LMFDB - Embedded newform 5520.2.a.by.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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +1.14637 q^{11} +0.853635 q^{13} -1.00000 q^{15} +1.34292 q^{17} +3.83221 q^{19} +4.48929 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.02877 q^{29} -2.19656 q^{31} -1.14637 q^{33} -4.48929 q^{35} -2.48929 q^{37} -0.853635 q^{39} +11.3001 q^{41} -10.6858 q^{43} +1.00000 q^{45} -1.53948 q^{47} +13.1537 q^{49} -1.34292 q^{51} +4.02877 q^{53} +1.14637 q^{55} -3.83221 q^{57} +15.0073 q^{59} -5.83221 q^{61} -4.48929 q^{63} +0.853635 q^{65} +11.5682 q^{67} +1.00000 q^{69} -4.32150 q^{71} -13.1035 q^{73} -1.00000 q^{75} -5.14637 q^{77} -0.585462 q^{79} +1.00000 q^{81} -5.63565 q^{83} +1.34292 q^{85} +8.02877 q^{87} -0.100384 q^{89} -3.83221 q^{91} +2.19656 q^{93} +3.83221 q^{95} +11.5640 q^{97} +1.14637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^5 - 6 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 2 q^{19} + 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{39} - 2 q^{41} - 20 q^{43} + 3 q^{45} + 6 q^{47} + 5 q^{49} + 2 q^{51} - 6 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 4 q^{61} - 6 q^{63} + 4 q^{65} + 6 q^{67} + 3 q^{69} + 8 q^{71} - 8 q^{73} - 3 q^{75} - 14 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{85} + 6 q^{87} + 6 q^{89} + 2 q^{91} + 2 q^{93} - 2 q^{95} + 14 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 6 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 2 * q^19 + 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^39 - 2 * q^41 - 20 * q^43 + 3 * q^45 + 6 * q^47 + 5 * q^49 + 2 * q^51 - 6 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 4 * q^61 - 6 * q^63 + 4 * q^65 + 6 * q^67 + 3 * q^69 + 8 * q^71 - 8 * q^73 - 3 * q^75 - 14 * q^77 + 4 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^85 + 6 * q^87 + 6 * q^89 + 2 * q^91 + 2 * q^93 - 2 * q^95 + 14 * 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
$7$	−4.48929	−1.69679	−0.848396	−	0.529362i	$-0.822431\pi$
$7$	−4.48929	−1.69679	−0.848396	+	0.529362i	$0.822431\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14637	0.345642	0.172821	−	0.984953i	$-0.444712\pi$
$11$	1.14637	0.345642	0.172821	+	0.984953i	$0.444712\pi$
$12$	0	0
$13$	0.853635	0.236756	0.118378	−	0.992969i	$-0.462231\pi$
$13$	0.853635	0.236756	0.118378	+	0.992969i	$0.462231\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.34292	0.325707	0.162853	−	0.986650i	$-0.447930\pi$
$17$	1.34292	0.325707	0.162853	+	0.986650i	$0.447930\pi$
$18$	0	0
$19$	3.83221	0.879170	0.439585	−	0.898201i	$-0.355126\pi$
$19$	3.83221	0.879170	0.439585	+	0.898201i	$0.355126\pi$
$20$	0	0
$21$	4.48929	0.979643
$22$	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
$29$	−8.02877	−1.49091	−0.745453	−	0.666559i	$-0.767767\pi$
$29$	−8.02877	−1.49091	−0.745453	+	0.666559i	$0.767767\pi$
$30$	0	0
$31$	−2.19656	−0.394513	−0.197257	−	0.980352i	$-0.563203\pi$
$31$	−2.19656	−0.394513	−0.197257	+	0.980352i	$0.563203\pi$
$32$	0	0
$33$	−1.14637	−0.199557
$34$	0	0
$35$	−4.48929	−0.758828
$36$	0	0
$37$	−2.48929	−0.409237	−0.204618	−	0.978842i	$-0.565595\pi$
$37$	−2.48929	−0.409237	−0.204618	+	0.978842i	$0.565595\pi$
$38$	0	0
$39$	−0.853635	−0.136691
$40$	0	0
$41$	11.3001	1.76478	0.882388	−	0.470523i	$-0.155935\pi$
$41$	11.3001	1.76478	0.882388	+	0.470523i	$0.155935\pi$
$42$	0	0
$43$	−10.6858	−1.62958	−0.814788	−	0.579759i	$-0.803147\pi$
$43$	−10.6858	−1.62958	−0.814788	+	0.579759i	$0.803147\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−1.53948	−0.224556	−0.112278	−	0.993677i	$-0.535815\pi$
$47$	−1.53948	−0.224556	−0.112278	+	0.993677i	$0.535815\pi$
$48$	0	0
$49$	13.1537	1.87910
$50$	0	0
$51$	−1.34292	−0.188047
$52$	0	0
$53$	4.02877	0.553394	0.276697	−	0.960957i	$-0.410760\pi$
$53$	4.02877	0.553394	0.276697	+	0.960957i	$0.410760\pi$
$54$	0	0
$55$	1.14637	0.154576
$56$	0	0
$57$	−3.83221	−0.507589
$58$	0	0
$59$	15.0073	1.95379	0.976895	−	0.213720i	$-0.0685579\pi$
$59$	15.0073	1.95379	0.976895	+	0.213720i	$0.0685579\pi$
$60$	0	0
$61$	−5.83221	−0.746738	−0.373369	−	0.927683i	$-0.621797\pi$
$61$	−5.83221	−0.746738	−0.373369	+	0.927683i	$0.621797\pi$
$62$	0	0
$63$	−4.48929	−0.565597
$64$	0	0
$65$	0.853635	0.105880
$66$	0	0
$67$	11.5682	1.41329	0.706643	−	0.707570i	$-0.250209\pi$
$67$	11.5682	1.41329	0.706643	+	0.707570i	$0.250209\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.32150	−0.512868	−0.256434	−	0.966562i	$-0.582548\pi$
$71$	−4.32150	−0.512868	−0.256434	+	0.966562i	$0.582548\pi$
$72$	0	0
$73$	−13.1035	−1.53365	−0.766825	−	0.641856i	$-0.778165\pi$
$73$	−13.1035	−1.53365	−0.766825	+	0.641856i	$0.778165\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.14637	−0.586483
