LMFDB - Embedded newform 1932.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,2,Mod(1,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, 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("1932.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1932.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:	$15.4270976705$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.551929$ of defining polynomial
Character	$\chi$	$=$	1932.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.551929 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.551929 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.103859 q^{11} +5.69537 q^{13} +0.551929 q^{15} -4.24730 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.69537 q^{25} -1.00000 q^{27} +1.14344 q^{29} +6.24730 q^{31} +0.103859 q^{33} -0.551929 q^{35} +3.35116 q^{37} -5.69537 q^{39} -9.28689 q^{41} -0.799233 q^{43} -0.551929 q^{45} +8.14344 q^{47} +1.00000 q^{49} +4.24730 q^{51} +0.695374 q^{53} +0.0573228 q^{55} -1.00000 q^{57} -0.695374 q^{59} +7.55193 q^{61} +1.00000 q^{63} -3.14344 q^{65} -0.551929 q^{67} -1.00000 q^{69} +12.1900 q^{71} -3.14344 q^{73} +4.69537 q^{75} -0.103859 q^{77} +8.45502 q^{79} +1.00000 q^{81} -1.14344 q^{83} +2.34421 q^{85} -1.14344 q^{87} +9.69537 q^{89} +5.69537 q^{91} -6.24730 q^{93} -0.551929 q^{95} +17.6381 q^{97} -0.103859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 10 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} - 3 q^{39} - q^{41} + 13 q^{43} - q^{45} + 11 q^{47} + 3 q^{49} - 2 q^{51} - 12 q^{53} + 29 q^{55} - 3 q^{57} + 12 q^{59} + 22 q^{61} + 3 q^{63} + 4 q^{65} - q^{67} - 3 q^{69} - 7 q^{71} + 4 q^{73} + q^{77} + 8 q^{79} + 3 q^{81} + 10 q^{83} + 9 q^{85} + 10 q^{87} + 15 q^{89} + 3 q^{91} - 4 q^{93} - q^{95} + 10 q^{97} + q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + q^15 + 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 10 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 - 3 * q^39 - q^41 + 13 * q^43 - q^45 + 11 * q^47 + 3 * q^49 - 2 * q^51 - 12 * q^53 + 29 * q^55 - 3 * q^57 + 12 * q^59 + 22 * q^61 + 3 * q^63 + 4 * q^65 - q^67 - 3 * q^69 - 7 * q^71 + 4 * q^73 + q^77 + 8 * q^79 + 3 * q^81 + 10 * q^83 + 9 * q^85 + 10 * q^87 + 15 * q^89 + 3 * q^91 - 4 * q^93 - q^95 + 10 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.551929	−0.246830	−0.123415	−	0.992355i	$-0.539385\pi$
$5$	−0.551929	−0.246830	−0.123415	+	0.992355i	$0.539385\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.103859	−0.0313146	−0.0156573	−	0.999877i	$-0.504984\pi$
$11$	−0.103859	−0.0313146	−0.0156573	+	0.999877i	$0.504984\pi$
$12$	0	0
$13$	5.69537	1.57961	0.789806	−	0.613356i	$-0.210181\pi$
$13$	5.69537	1.57961	0.789806	+	0.613356i	$0.210181\pi$
$14$	0	0
$15$	0.551929	0.142508
$16$	0	0
$17$	−4.24730	−1.03012	−0.515061	−	0.857153i	$-0.672231\pi$
$17$	−4.24730	−1.03012	−0.515061	+	0.857153i	$0.672231\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.69537	−0.939075
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.14344	0.212332	0.106166	−	0.994348i	$-0.466142\pi$
$29$	1.14344	0.212332	0.106166	+	0.994348i	$0.466142\pi$
$30$	0	0
$31$	6.24730	1.12205	0.561024	−	0.827799i	$-0.310407\pi$
$31$	6.24730	1.12205	0.561024	+	0.827799i	$0.310407\pi$
$32$	0	0
$33$	0.103859	0.0180795
$34$	0	0
$35$	−0.551929	−0.0932931
$36$	0	0
$37$	3.35116	0.550928	0.275464	−	0.961311i	$-0.411169\pi$
$37$	3.35116	0.550928	0.275464	+	0.961311i	$0.411169\pi$
$38$	0	0
$39$	−5.69537	−0.911990
$40$	0	0
$41$	−9.28689	−1.45037	−0.725184	−	0.688555i	$-0.758245\pi$
$41$	−9.28689	−1.45037	−0.725184	+	0.688555i	$0.758245\pi$
$42$	0	0
$43$	−0.799233	−0.121882	−0.0609409	−	0.998141i	$-0.519410\pi$
$43$	−0.799233	−0.121882	−0.0609409	+	0.998141i	$0.519410\pi$
$44$	0	0
$45$	−0.551929	−0.0822768
$46$	0	0
$47$	8.14344	1.18784	0.593922	−	0.804523i	$-0.297579\pi$
$47$	8.14344	1.18784	0.593922	+	0.804523i	$0.297579\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.24730	0.594741
$52$	0	0
$53$	0.695374	0.0955170	0.0477585	−	0.998859i	$-0.484792\pi$
$53$	0.695374	0.0955170	0.0477585	+	0.998859i	$0.484792\pi$
$54$	0	0
$55$	0.0573228	0.00772940
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−0.695374	−0.0905300	−0.0452650	−	0.998975i	$-0.514413\pi$
$59$	−0.695374	−0.0905300	−0.0452650	+	0.998975i	$0.514413\pi$
$60$	0	0
$61$	7.55193	0.966925	0.483463	−	0.875365i	$-0.339379\pi$
$61$	7.55193	0.966925	0.483463	+	0.875365i	$0.339379\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−3.14344	−0.389896
$66$	0	0
$67$	−0.551929	−0.0674289	−0.0337145	−	0.999432i	$-0.510734\pi$
$67$	−0.551929	−0.0674289	−0.0337145	+	0.999432i	$0.510734\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	12.1900	1.44668	0.723342	−	0.690490i	$-0.242605\pi$
