LMFDB - Embedded newform 1503.2.c.a.1502.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1502,1503)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1503.1502");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1502.8
Character	$\chi$	$=$	1503.1502
Dual form			1503.2.c.a.1502.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52227i q^{2} -4.36184 q^{4} +2.54201 q^{5} +2.74873 q^{7} -5.95720i q^{8} +O(q^{10})$	\(q+2.52227i q^{2} -4.36184 q^{4} +2.54201 q^{5} +2.74873 q^{7} -5.95720i q^{8} +6.41163i q^{10} +3.94023i q^{11} -6.61496i q^{13} +6.93304i q^{14} +6.30199 q^{16} +2.82153 q^{17} -0.505048 q^{19} -11.0878 q^{20} -9.93831 q^{22} +3.15028 q^{23} +1.46181 q^{25} +16.6847 q^{26} -11.9895 q^{28} +1.58954i q^{29} +9.87382 q^{31} +3.98090i q^{32} +7.11665i q^{34} +6.98729 q^{35} +4.16421i q^{37} -1.27387i q^{38} -15.1433i q^{40} +8.23743 q^{41} +10.2943i q^{43} -17.1866i q^{44} +7.94585i q^{46} -0.0716296i q^{47} +0.555512 q^{49} +3.68707i q^{50} +28.8534i q^{52} -5.47522 q^{53} +10.0161i q^{55} -16.3747i q^{56} -4.00925 q^{58} +3.92043 q^{59} -13.2706 q^{61} +24.9044i q^{62} +2.56307 q^{64} -16.8153i q^{65} +7.06477i q^{67} -12.3071 q^{68} +17.6238i q^{70} +9.81753 q^{71} -4.73281i q^{73} -10.5032 q^{74} +2.20294 q^{76} +10.8306i q^{77} -7.97708i q^{79} +16.0197 q^{80} +20.7770i q^{82} -13.6112 q^{83} +7.17234 q^{85} -25.9650 q^{86} +23.4727 q^{88} +3.41121i q^{89} -18.1827i q^{91} -13.7410 q^{92} +0.180669 q^{94} -1.28384 q^{95} -7.74095 q^{97} +1.40115i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 48 q^{4}+O(q^{10}) $ 56 * q - 48 * q^4	$ 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) $ 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1503\mathbb{Z}\right)^\times$.

$n$	$172$	$335$
$\chi(n)$	$-1$	$-1$

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 $

$2$			2.52227i			1.78351i	0.452515	+	0.891757i	$0.350527\pi$
$2$			2.52227i			1.78351i	−0.452515	+	0.891757i	$0.649473\pi$
$3$		0			0
$3$		0			0
$4$	−4.36184			−2.18092
$5$	2.54201			1.13682			0.568410	−	0.822745i	$-0.307559\pi$
$5$	2.54201			1.13682			0.568410	+	0.822745i	$0.307559\pi$
$6$		0			0
$7$	2.74873			1.03892			0.519461	−	0.854494i	$-0.326133\pi$
$7$	2.74873			1.03892			0.519461	+	0.854494i	$0.326133\pi$
$8$		−	5.95720i		−	2.10619i
$9$		0			0
$10$			6.41163i			2.02754i
$11$			3.94023i			1.18802i	0.804457	+	0.594011i	$0.202457\pi$
$11$			3.94023i			1.18802i	−0.804457	+	0.594011i	$0.797543\pi$
$12$		0			0
$13$		−	6.61496i		−	1.83466i	−0.398128	−	0.917330i	$-0.630340\pi$
$13$		−	6.61496i		−	1.83466i	0.398128	−	0.917330i	$-0.369660\pi$
$14$			6.93304i			1.85293i
$15$		0			0
$16$	6.30199			1.57550
$17$	2.82153			0.684321			0.342160	−	0.939642i	$-0.388841\pi$
$17$	2.82153			0.684321			0.342160	+	0.939642i	$0.388841\pi$
$18$		0			0
$19$	−0.505048			−0.115866			−0.0579330	−	0.998320i	$-0.518451\pi$
$19$	−0.505048			−0.115866			−0.0579330	+	0.998320i	$0.518451\pi$
$20$	−11.0878			−2.47932
$21$		0			0
$22$	−9.93831			−2.11885
$23$	3.15028			0.656878			0.328439	−	0.944525i	$-0.393477\pi$
$23$	3.15028			0.656878			0.328439	+	0.944525i	$0.393477\pi$
$24$		0			0
$25$	1.46181			0.292361
$26$	16.6847			3.27214
$27$		0			0
$28$	−11.9895			−2.26581
$29$			1.58954i			0.295171i	0.989049	+	0.147585i	$0.0471501\pi$
$29$			1.58954i			0.295171i	−0.989049	+	0.147585i	$0.952850\pi$
$30$		0			0
$31$	9.87382			1.77339			0.886695	−	0.462355i	$-0.152995\pi$
$31$	9.87382			1.77339			0.886695	+	0.462355i	$0.152995\pi$
$32$			3.98090i			0.703731i
$33$		0			0
$34$			7.11665i			1.22050i
$35$	6.98729			1.18107
$36$		0			0
$37$			4.16421i			0.684591i	0.939592	+	0.342296i	$0.111204\pi$
$37$			4.16421i			0.684591i	−0.939592	+	0.342296i	$0.888796\pi$
$38$		−	1.27387i		−	0.206649i
$39$		0			0
$40$		−	15.1433i		−	2.39436i
$41$	8.23743			1.28647			0.643235	−	0.765669i	$-0.277592\pi$
$41$	8.23743			1.28647			0.643235	+	0.765669i	$0.277592\pi$
$42$		0			0
$43$			10.2943i			1.56986i	0.619582	+	0.784932i	$0.287302\pi$
$43$			10.2943i			1.56986i	−0.619582	+	0.784932i	$0.712698\pi$
$44$		−	17.1866i		−	2.59098i
$45$		0			0
$46$			7.94585i			1.17155i
$47$		−	0.0716296i		−	0.0104483i	−0.999986	−	0.00522413i	$-0.998337\pi$
$47$		−	0.0716296i		−	0.0104483i	0.999986	−	0.00522413i	$-0.00166290\pi$
$48$		0			0
$49$	0.555512			0.0793588
$50$			3.68707i			0.521430i
$51$		0			0
$52$			28.8534i			4.00125i
$53$	−5.47522			−0.752079			−0.376039	−	0.926604i	$-0.622714\pi$
$53$	−5.47522			−0.752079			−0.376039	+	0.926604i	$0.622714\pi$
$54$		0			0
$55$			10.0161i			1.35057i
$56$		−	16.3747i		−	2.18817i
$57$		0			0
$58$	−4.00925			−0.526441
$59$	3.92043			0.510397			0.255198	−	0.966889i	$-0.417859\pi$
$59$	3.92043			0.510397			0.255198	+	0.966889i	$0.417859\pi$
$60$		0			0
$61$	−13.2706			−1.69913			−0.849565	−	0.527485i	$-0.823135\pi$
$61$	−13.2706			−1.69913			−0.849565	+	0.527485i	$0.823135\pi$
$62$			24.9044i			3.16287i
$63$		0			0
$64$	2.56307			0.320383
$65$		−	16.8153i		−	2.08568i
$66$		0			0
$67$			7.06477i			0.863099i	0.902089	+	0.431550i	$0.142033\pi$
$67$			7.06477i			0.863099i	−0.902089	+	0.431550i	$0.857967\pi$
$68$	−12.3071			−1.49245
$69$		0			0
$70$			17.6238i			2.10645i
$71$	9.81753			1.16513			0.582563	−	0.812786i	$-0.302050\pi$
$71$	9.81753			1.16513			0.582563	+	0.812786i	$0.302050\pi$
$72$		0			0
$73$		−	4.73281i		−	0.553934i	−0.960879	−	0.276967i	$-0.910671\pi$
$73$		−	4.73281i		−	0.553934i	0.960879	−	0.276967i	$-0.0893293\pi$
$74$	−10.5032			−1.22098
$75$		0			0
$76$	2.20294			0.252695
