LMFDB - Embedded newform 6036.2.a.i.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.970233 q^{5} +3.74274 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.970233 q^{5} +3.74274 q^{7} +1.00000 q^{9} -5.32343 q^{11} -0.298043 q^{13} +0.970233 q^{15} -7.82451 q^{17} -3.73515 q^{19} -3.74274 q^{21} +1.49696 q^{23} -4.05865 q^{25} -1.00000 q^{27} +3.78584 q^{29} -2.76588 q^{31} +5.32343 q^{33} -3.63133 q^{35} +0.959694 q^{37} +0.298043 q^{39} +1.24222 q^{41} +0.457677 q^{43} -0.970233 q^{45} +8.90957 q^{47} +7.00812 q^{49} +7.82451 q^{51} +6.31057 q^{53} +5.16497 q^{55} +3.73515 q^{57} -3.00315 q^{59} -2.73984 q^{61} +3.74274 q^{63} +0.289171 q^{65} +14.8495 q^{67} -1.49696 q^{69} -6.26414 q^{71} +7.46514 q^{73} +4.05865 q^{75} -19.9242 q^{77} +5.85609 q^{79} +1.00000 q^{81} +4.58704 q^{83} +7.59160 q^{85} -3.78584 q^{87} +9.45136 q^{89} -1.11550 q^{91} +2.76588 q^{93} +3.62396 q^{95} +17.7241 q^{97} -5.32343 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.970233	−0.433901	−0.216951	−	0.976183i	$-0.569611\pi$
$5$	−0.970233	−0.433901	−0.216951	+	0.976183i	$0.569611\pi$
$6$	0	0
$7$	3.74274	1.41462	0.707312	−	0.706902i	$-0.249908\pi$
$7$	3.74274	1.41462	0.707312	+	0.706902i	$0.249908\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.32343	−1.60507	−0.802537	−	0.596603i	$-0.796517\pi$
$11$	−5.32343	−1.60507	−0.802537	+	0.596603i	$0.796517\pi$
$12$	0	0
$13$	−0.298043	−0.0826622	−0.0413311	−	0.999146i	$-0.513160\pi$
$13$	−0.298043	−0.0826622	−0.0413311	+	0.999146i	$0.513160\pi$
$14$	0	0
$15$	0.970233	0.250513
$16$	0	0
$17$	−7.82451	−1.89772	−0.948861	−	0.315694i	$-0.897763\pi$
$17$	−7.82451	−1.89772	−0.948861	+	0.315694i	$0.897763\pi$
$18$	0	0
$19$	−3.73515	−0.856902	−0.428451	−	0.903565i	$-0.640940\pi$
$19$	−3.73515	−0.856902	−0.428451	+	0.903565i	$0.640940\pi$
$20$	0	0
$21$	−3.74274	−0.816733
$22$	0	0
$23$	1.49696	0.312137	0.156068	−	0.987746i	$-0.450118\pi$
$23$	1.49696	0.312137	0.156068	+	0.987746i	$0.450118\pi$
$24$	0	0
$25$	−4.05865	−0.811730
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.78584	0.703013	0.351507	−	0.936185i	$-0.385669\pi$
$29$	3.78584	0.703013	0.351507	+	0.936185i	$0.385669\pi$
$30$	0	0
$31$	−2.76588	−0.496767	−0.248384	−	0.968662i	$-0.579899\pi$
$31$	−2.76588	−0.496767	−0.248384	+	0.968662i	$0.579899\pi$
$32$	0	0
$33$	5.32343	0.926690
$34$	0	0
$35$	−3.63133	−0.613807
$36$	0	0
$37$	0.959694	0.157773	0.0788864	−	0.996884i	$-0.474864\pi$
$37$	0.959694	0.157773	0.0788864	+	0.996884i	$0.474864\pi$
$38$	0	0
$39$	0.298043	0.0477250
$40$	0	0
$41$	1.24222	0.194002	0.0970012	−	0.995284i	$-0.469075\pi$
$41$	1.24222	0.194002	0.0970012	+	0.995284i	$0.469075\pi$
$42$	0	0
$43$	0.457677	0.0697951	0.0348975	−	0.999391i	$-0.488890\pi$
$43$	0.457677	0.0697951	0.0348975	+	0.999391i	$0.488890\pi$
$44$	0	0
$45$	−0.970233	−0.144634
$46$	0	0
$47$	8.90957	1.29959	0.649797	−	0.760108i	$-0.274854\pi$
$47$	8.90957	1.29959	0.649797	+	0.760108i	$0.274854\pi$
$48$	0	0
$49$	7.00812	1.00116
$50$	0	0
$51$	7.82451	1.09565
$52$	0	0
$53$	6.31057	0.866823	0.433411	−	0.901196i	$-0.357310\pi$
$53$	6.31057	0.866823	0.433411	+	0.901196i	$0.357310\pi$
$54$	0	0
$55$	5.16497	0.696444
$56$	0	0
$57$	3.73515	0.494732
$58$	0	0
$59$	−3.00315	−0.390976	−0.195488	−	0.980706i	$-0.562629\pi$
$59$	−3.00315	−0.390976	−0.195488	+	0.980706i	$0.562629\pi$
$60$	0	0
$61$	−2.73984	−0.350801	−0.175400	−	0.984497i	$-0.556122\pi$
$61$	−2.73984	−0.350801	−0.175400	+	0.984497i	$0.556122\pi$
$62$	0	0
$63$	3.74274	0.471541
$64$	0	0
$65$	0.289171	0.0358672
$66$	0	0
$67$	14.8495	1.81416	0.907079	−	0.420961i	$-0.138307\pi$
$67$	14.8495	1.81416	0.907079	+	0.420961i	$0.138307\pi$
$68$	0	0
$69$	−1.49696	−0.180212
$70$	0	0
$71$	−6.26414	−0.743416	−0.371708	−	0.928350i	$-0.621228\pi$
$71$	−6.26414	−0.743416	−0.371708	+	0.928350i	$0.621228\pi$
$72$	0	0
$73$	7.46514	0.873728	0.436864	−	0.899528i	$-0.356089\pi$
$73$	7.46514	0.873728	0.436864	+	0.899528i	$0.356089\pi$
$74$	0	0
$75$	4.05865	0.468652
$76$	0	0
$77$	−19.9242	−2.27058
$78$	0	0
$79$	5.85609	0.658862	0.329431	−	0.944180i	$-0.393143\pi$
$79$	5.85609	0.658862	0.329431	+	0.944180i	$0.393143\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.58704	0.503493	0.251746	−	0.967793i	$-0.418995\pi$
$83$	4.58704	0.503493	0.251746	+	0.967793i	$0.418995\pi$
$84$	0	0
$85$	7.59160	0.823424
$86$	0	0
$87$	−3.78584	−0.405885
$88$	0	0
