LMFDB - Embedded newform 9200.2.a.dd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.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:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.69755$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69755 q^{3} +4.22860 q^{7} -0.118308 q^{9} +O(q^{10})$	$q-1.69755 q^{3} +4.22860 q^{7} -0.118308 q^{9} -4.59157 q^{11} +0.978254 q^{13} +3.04005 q^{17} -1.91463 q^{19} -7.17829 q^{21} +1.00000 q^{23} +5.29350 q^{27} -0.737607 q^{29} -2.97939 q^{31} +7.79445 q^{33} -8.93637 q^{37} -1.66064 q^{39} +9.08240 q^{41} -6.97873 q^{43} +2.58560 q^{47} +10.8811 q^{49} -5.16065 q^{51} +2.71400 q^{53} +3.25019 q^{57} -7.13514 q^{59} -0.731629 q^{61} -0.500277 q^{63} -7.16679 q^{67} -1.69755 q^{69} +6.08775 q^{71} +5.96311 q^{73} -19.4159 q^{77} +2.06100 q^{79} -8.63108 q^{81} +4.39408 q^{83} +1.25213 q^{87} -7.25034 q^{89} +4.13665 q^{91} +5.05767 q^{93} +7.04560 q^{97} +0.543219 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51} - 4 q^{53} - 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} - 15 q^{71} - 6 q^{73} - 18 q^{77} + 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} + 35 q^{91} - 20 q^{93} - 3 q^{97} - 61 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 - 7 * q^11 - 11 * q^13 + 7 * q^17 - 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 - 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 + 19 * q^51 - 4 * q^53 - 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 - 15 * q^71 - 6 * q^73 - 18 * q^77 + 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 + 35 * q^91 - 20 * q^93 - 3 * q^97 - 61 * 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
$3$	−1.69755	−0.980084	−0.490042	−	0.871699i	$-0.663019\pi$
$3$	−1.69755	−0.980084	−0.490042	+	0.871699i	$0.663019\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.22860	1.59826	0.799131	−	0.601157i	$-0.205293\pi$
$7$	4.22860	1.59826	0.799131	+	0.601157i	$0.205293\pi$
$8$	0	0
$9$	−0.118308	−0.0394359
$10$	0	0
$11$	−4.59157	−1.38441	−0.692206	−	0.721700i	$-0.743361\pi$
$11$	−4.59157	−1.38441	−0.692206	+	0.721700i	$0.743361\pi$
$12$	0	0
$13$	0.978254	0.271319	0.135659	−	0.990756i	$-0.456685\pi$
$13$	0.978254	0.271319	0.135659	+	0.990756i	$0.456685\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.04005	0.737321	0.368660	−	0.929564i	$-0.379817\pi$
$17$	3.04005	0.737321	0.368660	+	0.929564i	$0.379817\pi$
$18$	0	0
$19$	−1.91463	−0.439247	−0.219623	−	0.975585i	$-0.570483\pi$
$19$	−1.91463	−0.439247	−0.219623	+	0.975585i	$0.570483\pi$
$20$	0	0
$21$	−7.17829	−1.56643
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.29350	1.01873
$28$	0	0
$29$	−0.737607	−0.136970	−0.0684851	−	0.997652i	$-0.521817\pi$
$29$	−0.737607	−0.136970	−0.0684851	+	0.997652i	$0.521817\pi$
$30$	0	0
$31$	−2.97939	−0.535114	−0.267557	−	0.963542i	$-0.586216\pi$
$31$	−2.97939	−0.535114	−0.267557	+	0.963542i	$0.586216\pi$
$32$	0	0
$33$	7.79445	1.35684
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.93637	−1.46913	−0.734565	−	0.678539i	$-0.762614\pi$
$37$	−8.93637	−1.46913	−0.734565	+	0.678539i	$0.762614\pi$
$38$	0	0
$39$	−1.66064	−0.265915
$40$	0	0
$41$	9.08240	1.41843	0.709216	−	0.704991i	$-0.249049\pi$
$41$	9.08240	1.41843	0.709216	+	0.704991i	$0.249049\pi$
$42$	0	0
$43$	−6.97873	−1.06425	−0.532123	−	0.846667i	$-0.678606\pi$
$43$	−6.97873	−1.06425	−0.532123	+	0.846667i	$0.678606\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.58560	0.377148	0.188574	−	0.982059i	$-0.439613\pi$
$47$	2.58560	0.377148	0.188574	+	0.982059i	$0.439613\pi$
$48$	0	0
$49$	10.8811	1.55444
$50$	0	0
$51$	−5.16065	−0.722636
$52$	0	0
$53$	2.71400	0.372796	0.186398	−	0.982474i	$-0.440319\pi$
$53$	2.71400	0.372796	0.186398	+	0.982474i	$0.440319\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.25019	0.430499
$58$	0	0
$59$	−7.13514	−0.928916	−0.464458	−	0.885595i	$-0.653751\pi$
$59$	−7.13514	−0.928916	−0.464458	+	0.885595i	$0.653751\pi$
$60$	0	0
$61$	−0.731629	−0.0936755	−0.0468377	−	0.998903i	$-0.514914\pi$
$61$	−0.731629	−0.0936755	−0.0468377	+	0.998903i	$0.514914\pi$
$62$	0	0
$63$	−0.500277	−0.0630290
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.16679	−0.875563	−0.437781	−	0.899081i	$-0.644236\pi$
$67$	−7.16679	−0.875563	−0.437781	+	0.899081i	$0.644236\pi$
$68$	0	0
$69$	−1.69755	−0.204362
$70$	0	0
$71$	6.08775	0.722483	0.361241	−	0.932472i	$-0.382353\pi$
$71$	6.08775	0.722483	0.361241	+	0.932472i	$0.382353\pi$
$72$	0	0
$73$	5.96311	0.697930	0.348965	−	0.937136i	$-0.386533\pi$
$73$	5.96311	0.697930	0.348965	+	0.937136i	$0.386533\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−19.4159	−2.21265
