LMFDB - Embedded newform 9200.2.a.ca.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q+3.30278 q^{3} -0.302776 q^{7} +7.90833 q^{9} +O(q^{10})$	$q+3.30278 q^{3} -0.302776 q^{7} +7.90833 q^{9} +5.30278 q^{11} +0.302776 q^{13} +3.90833 q^{17} +4.90833 q^{19} -1.00000 q^{21} -1.00000 q^{23} +16.2111 q^{27} +4.60555 q^{29} -2.90833 q^{31} +17.5139 q^{33} -8.00000 q^{37} +1.00000 q^{39} -9.90833 q^{41} +5.21110 q^{43} +4.60555 q^{47} -6.90833 q^{49} +12.9083 q^{51} -3.21110 q^{53} +16.2111 q^{57} +10.6056 q^{59} -6.51388 q^{61} -2.39445 q^{63} -4.00000 q^{67} -3.30278 q^{69} +12.6972 q^{71} -15.8167 q^{73} -1.60555 q^{77} -14.4222 q^{79} +29.8167 q^{81} -3.21110 q^{83} +15.2111 q^{87} -0.0916731 q^{91} -9.60555 q^{93} -2.69722 q^{97} +41.9361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 3 * q^7 + 5 * q^9	$ 2 q + 3 q^{3} + 3 q^{7} + 5 q^{9} + 7 q^{11} - 3 q^{13} - 3 q^{17} - q^{19} - 2 q^{21} - 2 q^{23} + 18 q^{27} + 2 q^{29} + 5 q^{31} + 17 q^{33} - 16 q^{37} + 2 q^{39} - 9 q^{41} - 4 q^{43} + 2 q^{47} - 3 q^{49} + 15 q^{51} + 8 q^{53} + 18 q^{57} + 14 q^{59} + 5 q^{61} - 12 q^{63} - 8 q^{67} - 3 q^{69} + 29 q^{71} - 10 q^{73} + 4 q^{77} + 38 q^{81} + 8 q^{83} + 16 q^{87} - 11 q^{91} - 12 q^{93} - 9 q^{97} + 37 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 3 * q^7 + 5 * q^9 + 7 * q^11 - 3 * q^13 - 3 * q^17 - q^19 - 2 * q^21 - 2 * q^23 + 18 * q^27 + 2 * q^29 + 5 * q^31 + 17 * q^33 - 16 * q^37 + 2 * q^39 - 9 * q^41 - 4 * q^43 + 2 * q^47 - 3 * q^49 + 15 * q^51 + 8 * q^53 + 18 * q^57 + 14 * q^59 + 5 * q^61 - 12 * q^63 - 8 * q^67 - 3 * q^69 + 29 * q^71 - 10 * q^73 + 4 * q^77 + 38 * q^81 + 8 * q^83 + 16 * q^87 - 11 * q^91 - 12 * q^93 - 9 * q^97 + 37 * 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
$3$	3.30278	1.90686	0.953429	−	0.301617i	$-0.0975264\pi$
$3$	3.30278	1.90686	0.953429	+	0.301617i	$0.0975264\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.302776	−0.114438	−0.0572192	−	0.998362i	$-0.518223\pi$
$7$	−0.302776	−0.114438	−0.0572192	+	0.998362i	$0.518223\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	5.30278	1.59885	0.799424	−	0.600768i	$-0.205138\pi$
$11$	5.30278	1.59885	0.799424	+	0.600768i	$0.205138\pi$
$12$	0	0
$13$	0.302776	0.0839749	0.0419874	−	0.999118i	$-0.486631\pi$
$13$	0.302776	0.0839749	0.0419874	+	0.999118i	$0.486631\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.90833	0.947909	0.473954	−	0.880549i	$-0.342826\pi$
$17$	3.90833	0.947909	0.473954	+	0.880549i	$0.342826\pi$
$18$	0	0
$19$	4.90833	1.12605	0.563024	−	0.826441i	$-0.309638\pi$
$19$	4.90833	1.12605	0.563024	+	0.826441i	$0.309638\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	16.2111	3.11983
$28$	0	0
$29$	4.60555	0.855229	0.427615	−	0.903961i	$-0.359354\pi$
$29$	4.60555	0.855229	0.427615	+	0.903961i	$0.359354\pi$
$30$	0	0
$31$	−2.90833	−0.522351	−0.261175	−	0.965291i	$-0.584110\pi$
$31$	−2.90833	−0.522351	−0.261175	+	0.965291i	$0.584110\pi$
$32$	0	0
$33$	17.5139	3.04877
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.90833	−1.54742	−0.773710	−	0.633540i	$-0.781601\pi$
$41$	−9.90833	−1.54742	−0.773710	+	0.633540i	$0.781601\pi$
$42$	0	0
$43$	5.21110	0.794686	0.397343	−	0.917670i	$-0.369932\pi$
$43$	5.21110	0.794686	0.397343	+	0.917670i	$0.369932\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.60555	0.671789	0.335894	−	0.941900i	$-0.390961\pi$
$47$	4.60555	0.671789	0.335894	+	0.941900i	$0.390961\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	12.9083	1.80753
$52$	0	0
$53$	−3.21110	−0.441079	−0.220539	−	0.975378i	$-0.570782\pi$
$53$	−3.21110	−0.441079	−0.220539	+	0.975378i	$0.570782\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	16.2111	2.14721
$58$	0	0
$59$	10.6056	1.38073	0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056	1.38073	0.690363	+	0.723464i	$0.257451\pi$
$60$	0	0
$61$	−6.51388	−0.834017	−0.417008	−	0.908903i	$-0.636921\pi$
$61$	−6.51388	−0.834017	−0.417008	+	0.908903i	$0.636921\pi$
$62$	0	0
$63$	−2.39445	−0.301672
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	−3.30278	−0.397607
$70$	0	0
$71$	12.6972	1.50688	0.753442	−	0.657515i	$-0.228392\pi$
$71$	12.6972	1.50688	0.753442	+	0.657515i	$0.228392\pi$
$72$	0	0
$73$	−15.8167	−1.85120	−0.925600	−	0.378504i	$-0.876439\pi$
$73$	−15.8167	−1.85120	−0.925600	+	0.378504i	$0.876439\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.60555	−0.182970
$78$	0	0
