LMFDB - Embedded newform 2240.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,2,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2240.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:	$17.8864900528$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.561553 q^{9} -5.56155 q^{11} +3.56155 q^{13} +1.56155 q^{15} -6.68466 q^{17} -4.00000 q^{19} -1.56155 q^{21} +1.00000 q^{25} -5.56155 q^{27} -0.438447 q^{29} -3.12311 q^{31} -8.68466 q^{33} -1.00000 q^{35} -9.12311 q^{37} +5.56155 q^{39} +8.24621 q^{41} -0.561553 q^{45} +2.43845 q^{47} +1.00000 q^{49} -10.4384 q^{51} -1.12311 q^{53} -5.56155 q^{55} -6.24621 q^{57} +2.24621 q^{59} -8.24621 q^{61} +0.561553 q^{63} +3.56155 q^{65} -14.2462 q^{67} +14.2462 q^{71} +8.24621 q^{73} +1.56155 q^{75} +5.56155 q^{77} -5.56155 q^{79} -7.00000 q^{81} -10.2462 q^{83} -6.68466 q^{85} -0.684658 q^{87} +0.246211 q^{89} -3.56155 q^{91} -4.87689 q^{93} -4.00000 q^{95} -6.68466 q^{97} +3.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	$ 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} - 7 q^{11} + 3 q^{13} - q^{15} - q^{17} - 8 q^{19} + q^{21} + 2 q^{25} - 7 q^{27} - 5 q^{29} + 2 q^{31} - 5 q^{33} - 2 q^{35} - 10 q^{37} + 7 q^{39} + 3 q^{45} + 9 q^{47} + 2 q^{49} - 25 q^{51} + 6 q^{53} - 7 q^{55} + 4 q^{57} - 12 q^{59} - 3 q^{63} + 3 q^{65} - 12 q^{67} + 12 q^{71} - q^{75} + 7 q^{77} - 7 q^{79} - 14 q^{81} - 4 q^{83} - q^{85} + 11 q^{87} - 16 q^{89} - 3 q^{91} - 18 q^{93} - 8 q^{95} - q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 - 7 * q^11 + 3 * q^13 - q^15 - q^17 - 8 * q^19 + q^21 + 2 * q^25 - 7 * q^27 - 5 * q^29 + 2 * q^31 - 5 * q^33 - 2 * q^35 - 10 * q^37 + 7 * q^39 + 3 * q^45 + 9 * q^47 + 2 * q^49 - 25 * q^51 + 6 * q^53 - 7 * q^55 + 4 * q^57 - 12 * q^59 - 3 * q^63 + 3 * q^65 - 12 * q^67 + 12 * q^71 - q^75 + 7 * q^77 - 7 * q^79 - 14 * q^81 - 4 * q^83 - q^85 + 11 * q^87 - 16 * q^89 - 3 * q^91 - 18 * q^93 - 8 * q^95 - 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−5.56155	−1.67687	−0.838436	−	0.545001i	$-0.816529\pi$
$11$	−5.56155	−1.67687	−0.838436	+	0.545001i	$0.816529\pi$
$12$	0	0
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	0	0
$15$	1.56155	0.403191
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−0.438447	−0.0814176	−0.0407088	−	0.999171i	$-0.512962\pi$
$29$	−0.438447	−0.0814176	−0.0407088	+	0.999171i	$0.512962\pi$
$30$	0	0
$31$	−3.12311	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$31$	−3.12311	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$32$	0	0
$33$	−8.68466	−1.51180
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−9.12311	−1.49983	−0.749915	−	0.661535i	$-0.769905\pi$
$37$	−9.12311	−1.49983	−0.749915	+	0.661535i	$0.769905\pi$
$38$	0	0
$39$	5.56155	0.890561
$40$	0	0
$41$	8.24621	1.28784	0.643921	−	0.765092i	$-0.277307\pi$
$41$	8.24621	1.28784	0.643921	+	0.765092i	$0.277307\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	2.43845	0.355684	0.177842	−	0.984059i	$-0.443088\pi$
$47$	2.43845	0.355684	0.177842	+	0.984059i	$0.443088\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−10.4384	−1.46167
$52$	0	0
$53$	−1.12311	−0.154270	−0.0771352	−	0.997021i	$-0.524577\pi$
$53$	−1.12311	−0.154270	−0.0771352	+	0.997021i	$0.524577\pi$
$54$	0	0
$55$	−5.56155	−0.749920
$56$	0	0
$57$	−6.24621	−0.827331
$58$	0	0
$59$	2.24621	0.292432	0.146216	−	0.989253i	$-0.453291\pi$
$59$	2.24621	0.292432	0.146216	+	0.989253i	$0.453291\pi$
$60$	0	0
$61$	−8.24621	−1.05582	−0.527910	−	0.849301i	$-0.677024\pi$
$61$	−8.24621	−1.05582	−0.527910	+	0.849301i	$0.677024\pi$
$62$	0	0
$63$	0.561553	0.0707490
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	−14.2462	−1.74045	−0.870226	−	0.492653i	$-0.836027\pi$
$67$	−14.2462	−1.74045	−0.870226	+	0.492653i	$0.836027\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	14.2462	1.69071	0.845357	−	0.534202i	$-0.179388\pi$
$71$	14.2462	1.69071	0.845357	+	0.534202i	$0.179388\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	5.56155	0.633798
$78$	0	0
$79$	−5.56155	−0.625724	−0.312862	−	0.949799i	$-0.601288\pi$
$79$	−5.56155	−0.625724	−0.312862	+	0.949799i	$0.601288\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−10.2462	−1.12467	−0.562334	−	0.826910i	$-0.690096\pi$
$83$	−10.2462	−1.12467	−0.562334	+	0.826910i	$0.690096\pi$
$84$	0	0
$85$	−6.68466	−0.725053
$86$	0	0
$87$	−0.684658	−0.0734031
$88$	0	0
