LMFDB - Embedded newform 5568.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5568,2,Mod(1,5568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5568, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5568.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5568 = 2^{6} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5568.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:	$44.4607038454$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	5568.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.508203 q^{5} -3.68133 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.508203 q^{5} -3.68133 q^{7} +1.00000 q^{9} -0.318669 q^{11} -4.18953 q^{13} +0.508203 q^{15} +3.17313 q^{17} -5.87086 q^{19} +3.68133 q^{21} -2.50820 q^{23} -4.74173 q^{25} -1.00000 q^{27} -1.00000 q^{29} -2.50820 q^{31} +0.318669 q^{33} +1.87086 q^{35} -7.87086 q^{37} +4.18953 q^{39} +8.72532 q^{41} -10.7253 q^{43} -0.508203 q^{45} +11.0440 q^{47} +6.55220 q^{49} -3.17313 q^{51} +8.24993 q^{53} +0.161949 q^{55} +5.87086 q^{57} -11.3627 q^{59} +3.87086 q^{61} -3.68133 q^{63} +2.12914 q^{65} +7.04399 q^{67} +2.50820 q^{69} -6.24993 q^{71} -7.87086 q^{73} +4.74173 q^{75} +1.17313 q^{77} +4.85446 q^{79} +1.00000 q^{81} -8.37907 q^{83} -1.61259 q^{85} +1.00000 q^{87} -15.9313 q^{89} +15.4231 q^{91} +2.50820 q^{93} +2.98359 q^{95} +11.2335 q^{97} -0.318669 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{7} + 3 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 4 q^{21} - 6 q^{23} + 17 q^{25} - 3 q^{27} - 3 q^{29} - 6 q^{31} + 8 q^{33} - 10 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} - 4 q^{43} + 12 q^{47} - 3 q^{49} - 4 q^{51} - 8 q^{53} + 10 q^{55} + 2 q^{57} - 20 q^{59} - 4 q^{61} - 4 q^{63} + 22 q^{65} + 6 q^{69} + 14 q^{71} - 8 q^{73} - 17 q^{75} - 2 q^{77} + 2 q^{79} + 3 q^{81} - 8 q^{83} + 42 q^{85} + 3 q^{87} - 8 q^{89} + 8 q^{91} + 6 q^{93} + 12 q^{95} + 4 q^{97} - 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^7 + 3 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 4 * q^21 - 6 * q^23 + 17 * q^25 - 3 * q^27 - 3 * q^29 - 6 * q^31 + 8 * q^33 - 10 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 - 4 * q^43 + 12 * q^47 - 3 * q^49 - 4 * q^51 - 8 * q^53 + 10 * q^55 + 2 * q^57 - 20 * q^59 - 4 * q^61 - 4 * q^63 + 22 * q^65 + 6 * q^69 + 14 * q^71 - 8 * q^73 - 17 * q^75 - 2 * q^77 + 2 * q^79 + 3 * q^81 - 8 * q^83 + 42 * q^85 + 3 * q^87 - 8 * q^89 + 8 * q^91 + 6 * q^93 + 12 * q^95 + 4 * q^97 - 8 * 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
$5$	−0.508203	−0.227275	−0.113638	−	0.993522i	$-0.536250\pi$
$5$	−0.508203	−0.227275	−0.113638	+	0.993522i	$0.536250\pi$
$6$	0	0
$7$	−3.68133	−1.39141	−0.695706	−	0.718327i	$-0.744908\pi$
$7$	−3.68133	−1.39141	−0.695706	+	0.718327i	$0.744908\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.318669	−0.0960824	−0.0480412	−	0.998845i	$-0.515298\pi$
$11$	−0.318669	−0.0960824	−0.0480412	+	0.998845i	$0.515298\pi$
$12$	0	0
$13$	−4.18953	−1.16197	−0.580984	−	0.813915i	$-0.697332\pi$
$13$	−4.18953	−1.16197	−0.580984	+	0.813915i	$0.697332\pi$
$14$	0	0
$15$	0.508203	0.131218
$16$	0	0
$17$	3.17313	0.769596	0.384798	−	0.923001i	$-0.374271\pi$
$17$	3.17313	0.769596	0.384798	+	0.923001i	$0.374271\pi$
$18$	0	0
$19$	−5.87086	−1.34687	−0.673434	−	0.739247i	$-0.735182\pi$
$19$	−5.87086	−1.34687	−0.673434	+	0.739247i	$0.735182\pi$
$20$	0	0
$21$	3.68133	0.803332
$22$	0	0
$23$	−2.50820	−0.522997	−0.261498	−	0.965204i	$-0.584217\pi$
$23$	−2.50820	−0.522997	−0.261498	+	0.965204i	$0.584217\pi$
$24$	0	0
$25$	−4.74173	−0.948346
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.50820	−0.450487	−0.225243	−	0.974303i	$-0.572318\pi$
$31$	−2.50820	−0.450487	−0.225243	+	0.974303i	$0.572318\pi$
$32$	0	0
$33$	0.318669	0.0554732
$34$	0	0
$35$	1.87086	0.316234
$36$	0	0
$37$	−7.87086	−1.29396	−0.646981	−	0.762506i	$-0.723969\pi$
$37$	−7.87086	−1.29396	−0.646981	+	0.762506i	$0.723969\pi$
$38$	0	0
$39$	4.18953	0.670862
$40$	0	0
$41$	8.72532	1.36267	0.681333	−	0.731973i	$-0.261400\pi$
$41$	8.72532	1.36267	0.681333	+	0.731973i	$0.261400\pi$
$42$	0	0
$43$	−10.7253	−1.63560	−0.817798	−	0.575505i	$-0.804805\pi$
$43$	−10.7253	−1.63560	−0.817798	+	0.575505i	$0.804805\pi$
$44$	0	0
$45$	−0.508203	−0.0757585
$46$	0	0
$47$	11.0440	1.61093	0.805466	−	0.592642i	$-0.201915\pi$
$47$	11.0440	1.61093	0.805466	+	0.592642i	$0.201915\pi$
$48$	0	0
$49$	6.55220	0.936028
$50$	0	0
$51$	−3.17313	−0.444327
$52$	0	0
$53$	8.24993	1.13322	0.566608	−	0.823988i	$-0.308256\pi$
$53$	8.24993	1.13322	0.566608	+	0.823988i	$0.308256\pi$
$54$	0	0
$55$	0.161949	0.0218372
$56$	0	0
$57$	5.87086	0.777615
$58$	0	0
$59$	−11.3627	−1.47929	−0.739646	−	0.672996i	$-0.765007\pi$
$59$	−11.3627	−1.47929	−0.739646	+	0.672996i	$0.765007\pi$
$60$	0	0
$61$	3.87086	0.495613	0.247807	−	0.968809i	$-0.420290\pi$