$71$	12.1900	1.44668	0.723342	+	0.690490i	$0.242605\pi$
$72$	0	0
$73$	−3.14344	−0.367912	−0.183956	−	0.982934i	$-0.558890\pi$
$73$	−3.14344	−0.367912	−0.183956	+	0.982934i	$0.558890\pi$
$74$	0	0
$75$	4.69537	0.542175
$76$	0	0
$77$	−0.103859	−0.0118358
$78$	0	0
$79$	8.45502	0.951264	0.475632	−	0.879644i	$-0.342219\pi$
$79$	8.45502	0.951264	0.475632	+	0.879644i	$0.342219\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.14344	−0.125509	−0.0627547	−	0.998029i	$-0.519989\pi$
$83$	−1.14344	−0.125509	−0.0627547	+	0.998029i	$0.519989\pi$
$84$	0	0
$85$	2.34421	0.254265
$86$	0	0
$87$	−1.14344	−0.122590
$88$	0	0
$89$	9.69537	1.02771	0.513854	−	0.857878i	$-0.328217\pi$
$89$	9.69537	1.02771	0.513854	+	0.857878i	$0.328217\pi$
$90$	0	0
$91$	5.69537	0.597037
$92$	0	0
$93$	−6.24730	−0.647815
$94$	0	0
$95$	−0.551929	−0.0566268
$96$	0	0
$97$	17.6381	1.79087	0.895436	−	0.445189i	$-0.146864\pi$
$97$	17.6381	1.79087	0.895436	+	0.445189i	$0.146864\pi$
$98$	0	0
$99$	−0.103859	−0.0104382
$100$	0	0
$101$	10.2938	1.02428	0.512138	−	0.858903i	$-0.328854\pi$
$101$	10.2938	1.02428	0.512138	+	0.858903i	$0.328854\pi$
$102$	0	0
$103$	15.2473	1.50236	0.751181	−	0.660097i	$-0.229485\pi$
$103$	15.2473	1.50236	0.751181	+	0.660097i	$0.229485\pi$
$104$	0	0
$105$	0.551929	0.0538628
$106$	0	0
$107$	12.7992	1.23735	0.618674	−	0.785648i	$-0.287670\pi$
$107$	12.7992	1.23735	0.618674	+	0.785648i	$0.287670\pi$
$108$	0	0
$109$	−7.94268	−0.760771	−0.380385	−	0.924828i	$-0.624209\pi$
$109$	−7.94268	−0.760771	−0.380385	+	0.924828i	$0.624209\pi$
$110$	0	0
$111$	−3.35116	−0.318078
$112$	0	0
$113$	−8.15039	−0.766725	−0.383362	−	0.923598i	$-0.625234\pi$
$113$	−8.15039	−0.766725	−0.383362	+	0.923598i	$0.625234\pi$
$114$	0	0
$115$	−0.551929	−0.0514677
$116$	0	0
$117$	5.69537	0.526538
$118$	0	0
$119$	−4.24730	−0.389350
$120$	0	0
$121$	−10.9892	−0.999019
$122$	0	0
$123$	9.28689	0.837371
$124$	0	0
$125$	5.35116	0.478622
$126$	0	0
$127$	−6.73496	−0.597631	−0.298816	−	0.954311i	$-0.596592\pi$
$127$	−6.73496	−0.597631	−0.298816	+	0.954311i	$0.596592\pi$
$128$	0	0
$129$	0.799233	0.0703685
$130$	0	0
$131$	−3.49461	−0.305325	−0.152663	−	0.988278i	$-0.548785\pi$
$131$	−3.49461	−0.305325	−0.152663	+	0.988278i	$0.548785\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0.551929	0.0475025
$136$	0	0
$137$	4.39075	0.375127	0.187563	−	0.982252i	$-0.439941\pi$
$137$	4.39075	0.375127	0.187563	+	0.982252i	$0.439941\pi$
$138$	0	0
$139$	6.34421	0.538109	0.269055	−	0.963125i	$-0.413289\pi$
$139$	6.34421	0.538109	0.269055	+	0.963125i	$0.413289\pi$
$140$	0	0
$141$	−8.14344	−0.685802
$142$	0	0
$143$	−0.591515	−0.0494650
$144$	0	0
$145$	−0.631101	−0.0524101
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−5.93573	−0.486274	−0.243137	−	0.969992i	$-0.578176\pi$
$149$	−5.93573	−0.486274	−0.243137	+	0.969992i	$0.578176\pi$
$150$	0	0
$151$	1.64884	0.134181	0.0670903	−	0.997747i	$-0.478628\pi$
$151$	1.64884	0.134181	0.0670903	+	0.997747i	$0.478628\pi$
$152$	0	0
$153$	−4.24730	−0.343374
$154$	0	0
$155$	−3.44807	−0.276956
$156$	0	0
$157$	2.14344	0.171065	0.0855327	−	0.996335i	$-0.472741\pi$
$157$	2.14344	0.171065	0.0855327	+	0.996335i	$0.472741\pi$
$158$	0	0
$159$	−0.695374	−0.0551467
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−6.29384	−0.492972	−0.246486	−	0.969146i	$-0.579276\pi$
$163$	−6.29384	−0.492972	−0.246486	+	0.969146i	$0.579276\pi$
$164$	0	0
$165$	−0.0573228	−0.00446257
$166$	0	0
$167$	9.00000	0.696441	0.348220	−	0.937413i	$-0.386786\pi$
$167$	9.00000	0.696441	0.348220	+	0.937413i	$0.386786\pi$
$168$	0	0
$169$	19.4373	1.49518
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−17.0288	−1.29468	−0.647338	−	0.762203i	$-0.724118\pi$
$173$	−17.0288	−1.29468	−0.647338	+	0.762203i	$0.724118\pi$
$174$	0	0
$175$	−4.69537	−0.354937
$176$	0	0
$177$	0.695374	0.0522675
$178$	0	0
$179$	−1.24035	−0.0927083	−0.0463542	−	0.998925i	$-0.514760\pi$
$179$	−1.24035	−0.0927083	−0.0463542	+	0.998925i	$0.514760\pi$
$180$	0	0
$181$	19.4303	1.44425	0.722123	−	0.691765i	$-0.243167\pi$
$181$	19.4303	1.44425	0.722123	+	0.691765i	$0.243167\pi$
$182$	0	0
$183$	−7.55193	−0.558255
$184$	0	0
$185$	−1.84961	−0.135986
$186$	0	0
$187$	0.441120	0.0322579
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	16.4303	1.18886	0.594429	−	0.804148i	$-0.297378\pi$
$191$	16.4303	1.18886	0.594429	+	0.804148i	$0.297378\pi$
$192$	0	0
$193$	16.6381	1.19763	0.598817	−	0.800886i	$-0.295638\pi$
$193$	16.6381	1.19763	0.598817	+	0.800886i	$0.295638\pi$
$194$	0	0
$195$	3.14344	0.225107
$196$	0	0
$197$	−10.7992	−0.769413	−0.384707	−	0.923039i	$-0.625697\pi$
$197$	−10.7992	−0.769413	−0.384707	+	0.923039i	$0.625697\pi$
$198$	0	0
