LMFDB - Embedded newform 4005.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, 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("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.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:	$31.9800860095$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 8x^{5} + 6x^{4} + 19x^{3} - 10x^{2} - 12x + 6 $ x^7 - x^6 - 8x^5 + 6x^4 + 19x^3 - 10x^2 - 12*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 445)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.49803$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} +O(q^{10})$	\(q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} -0.755898 q^{10} -4.45116 q^{11} -2.43229 q^{13} +0.0376612 q^{14} +0.898186 q^{16} +2.48065 q^{17} -5.16842 q^{19} +1.42862 q^{20} -3.36463 q^{22} +4.99050 q^{23} +1.00000 q^{25} -1.83856 q^{26} -0.0711781 q^{28} -2.59152 q^{29} -7.31064 q^{31} +5.86231 q^{32} +1.87512 q^{34} -0.0498231 q^{35} +5.13825 q^{37} -3.90680 q^{38} +2.59169 q^{40} -9.11073 q^{41} -0.543007 q^{43} +6.35901 q^{44} +3.77231 q^{46} +9.63395 q^{47} -6.99752 q^{49} +0.755898 q^{50} +3.47482 q^{52} +10.7093 q^{53} +4.45116 q^{55} -0.129126 q^{56} -1.95893 q^{58} +12.6337 q^{59} -2.47399 q^{61} -5.52610 q^{62} +2.63494 q^{64} +2.43229 q^{65} -5.36128 q^{67} -3.54390 q^{68} -0.0376612 q^{70} -9.58146 q^{71} +2.86369 q^{73} +3.88400 q^{74} +7.38370 q^{76} -0.221771 q^{77} +6.62473 q^{79} -0.898186 q^{80} -6.88678 q^{82} +12.0061 q^{83} -2.48065 q^{85} -0.410458 q^{86} +11.5360 q^{88} -1.00000 q^{89} -0.121184 q^{91} -7.12952 q^{92} +7.28229 q^{94} +5.16842 q^{95} +4.82051 q^{97} -5.28941 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8}+O(q^{10}) $ 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8	$ 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8} - 4 q^{10} + 10 q^{11} - 7 q^{13} - 3 q^{14} + 10 q^{16} + 13 q^{17} - 7 q^{19} - 8 q^{20} + 2 q^{22} + 13 q^{23} + 7 q^{25} - q^{26} - 21 q^{28} + 4 q^{29} + q^{31} + 13 q^{32} + 10 q^{34} + 16 q^{35} - 5 q^{37} + 40 q^{38} - 12 q^{40} - 5 q^{41} - 31 q^{43} + 21 q^{44} + 16 q^{46} + 14 q^{47} + 19 q^{49} + 4 q^{50} + 13 q^{53} - 10 q^{55} + q^{56} + 17 q^{58} + 14 q^{59} + 3 q^{61} - 26 q^{62} + 14 q^{64} + 7 q^{65} + q^{67} + 35 q^{68} + 3 q^{70} + 8 q^{71} + 9 q^{73} + 35 q^{74} + 40 q^{76} - 42 q^{77} + 9 q^{79} - 10 q^{80} + 29 q^{82} + 42 q^{83} - 13 q^{85} - 35 q^{86} + 30 q^{88} - 7 q^{89} + 31 q^{91} - 19 q^{92} + 37 q^{94} + 7 q^{95} - 7 q^{97} - 9 q^{98}+O(q^{100}) $ 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8 - 4 * q^10 + 10 * q^11 - 7 * q^13 - 3 * q^14 + 10 * q^16 + 13 * q^17 - 7 * q^19 - 8 * q^20 + 2 * q^22 + 13 * q^23 + 7 * q^25 - q^26 - 21 * q^28 + 4 * q^29 + q^31 + 13 * q^32 + 10 * q^34 + 16 * q^35 - 5 * q^37 + 40 * q^38 - 12 * q^40 - 5 * q^41 - 31 * q^43 + 21 * q^44 + 16 * q^46 + 14 * q^47 + 19 * q^49 + 4 * q^50 + 13 * q^53 - 10 * q^55 + q^56 + 17 * q^58 + 14 * q^59 + 3 * q^61 - 26 * q^62 + 14 * q^64 + 7 * q^65 + q^67 + 35 * q^68 + 3 * q^70 + 8 * q^71 + 9 * q^73 + 35 * q^74 + 40 * q^76 - 42 * q^77 + 9 * q^79 - 10 * q^80 + 29 * q^82 + 42 * q^83 - 13 * q^85 - 35 * q^86 + 30 * q^88 - 7 * q^89 + 31 * q^91 - 19 * q^92 + 37 * q^94 + 7 * q^95 - 7 * q^97 - 9 * q^98

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 $

$2$	0.755898	0.534501	0.267250	−	0.963627i	$-0.413885\pi$
$2$	0.755898	0.534501	0.267250	+	0.963627i	$0.413885\pi$
$3$	0	0
$3$	0	0
$4$	−1.42862	−0.714309
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.0498231	0.0188314	0.00941568	−	0.999956i	$-0.497003\pi$
$7$	0.0498231	0.0188314	0.00941568	+	0.999956i	$0.497003\pi$
$8$	−2.59169	−0.916299
$9$	0	0
$10$	−0.755898	−0.239036
$11$	−4.45116	−1.34208	−0.671038	−	0.741423i	$-0.734151\pi$
$11$	−4.45116	−1.34208	−0.671038	+	0.741423i	$0.734151\pi$
$12$	0	0
$13$	−2.43229	−0.674596	−0.337298	−	0.941398i	$-0.609513\pi$
$13$	−2.43229	−0.674596	−0.337298	+	0.941398i	$0.609513\pi$
$14$	0.0376612	0.0100654
$15$	0	0
$16$	0.898186	0.224546
$17$	2.48065	0.601646	0.300823	−	0.953680i	$-0.402739\pi$
$17$	2.48065	0.601646	0.300823	+	0.953680i	$0.402739\pi$
$18$	0	0
$19$	−5.16842	−1.18572	−0.592858	−	0.805307i	$-0.702001\pi$
$19$	−5.16842	−1.18572	−0.592858	+	0.805307i	$0.702001\pi$
$20$	1.42862	0.319449
$21$	0	0
$22$	−3.36463	−0.717341
$23$	4.99050	1.04059	0.520295	−	0.853986i	$-0.325822\pi$
$23$	4.99050	1.04059	0.520295	+	0.853986i	$0.325822\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.83856	−0.360572
$27$	0	0
$28$	−0.0711781	−0.0134514
$29$	−2.59152	−0.481234	−0.240617	−	0.970620i	$-0.577350\pi$
$29$	−2.59152	−0.481234	−0.240617	+	0.970620i	$0.577350\pi$
$30$	0	0
$31$	−7.31064	−1.31303	−0.656515	−	0.754313i	$-0.727970\pi$
$31$	−7.31064	−1.31303	−0.656515	+	0.754313i	$0.727970\pi$
$32$	5.86231	1.03632
$33$	0	0
$34$	1.87512	0.321580
$35$	−0.0498231	−0.00842164
$36$	0	0
$37$	5.13825	0.844724	0.422362	−	0.906427i	$-0.361201\pi$
$37$	5.13825	0.844724	0.422362	+	0.906427i	$0.361201\pi$
$38$	−3.90680	−0.633766
$39$	0	0
$40$	2.59169	0.409782
$41$	−9.11073	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$41$	−9.11073	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$42$	0	0
$43$	−0.543007	−0.0828078	−0.0414039	−	0.999142i	$-0.513183\pi$