$61$	3.87086	0.495613	0.247807	+	0.968809i	$0.420290\pi$
$62$	0	0
$63$	−3.68133	−0.463804
$64$	0	0
$65$	2.12914	0.264087
$66$	0	0
$67$	7.04399	0.860561	0.430280	−	0.902695i	$-0.358415\pi$
$67$	7.04399	0.860561	0.430280	+	0.902695i	$0.358415\pi$
$68$	0	0
$69$	2.50820	0.301952
$70$	0	0
$71$	−6.24993	−0.741731	−0.370865	−	0.928687i	$-0.620939\pi$
$71$	−6.24993	−0.741731	−0.370865	+	0.928687i	$0.620939\pi$
$72$	0	0
$73$	−7.87086	−0.921215	−0.460608	−	0.887604i	$-0.652368\pi$
$73$	−7.87086	−0.921215	−0.460608	+	0.887604i	$0.652368\pi$
$74$	0	0
$75$	4.74173	0.547528
$76$	0	0
$77$	1.17313	0.133690
$78$	0	0
$79$	4.85446	0.546169	0.273085	−	0.961990i	$-0.411956\pi$
$79$	4.85446	0.546169	0.273085	+	0.961990i	$0.411956\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.37907	−0.919722	−0.459861	−	0.887991i	$-0.652101\pi$
$83$	−8.37907	−0.919722	−0.459861	+	0.887991i	$0.652101\pi$
$84$	0	0
$85$	−1.61259	−0.174910
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	−15.9313	−1.68871	−0.844355	−	0.535784i	$-0.820016\pi$
$89$	−15.9313	−1.68871	−0.844355	+	0.535784i	$0.820016\pi$
$90$	0	0
$91$	15.4231	1.61678
$92$	0	0
$93$	2.50820	0.260089
$94$	0	0
$95$	2.98359	0.306110
$96$	0	0
$97$	11.2335	1.14059	0.570296	−	0.821439i	$-0.306829\pi$
$97$	11.2335	1.14059	0.570296	+	0.821439i	$0.306829\pi$
$98$	0	0
$99$	−0.318669	−0.0320275
$100$	0	0
$101$	−10.9149	−1.08607	−0.543034	−	0.839710i	$-0.682725\pi$
$101$	−10.9149	−1.08607	−0.543034	+	0.839710i	$0.682725\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	−1.87086	−0.182578
$106$	0	0
$107$	3.83805	0.371038	0.185519	−	0.982641i	$-0.440603\pi$
$107$	3.83805	0.371038	0.185519	+	0.982641i	$0.440603\pi$
$108$	0	0
$109$	8.18953	0.784415	0.392208	−	0.919877i	$-0.371712\pi$
$109$	8.18953	0.784415	0.392208	+	0.919877i	$0.371712\pi$
$110$	0	0
$111$	7.87086	0.747069
$112$	0	0
$113$	13.2059	1.24231	0.621155	−	0.783688i	$-0.286664\pi$
$113$	13.2059	1.24231	0.621155	+	0.783688i	$0.286664\pi$
$114$	0	0
$115$	1.27468	0.118864
$116$	0	0
$117$	−4.18953	−0.387323
$118$	0	0
$119$	−11.6813	−1.07083
$120$	0	0
$121$	−10.8984	−0.990768
$122$	0	0
$123$	−8.72532	−0.786736
$124$	0	0
$125$	4.95078	0.442811
$126$	0	0
$127$	−9.39547	−0.833714	−0.416857	−	0.908972i	$-0.636868\pi$
$127$	−9.39547	−0.833714	−0.416857	+	0.908972i	$0.636868\pi$
$128$	0	0
$129$	10.7253	0.944312
$130$	0	0
$131$	−17.0768	−1.49201	−0.746004	−	0.665942i	$-0.768030\pi$
$131$	−17.0768	−1.49201	−0.746004	+	0.665942i	$0.768030\pi$
$132$	0	0
$133$	21.6126	1.87405
$134$	0	0
$135$	0.508203	0.0437392
$136$	0	0
$137$	15.0164	1.28294	0.641469	−	0.767149i	$-0.278325\pi$
$137$	15.0164	1.28294	0.641469	+	0.767149i	$0.278325\pi$
$138$	0	0
$139$	7.68133	0.651522	0.325761	−	0.945452i	$-0.394380\pi$
$139$	7.68133	0.651522	0.325761	+	0.945452i	$0.394380\pi$
$140$	0	0
$141$	−11.0440	−0.930072
$142$	0	0
$143$	1.33508	0.111645
$144$	0	0
$145$	0.508203	0.0422040
$146$	0	0
$147$	−6.55220	−0.540416
$148$	0	0
$149$	11.3955	0.933554	0.466777	−	0.884375i	$-0.345415\pi$
$149$	11.3955	0.933554	0.466777	+	0.884375i	$0.345415\pi$
$150$	0	0
$151$	2.03281	0.165428	0.0827140	−	0.996573i	$-0.473641\pi$
$151$	2.03281	0.165428	0.0827140	+	0.996573i	$0.473641\pi$
$152$	0	0
$153$	3.17313	0.256532
$154$	0	0
$155$	1.27468	0.102385
$156$	0	0
$157$	10.7581	0.858593	0.429296	−	0.903164i	$-0.358762\pi$
$157$	10.7581	0.858593	0.429296	+	0.903164i	$0.358762\pi$
$158$	0	0
$159$	−8.24993	−0.654262
$160$	0	0
$161$	9.23353	0.727704
$162$	0	0
$163$	21.8297	1.70984	0.854918	−	0.518764i	$-0.173608\pi$
$163$	21.8297	1.70984	0.854918	+	0.518764i	$0.173608\pi$
$164$	0	0
$165$	−0.161949	−0.0126077
$166$	0	0
$167$	20.4999	1.58633	0.793164	−	0.609009i	$-0.208433\pi$
$167$	20.4999	1.58633	0.793164	+	0.609009i	$0.208433\pi$
$168$	0	0
$169$	4.55220	0.350169
$170$	0	0
$171$	−5.87086	−0.448956
$172$	0	0
$173$	−1.62093	−0.123237	−0.0616186	−	0.998100i	$-0.519626\pi$
$173$	−1.62093	−0.123237	−0.0616186	+	0.998100i	$0.519626\pi$
$174$	0	0
$175$	17.4559	1.31954
$176$	0	0
$177$	11.3627	0.854070
$178$	0	0
$179$	4.47539	0.334506	0.167253	−	0.985914i	$-0.446510\pi$
$179$	4.47539	0.334506	0.167253	+	0.985914i	$0.446510\pi$
$180$	0	0
$181$	−1.20594	−0.0896369	−0.0448184	−	0.998995i	$-0.514271\pi$
$181$	−1.20594	−0.0896369	−0.0448184	+	0.998995i	$0.514271\pi$
$182$	0	0
$183$	−3.87086	−0.286143
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	−1.01118	−0.0739447
$188$	0	0
$189$	3.68133	0.267777
$190$	0	0
$191$	−0.758136	−0.0548568	−0.0274284	−	0.999624i	$-0.508732\pi$
$191$	−0.758136	−0.0548568	−0.0274284	+	0.999624i	$0.508732\pi$
$192$	0	0
$193$	−16.4671	−1.18532	−0.592662	−	0.805451i	$-0.701923\pi$
$193$	−16.4671	−1.18532	−0.592662	+	0.805451i	$0.701923\pi$
$194$	0	0
$195$	−2.12914	−0.152471
$196$	0	0
$197$	17.9588	1.27951	0.639757	−	0.768577i	$-0.279035\pi$
