LMFDB - Embedded newform 9200.2.a.ck.1.1

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:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.291367$ 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)

$f(q)$	$=$	$q-3.14073 q^{3} +1.20647 q^{7} +6.86420 q^{9} +O(q^{10})$	$q-3.14073 q^{3} +1.20647 q^{7} +6.86420 q^{9} -3.65773 q^{11} +0.859268 q^{13} +6.72347 q^{17} +1.51699 q^{19} -3.78921 q^{21} +1.00000 q^{23} -12.1364 q^{27} +0.548747 q^{29} +5.99568 q^{31} +11.4879 q^{33} +2.04100 q^{37} -2.69873 q^{39} -7.14998 q^{41} +10.0799 q^{43} +9.17040 q^{47} -5.54442 q^{49} -21.1166 q^{51} +5.37896 q^{53} -4.76447 q^{57} +0.582734 q^{59} -8.83244 q^{61} +8.28146 q^{63} +3.20647 q^{67} -3.14073 q^{69} +12.5784 q^{71} +8.62597 q^{73} -4.41294 q^{77} -0.0700619 q^{79} +17.5246 q^{81} -6.74197 q^{83} -1.72347 q^{87} +4.96393 q^{89} +1.03668 q^{91} -18.8308 q^{93} +11.3380 q^{97} -25.1074 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9	$ 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9} - 2 q^{11} + 14 q^{13} + 14 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} - 14 q^{27} + 4 q^{29} + 14 q^{33} + 2 q^{37} + 8 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} - 10 q^{51} + 4 q^{53} - 8 q^{61} + 12 q^{63} + 2 q^{67} - 2 q^{69} + 24 q^{71} + 18 q^{73} + 4 q^{77} - 24 q^{79} + 8 q^{81} + 6 q^{83} + 6 q^{87} - 8 q^{89} - 26 q^{91} + 2 q^{93} + 34 q^{97} - 36 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9 - 2 * q^11 + 14 * q^13 + 14 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 - 14 * q^27 + 4 * q^29 + 14 * q^33 + 2 * q^37 + 8 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 - 10 * q^51 + 4 * q^53 - 8 * q^61 + 12 * q^63 + 2 * q^67 - 2 * q^69 + 24 * q^71 + 18 * q^73 + 4 * q^77 - 24 * q^79 + 8 * q^81 + 6 * q^83 + 6 * q^87 - 8 * q^89 - 26 * q^91 + 2 * q^93 + 34 * q^97 - 36 * 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
$3$	−3.14073	−1.81330	−0.906651	−	0.421881i	$-0.861370\pi$
$3$	−3.14073	−1.81330	−0.906651	+	0.421881i	$0.861370\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.20647	0.456004	0.228002	−	0.973661i	$-0.426781\pi$
$7$	1.20647	0.456004	0.228002	+	0.973661i	$0.426781\pi$
$8$	0	0
$9$	6.86420	2.28807
$10$	0	0
$11$	−3.65773	−1.10285	−0.551423	−	0.834226i	$-0.685915\pi$
$11$	−3.65773	−1.10285	−0.551423	+	0.834226i	$0.685915\pi$
$12$	0	0
$13$	0.859268	0.238318	0.119159	−	0.992875i	$-0.461980\pi$
$13$	0.859268	0.238318	0.119159	+	0.992875i	$0.461980\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.72347	1.63068	0.815340	−	0.578982i	$-0.196550\pi$
$17$	6.72347	1.63068	0.815340	+	0.578982i	$0.196550\pi$
$18$	0	0
$19$	1.51699	0.348022	0.174011	−	0.984744i	$-0.444327\pi$
$19$	1.51699	0.348022	0.174011	+	0.984744i	$0.444327\pi$
$20$	0	0
$21$	−3.78921	−0.826873
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−12.1364	−2.33565
$28$	0	0
$29$	0.548747	0.101900	0.0509498	−	0.998701i	$-0.483775\pi$
$29$	0.548747	0.101900	0.0509498	+	0.998701i	$0.483775\pi$
$30$	0	0
$31$	5.99568	1.07686	0.538428	−	0.842672i	$-0.319018\pi$
$31$	5.99568	1.07686	0.538428	+	0.842672i	$0.319018\pi$
$32$	0	0
$33$	11.4879	1.99979
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.04100	0.335539	0.167770	−	0.985826i	$-0.446344\pi$
$37$	2.04100	0.335539	0.167770	+	0.985826i	$0.446344\pi$
$38$	0	0
$39$	−2.69873	−0.432143
$40$	0	0
$41$	−7.14998	−1.11664	−0.558320	−	0.829626i	$-0.688554\pi$
$41$	−7.14998	−1.11664	−0.558320	+	0.829626i	$0.688554\pi$
$42$	0	0
$43$	10.0799	1.53717	0.768587	−	0.639745i	$-0.220960\pi$
$43$	10.0799	1.53717	0.768587	+	0.639745i	$0.220960\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.17040	1.33764	0.668820	−	0.743424i	$-0.266800\pi$
$47$	9.17040	1.33764	0.668820	+	0.743424i	$0.266800\pi$
$48$	0	0
$49$	−5.54442	−0.792061
$50$	0	0
$51$	−21.1166	−2.95692
$52$	0	0
$53$	5.37896	0.738857	0.369428	−	0.929259i	$-0.379554\pi$
$53$	5.37896	0.738857	0.369428	+	0.929259i	$0.379554\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.76447	−0.631070
$58$	0	0
$59$	0.582734	0.0758655	0.0379327	−	0.999280i	$-0.487923\pi$
$59$	0.582734	0.0758655	0.0379327	+	0.999280i	$0.487923\pi$
$60$	0	0
$61$	−8.83244	−1.13088	−0.565439	−	0.824790i	$-0.691293\pi$
$61$	−8.83244	−1.13088	−0.565439	+	0.824790i	$0.691293\pi$
$62$	0	0
$63$	8.28146	1.04337
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.20647	0.391733	0.195866	−	0.980631i	$-0.437248\pi$
$67$	3.20647	0.391733	0.195866	+	0.980631i	$0.437248\pi$
$68$	0	0
$69$	−3.14073	−0.378100
$70$	0	0
$71$	12.5784	1.49278	0.746391	−	0.665507i	$-0.231785\pi$
$71$	12.5784	1.49278	0.746391	+	0.665507i	$0.231785\pi$
$72$	0	0
