LMFDB - Embedded newform 9200.2.a.de.1.5

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

Embedding invariants

Embedding label			1.5
Root			$0.493532$ 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)

$f(q)$	$=$	$q+0.493532 q^{3} +4.54439 q^{7} -2.75643 q^{9} +O(q^{10})$	$q+0.493532 q^{3} +4.54439 q^{7} -2.75643 q^{9} +4.61297 q^{11} +5.54274 q^{13} -7.16998 q^{17} -1.35966 q^{19} +2.24280 q^{21} -1.00000 q^{23} -2.84098 q^{27} -3.66351 q^{29} +4.46616 q^{31} +2.27665 q^{33} -3.32432 q^{37} +2.73552 q^{39} +8.95216 q^{41} +8.68458 q^{43} +9.59028 q^{47} +13.6515 q^{49} -3.53861 q^{51} -10.4676 q^{53} -0.671035 q^{57} +2.08543 q^{59} -0.686197 q^{61} -12.5263 q^{63} +8.84324 q^{67} -0.493532 q^{69} -15.9040 q^{71} -4.44647 q^{73} +20.9631 q^{77} +8.65227 q^{79} +6.86717 q^{81} -4.43591 q^{83} -1.80806 q^{87} +13.2284 q^{89} +25.1884 q^{91} +2.20419 q^{93} +1.21785 q^{97} -12.7153 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 - 7 * q^11 + 11 * q^13 - 7 * q^17 - 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 - 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 + 19 * q^51 + 4 * q^53 - 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 - 15 * q^71 + 6 * q^73 + 18 * q^77 + 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 + 35 * q^91 + 20 * q^93 + 3 * q^97 - 61 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.493532	0.284941	0.142470	−	0.989799i	$-0.454495\pi$
$3$	0.493532	0.284941	0.142470	+	0.989799i	$0.454495\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.54439	1.71762	0.858809	−	0.512297i	$-0.171205\pi$
$7$	4.54439	1.71762	0.858809	+	0.512297i	$0.171205\pi$
$8$	0	0
$9$	−2.75643	−0.918809
$10$	0	0
$11$	4.61297	1.39086	0.695431	−	0.718593i	$-0.255213\pi$
$11$	4.61297	1.39086	0.695431	+	0.718593i	$0.255213\pi$
$12$	0	0
$13$	5.54274	1.53728	0.768640	−	0.639682i	$-0.220934\pi$
$13$	5.54274	1.53728	0.768640	+	0.639682i	$0.220934\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.16998	−1.73897	−0.869487	−	0.493955i	$-0.835551\pi$
$17$	−7.16998	−1.73897	−0.869487	+	0.493955i	$0.835551\pi$
$18$	0	0
$19$	−1.35966	−0.311928	−0.155964	−	0.987763i	$-0.549848\pi$
$19$	−1.35966	−0.311928	−0.155964	+	0.987763i	$0.549848\pi$
$20$	0	0
$21$	2.24280	0.489419
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.84098	−0.546747
$28$	0	0
$29$	−3.66351	−0.680297	−0.340148	−	0.940372i	$-0.610477\pi$
$29$	−3.66351	−0.680297	−0.340148	+	0.940372i	$0.610477\pi$
$30$	0	0
$31$	4.46616	0.802146	0.401073	−	0.916046i	$-0.368637\pi$
$31$	4.46616	0.802146	0.401073	+	0.916046i	$0.368637\pi$
$32$	0	0
$33$	2.27665	0.396313
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.32432	−0.546514	−0.273257	−	0.961941i	$-0.588101\pi$
$37$	−3.32432	−0.546514	−0.273257	+	0.961941i	$0.588101\pi$
$38$	0	0
$39$	2.73552	0.438033
$40$	0	0
$41$	8.95216	1.39809	0.699046	−	0.715076i	$-0.253608\pi$
$41$	8.95216	1.39809	0.699046	+	0.715076i	$0.253608\pi$
$42$	0	0
$43$	8.68458	1.32439	0.662193	−	0.749333i	$-0.269626\pi$
$43$	8.68458	1.32439	0.662193	+	0.749333i	$0.269626\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.59028	1.39889	0.699443	−	0.714688i	$-0.253431\pi$
$47$	9.59028	1.39889	0.699443	+	0.714688i	$0.253431\pi$
$48$	0	0
$49$	13.6515	1.95021
$50$	0	0
$51$	−3.53861	−0.495505
$52$	0	0
$53$	−10.4676	−1.43784	−0.718918	−	0.695095i	$-0.755362\pi$
$53$	−10.4676	−1.43784	−0.718918	+	0.695095i	$0.755362\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.671035	−0.0888808
$58$	0	0
$59$	2.08543	0.271499	0.135750	−	0.990743i	$-0.456656\pi$
$59$	2.08543	0.271499	0.135750	+	0.990743i	$0.456656\pi$
$60$	0	0
$61$	−0.686197	−0.0878586	−0.0439293	−	0.999035i	$-0.513988\pi$
$61$	−0.686197	−0.0878586	−0.0439293	+	0.999035i	$0.513988\pi$
$62$	0	0
$63$	−12.5263	−1.57816
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.84324	1.08037	0.540187	−	0.841545i	$-0.318354\pi$
$67$	8.84324	1.08037	0.540187	+	0.841545i	$0.318354\pi$
$68$	0	0
$69$	−0.493532	−0.0594142
$70$	0	0
$71$	−15.9040	−1.88746	−0.943729	−	0.330721i	$-0.892708\pi$
$71$	−15.9040	−1.88746	−0.943729	+	0.330721i	$0.892708\pi$
$72$	0	0
$73$	−4.44647	−0.520420	−0.260210	−	0.965552i	$-0.583792\pi$
$73$	−4.44647	−0.520420	−0.260210	+	0.965552i	$0.583792\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	20.9631	2.38897
$78$	0	0
$79$	8.65227	0.973457	0.486728	−	0.873553i	$-0.338190\pi$
$79$	8.65227	0.973457	0.486728	+	0.873553i	$0.338190\pi$
$80$	0	0
$81$	6.86717	0.763019
$82$	0	0
$83$	−4.43591	−0.486904	−0.243452	−	0.969913i	$-0.578280\pi$
