LMFDB - Embedded newform 1932.2.a.e.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:	$1$
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			1.1
Root			$-1.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.60555 q^{11} -4.30278 q^{13} +1.30278 q^{15} -3.00000 q^{17} -0.394449 q^{19} -1.00000 q^{21} -1.00000 q^{23} -3.30278 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} -3.60555 q^{33} -1.30278 q^{35} +6.21110 q^{37} +4.30278 q^{39} +1.60555 q^{41} +2.90833 q^{43} -1.30278 q^{45} -3.39445 q^{47} +1.00000 q^{49} +3.00000 q^{51} +2.09167 q^{53} -4.69722 q^{55} +0.394449 q^{57} +0.0916731 q^{59} -9.51388 q^{61} +1.00000 q^{63} +5.60555 q^{65} -15.1194 q^{67} +1.00000 q^{69} -1.09167 q^{71} -10.8167 q^{73} +3.30278 q^{75} +3.60555 q^{77} -13.6056 q^{79} +1.00000 q^{81} -7.60555 q^{83} +3.90833 q^{85} -1.00000 q^{87} -0.302776 q^{89} -4.30278 q^{91} -1.00000 q^{93} +0.513878 q^{95} -11.0000 q^{97} +3.60555 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	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	$-0.317077\pi$
$11$	3.60555	1.08711	0.543557	+	0.839372i	$0.317077\pi$
$12$	0	0
$13$	−4.30278	−1.19338	−0.596688	−	0.802474i	$-0.703517\pi$
$13$	−4.30278	−1.19338	−0.596688	+	0.802474i	$0.703517\pi$
$14$	0	0
$15$	1.30278	0.336375
$16$	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
$19$	−0.394449	−0.0904927	−0.0452464	−	0.998976i	$-0.514407\pi$
$19$	−0.394449	−0.0904927	−0.0452464	+	0.998976i	$0.514407\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.30278	−0.660555
$26$	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
$35$	−1.30278	−0.220209
$36$	0	0
$37$	6.21110	1.02110	0.510549	−	0.859848i	$-0.329442\pi$
$37$	6.21110	1.02110	0.510549	+	0.859848i	$0.329442\pi$
$38$	0	0
$39$	4.30278	0.688996
$40$	0	0
$41$	1.60555	0.250745	0.125372	−	0.992110i	$-0.459987\pi$
$41$	1.60555	0.250745	0.125372	+	0.992110i	$0.459987\pi$
$42$	0	0
$43$	2.90833	0.443516	0.221758	−	0.975102i	$-0.428821\pi$
$43$	2.90833	0.443516	0.221758	+	0.975102i	$0.428821\pi$
$44$	0	0
$45$	−1.30278	−0.194206
$46$	0	0
$47$	−3.39445	−0.495131	−0.247566	−	0.968871i	$-0.579631\pi$
$47$	−3.39445	−0.495131	−0.247566	+	0.968871i	$0.579631\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	2.09167	0.287313	0.143657	−	0.989628i	$-0.454114\pi$
$53$	2.09167	0.287313	0.143657	+	0.989628i	$0.454114\pi$
$54$	0	0
$55$	−4.69722	−0.633374
$56$	0	0
$57$	0.394449	0.0522460
$58$	0	0
$59$	0.0916731	0.0119348	0.00596741	−	0.999982i	$-0.498101\pi$
$59$	0.0916731	0.0119348	0.00596741	+	0.999982i	$0.498101\pi$
$60$	0	0
$61$	−9.51388	−1.21813	−0.609064	−	0.793121i	$-0.708455\pi$
$61$	−9.51388	−1.21813	−0.609064	+	0.793121i	$0.708455\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.60555	0.695283
$66$	0	0
$67$	−15.1194	−1.84713	−0.923566	−	0.383439i	$-0.874740\pi$
$67$	−15.1194	−1.84713	−0.923566	+	0.383439i	$0.874740\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−1.09167	−0.129558	−0.0647789	−	0.997900i	$-0.520634\pi$
$71$	−1.09167	−0.129558	−0.0647789	+	0.997900i	$0.520634\pi$
$72$	0	0
$73$	−10.8167	−1.26599	−0.632997	−	0.774154i	$-0.718175\pi$
$73$	−10.8167	−1.26599	−0.632997	+	0.774154i	$0.718175\pi$
$74$	0	0
$75$	3.30278	0.381372
$76$	0	0
$77$	3.60555	0.410891
$78$	0	0
$79$	−13.6056	−1.53074	−0.765372	−	0.643588i	$-0.777445\pi$
$79$	−13.6056	−1.53074	−0.765372	+	0.643588i	$0.777445\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.60555	−0.834818	−0.417409	−	0.908719i	$-0.637062\pi$
$83$	−7.60555	−0.834818	−0.417409	+	0.908719i	$0.637062\pi$
$84$	0	0
$85$	3.90833	0.423918
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	−0.302776	−0.0320942	−0.0160471	−	0.999871i	$-0.505108\pi$
$89$	−0.302776	−0.0320942	−0.0160471	+	0.999871i	$0.505108\pi$
$90$	0	0
$91$	−4.30278	−0.451053
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0.513878	0.0527228
$96$	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
$101$	8.51388	0.847163	0.423581	−	0.905858i	$-0.360773\pi$
$101$	8.51388	0.847163	0.423581	+	0.905858i	$0.360773\pi$
$102$	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
$105$	1.30278	0.127138
$106$	0	0
$107$	−5.51388	−0.533047	−0.266523	−	0.963828i	$-0.585875\pi$
$107$	−5.51388	−0.533047	−0.266523	+	0.963828i	$0.585875\pi$
