LMFDB - Embedded newform 1125.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1125,2,Mod(1,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1125.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,2,0,0,0,0,0,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.98317022739$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.17557$ of defining polynomial
Character	$\chi$	$=$	1125.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17557 q^{2} -0.618034 q^{4} +2.62866 q^{7} -3.07768 q^{8} -5.00000 q^{11} -6.88191 q^{13} +3.09017 q^{14} -2.38197 q^{16} -4.97980 q^{17} +3.47214 q^{19} -5.87785 q^{22} +5.42882 q^{23} -8.09017 q^{26} -1.62460 q^{28} -5.00000 q^{29} -2.85410 q^{31} +3.35520 q^{32} -5.85410 q^{34} -1.62460 q^{37} +4.08174 q^{38} -8.09017 q^{41} -1.00406 q^{43} +3.09017 q^{44} +6.38197 q^{46} +8.33499 q^{47} -0.0901699 q^{49} +4.25325 q^{52} -7.15942 q^{53} -8.09017 q^{56} -5.87785 q^{58} -1.90983 q^{59} +7.32624 q^{61} -3.35520 q^{62} +8.70820 q^{64} +8.50651 q^{67} +3.07768 q^{68} -13.0902 q^{71} +4.25325 q^{73} -1.90983 q^{74} -2.14590 q^{76} -13.1433 q^{77} -8.47214 q^{79} -9.51057 q^{82} -8.22899 q^{83} -1.18034 q^{86} +15.3884 q^{88} -11.1803 q^{89} -18.0902 q^{91} -3.35520 q^{92} +9.79837 q^{94} +14.3844 q^{97} -0.106001 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 20 q^{11} - 10 q^{14} - 14 q^{16} - 4 q^{19} - 10 q^{26} - 20 q^{29} + 2 q^{31} - 10 q^{34} - 10 q^{41} - 10 q^{44} + 30 q^{46} + 22 q^{49} - 10 q^{56} - 30 q^{59} - 2 q^{61} + 8 q^{64}+ \cdots - 10 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 - 20 * q^11 - 10 * q^14 - 14 * q^16 - 4 * q^19 - 10 * q^26 - 20 * q^29 + 2 * q^31 - 10 * q^34 - 10 * q^41 - 10 * q^44 + 30 * q^46 + 22 * q^49 - 10 * q^56 - 30 * q^59 - 2 * q^61 + 8 * q^64 - 30 * q^71 - 30 * q^74 - 22 * q^76 - 16 * q^79 + 40 * q^86 - 50 * q^91 - 10 * q^94

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 $

$2$	1.17557	0.831254	0.415627	−	0.909535i	$-0.363562\pi$
$2$	1.17557	0.831254	0.415627	+	0.909535i	$0.363562\pi$
$3$	0	0
$3$	0	0
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.62866	0.993538	0.496769	−	0.867883i	$-0.334520\pi$
$7$	2.62866	0.993538	0.496769	+	0.867883i	$0.334520\pi$
$8$	−3.07768	−1.08813
$9$	0	0
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	−6.88191	−1.90870	−0.954349	−	0.298693i	$-0.903449\pi$
$13$	−6.88191	−1.90870	−0.954349	+	0.298693i	$0.903449\pi$
$14$	3.09017	0.825883
$15$	0	0
$16$	−2.38197	−0.595492
$17$	−4.97980	−1.20778	−0.603889	−	0.797068i	$-0.706383\pi$
$17$	−4.97980	−1.20778	−0.603889	+	0.797068i	$0.706383\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	0	0
$21$	0	0
$22$	−5.87785	−1.25316
$23$	5.42882	1.13199	0.565994	−	0.824409i	$-0.308492\pi$
$23$	5.42882	1.13199	0.565994	+	0.824409i	$0.308492\pi$
$24$	0	0
$25$	0	0
$26$	−8.09017	−1.58661
$27$	0	0
$28$	−1.62460	−0.307020
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−2.85410	−0.512612	−0.256306	−	0.966596i	$-0.582505\pi$
$31$	−2.85410	−0.512612	−0.256306	+	0.966596i	$0.582505\pi$
$32$	3.35520	0.593121
$33$	0	0
$34$	−5.85410	−1.00397
$35$	0	0
$36$	0	0
$37$	−1.62460	−0.267082	−0.133541	−	0.991043i	$-0.542635\pi$
$37$	−1.62460	−0.267082	−0.133541	+	0.991043i	$0.542635\pi$
$38$	4.08174	0.662146
$39$	0	0
$40$	0	0
$41$	−8.09017	−1.26347	−0.631736	−	0.775183i	$-0.717657\pi$
$41$	−8.09017	−1.26347	−0.631736	+	0.775183i	$0.717657\pi$
$42$	0	0
$43$	−1.00406	−0.153117	−0.0765586	−	0.997065i	$-0.524393\pi$
$43$	−1.00406	−0.153117	−0.0765586	+	0.997065i	$0.524393\pi$
$44$	3.09017	0.465861
$45$	0	0
$46$	6.38197	0.940970
$47$	8.33499	1.21578	0.607892	−	0.794020i	$-0.292015\pi$
$47$	8.33499	1.21578	0.607892	+	0.794020i	$0.292015\pi$
$48$	0	0
$49$	−0.0901699	−0.0128814
$50$	0	0
$51$	0	0
$52$	4.25325	0.589820
$53$	−7.15942	−0.983423	−0.491711	−	0.870758i	$-0.663628\pi$
$53$	−7.15942	−0.983423	−0.491711	+	0.870758i	$0.663628\pi$
$54$	0	0
$55$	0	0
$56$	−8.09017	−1.08109
$57$	0	0
$58$	−5.87785	−0.771800
$59$	−1.90983	−0.248639	−0.124319	−	0.992242i	$-0.539675\pi$
$59$	−1.90983	−0.248639	−0.124319	+	0.992242i	$0.539675\pi$
$60$	0	0
$61$	7.32624	0.938029	0.469014	−	0.883191i	$-0.344609\pi$
$61$	7.32624	0.938029	0.469014	+	0.883191i	$0.344609\pi$
$62$	−3.35520	−0.426111
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	0	0
$67$	8.50651	1.03924	0.519618	−	0.854399i	$-0.326074\pi$
$67$	8.50651	1.03924	0.519618	+	0.854399i	$0.326074\pi$
$68$	3.07768	0.373224
$69$	0	0
$70$	0	0
$71$	−13.0902	−1.55352	−0.776759	−	0.629798i	$-0.783138\pi$
$71$	−13.0902	−1.55352	−0.776759	+	0.629798i	$0.783138\pi$
$72$	0	0
