LMFDB - Embedded newform 8880.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8880,2,Mod(1,8880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8880, 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("8880.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8880.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:	$70.9071569949$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	8880.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.12311 q^{11} +5.68466 q^{13} -1.00000 q^{15} -5.00000 q^{17} -0.561553 q^{19} +1.00000 q^{21} +6.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.68466 q^{29} +3.56155 q^{31} +6.12311 q^{33} -1.00000 q^{35} +1.00000 q^{37} +5.68466 q^{39} +8.68466 q^{41} +12.6847 q^{43} -1.00000 q^{45} +8.00000 q^{47} -6.00000 q^{49} -5.00000 q^{51} +5.24621 q^{53} -6.12311 q^{55} -0.561553 q^{57} -4.24621 q^{59} -13.8078 q^{61} +1.00000 q^{63} -5.68466 q^{65} -13.1231 q^{67} +6.56155 q^{69} -1.12311 q^{71} -6.56155 q^{73} +1.00000 q^{75} +6.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} +7.43845 q^{83} +5.00000 q^{85} -6.68466 q^{87} +5.68466 q^{89} +5.68466 q^{91} +3.56155 q^{93} +0.561553 q^{95} +3.31534 q^{97} +6.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} - q^{13} - 2 q^{15} - 10 q^{17} + 3 q^{19} + 2 q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} - q^{29} + 3 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} - q^{39} + 5 q^{41} + 13 q^{43} - 2 q^{45} + 16 q^{47} - 12 q^{49} - 10 q^{51} - 6 q^{53} - 4 q^{55} + 3 q^{57} + 8 q^{59} - 7 q^{61} + 2 q^{63} + q^{65} - 18 q^{67} + 9 q^{69} + 6 q^{71} - 9 q^{73} + 2 q^{75} + 4 q^{77} - 12 q^{79} + 2 q^{81} + 19 q^{83} + 10 q^{85} - q^{87} - q^{89} - q^{91} + 3 q^{93} - 3 q^{95} + 19 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 - q^13 - 2 * q^15 - 10 * q^17 + 3 * q^19 + 2 * q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 - q^29 + 3 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 - q^39 + 5 * q^41 + 13 * q^43 - 2 * q^45 + 16 * q^47 - 12 * q^49 - 10 * q^51 - 6 * q^53 - 4 * q^55 + 3 * q^57 + 8 * q^59 - 7 * q^61 + 2 * q^63 + q^65 - 18 * q^67 + 9 * q^69 + 6 * q^71 - 9 * q^73 + 2 * q^75 + 4 * q^77 - 12 * q^79 + 2 * q^81 + 19 * q^83 + 10 * q^85 - q^87 - q^89 - q^91 + 3 * q^93 - 3 * q^95 + 19 * q^97 + 4 * 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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.12311	1.84619	0.923093	−	0.384577i	$-0.125653\pi$
$11$	6.12311	1.84619	0.923093	+	0.384577i	$0.125653\pi$
$12$	0	0
$13$	5.68466	1.57664	0.788320	−	0.615265i	$-0.210951\pi$
$13$	5.68466	1.57664	0.788320	+	0.615265i	$0.210951\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−0.561553	−0.128829	−0.0644145	−	0.997923i	$-0.520518\pi$
$19$	−0.561553	−0.128829	−0.0644145	+	0.997923i	$0.520518\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	3.56155	0.639674	0.319837	−	0.947473i	$-0.396372\pi$
$31$	3.56155	0.639674	0.319837	+	0.947473i	$0.396372\pi$
$32$	0	0
$33$	6.12311	1.06590
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	5.68466	0.910274
$40$	0	0
$41$	8.68466	1.35632	0.678158	−	0.734916i	$-0.262779\pi$
$41$	8.68466	1.35632	0.678158	+	0.734916i	$0.262779\pi$
$42$	0	0
$43$	12.6847	1.93439	0.967196	−	0.254031i	$-0.0817565\pi$
$43$	12.6847	1.93439	0.967196	+	0.254031i	$0.0817565\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−5.00000	−0.700140
$52$	0	0
$53$	5.24621	0.720623	0.360311	−	0.932832i	$-0.382670\pi$
$53$	5.24621	0.720623	0.360311	+	0.932832i	$0.382670\pi$
$54$	0	0
$55$	−6.12311	−0.825639
$56$	0	0
$57$	−0.561553	−0.0743795
$58$	0	0
$59$	−4.24621	−0.552810	−0.276405	−	0.961041i	$-0.589143\pi$
$59$	−4.24621	−0.552810	−0.276405	+	0.961041i	$0.589143\pi$
$60$	0	0
$61$	−13.8078	−1.76790	−0.883952	−	0.467579i	$-0.845126\pi$
$61$	−13.8078	−1.76790	−0.883952	+	0.467579i	$0.845126\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−5.68466	−0.705095
$66$	0	0
$67$	−13.1231	−1.60324	−0.801621	−	0.597832i	$-0.796029\pi$
$67$	−13.1231	−1.60324	−0.801621	+	0.597832i	$0.796029\pi$
$68$	0	0
$69$	6.56155	0.789918
$70$	0	0
$71$	−1.12311	−0.133288	−0.0666441	−	0.997777i	$-0.521229\pi$
$71$	−1.12311	−0.133288	−0.0666441	+	0.997777i	$0.521229\pi$
$72$	0	0
$73$	−6.56155	−0.767972	−0.383986	−	0.923339i	$-0.625449\pi$
$73$	−6.56155	−0.767972	−0.383986	+	0.923339i	$0.625449\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.12311	0.697793
$78$	0	0
$79$	−14.2462	−1.60282	−0.801412	−	0.598113i	$-0.795918\pi$