$108$	0	0
$109$	−0.0916731	−0.00878069	−0.00439034	−	0.999990i	$-0.501397\pi$
$109$	−0.0916731	−0.00878069	−0.00439034	+	0.999990i	$0.501397\pi$
$110$	0	0
$111$	−6.21110	−0.589532
$112$	0	0
$113$	−14.5139	−1.36535	−0.682675	−	0.730722i	$-0.739184\pi$
$113$	−14.5139	−1.36535	−0.682675	+	0.730722i	$0.739184\pi$
$114$	0	0
$115$	1.30278	0.121484
$116$	0	0
$117$	−4.30278	−0.397792
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	−1.60555	−0.144768
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−0.302776	−0.0268670	−0.0134335	−	0.999910i	$-0.504276\pi$
$127$	−0.302776	−0.0268670	−0.0134335	+	0.999910i	$0.504276\pi$
$128$	0	0
$129$	−2.90833	−0.256064
$130$	0	0
$131$	−16.8167	−1.46928	−0.734639	−	0.678458i	$-0.762648\pi$
$131$	−16.8167	−1.46928	−0.734639	+	0.678458i	$0.762648\pi$
$132$	0	0
$133$	−0.394449	−0.0342030
$134$	0	0
$135$	1.30278	0.112125
$136$	0	0
$137$	20.0278	1.71109	0.855543	−	0.517731i	$-0.173223\pi$
$137$	20.0278	1.71109	0.855543	+	0.517731i	$0.173223\pi$
$138$	0	0
$139$	−6.69722	−0.568051	−0.284026	−	0.958817i	$-0.591670\pi$
$139$	−6.69722	−0.568051	−0.284026	+	0.958817i	$0.591670\pi$
$140$	0	0
$141$	3.39445	0.285864
$142$	0	0
$143$	−15.5139	−1.29734
$144$	0	0
$145$	−1.30278	−0.108190
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−1.81665	−0.148826	−0.0744130	−	0.997228i	$-0.523708\pi$
$149$	−1.81665	−0.148826	−0.0744130	+	0.997228i	$0.523708\pi$
$150$	0	0
$151$	0.605551	0.0492791	0.0246395	−	0.999696i	$-0.492156\pi$
$151$	0.605551	0.0492791	0.0246395	+	0.999696i	$0.492156\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−1.30278	−0.104641
$156$	0	0
$157$	−13.0278	−1.03973	−0.519864	−	0.854249i	$-0.674017\pi$
$157$	−13.0278	−1.03973	−0.519864	+	0.854249i	$0.674017\pi$
$158$	0	0
$159$	−2.09167	−0.165880
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.48612	−0.429706	−0.214853	−	0.976646i	$-0.568927\pi$
$163$	−5.48612	−0.429706	−0.214853	+	0.976646i	$0.568927\pi$
$164$	0	0
$165$	4.69722	0.365678
$166$	0	0
$167$	4.81665	0.372724	0.186362	−	0.982481i	$-0.440330\pi$
$167$	4.81665	0.372724	0.186362	+	0.982481i	$0.440330\pi$
$168$	0	0
$169$	5.51388	0.424144
$170$	0	0
$171$	−0.394449	−0.0301642
$172$	0	0
$173$	18.0278	1.37062	0.685312	−	0.728249i	$-0.259666\pi$
$173$	18.0278	1.37062	0.685312	+	0.728249i	$0.259666\pi$
$174$	0	0
$175$	−3.30278	−0.249666
$176$	0	0
$177$	−0.0916731	−0.00689057
$178$	0	0
$179$	9.90833	0.740583	0.370292	−	0.928916i	$-0.379258\pi$
$179$	9.90833	0.740583	0.370292	+	0.928916i	$0.379258\pi$
$180$	0	0
$181$	19.4222	1.44364	0.721821	−	0.692080i	$-0.243306\pi$
$181$	19.4222	1.44364	0.721821	+	0.692080i	$0.243306\pi$
$182$	0	0
$183$	9.51388	0.703286
$184$	0	0
$185$	−8.09167	−0.594912
$186$	0	0
$187$	−10.8167	−0.790992
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−15.0278	−1.08737	−0.543685	−	0.839289i	$-0.682971\pi$
$191$	−15.0278	−1.08737	−0.543685	+	0.839289i	$0.682971\pi$
$192$	0	0
$193$	−11.8167	−0.850581	−0.425291	−	0.905057i	$-0.639828\pi$
$193$	−11.8167	−0.850581	−0.425291	+	0.905057i	$0.639828\pi$
$194$	0	0
$195$	−5.60555	−0.401422
$196$	0	0
$197$	19.6972	1.40337	0.701685	−	0.712488i	$-0.252432\pi$
$197$	19.6972	1.40337	0.701685	+	0.712488i	$0.252432\pi$
$198$	0	0
$199$	23.3028	1.65189	0.825945	−	0.563751i	$-0.190642\pi$
$199$	23.3028	1.65189	0.825945	+	0.563751i	$0.190642\pi$
$200$	0	0
$201$	15.1194	1.06644
$202$	0	0
$203$	1.00000	0.0701862
$204$	0	0
$205$	−2.09167	−0.146089
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−1.42221	−0.0983760
$210$	0	0
$211$	−6.39445	−0.440212	−0.220106	−	0.975476i	$-0.570640\pi$
$211$	−6.39445	−0.440212	−0.220106	+	0.975476i	$0.570640\pi$
$212$	0	0
$213$	1.09167	0.0748002
$214$	0	0
$215$	−3.78890	−0.258401
$216$	0	0
$217$	1.00000	0.0678844
$218$	0	0
$219$	10.8167	0.730922
$220$	0	0
$221$	12.9083	0.868308
$222$	0	0
$223$	−9.48612	−0.635238	−0.317619	−	0.948218i	$-0.602883\pi$
$223$	−9.48612	−0.635238	−0.317619	+	0.948218i	$0.602883\pi$
$224$	0	0
$225$	−3.30278	−0.220185
$226$	0	0
$227$	23.3028	1.54666	0.773330	−	0.634004i	$-0.218590\pi$
$227$	23.3028	1.54666	0.773330	+	0.634004i	$0.218590\pi$
$228$	0	0
$229$	−15.3028	−1.01124	−0.505618	−	0.862757i	$-0.668735\pi$
$229$	−15.3028	−1.01124	−0.505618	+	0.862757i	$0.668735\pi$