$73$	4.25325	0.497806	0.248903	−	0.968528i	$-0.419930\pi$
$73$	4.25325	0.497806	0.248903	+	0.968528i	$0.419930\pi$
$74$	−1.90983	−0.222013
$75$	0	0
$76$	−2.14590	−0.246151
$77$	−13.1433	−1.49782
$78$	0	0
$79$	−8.47214	−0.953190	−0.476595	−	0.879123i	$-0.658129\pi$
$79$	−8.47214	−0.953190	−0.476595	+	0.879123i	$0.658129\pi$
$80$	0	0
$81$	0	0
$82$	−9.51057	−1.05027
$83$	−8.22899	−0.903249	−0.451625	−	0.892208i	$-0.649155\pi$
$83$	−8.22899	−0.903249	−0.451625	+	0.892208i	$0.649155\pi$
$84$	0	0
$85$	0	0
$86$	−1.18034	−0.127279
$87$	0	0
$88$	15.3884	1.64041
$89$	−11.1803	−1.18511	−0.592557	−	0.805529i	$-0.701881\pi$
$89$	−11.1803	−1.18511	−0.592557	+	0.805529i	$0.701881\pi$
$90$	0	0
$91$	−18.0902	−1.89637
$92$	−3.35520	−0.349804
$93$	0	0
$94$	9.79837	1.01063
$95$	0	0
$96$	0	0
$97$	14.3844	1.46051	0.730255	−	0.683174i	$-0.239401\pi$
$97$	14.3844	1.46051	0.730255	+	0.683174i	$0.239401\pi$
$98$	−0.106001	−0.0107077
$99$	0	0
$100$	0	0
$101$	6.18034	0.614967	0.307483	−	0.951553i	$-0.400513\pi$
$101$	6.18034	0.614967	0.307483	+	0.951553i	$0.400513\pi$
$102$	0	0
$103$	12.7598	1.25726	0.628628	−	0.777706i	$-0.283617\pi$
$103$	12.7598	1.25726	0.628628	+	0.777706i	$0.283617\pi$
$104$	21.1803	2.07690
$105$	0	0
$106$	−8.41641	−0.817474
$107$	8.61251	0.832603	0.416301	−	0.909227i	$-0.363326\pi$
$107$	8.61251	0.832603	0.416301	+	0.909227i	$0.363326\pi$
$108$	0	0
$109$	−10.7082	−1.02566	−0.512830	−	0.858490i	$-0.671403\pi$
$109$	−10.7082	−1.02566	−0.512830	+	0.858490i	$0.671403\pi$
$110$	0	0
$111$	0	0
$112$	−6.26137	−0.591644
$113$	−9.40456	−0.884707	−0.442353	−	0.896841i	$-0.645856\pi$
$113$	−9.40456	−0.884707	−0.442353	+	0.896841i	$0.645856\pi$
$114$	0	0
$115$	0	0
$116$	3.09017	0.286915
$117$	0	0
$118$	−2.24514	−0.206682
$119$	−13.0902	−1.19997
$120$	0	0
$121$	14.0000	1.27273
$122$	8.61251	0.779740
$123$	0	0
$124$	1.76393	0.158406
$125$	0	0
$126$	0	0
$127$	−12.1392	−1.07718	−0.538591	−	0.842567i	$-0.681043\pi$
$127$	−12.1392	−1.07718	−0.538591	+	0.842567i	$0.681043\pi$
$128$	3.52671	0.311720
$129$	0	0
$130$	0	0
$131$	−1.90983	−0.166863	−0.0834313	−	0.996514i	$-0.526588\pi$
$131$	−1.90983	−0.166863	−0.0834313	+	0.996514i	$0.526588\pi$
$132$	0	0
$133$	9.12705	0.791416
$134$	10.0000	0.863868
$135$	0	0
$136$	15.3262	1.31421
$137$	15.3884	1.31472	0.657361	−	0.753576i	$-0.271673\pi$
$137$	15.3884	1.31472	0.657361	+	0.753576i	$0.271673\pi$
$138$	0	0
$139$	0.708204	0.0600691	0.0300345	−	0.999549i	$-0.490438\pi$
$139$	0.708204	0.0600691	0.0300345	+	0.999549i	$0.490438\pi$
$140$	0	0
$141$	0	0
$142$	−15.3884	−1.29137
$143$	34.4095	2.87747
$144$	0	0
$145$	0	0
$146$	5.00000	0.413803
$147$	0	0
$148$	1.00406	0.0825330
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	6.94427	0.565117	0.282558	−	0.959250i	$-0.408817\pi$
$151$	6.94427	0.565117	0.282558	+	0.959250i	$0.408817\pi$
$152$	−10.6861	−0.866760
$153$	0	0
$154$	−15.4508	−1.24506
$155$	0	0
$156$	0	0
$157$	5.87785	0.469104	0.234552	−	0.972104i	$-0.424638\pi$
$157$	5.87785	0.469104	0.234552	+	0.972104i	$0.424638\pi$
$158$	−9.95959	−0.792343
$159$	0	0
$160$	0	0
$161$	14.2705	1.12467
$162$	0	0
$163$	3.63271	0.284536	0.142268	−	0.989828i	$-0.454561\pi$
$163$	3.63271	0.284536	0.142268	+	0.989828i	$0.454561\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−9.67376	−0.750830
$167$	−7.05342	−0.545810	−0.272905	−	0.962041i	$-0.587985\pi$
$167$	−7.05342	−0.545810	−0.272905	+	0.962041i	$0.587985\pi$
$168$	0	0
$169$	34.3607	2.64313
$170$	0	0
$171$	0	0
$172$	0.620541	0.0473158
$173$	5.87785	0.446885	0.223442	−	0.974717i	$-0.428271\pi$
$173$	5.87785	0.446885	0.223442	+	0.974717i	$0.428271\pi$
$174$	0	0
$175$	0	0
$176$	11.9098	0.897737
$177$	0	0
$178$	−13.1433	−0.985130
$179$	−21.1803	−1.58309	−0.791546	−	0.611109i	$-0.790724\pi$
$179$	−21.1803	−1.58309	−0.791546	+	0.611109i	$0.790724\pi$
$180$	0	0
$181$	−9.23607	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$181$	−9.23607	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$182$	−21.2663	−1.57636
$183$	0	0
$184$	−16.7082	−1.23175
$185$	0	0
$186$	0	0
$187$	24.8990	1.82079
$188$	−5.15131	−0.375698
$189$	0	0
$190$	0	0
$191$	11.1803	0.808981	0.404491	−	0.914542i	$-0.367449\pi$
$191$	11.1803	0.808981	0.404491	+	0.914542i	$0.367449\pi$
$192$	0	0
$193$	15.3884	1.10768	0.553841	−	0.832622i	$-0.313161\pi$
$193$	15.3884	1.10768	0.553841	+	0.832622i	$0.313161\pi$
$194$	16.9098	1.21406
$195$	0	0
$196$	0.0557281	0.00398058
$197$	4.53077	0.322804	0.161402	−	0.986889i	$-0.448398\pi$
$197$	4.53077	0.322804	0.161402	+	0.986889i	$0.448398\pi$
$198$	0	0
$199$	5.47214	0.387909	0.193955	−	0.981010i	$-0.437869\pi$
$199$	5.47214	0.387909	0.193955	+	0.981010i	$0.437869\pi$