$43$	−0.543007	−0.0828078	−0.0414039	+	0.999142i	$0.513183\pi$
$44$	6.35901	0.958657
$45$	0	0
$46$	3.77231	0.556196
$47$	9.63395	1.40526	0.702628	−	0.711557i	$-0.252010\pi$
$47$	9.63395	1.40526	0.702628	+	0.711557i	$0.252010\pi$
$48$	0	0
$49$	−6.99752	−0.999645
$50$	0.755898	0.106900
$51$	0	0
$52$	3.47482	0.481870
$53$	10.7093	1.47104	0.735519	−	0.677504i	$-0.236938\pi$
$53$	10.7093	1.47104	0.735519	+	0.677504i	$0.236938\pi$
$54$	0	0
$55$	4.45116	0.600195
$56$	−0.129126	−0.0172552
$57$	0	0
$58$	−1.95893	−0.257220
$59$	12.6337	1.64477	0.822383	−	0.568934i	$-0.192644\pi$
$59$	12.6337	1.64477	0.822383	+	0.568934i	$0.192644\pi$
$60$	0	0
$61$	−2.47399	−0.316762	−0.158381	−	0.987378i	$-0.550627\pi$
$61$	−2.47399	−0.316762	−0.158381	+	0.987378i	$0.550627\pi$
$62$	−5.52610	−0.701816
$63$	0	0
$64$	2.63494	0.329367
$65$	2.43229	0.301689
$66$	0	0
$67$	−5.36128	−0.654984	−0.327492	−	0.944854i	$-0.606203\pi$
$67$	−5.36128	−0.654984	−0.327492	+	0.944854i	$0.606203\pi$
$68$	−3.54390	−0.429761
$69$	0	0
$70$	−0.0376612	−0.00450137
$71$	−9.58146	−1.13711	−0.568555	−	0.822645i	$-0.692497\pi$
$71$	−9.58146	−1.13711	−0.568555	+	0.822645i	$0.692497\pi$
$72$	0	0
$73$	2.86369	0.335169	0.167585	−	0.985858i	$-0.446403\pi$
$73$	2.86369	0.335169	0.167585	+	0.985858i	$0.446403\pi$
$74$	3.88400	0.451505
$75$	0	0
$76$	7.38370	0.846968
$77$	−0.221771	−0.0252731
$78$	0	0
$79$	6.62473	0.745340	0.372670	−	0.927964i	$-0.378442\pi$
$79$	6.62473	0.745340	0.372670	+	0.927964i	$0.378442\pi$
$80$	−0.898186	−0.100420
$81$	0	0
$82$	−6.88678	−0.760518
$83$	12.0061	1.31784	0.658919	−	0.752214i	$-0.271014\pi$
$83$	12.0061	1.31784	0.658919	+	0.752214i	$0.271014\pi$
$84$	0	0
$85$	−2.48065	−0.269064
$86$	−0.410458	−0.0442608
$87$	0	0
$88$	11.5360	1.22974
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−0.121184	−0.0127036
$92$	−7.12952	−0.743303
$93$	0	0
$94$	7.28229	0.751111
$95$	5.16842	0.530269
$96$	0	0
$97$	4.82051	0.489449	0.244724	−	0.969593i	$-0.421303\pi$
$97$	4.82051	0.489449	0.244724	+	0.969593i	$0.421303\pi$
$98$	−5.28941	−0.534311
$99$	0	0
$100$	−1.42862	−0.142862
$101$	11.5561	1.14988	0.574938	−	0.818197i	$-0.305026\pi$
$101$	11.5561	1.14988	0.574938	+	0.818197i	$0.305026\pi$
$102$	0	0
$103$	−2.92606	−0.288314	−0.144157	−	0.989555i	$-0.546047\pi$
$103$	−2.92606	−0.288314	−0.144157	+	0.989555i	$0.546047\pi$
$104$	6.30374	0.618132
$105$	0	0
$106$	8.09516	0.786271
$107$	3.97982	0.384744	0.192372	−	0.981322i	$-0.438382\pi$
$107$	3.97982	0.384744	0.192372	+	0.981322i	$0.438382\pi$
$108$	0	0
$109$	5.30200	0.507840	0.253920	−	0.967225i	$-0.418280\pi$
$109$	5.30200	0.507840	0.253920	+	0.967225i	$0.418280\pi$
$110$	3.36463	0.320805
$111$	0	0
$112$	0.0447504	0.00422851
$113$	10.3888	0.977296	0.488648	−	0.872481i	$-0.337490\pi$
$113$	10.3888	0.977296	0.488648	+	0.872481i	$0.337490\pi$
$114$	0	0
$115$	−4.99050	−0.465366
$116$	3.70230	0.343750
$117$	0	0
$118$	9.54978	0.879129
$119$	0.123594	0.0113298
$120$	0	0
$121$	8.81287	0.801170
$122$	−1.87008	−0.169310
$123$	0	0
$124$	10.4441	0.937909
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.31244	0.471403	0.235701	−	0.971826i	$-0.424261\pi$
$127$	5.31244	0.471403	0.235701	+	0.971826i	$0.424261\pi$
$128$	−9.73287	−0.860273
$129$	0	0
$130$	1.83856	0.161253
$131$	22.2268	1.94197	0.970984	−	0.239145i	$-0.0768671\pi$
$131$	22.2268	1.94197	0.970984	+	0.239145i	$0.0768671\pi$
$132$	0	0
$133$	−0.257507	−0.0223287
$134$	−4.05258	−0.350089
$135$	0	0
$136$	−6.42906	−0.551288
$137$	−8.08077	−0.690387	−0.345193	−	0.938532i	$-0.612187\pi$
$137$	−8.08077	−0.690387	−0.345193	+	0.938532i	$0.612187\pi$
$138$	0	0
$139$	0.763404	0.0647511	0.0323755	−	0.999476i	$-0.489693\pi$
$139$	0.763404	0.0647511	0.0323755	+	0.999476i	$0.489693\pi$
$140$	0.0711781	0.00601565
$141$	0	0
$142$	−7.24261	−0.607786
$143$	10.8265	0.905360
$144$	0	0
$145$	2.59152	0.215214
$146$	2.16466	0.179148
$147$	0	0
$148$	−7.34060	−0.603394
$149$	0.645836	0.0529089	0.0264545	−	0.999650i	$-0.491578\pi$
$149$	0.645836	0.0529089	0.0264545	+	0.999650i	$0.491578\pi$
$150$	0	0
$151$	−16.4449	−1.33827	−0.669134	−	0.743142i	$-0.733335\pi$
$151$	−16.4449	−1.33827	−0.669134	+	0.743142i	$0.733335\pi$
$152$	13.3949	1.08647
$153$	0	0
$154$	−0.167636	−0.0135085
$155$	7.31064	0.587205
$156$	0	0
$157$	8.23439	0.657176	0.328588	−	0.944473i	$-0.393427\pi$
$157$	8.23439	0.657176	0.328588	+	0.944473i	$0.393427\pi$
$158$	5.00762	0.398385
$159$	0	0
$160$	−5.86231	−0.463456
$161$	0.248642	0.0195957
$162$	0	0
$163$	1.57747	0.123557	0.0617786	−	0.998090i	$-0.480323\pi$
$163$	1.57747	0.123557	0.0617786	+	0.998090i	$0.480323\pi$
$164$	13.0157	1.01636
$165$	0	0
$166$	9.07537	0.704385
$167$	−1.78625	−0.138224	−0.0691121	−	0.997609i	$-0.522017\pi$
$167$	−1.78625	−0.138224	−0.0691121	+	0.997609i	$0.522017\pi$
$168$	0	0
$169$	−7.08396	−0.544920
$170$	−1.87512	−0.143815
$171$	0	0
$172$	0.775750	0.0591504
$173$	−23.0808	−1.75480	−0.877399	−	0.479761i	$-0.840723\pi$
$173$	−23.0808	−1.75480	−0.877399	+	0.479761i	$0.840723\pi$
$174$	0	0
$175$	0.0498231	0.00376627
$176$	−3.99797	−0.301359
$177$	0	0
$178$	−0.755898	−0.0566570
$179$	9.63601	0.720229	0.360115	−	0.932908i	$-0.382738\pi$
$179$	9.63601	0.720229	0.360115	+	0.932908i	$0.382738\pi$
$180$	0	0
$181$	−7.10071	−0.527792	−0.263896	−	0.964551i	$-0.585008\pi$
