LMFDB - Embedded newform 5520.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -3.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +0.438447 q^{17} +7.12311 q^{19} +1.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.43845 q^{29} -8.68466 q^{31} +3.12311 q^{33} +1.56155 q^{35} -3.56155 q^{37} -2.00000 q^{39} +7.56155 q^{41} -10.2462 q^{43} -1.00000 q^{45} -8.00000 q^{47} -4.56155 q^{49} -0.438447 q^{51} -3.56155 q^{53} +3.12311 q^{55} -7.12311 q^{57} -2.43845 q^{59} -11.3693 q^{61} -1.56155 q^{63} -2.00000 q^{65} -1.56155 q^{67} -1.00000 q^{69} +0.684658 q^{71} +2.00000 q^{73} -1.00000 q^{75} +4.87689 q^{77} +6.24621 q^{79} +1.00000 q^{81} +12.6847 q^{83} -0.438447 q^{85} -4.43845 q^{87} +5.12311 q^{89} -3.12311 q^{91} +8.68466 q^{93} -7.12311 q^{95} -6.00000 q^{97} -3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 5 q^{17} + 6 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 13 q^{29} - 5 q^{31} - 2 q^{33} - q^{35} - 3 q^{37} - 4 q^{39} + 11 q^{41} - 4 q^{43} - 2 q^{45} - 16 q^{47} - 5 q^{49} - 5 q^{51} - 3 q^{53} - 2 q^{55} - 6 q^{57} - 9 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} + 4 q^{73} - 2 q^{75} + 18 q^{77} - 4 q^{79} + 2 q^{81} + 13 q^{83} - 5 q^{85} - 13 q^{87} + 2 q^{89} + 2 q^{91} + 5 q^{93} - 6 q^{95} - 12 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 5 * q^17 + 6 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 13 * q^29 - 5 * q^31 - 2 * q^33 - q^35 - 3 * q^37 - 4 * q^39 + 11 * q^41 - 4 * q^43 - 2 * q^45 - 16 * q^47 - 5 * q^49 - 5 * q^51 - 3 * q^53 - 2 * q^55 - 6 * q^57 - 9 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + q^67 - 2 * q^69 - 11 * q^71 + 4 * q^73 - 2 * q^75 + 18 * q^77 - 4 * q^79 + 2 * q^81 + 13 * q^83 - 5 * q^85 - 13 * q^87 + 2 * q^89 + 2 * q^91 + 5 * q^93 - 6 * q^95 - 12 * q^97 + 2 * 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
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.438447	0.106339	0.0531695	−	0.998586i	$-0.483068\pi$
$17$	0.438447	0.106339	0.0531695	+	0.998586i	$0.483068\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	0	0
$21$	1.56155	0.340759
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.43845	0.824199	0.412099	−	0.911139i	$-0.364796\pi$
$29$	4.43845	0.824199	0.412099	+	0.911139i	$0.364796\pi$
$30$	0	0
$31$	−8.68466	−1.55981	−0.779905	−	0.625897i	$-0.784733\pi$
$31$	−8.68466	−1.55981	−0.779905	+	0.625897i	$0.784733\pi$
$32$	0	0
$33$	3.12311	0.543663
$34$	0	0
$35$	1.56155	0.263951
$36$	0	0
$37$	−3.56155	−0.585516	−0.292758	−	0.956187i	$-0.594573\pi$
$37$	−3.56155	−0.585516	−0.292758	+	0.956187i	$0.594573\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	7.56155	1.18092	0.590458	−	0.807068i	$-0.298947\pi$
$41$	7.56155	1.18092	0.590458	+	0.807068i	$0.298947\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−0.438447	−0.0613949
$52$	0	0
$53$	−3.56155	−0.489217	−0.244608	−	0.969622i	$-0.578659\pi$
$53$	−3.56155	−0.489217	−0.244608	+	0.969622i	$0.578659\pi$
$54$	0	0
$55$	3.12311	0.421119
$56$	0	0
$57$	−7.12311	−0.943478
$58$	0	0
$59$	−2.43845	−0.317459	−0.158729	−	0.987322i	$-0.550740\pi$
$59$	−2.43845	−0.317459	−0.158729	+	0.987322i	$0.550740\pi$
$60$	0	0
$61$	−11.3693	−1.45569	−0.727846	−	0.685741i	$-0.759478\pi$
$61$	−11.3693	−1.45569	−0.727846	+	0.685741i	$0.759478\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−1.56155	−0.190774	−0.0953870	−	0.995440i	$-0.530409\pi$
$67$	−1.56155	−0.190774	−0.0953870	+	0.995440i	$0.530409\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	0.684658	0.0812540	0.0406270	−	0.999174i	$-0.487064\pi$
$71$	0.684658	0.0812540	0.0406270	+	0.999174i	$0.487064\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	4.87689	0.555774
$78$	0	0
$79$	6.24621	0.702754	0.351377	−	0.936234i	$-0.385714\pi$
$79$	6.24621	0.702754	0.351377	+	0.936234i	$0.385714\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.6847	1.39232	0.696161	−	0.717886i	$-0.254890\pi$
$83$	12.6847	1.39232	0.696161	+	0.717886i	$0.254890\pi$
$84$	0	0
$85$	−0.438447	−0.0475563
$86$	0	0
$87$	−4.43845	−0.475851
$88$	0	0
$89$	5.12311	0.543048	0.271524	−	0.962432i	$-0.412472\pi$
$89$	5.12311	0.543048	0.271524	+	0.962432i	$0.412472\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	8.68466	0.900557
