LMFDB - Embedded newform 9200.2.a.bp.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1150)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +O(q^{10})$	$q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} +3.56155 q^{13} +1.43845 q^{17} -3.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} -1.43845 q^{27} +5.56155 q^{29} -3.12311 q^{31} -2.56155 q^{33} -5.12311 q^{37} -9.12311 q^{39} -10.1231 q^{41} +4.68466 q^{43} -6.00000 q^{47} -4.56155 q^{49} -3.68466 q^{51} +0.876894 q^{53} +7.68466 q^{57} -4.00000 q^{59} -14.2462 q^{61} +5.56155 q^{63} +1.43845 q^{67} +2.56155 q^{69} +7.12311 q^{71} +2.12311 q^{73} +1.56155 q^{77} +12.9309 q^{79} -7.00000 q^{81} +2.12311 q^{83} -14.2462 q^{87} -12.5616 q^{89} +5.56155 q^{91} +8.00000 q^{93} -8.24621 q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - q^7 + 3 * q^9	$ 2 q - q^{3} - q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + 7 q^{17} - 6 q^{19} - 8 q^{21} - 2 q^{23} - 7 q^{27} + 7 q^{29} + 2 q^{31} - q^{33} - 2 q^{37} - 10 q^{39} - 12 q^{41} - 3 q^{43} - 12 q^{47} - 5 q^{49} + 5 q^{51} + 10 q^{53} + 3 q^{57} - 8 q^{59} - 12 q^{61} + 7 q^{63} + 7 q^{67} + q^{69} + 6 q^{71} - 4 q^{73} - q^{77} - 3 q^{79} - 14 q^{81} - 4 q^{83} - 12 q^{87} - 21 q^{89} + 7 q^{91} + 16 q^{93} + 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^7 + 3 * q^9 + 2 * q^11 + 3 * q^13 + 7 * q^17 - 6 * q^19 - 8 * q^21 - 2 * q^23 - 7 * q^27 + 7 * q^29 + 2 * q^31 - q^33 - 2 * q^37 - 10 * q^39 - 12 * q^41 - 3 * q^43 - 12 * q^47 - 5 * q^49 + 5 * q^51 + 10 * q^53 + 3 * q^57 - 8 * q^59 - 12 * q^61 + 7 * q^63 + 7 * q^67 + q^69 + 6 * q^71 - 4 * q^73 - q^77 - 3 * q^79 - 14 * q^81 - 4 * q^83 - 12 * q^87 - 21 * q^89 + 7 * q^91 + 16 * q^93 + 3 * 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
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.56155	0.590211	0.295106	−	0.955465i	$-0.404645\pi$
$7$	1.56155	0.590211	0.295106	+	0.955465i	$0.404645\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.43845	0.348875	0.174437	−	0.984668i	$-0.444189\pi$
$17$	1.43845	0.348875	0.174437	+	0.984668i	$0.444189\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	5.56155	1.03275	0.516377	−	0.856361i	$-0.327280\pi$
$29$	5.56155	1.03275	0.516377	+	0.856361i	$0.327280\pi$
$30$	0	0
$31$	−3.12311	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$31$	−3.12311	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$32$	0	0
$33$	−2.56155	−0.445909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.12311	−0.842233	−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311	−0.842233	−0.421117	+	0.907006i	$0.638362\pi$
$38$	0	0
$39$	−9.12311	−1.46087
$40$	0	0
$41$	−10.1231	−1.58096	−0.790482	−	0.612486i	$-0.790170\pi$
$41$	−10.1231	−1.58096	−0.790482	+	0.612486i	$0.790170\pi$
$42$	0	0
$43$	4.68466	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−3.68466	−0.515955
$52$	0	0
$53$	0.876894	0.120451	0.0602254	−	0.998185i	$-0.480818\pi$
$53$	0.876894	0.120451	0.0602254	+	0.998185i	$0.480818\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	7.68466	1.01786
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−14.2462	−1.82404	−0.912020	−	0.410145i	$-0.865478\pi$
$61$	−14.2462	−1.82404	−0.912020	+	0.410145i	$0.865478\pi$
$62$	0	0
$63$	5.56155	0.700690
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.43845	0.175734	0.0878671	−	0.996132i	$-0.471995\pi$
$67$	1.43845	0.175734	0.0878671	+	0.996132i	$0.471995\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	7.12311	0.845357	0.422679	−	0.906280i	$-0.361090\pi$
$71$	7.12311	0.845357	0.422679	+	0.906280i	$0.361090\pi$
$72$	0	0
$73$	2.12311	0.248491	0.124245	−	0.992252i	$-0.460349\pi$
$73$	2.12311	0.248491	0.124245	+	0.992252i	$0.460349\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.56155	0.177955
$78$	0	0
$79$	12.9309	1.45484	0.727418	−	0.686194i	$-0.240720\pi$
$79$	12.9309	1.45484	0.727418	+	0.686194i	$0.240720\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.12311	0.233041	0.116521	−	0.993188i	$-0.462826\pi$
$83$	2.12311	0.233041	0.116521	+	0.993188i	$0.462826\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−14.2462	−1.52735
$88$	0	0
$89$	−12.5616	−1.33152	−0.665761	−	0.746165i	$-0.731893\pi$
$89$	−12.5616	−1.33152	−0.665761	+	0.746165i	$0.731893\pi$
$90$	0	0
$91$	5.56155	0.583009
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.24621	−0.837276	−0.418638	−	0.908153i	$-0.637492\pi$