$89$	0.246211	0.0260983	0.0130492	−	0.999915i	$-0.495846\pi$
$89$	0.246211	0.0260983	0.0130492	+	0.999915i	$0.495846\pi$
$90$	0	0
$91$	−3.56155	−0.373352
$92$	0	0
$93$	−4.87689	−0.505710
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−6.68466	−0.678724	−0.339362	−	0.940656i	$-0.610211\pi$
$97$	−6.68466	−0.678724	−0.339362	+	0.940656i	$0.610211\pi$
$98$	0	0
$99$	3.12311	0.313884
$100$	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
$103$	−13.5616	−1.33626	−0.668130	−	0.744045i	$-0.732905\pi$
$103$	−13.5616	−1.33626	−0.668130	+	0.744045i	$0.732905\pi$
$104$	0	0
$105$	−1.56155	−0.152392
$106$	0	0
$107$	9.36932	0.905766	0.452883	−	0.891570i	$-0.350396\pi$
$107$	9.36932	0.905766	0.452883	+	0.891570i	$0.350396\pi$
$108$	0	0
$109$	16.9309	1.62168	0.810842	−	0.585266i	$-0.199010\pi$
$109$	16.9309	1.62168	0.810842	+	0.585266i	$0.199010\pi$
$110$	0	0
$111$	−14.2462	−1.35219
$112$	0	0
$113$	19.3693	1.82211	0.911056	−	0.412283i	$-0.135268\pi$
$113$	19.3693	1.82211	0.911056	+	0.412283i	$0.135268\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	6.68466	0.612782
$120$	0	0
$121$	19.9309	1.81190
$122$	0	0
$123$	12.8769	1.16107
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.4924	1.10852	0.554262	−	0.832343i	$-0.313001\pi$
$127$	12.4924	1.10852	0.554262	+	0.832343i	$0.313001\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−5.56155	−0.478662
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−19.6155	−1.66377	−0.831884	−	0.554950i	$-0.812737\pi$
$139$	−19.6155	−1.66377	−0.831884	+	0.554950i	$0.812737\pi$
$140$	0	0
$141$	3.80776	0.320672
$142$	0	0
$143$	−19.8078	−1.65641
$144$	0	0
$145$	−0.438447	−0.0364111
$146$	0	0
$147$	1.56155	0.128795
$148$	0	0
$149$	−7.75379	−0.635215	−0.317608	−	0.948222i	$-0.602879\pi$
$149$	−7.75379	−0.635215	−0.317608	+	0.948222i	$0.602879\pi$
$150$	0	0
$151$	−13.5616	−1.10362	−0.551812	−	0.833969i	$-0.686063\pi$
$151$	−13.5616	−1.10362	−0.551812	+	0.833969i	$0.686063\pi$
$152$	0	0
$153$	3.75379	0.303476
$154$	0	0
$155$	−3.12311	−0.250854
$156$	0	0
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	0	0
$159$	−1.75379	−0.139084
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.3693	1.36047	0.680235	−	0.732994i	$-0.261878\pi$
$163$	17.3693	1.36047	0.680235	+	0.732994i	$0.261878\pi$
$164$	0	0
$165$	−8.68466	−0.676100
$166$	0	0
$167$	18.4384	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$167$	18.4384	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	2.24621	0.171772
$172$	0	0
$173$	4.93087	0.374887	0.187444	−	0.982275i	$-0.439980\pi$
$173$	4.93087	0.374887	0.187444	+	0.982275i	$0.439980\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	3.50758	0.263646
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−6.87689	−0.511156	−0.255578	−	0.966789i	$-0.582266\pi$
$181$	−6.87689	−0.511156	−0.255578	+	0.966789i	$0.582266\pi$
$182$	0	0
$183$	−12.8769	−0.951887
$184$	0	0
$185$	−9.12311	−0.670744
$186$	0	0
$187$	37.1771	2.71866
$188$	0	0
$189$	5.56155	0.404543
$190$	0	0
$191$	11.8078	0.854380	0.427190	−	0.904162i	$-0.359504\pi$
$191$	11.8078	0.854380	0.427190	+	0.904162i	$0.359504\pi$
$192$	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
$195$	5.56155	0.398271
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	1.75379	0.124323	0.0621614	−	0.998066i	$-0.480201\pi$
$199$	1.75379	0.124323	0.0621614	+	0.998066i	$0.480201\pi$
$200$	0	0
$201$	−22.2462	−1.56913
$202$	0	0
$203$	0.438447	0.0307730
$204$	0	0
$205$	8.24621	0.575940
$206$	0	0
$207$	0	0
$208$	0	0
$209$	22.2462	1.53880
$210$	0	0
$211$	−18.4384	−1.26936	−0.634678	−	0.772777i	$-0.718867\pi$
$211$	−18.4384	−1.26936	−0.634678	+	0.772777i	$0.718867\pi$
$212$	0	0
$213$	22.2462	1.52429
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.12311	0.212010
$218$	0	0
$219$	12.8769	0.870140
$220$	0	0
$221$	−23.8078	−1.60148
$222$	0	0
$223$	−21.5616	−1.44387	−0.721934	−	0.691962i	$-0.756747\pi$
$223$	−21.5616	−1.44387	−0.721934	+	0.691962i	$0.756747\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−18.9309	−1.25649	−0.628243	−	0.778017i	$-0.716226\pi$
$227$	−18.9309	−1.25649	−0.628243	+	0.778017i	$0.716226\pi$
$228$	0	0
$229$	20.2462	1.33791	0.668954	−	0.743304i	$-0.266742\pi$
$229$	20.2462	1.33791	0.668954	+	0.743304i	$0.266742\pi$
$230$	0	0
$231$	8.68466	0.571409
$232$	0	0
$233$	5.12311	0.335626	0.167813	−	0.985819i	$-0.446330\pi$
