LMFDB - Embedded newform 392.4.i.o.361.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [392,4,Mod(177,392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(392, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("392.177");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 392 = 2^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	392.i (of order $3$, degree $2$, not minimal)

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:	no
Analytic conductor:	$23.1287487223$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.54095201243136.19
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 102x^{6} + 320x^{5} + 4283x^{4} - 9104x^{3} - 85298x^{2} + 89904x + 714364 $ x^8 - 4x^7 - 102x^6 + 320x^5 + 4283x^4 - 9104x^3 - 85298x^2 + 89904*x + 714364
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.4
Root			$6.52218 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	392.361
Dual form			392.4.i.o.177.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.81898 - 8.34673i) q^{3} +(-2.95919 - 5.12547i) q^{5} +(-32.9452 - 57.0628i) q^{9} +O(q^{10})$	\(q+(4.81898 - 8.34673i) q^{3} +(-2.95919 - 5.12547i) q^{5} +(-32.9452 - 57.0628i) q^{9} +(9.18493 - 15.9088i) q^{11} -90.9844 q^{13} -57.0412 q^{15} +(-45.9619 + 79.6084i) q^{17} +(62.1237 + 107.601i) q^{19} +(-32.5206 - 56.3273i) q^{23} +(44.9864 - 77.9187i) q^{25} -374.825 q^{27} +83.4110 q^{29} +(82.6540 - 143.161i) q^{31} +(-88.5240 - 153.328i) q^{33} +(-155.486 - 269.310i) q^{37} +(-438.452 + 759.421i) q^{39} +181.504 q^{41} +82.3699 q^{43} +(-194.982 + 337.719i) q^{45} +(-183.954 - 318.618i) q^{47} +(442.980 + 767.263i) q^{51} +(-26.3287 + 45.6026i) q^{53} -108.720 q^{55} +1197.49 q^{57} +(-280.639 + 486.081i) q^{59} +(-138.152 - 239.287i) q^{61} +(269.240 + 466.337i) q^{65} +(30.0684 - 52.0800i) q^{67} -626.865 q^{69} -155.534 q^{71} +(-365.739 + 633.479i) q^{73} +(-433.577 - 750.978i) q^{75} +(-145.192 - 251.480i) q^{79} +(-916.754 + 1587.86i) q^{81} -43.5791 q^{83} +544.041 q^{85} +(401.956 - 696.209i) q^{87} +(196.547 + 340.429i) q^{89} +(-796.617 - 1379.78i) q^{93} +(367.672 - 636.827i) q^{95} +521.748 q^{97} -1210.40 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 136 q^{9}+O(q^{10}) $ 8 * q - 136 * q^9	$ 8 q - 136 q^{9} + 116 q^{11} + 224 q^{15} + 80 q^{23} - 448 q^{25} + 72 q^{29} - 436 q^{37} - 2232 q^{39} + 744 q^{43} + 780 q^{51} - 976 q^{53} + 7624 q^{57} - 780 q^{65} + 1176 q^{67} - 4816 q^{71} - 56 q^{79} - 2104 q^{81} + 9880 q^{85} - 2376 q^{93} - 4032 q^{95} - 5176 q^{99}+O(q^{100}) $ 8 * q - 136 * q^9 + 116 * q^11 + 224 * q^15 + 80 * q^23 - 448 * q^25 + 72 * q^29 - 436 * q^37 - 2232 * q^39 + 744 * q^43 + 780 * q^51 - 976 * q^53 + 7624 * q^57 - 780 * q^65 + 1176 * q^67 - 4816 * q^71 - 56 * q^79 - 2104 * q^81 + 9880 * q^85 - 2376 * q^93 - 4032 * q^95 - 5176 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/392\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$297$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	4.81898	−	8.34673i	0.927414	−	1.60633i	0.139782	−	0.990182i	$-0.455360\pi$
$3$	4.81898	−	8.34673i	0.927414	−	1.60633i	0.787632	−	0.616146i	$-0.211307\pi$
$4$		0			0
$5$	−2.95919	−	5.12547i	−0.264678	−	0.458436i	0.702801	−	0.711386i	$-0.251932\pi$
$5$	−2.95919	−	5.12547i	−0.264678	−	0.458436i	−0.967479	+	0.252951i	$0.918599\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−32.9452	−	57.0628i	−1.22019	−	2.11344i
$10$		0			0
$11$	9.18493	−	15.9088i	0.251760	−	0.436061i	−0.712250	−	0.701925i	$-0.752324\pi$
$11$	9.18493	−	15.9088i	0.251760	−	0.436061i	0.964010	+	0.265864i	$0.0856573\pi$
$12$		0			0
$13$	−90.9844			−1.94112			−0.970558	−	0.240866i	$-0.922569\pi$
$13$	−90.9844			−1.94112			−0.970558	+	0.240866i	$0.922569\pi$
$14$		0			0
$15$	−57.0412			−0.981864
$16$		0			0
$17$	−45.9619	+	79.6084i	−0.655730	+	1.13576i	0.325980	+	0.945377i	$0.394306\pi$
$17$	−45.9619	+	79.6084i	−0.655730	+	1.13576i	−0.981710	+	0.190381i	$0.939028\pi$
$18$		0			0
$19$	62.1237	+	107.601i	0.750114	+	1.29924i	0.947767	+	0.318963i	$0.103335\pi$
$19$	62.1237	+	107.601i	0.750114	+	1.29924i	−0.197653	+	0.980272i	$0.563332\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−32.5206	−	56.3273i	−0.294827	−	0.510655i	0.680118	−	0.733103i	$-0.261929\pi$
$23$	−32.5206	−	56.3273i	−0.294827	−	0.510655i	−0.974945	+	0.222448i	$0.928595\pi$
$24$		0			0
$25$	44.9864	−	77.9187i	0.359891	−	0.623350i
$26$		0			0
$27$	−374.825			−2.67167
$28$		0			0
$29$	83.4110			0.534105			0.267052	−	0.963682i	$-0.413950\pi$
$29$	83.4110			0.534105			0.267052	+	0.963682i	$0.413950\pi$
$30$		0			0
$31$	82.6540	−	143.161i	0.478874	−	0.829434i	−0.520832	−	0.853659i	$-0.674378\pi$
$31$	82.6540	−	143.161i	0.478874	−	0.829434i	0.999707	+	0.0242245i	$0.00771166\pi$
$32$		0			0
$33$	−88.5240	−	153.328i	−0.466971	−	0.808818i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−155.486	−	269.310i	−0.690860	−	1.19660i	−0.971557	−	0.236807i	$-0.923899\pi$
$37$	−155.486	−	269.310i	−0.690860	−	1.19660i	0.280697	−	0.959796i	$-0.409434\pi$
$38$		0			0
$39$	−438.452	+	759.421i	−1.80022	+	3.11807i
$40$		0			0
$41$	181.504			0.691369			0.345685	−	0.938351i	$-0.387647\pi$
$41$	181.504			0.691369			0.345685	+	0.938351i	$0.387647\pi$
$42$		0			0
$43$	82.3699			0.292123			0.146061	−	0.989276i	$-0.453340\pi$
$43$	82.3699			0.292123			0.146061	+	0.989276i	$0.453340\pi$
$44$		0			0
$45$	−194.982	+	337.719i	−0.645917	+	1.11876i
$46$		0			0
$47$	−183.954	−	318.618i	−0.570904	−	0.988835i	−0.996473	−	0.0839092i	$-0.973259\pi$
$47$	−183.954	−	318.618i	−0.570904	−	0.988835i	0.425569	−	0.904926i	$-0.360074\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	442.980	+	767.263i	1.21627	+	2.10663i
$52$		0			0
$53$	−26.3287	+	45.6026i	−0.0682363	+	0.118189i	−0.898125	−	0.439740i	$-0.855071\pi$
$53$	−26.3287	+	45.6026i	−0.0682363	+	0.118189i	0.829889	+	0.557929i	$0.188404\pi$
$54$		0			0
$55$	−108.720			−0.266541
$56$		0			0
$57$	1197.49			2.78266
$58$		0			0
$59$	−280.639	+	486.081i	−0.619256	+	1.07258i	0.370366	+	0.928886i	$0.379232\pi$
$59$	−280.639	+	486.081i	−0.619256	+	1.07258i	−0.989622	+	0.143696i	$0.954101\pi$
$60$		0			0
$61$	−138.152	−	239.287i	−0.289977	−	0.502255i	0.683827	−	0.729644i	$-0.260314\pi$
