LMFDB - Embedded newform 1232.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,2,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, 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("1232.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1232.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:	$9.83756952902$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11348.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 2 $ x^4 - x^3 - 5*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.589216$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.39434 q^{3} -0.215911 q^{5} +1.00000 q^{7} +8.52156 q^{9} +O(q^{10})$	$q-3.39434 q^{3} -0.215911 q^{5} +1.00000 q^{7} +8.52156 q^{9} -1.00000 q^{11} -2.12722 q^{13} +0.732876 q^{15} +0.821569 q^{17} -2.94879 q^{19} -3.39434 q^{21} +0.732876 q^{23} -4.95338 q^{25} -18.7421 q^{27} -4.35686 q^{29} -1.91131 q^{31} +3.39434 q^{33} -0.215911 q^{35} +9.52156 q^{37} +7.22051 q^{39} +0.821569 q^{41} +11.2205 q^{43} -1.83990 q^{45} +12.3989 q^{47} +1.00000 q^{49} -2.78868 q^{51} +2.00000 q^{53} +0.215911 q^{55} +10.0092 q^{57} +5.75121 q^{59} -8.91590 q^{61} +8.52156 q^{63} +0.459290 q^{65} +14.7421 q^{67} -2.48763 q^{69} -14.3102 q^{71} -7.43287 q^{73} +16.8135 q^{75} -1.00000 q^{77} +5.21132 q^{79} +38.0523 q^{81} -9.48304 q^{83} -0.177386 q^{85} +14.7887 q^{87} +5.08974 q^{89} -2.12722 q^{91} +6.48763 q^{93} +0.636675 q^{95} +12.7421 q^{97} -8.52156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9	$ 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9} - 4 q^{11} + 4 q^{13} + 3 q^{15} + 10 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 17 q^{25} - 13 q^{27} - 4 q^{29} - q^{31} + q^{33} + 5 q^{35} + 13 q^{37} - 8 q^{39} + 10 q^{41} + 8 q^{43} + 12 q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} + 8 q^{53} - 5 q^{55} - 22 q^{57} - 3 q^{59} + 2 q^{61} + 9 q^{63} - 3 q^{67} + 27 q^{69} - 7 q^{71} + 2 q^{73} + 18 q^{75} - 4 q^{77} + 46 q^{79} + 40 q^{81} - 32 q^{83} - 14 q^{85} + 34 q^{87} + 7 q^{89} + 4 q^{91} - 11 q^{93} + 14 q^{95} - 11 q^{97} - 9 q^{99}+O(q^{100}) $ 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9 - 4 * q^11 + 4 * q^13 + 3 * q^15 + 10 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 17 * q^25 - 13 * q^27 - 4 * q^29 - q^31 + q^33 + 5 * q^35 + 13 * q^37 - 8 * q^39 + 10 * q^41 + 8 * q^43 + 12 * q^45 + 6 * q^47 + 4 * q^49 + 14 * q^51 + 8 * q^53 - 5 * q^55 - 22 * q^57 - 3 * q^59 + 2 * q^61 + 9 * q^63 - 3 * q^67 + 27 * q^69 - 7 * q^71 + 2 * q^73 + 18 * q^75 - 4 * q^77 + 46 * q^79 + 40 * q^81 - 32 * q^83 - 14 * q^85 + 34 * q^87 + 7 * q^89 + 4 * q^91 - 11 * q^93 + 14 * q^95 - 11 * q^97 - 9 * 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
$3$	−3.39434	−1.95972	−0.979862	−	0.199675i	$-0.936011\pi$
$3$	−3.39434	−1.95972	−0.979862	+	0.199675i	$0.936011\pi$
$4$	0	0
$5$	−0.215911	−0.0965583	−0.0482792	−	0.998834i	$-0.515374\pi$
$5$	−0.215911	−0.0965583	−0.0482792	+	0.998834i	$0.515374\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	8.52156	2.84052
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.12722	−0.589984	−0.294992	−	0.955500i	$-0.595317\pi$
$13$	−2.12722	−0.589984	−0.294992	+	0.955500i	$0.595317\pi$
$14$	0	0
$15$	0.732876	0.189228
$16$	0	0
$17$	0.821569	0.199260	0.0996298	−	0.995025i	$-0.468234\pi$
$17$	0.821569	0.199260	0.0996298	+	0.995025i	$0.468234\pi$
$18$	0	0
$19$	−2.94879	−0.676498	−0.338249	−	0.941057i	$-0.609835\pi$
$19$	−2.94879	−0.676498	−0.338249	+	0.941057i	$0.609835\pi$
$20$	0	0
$21$	−3.39434	−0.740706
$22$	0	0
$23$	0.732876	0.152815	0.0764076	−	0.997077i	$-0.475655\pi$
$23$	0.732876	0.152815	0.0764076	+	0.997077i	$0.475655\pi$
$24$	0	0
$25$	−4.95338	−0.990676
$26$	0	0
$27$	−18.7421	−3.60691
$28$	0	0
$29$	−4.35686	−0.809049	−0.404525	−	0.914527i	$-0.632563\pi$
$29$	−4.35686	−0.809049	−0.404525	+	0.914527i	$0.632563\pi$
$30$	0	0
$31$	−1.91131	−0.343281	−0.171640	−	0.985160i	$-0.554907\pi$
$31$	−1.91131	−0.343281	−0.171640	+	0.985160i	$0.554907\pi$
$32$	0	0
$33$	3.39434	0.590879
$34$	0	0
$35$	−0.215911	−0.0364956
$36$	0	0
$37$	9.52156	1.56533	0.782667	−	0.622440i	$-0.213859\pi$
$37$	9.52156	1.56533	0.782667	+	0.622440i	$0.213859\pi$
$38$	0	0
$39$	7.22051	1.15621
$40$	0	0
$41$	0.821569	0.128307	0.0641537	−	0.997940i	$-0.479565\pi$
$41$	0.821569	0.128307	0.0641537	+	0.997940i	$0.479565\pi$
$42$	0	0
$43$	11.2205	1.71111	0.855556	−	0.517711i	$-0.173216\pi$
$43$	11.2205	1.71111	0.855556	+	0.517711i	$0.173216\pi$
$44$	0	0
$45$	−1.83990	−0.274276
$46$	0	0
$47$	12.3989	1.80857	0.904286	−	0.426928i	$-0.140404\pi$
$47$	12.3989	1.80857	0.904286	+	0.426928i	$0.140404\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.78868	−0.390494
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0.215911	0.0291134
$56$	0	0
$57$	10.0092	1.32575
$58$	0	0