$73$	8.62597	1.00959	0.504797	−	0.863238i	$-0.331567\pi$
$73$	8.62597	1.00959	0.504797	+	0.863238i	$0.331567\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.41294	−0.502902
$78$	0	0
$79$	−0.0700619	−0.00788258	−0.00394129	−	0.999992i	$-0.501255\pi$
$79$	−0.0700619	−0.00788258	−0.00394129	+	0.999992i	$0.501255\pi$
$80$	0	0
$81$	17.5246	1.94718
$82$	0	0
$83$	−6.74197	−0.740027	−0.370014	−	0.929026i	$-0.620647\pi$
$83$	−6.74197	−0.740027	−0.370014	+	0.929026i	$0.620647\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.72347	−0.184775
$88$	0	0
$89$	4.96393	0.526175	0.263088	−	0.964772i	$-0.415259\pi$
$89$	4.96393	0.526175	0.263088	+	0.964772i	$0.415259\pi$
$90$	0	0
$91$	1.03668	0.108674
$92$	0	0
$93$	−18.8308	−1.95266
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.3380	1.15119	0.575597	−	0.817733i	$-0.304770\pi$
$97$	11.3380	1.15119	0.575597	+	0.817733i	$0.304770\pi$
$98$	0	0
$99$	−25.1074	−2.52338
$100$	0	0
$101$	3.44693	0.342983	0.171491	−	0.985186i	$-0.445141\pi$
$101$	3.44693	0.342983	0.171491	+	0.985186i	$0.445141\pi$
$102$	0	0
$103$	−9.13641	−0.900237	−0.450119	−	0.892969i	$-0.648618\pi$
$103$	−9.13641	−0.900237	−0.450119	+	0.892969i	$0.648618\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.24539	−0.893786	−0.446893	−	0.894588i	$-0.647469\pi$
$107$	−9.24539	−0.893786	−0.446893	+	0.894588i	$0.647469\pi$
$108$	0	0
$109$	1.64847	0.157895	0.0789476	−	0.996879i	$-0.474844\pi$
$109$	1.64847	0.157895	0.0789476	+	0.996879i	$0.474844\pi$
$110$	0	0
$111$	−6.41025	−0.608434
$112$	0	0
$113$	1.20647	0.113495	0.0567477	−	0.998389i	$-0.481927\pi$
$113$	1.20647	0.113495	0.0567477	+	0.998389i	$0.481927\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	5.89819	0.545287
$118$	0	0
$119$	8.11167	0.743596
$120$	0	0
$121$	2.37896	0.216269
$122$	0	0
$123$	22.4562	2.02481
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15.3542	−1.36247	−0.681233	−	0.732066i	$-0.738556\pi$
$127$	−15.3542	−1.36247	−0.681233	+	0.732066i	$0.738556\pi$
$128$	0	0
$129$	−31.6583	−2.78736
$130$	0	0
$131$	−13.0734	−1.14223	−0.571113	−	0.820872i	$-0.693488\pi$
$131$	−13.0734	−1.14223	−0.571113	+	0.820872i	$0.693488\pi$
$132$	0	0
$133$	1.83021	0.158699
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.1840	−1.12638	−0.563191	−	0.826327i	$-0.690427\pi$
$137$	−13.1840	−1.12638	−0.563191	+	0.826327i	$0.690427\pi$
$138$	0	0
$139$	14.1160	1.19730	0.598652	−	0.801010i	$-0.295703\pi$
$139$	14.1160	1.19730	0.598652	+	0.801010i	$0.295703\pi$
$140$	0	0
$141$	−28.8018	−2.42555
$142$	0	0
$143$	−3.14297	−0.262828
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.4136	1.43625
$148$	0	0
$149$	−6.54442	−0.536140	−0.268070	−	0.963399i	$-0.586386\pi$
$149$	−6.54442	−0.536140	−0.268070	+	0.963399i	$0.586386\pi$
$150$	0	0
$151$	−2.61672	−0.212946	−0.106473	−	0.994316i	$-0.533956\pi$
$151$	−2.61672	−0.212946	−0.106473	+	0.994316i	$0.533956\pi$
$152$	0	0
$153$	46.1512	3.73110
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.81394	0.703429	0.351715	−	0.936107i	$-0.385599\pi$
$157$	8.81394	0.703429	0.351715	+	0.936107i	$0.385599\pi$
$158$	0	0
$159$	−16.8939	−1.33977
$160$	0	0
$161$	1.20647	0.0950833
$162$	0	0
$163$	10.1747	0.796946	0.398473	−	0.917180i	$-0.369540\pi$
$163$	10.1747	0.796946	0.398473	+	0.917180i	$0.369540\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.6647	0.980027	0.490014	−	0.871715i	$-0.336992\pi$
$167$	12.6647	0.980027	0.490014	+	0.871715i	$0.336992\pi$
$168$	0	0
$169$	−12.2617	−0.943205
$170$	0	0
$171$	10.4129	0.796298
$172$	0	0
$173$	−15.6901	−1.19290	−0.596448	−	0.802652i	$-0.703422\pi$
$173$	−15.6901	−1.19290	−0.596448	+	0.802652i	$0.703422\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.83021	−0.137567
$178$	0	0
$179$	2.39876	0.179292	0.0896460	−	0.995974i	$-0.471426\pi$
$179$	2.39876	0.179292	0.0896460	+	0.995974i	$0.471426\pi$
$180$	0	0
$181$	−9.87122	−0.733722	−0.366861	−	0.930276i	$-0.619567\pi$
$181$	−9.87122	−0.733722	−0.366861	+	0.930276i	$0.619567\pi$
$182$	0	0
$183$	27.7403	2.05063
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−24.5926	−1.79839
$188$	0	0
$189$	−14.6422	−1.06507
$190$	0	0
$191$	9.60617	0.695078	0.347539	−	0.937666i	$-0.387017\pi$
$191$	9.60617	0.695078	0.347539	+	0.937666i	$0.387017\pi$
$192$	0	0
$193$	24.9709	1.79745	0.898724	−	0.438515i	$-0.144495\pi$
$193$	24.9709	1.79745	0.898724	+	0.438515i	$0.144495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.92292	−0.706979	−0.353489	−	0.935439i	$-0.615005\pi$
$197$	−9.92292	−0.706979	−0.353489	+	0.935439i	$0.615005\pi$
$198$	0	0
$199$	−7.01818	−0.497506	−0.248753	−	0.968567i	$-0.580021\pi$
$199$	−7.01818	−0.497506	−0.248753	+	0.968567i	$0.580021\pi$
