LMFDB - Embedded newform 592.2.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,2,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, 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("592.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	592.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:	$4.72714379966$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.48389.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 10 $ x^4 - 2x^3 - 8x^2 + 3*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.23838$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.238381 q^{3} +3.65512 q^{5} -2.28805 q^{7} -2.94317 q^{9} +O(q^{10})$	$q+0.238381 q^{3} +3.65512 q^{5} -2.28805 q^{7} -2.94317 q^{9} +3.94317 q^{11} +4.52643 q^{13} +0.871312 q^{15} +2.00000 q^{17} +2.83348 q^{19} -0.545429 q^{21} -5.83668 q^{23} +8.35992 q^{25} -1.41674 q^{27} -0.0496716 q^{29} +2.92098 q^{31} +0.939978 q^{33} -8.36311 q^{35} +1.00000 q^{37} +1.07902 q^{39} -10.1247 q^{41} +3.31024 q^{43} -10.7577 q^{45} +13.1215 q^{47} -1.76481 q^{49} +0.476762 q^{51} -0.188710 q^{53} +14.4128 q^{55} +0.675448 q^{57} -10.7198 q^{59} +6.96537 q^{61} +6.73414 q^{63} +16.5447 q^{65} +1.17836 q^{67} -1.39135 q^{69} +14.5518 q^{71} -12.5637 q^{73} +1.99285 q^{75} -9.02219 q^{77} -14.7902 q^{79} +8.49180 q^{81} +2.38740 q^{83} +7.31024 q^{85} -0.0118408 q^{87} -13.4096 q^{89} -10.3567 q^{91} +0.696307 q^{93} +10.3567 q^{95} -1.62258 q^{97} -11.6055 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 5 * q^5 + q^7 + 8 * q^9	$ 4 q - 2 q^{3} + 5 q^{5} + q^{7} + 8 q^{9} - 4 q^{11} + 5 q^{13} + 8 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 7 q^{25} + q^{27} + 7 q^{29} + q^{31} + 3 q^{33} + 12 q^{35} + 4 q^{37} + 15 q^{39} + 2 q^{41} - 6 q^{43} + 29 q^{47} + 9 q^{49} - 4 q^{51} - 5 q^{53} + 5 q^{55} - 32 q^{57} + 10 q^{59} - q^{61} + 28 q^{63} - 2 q^{65} + q^{67} - 27 q^{69} + 17 q^{71} + 8 q^{73} - 19 q^{75} - 27 q^{77} - 15 q^{79} - 8 q^{81} - 15 q^{83} + 10 q^{85} + 17 q^{87} - 20 q^{89} - 34 q^{91} - 6 q^{93} + 34 q^{95} + 2 q^{97} - 44 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 5 * q^5 + q^7 + 8 * q^9 - 4 * q^11 + 5 * q^13 + 8 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 7 * q^25 + q^27 + 7 * q^29 + q^31 + 3 * q^33 + 12 * q^35 + 4 * q^37 + 15 * q^39 + 2 * q^41 - 6 * q^43 + 29 * q^47 + 9 * q^49 - 4 * q^51 - 5 * q^53 + 5 * q^55 - 32 * q^57 + 10 * q^59 - q^61 + 28 * q^63 - 2 * q^65 + q^67 - 27 * q^69 + 17 * q^71 + 8 * q^73 - 19 * q^75 - 27 * q^77 - 15 * q^79 - 8 * q^81 - 15 * q^83 + 10 * q^85 + 17 * q^87 - 20 * q^89 - 34 * q^91 - 6 * q^93 + 34 * q^95 + 2 * q^97 - 44 * 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
$3$	0.238381	0.137629	0.0688147	−	0.997629i	$-0.478078\pi$
$3$	0.238381	0.137629	0.0688147	+	0.997629i	$0.478078\pi$
$4$	0	0
$5$	3.65512	1.63462	0.817310	−	0.576198i	$-0.195464\pi$
$5$	3.65512	1.63462	0.817310	+	0.576198i	$0.195464\pi$
$6$	0	0
$7$	−2.28805	−0.864803	−0.432401	−	0.901681i	$-0.642334\pi$
$7$	−2.28805	−0.864803	−0.432401	+	0.901681i	$0.642334\pi$
$8$	0	0
$9$	−2.94317	−0.981058
$10$	0	0
$11$	3.94317	1.18891	0.594456	−	0.804128i	$-0.297367\pi$
$11$	3.94317	1.18891	0.594456	+	0.804128i	$0.297367\pi$
$12$	0	0
$13$	4.52643	1.25541	0.627703	−	0.778453i	$-0.283995\pi$
$13$	4.52643	1.25541	0.627703	+	0.778453i	$0.283995\pi$
$14$	0	0
$15$	0.871312	0.224972
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.83348	0.650045	0.325023	−	0.945706i	$-0.394628\pi$
$19$	2.83348	0.650045	0.325023	+	0.945706i	$0.394628\pi$
$20$	0	0
$21$	−0.545429	−0.119022
$22$	0	0
$23$	−5.83668	−1.21703	−0.608516	−	0.793542i	$-0.708235\pi$
$23$	−5.83668	−1.21703	−0.608516	+	0.793542i	$0.708235\pi$
$24$	0	0
$25$	8.35992	1.67198
$26$	0	0
$27$	−1.41674	−0.272652
$28$	0	0
$29$	−0.0496716	−0.00922378	−0.00461189	−	0.999989i	$-0.501468\pi$
$29$	−0.0496716	−0.00922378	−0.00461189	+	0.999989i	$0.501468\pi$
$30$	0	0
$31$	2.92098	0.524624	0.262312	−	0.964983i	$-0.415515\pi$
$31$	2.92098	0.524624	0.262312	+	0.964983i	$0.415515\pi$
$32$	0	0
$33$	0.939978	0.163629
$34$	0	0
$35$	−8.36311	−1.41362
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	1.07902	0.172781
$40$	0	0
$41$	−10.1247	−1.58122	−0.790609	−	0.612322i	$-0.790236\pi$
$41$	−10.1247	−1.58122	−0.790609	+	0.612322i	$0.790236\pi$
$42$	0	0
$43$	3.31024	0.504807	0.252404	−	0.967622i	$-0.418779\pi$
$43$	3.31024	0.504807	0.252404	+	0.967622i	$0.418779\pi$
$44$	0	0
$45$	−10.7577	−1.60366
$46$	0	0
$47$	13.1215	1.91397	0.956986	−	0.290133i	$-0.0936997\pi$
$47$	13.1215	1.91397	0.956986	+	0.290133i	$0.0936997\pi$
$48$	0	0
$49$	−1.76481	−0.252116
$50$	0	0
$51$	0.476762	0.0667601
$52$	0	0
$53$	−0.188710	−0.0259213	−0.0129606	−	0.999916i	$-0.504126\pi$
$53$	−0.188710	−0.0259213	−0.0129606	+	0.999916i	$0.504126\pi$
$54$	0	0