$89$	9.45136	1.00184	0.500921	−	0.865493i	$-0.332995\pi$
$89$	9.45136	1.00184	0.500921	+	0.865493i	$0.332995\pi$
$90$	0	0
$91$	−1.11550	−0.116936
$92$	0	0
$93$	2.76588	0.286809
$94$	0	0
$95$	3.62396	0.371811
$96$	0	0
$97$	17.7241	1.79961	0.899803	−	0.436296i	$-0.143710\pi$
$97$	17.7241	1.79961	0.899803	+	0.436296i	$0.143710\pi$
$98$	0	0
$99$	−5.32343	−0.535025
$100$	0	0
$101$	−7.92972	−0.789036	−0.394518	−	0.918888i	$-0.629088\pi$
$101$	−7.92972	−0.789036	−0.394518	+	0.918888i	$0.629088\pi$
$102$	0	0
$103$	3.52201	0.347034	0.173517	−	0.984831i	$-0.444487\pi$
$103$	3.52201	0.347034	0.173517	+	0.984831i	$0.444487\pi$
$104$	0	0
$105$	3.63133	0.354382
$106$	0	0
$107$	−11.3582	−1.09804	−0.549021	−	0.835808i	$-0.684999\pi$
$107$	−11.3582	−1.09804	−0.549021	+	0.835808i	$0.684999\pi$
$108$	0	0
$109$	3.06134	0.293224	0.146612	−	0.989194i	$-0.453163\pi$
$109$	3.06134	0.293224	0.146612	+	0.989194i	$0.453163\pi$
$110$	0	0
$111$	−0.959694	−0.0910902
$112$	0	0
$113$	−18.9256	−1.78038	−0.890188	−	0.455594i	$-0.849427\pi$
$113$	−18.9256	−1.78038	−0.890188	+	0.455594i	$0.849427\pi$
$114$	0	0
$115$	−1.45240	−0.135437
$116$	0	0
$117$	−0.298043	−0.0275541
$118$	0	0
$119$	−29.2851	−2.68456
$120$	0	0
$121$	17.3389	1.57626
$122$	0	0
$123$	−1.24222	−0.112007
$124$	0	0
$125$	8.78900	0.786112
$126$	0	0
$127$	10.0897	0.895312	0.447656	−	0.894206i	$-0.352259\pi$
$127$	10.0897	0.895312	0.447656	+	0.894206i	$0.352259\pi$
$128$	0	0
$129$	−0.457677	−0.0402962
$130$	0	0
$131$	2.69545	0.235503	0.117751	−	0.993043i	$-0.462431\pi$
$131$	2.69545	0.235503	0.117751	+	0.993043i	$0.462431\pi$
$132$	0	0
$133$	−13.9797	−1.21219
$134$	0	0
$135$	0.970233	0.0835044
$136$	0	0
$137$	14.4558	1.23504	0.617521	−	0.786554i	$-0.288137\pi$
$137$	14.4558	1.23504	0.617521	+	0.786554i	$0.288137\pi$
$138$	0	0
$139$	−2.63180	−0.223226	−0.111613	−	0.993752i	$-0.535602\pi$
$139$	−2.63180	−0.223226	−0.111613	+	0.993752i	$0.535602\pi$
$140$	0	0
$141$	−8.90957	−0.750321
$142$	0	0
$143$	1.58661	0.132679
$144$	0	0
$145$	−3.67315	−0.305038
$146$	0	0
$147$	−7.00812	−0.578020
$148$	0	0
$149$	−14.9613	−1.22568	−0.612839	−	0.790208i	$-0.709973\pi$
$149$	−14.9613	−1.22568	−0.612839	+	0.790208i	$0.709973\pi$
$150$	0	0
$151$	20.1181	1.63719	0.818595	−	0.574372i	$-0.194754\pi$
$151$	20.1181	1.63719	0.818595	+	0.574372i	$0.194754\pi$
$152$	0	0
$153$	−7.82451	−0.632574
$154$	0	0
$155$	2.68355	0.215548
$156$	0	0
$157$	6.89091	0.549954	0.274977	−	0.961451i	$-0.411330\pi$
$157$	6.89091	0.549954	0.274977	+	0.961451i	$0.411330\pi$
$158$	0	0
$159$	−6.31057	−0.500460
$160$	0	0
$161$	5.60272	0.441556
$162$	0	0
$163$	−11.4193	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$163$	−11.4193	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$164$	0	0
$165$	−5.16497	−0.402092
$166$	0	0
$167$	13.4307	1.03930	0.519650	−	0.854379i	$-0.326062\pi$
$167$	13.4307	1.03930	0.519650	+	0.854379i	$0.326062\pi$
$168$	0	0
$169$	−12.9112	−0.993167
$170$	0	0
$171$	−3.73515	−0.285634
$172$	0	0
$173$	−6.82988	−0.519266	−0.259633	−	0.965707i	$-0.583602\pi$
$173$	−6.82988	−0.519266	−0.259633	+	0.965707i	$0.583602\pi$
$174$	0	0
$175$	−15.1905	−1.14829
$176$	0	0
$177$	3.00315	0.225730
$178$	0	0
$179$	2.56472	0.191696	0.0958482	−	0.995396i	$-0.469444\pi$
$179$	2.56472	0.191696	0.0958482	+	0.995396i	$0.469444\pi$
$180$	0	0
$181$	21.9217	1.62943	0.814715	−	0.579862i	$-0.196893\pi$
$181$	21.9217	1.62943	0.814715	+	0.579862i	$0.196893\pi$
$182$	0	0
$183$	2.73984	0.202535
$184$	0	0
$185$	−0.931127	−0.0684578
$186$	0	0
$187$	41.6532	3.04598
$188$	0	0
$189$	−3.74274	−0.272244
$190$	0	0
$191$	−5.26149	−0.380708	−0.190354	−	0.981715i	$-0.560964\pi$
$191$	−5.26149	−0.380708	−0.190354	+	0.981715i	$0.560964\pi$
$192$	0	0
$193$	0.751439	0.0540898	0.0270449	−	0.999634i	$-0.491390\pi$
$193$	0.751439	0.0540898	0.0270449	+	0.999634i	$0.491390\pi$
$194$	0	0
$195$	−0.289171	−0.0207080
$196$	0	0
$197$	−17.2322	−1.22775	−0.613873	−	0.789405i	$-0.710389\pi$
$197$	−17.2322	−1.22775	−0.613873	+	0.789405i	$0.710389\pi$
$198$	0	0
$199$	−15.0929	−1.06991	−0.534954	−	0.844881i	$-0.679671\pi$
$199$	−15.0929	−1.06991	−0.534954	+	0.844881i	$0.679671\pi$
$200$	0	0
$201$	−14.8495	−1.04740
$202$	0	0
$203$	14.1694	0.994499
$204$	0	0
$205$	−1.20524	−0.0841780
$206$	0	0
$207$	1.49696	0.104046
$208$	0	0
$209$	19.8838	1.37539
$210$	0	0
$211$	5.52859	0.380604	0.190302	−	0.981726i	$-0.439053\pi$
$211$	5.52859	0.380604	0.190302	+	0.981726i	$0.439053\pi$