$59$	5.75121	0.748743	0.374372	−	0.927279i	$-0.377858\pi$
$59$	5.75121	0.748743	0.374372	+	0.927279i	$0.377858\pi$
$60$	0	0
$61$	−8.91590	−1.14156	−0.570782	−	0.821102i	$-0.693360\pi$
$61$	−8.91590	−1.14156	−0.570782	+	0.821102i	$0.693360\pi$
$62$	0	0
$63$	8.52156	1.07362
$64$	0	0
$65$	0.459290	0.0569679
$66$	0	0
$67$	14.7421	1.80103	0.900515	−	0.434825i	$-0.143190\pi$
$67$	14.7421	1.80103	0.900515	+	0.434825i	$0.143190\pi$
$68$	0	0
$69$	−2.48763	−0.299476
$70$	0	0
$71$	−14.3102	−1.69831	−0.849157	−	0.528141i	$-0.822889\pi$
$71$	−14.3102	−1.69831	−0.849157	+	0.528141i	$0.822889\pi$
$72$	0	0
$73$	−7.43287	−0.869951	−0.434976	−	0.900442i	$-0.643243\pi$
$73$	−7.43287	−0.869951	−0.434976	+	0.900442i	$0.643243\pi$
$74$	0	0
$75$	16.8135	1.94145
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.21132	0.586319	0.293159	−	0.956064i	$-0.405293\pi$
$79$	5.21132	0.586319	0.293159	+	0.956064i	$0.405293\pi$
$80$	0	0
$81$	38.0523	4.22803
$82$	0	0
$83$	−9.48304	−1.04090	−0.520449	−	0.853893i	$-0.674235\pi$
$83$	−9.48304	−1.04090	−0.520449	+	0.853893i	$0.674235\pi$
$84$	0	0
$85$	−0.177386	−0.0192402
$86$	0	0
$87$	14.7887	1.58551
$88$	0	0
$89$	5.08974	0.539511	0.269756	−	0.962929i	$-0.413057\pi$
$89$	5.08974	0.539511	0.269756	+	0.962929i	$0.413057\pi$
$90$	0	0
$91$	−2.12722	−0.222993
$92$	0	0
$93$	6.48763	0.672736
$94$	0	0
$95$	0.636675	0.0653215
$96$	0	0
$97$	12.7421	1.29376	0.646880	−	0.762591i	$-0.276073\pi$
$97$	12.7421	1.29376	0.646880	+	0.762591i	$0.276073\pi$
$98$	0	0
$99$	−8.52156	−0.856449
$100$	0	0
$101$	8.48408	0.844198	0.422099	−	0.906550i	$-0.361294\pi$
$101$	8.48408	0.844198	0.422099	+	0.906550i	$0.361294\pi$
$102$	0	0
$103$	11.5353	1.13661	0.568303	−	0.822819i	$-0.307600\pi$
$103$	11.5353	1.13661	0.568303	+	0.822819i	$0.307600\pi$
$104$	0	0
$105$	0.732876	0.0715213
$106$	0	0
$107$	13.1455	1.27083	0.635414	−	0.772172i	$-0.280829\pi$
$107$	13.1455	1.27083	0.635414	+	0.772172i	$0.280829\pi$
$108$	0	0
$109$	13.6523	1.30766	0.653828	−	0.756643i	$-0.273162\pi$
$109$	13.6523	1.30766	0.653828	+	0.756643i	$0.273162\pi$
$110$	0	0
$111$	−32.3194	−3.06763
$112$	0	0
$113$	5.95338	0.560047	0.280024	−	0.959993i	$-0.409658\pi$
$113$	5.95338	0.560047	0.280024	+	0.959993i	$0.409658\pi$
$114$	0	0
$115$	−0.158236	−0.0147556
$116$	0	0
$117$	−18.1272	−1.67586
$118$	0	0
$119$	0.821569	0.0753131
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.78868	−0.251447
$124$	0	0
$125$	2.14904	0.192216
$126$	0	0
$127$	−1.14555	−0.101651	−0.0508255	−	0.998708i	$-0.516185\pi$
$127$	−1.14555	−0.101651	−0.0508255	+	0.998708i	$0.516185\pi$
$128$	0	0
$129$	−38.0862	−3.35331
$130$	0	0
$131$	−14.1693	−1.23798	−0.618988	−	0.785400i	$-0.712457\pi$
$131$	−14.1693	−1.23798	−0.618988	+	0.785400i	$0.712457\pi$
$132$	0	0
$133$	−2.94879	−0.255692
$134$	0	0
$135$	4.04662	0.348277
$136$	0	0
$137$	−10.3852	−0.887268	−0.443634	−	0.896208i	$-0.646311\pi$
$137$	−10.3852	−0.887268	−0.443634	+	0.896208i	$0.646311\pi$
$138$	0	0
$139$	16.5262	1.40173	0.700865	−	0.713294i	$-0.252797\pi$
$139$	16.5262	1.40173	0.700865	+	0.713294i	$0.252797\pi$
$140$	0	0
$141$	−42.0862	−3.54430
$142$	0	0
$143$	2.12722	0.177887
$144$	0	0
$145$	0.940694	0.0781204
$146$	0	0
$147$	−3.39434	−0.279961
$148$	0	0
$149$	−6.71373	−0.550010	−0.275005	−	0.961443i	$-0.588679\pi$
$149$	−6.71373	−0.550010	−0.275005	+	0.961443i	$0.588679\pi$
$150$	0	0
$151$	21.4000	1.74151	0.870753	−	0.491721i	$-0.163632\pi$
$151$	21.4000	1.74151	0.870753	+	0.491721i	$0.163632\pi$
$152$	0	0
$153$	7.00105	0.566001
$154$	0	0
$155$	0.412672	0.0331466
$156$	0	0
$157$	−3.18198	−0.253950	−0.126975	−	0.991906i	$-0.540527\pi$
$157$	−3.18198	−0.253950	−0.126975	+	0.991906i	$0.540527\pi$
$158$	0	0
$159$	−6.78868	−0.538378
$160$	0	0
$161$	0.732876	0.0577587
$162$	0	0
$163$	9.32293	0.730229	0.365114	−	0.930963i	$-0.381030\pi$
$163$	9.32293	0.730229	0.365114	+	0.930963i	$0.381030\pi$
$164$	0	0
$165$	−0.732876	−0.0570543
$166$	0	0
$167$	−3.56818	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$167$	−3.56818	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$168$	0	0
$169$	−8.47494	−0.651919
$170$	0	0
$171$	−25.1283	−1.92161
$172$	0	0
$173$	19.9590	1.51746	0.758728	−	0.651407i	$-0.225821\pi$
$173$	19.9590	1.51746	0.758728	+	0.651407i	$0.225821\pi$
$174$	0	0
$175$	−4.95338	−0.374441
$176$	0	0
$177$	−19.5216	−1.46733
$178$	0	0
$179$	0.555490	0.0415193	0.0207596	−	0.999784i	$-0.493392\pi$
$179$	0.555490	0.0415193	0.0207596	+	0.999784i	$0.493392\pi$
$180$	0	0
$181$	1.85905	0.138182	0.0690910	−	0.997610i	$-0.477990\pi$
$181$	1.85905	0.138182	0.0690910	+	0.997610i	$0.477990\pi$
$182$	0	0
$183$	30.2636	2.23715
$184$	0	0
$185$	−2.05581	−0.151146
$186$	0	0
$187$	−0.821569	−0.0600790
$188$	0	0
$189$	−18.7421	−1.36328
$190$	0	0
$191$	−15.8035	−1.14350	−0.571749	−	0.820428i	$-0.693735\pi$