$61$	−138.152	−	239.287i	−0.289977	−	0.502255i	−0.973804	+	0.227390i	$0.926981\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	269.240	+	466.337i	0.513771	+	0.889877i
$66$		0			0
$67$	30.0684	−	52.0800i	0.0548275	−	0.0949639i	−0.837309	−	0.546730i	$-0.815872\pi$
$67$	30.0684	−	52.0800i	0.0548275	−	0.0949639i	0.892136	+	0.451766i	$0.149206\pi$
$68$		0			0
$69$	−626.865			−1.09370
$70$		0			0
$71$	−155.534			−0.259979			−0.129989	−	0.991515i	$-0.541494\pi$
$71$	−155.534			−0.259979			−0.129989	+	0.991515i	$0.541494\pi$
$72$		0			0
$73$	−365.739	+	633.479i	−0.586391	+	1.01566i	0.408310	+	0.912843i	$0.366118\pi$
$73$	−365.739	+	633.479i	−0.586391	+	1.01566i	−0.994701	+	0.102815i	$0.967215\pi$
$74$		0			0
$75$	−433.577	−	750.978i	−0.667536	−	1.15621i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−145.192	−	251.480i	−0.206777	−	0.358148i	0.743921	−	0.668268i	$-0.232964\pi$
$79$	−145.192	−	251.480i	−0.206777	−	0.358148i	−0.950697	+	0.310120i	$0.899631\pi$
$80$		0			0
$81$	−916.754	+	1587.86i	−1.25755	+	2.17814i
$82$		0			0
$83$	−43.5791			−0.0576317			−0.0288158	−	0.999585i	$-0.509174\pi$
$83$	−43.5791			−0.0576317			−0.0288158	+	0.999585i	$0.509174\pi$
$84$		0			0
$85$	544.041			0.694229
$86$		0			0
$87$	401.956	−	696.209i	0.495336	−	0.857947i
$88$		0			0
$89$	196.547	+	340.429i	0.234089	+	0.405454i	0.959007	−	0.283381i	$-0.0914560\pi$
$89$	196.547	+	340.429i	0.234089	+	0.405454i	−0.724919	+	0.688834i	$0.758123\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−796.617	−	1379.78i	−0.888229	−	1.53846i
$94$		0			0
$95$	367.672	−	636.827i	0.397077	−	0.687758i
$96$		0			0
$97$	521.748			0.546139			0.273070	−	0.961994i	$-0.411961\pi$
$97$	521.748			0.546139			0.273070	+	0.961994i	$0.411961\pi$
$98$		0			0
$99$	−1210.40			−1.22878
$100$		0			0
$101$	376.544	−	652.194i	0.370966	−	0.642532i	−0.618748	−	0.785589i	$-0.712360\pi$
$101$	376.544	−	652.194i	0.370966	−	0.642532i	0.989714	+	0.143057i	$0.0456933\pi$
$102$		0			0
$103$	−465.461	−	806.202i	−0.445274	−	0.771237i	0.552797	−	0.833316i	$-0.313560\pi$
$103$	−465.461	−	806.202i	−0.445274	−	0.771237i	−0.998071	+	0.0620787i	$0.980227\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−841.521	−	1457.56i	−0.760308	−	1.31689i	−0.942692	−	0.333664i	$-0.891715\pi$
$107$	−841.521	−	1457.56i	−0.760308	−	1.31689i	0.182384	−	0.983227i	$-0.441619\pi$
$108$		0			0
$109$	391.953	−	678.882i	0.344424	−	0.596560i	−0.640825	−	0.767687i	$-0.721407\pi$
$109$	391.953	−	678.882i	0.344424	−	0.596560i	0.985249	+	0.171127i	$0.0547408\pi$
$110$		0			0
$111$	−2997.15			−2.56285
$112$		0			0
$113$	−628.205			−0.522979			−0.261489	−	0.965206i	$-0.584214\pi$
$113$	−628.205			−0.522979			−0.261489	+	0.965206i	$0.584214\pi$
$114$		0			0
$115$	−192.469	+	333.366i	−0.156068	+	0.270318i
$116$		0			0
$117$	2997.50	+	5191.82i	2.36854	+	4.10243i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	496.774	+	860.438i	0.373234	+	0.646460i
$122$		0			0
$123$	874.664	−	1514.96i	0.641186	−	1.11057i
$124$		0			0
$125$	−1272.29			−0.910377
$126$		0			0
$127$	328.466			0.229501			0.114751	−	0.993394i	$-0.463393\pi$
$127$	328.466			0.229501			0.114751	+	0.993394i	$0.463393\pi$
$128$		0			0
$129$	396.939	−	687.519i	0.270919	−	0.469245i
$130$		0			0
$131$	−1206.40	−	2089.54i	−0.804606	−	1.39362i	−0.916557	−	0.399905i	$-0.869043\pi$
$131$	−1206.40	−	2089.54i	−0.804606	−	1.39362i	0.111950	−	0.993714i	$-0.464290\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1109.18	+	1921.15i	0.707132	+	1.22479i
$136$		0			0
$137$	87.8493	−	152.159i	0.0547845	−	0.0948895i	−0.837333	−	0.546694i	$-0.815886\pi$
$137$	87.8493	−	152.159i	0.0547845	−	0.0948895i	0.892117	+	0.451804i	$0.149219\pi$
$138$		0			0
$139$	1489.47			0.908885			0.454442	−	0.890776i	$-0.349839\pi$
$139$	1489.47			0.908885			0.454442	+	0.890776i	$0.349839\pi$
$140$		0			0
$141$	−3545.89			−2.11786
$142$		0			0
$143$	−835.685	+	1447.45i	−0.488696	+	0.846446i
$144$		0			0
$145$	−246.829	−	427.521i	−0.141366	−	0.244853i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1517.16	−	2627.81i	−0.834168	−	1.44482i	−0.894706	−	0.446655i	$-0.852615\pi$
$149$	−1517.16	−	2627.81i	−0.834168	−	1.44482i	0.0605388	−	0.998166i	$-0.480718\pi$
$150$		0			0
$151$	1057.41	−	1831.49i	0.569873	−	0.987050i	−0.426704	−	0.904391i	$-0.640326\pi$
$151$	1057.41	−	1831.49i	0.569873	−	0.987050i	0.996578	−	0.0826586i	$-0.0263411\pi$
$152$		0			0
$153$	6056.90			3.20047
$154$		0			0
$155$	−978.356			−0.506990
$156$		0			0
$157$	137.852	−	238.767i	0.0700752	−	0.121374i	−0.828859	−	0.559458i	$-0.811009\pi$
$157$	137.852	−	238.767i	0.0700752	−	0.121374i	0.898934	+	0.438084i	$0.144343\pi$
$158$		0			0
$159$	253.755	+	439.517i	0.126567	+	0.219220i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−609.623	−	1055.90i	−0.292941	−	0.507389i	0.681563	−	0.731760i	$-0.261301\pi$
$163$	−609.623	−	1055.90i	−0.292941	−	0.507389i	−0.974504	+	0.224371i	$0.927967\pi$
$164$		0			0
$165$	−523.919	+	907.454i	−0.247194	+	0.428153i
$166$		0			0
$167$	2781.99			1.28908			0.644542	−	0.764569i	$-0.277048\pi$
$167$	2781.99			1.28908			0.644542	+	0.764569i	$0.277048\pi$
$168$		0			0
$169$	6081.15			2.76794
$170$		0			0
$171$	4093.36	−	7089.91i	1.83057	−	3.17064i
$172$		0			0
$173$	−1353.52	−	2344.37i	−0.594835	−	1.03029i	−0.993570	−	0.113219i	$-0.963884\pi$
$173$	−1353.52	−	2344.37i	−0.594835	−	1.03029i	0.398735	−	0.917066i	$-0.369449\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	2704.79	+	4684.83i	1.14861	+	1.98946i
$178$		0			0
$179$	860.863	−	1491.06i	0.359463	−	0.622608i	−0.628408	−	0.777884i	$-0.716293\pi$
$179$	860.863	−	1491.06i	0.359463	−	0.622608i	0.987871	+	0.155276i	$0.0496265\pi$
$180$		0			0
$181$	624.549			0.256477			0.128238	−	0.991743i	$-0.459068\pi$
$181$	624.549			0.256477			0.128238	+	0.991743i	$0.459068\pi$
$182$		0			0
$183$	−2663.01			−1.07571
$184$		0			0
$185$	−920.228	+	1593.88i	−0.365711	+	0.633430i
$186$		0			0
$187$	844.314	+	1462.40i	0.330173	+	0.571877i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2394.89	+	4148.07i	0.907269	+	1.57144i	0.817843	+	0.575442i	$0.195170\pi$
$191$	2394.89	+	4148.07i	0.907269	+	1.57144i	0.0894257	+	0.995994i	$0.471497\pi$
$192$		0			0
