12.4633	−0.422301
$872$	0	0
$873$	−21.1308	−0.715170
$874$	0	0
$875$	−0.635552	−0.0214856
$876$	0	0
$877$	29.6519	1.00127	0.500637	−	0.865657i	$-0.333099\pi$
$877$	29.6519	1.00127	0.500637	+	0.865657i	$0.333099\pi$
$878$	0	0
$879$	4.88702	0.164835
$880$	0	0
$881$	43.2820	1.45821	0.729104	−	0.684403i	$-0.239937\pi$
$881$	43.2820	1.45821	0.729104	+	0.684403i	$0.239937\pi$
$882$	0	0
$883$	56.9660	1.91706	0.958529	−	0.284995i	$-0.0919920\pi$
$883$	56.9660	1.91706	0.958529	+	0.284995i	$0.0919920\pi$
$884$	0	0
$885$	3.31604	0.111468
$886$	0	0
$887$	−20.2316	−0.679312	−0.339656	−	0.940550i	$-0.610311\pi$
$887$	−20.2316	−0.679312	−0.339656	+	0.940550i	$0.610311\pi$
$888$	0	0
$889$	−10.3446	−0.346947
$890$	0	0
$891$	12.7937	0.428604
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−0.324970	−0.0108625
$896$	0	0
$897$	0.316041	0.0105523
$898$	0	0
$899$	16.9891	0.566618
$900$	0	0
$901$	−65.0846	−2.16828
$902$	0	0
$903$	0.917359	0.0305278
$904$	0	0
$905$	8.81331	0.292964
$906$	0	0
$907$	10.0000	0.332045	0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000	0.332045	0.166022	+	0.986122i	$0.446908\pi$
$908$	0	0
$909$	26.7363	0.886789
$910$	0	0
$911$	55.0109	1.82259	0.911297	−	0.411751i	$-0.135083\pi$
$911$	55.0109	1.82259	0.911297	+	0.411751i	$0.135083\pi$
$912$	0	0
$913$	−25.9516	−0.858872
$914$	0	0
$915$	−0.827699	−0.0273629
$916$	0	0
$917$	−6.75945	−0.223217
$918$	0	0
$919$	48.6159	1.60369	0.801846	−	0.597531i	$-0.203852\pi$
$919$	48.6159	1.60369	0.801846	+	0.597531i	$0.203852\pi$
$920$	0	0
$921$	−11.7376	−0.386767
$922$	0	0
$923$	−2.50273	−0.0823783
$924$	0	0
$925$	−0.635552	−0.0208968
$926$	0	0
$927$	−31.1581	−1.02337
$928$	0	0
$929$	11.3215	0.371446	0.185723	−	0.982602i	$-0.440537\pi$
$929$	11.3215	0.371446	0.185723	+	0.982602i	$0.440537\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.50819	0.147591
$934$	0	0
$935$	10.7882	0.352813
$936$	0	0
$937$	−10.4973	−0.342931	−0.171465	−	0.985190i	$-0.554850\pi$
$937$	−10.4973	−0.342931	−0.171465	+	0.985190i	$0.554850\pi$
$938$	0	0
$939$	9.03402	0.294814
$940$	0	0
$941$	2.81877	0.0918893	0.0459447	−	0.998944i	$-0.485370\pi$
$941$	2.81877	0.0918893	0.0459447	+	0.998944i	$0.485370\pi$
$942$	0	0
$943$	−1.65341	−0.0538424
$944$	0	0
$945$	1.35899	0.0442079
$946$	0	0
$947$	−45.5136	−1.47899	−0.739497	−	0.673159i	$-0.764937\pi$
$947$	−45.5136	−1.47899	−0.739497	+	0.673159i	$0.764937\pi$
$948$	0	0
$949$	−0.403926	−0.0131120
$950$	0	0
$951$	−2.66433	−0.0863967
$952$	0	0
$953$	20.7200	0.671186	0.335593	−	0.942007i	$-0.391063\pi$
$953$	20.7200	0.671186	0.335593	+	0.942007i	$0.391063\pi$
$954$	0	0
$955$	−25.1921	−0.815199
$956$	0	0
$957$	5.42356	0.175319
$958$	0	0
$959$	2.22270	0.0717746
$960$	0	0
$961$	−27.5136	−0.887537
$962$	0	0
$963$	32.4864	1.04686
$964$	0	0
$965$	−18.8726	−0.607532
$966$	0	0
$967$	−13.4687	−0.433125	−0.216562	−	0.976269i	$-0.569484\pi$
$967$	−13.4687	−0.433125	−0.216562	+	0.976269i	$0.569484\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	47.0504	1.50992	0.754960	−	0.655771i	$-0.227656\pi$
$971$	47.0504	1.50992	0.754960	+	0.655771i	$0.227656\pi$
$972$	0	0
$973$	0.466721	0.0149624
$974$	0	0
$975$	0.364448	0.0116717
$976$	0	0
$977$	−43.0648	−1.37776	−0.688882	−	0.724873i	$-0.741898\pi$
$977$	−43.0648	−1.37776	−0.688882	+	0.724873i	$0.741898\pi$
$978$	0	0
$979$	−0.886809	−0.0283425
$980$	0	0
$981$	37.9374	1.21125
$982$	0	0
$983$	54.0792	1.72486	0.862429	−	0.506178i	$-0.168942\pi$
$983$	54.0792	1.72486	0.862429	+	0.506178i	$0.168942\pi$
$984$	0	0
$985$	7.36445	0.234651
$986$	0	0