$78$	0	0
$79$	−0.585462	−0.0658696	−0.0329348	−	0.999458i	$-0.510485\pi$
$79$	−0.585462	−0.0658696	−0.0329348	+	0.999458i	$0.510485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.63565	−0.618593	−0.309297	−	0.950966i	$-0.600094\pi$
$83$	−5.63565	−0.618593	−0.309297	+	0.950966i	$0.600094\pi$
$84$	0	0
$85$	1.34292	0.145660
$86$	0	0
$87$	8.02877	0.860774
$88$	0	0
$89$	−0.100384	−0.0106407	−0.00532035	−	0.999986i	$-0.501694\pi$
$89$	−0.100384	−0.0106407	−0.00532035	+	0.999986i	$0.501694\pi$
$90$	0	0
$91$	−3.83221	−0.401725
$92$	0	0
$93$	2.19656	0.227772
$94$	0	0
$95$	3.83221	0.393177
$96$	0	0
$97$	11.5640	1.17415	0.587075	−	0.809532i	$-0.300279\pi$
$97$	11.5640	1.17415	0.587075	+	0.809532i	$0.300279\pi$
$98$	0	0
$99$	1.14637	0.115214
$100$	0	0
$101$	11.6932	1.16352	0.581758	−	0.813362i	$-0.302365\pi$
$101$	11.6932	1.16352	0.581758	+	0.813362i	$0.302365\pi$
$102$	0	0
$103$	19.7220	1.94326	0.971631	−	0.236501i	$-0.0760006\pi$
$103$	19.7220	1.94326	0.971631	+	0.236501i	$0.0760006\pi$
$104$	0	0
$105$	4.48929	0.438110
$106$	0	0
$107$	−5.44331	−0.526224	−0.263112	−	0.964765i	$-0.584749\pi$
$107$	−5.44331	−0.526224	−0.263112	+	0.964765i	$0.584749\pi$
$108$	0	0
$109$	−12.5181	−1.19901	−0.599506	−	0.800370i	$-0.704636\pi$
$109$	−12.5181	−1.19901	−0.599506	+	0.800370i	$0.704636\pi$
$110$	0	0
$111$	2.48929	0.236273
$112$	0	0
$113$	−14.1292	−1.32916	−0.664579	−	0.747218i	$-0.731389\pi$
$113$	−14.1292	−1.32916	−0.664579	+	0.747218i	$0.731389\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0.853635	0.0789185
$118$	0	0
$119$	−6.02877	−0.552656
$120$	0	0
$121$	−9.68585	−0.880531
$122$	0	0
$123$	−11.3001	−1.01889
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.48194	−0.486444	−0.243222	−	0.969971i	$-0.578204\pi$
$127$	−5.48194	−0.486444	−0.243222	+	0.969971i	$0.578204\pi$
$128$	0	0
$129$	10.6858	0.940836
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−17.2039	−1.49177
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.7648	1.00514	0.502568	−	0.864538i	$-0.332389\pi$
$137$	11.7648	1.00514	0.502568	+	0.864538i	$0.332389\pi$
$138$	0	0
$139$	−8.15371	−0.691589	−0.345794	−	0.938310i	$-0.612391\pi$
$139$	−8.15371	−0.691589	−0.345794	+	0.938310i	$0.612391\pi$
$140$	0	0
$141$	1.53948	0.129648
$142$	0	0
$143$	0.978577	0.0818327
$144$	0	0
$145$	−8.02877	−0.666753
$146$	0	0
$147$	−13.1537	−1.08490
$148$	0	0
$149$	−3.73183	−0.305723	−0.152862	−	0.988248i	$-0.548849\pi$
$149$	−3.73183	−0.305723	−0.152862	+	0.988248i	$0.548849\pi$
$150$	0	0
$151$	−16.6430	−1.35439	−0.677194	−	0.735804i	$-0.736804\pi$
$151$	−16.6430	−1.35439	−0.677194	+	0.735804i	$0.736804\pi$
$152$	0	0
$153$	1.34292	0.108569
$154$	0	0
$155$	−2.19656	−0.176432
$156$	0	0
$157$	−13.2327	−1.05608	−0.528041	−	0.849219i	$-0.677073\pi$
$157$	−13.2327	−1.05608	−0.528041	+	0.849219i	$0.677073\pi$
$158$	0	0
$159$	−4.02877	−0.319502
$160$	0	0
$161$	4.48929	0.353806
$162$	0	0
$163$	−16.3931	−1.28401	−0.642004	−	0.766701i	$-0.721897\pi$
$163$	−16.3931	−1.28401	−0.642004	+	0.766701i	$0.721897\pi$
$164$	0	0
$165$	−1.14637	−0.0892444
$166$	0	0
$167$	−3.04598	−0.235705	−0.117853	−	0.993031i	$-0.537601\pi$
$167$	−3.04598	−0.235705	−0.117853	+	0.993031i	$0.537601\pi$
$168$	0	0
$169$	−12.2713	−0.943947
$170$	0	0
$171$	3.83221	0.293057
$172$	0	0
$173$	−24.9357	−1.89583	−0.947914	−	0.318526i	$-0.896812\pi$
$173$	−24.9357	−1.89583	−0.947914	+	0.318526i	$0.896812\pi$
$174$	0	0
$175$	−4.48929	−0.339358
$176$	0	0
$177$	−15.0073	−1.12802
$178$	0	0
$179$	6.68585	0.499724	0.249862	−	0.968282i	$-0.419615\pi$
$179$	6.68585	0.499724	0.249862	+	0.968282i	$0.419615\pi$
$180$	0	0
$181$	−20.6002	−1.53120	−0.765599	−	0.643318i	$-0.777557\pi$
$181$	−20.6002	−1.53120	−0.765599	+	0.643318i	$0.777557\pi$
$182$	0	0
$183$	5.83221	0.431129
$184$	0	0
$185$	−2.48929	−0.183016
$186$	0	0
$187$	1.53948	0.112578
$188$	0	0
$189$	4.48929	0.326548
$190$	0	0
$191$	−4.95402	−0.358460	−0.179230	−	0.983807i	$-0.557361\pi$
$191$	−4.95402	−0.358460	−0.179230	+	0.983807i	$0.557361\pi$
$192$	0	0
$193$	6.05754	0.436031	0.218016	−	0.975945i	$-0.430042\pi$
$193$	6.05754	0.436031	0.218016	+	0.975945i	$0.430042\pi$
$194$	0	0
$195$	−0.853635	−0.0611300
$196$	0	0
$197$	−24.4078	−1.73898	−0.869492	−	0.493947i	$-0.835554\pi$
$197$	−24.4078	−1.73898	−0.869492	+	0.493947i	$0.835554\pi$
$198$	0	0
$199$	3.85677	0.273399	0.136700	−	0.990613i	$-0.456350\pi$
$199$	3.85677	0.273399	0.136700	+	0.990613i	$0.456350\pi$
$200$	0	0
$201$	−11.5682	−0.815961
$202$	0	0
$203$	36.0435	2.52976
$204$	0	0
$205$	11.3001	0.789232
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	4.39312	0.303878
$210$	0	0
