LMFDB - Embedded newform 6036.2.a.f.1.7

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - 12 x^{12} + 112 x^{11} + 7 x^{10} - 710 x^{9} + 281 x^{8} + 1850 x^{7} - 830 x^{6} + \cdots + 1 $ x^14 - 6x^13 - 12x^12 + 112x^11 + 7x^10 - 710x^9 + 281x^8 + 1850x^7 - 830x^6 - 1918x^5 + 623x^4 + 480x^3 - 29x^2 - 17*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.176671$ of defining polynomial
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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.176671 q^{5} +4.02875 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.176671 q^{5} +4.02875 q^{7} +1.00000 q^{9} +0.188946 q^{11} -3.04805 q^{13} +0.176671 q^{15} -0.150704 q^{17} +1.58572 q^{19} -4.02875 q^{21} -1.13710 q^{23} -4.96879 q^{25} -1.00000 q^{27} +3.28927 q^{29} -3.96658 q^{31} -0.188946 q^{33} -0.711764 q^{35} -0.860056 q^{37} +3.04805 q^{39} -5.72779 q^{41} -10.3028 q^{43} -0.176671 q^{45} -6.76140 q^{47} +9.23082 q^{49} +0.150704 q^{51} +7.94502 q^{53} -0.0333814 q^{55} -1.58572 q^{57} -1.11519 q^{59} -2.63747 q^{61} +4.02875 q^{63} +0.538502 q^{65} -4.54206 q^{67} +1.13710 q^{69} -15.1173 q^{71} -7.90767 q^{73} +4.96879 q^{75} +0.761217 q^{77} -5.98721 q^{79} +1.00000 q^{81} -3.05730 q^{83} +0.0266251 q^{85} -3.28927 q^{87} -1.66072 q^{89} -12.2798 q^{91} +3.96658 q^{93} -0.280151 q^{95} -0.601859 q^{97} +0.188946 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + 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
$5$	−0.176671	−0.0790098	−0.0395049	−	0.999219i	$-0.512578\pi$
$5$	−0.176671	−0.0790098	−0.0395049	+	0.999219i	$0.512578\pi$
$6$	0	0
$7$	4.02875	1.52272	0.761362	−	0.648327i	$-0.224531\pi$
$7$	4.02875	1.52272	0.761362	+	0.648327i	$0.224531\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.188946	0.0569694	0.0284847	−	0.999594i	$-0.490932\pi$
$11$	0.188946	0.0569694	0.0284847	+	0.999594i	$0.490932\pi$
$12$	0	0
$13$	−3.04805	−0.845376	−0.422688	−	0.906275i	$-0.638913\pi$
$13$	−3.04805	−0.845376	−0.422688	+	0.906275i	$0.638913\pi$
$14$	0	0
$15$	0.176671	0.0456163
$16$	0	0
$17$	−0.150704	−0.0365512	−0.0182756	−	0.999833i	$-0.505818\pi$
$17$	−0.150704	−0.0365512	−0.0182756	+	0.999833i	$0.505818\pi$
$18$	0	0
$19$	1.58572	0.363789	0.181894	−	0.983318i	$-0.441777\pi$
$19$	1.58572	0.363789	0.181894	+	0.983318i	$0.441777\pi$
$20$	0	0
$21$	−4.02875	−0.879145
$22$	0	0
$23$	−1.13710	−0.237102	−0.118551	−	0.992948i	$-0.537825\pi$
$23$	−1.13710	−0.237102	−0.118551	+	0.992948i	$0.537825\pi$
$24$	0	0
$25$	−4.96879	−0.993757
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.28927	0.610803	0.305401	−	0.952224i	$-0.401209\pi$
$29$	3.28927	0.610803	0.305401	+	0.952224i	$0.401209\pi$
$30$	0	0
$31$	−3.96658	−0.712420	−0.356210	−	0.934406i	$-0.615931\pi$
$31$	−3.96658	−0.712420	−0.356210	+	0.934406i	$0.615931\pi$
$32$	0	0
$33$	−0.188946	−0.0328913
$34$	0	0
$35$	−0.711764	−0.120310
$36$	0	0
$37$	−0.860056	−0.141392	−0.0706962	−	0.997498i	$-0.522522\pi$
$37$	−0.860056	−0.141392	−0.0706962	+	0.997498i	$0.522522\pi$
$38$	0	0
$39$	3.04805	0.488078
$40$	0	0
$41$	−5.72779	−0.894531	−0.447266	−	0.894401i	$-0.647602\pi$
$41$	−5.72779	−0.894531	−0.447266	+	0.894401i	$0.647602\pi$
$42$	0	0
$43$	−10.3028	−1.57116	−0.785579	−	0.618762i	$-0.787635\pi$
$43$	−10.3028	−1.57116	−0.785579	+	0.618762i	$0.787635\pi$
$44$	0	0
$45$	−0.176671	−0.0263366
$46$	0	0
$47$	−6.76140	−0.986252	−0.493126	−	0.869958i	$-0.664146\pi$
$47$	−6.76140	−0.986252	−0.493126	+	0.869958i	$0.664146\pi$
$48$	0	0
$49$	9.23082	1.31869
$50$	0	0
$51$	0.150704	0.0211028
$52$	0	0
$53$	7.94502	1.09133	0.545667	−	0.838002i	$-0.316277\pi$
$53$	7.94502	1.09133	0.545667	+	0.838002i	$0.316277\pi$
$54$	0	0
$55$	−0.0333814	−0.00450114
$56$	0	0
$57$	−1.58572	−0.210033
$58$	0	0
$59$	−1.11519	−0.145185	−0.0725926	−	0.997362i	$-0.523127\pi$
$59$	−1.11519	−0.145185	−0.0725926	+	0.997362i	$0.523127\pi$
$60$	0	0
$61$	−2.63747	−0.337694	−0.168847	−	0.985642i	$-0.554004\pi$
$61$	−2.63747	−0.337694	−0.168847	+	0.985642i	$0.554004\pi$
$62$	0	0
$63$	4.02875	0.507575
$64$	0	0
$65$	0.538502	0.0667930
$66$	0	0
$67$	−4.54206	−0.554900	−0.277450	−	0.960740i	$-0.589489\pi$
$67$	−4.54206	−0.554900	−0.277450	+	0.960740i	$0.589489\pi$
$68$	0	0
$69$	1.13710	0.136891
$70$	0	0
$71$	−15.1173	−1.79409	−0.897044	−	0.441941i	$-0.854290\pi$
$71$	−15.1173	−1.79409	−0.897044	+	0.441941i	$0.854290\pi$
$72$	0	0
$73$	−7.90767	−0.925523	−0.462761	−	0.886483i	$-0.653141\pi$
$73$	−7.90767	−0.925523	−0.462761	+	0.886483i	$0.653141\pi$
$74$	0	0
$75$	4.96879	0.573746
$76$	0	0
$77$	0.761217	0.0867488
$78$	0	0
$79$	−5.98721	−0.673614	−0.336807	−	0.941574i	$-0.609347\pi$
