LMFDB - Embedded newform 1120.2.n.b.559.23

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1120,2,Mod(559,1120)] 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("1120.559"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.n (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$8.94324502638$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			559.23
Character	$\chi$	$=$	1120.559
Dual form			1120.2.n.b.559.22

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.857983 q^{3} +(1.33029 - 1.79731i) q^{5} +(-2.41097 + 1.08959i) q^{7} -2.26387 q^{9} -3.05732 q^{11} -3.18505i q^{13} +(1.14136 - 1.54206i) q^{15} -7.44976 q^{17} +4.61892i q^{19} +(-2.06857 + 0.934853i) q^{21} +0.708794 q^{23} +(-1.46068 - 4.78189i) q^{25} -4.51631 q^{27} -2.41421i q^{29} -5.14028 q^{31} -2.62312 q^{33} +(-1.24894 + 5.78275i) q^{35} +2.07192 q^{37} -2.73272i q^{39} +5.51396i q^{41} -4.21172i q^{43} +(-3.01159 + 4.06888i) q^{45} -6.55463i q^{47} +(4.62557 - 5.25396i) q^{49} -6.39177 q^{51} +3.75594 q^{53} +(-4.06711 + 5.49496i) q^{55} +3.96296i q^{57} +3.11915i q^{59} +9.55219 q^{61} +(5.45812 - 2.46669i) q^{63} +(-5.72454 - 4.23703i) q^{65} -0.519160i q^{67} +0.608133 q^{69} +10.9754i q^{71} -2.32017 q^{73} +(-1.25323 - 4.10277i) q^{75} +(7.37110 - 3.33123i) q^{77} -9.98693i q^{79} +2.91669 q^{81} -8.73474 q^{83} +(-9.91032 + 13.3896i) q^{85} -2.07135i q^{87} +18.5697i q^{89} +(3.47042 + 7.67907i) q^{91} -4.41027 q^{93} +(8.30166 + 6.14449i) q^{95} +1.58285 q^{97} +6.92135 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) $ 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$

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
$3$	0.857983			0.495356			0.247678	−	0.968842i	$-0.420332\pi$
$3$	0.857983			0.495356			0.247678	+	0.968842i	$0.420332\pi$
$4$		0			0
$5$	1.33029	−	1.79731i	0.594922	−	0.803783i
$5$	1.33029	−	1.79731i	0.594922	−	0.803783i
$6$		0			0
$7$	−2.41097	+	1.08959i	−0.911262	+	0.411828i
$7$	−2.41097	+	1.08959i	−0.911262	+	0.411828i
$8$		0			0
$9$	−2.26387			−0.754622
$10$		0			0
$11$	−3.05732			−0.921815			−0.460908	−	0.887448i	$-0.652476\pi$
$11$	−3.05732			−0.921815			−0.460908	+	0.887448i	$0.652476\pi$
$12$		0			0
$13$		−	3.18505i		−	0.883375i	−0.897169	−	0.441687i	$-0.854380\pi$
$13$		−	3.18505i		−	0.883375i	0.897169	−	0.441687i	$-0.145620\pi$
$14$		0			0
$15$	1.14136	−	1.54206i	0.294699	−	0.398159i
$16$		0			0
$17$	−7.44976			−1.80683			−0.903417	−	0.428764i	$-0.858949\pi$
$17$	−7.44976			−1.80683			−0.903417	+	0.428764i	$0.858949\pi$
$18$		0			0
$19$			4.61892i			1.05965i	0.848106	+	0.529827i	$0.177743\pi$
$19$			4.61892i			1.05965i	−0.848106	+	0.529827i	$0.822257\pi$
$20$		0			0
$21$	−2.06857	+	0.934853i	−0.451399	+	0.204002i
$22$		0			0
$23$	0.708794			0.147794			0.0738969	−	0.997266i	$-0.476456\pi$
$23$	0.708794			0.147794			0.0738969	+	0.997266i	$0.476456\pi$
$24$		0			0
$25$	−1.46068	−	4.78189i	−0.292135	−	0.956377i
$26$		0			0
$27$	−4.51631			−0.869163
$28$		0			0
$29$		−	2.41421i		−	0.448307i	−0.974554	−	0.224154i	$-0.928038\pi$
$29$		−	2.41421i		−	0.448307i	0.974554	−	0.224154i	$-0.0719617\pi$
$30$		0			0
$31$	−5.14028			−0.923221			−0.461610	−	0.887083i	$-0.652728\pi$
$31$	−5.14028			−0.923221			−0.461610	+	0.887083i	$0.652728\pi$
$32$		0			0
$33$	−2.62312			−0.456627
$34$		0			0
$35$	−1.24894	+	5.78275i	−0.211109	+	0.977462i
$36$		0			0
$37$	2.07192			0.340621			0.170311	−	0.985390i	$-0.445523\pi$
$37$	2.07192			0.340621			0.170311	+	0.985390i	$0.445523\pi$
$38$		0			0
$39$		−	2.73272i		−	0.437586i
$40$		0			0
$41$			5.51396i			0.861136i	0.902558	+	0.430568i	$0.141687\pi$
$41$			5.51396i			0.861136i	−0.902558	+	0.430568i	$0.858313\pi$
$42$		0			0
$43$		−	4.21172i		−	0.642282i	−0.947031	−	0.321141i	$-0.895934\pi$
$43$		−	4.21172i		−	0.642282i	0.947031	−	0.321141i	$-0.104066\pi$
$44$		0			0
$45$	−3.01159	+	4.06888i	−0.448941	+	0.606553i
$46$		0			0
$47$		−	6.55463i		−	0.956091i	−0.878335	−	0.478045i	$-0.841345\pi$
$47$		−	6.55463i		−	0.956091i	0.878335	−	0.478045i	$-0.158655\pi$
$48$		0			0
$49$	4.62557	−	5.25396i	0.660796	−	0.750566i
$50$		0			0
$51$	−6.39177			−0.895027
$52$		0			0
$53$	3.75594			0.515918			0.257959	−	0.966156i	$-0.416950\pi$
$53$	3.75594			0.515918			0.257959	+	0.966156i	$0.416950\pi$
$54$		0			0
$55$	−4.06711	+	5.49496i	−0.548408	+	0.740940i
$56$		0			0
$57$			3.96296i			0.524906i
$58$		0			0
$59$			3.11915i			0.406079i	0.979171	+	0.203040i	$0.0650820\pi$
$59$			3.11915i			0.406079i	−0.979171	+	0.203040i	$0.934918\pi$
$60$		0			0
$61$	9.55219			1.22303			0.611516	−	0.791232i	$-0.290560\pi$
$61$	9.55219			1.22303			0.611516	+	0.791232i	$0.290560\pi$
$62$		0			0
$63$	5.45812	−	2.46669i	0.687658	−	0.310774i
$64$		0			0
$65$	−5.72454	−	4.23703i	−0.710042	−	0.525539i
$66$		0			0
$67$		−	0.519160i		−	0.0634255i	−0.999497	−	0.0317128i	$-0.989904\pi$
$67$		−	0.519160i		−	0.0634255i	0.999497	−	0.0317128i	$-0.0100962\pi$
$68$		0			0
$69$	0.608133			0.0732107
$70$		0			0
$71$			10.9754i			1.30254i	0.758848	+	0.651268i	$0.225763\pi$
$71$			10.9754i			1.30254i	−0.758848	+	0.651268i	$0.774237\pi$
$72$		0			0
$73$	−2.32017			−0.271555			−0.135778	−	0.990739i	$-0.543353\pi$
$73$	−2.32017			−0.271555			−0.135778	+	0.990739i	$0.543353\pi$
$74$		0			0
$75$	−1.25323	−	4.10277i	−0.144711	−	0.473748i
$76$		0			0
$77$	7.37110	−	3.33123i	0.840015	−	0.379629i
$78$		0			0
$79$		−	9.98693i		−	1.12362i	−0.827267	−	0.561808i	$-0.810106\pi$
$79$		−	9.98693i		−	1.12362i	0.827267	−	0.561808i	$-0.189894\pi$
$80$		0			0
$81$	2.91669			0.324076
$82$		0			0
$83$	−8.73474			−0.958762			−0.479381	−	0.877607i	$-0.659139\pi$
$83$	−8.73474			−0.958762			−0.479381	+	0.877607i	$0.659139\pi$