$200$	0	0
$201$	0	0
$202$	7.26543	0.511194
$203$	−13.1433	−0.922477
$204$	0	0
$205$	0	0
$206$	15.0000	1.04510
$207$	0	0
$208$	16.3925	1.13661
$209$	−17.3607	−1.20086
$210$	0	0
$211$	14.7639	1.01639	0.508195	−	0.861242i	$-0.330313\pi$
$211$	14.7639	1.01639	0.508195	+	0.861242i	$0.330313\pi$
$212$	4.42477	0.303894
$213$	0	0
$214$	10.1246	0.692104
$215$	0	0
$216$	0	0
$217$	−7.50245	−0.509300
$218$	−12.5882	−0.852584
$219$	0	0
$220$	0	0
$221$	34.2705	2.30528
$222$	0	0
$223$	−25.9030	−1.73460	−0.867298	−	0.497789i	$-0.834145\pi$
$223$	−25.9030	−1.73460	−0.867298	+	0.497789i	$0.834145\pi$
$224$	8.81966	0.589288
$225$	0	0
$226$	−11.0557	−0.735416
$227$	−8.16348	−0.541829	−0.270915	−	0.962603i	$-0.587326\pi$
$227$	−8.16348	−0.541829	−0.270915	+	0.962603i	$0.587326\pi$
$228$	0	0
$229$	−26.6525	−1.76125	−0.880623	−	0.473818i	$-0.842875\pi$
$229$	−26.6525	−1.76125	−0.880623	+	0.473818i	$0.842875\pi$
$230$	0	0
$231$	0	0
$232$	15.3884	1.01030
$233$	−2.24514	−0.147084	−0.0735420	−	0.997292i	$-0.523430\pi$
$233$	−2.24514	−0.147084	−0.0735420	+	0.997292i	$0.523430\pi$
$234$	0	0
$235$	0	0
$236$	1.18034	0.0768336
$237$	0	0
$238$	−15.3884	−0.997483
$239$	14.2705	0.923083	0.461541	−	0.887119i	$-0.347297\pi$
$239$	14.2705	0.923083	0.461541	+	0.887119i	$0.347297\pi$
$240$	0	0
$241$	−27.5967	−1.77766	−0.888831	−	0.458234i	$-0.848482\pi$
$241$	−27.5967	−1.77766	−0.888831	+	0.458234i	$0.848482\pi$
$242$	16.4580	1.05796
$243$	0	0
$244$	−4.52786	−0.289867
$245$	0	0
$246$	0	0
$247$	−23.8949	−1.52040
$248$	8.78402	0.557786
$249$	0	0
$250$	0	0
$251$	−3.09017	−0.195050	−0.0975249	−	0.995233i	$-0.531093\pi$
$251$	−3.09017	−0.195050	−0.0975249	+	0.995233i	$0.531093\pi$
$252$	0	0
$253$	−27.1441	−1.70654
$254$	−14.2705	−0.895411
$255$	0	0
$256$	−13.2705	−0.829407
$257$	22.4418	1.39988	0.699942	−	0.714200i	$-0.253209\pi$
$257$	22.4418	1.39988	0.699942	+	0.714200i	$0.253209\pi$
$258$	0	0
$259$	−4.27051	−0.265357
$260$	0	0
$261$	0	0
$262$	−2.24514	−0.138705
$263$	−4.08174	−0.251691	−0.125845	−	0.992050i	$-0.540164\pi$
$263$	−4.08174	−0.251691	−0.125845	+	0.992050i	$0.540164\pi$
$264$	0	0
$265$	0	0
$266$	10.7295	0.657867
$267$	0	0
$268$	−5.25731	−0.321141
$269$	−8.09017	−0.493266	−0.246633	−	0.969109i	$-0.579324\pi$
$269$	−8.09017	−0.493266	−0.246633	+	0.969109i	$0.579324\pi$
$270$	0	0
$271$	17.8541	1.08456	0.542280	−	0.840198i	$-0.317561\pi$
$271$	17.8541	1.08456	0.542280	+	0.840198i	$0.317561\pi$
$272$	11.8617	0.719222
$273$	0	0
$274$	18.0902	1.09287
$275$	0	0
$276$	0	0
$277$	6.26137	0.376209	0.188105	−	0.982149i	$-0.439766\pi$
$277$	6.26137	0.376209	0.188105	+	0.982149i	$0.439766\pi$
$278$	0.832544	0.0499327
$279$	0	0
$280$	0	0
$281$	29.2705	1.74613	0.873066	−	0.487602i	$-0.162128\pi$
$281$	29.2705	1.74613	0.873066	+	0.487602i	$0.162128\pi$
$282$	0	0
$283$	−15.3884	−0.914746	−0.457373	−	0.889275i	$-0.651210\pi$
$283$	−15.3884	−0.914746	−0.457373	+	0.889275i	$0.651210\pi$
$284$	8.09017	0.480063
$285$	0	0
$286$	40.4508	2.39191
$287$	−21.2663	−1.25531
$288$	0	0
$289$	7.79837	0.458728
$290$	0	0
$291$	0	0
$292$	−2.62866	−0.153830
$293$	−9.06154	−0.529381	−0.264690	−	0.964333i	$-0.585270\pi$
$293$	−9.06154	−0.529381	−0.264690	+	0.964333i	$0.585270\pi$
$294$	0	0
$295$	0	0
$296$	5.00000	0.290619
$297$	0	0
$298$	5.87785	0.340495
$299$	−37.3607	−2.16062
$300$	0	0
$301$	−2.63932	−0.152128
$302$	8.16348	0.469756
$303$	0	0
$304$	−8.27051	−0.474346
$305$	0	0
$306$	0	0
$307$	−21.8868	−1.24915	−0.624573	−	0.780966i	$-0.714727\pi$
$307$	−21.8868	−1.24915	−0.624573	+	0.780966i	$0.714727\pi$
$308$	8.12299	0.462850
$309$	0	0
$310$	0	0
$311$	−9.27051	−0.525682	−0.262841	−	0.964839i	$-0.584660\pi$
$311$	−9.27051	−0.525682	−0.262841	+	0.964839i	$0.584660\pi$
$312$	0	0
$313$	−7.88597	−0.445741	−0.222871	−	0.974848i	$-0.571543\pi$
$313$	−7.88597	−0.445741	−0.222871	+	0.974848i	$0.571543\pi$
$314$	6.90983	0.389944
$315$	0	0
$316$	5.23607	0.294552
$317$	−27.1441	−1.52457	−0.762283	−	0.647244i	$-0.775921\pi$
$317$	−27.1441	−1.52457	−0.762283	+	0.647244i	$0.775921\pi$
$318$	0	0
$319$	25.0000	1.39973
$320$	0	0
$321$	0	0
$322$	16.7760	0.934889
$323$	−17.2905	−0.962071
$324$	0	0
$325$	0	0
$326$	4.27051	0.236522
$327$	0	0
$328$	24.8990	1.37482
$329$	21.9098	1.20793
$330$	0	0
$331$	2.32624	0.127862	0.0639308	−	0.997954i	$-0.479636\pi$
$331$	2.32624	0.127862	0.0639308	+	0.997954i	$0.479636\pi$
$332$	5.08580	0.279119
$333$	0	0
$334$	−8.29180	−0.453707
$335$	0	0
$336$	0	0
$337$	−22.2703	−1.21314	−0.606571	−	0.795029i	$-0.707455\pi$
$337$	−22.2703	−1.21314	−0.606571	+	0.795029i	$0.707455\pi$
$338$	40.3934	2.19711
$339$	0	0
$340$	0	0