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	−3.21110	−0.352464	−0.176232	−	0.984349i	$-0.556391\pi$
$83$	−3.21110	−0.352464	−0.176232	+	0.984349i	$0.556391\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	15.2111	1.63080
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−0.0916731	−0.00960995
$92$	0	0
$93$	−9.60555	−0.996049
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.69722	−0.273862	−0.136931	−	0.990581i	$-0.543724\pi$
$97$	−2.69722	−0.273862	−0.136931	+	0.990581i	$0.543724\pi$
$98$	0	0
$99$	41.9361	4.21473
$100$	0	0
$101$	−4.60555	−0.458269	−0.229135	−	0.973395i	$-0.573590\pi$
$101$	−4.60555	−0.458269	−0.229135	+	0.973395i	$0.573590\pi$
$102$	0	0
$103$	−17.1194	−1.68683	−0.843414	−	0.537265i	$-0.819458\pi$
$103$	−17.1194	−1.68683	−0.843414	+	0.537265i	$0.819458\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.60555	0.445235	0.222618	−	0.974906i	$-0.428540\pi$
$107$	4.60555	0.445235	0.222618	+	0.974906i	$0.428540\pi$
$108$	0	0
$109$	19.5139	1.86909	0.934545	−	0.355844i	$-0.115807\pi$
$109$	19.5139	1.86909	0.934545	+	0.355844i	$0.115807\pi$
$110$	0	0
$111$	−26.4222	−2.50788
$112$	0	0
$113$	−12.4222	−1.16858	−0.584291	−	0.811544i	$-0.698627\pi$
$113$	−12.4222	−1.16858	−0.584291	+	0.811544i	$0.698627\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.39445	0.221367
$118$	0	0
$119$	−1.18335	−0.108477
$120$	0	0
$121$	17.1194	1.55631
$122$	0	0
$123$	−32.7250	−2.95071
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.8167	−1.04856	−0.524279	−	0.851546i	$-0.675665\pi$
$127$	−11.8167	−1.04856	−0.524279	+	0.851546i	$0.675665\pi$
$128$	0	0
$129$	17.2111	1.51535
$130$	0	0
$131$	−3.21110	−0.280555	−0.140278	−	0.990112i	$-0.544800\pi$
$131$	−3.21110	−0.280555	−0.140278	+	0.990112i	$0.544800\pi$
$132$	0	0
$133$	−1.48612	−0.128863
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.90833	0.590218	0.295109	−	0.955464i	$-0.404644\pi$
$137$	6.90833	0.590218	0.295109	+	0.955464i	$0.404644\pi$
$138$	0	0
$139$	5.39445	0.457551	0.228776	−	0.973479i	$-0.426528\pi$
$139$	5.39445	0.457551	0.228776	+	0.973479i	$0.426528\pi$
$140$	0	0
$141$	15.2111	1.28101
$142$	0	0
$143$	1.60555	0.134263
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−22.8167	−1.88189
$148$	0	0
$149$	9.69722	0.794428	0.397214	−	0.917726i	$-0.369977\pi$
$149$	9.69722	0.794428	0.397214	+	0.917726i	$0.369977\pi$
$150$	0	0
$151$	1.90833	0.155297	0.0776487	−	0.996981i	$-0.475259\pi$
$151$	1.90833	0.155297	0.0776487	+	0.996981i	$0.475259\pi$
$152$	0	0
$153$	30.9083	2.49879
$154$	0	0
$155$	0	0
$156$	0	0
$157$	11.3944	0.909376	0.454688	−	0.890651i	$-0.349751\pi$
$157$	11.3944	0.909376	0.454688	+	0.890651i	$0.349751\pi$
$158$	0	0
$159$	−10.6056	−0.841075
$160$	0	0
$161$	0.302776	0.0238621
$162$	0	0
$163$	5.69722	0.446241	0.223121	−	0.974791i	$-0.428376\pi$
$163$	5.69722	0.446241	0.223121	+	0.974791i	$0.428376\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.2111	1.64136	0.820682	−	0.571385i	$-0.193594\pi$
$167$	21.2111	1.64136	0.820682	+	0.571385i	$0.193594\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	0	0
$171$	38.8167	2.96838
$172$	0	0
$173$	−23.3028	−1.77168	−0.885839	−	0.463993i	$-0.846416\pi$
$173$	−23.3028	−1.77168	−0.885839	+	0.463993i	$0.846416\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	35.0278	2.63285
$178$	0	0
$179$	−16.6056	−1.24116	−0.620579	−	0.784144i	$-0.713102\pi$
$179$	−16.6056	−1.24116	−0.620579	+	0.784144i	$0.713102\pi$
$180$	0	0
$181$	−8.11943	−0.603512	−0.301756	−	0.953385i	$-0.597573\pi$
$181$	−8.11943	−0.603512	−0.301756	+	0.953385i	$0.597573\pi$
$182$	0	0
$183$	−21.5139	−1.59035
$184$	0	0
$185$	0	0
$186$	0	0
$187$	20.7250	1.51556
$188$	0	0
$189$	−4.90833	−0.357028
$190$	0	0
$191$	1.39445	0.100899	0.0504494	−	0.998727i	$-0.483935\pi$
$191$	1.39445	0.100899	0.0504494	+	0.998727i	$0.483935\pi$
$192$	0	0
$193$	−3.81665	−0.274729	−0.137364	−	0.990521i	$-0.543863\pi$
$193$	−3.81665	−0.274729	−0.137364	+	0.990521i	$0.543863\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.697224	−0.0496752	−0.0248376	−	0.999691i	$-0.507907\pi$
$197$	−0.697224	−0.0496752	−0.0248376	+	0.999691i	$0.507907\pi$
$198$	0	0
$199$	−8.42221	−0.597034	−0.298517	−	0.954404i	$-0.596492\pi$
$199$	−8.42221	−0.597034	−0.298517	+	0.954404i	$0.596492\pi$
$200$	0	0
$201$	−13.2111	−0.931839
$202$	0	0
$203$	−1.39445	−0.0978711
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.90833	−0.549667