$55$	14.4128	1.94342
$56$	0	0
$57$	0.675448	0.0894653
$58$	0	0
$59$	−10.7198	−1.39560	−0.697802	−	0.716291i	$-0.745838\pi$
$59$	−10.7198	−1.39560	−0.697802	+	0.716291i	$0.745838\pi$
$60$	0	0
$61$	6.96537	0.891824	0.445912	−	0.895077i	$-0.352879\pi$
$61$	6.96537	0.891824	0.445912	+	0.895077i	$0.352879\pi$
$62$	0	0
$63$	6.73414	0.848422
$64$	0	0
$65$	16.5447	2.05211
$66$	0	0
$67$	1.17836	0.143960	0.0719798	−	0.997406i	$-0.477068\pi$
$67$	1.17836	0.143960	0.0719798	+	0.997406i	$0.477068\pi$
$68$	0	0
$69$	−1.39135	−0.167499
$70$	0	0
$71$	14.5518	1.72698	0.863492	−	0.504363i	$-0.168273\pi$
$71$	14.5518	1.72698	0.863492	+	0.504363i	$0.168273\pi$
$72$	0	0
$73$	−12.5637	−1.47047	−0.735233	−	0.677815i	$-0.762927\pi$
$73$	−12.5637	−1.47047	−0.735233	+	0.677815i	$0.762927\pi$
$74$	0	0
$75$	1.99285	0.230114
$76$	0	0
$77$	−9.02219	−1.02817
$78$	0	0
$79$	−14.7902	−1.66403	−0.832014	−	0.554755i	$-0.812812\pi$
$79$	−14.7902	−1.66403	−0.832014	+	0.554755i	$0.812812\pi$
$80$	0	0
$81$	8.49180	0.943533
$82$	0	0
$83$	2.38740	0.262051	0.131025	−	0.991379i	$-0.458173\pi$
$83$	2.38740	0.262051	0.131025	+	0.991379i	$0.458173\pi$
$84$	0	0
$85$	7.31024	0.792907
$86$	0	0
$87$	−0.0118408	−0.00126946
$88$	0	0
$89$	−13.4096	−1.42141	−0.710707	−	0.703488i	$-0.751625\pi$
$89$	−13.4096	−1.42141	−0.710707	+	0.703488i	$0.751625\pi$
$90$	0	0
$91$	−10.3567	−1.08568
$92$	0	0
$93$	0.696307	0.0722037
$94$	0	0
$95$	10.3567	1.06258
$96$	0	0
$97$	−1.62258	−0.164748	−0.0823741	−	0.996601i	$-0.526250\pi$
$97$	−1.62258	−0.164748	−0.0823741	+	0.996601i	$0.526250\pi$
$98$	0	0
$99$	−11.6055	−1.16639
$100$	0	0
$101$	−11.3853	−1.13288	−0.566440	−	0.824103i	$-0.691680\pi$
$101$	−11.3853	−1.13288	−0.566440	+	0.824103i	$0.691680\pi$
$102$	0	0
$103$	−14.2312	−1.40224	−0.701122	−	0.713041i	$-0.747317\pi$
$103$	−14.2312	−1.40224	−0.701122	+	0.713041i	$0.747317\pi$
$104$	0	0
$105$	−1.99361	−0.194556
$106$	0	0
$107$	−2.04967	−0.198149	−0.0990746	−	0.995080i	$-0.531588\pi$
$107$	−2.04967	−0.198149	−0.0990746	+	0.995080i	$0.531588\pi$
$108$	0	0
$109$	−7.66696	−0.734362	−0.367181	−	0.930150i	$-0.619677\pi$
$109$	−7.66696	−0.734362	−0.367181	+	0.930150i	$0.619677\pi$
$110$	0	0
$111$	0.238381	0.0226261
$112$	0	0
$113$	1.04648	0.0984441	0.0492221	−	0.998788i	$-0.484326\pi$
$113$	1.04648	0.0984441	0.0492221	+	0.998788i	$0.484326\pi$
$114$	0	0
$115$	−21.3338	−1.98938
$116$	0	0
$117$	−13.3221	−1.23163
$118$	0	0
$119$	−4.57611	−0.419491
$120$	0	0
$121$	4.54862	0.413511
$122$	0	0
$123$	−2.41354	−0.217622
$124$	0	0
$125$	12.2809	1.09844
$126$	0	0
$127$	−0.0686663	−0.00609315	−0.00304658	−	0.999995i	$-0.500970\pi$
$127$	−0.0686663	−0.00609315	−0.00304658	+	0.999995i	$0.500970\pi$
$128$	0	0
$129$	0.789100	0.0694763
$130$	0	0
$131$	6.14372	0.536780	0.268390	−	0.963310i	$-0.413508\pi$
$131$	6.14372	0.536780	0.268390	+	0.963310i	$0.413508\pi$
$132$	0	0
$133$	−6.48315	−0.562161
$134$	0	0
$135$	−5.17836	−0.445682
$136$	0	0
$137$	−21.2841	−1.81842	−0.909211	−	0.416335i	$-0.863314\pi$
$137$	−21.2841	−1.81842	−0.909211	+	0.416335i	$0.863314\pi$
$138$	0	0
$139$	−20.5708	−1.74479	−0.872397	−	0.488798i	$-0.837435\pi$
$139$	−20.5708	−1.74479	−0.872397	+	0.488798i	$0.837435\pi$
$140$	0	0
$141$	3.12793	0.263419
$142$	0	0
$143$	17.8485	1.49257
$144$	0	0
$145$	−0.181556	−0.0150774
$146$	0	0
$147$	−0.420699	−0.0346986
$148$	0	0
$149$	16.6512	1.36412	0.682058	−	0.731298i	$-0.261085\pi$
$149$	16.6512	1.36412	0.682058	+	0.731298i	$0.261085\pi$
$150$	0	0
$151$	4.26377	0.346981	0.173490	−	0.984836i	$-0.444495\pi$
$151$	4.26377	0.346981	0.173490	+	0.984836i	$0.444495\pi$
$152$	0	0
$153$	−5.88635	−0.475883
$154$	0	0
$155$	10.6766	0.857561
$156$	0	0
$157$	18.0544	1.44089	0.720447	−	0.693510i	$-0.243937\pi$
$157$	18.0544	1.44089	0.720447	+	0.693510i	$0.243937\pi$
$158$	0	0
$159$	−0.0449848	−0.00356753
$160$	0	0
$161$	13.3546	1.05249
$162$	0	0
$163$	−16.3631	−1.28166	−0.640829	−	0.767684i	$-0.721409\pi$
$163$	−16.3631	−1.28166	−0.640829	+	0.767684i	$0.721409\pi$
$164$	0	0
$165$	3.43574	0.267472
$166$	0	0
$167$	14.3449	1.11004	0.555020	−	0.831837i	$-0.312711\pi$
$167$	14.3449	1.11004	0.555020	+	0.831837i	$0.312711\pi$
$168$	0	0
$169$	7.48860	0.576046
$170$	0	0
$171$	−8.33943	−0.637732
$172$	0	0
$173$	6.85777	0.521386	0.260693	−	0.965422i	$-0.416049\pi$
$173$	6.85777	0.521386	0.260693	+	0.965422i	$0.416049\pi$
$174$	0	0
$175$	−19.1279	−1.44594
$176$	0	0
$177$	−2.55541	−0.192076
$178$	0	0
$179$	1.26586	0.0946150	0.0473075	−	0.998880i	$-0.484936\pi$
$179$	1.26586	0.0946150	0.0473075	+	0.998880i	$0.484936\pi$
$180$	0	0
$181$	0.600390	0.0446266	0.0223133	−	0.999751i	$-0.492897\pi$
$181$	0.600390	0.0446266	0.0223133	+	0.999751i	$0.492897\pi$
$182$	0	0
$183$	1.66041	0.122741
$184$	0	0
$185$	3.65512	0.268730
$186$	0	0
$187$	7.88635	0.576707
$188$	0	0
$189$	3.24158	0.235790
