LMFDB - Embedded newform 1380.2.n.a.689.44

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,2,Mod(689,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("1380.689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1380.n (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:	$11.0193554789$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			689.44
Character	$\chi$	$=$	1380.689
Dual form			1380.2.n.a.689.42

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.63297 + 0.577419i) q^{3} +(2.15297 + 0.603919i) q^{5} +1.63712 q^{7} +(2.33317 + 1.88581i) q^{9} +O(q^{10})$	\(q+(1.63297 + 0.577419i) q^{3} +(2.15297 + 0.603919i) q^{5} +1.63712 q^{7} +(2.33317 + 1.88581i) q^{9} +0.207049 q^{11} -1.70340i q^{13} +(3.16702 + 2.22935i) q^{15} -5.57481i q^{17} -2.31904i q^{19} +(2.67337 + 0.945305i) q^{21} +(2.87814 - 3.83619i) q^{23} +(4.27056 + 2.60044i) q^{25} +(2.72110 + 4.42670i) q^{27} +4.70418i q^{29} -2.76252 q^{31} +(0.338104 + 0.119554i) q^{33} +(3.52467 + 0.988689i) q^{35} -1.83999 q^{37} +(0.983573 - 2.78159i) q^{39} +7.81253i q^{41} -3.48158 q^{43} +(3.88438 + 5.46915i) q^{45} -6.72000 q^{47} -4.31984 q^{49} +(3.21900 - 9.10348i) q^{51} +4.72554i q^{53} +(0.445770 + 0.125041i) q^{55} +(1.33906 - 3.78691i) q^{57} +11.3660i q^{59} -8.69994i q^{61} +(3.81969 + 3.08731i) q^{63} +(1.02871 - 3.66736i) q^{65} -9.87241 q^{67} +(6.91499 - 4.60248i) q^{69} -0.717119i q^{71} -12.2596i q^{73} +(5.47215 + 6.71234i) q^{75} +0.338964 q^{77} +4.32679i q^{79} +(1.88740 + 8.79987i) q^{81} -1.66726i q^{83} +(3.36673 - 12.0024i) q^{85} +(-2.71628 + 7.68178i) q^{87} +14.6958 q^{89} -2.78866i q^{91} +(-4.51110 - 1.59513i) q^{93} +(1.40051 - 4.99282i) q^{95} -9.51210 q^{97} +(0.483081 + 0.390456i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) $ 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	1.63297	+	0.577419i	0.942795	+	0.333373i
$3$	1.63297	+	0.577419i	0.942795	+	0.333373i
$4$		0			0
$5$	2.15297	+	0.603919i	0.962838	+	0.270081i
$5$	2.15297	+	0.603919i	0.962838	+	0.270081i
$6$		0			0
$7$	1.63712			0.618773			0.309387	−	0.950936i	$-0.399876\pi$
$7$	1.63712			0.618773			0.309387	+	0.950936i	$0.399876\pi$
$8$		0			0
$9$	2.33317	+	1.88581i	0.777725	+	0.628605i
$10$		0			0
$11$	0.207049			0.0624276			0.0312138	−	0.999513i	$-0.490063\pi$
$11$	0.207049			0.0624276			0.0312138	+	0.999513i	$0.490063\pi$
$12$		0			0
$13$		−	1.70340i		−	0.472437i	−0.971700	−	0.236218i	$-0.924092\pi$
$13$		−	1.70340i		−	0.472437i	0.971700	−	0.236218i	$-0.0759081\pi$
$14$		0			0
$15$	3.16702	+	2.22935i	0.817721	+	0.575615i
$16$		0			0
$17$		−	5.57481i		−	1.35209i	−0.736861	−	0.676044i	$-0.763693\pi$
$17$		−	5.57481i		−	1.35209i	0.736861	−	0.676044i	$-0.236307\pi$
