LMFDB - Embedded newform 1932.2.a.e.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			$2.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.60555 q^{11} -0.697224 q^{13} -2.30278 q^{15} -3.00000 q^{17} -7.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.302776 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} +3.60555 q^{33} +2.30278 q^{35} -8.21110 q^{37} +0.697224 q^{39} -5.60555 q^{41} -7.90833 q^{43} +2.30278 q^{45} -10.6056 q^{47} +1.00000 q^{49} +3.00000 q^{51} +12.9083 q^{53} -8.30278 q^{55} +7.60555 q^{57} +10.9083 q^{59} +8.51388 q^{61} +1.00000 q^{63} -1.60555 q^{65} +10.1194 q^{67} +1.00000 q^{69} -11.9083 q^{71} +10.8167 q^{73} -0.302776 q^{75} -3.60555 q^{77} -6.39445 q^{79} +1.00000 q^{81} -0.394449 q^{83} -6.90833 q^{85} -1.00000 q^{87} +3.30278 q^{89} -0.697224 q^{91} -1.00000 q^{93} -17.5139 q^{95} -11.0000 q^{97} -3.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 5 q^{13} - q^{15} - 6 q^{17} - 8 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + q^{35} - 2 q^{37} + 5 q^{39} - 4 q^{41} - 5 q^{43} + q^{45} - 14 q^{47} + 2 q^{49} + 6 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 5 q^{67} + 2 q^{69} - 13 q^{71} + 3 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} - 3 q^{85} - 2 q^{87} + 3 q^{89} - 5 q^{91} - 2 q^{93} - 17 q^{95} - 22 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 5 * q^13 - q^15 - 6 * q^17 - 8 * q^19 - 2 * q^21 - 2 * q^23 - 3 * q^25 - 2 * q^27 + 2 * q^29 + 2 * q^31 + q^35 - 2 * q^37 + 5 * q^39 - 4 * q^41 - 5 * q^43 + q^45 - 14 * q^47 + 2 * q^49 + 6 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 5 * q^67 + 2 * q^69 - 13 * q^71 + 3 * q^75 - 20 * q^79 + 2 * q^81 - 8 * q^83 - 3 * q^85 - 2 * q^87 + 3 * q^89 - 5 * q^91 - 2 * q^93 - 17 * q^95 - 22 * q^97

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
$5$	2.30278	1.02983	0.514916	−	0.857240i	$-0.327823\pi$
$5$	2.30278	1.02983	0.514916	+	0.857240i	$0.327823\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.60555	−1.08711	−0.543557	−	0.839372i	$-0.682923\pi$
$11$	−3.60555	−1.08711	−0.543557	+	0.839372i	$0.682923\pi$
$12$	0	0
$13$	−0.697224	−0.193375	−0.0966876	−	0.995315i	$-0.530825\pi$
$13$	−0.697224	−0.193375	−0.0966876	+	0.995315i	$0.530825\pi$
$14$	0	0
$15$	−2.30278	−0.594574
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−7.60555	−1.74483	−0.872417	−	0.488763i	$-0.837448\pi$
$19$	−7.60555	−1.74483	−0.872417	+	0.488763i	$0.837448\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0.302776	0.0605551
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	3.60555	0.627646
$34$	0	0
$35$	2.30278	0.389240
$36$	0	0
$37$	−8.21110	−1.34990	−0.674948	−	0.737865i	$-0.735834\pi$
$37$	−8.21110	−1.34990	−0.674948	+	0.737865i	$0.735834\pi$
$38$	0	0
$39$	0.697224	0.111645
$40$	0	0
$41$	−5.60555	−0.875440	−0.437720	−	0.899111i	$-0.644214\pi$
$41$	−5.60555	−0.875440	−0.437720	+	0.899111i	$0.644214\pi$
$42$	0	0
$43$	−7.90833	−1.20601	−0.603004	−	0.797738i	$-0.706030\pi$
$43$	−7.90833	−1.20601	−0.603004	+	0.797738i	$0.706030\pi$
$44$	0	0
$45$	2.30278	0.343278
$46$	0	0
$47$	−10.6056	−1.54698	−0.773489	−	0.633809i	$-0.781490\pi$
$47$	−10.6056	−1.54698	−0.773489	+	0.633809i	$0.781490\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	12.9083	1.77310	0.886548	−	0.462638i	$-0.153097\pi$
$53$	12.9083	1.77310	0.886548	+	0.462638i	$0.153097\pi$
$54$	0	0
$55$	−8.30278	−1.11955
$56$	0	0
$57$	7.60555	1.00738
$58$	0	0
$59$	10.9083	1.42014	0.710072	−	0.704129i	$-0.248663\pi$
$59$	10.9083	1.42014	0.710072	+	0.704129i	$0.248663\pi$
$60$	0	0
$61$	8.51388	1.09009	0.545045	−	0.838407i	$-0.316512\pi$
$61$	8.51388	1.09009	0.545045	+	0.838407i	$0.316512\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.60555	−0.199144
$66$	0	0
$67$	10.1194	1.23629	0.618143	−	0.786066i	$-0.287885\pi$
$67$	10.1194	1.23629	0.618143	+	0.786066i	$0.287885\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.9083	−1.41326	−0.706629	−	0.707584i	$-0.749785\pi$
$71$	−11.9083	−1.41326	−0.706629	+	0.707584i	$0.749785\pi$
$72$	0	0
$73$	10.8167	1.26599	0.632997	−	0.774154i	$-0.281825\pi$
$73$	10.8167	1.26599	0.632997	+	0.774154i	$0.281825\pi$
$74$	0	0
$75$	−0.302776	−0.0349615
$76$	0	0
$77$	−3.60555	−0.410891
$78$	0	0
$79$	−6.39445	−0.719432	−0.359716	−	0.933062i	$-0.617126\pi$
$79$	−6.39445	−0.719432	−0.359716	+	0.933062i	$0.617126\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.394449	−0.0432964	−0.0216482	−	0.999766i	$-0.506891\pi$
$83$	−0.394449	−0.0432964	−0.0216482	+	0.999766i	$0.506891\pi$