$94$	0	0
$95$	−7.12311	−0.730815
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	−3.12311	−0.313884
$100$	0	0
$101$	18.6847	1.85919	0.929597	−	0.368579i	$-0.120156\pi$
$101$	18.6847	1.85919	0.929597	+	0.368579i	$0.120156\pi$
$102$	0	0
$103$	2.24621	0.221326	0.110663	−	0.993858i	$-0.464703\pi$
$103$	2.24621	0.221326	0.110663	+	0.993858i	$0.464703\pi$
$104$	0	0
$105$	−1.56155	−0.152392
$106$	0	0
$107$	4.68466	0.452883	0.226442	−	0.974025i	$-0.427291\pi$
$107$	4.68466	0.452883	0.226442	+	0.974025i	$0.427291\pi$
$108$	0	0
$109$	12.2462	1.17297	0.586487	−	0.809959i	$-0.300510\pi$
$109$	12.2462	1.17297	0.586487	+	0.809959i	$0.300510\pi$
$110$	0	0
$111$	3.56155	0.338048
$112$	0	0
$113$	14.6847	1.38142	0.690708	−	0.723134i	$-0.257299\pi$
$113$	14.6847	1.38142	0.690708	+	0.723134i	$0.257299\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−0.684658	−0.0627625
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	−7.56155	−0.681802
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.24621	0.199319	0.0996595	−	0.995022i	$-0.468225\pi$
$127$	2.24621	0.199319	0.0996595	+	0.995022i	$0.468225\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	−6.24621	−0.545734	−0.272867	−	0.962052i	$-0.587972\pi$
$131$	−6.24621	−0.545734	−0.272867	+	0.962052i	$0.587972\pi$
$132$	0	0
$133$	−11.1231	−0.964496
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	−7.80776	−0.662246	−0.331123	−	0.943588i	$-0.607427\pi$
$139$	−7.80776	−0.662246	−0.331123	+	0.943588i	$0.607427\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−6.24621	−0.522334
$144$	0	0
$145$	−4.43845	−0.368593
$146$	0	0
$147$	4.56155	0.376231
$148$	0	0
$149$	11.3693	0.931411	0.465705	−	0.884940i	$-0.345801\pi$
$149$	11.3693	0.931411	0.465705	+	0.884940i	$0.345801\pi$
$150$	0	0
$151$	6.24621	0.508309	0.254155	−	0.967164i	$-0.418203\pi$
$151$	6.24621	0.508309	0.254155	+	0.967164i	$0.418203\pi$
$152$	0	0
$153$	0.438447	0.0354464
$154$	0	0
$155$	8.68466	0.697569
$156$	0	0
$157$	20.0540	1.60048	0.800241	−	0.599679i	$-0.204705\pi$
$157$	20.0540	1.60048	0.800241	+	0.599679i	$0.204705\pi$
$158$	0	0
$159$	3.56155	0.282450
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	5.36932	0.420557	0.210279	−	0.977641i	$-0.432563\pi$
$163$	5.36932	0.420557	0.210279	+	0.977641i	$0.432563\pi$
$164$	0	0
$165$	−3.12311	−0.243133
$166$	0	0
$167$	−20.4924	−1.58575	−0.792876	−	0.609383i	$-0.791417\pi$
$167$	−20.4924	−1.58575	−0.792876	+	0.609383i	$0.791417\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	7.12311	0.544718
$172$	0	0
$173$	6.87689	0.522841	0.261420	−	0.965225i	$-0.415809\pi$
$173$	6.87689	0.522841	0.261420	+	0.965225i	$0.415809\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	2.43845	0.183285
$178$	0	0
$179$	20.4924	1.53168	0.765838	−	0.643034i	$-0.222325\pi$
$179$	20.4924	1.53168	0.765838	+	0.643034i	$0.222325\pi$
$180$	0	0
$181$	9.12311	0.678115	0.339058	−	0.940766i	$-0.389892\pi$
$181$	9.12311	0.678115	0.339058	+	0.940766i	$0.389892\pi$
$182$	0	0
$183$	11.3693	0.840444
$184$	0	0
$185$	3.56155	0.261851
$186$	0	0
$187$	−1.36932	−0.100134
$188$	0	0
$189$	1.56155	0.113586
$190$	0	0
$191$	−20.4924	−1.48278	−0.741390	−	0.671075i	$-0.765833\pi$
$191$	−20.4924	−1.48278	−0.741390	+	0.671075i	$0.765833\pi$
$192$	0	0
$193$	11.3693	0.818381	0.409191	−	0.912449i	$-0.365811\pi$
$193$	11.3693	0.818381	0.409191	+	0.912449i	$0.365811\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	3.75379	0.267446	0.133723	−	0.991019i	$-0.457307\pi$
$197$	3.75379	0.267446	0.133723	+	0.991019i	$0.457307\pi$
$198$	0	0
$199$	11.1231	0.788496	0.394248	−	0.919004i	$-0.371005\pi$
$199$	11.1231	0.788496	0.394248	+	0.919004i	$0.371005\pi$
$200$	0	0
$201$	1.56155	0.110143
$202$	0	0
$203$	−6.93087	−0.486452
$204$	0	0
$205$	−7.56155	−0.528122
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−22.2462	−1.53880
$210$	0	0
$211$	17.5616	1.20899	0.604494	−	0.796610i	$-0.293376\pi$
$211$	17.5616	1.20899	0.604494	+	0.796610i	$0.293376\pi$
$212$	0	0
$213$	−0.684658	−0.0469120
$214$	0	0
$215$	10.2462	0.698786
$216$	0	0
$217$	13.5616	0.920618
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	0.876894	0.0589863
$222$	0	0