$79$	−5.98721	−0.673614	−0.336807	+	0.941574i	$0.609347\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.05730	−0.335582	−0.167791	−	0.985823i	$-0.553663\pi$
$83$	−3.05730	−0.335582	−0.167791	+	0.985823i	$0.553663\pi$
$84$	0	0
$85$	0.0266251	0.00288790
$86$	0	0
$87$	−3.28927	−0.352647
$88$	0	0
$89$	−1.66072	−0.176036	−0.0880182	−	0.996119i	$-0.528053\pi$
$89$	−1.66072	−0.176036	−0.0880182	+	0.996119i	$0.528053\pi$
$90$	0	0
$91$	−12.2798	−1.28727
$92$	0	0
$93$	3.96658	0.411316
$94$	0	0
$95$	−0.280151	−0.0287429
$96$	0	0
$97$	−0.601859	−0.0611096	−0.0305548	−	0.999533i	$-0.509727\pi$
$97$	−0.601859	−0.0611096	−0.0305548	+	0.999533i	$0.509727\pi$
$98$	0	0
$99$	0.188946	0.0189898
$100$	0	0
$101$	16.8212	1.67377	0.836887	−	0.547376i	$-0.184373\pi$
$101$	16.8212	1.67377	0.836887	+	0.547376i	$0.184373\pi$
$102$	0	0
$103$	18.6027	1.83298	0.916491	−	0.400056i	$-0.131009\pi$
$103$	18.6027	1.83298	0.916491	+	0.400056i	$0.131009\pi$
$104$	0	0
$105$	0.711764	0.0694611
$106$	0	0
$107$	−1.20367	−0.116363	−0.0581816	−	0.998306i	$-0.518530\pi$
$107$	−1.20367	−0.116363	−0.0581816	+	0.998306i	$0.518530\pi$
$108$	0	0
$109$	−4.35003	−0.416657	−0.208329	−	0.978059i	$-0.566802\pi$
$109$	−4.35003	−0.416657	−0.208329	+	0.978059i	$0.566802\pi$
$110$	0	0
$111$	0.860056	0.0816329
$112$	0	0
$113$	16.0590	1.51070	0.755352	−	0.655319i	$-0.227466\pi$
$113$	16.0590	1.51070	0.755352	+	0.655319i	$0.227466\pi$
$114$	0	0
$115$	0.200893	0.0187334
$116$	0	0
$117$	−3.04805	−0.281792
$118$	0	0
$119$	−0.607150	−0.0556574
$120$	0	0
$121$	−10.9643	−0.996754
$122$	0	0
$123$	5.72779	0.516458
$124$	0	0
$125$	1.76120	0.157526
$126$	0	0
$127$	−2.81545	−0.249831	−0.124916	−	0.992167i	$-0.539866\pi$
$127$	−2.81545	−0.249831	−0.124916	+	0.992167i	$0.539866\pi$
$128$	0	0
$129$	10.3028	0.907108
$130$	0	0
$131$	18.5011	1.61645	0.808225	−	0.588874i	$-0.200428\pi$
$131$	18.5011	1.61645	0.808225	+	0.588874i	$0.200428\pi$
$132$	0	0
$133$	6.38846	0.553950
$134$	0	0
$135$	0.176671	0.0152054
$136$	0	0
$137$	−11.6470	−0.995067	−0.497534	−	0.867445i	$-0.665761\pi$
$137$	−11.6470	−0.995067	−0.497534	+	0.867445i	$0.665761\pi$
$138$	0	0
$139$	−13.1245	−1.11321	−0.556604	−	0.830778i	$-0.687896\pi$
$139$	−13.1245	−1.11321	−0.556604	+	0.830778i	$0.687896\pi$
$140$	0	0
$141$	6.76140	0.569413
$142$	0	0
$143$	−0.575917	−0.0481606
$144$	0	0
$145$	−0.581120	−0.0482594
$146$	0	0
$147$	−9.23082	−0.761345
$148$	0	0
$149$	−3.36795	−0.275913	−0.137957	−	0.990438i	$-0.544053\pi$
$149$	−3.36795	−0.275913	−0.137957	+	0.990438i	$0.544053\pi$
$150$	0	0
$151$	−4.58305	−0.372963	−0.186482	−	0.982458i	$-0.559708\pi$
$151$	−4.58305	−0.372963	−0.186482	+	0.982458i	$0.559708\pi$
$152$	0	0
$153$	−0.150704	−0.0121837
$154$	0	0
$155$	0.700782	0.0562881
$156$	0	0
$157$	−18.5118	−1.47740	−0.738701	−	0.674034i	$-0.764560\pi$
$157$	−18.5118	−1.47740	−0.738701	+	0.674034i	$0.764560\pi$
$158$	0	0
$159$	−7.94502	−0.630081
$160$	0	0
$161$	−4.58109	−0.361041
$162$	0	0
$163$	11.3842	0.891678	0.445839	−	0.895113i	$-0.352905\pi$
$163$	11.3842	0.891678	0.445839	+	0.895113i	$0.352905\pi$
$164$	0	0
$165$	0.0333814	0.00259874
$166$	0	0
$167$	8.85773	0.685431	0.342716	−	0.939439i	$-0.388653\pi$
$167$	8.85773	0.685431	0.342716	+	0.939439i	$0.388653\pi$
$168$	0	0
$169$	−3.70941	−0.285339
$170$	0	0
$171$	1.58572	0.121263
$172$	0	0
$173$	−7.41109	−0.563455	−0.281727	−	0.959495i	$-0.590907\pi$
$173$	−7.41109	−0.563455	−0.281727	+	0.959495i	$0.590907\pi$
$174$	0	0
$175$	−20.0180	−1.51322
$176$	0	0
$177$	1.11519	0.0838227
$178$	0	0
$179$	7.51689	0.561838	0.280919	−	0.959731i	$-0.409361\pi$
$179$	7.51689	0.561838	0.280919	+	0.959731i	$0.409361\pi$
$180$	0	0
$181$	−6.14123	−0.456474	−0.228237	−	0.973606i	$-0.573296\pi$
$181$	−6.14123	−0.456474	−0.228237	+	0.973606i	$0.573296\pi$
$182$	0	0
$183$	2.63747	0.194967
$184$	0	0
$185$	0.151947	0.0111714
$186$	0	0
$187$	−0.0284750	−0.00208230
$188$	0	0
$189$	−4.02875	−0.293048
$190$	0	0
$191$	−0.0680958	−0.00492724	−0.00246362	−	0.999997i	$-0.500784\pi$
$191$	−0.0680958	−0.00492724	−0.00246362	+	0.999997i	$0.500784\pi$
$192$	0	0
$193$	6.34313	0.456588	0.228294	−	0.973592i	$-0.426685\pi$
$193$	6.34313	0.456588	0.228294	+	0.973592i	$0.426685\pi$
$194$	0	0
$195$	−0.538502	−0.0385629
$196$	0	0
$197$	17.5068	1.24731	0.623655	−	0.781700i	$-0.285647\pi$
$197$	17.5068	1.24731	0.623655	+	0.781700i	$0.285647\pi$
$198$	0	0
$199$	12.6959	0.899991	0.449995	−	0.893031i	$-0.351426\pi$
$199$	12.6959	0.899991	0.449995	+	0.893031i	$0.351426\pi$
$200$	0	0
$201$	4.54206	0.320372
$202$	0	0
$203$	13.2517	0.930084
$204$	0	0
$205$	1.01194	0.0706767
$206$	0	0
$207$	−1.13710	−0.0790339
$208$	0	0