$181$	−7.10071	−0.527792	−0.263896	+	0.964551i	$0.585008\pi$
$182$	−0.0916030	−0.00679006
$183$	0	0
$184$	−12.9338	−0.953493
$185$	−5.13825	−0.377772
$186$	0	0
$187$	−11.0418	−0.807455
$188$	−13.7632	−1.00379
$189$	0	0
$190$	3.90680	0.283429
$191$	−9.47202	−0.685371	−0.342686	−	0.939450i	$-0.611337\pi$
$191$	−9.47202	−0.685371	−0.342686	+	0.939450i	$0.611337\pi$
$192$	0	0
$193$	11.7250	0.843985	0.421992	−	0.906599i	$-0.361331\pi$
$193$	11.7250	0.843985	0.421992	+	0.906599i	$0.361331\pi$
$194$	3.64382	0.261611
$195$	0	0
$196$	9.99678	0.714056
$197$	18.2857	1.30280	0.651402	−	0.758733i	$-0.274181\pi$
$197$	18.2857	1.30280	0.651402	+	0.758733i	$0.274181\pi$
$198$	0	0
$199$	25.0469	1.77553	0.887763	−	0.460302i	$-0.152259\pi$
$199$	25.0469	1.77553	0.887763	+	0.460302i	$0.152259\pi$
$200$	−2.59169	−0.183260
$201$	0	0
$202$	8.73524	0.614609
$203$	−0.129118	−0.00906229
$204$	0	0
$205$	9.11073	0.636321
$206$	−2.21181	−0.154104
$207$	0	0
$208$	−2.18465	−0.151478
$209$	23.0055	1.59132
$210$	0	0
$211$	−17.6282	−1.21358	−0.606789	−	0.794863i	$-0.707543\pi$
$211$	−17.6282	−1.21358	−0.606789	+	0.794863i	$0.707543\pi$
$212$	−15.2995	−1.05078
$213$	0	0
$214$	3.00834	0.205646
$215$	0.543007	0.0370328
$216$	0	0
$217$	−0.364239	−0.0247261
$218$	4.00777	0.271441
$219$	0	0
$220$	−6.35901	−0.428725
$221$	−6.03366	−0.405868
$222$	0	0
$223$	27.8359	1.86403	0.932014	−	0.362421i	$-0.118050\pi$
$223$	27.8359	1.86403	0.932014	+	0.362421i	$0.118050\pi$
$224$	0.292078	0.0195153
$225$	0	0
$226$	7.85288	0.522366
$227$	−10.7150	−0.711181	−0.355591	−	0.934642i	$-0.615720\pi$
$227$	−10.7150	−0.711181	−0.355591	+	0.934642i	$0.615720\pi$
$228$	0	0
$229$	−20.0837	−1.32717	−0.663584	−	0.748102i	$-0.730966\pi$
$229$	−20.0837	−1.32717	−0.663584	+	0.748102i	$0.730966\pi$
$230$	−3.77231	−0.248739
$231$	0	0
$232$	6.71642	0.440954
$233$	−1.64318	−0.107648	−0.0538242	−	0.998550i	$-0.517141\pi$
$233$	−1.64318	−0.107648	−0.0538242	+	0.998550i	$0.517141\pi$
$234$	0	0
$235$	−9.63395	−0.628450
$236$	−18.0487	−1.17487
$237$	0	0
$238$	0.0934241	0.00605579
$239$	12.3117	0.796379	0.398189	−	0.917303i	$-0.369639\pi$
$239$	12.3117	0.796379	0.398189	+	0.917303i	$0.369639\pi$
$240$	0	0
$241$	−10.7917	−0.695152	−0.347576	−	0.937652i	$-0.612995\pi$
$241$	−10.7917	−0.695152	−0.347576	+	0.937652i	$0.612995\pi$
$242$	6.66163	0.428226
$243$	0	0
$244$	3.53439	0.226266
$245$	6.99752	0.447055
$246$	0	0
$247$	12.5711	0.799880
$248$	18.9469	1.20313
$249$	0	0
$250$	−0.755898	−0.0478072
$251$	31.0270	1.95840	0.979202	−	0.202888i	$-0.0650326\pi$
$251$	31.0270	1.95840	0.979202	+	0.202888i	$0.0650326\pi$
$252$	0	0
$253$	−22.2135	−1.39655
$254$	4.01566	0.251965
$255$	0	0
$256$	−12.6269	−0.789183
$257$	−16.5586	−1.03290	−0.516449	−	0.856318i	$-0.672747\pi$
$257$	−16.5586	−1.03290	−0.516449	+	0.856318i	$0.672747\pi$
$258$	0	0
$259$	0.256004	0.0159073
$260$	−3.47482	−0.215499
$261$	0	0
$262$	16.8012	1.03798
$263$	12.0270	0.741619	0.370810	−	0.928709i	$-0.379080\pi$
$263$	12.0270	0.741619	0.370810	+	0.928709i	$0.379080\pi$
$264$	0	0
$265$	−10.7093	−0.657868
$266$	−0.194649	−0.0119347
$267$	0	0
$268$	7.65922	0.467861
$269$	22.0871	1.34667	0.673336	−	0.739336i	$-0.264861\pi$
$269$	22.0871	1.34667	0.673336	+	0.739336i	$0.264861\pi$
$270$	0	0
$271$	13.8002	0.838302	0.419151	−	0.907917i	$-0.362328\pi$
$271$	13.8002	0.838302	0.419151	+	0.907917i	$0.362328\pi$
$272$	2.22808	0.135097
$273$	0	0
$274$	−6.10824	−0.369012
$275$	−4.45116	−0.268415
$276$	0	0
$277$	3.04290	0.182830	0.0914151	−	0.995813i	$-0.470861\pi$
$277$	3.04290	0.182830	0.0914151	+	0.995813i	$0.470861\pi$
$278$	0.577056	0.0346095
$279$	0	0
$280$	0.129126	0.00771674
$281$	−21.1218	−1.26002	−0.630011	−	0.776586i	$-0.716950\pi$
$281$	−21.1218	−1.26002	−0.630011	+	0.776586i	$0.716950\pi$
$282$	0	0
$283$	32.4871	1.93116	0.965579	−	0.260111i	$-0.0837590\pi$
$283$	32.4871	1.93116	0.965579	+	0.260111i	$0.0837590\pi$
$284$	13.6882	0.812248
$285$	0	0
$286$	8.18376	0.483916
$287$	−0.453924	−0.0267943
$288$	0	0
$289$	−10.8464	−0.638022
$290$	1.95893	0.115032
$291$	0	0
$292$	−4.09112	−0.239414
$293$	−32.0101	−1.87005	−0.935026	−	0.354578i	$-0.884624\pi$
$293$	−32.0101	−1.87005	−0.935026	+	0.354578i	$0.884624\pi$
$294$	0	0
$295$	−12.6337	−0.735562
$296$	−13.3167	−0.774020
$297$	0	0
$298$	0.488186	0.0282798
$299$	−12.1383	−0.701979
$300$	0	0
$301$	−0.0270543	−0.00155938
$302$	−12.4307	−0.715305
$303$	0	0
$304$	−4.64220	−0.266248
$305$	2.47399	0.141660
$306$	0	0
$307$	−19.6188	−1.11970	−0.559852	−	0.828593i	$-0.689142\pi$
$307$	−19.6188	−1.11970	−0.559852	+	0.828593i	$0.689142\pi$
$308$	0.316826	0.0180528
$309$	0	0
$310$	5.52610	0.313861
$311$	−24.8863	−1.41117	−0.705586	−	0.708625i	$-0.749316\pi$
$311$	−24.8863	−1.41117	−0.705586	+	0.708625i	$0.749316\pi$
$312$	0	0
$313$	−6.17328	−0.348934	−0.174467	−	0.984663i	$-0.555820\pi$
$313$	−6.17328	−0.348934	−0.174467	+	0.984663i	$0.555820\pi$
$314$	6.22436	0.351261
$315$	0	0
$316$	−9.46420	−0.532403
$317$	32.9455	1.85041	0.925203	−	0.379473i	$-0.123895\pi$
$317$	32.9455	1.85041	0.925203	+	0.379473i	$0.123895\pi$
$318$	0	0
$319$	11.5353	0.645853
$320$	−2.63494	−0.147297
$321$	0	0
$322$	0.187948	0.0104739
$323$	−12.8210	−0.713381
$324$	0	0
$325$	−2.43229	−0.134919
$326$	1.19241	0.0660414
$327$	0	0
$328$	23.6121	1.30376