$223$	−5.36932	−0.359556	−0.179778	−	0.983707i	$-0.557538\pi$
$223$	−5.36932	−0.359556	−0.179778	+	0.983707i	$0.557538\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	24.4924	1.62562	0.812810	−	0.582529i	$-0.197937\pi$
$227$	24.4924	1.62562	0.812810	+	0.582529i	$0.197937\pi$
$228$	0	0
$229$	1.12311	0.0742169	0.0371085	−	0.999311i	$-0.488185\pi$
$229$	1.12311	0.0742169	0.0371085	+	0.999311i	$0.488185\pi$
$230$	0	0
$231$	−4.87689	−0.320876
$232$	0	0
$233$	−11.3693	−0.744829	−0.372414	−	0.928067i	$-0.621470\pi$
$233$	−11.3693	−0.744829	−0.372414	+	0.928067i	$0.621470\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−6.24621	−0.405735
$238$	0	0
$239$	7.31534	0.473190	0.236595	−	0.971608i	$-0.423969\pi$
$239$	7.31534	0.473190	0.236595	+	0.971608i	$0.423969\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.56155	0.291427
$246$	0	0
$247$	14.2462	0.906465
$248$	0	0
$249$	−12.6847	−0.803858
$250$	0	0
$251$	20.8769	1.31774	0.658869	−	0.752258i	$-0.271035\pi$
$251$	20.8769	1.31774	0.658869	+	0.752258i	$0.271035\pi$
$252$	0	0
$253$	−3.12311	−0.196348
$254$	0	0
$255$	0.438447	0.0274566
$256$	0	0
$257$	−17.6155	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$257$	−17.6155	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$258$	0	0
$259$	5.56155	0.345578
$260$	0	0
$261$	4.43845	0.274733
$262$	0	0
$263$	−26.4384	−1.63026	−0.815132	−	0.579275i	$-0.803336\pi$
$263$	−26.4384	−1.63026	−0.815132	+	0.579275i	$0.803336\pi$
$264$	0	0
$265$	3.56155	0.218784
$266$	0	0
$267$	−5.12311	−0.313529
$268$	0	0
$269$	15.5616	0.948805	0.474402	−	0.880308i	$-0.342664\pi$
$269$	15.5616	0.948805	0.474402	+	0.880308i	$0.342664\pi$
$270$	0	0
$271$	0.684658	0.0415900	0.0207950	−	0.999784i	$-0.493380\pi$
$271$	0.684658	0.0415900	0.0207950	+	0.999784i	$0.493380\pi$
$272$	0	0
$273$	3.12311	0.189019
$274$	0	0
$275$	−3.12311	−0.188330
$276$	0	0
$277$	8.24621	0.495467	0.247733	−	0.968828i	$-0.420314\pi$
$277$	8.24621	0.495467	0.247733	+	0.968828i	$0.420314\pi$
$278$	0	0
$279$	−8.68466	−0.519937
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	1.56155	0.0928247	0.0464123	−	0.998922i	$-0.485221\pi$
$283$	1.56155	0.0928247	0.0464123	+	0.998922i	$0.485221\pi$
$284$	0	0
$285$	7.12311	0.421936
$286$	0	0
$287$	−11.8078	−0.696990
$288$	0	0
$289$	−16.8078	−0.988692
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−20.9309	−1.22279	−0.611397	−	0.791324i	$-0.709392\pi$
$293$	−20.9309	−1.22279	−0.611397	+	0.791324i	$0.709392\pi$
$294$	0	0
$295$	2.43845	0.141972
$296$	0	0
$297$	3.12311	0.181221
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	−18.6847	−1.07341
$304$	0	0
$305$	11.3693	0.651005
$306$	0	0
$307$	−11.6155	−0.662933	−0.331467	−	0.943467i	$-0.607543\pi$
$307$	−11.6155	−0.662933	−0.331467	+	0.943467i	$0.607543\pi$
$308$	0	0
$309$	−2.24621	−0.127782
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	7.56155	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$313$	7.56155	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$314$	0	0
$315$	1.56155	0.0879835
$316$	0	0
$317$	−2.87689	−0.161582	−0.0807912	−	0.996731i	$-0.525745\pi$
$317$	−2.87689	−0.161582	−0.0807912	+	0.996731i	$0.525745\pi$
$318$	0	0
$319$	−13.8617	−0.776108
$320$	0	0
$321$	−4.68466	−0.261472
$322$	0	0
$323$	3.12311	0.173774
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	−12.2462	−0.677217
$328$	0	0
$329$	12.4924	0.688730
$330$	0	0
$331$	23.8078	1.30859	0.654297	−	0.756238i	$-0.272965\pi$
$331$	23.8078	1.30859	0.654297	+	0.756238i	$0.272965\pi$
$332$	0	0
$333$	−3.56155	−0.195172
$334$	0	0
$335$	1.56155	0.0853167
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−14.6847	−0.797561
$340$	0	0
$341$	27.1231	1.46880
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	22.7386	1.22067	0.610337	−	0.792142i	$-0.291034\pi$
$347$	22.7386	1.22067	0.610337	+	0.792142i	$0.291034\pi$
$348$	0	0
$349$	−36.0540	−1.92993	−0.964963	−	0.262388i	$-0.915490\pi$
$349$	−36.0540	−1.92993	−0.964963	+	0.262388i	$0.915490\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	2.87689	0.153122	0.0765608	−	0.997065i	$-0.475606\pi$
$353$	2.87689	0.153122	0.0765608	+	0.997065i	$0.475606\pi$
$354$	0	0
$355$	−0.684658	−0.0363379
$356$	0	0
$357$	0.684658	0.0362360