$83$	−4.43591	−0.486904	−0.243452	+	0.969913i	$0.578280\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.80806	−0.193844
$88$	0	0
$89$	13.2284	1.40220	0.701102	−	0.713061i	$-0.252692\pi$
$89$	13.2284	1.40220	0.701102	+	0.713061i	$0.252692\pi$
$90$	0	0
$91$	25.1884	2.64046
$92$	0	0
$93$	2.20419	0.228564
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.21785	0.123654	0.0618270	−	0.998087i	$-0.480307\pi$
$97$	1.21785	0.123654	0.0618270	+	0.998087i	$0.480307\pi$
$98$	0	0
$99$	−12.7153	−1.27794
$100$	0	0
$101$	10.4999	1.04478	0.522390	−	0.852706i	$-0.325040\pi$
$101$	10.4999	1.04478	0.522390	+	0.852706i	$0.325040\pi$
$102$	0	0
$103$	8.32614	0.820399	0.410200	−	0.911996i	$-0.365459\pi$
$103$	8.32614	0.820399	0.410200	+	0.911996i	$0.365459\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.60096	−0.928160	−0.464080	−	0.885793i	$-0.653615\pi$
$107$	−9.60096	−0.928160	−0.464080	+	0.885793i	$0.653615\pi$
$108$	0	0
$109$	10.2997	0.986534	0.493267	−	0.869878i	$-0.335803\pi$
$109$	10.2997	0.986534	0.493267	+	0.869878i	$0.335803\pi$
$110$	0	0
$111$	−1.64066	−0.155724
$112$	0	0
$113$	16.2479	1.52848	0.764239	−	0.644933i	$-0.223115\pi$
$113$	16.2479	1.52848	0.764239	+	0.644933i	$0.223115\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.2782	−1.41247
$118$	0	0
$119$	−32.5832	−2.98689
$120$	0	0
$121$	10.2795	0.934498
$122$	0	0
$123$	4.41817	0.398373
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.58555	−0.140695	−0.0703473	−	0.997523i	$-0.522411\pi$
$127$	−1.58555	−0.140695	−0.0703473	+	0.997523i	$0.522411\pi$
$128$	0	0
$129$	4.28612	0.377371
$130$	0	0
$131$	−4.69595	−0.410287	−0.205143	−	0.978732i	$-0.565766\pi$
$131$	−4.69595	−0.410287	−0.205143	+	0.978732i	$0.565766\pi$
$132$	0	0
$133$	−6.17882	−0.535772
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.3059	0.880494	0.440247	−	0.897877i	$-0.354891\pi$
$137$	10.3059	0.880494	0.440247	+	0.897877i	$0.354891\pi$
$138$	0	0
$139$	−0.673372	−0.0571147	−0.0285573	−	0.999592i	$-0.509091\pi$
$139$	−0.673372	−0.0571147	−0.0285573	+	0.999592i	$0.509091\pi$
$140$	0	0
$141$	4.73311	0.398599
$142$	0	0
$143$	25.5685	2.13814
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.73743	0.555694
$148$	0	0
$149$	−4.74118	−0.388413	−0.194206	−	0.980961i	$-0.562213\pi$
$149$	−4.74118	−0.388413	−0.194206	+	0.980961i	$0.562213\pi$
$150$	0	0
$151$	−3.16664	−0.257698	−0.128849	−	0.991664i	$-0.541128\pi$
$151$	−3.16664	−0.257698	−0.128849	+	0.991664i	$0.541128\pi$
$152$	0	0
$153$	19.7635	1.59779
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.07863	0.644745	0.322373	−	0.946613i	$-0.395520\pi$
$157$	8.07863	0.644745	0.322373	+	0.946613i	$0.395520\pi$
$158$	0	0
$159$	−5.16609	−0.409698
$160$	0	0
$161$	−4.54439	−0.358148
$162$	0	0
$163$	−2.25691	−0.176775	−0.0883876	−	0.996086i	$-0.528171\pi$
$163$	−2.25691	−0.176775	−0.0883876	+	0.996086i	$0.528171\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.01566	−0.155976	−0.0779881	−	0.996954i	$-0.524850\pi$
$167$	−2.01566	−0.155976	−0.0779881	+	0.996954i	$0.524850\pi$
$168$	0	0
$169$	17.7220	1.36323
$170$	0	0
$171$	3.74780	0.286602
$172$	0	0
$173$	−8.17720	−0.621701	−0.310850	−	0.950459i	$-0.600614\pi$
$173$	−8.17720	−0.621701	−0.310850	+	0.950459i	$0.600614\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.02922	0.0773612
$178$	0	0
$179$	1.48152	0.110734	0.0553670	−	0.998466i	$-0.482367\pi$
$179$	1.48152	0.110734	0.0553670	+	0.998466i	$0.482367\pi$
$180$	0	0
$181$	−7.93951	−0.590139	−0.295069	−	0.955476i	$-0.595343\pi$
$181$	−7.93951	−0.590139	−0.295069	+	0.955476i	$0.595343\pi$
$182$	0	0
$183$	−0.338660	−0.0250345
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−33.0749	−2.41867
$188$	0	0
$189$	−12.9105	−0.939101
$190$	0	0
$191$	−18.2987	−1.32405	−0.662023	−	0.749484i	$-0.730302\pi$
$191$	−18.2987	−1.32405	−0.662023	+	0.749484i	$0.730302\pi$
$192$	0	0
$193$	−3.65357	−0.262990	−0.131495	−	0.991317i	$-0.541978\pi$
$193$	−3.65357	−0.262990	−0.131495	+	0.991317i	$0.541978\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.14216	0.651352	0.325676	−	0.945481i	$-0.394408\pi$
$197$	9.14216	0.651352	0.325676	+	0.945481i	$0.394408\pi$
$198$	0	0
$199$	0.0416302	0.00295109	0.00147554	−	0.999999i	$-0.499530\pi$
$199$	0.0416302	0.00295109	0.00147554	+	0.999999i	$0.499530\pi$
$200$	0	0
$201$	4.36442	0.307842
$202$	0	0
$203$	−16.6484	−1.16849
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.75643	0.191585
$208$	0	0
$209$	−6.27207	−0.433848
$210$	0	0