$191$	−15.8035	−1.14350	−0.571749	+	0.820428i	$0.693735\pi$
$192$	0	0
$193$	21.4749	1.54580	0.772900	−	0.634528i	$-0.218805\pi$
$193$	21.4749	1.54580	0.772900	+	0.634528i	$0.218805\pi$
$194$	0	0
$195$	−1.55899	−0.111641
$196$	0	0
$197$	0.611299	0.0435533	0.0217766	−	0.999763i	$-0.493068\pi$
$197$	0.611299	0.0435533	0.0217766	+	0.999763i	$0.493068\pi$
$198$	0	0
$199$	4.39894	0.311832	0.155916	−	0.987770i	$-0.450167\pi$
$199$	4.39894	0.311832	0.155916	+	0.987770i	$0.450167\pi$
$200$	0	0
$201$	−50.0396	−3.52952
$202$	0	0
$203$	−4.35686	−0.305792
$204$	0	0
$205$	−0.177386	−0.0123892
$206$	0	0
$207$	6.24524	0.434075
$208$	0	0
$209$	2.94879	0.203972
$210$	0	0
$211$	−18.7229	−1.28894	−0.644469	−	0.764630i	$-0.722922\pi$
$211$	−18.7229	−1.28894	−0.644469	+	0.764630i	$0.722922\pi$
$212$	0	0
$213$	48.5739	3.32823
$214$	0	0
$215$	−2.42263	−0.165222
$216$	0	0
$217$	−1.91131	−0.129748
$218$	0	0
$219$	25.2297	1.70487
$220$	0	0
$221$	−1.74766	−0.117560
$222$	0	0
$223$	5.47949	0.366933	0.183467	−	0.983026i	$-0.441268\pi$
$223$	5.47949	0.366933	0.183467	+	0.983026i	$0.441268\pi$
$224$	0	0
$225$	−42.2105	−2.81404
$226$	0	0
$227$	−17.4173	−1.15602	−0.578012	−	0.816028i	$-0.696171\pi$
$227$	−17.4173	−1.15602	−0.578012	+	0.816028i	$0.696171\pi$
$228$	0	0
$229$	−0.563582	−0.0372426	−0.0186213	−	0.999827i	$-0.505928\pi$
$229$	−0.563582	−0.0372426	−0.0186213	+	0.999827i	$0.505928\pi$
$230$	0	0
$231$	3.39434	0.223331
$232$	0	0
$233$	−8.35686	−0.547476	−0.273738	−	0.961804i	$-0.588260\pi$
$233$	−8.35686	−0.547476	−0.273738	+	0.961804i	$0.588260\pi$
$234$	0	0
$235$	−2.67707	−0.174633
$236$	0	0
$237$	−17.6890	−1.14902
$238$	0	0
$239$	−18.9661	−1.22681	−0.613407	−	0.789767i	$-0.710201\pi$
$239$	−18.9661	−1.22681	−0.613407	+	0.789767i	$0.710201\pi$
$240$	0	0
$241$	16.7283	1.07757	0.538783	−	0.842444i	$-0.318884\pi$
$241$	16.7283	1.07757	0.538783	+	0.842444i	$0.318884\pi$
$242$	0	0
$243$	−72.9364	−4.67887
$244$	0	0
$245$	−0.215911	−0.0137940
$246$	0	0
$247$	6.27271	0.399123
$248$	0	0
$249$	32.1887	2.03987
$250$	0	0
$251$	14.8693	0.938541	0.469270	−	0.883054i	$-0.344517\pi$
$251$	14.8693	0.938541	0.469270	+	0.883054i	$0.344517\pi$
$252$	0	0
$253$	−0.732876	−0.0460755
$254$	0	0
$255$	0.602108	0.0377054
$256$	0	0
$257$	−6.50887	−0.406012	−0.203006	−	0.979177i	$-0.565071\pi$
$257$	−6.50887	−0.406012	−0.203006	+	0.979177i	$0.565071\pi$
$258$	0	0
$259$	9.52156	0.591641
$260$	0	0
$261$	−37.1273	−2.29812
$262$	0	0
$263$	28.6205	1.76482	0.882408	−	0.470486i	$-0.155921\pi$
$263$	28.6205	1.76482	0.882408	+	0.470486i	$0.155921\pi$
$264$	0	0
$265$	−0.431822	−0.0265266
$266$	0	0
$267$	−17.2763	−1.05729
$268$	0	0
$269$	19.1283	1.16627	0.583135	−	0.812375i	$-0.301826\pi$
$269$	19.1283	1.16627	0.583135	+	0.812375i	$0.301826\pi$
$270$	0	0
$271$	27.7294	1.68444	0.842220	−	0.539134i	$-0.181248\pi$
$271$	27.7294	1.68444	0.842220	+	0.539134i	$0.181248\pi$
$272$	0	0
$273$	7.22051	0.437005
$274$	0	0
$275$	4.95338	0.298700
$276$	0	0
$277$	−25.0431	−1.50470	−0.752348	−	0.658766i	$-0.771079\pi$
$277$	−25.0431	−1.50470	−0.752348	+	0.658766i	$0.771079\pi$
$278$	0	0
$279$	−16.2873	−0.975096
$280$	0	0
$281$	11.7548	0.701230	0.350615	−	0.936520i	$-0.385973\pi$
$281$	11.7548	0.701230	0.350615	+	0.936520i	$0.385973\pi$
$282$	0	0
$283$	−28.3488	−1.68516	−0.842580	−	0.538572i	$-0.818964\pi$
$283$	−28.3488	−1.68516	−0.842580	+	0.538572i	$0.818964\pi$
$284$	0	0
$285$	−2.16109	−0.128012
$286$	0	0
$287$	0.821569	0.0484957
$288$	0	0
$289$	−16.3250	−0.960296
$290$	0	0
$291$	−43.2509	−2.53541
$292$	0	0
$293$	26.2388	1.53289	0.766445	−	0.642310i	$-0.222024\pi$
$293$	26.2388	1.53289	0.766445	+	0.642310i	$0.222024\pi$
$294$	0	0
$295$	−1.24175	−0.0722974
$296$	0	0
$297$	18.7421	1.08753
$298$	0	0
$299$	−1.55899	−0.0901585
$300$	0	0
$301$	11.2205	0.646739
$302$	0	0
$303$	−28.7979	−1.65439
$304$	0	0
$305$	1.92504	0.110228
$306$	0	0
$307$	−19.8399	−1.13232	−0.566161	−	0.824294i	$-0.691572\pi$
$307$	−19.8399	−1.13232	−0.566161	+	0.824294i	$0.691572\pi$
$308$	0	0
$309$	−39.1547	−2.22744
$310$	0	0
$311$	−18.2307	−1.03377	−0.516885	−	0.856055i	$-0.672909\pi$
$311$	−18.2307	−1.03377	−0.516885	+	0.856055i	$0.672909\pi$
$312$	0	0
$313$	12.7695	0.721777	0.360888	−	0.932609i	$-0.382474\pi$
$313$	12.7695	0.721777	0.360888	+	0.932609i	$0.382474\pi$
$314$	0	0
$315$	−1.83990	−0.103667
$316$	0	0
$317$	23.5308	1.32162	0.660809	−	0.750554i	$-0.270213\pi$
$317$	23.5308	1.32162	0.660809	+	0.750554i	$0.270213\pi$
$318$	0	0
$319$	4.35686	0.243937
$320$	0	0
$321$	−44.6205	−2.49047
$322$	0	0
$323$	−2.42263	−0.134799
$324$	0	0
$325$	10.5369	0.584483
$326$	0	0
$327$	−46.3407	−2.56265
$328$	0	0
$329$	12.3989	0.683576
$330$	0	0
$331$	−9.77391	−0.537222	−0.268611	−	0.963249i	$-0.586565\pi$
$331$	−9.77391	−0.537222	−0.268611	+	0.963249i	$0.586565\pi$
$332$	0	0
$333$	81.1386	4.44637
$334$	0	0