$77$			10.8306i			1.23426i
$78$		0			0
$79$		−	7.97708i		−	0.897492i	−0.893659	−	0.448746i	$-0.851871\pi$
$79$		−	7.97708i		−	0.897492i	0.893659	−	0.448746i	$-0.148129\pi$
$80$	16.0197			1.79106
$81$		0			0
$82$			20.7770i			2.29444i
$83$	−13.6112			−1.49402			−0.747012	−	0.664810i	$-0.768512\pi$
$83$	−13.6112			−1.49402			−0.747012	+	0.664810i	$0.768512\pi$
$84$		0			0
$85$	7.17234			0.777950
$86$	−25.9650			−2.79987
$87$		0			0
$88$	23.4727			2.50220
$89$			3.41121i			0.361587i	0.983521	+	0.180794i	$0.0578666\pi$
$89$			3.41121i			0.361587i	−0.983521	+	0.180794i	$0.942133\pi$
$90$		0			0
$91$		−	18.1827i		−	1.90607i
$92$	−13.7410			−1.43260
$93$		0			0
$94$	0.180669			0.0186346
$95$	−1.28384			−0.131719
$96$		0			0
$97$	−7.74095			−0.785974			−0.392987	−	0.919544i	$-0.628558\pi$
$97$	−7.74095			−0.785974			−0.392987	+	0.919544i	$0.628558\pi$
$98$			1.40115i			0.141538i
$99$		0			0
$100$	−6.37617			−0.637617
$101$	1.47714			0.146981			0.0734906	−	0.997296i	$-0.476586\pi$
$101$	1.47714			0.146981			0.0734906	+	0.997296i	$0.476586\pi$
$102$		0			0
$103$		−	4.08851i		−	0.402853i	−0.979504	−	0.201427i	$-0.935442\pi$
$103$		−	4.08851i		−	0.402853i	0.979504	−	0.201427i	$-0.0645578\pi$
$104$	−39.4067			−3.86414
$105$		0			0
$106$		−	13.8100i		−	1.34134i
$107$		−	17.2615i		−	1.66873i	−0.551211	−	0.834366i	$-0.685834\pi$
$107$		−	17.2615i		−	1.66873i	0.551211	−	0.834366i	$-0.314166\pi$
$108$		0			0
$109$		−	2.74302i		−	0.262733i	−0.991334	−	0.131367i	$-0.958063\pi$
$109$		−	2.74302i		−	0.262733i	0.991334	−	0.131367i	$-0.0419365\pi$
$110$	−25.2633			−2.40876
$111$		0			0
$112$	17.3225			1.63682
$113$	14.6215			1.37547			0.687737	−	0.725960i	$-0.258604\pi$
$113$	14.6215			1.37547			0.687737	+	0.725960i	$0.258604\pi$
$114$		0			0
$115$	8.00803			0.746753
$116$		−	6.93333i		−	0.643744i
$117$		0			0
$118$			9.88839i			0.910300i
$119$	7.75561			0.710956
$120$		0			0
$121$	−4.52538			−0.411398
$122$		−	33.4721i		−	3.03042i
$123$		0			0
$124$	−43.0680			−3.86762
$125$	−8.99412			−0.804458
$126$		0			0
$127$	−15.5361			−1.37860			−0.689302	−	0.724474i	$-0.742083\pi$
$127$	−15.5361			−1.37860			−0.689302	+	0.724474i	$0.742083\pi$
$128$			14.4265i			1.27514i
$129$		0			0
$130$	42.4127			3.71984
$131$	−9.93299			−0.867849			−0.433924	−	0.900949i	$-0.642872\pi$
$131$	−9.93299			−0.867849			−0.433924	+	0.900949i	$0.642872\pi$
$132$		0			0
$133$	−1.38824			−0.120376
$134$	−17.8193			−1.53935
$135$		0			0
$136$		−	16.8084i		−	1.44131i
$137$			17.4198i			1.48827i	0.668029	+	0.744135i	$0.267138\pi$
$137$			17.4198i			1.48827i	−0.668029	+	0.744135i	$0.732862\pi$
$138$		0			0
$139$			19.4707i			1.65149i	0.564047	+	0.825743i	$0.309244\pi$
$139$			19.4707i			1.65149i	−0.564047	+	0.825743i	$0.690756\pi$
$140$	−30.4775			−2.57582
$141$		0			0
$142$			24.7624i			2.07802i
$143$	26.0644			2.17962
$144$		0			0
$145$			4.04063i			0.335556i
$146$	11.9374			0.987949
$147$		0			0
$148$		−	18.1636i		−	1.49304i
$149$	−13.8855			−1.13754			−0.568772	−	0.822495i	$-0.692581\pi$
$149$	−13.8855			−1.13754			−0.568772	+	0.822495i	$0.692581\pi$
$150$		0			0
$151$		−	5.44339i		−	0.442977i	−0.975163	−	0.221489i	$-0.928908\pi$
$151$		−	5.44339i		−	0.442977i	0.975163	−	0.221489i	$-0.0710915\pi$
$152$			3.00868i			0.244036i
$153$		0			0
$154$	−27.3177			−2.20132
$155$	25.0993			2.01603
$156$		0			0
$157$	−4.19686			−0.334946			−0.167473	−	0.985877i	$-0.553561\pi$
$157$	−4.19686			−0.334946			−0.167473	+	0.985877i	$0.553561\pi$
$158$	20.1204			1.60069
$159$		0			0
$160$			10.1195i			0.800016i
$161$	8.65926			0.682445
$162$		0			0
$163$		−	5.79840i		−	0.454166i	−0.973875	−	0.227083i	$-0.927081\pi$
$163$		−	5.79840i		−	0.454166i	0.973875	−	0.227083i	$-0.0729188\pi$
$164$	−35.9304			−2.80569
$165$		0			0
$166$		−	34.3311i		−	2.66461i
$167$	12.6273	−	2.74786i	0.977132	−	0.212636i
$167$	12.6273	−	2.74786i	0.977132	−	0.212636i
$168$		0			0
$169$	−30.7577			−2.36598
$170$			18.0906i			1.38748i
$171$		0			0
$172$		−	44.9021i		−	3.42375i
$173$			0.601310i			0.0457167i	0.999739	+	0.0228584i	$0.00727668\pi$
$173$			0.601310i			0.0457167i	−0.999739	+	0.0228584i	$0.992723\pi$
$174$		0			0
$175$	4.01811			0.303741
$176$			24.8313i			1.87173i
$177$		0			0
$178$	−8.60398			−0.644896
$179$		−	12.2844i		−	0.918183i	−0.888389	−	0.459091i	$-0.848175\pi$
$179$		−	12.2844i		−	0.918183i	0.888389	−	0.459091i	$-0.151825\pi$
$180$		0			0
$181$	−11.4242			−0.849152			−0.424576	−	0.905392i	$-0.639577\pi$
$181$	−11.4242			−0.849152			−0.424576	+	0.905392i	$0.639577\pi$
$182$	45.8617			3.39950
$183$		0			0
$184$		−	18.7669i		−	1.38351i
$185$			10.5854i			0.778257i
$186$		0			0
$187$			11.1175i			0.812989i
$188$			0.312437i			0.0227868i
$189$		0			0
$190$		−	3.23818i		−	0.234922i
$191$			4.96770i			0.359450i	0.983717	+	0.179725i	$0.0575208\pi$
$191$			4.96770i			0.359450i	−0.983717	+	0.179725i	$0.942479\pi$
$192$		0			0
$193$		−	23.1024i		−	1.66295i	−0.555562	−	0.831475i	$-0.687497\pi$
$193$		−	23.1024i		−	1.66295i	0.555562	−	0.831475i	$-0.312503\pi$
$194$		−	19.5248i		−	1.40180i
$195$		0			0
$196$	−2.42305			−0.173075
$197$	17.9183			1.27662			0.638312	−	0.769777i	$-0.279633\pi$
$197$	17.9183			1.27662			0.638312	+	0.769777i	$0.279633\pi$
$198$		0			0
$199$	19.5767			1.38775			0.693877	−	0.720094i	$-0.255901\pi$
$199$	19.5767			1.38775			0.693877	+	0.720094i	$0.255901\pi$
$200$		−	8.70828i		−	0.615768i
$201$		0			0
$202$			3.72575i			0.262143i
$203$			4.36922i			0.306659i
$204$		0			0
$205$	20.9396			1.46249
$206$	10.3123			0.718494
$207$		0			0
