LMFDB - Embedded newform 4560.2.a.br.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.67282 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -4.67282 q^{7} +1.00000 q^{9} -5.81681 q^{11} -2.67282 q^{13} -1.00000 q^{15} -6.48963 q^{17} +1.00000 q^{19} +4.67282 q^{21} -8.20166 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.81681 q^{29} +6.20166 q^{31} +5.81681 q^{33} -4.67282 q^{35} -12.0185 q^{37} +2.67282 q^{39} +0.183190 q^{41} +6.96080 q^{43} +1.00000 q^{45} +4.20166 q^{47} +14.8353 q^{49} +6.48963 q^{51} +10.7776 q^{53} -5.81681 q^{55} -1.00000 q^{57} +7.05767 q^{59} +5.14399 q^{61} -4.67282 q^{63} -2.67282 q^{65} -10.6913 q^{67} +8.20166 q^{69} -11.0577 q^{71} -4.28797 q^{73} -1.00000 q^{75} +27.1809 q^{77} +6.28797 q^{79} +1.00000 q^{81} -1.14399 q^{83} -6.48963 q^{85} +3.81681 q^{87} -3.81681 q^{89} +12.4896 q^{91} -6.20166 q^{93} +1.00000 q^{95} -6.67282 q^{97} -5.81681 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 + 3 * q^19 + 4 * q^21 - 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^33 - 4 * q^35 - 6 * q^37 - 2 * q^39 + 12 * q^41 + 8 * q^43 + 3 * q^45 - 6 * q^47 + 3 * q^49 - 2 * q^51 + 8 * q^53 - 6 * q^55 - 3 * q^57 + 4 * q^59 + 14 * q^61 - 4 * q^63 + 2 * q^65 + 8 * q^67 + 6 * q^69 - 16 * q^71 - 10 * q^73 - 3 * q^75 + 20 * q^77 + 16 * q^79 + 3 * q^81 - 2 * q^83 + 2 * q^85 + 16 * q^91 + 3 * q^95 - 10 * q^97 - 6 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.67282	−1.76616	−0.883081	−	0.469221i	$-0.844535\pi$
$7$	−4.67282	−1.76616	−0.883081	+	0.469221i	$0.844535\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.81681	−1.75383	−0.876917	−	0.480642i	$-0.840404\pi$
$11$	−5.81681	−1.75383	−0.876917	+	0.480642i	$0.840404\pi$
$12$	0	0
$13$	−2.67282	−0.741308	−0.370654	−	0.928771i	$-0.620866\pi$
$13$	−2.67282	−0.741308	−0.370654	+	0.928771i	$0.620866\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.48963	−1.57397	−0.786984	−	0.616974i	$-0.788358\pi$
$17$	−6.48963	−1.57397	−0.786984	+	0.616974i	$0.788358\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.67282	1.01969
$22$	0	0
$23$	−8.20166	−1.71016	−0.855082	−	0.518492i	$-0.826493\pi$
$23$	−8.20166	−1.71016	−0.855082	+	0.518492i	$0.826493\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.81681	−0.708764	−0.354382	−	0.935101i	$-0.615309\pi$
$29$	−3.81681	−0.708764	−0.354382	+	0.935101i	$0.615309\pi$
$30$	0	0
$31$	6.20166	1.11385	0.556926	−	0.830562i	$-0.311981\pi$
$31$	6.20166	1.11385	0.556926	+	0.830562i	$0.311981\pi$
$32$	0	0
$33$	5.81681	1.01258
$34$	0	0
$35$	−4.67282	−0.789851
$36$	0	0
$37$	−12.0185	−1.97582	−0.987912	−	0.155014i	$-0.950458\pi$
$37$	−12.0185	−1.97582	−0.987912	+	0.155014i	$0.950458\pi$
$38$	0	0
$39$	2.67282	0.427994
$40$	0	0
$41$	0.183190	0.0286094	0.0143047	−	0.999898i	$-0.495447\pi$
$41$	0.183190	0.0286094	0.0143047	+	0.999898i	$0.495447\pi$
$42$	0	0
$43$	6.96080	1.06151	0.530756	−	0.847525i	$-0.321908\pi$
$43$	6.96080	1.06151	0.530756	+	0.847525i	$0.321908\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.20166	0.612875	0.306438	−	0.951891i	$-0.400863\pi$
$47$	4.20166	0.612875	0.306438	+	0.951891i	$0.400863\pi$
$48$	0	0
$49$	14.8353	2.11933
$50$	0	0
$51$	6.48963	0.908731
$52$	0	0
$53$	10.7776	1.48042	0.740209	−	0.672377i	$-0.234727\pi$
$53$	10.7776	1.48042	0.740209	+	0.672377i	$0.234727\pi$
$54$	0	0
$55$	−5.81681	−0.784339
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	7.05767	0.918831	0.459415	−	0.888221i	$-0.348059\pi$
$59$	7.05767	0.918831	0.459415	+	0.888221i	$0.348059\pi$
$60$	0	0
$61$	5.14399	0.658620	0.329310	−	0.944222i	$-0.393184\pi$
$61$	5.14399	0.658620	0.329310	+	0.944222i	$0.393184\pi$
$62$	0	0
$63$	−4.67282	−0.588720
$64$	0	0
$65$	−2.67282	−0.331523
$66$	0	0
$67$	−10.6913	−1.30615	−0.653075	−	0.757293i	$-0.726521\pi$
$67$	−10.6913	−1.30615	−0.653075	+	0.757293i	$0.726521\pi$
$68$	0	0
$69$	8.20166	0.987364
$70$	0	0
$71$	−11.0577	−1.31230	−0.656152	−	0.754629i	$-0.727817\pi$
$71$	−11.0577	−1.31230	−0.656152	+	0.754629i	$0.727817\pi$
$72$	0	0
$73$	−4.28797	−0.501869	−0.250935	−	0.968004i	$-0.580738\pi$
$73$	−4.28797	−0.501869	−0.250935	+	0.968004i	$0.580738\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	27.1809	3.09755
$78$	0	0
$79$	6.28797	0.707452	0.353726	−	0.935349i	$-0.384914\pi$
$79$	6.28797	0.707452	0.353726	+	0.935349i	$0.384914\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.14399	−0.125569	−0.0627844	−	0.998027i	$-0.519998\pi$
$83$	−1.14399	−0.125569	−0.0627844	+	0.998027i	$0.519998\pi$
$84$	0	0
$85$	−6.48963	−0.703900
$86$	0	0