$97$	−8.24621	−0.837276	−0.418638	+	0.908153i	$0.637492\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	−13.3693	−1.33030	−0.665148	−	0.746711i	$-0.731632\pi$
$101$	−13.3693	−1.33030	−0.665148	+	0.746711i	$0.731632\pi$
$102$	0	0
$103$	−6.68466	−0.658659	−0.329329	−	0.944215i	$-0.606823\pi$
$103$	−6.68466	−0.658659	−0.329329	+	0.944215i	$0.606823\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.80776	0.464784	0.232392	−	0.972622i	$-0.425345\pi$
$107$	4.80776	0.464784	0.232392	+	0.972622i	$0.425345\pi$
$108$	0	0
$109$	15.3693	1.47211	0.736057	−	0.676920i	$-0.236686\pi$
$109$	15.3693	1.47211	0.736057	+	0.676920i	$0.236686\pi$
$110$	0	0
$111$	13.1231	1.24559
$112$	0	0
$113$	19.9309	1.87494	0.937469	−	0.348068i	$-0.113162\pi$
$113$	19.9309	1.87494	0.937469	+	0.348068i	$0.113162\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.6847	1.17270
$118$	0	0
$119$	2.24621	0.205910
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	25.9309	2.33811
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.75379	0.155624	0.0778118	−	0.996968i	$-0.475207\pi$
$127$	1.75379	0.155624	0.0778118	+	0.996968i	$0.475207\pi$
$128$	0	0
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−7.12311	−0.622349	−0.311174	−	0.950353i	$-0.600722\pi$
$131$	−7.12311	−0.622349	−0.311174	+	0.950353i	$0.600722\pi$
$132$	0	0
$133$	−4.68466	−0.406211
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.8078	0.923370	0.461685	−	0.887044i	$-0.347245\pi$
$137$	10.8078	0.923370	0.461685	+	0.887044i	$0.347245\pi$
$138$	0	0
$139$	5.43845	0.461283	0.230642	−	0.973039i	$-0.425918\pi$
$139$	5.43845	0.461283	0.230642	+	0.973039i	$0.425918\pi$
$140$	0	0
$141$	15.3693	1.29433
$142$	0	0
$143$	3.56155	0.297832
$144$	0	0
$145$	0	0
$146$	0	0
$147$	11.6847	0.963734
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−21.3693	−1.73901	−0.869505	−	0.493924i	$-0.835562\pi$
$151$	−21.3693	−1.73901	−0.869505	+	0.493924i	$0.835562\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	−2.24621	−0.178136
$160$	0	0
$161$	−1.56155	−0.123068
$162$	0	0
$163$	−6.80776	−0.533225	−0.266613	−	0.963804i	$-0.585904\pi$
$163$	−6.80776	−0.533225	−0.266613	+	0.963804i	$0.585904\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	−10.6847	−0.817076
$172$	0	0
$173$	22.0540	1.67673	0.838366	−	0.545107i	$-0.183511\pi$
$173$	22.0540	1.67673	0.838366	+	0.545107i	$0.183511\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.2462	0.770152
$178$	0	0
$179$	14.8078	1.10678	0.553392	−	0.832921i	$-0.313333\pi$
$179$	14.8078	1.10678	0.553392	+	0.832921i	$0.313333\pi$
$180$	0	0
$181$	19.3693	1.43971	0.719855	−	0.694124i	$-0.244208\pi$
$181$	19.3693	1.43971	0.719855	+	0.694124i	$0.244208\pi$
$182$	0	0
$183$	36.4924	2.69760
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.43845	0.105190
$188$	0	0
$189$	−2.24621	−0.163388
$190$	0	0
$191$	−22.0540	−1.59577	−0.797885	−	0.602810i	$-0.794048\pi$
$191$	−22.0540	−1.59577	−0.797885	+	0.602810i	$0.794048\pi$
$192$	0	0
$193$	15.9309	1.14673	0.573365	−	0.819300i	$-0.305638\pi$
$193$	15.9309	1.14673	0.573365	+	0.819300i	$0.305638\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.684658	−0.0487799	−0.0243899	−	0.999703i	$-0.507764\pi$
$197$	−0.684658	−0.0487799	−0.0243899	+	0.999703i	$0.507764\pi$
$198$	0	0
$199$	14.6847	1.04097	0.520484	−	0.853871i	$-0.325752\pi$
$199$	14.6847	1.04097	0.520484	+	0.853871i	$0.325752\pi$
$200$	0	0
$201$	−3.68466	−0.259896
$202$	0	0
$203$	8.68466	0.609544
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.56155	−0.247545
$208$	0	0
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−6.31534	−0.434766	−0.217383	−	0.976086i	$-0.569752\pi$
$211$	−6.31534	−0.434766	−0.217383	+	0.976086i	$0.569752\pi$
$212$	0	0
$213$	−18.2462	−1.25021
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.87689	−0.331065
$218$	0	0
$219$	−5.43845	−0.367496
$220$	0	0
$221$	5.12311	0.344617
$222$	0	0
$223$	−23.6155	−1.58141	−0.790706	−	0.612196i	$-0.790287\pi$
$223$	−23.6155	−1.58141	−0.790706	+	0.612196i	$0.790287\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−9.75379	−0.647382	−0.323691	−	0.946163i	$-0.604924\pi$
$227$	−9.75379	−0.647382	−0.323691	+	0.946163i	$0.604924\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	−9.31534	−0.610268	−0.305134	−	0.952309i	$-0.598701\pi$