$78$	0	0
$79$	2.06100	0.231880	0.115940	−	0.993256i	$-0.463012\pi$
$79$	2.06100	0.231880	0.115940	+	0.993256i	$0.463012\pi$
$80$	0	0
$81$	−8.63108	−0.959009
$82$	0	0
$83$	4.39408	0.482312	0.241156	−	0.970486i	$-0.422473\pi$
$83$	4.39408	0.482312	0.241156	+	0.970486i	$0.422473\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.25213	0.134242
$88$	0	0
$89$	−7.25034	−0.768534	−0.384267	−	0.923222i	$-0.625546\pi$
$89$	−7.25034	−0.768534	−0.384267	+	0.923222i	$0.625546\pi$
$90$	0	0
$91$	4.13665	0.433638
$92$	0	0
$93$	5.05767	0.524456
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.04560	0.715372	0.357686	−	0.933842i	$-0.383566\pi$
$97$	7.04560	0.715372	0.357686	+	0.933842i	$0.383566\pi$
$98$	0	0
$99$	0.543219	0.0545956
$100$	0	0
$101$	7.63173	0.759385	0.379693	−	0.925113i	$-0.376030\pi$
$101$	7.63173	0.759385	0.379693	+	0.925113i	$0.376030\pi$
$102$	0	0
$103$	−16.7855	−1.65393	−0.826964	−	0.562254i	$-0.809934\pi$
$103$	−16.7855	−1.65393	−0.826964	+	0.562254i	$0.809934\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.3282	−1.48183	−0.740915	−	0.671599i	$-0.765608\pi$
$107$	−15.3282	−1.48183	−0.740915	+	0.671599i	$0.765608\pi$
$108$	0	0
$109$	14.9619	1.43309	0.716544	−	0.697542i	$-0.245723\pi$
$109$	14.9619	1.43309	0.716544	+	0.697542i	$0.245723\pi$
$110$	0	0
$111$	15.1700	1.43987
$112$	0	0
$113$	−19.5478	−1.83890	−0.919450	−	0.393207i	$-0.871366\pi$
$113$	−19.5478	−1.83890	−0.919450	+	0.393207i	$0.871366\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.115735	−0.0106997
$118$	0	0
$119$	12.8552	1.17843
$120$	0	0
$121$	10.0826	0.916596
$122$	0	0
$123$	−15.4179	−1.39018
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−20.6469	−1.83212	−0.916060	−	0.401042i	$-0.868648\pi$
$127$	−20.6469	−1.83212	−0.916060	+	0.401042i	$0.868648\pi$
$128$	0	0
$129$	11.8468	1.04305
$130$	0	0
$131$	−10.1208	−0.884255	−0.442127	−	0.896952i	$-0.645776\pi$
$131$	−10.1208	−0.884255	−0.442127	+	0.896952i	$0.645776\pi$
$132$	0	0
$133$	−8.09622	−0.702031
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.40734	0.461980	0.230990	−	0.972956i	$-0.425804\pi$
$137$	5.40734	0.461980	0.230990	+	0.972956i	$0.425804\pi$
$138$	0	0
$139$	−6.41028	−0.543713	−0.271856	−	0.962338i	$-0.587638\pi$
$139$	−6.41028	−0.543713	−0.271856	+	0.962338i	$0.587638\pi$
$140$	0	0
$141$	−4.38919	−0.369637
$142$	0	0
$143$	−4.49172	−0.375617
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−18.4712	−1.52348
$148$	0	0
$149$	13.3334	1.09232	0.546159	−	0.837682i	$-0.316089\pi$
$149$	13.3334	1.09232	0.546159	+	0.837682i	$0.316089\pi$
$150$	0	0
$151$	−19.4208	−1.58044	−0.790222	−	0.612821i	$-0.790035\pi$
$151$	−19.4208	−1.58044	−0.790222	+	0.612821i	$0.790035\pi$
$152$	0	0
$153$	−0.359662	−0.0290769
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2879	1.22011	0.610053	−	0.792361i	$-0.291148\pi$
$157$	15.2879	1.22011	0.610053	+	0.792361i	$0.291148\pi$
$158$	0	0
$159$	−4.60716	−0.365371
$160$	0	0
$161$	4.22860	0.333261
$162$	0	0
$163$	14.4308	1.13031	0.565153	−	0.824986i	$-0.308817\pi$
$163$	14.4308	1.13031	0.565153	+	0.824986i	$0.308817\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.7935	1.84120	0.920598	−	0.390511i	$-0.127702\pi$
$167$	23.7935	1.84120	0.920598	+	0.390511i	$0.127702\pi$
$168$	0	0
$169$	−12.0430	−0.926386
$170$	0	0
$171$	0.226516	0.0173221
$172$	0	0
$173$	1.54396	0.117385	0.0586927	−	0.998276i	$-0.481307\pi$
$173$	1.54396	0.117385	0.0586927	+	0.998276i	$0.481307\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.1123	0.910415
$178$	0	0
$179$	−13.0390	−0.974583	−0.487292	−	0.873239i	$-0.662015\pi$
$179$	−13.0390	−0.974583	−0.487292	+	0.873239i	$0.662015\pi$
$180$	0	0
$181$	1.77458	0.131903	0.0659517	−	0.997823i	$-0.478992\pi$
$181$	1.77458	0.131903	0.0659517	+	0.997823i	$0.478992\pi$
$182$	0	0
$183$	1.24198	0.0918098
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−13.9586	−1.02076
$188$	0	0
$189$	22.3841	1.62820
$190$	0	0
$191$	−11.8332	−0.856222	−0.428111	−	0.903726i	$-0.640821\pi$
$191$	−11.8332	−0.856222	−0.428111	+	0.903726i	$0.640821\pi$
$192$	0	0
$193$	−20.9549	−1.50837	−0.754184	−	0.656663i	$-0.771967\pi$
$193$	−20.9549	−1.50837	−0.754184	+	0.656663i	$0.771967\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.1752	1.00994	0.504970	−	0.863137i	$-0.331504\pi$
$197$	14.1752	1.00994	0.504970	+	0.863137i	$0.331504\pi$
$198$	0	0
$199$	20.6813	1.46606	0.733030	−	0.680197i	$-0.238106\pi$
$199$	20.6813	1.46606	0.733030	+	0.680197i	$0.238106\pi$
$200$	0	0
$201$	12.1660	0.858125
$202$	0	0
$203$	−3.11905	−0.218914
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−0.118308	−0.00822296