$200$	0	0
$201$	−10.0707	−0.710330
$202$	0	0
$203$	0.662047	0.0464666
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.86420	0.477095
$208$	0	0
$209$	−5.54875	−0.383815
$210$	0	0
$211$	−7.44693	−0.512668	−0.256334	−	0.966588i	$-0.582515\pi$
$211$	−7.44693	−0.512668	−0.256334	+	0.966588i	$0.582515\pi$
$212$	0	0
$213$	−39.5054	−2.70687
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.23362	0.491050
$218$	0	0
$219$	−27.0919	−1.83070
$220$	0	0
$221$	5.77726	0.388620
$222$	0	0
$223$	10.2180	0.684245	0.342123	−	0.939655i	$-0.388854\pi$
$223$	10.2180	0.684245	0.342123	+	0.939655i	$0.388854\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.83291	0.387144	0.193572	−	0.981086i	$-0.437993\pi$
$227$	5.83291	0.387144	0.193572	+	0.981086i	$0.437993\pi$
$228$	0	0
$229$	5.87061	0.387941	0.193970	−	0.981007i	$-0.437863\pi$
$229$	5.87061	0.387941	0.193970	+	0.981007i	$0.437863\pi$
$230$	0	0
$231$	13.8599	0.911913
$232$	0	0
$233$	−0.839462	−0.0549950	−0.0274975	−	0.999622i	$-0.508754\pi$
$233$	−0.839462	−0.0549950	−0.0274975	+	0.999622i	$0.508754\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.220046	0.0142935
$238$	0	0
$239$	21.3296	1.37970	0.689850	−	0.723953i	$-0.257677\pi$
$239$	21.3296	1.37970	0.689850	+	0.723953i	$0.257677\pi$
$240$	0	0
$241$	−20.7037	−1.33364	−0.666820	−	0.745219i	$-0.732345\pi$
$241$	−20.7037	−1.33364	−0.666820	+	0.745219i	$0.732345\pi$
$242$	0	0
$243$	−18.6309	−1.19517
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.30350	0.0829400
$248$	0	0
$249$	21.1747	1.34189
$250$	0	0
$251$	−23.7290	−1.49776	−0.748881	−	0.662705i	$-0.769408\pi$
$251$	−23.7290	−1.49776	−0.748881	+	0.662705i	$0.769408\pi$
$252$	0	0
$253$	−3.65773	−0.229959
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.2481	−1.13828	−0.569142	−	0.822239i	$-0.692725\pi$
$257$	−18.2481	−1.13828	−0.569142	+	0.822239i	$0.692725\pi$
$258$	0	0
$259$	2.46242	0.153007
$260$	0	0
$261$	3.76670	0.233153
$262$	0	0
$263$	−25.8718	−1.59532	−0.797662	−	0.603104i	$-0.793930\pi$
$263$	−25.8718	−1.59532	−0.797662	+	0.603104i	$0.793930\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.5904	−0.954115
$268$	0	0
$269$	7.34497	0.447831	0.223915	−	0.974609i	$-0.428116\pi$
$269$	7.34497	0.447831	0.223915	+	0.974609i	$0.428116\pi$
$270$	0	0
$271$	−30.2278	−1.83621	−0.918105	−	0.396338i	$-0.870281\pi$
$271$	−30.2278	−1.83621	−0.918105	+	0.396338i	$0.870281\pi$
$272$	0	0
$273$	−3.25594	−0.197059
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.50983	0.150801	0.0754005	−	0.997153i	$-0.475976\pi$
$277$	2.50983	0.150801	0.0754005	+	0.997153i	$0.475976\pi$
$278$	0	0
$279$	41.1555	2.46392
$280$	0	0
$281$	−10.4836	−0.625400	−0.312700	−	0.949852i	$-0.601233\pi$
$281$	−10.4836	−0.625400	−0.312700	+	0.949852i	$0.601233\pi$
$282$	0	0
$283$	−3.83946	−0.228232	−0.114116	−	0.993467i	$-0.536404\pi$
$283$	−3.83946	−0.228232	−0.114116	+	0.993467i	$0.536404\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.62626	−0.509192
$288$	0	0
$289$	28.2050	1.65912
$290$	0	0
$291$	−35.6095	−2.08746
$292$	0	0
$293$	0.544425	0.0318056	0.0159028	−	0.999874i	$-0.494938\pi$
$293$	0.544425	0.0318056	0.0159028	+	0.999874i	$0.494938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	44.3917	2.57587
$298$	0	0
$299$	0.859268	0.0496927
$300$	0	0
$301$	12.1611	0.700957
$302$	0	0
$303$	−10.8259	−0.621931
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−16.1850	−0.923729	−0.461865	−	0.886950i	$-0.652819\pi$
$307$	−16.1850	−0.923729	−0.461865	+	0.886950i	$0.652819\pi$
$308$	0	0
$309$	28.6950	1.63240
$310$	0	0
$311$	7.51923	0.426376	0.213188	−	0.977011i	$-0.431615\pi$
$311$	7.51923	0.426376	0.213188	+	0.977011i	$0.431615\pi$
$312$	0	0
$313$	25.2475	1.42707	0.713536	−	0.700619i	$-0.247093\pi$
$313$	25.2475	1.42707	0.713536	+	0.700619i	$0.247093\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.4173	−0.753589	−0.376794	−	0.926297i	$-0.622974\pi$
$317$	−13.4173	−0.753589	−0.376794	+	0.926297i	$0.622974\pi$
$318$	0	0
$319$	−2.00716	−0.112380
$320$	0	0
$321$	29.0373	1.62070
$322$	0	0
$323$	10.1995	0.567513
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.17741	−0.286312
$328$	0	0
$329$	11.0638	0.609969
$330$	0	0
$331$	24.3748	1.33976	0.669880	−	0.742470i	$-0.266346\pi$
$331$	24.3748	1.33976	0.669880	+	0.742470i	$0.266346\pi$
$332$	0	0
$333$	14.0099	0.767736
$334$	0	0
$335$	0	0
$336$	0	0
$337$	19.2454	1.04836	0.524182	−	0.851607i	$-0.324371\pi$
$337$	19.2454	1.04836	0.524182	+	0.851607i	$0.324371\pi$
$338$	0	0
$339$	−3.78921	−0.205801
$340$	0	0
$341$	−21.9305	−1.18761
$342$	0	0
$343$	−15.1345	−0.817186
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.51044	0.403181	0.201591	−	0.979470i	$-0.435389\pi$