$233$	−9.31534	−0.610268	−0.305134	+	0.952309i	$0.598701\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−33.1231	−2.15158
$238$	0	0
$239$	−16.2462	−1.05088	−0.525440	−	0.850831i	$-0.676099\pi$
$239$	−16.2462	−1.05088	−0.525440	+	0.850831i	$0.676099\pi$
$240$	0	0
$241$	−13.6847	−0.881506	−0.440753	−	0.897628i	$-0.645289\pi$
$241$	−13.6847	−0.881506	−0.440753	+	0.897628i	$0.645289\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.6847	−0.679849
$248$	0	0
$249$	−5.43845	−0.344648
$250$	0	0
$251$	11.6847	0.737529	0.368765	−	0.929523i	$-0.379781\pi$
$251$	11.6847	0.737529	0.368765	+	0.929523i	$0.379781\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.4924	−0.904012	−0.452006	−	0.892015i	$-0.649292\pi$
$257$	−14.4924	−0.904012	−0.452006	+	0.892015i	$0.649292\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	19.8078	1.22607
$262$	0	0
$263$	5.75379	0.354794	0.177397	−	0.984139i	$-0.443232\pi$
$263$	5.75379	0.354794	0.177397	+	0.984139i	$0.443232\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	32.1771	1.96921
$268$	0	0
$269$	15.1771	0.925363	0.462681	−	0.886525i	$-0.346887\pi$
$269$	15.1771	0.925363	0.462681	+	0.886525i	$0.346887\pi$
$270$	0	0
$271$	−3.75379	−0.228026	−0.114013	−	0.993479i	$-0.536371\pi$
$271$	−3.75379	−0.228026	−0.114013	+	0.993479i	$0.536371\pi$
$272$	0	0
$273$	−14.2462	−0.862220
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.05398	0.363748	0.181874	−	0.983322i	$-0.441784\pi$
$277$	6.05398	0.363748	0.181874	+	0.983322i	$0.441784\pi$
$278$	0	0
$279$	−11.1231	−0.665923
$280$	0	0
$281$	24.2462	1.44641	0.723204	−	0.690635i	$-0.242669\pi$
$281$	24.2462	1.44641	0.723204	+	0.690635i	$0.242669\pi$
$282$	0	0
$283$	24.1771	1.43718	0.718589	−	0.695435i	$-0.244788\pi$
$283$	24.1771	1.43718	0.718589	+	0.695435i	$0.244788\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.8078	−0.933103
$288$	0	0
$289$	−14.9309	−0.878286
$290$	0	0
$291$	21.1231	1.23826
$292$	0	0
$293$	20.4924	1.19718	0.598590	−	0.801056i	$-0.295728\pi$
$293$	20.4924	1.19718	0.598590	+	0.801056i	$0.295728\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.43845	−0.0834672
$298$	0	0
$299$	−3.56155	−0.205970
$300$	0	0
$301$	7.31534	0.421649
$302$	0	0
$303$	34.2462	1.96739
$304$	0	0
$305$	0	0
$306$	0	0
$307$	29.9309	1.70824	0.854122	−	0.520072i	$-0.174095\pi$
$307$	29.9309	1.70824	0.854122	+	0.520072i	$0.174095\pi$
$308$	0	0
$309$	17.1231	0.974099
$310$	0	0
$311$	−0.630683	−0.0357628	−0.0178814	−	0.999840i	$-0.505692\pi$
$311$	−0.630683	−0.0357628	−0.0178814	+	0.999840i	$0.505692\pi$
$312$	0	0
$313$	21.1231	1.19395	0.596974	−	0.802260i	$-0.296369\pi$
$313$	21.1231	1.19395	0.596974	+	0.802260i	$0.296369\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.5616	0.986355	0.493178	−	0.869929i	$-0.335835\pi$
$317$	17.5616	0.986355	0.493178	+	0.869929i	$0.335835\pi$
$318$	0	0
$319$	5.56155	0.311387
$320$	0	0
$321$	−12.3153	−0.687375
$322$	0	0
$323$	−4.31534	−0.240112
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−39.3693	−2.17713
$328$	0	0
$329$	−9.36932	−0.516547
$330$	0	0
$331$	23.3002	1.28069	0.640347	−	0.768086i	$-0.278791\pi$
$331$	23.3002	1.28069	0.640347	+	0.768086i	$0.278791\pi$
$332$	0	0
$333$	−18.2462	−0.999886
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−28.5616	−1.55585	−0.777923	−	0.628359i	$-0.783727\pi$
$337$	−28.5616	−1.55585	−0.777923	+	0.628359i	$0.783727\pi$
$338$	0	0
$339$	−51.0540	−2.77287
$340$	0	0
$341$	−3.12311	−0.169126
$342$	0	0
$343$	−18.0540	−0.974823
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.5616	−1.64063	−0.820315	−	0.571912i	$-0.806202\pi$
$347$	−30.5616	−1.64063	−0.820315	+	0.571912i	$0.806202\pi$
$348$	0	0
$349$	−16.0540	−0.859350	−0.429675	−	0.902984i	$-0.641372\pi$
$349$	−16.0540	−0.859350	−0.429675	+	0.902984i	$0.641372\pi$
$350$	0	0
$351$	−5.12311	−0.273451
$352$	0	0
$353$	−29.4233	−1.56604	−0.783022	−	0.621994i	$-0.786323\pi$
$353$	−29.4233	−1.56604	−0.783022	+	0.621994i	$0.786323\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.75379	−0.304523
$358$	0	0
$359$	17.5616	0.926863	0.463432	−	0.886133i	$-0.346618\pi$
$359$	17.5616	0.926863	0.463432	+	0.886133i	$0.346618\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	25.6155	1.34447
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.4384	0.649282	0.324641	−	0.945837i	$-0.394757\pi$
$367$	12.4384	0.649282	0.324641	+	0.945837i	$0.394757\pi$
$368$	0	0
$369$	−36.0540	−1.87689
$370$	0	0