$84$	0	0
$85$	−6.90833	−0.749313
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	3.30278	0.350094	0.175047	−	0.984560i	$-0.443992\pi$
$89$	3.30278	0.350094	0.175047	+	0.984560i	$0.443992\pi$
$90$	0	0
$91$	−0.697224	−0.0730890
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−17.5139	−1.79689
$96$	0	0
$97$	−11.0000	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$97$	−11.0000	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$98$	0	0
$99$	−3.60555	−0.362372
$100$	0	0
$101$	−9.51388	−0.946666	−0.473333	−	0.880884i	$-0.656949\pi$
$101$	−9.51388	−0.946666	−0.473333	+	0.880884i	$0.656949\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−2.30278	−0.224728
$106$	0	0
$107$	12.5139	1.20976	0.604881	−	0.796316i	$-0.293221\pi$
$107$	12.5139	1.20976	0.604881	+	0.796316i	$0.293221\pi$
$108$	0	0
$109$	−10.9083	−1.04483	−0.522414	−	0.852692i	$-0.674968\pi$
$109$	−10.9083	−1.04483	−0.522414	+	0.852692i	$0.674968\pi$
$110$	0	0
$111$	8.21110	0.779363
$112$	0	0
$113$	3.51388	0.330558	0.165279	−	0.986247i	$-0.447148\pi$
$113$	3.51388	0.330558	0.165279	+	0.986247i	$0.447148\pi$
$114$	0	0
$115$	−2.30278	−0.214735
$116$	0	0
$117$	−0.697224	−0.0644584
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	5.60555	0.505436
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	3.30278	0.293074	0.146537	−	0.989205i	$-0.453187\pi$
$127$	3.30278	0.293074	0.146537	+	0.989205i	$0.453187\pi$
$128$	0	0
$129$	7.90833	0.696289
$130$	0	0
$131$	4.81665	0.420833	0.210416	−	0.977612i	$-0.432518\pi$
$131$	4.81665	0.420833	0.210416	+	0.977612i	$0.432518\pi$
$132$	0	0
$133$	−7.60555	−0.659485
$134$	0	0
$135$	−2.30278	−0.198191
$136$	0	0
$137$	−16.0278	−1.36934	−0.684672	−	0.728851i	$-0.740054\pi$
$137$	−16.0278	−1.36934	−0.684672	+	0.728851i	$0.740054\pi$
$138$	0	0
$139$	−10.3028	−0.873870	−0.436935	−	0.899493i	$-0.643936\pi$
$139$	−10.3028	−0.873870	−0.436935	+	0.899493i	$0.643936\pi$
$140$	0	0
$141$	10.6056	0.893149
$142$	0	0
$143$	2.51388	0.210221
$144$	0	0
$145$	2.30278	0.191235
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	19.8167	1.62344	0.811722	−	0.584044i	$-0.198531\pi$
$149$	19.8167	1.62344	0.811722	+	0.584044i	$0.198531\pi$
$150$	0	0
$151$	−6.60555	−0.537552	−0.268776	−	0.963203i	$-0.586619\pi$
$151$	−6.60555	−0.537552	−0.268776	+	0.963203i	$0.586619\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	2.30278	0.184963
$156$	0	0
$157$	23.0278	1.83782	0.918908	−	0.394473i	$-0.129073\pi$
$157$	23.0278	1.83782	0.918908	+	0.394473i	$0.129073\pi$
$158$	0	0
$159$	−12.9083	−1.02370
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−23.5139	−1.84175	−0.920875	−	0.389859i	$-0.872524\pi$
$163$	−23.5139	−1.84175	−0.920875	+	0.389859i	$0.872524\pi$
$164$	0	0
$165$	8.30278	0.646370
$166$	0	0
$167$	−16.8167	−1.30131	−0.650656	−	0.759373i	$-0.725506\pi$
$167$	−16.8167	−1.30131	−0.650656	+	0.759373i	$0.725506\pi$
$168$	0	0
$169$	−12.5139	−0.962606
$170$	0	0
$171$	−7.60555	−0.581611
$172$	0	0
$173$	−18.0278	−1.37062	−0.685312	−	0.728249i	$-0.740334\pi$
$173$	−18.0278	−1.37062	−0.685312	+	0.728249i	$0.740334\pi$
$174$	0	0
$175$	0.302776	0.0228877
$176$	0	0
$177$	−10.9083	−0.819920
$178$	0	0
$179$	−0.908327	−0.0678915	−0.0339458	−	0.999424i	$-0.510807\pi$
$179$	−0.908327	−0.0678915	−0.0339458	+	0.999424i	$0.510807\pi$
$180$	0	0
$181$	−9.42221	−0.700347	−0.350173	−	0.936685i	$-0.613877\pi$
$181$	−9.42221	−0.700347	−0.350173	+	0.936685i	$0.613877\pi$
$182$	0	0
$183$	−8.51388	−0.629364
$184$	0	0
$185$	−18.9083	−1.39017
$186$	0	0
$187$	10.8167	0.790992
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	21.0278	1.52152	0.760758	−	0.649036i	$-0.224828\pi$
$191$	21.0278	1.52152	0.760758	+	0.649036i	$0.224828\pi$
$192$	0	0
$193$	9.81665	0.706618	0.353309	−	0.935507i	$-0.385056\pi$
$193$	9.81665	0.706618	0.353309	+	0.935507i	$0.385056\pi$
$194$	0	0
$195$	1.60555	0.114976
$196$	0	0
$197$	23.3028	1.66025	0.830127	−	0.557574i	$-0.188268\pi$
$197$	23.3028	1.66025	0.830127	+	0.557574i	$0.188268\pi$
$198$	0	0
$199$	19.6972	1.39630	0.698150	−	0.715952i	$-0.254007\pi$
$199$	19.6972	1.39630	0.698150	+	0.715952i	$0.254007\pi$
$200$	0	0
$201$	−10.1194	−0.713770
$202$	0	0
$203$	1.00000	0.0701862
$204$	0	0
$205$	−12.9083	−0.901557
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	27.4222	1.89683
$210$	0	0
$211$	−13.6056	−0.936645	−0.468322	−	0.883558i	$-0.655141\pi$
$211$	−13.6056	−0.936645	−0.468322	+	0.883558i	$0.655141\pi$
$212$	0	0
$213$	11.9083	0.815945
$214$	0	0
$215$	−18.2111	−1.24199