$208$	0	0
$209$	8.79118	0.608098
$210$	0	0
$211$	−4.31377	−0.296972	−0.148486	−	0.988914i	$-0.547440\pi$
$211$	−4.31377	−0.296972	−0.148486	+	0.988914i	$0.547440\pi$
$212$	0	0
$213$	−10.3343	−0.708094
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.5987	−0.855252
$218$	0	0
$219$	−10.1227	−0.684030
$220$	0	0
$221$	2.97394	0.200049
$222$	0	0
$223$	3.78248	0.253293	0.126647	−	0.991948i	$-0.459579\pi$
$223$	3.78248	0.253293	0.126647	+	0.991948i	$0.459579\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.4058	−0.956148	−0.478074	−	0.878320i	$-0.658665\pi$
$227$	−14.4058	−0.956148	−0.478074	+	0.878320i	$0.658665\pi$
$228$	0	0
$229$	22.6701	1.49808	0.749041	−	0.662524i	$-0.230515\pi$
$229$	22.6701	1.49808	0.749041	+	0.662524i	$0.230515\pi$
$230$	0	0
$231$	32.9596	2.16858
$232$	0	0
$233$	0.433977	0.0284308	0.0142154	−	0.999899i	$-0.495475\pi$
$233$	0.433977	0.0284308	0.0142154	+	0.999899i	$0.495475\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.49866	−0.227262
$238$	0	0
$239$	−27.3045	−1.76618	−0.883090	−	0.469204i	$-0.844541\pi$
$239$	−27.3045	−1.76618	−0.883090	+	0.469204i	$0.844541\pi$
$240$	0	0
$241$	23.9763	1.54445	0.772226	−	0.635348i	$-0.219143\pi$
$241$	23.9763	1.54445	0.772226	+	0.635348i	$0.219143\pi$
$242$	0	0
$243$	−1.22876	−0.0788253
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.87300	−0.119176
$248$	0	0
$249$	−7.45918	−0.472707
$250$	0	0
$251$	5.25761	0.331857	0.165929	−	0.986138i	$-0.446938\pi$
$251$	5.25761	0.331857	0.165929	+	0.986138i	$0.446938\pi$
$252$	0	0
$253$	−4.59157	−0.288670
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.6630	−1.28892	−0.644461	−	0.764637i	$-0.722918\pi$
$257$	−20.6630	−1.28892	−0.644461	+	0.764637i	$0.722918\pi$
$258$	0	0
$259$	−37.7884	−2.34805
$260$	0	0
$261$	0.0872647	0.00540155
$262$	0	0
$263$	4.88893	0.301464	0.150732	−	0.988575i	$-0.451837\pi$
$263$	4.88893	0.301464	0.150732	+	0.988575i	$0.451837\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	12.3078	0.753228
$268$	0	0
$269$	2.12989	0.129862	0.0649309	−	0.997890i	$-0.479317\pi$
$269$	2.12989	0.129862	0.0649309	+	0.997890i	$0.479317\pi$
$270$	0	0
$271$	25.6680	1.55922	0.779608	−	0.626267i	$-0.215418\pi$
$271$	25.6680	1.55922	0.779608	+	0.626267i	$0.215418\pi$
$272$	0	0
$273$	−7.02218	−0.425002
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.0110	−0.841837	−0.420918	−	0.907099i	$-0.638292\pi$
$277$	−14.0110	−0.841837	−0.420918	+	0.907099i	$0.638292\pi$
$278$	0	0
$279$	0.352485	0.0211027
$280$	0	0
$281$	−12.4269	−0.741324	−0.370662	−	0.928768i	$-0.620869\pi$
$281$	−12.4269	−0.741324	−0.370662	+	0.928768i	$0.620869\pi$
$282$	0	0
$283$	8.94768	0.531884	0.265942	−	0.963989i	$-0.414317\pi$
$283$	8.94768	0.531884	0.265942	+	0.963989i	$0.414317\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	38.4059	2.26703
$288$	0	0
$289$	−7.75808	−0.456358
$290$	0	0
$291$	−11.9603	−0.701125
$292$	0	0
$293$	−30.7264	−1.79506	−0.897529	−	0.440956i	$-0.854640\pi$
$293$	−30.7264	−1.79506	−0.897529	+	0.440956i	$0.854640\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−24.3055	−1.41035
$298$	0	0
$299$	0.978254	0.0565739
$300$	0	0
$301$	−29.5103	−1.70094
$302$	0	0
$303$	−12.9553	−0.744261
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.6480	0.778933	0.389467	−	0.921041i	$-0.372659\pi$
$307$	13.6480	0.778933	0.389467	+	0.921041i	$0.372659\pi$
$308$	0	0
$309$	28.4944	1.62099
$310$	0	0
$311$	28.7646	1.63109	0.815545	−	0.578693i	$-0.196437\pi$
$311$	28.7646	1.63109	0.815545	+	0.578693i	$0.196437\pi$
$312$	0	0
$313$	−11.4851	−0.649176	−0.324588	−	0.945856i	$-0.605226\pi$
$313$	−11.4851	−0.649176	−0.324588	+	0.945856i	$0.605226\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.51108	0.253368	0.126684	−	0.991943i	$-0.459567\pi$
$317$	4.51108	0.253368	0.126684	+	0.991943i	$0.459567\pi$
$318$	0	0
$319$	3.38678	0.189623
$320$	0	0
$321$	26.0204	1.45232
$322$	0	0
$323$	−5.82058	−0.323866
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−25.3986	−1.40455
$328$	0	0
$329$	10.9335	0.602781
$330$	0	0
$331$	−32.0738	−1.76293	−0.881467	−	0.472246i	$-0.843443\pi$
$331$	−32.0738	−1.76293	−0.881467	+	0.472246i	$0.843443\pi$
$332$	0	0
$333$	1.05724	0.0579365
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.9334	1.79399	0.896997	−	0.442036i	$-0.145744\pi$
$337$	32.9334	1.79399	0.896997	+	0.442036i	$0.145744\pi$
$338$	0	0
$339$	33.1834	1.80228
$340$	0	0
$341$	13.6801	0.740818
$342$	0	0
$343$	16.4116	0.886143
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.9194	0.639867	0.319933	−	0.947440i	$-0.396339\pi$
$347$	11.9194	0.639867	0.319933	+	0.947440i	$0.396339\pi$