$329$	0.479993	0.0264629
$330$	0	0
$331$	30.9785	1.70273	0.851367	−	0.524570i	$-0.175774\pi$
$331$	30.9785	1.70273	0.851367	+	0.524570i	$0.175774\pi$
$332$	−17.1521	−0.941343
$333$	0	0
$334$	−1.35022	−0.0738809
$335$	5.36128	0.292918
$336$	0	0
$337$	−3.53709	−0.192678	−0.0963389	−	0.995349i	$-0.530713\pi$
$337$	−3.53709	−0.192678	−0.0963389	+	0.995349i	$0.530713\pi$
$338$	−5.35475	−0.291260
$339$	0	0
$340$	3.54390	0.192195
$341$	32.5409	1.76219
$342$	0	0
$343$	−0.697399	−0.0376560
$344$	1.40730	0.0758768
$345$	0	0
$346$	−17.4467	−0.937941
$347$	13.0198	0.698939	0.349470	−	0.936948i	$-0.386362\pi$
$347$	13.0198	0.698939	0.349470	+	0.936948i	$0.386362\pi$
$348$	0	0
$349$	27.6562	1.48040	0.740201	−	0.672385i	$-0.234730\pi$
$349$	27.6562	1.48040	0.740201	+	0.672385i	$0.234730\pi$
$350$	0.0376612	0.00201307
$351$	0	0
$352$	−26.0941	−1.39082
$353$	26.3090	1.40029	0.700144	−	0.714002i	$-0.253119\pi$
$353$	26.3090	1.40029	0.700144	+	0.714002i	$0.253119\pi$
$354$	0	0
$355$	9.58146	0.508531
$356$	1.42862	0.0757166
$357$	0	0
$358$	7.28384	0.384963
$359$	−14.0008	−0.738934	−0.369467	−	0.929244i	$-0.620460\pi$
$359$	−14.0008	−0.738934	−0.369467	+	0.929244i	$0.620460\pi$
$360$	0	0
$361$	7.71256	0.405924
$362$	−5.36741	−0.282105
$363$	0	0
$364$	0.173126	0.00907427
$365$	−2.86369	−0.149892
$366$	0	0
$367$	22.0304	1.14998	0.574989	−	0.818161i	$-0.305006\pi$
$367$	22.0304	1.14998	0.574989	+	0.818161i	$0.305006\pi$
$368$	4.48239	0.233661
$369$	0	0
$370$	−3.88400	−0.201919
$371$	0.533571	0.0277016
$372$	0	0
$373$	13.6705	0.707830	0.353915	−	0.935278i	$-0.384850\pi$
$373$	13.6705	0.707830	0.353915	+	0.935278i	$0.384850\pi$
$374$	−8.34646	−0.431585
$375$	0	0
$376$	−24.9682	−1.28764
$377$	6.30334	0.324639
$378$	0	0
$379$	−15.1907	−0.780292	−0.390146	−	0.920753i	$-0.627575\pi$
$379$	−15.1907	−0.780292	−0.390146	+	0.920753i	$0.627575\pi$
$380$	−7.38370	−0.378776
$381$	0	0
$382$	−7.15988	−0.366331
$383$	0.297784	0.0152160	0.00760802	−	0.999971i	$-0.497578\pi$
$383$	0.297784	0.0152160	0.00760802	+	0.999971i	$0.497578\pi$
$384$	0	0
$385$	0.221771	0.0113025
$386$	8.86291	0.451110
$387$	0	0
$388$	−6.88667	−0.349618
$389$	2.25933	0.114553	0.0572763	−	0.998358i	$-0.481758\pi$
$389$	2.25933	0.114553	0.0572763	+	0.998358i	$0.481758\pi$
$390$	0	0
$391$	12.3797	0.626067
$392$	18.1354	0.915974
$393$	0	0
$394$	13.8221	0.696349
$395$	−6.62473	−0.333326
$396$	0	0
$397$	−23.8373	−1.19636	−0.598180	−	0.801361i	$-0.704109\pi$
$397$	−23.8373	−1.19636	−0.598180	+	0.801361i	$0.704109\pi$
$398$	18.9329	0.949019
$399$	0	0
$400$	0.898186	0.0449093
$401$	−26.1681	−1.30677	−0.653386	−	0.757025i	$-0.726652\pi$
$401$	−26.1681	−1.30677	−0.653386	+	0.757025i	$0.726652\pi$
$402$	0	0
$403$	17.7816	0.885766
$404$	−16.5093	−0.821366
$405$	0	0
$406$	−0.0975998	−0.00484380
$407$	−22.8712	−1.13368
$408$	0	0
$409$	−22.9348	−1.13406	−0.567028	−	0.823699i	$-0.691907\pi$
$409$	−22.9348	−1.13406	−0.567028	+	0.823699i	$0.691907\pi$
$410$	6.88678	0.340114
$411$	0	0
$412$	4.18023	0.205945
$413$	0.629449	0.0309732
$414$	0	0
$415$	−12.0061	−0.589355
$416$	−14.2588	−0.699097
$417$	0	0
$418$	17.3898	0.850563
$419$	−5.25412	−0.256681	−0.128340	−	0.991730i	$-0.540965\pi$
$419$	−5.25412	−0.256681	−0.128340	+	0.991730i	$0.540965\pi$
$420$	0	0
$421$	18.0800	0.881166	0.440583	−	0.897712i	$-0.354772\pi$
$421$	18.0800	0.881166	0.440583	+	0.897712i	$0.354772\pi$
$422$	−13.3251	−0.648658
$423$	0	0
$424$	−27.7552	−1.34791
$425$	2.48065	0.120329
$426$	0	0
$427$	−0.123262	−0.00596506
$428$	−5.68564	−0.274826
$429$	0	0
$430$	0.410458	0.0197940
$431$	−5.55910	−0.267773	−0.133886	−	0.990997i	$-0.542746\pi$
$431$	−5.55910	−0.267773	−0.133886	+	0.990997i	$0.542746\pi$
$432$	0	0
$433$	20.5795	0.988986	0.494493	−	0.869182i	$-0.335354\pi$
$433$	20.5795	0.988986	0.494493	+	0.869182i	$0.335354\pi$
$434$	−0.275327	−0.0132161
$435$	0	0
$436$	−7.57454	−0.362755
$437$	−25.7930	−1.23385
$438$	0	0
$439$	26.6104	1.27004	0.635022	−	0.772494i	$-0.280991\pi$
$439$	26.6104	1.27004	0.635022	+	0.772494i	$0.280991\pi$
$440$	−11.5360	−0.549958
$441$	0	0
$442$	−4.56083	−0.216937
$443$	−20.4411	−0.971188	−0.485594	−	0.874185i	$-0.661397\pi$
$443$	−20.4411	−0.971188	−0.485594	+	0.874185i	$0.661397\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	21.0411	0.996325
$447$	0	0
$448$	0.131281	0.00620243
$449$	−5.61381	−0.264932	−0.132466	−	0.991188i	$-0.542290\pi$
$449$	−5.61381	−0.264932	−0.132466	+	0.991188i	$0.542290\pi$
$450$	0	0
$451$	40.5533	1.90958
$452$	−14.8416	−0.698092
$453$	0	0
$454$	−8.09947	−0.380127
$455$	0.121184	0.00568121
$456$	0	0
$457$	−7.13528	−0.333774	−0.166887	−	0.985976i	$-0.553372\pi$
$457$	−7.13528	−0.333774	−0.166887	+	0.985976i	$0.553372\pi$
$458$	−15.1812	−0.709372
$459$	0	0
$460$	7.12952	0.332415
$461$	24.0844	1.12172	0.560860	−	0.827911i	$-0.310471\pi$
$461$	24.0844	1.12172	0.560860	+	0.827911i	$0.310471\pi$
$462$	0	0
$463$	−22.2301	−1.03312	−0.516561	−	0.856250i	$-0.672788\pi$
$463$	−22.2301	−1.03312	−0.516561	+	0.856250i	$0.672788\pi$
$464$	−2.32767	−0.108059
$465$	0	0
$466$	−1.24208	−0.0575382
$467$	−10.4176	−0.482069	−0.241034	−	0.970517i	$-0.577487\pi$
$467$	−10.4176	−0.482069	−0.241034	+	0.970517i	$0.577487\pi$
$468$	0	0
$469$	−0.267115	−0.0123342
$470$	−7.28229	−0.335907
$471$	0	0
$472$	−32.7426	−1.50710