$211$	1.08537	0.0747202	0.0373601	−	0.999302i	$-0.488105\pi$
$211$	1.08537	0.0747202	0.0373601	+	0.999302i	$0.488105\pi$
$212$	0	0
$213$	−7.84912	−0.537813
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.2960	1.37778
$218$	0	0
$219$	−2.19447	−0.148289
$220$	0	0
$221$	−39.7413	−2.67329
$222$	0	0
$223$	−12.2417	−0.819766	−0.409883	−	0.912138i	$-0.634430\pi$
$223$	−12.2417	−0.819766	−0.409883	+	0.912138i	$0.634430\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.60662	−0.438497	−0.219248	−	0.975669i	$-0.570361\pi$
$227$	−6.60662	−0.438497	−0.219248	+	0.975669i	$0.570361\pi$
$228$	0	0
$229$	−25.5008	−1.68514	−0.842571	−	0.538585i	$-0.818959\pi$
$229$	−25.5008	−1.68514	−0.842571	+	0.538585i	$0.818959\pi$
$230$	0	0
$231$	10.3460	0.680714
$232$	0	0
$233$	5.15810	0.337918	0.168959	−	0.985623i	$-0.445959\pi$
$233$	5.15810	0.337918	0.168959	+	0.985623i	$0.445959\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.27017	0.277377
$238$	0	0
$239$	−13.8196	−0.893917	−0.446959	−	0.894555i	$-0.647493\pi$
$239$	−13.8196	−0.893917	−0.446959	+	0.894555i	$0.647493\pi$
$240$	0	0
$241$	−14.9170	−0.960890	−0.480445	−	0.877025i	$-0.659525\pi$
$241$	−14.9170	−0.960890	−0.480445	+	0.877025i	$0.659525\pi$
$242$	0	0
$243$	11.9121	0.764161
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−7.53624	−0.479520
$248$	0	0
$249$	−2.18926	−0.138739
$250$	0	0
$251$	−12.8472	−0.810905	−0.405453	−	0.914116i	$-0.632886\pi$
$251$	−12.8472	−0.810905	−0.405453	+	0.914116i	$0.632886\pi$
$252$	0	0
$253$	−4.61297	−0.290015
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.2196	−0.762235	−0.381118	−	0.924527i	$-0.624461\pi$
$257$	−12.2196	−0.762235	−0.381118	+	0.924527i	$0.624461\pi$
$258$	0	0
$259$	−15.1070	−0.938702
$260$	0	0
$261$	10.0982	0.625062
$262$	0	0
$263$	25.2134	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$263$	25.2134	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.52862	0.399545
$268$	0	0
$269$	−5.85220	−0.356815	−0.178407	−	0.983957i	$-0.557094\pi$
$269$	−5.85220	−0.356815	−0.178407	+	0.983957i	$0.557094\pi$
$270$	0	0
$271$	12.8405	0.780008	0.390004	−	0.920813i	$-0.372474\pi$
$271$	12.8405	0.780008	0.390004	+	0.920813i	$0.372474\pi$
$272$	0	0
$273$	12.4312	0.752373
$274$	0	0
$275$	0	0
$276$	0	0
$277$	31.6190	1.89980	0.949902	−	0.312547i	$-0.101182\pi$
$277$	31.6190	1.89980	0.949902	+	0.312547i	$0.101182\pi$
$278$	0	0
$279$	−12.3106	−0.737019
$280$	0	0
$281$	−22.4411	−1.33873	−0.669363	−	0.742935i	$-0.733433\pi$
$281$	−22.4411	−1.33873	−0.669363	+	0.742935i	$0.733433\pi$
$282$	0	0
$283$	1.10282	0.0655559	0.0327779	−	0.999463i	$-0.489565\pi$
$283$	1.10282	0.0655559	0.0327779	+	0.999463i	$0.489565\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	40.6821	2.40139
$288$	0	0
$289$	34.4086	2.02403
$290$	0	0
$291$	0.601048	0.0352340
$292$	0	0
$293$	24.5092	1.43185	0.715923	−	0.698180i	$-0.246006\pi$
$293$	24.5092	1.43185	0.715923	+	0.698180i	$0.246006\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−13.1053	−0.760449
$298$	0	0
$299$	−5.54274	−0.320545
$300$	0	0
$301$	39.4661	2.27479
$302$	0	0
$303$	5.18204	0.297700
$304$	0	0
$305$	0	0
$306$	0	0
$307$	34.3319	1.95943	0.979714	−	0.200401i	$-0.0642246\pi$
$307$	34.3319	1.95943	0.979714	+	0.200401i	$0.0642246\pi$
$308$	0	0
$309$	4.10921	0.233765
$310$	0	0
$311$	−13.6914	−0.776369	−0.388185	−	0.921582i	$-0.626898\pi$
$311$	−13.6914	−0.776369	−0.388185	+	0.921582i	$0.626898\pi$
$312$	0	0
$313$	−4.65568	−0.263155	−0.131577	−	0.991306i	$-0.542004\pi$
$313$	−4.65568	−0.263155	−0.131577	+	0.991306i	$0.542004\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.44552	−0.0811884	−0.0405942	−	0.999176i	$-0.512925\pi$
$317$	−1.44552	−0.0811884	−0.0405942	+	0.999176i	$0.512925\pi$
$318$	0	0
$319$	−16.8997	−0.946199
$320$	0	0
$321$	−4.73838	−0.264470
$322$	0	0
$323$	9.74873	0.542434
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.08324	0.281104
$328$	0	0
$329$	43.5820	2.40275
$330$	0	0
$331$	17.9258	0.985292	0.492646	−	0.870230i	$-0.336030\pi$
$331$	17.9258	0.985292	0.492646	+	0.870230i	$0.336030\pi$
$332$	0	0
$333$	9.16323	0.502142
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11.2720	−0.614026	−0.307013	−	0.951705i	$-0.599330\pi$
$337$	−11.2720	−0.614026	−0.307013	+	0.951705i	$0.599330\pi$
$338$	0	0
$339$	8.01887	0.435525
$340$	0	0
$341$	20.6023	1.11567
$342$	0	0
$343$	30.2268	1.63209
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.8228	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$347$	20.8228	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$348$	0	0