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	1.24621	0.0654091
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	36.3002	1.89485	0.947427	−	0.319972i	$-0.103673\pi$
$367$	36.3002	1.89485	0.947427	+	0.319972i	$0.103673\pi$
$368$	0	0
$369$	7.56155	0.393639
$370$	0	0
$371$	5.56155	0.288741
$372$	0	0
$373$	−36.2462	−1.87676	−0.938379	−	0.345608i	$-0.887673\pi$
$373$	−36.2462	−1.87676	−0.938379	+	0.345608i	$0.887673\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	8.87689	0.457183
$378$	0	0
$379$	18.2462	0.937245	0.468622	−	0.883399i	$-0.344750\pi$
$379$	18.2462	0.937245	0.468622	+	0.883399i	$0.344750\pi$
$380$	0	0
$381$	−2.24621	−0.115077
$382$	0	0
$383$	−6.93087	−0.354151	−0.177075	−	0.984197i	$-0.556664\pi$
$383$	−6.93087	−0.354151	−0.177075	+	0.984197i	$0.556664\pi$
$384$	0	0
$385$	−4.87689	−0.248550
$386$	0	0
$387$	−10.2462	−0.520844
$388$	0	0
$389$	−2.49242	−0.126371	−0.0631854	−	0.998002i	$-0.520126\pi$
$389$	−2.49242	−0.126371	−0.0631854	+	0.998002i	$0.520126\pi$
$390$	0	0
$391$	0.438447	0.0221732
$392$	0	0
$393$	6.24621	0.315080
$394$	0	0
$395$	−6.24621	−0.314281
$396$	0	0
$397$	17.6155	0.884098	0.442049	−	0.896991i	$-0.354252\pi$
$397$	17.6155	0.884098	0.442049	+	0.896991i	$0.354252\pi$
$398$	0	0
$399$	11.1231	0.556852
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	−17.3693	−0.865227
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	11.1231	0.551352
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	3.80776	0.187368
$414$	0	0
$415$	−12.6847	−0.622665
$416$	0	0
$417$	7.80776	0.382348
$418$	0	0
$419$	35.1231	1.71588	0.857938	−	0.513753i	$-0.171745\pi$
$419$	35.1231	1.71588	0.857938	+	0.513753i	$0.171745\pi$
$420$	0	0
$421$	−6.49242	−0.316421	−0.158211	−	0.987405i	$-0.550573\pi$
$421$	−6.49242	−0.316421	−0.158211	+	0.987405i	$0.550573\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	0.438447	0.0212678
$426$	0	0
$427$	17.7538	0.859166
$428$	0	0
$429$	6.24621	0.301570
$430$	0	0
$431$	12.4924	0.601739	0.300869	−	0.953665i	$-0.402723\pi$
$431$	12.4924	0.601739	0.300869	+	0.953665i	$0.402723\pi$
$432$	0	0
$433$	−11.5616	−0.555613	−0.277806	−	0.960637i	$-0.589607\pi$
$433$	−11.5616	−0.555613	−0.277806	+	0.960637i	$0.589607\pi$
$434$	0	0
$435$	4.43845	0.212807
$436$	0	0
$437$	7.12311	0.340744
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	16.4924	0.783579	0.391789	−	0.920055i	$-0.371856\pi$
$443$	16.4924	0.783579	0.391789	+	0.920055i	$0.371856\pi$
$444$	0	0
$445$	−5.12311	−0.242858
$446$	0	0
$447$	−11.3693	−0.537750
$448$	0	0
$449$	21.4233	1.01103	0.505514	−	0.862818i	$-0.331303\pi$
$449$	21.4233	1.01103	0.505514	+	0.862818i	$0.331303\pi$
$450$	0	0
$451$	−23.6155	−1.11201
$452$	0	0
$453$	−6.24621	−0.293473
$454$	0	0
$455$	3.12311	0.146413
$456$	0	0
$457$	−22.6847	−1.06114	−0.530572	−	0.847640i	$-0.678023\pi$
$457$	−22.6847	−1.06114	−0.530572	+	0.847640i	$0.678023\pi$
$458$	0	0
$459$	−0.438447	−0.0204650
$460$	0	0
$461$	3.75379	0.174831	0.0874157	−	0.996172i	$-0.472139\pi$
$461$	3.75379	0.174831	0.0874157	+	0.996172i	$0.472139\pi$
$462$	0	0
$463$	−29.3693	−1.36491	−0.682454	−	0.730929i	$-0.739087\pi$
$463$	−29.3693	−1.36491	−0.682454	+	0.730929i	$0.739087\pi$
$464$	0	0
$465$	−8.68466	−0.402741
$466$	0	0
$467$	−19.3153	−0.893807	−0.446904	−	0.894582i	$-0.647473\pi$
$467$	−19.3153	−0.893807	−0.446904	+	0.894582i	$0.647473\pi$
$468$	0	0
$469$	2.43845	0.112597
$470$	0	0
$471$	−20.0540	−0.924038
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	7.12311	0.326831
$476$	0	0
$477$	−3.56155	−0.163072
$478$	0	0
$479$	26.7386	1.22172	0.610860	−	0.791739i	$-0.290824\pi$
$479$	26.7386	1.22172	0.610860	+	0.791739i	$0.290824\pi$
$480$	0	0
$481$	−7.12311	−0.324786
$482$	0	0
$483$	1.56155	0.0710531
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	30.7386	1.39290	0.696450	−	0.717605i	$-0.254762\pi$
$487$	30.7386	1.39290	0.696450	+	0.717605i	$0.254762\pi$
$488$	0	0
$489$	−5.36932	−0.242809
$490$	0	0
$491$	16.6847	0.752968	0.376484	−	0.926423i	$-0.377133\pi$
$491$	16.6847	0.752968	0.376484	+	0.926423i	$0.377133\pi$
$492$	0	0
$493$	1.94602	0.0876445
$494$	0	0
$495$	3.12311	0.140373
$496$	0	0
$497$	−1.06913	−0.0479570
$498$	0	0
$499$	39.4233	1.76483	0.882414	−	0.470473i	$-0.155917\pi$