$212$	0	0
$213$	6.26414	0.429212
$214$	0	0
$215$	−0.444053	−0.0302842
$216$	0	0
$217$	−10.3520	−0.702739
$218$	0	0
$219$	−7.46514	−0.504447
$220$	0	0
$221$	2.33204	0.156870
$222$	0	0
$223$	−12.3135	−0.824575	−0.412288	−	0.911054i	$-0.635270\pi$
$223$	−12.3135	−0.824575	−0.412288	+	0.911054i	$0.635270\pi$
$224$	0	0
$225$	−4.05865	−0.270577
$226$	0	0
$227$	7.64652	0.507518	0.253759	−	0.967268i	$-0.418333\pi$
$227$	7.64652	0.507518	0.253759	+	0.967268i	$0.418333\pi$
$228$	0	0
$229$	5.52010	0.364778	0.182389	−	0.983226i	$-0.441617\pi$
$229$	5.52010	0.364778	0.182389	+	0.983226i	$0.441617\pi$
$230$	0	0
$231$	19.9242	1.31092
$232$	0	0
$233$	12.6031	0.825658	0.412829	−	0.910808i	$-0.364541\pi$
$233$	12.6031	0.825658	0.412829	+	0.910808i	$0.364541\pi$
$234$	0	0
$235$	−8.64436	−0.563896
$236$	0	0
$237$	−5.85609	−0.380394
$238$	0	0
$239$	−13.6786	−0.884795	−0.442398	−	0.896819i	$-0.645872\pi$
$239$	−13.6786	−0.884795	−0.442398	+	0.896819i	$0.645872\pi$
$240$	0	0
$241$	15.4794	0.997118	0.498559	−	0.866856i	$-0.333863\pi$
$241$	15.4794	0.997118	0.498559	+	0.866856i	$0.333863\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−6.79951	−0.434405
$246$	0	0
$247$	1.11323	0.0708334
$248$	0	0
$249$	−4.58704	−0.290692
$250$	0	0
$251$	0.134773	0.00850677	0.00425338	−	0.999991i	$-0.498646\pi$
$251$	0.134773	0.00850677	0.00425338	+	0.999991i	$0.498646\pi$
$252$	0	0
$253$	−7.96893	−0.501002
$254$	0	0
$255$	−7.59160	−0.475404
$256$	0	0
$257$	13.5452	0.844924	0.422462	−	0.906381i	$-0.361166\pi$
$257$	13.5452	0.844924	0.422462	+	0.906381i	$0.361166\pi$
$258$	0	0
$259$	3.59189	0.223189
$260$	0	0
$261$	3.78584	0.234338
$262$	0	0
$263$	−15.5675	−0.959931	−0.479965	−	0.877287i	$-0.659351\pi$
$263$	−15.5675	−0.959931	−0.479965	+	0.877287i	$0.659351\pi$
$264$	0	0
$265$	−6.12272	−0.376116
$266$	0	0
$267$	−9.45136	−0.578414
$268$	0	0
$269$	−25.0315	−1.52620	−0.763098	−	0.646283i	$-0.776323\pi$
$269$	−25.0315	−1.52620	−0.763098	+	0.646283i	$0.776323\pi$
$270$	0	0
$271$	−1.28841	−0.0782651	−0.0391325	−	0.999234i	$-0.512459\pi$
$271$	−1.28841	−0.0782651	−0.0391325	+	0.999234i	$0.512459\pi$
$272$	0	0
$273$	1.11550	0.0675130
$274$	0	0
$275$	21.6059	1.30289
$276$	0	0
$277$	−2.22897	−0.133926	−0.0669629	−	0.997755i	$-0.521331\pi$
$277$	−2.22897	−0.133926	−0.0669629	+	0.997755i	$0.521331\pi$
$278$	0	0
$279$	−2.76588	−0.165589
$280$	0	0
$281$	21.5713	1.28683	0.643417	−	0.765516i	$-0.277516\pi$
$281$	21.5713	1.28683	0.643417	+	0.765516i	$0.277516\pi$
$282$	0	0
$283$	6.93160	0.412041	0.206020	−	0.978548i	$-0.433949\pi$
$283$	6.93160	0.412041	0.206020	+	0.978548i	$0.433949\pi$
$284$	0	0
$285$	−3.62396	−0.214665
$286$	0	0
$287$	4.64932	0.274441
$288$	0	0
$289$	44.2229	2.60135
$290$	0	0
$291$	−17.7241	−1.03900
$292$	0	0
$293$	19.4521	1.13640	0.568202	−	0.822889i	$-0.307639\pi$
$293$	19.4521	1.13640	0.568202	+	0.822889i	$0.307639\pi$
$294$	0	0
$295$	2.91375	0.169645
$296$	0	0
$297$	5.32343	0.308897
$298$	0	0
$299$	−0.446157	−0.0258019
$300$	0	0
$301$	1.71297	0.0987338
$302$	0	0
$303$	7.92972	0.455550
$304$	0	0
$305$	2.65828	0.152213
$306$	0	0
$307$	−11.0657	−0.631551	−0.315775	−	0.948834i	$-0.602265\pi$
$307$	−11.0657	−0.631551	−0.315775	+	0.948834i	$0.602265\pi$
$308$	0	0
$309$	−3.52201	−0.200360
$310$	0	0
$311$	3.72332	0.211130	0.105565	−	0.994412i	$-0.466335\pi$
$311$	3.72332	0.211130	0.105565	+	0.994412i	$0.466335\pi$
$312$	0	0
$313$	18.1184	1.02411	0.512057	−	0.858952i	$-0.328884\pi$
$313$	18.1184	1.02411	0.512057	+	0.858952i	$0.328884\pi$
$314$	0	0
$315$	−3.63133	−0.204602
$316$	0	0
$317$	10.7581	0.604233	0.302116	−	0.953271i	$-0.402307\pi$
$317$	10.7581	0.604233	0.302116	+	0.953271i	$0.402307\pi$
$318$	0	0
$319$	−20.1537	−1.12839
$320$	0	0
$321$	11.3582	0.633955
$322$	0	0
$323$	29.2257	1.62616
$324$	0	0
$325$	1.20965	0.0670993
$326$	0	0
$327$	−3.06134	−0.169293
$328$	0	0
$329$	33.3462	1.83844
$330$	0	0
$331$	19.8675	1.09201	0.546007	−	0.837781i	$-0.316147\pi$
$331$	19.8675	1.09201	0.546007	+	0.837781i	$0.316147\pi$
$332$	0	0
$333$	0.959694	0.0525909
$334$	0	0
$335$	−14.4075	−0.787166
$336$	0	0
$337$	10.7228	0.584107	0.292053	−	0.956402i	$-0.405661\pi$
$337$	10.7228	0.584107	0.292053	+	0.956402i	$0.405661\pi$
$338$	0	0
$339$	18.9256	1.02790
$340$	0	0
$341$	14.7240	0.797348
$342$	0	0
$343$	0.0303972	0.00164129
$344$	0	0
$345$	1.45240	0.0781943
$346$	0	0