$199$	16.5016	1.16976	0.584882	−	0.811118i	$-0.301141\pi$
$199$	16.5016	1.16976	0.584882	+	0.811118i	$0.301141\pi$
$200$	0	0
$201$	0.551929	0.0389301
$202$	0	0
$203$	1.14344	0.0802541
$204$	0	0
$205$	5.12571	0.357995
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−0.103859	−0.00718407
$210$	0	0
$211$	−11.8961	−0.818964	−0.409482	−	0.912318i	$-0.634291\pi$
$211$	−11.8961	−0.818964	−0.409482	+	0.912318i	$0.634291\pi$
$212$	0	0
$213$	−12.1900	−0.835244
$214$	0	0
$215$	0.441120	0.0300841
$216$	0	0
$217$	6.24730	0.424095
$218$	0	0
$219$	3.14344	0.212414
$220$	0	0
$221$	−24.1900	−1.62719
$222$	0	0
$223$	12.4234	0.831931	0.415966	−	0.909380i	$-0.363444\pi$
$223$	12.4234	0.831931	0.415966	+	0.909380i	$0.363444\pi$
$224$	0	0
$225$	−4.69537	−0.313025
$226$	0	0
$227$	−1.44807	−0.0961118	−0.0480559	−	0.998845i	$-0.515303\pi$
$227$	−1.44807	−0.0961118	−0.0480559	+	0.998845i	$0.515303\pi$
$228$	0	0
$229$	−9.87840	−0.652783	−0.326392	−	0.945235i	$-0.605833\pi$
$229$	−9.87840	−0.652783	−0.326392	+	0.945235i	$0.605833\pi$
$230$	0	0
$231$	0.103859	0.00683341
$232$	0	0
$233$	−9.04654	−0.592658	−0.296329	−	0.955086i	$-0.595763\pi$
$233$	−9.04654	−0.592658	−0.296329	+	0.955086i	$0.595763\pi$
$234$	0	0
$235$	−4.49461	−0.293196
$236$	0	0
$237$	−8.45502	−0.549213
$238$	0	0
$239$	−29.6203	−1.91598	−0.957989	−	0.286804i	$-0.907407\pi$
$239$	−29.6203	−1.91598	−0.957989	+	0.286804i	$0.907407\pi$
$240$	0	0
$241$	−10.3907	−0.669327	−0.334663	−	0.942338i	$-0.608623\pi$
$241$	−10.3907	−0.669327	−0.334663	+	0.942338i	$0.608623\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.551929	−0.0352615
$246$	0	0
$247$	5.69537	0.362388
$248$	0	0
$249$	1.14344	0.0724628
$250$	0	0
$251$	7.59847	0.479611	0.239805	−	0.970821i	$-0.422916\pi$
$251$	7.59847	0.479611	0.239805	+	0.970821i	$0.422916\pi$
$252$	0	0
$253$	−0.103859	−0.00652955
$254$	0	0
$255$	−2.34421	−0.146800
$256$	0	0
$257$	13.1327	0.819193	0.409596	−	0.912267i	$-0.365670\pi$
$257$	13.1327	0.819193	0.409596	+	0.912267i	$0.365670\pi$
$258$	0	0
$259$	3.35116	0.208231
$260$	0	0
$261$	1.14344	0.0707774
$262$	0	0
$263$	11.7023	0.721596	0.360798	−	0.932644i	$-0.382504\pi$
$263$	11.7023	0.721596	0.360798	+	0.932644i	$0.382504\pi$
$264$	0	0
$265$	−0.383797	−0.0235765
$266$	0	0
$267$	−9.69537	−0.593347
$268$	0	0
$269$	−23.7093	−1.44558	−0.722790	−	0.691068i	$-0.757141\pi$
$269$	−23.7093	−1.44558	−0.722790	+	0.691068i	$0.757141\pi$
$270$	0	0
$271$	25.9249	1.57483	0.787414	−	0.616425i	$-0.211419\pi$
$271$	25.9249	1.57483	0.787414	+	0.616425i	$0.211419\pi$
$272$	0	0
$273$	−5.69537	−0.344700
$274$	0	0
$275$	0.487656	0.0294068
$276$	0	0
$277$	−2.16118	−0.129853	−0.0649264	−	0.997890i	$-0.520681\pi$
$277$	−2.16118	−0.129853	−0.0649264	+	0.997890i	$0.520681\pi$
$278$	0	0
$279$	6.24730	0.374016
$280$	0	0
$281$	−14.6093	−0.871515	−0.435757	−	0.900064i	$-0.643519\pi$
$281$	−14.6093	−0.871515	−0.435757	+	0.900064i	$0.643519\pi$
$282$	0	0
$283$	9.04654	0.537761	0.268880	−	0.963174i	$-0.413346\pi$
$283$	9.04654	0.537761	0.268880	+	0.963174i	$0.413346\pi$
$284$	0	0
$285$	0.551929	0.0326935
$286$	0	0
$287$	−9.28689	−0.548188
$288$	0	0
$289$	1.03959	0.0611521
$290$	0	0
$291$	−17.6381	−1.03396
$292$	0	0
$293$	12.9892	0.758838	0.379419	−	0.925225i	$-0.376124\pi$
$293$	12.9892	0.758838	0.379419	+	0.925225i	$0.376124\pi$
$294$	0	0
$295$	0.383797	0.0223455
$296$	0	0
$297$	0.103859	0.00602650
$298$	0	0
$299$	5.69537	0.329372
$300$	0	0
$301$	−0.799233	−0.0460670
$302$	0	0
$303$	−10.2938	−0.591366
$304$	0	0
$305$	−4.16813	−0.238667
$306$	0	0
$307$	−2.53419	−0.144634	−0.0723170	−	0.997382i	$-0.523039\pi$
$307$	−2.53419	−0.144634	−0.0723170	+	0.997382i	$0.523039\pi$
$308$	0	0
$309$	−15.2473	−0.867389
$310$	0	0
$311$	6.94268	0.393683	0.196842	−	0.980435i	$-0.436932\pi$
$311$	6.94268	0.393683	0.196842	+	0.980435i	$0.436932\pi$
$312$	0	0
$313$	−7.53419	−0.425858	−0.212929	−	0.977068i	$-0.568300\pi$
$313$	−7.53419	−0.425858	−0.212929	+	0.977068i	$0.568300\pi$
$314$	0	0
$315$	−0.551929	−0.0310977
$316$	0	0
$317$	−11.2404	−0.631321	−0.315661	−	0.948872i	$-0.602226\pi$
$317$	−11.2404	−0.631321	−0.315661	+	0.948872i	$0.602226\pi$
$318$	0	0
$319$	−0.118757	−0.00664911
$320$	0	0
$321$	−12.7992	−0.714384
$322$	0	0
$323$	−4.24730	−0.236326
$324$	0	0
$325$	−26.7419	−1.48337
$326$	0	0
$327$	7.94268	0.439231
$328$	0	0
$329$	8.14344	0.448963
$330$	0	0
$331$	−1.24730	−0.0685580	−0.0342790	−	0.999412i	$-0.510913\pi$
$331$	−1.24730	−0.0685580	−0.0342790	+	0.999412i	$0.510913\pi$
$332$	0	0
$333$	3.35116	0.183643
$334$	0	0
$335$	0.304626	0.0166435
$336$	0	0
$337$	5.20077	0.283304	0.141652	−	0.989917i	$-0.454759\pi$
$337$	5.20077	0.283304	0.141652	+	0.989917i	$0.454759\pi$
$338$	0	0
$339$	8.15039	0.442669