$499$	39.4233	1.76483	0.882414	+	0.470473i	$0.155917\pi$
$500$	0	0
$501$	20.4924	0.915534
$502$	0	0
$503$	−24.6847	−1.10063	−0.550317	−	0.834956i	$-0.685493\pi$
$503$	−24.6847	−1.10063	−0.550317	+	0.834956i	$0.685493\pi$
$504$	0	0
$505$	−18.6847	−0.831456
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	−26.4924	−1.17426	−0.587128	−	0.809494i	$-0.699741\pi$
$509$	−26.4924	−1.17426	−0.587128	+	0.809494i	$0.699741\pi$
$510$	0	0
$511$	−3.12311	−0.138158
$512$	0	0
$513$	−7.12311	−0.314493
$514$	0	0
$515$	−2.24621	−0.0989799
$516$	0	0
$517$	24.9848	1.09883
$518$	0	0
$519$	−6.87689	−0.301862
$520$	0	0
$521$	38.4924	1.68638	0.843192	−	0.537613i	$-0.180674\pi$
$521$	38.4924	1.68638	0.843192	+	0.537613i	$0.180674\pi$
$522$	0	0
$523$	32.4924	1.42079	0.710397	−	0.703801i	$-0.248515\pi$
$523$	32.4924	1.42079	0.710397	+	0.703801i	$0.248515\pi$
$524$	0	0
$525$	1.56155	0.0681518
$526$	0	0
$527$	−3.80776	−0.165869
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.43845	−0.105820
$532$	0	0
$533$	15.1231	0.655054
$534$	0	0
$535$	−4.68466	−0.202535
$536$	0	0
$537$	−20.4924	−0.884313
$538$	0	0
$539$	14.2462	0.613628
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−9.12311	−0.391510
$544$	0	0
$545$	−12.2462	−0.524570
$546$	0	0
$547$	−26.2462	−1.12221	−0.561103	−	0.827746i	$-0.689623\pi$
$547$	−26.2462	−1.12221	−0.561103	+	0.827746i	$0.689623\pi$
$548$	0	0
$549$	−11.3693	−0.485231
$550$	0	0
$551$	31.6155	1.34687
$552$	0	0
$553$	−9.75379	−0.414773
$554$	0	0
$555$	−3.56155	−0.151179
$556$	0	0
$557$	16.9309	0.717384	0.358692	−	0.933456i	$-0.383223\pi$
$557$	16.9309	0.717384	0.358692	+	0.933456i	$0.383223\pi$
$558$	0	0
$559$	−20.4924	−0.866737
$560$	0	0
$561$	1.36932	0.0578126
$562$	0	0
$563$	3.31534	0.139725	0.0698625	−	0.997557i	$-0.477744\pi$
$563$	3.31534	0.139725	0.0698625	+	0.997557i	$0.477744\pi$
$564$	0	0
$565$	−14.6847	−0.617788
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	20.4924	0.856083
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−11.8617	−0.493811	−0.246905	−	0.969040i	$-0.579414\pi$
$577$	−11.8617	−0.493811	−0.246905	+	0.969040i	$0.579414\pi$
$578$	0	0
$579$	−11.3693	−0.472493
$580$	0	0
$581$	−19.8078	−0.821765
$582$	0	0
$583$	11.1231	0.460672
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−17.8617	−0.737233	−0.368616	−	0.929582i	$-0.620168\pi$
$587$	−17.8617	−0.737233	−0.368616	+	0.929582i	$0.620168\pi$
$588$	0	0
$589$	−61.8617	−2.54897
$590$	0	0
$591$	−3.75379	−0.154410
$592$	0	0
$593$	−21.1231	−0.867422	−0.433711	−	0.901052i	$-0.642796\pi$
$593$	−21.1231	−0.867422	−0.433711	+	0.901052i	$0.642796\pi$
$594$	0	0
$595$	0.684658	0.0280683
$596$	0	0
$597$	−11.1231	−0.455238
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−3.56155	−0.145279	−0.0726394	−	0.997358i	$-0.523142\pi$
$601$	−3.56155	−0.145279	−0.0726394	+	0.997358i	$0.523142\pi$
$602$	0	0
$603$	−1.56155	−0.0635913
$604$	0	0
$605$	1.24621	0.0506657
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	6.93087	0.280853
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	3.75379	0.151614	0.0758070	−	0.997123i	$-0.475847\pi$
$613$	3.75379	0.151614	0.0758070	+	0.997123i	$0.475847\pi$
$614$	0	0
$615$	7.56155	0.304911
$616$	0	0
$617$	13.3153	0.536055	0.268028	−	0.963411i	$-0.413628\pi$
$617$	13.3153	0.536055	0.268028	+	0.963411i	$0.413628\pi$
$618$	0	0
$619$	−35.2311	−1.41606	−0.708028	−	0.706185i	$-0.750415\pi$
$619$	−35.2311	−1.41606	−0.708028	+	0.706185i	$0.750415\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	22.2462	0.888428
$628$	0	0
$629$	−1.56155	−0.0622632
$630$	0	0
$631$	1.75379	0.0698172	0.0349086	−	0.999391i	$-0.488886\pi$
$631$	1.75379	0.0698172	0.0349086	+	0.999391i	$0.488886\pi$
$632$	0	0
$633$	−17.5616	−0.698009
$634$	0	0
$635$	−2.24621	−0.0891382
$636$	0	0
$637$	−9.12311	−0.361471
$638$	0	0
$639$	0.684658	0.0270847
$640$	0	0
$641$	−15.7538	−0.622237	−0.311119	−	0.950371i	$-0.600704\pi$
$641$	−15.7538	−0.622237	−0.311119	+	0.950371i	$0.600704\pi$
$642$	0	0
$643$	−20.3002	−0.800561	−0.400281	−	0.916393i	$-0.631087\pi$
$643$	−20.3002	−0.800561	−0.400281	+	0.916393i	$0.631087\pi$
$644$	0	0
$645$	−10.2462	−0.403444
$646$	0	0
$647$	−43.1231	−1.69534	−0.847672	−	0.530520i	$-0.821997\pi$