$87$	3.81681	0.409205
$88$	0	0
$89$	−3.81681	−0.404581	−0.202291	−	0.979326i	$-0.564839\pi$
$89$	−3.81681	−0.404581	−0.202291	+	0.979326i	$0.564839\pi$
$90$	0	0
$91$	12.4896	1.30927
$92$	0	0
$93$	−6.20166	−0.643082
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−6.67282	−0.677523	−0.338761	−	0.940872i	$-0.610008\pi$
$97$	−6.67282	−0.677523	−0.338761	+	0.940872i	$0.610008\pi$
$98$	0	0
$99$	−5.81681	−0.584611
$100$	0	0
$101$	0.654353	0.0651105	0.0325553	−	0.999470i	$-0.489636\pi$
$101$	0.654353	0.0651105	0.0325553	+	0.999470i	$0.489636\pi$
$102$	0	0
$103$	1.71203	0.168691	0.0843455	−	0.996437i	$-0.473120\pi$
$103$	1.71203	0.168691	0.0843455	+	0.996437i	$0.473120\pi$
$104$	0	0
$105$	4.67282	0.456021
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−9.05767	−0.867568	−0.433784	−	0.901017i	$-0.642822\pi$
$109$	−9.05767	−0.867568	−0.433784	+	0.901017i	$0.642822\pi$
$110$	0	0
$111$	12.0185	1.14074
$112$	0	0
$113$	11.5473	1.08628	0.543140	−	0.839642i	$-0.317235\pi$
$113$	11.5473	1.08628	0.543140	+	0.839642i	$0.317235\pi$
$114$	0	0
$115$	−8.20166	−0.764809
$116$	0	0
$117$	−2.67282	−0.247103
$118$	0	0
$119$	30.3249	2.77988
$120$	0	0
$121$	22.8353	2.07593
$122$	0	0
$123$	−0.183190	−0.0165177
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	−6.96080	−0.612864
$130$	0	0
$131$	−2.18319	−0.190746	−0.0953731	−	0.995442i	$-0.530404\pi$
$131$	−2.18319	−0.190746	−0.0953731	+	0.995442i	$0.530404\pi$
$132$	0	0
$133$	−4.67282	−0.405185
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	4.36638	0.370351	0.185176	−	0.982705i	$-0.440715\pi$
$139$	4.36638	0.370351	0.185176	+	0.982705i	$0.440715\pi$
$140$	0	0
$141$	−4.20166	−0.353844
$142$	0	0
$143$	15.5473	1.30013
$144$	0	0
$145$	−3.81681	−0.316969
$146$	0	0
$147$	−14.8353	−1.22359
$148$	0	0
$149$	4.28797	0.351284	0.175642	−	0.984454i	$-0.443800\pi$
$149$	4.28797	0.351284	0.175642	+	0.984454i	$0.443800\pi$
$150$	0	0
$151$	−16.8930	−1.37473	−0.687365	−	0.726313i	$-0.741233\pi$
$151$	−16.8930	−1.37473	−0.687365	+	0.726313i	$0.741233\pi$
$152$	0	0
$153$	−6.48963	−0.524656
$154$	0	0
$155$	6.20166	0.498129
$156$	0	0
$157$	16.3249	1.30287	0.651435	−	0.758704i	$-0.274167\pi$
$157$	16.3249	1.30287	0.651435	+	0.758704i	$0.274167\pi$
$158$	0	0
$159$	−10.7776	−0.854720
$160$	0	0
$161$	38.3249	3.02043
$162$	0	0
$163$	18.0185	1.41132	0.705658	−	0.708553i	$-0.250652\pi$
$163$	18.0185	1.41132	0.705658	+	0.708553i	$0.250652\pi$
$164$	0	0
$165$	5.81681	0.452838
$166$	0	0
$167$	7.14399	0.552818	0.276409	−	0.961040i	$-0.410856\pi$
$167$	7.14399	0.552818	0.276409	+	0.961040i	$0.410856\pi$
$168$	0	0
$169$	−5.85601	−0.450463
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−22.4112	−1.70389	−0.851947	−	0.523628i	$-0.824578\pi$
$173$	−22.4112	−1.70389	−0.851947	+	0.523628i	$0.824578\pi$
$174$	0	0
$175$	−4.67282	−0.353232
$176$	0	0
$177$	−7.05767	−0.530487
$178$	0	0
$179$	−16.0369	−1.19866	−0.599329	−	0.800503i	$-0.704566\pi$
$179$	−16.0369	−1.19866	−0.599329	+	0.800503i	$0.704566\pi$
$180$	0	0
$181$	−11.9216	−0.886125	−0.443063	−	0.896491i	$-0.646108\pi$
$181$	−11.9216	−0.886125	−0.443063	+	0.896491i	$0.646108\pi$
$182$	0	0
$183$	−5.14399	−0.380254
$184$	0	0
$185$	−12.0185	−0.883616
$186$	0	0
$187$	37.7490	2.76048
$188$	0	0
$189$	4.67282	0.339898
$190$	0	0
$191$	−8.47116	−0.612952	−0.306476	−	0.951878i	$-0.599150\pi$
$191$	−8.47116	−0.612952	−0.306476	+	0.951878i	$0.599150\pi$
$192$	0	0
$193$	−13.7305	−0.988343	−0.494171	−	0.869364i	$-0.664528\pi$
$193$	−13.7305	−0.988343	−0.494171	+	0.869364i	$0.664528\pi$
$194$	0	0
$195$	2.67282	0.191405
$196$	0	0
$197$	−19.4689	−1.38710	−0.693551	−	0.720408i	$-0.743955\pi$
$197$	−19.4689	−1.38710	−0.693551	+	0.720408i	$0.743955\pi$
$198$	0	0
$199$	0.942326	0.0667997	0.0333998	−	0.999442i	$-0.489367\pi$
$199$	0.942326	0.0667997	0.0333998	+	0.999442i	$0.489367\pi$
$200$	0	0
$201$	10.6913	0.754106
$202$	0	0
$203$	17.8353	1.25179
$204$	0	0
$205$	0.183190	0.0127945
$206$	0	0
$207$	−8.20166	−0.570055
$208$	0	0
$209$	−5.81681	−0.402357
$210$	0	0
$211$	5.71203	0.393232	0.196616	−	0.980481i	$-0.437005\pi$
$211$	5.71203	0.393232	0.196616	+	0.980481i	$0.437005\pi$
$212$	0	0
$213$	11.0577	0.757659
$214$	0	0
$215$	6.96080	0.474722
$216$	0	0
$217$	−28.9793	−1.96724
$218$	0	0
$219$	4.28797	0.289754
$220$	0	0
$221$	17.3456	1.16679
$222$	0	0
$223$	1.71203	0.114646	0.0573229	−	0.998356i	$-0.481744\pi$
$223$	1.71203	0.114646	0.0573229	+	0.998356i	$0.481744\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.452692	0.0300462	0.0150231	−	0.999887i	$-0.495218\pi$