$193$	718.117	−	1243.81i	0.267830	−	0.463895i	−0.700471	−	0.713681i	$-0.747027\pi$
$193$	718.117	−	1243.81i	0.267830	−	0.463895i	0.968301	+	0.249786i	$0.0803602\pi$
$194$		0			0
$195$	5189.85			1.90591
$196$		0			0
$197$	2508.66			0.907282			0.453641	−	0.891184i	$-0.350125\pi$
$197$	2508.66			0.907282			0.453641	+	0.891184i	$0.350125\pi$
$198$		0			0
$199$	−2046.88	+	3545.30i	−0.729143	+	1.26291i	0.228102	+	0.973637i	$0.426748\pi$
$199$	−2046.88	+	3545.30i	−0.729143	+	1.26291i	−0.957246	+	0.289276i	$0.906585\pi$
$200$		0			0
$201$	−289.798	−	501.945i	−0.101695	−	0.176142i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−537.105	−	930.292i	−0.182990	−	0.316948i
$206$		0			0
$207$	−2142.80	+	3711.43i	−0.719491	+	1.24619i
$208$		0			0
$209$	2282.41			0.755395
$210$		0			0
$211$	−2308.68			−0.753253			−0.376626	−	0.926365i	$-0.622916\pi$
$211$	−2308.68			−0.753253			−0.376626	+	0.926365i	$0.622916\pi$
$212$		0			0
$213$	−749.515	+	1298.20i	−0.241108	+	0.417611i
$214$		0			0
$215$	−243.748	−	422.184i	−0.0773185	−	0.133920i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3524.98	+	6105.45i	1.08765	+	1.88387i
$220$		0			0
$221$	4181.82	−	7243.12i	1.27285	−	2.20464i
$222$		0			0
$223$	−775.802			−0.232967			−0.116483	−	0.993193i	$-0.537162\pi$
$223$	−775.802			−0.232967			−0.116483	+	0.993193i	$0.537162\pi$
$224$		0			0
$225$	−5928.35			−1.75655
$226$		0			0
$227$	89.1192	−	154.359i	0.0260575	−	0.0451329i	−0.852703	−	0.522397i	$-0.825038\pi$
$227$	89.1192	−	154.359i	0.0260575	−	0.0451329i	0.878760	+	0.477264i	$0.158371\pi$
$228$		0			0
$229$	1304.91	+	2260.17i	0.376553	+	0.652209i	0.990558	−	0.137093i	$-0.0437759\pi$
$229$	1304.91	+	2260.17i	0.376553	+	0.652209i	−0.614005	+	0.789302i	$0.710443\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−787.919	−	1364.72i	−0.221538	−	0.383715i	0.733737	−	0.679433i	$-0.237774\pi$
$233$	−787.919	−	1364.72i	−0.221538	−	0.383715i	−0.955275	+	0.295719i	$0.904441\pi$
$234$		0			0
$235$	−1088.71	+	1885.70i	−0.302212	+	0.523446i
$236$		0			0
$237$	−2798.71			−0.767071
$238$		0			0
$239$	620.547			0.167949			0.0839746	−	0.996468i	$-0.473239\pi$
$239$	620.547			0.167949			0.0839746	+	0.996468i	$0.473239\pi$
$240$		0			0
$241$	1276.40	−	2210.78i	0.341161	−	0.590909i	−0.643487	−	0.765457i	$-0.722513\pi$
$241$	1276.40	−	2210.78i	0.341161	−	0.590909i	0.984649	+	0.174548i	$0.0558464\pi$
$242$		0			0
$243$	3775.51	+	6539.38i	0.996704	+	1.72634i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5652.29	−	9790.05i	−1.45606	−	2.52197i
$248$		0			0
$249$	−210.007	+	363.743i	−0.0534484	+	0.0925753i
$250$		0			0
$251$	−4057.62			−1.02038			−0.510189	−	0.860062i	$-0.670425\pi$
$251$	−4057.62			−1.02038			−0.510189	+	0.860062i	$0.670425\pi$
$252$		0			0
$253$	−1194.80			−0.296902
$254$		0			0
$255$	2621.72	−	4540.96i	0.643838	−	1.11516i
$256$		0			0
$257$	−2580.56	−	4469.67i	−0.626347	−	1.08486i	−0.988279	−	0.152660i	$-0.951216\pi$
$257$	−2580.56	−	4469.67i	−0.626347	−	1.08486i	0.361932	−	0.932204i	$-0.382117\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−2747.99	−	4759.67i	−0.651711	−	1.12880i
$262$		0			0
$263$	−3559.82	+	6165.80i	−0.834632	+	1.44562i	0.0596982	+	0.998216i	$0.480986\pi$
$263$	−3559.82	+	6165.80i	−0.834632	+	1.44562i	−0.894330	+	0.447408i	$0.852347\pi$
$264$		0			0
$265$	311.646			0.0722426
$266$		0			0
$267$	3788.62			0.868389
$268$		0			0
$269$	3001.35	−	5198.50i	0.680282	−	1.17828i	−0.294613	−	0.955617i	$-0.595191\pi$
$269$	3001.35	−	5198.50i	0.680282	−	1.17828i	0.974895	−	0.222666i	$-0.0714759\pi$
$270$		0			0
$271$	396.542	+	686.831i	0.0888864	+	0.153956i	0.907041	−	0.421043i	$-0.138336\pi$
$271$	396.542	+	686.831i	0.0888864	+	0.153956i	−0.818154	+	0.574999i	$0.805003\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−826.393	−	1431.36i	−0.181212	−	0.313869i
$276$		0			0
$277$	2027.17	−	3511.15i	0.439713	−	0.761606i	−0.557954	−	0.829872i	$-0.688413\pi$
$277$	2027.17	−	3511.15i	0.439713	−	0.761606i	0.997667	+	0.0682662i	$0.0217467\pi$
$278$		0			0
$279$	−10892.2			−2.33728
$280$		0			0
$281$	2990.91			0.634955			0.317478	−	0.948266i	$-0.397164\pi$
$281$	2990.91			0.634955			0.317478	+	0.948266i	$0.397164\pi$
$282$		0			0
$283$	−2858.05	+	4950.30i	−0.600331	+	1.03980i	0.392440	+	0.919778i	$0.371631\pi$
$283$	−2858.05	+	4950.30i	−0.600331	+	1.03980i	−0.992771	+	0.120026i	$0.961702\pi$
$284$		0			0
$285$	−3543.61	−	6137.71i	−0.736510	−	1.27567i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1768.50	−	3063.13i	−0.359963	−	0.623475i
$290$		0			0
$291$	2514.30	−	4354.89i	0.506497	−	0.877279i
$292$		0			0
$293$	−5301.47			−1.05705			−0.528524	−	0.848918i	$-0.677254\pi$
$293$	−5301.47			−1.05705			−0.528524	+	0.848918i	$0.677254\pi$
$294$		0			0
$295$	3321.86			0.655613
$296$		0			0
$297$	−3442.74	+	5963.00i	−0.672619	+	1.16501i
$298$		0			0
$299$	2958.86	+	5124.90i	0.572293	+	0.991240i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−3629.12	−	6285.83i	−0.688078	−	1.19179i
$304$		0			0
$305$	−817.638	+	1416.19i	−0.153501	+	0.265872i
$306$		0			0
$307$	5978.40			1.11142			0.555709	−	0.831377i	$-0.312447\pi$
$307$	5978.40			1.11142			0.555709	+	0.831377i	$0.312447\pi$
$308$		0			0
$309$	−8972.19			−1.65181
$310$		0			0
$311$	−2425.36	+	4200.84i	−0.442217	+	0.765942i	−0.997854	−	0.0654832i	$-0.979141\pi$
$311$	−2425.36	+	4200.84i	−0.442217	+	0.765942i	0.555637	+	0.831425i	$0.312474\pi$
$312$		0			0
$313$	2227.67	+	3858.44i	0.402285	+	0.696779i	0.994001	−	0.109368i	$-0.0348826\pi$
$313$	2227.67	+	3858.44i	0.402285	+	0.696779i	−0.591716	+	0.806147i	$0.701549\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5130.93	+	8887.04i	0.909091	+	1.57459i	0.815329	+	0.578998i	$0.196556\pi$
$317$	5130.93	+	8887.04i	0.909091	+	1.57459i	0.0937624	+	0.995595i	$0.470111\pi$
$318$		0			0
$319$	766.124	−	1326.97i	0.134466	−	0.232902i
$320$		0			0
$321$	−16221.1			−2.82048
$322$		0			0
$323$	−11421.3			−1.96749
$324$		0			0
$325$	−4093.06	+	7089.38i	−0.698591	+	1.20999i
$326$		0			0
$327$	−3777.63	−	6543.04i	−0.638848	−	1.10652i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1543.56	+	2673.52i	0.256319	+	0.443958i	0.965253	−	0.261317i	$-0.0841569\pi$
$331$	1543.56	+	2673.52i	0.256319	+	0.443958i	−0.708934	+	0.705275i	$0.750824\pi$
