LMFDB - Embedded newform 927.2.a.h.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,2,Mod(1,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("927.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	927.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:	$7.40213226737$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 24x^{12} + 221x^{10} - 980x^{8} + 2160x^{6} - 2203x^{4} + 808x^{2} - 75 $ x^14 - 24x^12 + 221x^10 - 980x^8 + 2160x^6 - 2203x^4 + 808x^2 - 75
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			1.5
Root			$-1.46308$ of defining polynomial
Character	$\chi$	$=$	927.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.46308 q^{2} +0.140605 q^{4} -0.979247 q^{5} +4.33523 q^{7} +2.72044 q^{8} +O(q^{10})$	\(q-1.46308 q^{2} +0.140605 q^{4} -0.979247 q^{5} +4.33523 q^{7} +2.72044 q^{8} +1.43272 q^{10} -3.10803 q^{11} -1.57505 q^{13} -6.34279 q^{14} -4.26144 q^{16} +3.47437 q^{17} +8.01333 q^{19} -0.137687 q^{20} +4.54730 q^{22} +0.212350 q^{23} -4.04108 q^{25} +2.30442 q^{26} +0.609555 q^{28} -8.02086 q^{29} +4.88887 q^{31} +0.793941 q^{32} -5.08328 q^{34} -4.24526 q^{35} -7.21775 q^{37} -11.7241 q^{38} -2.66399 q^{40} +9.99565 q^{41} +7.71738 q^{43} -0.437004 q^{44} -0.310686 q^{46} -9.94129 q^{47} +11.7942 q^{49} +5.91242 q^{50} -0.221459 q^{52} +4.48025 q^{53} +3.04353 q^{55} +11.7938 q^{56} +11.7352 q^{58} +10.5542 q^{59} +0.834851 q^{61} -7.15281 q^{62} +7.36128 q^{64} +1.54236 q^{65} +2.10798 q^{67} +0.488513 q^{68} +6.21116 q^{70} +15.3566 q^{71} +6.99789 q^{73} +10.5602 q^{74} +1.12671 q^{76} -13.4740 q^{77} +2.80934 q^{79} +4.17300 q^{80} -14.6244 q^{82} +5.64397 q^{83} -3.40226 q^{85} -11.2911 q^{86} -8.45522 q^{88} +15.1339 q^{89} -6.82819 q^{91} +0.0298575 q^{92} +14.5449 q^{94} -7.84702 q^{95} +2.98706 q^{97} -17.2559 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 20 q^{4} + 12 q^{7}+O(q^{10}) $ 14 * q + 20 * q^4 + 12 * q^7	$ 14 q + 20 q^{4} + 12 q^{7} + 4 q^{10} + 20 q^{13} + 36 q^{16} + 8 q^{19} + 32 q^{22} + 58 q^{25} + 26 q^{28} - 4 q^{31} - 16 q^{34} + 28 q^{37} - 32 q^{40} + 40 q^{43} + 14 q^{46} + 26 q^{49} + 28 q^{55} - 4 q^{58} + 24 q^{61} + 64 q^{64} - 8 q^{67} - 64 q^{70} + 56 q^{73} + 4 q^{79} - 26 q^{82} + 24 q^{85} + 28 q^{88} + 44 q^{91} - 60 q^{94} + 12 q^{97}+O(q^{100}) $ 14 * q + 20 * q^4 + 12 * q^7 + 4 * q^10 + 20 * q^13 + 36 * q^16 + 8 * q^19 + 32 * q^22 + 58 * q^25 + 26 * q^28 - 4 * q^31 - 16 * q^34 + 28 * q^37 - 32 * q^40 + 40 * q^43 + 14 * q^46 + 26 * q^49 + 28 * q^55 - 4 * q^58 + 24 * q^61 + 64 * q^64 - 8 * q^67 - 64 * q^70 + 56 * q^73 + 4 * q^79 - 26 * q^82 + 24 * q^85 + 28 * q^88 + 44 * q^91 - 60 * q^94 + 12 * 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 $

$2$	−1.46308	−1.03455	−0.517277	−	0.855818i	$-0.673054\pi$
$2$	−1.46308	−1.03455	−0.517277	+	0.855818i	$0.673054\pi$
$3$	0	0
$3$	0	0
$4$	0.140605	0.0703025
$5$	−0.979247	−0.437932	−0.218966	−	0.975732i	$-0.570268\pi$
$5$	−0.979247	−0.437932	−0.218966	+	0.975732i	$0.570268\pi$
$6$	0	0
$7$	4.33523	1.63856	0.819281	−	0.573392i	$-0.194373\pi$
$7$	4.33523	1.63856	0.819281	+	0.573392i	$0.194373\pi$
$8$	2.72044	0.961822
$9$	0	0
$10$	1.43272	0.453065
$11$	−3.10803	−0.937106	−0.468553	−	0.883435i	$-0.655224\pi$
$11$	−3.10803	−0.937106	−0.468553	+	0.883435i	$0.655224\pi$
$12$	0	0
$13$	−1.57505	−0.436839	−0.218420	−	0.975855i	$-0.570090\pi$
$13$	−1.57505	−0.436839	−0.218420	+	0.975855i	$0.570090\pi$
$14$	−6.34279	−1.69518
$15$	0	0
$16$	−4.26144	−1.06536
$17$	3.47437	0.842658	0.421329	−	0.906908i	$-0.361564\pi$
$17$	3.47437	0.842658	0.421329	+	0.906908i	$0.361564\pi$
$18$	0	0
$19$	8.01333	1.83838	0.919191	−	0.393811i	$-0.128844\pi$
$19$	8.01333	1.83838	0.919191	+	0.393811i	$0.128844\pi$
$20$	−0.137687	−0.0307877
$21$	0	0
$22$	4.54730	0.969487
$23$	0.212350	0.0442781	0.0221391	−	0.999755i	$-0.492952\pi$
$23$	0.212350	0.0442781	0.0221391	+	0.999755i	$0.492952\pi$
$24$	0	0
$25$	−4.04108	−0.808215
$26$	2.30442	0.451934
$27$	0	0
$28$	0.609555	0.115195
$29$	−8.02086	−1.48944	−0.744718	−	0.667379i	$-0.767416\pi$
$29$	−8.02086	−1.48944	−0.744718	+	0.667379i	$0.767416\pi$
$30$	0	0
$31$	4.88887	0.878067	0.439034	−	0.898471i	$-0.355321\pi$
$31$	4.88887	0.878067	0.439034	+	0.898471i	$0.355321\pi$
$32$	0.793941	0.140350
$33$	0	0
$34$	−5.08328	−0.871775
$35$	−4.24526	−0.717580
$36$	0	0
$37$	−7.21775	−1.18659	−0.593296	−	0.804985i	$-0.702174\pi$
$37$	−7.21775	−1.18659	−0.593296	+	0.804985i	$0.702174\pi$
$38$	−11.7241	−1.90191
$39$	0	0
$40$	−2.66399	−0.421213
$41$	9.99565	1.56106	0.780529	−	0.625119i	$-0.214950\pi$
$41$	9.99565	1.56106	0.780529	+	0.625119i	$0.214950\pi$
$42$	0	0
$43$	7.71738	1.17689	0.588444	−	0.808538i	$-0.299741\pi$
$43$	7.71738	1.17689	0.588444	+	0.808538i	$0.299741\pi$
$44$	−0.437004	−0.0658809
$45$	0	0
$46$	−0.310686	−0.0458081
$47$	−9.94129	−1.45009	−0.725043	−	0.688704i	$-0.758180\pi$
$47$	−9.94129	−1.45009	−0.725043	+	0.688704i	$0.758180\pi$
$48$	0	0
$49$	11.7942	1.68489
$50$	5.91242	0.836142
$51$	0	0
$52$	−0.221459	−0.0307109
$53$	4.48025	0.615410	0.307705	−	0.951482i	$-0.400439\pi$