$190$	0	0
$191$	−26.5187	−1.91882	−0.959412	−	0.282008i	$-0.909000\pi$
$191$	−26.5187	−1.91882	−0.959412	+	0.282008i	$0.909000\pi$
$192$	0	0
$193$	8.31873	0.598795	0.299398	−	0.954128i	$-0.403214\pi$
$193$	8.31873	0.598795	0.299398	+	0.954128i	$0.403214\pi$
$194$	0	0
$195$	3.94394	0.282431
$196$	0	0
$197$	−5.71834	−0.407415	−0.203707	−	0.979032i	$-0.565299\pi$
$197$	−5.71834	−0.407415	−0.203707	+	0.979032i	$0.565299\pi$
$198$	0	0
$199$	−0.803408	−0.0569521	−0.0284760	−	0.999594i	$-0.509065\pi$
$199$	−0.803408	−0.0569521	−0.0284760	+	0.999594i	$0.509065\pi$
$200$	0	0
$201$	0.280899	0.0198131
$202$	0	0
$203$	0.113651	0.00797675
$204$	0	0
$205$	−37.0071	−2.58469
$206$	0	0
$207$	17.1784	1.19398
$208$	0	0
$209$	11.1729	0.772846
$210$	0	0
$211$	9.58646	0.659959	0.329979	−	0.943988i	$-0.392958\pi$
$211$	9.58646	0.659959	0.329979	+	0.943988i	$0.392958\pi$
$212$	0	0
$213$	3.46888	0.237684
$214$	0	0
$215$	12.0993	0.825168
$216$	0	0
$217$	−6.68336	−0.453696
$218$	0	0
$219$	−2.99494	−0.202379
$220$	0	0
$221$	9.05287	0.608962
$222$	0	0
$223$	−25.2717	−1.69231	−0.846157	−	0.532933i	$-0.821090\pi$
$223$	−25.2717	−1.69231	−0.846157	+	0.532933i	$0.821090\pi$
$224$	0	0
$225$	−24.6047	−1.64031
$226$	0	0
$227$	−18.8843	−1.25339	−0.626696	−	0.779264i	$-0.715593\pi$
$227$	−18.8843	−1.25339	−0.626696	+	0.779264i	$0.715593\pi$
$228$	0	0
$229$	0.0893665	0.00590550	0.00295275	−	0.999996i	$-0.499060\pi$
$229$	0.0893665	0.00590550	0.00295275	+	0.999996i	$0.499060\pi$
$230$	0	0
$231$	−2.15072	−0.141507
$232$	0	0
$233$	29.4657	1.93036	0.965179	−	0.261589i	$-0.0842464\pi$
$233$	29.4657	1.93036	0.965179	+	0.261589i	$0.0842464\pi$
$234$	0	0
$235$	47.9608	3.12862
$236$	0	0
$237$	−3.52571	−0.229019
$238$	0	0
$239$	13.9189	0.900338	0.450169	−	0.892943i	$-0.351364\pi$
$239$	13.9189	0.900338	0.450169	+	0.892943i	$0.351364\pi$
$240$	0	0
$241$	−2.87786	−0.185379	−0.0926897	−	0.995695i	$-0.529546\pi$
$241$	−2.87786	−0.185379	−0.0926897	+	0.995695i	$0.529546\pi$
$242$	0	0
$243$	6.27451	0.402510
$244$	0	0
$245$	−6.45061	−0.412115
$246$	0	0
$247$	12.8256	0.816071
$248$	0	0
$249$	0.569110	0.0360659
$250$	0	0
$251$	−8.19869	−0.517496	−0.258748	−	0.965945i	$-0.583310\pi$
$251$	−8.19869	−0.517496	−0.258748	+	0.965945i	$0.583310\pi$
$252$	0	0
$253$	−23.0150	−1.44694
$254$	0	0
$255$	1.74262	0.109127
$256$	0	0
$257$	−8.24307	−0.514188	−0.257094	−	0.966386i	$-0.582765\pi$
$257$	−8.24307	−0.514188	−0.257094	+	0.966386i	$0.582765\pi$
$258$	0	0
$259$	−2.28805	−0.142173
$260$	0	0
$261$	0.146192	0.00904906
$262$	0	0
$263$	17.5603	1.08281	0.541407	−	0.840760i	$-0.317892\pi$
$263$	17.5603	1.08281	0.541407	+	0.840760i	$0.317892\pi$
$264$	0	0
$265$	−0.689756	−0.0423714
$266$	0	0
$267$	−3.19659	−0.195628
$268$	0	0
$269$	−0.257376	−0.0156925	−0.00784624	−	0.999969i	$-0.502498\pi$
$269$	−0.257376	−0.0156925	−0.00784624	+	0.999969i	$0.502498\pi$
$270$	0	0
$271$	−24.2045	−1.47032	−0.735159	−	0.677895i	$-0.762892\pi$
$271$	−24.2045	−1.47032	−0.735159	+	0.677895i	$0.762892\pi$
$272$	0	0
$273$	−2.46885	−0.149421
$274$	0	0
$275$	32.9646	1.98784
$276$	0	0
$277$	3.65512	0.219615	0.109807	−	0.993953i	$-0.464977\pi$
$277$	3.65512	0.219615	0.109807	+	0.993953i	$0.464977\pi$
$278$	0	0
$279$	−8.59696	−0.514687
$280$	0	0
$281$	−8.83348	−0.526961	−0.263481	−	0.964665i	$-0.584871\pi$
$281$	−8.83348	−0.526961	−0.263481	+	0.964665i	$0.584871\pi$
$282$	0	0
$283$	22.3695	1.32973	0.664864	−	0.746964i	$-0.268489\pi$
$283$	22.3695	1.32973	0.664864	+	0.746964i	$0.268489\pi$
$284$	0	0
$285$	2.46885	0.146242
$286$	0	0
$287$	23.1659	1.36744
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−0.386793	−0.0226742
$292$	0	0
$293$	26.6269	1.55556	0.777779	−	0.628537i	$-0.216346\pi$
$293$	26.6269	1.55556	0.777779	+	0.628537i	$0.216346\pi$
$294$	0	0
$295$	−39.1823	−2.28128
$296$	0	0
$297$	−5.58646	−0.324159
$298$	0	0
$299$	−26.4193	−1.52787
$300$	0	0
$301$	−7.57401	−0.436559
$302$	0	0
$303$	−2.71404	−0.155918
$304$	0	0
$305$	25.4593	1.45779
$306$	0	0
$307$	−0.437067	−0.0249448	−0.0124724	−	0.999922i	$-0.503970\pi$
$307$	−0.437067	−0.0249448	−0.0124724	+	0.999922i	$0.503970\pi$
$308$	0	0
$309$	−3.39246	−0.192990
$310$	0	0
$311$	9.73733	0.552154	0.276077	−	0.961136i	$-0.410966\pi$
$311$	9.73733	0.552154	0.276077	+	0.961136i	$0.410966\pi$
$312$	0	0
$313$	19.9921	1.13002	0.565010	−	0.825084i	$-0.308872\pi$
$313$	19.9921	1.13002	0.565010	+	0.825084i	$0.308872\pi$
$314$	0	0
$315$	24.6141	1.38685
$316$	0	0
$317$	14.1644	0.795553	0.397777	−	0.917482i	$-0.369782\pi$
$317$	14.1644	0.795553	0.397777	+	0.917482i	$0.369782\pi$
$318$	0	0
$319$	−0.195864	−0.0109663
$320$	0	0
$321$	−0.488603	−0.0272712
$322$	0	0
$323$	5.66696	0.315318
$324$	0	0
$325$	37.8406	2.09902
$326$	0	0
$327$	−1.82766	−0.101070
$328$	0	0
$329$	−30.0228	−1.65521
$330$	0	0