$211$	0.824865	0.0567861	0.0283930	−	0.999597i	$-0.490961\pi$
$211$	0.824865	0.0567861	0.0283930	+	0.999597i	$0.490961\pi$
$212$	0	0
$213$	4.32150	0.296104
$214$	0	0
$215$	−10.6858	−0.728769
$216$	0	0
$217$	9.86098	0.669407
$218$	0	0
$219$	13.1035	0.885454
$220$	0	0
$221$	1.14637	0.0771129
$222$	0	0
$223$	−5.56404	−0.372596	−0.186298	−	0.982493i	$-0.559649\pi$
$223$	−5.56404	−0.372596	−0.186298	+	0.982493i	$0.559649\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−16.5855	−1.10082	−0.550408	−	0.834896i	$-0.685528\pi$
$227$	−16.5855	−1.10082	−0.550408	+	0.834896i	$0.685528\pi$
$228$	0	0
$229$	6.64300	0.438982	0.219491	−	0.975615i	$-0.429560\pi$
$229$	6.64300	0.438982	0.219491	+	0.975615i	$0.429560\pi$
$230$	0	0
$231$	5.14637	0.338606
$232$	0	0
$233$	−14.3931	−0.942924	−0.471462	−	0.881886i	$-0.656274\pi$
$233$	−14.3931	−0.942924	−0.471462	+	0.881886i	$0.656274\pi$
$234$	0	0
$235$	−1.53948	−0.100425
$236$	0	0
$237$	0.585462	0.0380298
$238$	0	0
$239$	13.1077	0.847869	0.423934	−	0.905693i	$-0.360649\pi$
$239$	13.1077	0.847869	0.423934	+	0.905693i	$0.360649\pi$
$240$	0	0
$241$	12.5181	0.806359	0.403179	−	0.915121i	$-0.367905\pi$
$241$	12.5181	0.806359	0.403179	+	0.915121i	$0.367905\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.1537	0.840360
$246$	0	0
$247$	3.27131	0.208148
$248$	0	0
$249$	5.63565	0.357145
$250$	0	0
$251$	26.3503	1.66321	0.831607	−	0.555364i	$-0.187421\pi$
$251$	26.3503	1.66321	0.831607	+	0.555364i	$0.187421\pi$
$252$	0	0
$253$	−1.14637	−0.0720714
$254$	0	0
$255$	−1.34292	−0.0840971
$256$	0	0
$257$	−27.0116	−1.68493	−0.842467	−	0.538747i	$-0.818898\pi$
$257$	−27.0116	−1.68493	−0.842467	+	0.538747i	$0.818898\pi$
$258$	0	0
$259$	11.1751	0.694389
$260$	0	0
$261$	−8.02877	−0.496968
$262$	0	0
$263$	−21.8855	−1.34952	−0.674760	−	0.738037i	$-0.735753\pi$
$263$	−21.8855	−1.34952	−0.674760	+	0.738037i	$0.735753\pi$
$264$	0	0
$265$	4.02877	0.247485
$266$	0	0
$267$	0.100384	0.00614341
$268$	0	0
$269$	3.69319	0.225178	0.112589	−	0.993642i	$-0.464086\pi$
$269$	3.69319	0.225178	0.112589	+	0.993642i	$0.464086\pi$
$270$	0	0
$271$	−0.288520	−0.0175264	−0.00876318	−	0.999962i	$-0.502789\pi$
$271$	−0.288520	−0.0175264	−0.00876318	+	0.999962i	$0.502789\pi$
$272$	0	0
$273$	3.83221	0.231936
$274$	0	0
$275$	1.14637	0.0691284
$276$	0	0
$277$	21.3288	1.28153	0.640763	−	0.767739i	$-0.278618\pi$
$277$	21.3288	1.28153	0.640763	+	0.767739i	$0.278618\pi$
$278$	0	0
$279$	−2.19656	−0.131504
$280$	0	0
$281$	−26.6676	−1.59085	−0.795427	−	0.606050i	$-0.792753\pi$
$281$	−26.6676	−1.59085	−0.795427	+	0.606050i	$0.792753\pi$
$282$	0	0
$283$	−25.9185	−1.54070	−0.770348	−	0.637624i	$-0.779918\pi$
$283$	−25.9185	−1.54070	−0.770348	+	0.637624i	$0.779918\pi$
$284$	0	0
$285$	−3.83221	−0.227001
$286$	0	0
$287$	−50.7293	−2.99446
$288$	0	0
$289$	−15.1966	−0.893915
$290$	0	0
$291$	−11.5640	−0.677896
$292$	0	0
$293$	13.6503	0.797462	0.398731	−	0.917068i	$-0.369451\pi$
$293$	13.6503	0.797462	0.398731	+	0.917068i	$0.369451\pi$
$294$	0	0
$295$	15.0073	0.873761
$296$	0	0
$297$	−1.14637	−0.0665189
$298$	0	0
$299$	−0.853635	−0.0493670
$300$	0	0
$301$	47.9718	2.76505
$302$	0	0
$303$	−11.6932	−0.671756
$304$	0	0
$305$	−5.83221	−0.333951
$306$	0	0
$307$	9.78937	0.558709	0.279354	−	0.960188i	$-0.409880\pi$
$307$	9.78937	0.558709	0.279354	+	0.960188i	$0.409880\pi$
$308$	0	0
$309$	−19.7220	−1.12194
$310$	0	0
$311$	3.32885	0.188762	0.0943808	−	0.995536i	$-0.469913\pi$
$311$	3.32885	0.188762	0.0943808	+	0.995536i	$0.469913\pi$
$312$	0	0
$313$	−18.8824	−1.06730	−0.533648	−	0.845707i	$-0.679179\pi$
$313$	−18.8824	−1.06730	−0.533648	+	0.845707i	$0.679179\pi$
$314$	0	0
$315$	−4.48929	−0.252943
$316$	0	0
$317$	−8.06740	−0.453111	−0.226555	−	0.973998i	$-0.572746\pi$
$317$	−8.06740	−0.453111	−0.226555	+	0.973998i	$0.572746\pi$
$318$	0	0
$319$	−9.20390	−0.515320
$320$	0	0
$321$	5.44331	0.303816
$322$	0	0
$323$	5.14637	0.286351
$324$	0	0
$325$	0.853635	0.0473511
$326$	0	0
$327$	12.5181	0.692250
$328$	0	0
$329$	6.91117	0.381025
$330$	0	0
$331$	26.6388	1.46420	0.732100	−	0.681197i	$-0.238540\pi$
$331$	26.6388	1.46420	0.732100	+	0.681197i	$0.238540\pi$
$332$	0	0
$333$	−2.48929	−0.136412
$334$	0	0
$335$	11.5682	0.632041
$336$	0	0
$337$	−8.54262	−0.465346	−0.232673	−	0.972555i	$-0.574747\pi$
$337$	−8.54262	−0.465346	−0.232673	+	0.972555i	$0.574747\pi$
$338$	0	0
$339$	14.1292	0.767390
$340$	0	0
$341$	−2.51806	−0.136360
$342$	0	0
$343$	−27.6258	−1.49165
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	16.2499	0.872340	0.436170	−	0.899864i	$-0.356335\pi$
$347$	16.2499	0.872340	0.436170	+	0.899864i	$0.356335\pi$
$348$	0	0
$349$	23.1898	1.24132	0.620662	−	0.784079i	$-0.286864\pi$
