LMFDB - Embedded newform 4864.2.a.bt.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4864,2,Mod(1,4864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4864 = 2^{8} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4864.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:	$38.8392355432$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 23x^{8} + 44x^{7} + 167x^{6} - 266x^{5} - 491x^{4} + 460x^{3} + 546x^{2} + 56x - 8 $ x^10 - 2x^9 - 23x^8 + 44x^7 + 167x^6 - 266x^5 - 491x^4 + 460x^3 + 546x^2 + 56*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 2432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.30294$ of defining polynomial
Character	$\chi$	$=$	4864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.458177 q^{3} +3.00397 q^{5} -2.21624 q^{7} -2.79007 q^{9} +O(q^{10})$	$q+0.458177 q^{3} +3.00397 q^{5} -2.21624 q^{7} -2.79007 q^{9} +5.02382 q^{11} -5.22021 q^{13} +1.37635 q^{15} +7.73025 q^{17} +1.00000 q^{19} -1.01543 q^{21} +6.51613 q^{23} +4.02382 q^{25} -2.65288 q^{27} +2.23562 q^{29} +3.33566 q^{31} +2.30180 q^{33} -6.65751 q^{35} +1.45208 q^{37} -2.39178 q^{39} +3.16461 q^{41} -2.16101 q^{43} -8.38129 q^{45} -9.95256 q^{47} -2.08828 q^{49} +3.54182 q^{51} -6.66810 q^{53} +15.0914 q^{55} +0.458177 q^{57} -3.89126 q^{59} -7.05259 q^{61} +6.18347 q^{63} -15.6813 q^{65} +13.2767 q^{67} +2.98554 q^{69} +5.72840 q^{71} -5.89754 q^{73} +1.84362 q^{75} -11.1340 q^{77} +15.9369 q^{79} +7.15473 q^{81} +8.71734 q^{83} +23.2214 q^{85} +1.02431 q^{87} -3.79938 q^{89} +11.5692 q^{91} +1.52833 q^{93} +3.00397 q^{95} -3.94645 q^{97} -14.0168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{3} + 14 q^{9}+O(q^{10}) $ 10 * q + 4 * q^3 + 14 * q^9	$ 10 q + 4 q^{3} + 14 q^{9} + 20 q^{11} + 4 q^{17} + 10 q^{19} + 10 q^{25} + 28 q^{27} - 8 q^{33} + 36 q^{35} - 12 q^{41} - 4 q^{43} + 26 q^{49} + 36 q^{51} + 4 q^{57} + 52 q^{59} - 24 q^{65} + 12 q^{67} + 12 q^{73} - 12 q^{75} + 34 q^{81} + 16 q^{83} - 20 q^{89} + 60 q^{91} - 28 q^{97} + 60 q^{99}+O(q^{100}) $ 10 * q + 4 * q^3 + 14 * q^9 + 20 * q^11 + 4 * q^17 + 10 * q^19 + 10 * q^25 + 28 * q^27 - 8 * q^33 + 36 * q^35 - 12 * q^41 - 4 * q^43 + 26 * q^49 + 36 * q^51 + 4 * q^57 + 52 * q^59 - 24 * q^65 + 12 * q^67 + 12 * q^73 - 12 * q^75 + 34 * q^81 + 16 * q^83 - 20 * q^89 + 60 * q^91 - 28 * q^97 + 60 * 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
$3$	0.458177	0.264529	0.132264	−	0.991214i	$-0.457775\pi$
$3$	0.458177	0.264529	0.132264	+	0.991214i	$0.457775\pi$
$4$	0	0
$5$	3.00397	1.34341	0.671707	−	0.740817i	$-0.265561\pi$
$5$	3.00397	1.34341	0.671707	+	0.740817i	$0.265561\pi$
$6$	0	0
$7$	−2.21624	−0.837660	−0.418830	−	0.908065i	$-0.637560\pi$
$7$	−2.21624	−0.837660	−0.418830	+	0.908065i	$0.637560\pi$
$8$	0	0
$9$	−2.79007	−0.930025
$10$	0	0
$11$	5.02382	1.51474	0.757369	−	0.652987i	$-0.226484\pi$
$11$	5.02382	1.51474	0.757369	+	0.652987i	$0.226484\pi$
$12$	0	0
$13$	−5.22021	−1.44783	−0.723913	−	0.689892i	$-0.757658\pi$
$13$	−5.22021	−1.44783	−0.723913	+	0.689892i	$0.757658\pi$
$14$	0	0
$15$	1.37635	0.355372
$16$	0	0
$17$	7.73025	1.87486	0.937430	−	0.348174i	$-0.113198\pi$
$17$	7.73025	1.87486	0.937430	+	0.348174i	$0.113198\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.01543	−0.221585
$22$	0	0
$23$	6.51613	1.35871	0.679353	−	0.733811i	$-0.262260\pi$
$23$	6.51613	1.35871	0.679353	+	0.733811i	$0.262260\pi$
$24$	0	0
$25$	4.02382	0.804764
$26$	0	0
$27$	−2.65288	−0.510547
$28$	0	0
$29$	2.23562	0.415145	0.207572	−	0.978220i	$-0.433444\pi$
$29$	2.23562	0.415145	0.207572	+	0.978220i	$0.433444\pi$
$30$	0	0
$31$	3.33566	0.599103	0.299551	−	0.954080i	$-0.403163\pi$
$31$	3.33566	0.599103	0.299551	+	0.954080i	$0.403163\pi$
$32$	0	0
$33$	2.30180	0.400692
$34$	0	0
$35$	−6.65751	−1.12533
$36$	0	0
$37$	1.45208	0.238720	0.119360	−	0.992851i	$-0.461916\pi$
$37$	1.45208	0.238720	0.119360	+	0.992851i	$0.461916\pi$
$38$	0	0
$39$	−2.39178	−0.382991
$40$	0	0
$41$	3.16461	0.494228	0.247114	−	0.968986i	$-0.420518\pi$
$41$	3.16461	0.494228	0.247114	+	0.968986i	$0.420518\pi$
$42$	0	0
$43$	−2.16101	−0.329551	−0.164776	−	0.986331i	$-0.552690\pi$
$43$	−2.16101	−0.329551	−0.164776	+	0.986331i	$0.552690\pi$
$44$	0	0
$45$	−8.38129	−1.24941
$46$	0	0
$47$	−9.95256	−1.45173	−0.725865	−	0.687837i	$-0.758560\pi$
$47$	−9.95256	−1.45173	−0.725865	+	0.687837i	$0.758560\pi$
$48$	0	0
$49$	−2.08828	−0.298325
$50$	0	0
$51$	3.54182	0.495954
$52$	0	0
$53$	−6.66810	−0.915935	−0.457967	−	0.888969i	$-0.651422\pi$
$53$	−6.66810	−0.915935	−0.457967	+	0.888969i	$0.651422\pi$
$54$	0	0
$55$	15.0914	2.03492
$56$	0	0
$57$	0.458177	0.0606871
$58$	0	0
$59$	−3.89126	−0.506599	−0.253299	−	0.967388i	$-0.581516\pi$
$59$	−3.89126	−0.506599	−0.253299	+	0.967388i	$0.581516\pi$
$60$	0	0