$208$		−	41.6874i		−	2.89050i
$209$		−	1.99000i		−	0.137651i
$210$		0			0
$211$	3.74239			0.257637			0.128818	−	0.991668i	$-0.458882\pi$
$211$	3.74239			0.257637			0.128818	+	0.991668i	$0.458882\pi$
$212$	23.8820			1.64022
$213$		0			0
$214$	43.5381			2.97621
$215$			26.1682i			1.78465i
$216$		0			0
$217$	27.1404			1.84241
$218$	6.91862			0.468588
$219$		0			0
$220$		−	43.6886i		−	2.94548i
$221$		−	18.6643i		−	1.25550i
$222$		0			0
$223$	8.47735			0.567685			0.283843	−	0.958871i	$-0.408391\pi$
$223$	8.47735			0.567685			0.283843	+	0.958871i	$0.408391\pi$
$224$			10.9424i			0.731121i
$225$		0			0
$226$			36.8793i			2.45318i
$227$	−28.1929			−1.87123			−0.935614	−	0.353024i	$-0.885154\pi$
$227$	−28.1929			−1.87123			−0.935614	+	0.353024i	$0.885154\pi$
$228$		0			0
$229$	10.9277			0.722125			0.361063	−	0.932542i	$-0.382414\pi$
$229$	10.9277			0.722125			0.361063	+	0.932542i	$0.382414\pi$
$230$			20.1984i			1.33184i
$231$		0			0
$232$	9.46923			0.621685
$233$		−	11.0531i		−	0.724113i	−0.932156	−	0.362057i	$-0.882075\pi$
$233$		−	11.0531i		−	0.724113i	0.932156	−	0.362057i	$-0.117925\pi$
$234$		0			0
$235$		−	0.182083i		−	0.0118778i
$236$	−17.1003			−1.11314
$237$		0			0
$238$			19.5617i			1.26800i
$239$		−	11.1714i		−	0.722619i	−0.932446	−	0.361309i	$-0.882330\pi$
$239$		−	11.1714i		−	0.722619i	0.932446	−	0.361309i	$-0.117670\pi$
$240$		0			0
$241$		−	1.51339i		−	0.0974858i	−0.998811	−	0.0487429i	$-0.984479\pi$
$241$		−	1.51339i		−	0.0974858i	0.998811	−	0.0487429i	$-0.0155215\pi$
$242$		−	11.4142i		−	0.733734i
$243$		0			0
$244$	57.8844			3.70567
$245$	1.41212			0.0902167
$246$		0			0
$247$			3.34087i			0.212575i
$248$		−	58.8203i		−	3.73510i
$249$		0			0
$250$		−	22.6856i		−	1.43476i
$251$			9.24106i			0.583290i	0.956527	+	0.291645i	$0.0942026\pi$
$251$			9.24106i			0.583290i	−0.956527	+	0.291645i	$0.905797\pi$
$252$		0			0
$253$			12.4128i			0.780387i
$254$		−	39.1862i		−	2.45876i
$255$		0			0
$256$	−31.2615			−1.95384
$257$	−15.6750			−0.977778			−0.488889	−	0.872346i	$-0.662598\pi$
$257$	−15.6750			−0.977778			−0.488889	+	0.872346i	$0.662598\pi$
$258$		0			0
$259$			11.4463i			0.711237i
$260$			73.3456i			4.54870i
$261$		0			0
$262$		−	25.0537i		−	1.54782i
$263$			19.0309i			1.17349i	0.809771	+	0.586746i	$0.199591\pi$
$263$			19.0309i			1.17349i	−0.809771	+	0.586746i	$0.800409\pi$
$264$		0			0
$265$	−13.9180			−0.854979
$266$		−	3.50152i		−	0.214692i
$267$		0			0
$268$		−	30.8154i		−	1.88235i
$269$	16.7370			1.02047			0.510236	−	0.860034i	$-0.329558\pi$
$269$	16.7370			1.02047			0.510236	+	0.860034i	$0.329558\pi$
$270$		0			0
$271$		−	29.8710i		−	1.81453i	−0.420557	−	0.907266i	$-0.638165\pi$
$271$		−	29.8710i		−	1.81453i	0.420557	−	0.907266i	$-0.361835\pi$
$272$	17.7812			1.07815
$273$		0			0
$274$	−43.9373			−2.65435
$275$			5.75985i			0.347332i
$276$		0			0
$277$			6.99335i			0.420189i	0.977681	+	0.210095i	$0.0673772\pi$
$277$			6.99335i			0.420189i	−0.977681	+	0.210095i	$0.932623\pi$
$278$	−49.1104			−2.94545
$279$		0			0
$280$		−	41.6247i		−	2.48755i
$281$			6.50479i			0.388043i	0.980997	+	0.194022i	$0.0621532\pi$
$281$			6.50479i			0.388043i	−0.980997	+	0.194022i	$0.937847\pi$
$282$		0			0
$283$	−10.6241			−0.631538			−0.315769	−	0.948836i	$-0.602263\pi$
$283$	−10.6241			−0.631538			−0.315769	+	0.948836i	$0.602263\pi$
$284$	−42.8225			−2.54105
$285$		0			0
$286$			65.7415i			3.88738i
$287$	22.6425			1.33654
$288$		0			0
$289$	−9.03899			−0.531705
$290$	−10.1916			−0.598469
$291$		0			0
$292$			20.6438i			1.20809i
$293$			28.6123i			1.67155i	0.549075	+	0.835773i	$0.314980\pi$
$293$			28.6123i			1.67155i	−0.549075	+	0.835773i	$0.685020\pi$
$294$		0			0
$295$	9.96577			0.580230
$296$	24.8070			1.44188
$297$		0			0
$298$		−	35.0230i		−	2.02883i
$299$		−	20.8390i		−	1.20515i
$300$		0			0
$301$			28.2962i			1.63097i
$302$	13.7297			0.790056
$303$		0			0
$304$	−3.18281			−0.182547
$305$	−33.7340			−1.93161
$306$		0			0
$307$			8.61615i			0.491750i	0.969302	+	0.245875i	$0.0790752\pi$
$307$			8.61615i			0.491750i	−0.969302	+	0.245875i	$0.920925\pi$
$308$		−	47.2414i		−	2.69183i
$309$		0			0
$310$			63.3073i			3.59561i
$311$		−	9.89755i		−	0.561238i	−0.959819	−	0.280619i	$-0.909460\pi$
$311$		−	9.89755i		−	0.561238i	0.959819	−	0.280619i	$-0.0905398\pi$
$312$		0			0
$313$		−	21.2108i		−	1.19891i	−0.800410	−	0.599453i	$-0.795385\pi$
$313$		−	21.2108i		−	1.19891i	0.800410	−	0.599453i	$-0.204615\pi$
$314$		−	10.5856i		−	0.597381i
$315$		0			0
$316$			34.7948i			1.95736i
$317$			31.7208i			1.78162i	0.454378	+	0.890809i	$0.349862\pi$
$317$			31.7208i			1.78162i	−0.454378	+	0.890809i	$0.650138\pi$
$318$		0			0
$319$	−6.26316			−0.350669
$320$	6.51534			0.364218
$321$		0			0
$322$			21.8410i			1.21715i
$323$	−1.42501			−0.0792895
$324$		0			0
$325$		−	9.66979i		−	0.536384i
$326$	14.6251			0.810011
$327$		0			0
$328$		−	49.0721i		−	2.70955i
$329$		−	0.196890i		−	0.0108549i
$330$		0			0
$331$		−	23.7009i		−	1.30272i	−0.758770	−	0.651358i	$-0.774199\pi$
$331$		−	23.7009i		−	1.30272i	0.758770	−	0.651358i	$-0.225801\pi$
$332$	59.3700			3.25835
$333$		0			0
$334$	6.93084	+	31.8495i	0.379239	+	1.74273i
$335$			17.9587i			0.981189i
$336$		0			0
$337$	−5.10594			−0.278138			−0.139069	−	0.990283i	$-0.544411\pi$
$337$	−5.10594			−0.278138			−0.139069	+	0.990283i	$0.544411\pi$
$338$		−	77.5792i		−	4.21975i
$339$		0			0
$340$	−31.2846			−1.69665
$341$			38.9051i			2.10683i
$342$		0			0
$343$	−17.7142			−0.956474
$344$	61.3252			3.30643
$345$		0			0
$346$	−1.51667			−0.0815364