$332$		0			0
$333$	−10245.1	+	17745.0i	−1.68596	+	2.92018i
$334$		0			0
$335$	−355.912			−0.0580465
$336$		0			0
$337$	4588.57			0.741707			0.370853	−	0.928691i	$-0.379065\pi$
$337$	4588.57			0.741707			0.370853	+	0.928691i	$0.379065\pi$
$338$		0			0
$339$	−3027.31	+	5243.46i	−0.485018	+	0.840075i
$340$		0			0
$341$	−1518.34	−	2629.85i	−0.241123	−	0.417637i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	1855.01	+	3212.98i	0.289480	+	0.501394i
$346$		0			0
$347$	4822.79	−	8353.32i	0.746112	−	1.29230i	−0.203561	−	0.979062i	$-0.565252\pi$
$347$	4822.79	−	8353.32i	0.746112	−	1.29230i	0.949673	−	0.313242i	$-0.101415\pi$
$348$		0			0
$349$	2612.70			0.400730			0.200365	−	0.979721i	$-0.435787\pi$
$349$	2612.70			0.400730			0.200365	+	0.979721i	$0.435787\pi$
$350$		0			0
$351$	34103.2			5.18602
$352$		0			0
$353$	4344.04	−	7524.09i	0.654985	−	1.13447i	−0.326912	−	0.945055i	$-0.606008\pi$
$353$	4344.04	−	7524.09i	0.654985	−	1.13447i	0.981897	−	0.189413i	$-0.0606585\pi$
$354$		0			0
$355$	460.254	+	797.184i	0.0688106	+	0.119183i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1369.41	+	2371.89i	0.201322	+	0.348701i	0.948955	−	0.315412i	$-0.102143\pi$
$359$	1369.41	+	2371.89i	0.201322	+	0.348701i	−0.747632	+	0.664113i	$0.768809\pi$
$360$		0			0
$361$	−4289.22	+	7429.14i	−0.625342	+	1.08312i
$362$		0			0
$363$	9575.79			1.38457
$364$		0			0
$365$	4329.17			0.620819
$366$		0			0
$367$	984.294	−	1704.85i	0.139999	−	0.242486i	−0.787497	−	0.616319i	$-0.788623\pi$
$367$	984.294	−	1704.85i	0.139999	−	0.242486i	0.927496	+	0.373833i	$0.121957\pi$
$368$		0			0
$369$	−5979.69	−	10357.1i	−0.843604	−	1.46117i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4957.80	+	8587.16i	0.688217	+	1.19203i	0.972414	+	0.233262i	$0.0749398\pi$
$373$	4957.80	+	8587.16i	0.688217	+	1.19203i	−0.284197	+	0.958766i	$0.591727\pi$
$374$		0			0
$375$	−6131.15	+	10619.5i	−0.844296	+	1.46236i
$376$		0			0
$377$	−7589.10			−1.03676
$378$		0			0
$379$	10933.4			1.48183			0.740913	−	0.671601i	$-0.234393\pi$
$379$	10933.4			1.48183			0.740913	+	0.671601i	$0.234393\pi$
$380$		0			0
$381$	1582.87	−	2741.62i	0.212843	−	0.368654i
$382$		0			0
$383$	−6004.84	−	10400.7i	−0.801131	−	1.38760i	−0.918872	−	0.394555i	$-0.870899\pi$
$383$	−6004.84	−	10400.7i	−0.801131	−	1.38760i	0.117742	−	0.993044i	$-0.462435\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2713.69	−	4700.25i	−0.356446	−	0.617383i
$388$		0			0
$389$	−4621.13	+	8004.04i	−0.602316	+	1.04324i	0.390154	+	0.920750i	$0.372422\pi$
$389$	−4621.13	+	8004.04i	−0.602316	+	1.04324i	−0.992470	+	0.122492i	$0.960912\pi$
$390$		0			0
$391$	5978.84			0.773306
$392$		0			0
$393$	−23254.4			−2.98481
$394$		0			0
$395$	−859.301	+	1488.35i	−0.109459	+	0.189588i
$396$		0			0
$397$	1851.23	+	3206.42i	0.234032	+	0.405354i	0.958991	−	0.283437i	$-0.0914748\pi$
$397$	1851.23	+	3206.42i	0.234032	+	0.405354i	−0.724959	+	0.688792i	$0.758141\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1597.39	−	2766.76i	−0.198927	−	0.344552i	0.749254	−	0.662283i	$-0.230412\pi$
$401$	−1597.39	−	2766.76i	−0.198927	−	0.344552i	−0.948181	+	0.317731i	$0.897079\pi$
$402$		0			0
$403$	−7520.22	+	13025.4i	−0.929551	+	1.61003i
$404$		0			0
$405$	10851.4			1.33138
$406$		0			0
$407$	−5712.52			−0.695723
$408$		0			0
$409$	3108.79	−	5384.59i	0.375843	−	0.650980i	−0.614610	−	0.788831i	$-0.710686\pi$
$409$	3108.79	−	5384.59i	0.375843	−	0.650980i	0.990453	+	0.137852i	$0.0440197\pi$
$410$		0			0
$411$	−846.689	−	1466.51i	−0.101616	−	0.176004i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	128.959	+	223.363i	0.0152538	+	0.0264204i
$416$		0			0
$417$	7177.72	−	12432.2i	0.842913	−	1.45997i
$418$		0			0
$419$	290.002			0.0338127			0.0169063	−	0.999857i	$-0.494618\pi$
$419$	290.002			0.0338127			0.0169063	+	0.999857i	$0.494618\pi$
$420$		0			0
$421$	16109.3			1.86489			0.932445	−	0.361311i	$-0.117671\pi$
$421$	16109.3			1.86489			0.932445	+	0.361311i	$0.117671\pi$
$422$		0			0
$423$	−12120.8	+	20993.9i	−1.39323	+	2.41314i
$424$		0			0
$425$	4135.32	+	7162.59i	0.471983	+	0.817498i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	8054.30	+	13950.5i	0.906446	+	1.57001i
$430$		0			0
$431$	35.6998	−	61.8339i	0.00398979	−	0.00691052i	−0.864024	−	0.503451i	$-0.832063\pi$
$431$	35.6998	−	61.8339i	0.00398979	−	0.00691052i	0.868013	+	0.496541i	$0.165397\pi$
$432$		0			0
$433$	2473.42			0.274515			0.137258	−	0.990535i	$-0.456171\pi$
$433$	2473.42			0.274515			0.137258	+	0.990535i	$0.456171\pi$
$434$		0			0
$435$	−4757.86			−0.524418
$436$		0			0
$437$	4040.60	−	6998.53i	0.442307	−	0.766098i
$438$		0			0
$439$	6525.92	+	11303.2i	0.709488	+	1.22887i	0.965047	+	0.262076i	$0.0844070\pi$
$439$	6525.92	+	11303.2i	0.709488	+	1.22887i	−0.255560	+	0.966793i	$0.582260\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2552.18	−	4420.50i	−0.273719	−	0.474096i	0.696092	−	0.717953i	$-0.254921\pi$
$443$	−2552.18	−	4420.50i	−0.273719	−	0.474096i	−0.969811	+	0.243857i	$0.921587\pi$
$444$		0			0
$445$	1163.24	−	2014.79i	0.123916	−	0.214629i
$446$		0			0
$447$	−29244.8			−3.09447
$448$		0			0
$449$	−7106.77			−0.746969			−0.373485	−	0.927636i	$-0.621837\pi$
$449$	−7106.77			−0.746969			−0.373485	+	0.927636i	$0.621837\pi$
$450$		0			0
$451$	1667.10	−	2887.50i	0.174059	−	0.301479i
$452$		0			0
$453$	−10191.3	−	17651.8i	−1.05702	−	1.83081i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−471.474	−	816.618i	−0.0482596	−	0.0835881i	0.840887	−	0.541211i	$-0.182034\pi$
$457$	−471.474	−	816.618i	−0.0482596	−	0.0835881i	−0.889146	+	0.457623i	$0.848701\pi$
$458$		0			0
$459$	17227.7	−	29839.2i	1.75189	−	3.03437i
$460$		0			0
$461$	1906.71			0.192634			0.0963172	−	0.995351i	$-0.469294\pi$
$461$	1906.71			0.192634			0.0963172	+	0.995351i	$0.469294\pi$
$462$		0			0
$463$	−4896.83			−0.491523			−0.245761	−	0.969330i	$-0.579038\pi$
$463$	−4896.83			−0.491523			−0.245761	+	0.969330i	$0.579038\pi$
$464$		0			0
$465$	−4714.68	+	8166.07i	−0.470190	+	0.814392i
$466$		0			0
$467$	181.948	+	315.143i	0.0180290	+	0.0312271i	0.874899	−	0.484305i	$-0.160928\pi$
$467$	181.948	+	315.143i	0.0180290	+	0.0312271i	−0.856870	+	0.515532i	$0.827594\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1328.62	−	2301.23i	−0.129977	−	0.225128i