$349$	−8.58121	−0.459342	−0.229671	−	0.973268i	$-0.573765\pi$
$349$	−8.58121	−0.459342	−0.229671	+	0.973268i	$0.573765\pi$
$350$	0	0
$351$	−15.7468	−0.840502
$352$	0	0
$353$	−7.31628	−0.389406	−0.194703	−	0.980862i	$-0.562374\pi$
$353$	−7.31628	−0.389406	−0.194703	+	0.980862i	$0.562374\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−16.0808	−0.851087
$358$	0	0
$359$	33.9028	1.78932	0.894660	−	0.446747i	$-0.147418\pi$
$359$	33.9028	1.78932	0.894660	+	0.446747i	$0.147418\pi$
$360$	0	0
$361$	−17.1513	−0.902701
$362$	0	0
$363$	5.07325	0.266276
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−11.3121	−0.590488	−0.295244	−	0.955422i	$-0.595401\pi$
$367$	−11.3121	−0.590488	−0.295244	+	0.955422i	$0.595401\pi$
$368$	0	0
$369$	−24.6760	−1.28458
$370$	0	0
$371$	−47.5689	−2.46965
$372$	0	0
$373$	22.1518	1.14697	0.573487	−	0.819214i	$-0.305590\pi$
$373$	22.1518	1.14697	0.573487	+	0.819214i	$0.305590\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−20.3059	−1.04581
$378$	0	0
$379$	8.82898	0.453514	0.226757	−	0.973951i	$-0.427188\pi$
$379$	8.82898	0.453514	0.226757	+	0.973951i	$0.427188\pi$
$380$	0	0
$381$	−0.782518	−0.0400896
$382$	0	0
$383$	21.2152	1.08405	0.542023	−	0.840364i	$-0.317659\pi$
$383$	21.2152	1.08405	0.542023	+	0.840364i	$0.317659\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23.9384	−1.21686
$388$	0	0
$389$	27.8793	1.41354	0.706768	−	0.707445i	$-0.250152\pi$
$389$	27.8793	1.41354	0.706768	+	0.707445i	$0.250152\pi$
$390$	0	0
$391$	7.16998	0.362601
$392$	0	0
$393$	−2.31760	−0.116907
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.8676	−1.14769	−0.573845	−	0.818964i	$-0.694549\pi$
$397$	−22.8676	−1.14769	−0.573845	+	0.818964i	$0.694549\pi$
$398$	0	0
$399$	−3.04944	−0.152663
$400$	0	0
$401$	−2.37194	−0.118449	−0.0592246	−	0.998245i	$-0.518863\pi$
$401$	−2.37194	−0.118449	−0.0592246	+	0.998245i	$0.518863\pi$
$402$	0	0
$403$	24.7548	1.23312
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.3350	−0.760126
$408$	0	0
$409$	−18.3125	−0.905494	−0.452747	−	0.891639i	$-0.649556\pi$
$409$	−18.3125	−0.905494	−0.452747	+	0.891639i	$0.649556\pi$
$410$	0	0
$411$	5.08630	0.250888
$412$	0	0
$413$	9.47698	0.466332
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−0.332331	−0.0162743
$418$	0	0
$419$	−22.0541	−1.07741	−0.538706	−	0.842494i	$-0.681087\pi$
$419$	−22.0541	−1.07741	−0.538706	+	0.842494i	$0.681087\pi$
$420$	0	0
$421$	37.9042	1.84734	0.923670	−	0.383190i	$-0.125174\pi$
$421$	37.9042	1.84734	0.923670	+	0.383190i	$0.125174\pi$
$422$	0	0
$423$	−26.4349	−1.28531
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.11835	−0.150907
$428$	0	0
$429$	12.6189	0.609244
$430$	0	0
$431$	−33.8877	−1.63231	−0.816156	−	0.577831i	$-0.803899\pi$
$431$	−33.8877	−1.63231	−0.816156	+	0.577831i	$0.803899\pi$
$432$	0	0
$433$	5.57737	0.268031	0.134016	−	0.990979i	$-0.457213\pi$
$433$	5.57737	0.268031	0.134016	+	0.990979i	$0.457213\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.35966	0.0650414
$438$	0	0
$439$	15.2124	0.726046	0.363023	−	0.931780i	$-0.381745\pi$
$439$	15.2124	0.726046	0.363023	+	0.931780i	$0.381745\pi$
$440$	0	0
$441$	−37.6292	−1.79187
$442$	0	0
$443$	11.7086	0.556291	0.278145	−	0.960539i	$-0.410280\pi$
$443$	11.7086	0.556291	0.278145	+	0.960539i	$0.410280\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2.33992	−0.110675
$448$	0	0
$449$	−14.1523	−0.667889	−0.333945	−	0.942593i	$-0.608380\pi$
$449$	−14.1523	−0.667889	−0.333945	+	0.942593i	$0.608380\pi$
$450$	0	0
$451$	41.2960	1.94455
$452$	0	0
$453$	−1.56284	−0.0734285
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.4278	−0.487793	−0.243896	−	0.969801i	$-0.578426\pi$
$457$	−10.4278	−0.487793	−0.243896	+	0.969801i	$0.578426\pi$
$458$	0	0
$459$	20.3697	0.950778
$460$	0	0
$461$	39.5905	1.84391	0.921957	−	0.387291i	$-0.126589\pi$
$461$	39.5905	1.84391	0.921957	+	0.387291i	$0.126589\pi$
$462$	0	0
$463$	−10.5954	−0.492408	−0.246204	−	0.969218i	$-0.579183\pi$
$463$	−10.5954	−0.492408	−0.246204	+	0.969218i	$0.579183\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	34.0553	1.57589	0.787945	−	0.615746i	$-0.211145\pi$
$467$	34.0553	1.57589	0.787945	+	0.615746i	$0.211145\pi$
$468$	0	0
$469$	40.1871	1.85567
$470$	0	0
$471$	3.98706	0.183714
$472$	0	0
$473$	40.0617	1.84204
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.8532	1.32110
$478$	0	0
$479$	−11.8776	−0.542702	−0.271351	−	0.962480i	$-0.587470\pi$
$479$	−11.8776	−0.542702	−0.271351	+	0.962480i	$0.587470\pi$