$208$	0	0
$209$	26.0278	1.80038
$210$	0	0
$211$	7.21110	0.496433	0.248216	−	0.968705i	$-0.420156\pi$
$211$	7.21110	0.496433	0.248216	+	0.968705i	$0.420156\pi$
$212$	0	0
$213$	41.9361	2.87341
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.880571	0.0597770
$218$	0	0
$219$	−52.2389	−3.52997
$220$	0	0
$221$	1.18335	0.0796005
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.39445	−0.490787	−0.245393	−	0.969424i	$-0.578917\pi$
$227$	−7.39445	−0.490787	−0.245393	+	0.969424i	$0.578917\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−5.30278	−0.348897
$232$	0	0
$233$	−4.18335	−0.274060	−0.137030	−	0.990567i	$-0.543756\pi$
$233$	−4.18335	−0.274060	−0.137030	+	0.990567i	$0.543756\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−47.6333	−3.09412
$238$	0	0
$239$	9.21110	0.595817	0.297908	−	0.954594i	$-0.403711\pi$
$239$	9.21110	0.595817	0.297908	+	0.954594i	$0.403711\pi$
$240$	0	0
$241$	14.4222	0.929016	0.464508	−	0.885569i	$-0.346231\pi$
$241$	14.4222	0.929016	0.464508	+	0.885569i	$0.346231\pi$
$242$	0	0
$243$	49.8444	3.19752
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.48612	0.0945597
$248$	0	0
$249$	−10.6056	−0.672100
$250$	0	0
$251$	5.51388	0.348033	0.174016	−	0.984743i	$-0.444325\pi$
$251$	5.51388	0.348033	0.174016	+	0.984743i	$0.444325\pi$
$252$	0	0
$253$	−5.30278	−0.333383
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.8167	−1.23613	−0.618064	−	0.786127i	$-0.712083\pi$
$257$	−19.8167	−1.23613	−0.618064	+	0.786127i	$0.712083\pi$
$258$	0	0
$259$	2.42221	0.150509
$260$	0	0
$261$	36.4222	2.25448
$262$	0	0
$263$	−14.5139	−0.894964	−0.447482	−	0.894293i	$-0.647679\pi$
$263$	−14.5139	−0.894964	−0.447482	+	0.894293i	$0.647679\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.8167	1.57407	0.787035	−	0.616909i	$-0.211615\pi$
$269$	25.8167	1.57407	0.787035	+	0.616909i	$0.211615\pi$
$270$	0	0
$271$	6.30278	0.382866	0.191433	−	0.981506i	$-0.438686\pi$
$271$	6.30278	0.382866	0.191433	+	0.981506i	$0.438686\pi$
$272$	0	0
$273$	−0.302776	−0.0183248
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.7889	0.768410	0.384205	−	0.923248i	$-0.374476\pi$
$277$	12.7889	0.768410	0.384205	+	0.923248i	$0.374476\pi$
$278$	0	0
$279$	−23.0000	−1.37697
$280$	0	0
$281$	−19.3944	−1.15698	−0.578488	−	0.815691i	$-0.696357\pi$
$281$	−19.3944	−1.15698	−0.578488	+	0.815691i	$0.696357\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.00000	0.177084
$288$	0	0
$289$	−1.72498	−0.101469
$290$	0	0
$291$	−8.90833	−0.522215
$292$	0	0
$293$	−8.78890	−0.513453	−0.256726	−	0.966484i	$-0.582644\pi$
$293$	−8.78890	−0.513453	−0.256726	+	0.966484i	$0.582644\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	85.9638	4.98813
$298$	0	0
$299$	−0.302776	−0.0175100
$300$	0	0
$301$	−1.57779	−0.0909426
$302$	0	0
$303$	−15.2111	−0.873855
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.3028	−0.873376	−0.436688	−	0.899613i	$-0.643849\pi$
$307$	−15.3028	−0.873376	−0.436688	+	0.899613i	$0.643849\pi$
$308$	0	0
$309$	−56.5416	−3.21654
$310$	0	0
$311$	6.42221	0.364170	0.182085	−	0.983283i	$-0.441715\pi$
$311$	6.42221	0.364170	0.182085	+	0.983283i	$0.441715\pi$
$312$	0	0
$313$	12.7250	0.719258	0.359629	−	0.933095i	$-0.382903\pi$
$313$	12.7250	0.719258	0.359629	+	0.933095i	$0.382903\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.7250	0.827037	0.413519	−	0.910496i	$-0.364300\pi$
$317$	14.7250	0.827037	0.413519	+	0.910496i	$0.364300\pi$
$318$	0	0
$319$	24.4222	1.36738
$320$	0	0
$321$	15.2111	0.849001
$322$	0	0
$323$	19.1833	1.06739
$324$	0	0
$325$	0	0
$326$	0	0
$327$	64.4500	3.56409
$328$	0	0
$329$	−1.39445	−0.0768784
$330$	0	0
$331$	−9.39445	−0.516366	−0.258183	−	0.966096i	$-0.583124\pi$
$331$	−9.39445	−0.516366	−0.258183	+	0.966096i	$0.583124\pi$
$332$	0	0
$333$	−63.2666	−3.46699
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.48612	0.244375	0.122187	−	0.992507i	$-0.461009\pi$
$337$	4.48612	0.244375	0.122187	+	0.992507i	$0.461009\pi$
$338$	0	0
$339$	−41.0278	−2.22832
$340$	0	0
$341$	−15.4222	−0.835159
$342$	0	0
$343$	4.21110	0.227378
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.5416	−1.37115	−0.685573	−	0.728004i	$-0.740448\pi$
$347$	−25.5416	−1.37115	−0.685573	+	0.728004i	$0.740448\pi$
$348$	0	0
$349$	−12.7889	−0.684574	−0.342287	−	0.939595i	$-0.611202\pi$
$349$	−12.7889	−0.684574	−0.342287	+	0.939595i	$0.611202\pi$
$350$	0	0
$351$	4.90833	0.261987
$352$	0	0