$331$	21.7283	1.19430	0.597148	−	0.802131i	$-0.296300\pi$
$331$	21.7283	1.19430	0.597148	+	0.802131i	$0.296300\pi$
$332$	0	0
$333$	−2.94317	−0.161285
$334$	0	0
$335$	4.30705	0.235319
$336$	0	0
$337$	17.8295	0.971236	0.485618	−	0.874171i	$-0.338595\pi$
$337$	17.8295	0.971236	0.485618	+	0.874171i	$0.338595\pi$
$338$	0	0
$339$	0.249460	0.0135488
$340$	0	0
$341$	11.5179	0.623732
$342$	0	0
$343$	20.0544	1.08283
$344$	0	0
$345$	−5.08557	−0.273798
$346$	0	0
$347$	0.113651	0.00610111	0.00305056	−	0.999995i	$-0.499029\pi$
$347$	0.113651	0.00610111	0.00305056	+	0.999995i	$0.499029\pi$
$348$	0	0
$349$	−24.2051	−1.29567	−0.647834	−	0.761781i	$-0.724325\pi$
$349$	−24.2051	−1.29567	−0.647834	+	0.761781i	$0.724325\pi$
$350$	0	0
$351$	−6.41278	−0.342289
$352$	0	0
$353$	18.8128	1.00130	0.500652	−	0.865649i	$-0.333094\pi$
$353$	18.8128	1.00130	0.500652	+	0.865649i	$0.333094\pi$
$354$	0	0
$355$	53.1887	2.82296
$356$	0	0
$357$	−1.09086	−0.0577343
$358$	0	0
$359$	13.5983	0.717691	0.358845	−	0.933397i	$-0.383171\pi$
$359$	13.5983	0.717691	0.358845	+	0.933397i	$0.383171\pi$
$360$	0	0
$361$	−10.9714	−0.577441
$362$	0	0
$363$	1.08431	0.0569113
$364$	0	0
$365$	−45.9217	−2.40365
$366$	0	0
$367$	14.2051	0.741499	0.370750	−	0.928733i	$-0.379101\pi$
$367$	14.2051	0.741499	0.370750	+	0.928733i	$0.379101\pi$
$368$	0	0
$369$	29.7988	1.55127
$370$	0	0
$371$	0.431777	0.0224168
$372$	0	0
$373$	5.61900	0.290941	0.145470	−	0.989363i	$-0.453530\pi$
$373$	5.61900	0.290941	0.145470	+	0.989363i	$0.453530\pi$
$374$	0	0
$375$	2.92753	0.151177
$376$	0	0
$377$	−0.224835	−0.0115796
$378$	0	0
$379$	−24.0111	−1.23337	−0.616683	−	0.787211i	$-0.711524\pi$
$379$	−24.0111	−1.23337	−0.616683	+	0.787211i	$0.711524\pi$
$380$	0	0
$381$	−0.0163688	−0.000838597 0
$382$	0	0
$383$	−12.7262	−0.650280	−0.325140	−	0.945666i	$-0.605411\pi$
$383$	−12.7262	−0.650280	−0.325140	+	0.945666i	$0.605411\pi$
$384$	0	0
$385$	−32.9772	−1.68067
$386$	0	0
$387$	−9.74262	−0.495245
$388$	0	0
$389$	36.4978	1.85051	0.925256	−	0.379342i	$-0.123850\pi$
$389$	36.4978	1.85051	0.925256	+	0.379342i	$0.123850\pi$
$390$	0	0
$391$	−11.6734	−0.590347
$392$	0	0
$393$	1.46455	0.0738767
$394$	0	0
$395$	−54.0600	−2.72005
$396$	0	0
$397$	−16.6655	−0.836416	−0.418208	−	0.908351i	$-0.637342\pi$
$397$	−16.6655	−0.836416	−0.418208	+	0.908351i	$0.637342\pi$
$398$	0	0
$399$	−1.54546	−0.0773699
$400$	0	0
$401$	−3.78061	−0.188795	−0.0943974	−	0.995535i	$-0.530092\pi$
$401$	−3.78061	−0.188795	−0.0943974	+	0.995535i	$0.530092\pi$
$402$	0	0
$403$	13.2216	0.658617
$404$	0	0
$405$	31.0386	1.54232
$406$	0	0
$407$	3.94317	0.195456
$408$	0	0
$409$	32.3424	1.59923	0.799615	−	0.600513i	$-0.205037\pi$
$409$	32.3424	1.59923	0.799615	+	0.600513i	$0.205037\pi$
$410$	0	0
$411$	−5.07373	−0.250268
$412$	0	0
$413$	24.5275	1.20692
$414$	0	0
$415$	8.72622	0.428353
$416$	0	0
$417$	−4.90369	−0.240135
$418$	0	0
$419$	30.7689	1.50316	0.751580	−	0.659642i	$-0.229292\pi$
$419$	30.7689	1.50316	0.751580	+	0.659642i	$0.229292\pi$
$420$	0	0
$421$	28.5772	1.39277	0.696384	−	0.717669i	$-0.254791\pi$
$421$	28.5772	1.39277	0.696384	+	0.717669i	$0.254791\pi$
$422$	0	0
$423$	−38.6190	−1.87772
$424$	0	0
$425$	16.7198	0.811031
$426$	0	0
$427$	−15.9371	−0.771251
$428$	0	0
$429$	4.25475	0.205421
$430$	0	0
$431$	−3.04648	−0.146744	−0.0733718	−	0.997305i	$-0.523376\pi$
$431$	−3.04648	−0.146744	−0.0733718	+	0.997305i	$0.523376\pi$
$432$	0	0
$433$	2.63948	0.126845	0.0634227	−	0.997987i	$-0.479798\pi$
$433$	2.63948	0.126845	0.0634227	+	0.997987i	$0.479798\pi$
$434$	0	0
$435$	−0.0432794	−0.00207509
$436$	0	0
$437$	−16.5381	−0.791125
$438$	0	0
$439$	−17.4593	−0.833285	−0.416642	−	0.909070i	$-0.636793\pi$
$439$	−17.4593	−0.833285	−0.416642	+	0.909070i	$0.636793\pi$
$440$	0	0
$441$	5.19416	0.247341
$442$	0	0
$443$	−22.2762	−1.05837	−0.529187	−	0.848505i	$-0.677503\pi$
$443$	−22.2762	−1.05837	−0.529187	+	0.848505i	$0.677503\pi$
$444$	0	0
$445$	−49.0137	−2.32347
$446$	0	0
$447$	3.96932	0.187743
$448$	0	0
$449$	−4.62049	−0.218054	−0.109027	−	0.994039i	$-0.534774\pi$
$449$	−4.62049	−0.218054	−0.109027	+	0.994039i	$0.534774\pi$
$450$	0	0
$451$	−39.9236	−1.87993
$452$	0	0
$453$	1.01640	0.0477547
$454$	0	0
$455$	−37.8551	−1.77467
$456$	0	0
$457$	−19.9793	−0.934592	−0.467296	−	0.884101i	$-0.654772\pi$
$457$	−19.9793	−0.934592	−0.467296	+	0.884101i	$0.654772\pi$
$458$	0	0
$459$	−2.83348	−0.132256
$460$	0	0
$461$	−35.5898	−1.65758	−0.828791	−	0.559559i	$-0.810971\pi$
$461$	−35.5898	−1.65758	−0.828791	+	0.559559i	$0.810971\pi$
$462$	0	0
$463$	−13.6094	−0.632481	−0.316241	−	0.948679i	$-0.602421\pi$
$463$	−13.6094	−0.632481	−0.316241	+	0.948679i	$0.602421\pi$
$464$	0	0
$465$	2.54509	0.118026
$466$	0	0
$467$	−19.6226	−0.908025	−0.454012	−	0.890995i	$-0.650008\pi$
$467$	−19.6226	−0.908025	−0.454012	+	0.890995i	$0.650008\pi$
$468$	0	0
$469$	−2.69615	−0.124497