$61$	−7.05259	−0.902991	−0.451496	−	0.892273i	$-0.649109\pi$
$61$	−7.05259	−0.902991	−0.451496	+	0.892273i	$0.649109\pi$
$62$	0	0
$63$	6.18347	0.779044
$64$	0	0
$65$	−15.6813	−1.94503
$66$	0	0
$67$	13.2767	1.62201	0.811004	−	0.585041i	$-0.198922\pi$
$67$	13.2767	1.62201	0.811004	+	0.585041i	$0.198922\pi$
$68$	0	0
$69$	2.98554	0.359417
$70$	0	0
$71$	5.72840	0.679836	0.339918	−	0.940455i	$-0.389601\pi$
$71$	5.72840	0.679836	0.339918	+	0.940455i	$0.389601\pi$
$72$	0	0
$73$	−5.89754	−0.690254	−0.345127	−	0.938556i	$-0.612164\pi$
$73$	−5.89754	−0.690254	−0.345127	+	0.938556i	$0.612164\pi$
$74$	0	0
$75$	1.84362	0.212883
$76$	0	0
$77$	−11.1340	−1.26884
$78$	0	0
$79$	15.9369	1.79304	0.896522	−	0.442998i	$-0.146085\pi$
$79$	15.9369	1.79304	0.896522	+	0.442998i	$0.146085\pi$
$80$	0	0
$81$	7.15473	0.794970
$82$	0	0
$83$	8.71734	0.956852	0.478426	−	0.878128i	$-0.341207\pi$
$83$	8.71734	0.956852	0.478426	+	0.878128i	$0.341207\pi$
$84$	0	0
$85$	23.2214	2.51871
$86$	0	0
$87$	1.02431	0.109818
$88$	0	0
$89$	−3.79938	−0.402734	−0.201367	−	0.979516i	$-0.564538\pi$
$89$	−3.79938	−0.402734	−0.201367	+	0.979516i	$0.564538\pi$
$90$	0	0
$91$	11.5692	1.21279
$92$	0	0
$93$	1.52833	0.158480
$94$	0	0
$95$	3.00397	0.308201
$96$	0	0
$97$	−3.94645	−0.400701	−0.200351	−	0.979724i	$-0.564208\pi$
$97$	−3.94645	−0.400701	−0.200351	+	0.979724i	$0.564208\pi$
$98$	0	0
$99$	−14.0168	−1.40874
$100$	0	0
$101$	6.32025	0.628888	0.314444	−	0.949276i	$-0.398182\pi$
$101$	6.32025	0.628888	0.314444	+	0.949276i	$0.398182\pi$
$102$	0	0
$103$	−4.71201	−0.464288	−0.232144	−	0.972681i	$-0.574574\pi$
$103$	−4.71201	−0.464288	−0.232144	+	0.972681i	$0.574574\pi$
$104$	0	0
$105$	−3.05032	−0.297681
$106$	0	0
$107$	18.5550	1.79378	0.896892	−	0.442249i	$-0.145819\pi$
$107$	18.5550	1.79378	0.896892	+	0.442249i	$0.145819\pi$
$108$	0	0
$109$	9.25995	0.886942	0.443471	−	0.896289i	$-0.353747\pi$
$109$	9.25995	0.886942	0.443471	+	0.896289i	$0.353747\pi$
$110$	0	0
$111$	0.665309	0.0631483
$112$	0	0
$113$	−7.97386	−0.750118	−0.375059	−	0.927001i	$-0.622378\pi$
$113$	−7.97386	−0.750118	−0.375059	+	0.927001i	$0.622378\pi$
$114$	0	0
$115$	19.5742	1.82531
$116$	0	0
$117$	14.5648	1.34651
$118$	0	0
$119$	−17.1321	−1.57050
$120$	0	0
$121$	14.2387	1.29443
$122$	0	0
$123$	1.44995	0.130738
$124$	0	0
$125$	−2.93242	−0.262284
$126$	0	0
$127$	8.32912	0.739090	0.369545	−	0.929213i	$-0.379514\pi$
$127$	8.32912	0.739090	0.369545	+	0.929213i	$0.379514\pi$
$128$	0	0
$129$	−0.990126	−0.0871758
$130$	0	0
$131$	16.3831	1.43140	0.715700	−	0.698408i	$-0.246108\pi$
$131$	16.3831	1.43140	0.715700	+	0.698408i	$0.246108\pi$
$132$	0	0
$133$	−2.21624	−0.192172
$134$	0	0
$135$	−7.96916	−0.685876
$136$	0	0
$137$	16.2267	1.38634	0.693172	−	0.720772i	$-0.256212\pi$
$137$	16.2267	1.38634	0.693172	+	0.720772i	$0.256212\pi$
$138$	0	0
$139$	4.38904	0.372273	0.186137	−	0.982524i	$-0.440403\pi$
$139$	4.38904	0.372273	0.186137	+	0.982524i	$0.440403\pi$
$140$	0	0
$141$	−4.56004	−0.384024
$142$	0	0
$143$	−26.2254	−2.19308
$144$	0	0
$145$	6.71574	0.557712
$146$	0	0
$147$	−0.956802	−0.0789157
$148$	0	0
$149$	1.28542	0.105306	0.0526528	−	0.998613i	$-0.483232\pi$
$149$	1.28542	0.105306	0.0526528	+	0.998613i	$0.483232\pi$
$150$	0	0
$151$	9.07295	0.738346	0.369173	−	0.929361i	$-0.379641\pi$
$151$	9.07295	0.738346	0.369173	+	0.929361i	$0.379641\pi$
$152$	0	0
$153$	−21.5680	−1.74367
$154$	0	0
$155$	10.0202	0.804844
$156$	0	0
$157$	3.13656	0.250325	0.125162	−	0.992136i	$-0.460055\pi$
$157$	3.13656	0.250325	0.125162	+	0.992136i	$0.460055\pi$
$158$	0	0
$159$	−3.05517	−0.242291
$160$	0	0
$161$	−14.4413	−1.13813
$162$	0	0
$163$	14.9640	1.17207	0.586035	−	0.810286i	$-0.300688\pi$
$163$	14.9640	1.17207	0.586035	+	0.810286i	$0.300688\pi$
$164$	0	0
$165$	6.91453	0.538295
$166$	0	0
$167$	−21.0799	−1.63121	−0.815607	−	0.578606i	$-0.803597\pi$
$167$	−21.0799	−1.63121	−0.815607	+	0.578606i	$0.803597\pi$
$168$	0	0
$169$	14.2506	1.09620
$170$	0	0
$171$	−2.79007	−0.213362
$172$	0	0
$173$	−23.7808	−1.80802	−0.904011	−	0.427510i	$-0.859391\pi$
$173$	−23.7808	−1.80802	−0.904011	+	0.427510i	$0.859391\pi$
$174$	0	0
$175$	−8.91775	−0.674118
$176$	0	0
$177$	−1.78289	−0.134010
$178$	0	0
$179$	14.3018	1.06897	0.534483	−	0.845179i	$-0.320506\pi$
$179$	14.3018	1.06897	0.534483	+	0.845179i	$0.320506\pi$
$180$	0	0
$181$	−13.9014	−1.03328	−0.516640	−	0.856203i	$-0.672818\pi$
$181$	−13.9014	−1.03328	−0.516640	+	0.856203i	$0.672818\pi$
$182$	0	0
$183$	−3.23133	−0.238867
$184$	0	0
$185$	4.36199	0.320700
$186$	0	0
$187$	38.8353	2.83992
$188$	0	0
$189$	5.87942	0.427665
$190$	0	0
$191$	−2.17747	−0.157557	−0.0787783	−	0.996892i	$-0.525102\pi$
$191$	−2.17747	−0.157557	−0.0787783	+	0.996892i	$0.525102\pi$
$192$	0	0
$193$	26.6410	1.91766	0.958831	−	0.283978i	$-0.0916541\pi$