$348$	0	0
$349$	−4.63550	−0.248132	−0.124066	−	0.992274i	$-0.539594\pi$
$349$	−4.63550	−0.248132	−0.124066	+	0.992274i	$0.539594\pi$
$350$	0	0
$351$	5.17838	0.276402
$352$	0	0
$353$	−16.2153	−0.863055	−0.431528	−	0.902100i	$-0.642025\pi$
$353$	−16.2153	−0.863055	−0.431528	+	0.902100i	$0.642025\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−21.8224	−1.15496
$358$	0	0
$359$	33.5549	1.77096	0.885481	−	0.464676i	$-0.153829\pi$
$359$	33.5549	1.77096	0.885481	+	0.464676i	$0.153829\pi$
$360$	0	0
$361$	−15.3342	−0.807062
$362$	0	0
$363$	−17.1157	−0.898341
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.44019	−0.231776	−0.115888	−	0.993262i	$-0.536971\pi$
$367$	−4.44019	−0.231776	−0.115888	+	0.993262i	$0.536971\pi$
$368$	0	0
$369$	−1.07452	−0.0559372
$370$	0	0
$371$	11.4764	0.595826
$372$	0	0
$373$	−29.3416	−1.51925	−0.759624	−	0.650362i	$-0.774617\pi$
$373$	−29.3416	−1.51925	−0.759624	+	0.650362i	$0.774617\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.721567	−0.0371626
$378$	0	0
$379$	−18.7557	−0.963415	−0.481708	−	0.876332i	$-0.659983\pi$
$379$	−18.7557	−0.963415	−0.481708	+	0.876332i	$0.659983\pi$
$380$	0	0
$381$	35.0493	1.79563
$382$	0	0
$383$	−29.1684	−1.49043	−0.745217	−	0.666822i	$-0.767654\pi$
$383$	−29.1684	−1.49043	−0.745217	+	0.666822i	$0.767654\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.825638	0.0419695
$388$	0	0
$389$	−27.2943	−1.38388	−0.691939	−	0.721956i	$-0.743243\pi$
$389$	−27.2943	−1.38388	−0.691939	+	0.721956i	$0.743243\pi$
$390$	0	0
$391$	3.04005	0.153742
$392$	0	0
$393$	17.1805	0.866644
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.8548	−1.14705	−0.573525	−	0.819188i	$-0.694424\pi$
$397$	−22.8548	−1.14705	−0.573525	+	0.819188i	$0.694424\pi$
$398$	0	0
$399$	13.7438	0.688049
$400$	0	0
$401$	19.6913	0.983337	0.491668	−	0.870783i	$-0.336387\pi$
$401$	19.6913	0.983337	0.491668	+	0.870783i	$0.336387\pi$
$402$	0	0
$403$	−2.91460	−0.145186
$404$	0	0
$405$	0	0
$406$	0	0
$407$	41.0320	2.03388
$408$	0	0
$409$	−33.8863	−1.67557	−0.837785	−	0.546001i	$-0.816149\pi$
$409$	−33.8863	−1.67557	−0.837785	+	0.546001i	$0.816149\pi$
$410$	0	0
$411$	−9.17925	−0.452779
$412$	0	0
$413$	−30.1717	−1.48465
$414$	0	0
$415$	0	0
$416$	0	0
$417$	10.8818	0.532884
$418$	0	0
$419$	−15.8605	−0.774837	−0.387419	−	0.921904i	$-0.626633\pi$
$419$	−15.8605	−0.774837	−0.387419	+	0.921904i	$0.626633\pi$
$420$	0	0
$421$	−13.9043	−0.677652	−0.338826	−	0.940849i	$-0.610030\pi$
$421$	−13.9043	−0.677652	−0.338826	+	0.940849i	$0.610030\pi$
$422$	0	0
$423$	−0.305896	−0.0148732
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.09377	−0.149718
$428$	0	0
$429$	7.62495	0.368136
$430$	0	0
$431$	0.736783	0.0354896	0.0177448	−	0.999843i	$-0.494351\pi$
$431$	0.736783	0.0354896	0.0177448	+	0.999843i	$0.494351\pi$
$432$	0	0
$433$	−3.79670	−0.182458	−0.0912288	−	0.995830i	$-0.529079\pi$
$433$	−3.79670	−0.182458	−0.0912288	+	0.995830i	$0.529079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.91463	−0.0915893
$438$	0	0
$439$	5.74582	0.274233	0.137117	−	0.990555i	$-0.456217\pi$
$439$	5.74582	0.274233	0.137117	+	0.990555i	$0.456217\pi$
$440$	0	0
$441$	−1.28732	−0.0613009
$442$	0	0
$443$	−6.53509	−0.310492	−0.155246	−	0.987876i	$-0.549617\pi$
$443$	−6.53509	−0.310492	−0.155246	+	0.987876i	$0.549617\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−22.6342	−1.07056
$448$	0	0
$449$	4.53781	0.214152	0.107076	−	0.994251i	$-0.465851\pi$
$449$	4.53781	0.214152	0.107076	+	0.994251i	$0.465851\pi$
$450$	0	0
$451$	−41.7025	−1.96369
$452$	0	0
$453$	32.9679	1.54897
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15.7911	−0.738676	−0.369338	−	0.929295i	$-0.620416\pi$
$457$	−15.7911	−0.738676	−0.369338	+	0.929295i	$0.620416\pi$
$458$	0	0
$459$	16.0925	0.751134
$460$	0	0
$461$	15.6324	0.728074	0.364037	−	0.931384i	$-0.381398\pi$
$461$	15.6324	0.728074	0.364037	+	0.931384i	$0.381398\pi$
$462$	0	0
$463$	−4.29485	−0.199598	−0.0997992	−	0.995008i	$-0.531820\pi$
$463$	−4.29485	−0.199598	−0.0997992	+	0.995008i	$0.531820\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	29.7617	1.37720	0.688602	−	0.725139i	$-0.258225\pi$
$467$	29.7617	1.37720	0.688602	+	0.725139i	$0.258225\pi$
$468$	0	0
$469$	−30.3055	−1.39938
$470$	0	0
$471$	−25.9520	−1.19581
$472$	0	0
$473$	32.0433	1.47335
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.321087	−0.0147016
$478$	0	0
$479$	−36.8371	−1.68313	−0.841564	−	0.540157i	$-0.818365\pi$
$479$	−36.8371	−1.68313	−0.841564	+	0.540157i	$0.818365\pi$
$480$	0	0
$481$	−8.74203	−0.398602
$482$	0	0
$483$	−7.17829	−0.326623