$227$	0.452692	0.0300462	0.0150231	+	0.999887i	$0.495218\pi$
$228$	0	0
$229$	−14.1233	−0.933291	−0.466645	−	0.884444i	$-0.654538\pi$
$229$	−14.1233	−0.933291	−0.466645	+	0.884444i	$0.654538\pi$
$230$	0	0
$231$	−27.1809	−1.78837
$232$	0	0
$233$	−11.7120	−0.767280	−0.383640	−	0.923483i	$-0.625330\pi$
$233$	−11.7120	−0.767280	−0.383640	+	0.923483i	$0.625330\pi$
$234$	0	0
$235$	4.20166	0.274086
$236$	0	0
$237$	−6.28797	−0.408448
$238$	0	0
$239$	−25.4874	−1.64864	−0.824321	−	0.566123i	$-0.808443\pi$
$239$	−25.4874	−1.64864	−0.824321	+	0.566123i	$0.808443\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	14.8353	0.947791
$246$	0	0
$247$	−2.67282	−0.170068
$248$	0	0
$249$	1.14399	0.0724972
$250$	0	0
$251$	−3.12552	−0.197281	−0.0986404	−	0.995123i	$-0.531449\pi$
$251$	−3.12552	−0.197281	−0.0986404	+	0.995123i	$0.531449\pi$
$252$	0	0
$253$	47.7075	2.99935
$254$	0	0
$255$	6.48963	0.406397
$256$	0	0
$257$	−14.5680	−0.908729	−0.454365	−	0.890816i	$-0.650134\pi$
$257$	−14.5680	−0.908729	−0.454365	+	0.890816i	$0.650134\pi$
$258$	0	0
$259$	56.1602	3.48962
$260$	0	0
$261$	−3.81681	−0.236255
$262$	0	0
$263$	−5.54731	−0.342062	−0.171031	−	0.985266i	$-0.554710\pi$
$263$	−5.54731	−0.342062	−0.171031	+	0.985266i	$0.554710\pi$
$264$	0	0
$265$	10.7776	0.662063
$266$	0	0
$267$	3.81681	0.233585
$268$	0	0
$269$	29.1625	1.77807	0.889033	−	0.457843i	$-0.151378\pi$
$269$	29.1625	1.77807	0.889033	+	0.457843i	$0.151378\pi$
$270$	0	0
$271$	−1.92159	−0.116728	−0.0583642	−	0.998295i	$-0.518588\pi$
$271$	−1.92159	−0.116728	−0.0583642	+	0.998295i	$0.518588\pi$
$272$	0	0
$273$	−12.4896	−0.755907
$274$	0	0
$275$	−5.81681	−0.350767
$276$	0	0
$277$	−26.9793	−1.62103	−0.810514	−	0.585720i	$-0.800812\pi$
$277$	−26.9793	−1.62103	−0.810514	+	0.585720i	$0.800812\pi$
$278$	0	0
$279$	6.20166	0.371284
$280$	0	0
$281$	4.39276	0.262050	0.131025	−	0.991379i	$-0.458173\pi$
$281$	4.39276	0.262050	0.131025	+	0.991379i	$0.458173\pi$
$282$	0	0
$283$	24.8824	1.47910	0.739552	−	0.673099i	$-0.235037\pi$
$283$	24.8824	1.47910	0.739552	+	0.673099i	$0.235037\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−0.856013	−0.0505289
$288$	0	0
$289$	25.1153	1.47737
$290$	0	0
$291$	6.67282	0.391168
$292$	0	0
$293$	1.83528	0.107218	0.0536091	−	0.998562i	$-0.482927\pi$
$293$	1.83528	0.107218	0.0536091	+	0.998562i	$0.482927\pi$
$294$	0	0
$295$	7.05767	0.410914
$296$	0	0
$297$	5.81681	0.337526
$298$	0	0
$299$	21.9216	1.26776
$300$	0	0
$301$	−32.5266	−1.87480
$302$	0	0
$303$	−0.654353	−0.0375916
$304$	0	0
$305$	5.14399	0.294544
$306$	0	0
$307$	12.3664	0.705787	0.352893	−	0.935664i	$-0.385198\pi$
$307$	12.3664	0.705787	0.352893	+	0.935664i	$0.385198\pi$
$308$	0	0
$309$	−1.71203	−0.0973938
$310$	0	0
$311$	6.79608	0.385370	0.192685	−	0.981261i	$-0.438280\pi$
$311$	6.79608	0.385370	0.192685	+	0.981261i	$0.438280\pi$
$312$	0	0
$313$	5.23030	0.295634	0.147817	−	0.989015i	$-0.452775\pi$
$313$	5.23030	0.295634	0.147817	+	0.989015i	$0.452775\pi$
$314$	0	0
$315$	−4.67282	−0.263284
$316$	0	0
$317$	31.7569	1.78364	0.891822	−	0.452387i	$-0.149427\pi$
$317$	31.7569	1.78364	0.891822	+	0.452387i	$0.149427\pi$
$318$	0	0
$319$	22.2017	1.24305
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.48963	−0.361093
$324$	0	0
$325$	−2.67282	−0.148262
$326$	0	0
$327$	9.05767	0.500891
$328$	0	0
$329$	−19.6336	−1.08244
$330$	0	0
$331$	−12.8930	−0.708661	−0.354330	−	0.935120i	$-0.615291\pi$
$331$	−12.8930	−0.708661	−0.354330	+	0.935120i	$0.615291\pi$
$332$	0	0
$333$	−12.0185	−0.658608
$334$	0	0
$335$	−10.6913	−0.584128
$336$	0	0
$337$	−24.2280	−1.31979	−0.659893	−	0.751360i	$-0.729398\pi$
$337$	−24.2280	−1.31979	−0.659893	+	0.751360i	$0.729398\pi$
$338$	0	0
$339$	−11.5473	−0.627164
$340$	0	0
$341$	−36.0739	−1.95351
$342$	0	0
$343$	−36.6129	−1.97691
$344$	0	0
$345$	8.20166	0.441563
$346$	0	0
$347$	6.45269	0.346399	0.173199	−	0.984887i	$-0.444590\pi$
$347$	6.45269	0.346399	0.173199	+	0.984887i	$0.444590\pi$
$348$	0	0
$349$	11.3087	0.605341	0.302671	−	0.953095i	$-0.402122\pi$
$349$	11.3087	0.605341	0.302671	+	0.953095i	$0.402122\pi$
$350$	0	0
$351$	2.67282	0.142665
$352$	0	0
$353$	−32.6913	−1.73998	−0.869991	−	0.493068i	$-0.835876\pi$
$353$	−32.6913	−1.73998	−0.869991	+	0.493068i	$0.835876\pi$
$354$	0	0
$355$	−11.0577	−0.586880
$356$	0	0
$357$	−30.3249	−1.60496
$358$	0	0
$359$	−5.04711	−0.266376	−0.133188	−	0.991091i	$-0.542521\pi$
$359$	−5.04711	−0.266376	−0.133188	+	0.991091i	$0.542521\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−22.8353	−1.19854
$364$	0	0
$365$	−4.28797	−0.224443