$193$	26.6410	1.91766	0.958831	+	0.283978i	$0.0916541\pi$
$194$	0	0
$195$	−7.18483	−0.514516
$196$	0	0
$197$	20.8500	1.48550	0.742749	−	0.669570i	$-0.233521\pi$
$197$	20.8500	1.48550	0.742749	+	0.669570i	$0.233521\pi$
$198$	0	0
$199$	4.24710	0.301069	0.150535	−	0.988605i	$-0.451901\pi$
$199$	4.24710	0.301069	0.150535	+	0.988605i	$0.451901\pi$
$200$	0	0
$201$	6.08308	0.429068
$202$	0	0
$203$	−4.95468	−0.347750
$204$	0	0
$205$	9.50637	0.663954
$206$	0	0
$207$	−18.1805	−1.26363
$208$	0	0
$209$	5.02382	0.347505
$210$	0	0
$211$	−18.1134	−1.24698	−0.623488	−	0.781833i	$-0.714285\pi$
$211$	−18.1134	−1.24698	−0.623488	+	0.781833i	$0.714285\pi$
$212$	0	0
$213$	2.62462	0.179836
$214$	0	0
$215$	−6.49161	−0.442724
$216$	0	0
$217$	−7.39263	−0.501845
$218$	0	0
$219$	−2.70212	−0.182592
$220$	0	0
$221$	−40.3535	−2.71447
$222$	0	0
$223$	3.71952	0.249078	0.124539	−	0.992215i	$-0.460255\pi$
$223$	3.71952	0.249078	0.124539	+	0.992215i	$0.460255\pi$
$224$	0	0
$225$	−11.2267	−0.748450
$226$	0	0
$227$	−17.8318	−1.18354	−0.591769	−	0.806108i	$-0.701570\pi$
$227$	−17.8318	−1.18354	−0.591769	+	0.806108i	$0.701570\pi$
$228$	0	0
$229$	13.5159	0.893158	0.446579	−	0.894744i	$-0.352642\pi$
$229$	13.5159	0.893158	0.446579	+	0.894744i	$0.352642\pi$
$230$	0	0
$231$	−5.10134	−0.335644
$232$	0	0
$233$	−18.5061	−1.21238	−0.606189	−	0.795321i	$-0.707302\pi$
$233$	−18.5061	−1.21238	−0.606189	+	0.795321i	$0.707302\pi$
$234$	0	0
$235$	−29.8972	−1.95028
$236$	0	0
$237$	7.30194	0.474312
$238$	0	0
$239$	−11.6874	−0.755995	−0.377997	−	0.925807i	$-0.623387\pi$
$239$	−11.6874	−0.755995	−0.377997	+	0.925807i	$0.623387\pi$
$240$	0	0
$241$	10.2701	0.661554	0.330777	−	0.943709i	$-0.392689\pi$
$241$	10.2701	0.661554	0.330777	+	0.943709i	$0.392689\pi$
$242$	0	0
$243$	11.2368	0.720839
$244$	0	0
$245$	−6.27312	−0.400775
$246$	0	0
$247$	−5.22021	−0.332154
$248$	0	0
$249$	3.99409	0.253115
$250$	0	0
$251$	25.1064	1.58470	0.792350	−	0.610066i	$-0.208857\pi$
$251$	25.1064	1.58470	0.792350	+	0.610066i	$0.208857\pi$
$252$	0	0
$253$	32.7358	2.05808
$254$	0	0
$255$	10.6395	0.666273
$256$	0	0
$257$	12.5905	0.785374	0.392687	−	0.919672i	$-0.371546\pi$
$257$	12.5905	0.785374	0.392687	+	0.919672i	$0.371546\pi$
$258$	0	0
$259$	−3.21815	−0.199966
$260$	0	0
$261$	−6.23755	−0.386095
$262$	0	0
$263$	22.7749	1.40436	0.702181	−	0.711999i	$-0.252210\pi$
$263$	22.7749	1.40436	0.702181	+	0.711999i	$0.252210\pi$
$264$	0	0
$265$	−20.0308	−1.23048
$266$	0	0
$267$	−1.74079	−0.106535
$268$	0	0
$269$	17.0271	1.03816	0.519081	−	0.854725i	$-0.326274\pi$
$269$	17.0271	1.03816	0.519081	+	0.854725i	$0.326274\pi$
$270$	0	0
$271$	−8.19591	−0.497866	−0.248933	−	0.968521i	$-0.580080\pi$
$271$	−8.19591	−0.497866	−0.248933	+	0.968521i	$0.580080\pi$
$272$	0	0
$273$	5.30076	0.320817
$274$	0	0
$275$	20.2149	1.21901
$276$	0	0
$277$	−18.7174	−1.12462	−0.562309	−	0.826927i	$-0.690087\pi$
$277$	−18.7174	−1.12462	−0.562309	+	0.826927i	$0.690087\pi$
$278$	0	0
$279$	−9.30675	−0.557180
$280$	0	0
$281$	0.189116	0.0112817	0.00564087	−	0.999984i	$-0.498204\pi$
$281$	0.189116	0.0112817	0.00564087	+	0.999984i	$0.498204\pi$
$282$	0	0
$283$	10.6357	0.632226	0.316113	−	0.948722i	$-0.397622\pi$
$283$	10.6357	0.632226	0.316113	+	0.948722i	$0.397622\pi$
$284$	0	0
$285$	1.37635	0.0815279
$286$	0	0
$287$	−7.01353	−0.413995
$288$	0	0
$289$	42.7567	2.51510
$290$	0	0
$291$	−1.80817	−0.105997
$292$	0	0
$293$	−27.6765	−1.61688	−0.808439	−	0.588580i	$-0.799687\pi$
$293$	−27.6765	−1.61688	−0.808439	+	0.588580i	$0.799687\pi$
$294$	0	0
$295$	−11.6892	−0.680572
$296$	0	0
$297$	−13.3276	−0.773345
$298$	0	0
$299$	−34.0155	−1.96717
$300$	0	0
$301$	4.78932	0.276052
$302$	0	0
$303$	2.89579	0.166359
$304$	0	0
$305$	−21.1857	−1.21309
$306$	0	0
$307$	8.14707	0.464978	0.232489	−	0.972599i	$-0.425313\pi$
$307$	8.14707	0.464978	0.232489	+	0.972599i	$0.425313\pi$
$308$	0	0
$309$	−2.15894	−0.122818
$310$	0	0
$311$	−11.1283	−0.631030	−0.315515	−	0.948921i	$-0.602177\pi$
$311$	−11.1283	−0.631030	−0.315515	+	0.948921i	$0.602177\pi$
$312$	0	0
$313$	−10.2647	−0.580198	−0.290099	−	0.956997i	$-0.593688\pi$
$313$	−10.2647	−0.580198	−0.290099	+	0.956997i	$0.593688\pi$
$314$	0	0
$315$	18.5750	1.04658
$316$	0	0
$317$	7.14866	0.401509	0.200754	−	0.979642i	$-0.435661\pi$
$317$	7.14866	0.401509	0.200754	+	0.979642i	$0.435661\pi$
$318$	0	0
$319$	11.2314	0.628836
$320$	0	0
$321$	8.50150	0.474508
$322$	0	0
$323$	7.73025	0.430122
$324$	0	0
$325$	−21.0052	−1.16516
$326$	0	0
$327$	4.24270	0.234622
$328$	0	0
$329$	22.0573	1.21606
$330$	0	0
$331$	−28.4370	−1.56304	−0.781519	−	0.623881i	$-0.785555\pi$
$331$	−28.4370	−1.56304	−0.781519	+	0.623881i	$0.785555\pi$
$332$	0	0
$333$	−4.05140	−0.222016
$334$	0	0
$335$	39.8828	2.17903
$336$	0	0