$233$	5.12311	0.335626	0.167813	+	0.985819i	$0.446330\pi$
$234$	0	0
$235$	2.43845	0.159067
$236$	0	0
$237$	−8.68466	−0.564129
$238$	0	0
$239$	−26.0540	−1.68529	−0.842646	−	0.538468i	$-0.819003\pi$
$239$	−26.0540	−1.68529	−0.842646	+	0.538468i	$0.819003\pi$
$240$	0	0
$241$	−23.3693	−1.50535	−0.752675	−	0.658392i	$-0.771237\pi$
$241$	−23.3693	−1.50535	−0.752675	+	0.658392i	$0.771237\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−14.2462	−0.906465
$248$	0	0
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−13.3693	−0.843864	−0.421932	−	0.906628i	$-0.638648\pi$
$251$	−13.3693	−0.843864	−0.421932	+	0.906628i	$0.638648\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−10.4384	−0.653681
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	9.12311	0.566882
$260$	0	0
$261$	0.246211	0.0152401
$262$	0	0
$263$	−3.12311	−0.192579	−0.0962895	−	0.995353i	$-0.530697\pi$
$263$	−3.12311	−0.192579	−0.0962895	+	0.995353i	$0.530697\pi$
$264$	0	0
$265$	−1.12311	−0.0689918
$266$	0	0
$267$	0.384472	0.0235293
$268$	0	0
$269$	12.2462	0.746665	0.373332	−	0.927698i	$-0.378215\pi$
$269$	12.2462	0.746665	0.373332	+	0.927698i	$0.378215\pi$
$270$	0	0
$271$	−4.87689	−0.296250	−0.148125	−	0.988969i	$-0.547324\pi$
$271$	−4.87689	−0.296250	−0.148125	+	0.988969i	$0.547324\pi$
$272$	0	0
$273$	−5.56155	−0.336600
$274$	0	0
$275$	−5.56155	−0.335374
$276$	0	0
$277$	−12.2462	−0.735804	−0.367902	−	0.929865i	$-0.619924\pi$
$277$	−12.2462	−0.735804	−0.367902	+	0.929865i	$0.619924\pi$
$278$	0	0
$279$	1.75379	0.104997
$280$	0	0
$281$	18.6847	1.11463	0.557317	−	0.830300i	$-0.311831\pi$
$281$	18.6847	1.11463	0.557317	+	0.830300i	$0.311831\pi$
$282$	0	0
$283$	14.4384	0.858277	0.429138	−	0.903239i	$-0.358817\pi$
$283$	14.4384	0.858277	0.429138	+	0.903239i	$0.358817\pi$
$284$	0	0
$285$	−6.24621	−0.369994
$286$	0	0
$287$	−8.24621	−0.486758
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	−10.4384	−0.611913
$292$	0	0
$293$	−13.8078	−0.806658	−0.403329	−	0.915055i	$-0.632147\pi$
$293$	−13.8078	−0.806658	−0.403329	+	0.915055i	$0.632147\pi$
$294$	0	0
$295$	2.24621	0.130779
$296$	0	0
$297$	30.9309	1.79479
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	16.9848	0.975754
$304$	0	0
$305$	−8.24621	−0.472177
$306$	0	0
$307$	20.6847	1.18054	0.590268	−	0.807207i	$-0.299022\pi$
$307$	20.6847	1.18054	0.590268	+	0.807207i	$0.299022\pi$
$308$	0	0
$309$	−21.1771	−1.20472
$310$	0	0
$311$	−6.24621	−0.354190	−0.177095	−	0.984194i	$-0.556670\pi$
$311$	−6.24621	−0.354190	−0.177095	+	0.984194i	$0.556670\pi$
$312$	0	0
$313$	−13.3153	−0.752628	−0.376314	−	0.926492i	$-0.622809\pi$
$313$	−13.3153	−0.752628	−0.376314	+	0.926492i	$0.622809\pi$
$314$	0	0
$315$	0.561553	0.0316399
$316$	0	0
$317$	−32.7386	−1.83878	−0.919392	−	0.393342i	$-0.871319\pi$
$317$	−32.7386	−1.83878	−0.919392	+	0.393342i	$0.871319\pi$
$318$	0	0
$319$	2.43845	0.136527
$320$	0	0
$321$	14.6307	0.816605
$322$	0	0
$323$	26.7386	1.48778
$324$	0	0
$325$	3.56155	0.197559
$326$	0	0
$327$	26.4384	1.46205
$328$	0	0
$329$	−2.43845	−0.134436
$330$	0	0
$331$	−1.75379	−0.0963969	−0.0481985	−	0.998838i	$-0.515348\pi$
$331$	−1.75379	−0.0963969	−0.0481985	+	0.998838i	$0.515348\pi$
$332$	0	0
$333$	5.12311	0.280744
$334$	0	0
$335$	−14.2462	−0.778354
$336$	0	0
$337$	8.24621	0.449200	0.224600	−	0.974451i	$-0.427892\pi$
$337$	8.24621	0.449200	0.224600	+	0.974451i	$0.427892\pi$
$338$	0	0
$339$	30.2462	1.64275
$340$	0	0
$341$	17.3693	0.940601
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.4924	−0.670628	−0.335314	−	0.942106i	$-0.608842\pi$
$347$	−12.4924	−0.670628	−0.335314	+	0.942106i	$0.608842\pi$
$348$	0	0
$349$	10.8769	0.582227	0.291113	−	0.956689i	$-0.405974\pi$
$349$	10.8769	0.582227	0.291113	+	0.956689i	$0.405974\pi$
$350$	0	0
$351$	−19.8078	−1.05726
$352$	0	0
$353$	4.05398	0.215771	0.107886	−	0.994163i	$-0.465592\pi$
$353$	4.05398	0.215771	0.107886	+	0.994163i	$0.465592\pi$
$354$	0	0
$355$	14.2462	0.756110
$356$	0	0
$357$	10.4384	0.552461
$358$	0	0
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	31.1231	1.63354
$364$	0	0
$365$	8.24621	0.431626
$366$	0	0
$367$	13.1771	0.687838	0.343919	−	0.938999i	$-0.388245\pi$
$367$	13.1771	0.687838	0.343919	+	0.938999i	$0.388245\pi$
$368$	0	0
$369$	−4.63068	−0.241064
$370$	0	0
$371$	1.12311	0.0583087
$372$	0	0
$373$	−34.4924	−1.78595	−0.892975	−	0.450106i	$-0.851386\pi$
$373$	−34.4924	−1.78595	−0.892975	+	0.450106i	$0.851386\pi$