$349$	23.1898	1.24132	0.620662	+	0.784079i	$0.286864\pi$
$350$	0	0
$351$	−0.853635	−0.0455636
$352$	0	0
$353$	−7.48194	−0.398224	−0.199112	−	0.979977i	$-0.563806\pi$
$353$	−7.48194	−0.398224	−0.199112	+	0.979977i	$0.563806\pi$
$354$	0	0
$355$	−4.32150	−0.229361
$356$	0	0
$357$	6.02877	0.319076
$358$	0	0
$359$	−1.87506	−0.0989617	−0.0494809	−	0.998775i	$-0.515757\pi$
$359$	−1.87506	−0.0989617	−0.0494809	+	0.998775i	$0.515757\pi$
$360$	0	0
$361$	−4.31415	−0.227061
$362$	0	0
$363$	9.68585	0.508375
$364$	0	0
$365$	−13.1035	−0.685870
$366$	0	0
$367$	4.68164	0.244379	0.122190	−	0.992507i	$-0.461008\pi$
$367$	4.68164	0.244379	0.122190	+	0.992507i	$0.461008\pi$
$368$	0	0
$369$	11.3001	0.588259
$370$	0	0
$371$	−18.0863	−0.938994
$372$	0	0
$373$	18.6430	0.965298	0.482649	−	0.875814i	$-0.339675\pi$
$373$	18.6430	0.965298	0.482649	+	0.875814i	$0.339675\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−6.85363	−0.352980
$378$	0	0
$379$	−29.8715	−1.53439	−0.767197	−	0.641412i	$-0.778349\pi$
$379$	−29.8715	−1.53439	−0.767197	+	0.641412i	$0.778349\pi$
$380$	0	0
$381$	5.48194	0.280848
$382$	0	0
$383$	−20.5714	−1.05115	−0.525574	−	0.850748i	$-0.676150\pi$
$383$	−20.5714	−1.05115	−0.525574	+	0.850748i	$0.676150\pi$
$384$	0	0
$385$	−5.14637	−0.262283
$386$	0	0
$387$	−10.6858	−0.543192
$388$	0	0
$389$	−2.33558	−0.118418	−0.0592092	−	0.998246i	$-0.518858\pi$
$389$	−2.33558	−0.118418	−0.0592092	+	0.998246i	$0.518858\pi$
$390$	0	0
$391$	−1.34292	−0.0679145
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.585462	−0.0294578
$396$	0	0
$397$	29.1365	1.46232	0.731160	−	0.682207i	$-0.238980\pi$
$397$	29.1365	1.46232	0.731160	+	0.682207i	$0.238980\pi$
$398$	0	0
$399$	17.2039	0.861272
$400$	0	0
$401$	−14.8353	−0.740842	−0.370421	−	0.928864i	$-0.620787\pi$
$401$	−14.8353	−0.740842	−0.370421	+	0.928864i	$0.620787\pi$
$402$	0	0
$403$	−1.87506	−0.0934033
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−2.85363	−0.141449
$408$	0	0
$409$	−38.0189	−1.87991	−0.939957	−	0.341293i	$-0.889135\pi$
$409$	−38.0189	−1.87991	−0.939957	+	0.341293i	$0.889135\pi$
$410$	0	0
$411$	−11.7648	−0.580315
$412$	0	0
$413$	−67.3723	−3.31517
$414$	0	0
$415$	−5.63565	−0.276643
$416$	0	0
$417$	8.15371	0.399289
$418$	0	0
$419$	−21.8469	−1.06729	−0.533646	−	0.845708i	$-0.679178\pi$
$419$	−21.8469	−1.06729	−0.533646	+	0.845708i	$0.679178\pi$
$420$	0	0
$421$	21.4391	1.04488	0.522439	−	0.852677i	$-0.325022\pi$
$421$	21.4391	1.04488	0.522439	+	0.852677i	$0.325022\pi$
$422$	0	0
$423$	−1.53948	−0.0748521
$424$	0	0
$425$	1.34292	0.0651413
$426$	0	0
$427$	26.1825	1.26706
$428$	0	0
$429$	−0.978577	−0.0472461
$430$	0	0
$431$	−29.8223	−1.43649	−0.718246	−	0.695789i	$-0.755055\pi$
$431$	−29.8223	−1.43649	−0.718246	+	0.695789i	$0.755055\pi$
$432$	0	0
$433$	−0.925249	−0.0444647	−0.0222323	−	0.999753i	$-0.507077\pi$
$433$	−0.925249	−0.0444647	−0.0222323	+	0.999753i	$0.507077\pi$
$434$	0	0
$435$	8.02877	0.384950
$436$	0	0
$437$	−3.83221	−0.183320
$438$	0	0
$439$	21.6791	1.03469	0.517344	−	0.855778i	$-0.326921\pi$
$439$	21.6791	1.03469	0.517344	+	0.855778i	$0.326921\pi$
$440$	0	0
$441$	13.1537	0.626367
$442$	0	0
$443$	−5.20390	−0.247245	−0.123622	−	0.992329i	$-0.539451\pi$
$443$	−5.20390	−0.247245	−0.123622	+	0.992329i	$0.539451\pi$
$444$	0	0
$445$	−0.100384	−0.00475867
$446$	0	0
$447$	3.73183	0.176509
$448$	0	0
$449$	−10.8578	−0.512413	−0.256207	−	0.966622i	$-0.582473\pi$
$449$	−10.8578	−0.512413	−0.256207	+	0.966622i	$0.582473\pi$
$450$	0	0
$451$	12.9540	0.609981
$452$	0	0
$453$	16.6430	0.781956
$454$	0	0
$455$	−3.83221	−0.179657
$456$	0	0
$457$	27.2755	1.27589	0.637947	−	0.770080i	$-0.279784\pi$
$457$	27.2755	1.27589	0.637947	+	0.770080i	$0.279784\pi$
$458$	0	0
$459$	−1.34292	−0.0626823
$460$	0	0
$461$	20.2070	0.941136	0.470568	−	0.882364i	$-0.344049\pi$
$461$	20.2070	0.941136	0.470568	+	0.882364i	$0.344049\pi$
$462$	0	0
$463$	−8.03298	−0.373324	−0.186662	−	0.982424i	$-0.559767\pi$
$463$	−8.03298	−0.373324	−0.186662	+	0.982424i	$0.559767\pi$
$464$	0	0
$465$	2.19656	0.101863
$466$	0	0
$467$	19.4580	0.900409	0.450204	−	0.892926i	$-0.351351\pi$
$467$	19.4580	0.900409	0.450204	+	0.892926i	$0.351351\pi$
$468$	0	0
$469$	−51.9332	−2.39805
$470$	0	0
$471$	13.2327	0.609729
$472$	0	0
$473$	−12.2499	−0.563250
$474$	0	0
$475$	3.83221	0.175834
$476$	0	0
$477$	4.02877	0.184465
$478$	0	0
$479$	−28.9540	−1.32294	−0.661471	−	0.749970i	$-0.730068\pi$
$479$	−28.9540	−1.32294	−0.661471	+	0.749970i	$0.730068\pi$
$480$	0	0
$481$	−2.12494	−0.0968890
$482$	0	0
$483$	−4.48929	−0.204270
$484$	0	0
$485$	11.5640	0.525096
$486$	0	0
$487$	−20.8108	−0.943027	−0.471513	−	0.881859i	$-0.656292\pi$
$487$	−20.8108	−0.943027	−0.471513	+	0.881859i	$0.656292\pi$