$79$	−14.2462	−1.60282	−0.801412	+	0.598113i	$0.795918\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.43845	0.816476	0.408238	−	0.912876i	$-0.366143\pi$
$83$	7.43845	0.816476	0.408238	+	0.912876i	$0.366143\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	0	0
$87$	−6.68466	−0.716671
$88$	0	0
$89$	5.68466	0.602573	0.301286	−	0.953534i	$-0.402584\pi$
$89$	5.68466	0.602573	0.301286	+	0.953534i	$0.402584\pi$
$90$	0	0
$91$	5.68466	0.595914
$92$	0	0
$93$	3.56155	0.369316
$94$	0	0
$95$	0.561553	0.0576141
$96$	0	0
$97$	3.31534	0.336622	0.168311	−	0.985734i	$-0.446169\pi$
$97$	3.31534	0.336622	0.168311	+	0.985734i	$0.446169\pi$
$98$	0	0
$99$	6.12311	0.615395
$100$	0	0
$101$	10.8769	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$102$	0	0
$103$	16.4924	1.62505	0.812523	−	0.582929i	$-0.198093\pi$
$103$	16.4924	1.62505	0.812523	+	0.582929i	$0.198093\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	10.8078	1.04483	0.522413	−	0.852693i	$-0.325032\pi$
$107$	10.8078	1.04483	0.522413	+	0.852693i	$0.325032\pi$
$108$	0	0
$109$	−2.12311	−0.203357	−0.101678	−	0.994817i	$-0.532421\pi$
$109$	−2.12311	−0.203357	−0.101678	+	0.994817i	$0.532421\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	0	0
$113$	−10.6847	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$113$	−10.6847	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$114$	0	0
$115$	−6.56155	−0.611868
$116$	0	0
$117$	5.68466	0.525547
$118$	0	0
$119$	−5.00000	−0.458349
$120$	0	0
$121$	26.4924	2.40840
$122$	0	0
$123$	8.68466	0.783069
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.5616	−1.29213	−0.646064	−	0.763283i	$-0.723586\pi$
$127$	−14.5616	−1.29213	−0.646064	+	0.763283i	$0.723586\pi$
$128$	0	0
$129$	12.6847	1.11682
$130$	0	0
$131$	−7.12311	−0.622349	−0.311174	−	0.950353i	$-0.600722\pi$
$131$	−7.12311	−0.622349	−0.311174	+	0.950353i	$0.600722\pi$
$132$	0	0
$133$	−0.561553	−0.0486928
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−4.24621	−0.362778	−0.181389	−	0.983411i	$-0.558059\pi$
$137$	−4.24621	−0.362778	−0.181389	+	0.983411i	$0.558059\pi$
$138$	0	0
$139$	−13.8078	−1.17116	−0.585580	−	0.810615i	$-0.699133\pi$
$139$	−13.8078	−1.17116	−0.585580	+	0.810615i	$0.699133\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	34.8078	2.91077
$144$	0	0
$145$	6.68466	0.555131
$146$	0	0
$147$	−6.00000	−0.494872
$148$	0	0
$149$	7.75379	0.635215	0.317608	−	0.948222i	$-0.397121\pi$
$149$	7.75379	0.635215	0.317608	+	0.948222i	$0.397121\pi$
$150$	0	0
$151$	15.4384	1.25636	0.628182	−	0.778067i	$-0.283800\pi$
$151$	15.4384	1.25636	0.628182	+	0.778067i	$0.283800\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−3.56155	−0.286071
$156$	0	0
$157$	−18.0540	−1.44086	−0.720432	−	0.693526i	$-0.756056\pi$
$157$	−18.0540	−1.44086	−0.720432	+	0.693526i	$0.756056\pi$
$158$	0	0
$159$	5.24621	0.416052
$160$	0	0
$161$	6.56155	0.517123
$162$	0	0
$163$	−19.4924	−1.52676	−0.763382	−	0.645947i	$-0.776463\pi$
$163$	−19.4924	−1.52676	−0.763382	+	0.645947i	$0.776463\pi$
$164$	0	0
$165$	−6.12311	−0.476683
$166$	0	0
$167$	−19.0540	−1.47444	−0.737220	−	0.675652i	$-0.763862\pi$
$167$	−19.0540	−1.47444	−0.737220	+	0.675652i	$0.763862\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	0	0
$171$	−0.561553	−0.0429430
$172$	0	0
$173$	−2.12311	−0.161417	−0.0807084	−	0.996738i	$-0.525718\pi$
$173$	−2.12311	−0.161417	−0.0807084	+	0.996738i	$0.525718\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.24621	−0.319165
$178$	0	0
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	0	0
$183$	−13.8078	−1.02070
$184$	0	0
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	−30.6155	−2.23883
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−14.3693	−1.03973	−0.519864	−	0.854249i	$-0.674017\pi$
$191$	−14.3693	−1.03973	−0.519864	+	0.854249i	$0.674017\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	−5.68466	−0.407087
$196$	0	0
$197$	−2.31534	−0.164961	−0.0824806	−	0.996593i	$-0.526284\pi$
$197$	−2.31534	−0.164961	−0.0824806	+	0.996593i	$0.526284\pi$
$198$	0	0
$199$	16.4924	1.16912	0.584558	−	0.811352i	$-0.301268\pi$
$199$	16.4924	1.16912	0.584558	+	0.811352i	$0.301268\pi$
$200$	0	0
$201$	−13.1231	−0.925633
$202$	0	0
$203$	−6.68466	−0.469171
$204$	0	0
$205$	−8.68466	−0.606563
$206$	0	0
$207$	6.56155	0.456059
$208$	0	0
$209$	−3.43845	−0.237842
$210$	0	0
$211$	−4.43845	−0.305555	−0.152778	−	0.988261i	$-0.548822\pi$
$211$	−4.43845	−0.305555	−0.152778	+	0.988261i	$0.548822\pi$
$212$	0	0
$213$	−1.12311	−0.0769539
$214$	0	0
$215$	−12.6847	−0.865087
$216$	0	0
$217$	3.56155	0.241774
$218$	0	0
$219$	−6.56155	−0.443389