$472$		0			0
$473$	756.561	−	1310.40i	0.0735449	−	0.127383i
$474$		0			0
$475$	11178.9			1.07984
$476$		0			0
$477$	3469.62			0.333046
$478$		0			0
$479$	−2611.92	+	4523.98i	−0.249148	+	0.431537i	−0.963290	−	0.268464i	$-0.913484\pi$
$479$	−2611.92	+	4523.98i	−0.249148	+	0.431537i	0.714142	+	0.700001i	$0.246817\pi$
$480$		0			0
$481$	14146.8	+	24503.0i	1.34104	+	2.32275i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1543.95	−	2674.20i	−0.144551	−	0.250370i
$486$		0			0
$487$	−6066.58	+	10507.6i	−0.564482	+	0.977712i	0.432615	+	0.901579i	$0.357591\pi$
$487$	−6066.58	+	10507.6i	−0.564482	+	0.977712i	−0.997098	+	0.0761334i	$0.975743\pi$
$488$		0			0
$489$	−11751.1			−1.08671
$490$		0			0
$491$	2875.76			0.264320			0.132160	−	0.991228i	$-0.457809\pi$
$491$	2875.76			0.264320			0.132160	+	0.991228i	$0.457809\pi$
$492$		0			0
$493$	−3833.73	+	6640.22i	−0.350228	+	0.606613i
$494$		0			0
$495$	3581.80	+	6203.85i	0.325232	+	0.563318i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−7821.93	−	13548.0i	−0.701719	−	1.21541i	−0.967863	−	0.251480i	$-0.919083\pi$
$499$	−7821.93	−	13548.0i	−0.701719	−	1.21541i	0.266143	−	0.963933i	$-0.414251\pi$
$500$		0			0
$501$	13406.4	−	23220.5i	1.19551	−	2.07069i
$502$		0			0
$503$	3530.58			0.312964			0.156482	−	0.987681i	$-0.449985\pi$
$503$	3530.58			0.312964			0.156482	+	0.987681i	$0.449985\pi$
$504$		0			0
$505$	−4457.07			−0.392746
$506$		0			0
$507$	29305.0	−	50757.7i	2.56702	−	4.44621i
$508$		0			0
$509$	−3026.80	−	5242.57i	−0.263577	−	0.456528i	0.703613	−	0.710583i	$-0.251569\pi$
$509$	−3026.80	−	5242.57i	−0.263577	−	0.456528i	−0.967190	+	0.254055i	$0.918236\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−23285.5	−	40331.7i	−2.00406	−	3.47113i
$514$		0			0
$515$	−2754.77	+	4771.41i	−0.235708	+	0.408259i
$516$		0			0
$517$	−6758.43			−0.574923
$518$		0			0
$519$	−26090.4			−2.20663
$520$		0			0
$521$	10438.2	−	18079.5i	0.877746	−	1.52030i	0.0239383	−	0.999713i	$-0.492379\pi$
$521$	10438.2	−	18079.5i	0.877746	−	1.52030i	0.853808	−	0.520588i	$-0.174287\pi$
$522$		0			0
$523$	9508.68	+	16469.5i	0.795001	+	1.37698i	0.922838	+	0.385187i	$0.125863\pi$
$523$	9508.68	+	16469.5i	0.795001	+	1.37698i	−0.127837	+	0.991795i	$0.540804\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7597.88	+	13159.9i	0.628024	+	1.08777i
$528$		0			0
$529$	3968.32	−	6873.34i	0.326155	−	0.564916i
$530$		0			0
$531$	36982.9			3.02245
$532$		0			0
$533$	−16514.0			−1.34203
$534$		0			0
$535$	−4980.44	+	8626.38i	−0.402474	+	0.697105i
$536$		0			0
$537$	−8296.97	−	14370.8i	−0.666742	−	1.15483i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7035.33	−	12185.6i	−0.559099	−	0.968388i	−0.997572	−	0.0696433i	$-0.977814\pi$
$541$	−7035.33	−	12185.6i	−0.559099	−	0.968388i	0.438473	−	0.898744i	$-0.355519\pi$
$542$		0			0
$543$	3009.69	−	5212.94i	0.237860	−	0.411986i
$544$		0			0
$545$	−4639.45			−0.364646
$546$		0			0
$547$	−16099.5			−1.25844			−0.629218	−	0.777229i	$-0.716625\pi$
$547$	−16099.5			−1.25844			−0.629218	+	0.777229i	$0.716625\pi$
$548$		0			0
$549$	−9102.91	+	15766.7i	−0.707656	+	1.22570i
$550$		0			0
$551$	5181.80	+	8975.15i	0.400639	+	0.693928i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8869.12	+	15361.8i	0.678330	+	1.17490i
$556$		0			0
$557$	−6163.61	+	10675.7i	−0.468870	+	0.812107i	−0.999367	−	0.0355800i	$-0.988672\pi$
$557$	−6163.61	+	10675.7i	−0.468870	+	0.812107i	0.530497	+	0.847687i	$0.322005\pi$
$558$		0			0
$559$	−7494.37			−0.567045
$560$		0			0
$561$	16274.9			1.22483
$562$		0			0
$563$	−2246.17	+	3890.48i	−0.168143	+	0.291233i	−0.937767	−	0.347265i	$-0.887110\pi$
$563$	−2246.17	+	3890.48i	−0.168143	+	0.291233i	0.769624	+	0.638498i	$0.220444\pi$
$564$		0			0
$565$	1858.98	+	3219.85i	0.138421	+	0.239752i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8447.56	+	14631.6i	0.622390	+	1.07801i	0.989039	+	0.147652i	$0.0471716\pi$
$569$	8447.56	+	14631.6i	0.622390	+	1.07801i	−0.366649	+	0.930359i	$0.619495\pi$
$570$		0			0
$571$	44.0576	−	76.3099i	0.00322899	−	0.00559277i	−0.864406	−	0.502794i	$-0.832306\pi$
$571$	44.0576	−	76.3099i	0.00322899	−	0.00559277i	0.867635	+	0.497201i	$0.165639\pi$
$572$		0			0
$573$	46163.8			3.36565
$574$		0			0
$575$	−5851.93			−0.424422
$576$		0			0
$577$	−4086.46	+	7077.96i	−0.294838	+	0.510674i	−0.974947	−	0.222436i	$-0.928599\pi$
$577$	−4086.46	+	7077.96i	−0.294838	+	0.510674i	0.680109	+	0.733111i	$0.261932\pi$
$578$		0			0
$579$	−6921.18	−	11987.8i	−0.496778	−	0.860445i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	483.654	+	837.714i	0.0343583	+	0.0595104i
$584$		0			0
$585$	17740.3	−	30727.2i	1.25380	−	2.17165i
$586$		0			0
$587$	−18301.4			−1.28685			−0.643425	−	0.765509i	$-0.722487\pi$
$587$	−18301.4			−1.28685			−0.643425	+	0.765509i	$0.722487\pi$
$588$		0			0
$589$	20539.1			1.43684
$590$		0			0
$591$	12089.2	−	20939.1i	0.841426	−	1.45739i
$592$		0			0
$593$	1034.59	+	1791.97i	0.0716452	+	0.124093i	0.899622	−	0.436669i	$-0.143842\pi$
$593$	1034.59	+	1791.97i	0.0716452	+	0.124093i	−0.827977	+	0.560762i	$0.810508\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19727.8	+	34169.5i	1.35244	+	2.34249i
$598$		0			0
$599$	−6807.81	+	11791.5i	−0.464373	+	0.804318i	−0.999173	−	0.0406606i	$-0.987054\pi$
$599$	−6807.81	+	11791.5i	−0.464373	+	0.804318i	0.534800	+	0.844979i	$0.320387\pi$
$600$		0			0
$601$	−1041.12			−0.0706624			−0.0353312	−	0.999376i	$-0.511249\pi$
$601$	−1041.12			−0.0706624			−0.0353312	+	0.999376i	$0.511249\pi$
$602$		0			0
$603$	−3962.44			−0.267600
$604$		0			0
$605$	2940.10	−	5092.40i	0.197574	−	0.342207i
$606$		0			0
$607$	7747.98	+	13419.9i	0.518090	+	0.897358i	0.999779	+	0.0210160i	$0.00669010\pi$
$607$	7747.98	+	13419.9i	0.518090	+	0.897358i	−0.481689	+	0.876342i	$0.659977\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	16737.0	+	28989.3i	1.10819	+	1.91944i
$612$		0			0
$613$	−468.229	+	810.996i	−0.0308509	+	0.0534353i	−0.881039	−	0.473044i	$-0.843155\pi$
$613$	−468.229	+	810.996i	−0.0308509	+	0.0534353i	0.850188	+	0.526480i	$0.176488\pi$
$614$		0			0
$615$	−10353.2			−0.678831
$616$		0			0
$617$	−798.142			−0.0520778			−0.0260389	−	0.999661i	$-0.508289\pi$