$53$	4.48025	0.615410	0.307705	+	0.951482i	$0.400439\pi$
$54$	0	0
$55$	3.04353	0.410389
$56$	11.7938	1.57601
$57$	0	0
$58$	11.7352	1.54090
$59$	10.5542	1.37404	0.687020	−	0.726639i	$-0.258919\pi$
$59$	10.5542	1.37404	0.687020	+	0.726639i	$0.258919\pi$
$60$	0	0
$61$	0.834851	0.106892	0.0534459	−	0.998571i	$-0.482980\pi$
$61$	0.834851	0.106892	0.0534459	+	0.998571i	$0.482980\pi$
$62$	−7.15281	−0.908408
$63$	0	0
$64$	7.36128	0.920160
$65$	1.54236	0.191306
$66$	0	0
$67$	2.10798	0.257531	0.128765	−	0.991675i	$-0.458899\pi$
$67$	2.10798	0.257531	0.128765	+	0.991675i	$0.458899\pi$
$68$	0.488513	0.0592409
$69$	0	0
$70$	6.21116	0.742375
$71$	15.3566	1.82249	0.911247	−	0.411861i	$-0.135121\pi$
$71$	15.3566	1.82249	0.911247	+	0.411861i	$0.135121\pi$
$72$	0	0
$73$	6.99789	0.819041	0.409520	−	0.912301i	$-0.365696\pi$
$73$	6.99789	0.819041	0.409520	+	0.912301i	$0.365696\pi$
$74$	10.5602	1.22759
$75$	0	0
$76$	1.12671	0.129243
$77$	−13.4740	−1.53551
$78$	0	0
$79$	2.80934	0.316075	0.158038	−	0.987433i	$-0.449483\pi$
$79$	2.80934	0.316075	0.158038	+	0.987433i	$0.449483\pi$
$80$	4.17300	0.466556
$81$	0	0
$82$	−14.6244	−1.61500
$83$	5.64397	0.619506	0.309753	−	0.950817i	$-0.399754\pi$
$83$	5.64397	0.619506	0.309753	+	0.950817i	$0.399754\pi$
$84$	0	0
$85$	−3.40226	−0.369027
$86$	−11.2911	−1.21756
$87$	0	0
$88$	−8.45522	−0.901329
$89$	15.1339	1.60419	0.802096	−	0.597195i	$-0.203718\pi$
$89$	15.1339	1.60419	0.802096	+	0.597195i	$0.203718\pi$
$90$	0	0
$91$	−6.82819	−0.715789
$92$	0.0298575	0.00311286
$93$	0	0
$94$	14.5449	1.50019
$95$	−7.84702	−0.805087
$96$	0	0
$97$	2.98706	0.303290	0.151645	−	0.988435i	$-0.451543\pi$
$97$	2.98706	0.303290	0.151645	+	0.988435i	$0.451543\pi$
$98$	−17.2559	−1.74311
$99$	0	0
$100$	−0.568196	−0.0568196
$101$	4.39525	0.437344	0.218672	−	0.975798i	$-0.429828\pi$
$101$	4.39525	0.437344	0.218672	+	0.975798i	$0.429828\pi$
$102$	0	0
$103$	1.00000	0.0985329
$103$	1.00000	0.0985329
$104$	−4.28483	−0.420162
$105$	0	0
$106$	−6.55497	−0.636675
$107$	8.27003	0.799494	0.399747	−	0.916625i	$-0.369098\pi$
$107$	8.27003	0.799494	0.399747	+	0.916625i	$0.369098\pi$
$108$	0	0
$109$	−15.6360	−1.49766	−0.748831	−	0.662761i	$-0.769384\pi$
$109$	−15.6360	−1.49766	−0.748831	+	0.662761i	$0.769384\pi$
$110$	−4.45292	−0.424570
$111$	0	0
$112$	−18.4743	−1.74566
$113$	−11.6331	−1.09435	−0.547173	−	0.837019i	$-0.684296\pi$
$113$	−11.6331	−1.09435	−0.547173	+	0.837019i	$0.684296\pi$
$114$	0	0
$115$	−0.207943	−0.0193908
$116$	−1.12777	−0.104711
$117$	0	0
$118$	−15.4416	−1.42152
$119$	15.0622	1.38075
$120$	0	0
$121$	−1.34016	−0.121833
$122$	−1.22145	−0.110585
$123$	0	0
$124$	0.687400	0.0617303
$125$	8.85344	0.791876
$126$	0	0
$127$	1.71393	0.152086	0.0760432	−	0.997105i	$-0.475771\pi$
$127$	1.71393	0.152086	0.0760432	+	0.997105i	$0.475771\pi$
$128$	−12.3580	−1.09231
$129$	0	0
$130$	−2.25660	−0.197917
$131$	−2.06767	−0.180653	−0.0903267	−	0.995912i	$-0.528791\pi$
$131$	−2.06767	−0.180653	−0.0903267	+	0.995912i	$0.528791\pi$
$132$	0	0
$133$	34.7396	3.01231
$134$	−3.08415	−0.266430
$135$	0	0
$136$	9.45182	0.810487
$137$	21.9363	1.87414	0.937070	−	0.349141i	$-0.113526\pi$
$137$	21.9363	1.87414	0.937070	+	0.349141i	$0.113526\pi$
$138$	0	0
$139$	−14.2602	−1.20954	−0.604769	−	0.796401i	$-0.706735\pi$
$139$	−14.2602	−1.20954	−0.604769	+	0.796401i	$0.706735\pi$
$140$	−0.596905	−0.0504477
$141$	0	0
$142$	−22.4680	−1.88547
$143$	4.89529	0.409365
$144$	0	0
$145$	7.85440	0.652272
$146$	−10.2385	−0.847342
$147$	0	0
$148$	−1.01485	−0.0834204
$149$	−3.03336	−0.248503	−0.124251	−	0.992251i	$-0.539653\pi$
$149$	−3.03336	−0.248503	−0.124251	+	0.992251i	$0.539653\pi$
$150$	0	0
$151$	−19.9542	−1.62385	−0.811924	−	0.583763i	$-0.801580\pi$
$151$	−19.9542	−1.62385	−0.811924	+	0.583763i	$0.801580\pi$
$152$	21.7998	1.76820
$153$	0	0
$154$	19.7136	1.58856
$155$	−4.78741	−0.384534
$156$	0	0
$157$	10.6062	0.846471	0.423235	−	0.906020i	$-0.360894\pi$
$157$	10.6062	0.846471	0.423235	+	0.906020i	$0.360894\pi$
$158$	−4.11029	−0.326997
$159$	0	0
$160$	−0.777464	−0.0614639
$161$	0.920587	0.0725525
$162$	0	0
$163$	6.48743	0.508135	0.254068	−	0.967186i	$-0.418231\pi$
$163$	6.48743	0.508135	0.254068	+	0.967186i	$0.418231\pi$
$164$	1.40544	0.109746
$165$	0	0
$166$	−8.25758	−0.640913
$167$	23.5567	1.82287	0.911437	−	0.411439i	$-0.134974\pi$
$167$	23.5567	1.82287	0.911437	+	0.411439i	$0.134974\pi$
$168$	0	0
$169$	−10.5192	−0.809171
$170$	4.97778	0.381779
$171$	0	0
$172$	1.08510	0.0827382
$173$	0.491094	0.0373372	0.0186686	−	0.999826i	$-0.494057\pi$
$173$	0.491094	0.0373372	0.0186686	+	0.999826i	$0.494057\pi$
$174$	0	0
$175$	−17.5190	−1.32431
$176$	13.2447	0.998355
$177$	0	0
$178$	−22.1421	−1.65962
$179$	−22.7912	−1.70349	−0.851747	−	0.523953i	$-0.824457\pi$
$179$	−22.7912	−1.70349	−0.851747	+	0.523953i	$0.824457\pi$
$180$	0	0
$181$	−7.97860	−0.593045	−0.296522	−	0.955026i	$-0.595827\pi$
$181$	−7.97860	−0.593045	−0.296522	+	0.955026i	$0.595827\pi$
$182$	9.99019	0.740522
$183$	0	0
$184$	0.577687	0.0425877
$185$	7.06796	0.519647
$186$	0	0
$187$	−10.7984	−0.789659
$188$	−1.39779	−0.101945
$189$	0	0