$647$	−43.1231	−1.69534	−0.847672	+	0.530520i	$0.821997\pi$
$648$	0	0
$649$	7.61553	0.298936
$650$	0	0
$651$	−13.5616	−0.531519
$652$	0	0
$653$	−1.50758	−0.0589961	−0.0294980	−	0.999565i	$-0.509391\pi$
$653$	−1.50758	−0.0589961	−0.0294980	+	0.999565i	$0.509391\pi$
$654$	0	0
$655$	6.24621	0.244060
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	−9.36932	−0.364977	−0.182488	−	0.983208i	$-0.558415\pi$
$659$	−9.36932	−0.364977	−0.182488	+	0.983208i	$0.558415\pi$
$660$	0	0
$661$	32.7386	1.27339	0.636693	−	0.771118i	$-0.280302\pi$
$661$	32.7386	1.27339	0.636693	+	0.771118i	$0.280302\pi$
$662$	0	0
$663$	−0.876894	−0.0340558
$664$	0	0
$665$	11.1231	0.431336
$666$	0	0
$667$	4.43845	0.171857
$668$	0	0
$669$	5.36932	0.207590
$670$	0	0
$671$	35.5076	1.37075
$672$	0	0
$673$	43.3693	1.67176	0.835882	−	0.548909i	$-0.184957\pi$
$673$	43.3693	1.67176	0.835882	+	0.548909i	$0.184957\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	42.3002	1.62573	0.812864	−	0.582453i	$-0.197907\pi$
$677$	42.3002	1.62573	0.812864	+	0.582453i	$0.197907\pi$
$678$	0	0
$679$	9.36932	0.359561
$680$	0	0
$681$	−24.4924	−0.938552
$682$	0	0
$683$	−10.2462	−0.392060	−0.196030	−	0.980598i	$-0.562805\pi$
$683$	−10.2462	−0.392060	−0.196030	+	0.980598i	$0.562805\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	−1.12311	−0.0428492
$688$	0	0
$689$	−7.12311	−0.271369
$690$	0	0
$691$	40.4924	1.54040	0.770202	−	0.637800i	$-0.220155\pi$
$691$	40.4924	1.54040	0.770202	+	0.637800i	$0.220155\pi$
$692$	0	0
$693$	4.87689	0.185258
$694$	0	0
$695$	7.80776	0.296165
$696$	0	0
$697$	3.31534	0.125578
$698$	0	0
$699$	11.3693	0.430027
$700$	0	0
$701$	−19.8617	−0.750168	−0.375084	−	0.926991i	$-0.622386\pi$
$701$	−19.8617	−0.750168	−0.375084	+	0.926991i	$0.622386\pi$
$702$	0	0
$703$	−25.3693	−0.956822
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−29.1771	−1.09732
$708$	0	0
$709$	−37.1231	−1.39419	−0.697094	−	0.716980i	$-0.745524\pi$
$709$	−37.1231	−1.39419	−0.697094	+	0.716980i	$0.745524\pi$
$710$	0	0
$711$	6.24621	0.234251
$712$	0	0
$713$	−8.68466	−0.325243
$714$	0	0
$715$	6.24621	0.233595
$716$	0	0
$717$	−7.31534	−0.273196
$718$	0	0
$719$	40.3002	1.50294	0.751472	−	0.659765i	$-0.229344\pi$
$719$	40.3002	1.50294	0.751472	+	0.659765i	$0.229344\pi$
$720$	0	0
$721$	−3.50758	−0.130629
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	4.43845	0.164840
$726$	0	0
$727$	7.80776	0.289574	0.144787	−	0.989463i	$-0.453750\pi$
$727$	7.80776	0.289574	0.144787	+	0.989463i	$0.453750\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.49242	−0.166158
$732$	0	0
$733$	0.930870	0.0343825	0.0171912	−	0.999852i	$-0.494528\pi$
$733$	0.930870	0.0343825	0.0171912	+	0.999852i	$0.494528\pi$
$734$	0	0
$735$	−4.56155	−0.168255
$736$	0	0
$737$	4.87689	0.179643
$738$	0	0
$739$	−2.93087	−0.107814	−0.0539069	−	0.998546i	$-0.517167\pi$
$739$	−2.93087	−0.107814	−0.0539069	+	0.998546i	$0.517167\pi$
$740$	0	0
$741$	−14.2462	−0.523348
$742$	0	0
$743$	−19.5076	−0.715664	−0.357832	−	0.933786i	$-0.616484\pi$
$743$	−19.5076	−0.715664	−0.357832	+	0.933786i	$0.616484\pi$
$744$	0	0
$745$	−11.3693	−0.416540
$746$	0	0
$747$	12.6847	0.464107
$748$	0	0
$749$	−7.31534	−0.267297
$750$	0	0
$751$	39.6155	1.44559	0.722796	−	0.691062i	$-0.242857\pi$
$751$	39.6155	1.44559	0.722796	+	0.691062i	$0.242857\pi$
$752$	0	0
$753$	−20.8769	−0.760796
$754$	0	0
$755$	−6.24621	−0.227323
$756$	0	0
$757$	−43.1771	−1.56930	−0.784649	−	0.619940i	$-0.787157\pi$
$757$	−43.1771	−1.56930	−0.784649	+	0.619940i	$0.787157\pi$
$758$	0	0
$759$	3.12311	0.113362
$760$	0	0
$761$	21.8078	0.790531	0.395265	−	0.918567i	$-0.370653\pi$
$761$	21.8078	0.790531	0.395265	+	0.918567i	$0.370653\pi$
$762$	0	0
$763$	−19.1231	−0.692303
$764$	0	0
$765$	−0.438447	−0.0158521
$766$	0	0
$767$	−4.87689	−0.176094
$768$	0	0
$769$	0.630683	0.0227430	0.0113715	−	0.999935i	$-0.496380\pi$
$769$	0.630683	0.0227430	0.0113715	+	0.999935i	$0.496380\pi$
$770$	0	0
$771$	17.6155	0.634408
$772$	0	0
$773$	−3.26137	−0.117303	−0.0586516	−	0.998279i	$-0.518680\pi$
$773$	−3.26137	−0.117303	−0.0586516	+	0.998279i	$0.518680\pi$
$774$	0	0
$775$	−8.68466	−0.311962
$776$	0	0
$777$	−5.56155	−0.199520
$778$	0	0
$779$	53.8617	1.92980
$780$	0	0