$473$	2.41701	0.111134
$474$	0	0
$475$	−5.16842	−0.237143
$476$	−0.176568	−0.00809298
$477$	0	0
$478$	9.30640	0.425665
$479$	20.6366	0.942912	0.471456	−	0.881889i	$-0.343729\pi$
$479$	20.6366	0.942912	0.471456	+	0.881889i	$0.343729\pi$
$480$	0	0
$481$	−12.4977	−0.569848
$482$	−8.15740	−0.371559
$483$	0	0
$484$	−12.5902	−0.572283
$485$	−4.82051	−0.218888
$486$	0	0
$487$	−29.5496	−1.33902	−0.669510	−	0.742803i	$-0.733496\pi$
$487$	−29.5496	−1.33902	−0.669510	+	0.742803i	$0.733496\pi$
$488$	6.41181	0.290249
$489$	0	0
$490$	5.28941	0.238951
$491$	−37.8108	−1.70638	−0.853189	−	0.521602i	$-0.825335\pi$
$491$	−37.8108	−1.70638	−0.853189	+	0.521602i	$0.825335\pi$
$492$	0	0
$493$	−6.42866	−0.289532
$494$	9.50247	0.427537
$495$	0	0
$496$	−6.56632	−0.294836
$497$	−0.477378	−0.0214133
$498$	0	0
$499$	7.09756	0.317730	0.158865	−	0.987300i	$-0.449217\pi$
$499$	7.09756	0.317730	0.158865	+	0.987300i	$0.449217\pi$
$500$	1.42862	0.0638897
$501$	0	0
$502$	23.4532	1.04677
$503$	25.0615	1.11744	0.558718	−	0.829358i	$-0.311294\pi$
$503$	25.0615	1.11744	0.558718	+	0.829358i	$0.311294\pi$
$504$	0	0
$505$	−11.5561	−0.514240
$506$	−16.7912	−0.746458
$507$	0	0
$508$	−7.58945	−0.336727
$509$	1.59934	0.0708895	0.0354447	−	0.999372i	$-0.488715\pi$
$509$	1.59934	0.0708895	0.0354447	+	0.999372i	$0.488715\pi$
$510$	0	0
$511$	0.142678	0.00631169
$512$	9.92107	0.438454
$513$	0	0
$514$	−12.5166	−0.552085
$515$	2.92606	0.128938
$516$	0	0
$517$	−42.8823	−1.88596
$518$	0.193513	0.00850246
$519$	0	0
$520$	−6.30374	−0.276437
$521$	−25.2660	−1.10693	−0.553463	−	0.832874i	$-0.686694\pi$
$521$	−25.2660	−1.10693	−0.553463	+	0.832874i	$0.686694\pi$
$522$	0	0
$523$	−1.75283	−0.0766458	−0.0383229	−	0.999265i	$-0.512202\pi$
$523$	−1.75283	−0.0766458	−0.0383229	+	0.999265i	$0.512202\pi$
$524$	−31.7537	−1.38717
$525$	0	0
$526$	9.09122	0.396396
$527$	−18.1351	−0.789979
$528$	0	0
$529$	1.90507	0.0828292
$530$	−8.09516	−0.351631
$531$	0	0
$532$	0.367879	0.0159496
$533$	22.1599	0.959854
$534$	0	0
$535$	−3.97982	−0.172063
$536$	13.8947	0.600161
$537$	0	0
$538$	16.6956	0.719797
$539$	31.1471	1.34160
$540$	0	0
$541$	5.15589	0.221669	0.110835	−	0.993839i	$-0.464648\pi$
$541$	5.15589	0.221669	0.110835	+	0.993839i	$0.464648\pi$
$542$	10.4315	0.448073
$543$	0	0
$544$	14.5423	0.623497
$545$	−5.30200	−0.227113
$546$	0	0
$547$	13.3060	0.568925	0.284462	−	0.958687i	$-0.408185\pi$
$547$	13.3060	0.568925	0.284462	+	0.958687i	$0.408185\pi$
$548$	11.5443	0.493149
$549$	0	0
$550$	−3.36463	−0.143468
$551$	13.3941	0.570607
$552$	0	0
$553$	0.330064	0.0140358
$554$	2.30012	0.0977229
$555$	0	0
$556$	−1.09061	−0.0462523
$557$	32.2911	1.36822	0.684109	−	0.729380i	$-0.260191\pi$
$557$	32.2911	1.36822	0.684109	+	0.729380i	$0.260191\pi$
$558$	0	0
$559$	1.32075	0.0558619
$560$	−0.0447504	−0.00189105
$561$	0	0
$562$	−15.9659	−0.673482
$563$	1.17102	0.0493525	0.0246762	−	0.999695i	$-0.492145\pi$
$563$	1.17102	0.0493525	0.0246762	+	0.999695i	$0.492145\pi$
$564$	0	0
$565$	−10.3888	−0.437060
$566$	24.5569	1.03221
$567$	0	0
$568$	24.8321	1.04193
$569$	−20.9355	−0.877662	−0.438831	−	0.898570i	$-0.644607\pi$
$569$	−20.9355	−0.877662	−0.438831	+	0.898570i	$0.644607\pi$
$570$	0	0
$571$	−44.5105	−1.86271	−0.931354	−	0.364116i	$-0.881371\pi$
$571$	−44.5105	−1.86271	−0.931354	+	0.364116i	$0.881371\pi$
$572$	−15.4670	−0.646707
$573$	0	0
$574$	−0.343121	−0.0143216
$575$	4.99050	0.208118
$576$	0	0
$577$	−44.7453	−1.86277	−0.931386	−	0.364034i	$-0.881399\pi$
$577$	−44.7453	−1.86277	−0.931386	+	0.364034i	$0.881399\pi$
$578$	−8.19876	−0.341023
$579$	0	0
$580$	−3.70230	−0.153730
$581$	0.598180	0.0248167
$582$	0	0
$583$	−47.6690	−1.97425
$584$	−7.42178	−0.307115
$585$	0	0
$586$	−24.1964	−0.999544
$587$	−14.8194	−0.611663	−0.305832	−	0.952086i	$-0.598934\pi$
$587$	−14.8194	−0.611663	−0.305832	+	0.952086i	$0.598934\pi$
$588$	0	0
$589$	37.7845	1.55688
$590$	−9.54978	−0.393158
$591$	0	0
$592$	4.61511	0.189680
$593$	23.3625	0.959384	0.479692	−	0.877437i	$-0.340748\pi$
$593$	23.3625	0.959384	0.479692	+	0.877437i	$0.340748\pi$
$594$	0	0
$595$	−0.123594	−0.00506684
$596$	−0.922652	−0.0377933
$597$	0	0
$598$	−9.17535	−0.375208
$599$	2.73683	0.111824	0.0559120	−	0.998436i	$-0.482193\pi$
$599$	2.73683	0.111824	0.0559120	+	0.998436i	$0.482193\pi$
$600$	0	0
$601$	−14.7244	−0.600622	−0.300311	−	0.953841i	$-0.597090\pi$
$601$	−14.7244	−0.600622	−0.300311	+	0.953841i	$0.597090\pi$
$602$	−0.0204503	−0.000833491 0
$603$	0	0
$604$	23.4935	0.955937
$605$	−8.81287	−0.358294
$606$	0	0
$607$	−7.93966	−0.322261	−0.161131	−	0.986933i	$-0.551514\pi$
$607$	−7.93966	−0.322261	−0.161131	+	0.986933i	$0.551514\pi$
$608$	−30.2989	−1.22878
$609$	0	0
$610$	1.87008	0.0757175
$611$	−23.4326	−0.947981
$612$	0	0
$613$	−5.63827	−0.227728	−0.113864	−	0.993496i	$-0.536323\pi$
$613$	−5.63827	−0.227728	−0.113864	+	0.993496i	$0.536323\pi$
$614$	−14.8298	−0.598483
$615$	0	0
$616$	0.574760	0.0231577
$617$	−24.0408	−0.967848	−0.483924	−	0.875110i	$-0.660789\pi$
$617$	−24.0408	−0.967848	−0.483924	+	0.875110i	$0.660789\pi$
$618$	0	0
$619$	18.0743	0.726469	0.363234	−	0.931698i	$-0.381672\pi$
$619$	18.0743	0.726469	0.363234	+	0.931698i	$0.381672\pi$
$620$	−10.4441	−0.419446
$621$	0	0
$622$	−18.8115	−0.754272
$623$	−0.0498231	−0.00199612