$347$	−15.1013	−0.810681	−0.405340	−	0.914166i	$-0.632847\pi$
$347$	−15.1013	−0.810681	−0.405340	+	0.914166i	$0.632847\pi$
$348$	0	0
$349$	19.3068	1.03347	0.516734	−	0.856146i	$-0.327148\pi$
$349$	19.3068	1.03347	0.516734	+	0.856146i	$0.327148\pi$
$350$	0	0
$351$	0.298043	0.0159083
$352$	0	0
$353$	30.2247	1.60870	0.804350	−	0.594156i	$-0.202514\pi$
$353$	30.2247	1.60870	0.804350	+	0.594156i	$0.202514\pi$
$354$	0	0
$355$	6.07767	0.322569
$356$	0	0
$357$	29.2851	1.54993
$358$	0	0
$359$	−3.49088	−0.184241	−0.0921207	−	0.995748i	$-0.529365\pi$
$359$	−3.49088	−0.184241	−0.0921207	+	0.995748i	$0.529365\pi$
$360$	0	0
$361$	−5.04867	−0.265720
$362$	0	0
$363$	−17.3389	−0.910055
$364$	0	0
$365$	−7.24292	−0.379112
$366$	0	0
$367$	21.7768	1.13674	0.568370	−	0.822773i	$-0.307574\pi$
$367$	21.7768	1.13674	0.568370	+	0.822773i	$0.307574\pi$
$368$	0	0
$369$	1.24222	0.0646675
$370$	0	0
$371$	23.6188	1.22623
$372$	0	0
$373$	−10.9810	−0.568573	−0.284287	−	0.958739i	$-0.591757\pi$
$373$	−10.9810	−0.568573	−0.284287	+	0.958739i	$0.591757\pi$
$374$	0	0
$375$	−8.78900	−0.453862
$376$	0	0
$377$	−1.12834	−0.0581126
$378$	0	0
$379$	−24.6997	−1.26874	−0.634370	−	0.773030i	$-0.718740\pi$
$379$	−24.6997	−1.26874	−0.634370	+	0.773030i	$0.718740\pi$
$380$	0	0
$381$	−10.0897	−0.516909
$382$	0	0
$383$	21.0750	1.07688	0.538442	−	0.842663i	$-0.319013\pi$
$383$	21.0750	1.07688	0.538442	+	0.842663i	$0.319013\pi$
$384$	0	0
$385$	19.3311	0.985206
$386$	0	0
$387$	0.457677	0.0232650
$388$	0	0
$389$	−8.39614	−0.425701	−0.212851	−	0.977085i	$-0.568275\pi$
$389$	−8.39614	−0.425701	−0.212851	+	0.977085i	$0.568275\pi$
$390$	0	0
$391$	−11.7129	−0.592349
$392$	0	0
$393$	−2.69545	−0.135967
$394$	0	0
$395$	−5.68177	−0.285881
$396$	0	0
$397$	9.59364	0.481491	0.240745	−	0.970588i	$-0.422608\pi$
$397$	9.59364	0.481491	0.240745	+	0.970588i	$0.422608\pi$
$398$	0	0
$399$	13.9797	0.699860
$400$	0	0
$401$	8.30491	0.414727	0.207364	−	0.978264i	$-0.433512\pi$
$401$	8.30491	0.414727	0.207364	+	0.978264i	$0.433512\pi$
$402$	0	0
$403$	0.824351	0.0410639
$404$	0	0
$405$	−0.970233	−0.0482113
$406$	0	0
$407$	−5.10886	−0.253237
$408$	0	0
$409$	−14.4707	−0.715529	−0.357764	−	0.933812i	$-0.616461\pi$
$409$	−14.4707	−0.715529	−0.357764	+	0.933812i	$0.616461\pi$
$410$	0	0
$411$	−14.4558	−0.713052
$412$	0	0
$413$	−11.2400	−0.553084
$414$	0	0
$415$	−4.45050	−0.218466
$416$	0	0
$417$	2.63180	0.128880
$418$	0	0
$419$	−7.40307	−0.361664	−0.180832	−	0.983514i	$-0.557879\pi$
$419$	−7.40307	−0.361664	−0.180832	+	0.983514i	$0.557879\pi$
$420$	0	0
$421$	−9.57152	−0.466487	−0.233244	−	0.972418i	$-0.574934\pi$
$421$	−9.57152	−0.466487	−0.233244	+	0.972418i	$0.574934\pi$
$422$	0	0
$423$	8.90957	0.433198
$424$	0	0
$425$	31.7569	1.54044
$426$	0	0
$427$	−10.2545	−0.496251
$428$	0	0
$429$	−1.58661	−0.0766022
$430$	0	0
$431$	−15.8802	−0.764922	−0.382461	−	0.923972i	$-0.624923\pi$
$431$	−15.8802	−0.764922	−0.382461	+	0.923972i	$0.624923\pi$
$432$	0	0
$433$	−5.04520	−0.242457	−0.121228	−	0.992625i	$-0.538683\pi$
$433$	−5.04520	−0.242457	−0.121228	+	0.992625i	$0.538683\pi$
$434$	0	0
$435$	3.67315	0.176114
$436$	0	0
$437$	−5.59135	−0.267470
$438$	0	0
$439$	21.0761	1.00591	0.502953	−	0.864314i	$-0.332247\pi$
$439$	21.0761	1.00591	0.502953	+	0.864314i	$0.332247\pi$
$440$	0	0
$441$	7.00812	0.333720
$442$	0	0
$443$	−22.4139	−1.06492	−0.532458	−	0.846457i	$-0.678731\pi$
$443$	−22.4139	−1.06492	−0.532458	+	0.846457i	$0.678731\pi$
$444$	0	0
$445$	−9.17003	−0.434701
$446$	0	0
$447$	14.9613	0.707645
$448$	0	0
$449$	25.1571	1.18724	0.593618	−	0.804747i	$-0.297699\pi$
$449$	25.1571	1.18724	0.593618	+	0.804747i	$0.297699\pi$
$450$	0	0
$451$	−6.61288	−0.311388
$452$	0	0
$453$	−20.1181	−0.945232
$454$	0	0
$455$	1.08229	0.0507387
$456$	0	0
$457$	26.3626	1.23319	0.616596	−	0.787280i	$-0.288511\pi$
$457$	26.3626	1.23319	0.616596	+	0.787280i	$0.288511\pi$
$458$	0	0
$459$	7.82451	0.365217
$460$	0	0
$461$	−24.9810	−1.16348	−0.581740	−	0.813375i	$-0.697628\pi$
$461$	−24.9810	−1.16348	−0.581740	+	0.813375i	$0.697628\pi$
$462$	0	0
$463$	27.9821	1.30044	0.650218	−	0.759748i	$-0.274678\pi$
$463$	27.9821	1.30044	0.650218	+	0.759748i	$0.274678\pi$
$464$	0	0
$465$	−2.68355	−0.124447
$466$	0	0
$467$	29.2586	1.35393	0.676964	−	0.736016i	$-0.263295\pi$
$467$	29.2586	1.35393	0.676964	+	0.736016i	$0.263295\pi$
$468$	0	0
$469$	55.5779	2.56635
$470$	0	0
$471$	−6.89091	−0.317516
$472$	0	0
$473$	−2.43641	−0.112026