$340$	0	0
$341$	−0.648838	−0.0351365
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.551929	0.0297149
$346$	0	0
$347$	−7.62415	−0.409286	−0.204643	−	0.978837i	$-0.565603\pi$
$347$	−7.62415	−0.409286	−0.204643	+	0.978837i	$0.565603\pi$
$348$	0	0
$349$	−32.0357	−1.71483	−0.857417	−	0.514622i	$-0.827932\pi$
$349$	−32.0357	−1.71483	−0.857417	+	0.514622i	$0.827932\pi$
$350$	0	0
$351$	−5.69537	−0.303997
$352$	0	0
$353$	−26.7419	−1.42333	−0.711664	−	0.702520i	$-0.752058\pi$
$353$	−26.7419	−1.42333	−0.711664	+	0.702520i	$0.752058\pi$
$354$	0	0
$355$	−6.72801	−0.357086
$356$	0	0
$357$	4.24730	0.224791
$358$	0	0
$359$	−8.63110	−0.455532	−0.227766	−	0.973716i	$-0.573142\pi$
$359$	−8.63110	−0.455532	−0.227766	+	0.973716i	$0.573142\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	10.9892	0.576784
$364$	0	0
$365$	1.73496	0.0908119
$366$	0	0
$367$	−36.1257	−1.88575	−0.942873	−	0.333152i	$-0.891888\pi$
$367$	−36.1257	−1.88575	−0.942873	+	0.333152i	$0.891888\pi$
$368$	0	0
$369$	−9.28689	−0.483456
$370$	0	0
$371$	0.695374	0.0361020
$372$	0	0
$373$	−13.6381	−0.706152	−0.353076	−	0.935595i	$-0.614864\pi$
$373$	−13.6381	−0.706152	−0.353076	+	0.935595i	$0.614864\pi$
$374$	0	0
$375$	−5.35116	−0.276333
$376$	0	0
$377$	6.51234	0.335403
$378$	0	0
$379$	0.702324	0.0360760	0.0180380	−	0.999837i	$-0.494258\pi$
$379$	0.702324	0.0360760	0.0180380	+	0.999837i	$0.494258\pi$
$380$	0	0
$381$	6.73496	0.345042
$382$	0	0
$383$	31.3512	1.60197	0.800985	−	0.598685i	$-0.204310\pi$
$383$	31.3512	1.60197	0.800985	+	0.598685i	$0.204310\pi$
$384$	0	0
$385$	0.0573228	0.00292144
$386$	0	0
$387$	−0.799233	−0.0406273
$388$	0	0
$389$	−30.2473	−1.53360	−0.766800	−	0.641887i	$-0.778152\pi$
$389$	−30.2473	−1.53360	−0.766800	+	0.641887i	$0.778152\pi$
$390$	0	0
$391$	−4.24730	−0.214795
$392$	0	0
$393$	3.49461	0.176280
$394$	0	0
$395$	−4.66657	−0.234801
$396$	0	0
$397$	16.0931	0.807688	0.403844	−	0.914828i	$-0.367674\pi$
$397$	16.0931	0.807688	0.403844	+	0.914828i	$0.367674\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	7.00000	0.349563	0.174782	−	0.984607i	$-0.444078\pi$
$401$	7.00000	0.349563	0.174782	+	0.984607i	$0.444078\pi$
$402$	0	0
$403$	35.5807	1.77240
$404$	0	0
$405$	−0.551929	−0.0274256
$406$	0	0
$407$	−0.348048	−0.0172521
$408$	0	0
$409$	−20.5342	−1.01535	−0.507675	−	0.861549i	$-0.669495\pi$
$409$	−20.5342	−1.01535	−0.507675	+	0.861549i	$0.669495\pi$
$410$	0	0
$411$	−4.39075	−0.216580
$412$	0	0
$413$	−0.695374	−0.0342171
$414$	0	0
$415$	0.631101	0.0309795
$416$	0	0
$417$	−6.34421	−0.310677
$418$	0	0
$419$	1.86351	0.0910382	0.0455191	−	0.998963i	$-0.485506\pi$
$419$	1.86351	0.0910382	0.0455191	+	0.998963i	$0.485506\pi$
$420$	0	0
$421$	−13.9823	−0.681454	−0.340727	−	0.940162i	$-0.610673\pi$
$421$	−13.9823	−0.681454	−0.340727	+	0.940162i	$0.610673\pi$
$422$	0	0
$423$	8.14344	0.395948
$424$	0	0
$425$	19.9427	0.967362
$426$	0	0
$427$	7.55193	0.365463
$428$	0	0
$429$	0.591515	0.0285586
$430$	0	0
$431$	−5.83882	−0.281246	−0.140623	−	0.990063i	$-0.544911\pi$
$431$	−5.83882	−0.281246	−0.140623	+	0.990063i	$0.544911\pi$
$432$	0	0
$433$	33.0535	1.58845	0.794225	−	0.607624i	$-0.207877\pi$
$433$	33.0535	1.58845	0.794225	+	0.607624i	$0.207877\pi$
$434$	0	0
$435$	0.631101	0.0302590
$436$	0	0
$437$	1.00000	0.0478365
$438$	0	0
$439$	20.3265	0.970130	0.485065	−	0.874478i	$-0.338796\pi$
$439$	20.3265	0.970130	0.485065	+	0.874478i	$0.338796\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	16.4154	0.779921	0.389960	−	0.920832i	$-0.372489\pi$
$443$	16.4154	0.779921	0.389960	+	0.920832i	$0.372489\pi$
$444$	0	0
$445$	−5.35116	−0.253669
$446$	0	0
$447$	5.93573	0.280750
$448$	0	0
$449$	1.33343	0.0629282	0.0314641	−	0.999505i	$-0.489983\pi$
$449$	1.33343	0.0629282	0.0314641	+	0.999505i	$0.489983\pi$
$450$	0	0
$451$	0.964526	0.0454177
$452$	0	0
$453$	−1.64884	−0.0774692
$454$	0	0
$455$	−3.14344	−0.147367
$456$	0	0
$457$	11.2404	0.525802	0.262901	−	0.964823i	$-0.415321\pi$
$457$	11.2404	0.525802	0.262901	+	0.964823i	$0.415321\pi$
$458$	0	0
$459$	4.24730	0.198247
$460$	0	0
$461$	−33.9427	−1.58087	−0.790434	−	0.612547i	$-0.790145\pi$
$461$	−33.9427	−1.58087	−0.790434	+	0.612547i	$0.790145\pi$
$462$	0	0
$463$	25.8607	1.20185	0.600924	−	0.799306i	$-0.294800\pi$
$463$	25.8607	1.20185	0.600924	+	0.799306i	$0.294800\pi$
$464$	0	0
$465$	3.44807	0.159900
$466$	0	0
$467$	−32.8350	−1.51942	−0.759711	−	0.650261i	$-0.774660\pi$
$467$	−32.8350	−1.51942	−0.759711	+	0.650261i	$0.774660\pi$
$468$	0	0
$469$	−0.551929	−0.0254857
$470$	0	0
$471$	−2.14344	−0.0987647
$472$	0	0
$473$	0.0830074	0.00381668
$474$	0	0
$475$	−4.69537	−0.215439
$476$	0	0
$477$	0.695374	0.0318390
$478$	0	0
$479$	−27.3651	−1.25034	−0.625171	−	0.780488i	$-0.714971\pi$