$230$	0	0
$231$	−3.60555	−0.237228
$232$	0	0
$233$	−18.3305	−1.20087	−0.600437	−	0.799672i	$-0.705006\pi$
$233$	−18.3305	−1.20087	−0.600437	+	0.799672i	$0.705006\pi$
$234$	0	0
$235$	4.42221	0.288473
$236$	0	0
$237$	13.6056	0.883776
$238$	0	0
$239$	13.3028	0.860485	0.430243	−	0.902713i	$-0.358428\pi$
$239$	13.3028	0.860485	0.430243	+	0.902713i	$0.358428\pi$
$240$	0	0
$241$	20.2111	1.30191	0.650956	−	0.759116i	$-0.274368\pi$
$241$	20.2111	1.30191	0.650956	+	0.759116i	$0.274368\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.30278	−0.0832313
$246$	0	0
$247$	1.69722	0.107992
$248$	0	0
$249$	7.60555	0.481982
$250$	0	0
$251$	−10.6056	−0.669416	−0.334708	−	0.942322i	$-0.608638\pi$
$251$	−10.6056	−0.669416	−0.334708	+	0.942322i	$0.608638\pi$
$252$	0	0
$253$	−3.60555	−0.226679
$254$	0	0
$255$	−3.90833	−0.244749
$256$	0	0
$257$	−23.0278	−1.43643	−0.718216	−	0.695820i	$-0.755041\pi$
$257$	−23.0278	−1.43643	−0.718216	+	0.695820i	$0.755041\pi$
$258$	0	0
$259$	6.21110	0.385939
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	−24.6333	−1.51895	−0.759477	−	0.650534i	$-0.774545\pi$
$263$	−24.6333	−1.51895	−0.759477	+	0.650534i	$0.774545\pi$
$264$	0	0
$265$	−2.72498	−0.167394
$266$	0	0
$267$	0.302776	0.0185296
$268$	0	0
$269$	24.7250	1.50751	0.753754	−	0.657156i	$-0.228241\pi$
$269$	24.7250	1.50751	0.753754	+	0.657156i	$0.228241\pi$
$270$	0	0
$271$	10.8167	0.657065	0.328532	−	0.944493i	$-0.393446\pi$
$271$	10.8167	0.657065	0.328532	+	0.944493i	$0.393446\pi$
$272$	0	0
$273$	4.30278	0.260416
$274$	0	0
$275$	−11.9083	−0.718099
$276$	0	0
$277$	23.3305	1.40180	0.700898	−	0.713262i	$-0.252783\pi$
$277$	23.3305	1.40180	0.700898	+	0.713262i	$0.252783\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−3.81665	−0.227682	−0.113841	−	0.993499i	$-0.536316\pi$
$281$	−3.81665	−0.227682	−0.113841	+	0.993499i	$0.536316\pi$
$282$	0	0
$283$	9.30278	0.552993	0.276496	−	0.961015i	$-0.410827\pi$
$283$	9.30278	0.552993	0.276496	+	0.961015i	$0.410827\pi$
$284$	0	0
$285$	−0.513878	−0.0304395
$286$	0	0
$287$	1.60555	0.0947727
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	11.0000	0.644831
$292$	0	0
$293$	1.21110	0.0707534	0.0353767	−	0.999374i	$-0.488737\pi$
$293$	1.21110	0.0707534	0.0353767	+	0.999374i	$0.488737\pi$
$294$	0	0
$295$	−0.119429	−0.00695345
$296$	0	0
$297$	−3.60555	−0.209215
$298$	0	0
$299$	4.30278	0.248836
$300$	0	0
$301$	2.90833	0.167633
$302$	0	0
$303$	−8.51388	−0.489110
$304$	0	0
$305$	12.3944	0.709704
$306$	0	0
$307$	−15.4222	−0.880192	−0.440096	−	0.897951i	$-0.645056\pi$
$307$	−15.4222	−0.880192	−0.440096	+	0.897951i	$0.645056\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	−4.90833	−0.278326	−0.139163	−	0.990270i	$-0.544441\pi$
$311$	−4.90833	−0.278326	−0.139163	+	0.990270i	$0.544441\pi$
$312$	0	0
$313$	−10.7889	−0.609825	−0.304912	−	0.952380i	$-0.598627\pi$
$313$	−10.7889	−0.609825	−0.304912	+	0.952380i	$0.598627\pi$
$314$	0	0
$315$	−1.30278	−0.0734031
$316$	0	0
$317$	−1.30278	−0.0731712	−0.0365856	−	0.999331i	$-0.511648\pi$
$317$	−1.30278	−0.0731712	−0.0365856	+	0.999331i	$0.511648\pi$
$318$	0	0
$319$	3.60555	0.201872
$320$	0	0
$321$	5.51388	0.307755
$322$	0	0
$323$	1.18335	0.0658431
$324$	0	0
$325$	14.2111	0.788290
$326$	0	0
$327$	0.0916731	0.00506953
$328$	0	0
$329$	−3.39445	−0.187142
$330$	0	0
$331$	4.78890	0.263222	0.131611	−	0.991301i	$-0.457985\pi$
$331$	4.78890	0.263222	0.131611	+	0.991301i	$0.457985\pi$
$332$	0	0
$333$	6.21110	0.340366
$334$	0	0
$335$	19.6972	1.07617
$336$	0	0
$337$	−5.51388	−0.300360	−0.150180	−	0.988659i	$-0.547985\pi$
$337$	−5.51388	−0.300360	−0.150180	+	0.988659i	$0.547985\pi$
$338$	0	0
$339$	14.5139	0.788286
$340$	0	0
$341$	3.60555	0.195252
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.30278	−0.0701391
$346$	0	0
$347$	7.18335	0.385622	0.192811	−	0.981236i	$-0.438240\pi$
$347$	7.18335	0.385622	0.192811	+	0.981236i	$0.438240\pi$
$348$	0	0
$349$	−22.5139	−1.20514	−0.602570	−	0.798066i	$-0.705857\pi$
$349$	−22.5139	−1.20514	−0.602570	+	0.798066i	$0.705857\pi$
$350$	0	0
$351$	4.30278	0.229665
$352$	0	0
$353$	4.39445	0.233893	0.116946	−	0.993138i	$-0.462689\pi$
$353$	4.39445	0.233893	0.116946	+	0.993138i	$0.462689\pi$
$354$	0	0
$355$	1.42221	0.0754828