$18$		0			0
$19$		−	2.31904i		−	0.532023i	−0.963970	−	0.266012i	$-0.914294\pi$
$19$		−	2.31904i		−	0.532023i	0.963970	−	0.266012i	$-0.0857060\pi$
$20$		0			0
$21$	2.67337	+	0.945305i	0.583377	+	0.206282i
$22$		0			0
$23$	2.87814	−	3.83619i	0.600133	−	0.799900i
$23$	2.87814	−	3.83619i	0.600133	−	0.799900i
$24$		0			0
$25$	4.27056	+	2.60044i	0.854113	+	0.520088i
$26$		0			0
$27$	2.72110	+	4.42670i	0.523675	+	0.851918i
$28$		0			0
$29$			4.70418i			0.873544i	0.899572	+	0.436772i	$0.143878\pi$
$29$			4.70418i			0.873544i	−0.899572	+	0.436772i	$0.856122\pi$
$30$		0			0
$31$	−2.76252			−0.496163			−0.248081	−	0.968739i	$-0.579800\pi$
$31$	−2.76252			−0.496163			−0.248081	+	0.968739i	$0.579800\pi$
$32$		0			0
$33$	0.338104	+	0.119554i	0.0588564	+	0.0208117i
$34$		0			0
$35$	3.52467	+	0.988689i	0.595778	+	0.167119i
$36$		0			0
$37$	−1.83999			−0.302493			−0.151246	−	0.988496i	$-0.548329\pi$
$37$	−1.83999			−0.302493			−0.151246	+	0.988496i	$0.548329\pi$
$38$		0			0
$39$	0.983573	−	2.78159i	0.157498	−	0.445411i
$40$		0			0
$41$			7.81253i			1.22011i	0.792358	+	0.610056i	$0.208853\pi$
$41$			7.81253i			1.22011i	−0.792358	+	0.610056i	$0.791147\pi$
$42$		0			0
$43$	−3.48158			−0.530937			−0.265468	−	0.964120i	$-0.585527\pi$
$43$	−3.48158			−0.530937			−0.265468	+	0.964120i	$0.585527\pi$
$44$		0			0
$45$	3.88438	+	5.46915i	0.579048	+	0.815293i
$46$		0			0
$47$	−6.72000			−0.980213			−0.490107	−	0.871662i	$-0.663042\pi$
$47$	−6.72000			−0.980213			−0.490107	+	0.871662i	$0.663042\pi$
$48$		0			0
$49$	−4.31984			−0.617119
$50$		0			0
$51$	3.21900	−	9.10348i	0.450750	−	1.27474i
$52$		0			0
$53$			4.72554i			0.649103i	0.945868	+	0.324552i	$0.105213\pi$
$53$			4.72554i			0.649103i	−0.945868	+	0.324552i	$0.894787\pi$
$54$		0			0
$55$	0.445770	+	0.125041i	0.0601076	+	0.0168605i
$56$		0			0
$57$	1.33906	−	3.78691i	0.177362	−	0.501589i
$58$		0			0
$59$			11.3660i			1.47973i	0.672755	+	0.739866i	$0.265111\pi$
$59$			11.3660i			1.47973i	−0.672755	+	0.739866i	$0.734889\pi$
$60$		0			0
$61$		−	8.69994i		−	1.11391i	−0.830542	−	0.556957i	$-0.811969\pi$
$61$		−	8.69994i		−	1.11391i	0.830542	−	0.556957i	$-0.188031\pi$
$62$		0			0
$63$	3.81969	+	3.08731i	0.481235	+	0.388964i
$64$		0			0
$65$	1.02871	−	3.66736i	0.127596	−	0.454880i
$66$		0			0
$67$	−9.87241			−1.20611			−0.603054	−	0.797701i	$-0.706049\pi$
$67$	−9.87241			−1.20611			−0.603054	+	0.797701i	$0.706049\pi$
$68$		0			0
$69$	6.91499	−	4.60248i	0.832467	−	0.554074i
$70$		0			0
$71$		−	0.717119i		−	0.0851064i	−0.999094	−	0.0425532i	$-0.986451\pi$
$71$		−	0.717119i		−	0.0851064i	0.999094	−	0.0425532i	$-0.0135492\pi$
$72$		0			0
$73$		−	12.2596i		−	1.43488i	−0.696621	−	0.717439i	$-0.745314\pi$