$220$	0	0
$221$	−28.4233	−1.91196
$222$	0	0
$223$	19.8078	1.32643	0.663213	−	0.748431i	$-0.269192\pi$
$223$	19.8078	1.32643	0.663213	+	0.748431i	$0.269192\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	10.9309	0.725507	0.362754	−	0.931885i	$-0.381837\pi$
$227$	10.9309	0.725507	0.362754	+	0.931885i	$0.381837\pi$
$228$	0	0
$229$	−5.12311	−0.338544	−0.169272	−	0.985569i	$-0.554142\pi$
$229$	−5.12311	−0.338544	−0.169272	+	0.985569i	$0.554142\pi$
$230$	0	0
$231$	6.12311	0.402871
$232$	0	0
$233$	11.3693	0.744829	0.372414	−	0.928067i	$-0.378530\pi$
$233$	11.3693	0.744829	0.372414	+	0.928067i	$0.378530\pi$
$234$	0	0
$235$	−8.00000	−0.521862
$236$	0	0
$237$	−14.2462	−0.925391
$238$	0	0
$239$	−7.80776	−0.505042	−0.252521	−	0.967591i	$-0.581260\pi$
$239$	−7.80776	−0.505042	−0.252521	+	0.967591i	$0.581260\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−3.19224	−0.203117
$248$	0	0
$249$	7.43845	0.471392
$250$	0	0
$251$	−25.6155	−1.61684	−0.808419	−	0.588608i	$-0.799676\pi$
$251$	−25.6155	−1.61684	−0.808419	+	0.588608i	$0.799676\pi$
$252$	0	0
$253$	40.1771	2.52591
$254$	0	0
$255$	5.00000	0.313112
$256$	0	0
$257$	10.8078	0.674170	0.337085	−	0.941474i	$-0.390559\pi$
$257$	10.8078	0.674170	0.337085	+	0.941474i	$0.390559\pi$
$258$	0	0
$259$	1.00000	0.0621370
$260$	0	0
$261$	−6.68466	−0.413770
$262$	0	0
$263$	6.43845	0.397012	0.198506	−	0.980100i	$-0.436391\pi$
$263$	6.43845	0.397012	0.198506	+	0.980100i	$0.436391\pi$
$264$	0	0
$265$	−5.24621	−0.322272
$266$	0	0
$267$	5.68466	0.347895
$268$	0	0
$269$	−16.8078	−1.02479	−0.512394	−	0.858751i	$-0.671241\pi$
$269$	−16.8078	−1.02479	−0.512394	+	0.858751i	$0.671241\pi$
$270$	0	0
$271$	16.4924	1.00184	0.500922	−	0.865493i	$-0.332994\pi$
$271$	16.4924	1.00184	0.500922	+	0.865493i	$0.332994\pi$
$272$	0	0
$273$	5.68466	0.344051
$274$	0	0
$275$	6.12311	0.369237
$276$	0	0
$277$	27.6847	1.66341	0.831705	−	0.555218i	$-0.187365\pi$
$277$	27.6847	1.66341	0.831705	+	0.555218i	$0.187365\pi$
$278$	0	0
$279$	3.56155	0.213225
$280$	0	0
$281$	22.1771	1.32297	0.661487	−	0.749957i	$-0.269926\pi$
$281$	22.1771	1.32297	0.661487	+	0.749957i	$0.269926\pi$
$282$	0	0
$283$	−9.93087	−0.590329	−0.295164	−	0.955446i	$-0.595374\pi$
$283$	−9.93087	−0.590329	−0.295164	+	0.955446i	$0.595374\pi$
$284$	0	0
$285$	0.561553	0.0332635
$286$	0	0
$287$	8.68466	0.512639
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	3.31534	0.194349
$292$	0	0
$293$	−16.6155	−0.970690	−0.485345	−	0.874323i	$-0.661306\pi$
$293$	−16.6155	−0.970690	−0.485345	+	0.874323i	$0.661306\pi$
$294$	0	0
$295$	4.24621	0.247224
$296$	0	0
$297$	6.12311	0.355299
$298$	0	0
$299$	37.3002	2.15713
$300$	0	0
$301$	12.6847	0.731132
$302$	0	0
$303$	10.8769	0.624861
$304$	0	0
$305$	13.8078	0.790630
$306$	0	0
$307$	0.246211	0.0140520	0.00702601	−	0.999975i	$-0.497764\pi$
$307$	0.246211	0.0140520	0.00702601	+	0.999975i	$0.497764\pi$
$308$	0	0
$309$	16.4924	0.938221
$310$	0	0
$311$	21.1771	1.20084	0.600421	−	0.799684i	$-0.295000\pi$
$311$	21.1771	1.20084	0.600421	+	0.799684i	$0.295000\pi$
$312$	0	0
$313$	−24.8769	−1.40613	−0.703063	−	0.711128i	$-0.748185\pi$
$313$	−24.8769	−1.40613	−0.703063	+	0.711128i	$0.748185\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	16.9309	0.950932	0.475466	−	0.879734i	$-0.342279\pi$
$317$	16.9309	0.950932	0.475466	+	0.879734i	$0.342279\pi$
$318$	0	0
$319$	−40.9309	−2.29169
$320$	0	0
$321$	10.8078	0.603231
$322$	0	0
$323$	2.80776	0.156228
$324$	0	0
$325$	5.68466	0.315328
$326$	0	0
$327$	−2.12311	−0.117408
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	1.00000	0.0547997
$334$	0	0
$335$	13.1231	0.716992
$336$	0	0
$337$	4.56155	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$337$	4.56155	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$338$	0	0
$339$	−10.6847	−0.580311
$340$	0	0
$341$	21.8078	1.18096
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−6.56155	−0.353262
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	14.2462	0.762582	0.381291	−	0.924455i	$-0.375480\pi$
$349$	14.2462	0.762582	0.381291	+	0.924455i	$0.375480\pi$
$350$	0	0
$351$	5.68466	0.303425
$352$	0	0
$353$	−29.8078	−1.58651	−0.793254	−	0.608891i	$-0.791615\pi$
$353$	−29.8078	−1.58651	−0.793254	+	0.608891i	$0.791615\pi$
$354$	0	0
$355$	1.12311	0.0596083
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	20.8769	1.10184	0.550920	−	0.834558i	$-0.314277\pi$
$359$	20.8769	1.10184	0.550920	+	0.834558i	$0.314277\pi$
$360$	0	0
$361$	−18.6847	−0.983403
$362$	0	0
$363$	26.4924	1.39049
$364$	0	0
$365$	6.56155	0.343447