$356$	0	0
$357$	3.00000	0.158777
$358$	0	0
$359$	−21.9083	−1.15628	−0.578139	−	0.815939i	$-0.696221\pi$
$359$	−21.9083	−1.15628	−0.578139	+	0.815939i	$0.696221\pi$
$360$	0	0
$361$	−18.8444	−0.991811
$362$	0	0
$363$	−2.00000	−0.104973
$364$	0	0
$365$	14.0917	0.737592
$366$	0	0
$367$	2.72498	0.142243	0.0711214	−	0.997468i	$-0.477342\pi$
$367$	2.72498	0.142243	0.0711214	+	0.997468i	$0.477342\pi$
$368$	0	0
$369$	1.60555	0.0835817
$370$	0	0
$371$	2.09167	0.108594
$372$	0	0
$373$	2.57779	0.133473	0.0667366	−	0.997771i	$-0.478741\pi$
$373$	2.57779	0.133473	0.0667366	+	0.997771i	$0.478741\pi$
$374$	0	0
$375$	−10.8167	−0.558570
$376$	0	0
$377$	−4.30278	−0.221604
$378$	0	0
$379$	9.21110	0.473143	0.236571	−	0.971614i	$-0.423976\pi$
$379$	9.21110	0.473143	0.236571	+	0.971614i	$0.423976\pi$
$380$	0	0
$381$	0.302776	0.0155117
$382$	0	0
$383$	6.02776	0.308004	0.154002	−	0.988071i	$-0.450784\pi$
$383$	6.02776	0.308004	0.154002	+	0.988071i	$0.450784\pi$
$384$	0	0
$385$	−4.69722	−0.239393
$386$	0	0
$387$	2.90833	0.147839
$388$	0	0
$389$	30.2111	1.53176	0.765882	−	0.642981i	$-0.222303\pi$
$389$	30.2111	1.53176	0.765882	+	0.642981i	$0.222303\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	16.8167	0.848288
$394$	0	0
$395$	17.7250	0.891841
$396$	0	0
$397$	−3.39445	−0.170362	−0.0851812	−	0.996365i	$-0.527147\pi$
$397$	−3.39445	−0.170362	−0.0851812	+	0.996365i	$0.527147\pi$
$398$	0	0
$399$	0.394449	0.0197471
$400$	0	0
$401$	13.4222	0.670273	0.335136	−	0.942170i	$-0.391218\pi$
$401$	13.4222	0.670273	0.335136	+	0.942170i	$0.391218\pi$
$402$	0	0
$403$	−4.30278	−0.214337
$404$	0	0
$405$	−1.30278	−0.0647354
$406$	0	0
$407$	22.3944	1.11005
$408$	0	0
$409$	7.42221	0.367004	0.183502	−	0.983019i	$-0.441257\pi$
$409$	7.42221	0.367004	0.183502	+	0.983019i	$0.441257\pi$
$410$	0	0
$411$	−20.0278	−0.987896
$412$	0	0
$413$	0.0916731	0.00451094
$414$	0	0
$415$	9.90833	0.486381
$416$	0	0
$417$	6.69722	0.327964
$418$	0	0
$419$	−8.51388	−0.415930	−0.207965	−	0.978136i	$-0.566684\pi$
$419$	−8.51388	−0.415930	−0.207965	+	0.978136i	$0.566684\pi$
$420$	0	0
$421$	1.09167	0.0532049	0.0266024	−	0.999646i	$-0.491531\pi$
$421$	1.09167	0.0532049	0.0266024	+	0.999646i	$0.491531\pi$
$422$	0	0
$423$	−3.39445	−0.165044
$424$	0	0
$425$	9.90833	0.480624
$426$	0	0
$427$	−9.51388	−0.460409
$428$	0	0
$429$	15.5139	0.749017
$430$	0	0
$431$	−12.9083	−0.621772	−0.310886	−	0.950447i	$-0.600626\pi$
$431$	−12.9083	−0.621772	−0.310886	+	0.950447i	$0.600626\pi$
$432$	0	0
$433$	1.02776	0.0493908	0.0246954	−	0.999695i	$-0.492138\pi$
$433$	1.02776	0.0493908	0.0246954	+	0.999695i	$0.492138\pi$
$434$	0	0
$435$	1.30278	0.0624633
$436$	0	0
$437$	0.394449	0.0188690
$438$	0	0
$439$	−10.6333	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$439$	−10.6333	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	19.6333	0.932807	0.466403	−	0.884572i	$-0.345550\pi$
$443$	19.6333	0.932807	0.466403	+	0.884572i	$0.345550\pi$
$444$	0	0
$445$	0.394449	0.0186987
$446$	0	0
$447$	1.81665	0.0859248
$448$	0	0
$449$	33.1194	1.56300	0.781501	−	0.623904i	$-0.214454\pi$
$449$	33.1194	1.56300	0.781501	+	0.623904i	$0.214454\pi$
$450$	0	0
$451$	5.78890	0.272589
$452$	0	0
$453$	−0.605551	−0.0284513
$454$	0	0
$455$	5.60555	0.262792
$456$	0	0
$457$	31.7250	1.48403	0.742016	−	0.670382i	$-0.233870\pi$
$457$	31.7250	1.48403	0.742016	+	0.670382i	$0.233870\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	−25.5416	−1.18959	−0.594796	−	0.803876i	$-0.702767\pi$
$461$	−25.5416	−1.18959	−0.594796	+	0.803876i	$0.702767\pi$
$462$	0	0
$463$	11.7889	0.547877	0.273938	−	0.961747i	$-0.411674\pi$
$463$	11.7889	0.547877	0.273938	+	0.961747i	$0.411674\pi$
$464$	0	0
$465$	1.30278	0.0604148
$466$	0	0
$467$	−12.3944	−0.573547	−0.286773	−	0.957998i	$-0.592583\pi$
$467$	−12.3944	−0.573547	−0.286773	+	0.957998i	$0.592583\pi$
$468$	0	0
$469$	−15.1194	−0.698150
$470$	0	0
$471$	13.0278	0.600287
$472$	0	0
$473$	10.4861	0.482152
$474$	0	0
$475$	1.30278	0.0597754
$476$	0	0
$477$	2.09167	0.0957711
$478$	0	0
$479$	13.4222	0.613276	0.306638	−	0.951826i	$-0.400796\pi$
$479$	13.4222	0.613276	0.306638	+	0.951826i	$0.400796\pi$
$480$	0	0
$481$	−26.7250	−1.21855
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	14.3305	0.650716
$486$	0	0