$84$		0			0
$85$	−9.91032	+	13.3896i	−1.07493	+	1.45230i
$86$		0			0
$87$		−	2.07135i		−	0.222072i
$88$		0			0
$89$			18.5697i			1.96839i	0.177093	+	0.984194i	$0.443331\pi$
$89$			18.5697i			1.96839i	−0.177093	+	0.984194i	$0.556669\pi$
$90$		0			0
$91$	3.47042	+	7.67907i	0.363798	+	0.804986i
$92$		0			0
$93$	−4.41027			−0.457323
$94$		0			0
$95$	8.30166	+	6.14449i	0.851732	+	0.630411i
$96$		0			0
$97$	1.58285			0.160714			0.0803568	−	0.996766i	$-0.474394\pi$
$97$	1.58285			0.160714			0.0803568	+	0.996766i	$0.474394\pi$
$98$		0			0
$99$	6.92135			0.695622
$100$		0			0
$101$	−13.5493			−1.34820			−0.674101	−	0.738639i	$-0.735469\pi$
$101$	−13.5493			−1.34820			−0.674101	+	0.738639i	$0.735469\pi$
$102$		0			0
$103$		−	11.0978i		−	1.09349i	−0.837298	−	0.546747i	$-0.815866\pi$
$103$		−	11.0978i		−	1.09349i	0.837298	−	0.546747i	$-0.184134\pi$
$104$		0			0
$105$	−1.07157	+	4.96150i	−0.104574	+	0.484192i
$106$		0			0
$107$		−	17.5187i		−	1.69359i	−0.531916	−	0.846797i	$-0.678528\pi$
$107$		−	17.5187i		−	1.69359i	0.531916	−	0.846797i	$-0.321472\pi$
$108$		0			0
$109$		−	15.6899i		−	1.50282i	−0.659834	−	0.751412i	$-0.729373\pi$
$109$		−	15.6899i		−	1.50282i	0.659834	−	0.751412i	$-0.270627\pi$
$110$		0			0
$111$	1.77767			0.168729
$112$		0			0
$113$		−	12.3864i		−	1.16521i	−0.812754	−	0.582607i	$-0.802033\pi$
$113$		−	12.3864i		−	1.16521i	0.812754	−	0.582607i	$-0.197967\pi$
$114$		0			0
$115$	0.942900	−	1.27393i	0.0879258	−	0.118794i
$116$		0			0
$117$			7.21053i			0.666614i
$118$		0			0
$119$	17.9612	−	8.11722i	1.64650	−	0.744104i
$120$		0			0
$121$	−1.65282			−0.150256
$122$		0			0
$123$			4.73088i			0.426569i
$124$		0			0
$125$	−10.5377	−	3.73598i	−0.942518	−	0.334156i
$126$		0			0
$127$	−17.2814			−1.53348			−0.766739	−	0.641959i	$-0.778122\pi$
$127$	−17.2814			−1.53348			−0.766739	+	0.641959i	$0.778122\pi$
$128$		0			0
$129$		−	3.61358i		−	0.318158i
$130$		0			0
$131$		−	12.6396i		−	1.10433i	−0.833736	−	0.552164i	$-0.813802\pi$
$131$		−	12.6396i		−	1.10433i	0.833736	−	0.552164i	$-0.186198\pi$
$132$		0			0
$133$	−5.03275	−	11.1361i	−0.436395	−	0.965622i
$134$		0			0
$135$	−6.00798	+	8.11722i	−0.517085	+	0.698619i
$136$		0			0
$137$			11.3987i			0.973857i	0.873442	+	0.486928i	$0.161883\pi$
$137$			11.3987i			0.973857i	−0.873442	+	0.486928i	$0.838117\pi$
$138$		0			0
$139$		−	7.90877i		−	0.670813i	−0.942073	−	0.335407i	$-0.891126\pi$
$139$		−	7.90877i		−	0.670813i	0.942073	−	0.335407i	$-0.108874\pi$
$140$		0			0
$141$		−	5.62376i		−	0.473606i
$142$		0			0
$143$			9.73772i			0.814309i
$144$		0			0
$145$	−4.33909	−	3.21159i	−0.360342	−	0.266708i
$146$		0			0
$147$	3.96866	−	4.50781i	0.327329	−	0.371798i
$148$		0			0
$149$		−	7.33515i		−	0.600919i	−0.953795	−	0.300459i	$-0.902860\pi$
$149$		−	7.33515i		−	0.600919i	0.953795	−	0.300459i	$-0.0971400\pi$
$150$		0			0
$151$		−	8.24264i		−	0.670776i	−0.942080	−	0.335388i	$-0.891133\pi$
$151$		−	8.24264i		−	0.670776i	0.942080	−	0.335388i	$-0.108867\pi$
$152$		0			0
$153$	16.8653			1.36348
$154$		0			0
$155$	−6.83804	+	9.23869i	−0.549244	+	0.742069i
$156$		0			0
$157$			16.8886i			1.34786i	0.738795	+	0.673930i	$0.235395\pi$
$157$			16.8886i			1.34786i	−0.738795	+	0.673930i	$0.764605\pi$
$158$		0			0
$159$	3.22253			0.255563
$160$		0			0
$161$	−1.70888	+	0.772298i	−0.134679	+	0.0608656i
$162$		0			0
$163$			17.5187i			1.37217i	0.727522	+	0.686084i	$0.240672\pi$
$163$			17.5187i			1.37217i	−0.727522	+	0.686084i	$0.759328\pi$
$164$		0			0
$165$	−3.48951	+	4.71458i	−0.271658	+	0.367029i
$166$		0			0
$167$			11.0978i			0.858770i	0.903122	+	0.429385i	$0.141270\pi$
$167$			11.0978i			0.858770i	−0.903122	+	0.429385i	$0.858730\pi$
$168$		0			0
$169$	2.85543			0.219649
$170$		0			0
$171$		−	10.4566i		−	0.799638i
$172$		0			0
$173$			0.430852i			0.0327571i	0.999866	+	0.0163785i	$0.00521369\pi$
$173$			0.430852i			0.0327571i	−0.999866	+	0.0163785i	$0.994786\pi$
$174$		0			0
$175$	8.73196	+	9.93745i	0.660074	+	0.751200i
$176$		0			0
$177$			2.67618i			0.201154i
$178$		0			0
$179$	14.7547			1.10282			0.551410	−	0.834234i	$-0.314090\pi$
$179$	14.7547			1.10282			0.551410	+	0.834234i	$0.314090\pi$
$180$		0			0
$181$	−2.02611			−0.150600			−0.0752999	−	0.997161i	$-0.523991\pi$
$181$	−2.02611			−0.150600			−0.0752999	+	0.997161i	$0.523991\pi$
$182$		0			0
$183$	8.19561			0.605837
$184$		0			0
$185$	2.75624	−	3.72389i	0.202643	−	0.273786i
$186$		0			0
$187$	22.7763			1.66557
$188$		0			0
$189$	10.8887	−	4.92094i	0.792035	−	0.357946i
$190$		0			0
$191$			9.98693i			0.722629i	0.932444	+	0.361314i	$0.117672\pi$
$191$			9.98693i			0.722629i	−0.932444	+	0.361314i	$0.882328\pi$
$192$		0			0
$193$			6.58851i			0.474252i	0.971479	+	0.237126i	$0.0762053\pi$
$193$			6.58851i			0.474252i	−0.971479	+	0.237126i	$0.923795\pi$
$194$		0			0
$195$	−4.91156	−	3.63530i	−0.351724	−	0.260329i
$196$		0			0
$197$	12.1374			0.864755			0.432378	−	0.901693i	$-0.357675\pi$
$197$	12.1374			0.864755			0.432378	+	0.901693i	$0.357675\pi$
$198$		0			0
$199$	−16.1512			−1.14493			−0.572465	−	0.819929i	$-0.694013\pi$
$199$	−16.1512			−1.14493			−0.572465	+	0.819929i	$0.694013\pi$
$200$		0			0
$201$		−	0.445430i		−	0.0314182i
$202$		0			0
$203$	2.63051	+	5.82059i	0.184625	+	0.408525i
$204$		0			0
$205$	9.91032	+	7.33515i	0.692167	+	0.512309i
$206$		0			0
$207$	−1.60462			−0.111528
$208$		0			0
$209$		−	14.1215i		−	0.976805i
$210$		0			0
$211$	−1.35578			−0.0933360			−0.0466680	−	0.998910i	$-0.514860\pi$
$211$	−1.35578			−0.0933360			−0.0466680	+	0.998910i	$0.514860\pi$
$212$		0			0
$213$			9.41667i			0.645219i
$214$		0			0
$215$	−7.56979	−	5.60280i	−0.516255	−	0.382108i
$216$		0			0