$366$	0	0
$367$	34.2280	1.78669	0.893345	−	0.449372i	$-0.148352\pi$
$367$	34.2280	1.78669	0.893345	+	0.449372i	$0.148352\pi$
$368$	0	0
$369$	0.183190	0.00953648
$370$	0	0
$371$	−50.3619	−2.61466
$372$	0	0
$373$	26.3064	1.36210	0.681048	−	0.732239i	$-0.261524\pi$
$373$	26.3064	1.36210	0.681048	+	0.732239i	$0.261524\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	10.2017	0.525412
$378$	0	0
$379$	38.6050	1.98300	0.991502	−	0.130089i	$-0.0415262\pi$
$379$	38.6050	1.98300	0.991502	+	0.130089i	$0.0415262\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−8.40332	−0.429390	−0.214695	−	0.976681i	$-0.568876\pi$
$383$	−8.40332	−0.429390	−0.214695	+	0.976681i	$0.568876\pi$
$384$	0	0
$385$	27.1809	1.38527
$386$	0	0
$387$	6.96080	0.353837
$388$	0	0
$389$	12.2880	0.623025	0.311512	−	0.950242i	$-0.399164\pi$
$389$	12.2880	0.623025	0.311512	+	0.950242i	$0.399164\pi$
$390$	0	0
$391$	53.2258	2.69174
$392$	0	0
$393$	2.18319	0.110127
$394$	0	0
$395$	6.28797	0.316382
$396$	0	0
$397$	34.6498	1.73903	0.869513	−	0.493911i	$-0.164433\pi$
$397$	34.6498	1.73903	0.869513	+	0.493911i	$0.164433\pi$
$398$	0	0
$399$	4.67282	0.233934
$400$	0	0
$401$	−5.73840	−0.286562	−0.143281	−	0.989682i	$-0.545765\pi$
$401$	−5.73840	−0.286562	−0.143281	+	0.989682i	$0.545765\pi$
$402$	0	0
$403$	−16.5759	−0.825707
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	69.9092	3.46527
$408$	0	0
$409$	28.9009	1.42906	0.714528	−	0.699607i	$-0.246642\pi$
$409$	28.9009	1.42906	0.714528	+	0.699607i	$0.246642\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−32.9793	−1.62280
$414$	0	0
$415$	−1.14399	−0.0561561
$416$	0	0
$417$	−4.36638	−0.213823
$418$	0	0
$419$	4.87448	0.238134	0.119067	−	0.992886i	$-0.462010\pi$
$419$	4.87448	0.238134	0.119067	+	0.992886i	$0.462010\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	4.20166	0.204292
$424$	0	0
$425$	−6.48963	−0.314793
$426$	0	0
$427$	−24.0369	−1.16323
$428$	0	0
$429$	−15.5473	−0.750631
$430$	0	0
$431$	−10.6544	−0.513202	−0.256601	−	0.966517i	$-0.582603\pi$
$431$	−10.6544	−0.513202	−0.256601	+	0.966517i	$0.582603\pi$
$432$	0	0
$433$	−6.88239	−0.330747	−0.165373	−	0.986231i	$-0.552883\pi$
$433$	−6.88239	−0.330747	−0.165373	+	0.986231i	$0.552883\pi$
$434$	0	0
$435$	3.81681	0.183002
$436$	0	0
$437$	−8.20166	−0.392339
$438$	0	0
$439$	−40.0739	−1.91262	−0.956311	−	0.292351i	$-0.905562\pi$
$439$	−40.0739	−1.91262	−0.956311	+	0.292351i	$0.905562\pi$
$440$	0	0
$441$	14.8353	0.706442
$442$	0	0
$443$	30.8930	1.46777	0.733884	−	0.679274i	$-0.237705\pi$
$443$	30.8930	1.46777	0.733884	+	0.679274i	$0.237705\pi$
$444$	0	0
$445$	−3.81681	−0.180934
$446$	0	0
$447$	−4.28797	−0.202814
$448$	0	0
$449$	−12.7961	−0.603884	−0.301942	−	0.953326i	$-0.597635\pi$
$449$	−12.7961	−0.603884	−0.301942	+	0.953326i	$0.597635\pi$
$450$	0	0
$451$	−1.06558	−0.0501762
$452$	0	0
$453$	16.8930	0.793700
$454$	0	0
$455$	12.4896	0.585523
$456$	0	0
$457$	28.3249	1.32498	0.662492	−	0.749069i	$-0.269499\pi$
$457$	28.3249	1.32498	0.662492	+	0.749069i	$0.269499\pi$
$458$	0	0
$459$	6.48963	0.302910
$460$	0	0
$461$	1.42405	0.0663248	0.0331624	−	0.999450i	$-0.489442\pi$
$461$	1.42405	0.0663248	0.0331624	+	0.999450i	$0.489442\pi$
$462$	0	0
$463$	−31.7305	−1.47464	−0.737321	−	0.675543i	$-0.763909\pi$
$463$	−31.7305	−1.47464	−0.737321	+	0.675543i	$0.763909\pi$
$464$	0	0
$465$	−6.20166	−0.287595
$466$	0	0
$467$	14.4896	0.670500	0.335250	−	0.942129i	$-0.391179\pi$
$467$	14.4896	0.670500	0.335250	+	0.942129i	$0.391179\pi$
$468$	0	0
$469$	49.9585	2.30687
$470$	0	0
$471$	−16.3249	−0.752212
$472$	0	0
$473$	−40.4896	−1.86172
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	10.7776	0.493473
$478$	0	0
$479$	31.1994	1.42554	0.712768	−	0.701399i	$-0.247441\pi$
$479$	31.1994	1.42554	0.712768	+	0.701399i	$0.247441\pi$
$480$	0	0
$481$	32.1233	1.46469
$482$	0	0
$483$	−38.3249	−1.74384
$484$	0	0
$485$	−6.67282	−0.302997
$486$	0	0
$487$	−38.3249	−1.73667	−0.868334	−	0.495980i	$-0.834809\pi$
$487$	−38.3249	−1.73667	−0.868334	+	0.495980i	$0.834809\pi$
$488$	0	0
$489$	−18.0185	−0.814823
$490$	0	0
$491$	−15.1625	−0.684272	−0.342136	−	0.939650i	$-0.611150\pi$
$491$	−15.1625	−0.684272	−0.342136	+	0.939650i	$0.611150\pi$
$492$	0	0
$493$	24.7697	1.11557
$494$	0	0
$495$	−5.81681	−0.261446
$496$	0	0
$497$	51.6706	2.31774
$498$	0	0
$499$	26.9009	1.20425	0.602124	−	0.798403i	$-0.294321\pi$
$499$	26.9009	1.20425	0.602124	+	0.798403i	$0.294321\pi$
$500$	0	0
$501$	−7.14399	−0.319170
$502$	0	0
$503$	−34.8560	−1.55415	−0.777076	−	0.629406i	$-0.783298\pi$