$197$	17.9588	1.27951	0.639757	+	0.768577i	$0.279035\pi$
$198$	0	0
$199$	−1.33508	−0.0946410	−0.0473205	−	0.998880i	$-0.515068\pi$
$199$	−1.33508	−0.0946410	−0.0473205	+	0.998880i	$0.515068\pi$
$200$	0	0
$201$	−7.04399	−0.496845
$202$	0	0
$203$	3.68133	0.258379
$204$	0	0
$205$	−4.43424	−0.309701
$206$	0	0
$207$	−2.50820	−0.174332
$208$	0	0
$209$	1.87086	0.129410
$210$	0	0
$211$	8.37907	0.576839	0.288419	−	0.957504i	$-0.406870\pi$
$211$	8.37907	0.576839	0.288419	+	0.957504i	$0.406870\pi$
$212$	0	0
$213$	6.24993	0.428238
$214$	0	0
$215$	5.45065	0.371731
$216$	0	0
$217$	9.23353	0.626813
$218$	0	0
$219$	7.87086	0.531864
$220$	0	0
$221$	−13.2939	−0.894246
$222$	0	0
$223$	6.66492	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$223$	6.66492	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$224$	0	0
$225$	−4.74173	−0.316115
$226$	0	0
$227$	0.379068	0.0251596	0.0125798	−	0.999921i	$-0.495996\pi$
$227$	0.379068	0.0251596	0.0125798	+	0.999921i	$0.495996\pi$
$228$	0	0
$229$	−9.26634	−0.612337	−0.306168	−	0.951977i	$-0.599047\pi$
$229$	−9.26634	−0.612337	−0.306168	+	0.951977i	$0.599047\pi$
$230$	0	0
$231$	−1.17313	−0.0771861
$232$	0	0
$233$	13.7417	0.900251	0.450125	−	0.892965i	$-0.351379\pi$
$233$	13.7417	0.900251	0.450125	+	0.892965i	$0.351379\pi$
$234$	0	0
$235$	−5.61259	−0.366125
$236$	0	0
$237$	−4.85446	−0.315331
$238$	0	0
$239$	−17.8709	−1.15597	−0.577985	−	0.816047i	$-0.696161\pi$
$239$	−17.8709	−1.15597	−0.577985	+	0.816047i	$0.696161\pi$
$240$	0	0
$241$	−14.2223	−0.916142	−0.458071	−	0.888916i	$-0.651459\pi$
$241$	−14.2223	−0.916142	−0.458071	+	0.888916i	$0.651459\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.32985	−0.212736
$246$	0	0
$247$	24.5962	1.56502
$248$	0	0
$249$	8.37907	0.531002
$250$	0	0
$251$	8.43947	0.532694	0.266347	−	0.963877i	$-0.414183\pi$
$251$	8.43947	0.532694	0.266347	+	0.963877i	$0.414183\pi$
$252$	0	0
$253$	0.799288	0.0502508
$254$	0	0
$255$	1.61259	0.100985
$256$	0	0
$257$	25.3627	1.58208	0.791040	−	0.611765i	$-0.209540\pi$
$257$	25.3627	1.58208	0.791040	+	0.611765i	$0.209540\pi$
$258$	0	0
$259$	28.9753	1.80043
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	−23.4835	−1.44805	−0.724026	−	0.689773i	$-0.757710\pi$
$263$	−23.4835	−1.44805	−0.724026	+	0.689773i	$0.757710\pi$
$264$	0	0
$265$	−4.19264	−0.257552
$266$	0	0
$267$	15.9313	0.974977
$268$	0	0
$269$	2.10155	0.128134	0.0640669	−	0.997946i	$-0.479593\pi$
$269$	2.10155	0.128134	0.0640669	+	0.997946i	$0.479593\pi$
$270$	0	0
$271$	−3.10439	−0.188578	−0.0942892	−	0.995545i	$-0.530058\pi$
$271$	−3.10439	−0.188578	−0.0942892	+	0.995545i	$0.530058\pi$
$272$	0	0
$273$	−15.4231	−0.933446
$274$	0	0
$275$	1.51104	0.0911194
$276$	0	0
$277$	13.9641	0.839020	0.419510	−	0.907751i	$-0.362202\pi$
$277$	13.9641	0.839020	0.419510	+	0.907751i	$0.362202\pi$
$278$	0	0
$279$	−2.50820	−0.150162
$280$	0	0
$281$	−14.4342	−0.861074	−0.430537	−	0.902573i	$-0.641676\pi$
$281$	−14.4342	−0.861074	−0.430537	+	0.902573i	$0.641676\pi$
$282$	0	0
$283$	8.25827	0.490903	0.245452	−	0.969409i	$-0.421064\pi$
$283$	8.25827	0.490903	0.245452	+	0.969409i	$0.421064\pi$
$284$	0	0
$285$	−2.98359	−0.176733
$286$	0	0
$287$	−32.1208	−1.89603
$288$	0	0
$289$	−6.93126	−0.407721
$290$	0	0
$291$	−11.2335	−0.658521
$292$	0	0
$293$	12.4478	0.727209	0.363604	−	0.931554i	$-0.381546\pi$
$293$	12.4478	0.727209	0.363604	+	0.931554i	$0.381546\pi$
$294$	0	0
$295$	5.77454	0.336207
$296$	0	0
$297$	0.318669	0.0184911
$298$	0	0
$299$	10.5082	0.607705
$300$	0	0
$301$	39.4835	2.27579
$302$	0	0
$303$	10.9149	0.627042
$304$	0	0
$305$	−1.96719	−0.112641
$306$	0	0
$307$	−21.9917	−1.25513	−0.627565	−	0.778564i	$-0.715948\pi$
$307$	−21.9917	−1.25513	−0.627565	+	0.778564i	$0.715948\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	19.0440	1.07989	0.539943	−	0.841702i	$-0.318446\pi$
$311$	19.0440	1.07989	0.539943	+	0.841702i	$0.318446\pi$
$312$	0	0
$313$	9.89845	0.559493	0.279747	−	0.960074i	$-0.409750\pi$
$313$	9.89845	0.559493	0.279747	+	0.960074i	$0.409750\pi$
$314$	0	0
$315$	1.87086	0.105411
$316$	0	0
$317$	−0.568602	−0.0319359	−0.0159679	−	0.999873i	$-0.505083\pi$
$317$	−0.568602	−0.0319359	−0.0159679	+	0.999873i	$0.505083\pi$
$318$	0	0
$319$	0.318669	0.0178421
$320$	0	0
$321$	−3.83805	−0.214219
$322$	0	0
$323$	−18.6290	−1.03655
$324$	0	0
$325$	19.8656	1.10195
$326$	0	0
$327$	−8.18953	−0.452882
$328$	0	0
$329$	−40.6566	−2.24147
$330$	0	0
$331$	1.17836	0.0647683	0.0323841	−	0.999475i	$-0.489690\pi$
$331$	1.17836	0.0647683	0.0323841	+	0.999475i	$0.489690\pi$
$332$	0	0
$333$	−7.87086	−0.431321
$334$	0	0
$335$	−3.57978	−0.195584
$336$	0	0
$337$	17.2007	0.936983	0.468491	−	0.883468i	$-0.344798\pi$
$337$	17.2007	0.936983	0.468491	+	0.883468i	$0.344798\pi$
$338$	0	0
$339$	−13.2059	−0.717248
$340$	0	0
$341$	0.799288	0.0432838
