LMFDB - Embedded newform 1932.2.a.h.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -2.61803 q^{13} -2.85410 q^{15} +7.47214 q^{17} -4.23607 q^{19} +1.00000 q^{21} -1.00000 q^{23} +3.14590 q^{25} +1.00000 q^{27} -5.00000 q^{29} +10.2361 q^{31} +3.00000 q^{33} -2.85410 q^{35} +0.236068 q^{37} -2.61803 q^{39} +3.00000 q^{41} +0.618034 q^{43} -2.85410 q^{45} +1.23607 q^{47} +1.00000 q^{49} +7.47214 q^{51} +13.3262 q^{53} -8.56231 q^{55} -4.23607 q^{57} +6.56231 q^{59} +1.85410 q^{61} +1.00000 q^{63} +7.47214 q^{65} -13.3262 q^{67} -1.00000 q^{69} +5.38197 q^{71} +11.9443 q^{73} +3.14590 q^{75} +3.00000 q^{77} -0.527864 q^{79} +1.00000 q^{81} +7.76393 q^{83} -21.3262 q^{85} -5.00000 q^{87} +13.5623 q^{89} -2.61803 q^{91} +10.2361 q^{93} +12.0902 q^{95} +15.9443 q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 13 q^{25} + 2 q^{27} - 10 q^{29} + 16 q^{31} + 6 q^{33} + q^{35} - 4 q^{37} - 3 q^{39} + 6 q^{41} - q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 6 q^{51} + 11 q^{53} + 3 q^{55} - 4 q^{57} - 7 q^{59} - 3 q^{61} + 2 q^{63} + 6 q^{65} - 11 q^{67} - 2 q^{69} + 13 q^{71} + 6 q^{73} + 13 q^{75} + 6 q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} - 27 q^{85} - 10 q^{87} + 7 q^{89} - 3 q^{91} + 16 q^{93} + 13 q^{95} + 14 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 3 * q^13 + q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 13 * q^25 + 2 * q^27 - 10 * q^29 + 16 * q^31 + 6 * q^33 + q^35 - 4 * q^37 - 3 * q^39 + 6 * q^41 - q^43 + q^45 - 2 * q^47 + 2 * q^49 + 6 * q^51 + 11 * q^53 + 3 * q^55 - 4 * q^57 - 7 * q^59 - 3 * q^61 + 2 * q^63 + 6 * q^65 - 11 * q^67 - 2 * q^69 + 13 * q^71 + 6 * q^73 + 13 * q^75 + 6 * q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 - 27 * q^85 - 10 * q^87 + 7 * q^89 - 3 * q^91 + 16 * q^93 + 13 * q^95 + 14 * q^97 + 6 * 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
$5$	−2.85410	−1.27639	−0.638197	−	0.769873i	$-0.720319\pi$
$5$	−2.85410	−1.27639	−0.638197	+	0.769873i	$0.720319\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−2.61803	−0.726112	−0.363056	−	0.931767i	$-0.618267\pi$
$13$	−2.61803	−0.726112	−0.363056	+	0.931767i	$0.618267\pi$
$14$	0	0
$15$	−2.85410	−0.736926
$16$	0	0
$17$	7.47214	1.81226	0.906130	−	0.423000i	$-0.139023\pi$
$17$	7.47214	1.81226	0.906130	+	0.423000i	$0.139023\pi$
$18$	0	0
$19$	−4.23607	−0.971821	−0.485910	−	0.874009i	$-0.661512\pi$
$19$	−4.23607	−0.971821	−0.485910	+	0.874009i	$0.661512\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.14590	0.629180
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	10.2361	1.83845	0.919226	−	0.393730i	$-0.128816\pi$
$31$	10.2361	1.83845	0.919226	+	0.393730i	$0.128816\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−2.85410	−0.482431
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−2.61803	−0.419221
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	0.618034	0.0942493	0.0471246	−	0.998889i	$-0.484994\pi$
$43$	0.618034	0.0942493	0.0471246	+	0.998889i	$0.484994\pi$
$44$	0	0
$45$	−2.85410	−0.425464
$46$	0	0
$47$	1.23607	0.180299	0.0901495	−	0.995928i	$-0.471266\pi$
$47$	1.23607	0.180299	0.0901495	+	0.995928i	$0.471266\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.47214	1.04631
$52$	0	0
$53$	13.3262	1.83050	0.915250	−	0.402887i	$-0.131993\pi$
$53$	13.3262	1.83050	0.915250	+	0.402887i	$0.131993\pi$
$54$	0	0
$55$	−8.56231	−1.15454
$56$	0	0
$57$	−4.23607	−0.561081
$58$	0	0
$59$	6.56231	0.854339	0.427170	−	0.904171i	$-0.359511\pi$
$59$	6.56231	0.854339	0.427170	+	0.904171i	$0.359511\pi$
$60$	0	0
$61$	1.85410	0.237393	0.118697	−	0.992931i	$-0.462128\pi$
$61$	1.85410	0.237393	0.118697	+	0.992931i	$0.462128\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	7.47214	0.926804
$66$	0	0
$67$	−13.3262	−1.62806	−0.814030	−	0.580823i	$-0.802731\pi$
$67$	−13.3262	−1.62806	−0.814030	+	0.580823i	$0.802731\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	5.38197	0.638722	0.319361	−	0.947633i	$-0.396532\pi$
$71$	5.38197	0.638722	0.319361	+	0.947633i	$0.396532\pi$
$72$	0	0
$73$	11.9443	1.39797	0.698986	−	0.715136i	$-0.253635\pi$
$73$	11.9443	1.39797	0.698986	+	0.715136i	$0.253635\pi$
$74$	0	0
$75$	3.14590	0.363257
$76$	0	0
$77$	3.00000	0.341882
$78$	0	0
$79$	−0.527864	−0.0593893	−0.0296947	−	0.999559i	$-0.509453\pi$
$79$	−0.527864	−0.0593893	−0.0296947	+	0.999559i	$0.509453\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.76393	0.852202	0.426101	−	0.904676i	$-0.359887\pi$
$83$	7.76393	0.852202	0.426101	+	0.904676i	$0.359887\pi$
$84$	0	0
$85$	−21.3262	−2.31316
$86$	0	0