$480$	0	0
$481$	−18.4258	−0.840145
$482$	0	0
$483$	−2.24280	−0.102051
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.7980	1.12371	0.561853	−	0.827237i	$-0.310089\pi$
$487$	24.7980	1.12371	0.561853	+	0.827237i	$0.310089\pi$
$488$	0	0
$489$	−1.11386	−0.0503704
$490$	0	0
$491$	−25.6220	−1.15630	−0.578152	−	0.815929i	$-0.696226\pi$
$491$	−25.6220	−1.15630	−0.578152	+	0.815929i	$0.696226\pi$
$492$	0	0
$493$	26.2673	1.18302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−72.2739	−3.24193
$498$	0	0
$499$	−28.7020	−1.28488	−0.642438	−	0.766337i	$-0.722077\pi$
$499$	−28.7020	−1.28488	−0.642438	+	0.766337i	$0.722077\pi$
$500$	0	0
$501$	−0.994790	−0.0444439
$502$	0	0
$503$	−14.8193	−0.660760	−0.330380	−	0.943848i	$-0.607177\pi$
$503$	−14.8193	−0.660760	−0.330380	+	0.943848i	$0.607177\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	8.74635	0.388439
$508$	0	0
$509$	−22.5059	−0.997555	−0.498777	−	0.866730i	$-0.666217\pi$
$509$	−22.5059	−0.997555	−0.498777	+	0.866730i	$0.666217\pi$
$510$	0	0
$511$	−20.2065	−0.893883
$512$	0	0
$513$	3.86277	0.170545
$514$	0	0
$515$	0	0
$516$	0	0
$517$	44.2397	1.94566
$518$	0	0
$519$	−4.03570	−0.177148
$520$	0	0
$521$	−21.1227	−0.925404	−0.462702	−	0.886514i	$-0.653120\pi$
$521$	−21.1227	−0.925404	−0.462702	+	0.886514i	$0.653120\pi$
$522$	0	0
$523$	0.617011	0.0269800	0.0134900	−	0.999909i	$-0.495706\pi$
$523$	0.617011	0.0269800	0.0134900	+	0.999909i	$0.495706\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.0223	−1.39491
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−5.74832	−0.249456
$532$	0	0
$533$	49.6195	2.14926
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.731177	0.0315526
$538$	0	0
$539$	62.9738	2.71247
$540$	0	0
$541$	13.5801	0.583856	0.291928	−	0.956440i	$-0.405703\pi$
$541$	13.5801	0.583856	0.291928	+	0.956440i	$0.405703\pi$
$542$	0	0
$543$	−3.91840	−0.168154
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.4825	−1.08955	−0.544777	−	0.838581i	$-0.683386\pi$
$547$	−25.4825	−1.08955	−0.544777	+	0.838581i	$0.683386\pi$
$548$	0	0
$549$	1.89145	0.0807252
$550$	0	0
$551$	4.98113	0.212203
$552$	0	0
$553$	39.3193	1.67203
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.2051	0.601891	0.300945	−	0.953641i	$-0.402698\pi$
$557$	14.2051	0.601891	0.300945	+	0.953641i	$0.402698\pi$
$558$	0	0
$559$	48.1364	2.03595
$560$	0	0
$561$	−16.3235	−0.689179
$562$	0	0
$563$	6.10564	0.257322	0.128661	−	0.991689i	$-0.458932\pi$
$563$	6.10564	0.257322	0.128661	+	0.991689i	$0.458932\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	31.2071	1.31057
$568$	0	0
$569$	9.28182	0.389114	0.194557	−	0.980891i	$-0.437673\pi$
$569$	9.28182	0.389114	0.194557	+	0.980891i	$0.437673\pi$
$570$	0	0
$571$	6.76185	0.282975	0.141487	−	0.989940i	$-0.454812\pi$
$571$	6.76185	0.282975	0.141487	+	0.989940i	$0.454812\pi$
$572$	0	0
$573$	−9.03097	−0.377274
$574$	0	0
$575$	0	0
$576$	0	0
$577$	31.2132	1.29942	0.649711	−	0.760182i	$-0.274890\pi$
$577$	31.2132	1.29942	0.649711	+	0.760182i	$0.274890\pi$
$578$	0	0
$579$	−1.80315	−0.0749365
$580$	0	0
$581$	−20.1585	−0.836315
$582$	0	0
$583$	−48.2867	−1.99983
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.7138	−1.02005	−0.510023	−	0.860161i	$-0.670363\pi$
$587$	−24.7138	−1.02005	−0.510023	+	0.860161i	$0.670363\pi$
$588$	0	0
$589$	−6.07246	−0.250211
$590$	0	0
$591$	4.51195	0.185597
$592$	0	0
$593$	−20.3119	−0.834109	−0.417054	−	0.908882i	$-0.636938\pi$
$593$	−20.3119	−0.834109	−0.417054	+	0.908882i	$0.636938\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.0205458	0.000840885 0
$598$	0	0
$599$	−17.2979	−0.706772	−0.353386	−	0.935478i	$-0.614970\pi$
$599$	−17.2979	−0.706772	−0.353386	+	0.935478i	$0.614970\pi$
$600$	0	0
$601$	32.9939	1.34585	0.672924	−	0.739711i	$-0.265038\pi$
$601$	32.9939	1.34585	0.672924	+	0.739711i	$0.265038\pi$
$602$	0	0
$603$	−24.3757	−0.992657
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−35.8434	−1.45484	−0.727419	−	0.686193i	$-0.759281\pi$
$607$	−35.8434	−1.45484	−0.727419	+	0.686193i	$0.759281\pi$
$608$	0	0
$609$	−8.21651	−0.332950
$610$	0	0
$611$	53.1564	2.15048
$612$	0	0
$613$	−30.1971	−1.21965	−0.609825	−	0.792536i	$-0.708760\pi$
$613$	−30.1971	−1.21965	−0.609825	+	0.792536i	$0.708760\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.34282	−0.215094	−0.107547	−	0.994200i	$-0.534300\pi$
$617$	−5.34282	−0.215094	−0.107547	+	0.994200i	$0.534300\pi$
$618$	0	0
$619$	−2.48462	−0.0998652	−0.0499326	−	0.998753i	$-0.515901\pi$