$374$	0	0
$375$	1.56155	0.0806382
$376$	0	0
$377$	−1.56155	−0.0804241
$378$	0	0
$379$	4.49242	0.230760	0.115380	−	0.993321i	$-0.463191\pi$
$379$	4.49242	0.230760	0.115380	+	0.993321i	$0.463191\pi$
$380$	0	0
$381$	19.5076	0.999403
$382$	0	0
$383$	22.2462	1.13673	0.568364	−	0.822777i	$-0.307576\pi$
$383$	22.2462	1.13673	0.568364	+	0.822777i	$0.307576\pi$
$384$	0	0
$385$	5.56155	0.283443
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.8078	−1.30851	−0.654253	−	0.756276i	$-0.727017\pi$
$389$	−25.8078	−1.30851	−0.654253	+	0.756276i	$0.727017\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−31.2311	−1.57540
$394$	0	0
$395$	−5.56155	−0.279832
$396$	0	0
$397$	2.19224	0.110025	0.0550126	−	0.998486i	$-0.482480\pi$
$397$	2.19224	0.110025	0.0550126	+	0.998486i	$0.482480\pi$
$398$	0	0
$399$	6.24621	0.312702
$400$	0	0
$401$	−19.1771	−0.957658	−0.478829	−	0.877908i	$-0.658939\pi$
$401$	−19.1771	−0.957658	−0.478829	+	0.877908i	$0.658939\pi$
$402$	0	0
$403$	−11.1231	−0.554081
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	50.7386	2.51502
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	−9.36932	−0.462154
$412$	0	0
$413$	−2.24621	−0.110529
$414$	0	0
$415$	−10.2462	−0.502967
$416$	0	0
$417$	−30.6307	−1.49999
$418$	0	0
$419$	−8.49242	−0.414882	−0.207441	−	0.978248i	$-0.566513\pi$
$419$	−8.49242	−0.414882	−0.207441	+	0.978248i	$0.566513\pi$
$420$	0	0
$421$	−5.31534	−0.259054	−0.129527	−	0.991576i	$-0.541346\pi$
$421$	−5.31534	−0.259054	−0.129527	+	0.991576i	$0.541346\pi$
$422$	0	0
$423$	−1.36932	−0.0665785
$424$	0	0
$425$	−6.68466	−0.324254
$426$	0	0
$427$	8.24621	0.399062
$428$	0	0
$429$	−30.9309	−1.49336
$430$	0	0
$431$	1.06913	0.0514982	0.0257491	−	0.999668i	$-0.491803\pi$
$431$	1.06913	0.0514982	0.0257491	+	0.999668i	$0.491803\pi$
$432$	0	0
$433$	24.2462	1.16520	0.582599	−	0.812760i	$-0.302036\pi$
$433$	24.2462	1.16520	0.582599	+	0.812760i	$0.302036\pi$
$434$	0	0
$435$	−0.684658	−0.0328269
$436$	0	0
$437$	0	0
$438$	0	0
$439$	23.6155	1.12711	0.563554	−	0.826079i	$-0.309434\pi$
$439$	23.6155	1.12711	0.563554	+	0.826079i	$0.309434\pi$
$440$	0	0
$441$	−0.561553	−0.0267406
$442$	0	0
$443$	36.4924	1.73381	0.866904	−	0.498476i	$-0.166107\pi$
$443$	36.4924	1.73381	0.866904	+	0.498476i	$0.166107\pi$
$444$	0	0
$445$	0.246211	0.0116715
$446$	0	0
$447$	−12.1080	−0.572686
$448$	0	0
$449$	−30.3002	−1.42995	−0.714977	−	0.699148i	$-0.753563\pi$
$449$	−30.3002	−1.42995	−0.714977	+	0.699148i	$0.753563\pi$
$450$	0	0
$451$	−45.8617	−2.15954
$452$	0	0
$453$	−21.1771	−0.994986
$454$	0	0
$455$	−3.56155	−0.166968
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	0	0
$459$	37.1771	1.73528
$460$	0	0
$461$	23.3693	1.08842	0.544209	−	0.838950i	$-0.316830\pi$
$461$	23.3693	1.08842	0.544209	+	0.838950i	$0.316830\pi$
$462$	0	0
$463$	−15.6155	−0.725715	−0.362858	−	0.931845i	$-0.618199\pi$
$463$	−15.6155	−0.725715	−0.362858	+	0.931845i	$0.618199\pi$
$464$	0	0
$465$	−4.87689	−0.226161
$466$	0	0
$467$	10.9309	0.505820	0.252910	−	0.967490i	$-0.418612\pi$
$467$	10.9309	0.505820	0.252910	+	0.967490i	$0.418612\pi$
$468$	0	0
$469$	14.2462	0.657829
$470$	0	0
$471$	9.36932	0.431715
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0.630683	0.0288770
$478$	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
$481$	−32.4924	−1.48153
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.68466	−0.303535
$486$	0	0
$487$	4.87689	0.220993	0.110497	−	0.993877i	$-0.464756\pi$
$487$	4.87689	0.220993	0.110497	+	0.993877i	$0.464756\pi$
$488$	0	0
$489$	27.1231	1.22655
$490$	0	0
$491$	10.0540	0.453730	0.226865	−	0.973926i	$-0.427152\pi$
$491$	10.0540	0.453730	0.226865	+	0.973926i	$0.427152\pi$
$492$	0	0
$493$	2.93087	0.132000
$494$	0	0
$495$	3.12311	0.140373
$496$	0	0
$497$	−14.2462	−0.639030
$498$	0	0
$499$	−38.5464	−1.72557	−0.862787	−	0.505568i	$-0.831283\pi$
$499$	−38.5464	−1.72557	−0.862787	+	0.505568i	$0.831283\pi$
$500$	0	0
$501$	28.7926	1.28636
$502$	0	0
$503$	34.0540	1.51839	0.759196	−	0.650862i	$-0.225592\pi$
$503$	34.0540	1.51839	0.759196	+	0.650862i	$0.225592\pi$
$504$	0	0
$505$	10.8769	0.484015
$506$	0	0
$507$	−0.492423	−0.0218693
$508$	0	0
$509$	−38.4924	−1.70615	−0.853073	−	0.521791i	$-0.825264\pi$
$509$	−38.4924	−1.70615	−0.853073	+	0.521791i	$0.825264\pi$
$510$	0	0
$511$	−8.24621	−0.364791
$512$	0	0
$513$	22.2462	0.982194
$514$	0	0
$515$	−13.5616	−0.597593