$479$	−27.3651	−1.25034	−0.625171	+	0.780488i	$0.714971\pi$
$480$	0	0
$481$	19.0861	0.870252
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−9.73496	−0.442042
$486$	0	0
$487$	−8.24730	−0.373721	−0.186860	−	0.982386i	$-0.559831\pi$
$487$	−8.24730	−0.373721	−0.186860	+	0.982386i	$0.559831\pi$
$488$	0	0
$489$	6.29384	0.284617
$490$	0	0
$491$	−35.3730	−1.59636	−0.798181	−	0.602418i	$-0.794204\pi$
$491$	−35.3730	−1.59636	−0.798181	+	0.602418i	$0.794204\pi$
$492$	0	0
$493$	−4.85656	−0.218728
$494$	0	0
$495$	0.0573228	0.00257647
$496$	0	0
$497$	12.1900	0.546795
$498$	0	0
$499$	−20.6311	−0.923575	−0.461787	−	0.886991i	$-0.652792\pi$
$499$	−20.6311	−0.923575	−0.461787	+	0.886991i	$0.652792\pi$
$500$	0	0
$501$	−9.00000	−0.402090
$502$	0	0
$503$	8.05732	0.359258	0.179629	−	0.983734i	$-0.442510\pi$
$503$	8.05732	0.359258	0.179629	+	0.983734i	$0.442510\pi$
$504$	0	0
$505$	−5.68147	−0.252822
$506$	0	0
$507$	−19.4373	−0.863240
$508$	0	0
$509$	33.7064	1.49401	0.747006	−	0.664818i	$-0.231491\pi$
$509$	33.7064	1.49401	0.747006	+	0.664818i	$0.231491\pi$
$510$	0	0
$511$	−3.14344	−0.139058
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−8.41544	−0.370828
$516$	0	0
$517$	−0.845769	−0.0371969
$518$	0	0
$519$	17.0288	0.747481
$520$	0	0
$521$	32.7815	1.43618	0.718092	−	0.695948i	$-0.245016\pi$
$521$	32.7815	1.43618	0.718092	+	0.695948i	$0.245016\pi$
$522$	0	0
$523$	−9.53419	−0.416901	−0.208451	−	0.978033i	$-0.566842\pi$
$523$	−9.53419	−0.416901	−0.208451	+	0.978033i	$0.566842\pi$
$524$	0	0
$525$	4.69537	0.204923
$526$	0	0
$527$	−26.5342	−1.15585
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−0.695374	−0.0301767
$532$	0	0
$533$	−52.8923	−2.29102
$534$	0	0
$535$	−7.06427	−0.305415
$536$	0	0
$537$	1.24035	0.0535252
$538$	0	0
$539$	−0.103859	−0.00447352
$540$	0	0
$541$	28.0149	1.20445	0.602227	−	0.798325i	$-0.294280\pi$
$541$	28.0149	1.20445	0.602227	+	0.798325i	$0.294280\pi$
$542$	0	0
$543$	−19.4303	−0.833835
$544$	0	0
$545$	4.38380	0.187781
$546$	0	0
$547$	−23.2543	−0.994280	−0.497140	−	0.867670i	$-0.665616\pi$
$547$	−23.2543	−0.994280	−0.497140	+	0.867670i	$0.665616\pi$
$548$	0	0
$549$	7.55193	0.322308
$550$	0	0
$551$	1.14344	0.0487124
$552$	0	0
$553$	8.45502	0.359544
$554$	0	0
$555$	1.84961	0.0785114
$556$	0	0
$557$	−37.8062	−1.60190	−0.800950	−	0.598732i	$-0.795672\pi$
$557$	−37.8062	−1.60190	−0.800950	+	0.598732i	$0.795672\pi$
$558$	0	0
$559$	−4.55193	−0.192526
$560$	0	0
$561$	−0.441120	−0.0186241
$562$	0	0
$563$	−8.18998	−0.345167	−0.172583	−	0.984995i	$-0.555211\pi$
$563$	−8.18998	−0.345167	−0.172583	+	0.984995i	$0.555211\pi$
$564$	0	0
$565$	4.49844	0.189251
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	14.0288	0.588118	0.294059	−	0.955787i	$-0.404994\pi$
$569$	14.0288	0.588118	0.294059	+	0.955787i	$0.404994\pi$
$570$	0	0
$571$	16.6627	0.697314	0.348657	−	0.937250i	$-0.386638\pi$
$571$	16.6627	0.697314	0.348657	+	0.937250i	$0.386638\pi$
$572$	0	0
$573$	−16.4303	−0.686387
$574$	0	0
$575$	−4.69537	−0.195811
$576$	0	0
$577$	−21.1969	−0.882440	−0.441220	−	0.897399i	$-0.645454\pi$
$577$	−21.1969	−0.882440	−0.441220	+	0.897399i	$0.645454\pi$
$578$	0	0
$579$	−16.6381	−0.691454
$580$	0	0
$581$	−1.14344	−0.0474381
$582$	0	0
$583$	−0.0722207	−0.00299108
$584$	0	0
$585$	−3.14344	−0.129965
$586$	0	0
$587$	0.368899	0.0152261	0.00761305	−	0.999971i	$-0.497577\pi$
$587$	0.368899	0.0152261	0.00761305	+	0.999971i	$0.497577\pi$
$588$	0	0
$589$	6.24730	0.257416
$590$	0	0
$591$	10.7992	0.444221
$592$	0	0
$593$	−1.39075	−0.0571112	−0.0285556	−	0.999592i	$-0.509091\pi$
$593$	−1.39075	−0.0571112	−0.0285556	+	0.999592i	$0.509091\pi$
$594$	0	0
$595$	2.34421	0.0961033
$596$	0	0
$597$	−16.5016	−0.675364
$598$	0	0
$599$	40.5560	1.65707	0.828537	−	0.559934i	$-0.189173\pi$
$599$	40.5560	1.65707	0.828537	+	0.559934i	$0.189173\pi$
$600$	0	0
$601$	8.47276	0.345611	0.172806	−	0.984956i	$-0.444717\pi$
$601$	8.47276	0.345611	0.172806	+	0.984956i	$0.444717\pi$
$602$	0	0
$603$	−0.551929	−0.0224763
$604$	0	0
$605$	6.06527	0.246588
$606$	0	0
$607$	−2.51234	−0.101973	−0.0509864	−	0.998699i	$-0.516237\pi$
$607$	−2.51234	−0.101973	−0.0509864	+	0.998699i	$0.516237\pi$
$608$	0	0
$609$	−1.14344	−0.0463347
$610$	0	0
$611$	46.3800	1.87633
$612$	0	0
$613$	−6.32647	−0.255524	−0.127762	−	0.991805i	$-0.540779\pi$
$613$	−6.32647	−0.255524	−0.127762	+	0.991805i	$0.540779\pi$
$614$	0	0
$615$	−5.12571	−0.206688
$616$	0	0
$617$	−27.2584	−1.09738	−0.548690	−	0.836026i	$-0.684873\pi$
$617$	−27.2584	−1.09738	−0.548690	+	0.836026i	$0.684873\pi$
$618$	0	0
$619$	−39.5016	−1.58770	−0.793851	−	0.608113i	$-0.791927\pi$
$619$	−39.5016	−1.58770	−0.793851	+	0.608113i	$0.791927\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	9.69537	0.388437
