LMFDB - Embedded newform 4900.2.e.u.2549.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4900,2,Mod(2549,4900)] 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("4900.2549"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,16, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.1266969904$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2549.2
Root			$-1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	4900.2549
Dual form			4900.2.e.u.2549.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.28825i q^{3} -2.23607 q^{9} -5.47214 q^{11} +0.874032i q^{13} +4.57649i q^{17} +5.99070 q^{19} -3.47214i q^{23} -1.74806i q^{27} -0.236068 q^{29} -8.27895 q^{31} +12.5216i q^{33} -4.23607i q^{37} +2.00000 q^{39} -5.11667 q^{41} +3.76393i q^{43} +4.91034i q^{47} +10.4721 q^{51} +11.7082i q^{53} -13.7082i q^{57} -1.95440 q^{59} -7.53244 q^{61} +13.9443i q^{67} -7.94510 q^{69} +16.7082 q^{71} +7.53244i q^{73} +11.4721 q^{79} -10.7082 q^{81} -12.1877i q^{83} +0.540182i q^{87} +5.86319 q^{89} +18.9443i q^{93} +16.3516i q^{97} +12.2361 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{11} + 16 q^{29} + 16 q^{39} + 48 q^{51} + 80 q^{71} + 56 q^{79} - 32 q^{81} + 80 q^{99}+O(q^{100}) $ 8 * q - 8 * q^11 + 16 * q^29 + 16 * q^39 + 48 * q^51 + 80 * q^71 + 56 * q^79 - 32 * q^81 + 80 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4900\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$	$2451$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.28825i	− 1.32112i	−0.750774	−	0.660560i	$-0.770319\pi$
$3$	− 2.28825i	− 1.32112i	0.750774	−	0.660560i	$-0.229681\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.23607	−0.745356
$10$	0	0
$11$	−5.47214	−1.64991	−0.824956	−	0.565198i	$-0.808800\pi$
$11$	−5.47214	−1.64991	−0.824956	+	0.565198i	$0.808800\pi$
$12$	0	0
$13$	0.874032i	0.242413i	0.992627	+	0.121206i	$0.0386763\pi$
$13$	0.874032i	0.242413i	−0.992627	+	0.121206i	$0.961324\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.57649i	1.10996i	0.831863	+	0.554981i	$0.187275\pi$
$17$	4.57649i	1.10996i	−0.831863	+	0.554981i	$0.812725\pi$
$18$	0	0
$19$	5.99070	1.37436	0.687181	−	0.726486i	$-0.258848\pi$
$19$	5.99070	1.37436	0.687181	+	0.726486i	$0.258848\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.47214i	− 0.723990i	−0.932180	−	0.361995i	$-0.882096\pi$
$23$	− 3.47214i	− 0.723990i	0.932180	−	0.361995i	$-0.117904\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.74806i	− 0.336415i
$28$	0	0
$29$	−0.236068	−0.0438367	−0.0219184	−	0.999760i	$-0.506977\pi$
$29$	−0.236068	−0.0438367	−0.0219184	+	0.999760i	$0.506977\pi$
$30$	0	0
$31$	−8.27895	−1.48694	−0.743472	−	0.668767i	$-0.766822\pi$
$31$	−8.27895	−1.48694	−0.743472	+	0.668767i	$0.766822\pi$
$32$	0	0
$33$	12.5216i	2.17973i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 4.23607i	− 0.696405i	−0.937419	−	0.348203i	$-0.886792\pi$
$37$	− 4.23607i	− 0.696405i	0.937419	−	0.348203i	$-0.113208\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−5.11667	−0.799090	−0.399545	−	0.916714i	$-0.630832\pi$
$41$	−5.11667	−0.799090	−0.399545	+	0.916714i	$0.630832\pi$
$42$	0	0
$43$	3.76393i	0.573994i	0.957931	+	0.286997i	$0.0926570\pi$
$43$	3.76393i	0.573994i	−0.957931	+	0.286997i	$0.907343\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.91034i	0.716247i	0.933674	+	0.358123i	$0.116583\pi$
$47$	4.91034i	0.716247i	−0.933674	+	0.358123i	$0.883417\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	10.4721	1.46639
$52$	0	0
$53$	11.7082i	1.60825i	0.594463	+	0.804123i	$0.297365\pi$
$53$	11.7082i	1.60825i	−0.594463	+	0.804123i	$0.702635\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 13.7082i	− 1.81570i
$58$	0	0
$59$	−1.95440	−0.254441	−0.127220	−	0.991874i	$-0.540606\pi$
$59$	−1.95440	−0.254441	−0.127220	+	0.991874i	$0.540606\pi$
$60$	0	0
$61$	−7.53244	−0.964430	−0.482215	−	0.876053i	$-0.660168\pi$
$61$	−7.53244	−0.964430	−0.482215	+	0.876053i	$0.660168\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.9443i	1.70356i	0.523896	+	0.851782i	$0.324478\pi$
$67$	13.9443i	1.70356i	−0.523896	+	0.851782i	$0.675522\pi$
$68$	0	0
$69$	−7.94510	−0.956478
$70$	0	0
$71$	16.7082	1.98290	0.991449	−	0.130491i	$-0.0416554\pi$
$71$	16.7082	1.98290	0.991449	+	0.130491i	$0.0416554\pi$
$72$	0	0
$73$	7.53244i	0.881605i	0.897604	+	0.440803i	$0.145306\pi$
$73$	7.53244i	0.881605i	−0.897604	+	0.440803i	$0.854694\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.4721	1.29072	0.645358	−	0.763880i	$-0.276708\pi$