$474$	0	0
$475$	15.1596	0.695572
$476$	0	0
$477$	6.31057	0.288941
$478$	0	0
$479$	18.5009	0.845327	0.422663	−	0.906287i	$-0.361095\pi$
$479$	18.5009	0.845327	0.422663	+	0.906287i	$0.361095\pi$
$480$	0	0
$481$	−0.286030	−0.0130418
$482$	0	0
$483$	−5.60272	−0.254932
$484$	0	0
$485$	−17.1965	−0.780852
$486$	0	0
$487$	−10.5977	−0.480225	−0.240113	−	0.970745i	$-0.577184\pi$
$487$	−10.5977	−0.480225	−0.240113	+	0.970745i	$0.577184\pi$
$488$	0	0
$489$	11.4193	0.516398
$490$	0	0
$491$	−7.28208	−0.328636	−0.164318	−	0.986407i	$-0.552542\pi$
$491$	−7.28208	−0.328636	−0.164318	+	0.986407i	$0.552542\pi$
$492$	0	0
$493$	−29.6223	−1.33412
$494$	0	0
$495$	5.16497	0.232148
$496$	0	0
$497$	−23.4451	−1.05165
$498$	0	0
$499$	−7.94263	−0.355561	−0.177781	−	0.984070i	$-0.556892\pi$
$499$	−7.94263	−0.355561	−0.177781	+	0.984070i	$0.556892\pi$
$500$	0	0
$501$	−13.4307	−0.600041
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	7.69367	0.342364
$506$	0	0
$507$	12.9112	0.573405
$508$	0	0
$509$	36.8195	1.63199	0.815997	−	0.578056i	$-0.196189\pi$
$509$	36.8195	1.63199	0.815997	+	0.578056i	$0.196189\pi$
$510$	0	0
$511$	27.9401	1.23600
$512$	0	0
$513$	3.73515	0.164911
$514$	0	0
$515$	−3.41717	−0.150579
$516$	0	0
$517$	−47.4294	−2.08594
$518$	0	0
$519$	6.82988	0.299798
$520$	0	0
$521$	−14.1581	−0.620277	−0.310139	−	0.950691i	$-0.600375\pi$
$521$	−14.1581	−0.620277	−0.310139	+	0.950691i	$0.600375\pi$
$522$	0	0
$523$	0.736288	0.0321956	0.0160978	−	0.999870i	$-0.494876\pi$
$523$	0.736288	0.0321956	0.0160978	+	0.999870i	$0.494876\pi$
$524$	0	0
$525$	15.1905	0.662967
$526$	0	0
$527$	21.6417	0.942726
$528$	0	0
$529$	−20.7591	−0.902571
$530$	0	0
$531$	−3.00315	−0.130325
$532$	0	0
$533$	−0.370235	−0.0160367
$534$	0	0
$535$	11.0201	0.476442
$536$	0	0
$537$	−2.56472	−0.110676
$538$	0	0
$539$	−37.3072	−1.60694
$540$	0	0
$541$	31.7599	1.36547	0.682733	−	0.730668i	$-0.260791\pi$
$541$	31.7599	1.36547	0.682733	+	0.730668i	$0.260791\pi$
$542$	0	0
$543$	−21.9217	−0.940752
$544$	0	0
$545$	−2.97022	−0.127230
$546$	0	0
$547$	−4.83682	−0.206808	−0.103404	−	0.994639i	$-0.532973\pi$
$547$	−4.83682	−0.206808	−0.103404	+	0.994639i	$0.532973\pi$
$548$	0	0
$549$	−2.73984	−0.116934
$550$	0	0
$551$	−14.1407	−0.602413
$552$	0	0
$553$	21.9178	0.932041
$554$	0	0
$555$	0.931127	0.0395242
$556$	0	0
$557$	−3.19223	−0.135259	−0.0676296	−	0.997710i	$-0.521544\pi$
$557$	−3.19223	−0.135259	−0.0676296	+	0.997710i	$0.521544\pi$
$558$	0	0
$559$	−0.136407	−0.00576941
$560$	0	0
$561$	−41.6532	−1.75860
$562$	0	0
$563$	18.9982	0.800679	0.400340	−	0.916367i	$-0.368892\pi$
$563$	18.9982	0.800679	0.400340	+	0.916367i	$0.368892\pi$
$564$	0	0
$565$	18.3623	0.772507
$566$	0	0
$567$	3.74274	0.157180
$568$	0	0
$569$	15.6807	0.657369	0.328684	−	0.944440i	$-0.393395\pi$
$569$	15.6807	0.657369	0.328684	+	0.944440i	$0.393395\pi$
$570$	0	0
$571$	−34.1347	−1.42849	−0.714247	−	0.699894i	$-0.753231\pi$
$571$	−34.1347	−1.42849	−0.714247	+	0.699894i	$0.753231\pi$
$572$	0	0
$573$	5.26149	0.219802
$574$	0	0
$575$	−6.07561	−0.253371
$576$	0	0
$577$	19.8710	0.827240	0.413620	−	0.910450i	$-0.364264\pi$
$577$	19.8710	0.827240	0.413620	+	0.910450i	$0.364264\pi$
$578$	0	0
$579$	−0.751439	−0.0312287
$580$	0	0
$581$	17.1681	0.712253
$582$	0	0
$583$	−33.5938	−1.39131
$584$	0	0
$585$	0.289171	0.0119557
$586$	0	0
$587$	20.6230	0.851203	0.425602	−	0.904911i	$-0.360063\pi$
$587$	20.6230	0.851203	0.425602	+	0.904911i	$0.360063\pi$
$588$	0	0
$589$	10.3310	0.425681
$590$	0	0
$591$	17.2322	0.708839
$592$	0	0
$593$	22.0694	0.906280	0.453140	−	0.891439i	$-0.350304\pi$
$593$	22.0694	0.906280	0.453140	+	0.891439i	$0.350304\pi$
$594$	0	0
$595$	28.4134	1.16484
$596$	0	0
$597$	15.0929	0.617712
$598$	0	0
$599$	31.1111	1.27116	0.635582	−	0.772033i	$-0.280760\pi$
$599$	31.1111	1.27116	0.635582	+	0.772033i	$0.280760\pi$
$600$	0	0
$601$	−24.6052	−1.00367	−0.501833	−	0.864965i	$-0.667341\pi$
$601$	−24.6052	−1.00367	−0.501833	+	0.864965i	$0.667341\pi$
$602$	0	0
$603$	14.8495	0.604719
$604$	0	0
$605$	−16.8228	−0.683942
$606$	0	0
$607$	5.52096	0.224089	0.112044	−	0.993703i	$-0.464260\pi$
$607$	5.52096	0.224089	0.112044	+	0.993703i	$0.464260\pi$
$608$	0	0
$609$	−14.1694	−0.574174
$610$	0	0
$611$	−2.65543	−0.107427
$612$	0	0
$613$	−29.0268	−1.17238	−0.586190	−	0.810174i	$-0.699373\pi$
$613$	−29.0268	−1.17238	−0.586190	+	0.810174i	$0.699373\pi$
$614$	0	0
$615$	1.20524	0.0486002