$624$	0	0
$625$	20.5234	0.820936
$626$	0	0
$627$	0.103859	0.00414772
$628$	0	0
$629$	−14.2334	−0.567523
$630$	0	0
$631$	−38.2118	−1.52119	−0.760594	−	0.649227i	$-0.775092\pi$
$631$	−38.2118	−1.52119	−0.760594	+	0.649227i	$0.775092\pi$
$632$	0	0
$633$	11.8961	0.472829
$634$	0	0
$635$	3.71722	0.147513
$636$	0	0
$637$	5.69537	0.225659
$638$	0	0
$639$	12.1900	0.482228
$640$	0	0
$641$	−18.8141	−0.743113	−0.371557	−	0.928410i	$-0.621176\pi$
$641$	−18.8141	−0.743113	−0.371557	+	0.928410i	$0.621176\pi$
$642$	0	0
$643$	19.3581	0.763409	0.381705	−	0.924284i	$-0.375337\pi$
$643$	19.3581	0.763409	0.381705	+	0.924284i	$0.375337\pi$
$644$	0	0
$645$	−0.441120	−0.0173691
$646$	0	0
$647$	−8.43728	−0.331704	−0.165852	−	0.986151i	$-0.553037\pi$
$647$	−8.43728	−0.331704	−0.165852	+	0.986151i	$0.553037\pi$
$648$	0	0
$649$	0.0722207	0.00283491
$650$	0	0
$651$	−6.24730	−0.244851
$652$	0	0
$653$	47.1396	1.84472	0.922358	−	0.386337i	$-0.126260\pi$
$653$	47.1396	1.84472	0.922358	+	0.386337i	$0.126260\pi$
$654$	0	0
$655$	1.92878	0.0753635
$656$	0	0
$657$	−3.14344	−0.122637
$658$	0	0
$659$	−2.64884	−0.103184	−0.0515920	−	0.998668i	$-0.516430\pi$
$659$	−2.64884	−0.103184	−0.0515920	+	0.998668i	$0.516430\pi$
$660$	0	0
$661$	3.37996	0.131465	0.0657326	−	0.997837i	$-0.479062\pi$
$661$	3.37996	0.131465	0.0657326	+	0.997837i	$0.479062\pi$
$662$	0	0
$663$	24.1900	0.939461
$664$	0	0
$665$	−0.551929	−0.0214029
$666$	0	0
$667$	1.14344	0.0442743
$668$	0	0
$669$	−12.4234	−0.480316
$670$	0	0
$671$	−0.784335	−0.0302789
$672$	0	0
$673$	19.2077	0.740403	0.370202	−	0.928951i	$-0.379289\pi$
$673$	19.2077	0.740403	0.370202	+	0.928951i	$0.379289\pi$
$674$	0	0
$675$	4.69537	0.180725
$676$	0	0
$677$	−47.0506	−1.80830	−0.904152	−	0.427212i	$-0.859496\pi$
$677$	−47.0506	−1.80830	−0.904152	+	0.427212i	$0.859496\pi$
$678$	0	0
$679$	17.6381	0.676886
$680$	0	0
$681$	1.44807	0.0554902
$682$	0	0
$683$	23.3651	0.894039	0.447020	−	0.894524i	$-0.352485\pi$
$683$	23.3651	0.894039	0.447020	+	0.894524i	$0.352485\pi$
$684$	0	0
$685$	−2.42338	−0.0925927
$686$	0	0
$687$	9.87840	0.376885
$688$	0	0
$689$	3.96041	0.150880
$690$	0	0
$691$	−41.1000	−1.56352	−0.781759	−	0.623580i	$-0.785677\pi$
$691$	−41.1000	−1.56352	−0.781759	+	0.623580i	$0.785677\pi$
$692$	0	0
$693$	−0.103859	−0.00394527
$694$	0	0
$695$	−3.50156	−0.132822
$696$	0	0
$697$	39.4442	1.49406
$698$	0	0
$699$	9.04654	0.342171
$700$	0	0
$701$	39.2039	1.48071	0.740355	−	0.672216i	$-0.234657\pi$
$701$	39.2039	1.48071	0.740355	+	0.672216i	$0.234657\pi$
$702$	0	0
$703$	3.35116	0.126391
$704$	0	0
$705$	4.49461	0.169277
$706$	0	0
$707$	10.2938	0.387140
$708$	0	0
$709$	2.03164	0.0762998	0.0381499	−	0.999272i	$-0.487854\pi$
$709$	2.03164	0.0762998	0.0381499	+	0.999272i	$0.487854\pi$
$710$	0	0
$711$	8.45502	0.317088
$712$	0	0
$713$	6.24730	0.233963
$714$	0	0
$715$	0.326475	0.0122095
$716$	0	0
$717$	29.6203	1.10619
$718$	0	0
$719$	33.4303	1.24674	0.623371	−	0.781927i	$-0.285763\pi$
$719$	33.4303	1.24674	0.623371	+	0.781927i	$0.285763\pi$
$720$	0	0
$721$	15.2473	0.567839
$722$	0	0
$723$	10.3907	0.386436
$724$	0	0
$725$	−5.36890	−0.199396
$726$	0	0
$727$	−10.2185	−0.378983	−0.189492	−	0.981882i	$-0.560684\pi$
$727$	−10.2185	−0.378983	−0.189492	+	0.981882i	$0.560684\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.39458	0.125553
$732$	0	0
$733$	42.3800	1.56534	0.782670	−	0.622437i	$-0.213857\pi$
$733$	42.3800	1.56534	0.782670	+	0.622437i	$0.213857\pi$
$734$	0	0
$735$	0.551929	0.0203582
$736$	0	0
$737$	0.0573228	0.00211151
$738$	0	0
$739$	37.4303	1.37690	0.688449	−	0.725285i	$-0.258292\pi$
$739$	37.4303	1.37690	0.688449	+	0.725285i	$0.258292\pi$
$740$	0	0
$741$	−5.69537	−0.209225
$742$	0	0
$743$	−17.5123	−0.642466	−0.321233	−	0.947000i	$-0.604097\pi$
$743$	−17.5123	−0.642466	−0.321233	+	0.947000i	$0.604097\pi$
$744$	0	0
$745$	3.27610	0.120027
$746$	0	0
$747$	−1.14344	−0.0418364
$748$	0	0
$749$	12.7992	0.467674
$750$	0	0
$751$	−18.7242	−0.683255	−0.341627	−	0.939835i	$-0.610978\pi$
$751$	−18.7242	−0.683255	−0.341627	+	0.939835i	$0.610978\pi$
$752$	0	0
$753$	−7.59847	−0.276903
$754$	0	0
$755$	−0.910042	−0.0331198
$756$	0	0
$757$	−4.85656	−0.176515	−0.0882573	−	0.996098i	$-0.528130\pi$
$757$	−4.85656	−0.176515	−0.0882573	+	0.996098i	$0.528130\pi$
$758$	0	0
$759$	0.103859	0.00376984
$760$	0	0
$761$	8.46581	0.306885	0.153443	−	0.988158i	$-0.450964\pi$
$761$	8.46581	0.306885	0.153443	+	0.988158i	$0.450964\pi$
$762$	0	0
$763$	−7.94268	−0.287544
$764$	0	0
$765$	2.34421	0.0847552
$766$	0	0
$767$	−3.96041	−0.143002
$768$	0	0
$769$	−4.58768	−0.165436	−0.0827180	−	0.996573i	$-0.526360\pi$
$769$	−4.58768	−0.165436	−0.0827180	+	0.996573i	$0.526360\pi$
$770$	0	0