$353$	−18.4222	−0.980515	−0.490258	−	0.871578i	$-0.663097\pi$
$353$	−18.4222	−0.980515	−0.490258	+	0.871578i	$0.663097\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.90833	−0.206851
$358$	0	0
$359$	3.21110	0.169476	0.0847378	−	0.996403i	$-0.472995\pi$
$359$	3.21110	0.169476	0.0847378	+	0.996403i	$0.472995\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	0	0
$363$	56.5416	2.96767
$364$	0	0
$365$	0	0
$366$	0	0
$367$	29.2111	1.52481	0.762404	−	0.647102i	$-0.224019\pi$
$367$	29.2111	1.52481	0.762404	+	0.647102i	$0.224019\pi$
$368$	0	0
$369$	−78.3583	−4.07917
$370$	0	0
$371$	0.972244	0.0504764
$372$	0	0
$373$	2.60555	0.134910	0.0674552	−	0.997722i	$-0.478512\pi$
$373$	2.60555	0.134910	0.0674552	+	0.997722i	$0.478512\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.39445	0.0718178
$378$	0	0
$379$	−4.09167	−0.210175	−0.105088	−	0.994463i	$-0.533512\pi$
$379$	−4.09167	−0.210175	−0.105088	+	0.994463i	$0.533512\pi$
$380$	0	0
$381$	−39.0278	−1.99945
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	41.2111	2.09488
$388$	0	0
$389$	20.9361	1.06150	0.530751	−	0.847528i	$-0.321910\pi$
$389$	20.9361	1.06150	0.530751	+	0.847528i	$0.321910\pi$
$390$	0	0
$391$	−3.90833	−0.197653
$392$	0	0
$393$	−10.6056	−0.534979
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.7250	1.09035	0.545173	−	0.838324i	$-0.316464\pi$
$397$	21.7250	1.09035	0.545173	+	0.838324i	$0.316464\pi$
$398$	0	0
$399$	−4.90833	−0.245724
$400$	0	0
$401$	−1.39445	−0.0696354	−0.0348177	−	0.999394i	$-0.511085\pi$
$401$	−1.39445	−0.0696354	−0.0348177	+	0.999394i	$0.511085\pi$
$402$	0	0
$403$	−0.880571	−0.0438643
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−42.4222	−2.10279
$408$	0	0
$409$	−15.0917	−0.746235	−0.373118	−	0.927784i	$-0.621711\pi$
$409$	−15.0917	−0.746235	−0.373118	+	0.927784i	$0.621711\pi$
$410$	0	0
$411$	22.8167	1.12546
$412$	0	0
$413$	−3.21110	−0.158008
$414$	0	0
$415$	0	0
$416$	0	0
$417$	17.8167	0.872485
$418$	0	0
$419$	39.6333	1.93621	0.968107	−	0.250538i	$-0.0806073\pi$
$419$	39.6333	1.93621	0.968107	+	0.250538i	$0.0806073\pi$
$420$	0	0
$421$	34.3028	1.67181	0.835907	−	0.548870i	$-0.184942\pi$
$421$	34.3028	1.67181	0.835907	+	0.548870i	$0.184942\pi$
$422$	0	0
$423$	36.4222	1.77091
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.97224	0.0954436
$428$	0	0
$429$	5.30278	0.256020
$430$	0	0
$431$	−20.2389	−0.974872	−0.487436	−	0.873159i	$-0.662068\pi$
$431$	−20.2389	−0.974872	−0.487436	+	0.873159i	$0.662068\pi$
$432$	0	0
$433$	34.9083	1.67759	0.838794	−	0.544450i	$-0.183261\pi$
$433$	34.9083	1.67759	0.838794	+	0.544450i	$0.183261\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.90833	−0.234797
$438$	0	0
$439$	18.3028	0.873544	0.436772	−	0.899572i	$-0.356122\pi$
$439$	18.3028	0.873544	0.436772	+	0.899572i	$0.356122\pi$
$440$	0	0
$441$	−54.6333	−2.60159
$442$	0	0
$443$	35.5139	1.68732	0.843658	−	0.536882i	$-0.180398\pi$
$443$	35.5139	1.68732	0.843658	+	0.536882i	$0.180398\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	32.0278	1.51486
$448$	0	0
$449$	−12.9083	−0.609182	−0.304591	−	0.952483i	$-0.598520\pi$
$449$	−12.9083	−0.609182	−0.304591	+	0.952483i	$0.598520\pi$
$450$	0	0
$451$	−52.5416	−2.47409
$452$	0	0
$453$	6.30278	0.296130
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.57779	0.167362	0.0836811	−	0.996493i	$-0.473332\pi$
$457$	3.57779	0.167362	0.0836811	+	0.996493i	$0.473332\pi$
$458$	0	0
$459$	63.3583	2.95731
$460$	0	0
$461$	31.8167	1.48185	0.740925	−	0.671588i	$-0.234388\pi$
$461$	31.8167	1.48185	0.740925	+	0.671588i	$0.234388\pi$
$462$	0	0
$463$	−25.6333	−1.19128	−0.595640	−	0.803251i	$-0.703102\pi$
$463$	−25.6333	−1.19128	−0.595640	+	0.803251i	$0.703102\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.8167	0.917005	0.458503	−	0.888693i	$-0.348386\pi$
$467$	19.8167	0.917005	0.458503	+	0.888693i	$0.348386\pi$
$468$	0	0
$469$	1.21110	0.0559235
$470$	0	0
$471$	37.6333	1.73405
$472$	0	0
$473$	27.6333	1.27058
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−25.3944	−1.16273
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	−2.42221	−0.110443
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.8167	−0.535464	−0.267732	−	0.963493i	$-0.586274\pi$
$487$	−11.8167	−0.535464	−0.267732	+	0.963493i	$0.586274\pi$
$488$	0	0
$489$	18.8167	0.850918
$490$	0	0
$491$	25.8167	1.16509	0.582545	−	0.812799i	$-0.302057\pi$
$491$	25.8167	1.16509	0.582545	+	0.812799i	$0.302057\pi$