$616$	0	0
$617$	14.5378	0.585270	0.292635	−	0.956224i	$-0.405468\pi$
$617$	14.5378	0.585270	0.292635	+	0.956224i	$0.405468\pi$
$618$	0	0
$619$	−26.5159	−1.06576	−0.532882	−	0.846190i	$-0.678891\pi$
$619$	−26.5159	−1.06576	−0.532882	+	0.846190i	$0.678891\pi$
$620$	0	0
$621$	−1.49696	−0.0600707
$622$	0	0
$623$	35.3740	1.41723
$624$	0	0
$625$	11.7659	0.470634
$626$	0	0
$627$	−19.8838	−0.794082
$628$	0	0
$629$	−7.50914	−0.299409
$630$	0	0
$631$	16.4460	0.654707	0.327353	−	0.944902i	$-0.393843\pi$
$631$	16.4460	0.654707	0.327353	+	0.944902i	$0.393843\pi$
$632$	0	0
$633$	−5.52859	−0.219742
$634$	0	0
$635$	−9.78932	−0.388477
$636$	0	0
$637$	−2.08872	−0.0827581
$638$	0	0
$639$	−6.26414	−0.247805
$640$	0	0
$641$	−17.0432	−0.673166	−0.336583	−	0.941654i	$-0.609271\pi$
$641$	−17.0432	−0.673166	−0.336583	+	0.941654i	$0.609271\pi$
$642$	0	0
$643$	−22.7389	−0.896736	−0.448368	−	0.893849i	$-0.647995\pi$
$643$	−22.7389	−0.896736	−0.448368	+	0.893849i	$0.647995\pi$
$644$	0	0
$645$	0.444053	0.0174846
$646$	0	0
$647$	−36.2181	−1.42388	−0.711940	−	0.702240i	$-0.752183\pi$
$647$	−36.2181	−1.42388	−0.711940	+	0.702240i	$0.752183\pi$
$648$	0	0
$649$	15.9870	0.627546
$650$	0	0
$651$	10.3520	0.405726
$652$	0	0
$653$	−5.28226	−0.206711	−0.103355	−	0.994644i	$-0.532958\pi$
$653$	−5.28226	−0.206711	−0.103355	+	0.994644i	$0.532958\pi$
$654$	0	0
$655$	−2.61522	−0.102185
$656$	0	0
$657$	7.46514	0.291243
$658$	0	0
$659$	−36.4887	−1.42140	−0.710699	−	0.703496i	$-0.751621\pi$
$659$	−36.4887	−1.42140	−0.710699	+	0.703496i	$0.751621\pi$
$660$	0	0
$661$	−1.99609	−0.0776390	−0.0388195	−	0.999246i	$-0.512360\pi$
$661$	−1.99609	−0.0776390	−0.0388195	+	0.999246i	$0.512360\pi$
$662$	0	0
$663$	−2.33204	−0.0905689
$664$	0	0
$665$	13.5636	0.525972
$666$	0	0
$667$	5.66723	0.219436
$668$	0	0
$669$	12.3135	0.476069
$670$	0	0
$671$	14.5853	0.563061
$672$	0	0
$673$	39.8922	1.53773	0.768865	−	0.639411i	$-0.220822\pi$
$673$	39.8922	1.53773	0.768865	+	0.639411i	$0.220822\pi$
$674$	0	0
$675$	4.05865	0.156217
$676$	0	0
$677$	12.0538	0.463267	0.231634	−	0.972803i	$-0.425593\pi$
$677$	12.0538	0.463267	0.231634	+	0.972803i	$0.425593\pi$
$678$	0	0
$679$	66.3366	2.54577
$680$	0	0
$681$	−7.64652	−0.293015
$682$	0	0
$683$	11.7370	0.449103	0.224552	−	0.974462i	$-0.427908\pi$
$683$	11.7370	0.449103	0.224552	+	0.974462i	$0.427908\pi$
$684$	0	0
$685$	−14.0255	−0.535887
$686$	0	0
$687$	−5.52010	−0.210605
$688$	0	0
$689$	−1.88082	−0.0716535
$690$	0	0
$691$	4.15214	0.157955	0.0789774	−	0.996876i	$-0.474834\pi$
$691$	4.15214	0.157955	0.0789774	+	0.996876i	$0.474834\pi$
$692$	0	0
$693$	−19.9242	−0.756858
$694$	0	0
$695$	2.55346	0.0968581
$696$	0	0
$697$	−9.71978	−0.368163
$698$	0	0
$699$	−12.6031	−0.476694
$700$	0	0
$701$	−32.6446	−1.23297	−0.616485	−	0.787367i	$-0.711444\pi$
$701$	−32.6446	−1.23297	−0.616485	+	0.787367i	$0.711444\pi$
$702$	0	0
$703$	−3.58460	−0.135196
$704$	0	0
$705$	8.64436	0.325565
$706$	0	0
$707$	−29.6789	−1.11619
$708$	0	0
$709$	22.1300	0.831110	0.415555	−	0.909568i	$-0.363587\pi$
$709$	22.1300	0.831110	0.415555	+	0.909568i	$0.363587\pi$
$710$	0	0
$711$	5.85609	0.219621
$712$	0	0
$713$	−4.14040	−0.155059
$714$	0	0
$715$	−1.53938	−0.0575696
$716$	0	0
$717$	13.6786	0.510837
$718$	0	0
$719$	−36.3052	−1.35395	−0.676977	−	0.736004i	$-0.736710\pi$
$719$	−36.3052	−1.35395	−0.676977	+	0.736004i	$0.736710\pi$
$720$	0	0
$721$	13.1820	0.490923
$722$	0	0
$723$	−15.4794	−0.575686
$724$	0	0
$725$	−15.3654	−0.570657
$726$	0	0
$727$	−25.8308	−0.958011	−0.479006	−	0.877812i	$-0.659003\pi$
$727$	−25.8308	−0.958011	−0.479006	+	0.877812i	$0.659003\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.58110	−0.132452
$732$	0	0
$733$	17.0373	0.629286	0.314643	−	0.949210i	$-0.398115\pi$
$733$	17.0373	0.629286	0.314643	+	0.949210i	$0.398115\pi$
$734$	0	0
$735$	6.79951	0.250804
$736$	0	0
$737$	−79.0503	−2.91186
$738$	0	0
$739$	−25.2451	−0.928657	−0.464328	−	0.885663i	$-0.653704\pi$
$739$	−25.2451	−0.928657	−0.464328	+	0.885663i	$0.653704\pi$
$740$	0	0
$741$	−1.11323	−0.0408957
$742$	0	0
$743$	−39.1984	−1.43805	−0.719024	−	0.694985i	$-0.755411\pi$
$743$	−39.1984	−1.43805	−0.719024	+	0.694985i	$0.755411\pi$
$744$	0	0
$745$	14.5160	0.531823
$746$	0	0
$747$	4.58704	0.167831
$748$	0	0
$749$	−42.5110	−1.55332
$750$	0	0
$751$	4.18296	0.152638	0.0763192	−	0.997083i	$-0.475683\pi$
$751$	4.18296	0.152638	0.0763192	+	0.997083i	$0.475683\pi$