$341$	14.2705	0.772791
$342$	0	0
$343$	−18.6376	−1.00634
$344$	3.09017	0.166611
$345$	0	0
$346$	6.90983	0.371475
$347$	−33.2340	−1.78409	−0.892047	−	0.451943i	$-0.850731\pi$
$347$	−33.2340	−1.78409	−0.892047	+	0.451943i	$0.850731\pi$
$348$	0	0
$349$	−23.5623	−1.26126	−0.630631	−	0.776083i	$-0.717204\pi$
$349$	−23.5623	−1.26126	−0.630631	+	0.776083i	$0.717204\pi$
$350$	0	0
$351$	0	0
$352$	−16.7760	−0.894163
$353$	15.8374	0.842942	0.421471	−	0.906842i	$-0.361514\pi$
$353$	15.8374	0.842942	0.421471	+	0.906842i	$0.361514\pi$
$354$	0	0
$355$	0	0
$356$	6.90983	0.366220
$357$	0	0
$358$	−24.8990	−1.31595
$359$	6.18034	0.326186	0.163093	−	0.986611i	$-0.447853\pi$
$359$	6.18034	0.326186	0.163093	+	0.986611i	$0.447853\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	−10.8576	−0.570665
$363$	0	0
$364$	11.1803	0.586009
$365$	0	0
$366$	0	0
$367$	26.5236	1.38452	0.692260	−	0.721648i	$-0.256615\pi$
$367$	26.5236	1.38452	0.692260	+	0.721648i	$0.256615\pi$
$368$	−12.9313	−0.674089
$369$	0	0
$370$	0	0
$371$	−18.8197	−0.977068
$372$	0	0
$373$	16.7760	0.868628	0.434314	−	0.900762i	$-0.356991\pi$
$373$	16.7760	0.868628	0.434314	+	0.900762i	$0.356991\pi$
$374$	29.2705	1.51354
$375$	0	0
$376$	−25.6525	−1.32293
$377$	34.4095	1.77218
$378$	0	0
$379$	−14.0902	−0.723763	−0.361882	−	0.932224i	$-0.617866\pi$
$379$	−14.0902	−0.723763	−0.361882	+	0.932224i	$0.617866\pi$
$380$	0	0
$381$	0	0
$382$	13.1433	0.672469
$383$	14.9394	0.763367	0.381684	−	0.924293i	$-0.375344\pi$
$383$	14.9394	0.763367	0.381684	+	0.924293i	$0.375344\pi$
$384$	0	0
$385$	0	0
$386$	18.0902	0.920765
$387$	0	0
$388$	−8.89002	−0.451323
$389$	−35.4508	−1.79743	−0.898715	−	0.438534i	$-0.855498\pi$
$389$	−35.4508	−1.79743	−0.898715	+	0.438534i	$0.855498\pi$
$390$	0	0
$391$	−27.0344	−1.36719
$392$	0.277515	0.0140166
$393$	0	0
$394$	5.32624	0.268332
$395$	0	0
$396$	0	0
$397$	5.25731	0.263857	0.131928	−	0.991259i	$-0.457883\pi$
$397$	5.25731	0.263857	0.131928	+	0.991259i	$0.457883\pi$
$398$	6.43288	0.322451
$399$	0	0
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	19.6417	0.978421
$404$	−3.81966	−0.190035
$405$	0	0
$406$	−15.4508	−0.766813
$407$	8.12299	0.402642
$408$	0	0
$409$	−1.52786	−0.0755480	−0.0377740	−	0.999286i	$-0.512027\pi$
$409$	−1.52786	−0.0755480	−0.0377740	+	0.999286i	$0.512027\pi$
$410$	0	0
$411$	0	0
$412$	−7.88597	−0.388514
$413$	−5.02029	−0.247032
$414$	0	0
$415$	0	0
$416$	−23.0902	−1.13209
$417$	0	0
$418$	−20.4087	−0.998222
$419$	19.2705	0.941426	0.470713	−	0.882286i	$-0.343997\pi$
$419$	19.2705	0.941426	0.470713	+	0.882286i	$0.343997\pi$
$420$	0	0
$421$	0.763932	0.0372318	0.0186159	−	0.999827i	$-0.494074\pi$
$421$	0.763932	0.0372318	0.0186159	+	0.999827i	$0.494074\pi$
$422$	17.3560	0.844879
$423$	0	0
$424$	22.0344	1.07009
$425$	0	0
$426$	0	0
$427$	19.2582	0.931967
$428$	−5.32282	−0.257288
$429$	0	0
$430$	0	0
$431$	−41.1803	−1.98359	−0.991794	−	0.127849i	$-0.959193\pi$
$431$	−41.1803	−1.98359	−0.991794	+	0.127849i	$0.959193\pi$
$432$	0	0
$433$	−6.49839	−0.312293	−0.156146	−	0.987734i	$-0.549907\pi$
$433$	−6.49839	−0.312293	−0.156146	+	0.987734i	$0.549907\pi$
$434$	−8.81966	−0.423357
$435$	0	0
$436$	6.61803	0.316946
$437$	18.8496	0.901699
$438$	0	0
$439$	−27.0902	−1.29294	−0.646472	−	0.762938i	$-0.723756\pi$
$439$	−27.0902	−1.29294	−0.646472	+	0.762938i	$0.723756\pi$
$440$	0	0
$441$	0	0
$442$	40.2874	1.91628
$443$	−27.2501	−1.29469	−0.647346	−	0.762196i	$-0.724121\pi$
$443$	−27.2501	−1.29469	−0.647346	+	0.762196i	$0.724121\pi$
$444$	0	0
$445$	0	0
$446$	−30.4508	−1.44189
$447$	0	0
$448$	22.8909	1.08149
$449$	−0.729490	−0.0344268	−0.0172134	−	0.999852i	$-0.505479\pi$
$449$	−0.729490	−0.0344268	−0.0172134	+	0.999852i	$0.505479\pi$
$450$	0	0
$451$	40.4508	1.90476
$452$	5.81234	0.273389
$453$	0	0
$454$	−9.59675	−0.450398
$455$	0	0
$456$	0	0
$457$	−18.4006	−0.860743	−0.430372	−	0.902652i	$-0.641617\pi$
$457$	−18.4006	−0.860743	−0.430372	+	0.902652i	$0.641617\pi$
$458$	−31.3319	−1.46404
$459$	0	0
$460$	0	0
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	35.4136	1.64581	0.822905	−	0.568179i	$-0.192352\pi$
$463$	35.4136	1.64581	0.822905	+	0.568179i	$0.192352\pi$
$464$	11.9098	0.552900
$465$	0	0
$466$	−2.63932	−0.122264
$467$	16.5640	0.766490	0.383245	−	0.923647i	$-0.374807\pi$
$467$	16.5640	0.766490	0.383245	+	0.923647i	$0.374807\pi$
$468$	0	0
$469$	22.3607	1.03252
$470$	0	0
$471$	0	0
$472$	5.87785	0.270550
$473$	5.02029	0.230833
$474$	0	0
$475$	0	0
$476$	8.09017	0.370812
$477$	0	0
$478$	16.7760	0.767316
$479$	12.3607	0.564774	0.282387	−	0.959301i	$-0.408874\pi$
$479$	12.3607	0.564774	0.282387	+	0.959301i	$0.408874\pi$
$480$	0	0