$190$	11.4808	0.832907
$191$	−3.70508	−0.268091	−0.134045	−	0.990975i	$-0.542797\pi$
$191$	−3.70508	−0.268091	−0.134045	+	0.990975i	$0.542797\pi$
$192$	0	0
$193$	−18.6246	−1.34063	−0.670316	−	0.742076i	$-0.733841\pi$
$193$	−18.6246	−1.34063	−0.670316	+	0.742076i	$0.733841\pi$
$194$	−4.37030	−0.313770
$195$	0	0
$196$	1.65833	0.118452
$197$	−7.14218	−0.508859	−0.254430	−	0.967091i	$-0.581888\pi$
$197$	−7.14218	−0.508859	−0.254430	+	0.967091i	$0.581888\pi$
$198$	0	0
$199$	1.36675	0.0968861	0.0484431	−	0.998826i	$-0.484574\pi$
$199$	1.36675	0.0968861	0.0484431	+	0.998826i	$0.484574\pi$
$200$	−10.9935	−0.777360
$201$	0	0
$202$	−6.43060	−0.452456
$203$	−34.7723	−2.44053
$204$	0	0
$205$	−9.78821	−0.683638
$206$	−1.46308	−0.101938
$207$	0	0
$208$	6.71197	0.465391
$209$	−24.9056	−1.72276
$210$	0	0
$211$	16.0681	1.10617	0.553086	−	0.833124i	$-0.313450\pi$
$211$	16.0681	1.10617	0.553086	+	0.833124i	$0.313450\pi$
$212$	0.629946	0.0432649
$213$	0	0
$214$	−12.0997	−0.827120
$215$	−7.55722	−0.515398
$216$	0	0
$217$	21.1944	1.43877
$218$	22.8768	1.54941
$219$	0	0
$220$	0.427935	0.0288514
$221$	−5.47229	−0.368106
$222$	0	0
$223$	25.0765	1.67925	0.839623	−	0.543170i	$-0.182776\pi$
$223$	25.0765	1.67925	0.839623	+	0.543170i	$0.182776\pi$
$224$	3.44192	0.229973
$225$	0	0
$226$	17.0201	1.13216
$227$	12.6856	0.841972	0.420986	−	0.907067i	$-0.361684\pi$
$227$	12.6856	0.841972	0.420986	+	0.907067i	$0.361684\pi$
$228$	0	0
$229$	7.83561	0.517792	0.258896	−	0.965905i	$-0.416641\pi$
$229$	7.83561	0.517792	0.258896	+	0.965905i	$0.416641\pi$
$230$	0.304238	0.0200609
$231$	0	0
$232$	−21.8203	−1.43257
$233$	−24.3548	−1.59554	−0.797769	−	0.602963i	$-0.793987\pi$
$233$	−24.3548	−1.59554	−0.797769	+	0.602963i	$0.793987\pi$
$234$	0	0
$235$	9.73497	0.635040
$236$	1.48397	0.0965984
$237$	0	0
$238$	−22.0372	−1.42846
$239$	−7.16618	−0.463542	−0.231771	−	0.972770i	$-0.574452\pi$
$239$	−7.16618	−0.463542	−0.231771	+	0.972770i	$0.574452\pi$
$240$	0	0
$241$	3.54070	0.228076	0.114038	−	0.993476i	$-0.463621\pi$
$241$	3.54070	0.228076	0.114038	+	0.993476i	$0.463621\pi$
$242$	1.96076	0.126043
$243$	0	0
$244$	0.117384	0.00751476
$245$	−11.5494	−0.737867
$246$	0	0
$247$	−12.6214	−0.803078
$248$	13.2999	0.844545
$249$	0	0
$250$	−12.9533	−0.819239
$251$	−2.20675	−0.139289	−0.0696445	−	0.997572i	$-0.522186\pi$
$251$	−2.20675	−0.139289	−0.0696445	+	0.997572i	$0.522186\pi$
$252$	0	0
$253$	−0.659991	−0.0414933
$254$	−2.50761	−0.157342
$255$	0	0
$256$	3.35823	0.209890
$257$	−9.79114	−0.610755	−0.305377	−	0.952231i	$-0.598783\pi$
$257$	−9.79114	−0.610755	−0.305377	+	0.952231i	$0.598783\pi$
$258$	0	0
$259$	−31.2906	−1.94430
$260$	0.216863	0.0134493
$261$	0	0
$262$	3.02517	0.186896
$263$	−16.8549	−1.03932	−0.519660	−	0.854373i	$-0.673941\pi$
$263$	−16.8549	−1.03932	−0.519660	+	0.854373i	$0.673941\pi$
$264$	0	0
$265$	−4.38727	−0.269508
$266$	−50.8268	−3.11639
$267$	0	0
$268$	0.296393	0.0181051
$269$	−10.9935	−0.670286	−0.335143	−	0.942167i	$-0.608785\pi$
$269$	−10.9935	−0.670286	−0.335143	+	0.942167i	$0.608785\pi$
$270$	0	0
$271$	20.5156	1.24624	0.623118	−	0.782128i	$-0.285866\pi$
$271$	20.5156	1.24624	0.623118	+	0.782128i	$0.285866\pi$
$272$	−14.8058	−0.897734
$273$	0	0
$274$	−32.0945	−1.93890
$275$	12.5598	0.757383
$276$	0	0
$277$	8.05999	0.484278	0.242139	−	0.970242i	$-0.422151\pi$
$277$	8.05999	0.484278	0.242139	+	0.970242i	$0.422151\pi$
$278$	20.8639	1.25133
$279$	0	0
$280$	−11.5490	−0.690184
$281$	22.1078	1.31884	0.659420	−	0.751775i	$-0.270802\pi$
$281$	22.1078	1.31884	0.659420	+	0.751775i	$0.270802\pi$
$282$	0	0
$283$	−24.9634	−1.48392	−0.741960	−	0.670444i	$-0.766103\pi$
$283$	−24.9634	−1.48392	−0.741960	+	0.670444i	$0.766103\pi$
$284$	2.15922	0.128126
$285$	0	0
$286$	−7.16220	−0.423510
$287$	43.3334	2.55789
$288$	0	0
$289$	−4.92878	−0.289928
$290$	−11.4916	−0.674811
$291$	0	0
$292$	0.983938	0.0575806
$293$	9.34592	0.545994	0.272997	−	0.962015i	$-0.411985\pi$
$293$	9.34592	0.545994	0.272997	+	0.962015i	$0.411985\pi$
$294$	0	0
$295$	−10.3352	−0.601736
$296$	−19.6355	−1.14129
$297$	0	0
$298$	4.43806	0.257090
$299$	−0.334462	−0.0193424
$300$	0	0
$301$	33.4566	1.92841
$302$	29.1946	1.67996
$303$	0	0
$304$	−34.1483	−1.95854
$305$	−0.817525	−0.0468114
$306$	0	0
$307$	−10.3889	−0.592924	−0.296462	−	0.955045i	$-0.595807\pi$
$307$	−10.3889	−0.592924	−0.296462	+	0.955045i	$0.595807\pi$
$308$	−1.89451	−0.107950
$309$	0	0
$310$	7.00437	0.397821
$311$	10.9179	0.619099	0.309549	−	0.950883i	$-0.399822\pi$
$311$	10.9179	0.619099	0.309549	+	0.950883i	$0.399822\pi$
$312$	0	0
$313$	11.9374	0.674742	0.337371	−	0.941372i	$-0.390462\pi$
$313$	11.9374	0.674742	0.337371	+	0.941372i	$0.390462\pi$
$314$	−15.5178	−0.875720
$315$	0	0
$316$	0.395007	0.0222209
$317$	−30.9368	−1.73758	−0.868792	−	0.495177i	$-0.835103\pi$
$317$	−30.9368	−1.73758	−0.868792	+	0.495177i	$0.835103\pi$
$318$	0	0
$319$	24.9291	1.39576
$320$	−7.20851	−0.402968
$321$	0	0
$322$	−1.34689	−0.0750595
$323$	27.8412	1.54913
$324$	0	0
$325$	6.36488	0.353060
$326$	−9.49164	−0.525693
$327$	0	0
$328$	27.1926	1.50146
$329$	−43.0978	−2.37606
$330$	0	0