$209$	0.299615	0.0207248
$210$	0	0
$211$	1.86554	0.128429	0.0642145	−	0.997936i	$-0.479546\pi$
$211$	1.86554	0.128429	0.0642145	+	0.997936i	$0.479546\pi$
$212$	0	0
$213$	15.1173	1.03582
$214$	0	0
$215$	1.82020	0.124137
$216$	0	0
$217$	−15.9804	−1.08482
$218$	0	0
$219$	7.90767	0.534351
$220$	0	0
$221$	0.459354	0.0308995
$222$	0	0
$223$	−0.968533	−0.0648578	−0.0324289	−	0.999474i	$-0.510324\pi$
$223$	−0.968533	−0.0648578	−0.0324289	+	0.999474i	$0.510324\pi$
$224$	0	0
$225$	−4.96879	−0.331252
$226$	0	0
$227$	−5.41882	−0.359660	−0.179830	−	0.983698i	$-0.557555\pi$
$227$	−5.41882	−0.359660	−0.179830	+	0.983698i	$0.557555\pi$
$228$	0	0
$229$	−20.1638	−1.33246	−0.666231	−	0.745746i	$-0.732093\pi$
$229$	−20.1638	−1.33246	−0.666231	+	0.745746i	$0.732093\pi$
$230$	0	0
$231$	−0.761217	−0.0500844
$232$	0	0
$233$	−19.5988	−1.28396	−0.641979	−	0.766722i	$-0.721886\pi$
$233$	−19.5988	−1.28396	−0.641979	+	0.766722i	$0.721886\pi$
$234$	0	0
$235$	1.19455	0.0779235
$236$	0	0
$237$	5.98721	0.388911
$238$	0	0
$239$	−6.28088	−0.406276	−0.203138	−	0.979150i	$-0.565114\pi$
$239$	−6.28088	−0.406276	−0.203138	+	0.979150i	$0.565114\pi$
$240$	0	0
$241$	−2.85707	−0.184040	−0.0920201	−	0.995757i	$-0.529332\pi$
$241$	−2.85707	−0.184040	−0.0920201	+	0.995757i	$0.529332\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.63082	−0.104189
$246$	0	0
$247$	−4.83334	−0.307538
$248$	0	0
$249$	3.05730	0.193749
$250$	0	0
$251$	9.85763	0.622208	0.311104	−	0.950376i	$-0.399301\pi$
$251$	9.85763	0.622208	0.311104	+	0.950376i	$0.399301\pi$
$252$	0	0
$253$	−0.214851	−0.0135076
$254$	0	0
$255$	−0.0266251	−0.00166733
$256$	0	0
$257$	−12.8350	−0.800626	−0.400313	−	0.916379i	$-0.631099\pi$
$257$	−12.8350	−0.800626	−0.400313	+	0.916379i	$0.631099\pi$
$258$	0	0
$259$	−3.46495	−0.215301
$260$	0	0
$261$	3.28927	0.203601
$262$	0	0
$263$	22.5355	1.38960	0.694799	−	0.719204i	$-0.255493\pi$
$263$	22.5355	1.38960	0.694799	+	0.719204i	$0.255493\pi$
$264$	0	0
$265$	−1.40366	−0.0862260
$266$	0	0
$267$	1.66072	0.101635
$268$	0	0
$269$	−7.11704	−0.433933	−0.216967	−	0.976179i	$-0.569616\pi$
$269$	−7.11704	−0.433933	−0.216967	+	0.976179i	$0.569616\pi$
$270$	0	0
$271$	18.3911	1.11718	0.558591	−	0.829443i	$-0.311342\pi$
$271$	18.3911	1.11718	0.558591	+	0.829443i	$0.311342\pi$
$272$	0	0
$273$	12.2798	0.743208
$274$	0	0
$275$	−0.938834	−0.0566138
$276$	0	0
$277$	7.00797	0.421068	0.210534	−	0.977586i	$-0.432480\pi$
$277$	7.00797	0.421068	0.210534	+	0.977586i	$0.432480\pi$
$278$	0	0
$279$	−3.96658	−0.237473
$280$	0	0
$281$	−6.49560	−0.387495	−0.193748	−	0.981051i	$-0.562064\pi$
$281$	−6.49560	−0.387495	−0.193748	+	0.981051i	$0.562064\pi$
$282$	0	0
$283$	2.14424	0.127462	0.0637308	−	0.997967i	$-0.479700\pi$
$283$	2.14424	0.127462	0.0637308	+	0.997967i	$0.479700\pi$
$284$	0	0
$285$	0.280151	0.0165947
$286$	0	0
$287$	−23.0758	−1.36212
$288$	0	0
$289$	−16.9773	−0.998664
$290$	0	0
$291$	0.601859	0.0352816
$292$	0	0
$293$	−13.1782	−0.769880	−0.384940	−	0.922942i	$-0.625778\pi$
$293$	−13.1782	−0.769880	−0.384940	+	0.922942i	$0.625778\pi$
$294$	0	0
$295$	0.197022	0.0114711
$296$	0	0
$297$	−0.188946	−0.0109638
$298$	0	0
$299$	3.46593	0.200440
$300$	0	0
$301$	−41.5073	−2.39244
$302$	0	0
$303$	−16.8212	−0.966353
$304$	0	0
$305$	0.465965	0.0266811
$306$	0	0
$307$	−21.2510	−1.21286	−0.606430	−	0.795137i	$-0.707399\pi$
$307$	−21.2510	−1.21286	−0.606430	+	0.795137i	$0.707399\pi$
$308$	0	0
$309$	−18.6027	−1.05827
$310$	0	0
$311$	−24.8338	−1.40820	−0.704098	−	0.710103i	$-0.748649\pi$
$311$	−24.8338	−1.40820	−0.704098	+	0.710103i	$0.748649\pi$
$312$	0	0
$313$	−5.32724	−0.301113	−0.150557	−	0.988601i	$-0.548107\pi$
$313$	−5.32724	−0.301113	−0.150557	+	0.988601i	$0.548107\pi$
$314$	0	0
$315$	−0.711764	−0.0401034
$316$	0	0
$317$	19.5937	1.10049	0.550247	−	0.835002i	$-0.314533\pi$
$317$	19.5937	1.10049	0.550247	+	0.835002i	$0.314533\pi$
$318$	0	0
$319$	0.621496	0.0347971
$320$	0	0
$321$	1.20367	0.0671824
$322$	0	0
$323$	−0.238975	−0.0132969
$324$	0	0
$325$	15.1451	0.840099
$326$	0	0
$327$	4.35003	0.240557
$328$	0	0
$329$	−27.2400	−1.50179
$330$	0	0
$331$	19.1464	1.05238	0.526190	−	0.850367i	$-0.323620\pi$
$331$	19.1464	1.05238	0.526190	+	0.850367i	$0.323620\pi$
$332$	0	0
$333$	−0.860056	−0.0471308
$334$	0	0
$335$	0.802451	0.0438426
$336$	0	0
$337$	−33.3176	−1.81493	−0.907463	−	0.420132i	$-0.861984\pi$
$337$	−33.3176	−1.81493	−0.907463	+	0.420132i	$0.861984\pi$
$338$	0	0
$339$	−16.0590	−0.872205
$340$	0	0
$341$	−0.749471	−0.0405861
$342$	0	0
$343$	8.98743	0.485276
$344$	0	0
$345$	−0.200893	−0.0108157
$346$	0	0
$347$	−2.62464	−0.140898	−0.0704490	−	0.997515i	$-0.522443\pi$