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.66637	−0.186506
$627$	0	0
$628$	−11.7638	−0.469427
$629$	12.7462	0.508224
$630$	0	0
$631$	−3.58541	−0.142733	−0.0713665	−	0.997450i	$-0.522736\pi$
$631$	−3.58541	−0.142733	−0.0713665	+	0.997450i	$0.522736\pi$
$632$	−17.1692	−0.682955
$633$	0	0
$634$	24.9035	0.989043
$635$	−5.31244	−0.210818
$636$	0	0
$637$	17.0200	0.674357
$638$	8.71951	0.345209
$639$	0	0
$640$	9.73287	0.384726
$641$	19.3761	0.765311	0.382656	−	0.923891i	$-0.375010\pi$
$641$	19.3761	0.765311	0.382656	+	0.923891i	$0.375010\pi$
$642$	0	0
$643$	26.7275	1.05403	0.527015	−	0.849856i	$-0.323311\pi$
$643$	26.7275	1.05403	0.527015	+	0.849856i	$0.323311\pi$
$644$	−0.355214	−0.0139974
$645$	0	0
$646$	−9.69139	−0.381303
$647$	−20.6868	−0.813281	−0.406641	−	0.913588i	$-0.633300\pi$
$647$	−20.6868	−0.813281	−0.406641	+	0.913588i	$0.633300\pi$
$648$	0	0
$649$	−56.2346	−2.20740
$650$	−1.83856	−0.0721144
$651$	0	0
$652$	−2.25361	−0.0882580
$653$	33.7364	1.32021	0.660104	−	0.751174i	$-0.270512\pi$
$653$	33.7364	1.32021	0.660104	+	0.751174i	$0.270512\pi$
$654$	0	0
$655$	−22.2268	−0.868474
$656$	−8.18313	−0.319497
$657$	0	0
$658$	0.362826	0.0141444
$659$	9.44930	0.368092	0.184046	−	0.982918i	$-0.441080\pi$
$659$	9.44930	0.368092	0.184046	+	0.982918i	$0.441080\pi$
$660$	0	0
$661$	−36.6486	−1.42547	−0.712733	−	0.701435i	$-0.752543\pi$
$661$	−36.6486	−1.42547	−0.712733	+	0.701435i	$0.752543\pi$
$662$	23.4166	0.910113
$663$	0	0
$664$	−31.1160	−1.20753
$665$	0.257507	0.00998568
$666$	0	0
$667$	−12.9330	−0.500768
$668$	2.55187	0.0987348
$669$	0	0
$670$	4.05258	0.156565
$671$	11.0121	0.425119
$672$	0	0
$673$	42.5711	1.64099	0.820497	−	0.571651i	$-0.193697\pi$
$673$	42.5711	1.64099	0.820497	+	0.571651i	$0.193697\pi$
$674$	−2.67368	−0.102986
$675$	0	0
$676$	10.1203	0.389241
$677$	29.6301	1.13878	0.569389	−	0.822068i	$-0.307180\pi$
$677$	29.6301	1.13878	0.569389	+	0.822068i	$0.307180\pi$
$678$	0	0
$679$	0.240173	0.00921698
$680$	6.42906	0.246543
$681$	0	0
$682$	24.5976	0.941890
$683$	−18.4389	−0.705545	−0.352772	−	0.935709i	$-0.614761\pi$
$683$	−18.4389	−0.705545	−0.352772	+	0.935709i	$0.614761\pi$
$684$	0	0
$685$	8.08077	0.308750
$686$	−0.527163	−0.0201272
$687$	0	0
$688$	−0.487721	−0.0185942
$689$	−26.0482	−0.992357
$690$	0	0
$691$	−39.7294	−1.51138	−0.755689	−	0.654930i	$-0.772698\pi$
$691$	−39.7294	−1.51138	−0.755689	+	0.654930i	$0.772698\pi$
$692$	32.9736	1.25347
$693$	0	0
$694$	9.84164	0.373584
$695$	−0.763404	−0.0289576
$696$	0	0
$697$	−22.6005	−0.856056
$698$	20.9053	0.791276
$699$	0	0
$700$	−0.0711781	−0.00269028
$701$	8.13101	0.307104	0.153552	−	0.988141i	$-0.450929\pi$
$701$	8.13101	0.307104	0.153552	+	0.988141i	$0.450929\pi$
$702$	0	0
$703$	−26.5567	−1.00160
$704$	−11.7285	−0.442036
$705$	0	0
$706$	19.8869	0.748455
$707$	0.575761	0.0216537
$708$	0	0
$709$	24.2930	0.912343	0.456171	−	0.889892i	$-0.349220\pi$
$709$	24.2930	0.912343	0.456171	+	0.889892i	$0.349220\pi$
$710$	7.24261	0.271810
$711$	0	0
$712$	2.59169	0.0971275
$713$	−36.4838	−1.36633
$714$	0	0
$715$	−10.8265	−0.404889
$716$	−13.7662	−0.514466
$717$	0	0
$718$	−10.5832	−0.394961
$719$	−35.3355	−1.31779	−0.658896	−	0.752234i	$-0.728976\pi$
$719$	−35.3355	−1.31779	−0.658896	+	0.752234i	$0.728976\pi$
$720$	0	0
$721$	−0.145785	−0.00542933
$722$	5.82991	0.216967
$723$	0	0
$724$	10.1442	0.377006
$725$	−2.59152	−0.0962468
$726$	0	0
$727$	−27.6135	−1.02413	−0.512065	−	0.858947i	$-0.671119\pi$
$727$	−27.6135	−1.02413	−0.512065	+	0.858947i	$0.671119\pi$
$728$	0.314072	0.0116403
$729$	0	0
$730$	−2.16466	−0.0801175
$731$	−1.34701	−0.0498210
$732$	0	0
$733$	−19.3580	−0.715005	−0.357503	−	0.933912i	$-0.616372\pi$
$733$	−19.3580	−0.715005	−0.357503	+	0.933912i	$0.616372\pi$
$734$	16.6527	0.614664
$735$	0	0
$736$	29.2558	1.07838
$737$	23.8639	0.879039
$738$	0	0
$739$	28.8545	1.06143	0.530715	−	0.847550i	$-0.321923\pi$
$739$	28.8545	1.06143	0.530715	+	0.847550i	$0.321923\pi$
$740$	7.34060	0.269846
$741$	0	0
$742$	0.403326	0.0148065
$743$	−25.5760	−0.938292	−0.469146	−	0.883121i	$-0.655438\pi$
$743$	−25.5760	−0.938292	−0.469146	+	0.883121i	$0.655438\pi$
$744$	0	0
$745$	−0.645836	−0.0236616
$746$	10.3335	0.378336
$747$	0	0
$748$	15.7745	0.576772
$749$	0.198287	0.00724525
$750$	0	0
$751$	−26.6217	−0.971440	−0.485720	−	0.874114i	$-0.661443\pi$
$751$	−26.6217	−0.971440	−0.485720	+	0.874114i	$0.661443\pi$
$752$	8.65308	0.315545
$753$	0	0
$754$	4.76468	0.173520
$755$	16.4449	0.598492
$756$	0	0
$757$	16.8066	0.610848	0.305424	−	0.952216i	$-0.401202\pi$
$757$	16.8066	0.610848	0.305424	+	0.952216i	$0.401202\pi$
$758$	−11.4826	−0.417067
$759$	0	0
$760$	−13.3949	−0.485885
$761$	23.6421	0.857026	0.428513	−	0.903536i	$-0.359038\pi$
$761$	23.6421	0.857026	0.428513	+	0.903536i	$0.359038\pi$
$762$	0	0
$763$	0.264162	0.00956331
$764$	13.5319	0.489567
$765$	0	0
$766$	0.225094	0.00813299
$767$	−30.7288	−1.10955
$768$	0	0
$769$	20.1859	0.727921	0.363960	−	0.931414i	$-0.381424\pi$
$769$	20.1859	0.727921	0.363960	+	0.931414i	$0.381424\pi$
$770$	0.167636	0.00604118
$771$	0	0
$772$	−16.7506	−0.602866
$773$	14.8504	0.534132	0.267066	−	0.963678i	$-0.413946\pi$
$773$	14.8504	0.534132	0.267066	+	0.963678i	$0.413946\pi$
$774$	0	0
$775$	−7.31064	−0.262606
$776$	−12.4933	−0.448482
$777$	0	0
$778$	1.70782	0.0612285
$779$	47.0881	1.68710