$771$	−13.1327	−0.472961
$772$	0	0
$773$	42.3404	1.52288	0.761439	−	0.648237i	$-0.224493\pi$
$773$	42.3404	1.52288	0.761439	+	0.648237i	$0.224493\pi$
$774$	0	0
$775$	−29.3334	−1.05369
$776$	0	0
$777$	−3.35116	−0.120222
$778$	0	0
$779$	−9.28689	−0.332737
$780$	0	0
$781$	−1.26604	−0.0453024
$782$	0	0
$783$	−1.14344	−0.0408634
$784$	0	0
$785$	−1.18303	−0.0422242
$786$	0	0
$787$	−30.5668	−1.08959	−0.544795	−	0.838569i	$-0.683392\pi$
$787$	−30.5668	−1.08959	−0.544795	+	0.838569i	$0.683392\pi$
$788$	0	0
$789$	−11.7023	−0.416614
$790$	0	0
$791$	−8.15039	−0.289795
$792$	0	0
$793$	43.0111	1.52737
$794$	0	0
$795$	0.383797	0.0136119
$796$	0	0
$797$	26.6884	0.945352	0.472676	−	0.881236i	$-0.343288\pi$
$797$	26.6884	0.945352	0.472676	+	0.881236i	$0.343288\pi$
$798$	0	0
$799$	−34.5877	−1.22362
$800$	0	0
$801$	9.69537	0.342569
$802$	0	0
$803$	0.326475	0.0115210
$804$	0	0
$805$	−0.551929	−0.0194530
$806$	0	0
$807$	23.7093	0.834606
$808$	0	0
$809$	−46.6846	−1.64134	−0.820671	−	0.571401i	$-0.806400\pi$
$809$	−46.6846	−1.64134	−0.820671	+	0.571401i	$0.806400\pi$
$810$	0	0
$811$	−4.80718	−0.168803	−0.0844015	−	0.996432i	$-0.526898\pi$
$811$	−4.80718	−0.168803	−0.0844015	+	0.996432i	$0.526898\pi$
$812$	0	0
$813$	−25.9249	−0.909227
$814$	0	0
$815$	3.47376	0.121680
$816$	0	0
$817$	−0.799233	−0.0279616
$818$	0	0
$819$	5.69537	0.199012
$820$	0	0
$821$	−7.96453	−0.277964	−0.138982	−	0.990295i	$-0.544383\pi$
$821$	−7.96453	−0.277964	−0.138982	+	0.990295i	$0.544383\pi$
$822$	0	0
$823$	20.0070	0.697398	0.348699	−	0.937235i	$-0.386623\pi$
$823$	20.0070	0.697398	0.348699	+	0.937235i	$0.386623\pi$
$824$	0	0
$825$	−0.487656	−0.0169780
$826$	0	0
$827$	39.9715	1.38994	0.694972	−	0.719037i	$-0.255417\pi$
$827$	39.9715	1.38994	0.694972	+	0.719037i	$0.255417\pi$
$828$	0	0
$829$	−39.8103	−1.38267	−0.691334	−	0.722535i	$-0.742977\pi$
$829$	−39.8103	−1.38267	−0.691334	+	0.722535i	$0.742977\pi$
$830$	0	0
$831$	2.16118	0.0749706
$832$	0	0
$833$	−4.24730	−0.147160
$834$	0	0
$835$	−4.96736	−0.171903
$836$	0	0
$837$	−6.24730	−0.215938
$838$	0	0
$839$	24.7499	0.854460	0.427230	−	0.904143i	$-0.359489\pi$
$839$	24.7499	0.854460	0.427230	+	0.904143i	$0.359489\pi$
$840$	0	0
$841$	−27.6925	−0.954915
$842$	0	0
$843$	14.6093	0.503169
$844$	0	0
$845$	−10.7280	−0.369055
$846$	0	0
$847$	−10.9892	−0.377594
$848$	0	0
$849$	−9.04654	−0.310476
$850$	0	0
$851$	3.35116	0.114876
$852$	0	0
$853$	50.2514	1.72058	0.860288	−	0.509809i	$-0.170284\pi$
$853$	50.2514	1.72058	0.860288	+	0.509809i	$0.170284\pi$
$854$	0	0
$855$	−0.551929	−0.0188756
$856$	0	0
$857$	−9.31257	−0.318111	−0.159056	−	0.987270i	$-0.550845\pi$
$857$	−9.31257	−0.318111	−0.159056	+	0.987270i	$0.550845\pi$
$858$	0	0
$859$	−23.6637	−0.807396	−0.403698	−	0.914892i	$-0.632275\pi$
$859$	−23.6637	−0.807396	−0.403698	+	0.914892i	$0.632275\pi$
$860$	0	0
$861$	9.28689	0.316496
$862$	0	0
$863$	31.5985	1.07562	0.537812	−	0.843065i	$-0.319251\pi$
$863$	31.5985	1.07562	0.537812	+	0.843065i	$0.319251\pi$
$864$	0	0
$865$	9.39870	0.319565
$866$	0	0
$867$	−1.03959	−0.0353062
$868$	0	0
$869$	−0.878129	−0.0297885
$870$	0	0
$871$	−3.14344	−0.106512
$872$	0	0
$873$	17.6381	0.596958
$874$	0	0
$875$	5.35116	0.180902
$876$	0	0
$877$	−28.9388	−0.977195	−0.488598	−	0.872509i	$-0.662491\pi$
$877$	−28.9388	−0.977195	−0.488598	+	0.872509i	$0.662491\pi$
$878$	0	0
$879$	−12.9892	−0.438115
$880$	0	0
$881$	24.9753	0.841440	0.420720	−	0.907191i	$-0.361778\pi$
$881$	24.9753	0.841440	0.420720	+	0.907191i	$0.361778\pi$
$882$	0	0
$883$	−6.92778	−0.233138	−0.116569	−	0.993183i	$-0.537190\pi$
$883$	−6.92778	−0.233138	−0.116569	+	0.993183i	$0.537190\pi$
$884$	0	0
$885$	−0.383797	−0.0129012
$886$	0	0
$887$	24.9139	0.836526	0.418263	−	0.908326i	$-0.362639\pi$
$887$	24.9139	0.836526	0.418263	+	0.908326i	$0.362639\pi$
$888$	0	0
$889$	−6.73496	−0.225883
$890$	0	0
$891$	−0.103859	−0.00347940
$892$	0	0
$893$	8.14344	0.272510
$894$	0	0
$895$	0.684587	0.0228832
$896$	0	0
$897$	−5.69537	−0.190163
$898$	0	0
$899$	7.14344	0.238247
$900$	0	0
$901$	−2.95346	−0.0983941
$902$	0	0
$903$	0.799233	0.0265968
$904$	0	0
$905$	−10.7242	−0.356484
$906$	0	0
$907$	28.2435	0.937809	0.468904	−	0.883249i	$-0.344649\pi$
$907$	28.2435	0.937809	0.468904	+	0.883249i	$0.344649\pi$
$908$	0	0
$909$	10.2938	0.341425
$910$	0	0
$911$	−35.9645	−1.19156	−0.595779	−	0.803148i	$-0.703157\pi$
$911$	−35.9645	−1.19156	−0.595779	+	0.803148i	$0.703157\pi$
$912$	0	0
$913$	0.118757	0.00393028
$914$	0	0
$915$	4.16813	0.137794
$916$	0	0
$917$	−3.49461	−0.115402
$918$	0	0
$919$	−46.8566	−1.54566	−0.772828	−	0.634616i	$-0.781158\pi$
$919$	−46.8566	−1.54566	−0.772828	+	0.634616i	$0.781158\pi$
$920$	0	0
$921$	2.53419	0.0835045
$922$	0	0
$923$	69.4265	2.28520
$924$	0	0