$470$	0	0
$471$	4.30382	0.198309
$472$	0	0
$473$	13.0529	0.600171
$474$	0	0
$475$	23.6877	1.08686
$476$	0	0
$477$	0.555405	0.0254303
$478$	0	0
$479$	3.30495	0.151007	0.0755036	−	0.997146i	$-0.475944\pi$
$479$	3.30495	0.151007	0.0755036	+	0.997146i	$0.475944\pi$
$480$	0	0
$481$	4.52643	0.206388
$482$	0	0
$483$	3.18349	0.144854
$484$	0	0
$485$	−5.93073	−0.269301
$486$	0	0
$487$	−2.90914	−0.131826	−0.0659129	−	0.997825i	$-0.520996\pi$
$487$	−2.90914	−0.131826	−0.0659129	+	0.997825i	$0.520996\pi$
$488$	0	0
$489$	−3.90066	−0.176394
$490$	0	0
$491$	26.0497	1.17561	0.587803	−	0.809004i	$-0.299993\pi$
$491$	26.0497	1.17561	0.587803	+	0.809004i	$0.299993\pi$
$492$	0	0
$493$	−0.0993431	−0.00447419
$494$	0	0
$495$	−42.4193	−1.90661
$496$	0	0
$497$	−33.2953	−1.49350
$498$	0	0
$499$	−10.7198	−0.479886	−0.239943	−	0.970787i	$-0.577129\pi$
$499$	−10.7198	−0.479886	−0.239943	+	0.970787i	$0.577129\pi$
$500$	0	0
$501$	3.41955	0.152774
$502$	0	0
$503$	33.2291	1.48161	0.740807	−	0.671718i	$-0.234443\pi$
$503$	33.2291	1.48161	0.740807	+	0.671718i	$0.234443\pi$
$504$	0	0
$505$	−41.6147	−1.85183
$506$	0	0
$507$	1.78514	0.0792809
$508$	0	0
$509$	10.3667	0.459496	0.229748	−	0.973250i	$-0.426210\pi$
$509$	10.3667	0.459496	0.229748	+	0.973250i	$0.426210\pi$
$510$	0	0
$511$	28.7463	1.27166
$512$	0	0
$513$	−4.01431	−0.177236
$514$	0	0
$515$	−52.0169	−2.29214
$516$	0	0
$517$	51.7405	2.27554
$518$	0	0
$519$	1.63476	0.0717581
$520$	0	0
$521$	15.9614	0.699282	0.349641	−	0.936884i	$-0.386304\pi$
$521$	15.9614	0.699282	0.349641	+	0.936884i	$0.386304\pi$
$522$	0	0
$523$	0.614095	0.0268525	0.0134262	−	0.999910i	$-0.495726\pi$
$523$	0.614095	0.0268525	0.0134262	+	0.999910i	$0.495726\pi$
$524$	0	0
$525$	−4.55974	−0.199003
$526$	0	0
$527$	5.84197	0.254480
$528$	0	0
$529$	11.0668	0.481165
$530$	0	0
$531$	31.5503	1.36917
$532$	0	0
$533$	−45.8289	−1.98507
$534$	0	0
$535$	−7.49180	−0.323899
$536$	0	0
$537$	0.301758	0.0130218
$538$	0	0
$539$	−6.95897	−0.299744
$540$	0	0
$541$	9.26979	0.398539	0.199270	−	0.979945i	$-0.436143\pi$
$541$	9.26979	0.398539	0.199270	+	0.979945i	$0.436143\pi$
$542$	0	0
$543$	0.143122	0.00614194
$544$	0	0
$545$	−28.0237	−1.20040
$546$	0	0
$547$	42.6570	1.82388	0.911940	−	0.410324i	$-0.134584\pi$
$547$	42.6570	1.82388	0.911940	+	0.410324i	$0.134584\pi$
$548$	0	0
$549$	−20.5003	−0.874931
$550$	0	0
$551$	−0.140743	−0.00599587
$552$	0	0
$553$	33.8408	1.43906
$554$	0	0
$555$	0.871312	0.0369851
$556$	0	0
$557$	9.00975	0.381755	0.190878	−	0.981614i	$-0.438867\pi$
$557$	9.00975	0.381755	0.190878	+	0.981614i	$0.438867\pi$
$558$	0	0
$559$	14.9836	0.633739
$560$	0	0
$561$	1.87996	0.0793718
$562$	0	0
$563$	1.74262	0.0734428	0.0367214	−	0.999326i	$-0.488309\pi$
$563$	1.74262	0.0734428	0.0367214	+	0.999326i	$0.488309\pi$
$564$	0	0
$565$	3.82500	0.160919
$566$	0	0
$567$	−19.4297	−0.815970
$568$	0	0
$569$	10.6518	0.446545	0.223273	−	0.974756i	$-0.428326\pi$
$569$	10.6518	0.446545	0.223273	+	0.974756i	$0.428326\pi$
$570$	0	0
$571$	34.2828	1.43469	0.717344	−	0.696719i	$-0.245357\pi$
$571$	34.2828	1.43469	0.717344	+	0.696719i	$0.245357\pi$
$572$	0	0
$573$	−6.32155	−0.264087
$574$	0	0
$575$	−48.7941	−2.03486
$576$	0	0
$577$	−0.782707	−0.0325845	−0.0162923	−	0.999867i	$-0.505186\pi$
$577$	−0.782707	−0.0325845	−0.0162923	+	0.999867i	$0.505186\pi$
$578$	0	0
$579$	1.98303	0.0824118
$580$	0	0
$581$	−5.46249	−0.226622
$582$	0	0
$583$	−0.744115	−0.0308181
$584$	0	0
$585$	−48.6938	−2.01324
$586$	0	0
$587$	33.0280	1.36321	0.681605	−	0.731720i	$-0.261282\pi$
$587$	33.0280	1.36321	0.681605	+	0.731720i	$0.261282\pi$
$588$	0	0
$589$	8.27655	0.341029
$590$	0	0
$591$	−1.36314	−0.0560723
$592$	0	0
$593$	17.2048	0.706518	0.353259	−	0.935525i	$-0.385073\pi$
$593$	17.2048	0.706518	0.353259	+	0.935525i	$0.385073\pi$
$594$	0	0
$595$	−16.7262	−0.685708
$596$	0	0
$597$	−0.191517	−0.00783828
$598$	0	0
$599$	−33.9471	−1.38704	−0.693520	−	0.720437i	$-0.743941\pi$
$599$	−33.9471	−1.38704	−0.693520	+	0.720437i	$0.743941\pi$
$600$	0	0
$601$	−21.0568	−0.858924	−0.429462	−	0.903085i	$-0.641297\pi$
$601$	−21.0568	−0.858924	−0.429462	+	0.903085i	$0.641297\pi$
$602$	0	0
$603$	−3.46812	−0.141233
$604$	0	0
$605$	16.6258	0.675934
$606$	0	0
$607$	3.50364	0.142208	0.0711042	−	0.997469i	$-0.477348\pi$
$607$	3.50364	0.142208	0.0711042	+	0.997469i	$0.477348\pi$
$608$	0	0
$609$	0.0270923	0.00109783
$610$	0	0
$611$	59.3938	2.40281
$612$	0	0
$613$	14.9085	0.602150	0.301075	−	0.953600i	$-0.402655\pi$
$613$	14.9085	0.602150	0.301075	+	0.953600i	$0.402655\pi$
$614$	0	0
$615$	−8.82180	−0.355729
$616$	0	0
$617$	−19.6416	−0.790740	−0.395370	−	0.918522i	$-0.629384\pi$
$617$	−19.6416	−0.790740	−0.395370	+	0.918522i	$0.629384\pi$
$618$	0	0
$619$	−10.4786	−0.421172	−0.210586	−	0.977575i	$-0.567537\pi$
$619$	−10.4786	−0.421172	−0.210586	+	0.977575i	$0.567537\pi$