$492$	0	0
$493$	18.0000	0.810679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.84441	−0.172445
$498$	0	0
$499$	−11.6333	−0.520778	−0.260389	−	0.965504i	$-0.583851\pi$
$499$	−11.6333	−0.520778	−0.260389	+	0.965504i	$0.583851\pi$
$500$	0	0
$501$	70.0555	3.12985
$502$	0	0
$503$	−2.72498	−0.121501	−0.0607504	−	0.998153i	$-0.519349\pi$
$503$	−2.72498	−0.121501	−0.0607504	+	0.998153i	$0.519349\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−42.6333	−1.89341
$508$	0	0
$509$	29.4500	1.30535	0.652673	−	0.757639i	$-0.273647\pi$
$509$	29.4500	1.30535	0.652673	+	0.757639i	$0.273647\pi$
$510$	0	0
$511$	4.78890	0.211848
$512$	0	0
$513$	79.5694	3.51307
$514$	0	0
$515$	0	0
$516$	0	0
$517$	24.4222	1.07409
$518$	0	0
$519$	−76.9638	−3.37834
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	8.42221	0.368277	0.184139	−	0.982900i	$-0.441050\pi$
$523$	8.42221	0.368277	0.184139	+	0.982900i	$0.441050\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.3667	−0.495141
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	83.8722	3.63974
$532$	0	0
$533$	−3.00000	−0.129944
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−54.8444	−2.36671
$538$	0	0
$539$	−36.6333	−1.57791
$540$	0	0
$541$	−28.8444	−1.24012	−0.620059	−	0.784555i	$-0.712891\pi$
$541$	−28.8444	−1.24012	−0.620059	+	0.784555i	$0.712891\pi$
$542$	0	0
$543$	−26.8167	−1.15081
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.51388	0.321270	0.160635	−	0.987014i	$-0.448646\pi$
$547$	7.51388	0.321270	0.160635	+	0.987014i	$0.448646\pi$
$548$	0	0
$549$	−51.5139	−2.19856
$550$	0	0
$551$	22.6056	0.963029
$552$	0	0
$553$	4.36669	0.185691
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.42221	0.272118	0.136059	−	0.990701i	$-0.456556\pi$
$557$	6.42221	0.272118	0.136059	+	0.990701i	$0.456556\pi$
$558$	0	0
$559$	1.57779	0.0667336
$560$	0	0
$561$	68.4500	2.88996
$562$	0	0
$563$	39.6333	1.67034	0.835172	−	0.549988i	$-0.185368\pi$
$563$	39.6333	1.67034	0.835172	+	0.549988i	$0.185368\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−9.02776	−0.379130
$568$	0	0
$569$	−0.422205	−0.0176998	−0.00884988	−	0.999961i	$-0.502817\pi$
$569$	−0.422205	−0.0176998	−0.00884988	+	0.999961i	$0.502817\pi$
$570$	0	0
$571$	−9.11943	−0.381636	−0.190818	−	0.981625i	$-0.561114\pi$
$571$	−9.11943	−0.381636	−0.190818	+	0.981625i	$0.561114\pi$
$572$	0	0
$573$	4.60555	0.192400
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	−12.6056	−0.523869
$580$	0	0
$581$	0.972244	0.0403355
$582$	0	0
$583$	−17.0278	−0.705218
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.5416	−1.54951	−0.774755	−	0.632262i	$-0.782127\pi$
$587$	−37.5416	−1.54951	−0.774755	+	0.632262i	$0.782127\pi$
$588$	0	0
$589$	−14.2750	−0.588192
$590$	0	0
$591$	−2.30278	−0.0947235
$592$	0	0
$593$	−19.8167	−0.813772	−0.406886	−	0.913479i	$-0.633385\pi$
$593$	−19.8167	−0.813772	−0.406886	+	0.913479i	$0.633385\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−27.8167	−1.13846
$598$	0	0
$599$	−4.33053	−0.176941	−0.0884704	−	0.996079i	$-0.528198\pi$
$599$	−4.33053	−0.176941	−0.0884704	+	0.996079i	$0.528198\pi$
$600$	0	0
$601$	−3.93608	−0.160556	−0.0802781	−	0.996773i	$-0.525581\pi$
$601$	−3.93608	−0.160556	−0.0802781	+	0.996773i	$0.525581\pi$
$602$	0	0
$603$	−31.6333	−1.28821
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.0555	−1.05756	−0.528780	−	0.848759i	$-0.677350\pi$
$607$	−26.0555	−1.05756	−0.528780	+	0.848759i	$0.677350\pi$
$608$	0	0
$609$	−4.60555	−0.186626
$610$	0	0
$611$	1.39445	0.0564134
$612$	0	0
$613$	−32.4222	−1.30952	−0.654760	−	0.755837i	$-0.727230\pi$
$613$	−32.4222	−1.30952	−0.654760	+	0.755837i	$0.727230\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8.09167	−0.325758	−0.162879	−	0.986646i	$-0.552078\pi$
$617$	−8.09167	−0.325758	−0.162879	+	0.986646i	$0.552078\pi$
$618$	0	0
$619$	−27.3305	−1.09851	−0.549253	−	0.835656i	$-0.685088\pi$
$619$	−27.3305	−1.09851	−0.549253	+	0.835656i	$0.685088\pi$
$620$	0	0
$621$	−16.2111	−0.650529
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	85.9638	3.43307
$628$	0	0
$629$	−31.2666	−1.24668
$630$	0	0
$631$	−30.6056	−1.21839	−0.609194	−	0.793021i	$-0.708507\pi$
$631$	−30.6056	−1.21839	−0.609194	+	0.793021i	$0.708507\pi$
$632$	0	0
$633$	23.8167	0.946627
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.09167	−0.0828751
$638$	0	0
$639$	100.414	3.97231
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	0	0