$79$	11.4721	1.29072	0.645358	+	0.763880i	$0.276708\pi$
$80$	0	0
$81$	−10.7082	−1.18980
$82$	0	0
$83$	− 12.1877i	− 1.33778i	−0.743362	−	0.668889i	$-0.766770\pi$
$83$	− 12.1877i	− 1.33778i	0.743362	−	0.668889i	$-0.233230\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.540182i	0.0579135i
$88$	0	0
$89$	5.86319	0.621496	0.310748	−	0.950492i	$-0.399420\pi$
$89$	5.86319	0.621496	0.310748	+	0.950492i	$0.399420\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	18.9443i	1.96443i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	16.3516i	1.66025i	0.557577	+	0.830125i	$0.311731\pi$
$97$	16.3516i	1.66025i	−0.557577	+	0.830125i	$0.688269\pi$
$98$	0	0
$99$	12.2361	1.22977
$100$	0	0
$101$	−3.36861	−0.335189	−0.167595	−	0.985856i	$-0.553600\pi$
$101$	−3.36861	−0.335189	−0.167595	+	0.985856i	$0.553600\pi$
$102$	0	0
$103$	18.3848i	1.81151i	0.423806	+	0.905753i	$0.360694\pi$
$103$	18.3848i	1.81151i	−0.423806	+	0.905753i	$0.639306\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.4721i	− 1.20573i	−0.797844	−	0.602863i	$-0.794026\pi$
$107$	− 12.4721i	− 1.20573i	0.797844	−	0.602863i	$-0.205974\pi$
$108$	0	0
$109$	16.4164	1.57241	0.786203	−	0.617968i	$-0.212044\pi$
$109$	16.4164	1.57241	0.786203	+	0.617968i	$0.212044\pi$
$110$	0	0
$111$	−9.69316	−0.920034
$112$	0	0
$113$	13.7639i	1.29480i	0.762150	+	0.647401i	$0.224144\pi$
$113$	13.7639i	1.29480i	−0.762150	+	0.647401i	$0.775856\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.95440i	− 0.180684i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.9443	1.72221
$122$	0	0
$123$	11.7082i	1.05569i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	11.1803i	0.992095i	0.868295	+	0.496047i	$0.165216\pi$
$127$	11.1803i	0.992095i	−0.868295	+	0.496047i	$0.834784\pi$
$128$	0	0
$129$	8.61280	0.758315
$130$	0	0
$131$	2.16073	0.188784	0.0943918	−	0.995535i	$-0.469909\pi$
$131$	2.16073	0.188784	0.0943918	+	0.995535i	$0.469909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 10.0000i	− 0.854358i	−0.904167	−	0.427179i	$-0.859507\pi$
$137$	− 10.0000i	− 0.854358i	0.904167	−	0.427179i	$-0.140493\pi$
$138$	0	0
$139$	10.3609	0.878797	0.439399	−	0.898292i	$-0.355192\pi$
$139$	10.3609	0.878797	0.439399	+	0.898292i	$0.355192\pi$
$140$	0	0
$141$	11.2361	0.946248
$142$	0	0
$143$	− 4.78282i	− 0.399960i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.70820	−0.549557	−0.274779	−	0.961507i	$-0.588605\pi$
$149$	−6.70820	−0.549557	−0.274779	+	0.961507i	$0.588605\pi$
$150$	0	0
$151$	8.23607	0.670242	0.335121	−	0.942175i	$-0.391223\pi$
$151$	8.23607	0.670242	0.335121	+	0.942175i	$0.391223\pi$
$152$	0	0
$153$	− 10.2333i	− 0.827317i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	9.02546i	0.720310i	0.932892	+	0.360155i	$0.117276\pi$
$157$	9.02546i	0.720310i	−0.932892	+	0.360155i	$0.882724\pi$
$158$	0	0
$159$	26.7912	2.12468
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2.18034i	− 0.170777i	−0.996348	−	0.0853887i	$-0.972787\pi$
$163$	− 2.18034i	− 0.170777i	0.996348	−	0.0853887i	$-0.0272132\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.11512i	0.318438i	0.987243	+	0.159219i	$0.0508975\pi$
$167$	4.11512i	0.318438i	−0.987243	+	0.159219i	$0.949102\pi$
$168$	0	0
$169$	12.2361	0.941236
$170$	0	0
$171$	−13.3956	−1.02439
$172$	0	0
$173$	18.3848i	1.39777i	0.715235	+	0.698884i	$0.246320\pi$
$173$	18.3848i	1.39777i	−0.715235	+	0.698884i	$0.753680\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.47214i	0.336146i
$178$	0	0
$179$	−1.70820	−0.127677	−0.0638386	−	0.997960i	$-0.520334\pi$
$179$	−1.70820	−0.127677	−0.0638386	+	0.997960i	$0.520334\pi$
$180$	0	0
$181$	3.62365	0.269344	0.134672	−	0.990890i	$-0.457002\pi$
$181$	3.62365	0.269344	0.134672	+	0.990890i	$0.457002\pi$
$182$	0	0
$183$	17.2361i	1.27413i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 25.0432i	− 1.83134i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.1803	−0.881338	−0.440669	−	0.897670i	$-0.645259\pi$
$191$	−12.1803	−0.881338	−0.440669	+	0.897670i	$0.645259\pi$
$192$	0	0
$193$	− 15.6525i	− 1.12669i	−0.826222	−	0.563345i	$-0.809514\pi$
$193$	− 15.6525i	− 1.12669i	0.826222	−	0.563345i	$-0.190486\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19.4721i	− 1.38733i	−0.720297	−	0.693666i	$-0.755994\pi$
$197$	− 19.4721i	− 1.38733i	0.720297	−	0.693666i	$-0.244006\pi$
$198$	0	0
$199$	−21.6746	−1.53647	−0.768235	−	0.640168i	$-0.778865\pi$
$199$	−21.6746	−1.53647	−0.768235	+	0.640168i	$0.778865\pi$
$200$	0	0