$503$	−34.8560	−1.55415	−0.777076	+	0.629406i	$0.783298\pi$
$504$	0	0
$505$	0.654353	0.0291183
$506$	0	0
$507$	5.85601	0.260075
$508$	0	0
$509$	−12.7591	−0.565539	−0.282769	−	0.959188i	$-0.591253\pi$
$509$	−12.7591	−0.565539	−0.282769	+	0.959188i	$0.591253\pi$
$510$	0	0
$511$	20.0369	0.886382
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	1.71203	0.0754409
$516$	0	0
$517$	−24.4403	−1.07488
$518$	0	0
$519$	22.4112	0.983744
$520$	0	0
$521$	5.52884	0.242223	0.121111	−	0.992639i	$-0.461354\pi$
$521$	5.52884	0.242223	0.121111	+	0.992639i	$0.461354\pi$
$522$	0	0
$523$	−11.0577	−0.483518	−0.241759	−	0.970336i	$-0.577724\pi$
$523$	−11.0577	−0.483518	−0.241759	+	0.970336i	$0.577724\pi$
$524$	0	0
$525$	4.67282	0.203939
$526$	0	0
$527$	−40.2465	−1.75317
$528$	0	0
$529$	44.2672	1.92466
$530$	0	0
$531$	7.05767	0.306277
$532$	0	0
$533$	−0.489634	−0.0212084
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0369	0.692045
$538$	0	0
$539$	−86.2940	−3.71695
$540$	0	0
$541$	−23.9585	−1.03006	−0.515029	−	0.857173i	$-0.672219\pi$
$541$	−23.9585	−1.03006	−0.515029	+	0.857173i	$0.672219\pi$
$542$	0	0
$543$	11.9216	0.511605
$544$	0	0
$545$	−9.05767	−0.387988
$546$	0	0
$547$	−0.209567	−0.00896042	−0.00448021	−	0.999990i	$-0.501426\pi$
$547$	−0.209567	−0.00896042	−0.00448021	+	0.999990i	$0.501426\pi$
$548$	0	0
$549$	5.14399	0.219540
$550$	0	0
$551$	−3.81681	−0.162602
$552$	0	0
$553$	−29.3826	−1.24947
$554$	0	0
$555$	12.0185	0.510156
$556$	0	0
$557$	−14.9793	−0.634692	−0.317346	−	0.948310i	$-0.602792\pi$
$557$	−14.9793	−0.634692	−0.317346	+	0.948310i	$0.602792\pi$
$558$	0	0
$559$	−18.6050	−0.786907
$560$	0	0
$561$	−37.7490	−1.59376
$562$	0	0
$563$	−16.6992	−0.703787	−0.351894	−	0.936040i	$-0.614462\pi$
$563$	−16.6992	−0.703787	−0.351894	+	0.936040i	$0.614462\pi$
$564$	0	0
$565$	11.5473	0.485799
$566$	0	0
$567$	−4.67282	−0.196240
$568$	0	0
$569$	34.5450	1.44820	0.724102	−	0.689693i	$-0.242255\pi$
$569$	34.5450	1.44820	0.724102	+	0.689693i	$0.242255\pi$
$570$	0	0
$571$	5.92159	0.247811	0.123905	−	0.992294i	$-0.460458\pi$
$571$	5.92159	0.247811	0.123905	+	0.992294i	$0.460458\pi$
$572$	0	0
$573$	8.47116	0.353888
$574$	0	0
$575$	−8.20166	−0.342033
$576$	0	0
$577$	24.6544	1.02637	0.513187	−	0.858277i	$-0.328465\pi$
$577$	24.6544	1.02637	0.513187	+	0.858277i	$0.328465\pi$
$578$	0	0
$579$	13.7305	0.570620
$580$	0	0
$581$	5.34565	0.221775
$582$	0	0
$583$	−62.6913	−2.59641
$584$	0	0
$585$	−2.67282	−0.110508
$586$	0	0
$587$	−12.2017	−0.503616	−0.251808	−	0.967777i	$-0.581025\pi$
$587$	−12.2017	−0.503616	−0.251808	+	0.967777i	$0.581025\pi$
$588$	0	0
$589$	6.20166	0.255535
$590$	0	0
$591$	19.4689	0.800844
$592$	0	0
$593$	−1.02073	−0.0419164	−0.0209582	−	0.999780i	$-0.506672\pi$
$593$	−1.02073	−0.0419164	−0.0209582	+	0.999780i	$0.506672\pi$
$594$	0	0
$595$	30.3249	1.24320
$596$	0	0
$597$	−0.942326	−0.0385668
$598$	0	0
$599$	−7.42405	−0.303339	−0.151669	−	0.988431i	$-0.548465\pi$
$599$	−7.42405	−0.303339	−0.151669	+	0.988431i	$0.548465\pi$
$600$	0	0
$601$	19.3456	0.789125	0.394563	−	0.918869i	$-0.370896\pi$
$601$	19.3456	0.789125	0.394563	+	0.918869i	$0.370896\pi$
$602$	0	0
$603$	−10.6913	−0.435383
$604$	0	0
$605$	22.8353	0.928386
$606$	0	0
$607$	16.0369	0.650919	0.325460	−	0.945556i	$-0.394481\pi$
$607$	16.0369	0.650919	0.325460	+	0.945556i	$0.394481\pi$
$608$	0	0
$609$	−17.8353	−0.722722
$610$	0	0
$611$	−11.2303	−0.454329
$612$	0	0
$613$	−36.1523	−1.46018	−0.730089	−	0.683352i	$-0.760521\pi$
$613$	−36.1523	−1.46018	−0.730089	+	0.683352i	$0.760521\pi$
$614$	0	0
$615$	−0.183190	−0.00738692
$616$	0	0
$617$	−40.4482	−1.62838	−0.814191	−	0.580597i	$-0.802819\pi$
$617$	−40.4482	−1.62838	−0.814191	+	0.580597i	$0.802819\pi$
$618$	0	0
$619$	4.19376	0.168561	0.0842806	−	0.996442i	$-0.473141\pi$
$619$	4.19376	0.168561	0.0842806	+	0.996442i	$0.473141\pi$
$620$	0	0
$621$	8.20166	0.329121
$622$	0	0
$623$	17.8353	0.714555
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.81681	0.232301
$628$	0	0
$629$	77.9955	3.10988
$630$	0	0
$631$	−5.30871	−0.211336	−0.105668	−	0.994401i	$-0.533698\pi$
$631$	−5.30871	−0.211336	−0.105668	+	0.994401i	$0.533698\pi$
$632$	0	0
$633$	−5.71203	−0.227033
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	−39.6521	−1.57107
$638$	0	0
$639$	−11.0577	−0.437435
$640$	0	0
$641$	−7.24086	−0.285997	−0.142998	−	0.989723i	$-0.545674\pi$
$641$	−7.24086	−0.285997	−0.142998	+	0.989723i	$0.545674\pi$
$642$	0	0
$643$	−2.38485	−0.0940493	−0.0470247	−	0.998894i	$-0.514974\pi$
$643$	−2.38485	−0.0940493	−0.0470247	+	0.998894i	$0.514974\pi$
$644$	0	0