$481$	11.1803	0.509780
$482$	−32.4419	−1.47769
$483$	0	0
$484$	−8.65248	−0.393294
$485$	0	0
$486$	0	0
$487$	−19.0211	−0.861930	−0.430965	−	0.902369i	$-0.641827\pi$
$487$	−19.0211	−0.861930	−0.430965	+	0.902369i	$0.641827\pi$
$488$	−22.5478	−1.02069
$489$	0	0
$490$	0	0
$491$	−23.0902	−1.04204	−0.521022	−	0.853543i	$-0.674449\pi$
$491$	−23.0902	−1.04204	−0.521022	+	0.853543i	$0.674449\pi$
$492$	0	0
$493$	24.8990	1.12139
$494$	−28.0902	−1.26384
$495$	0	0
$496$	6.79837	0.305256
$497$	−34.4095	−1.54348
$498$	0	0
$499$	6.52786	0.292227	0.146114	−	0.989268i	$-0.453323\pi$
$499$	6.52786	0.292227	0.146114	+	0.989268i	$0.453323\pi$
$500$	0	0
$501$	0	0
$502$	−3.63271	−0.162136
$503$	9.51057	0.424055	0.212028	−	0.977264i	$-0.431993\pi$
$503$	9.51057	0.424055	0.212028	+	0.977264i	$0.431993\pi$
$504$	0	0
$505$	0	0
$506$	−31.9098	−1.41856
$507$	0	0
$508$	7.50245	0.332867
$509$	13.8197	0.612546	0.306273	−	0.951944i	$-0.400918\pi$
$509$	13.8197	0.612546	0.306273	+	0.951944i	$0.400918\pi$
$510$	0	0
$511$	11.1803	0.494589
$512$	−22.6538	−1.00117
$513$	0	0
$514$	26.3820	1.16366
$515$	0	0
$516$	0	0
$517$	−41.6750	−1.83286
$518$	−5.02029	−0.220579
$519$	0	0
$520$	0	0
$521$	6.90983	0.302725	0.151363	−	0.988478i	$-0.451634\pi$
$521$	6.90983	0.302725	0.151363	+	0.988478i	$0.451634\pi$
$522$	0	0
$523$	17.0130	0.743927	0.371964	−	0.928247i	$-0.378685\pi$
$523$	17.0130	0.743927	0.371964	+	0.928247i	$0.378685\pi$
$524$	1.18034	0.0515634
$525$	0	0
$526$	−4.79837	−0.209219
$527$	14.2128	0.619121
$528$	0	0
$529$	6.47214	0.281397
$530$	0	0
$531$	0	0
$532$	−5.64083	−0.244561
$533$	55.6758	2.41159
$534$	0	0
$535$	0	0
$536$	−26.1803	−1.13082
$537$	0	0
$538$	−9.51057	−0.410030
$539$	0.450850	0.0194195
$540$	0	0
$541$	−9.50658	−0.408720	−0.204360	−	0.978896i	$-0.565511\pi$
$541$	−9.50658	−0.408720	−0.204360	+	0.978896i	$0.565511\pi$
$542$	20.9888	0.901544
$543$	0	0
$544$	−16.7082	−0.716358
$545$	0	0
$546$	0	0
$547$	−38.6628	−1.65310	−0.826551	−	0.562862i	$-0.809700\pi$
$547$	−38.6628	−1.65310	−0.826551	+	0.562862i	$0.809700\pi$
$548$	−9.51057	−0.406271
$549$	0	0
$550$	0	0
$551$	−17.3607	−0.739590
$552$	0	0
$553$	−22.2703	−0.947031
$554$	7.36068	0.312725
$555$	0	0
$556$	−0.437694	−0.0185624
$557$	−40.2874	−1.70703	−0.853516	−	0.521067i	$-0.825534\pi$
$557$	−40.2874	−1.70703	−0.853516	+	0.521067i	$0.825534\pi$
$558$	0	0
$559$	6.90983	0.292255
$560$	0	0
$561$	0	0
$562$	34.4095	1.45148
$563$	−18.5721	−0.782721	−0.391360	−	0.920237i	$-0.627995\pi$
$563$	−18.5721	−0.782721	−0.391360	+	0.920237i	$0.627995\pi$
$564$	0	0
$565$	0	0
$566$	−18.0902	−0.760387
$567$	0	0
$568$	40.2874	1.69042
$569$	−17.3607	−0.727798	−0.363899	−	0.931438i	$-0.618555\pi$
$569$	−17.3607	−0.727798	−0.363899	+	0.931438i	$0.618555\pi$
$570$	0	0
$571$	−7.27051	−0.304261	−0.152131	−	0.988360i	$-0.548613\pi$
$571$	−7.27051	−0.304261	−0.152131	+	0.988360i	$0.548613\pi$
$572$	−21.2663	−0.889187
$573$	0	0
$574$	−25.0000	−1.04348
$575$	0	0
$576$	0	0
$577$	8.89002	0.370097	0.185048	−	0.982729i	$-0.440756\pi$
$577$	8.89002	0.370097	0.185048	+	0.982729i	$0.440756\pi$
$578$	9.16754	0.381319
$579$	0	0
$580$	0	0
$581$	−21.6312	−0.897413
$582$	0	0
$583$	35.7971	1.48257
$584$	−13.0902	−0.541675
$585$	0	0
$586$	−10.6525	−0.440050
$587$	33.5115	1.38317	0.691584	−	0.722296i	$-0.256913\pi$
$587$	33.5115	1.38317	0.691584	+	0.722296i	$0.256913\pi$
$588$	0	0
$589$	−9.90983	−0.408327
$590$	0	0
$591$	0	0
$592$	3.86974	0.159045
$593$	13.2493	0.544083	0.272041	−	0.962286i	$-0.412301\pi$
$593$	13.2493	0.544083	0.272041	+	0.962286i	$0.412301\pi$
$594$	0	0
$595$	0	0
$596$	−3.09017	−0.126578
$597$	0	0
$598$	−43.9201	−1.79603
$599$	14.2705	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$599$	14.2705	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$600$	0	0
$601$	−2.12461	−0.0866647	−0.0433324	−	0.999061i	$-0.513797\pi$
$601$	−2.12461	−0.0866647	−0.0433324	+	0.999061i	$0.513797\pi$
$602$	−3.10271	−0.126457
$603$	0	0
$604$	−4.29180	−0.174631
$605$	0	0
$606$	0	0
$607$	22.2703	0.903925	0.451962	−	0.892037i	$-0.350724\pi$
$607$	22.2703	0.903925	0.451962	+	0.892037i	$0.350724\pi$
$608$	11.6497	0.472458
$609$	0	0
$610$	0	0
$611$	−57.3607	−2.32056
$612$	0	0
$613$	−8.89002	−0.359065	−0.179532	−	0.983752i	$-0.557458\pi$
$613$	−8.89002	−0.359065	−0.179532	+	0.983752i	$0.557458\pi$
$614$	−25.7295	−1.03836
$615$	0	0
$616$	40.4508	1.62981
$617$	40.7769	1.64162	0.820808	−	0.571204i	$-0.193523\pi$
$617$	40.7769	1.64162	0.820808	+	0.571204i	$0.193523\pi$
$618$	0	0
$619$	6.65248	0.267386	0.133693	−	0.991023i	$-0.457316\pi$
$619$	6.65248	0.267386	0.133693	+	0.991023i	$0.457316\pi$
$620$	0	0
$621$	0	0