$217$	12.3931	−	5.60081i	0.841296	−	0.380208i
$218$		0			0
$219$	−1.99067			−0.134517
$220$		0			0
$221$			23.7279i			1.59611i
$222$		0			0
$223$			8.30493i			0.556139i	0.960561	+	0.278070i	$0.0896946\pi$
$223$			8.30493i			0.556139i	−0.960561	+	0.278070i	$0.910305\pi$
$224$		0			0
$225$	3.30678	+	10.8255i	0.220452	+	0.721703i
$226$		0			0
$227$	−6.28883			−0.417404			−0.208702	−	0.977979i	$-0.566924\pi$
$227$	−6.28883			−0.417404			−0.208702	+	0.977979i	$0.566924\pi$
$228$		0			0
$229$	23.8298			1.57472			0.787359	−	0.616495i	$-0.211448\pi$
$229$	23.8298			1.57472			0.787359	+	0.616495i	$0.211448\pi$
$230$		0			0
$231$	6.32428	−	2.85814i	0.416107	−	0.188052i
$232$		0			0
$233$			5.79789i			0.379832i	0.981800	+	0.189916i	$0.0608216\pi$
$233$			5.79789i			0.379832i	−0.981800	+	0.189916i	$0.939178\pi$
$234$		0			0
$235$	−11.7807	−	8.71953i	−0.768490	−	0.568799i
$236$		0			0
$237$		−	8.56861i		−	0.556591i
$238$		0			0
$239$		−	22.2759i		−	1.44091i	−0.693502	−	0.720454i	$-0.743933\pi$
$239$		−	22.2759i		−	1.44091i	0.693502	−	0.720454i	$-0.256067\pi$
$240$		0			0
$241$		−	22.0893i		−	1.42290i	−0.702739	−	0.711448i	$-0.748040\pi$
$241$		−	22.0893i		−	1.42290i	0.702739	−	0.711448i	$-0.251960\pi$
$242$		0			0
$243$	16.0514			1.02970
$244$		0			0
$245$	−3.28969	−	15.3029i	−0.210170	−	0.977665i
$246$		0			0
$247$	14.7115			0.936072
$248$		0			0
$249$	−7.49425			−0.474929
$250$		0			0
$251$			15.9294i			1.00546i	0.864444	+	0.502729i	$0.167671\pi$
$251$			15.9294i			1.00546i	−0.864444	+	0.502729i	$0.832329\pi$
$252$		0			0
$253$	−2.16701			−0.136239
$254$		0			0
$255$	−8.50288	+	11.4880i	−0.532471	+	0.719407i
$256$		0			0
$257$	−9.71846			−0.606221			−0.303110	−	0.952955i	$-0.598025\pi$
$257$	−9.71846			−0.606221			−0.303110	+	0.952955i	$0.598025\pi$
$258$		0			0
$259$	−4.99533	+	2.25755i	−0.310395	+	0.140277i
$260$		0			0
$261$			5.46544i			0.338302i
$262$		0			0
$263$	−8.90441			−0.549069			−0.274535	−	0.961577i	$-0.588524\pi$
$263$	−8.90441			−0.549069			−0.274535	+	0.961577i	$0.588524\pi$
$264$		0			0
$265$	4.99647	−	6.75060i	0.306931	−	0.414686i
$266$		0			0
$267$			15.9325i			0.975054i
$268$		0			0
$269$	−4.80691			−0.293083			−0.146541	−	0.989205i	$-0.546814\pi$
$269$	−4.80691			−0.293083			−0.146541	+	0.989205i	$0.546814\pi$
$270$		0			0
$271$	12.0056			0.729286			0.364643	−	0.931147i	$-0.381191\pi$
$271$	12.0056			0.729286			0.364643	+	0.931147i	$0.381191\pi$
$272$		0			0
$273$	2.97756	+	6.58851i	0.180210	+	0.398755i
$274$		0			0
$275$	4.46575	+	14.6197i	0.269295	+	0.881603i
$276$		0			0
$277$	1.32250			0.0794612			0.0397306	−	0.999210i	$-0.487350\pi$
$277$	1.32250			0.0794612			0.0397306	+	0.999210i	$0.487350\pi$
$278$		0			0
$279$	11.6369			0.696683
$280$		0			0
$281$	−13.6055			−0.811636			−0.405818	−	0.913954i	$-0.633013\pi$
$281$	−13.6055			−0.811636			−0.405818	+	0.913954i	$0.633013\pi$
$282$		0			0
$283$	4.03383			0.239786			0.119893	−	0.992787i	$-0.461745\pi$
$283$	4.03383			0.239786			0.119893	+	0.992787i	$0.461745\pi$
$284$		0			0
$285$	7.12268	+	5.27187i	0.421911	+	0.312278i
$286$		0			0
$287$	−6.00798	−	13.2940i	−0.354640	−	0.784720i
$288$		0			0
$289$	38.4990			2.26465
$290$		0			0
$291$	1.35805			0.0796106
$292$		0			0
$293$			15.1789i			0.886759i	0.896334	+	0.443379i	$0.146221\pi$
$293$			15.1789i			0.886759i	−0.896334	+	0.443379i	$0.853779\pi$
$294$		0			0
$295$	5.60610	+	4.14937i	0.326400	+	0.241585i
$296$		0			0
$297$	13.8078			0.801208
$298$		0			0
$299$		−	2.25755i		−	0.130557i
$300$		0			0
$301$	4.58907	+	10.1543i	0.264509	+	0.585287i
$302$		0			0
$303$	−11.6250			−0.667840
$304$		0			0
$305$	12.7071	−	17.1683i	0.727609	−	0.983053i
$306$		0			0
$307$	15.7575			0.899329			0.449664	−	0.893198i	$-0.351544\pi$
$307$	15.7575			0.899329			0.449664	+	0.893198i	$0.351544\pi$
$308$		0			0
$309$		−	9.52168i		−	0.541669i
$310$		0			0
$311$	−27.2794			−1.54687			−0.773436	−	0.633874i	$-0.781464\pi$
$311$	−27.2794			−1.54687			−0.773436	+	0.633874i	$0.781464\pi$
$312$		0			0
$313$	7.37828			0.417045			0.208523	−	0.978018i	$-0.433135\pi$
$313$	7.37828			0.417045			0.208523	+	0.978018i	$0.433135\pi$
$314$		0			0
$315$	2.82743	−	13.0914i	0.159308	−	0.737615i
$316$		0			0
$317$	−27.5478			−1.54724			−0.773619	−	0.633651i	$-0.781556\pi$
$317$	−27.5478			−1.54724			−0.773619	+	0.633651i	$0.781556\pi$
$318$		0			0
$319$			7.38099i			0.413256i
$320$		0			0
$321$		−	15.0307i		−	0.838933i
$322$		0			0
$323$		−	34.4099i		−	1.91462i
$324$		0			0
$325$	−15.2306	+	4.65233i	−0.844840	+	0.258065i
$326$		0			0
$327$		−	13.4617i		−	0.744433i
$328$		0			0
$329$	7.14188	+	15.8030i	0.393745	+	0.871249i
$330$		0			0
$331$	30.0660			1.65258			0.826288	−	0.563248i	$-0.190448\pi$
$331$	30.0660			1.65258			0.826288	+	0.563248i	$0.190448\pi$
$332$		0			0
$333$	−4.69054			−0.257040
$334$		0			0
$335$	−0.933094	−	0.690632i	−0.0509804	−	0.0377332i
$336$		0			0
$337$			2.07740i			0.113163i	0.998398	+	0.0565816i	$0.0180201\pi$
$337$			2.07740i			0.113163i	−0.998398	+	0.0565816i	$0.981980\pi$
$338$		0			0
$339$		−	10.6273i		−	0.577197i
$340$		0			0
$341$	15.7154			0.851039
$342$		0			0
$343$	−5.42743	+	17.7071i	−0.293054	+	0.956096i
$344$		0			0
$345$	0.808991	−	1.09301i	0.0435546	−	0.0588455i
$346$		0			0
$347$			1.58617i			0.0851499i	0.999093	+	0.0425749i	$0.0135561\pi$
$347$			1.58617i			0.0851499i	−0.999093	+	0.0425749i	$0.986444\pi$
$348$		0			0
$349$	0.0939008			0.00502640			0.00251320	−	0.999997i	$-0.499200\pi$
$349$	0.0939008			0.00502640			0.00251320	+	0.999997i	$0.499200\pi$
$350$		0			0
$351$			14.3847i			0.767797i
$352$		0			0
$353$	−16.6771			−0.887634			−0.443817	−	0.896117i	$-0.646376\pi$