$780$	0	0
$781$	42.6487	1.52609
$782$	9.35777	0.334633
$783$	0	0
$784$	−6.28507	−0.224467
$785$	−8.23439	−0.293898
$786$	0	0
$787$	22.9126	0.816746	0.408373	−	0.912815i	$-0.366096\pi$
$787$	22.9126	0.816746	0.408373	+	0.912815i	$0.366096\pi$
$788$	−26.1233	−0.930604
$789$	0	0
$790$	−5.00762	−0.178163
$791$	0.517602	0.0184038
$792$	0	0
$793$	6.01747	0.213687
$794$	−18.0186	−0.639456
$795$	0	0
$796$	−35.7824	−1.26827
$797$	−32.7951	−1.16166	−0.580831	−	0.814024i	$-0.697272\pi$
$797$	−32.7951	−1.16166	−0.580831	+	0.814024i	$0.697272\pi$
$798$	0	0
$799$	23.8985	0.845467
$800$	5.86231	0.207264
$801$	0	0
$802$	−19.7804	−0.698470
$803$	−12.7467	−0.449823
$804$	0	0
$805$	−0.248642	−0.00876348
$806$	13.4411	0.473442
$807$	0	0
$808$	−29.9498	−1.05363
$809$	25.8540	0.908978	0.454489	−	0.890752i	$-0.349822\pi$
$809$	25.8540	0.908978	0.454489	+	0.890752i	$0.349822\pi$
$810$	0	0
$811$	−5.12583	−0.179992	−0.0899962	−	0.995942i	$-0.528685\pi$
$811$	−5.12583	−0.179992	−0.0899962	+	0.995942i	$0.528685\pi$
$812$	0.184460	0.00647327
$813$	0	0
$814$	−17.2883	−0.605955
$815$	−1.57747	−0.0552565
$816$	0	0
$817$	2.80649	0.0981866
$818$	−17.3364	−0.606153
$819$	0	0
$820$	−13.0157	−0.454530
$821$	15.9798	0.557701	0.278850	−	0.960335i	$-0.410047\pi$
$821$	15.9798	0.557701	0.278850	+	0.960335i	$0.410047\pi$
$822$	0	0
$823$	37.5936	1.31043	0.655216	−	0.755442i	$-0.272578\pi$
$823$	37.5936	1.31043	0.655216	+	0.755442i	$0.272578\pi$
$824$	7.58344	0.264182
$825$	0	0
$826$	0.475799	0.0165552
$827$	53.0600	1.84508	0.922538	−	0.385906i	$-0.126111\pi$
$827$	53.0600	1.84508	0.922538	+	0.385906i	$0.126111\pi$
$828$	0	0
$829$	7.43536	0.258241	0.129120	−	0.991629i	$-0.458785\pi$
$829$	7.43536	0.258241	0.129120	+	0.991629i	$0.458785\pi$
$830$	−9.07537	−0.315011
$831$	0	0
$832$	−6.40893	−0.222190
$833$	−17.3584	−0.601432
$834$	0	0
$835$	1.78625	0.0618157
$836$	−32.8661	−1.13670
$837$	0	0
$838$	−3.97158	−0.137196
$839$	38.6435	1.33412	0.667061	−	0.745003i	$-0.267552\pi$
$839$	38.6435	1.33412	0.667061	+	0.745003i	$0.267552\pi$
$840$	0	0
$841$	−22.2840	−0.768414
$842$	13.6667	0.470984
$843$	0	0
$844$	25.1840	0.866869
$845$	7.08396	0.243695
$846$	0	0
$847$	0.439084	0.0150871
$848$	9.61896	0.330316
$849$	0	0
$850$	1.87512	0.0643160
$851$	25.6424	0.879012
$852$	0	0
$853$	34.8883	1.19455	0.597277	−	0.802035i	$-0.296249\pi$
$853$	34.8883	1.19455	0.597277	+	0.802035i	$0.296249\pi$
$854$	−0.0931734	−0.00318833
$855$	0	0
$856$	−10.3144	−0.352540
$857$	5.99133	0.204660	0.102330	−	0.994751i	$-0.467370\pi$
$857$	5.99133	0.204660	0.102330	+	0.994751i	$0.467370\pi$
$858$	0	0
$859$	36.8422	1.25704	0.628519	−	0.777794i	$-0.283661\pi$
$859$	36.8422	1.25704	0.628519	+	0.777794i	$0.283661\pi$
$860$	−0.775750	−0.0264529
$861$	0	0
$862$	−4.20212	−0.143125
$863$	−51.2726	−1.74534	−0.872669	−	0.488312i	$-0.837613\pi$
$863$	−51.2726	−1.74534	−0.872669	+	0.488312i	$0.837613\pi$
$864$	0	0
$865$	23.0808	0.784770
$866$	15.5560	0.528614
$867$	0	0
$868$	0.520358	0.0176621
$869$	−29.4877	−1.00030
$870$	0	0
$871$	13.0402	0.441850
$872$	−13.7411	−0.465333
$873$	0	0
$874$	−19.4969	−0.659491
$875$	−0.0498231	−0.00168433
$876$	0	0
$877$	−5.47506	−0.184880	−0.0924399	−	0.995718i	$-0.529467\pi$
$877$	−5.47506	−0.184880	−0.0924399	+	0.995718i	$0.529467\pi$
$878$	20.1147	0.678840
$879$	0	0
$880$	3.99797	0.134772
$881$	16.6965	0.562521	0.281260	−	0.959632i	$-0.409248\pi$
$881$	16.6965	0.562521	0.281260	+	0.959632i	$0.409248\pi$
$882$	0	0
$883$	−20.7620	−0.698696	−0.349348	−	0.936993i	$-0.613597\pi$
$883$	−20.7620	−0.698696	−0.349348	+	0.936993i	$0.613597\pi$
$884$	8.61980	0.289915
$885$	0	0
$886$	−15.4514	−0.519100
$887$	30.8479	1.03577	0.517886	−	0.855450i	$-0.326719\pi$
$887$	30.8479	1.03577	0.517886	+	0.855450i	$0.326719\pi$
$888$	0	0
$889$	0.264682	0.00887715
$890$	0.755898	0.0253378
$891$	0	0
$892$	−39.7669	−1.33149
$893$	−49.7923	−1.66624
$894$	0	0
$895$	−9.63601	−0.322096
$896$	−0.484922	−0.0162001
$897$	0	0
$898$	−4.24347	−0.141606
$899$	18.9457	0.631875
$900$	0	0
$901$	26.5661	0.885044
$902$	30.6542	1.02067
$903$	0	0
$904$	−26.9245	−0.895496
$905$	7.10071	0.236036
$906$	0	0
$907$	30.8544	1.02450	0.512251	−	0.858836i	$-0.328812\pi$
$907$	30.8544	1.02450	0.512251	+	0.858836i	$0.328812\pi$
$908$	15.3077	0.508003
$909$	0	0
$910$	0.0916030	0.00303661
$911$	−34.6247	−1.14717	−0.573584	−	0.819147i	$-0.694447\pi$
$911$	−34.6247	−1.14717	−0.573584	+	0.819147i	$0.694447\pi$
$912$	0	0
$913$	−53.4410	−1.76864
$914$	−5.39354	−0.178403
$915$	0	0
$916$	28.6919	0.948008
$917$	1.10741	0.0365699
$918$	0	0
$919$	−3.03636	−0.100160	−0.0500802	−	0.998745i	$-0.515948\pi$
$919$	−3.03636	−0.100160	−0.0500802	+	0.998745i	$0.515948\pi$
$920$	12.9338	0.426415
$921$	0	0
$922$	18.2053	0.599560
$923$	23.3049	0.767090
$924$	0	0
$925$	5.13825	0.168945
$926$	−16.8037	−0.552204
$927$	0	0
$928$	−15.1923	−0.498712
$929$	17.2687	0.566568	0.283284	−	0.959036i	$-0.408576\pi$
$929$	17.2687	0.566568	0.283284	+	0.959036i	$0.408576\pi$
$930$	0	0
$931$	36.1661	1.18530
$932$	2.34748	0.0768943
$933$	0	0
$934$	−7.87464	−0.257666
$935$	11.0418	0.361105
$936$	0	0
$937$	41.8282	1.36647	0.683234	−	0.730199i	$-0.260573\pi$
$937$	41.8282	1.36647	0.683234	+	0.730199i	$0.260573\pi$
$938$	−0.201912	−0.00659266
$939$	0	0
$940$	13.7632	0.448907