$488$	0	0
$489$	16.3931	0.741322
$490$	0	0
$491$	−35.1140	−1.58467	−0.792336	−	0.610084i	$-0.791135\pi$
$491$	−35.1140	−1.58467	−0.792336	+	0.610084i	$0.791135\pi$
$492$	0	0
$493$	−10.7820	−0.485598
$494$	0	0
$495$	1.14637	0.0515253
$496$	0	0
$497$	19.4005	0.870230
$498$	0	0
$499$	7.31836	0.327615	0.163807	−	0.986492i	$-0.447622\pi$
$499$	7.31836	0.327615	0.163807	+	0.986492i	$0.447622\pi$
$500$	0	0
$501$	3.04598	0.136084
$502$	0	0
$503$	22.0006	0.980959	0.490479	−	0.871453i	$-0.336822\pi$
$503$	22.0006	0.980959	0.490479	+	0.871453i	$0.336822\pi$
$504$	0	0
$505$	11.6932	0.520340
$506$	0	0
$507$	12.2713	0.544988
$508$	0	0
$509$	−35.4783	−1.57255	−0.786275	−	0.617877i	$-0.787993\pi$
$509$	−35.4783	−1.57255	−0.786275	+	0.617877i	$0.787993\pi$
$510$	0	0
$511$	58.8255	2.60229
$512$	0	0
$513$	−3.83221	−0.169196
$514$	0	0
$515$	19.7220	0.869053
$516$	0	0
$517$	−1.76481	−0.0776161
$518$	0	0
$519$	24.9357	1.09456
$520$	0	0
$521$	21.4966	0.941785	0.470892	−	0.882191i	$-0.343932\pi$
$521$	21.4966	0.941785	0.470892	+	0.882191i	$0.343932\pi$
$522$	0	0
$523$	11.3288	0.495376	0.247688	−	0.968840i	$-0.420329\pi$
$523$	11.3288	0.495376	0.247688	+	0.968840i	$0.420329\pi$
$524$	0	0
$525$	4.48929	0.195929
$526$	0	0
$527$	−2.94981	−0.128496
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	15.0073	0.651263
$532$	0	0
$533$	9.64614	0.417821
$534$	0	0
$535$	−5.44331	−0.235335
$536$	0	0
$537$	−6.68585	−0.288516
$538$	0	0
$539$	15.0790	0.649497
$540$	0	0
$541$	−7.67912	−0.330151	−0.165075	−	0.986281i	$-0.552787\pi$
$541$	−7.67912	−0.330151	−0.165075	+	0.986281i	$0.552787\pi$
$542$	0	0
$543$	20.6002	0.884037
$544$	0	0
$545$	−12.5181	−0.536215
$546$	0	0
$547$	−31.5296	−1.34811	−0.674054	−	0.738682i	$-0.735449\pi$
$547$	−31.5296	−1.34811	−0.674054	+	0.738682i	$0.735449\pi$
$548$	0	0
$549$	−5.83221	−0.248913
$550$	0	0
$551$	−30.7679	−1.31076
$552$	0	0
$553$	2.62831	0.111767
$554$	0	0
$555$	2.48929	0.105664
$556$	0	0
$557$	−10.1292	−0.429186	−0.214593	−	0.976704i	$-0.568843\pi$
$557$	−10.1292	−0.429186	−0.214593	+	0.976704i	$0.568843\pi$
$558$	0	0
$559$	−9.12181	−0.385811
$560$	0	0
$561$	−1.53948	−0.0649969
$562$	0	0
$563$	−40.7146	−1.71592	−0.857958	−	0.513719i	$-0.828267\pi$
$563$	−40.7146	−1.71592	−0.857958	+	0.513719i	$0.828267\pi$
$564$	0	0
$565$	−14.1292	−0.594418
$566$	0	0
$567$	−4.48929	−0.188532
$568$	0	0
$569$	1.66442	0.0697763	0.0348881	−	0.999391i	$-0.488893\pi$
$569$	1.66442	0.0697763	0.0348881	+	0.999391i	$0.488893\pi$
$570$	0	0
$571$	41.8469	1.75124	0.875619	−	0.483002i	$-0.160454\pi$
$571$	41.8469	1.75124	0.875619	+	0.483002i	$0.160454\pi$
$572$	0	0
$573$	4.95402	0.206957
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	21.6069	0.899506	0.449753	−	0.893153i	$-0.351512\pi$
$577$	21.6069	0.899506	0.449753	+	0.893153i	$0.351512\pi$
$578$	0	0
$579$	−6.05754	−0.251743
$580$	0	0
$581$	25.3001	1.04962
$582$	0	0
$583$	4.61844	0.191276
$584$	0	0
$585$	0.853635	0.0352934
$586$	0	0
$587$	−0.393115	−0.0162256	−0.00811280	−	0.999967i	$-0.502582\pi$
$587$	−0.393115	−0.0162256	−0.00811280	+	0.999967i	$0.502582\pi$
$588$	0	0
$589$	−8.41767	−0.346844
$590$	0	0
$591$	24.4078	1.00400
$592$	0	0
$593$	−8.31729	−0.341550	−0.170775	−	0.985310i	$-0.554627\pi$
$593$	−8.31729	−0.341550	−0.170775	+	0.985310i	$0.554627\pi$
$594$	0	0
$595$	−6.02877	−0.247155
$596$	0	0
$597$	−3.85677	−0.157847
$598$	0	0
$599$	−9.03612	−0.369206	−0.184603	−	0.982813i	$-0.559100\pi$
$599$	−9.03612	−0.369206	−0.184603	+	0.982813i	$0.559100\pi$
$600$	0	0
$601$	−27.6602	−1.12828	−0.564142	−	0.825678i	$-0.690793\pi$
$601$	−27.6602	−1.12828	−0.564142	+	0.825678i	$0.690793\pi$
$602$	0	0
$603$	11.5682	0.471096
$604$	0	0
$605$	−9.68585	−0.393786
$606$	0	0
$607$	−29.6461	−1.20330	−0.601650	−	0.798760i	$-0.705490\pi$
$607$	−29.6461	−1.20330	−0.601650	+	0.798760i	$0.705490\pi$
$608$	0	0
$609$	−36.0435	−1.46055
$610$	0	0
$611$	−1.31415	−0.0531650
$612$	0	0
$613$	39.4868	1.59486	0.797428	−	0.603414i	$-0.206193\pi$
$613$	39.4868	1.59486	0.797428	+	0.603414i	$0.206193\pi$
$614$	0	0
$615$	−11.3001	−0.455663
$616$	0	0
$617$	16.8641	0.678924	0.339462	−	0.940620i	$-0.389755\pi$
$617$	16.8641	0.678924	0.339462	+	0.940620i	$0.389755\pi$
$618$	0	0
$619$	−2.74338	−0.110266	−0.0551330	−	0.998479i	$-0.517558\pi$
$619$	−2.74338	−0.110266	−0.0551330	+	0.998479i	$0.517558\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	0.450654	0.0180551
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.39312	−0.175444
$628$	0	0
$629$	−3.34292	−0.133291
$630$	0	0
$631$	9.93260	0.395410	0.197705	−	0.980262i	$-0.436651\pi$
$631$	9.93260	0.395410	0.197705	+	0.980262i	$0.436651\pi$
$632$	0	0
$633$	−0.824865	−0.0327855
$634$	0	0