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.0340	−1.17971	−0.589857	−	0.807508i	$-0.700816\pi$
$487$	−26.0340	−1.17971	−0.589857	+	0.807508i	$0.700816\pi$
$488$	0	0
$489$	−24.4970	−1.10779
$490$	0	0
$491$	12.2553	0.553074	0.276537	−	0.961003i	$-0.410813\pi$
$491$	12.2553	0.553074	0.276537	+	0.961003i	$0.410813\pi$
$492$	0	0
$493$	−2.24236	−0.100991
$494$	0	0
$495$	0	0
$496$	0	0
$497$	25.7427	1.15472
$498$	0	0
$499$	−32.4229	−1.45145	−0.725724	−	0.687986i	$-0.758495\pi$
$499$	−32.4229	−1.45145	−0.725724	+	0.687986i	$0.758495\pi$
$500$	0	0
$501$	−40.3908	−1.80453
$502$	0	0
$503$	32.1726	1.43451	0.717253	−	0.696813i	$-0.245399\pi$
$503$	32.1726	1.43451	0.717253	+	0.696813i	$0.245399\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.4437	0.907936
$508$	0	0
$509$	25.1343	1.11406	0.557030	−	0.830492i	$-0.311941\pi$
$509$	25.1343	1.11406	0.557030	+	0.830492i	$0.311941\pi$
$510$	0	0
$511$	25.2156	1.11547
$512$	0	0
$513$	−10.1351	−0.447476
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−11.8720	−0.522128
$518$	0	0
$519$	−2.62097	−0.115048
$520$	0	0
$521$	9.45730	0.414332	0.207166	−	0.978306i	$-0.433576\pi$
$521$	9.45730	0.414332	0.207166	+	0.978306i	$0.433576\pi$
$522$	0	0
$523$	−39.9479	−1.74680	−0.873400	−	0.487003i	$-0.838090\pi$
$523$	−39.9479	−1.74680	−0.873400	+	0.487003i	$0.838090\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.05749	−0.394551
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.844143	0.0366327
$532$	0	0
$533$	8.88489	0.384847
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.1345	0.955173
$538$	0	0
$539$	−49.9613	−2.15199
$540$	0	0
$541$	−40.1829	−1.72760	−0.863800	−	0.503835i	$-0.831922\pi$
$541$	−40.1829	−1.72760	−0.863800	+	0.503835i	$0.831922\pi$
$542$	0	0
$543$	−3.01244	−0.129276
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.86104	−0.122329	−0.0611647	−	0.998128i	$-0.519481\pi$
$547$	−2.86104	−0.122329	−0.0611647	+	0.998128i	$0.519481\pi$
$548$	0	0
$549$	0.0865574	0.00369418
$550$	0	0
$551$	1.41225	0.0601637
$552$	0	0
$553$	8.71514	0.370606
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15.4417	−0.654286	−0.327143	−	0.944975i	$-0.606086\pi$
$557$	−15.4417	−0.654286	−0.327143	+	0.944975i	$0.606086\pi$
$558$	0	0
$559$	−6.82696	−0.288750
$560$	0	0
$561$	23.6955	1.00043
$562$	0	0
$563$	17.5808	0.740941	0.370470	−	0.928844i	$-0.379197\pi$
$563$	17.5808	0.740941	0.370470	+	0.928844i	$0.379197\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−36.4974	−1.53275
$568$	0	0
$569$	−43.1463	−1.80879	−0.904394	−	0.426698i	$-0.859677\pi$
$569$	−43.1463	−1.80879	−0.904394	+	0.426698i	$0.859677\pi$
$570$	0	0
$571$	13.9876	0.585365	0.292682	−	0.956210i	$-0.405452\pi$
$571$	13.9876	0.585365	0.292682	+	0.956210i	$0.405452\pi$
$572$	0	0
$573$	20.0875	0.839169
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−41.5627	−1.73028	−0.865139	−	0.501533i	$-0.832770\pi$
$577$	−41.5627	−1.73028	−0.865139	+	0.501533i	$0.832770\pi$
$578$	0	0
$579$	35.5721	1.47833
$580$	0	0
$581$	18.5808	0.770862
$582$	0	0
$583$	−12.4615	−0.516103
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.4457	1.66937	0.834687	−	0.550724i	$-0.185648\pi$
$587$	40.4457	1.66937	0.834687	+	0.550724i	$0.185648\pi$
$588$	0	0
$589$	5.70443	0.235047
$590$	0	0
$591$	−24.0631	−0.989825
$592$	0	0
$593$	−28.0011	−1.14987	−0.574934	−	0.818200i	$-0.694972\pi$
$593$	−28.0011	−1.14987	−0.574934	+	0.818200i	$0.694972\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−35.1077	−1.43686
$598$	0	0
$599$	6.61659	0.270347	0.135173	−	0.990822i	$-0.456841\pi$
$599$	6.61659	0.270347	0.135173	+	0.990822i	$0.456841\pi$
$600$	0	0
$601$	−44.3974	−1.81101	−0.905504	−	0.424337i	$-0.860507\pi$
$601$	−44.3974	−1.81101	−0.905504	+	0.424337i	$0.860507\pi$
$602$	0	0
$603$	0.847887	0.0345286
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.0131	−0.934074	−0.467037	−	0.884238i	$-0.654678\pi$
$607$	−23.0131	−0.934074	−0.467037	+	0.884238i	$0.654678\pi$
$608$	0	0
$609$	5.29475	0.214554
$610$	0	0
$611$	2.52937	0.102327
$612$	0	0
$613$	7.77554	0.314051	0.157025	−	0.987595i	$-0.449810\pi$
$613$	7.77554	0.314051	0.157025	+	0.987595i	$0.449810\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	14.6868	0.591268	0.295634	−	0.955301i	$-0.404469\pi$
$617$	14.6868	0.591268	0.295634	+	0.955301i	$0.404469\pi$
$618$	0	0
$619$	−18.1633	−0.730045	−0.365022	−	0.930999i	$-0.618939\pi$
$619$	−18.1633	−0.730045	−0.365022	+	0.930999i	$0.618939\pi$
$620$	0	0
$621$	5.29350	0.212421
$622$	0	0
$623$	−30.6588	−1.22832
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−14.9235	−0.595987
$628$	0	0
$629$	−27.1670	−1.08322
$630$	0	0