$201$	31.9079	2.25061
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	7.76393i	0.539631i
$208$	0	0
$209$	−32.7820	−2.26757
$210$	0	0
$211$	9.52786	0.655925	0.327963	−	0.944691i	$-0.393638\pi$
$211$	9.52786	0.655925	0.327963	+	0.944691i	$0.393638\pi$
$212$	0	0
$213$	− 38.2325i	− 2.61965i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	17.2361	1.16471
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 11.8539i	− 0.793795i	−0.917863	−	0.396898i	$-0.870087\pi$
$223$	− 11.8539i	− 0.793795i	0.917863	−	0.396898i	$-0.129913\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6.40337i	− 0.425006i	−0.977160	−	0.212503i	$-0.931838\pi$
$227$	− 6.40337i	− 0.425006i	0.977160	−	0.212503i	$-0.0681616\pi$
$228$	0	0
$229$	−5.57804	−0.368607	−0.184304	−	0.982869i	$-0.559003\pi$
$229$	−5.57804	−0.368607	−0.184304	+	0.982869i	$0.559003\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.5279i	0.689703i	0.938657	+	0.344852i	$0.112071\pi$
$233$	10.5279i	0.689703i	−0.938657	+	0.344852i	$0.887929\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 26.2511i	− 1.70519i
$238$	0	0
$239$	−8.65248	−0.559682	−0.279841	−	0.960046i	$-0.590282\pi$
$239$	−8.65248	−0.559682	−0.279841	+	0.960046i	$0.590282\pi$
$240$	0	0
$241$	1.41421	0.0910975	0.0455488	−	0.998962i	$-0.485496\pi$
$241$	1.41421	0.0910975	0.0455488	+	0.998962i	$0.485496\pi$
$242$	0	0
$243$	19.2588i	1.23545i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.23607i	0.333163i
$248$	0	0
$249$	−27.8885	−1.76736
$250$	0	0
$251$	−1.74806	−0.110337	−0.0551684	−	0.998477i	$-0.517570\pi$
$251$	−1.74806	−0.110337	−0.0551684	+	0.998477i	$0.517570\pi$
$252$	0	0
$253$	19.0000i	1.19452i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.27895i	0.516427i	0.966088	+	0.258213i	$0.0831338\pi$
$257$	8.27895i	0.516427i	−0.966088	+	0.258213i	$0.916866\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.527864	0.0326740
$262$	0	0
$263$	− 10.7082i	− 0.660296i	−0.943929	−	0.330148i	$-0.892901\pi$
$263$	− 10.7082i	− 0.660296i	0.943929	−	0.330148i	$-0.107099\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 13.4164i	− 0.821071i
$268$	0	0
$269$	−32.2418	−1.96582	−0.982908	−	0.184099i	$-0.941063\pi$
$269$	−32.2418	−1.96582	−0.982908	+	0.184099i	$0.941063\pi$
$270$	0	0
$271$	24.8369	1.50873	0.754366	−	0.656454i	$-0.227945\pi$
$271$	24.8369	1.50873	0.754366	+	0.656454i	$0.227945\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 20.9443i	− 1.25842i	−0.777236	−	0.629210i	$-0.783379\pi$
$277$	− 20.9443i	− 1.25842i	0.777236	−	0.629210i	$-0.216621\pi$
$278$	0	0
$279$	18.5123	1.10830
$280$	0	0
$281$	−24.1246	−1.43915	−0.719577	−	0.694413i	$-0.755664\pi$
$281$	−24.1246	−1.43915	−0.719577	+	0.694413i	$0.755664\pi$
$282$	0	0
$283$	− 10.5672i	− 0.628155i	−0.949397	−	0.314077i	$-0.898305\pi$
$283$	− 10.5672i	− 0.628155i	0.949397	−	0.314077i	$-0.101695\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−3.94427	−0.232016
$290$	0	0
$291$	37.4164	2.19339
$292$	0	0
$293$	19.8477i	1.15951i	0.814789	+	0.579757i	$0.196853\pi$
$293$	19.8477i	1.15951i	−0.814789	+	0.579757i	$0.803147\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	9.56564i	0.555055i
$298$	0	0
$299$	3.03476	0.175505
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.70820i	0.442825i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7186i	1.06833i	0.845381	+	0.534164i	$0.179374\pi$
$307$	18.7186i	1.06833i	−0.845381	+	0.534164i	$0.820626\pi$
$308$	0	0
$309$	42.0689	2.39322
$310$	0	0
$311$	9.40802	0.533480	0.266740	−	0.963769i	$-0.414054\pi$
$311$	9.40802	0.533480	0.266740	+	0.963769i	$0.414054\pi$
$312$	0	0
$313$	3.57494i	0.202068i	0.994883	+	0.101034i	$0.0322150\pi$
$313$	3.57494i	0.202068i	−0.994883	+	0.101034i	$0.967785\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.416408i	0.0233878i	0.999932	+	0.0116939i	$0.00372237\pi$
$317$	0.416408i	0.0233878i	−0.999932	+	0.0116939i	$0.996278\pi$
$318$	0	0
$319$	1.29180	0.0723267
$320$	0	0
$321$	−28.5393	−1.59291
$322$	0	0
$323$	27.4164i	1.52549i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 37.5648i	− 2.07734i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.3607	−0.954229	−0.477115	−	0.878841i	$-0.658317\pi$
$331$	−17.3607	−0.954229	−0.477115	+	0.878841i	$0.658317\pi$
$332$	0	0
$333$	9.47214i	0.519070i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.1246i	0.932837i	0.884564	+	0.466419i	$0.154456\pi$
$337$	17.1246i	0.932837i	−0.884564	+	0.466419i	$0.845544\pi$
$338$	0	0
$339$	31.4953	1.71059