$635$	−5.48194	−0.217544
$636$	0	0
$637$	11.2285	0.444888
$638$	0	0
$639$	−4.32150	−0.170956
$640$	0	0
$641$	−10.6184	−0.419403	−0.209702	−	0.977765i	$-0.567249\pi$
$641$	−10.6184	−0.419403	−0.209702	+	0.977765i	$0.567249\pi$
$642$	0	0
$643$	−20.5959	−0.812225	−0.406112	−	0.913823i	$-0.633116\pi$
$643$	−20.5959	−0.812225	−0.406112	+	0.913823i	$0.633116\pi$
$644$	0	0
$645$	10.6858	0.420755
$646$	0	0
$647$	6.71883	0.264144	0.132072	−	0.991240i	$-0.457837\pi$
$647$	6.71883	0.264144	0.132072	+	0.991240i	$0.457837\pi$
$648$	0	0
$649$	17.2039	0.675312
$650$	0	0
$651$	−9.86098	−0.386482
$652$	0	0
$653$	−4.07583	−0.159499	−0.0797497	−	0.996815i	$-0.525412\pi$
$653$	−4.07583	−0.159499	−0.0797497	+	0.996815i	$0.525412\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−13.1035	−0.511217
$658$	0	0
$659$	29.0607	1.13204	0.566022	−	0.824390i	$-0.308482\pi$
$659$	29.0607	1.13204	0.566022	+	0.824390i	$0.308482\pi$
$660$	0	0
$661$	31.2369	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$661$	31.2369	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$662$	0	0
$663$	−1.14637	−0.0445211
$664$	0	0
$665$	−17.2039	−0.667139
$666$	0	0
$667$	8.02877	0.310875
$668$	0	0
$669$	5.56404	0.215118
$670$	0	0
$671$	−6.68585	−0.258104
$672$	0	0
$673$	9.74652	0.375701	0.187850	−	0.982198i	$-0.439848\pi$
$673$	9.74652	0.375701	0.187850	+	0.982198i	$0.439848\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	6.80031	0.261357	0.130679	−	0.991425i	$-0.458284\pi$
$677$	6.80031	0.261357	0.130679	+	0.991425i	$0.458284\pi$
$678$	0	0
$679$	−51.9143	−1.99229
$680$	0	0
$681$	16.5855	0.635556
$682$	0	0
$683$	41.5029	1.58806	0.794032	−	0.607876i	$-0.207978\pi$
$683$	41.5029	1.58806	0.794032	+	0.607876i	$0.207978\pi$
$684$	0	0
$685$	11.7648	0.449510
$686$	0	0
$687$	−6.64300	−0.253446
$688$	0	0
$689$	3.43910	0.131019
$690$	0	0
$691$	31.2713	1.18962	0.594808	−	0.803868i	$-0.297228\pi$
$691$	31.2713	1.18962	0.594808	+	0.803868i	$0.297228\pi$
$692$	0	0
$693$	−5.14637	−0.195494
$694$	0	0
$695$	−8.15371	−0.309288
$696$	0	0
$697$	15.1751	0.574799
$698$	0	0
$699$	14.3931	0.544398
$700$	0	0
$701$	−6.02456	−0.227544	−0.113772	−	0.993507i	$-0.536293\pi$
$701$	−6.02456	−0.227544	−0.113772	+	0.993507i	$0.536293\pi$
$702$	0	0
$703$	−9.53948	−0.359788
$704$	0	0
$705$	1.53948	0.0579802
$706$	0	0
$707$	−52.4941	−1.97424
$708$	0	0
$709$	37.6974	1.41576	0.707878	−	0.706335i	$-0.249653\pi$
$709$	37.6974	1.41576	0.707878	+	0.706335i	$0.249653\pi$
$710$	0	0
$711$	−0.585462	−0.0219565
$712$	0	0
$713$	2.19656	0.0822617
$714$	0	0
$715$	0.978577	0.0365967
$716$	0	0
$717$	−13.1077	−0.489517
$718$	0	0
$719$	−19.8708	−0.741058	−0.370529	−	0.928821i	$-0.620824\pi$
$719$	−19.8708	−0.741058	−0.370529	+	0.928821i	$0.620824\pi$
$720$	0	0
$721$	−88.5376	−3.29731
$722$	0	0
$723$	−12.5181	−0.465552
$724$	0	0
$725$	−8.02877	−0.298181
$726$	0	0
$727$	−15.8757	−0.588796	−0.294398	−	0.955683i	$-0.595119\pi$
$727$	−15.8757	−0.588796	−0.294398	+	0.955683i	$0.595119\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−14.3503	−0.530764
$732$	0	0
$733$	37.1751	1.37309	0.686547	−	0.727085i	$-0.259125\pi$
$733$	37.1751	1.37309	0.686547	+	0.727085i	$0.259125\pi$
$734$	0	0
$735$	−13.1537	−0.485182
$736$	0	0
$737$	13.2614	0.488492
$738$	0	0
$739$	−44.0189	−1.61926	−0.809631	−	0.586940i	$-0.800333\pi$
$739$	−44.0189	−1.61926	−0.809631	+	0.586940i	$0.800333\pi$
$740$	0	0
$741$	−3.27131	−0.120175
$742$	0	0
$743$	−51.5296	−1.89044	−0.945219	−	0.326437i	$-0.894152\pi$
$743$	−51.5296	−1.89044	−0.945219	+	0.326437i	$0.894152\pi$
$744$	0	0
$745$	−3.73183	−0.136724
$746$	0	0
$747$	−5.63565	−0.206198
$748$	0	0
$749$	24.4366	0.892893
$750$	0	0
$751$	3.91790	0.142966	0.0714832	−	0.997442i	$-0.477227\pi$
$751$	3.91790	0.142966	0.0714832	+	0.997442i	$0.477227\pi$
$752$	0	0
$753$	−26.3503	−0.960257
$754$	0	0
$755$	−16.6430	−0.605701
$756$	0	0
$757$	25.7606	0.936285	0.468142	−	0.883653i	$-0.344923\pi$
$757$	25.7606	0.936285	0.468142	+	0.883653i	$0.344923\pi$
$758$	0	0
$759$	1.14637	0.0416104
$760$	0	0
$761$	−15.7507	−0.570964	−0.285482	−	0.958384i	$-0.592154\pi$
$761$	−15.7507	−0.570964	−0.285482	+	0.958384i	$0.592154\pi$
$762$	0	0
$763$	56.1972	2.03447
$764$	0	0
$765$	1.34292	0.0485535
$766$	0	0
$767$	12.8108	0.462571
$768$	0	0
$769$	−19.0031	−0.685271	−0.342635	−	0.939468i	$-0.611320\pi$
$769$	−19.0031	−0.685271	−0.342635	+	0.939468i	$0.611320\pi$
$770$	0	0
$771$	27.0116	0.972797
$772$	0	0
$773$	−41.7795	−1.50270	−0.751352	−	0.659902i	$-0.770598\pi$
$773$	−41.7795	−1.50270	−0.751352	+	0.659902i	$0.770598\pi$
$774$	0	0
$775$	−2.19656	−0.0789027
$776$	0	0
$777$	−11.1751	−0.400906
$778$	0	0
$779$	43.3043	1.55154
$780$	0	0