$216$	0	0
$217$	1.00000	0.0678844
$218$	0	0
$219$	−10.8167	−0.730922
$220$	0	0
$221$	2.09167	0.140701
$222$	0	0
$223$	−27.5139	−1.84247	−0.921233	−	0.389012i	$-0.872817\pi$
$223$	−27.5139	−1.84247	−0.921233	+	0.389012i	$0.872817\pi$
$224$	0	0
$225$	0.302776	0.0201850
$226$	0	0
$227$	19.6972	1.30735	0.653675	−	0.756775i	$-0.273226\pi$
$227$	19.6972	1.30735	0.653675	+	0.756775i	$0.273226\pi$
$228$	0	0
$229$	−11.6972	−0.772974	−0.386487	−	0.922295i	$-0.626312\pi$
$229$	−11.6972	−0.772974	−0.386487	+	0.922295i	$0.626312\pi$
$230$	0	0
$231$	3.60555	0.237228
$232$	0	0
$233$	21.3305	1.39741	0.698705	−	0.715410i	$-0.253760\pi$
$233$	21.3305	1.39741	0.698705	+	0.715410i	$0.253760\pi$
$234$	0	0
$235$	−24.4222	−1.59313
$236$	0	0
$237$	6.39445	0.415364
$238$	0	0
$239$	9.69722	0.627261	0.313631	−	0.949545i	$-0.398455\pi$
$239$	9.69722	0.627261	0.313631	+	0.949545i	$0.398455\pi$
$240$	0	0
$241$	5.78890	0.372896	0.186448	−	0.982465i	$-0.440302\pi$
$241$	5.78890	0.372896	0.186448	+	0.982465i	$0.440302\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.30278	0.147119
$246$	0	0
$247$	5.30278	0.337408
$248$	0	0
$249$	0.394449	0.0249972
$250$	0	0
$251$	−3.39445	−0.214256	−0.107128	−	0.994245i	$-0.534165\pi$
$251$	−3.39445	−0.214256	−0.107128	+	0.994245i	$0.534165\pi$
$252$	0	0
$253$	3.60555	0.226679
$254$	0	0
$255$	6.90833	0.432616
$256$	0	0
$257$	13.0278	0.812649	0.406325	−	0.913729i	$-0.366810\pi$
$257$	13.0278	0.812649	0.406325	+	0.913729i	$0.366810\pi$
$258$	0	0
$259$	−8.21110	−0.510213
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	18.6333	1.14898	0.574489	−	0.818512i	$-0.305201\pi$
$263$	18.6333	1.14898	0.574489	+	0.818512i	$0.305201\pi$
$264$	0	0
$265$	29.7250	1.82599
$266$	0	0
$267$	−3.30278	−0.202127
$268$	0	0
$269$	−7.72498	−0.471000	−0.235500	−	0.971874i	$-0.575673\pi$
$269$	−7.72498	−0.471000	−0.235500	+	0.971874i	$0.575673\pi$
$270$	0	0
$271$	−10.8167	−0.657065	−0.328532	−	0.944493i	$-0.606554\pi$
$271$	−10.8167	−0.657065	−0.328532	+	0.944493i	$0.606554\pi$
$272$	0	0
$273$	0.697224	0.0421979
$274$	0	0
$275$	−1.09167	−0.0658304
$276$	0	0
$277$	−16.3305	−0.981207	−0.490603	−	0.871383i	$-0.663224\pi$
$277$	−16.3305	−0.981207	−0.490603	+	0.871383i	$0.663224\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	17.8167	1.06285	0.531426	−	0.847105i	$-0.321656\pi$
$281$	17.8167	1.06285	0.531426	+	0.847105i	$0.321656\pi$
$282$	0	0
$283$	5.69722	0.338665	0.169332	−	0.985559i	$-0.445839\pi$
$283$	5.69722	0.338665	0.169332	+	0.985559i	$0.445839\pi$
$284$	0	0
$285$	17.5139	1.03743
$286$	0	0
$287$	−5.60555	−0.330885
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	11.0000	0.644831
$292$	0	0
$293$	−13.2111	−0.771801	−0.385900	−	0.922540i	$-0.626109\pi$
$293$	−13.2111	−0.771801	−0.385900	+	0.922540i	$0.626109\pi$
$294$	0	0
$295$	25.1194	1.46251
$296$	0	0
$297$	3.60555	0.209215
$298$	0	0
$299$	0.697224	0.0403215
$300$	0	0
$301$	−7.90833	−0.455828
$302$	0	0
$303$	9.51388	0.546558
$304$	0	0
$305$	19.6056	1.12261
$306$	0	0
$307$	13.4222	0.766046	0.383023	−	0.923739i	$-0.374883\pi$
$307$	13.4222	0.766046	0.383023	+	0.923739i	$0.374883\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	5.90833	0.335030	0.167515	−	0.985869i	$-0.446426\pi$
$311$	5.90833	0.335030	0.167515	+	0.985869i	$0.446426\pi$
$312$	0	0
$313$	−25.2111	−1.42502	−0.712508	−	0.701664i	$-0.752441\pi$
$313$	−25.2111	−1.42502	−0.712508	+	0.701664i	$0.752441\pi$
$314$	0	0
$315$	2.30278	0.129747
$316$	0	0
$317$	2.30278	0.129337	0.0646684	−	0.997907i	$-0.479401\pi$
$317$	2.30278	0.129337	0.0646684	+	0.997907i	$0.479401\pi$
$318$	0	0
$319$	−3.60555	−0.201872
$320$	0	0
$321$	−12.5139	−0.698457
$322$	0	0
$323$	22.8167	1.26955
$324$	0	0
$325$	−0.211103	−0.0117099
$326$	0	0
$327$	10.9083	0.603232
$328$	0	0
$329$	−10.6056	−0.584703
$330$	0	0
$331$	19.2111	1.05594	0.527969	−	0.849264i	$-0.322954\pi$
$331$	19.2111	1.05594	0.527969	+	0.849264i	$0.322954\pi$
$332$	0	0
$333$	−8.21110	−0.449966
$334$	0	0
$335$	23.3028	1.27317
$336$	0	0
$337$	12.5139	0.681674	0.340837	−	0.940122i	$-0.389290\pi$
$337$	12.5139	0.681674	0.340837	+	0.940122i	$0.389290\pi$
$338$	0	0
$339$	−3.51388	−0.190848
$340$	0	0
$341$	−3.60555	−0.195252
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	2.30278	0.123977
$346$	0	0
$347$	28.8167	1.54696	0.773480	−	0.633821i	$-0.218515\pi$
$347$	28.8167	1.54696	0.773480	+	0.633821i	$0.218515\pi$
$348$	0	0
$349$	−4.48612	−0.240137	−0.120068	−	0.992766i	$-0.538311\pi$
$349$	−4.48612	−0.240137	−0.120068	+	0.992766i	$0.538311\pi$