$337$	−18.9197	−1.03062	−0.515311	−	0.857003i	$-0.672324\pi$
$337$	−18.9197	−1.03062	−0.515311	+	0.857003i	$0.672324\pi$
$338$	0	0
$339$	−3.65344	−0.198428
$340$	0	0
$341$	16.7578	0.907484
$342$	0	0
$343$	20.1418	1.08756
$344$	0	0
$345$	8.96847	0.482846
$346$	0	0
$347$	2.34140	0.125693	0.0628466	−	0.998023i	$-0.479982\pi$
$347$	2.34140	0.125693	0.0628466	+	0.998023i	$0.479982\pi$
$348$	0	0
$349$	−14.8506	−0.794935	−0.397468	−	0.917616i	$-0.630111\pi$
$349$	−14.8506	−0.794935	−0.397468	+	0.917616i	$0.630111\pi$
$350$	0	0
$351$	13.8486	0.739183
$352$	0	0
$353$	−16.9690	−0.903168	−0.451584	−	0.892229i	$-0.649141\pi$
$353$	−16.9690	−0.903168	−0.451584	+	0.892229i	$0.649141\pi$
$354$	0	0
$355$	17.2079	0.913302
$356$	0	0
$357$	−7.84953	−0.415441
$358$	0	0
$359$	−32.2107	−1.70002	−0.850008	−	0.526770i	$-0.823403\pi$
$359$	−32.2107	−1.70002	−0.850008	+	0.526770i	$0.823403\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	6.52387	0.342414
$364$	0	0
$365$	−17.7160	−0.927298
$366$	0	0
$367$	−10.2366	−0.534347	−0.267173	−	0.963648i	$-0.586090\pi$
$367$	−10.2366	−0.534347	−0.267173	+	0.963648i	$0.586090\pi$
$368$	0	0
$369$	−8.82948	−0.459644
$370$	0	0
$371$	14.7781	0.767242
$372$	0	0
$373$	−30.1908	−1.56322	−0.781610	−	0.623767i	$-0.785601\pi$
$373$	−30.1908	−1.56322	−0.781610	+	0.623767i	$0.785601\pi$
$374$	0	0
$375$	−1.34357	−0.0693815
$376$	0	0
$377$	−11.6704	−0.601057
$378$	0	0
$379$	−15.9865	−0.821173	−0.410586	−	0.911822i	$-0.634676\pi$
$379$	−15.9865	−0.821173	−0.410586	+	0.911822i	$0.634676\pi$
$380$	0	0
$381$	3.81622	0.195511
$382$	0	0
$383$	−22.6961	−1.15971	−0.579857	−	0.814718i	$-0.696892\pi$
$383$	−22.6961	−1.15971	−0.579857	+	0.814718i	$0.696892\pi$
$384$	0	0
$385$	−33.4461	−1.70457
$386$	0	0
$387$	6.02938	0.306491
$388$	0	0
$389$	12.1956	0.618341	0.309170	−	0.951007i	$-0.399949\pi$
$389$	12.1956	0.618341	0.309170	+	0.951007i	$0.399949\pi$
$390$	0	0
$391$	50.3713	2.54738
$392$	0	0
$393$	7.50638	0.378647
$394$	0	0
$395$	47.8740	2.40880
$396$	0	0
$397$	10.7422	0.539137	0.269568	−	0.962981i	$-0.413119\pi$
$397$	10.7422	0.539137	0.269568	+	0.962981i	$0.413119\pi$
$398$	0	0
$399$	−1.01543	−0.0508351
$400$	0	0
$401$	−5.65983	−0.282639	−0.141319	−	0.989964i	$-0.545134\pi$
$401$	−5.65983	−0.282639	−0.141319	+	0.989964i	$0.545134\pi$
$402$	0	0
$403$	−17.4129	−0.867396
$404$	0	0
$405$	21.4926	1.06797
$406$	0	0
$407$	7.29497	0.361598
$408$	0	0
$409$	−27.6645	−1.36792	−0.683960	−	0.729520i	$-0.739744\pi$
$409$	−27.6645	−1.36792	−0.683960	+	0.729520i	$0.739744\pi$
$410$	0	0
$411$	7.43473	0.366728
$412$	0	0
$413$	8.62396	0.424357
$414$	0	0
$415$	26.1866	1.28545
$416$	0	0
$417$	2.01096	0.0984770
$418$	0	0
$419$	−6.91763	−0.337948	−0.168974	−	0.985620i	$-0.554045\pi$
$419$	−6.91763	−0.337948	−0.168974	+	0.985620i	$0.554045\pi$
$420$	0	0
$421$	−3.07015	−0.149630	−0.0748150	−	0.997197i	$-0.523837\pi$
$421$	−3.07015	−0.149630	−0.0748150	+	0.997197i	$0.523837\pi$
$422$	0	0
$423$	27.7684	1.35014
$424$	0	0
$425$	31.1051	1.50882
$426$	0	0
$427$	15.6302	0.756400
$428$	0	0
$429$	−12.0159	−0.580132
$430$	0	0
$431$	13.2015	0.635893	0.317946	−	0.948109i	$-0.397007\pi$
$431$	13.2015	0.635893	0.317946	+	0.948109i	$0.397007\pi$
$432$	0	0
$433$	−20.1544	−0.968559	−0.484279	−	0.874913i	$-0.660918\pi$
$433$	−20.1544	−0.968559	−0.484279	+	0.874913i	$0.660918\pi$
$434$	0	0
$435$	3.07700	0.147531
$436$	0	0
$437$	6.51613	0.311709
$438$	0	0
$439$	−1.72144	−0.0821601	−0.0410800	−	0.999156i	$-0.513080\pi$
$439$	−1.72144	−0.0821601	−0.0410800	+	0.999156i	$0.513080\pi$
$440$	0	0
$441$	5.82645	0.277450
$442$	0	0
$443$	38.0168	1.80623	0.903117	−	0.429395i	$-0.141273\pi$
$443$	38.0168	1.80623	0.903117	+	0.429395i	$0.141273\pi$
$444$	0	0
$445$	−11.4132	−0.541039
$446$	0	0
$447$	0.588950	0.0278564
$448$	0	0
$449$	−27.1861	−1.28299	−0.641496	−	0.767127i	$-0.721686\pi$
$449$	−27.1861	−1.28299	−0.641496	+	0.767127i	$0.721686\pi$
$450$	0	0
$451$	15.8984	0.748626
$452$	0	0
$453$	4.15702	0.195314
$454$	0	0
$455$	34.7536	1.62927
$456$	0	0
$457$	15.8069	0.739415	0.369708	−	0.929148i	$-0.379458\pi$
$457$	15.8069	0.739415	0.369708	+	0.929148i	$0.379458\pi$
$458$	0	0
$459$	−20.5074	−0.957204
$460$	0	0
$461$	29.7741	1.38672	0.693358	−	0.720593i	$-0.256130\pi$
$461$	29.7741	1.38672	0.693358	+	0.720593i	$0.256130\pi$
$462$	0	0
$463$	−11.8339	−0.549967	−0.274984	−	0.961449i	$-0.588672\pi$
$463$	−11.8339	−0.549967	−0.274984	+	0.961449i	$0.588672\pi$
$464$	0	0
$465$	4.59104	0.212904
$466$	0	0
$467$	30.6052	1.41624	0.708121	−	0.706091i	$-0.249543\pi$
$467$	30.6052	1.41624	0.708121	+	0.706091i	$0.249543\pi$
$468$	0	0
$469$	−29.4244	−1.35869
$470$	0	0
$471$	1.43710	0.0662181
$472$	0	0
$473$	−10.8565	−0.499184
$474$	0	0
$475$	4.02382	0.184625
$476$	0	0