$87$	−5.00000	−0.536056
$88$	0	0
$89$	13.5623	1.43760	0.718801	−	0.695216i	$-0.244691\pi$
$89$	13.5623	1.43760	0.718801	+	0.695216i	$0.244691\pi$
$90$	0	0
$91$	−2.61803	−0.274445
$92$	0	0
$93$	10.2361	1.06143
$94$	0	0
$95$	12.0902	1.24043
$96$	0	0
$97$	15.9443	1.61890	0.809448	−	0.587192i	$-0.199767\pi$
$97$	15.9443	1.61890	0.809448	+	0.587192i	$0.199767\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	12.5623	1.25000	0.624998	−	0.780626i	$-0.285100\pi$
$101$	12.5623	1.25000	0.624998	+	0.780626i	$0.285100\pi$
$102$	0	0
$103$	−4.94427	−0.487174	−0.243587	−	0.969879i	$-0.578324\pi$
$103$	−4.94427	−0.487174	−0.243587	+	0.969879i	$0.578324\pi$
$104$	0	0
$105$	−2.85410	−0.278532
$106$	0	0
$107$	−7.56231	−0.731076	−0.365538	−	0.930796i	$-0.619115\pi$
$107$	−7.56231	−0.731076	−0.365538	+	0.930796i	$0.619115\pi$
$108$	0	0
$109$	−16.0902	−1.54116	−0.770579	−	0.637344i	$-0.780033\pi$
$109$	−16.0902	−1.54116	−0.770579	+	0.637344i	$0.780033\pi$
$110$	0	0
$111$	0.236068	0.0224066
$112$	0	0
$113$	21.0344	1.97875	0.989377	−	0.145373i	$-0.0464382\pi$
$113$	21.0344	1.97875	0.989377	+	0.145373i	$0.0464382\pi$
$114$	0	0
$115$	2.85410	0.266146
$116$	0	0
$117$	−2.61803	−0.242037
$118$	0	0
$119$	7.47214	0.684970
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	3.00000	0.270501
$124$	0	0
$125$	5.29180	0.473313
$126$	0	0
$127$	−12.3262	−1.09378	−0.546888	−	0.837206i	$-0.684188\pi$
$127$	−12.3262	−1.09378	−0.546888	+	0.837206i	$0.684188\pi$
$128$	0	0
$129$	0.618034	0.0544149
$130$	0	0
$131$	3.00000	0.262111	0.131056	−	0.991375i	$-0.458163\pi$
$131$	3.00000	0.262111	0.131056	+	0.991375i	$0.458163\pi$
$132$	0	0
$133$	−4.23607	−0.367314
$134$	0	0
$135$	−2.85410	−0.245642
$136$	0	0
$137$	−18.4164	−1.57342	−0.786710	−	0.617323i	$-0.788217\pi$
$137$	−18.4164	−1.57342	−0.786710	+	0.617323i	$0.788217\pi$
$138$	0	0
$139$	13.7984	1.17036	0.585181	−	0.810902i	$-0.301023\pi$
$139$	13.7984	1.17036	0.585181	+	0.810902i	$0.301023\pi$
$140$	0	0
$141$	1.23607	0.104096
$142$	0	0
$143$	−7.85410	−0.656793
$144$	0	0
$145$	14.2705	1.18510
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−6.18034	−0.506313	−0.253157	−	0.967425i	$-0.581469\pi$
$149$	−6.18034	−0.506313	−0.253157	+	0.967425i	$0.581469\pi$
$150$	0	0
$151$	8.65248	0.704128	0.352064	−	0.935976i	$-0.385480\pi$
$151$	8.65248	0.704128	0.352064	+	0.935976i	$0.385480\pi$
$152$	0	0
$153$	7.47214	0.604086
$154$	0	0
$155$	−29.2148	−2.34659
$156$	0	0
$157$	−11.2361	−0.896736	−0.448368	−	0.893849i	$-0.647995\pi$
$157$	−11.2361	−0.896736	−0.448368	+	0.893849i	$0.647995\pi$
$158$	0	0
$159$	13.3262	1.05684
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	21.3262	1.67040	0.835200	−	0.549946i	$-0.185352\pi$
$163$	21.3262	1.67040	0.835200	+	0.549946i	$0.185352\pi$
$164$	0	0
$165$	−8.56231	−0.666575
$166$	0	0
$167$	−18.4164	−1.42510	−0.712552	−	0.701619i	$-0.752461\pi$
$167$	−18.4164	−1.42510	−0.712552	+	0.701619i	$0.752461\pi$
$168$	0	0
$169$	−6.14590	−0.472761
$170$	0	0
$171$	−4.23607	−0.323940
$172$	0	0
$173$	−8.52786	−0.648361	−0.324181	−	0.945995i	$-0.605089\pi$
$173$	−8.52786	−0.648361	−0.324181	+	0.945995i	$0.605089\pi$
$174$	0	0
$175$	3.14590	0.237808
$176$	0	0
$177$	6.56231	0.493253
$178$	0	0
$179$	1.43769	0.107458	0.0537292	−	0.998556i	$-0.482889\pi$
$179$	1.43769	0.107458	0.0537292	+	0.998556i	$0.482889\pi$
$180$	0	0
$181$	−20.4164	−1.51754	−0.758770	−	0.651359i	$-0.774199\pi$
$181$	−20.4164	−1.51754	−0.758770	+	0.651359i	$0.774199\pi$
$182$	0	0
$183$	1.85410	0.137059
$184$	0	0
$185$	−0.673762	−0.0495360
$186$	0	0
$187$	22.4164	1.63925
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	17.2361	1.24716	0.623579	−	0.781760i	$-0.285678\pi$
$191$	17.2361	1.24716	0.623579	+	0.781760i	$0.285678\pi$
$192$	0	0
$193$	−6.65248	−0.478856	−0.239428	−	0.970914i	$-0.576960\pi$
$193$	−6.65248	−0.478856	−0.239428	+	0.970914i	$0.576960\pi$
$194$	0	0
$195$	7.47214	0.535091
$196$	0	0
$197$	−7.09017	−0.505154	−0.252577	−	0.967577i	$-0.581278\pi$
$197$	−7.09017	−0.505154	−0.252577	+	0.967577i	$0.581278\pi$
$198$	0	0
$199$	−13.7984	−0.978141	−0.489070	−	0.872244i	$-0.662664\pi$
$199$	−13.7984	−0.978141	−0.489070	+	0.872244i	$0.662664\pi$
$200$	0	0
$201$	−13.3262	−0.939960
$202$	0	0
$203$	−5.00000	−0.350931
$204$	0	0
$205$	−8.56231	−0.598017
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−12.7082	−0.879045
$210$	0	0
$211$	−0.236068	−0.0162516	−0.00812579	−	0.999967i	$-0.502587\pi$
$211$	−0.236068	−0.0162516	−0.00812579	+	0.999967i	$0.502587\pi$
$212$	0	0
$213$	5.38197	0.368766
$214$	0	0