$366$	0	0
$367$	17.0000	0.887393	0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000	0.887393	0.443696	+	0.896177i	$0.353667\pi$
$368$	0	0
$369$	8.68466	0.452105
$370$	0	0
$371$	5.24621	0.272370
$372$	0	0
$373$	−30.4924	−1.57884	−0.789419	−	0.613855i	$-0.789618\pi$
$373$	−30.4924	−1.57884	−0.789419	+	0.613855i	$0.789618\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−38.0000	−1.95710
$378$	0	0
$379$	2.87689	0.147776	0.0738881	−	0.997267i	$-0.476459\pi$
$379$	2.87689	0.147776	0.0738881	+	0.997267i	$0.476459\pi$
$380$	0	0
$381$	−14.5616	−0.746011
$382$	0	0
$383$	27.6847	1.41462	0.707310	−	0.706904i	$-0.249909\pi$
$383$	27.6847	1.41462	0.707310	+	0.706904i	$0.249909\pi$
$384$	0	0
$385$	−6.12311	−0.312062
$386$	0	0
$387$	12.6847	0.644797
$388$	0	0
$389$	20.9309	1.06124	0.530619	−	0.847611i	$-0.321960\pi$
$389$	20.9309	1.06124	0.530619	+	0.847611i	$0.321960\pi$
$390$	0	0
$391$	−32.8078	−1.65916
$392$	0	0
$393$	−7.12311	−0.359313
$394$	0	0
$395$	14.2462	0.716805
$396$	0	0
$397$	27.1231	1.36127	0.680635	−	0.732623i	$-0.261704\pi$
$397$	27.1231	1.36127	0.680635	+	0.732623i	$0.261704\pi$
$398$	0	0
$399$	−0.561553	−0.0281128
$400$	0	0
$401$	−14.8078	−0.739464	−0.369732	−	0.929138i	$-0.620551\pi$
$401$	−14.8078	−0.739464	−0.369732	+	0.929138i	$0.620551\pi$
$402$	0	0
$403$	20.2462	1.00854
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.12311	0.303511
$408$	0	0
$409$	−0.876894	−0.0433596	−0.0216798	−	0.999765i	$-0.506901\pi$
$409$	−0.876894	−0.0433596	−0.0216798	+	0.999765i	$0.506901\pi$
$410$	0	0
$411$	−4.24621	−0.209450
$412$	0	0
$413$	−4.24621	−0.208942
$414$	0	0
$415$	−7.43845	−0.365139
$416$	0	0
$417$	−13.8078	−0.676169
$418$	0	0
$419$	10.5616	0.515966	0.257983	−	0.966150i	$-0.416942\pi$
$419$	10.5616	0.515966	0.257983	+	0.966150i	$0.416942\pi$
$420$	0	0
$421$	30.9848	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$421$	30.9848	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	−5.00000	−0.242536
$426$	0	0
$427$	−13.8078	−0.668205
$428$	0	0
$429$	34.8078	1.68053
$430$	0	0
$431$	29.0000	1.39688	0.698440	−	0.715668i	$-0.253878\pi$
$431$	29.0000	1.39688	0.698440	+	0.715668i	$0.253878\pi$
$432$	0	0
$433$	−30.4233	−1.46205	−0.731025	−	0.682351i	$-0.760958\pi$
$433$	−30.4233	−1.46205	−0.731025	+	0.682351i	$0.760958\pi$
$434$	0	0
$435$	6.68466	0.320505
$436$	0	0
$437$	−3.68466	−0.176261
$438$	0	0
$439$	14.0540	0.670760	0.335380	−	0.942083i	$-0.391135\pi$
$439$	14.0540	0.670760	0.335380	+	0.942083i	$0.391135\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	−33.8617	−1.60882	−0.804410	−	0.594075i	$-0.797518\pi$
$443$	−33.8617	−1.60882	−0.804410	+	0.594075i	$0.797518\pi$
$444$	0	0
$445$	−5.68466	−0.269479
$446$	0	0
$447$	7.75379	0.366742
$448$	0	0
$449$	−1.61553	−0.0762415	−0.0381207	−	0.999273i	$-0.512137\pi$
$449$	−1.61553	−0.0762415	−0.0381207	+	0.999273i	$0.512137\pi$
$450$	0	0
$451$	53.1771	2.50401
$452$	0	0
$453$	15.4384	0.725362
$454$	0	0
$455$	−5.68466	−0.266501
$456$	0	0
$457$	−30.9309	−1.44689	−0.723443	−	0.690385i	$-0.757441\pi$
$457$	−30.9309	−1.44689	−0.723443	+	0.690385i	$0.757441\pi$
$458$	0	0
$459$	−5.00000	−0.233380
$460$	0	0
$461$	0.438447	0.0204205	0.0102103	−	0.999948i	$-0.496750\pi$
$461$	0.438447	0.0204205	0.0102103	+	0.999948i	$0.496750\pi$
$462$	0	0
$463$	−4.87689	−0.226649	−0.113324	−	0.993558i	$-0.536150\pi$
$463$	−4.87689	−0.226649	−0.113324	+	0.993558i	$0.536150\pi$
$464$	0	0
$465$	−3.56155	−0.165163
$466$	0	0
$467$	−2.43845	−0.112838	−0.0564189	−	0.998407i	$-0.517968\pi$
$467$	−2.43845	−0.112838	−0.0564189	+	0.998407i	$0.517968\pi$
$468$	0	0
$469$	−13.1231	−0.605969
$470$	0	0
$471$	−18.0540	−0.831883
$472$	0	0
$473$	77.6695	3.57125
$474$	0	0
$475$	−0.561553	−0.0257658
$476$	0	0
$477$	5.24621	0.240208
$478$	0	0
$479$	6.56155	0.299805	0.149903	−	0.988701i	$-0.452104\pi$
$479$	6.56155	0.299805	0.149903	+	0.988701i	$0.452104\pi$
$480$	0	0
$481$	5.68466	0.259198
$482$	0	0
$483$	6.56155	0.298561
$484$	0	0
$485$	−3.31534	−0.150542
$486$	0	0
$487$	−16.7386	−0.758500	−0.379250	−	0.925294i	$-0.623818\pi$
$487$	−16.7386	−0.758500	−0.379250	+	0.925294i	$0.623818\pi$
$488$	0	0
$489$	−19.4924	−0.881478
$490$	0	0
$491$	12.1771	0.549544	0.274772	−	0.961509i	$-0.411398\pi$
$491$	12.1771	0.549544	0.274772	+	0.961509i	$0.411398\pi$
$492$	0	0
$493$	33.4233	1.50531
$494$	0	0
$495$	−6.12311	−0.275213
$496$	0	0
$497$	−1.12311	−0.0503782
$498$	0	0
$499$	−32.8078	−1.46868	−0.734339	−	0.678783i	$-0.762508\pi$
$499$	−32.8078	−1.46868	−0.734339	+	0.678783i	$0.762508\pi$