$347$	7.51044	0.403181	0.201591	+	0.979470i	$0.435389\pi$
$348$	0	0
$349$	12.3208	0.659520	0.329760	−	0.944065i	$-0.393032\pi$
$349$	12.3208	0.659520	0.329760	+	0.944065i	$0.393032\pi$
$350$	0	0
$351$	−10.4284	−0.556628
$352$	0	0
$353$	−11.7333	−0.624502	−0.312251	−	0.950000i	$-0.601083\pi$
$353$	−11.7333	−0.624502	−0.312251	+	0.950000i	$0.601083\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−25.4766	−1.34836
$358$	0	0
$359$	18.6879	0.986307	0.493154	−	0.869942i	$-0.335844\pi$
$359$	18.6879	0.986307	0.493154	+	0.869942i	$0.335844\pi$
$360$	0	0
$361$	−16.6987	−0.878881
$362$	0	0
$363$	−7.47167	−0.392161
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.0103	1.20113	0.600564	−	0.799576i	$-0.294943\pi$
$367$	23.0103	1.20113	0.600564	+	0.799576i	$0.294943\pi$
$368$	0	0
$369$	−49.0789	−2.55495
$370$	0	0
$371$	6.48956	0.336921
$372$	0	0
$373$	35.7402	1.85056	0.925278	−	0.379290i	$-0.123832\pi$
$373$	35.7402	1.85056	0.925278	+	0.379290i	$0.123832\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.471520	0.0242845
$378$	0	0
$379$	−9.65178	−0.495779	−0.247889	−	0.968788i	$-0.579737\pi$
$379$	−9.65178	−0.495779	−0.247889	+	0.968788i	$0.579737\pi$
$380$	0	0
$381$	48.2235	2.47056
$382$	0	0
$383$	25.0954	1.28232	0.641158	−	0.767409i	$-0.278454\pi$
$383$	25.0954	1.28232	0.641158	+	0.767409i	$0.278454\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	69.1906	3.51715
$388$	0	0
$389$	16.8259	0.853106	0.426553	−	0.904462i	$-0.359728\pi$
$389$	16.8259	0.853106	0.426553	+	0.904462i	$0.359728\pi$
$390$	0	0
$391$	6.72347	0.340020
$392$	0	0
$393$	41.0599	2.07120
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.8080	1.39564	0.697822	−	0.716272i	$-0.254153\pi$
$397$	27.8080	1.39564	0.697822	+	0.716272i	$0.254153\pi$
$398$	0	0
$399$	−5.74820	−0.287770
$400$	0	0
$401$	1.46483	0.0731500	0.0365750	−	0.999331i	$-0.488355\pi$
$401$	1.46483	0.0731500	0.0365750	+	0.999331i	$0.488355\pi$
$402$	0	0
$403$	5.15189	0.256634
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.46544	−0.370048
$408$	0	0
$409$	0.514906	0.0254605	0.0127302	−	0.999919i	$-0.495948\pi$
$409$	0.514906	0.0254605	0.0127302	+	0.999919i	$0.495948\pi$
$410$	0	0
$411$	41.4073	2.04247
$412$	0	0
$413$	0.703052	0.0345949
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−44.3346	−2.17107
$418$	0	0
$419$	9.65387	0.471622	0.235811	−	0.971799i	$-0.424225\pi$
$419$	9.65387	0.471622	0.235811	+	0.971799i	$0.424225\pi$
$420$	0	0
$421$	14.2788	0.695905	0.347952	−	0.937512i	$-0.386877\pi$
$421$	14.2788	0.695905	0.347952	+	0.937512i	$0.386877\pi$
$422$	0	0
$423$	62.9474	3.06061
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.6561	−0.515685
$428$	0	0
$429$	9.87122	0.476587
$430$	0	0
$431$	−5.71198	−0.275136	−0.137568	−	0.990492i	$-0.543929\pi$
$431$	−5.71198	−0.275136	−0.137568	+	0.990492i	$0.543929\pi$
$432$	0	0
$433$	−30.3302	−1.45758	−0.728789	−	0.684738i	$-0.759917\pi$
$433$	−30.3302	−1.45758	−0.728789	+	0.684738i	$0.759917\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.51699	0.0725676
$438$	0	0
$439$	19.7579	0.942994	0.471497	−	0.881868i	$-0.343714\pi$
$439$	19.7579	0.942994	0.471497	+	0.881868i	$0.343714\pi$
$440$	0	0
$441$	−38.0580	−1.81229
$442$	0	0
$443$	−16.0643	−0.763236	−0.381618	−	0.924320i	$-0.624633\pi$
$443$	−16.0643	−0.763236	−0.381618	+	0.924320i	$0.624633\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	20.5543	0.972184
$448$	0	0
$449$	28.5881	1.34916	0.674579	−	0.738203i	$-0.264325\pi$
$449$	28.5881	1.34916	0.674579	+	0.738203i	$0.264325\pi$
$450$	0	0
$451$	26.1527	1.23148
$452$	0	0
$453$	8.21842	0.386135
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.1209	0.473437	0.236718	−	0.971578i	$-0.423928\pi$
$457$	10.1209	0.473437	0.236718	+	0.971578i	$0.423928\pi$
$458$	0	0
$459$	−81.5987	−3.80870
$460$	0	0
$461$	15.6931	0.730901	0.365450	−	0.930831i	$-0.380915\pi$
$461$	15.6931	0.730901	0.365450	+	0.930831i	$0.380915\pi$
$462$	0	0
$463$	−12.9124	−0.600089	−0.300044	−	0.953925i	$-0.597001\pi$
$463$	−12.9124	−0.600089	−0.300044	+	0.953925i	$0.597001\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.10196	0.328640	0.164320	−	0.986407i	$-0.447457\pi$
$467$	7.10196	0.328640	0.164320	+	0.986407i	$0.447457\pi$
$468$	0	0
$469$	3.86852	0.178632
$470$	0	0
$471$	−27.6822	−1.27553
$472$	0	0
$473$	−36.8696	−1.69527
$474$	0	0
$475$	0	0
$476$	0	0
$477$	36.9222	1.69055
$478$	0	0
$479$	−16.7758	−0.766506	−0.383253	−	0.923643i	$-0.625196\pi$
$479$	−16.7758	−0.766506	−0.383253	+	0.923643i	$0.625196\pi$
$480$	0	0
$481$	1.75377	0.0799650
$482$	0	0
$483$	−3.78921	−0.172415
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7.02488	−0.318328	−0.159164	−	0.987252i	$-0.550880\pi$