$477$	18.6045	0.851842
$478$	0	0
$479$	−12.3908	−0.566150	−0.283075	−	0.959098i	$-0.591355\pi$
$479$	−12.3908	−0.566150	−0.283075	+	0.959098i	$0.591355\pi$
$480$	0	0
$481$	−7.58015	−0.345625
$482$	0	0
$483$	−6.61668	−0.301069
$484$	0	0
$485$	−11.8550	−0.538308
$486$	0	0
$487$	−7.61617	−0.345122	−0.172561	−	0.984999i	$-0.555204\pi$
$487$	−7.61617	−0.345122	−0.172561	+	0.984999i	$0.555204\pi$
$488$	0	0
$489$	6.85616	0.310046
$490$	0	0
$491$	−3.09291	−0.139581	−0.0697906	−	0.997562i	$-0.522233\pi$
$491$	−3.09291	−0.139581	−0.0697906	+	0.997562i	$0.522233\pi$
$492$	0	0
$493$	17.2819	0.778338
$494$	0	0
$495$	−42.1061	−1.89253
$496$	0	0
$497$	−12.6955	−0.569472
$498$	0	0
$499$	−6.55042	−0.293237	−0.146618	−	0.989193i	$-0.546839\pi$
$499$	−6.55042	−0.293237	−0.146618	+	0.989193i	$0.546839\pi$
$500$	0	0
$501$	−9.65835	−0.431503
$502$	0	0
$503$	−34.9332	−1.55760	−0.778798	−	0.627275i	$-0.784170\pi$
$503$	−34.9332	−1.55760	−0.778798	+	0.627275i	$0.784170\pi$
$504$	0	0
$505$	18.9858	0.844858
$506$	0	0
$507$	6.52928	0.289976
$508$	0	0
$509$	−10.6461	−0.471882	−0.235941	−	0.971767i	$-0.575817\pi$
$509$	−10.6461	−0.471882	−0.235941	+	0.971767i	$0.575817\pi$
$510$	0	0
$511$	13.0704	0.578199
$512$	0	0
$513$	−2.65288	−0.117128
$514$	0	0
$515$	−14.1547	−0.623732
$516$	0	0
$517$	−49.9999	−2.19899
$518$	0	0
$519$	−10.8958	−0.478274
$520$	0	0
$521$	−33.7087	−1.47681	−0.738403	−	0.674359i	$-0.764420\pi$
$521$	−33.7087	−1.47681	−0.738403	+	0.674359i	$0.764420\pi$
$522$	0	0
$523$	−3.61224	−0.157952	−0.0789761	−	0.996877i	$-0.525165\pi$
$523$	−3.61224	−0.157952	−0.0789761	+	0.996877i	$0.525165\pi$
$524$	0	0
$525$	−4.08591	−0.178324
$526$	0	0
$527$	25.7855	1.12323
$528$	0	0
$529$	19.4599	0.846084
$530$	0	0
$531$	10.8569	0.471149
$532$	0	0
$533$	−16.5199	−0.715556
$534$	0	0
$535$	55.7388	2.40980
$536$	0	0
$537$	6.55276	0.282772
$538$	0	0
$539$	−10.4911	−0.451885
$540$	0	0
$541$	45.1835	1.94259	0.971295	−	0.237876i	$-0.0764513\pi$
$541$	45.1835	1.94259	0.971295	+	0.237876i	$0.0764513\pi$
$542$	0	0
$543$	−6.36929	−0.273332
$544$	0	0
$545$	27.8166	1.19153
$546$	0	0
$547$	−25.9716	−1.11047	−0.555233	−	0.831695i	$-0.687371\pi$
$547$	−25.9716	−1.11047	−0.555233	+	0.831695i	$0.687371\pi$
$548$	0	0
$549$	19.6772	0.839804
$550$	0	0
$551$	2.23562	0.0952408
$552$	0	0
$553$	−35.3201	−1.50196
$554$	0	0
$555$	1.99857	0.0848344
$556$	0	0
$557$	27.5515	1.16739	0.583697	−	0.811972i	$-0.301606\pi$
$557$	27.5515	1.16739	0.583697	+	0.811972i	$0.301606\pi$
$558$	0	0
$559$	11.2809	0.477132
$560$	0	0
$561$	17.7935	0.751241
$562$	0	0
$563$	−14.8531	−0.625985	−0.312992	−	0.949756i	$-0.601331\pi$
$563$	−14.8531	−0.625985	−0.312992	+	0.949756i	$0.601331\pi$
$564$	0	0
$565$	−23.9532	−1.00772
$566$	0	0
$567$	−15.8566	−0.665915
$568$	0	0
$569$	−4.86281	−0.203859	−0.101930	−	0.994792i	$-0.532502\pi$
$569$	−4.86281	−0.203859	−0.101930	+	0.994792i	$0.532502\pi$
$570$	0	0
$571$	21.6960	0.907948	0.453974	−	0.891015i	$-0.350006\pi$
$571$	21.6960	0.907948	0.453974	+	0.891015i	$0.350006\pi$
$572$	0	0
$573$	−0.997669	−0.0416782
$574$	0	0
$575$	26.2197	1.09344
$576$	0	0
$577$	28.5083	1.18682	0.593409	−	0.804901i	$-0.297782\pi$
$577$	28.5083	1.18682	0.593409	+	0.804901i	$0.297782\pi$
$578$	0	0
$579$	12.2063	0.507277
$580$	0	0
$581$	−19.3197	−0.801517
$582$	0	0
$583$	−33.4993	−1.38740
$584$	0	0
$585$	43.7521	1.80893
$586$	0	0
$587$	−22.5133	−0.929224	−0.464612	−	0.885514i	$-0.653806\pi$
$587$	−22.5133	−0.929224	−0.464612	+	0.885514i	$0.653806\pi$
$588$	0	0
$589$	3.33566	0.137444
$590$	0	0
$591$	9.55298	0.392957
$592$	0	0
$593$	−11.0209	−0.452574	−0.226287	−	0.974061i	$-0.572659\pi$
$593$	−11.0209	−0.452574	−0.226287	+	0.974061i	$0.572659\pi$
$594$	0	0
$595$	−51.4642	−2.10983
$596$	0	0
$597$	1.94593	0.0796414
$598$	0	0
$599$	18.7308	0.765319	0.382659	−	0.923890i	$-0.375008\pi$
$599$	18.7308	0.765319	0.382659	+	0.923890i	$0.375008\pi$
$600$	0	0
$601$	14.8859	0.607209	0.303604	−	0.952798i	$-0.401810\pi$
$601$	14.8859	0.607209	0.303604	+	0.952798i	$0.401810\pi$
$602$	0	0
$603$	−37.0430	−1.50851
$604$	0	0
$605$	42.7727	1.73896
$606$	0	0
$607$	−37.9701	−1.54116	−0.770579	−	0.637345i	$-0.780033\pi$
$607$	−37.9701	−1.54116	−0.770579	+	0.637345i	$0.780033\pi$
$608$	0	0
$609$	−2.27012	−0.0919900
$610$	0	0
$611$	51.9544	2.10185
$612$	0	0
$613$	28.5379	1.15264	0.576318	−	0.817226i	$-0.304489\pi$
$613$	28.5379	1.15264	0.576318	+	0.817226i	$0.304489\pi$
$614$	0	0
$615$	4.35560	0.175635
$616$	0	0
$617$	3.91835	0.157747	0.0788733	−	0.996885i	$-0.474868\pi$
$617$	3.91835	0.157747	0.0788733	+	0.996885i	$0.474868\pi$
$618$	0	0
$619$	−5.37059	−0.215862	−0.107931	−	0.994158i	$-0.534423\pi$
$619$	−5.37059	−0.215862	−0.107931	+	0.994158i	$0.534423\pi$
$620$	0	0
$621$	−17.2865	−0.693684