$622$	−10.8981	−0.436976
$623$	−29.3893	−1.17746
$624$	0	0
$625$	0	0
$626$	−9.27051	−0.370524
$627$	0	0
$628$	−3.63271	−0.144961
$629$	8.09017	0.322576
$630$	0	0
$631$	−29.1803	−1.16165	−0.580825	−	0.814028i	$-0.697270\pi$
$631$	−29.1803	−1.16165	−0.580825	+	0.814028i	$0.697270\pi$
$632$	26.0746	1.03719
$633$	0	0
$634$	−31.9098	−1.26730
$635$	0	0
$636$	0	0
$637$	0.620541	0.0245867
$638$	29.3893	1.16353
$639$	0	0
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	−6.49839	−0.256272	−0.128136	−	0.991757i	$-0.540899\pi$
$643$	−6.49839	−0.256272	−0.128136	+	0.991757i	$0.540899\pi$
$644$	−8.81966	−0.347543
$645$	0	0
$646$	−20.3262	−0.799725
$647$	−1.38757	−0.0545511	−0.0272756	−	0.999628i	$-0.508683\pi$
$647$	−1.38757	−0.0545511	−0.0272756	+	0.999628i	$0.508683\pi$
$648$	0	0
$649$	9.54915	0.374837
$650$	0	0
$651$	0	0
$652$	−2.24514	−0.0879265
$653$	−2.35114	−0.0920073	−0.0460036	−	0.998941i	$-0.514649\pi$
$653$	−2.35114	−0.0920073	−0.0460036	+	0.998941i	$0.514649\pi$
$654$	0	0
$655$	0	0
$656$	19.2705	0.752387
$657$	0	0
$658$	25.7565	1.00409
$659$	−8.81966	−0.343565	−0.171783	−	0.985135i	$-0.554953\pi$
$659$	−8.81966	−0.343565	−0.171783	+	0.985135i	$0.554953\pi$
$660$	0	0
$661$	−32.8541	−1.27788	−0.638938	−	0.769258i	$-0.720626\pi$
$661$	−32.8541	−1.27788	−0.638938	+	0.769258i	$0.720626\pi$
$662$	2.73466	0.106285
$663$	0	0
$664$	25.3262	0.982849
$665$	0	0
$666$	0	0
$667$	−27.1441	−1.05102
$668$	4.35926	0.168665
$669$	0	0
$670$	0	0
$671$	−36.6312	−1.41413
$672$	0	0
$673$	7.26543	0.280062	0.140031	−	0.990147i	$-0.455280\pi$
$673$	7.26543	0.280062	0.140031	+	0.990147i	$0.455280\pi$
$674$	−26.1803	−1.00843
$675$	0	0
$676$	−21.2361	−0.816772
$677$	−4.70228	−0.180723	−0.0903617	−	0.995909i	$-0.528802\pi$
$677$	−4.70228	−0.180723	−0.0903617	+	0.995909i	$0.528802\pi$
$678$	0	0
$679$	37.8115	1.45107
$680$	0	0
$681$	0	0
$682$	16.7760	0.642386
$683$	44.3691	1.69774	0.848869	−	0.528603i	$-0.177284\pi$
$683$	44.3691	1.69774	0.848869	+	0.528603i	$0.177284\pi$
$684$	0	0
$685$	0	0
$686$	−21.9098	−0.836521
$687$	0	0
$688$	2.39163	0.0911800
$689$	49.2705	1.87706
$690$	0	0
$691$	−5.94427	−0.226131	−0.113065	−	0.993588i	$-0.536067\pi$
$691$	−5.94427	−0.226131	−0.113065	+	0.993588i	$0.536067\pi$
$692$	−3.63271	−0.138095
$693$	0	0
$694$	−39.0689	−1.48303
$695$	0	0
$696$	0	0
$697$	40.2874	1.52599
$698$	−27.6992	−1.04843
$699$	0	0
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	0	0
$703$	−5.64083	−0.212748
$704$	−43.5410	−1.64101
$705$	0	0
$706$	18.6180	0.700699
$707$	16.2460	0.610993
$708$	0	0
$709$	37.2918	1.40052	0.700261	−	0.713887i	$-0.253067\pi$
$709$	37.2918	1.40052	0.700261	+	0.713887i	$0.253067\pi$
$710$	0	0
$711$	0	0
$712$	34.4095	1.28955
$713$	−15.4944	−0.580271
$714$	0	0
$715$	0	0
$716$	13.0902	0.489203
$717$	0	0
$718$	7.26543	0.271143
$719$	37.3607	1.39332	0.696659	−	0.717402i	$-0.254669\pi$
$719$	37.3607	1.39332	0.696659	+	0.717402i	$0.254669\pi$
$720$	0	0
$721$	33.5410	1.24913
$722$	−8.16348	−0.303813
$723$	0	0
$724$	5.70820	0.212144
$725$	0	0
$726$	0	0
$727$	−18.6376	−0.691231	−0.345615	−	0.938376i	$-0.612330\pi$
$727$	−18.6376	−0.691231	−0.345615	+	0.938376i	$0.612330\pi$
$728$	55.6758	2.06348
$729$	0	0
$730$	0	0
$731$	5.00000	0.184932
$732$	0	0
$733$	−7.26543	−0.268355	−0.134177	−	0.990957i	$-0.542839\pi$
$733$	−7.26543	−0.268355	−0.134177	+	0.990957i	$0.542839\pi$
$734$	31.1803	1.15089
$735$	0	0
$736$	18.2148	0.671406
$737$	−42.5325	−1.56671
$738$	0	0
$739$	−37.1033	−1.36487	−0.682434	−	0.730947i	$-0.739079\pi$
$739$	−37.1033	−1.36487	−0.682434	+	0.730947i	$0.739079\pi$
$740$	0	0
$741$	0	0
$742$	−22.1238	−0.812192
$743$	−38.0423	−1.39564	−0.697818	−	0.716276i	$-0.745845\pi$
$743$	−38.0423	−1.39564	−0.697818	+	0.716276i	$0.745845\pi$
$744$	0	0
$745$	0	0
$746$	19.7214	0.722050
$747$	0	0
$748$	−15.3884	−0.562656
$749$	22.6393	0.827223
$750$	0	0
$751$	25.0344	0.913520	0.456760	−	0.889590i	$-0.349010\pi$
$751$	25.0344	0.913520	0.456760	+	0.889590i	$0.349010\pi$
$752$	−19.8537	−0.723989
$753$	0	0
$754$	40.4508	1.47313
$755$	0	0
$756$	0	0
$757$	37.6587	1.36873	0.684365	−	0.729139i	$-0.260079\pi$
$757$	37.6587	1.36873	0.684365	+	0.729139i	$0.260079\pi$
$758$	−16.5640	−0.601631
$759$	0	0
$760$	0	0
$761$	43.5410	1.57836	0.789180	−	0.614162i	$-0.210506\pi$
$761$	43.5410	1.57836	0.789180	+	0.614162i	$0.210506\pi$
$762$	0	0
$763$	−28.1482	−1.01903
$764$	−6.90983	−0.249989
$765$	0	0
$766$	17.5623	0.634552
$767$	13.1433	0.474576
$768$	0	0
$769$	−14.6525	−0.528382	−0.264191	−	0.964470i	$-0.585105\pi$
$769$	−14.6525	−0.528382	−0.264191	+	0.964470i	$0.585105\pi$
$770$	0	0
$771$	0	0
$772$	−9.51057	−0.342293