$487$	−34.2111	−1.55025	−0.775127	−	0.631806i	$-0.782314\pi$
$487$	−34.2111	−1.55025	−0.775127	+	0.631806i	$0.782314\pi$
$488$	0	0
$489$	5.48612	0.248091
$490$	0	0
$491$	27.3305	1.23341	0.616705	−	0.787194i	$-0.288467\pi$
$491$	27.3305	1.23341	0.616705	+	0.787194i	$0.288467\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	−4.69722	−0.211125
$496$	0	0
$497$	−1.09167	−0.0489682
$498$	0	0
$499$	−17.5416	−0.785271	−0.392636	−	0.919694i	$-0.628437\pi$
$499$	−17.5416	−0.785271	−0.392636	+	0.919694i	$0.628437\pi$
$500$	0	0
$501$	−4.81665	−0.215192
$502$	0	0
$503$	−12.6972	−0.566141	−0.283071	−	0.959099i	$-0.591353\pi$
$503$	−12.6972	−0.566141	−0.283071	+	0.959099i	$0.591353\pi$
$504$	0	0
$505$	−11.0917	−0.493573
$506$	0	0
$507$	−5.51388	−0.244880
$508$	0	0
$509$	−8.60555	−0.381434	−0.190717	−	0.981645i	$-0.561081\pi$
$509$	−8.60555	−0.381434	−0.190717	+	0.981645i	$0.561081\pi$
$510$	0	0
$511$	−10.8167	−0.478501
$512$	0	0
$513$	0.394449	0.0174153
$514$	0	0
$515$	20.8444	0.918514
$516$	0	0
$517$	−12.2389	−0.538264
$518$	0	0
$519$	−18.0278	−0.791331
$520$	0	0
$521$	39.2111	1.71787	0.858935	−	0.512085i	$-0.171127\pi$
$521$	39.2111	1.71787	0.858935	+	0.512085i	$0.171127\pi$
$522$	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
$525$	3.30278	0.144145
$526$	0	0
$527$	−3.00000	−0.130682
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.0916731	0.00397827
$532$	0	0
$533$	−6.90833	−0.299233
$534$	0	0
$535$	7.18335	0.310563
$536$	0	0
$537$	−9.90833	−0.427576
$538$	0	0
$539$	3.60555	0.155302
$540$	0	0
$541$	−8.60555	−0.369982	−0.184991	−	0.982740i	$-0.559226\pi$
$541$	−8.60555	−0.369982	−0.184991	+	0.982740i	$0.559226\pi$
$542$	0	0
$543$	−19.4222	−0.833487
$544$	0	0
$545$	0.119429	0.00511580
$546$	0	0
$547$	16.5139	0.706082	0.353041	−	0.935608i	$-0.385148\pi$
$547$	16.5139	0.706082	0.353041	+	0.935608i	$0.385148\pi$
$548$	0	0
$549$	−9.51388	−0.406042
$550$	0	0
$551$	−0.394449	−0.0168041
$552$	0	0
$553$	−13.6056	−0.578567
$554$	0	0
$555$	8.09167	0.343472
$556$	0	0
$557$	35.4500	1.50206	0.751032	−	0.660266i	$-0.229557\pi$
$557$	35.4500	1.50206	0.751032	+	0.660266i	$0.229557\pi$
$558$	0	0
$559$	−12.5139	−0.529281
$560$	0	0
$561$	10.8167	0.456679
$562$	0	0
$563$	−18.1194	−0.763643	−0.381821	−	0.924236i	$-0.624703\pi$
$563$	−18.1194	−0.763643	−0.381821	+	0.924236i	$0.624703\pi$
$564$	0	0
$565$	18.9083	0.795479
$566$	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
$571$	−0.816654	−0.0341759	−0.0170879	−	0.999854i	$-0.505440\pi$
$571$	−0.816654	−0.0341759	−0.0170879	+	0.999854i	$0.505440\pi$
$572$	0	0
$573$	15.0278	0.627794
$574$	0	0
$575$	3.30278	0.137735
$576$	0	0
$577$	25.6333	1.06713	0.533564	−	0.845760i	$-0.320852\pi$
$577$	25.6333	1.06713	0.533564	+	0.845760i	$0.320852\pi$
$578$	0	0
$579$	11.8167	0.491083
$580$	0	0
$581$	−7.60555	−0.315531
$582$	0	0
$583$	7.54163	0.312343
$584$	0	0
$585$	5.60555	0.231761
$586$	0	0
$587$	2.30278	0.0950457	0.0475229	−	0.998870i	$-0.484867\pi$
$587$	2.30278	0.0950457	0.0475229	+	0.998870i	$0.484867\pi$
$588$	0	0
$589$	−0.394449	−0.0162530
$590$	0	0
$591$	−19.6972	−0.810236
$592$	0	0
$593$	−25.0278	−1.02777	−0.513883	−	0.857860i	$-0.671794\pi$
$593$	−25.0278	−1.02777	−0.513883	+	0.857860i	$0.671794\pi$
$594$	0	0
$595$	3.90833	0.160226
$596$	0	0
$597$	−23.3028	−0.953719
$598$	0	0
$599$	−14.7250	−0.601646	−0.300823	−	0.953680i	$-0.597261\pi$
$599$	−14.7250	−0.601646	−0.300823	+	0.953680i	$0.597261\pi$
$600$	0	0
$601$	10.9361	0.446092	0.223046	−	0.974808i	$-0.428400\pi$
$601$	10.9361	0.446092	0.223046	+	0.974808i	$0.428400\pi$
$602$	0	0
$603$	−15.1194	−0.615711
$604$	0	0
$605$	−2.60555	−0.105931
$606$	0	0
$607$	28.1194	1.14133	0.570666	−	0.821182i	$-0.306685\pi$
$607$	28.1194	1.14133	0.570666	+	0.821182i	$0.306685\pi$
$608$	0	0
$609$	−1.00000	−0.0405220
$610$	0	0
$611$	14.6056	0.590877
$612$	0	0
$613$	20.4500	0.825966	0.412983	−	0.910739i	$-0.364487\pi$
$613$	20.4500	0.825966	0.412983	+	0.910739i	$0.364487\pi$
$614$	0	0
$615$	2.09167	0.0843444
$616$	0	0
$617$	18.1194	0.729461	0.364730	−	0.931113i	$-0.381161\pi$
$617$	18.1194	0.729461	0.364730	+	0.931113i	$0.381161\pi$
$618$	0	0
$619$	−7.33053	−0.294639	−0.147319	−	0.989089i	$-0.547065\pi$