$215$	−1.76393	−0.120299
$216$	0	0
$217$	10.2361	0.694870
$218$	0	0
$219$	11.9443	0.807119
$220$	0	0
$221$	−19.5623	−1.31590
$222$	0	0
$223$	11.3262	0.758461	0.379230	−	0.925302i	$-0.376189\pi$
$223$	11.3262	0.758461	0.379230	+	0.925302i	$0.376189\pi$
$224$	0	0
$225$	3.14590	0.209727
$226$	0	0
$227$	−1.61803	−0.107393	−0.0536963	−	0.998557i	$-0.517100\pi$
$227$	−1.61803	−0.107393	−0.0536963	+	0.998557i	$0.517100\pi$
$228$	0	0
$229$	−22.5623	−1.49096	−0.745480	−	0.666528i	$-0.767779\pi$
$229$	−22.5623	−1.49096	−0.745480	+	0.666528i	$0.767779\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	−22.8541	−1.49722	−0.748611	−	0.663009i	$-0.769279\pi$
$233$	−22.8541	−1.49722	−0.748611	+	0.663009i	$0.769279\pi$
$234$	0	0
$235$	−3.52786	−0.230132
$236$	0	0
$237$	−0.527864	−0.0342885
$238$	0	0
$239$	−21.3262	−1.37948	−0.689740	−	0.724057i	$-0.742275\pi$
$239$	−21.3262	−1.37948	−0.689740	+	0.724057i	$0.742275\pi$
$240$	0	0
$241$	−19.9443	−1.28472	−0.642362	−	0.766402i	$-0.722045\pi$
$241$	−19.9443	−1.28472	−0.642362	+	0.766402i	$0.722045\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.85410	−0.182342
$246$	0	0
$247$	11.0902	0.705651
$248$	0	0
$249$	7.76393	0.492019
$250$	0	0
$251$	8.76393	0.553174	0.276587	−	0.960989i	$-0.410797\pi$
$251$	8.76393	0.553174	0.276587	+	0.960989i	$0.410797\pi$
$252$	0	0
$253$	−3.00000	−0.188608
$254$	0	0
$255$	−21.3262	−1.33550
$256$	0	0
$257$	−5.23607	−0.326617	−0.163308	−	0.986575i	$-0.552217\pi$
$257$	−5.23607	−0.326617	−0.163308	+	0.986575i	$0.552217\pi$
$258$	0	0
$259$	0.236068	0.0146686
$260$	0	0
$261$	−5.00000	−0.309492
$262$	0	0
$263$	30.7082	1.89355	0.946774	−	0.321898i	$-0.104321\pi$
$263$	30.7082	1.89355	0.946774	+	0.321898i	$0.104321\pi$
$264$	0	0
$265$	−38.0344	−2.33644
$266$	0	0
$267$	13.5623	0.830000
$268$	0	0
$269$	24.3262	1.48320	0.741598	−	0.670844i	$-0.234068\pi$
$269$	24.3262	1.48320	0.741598	+	0.670844i	$0.234068\pi$
$270$	0	0
$271$	−2.41641	−0.146786	−0.0733932	−	0.997303i	$-0.523383\pi$
$271$	−2.41641	−0.146786	−0.0733932	+	0.997303i	$0.523383\pi$
$272$	0	0
$273$	−2.61803	−0.158451
$274$	0	0
$275$	9.43769	0.569114
$276$	0	0
$277$	−12.3262	−0.740612	−0.370306	−	0.928910i	$-0.620747\pi$
$277$	−12.3262	−0.740612	−0.370306	+	0.928910i	$0.620747\pi$
$278$	0	0
$279$	10.2361	0.612817
$280$	0	0
$281$	21.1246	1.26019	0.630094	−	0.776519i	$-0.283016\pi$
$281$	21.1246	1.26019	0.630094	+	0.776519i	$0.283016\pi$
$282$	0	0
$283$	3.03444	0.180379	0.0901894	−	0.995925i	$-0.471253\pi$
$283$	3.03444	0.180379	0.0901894	+	0.995925i	$0.471253\pi$
$284$	0	0
$285$	12.0902	0.716160
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	38.8328	2.28428
$290$	0	0
$291$	15.9443	0.934670
$292$	0	0
$293$	−4.94427	−0.288847	−0.144424	−	0.989516i	$-0.546133\pi$
$293$	−4.94427	−0.288847	−0.144424	+	0.989516i	$0.546133\pi$
$294$	0	0
$295$	−18.7295	−1.09047
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	2.61803	0.151405
$300$	0	0
$301$	0.618034	0.0356229
$302$	0	0
$303$	12.5623	0.721686
$304$	0	0
$305$	−5.29180	−0.303007
$306$	0	0
$307$	7.65248	0.436750	0.218375	−	0.975865i	$-0.429924\pi$
$307$	7.65248	0.436750	0.218375	+	0.975865i	$0.429924\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	0	0
$311$	−18.6180	−1.05573	−0.527866	−	0.849328i	$-0.677008\pi$
$311$	−18.6180	−1.05573	−0.527866	+	0.849328i	$0.677008\pi$
$312$	0	0
$313$	−12.0000	−0.678280	−0.339140	−	0.940736i	$-0.610136\pi$
$313$	−12.0000	−0.678280	−0.339140	+	0.940736i	$0.610136\pi$
$314$	0	0
$315$	−2.85410	−0.160810
$316$	0	0
$317$	0.965558	0.0542311	0.0271156	−	0.999632i	$-0.491368\pi$
$317$	0.965558	0.0542311	0.0271156	+	0.999632i	$0.491368\pi$
$318$	0	0
$319$	−15.0000	−0.839839
$320$	0	0
$321$	−7.56231	−0.422087
$322$	0	0
$323$	−31.6525	−1.76119
$324$	0	0
$325$	−8.23607	−0.456855
$326$	0	0
$327$	−16.0902	−0.889788
$328$	0	0
$329$	1.23607	0.0681466
$330$	0	0
$331$	26.3607	1.44891	0.724457	−	0.689320i	$-0.242091\pi$
$331$	26.3607	1.44891	0.724457	+	0.689320i	$0.242091\pi$
$332$	0	0
$333$	0.236068	0.0129364
$334$	0	0
$335$	38.0344	2.07804
$336$	0	0
$337$	−28.7984	−1.56875	−0.784374	−	0.620289i	$-0.787015\pi$
$337$	−28.7984	−1.56875	−0.784374	+	0.620289i	$0.787015\pi$
$338$	0	0
$339$	21.0344	1.14243
$340$	0	0
$341$	30.7082	1.66294
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.85410	0.153660
$346$	0	0
$347$	20.5967	1.10569	0.552846	−	0.833284i	$-0.313542\pi$
$347$	20.5967	1.10569	0.552846	+	0.833284i	$0.313542\pi$
$348$	0	0
$349$	28.4508	1.52294	0.761470	−	0.648201i	$-0.224478\pi$
$349$	28.4508	1.52294	0.761470	+	0.648201i	$0.224478\pi$