$500$	0	0
$501$	−19.0540	−0.851269
$502$	0	0
$503$	18.2462	0.813558	0.406779	−	0.913527i	$-0.366652\pi$
$503$	18.2462	0.813558	0.406779	+	0.913527i	$0.366652\pi$
$504$	0	0
$505$	−10.8769	−0.484015
$506$	0	0
$507$	19.3153	0.857824
$508$	0	0
$509$	23.0540	1.02185	0.510925	−	0.859625i	$-0.329303\pi$
$509$	23.0540	1.02185	0.510925	+	0.859625i	$0.329303\pi$
$510$	0	0
$511$	−6.56155	−0.290266
$512$	0	0
$513$	−0.561553	−0.0247932
$514$	0	0
$515$	−16.4924	−0.726743
$516$	0	0
$517$	48.9848	2.15435
$518$	0	0
$519$	−2.12311	−0.0931940
$520$	0	0
$521$	−18.4384	−0.807803	−0.403902	−	0.914802i	$-0.632346\pi$
$521$	−18.4384	−0.807803	−0.403902	+	0.914802i	$0.632346\pi$
$522$	0	0
$523$	−30.7386	−1.34411	−0.672053	−	0.740503i	$-0.734587\pi$
$523$	−30.7386	−1.34411	−0.672053	+	0.740503i	$0.734587\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−17.8078	−0.775718
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	−4.24621	−0.184270
$532$	0	0
$533$	49.3693	2.13842
$534$	0	0
$535$	−10.8078	−0.467260
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	−36.7386	−1.58244
$540$	0	0
$541$	−20.4233	−0.878066	−0.439033	−	0.898471i	$-0.644679\pi$
$541$	−20.4233	−0.878066	−0.439033	+	0.898471i	$0.644679\pi$
$542$	0	0
$543$	20.2462	0.868848
$544$	0	0
$545$	2.12311	0.0909439
$546$	0	0
$547$	39.7386	1.69910	0.849551	−	0.527507i	$-0.176873\pi$
$547$	39.7386	1.69910	0.849551	+	0.527507i	$0.176873\pi$
$548$	0	0
$549$	−13.8078	−0.589301
$550$	0	0
$551$	3.75379	0.159917
$552$	0	0
$553$	−14.2462	−0.605811
$554$	0	0
$555$	−1.00000	−0.0424476
$556$	0	0
$557$	7.61553	0.322680	0.161340	−	0.986899i	$-0.448418\pi$
$557$	7.61553	0.322680	0.161340	+	0.986899i	$0.448418\pi$
$558$	0	0
$559$	72.1080	3.04984
$560$	0	0
$561$	−30.6155	−1.29259
$562$	0	0
$563$	−4.93087	−0.207811	−0.103906	−	0.994587i	$-0.533134\pi$
$563$	−4.93087	−0.207811	−0.103906	+	0.994587i	$0.533134\pi$
$564$	0	0
$565$	10.6847	0.449507
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	2.31534	0.0970642	0.0485321	−	0.998822i	$-0.484546\pi$
$569$	2.31534	0.0970642	0.0485321	+	0.998822i	$0.484546\pi$
$570$	0	0
$571$	5.80776	0.243047	0.121524	−	0.992589i	$-0.461222\pi$
$571$	5.80776	0.243047	0.121524	+	0.992589i	$0.461222\pi$
$572$	0	0
$573$	−14.3693	−0.600287
$574$	0	0
$575$	6.56155	0.273636
$576$	0	0
$577$	−14.4924	−0.603327	−0.301664	−	0.953414i	$-0.597542\pi$
$577$	−14.4924	−0.603327	−0.301664	+	0.953414i	$0.597542\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	7.43845	0.308599
$582$	0	0
$583$	32.1231	1.33040
$584$	0	0
$585$	−5.68466	−0.235032
$586$	0	0
$587$	12.3002	0.507683	0.253842	−	0.967246i	$-0.418306\pi$
$587$	12.3002	0.507683	0.253842	+	0.967246i	$0.418306\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−2.31534	−0.0952404
$592$	0	0
$593$	4.87689	0.200270	0.100135	−	0.994974i	$-0.468073\pi$
$593$	4.87689	0.200270	0.100135	+	0.994974i	$0.468073\pi$
$594$	0	0
$595$	5.00000	0.204980
$596$	0	0
$597$	16.4924	0.674990
$598$	0	0
$599$	−26.7386	−1.09251	−0.546255	−	0.837619i	$-0.683947\pi$
$599$	−26.7386	−1.09251	−0.546255	+	0.837619i	$0.683947\pi$
$600$	0	0
$601$	−42.8617	−1.74837	−0.874183	−	0.485596i	$-0.838603\pi$
$601$	−42.8617	−1.74837	−0.874183	+	0.485596i	$0.838603\pi$
$602$	0	0
$603$	−13.1231	−0.534414
$604$	0	0
$605$	−26.4924	−1.07707
$606$	0	0
$607$	26.4924	1.07529	0.537647	−	0.843170i	$-0.319313\pi$
$607$	26.4924	1.07529	0.537647	+	0.843170i	$0.319313\pi$
$608$	0	0
$609$	−6.68466	−0.270876
$610$	0	0
$611$	45.4773	1.83981
$612$	0	0
$613$	8.43845	0.340826	0.170413	−	0.985373i	$-0.445490\pi$
$613$	8.43845	0.340826	0.170413	+	0.985373i	$0.445490\pi$
$614$	0	0
$615$	−8.68466	−0.350199
$616$	0	0
$617$	14.4924	0.583443	0.291721	−	0.956503i	$-0.405772\pi$
$617$	14.4924	0.583443	0.291721	+	0.956503i	$0.405772\pi$
$618$	0	0
$619$	35.8078	1.43924	0.719618	−	0.694370i	$-0.244317\pi$
$619$	35.8078	1.43924	0.719618	+	0.694370i	$0.244317\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	0	0
$623$	5.68466	0.227751
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.43845	−0.137318
$628$	0	0
$629$	−5.00000	−0.199363
$630$	0	0
$631$	−15.3153	−0.609694	−0.304847	−	0.952401i	$-0.598605\pi$
$631$	−15.3153	−0.609694	−0.304847	+	0.952401i	$0.598605\pi$
$632$	0	0
$633$	−4.43845	−0.176412
$634$	0	0
$635$	14.5616	0.577858
$636$	0	0
$637$	−34.1080	−1.35141
$638$	0	0
$639$	−1.12311	−0.0444294
$640$	0	0
$641$	1.94602	0.0768634	0.0384317	−	0.999261i	$-0.487764\pi$
$641$	1.94602	0.0768634	0.0384317	+	0.999261i	$0.487764\pi$
$642$	0	0