$643$	16.2389	0.640398	0.320199	−	0.947350i	$-0.396250\pi$
$643$	16.2389	0.640398	0.320199	+	0.947350i	$0.396250\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30.8444	−1.21262	−0.606309	−	0.795229i	$-0.707351\pi$
$647$	−30.8444	−1.21262	−0.606309	+	0.795229i	$0.707351\pi$
$648$	0	0
$649$	56.2389	2.20757
$650$	0	0
$651$	2.90833	0.113986
$652$	0	0
$653$	−9.27502	−0.362960	−0.181480	−	0.983395i	$-0.558089\pi$
$653$	−9.27502	−0.362960	−0.181480	+	0.983395i	$0.558089\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−125.083	−4.87996
$658$	0	0
$659$	27.6333	1.07644	0.538220	−	0.842804i	$-0.319097\pi$
$659$	27.6333	1.07644	0.538220	+	0.842804i	$0.319097\pi$
$660$	0	0
$661$	−24.0917	−0.937057	−0.468529	−	0.883448i	$-0.655216\pi$
$661$	−24.0917	−0.937057	−0.468529	+	0.883448i	$0.655216\pi$
$662$	0	0
$663$	3.90833	0.151787
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.60555	−0.178328
$668$	0	0
$669$	−13.2111	−0.510771
$670$	0	0
$671$	−34.5416	−1.33347
$672$	0	0
$673$	−5.63331	−0.217148	−0.108574	−	0.994088i	$-0.534628\pi$
$673$	−5.63331	−0.217148	−0.108574	+	0.994088i	$0.534628\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.4222	0.477424	0.238712	−	0.971090i	$-0.423275\pi$
$677$	12.4222	0.477424	0.238712	+	0.971090i	$0.423275\pi$
$678$	0	0
$679$	0.816654	0.0313403
$680$	0	0
$681$	−24.4222	−0.935861
$682$	0	0
$683$	−32.7250	−1.25219	−0.626093	−	0.779748i	$-0.715347\pi$
$683$	−32.7250	−1.25219	−0.626093	+	0.779748i	$0.715347\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.60555	0.252018
$688$	0	0
$689$	−0.972244	−0.0370395
$690$	0	0
$691$	−30.1833	−1.14823	−0.574114	−	0.818775i	$-0.694653\pi$
$691$	−30.1833	−1.14823	−0.574114	+	0.818775i	$0.694653\pi$
$692$	0	0
$693$	−12.6972	−0.482328
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−38.7250	−1.46681
$698$	0	0
$699$	−13.8167	−0.522594
$700$	0	0
$701$	42.9083	1.62063	0.810313	−	0.585998i	$-0.199297\pi$
$701$	42.9083	1.62063	0.810313	+	0.585998i	$0.199297\pi$
$702$	0	0
$703$	−39.2666	−1.48097
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.39445	0.0524436
$708$	0	0
$709$	−41.1194	−1.54427	−0.772136	−	0.635457i	$-0.780812\pi$
$709$	−41.1194	−1.54427	−0.772136	+	0.635457i	$0.780812\pi$
$710$	0	0
$711$	−114.056	−4.27742
$712$	0	0
$713$	2.90833	0.108918
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30.4222	1.13614
$718$	0	0
$719$	−14.3028	−0.533404	−0.266702	−	0.963779i	$-0.585934\pi$
$719$	−14.3028	−0.533404	−0.266702	+	0.963779i	$0.585934\pi$
$720$	0	0
$721$	5.18335	0.193038
$722$	0	0
$723$	47.6333	1.77150
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−7.90833	−0.293304	−0.146652	−	0.989188i	$-0.546850\pi$
$727$	−7.90833	−0.293304	−0.146652	+	0.989188i	$0.546850\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	20.3667	0.753289
$732$	0	0
$733$	13.6333	0.503558	0.251779	−	0.967785i	$-0.418984\pi$
$733$	13.6333	0.503558	0.251779	+	0.967785i	$0.418984\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−21.2111	−0.781321
$738$	0	0
$739$	7.63331	0.280796	0.140398	−	0.990095i	$-0.455162\pi$
$739$	7.63331	0.280796	0.140398	+	0.990095i	$0.455162\pi$
$740$	0	0
$741$	4.90833	0.180312
$742$	0	0
$743$	7.33053	0.268931	0.134466	−	0.990918i	$-0.457068\pi$
$743$	7.33053	0.268931	0.134466	+	0.990918i	$0.457068\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−25.3944	−0.929134
$748$	0	0
$749$	−1.39445	−0.0509520
$750$	0	0
$751$	−0.183346	−0.00669040	−0.00334520	−	0.999994i	$-0.501065\pi$
$751$	−0.183346	−0.00669040	−0.00334520	+	0.999994i	$0.501065\pi$
$752$	0	0
$753$	18.2111	0.663649
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.21110	0.0440183	0.0220091	−	0.999758i	$-0.492994\pi$
$757$	1.21110	0.0440183	0.0220091	+	0.999758i	$0.492994\pi$
$758$	0	0
$759$	−17.5139	−0.635714
$760$	0	0
$761$	4.54163	0.164634	0.0823171	−	0.996606i	$-0.473768\pi$
$761$	4.54163	0.164634	0.0823171	+	0.996606i	$0.473768\pi$
$762$	0	0
$763$	−5.90833	−0.213896
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.21110	0.115946
$768$	0	0
$769$	−41.2666	−1.48811	−0.744056	−	0.668117i	$-0.767100\pi$
$769$	−41.2666	−1.48811	−0.744056	+	0.668117i	$0.767100\pi$
$770$	0	0
$771$	−65.4500	−2.35712
$772$	0	0
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	−48.6333	−1.74247
$780$	0	0
$781$	67.3305	2.40928
$782$	0	0
$783$	74.6611	2.66817