$353$	−16.6771			−0.887634			−0.443817	+	0.896117i	$0.646376\pi$
$354$		0			0
$355$	19.7262	+	14.6004i	1.04696	+	0.774907i
$356$		0			0
$357$	15.4104	−	6.96443i	0.815603	−	0.368597i
$358$		0			0
$359$		−	19.7492i		−	1.04232i	−0.853458	−	0.521162i	$-0.825499\pi$
$359$		−	19.7492i		−	1.04232i	0.853458	−	0.521162i	$-0.174501\pi$
$360$		0			0
$361$	−2.33445			−0.122866
$362$		0			0
$363$	−1.41809			−0.0744305
$364$		0			0
$365$	−3.08649	+	4.17008i	−0.161554	+	0.218272i
$366$		0			0
$367$			28.2305i			1.47362i	0.676101	+	0.736809i	$0.263668\pi$
$367$			28.2305i			1.47362i	−0.676101	+	0.736809i	$0.736332\pi$
$368$		0			0
$369$		−	12.4829i		−	0.649832i
$370$		0			0
$371$	−9.05546	+	4.09245i	−0.470136	+	0.212469i
$372$		0			0
$373$	11.2527			0.582643			0.291321	−	0.956625i	$-0.405905\pi$
$373$	11.2527			0.582643			0.291321	+	0.956625i	$0.405905\pi$
$374$		0			0
$375$	−9.04114	−	3.20541i	−0.466882	−	0.165527i
$376$		0			0
$377$	−7.68938			−0.396023
$378$		0			0
$379$	−6.92135			−0.355526			−0.177763	−	0.984073i	$-0.556886\pi$
$379$	−6.92135			−0.355526			−0.177763	+	0.984073i	$0.556886\pi$
$380$		0			0
$381$	−14.8272			−0.759618
$382$		0			0
$383$		−	11.7492i		−	0.600357i	−0.953883	−	0.300178i	$-0.902954\pi$
$383$		−	11.7492i		−	0.600357i	0.953883	−	0.300178i	$-0.0970462\pi$
$384$		0			0
$385$	3.81840	−	17.6797i	0.194604	−	0.901040i
$386$		0			0
$387$			9.53477i			0.484680i
$388$		0			0
$389$		−	37.4556i		−	1.89907i	−0.313654	−	0.949537i	$-0.601553\pi$
$389$		−	37.4556i		−	1.89907i	0.313654	−	0.949537i	$-0.398447\pi$
$390$		0			0
$391$	−5.28035			−0.267039
$392$		0			0
$393$		−	10.8446i		−	0.547036i
$394$		0			0
$395$	−17.9496	−	13.2855i	−0.903145	−	0.668465i
$396$		0			0
$397$		−	26.0150i		−	1.30565i	−0.757507	−	0.652827i	$-0.773583\pi$
$397$		−	26.0150i		−	1.30565i	0.757507	−	0.652827i	$-0.226417\pi$
$398$		0			0
$399$	−4.31801	−	9.55457i	−0.216171	−	0.478327i
$400$		0			0
$401$	−10.3832			−0.518510			−0.259255	−	0.965809i	$-0.583477\pi$
$401$	−10.3832			−0.518510			−0.259255	+	0.965809i	$0.583477\pi$
$402$		0			0
$403$			16.3721i			0.815550i
$404$		0			0
$405$	3.88003	−	5.24220i	0.192800	−	0.260487i
$406$		0			0
$407$	−6.33450			−0.313990
$408$		0			0
$409$			7.50838i			0.371265i	0.982619	+	0.185633i	$0.0594335\pi$
$409$			7.50838i			0.371265i	−0.982619	+	0.185633i	$0.940567\pi$
$410$		0			0
$411$			9.77989i			0.482406i
$412$		0			0
$413$	−3.39861	−	7.52019i	−0.167235	−	0.370044i
$414$		0			0
$415$	−11.6197	+	15.6991i	−0.570389	+	0.770637i
$416$		0			0
$417$		−	6.78559i		−	0.332292i
$418$		0			0
$419$			22.3526i			1.09200i	0.837786	+	0.545999i	$0.183850\pi$
$419$			22.3526i			1.09200i	−0.837786	+	0.545999i	$0.816150\pi$
$420$		0			0
$421$			1.24732i			0.0607905i	0.999538	+	0.0303953i	$0.00967660\pi$
$421$			1.24732i			0.0607905i	−0.999538	+	0.0303953i	$0.990323\pi$
$422$		0			0
$423$			14.8388i			0.721487i
$424$		0			0
$425$	10.8817	+	35.6239i	0.527840	+	1.72801i
$426$		0			0
$427$	−23.0301	+	10.4080i	−1.11450	+	0.503679i
$428$		0			0
$429$			8.35479i			0.403373i
$430$		0			0
$431$		−	8.69462i		−	0.418805i	−0.977830	−	0.209402i	$-0.932848\pi$
$431$		−	8.69462i		−	0.418805i	0.977830	−	0.209402i	$-0.0671519\pi$
$432$		0			0
$433$	−25.0439			−1.20353			−0.601767	−	0.798672i	$-0.705536\pi$
$433$	−25.0439			−1.20353			−0.601767	+	0.798672i	$0.705536\pi$
$434$		0			0
$435$	−3.72286	−	2.75549i	−0.178498	−	0.132115i
$436$		0			0
$437$			3.27387i			0.156610i
$438$		0			0
$439$	37.3313			1.78173			0.890864	−	0.454270i	$-0.150100\pi$
$439$	37.3313			1.78173			0.890864	+	0.454270i	$0.150100\pi$
$440$		0			0
$441$	−10.4717	+	11.8943i	−0.498651	+	0.566393i
$442$		0			0
$443$		−	25.4673i		−	1.20999i	−0.796230	−	0.604994i	$-0.793176\pi$
$443$		−	25.4673i		−	1.20999i	0.796230	−	0.604994i	$-0.206824\pi$
$444$		0			0
$445$	33.3756	+	24.7031i	1.58216	+	1.17104i
$446$		0			0
$447$		−	6.29343i		−	0.297669i
$448$		0			0
$449$	−4.73840			−0.223619			−0.111810	−	0.993730i	$-0.535665\pi$
$449$	−4.73840			−0.223619			−0.111810	+	0.993730i	$0.535665\pi$
$450$		0			0
$451$		−	16.8579i		−	0.793808i
$452$		0			0
$453$		−	7.07204i		−	0.332273i
$454$		0			0
$455$	18.4184	+	3.97794i	0.863466	+	0.186489i
$456$		0			0
$457$			11.9469i			0.558850i	0.960167	+	0.279425i	$0.0901439\pi$
$457$			11.9469i			0.558850i	−0.960167	+	0.279425i	$0.909856\pi$
$458$		0			0
$459$	33.6454			1.57043
$460$		0			0
$461$	−32.0778			−1.49401			−0.747005	−	0.664818i	$-0.768509\pi$
$461$	−32.0778			−1.49401			−0.747005	+	0.664818i	$0.768509\pi$
$462$		0			0
$463$	21.3332			0.991438			0.495719	−	0.868483i	$-0.334905\pi$
$463$	21.3332			0.991438			0.495719	+	0.868483i	$0.334905\pi$
$464$		0			0
$465$	−5.86692	+	7.92664i	−0.272072	+	0.367589i
$466$		0			0
$467$	−38.4098			−1.77739			−0.888696	−	0.458497i	$-0.848388\pi$
$467$	−38.4098			−1.77739			−0.888696	+	0.458497i	$0.848388\pi$
$468$		0			0
$469$	0.565674	+	1.25168i	0.0261204	+	0.0577972i
$470$		0			0
$471$			14.4902i			0.667671i
$472$		0			0
$473$			12.8766i			0.592065i
$474$		0			0
$475$	22.0872	−	6.74675i	1.01343	−	0.309562i
$476$		0			0
$477$	−8.50294			−0.389323
$478$		0			0
$479$	−2.77227			−0.126668			−0.0633342	−	0.997992i	$-0.520173\pi$
$479$	−2.77227			−0.126668			−0.0633342	+	0.997992i	$0.520173\pi$
$480$		0			0
$481$		−	6.59917i		−	0.300896i
$482$		0			0
$483$	−1.46619	+	0.662619i	−0.0667141	+	0.0301502i
$484$		0			0
$485$	2.10564	−	2.84487i	0.0956121	−	0.129179i
$486$		0			0
$487$	15.3310			0.694712			0.347356	−	0.937733i	$-0.387079\pi$
$487$	15.3310			0.694712			0.347356	+	0.937733i	$0.387079\pi$
$488$		0			0
$489$			15.0307i			0.679713i
$490$		0			0