$617$	−798.142			−0.0520778			−0.0260389	+	0.999661i	$0.508289\pi$
$618$		0			0
$619$	499.517	−	865.189i	0.0324350	−	0.0561791i	−0.849352	−	0.527826i	$-0.823007\pi$
$619$	499.517	−	865.189i	0.0324350	−	0.0561791i	0.881787	+	0.471647i	$0.156340\pi$
$620$		0			0
$621$	12189.5	+	21112.9i	0.787679	+	1.36430i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1858.35	−	3218.75i	−0.118934	−	0.206000i
$626$		0			0
$627$	10998.9	−	19050.6i	0.700564	−	1.21341i
$628$		0			0
$629$	28585.8			1.81207
$630$		0			0
$631$	25618.7			1.61626			0.808132	−	0.589001i	$-0.200479\pi$
$631$	25618.7			1.61626			0.808132	+	0.589001i	$0.200479\pi$
$632$		0			0
$633$	−11125.5	+	19270.0i	−0.698577	+	1.20997i
$634$		0			0
$635$	−971.994	−	1683.54i	−0.0607439	−	0.105212i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	5124.10	+	8875.20i	0.317224	+	0.549448i
$640$		0			0
$641$	−9175.68	+	15892.7i	−0.565394	+	0.979291i	0.431619	+	0.902056i	$0.357943\pi$
$641$	−9175.68	+	15892.7i	−0.565394	+	0.979291i	−0.997013	+	0.0772350i	$0.975391\pi$
$642$		0			0
$643$	22068.9			1.35352			0.676759	−	0.736205i	$-0.263384\pi$
$643$	22068.9			1.35352			0.676759	+	0.736205i	$0.263384\pi$
$644$		0			0
$645$	−4698.47			−0.286825
$646$		0			0
$647$	−3204.75	+	5550.79i	−0.194732	+	0.337286i	−0.946813	−	0.321785i	$-0.895717\pi$
$647$	−3204.75	+	5550.79i	−0.194732	+	0.337286i	0.752081	+	0.659071i	$0.229050\pi$
$648$		0			0
$649$	5155.30	+	8929.24i	0.311808	+	0.540067i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−10549.4	−	18272.1i	−0.632206	−	1.09501i	−0.987100	−	0.160106i	$-0.948816\pi$
$653$	−10549.4	−	18272.1i	−0.632206	−	1.09501i	0.354894	−	0.934907i	$-0.384517\pi$
$654$		0			0
$655$	−7139.92	+	12366.7i	−0.425923	+	0.737721i
$656$		0			0
$657$	48197.4			2.86204
$658$		0			0
$659$	−29509.0			−1.74432			−0.872159	−	0.489222i	$-0.837281\pi$
$659$	−29509.0			−1.74432			−0.872159	+	0.489222i	$0.837281\pi$
$660$		0			0
$661$	10858.9	−	18808.2i	0.638976	−	1.10674i	−0.346682	−	0.937983i	$-0.612692\pi$
$661$	10858.9	−	18808.2i	0.638976	−	1.10674i	0.985658	−	0.168756i	$-0.0539751\pi$
$662$		0			0
$663$	−40304.2	−	69809.0i	−2.36091	−	4.08922i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2712.58	−	4698.32i	−0.157468	−	0.272743i
$668$		0			0
$669$	−3738.58	+	6475.41i	−0.216057	+	0.374221i
$670$		0			0
$671$	−5075.67			−0.292018
$672$		0			0
$673$	3168.78			0.181497			0.0907486	−	0.995874i	$-0.471074\pi$
$673$	3168.78			0.181497			0.0907486	+	0.995874i	$0.471074\pi$
$674$		0			0
$675$	−16862.0	+	29205.9i	−0.961510	+	1.66538i
$676$		0			0
$677$	−11293.4	−	19560.8i	−0.641124	−	1.11046i	−0.985182	−	0.171511i	$-0.945135\pi$
$677$	−11293.4	−	19560.8i	−0.641124	−	1.11046i	0.344058	−	0.938948i	$-0.388198\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−858.928	−	1487.71i	−0.0483321	−	0.0837137i
$682$		0			0
$683$	3565.41	−	6175.47i	0.199746	−	0.345970i	−0.748700	−	0.662909i	$-0.769322\pi$
$683$	3565.41	−	6175.47i	0.199746	−	0.345970i	0.948446	+	0.316939i	$0.102655\pi$
$684$		0			0
$685$	−1039.85			−0.0580010
$686$		0			0
$687$	25153.3			1.39688
$688$		0			0
$689$	2395.50	−	4149.13i	0.132455	−	0.229418i
$690$		0			0
$691$	−5707.28	−	9885.29i	−0.314204	−	0.544217i	0.665064	−	0.746786i	$-0.268404\pi$
$691$	−5707.28	−	9885.29i	−0.314204	−	0.544217i	−0.979268	+	0.202569i	$0.935071\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4407.62	−	7634.22i	−0.240562	−	0.416665i
$696$		0			0
$697$	−8342.27	+	14449.2i	−0.453352	+	0.785228i
$698$		0			0
$699$	−15187.9			−0.821829
$700$		0			0
$701$	10005.8			0.539106			0.269553	−	0.962986i	$-0.413124\pi$
$701$	10005.8			0.539106			0.269553	+	0.962986i	$0.413124\pi$
$702$		0			0
$703$	19318.8	−	33461.1i	1.03645	−	1.79518i
$704$		0			0
$705$	10493.0	+	18174.4i	0.560551	+	0.970902i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4770.10	+	8262.06i	0.252673	+	0.437642i	0.964261	−	0.264955i	$-0.0853571\pi$
$709$	4770.10	+	8262.06i	0.252673	+	0.437642i	−0.711588	+	0.702597i	$0.752024\pi$
$710$		0			0
$711$	−9566.76	+	16570.1i	−0.504615	+	0.874019i
$712$		0			0
$713$	−10751.8			−0.564739
$714$		0			0
$715$	9891.80			0.517388
$716$		0			0
$717$	2990.41	−	5179.54i	0.155758	−	0.269782i
$718$		0			0
$719$	2856.77	+	4948.07i	0.148177	+	0.256651i	0.930554	−	0.366155i	$-0.119326\pi$
$719$	2856.77	+	4948.07i	0.148177	+	0.256651i	−0.782377	+	0.622806i	$0.785993\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−12301.9	−	21307.5i	−0.632796	−	1.09603i
$724$		0			0
$725$	3752.36	−	6499.28i	0.192220	−	0.332934i
$726$		0			0
$727$	18615.1			0.949649			0.474824	−	0.880081i	$-0.342512\pi$
$727$	18615.1			0.949649			0.474824	+	0.880081i	$0.342512\pi$
$728$		0			0
$729$	23271.8			1.18233
$730$		0			0
$731$	−3785.88	+	6557.33i	−0.191554	+	0.331781i
$732$		0			0
$733$	13226.1	+	22908.3i	0.666463	+	1.15435i	0.978887	+	0.204404i	$0.0655258\pi$
$733$	13226.1	+	22908.3i	0.666463	+	1.15435i	−0.312424	+	0.949943i	$0.601141\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−552.352	−	956.702i	−0.0276067	−	0.0478162i
$738$		0			0
$739$	11458.0	−	19845.9i	0.570353	−	0.987880i	−0.426177	−	0.904640i	$-0.640140\pi$
$739$	11458.0	−	19845.9i	0.570353	−	0.987880i	0.996530	−	0.0832398i	$-0.0265267\pi$
$740$		0			0
$741$	−108953.			−5.40148
$742$		0			0
$743$	−24313.3			−1.20050			−0.600248	−	0.799814i	$-0.704931\pi$
$743$	−24313.3			−1.20050			−0.600248	+	0.799814i	$0.704931\pi$
$744$		0			0
$745$	−8979.16	+	15552.4i	−0.441572	+	0.764825i
$746$		0			0
$747$	1435.72	+	2486.74i	0.0703218	+	0.121801i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2809.24	+	4865.74i	0.136499	+	0.236423i	0.926169	−	0.377109i	$-0.123082\pi$
$751$	2809.24	+	4865.74i	0.136499	+	0.236423i	−0.789670	+	0.613531i	$0.789748\pi$
$752$		0			0
$753$	−19553.6	+	33867.9i	−0.946313	+	1.63906i
$754$		0			0
$755$	−12516.3			−0.603332
$756$		0			0
$757$	−2786.08			−0.133767			−0.0668837	−	0.997761i	$-0.521306\pi$
$757$	−2786.08			−0.133767			−0.0668837	+	0.997761i	$0.521306\pi$
$758$		0			0
$759$	−5757.71	+	9972.64i	−0.275351	+	0.476922i
$760$		0			0
$761$	−3407.22	−	5901.47i	−0.162302	−	0.281115i	0.773392	−	0.633928i	$-0.218558\pi$
$761$	−3407.22	−	5901.47i	−0.162302	−	0.281115i	−0.935694	+	0.352813i	$0.885225\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−17923.5	−	31044.5i	−0.847094	−	1.46721i