$643$	34.1231	1.34568	0.672842	−	0.739786i	$-0.265073\pi$
$643$	34.1231	1.34568	0.672842	+	0.739786i	$0.265073\pi$
$644$	0	0
$645$	−12.6847	−0.499458
$646$	0	0
$647$	36.8078	1.44706	0.723531	−	0.690292i	$-0.242518\pi$
$647$	36.8078	1.44706	0.723531	+	0.690292i	$0.242518\pi$
$648$	0	0
$649$	−26.0000	−1.02059
$650$	0	0
$651$	3.56155	0.139588
$652$	0	0
$653$	−28.4924	−1.11499	−0.557497	−	0.830179i	$-0.688238\pi$
$653$	−28.4924	−1.11499	−0.557497	+	0.830179i	$0.688238\pi$
$654$	0	0
$655$	7.12311	0.278323
$656$	0	0
$657$	−6.56155	−0.255991
$658$	0	0
$659$	18.2462	0.710771	0.355386	−	0.934720i	$-0.384350\pi$
$659$	18.2462	0.710771	0.355386	+	0.934720i	$0.384350\pi$
$660$	0	0
$661$	32.8617	1.27817	0.639087	−	0.769135i	$-0.279312\pi$
$661$	32.8617	1.27817	0.639087	+	0.769135i	$0.279312\pi$
$662$	0	0
$663$	−28.4233	−1.10387
$664$	0	0
$665$	0.561553	0.0217761
$666$	0	0
$667$	−43.8617	−1.69833
$668$	0	0
$669$	19.8078	0.765812
$670$	0	0
$671$	−84.5464	−3.26388
$672$	0	0
$673$	−18.5616	−0.715495	−0.357748	−	0.933818i	$-0.616455\pi$
$673$	−18.5616	−0.715495	−0.357748	+	0.933818i	$0.616455\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	35.9309	1.38094	0.690468	−	0.723363i	$-0.257405\pi$
$677$	35.9309	1.38094	0.690468	+	0.723363i	$0.257405\pi$
$678$	0	0
$679$	3.31534	0.127231
$680$	0	0
$681$	10.9309	0.418872
$682$	0	0
$683$	30.4384	1.16469	0.582347	−	0.812940i	$-0.302134\pi$
$683$	30.4384	1.16469	0.582347	+	0.812940i	$0.302134\pi$
$684$	0	0
$685$	4.24621	0.162239
$686$	0	0
$687$	−5.12311	−0.195459
$688$	0	0
$689$	29.8229	1.13616
$690$	0	0
$691$	5.31534	0.202205	0.101103	−	0.994876i	$-0.467763\pi$
$691$	5.31534	0.202205	0.101103	+	0.994876i	$0.467763\pi$
$692$	0	0
$693$	6.12311	0.232598
$694$	0	0
$695$	13.8078	0.523758
$696$	0	0
$697$	−43.4233	−1.64477
$698$	0	0
$699$	11.3693	0.430027
$700$	0	0
$701$	9.36932	0.353874	0.176937	−	0.984222i	$-0.443381\pi$
$701$	9.36932	0.353874	0.176937	+	0.984222i	$0.443381\pi$
$702$	0	0
$703$	−0.561553	−0.0211794
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	10.8769	0.409068
$708$	0	0
$709$	−4.50758	−0.169286	−0.0846428	−	0.996411i	$-0.526975\pi$
$709$	−4.50758	−0.169286	−0.0846428	+	0.996411i	$0.526975\pi$
$710$	0	0
$711$	−14.2462	−0.534275
$712$	0	0
$713$	23.3693	0.875188
$714$	0	0
$715$	−34.8078	−1.30174
$716$	0	0
$717$	−7.80776	−0.291586
$718$	0	0
$719$	2.00000	0.0745874	0.0372937	−	0.999304i	$-0.488126\pi$
$719$	2.00000	0.0745874	0.0372937	+	0.999304i	$0.488126\pi$
$720$	0	0
$721$	16.4924	0.614210
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−6.68466	−0.248262
$726$	0	0
$727$	−50.9848	−1.89092	−0.945462	−	0.325734i	$-0.894389\pi$
$727$	−50.9848	−1.89092	−0.945462	+	0.325734i	$0.894389\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−63.4233	−2.34580
$732$	0	0
$733$	−22.3002	−0.823676	−0.411838	−	0.911257i	$-0.635113\pi$
$733$	−22.3002	−0.823676	−0.411838	+	0.911257i	$0.635113\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	−80.3542	−2.95988
$738$	0	0
$739$	39.1771	1.44115	0.720576	−	0.693376i	$-0.243877\pi$
$739$	39.1771	1.44115	0.720576	+	0.693376i	$0.243877\pi$
$740$	0	0
$741$	−3.19224	−0.117270
$742$	0	0
$743$	−21.4233	−0.785944	−0.392972	−	0.919550i	$-0.628553\pi$
$743$	−21.4233	−0.785944	−0.392972	+	0.919550i	$0.628553\pi$
$744$	0	0
$745$	−7.75379	−0.284077
$746$	0	0
$747$	7.43845	0.272159
$748$	0	0
$749$	10.8078	0.394907
$750$	0	0
$751$	−22.7386	−0.829745	−0.414872	−	0.909880i	$-0.636174\pi$
$751$	−22.7386	−0.829745	−0.414872	+	0.909880i	$0.636174\pi$
$752$	0	0
$753$	−25.6155	−0.933482
$754$	0	0
$755$	−15.4384	−0.561863
$756$	0	0
$757$	−14.9460	−0.543223	−0.271611	−	0.962407i	$-0.587556\pi$
$757$	−14.9460	−0.543223	−0.271611	+	0.962407i	$0.587556\pi$
$758$	0	0
$759$	40.1771	1.45834
$760$	0	0
$761$	17.3153	0.627681	0.313840	−	0.949476i	$-0.398384\pi$
$761$	17.3153	0.627681	0.313840	+	0.949476i	$0.398384\pi$
$762$	0	0
$763$	−2.12311	−0.0768616
$764$	0	0
$765$	5.00000	0.180775
$766$	0	0
$767$	−24.1383	−0.871582
$768$	0	0
$769$	−9.50758	−0.342852	−0.171426	−	0.985197i	$-0.554837\pi$
$769$	−9.50758	−0.342852	−0.171426	+	0.985197i	$0.554837\pi$
$770$	0	0
$771$	10.8078	0.389232
$772$	0	0
$773$	−3.49242	−0.125614	−0.0628069	−	0.998026i	$-0.520005\pi$
$773$	−3.49242	−0.125614	−0.0628069	+	0.998026i	$0.520005\pi$
$774$	0	0
$775$	3.56155	0.127935
$776$	0	0
$777$	1.00000	0.0358748
$778$	0	0
$779$	−4.87689	−0.174733
$780$	0	0
$781$	−6.87689	−0.246075
$782$	0	0
$783$	−6.68466	−0.238890
$784$	0	0
$785$	18.0540	0.644374
$786$	0	0
$787$	−26.8769	−0.958058	−0.479029	−	0.877799i	$-0.659011\pi$