$335$	−3.18297	−0.173904
$336$	0	0
$337$	−6.60921	−0.360026	−0.180013	−	0.983664i	$-0.557614\pi$
$337$	−6.60921	−0.360026	−0.180013	+	0.983664i	$0.557614\pi$
$338$	0	0
$339$	−20.2078	−1.09754
$340$	0	0
$341$	1.91131	0.103503
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.537107	0.0289169
$346$	0	0
$347$	−8.89111	−0.477300	−0.238650	−	0.971106i	$-0.576705\pi$
$347$	−8.89111	−0.477300	−0.238650	+	0.971106i	$0.576705\pi$
$348$	0	0
$349$	29.7138	1.59054	0.795271	−	0.606254i	$-0.207329\pi$
$349$	29.7138	1.59054	0.795271	+	0.606254i	$0.207329\pi$
$350$	0	0
$351$	39.8685	2.12802
$352$	0	0
$353$	−25.8601	−1.37640	−0.688198	−	0.725523i	$-0.741598\pi$
$353$	−25.8601	−1.37640	−0.688198	+	0.725523i	$0.741598\pi$
$354$	0	0
$355$	3.08974	0.163986
$356$	0	0
$357$	−2.78868	−0.147593
$358$	0	0
$359$	20.7979	1.09767	0.548835	−	0.835931i	$-0.315072\pi$
$359$	20.7979	1.09767	0.548835	+	0.835931i	$0.315072\pi$
$360$	0	0
$361$	−10.3047	−0.542350
$362$	0	0
$363$	−3.39434	−0.178157
$364$	0	0
$365$	1.60484	0.0840010
$366$	0	0
$367$	14.3431	0.748705	0.374353	−	0.927286i	$-0.377865\pi$
$367$	14.3431	0.748705	0.374353	+	0.927286i	$0.377865\pi$
$368$	0	0
$369$	7.00105	0.364460
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	7.14555	0.369982	0.184991	−	0.982740i	$-0.440774\pi$
$373$	7.14555	0.369982	0.184991	+	0.982740i	$0.440774\pi$
$374$	0	0
$375$	−7.29459	−0.376691
$376$	0	0
$377$	9.26800	0.477326
$378$	0	0
$379$	12.9215	0.663735	0.331868	−	0.943326i	$-0.392321\pi$
$379$	12.9215	0.663735	0.331868	+	0.943326i	$0.392321\pi$
$380$	0	0
$381$	3.88838	0.199208
$382$	0	0
$383$	−24.3523	−1.24435	−0.622173	−	0.782880i	$-0.713750\pi$
$383$	−24.3523	−1.24435	−0.622173	+	0.782880i	$0.713750\pi$
$384$	0	0
$385$	0.215911	0.0110038
$386$	0	0
$387$	95.6162	4.86045
$388$	0	0
$389$	−7.09893	−0.359930	−0.179965	−	0.983673i	$-0.557598\pi$
$389$	−7.09893	−0.359930	−0.179965	+	0.983673i	$0.557598\pi$
$390$	0	0
$391$	0.602108	0.0304499
$392$	0	0
$393$	48.0954	2.42609
$394$	0	0
$395$	−1.12518	−0.0566139
$396$	0	0
$397$	28.7898	1.44492	0.722459	−	0.691414i	$-0.243012\pi$
$397$	28.7898	1.44492	0.722459	+	0.691414i	$0.243012\pi$
$398$	0	0
$399$	10.0092	0.501086
$400$	0	0
$401$	5.79515	0.289396	0.144698	−	0.989476i	$-0.453779\pi$
$401$	5.79515	0.289396	0.144698	+	0.989476i	$0.453779\pi$
$402$	0	0
$403$	4.06577	0.202530
$404$	0	0
$405$	−8.21591	−0.408252
$406$	0	0
$407$	−9.52156	−0.471966
$408$	0	0
$409$	−4.82157	−0.238411	−0.119206	−	0.992870i	$-0.538035\pi$
$409$	−4.82157	−0.238411	−0.119206	+	0.992870i	$0.538035\pi$
$410$	0	0
$411$	35.2509	1.73880
$412$	0	0
$413$	5.75121	0.282998
$414$	0	0
$415$	2.04749	0.100507
$416$	0	0
$417$	−56.0954	−2.74701
$418$	0	0
$419$	−2.72723	−0.133234	−0.0666171	−	0.997779i	$-0.521221\pi$
$419$	−2.72723	−0.133234	−0.0666171	+	0.997779i	$0.521221\pi$
$420$	0	0
$421$	28.2911	1.37882	0.689412	−	0.724370i	$-0.257869\pi$
$421$	28.2911	1.37882	0.689412	+	0.724370i	$0.257869\pi$
$422$	0	0
$423$	105.658	5.13728
$424$	0	0
$425$	−4.06954	−0.197402
$426$	0	0
$427$	−8.91590	−0.431471
$428$	0	0
$429$	−7.22051	−0.348609
$430$	0	0
$431$	−27.5795	−1.32846	−0.664228	−	0.747530i	$-0.731240\pi$
$431$	−27.5795	−1.32846	−0.664228	+	0.747530i	$0.731240\pi$
$432$	0	0
$433$	−1.34417	−0.0645969	−0.0322985	−	0.999478i	$-0.510283\pi$
$433$	−1.34417	−0.0645969	−0.0322985	+	0.999478i	$0.510283\pi$
$434$	0	0
$435$	−3.19304	−0.153095
$436$	0	0
$437$	−2.16109	−0.103379
$438$	0	0
$439$	21.5499	1.02852	0.514260	−	0.857634i	$-0.328067\pi$
$439$	21.5499	1.02852	0.514260	+	0.857634i	$0.328067\pi$
$440$	0	0
$441$	8.52156	0.405789
$442$	0	0
$443$	5.97876	0.284059	0.142030	−	0.989862i	$-0.454637\pi$
$443$	5.97876	0.284059	0.142030	+	0.989862i	$0.454637\pi$
$444$	0	0
$445$	−1.09893	−0.0520943
$446$	0	0
$447$	22.7887	1.07787
$448$	0	0
$449$	−8.05581	−0.380177	−0.190089	−	0.981767i	$-0.560878\pi$
$449$	−8.05581	−0.380177	−0.190089	+	0.981767i	$0.560878\pi$
$450$	0	0
$451$	−0.821569	−0.0386862
$452$	0	0
$453$	−72.6389	−3.41287
$454$	0	0
$455$	0.459290	0.0215318
$456$	0	0
$457$	−4.72292	−0.220929	−0.110464	−	0.993880i	$-0.535234\pi$
$457$	−4.72292	−0.220929	−0.110464	+	0.993880i	$0.535234\pi$
$458$	0	0
$459$	−15.3979	−0.718712
$460$	0	0
$461$	6.40912	0.298503	0.149251	−	0.988799i	$-0.452314\pi$
$461$	6.40912	0.298503	0.149251	+	0.988799i	$0.452314\pi$
$462$	0	0
$463$	3.98085	0.185006	0.0925029	−	0.995712i	$-0.470513\pi$
$463$	3.98085	0.185006	0.0925029	+	0.995712i	$0.470513\pi$
$464$	0	0
$465$	−1.40075	−0.0649583
$466$	0	0
$467$	−9.31938	−0.431250	−0.215625	−	0.976476i	$-0.569179\pi$
$467$	−9.31938	−0.431250	−0.215625	+	0.976476i	$0.569179\pi$
$468$	0	0
$469$	14.7421	0.680725
$470$	0	0
$471$	10.8007	0.497672
$472$	0	0
$473$	−11.2205	−0.515919
$474$	0	0
$475$	14.6065	0.670191
$476$	0	0
$477$	17.0431	0.780351
$478$	0	0