$347$	−2.62464	−0.140898	−0.0704490	+	0.997515i	$0.522443\pi$
$348$	0	0
$349$	−19.8929	−1.06485	−0.532423	−	0.846479i	$-0.678718\pi$
$349$	−19.8929	−1.06485	−0.532423	+	0.846479i	$0.678718\pi$
$350$	0	0
$351$	3.04805	0.162693
$352$	0	0
$353$	29.3094	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$353$	29.3094	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$354$	0	0
$355$	2.67079	0.141751
$356$	0	0
$357$	0.607150	0.0321338
$358$	0	0
$359$	−15.5358	−0.819951	−0.409975	−	0.912097i	$-0.634463\pi$
$359$	−15.5358	−0.819951	−0.409975	+	0.912097i	$0.634463\pi$
$360$	0	0
$361$	−16.4855	−0.867658
$362$	0	0
$363$	10.9643	0.575476
$364$	0	0
$365$	1.39706	0.0731254
$366$	0	0
$367$	−6.04554	−0.315575	−0.157787	−	0.987473i	$-0.550436\pi$
$367$	−6.04554	−0.315575	−0.157787	+	0.987473i	$0.550436\pi$
$368$	0	0
$369$	−5.72779	−0.298177
$370$	0	0
$371$	32.0085	1.66180
$372$	0	0
$373$	16.5744	0.858190	0.429095	−	0.903259i	$-0.358832\pi$
$373$	16.5744	0.858190	0.429095	+	0.903259i	$0.358832\pi$
$374$	0	0
$375$	−1.76120	−0.0909479
$376$	0	0
$377$	−10.0259	−0.516358
$378$	0	0
$379$	30.7763	1.58087	0.790437	−	0.612543i	$-0.209853\pi$
$379$	30.7763	1.58087	0.790437	+	0.612543i	$0.209853\pi$
$380$	0	0
$381$	2.81545	0.144240
$382$	0	0
$383$	−17.3433	−0.886202	−0.443101	−	0.896472i	$-0.646122\pi$
$383$	−17.3433	−0.886202	−0.443101	+	0.896472i	$0.646122\pi$
$384$	0	0
$385$	−0.134485	−0.00685400
$386$	0	0
$387$	−10.3028	−0.523719
$388$	0	0
$389$	−18.3057	−0.928135	−0.464068	−	0.885800i	$-0.653611\pi$
$389$	−18.3057	−0.928135	−0.464068	+	0.885800i	$0.653611\pi$
$390$	0	0
$391$	0.171366	0.00866635
$392$	0	0
$393$	−18.5011	−0.933258
$394$	0	0
$395$	1.05777	0.0532221
$396$	0	0
$397$	−37.1308	−1.86354	−0.931770	−	0.363049i	$-0.881736\pi$
$397$	−37.1308	−1.86354	−0.931770	+	0.363049i	$0.881736\pi$
$398$	0	0
$399$	−6.38846	−0.319823
$400$	0	0
$401$	0.963260	0.0481029	0.0240515	−	0.999711i	$-0.492343\pi$
$401$	0.963260	0.0481029	0.0240515	+	0.999711i	$0.492343\pi$
$402$	0	0
$403$	12.0903	0.602262
$404$	0	0
$405$	−0.176671	−0.00877887
$406$	0	0
$407$	−0.162504	−0.00805504
$408$	0	0
$409$	7.14085	0.353092	0.176546	−	0.984292i	$-0.443508\pi$
$409$	7.14085	0.353092	0.176546	+	0.984292i	$0.443508\pi$
$410$	0	0
$411$	11.6470	0.574502
$412$	0	0
$413$	−4.49282	−0.221077
$414$	0	0
$415$	0.540137	0.0265143
$416$	0	0
$417$	13.1245	0.642710
$418$	0	0
$419$	−16.0268	−0.782959	−0.391479	−	0.920187i	$-0.628037\pi$
$419$	−16.0268	−0.782959	−0.391479	+	0.920187i	$0.628037\pi$
$420$	0	0
$421$	0.538687	0.0262540	0.0131270	−	0.999914i	$-0.495821\pi$
$421$	0.538687	0.0262540	0.0131270	+	0.999914i	$0.495821\pi$
$422$	0	0
$423$	−6.76140	−0.328751
$424$	0	0
$425$	0.748818	0.0363230
$426$	0	0
$427$	−10.6257	−0.514214
$428$	0	0
$429$	0.575917	0.0278055
$430$	0	0
$431$	−7.81354	−0.376365	−0.188182	−	0.982134i	$-0.560260\pi$
$431$	−7.81354	−0.376365	−0.188182	+	0.982134i	$0.560260\pi$
$432$	0	0
$433$	−37.2809	−1.79161	−0.895804	−	0.444450i	$-0.853399\pi$
$433$	−37.2809	−1.79161	−0.895804	+	0.444450i	$0.853399\pi$
$434$	0	0
$435$	0.581120	0.0278626
$436$	0	0
$437$	−1.80312	−0.0862549
$438$	0	0
$439$	−16.5065	−0.787812	−0.393906	−	0.919151i	$-0.628877\pi$
$439$	−16.5065	−0.787812	−0.393906	+	0.919151i	$0.628877\pi$
$440$	0	0
$441$	9.23082	0.439563
$442$	0	0
$443$	−9.01454	−0.428294	−0.214147	−	0.976801i	$-0.568697\pi$
$443$	−9.01454	−0.428294	−0.214147	+	0.976801i	$0.568697\pi$
$444$	0	0
$445$	0.293402	0.0139086
$446$	0	0
$447$	3.36795	0.159299
$448$	0	0
$449$	−18.2900	−0.863158	−0.431579	−	0.902075i	$-0.642043\pi$
$449$	−18.2900	−0.863158	−0.431579	+	0.902075i	$0.642043\pi$
$450$	0	0
$451$	−1.08225	−0.0509609
$452$	0	0
$453$	4.58305	0.215330
$454$	0	0
$455$	2.16949	0.101707
$456$	0	0
$457$	31.5342	1.47511	0.737553	−	0.675289i	$-0.235981\pi$
$457$	31.5342	1.47511	0.737553	+	0.675289i	$0.235981\pi$
$458$	0	0
$459$	0.150704	0.00703428
$460$	0	0
$461$	11.6346	0.541876	0.270938	−	0.962597i	$-0.412666\pi$
$461$	11.6346	0.541876	0.270938	+	0.962597i	$0.412666\pi$
$462$	0	0
$463$	11.5472	0.536643	0.268322	−	0.963329i	$-0.413531\pi$
$463$	11.5472	0.536643	0.268322	+	0.963329i	$0.413531\pi$
$464$	0	0
$465$	−0.700782	−0.0324980
$466$	0	0
$467$	29.5703	1.36835	0.684176	−	0.729317i	$-0.260162\pi$
$467$	29.5703	1.36835	0.684176	+	0.729317i	$0.260162\pi$
$468$	0	0
$469$	−18.2988	−0.844960
$470$	0	0
$471$	18.5118	0.852978
$472$	0	0
$473$	−1.94667	−0.0895080
$474$	0	0
$475$	−7.87909	−0.361518
$476$	0	0
$477$	7.94502	0.363778
$478$	0	0
$479$	10.5586	0.482434	0.241217	−	0.970471i	$-0.422453\pi$
$479$	10.5586	0.482434	0.241217	+	0.970471i	$0.422453\pi$
$480$	0	0
$481$	2.62149	0.119530
$482$	0	0