$987$	2.10752	0.0670830
$988$	0	0
$989$	3.43448	0.109210
$990$	0	0
$991$	−41.2711	−1.31102	−0.655510	−	0.755187i	$-0.727546\pi$
$991$	−41.2711	−1.31102	−0.655510	+	0.755187i	$0.727546\pi$
$992$	0	0
$993$	−6.73215	−0.213638
$994$	0	0
$995$	−2.76837	−0.0877634
$996$	0	0
$997$	−1.08987	−0.0345167	−0.0172583	−	0.999851i	$-0.505494\pi$
$997$	−1.08987	−0.0345167	−0.0172583	+	0.999851i	$0.505494\pi$
$998$	0	0
$999$	1.35899	0.0429965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.o.1.2		3
19.8	odd	6		380.2.i.b.121.2	✓	6
19.12	odd	6		380.2.i.b.201.2	yes	6
19.18	odd	2		7220.2.a.n.1.2		3
57.8	even	6		3420.2.t.v.1261.2		6
57.50	even	6		3420.2.t.v.3241.2		6
76.27	even	6		1520.2.q.i.881.2		6
76.31	even	6		1520.2.q.i.961.2		6
95.8	even	12		1900.2.s.c.349.3		12
95.12	even	12		1900.2.s.c.49.3		12
95.27	even	12		1900.2.s.c.349.4		12
95.69	odd	6		1900.2.i.c.201.2		6
95.84	odd	6		1900.2.i.c.501.2		6
95.88	even	12		1900.2.s.c.49.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.b.121.2	✓	6	19.8	odd	6
380.2.i.b.201.2	yes	6	19.12	odd	6
1520.2.q.i.881.2		6	76.27	even	6
1520.2.q.i.961.2		6	76.31	even	6
1900.2.i.c.201.2		6	95.69	odd	6
1900.2.i.c.501.2		6	95.84	odd	6
1900.2.s.c.49.3		12	95.12	even	12
1900.2.s.c.49.4		12	95.88	even	12
1900.2.s.c.349.3		12	95.8	even	12
1900.2.s.c.349.4		12	95.27	even	12
3420.2.t.v.1261.2		6	57.8	even	6
3420.2.t.v.3241.2		6	57.50	even	6
7220.2.a.n.1.2		3	19.18	odd	2
7220.2.a.o.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q+0.364448 q^{3} -1.00000 q^{5} +0.635552 q^{7} -2.86718 q^{9} +O(q^{10})\)	\(q+0.364448 q^{3} -1.00000 q^{5} +0.635552 q^{7} -2.86718 q^{9} +1.63555 q^{11} +1.00000 q^{13} -0.364448 q^{15} -6.59607 q^{17} +0.231626 q^{21} +0.867178 q^{23} +1.00000 q^{25} -2.13828 q^{27} +9.09880 q^{29} +1.86718 q^{31} +0.596074 q^{33} -0.635552 q^{35} -0.635552 q^{37} +0.364448 q^{39} -1.90666 q^{41} +3.96052 q^{43} +2.86718 q^{45} +9.09880 q^{47} -6.59607 q^{49} -2.40393 q^{51} +9.86718 q^{53} -1.63555 q^{55} -9.09880 q^{59} +2.27110 q^{61} -1.82224 q^{63} -1.00000 q^{65} -12.4633 q^{67} +0.316041 q^{69} -2.50273 q^{71} -0.403926 q^{73} +0.364448 q^{75} +1.03948 q^{77} -10.7289 q^{79} +7.82224 q^{81} -15.8672 q^{83} +6.59607 q^{85} +3.31604 q^{87} -0.542208 q^{89} +0.635552 q^{91} +0.680489 q^{93} +7.36991 q^{97} -4.68942 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9}+O(q^{10}) \) 3 * q + q^3 - 3 * q^5 + 2 * q^7 + 8 * q^9	\( 3 q + q^{3} - 3 q^{5} + 2 q^{7} + 8 q^{9} + 5 q^{11} + 3 q^{13} - q^{15} - 3 q^{17} - 16 q^{21} - 14 q^{23} + 3 q^{25} + 10 q^{27} - 6 q^{29} - 11 q^{31} - 15 q^{33} - 2 q^{35} - 2 q^{37} + q^{39} - 6 q^{41} - 5 q^{43} - 8 q^{45} - 6 q^{47} - 3 q^{49} - 24 q^{51} + 13 q^{53} - 5 q^{55} + 6 q^{59} + 7 q^{61} - 5 q^{63} - 3 q^{65} - 4 q^{67} - 15 q^{69} + 9 q^{71} - 18 q^{73} + q^{75} + 20 q^{77} - 32 q^{79} + 23 q^{81} - 31 q^{83} + 3 q^{85} - 6 q^{87} - 2 q^{89} + 2 q^{91} - 14 q^{93} - 11 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 + 2 * q^7 + 8 * q^9 + 5 * q^11 + 3 * q^13 - q^15 - 3 * q^17 - 16 * q^21 - 14 * q^23 + 3 * q^25 + 10 * q^27 - 6 * q^29 - 11 * q^31 - 15 * q^33 - 2 * q^35 - 2 * q^37 + q^39 - 6 * q^41 - 5 * q^43 - 8 * q^45 - 6 * q^47 - 3 * q^49 - 24 * q^51 + 13 * q^53 - 5 * q^55 + 6 * q^59 + 7 * q^61 - 5 * q^63 - 3 * q^65 - 4 * q^67 - 15 * q^69 + 9 * q^71 - 18 * q^73 + q^75 + 20 * q^77 - 32 * q^79 + 23 * q^81 - 31 * q^83 + 3 * q^85 - 6 * q^87 - 2 * q^89 + 2 * q^91 - 14 * q^93 - 11 * q^97 + 3 * q^99

Introduction

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