$631$	−11.0472	−0.439780	−0.219890	−	0.975525i	$-0.570570\pi$
$631$	−11.0472	−0.439780	−0.219890	+	0.975525i	$0.570570\pi$
$632$	0	0
$633$	7.32286	0.291058
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.6445	0.421749
$638$	0	0
$639$	−0.720228	−0.0284918
$640$	0	0
$641$	44.7133	1.76607	0.883035	−	0.469306i	$-0.155496\pi$
$641$	44.7133	1.76607	0.883035	+	0.469306i	$0.155496\pi$
$642$	0	0
$643$	−8.30391	−0.327474	−0.163737	−	0.986504i	$-0.552355\pi$
$643$	−8.30391	−0.327474	−0.163737	+	0.986504i	$0.552355\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−50.0603	−1.96807	−0.984036	−	0.177971i	$-0.943047\pi$
$647$	−50.0603	−1.96807	−0.984036	+	0.177971i	$0.943047\pi$
$648$	0	0
$649$	32.7615	1.28600
$650$	0	0
$651$	21.3869	0.838219
$652$	0	0
$653$	22.4150	0.877166	0.438583	−	0.898691i	$-0.355481\pi$
$653$	22.4150	0.877166	0.438583	+	0.898691i	$0.355481\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−0.705483	−0.0275235
$658$	0	0
$659$	32.3712	1.26100	0.630502	−	0.776187i	$-0.282849\pi$
$659$	32.3712	1.26100	0.630502	+	0.776187i	$0.282849\pi$
$660$	0	0
$661$	−23.2386	−0.903875	−0.451938	−	0.892050i	$-0.649267\pi$
$661$	−23.2386	−0.903875	−0.451938	+	0.892050i	$0.649267\pi$
$662$	0	0
$663$	−5.04843	−0.196065
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−0.737607	−0.0285602
$668$	0	0
$669$	−6.42096	−0.248249
$670$	0	0
$671$	3.35933	0.129685
$672$	0	0
$673$	−3.02445	−0.116584	−0.0582919	−	0.998300i	$-0.518565\pi$
$673$	−3.02445	−0.116584	−0.0582919	+	0.998300i	$0.518565\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.34663	0.205488	0.102744	−	0.994708i	$-0.467238\pi$
$677$	5.34663	0.205488	0.102744	+	0.994708i	$0.467238\pi$
$678$	0	0
$679$	29.7931	1.14335
$680$	0	0
$681$	24.4547	0.937105
$682$	0	0
$683$	−21.7778	−0.833305	−0.416653	−	0.909066i	$-0.636797\pi$
$683$	−21.7778	−0.833305	−0.416653	+	0.909066i	$0.636797\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−38.4837	−1.46825
$688$	0	0
$689$	2.65498	0.101147
$690$	0	0
$691$	−17.1839	−0.653705	−0.326852	−	0.945075i	$-0.605988\pi$
$691$	−17.1839	−0.653705	−0.326852	+	0.945075i	$0.605988\pi$
$692$	0	0
$693$	2.29706	0.0872580
$694$	0	0
$695$	0	0
$696$	0	0
$697$	27.6110	1.04584
$698$	0	0
$699$	−0.736700	−0.0278646
$700$	0	0
$701$	−10.9444	−0.413363	−0.206682	−	0.978408i	$-0.566266\pi$
$701$	−10.9444	−0.413363	−0.206682	+	0.978408i	$0.566266\pi$
$702$	0	0
$703$	17.1099	0.645310
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32.2715	1.21370
$708$	0	0
$709$	−35.1463	−1.31995	−0.659973	−	0.751290i	$-0.729432\pi$
$709$	−35.1463	−1.31995	−0.659973	+	0.751290i	$0.729432\pi$
$710$	0	0
$711$	−0.243832	−0.00914442
$712$	0	0
$713$	−2.97939	−0.111579
$714$	0	0
$715$	0	0
$716$	0	0
$717$	46.3508	1.73100
$718$	0	0
$719$	−7.07135	−0.263717	−0.131858	−	0.991269i	$-0.542094\pi$
$719$	−7.07135	−0.263717	−0.131858	+	0.991269i	$0.542094\pi$
$720$	0	0
$721$	−70.9794	−2.64341
$722$	0	0
$723$	−40.7012	−1.51369
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.75083	−0.0649345	−0.0324673	−	0.999473i	$-0.510336\pi$
$727$	−1.75083	−0.0649345	−0.0324673	+	0.999473i	$0.510336\pi$
$728$	0	0
$729$	27.9791	1.03626
$730$	0	0
$731$	−21.2157	−0.784691
$732$	0	0
$733$	−29.8141	−1.10121	−0.550605	−	0.834766i	$-0.685603\pi$
$733$	−29.8141	−1.10121	−0.550605	+	0.834766i	$0.685603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.9069	1.21214
$738$	0	0
$739$	−14.2585	−0.524508	−0.262254	−	0.964999i	$-0.584466\pi$
$739$	−14.2585	−0.524508	−0.262254	+	0.964999i	$0.584466\pi$
$740$	0	0
$741$	3.17951	0.116802
$742$	0	0
$743$	−1.02428	−0.0375772	−0.0187886	−	0.999823i	$-0.505981\pi$
$743$	−1.02428	−0.0375772	−0.0187886	+	0.999823i	$0.505981\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.519854	−0.0190204
$748$	0	0
$749$	−64.8167	−2.36835
$750$	0	0
$751$	−21.1173	−0.770582	−0.385291	−	0.922795i	$-0.625899\pi$
$751$	−21.1173	−0.770582	−0.385291	+	0.922795i	$0.625899\pi$
$752$	0	0
$753$	−8.92508	−0.325248
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.6048	−0.748895	−0.374447	−	0.927248i	$-0.622168\pi$
$757$	−20.6048	−0.748895	−0.374447	+	0.927248i	$0.622168\pi$
$758$	0	0
$759$	7.79445	0.282921
$760$	0	0
$761$	−42.4576	−1.53909	−0.769543	−	0.638595i	$-0.779516\pi$
$761$	−42.4576	−1.53909	−0.769543	+	0.638595i	$0.779516\pi$
$762$	0	0
$763$	63.2679	2.29045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.97997	−0.252032
$768$	0	0
$769$	14.2501	0.513872	0.256936	−	0.966428i	$-0.417287\pi$
$769$	14.2501	0.513872	0.256936	+	0.966428i	$0.417287\pi$
$770$	0	0
$771$	35.0766	1.26325
$772$	0	0
$773$	8.71017	0.313283	0.156642	−	0.987656i	$-0.449933\pi$