$342$	0	0
$343$	1.64852	0.0890116
$344$	0	0
$345$	−1.27468	−0.0686263
$346$	0	0
$347$	15.2663	0.819540	0.409770	−	0.912189i	$-0.365609\pi$
$347$	15.2663	0.819540	0.409770	+	0.912189i	$0.365609\pi$
$348$	0	0
$349$	27.7089	1.48322	0.741612	−	0.670829i	$-0.234062\pi$
$349$	27.7089	1.48322	0.741612	+	0.670829i	$0.234062\pi$
$350$	0	0
$351$	4.18953	0.223621
$352$	0	0
$353$	2.79929	0.148991	0.0744955	−	0.997221i	$-0.476265\pi$
$353$	2.79929	0.148991	0.0744955	+	0.997221i	$0.476265\pi$
$354$	0	0
$355$	3.17624	0.168577
$356$	0	0
$357$	11.6813	0.618242
$358$	0	0
$359$	−12.2583	−0.646967	−0.323483	−	0.946234i	$-0.604854\pi$
$359$	−12.2583	−0.646967	−0.323483	+	0.946234i	$0.604854\pi$
$360$	0	0
$361$	15.4671	0.814055
$362$	0	0
$363$	10.8984	0.572020
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	4.25827	0.222280	0.111140	−	0.993805i	$-0.464550\pi$
$367$	4.25827	0.222280	0.111140	+	0.993805i	$0.464550\pi$
$368$	0	0
$369$	8.72532	0.454222
$370$	0	0
$371$	−30.3707	−1.57677
$372$	0	0
$373$	−25.7417	−1.33286	−0.666428	−	0.745569i	$-0.732178\pi$
$373$	−25.7417	−1.33286	−0.666428	+	0.745569i	$0.732178\pi$
$374$	0	0
$375$	−4.95078	−0.255657
$376$	0	0
$377$	4.18953	0.215772
$378$	0	0
$379$	19.6454	1.00912	0.504558	−	0.863378i	$-0.331655\pi$
$379$	19.6454	1.00912	0.504558	+	0.863378i	$0.331655\pi$
$380$	0	0
$381$	9.39547	0.481345
$382$	0	0
$383$	22.5634	1.15293	0.576467	−	0.817120i	$-0.304431\pi$
$383$	22.5634	1.15293	0.576467	+	0.817120i	$0.304431\pi$
$384$	0	0
$385$	−0.596187	−0.0303845
$386$	0	0
$387$	−10.7253	−0.545199
$388$	0	0
$389$	−15.8105	−0.801622	−0.400811	−	0.916161i	$-0.631272\pi$
$389$	−15.8105	−0.801622	−0.400811	+	0.916161i	$0.631272\pi$
$390$	0	0
$391$	−7.95885	−0.402496
$392$	0	0
$393$	17.0768	0.861411
$394$	0	0
$395$	−2.46705	−0.124131
$396$	0	0
$397$	3.27468	0.164351	0.0821757	−	0.996618i	$-0.473813\pi$
$397$	3.27468	0.164351	0.0821757	+	0.996618i	$0.473813\pi$
$398$	0	0
$399$	−21.6126	−1.08198
$400$	0	0
$401$	2.47539	0.123615	0.0618075	−	0.998088i	$-0.480314\pi$
$401$	2.47539	0.123615	0.0618075	+	0.998088i	$0.480314\pi$
$402$	0	0
$403$	10.5082	0.523451
$404$	0	0
$405$	−0.508203	−0.0252528
$406$	0	0
$407$	2.50820	0.124327
$408$	0	0
$409$	15.0164	0.742514	0.371257	−	0.928530i	$-0.378927\pi$
$409$	15.0164	0.742514	0.371257	+	0.928530i	$0.378927\pi$
$410$	0	0
$411$	−15.0164	−0.740705
$412$	0	0
$413$	41.8297	2.05831
$414$	0	0
$415$	4.25827	0.209030
$416$	0	0
$417$	−7.68133	−0.376156
$418$	0	0
$419$	−13.8709	−0.677636	−0.338818	−	0.940852i	$-0.610027\pi$
$419$	−13.8709	−0.677636	−0.338818	+	0.940852i	$0.610027\pi$
$420$	0	0
$421$	7.45065	0.363122	0.181561	−	0.983380i	$-0.441885\pi$
$421$	7.45065	0.363122	0.181561	+	0.983380i	$0.441885\pi$
$422$	0	0
$423$	11.0440	0.536977
$424$	0	0
$425$	−15.0461	−0.729844
$426$	0	0
$427$	−14.2499	−0.689603
$428$	0	0
$429$	−1.33508	−0.0644581
$430$	0	0
$431$	−14.7253	−0.709294	−0.354647	−	0.935000i	$-0.615399\pi$
$431$	−14.7253	−0.709294	−0.354647	+	0.935000i	$0.615399\pi$
$432$	0	0
$433$	1.52461	0.0732681	0.0366340	−	0.999329i	$-0.488336\pi$
$433$	1.52461	0.0732681	0.0366340	+	0.999329i	$0.488336\pi$
$434$	0	0
$435$	−0.508203	−0.0243665
$436$	0	0
$437$	14.7253	0.704408
$438$	0	0
$439$	−23.3023	−1.11216	−0.556078	−	0.831130i	$-0.687694\pi$
$439$	−23.3023	−1.11216	−0.556078	+	0.831130i	$0.687694\pi$
$440$	0	0
$441$	6.55220	0.312009
$442$	0	0
$443$	23.0440	1.09485	0.547427	−	0.836854i	$-0.315608\pi$
$443$	23.0440	1.09485	0.547427	+	0.836854i	$0.315608\pi$
$444$	0	0
$445$	8.09632	0.383802
$446$	0	0
$447$	−11.3955	−0.538987
$448$	0	0
$449$	−6.53579	−0.308443	−0.154221	−	0.988036i	$-0.549287\pi$
$449$	−6.53579	−0.308443	−0.154221	+	0.988036i	$0.549287\pi$
$450$	0	0
$451$	−2.78049	−0.130928
$452$	0	0
$453$	−2.03281	−0.0955099
$454$	0	0
$455$	−7.83805	−0.367454
$456$	0	0
$457$	10.2223	0.478181	0.239091	−	0.970997i	$-0.423151\pi$
$457$	10.2223	0.478181	0.239091	+	0.970997i	$0.423151\pi$
$458$	0	0
$459$	−3.17313	−0.148109
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	24.1812	1.12380	0.561898	−	0.827207i	$-0.310071\pi$
$463$	24.1812	1.12380	0.561898	+	0.827207i	$0.310071\pi$
$464$	0	0
$465$	−1.27468	−0.0591118
$466$	0	0
$467$	−9.96719	−0.461226	−0.230613	−	0.973046i	$-0.574073\pi$
$467$	−9.96719	−0.461226	−0.230613	+	0.973046i	$0.574073\pi$
$468$	0	0
$469$	−25.9313	−1.19739
$470$	0	0
$471$	−10.7581	−0.495709
$472$	0	0
$473$	3.41783	0.157152
$474$	0	0
$475$	27.8381	1.27730
$476$	0	0
$477$	8.24993	0.377738
$478$	0	0
$479$	21.7745	0.994904	0.497452	−	0.867491i	$-0.334269\pi$
$479$	21.7745	0.994904	0.497452	+	0.867491i	$0.334269\pi$
$480$	0	0
$481$	32.9753	1.50354
$482$	0	0
$483$	−9.23353	−0.420140
$484$	0	0
$485$	−5.70892	−0.259229
$486$	0	0
$487$	−37.4506	−1.69705	−0.848525	−	0.529155i	$-0.822509\pi$
$487$	−37.4506	−1.69705	−0.848525	+	0.529155i	$0.822509\pi$