$787$	−26.8769	−0.958058	−0.479029	+	0.877799i	$0.659011\pi$
$788$	0	0
$789$	6.43845	0.229215
$790$	0	0
$791$	−10.6847	−0.379903
$792$	0	0
$793$	−78.4924	−2.78735
$794$	0	0
$795$	−5.24621	−0.186064
$796$	0	0
$797$	−25.6155	−0.907349	−0.453674	−	0.891168i	$-0.649887\pi$
$797$	−25.6155	−0.907349	−0.453674	+	0.891168i	$0.649887\pi$
$798$	0	0
$799$	−40.0000	−1.41510
$800$	0	0
$801$	5.68466	0.200858
$802$	0	0
$803$	−40.1771	−1.41782
$804$	0	0
$805$	−6.56155	−0.231264
$806$	0	0
$807$	−16.8078	−0.591661
$808$	0	0
$809$	−25.4384	−0.894368	−0.447184	−	0.894442i	$-0.647573\pi$
$809$	−25.4384	−0.894368	−0.447184	+	0.894442i	$0.647573\pi$
$810$	0	0
$811$	−2.73863	−0.0961664	−0.0480832	−	0.998843i	$-0.515311\pi$
$811$	−2.73863	−0.0961664	−0.0480832	+	0.998843i	$0.515311\pi$
$812$	0	0
$813$	16.4924	0.578415
$814$	0	0
$815$	19.4924	0.682790
$816$	0	0
$817$	−7.12311	−0.249206
$818$	0	0
$819$	5.68466	0.198638
$820$	0	0
$821$	23.4384	0.818007	0.409004	−	0.912533i	$-0.365876\pi$
$821$	23.4384	0.818007	0.409004	+	0.912533i	$0.365876\pi$
$822$	0	0
$823$	−31.5464	−1.09964	−0.549819	−	0.835284i	$-0.685303\pi$
$823$	−31.5464	−1.09964	−0.549819	+	0.835284i	$0.685303\pi$
$824$	0	0
$825$	6.12311	0.213179
$826$	0	0
$827$	10.4384	0.362980	0.181490	−	0.983393i	$-0.441908\pi$
$827$	10.4384	0.362980	0.181490	+	0.983393i	$0.441908\pi$
$828$	0	0
$829$	−22.1231	−0.768367	−0.384184	−	0.923257i	$-0.625517\pi$
$829$	−22.1231	−0.768367	−0.384184	+	0.923257i	$0.625517\pi$
$830$	0	0
$831$	27.6847	0.960370
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	19.0540	0.659390
$836$	0	0
$837$	3.56155	0.123105
$838$	0	0
$839$	−28.2462	−0.975168	−0.487584	−	0.873076i	$-0.662122\pi$
$839$	−28.2462	−0.975168	−0.487584	+	0.873076i	$0.662122\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	22.1771	0.763819
$844$	0	0
$845$	−19.3153	−0.664468
$846$	0	0
$847$	26.4924	0.910290
$848$	0	0
$849$	−9.93087	−0.340827
$850$	0	0
$851$	6.56155	0.224927
$852$	0	0
$853$	7.93087	0.271548	0.135774	−	0.990740i	$-0.456648\pi$
$853$	7.93087	0.271548	0.135774	+	0.990740i	$0.456648\pi$
$854$	0	0
$855$	0.561553	0.0192047
$856$	0	0
$857$	−8.75379	−0.299024	−0.149512	−	0.988760i	$-0.547770\pi$
$857$	−8.75379	−0.299024	−0.149512	+	0.988760i	$0.547770\pi$
$858$	0	0
$859$	2.80776	0.0957997	0.0478998	−	0.998852i	$-0.484747\pi$
$859$	2.80776	0.0957997	0.0478998	+	0.998852i	$0.484747\pi$
$860$	0	0
$861$	8.68466	0.295972
$862$	0	0
$863$	−20.1922	−0.687352	−0.343676	−	0.939088i	$-0.611672\pi$
$863$	−20.1922	−0.687352	−0.343676	+	0.939088i	$0.611672\pi$
$864$	0	0
$865$	2.12311	0.0721878
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−87.2311	−2.95911
$870$	0	0
$871$	−74.6004	−2.52774
$872$	0	0
$873$	3.31534	0.112207
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	46.9309	1.58474	0.792371	−	0.610039i	$-0.208846\pi$
$877$	46.9309	1.58474	0.792371	+	0.610039i	$0.208846\pi$
$878$	0	0
$879$	−16.6155	−0.560428
$880$	0	0
$881$	34.9309	1.17685	0.588425	−	0.808551i	$-0.299748\pi$
$881$	34.9309	1.17685	0.588425	+	0.808551i	$0.299748\pi$
$882$	0	0
$883$	14.3693	0.483566	0.241783	−	0.970330i	$-0.422268\pi$
$883$	14.3693	0.483566	0.241783	+	0.970330i	$0.422268\pi$
$884$	0	0
$885$	4.24621	0.142735
$886$	0	0
$887$	17.8078	0.597926	0.298963	−	0.954265i	$-0.403359\pi$
$887$	17.8078	0.597926	0.298963	+	0.954265i	$0.403359\pi$
$888$	0	0
$889$	−14.5616	−0.488379
$890$	0	0
$891$	6.12311	0.205132
$892$	0	0
$893$	−4.49242	−0.150333
$894$	0	0
$895$	−4.00000	−0.133705
$896$	0	0
$897$	37.3002	1.24542
$898$	0	0
$899$	−23.8078	−0.794033
$900$	0	0
$901$	−26.2311	−0.873883
$902$	0	0
$903$	12.6847	0.422119
$904$	0	0
$905$	−20.2462	−0.673007
$906$	0	0
$907$	−45.4384	−1.50876	−0.754379	−	0.656439i	$-0.772062\pi$
$907$	−45.4384	−1.50876	−0.754379	+	0.656439i	$0.772062\pi$
$908$	0	0
$909$	10.8769	0.360764
$910$	0	0
$911$	−20.4924	−0.678944	−0.339472	−	0.940616i	$-0.610248\pi$
$911$	−20.4924	−0.678944	−0.339472	+	0.940616i	$0.610248\pi$
$912$	0	0
$913$	45.5464	1.50737
$914$	0	0
$915$	13.8078	0.456471
$916$	0	0
$917$	−7.12311	−0.235226
$918$	0	0
$919$	−6.73863	−0.222287	−0.111144	−	0.993804i	$-0.535451\pi$
$919$	−6.73863	−0.222287	−0.111144	+	0.993804i	$0.535451\pi$
$920$	0	0
$921$	0.246211	0.00811294
$922$	0	0
$923$	−6.38447	−0.210147
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	16.4924	0.541682
$928$	0	0
$929$	−5.80776	−0.190547	−0.0952733	−	0.995451i	$-0.530372\pi$
$929$	−5.80776	−0.190547	−0.0952733	+	0.995451i	$0.530372\pi$
$930$	0	0
$931$	3.36932	0.110425
$932$	0	0