$479$	−33.1181	−1.51320	−0.756602	−	0.653876i	$-0.773142\pi$
$479$	−33.1181	−1.51320	−0.756602	+	0.653876i	$0.773142\pi$
$480$	0	0
$481$	−20.2544	−0.923523
$482$	0	0
$483$	−2.48763	−0.113191
$484$	0	0
$485$	−2.75115	−0.124923
$486$	0	0
$487$	−12.4897	−0.565963	−0.282982	−	0.959125i	$-0.591324\pi$
$487$	−12.4897	−0.565963	−0.282982	+	0.959125i	$0.591324\pi$
$488$	0	0
$489$	−31.6452	−1.43105
$490$	0	0
$491$	0.347671	0.0156902	0.00784509	−	0.999969i	$-0.497503\pi$
$491$	0.347671	0.0156902	0.00784509	+	0.999969i	$0.497503\pi$
$492$	0	0
$493$	−3.57946	−0.161211
$494$	0	0
$495$	1.83990	0.0826973
$496$	0	0
$497$	−14.3102	−0.641902
$498$	0	0
$499$	21.1181	0.945375	0.472688	−	0.881230i	$-0.343284\pi$
$499$	21.1181	0.945375	0.472688	+	0.881230i	$0.343284\pi$
$500$	0	0
$501$	12.1116	0.541107
$502$	0	0
$503$	−40.4431	−1.80327	−0.901634	−	0.432499i	$-0.857632\pi$
$503$	−40.4431	−1.80327	−0.901634	+	0.432499i	$0.857632\pi$
$504$	0	0
$505$	−1.83181	−0.0815143
$506$	0	0
$507$	28.7669	1.27758
$508$	0	0
$509$	0.995404	0.0441205	0.0220603	−	0.999757i	$-0.492977\pi$
$509$	0.995404	0.0441205	0.0220603	+	0.999757i	$0.492977\pi$
$510$	0	0
$511$	−7.43287	−0.328811
$512$	0	0
$513$	55.2664	2.44007
$514$	0	0
$515$	−2.49060	−0.109749
$516$	0	0
$517$	−12.3989	−0.545305
$518$	0	0
$519$	−67.7478	−2.97380
$520$	0	0
$521$	34.3194	1.50356	0.751781	−	0.659413i	$-0.229195\pi$
$521$	34.3194	1.50356	0.751781	+	0.659413i	$0.229195\pi$
$522$	0	0
$523$	−10.9763	−0.479958	−0.239979	−	0.970778i	$-0.577141\pi$
$523$	−10.9763	−0.479958	−0.239979	+	0.970778i	$0.577141\pi$
$524$	0	0
$525$	16.8135	0.733800
$526$	0	0
$527$	−1.57027	−0.0684020
$528$	0	0
$529$	−22.4629	−0.976648
$530$	0	0
$531$	49.0092	2.12682
$532$	0	0
$533$	−1.74766	−0.0756994
$534$	0	0
$535$	−2.83827	−0.122709
$536$	0	0
$537$	−1.88552	−0.0813663
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	18.9407	0.814324	0.407162	−	0.913356i	$-0.366518\pi$
$541$	18.9407	0.814324	0.407162	+	0.913356i	$0.366518\pi$
$542$	0	0
$543$	−6.31025	−0.270799
$544$	0	0
$545$	−2.94769	−0.126265
$546$	0	0
$547$	16.6588	0.712278	0.356139	−	0.934433i	$-0.384093\pi$
$547$	16.6588	0.712278	0.356139	+	0.934433i	$0.384093\pi$
$548$	0	0
$549$	−75.9774	−3.24264
$550$	0	0
$551$	12.8475	0.547320
$552$	0	0
$553$	5.21132	0.221608
$554$	0	0
$555$	6.97812	0.296205
$556$	0	0
$557$	−43.5612	−1.84575	−0.922873	−	0.385104i	$-0.874166\pi$
$557$	−43.5612	−1.84575	−0.922873	+	0.385104i	$0.874166\pi$
$558$	0	0
$559$	−23.8685	−1.00953
$560$	0	0
$561$	2.78868	0.117738
$562$	0	0
$563$	12.4145	0.523210	0.261605	−	0.965175i	$-0.415748\pi$
$563$	12.4145	0.523210	0.261605	+	0.965175i	$0.415748\pi$
$564$	0	0
$565$	−1.28540	−0.0540772
$566$	0	0
$567$	38.0523	1.59805
$568$	0	0
$569$	2.60921	0.109384	0.0546918	−	0.998503i	$-0.482582\pi$
$569$	2.60921	0.109384	0.0546918	+	0.998503i	$0.482582\pi$
$570$	0	0
$571$	−13.4275	−0.561921	−0.280961	−	0.959719i	$-0.590653\pi$
$571$	−13.4275	−0.561921	−0.280961	+	0.959719i	$0.590653\pi$
$572$	0	0
$573$	53.6424	2.24094
$574$	0	0
$575$	−3.63021	−0.151390
$576$	0	0
$577$	4.70750	0.195976	0.0979879	−	0.995188i	$-0.468759\pi$
$577$	4.70750	0.195976	0.0979879	+	0.995188i	$0.468759\pi$
$578$	0	0
$579$	−72.8933	−3.02934
$580$	0	0
$581$	−9.48304	−0.393423
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	3.91386	0.161818
$586$	0	0
$587$	−15.8841	−0.655606	−0.327803	−	0.944746i	$-0.606308\pi$
$587$	−15.8841	−0.655606	−0.327803	+	0.944746i	$0.606308\pi$
$588$	0	0
$589$	5.63604	0.232229
$590$	0	0
$591$	−2.07496	−0.0853524
$592$	0	0
$593$	9.07600	0.372707	0.186353	−	0.982483i	$-0.440333\pi$
$593$	9.07600	0.372707	0.186353	+	0.982483i	$0.440333\pi$
$594$	0	0
$595$	−0.177386	−0.00727210
$596$	0	0
$597$	−14.9315	−0.611106
$598$	0	0
$599$	33.3067	1.36088	0.680438	−	0.732805i	$-0.261789\pi$
$599$	33.3067	1.36088	0.680438	+	0.732805i	$0.261789\pi$
$600$	0	0
$601$	−11.6103	−0.473592	−0.236796	−	0.971559i	$-0.576097\pi$
$601$	−11.6103	−0.473592	−0.236796	+	0.971559i	$0.576097\pi$
$602$	0	0
$603$	125.625	5.11586
$604$	0	0
$605$	−0.215911	−0.00877803
$606$	0	0
$607$	30.1795	1.22495	0.612474	−	0.790491i	$-0.290175\pi$
$607$	30.1795	1.22495	0.612474	+	0.790491i	$0.290175\pi$
$608$	0	0
$609$	14.7887	0.599268
$610$	0	0
$611$	−26.3752	−1.06703
$612$	0	0
$613$	−3.75476	−0.151653	−0.0758266	−	0.997121i	$-0.524160\pi$
$613$	−3.75476	−0.151653	−0.0758266	+	0.997121i	$0.524160\pi$
$614$	0	0
$615$	0.602108	0.0242793
$616$	0	0
$617$	−34.5434	−1.39067	−0.695333	−	0.718687i	$-0.744743\pi$
$617$	−34.5434	−1.39067	−0.695333	+	0.718687i	$0.744743\pi$
$618$	0	0
$619$	−21.0762	−0.847125	−0.423563	−	0.905867i	$-0.639221\pi$
$619$	−21.0762	−0.847125	−0.423563	+	0.905867i	$0.639221\pi$
$620$	0	0
$621$	−13.7356	−0.551191
$622$	0	0
$623$	5.08974	0.203916
$624$	0	0
$625$	24.3029	0.972116
$626$	0	0
$627$	−10.0092	−0.399729