$619$	−7.33053	−0.294639	−0.147319	+	0.989089i	$0.547065\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−0.302776	−0.0121304
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	1.42221	0.0567974
$628$	0	0
$629$	−18.6333	−0.742959
$630$	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
$633$	6.39445	0.254157
$634$	0	0
$635$	0.394449	0.0156532
$636$	0	0
$637$	−4.30278	−0.170482
$638$	0	0
$639$	−1.09167	−0.0431859
$640$	0	0
$641$	−39.1472	−1.54622	−0.773110	−	0.634271i	$-0.781300\pi$
$641$	−39.1472	−1.54622	−0.773110	+	0.634271i	$0.781300\pi$
$642$	0	0
$643$	−17.2750	−0.681260	−0.340630	−	0.940197i	$-0.610640\pi$
$643$	−17.2750	−0.681260	−0.340630	+	0.940197i	$0.610640\pi$
$644$	0	0
$645$	3.78890	0.149188
$646$	0	0
$647$	−33.9083	−1.33307	−0.666537	−	0.745472i	$-0.732224\pi$
$647$	−33.9083	−1.33307	−0.666537	+	0.745472i	$0.732224\pi$
$648$	0	0
$649$	0.330532	0.0129745
$650$	0	0
$651$	−1.00000	−0.0391931
$652$	0	0
$653$	10.5139	0.411440	0.205720	−	0.978611i	$-0.434046\pi$
$653$	10.5139	0.411440	0.205720	+	0.978611i	$0.434046\pi$
$654$	0	0
$655$	21.9083	0.856029
$656$	0	0
$657$	−10.8167	−0.421998
$658$	0	0
$659$	27.6056	1.07536	0.537680	−	0.843149i	$-0.319301\pi$
$659$	27.6056	1.07536	0.537680	+	0.843149i	$0.319301\pi$
$660$	0	0
$661$	6.21110	0.241584	0.120792	−	0.992678i	$-0.461457\pi$
$661$	6.21110	0.241584	0.120792	+	0.992678i	$0.461457\pi$
$662$	0	0
$663$	−12.9083	−0.501318
$664$	0	0
$665$	0.513878	0.0199273
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	0	0
$669$	9.48612	0.366755
$670$	0	0
$671$	−34.3028	−1.32424
$672$	0	0
$673$	−33.4222	−1.28833	−0.644166	−	0.764886i	$-0.722795\pi$
$673$	−33.4222	−1.28833	−0.644166	+	0.764886i	$0.722795\pi$
$674$	0	0
$675$	3.30278	0.127124
$676$	0	0
$677$	34.3028	1.31836	0.659181	−	0.751984i	$-0.270903\pi$
$677$	34.3028	1.31836	0.659181	+	0.751984i	$0.270903\pi$
$678$	0	0
$679$	−11.0000	−0.422141
$680$	0	0
$681$	−23.3028	−0.892964
$682$	0	0
$683$	−40.3944	−1.54565	−0.772825	−	0.634619i	$-0.781157\pi$
$683$	−40.3944	−1.54565	−0.772825	+	0.634619i	$0.781157\pi$
$684$	0	0
$685$	−26.0917	−0.996912
$686$	0	0
$687$	15.3028	0.583837
$688$	0	0
$689$	−9.00000	−0.342873
$690$	0	0
$691$	37.9361	1.44316	0.721578	−	0.692333i	$-0.243417\pi$
$691$	37.9361	1.44316	0.721578	+	0.692333i	$0.243417\pi$
$692$	0	0
$693$	3.60555	0.136964
$694$	0	0
$695$	8.72498	0.330957
$696$	0	0
$697$	−4.81665	−0.182444
$698$	0	0
$699$	18.3305	0.693324
$700$	0	0
$701$	27.1194	1.02429	0.512143	−	0.858900i	$-0.328852\pi$
$701$	27.1194	1.02429	0.512143	+	0.858900i	$0.328852\pi$
$702$	0	0
$703$	−2.44996	−0.0924020
$704$	0	0
$705$	−4.42221	−0.166550
$706$	0	0
$707$	8.51388	0.320197
$708$	0	0
$709$	16.9083	0.635006	0.317503	−	0.948257i	$-0.397156\pi$
$709$	16.9083	0.635006	0.317503	+	0.948257i	$0.397156\pi$
$710$	0	0
$711$	−13.6056	−0.510248
$712$	0	0
$713$	−1.00000	−0.0374503
$714$	0	0
$715$	20.2111	0.755852
$716$	0	0
$717$	−13.3028	−0.496801
$718$	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
$723$	−20.2111	−0.751659
$724$	0	0
$725$	−3.30278	−0.122662
$726$	0	0
$727$	−3.60555	−0.133722	−0.0668612	−	0.997762i	$-0.521298\pi$
$727$	−3.60555	−0.133722	−0.0668612	+	0.997762i	$0.521298\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.72498	−0.322705
$732$	0	0
$733$	−13.0278	−0.481191	−0.240596	−	0.970625i	$-0.577343\pi$
$733$	−13.0278	−0.481191	−0.240596	+	0.970625i	$0.577343\pi$
$734$	0	0
$735$	1.30278	0.0480536
$736$	0	0
$737$	−54.5139	−2.00804
$738$	0	0
$739$	13.9722	0.513977	0.256989	−	0.966414i	$-0.417270\pi$
$739$	13.9722	0.513977	0.256989	+	0.966414i	$0.417270\pi$
$740$	0	0
$741$	−1.69722	−0.0623491
$742$	0	0
$743$	25.7250	0.943758	0.471879	−	0.881663i	$-0.343576\pi$
$743$	25.7250	0.943758	0.471879	+	0.881663i	$0.343576\pi$
$744$	0	0
$745$	2.36669	0.0867089
$746$	0	0
$747$	−7.60555	−0.278273
$748$	0	0
$749$	−5.51388	−0.201473
$750$	0	0
$751$	17.4861	0.638078	0.319039	−	0.947742i	$-0.396640\pi$
$751$	17.4861	0.638078	0.319039	+	0.947742i	$0.396640\pi$
$752$	0	0
$753$	10.6056	0.386488
$754$	0	0
$755$	−0.788897	−0.0287109
$756$	0	0
$757$	−11.4222	−0.415147	−0.207574	−	0.978219i	$-0.566557\pi$
$757$	−11.4222	−0.415147	−0.207574	+	0.978219i	$0.566557\pi$
$758$	0	0
$759$	3.60555	0.130873
$760$	0	0
$761$	1.21110	0.0439024	0.0219512	−	0.999759i	$-0.493012\pi$