$487$	−7.02488	−0.318328	−0.159164	+	0.987252i	$0.550880\pi$
$488$	0	0
$489$	−31.9561	−1.44510
$490$	0	0
$491$	−11.1500	−0.503192	−0.251596	−	0.967832i	$-0.580955\pi$
$491$	−11.1500	−0.503192	−0.251596	+	0.967832i	$0.580955\pi$
$492$	0	0
$493$	3.68948	0.166166
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.1755	0.680714
$498$	0	0
$499$	−14.8347	−0.664091	−0.332046	−	0.943263i	$-0.607739\pi$
$499$	−14.8347	−0.664091	−0.332046	+	0.943263i	$0.607739\pi$
$500$	0	0
$501$	−39.7766	−1.77709
$502$	0	0
$503$	3.10196	0.138310	0.0691548	−	0.997606i	$-0.477970\pi$
$503$	3.10196	0.138310	0.0691548	+	0.997606i	$0.477970\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	38.5106	1.71032
$508$	0	0
$509$	−22.3353	−0.989993	−0.494996	−	0.868895i	$-0.664831\pi$
$509$	−22.3353	−0.989993	−0.494996	+	0.868895i	$0.664831\pi$
$510$	0	0
$511$	10.4070	0.460378
$512$	0	0
$513$	−18.4109	−0.812859
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−33.5428	−1.47521
$518$	0	0
$519$	49.2784	2.16308
$520$	0	0
$521$	−14.2147	−0.622758	−0.311379	−	0.950286i	$-0.600791\pi$
$521$	−14.2147	−0.622758	−0.311379	+	0.950286i	$0.600791\pi$
$522$	0	0
$523$	−4.09494	−0.179059	−0.0895297	−	0.995984i	$-0.528536\pi$
$523$	−4.09494	−0.179059	−0.0895297	+	0.995984i	$0.528536\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.3117	1.75601
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	−6.14375	−0.266115
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−7.53387	−0.325111
$538$	0	0
$539$	20.2800	0.873521
$540$	0	0
$541$	33.6902	1.44846	0.724228	−	0.689560i	$-0.242196\pi$
$541$	33.6902	1.44846	0.724228	+	0.689560i	$0.242196\pi$
$542$	0	0
$543$	31.0028	1.33046
$544$	0	0
$545$	0	0
$546$	0	0
$547$	32.9358	1.40823	0.704117	−	0.710084i	$-0.251343\pi$
$547$	32.9358	1.40823	0.704117	+	0.710084i	$0.251343\pi$
$548$	0	0
$549$	−60.6277	−2.58753
$550$	0	0
$551$	0.832445	0.0354633
$552$	0	0
$553$	−0.0845278	−0.00359449
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.8337	0.882751	0.441375	−	0.897323i	$-0.354491\pi$
$557$	20.8337	0.882751	0.441375	+	0.897323i	$0.354491\pi$
$558$	0	0
$559$	8.66135	0.366336
$560$	0	0
$561$	77.2387	3.26102
$562$	0	0
$563$	30.4368	1.28276	0.641380	−	0.767223i	$-0.278362\pi$
$563$	30.4368	1.28276	0.641380	+	0.767223i	$0.278362\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	21.1430	0.887921
$568$	0	0
$569$	39.4801	1.65509	0.827545	−	0.561399i	$-0.189737\pi$
$569$	39.4801	1.65509	0.827545	+	0.561399i	$0.189737\pi$
$570$	0	0
$571$	−27.0301	−1.13118	−0.565588	−	0.824688i	$-0.691351\pi$
$571$	−27.0301	−1.13118	−0.565588	+	0.824688i	$0.691351\pi$
$572$	0	0
$573$	−30.1704	−1.26039
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.68818	−0.195171	−0.0975857	−	0.995227i	$-0.531112\pi$
$577$	−4.68818	−0.195171	−0.0975857	+	0.995227i	$0.531112\pi$
$578$	0	0
$579$	−78.4270	−3.25932
$580$	0	0
$581$	−8.13400	−0.337455
$582$	0	0
$583$	−19.6747	−0.814845
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.61718	0.396944	0.198472	−	0.980107i	$-0.436402\pi$
$587$	9.61718	0.396944	0.198472	+	0.980107i	$0.436402\pi$
$588$	0	0
$589$	9.09541	0.374770
$590$	0	0
$591$	31.1652	1.28197
$592$	0	0
$593$	−8.68902	−0.356815	−0.178408	−	0.983957i	$-0.557095\pi$
$593$	−8.68902	−0.356815	−0.178408	+	0.983957i	$0.557095\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.0422	0.902128
$598$	0	0
$599$	−11.7581	−0.480421	−0.240211	−	0.970721i	$-0.577217\pi$
$599$	−11.7581	−0.480421	−0.240211	+	0.970721i	$0.577217\pi$
$600$	0	0
$601$	37.9388	1.54756	0.773778	−	0.633457i	$-0.218365\pi$
$601$	37.9388	1.54756	0.773778	+	0.633457i	$0.218365\pi$
$602$	0	0
$603$	22.0099	0.896311
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.01418	0.365874	0.182937	−	0.983125i	$-0.441440\pi$
$607$	9.01418	0.365874	0.182937	+	0.983125i	$0.441440\pi$
$608$	0	0
$609$	−2.07931	−0.0842580
$610$	0	0
$611$	7.87983	0.318784
$612$	0	0
$613$	−42.5846	−1.71997	−0.859987	−	0.510316i	$-0.829528\pi$
$613$	−42.5846	−1.71997	−0.859987	+	0.510316i	$0.829528\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.6190	0.508020	0.254010	−	0.967202i	$-0.418250\pi$
$617$	12.6190	0.508020	0.254010	+	0.967202i	$0.418250\pi$
$618$	0	0
$619$	26.0638	1.04759	0.523796	−	0.851844i	$-0.324515\pi$
$619$	26.0638	1.04759	0.523796	+	0.851844i	$0.324515\pi$
$620$	0	0
$621$	−12.1364	−0.487017
$622$	0	0
$623$	5.98884	0.239938
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.4271	0.695972
$628$	0	0
$629$	13.7226	0.547157
$630$	0	0
$631$	46.1083	1.83554	0.917771	−	0.397110i	$-0.129987\pi$
$631$	46.1083	1.83554	0.917771	+	0.397110i	$0.129987\pi$
$632$	0	0