$628$	0	0
$629$	7.82261	0.311908
$630$	0	0
$631$	−20.7787	−0.827188	−0.413594	−	0.910461i	$-0.635727\pi$
$631$	−20.7787	−0.827188	−0.413594	+	0.910461i	$0.635727\pi$
$632$	0	0
$633$	63.5520	2.52596
$634$	0	0
$635$	0.247336	0.00981524
$636$	0	0
$637$	−2.12722	−0.0842835
$638$	0	0
$639$	−121.946	−4.82409
$640$	0	0
$641$	−30.4968	−1.20455	−0.602276	−	0.798288i	$-0.705739\pi$
$641$	−30.4968	−1.20455	−0.602276	+	0.798288i	$0.705739\pi$
$642$	0	0
$643$	−22.5216	−0.888166	−0.444083	−	0.895986i	$-0.646470\pi$
$643$	−22.5216	−0.888166	−0.444083	+	0.895986i	$0.646470\pi$
$644$	0	0
$645$	8.22324	0.323790
$646$	0	0
$647$	39.3680	1.54771	0.773857	−	0.633360i	$-0.218325\pi$
$647$	39.3680	1.54771	0.773857	+	0.633360i	$0.218325\pi$
$648$	0	0
$649$	−5.75121	−0.225755
$650$	0	0
$651$	6.48763	0.254270
$652$	0	0
$653$	−25.4283	−0.995087	−0.497544	−	0.867439i	$-0.665765\pi$
$653$	−25.4283	−0.995087	−0.497544	+	0.867439i	$0.665765\pi$
$654$	0	0
$655$	3.05931	0.119537
$656$	0	0
$657$	−63.3396	−2.47111
$658$	0	0
$659$	−33.4000	−1.30108	−0.650539	−	0.759473i	$-0.725457\pi$
$659$	−33.4000	−1.30108	−0.650539	+	0.759473i	$0.725457\pi$
$660$	0	0
$661$	−1.07955	−0.0419898	−0.0209949	−	0.999780i	$-0.506683\pi$
$661$	−1.07955	−0.0419898	−0.0209949	+	0.999780i	$0.506683\pi$
$662$	0	0
$663$	5.93214	0.230385
$664$	0	0
$665$	0.636675	0.0246892
$666$	0	0
$667$	−3.19304	−0.123635
$668$	0	0
$669$	−18.5992	−0.719089
$670$	0	0
$671$	8.91590	0.344195
$672$	0	0
$673$	−23.4000	−0.902003	−0.451002	−	0.892523i	$-0.648933\pi$
$673$	−23.4000	−0.902003	−0.451002	+	0.892523i	$0.648933\pi$
$674$	0	0
$675$	92.8366	3.57328
$676$	0	0
$677$	25.6296	0.985027	0.492513	−	0.870305i	$-0.336078\pi$
$677$	25.6296	0.985027	0.492513	+	0.870305i	$0.336078\pi$
$678$	0	0
$679$	12.7421	0.488996
$680$	0	0
$681$	59.1202	2.26549
$682$	0	0
$683$	30.6459	1.17263	0.586316	−	0.810083i	$-0.300578\pi$
$683$	30.6459	1.17263	0.586316	+	0.810083i	$0.300578\pi$
$684$	0	0
$685$	2.24228	0.0856731
$686$	0	0
$687$	1.91299	0.0729852
$688$	0	0
$689$	−4.25444	−0.162081
$690$	0	0
$691$	−1.55345	−0.0590961	−0.0295480	−	0.999563i	$-0.509407\pi$
$691$	−1.55345	−0.0590961	−0.0295480	+	0.999563i	$0.509407\pi$
$692$	0	0
$693$	−8.52156	−0.323707
$694$	0	0
$695$	−3.56818	−0.135349
$696$	0	0
$697$	0.674975	0.0255665
$698$	0	0
$699$	28.3661	1.07290
$700$	0	0
$701$	8.93860	0.337606	0.168803	−	0.985650i	$-0.446010\pi$
$701$	8.93860	0.337606	0.168803	+	0.985650i	$0.446010\pi$
$702$	0	0
$703$	−28.0771	−1.05895
$704$	0	0
$705$	9.08688	0.342232
$706$	0	0
$707$	8.48408	0.319077
$708$	0	0
$709$	5.52156	0.207367	0.103683	−	0.994610i	$-0.466937\pi$
$709$	5.52156	0.207367	0.103683	+	0.994610i	$0.466937\pi$
$710$	0	0
$711$	44.4085	1.66545
$712$	0	0
$713$	−1.40075	−0.0524585
$714$	0	0
$715$	−0.459290	−0.0171765
$716$	0	0
$717$	64.3773	2.40422
$718$	0	0
$719$	0.379789	0.0141637	0.00708186	−	0.999975i	$-0.497746\pi$
$719$	0.379789	0.0141637	0.00708186	+	0.999975i	$0.497746\pi$
$720$	0	0
$721$	11.5353	0.429597
$722$	0	0
$723$	−56.7817	−2.11173
$724$	0	0
$725$	21.5812	0.801506
$726$	0	0
$727$	29.2250	1.08390	0.541949	−	0.840412i	$-0.317687\pi$
$727$	29.2250	1.08390	0.541949	+	0.840412i	$0.317687\pi$
$728$	0	0
$729$	133.414	4.94126
$730$	0	0
$731$	9.21842	0.340955
$732$	0	0
$733$	−16.8227	−0.621359	−0.310680	−	0.950515i	$-0.600557\pi$
$733$	−16.8227	−0.621359	−0.310680	+	0.950515i	$0.600557\pi$
$734$	0	0
$735$	0.732876	0.0270325
$736$	0	0
$737$	−14.7421	−0.543031
$738$	0	0
$739$	14.8383	0.545835	0.272917	−	0.962038i	$-0.412011\pi$
$739$	14.8383	0.545835	0.272917	+	0.962038i	$0.412011\pi$
$740$	0	0
$741$	−21.2917	−0.782171
$742$	0	0
$743$	−22.1208	−0.811534	−0.405767	−	0.913977i	$-0.632996\pi$
$743$	−22.1208	−0.811534	−0.405767	+	0.913977i	$0.632996\pi$
$744$	0	0
$745$	1.44957	0.0531080
$746$	0	0
$747$	−80.8103	−2.95669
$748$	0	0
$749$	13.1455	0.480328
$750$	0	0
$751$	−4.88279	−0.178176	−0.0890878	−	0.996024i	$-0.528395\pi$
$751$	−4.88279	−0.178176	−0.0890878	+	0.996024i	$0.528395\pi$
$752$	0	0
$753$	−50.4714	−1.83928
$754$	0	0
$755$	−4.62049	−0.168157
$756$	0	0
$757$	35.2022	1.27945	0.639723	−	0.768605i	$-0.279049\pi$
$757$	35.2022	1.27945	0.639723	+	0.768605i	$0.279049\pi$
$758$	0	0
$759$	2.48763	0.0902953
$760$	0	0
$761$	−19.7332	−0.715326	−0.357663	−	0.933851i	$-0.616426\pi$
$761$	−19.7332	−0.715326	−0.357663	+	0.933851i	$0.616426\pi$
$762$	0	0
$763$	13.6523	0.494247
$764$	0	0
$765$	−1.51160	−0.0546521
$766$	0	0
$767$	−12.2341	−0.441747
$768$	0	0
$769$	28.0038	1.00984	0.504921	−	0.863166i	$-0.331522\pi$
$769$	28.0038	1.00984	0.504921	+	0.863166i	$0.331522\pi$
$770$	0	0
$771$	22.0933	0.795672
$772$	0	0
$773$	−43.1466	−1.55188	−0.775939	−	0.630809i	$-0.782723\pi$
$773$	−43.1466	−1.55188	−0.775939	+	0.630809i	$0.782723\pi$
$774$	0	0
$775$	9.46744	0.340080
$776$	0	0