$73$		−	12.2596i		−	1.43488i	0.696621	−	0.717439i	$-0.254686\pi$
$74$		0			0
$75$	5.47215	+	6.71234i	0.631870	+	0.775075i
$76$		0			0
$77$	0.338964			0.0386285
$78$		0			0
$79$			4.32679i			0.486802i	0.969926	+	0.243401i	$0.0782630\pi$
$79$			4.32679i			0.486802i	−0.969926	+	0.243401i	$0.921737\pi$
$80$		0			0
$81$	1.88740	+	8.79987i	0.209712	+	0.977763i
$82$		0			0
$83$		−	1.66726i		−	0.183005i	−0.995805	−	0.0915026i	$-0.970833\pi$
$83$		−	1.66726i		−	0.183005i	0.995805	−	0.0915026i	$-0.0291670\pi$
$84$		0			0
$85$	3.36673	−	12.0024i	0.365173	−	1.30184i
$86$		0			0
$87$	−2.71628	+	7.68178i	−0.291216	+	0.823573i
$88$		0			0
$89$	14.6958			1.55775			0.778877	−	0.627177i	$-0.215790\pi$
$89$	14.6958			1.55775			0.778877	+	0.627177i	$0.215790\pi$
$90$		0			0
$91$		−	2.78866i		−	0.292331i
$92$		0			0
$93$	−4.51110	−	1.59513i	−0.467780	−	0.165407i
$94$		0			0
$95$	1.40051	−	4.99282i	0.143689	−	0.512252i
$96$		0			0
$97$	−9.51210			−0.965807			−0.482904	−	0.875673i	$-0.660418\pi$
$97$	−9.51210			−0.965807			−0.482904	+	0.875673i	$0.660418\pi$
$98$		0			0
$99$	0.483081	+	0.390456i	0.0485515	+	0.0392423i
$100$		0			0
$101$			13.3757i			1.33093i	0.746428	+	0.665467i	$0.231767\pi$
$101$			13.3757i			1.33093i	−0.746428	+	0.665467i	$0.768233\pi$
$102$		0			0
$103$	−4.15779			−0.409679			−0.204840	−	0.978796i	$-0.565667\pi$
$103$	−4.15779			−0.409679			−0.204840	+	0.978796i	$0.565667\pi$
$104$		0			0
$105$	5.18479	+	3.64971i	0.505984	+	0.356175i
$106$		0			0
$107$		−	4.39378i		−	0.424763i	−0.977187	−	0.212381i	$-0.931878\pi$
$107$		−	4.39378i		−	0.424763i	0.977187	−	0.212381i	$-0.0681219\pi$
$108$		0			0
$109$			13.4518i			1.28844i	0.764839	+	0.644222i	$0.222819\pi$
$109$			13.4518i			1.28844i	−0.764839	+	0.644222i	$0.777181\pi$
$110$		0			0
$111$	−3.00465	−	1.06245i	−0.285189	−	0.100843i
$112$		0			0
$113$			17.2720i			1.62482i	0.583089	+	0.812408i	$0.301844\pi$
$113$			17.2720i			1.62482i	−0.583089	+	0.812408i	$0.698156\pi$
$114$		0			0
$115$	8.51329	−	6.52104i	0.793868	−	0.608090i
$116$		0			0
$117$	3.21229	−	3.97432i	0.296976	−	0.367426i
$118$		0			0
$119$		−	9.12663i		−	0.836637i
$120$		0			0
$121$	−10.9571			−0.996103
$122$		0			0
$123$	−4.51110	+	12.7576i	−0.406753	+	1.15032i
$124$		0			0
$125$	7.62394	+	8.17775i	0.681906	+	0.731440i
$126$		0			0
$127$		−	9.86420i		−	0.875306i	−0.899144	−	0.437653i	$-0.855810\pi$
$127$		−	9.86420i		−	0.875306i	0.899144	−	0.437653i	$-0.144190\pi$
$128$		0			0
$129$	−5.68532	−	2.01033i	−0.500564	−	0.177000i
$130$		0			0
$131$		−	6.01103i		−	0.525186i	−0.964907	−	0.262593i	$-0.915422\pi$
$131$		−	6.01103i		−	0.525186i	0.964907	−	0.262593i	$-0.0845777\pi$
$132$		0			0
$133$		−	3.79654i		−	0.329202i