$622$	0	0
$623$	8.42035	0.337354
$624$	0	0
$625$	−28.9280	−1.15712
$626$	0	0
$627$	2.30180	0.0919250
$628$	0	0
$629$	11.2249	0.447567
$630$	0	0
$631$	−18.6483	−0.742377	−0.371189	−	0.928557i	$-0.621050\pi$
$631$	−18.6483	−0.742377	−0.371189	+	0.928557i	$0.621050\pi$
$632$	0	0
$633$	−8.29913	−0.329861
$634$	0	0
$635$	25.0204	0.992905
$636$	0	0
$637$	10.9012	0.431923
$638$	0	0
$639$	−15.9827	−0.632264
$640$	0	0
$641$	42.2857	1.67018	0.835092	−	0.550111i	$-0.185415\pi$
$641$	42.2857	1.67018	0.835092	+	0.550111i	$0.185415\pi$
$642$	0	0
$643$	−18.4426	−0.727305	−0.363652	−	0.931535i	$-0.618470\pi$
$643$	−18.4426	−0.727305	−0.363652	+	0.931535i	$0.618470\pi$
$644$	0	0
$645$	−2.97431	−0.117113
$646$	0	0
$647$	−2.19012	−0.0861024	−0.0430512	−	0.999073i	$-0.513708\pi$
$647$	−2.19012	−0.0861024	−0.0430512	+	0.999073i	$0.513708\pi$
$648$	0	0
$649$	−19.5490	−0.767364
$650$	0	0
$651$	−3.38714	−0.132752
$652$	0	0
$653$	43.8208	1.71484	0.857420	−	0.514617i	$-0.172066\pi$
$653$	43.8208	1.71484	0.857420	+	0.514617i	$0.172066\pi$
$654$	0	0
$655$	49.2144	1.92296
$656$	0	0
$657$	16.4546	0.641954
$658$	0	0
$659$	−11.0297	−0.429657	−0.214829	−	0.976652i	$-0.568919\pi$
$659$	−11.0297	−0.429657	−0.214829	+	0.976652i	$0.568919\pi$
$660$	0	0
$661$	7.49137	0.291381	0.145690	−	0.989330i	$-0.453460\pi$
$661$	7.49137	0.291381	0.145690	+	0.989330i	$0.453460\pi$
$662$	0	0
$663$	−18.4890	−0.718055
$664$	0	0
$665$	−6.65751	−0.258167
$666$	0	0
$667$	14.5676	0.564060
$668$	0	0
$669$	1.70420	0.0658882
$670$	0	0
$671$	−35.4309	−1.36780
$672$	0	0
$673$	−34.6195	−1.33448	−0.667242	−	0.744841i	$-0.732525\pi$
$673$	−34.6195	−1.33448	−0.667242	+	0.744841i	$0.732525\pi$
$674$	0	0
$675$	−10.6747	−0.410870
$676$	0	0
$677$	−29.0201	−1.11533	−0.557666	−	0.830066i	$-0.688303\pi$
$677$	−29.0201	−1.11533	−0.557666	+	0.830066i	$0.688303\pi$
$678$	0	0
$679$	8.74629	0.335652
$680$	0	0
$681$	−8.17012	−0.313080
$682$	0	0
$683$	−13.3799	−0.511966	−0.255983	−	0.966681i	$-0.582399\pi$
$683$	−13.3799	−0.511966	−0.255983	+	0.966681i	$0.582399\pi$
$684$	0	0
$685$	48.7446	1.86244
$686$	0	0
$687$	6.19269	0.236266
$688$	0	0
$689$	34.8089	1.32611
$690$	0	0
$691$	−40.7415	−1.54988	−0.774940	−	0.632035i	$-0.782220\pi$
$691$	−40.7415	−1.54988	−0.774940	+	0.632035i	$0.782220\pi$
$692$	0	0
$693$	31.0646	1.18005
$694$	0	0
$695$	13.1845	0.500118
$696$	0	0
$697$	24.4632	0.926609
$698$	0	0
$699$	−8.47909	−0.320709
$700$	0	0
$701$	−43.7760	−1.65340	−0.826698	−	0.562645i	$-0.809784\pi$
$701$	−43.7760	−1.65340	−0.826698	+	0.562645i	$0.809784\pi$
$702$	0	0
$703$	1.45208	0.0547662
$704$	0	0
$705$	−13.6982	−0.515904
$706$	0	0
$707$	−14.0072	−0.526795
$708$	0	0
$709$	26.0155	0.977031	0.488516	−	0.872555i	$-0.337538\pi$
$709$	26.0155	0.977031	0.488516	+	0.872555i	$0.337538\pi$
$710$	0	0
$711$	−44.4652	−1.66758
$712$	0	0
$713$	21.7356	0.814005
$714$	0	0
$715$	−78.7801	−2.94621
$716$	0	0
$717$	−5.35490	−0.199982
$718$	0	0
$719$	31.3051	1.16748	0.583741	−	0.811940i	$-0.301588\pi$
$719$	31.3051	1.16748	0.583741	+	0.811940i	$0.301588\pi$
$720$	0	0
$721$	10.4430	0.388916
$722$	0	0
$723$	4.70552	0.175000
$724$	0	0
$725$	8.99574	0.334093
$726$	0	0
$727$	−7.31403	−0.271262	−0.135631	−	0.990759i	$-0.543306\pi$
$727$	−7.31403	−0.271262	−0.135631	+	0.990759i	$0.543306\pi$
$728$	0	0
$729$	−16.3158	−0.604287
$730$	0	0
$731$	−16.7051	−0.617862
$732$	0	0
$733$	−13.0935	−0.483620	−0.241810	−	0.970324i	$-0.577741\pi$
$733$	−13.0935	−0.483620	−0.241810	+	0.970324i	$0.577741\pi$
$734$	0	0
$735$	−2.87420	−0.106016
$736$	0	0
$737$	66.6997	2.45692
$738$	0	0
$739$	−10.5782	−0.389124	−0.194562	−	0.980890i	$-0.562328\pi$
$739$	−10.5782	−0.389124	−0.194562	+	0.980890i	$0.562328\pi$
$740$	0	0
$741$	−2.39178	−0.0878642
$742$	0	0
$743$	−23.6180	−0.866460	−0.433230	−	0.901283i	$-0.642626\pi$
$743$	−23.6180	−0.866460	−0.433230	+	0.901283i	$0.642626\pi$
$744$	0	0
$745$	3.86136	0.141469
$746$	0	0
$747$	−24.3220	−0.889896
$748$	0	0
$749$	−41.1225	−1.50258
$750$	0	0
$751$	24.5994	0.897644	0.448822	−	0.893621i	$-0.351844\pi$
$751$	24.5994	0.897644	0.448822	+	0.893621i	$0.351844\pi$
$752$	0	0
$753$	11.5032	0.419199
$754$	0	0
$755$	27.2548	0.991905
$756$	0	0
$757$	3.03483	0.110303	0.0551514	−	0.998478i	$-0.482436\pi$
$757$	3.03483	0.110303	0.0551514	+	0.998478i	$0.482436\pi$
$758$	0	0
$759$	14.9988	0.544423
$760$	0	0
$761$	−8.15330	−0.295557	−0.147778	−	0.989020i	$-0.547212\pi$
$761$	−8.15330	−0.295557	−0.147778	+	0.989020i	$0.547212\pi$
$762$	0	0
$763$	−20.5223	−0.742956
$764$	0	0
$765$	−64.7894	−2.34247
$766$	0	0
$767$	20.3132	0.733466
$768$	0	0
$769$	−14.2744	−0.514747	−0.257374	−	0.966312i	$-0.582857\pi$
$769$	−14.2744	−0.514747	−0.257374	+	0.966312i	$0.582857\pi$
$770$	0	0
$771$	5.76868	0.207754
$772$	0	0