$777$	−32.3194	−1.15945
$778$	0	0
$779$	−2.42263	−0.0867997
$780$	0	0
$781$	14.3102	0.512061
$782$	0	0
$783$	81.6566	2.91817
$784$	0	0
$785$	0.687025	0.0245210
$786$	0	0
$787$	−3.30356	−0.117759	−0.0588796	−	0.998265i	$-0.518753\pi$
$787$	−3.30356	−0.117759	−0.0588796	+	0.998265i	$0.518753\pi$
$788$	0	0
$789$	−97.1477	−3.45855
$790$	0	0
$791$	5.95338	0.211678
$792$	0	0
$793$	18.9661	0.673505
$794$	0	0
$795$	1.46575	0.0519848
$796$	0	0
$797$	29.9049	1.05929	0.529643	−	0.848221i	$-0.322326\pi$
$797$	29.9049	1.05929	0.529643	+	0.848221i	$0.322326\pi$
$798$	0	0
$799$	10.1866	0.360375
$800$	0	0
$801$	43.3725	1.53249
$802$	0	0
$803$	7.43287	0.262300
$804$	0	0
$805$	−0.158236	−0.00557708
$806$	0	0
$807$	−64.9279	−2.28557
$808$	0	0
$809$	−43.8976	−1.54336	−0.771678	−	0.636013i	$-0.780582\pi$
$809$	−43.8976	−1.54336	−0.771678	+	0.636013i	$0.780582\pi$
$810$	0	0
$811$	38.8668	1.36480	0.682400	−	0.730979i	$-0.260936\pi$
$811$	38.8668	1.36480	0.682400	+	0.730979i	$0.260936\pi$
$812$	0	0
$813$	−94.1230	−3.30104
$814$	0	0
$815$	−2.01292	−0.0705096
$816$	0	0
$817$	−33.0869	−1.15756
$818$	0	0
$819$	−18.1272	−0.633416
$820$	0	0
$821$	−20.2911	−0.708164	−0.354082	−	0.935214i	$-0.615207\pi$
$821$	−20.2911	−0.708164	−0.354082	+	0.935214i	$0.615207\pi$
$822$	0	0
$823$	45.8897	1.59961	0.799807	−	0.600257i	$-0.204935\pi$
$823$	45.8897	1.59961	0.799807	+	0.600257i	$0.204935\pi$
$824$	0	0
$825$	−16.8135	−0.585370
$826$	0	0
$827$	−52.7817	−1.83540	−0.917700	−	0.397275i	$-0.869956\pi$
$827$	−52.7817	−1.83540	−0.917700	+	0.397275i	$0.869956\pi$
$828$	0	0
$829$	−31.8611	−1.10658	−0.553292	−	0.832988i	$-0.686628\pi$
$829$	−31.8611	−1.10658	−0.553292	+	0.832988i	$0.686628\pi$
$830$	0	0
$831$	85.0049	2.94879
$832$	0	0
$833$	0.821569	0.0284657
$834$	0	0
$835$	0.770409	0.0266611
$836$	0	0
$837$	35.8218	1.23818
$838$	0	0
$839$	18.4476	0.636884	0.318442	−	0.947942i	$-0.396840\pi$
$839$	18.4476	0.636884	0.318442	+	0.947942i	$0.396840\pi$
$840$	0	0
$841$	−10.0177	−0.345439
$842$	0	0
$843$	−39.8997	−1.37422
$844$	0	0
$845$	1.82983	0.0629482
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	96.2254	3.30245
$850$	0	0
$851$	6.97812	0.239207
$852$	0	0
$853$	23.2070	0.794593	0.397296	−	0.917690i	$-0.369949\pi$
$853$	23.2070	0.794593	0.397296	+	0.917690i	$0.369949\pi$
$854$	0	0
$855$	5.42547	0.185547
$856$	0	0
$857$	−32.7467	−1.11861	−0.559303	−	0.828963i	$-0.688931\pi$
$857$	−32.7467	−1.11861	−0.559303	+	0.828963i	$0.688931\pi$
$858$	0	0
$859$	39.6014	1.35118	0.675591	−	0.737277i	$-0.263889\pi$
$859$	39.6014	1.35118	0.675591	+	0.737277i	$0.263889\pi$
$860$	0	0
$861$	−2.78868	−0.0950381
$862$	0	0
$863$	38.0862	1.29647	0.648235	−	0.761440i	$-0.275507\pi$
$863$	38.0862	1.29647	0.648235	+	0.761440i	$0.275507\pi$
$864$	0	0
$865$	−4.30937	−0.146523
$866$	0	0
$867$	55.4127	1.88191
$868$	0	0
$869$	−5.21132	−0.176782
$870$	0	0
$871$	−31.3596	−1.06258
$872$	0	0
$873$	108.582	3.67495
$874$	0	0
$875$	2.14904	0.0726510
$876$	0	0
$877$	−30.0496	−1.01470	−0.507351	−	0.861739i	$-0.669375\pi$
$877$	−30.0496	−1.01470	−0.507351	+	0.861739i	$0.669375\pi$
$878$	0	0
$879$	−89.0636	−3.00404
$880$	0	0
$881$	22.9986	0.774842	0.387421	−	0.921903i	$-0.373366\pi$
$881$	22.9986	0.774842	0.387421	+	0.921903i	$0.373366\pi$
$882$	0	0
$883$	−31.7951	−1.06999	−0.534996	−	0.844855i	$-0.679687\pi$
$883$	−31.7951	−1.06999	−0.534996	+	0.844855i	$0.679687\pi$
$884$	0	0
$885$	4.21492	0.141683
$886$	0	0
$887$	−30.7337	−1.03194	−0.515969	−	0.856607i	$-0.672568\pi$
$887$	−30.7337	−1.03194	−0.515969	+	0.856607i	$0.672568\pi$
$888$	0	0
$889$	−1.14555	−0.0384204
$890$	0	0
$891$	−38.0523	−1.27480
$892$	0	0
$893$	−36.5618	−1.22349
$894$	0	0
$895$	−0.119936	−0.00400903
$896$	0	0
$897$	5.29173	0.176686
$898$	0	0
$899$	8.32730	0.277731
$900$	0	0
$901$	1.64314	0.0547408
$902$	0	0
$903$	−38.0862	−1.26743
$904$	0	0
$905$	−0.401389	−0.0133426
$906$	0	0
$907$	−10.6954	−0.355137	−0.177568	−	0.984108i	$-0.556823\pi$
$907$	−10.6954	−0.355137	−0.177568	+	0.984108i	$0.556823\pi$
$908$	0	0
$909$	72.2976	2.39796
$910$	0	0
$911$	−15.0706	−0.499311	−0.249655	−	0.968335i	$-0.580317\pi$
$911$	−15.0706	−0.499311	−0.249655	+	0.968335i	$0.580317\pi$
$912$	0	0
$913$	9.48304	0.313843
$914$	0	0
$915$	−6.53425	−0.216016
$916$	0	0
$917$	−14.1693	−0.467911
$918$	0	0
$919$	−25.3725	−0.836962	−0.418481	−	0.908226i	$-0.637437\pi$
$919$	−25.3725	−0.836962	−0.418481	+	0.908226i	$0.637437\pi$
$920$	0	0
$921$	67.3434	2.21904
$922$	0	0
$923$	30.4410	1.00198
$924$	0	0
$925$	−47.1639	−1.55074
$926$	0	0
$927$	98.2987	3.22855
$928$	0	0
$929$	39.1914	1.28583	0.642914	−	0.765938i	$-0.277725\pi$
$929$	39.1914	1.28583	0.642914	+	0.765938i	$0.277725\pi$
$930$	0	0
$931$	−2.94879	−0.0966426
$932$	0	0
$933$	61.8814	2.02591
$934$	0	0
$935$	0.177386	0.00580113
$936$	0	0
$937$	11.9692	0.391017	0.195508	−	0.980702i	$-0.437364\pi$