$331$	−15.1740	−0.834038	−0.417019	−	0.908898i	$-0.636925\pi$
$331$	−15.1740	−0.834038	−0.417019	+	0.908898i	$0.636925\pi$
$332$	0.793570	0.0435528
$333$	0	0
$334$	−34.4654	−1.88586
$335$	−2.06423	−0.112781
$336$	0	0
$337$	22.0575	1.20155	0.600776	−	0.799418i	$-0.294859\pi$
$337$	22.0575	1.20155	0.600776	+	0.799418i	$0.294859\pi$
$338$	15.3905	0.837132
$339$	0	0
$340$	−0.478375	−0.0259435
$341$	−15.1947	−0.822842
$342$	0	0
$343$	20.7840	1.12223
$344$	20.9947	1.13196
$345$	0	0
$346$	−0.718510	−0.0386273
$347$	−8.28373	−0.444694	−0.222347	−	0.974968i	$-0.571372\pi$
$347$	−8.28373	−0.444694	−0.222347	+	0.974968i	$0.571372\pi$
$348$	0	0
$349$	−0.479634	−0.0256742	−0.0128371	−	0.999918i	$-0.504086\pi$
$349$	−0.479634	−0.0256742	−0.0128371	+	0.999918i	$0.504086\pi$
$350$	25.6317	1.37007
$351$	0	0
$352$	−2.46759	−0.131523
$353$	10.0710	0.536024	0.268012	−	0.963416i	$-0.413633\pi$
$353$	10.0710	0.536024	0.268012	+	0.963416i	$0.413633\pi$
$354$	0	0
$355$	−15.0379	−0.798129
$356$	2.12790	0.112779
$357$	0	0
$358$	33.3454	1.76236
$359$	−11.4814	−0.605964	−0.302982	−	0.952996i	$-0.597982\pi$
$359$	−11.4814	−0.605964	−0.302982	+	0.952996i	$0.597982\pi$
$360$	0	0
$361$	45.2134	2.37965
$362$	11.6733	0.613537
$363$	0	0
$364$	−0.960078	−0.0503217
$365$	−6.85266	−0.358685
$366$	0	0
$367$	5.34573	0.279045	0.139523	−	0.990219i	$-0.455443\pi$
$367$	5.34573	0.279045	0.139523	+	0.990219i	$0.455443\pi$
$368$	−0.904918	−0.0471721
$369$	0	0
$370$	−10.3410	−0.537603
$371$	19.4229	1.00839
$372$	0	0
$373$	−5.54266	−0.286988	−0.143494	−	0.989651i	$-0.545834\pi$
$373$	−5.54266	−0.286988	−0.143494	+	0.989651i	$0.545834\pi$
$374$	15.7990	0.816945
$375$	0	0
$376$	−27.0447	−1.39473
$377$	12.6332	0.650644
$378$	0	0
$379$	21.6803	1.11364	0.556820	−	0.830633i	$-0.312021\pi$
$379$	21.6803	1.11364	0.556820	+	0.830633i	$0.312021\pi$
$380$	−1.10333	−0.0565997
$381$	0	0
$382$	5.42084	0.277354
$383$	−20.5723	−1.05120	−0.525598	−	0.850733i	$-0.676158\pi$
$383$	−20.5723	−1.05120	−0.525598	+	0.850733i	$0.676158\pi$
$384$	0	0
$385$	13.1944	0.672448
$386$	27.2494	1.38696
$387$	0	0
$388$	0.419995	0.0213220
$389$	31.0468	1.57413	0.787067	−	0.616868i	$-0.211599\pi$
$389$	31.0468	1.57413	0.787067	+	0.616868i	$0.211599\pi$
$390$	0	0
$391$	0.737783	0.0373113
$392$	32.0855	1.62056
$393$	0	0
$394$	10.4496	0.526442
$395$	−2.75104	−0.138420
$396$	0	0
$397$	−9.94041	−0.498895	−0.249448	−	0.968388i	$-0.580249\pi$
$397$	−9.94041	−0.498895	−0.249448	+	0.968388i	$0.580249\pi$
$398$	−1.99966	−0.100234
$399$	0	0
$400$	17.2208	0.861040
$401$	3.11063	0.155338	0.0776688	−	0.996979i	$-0.475252\pi$
$401$	3.11063	0.155338	0.0776688	+	0.996979i	$0.475252\pi$
$402$	0	0
$403$	−7.70020	−0.383574
$404$	0.617994	0.0307464
$405$	0	0
$406$	50.8746	2.52487
$407$	22.4330	1.11196
$408$	0	0
$409$	−19.7133	−0.974758	−0.487379	−	0.873190i	$-0.662047\pi$
$409$	−19.7133	−0.974758	−0.487379	+	0.873190i	$0.662047\pi$
$410$	14.3209	0.707261
$411$	0	0
$412$	0.140605	0.00692711
$413$	45.7549	2.25145
$414$	0	0
$415$	−5.52684	−0.271302
$416$	−1.25049	−0.0613105
$417$	0	0
$418$	36.4390	1.78229
$419$	−10.3921	−0.507686	−0.253843	−	0.967245i	$-0.581695\pi$
$419$	−10.3921	−0.507686	−0.253843	+	0.967245i	$0.581695\pi$
$420$	0	0
$421$	−13.6436	−0.664946	−0.332473	−	0.943113i	$-0.607883\pi$
$421$	−13.6436	−0.664946	−0.332473	+	0.943113i	$0.607883\pi$
$422$	−23.5089	−1.14439
$423$	0	0
$424$	12.1883	0.591915
$425$	−14.0402	−0.681049
$426$	0	0
$427$	3.61927	0.175149
$428$	1.16281	0.0562065
$429$	0	0
$430$	11.0568	0.533207
$431$	−17.8981	−0.862122	−0.431061	−	0.902323i	$-0.641861\pi$
$431$	−17.8981	−0.862122	−0.431061	+	0.902323i	$0.641861\pi$
$432$	0	0
$433$	24.4912	1.17697	0.588485	−	0.808508i	$-0.299725\pi$
$433$	24.4912	1.17697	0.588485	+	0.808508i	$0.299725\pi$
$434$	−31.0091	−1.48848
$435$	0	0
$436$	−2.19851	−0.105289
$437$	1.70163	0.0814001
$438$	0	0
$439$	−28.5922	−1.36463	−0.682314	−	0.731059i	$-0.739026\pi$
$439$	−28.5922	−1.36463	−0.682314	+	0.731059i	$0.739026\pi$
$440$	8.27975	0.394721
$441$	0	0
$442$	8.00640	0.380826
$443$	−11.1639	−0.530415	−0.265207	−	0.964191i	$-0.585440\pi$
$443$	−11.1639	−0.530415	−0.265207	+	0.964191i	$0.585440\pi$
$444$	0	0
$445$	−14.8198	−0.702528
$446$	−36.6889	−1.73727
$447$	0	0
$448$	31.9128	1.50774
$449$	2.83705	0.133889	0.0669444	−	0.997757i	$-0.478675\pi$
$449$	2.83705	0.133889	0.0669444	+	0.997757i	$0.478675\pi$
$450$	0	0
$451$	−31.0668	−1.46288
$452$	−1.63567	−0.0769353
$453$	0	0
$454$	−18.5600	−0.871065
$455$	6.68648	0.313467
$456$	0	0
$457$	35.2362	1.64828	0.824139	−	0.566388i	$-0.191660\pi$
$457$	35.2362	1.64828	0.824139	+	0.566388i	$0.191660\pi$
$458$	−11.4641	−0.535683
$459$	0	0
$460$	−0.0292379	−0.00136322
$461$	35.0373	1.63185	0.815925	−	0.578158i	$-0.196228\pi$
$461$	35.0373	1.63185	0.815925	+	0.578158i	$0.196228\pi$
$462$	0	0
$463$	−28.5134	−1.32513	−0.662565	−	0.749005i	$-0.730532\pi$
$463$	−28.5134	−1.32513	−0.662565	+	0.749005i	$0.730532\pi$
$464$	34.1804	1.58679
$465$	0	0
$466$	35.6331	1.65067
$467$	−18.7973	−0.869835	−0.434918	−	0.900470i	$-0.643222\pi$