$620$	0	0
$621$	8.26906	0.331826
$622$	0	0
$623$	30.6818	1.22924
$624$	0	0
$625$	3.08860	0.123544
$626$	0	0
$627$	2.66341	0.106366
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	−23.9424	−0.953132	−0.476566	−	0.879139i	$-0.658119\pi$
$631$	−23.9424	−0.953132	−0.476566	+	0.879139i	$0.658119\pi$
$632$	0	0
$633$	2.28523	0.0908297
$634$	0	0
$635$	−0.250984	−0.00995999
$636$	0	0
$637$	−7.98832	−0.316509
$638$	0	0
$639$	−42.8285	−1.69427
$640$	0	0
$641$	27.8425	1.09971	0.549856	−	0.835260i	$-0.314683\pi$
$641$	27.8425	1.09971	0.549856	+	0.835260i	$0.314683\pi$
$642$	0	0
$643$	9.85628	0.388694	0.194347	−	0.980933i	$-0.437741\pi$
$643$	9.85628	0.388694	0.194347	+	0.980933i	$0.437741\pi$
$644$	0	0
$645$	2.88426	0.113567
$646$	0	0
$647$	27.5036	1.08128	0.540640	−	0.841254i	$-0.318182\pi$
$647$	27.5036	1.08128	0.540640	+	0.841254i	$0.318182\pi$
$648$	0	0
$649$	−42.2702	−1.65925
$650$	0	0
$651$	−1.59319	−0.0624420
$652$	0	0
$653$	14.9068	0.583350	0.291675	−	0.956518i	$-0.405788\pi$
$653$	14.9068	0.583350	0.291675	+	0.956518i	$0.405788\pi$
$654$	0	0
$655$	22.4561	0.877431
$656$	0	0
$657$	36.9770	1.44261
$658$	0	0
$659$	−50.2541	−1.95762	−0.978812	−	0.204762i	$-0.934358\pi$
$659$	−50.2541	−1.95762	−0.978812	+	0.204762i	$0.934358\pi$
$660$	0	0
$661$	−19.5243	−0.759409	−0.379704	−	0.925108i	$-0.623974\pi$
$661$	−19.5243	−0.759409	−0.379704	+	0.925108i	$0.623974\pi$
$662$	0	0
$663$	2.15803	0.0838110
$664$	0	0
$665$	−23.6967	−0.918919
$666$	0	0
$667$	0.289917	0.0112256
$668$	0	0
$669$	−6.02428	−0.232912
$670$	0	0
$671$	27.4657	1.06030
$672$	0	0
$673$	−28.2305	−1.08820	−0.544102	−	0.839019i	$-0.683130\pi$
$673$	−28.2305	−1.08820	−0.544102	+	0.839019i	$0.683130\pi$
$674$	0	0
$675$	−11.8438	−0.455869
$676$	0	0
$677$	−42.8170	−1.64559	−0.822796	−	0.568336i	$-0.807587\pi$
$677$	−42.8170	−1.64559	−0.822796	+	0.568336i	$0.807587\pi$
$678$	0	0
$679$	3.71255	0.142475
$680$	0	0
$681$	−4.50165	−0.172504
$682$	0	0
$683$	15.9714	0.611128	0.305564	−	0.952172i	$-0.401155\pi$
$683$	15.9714	0.611128	0.305564	+	0.952172i	$0.401155\pi$
$684$	0	0
$685$	−77.7960	−2.97243
$686$	0	0
$687$	0.0213033	0.000812771 0
$688$	0	0
$689$	−0.854181	−0.0325417
$690$	0	0
$691$	−36.7262	−1.39713	−0.698566	−	0.715546i	$-0.746178\pi$
$691$	−36.7262	−1.39713	−0.698566	+	0.715546i	$0.746178\pi$
$692$	0	0
$693$	26.5539	1.00870
$694$	0	0
$695$	−75.1888	−2.85208
$696$	0	0
$697$	−20.2495	−0.767003
$698$	0	0
$699$	7.02405	0.265674
$700$	0	0
$701$	0.595862	0.0225054	0.0112527	−	0.999937i	$-0.496418\pi$
$701$	0.595862	0.0225054	0.0112527	+	0.999937i	$0.496418\pi$
$702$	0	0
$703$	2.83348	0.106867
$704$	0	0
$705$	11.4330	0.430590
$706$	0	0
$707$	26.0502	0.979718
$708$	0	0
$709$	−21.3834	−0.803072	−0.401536	−	0.915843i	$-0.631524\pi$
$709$	−21.3834	−0.803072	−0.401536	+	0.915843i	$0.631524\pi$
$710$	0	0
$711$	43.5301	1.63251
$712$	0	0
$713$	−17.0488	−0.638484
$714$	0	0
$715$	65.2385	2.43978
$716$	0	0
$717$	3.31800	0.123913
$718$	0	0
$719$	11.9170	0.444430	0.222215	−	0.974998i	$-0.428671\pi$
$719$	11.9170	0.444430	0.222215	+	0.974998i	$0.428671\pi$
$720$	0	0
$721$	32.5618	1.21266
$722$	0	0
$723$	−0.686028	−0.0255137
$724$	0	0
$725$	−0.415250	−0.0154220
$726$	0	0
$727$	−22.6451	−0.839861	−0.419930	−	0.907556i	$-0.637946\pi$
$727$	−22.6451	−0.839861	−0.419930	+	0.907556i	$0.637946\pi$
$728$	0	0
$729$	−23.9797	−0.888136
$730$	0	0
$731$	6.62049	0.244868
$732$	0	0
$733$	−43.9128	−1.62196	−0.810979	−	0.585075i	$-0.801065\pi$
$733$	−43.9128	−1.62196	−0.810979	+	0.585075i	$0.801065\pi$
$734$	0	0
$735$	−1.53770	−0.0567191
$736$	0	0
$737$	4.64648	0.171155
$738$	0	0
$739$	−7.19206	−0.264564	−0.132282	−	0.991212i	$-0.542231\pi$
$739$	−7.19206	−0.264564	−0.132282	+	0.991212i	$0.542231\pi$
$740$	0	0
$741$	3.05737	0.112315
$742$	0	0
$743$	36.0021	1.32079	0.660394	−	0.750919i	$-0.270389\pi$
$743$	36.0021	1.32079	0.660394	+	0.750919i	$0.270389\pi$
$744$	0	0
$745$	60.8620	2.22981
$746$	0	0
$747$	−7.02652	−0.257087
$748$	0	0
$749$	4.68976	0.171360
$750$	0	0
$751$	−19.7526	−0.720783	−0.360391	−	0.932801i	$-0.617357\pi$
$751$	−19.7526	−0.720783	−0.360391	+	0.932801i	$0.617357\pi$
$752$	0	0
$753$	−1.95441	−0.0712227
$754$	0	0
$755$	15.5846	0.567181
$756$	0	0
$757$	7.27770	0.264513	0.132256	−	0.991216i	$-0.457778\pi$
$757$	7.27770	0.264513	0.132256	+	0.991216i	$0.457778\pi$
$758$	0	0
$759$	−5.48635	−0.199142
$760$	0	0
$761$	14.7494	0.534666	0.267333	−	0.963604i	$-0.413858\pi$
$761$	14.7494	0.534666	0.267333	+	0.963604i	$0.413858\pi$
$762$	0	0
$763$	17.5424	0.635078
$764$	0	0
$765$	−21.5153	−0.777888
$766$	0	0
$767$	−48.5226	−1.75205
$768$	0	0
$769$	−38.3002	−1.38114	−0.690571	−	0.723265i	$-0.742641\pi$
$769$	−38.3002	−1.38114	−0.690571	+	0.723265i	$0.742641\pi$
$770$	0	0
$771$	−1.96499	−0.0707675
$772$	0	0