$937$	11.9692	0.391017	0.195508	+	0.980702i	$0.437364\pi$
$938$	0	0
$939$	−43.3442	−1.41448
$940$	0	0
$941$	−9.42478	−0.307239	−0.153619	−	0.988130i	$-0.549093\pi$
$941$	−9.42478	−0.307239	−0.153619	+	0.988130i	$0.549093\pi$
$942$	0	0
$943$	0.602108	0.0196073
$944$	0	0
$945$	4.04662	0.131636
$946$	0	0
$947$	49.5874	1.61138	0.805688	−	0.592341i	$-0.201796\pi$
$947$	49.5874	1.61138	0.805688	+	0.592341i	$0.201796\pi$
$948$	0	0
$949$	15.8113	0.513258
$950$	0	0
$951$	−79.8714	−2.59001
$952$	0	0
$953$	5.88838	0.190743	0.0953717	−	0.995442i	$-0.469596\pi$
$953$	5.88838	0.190743	0.0953717	+	0.995442i	$0.469596\pi$
$954$	0	0
$955$	3.41214	0.110414
$956$	0	0
$957$	−14.7887	−0.478050
$958$	0	0
$959$	−10.3852	−0.335356
$960$	0	0
$961$	−27.3469	−0.882158
$962$	0	0
$963$	112.021	3.60981
$964$	0	0
$965$	−4.63668	−0.149260
$966$	0	0
$967$	4.53634	0.145879	0.0729394	−	0.997336i	$-0.476762\pi$
$967$	4.53634	0.145879	0.0729394	+	0.997336i	$0.476762\pi$
$968$	0	0
$969$	8.22324	0.264168
$970$	0	0
$971$	1.41972	0.0455609	0.0227805	−	0.999740i	$-0.492748\pi$
$971$	1.41972	0.0455609	0.0227805	+	0.999740i	$0.492748\pi$
$972$	0	0
$973$	16.5262	0.529804
$974$	0	0
$975$	−35.7659	−1.14543
$976$	0	0
$977$	−42.9103	−1.37282	−0.686410	−	0.727214i	$-0.740815\pi$
$977$	−42.9103	−1.37282	−0.686410	+	0.727214i	$0.740815\pi$
$978$	0	0
$979$	−5.08974	−0.162669
$980$	0	0
$981$	116.339	3.71442
$982$	0	0
$983$	27.3771	0.873193	0.436596	−	0.899658i	$-0.356184\pi$
$983$	27.3771	0.873193	0.436596	+	0.899658i	$0.356184\pi$
$984$	0	0
$985$	−0.131986	−0.00420543
$986$	0	0
$987$	−42.0862	−1.33962
$988$	0	0
$989$	8.22324	0.261484
$990$	0	0
$991$	5.00273	0.158917	0.0794585	−	0.996838i	$-0.474681\pi$
$991$	5.00273	0.158917	0.0794585	+	0.996838i	$0.474681\pi$
$992$	0	0
$993$	33.1760	1.05281
$994$	0	0
$995$	−0.949779	−0.0301100
$996$	0	0
$997$	−26.6707	−0.844668	−0.422334	−	0.906440i	$-0.638789\pi$
$997$	−26.6707	−0.844668	−0.422334	+	0.906440i	$0.638789\pi$
$998$	0	0
$999$	−178.454	−5.64603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.2.a.s.1.1		4
4.3	odd	2		616.2.a.h.1.4	✓	4
7.6	odd	2		8624.2.a.cy.1.4		4
8.3	odd	2		4928.2.a.cc.1.1		4
8.5	even	2		4928.2.a.ch.1.4		4
12.11	even	2		5544.2.a.bm.1.3		4
28.27	even	2		4312.2.a.z.1.1		4
44.43	even	2		6776.2.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.h.1.4	✓	4	4.3	odd	2
1232.2.a.s.1.1		4	1.1	even	1	trivial
4312.2.a.z.1.1		4	28.27	even	2
4928.2.a.cc.1.1		4	8.3	odd	2
4928.2.a.ch.1.4		4	8.5	even	2
5544.2.a.bm.1.3		4	12.11	even	2
6776.2.a.bb.1.4		4	44.43	even	2
8624.2.a.cy.1.4		4	7.6	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:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q-3.39434 q^{3} -0.215911 q^{5} +1.00000 q^{7} +8.52156 q^{9} +O(q^{10})\)	\(q-3.39434 q^{3} -0.215911 q^{5} +1.00000 q^{7} +8.52156 q^{9} -1.00000 q^{11} -2.12722 q^{13} +0.732876 q^{15} +0.821569 q^{17} -2.94879 q^{19} -3.39434 q^{21} +0.732876 q^{23} -4.95338 q^{25} -18.7421 q^{27} -4.35686 q^{29} -1.91131 q^{31} +3.39434 q^{33} -0.215911 q^{35} +9.52156 q^{37} +7.22051 q^{39} +0.821569 q^{41} +11.2205 q^{43} -1.83990 q^{45} +12.3989 q^{47} +1.00000 q^{49} -2.78868 q^{51} +2.00000 q^{53} +0.215911 q^{55} +10.0092 q^{57} +5.75121 q^{59} -8.91590 q^{61} +8.52156 q^{63} +0.459290 q^{65} +14.7421 q^{67} -2.48763 q^{69} -14.3102 q^{71} -7.43287 q^{73} +16.8135 q^{75} -1.00000 q^{77} +5.21132 q^{79} +38.0523 q^{81} -9.48304 q^{83} -0.177386 q^{85} +14.7887 q^{87} +5.08974 q^{89} -2.12722 q^{91} +6.48763 q^{93} +0.636675 q^{95} +12.7421 q^{97} -8.52156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9	\( 4 q - q^{3} + 5 q^{5} + 4 q^{7} + 9 q^{9} - 4 q^{11} + 4 q^{13} + 3 q^{15} + 10 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 17 q^{25} - 13 q^{27} - 4 q^{29} - q^{31} + q^{33} + 5 q^{35} + 13 q^{37} - 8 q^{39} + 10 q^{41} + 8 q^{43} + 12 q^{45} + 6 q^{47} + 4 q^{49} + 14 q^{51} + 8 q^{53} - 5 q^{55} - 22 q^{57} - 3 q^{59} + 2 q^{61} + 9 q^{63} - 3 q^{67} + 27 q^{69} - 7 q^{71} + 2 q^{73} + 18 q^{75} - 4 q^{77} + 46 q^{79} + 40 q^{81} - 32 q^{83} - 14 q^{85} + 34 q^{87} + 7 q^{89} + 4 q^{91} - 11 q^{93} + 14 q^{95} - 11 q^{97} - 9 q^{99}+O(q^{100}) \) 4 * q - q^3 + 5 * q^5 + 4 * q^7 + 9 * q^9 - 4 * q^11 + 4 * q^13 + 3 * q^15 + 10 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 17 * q^25 - 13 * q^27 - 4 * q^29 - q^31 + q^33 + 5 * q^35 + 13 * q^37 - 8 * q^39 + 10 * q^41 + 8 * q^43 + 12 * q^45 + 6 * q^47 + 4 * q^49 + 14 * q^51 + 8 * q^53 - 5 * q^55 - 22 * q^57 - 3 * q^59 + 2 * q^61 + 9 * q^63 - 3 * q^67 + 27 * q^69 - 7 * q^71 + 2 * q^73 + 18 * q^75 - 4 * q^77 + 46 * q^79 + 40 * q^81 - 32 * q^83 - 14 * q^85 + 34 * q^87 + 7 * q^89 + 4 * q^91 - 11 * q^93 + 14 * q^95 - 11 * q^97 - 9 * q^99

Introduction

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