$491$	−3.59438			−0.162212			−0.0811061	−	0.996705i	$-0.525845\pi$
$491$	−3.59438			−0.162212			−0.0811061	+	0.996705i	$0.525845\pi$
$492$		0			0
$493$			17.9853i			0.810016i
$494$		0			0
$495$	9.20738	−	12.4398i	0.413841	−	0.559129i
$496$		0			0
$497$	−11.9587	−	26.4613i	−0.536420	−	1.18695i
$498$		0			0
$499$	−14.2057			−0.635933			−0.317966	−	0.948102i	$-0.603000\pi$
$499$	−14.2057			−0.635933			−0.317966	+	0.948102i	$0.603000\pi$
$500$		0			0
$501$			9.52168i			0.425397i
$502$		0			0
$503$		−	19.0792i		−	0.850698i	−0.905029	−	0.425349i	$-0.860151\pi$
$503$		−	19.0792i		−	0.850698i	0.905029	−	0.425349i	$-0.139849\pi$
$504$		0			0
$505$	−18.0244	+	24.3523i	−0.802075	+	1.08366i
$506$		0			0
$507$	2.44991			0.108804
$508$		0			0
$509$	5.06084			0.224318			0.112159	−	0.993690i	$-0.464223\pi$
$509$	5.06084			0.224318			0.112159	+	0.993690i	$0.464223\pi$
$510$		0			0
$511$	5.59387	−	2.52804i	0.247458	−	0.111834i
$512$		0			0
$513$		−	20.8605i		−	0.921012i
$514$		0			0
$515$	−19.9462	−	14.7632i	−0.878932	−	0.650544i
$516$		0			0
$517$			20.0396i			0.881339i
$518$		0			0
$519$			0.369664i			0.0162264i
$520$		0			0
$521$			36.4725i			1.59789i	0.601404	+	0.798945i	$0.294608\pi$
$521$			36.4725i			1.59789i	−0.601404	+	0.798945i	$0.705392\pi$
$522$		0			0
$523$	−23.4497			−1.02538			−0.512691	−	0.858573i	$-0.671351\pi$
$523$	−23.4497			−1.02538			−0.512691	+	0.858573i	$0.671351\pi$
$524$		0			0
$525$	7.49187	+	8.52616i	0.326972	+	0.372112i
$526$		0			0
$527$	38.2938			1.66811
$528$		0			0
$529$	−22.4976			−0.978157
$530$		0			0
$531$		−	7.06134i		−	0.306436i
$532$		0			0
$533$	17.5623			0.760706
$534$		0			0
$535$	−31.4866	−	23.3049i	−1.36128	−	1.00756i
$536$		0			0
$537$	12.6593			0.546289
$538$		0			0
$539$	−14.1418	+	16.0630i	−0.609132	+	0.691883i
$540$		0			0
$541$		−	25.4539i		−	1.09435i	−0.837018	−	0.547175i	$-0.815703\pi$
$541$		−	25.4539i		−	1.09435i	0.837018	−	0.547175i	$-0.184297\pi$
$542$		0			0
$543$	−1.73837			−0.0746006
$544$		0			0
$545$	−28.1997	−	20.8721i	−1.20794	−	0.894063i
$546$		0			0
$547$			13.6735i			0.584636i	0.956321	+	0.292318i	$0.0944266\pi$
$547$			13.6735i			0.584636i	−0.956321	+	0.292318i	$0.905573\pi$
$548$		0			0
$549$	−21.6249			−0.922927
$550$		0			0
$551$	11.1510			0.475050
$552$		0			0
$553$	10.8817	+	24.0782i	0.462737	+	1.02391i
$554$		0			0
$555$	2.36481	−	3.19503i	0.100381	−	0.135621i
$556$		0			0
$557$	−25.2870			−1.07144			−0.535722	−	0.844395i	$-0.679960\pi$
$557$	−25.2870			−1.07144			−0.535722	+	0.844395i	$0.679960\pi$
$558$		0			0
$559$	−13.4146			−0.567376
$560$		0			0
$561$	19.5417			0.825049
$562$		0			0
$563$	−17.9372			−0.755962			−0.377981	−	0.925813i	$-0.623382\pi$
$563$	−17.9372			−0.755962			−0.377981	+	0.925813i	$0.623382\pi$
$564$		0			0
$565$	−22.2623	−	16.4775i	−0.936580	−	0.693212i
$566$		0			0
$567$	−7.03205	+	3.17800i	−0.295318	+	0.133464i
$568$		0			0
$569$	−12.5418			−0.525781			−0.262891	−	0.964826i	$-0.584676\pi$
$569$	−12.5418			−0.525781			−0.262891	+	0.964826i	$0.584676\pi$
$570$		0			0
$571$	−27.0229			−1.13088			−0.565438	−	0.824791i	$-0.691293\pi$
$571$	−27.0229			−1.13088			−0.565438	+	0.824791i	$0.691293\pi$
$572$		0			0
$573$			8.56861i			0.357959i
$574$		0			0
$575$	−1.03532	−	3.38937i	−0.0431758	−	0.141347i
$576$		0			0
$577$	−13.9759			−0.581825			−0.290912	−	0.956750i	$-0.593959\pi$
$577$	−13.9759			−0.581825			−0.290912	+	0.956750i	$0.593959\pi$
$578$		0			0
$579$			5.65283i			0.234924i
$580$		0			0
$581$	21.0592	−	9.51732i	0.873683	−	0.394845i
$582$		0			0
$583$	−11.4831			−0.475581
$584$		0			0
$585$	12.9596	+	9.59208i	0.535813	+	0.396584i
$586$		0			0
$587$	−28.5757			−1.17945			−0.589723	−	0.807605i	$-0.700763\pi$
$587$	−28.5757			−1.17945			−0.589723	+	0.807605i	$0.700763\pi$
$588$		0			0
$589$		−	23.7425i		−	0.978294i
$590$		0			0
$591$	10.4137			0.428362
$592$		0			0
$593$	0.693990			0.0284987			0.0142494	−	0.999898i	$-0.495464\pi$
$593$	0.693990			0.0284987			0.0142494	+	0.999898i	$0.495464\pi$
$594$		0			0
$595$	9.30431	−	43.0801i	0.381439	−	1.76611i
$596$		0			0
$597$	−13.8575			−0.567149
$598$		0			0
$599$			30.4359i			1.24358i	0.783185	+	0.621789i	$0.213594\pi$
$599$			30.4359i			1.24358i	−0.783185	+	0.621789i	$0.786406\pi$
$600$		0			0
$601$			22.1088i			0.901835i	0.892566	+	0.450917i	$0.148903\pi$
$601$			22.1088i			0.901835i	−0.892566	+	0.450917i	$0.851097\pi$
$602$		0			0
$603$			1.17531i			0.0478623i
$604$		0			0
$605$	−2.19872	+	2.97064i	−0.0893908	+	0.120774i
$606$		0			0
$607$			0.411826i			0.0167155i	0.999965	+	0.00835774i	$0.00266038\pi$
$607$			0.411826i			0.0167155i	−0.999965	+	0.00835774i	$0.997340\pi$
$608$		0			0
$609$	2.25693	+	4.99396i	0.0914554	+	0.202366i
$610$		0			0
$611$	−20.8768			−0.844586
$612$		0			0
$613$	−38.0372			−1.53631			−0.768155	−	0.640264i	$-0.778825\pi$
$613$	−38.0372			−1.53631			−0.768155	+	0.640264i	$0.778825\pi$
$614$		0			0
$615$	8.50288	+	6.29343i	0.342869	+	0.253776i
$616$		0			0
$617$		−	11.2523i		−	0.452999i	−0.974011	−	0.226499i	$-0.927272\pi$
$617$		−	11.2523i		−	0.452999i	0.974011	−	0.226499i	$-0.0727281\pi$
$618$		0			0
$619$		−	15.2620i		−	0.613433i	−0.951801	−	0.306716i	$-0.900770\pi$
$619$		−	15.2620i		−	0.613433i	0.951801	−	0.306716i	$-0.0992303\pi$
$620$		0			0
$621$	−3.20113			−0.128457
$622$		0			0
$623$	−20.2335	−	44.7711i	−0.810637	−	1.79372i
$624$		0			0
$625$	−20.7328	+	13.9696i	−0.829314	+	0.558783i
$626$		0			0
$627$		−	12.1160i		−	0.483867i
$628$		0			0
$629$	−15.4353			−0.615445
$630$		0			0
$631$		−	41.5945i		−	1.65585i	−0.560839	−	0.827925i	$-0.689521\pi$
$631$		−	41.5945i		−	1.65585i	0.560839	−	0.827925i	$-0.310479\pi$
$632$		0			0
$633$	−1.16324			−0.0462346