$752$	0	0
$753$	−0.134773	−0.00491139
$754$	0	0
$755$	−19.5193	−0.710379
$756$	0	0
$757$	−11.4099	−0.414699	−0.207349	−	0.978267i	$-0.566484\pi$
$757$	−11.4099	−0.414699	−0.207349	+	0.978267i	$0.566484\pi$
$758$	0	0
$759$	7.96893	0.289254
$760$	0	0
$761$	−10.6881	−0.387445	−0.193722	−	0.981056i	$-0.562056\pi$
$761$	−10.6881	−0.387445	−0.193722	+	0.981056i	$0.562056\pi$
$762$	0	0
$763$	11.4578	0.414801
$764$	0	0
$765$	7.59160	0.274475
$766$	0	0
$767$	0.895066	0.0323190
$768$	0	0
$769$	−17.7015	−0.638332	−0.319166	−	0.947699i	$-0.603403\pi$
$769$	−17.7015	−0.638332	−0.319166	+	0.947699i	$0.603403\pi$
$770$	0	0
$771$	−13.5452	−0.487817
$772$	0	0
$773$	5.70124	0.205059	0.102530	−	0.994730i	$-0.467306\pi$
$773$	5.70124	0.205059	0.102530	+	0.994730i	$0.467306\pi$
$774$	0	0
$775$	11.2257	0.403241
$776$	0	0
$777$	−3.59189	−0.128858
$778$	0	0
$779$	−4.63988	−0.166241
$780$	0	0
$781$	33.3467	1.19324
$782$	0	0
$783$	−3.78584	−0.135295
$784$	0	0
$785$	−6.68579	−0.238626
$786$	0	0
$787$	−49.5086	−1.76479	−0.882396	−	0.470507i	$-0.844071\pi$
$787$	−49.5086	−1.76479	−0.882396	+	0.470507i	$0.844071\pi$
$788$	0	0
$789$	15.5675	0.554216
$790$	0	0
$791$	−70.8338	−2.51856
$792$	0	0
$793$	0.816590	0.0289979
$794$	0	0
$795$	6.12272	0.217150
$796$	0	0
$797$	−16.5838	−0.587429	−0.293714	−	0.955893i	$-0.594891\pi$
$797$	−16.5838	−0.587429	−0.293714	+	0.955893i	$0.594891\pi$
$798$	0	0
$799$	−69.7130	−2.46627
$800$	0	0
$801$	9.45136	0.333947
$802$	0	0
$803$	−39.7401	−1.40240
$804$	0	0
$805$	−5.43594	−0.191592
$806$	0	0
$807$	25.0315	0.881149
$808$	0	0
$809$	20.5951	0.724085	0.362042	−	0.932162i	$-0.382080\pi$
$809$	20.5951	0.724085	0.362042	+	0.932162i	$0.382080\pi$
$810$	0	0
$811$	36.3640	1.27691	0.638457	−	0.769657i	$-0.279573\pi$
$811$	36.3640	1.27691	0.638457	+	0.769657i	$0.279573\pi$
$812$	0	0
$813$	1.28841	0.0451864
$814$	0	0
$815$	11.0794	0.388093
$816$	0	0
$817$	−1.70949	−0.0598075
$818$	0	0
$819$	−1.11550	−0.0389786
$820$	0	0
$821$	12.0061	0.419014	0.209507	−	0.977807i	$-0.432814\pi$
$821$	12.0061	0.419014	0.209507	+	0.977807i	$0.432814\pi$
$822$	0	0
$823$	−41.7419	−1.45503	−0.727516	−	0.686091i	$-0.759325\pi$
$823$	−41.7419	−1.45503	−0.727516	+	0.686091i	$0.759325\pi$
$824$	0	0
$825$	−21.6059	−0.752221
$826$	0	0
$827$	35.1880	1.22361	0.611803	−	0.791010i	$-0.290445\pi$
$827$	35.1880	1.22361	0.611803	+	0.791010i	$0.290445\pi$
$828$	0	0
$829$	26.2602	0.912053	0.456026	−	0.889966i	$-0.349272\pi$
$829$	26.2602	0.912053	0.456026	+	0.889966i	$0.349272\pi$
$830$	0	0
$831$	2.22897	0.0773222
$832$	0	0
$833$	−54.8351	−1.89992
$834$	0	0
$835$	−13.0309	−0.450954
$836$	0	0
$837$	2.76588	0.0956029
$838$	0	0
$839$	−11.3329	−0.391256	−0.195628	−	0.980678i	$-0.562675\pi$
$839$	−11.3329	−0.391256	−0.195628	+	0.980678i	$0.562675\pi$
$840$	0	0
$841$	−14.6674	−0.505773
$842$	0	0
$843$	−21.5713	−0.742953
$844$	0	0
$845$	12.5268	0.430937
$846$	0	0
$847$	64.8950	2.22982
$848$	0	0
$849$	−6.93160	−0.237892
$850$	0	0
$851$	1.43662	0.0492467
$852$	0	0
$853$	−40.5764	−1.38931	−0.694655	−	0.719343i	$-0.744443\pi$
$853$	−40.5764	−1.38931	−0.694655	+	0.719343i	$0.744443\pi$
$854$	0	0
$855$	3.62396	0.123937
$856$	0	0
$857$	−3.32006	−0.113411	−0.0567055	−	0.998391i	$-0.518060\pi$
$857$	−3.32006	−0.113411	−0.0567055	+	0.998391i	$0.518060\pi$
$858$	0	0
$859$	−24.4698	−0.834897	−0.417449	−	0.908701i	$-0.637076\pi$
$859$	−24.4698	−0.834897	−0.417449	+	0.908701i	$0.637076\pi$
$860$	0	0
$861$	−4.64932	−0.158448
$862$	0	0
$863$	48.0824	1.63674	0.818372	−	0.574689i	$-0.194877\pi$
$863$	48.0824	1.63674	0.818372	+	0.574689i	$0.194877\pi$
$864$	0	0
$865$	6.62658	0.225310
$866$	0	0
$867$	−44.2229	−1.50189
$868$	0	0
$869$	−31.1745	−1.05752
$870$	0	0
$871$	−4.42579	−0.149962
$872$	0	0
$873$	17.7241	0.599869
$874$	0	0
$875$	32.8950	1.11205
$876$	0	0
$877$	42.1000	1.42162	0.710809	−	0.703385i	$-0.248329\pi$
$877$	42.1000	1.42162	0.710809	+	0.703385i	$0.248329\pi$
$878$	0	0
$879$	−19.4521	−0.656103
$880$	0	0
$881$	41.3419	1.39284	0.696422	−	0.717633i	$-0.254774\pi$
$881$	41.3419	1.39284	0.696422	+	0.717633i	$0.254774\pi$
$882$	0	0
$883$	8.99926	0.302849	0.151425	−	0.988469i	$-0.451614\pi$
$883$	8.99926	0.302849	0.151425	+	0.988469i	$0.451614\pi$
$884$	0	0
$885$	−2.91375	−0.0979447
$886$	0	0
$887$	5.62010	0.188705	0.0943523	−	0.995539i	$-0.469922\pi$
$887$	5.62010	0.188705	0.0943523	+	0.995539i	$0.469922\pi$
$888$	0	0