$371$	1.36932	0.0710914
$372$	0	0
$373$	−16.4924	−0.853945	−0.426973	−	0.904265i	$-0.640420\pi$
$373$	−16.4924	−0.853945	−0.426973	+	0.904265i	$0.640420\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	19.8078	1.02015
$378$	0	0
$379$	−12.8078	−0.657891	−0.328945	−	0.944349i	$-0.606693\pi$
$379$	−12.8078	−0.657891	−0.328945	+	0.944349i	$0.606693\pi$
$380$	0	0
$381$	−4.49242	−0.230154
$382$	0	0
$383$	−17.8078	−0.909934	−0.454967	−	0.890508i	$-0.650349\pi$
$383$	−17.8078	−0.909934	−0.454967	+	0.890508i	$0.650349\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.6847	0.848129
$388$	0	0
$389$	−31.1231	−1.57800	−0.789002	−	0.614391i	$-0.789402\pi$
$389$	−31.1231	−1.57800	−0.789002	+	0.614391i	$0.789402\pi$
$390$	0	0
$391$	−1.43845	−0.0727454
$392$	0	0
$393$	18.2462	0.920400
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15.1231	−0.759007	−0.379503	−	0.925190i	$-0.623905\pi$
$397$	−15.1231	−0.759007	−0.379503	+	0.925190i	$0.623905\pi$
$398$	0	0
$399$	12.0000	0.600751
$400$	0	0
$401$	15.6847	0.783254	0.391627	−	0.920124i	$-0.371912\pi$
$401$	15.6847	0.783254	0.391627	+	0.920124i	$0.371912\pi$
$402$	0	0
$403$	−11.1231	−0.554081
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.12311	−0.253943
$408$	0	0
$409$	−12.7538	−0.630634	−0.315317	−	0.948986i	$-0.602111\pi$
$409$	−12.7538	−0.630634	−0.315317	+	0.948986i	$0.602111\pi$
$410$	0	0
$411$	−27.6847	−1.36558
$412$	0	0
$413$	−6.24621	−0.307356
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−13.9309	−0.682198
$418$	0	0
$419$	−11.8769	−0.580224	−0.290112	−	0.956993i	$-0.593693\pi$
$419$	−11.8769	−0.580224	−0.290112	+	0.956993i	$0.593693\pi$
$420$	0	0
$421$	−15.7538	−0.767793	−0.383896	−	0.923376i	$-0.625418\pi$
$421$	−15.7538	−0.767793	−0.383896	+	0.923376i	$0.625418\pi$
$422$	0	0
$423$	−21.3693	−1.03901
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−22.2462	−1.07657
$428$	0	0
$429$	−9.12311	−0.440468
$430$	0	0
$431$	−18.2462	−0.878889	−0.439445	−	0.898270i	$-0.644825\pi$
$431$	−18.2462	−0.878889	−0.439445	+	0.898270i	$0.644825\pi$
$432$	0	0
$433$	−25.3002	−1.21585	−0.607925	−	0.793995i	$-0.707998\pi$
$433$	−25.3002	−1.21585	−0.607925	+	0.793995i	$0.707998\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	−30.8769	−1.47367	−0.736837	−	0.676071i	$-0.763681\pi$
$439$	−30.8769	−1.47367	−0.736837	+	0.676071i	$0.763681\pi$
$440$	0	0
$441$	−16.2462	−0.773629
$442$	0	0
$443$	−2.80776	−0.133401	−0.0667004	−	0.997773i	$-0.521247\pi$
$443$	−2.80776	−0.133401	−0.0667004	+	0.997773i	$0.521247\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	46.1080	2.18083
$448$	0	0
$449$	24.4233	1.15261	0.576303	−	0.817236i	$-0.304495\pi$
$449$	24.4233	1.15261	0.576303	+	0.817236i	$0.304495\pi$
$450$	0	0
$451$	−10.1231	−0.476678
$452$	0	0
$453$	54.7386	2.57185
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.1771	−1.22451	−0.612256	−	0.790660i	$-0.709738\pi$
$457$	−26.1771	−1.22451	−0.612256	+	0.790660i	$0.709738\pi$
$458$	0	0
$459$	−2.06913	−0.0965787
$460$	0	0
$461$	−13.1771	−0.613718	−0.306859	−	0.951755i	$-0.599278\pi$
$461$	−13.1771	−0.613718	−0.306859	+	0.951755i	$0.599278\pi$
$462$	0	0
$463$	−4.63068	−0.215206	−0.107603	−	0.994194i	$-0.534318\pi$
$463$	−4.63068	−0.215206	−0.107603	+	0.994194i	$0.534318\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.68466	−0.216780	−0.108390	−	0.994108i	$-0.534570\pi$
$467$	−4.68466	−0.216780	−0.108390	+	0.994108i	$0.534570\pi$
$468$	0	0
$469$	2.24621	0.103720
$470$	0	0
$471$	15.3693	0.708181
$472$	0	0
$473$	4.68466	0.215401
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.12311	0.142997
$478$	0	0
$479$	2.93087	0.133915	0.0669574	−	0.997756i	$-0.478671\pi$
$479$	2.93087	0.133915	0.0669574	+	0.997756i	$0.478671\pi$
$480$	0	0
$481$	−18.2462	−0.831956
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.24621	−0.192414	−0.0962071	−	0.995361i	$-0.530671\pi$
$487$	−4.24621	−0.192414	−0.0962071	+	0.995361i	$0.530671\pi$
$488$	0	0
$489$	17.4384	0.788594
$490$	0	0
$491$	−6.24621	−0.281888	−0.140944	−	0.990018i	$-0.545014\pi$
$491$	−6.24621	−0.281888	−0.140944	+	0.990018i	$0.545014\pi$
$492$	0	0
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11.1231	0.498939
$498$	0	0
$499$	−33.8617	−1.51586	−0.757930	−	0.652336i	$-0.773789\pi$
$499$	−33.8617	−1.51586	−0.757930	+	0.652336i	$0.773789\pi$
$500$	0	0
$501$	25.6155	1.14442
$502$	0	0