$634$		0			0
$635$	−22.9892	+	31.0601i	−0.912300	+	1.23258i
$636$		0			0
$637$	−16.7341	−	14.7327i	−0.663031	−	0.583730i
$638$		0			0
$639$		−	24.8467i		−	0.982922i
$640$		0			0
$641$	19.7501			0.780080			0.390040	−	0.920798i	$-0.372461\pi$
$641$	19.7501			0.780080			0.390040	+	0.920798i	$0.372461\pi$
$642$		0			0
$643$	−1.64849			−0.0650100			−0.0325050	−	0.999472i	$-0.510348\pi$
$643$	−1.64849			−0.0650100			−0.0325050	+	0.999472i	$0.510348\pi$
$644$		0			0
$645$	−6.49475	−	4.80710i	−0.255730	−	0.189279i
$646$		0			0
$647$		−	8.32199i		−	0.327171i	−0.986529	−	0.163586i	$-0.947694\pi$
$647$		−	8.32199i		−	0.327171i	0.986529	−	0.163586i	$-0.0523060\pi$
$648$		0			0
$649$		−	9.53623i		−	0.374330i
$650$		0			0
$651$	10.6330	−	4.80540i	0.416741	−	0.188339i
$652$		0			0
$653$	47.2746			1.85000			0.924999	−	0.379968i	$-0.124065\pi$
$653$	47.2746			1.85000			0.924999	+	0.379968i	$0.124065\pi$
$654$		0			0
$655$	−22.7173	−	16.8143i	−0.887640	−	0.656989i
$656$		0			0
$657$	5.25256			0.204922
$658$		0			0
$659$	23.0319			0.897194			0.448597	−	0.893734i	$-0.351924\pi$
$659$	23.0319			0.897194			0.448597	+	0.893734i	$0.351924\pi$
$660$		0			0
$661$	14.2975			0.556107			0.278054	−	0.960566i	$-0.410311\pi$
$661$	14.2975			0.556107			0.278054	+	0.960566i	$0.410311\pi$
$662$		0			0
$663$			20.3581i			0.790644i
$664$		0			0
$665$	−26.7101	−	5.76876i	−1.03577	−	0.223703i
$666$		0			0
$667$		−	1.71118i		−	0.0662570i
$668$		0			0
$669$			7.12548i			0.275487i
$670$		0			0
$671$	−29.2041			−1.12741
$672$		0			0
$673$		−	27.1433i		−	1.04630i	−0.852242	−	0.523149i	$-0.824757\pi$
$673$		−	27.1433i		−	1.04630i	0.852242	−	0.523149i	$-0.175243\pi$
$674$		0			0
$675$	6.59686	+	21.5965i	0.253913	+	0.831248i
$676$		0			0
$677$		−	11.9050i		−	0.457546i	−0.973480	−	0.228773i	$-0.926529\pi$
$677$		−	11.9050i		−	0.457546i	0.973480	−	0.228773i	$-0.0734714\pi$
$678$		0			0
$679$	−3.81620	+	1.72466i	−0.146452	+	0.0661864i
$680$		0			0
$681$	−5.39571			−0.206764
$682$		0			0
$683$			10.0550i			0.384744i	0.981322	+	0.192372i	$0.0616180\pi$
$683$			10.0550i			0.384744i	−0.981322	+	0.192372i	$0.938382\pi$
$684$		0			0
$685$	20.4871	+	15.1635i	0.782770	+	0.579369i
$686$		0			0
$687$	20.4456			0.780047
$688$		0			0
$689$		−	11.9629i		−	0.455749i
$690$		0			0
$691$			6.60195i			0.251150i	0.992084	+	0.125575i	$0.0400776\pi$
$691$			6.60195i			0.251150i	−0.992084	+	0.125575i	$0.959922\pi$
$692$		0			0
$693$	−16.6872	+	7.54147i	−0.633894	+	0.286477i
$694$		0			0
$695$	−14.2145	−	10.5209i	−0.539188	−	0.399082i
$696$		0			0
$697$		−	41.0777i		−	1.55593i
$698$		0			0
$699$			4.97449i			0.188152i
$700$		0			0
$701$			2.49209i			0.0941251i	0.998892	+	0.0470625i	$0.0149860\pi$
$701$			2.49209i			0.0941251i	−0.998892	+	0.0470625i	$0.985014\pi$
$702$		0			0
$703$			9.57003i			0.360940i
$704$		0			0
$705$	−10.1077	−	7.48121i	−0.380676	−	0.281758i
$706$		0			0
$707$	32.6669	−	14.7632i	1.22856	−	0.555227i
$708$		0			0
$709$			39.3253i			1.47689i	0.674312	+	0.738447i	$0.264440\pi$
$709$			39.3253i			1.47689i	−0.674312	+	0.738447i	$0.735560\pi$
$710$		0			0
$711$			22.6091i			0.847906i
$712$		0			0
$713$	−3.64340			−0.136446
$714$		0			0
$715$	17.5017	+	12.9539i	0.654528	+	0.484450i
$716$		0			0
$717$		−	19.1123i		−	0.713763i
$718$		0			0
$719$	−25.2731			−0.942528			−0.471264	−	0.881992i	$-0.656202\pi$
$719$	−25.2731			−0.942528			−0.471264	+	0.881992i	$0.656202\pi$
$720$		0			0
$721$	12.0920	+	26.7564i	0.450331	+	0.996459i
$722$		0			0
$723$		−	18.9522i		−	0.704841i
$724$		0			0
$725$	−11.5445	+	3.52638i	−0.428751	+	0.130966i
$726$		0			0
$727$		−	7.09135i		−	0.263004i	−0.991316	−	0.131502i	$-0.958020\pi$
$727$		−	7.09135i		−	0.263004i	0.991316	−	0.131502i	$-0.0419799\pi$
$728$		0			0
$729$	5.02175			0.185991
$730$		0			0
$731$			31.3763i			1.16050i
$732$		0			0
$733$		−	6.53560i		−	0.241398i	−0.992689	−	0.120699i	$-0.961486\pi$
$733$		−	6.53560i		−	0.241398i	0.992689	−	0.120699i	$-0.0385136\pi$
$734$		0			0
$735$	−2.82249	−	13.1296i	−0.104109	−	0.484293i
$736$		0			0
$737$			1.58724i			0.0584666i
$738$		0			0
$739$	−14.1221			−0.519490			−0.259745	−	0.965677i	$-0.583639\pi$
$739$	−14.1221			−0.519490			−0.259745	+	0.965677i	$0.583639\pi$
$740$		0			0
$741$	12.6222			0.463689
$742$		0			0
$743$	−2.66196			−0.0976578			−0.0488289	−	0.998807i	$-0.515549\pi$
$743$	−2.66196			−0.0976578			−0.0488289	+	0.998807i	$0.515549\pi$
$744$		0			0
$745$	−13.1836	−	9.75785i	−0.483008	−	0.357500i
$746$		0			0
$747$	19.7743			0.723503
$748$		0			0
$749$	19.0882	+	42.2370i	0.697469	+	1.54331i
$750$		0			0
$751$		−	7.53982i		−	0.275132i	−0.990493	−	0.137566i	$-0.956072\pi$
$751$		−	7.53982i		−	0.275132i	0.990493	−	0.137566i	$-0.0439279\pi$
$752$		0			0
$753$			13.6672i			0.498060i
$754$		0			0
$755$	−14.8146	−	10.9651i	−0.539159	−	0.399060i
$756$		0			0
$757$	42.4127			1.54152			0.770759	−	0.637127i	$-0.219877\pi$
$757$	42.4127			1.54152			0.770759	+	0.637127i	$0.219877\pi$
$758$		0			0
$759$	−1.85926			−0.0674867
$760$		0			0
$761$		−	21.4947i		−	0.779183i	−0.920988	−	0.389591i	$-0.872616\pi$
$761$		−	21.4947i		−	0.779183i	0.920988	−	0.389591i	$-0.127384\pi$
$762$		0			0
$763$	17.0957	+	37.8280i	0.618905	+	1.36947i
$764$		0			0
$765$	22.4356	−	30.3122i	0.811162	−	1.09594i
$766$		0			0
$767$	9.93467			0.358720
$768$		0			0
$769$			19.1155i			0.689321i	0.938727	+	0.344661i	$0.112006\pi$
$769$			19.1155i			0.689321i	−0.938727	+	0.344661i	$0.887994\pi$
$770$		0			0
$771$	−8.33827			−0.300295
$772$		0			0
$773$		−	25.8616i		−	0.930177i	−0.885264	−	0.465089i	$-0.846022\pi$
$773$		−	25.8616i		−	0.930177i	0.885264	−	0.465089i	$-0.153978\pi$
$774$		0			0