$350$	0	0
$351$	−2.61803	−0.139740
$352$	0	0
$353$	3.58359	0.190735	0.0953677	−	0.995442i	$-0.469597\pi$
$353$	3.58359	0.190735	0.0953677	+	0.995442i	$0.469597\pi$
$354$	0	0
$355$	−15.3607	−0.815260
$356$	0	0
$357$	7.47214	0.395467
$358$	0	0
$359$	−17.1459	−0.904926	−0.452463	−	0.891783i	$-0.649455\pi$
$359$	−17.1459	−0.904926	−0.452463	+	0.891783i	$0.649455\pi$
$360$	0	0
$361$	−1.05573	−0.0555646
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	−34.0902	−1.78436
$366$	0	0
$367$	21.5623	1.12554	0.562772	−	0.826612i	$-0.309735\pi$
$367$	21.5623	1.12554	0.562772	+	0.826612i	$0.309735\pi$
$368$	0	0
$369$	3.00000	0.156174
$370$	0	0
$371$	13.3262	0.691864
$372$	0	0
$373$	−11.7639	−0.609113	−0.304557	−	0.952494i	$-0.598508\pi$
$373$	−11.7639	−0.609113	−0.304557	+	0.952494i	$0.598508\pi$
$374$	0	0
$375$	5.29180	0.273267
$376$	0	0
$377$	13.0902	0.674178
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−12.3262	−0.631492
$382$	0	0
$383$	10.7082	0.547164	0.273582	−	0.961849i	$-0.411792\pi$
$383$	10.7082	0.547164	0.273582	+	0.961849i	$0.411792\pi$
$384$	0	0
$385$	−8.56231	−0.436376
$386$	0	0
$387$	0.618034	0.0314164
$388$	0	0
$389$	7.18034	0.364058	0.182029	−	0.983293i	$-0.441734\pi$
$389$	7.18034	0.364058	0.182029	+	0.983293i	$0.441734\pi$
$390$	0	0
$391$	−7.47214	−0.377882
$392$	0	0
$393$	3.00000	0.151330
$394$	0	0
$395$	1.50658	0.0758042
$396$	0	0
$397$	9.23607	0.463545	0.231772	−	0.972770i	$-0.425548\pi$
$397$	9.23607	0.463545	0.231772	+	0.972770i	$0.425548\pi$
$398$	0	0
$399$	−4.23607	−0.212069
$400$	0	0
$401$	−31.5410	−1.57508	−0.787542	−	0.616261i	$-0.788646\pi$
$401$	−31.5410	−1.57508	−0.787542	+	0.616261i	$0.788646\pi$
$402$	0	0
$403$	−26.7984	−1.33492
$404$	0	0
$405$	−2.85410	−0.141821
$406$	0	0
$407$	0.708204	0.0351044
$408$	0	0
$409$	−21.1803	−1.04730	−0.523650	−	0.851933i	$-0.675430\pi$
$409$	−21.1803	−1.04730	−0.523650	+	0.851933i	$0.675430\pi$
$410$	0	0
$411$	−18.4164	−0.908414
$412$	0	0
$413$	6.56231	0.322910
$414$	0	0
$415$	−22.1591	−1.08775
$416$	0	0
$417$	13.7984	0.675709
$418$	0	0
$419$	−31.6869	−1.54801	−0.774004	−	0.633181i	$-0.781749\pi$
$419$	−31.6869	−1.54801	−0.774004	+	0.633181i	$0.781749\pi$
$420$	0	0
$421$	−2.72949	−0.133027	−0.0665136	−	0.997786i	$-0.521188\pi$
$421$	−2.72949	−0.133027	−0.0665136	+	0.997786i	$0.521188\pi$
$422$	0	0
$423$	1.23607	0.0600997
$424$	0	0
$425$	23.5066	1.14024
$426$	0	0
$427$	1.85410	0.0897263
$428$	0	0
$429$	−7.85410	−0.379200
$430$	0	0
$431$	−1.96556	−0.0946776	−0.0473388	−	0.998879i	$-0.515074\pi$
$431$	−1.96556	−0.0946776	−0.0473388	+	0.998879i	$0.515074\pi$
$432$	0	0
$433$	−2.29180	−0.110137	−0.0550683	−	0.998483i	$-0.517538\pi$
$433$	−2.29180	−0.110137	−0.0550683	+	0.998483i	$0.517538\pi$
$434$	0	0
$435$	14.2705	0.684219
$436$	0	0
$437$	4.23607	0.202639
$438$	0	0
$439$	−5.18034	−0.247244	−0.123622	−	0.992329i	$-0.539451\pi$
$439$	−5.18034	−0.247244	−0.123622	+	0.992329i	$0.539451\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0.583592	0.0277273	0.0138636	−	0.999904i	$-0.495587\pi$
$443$	0.583592	0.0277273	0.0138636	+	0.999904i	$0.495587\pi$
$444$	0	0
$445$	−38.7082	−1.83494
$446$	0	0
$447$	−6.18034	−0.292320
$448$	0	0
$449$	6.27051	0.295924	0.147962	−	0.988993i	$-0.452729\pi$
$449$	6.27051	0.295924	0.147962	+	0.988993i	$0.452729\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	8.65248	0.406529
$454$	0	0
$455$	7.47214	0.350299
$456$	0	0
$457$	41.9787	1.96368	0.981841	−	0.189707i	$-0.0607539\pi$
$457$	41.9787	1.96368	0.981841	+	0.189707i	$0.0607539\pi$
$458$	0	0
$459$	7.47214	0.348769
$460$	0	0
$461$	16.5623	0.771383	0.385692	−	0.922628i	$-0.373963\pi$
$461$	16.5623	0.771383	0.385692	+	0.922628i	$0.373963\pi$
$462$	0	0
$463$	25.3607	1.17861	0.589305	−	0.807910i	$-0.299401\pi$
$463$	25.3607	1.17861	0.589305	+	0.807910i	$0.299401\pi$
$464$	0	0
$465$	−29.2148	−1.35480
$466$	0	0
$467$	−13.6525	−0.631761	−0.315881	−	0.948799i	$-0.602300\pi$
$467$	−13.6525	−0.631761	−0.315881	+	0.948799i	$0.602300\pi$
$468$	0	0
$469$	−13.3262	−0.615348
$470$	0	0
$471$	−11.2361	−0.517731
$472$	0	0
$473$	1.85410	0.0852517
$474$	0	0
$475$	−13.3262	−0.611450
$476$	0	0
$477$	13.3262	0.610167
$478$	0	0
$479$	−37.9443	−1.73372	−0.866859	−	0.498553i	$-0.833865\pi$
$479$	−37.9443	−1.73372	−0.866859	+	0.498553i	$0.833865\pi$
$480$	0	0
$481$	−0.618034	−0.0281799
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−45.5066	−2.06635
$486$	0	0
$487$	29.8328	1.35185	0.675927	−	0.736969i	$-0.263743\pi$
$487$	29.8328	1.35185	0.675927	+	0.736969i	$0.263743\pi$
$488$	0	0