$773$	8.71017	0.313283	0.156642	+	0.987656i	$0.449933\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	64.1478	2.30129
$778$	0	0
$779$	−17.3895	−0.623042
$780$	0	0
$781$	−27.9524	−1.00021
$782$	0	0
$783$	−3.90452	−0.139536
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−19.5185	−0.695760	−0.347880	−	0.937539i	$-0.613098\pi$
$787$	−19.5185	−0.695760	−0.347880	+	0.937539i	$0.613098\pi$
$788$	0	0
$789$	−8.29922	−0.295460
$790$	0	0
$791$	−82.6598	−2.93904
$792$	0	0
$793$	−0.715718	−0.0254159
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.8913	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$797$	−26.8913	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$798$	0	0
$799$	7.86035	0.278079
$800$	0	0
$801$	0.857772	0.0303079
$802$	0	0
$803$	−27.3801	−0.966222
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.61561	−0.127275
$808$	0	0
$809$	16.5468	0.581755	0.290878	−	0.956760i	$-0.406053\pi$
$809$	16.5468	0.581755	0.290878	+	0.956760i	$0.406053\pi$
$810$	0	0
$811$	25.3327	0.889551	0.444775	−	0.895642i	$-0.353284\pi$
$811$	25.3327	0.889551	0.444775	+	0.895642i	$0.353284\pi$
$812$	0	0
$813$	−43.5728	−1.52816
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13.3617	0.467466
$818$	0	0
$819$	−0.489398	−0.0171009
$820$	0	0
$821$	41.5845	1.45131	0.725655	−	0.688059i	$-0.241537\pi$
$821$	41.5845	1.45131	0.725655	+	0.688059i	$0.241537\pi$
$822$	0	0
$823$	−43.2049	−1.50603	−0.753015	−	0.658003i	$-0.771401\pi$
$823$	−43.2049	−1.50603	−0.753015	+	0.658003i	$0.771401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.6851	−0.893159	−0.446580	−	0.894744i	$-0.647358\pi$
$827$	−25.6851	−0.893159	−0.446580	+	0.894744i	$0.647358\pi$
$828$	0	0
$829$	39.3428	1.36643	0.683215	−	0.730217i	$-0.260581\pi$
$829$	39.3428	1.36643	0.683215	+	0.730217i	$0.260581\pi$
$830$	0	0
$831$	23.7844	0.825070
$832$	0	0
$833$	33.0791	1.14612
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−15.7714	−0.545139
$838$	0	0
$839$	−32.7583	−1.13094	−0.565471	−	0.824768i	$-0.691306\pi$
$839$	−32.7583	−1.13094	−0.565471	+	0.824768i	$0.691306\pi$
$840$	0	0
$841$	−28.4559	−0.981239
$842$	0	0
$843$	21.0953	0.726560
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.6351	1.46496
$848$	0	0
$849$	−15.1892	−0.521291
$850$	0	0
$851$	−8.93637	−0.306335
$852$	0	0
$853$	2.15477	0.0737778	0.0368889	−	0.999319i	$-0.488255\pi$
$853$	2.15477	0.0737778	0.0368889	+	0.999319i	$0.488255\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.0312	−1.06000	−0.530002	−	0.847996i	$-0.677809\pi$
$857$	−31.0312	−1.06000	−0.530002	+	0.847996i	$0.677809\pi$
$858$	0	0
$859$	23.0328	0.785868	0.392934	−	0.919567i	$-0.371460\pi$
$859$	23.0328	0.785868	0.392934	+	0.919567i	$0.371460\pi$
$860$	0	0
$861$	−65.1961	−2.22188
$862$	0	0
$863$	−18.9638	−0.645536	−0.322768	−	0.946478i	$-0.604613\pi$
$863$	−18.9638	−0.645536	−0.322768	+	0.946478i	$0.604613\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	13.1698	0.447269
$868$	0	0
$869$	−9.46322	−0.321018
$870$	0	0
$871$	−7.01094	−0.237557
$872$	0	0
$873$	−0.833550	−0.0282114
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.9808	−0.911076	−0.455538	−	0.890216i	$-0.650553\pi$
$877$	−26.9808	−0.911076	−0.455538	+	0.890216i	$0.650553\pi$
$878$	0	0
$879$	52.1598	1.75931
$880$	0	0
$881$	8.64313	0.291195	0.145597	−	0.989344i	$-0.453490\pi$
$881$	8.64313	0.291195	0.145597	+	0.989344i	$0.453490\pi$
$882$	0	0
$883$	−19.2554	−0.647996	−0.323998	−	0.946058i	$-0.605027\pi$
$883$	−19.2554	−0.647996	−0.323998	+	0.946058i	$0.605027\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.0027	1.27600	0.638002	−	0.770034i	$-0.279761\pi$
$887$	38.0027	1.27600	0.638002	+	0.770034i	$0.279761\pi$
$888$	0	0
$889$	−87.3077	−2.92821
$890$	0	0
$891$	39.6302	1.32766
$892$	0	0
$893$	−4.95047	−0.165661
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.66064	−0.0554471
$898$	0	0
$899$	2.19762	0.0732946
$900$	0	0
$901$	8.25069	0.274870
$902$	0	0
$903$	50.0953	1.66707
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−33.1430	−1.10050	−0.550248	−	0.835001i	$-0.685467\pi$
$907$	−33.1430	−1.10050	−0.550248	+	0.835001i	$0.685467\pi$
$908$	0	0
$909$	−0.902893	−0.0299471
$910$	0	0
$911$	−14.9983	−0.496916	−0.248458	−	0.968643i	$-0.579924\pi$
$911$	−14.9983	−0.496916	−0.248458	+	0.968643i	$0.579924\pi$
$912$	0	0
$913$	−20.1757	−0.667719
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−42.7967	−1.41327
$918$	0	0
$919$	21.9030	0.722513	0.361257	−	0.932466i	$-0.382348\pi$
$919$	21.9030	0.722513	0.361257	+	0.932466i	$0.382348\pi$
$920$	0	0
$921$	−23.1682	−0.763420
$922$	0	0
$923$	5.95536	0.196023