$645$	−6.96080	−0.274081
$646$	0	0
$647$	−1.47342	−0.0579263	−0.0289631	−	0.999580i	$-0.509221\pi$
$647$	−1.47342	−0.0579263	−0.0289631	+	0.999580i	$0.509221\pi$
$648$	0	0
$649$	−41.0532	−1.61148
$650$	0	0
$651$	28.9793	1.13579
$652$	0	0
$653$	−25.9137	−1.01408	−0.507040	−	0.861922i	$-0.669261\pi$
$653$	−25.9137	−1.01408	−0.507040	+	0.861922i	$0.669261\pi$
$654$	0	0
$655$	−2.18319	−0.0853043
$656$	0	0
$657$	−4.28797	−0.167290
$658$	0	0
$659$	24.9793	0.973054	0.486527	−	0.873666i	$-0.338264\pi$
$659$	24.9793	0.973054	0.486527	+	0.873666i	$0.338264\pi$
$660$	0	0
$661$	39.4195	1.53324	0.766621	−	0.642100i	$-0.221937\pi$
$661$	39.4195	1.53324	0.766621	+	0.642100i	$0.221937\pi$
$662$	0	0
$663$	−17.3456	−0.673649
$664$	0	0
$665$	−4.67282	−0.181204
$666$	0	0
$667$	31.3042	1.21210
$668$	0	0
$669$	−1.71203	−0.0661908
$670$	0	0
$671$	−29.9216	−1.15511
$672$	0	0
$673$	12.0185	0.463278	0.231639	−	0.972802i	$-0.425591\pi$
$673$	12.0185	0.463278	0.231639	+	0.972802i	$0.425591\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	22.2017	0.853279	0.426640	−	0.904422i	$-0.359697\pi$
$677$	22.2017	0.853279	0.426640	+	0.904422i	$0.359697\pi$
$678$	0	0
$679$	31.1809	1.19661
$680$	0	0
$681$	−0.452692	−0.0173472
$682$	0	0
$683$	32.0739	1.22727	0.613637	−	0.789589i	$-0.289706\pi$
$683$	32.0739	1.22727	0.613637	+	0.789589i	$0.289706\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	14.1233	0.538836
$688$	0	0
$689$	−28.8066	−1.09745
$690$	0	0
$691$	−44.0369	−1.67524	−0.837622	−	0.546250i	$-0.816055\pi$
$691$	−44.0369	−1.67524	−0.837622	+	0.546250i	$0.816055\pi$
$692$	0	0
$693$	27.1809	1.03252
$694$	0	0
$695$	4.36638	0.165626
$696$	0	0
$697$	−1.18883	−0.0450303
$698$	0	0
$699$	11.7120	0.442990
$700$	0	0
$701$	−18.9793	−0.716837	−0.358419	−	0.933561i	$-0.616684\pi$
$701$	−18.9793	−0.716837	−0.358419	+	0.933561i	$0.616684\pi$
$702$	0	0
$703$	−12.0185	−0.453285
$704$	0	0
$705$	−4.20166	−0.158244
$706$	0	0
$707$	−3.05767	−0.114996
$708$	0	0
$709$	−2.12325	−0.0797405	−0.0398702	−	0.999205i	$-0.512694\pi$
$709$	−2.12325	−0.0797405	−0.0398702	+	0.999205i	$0.512694\pi$
$710$	0	0
$711$	6.28797	0.235817
$712$	0	0
$713$	−50.8639	−1.90487
$714$	0	0
$715$	15.5473	0.581436
$716$	0	0
$717$	25.4874	0.951843
$718$	0	0
$719$	11.5658	0.431331	0.215665	−	0.976467i	$-0.430808\pi$
$719$	11.5658	0.431331	0.215665	+	0.976467i	$0.430808\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−14.0000	−0.520666
$724$	0	0
$725$	−3.81681	−0.141753
$726$	0	0
$727$	−32.3064	−1.19818	−0.599090	−	0.800682i	$-0.704471\pi$
$727$	−32.3064	−1.19818	−0.599090	+	0.800682i	$0.704471\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−45.1730	−1.67078
$732$	0	0
$733$	18.4033	0.679742	0.339871	−	0.940472i	$-0.389617\pi$
$733$	18.4033	0.679742	0.339871	+	0.940472i	$0.389617\pi$
$734$	0	0
$735$	−14.8353	−0.547208
$736$	0	0
$737$	62.1892	2.29077
$738$	0	0
$739$	28.0000	1.03000	0.514998	−	0.857191i	$-0.327793\pi$
$739$	28.0000	1.03000	0.514998	+	0.857191i	$0.327793\pi$
$740$	0	0
$741$	2.67282	0.0981886
$742$	0	0
$743$	−33.5552	−1.23102	−0.615511	−	0.788129i	$-0.711050\pi$
$743$	−33.5552	−1.23102	−0.615511	+	0.788129i	$0.711050\pi$
$744$	0	0
$745$	4.28797	0.157099
$746$	0	0
$747$	−1.14399	−0.0418563
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−9.06558	−0.330808	−0.165404	−	0.986226i	$-0.552893\pi$
$751$	−9.06558	−0.330808	−0.165404	+	0.986226i	$0.552893\pi$
$752$	0	0
$753$	3.12552	0.113900
$754$	0	0
$755$	−16.8930	−0.614798
$756$	0	0
$757$	49.1316	1.78572	0.892858	−	0.450337i	$-0.148696\pi$
$757$	49.1316	1.78572	0.892858	+	0.450337i	$0.148696\pi$
$758$	0	0
$759$	−47.7075	−1.73167
$760$	0	0
$761$	−26.4033	−0.957120	−0.478560	−	0.878055i	$-0.658841\pi$
$761$	−26.4033	−0.957120	−0.478560	+	0.878055i	$0.658841\pi$
$762$	0	0
$763$	42.3249	1.53226
$764$	0	0
$765$	−6.48963	−0.234633
$766$	0	0
$767$	−18.8639	−0.681137
$768$	0	0
$769$	16.6913	0.601903	0.300952	−	0.953639i	$-0.402696\pi$
$769$	16.6913	0.601903	0.300952	+	0.953639i	$0.402696\pi$
$770$	0	0
$771$	14.5680	0.524655
$772$	0	0
$773$	37.8722	1.36217	0.681085	−	0.732205i	$-0.261509\pi$
$773$	37.8722	1.36217	0.681085	+	0.732205i	$0.261509\pi$
$774$	0	0
$775$	6.20166	0.222770
$776$	0	0
$777$	−56.1602	−2.01474
$778$	0	0
$779$	0.183190	0.00656345
$780$	0	0
$781$	64.3204	2.30156
$782$	0	0
$783$	3.81681	0.136402
$784$	0	0
$785$	16.3249	0.582661
$786$	0	0
$787$	14.4817	0.516218	0.258109	−	0.966116i	$-0.416901\pi$
$787$	14.4817	0.516218	0.258109	+	0.966116i	$0.416901\pi$
$788$	0	0
$789$	5.54731	0.197489
$790$	0	0