$773$	−45.2404	−1.62718	−0.813591	−	0.581437i	$-0.802491\pi$
$773$	−45.2404	−1.62718	−0.813591	+	0.581437i	$0.802491\pi$
$774$	0	0
$775$	13.4221	0.482136
$776$	0	0
$777$	−1.47448	−0.0528968
$778$	0	0
$779$	3.16461	0.113384
$780$	0	0
$781$	28.7784	1.02977
$782$	0	0
$783$	−5.93084	−0.211951
$784$	0	0
$785$	9.42212	0.336290
$786$	0	0
$787$	50.1656	1.78821	0.894106	−	0.447856i	$-0.147812\pi$
$787$	50.1656	1.78821	0.894106	+	0.447856i	$0.147812\pi$
$788$	0	0
$789$	10.4349	0.371494
$790$	0	0
$791$	17.6720	0.628344
$792$	0	0
$793$	36.8160	1.30737
$794$	0	0
$795$	−9.17764	−0.325497
$796$	0	0
$797$	−21.7396	−0.770056	−0.385028	−	0.922905i	$-0.625808\pi$
$797$	−21.7396	−0.770056	−0.385028	+	0.922905i	$0.625808\pi$
$798$	0	0
$799$	−76.9357	−2.72179
$800$	0	0
$801$	10.6006	0.374552
$802$	0	0
$803$	−29.6281	−1.04555
$804$	0	0
$805$	−43.3812	−1.52899
$806$	0	0
$807$	7.80144	0.274624
$808$	0	0
$809$	−9.48695	−0.333543	−0.166772	−	0.985996i	$-0.553334\pi$
$809$	−9.48695	−0.333543	−0.166772	+	0.985996i	$0.553334\pi$
$810$	0	0
$811$	−22.6529	−0.795450	−0.397725	−	0.917505i	$-0.630200\pi$
$811$	−22.6529	−0.795450	−0.397725	+	0.917505i	$0.630200\pi$
$812$	0	0
$813$	−3.75518	−0.131700
$814$	0	0
$815$	44.9513	1.57458
$816$	0	0
$817$	−2.16101	−0.0756042
$818$	0	0
$819$	−32.2790	−1.12792
$820$	0	0
$821$	26.4552	0.923292	0.461646	−	0.887064i	$-0.347259\pi$
$821$	26.4552	0.923292	0.461646	+	0.887064i	$0.347259\pi$
$822$	0	0
$823$	35.1129	1.22396	0.611980	−	0.790873i	$-0.290373\pi$
$823$	35.1129	1.22396	0.611980	+	0.790873i	$0.290373\pi$
$824$	0	0
$825$	9.26202	0.322462
$826$	0	0
$827$	27.4003	0.952803	0.476402	−	0.879228i	$-0.341941\pi$
$827$	27.4003	0.952803	0.476402	+	0.879228i	$0.341941\pi$
$828$	0	0
$829$	25.2213	0.875972	0.437986	−	0.898982i	$-0.355692\pi$
$829$	25.2213	0.875972	0.437986	+	0.898982i	$0.355692\pi$
$830$	0	0
$831$	−8.57588	−0.297494
$832$	0	0
$833$	−16.1429	−0.559319
$834$	0	0
$835$	−63.3234	−2.19140
$836$	0	0
$837$	−8.84912	−0.305870
$838$	0	0
$839$	36.8869	1.27348	0.636738	−	0.771080i	$-0.280283\pi$
$839$	36.8869	1.27348	0.636738	+	0.771080i	$0.280283\pi$
$840$	0	0
$841$	−24.0020	−0.827655
$842$	0	0
$843$	0.0866488	0.00298434
$844$	0	0
$845$	42.8082	1.47265
$846$	0	0
$847$	−31.5565	−1.08429
$848$	0	0
$849$	4.87303	0.167242
$850$	0	0
$851$	9.46193	0.324351
$852$	0	0
$853$	3.36746	0.115300	0.0576499	−	0.998337i	$-0.481639\pi$
$853$	3.36746	0.115300	0.0576499	+	0.998337i	$0.481639\pi$
$854$	0	0
$855$	−8.38129	−0.286634
$856$	0	0
$857$	−16.0416	−0.547970	−0.273985	−	0.961734i	$-0.588342\pi$
$857$	−16.0416	−0.547970	−0.273985	+	0.961734i	$0.588342\pi$
$858$	0	0
$859$	14.7310	0.502614	0.251307	−	0.967907i	$-0.419140\pi$
$859$	14.7310	0.502614	0.251307	+	0.967907i	$0.419140\pi$
$860$	0	0
$861$	−3.21344	−0.109514
$862$	0	0
$863$	−41.3034	−1.40598	−0.702991	−	0.711198i	$-0.748153\pi$
$863$	−41.3034	−1.40598	−0.702991	+	0.711198i	$0.748153\pi$
$864$	0	0
$865$	−71.4368	−2.42892
$866$	0	0
$867$	19.5901	0.665316
$868$	0	0
$869$	80.0642	2.71599
$870$	0	0
$871$	−69.3071	−2.34838
$872$	0	0
$873$	11.0109	0.372662
$874$	0	0
$875$	6.49895	0.219704
$876$	0	0
$877$	−3.84438	−0.129815	−0.0649077	−	0.997891i	$-0.520675\pi$
$877$	−3.84438	−0.129815	−0.0649077	+	0.997891i	$0.520675\pi$
$878$	0	0
$879$	−12.6807	−0.427711
$880$	0	0
$881$	−4.57677	−0.154195	−0.0770976	−	0.997024i	$-0.524565\pi$
$881$	−4.57677	−0.154195	−0.0770976	+	0.997024i	$0.524565\pi$
$882$	0	0
$883$	−2.34140	−0.0787945	−0.0393972	−	0.999224i	$-0.512544\pi$
$883$	−2.34140	−0.0787945	−0.0393972	+	0.999224i	$0.512544\pi$
$884$	0	0
$885$	−5.35573	−0.180031
$886$	0	0
$887$	−48.1958	−1.61826	−0.809129	−	0.587631i	$-0.800061\pi$
$887$	−48.1958	−1.61826	−0.809129	+	0.587631i	$0.800061\pi$
$888$	0	0
$889$	−18.4593	−0.619106
$890$	0	0
$891$	35.9441	1.20417
$892$	0	0
$893$	−9.95256	−0.333050
$894$	0	0
$895$	42.9621	1.43607
$896$	0	0
$897$	−15.5851	−0.520373
$898$	0	0
$899$	7.45729	0.248714
$900$	0	0
$901$	−51.5461	−1.71725
$902$	0	0
$903$	2.19436	0.0730237
$904$	0	0
$905$	−41.7593	−1.38812
$906$	0	0
$907$	−4.08133	−0.135518	−0.0677592	−	0.997702i	$-0.521585\pi$
$907$	−4.08133	−0.135518	−0.0677592	+	0.997702i	$0.521585\pi$
$908$	0	0
$909$	−17.6340	−0.584881
$910$	0	0
$911$	−54.1174	−1.79299	−0.896495	−	0.443054i	$-0.853895\pi$
$911$	−54.1174	−1.79299	−0.896495	+	0.443054i	$0.853895\pi$
$912$	0	0
$913$	43.7943	1.44938
$914$	0	0
$915$	−9.70682	−0.320898
$916$	0	0
$917$	−36.3089	−1.19903
$918$	0	0
$919$	−32.1252	−1.05971	−0.529857	−	0.848087i	$-0.677754\pi$
$919$	−32.1252	−1.05971	−0.529857	+	0.848087i	$0.677754\pi$
$920$	0	0
$921$	3.73280	0.123000
$922$	0	0
$923$	−29.9034	−0.984284
$924$	0	0
$925$	5.84290	0.192113
$926$	0	0
$927$	13.1469	0.431800
$928$	0	0