$340$	0	0
$341$	45.3035	2.45332
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 25.3607i	− 1.36143i	−0.732547	−	0.680716i	$-0.761669\pi$
$347$	− 25.3607i	− 1.36143i	0.732547	−	0.680716i	$-0.238331\pi$
$348$	0	0
$349$	−34.9427	−1.87044	−0.935219	−	0.354069i	$-0.884798\pi$
$349$	−34.9427	−1.87044	−0.935219	+	0.354069i	$0.884798\pi$
$350$	0	0
$351$	1.52786	0.0815513
$352$	0	0
$353$	11.7751i	0.626724i	0.949634	+	0.313362i	$0.101455\pi$
$353$	11.7751i	0.626724i	−0.949634	+	0.313362i	$0.898545\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	29.0689	1.53420	0.767099	−	0.641529i	$-0.221700\pi$
$359$	29.0689	1.53420	0.767099	+	0.641529i	$0.221700\pi$
$360$	0	0
$361$	16.8885	0.888871
$362$	0	0
$363$	− 43.3491i	− 2.27524i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5.78437i	− 0.301942i	−0.988538	−	0.150971i	$-0.951760\pi$
$367$	− 5.78437i	− 0.301942i	0.988538	−	0.150971i	$-0.0482400\pi$
$368$	0	0
$369$	11.4412	0.595607
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30.1246i	1.55979i	0.625908	+	0.779897i	$0.284728\pi$
$373$	30.1246i	1.55979i	−0.625908	+	0.779897i	$0.715272\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 0.206331i	− 0.0106266i
$378$	0	0
$379$	−10.1246	−0.520066	−0.260033	−	0.965600i	$-0.583734\pi$
$379$	−10.1246	−0.520066	−0.260033	+	0.965600i	$0.583734\pi$
$380$	0	0
$381$	25.5834	1.31068
$382$	0	0
$383$	− 21.5958i	− 1.10349i	−0.834012	−	0.551746i	$-0.813962\pi$
$383$	− 21.5958i	− 1.10349i	0.834012	−	0.551746i	$-0.186038\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 8.41641i	− 0.427830i
$388$	0	0
$389$	3.18034	0.161250	0.0806248	−	0.996745i	$-0.474308\pi$
$389$	3.18034	0.161250	0.0806248	+	0.996745i	$0.474308\pi$
$390$	0	0
$391$	15.8902	0.803602
$392$	0	0
$393$	− 4.94427i	− 0.249406i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.73877i	0.388398i	0.980962	+	0.194199i	$0.0622107\pi$
$397$	7.73877i	0.388398i	−0.980962	+	0.194199i	$0.937789\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.708204	0.0353660	0.0176830	−	0.999844i	$-0.494371\pi$
$401$	0.708204	0.0353660	0.0176830	+	0.999844i	$0.494371\pi$
$402$	0	0
$403$	− 7.23607i	− 0.360454i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	23.1803i	1.14901i
$408$	0	0
$409$	9.89949	0.489499	0.244749	−	0.969586i	$-0.421294\pi$
$409$	9.89949	0.489499	0.244749	+	0.969586i	$0.421294\pi$
$410$	0	0
$411$	−22.8825	−1.12871
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 23.7082i	− 1.16100i
$418$	0	0
$419$	22.4211	1.09534	0.547671	−	0.836694i	$-0.315515\pi$
$419$	22.4211	1.09534	0.547671	+	0.836694i	$0.315515\pi$
$420$	0	0
$421$	30.5967	1.49119	0.745597	−	0.666397i	$-0.232164\pi$
$421$	30.5967	1.49119	0.745597	+	0.666397i	$0.232164\pi$
$422$	0	0
$423$	− 10.9799i	− 0.533859i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.9443	−0.528394
$430$	0	0
$431$	14.6525	0.705785	0.352892	−	0.935664i	$-0.385198\pi$
$431$	14.6525	0.705785	0.352892	+	0.935664i	$0.385198\pi$
$432$	0	0
$433$	− 2.41577i	− 0.116094i	−0.998314	−	0.0580471i	$-0.981513\pi$
$433$	− 2.41577i	− 0.116094i	0.998314	−	0.0580471i	$-0.0184874\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.8005i	− 0.995025i
$438$	0	0
$439$	−23.7565	−1.13384	−0.566918	−	0.823774i	$-0.691864\pi$
$439$	−23.7565	−1.13384	−0.566918	+	0.823774i	$0.691864\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.1246i	1.85887i	0.368990	+	0.929433i	$0.379704\pi$
$443$	39.1246i	1.85887i	−0.368990	+	0.929433i	$0.620296\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.3500i	0.726031i
$448$	0	0
$449$	−9.76393	−0.460788	−0.230394	−	0.973097i	$-0.574002\pi$
$449$	−9.76393	−0.460788	−0.230394	+	0.973097i	$0.574002\pi$
$450$	0	0
$451$	27.9991	1.31843
$452$	0	0
$453$	− 18.8461i	− 0.885469i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	25.6525i	1.19997i	0.800010	+	0.599986i	$0.204827\pi$
$457$	25.6525i	1.19997i	−0.800010	+	0.599986i	$0.795173\pi$
$458$	0	0
$459$	8.00000	0.373408
$460$	0	0
$461$	17.0193	0.792666	0.396333	−	0.918107i	$-0.370282\pi$
$461$	17.0193	0.792666	0.396333	+	0.918107i	$0.370282\pi$
$462$	0	0
$463$	13.4164i	0.623513i	0.950162	+	0.311757i	$0.100917\pi$
$463$	13.4164i	0.623513i	−0.950162	+	0.311757i	$0.899083\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.5595i	− 0.812555i	−0.913750	−	0.406277i	$-0.866827\pi$
$467$	− 17.5595i	− 0.812555i	0.913750	−	0.406277i	$-0.133173\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.6525	0.951616
$472$	0	0