$761$	1.21110	0.0439024	0.0219512	+	0.999759i	$0.493012\pi$
$762$	0	0
$763$	−0.0916731	−0.00331879
$764$	0	0
$765$	3.90833	0.141306
$766$	0	0
$767$	−0.394449	−0.0142427
$768$	0	0
$769$	36.6611	1.32203	0.661016	−	0.750372i	$-0.270126\pi$
$769$	36.6611	1.32203	0.661016	+	0.750372i	$0.270126\pi$
$770$	0	0
$771$	23.0278	0.829324
$772$	0	0
$773$	12.0278	0.432608	0.216304	−	0.976326i	$-0.430600\pi$
$773$	12.0278	0.432608	0.216304	+	0.976326i	$0.430600\pi$
$774$	0	0
$775$	−3.30278	−0.118639
$776$	0	0
$777$	−6.21110	−0.222822
$778$	0	0
$779$	−0.633308	−0.0226906
$780$	0	0
$781$	−3.93608	−0.140844
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	16.9722	0.605765
$786$	0	0
$787$	41.3583	1.47426	0.737132	−	0.675749i	$-0.236180\pi$
$787$	41.3583	1.47426	0.737132	+	0.675749i	$0.236180\pi$
$788$	0	0
$789$	24.6333	0.876969
$790$	0	0
$791$	−14.5139	−0.516054
$792$	0	0
$793$	40.9361	1.45368
$794$	0	0
$795$	2.72498	0.0966451
$796$	0	0
$797$	−7.39445	−0.261925	−0.130962	−	0.991387i	$-0.541807\pi$
$797$	−7.39445	−0.261925	−0.130962	+	0.991387i	$0.541807\pi$
$798$	0	0
$799$	10.1833	0.360261
$800$	0	0
$801$	−0.302776	−0.0106981
$802$	0	0
$803$	−39.0000	−1.37628
$804$	0	0
$805$	1.30278	0.0459168
$806$	0	0
$807$	−24.7250	−0.870361
$808$	0	0
$809$	−46.5416	−1.63632	−0.818158	−	0.574993i	$-0.805005\pi$
$809$	−46.5416	−1.63632	−0.818158	+	0.574993i	$0.805005\pi$
$810$	0	0
$811$	−25.2389	−0.886256	−0.443128	−	0.896458i	$-0.646131\pi$
$811$	−25.2389	−0.886256	−0.443128	+	0.896458i	$0.646131\pi$
$812$	0	0
$813$	−10.8167	−0.379357
$814$	0	0
$815$	7.14719	0.250355
$816$	0	0
$817$	−1.14719	−0.0401350
$818$	0	0
$819$	−4.30278	−0.150351
$820$	0	0
$821$	−13.8167	−0.482205	−0.241102	−	0.970500i	$-0.577509\pi$
$821$	−13.8167	−0.482205	−0.241102	+	0.970500i	$0.577509\pi$
$822$	0	0
$823$	−11.6695	−0.406772	−0.203386	−	0.979099i	$-0.565195\pi$
$823$	−11.6695	−0.406772	−0.203386	+	0.979099i	$0.565195\pi$
$824$	0	0
$825$	11.9083	0.414595
$826$	0	0
$827$	−19.5416	−0.679529	−0.339765	−	0.940511i	$-0.610347\pi$
$827$	−19.5416	−0.679529	−0.339765	+	0.940511i	$0.610347\pi$
$828$	0	0
$829$	−37.8444	−1.31439	−0.657195	−	0.753720i	$-0.728257\pi$
$829$	−37.8444	−1.31439	−0.657195	+	0.753720i	$0.728257\pi$
$830$	0	0
$831$	−23.3305	−0.809327
$832$	0	0
$833$	−3.00000	−0.103944
$834$	0	0
$835$	−6.27502	−0.217156
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	34.3028	1.18426	0.592132	−	0.805841i	$-0.298287\pi$
$839$	34.3028	1.18426	0.592132	+	0.805841i	$0.298287\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	0	0
$843$	3.81665	0.131453
$844$	0	0
$845$	−7.18335	−0.247115
$846$	0	0
$847$	2.00000	0.0687208
$848$	0	0
$849$	−9.30278	−0.319270
$850$	0	0
$851$	−6.21110	−0.212914
$852$	0	0
$853$	0.183346	0.00627765	0.00313883	−	0.999995i	$-0.499001\pi$
$853$	0.183346	0.00627765	0.00313883	+	0.999995i	$0.499001\pi$
$854$	0	0
$855$	0.513878	0.0175743
$856$	0	0
$857$	−4.18335	−0.142900	−0.0714502	−	0.997444i	$-0.522763\pi$
$857$	−4.18335	−0.142900	−0.0714502	+	0.997444i	$0.522763\pi$
$858$	0	0
$859$	4.78890	0.163395	0.0816975	−	0.996657i	$-0.473966\pi$
$859$	4.78890	0.163395	0.0816975	+	0.996657i	$0.473966\pi$
$860$	0	0
$861$	−1.60555	−0.0547170
$862$	0	0
$863$	−26.6056	−0.905663	−0.452832	−	0.891596i	$-0.649586\pi$
$863$	−26.6056	−0.905663	−0.452832	+	0.891596i	$0.649586\pi$
$864$	0	0
$865$	−23.4861	−0.798552
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−49.0555	−1.66409
$870$	0	0
$871$	65.0555	2.20432
$872$	0	0
$873$	−11.0000	−0.372294
$874$	0	0
$875$	10.8167	0.365670
$876$	0	0
$877$	−2.78890	−0.0941744	−0.0470872	−	0.998891i	$-0.514994\pi$
$877$	−2.78890	−0.0941744	−0.0470872	+	0.998891i	$0.514994\pi$
$878$	0	0
$879$	−1.21110	−0.0408495
$880$	0	0
$881$	−32.6611	−1.10038	−0.550190	−	0.835040i	$-0.685445\pi$
$881$	−32.6611	−1.10038	−0.550190	+	0.835040i	$0.685445\pi$
$882$	0	0
$883$	−12.1194	−0.407851	−0.203926	−	0.978986i	$-0.565370\pi$
$883$	−12.1194	−0.407851	−0.203926	+	0.978986i	$0.565370\pi$
$884$	0	0
$885$	0.119429	0.00401458
$886$	0	0
$887$	53.5139	1.79682	0.898410	−	0.439158i	$-0.144723\pi$
$887$	53.5139	1.79682	0.898410	+	0.439158i	$0.144723\pi$
$888$	0	0
$889$	−0.302776	−0.0101548
$890$	0	0
$891$	3.60555	0.120791
$892$	0	0