$483$	4.58109	0.208447
$484$	0	0
$485$	0.106331	0.00482825
$486$	0	0
$487$	−40.3885	−1.83018	−0.915090	−	0.403250i	$-0.867880\pi$
$487$	−40.3885	−1.83018	−0.915090	+	0.403250i	$0.867880\pi$
$488$	0	0
$489$	−11.3842	−0.514811
$490$	0	0
$491$	−40.6857	−1.83612	−0.918060	−	0.396442i	$-0.870245\pi$
$491$	−40.6857	−1.83612	−0.918060	+	0.396442i	$0.870245\pi$
$492$	0	0
$493$	−0.495708	−0.0223256
$494$	0	0
$495$	−0.0333814	−0.00150038
$496$	0	0
$497$	−60.9037	−2.73190
$498$	0	0
$499$	6.68499	0.299261	0.149631	−	0.988742i	$-0.452192\pi$
$499$	6.68499	0.299261	0.149631	+	0.988742i	$0.452192\pi$
$500$	0	0
$501$	−8.85773	−0.395734
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−2.97183	−0.132244
$506$	0	0
$507$	3.70941	0.164741
$508$	0	0
$509$	−8.13051	−0.360378	−0.180189	−	0.983632i	$-0.557671\pi$
$509$	−8.13051	−0.360378	−0.180189	+	0.983632i	$0.557671\pi$
$510$	0	0
$511$	−31.8580	−1.40932
$512$	0	0
$513$	−1.58572	−0.0700112
$514$	0	0
$515$	−3.28657	−0.144823
$516$	0	0
$517$	−1.27754	−0.0561862
$518$	0	0
$519$	7.41109	0.325311
$520$	0	0
$521$	1.93602	0.0848184	0.0424092	−	0.999100i	$-0.486497\pi$
$521$	1.93602	0.0848184	0.0424092	+	0.999100i	$0.486497\pi$
$522$	0	0
$523$	−22.5857	−0.987603	−0.493802	−	0.869575i	$-0.664393\pi$
$523$	−22.5857	−0.987603	−0.493802	+	0.869575i	$0.664393\pi$
$524$	0	0
$525$	20.0180	0.873657
$526$	0	0
$527$	0.597782	0.0260398
$528$	0	0
$529$	−21.7070	−0.943783
$530$	0	0
$531$	−1.11519	−0.0483951
$532$	0	0
$533$	17.4586	0.756215
$534$	0	0
$535$	0.212654	0.00919384
$536$	0	0
$537$	−7.51689	−0.324378
$538$	0	0
$539$	1.74413	0.0751250
$540$	0	0
$541$	−5.40945	−0.232570	−0.116285	−	0.993216i	$-0.537099\pi$
$541$	−5.40945	−0.232570	−0.116285	+	0.993216i	$0.537099\pi$
$542$	0	0
$543$	6.14123	0.263545
$544$	0	0
$545$	0.768525	0.0329200
$546$	0	0
$547$	7.57570	0.323913	0.161957	−	0.986798i	$-0.448219\pi$
$547$	7.57570	0.323913	0.161957	+	0.986798i	$0.448219\pi$
$548$	0	0
$549$	−2.63747	−0.112565
$550$	0	0
$551$	5.21586	0.222203
$552$	0	0
$553$	−24.1210	−1.02573
$554$	0	0
$555$	−0.151947	−0.00644980
$556$	0	0
$557$	42.5292	1.80202	0.901009	−	0.433800i	$-0.142828\pi$
$557$	42.5292	1.80202	0.901009	+	0.433800i	$0.142828\pi$
$558$	0	0
$559$	31.4033	1.32822
$560$	0	0
$561$	0.0284750	0.00120222
$562$	0	0
$563$	−33.8191	−1.42531	−0.712654	−	0.701516i	$-0.752507\pi$
$563$	−33.8191	−1.42531	−0.712654	+	0.701516i	$0.752507\pi$
$564$	0	0
$565$	−2.83717	−0.119360
$566$	0	0
$567$	4.02875	0.169192
$568$	0	0
$569$	−8.18425	−0.343101	−0.171551	−	0.985175i	$-0.554878\pi$
$569$	−8.18425	−0.343101	−0.171551	+	0.985175i	$0.554878\pi$
$570$	0	0
$571$	9.80629	0.410381	0.205190	−	0.978722i	$-0.434219\pi$
$571$	9.80629	0.410381	0.205190	+	0.978722i	$0.434219\pi$
$572$	0	0
$573$	0.0680958	0.00284474
$574$	0	0
$575$	5.65001	0.235622
$576$	0	0
$577$	−25.8255	−1.07513	−0.537565	−	0.843222i	$-0.680656\pi$
$577$	−25.8255	−1.07513	−0.537565	+	0.843222i	$0.680656\pi$
$578$	0	0
$579$	−6.34313	−0.263611
$580$	0	0
$581$	−12.3171	−0.510999
$582$	0	0
$583$	1.50118	0.0621726
$584$	0	0
$585$	0.538502	0.0222643
$586$	0	0
$587$	36.9081	1.52336	0.761681	−	0.647952i	$-0.224374\pi$
$587$	36.9081	1.52336	0.761681	+	0.647952i	$0.224374\pi$
$588$	0	0
$589$	−6.28988	−0.259170
$590$	0	0
$591$	−17.5068	−0.720134
$592$	0	0
$593$	−26.0837	−1.07113	−0.535565	−	0.844494i	$-0.679901\pi$
$593$	−26.0837	−1.07113	−0.535565	+	0.844494i	$0.679901\pi$
$594$	0	0
$595$	0.107266	0.00439748
$596$	0	0
$597$	−12.6959	−0.519610
$598$	0	0
$599$	−28.0873	−1.14761	−0.573807	−	0.818990i	$-0.694534\pi$
$599$	−28.0873	−1.14761	−0.573807	+	0.818990i	$0.694534\pi$
$600$	0	0
$601$	−27.8853	−1.13746	−0.568732	−	0.822523i	$-0.692566\pi$
$601$	−27.8853	−1.13746	−0.568732	+	0.822523i	$0.692566\pi$
$602$	0	0
$603$	−4.54206	−0.184967
$604$	0	0
$605$	1.93708	0.0787534
$606$	0	0
$607$	37.6075	1.52644	0.763221	−	0.646137i	$-0.223617\pi$
$607$	37.6075	1.52644	0.763221	+	0.646137i	$0.223617\pi$
$608$	0	0
$609$	−13.2517	−0.536984
$610$	0	0
$611$	20.6091	0.833753
$612$	0	0
$613$	47.1518	1.90444	0.952222	−	0.305407i	$-0.0987924\pi$
$613$	47.1518	1.90444	0.952222	+	0.305407i	$0.0987924\pi$
$614$	0	0
$615$	−1.01194	−0.0408052
$616$	0	0
$617$	18.7580	0.755167	0.377583	−	0.925976i	$-0.376755\pi$
$617$	18.7580	0.755167	0.377583	+	0.925976i	$0.376755\pi$
$618$	0	0
$619$	−16.0433	−0.644834	−0.322417	−	0.946598i	$-0.604495\pi$
$619$	−16.0433	−0.644834	−0.322417	+	0.946598i	$0.604495\pi$
$620$	0	0
$621$	1.13710	0.0456303
$622$	0	0
$623$	−6.69064	−0.268055
$624$	0	0
$625$	24.5328	0.981311
$626$	0	0
$627$	−0.299615	−0.0119655
$628$	0	0
$629$	0.129614	0.00516805
$630$	0	0