$347$	−5.98786			−0.321445			−0.160722	−	0.987000i	$-0.551382\pi$
$347$	−5.98786			−0.321445			−0.160722	+	0.987000i	$0.551382\pi$
$348$		0			0
$349$			9.02963i			0.483345i	0.970358	+	0.241673i	$0.0776960\pi$
$349$			9.02963i			0.483345i	−0.970358	+	0.241673i	$0.922304\pi$
$350$			10.1348i			0.541726i
$351$		0			0
$352$	−15.6857			−0.836048
$353$			5.74260i			0.305648i	0.988253	+	0.152824i	$0.0488367\pi$
$353$			5.74260i			0.305648i	−0.988253	+	0.152824i	$0.951163\pi$
$354$		0			0
$355$	24.9562			1.32454
$356$		−	14.8792i		−	0.788593i
$357$		0			0
$358$	30.9847			1.63759
$359$		−	18.8472i		−	0.994718i	−0.867545	−	0.497359i	$-0.834303\pi$
$359$		−	18.8472i		−	0.994718i	0.867545	−	0.497359i	$-0.165697\pi$
$360$		0			0
$361$	−18.7449			−0.986575
$362$		−	28.8148i		−	1.51447i
$363$		0			0
$364$			79.3102i			4.15699i
$365$		−	12.0309i		−	0.629723i
$366$		0			0
$367$	−5.61053			−0.292867			−0.146434	−	0.989220i	$-0.546780\pi$
$367$	−5.61053			−0.292867			−0.146434	+	0.989220i	$0.546780\pi$
$368$	19.8530			1.03491
$369$		0			0
$370$	−26.6993			−1.38803
$371$	−15.0499			−0.781351
$372$		0			0
$373$		−	17.3730i		−	0.899542i	−0.893144	−	0.449771i	$-0.851506\pi$
$373$		−	17.3730i		−	0.899542i	0.893144	−	0.449771i	$-0.148494\pi$
$374$	−28.0412			−1.44998
$375$		0			0
$376$	−0.426712			−0.0220060
$377$	10.5148			0.541538
$378$		0			0
$379$		−	22.2726i		−	1.14407i	−0.820231	−	0.572033i	$-0.806155\pi$
$379$		−	22.2726i		−	1.14407i	0.820231	−	0.572033i	$-0.193845\pi$
$380$	5.59989			0.287269
$381$		0			0
$382$	−12.5299			−0.641084
$383$		−	27.6214i		−	1.41139i	−0.708516	−	0.705695i	$-0.750635\pi$
$383$		−	27.6214i		−	1.41139i	0.708516	−	0.705695i	$-0.249365\pi$
$384$		0			0
$385$			27.5315i			1.40314i
$386$	58.2706			2.96589
$387$		0			0
$388$	33.7648			1.71415
$389$	10.6750			0.541243			0.270621	−	0.962686i	$-0.412771\pi$
$389$	10.6750			0.541243			0.270621	+	0.962686i	$0.412771\pi$
$390$		0			0
$391$	8.88859			0.449516
$392$		−	3.30930i		−	0.167145i
$393$		0			0
$394$			45.1947i			2.27688i
$395$		−	20.2778i		−	1.02029i
$396$		0			0
$397$	39.2421			1.96951			0.984753	−	0.173958i	$-0.0556557\pi$
$397$	39.2421			1.96951			0.984753	+	0.173958i	$0.0556557\pi$
$398$			49.3777i			2.47508i
$399$		0			0
$400$	9.21229			0.460614
$401$	22.3061			1.11391			0.556957	−	0.830541i	$-0.311969\pi$
$401$	22.3061			1.11391			0.556957	+	0.830541i	$0.311969\pi$
$402$		0			0
$403$		−	65.3149i		−	3.25357i
$404$	−6.44307			−0.320555
$405$		0			0
$406$	−11.0204			−0.546931
$407$	−16.4079			−0.813310
$408$		0			0
$409$	23.2743			1.15084			0.575421	−	0.817858i	$-0.304838\pi$
$409$	23.2743			1.15084			0.575421	+	0.817858i	$0.304838\pi$
$410$			52.8154i			2.60836i
$411$		0			0
$412$			17.8335i			0.878591i
$413$	10.7762			0.530262
$414$		0			0
$415$	−34.5998			−1.69844
$416$	26.3335			1.29111
$417$		0			0
$418$	5.01933			0.245503
$419$			37.6687i			1.84024i	0.391640	+	0.920119i	$0.371908\pi$
$419$			37.6687i			1.84024i	−0.391640	+	0.920119i	$0.628092\pi$
$420$		0			0
$421$	32.9903			1.60785			0.803924	−	0.594732i	$-0.202742\pi$
$421$	32.9903			1.60785			0.803924	+	0.594732i	$0.202742\pi$
$422$			9.43932i			0.459499i
$423$		0			0
$424$			32.6170i			1.58402i
$425$	4.12453			0.200069
$426$		0			0
$427$	−36.4773			−1.76526
$428$			75.2919i			3.63937i
$429$		0			0
$430$	−66.0032			−3.18296
$431$		−	8.26870i		−	0.398289i	−0.979970	−	0.199145i	$-0.936184\pi$
$431$		−	8.26870i		−	0.398289i	0.979970	−	0.199145i	$-0.0638164\pi$
$432$		0			0
$433$	−36.8619			−1.77147			−0.885735	−	0.464191i	$-0.846345\pi$
$433$	−36.8619			−1.77147			−0.885735	+	0.464191i	$0.846345\pi$
$434$			68.4555i			3.28597i
$435$		0			0
$436$			11.9646i			0.573001i
$437$	−1.59104			−0.0761099
$438$		0			0
$439$			36.3974i			1.73715i	0.495556	+	0.868576i	$0.334964\pi$
$439$			36.3974i			1.73715i	−0.495556	+	0.868576i	$0.665036\pi$
$440$	59.6679			2.84455
$441$		0			0
$442$	47.0764			2.23919
$443$	27.5875			1.31072			0.655361	−	0.755316i	$-0.272517\pi$
$443$	27.5875			1.31072			0.655361	+	0.755316i	$0.272517\pi$
$444$		0			0
$445$			8.67132i			0.411060i
$446$			21.3822i			1.01247i
$447$		0			0
$448$	7.04517			0.332853
$449$			0.736607i			0.0347626i	0.999849	+	0.0173813i	$0.00553292\pi$
$449$			0.736607i			0.0347626i	−0.999849	+	0.0173813i	$0.994467\pi$
$450$		0			0
$451$			32.4573i			1.52836i
$452$	−63.7766			−2.99980
$453$		0			0
$454$		−	71.1101i		−	3.33736i
$455$		−	46.2207i		−	2.16686i
$456$		0			0
$457$			9.75417i			0.456281i	0.973628	+	0.228140i	$0.0732645\pi$
$457$			9.75417i			0.456281i	−0.973628	+	0.228140i	$0.926735\pi$
$458$			27.5627i			1.28792i
$459$		0			0
$460$	−34.9298			−1.62861
$461$		−	12.5391i		−	0.584003i	−0.956418	−	0.292002i	$-0.905679\pi$
$461$		−	12.5391i		−	0.584003i	0.956418	−	0.292002i	$-0.0943213\pi$
$462$		0			0
$463$		−	38.8395i		−	1.80503i	−0.430664	−	0.902513i	$-0.641720\pi$
$463$		−	38.8395i		−	1.80503i	0.430664	−	0.902513i	$-0.358280\pi$
$464$			10.0173i			0.465040i
$465$		0			0
$466$	27.8789			1.29147
$467$		−	14.5213i		−	0.671966i	−0.941868	−	0.335983i	$-0.890932\pi$
$467$		−	14.5213i		−	0.671966i	0.941868	−	0.335983i	$-0.109068\pi$
$468$		0			0
$469$			19.4191i			0.896693i
$470$	0.459263			0.0211842
$471$		0			0
$472$		−	23.3548i		−	1.07499i
$473$	−40.5618			−1.86503
$474$		0			0
$475$	−0.738283			−0.0338747
$476$	−33.8288			−1.55054
$477$		0			0
$478$	28.1773			1.28880
$479$	10.9811			0.501737			0.250869	−	0.968021i	$-0.419284\pi$
$479$	10.9811			0.501737			0.250869	+	0.968021i	$0.419284\pi$
$480$		0			0