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.98586	0.0652242
$928$	0	0
$929$	−13.5384	−0.444181	−0.222091	−	0.975026i	$-0.571288\pi$
$929$	−13.5384	−0.444181	−0.222091	+	0.975026i	$0.571288\pi$
$930$	0	0
$931$	−20.8333	−0.682783
$932$	0	0
$933$	−48.8295	−1.59861
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.6360	0.543474	0.271737	−	0.962372i	$-0.412402\pi$
$937$	16.6360	0.543474	0.271737	+	0.962372i	$0.412402\pi$
$938$	0	0
$939$	19.4966	0.636247
$940$	0	0
$941$	40.4273	1.31789	0.658945	−	0.752191i	$-0.271003\pi$
$941$	40.4273	1.31789	0.658945	+	0.752191i	$0.271003\pi$
$942$	0	0
$943$	9.08240	0.295764
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−48.1251	−1.56386	−0.781928	−	0.623369i	$-0.785763\pi$
$947$	−48.1251	−1.56386	−0.781928	+	0.623369i	$0.785763\pi$
$948$	0	0
$949$	5.83344	0.189361
$950$	0	0
$951$	−7.65781	−0.248322
$952$	0	0
$953$	−38.1482	−1.23574	−0.617871	−	0.786279i	$-0.712005\pi$
$953$	−38.1482	−1.23574	−0.617871	+	0.786279i	$0.712005\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.74924	−0.185846
$958$	0	0
$959$	22.8655	0.738365
$960$	0	0
$961$	−22.1232	−0.713653
$962$	0	0
$963$	1.81344	0.0584373
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.0361	0.933737	0.466869	−	0.884327i	$-0.345382\pi$
$967$	29.0361	0.933737	0.466869	+	0.884327i	$0.345382\pi$
$968$	0	0
$969$	9.88076	0.317416
$970$	0	0
$971$	−27.8281	−0.893046	−0.446523	−	0.894772i	$-0.647338\pi$
$971$	−27.8281	−0.893046	−0.446523	+	0.894772i	$0.647338\pi$
$972$	0	0
$973$	−27.1065	−0.868995
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.1406	1.76411	0.882053	−	0.471151i	$-0.156161\pi$
$977$	55.1406	1.76411	0.882053	+	0.471151i	$0.156161\pi$
$978$	0	0
$979$	33.2905	1.06397
$980$	0	0
$981$	−1.77011	−0.0565152
$982$	0	0
$983$	11.1171	0.354581	0.177290	−	0.984159i	$-0.443267\pi$
$983$	11.1171	0.354581	0.177290	+	0.984159i	$0.443267\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−18.5602	−0.590776
$988$	0	0
$989$	−6.97873	−0.221911
$990$	0	0
$991$	40.3976	1.28327	0.641637	−	0.767009i	$-0.278256\pi$
$991$	40.3976	1.28327	0.641637	+	0.767009i	$0.278256\pi$
$992$	0	0
$993$	54.4470	1.72782
$994$	0	0
$995$	0	0
$996$	0	0
$997$	8.01261	0.253762	0.126881	−	0.991918i	$-0.459503\pi$
$997$	8.01261	0.253762	0.126881	+	0.991918i	$0.459503\pi$
$998$	0	0
$999$	−47.3046	−1.49665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.dd.1.3		8
4.3	odd	2		4600.2.a.bk.1.6		8
5.2	odd	4		1840.2.e.h.369.12		16
5.3	odd	4		1840.2.e.h.369.5		16
5.4	even	2		9200.2.a.de.1.6		8
20.3	even	4		920.2.e.c.369.12	yes	16
20.7	even	4		920.2.e.c.369.5	✓	16
20.19	odd	2		4600.2.a.bj.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.5	✓	16	20.7	even	4
920.2.e.c.369.12	yes	16	20.3	even	4
1840.2.e.h.369.5		16	5.3	odd	4
1840.2.e.h.369.12		16	5.2	odd	4
4600.2.a.bj.1.3		8	20.19	odd	2
4600.2.a.bk.1.6		8	4.3	odd	2
9200.2.a.dd.1.3		8	1.1	even	1	trivial
9200.2.a.de.1.6		8	5.4	even	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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69755 q^{3} +4.22860 q^{7} -0.118308 q^{9} +O(q^{10})\)	\(q-1.69755 q^{3} +4.22860 q^{7} -0.118308 q^{9} -4.59157 q^{11} +0.978254 q^{13} +3.04005 q^{17} -1.91463 q^{19} -7.17829 q^{21} +1.00000 q^{23} +5.29350 q^{27} -0.737607 q^{29} -2.97939 q^{31} +7.79445 q^{33} -8.93637 q^{37} -1.66064 q^{39} +9.08240 q^{41} -6.97873 q^{43} +2.58560 q^{47} +10.8811 q^{49} -5.16065 q^{51} +2.71400 q^{53} +3.25019 q^{57} -7.13514 q^{59} -0.731629 q^{61} -0.500277 q^{63} -7.16679 q^{67} -1.69755 q^{69} +6.08775 q^{71} +5.96311 q^{73} -19.4159 q^{77} +2.06100 q^{79} -8.63108 q^{81} +4.39408 q^{83} +1.25213 q^{87} -7.25034 q^{89} +4.13665 q^{91} +5.05767 q^{93} +7.04560 q^{97} +0.543219 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} - 7 q^{11} - 11 q^{13} + 7 q^{17} - 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} - 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} + 19 q^{51} - 4 q^{53} - 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} - 15 q^{71} - 6 q^{73} - 18 q^{77} + 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} + 35 q^{91} - 20 q^{93} - 3 q^{97} - 61 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 - 7 * q^11 - 11 * q^13 + 7 * q^17 - 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 - 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 + 19 * q^51 - 4 * q^53 - 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 - 15 * q^71 - 6 * q^73 - 18 * q^77 + 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 + 35 * q^91 - 20 * q^93 - 3 * q^97 - 61 * q^99

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