$781$	−4.95402	−0.177269
$782$	0	0
$783$	8.02877	0.286925
$784$	0	0
$785$	−13.2327	−0.472294
$786$	0	0
$787$	21.8610	0.779260	0.389630	−	0.920972i	$-0.372603\pi$
$787$	21.8610	0.779260	0.389630	+	0.920972i	$0.372603\pi$
$788$	0	0
$789$	21.8855	0.779146
$790$	0	0
$791$	63.4298	2.25531
$792$	0	0
$793$	−4.97858	−0.176794
$794$	0	0
$795$	−4.02877	−0.142886
$796$	0	0
$797$	−27.8280	−0.985718	−0.492859	−	0.870109i	$-0.664048\pi$
$797$	−27.8280	−0.985718	−0.492859	+	0.870109i	$0.664048\pi$
$798$	0	0
$799$	−2.06740	−0.0731395
$800$	0	0
$801$	−0.100384	−0.00354690
$802$	0	0
$803$	−15.0214	−0.530095
$804$	0	0
$805$	4.48929	0.158227
$806$	0	0
$807$	−3.69319	−0.130007
$808$	0	0
$809$	−21.0565	−0.740306	−0.370153	−	0.928971i	$-0.620695\pi$
$809$	−21.0565	−0.740306	−0.370153	+	0.928971i	$0.620695\pi$
$810$	0	0
$811$	16.8249	0.590801	0.295400	−	0.955374i	$-0.404547\pi$
$811$	16.8249	0.590801	0.295400	+	0.955374i	$0.404547\pi$
$812$	0	0
$813$	0.288520	0.0101188
$814$	0	0
$815$	−16.3931	−0.574226
$816$	0	0
$817$	−40.9504	−1.43267
$818$	0	0
$819$	−3.83221	−0.133908
$820$	0	0
$821$	−2.59388	−0.0905272	−0.0452636	−	0.998975i	$-0.514413\pi$
$821$	−2.59388	−0.0905272	−0.0452636	+	0.998975i	$0.514413\pi$
$822$	0	0
$823$	−0.786230	−0.0274063	−0.0137031	−	0.999906i	$-0.504362\pi$
$823$	−0.786230	−0.0274063	−0.0137031	+	0.999906i	$0.504362\pi$
$824$	0	0
$825$	−1.14637	−0.0399113
$826$	0	0
$827$	−10.2211	−0.355423	−0.177712	−	0.984083i	$-0.556869\pi$
$827$	−10.2211	−0.355423	−0.177712	+	0.984083i	$0.556869\pi$
$828$	0	0
$829$	−8.97437	−0.311693	−0.155846	−	0.987781i	$-0.549810\pi$
$829$	−8.97437	−0.311693	−0.155846	+	0.987781i	$0.549810\pi$
$830$	0	0
$831$	−21.3288	−0.739889
$832$	0	0
$833$	17.6644	0.612036
$834$	0	0
$835$	−3.04598	−0.105411
$836$	0	0
$837$	2.19656	0.0759241
$838$	0	0
$839$	−19.6069	−0.676905	−0.338452	−	0.940984i	$-0.609903\pi$
$839$	−19.6069	−0.676905	−0.338452	+	0.940984i	$0.609903\pi$
$840$	0	0
$841$	35.4611	1.22280
$842$	0	0
$843$	26.6676	0.918480
$844$	0	0
$845$	−12.2713	−0.422146
$846$	0	0
$847$	43.4826	1.49408
$848$	0	0
$849$	25.9185	0.889521
$850$	0	0
$851$	2.48929	0.0853317
$852$	0	0
$853$	−14.3650	−0.491847	−0.245923	−	0.969289i	$-0.579091\pi$
$853$	−14.3650	−0.491847	−0.245923	+	0.969289i	$0.579091\pi$
$854$	0	0
$855$	3.83221	0.131059
$856$	0	0
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	0	0
$859$	4.85425	0.165625	0.0828125	−	0.996565i	$-0.473610\pi$
$859$	4.85425	0.165625	0.0828125	+	0.996565i	$0.473610\pi$
$860$	0	0
$861$	50.7293	1.72885
$862$	0	0
$863$	−41.8139	−1.42336	−0.711681	−	0.702503i	$-0.752066\pi$
$863$	−41.8139	−1.42336	−0.711681	+	0.702503i	$0.752066\pi$
$864$	0	0
$865$	−24.9357	−0.847840
$866$	0	0
$867$	15.1966	0.516102
$868$	0	0
$869$	−0.671153	−0.0227673
$870$	0	0
$871$	9.87506	0.334604
$872$	0	0
$873$	11.5640	0.391383
$874$	0	0
$875$	−4.48929	−0.151766
$876$	0	0
$877$	−1.27973	−0.0432134	−0.0216067	−	0.999767i	$-0.506878\pi$
$877$	−1.27973	−0.0432134	−0.0216067	+	0.999767i	$0.506878\pi$
$878$	0	0
$879$	−13.6503	−0.460415
$880$	0	0
$881$	52.9687	1.78456	0.892281	−	0.451481i	$-0.149104\pi$
$881$	52.9687	1.78456	0.892281	+	0.451481i	$0.149104\pi$
$882$	0	0
$883$	17.5479	0.590534	0.295267	−	0.955415i	$-0.404591\pi$
$883$	17.5479	0.590534	0.295267	+	0.955415i	$0.404591\pi$
$884$	0	0
$885$	−15.0073	−0.504466
$886$	0	0
$887$	11.2713	0.378453	0.189227	−	0.981933i	$-0.439402\pi$
$887$	11.2713	0.378453	0.189227	+	0.981933i	$0.439402\pi$
$888$	0	0
$889$	24.6100	0.825394
$890$	0	0
$891$	1.14637	0.0384047
$892$	0	0
$893$	−5.89962	−0.197423
$894$	0	0
$895$	6.68585	0.223483
$896$	0	0
$897$	0.853635	0.0285020
$898$	0	0
$899$	17.6357	0.588182
$900$	0	0
$901$	5.41033	0.180244
$902$	0	0
$903$	−47.9718	−1.59640
$904$	0	0
$905$	−20.6002	−0.684772
$906$	0	0
$907$	−27.2045	−0.903311	−0.451656	−	0.892192i	$-0.649166\pi$
$907$	−27.2045	−0.903311	−0.451656	+	0.892192i	$0.649166\pi$
$908$	0	0
$909$	11.6932	0.387839
$910$	0	0
$911$	3.41454	0.113129	0.0565643	−	0.998399i	$-0.481985\pi$
$911$	3.41454	0.113129	0.0565643	+	0.998399i	$0.481985\pi$
$912$	0	0
$913$	−6.46052	−0.213812
$914$	0	0
$915$	5.83221	0.192807
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−20.4998	−0.676225	−0.338113	−	0.941106i	$-0.609788\pi$
$919$	−20.4998	−0.676225	−0.338113	+	0.941106i	$0.609788\pi$
$920$	0	0
$921$	−9.78937	−0.322571
$922$	0	0
$923$	−3.68898	−0.121424
$924$	0	0
$925$	−2.48929	−0.0818473
$926$	0	0
$927$	19.7220	0.647754
$928$	0	0
$929$	38.2640	1.25540	0.627700	−	0.778455i	$-0.283996\pi$
$929$	38.2640	1.25540	0.627700	+	0.778455i	$0.283996\pi$
$930$	0	0
$931$	50.4078	1.65205
$932$	0	0
$933$	−3.32885	−0.108982
$934$	0	0
$935$	1.53948	0.0503464
$936$	0	0