$488$	0	0
$489$	−21.8297	−0.987174
$490$	0	0
$491$	−16.6925	−0.753322	−0.376661	−	0.926351i	$-0.622928\pi$
$491$	−16.6925	−0.753322	−0.376661	+	0.926351i	$0.622928\pi$
$492$	0	0
$493$	−3.17313	−0.142910
$494$	0	0
$495$	0.161949	0.00727906
$496$	0	0
$497$	23.0081	1.03205
$498$	0	0
$499$	43.8573	1.96332	0.981661	−	0.190634i	$-0.0610544\pi$
$499$	43.8573	1.96332	0.981661	+	0.190634i	$0.0610544\pi$
$500$	0	0
$501$	−20.4999	−0.915866
$502$	0	0
$503$	−26.4067	−1.17741	−0.588707	−	0.808346i	$-0.700363\pi$
$503$	−26.4067	−1.17741	−0.588707	+	0.808346i	$0.700363\pi$
$504$	0	0
$505$	5.54697	0.246837
$506$	0	0
$507$	−4.55220	−0.202170
$508$	0	0
$509$	36.1432	1.60202	0.801009	−	0.598653i	$-0.204297\pi$
$509$	36.1432	1.60202	0.801009	+	0.598653i	$0.204297\pi$
$510$	0	0
$511$	28.9753	1.28179
$512$	0	0
$513$	5.87086	0.259205
$514$	0	0
$515$	4.06563	0.179153
$516$	0	0
$517$	−3.51938	−0.154782
$518$	0	0
$519$	1.62093	0.0711510
$520$	0	0
$521$	26.1208	1.14437	0.572186	−	0.820124i	$-0.306095\pi$
$521$	26.1208	1.14437	0.572186	+	0.820124i	$0.306095\pi$
$522$	0	0
$523$	−2.35148	−0.102823	−0.0514116	−	0.998678i	$-0.516372\pi$
$523$	−2.35148	−0.102823	−0.0514116	+	0.998678i	$0.516372\pi$
$524$	0	0
$525$	−17.4559	−0.761837
$526$	0	0
$527$	−7.95885	−0.346693
$528$	0	0
$529$	−16.7089	−0.726475
$530$	0	0
$531$	−11.3627	−0.493097
$532$	0	0
$533$	−36.5550	−1.58337
$534$	0	0
$535$	−1.95051	−0.0843279
$536$	0	0
$537$	−4.47539	−0.193127
$538$	0	0
$539$	−2.08798	−0.0899358
$540$	0	0
$541$	−12.2499	−0.526666	−0.263333	−	0.964705i	$-0.584822\pi$
$541$	−12.2499	−0.526666	−0.263333	+	0.964705i	$0.584822\pi$
$542$	0	0
$543$	1.20594	0.0517519
$544$	0	0
$545$	−4.16195	−0.178278
$546$	0	0
$547$	−11.3023	−0.483250	−0.241625	−	0.970370i	$-0.577680\pi$
$547$	−11.3023	−0.483250	−0.241625	+	0.970370i	$0.577680\pi$
$548$	0	0
$549$	3.87086	0.165204
$550$	0	0
$551$	5.87086	0.250107
$552$	0	0
$553$	−17.8709	−0.759946
$554$	0	0
$555$	−4.00000	−0.169791
$556$	0	0
$557$	−32.4671	−1.37567	−0.687837	−	0.725866i	$-0.741439\pi$
$557$	−32.4671	−1.37567	−0.687837	+	0.725866i	$0.741439\pi$
$558$	0	0
$559$	44.9341	1.90051
$560$	0	0
$561$	1.01118	0.0426920
$562$	0	0
$563$	11.5439	0.486516	0.243258	−	0.969962i	$-0.421784\pi$
$563$	11.5439	0.486516	0.243258	+	0.969962i	$0.421784\pi$
$564$	0	0
$565$	−6.71130	−0.282347
$566$	0	0
$567$	−3.68133	−0.154601
$568$	0	0
$569$	34.5358	1.44782	0.723908	−	0.689897i	$-0.242344\pi$
$569$	34.5358	1.44782	0.723908	+	0.689897i	$0.242344\pi$
$570$	0	0
$571$	21.7089	0.908490	0.454245	−	0.890877i	$-0.349909\pi$
$571$	21.7089	0.908490	0.454245	+	0.890877i	$0.349909\pi$
$572$	0	0
$573$	0.758136	0.0316716
$574$	0	0
$575$	11.8932	0.495982
$576$	0	0
$577$	−3.13720	−0.130604	−0.0653018	−	0.997866i	$-0.520801\pi$
$577$	−3.13720	−0.130604	−0.0653018	+	0.997866i	$0.520801\pi$
$578$	0	0
$579$	16.4671	0.684347
$580$	0	0
$581$	30.8461	1.27971
$582$	0	0
$583$	−2.62900	−0.108882
$584$	0	0
$585$	2.12914	0.0880289
$586$	0	0
$587$	31.5386	1.30174	0.650869	−	0.759190i	$-0.274405\pi$
$587$	31.5386	1.30174	0.650869	+	0.759190i	$0.274405\pi$
$588$	0	0
$589$	14.7253	0.606746
$590$	0	0
$591$	−17.9588	−0.738728
$592$	0	0
$593$	−26.9753	−1.10774	−0.553870	−	0.832603i	$-0.686850\pi$
$593$	−26.9753	−1.10774	−0.553870	+	0.832603i	$0.686850\pi$
$594$	0	0
$595$	5.93649	0.243372
$596$	0	0
$597$	1.33508	0.0546410
$598$	0	0
$599$	20.2364	0.826836	0.413418	−	0.910541i	$-0.364335\pi$
$599$	20.2364	0.826836	0.413418	+	0.910541i	$0.364335\pi$
$600$	0	0
$601$	−42.7826	−1.74514	−0.872570	−	0.488490i	$-0.837548\pi$
$601$	−42.7826	−1.74514	−0.872570	+	0.488490i	$0.837548\pi$
$602$	0	0
$603$	7.04399	0.286854
$604$	0	0
$605$	5.53863	0.225177
$606$	0	0
$607$	−3.42829	−0.139150	−0.0695750	−	0.997577i	$-0.522164\pi$
$607$	−3.42829	−0.139150	−0.0695750	+	0.997577i	$0.522164\pi$
$608$	0	0
$609$	−3.68133	−0.149175
$610$	0	0
$611$	−46.2692	−1.87185
$612$	0	0
$613$	21.3267	0.861379	0.430689	−	0.902500i	$-0.358270\pi$
$613$	21.3267	0.861379	0.430689	+	0.902500i	$0.358270\pi$
$614$	0	0
$615$	4.43424	0.178806
$616$	0	0
$617$	−29.8074	−1.20000	−0.599999	−	0.800000i	$-0.704833\pi$
$617$	−29.8074	−1.20000	−0.599999	+	0.800000i	$0.704833\pi$
$618$	0	0
$619$	5.87086	0.235970	0.117985	−	0.993015i	$-0.462357\pi$
$619$	5.87086	0.235970	0.117985	+	0.993015i	$0.462357\pi$
$620$	0	0
$621$	2.50820	0.100651
$622$	0	0
$623$	58.6482	2.34969
$624$	0	0
$625$	21.1926	0.847706
$626$	0	0
$627$	−1.87086	−0.0747151
$628$	0	0
$629$	−24.9753	−0.995829
$630$	0	0
$631$	−20.7529	−0.826160	−0.413080	−	0.910695i	$-0.635547\pi$
$631$	−20.7529	−0.826160	−0.413080	+	0.910695i	$0.635547\pi$
$632$	0	0
$633$	−8.37907	−0.333038
$634$	0	0
$635$	4.77481	0.189483
$636$	0	0
$637$	−27.4506	−1.08763
$638$	0	0
$639$	−6.24993	−0.247244
$640$	0	0