$791$	−53.9585	−1.91854
$792$	0	0
$793$	−13.7490	−0.488240
$794$	0	0
$795$	−10.7776	−0.382242
$796$	0	0
$797$	−42.6419	−1.51045	−0.755227	−	0.655463i	$-0.772473\pi$
$797$	−42.6419	−1.51045	−0.755227	+	0.655463i	$0.772473\pi$
$798$	0	0
$799$	−27.2672	−0.964646
$800$	0	0
$801$	−3.81681	−0.134860
$802$	0	0
$803$	24.9423	0.880196
$804$	0	0
$805$	38.3249	1.35078
$806$	0	0
$807$	−29.1625	−1.02657
$808$	0	0
$809$	−41.6706	−1.46506	−0.732529	−	0.680735i	$-0.761660\pi$
$809$	−41.6706	−1.46506	−0.732529	+	0.680735i	$0.761660\pi$
$810$	0	0
$811$	−53.9585	−1.89474	−0.947370	−	0.320140i	$-0.896270\pi$
$811$	−53.9585	−1.89474	−0.947370	+	0.320140i	$0.896270\pi$
$812$	0	0
$813$	1.92159	0.0673932
$814$	0	0
$815$	18.0185	0.631160
$816$	0	0
$817$	6.96080	0.243527
$818$	0	0
$819$	12.4896	0.436423
$820$	0	0
$821$	7.74897	0.270441	0.135220	−	0.990816i	$-0.456826\pi$
$821$	7.74897	0.270441	0.135220	+	0.990816i	$0.456826\pi$
$822$	0	0
$823$	38.0554	1.32653	0.663264	−	0.748385i	$-0.269171\pi$
$823$	38.0554	1.32653	0.663264	+	0.748385i	$0.269171\pi$
$824$	0	0
$825$	5.81681	0.202515
$826$	0	0
$827$	48.9299	1.70146	0.850730	−	0.525604i	$-0.176160\pi$
$827$	48.9299	1.70146	0.850730	+	0.525604i	$0.176160\pi$
$828$	0	0
$829$	−18.0369	−0.626449	−0.313224	−	0.949679i	$-0.601409\pi$
$829$	−18.0369	−0.626449	−0.313224	+	0.949679i	$0.601409\pi$
$830$	0	0
$831$	26.9793	0.935900
$832$	0	0
$833$	−96.2755	−3.33575
$834$	0	0
$835$	7.14399	0.247228
$836$	0	0
$837$	−6.20166	−0.214361
$838$	0	0
$839$	−27.0577	−0.934135	−0.467067	−	0.884222i	$-0.654689\pi$
$839$	−27.0577	−0.934135	−0.467067	+	0.884222i	$0.654689\pi$
$840$	0	0
$841$	−14.4320	−0.497654
$842$	0	0
$843$	−4.39276	−0.151295
$844$	0	0
$845$	−5.85601	−0.201453
$846$	0	0
$847$	−106.705	−3.66644
$848$	0	0
$849$	−24.8824	−0.853961
$850$	0	0
$851$	98.5714	3.37898
$852$	0	0
$853$	−41.6336	−1.42551	−0.712754	−	0.701414i	$-0.752552\pi$
$853$	−41.6336	−1.42551	−0.712754	+	0.701414i	$0.752552\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	45.8722	1.56697	0.783483	−	0.621414i	$-0.213441\pi$
$857$	45.8722	1.56697	0.783483	+	0.621414i	$0.213441\pi$
$858$	0	0
$859$	−30.5187	−1.04128	−0.520642	−	0.853775i	$-0.674307\pi$
$859$	−30.5187	−1.04128	−0.520642	+	0.853775i	$0.674307\pi$
$860$	0	0
$861$	0.856013	0.0291729
$862$	0	0
$863$	−7.42405	−0.252718	−0.126359	−	0.991985i	$-0.540329\pi$
$863$	−7.42405	−0.252718	−0.126359	+	0.991985i	$0.540329\pi$
$864$	0	0
$865$	−22.4112	−0.762005
$866$	0	0
$867$	−25.1153	−0.852962
$868$	0	0
$869$	−36.5759	−1.24075
$870$	0	0
$871$	28.5759	0.968259
$872$	0	0
$873$	−6.67282	−0.225841
$874$	0	0
$875$	−4.67282	−0.157970
$876$	0	0
$877$	−24.0554	−0.812294	−0.406147	−	0.913808i	$-0.633128\pi$
$877$	−24.0554	−0.812294	−0.406147	+	0.913808i	$0.633128\pi$
$878$	0	0
$879$	−1.83528	−0.0619025
$880$	0	0
$881$	−0.481728	−0.0162298	−0.00811492	−	0.999967i	$-0.502583\pi$
$881$	−0.481728	−0.0162298	−0.00811492	+	0.999967i	$0.502583\pi$
$882$	0	0
$883$	−25.4056	−0.854966	−0.427483	−	0.904023i	$-0.640600\pi$
$883$	−25.4056	−0.854966	−0.427483	+	0.904023i	$0.640600\pi$
$884$	0	0
$885$	−7.05767	−0.237241
$886$	0	0
$887$	−28.2307	−0.947894	−0.473947	−	0.880553i	$-0.657171\pi$
$887$	−28.2307	−0.947894	−0.473947	+	0.880553i	$0.657171\pi$
$888$	0	0
$889$	−18.6913	−0.626886
$890$	0	0
$891$	−5.81681	−0.194870
$892$	0	0
$893$	4.20166	0.140603
$894$	0	0
$895$	−16.0369	−0.536056
$896$	0	0
$897$	−21.9216	−0.731941
$898$	0	0
$899$	−23.6706	−0.789457
$900$	0	0
$901$	−69.9427	−2.33013
$902$	0	0
$903$	32.5266	1.08242
$904$	0	0
$905$	−11.9216	−0.396287
$906$	0	0
$907$	37.1888	1.23483	0.617417	−	0.786636i	$-0.288179\pi$
$907$	37.1888	1.23483	0.617417	+	0.786636i	$0.288179\pi$
$908$	0	0
$909$	0.654353	0.0217035
$910$	0	0
$911$	−18.5187	−0.613551	−0.306775	−	0.951782i	$-0.599250\pi$
$911$	−18.5187	−0.613551	−0.306775	+	0.951782i	$0.599250\pi$
$912$	0	0
$913$	6.65435	0.220227
$914$	0	0
$915$	−5.14399	−0.170055
$916$	0	0
$917$	10.2017	0.336889
$918$	0	0
$919$	−12.3294	−0.406711	−0.203355	−	0.979105i	$-0.565185\pi$
$919$	−12.3294	−0.406711	−0.203355	+	0.979105i	$0.565185\pi$
$920$	0	0
$921$	−12.3664	−0.407486
$922$	0	0
$923$	29.5552	0.972822
$924$	0	0
$925$	−12.0185	−0.395165
$926$	0	0
$927$	1.71203	0.0562303
$928$	0	0
$929$	−7.67508	−0.251811	−0.125906	−	0.992042i	$-0.540184\pi$
$929$	−7.67508	−0.251811	−0.125906	+	0.992042i	$0.540184\pi$
$930$	0	0
$931$	14.8353	0.486207
$932$	0	0
$933$	−6.79608	−0.222494
$934$	0	0
$935$	37.7490	1.23452
$936$	0	0
$937$	−9.05767	−0.295901	−0.147951	−	0.988995i	$-0.547268\pi$