$781$	−2.13826	−0.0765130
$782$	0	0
$783$	−4.43845	−0.158617
$784$	0	0
$785$	−20.0540	−0.715757
$786$	0	0
$787$	45.6695	1.62794	0.813971	−	0.580906i	$-0.197301\pi$
$787$	45.6695	1.62794	0.813971	+	0.580906i	$0.197301\pi$
$788$	0	0
$789$	26.4384	0.941234
$790$	0	0
$791$	−22.9309	−0.815328
$792$	0	0
$793$	−22.7386	−0.807473
$794$	0	0
$795$	−3.56155	−0.126315
$796$	0	0
$797$	30.7926	1.09073	0.545365	−	0.838199i	$-0.316391\pi$
$797$	30.7926	1.09073	0.545365	+	0.838199i	$0.316391\pi$
$798$	0	0
$799$	−3.50758	−0.124089
$800$	0	0
$801$	5.12311	0.181016
$802$	0	0
$803$	−6.24621	−0.220424
$804$	0	0
$805$	1.56155	0.0550375
$806$	0	0
$807$	−15.5616	−0.547793
$808$	0	0
$809$	−41.8078	−1.46988	−0.734941	−	0.678131i	$-0.762790\pi$
$809$	−41.8078	−1.46988	−0.734941	+	0.678131i	$0.762790\pi$
$810$	0	0
$811$	50.5464	1.77492	0.887462	−	0.460881i	$-0.152466\pi$
$811$	50.5464	1.77492	0.887462	+	0.460881i	$0.152466\pi$
$812$	0	0
$813$	−0.684658	−0.0240120
$814$	0	0
$815$	−5.36932	−0.188079
$816$	0	0
$817$	−72.9848	−2.55342
$818$	0	0
$819$	−3.12311	−0.109130
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	0	0
$825$	3.12311	0.108733
$826$	0	0
$827$	46.0540	1.60145	0.800727	−	0.599030i	$-0.204447\pi$
$827$	46.0540	1.60145	0.800727	+	0.599030i	$0.204447\pi$
$828$	0	0
$829$	25.8078	0.896341	0.448170	−	0.893948i	$-0.352076\pi$
$829$	25.8078	0.896341	0.448170	+	0.893948i	$0.352076\pi$
$830$	0	0
$831$	−8.24621	−0.286058
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	20.4924	0.709170
$836$	0	0
$837$	8.68466	0.300186
$838$	0	0
$839$	−3.12311	−0.107822	−0.0539108	−	0.998546i	$-0.517169\pi$
$839$	−3.12311	−0.107822	−0.0539108	+	0.998546i	$0.517169\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	1.94602	0.0668662
$848$	0	0
$849$	−1.56155	−0.0535924
$850$	0	0
$851$	−3.56155	−0.122088
$852$	0	0
$853$	33.2311	1.13781	0.568905	−	0.822403i	$-0.307367\pi$
$853$	33.2311	1.13781	0.568905	+	0.822403i	$0.307367\pi$
$854$	0	0
$855$	−7.12311	−0.243605
$856$	0	0
$857$	9.50758	0.324773	0.162386	−	0.986727i	$-0.448081\pi$
$857$	9.50758	0.324773	0.162386	+	0.986727i	$0.448081\pi$
$858$	0	0
$859$	−25.1771	−0.859031	−0.429515	−	0.903060i	$-0.641316\pi$
$859$	−25.1771	−0.859031	−0.429515	+	0.903060i	$0.641316\pi$
$860$	0	0
$861$	11.8078	0.402408
$862$	0	0
$863$	−45.8617	−1.56115	−0.780576	−	0.625061i	$-0.785074\pi$
$863$	−45.8617	−1.56115	−0.780576	+	0.625061i	$0.785074\pi$
$864$	0	0
$865$	−6.87689	−0.233821
$866$	0	0
$867$	16.8078	0.570822
$868$	0	0
$869$	−19.5076	−0.661749
$870$	0	0
$871$	−3.12311	−0.105822
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	1.56155	0.0527901
$876$	0	0
$877$	−29.2311	−0.987063	−0.493531	−	0.869728i	$-0.664294\pi$
$877$	−29.2311	−0.987063	−0.493531	+	0.869728i	$0.664294\pi$
$878$	0	0
$879$	20.9309	0.705981
$880$	0	0
$881$	51.3693	1.73068	0.865338	−	0.501189i	$-0.167104\pi$
$881$	51.3693	1.73068	0.865338	+	0.501189i	$0.167104\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−2.43845	−0.0819675
$886$	0	0
$887$	12.8769	0.432364	0.216182	−	0.976353i	$-0.430640\pi$
$887$	12.8769	0.432364	0.216182	+	0.976353i	$0.430640\pi$
$888$	0	0
$889$	−3.50758	−0.117640
$890$	0	0
$891$	−3.12311	−0.104628
$892$	0	0
$893$	−56.9848	−1.90693
$894$	0	0
$895$	−20.4924	−0.684986
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	−38.5464	−1.28559
$900$	0	0
$901$	−1.56155	−0.0520229
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−9.12311	−0.303262
$906$	0	0
$907$	42.9309	1.42550	0.712748	−	0.701420i	$-0.247450\pi$
$907$	42.9309	1.42550	0.712748	+	0.701420i	$0.247450\pi$
$908$	0	0
$909$	18.6847	0.619731
$910$	0	0
$911$	18.7386	0.620839	0.310419	−	0.950600i	$-0.399531\pi$
$911$	18.7386	0.620839	0.310419	+	0.950600i	$0.399531\pi$
$912$	0	0
$913$	−39.6155	−1.31108
$914$	0	0
$915$	−11.3693	−0.375858
$916$	0	0
$917$	9.75379	0.322098
$918$	0	0
$919$	−16.9848	−0.560278	−0.280139	−	0.959959i	$-0.590381\pi$
$919$	−16.9848	−0.560278	−0.280139	+	0.959959i	$0.590381\pi$
$920$	0	0
$921$	11.6155	0.382745
$922$	0	0
$923$	1.36932	0.0450716
$924$	0	0
$925$	−3.56155	−0.117103
$926$	0	0
$927$	2.24621	0.0737753
$928$	0	0
$929$	−16.0540	−0.526714	−0.263357	−	0.964698i	$-0.584830\pi$
$929$	−16.0540	−0.526714	−0.263357	+	0.964698i	$0.584830\pi$