$641$	−18.9149	−0.747092	−0.373546	−	0.927612i	$-0.621858\pi$
$641$	−18.9149	−0.747092	−0.373546	+	0.927612i	$0.621858\pi$
$642$	0	0
$643$	−47.1648	−1.86000	−0.929999	−	0.367562i	$-0.880192\pi$
$643$	−47.1648	−1.86000	−0.929999	+	0.367562i	$0.880192\pi$
$644$	0	0
$645$	−5.45065	−0.214619
$646$	0	0
$647$	−7.20071	−0.283089	−0.141545	−	0.989932i	$-0.545207\pi$
$647$	−7.20071	−0.283089	−0.141545	+	0.989932i	$0.545207\pi$
$648$	0	0
$649$	3.62093	0.142134
$650$	0	0
$651$	−9.23353	−0.361890
$652$	0	0
$653$	−42.5191	−1.66390	−0.831951	−	0.554850i	$-0.812776\pi$
$653$	−42.5191	−1.66390	−0.831951	+	0.554850i	$0.812776\pi$
$654$	0	0
$655$	8.67849	0.339097
$656$	0	0
$657$	−7.87086	−0.307072
$658$	0	0
$659$	−1.90679	−0.0742779	−0.0371390	−	0.999310i	$-0.511824\pi$
$659$	−1.90679	−0.0742779	−0.0371390	+	0.999310i	$0.511824\pi$
$660$	0	0
$661$	−35.9969	−1.40012	−0.700058	−	0.714086i	$-0.746843\pi$
$661$	−35.9969	−1.40012	−0.700058	+	0.714086i	$0.746843\pi$
$662$	0	0
$663$	13.2939	0.516293
$664$	0	0
$665$	−10.9836	−0.425925
$666$	0	0
$667$	2.50820	0.0971180
$668$	0	0
$669$	−6.66492	−0.257681
$670$	0	0
$671$	−1.23353	−0.0476197
$672$	0	0
$673$	9.46421	0.364819	0.182409	−	0.983223i	$-0.441610\pi$
$673$	9.46421	0.364819	0.182409	+	0.983223i	$0.441610\pi$
$674$	0	0
$675$	4.74173	0.182509
$676$	0	0
$677$	3.81047	0.146448	0.0732241	−	0.997316i	$-0.476671\pi$
$677$	3.81047	0.146448	0.0732241	+	0.997316i	$0.476671\pi$
$678$	0	0
$679$	−41.3543	−1.58703
$680$	0	0
$681$	−0.379068	−0.0145259
$682$	0	0
$683$	32.9341	1.26019	0.630094	−	0.776519i	$-0.283016\pi$
$683$	32.9341	1.26019	0.630094	+	0.776519i	$0.283016\pi$
$684$	0	0
$685$	−7.63139	−0.291580
$686$	0	0
$687$	9.26634	0.353533
$688$	0	0
$689$	−34.5634	−1.31676
$690$	0	0
$691$	0.752908	0.0286420	0.0143210	−	0.999897i	$-0.495441\pi$
$691$	0.752908	0.0286420	0.0143210	+	0.999897i	$0.495441\pi$
$692$	0	0
$693$	1.17313	0.0445634
$694$	0	0
$695$	−3.90368	−0.148075
$696$	0	0
$697$	27.6866	1.04870
$698$	0	0
$699$	−13.7417	−0.519760
$700$	0	0
$701$	27.3543	1.03316	0.516579	−	0.856239i	$-0.327205\pi$
$701$	27.3543	1.03316	0.516579	+	0.856239i	$0.327205\pi$
$702$	0	0
$703$	46.2088	1.74280
$704$	0	0
$705$	5.61259	0.211383
$706$	0	0
$707$	40.1812	1.51117
$708$	0	0
$709$	−50.1760	−1.88440	−0.942199	−	0.335054i	$-0.891246\pi$
$709$	−50.1760	−1.88440	−0.942199	+	0.335054i	$0.891246\pi$
$710$	0	0
$711$	4.85446	0.182056
$712$	0	0
$713$	6.29108	0.235603
$714$	0	0
$715$	−0.678490	−0.0253741
$716$	0	0
$717$	17.8709	0.667400
$718$	0	0
$719$	−30.4014	−1.13378	−0.566891	−	0.823793i	$-0.691854\pi$
$719$	−30.4014	−1.13378	−0.566891	+	0.823793i	$0.691854\pi$
$720$	0	0
$721$	29.4506	1.09680
$722$	0	0
$723$	14.2223	0.528935
$724$	0	0
$725$	4.74173	0.176103
$726$	0	0
$727$	46.1676	1.71226	0.856131	−	0.516758i	$-0.172861\pi$
$727$	46.1676	1.71226	0.856131	+	0.516758i	$0.172861\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−34.0328	−1.25875
$732$	0	0
$733$	34.1208	1.26028	0.630140	−	0.776481i	$-0.282997\pi$
$733$	34.1208	1.26028	0.630140	+	0.776481i	$0.282997\pi$
$734$	0	0
$735$	3.32985	0.122823
$736$	0	0
$737$	−2.24470	−0.0826847
$738$	0	0
$739$	−40.7826	−1.50021	−0.750106	−	0.661317i	$-0.769998\pi$
$739$	−40.7826	−1.50021	−0.750106	+	0.661317i	$0.769998\pi$
$740$	0	0
$741$	−24.5962	−0.903564
$742$	0	0
$743$	14.3515	0.526505	0.263252	−	0.964727i	$-0.415205\pi$
$743$	14.3515	0.526505	0.263252	+	0.964727i	$0.415205\pi$
$744$	0	0
$745$	−5.79122	−0.212174
$746$	0	0
$747$	−8.37907	−0.306574
$748$	0	0
$749$	−14.1291	−0.516267
$750$	0	0
$751$	21.4506	0.782745	0.391373	−	0.920232i	$-0.372000\pi$
$751$	21.4506	0.782745	0.391373	+	0.920232i	$0.372000\pi$
$752$	0	0
$753$	−8.43947	−0.307551
$754$	0	0
$755$	−1.03308	−0.0375977
$756$	0	0
$757$	−0.862796	−0.0313588	−0.0156794	−	0.999877i	$-0.504991\pi$
$757$	−0.862796	−0.0313588	−0.0156794	+	0.999877i	$0.504991\pi$
$758$	0	0
$759$	−0.799288	−0.0290123
$760$	0	0
$761$	25.4423	0.922283	0.461141	−	0.887327i	$-0.347440\pi$
$761$	25.4423	0.922283	0.461141	+	0.887327i	$0.347440\pi$
$762$	0	0
$763$	−30.1484	−1.09144
$764$	0	0
$765$	−1.61259	−0.0583035
$766$	0	0
$767$	47.6043	1.71889
$768$	0	0
$769$	−0.194762	−0.00702331	−0.00351165	−	0.999994i	$-0.501118\pi$
$769$	−0.194762	−0.00702331	−0.00351165	+	0.999994i	$0.501118\pi$
$770$	0	0
$771$	−25.3627	−0.913414
$772$	0	0
$773$	−44.5327	−1.60173	−0.800865	−	0.598846i	$-0.795626\pi$
$773$	−44.5327	−1.60173	−0.800865	+	0.598846i	$0.795626\pi$
$774$	0	0
$775$	11.8932	0.427217
$776$	0	0
$777$	−28.9753	−1.03948
$778$	0	0
$779$	−51.2252	−1.83533
$780$	0	0
$781$	1.99166	0.0712673
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−5.46732	−0.195137
$786$	0	0
$787$	−24.4999	−0.873326	−0.436663	−	0.899625i	$-0.643840\pi$
$787$	−24.4999	−0.873326	−0.436663	+	0.899625i	$0.643840\pi$
$788$	0	0
$789$	23.4835	0.836033
$790$	0	0