$516$	0	0
$517$	−13.5616	−0.596436
$518$	0	0
$519$	7.69981	0.337984
$520$	0	0
$521$	−31.3693	−1.37431	−0.687157	−	0.726509i	$-0.741142\pi$
$521$	−31.3693	−1.37431	−0.687157	+	0.726509i	$0.741142\pi$
$522$	0	0
$523$	−16.4924	−0.721163	−0.360582	−	0.932728i	$-0.617422\pi$
$523$	−16.4924	−0.721163	−0.360582	+	0.932728i	$0.617422\pi$
$524$	0	0
$525$	−1.56155	−0.0681518
$526$	0	0
$527$	20.8769	0.909412
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−1.26137	−0.0547386
$532$	0	0
$533$	29.3693	1.27213
$534$	0	0
$535$	9.36932	0.405071
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.56155	−0.239553
$540$	0	0
$541$	7.56155	0.325097	0.162548	−	0.986701i	$-0.448029\pi$
$541$	7.56155	0.325097	0.162548	+	0.986701i	$0.448029\pi$
$542$	0	0
$543$	−10.7386	−0.460839
$544$	0	0
$545$	16.9309	0.725239
$546$	0	0
$547$	4.87689	0.208521	0.104260	−	0.994550i	$-0.466752\pi$
$547$	4.87689	0.208521	0.104260	+	0.994550i	$0.466752\pi$
$548$	0	0
$549$	4.63068	0.197633
$550$	0	0
$551$	1.75379	0.0747139
$552$	0	0
$553$	5.56155	0.236501
$554$	0	0
$555$	−14.2462	−0.604718
$556$	0	0
$557$	9.61553	0.407423	0.203712	−	0.979031i	$-0.434700\pi$
$557$	9.61553	0.407423	0.203712	+	0.979031i	$0.434700\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	58.0540	2.45104
$562$	0	0
$563$	−36.9848	−1.55873	−0.779363	−	0.626573i	$-0.784457\pi$
$563$	−36.9848	−1.55873	−0.779363	+	0.626573i	$0.784457\pi$
$564$	0	0
$565$	19.3693	0.814873
$566$	0	0
$567$	7.00000	0.293972
$568$	0	0
$569$	−40.7386	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$569$	−40.7386	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$570$	0	0
$571$	−14.2462	−0.596185	−0.298093	−	0.954537i	$-0.596350\pi$
$571$	−14.2462	−0.596185	−0.298093	+	0.954537i	$0.596350\pi$
$572$	0	0
$573$	18.4384	0.770277
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.4384	0.517819	0.258910	−	0.965902i	$-0.416637\pi$
$577$	12.4384	0.517819	0.258910	+	0.965902i	$0.416637\pi$
$578$	0	0
$579$	−21.8617	−0.908543
$580$	0	0
$581$	10.2462	0.425084
$582$	0	0
$583$	6.24621	0.258692
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	16.4924	0.680715	0.340358	−	0.940296i	$-0.389452\pi$
$587$	16.4924	0.680715	0.340358	+	0.940296i	$0.389452\pi$
$588$	0	0
$589$	12.4924	0.514741
$590$	0	0
$591$	15.6155	0.642337
$592$	0	0
$593$	8.93087	0.366747	0.183373	−	0.983043i	$-0.441298\pi$
$593$	8.93087	0.366747	0.183373	+	0.983043i	$0.441298\pi$
$594$	0	0
$595$	6.68466	0.274044
$596$	0	0
$597$	2.73863	0.112085
$598$	0	0
$599$	21.5616	0.880981	0.440491	−	0.897757i	$-0.354805\pi$
$599$	21.5616	0.880981	0.440491	+	0.897757i	$0.354805\pi$
$600$	0	0
$601$	32.2462	1.31535	0.657675	−	0.753302i	$-0.271540\pi$
$601$	32.2462	1.31535	0.657675	+	0.753302i	$0.271540\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	19.9309	0.810305
$606$	0	0
$607$	−6.93087	−0.281315	−0.140658	−	0.990058i	$-0.544922\pi$
$607$	−6.93087	−0.281315	−0.140658	+	0.990058i	$0.544922\pi$
$608$	0	0
$609$	0.684658	0.0277438
$610$	0	0
$611$	8.68466	0.351344
$612$	0	0
$613$	−20.2462	−0.817737	−0.408868	−	0.912593i	$-0.634076\pi$
$613$	−20.2462	−0.817737	−0.408868	+	0.912593i	$0.634076\pi$
$614$	0	0
$615$	12.8769	0.519246
$616$	0	0
$617$	28.7386	1.15697	0.578487	−	0.815692i	$-0.303643\pi$
$617$	28.7386	1.15697	0.578487	+	0.815692i	$0.303643\pi$
$618$	0	0
$619$	14.7386	0.592396	0.296198	−	0.955127i	$-0.404281\pi$
$619$	14.7386	0.592396	0.296198	+	0.955127i	$0.404281\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.246211	−0.00986425
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	34.7386	1.38733
$628$	0	0
$629$	60.9848	2.43163
$630$	0	0
$631$	−10.4384	−0.415548	−0.207774	−	0.978177i	$-0.566622\pi$
$631$	−10.4384	−0.415548	−0.207774	+	0.978177i	$0.566622\pi$
$632$	0	0
$633$	−28.7926	−1.14440
$634$	0	0
$635$	12.4924	0.495747
$636$	0	0
$637$	3.56155	0.141114
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	8.24621	0.325706	0.162853	−	0.986650i	$-0.447930\pi$
$641$	8.24621	0.325706	0.162853	+	0.986650i	$0.447930\pi$
$642$	0	0
$643$	−30.4384	−1.20038	−0.600188	−	0.799859i	$-0.704907\pi$
$643$	−30.4384	−1.20038	−0.600188	+	0.799859i	$0.704907\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.2462	1.50361	0.751807	−	0.659383i	$-0.229183\pi$
$647$	38.2462	1.50361	0.751807	+	0.659383i	$0.229183\pi$
$648$	0	0
$649$	−12.4924	−0.490370
$650$	0	0
$651$	4.87689	0.191141