$481$	27.5460			1.25599
$482$	3.81717			0.173867
$483$		0			0
$484$	19.7390			0.897227
$485$	−19.6776			−0.893512
$486$		0			0
$487$		−	10.8533i		−	0.491808i	−0.969294	−	0.245904i	$-0.920915\pi$
$487$		−	10.8533i		−	0.491808i	0.969294	−	0.245904i	$-0.0790848\pi$
$488$			79.0558i			3.57869i
$489$		0			0
$490$			3.56173i			0.160903i
$491$			15.4303i			0.696360i	0.937428	+	0.348180i	$0.113200\pi$
$491$			15.4303i			0.696360i	−0.937428	+	0.348180i	$0.886800\pi$
$492$		0			0
$493$			4.48494i			0.201991i
$494$	−8.42658			−0.379130
$495$		0			0
$496$	62.2247			2.79397
$497$	26.9857			1.21048
$498$		0			0
$499$		−	4.87453i		−	0.218214i	−0.994030	−	0.109107i	$-0.965201\pi$
$499$		−	4.87453i		−	0.218214i	0.994030	−	0.109107i	$-0.0347991\pi$
$500$	39.2309			1.75446
$501$		0			0
$502$	−23.3084			−1.04031
$503$			16.4720i			0.734451i	0.930132	+	0.367226i	$0.119692\pi$
$503$			16.4720i			0.734451i	−0.930132	+	0.367226i	$0.880308\pi$
$504$		0			0
$505$	3.75491			0.167091
$506$	−31.3084			−1.39183
$507$		0			0
$508$	67.7660			3.00663
$509$		−	38.4937i		−	1.70620i	−0.521745	−	0.853102i	$-0.674719\pi$
$509$		−	38.4937i		−	1.70620i	0.521745	−	0.853102i	$-0.325281\pi$
$510$		0			0
$511$		−	13.0092i		−	0.575494i
$512$		−	49.9969i		−	2.20957i
$513$		0			0
$514$		−	39.5365i		−	1.74388i
$515$		−	10.3930i		−	0.457972i
$516$		0			0
$517$	0.282237			0.0124128
$518$	−28.8706			−1.26850
$519$		0			0
$520$	−100.172			−4.39284
$521$	−2.56632			−0.112432			−0.0562162	−	0.998419i	$-0.517904\pi$
$521$	−2.56632			−0.112432			−0.0562162	+	0.998419i	$0.517904\pi$
$522$		0			0
$523$	−17.6529			−0.771907			−0.385953	−	0.922518i	$-0.626127\pi$
$523$	−17.6529			−0.771907			−0.385953	+	0.922518i	$0.626127\pi$
$524$	43.3261			1.89271
$525$		0			0
$526$	−48.0009			−2.09294
$527$	27.8592			1.21357
$528$		0			0
$529$	−13.0757			−0.568511
$530$		−	35.1051i		−	1.52487i
$531$		0			0
$532$	6.05529			0.262530
$533$		−	54.4903i		−	2.36024i
$534$		0			0
$535$		−	43.8789i		−	1.89705i
$536$	42.0863			1.81785
$537$		0			0
$538$			42.2152i			1.82003i
$539$			2.18884i			0.0942800i
$540$		0			0
$541$		−	14.8544i		−	0.638640i	−0.947647	−	0.319320i	$-0.896546\pi$
$541$		−	14.8544i		−	0.638640i	0.947647	−	0.319320i	$-0.103454\pi$
$542$	75.3426			3.23624
$543$		0			0
$544$			11.2322i			0.481578i
$545$		−	6.97277i		−	0.298681i
$546$		0			0
$547$		−	21.9035i		−	0.936526i	−0.883589	−	0.468263i	$-0.844880\pi$
$547$		−	21.9035i		−	0.936526i	0.883589	−	0.468263i	$-0.155120\pi$
$548$		−	75.9822i		−	3.24580i
$549$		0			0
$550$	−14.5279			−0.619471
$551$		−	0.802796i		−	0.0342002i
$552$		0			0
$553$		−	21.9268i		−	0.932424i
$554$	−17.6391			−0.749414
$555$		0			0
$556$		−	84.9283i		−	3.60176i
$557$		−	6.31134i		−	0.267420i	−0.991021	−	0.133710i	$-0.957311\pi$
$557$		−	6.31134i		−	0.267420i	0.991021	−	0.133710i	$-0.0426891\pi$
$558$		0			0
$559$	68.0963			2.88017
$560$	44.0338			1.86077
$561$		0			0
$562$	−16.4068			−0.692080
$563$			23.8039i			1.00321i	0.865096	+	0.501607i	$0.167258\pi$
$563$			23.8039i			1.00321i	−0.865096	+	0.501607i	$0.832742\pi$
$564$		0			0
$565$	37.1679			1.56367
$566$		−	26.7969i		−	1.12636i
$567$		0			0
$568$		−	58.4850i		−	2.45398i
$569$	22.8608			0.958374			0.479187	−	0.877713i	$-0.340932\pi$
$569$	22.8608			0.958374			0.479187	+	0.877713i	$0.340932\pi$
$570$		0			0
$571$		−	34.8835i		−	1.45983i	−0.683538	−	0.729915i	$-0.739560\pi$
$571$		−	34.8835i		−	1.45983i	0.683538	−	0.729915i	$-0.260440\pi$
$572$	−113.689			−4.75357
$573$		0			0
$574$			57.1104i			2.38374i
$575$	4.60510			0.192046
$576$		0			0
$577$	−29.0071			−1.20758			−0.603790	−	0.797143i	$-0.706343\pi$
$577$	−29.0071			−1.20758			−0.603790	+	0.797143i	$0.706343\pi$
$578$		−	22.7988i		−	0.948303i
$579$		0			0
$580$		−	17.6246i		−	0.731821i
$581$	−37.4135			−1.55218
$582$		0			0
$583$		−	21.5736i		−	0.893487i
$584$	−28.1943			−1.16669
$585$		0			0
$586$	−72.1678			−2.98122
$587$	−4.62033			−0.190702			−0.0953508	−	0.995444i	$-0.530397\pi$
$587$	−4.62033			−0.190702			−0.0953508	+	0.995444i	$0.530397\pi$
$588$		0			0
$589$	−4.98675			−0.205476
$590$			25.1364i			1.03485i
$591$		0			0
$592$			26.2428i			1.07857i
$593$	−24.7894			−1.01798			−0.508990	−	0.860772i	$-0.669981\pi$
$593$	−24.7894			−1.01798			−0.508990	+	0.860772i	$0.669981\pi$
$594$		0			0
$595$	19.7148			0.808229
$596$	60.5663			2.48089
$597$		0			0
$598$	52.5615			2.14940
$599$			12.1460i			0.496272i	0.968725	+	0.248136i	$0.0798180\pi$
$599$			12.1460i			0.496272i	−0.968725	+	0.248136i	$0.920182\pi$
$600$		0			0
$601$	−12.8345			−0.523531			−0.261766	−	0.965131i	$-0.584305\pi$
$601$	−12.8345			−0.523531			−0.261766	+	0.965131i	$0.584305\pi$
$602$	−71.3707			−2.90885
$603$		0			0
$604$			23.7432i			0.966098i
$605$	−11.5035			−0.467686
$606$		0			0
$607$			26.9540i			1.09403i	0.837123	+	0.547015i	$0.184236\pi$
$607$			26.9540i			1.09403i	−0.837123	+	0.547015i	$0.815764\pi$
$608$		−	2.01055i		−	0.0815385i
$609$		0			0
$610$		−	85.0863i		−	3.44504i
$611$	−0.473827			−0.0191690
$612$		0			0
$613$	−13.3455			−0.539020			−0.269510	−	0.962998i	$-0.586862\pi$
$613$	−13.3455			−0.539020			−0.269510	+	0.962998i	$0.586862\pi$
$614$	−21.7322			−0.877042
$615$		0			0
$616$	64.5202			2.59959
$617$			9.73362i			0.391861i	0.980618	+	0.195930i	$0.0627727\pi$
$617$			9.73362i			0.391861i	−0.980618	+	0.195930i	$0.937227\pi$
$618$		0			0
$619$		−	5.81966i		−	0.233912i	−0.993137	−	0.116956i	$-0.962686\pi$
$619$		−	5.81966i		−	0.233912i	0.993137	−	0.116956i	$-0.0373136\pi$
$620$	−109.479			−4.39679