$633$	23.3888	0.929622
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.76415	−0.188762
$638$	0	0
$639$	86.3407	3.41559
$640$	0	0
$641$	19.3319	0.763563	0.381781	−	0.924253i	$-0.375311\pi$
$641$	19.3319	0.763563	0.381781	+	0.924253i	$0.375311\pi$
$642$	0	0
$643$	−2.27891	−0.0898713	−0.0449356	−	0.998990i	$-0.514308\pi$
$643$	−2.27891	−0.0898713	−0.0449356	+	0.998990i	$0.514308\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−47.5054	−1.86763	−0.933815	−	0.357755i	$-0.883542\pi$
$647$	−47.5054	−1.86763	−0.933815	+	0.357755i	$0.883542\pi$
$648$	0	0
$649$	−2.13148	−0.0836679
$650$	0	0
$651$	−22.7189	−0.890422
$652$	0	0
$653$	2.73226	0.106921	0.0534607	−	0.998570i	$-0.482975\pi$
$653$	2.73226	0.106921	0.0534607	+	0.998570i	$0.482975\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	59.2104	2.31002
$658$	0	0
$659$	12.1375	0.472809	0.236405	−	0.971655i	$-0.424031\pi$
$659$	12.1375	0.472809	0.236405	+	0.971655i	$0.424031\pi$
$660$	0	0
$661$	41.1992	1.60246	0.801232	−	0.598354i	$-0.204178\pi$
$661$	41.1992	1.60246	0.801232	+	0.598354i	$0.204178\pi$
$662$	0	0
$663$	−18.1448	−0.704686
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.548747	0.0212476
$668$	0	0
$669$	−32.0919	−1.24074
$670$	0	0
$671$	32.3067	1.24718
$672$	0	0
$673$	24.2175	0.933516	0.466758	−	0.884385i	$-0.345422\pi$
$673$	24.2175	0.933516	0.466758	+	0.884385i	$0.345422\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	48.1512	1.85060	0.925300	−	0.379235i	$-0.123813\pi$
$677$	48.1512	1.85060	0.925300	+	0.379235i	$0.123813\pi$
$678$	0	0
$679$	13.6789	0.524949
$680$	0	0
$681$	−18.3196	−0.702008
$682$	0	0
$683$	−12.1351	−0.464337	−0.232169	−	0.972676i	$-0.574582\pi$
$683$	−12.1351	−0.464337	−0.232169	+	0.972676i	$0.574582\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−18.4380	−0.703454
$688$	0	0
$689$	4.62197	0.176083
$690$	0	0
$691$	13.8289	0.526076	0.263038	−	0.964785i	$-0.415275\pi$
$691$	13.8289	0.526076	0.263038	+	0.964785i	$0.415275\pi$
$692$	0	0
$693$	−30.2913	−1.15067
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−48.0727	−1.82088
$698$	0	0
$699$	2.63653	0.0997226
$700$	0	0
$701$	−15.2263	−0.575089	−0.287544	−	0.957767i	$-0.592839\pi$
$701$	−15.2263	−0.575089	−0.287544	+	0.957767i	$0.592839\pi$
$702$	0	0
$703$	3.09619	0.116775
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.15863	0.156401
$708$	0	0
$709$	15.2317	0.572037	0.286019	−	0.958224i	$-0.407668\pi$
$709$	15.2317	0.572037	0.286019	+	0.958224i	$0.407668\pi$
$710$	0	0
$711$	−0.480919	−0.0180359
$712$	0	0
$713$	5.99568	0.224540
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−66.9907	−2.50181
$718$	0	0
$719$	−4.58943	−0.171157	−0.0855784	−	0.996331i	$-0.527274\pi$
$719$	−4.58943	−0.171157	−0.0855784	+	0.996331i	$0.527274\pi$
$720$	0	0
$721$	−11.0228	−0.410511
$722$	0	0
$723$	65.0246	2.41829
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21.2414	−0.787800	−0.393900	−	0.919153i	$-0.628874\pi$
$727$	−21.2414	−0.787800	−0.393900	+	0.919153i	$0.628874\pi$
$728$	0	0
$729$	5.94082	0.220030
$730$	0	0
$731$	67.7720	2.50664
$732$	0	0
$733$	40.5297	1.49700	0.748499	−	0.663136i	$-0.230775\pi$
$733$	40.5297	1.49700	0.748499	+	0.663136i	$0.230775\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.7284	−0.432021
$738$	0	0
$739$	−30.6476	−1.12739	−0.563695	−	0.825983i	$-0.690621\pi$
$739$	−30.6476	−1.12739	−0.563695	+	0.825983i	$0.690621\pi$
$740$	0	0
$741$	−4.09396	−0.150395
$742$	0	0
$743$	−1.01357	−0.0371844	−0.0185922	−	0.999827i	$-0.505918\pi$
$743$	−1.01357	−0.0371844	−0.0185922	+	0.999827i	$0.505918\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−46.2782	−1.69323
$748$	0	0
$749$	−11.1543	−0.407569
$750$	0	0
$751$	21.0644	0.768652	0.384326	−	0.923197i	$-0.374434\pi$
$751$	21.0644	0.768652	0.384326	+	0.923197i	$0.374434\pi$
$752$	0	0
$753$	74.5264	2.71589
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−13.2109	−0.480160	−0.240080	−	0.970753i	$-0.577174\pi$
$757$	−13.2109	−0.480160	−0.240080	+	0.970753i	$0.577174\pi$
$758$	0	0
$759$	11.4879	0.416986
$760$	0	0
$761$	−11.9759	−0.434125	−0.217063	−	0.976158i	$-0.569648\pi$
$761$	−11.9759	−0.434125	−0.217063	+	0.976158i	$0.569648\pi$
$762$	0	0
$763$	1.98884	0.0720008
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.500724	0.0180801
$768$	0	0
$769$	39.6569	1.43006	0.715031	−	0.699092i	$-0.246412\pi$
$769$	39.6569	1.43006	0.715031	+	0.699092i	$0.246412\pi$
$770$	0	0
$771$	57.3123	2.06405
$772$	0	0
$773$	16.0305	0.576575	0.288288	−	0.957544i	$-0.406914\pi$
$773$	16.0305	0.576575	0.288288	+	0.957544i	$0.406914\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.73379	−0.277448
$778$	0	0
$779$	−10.8465	−0.388615