$652$	0	0
$653$	1.61553	0.0632205	0.0316103	−	0.999500i	$-0.489936\pi$
$653$	1.61553	0.0632205	0.0316103	+	0.999500i	$0.489936\pi$
$654$	0	0
$655$	−20.0000	−0.781465
$656$	0	0
$657$	−4.63068	−0.180660
$658$	0	0
$659$	41.6695	1.62321	0.811607	−	0.584204i	$-0.198593\pi$
$659$	41.6695	1.62321	0.811607	+	0.584204i	$0.198593\pi$
$660$	0	0
$661$	33.1231	1.28834	0.644170	−	0.764883i	$-0.277203\pi$
$661$	33.1231	1.28834	0.644170	+	0.764883i	$0.277203\pi$
$662$	0	0
$663$	−37.1771	−1.44384
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−33.6695	−1.30174
$670$	0	0
$671$	45.8617	1.77047
$672$	0	0
$673$	14.4924	0.558642	0.279321	−	0.960198i	$-0.409891\pi$
$673$	14.4924	0.558642	0.279321	+	0.960198i	$0.409891\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	14.3002	0.549601	0.274800	−	0.961501i	$-0.411388\pi$
$677$	14.3002	0.549601	0.274800	+	0.961501i	$0.411388\pi$
$678$	0	0
$679$	6.68466	0.256534
$680$	0	0
$681$	−29.5616	−1.13280
$682$	0	0
$683$	19.1231	0.731725	0.365863	−	0.930669i	$-0.380774\pi$
$683$	19.1231	0.731725	0.365863	+	0.930669i	$0.380774\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	31.6155	1.20621
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	2.24621	0.0854499	0.0427250	−	0.999087i	$-0.486396\pi$
$691$	2.24621	0.0854499	0.0427250	+	0.999087i	$0.486396\pi$
$692$	0	0
$693$	−3.12311	−0.118637
$694$	0	0
$695$	−19.6155	−0.744059
$696$	0	0
$697$	−55.1231	−2.08794
$698$	0	0
$699$	8.00000	0.302588
$700$	0	0
$701$	−41.4233	−1.56454	−0.782268	−	0.622942i	$-0.785937\pi$
$701$	−41.4233	−1.56454	−0.782268	+	0.622942i	$0.785937\pi$
$702$	0	0
$703$	36.4924	1.37634
$704$	0	0
$705$	3.80776	0.143409
$706$	0	0
$707$	−10.8769	−0.409068
$708$	0	0
$709$	−24.4384	−0.917805	−0.458903	−	0.888487i	$-0.651757\pi$
$709$	−24.4384	−0.917805	−0.458903	+	0.888487i	$0.651757\pi$
$710$	0	0
$711$	3.12311	0.117126
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−19.8078	−0.740768
$716$	0	0
$717$	−40.6847	−1.51940
$718$	0	0
$719$	−11.5076	−0.429160	−0.214580	−	0.976706i	$-0.568838\pi$
$719$	−11.5076	−0.429160	−0.214580	+	0.976706i	$0.568838\pi$
$720$	0	0
$721$	13.5616	0.505059
$722$	0	0
$723$	−36.4924	−1.35717
$724$	0	0
$725$	−0.438447	−0.0162835
$726$	0	0
$727$	−48.0000	−1.78022	−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000	−1.78022	−0.890111	+	0.455744i	$0.849373\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	0	0
$732$	0	0
$733$	17.8078	0.657745	0.328872	−	0.944374i	$-0.393331\pi$
$733$	17.8078	0.657745	0.328872	+	0.944374i	$0.393331\pi$
$734$	0	0
$735$	1.56155	0.0575987
$736$	0	0
$737$	79.2311	2.91851
$738$	0	0
$739$	35.4233	1.30307	0.651533	−	0.758620i	$-0.274126\pi$
$739$	35.4233	1.30307	0.651533	+	0.758620i	$0.274126\pi$
$740$	0	0
$741$	−22.2462	−0.817235
$742$	0	0
$743$	32.0000	1.17397	0.586983	−	0.809599i	$-0.300316\pi$
$743$	32.0000	1.17397	0.586983	+	0.809599i	$0.300316\pi$
$744$	0	0
$745$	−7.75379	−0.284077
$746$	0	0
$747$	5.75379	0.210520
$748$	0	0
$749$	−9.36932	−0.342347
$750$	0	0
$751$	−39.3153	−1.43464	−0.717319	−	0.696745i	$-0.754631\pi$
$751$	−39.3153	−1.43464	−0.717319	+	0.696745i	$0.754631\pi$
$752$	0	0
$753$	−20.8769	−0.760796
$754$	0	0
$755$	−13.5616	−0.493555
$756$	0	0
$757$	47.8617	1.73956	0.869782	−	0.493436i	$-0.164259\pi$
$757$	47.8617	1.73956	0.869782	+	0.493436i	$0.164259\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.8617	1.15499	0.577494	−	0.816395i	$-0.304031\pi$
$761$	31.8617	1.15499	0.577494	+	0.816395i	$0.304031\pi$
$762$	0	0
$763$	−16.9309	−0.612939
$764$	0	0
$765$	3.75379	0.135719
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−8.73863	−0.315123	−0.157562	−	0.987509i	$-0.550363\pi$
$769$	−8.73863	−0.315123	−0.157562	+	0.987509i	$0.550363\pi$
$770$	0	0
$771$	−9.36932	−0.337428
$772$	0	0
$773$	13.3153	0.478920	0.239460	−	0.970906i	$-0.423030\pi$
$773$	13.3153	0.478920	0.239460	+	0.970906i	$0.423030\pi$
$774$	0	0
$775$	−3.12311	−0.112185
$776$	0	0
$777$	14.2462	0.511080
$778$	0	0
$779$	−32.9848	−1.18180
$780$	0	0
$781$	−79.2311	−2.83511
$782$	0	0
$783$	2.43845	0.0871430
$784$	0	0
$785$	6.00000	0.214149
$786$	0	0
$787$	13.0691	0.465864	0.232932	−	0.972493i	$-0.425168\pi$
$787$	13.0691	0.465864	0.232932	+	0.972493i	$0.425168\pi$
$788$	0	0
$789$	−4.87689	−0.173622
$790$	0	0
$791$	−19.3693	−0.688694
$792$	0	0
$793$	−29.3693	−1.04294
$794$	0	0
$795$	−1.75379	−0.0622005
$796$	0	0