$467$	−18.7973	−0.869835	−0.434918	+	0.900470i	$0.643222\pi$
$468$	0	0
$469$	9.13858	0.421980
$470$	−14.2430	−0.656983
$471$	0	0
$472$	28.7121	1.32158
$473$	−23.9858	−1.10287
$474$	0	0
$475$	−32.3825	−1.48581
$476$	2.11782	0.0970700
$477$	0	0
$478$	10.4847	0.479559
$479$	−21.1909	−0.968236	−0.484118	−	0.875003i	$-0.660859\pi$
$479$	−21.1909	−0.968236	−0.484118	+	0.875003i	$0.660859\pi$
$480$	0	0
$481$	11.3683	0.518350
$482$	−5.18033	−0.235957
$483$	0	0
$484$	−0.188433	−0.00856515
$485$	−2.92507	−0.132820
$486$	0	0
$487$	−15.3700	−0.696479	−0.348240	−	0.937406i	$-0.613220\pi$
$487$	−15.3700	−0.696479	−0.348240	+	0.937406i	$0.613220\pi$
$488$	2.27117	0.102811
$489$	0	0
$490$	16.8978	0.763363
$491$	−17.7551	−0.801278	−0.400639	−	0.916236i	$-0.631212\pi$
$491$	−17.7551	−0.801278	−0.400639	+	0.916236i	$0.631212\pi$
$492$	0	0
$493$	−27.8674	−1.25508
$494$	18.4661	0.830828
$495$	0	0
$496$	−20.8336	−0.935458
$497$	66.5744	2.98627
$498$	0	0
$499$	8.61856	0.385820	0.192910	−	0.981216i	$-0.438207\pi$
$499$	8.61856	0.385820	0.192910	+	0.981216i	$0.438207\pi$
$500$	1.24484	0.0556709
$501$	0	0
$502$	3.22866	0.144102
$503$	−9.97468	−0.444749	−0.222374	−	0.974961i	$-0.571381\pi$
$503$	−9.97468	−0.444749	−0.222374	+	0.974961i	$0.571381\pi$
$504$	0	0
$505$	−4.30403	−0.191527
$506$	0.965620	0.0429270
$507$	0	0
$508$	0.240987	0.0106921
$509$	−20.3784	−0.903258	−0.451629	−	0.892206i	$-0.649157\pi$
$509$	−20.3784	−0.903258	−0.451629	+	0.892206i	$0.649157\pi$
$510$	0	0
$511$	30.3375	1.34205
$512$	19.8027	0.875164
$513$	0	0
$514$	14.3252	0.631859
$515$	−0.979247	−0.0431508
$516$	0	0
$517$	30.8978	1.35888
$518$	45.7807	2.01149
$519$	0	0
$520$	4.19590	0.184003
$521$	27.7767	1.21692	0.608460	−	0.793585i	$-0.291788\pi$
$521$	27.7767	1.21692	0.608460	+	0.793585i	$0.291788\pi$
$522$	0	0
$523$	−34.5219	−1.50954	−0.754768	−	0.655992i	$-0.772251\pi$
$523$	−34.5219	−1.50954	−0.754768	+	0.655992i	$0.772251\pi$
$524$	−0.290725	−0.0127004
$525$	0	0
$526$	24.6601	1.07523
$527$	16.9857	0.739910
$528$	0	0
$529$	−22.9549	−0.998039
$530$	6.41893	0.278821
$531$	0	0
$532$	4.88456	0.211773
$533$	−15.7436	−0.681932
$534$	0	0
$535$	−8.09840	−0.350125
$536$	5.73465	0.247699
$537$	0	0
$538$	16.0844	0.693447
$539$	−36.6568	−1.57892
$540$	0	0
$541$	−2.65994	−0.114360	−0.0571800	−	0.998364i	$-0.518211\pi$
$541$	−2.65994	−0.114360	−0.0571800	+	0.998364i	$0.518211\pi$
$542$	−30.0160	−1.28930
$543$	0	0
$544$	2.75844	0.118267
$545$	15.3115	0.655875
$546$	0	0
$547$	−19.7884	−0.846091	−0.423046	−	0.906108i	$-0.639039\pi$
$547$	−19.7884	−0.846091	−0.423046	+	0.906108i	$0.639039\pi$
$548$	3.08435	0.131757
$549$	0	0
$550$	−18.3760	−0.783554
$551$	−64.2737	−2.73815
$552$	0	0
$553$	12.1791	0.517909
$554$	−11.7924	−0.501012
$555$	0	0
$556$	−2.00506	−0.0850336
$557$	0.506629	0.0214666	0.0107333	−	0.999942i	$-0.496583\pi$
$557$	0.506629	0.0214666	0.0107333	+	0.999942i	$0.496583\pi$
$558$	0	0
$559$	−12.1552	−0.514111
$560$	18.0909	0.764481
$561$	0	0
$562$	−32.3455	−1.36441
$563$	−34.8259	−1.46774	−0.733868	−	0.679292i	$-0.762287\pi$
$563$	−34.8259	−1.46774	−0.733868	+	0.679292i	$0.762287\pi$
$564$	0	0
$565$	11.3916	0.479250
$566$	36.5235	1.53520
$567$	0	0
$568$	41.7768	1.75292
$569$	−26.0444	−1.09184	−0.545919	−	0.837838i	$-0.683819\pi$
$569$	−26.0444	−1.09184	−0.545919	+	0.837838i	$0.683819\pi$
$570$	0	0
$571$	−8.04232	−0.336560	−0.168280	−	0.985739i	$-0.553821\pi$
$571$	−8.04232	−0.336560	−0.168280	+	0.985739i	$0.553821\pi$
$572$	0.688302	0.0287794
$573$	0	0
$574$	−63.4003	−2.64628
$575$	−0.858124	−0.0357862
$576$	0	0
$577$	21.8734	0.910602	0.455301	−	0.890338i	$-0.349532\pi$
$577$	21.8734	0.910602	0.455301	+	0.890338i	$0.349532\pi$
$578$	7.21120	0.299947
$579$	0	0
$580$	1.10437	0.0458564
$581$	24.4679	1.01510
$582$	0	0
$583$	−13.9247	−0.576704
$584$	19.0374	0.787772
$585$	0	0
$586$	−13.6738	−0.564861
$587$	42.2010	1.74182	0.870911	−	0.491441i	$-0.163530\pi$
$587$	42.2010	1.74182	0.870911	+	0.491441i	$0.163530\pi$
$588$	0	0
$589$	39.1761	1.61422
$590$	15.1212	0.622529
$591$	0	0
$592$	30.7580	1.26415
$593$	13.4590	0.552695	0.276347	−	0.961058i	$-0.410876\pi$
$593$	13.4590	0.552695	0.276347	+	0.961058i	$0.410876\pi$
$594$	0	0
$595$	−14.7496	−0.604674
$596$	−0.426506	−0.0174704
$597$	0	0
$598$	0.489344	0.0200108
$599$	−16.7357	−0.683802	−0.341901	−	0.939736i	$-0.611071\pi$
$599$	−16.7357	−0.683802	−0.341901	+	0.939736i	$0.611071\pi$
$600$	0	0
$601$	−2.09244	−0.0853523	−0.0426762	−	0.999089i	$-0.513588\pi$
$601$	−2.09244	−0.0853523	−0.0426762	+	0.999089i	$0.513588\pi$
$602$	−48.9497	−1.99504
$603$	0	0
$604$	−2.80566	−0.114161
$605$	1.31235	0.0533546
$606$	0	0
$607$	40.1523	1.62973	0.814866	−	0.579649i	$-0.196810\pi$
$607$	40.1523	1.62973	0.814866	+	0.579649i	$0.196810\pi$
$608$	6.36211	0.258018
$609$	0	0
$610$	1.19611	0.0484289
$611$	15.6580	0.633454
$612$	0	0
$613$	−4.75493	−0.192050	−0.0960250	−	0.995379i	$-0.530613\pi$
$613$	−4.75493	−0.192050	−0.0960250	+	0.995379i	$0.530613\pi$
$614$	15.1998	0.613412
$615$	0	0
$616$	−36.6553	−1.47688
$617$	−24.9336	−1.00379	−0.501895	−	0.864929i	$-0.667364\pi$