$766$		0			0
$767$	25533.8	−	44225.8i	1.20205	−	2.08201i
$768$		0			0
$769$	−13274.3			−0.622474			−0.311237	−	0.950332i	$-0.600743\pi$
$769$	−13274.3			−0.622474			−0.311237	+	0.950332i	$0.600743\pi$
$770$		0			0
$771$	−49742.8			−2.32353
$772$		0			0
$773$	−9325.70	+	16152.6i	−0.433922	+	0.751576i	−0.997207	−	0.0746872i	$-0.976204\pi$
$773$	−9325.70	+	16152.6i	−0.433922	+	0.751576i	0.563285	+	0.826263i	$0.309537\pi$
$774$		0			0
$775$	−7436.61	−	12880.6i	−0.344685	−	0.597012i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	11275.7	+	19530.1i	0.518606	+	0.898252i
$780$		0			0
$781$	−1428.57	+	2474.35i	−0.0654522	+	0.113367i
$782$		0			0
$783$	−31264.5			−1.42695
$784$		0			0
$785$	−1631.72			−0.0741895
$786$		0			0
$787$	−17514.2	+	30335.5i	−0.793284	+	1.37401i	0.130639	+	0.991430i	$0.458297\pi$
$787$	−17514.2	+	30335.5i	−0.793284	+	1.37401i	−0.923923	+	0.382578i	$0.875036\pi$
$788$		0			0
$789$	34309.5	+	59425.7i	1.54810	+	2.68138i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12569.7	+	21771.4i	0.562879	+	0.974935i
$794$		0			0
$795$	1501.82	−	2601.23i	0.0669988	−	0.116045i
$796$		0			0
$797$	1170.81			0.0520352			0.0260176	−	0.999661i	$-0.491717\pi$
$797$	1170.81			0.0520352			0.0260176	+	0.999661i	$0.491717\pi$
$798$		0			0
$799$	33819.6			1.49744
$800$		0			0
$801$	12950.5	−	22431.0i	0.571267	−	0.989463i
$802$		0			0
$803$	6718.57	+	11636.9i	0.295259	+	0.511404i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−28927.0	−	50103.0i	−1.26181	−	2.18551i
$808$		0			0
$809$	−6677.38	+	11565.6i	−0.290190	+	0.502625i	−0.973855	−	0.227172i	$-0.927052\pi$
$809$	−6677.38	+	11565.6i	−0.290190	+	0.502625i	0.683664	+	0.729797i	$0.260385\pi$
$810$		0			0
$811$	−12327.2			−0.533745			−0.266872	−	0.963732i	$-0.585990\pi$
$811$	−12327.2			−0.533745			−0.266872	+	0.963732i	$0.585990\pi$
$812$		0			0
$813$	7643.72			0.329738
$814$		0			0
$815$	−3607.98	+	6249.21i	−0.155070	+	0.268589i
$816$		0			0
$817$	5117.12	+	8863.12i	0.219125	+	0.379536i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22726.9	−	39364.2i	−0.966109	−	1.67335i	−0.706605	−	0.707608i	$-0.749774\pi$
$821$	−22726.9	−	39364.2i	−0.966109	−	1.67335i	−0.259504	−	0.965742i	$-0.583559\pi$
$822$		0			0
$823$	−4643.12	+	8042.13i	−0.196657	+	0.340621i	−0.947443	−	0.319926i	$-0.896342\pi$
$823$	−4643.12	+	8042.13i	−0.196657	+	0.340621i	0.750785	+	0.660547i	$0.229675\pi$
$824$		0			0
$825$	−15929.5			−0.672235
$826$		0			0
$827$	22224.4			0.934485			0.467242	−	0.884129i	$-0.345248\pi$
$827$	22224.4			0.934485			0.467242	+	0.884129i	$0.345248\pi$
$828$		0			0
$829$	−22018.4	+	38136.9i	−0.922473	+	1.59777i	−0.126896	+	0.991916i	$0.540501\pi$
$829$	−22018.4	+	38136.9i	−0.922473	+	1.59777i	−0.795576	+	0.605853i	$0.792832\pi$
$830$		0			0
$831$	−19537.8	−	33840.4i	−0.815593	−	1.41265i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8232.44	−	14259.0i	−0.341192	−	0.590962i
$836$		0			0
$837$	−30980.8	+	53660.3i	−1.27939	+	2.21597i
$838$		0			0
$839$	−37521.9			−1.54398			−0.771991	−	0.635633i	$-0.780739\pi$
$839$	−37521.9			−1.54398			−0.771991	+	0.635633i	$0.780739\pi$
$840$		0			0
$841$	−17431.6			−0.714732
$842$		0			0
$843$	14413.1	−	24964.3i	0.588867	−	1.01995i
$844$		0			0
$845$	−17995.3	−	31168.8i	−0.732612	−	1.26892i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	27545.8	+	47710.8i	1.11351	+	1.92866i
$850$		0			0
$851$	−10113.0	+	17516.3i	−0.407367	+	0.705581i
$852$		0			0
$853$	34441.3			1.38247			0.691235	−	0.722630i	$-0.257067\pi$
$853$	34441.3			1.38247			0.691235	+	0.722630i	$0.257067\pi$
$854$		0			0
$855$	−48452.1			−1.93804
$856$		0			0
$857$	12485.4	−	21625.4i	0.497659	−	0.861970i	−0.502338	−	0.864672i	$-0.667527\pi$
$857$	12485.4	−	21625.4i	0.497659	−	0.861970i	0.999996	+	0.00270129i	$0.000859847\pi$
$858$		0			0
$859$	−8677.77	−	15030.3i	−0.344682	−	0.597006i	0.640614	−	0.767863i	$-0.278680\pi$
$859$	−8677.77	−	15030.3i	−0.344682	−	0.597006i	−0.985296	+	0.170857i	$0.945347\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−13284.0	−	23008.5i	−0.523977	−	0.907555i	−0.999610	−	0.0279113i	$-0.991114\pi$
$863$	−13284.0	−	23008.5i	−0.523977	−	0.907555i	0.475633	−	0.879644i	$-0.342219\pi$
$864$		0			0
$865$	−8010.67	+	13874.9i	−0.314880	+	0.545388i
$866$		0			0
$867$	−34089.5			−1.33534
$868$		0			0
$869$	−5334.31			−0.208232
$870$		0			0
$871$	−2735.75	+	4738.46i	−0.106426	+	0.184336i
$872$		0			0
$873$	−17189.1	−	29772.4i	−0.666395	−	1.15423i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−23617.7	−	40907.0i	−0.909363	−	1.57506i	−0.814951	−	0.579530i	$-0.803236\pi$
$877$	−23617.7	−	40907.0i	−0.909363	−	1.57506i	−0.0944124	−	0.995533i	$-0.530097\pi$
$878$		0			0
$879$	−25547.7	+	44249.9i	−0.980321	+	1.69797i
$880$		0			0
$881$	29598.0			1.13188			0.565938	−	0.824448i	$-0.308514\pi$
$881$	29598.0			1.13188			0.565938	+	0.824448i	$0.308514\pi$
$882$		0			0
$883$	42952.5			1.63700			0.818498	−	0.574509i	$-0.194807\pi$
$883$	42952.5			1.63700			0.818498	+	0.574509i	$0.194807\pi$
$884$		0			0
$885$	16008.0	−	27726.6i	0.608025	−	1.05313i
$886$		0			0
$887$	16748.9	+	29010.0i	0.634018	+	1.09815i	0.986722	+	0.162417i	$0.0519290\pi$
$887$	16748.9	+	29010.0i	0.634018	+	1.09815i	−0.352704	+	0.935735i	$0.614738\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	16840.6	+	29168.8i	0.633202	+	1.09674i
$892$		0			0
$893$	22855.9	−	39587.5i	0.856486	−	1.48348i
$894$		0			0
$895$	−10189.8			−0.380568
$896$		0			0
$897$	57034.9			2.12301
$898$		0			0
$899$	6894.26	−	11941.2i	0.255769	−	0.443005i
$900$		0			0
$901$	−2420.24	−	4191.97i	−0.0894891	−	0.155000i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1848.16	−	3201.10i	−0.0678838	−	0.117578i
$906$		0			0
$907$	−6875.57	+	11908.8i	−0.251708	+	0.435972i	−0.963996	−	0.265916i	$-0.914326\pi$
$907$	−6875.57	+	11908.8i	−0.251708	+	0.435972i	0.712288	+	0.701887i	$0.247659\pi$
$908$		0			0
$909$	−49621.4			−1.81060
$910$		0			0
$911$	38706.9			1.40770			0.703851	−	0.710348i	$-0.251462\pi$
$911$	38706.9			1.40770			0.703851	+	0.710348i	$0.251462\pi$
$912$		0			0
$913$	−400.271	+	693.289i	−0.0145093	+	0.0251309i
$914$		0			0