$773$	−51.0795	−1.83720	−0.918602	−	0.395185i	$-0.870681\pi$
$773$	−51.0795	−1.83720	−0.918602	+	0.395185i	$0.870681\pi$
$774$	0	0
$775$	0	0
$776$	−44.2705	−1.58922
$777$	0	0
$778$	−41.6750	−1.49412
$779$	−28.0902	−1.00643
$780$	0	0
$781$	65.4508	2.34202
$782$	−31.7809	−1.13648
$783$	0	0
$784$	0.214782	0.00767078
$785$	0	0
$786$	0	0
$787$	2.24514	0.0800306	0.0400153	−	0.999199i	$-0.487259\pi$
$787$	2.24514	0.0800306	0.0400153	+	0.999199i	$0.487259\pi$
$788$	−2.80017	−0.0997519
$789$	0	0
$790$	0	0
$791$	−24.7214	−0.878990
$792$	0	0
$793$	−50.4185	−1.79041
$794$	6.18034	0.219332
$795$	0	0
$796$	−3.38197	−0.119871
$797$	−29.4298	−1.04246	−0.521228	−	0.853418i	$-0.674526\pi$
$797$	−29.4298	−1.04246	−0.521228	+	0.853418i	$0.674526\pi$
$798$	0	0
$799$	−41.5066	−1.46840
$800$	0	0
$801$	0	0
$802$	−5.87785	−0.207554
$803$	−21.2663	−0.750470
$804$	0	0
$805$	0	0
$806$	23.0902	0.813317
$807$	0	0
$808$	−19.0211	−0.669161
$809$	−21.9098	−0.770309	−0.385154	−	0.922852i	$-0.625852\pi$
$809$	−21.9098	−0.770309	−0.385154	+	0.922852i	$0.625852\pi$
$810$	0	0
$811$	−54.6869	−1.92032	−0.960159	−	0.279455i	$-0.909846\pi$
$811$	−54.6869	−1.92032	−0.960159	+	0.279455i	$0.909846\pi$
$812$	8.12299	0.285061
$813$	0	0
$814$	9.54915	0.334698
$815$	0	0
$816$	0	0
$817$	−3.48622	−0.121967
$818$	−1.79611	−0.0627996
$819$	0	0
$820$	0	0
$821$	33.5410	1.17059	0.585295	−	0.810821i	$-0.300979\pi$
$821$	33.5410	1.17059	0.585295	+	0.810821i	$0.300979\pi$
$822$	0	0
$823$	−29.5358	−1.02955	−0.514776	−	0.857325i	$-0.672125\pi$
$823$	−29.5358	−1.02955	−0.514776	+	0.857325i	$0.672125\pi$
$824$	−39.2705	−1.36805
$825$	0	0
$826$	−5.90170	−0.205346
$827$	20.1967	0.702308	0.351154	−	0.936318i	$-0.385789\pi$
$827$	20.1967	0.702308	0.351154	+	0.936318i	$0.385789\pi$
$828$	0	0
$829$	−27.7082	−0.962346	−0.481173	−	0.876626i	$-0.659789\pi$
$829$	−27.7082	−0.962346	−0.481173	+	0.876626i	$0.659789\pi$
$830$	0	0
$831$	0	0
$832$	−59.9291	−2.07767
$833$	0.449028	0.0155579
$834$	0	0
$835$	0	0
$836$	10.7295	0.371087
$837$	0	0
$838$	22.6538	0.782564
$839$	25.0000	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$839$	25.0000	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	0.898056	0.0309491
$843$	0	0
$844$	−9.12461	−0.314082
$845$	0	0
$846$	0	0
$847$	36.8012	1.26450
$848$	17.0535	0.585620
$849$	0	0
$850$	0	0
$851$	−8.81966	−0.302334
$852$	0	0
$853$	27.9112	0.955660	0.477830	−	0.878452i	$-0.341424\pi$
$853$	27.9112	0.955660	0.477830	+	0.878452i	$0.341424\pi$
$854$	22.6393	0.774702
$855$	0	0
$856$	−26.5066	−0.905976
$857$	14.2128	0.485502	0.242751	−	0.970089i	$-0.421950\pi$
$857$	14.2128	0.485502	0.242751	+	0.970089i	$0.421950\pi$
$858$	0	0
$859$	39.5410	1.34912	0.674561	−	0.738219i	$-0.264333\pi$
$859$	39.5410	1.34912	0.674561	+	0.738219i	$0.264333\pi$
$860$	0	0
$861$	0	0
$862$	−48.4104	−1.64886
$863$	−16.7760	−0.571061	−0.285531	−	0.958370i	$-0.592170\pi$
$863$	−16.7760	−0.571061	−0.285531	+	0.958370i	$0.592170\pi$
$864$	0	0
$865$	0	0
$866$	−7.63932	−0.259595
$867$	0	0
$868$	4.63677	0.157382
$869$	42.3607	1.43699
$870$	0	0
$871$	−58.5410	−1.98359
$872$	32.9565	1.11605
$873$	0	0
$874$	22.1591	0.749541
$875$	0	0
$876$	0	0
$877$	−38.8998	−1.31355	−0.656777	−	0.754085i	$-0.728081\pi$
$877$	−38.8998	−1.31355	−0.656777	+	0.754085i	$0.728081\pi$
$878$	−31.8464	−1.07476
$879$	0	0
$880$	0	0
$881$	42.3607	1.42717	0.713584	−	0.700570i	$-0.247071\pi$
$881$	42.3607	1.42717	0.713584	+	0.700570i	$0.247071\pi$
$882$	0	0
$883$	−42.9161	−1.44424	−0.722120	−	0.691768i	$-0.756832\pi$
$883$	−42.9161	−1.44424	−0.722120	+	0.691768i	$0.756832\pi$
$884$	−21.1803	−0.712372
$885$	0	0
$886$	−32.0344	−1.07622
$887$	20.0907	0.674580	0.337290	−	0.941401i	$-0.390490\pi$
$887$	20.0907	0.674580	0.337290	+	0.941401i	$0.390490\pi$
$888$	0	0
$889$	−31.9098	−1.07022
$890$	0	0
$891$	0	0
$892$	16.0090	0.536020
$893$	28.9402	0.968448
$894$	0	0
$895$	0	0
$896$	9.27051	0.309706
$897$	0	0
$898$	−0.857567	−0.0286174
$899$	14.2705	0.475948
$900$	0	0
$901$	35.6525	1.18776
$902$	47.5528	1.58334
$903$	0	0
$904$	28.9443	0.962672
$905$	0	0
$906$	0	0
$907$	−29.1522	−0.967984	−0.483992	−	0.875072i	$-0.660814\pi$
$907$	−29.1522	−0.967984	−0.483992	+	0.875072i	$0.660814\pi$
$908$	5.04531	0.167434
$909$	0	0
$910$	0	0
$911$	11.9098	0.394590	0.197295	−	0.980344i	$-0.436784\pi$
$911$	11.9098	0.394590	0.197295	+	0.980344i	$0.436784\pi$
$912$	0	0
$913$	41.1450	1.36170
$914$	−21.6312	−0.715496
$915$	0	0
$916$	16.4721	0.544255
$917$	−5.02029	−0.165784
$918$	0	0
$919$	−27.3820	−0.903248	−0.451624	−	0.892208i	$-0.649155\pi$
$919$	−27.3820	−0.903248	−0.451624	+	0.892208i	$0.649155\pi$