$621$		0			0
$622$	24.9643			1.00098
$623$			9.37649i			0.375661i
$624$		0			0
$625$	−30.1722			−1.20689
$626$	53.4994			2.13826
$627$		0			0
$628$	18.3061			0.730491
$629$			11.7494i			0.468480i
$630$		0			0
$631$	−3.04553			−0.121241			−0.0606203	−	0.998161i	$-0.519308\pi$
$631$	−3.04553			−0.121241			−0.0606203	+	0.998161i	$0.519308\pi$
$632$	−47.5211			−1.89029
$633$		0			0
$634$	−80.0084			−3.17754
$635$	−39.4929			−1.56723
$636$		0			0
$637$		−	3.67469i		−	0.145596i
$638$		−	15.7974i		−	0.625424i
$639$		0			0
$640$			36.6724i			1.44960i
$641$	−31.8331			−1.25733			−0.628666	−	0.777675i	$-0.716399\pi$
$641$	−31.8331			−1.25733			−0.628666	+	0.777675i	$0.716399\pi$
$642$		0			0
$643$		−	5.04946i		−	0.199131i	−0.995031	−	0.0995657i	$-0.968255\pi$
$643$		−	5.04946i		−	0.199131i	0.995031	−	0.0995657i	$-0.0317453\pi$
$644$	−37.7703			−1.48836
$645$		0			0
$646$		−	3.59425i		−	0.141414i
$647$	9.13335			0.359069			0.179535	−	0.983752i	$-0.442541\pi$
$647$	9.13335			0.359069			0.179535	+	0.983752i	$0.442541\pi$
$648$		0			0
$649$			15.4474i			0.606363i
$650$	24.3898			0.956647
$651$		0			0
$652$			25.2917i			0.990500i
$653$		−	5.87050i		−	0.229731i	−0.993381	−	0.114865i	$-0.963356\pi$
$653$		−	5.87050i		−	0.229731i	0.993381	−	0.114865i	$-0.0366436\pi$
$654$		0			0
$655$	−25.2497			−0.986589
$656$	51.9122			2.02683
$657$		0			0
$658$	0.496611			0.0193599
$659$	16.9150			0.658913			0.329457	−	0.944171i	$-0.393134\pi$
$659$	16.9150			0.658913			0.329457	+	0.944171i	$0.393134\pi$
$660$		0			0
$661$			42.2984i			1.64522i	0.568608	+	0.822609i	$0.307482\pi$
$661$			42.2984i			1.64522i	−0.568608	+	0.822609i	$0.692518\pi$
$662$	59.7799			2.32341
$663$		0			0
$664$			81.0848i			3.14670i
$665$	−3.52892			−0.136846
$666$		0			0
$667$			5.00750i			0.193891i
$668$	−55.0784	+	11.9857i	−2.13105	+	0.463742i
$669$		0			0
$670$	−45.2967			−1.74996
$671$		−	52.2892i		−	2.01860i
$672$		0			0
$673$		−	15.5591i		−	0.599759i	−0.953977	−	0.299880i	$-0.903053\pi$
$673$		−	15.5591i		−	0.599759i	0.953977	−	0.299880i	$-0.0969465\pi$
$674$		−	12.8786i		−	0.496063i
$675$		0			0
$676$	134.160			5.16001
$677$			9.49044i			0.364747i	0.983229	+	0.182374i	$0.0583780\pi$
$677$			9.49044i			0.364747i	−0.983229	+	0.182374i	$0.941622\pi$
$678$		0			0
$679$	−21.2778			−0.816566
$680$		−	42.7271i		−	1.63851i
$681$		0			0
$682$	−98.1291			−3.75756
$683$	−21.5792			−0.825704			−0.412852	−	0.910798i	$-0.635467\pi$
$683$	−21.5792			−0.825704			−0.412852	+	0.910798i	$0.635467\pi$
$684$		0			0
$685$			44.2812i			1.69190i
$686$		−	44.6799i		−	1.70589i
$687$		0			0
$688$			64.8745i			2.47332i
$689$			36.2183i			1.37981i
$690$		0			0
$691$			8.75013i			0.332871i	0.986052	+	0.166435i	$0.0532257\pi$
$691$			8.75013i			0.332871i	−0.986052	+	0.166435i	$0.946774\pi$
$692$		−	2.62282i		−	0.0997046i
$693$		0			0
$694$		−	15.1030i		−	0.573301i
$695$			49.4948i			1.87744i
$696$		0			0
$697$	23.2421			0.880358
$698$	−22.7752			−0.862053
$699$		0			0
$700$	−17.5264			−0.662434
$701$		−	33.5827i		−	1.26840i	−0.773169	−	0.634200i	$-0.781329\pi$
$701$		−	33.5827i		−	1.26840i	0.773169	−	0.634200i	$-0.218671\pi$
$702$		0			0
$703$		−	2.10312i		−	0.0793208i
$704$			10.0991i			0.380623i
$705$		0			0
$706$	−14.4844			−0.545127
$707$	4.06027			0.152702
$708$		0			0
$709$		−	35.1411i		−	1.31975i	−0.751375	−	0.659875i	$-0.770609\pi$
$709$		−	35.1411i		−	1.31975i	0.751375	−	0.659875i	$-0.229391\pi$
$710$			62.9463i			2.36233i
$711$		0			0
$712$	20.3213			0.761571
$713$	31.1053			1.16490
$714$		0			0
$715$	66.2560			2.47783
$716$			53.5828i			2.00248i
$717$		0			0
$718$	47.5377			1.77409
$719$	−20.7402			−0.773478			−0.386739	−	0.922189i	$-0.626399\pi$
$719$	−20.7402			−0.773478			−0.386739	+	0.922189i	$0.626399\pi$
$720$		0			0
$721$		−	11.2382i		−	0.418533i
$722$		−	47.2798i		−	1.75957i
$723$		0			0
$724$	49.8304			1.85193
$725$			2.32360i			0.0862965i
$726$		0			0
$727$		−	14.4249i		−	0.534989i	−0.963559	−	0.267494i	$-0.913804\pi$
$727$		−	14.4249i		−	0.534989i	0.963559	−	0.267494i	$-0.0861957\pi$
$728$	−108.318			−4.01454
$729$		0			0
$730$	30.3450			1.12312
$731$			29.0456i			1.07429i
$732$		0			0
$733$	33.3499			1.23181			0.615903	−	0.787822i	$-0.288791\pi$
$733$	33.3499			1.23181			0.615903	+	0.787822i	$0.288791\pi$
$734$		−	14.1513i		−	0.522333i
$735$		0			0
$736$			12.5410i			0.462266i
$737$	−27.8368			−1.02538
$738$		0			0
$739$			24.5926i			0.904655i	0.891852	+	0.452327i	$0.149406\pi$
$739$			24.5926i			0.904655i	−0.891852	+	0.452327i	$0.850594\pi$
$740$		−	46.1720i		−	1.69732i
$741$		0			0
$742$		−	37.9599i		−	1.39355i
$743$			12.5500i			0.460416i	0.973141	+	0.230208i	$0.0739406\pi$
$743$			12.5500i			0.460416i	−0.973141	+	0.230208i	$0.926059\pi$
$744$		0			0
$745$	−35.2970			−1.29318
$746$	43.8195			1.60435
$747$		0			0
$748$		−	48.4926i		−	1.77306i
$749$		−	47.4472i		−	1.73368i
$750$		0			0
$751$		−	14.5037i		−	0.529246i	−0.964352	−	0.264623i	$-0.914753\pi$
$751$		−	14.5037i		−	0.529246i	0.964352	−	0.264623i	$-0.0852475\pi$
$752$		−	0.451409i		−	0.0164612i
$753$		0			0
$754$			26.5211i			0.965840i
$755$		−	13.8371i		−	0.503585i
$756$		0			0
$757$	−33.6840			−1.22427			−0.612133	−	0.790755i	$-0.709688\pi$
$757$	−33.6840			−1.22427			−0.612133	+	0.790755i	$0.709688\pi$
$758$	56.1775			2.04046
$759$		0			0
$760$			7.64808i			0.277425i
$761$			22.4950i			0.815445i	0.913106	+	0.407722i	$0.133677\pi$
$761$			22.4950i			0.815445i	−0.913106	+	0.407722i	$0.866323\pi$
$762$		0			0
$763$		−	7.53981i		−	0.272959i