$489$	21.3262	0.964406
$490$	0	0
$491$	−20.0344	−0.904142	−0.452071	−	0.891982i	$-0.649315\pi$
$491$	−20.0344	−0.904142	−0.452071	+	0.891982i	$0.649315\pi$
$492$	0	0
$493$	−37.3607	−1.68264
$494$	0	0
$495$	−8.56231	−0.384847
$496$	0	0
$497$	5.38197	0.241414
$498$	0	0
$499$	−20.4508	−0.915506	−0.457753	−	0.889079i	$-0.651346\pi$
$499$	−20.4508	−0.915506	−0.457753	+	0.889079i	$0.651346\pi$
$500$	0	0
$501$	−18.4164	−0.822784
$502$	0	0
$503$	−29.0344	−1.29458	−0.647291	−	0.762243i	$-0.724098\pi$
$503$	−29.0344	−1.29458	−0.647291	+	0.762243i	$0.724098\pi$
$504$	0	0
$505$	−35.8541	−1.59549
$506$	0	0
$507$	−6.14590	−0.272949
$508$	0	0
$509$	−12.2918	−0.544824	−0.272412	−	0.962181i	$-0.587821\pi$
$509$	−12.2918	−0.544824	−0.272412	+	0.962181i	$0.587821\pi$
$510$	0	0
$511$	11.9443	0.528383
$512$	0	0
$513$	−4.23607	−0.187027
$514$	0	0
$515$	14.1115	0.621825
$516$	0	0
$517$	3.70820	0.163087
$518$	0	0
$519$	−8.52786	−0.374332
$520$	0	0
$521$	−14.5836	−0.638919	−0.319459	−	0.947600i	$-0.603501\pi$
$521$	−14.5836	−0.638919	−0.319459	+	0.947600i	$0.603501\pi$
$522$	0	0
$523$	19.5279	0.853894	0.426947	−	0.904277i	$-0.359589\pi$
$523$	19.5279	0.853894	0.426947	+	0.904277i	$0.359589\pi$
$524$	0	0
$525$	3.14590	0.137298
$526$	0	0
$527$	76.4853	3.33175
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	6.56231	0.284780
$532$	0	0
$533$	−7.85410	−0.340199
$534$	0	0
$535$	21.5836	0.933140
$536$	0	0
$537$	1.43769	0.0620411
$538$	0	0
$539$	3.00000	0.129219
$540$	0	0
$541$	−19.7082	−0.847322	−0.423661	−	0.905821i	$-0.639255\pi$
$541$	−19.7082	−0.847322	−0.423661	+	0.905821i	$0.639255\pi$
$542$	0	0
$543$	−20.4164	−0.876152
$544$	0	0
$545$	45.9230	1.96712
$546$	0	0
$547$	−11.1459	−0.476564	−0.238282	−	0.971196i	$-0.576584\pi$
$547$	−11.1459	−0.476564	−0.238282	+	0.971196i	$0.576584\pi$
$548$	0	0
$549$	1.85410	0.0791311
$550$	0	0
$551$	21.1803	0.902313
$552$	0	0
$553$	−0.527864	−0.0224471
$554$	0	0
$555$	−0.673762	−0.0285996
$556$	0	0
$557$	−20.7639	−0.879796	−0.439898	−	0.898048i	$-0.644985\pi$
$557$	−20.7639	−0.879796	−0.439898	+	0.898048i	$0.644985\pi$
$558$	0	0
$559$	−1.61803	−0.0684355
$560$	0	0
$561$	22.4164	0.946421
$562$	0	0
$563$	−34.5066	−1.45428	−0.727139	−	0.686490i	$-0.759151\pi$
$563$	−34.5066	−1.45428	−0.727139	+	0.686490i	$0.759151\pi$
$564$	0	0
$565$	−60.0344	−2.52567
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−13.8885	−0.582238	−0.291119	−	0.956687i	$-0.594028\pi$
$569$	−13.8885	−0.582238	−0.291119	+	0.956687i	$0.594028\pi$
$570$	0	0
$571$	33.4721	1.40077	0.700383	−	0.713768i	$-0.253013\pi$
$571$	33.4721	1.40077	0.700383	+	0.713768i	$0.253013\pi$
$572$	0	0
$573$	17.2361	0.720047
$574$	0	0
$575$	−3.14590	−0.131193
$576$	0	0
$577$	−34.3607	−1.43045	−0.715227	−	0.698892i	$-0.753677\pi$
$577$	−34.3607	−1.43045	−0.715227	+	0.698892i	$0.753677\pi$
$578$	0	0
$579$	−6.65248	−0.276467
$580$	0	0
$581$	7.76393	0.322102
$582$	0	0
$583$	39.9787	1.65575
$584$	0	0
$585$	7.47214	0.308935
$586$	0	0
$587$	32.3262	1.33425	0.667123	−	0.744947i	$-0.267525\pi$
$587$	32.3262	1.33425	0.667123	+	0.744947i	$0.267525\pi$
$588$	0	0
$589$	−43.3607	−1.78665
$590$	0	0
$591$	−7.09017	−0.291651
$592$	0	0
$593$	−1.12461	−0.0461823	−0.0230911	−	0.999733i	$-0.507351\pi$
$593$	−1.12461	−0.0461823	−0.0230911	+	0.999733i	$0.507351\pi$
$594$	0	0
$595$	−21.3262	−0.874291
$596$	0	0
$597$	−13.7984	−0.564730
$598$	0	0
$599$	23.7426	0.970098	0.485049	−	0.874487i	$-0.338802\pi$
$599$	23.7426	0.970098	0.485049	+	0.874487i	$0.338802\pi$
$600$	0	0
$601$	11.4377	0.466553	0.233277	−	0.972410i	$-0.425055\pi$
$601$	11.4377	0.466553	0.233277	+	0.972410i	$0.425055\pi$
$602$	0	0
$603$	−13.3262	−0.542686
$604$	0	0
$605$	5.70820	0.232071
$606$	0	0
$607$	−2.50658	−0.101739	−0.0508694	−	0.998705i	$-0.516199\pi$
$607$	−2.50658	−0.101739	−0.0508694	+	0.998705i	$0.516199\pi$
$608$	0	0
$609$	−5.00000	−0.202610
$610$	0	0
$611$	−3.23607	−0.130917
$612$	0	0
$613$	−11.5836	−0.467857	−0.233928	−	0.972254i	$-0.575158\pi$
$613$	−11.5836	−0.467857	−0.233928	+	0.972254i	$0.575158\pi$
$614$	0	0
$615$	−8.56231	−0.345265
$616$	0	0
$617$	−15.5623	−0.626515	−0.313257	−	0.949668i	$-0.601420\pi$
$617$	−15.5623	−0.626515	−0.313257	+	0.949668i	$0.601420\pi$
$618$	0	0
$619$	−12.3262	−0.495433	−0.247717	−	0.968833i	$-0.579680\pi$
$619$	−12.3262	−0.495433	−0.247717	+	0.968833i	$0.579680\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	13.5623	0.543362
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	−12.7082	−0.507517
$628$	0	0
$629$	1.76393	0.0703326
$630$	0	0
$631$	19.6525	0.782353	0.391176	−	0.920316i	$-0.372068\pi$