$937$	35.6791	1.16559	0.582793	−	0.812621i	$-0.301960\pi$
$937$	35.6791	1.16559	0.582793	+	0.812621i	$0.301960\pi$
$938$	0	0
$939$	18.8824	0.616204
$940$	0	0
$941$	15.2321	0.496551	0.248275	−	0.968689i	$-0.420136\pi$
$941$	15.2321	0.496551	0.248275	+	0.968689i	$0.420136\pi$
$942$	0	0
$943$	−11.3001	−0.367981
$944$	0	0
$945$	4.48929	0.146037
$946$	0	0
$947$	−6.29273	−0.204486	−0.102243	−	0.994759i	$-0.532602\pi$
$947$	−6.29273	−0.204486	−0.102243	+	0.994759i	$0.532602\pi$
$948$	0	0
$949$	−11.1856	−0.363100
$950$	0	0
$951$	8.06740	0.261604
$952$	0	0
$953$	20.3418	0.658937	0.329469	−	0.944167i	$-0.393130\pi$
$953$	20.3418	0.658937	0.329469	+	0.944167i	$0.393130\pi$
$954$	0	0
$955$	−4.95402	−0.160308
$956$	0	0
$957$	9.20390	0.297520
$958$	0	0
$959$	−52.8156	−1.70551
$960$	0	0
$961$	−26.1751	−0.844359
$962$	0	0
$963$	−5.44331	−0.175408
$964$	0	0
$965$	6.05754	0.194999
$966$	0	0
$967$	37.6461	1.21062	0.605309	−	0.795991i	$-0.293050\pi$
$967$	37.6461	1.21062	0.605309	+	0.795991i	$0.293050\pi$
$968$	0	0
$969$	−5.14637	−0.165325
$970$	0	0
$971$	29.6216	0.950602	0.475301	−	0.879823i	$-0.342339\pi$
$971$	29.6216	0.950602	0.475301	+	0.879823i	$0.342339\pi$
$972$	0	0
$973$	36.6044	1.17348
$974$	0	0
$975$	−0.853635	−0.0273382
$976$	0	0
$977$	21.0158	0.672354	0.336177	−	0.941799i	$-0.390866\pi$
$977$	21.0158	0.672354	0.336177	+	0.941799i	$0.390866\pi$
$978$	0	0
$979$	−0.115077	−0.00367788
$980$	0	0
$981$	−12.5181	−0.399671
$982$	0	0
$983$	27.5647	0.879176	0.439588	−	0.898200i	$-0.355124\pi$
$983$	27.5647	0.879176	0.439588	+	0.898200i	$0.355124\pi$
$984$	0	0
$985$	−24.4078	−0.777697
$986$	0	0
$987$	−6.91117	−0.219985
$988$	0	0
$989$	10.6858	0.339790
$990$	0	0
$991$	20.1537	0.640204	0.320102	−	0.947383i	$-0.396283\pi$
$991$	20.1537	0.640204	0.320102	+	0.947383i	$0.396283\pi$
$992$	0	0
$993$	−26.6388	−0.845356
$994$	0	0
$995$	3.85677	0.122268
$996$	0	0
$997$	14.9295	0.472821	0.236410	−	0.971653i	$-0.424029\pi$
$997$	14.9295	0.472821	0.236410	+	0.971653i	$0.424029\pi$
$998$	0	0
$999$	2.48929	0.0787576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.by.1.1		3
4.3	odd	2		345.2.a.j.1.1	✓	3
12.11	even	2		1035.2.a.n.1.3		3
20.3	even	4		1725.2.b.u.1174.6		6
20.7	even	4		1725.2.b.u.1174.1		6
20.19	odd	2		1725.2.a.bi.1.3		3
60.59	even	2		5175.2.a.br.1.1		3
92.91	even	2		7935.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.j.1.1	✓	3	4.3	odd	2
1035.2.a.n.1.3		3	12.11	even	2
1725.2.a.bi.1.3		3	20.19	odd	2
1725.2.b.u.1174.1		6	20.7	even	4
1725.2.b.u.1174.6		6	20.3	even	4
5175.2.a.br.1.1		3	60.59	even	2
5520.2.a.by.1.1		3	1.1	even	1	trivial
7935.2.a.u.1.1		3	92.91	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.48929 q^{7} +1.00000 q^{9} +1.14637 q^{11} +0.853635 q^{13} -1.00000 q^{15} +1.34292 q^{17} +3.83221 q^{19} +4.48929 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.02877 q^{29} -2.19656 q^{31} -1.14637 q^{33} -4.48929 q^{35} -2.48929 q^{37} -0.853635 q^{39} +11.3001 q^{41} -10.6858 q^{43} +1.00000 q^{45} -1.53948 q^{47} +13.1537 q^{49} -1.34292 q^{51} +4.02877 q^{53} +1.14637 q^{55} -3.83221 q^{57} +15.0073 q^{59} -5.83221 q^{61} -4.48929 q^{63} +0.853635 q^{65} +11.5682 q^{67} +1.00000 q^{69} -4.32150 q^{71} -13.1035 q^{73} -1.00000 q^{75} -5.14637 q^{77} -0.585462 q^{79} +1.00000 q^{81} -5.63565 q^{83} +1.34292 q^{85} +8.02877 q^{87} -0.100384 q^{89} -3.83221 q^{91} +2.19656 q^{93} +3.83221 q^{95} +11.5640 q^{97} +1.14637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 6 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 6 q^{7} + 3 q^{9} + 2 q^{11} + 4 q^{13} - 3 q^{15} - 2 q^{17} - 2 q^{19} + 6 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 6 q^{29} - 2 q^{31} - 2 q^{33} - 6 q^{35} - 4 q^{39} - 2 q^{41} - 20 q^{43} + 3 q^{45} + 6 q^{47} + 5 q^{49} + 2 q^{51} - 6 q^{53} + 2 q^{55} + 2 q^{57} + 12 q^{59} - 4 q^{61} - 6 q^{63} + 4 q^{65} + 6 q^{67} + 3 q^{69} + 8 q^{71} - 8 q^{73} - 3 q^{75} - 14 q^{77} + 4 q^{79} + 3 q^{81} - 8 q^{83} - 2 q^{85} + 6 q^{87} + 6 q^{89} + 2 q^{91} + 2 q^{93} - 2 q^{95} + 14 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 6 * q^7 + 3 * q^9 + 2 * q^11 + 4 * q^13 - 3 * q^15 - 2 * q^17 - 2 * q^19 + 6 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 6 * q^29 - 2 * q^31 - 2 * q^33 - 6 * q^35 - 4 * q^39 - 2 * q^41 - 20 * q^43 + 3 * q^45 + 6 * q^47 + 5 * q^49 + 2 * q^51 - 6 * q^53 + 2 * q^55 + 2 * q^57 + 12 * q^59 - 4 * q^61 - 6 * q^63 + 4 * q^65 + 6 * q^67 + 3 * q^69 + 8 * q^71 - 8 * q^73 - 3 * q^75 - 14 * q^77 + 4 * q^79 + 3 * q^81 - 8 * q^83 - 2 * q^85 + 6 * q^87 + 6 * q^89 + 2 * q^91 + 2 * q^93 - 2 * q^95 + 14 * q^97 + 2 * q^99

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