$791$	−48.6154	−1.72857
$792$	0	0
$793$	−16.2171	−0.575887
$794$	0	0
$795$	4.19264	0.148698
$796$	0	0
$797$	−12.7253	−0.450754	−0.225377	−	0.974272i	$-0.572361\pi$
$797$	−12.7253	−0.450754	−0.225377	+	0.974272i	$0.572361\pi$
$798$	0	0
$799$	35.0440	1.23977
$800$	0	0
$801$	−15.9313	−0.562904
$802$	0	0
$803$	2.50820	0.0885126
$804$	0	0
$805$	−4.69251	−0.165389
$806$	0	0
$807$	−2.10155	−0.0739781
$808$	0	0
$809$	13.6402	0.479563	0.239782	−	0.970827i	$-0.422924\pi$
$809$	13.6402	0.479563	0.239782	+	0.970827i	$0.422924\pi$
$810$	0	0
$811$	15.9948	0.561652	0.280826	−	0.959759i	$-0.409392\pi$
$811$	15.9948	0.561652	0.280826	+	0.959759i	$0.409392\pi$
$812$	0	0
$813$	3.10439	0.108876
$814$	0	0
$815$	−11.0939	−0.388604
$816$	0	0
$817$	62.9669	2.20293
$818$	0	0
$819$	15.4231	0.538925
$820$	0	0
$821$	3.70892	0.129442	0.0647210	−	0.997903i	$-0.479384\pi$
$821$	3.70892	0.129442	0.0647210	+	0.997903i	$0.479384\pi$
$822$	0	0
$823$	48.8685	1.70345	0.851724	−	0.523991i	$-0.175557\pi$
$823$	48.8685	1.70345	0.851724	+	0.523991i	$0.175557\pi$
$824$	0	0
$825$	−1.51104	−0.0526078
$826$	0	0
$827$	1.20905	0.0420428	0.0210214	−	0.999779i	$-0.493308\pi$
$827$	1.20905	0.0420428	0.0210214	+	0.999779i	$0.493308\pi$
$828$	0	0
$829$	4.67015	0.162201	0.0811005	−	0.996706i	$-0.474157\pi$
$829$	4.67015	0.162201	0.0811005	+	0.996706i	$0.474157\pi$
$830$	0	0
$831$	−13.9641	−0.484408
$832$	0	0
$833$	20.7909	0.720364
$834$	0	0
$835$	−10.4181	−0.360533
$836$	0	0
$837$	2.50820	0.0866962
$838$	0	0
$839$	10.8513	0.374630	0.187315	−	0.982300i	$-0.440021\pi$
$839$	10.8513	0.374630	0.187315	+	0.982300i	$0.440021\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	14.4342	0.497142
$844$	0	0
$845$	−2.31344	−0.0795848
$846$	0	0
$847$	40.1208	1.37857
$848$	0	0
$849$	−8.25827	−0.283423
$850$	0	0
$851$	19.7417	0.676738
$852$	0	0
$853$	−44.3296	−1.51782	−0.758908	−	0.651198i	$-0.774267\pi$
$853$	−44.3296	−1.51782	−0.758908	+	0.651198i	$0.774267\pi$
$854$	0	0
$855$	2.98359	0.102037
$856$	0	0
$857$	3.23353	0.110455	0.0552276	−	0.998474i	$-0.482412\pi$
$857$	3.23353	0.110455	0.0552276	+	0.998474i	$0.482412\pi$
$858$	0	0
$859$	38.3463	1.30836	0.654179	−	0.756340i	$-0.273014\pi$
$859$	38.3463	1.30836	0.654179	+	0.756340i	$0.273014\pi$
$860$	0	0
$861$	32.1208	1.09467
$862$	0	0
$863$	−41.4751	−1.41183	−0.705915	−	0.708297i	$-0.749464\pi$
$863$	−41.4751	−1.41183	−0.705915	+	0.708297i	$0.749464\pi$
$864$	0	0
$865$	0.823763	0.0280088
$866$	0	0
$867$	6.93126	0.235398
$868$	0	0
$869$	−1.54697	−0.0524773
$870$	0	0
$871$	−29.5110	−0.999944
$872$	0	0
$873$	11.2335	0.380197
$874$	0	0
$875$	−18.2255	−0.616133
$876$	0	0
$877$	−13.0492	−0.440641	−0.220320	−	0.975428i	$-0.570710\pi$
$877$	−13.0492	−0.440641	−0.220320	+	0.975428i	$0.570710\pi$
$878$	0	0
$879$	−12.4478	−0.419854
$880$	0	0
$881$	−39.6074	−1.33441	−0.667203	−	0.744876i	$-0.732509\pi$
$881$	−39.6074	−1.33441	−0.667203	+	0.744876i	$0.732509\pi$
$882$	0	0
$883$	29.9505	1.00791	0.503957	−	0.863728i	$-0.331877\pi$
$883$	29.9505	1.00791	0.503957	+	0.863728i	$0.331877\pi$
$884$	0	0
$885$	−5.77454	−0.194109
$886$	0	0
$887$	−47.5439	−1.59637	−0.798183	−	0.602415i	$-0.794205\pi$
$887$	−47.5439	−1.59637	−0.798183	+	0.602415i	$0.794205\pi$
$888$	0	0
$889$	34.5878	1.16004
$890$	0	0
$891$	−0.318669	−0.0106758
$892$	0	0
$893$	−64.8378	−2.16971
$894$	0	0
$895$	−2.27441	−0.0760251
$896$	0	0
$897$	−10.5082	−0.350859
$898$	0	0
$899$	2.50820	0.0836533
$900$	0	0
$901$	26.1781	0.872119
$902$	0	0
$903$	−39.4835	−1.31393
$904$	0	0
$905$	0.612863	0.0203723
$906$	0	0
$907$	14.8628	0.493511	0.246756	−	0.969078i	$-0.420636\pi$
$907$	14.8628	0.493511	0.246756	+	0.969078i	$0.420636\pi$
$908$	0	0
$909$	−10.9149	−0.362023
$910$	0	0
$911$	28.5603	0.946244	0.473122	−	0.880997i	$-0.343127\pi$
$911$	28.5603	0.946244	0.473122	+	0.880997i	$0.343127\pi$
$912$	0	0
$913$	2.67015	0.0883691
$914$	0	0
$915$	1.96719	0.0650332
$916$	0	0
$917$	62.8654	2.07600
$918$	0	0
$919$	9.32462	0.307591	0.153795	−	0.988103i	$-0.450850\pi$
$919$	9.32462	0.307591	0.153795	+	0.988103i	$0.450850\pi$
$920$	0	0
$921$	21.9917	0.724650
$922$	0	0
$923$	26.1843	0.861867
$924$	0	0
$925$	37.3215	1.22712
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	−4.29108	−0.140786	−0.0703930	−	0.997519i	$-0.522425\pi$
$929$	−4.29108	−0.140786	−0.0703930	+	0.997519i	$0.522425\pi$
$930$	0	0
$931$	−38.4671	−1.26071
$932$	0	0
$933$	−19.0440	−0.623472
$934$	0	0
$935$	0.513884	0.0168058
$936$	0	0
$937$	−4.94767	−0.161633	−0.0808167	−	0.996729i	$-0.525753\pi$
$937$	−4.94767	−0.161633	−0.0808167	+	0.996729i	$0.525753\pi$
$938$	0	0
$939$	−9.89845	−0.323024
$940$	0	0
$941$	17.7969	0.580162	0.290081	−	0.957002i	$-0.406318\pi$
$941$	17.7969	0.580162	0.290081	+	0.957002i	$0.406318\pi$
$942$	0	0
$943$	−21.8849	−0.712670
$944$	0	0
$945$	−1.87086	−0.0608592