$775$	7.50828	+	24.5802i	0.269705	+	0.882947i
$776$		0			0
$777$	−4.28591	+	1.93694i	−0.153756	+	0.0694872i
$778$		0			0
$779$	−25.4686			−0.912506
$780$		0			0
$781$		−	33.5551i		−	1.20070i
$782$		0			0
$783$			10.9033i			0.389652i
$784$		0			0
$785$	30.3542	+	22.4667i	1.08339	+	0.801872i
$786$		0			0
$787$	−9.23137			−0.329063			−0.164531	−	0.986372i	$-0.552611\pi$
$787$	−9.23137			−0.329063			−0.164531	+	0.986372i	$0.552611\pi$
$788$		0			0
$789$	−7.63983			−0.271985
$790$		0			0
$791$	13.4962	+	29.8633i	0.479868	+	1.06182i
$792$		0			0
$793$		−	30.4242i		−	1.08040i
$794$		0			0
$795$	4.28689	−	5.79190i	0.152040	−	0.205418i
$796$		0			0
$797$		−	1.25393i		−	0.0444165i	−0.999753	−	0.0222083i	$-0.992930\pi$
$797$		−	1.25393i		−	0.0444165i	0.999753	−	0.0222083i	$-0.00706969\pi$
$798$		0			0
$799$			48.8304i			1.72750i
$800$		0			0
$801$		−	42.0394i		−	1.48539i
$802$		0			0
$803$	7.09349			0.250324
$804$		0			0
$805$	−0.885242	+	4.09878i	−0.0312007	+	0.144463i
$806$		0			0
$807$	−4.12425			−0.145180
$808$		0			0
$809$	−41.6114			−1.46298			−0.731489	−	0.681854i	$-0.761174\pi$
$809$	−41.6114			−1.46298			−0.731489	+	0.681854i	$0.761174\pi$
$810$		0			0
$811$		−	14.9717i		−	0.525728i	−0.964833	−	0.262864i	$-0.915333\pi$
$811$		−	14.9717i		−	0.525728i	0.964833	−	0.262864i	$-0.0846671\pi$
$812$		0			0
$813$	10.3006			0.361257
$814$		0			0
$815$	31.4866	+	23.3049i	1.10293	+	0.816334i
$816$		0			0
$817$	19.4536			0.680596
$818$		0			0
$819$	−7.85656	−	17.3844i	−0.274530	−	0.607460i
$820$		0			0
$821$		−	0.316827i		−	0.0110574i	−0.999985	−	0.00552868i	$-0.998240\pi$
$821$		−	0.316827i		−	0.0110574i	0.999985	−	0.00552868i	$-0.00175984\pi$
$822$		0			0
$823$	5.40508			0.188409			0.0942046	−	0.995553i	$-0.469969\pi$
$823$	5.40508			0.188409			0.0942046	+	0.995553i	$0.469969\pi$
$824$		0			0
$825$	3.83154	+	12.5435i	0.133397	+	0.436708i
$826$		0			0
$827$			35.9961i			1.25171i	0.779941	+	0.625853i	$0.215249\pi$
$827$			35.9961i			1.25171i	−0.779941	+	0.625853i	$0.784751\pi$
$828$		0			0
$829$	−22.2653			−0.773305			−0.386652	−	0.922225i	$-0.626369\pi$
$829$	−22.2653			−0.773305			−0.386652	+	0.922225i	$0.626369\pi$
$830$		0			0
$831$	1.13468			0.0393616
$832$		0			0
$833$	−34.4594	+	39.1408i	−1.19395	+	1.35615i
$834$		0			0
$835$	19.9462	+	14.7632i	0.690265	+	0.510901i
$836$		0			0
$837$	23.2151			0.802430
$838$		0			0
$839$	−10.8709			−0.375305			−0.187652	−	0.982236i	$-0.560088\pi$
$839$	−10.8709			−0.375305			−0.187652	+	0.982236i	$0.560088\pi$
$840$		0			0
$841$	23.1716			0.799021
$842$		0			0
$843$	−11.6733			−0.402049
$844$		0			0
$845$	3.79854	−	5.13211i	0.130674	−	0.176550i
$846$		0			0
$847$	3.98490	−	1.80090i	0.136923	−	0.0618798i
$848$		0			0
$849$	3.46096			0.118780
$850$		0			0
$851$	1.46856			0.0503417
$852$		0			0
$853$		−	24.9401i		−	0.853931i	−0.904268	−	0.426966i	$-0.859583\pi$
$853$		−	24.9401i		−	0.853931i	0.904268	−	0.426966i	$-0.140417\pi$
$854$		0			0
$855$	−18.7938	−	13.9103i	−0.642736	−	0.475722i
$856$		0			0
$857$	14.7132			0.502594			0.251297	−	0.967910i	$-0.419143\pi$
$857$	14.7132			0.502594			0.251297	+	0.967910i	$0.419143\pi$
$858$		0			0
$859$			21.0920i			0.719651i	0.933020	+	0.359825i	$0.117164\pi$
$859$			21.0920i			0.719651i	−0.933020	+	0.359825i	$0.882836\pi$
$860$		0			0
$861$	−5.15474	−	11.4060i	−0.175673	−	0.388716i
$862$		0			0
$863$	13.6813			0.465715			0.232858	−	0.972511i	$-0.425192\pi$
$863$	13.6813			0.465715			0.232858	+	0.972511i	$0.425192\pi$
$864$		0			0
$865$	0.774377	+	0.573157i	0.0263296	+	0.0194879i
$866$		0			0
$867$	33.0315			1.12181
$868$		0			0
$869$			30.5332i			1.03577i
$870$		0			0
$871$	−1.65355			−0.0560285
$872$		0			0
$873$	−3.58335			−0.121278
$874$		0			0
$875$	29.4767	−	2.47443i	0.996495	−	0.0836511i
$876$		0			0
$877$	−41.6633			−1.40687			−0.703435	−	0.710759i	$-0.748351\pi$
$877$	−41.6633			−1.40687			−0.703435	+	0.710759i	$0.748351\pi$
$878$		0			0
$879$			13.0232i			0.439262i
$880$		0			0
$881$			5.53343i			0.186426i	0.995646	+	0.0932131i	$0.0297138\pi$
$881$			5.53343i			0.186426i	−0.995646	+	0.0932131i	$0.970286\pi$
$882$		0			0
$883$		−	36.9558i		−	1.24366i	−0.783151	−	0.621831i	$-0.786389\pi$
$883$		−	36.9558i		−	1.24366i	0.783151	−	0.621831i	$-0.213611\pi$
$884$		0			0
$885$	4.80993	+	3.56008i	0.161684	+	0.119671i
$886$		0			0
$887$			52.2816i			1.75545i	0.479169	+	0.877723i	$0.340938\pi$
$887$			52.2816i			1.75545i	−0.479169	+	0.877723i	$0.659062\pi$
$888$		0			0
$889$	41.6650	−	18.8297i	1.39740	−	0.631529i
$890$		0			0
$891$	−8.91723			−0.298738
$892$		0			0
$893$	30.2753			1.01312
$894$		0			0
$895$	19.6280	−	26.5189i	0.656092	−	0.886428i
$896$		0			0
$897$		−	1.93694i		−	0.0646725i
$898$		0			0
$899$			12.4097i			0.413886i
$900$		0			0
$901$	−27.9809			−0.932178
$902$		0			0
$903$	3.93734	+	8.71225i	0.131026	+	0.289926i
$904$		0			0
$905$	−2.69531	+	3.64156i	−0.0895951	+	0.121050i
$906$		0			0
$907$		−	37.0631i		−	1.23066i	−0.788270	−	0.615329i	$-0.789023\pi$
$907$		−	37.0631i		−	1.23066i	0.788270	−	0.615329i	$-0.210977\pi$
$908$		0			0
$909$	30.6737			1.01738
$910$		0			0
$911$			2.17495i			0.0720593i	0.999351	+	0.0360297i	$0.0114711\pi$
$911$			2.17495i			0.0720593i	−0.999351	+	0.0360297i	$0.988529\pi$
$912$		0			0
$913$	26.7048			0.883801
$914$		0			0
$915$	10.9025	−	14.7301i	0.360426	−	0.486962i
$916$		0			0
$917$	13.7720	+	30.4737i	0.454793	+	1.00633i
$918$		0			0
$919$			13.2512i			0.437117i	0.975824	+	0.218558i	$0.0701354\pi$
$919$			13.2512i			0.437117i	−0.975824	+	0.218558i	$0.929865\pi$
$920$		0			0
$921$	13.5197			0.445488
$922$		0			0
$923$	34.9571			1.15063
$924$		0			0
$925$	−3.02640	−	9.90767i	−0.0995074	−	0.325762i