$473$	− 20.5967i	− 0.947039i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 26.1803i	− 1.19872i
$478$	0	0
$479$	−27.4102	−1.25241	−0.626203	−	0.779660i	$-0.715392\pi$
$479$	−27.4102	−1.25241	−0.626203	+	0.779660i	$0.715392\pi$
$480$	0	0
$481$	3.70246	0.168818
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	31.4721i	1.42614i	0.701094	+	0.713069i	$0.252696\pi$
$487$	31.4721i	1.42614i	−0.701094	+	0.713069i	$0.747304\pi$
$488$	0	0
$489$	−4.98915	−0.225617
$490$	0	0
$491$	1.76393	0.0796051	0.0398026	−	0.999208i	$-0.487327\pi$
$491$	1.76393	0.0796051	0.0398026	+	0.999208i	$0.487327\pi$
$492$	0	0
$493$	− 1.08036i	− 0.0486571i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.9443	1.20619	0.603096	−	0.797669i	$-0.293934\pi$
$499$	26.9443	1.20619	0.603096	+	0.797669i	$0.293934\pi$
$500$	0	0
$501$	9.41641	0.420694
$502$	0	0
$503$	− 12.3153i	− 0.549110i	−0.961571	−	0.274555i	$-0.911469\pi$
$503$	− 12.3153i	− 0.549110i	0.961571	−	0.274555i	$-0.0885306\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 27.9991i	− 1.24348i
$508$	0	0
$509$	15.0162	0.665580	0.332790	−	0.943001i	$-0.392010\pi$
$509$	15.0162	0.665580	0.332790	+	0.943001i	$0.392010\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	− 10.4721i	− 0.462356i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 26.8701i	− 1.18174i
$518$	0	0
$519$	42.0689	1.84662
$520$	0	0
$521$	−34.5300	−1.51279	−0.756394	−	0.654117i	$-0.773041\pi$
$521$	−34.5300	−1.51279	−0.756394	+	0.654117i	$0.773041\pi$
$522$	0	0
$523$	− 3.90879i	− 0.170919i	−0.996342	−	0.0854597i	$-0.972764\pi$
$523$	− 3.90879i	− 0.170919i	0.996342	−	0.0854597i	$-0.0272359\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 37.8885i	− 1.65045i
$528$	0	0
$529$	10.9443	0.475838
$530$	0	0
$531$	4.37016	0.189649
$532$	0	0
$533$	− 4.47214i	− 0.193710i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.90879i	0.168677i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−10.7082	−0.460382	−0.230191	−	0.973146i	$-0.573935\pi$
$541$	−10.7082	−0.460382	−0.230191	+	0.973146i	$0.573935\pi$
$542$	0	0
$543$	− 8.29180i	− 0.355835i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.29180i	0.140747i	0.997521	+	0.0703735i	$0.0224191\pi$
$547$	3.29180i	0.140747i	−0.997521	+	0.0703735i	$0.977581\pi$
$548$	0	0
$549$	16.8430	0.718844
$550$	0	0
$551$	−1.41421	−0.0602475
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 14.3475i	− 0.607924i	−0.952684	−	0.303962i	$-0.901690\pi$
$557$	− 14.3475i	− 0.607924i	0.952684	−	0.303962i	$-0.0983096\pi$
$558$	0	0
$559$	−3.28980	−0.139144
$560$	0	0
$561$	−57.3050	−2.41942
$562$	0	0
$563$	41.6799i	1.75660i	0.478112	+	0.878299i	$0.341321\pi$
$563$	41.6799i	1.75660i	−0.478112	+	0.878299i	$0.658679\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.9443	−1.00380	−0.501898	−	0.864927i	$-0.667365\pi$
$569$	−23.9443	−1.00380	−0.501898	+	0.864927i	$0.667365\pi$
$570$	0	0
$571$	4.34752	0.181938	0.0909691	−	0.995854i	$-0.471004\pi$
$571$	4.34752	0.181938	0.0909691	+	0.995854i	$0.471004\pi$
$572$	0	0
$573$	27.8716i	1.16435i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	39.5192i	1.64520i	0.568617	+	0.822602i	$0.307479\pi$
$577$	39.5192i	1.64520i	−0.568617	+	0.822602i	$0.692521\pi$
$578$	0	0
$579$	−35.8167	−1.48849
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 64.0689i	− 2.65346i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.8971i	1.52291i	0.648221	+	0.761453i	$0.275513\pi$
$587$	36.8971i	1.52291i	−0.648221	+	0.761453i	$0.724487\pi$
$588$	0	0
$589$	−49.5967	−2.04360
$590$	0	0
$591$	−44.5570	−1.83283
$592$	0	0
$593$	30.5424i	1.25423i	0.778928	+	0.627113i	$0.215764\pi$
$593$	30.5424i	1.25423i	−0.778928	+	0.627113i	$0.784236\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	49.5967i	2.02986i
$598$	0	0
$599$	−8.52786	−0.348439	−0.174220	−	0.984707i	$-0.555740\pi$
$599$	−8.52786	−0.348439	−0.174220	+	0.984707i	$0.555740\pi$
$600$	0	0
$601$	−32.4481	−1.32359	−0.661793	−	0.749687i	$-0.730204\pi$
$601$	−32.4481	−1.32359	−0.661793	+	0.749687i	$0.730204\pi$
$602$	0	0
$603$	− 31.1803i	− 1.26976i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5.03786i	0.204480i	0.994760	+	0.102240i	$0.0326010\pi$
$607$	5.03786i	0.204480i	−0.994760	+	0.102240i	$0.967399\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.29180	−0.173627
$612$	0	0
$613$	26.8885i	1.08602i	0.839727	+	0.543009i	$0.182715\pi$
$613$	26.8885i	1.08602i	−0.839727	+	0.543009i	$0.817285\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15.3607i	0.618398i	0.950997	+	0.309199i	$0.100061\pi$