$631$	19.6525	0.782353	0.391176	+	0.920316i	$0.372068\pi$
$632$	0	0
$633$	−0.236068	−0.00938286
$634$	0	0
$635$	35.1803	1.39609
$636$	0	0
$637$	−2.61803	−0.103730
$638$	0	0
$639$	5.38197	0.212907
$640$	0	0
$641$	−32.9787	−1.30258	−0.651290	−	0.758829i	$-0.725772\pi$
$641$	−32.9787	−1.30258	−0.651290	+	0.758829i	$0.725772\pi$
$642$	0	0
$643$	−31.7426	−1.25181	−0.625904	−	0.779900i	$-0.715270\pi$
$643$	−31.7426	−1.25181	−0.625904	+	0.779900i	$0.715270\pi$
$644$	0	0
$645$	−1.76393	−0.0694548
$646$	0	0
$647$	−13.3262	−0.523908	−0.261954	−	0.965080i	$-0.584367\pi$
$647$	−13.3262	−0.523908	−0.261954	+	0.965080i	$0.584367\pi$
$648$	0	0
$649$	19.6869	0.772779
$650$	0	0
$651$	10.2361	0.401183
$652$	0	0
$653$	8.56231	0.335069	0.167534	−	0.985866i	$-0.446419\pi$
$653$	8.56231	0.335069	0.167534	+	0.985866i	$0.446419\pi$
$654$	0	0
$655$	−8.56231	−0.334557
$656$	0	0
$657$	11.9443	0.465990
$658$	0	0
$659$	−13.0000	−0.506408	−0.253204	−	0.967413i	$-0.581484\pi$
$659$	−13.0000	−0.506408	−0.253204	+	0.967413i	$0.581484\pi$
$660$	0	0
$661$	7.58359	0.294968	0.147484	−	0.989064i	$-0.452883\pi$
$661$	7.58359	0.294968	0.147484	+	0.989064i	$0.452883\pi$
$662$	0	0
$663$	−19.5623	−0.759737
$664$	0	0
$665$	12.0902	0.468837
$666$	0	0
$667$	5.00000	0.193601
$668$	0	0
$669$	11.3262	0.437898
$670$	0	0
$671$	5.56231	0.214730
$672$	0	0
$673$	4.41641	0.170240	0.0851200	−	0.996371i	$-0.472873\pi$
$673$	4.41641	0.170240	0.0851200	+	0.996371i	$0.472873\pi$
$674$	0	0
$675$	3.14590	0.121086
$676$	0	0
$677$	−41.0902	−1.57922	−0.789612	−	0.613607i	$-0.789718\pi$
$677$	−41.0902	−1.57922	−0.789612	+	0.613607i	$0.789718\pi$
$678$	0	0
$679$	15.9443	0.611885
$680$	0	0
$681$	−1.61803	−0.0620032
$682$	0	0
$683$	25.0689	0.959234	0.479617	−	0.877478i	$-0.340776\pi$
$683$	25.0689	0.959234	0.479617	+	0.877478i	$0.340776\pi$
$684$	0	0
$685$	52.5623	2.00830
$686$	0	0
$687$	−22.5623	−0.860806
$688$	0	0
$689$	−34.8885	−1.32915
$690$	0	0
$691$	−0.326238	−0.0124107	−0.00620534	−	0.999981i	$-0.501975\pi$
$691$	−0.326238	−0.0124107	−0.00620534	+	0.999981i	$0.501975\pi$
$692$	0	0
$693$	3.00000	0.113961
$694$	0	0
$695$	−39.3820	−1.49384
$696$	0	0
$697$	22.4164	0.849082
$698$	0	0
$699$	−22.8541	−0.864422
$700$	0	0
$701$	20.5623	0.776628	0.388314	−	0.921527i	$-0.373058\pi$
$701$	20.5623	0.776628	0.388314	+	0.921527i	$0.373058\pi$
$702$	0	0
$703$	−1.00000	−0.0377157
$704$	0	0
$705$	−3.52786	−0.132867
$706$	0	0
$707$	12.5623	0.472454
$708$	0	0
$709$	−50.3262	−1.89004	−0.945021	−	0.327010i	$-0.893959\pi$
$709$	−50.3262	−1.89004	−0.945021	+	0.327010i	$0.893959\pi$
$710$	0	0
$711$	−0.527864	−0.0197964
$712$	0	0
$713$	−10.2361	−0.383344
$714$	0	0
$715$	22.4164	0.838326
$716$	0	0
$717$	−21.3262	−0.796443
$718$	0	0
$719$	23.7639	0.886245	0.443123	−	0.896461i	$-0.353871\pi$
$719$	23.7639	0.886245	0.443123	+	0.896461i	$0.353871\pi$
$720$	0	0
$721$	−4.94427	−0.184134
$722$	0	0
$723$	−19.9443	−0.741735
$724$	0	0
$725$	−15.7295	−0.584179
$726$	0	0
$727$	−41.0689	−1.52316	−0.761580	−	0.648071i	$-0.775576\pi$
$727$	−41.0689	−1.52316	−0.761580	+	0.648071i	$0.775576\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.61803	0.170804
$732$	0	0
$733$	39.5967	1.46254	0.731270	−	0.682089i	$-0.238928\pi$
$733$	39.5967	1.46254	0.731270	+	0.682089i	$0.238928\pi$
$734$	0	0
$735$	−2.85410	−0.105275
$736$	0	0
$737$	−39.9787	−1.47263
$738$	0	0
$739$	−12.8197	−0.471579	−0.235789	−	0.971804i	$-0.575768\pi$
$739$	−12.8197	−0.471579	−0.235789	+	0.971804i	$0.575768\pi$
$740$	0	0
$741$	11.0902	0.407408
$742$	0	0
$743$	14.8541	0.544944	0.272472	−	0.962164i	$-0.412159\pi$
$743$	14.8541	0.544944	0.272472	+	0.962164i	$0.412159\pi$
$744$	0	0
$745$	17.6393	0.646255
$746$	0	0
$747$	7.76393	0.284067
$748$	0	0
$749$	−7.56231	−0.276321
$750$	0	0
$751$	−35.5066	−1.29565	−0.647827	−	0.761788i	$-0.724322\pi$
$751$	−35.5066	−1.29565	−0.647827	+	0.761788i	$0.724322\pi$
$752$	0	0
$753$	8.76393	0.319375
$754$	0	0
$755$	−24.6950	−0.898745
$756$	0	0
$757$	19.1803	0.697121	0.348561	−	0.937286i	$-0.386671\pi$
$757$	19.1803	0.697121	0.348561	+	0.937286i	$0.386671\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	46.4721	1.68461	0.842307	−	0.538998i	$-0.181197\pi$
$761$	46.4721	1.68461	0.842307	+	0.538998i	$0.181197\pi$
$762$	0	0
$763$	−16.0902	−0.582503
$764$	0	0
$765$	−21.3262	−0.771052
$766$	0	0
$767$	−17.1803	−0.620346
$768$	0	0
$769$	22.0689	0.795824	0.397912	−	0.917424i	$-0.369735\pi$
$769$	22.0689	0.795824	0.397912	+	0.917424i	$0.369735\pi$
$770$	0	0
$771$	−5.23607	−0.188572
$772$	0	0