$797$	8.43845	0.298905	0.149453	−	0.988769i	$-0.452249\pi$
$797$	8.43845	0.298905	0.149453	+	0.988769i	$0.452249\pi$
$798$	0	0
$799$	−16.3002	−0.576659
$800$	0	0
$801$	−0.138261	−0.00488520
$802$	0	0
$803$	−45.8617	−1.61843
$804$	0	0
$805$	0	0
$806$	0	0
$807$	19.1231	0.673165
$808$	0	0
$809$	−4.93087	−0.173360	−0.0866801	−	0.996236i	$-0.527626\pi$
$809$	−4.93087	−0.173360	−0.0866801	+	0.996236i	$0.527626\pi$
$810$	0	0
$811$	−43.6155	−1.53155	−0.765774	−	0.643110i	$-0.777644\pi$
$811$	−43.6155	−1.53155	−0.765774	+	0.643110i	$0.777644\pi$
$812$	0	0
$813$	−7.61553	−0.267088
$814$	0	0
$815$	17.3693	0.608421
$816$	0	0
$817$	0	0
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−46.3002	−1.61589	−0.807944	−	0.589260i	$-0.799420\pi$
$821$	−46.3002	−1.61589	−0.807944	+	0.589260i	$0.799420\pi$
$822$	0	0
$823$	−12.8769	−0.448860	−0.224430	−	0.974490i	$-0.572052\pi$
$823$	−12.8769	−0.448860	−0.224430	+	0.974490i	$0.572052\pi$
$824$	0	0
$825$	−8.68466	−0.302361
$826$	0	0
$827$	−21.8617	−0.760207	−0.380104	−	0.924944i	$-0.624112\pi$
$827$	−21.8617	−0.760207	−0.380104	+	0.924944i	$0.624112\pi$
$828$	0	0
$829$	−8.24621	−0.286403	−0.143201	−	0.989694i	$-0.545740\pi$
$829$	−8.24621	−0.286403	−0.143201	+	0.989694i	$0.545740\pi$
$830$	0	0
$831$	−19.1231	−0.663373
$832$	0	0
$833$	−6.68466	−0.231610
$834$	0	0
$835$	18.4384	0.638089
$836$	0	0
$837$	17.3693	0.600371
$838$	0	0
$839$	4.49242	0.155096	0.0775478	−	0.996989i	$-0.475291\pi$
$839$	4.49242	0.155096	0.0775478	+	0.996989i	$0.475291\pi$
$840$	0	0
$841$	−28.8078	−0.993371
$842$	0	0
$843$	29.1771	1.00491
$844$	0	0
$845$	−0.315342	−0.0108481
$846$	0	0
$847$	−19.9309	−0.684833
$848$	0	0
$849$	22.5464	0.773790
$850$	0	0
$851$	0	0
$852$	0	0
$853$	7.75379	0.265485	0.132742	−	0.991151i	$-0.457622\pi$
$853$	7.75379	0.265485	0.132742	+	0.991151i	$0.457622\pi$
$854$	0	0
$855$	2.24621	0.0768188
$856$	0	0
$857$	19.7538	0.674777	0.337388	−	0.941366i	$-0.390456\pi$
$857$	19.7538	0.674777	0.337388	+	0.941366i	$0.390456\pi$
$858$	0	0
$859$	−15.5076	−0.529112	−0.264556	−	0.964370i	$-0.585225\pi$
$859$	−15.5076	−0.529112	−0.264556	+	0.964370i	$0.585225\pi$
$860$	0	0
$861$	−12.8769	−0.438843
$862$	0	0
$863$	41.3693	1.40823	0.704114	−	0.710087i	$-0.251344\pi$
$863$	41.3693	1.40823	0.704114	+	0.710087i	$0.251344\pi$
$864$	0	0
$865$	4.93087	0.167655
$866$	0	0
$867$	43.2311	1.46820
$868$	0	0
$869$	30.9309	1.04926
$870$	0	0
$871$	−50.7386	−1.71921
$872$	0	0
$873$	3.75379	0.127047
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	58.6004	1.97879	0.989397	−	0.145236i	$-0.0463942\pi$
$877$	58.6004	1.97879	0.989397	+	0.145236i	$0.0463942\pi$
$878$	0	0
$879$	−21.5616	−0.727253
$880$	0	0
$881$	−15.3693	−0.517805	−0.258903	−	0.965903i	$-0.583361\pi$
$881$	−15.3693	−0.517805	−0.258903	+	0.965903i	$0.583361\pi$
$882$	0	0
$883$	5.86174	0.197263	0.0986316	−	0.995124i	$-0.468553\pi$
$883$	5.86174	0.197263	0.0986316	+	0.995124i	$0.468553\pi$
$884$	0	0
$885$	3.50758	0.117906
$886$	0	0
$887$	−12.4924	−0.419454	−0.209727	−	0.977760i	$-0.567258\pi$
$887$	−12.4924	−0.419454	−0.209727	+	0.977760i	$0.567258\pi$
$888$	0	0
$889$	−12.4924	−0.418982
$890$	0	0
$891$	38.9309	1.30423
$892$	0	0
$893$	−9.75379	−0.326398
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.36932	0.0456693
$900$	0	0
$901$	7.50758	0.250114
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.87689	−0.228596
$906$	0	0
$907$	11.5076	0.382103	0.191051	−	0.981580i	$-0.438810\pi$
$907$	11.5076	0.382103	0.191051	+	0.981580i	$0.438810\pi$
$908$	0	0
$909$	−6.10795	−0.202588
$910$	0	0
$911$	−56.9848	−1.88799	−0.943996	−	0.329957i	$-0.892966\pi$
$911$	−56.9848	−1.88799	−0.943996	+	0.329957i	$0.892966\pi$
$912$	0	0
$913$	56.9848	1.88592
$914$	0	0
$915$	−12.8769	−0.425697
$916$	0	0
$917$	20.0000	0.660458
$918$	0	0
$919$	−36.7926	−1.21368	−0.606838	−	0.794825i	$-0.707562\pi$
$919$	−36.7926	−1.21368	−0.606838	+	0.794825i	$0.707562\pi$
$920$	0	0
$921$	32.3002	1.06433
$922$	0	0
$923$	50.7386	1.67008
$924$	0	0
$925$	−9.12311	−0.299966
$926$	0	0
$927$	7.61553	0.250127
$928$	0	0
$929$	−32.3542	−1.06151	−0.530753	−	0.847527i	$-0.678091\pi$
$929$	−32.3542	−1.06151	−0.530753	+	0.847527i	$0.678091\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	0	0
$933$	−9.75379	−0.319325
$934$	0	0
$935$	37.1771	1.21582
$936$	0	0
$937$	−3.94602	−0.128911	−0.0644555	−	0.997921i	$-0.520531\pi$