$617$	−24.9336	−1.00379	−0.501895	+	0.864929i	$0.667364\pi$
$618$	0	0
$619$	9.85141	0.395962	0.197981	−	0.980206i	$-0.436562\pi$
$619$	9.85141	0.395962	0.197981	+	0.980206i	$0.436562\pi$
$620$	−0.673134	−0.0270337
$621$	0	0
$622$	−15.9738	−0.640491
$623$	65.6090	2.62857
$624$	0	0
$625$	11.5357	0.461427
$626$	−17.4654	−0.698057
$627$	0	0
$628$	1.49129	0.0595090
$629$	−25.0771	−0.999890
$630$	0	0
$631$	23.7231	0.944403	0.472201	−	0.881491i	$-0.343460\pi$
$631$	23.7231	0.944403	0.472201	+	0.881491i	$0.343460\pi$
$632$	7.64265	0.304008
$633$	0	0
$634$	45.2630	1.79763
$635$	−1.67836	−0.0666036
$636$	0	0
$637$	−18.5764	−0.736025
$638$	−36.4732	−1.44399
$639$	0	0
$640$	12.1016	0.478356
$641$	−27.7679	−1.09677	−0.548384	−	0.836227i	$-0.684757\pi$
$641$	−27.7679	−1.09677	−0.548384	+	0.836227i	$0.684757\pi$
$642$	0	0
$643$	25.6986	1.01345	0.506727	−	0.862107i	$-0.330855\pi$
$643$	25.6986	1.01345	0.506727	+	0.862107i	$0.330855\pi$
$644$	0.129439	0.00510062
$645$	0	0
$646$	−40.7340	−1.60266
$647$	−10.6804	−0.419890	−0.209945	−	0.977713i	$-0.567329\pi$
$647$	−10.6804	−0.419890	−0.209945	+	0.977713i	$0.567329\pi$
$648$	0	0
$649$	−32.8027	−1.28762
$650$	−9.31234	−0.365260
$651$	0	0
$652$	0.912166	0.0357232
$653$	−29.3551	−1.14875	−0.574377	−	0.818591i	$-0.694756\pi$
$653$	−29.3551	−1.14875	−0.574377	+	0.818591i	$0.694756\pi$
$654$	0	0
$655$	2.02476	0.0791140
$656$	−42.5959	−1.66309
$657$	0	0
$658$	63.0555	2.45816
$659$	8.85132	0.344799	0.172399	−	0.985027i	$-0.444848\pi$
$659$	8.85132	0.344799	0.172399	+	0.985027i	$0.444848\pi$
$660$	0	0
$661$	−13.4372	−0.522645	−0.261323	−	0.965252i	$-0.584159\pi$
$661$	−13.4372	−0.522645	−0.261323	+	0.965252i	$0.584159\pi$
$662$	22.2008	0.862857
$663$	0	0
$664$	15.3541	0.595855
$665$	−34.0186	−1.31919
$666$	0	0
$667$	−1.70323	−0.0659494
$668$	3.31219	0.128153
$669$	0	0
$670$	3.02014	0.116678
$671$	−2.59474	−0.100169
$672$	0	0
$673$	34.9587	1.34756	0.673780	−	0.738932i	$-0.264670\pi$
$673$	34.9587	1.34756	0.673780	+	0.738932i	$0.264670\pi$
$674$	−32.2720	−1.24307
$675$	0	0
$676$	−1.47906	−0.0568868
$677$	−13.6610	−0.525035	−0.262518	−	0.964927i	$-0.584553\pi$
$677$	−13.6610	−0.525035	−0.262518	+	0.964927i	$0.584553\pi$
$678$	0	0
$679$	12.9496	0.496959
$680$	−9.25566	−0.354939
$681$	0	0
$682$	22.2311	0.851274
$683$	−14.8886	−0.569696	−0.284848	−	0.958573i	$-0.591943\pi$
$683$	−14.8886	−0.569696	−0.284848	+	0.958573i	$0.591943\pi$
$684$	0	0
$685$	−21.4810	−0.820747
$686$	−30.4087	−1.16101
$687$	0	0
$688$	−32.8871	−1.25381
$689$	−7.05661	−0.268835
$690$	0	0
$691$	−31.5534	−1.20035	−0.600175	−	0.799869i	$-0.704902\pi$
$691$	−31.5534	−1.20035	−0.600175	+	0.799869i	$0.704902\pi$
$692$	0.0690503	0.00262490
$693$	0	0
$694$	12.1198	0.460060
$695$	13.9643	0.529696
$696$	0	0
$697$	34.7286	1.31544
$698$	0.701744	0.0265614
$699$	0	0
$700$	−2.46326	−0.0931024
$701$	13.0497	0.492880	0.246440	−	0.969158i	$-0.420739\pi$
$701$	13.0497	0.492880	0.246440	+	0.969158i	$0.420739\pi$
$702$	0	0
$703$	−57.8382	−2.18141
$704$	−22.8791	−0.862287
$705$	0	0
$706$	−14.7347	−0.554546
$707$	19.0544	0.716615
$708$	0	0
$709$	32.7328	1.22931	0.614654	−	0.788797i	$-0.289296\pi$
$709$	32.7328	1.22931	0.614654	+	0.788797i	$0.289296\pi$
$710$	22.0017	0.825708
$711$	0	0
$712$	41.1710	1.54295
$713$	1.03815	0.0388791
$714$	0	0
$715$	−4.79370	−0.179274
$716$	−3.20456	−0.119760
$717$	0	0
$718$	16.7982	0.626903
$719$	−24.6578	−0.919579	−0.459790	−	0.888028i	$-0.652075\pi$
$719$	−24.6578	−0.919579	−0.459790	+	0.888028i	$0.652075\pi$
$720$	0	0
$721$	4.33523	0.161452
$722$	−66.1508	−2.46188
$723$	0	0
$724$	−1.12183	−0.0416925
$725$	32.4129	1.20378
$726$	0	0
$727$	16.6030	0.615773	0.307886	−	0.951423i	$-0.400378\pi$
$727$	16.6030	0.615773	0.307886	+	0.951423i	$0.400378\pi$
$728$	−18.5757	−0.688462
$729$	0	0
$730$	10.0260	0.371079
$731$	26.8130	0.991714
$732$	0	0
$733$	−24.4874	−0.904464	−0.452232	−	0.891900i	$-0.649372\pi$
$733$	−24.4874	−0.904464	−0.452232	+	0.891900i	$0.649372\pi$
$734$	−7.82124	−0.288687
$735$	0	0
$736$	0.168594	0.00621445
$737$	−6.55166	−0.241334
$738$	0	0
$739$	−6.79794	−0.250066	−0.125033	−	0.992153i	$-0.539904\pi$
$739$	−6.79794	−0.250066	−0.125033	+	0.992153i	$0.539904\pi$
$740$	0.993791	0.0365325
$741$	0	0
$742$	−28.4173	−1.04323
$743$	−3.60079	−0.132100	−0.0660501	−	0.997816i	$-0.521040\pi$
$743$	−3.60079	−0.132100	−0.0660501	+	0.997816i	$0.521040\pi$
$744$	0	0
$745$	2.97041	0.108827
$746$	8.10937	0.296905
$747$	0	0
$748$	−1.51831	−0.0555150
$749$	35.8525	1.31002
$750$	0	0
$751$	−9.43327	−0.344225	−0.172112	−	0.985077i	$-0.555059\pi$
$751$	−9.43327	−0.344225	−0.172112	+	0.985077i	$0.555059\pi$
$752$	42.3642	1.54486
$753$	0	0
$754$	−18.4834	−0.673127
$755$	19.5401	0.711136
$756$	0	0
$757$	3.03704	0.110383	0.0551916	−	0.998476i	$-0.482423\pi$
$757$	3.03704	0.110383	0.0551916	+	0.998476i	$0.482423\pi$
$758$	−31.7200	−1.15212
$759$	0	0
$760$	−21.3474	−0.774351
$761$	42.5238	1.54149	0.770743	−	0.637146i	$-0.219885\pi$
$761$	42.5238	1.54149	0.770743	+	0.637146i	$0.219885\pi$
$762$	0	0
$763$	−67.7858	−2.45401
$764$	−0.520954	−0.0188474
$765$	0	0
$766$	30.0989	1.08752