$915$	7880.37	+	13649.2i	0.284718	+	0.493146i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	11646.2	+	20171.8i	0.418034	+	0.724056i	0.995742	−	0.0921874i	$-0.0293859\pi$
$919$	11646.2	+	20171.8i	0.418034	+	0.724056i	−0.577707	+	0.816244i	$0.696053\pi$
$920$		0			0
$921$	28809.8	−	49900.0i	1.03074	−	1.78530i
$922$		0			0
$923$	14151.1			0.504649
$924$		0			0
$925$	−27979.1			−0.994537
$926$		0			0
$927$	−30669.4	+	53121.0i	−1.08664	+	1.88212i
$928$		0			0
$929$	−21462.7	−	37174.4i	−0.757984	−	1.31287i	−0.943877	−	0.330297i	$-0.892851\pi$
$929$	−21462.7	−	37174.4i	−0.757984	−	1.31287i	0.185893	−	0.982570i	$-0.440482\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	23375.5	+	40487.6i	0.820236	+	1.42069i
$934$		0			0
$935$	4996.97	−	8655.01i	0.174779	−	0.302726i
$936$		0			0
$937$	44869.6			1.56438			0.782191	−	0.623039i	$-0.214102\pi$
$937$	44869.6			1.56438			0.782191	+	0.623039i	$0.214102\pi$
$938$		0			0
$939$	42940.4			1.49234
$940$		0			0
$941$	6121.69	−	10603.1i	0.212074	−	0.367323i	−0.740290	−	0.672288i	$-0.765312\pi$
$941$	6121.69	−	10603.1i	0.212074	−	0.367323i	0.952363	+	0.304966i	$0.0986449\pi$
$942$		0			0
$943$	−5902.61	−	10223.6i	−0.203834	−	0.353051i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	21013.5	+	36396.5i	0.721065	+	1.24892i	0.960573	+	0.278027i	$0.0896803\pi$
$947$	21013.5	+	36396.5i	0.721065	+	1.24892i	−0.239509	+	0.970894i	$0.576986\pi$
$948$		0			0
$949$	33276.5	−	57636.6i	1.13825	−	1.97151i
$950$		0			0
$951$	98903.6			3.37242
$952$		0			0
$953$	15757.3			0.535601			0.267801	−	0.963474i	$-0.413703\pi$
$953$	15757.3			0.535601			0.267801	+	0.963474i	$0.413703\pi$
$954$		0			0
$955$	14173.9	−	24549.9i	0.480268	−	0.831849i
$956$		0			0
$957$	−7383.88	−	12789.3i	−0.249412	−	0.431994i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1232.13	+	2134.10i	0.0413590	+	0.0716359i
$962$		0			0
$963$	−55448.2	+	96039.1i	−1.85544	+	3.21372i
$964$		0			0
$965$	−8500.17			−0.283555
$966$		0			0
$967$	−25863.4			−0.860093			−0.430046	−	0.902807i	$-0.641503\pi$
$967$	−25863.4			−0.860093			−0.430046	+	0.902807i	$0.641503\pi$
$968$		0			0
$969$	−55039.1	+	95330.5i	−1.82468	+	3.16043i
$970$		0			0
$971$	−23922.8	−	41435.5i	−0.790647	−	1.36944i	−0.925567	−	0.378585i	$-0.876411\pi$
$971$	−23922.8	−	41435.5i	−0.790647	−	1.36944i	0.134919	−	0.990857i	$-0.456923\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	39448.8	+	68327.3i	1.29577	+	2.24433i
$976$		0			0
$977$	−27361.3	+	47391.2i	−0.895973	+	1.55187i	−0.0633779	+	0.997990i	$0.520187\pi$
$977$	−27361.3	+	47391.2i	−0.895973	+	1.55187i	−0.832595	+	0.553882i	$0.813146\pi$
$978$		0			0
$979$	7221.06			0.235737
$980$		0			0
$981$	−51651.8			−1.68106
$982$		0			0
$983$	−15031.3	+	26035.0i	−0.487715	+	0.844747i	−0.999900	−	0.0141280i	$-0.995503\pi$
$983$	−15031.3	+	26035.0i	−0.487715	+	0.844747i	0.512185	+	0.858875i	$0.328836\pi$
$984$		0			0
$985$	−7423.60	−	12858.1i	−0.240138	−	0.415931i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2678.72	−	4639.67i	−0.0861256	−	0.149174i
$990$		0			0
$991$	18643.3	−	32291.2i	0.597603	−	1.03508i	−0.395571	−	0.918435i	$-0.629453\pi$
$991$	18643.3	−	32291.2i	0.597603	−	1.03508i	0.993174	−	0.116643i	$-0.0372133\pi$
$992$		0			0
$993$	29753.5			0.950855
$994$		0			0
$995$	24228.4			0.771953
$996$		0			0
$997$	−29467.2	+	51038.7i	−0.936045	+	1.62128i	−0.163285	+	0.986579i	$0.552209\pi$
$997$	−29467.2	+	51038.7i	−0.936045	+	1.62128i	−0.772760	+	0.634698i	$0.781124\pi$
$998$		0			0
$999$	58280.2	+	100944.i	1.84575	+	3.19693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	392.4.i.o.361.4		8
7.2	even	3	inner	392.4.i.o.177.4		8
7.3	odd	6		392.4.a.n.1.4	yes	4
7.4	even	3		392.4.a.n.1.1	✓	4
7.5	odd	6	inner	392.4.i.o.177.1		8
7.6	odd	2	inner	392.4.i.o.361.1		8
28.3	even	6		784.4.a.bh.1.1		4
28.11	odd	6		784.4.a.bh.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
392.4.a.n.1.1	✓	4	7.4	even	3
392.4.a.n.1.4	yes	4	7.3	odd	6
392.4.i.o.177.1		8	7.5	odd	6	inner
392.4.i.o.177.4		8	7.2	even	3	inner
392.4.i.o.361.1		8	7.6	odd	2	inner
392.4.i.o.361.4		8	1.1	even	1	trivial
784.4.a.bh.1.1		4	28.3	even	6
784.4.a.bh.1.4		4	28.11	odd	6

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:	\( N \)	\(=\)	\( 392 = 2^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	392.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.81898 - 8.34673i) q^{3} +(-2.95919 - 5.12547i) q^{5} +(-32.9452 - 57.0628i) q^{9} +O(q^{10})\)	\(q+(4.81898 - 8.34673i) q^{3} +(-2.95919 - 5.12547i) q^{5} +(-32.9452 - 57.0628i) q^{9} +(9.18493 - 15.9088i) q^{11} -90.9844 q^{13} -57.0412 q^{15} +(-45.9619 + 79.6084i) q^{17} +(62.1237 + 107.601i) q^{19} +(-32.5206 - 56.3273i) q^{23} +(44.9864 - 77.9187i) q^{25} -374.825 q^{27} +83.4110 q^{29} +(82.6540 - 143.161i) q^{31} +(-88.5240 - 153.328i) q^{33} +(-155.486 - 269.310i) q^{37} +(-438.452 + 759.421i) q^{39} +181.504 q^{41} +82.3699 q^{43} +(-194.982 + 337.719i) q^{45} +(-183.954 - 318.618i) q^{47} +(442.980 + 767.263i) q^{51} +(-26.3287 + 45.6026i) q^{53} -108.720 q^{55} +1197.49 q^{57} +(-280.639 + 486.081i) q^{59} +(-138.152 - 239.287i) q^{61} +(269.240 + 466.337i) q^{65} +(30.0684 - 52.0800i) q^{67} -626.865 q^{69} -155.534 q^{71} +(-365.739 + 633.479i) q^{73} +(-433.577 - 750.978i) q^{75} +(-145.192 - 251.480i) q^{79} +(-916.754 + 1587.86i) q^{81} -43.5791 q^{83} +544.041 q^{85} +(401.956 - 696.209i) q^{87} +(196.547 + 340.429i) q^{89} +(-796.617 - 1379.78i) q^{93} +(367.672 - 636.827i) q^{95} +521.748 q^{97} -1210.40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 136 q^{9}+O(q^{10}) \) 8 * q - 136 * q^9	\( 8 q - 136 q^{9} + 116 q^{11} + 224 q^{15} + 80 q^{23} - 448 q^{25} + 72 q^{29} - 436 q^{37} - 2232 q^{39} + 744 q^{43} + 780 q^{51} - 976 q^{53} + 7624 q^{57} - 780 q^{65} + 1176 q^{67} - 4816 q^{71} - 56 q^{79} - 2104 q^{81} + 9880 q^{85} - 2376 q^{93} - 4032 q^{95} - 5176 q^{99}+O(q^{100}) \) 8 * q - 136 * q^9 + 116 * q^11 + 224 * q^15 + 80 * q^23 - 448 * q^25 + 72 * q^29 - 436 * q^37 - 2232 * q^39 + 744 * q^43 + 780 * q^51 - 976 * q^53 + 7624 * q^57 - 780 * q^65 + 1176 * q^67 - 4816 * q^71 - 56 * q^79 - 2104 * q^81 + 9880 * q^85 - 2376 * q^93 - 4032 * q^95 - 5176 * q^99

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