$503$	−21.5616	−0.961382	−0.480691	−	0.876890i	$-0.659614\pi$
$503$	−21.5616	−0.961382	−0.480691	+	0.876890i	$0.659614\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.807764	0.0358741
$508$	0	0
$509$	−16.8769	−0.748055	−0.374028	−	0.927418i	$-0.622023\pi$
$509$	−16.8769	−0.748055	−0.374028	+	0.927418i	$0.622023\pi$
$510$	0	0
$511$	3.31534	0.146662
$512$	0	0
$513$	4.31534	0.190527
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−6.00000	−0.263880
$518$	0	0
$519$	−56.4924	−2.47974
$520$	0	0
$521$	1.93087	0.0845929	0.0422965	−	0.999105i	$-0.486533\pi$
$521$	1.93087	0.0845929	0.0422965	+	0.999105i	$0.486533\pi$
$522$	0	0
$523$	−35.1080	−1.53516	−0.767582	−	0.640951i	$-0.778540\pi$
$523$	−35.1080	−1.53516	−0.767582	+	0.640951i	$0.778540\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.49242	−0.195693
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−14.2462	−0.618233
$532$	0	0
$533$	−36.0540	−1.56167
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−37.9309	−1.63684
$538$	0	0
$539$	−4.56155	−0.196480
$540$	0	0
$541$	−8.68466	−0.373383	−0.186691	−	0.982419i	$-0.559776\pi$
$541$	−8.68466	−0.373383	−0.186691	+	0.982419i	$0.559776\pi$
$542$	0	0
$543$	−49.6155	−2.12921
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.5616	0.537093	0.268547	−	0.963267i	$-0.413457\pi$
$547$	12.5616	0.537093	0.268547	+	0.963267i	$0.413457\pi$
$548$	0	0
$549$	−50.7386	−2.16547
$550$	0	0
$551$	−16.6847	−0.710790
$552$	0	0
$553$	20.1922	0.858661
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.3693	−0.481733	−0.240867	−	0.970558i	$-0.577432\pi$
$557$	−11.3693	−0.481733	−0.240867	+	0.970558i	$0.577432\pi$
$558$	0	0
$559$	16.6847	0.705686
$560$	0	0
$561$	−3.68466	−0.155566
$562$	0	0
$563$	−15.8078	−0.666218	−0.333109	−	0.942888i	$-0.608098\pi$
$563$	−15.8078	−0.666218	−0.333109	+	0.942888i	$0.608098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−10.9309	−0.459053
$568$	0	0
$569$	35.6847	1.49598	0.747989	−	0.663711i	$-0.231019\pi$
$569$	35.6847	1.49598	0.747989	+	0.663711i	$0.231019\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	56.4924	2.36000
$574$	0	0
$575$	0	0
$576$	0	0
$577$	35.8769	1.49357	0.746787	−	0.665063i	$-0.231595\pi$
$577$	35.8769	1.49357	0.746787	+	0.665063i	$0.231595\pi$
$578$	0	0
$579$	−40.8078	−1.69591
$580$	0	0
$581$	3.31534	0.137544
$582$	0	0
$583$	0.876894	0.0363173
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.8078	1.51922	0.759610	−	0.650379i	$-0.225390\pi$
$587$	36.8078	1.51922	0.759610	+	0.650379i	$0.225390\pi$
$588$	0	0
$589$	9.36932	0.386056
$590$	0	0
$591$	1.75379	0.0721412
$592$	0	0
$593$	4.75379	0.195215	0.0976074	−	0.995225i	$-0.468881\pi$
$593$	4.75379	0.195215	0.0976074	+	0.995225i	$0.468881\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37.6155	−1.53950
$598$	0	0
$599$	31.1231	1.27166	0.635828	−	0.771831i	$-0.280659\pi$
$599$	31.1231	1.27166	0.635828	+	0.771831i	$0.280659\pi$
$600$	0	0
$601$	−25.6847	−1.04770	−0.523850	−	0.851811i	$-0.675505\pi$
$601$	−25.6847	−1.04770	−0.523850	+	0.851811i	$0.675505\pi$
$602$	0	0
$603$	5.12311	0.208629
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−30.9848	−1.25764	−0.628818	−	0.777552i	$-0.716461\pi$
$607$	−30.9848	−1.25764	−0.628818	+	0.777552i	$0.716461\pi$
$608$	0	0
$609$	−22.2462	−0.901462
$610$	0	0
$611$	−21.3693	−0.864510
$612$	0	0
$613$	−4.49242	−0.181447	−0.0907236	−	0.995876i	$-0.528918\pi$
$613$	−4.49242	−0.181447	−0.0907236	+	0.995876i	$0.528918\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.8617	1.28271	0.641353	−	0.767246i	$-0.278374\pi$
$617$	31.8617	1.28271	0.641353	+	0.767246i	$0.278374\pi$
$618$	0	0
$619$	−28.9848	−1.16500	−0.582500	−	0.812831i	$-0.697925\pi$
$619$	−28.9848	−1.16500	−0.582500	+	0.812831i	$0.697925\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	−19.6155	−0.785880
$624$	0	0
$625$	0	0
$626$	0	0
$627$	7.68466	0.306896
$628$	0	0
$629$	−7.36932	−0.293834
$630$	0	0
$631$	−43.1771	−1.71885	−0.859426	−	0.511260i	$-0.829179\pi$
$631$	−43.1771	−1.71885	−0.859426	+	0.511260i	$0.829179\pi$
$632$	0	0
$633$	16.1771	0.642981
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−16.2462	−0.643698
$638$	0	0
$639$	25.3693	1.00359
$640$	0	0
$641$	27.8617	1.10047	0.550236	−	0.835009i	$-0.314538\pi$
$641$	27.8617	1.10047	0.550236	+	0.835009i	$0.314538\pi$
$642$	0	0
$643$	−50.0540	−1.97394	−0.986968	−	0.160916i	$-0.948555\pi$