$929$	−10.9339	−0.358730	−0.179365	−	0.983783i	$-0.557404\pi$
$929$	−10.9339	−0.358730	−0.179365	+	0.983783i	$0.557404\pi$
$930$	0	0
$931$	−2.08828	−0.0684406
$932$	0	0
$933$	−5.09875	−0.166926
$934$	0	0
$935$	116.660	3.81519
$936$	0	0
$937$	−46.5172	−1.51965	−0.759825	−	0.650127i	$-0.774716\pi$
$937$	−46.5172	−1.51965	−0.759825	+	0.650127i	$0.774716\pi$
$938$	0	0
$939$	−4.70307	−0.153479
$940$	0	0
$941$	−39.9678	−1.30291	−0.651457	−	0.758686i	$-0.725842\pi$
$941$	−39.9678	−1.30291	−0.651457	+	0.758686i	$0.725842\pi$
$942$	0	0
$943$	20.6210	0.671511
$944$	0	0
$945$	17.6616	0.574531
$946$	0	0
$947$	33.7011	1.09514	0.547569	−	0.836761i	$-0.315553\pi$
$947$	33.7011	1.09514	0.547569	+	0.836761i	$0.315553\pi$
$948$	0	0
$949$	30.7864	0.999368
$950$	0	0
$951$	3.27535	0.106211
$952$	0	0
$953$	−30.1472	−0.976564	−0.488282	−	0.872686i	$-0.662376\pi$
$953$	−30.1472	−0.976564	−0.488282	+	0.872686i	$0.662376\pi$
$954$	0	0
$955$	−6.54106	−0.211664
$956$	0	0
$957$	5.14596	0.166345
$958$	0	0
$959$	−35.9624	−1.16129
$960$	0	0
$961$	−19.8733	−0.641076
$962$	0	0
$963$	−51.7700	−1.66826
$964$	0	0
$965$	80.0287	2.57622
$966$	0	0
$967$	−35.1010	−1.12877	−0.564385	−	0.825511i	$-0.690887\pi$
$967$	−35.1010	−1.12877	−0.564385	+	0.825511i	$0.690887\pi$
$968$	0	0
$969$	3.54182	0.113780
$970$	0	0
$971$	−38.1884	−1.22552	−0.612762	−	0.790268i	$-0.709941\pi$
$971$	−38.1884	−1.22552	−0.612762	+	0.790268i	$0.709941\pi$
$972$	0	0
$973$	−9.72716	−0.311839
$974$	0	0
$975$	−9.62409	−0.308217
$976$	0	0
$977$	22.6596	0.724946	0.362473	−	0.931994i	$-0.381933\pi$
$977$	22.6596	0.724946	0.362473	+	0.931994i	$0.381933\pi$
$978$	0	0
$979$	−19.0874	−0.610036
$980$	0	0
$981$	−25.8359	−0.824878
$982$	0	0
$983$	−1.47180	−0.0469430	−0.0234715	−	0.999725i	$-0.507472\pi$
$983$	−1.47180	−0.0469430	−0.0234715	+	0.999725i	$0.507472\pi$
$984$	0	0
$985$	62.6326	1.99564
$986$	0	0
$987$	10.1061	0.321682
$988$	0	0
$989$	−14.0814	−0.447763
$990$	0	0
$991$	30.6228	0.972767	0.486383	−	0.873746i	$-0.338316\pi$
$991$	30.6228	0.972767	0.486383	+	0.873746i	$0.338316\pi$
$992$	0	0
$993$	−13.0292	−0.413469
$994$	0	0
$995$	12.7582	0.404461
$996$	0	0
$997$	−7.61232	−0.241085	−0.120542	−	0.992708i	$-0.538463\pi$
$997$	−7.61232	−0.241085	−0.120542	+	0.992708i	$0.538463\pi$
$998$	0	0
$999$	−3.85219	−0.121878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4864.2.a.bt.1.6		10
4.3	odd	2		4864.2.a.bs.1.6		10
8.3	odd	2	inner	4864.2.a.bt.1.5		10
8.5	even	2		4864.2.a.bs.1.5		10
16.3	odd	4		2432.2.c.j.1217.9	✓	20
16.5	even	4		2432.2.c.j.1217.10	yes	20
16.11	odd	4		2432.2.c.j.1217.12	yes	20
16.13	even	4		2432.2.c.j.1217.11	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.j.1217.9	✓	20	16.3	odd	4
2432.2.c.j.1217.10	yes	20	16.5	even	4
2432.2.c.j.1217.11	yes	20	16.13	even	4
2432.2.c.j.1217.12	yes	20	16.11	odd	4
4864.2.a.bs.1.5		10	8.5	even	2
4864.2.a.bs.1.6		10	4.3	odd	2
4864.2.a.bt.1.5		10	8.3	odd	2	inner
4864.2.a.bt.1.6		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4864 = 2^{8} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4864.a (trivial)

\(f(q)\)	\(=\)	\(q+0.458177 q^{3} +3.00397 q^{5} -2.21624 q^{7} -2.79007 q^{9} +O(q^{10})\)	\(q+0.458177 q^{3} +3.00397 q^{5} -2.21624 q^{7} -2.79007 q^{9} +5.02382 q^{11} -5.22021 q^{13} +1.37635 q^{15} +7.73025 q^{17} +1.00000 q^{19} -1.01543 q^{21} +6.51613 q^{23} +4.02382 q^{25} -2.65288 q^{27} +2.23562 q^{29} +3.33566 q^{31} +2.30180 q^{33} -6.65751 q^{35} +1.45208 q^{37} -2.39178 q^{39} +3.16461 q^{41} -2.16101 q^{43} -8.38129 q^{45} -9.95256 q^{47} -2.08828 q^{49} +3.54182 q^{51} -6.66810 q^{53} +15.0914 q^{55} +0.458177 q^{57} -3.89126 q^{59} -7.05259 q^{61} +6.18347 q^{63} -15.6813 q^{65} +13.2767 q^{67} +2.98554 q^{69} +5.72840 q^{71} -5.89754 q^{73} +1.84362 q^{75} -11.1340 q^{77} +15.9369 q^{79} +7.15473 q^{81} +8.71734 q^{83} +23.2214 q^{85} +1.02431 q^{87} -3.79938 q^{89} +11.5692 q^{91} +1.52833 q^{93} +3.00397 q^{95} -3.94645 q^{97} -14.0168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{3} + 14 q^{9}+O(q^{10}) \) 10 * q + 4 * q^3 + 14 * q^9	\( 10 q + 4 q^{3} + 14 q^{9} + 20 q^{11} + 4 q^{17} + 10 q^{19} + 10 q^{25} + 28 q^{27} - 8 q^{33} + 36 q^{35} - 12 q^{41} - 4 q^{43} + 26 q^{49} + 36 q^{51} + 4 q^{57} + 52 q^{59} - 24 q^{65} + 12 q^{67} + 12 q^{73} - 12 q^{75} + 34 q^{81} + 16 q^{83} - 20 q^{89} + 60 q^{91} - 28 q^{97} + 60 q^{99}+O(q^{100}) \) 10 * q + 4 * q^3 + 14 * q^9 + 20 * q^11 + 4 * q^17 + 10 * q^19 + 10 * q^25 + 28 * q^27 - 8 * q^33 + 36 * q^35 - 12 * q^41 - 4 * q^43 + 26 * q^49 + 36 * q^51 + 4 * q^57 + 52 * q^59 - 24 * q^65 + 12 * q^67 + 12 * q^73 - 12 * q^75 + 34 * q^81 + 16 * q^83 - 20 * q^89 + 60 * q^91 - 28 * q^97 + 60 * q^99

Introduction

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