$631$	−7.44205	−0.296263	−0.148132	−	0.988968i	$-0.547326\pi$
$631$	−7.44205	−0.296263	−0.148132	+	0.988968i	$0.547326\pi$
$632$	0	0
$633$	−1.86554	−0.0741485
$634$	0	0
$635$	0.497410	0.0197391
$636$	0	0
$637$	−28.1360	−1.11479
$638$	0	0
$639$	−15.1173	−0.598030
$640$	0	0
$641$	40.3170	1.59242	0.796212	−	0.605017i	$-0.206834\pi$
$641$	40.3170	1.59242	0.796212	+	0.605017i	$0.206834\pi$
$642$	0	0
$643$	−23.0881	−0.910505	−0.455253	−	0.890362i	$-0.650451\pi$
$643$	−23.0881	−0.910505	−0.455253	+	0.890362i	$0.650451\pi$
$644$	0	0
$645$	−1.82020	−0.0716704
$646$	0	0
$647$	31.5866	1.24180	0.620898	−	0.783891i	$-0.286768\pi$
$647$	31.5866	1.24180	0.620898	+	0.783891i	$0.286768\pi$
$648$	0	0
$649$	−0.210711	−0.00827112
$650$	0	0
$651$	15.9804	0.626320
$652$	0	0
$653$	−33.0043	−1.29156	−0.645780	−	0.763524i	$-0.723468\pi$
$653$	−33.0043	−1.29156	−0.645780	+	0.763524i	$0.723468\pi$
$654$	0	0
$655$	−3.26862	−0.127715
$656$	0	0
$657$	−7.90767	−0.308508
$658$	0	0
$659$	37.7172	1.46925	0.734627	−	0.678471i	$-0.237357\pi$
$659$	37.7172	1.46925	0.734627	+	0.678471i	$0.237357\pi$
$660$	0	0
$661$	17.8456	0.694114	0.347057	−	0.937844i	$-0.387181\pi$
$661$	17.8456	0.694114	0.347057	+	0.937844i	$0.387181\pi$
$662$	0	0
$663$	−0.459354	−0.0178398
$664$	0	0
$665$	−1.12866	−0.0437675
$666$	0	0
$667$	−3.74023	−0.144822
$668$	0	0
$669$	0.968533	0.0374457
$670$	0	0
$671$	−0.498340	−0.0192382
$672$	0	0
$673$	20.9044	0.805806	0.402903	−	0.915243i	$-0.368001\pi$
$673$	20.9044	0.805806	0.402903	+	0.915243i	$0.368001\pi$
$674$	0	0
$675$	4.96879	0.191249
$676$	0	0
$677$	14.0459	0.539827	0.269914	−	0.962885i	$-0.413005\pi$
$677$	14.0459	0.539827	0.269914	+	0.962885i	$0.413005\pi$
$678$	0	0
$679$	−2.42474	−0.0930530
$680$	0	0
$681$	5.41882	0.207650
$682$	0	0
$683$	6.94315	0.265672	0.132836	−	0.991138i	$-0.457592\pi$
$683$	6.94315	0.265672	0.132836	+	0.991138i	$0.457592\pi$
$684$	0	0
$685$	2.05768	0.0786201
$686$	0	0
$687$	20.1638	0.769297
$688$	0	0
$689$	−24.2168	−0.922587
$690$	0	0
$691$	51.8501	1.97247	0.986235	−	0.165352i	$-0.0528761\pi$
$691$	51.8501	1.97247	0.986235	+	0.165352i	$0.0528761\pi$
$692$	0	0
$693$	0.761217	0.0289163
$694$	0	0
$695$	2.31873	0.0879543
$696$	0	0
$697$	0.863203	0.0326962
$698$	0	0
$699$	19.5988	0.741294
$700$	0	0
$701$	38.7254	1.46264	0.731320	−	0.682035i	$-0.238905\pi$
$701$	38.7254	1.46264	0.731320	+	0.682035i	$0.238905\pi$
$702$	0	0
$703$	−1.36381	−0.0514369
$704$	0	0
$705$	−1.19455	−0.0449892
$706$	0	0
$707$	67.7685	2.54870
$708$	0	0
$709$	21.0263	0.789658	0.394829	−	0.918755i	$-0.370804\pi$
$709$	21.0263	0.789658	0.394829	+	0.918755i	$0.370804\pi$
$710$	0	0
$711$	−5.98721	−0.224538
$712$	0	0
$713$	4.51040	0.168916
$714$	0	0
$715$	0.101748	0.00380516
$716$	0	0
$717$	6.28088	0.234564
$718$	0	0
$719$	0.330004	0.0123071	0.00615354	−	0.999981i	$-0.498041\pi$
$719$	0.330004	0.0123071	0.00615354	+	0.999981i	$0.498041\pi$
$720$	0	0
$721$	74.9457	2.79112
$722$	0	0
$723$	2.85707	0.106256
$724$	0	0
$725$	−16.3437	−0.606990
$726$	0	0
$727$	−26.0298	−0.965392	−0.482696	−	0.875788i	$-0.660342\pi$
$727$	−26.0298	−0.965392	−0.482696	+	0.875788i	$0.660342\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.55267	0.0574277
$732$	0	0
$733$	−20.4418	−0.755036	−0.377518	−	0.926002i	$-0.623222\pi$
$733$	−20.4418	−0.755036	−0.377518	+	0.926002i	$0.623222\pi$
$734$	0	0
$735$	1.63082	0.0601538
$736$	0	0
$737$	−0.858205	−0.0316124
$738$	0	0
$739$	41.1300	1.51299	0.756495	−	0.654000i	$-0.226910\pi$
$739$	41.1300	1.51299	0.756495	+	0.654000i	$0.226910\pi$
$740$	0	0
$741$	4.83334	0.177557
$742$	0	0
$743$	22.8866	0.839629	0.419815	−	0.907610i	$-0.362095\pi$
$743$	22.8866	0.839629	0.419815	+	0.907610i	$0.362095\pi$
$744$	0	0
$745$	0.595020	0.0217998
$746$	0	0
$747$	−3.05730	−0.111861
$748$	0	0
$749$	−4.84929	−0.177189
$750$	0	0
$751$	10.2961	0.375709	0.187854	−	0.982197i	$-0.439847\pi$
$751$	10.2961	0.375709	0.187854	+	0.982197i	$0.439847\pi$
$752$	0	0
$753$	−9.85763	−0.359232
$754$	0	0
$755$	0.809693	0.0294677
$756$	0	0
$757$	22.2268	0.807847	0.403924	−	0.914793i	$-0.367646\pi$
$757$	22.2268	0.807847	0.403924	+	0.914793i	$0.367646\pi$
$758$	0	0
$759$	0.214851	0.00779859
$760$	0	0
$761$	−25.6479	−0.929734	−0.464867	−	0.885380i	$-0.653898\pi$
$761$	−25.6479	−0.929734	−0.464867	+	0.885380i	$0.653898\pi$
$762$	0	0
$763$	−17.5252	−0.634454
$764$	0	0
$765$	0.0266251	0.000962634 0
$766$	0	0
$767$	3.39915	0.122736
$768$	0	0
$769$	13.3777	0.482413	0.241207	−	0.970474i	$-0.422457\pi$
$769$	13.3777	0.482413	0.241207	+	0.970474i	$0.422457\pi$
$770$	0	0
$771$	12.8350	0.462242
$772$	0	0
$773$	−9.00800	−0.323995	−0.161998	−	0.986791i	$-0.551794\pi$