$617$	15.3607i	0.618398i	−0.950997	+	0.309199i	$0.899939\pi$
$618$	0	0
$619$	−26.1723	−1.05195	−0.525976	−	0.850500i	$-0.676300\pi$
$619$	−26.1723	−1.05195	−0.525976	+	0.850500i	$0.676300\pi$
$620$	0	0
$621$	−6.06952	−0.243561
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	75.0132i	2.99574i
$628$	0	0
$629$	19.3863	0.772984
$630$	0	0
$631$	−1.00000	−0.0398094	−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000	−0.0398094	−0.0199047	+	0.999802i	$0.506336\pi$
$632$	0	0
$633$	− 21.8021i	− 0.866555i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−37.3607	−1.47797
$640$	0	0
$641$	−39.3607	−1.55465	−0.777327	−	0.629097i	$-0.783425\pi$
$641$	−39.3607	−1.55465	−0.777327	+	0.629097i	$0.783425\pi$
$642$	0	0
$643$	26.3786i	1.04027i	0.854084	+	0.520135i	$0.174118\pi$
$643$	26.3786i	1.04027i	−0.854084	+	0.520135i	$0.825882\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 12.4729i	− 0.490360i	−0.969478	−	0.245180i	$-0.921153\pi$
$647$	− 12.4729i	− 0.490360i	0.969478	−	0.245180i	$-0.0788470\pi$
$648$	0	0
$649$	10.6947	0.419804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 35.3050i	− 1.38159i	−0.723051	−	0.690795i	$-0.757261\pi$
$653$	− 35.3050i	− 1.38159i	0.723051	−	0.690795i	$-0.242739\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 16.8430i	− 0.657110i
$658$	0	0
$659$	−16.4721	−0.641663	−0.320832	−	0.947136i	$-0.603962\pi$
$659$	−16.4721	−0.641663	−0.320832	+	0.947136i	$0.603962\pi$
$660$	0	0
$661$	44.0957	1.71512	0.857561	−	0.514382i	$-0.171979\pi$
$661$	44.0957	1.71512	0.857561	+	0.514382i	$0.171979\pi$
$662$	0	0
$663$	9.15298i	0.355472i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.819660i	0.0317374i
$668$	0	0
$669$	−27.1246	−1.04870
$670$	0	0
$671$	41.2185	1.59122
$672$	0	0
$673$	− 8.29180i	− 0.319625i	−0.987147	−	0.159813i	$-0.948911\pi$
$673$	− 8.29180i	− 0.319625i	0.987147	−	0.159813i	$-0.0510890\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30.7788i	1.18293i	0.806332	+	0.591464i	$0.201450\pi$
$677$	30.7788i	1.18293i	−0.806332	+	0.591464i	$0.798550\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.6525	−0.561484
$682$	0	0
$683$	9.47214i	0.362441i	0.983442	+	0.181221i	$0.0580048\pi$
$683$	9.47214i	0.362441i	−0.983442	+	0.181221i	$0.941995\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.7639i	0.486974i
$688$	0	0
$689$	−10.2333	−0.389859
$690$	0	0
$691$	−27.0764	−1.03003	−0.515017	−	0.857180i	$-0.672214\pi$
$691$	−27.0764	−1.03003	−0.515017	+	0.857180i	$0.672214\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 23.4164i	− 0.886960i
$698$	0	0
$699$	24.0903	0.911180
$700$	0	0
$701$	34.2492	1.29358	0.646788	−	0.762670i	$-0.276112\pi$
$701$	34.2492	1.29358	0.646788	+	0.762670i	$0.276112\pi$
$702$	0	0
$703$	− 25.3770i	− 0.957113i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15.8885	0.596707	0.298353	−	0.954455i	$-0.403563\pi$
$709$	15.8885	0.596707	0.298353	+	0.954455i	$0.403563\pi$
$710$	0	0
$711$	−25.6525	−0.962043
$712$	0	0
$713$	28.7456i	1.07653i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.7990i	0.739407i
$718$	0	0
$719$	23.6777	0.883028	0.441514	−	0.897254i	$-0.354441\pi$
$719$	23.6777	0.883028	0.441514	+	0.897254i	$0.354441\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 3.23607i	− 0.120351i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.9473i	− 1.40739i	−0.710503	−	0.703694i	$-0.751532\pi$
$727$	− 37.9473i	− 1.40739i	0.710503	−	0.703694i	$-0.248468\pi$
$728$	0	0
$729$	11.9443	0.442380
$730$	0	0
$731$	−17.2256	−0.637112
$732$	0	0
$733$	− 19.1313i	− 0.706630i	−0.935504	−	0.353315i	$-0.885054\pi$
$733$	− 19.1313i	− 0.706630i	0.935504	−	0.353315i	$-0.114946\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 76.3050i	− 2.81073i
$738$	0	0
$739$	18.0557	0.664191	0.332095	−	0.943246i	$-0.392244\pi$
$739$	18.0557	0.664191	0.332095	+	0.943246i	$0.392244\pi$
$740$	0	0
$741$	11.9814	0.440148
$742$	0	0
$743$	− 46.1803i	− 1.69419i	−0.531440	−	0.847096i	$-0.678349\pi$
$743$	− 46.1803i	− 1.69419i	0.531440	−	0.847096i	$-0.321651\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	27.2526i	0.997121i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.94427	0.180419	0.0902095	−	0.995923i	$-0.471246\pi$
$751$	4.94427	0.180419	0.0902095	+	0.995923i	$0.471246\pi$
$752$	0	0
$753$	4.00000i	0.145768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 49.7214i	− 1.80715i	−0.428426	−	0.903577i	$-0.640932\pi$