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−27.4500	−0.978485	−0.489243	−	0.872148i	$-0.662727\pi$
$787$	−27.4500	−0.978485	−0.489243	+	0.872148i	$0.662727\pi$
$788$	0	0
$789$	−47.9361	−1.70657
$790$	0	0
$791$	3.76114	0.133731
$792$	0	0
$793$	−1.97224	−0.0700364
$794$	0	0
$795$	0	0
$796$	0	0
$797$	19.8167	0.701942	0.350971	−	0.936386i	$-0.385852\pi$
$797$	19.8167	0.701942	0.350971	+	0.936386i	$0.385852\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−83.8722	−2.95978
$804$	0	0
$805$	0	0
$806$	0	0
$807$	85.2666	3.00153
$808$	0	0
$809$	18.2750	0.642515	0.321258	−	0.946992i	$-0.395894\pi$
$809$	18.2750	0.642515	0.321258	+	0.946992i	$0.395894\pi$
$810$	0	0
$811$	4.97224	0.174599	0.0872995	−	0.996182i	$-0.472176\pi$
$811$	4.97224	0.174599	0.0872995	+	0.996182i	$0.472176\pi$
$812$	0	0
$813$	20.8167	0.730072
$814$	0	0
$815$	0	0
$816$	0	0
$817$	25.5778	0.894854
$818$	0	0
$819$	−0.724981	−0.0253329
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−0.788897	−0.0274992	−0.0137496	−	0.999905i	$-0.504377\pi$
$823$	−0.788897	−0.0274992	−0.0137496	+	0.999905i	$0.504377\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.4500	1.23272	0.616358	−	0.787466i	$-0.288607\pi$
$827$	35.4500	1.23272	0.616358	+	0.787466i	$0.288607\pi$
$828$	0	0
$829$	16.7889	0.583103	0.291551	−	0.956555i	$-0.405829\pi$
$829$	16.7889	0.583103	0.291551	+	0.956555i	$0.405829\pi$
$830$	0	0
$831$	42.2389	1.46525
$832$	0	0
$833$	−27.0000	−0.935495
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−47.1472	−1.62965
$838$	0	0
$839$	22.1833	0.765854	0.382927	−	0.923779i	$-0.374916\pi$
$839$	22.1833	0.765854	0.382927	+	0.923779i	$0.374916\pi$
$840$	0	0
$841$	−7.78890	−0.268583
$842$	0	0
$843$	−64.0555	−2.20619
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.18335	−0.178102
$848$	0	0
$849$	6.60555	0.226702
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−10.7250	−0.367216	−0.183608	−	0.983000i	$-0.558778\pi$
$853$	−10.7250	−0.367216	−0.183608	+	0.983000i	$0.558778\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	33.6333	1.14889	0.574446	−	0.818543i	$-0.305218\pi$
$857$	33.6333	1.14889	0.574446	+	0.818543i	$0.305218\pi$
$858$	0	0
$859$	14.1833	0.483930	0.241965	−	0.970285i	$-0.422208\pi$
$859$	14.1833	0.483930	0.241965	+	0.970285i	$0.422208\pi$
$860$	0	0
$861$	9.90833	0.337675
$862$	0	0
$863$	23.4500	0.798246	0.399123	−	0.916897i	$-0.369315\pi$
$863$	23.4500	0.798246	0.399123	+	0.916897i	$0.369315\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−5.69722	−0.193488
$868$	0	0
$869$	−76.4777	−2.59433
$870$	0	0
$871$	−1.21110	−0.0410366
$872$	0	0
$873$	−21.3305	−0.721929
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.1749	−1.66052	−0.830260	−	0.557376i	$-0.811808\pi$
$877$	−49.1749	−1.66052	−0.830260	+	0.557376i	$0.811808\pi$
$878$	0	0
$879$	−29.0278	−0.979082
$880$	0	0
$881$	−31.2666	−1.05340	−0.526700	−	0.850052i	$-0.676571\pi$
$881$	−31.2666	−1.05340	−0.526700	+	0.850052i	$0.676571\pi$
$882$	0	0
$883$	40.7250	1.37050	0.685252	−	0.728306i	$-0.259692\pi$
$883$	40.7250	1.37050	0.685252	+	0.728306i	$0.259692\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.6333	−0.524915	−0.262458	−	0.964944i	$-0.584533\pi$
$887$	−15.6333	−0.524915	−0.262458	+	0.964944i	$0.584533\pi$
$888$	0	0
$889$	3.57779	0.119995
$890$	0	0
$891$	158.111	5.29692
$892$	0	0
$893$	22.6056	0.756466
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.00000	−0.0333890
$898$	0	0
$899$	−13.3944	−0.446730
$900$	0	0
$901$	−12.5500	−0.418102
$902$	0	0
$903$	−5.21110	−0.173415
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−30.6611	−1.01808	−0.509042	−	0.860742i	$-0.670000\pi$
$907$	−30.6611	−1.01808	−0.509042	+	0.860742i	$0.670000\pi$
$908$	0	0
$909$	−36.4222	−1.20805
$910$	0	0
$911$	25.8167	0.855344	0.427672	−	0.903934i	$-0.359334\pi$
$911$	25.8167	0.855344	0.427672	+	0.903934i	$0.359334\pi$
$912$	0	0
$913$	−17.0278	−0.563536
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0.972244	0.0321063
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	−50.5416	−1.66540
$922$	0	0
$923$	3.84441	0.126540
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−135.386	−4.44666
$928$	0	0
$929$	−57.6333	−1.89089	−0.945444	−	0.325785i	$-0.894371\pi$
$929$	−57.6333	−1.89089	−0.945444	+	0.325785i	$0.894371\pi$
$930$	0	0
$931$	−33.9083	−1.11130
$932$	0	0
$933$	21.2111	0.694420
$934$	0	0
$935$	0	0
$936$	0	0
$937$	44.9638	1.46890	0.734452	−	0.678660i	$-0.237439\pi$