$941$	−55.0411	−1.79429	−0.897145	−	0.441736i	$-0.854363\pi$
$941$	−55.0411	−1.79429	−0.897145	+	0.441736i	$0.854363\pi$
$942$	0	0
$943$	−45.4671	−1.48061
$944$	11.3474	0.369326
$945$	0	0
$946$	1.82702	0.0594014
$947$	−5.20266	−0.169064	−0.0845318	−	0.996421i	$-0.526939\pi$
$947$	−5.20266	−0.169064	−0.0845318	+	0.996421i	$0.526939\pi$
$948$	0	0
$949$	−6.96532	−0.226104
$950$	−3.90680	−0.126753
$951$	0	0
$952$	−0.320316	−0.0103815
$953$	−23.9232	−0.774950	−0.387475	−	0.921880i	$-0.626653\pi$
$953$	−23.9232	−0.774950	−0.387475	+	0.921880i	$0.626653\pi$
$954$	0	0
$955$	9.47202	0.306507
$956$	−17.5887	−0.568860
$957$	0	0
$958$	15.5992	0.503987
$959$	−0.402609	−0.0130009
$960$	0	0
$961$	22.4455	0.724049
$962$	−9.44701	−0.304584
$963$	0	0
$964$	15.4172	0.496553
$965$	−11.7250	−0.377441
$966$	0	0
$967$	47.9852	1.54310	0.771550	−	0.636168i	$-0.219482\pi$
$967$	47.9852	1.54310	0.771550	+	0.636168i	$0.219482\pi$
$968$	−22.8402	−0.734111
$969$	0	0
$970$	−3.64382	−0.116996
$971$	−4.06546	−0.130467	−0.0652334	−	0.997870i	$-0.520779\pi$
$971$	−4.06546	−0.130467	−0.0652334	+	0.997870i	$0.520779\pi$
$972$	0	0
$973$	0.0380351	0.00121935
$974$	−22.3365	−0.715707
$975$	0	0
$976$	−2.22210	−0.0711278
$977$	−4.40304	−0.140866	−0.0704328	−	0.997517i	$-0.522438\pi$
$977$	−4.40304	−0.140866	−0.0704328	+	0.997517i	$0.522438\pi$
$978$	0	0
$979$	4.45116	0.142260
$980$	−9.99678	−0.319335
$981$	0	0
$982$	−28.5811	−0.912060
$983$	8.02508	0.255960	0.127980	−	0.991777i	$-0.459151\pi$
$983$	8.02508	0.255960	0.127980	+	0.991777i	$0.459151\pi$
$984$	0	0
$985$	−18.2857	−0.582631
$986$	−4.85941	−0.154755
$987$	0	0
$988$	−17.9593	−0.571362
$989$	−2.70988	−0.0861691
$990$	0	0
$991$	28.4040	0.902282	0.451141	−	0.892453i	$-0.351017\pi$
$991$	28.4040	0.902282	0.451141	+	0.892453i	$0.351017\pi$
$992$	−42.8572	−1.36072
$993$	0	0
$994$	−0.360849	−0.0114454
$995$	−25.0469	−0.794039
$996$	0	0
$997$	−6.03403	−0.191100	−0.0955499	−	0.995425i	$-0.530461\pi$
$997$	−6.03403	−0.191100	−0.0955499	+	0.995425i	$0.530461\pi$
$998$	5.36503	0.169827
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.o.1.4		7
3.2	odd	2		445.2.a.f.1.4	✓	7
12.11	even	2		7120.2.a.bj.1.6		7
15.14	odd	2		2225.2.a.k.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
445.2.a.f.1.4	✓	7	3.2	odd	2
2225.2.a.k.1.4		7	15.14	odd	2
4005.2.a.o.1.4		7	1.1	even	1	trivial
7120.2.a.bj.1.6		7	12.11	even	2

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:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} +O(q^{10})\)	\(q+0.755898 q^{2} -1.42862 q^{4} -1.00000 q^{5} +0.0498231 q^{7} -2.59169 q^{8} -0.755898 q^{10} -4.45116 q^{11} -2.43229 q^{13} +0.0376612 q^{14} +0.898186 q^{16} +2.48065 q^{17} -5.16842 q^{19} +1.42862 q^{20} -3.36463 q^{22} +4.99050 q^{23} +1.00000 q^{25} -1.83856 q^{26} -0.0711781 q^{28} -2.59152 q^{29} -7.31064 q^{31} +5.86231 q^{32} +1.87512 q^{34} -0.0498231 q^{35} +5.13825 q^{37} -3.90680 q^{38} +2.59169 q^{40} -9.11073 q^{41} -0.543007 q^{43} +6.35901 q^{44} +3.77231 q^{46} +9.63395 q^{47} -6.99752 q^{49} +0.755898 q^{50} +3.47482 q^{52} +10.7093 q^{53} +4.45116 q^{55} -0.129126 q^{56} -1.95893 q^{58} +12.6337 q^{59} -2.47399 q^{61} -5.52610 q^{62} +2.63494 q^{64} +2.43229 q^{65} -5.36128 q^{67} -3.54390 q^{68} -0.0376612 q^{70} -9.58146 q^{71} +2.86369 q^{73} +3.88400 q^{74} +7.38370 q^{76} -0.221771 q^{77} +6.62473 q^{79} -0.898186 q^{80} -6.88678 q^{82} +12.0061 q^{83} -2.48065 q^{85} -0.410458 q^{86} +11.5360 q^{88} -1.00000 q^{89} -0.121184 q^{91} -7.12952 q^{92} +7.28229 q^{94} +5.16842 q^{95} +4.82051 q^{97} -5.28941 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8}+O(q^{10}) \) 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8	\( 7 q + 4 q^{2} + 8 q^{4} - 7 q^{5} - 16 q^{7} + 12 q^{8} - 4 q^{10} + 10 q^{11} - 7 q^{13} - 3 q^{14} + 10 q^{16} + 13 q^{17} - 7 q^{19} - 8 q^{20} + 2 q^{22} + 13 q^{23} + 7 q^{25} - q^{26} - 21 q^{28} + 4 q^{29} + q^{31} + 13 q^{32} + 10 q^{34} + 16 q^{35} - 5 q^{37} + 40 q^{38} - 12 q^{40} - 5 q^{41} - 31 q^{43} + 21 q^{44} + 16 q^{46} + 14 q^{47} + 19 q^{49} + 4 q^{50} + 13 q^{53} - 10 q^{55} + q^{56} + 17 q^{58} + 14 q^{59} + 3 q^{61} - 26 q^{62} + 14 q^{64} + 7 q^{65} + q^{67} + 35 q^{68} + 3 q^{70} + 8 q^{71} + 9 q^{73} + 35 q^{74} + 40 q^{76} - 42 q^{77} + 9 q^{79} - 10 q^{80} + 29 q^{82} + 42 q^{83} - 13 q^{85} - 35 q^{86} + 30 q^{88} - 7 q^{89} + 31 q^{91} - 19 q^{92} + 37 q^{94} + 7 q^{95} - 7 q^{97} - 9 q^{98}+O(q^{100}) \) 7 * q + 4 * q^2 + 8 * q^4 - 7 * q^5 - 16 * q^7 + 12 * q^8 - 4 * q^10 + 10 * q^11 - 7 * q^13 - 3 * q^14 + 10 * q^16 + 13 * q^17 - 7 * q^19 - 8 * q^20 + 2 * q^22 + 13 * q^23 + 7 * q^25 - q^26 - 21 * q^28 + 4 * q^29 + q^31 + 13 * q^32 + 10 * q^34 + 16 * q^35 - 5 * q^37 + 40 * q^38 - 12 * q^40 - 5 * q^41 - 31 * q^43 + 21 * q^44 + 16 * q^46 + 14 * q^47 + 19 * q^49 + 4 * q^50 + 13 * q^53 - 10 * q^55 + q^56 + 17 * q^58 + 14 * q^59 + 3 * q^61 - 26 * q^62 + 14 * q^64 + 7 * q^65 + q^67 + 35 * q^68 + 3 * q^70 + 8 * q^71 + 9 * q^73 + 35 * q^74 + 40 * q^76 - 42 * q^77 + 9 * q^79 - 10 * q^80 + 29 * q^82 + 42 * q^83 - 13 * q^85 - 35 * q^86 + 30 * q^88 - 7 * q^89 + 31 * q^91 - 19 * q^92 + 37 * q^94 + 7 * q^95 - 7 * q^97 - 9 * q^98

Introduction

L-functions

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