$767$	−16.6234	−0.600234
$768$	0	0
$769$	−24.2938	−0.876056	−0.438028	−	0.898961i	$-0.644323\pi$
$769$	−24.2938	−0.876056	−0.438028	+	0.898961i	$0.644323\pi$
$770$	−19.3044	−0.695684
$771$	0	0
$772$	−2.61872	−0.0942497
$773$	22.0091	0.791614	0.395807	−	0.918334i	$-0.370465\pi$
$773$	22.0091	0.791614	0.395807	+	0.918334i	$0.370465\pi$
$774$	0	0
$775$	−19.7563	−0.709667
$776$	8.12612	0.291711
$777$	0	0
$778$	−45.4239	−1.62853
$779$	80.0984	2.86982
$780$	0	0
$781$	−47.7288	−1.70787
$782$	−1.07944	−0.0386005
$783$	0	0
$784$	−50.2603	−1.79501
$785$	−10.3861	−0.370697
$786$	0	0
$787$	1.72444	0.0614697	0.0307349	−	0.999528i	$-0.490215\pi$
$787$	1.72444	0.0614697	0.0307349	+	0.999528i	$0.490215\pi$
$788$	−1.00423	−0.0357741
$789$	0	0
$790$	4.02499	0.143203
$791$	−50.4320	−1.79316
$792$	0	0
$793$	−1.31493	−0.0466945
$794$	14.5436	0.516134
$795$	0	0
$796$	0.192171	0.00681134
$797$	11.6138	0.411383	0.205692	−	0.978617i	$-0.434056\pi$
$797$	11.6138	0.411383	0.205692	+	0.978617i	$0.434056\pi$
$798$	0	0
$799$	−34.5397	−1.22193
$800$	−3.20838	−0.113433
$801$	0	0
$802$	−4.55111	−0.160705
$803$	−21.7496	−0.767528
$804$	0	0
$805$	−0.901482	−0.0317731
$806$	11.2660	0.396828
$807$	0	0
$808$	11.9570	0.420647
$809$	−29.9527	−1.05308	−0.526541	−	0.850150i	$-0.676511\pi$
$809$	−29.9527	−1.05308	−0.526541	+	0.850150i	$0.676511\pi$
$810$	0	0
$811$	−34.3973	−1.20785	−0.603926	−	0.797041i	$-0.706398\pi$
$811$	−34.3973	−1.20785	−0.603926	+	0.797041i	$0.706398\pi$
$812$	−4.88915	−0.171576
$813$	0	0
$814$	−32.8213	−1.15038
$815$	−6.35280	−0.222529
$816$	0	0
$817$	61.8418	2.16357
$818$	28.8421	1.00844
$819$	0	0
$820$	−1.37627	−0.0480615
$821$	5.39340	0.188231	0.0941155	−	0.995561i	$-0.469998\pi$
$821$	5.39340	0.188231	0.0941155	+	0.995561i	$0.469998\pi$
$822$	0	0
$823$	23.7578	0.828145	0.414072	−	0.910244i	$-0.364106\pi$
$823$	23.7578	0.828145	0.414072	+	0.910244i	$0.364106\pi$
$824$	2.72044	0.0947712
$825$	0	0
$826$	−66.9430	−2.32925
$827$	−33.7726	−1.17439	−0.587194	−	0.809446i	$-0.699767\pi$
$827$	−33.7726	−1.17439	−0.587194	+	0.809446i	$0.699767\pi$
$828$	0	0
$829$	−46.4676	−1.61389	−0.806944	−	0.590628i	$-0.798880\pi$
$829$	−46.4676	−1.61389	−0.806944	+	0.590628i	$0.798880\pi$
$830$	8.08621	0.280676
$831$	0	0
$832$	−11.5944	−0.401962
$833$	40.9774	1.41978
$834$	0	0
$835$	−23.0678	−0.798296
$836$	−3.50186	−0.121114
$837$	0	0
$838$	15.2044	0.525229
$839$	7.72365	0.266650	0.133325	−	0.991072i	$-0.457435\pi$
$839$	7.72365	0.266650	0.133325	+	0.991072i	$0.457435\pi$
$840$	0	0
$841$	35.3342	1.21842
$842$	19.9616	0.687923
$843$	0	0
$844$	2.25925	0.0777666
$845$	10.3009	0.354362
$846$	0	0
$847$	−5.80991	−0.199631
$848$	−19.0923	−0.655633
$849$	0	0
$850$	20.5419	0.704582
$851$	−1.53269	−0.0525400
$852$	0	0
$853$	13.0384	0.446425	0.223213	−	0.974770i	$-0.428346\pi$
$853$	13.0384	0.446425	0.223213	+	0.974770i	$0.428346\pi$
$854$	−5.29529	−0.181201
$855$	0	0
$856$	22.4982	0.768972
$857$	17.4262	0.595267	0.297634	−	0.954680i	$-0.403803\pi$
$857$	17.4262	0.595267	0.297634	+	0.954680i	$0.403803\pi$
$858$	0	0
$859$	−50.1030	−1.70949	−0.854747	−	0.519045i	$-0.826288\pi$
$859$	−50.1030	−1.70949	−0.854747	+	0.519045i	$0.826288\pi$
$860$	−1.06258	−0.0362338
$861$	0	0
$862$	26.1864	0.891911
$863$	20.3805	0.693762	0.346881	−	0.937909i	$-0.387241\pi$
$863$	20.3805	0.693762	0.346881	+	0.937909i	$0.387241\pi$
$864$	0	0
$865$	−0.480902	−0.0163512
$866$	−35.8325	−1.21764
$867$	0	0
$868$	2.98004	0.101149
$869$	−8.73151	−0.296196
$870$	0	0
$871$	−3.32017	−0.112500
$872$	−42.5370	−1.44048
$873$	0	0
$874$	−2.48963	−0.0842128
$875$	38.3817	1.29754
$876$	0	0
$877$	−29.0756	−0.981814	−0.490907	−	0.871212i	$-0.663335\pi$
$877$	−29.0756	−0.981814	−0.490907	+	0.871212i	$0.663335\pi$
$878$	41.8326	1.41178
$879$	0	0
$880$	−12.9698	−0.437212
$881$	5.99800	0.202078	0.101039	−	0.994882i	$-0.467783\pi$
$881$	5.99800	0.202078	0.101039	+	0.994882i	$0.467783\pi$
$882$	0	0
$883$	16.5048	0.555431	0.277715	−	0.960663i	$-0.410423\pi$
$883$	16.5048	0.555431	0.277715	+	0.960663i	$0.410423\pi$
$884$	−0.769431	−0.0258788
$885$	0	0
$886$	16.3337	0.548743
$887$	1.75149	0.0588093	0.0294047	−	0.999568i	$-0.490639\pi$
$887$	1.75149	0.0588093	0.0294047	+	0.999568i	$0.490639\pi$
$888$	0	0
$889$	7.43027	0.249203
$890$	21.6826	0.726803
$891$	0	0
$892$	3.52588	0.118055
$893$	−79.6628	−2.66581
$894$	0	0
$895$	22.3182	0.746015
$896$	−53.5749	−1.78981
$897$	0	0
$898$	−4.15083	−0.138515
$899$	−39.2129	−1.30782
$900$	0	0
$901$	15.5660	0.518580
$902$	45.4532	1.51343
$903$	0	0
$904$	−31.6471	−1.05257
$905$	7.81302	0.259713
$906$	0	0
$907$	−21.1269	−0.701508	−0.350754	−	0.936468i	$-0.614075\pi$
$907$	−21.1269	−0.701508	−0.350754	+	0.936468i	$0.614075\pi$
$908$	1.78366	0.0591927
$909$	0	0
$910$	−9.78286	−0.324299
$911$	4.02983	0.133514	0.0667571	−	0.997769i	$-0.478735\pi$
$911$	4.02983	0.133514	0.0667571	+	0.997769i	$0.478735\pi$
$912$	0	0
$913$	−17.5416	−0.580543
$914$	−51.5533	−1.70523
$915$	0	0
$916$	1.10173	0.0364020
$917$	−8.96384	−0.296012
$918$	0	0
$919$	−36.1329	−1.19191	−0.595956	−	0.803017i	$-0.703227\pi$