$773$	−0.236068	−0.00849077	−0.00424539	−	0.999991i	$-0.501351\pi$
$773$	−0.236068	−0.00849077	−0.00424539	+	0.999991i	$0.501351\pi$
$774$	0	0
$775$	32.2016	1.15672
$776$	0	0
$777$	0.236068	0.00846889
$778$	0	0
$779$	−12.7082	−0.455319
$780$	0	0
$781$	16.1459	0.577746
$782$	0	0
$783$	−5.00000	−0.178685
$784$	0	0
$785$	32.0689	1.14459
$786$	0	0
$787$	42.5623	1.51718	0.758591	−	0.651567i	$-0.225888\pi$
$787$	42.5623	1.51718	0.758591	+	0.651567i	$0.225888\pi$
$788$	0	0
$789$	30.7082	1.09324
$790$	0	0
$791$	21.0344	0.747899
$792$	0	0
$793$	−4.85410	−0.172374
$794$	0	0
$795$	−38.0344	−1.34894
$796$	0	0
$797$	31.7082	1.12316	0.561581	−	0.827422i	$-0.310193\pi$
$797$	31.7082	1.12316	0.561581	+	0.827422i	$0.310193\pi$
$798$	0	0
$799$	9.23607	0.326749
$800$	0	0
$801$	13.5623	0.479201
$802$	0	0
$803$	35.8328	1.26451
$804$	0	0
$805$	2.85410	0.100594
$806$	0	0
$807$	24.3262	0.856324
$808$	0	0
$809$	19.2016	0.675093	0.337547	−	0.941309i	$-0.390403\pi$
$809$	19.2016	0.675093	0.337547	+	0.941309i	$0.390403\pi$
$810$	0	0
$811$	42.7771	1.50211	0.751053	−	0.660242i	$-0.229546\pi$
$811$	42.7771	1.50211	0.751053	+	0.660242i	$0.229546\pi$
$812$	0	0
$813$	−2.41641	−0.0847471
$814$	0	0
$815$	−60.8673	−2.13209
$816$	0	0
$817$	−2.61803	−0.0915934
$818$	0	0
$819$	−2.61803	−0.0914815
$820$	0	0
$821$	19.7082	0.687821	0.343911	−	0.939002i	$-0.388248\pi$
$821$	19.7082	0.687821	0.343911	+	0.939002i	$0.388248\pi$
$822$	0	0
$823$	13.2574	0.462122	0.231061	−	0.972939i	$-0.425780\pi$
$823$	13.2574	0.462122	0.231061	+	0.972939i	$0.425780\pi$
$824$	0	0
$825$	9.43769	0.328578
$826$	0	0
$827$	23.9098	0.831426	0.415713	−	0.909496i	$-0.363532\pi$
$827$	23.9098	0.831426	0.415713	+	0.909496i	$0.363532\pi$
$828$	0	0
$829$	30.7082	1.06654	0.533270	−	0.845945i	$-0.320963\pi$
$829$	30.7082	1.06654	0.533270	+	0.845945i	$0.320963\pi$
$830$	0	0
$831$	−12.3262	−0.427592
$832$	0	0
$833$	7.47214	0.258894
$834$	0	0
$835$	52.5623	1.81899
$836$	0	0
$837$	10.2361	0.353810
$838$	0	0
$839$	22.6869	0.783239	0.391620	−	0.920127i	$-0.371915\pi$
$839$	22.6869	0.783239	0.391620	+	0.920127i	$0.371915\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0	0
$843$	21.1246	0.727570
$844$	0	0
$845$	17.5410	0.603429
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	3.03444	0.104142
$850$	0	0
$851$	−0.236068	−0.00809231
$852$	0	0
$853$	31.5967	1.08185	0.540926	−	0.841070i	$-0.318074\pi$
$853$	31.5967	1.08185	0.540926	+	0.841070i	$0.318074\pi$
$854$	0	0
$855$	12.0902	0.413475
$856$	0	0
$857$	38.0689	1.30041	0.650204	−	0.759760i	$-0.274683\pi$
$857$	38.0689	1.30041	0.650204	+	0.759760i	$0.274683\pi$
$858$	0	0
$859$	43.5279	1.48515	0.742576	−	0.669762i	$-0.233604\pi$
$859$	43.5279	1.48515	0.742576	+	0.669762i	$0.233604\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	0	0
$863$	11.2361	0.382480	0.191240	−	0.981543i	$-0.438749\pi$
$863$	11.2361	0.382480	0.191240	+	0.981543i	$0.438749\pi$
$864$	0	0
$865$	24.3394	0.827564
$866$	0	0
$867$	38.8328	1.31883
$868$	0	0
$869$	−1.58359	−0.0537197
$870$	0	0
$871$	34.8885	1.18215
$872$	0	0
$873$	15.9443	0.539632
$874$	0	0
$875$	5.29180	0.178895
$876$	0	0
$877$	12.9443	0.437097	0.218549	−	0.975826i	$-0.429868\pi$
$877$	12.9443	0.437097	0.218549	+	0.975826i	$0.429868\pi$
$878$	0	0
$879$	−4.94427	−0.166766
$880$	0	0
$881$	−15.5967	−0.525468	−0.262734	−	0.964868i	$-0.584624\pi$
$881$	−15.5967	−0.525468	−0.262734	+	0.964868i	$0.584624\pi$
$882$	0	0
$883$	−42.9787	−1.44635	−0.723174	−	0.690665i	$-0.757318\pi$
$883$	−42.9787	−1.44635	−0.723174	+	0.690665i	$0.757318\pi$
$884$	0	0
$885$	−18.7295	−0.629585
$886$	0	0
$887$	21.7426	0.730047	0.365023	−	0.930998i	$-0.381061\pi$
$887$	21.7426	0.730047	0.365023	+	0.930998i	$0.381061\pi$
$888$	0	0
$889$	−12.3262	−0.413409
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	−5.23607	−0.175218
$894$	0	0
$895$	−4.10333	−0.137159
$896$	0	0
$897$	2.61803	0.0874136
$898$	0	0
$899$	−51.1803	−1.70696
$900$	0	0
$901$	99.5755	3.31734
$902$	0	0
$903$	0.618034	0.0205669
$904$	0	0
$905$	58.2705	1.93698
$906$	0	0
$907$	−5.61803	−0.186544	−0.0932719	−	0.995641i	$-0.529733\pi$
$907$	−5.61803	−0.186544	−0.0932719	+	0.995641i	$0.529733\pi$
$908$	0	0
$909$	12.5623	0.416665
$910$	0	0
$911$	−13.5967	−0.450480	−0.225240	−	0.974303i	$-0.572317\pi$
$911$	−13.5967	−0.450480	−0.225240	+	0.974303i	$0.572317\pi$
$912$	0	0
$913$	23.2918	0.770846
$914$	0	0
$915$	−5.29180	−0.174941
$916$	0	0
$917$	3.00000	0.0990687
$918$	0	0
$919$	−3.40325	−0.112263	−0.0561315	−	0.998423i	$-0.517877\pi$
$919$	−3.40325	−0.112263	−0.0561315	+	0.998423i	$0.517877\pi$