$773$	−9.00800	−0.323995	−0.161998	+	0.986791i	$0.551794\pi$
$774$	0	0
$775$	19.7091	0.707972
$776$	0	0
$777$	3.46495	0.124304
$778$	0	0
$779$	−9.08267	−0.325420
$780$	0	0
$781$	−2.85635	−0.102208
$782$	0	0
$783$	−3.28927	−0.117549
$784$	0	0
$785$	3.27050	0.116729
$786$	0	0
$787$	−4.16500	−0.148466	−0.0742331	−	0.997241i	$-0.523651\pi$
$787$	−4.16500	−0.148466	−0.0742331	+	0.997241i	$0.523651\pi$
$788$	0	0
$789$	−22.5355	−0.802285
$790$	0	0
$791$	64.6977	2.30039
$792$	0	0
$793$	8.03913	0.285478
$794$	0	0
$795$	1.40366	0.0497826
$796$	0	0
$797$	−12.6654	−0.448633	−0.224317	−	0.974516i	$-0.572015\pi$
$797$	−12.6654	−0.448633	−0.224317	+	0.974516i	$0.572015\pi$
$798$	0	0
$799$	1.01897	0.0360487
$800$	0	0
$801$	−1.66072	−0.0586788
$802$	0	0
$803$	−1.49412	−0.0527265
$804$	0	0
$805$	0.809347	0.0285257
$806$	0	0
$807$	7.11704	0.250532
$808$	0	0
$809$	18.5491	0.652152	0.326076	−	0.945344i	$-0.394274\pi$
$809$	18.5491	0.652152	0.326076	+	0.945344i	$0.394274\pi$
$810$	0	0
$811$	1.33167	0.0467611	0.0233806	−	0.999727i	$-0.492557\pi$
$811$	1.33167	0.0467611	0.0233806	+	0.999727i	$0.492557\pi$
$812$	0	0
$813$	−18.3911	−0.645006
$814$	0	0
$815$	−2.01126	−0.0704513
$816$	0	0
$817$	−16.3373	−0.571569
$818$	0	0
$819$	−12.2798	−0.429091
$820$	0	0
$821$	−36.8534	−1.28619	−0.643095	−	0.765786i	$-0.722350\pi$
$821$	−36.8534	−1.28619	−0.643095	+	0.765786i	$0.722350\pi$
$822$	0	0
$823$	−10.3020	−0.359107	−0.179553	−	0.983748i	$-0.557465\pi$
$823$	−10.3020	−0.359107	−0.179553	+	0.983748i	$0.557465\pi$
$824$	0	0
$825$	0.938834	0.0326860
$826$	0	0
$827$	−8.32192	−0.289382	−0.144691	−	0.989477i	$-0.546219\pi$
$827$	−8.32192	−0.289382	−0.144691	+	0.989477i	$0.546219\pi$
$828$	0	0
$829$	7.59333	0.263727	0.131864	−	0.991268i	$-0.457904\pi$
$829$	7.59333	0.263727	0.131864	+	0.991268i	$0.457904\pi$
$830$	0	0
$831$	−7.00797	−0.243104
$832$	0	0
$833$	−1.39113	−0.0481996
$834$	0	0
$835$	−1.56491	−0.0541558
$836$	0	0
$837$	3.96658	0.137105
$838$	0	0
$839$	45.6352	1.57550	0.787752	−	0.615993i	$-0.211245\pi$
$839$	45.6352	1.57550	0.787752	+	0.615993i	$0.211245\pi$
$840$	0	0
$841$	−18.1807	−0.626920
$842$	0	0
$843$	6.49560	0.223720
$844$	0	0
$845$	0.655347	0.0225446
$846$	0	0
$847$	−44.1724	−1.51778
$848$	0	0
$849$	−2.14424	−0.0735900
$850$	0	0
$851$	0.977969	0.0335244
$852$	0	0
$853$	7.06248	0.241815	0.120907	−	0.992664i	$-0.461420\pi$
$853$	7.06248	0.241815	0.120907	+	0.992664i	$0.461420\pi$
$854$	0	0
$855$	−0.280151	−0.00958096
$856$	0	0
$857$	−50.2740	−1.71733	−0.858663	−	0.512540i	$-0.828705\pi$
$857$	−50.2740	−1.71733	−0.858663	+	0.512540i	$0.828705\pi$
$858$	0	0
$859$	−4.16903	−0.142246	−0.0711228	−	0.997468i	$-0.522658\pi$
$859$	−4.16903	−0.142246	−0.0711228	+	0.997468i	$0.522658\pi$
$860$	0	0
$861$	23.0758	0.786423
$862$	0	0
$863$	19.6442	0.668697	0.334348	−	0.942450i	$-0.391484\pi$
$863$	19.6442	0.668697	0.334348	+	0.942450i	$0.391484\pi$
$864$	0	0
$865$	1.30933	0.0445184
$866$	0	0
$867$	16.9773	0.576579
$868$	0	0
$869$	−1.13126	−0.0383754
$870$	0	0
$871$	13.8444	0.469100
$872$	0	0
$873$	−0.601859	−0.0203699
$874$	0	0
$875$	7.09543	0.239869
$876$	0	0
$877$	10.0669	0.339936	0.169968	−	0.985450i	$-0.445634\pi$
$877$	10.0669	0.339936	0.169968	+	0.985450i	$0.445634\pi$
$878$	0	0
$879$	13.1782	0.444490
$880$	0	0
$881$	−22.3691	−0.753635	−0.376817	−	0.926288i	$-0.622982\pi$
$881$	−22.3691	−0.753635	−0.376817	+	0.926288i	$0.622982\pi$
$882$	0	0
$883$	50.2564	1.69126	0.845630	−	0.533769i	$-0.179225\pi$
$883$	50.2564	1.69126	0.845630	+	0.533769i	$0.179225\pi$
$884$	0	0
$885$	−0.197022	−0.00662282
$886$	0	0
$887$	−23.6912	−0.795473	−0.397736	−	0.917500i	$-0.630204\pi$
$887$	−23.6912	−0.795473	−0.397736	+	0.917500i	$0.630204\pi$
$888$	0	0
$889$	−11.3428	−0.380424
$890$	0	0
$891$	0.188946	0.00632994
$892$	0	0
$893$	−10.7217	−0.358787
$894$	0	0
$895$	−1.32802	−0.0443907
$896$	0	0
$897$	−3.46593	−0.115724
$898$	0	0
$899$	−13.0472	−0.435148
$900$	0	0
$901$	−1.19735	−0.0398895
$902$	0	0
$903$	41.5073	1.38128
$904$	0	0
$905$	1.08498	0.0360659
$906$	0	0
$907$	6.46269	0.214590	0.107295	−	0.994227i	$-0.465781\pi$
$907$	6.46269	0.214590	0.107295	+	0.994227i	$0.465781\pi$
$908$	0	0
$909$	16.8212	0.557924
$910$	0	0
$911$	8.03186	0.266107	0.133054	−	0.991109i	$-0.457522\pi$
$911$	8.03186	0.266107	0.133054	+	0.991109i	$0.457522\pi$
$912$	0	0
$913$	−0.577665	−0.0191179
$914$	0	0
$915$	−0.465965	−0.0154043
$916$	0	0
$917$	74.5364	2.46141
$918$	0	0
$919$	−15.3110	−0.505062	−0.252531	−	0.967589i	$-0.581263\pi$
$919$	−15.3110	−0.505062	−0.252531	+	0.967589i	$0.581263\pi$
$920$	0	0
$921$	21.2510	0.700245
$922$	0	0
$923$	46.0781	1.51668
$924$	0	0
$925$	4.27343	0.140510