$134$		0			0
$135$	3.18507	+	11.1739i	0.274127	+	0.961693i
$136$		0			0
$137$		−	9.09251i		−	0.776826i	−0.921485	−	0.388413i	$-0.873023\pi$
$137$		−	9.09251i		−	0.776826i	0.921485	−	0.388413i	$-0.126977\pi$
$138$		0			0
$139$	10.0862			0.855496			0.427748	−	0.903898i	$-0.359307\pi$
$139$	10.0862			0.855496			0.427748	+	0.903898i	$0.359307\pi$
$140$		0			0
$141$	−10.9736	−	3.88026i	−0.924140	−	0.326777i
$142$		0			0
$143$		−	0.352686i		−	0.0294931i
$144$		0			0
$145$	−2.84094	+	10.1280i	−0.235928	+	0.841081i
$146$		0			0
$147$	−7.05416	−	2.49436i	−0.581817	−	0.205731i
$148$		0			0
$149$	5.19016			0.425194			0.212597	−	0.977140i	$-0.431808\pi$
$149$	5.19016			0.425194			0.212597	+	0.977140i	$0.431808\pi$
$150$		0			0
$151$	−3.28461			−0.267297			−0.133649	−	0.991029i	$-0.542669\pi$
$151$	−3.28461			−0.267297			−0.133649	+	0.991029i	$0.542669\pi$
$152$		0			0
$153$	10.5131	−	13.0070i	0.849930	−	1.05155i
$154$		0			0
$155$	−5.94762	−	1.66834i	−0.477724	−	0.134004i
$156$		0			0
$157$	−4.39339			−0.350631			−0.175316	−	0.984512i	$-0.556095\pi$
$157$	−4.39339			−0.350631			−0.175316	+	0.984512i	$0.556095\pi$
$158$		0			0
$159$	−2.72862	+	7.71666i	−0.216394	+	0.611971i
$160$		0			0
$161$	4.71186	−	6.28030i	0.371346	−	0.494957i
$162$		0			0
$163$		−	5.41305i		−	0.423983i	−0.977272	−	0.211991i	$-0.932005\pi$
$163$		−	5.41305i		−	0.423983i	0.977272	−	0.211991i	$-0.0679949\pi$
$164$		0			0
$165$	0.655728	+	0.461584i	0.0510483	+	0.0359343i
$166$		0			0
$167$	−2.50559			−0.193889			−0.0969443	−	0.995290i	$-0.530907\pi$
$167$	−2.50559			−0.193889			−0.0969443	+	0.995290i	$0.530907\pi$
$168$		0			0
$169$	10.0984			0.776803
$170$		0			0
$171$	4.37327	−	5.41072i	0.334433	−	0.413768i
$172$		0			0
$173$	12.8056			0.973593			0.486797	−	0.873515i	$-0.338165\pi$
$173$	12.8056			0.973593			0.486797	+	0.873515i	$0.338165\pi$
$174$		0			0
$175$	6.99143	+	4.25724i	0.528502	+	0.321817i
$176$		0			0
$177$	−6.56296	+	18.5604i	−0.493303	+	1.39508i
$178$		0			0
$179$		−	2.19281i		−	0.163898i	−0.996637	−	0.0819491i	$-0.973886\pi$
$179$		−	2.19281i		−	0.163898i	0.996637	−	0.0819491i	$-0.0261145\pi$
$180$		0			0
$181$		−	18.2945i		−	1.35982i	−0.733295	−	0.679911i	$-0.762018\pi$
$181$		−	18.2945i		−	1.35982i	0.733295	−	0.679911i	$-0.237982\pi$
$182$		0			0
$183$	5.02351	−	14.2067i	0.371349	−	1.05019i
$184$		0			0
$185$	−3.96145	−	1.11121i	−0.291252	−	0.0816976i
$186$		0			0
$187$		−	1.15426i		−	0.0844076i
$188$		0			0
$189$	4.45476	+	7.24704i	0.324036	+	0.527144i
$190$		0			0
$191$	−20.8521			−1.50881			−0.754404	−	0.656411i	$-0.772074\pi$
$191$	−20.8521			−1.50881			−0.754404	+	0.656411i	$0.772074\pi$
$192$		0			0
$193$			12.8994i			0.928522i	0.885699	+	0.464261i	$0.153680\pi$