$919$	−36.1329	−1.19191	−0.595956	+	0.803017i	$0.703227\pi$
$920$	−0.565698	−0.0186505
$921$	0	0
$922$	−51.2624	−1.68824
$923$	−24.1874	−0.796137
$924$	0	0
$925$	29.1675	0.959021
$926$	41.7174	1.37092
$927$	0	0
$928$	−6.36809	−0.209043
$929$	−3.54245	−0.116224	−0.0581121	−	0.998310i	$-0.518508\pi$
$929$	−3.54245	−0.116224	−0.0581121	+	0.998310i	$0.518508\pi$
$930$	0	0
$931$	94.5109	3.09747
$932$	−3.42441	−0.112170
$933$	0	0
$934$	27.5020	0.899892
$935$	10.5743	0.345817
$936$	0	0
$937$	−20.6110	−0.673333	−0.336666	−	0.941624i	$-0.609299\pi$
$937$	−20.6110	−0.673333	−0.336666	+	0.941624i	$0.609299\pi$
$938$	−13.3705	−0.436562
$939$	0	0
$940$	1.36879	0.0446449
$941$	−34.8331	−1.13553	−0.567764	−	0.823191i	$-0.692191\pi$
$941$	−34.8331	−1.13553	−0.567764	+	0.823191i	$0.692191\pi$
$942$	0	0
$943$	2.12258	0.0691207
$944$	−44.9761	−1.46385
$945$	0	0
$946$	35.0932	1.14098
$947$	24.7436	0.804060	0.402030	−	0.915627i	$-0.368305\pi$
$947$	24.7436	0.804060	0.402030	+	0.915627i	$0.368305\pi$
$948$	0	0
$949$	−11.0220	−0.357789
$950$	47.3781	1.53715
$951$	0	0
$952$	40.9758	1.32803
$953$	43.2497	1.40099	0.700497	−	0.713655i	$-0.252962\pi$
$953$	43.2497	1.40099	0.700497	+	0.713655i	$0.252962\pi$
$954$	0	0
$955$	3.62819	0.117406
$956$	−1.00760	−0.0325882
$957$	0	0
$958$	31.0040	1.00169
$959$	95.0987	3.07090
$960$	0	0
$961$	−7.09894	−0.228998
$962$	−16.6327	−0.536261
$963$	0	0
$964$	0.497840	0.0160343
$965$	18.2381	0.587106
$966$	0	0
$967$	26.0346	0.837216	0.418608	−	0.908167i	$-0.362518\pi$
$967$	26.0346	0.837216	0.418608	+	0.908167i	$0.362518\pi$
$968$	−3.64583	−0.117182
$969$	0	0
$970$	4.27961	0.137410
$971$	42.3145	1.35794	0.678968	−	0.734168i	$-0.262428\pi$
$971$	42.3145	1.35794	0.678968	+	0.734168i	$0.262428\pi$
$972$	0	0
$973$	−61.8214	−1.98190
$974$	22.4875	0.720546
$975$	0	0
$976$	−3.55767	−0.113878
$977$	−9.09981	−0.291129	−0.145564	−	0.989349i	$-0.546500\pi$
$977$	−9.09981	−0.291129	−0.145564	+	0.989349i	$0.546500\pi$
$978$	0	0
$979$	−47.0366	−1.50330
$980$	−1.62391	−0.0518739
$981$	0	0
$982$	25.9772	0.828966
$983$	−5.72010	−0.182443	−0.0912214	−	0.995831i	$-0.529077\pi$
$983$	−5.72010	−0.182443	−0.0912214	+	0.995831i	$0.529077\pi$
$984$	0	0
$985$	6.99395	0.222846
$986$	40.7723	1.29845
$987$	0	0
$988$	−1.77463	−0.0564584
$989$	1.63879	0.0521104
$990$	0	0
$991$	−9.92547	−0.315293	−0.157646	−	0.987496i	$-0.550391\pi$
$991$	−9.92547	−0.315293	−0.157646	+	0.987496i	$0.550391\pi$
$992$	3.88148	0.123237
$993$	0	0
$994$	−97.4037	−3.08946
$995$	−1.33838	−0.0424296
$996$	0	0
$997$	1.90856	0.0604446	0.0302223	−	0.999543i	$-0.490378\pi$
$997$	1.90856	0.0604446	0.0302223	+	0.999543i	$0.490378\pi$
$998$	−12.6096	−0.399151
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.2.a.h.1.5	✓	14
3.2	odd	2	inner	927.2.a.h.1.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.2.a.h.1.5	✓	14	1.1	even	1	trivial
927.2.a.h.1.10	yes	14	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	927.a (trivial)

\(f(q)\)	\(=\)	\(q-1.46308 q^{2} +0.140605 q^{4} -0.979247 q^{5} +4.33523 q^{7} +2.72044 q^{8} +O(q^{10})\)	\(q-1.46308 q^{2} +0.140605 q^{4} -0.979247 q^{5} +4.33523 q^{7} +2.72044 q^{8} +1.43272 q^{10} -3.10803 q^{11} -1.57505 q^{13} -6.34279 q^{14} -4.26144 q^{16} +3.47437 q^{17} +8.01333 q^{19} -0.137687 q^{20} +4.54730 q^{22} +0.212350 q^{23} -4.04108 q^{25} +2.30442 q^{26} +0.609555 q^{28} -8.02086 q^{29} +4.88887 q^{31} +0.793941 q^{32} -5.08328 q^{34} -4.24526 q^{35} -7.21775 q^{37} -11.7241 q^{38} -2.66399 q^{40} +9.99565 q^{41} +7.71738 q^{43} -0.437004 q^{44} -0.310686 q^{46} -9.94129 q^{47} +11.7942 q^{49} +5.91242 q^{50} -0.221459 q^{52} +4.48025 q^{53} +3.04353 q^{55} +11.7938 q^{56} +11.7352 q^{58} +10.5542 q^{59} +0.834851 q^{61} -7.15281 q^{62} +7.36128 q^{64} +1.54236 q^{65} +2.10798 q^{67} +0.488513 q^{68} +6.21116 q^{70} +15.3566 q^{71} +6.99789 q^{73} +10.5602 q^{74} +1.12671 q^{76} -13.4740 q^{77} +2.80934 q^{79} +4.17300 q^{80} -14.6244 q^{82} +5.64397 q^{83} -3.40226 q^{85} -11.2911 q^{86} -8.45522 q^{88} +15.1339 q^{89} -6.82819 q^{91} +0.0298575 q^{92} +14.5449 q^{94} -7.84702 q^{95} +2.98706 q^{97} -17.2559 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 20 q^{4} + 12 q^{7}+O(q^{10}) \) 14 * q + 20 * q^4 + 12 * q^7	\( 14 q + 20 q^{4} + 12 q^{7} + 4 q^{10} + 20 q^{13} + 36 q^{16} + 8 q^{19} + 32 q^{22} + 58 q^{25} + 26 q^{28} - 4 q^{31} - 16 q^{34} + 28 q^{37} - 32 q^{40} + 40 q^{43} + 14 q^{46} + 26 q^{49} + 28 q^{55} - 4 q^{58} + 24 q^{61} + 64 q^{64} - 8 q^{67} - 64 q^{70} + 56 q^{73} + 4 q^{79} - 26 q^{82} + 24 q^{85} + 28 q^{88} + 44 q^{91} - 60 q^{94} + 12 q^{97}+O(q^{100}) \) 14 * q + 20 * q^4 + 12 * q^7 + 4 * q^10 + 20 * q^13 + 36 * q^16 + 8 * q^19 + 32 * q^22 + 58 * q^25 + 26 * q^28 - 4 * q^31 - 16 * q^34 + 28 * q^37 - 32 * q^40 + 40 * q^43 + 14 * q^46 + 26 * q^49 + 28 * q^55 - 4 * q^58 + 24 * q^61 + 64 * q^64 - 8 * q^67 - 64 * q^70 + 56 * q^73 + 4 * q^79 - 26 * q^82 + 24 * q^85 + 28 * q^88 + 44 * q^91 - 60 * q^94 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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