$773$	24.9906	0.898849	0.449425	−	0.893318i	$-0.351629\pi$
$773$	24.9906	0.898849	0.449425	+	0.893318i	$0.351629\pi$
$774$	0	0
$775$	24.4192	0.877163
$776$	0	0
$777$	−0.545429	−0.0195671
$778$	0	0
$779$	−28.6882	−1.02786
$780$	0	0
$781$	57.3804	2.05323
$782$	0	0
$783$	0.0703717	0.00251488
$784$	0	0
$785$	65.9909	2.35532
$786$	0	0
$787$	22.0100	0.784571	0.392285	−	0.919844i	$-0.371685\pi$
$787$	22.0100	0.784571	0.392285	+	0.919844i	$0.371685\pi$
$788$	0	0
$789$	4.18605	0.149027
$790$	0	0
$791$	−2.39439	−0.0851347
$792$	0	0
$793$	31.5283	1.11960
$794$	0	0
$795$	−0.164425	−0.00583155
$796$	0	0
$797$	20.4872	0.725695	0.362848	−	0.931848i	$-0.381805\pi$
$797$	20.4872	0.725695	0.362848	+	0.931848i	$0.381805\pi$
$798$	0	0
$799$	26.2431	0.928413
$800$	0	0
$801$	39.4668	1.39449
$802$	0	0
$803$	−49.5407	−1.74825
$804$	0	0
$805$	48.8128	1.72042
$806$	0	0
$807$	−0.0613536	−0.00215975
$808$	0	0
$809$	−41.8378	−1.47094	−0.735469	−	0.677558i	$-0.763038\pi$
$809$	−41.8378	−1.47094	−0.735469	+	0.677558i	$0.763038\pi$
$810$	0	0
$811$	12.3405	0.433335	0.216668	−	0.976245i	$-0.430481\pi$
$811$	12.3405	0.433335	0.216668	+	0.976245i	$0.430481\pi$
$812$	0	0
$813$	−5.76989	−0.202359
$814$	0	0
$815$	−59.8092	−2.09502
$816$	0	0
$817$	9.37951	0.328148
$818$	0	0
$819$	30.4816	1.06511
$820$	0	0
$821$	−6.08297	−0.212297	−0.106149	−	0.994350i	$-0.533852\pi$
$821$	−6.08297	−0.212297	−0.106149	+	0.994350i	$0.533852\pi$
$822$	0	0
$823$	29.2410	1.01928	0.509638	−	0.860389i	$-0.329779\pi$
$823$	29.2410	1.01928	0.509638	+	0.860389i	$0.329779\pi$
$824$	0	0
$825$	7.85814	0.273585
$826$	0	0
$827$	22.5411	0.783831	0.391915	−	0.920001i	$-0.371813\pi$
$827$	22.5411	0.783831	0.391915	+	0.920001i	$0.371813\pi$
$828$	0	0
$829$	26.3815	0.916267	0.458134	−	0.888883i	$-0.348518\pi$
$829$	26.3815	0.916267	0.458134	+	0.888883i	$0.348518\pi$
$830$	0	0
$831$	0.871312	0.0302255
$832$	0	0
$833$	−3.52963	−0.122294
$834$	0	0
$835$	52.4323	1.81449
$836$	0	0
$837$	−4.13828	−0.143040
$838$	0	0
$839$	−35.0829	−1.21120	−0.605599	−	0.795770i	$-0.707066\pi$
$839$	−35.0829	−1.21120	−0.605599	+	0.795770i	$0.707066\pi$
$840$	0	0
$841$	−28.9975	−0.999915
$842$	0	0
$843$	−2.10574	−0.0725254
$844$	0	0
$845$	27.3718	0.941617
$846$	0	0
$847$	−10.4075	−0.357606
$848$	0	0
$849$	5.33247	0.183010
$850$	0	0
$851$	−5.83668	−0.200079
$852$	0	0
$853$	2.79157	0.0955814	0.0477907	−	0.998857i	$-0.484782\pi$
$853$	2.79157	0.0955814	0.0477907	+	0.998857i	$0.484782\pi$
$854$	0	0
$855$	−30.4816	−1.04245
$856$	0	0
$857$	49.9714	1.70699	0.853495	−	0.521101i	$-0.174479\pi$
$857$	49.9714	1.70699	0.853495	+	0.521101i	$0.174479\pi$
$858$	0	0
$859$	−23.1209	−0.788876	−0.394438	−	0.918923i	$-0.629061\pi$
$859$	−23.1209	−0.788876	−0.394438	+	0.918923i	$0.629061\pi$
$860$	0	0
$861$	5.52232	0.188200
$862$	0	0
$863$	−6.42180	−0.218601	−0.109300	−	0.994009i	$-0.534861\pi$
$863$	−6.42180	−0.218601	−0.109300	+	0.994009i	$0.534861\pi$
$864$	0	0
$865$	25.0660	0.852269
$866$	0	0
$867$	−3.09895	−0.105246
$868$	0	0
$869$	−58.3203	−1.97838
$870$	0	0
$871$	5.33377	0.180728
$872$	0	0
$873$	4.77554	0.161627
$874$	0	0
$875$	−28.0993	−0.949931
$876$	0	0
$877$	−45.8328	−1.54767	−0.773833	−	0.633390i	$-0.781663\pi$
$877$	−45.8328	−1.54767	−0.773833	+	0.633390i	$0.781663\pi$
$878$	0	0
$879$	6.34735	0.214091
$880$	0	0
$881$	18.5123	0.623695	0.311847	−	0.950132i	$-0.399052\pi$
$881$	18.5123	0.623695	0.311847	+	0.950132i	$0.399052\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	−9.34032	−0.313971
$886$	0	0
$887$	9.91436	0.332892	0.166446	−	0.986051i	$-0.446771\pi$
$887$	9.91436	0.332892	0.166446	+	0.986051i	$0.446771\pi$
$888$	0	0
$889$	0.157112	0.00526937
$890$	0	0
$891$	33.4846	1.12178
$892$	0	0
$893$	37.1796	1.24417
$894$	0	0
$895$	4.62688	0.154660
$896$	0	0
$897$	−6.29787	−0.210280
$898$	0	0
$899$	−0.145090	−0.00483902
$900$	0	0
$901$	−0.377419	−0.0125737
$902$	0	0
$903$	−1.80550	−0.0600833
$904$	0	0
$905$	2.19450	0.0729476
$906$	0	0
$907$	−46.6050	−1.54749	−0.773746	−	0.633496i	$-0.781620\pi$
$907$	−46.6050	−1.54749	−0.773746	+	0.633496i	$0.781620\pi$
$908$	0	0
$909$	33.5089	1.11142
$910$	0	0
$911$	−32.2115	−1.06721	−0.533607	−	0.845733i	$-0.679164\pi$
$911$	−32.2115	−1.06721	−0.533607	+	0.845733i	$0.679164\pi$
$912$	0	0
$913$	9.41392	0.311555
$914$	0	0
$915$	6.06901	0.200635
$916$	0	0
$917$	−14.0572	−0.464209
$918$	0	0
$919$	35.5454	1.17253	0.586267	−	0.810118i	$-0.300597\pi$
$919$	35.5454	1.17253	0.586267	+	0.810118i	$0.300597\pi$
$920$	0	0
$921$	−0.104189	−0.00343313
$922$	0	0
$923$	65.8679	2.16807
$924$	0	0
$925$	8.35992	0.274872
$926$	0	0
$927$	41.8850	1.37568
$928$	0	0
$929$	−9.77517	−0.320713	−0.160356	−	0.987059i	$-0.551264\pi$
$929$	−9.77517	−0.320713	−0.160356	+	0.987059i	$0.551264\pi$
$930$	0	0
$931$	−5.00057	−0.163887
$932$	0	0
$933$	2.32120	0.0759926
$934$	0	0
$935$	28.8256	0.942697