$920$	0	0
$921$	7.65248	0.252158
$922$	0	0
$923$	−14.0902	−0.463784
$924$	0	0
$925$	0.742646	0.0244180
$926$	0	0
$927$	−4.94427	−0.162391
$928$	0	0
$929$	21.4377	0.703348	0.351674	−	0.936123i	$-0.385613\pi$
$929$	21.4377	0.703348	0.351674	+	0.936123i	$0.385613\pi$
$930$	0	0
$931$	−4.23607	−0.138832
$932$	0	0
$933$	−18.6180	−0.609527
$934$	0	0
$935$	−63.9787	−2.09233
$936$	0	0
$937$	−21.5410	−0.703714	−0.351857	−	0.936054i	$-0.614450\pi$
$937$	−21.5410	−0.703714	−0.351857	+	0.936054i	$0.614450\pi$
$938$	0	0
$939$	−12.0000	−0.391605
$940$	0	0
$941$	−43.4164	−1.41533	−0.707667	−	0.706546i	$-0.750252\pi$
$941$	−43.4164	−1.41533	−0.707667	+	0.706546i	$0.750252\pi$
$942$	0	0
$943$	−3.00000	−0.0976934
$944$	0	0
$945$	−2.85410	−0.0928439
$946$	0	0
$947$	35.5279	1.15450	0.577250	−	0.816567i	$-0.304126\pi$
$947$	35.5279	1.15450	0.577250	+	0.816567i	$0.304126\pi$
$948$	0	0
$949$	−31.2705	−1.01508
$950$	0	0
$951$	0.965558	0.0313104
$952$	0	0
$953$	−34.0344	−1.10248	−0.551242	−	0.834346i	$-0.685846\pi$
$953$	−34.0344	−1.10248	−0.551242	+	0.834346i	$0.685846\pi$
$954$	0	0
$955$	−49.1935	−1.59186
$956$	0	0
$957$	−15.0000	−0.484881
$958$	0	0
$959$	−18.4164	−0.594697
$960$	0	0
$961$	73.7771	2.37991
$962$	0	0
$963$	−7.56231	−0.243692
$964$	0	0
$965$	18.9868	0.611208
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	−31.6525	−1.01682
$970$	0	0
$971$	−1.27051	−0.0407726	−0.0203863	−	0.999792i	$-0.506490\pi$
$971$	−1.27051	−0.0407726	−0.0203863	+	0.999792i	$0.506490\pi$
$972$	0	0
$973$	13.7984	0.442356
$974$	0	0
$975$	−8.23607	−0.263765
$976$	0	0
$977$	−29.5623	−0.945782	−0.472891	−	0.881121i	$-0.656790\pi$
$977$	−29.5623	−0.945782	−0.472891	+	0.881121i	$0.656790\pi$
$978$	0	0
$979$	40.6869	1.30036
$980$	0	0
$981$	−16.0902	−0.513720
$982$	0	0
$983$	8.59675	0.274194	0.137097	−	0.990558i	$-0.456223\pi$
$983$	8.59675	0.274194	0.137097	+	0.990558i	$0.456223\pi$
$984$	0	0
$985$	20.2361	0.644775
$986$	0	0
$987$	1.23607	0.0393445
$988$	0	0
$989$	−0.618034	−0.0196523
$990$	0	0
$991$	−26.8673	−0.853467	−0.426733	−	0.904378i	$-0.640336\pi$
$991$	−26.8673	−0.853467	−0.426733	+	0.904378i	$0.640336\pi$
$992$	0	0
$993$	26.3607	0.836531
$994$	0	0
$995$	39.3820	1.24849
$996$	0	0
$997$	24.5967	0.778987	0.389493	−	0.921029i	$-0.372650\pi$
$997$	24.5967	0.778987	0.389493	+	0.921029i	$0.372650\pi$
$998$	0	0
$999$	0.236068	0.00746886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.2.a.h.1.1	✓	2
3.2	odd	2		5796.2.a.j.1.2		2
4.3	odd	2		7728.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.h.1.1	✓	2	1.1	even	1	trivial
5796.2.a.j.1.2		2	3.2	odd	2
7728.2.a.bb.1.1		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.85410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -2.61803 q^{13} -2.85410 q^{15} +7.47214 q^{17} -4.23607 q^{19} +1.00000 q^{21} -1.00000 q^{23} +3.14590 q^{25} +1.00000 q^{27} -5.00000 q^{29} +10.2361 q^{31} +3.00000 q^{33} -2.85410 q^{35} +0.236068 q^{37} -2.61803 q^{39} +3.00000 q^{41} +0.618034 q^{43} -2.85410 q^{45} +1.23607 q^{47} +1.00000 q^{49} +7.47214 q^{51} +13.3262 q^{53} -8.56231 q^{55} -4.23607 q^{57} +6.56231 q^{59} +1.85410 q^{61} +1.00000 q^{63} +7.47214 q^{65} -13.3262 q^{67} -1.00000 q^{69} +5.38197 q^{71} +11.9443 q^{73} +3.14590 q^{75} +3.00000 q^{77} -0.527864 q^{79} +1.00000 q^{81} +7.76393 q^{83} -21.3262 q^{85} -5.00000 q^{87} +13.5623 q^{89} -2.61803 q^{91} +10.2361 q^{93} +12.0902 q^{95} +15.9443 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 3 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + 2 q^{21} - 2 q^{23} + 13 q^{25} + 2 q^{27} - 10 q^{29} + 16 q^{31} + 6 q^{33} + q^{35} - 4 q^{37} - 3 q^{39} + 6 q^{41} - q^{43} + q^{45} - 2 q^{47} + 2 q^{49} + 6 q^{51} + 11 q^{53} + 3 q^{55} - 4 q^{57} - 7 q^{59} - 3 q^{61} + 2 q^{63} + 6 q^{65} - 11 q^{67} - 2 q^{69} + 13 q^{71} + 6 q^{73} + 13 q^{75} + 6 q^{77} - 10 q^{79} + 2 q^{81} + 20 q^{83} - 27 q^{85} - 10 q^{87} + 7 q^{89} - 3 q^{91} + 16 q^{93} + 13 q^{95} + 14 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 3 * q^13 + q^15 + 6 * q^17 - 4 * q^19 + 2 * q^21 - 2 * q^23 + 13 * q^25 + 2 * q^27 - 10 * q^29 + 16 * q^31 + 6 * q^33 + q^35 - 4 * q^37 - 3 * q^39 + 6 * q^41 - q^43 + q^45 - 2 * q^47 + 2 * q^49 + 6 * q^51 + 11 * q^53 + 3 * q^55 - 4 * q^57 - 7 * q^59 - 3 * q^61 + 2 * q^63 + 6 * q^65 - 11 * q^67 - 2 * q^69 + 13 * q^71 + 6 * q^73 + 13 * q^75 + 6 * q^77 - 10 * q^79 + 2 * q^81 + 20 * q^83 - 27 * q^85 - 10 * q^87 + 7 * q^89 - 3 * q^91 + 16 * q^93 + 13 * q^95 + 14 * q^97 + 6 * q^99

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