$937$	−9.05767	−0.295901	−0.147951	+	0.988995i	$0.547268\pi$
$938$	0	0
$939$	−5.23030	−0.170684
$940$	0	0
$941$	−56.2201	−1.83272	−0.916362	−	0.400351i	$-0.868888\pi$
$941$	−56.2201	−1.83272	−0.916362	+	0.400351i	$0.868888\pi$
$942$	0	0
$943$	−1.50246	−0.0489268
$944$	0	0
$945$	4.67282	0.152007
$946$	0	0
$947$	33.2179	1.07944	0.539718	−	0.841846i	$-0.318531\pi$
$947$	33.2179	1.07944	0.539718	+	0.841846i	$0.318531\pi$
$948$	0	0
$949$	11.4610	0.372040
$950$	0	0
$951$	−31.7569	−1.02979
$952$	0	0
$953$	6.81455	0.220745	0.110372	−	0.993890i	$-0.464796\pi$
$953$	6.81455	0.220745	0.110372	+	0.993890i	$0.464796\pi$
$954$	0	0
$955$	−8.47116	−0.274120
$956$	0	0
$957$	−22.2017	−0.717678
$958$	0	0
$959$	−9.34565	−0.301787
$960$	0	0
$961$	7.46060	0.240664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.7305	−0.442000
$966$	0	0
$967$	−45.2857	−1.45629	−0.728145	−	0.685423i	$-0.759617\pi$
$967$	−45.2857	−1.45629	−0.728145	+	0.685423i	$0.759617\pi$
$968$	0	0
$969$	6.48963	0.208477
$970$	0	0
$971$	−15.0207	−0.482038	−0.241019	−	0.970520i	$-0.577482\pi$
$971$	−15.0207	−0.482038	−0.241019	+	0.970520i	$0.577482\pi$
$972$	0	0
$973$	−20.4033	−0.654100
$974$	0	0
$975$	2.67282	0.0855989
$976$	0	0
$977$	6.77761	0.216835	0.108417	−	0.994105i	$-0.465422\pi$
$977$	6.77761	0.216835	0.108417	+	0.994105i	$0.465422\pi$
$978$	0	0
$979$	22.2017	0.709568
$980$	0	0
$981$	−9.05767	−0.289189
$982$	0	0
$983$	−33.1025	−1.05581	−0.527903	−	0.849305i	$-0.677022\pi$
$983$	−33.1025	−1.05581	−0.527903	+	0.849305i	$0.677022\pi$
$984$	0	0
$985$	−19.4689	−0.620331
$986$	0	0
$987$	19.6336	0.624945
$988$	0	0
$989$	−57.0901	−1.81536
$990$	0	0
$991$	47.2672	1.50149	0.750747	−	0.660590i	$-0.229694\pi$
$991$	47.2672	1.50149	0.750747	+	0.660590i	$0.229694\pi$
$992$	0	0
$993$	12.8930	0.409146
$994$	0	0
$995$	0.942326	0.0298737
$996$	0	0
$997$	4.09422	0.129665	0.0648326	−	0.997896i	$-0.479349\pi$
$997$	4.09422	0.129665	0.0648326	+	0.997896i	$0.479349\pi$
$998$	0	0
$999$	12.0185	0.380248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.br.1.1		3
4.3	odd	2		1140.2.a.g.1.3	✓	3
12.11	even	2		3420.2.a.m.1.3		3
20.3	even	4		5700.2.f.q.3649.4		6
20.7	even	4		5700.2.f.q.3649.3		6
20.19	odd	2		5700.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.g.1.3	✓	3	4.3	odd	2
3420.2.a.m.1.3		3	12.11	even	2
4560.2.a.br.1.1		3	1.1	even	1	trivial
5700.2.a.w.1.1		3	20.19	odd	2
5700.2.f.q.3649.3		6	20.7	even	4
5700.2.f.q.3649.4		6	20.3	even	4

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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.67282 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.67282 q^{7} +1.00000 q^{9} -5.81681 q^{11} -2.67282 q^{13} -1.00000 q^{15} -6.48963 q^{17} +1.00000 q^{19} +4.67282 q^{21} -8.20166 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.81681 q^{29} +6.20166 q^{31} +5.81681 q^{33} -4.67282 q^{35} -12.0185 q^{37} +2.67282 q^{39} +0.183190 q^{41} +6.96080 q^{43} +1.00000 q^{45} +4.20166 q^{47} +14.8353 q^{49} +6.48963 q^{51} +10.7776 q^{53} -5.81681 q^{55} -1.00000 q^{57} +7.05767 q^{59} +5.14399 q^{61} -4.67282 q^{63} -2.67282 q^{65} -10.6913 q^{67} +8.20166 q^{69} -11.0577 q^{71} -4.28797 q^{73} -1.00000 q^{75} +27.1809 q^{77} +6.28797 q^{79} +1.00000 q^{81} -1.14399 q^{83} -6.48963 q^{85} +3.81681 q^{87} -3.81681 q^{89} +12.4896 q^{91} -6.20166 q^{93} +1.00000 q^{95} -6.67282 q^{97} -5.81681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 4 q^{7} + 3 q^{9} - 6 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} + 3 q^{19} + 4 q^{21} - 6 q^{23} + 3 q^{25} - 3 q^{27} + 6 q^{33} - 4 q^{35} - 6 q^{37} - 2 q^{39} + 12 q^{41} + 8 q^{43} + 3 q^{45} - 6 q^{47} + 3 q^{49} - 2 q^{51} + 8 q^{53} - 6 q^{55} - 3 q^{57} + 4 q^{59} + 14 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{67} + 6 q^{69} - 16 q^{71} - 10 q^{73} - 3 q^{75} + 20 q^{77} + 16 q^{79} + 3 q^{81} - 2 q^{83} + 2 q^{85} + 16 q^{91} + 3 q^{95} - 10 q^{97} - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 4 * q^7 + 3 * q^9 - 6 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 + 3 * q^19 + 4 * q^21 - 6 * q^23 + 3 * q^25 - 3 * q^27 + 6 * q^33 - 4 * q^35 - 6 * q^37 - 2 * q^39 + 12 * q^41 + 8 * q^43 + 3 * q^45 - 6 * q^47 + 3 * q^49 - 2 * q^51 + 8 * q^53 - 6 * q^55 - 3 * q^57 + 4 * q^59 + 14 * q^61 - 4 * q^63 + 2 * q^65 + 8 * q^67 + 6 * q^69 - 16 * q^71 - 10 * q^73 - 3 * q^75 + 20 * q^77 + 16 * q^79 + 3 * q^81 - 2 * q^83 + 2 * q^85 + 16 * q^91 + 3 * q^95 - 10 * q^97 - 6 * q^99

Introduction

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