$936$	0	0
$937$	39.1904	1.28029	0.640147	−	0.768252i	$-0.278873\pi$
$937$	39.1904	1.28029	0.640147	+	0.768252i	$0.278873\pi$
$938$	0	0
$939$	4.76574	0.155524
$940$	0	0
$941$	26.8256	0.874488	0.437244	−	0.899343i	$-0.355955\pi$
$941$	26.8256	0.874488	0.437244	+	0.899343i	$0.355955\pi$
$942$	0	0
$943$	59.0948	1.92439
$944$	0	0
$945$	11.8484	0.385427
$946$	0	0
$947$	−44.5475	−1.44760	−0.723799	−	0.690010i	$-0.757606\pi$
$947$	−44.5475	−1.44760	−0.723799	+	0.690010i	$0.757606\pi$
$948$	0	0
$949$	−56.8686	−1.84603
$950$	0	0
$951$	3.37653	0.109492
$952$	0	0
$953$	−4.50231	−0.145844	−0.0729220	−	0.997338i	$-0.523232\pi$
$953$	−4.50231	−0.145844	−0.0729220	+	0.997338i	$0.523232\pi$
$954$	0	0
$955$	−96.9290	−3.13655
$956$	0	0
$957$	−0.0466902	−0.00150928
$958$	0	0
$959$	48.6991	1.57258
$960$	0	0
$961$	−22.4679	−0.724770
$962$	0	0
$963$	6.03254	0.194396
$964$	0	0
$965$	30.4060	0.978803
$966$	0	0
$967$	45.2656	1.45564	0.727822	−	0.685766i	$-0.240533\pi$
$967$	45.2656	1.45564	0.727822	+	0.685766i	$0.240533\pi$
$968$	0	0
$969$	1.35090	0.0433971
$970$	0	0
$971$	−46.6180	−1.49604	−0.748022	−	0.663674i	$-0.768996\pi$
$971$	−46.6180	−1.49604	−0.748022	+	0.663674i	$0.768996\pi$
$972$	0	0
$973$	47.0671	1.50890
$974$	0	0
$975$	9.02049	0.288887
$976$	0	0
$977$	13.9228	0.445430	0.222715	−	0.974884i	$-0.428508\pi$
$977$	13.9228	0.445430	0.222715	+	0.974884i	$0.428508\pi$
$978$	0	0
$979$	−52.8763	−1.68994
$980$	0	0
$981$	22.5652	0.720452
$982$	0	0
$983$	38.4889	1.22761	0.613803	−	0.789459i	$-0.289639\pi$
$983$	38.4889	1.22761	0.613803	+	0.789459i	$0.289639\pi$
$984$	0	0
$985$	−20.9012	−0.665968
$986$	0	0
$987$	−7.15686	−0.227805
$988$	0	0
$989$	−19.3208	−0.614366
$990$	0	0
$991$	−36.1685	−1.14893	−0.574465	−	0.818529i	$-0.694790\pi$
$991$	−36.1685	−1.14893	−0.574465	+	0.818529i	$0.694790\pi$
$992$	0	0
$993$	5.17962	0.164370
$994$	0	0
$995$	−2.93655	−0.0930950
$996$	0	0
$997$	31.1522	0.986600	0.493300	−	0.869859i	$-0.335790\pi$
$997$	31.1522	0.986600	0.493300	+	0.869859i	$0.335790\pi$
$998$	0	0
$999$	−1.41674	−0.0448237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.2.a.j.1.3		4
3.2	odd	2		5328.2.a.bp.1.1		4
4.3	odd	2		296.2.a.d.1.2	✓	4
8.3	odd	2		2368.2.a.bg.1.3		4
8.5	even	2		2368.2.a.bh.1.2		4
12.11	even	2		2664.2.a.r.1.1		4
20.19	odd	2		7400.2.a.n.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.d.1.2	✓	4	4.3	odd	2
592.2.a.j.1.3		4	1.1	even	1	trivial
2368.2.a.bg.1.3		4	8.3	odd	2
2368.2.a.bh.1.2		4	8.5	even	2
2664.2.a.r.1.1		4	12.11	even	2
5328.2.a.bp.1.1		4	3.2	odd	2
7400.2.a.n.1.3		4	20.19	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:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q+0.238381 q^{3} +3.65512 q^{5} -2.28805 q^{7} -2.94317 q^{9} +O(q^{10})\)	\(q+0.238381 q^{3} +3.65512 q^{5} -2.28805 q^{7} -2.94317 q^{9} +3.94317 q^{11} +4.52643 q^{13} +0.871312 q^{15} +2.00000 q^{17} +2.83348 q^{19} -0.545429 q^{21} -5.83668 q^{23} +8.35992 q^{25} -1.41674 q^{27} -0.0496716 q^{29} +2.92098 q^{31} +0.939978 q^{33} -8.36311 q^{35} +1.00000 q^{37} +1.07902 q^{39} -10.1247 q^{41} +3.31024 q^{43} -10.7577 q^{45} +13.1215 q^{47} -1.76481 q^{49} +0.476762 q^{51} -0.188710 q^{53} +14.4128 q^{55} +0.675448 q^{57} -10.7198 q^{59} +6.96537 q^{61} +6.73414 q^{63} +16.5447 q^{65} +1.17836 q^{67} -1.39135 q^{69} +14.5518 q^{71} -12.5637 q^{73} +1.99285 q^{75} -9.02219 q^{77} -14.7902 q^{79} +8.49180 q^{81} +2.38740 q^{83} +7.31024 q^{85} -0.0118408 q^{87} -13.4096 q^{89} -10.3567 q^{91} +0.696307 q^{93} +10.3567 q^{95} -1.62258 q^{97} -11.6055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 5 * q^5 + q^7 + 8 * q^9	\( 4 q - 2 q^{3} + 5 q^{5} + q^{7} + 8 q^{9} - 4 q^{11} + 5 q^{13} + 8 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 7 q^{25} + q^{27} + 7 q^{29} + q^{31} + 3 q^{33} + 12 q^{35} + 4 q^{37} + 15 q^{39} + 2 q^{41} - 6 q^{43} + 29 q^{47} + 9 q^{49} - 4 q^{51} - 5 q^{53} + 5 q^{55} - 32 q^{57} + 10 q^{59} - q^{61} + 28 q^{63} - 2 q^{65} + q^{67} - 27 q^{69} + 17 q^{71} + 8 q^{73} - 19 q^{75} - 27 q^{77} - 15 q^{79} - 8 q^{81} - 15 q^{83} + 10 q^{85} + 17 q^{87} - 20 q^{89} - 34 q^{91} - 6 q^{93} + 34 q^{95} + 2 q^{97} - 44 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 5 * q^5 + q^7 + 8 * q^9 - 4 * q^11 + 5 * q^13 + 8 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 7 * q^25 + q^27 + 7 * q^29 + q^31 + 3 * q^33 + 12 * q^35 + 4 * q^37 + 15 * q^39 + 2 * q^41 - 6 * q^43 + 29 * q^47 + 9 * q^49 - 4 * q^51 - 5 * q^53 + 5 * q^55 - 32 * q^57 + 10 * q^59 - q^61 + 28 * q^63 - 2 * q^65 + q^67 - 27 * q^69 + 17 * q^71 + 8 * q^73 - 19 * q^75 - 27 * q^77 - 15 * q^79 - 8 * q^81 - 15 * q^83 + 10 * q^85 + 17 * q^87 - 20 * q^89 - 34 * q^91 - 6 * q^93 + 34 * q^95 + 2 * q^97 - 44 * q^99

Introduction

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