$926$	0	0
$927$	18.6027	0.610994
$928$	0	0
$929$	4.47453	0.146805	0.0734023	−	0.997302i	$-0.476614\pi$
$929$	4.47453	0.146805	0.0734023	+	0.997302i	$0.476614\pi$
$930$	0	0
$931$	14.6375	0.479724
$932$	0	0
$933$	24.8338	0.813023
$934$	0	0
$935$	0.00503072	0.000164522 0
$936$	0	0
$937$	36.2222	1.18333	0.591665	−	0.806184i	$-0.298471\pi$
$937$	36.2222	1.18333	0.591665	+	0.806184i	$0.298471\pi$
$938$	0	0
$939$	5.32724	0.173848
$940$	0	0
$941$	27.2352	0.887841	0.443920	−	0.896066i	$-0.353587\pi$
$941$	27.2352	0.887841	0.443920	+	0.896066i	$0.353587\pi$
$942$	0	0
$943$	6.51307	0.212095
$944$	0	0
$945$	0.711764	0.0231537
$946$	0	0
$947$	−8.37982	−0.272307	−0.136154	−	0.990688i	$-0.543474\pi$
$947$	−8.37982	−0.272307	−0.136154	+	0.990688i	$0.543474\pi$
$948$	0	0
$949$	24.1029	0.782415
$950$	0	0
$951$	−19.5937	−0.635371
$952$	0	0
$953$	−37.8802	−1.22706	−0.613530	−	0.789671i	$-0.710251\pi$
$953$	−37.8802	−1.22706	−0.613530	+	0.789671i	$0.710251\pi$
$954$	0	0
$955$	0.0120306	0.000389300 0
$956$	0	0
$957$	−0.621496	−0.0200901
$958$	0	0
$959$	−46.9227	−1.51521
$960$	0	0
$961$	−15.2662	−0.492458
$962$	0	0
$963$	−1.20367	−0.0387878
$964$	0	0
$965$	−1.12065	−0.0360750
$966$	0	0
$967$	−47.3094	−1.52137	−0.760684	−	0.649122i	$-0.775136\pi$
$967$	−47.3094	−1.52137	−0.760684	+	0.649122i	$0.775136\pi$
$968$	0	0
$969$	0.238975	0.00767697
$970$	0	0
$971$	21.4559	0.688553	0.344276	−	0.938868i	$-0.388124\pi$
$971$	21.4559	0.688553	0.344276	+	0.938868i	$0.388124\pi$
$972$	0	0
$973$	−52.8754	−1.69511
$974$	0	0
$975$	−15.1451	−0.485031
$976$	0	0
$977$	−5.39689	−0.172662	−0.0863310	−	0.996267i	$-0.527514\pi$
$977$	−5.39689	−0.172662	−0.0863310	+	0.996267i	$0.527514\pi$
$978$	0	0
$979$	−0.313787	−0.0100287
$980$	0	0
$981$	−4.35003	−0.138886
$982$	0	0
$983$	2.57941	0.0822705	0.0411353	−	0.999154i	$-0.486903\pi$
$983$	2.57941	0.0822705	0.0411353	+	0.999154i	$0.486903\pi$
$984$	0	0
$985$	−3.09295	−0.0985496
$986$	0	0
$987$	27.2400	0.867058
$988$	0	0
$989$	11.7153	0.372524
$990$	0	0
$991$	12.8023	0.406677	0.203339	−	0.979108i	$-0.434821\pi$
$991$	12.8023	0.406677	0.203339	+	0.979108i	$0.434821\pi$
$992$	0	0
$993$	−19.1464	−0.607592
$994$	0	0
$995$	−2.24301	−0.0711081
$996$	0	0
$997$	−37.0666	−1.17391	−0.586956	−	0.809619i	$-0.699674\pi$
$997$	−37.0666	−1.17391	−0.586956	+	0.809619i	$0.699674\pi$
$998$	0	0
$999$	0.860056	0.0272110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.f.1.7	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.f.1.7	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.176671 q^{5} +4.02875 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.176671 q^{5} +4.02875 q^{7} +1.00000 q^{9} +0.188946 q^{11} -3.04805 q^{13} +0.176671 q^{15} -0.150704 q^{17} +1.58572 q^{19} -4.02875 q^{21} -1.13710 q^{23} -4.96879 q^{25} -1.00000 q^{27} +3.28927 q^{29} -3.96658 q^{31} -0.188946 q^{33} -0.711764 q^{35} -0.860056 q^{37} +3.04805 q^{39} -5.72779 q^{41} -10.3028 q^{43} -0.176671 q^{45} -6.76140 q^{47} +9.23082 q^{49} +0.150704 q^{51} +7.94502 q^{53} -0.0333814 q^{55} -1.58572 q^{57} -1.11519 q^{59} -2.63747 q^{61} +4.02875 q^{63} +0.538502 q^{65} -4.54206 q^{67} +1.13710 q^{69} -15.1173 q^{71} -7.90767 q^{73} +4.96879 q^{75} +0.761217 q^{77} -5.98721 q^{79} +1.00000 q^{81} -3.05730 q^{83} +0.0266251 q^{85} -3.28927 q^{87} -1.66072 q^{89} -12.2798 q^{91} +3.96658 q^{93} -0.280151 q^{95} -0.601859 q^{97} +0.188946 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 6 q^{5} - 7 q^{7} + 14 q^{9} + q^{11} - q^{13} + 6 q^{15} - 6 q^{17} + q^{19} + 7 q^{21} + 10 q^{23} - 10 q^{25} - 14 q^{27} - 6 q^{29} - 5 q^{31} - q^{33} + 17 q^{35} - 12 q^{37} + q^{39} - 21 q^{41} + 8 q^{43} - 6 q^{45} + 18 q^{47} - 19 q^{49} + 6 q^{51} - q^{53} - q^{57} + 14 q^{59} - 19 q^{61} - 7 q^{63} + 17 q^{65} + 17 q^{67} - 10 q^{69} - 13 q^{71} - 12 q^{73} + 10 q^{75} - 9 q^{77} - 8 q^{79} + 14 q^{81} + 11 q^{83} - 17 q^{85} + 6 q^{87} - 9 q^{89} - 5 q^{91} + 5 q^{93} + 8 q^{95} - 25 q^{97} + q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 6 * q^5 - 7 * q^7 + 14 * q^9 + q^11 - q^13 + 6 * q^15 - 6 * q^17 + q^19 + 7 * q^21 + 10 * q^23 - 10 * q^25 - 14 * q^27 - 6 * q^29 - 5 * q^31 - q^33 + 17 * q^35 - 12 * q^37 + q^39 - 21 * q^41 + 8 * q^43 - 6 * q^45 + 18 * q^47 - 19 * q^49 + 6 * q^51 - q^53 - q^57 + 14 * q^59 - 19 * q^61 - 7 * q^63 + 17 * q^65 + 17 * q^67 - 10 * q^69 - 13 * q^71 - 12 * q^73 + 10 * q^75 - 9 * q^77 - 8 * q^79 + 14 * q^81 + 11 * q^83 - 17 * q^85 + 6 * q^87 - 9 * q^89 - 5 * q^91 + 5 * q^93 + 8 * q^95 - 25 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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