$350$	0	0
$351$	0.697224	0.0372151
$352$	0	0
$353$	11.6056	0.617701	0.308851	−	0.951111i	$-0.400056\pi$
$353$	11.6056	0.617701	0.308851	+	0.951111i	$0.400056\pi$
$354$	0	0
$355$	−27.4222	−1.45542
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−11.0917	−0.585396	−0.292698	−	0.956205i	$-0.594553\pi$
$359$	−11.0917	−0.585396	−0.292698	+	0.956205i	$0.594553\pi$
$360$	0	0
$361$	38.8444	2.04444
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	24.9083	1.30376
$366$	0	0
$367$	−29.7250	−1.55163	−0.775816	−	0.630960i	$-0.782661\pi$
$367$	−29.7250	−1.55163	−0.775816	+	0.630960i	$0.782661\pi$
$368$	0	0
$369$	−5.60555	−0.291813
$370$	0	0
$371$	12.9083	0.670167
$372$	0	0
$373$	31.4222	1.62698	0.813490	−	0.581579i	$-0.197565\pi$
$373$	31.4222	1.62698	0.813490	+	0.581579i	$0.197565\pi$
$374$	0	0
$375$	10.8167	0.558570
$376$	0	0
$377$	−0.697224	−0.0359089
$378$	0	0
$379$	−5.21110	−0.267676	−0.133838	−	0.991003i	$-0.542730\pi$
$379$	−5.21110	−0.267676	−0.133838	+	0.991003i	$0.542730\pi$
$380$	0	0
$381$	−3.30278	−0.169206
$382$	0	0
$383$	−30.0278	−1.53435	−0.767173	−	0.641440i	$-0.778337\pi$
$383$	−30.0278	−1.53435	−0.767173	+	0.641440i	$0.778337\pi$
$384$	0	0
$385$	−8.30278	−0.423149
$386$	0	0
$387$	−7.90833	−0.402003
$388$	0	0
$389$	15.7889	0.800529	0.400264	−	0.916400i	$-0.368918\pi$
$389$	15.7889	0.800529	0.400264	+	0.916400i	$0.368918\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−4.81665	−0.242968
$394$	0	0
$395$	−14.7250	−0.740894
$396$	0	0
$397$	−10.6056	−0.532277	−0.266139	−	0.963935i	$-0.585748\pi$
$397$	−10.6056	−0.532277	−0.266139	+	0.963935i	$0.585748\pi$
$398$	0	0
$399$	7.60555	0.380754
$400$	0	0
$401$	−15.4222	−0.770148	−0.385074	−	0.922886i	$-0.625824\pi$
$401$	−15.4222	−0.770148	−0.385074	+	0.922886i	$0.625824\pi$
$402$	0	0
$403$	−0.697224	−0.0347312
$404$	0	0
$405$	2.30278	0.114426
$406$	0	0
$407$	29.6056	1.46749
$408$	0	0
$409$	−21.4222	−1.05926	−0.529630	−	0.848229i	$-0.677669\pi$
$409$	−21.4222	−1.05926	−0.529630	+	0.848229i	$0.677669\pi$
$410$	0	0
$411$	16.0278	0.790591
$412$	0	0
$413$	10.9083	0.536764
$414$	0	0
$415$	−0.908327	−0.0445880
$416$	0	0
$417$	10.3028	0.504529
$418$	0	0
$419$	9.51388	0.464783	0.232392	−	0.972622i	$-0.425345\pi$
$419$	9.51388	0.464783	0.232392	+	0.972622i	$0.425345\pi$
$420$	0	0
$421$	11.9083	0.580376	0.290188	−	0.956970i	$-0.406282\pi$
$421$	11.9083	0.580376	0.290188	+	0.956970i	$0.406282\pi$
$422$	0	0
$423$	−10.6056	−0.515660
$424$	0	0
$425$	−0.908327	−0.0440603
$426$	0	0
$427$	8.51388	0.412015
$428$	0	0
$429$	−2.51388	−0.121371
$430$	0	0
$431$	−2.09167	−0.100752	−0.0503762	−	0.998730i	$-0.516042\pi$
$431$	−2.09167	−0.100752	−0.0503762	+	0.998730i	$0.516042\pi$
$432$	0	0
$433$	−35.0278	−1.68333	−0.841663	−	0.540003i	$-0.818423\pi$
$433$	−35.0278	−1.68333	−0.841663	+	0.540003i	$0.818423\pi$
$434$	0	0
$435$	−2.30278	−0.110410
$436$	0	0
$437$	7.60555	0.363823
$438$	0	0
$439$	32.6333	1.55750	0.778751	−	0.627333i	$-0.215853\pi$
$439$	32.6333	1.55750	0.778751	+	0.627333i	$0.215853\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.6333	−1.12285	−0.561426	−	0.827527i	$-0.689747\pi$
$443$	−23.6333	−1.12285	−0.561426	+	0.827527i	$0.689747\pi$
$444$	0	0
$445$	7.60555	0.360538
$446$	0	0
$447$	−19.8167	−0.937296
$448$	0	0
$449$	7.88057	0.371907	0.185954	−	0.982559i	$-0.440463\pi$
$449$	7.88057	0.371907	0.185954	+	0.982559i	$0.440463\pi$
$450$	0	0
$451$	20.2111	0.951704
$452$	0	0
$453$	6.60555	0.310356
$454$	0	0
$455$	−1.60555	−0.0752694
$456$	0	0
$457$	−0.724981	−0.0339132	−0.0169566	−	0.999856i	$-0.505398\pi$
$457$	−0.724981	−0.0339132	−0.0169566	+	0.999856i	$0.505398\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	28.5416	1.32932	0.664658	−	0.747148i	$-0.268577\pi$
$461$	28.5416	1.32932	0.664658	+	0.747148i	$0.268577\pi$
$462$	0	0
$463$	26.2111	1.21813	0.609067	−	0.793119i	$-0.291544\pi$
$463$	26.2111	1.21813	0.609067	+	0.793119i	$0.291544\pi$
$464$	0	0
$465$	−2.30278	−0.106789
$466$	0	0
$467$	−19.6056	−0.907237	−0.453618	−	0.891196i	$-0.649867\pi$
$467$	−19.6056	−0.907237	−0.453618	+	0.891196i	$0.649867\pi$
$468$	0	0
$469$	10.1194	0.467272
$470$	0	0
$471$	−23.0278	−1.06106
$472$	0	0
$473$	28.5139	1.31107
$474$	0	0
$475$	−2.30278	−0.105659
$476$	0	0
$477$	12.9083	0.591032
$478$	0	0
$479$	−15.4222	−0.704659	−0.352329	−	0.935876i	$-0.614610\pi$
$479$	−15.4222	−0.704659	−0.352329	+	0.935876i	$0.614610\pi$
$480$	0	0
$481$	5.72498	0.261037
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−25.3305	−1.15020
$486$	0	0