$193$			12.8994i			0.928522i	−0.885699	+	0.464261i	$0.846320\pi$
$194$		0			0
$195$	3.79746	−	5.39469i	0.271942	−	0.386321i
$196$		0			0
$197$	−10.6294			−0.757311			−0.378655	−	0.925538i	$-0.623613\pi$
$197$	−10.6294			−0.757311			−0.378655	+	0.925538i	$0.623613\pi$
$198$		0			0
$199$			18.7525i			1.32933i	0.747141	+	0.664666i	$0.231426\pi$
$199$			18.7525i			1.32933i	−0.747141	+	0.664666i	$0.768574\pi$
$200$		0			0
$201$	−16.1213	−	5.70052i	−1.13711	−	0.402084i
$202$		0			0
$203$			7.70131i			0.540526i
$204$		0			0
$205$	−4.71814	+	16.8201i	−0.329529	+	1.17477i
$206$		0			0
$207$	13.9495	−	3.52286i	0.969560	−	0.244856i
$208$		0			0
$209$		−	0.480154i		−	0.0332129i
$210$		0			0
$211$	−16.6868			−1.14877			−0.574383	−	0.818587i	$-0.694758\pi$
$211$	−16.6868			−1.14877			−0.574383	+	0.818587i	$0.694758\pi$
$212$		0			0
$213$	0.414078	−	1.17103i	0.0283722	−	0.0802379i
$214$		0			0
$215$	−7.49575	−	2.10260i	−0.511206	−	0.143396i
$216$		0			0
$217$	−4.52257			−0.307012
$218$		0			0
$219$	7.07893	−	20.0195i	0.478350	−	1.35280i
$220$		0			0
$221$	−9.49610			−0.638777
$222$		0			0
$223$		−	8.00636i		−	0.536146i	−0.963399	−	0.268073i	$-0.913613\pi$
$223$		−	8.00636i		−	0.536146i	0.963399	−	0.268073i	$-0.0863868\pi$
$224$		0			0
$225$	5.06002	+	14.1208i	0.337335	+	0.941385i
$226$		0			0
$227$		−	2.69738i		−	0.179031i	−0.995985	−	0.0895156i	$-0.971468\pi$
$227$		−	2.69738i		−	0.179031i	0.995985	−	0.0895156i	$-0.0285319\pi$
$228$		0			0
$229$		−	20.0976i		−	1.32809i	−0.747695	−	0.664043i	$-0.768839\pi$
$229$		−	20.0976i		−	1.32809i	0.747695	−	0.664043i	$-0.231161\pi$
$230$		0			0
$231$	0.553518	+	0.195724i	0.0364188	+	0.0128777i
$232$		0			0
$233$	24.1114			1.57959			0.789794	−	0.613372i	$-0.210187\pi$
$233$	24.1114			1.57959			0.789794	+	0.613372i	$0.210187\pi$
$234$		0			0
$235$	−14.4680	−	4.05834i	−0.943786	−	0.264737i
$236$		0			0
$237$	−2.49837	+	7.06551i	−0.162287	+	0.458954i
$238$		0			0
$239$		−	20.6543i		−	1.33601i	−0.744156	−	0.668006i	$-0.767148\pi$
$239$		−	20.6543i		−	1.33601i	0.744156	−	0.668006i	$-0.232852\pi$
$240$		0			0
$241$		−	22.8862i		−	1.47423i	−0.675766	−	0.737116i	$-0.736187\pi$
$241$		−	22.8862i		−	1.47423i	0.675766	−	0.737116i	$-0.263813\pi$
$242$		0			0
$243$	−1.99914	+	15.4597i	−0.128245	+	0.991743i
$244$		0			0
$245$	−9.30048	−	2.60883i	−0.594186	−	0.166672i
$246$		0			0
$247$	−3.95024			−0.251347
$248$		0			0
$249$	0.962706	−	2.72258i	0.0610090	−	0.172536i
$250$		0			0
$251$	−12.2602			−0.773855			−0.386927	−	0.922110i	$-0.626464\pi$
$251$	−12.2602			−0.773855			−0.386927	+	0.922110i	$0.626464\pi$
$252$		0			0
$253$	0.595915	−	0.794278i	0.0374648	−	0.0499358i
$254$		0			0