$937$	44.9638	1.46890	0.734452	+	0.678660i	$0.237439\pi$
$938$	0	0
$939$	42.0278	1.37152
$940$	0	0
$941$	−20.9361	−0.682497	−0.341248	−	0.939973i	$-0.610850\pi$
$941$	−20.9361	−0.682497	−0.341248	+	0.939973i	$0.610850\pi$
$942$	0	0
$943$	9.90833	0.322660
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.9361	1.36274	0.681370	−	0.731939i	$-0.261385\pi$
$947$	41.9361	1.36274	0.681370	+	0.731939i	$0.261385\pi$
$948$	0	0
$949$	−4.78890	−0.155454
$950$	0	0
$951$	48.6333	1.57704
$952$	0	0
$953$	1.66947	0.0540794	0.0270397	−	0.999634i	$-0.491392\pi$
$953$	1.66947	0.0540794	0.0270397	+	0.999634i	$0.491392\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	80.6611	2.60740
$958$	0	0
$959$	−2.09167	−0.0675436
$960$	0	0
$961$	−22.5416	−0.727150
$962$	0	0
$963$	36.4222	1.17369
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−5.39445	−0.173474	−0.0867369	−	0.996231i	$-0.527644\pi$
$967$	−5.39445	−0.173474	−0.0867369	+	0.996231i	$0.527644\pi$
$968$	0	0
$969$	63.3583	2.03536
$970$	0	0
$971$	27.9083	0.895621	0.447810	−	0.894129i	$-0.352204\pi$
$971$	27.9083	0.895621	0.447810	+	0.894129i	$0.352204\pi$
$972$	0	0
$973$	−1.63331	−0.0523614
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.5139	0.368362	0.184181	−	0.982892i	$-0.441037\pi$
$977$	11.5139	0.368362	0.184181	+	0.982892i	$0.441037\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	154.322	4.92713
$982$	0	0
$983$	−19.5416	−0.623281	−0.311641	−	0.950200i	$-0.600879\pi$
$983$	−19.5416	−0.623281	−0.311641	+	0.950200i	$0.600879\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.60555	−0.146596
$988$	0	0
$989$	−5.21110	−0.165703
$990$	0	0
$991$	−24.3305	−0.772885	−0.386442	−	0.922314i	$-0.626296\pi$
$991$	−24.3305	−0.772885	−0.386442	+	0.922314i	$0.626296\pi$
$992$	0	0
$993$	−31.0278	−0.984636
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.2111	0.988466	0.494233	−	0.869330i	$-0.335449\pi$
$997$	31.2111	0.988466	0.494233	+	0.869330i	$0.335449\pi$
$998$	0	0
$999$	−129.689	−4.10317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ca.1.2		2
4.3	odd	2		1150.2.a.m.1.1		2
5.4	even	2		1840.2.a.j.1.1		2
20.3	even	4		1150.2.b.f.599.1		4
20.7	even	4		1150.2.b.f.599.4		4
20.19	odd	2		230.2.a.b.1.2	✓	2
40.19	odd	2		7360.2.a.bc.1.1		2
40.29	even	2		7360.2.a.bu.1.2		2
60.59	even	2		2070.2.a.w.1.1		2
460.459	even	2		5290.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.b.1.2	✓	2	20.19	odd	2
1150.2.a.m.1.1		2	4.3	odd	2
1150.2.b.f.599.1		4	20.3	even	4
1150.2.b.f.599.4		4	20.7	even	4
1840.2.a.j.1.1		2	5.4	even	2
2070.2.a.w.1.1		2	60.59	even	2
5290.2.a.j.1.2		2	460.459	even	2
7360.2.a.bc.1.1		2	40.19	odd	2
7360.2.a.bu.1.2		2	40.29	even	2
9200.2.a.ca.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+3.30278 q^{3} -0.302776 q^{7} +7.90833 q^{9} +O(q^{10})\)	\(q+3.30278 q^{3} -0.302776 q^{7} +7.90833 q^{9} +5.30278 q^{11} +0.302776 q^{13} +3.90833 q^{17} +4.90833 q^{19} -1.00000 q^{21} -1.00000 q^{23} +16.2111 q^{27} +4.60555 q^{29} -2.90833 q^{31} +17.5139 q^{33} -8.00000 q^{37} +1.00000 q^{39} -9.90833 q^{41} +5.21110 q^{43} +4.60555 q^{47} -6.90833 q^{49} +12.9083 q^{51} -3.21110 q^{53} +16.2111 q^{57} +10.6056 q^{59} -6.51388 q^{61} -2.39445 q^{63} -4.00000 q^{67} -3.30278 q^{69} +12.6972 q^{71} -15.8167 q^{73} -1.60555 q^{77} -14.4222 q^{79} +29.8167 q^{81} -3.21110 q^{83} +15.2111 q^{87} -0.0916731 q^{91} -9.60555 q^{93} -2.69722 q^{97} +41.9361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 3 * q^7 + 5 * q^9	\( 2 q + 3 q^{3} + 3 q^{7} + 5 q^{9} + 7 q^{11} - 3 q^{13} - 3 q^{17} - q^{19} - 2 q^{21} - 2 q^{23} + 18 q^{27} + 2 q^{29} + 5 q^{31} + 17 q^{33} - 16 q^{37} + 2 q^{39} - 9 q^{41} - 4 q^{43} + 2 q^{47} - 3 q^{49} + 15 q^{51} + 8 q^{53} + 18 q^{57} + 14 q^{59} + 5 q^{61} - 12 q^{63} - 8 q^{67} - 3 q^{69} + 29 q^{71} - 10 q^{73} + 4 q^{77} + 38 q^{81} + 8 q^{83} + 16 q^{87} - 11 q^{91} - 12 q^{93} - 9 q^{97} + 37 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 3 * q^7 + 5 * q^9 + 7 * q^11 - 3 * q^13 - 3 * q^17 - q^19 - 2 * q^21 - 2 * q^23 + 18 * q^27 + 2 * q^29 + 5 * q^31 + 17 * q^33 - 16 * q^37 + 2 * q^39 - 9 * q^41 - 4 * q^43 + 2 * q^47 - 3 * q^49 + 15 * q^51 + 8 * q^53 + 18 * q^57 + 14 * q^59 + 5 * q^61 - 12 * q^63 - 8 * q^67 - 3 * q^69 + 29 * q^71 - 10 * q^73 + 4 * q^77 + 38 * q^81 + 8 * q^83 + 16 * q^87 - 11 * q^91 - 12 * q^93 - 9 * q^97 + 37 * q^99

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