$619$	−2.48462	−0.0998652	−0.0499326	+	0.998753i	$0.515901\pi$
$620$	0	0
$621$	2.84098	0.114005
$622$	0	0
$623$	60.1148	2.40845
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.09546	−0.123621
$628$	0	0
$629$	23.8353	0.950375
$630$	0	0
$631$	−0.724918	−0.0288585	−0.0144293	−	0.999896i	$-0.504593\pi$
$631$	−0.724918	−0.0288585	−0.0144293	+	0.999896i	$0.504593\pi$
$632$	0	0
$633$	0.535666	0.0212908
$634$	0	0
$635$	0	0
$636$	0	0
$637$	75.6665	2.99802
$638$	0	0
$639$	43.8382	1.73421
$640$	0	0
$641$	26.8404	1.06013	0.530067	−	0.847956i	$-0.322167\pi$
$641$	26.8404	1.06013	0.530067	+	0.847956i	$0.322167\pi$
$642$	0	0
$643$	2.47661	0.0976679	0.0488340	−	0.998807i	$-0.484449\pi$
$643$	2.47661	0.0976679	0.0488340	+	0.998807i	$0.484449\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−35.7731	−1.40639	−0.703193	−	0.710999i	$-0.748243\pi$
$647$	−35.7731	−1.40639	−0.703193	+	0.710999i	$0.748243\pi$
$648$	0	0
$649$	9.62000	0.377618
$650$	0	0
$651$	10.0167	0.392585
$652$	0	0
$653$	14.5123	0.567909	0.283955	−	0.958838i	$-0.408354\pi$
$653$	14.5123	0.567909	0.283955	+	0.958838i	$0.408354\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.2564	0.478167
$658$	0	0
$659$	−8.05918	−0.313941	−0.156970	−	0.987603i	$-0.550173\pi$
$659$	−8.05918	−0.313941	−0.156970	+	0.987603i	$0.550173\pi$
$660$	0	0
$661$	−27.0801	−1.05329	−0.526647	−	0.850084i	$-0.676551\pi$
$661$	−27.0801	−1.05329	−0.526647	+	0.850084i	$0.676551\pi$
$662$	0	0
$663$	−19.6136	−0.761729
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.66351	0.141852
$668$	0	0
$669$	−6.04167	−0.233585
$670$	0	0
$671$	−3.16541	−0.122199
$672$	0	0
$673$	−20.1388	−0.776292	−0.388146	−	0.921598i	$-0.626884\pi$
$673$	−20.1388	−0.776292	−0.388146	+	0.921598i	$0.626884\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.8739	−1.03285	−0.516424	−	0.856333i	$-0.672737\pi$
$677$	−26.8739	−1.03285	−0.516424	+	0.856333i	$0.672737\pi$
$678$	0	0
$679$	5.53439	0.212390
$680$	0	0
$681$	−3.26058	−0.124946
$682$	0	0
$683$	40.0529	1.53258	0.766290	−	0.642495i	$-0.222100\pi$
$683$	40.0529	1.53258	0.766290	+	0.642495i	$0.222100\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−12.5855	−0.480166
$688$	0	0
$689$	−58.0192	−2.21036
$690$	0	0
$691$	−25.1106	−0.955254	−0.477627	−	0.878563i	$-0.658503\pi$
$691$	−25.1106	−0.955254	−0.477627	+	0.878563i	$0.658503\pi$
$692$	0	0
$693$	−57.7833	−2.19501
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−64.1868	−2.43125
$698$	0	0
$699$	2.54568	0.0962866
$700$	0	0
$701$	−33.6355	−1.27039	−0.635197	−	0.772350i	$-0.719081\pi$
$701$	−33.6355	−1.27039	−0.635197	+	0.772350i	$0.719081\pi$
$702$	0	0
$703$	4.51994	0.170473
$704$	0	0
$705$	0	0
$706$	0	0
$707$	47.7157	1.79453
$708$	0	0
$709$	6.25559	0.234934	0.117467	−	0.993077i	$-0.462523\pi$
$709$	6.25559	0.234934	0.117467	+	0.993077i	$0.462523\pi$
$710$	0	0
$711$	−23.8494	−0.894421
$712$	0	0
$713$	−4.46616	−0.167259
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.82042	−0.254713
$718$	0	0
$719$	45.8585	1.71024	0.855118	−	0.518434i	$-0.173485\pi$
$719$	45.8585	1.71024	0.855118	+	0.518434i	$0.173485\pi$
$720$	0	0
$721$	37.8372	1.40913
$722$	0	0
$723$	−7.36202	−0.273797
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−15.5296	−0.575961	−0.287981	−	0.957636i	$-0.592984\pi$
$727$	−15.5296	−0.575961	−0.287981	+	0.957636i	$0.592984\pi$
$728$	0	0
$729$	−14.7225	−0.545278
$730$	0	0
$731$	−62.2683	−2.30307
$732$	0	0
$733$	43.1452	1.59360	0.796802	−	0.604240i	$-0.206523\pi$
$733$	43.1452	1.59360	0.796802	+	0.604240i	$0.206523\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	40.7936	1.50265
$738$	0	0
$739$	33.5882	1.23556	0.617781	−	0.786350i	$-0.288032\pi$
$739$	33.5882	1.23556	0.617781	+	0.786350i	$0.288032\pi$
$740$	0	0
$741$	−3.71937	−0.136635
$742$	0	0
$743$	13.8916	0.509632	0.254816	−	0.966990i	$-0.417985\pi$
$743$	13.8916	0.509632	0.254816	+	0.966990i	$0.417985\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.2273	0.447372
$748$	0	0
$749$	−43.6305	−1.59422
$750$	0	0
$751$	38.4863	1.40439	0.702193	−	0.711987i	$-0.252204\pi$
$751$	38.4863	1.40439	0.702193	+	0.711987i	$0.252204\pi$
$752$	0	0
$753$	−6.34048	−0.231060
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4.03386	0.146613	0.0733066	−	0.997309i	$-0.476645\pi$
$757$	4.03386	0.146613	0.0733066	+	0.997309i	$0.476645\pi$
$758$	0	0
$759$	−2.27665	−0.0826370
$760$	0	0
$761$	−2.28005	−0.0826518	−0.0413259	−	0.999146i	$-0.513158\pi$
$761$	−2.28005	−0.0826518	−0.0413259	+	0.999146i	$0.513158\pi$
$762$	0	0
$763$	46.8059	1.69449