$764$		−	21.6683i		−	0.783932i
$765$		0			0
$766$	69.6687			2.51723
$767$		−	25.9335i		−	0.936404i
$768$		0			0
$769$			15.1229i			0.545346i	0.962107	+	0.272673i	$0.0879077\pi$
$769$			15.1229i			0.545346i	−0.962107	+	0.272673i	$0.912092\pi$
$770$	−69.4419			−2.50251
$771$		0			0
$772$			100.769i			3.62676i
$773$	−47.2175			−1.69829			−0.849147	−	0.528156i	$-0.822884\pi$
$773$	−47.2175			−1.69829			−0.849147	+	0.528156i	$0.822884\pi$
$774$		0			0
$775$	14.4336			0.518471
$776$			46.1144i			1.65541i
$777$		0			0
$778$			26.9252i			0.965314i
$779$	−4.16030			−0.149058
$780$		0			0
$781$			38.6833i			1.38420i
$782$			22.4194i			0.801717i
$783$		0			0
$784$	3.50083			0.125030
$785$	−10.6685			−0.380774
$786$		0			0
$787$		−	28.1022i		−	1.00174i	−0.865524	−	0.500868i	$-0.833014\pi$
$787$		−	28.1022i		−	1.00174i	0.865524	−	0.500868i	$-0.166986\pi$
$788$	−78.1567			−2.78422
$789$		0			0
$790$	51.1461			1.81970
$791$	40.1905			1.42901
$792$		0			0
$793$			87.7846i			3.11732i
$794$			98.9792i			3.51264i
$795$		0			0
$796$	−85.3904			−3.02658
$797$	−41.1361			−1.45711			−0.728557	−	0.684985i	$-0.759809\pi$
$797$	−41.1361			−1.45711			−0.728557	+	0.684985i	$0.759809\pi$
$798$		0			0
$799$		−	0.202105i		−	0.00714996i
$800$			5.81931i			0.205744i
$801$		0			0
$802$			56.2621i			1.98668i
$803$	18.6484			0.658086
$804$		0			0
$805$	22.0119			0.775818
$806$	164.742			5.80278
$807$		0			0
$808$		−	8.79964i		−	0.309570i
$809$		−	33.5291i		−	1.17882i	−0.807834	−	0.589409i	$-0.799360\pi$
$809$		−	33.5291i		−	1.17882i	0.807834	−	0.589409i	$-0.200640\pi$
$810$		0			0
$811$		−	38.8373i		−	1.36376i	−0.731464	−	0.681880i	$-0.761162\pi$
$811$		−	38.8373i		−	1.36376i	0.731464	−	0.681880i	$-0.238838\pi$
$812$		−	19.0579i		−	0.668800i
$813$		0			0
$814$		−	41.3852i		−	1.45055i
$815$		−	14.7396i		−	0.516305i
$816$		0			0
$817$		−	5.19911i		−	0.181894i
$818$			58.7041i			2.05254i
$819$		0			0
$820$	−91.3353			−3.18957
$821$	−32.2518			−1.12560			−0.562798	−	0.826594i	$-0.690275\pi$
$821$	−32.2518			−1.12560			−0.562798	+	0.826594i	$0.690275\pi$
$822$		0			0
$823$		−	34.8407i		−	1.21447i	−0.794523	−	0.607235i	$-0.792279\pi$
$823$		−	34.8407i		−	1.21447i	0.794523	−	0.607235i	$-0.207721\pi$
$824$	−24.3561			−0.848485
$825$		0			0
$826$			27.1805i			0.945730i
$827$	−6.78667			−0.235996			−0.117998	−	0.993014i	$-0.537648\pi$
$827$	−6.78667			−0.235996			−0.117998	+	0.993014i	$0.537648\pi$
$828$		0			0
$829$			25.8941i			0.899340i	0.893195	+	0.449670i	$0.148458\pi$
$829$			25.8941i			0.899340i	−0.893195	+	0.449670i	$0.851542\pi$
$830$		−	87.2701i		−	3.02919i
$831$		0			0
$832$		−	16.9546i		−	0.587794i
$833$	1.56739			0.0543069
$834$		0			0
$835$	32.0988	−	6.98508i	1.11082	−	0.241729i
$836$			8.68008i			0.300207i
$837$		0			0
$838$	−95.0106			−3.28209
$839$			20.9968i			0.724889i	0.932005	+	0.362445i	$0.118058\pi$
$839$			20.9968i			0.724889i	−0.932005	+	0.362445i	$0.881942\pi$
$840$		0			0
$841$	26.4734			0.912874
$842$			83.2104i			2.86762i
$843$		0			0
$844$	−16.3237			−0.561886
$845$	−78.1863			−2.68969
$846$		0			0
$847$	−12.4390			−0.427410
$848$	−34.5047			−1.18490
$849$		0			0
$850$			10.4032i			0.356826i
$851$			13.1184i			0.449693i
$852$		0			0
$853$	−24.0662			−0.824010			−0.412005	−	0.911182i	$-0.635171\pi$
$853$	−24.0662			−0.824010			−0.412005	+	0.911182i	$0.635171\pi$
$854$		−	92.0057i		−	3.14837i
$855$		0			0
$856$	−102.830			−3.51467
$857$			6.26235i			0.213918i	0.994263	+	0.106959i	$0.0341113\pi$
$857$			6.26235i			0.213918i	−0.994263	+	0.106959i	$0.965889\pi$
$858$		0			0
$859$	−33.3659			−1.13843			−0.569214	−	0.822189i	$-0.692753\pi$
$859$	−33.3659			−1.13843			−0.569214	+	0.822189i	$0.692753\pi$
$860$		−	114.141i		−	3.89219i
$861$		0			0
$862$	20.8559			0.710355
$863$		−	9.47164i		−	0.322418i	−0.986920	−	0.161209i	$-0.948461\pi$
$863$		−	9.47164i		−	0.322418i	0.986920	−	0.161209i	$-0.0515394\pi$
$864$		0			0
$865$			1.52853i			0.0519717i
$866$		−	92.9757i		−	3.15944i
$867$		0			0
$868$	−118.382			−4.01816
$869$	31.4315			1.06624
$870$		0			0
$871$	46.7332			1.58349
$872$	−16.3407			−0.553366
$873$		0			0
$874$		−	4.01304i		−	0.135743i
$875$	−24.7224			−0.835769
$876$		0			0
$877$	19.5180			0.659077			0.329539	−	0.944142i	$-0.393107\pi$
$877$	19.5180			0.659077			0.329539	+	0.944142i	$0.393107\pi$
$878$	−91.8040			−3.09823
$879$		0			0
$880$			63.1213i			2.12782i
$881$	−20.1693			−0.679519			−0.339760	−	0.940512i	$-0.610346\pi$
$881$	−20.1693			−0.679519			−0.339760	+	0.940512i	$0.610346\pi$
$882$		0			0
$883$	18.9321			0.637114			0.318557	−	0.947904i	$-0.396802\pi$
$883$	18.9321			0.637114			0.318557	+	0.947904i	$0.396802\pi$
$884$			81.4107i			2.73814i
$885$		0			0
$886$			69.5831i			2.33769i
$887$	−37.9692			−1.27488			−0.637440	−	0.770500i	$-0.720007\pi$
$887$	−37.9692			−1.27488			−0.637440	+	0.770500i	$0.720007\pi$
$888$		0			0
$889$	−42.7045			−1.43226
$890$	−21.8714			−0.733131
$891$		0			0
$892$	−36.9769			−1.23808
$893$			0.0361764i			0.00121060i
$894$		0			0
$895$		−	31.2272i		−	1.04381i
$896$			39.6547i			1.32477i
$897$		0			0
$898$	−1.85792			−0.0619996
$899$			15.6948i			0.523453i
$900$		0			0
$901$	−15.4485			−0.514663
$902$	−81.8661			−2.72584
$903$		0			0
$904$		−	87.1031i		−	2.89701i
$905$	−29.0403			−0.965333
$906$		0			0
$907$	13.4811			0.447633			0.223817	−	0.974631i	$-0.428148\pi$
$907$	13.4811			0.447633			0.223817	+	0.974631i	$0.428148\pi$
$908$	122.973			4.08100
$909$		0			0
$910$	116.581			3.86462
$911$			19.6587i			0.651322i	0.945487	+	0.325661i	$0.105587\pi$