$643$	−50.0540	−1.97394	−0.986968	+	0.160916i	$0.948555\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−37.3693	−1.46914	−0.734570	−	0.678533i	$-0.762616\pi$
$647$	−37.3693	−1.46914	−0.734570	+	0.678533i	$0.762616\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	12.4924	0.489617
$652$	0	0
$653$	19.0691	0.746233	0.373116	−	0.927785i	$-0.378289\pi$
$653$	19.0691	0.746233	0.373116	+	0.927785i	$0.378289\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	7.56155	0.295004
$658$	0	0
$659$	−27.7386	−1.08054	−0.540272	−	0.841491i	$-0.681679\pi$
$659$	−27.7386	−1.08054	−0.540272	+	0.841491i	$0.681679\pi$
$660$	0	0
$661$	29.8617	1.16149	0.580744	−	0.814087i	$-0.302762\pi$
$661$	29.8617	1.16149	0.580744	+	0.814087i	$0.302762\pi$
$662$	0	0
$663$	−13.1231	−0.509659
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.56155	−0.215344
$668$	0	0
$669$	60.4924	2.33877
$670$	0	0
$671$	−14.2462	−0.549969
$672$	0	0
$673$	−37.8078	−1.45738	−0.728691	−	0.684843i	$-0.759871\pi$
$673$	−37.8078	−1.45738	−0.728691	+	0.684843i	$0.759871\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.1231	−0.504362	−0.252181	−	0.967680i	$-0.581148\pi$
$677$	−13.1231	−0.504362	−0.252181	+	0.967680i	$0.581148\pi$
$678$	0	0
$679$	−12.8769	−0.494170
$680$	0	0
$681$	24.9848	0.957421
$682$	0	0
$683$	−5.68466	−0.217517	−0.108759	−	0.994068i	$-0.534688\pi$
$683$	−5.68466	−0.217517	−0.108759	+	0.994068i	$0.534688\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.6155	−0.977293
$688$	0	0
$689$	3.12311	0.118981
$690$	0	0
$691$	5.43845	0.206888	0.103444	−	0.994635i	$-0.467014\pi$
$691$	5.43845	0.206888	0.103444	+	0.994635i	$0.467014\pi$
$692$	0	0
$693$	5.56155	0.211266
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14.5616	−0.551558
$698$	0	0
$699$	23.8617	0.902534
$700$	0	0
$701$	17.7538	0.670551	0.335276	−	0.942120i	$-0.391171\pi$
$701$	17.7538	0.670551	0.335276	+	0.942120i	$0.391171\pi$
$702$	0	0
$703$	15.3693	0.579665
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.8769	−0.785156
$708$	0	0
$709$	−29.8617	−1.12148	−0.560741	−	0.827992i	$-0.689483\pi$
$709$	−29.8617	−1.12148	−0.560741	+	0.827992i	$0.689483\pi$
$710$	0	0
$711$	46.0540	1.72716
$712$	0	0
$713$	3.12311	0.116961
$714$	0	0
$715$	0	0
$716$	0	0
$717$	41.6155	1.55416
$718$	0	0
$719$	10.7386	0.400483	0.200242	−	0.979747i	$-0.435827\pi$
$719$	10.7386	0.400483	0.200242	+	0.979747i	$0.435827\pi$
$720$	0	0
$721$	−10.4384	−0.388748
$722$	0	0
$723$	35.0540	1.30367
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.24621	−0.0833074	−0.0416537	−	0.999132i	$-0.513263\pi$
$727$	−2.24621	−0.0833074	−0.0416537	+	0.999132i	$0.513263\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	6.73863	0.249237
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.43845	0.0529859
$738$	0	0
$739$	24.8769	0.915111	0.457556	−	0.889181i	$-0.348725\pi$
$739$	24.8769	0.915111	0.457556	+	0.889181i	$0.348725\pi$
$740$	0	0
$741$	27.3693	1.00544
$742$	0	0
$743$	35.6695	1.30859	0.654294	−	0.756241i	$-0.272966\pi$
$743$	35.6695	1.30859	0.654294	+	0.756241i	$0.272966\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.56155	0.276663
$748$	0	0
$749$	7.50758	0.274321
$750$	0	0
$751$	−11.8078	−0.430871	−0.215436	−	0.976518i	$-0.569117\pi$
$751$	−11.8078	−0.430871	−0.215436	+	0.976518i	$0.569117\pi$
$752$	0	0
$753$	−29.9309	−1.09074
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−38.2462	−1.39008	−0.695041	−	0.718970i	$-0.744614\pi$
$757$	−38.2462	−1.39008	−0.695041	+	0.718970i	$0.744614\pi$
$758$	0	0
$759$	2.56155	0.0929785
$760$	0	0
$761$	−3.49242	−0.126600	−0.0633001	−	0.997995i	$-0.520163\pi$
$761$	−3.49242	−0.126600	−0.0633001	+	0.997995i	$0.520163\pi$
$762$	0	0
$763$	24.0000	0.868858
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.2462	−0.514401
$768$	0	0
$769$	42.4233	1.52982	0.764912	−	0.644135i	$-0.222783\pi$
$769$	42.4233	1.52982	0.764912	+	0.644135i	$0.222783\pi$
$770$	0	0
$771$	37.1231	1.33696
$772$	0	0
$773$	5.75379	0.206949	0.103475	−	0.994632i	$-0.467004\pi$
$773$	5.75379	0.206949	0.103475	+	0.994632i	$0.467004\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.4924	0.735162
$778$	0	0
$779$	30.3693	1.08809
$780$	0	0
$781$	7.12311	0.254885
$782$	0	0
$783$	−8.00000	−0.285897
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.94602	−0.0693683	−0.0346841	−	0.999398i	$-0.511043\pi$
$787$	−1.94602	−0.0693683	−0.0346841	+	0.999398i	$0.511043\pi$