$930$	0	0
$931$	−32.4924	−1.06490
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	1.36932	0.0447815
$936$	0	0
$937$	−1.50758	−0.0492504	−0.0246252	−	0.999697i	$-0.507839\pi$
$937$	−1.50758	−0.0492504	−0.0246252	+	0.999697i	$0.507839\pi$
$938$	0	0
$939$	−7.56155	−0.246762
$940$	0	0
$941$	3.36932	0.109837	0.0549183	−	0.998491i	$-0.482510\pi$
$941$	3.36932	0.109837	0.0549183	+	0.998491i	$0.482510\pi$
$942$	0	0
$943$	7.56155	0.246238
$944$	0	0
$945$	−1.56155	−0.0507973
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	2.87689	0.0932897
$952$	0	0
$953$	36.2462	1.17413	0.587065	−	0.809540i	$-0.300283\pi$
$953$	36.2462	1.17413	0.587065	+	0.809540i	$0.300283\pi$
$954$	0	0
$955$	20.4924	0.663119
$956$	0	0
$957$	13.8617	0.448086
$958$	0	0
$959$	−21.8617	−0.705952
$960$	0	0
$961$	44.4233	1.43301
$962$	0	0
$963$	4.68466	0.150961
$964$	0	0
$965$	−11.3693	−0.365991
$966$	0	0
$967$	−52.9848	−1.70388	−0.851939	−	0.523641i	$-0.824573\pi$
$967$	−52.9848	−1.70388	−0.851939	+	0.523641i	$0.824573\pi$
$968$	0	0
$969$	−3.12311	−0.100329
$970$	0	0
$971$	2.73863	0.0878869	0.0439435	−	0.999034i	$-0.486008\pi$
$971$	2.73863	0.0878869	0.0439435	+	0.999034i	$0.486008\pi$
$972$	0	0
$973$	12.1922	0.390865
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−13.4233	−0.429449	−0.214725	−	0.976675i	$-0.568885\pi$
$977$	−13.4233	−0.429449	−0.214725	+	0.976675i	$0.568885\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	12.2462	0.390991
$982$	0	0
$983$	21.1771	0.675444	0.337722	−	0.941246i	$-0.390344\pi$
$983$	21.1771	0.675444	0.337722	+	0.941246i	$0.390344\pi$
$984$	0	0
$985$	−3.75379	−0.119606
$986$	0	0
$987$	−12.4924	−0.397638
$988$	0	0
$989$	−10.2462	−0.325811
$990$	0	0
$991$	50.0540	1.59002	0.795008	−	0.606598i	$-0.207466\pi$
$991$	50.0540	1.59002	0.795008	+	0.606598i	$0.207466\pi$
$992$	0	0
$993$	−23.8078	−0.755517
$994$	0	0
$995$	−11.1231	−0.352626
$996$	0	0
$997$	−19.8617	−0.629028	−0.314514	−	0.949253i	$-0.601841\pi$
$997$	−19.8617	−0.629028	−0.314514	+	0.949253i	$0.601841\pi$
$998$	0	0
$999$	3.56155	0.112683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bh.1.1		2
4.3	odd	2		2760.2.a.p.1.2	✓	2
12.11	even	2		8280.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.p.1.2	✓	2	4.3	odd	2
5520.2.a.bh.1.1		2	1.1	even	1	trivial
8280.2.a.be.1.2		2	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.56155 q^{7} +1.00000 q^{9} -3.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +0.438447 q^{17} +7.12311 q^{19} +1.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +4.43845 q^{29} -8.68466 q^{31} +3.12311 q^{33} +1.56155 q^{35} -3.56155 q^{37} -2.00000 q^{39} +7.56155 q^{41} -10.2462 q^{43} -1.00000 q^{45} -8.00000 q^{47} -4.56155 q^{49} -0.438447 q^{51} -3.56155 q^{53} +3.12311 q^{55} -7.12311 q^{57} -2.43845 q^{59} -11.3693 q^{61} -1.56155 q^{63} -2.00000 q^{65} -1.56155 q^{67} -1.00000 q^{69} +0.684658 q^{71} +2.00000 q^{73} -1.00000 q^{75} +4.87689 q^{77} +6.24621 q^{79} +1.00000 q^{81} +12.6847 q^{83} -0.438447 q^{85} -4.43845 q^{87} +5.12311 q^{89} -3.12311 q^{91} +8.68466 q^{93} -7.12311 q^{95} -6.00000 q^{97} -3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 5 q^{17} + 6 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 13 q^{29} - 5 q^{31} - 2 q^{33} - q^{35} - 3 q^{37} - 4 q^{39} + 11 q^{41} - 4 q^{43} - 2 q^{45} - 16 q^{47} - 5 q^{49} - 5 q^{51} - 3 q^{53} - 2 q^{55} - 6 q^{57} - 9 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} + 4 q^{73} - 2 q^{75} + 18 q^{77} - 4 q^{79} + 2 q^{81} + 13 q^{83} - 5 q^{85} - 13 q^{87} + 2 q^{89} + 2 q^{91} + 5 q^{93} - 6 q^{95} - 12 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 5 * q^17 + 6 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 13 * q^29 - 5 * q^31 - 2 * q^33 - q^35 - 3 * q^37 - 4 * q^39 + 11 * q^41 - 4 * q^43 - 2 * q^45 - 16 * q^47 - 5 * q^49 - 5 * q^51 - 3 * q^53 - 2 * q^55 - 6 * q^57 - 9 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + q^67 - 2 * q^69 - 11 * q^71 + 4 * q^73 - 2 * q^75 + 18 * q^77 - 4 * q^79 + 2 * q^81 + 13 * q^83 - 5 * q^85 - 13 * q^87 + 2 * q^89 + 2 * q^91 + 5 * q^93 - 6 * q^95 - 12 * q^97 + 2 * q^99

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