$757$	− 49.7214i	− 1.80715i	0.428426	−	0.903577i	$-0.359068\pi$
$758$	0	0
$759$	43.4767	1.57810
$760$	0	0
$761$	−49.9287	−1.80992	−0.904958	−	0.425501i	$-0.860098\pi$
$761$	−49.9287	−1.80992	−0.904958	+	0.425501i	$0.860098\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.70820i	− 0.0616797i
$768$	0	0
$769$	−16.6854	−0.601692	−0.300846	−	0.953673i	$-0.597269\pi$
$769$	−16.6854	−0.601692	−0.300846	+	0.953673i	$0.597269\pi$
$770$	0	0
$771$	18.9443	0.682261
$772$	0	0
$773$	6.94355i	0.249742i	0.992173	+	0.124871i	$0.0398517\pi$
$773$	6.94355i	0.249742i	−0.992173	+	0.124871i	$0.960148\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−30.6525	−1.09824
$780$	0	0
$781$	−91.4296	−3.27161
$782$	0	0
$783$	0.412662i	0.0147473i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	38.4875i	1.37193i	0.727634	+	0.685966i	$0.240620\pi$
$787$	38.4875i	1.37193i	−0.727634	+	0.685966i	$0.759380\pi$
$788$	0	0
$789$	−24.5030	−0.872330
$790$	0	0
$791$	0	0
$792$	0	0
$793$	− 6.58359i	− 0.233790i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.3972i	0.509974i	0.966944	+	0.254987i	$0.0820712\pi$
$797$	14.3972i	0.509974i	−0.966944	+	0.254987i	$0.917929\pi$
$798$	0	0
$799$	−22.4721	−0.795007
$800$	0	0
$801$	−13.1105	−0.463236
$802$	0	0
$803$	− 41.2185i	− 1.45457i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	73.7771i	2.59708i
$808$	0	0
$809$	−11.8754	−0.417516	−0.208758	−	0.977967i	$-0.566942\pi$
$809$	−11.8754	−0.417516	−0.208758	+	0.977967i	$0.566942\pi$
$810$	0	0
$811$	−10.1545	−0.356574	−0.178287	−	0.983979i	$-0.557056\pi$
$811$	−10.1545	−0.356574	−0.178287	+	0.983979i	$0.557056\pi$
$812$	0	0
$813$	− 56.8328i	− 1.99321i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	22.5486i	0.788876i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.5279	−0.681527	−0.340764	−	0.940149i	$-0.610686\pi$
$821$	−19.5279	−0.681527	−0.340764	+	0.940149i	$0.610686\pi$
$822$	0	0
$823$	− 16.0557i	− 0.559667i	−0.960048	−	0.279834i	$-0.909721\pi$
$823$	− 16.0557i	− 0.559667i	0.960048	−	0.279834i	$-0.0902793\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.6525i	1.65704i	0.559960	+	0.828519i	$0.310816\pi$
$827$	47.6525i	1.65704i	−0.559960	+	0.828519i	$0.689184\pi$
$828$	0	0
$829$	43.3491	1.50558	0.752789	−	0.658262i	$-0.228708\pi$
$829$	43.3491	1.50558	0.752789	+	0.658262i	$0.228708\pi$
$830$	0	0
$831$	−47.9256	−1.66252
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.4721i	0.500230i
$838$	0	0
$839$	23.5803	0.814081	0.407040	−	0.913410i	$-0.366561\pi$
$839$	23.5803	0.814081	0.407040	+	0.913410i	$0.366561\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	0	0
$843$	55.2030i	1.90129i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−24.1803	−0.829867
$850$	0	0
$851$	−14.7082	−0.504191
$852$	0	0
$853$	− 0.255039i	− 0.00873237i	−0.999990	−	0.00436619i	$-0.998610\pi$
$853$	− 0.255039i	− 0.00873237i	0.999990	−	0.00436619i	$-0.00138980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.03941i	0.206302i	0.994666	+	0.103151i	$0.0328926\pi$
$857$	6.03941i	0.206302i	−0.994666	+	0.103151i	$0.967107\pi$
$858$	0	0
$859$	10.4096	0.355170	0.177585	−	0.984105i	$-0.443172\pi$
$859$	10.4096	0.355170	0.177585	+	0.984105i	$0.443172\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.87539i	− 0.336162i	−0.985773	−	0.168081i	$-0.946243\pi$
$863$	− 9.87539i	− 0.336162i	0.985773	−	0.168081i	$-0.0537570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.02546i	0.306521i
$868$	0	0
$869$	−62.7771	−2.12957
$870$	0	0
$871$	−12.1877	−0.412966
$872$	0	0
$873$	− 36.5632i	− 1.23748i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 44.4296i	− 1.50028i	−0.661279	−	0.750140i	$-0.729986\pi$
$877$	− 44.4296i	− 1.50028i	0.661279	−	0.750140i	$-0.270014\pi$
$878$	0	0
$879$	45.4164	1.53186
$880$	0	0
$881$	−28.7456	−0.968465	−0.484233	−	0.874939i	$-0.660901\pi$
$881$	−28.7456	−0.968465	−0.484233	+	0.874939i	$0.660901\pi$
$882$	0	0
$883$	− 6.05573i	− 0.203791i	−0.994795	−	0.101896i	$-0.967509\pi$
$883$	− 6.05573i	− 0.203791i	0.994795	−	0.101896i	$-0.0324908\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48.4658i	1.62732i	0.581339	+	0.813661i	$0.302529\pi$
$887$	48.4658i	1.62732i	−0.581339	+	0.813661i	$0.697471\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	58.5967	1.96306
$892$	0	0
$893$	29.4164i	0.984383i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 6.94427i	− 0.231862i
$898$	0	0
$899$	1.95440	0.0651827
$900$	0	0
$901$	−53.5825	−1.78509