$487$	−19.7889	−0.896721	−0.448360	−	0.893853i	$-0.647992\pi$
$487$	−19.7889	−0.896721	−0.448360	+	0.893853i	$0.647992\pi$
$488$	0	0
$489$	23.5139	1.06333
$490$	0	0
$491$	−12.3305	−0.556469	−0.278235	−	0.960513i	$-0.589749\pi$
$491$	−12.3305	−0.556469	−0.278235	+	0.960513i	$0.589749\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−8.30278	−0.373182
$496$	0	0
$497$	−11.9083	−0.534161
$498$	0	0
$499$	36.5416	1.63583	0.817914	−	0.575340i	$-0.195130\pi$
$499$	36.5416	1.63583	0.817914	+	0.575340i	$0.195130\pi$
$500$	0	0
$501$	16.8167	0.751313
$502$	0	0
$503$	−16.3028	−0.726905	−0.363452	−	0.931613i	$-0.618402\pi$
$503$	−16.3028	−0.726905	−0.363452	+	0.931613i	$0.618402\pi$
$504$	0	0
$505$	−21.9083	−0.974908
$506$	0	0
$507$	12.5139	0.555761
$508$	0	0
$509$	−1.39445	−0.0618079	−0.0309039	−	0.999522i	$-0.509839\pi$
$509$	−1.39445	−0.0618079	−0.0309039	+	0.999522i	$0.509839\pi$
$510$	0	0
$511$	10.8167	0.478501
$512$	0	0
$513$	7.60555	0.335793
$514$	0	0
$515$	−36.8444	−1.62356
$516$	0	0
$517$	38.2389	1.68174
$518$	0	0
$519$	18.0278	0.791331
$520$	0	0
$521$	24.7889	1.08602	0.543011	−	0.839726i	$-0.317284\pi$
$521$	24.7889	1.08602	0.543011	+	0.839726i	$0.317284\pi$
$522$	0	0
$523$	2.00000	0.0874539	0.0437269	−	0.999044i	$-0.486077\pi$
$523$	2.00000	0.0874539	0.0437269	+	0.999044i	$0.486077\pi$
$524$	0	0
$525$	−0.302776	−0.0132142
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.9083	0.473381
$532$	0	0
$533$	3.90833	0.169288
$534$	0	0
$535$	28.8167	1.24585
$536$	0	0
$537$	0.908327	0.0391972
$538$	0	0
$539$	−3.60555	−0.155302
$540$	0	0
$541$	−1.39445	−0.0599520	−0.0299760	−	0.999551i	$-0.509543\pi$
$541$	−1.39445	−0.0599520	−0.0299760	+	0.999551i	$0.509543\pi$
$542$	0	0
$543$	9.42221	0.404346
$544$	0	0
$545$	−25.1194	−1.07600
$546$	0	0
$547$	−1.51388	−0.0647288	−0.0323644	−	0.999476i	$-0.510304\pi$
$547$	−1.51388	−0.0647288	−0.0323644	+	0.999476i	$0.510304\pi$
$548$	0	0
$549$	8.51388	0.363363
$550$	0	0
$551$	−7.60555	−0.324007
$552$	0	0
$553$	−6.39445	−0.271920
$554$	0	0
$555$	18.9083	0.802614
$556$	0	0
$557$	−29.4500	−1.24783	−0.623917	−	0.781490i	$-0.714460\pi$
$557$	−29.4500	−1.24783	−0.623917	+	0.781490i	$0.714460\pi$
$558$	0	0
$559$	5.51388	0.233212
$560$	0	0
$561$	−10.8167	−0.456679
$562$	0	0
$563$	7.11943	0.300048	0.150024	−	0.988682i	$-0.452065\pi$
$563$	7.11943	0.300048	0.150024	+	0.988682i	$0.452065\pi$
$564$	0	0
$565$	8.09167	0.340419
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	20.8167	0.871150	0.435575	−	0.900152i	$-0.356545\pi$
$571$	20.8167	0.871150	0.435575	+	0.900152i	$0.356545\pi$
$572$	0	0
$573$	−21.0278	−0.878447
$574$	0	0
$575$	−0.302776	−0.0126266
$576$	0	0
$577$	−17.6333	−0.734084	−0.367042	−	0.930204i	$-0.619630\pi$
$577$	−17.6333	−0.734084	−0.367042	+	0.930204i	$0.619630\pi$
$578$	0	0
$579$	−9.81665	−0.407966
$580$	0	0
$581$	−0.394449	−0.0163645
$582$	0	0
$583$	−46.5416	−1.92756
$584$	0	0
$585$	−1.60555	−0.0663814
$586$	0	0
$587$	−1.30278	−0.0537713	−0.0268857	−	0.999639i	$-0.508559\pi$
$587$	−1.30278	−0.0537713	−0.0268857	+	0.999639i	$0.508559\pi$
$588$	0	0
$589$	−7.60555	−0.313381
$590$	0	0
$591$	−23.3028	−0.958548
$592$	0	0
$593$	11.0278	0.452856	0.226428	−	0.974028i	$-0.427295\pi$
$593$	11.0278	0.452856	0.226428	+	0.974028i	$0.427295\pi$
$594$	0	0
$595$	−6.90833	−0.283214
$596$	0	0
$597$	−19.6972	−0.806154
$598$	0	0
$599$	17.7250	0.724223	0.362112	−	0.932135i	$-0.382056\pi$
$599$	17.7250	0.724223	0.362112	+	0.932135i	$0.382056\pi$
$600$	0	0
$601$	−35.9361	−1.46586	−0.732932	−	0.680302i	$-0.761849\pi$
$601$	−35.9361	−1.46586	−0.732932	+	0.680302i	$0.761849\pi$
$602$	0	0
$603$	10.1194	0.412095
$604$	0	0
$605$	4.60555	0.187242
$606$	0	0
$607$	2.88057	0.116919	0.0584594	−	0.998290i	$-0.481381\pi$
$607$	2.88057	0.116919	0.0584594	+	0.998290i	$0.481381\pi$
$608$	0	0
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	7.39445	0.299147
$612$	0	0
$613$	−44.4500	−1.79532	−0.897659	−	0.440692i	$-0.854733\pi$
$613$	−44.4500	−1.79532	−0.897659	+	0.440692i	$0.854733\pi$
$614$	0	0
$615$	12.9083	0.520514
$616$	0	0
$617$	−7.11943	−0.286617	−0.143309	−	0.989678i	$-0.545774\pi$
$617$	−7.11943	−0.286617	−0.143309	+	0.989678i	$0.545774\pi$
$618$	0	0
$619$	32.3305	1.29947	0.649737	−	0.760159i	$-0.274879\pi$
$619$	32.3305	1.29947	0.649737	+	0.760159i	$0.274879\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	3.30278	0.132323
$624$	0	0
$625$	−26.4222	−1.05689
$626$	0	0
$627$	−27.4222	−1.09514
$628$	0	0
$629$	24.6333	0.982194
$630$	0	0