$780$	0	0
$781$	−46.0084	−1.64631
$782$	0	0
$783$	−6.65981	−0.238002
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.0352	1.39145	0.695727	−	0.718306i	$-0.255082\pi$
$787$	39.0352	1.39145	0.695727	+	0.718306i	$0.255082\pi$
$788$	0	0
$789$	81.2565	2.89281
$790$	0	0
$791$	1.45558	0.0517543
$792$	0	0
$793$	−7.58944	−0.269509
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3.37687	−0.119615	−0.0598074	−	0.998210i	$-0.519049\pi$
$797$	−3.37687	−0.119615	−0.0598074	+	0.998210i	$0.519049\pi$
$798$	0	0
$799$	61.6569	2.18126
$800$	0	0
$801$	34.0734	1.20392
$802$	0	0
$803$	−31.5514	−1.11343
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.0686	−0.812053
$808$	0	0
$809$	−49.3406	−1.73472	−0.867361	−	0.497680i	$-0.834186\pi$
$809$	−49.3406	−1.73472	−0.867361	+	0.497680i	$0.834186\pi$
$810$	0	0
$811$	−18.3582	−0.644645	−0.322322	−	0.946630i	$-0.604463\pi$
$811$	−18.3582	−0.644645	−0.322322	+	0.946630i	$0.604463\pi$
$812$	0	0
$813$	94.9375	3.32960
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.2912	0.534971
$818$	0	0
$819$	7.11600	0.248653
$820$	0	0
$821$	19.4610	0.679192	0.339596	−	0.940571i	$-0.389710\pi$
$821$	19.4610	0.679192	0.339596	+	0.940571i	$0.389710\pi$
$822$	0	0
$823$	47.2973	1.64868	0.824341	−	0.566094i	$-0.191546\pi$
$823$	47.2973	1.64868	0.824341	+	0.566094i	$0.191546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.2901	−0.914197	−0.457098	−	0.889416i	$-0.651111\pi$
$827$	−26.2901	−0.914197	−0.457098	+	0.889416i	$0.651111\pi$
$828$	0	0
$829$	−20.0140	−0.695116	−0.347558	−	0.937658i	$-0.612989\pi$
$829$	−20.0140	−0.695116	−0.347558	+	0.937658i	$0.612989\pi$
$830$	0	0
$831$	−7.88270	−0.273448
$832$	0	0
$833$	−37.2778	−1.29160
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−72.7660	−2.51516
$838$	0	0
$839$	40.7210	1.40585	0.702923	−	0.711266i	$-0.251878\pi$
$839$	40.7210	1.40585	0.702923	+	0.711266i	$0.251878\pi$
$840$	0	0
$841$	−28.6989	−0.989616
$842$	0	0
$843$	32.9262	1.13404
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.87015	0.0986194
$848$	0	0
$849$	12.0587	0.413854
$850$	0	0
$851$	2.04100	0.0699647
$852$	0	0
$853$	5.34390	0.182972	0.0914858	−	0.995806i	$-0.470838\pi$
$853$	5.34390	0.182972	0.0914858	+	0.995806i	$0.470838\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.4603	1.65537	0.827686	−	0.561192i	$-0.189657\pi$
$857$	48.4603	1.65537	0.827686	+	0.561192i	$0.189657\pi$
$858$	0	0
$859$	51.3123	1.75075	0.875377	−	0.483440i	$-0.160613\pi$
$859$	51.3123	1.75075	0.875377	+	0.483440i	$0.160613\pi$
$860$	0	0
$861$	27.0928	0.923319
$862$	0	0
$863$	−24.5507	−0.835714	−0.417857	−	0.908513i	$-0.637219\pi$
$863$	−24.5507	−0.835714	−0.417857	+	0.908513i	$0.637219\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−88.5843	−3.00848
$868$	0	0
$869$	0.256267	0.00869327
$870$	0	0
$871$	2.75522	0.0933570
$872$	0	0
$873$	77.8260	2.63401
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30.1400	1.01776	0.508878	−	0.860838i	$-0.330060\pi$
$877$	30.1400	1.01776	0.508878	+	0.860838i	$0.330060\pi$
$878$	0	0
$879$	−1.70989	−0.0576732
$880$	0	0
$881$	36.4729	1.22880	0.614401	−	0.788994i	$-0.289398\pi$
$881$	36.4729	1.22880	0.614401	+	0.788994i	$0.289398\pi$
$882$	0	0
$883$	2.17426	0.0731696	0.0365848	−	0.999331i	$-0.488352\pi$
$883$	2.17426	0.0731696	0.0365848	+	0.999331i	$0.488352\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.1463	0.542139	0.271069	−	0.962560i	$-0.412623\pi$
$887$	16.1463	0.542139	0.271069	+	0.962560i	$0.412623\pi$
$888$	0	0
$889$	−18.5244	−0.621290
$890$	0	0
$891$	−64.1003	−2.14744
$892$	0	0
$893$	13.9114	0.465528
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.69873	−0.0901080
$898$	0	0
$899$	3.29011	0.109731
$900$	0	0
$901$	36.1652	1.20484
$902$	0	0
$903$	−38.1949	−1.27105
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.74475	0.157547	0.0787734	−	0.996893i	$-0.474900\pi$
$907$	4.74475	0.157547	0.0787734	+	0.996893i	$0.474900\pi$
$908$	0	0
$909$	23.6604	0.784767
$910$	0	0
$911$	−38.8650	−1.28765	−0.643827	−	0.765171i	$-0.722654\pi$
$911$	−38.8650	−1.28765	−0.643827	+	0.765171i	$0.722654\pi$
$912$	0	0
$913$	24.6603	0.816136
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.7727	−0.520859
$918$	0	0
$919$	−40.5516	−1.33767	−0.668837	−	0.743409i	$-0.733207\pi$
$919$	−40.5516	−1.33767	−0.668837	+	0.743409i	$0.733207\pi$
$920$	0	0
$921$	50.8329	1.67500
$922$	0	0
$923$	10.8082	0.355757
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−62.7141	−2.05980
$928$	0	0
$929$	19.4994	0.639755	0.319878	−	0.947459i	$-0.396358\pi$
$929$	19.4994	0.639755	0.319878	+	0.947459i	$0.396358\pi$
$930$	0	0
$931$	−8.41086	−0.275655
$932$	0	0
$933$	−23.6159	−0.773149