$764$	0	0
$765$	0	0
$766$	0	0
$767$	11.5590	0.417370
$768$	0	0
$769$	−48.0050	−1.73111	−0.865553	−	0.500818i	$-0.833033\pi$
$769$	−48.0050	−1.73111	−0.865553	+	0.500818i	$0.833033\pi$
$770$	0	0
$771$	−6.03074	−0.217192
$772$	0	0
$773$	−3.67822	−0.132297	−0.0661483	−	0.997810i	$-0.521071\pi$
$773$	−3.67822	−0.132297	−0.0661483	+	0.997810i	$0.521071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.45577	−0.267474
$778$	0	0
$779$	−12.1719	−0.436104
$780$	0	0
$781$	−73.3646	−2.62519
$782$	0	0
$783$	10.4079	0.371950
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13.9078	0.495761	0.247881	−	0.968791i	$-0.420266\pi$
$787$	13.9078	0.495761	0.247881	+	0.968791i	$0.420266\pi$
$788$	0	0
$789$	12.4436	0.443004
$790$	0	0
$791$	73.8369	2.62534
$792$	0	0
$793$	−3.80341	−0.135063
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.8982	0.492301	0.246150	−	0.969232i	$-0.420834\pi$
$797$	13.8982	0.492301	0.246150	+	0.969232i	$0.420834\pi$
$798$	0	0
$799$	−68.7621	−2.43263
$800$	0	0
$801$	−36.4630	−1.28836
$802$	0	0
$803$	−20.5114	−0.723833
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.88824	−0.101671
$808$	0	0
$809$	29.4828	1.03656	0.518279	−	0.855211i	$-0.326573\pi$
$809$	29.4828	1.03656	0.518279	+	0.855211i	$0.326573\pi$
$810$	0	0
$811$	6.73889	0.236634	0.118317	−	0.992976i	$-0.462250\pi$
$811$	6.73889	0.236634	0.118317	+	0.992976i	$0.462250\pi$
$812$	0	0
$813$	6.33722	0.222256
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−11.8081	−0.413113
$818$	0	0
$819$	−69.4299	−2.42608
$820$	0	0
$821$	−15.0521	−0.525322	−0.262661	−	0.964888i	$-0.584600\pi$
$821$	−15.0521	−0.525322	−0.262661	+	0.964888i	$0.584600\pi$
$822$	0	0
$823$	−23.5906	−0.822316	−0.411158	−	0.911564i	$-0.634876\pi$
$823$	−23.5906	−0.822316	−0.411158	+	0.911564i	$0.634876\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.2147	−1.53750	−0.768748	−	0.639552i	$-0.779120\pi$
$827$	−44.2147	−1.53750	−0.768748	+	0.639552i	$0.779120\pi$
$828$	0	0
$829$	−3.30736	−0.114870	−0.0574348	−	0.998349i	$-0.518292\pi$
$829$	−3.30736	−0.114870	−0.0574348	+	0.998349i	$0.518292\pi$
$830$	0	0
$831$	15.6050	0.541331
$832$	0	0
$833$	−97.8807	−3.39136
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−12.6883	−0.438571
$838$	0	0
$839$	27.5243	0.950245	0.475122	−	0.879920i	$-0.342404\pi$
$839$	27.5243	0.950245	0.475122	+	0.879920i	$0.342404\pi$
$840$	0	0
$841$	−15.5787	−0.537197
$842$	0	0
$843$	−11.0754	−0.381458
$844$	0	0
$845$	0	0
$846$	0	0
$847$	46.7139	1.60511
$848$	0	0
$849$	0.544277	0.0186795
$850$	0	0
$851$	3.32432	0.113956
$852$	0	0
$853$	−7.90006	−0.270493	−0.135246	−	0.990812i	$-0.543183\pi$
$853$	−7.90006	−0.270493	−0.135246	+	0.990812i	$0.543183\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	8.41234	0.287360	0.143680	−	0.989624i	$-0.454106\pi$
$857$	8.41234	0.287360	0.143680	+	0.989624i	$0.454106\pi$
$858$	0	0
$859$	6.63662	0.226439	0.113219	−	0.993570i	$-0.463884\pi$
$859$	6.63662	0.226439	0.113219	+	0.993570i	$0.463884\pi$
$860$	0	0
$861$	20.0779	0.684253
$862$	0	0
$863$	43.7797	1.49028	0.745140	−	0.666908i	$-0.232383\pi$
$863$	43.7797	1.49028	0.745140	+	0.666908i	$0.232383\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.9817	0.576729
$868$	0	0
$869$	39.9127	1.35394
$870$	0	0
$871$	49.0158	1.66084
$872$	0	0
$873$	−3.35692	−0.113614
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.93519	−0.0991142	−0.0495571	−	0.998771i	$-0.515781\pi$
$877$	−2.93519	−0.0991142	−0.0495571	+	0.998771i	$0.515781\pi$
$878$	0	0
$879$	12.0961	0.407991
$880$	0	0
$881$	−20.2151	−0.681064	−0.340532	−	0.940233i	$-0.610607\pi$
$881$	−20.2151	−0.681064	−0.340532	+	0.940233i	$0.610607\pi$
$882$	0	0
$883$	32.2407	1.08499	0.542494	−	0.840060i	$-0.317480\pi$
$883$	32.2407	1.08499	0.542494	+	0.840060i	$0.317480\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.2726	0.546381	0.273190	−	0.961960i	$-0.411921\pi$
$887$	16.2726	0.546381	0.273190	+	0.961960i	$0.411921\pi$
$888$	0	0
$889$	−7.20535	−0.241660
$890$	0	0
$891$	31.6780	1.06125
$892$	0	0
$893$	−13.0395	−0.436351
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.73552	−0.0913362
$898$	0	0
$899$	−16.3618	−0.545697
$900$	0	0
$901$	75.0525	2.50036
$902$	0	0
$903$	19.4778	0.648180
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.6897	0.985830	0.492915	−	0.870077i	$-0.335931\pi$
$907$	29.6897	0.985830	0.492915	+	0.870077i	$0.335931\pi$
$908$	0	0
$909$	−28.9422	−0.959954
$910$	0	0
$911$	11.7787	0.390246	0.195123	−	0.980779i	$-0.437489\pi$