$631$	25.0000	0.995234	0.497617	−	0.867397i	$-0.334208\pi$
$631$	25.0000	0.995234	0.497617	+	0.867397i	$0.334208\pi$
$632$	0	0
$633$	13.6056	0.540772
$634$	0	0
$635$	7.60555	0.301817
$636$	0	0
$637$	−0.697224	−0.0276250
$638$	0	0
$639$	−11.9083	−0.471086
$640$	0	0
$641$	22.1472	0.874761	0.437381	−	0.899276i	$-0.355906\pi$
$641$	22.1472	0.874761	0.437381	+	0.899276i	$0.355906\pi$
$642$	0	0
$643$	−49.7250	−1.96096	−0.980481	−	0.196614i	$-0.937005\pi$
$643$	−49.7250	−1.96096	−0.980481	+	0.196614i	$0.937005\pi$
$644$	0	0
$645$	18.2111	0.717061
$646$	0	0
$647$	−23.0917	−0.907827	−0.453914	−	0.891046i	$-0.649973\pi$
$647$	−23.0917	−0.907827	−0.453914	+	0.891046i	$0.649973\pi$
$648$	0	0
$649$	−39.3305	−1.54386
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	0	0
$653$	−7.51388	−0.294041	−0.147020	−	0.989133i	$-0.546968\pi$
$653$	−7.51388	−0.294041	−0.147020	+	0.989133i	$0.546968\pi$
$654$	0	0
$655$	11.0917	0.433388
$656$	0	0
$657$	10.8167	0.421998
$658$	0	0
$659$	20.3944	0.794455	0.397227	−	0.917720i	$-0.369972\pi$
$659$	20.3944	0.794455	0.397227	+	0.917720i	$0.369972\pi$
$660$	0	0
$661$	−8.21110	−0.319375	−0.159687	−	0.987168i	$-0.551049\pi$
$661$	−8.21110	−0.319375	−0.159687	+	0.987168i	$0.551049\pi$
$662$	0	0
$663$	−2.09167	−0.0812339
$664$	0	0
$665$	−17.5139	−0.679159
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	27.5139	1.06375
$670$	0	0
$671$	−30.6972	−1.18505
$672$	0	0
$673$	−4.57779	−0.176461	−0.0882305	−	0.996100i	$-0.528121\pi$
$673$	−4.57779	−0.176461	−0.0882305	+	0.996100i	$0.528121\pi$
$674$	0	0
$675$	−0.302776	−0.0116538
$676$	0	0
$677$	30.6972	1.17979	0.589895	−	0.807480i	$-0.299169\pi$
$677$	30.6972	1.17979	0.589895	+	0.807480i	$0.299169\pi$
$678$	0	0
$679$	−11.0000	−0.422141
$680$	0	0
$681$	−19.6972	−0.754799
$682$	0	0
$683$	−47.6056	−1.82158	−0.910788	−	0.412875i	$-0.864525\pi$
$683$	−47.6056	−1.82158	−0.910788	+	0.412875i	$0.864525\pi$
$684$	0	0
$685$	−36.9083	−1.41019
$686$	0	0
$687$	11.6972	0.446277
$688$	0	0
$689$	−9.00000	−0.342873
$690$	0	0
$691$	−8.93608	−0.339945	−0.169972	−	0.985449i	$-0.554368\pi$
$691$	−8.93608	−0.339945	−0.169972	+	0.985449i	$0.554368\pi$
$692$	0	0
$693$	−3.60555	−0.136964
$694$	0	0
$695$	−23.7250	−0.899940
$696$	0	0
$697$	16.8167	0.636976
$698$	0	0
$699$	−21.3305	−0.806795
$700$	0	0
$701$	1.88057	0.0710282	0.0355141	−	0.999369i	$-0.488693\pi$
$701$	1.88057	0.0710282	0.0355141	+	0.999369i	$0.488693\pi$
$702$	0	0
$703$	62.4500	2.35534
$704$	0	0
$705$	24.4222	0.919793
$706$	0	0
$707$	−9.51388	−0.357806
$708$	0	0
$709$	6.09167	0.228778	0.114389	−	0.993436i	$-0.463509\pi$
$709$	6.09167	0.228778	0.114389	+	0.993436i	$0.463509\pi$
$710$	0	0
$711$	−6.39445	−0.239811
$712$	0	0
$713$	−1.00000	−0.0374503
$714$	0	0
$715$	5.78890	0.216492
$716$	0	0
$717$	−9.69722	−0.362149
$718$	0	0
$719$	−25.0000	−0.932343	−0.466171	−	0.884694i	$-0.654367\pi$
$719$	−25.0000	−0.932343	−0.466171	+	0.884694i	$0.654367\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	−5.78890	−0.215291
$724$	0	0
$725$	0.302776	0.0112448
$726$	0	0
$727$	3.60555	0.133722	0.0668612	−	0.997762i	$-0.478702\pi$
$727$	3.60555	0.133722	0.0668612	+	0.997762i	$0.478702\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.7250	0.877500
$732$	0	0
$733$	23.0278	0.850550	0.425275	−	0.905064i	$-0.360177\pi$
$733$	23.0278	0.850550	0.425275	+	0.905064i	$0.360177\pi$
$734$	0	0
$735$	−2.30278	−0.0849392
$736$	0	0
$737$	−36.4861	−1.34398
$738$	0	0
$739$	50.0278	1.84030	0.920150	−	0.391565i	$-0.128066\pi$
$739$	50.0278	1.84030	0.920150	+	0.391565i	$0.128066\pi$
$740$	0	0
$741$	−5.30278	−0.194802
$742$	0	0
$743$	−6.72498	−0.246716	−0.123358	−	0.992362i	$-0.539366\pi$
$743$	−6.72498	−0.246716	−0.123358	+	0.992362i	$0.539366\pi$
$744$	0	0
$745$	45.6333	1.67188
$746$	0	0
$747$	−0.394449	−0.0144321
$748$	0	0
$749$	12.5139	0.457247
$750$	0	0
$751$	35.5139	1.29592	0.647960	−	0.761674i	$-0.275622\pi$
$751$	35.5139	1.29592	0.647960	+	0.761674i	$0.275622\pi$
$752$	0	0
$753$	3.39445	0.123701
$754$	0	0
$755$	−15.2111	−0.553589
$756$	0	0
$757$	17.4222	0.633221	0.316610	−	0.948556i	$-0.397455\pi$
$757$	17.4222	0.633221	0.316610	+	0.948556i	$0.397455\pi$
$758$	0	0
$759$	−3.60555	−0.130873
$760$	0	0
$761$	−13.2111	−0.478902	−0.239451	−	0.970908i	$-0.576967\pi$
$761$	−13.2111	−0.478902	−0.239451	+	0.970908i	$0.576967\pi$
$762$	0	0
$763$	−10.9083	−0.394908
$764$	0	0
$765$	−6.90833	−0.249771
$766$	0	0
$767$	−7.60555	−0.274621
$768$	0	0
$769$	−42.6611	−1.53840	−0.769199	−	0.639010i	$-0.779344\pi$
$769$	−42.6611	−1.53840	−0.769199	+	0.639010i	$0.779344\pi$