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.9361	−0.814626	−0.407313	−	0.913289i	$-0.633534\pi$
$937$	−24.9361	−0.814626	−0.407313	+	0.913289i	$0.633534\pi$
$938$	0	0
$939$	−79.2956	−2.58771
$940$	0	0
$941$	−41.9160	−1.36642	−0.683211	−	0.730221i	$-0.739417\pi$
$941$	−41.9160	−1.36642	−0.683211	+	0.730221i	$0.739417\pi$
$942$	0	0
$943$	−7.14998	−0.232836
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.3006	0.952141	0.476070	−	0.879407i	$-0.342061\pi$
$947$	29.3006	0.952141	0.476070	+	0.879407i	$0.342061\pi$
$948$	0	0
$949$	7.41202	0.240604
$950$	0	0
$951$	42.1400	1.36648
$952$	0	0
$953$	−24.9759	−0.809048	−0.404524	−	0.914527i	$-0.632563\pi$
$953$	−24.9759	−0.809048	−0.404524	+	0.914527i	$0.632563\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.30397	0.203778
$958$	0	0
$959$	−15.9061	−0.513635
$960$	0	0
$961$	4.94816	0.159618
$962$	0	0
$963$	−63.4622	−2.04504
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−0.405142	−0.0130285	−0.00651424	−	0.999979i	$-0.502074\pi$
$967$	−0.405142	−0.0130285	−0.00651424	+	0.999979i	$0.502074\pi$
$968$	0	0
$969$	−32.0338	−1.02907
$970$	0	0
$971$	−38.8032	−1.24525	−0.622627	−	0.782519i	$-0.713935\pi$
$971$	−38.8032	−1.24525	−0.622627	+	0.782519i	$0.713935\pi$
$972$	0	0
$973$	17.0306	0.545975
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−39.9940	−1.27952	−0.639760	−	0.768575i	$-0.720966\pi$
$977$	−39.9940	−1.27952	−0.639760	+	0.768575i	$0.720966\pi$
$978$	0	0
$979$	−18.1567	−0.580290
$980$	0	0
$981$	11.3155	0.361275
$982$	0	0
$983$	−21.3563	−0.681160	−0.340580	−	0.940216i	$-0.610623\pi$
$983$	−21.3563	−0.681160	−0.340580	+	0.940216i	$0.610623\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−34.7485	−1.10606
$988$	0	0
$989$	10.0799	0.320523
$990$	0	0
$991$	−4.39444	−0.139594	−0.0697970	−	0.997561i	$-0.522235\pi$
$991$	−4.39444	−0.139594	−0.0697970	+	0.997561i	$0.522235\pi$
$992$	0	0
$993$	−76.5547	−2.42939
$994$	0	0
$995$	0	0
$996$	0	0
$997$	28.9420	0.916603	0.458302	−	0.888797i	$-0.348458\pi$
$997$	28.9420	0.916603	0.458302	+	0.888797i	$0.348458\pi$
$998$	0	0
$999$	−24.7705	−0.783703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.ck.1.1		4
4.3	odd	2		575.2.a.j.1.3		4
5.2	odd	4		1840.2.e.d.369.8		8
5.3	odd	4		1840.2.e.d.369.1		8
5.4	even	2		9200.2.a.cq.1.4		4
12.11	even	2		5175.2.a.bw.1.2		4
20.3	even	4		115.2.b.b.24.4	✓	8
20.7	even	4		115.2.b.b.24.5	yes	8
20.19	odd	2		575.2.a.i.1.2		4
60.23	odd	4		1035.2.b.e.829.5		8
60.47	odd	4		1035.2.b.e.829.4		8
60.59	even	2		5175.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.b.b.24.4	✓	8	20.3	even	4
115.2.b.b.24.5	yes	8	20.7	even	4
575.2.a.i.1.2		4	20.19	odd	2
575.2.a.j.1.3		4	4.3	odd	2
1035.2.b.e.829.4		8	60.47	odd	4
1035.2.b.e.829.5		8	60.23	odd	4
1840.2.e.d.369.1		8	5.3	odd	4
1840.2.e.d.369.8		8	5.2	odd	4
5175.2.a.bv.1.3		4	60.59	even	2
5175.2.a.bw.1.2		4	12.11	even	2
9200.2.a.ck.1.1		4	1.1	even	1	trivial
9200.2.a.cq.1.4		4	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)

\(f(q)\)	\(=\)	\(q-3.14073 q^{3} +1.20647 q^{7} +6.86420 q^{9} +O(q^{10})\)	\(q-3.14073 q^{3} +1.20647 q^{7} +6.86420 q^{9} -3.65773 q^{11} +0.859268 q^{13} +6.72347 q^{17} +1.51699 q^{19} -3.78921 q^{21} +1.00000 q^{23} -12.1364 q^{27} +0.548747 q^{29} +5.99568 q^{31} +11.4879 q^{33} +2.04100 q^{37} -2.69873 q^{39} -7.14998 q^{41} +10.0799 q^{43} +9.17040 q^{47} -5.54442 q^{49} -21.1166 q^{51} +5.37896 q^{53} -4.76447 q^{57} +0.582734 q^{59} -8.83244 q^{61} +8.28146 q^{63} +3.20647 q^{67} -3.14073 q^{69} +12.5784 q^{71} +8.62597 q^{73} -4.41294 q^{77} -0.0700619 q^{79} +17.5246 q^{81} -6.74197 q^{83} -1.72347 q^{87} +4.96393 q^{89} +1.03668 q^{91} -18.8308 q^{93} +11.3380 q^{97} -25.1074 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9	\( 4 q - 2 q^{3} - 6 q^{7} + 4 q^{9} - 2 q^{11} + 14 q^{13} + 14 q^{17} + 4 q^{19} - 2 q^{21} + 4 q^{23} - 14 q^{27} + 4 q^{29} + 14 q^{33} + 2 q^{37} + 8 q^{39} - 8 q^{41} - 4 q^{43} - 2 q^{47} - 10 q^{51} + 4 q^{53} - 8 q^{61} + 12 q^{63} + 2 q^{67} - 2 q^{69} + 24 q^{71} + 18 q^{73} + 4 q^{77} - 24 q^{79} + 8 q^{81} + 6 q^{83} + 6 q^{87} - 8 q^{89} - 26 q^{91} + 2 q^{93} + 34 q^{97} - 36 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 6 * q^7 + 4 * q^9 - 2 * q^11 + 14 * q^13 + 14 * q^17 + 4 * q^19 - 2 * q^21 + 4 * q^23 - 14 * q^27 + 4 * q^29 + 14 * q^33 + 2 * q^37 + 8 * q^39 - 8 * q^41 - 4 * q^43 - 2 * q^47 - 10 * q^51 + 4 * q^53 - 8 * q^61 + 12 * q^63 + 2 * q^67 - 2 * q^69 + 24 * q^71 + 18 * q^73 + 4 * q^77 - 24 * q^79 + 8 * q^81 + 6 * q^83 + 6 * q^87 - 8 * q^89 - 26 * q^91 + 2 * q^93 + 34 * q^97 - 36 * q^99

Introduction

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