$911$	11.7787	0.390246	0.195123	+	0.980779i	$0.437489\pi$
$912$	0	0
$913$	−20.4627	−0.677216
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21.3402	−0.704716
$918$	0	0
$919$	−48.4606	−1.59857	−0.799284	−	0.600954i	$-0.794787\pi$
$919$	−48.4606	−1.59857	−0.799284	+	0.600954i	$0.794787\pi$
$920$	0	0
$921$	16.9439	0.558320
$922$	0	0
$923$	−88.1517	−2.90155
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−22.9504	−0.753790
$928$	0	0
$929$	9.51219	0.312085	0.156042	−	0.987750i	$-0.450126\pi$
$929$	9.51219	0.312085	0.156042	+	0.987750i	$0.450126\pi$
$930$	0	0
$931$	−18.5614	−0.608324
$932$	0	0
$933$	−6.75715	−0.221219
$934$	0	0
$935$	0	0
$936$	0	0
$937$	40.7865	1.33244	0.666219	−	0.745757i	$-0.267912\pi$
$937$	40.7865	1.33244	0.666219	+	0.745757i	$0.267912\pi$
$938$	0	0
$939$	−2.29772	−0.0749834
$940$	0	0
$941$	10.8016	0.352123	0.176062	−	0.984379i	$-0.443664\pi$
$941$	10.8016	0.352123	0.176062	+	0.984379i	$0.443664\pi$
$942$	0	0
$943$	−8.95216	−0.291522
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.00967	0.162792	0.0813962	−	0.996682i	$-0.474062\pi$
$947$	5.00967	0.162792	0.0813962	+	0.996682i	$0.474062\pi$
$948$	0	0
$949$	−24.6456	−0.800031
$950$	0	0
$951$	−0.713409	−0.0231339
$952$	0	0
$953$	−19.6780	−0.637434	−0.318717	−	0.947850i	$-0.603252\pi$
$953$	−19.6780	−0.637434	−0.318717	+	0.947850i	$0.603252\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−8.34051	−0.269610
$958$	0	0
$959$	46.8341	1.51235
$960$	0	0
$961$	−11.0534	−0.356562
$962$	0	0
$963$	26.4643	0.852801
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−26.7448	−0.860055	−0.430028	−	0.902816i	$-0.641496\pi$
$967$	−26.7448	−0.860055	−0.430028	+	0.902816i	$0.641496\pi$
$968$	0	0
$969$	4.81131	0.154561
$970$	0	0
$971$	−28.5685	−0.916806	−0.458403	−	0.888744i	$-0.651578\pi$
$971$	−28.5685	−0.916806	−0.458403	+	0.888744i	$0.651578\pi$
$972$	0	0
$973$	−3.06007	−0.0981012
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.0365	0.545047	0.272524	−	0.962149i	$-0.412142\pi$
$977$	17.0365	0.545047	0.272524	+	0.962149i	$0.412142\pi$
$978$	0	0
$979$	61.0220	1.95027
$980$	0	0
$981$	−28.3904	−0.906436
$982$	0	0
$983$	−33.7247	−1.07565	−0.537826	−	0.843056i	$-0.680754\pi$
$983$	−33.7247	−1.07565	−0.537826	+	0.843056i	$0.680754\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.5091	0.684641
$988$	0	0
$989$	−8.68458	−0.276154
$990$	0	0
$991$	1.34150	0.0426140	0.0213070	−	0.999773i	$-0.493217\pi$
$991$	1.34150	0.0426140	0.0213070	+	0.999773i	$0.493217\pi$
$992$	0	0
$993$	8.84696	0.280750
$994$	0	0
$995$	0	0
$996$	0	0
$997$	52.1926	1.65296	0.826478	−	0.562969i	$-0.190341\pi$
$997$	52.1926	1.65296	0.826478	+	0.562969i	$0.190341\pi$
$998$	0	0
$999$	9.44431	0.298805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9200.2.a.de.1.5		8
4.3	odd	2		4600.2.a.bj.1.4		8
5.2	odd	4		1840.2.e.h.369.7		16
5.3	odd	4		1840.2.e.h.369.10		16
5.4	even	2		9200.2.a.dd.1.4		8
20.3	even	4		920.2.e.c.369.7	✓	16
20.7	even	4		920.2.e.c.369.10	yes	16
20.19	odd	2		4600.2.a.bk.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.7	✓	16	20.3	even	4
920.2.e.c.369.10	yes	16	20.7	even	4
1840.2.e.h.369.7		16	5.2	odd	4
1840.2.e.h.369.10		16	5.3	odd	4
4600.2.a.bj.1.4		8	4.3	odd	2
4600.2.a.bk.1.5		8	20.19	odd	2
9200.2.a.dd.1.4		8	5.4	even	2
9200.2.a.de.1.5		8	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)

\(f(q)\)	\(=\)	\(q+0.493532 q^{3} +4.54439 q^{7} -2.75643 q^{9} +O(q^{10})\)	\(q+0.493532 q^{3} +4.54439 q^{7} -2.75643 q^{9} +4.61297 q^{11} +5.54274 q^{13} -7.16998 q^{17} -1.35966 q^{19} +2.24280 q^{21} -1.00000 q^{23} -2.84098 q^{27} -3.66351 q^{29} +4.46616 q^{31} +2.27665 q^{33} -3.32432 q^{37} +2.73552 q^{39} +8.95216 q^{41} +8.68458 q^{43} +9.59028 q^{47} +13.6515 q^{49} -3.53861 q^{51} -10.4676 q^{53} -0.671035 q^{57} +2.08543 q^{59} -0.686197 q^{61} -12.5263 q^{63} +8.84324 q^{67} -0.493532 q^{69} -15.9040 q^{71} -4.44647 q^{73} +20.9631 q^{77} +8.65227 q^{79} +6.86717 q^{81} -4.43591 q^{83} -1.80806 q^{87} +13.2284 q^{89} +25.1884 q^{91} +2.20419 q^{93} +1.21785 q^{97} -12.7153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 - 7 * q^11 + 11 * q^13 - 7 * q^17 - 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 - 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 + 19 * q^51 + 4 * q^53 - 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 - 15 * q^71 + 6 * q^73 + 18 * q^77 + 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 + 35 * q^91 + 20 * q^93 + 3 * q^97 - 61 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more