$946$	0	0
$947$	−53.7693	−1.74727	−0.873634	−	0.486584i	$-0.838243\pi$
$947$	−53.7693	−1.74727	−0.873634	+	0.486584i	$0.838243\pi$
$948$	0	0
$949$	32.9753	1.07042
$950$	0	0
$951$	0.568602	0.0184382
$952$	0	0
$953$	50.5550	1.63764	0.818819	−	0.574052i	$-0.194629\pi$
$953$	50.5550	1.63764	0.818819	+	0.574052i	$0.194629\pi$
$954$	0	0
$955$	0.385287	0.0124676
$956$	0	0
$957$	−0.318669	−0.0103011
$958$	0	0
$959$	−55.2804	−1.78510
$960$	0	0
$961$	−24.7089	−0.797062
$962$	0	0
$963$	3.83805	0.123679
$964$	0	0
$965$	8.36861	0.269395
$966$	0	0
$967$	13.1784	0.423787	0.211894	−	0.977293i	$-0.432037\pi$
$967$	13.1784	0.423787	0.211894	+	0.977293i	$0.432037\pi$
$968$	0	0
$969$	18.6290	0.598450
$970$	0	0
$971$	−16.3686	−0.525294	−0.262647	−	0.964892i	$-0.584595\pi$
$971$	−16.3686	−0.525294	−0.262647	+	0.964892i	$0.584595\pi$
$972$	0	0
$973$	−28.2775	−0.906536
$974$	0	0
$975$	−19.8656	−0.636210
$976$	0	0
$977$	−59.7229	−1.91071	−0.955353	−	0.295467i	$-0.904525\pi$
$977$	−59.7229	−1.91071	−0.955353	+	0.295467i	$0.904525\pi$
$978$	0	0
$979$	5.07681	0.162255
$980$	0	0
$981$	8.18953	0.261472
$982$	0	0
$983$	−55.4835	−1.76965	−0.884824	−	0.465926i	$-0.845721\pi$
$983$	−55.4835	−1.76965	−0.884824	+	0.465926i	$0.845721\pi$
$984$	0	0
$985$	−9.12675	−0.290802
$986$	0	0
$987$	40.6566	1.29411
$988$	0	0
$989$	26.9013	0.855411
$990$	0	0
$991$	−12.8737	−0.408947	−0.204473	−	0.978872i	$-0.565548\pi$
$991$	−12.8737	−0.408947	−0.204473	+	0.978872i	$0.565548\pi$
$992$	0	0
$993$	−1.17836	−0.0373940
$994$	0	0
$995$	0.678490	0.0215096
$996$	0	0
$997$	16.0880	0.509512	0.254756	−	0.967005i	$-0.418005\pi$
$997$	16.0880	0.509512	0.254756	+	0.967005i	$0.418005\pi$
$998$	0	0
$999$	7.87086	0.249023

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5568.2.a.bx.1.2		3
4.3	odd	2		5568.2.a.cb.1.2		3
8.3	odd	2		87.2.a.b.1.1	✓	3
8.5	even	2		1392.2.a.u.1.2		3
24.5	odd	2		4176.2.a.bx.1.2		3
24.11	even	2		261.2.a.e.1.3		3
40.3	even	4		2175.2.c.l.349.5		6
40.19	odd	2		2175.2.a.t.1.3		3
40.27	even	4		2175.2.c.l.349.2		6
56.27	even	2		4263.2.a.m.1.1		3
120.59	even	2		6525.2.a.bg.1.1		3
232.115	odd	2		2523.2.a.h.1.3		3
696.347	even	2		7569.2.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.a.b.1.1	✓	3	8.3	odd	2
261.2.a.e.1.3		3	24.11	even	2
1392.2.a.u.1.2		3	8.5	even	2
2175.2.a.t.1.3		3	40.19	odd	2
2175.2.c.l.349.2		6	40.27	even	4
2175.2.c.l.349.5		6	40.3	even	4
2523.2.a.h.1.3		3	232.115	odd	2
4176.2.a.bx.1.2		3	24.5	odd	2
4263.2.a.m.1.1		3	56.27	even	2
5568.2.a.bx.1.2		3	1.1	even	1	trivial
5568.2.a.cb.1.2		3	4.3	odd	2
6525.2.a.bg.1.1		3	120.59	even	2
7569.2.a.t.1.1		3	696.347	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:	\( N \)	\(=\)	\( 5568 = 2^{6} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5568.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.508203 q^{5} -3.68133 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.508203 q^{5} -3.68133 q^{7} +1.00000 q^{9} -0.318669 q^{11} -4.18953 q^{13} +0.508203 q^{15} +3.17313 q^{17} -5.87086 q^{19} +3.68133 q^{21} -2.50820 q^{23} -4.74173 q^{25} -1.00000 q^{27} -1.00000 q^{29} -2.50820 q^{31} +0.318669 q^{33} +1.87086 q^{35} -7.87086 q^{37} +4.18953 q^{39} +8.72532 q^{41} -10.7253 q^{43} -0.508203 q^{45} +11.0440 q^{47} +6.55220 q^{49} -3.17313 q^{51} +8.24993 q^{53} +0.161949 q^{55} +5.87086 q^{57} -11.3627 q^{59} +3.87086 q^{61} -3.68133 q^{63} +2.12914 q^{65} +7.04399 q^{67} +2.50820 q^{69} -6.24993 q^{71} -7.87086 q^{73} +4.74173 q^{75} +1.17313 q^{77} +4.85446 q^{79} +1.00000 q^{81} -8.37907 q^{83} -1.61259 q^{85} +1.00000 q^{87} -15.9313 q^{89} +15.4231 q^{91} +2.50820 q^{93} +2.98359 q^{95} +11.2335 q^{97} -0.318669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{7} + 3 q^{9} - 8 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 4 q^{21} - 6 q^{23} + 17 q^{25} - 3 q^{27} - 3 q^{29} - 6 q^{31} + 8 q^{33} - 10 q^{35} - 8 q^{37} + 4 q^{39} - 2 q^{41} - 4 q^{43} + 12 q^{47} - 3 q^{49} - 4 q^{51} - 8 q^{53} + 10 q^{55} + 2 q^{57} - 20 q^{59} - 4 q^{61} - 4 q^{63} + 22 q^{65} + 6 q^{69} + 14 q^{71} - 8 q^{73} - 17 q^{75} - 2 q^{77} + 2 q^{79} + 3 q^{81} - 8 q^{83} + 42 q^{85} + 3 q^{87} - 8 q^{89} + 8 q^{91} + 6 q^{93} + 12 q^{95} + 4 q^{97} - 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^7 + 3 * q^9 - 8 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 4 * q^21 - 6 * q^23 + 17 * q^25 - 3 * q^27 - 3 * q^29 - 6 * q^31 + 8 * q^33 - 10 * q^35 - 8 * q^37 + 4 * q^39 - 2 * q^41 - 4 * q^43 + 12 * q^47 - 3 * q^49 - 4 * q^51 - 8 * q^53 + 10 * q^55 + 2 * q^57 - 20 * q^59 - 4 * q^61 - 4 * q^63 + 22 * q^65 + 6 * q^69 + 14 * q^71 - 8 * q^73 - 17 * q^75 - 2 * q^77 + 2 * q^79 + 3 * q^81 - 8 * q^83 + 42 * q^85 + 3 * q^87 - 8 * q^89 + 8 * q^91 + 6 * q^93 + 12 * q^95 + 4 * q^97 - 8 * q^99

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