$911$			19.6587i			0.651322i	−0.945487	+	0.325661i	$0.894413\pi$
$912$		0			0
$913$		−	53.6313i		−	1.77494i
$914$	−24.6026			−0.813783
$915$		0			0
$916$	−47.6651			−1.57490
$917$	−27.3031			−0.901627
$918$		0			0
$919$	19.5169			0.643805			0.321902	−	0.946773i	$-0.395678\pi$
$919$	19.5169			0.643805			0.321902	+	0.946773i	$0.395678\pi$
$920$		−	47.7055i		−	1.57280i
$921$		0			0
$922$	31.6269			1.04158
$923$		−	64.9425i		−	2.13761i
$924$		0			0
$925$			6.08726i			0.200148i
$926$	97.9637			3.21929
$927$		0			0
$928$	−6.32781			−0.207721
$929$		−	10.9823i		−	0.360317i	−0.983638	−	0.180159i	$-0.942339\pi$
$929$		−	10.9823i		−	0.360317i	0.983638	−	0.180159i	$-0.0576611\pi$
$930$		0			0
$931$	−0.280560			−0.00919499
$932$			48.2119i			1.57923i
$933$		0			0
$934$	36.6266			1.19846
$935$			28.2607i			0.924222i
$936$		0			0
$937$		−	6.03651i		−	0.197204i	−0.995127	−	0.0986021i	$-0.968563\pi$
$937$		−	6.03651i		−	0.197204i	0.995127	−	0.0986021i	$-0.0314371\pi$
$938$	−48.9803			−1.59926
$939$		0			0
$940$			0.794218i			0.0259045i
$941$	33.7093			1.09889			0.549445	−	0.835530i	$-0.314839\pi$
$941$	33.7093			1.09889			0.549445	+	0.835530i	$0.314839\pi$
$942$		0			0
$943$	25.9502			0.845055
$944$	24.7065			0.804129
$945$		0			0
$946$		−	102.308i		−	3.32631i
$947$		−	10.7458i		−	0.349192i	−0.984640	−	0.174596i	$-0.944138\pi$
$947$		−	10.7458i		−	0.349192i	0.984640	−	0.174596i	$-0.0558619\pi$
$948$		0			0
$949$	−31.3074			−1.01628
$950$		−	1.86215i		−	0.0604161i
$951$		0			0
$952$		−	46.2018i		−	1.49741i
$953$	17.8779			0.579123			0.289562	−	0.957159i	$-0.406490\pi$
$953$	17.8779			0.579123			0.289562	+	0.957159i	$0.406490\pi$
$954$		0			0
$955$			12.6279i			0.408630i
$956$			48.7280i			1.57598i
$957$		0			0
$958$			27.6972i			0.894855i
$959$			47.8822i			1.54620i
$960$		0			0
$961$	66.4923			2.14491
$962$			69.4786i			2.24008i
$963$		0			0
$964$			6.60115i			0.212609i
$965$		−	58.7266i		−	1.89048i
$966$		0			0
$967$	12.5731			0.404324			0.202162	−	0.979352i	$-0.435203\pi$
$967$	12.5731			0.404324			0.202162	+	0.979352i	$0.435203\pi$
$968$			26.9586i			0.866482i
$969$		0			0
$970$		−	49.6321i		−	1.59359i
$971$	−3.14318			−0.100870			−0.0504348	−	0.998727i	$-0.516061\pi$
$971$	−3.14318			−0.100870			−0.0504348	+	0.998727i	$0.516061\pi$
$972$		0			0
$973$			53.5198i			1.71576i
$974$	27.3748			0.877146
$975$		0			0
$976$	−83.6313			−2.67697
$977$	−57.0419			−1.82493			−0.912466	−	0.409151i	$-0.865825\pi$
$977$	−57.0419			−1.82493			−0.912466	+	0.409151i	$0.865825\pi$
$978$		0			0
$979$	−13.4409			−0.429574
$980$	−6.15942			−0.196756
$981$		0			0
$982$	−38.9194			−1.24197
$983$	−2.44441			−0.0779645			−0.0389823	−	0.999240i	$-0.512412\pi$
$983$	−2.44441			−0.0779645			−0.0389823	+	0.999240i	$0.512412\pi$
$984$		0			0
$985$	45.5484			1.45129
$986$	−11.3122			−0.360254
$987$		0			0
$988$		−	14.5724i		−	0.463609i
$989$			32.4299i			1.03121i
$990$		0			0
$991$			45.3924i			1.44194i	0.692968	+	0.720969i	$0.256303\pi$
$991$			45.3924i			1.44194i	−0.692968	+	0.720969i	$0.743697\pi$
$992$			39.3067i			1.24799i
$993$		0			0
$994$			68.0653i			2.15890i
$995$	49.7641			1.57763
$996$		0			0
$997$	−28.2503			−0.894695			−0.447348	−	0.894360i	$-0.647631\pi$
$997$	−28.2503			−0.894695			−0.447348	+	0.894360i	$0.647631\pi$
$998$	12.2949			0.389187
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.c.a.1502.8	yes	56
3.2	odd	2	inner	1503.2.c.a.1502.49	yes	56
167.166	odd	2	inner	1503.2.c.a.1502.50	yes	56
501.500	even	2	inner	1503.2.c.a.1502.7	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1503.2.c.a.1502.7	✓	56	501.500	even	2	inner
1503.2.c.a.1502.8	yes	56	1.1	even	1	trivial
1503.2.c.a.1502.49	yes	56	3.2	odd	2	inner
1503.2.c.a.1502.50	yes	56	167.166	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.52227i q^{2} -4.36184 q^{4} +2.54201 q^{5} +2.74873 q^{7} -5.95720i q^{8} +O(q^{10})\)	\(q+2.52227i q^{2} -4.36184 q^{4} +2.54201 q^{5} +2.74873 q^{7} -5.95720i q^{8} +6.41163i q^{10} +3.94023i q^{11} -6.61496i q^{13} +6.93304i q^{14} +6.30199 q^{16} +2.82153 q^{17} -0.505048 q^{19} -11.0878 q^{20} -9.93831 q^{22} +3.15028 q^{23} +1.46181 q^{25} +16.6847 q^{26} -11.9895 q^{28} +1.58954i q^{29} +9.87382 q^{31} +3.98090i q^{32} +7.11665i q^{34} +6.98729 q^{35} +4.16421i q^{37} -1.27387i q^{38} -15.1433i q^{40} +8.23743 q^{41} +10.2943i q^{43} -17.1866i q^{44} +7.94585i q^{46} -0.0716296i q^{47} +0.555512 q^{49} +3.68707i q^{50} +28.8534i q^{52} -5.47522 q^{53} +10.0161i q^{55} -16.3747i q^{56} -4.00925 q^{58} +3.92043 q^{59} -13.2706 q^{61} +24.9044i q^{62} +2.56307 q^{64} -16.8153i q^{65} +7.06477i q^{67} -12.3071 q^{68} +17.6238i q^{70} +9.81753 q^{71} -4.73281i q^{73} -10.5032 q^{74} +2.20294 q^{76} +10.8306i q^{77} -7.97708i q^{79} +16.0197 q^{80} +20.7770i q^{82} -13.6112 q^{83} +7.17234 q^{85} -25.9650 q^{86} +23.4727 q^{88} +3.41121i q^{89} -18.1827i q^{91} -13.7410 q^{92} +0.180669 q^{94} -1.28384 q^{95} -7.74095 q^{97} +1.40115i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 48 q^{4}+O(q^{10}) \) 56 * q - 48 * q^4	\( 56 q - 48 q^{4} + 32 q^{16} - 8 q^{19} + 16 q^{22} + 64 q^{25} - 32 q^{28} + 40 q^{31} + 56 q^{49} - 32 q^{58} + 24 q^{61} - 56 q^{76} + 72 q^{88} - 56 q^{94} - 48 q^{97}+O(q^{100}) \) 56 * q - 48 * q^4 + 32 * q^16 - 8 * q^19 + 16 * q^22 + 64 * q^25 - 32 * q^28 + 40 * q^31 + 56 * q^49 - 32 * q^58 + 24 * q^61 - 56 * q^76 + 72 * q^88 - 56 * q^94 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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