$770$	0	0
$771$	−13.0278	−0.469183
$772$	0	0
$773$	−24.0278	−0.864218	−0.432109	−	0.901821i	$-0.642230\pi$
$773$	−24.0278	−0.864218	−0.432109	+	0.901821i	$0.642230\pi$
$774$	0	0
$775$	0.302776	0.0108760
$776$	0	0
$777$	8.21110	0.294572
$778$	0	0
$779$	42.6333	1.52750
$780$	0	0
$781$	42.9361	1.53637
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	53.0278	1.89264
$786$	0	0
$787$	−34.3583	−1.22474	−0.612370	−	0.790571i	$-0.709784\pi$
$787$	−34.3583	−1.22474	−0.612370	+	0.790571i	$0.709784\pi$
$788$	0	0
$789$	−18.6333	−0.663363
$790$	0	0
$791$	3.51388	0.124939
$792$	0	0
$793$	−5.93608	−0.210796
$794$	0	0
$795$	−29.7250	−1.05424
$796$	0	0
$797$	−14.6056	−0.517355	−0.258678	−	0.965964i	$-0.583287\pi$
$797$	−14.6056	−0.517355	−0.258678	+	0.965964i	$0.583287\pi$
$798$	0	0
$799$	31.8167	1.12559
$800$	0	0
$801$	3.30278	0.116698
$802$	0	0
$803$	−39.0000	−1.37628
$804$	0	0
$805$	−2.30278	−0.0811622
$806$	0	0
$807$	7.72498	0.271932
$808$	0	0
$809$	7.54163	0.265150	0.132575	−	0.991173i	$-0.457676\pi$
$809$	7.54163	0.265150	0.132575	+	0.991173i	$0.457676\pi$
$810$	0	0
$811$	25.2389	0.886256	0.443128	−	0.896458i	$-0.353869\pi$
$811$	25.2389	0.886256	0.443128	+	0.896458i	$0.353869\pi$
$812$	0	0
$813$	10.8167	0.379357
$814$	0	0
$815$	−54.1472	−1.89669
$816$	0	0
$817$	60.1472	2.10428
$818$	0	0
$819$	−0.697224	−0.0243630
$820$	0	0
$821$	7.81665	0.272803	0.136402	−	0.990654i	$-0.456446\pi$
$821$	7.81665	0.272803	0.136402	+	0.990654i	$0.456446\pi$
$822$	0	0
$823$	−51.3305	−1.78927	−0.894635	−	0.446798i	$-0.852564\pi$
$823$	−51.3305	−1.78927	−0.894635	+	0.446798i	$0.852564\pi$
$824$	0	0
$825$	1.09167	0.0380072
$826$	0	0
$827$	34.5416	1.20113	0.600565	−	0.799576i	$-0.294942\pi$
$827$	34.5416	1.20113	0.600565	+	0.799576i	$0.294942\pi$
$828$	0	0
$829$	19.8444	0.689225	0.344612	−	0.938745i	$-0.388010\pi$
$829$	19.8444	0.689225	0.344612	+	0.938745i	$0.388010\pi$
$830$	0	0
$831$	16.3305	0.566500
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−38.7250	−1.34013
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	30.6972	1.05979	0.529893	−	0.848065i	$-0.322232\pi$
$839$	30.6972	1.05979	0.529893	+	0.848065i	$0.322232\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	−17.8167	−0.613638
$844$	0	0
$845$	−28.8167	−0.991323
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−5.69722	−0.195528
$850$	0	0
$851$	8.21110	0.281473
$852$	0	0
$853$	21.8167	0.746988	0.373494	−	0.927633i	$-0.378160\pi$
$853$	21.8167	0.746988	0.373494	+	0.927633i	$0.378160\pi$
$854$	0	0
$855$	−17.5139	−0.598962
$856$	0	0
$857$	−25.8167	−0.881880	−0.440940	−	0.897537i	$-0.645355\pi$
$857$	−25.8167	−0.881880	−0.440940	+	0.897537i	$0.645355\pi$
$858$	0	0
$859$	19.2111	0.655474	0.327737	−	0.944769i	$-0.393714\pi$
$859$	19.2111	0.655474	0.327737	+	0.944769i	$0.393714\pi$
$860$	0	0
$861$	5.60555	0.191037
$862$	0	0
$863$	−19.3944	−0.660195	−0.330097	−	0.943947i	$-0.607082\pi$
$863$	−19.3944	−0.660195	−0.330097	+	0.943947i	$0.607082\pi$
$864$	0	0
$865$	−41.5139	−1.41151
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	23.0555	0.782105
$870$	0	0
$871$	−7.05551	−0.239067
$872$	0	0
$873$	−11.0000	−0.372294
$874$	0	0
$875$	−10.8167	−0.365670
$876$	0	0
$877$	−17.2111	−0.581178	−0.290589	−	0.956848i	$-0.593851\pi$
$877$	−17.2111	−0.581178	−0.290589	+	0.956848i	$0.593851\pi$
$878$	0	0
$879$	13.2111	0.445599
$880$	0	0
$881$	46.6611	1.57205	0.786026	−	0.618194i	$-0.212135\pi$
$881$	46.6611	1.57205	0.786026	+	0.618194i	$0.212135\pi$
$882$	0	0
$883$	13.1194	0.441504	0.220752	−	0.975330i	$-0.429149\pi$
$883$	13.1194	0.441504	0.220752	+	0.975330i	$0.429149\pi$
$884$	0	0
$885$	−25.1194	−0.844380
$886$	0	0
$887$	35.4861	1.19151	0.595754	−	0.803167i	$-0.296853\pi$
$887$	35.4861	1.19151	0.595754	+	0.803167i	$0.296853\pi$
$888$	0	0
$889$	3.30278	0.110772
$890$	0	0
$891$	−3.60555	−0.120791
$892$	0	0
$893$	80.6611	2.69922
$894$	0	0
$895$	−2.09167	−0.0699169
$896$	0	0
$897$	−0.697224	−0.0232796
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	−38.7250	−1.29012
$902$	0	0
$903$	7.90833	0.263173
$904$	0	0
$905$	−21.6972	−0.721240
$906$	0	0
$907$	−7.27502	−0.241563	−0.120782	−	0.992679i	$-0.538540\pi$
$907$	−7.27502	−0.241563	−0.120782	+	0.992679i	$0.538540\pi$
$908$	0	0
$909$	−9.51388	−0.315555
$910$	0	0
$911$	−14.6056	−0.483904	−0.241952	−	0.970288i	$-0.577788\pi$
$911$	−14.6056	−0.483904	−0.241952	+	0.970288i	$0.577788\pi$
$912$	0	0
$913$	1.42221	0.0470681
$914$	0	0
$915$	−19.6056	−0.648140
$916$	0	0
$917$	4.81665	0.159060