$925$	−15.7350	−0.517362
$926$	0	0
$927$	15.2473	0.500787
$928$	0	0
$929$	−44.5304	−1.46099	−0.730497	−	0.682916i	$-0.760711\pi$
$929$	−44.5304	−1.46099	−0.730497	+	0.682916i	$0.760711\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−6.94268	−0.227293
$934$	0	0
$935$	−0.243467	−0.00796223
$936$	0	0
$937$	43.3661	1.41671	0.708354	−	0.705858i	$-0.249438\pi$
$937$	43.3661	1.41671	0.708354	+	0.705858i	$0.249438\pi$
$938$	0	0
$939$	7.53419	0.245869
$940$	0	0
$941$	−14.3661	−0.468320	−0.234160	−	0.972198i	$-0.575234\pi$
$941$	−14.3661	−0.468320	−0.234160	+	0.972198i	$0.575234\pi$
$942$	0	0
$943$	−9.28689	−0.302423
$944$	0	0
$945$	0.551929	0.0179543
$946$	0	0
$947$	−7.50539	−0.243893	−0.121946	−	0.992537i	$-0.538914\pi$
$947$	−7.50539	−0.243893	−0.121946	+	0.992537i	$0.538914\pi$
$948$	0	0
$949$	−17.9031	−0.581159
$950$	0	0
$951$	11.2404	0.364493
$952$	0	0
$953$	40.4918	1.31166	0.655829	−	0.754910i	$-0.272319\pi$
$953$	40.4918	1.31166	0.655829	+	0.754910i	$0.272319\pi$
$954$	0	0
$955$	−9.06838	−0.293446
$956$	0	0
$957$	0.118757	0.00383886
$958$	0	0
$959$	4.39075	0.141785
$960$	0	0
$961$	8.02880	0.258994
$962$	0	0
$963$	12.7992	0.412450
$964$	0	0
$965$	−9.18303	−0.295612
$966$	0	0
$967$	−4.88224	−0.157002	−0.0785011	−	0.996914i	$-0.525013\pi$
$967$	−4.88224	−0.157002	−0.0785011	+	0.996914i	$0.525013\pi$
$968$	0	0
$969$	4.24730	0.136443
$970$	0	0
$971$	−1.50156	−0.0481873	−0.0240936	−	0.999710i	$-0.507670\pi$
$971$	−1.50156	−0.0481873	−0.0240936	+	0.999710i	$0.507670\pi$
$972$	0	0
$973$	6.34421	0.203386
$974$	0	0
$975$	26.7419	0.856427
$976$	0	0
$977$	9.47276	0.303060	0.151530	−	0.988453i	$-0.451580\pi$
$977$	9.47276	0.303060	0.151530	+	0.988453i	$0.451580\pi$
$978$	0	0
$979$	−1.00695	−0.0321823
$980$	0	0
$981$	−7.94268	−0.253590
$982$	0	0
$983$	12.3404	0.393597	0.196798	−	0.980444i	$-0.436946\pi$
$983$	12.3404	0.393597	0.196798	+	0.980444i	$0.436946\pi$
$984$	0	0
$985$	5.96041	0.189915
$986$	0	0
$987$	−8.14344	−0.259209
$988$	0	0
$989$	−0.799233	−0.0254141
$990$	0	0
$991$	18.6064	0.591052	0.295526	−	0.955335i	$-0.404505\pi$
$991$	18.6064	0.591052	0.295526	+	0.955335i	$0.404505\pi$
$992$	0	0
$993$	1.24730	0.0395820
$994$	0	0
$995$	−9.10770	−0.288733
$996$	0	0
$997$	11.0288	0.349286	0.174643	−	0.984632i	$-0.444123\pi$
$997$	11.0288	0.349286	0.174643	+	0.984632i	$0.444123\pi$
$998$	0	0
$999$	−3.35116	−0.106026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.i.1.2	✓	3
3.2	odd	2		5796.2.a.p.1.2		3
4.3	odd	2		7728.2.a.bv.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.i.1.2	✓	3	1.1	even	1	trivial
5796.2.a.p.1.2		3	3.2	odd	2
7728.2.a.bv.1.2		3	4.3	odd	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1932.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.551929 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.551929 q^{5} +1.00000 q^{7} +1.00000 q^{9} -0.103859 q^{11} +5.69537 q^{13} +0.551929 q^{15} -4.24730 q^{17} +1.00000 q^{19} -1.00000 q^{21} +1.00000 q^{23} -4.69537 q^{25} -1.00000 q^{27} +1.14344 q^{29} +6.24730 q^{31} +0.103859 q^{33} -0.551929 q^{35} +3.35116 q^{37} -5.69537 q^{39} -9.28689 q^{41} -0.799233 q^{43} -0.551929 q^{45} +8.14344 q^{47} +1.00000 q^{49} +4.24730 q^{51} +0.695374 q^{53} +0.0573228 q^{55} -1.00000 q^{57} -0.695374 q^{59} +7.55193 q^{61} +1.00000 q^{63} -3.14344 q^{65} -0.551929 q^{67} -1.00000 q^{69} +12.1900 q^{71} -3.14344 q^{73} +4.69537 q^{75} -0.103859 q^{77} +8.45502 q^{79} +1.00000 q^{81} -1.14344 q^{83} +2.34421 q^{85} -1.14344 q^{87} +9.69537 q^{89} +5.69537 q^{91} -6.24730 q^{93} -0.551929 q^{95} +17.6381 q^{97} -0.103859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} + q^{11} + 3 q^{13} + q^{15} + 2 q^{17} + 3 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} - 10 q^{29} + 4 q^{31} - q^{33} - q^{35} - 6 q^{37} - 3 q^{39} - q^{41} + 13 q^{43} - q^{45} + 11 q^{47} + 3 q^{49} - 2 q^{51} - 12 q^{53} + 29 q^{55} - 3 q^{57} + 12 q^{59} + 22 q^{61} + 3 q^{63} + 4 q^{65} - q^{67} - 3 q^{69} - 7 q^{71} + 4 q^{73} + q^{77} + 8 q^{79} + 3 q^{81} + 10 q^{83} + 9 q^{85} + 10 q^{87} + 15 q^{89} + 3 q^{91} - 4 q^{93} - q^{95} + 10 q^{97} + q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^5 + 3 * q^7 + 3 * q^9 + q^11 + 3 * q^13 + q^15 + 2 * q^17 + 3 * q^19 - 3 * q^21 + 3 * q^23 - 3 * q^27 - 10 * q^29 + 4 * q^31 - q^33 - q^35 - 6 * q^37 - 3 * q^39 - q^41 + 13 * q^43 - q^45 + 11 * q^47 + 3 * q^49 - 2 * q^51 - 12 * q^53 + 29 * q^55 - 3 * q^57 + 12 * q^59 + 22 * q^61 + 3 * q^63 + 4 * q^65 - q^67 - 3 * q^69 - 7 * q^71 + 4 * q^73 + q^77 + 8 * q^79 + 3 * q^81 + 10 * q^83 + 9 * q^85 + 10 * q^87 + 15 * q^89 + 3 * q^91 - 4 * q^93 - q^95 + 10 * q^97 + q^99

Introduction

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