$933$	21.1771	0.693307
$934$	0	0
$935$	30.6155	1.00123
$936$	0	0
$937$	−0.630683	−0.0206035	−0.0103018	−	0.999947i	$-0.503279\pi$
$937$	−0.630683	−0.0206035	−0.0103018	+	0.999947i	$0.503279\pi$
$938$	0	0
$939$	−24.8769	−0.811827
$940$	0	0
$941$	−8.24621	−0.268819	−0.134409	−	0.990926i	$-0.542914\pi$
$941$	−8.24621	−0.268819	−0.134409	+	0.990926i	$0.542914\pi$
$942$	0	0
$943$	56.9848	1.85568
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−16.3002	−0.529685	−0.264842	−	0.964292i	$-0.585320\pi$
$947$	−16.3002	−0.529685	−0.264842	+	0.964292i	$0.585320\pi$
$948$	0	0
$949$	−37.3002	−1.21082
$950$	0	0
$951$	16.9309	0.549021
$952$	0	0
$953$	−1.61553	−0.0523321	−0.0261660	−	0.999658i	$-0.508330\pi$
$953$	−1.61553	−0.0523321	−0.0261660	+	0.999658i	$0.508330\pi$
$954$	0	0
$955$	14.3693	0.464980
$956$	0	0
$957$	−40.9309	−1.32311
$958$	0	0
$959$	−4.24621	−0.137117
$960$	0	0
$961$	−18.3153	−0.590817
$962$	0	0
$963$	10.8078	0.348275
$964$	0	0
$965$	14.0000	0.450676
$966$	0	0
$967$	−32.6307	−1.04933	−0.524666	−	0.851308i	$-0.675810\pi$
$967$	−32.6307	−1.04933	−0.524666	+	0.851308i	$0.675810\pi$
$968$	0	0
$969$	2.80776	0.0901984
$970$	0	0
$971$	15.8078	0.507295	0.253648	−	0.967297i	$-0.418370\pi$
$971$	15.8078	0.507295	0.253648	+	0.967297i	$0.418370\pi$
$972$	0	0
$973$	−13.8078	−0.442657
$974$	0	0
$975$	5.68466	0.182055
$976$	0	0
$977$	−12.5076	−0.400153	−0.200076	−	0.979780i	$-0.564119\pi$
$977$	−12.5076	−0.400153	−0.200076	+	0.979780i	$0.564119\pi$
$978$	0	0
$979$	34.8078	1.11246
$980$	0	0
$981$	−2.12311	−0.0677855
$982$	0	0
$983$	24.9309	0.795171	0.397586	−	0.917565i	$-0.369848\pi$
$983$	24.9309	0.795171	0.397586	+	0.917565i	$0.369848\pi$
$984$	0	0
$985$	2.31534	0.0737729
$986$	0	0
$987$	8.00000	0.254643
$988$	0	0
$989$	83.2311	2.64659
$990$	0	0
$991$	−3.42329	−0.108744	−0.0543722	−	0.998521i	$-0.517316\pi$
$991$	−3.42329	−0.108744	−0.0543722	+	0.998521i	$0.517316\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	−16.4924	−0.522845
$996$	0	0
$997$	19.3002	0.611243	0.305622	−	0.952153i	$-0.401136\pi$
$997$	19.3002	0.611243	0.305622	+	0.952153i	$0.401136\pi$
$998$	0	0
$999$	1.00000	0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8880.2.a.bp.1.2		2
4.3	odd	2		1110.2.a.r.1.1	✓	2
12.11	even	2		3330.2.a.bc.1.2		2
20.19	odd	2		5550.2.a.bw.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.a.r.1.1	✓	2	4.3	odd	2
3330.2.a.bc.1.2		2	12.11	even	2
5550.2.a.bw.1.1		2	20.19	odd	2
8880.2.a.bp.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8880 = 2^{4} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8880.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.12311 q^{11} +5.68466 q^{13} -1.00000 q^{15} -5.00000 q^{17} -0.561553 q^{19} +1.00000 q^{21} +6.56155 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.68466 q^{29} +3.56155 q^{31} +6.12311 q^{33} -1.00000 q^{35} +1.00000 q^{37} +5.68466 q^{39} +8.68466 q^{41} +12.6847 q^{43} -1.00000 q^{45} +8.00000 q^{47} -6.00000 q^{49} -5.00000 q^{51} +5.24621 q^{53} -6.12311 q^{55} -0.561553 q^{57} -4.24621 q^{59} -13.8078 q^{61} +1.00000 q^{63} -5.68466 q^{65} -13.1231 q^{67} +6.56155 q^{69} -1.12311 q^{71} -6.56155 q^{73} +1.00000 q^{75} +6.12311 q^{77} -14.2462 q^{79} +1.00000 q^{81} +7.43845 q^{83} +5.00000 q^{85} -6.68466 q^{87} +5.68466 q^{89} +5.68466 q^{91} +3.56155 q^{93} +0.561553 q^{95} +3.31534 q^{97} +6.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{11} - q^{13} - 2 q^{15} - 10 q^{17} + 3 q^{19} + 2 q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} - q^{29} + 3 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} - q^{39} + 5 q^{41} + 13 q^{43} - 2 q^{45} + 16 q^{47} - 12 q^{49} - 10 q^{51} - 6 q^{53} - 4 q^{55} + 3 q^{57} + 8 q^{59} - 7 q^{61} + 2 q^{63} + q^{65} - 18 q^{67} + 9 q^{69} + 6 q^{71} - 9 q^{73} + 2 q^{75} + 4 q^{77} - 12 q^{79} + 2 q^{81} + 19 q^{83} + 10 q^{85} - q^{87} - q^{89} - q^{91} + 3 q^{93} - 3 q^{95} + 19 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^11 - q^13 - 2 * q^15 - 10 * q^17 + 3 * q^19 + 2 * q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 - q^29 + 3 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 - q^39 + 5 * q^41 + 13 * q^43 - 2 * q^45 + 16 * q^47 - 12 * q^49 - 10 * q^51 - 6 * q^53 - 4 * q^55 + 3 * q^57 + 8 * q^59 - 7 * q^61 + 2 * q^63 + q^65 - 18 * q^67 + 9 * q^69 + 6 * q^71 - 9 * q^73 + 2 * q^75 + 4 * q^77 - 12 * q^79 + 2 * q^81 + 19 * q^83 + 10 * q^85 - q^87 - q^89 - q^91 + 3 * q^93 - 3 * q^95 + 19 * q^97 + 4 * q^99

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