$937$	−3.94602	−0.128911	−0.0644555	+	0.997921i	$0.520531\pi$
$938$	0	0
$939$	−20.7926	−0.678541
$940$	0	0
$941$	52.2462	1.70318	0.851589	−	0.524210i	$-0.175639\pi$
$941$	52.2462	1.70318	0.851589	+	0.524210i	$0.175639\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	5.56155	0.180917
$946$	0	0
$947$	−1.36932	−0.0444968	−0.0222484	−	0.999752i	$-0.507082\pi$
$947$	−1.36932	−0.0444968	−0.0222484	+	0.999752i	$0.507082\pi$
$948$	0	0
$949$	29.3693	0.953368
$950$	0	0
$951$	−51.1231	−1.65778
$952$	0	0
$953$	0.246211	0.00797556	0.00398778	−	0.999992i	$-0.498731\pi$
$953$	0.246211	0.00797556	0.00398778	+	0.999992i	$0.498731\pi$
$954$	0	0
$955$	11.8078	0.382090
$956$	0	0
$957$	3.80776	0.123088
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	−5.26137	−0.169545
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−30.6307	−0.985016	−0.492508	−	0.870308i	$-0.663920\pi$
$967$	−30.6307	−0.985016	−0.492508	+	0.870308i	$0.663920\pi$
$968$	0	0
$969$	41.7538	1.34132
$970$	0	0
$971$	10.6307	0.341155	0.170577	−	0.985344i	$-0.445437\pi$
$971$	10.6307	0.341155	0.170577	+	0.985344i	$0.445437\pi$
$972$	0	0
$973$	19.6155	0.628845
$974$	0	0
$975$	5.56155	0.178112
$976$	0	0
$977$	−34.1080	−1.09121	−0.545605	−	0.838042i	$-0.683700\pi$
$977$	−34.1080	−1.09121	−0.545605	+	0.838042i	$0.683700\pi$
$978$	0	0
$979$	−1.36932	−0.0437636
$980$	0	0
$981$	−9.50758	−0.303554
$982$	0	0
$983$	−49.6695	−1.58421	−0.792106	−	0.610384i	$-0.791015\pi$
$983$	−49.6695	−1.58421	−0.792106	+	0.610384i	$0.791015\pi$
$984$	0	0
$985$	10.0000	0.318626
$986$	0	0
$987$	−3.80776	−0.121202
$988$	0	0
$989$	0	0
$990$	0	0
$991$	4.49242	0.142707	0.0713533	−	0.997451i	$-0.477268\pi$
$991$	4.49242	0.142707	0.0713533	+	0.997451i	$0.477268\pi$
$992$	0	0
$993$	−2.73863	−0.0869079
$994$	0	0
$995$	1.75379	0.0555988
$996$	0	0
$997$	−45.8078	−1.45075	−0.725373	−	0.688356i	$-0.758333\pi$
$997$	−45.8078	−1.45075	−0.725373	+	0.688356i	$0.758333\pi$
$998$	0	0
$999$	50.7386	1.60530

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.2.a.bf.1.2		2
4.3	odd	2		2240.2.a.bl.1.1		2
8.3	odd	2		1120.2.a.q.1.2	✓	2
8.5	even	2		1120.2.a.s.1.1	yes	2
40.19	odd	2		5600.2.a.bg.1.1		2
40.29	even	2		5600.2.a.bb.1.2		2
56.13	odd	2		7840.2.a.bd.1.2		2
56.27	even	2		7840.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.2.a.q.1.2	✓	2	8.3	odd	2
1120.2.a.s.1.1	yes	2	8.5	even	2
2240.2.a.bf.1.2		2	1.1	even	1	trivial
2240.2.a.bl.1.1		2	4.3	odd	2
5600.2.a.bb.1.2		2	40.29	even	2
5600.2.a.bg.1.1		2	40.19	odd	2
7840.2.a.bd.1.2		2	56.13	odd	2
7840.2.a.bi.1.1		2	56.27	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +1.00000 q^{5} -1.00000 q^{7} -0.561553 q^{9} -5.56155 q^{11} +3.56155 q^{13} +1.56155 q^{15} -6.68466 q^{17} -4.00000 q^{19} -1.56155 q^{21} +1.00000 q^{25} -5.56155 q^{27} -0.438447 q^{29} -3.12311 q^{31} -8.68466 q^{33} -1.00000 q^{35} -9.12311 q^{37} +5.56155 q^{39} +8.24621 q^{41} -0.561553 q^{45} +2.43845 q^{47} +1.00000 q^{49} -10.4384 q^{51} -1.12311 q^{53} -5.56155 q^{55} -6.24621 q^{57} +2.24621 q^{59} -8.24621 q^{61} +0.561553 q^{63} +3.56155 q^{65} -14.2462 q^{67} +14.2462 q^{71} +8.24621 q^{73} +1.56155 q^{75} +5.56155 q^{77} -5.56155 q^{79} -7.00000 q^{81} -10.2462 q^{83} -6.68466 q^{85} -0.684658 q^{87} +0.246211 q^{89} -3.56155 q^{91} -4.87689 q^{93} -4.00000 q^{95} -6.68466 q^{97} +3.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9	\( 2 q - q^{3} + 2 q^{5} - 2 q^{7} + 3 q^{9} - 7 q^{11} + 3 q^{13} - q^{15} - q^{17} - 8 q^{19} + q^{21} + 2 q^{25} - 7 q^{27} - 5 q^{29} + 2 q^{31} - 5 q^{33} - 2 q^{35} - 10 q^{37} + 7 q^{39} + 3 q^{45} + 9 q^{47} + 2 q^{49} - 25 q^{51} + 6 q^{53} - 7 q^{55} + 4 q^{57} - 12 q^{59} - 3 q^{63} + 3 q^{65} - 12 q^{67} + 12 q^{71} - q^{75} + 7 q^{77} - 7 q^{79} - 14 q^{81} - 4 q^{83} - q^{85} + 11 q^{87} - 16 q^{89} - 3 q^{91} - 18 q^{93} - 8 q^{95} - q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - 2 * q^7 + 3 * q^9 - 7 * q^11 + 3 * q^13 - q^15 - q^17 - 8 * q^19 + q^21 + 2 * q^25 - 7 * q^27 - 5 * q^29 + 2 * q^31 - 5 * q^33 - 2 * q^35 - 10 * q^37 + 7 * q^39 + 3 * q^45 + 9 * q^47 + 2 * q^49 - 25 * q^51 + 6 * q^53 - 7 * q^55 + 4 * q^57 - 12 * q^59 - 3 * q^63 + 3 * q^65 - 12 * q^67 + 12 * q^71 - q^75 + 7 * q^77 - 7 * q^79 - 14 * q^81 - 4 * q^83 - q^85 + 11 * q^87 - 16 * q^89 - 3 * q^91 - 18 * q^93 - 8 * q^95 - q^97 - 2 * q^99

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