$788$	0	0
$789$	−14.7386	−0.524709
$790$	0	0
$791$	31.1231	1.10661
$792$	0	0
$793$	−50.7386	−1.80178
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.63068	−0.164027	−0.0820136	−	0.996631i	$-0.526135\pi$
$797$	−4.63068	−0.164027	−0.0820136	+	0.996631i	$0.526135\pi$
$798$	0	0
$799$	−8.63068	−0.305332
$800$	0	0
$801$	−44.7386	−1.58076
$802$	0	0
$803$	2.12311	0.0749228
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−38.8769	−1.36853
$808$	0	0
$809$	11.0691	0.389170	0.194585	−	0.980886i	$-0.437664\pi$
$809$	11.0691	0.389170	0.194585	+	0.980886i	$0.437664\pi$
$810$	0	0
$811$	20.9848	0.736878	0.368439	−	0.929652i	$-0.379892\pi$
$811$	20.9848	0.736878	0.368439	+	0.929652i	$0.379892\pi$
$812$	0	0
$813$	9.61553	0.337231
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−14.0540	−0.491686
$818$	0	0
$819$	19.8078	0.692139
$820$	0	0
$821$	−16.3002	−0.568880	−0.284440	−	0.958694i	$-0.591808\pi$
$821$	−16.3002	−0.568880	−0.284440	+	0.958694i	$0.591808\pi$
$822$	0	0
$823$	−27.3693	−0.954034	−0.477017	−	0.878894i	$-0.658282\pi$
$823$	−27.3693	−0.954034	−0.477017	+	0.878894i	$0.658282\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21.0000	−0.730242	−0.365121	−	0.930960i	$-0.618972\pi$
$827$	−21.0000	−0.730242	−0.365121	+	0.930960i	$0.618972\pi$
$828$	0	0
$829$	−20.1922	−0.701305	−0.350653	−	0.936506i	$-0.614040\pi$
$829$	−20.1922	−0.701305	−0.350653	+	0.936506i	$0.614040\pi$
$830$	0	0
$831$	−15.5076	−0.537952
$832$	0	0
$833$	−6.56155	−0.227344
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.49242	0.155281
$838$	0	0
$839$	−16.9309	−0.584519	−0.292259	−	0.956339i	$-0.594407\pi$
$839$	−16.9309	−0.584519	−0.292259	+	0.956339i	$0.594407\pi$
$840$	0	0
$841$	1.93087	0.0665817
$842$	0	0
$843$	−62.1080	−2.13911
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−15.6155	−0.536556
$848$	0	0
$849$	−61.9309	−2.12546
$850$	0	0
$851$	5.12311	0.175618
$852$	0	0
$853$	0.192236	0.00658203	0.00329102	−	0.999995i	$-0.498952\pi$
$853$	0.192236	0.00658203	0.00329102	+	0.999995i	$0.498952\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.9460	0.373909	0.186955	−	0.982369i	$-0.440138\pi$
$857$	10.9460	0.373909	0.186955	+	0.982369i	$0.440138\pi$
$858$	0	0
$859$	40.1771	1.37082	0.685412	−	0.728155i	$-0.259622\pi$
$859$	40.1771	1.37082	0.685412	+	0.728155i	$0.259622\pi$
$860$	0	0
$861$	40.4924	1.37998
$862$	0	0
$863$	35.1231	1.19560	0.597802	−	0.801644i	$-0.296041\pi$
$863$	35.1231	1.19560	0.597802	+	0.801644i	$0.296041\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	38.2462	1.29891
$868$	0	0
$869$	12.9309	0.438650
$870$	0	0
$871$	5.12311	0.173590
$872$	0	0
$873$	−29.3693	−0.994001
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.49242	−0.0841631	−0.0420816	−	0.999114i	$-0.513399\pi$
$877$	−2.49242	−0.0841631	−0.0420816	+	0.999114i	$0.513399\pi$
$878$	0	0
$879$	−52.4924	−1.77053
$880$	0	0
$881$	−47.3693	−1.59591	−0.797956	−	0.602715i	$-0.794086\pi$
$881$	−47.3693	−1.59591	−0.797956	+	0.602715i	$0.794086\pi$
$882$	0	0
$883$	51.6847	1.73933	0.869664	−	0.493645i	$-0.164336\pi$
$883$	51.6847	1.73933	0.869664	+	0.493645i	$0.164336\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.00000	−0.134307	−0.0671534	−	0.997743i	$-0.521392\pi$
$887$	−4.00000	−0.134307	−0.0671534	+	0.997743i	$0.521392\pi$
$888$	0	0
$889$	2.73863	0.0918508
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	18.0000	0.602347
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.12311	0.304612
$898$	0	0
$899$	−17.3693	−0.579299
$900$	0	0
$901$	1.26137	0.0420222
$902$	0	0
$903$	−18.7386	−0.623583
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12.6847	−0.421187	−0.210594	−	0.977574i	$-0.567540\pi$
$907$	−12.6847	−0.421187	−0.210594	+	0.977574i	$0.567540\pi$
$908$	0	0
$909$	−47.6155	−1.57931
$910$	0	0
$911$	−52.5464	−1.74094	−0.870470	−	0.492222i	$-0.836185\pi$
$911$	−52.5464	−1.74094	−0.870470	+	0.492222i	$0.836185\pi$
$912$	0	0
$913$	2.12311	0.0702645
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11.1231	−0.367317
$918$	0	0
$919$	14.7386	0.486183	0.243091	−	0.970003i	$-0.421839\pi$
$919$	14.7386	0.486183	0.243091	+	0.970003i	$0.421839\pi$
$920$	0	0
$921$	−76.6695	−2.52635
$922$	0	0
$923$	25.3693	0.835041
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−23.8078	−0.781950
$928$	0	0
$929$	40.9309	1.34290	0.671449	−	0.741051i	$-0.265672\pi$
$929$	40.9309	1.34290	0.671449	+	0.741051i	$0.265672\pi$