$920$	0	0
$921$	0	0
$922$	17.6336	0.580730
$923$	90.0854	2.96520
$924$	0	0
$925$	0	0
$926$	41.6312	1.36809
$927$	0	0
$928$	−16.7760	−0.550699
$929$	11.9098	0.390749	0.195374	−	0.980729i	$-0.437408\pi$
$929$	11.9098	0.390749	0.195374	+	0.980729i	$0.437408\pi$
$930$	0	0
$931$	−0.313082	−0.0102609
$932$	1.38757	0.0454515
$933$	0	0
$934$	19.4721	0.637148
$935$	0	0
$936$	0	0
$937$	12.5227	0.409100	0.204550	−	0.978856i	$-0.434427\pi$
$937$	12.5227	0.409100	0.204550	+	0.978856i	$0.434427\pi$
$938$	26.2866	0.858286
$939$	0	0
$940$	0	0
$941$	38.0902	1.24170	0.620852	−	0.783928i	$-0.286787\pi$
$941$	38.0902	1.24170	0.620852	+	0.783928i	$0.286787\pi$
$942$	0	0
$943$	−43.9201	−1.43024
$944$	4.54915	0.148062
$945$	0	0
$946$	5.90170	0.191881
$947$	−28.0422	−0.911248	−0.455624	−	0.890172i	$-0.650584\pi$
$947$	−28.0422	−0.911248	−0.455624	+	0.890172i	$0.650584\pi$
$948$	0	0
$949$	−29.2705	−0.950161
$950$	0	0
$951$	0	0
$952$	40.2874	1.30572
$953$	11.3067	0.366259	0.183130	−	0.983089i	$-0.441377\pi$
$953$	11.3067	0.366259	0.183130	+	0.983089i	$0.441377\pi$
$954$	0	0
$955$	0	0
$956$	−8.81966	−0.285248
$957$	0	0
$958$	14.5309	0.469470
$959$	40.4508	1.30623
$960$	0	0
$961$	−22.8541	−0.737229
$962$	13.1433	0.423756
$963$	0	0
$964$	17.0557	0.549328
$965$	0	0
$966$	0	0
$967$	30.3933	0.977383	0.488692	−	0.872457i	$-0.337474\pi$
$967$	30.3933	0.977383	0.488692	+	0.872457i	$0.337474\pi$
$968$	−43.0876	−1.38489
$969$	0	0
$970$	0	0
$971$	−12.3607	−0.396673	−0.198337	−	0.980134i	$-0.563554\pi$
$971$	−12.3607	−0.396673	−0.198337	+	0.980134i	$0.563554\pi$
$972$	0	0
$973$	1.86162	0.0596809
$974$	−22.3607	−0.716482
$975$	0	0
$976$	−17.4508	−0.558588
$977$	−20.3682	−0.651637	−0.325818	−	0.945432i	$-0.605640\pi$
$977$	−20.3682	−0.651637	−0.325818	+	0.945432i	$0.605640\pi$
$978$	0	0
$979$	55.9017	1.78663
$980$	0	0
$981$	0	0
$982$	−27.1441	−0.866204
$983$	−30.6708	−0.978248	−0.489124	−	0.872214i	$-0.662683\pi$
$983$	−30.6708	−0.978248	−0.489124	+	0.872214i	$0.662683\pi$
$984$	0	0
$985$	0	0
$986$	29.2705	0.932163
$987$	0	0
$988$	14.7679	0.469829
$989$	−5.45085	−0.173327
$990$	0	0
$991$	48.5066	1.54086	0.770431	−	0.637523i	$-0.220041\pi$
$991$	48.5066	1.54086	0.770431	+	0.637523i	$0.220041\pi$
$992$	−9.57608	−0.304041
$993$	0	0
$994$	−40.4508	−1.28302
$995$	0	0
$996$	0	0
$997$	−8.89002	−0.281550	−0.140775	−	0.990042i	$-0.544959\pi$
$997$	−8.89002	−0.281550	−0.140775	+	0.990042i	$0.544959\pi$
$998$	7.67396	0.242915
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.2.a.i.1.3	yes	4
3.2	odd	2		1125.2.a.j.1.2	yes	4
5.2	odd	4		1125.2.b.c.874.3		4
5.3	odd	4		1125.2.b.c.874.2		4
5.4	even	2	inner	1125.2.a.i.1.2	✓	4
15.2	even	4		1125.2.b.d.874.2		4
15.8	even	4		1125.2.b.d.874.3		4
15.14	odd	2		1125.2.a.j.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1125.2.a.i.1.2	✓	4	5.4	even	2	inner
1125.2.a.i.1.3	yes	4	1.1	even	1	trivial
1125.2.a.j.1.2	yes	4	3.2	odd	2
1125.2.a.j.1.3	yes	4	15.14	odd	2
1125.2.b.c.874.2		4	5.3	odd	4
1125.2.b.c.874.3		4	5.2	odd	4
1125.2.b.d.874.2		4	15.2	even	4
1125.2.b.d.874.3		4	15.8	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17557 q^{2} -0.618034 q^{4} +2.62866 q^{7} -3.07768 q^{8} -5.00000 q^{11} -6.88191 q^{13} +3.09017 q^{14} -2.38197 q^{16} -4.97980 q^{17} +3.47214 q^{19} -5.87785 q^{22} +5.42882 q^{23} -8.09017 q^{26} -1.62460 q^{28} -5.00000 q^{29} -2.85410 q^{31} +3.35520 q^{32} -5.85410 q^{34} -1.62460 q^{37} +4.08174 q^{38} -8.09017 q^{41} -1.00406 q^{43} +3.09017 q^{44} +6.38197 q^{46} +8.33499 q^{47} -0.0901699 q^{49} +4.25325 q^{52} -7.15942 q^{53} -8.09017 q^{56} -5.87785 q^{58} -1.90983 q^{59} +7.32624 q^{61} -3.35520 q^{62} +8.70820 q^{64} +8.50651 q^{67} +3.07768 q^{68} -13.0902 q^{71} +4.25325 q^{73} -1.90983 q^{74} -2.14590 q^{76} -13.1433 q^{77} -8.47214 q^{79} -9.51057 q^{82} -8.22899 q^{83} -1.18034 q^{86} +15.3884 q^{88} -11.1803 q^{89} -18.0902 q^{91} -3.35520 q^{92} +9.79837 q^{94} +14.3844 q^{97} -0.106001 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 20 q^{11} - 10 q^{14} - 14 q^{16} - 4 q^{19} - 10 q^{26} - 20 q^{29} + 2 q^{31} - 10 q^{34} - 10 q^{41} - 10 q^{44} + 30 q^{46} + 22 q^{49} - 10 q^{56} - 30 q^{59} - 2 q^{61} + 8 q^{64}+ \cdots - 10 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 - 20 * q^11 - 10 * q^14 - 14 * q^16 - 4 * q^19 - 10 * q^26 - 20 * q^29 + 2 * q^31 - 10 * q^34 - 10 * q^41 - 10 * q^44 + 30 * q^46 + 22 * q^49 - 10 * q^56 - 30 * q^59 - 2 * q^61 + 8 * q^64 - 30 * q^71 - 30 * q^74 - 22 * q^76 - 16 * q^79 + 40 * q^86 - 50 * q^91 - 10 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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