$918$	0	0
$919$	52.2666	1.72412	0.862058	−	0.506809i	$-0.169175\pi$
$919$	52.2666	1.72412	0.862058	+	0.506809i	$0.169175\pi$
$920$	0	0
$921$	−13.4222	−0.442277
$922$	0	0
$923$	8.30278	0.273289
$924$	0	0
$925$	−2.48612	−0.0817432
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−6.90833	−0.226655	−0.113327	−	0.993558i	$-0.536151\pi$
$929$	−6.90833	−0.226655	−0.113327	+	0.993558i	$0.536151\pi$
$930$	0	0
$931$	−7.60555	−0.249262
$932$	0	0
$933$	−5.90833	−0.193430
$934$	0	0
$935$	24.9083	0.814589
$936$	0	0
$937$	48.8722	1.59658	0.798292	−	0.602271i	$-0.205737\pi$
$937$	48.8722	1.59658	0.798292	+	0.602271i	$0.205737\pi$
$938$	0	0
$939$	25.2111	0.822733
$940$	0	0
$941$	−48.8444	−1.59228	−0.796141	−	0.605111i	$-0.793129\pi$
$941$	−48.8444	−1.59228	−0.796141	+	0.605111i	$0.793129\pi$
$942$	0	0
$943$	5.60555	0.182542
$944$	0	0
$945$	−2.30278	−0.0749093
$946$	0	0
$947$	−48.4222	−1.57351	−0.786755	−	0.617265i	$-0.788241\pi$
$947$	−48.4222	−1.57351	−0.786755	+	0.617265i	$0.788241\pi$
$948$	0	0
$949$	−7.54163	−0.244812
$950$	0	0
$951$	−2.30278	−0.0746726
$952$	0	0
$953$	−32.3305	−1.04729	−0.523644	−	0.851937i	$-0.675428\pi$
$953$	−32.3305	−1.04729	−0.523644	+	0.851937i	$0.675428\pi$
$954$	0	0
$955$	48.4222	1.56691
$956$	0	0
$957$	3.60555	0.116551
$958$	0	0
$959$	−16.0278	−0.517563
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	12.5139	0.403254
$964$	0	0
$965$	22.6056	0.727698
$966$	0	0
$967$	6.36669	0.204739	0.102370	−	0.994746i	$-0.467358\pi$
$967$	6.36669	0.204739	0.102370	+	0.994746i	$0.467358\pi$
$968$	0	0
$969$	−22.8167	−0.732977
$970$	0	0
$971$	−22.2750	−0.714839	−0.357420	−	0.933944i	$-0.616343\pi$
$971$	−22.2750	−0.714839	−0.357420	+	0.933944i	$0.616343\pi$
$972$	0	0
$973$	−10.3028	−0.330292
$974$	0	0
$975$	0.211103	0.00676069
$976$	0	0
$977$	46.5139	1.48811	0.744055	−	0.668118i	$-0.232900\pi$
$977$	46.5139	1.48811	0.744055	+	0.668118i	$0.232900\pi$
$978$	0	0
$979$	−11.9083	−0.380592
$980$	0	0
$981$	−10.9083	−0.348276
$982$	0	0
$983$	1.23886	0.0395135	0.0197567	−	0.999805i	$-0.493711\pi$
$983$	1.23886	0.0395135	0.0197567	+	0.999805i	$0.493711\pi$
$984$	0	0
$985$	53.6611	1.70978
$986$	0	0
$987$	10.6056	0.337578
$988$	0	0
$989$	7.90833	0.251470
$990$	0	0
$991$	8.69722	0.276276	0.138138	−	0.990413i	$-0.455888\pi$
$991$	8.69722	0.276276	0.138138	+	0.990413i	$0.455888\pi$
$992$	0	0
$993$	−19.2111	−0.609646
$994$	0	0
$995$	45.3583	1.43795
$996$	0	0
$997$	12.5778	0.398343	0.199171	−	0.979965i	$-0.436175\pi$
$997$	12.5778	0.398343	0.199171	+	0.979965i	$0.436175\pi$
$998$	0	0
$999$	8.21110	0.259788

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.60555 q^{11} -0.697224 q^{13} -2.30278 q^{15} -3.00000 q^{17} -7.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} +0.302776 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} +3.60555 q^{33} +2.30278 q^{35} -8.21110 q^{37} +0.697224 q^{39} -5.60555 q^{41} -7.90833 q^{43} +2.30278 q^{45} -10.6056 q^{47} +1.00000 q^{49} +3.00000 q^{51} +12.9083 q^{53} -8.30278 q^{55} +7.60555 q^{57} +10.9083 q^{59} +8.51388 q^{61} +1.00000 q^{63} -1.60555 q^{65} +10.1194 q^{67} +1.00000 q^{69} -11.9083 q^{71} +10.8167 q^{73} -0.302776 q^{75} -3.60555 q^{77} -6.39445 q^{79} +1.00000 q^{81} -0.394449 q^{83} -6.90833 q^{85} -1.00000 q^{87} +3.30278 q^{89} -0.697224 q^{91} -1.00000 q^{93} -17.5139 q^{95} -11.0000 q^{97} -3.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 5 q^{13} - q^{15} - 6 q^{17} - 8 q^{19} - 2 q^{21} - 2 q^{23} - 3 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{31} + q^{35} - 2 q^{37} + 5 q^{39} - 4 q^{41} - 5 q^{43} + q^{45} - 14 q^{47} + 2 q^{49} + 6 q^{51} + 15 q^{53} - 13 q^{55} + 8 q^{57} + 11 q^{59} - q^{61} + 2 q^{63} + 4 q^{65} - 5 q^{67} + 2 q^{69} - 13 q^{71} + 3 q^{75} - 20 q^{79} + 2 q^{81} - 8 q^{83} - 3 q^{85} - 2 q^{87} + 3 q^{89} - 5 q^{91} - 2 q^{93} - 17 q^{95} - 22 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 5 * q^13 - q^15 - 6 * q^17 - 8 * q^19 - 2 * q^21 - 2 * q^23 - 3 * q^25 - 2 * q^27 + 2 * q^29 + 2 * q^31 + q^35 - 2 * q^37 + 5 * q^39 - 4 * q^41 - 5 * q^43 + q^45 - 14 * q^47 + 2 * q^49 + 6 * q^51 + 15 * q^53 - 13 * q^55 + 8 * q^57 + 11 * q^59 - q^61 + 2 * q^63 + 4 * q^65 - 5 * q^67 + 2 * q^69 - 13 * q^71 + 3 * q^75 - 20 * q^79 + 2 * q^81 - 8 * q^83 - 3 * q^85 - 2 * q^87 + 3 * q^89 - 5 * q^91 - 2 * q^93 - 17 * q^95 - 22 * q^97

Introduction

L-functions

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Database

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