$926$		0			0
$927$			25.1238i			0.825175i
$928$		0			0
$929$			7.96007i			0.261161i	0.991438	+	0.130581i	$0.0416842\pi$
$929$			7.96007i			0.261161i	−0.991438	+	0.130581i	$0.958316\pi$
$930$		0			0
$931$	24.2676	+	21.3652i	0.795340	+	0.700215i
$932$		0			0
$933$	−23.4052			−0.766253
$934$		0			0
$935$	30.2990	−	40.9361i	0.990883	−	1.33875i
$936$		0			0
$937$	−5.62068			−0.183620			−0.0918099	−	0.995777i	$-0.529265\pi$
$937$	−5.62068			−0.183620			−0.0918099	+	0.995777i	$0.529265\pi$
$938$		0			0
$939$	6.33044			0.206586
$940$		0			0
$941$	10.7839			0.351546			0.175773	−	0.984431i	$-0.443758\pi$
$941$	10.7839			0.351546			0.175773	+	0.984431i	$0.443758\pi$
$942$		0			0
$943$			3.90827i			0.127271i
$944$		0			0
$945$	5.64059	−	26.1166i	0.183489	−	0.849575i
$946$		0			0
$947$			32.8524i			1.06756i	0.845623	+	0.533780i	$0.179229\pi$
$947$			32.8524i			1.06756i	−0.845623	+	0.533780i	$0.820771\pi$
$948$		0			0
$949$			7.38987i			0.239885i
$950$		0			0
$951$	−23.6355			−0.766435
$952$		0			0
$953$		−	8.51329i		−	0.275772i	−0.990448	−	0.137886i	$-0.955969\pi$
$953$		−	8.51329i		−	0.275772i	0.990448	−	0.137886i	$-0.0440308\pi$
$954$		0			0
$955$	17.9496	+	13.2855i	0.580837	+	0.429908i
$956$		0			0
$957$			6.33276i			0.204709i
$958$		0			0
$959$	−12.4200	−	27.4820i	−0.401061	−	0.887438i
$960$		0			0
$961$	−4.57757			−0.147663
$962$		0			0
$963$			39.6599i			1.27802i
$964$		0			0
$965$	11.8416	+	8.76461i	0.381195	+	0.282143i
$966$		0			0
$967$	−34.3758			−1.10545			−0.552726	−	0.833363i	$-0.686412\pi$
$967$	−34.3758			−1.10545			−0.552726	+	0.833363i	$0.686412\pi$
$968$		0			0
$969$		−	29.5231i		−	0.948418i
$970$		0			0
$971$		−	57.5759i		−	1.84770i	−0.382755	−	0.923850i	$-0.625025\pi$
$971$		−	57.5759i		−	1.84770i	0.382755	−	0.923850i	$-0.374975\pi$
$972$		0			0
$973$	8.61735	+	19.0678i	0.276260	+	0.611286i
$974$		0			0
$975$	−13.0676	+	3.99162i	−0.418497	+	0.127834i
$976$		0			0
$977$		−	14.6164i		−	0.467621i	−0.972282	−	0.233810i	$-0.924880\pi$
$977$		−	14.6164i		−	0.467621i	0.972282	−	0.233810i	$-0.0751195\pi$
$978$		0			0
$979$		−	56.7735i		−	1.81449i
$980$		0			0
$981$			35.5199i			1.13406i
$982$		0			0
$983$		−	30.3916i		−	0.969342i	−0.874696	−	0.484671i	$-0.838939\pi$
$983$		−	30.3916i		−	0.969342i	0.874696	−	0.484671i	$-0.161061\pi$
$984$		0			0
$985$	16.1462	−	21.8148i	0.514462	−	0.695076i
$986$		0			0
$987$	6.12761	+	13.5587i	0.195044	+	0.431579i
$988$		0			0
$989$		−	2.98525i		−	0.0949253i
$990$		0			0
$991$		−	19.7017i		−	0.625845i	−0.949779	−	0.312923i	$-0.898692\pi$
$991$		−	19.7017i		−	0.625845i	0.949779	−	0.312923i	$-0.101308\pi$
$992$		0			0
$993$	25.7961			0.818614
$994$		0			0
$995$	−21.4858	+	29.0288i	−0.681145	+	0.920276i
$996$		0			0
$997$			36.7474i			1.16380i	0.813260	+	0.581901i	$0.197691\pi$
$997$			36.7474i			1.16380i	−0.813260	+	0.581901i	$0.802309\pi$
$998$		0			0
$999$	−9.35741			−0.296055

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.n.b.559.23		40
4.3	odd	2		280.2.n.b.139.13	✓	40
5.4	even	2	inner	1120.2.n.b.559.17		40
7.6	odd	2	inner	1120.2.n.b.559.19		40
8.3	odd	2	inner	1120.2.n.b.559.24		40
8.5	even	2		280.2.n.b.139.25	yes	40
20.19	odd	2		280.2.n.b.139.28	yes	40
28.27	even	2		280.2.n.b.139.14	yes	40
35.34	odd	2	inner	1120.2.n.b.559.21		40
40.19	odd	2	inner	1120.2.n.b.559.18		40
40.29	even	2		280.2.n.b.139.16	yes	40
56.13	odd	2		280.2.n.b.139.26	yes	40
56.27	even	2	inner	1120.2.n.b.559.20		40
140.139	even	2		280.2.n.b.139.27	yes	40
280.69	odd	2		280.2.n.b.139.15	yes	40
280.139	even	2	inner	1120.2.n.b.559.22		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.n.b.139.13	✓	40	4.3	odd	2
280.2.n.b.139.14	yes	40	28.27	even	2
280.2.n.b.139.15	yes	40	280.69	odd	2
280.2.n.b.139.16	yes	40	40.29	even	2
280.2.n.b.139.25	yes	40	8.5	even	2
280.2.n.b.139.26	yes	40	56.13	odd	2
280.2.n.b.139.27	yes	40	140.139	even	2
280.2.n.b.139.28	yes	40	20.19	odd	2
1120.2.n.b.559.17		40	5.4	even	2	inner
1120.2.n.b.559.18		40	40.19	odd	2	inner
1120.2.n.b.559.19		40	7.6	odd	2	inner
1120.2.n.b.559.20		40	56.27	even	2	inner
1120.2.n.b.559.21		40	35.34	odd	2	inner
1120.2.n.b.559.22		40	280.139	even	2	inner
1120.2.n.b.559.23		40	1.1	even	1	trivial
1120.2.n.b.559.24		40	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.n (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.857983 q^{3} +(1.33029 - 1.79731i) q^{5} +(-2.41097 + 1.08959i) q^{7} -2.26387 q^{9} -3.05732 q^{11} -3.18505i q^{13} +(1.14136 - 1.54206i) q^{15} -7.44976 q^{17} +4.61892i q^{19} +(-2.06857 + 0.934853i) q^{21} +0.708794 q^{23} +(-1.46068 - 4.78189i) q^{25} -4.51631 q^{27} -2.41421i q^{29} -5.14028 q^{31} -2.62312 q^{33} +(-1.24894 + 5.78275i) q^{35} +2.07192 q^{37} -2.73272i q^{39} +5.51396i q^{41} -4.21172i q^{43} +(-3.01159 + 4.06888i) q^{45} -6.55463i q^{47} +(4.62557 - 5.25396i) q^{49} -6.39177 q^{51} +3.75594 q^{53} +(-4.06711 + 5.49496i) q^{55} +3.96296i q^{57} +3.11915i q^{59} +9.55219 q^{61} +(5.45812 - 2.46669i) q^{63} +(-5.72454 - 4.23703i) q^{65} -0.519160i q^{67} +0.608133 q^{69} +10.9754i q^{71} -2.32017 q^{73} +(-1.25323 - 4.10277i) q^{75} +(7.37110 - 3.33123i) q^{77} -9.98693i q^{79} +2.91669 q^{81} -8.73474 q^{83} +(-9.91032 + 13.3896i) q^{85} -2.07135i q^{87} +18.5697i q^{89} +(3.47042 + 7.67907i) q^{91} -4.41027 q^{93} +(8.30166 + 6.14449i) q^{95} +1.58285 q^{97} +6.92135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 40 q^{9} + 16 q^{25} - 16 q^{35} + 8 q^{49} + 32 q^{51} - 24 q^{65} - 72 q^{81} + 128 q^{99}+O(q^{100}) \) 40 * q + 40 * q^9 + 16 * q^25 - 16 * q^35 + 8 * q^49 + 32 * q^51 - 24 * q^65 - 72 * q^81 + 128 * q^99

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