$893$	1.33894	0.0448058
$894$	0	0
$895$	−12.9083	−0.431478
$896$	0	0
$897$	−4.30278	−0.143665
$898$	0	0
$899$	1.00000	0.0333519
$900$	0	0
$901$	−6.27502	−0.209051
$902$	0	0
$903$	−2.90833	−0.0967831
$904$	0	0
$905$	−25.3028	−0.841093
$906$	0	0
$907$	−39.7250	−1.31905	−0.659523	−	0.751684i	$-0.729242\pi$
$907$	−39.7250	−1.31905	−0.659523	+	0.751684i	$0.729242\pi$
$908$	0	0
$909$	8.51388	0.282388
$910$	0	0
$911$	−7.39445	−0.244989	−0.122495	−	0.992469i	$-0.539089\pi$
$911$	−7.39445	−0.244989	−0.122495	+	0.992469i	$0.539089\pi$
$912$	0	0
$913$	−27.4222	−0.907543
$914$	0	0
$915$	−12.3944	−0.409748
$916$	0	0
$917$	−16.8167	−0.555335
$918$	0	0
$919$	−34.2666	−1.13035	−0.565176	−	0.824971i	$-0.691192\pi$
$919$	−34.2666	−1.13035	−0.565176	+	0.824971i	$0.691192\pi$
$920$	0	0
$921$	15.4222	0.508179
$922$	0	0
$923$	4.69722	0.154611
$924$	0	0
$925$	−20.5139	−0.674492
$926$	0	0
$927$	−16.0000	−0.525509
$928$	0	0
$929$	3.90833	0.128228	0.0641140	−	0.997943i	$-0.479578\pi$
$929$	3.90833	0.128228	0.0641140	+	0.997943i	$0.479578\pi$
$930$	0	0
$931$	−0.394449	−0.0129275
$932$	0	0
$933$	4.90833	0.160691
$934$	0	0
$935$	14.0917	0.460847
$936$	0	0
$937$	−44.8722	−1.46591	−0.732955	−	0.680277i	$-0.761859\pi$
$937$	−44.8722	−1.46591	−0.732955	+	0.680277i	$0.761859\pi$
$938$	0	0
$939$	10.7889	0.352082
$940$	0	0
$941$	8.84441	0.288320	0.144160	−	0.989554i	$-0.453952\pi$
$941$	8.84441	0.288320	0.144160	+	0.989554i	$0.453952\pi$
$942$	0	0
$943$	−1.60555	−0.0522839
$944$	0	0
$945$	1.30278	0.0423793
$946$	0	0
$947$	−19.5778	−0.636193	−0.318096	−	0.948058i	$-0.603044\pi$
$947$	−19.5778	−0.636193	−0.318096	+	0.948058i	$0.603044\pi$
$948$	0	0
$949$	46.5416	1.51081
$950$	0	0
$951$	1.30278	0.0422454
$952$	0	0
$953$	7.33053	0.237459	0.118730	−	0.992927i	$-0.462118\pi$
$953$	7.33053	0.237459	0.118730	+	0.992927i	$0.462118\pi$
$954$	0	0
$955$	19.5778	0.633523
$956$	0	0
$957$	−3.60555	−0.116551
$958$	0	0
$959$	20.0278	0.646730
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−5.51388	−0.177682
$964$	0	0
$965$	15.3944	0.495565
$966$	0	0
$967$	49.6333	1.59610	0.798050	−	0.602592i	$-0.205865\pi$
$967$	49.6333	1.59610	0.798050	+	0.602592i	$0.205865\pi$
$968$	0	0
$969$	−1.18335	−0.0380146
$970$	0	0
$971$	−54.7250	−1.75621	−0.878104	−	0.478470i	$-0.841192\pi$
$971$	−54.7250	−1.75621	−0.878104	+	0.478470i	$0.841192\pi$
$972$	0	0
$973$	−6.69722	−0.214703
$974$	0	0
$975$	−14.2111	−0.455120
$976$	0	0
$977$	28.4861	0.911352	0.455676	−	0.890146i	$-0.349398\pi$
$977$	28.4861	0.911352	0.455676	+	0.890146i	$0.349398\pi$
$978$	0	0
$979$	−1.09167	−0.0348900
$980$	0	0
$981$	−0.0916731	−0.00292690
$982$	0	0
$983$	−49.2389	−1.57048	−0.785238	−	0.619194i	$-0.787459\pi$
$983$	−49.2389	−1.57048	−0.785238	+	0.619194i	$0.787459\pi$
$984$	0	0
$985$	−25.6611	−0.817629
$986$	0	0
$987$	3.39445	0.108046
$988$	0	0
$989$	−2.90833	−0.0924794
$990$	0	0
$991$	12.3028	0.390811	0.195405	−	0.980723i	$-0.437398\pi$
$991$	12.3028	0.390811	0.195405	+	0.980723i	$0.437398\pi$
$992$	0	0
$993$	−4.78890	−0.151971
$994$	0	0
$995$	−30.3583	−0.962422
$996$	0	0
$997$	41.4222	1.31185	0.655927	−	0.754824i	$-0.272278\pi$
$997$	41.4222	1.31185	0.655927	+	0.754824i	$0.272278\pi$
$998$	0	0
$999$	−6.21110	−0.196511

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.30278 q^{5} +1.00000 q^{7} +1.00000 q^{9} +3.60555 q^{11} -4.30278 q^{13} +1.30278 q^{15} -3.00000 q^{17} -0.394449 q^{19} -1.00000 q^{21} -1.00000 q^{23} -3.30278 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.00000 q^{31} -3.60555 q^{33} -1.30278 q^{35} +6.21110 q^{37} +4.30278 q^{39} +1.60555 q^{41} +2.90833 q^{43} -1.30278 q^{45} -3.39445 q^{47} +1.00000 q^{49} +3.00000 q^{51} +2.09167 q^{53} -4.69722 q^{55} +0.394449 q^{57} +0.0916731 q^{59} -9.51388 q^{61} +1.00000 q^{63} +5.60555 q^{65} -15.1194 q^{67} +1.00000 q^{69} -1.09167 q^{71} -10.8167 q^{73} +3.30278 q^{75} +3.60555 q^{77} -13.6056 q^{79} +1.00000 q^{81} -7.60555 q^{83} +3.90833 q^{85} -1.00000 q^{87} -0.302776 q^{89} -4.30278 q^{91} -1.00000 q^{93} +0.513878 q^{95} -11.0000 q^{97} +3.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

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Database

Properties

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