$889$	37.7630	1.26653
$890$	0	0
$891$	−5.32343	−0.178342
$892$	0	0
$893$	−33.2786	−1.11362
$894$	0	0
$895$	−2.48838	−0.0831773
$896$	0	0
$897$	0.446157	0.0148967
$898$	0	0
$899$	−10.4712	−0.349234
$900$	0	0
$901$	−49.3771	−1.64499
$902$	0	0
$903$	−1.71297	−0.0570040
$904$	0	0
$905$	−21.2692	−0.707012
$906$	0	0
$907$	31.1705	1.03500	0.517500	−	0.855683i	$-0.326863\pi$
$907$	31.1705	1.03500	0.517500	+	0.855683i	$0.326863\pi$
$908$	0	0
$909$	−7.92972	−0.263012
$910$	0	0
$911$	−44.5048	−1.47451	−0.737255	−	0.675614i	$-0.763878\pi$
$911$	−44.5048	−1.47451	−0.737255	+	0.675614i	$0.763878\pi$
$912$	0	0
$913$	−24.4188	−0.808143
$914$	0	0
$915$	−2.65828	−0.0878802
$916$	0	0
$917$	10.0884	0.333148
$918$	0	0
$919$	29.6769	0.978949	0.489475	−	0.872018i	$-0.337189\pi$
$919$	29.6769	0.978949	0.489475	+	0.872018i	$0.337189\pi$
$920$	0	0
$921$	11.0657	0.364626
$922$	0	0
$923$	1.86698	0.0614524
$924$	0	0
$925$	−3.89506	−0.128069
$926$	0	0
$927$	3.52201	0.115678
$928$	0	0
$929$	20.6942	0.678954	0.339477	−	0.940614i	$-0.389750\pi$
$929$	20.6942	0.678954	0.339477	+	0.940614i	$0.389750\pi$
$930$	0	0
$931$	−26.1764	−0.857896
$932$	0	0
$933$	−3.72332	−0.121896
$934$	0	0
$935$	−40.4133	−1.32166
$936$	0	0
$937$	9.26828	0.302782	0.151391	−	0.988474i	$-0.451625\pi$
$937$	9.26828	0.302782	0.151391	+	0.988474i	$0.451625\pi$
$938$	0	0
$939$	−18.1184	−0.591272
$940$	0	0
$941$	37.7530	1.23071	0.615356	−	0.788249i	$-0.289012\pi$
$941$	37.7530	1.23071	0.615356	+	0.788249i	$0.289012\pi$
$942$	0	0
$943$	1.85955	0.0605553
$944$	0	0
$945$	3.63133	0.118127
$946$	0	0
$947$	−38.9781	−1.26662	−0.633310	−	0.773898i	$-0.718304\pi$
$947$	−38.9781	−1.26662	−0.633310	+	0.773898i	$0.718304\pi$
$948$	0	0
$949$	−2.22493	−0.0722243
$950$	0	0
$951$	−10.7581	−0.348854
$952$	0	0
$953$	14.4031	0.466563	0.233282	−	0.972409i	$-0.425054\pi$
$953$	14.4031	0.466563	0.233282	+	0.972409i	$0.425054\pi$
$954$	0	0
$955$	5.10488	0.165190
$956$	0	0
$957$	20.1537	0.651475
$958$	0	0
$959$	54.1043	1.74712
$960$	0	0
$961$	−23.3499	−0.753222
$962$	0	0
$963$	−11.3582	−0.366014
$964$	0	0
$965$	−0.729071	−0.0234696
$966$	0	0
$967$	−55.7370	−1.79238	−0.896191	−	0.443669i	$-0.853677\pi$
$967$	−55.7370	−1.79238	−0.896191	+	0.443669i	$0.853677\pi$
$968$	0	0
$969$	−29.2257	−0.938864
$970$	0	0
$971$	−13.8016	−0.442915	−0.221457	−	0.975170i	$-0.571081\pi$
$971$	−13.8016	−0.442915	−0.221457	+	0.975170i	$0.571081\pi$
$972$	0	0
$973$	−9.85013	−0.315781
$974$	0	0
$975$	−1.20965	−0.0387398
$976$	0	0
$977$	41.1936	1.31790	0.658950	−	0.752187i	$-0.271001\pi$
$977$	41.1936	1.31790	0.658950	+	0.752187i	$0.271001\pi$
$978$	0	0
$979$	−50.3136	−1.60803
$980$	0	0
$981$	3.06134	0.0977412
$982$	0	0
$983$	17.7452	0.565984	0.282992	−	0.959122i	$-0.408673\pi$
$983$	17.7452	0.565984	0.282992	+	0.959122i	$0.408673\pi$
$984$	0	0
$985$	16.7193	0.532721
$986$	0	0
$987$	−33.3462	−1.06142
$988$	0	0
$989$	0.685122	0.0217856
$990$	0	0
$991$	−37.7101	−1.19790	−0.598950	−	0.800786i	$-0.704415\pi$
$991$	−37.7101	−1.19790	−0.598950	+	0.800786i	$0.704415\pi$
$992$	0	0
$993$	−19.8675	−0.630474
$994$	0	0
$995$	14.6436	0.464235
$996$	0	0
$997$	−21.2510	−0.673026	−0.336513	−	0.941679i	$-0.609248\pi$
$997$	−21.2510	−0.673026	−0.336513	+	0.941679i	$0.609248\pi$
$998$	0	0
$999$	−0.959694	−0.0303634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.10	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.10	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.970233 q^{5} +3.74274 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.970233 q^{5} +3.74274 q^{7} +1.00000 q^{9} -5.32343 q^{11} -0.298043 q^{13} +0.970233 q^{15} -7.82451 q^{17} -3.73515 q^{19} -3.74274 q^{21} +1.49696 q^{23} -4.05865 q^{25} -1.00000 q^{27} +3.78584 q^{29} -2.76588 q^{31} +5.32343 q^{33} -3.63133 q^{35} +0.959694 q^{37} +0.298043 q^{39} +1.24222 q^{41} +0.457677 q^{43} -0.970233 q^{45} +8.90957 q^{47} +7.00812 q^{49} +7.82451 q^{51} +6.31057 q^{53} +5.16497 q^{55} +3.73515 q^{57} -3.00315 q^{59} -2.73984 q^{61} +3.74274 q^{63} +0.289171 q^{65} +14.8495 q^{67} -1.49696 q^{69} -6.26414 q^{71} +7.46514 q^{73} +4.05865 q^{75} -19.9242 q^{77} +5.85609 q^{79} +1.00000 q^{81} +4.58704 q^{83} +7.59160 q^{85} -3.78584 q^{87} +9.45136 q^{89} -1.11550 q^{91} +2.76588 q^{93} +3.62396 q^{95} +17.7241 q^{97} -5.32343 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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