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 27.7771i	− 0.922323i	−0.887316	−	0.461162i	$-0.847433\pi$
$907$	− 27.7771i	− 0.922323i	0.887316	−	0.461162i	$-0.152567\pi$
$908$	0	0
$909$	7.53244	0.249835
$910$	0	0
$911$	−7.11146	−0.235613	−0.117807	−	0.993037i	$-0.537586\pi$
$911$	−7.11146	−0.235613	−0.117807	+	0.993037i	$0.537586\pi$
$912$	0	0
$913$	66.6930i	2.20722i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.7082	−0.617127	−0.308563	−	0.951204i	$-0.599848\pi$
$919$	−18.7082	−0.617127	−0.308563	+	0.951204i	$0.599848\pi$
$920$	0	0
$921$	42.8328	1.41139
$922$	0	0
$923$	14.6035i	0.480680i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 41.1096i	− 1.35022i
$928$	0	0
$929$	−27.9991	−0.918622	−0.459311	−	0.888276i	$-0.651904\pi$
$929$	−27.9991	−0.918622	−0.459311	+	0.888276i	$0.651904\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 21.5279i	− 0.704791i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 55.6157i	− 1.81689i	−0.418009	−	0.908443i	$-0.637272\pi$
$937$	− 55.6157i	− 1.81689i	0.418009	−	0.908443i	$-0.362728\pi$
$938$	0	0
$939$	8.18034	0.266955
$940$	0	0
$941$	6.78593	0.221215	0.110607	−	0.993864i	$-0.464720\pi$
$941$	6.78593	0.221215	0.110607	+	0.993864i	$0.464720\pi$
$942$	0	0
$943$	17.7658i	0.578534i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.7771i	0.512686i	0.966586	+	0.256343i	$0.0825177\pi$
$947$	15.7771i	0.512686i	−0.966586	+	0.256343i	$0.917482\pi$
$948$	0	0
$949$	−6.58359	−0.213712
$950$	0	0
$951$	0.952843	0.0308981
$952$	0	0
$953$	55.9017i	1.81083i	0.424524	+	0.905417i	$0.360442\pi$
$953$	55.9017i	1.81083i	−0.424524	+	0.905417i	$0.639558\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.95595i	− 0.0955522i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	37.5410	1.21100
$962$	0	0
$963$	27.8885i	0.898696i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.58359i	0.147398i	0.997281	+	0.0736992i	$0.0234805\pi$
$967$	4.58359i	0.147398i	−0.997281	+	0.0736992i	$0.976520\pi$
$968$	0	0
$969$	62.7355	2.01535
$970$	0	0
$971$	−12.4729	−0.400274	−0.200137	−	0.979768i	$-0.564139\pi$
$971$	−12.4729	−0.400274	−0.200137	+	0.979768i	$0.564139\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	6.59675i	0.211049i	0.994417	+	0.105524i	$0.0336521\pi$
$977$	6.59675i	0.211049i	−0.994417	+	0.105524i	$0.966348\pi$
$978$	0	0
$979$	−32.0841	−1.02541
$980$	0	0
$981$	−36.7082	−1.17200
$982$	0	0
$983$	42.4751i	1.35475i	0.735640	+	0.677373i	$0.236882\pi$
$983$	42.4751i	1.35475i	−0.735640	+	0.677373i	$0.763118\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.0689	0.415566
$990$	0	0
$991$	4.34752	0.138104	0.0690518	−	0.997613i	$-0.478003\pi$
$991$	4.34752	0.138104	0.0690518	+	0.997613i	$0.478003\pi$
$992$	0	0
$993$	39.7255i	1.26065i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.23179i	0.292374i	0.989257	+	0.146187i	$0.0467001\pi$
$997$	9.23179i	0.292374i	−0.989257	+	0.146187i	$0.953300\pi$
$998$	0	0
$999$	−7.40492	−0.234281

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.e.u.2549.2		8
5.2	odd	4		4900.2.a.bi.1.1	yes	4
5.3	odd	4		4900.2.a.bg.1.4	yes	4
5.4	even	2	inner	4900.2.e.u.2549.8		8
7.6	odd	2	inner	4900.2.e.u.2549.7		8
35.13	even	4		4900.2.a.bg.1.1	✓	4
35.27	even	4		4900.2.a.bi.1.4	yes	4
35.34	odd	2	inner	4900.2.e.u.2549.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4900.2.a.bg.1.1	✓	4	35.13	even	4
4900.2.a.bg.1.4	yes	4	5.3	odd	4
4900.2.a.bi.1.1	yes	4	5.2	odd	4
4900.2.a.bi.1.4	yes	4	35.27	even	4
4900.2.e.u.2549.1		8	35.34	odd	2	inner
4900.2.e.u.2549.2		8	1.1	even	1	trivial
4900.2.e.u.2549.7		8	7.6	odd	2	inner
4900.2.e.u.2549.8		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.28825i q^{3} -2.23607 q^{9} -5.47214 q^{11} +0.874032i q^{13} +4.57649i q^{17} +5.99070 q^{19} -3.47214i q^{23} -1.74806i q^{27} -0.236068 q^{29} -8.27895 q^{31} +12.5216i q^{33} -4.23607i q^{37} +2.00000 q^{39} -5.11667 q^{41} +3.76393i q^{43} +4.91034i q^{47} +10.4721 q^{51} +11.7082i q^{53} -13.7082i q^{57} -1.95440 q^{59} -7.53244 q^{61} +13.9443i q^{67} -7.94510 q^{69} +16.7082 q^{71} +7.53244i q^{73} +11.4721 q^{79} -10.7082 q^{81} -12.1877i q^{83} +0.540182i q^{87} +5.86319 q^{89} +18.9443i q^{93} +16.3516i q^{97} +12.2361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{11} + 16 q^{29} + 16 q^{39} + 48 q^{51} + 80 q^{71} + 56 q^{79} - 32 q^{81} + 80 q^{99}+O(q^{100}) \) 8 * q - 8 * q^11 + 16 * q^29 + 16 * q^39 + 48 * q^51 + 80 * q^71 + 56 * q^79 - 32 * q^81 + 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more