$930$	0	0
$931$	13.6847	0.448497
$932$	0	0
$933$	1.61553	0.0528900
$934$	0	0
$935$	0	0
$936$	0	0
$937$	43.9309	1.43516	0.717579	−	0.696477i	$-0.245250\pi$
$937$	43.9309	1.43516	0.717579	+	0.696477i	$0.245250\pi$
$938$	0	0
$939$	−54.1080	−1.76575
$940$	0	0
$941$	33.8617	1.10386	0.551931	−	0.833890i	$-0.313891\pi$
$941$	33.8617	1.10386	0.551931	+	0.833890i	$0.313891\pi$
$942$	0	0
$943$	10.1231	0.329654
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−56.1080	−1.82326	−0.911632	−	0.411008i	$-0.865177\pi$
$947$	−56.1080	−1.82326	−0.911632	+	0.411008i	$0.865177\pi$
$948$	0	0
$949$	7.56155	0.245458
$950$	0	0
$951$	−44.9848	−1.45873
$952$	0	0
$953$	−46.5616	−1.50828	−0.754138	−	0.656716i	$-0.771945\pi$
$953$	−46.5616	−1.50828	−0.754138	+	0.656716i	$0.771945\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−14.2462	−0.460515
$958$	0	0
$959$	16.8769	0.544983
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	17.1231	0.551784
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.00000	0.128631	0.0643157	−	0.997930i	$-0.479514\pi$
$967$	4.00000	0.128631	0.0643157	+	0.997930i	$0.479514\pi$
$968$	0	0
$969$	11.0540	0.355105
$970$	0	0
$971$	−9.63068	−0.309063	−0.154532	−	0.987988i	$-0.549387\pi$
$971$	−9.63068	−0.309063	−0.154532	+	0.987988i	$0.549387\pi$
$972$	0	0
$973$	8.49242	0.272255
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.05398	0.161691	0.0808455	−	0.996727i	$-0.474238\pi$
$977$	5.05398	0.161691	0.0808455	+	0.996727i	$0.474238\pi$
$978$	0	0
$979$	−12.5616	−0.401469
$980$	0	0
$981$	54.7386	1.74767
$982$	0	0
$983$	9.31534	0.297113	0.148557	−	0.988904i	$-0.452537\pi$
$983$	9.31534	0.297113	0.148557	+	0.988904i	$0.452537\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	0	0
$989$	−4.68466	−0.148963
$990$	0	0
$991$	4.24621	0.134885	0.0674427	−	0.997723i	$-0.478516\pi$
$991$	4.24621	0.134885	0.0674427	+	0.997723i	$0.478516\pi$
$992$	0	0
$993$	−59.6847	−1.89404
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−0.822919	−0.0260621	−0.0130311	−	0.999915i	$-0.504148\pi$
$997$	−0.822919	−0.0260621	−0.0130311	+	0.999915i	$0.504148\pi$
$998$	0	0
$999$	7.36932	0.233155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.bp.1.1		2
4.3	odd	2		1150.2.a.p.1.2	yes	2
5.4	even	2		9200.2.a.bw.1.2		2
20.3	even	4		1150.2.b.h.599.2		4
20.7	even	4		1150.2.b.h.599.3		4
20.19	odd	2		1150.2.a.k.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1150.2.a.k.1.1	✓	2	20.19	odd	2
1150.2.a.p.1.2	yes	2	4.3	odd	2
1150.2.b.h.599.2		4	20.3	even	4
1150.2.b.h.599.3		4	20.7	even	4
9200.2.a.bp.1.1		2	1.1	even	1	trivial
9200.2.a.bw.1.2		2	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)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q-2.56155 q^{3} +1.56155 q^{7} +3.56155 q^{9} +1.00000 q^{11} +3.56155 q^{13} +1.43845 q^{17} -3.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} -1.43845 q^{27} +5.56155 q^{29} -3.12311 q^{31} -2.56155 q^{33} -5.12311 q^{37} -9.12311 q^{39} -10.1231 q^{41} +4.68466 q^{43} -6.00000 q^{47} -4.56155 q^{49} -3.68466 q^{51} +0.876894 q^{53} +7.68466 q^{57} -4.00000 q^{59} -14.2462 q^{61} +5.56155 q^{63} +1.43845 q^{67} +2.56155 q^{69} +7.12311 q^{71} +2.12311 q^{73} +1.56155 q^{77} +12.9309 q^{79} -7.00000 q^{81} +2.12311 q^{83} -14.2462 q^{87} -12.5616 q^{89} +5.56155 q^{91} +8.00000 q^{93} -8.24621 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - q^7 + 3 * q^9	\( 2 q - q^{3} - q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{13} + 7 q^{17} - 6 q^{19} - 8 q^{21} - 2 q^{23} - 7 q^{27} + 7 q^{29} + 2 q^{31} - q^{33} - 2 q^{37} - 10 q^{39} - 12 q^{41} - 3 q^{43} - 12 q^{47} - 5 q^{49} + 5 q^{51} + 10 q^{53} + 3 q^{57} - 8 q^{59} - 12 q^{61} + 7 q^{63} + 7 q^{67} + q^{69} + 6 q^{71} - 4 q^{73} - q^{77} - 3 q^{79} - 14 q^{81} - 4 q^{83} - 12 q^{87} - 21 q^{89} + 7 q^{91} + 16 q^{93} + 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^7 + 3 * q^9 + 2 * q^11 + 3 * q^13 + 7 * q^17 - 6 * q^19 - 8 * q^21 - 2 * q^23 - 7 * q^27 + 7 * q^29 + 2 * q^31 - q^33 - 2 * q^37 - 10 * q^39 - 12 * q^41 - 3 * q^43 - 12 * q^47 - 5 * q^49 + 5 * q^51 + 10 * q^53 + 3 * q^57 - 8 * q^59 - 12 * q^61 + 7 * q^63 + 7 * q^67 + q^69 + 6 * q^71 - 4 * q^73 - q^77 - 3 * q^79 - 14 * q^81 - 4 * q^83 - 12 * q^87 - 21 * q^89 + 7 * q^91 + 16 * q^93 + 3 * q^99

Introduction

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