$255$	12.4282	−	17.6555i	0.778283	−	1.10563i
$256$		0			0
$257$	20.8607			1.30125			0.650627	−	0.759397i	$-0.274506\pi$
$257$	20.8607			1.30125			0.650627	+	0.759397i	$0.274506\pi$
$258$		0			0
$259$	−3.01229			−0.187175
$260$		0			0
$261$	−8.87121	+	10.9757i	−0.549114	+	0.679377i
$262$		0			0
$263$			0.342111i			0.0210954i	0.999944	+	0.0105477i	$0.00335751\pi$
$263$			0.342111i			0.0210954i	−0.999944	+	0.0105477i	$0.996642\pi$
$264$		0			0
$265$	−2.85385	+	10.1740i	−0.175310	+	0.624981i
$266$		0			0
$267$	23.9978	+	8.48565i	1.46864	+	0.519313i
$268$		0			0
$269$			4.63063i			0.282335i	0.989986	+	0.141167i	$0.0450856\pi$
$269$			4.63063i			0.282335i	−0.989986	+	0.141167i	$0.954914\pi$
$270$		0			0
$271$	−15.3798			−0.934258			−0.467129	−	0.884189i	$-0.654712\pi$
$271$	−15.3798			−0.934258			−0.467129	+	0.884189i	$0.654712\pi$
$272$		0			0
$273$	1.61023	−	4.55380i	0.0974554	−	0.275609i
$274$		0			0
$275$	0.884215	+	0.538418i	0.0533202	+	0.0324678i
$276$		0			0
$277$			13.0539i			0.784332i	0.919895	+	0.392166i	$0.128274\pi$
$277$			13.0539i			0.784332i	−0.919895	+	0.392166i	$0.871726\pi$
$278$		0			0
$279$	−6.44543	−	5.20960i	−0.385878	−	0.311890i
$280$		0			0
$281$	10.2884			0.613756			0.306878	−	0.951749i	$-0.400716\pi$
$281$	10.2884			0.613756			0.306878	+	0.951749i	$0.400716\pi$
$282$		0			0
$283$	2.24219			0.133284			0.0666422	−	0.997777i	$-0.478771\pi$
$283$	2.24219			0.133284			0.0666422	+	0.997777i	$0.478771\pi$
$284$		0			0
$285$	5.16994	−	7.34443i	0.306241	−	0.435047i
$286$		0			0
$287$			12.7901i			0.754973i
$288$		0			0
$289$	−14.0785			−0.828144
$290$		0			0
$291$	−15.5330	−	5.49247i	−0.910558	−	0.321974i
$292$		0			0
$293$			5.95505i			0.347898i	0.984755	+	0.173949i	$0.0556528\pi$
$293$			5.95505i			0.347898i	−0.984755	+	0.173949i	$0.944347\pi$
$294$		0			0
$295$	−6.86416	+	24.4707i	−0.399647	+	1.42474i
$296$		0			0
$297$	0.563400	+	0.916543i	0.0326918	+	0.0531832i
$298$		0			0
$299$	−6.53454	−	4.90260i	−0.377902	−	0.283525i
$300$		0			0
$301$	−5.69977			−0.328529
$302$		0			0
$303$	−7.72339	+	21.8421i	−0.443697	+	1.25480i
$304$		0			0
$305$	5.25406	−	18.7307i	0.300847	−	1.07252i
$306$		0			0
$307$		−	25.5158i		−	1.45626i	−0.685437	−	0.728132i	$-0.740389\pi$
$307$		−	25.5158i		−	1.45626i	0.685437	−	0.728132i	$-0.259611\pi$
$308$		0			0
$309$	−6.78955	−	2.40079i	−0.386244	−	0.136576i
$310$		0			0
$311$		−	32.5910i		−	1.84806i	−0.382315	−	0.924032i	$-0.624873\pi$
$311$		−	32.5910i		−	1.84806i	0.382315	−	0.924032i	$-0.375127\pi$
$312$		0			0
$313$	21.1347			1.19460			0.597301	−	0.802017i	$-0.296240\pi$
$313$	21.1347			1.19460			0.597301	+	0.802017i	$0.296240\pi$
$314$		0			0
$315$	6.35919	+	8.95366i	0.358300	+	0.504482i
$316$		0			0
$317$	−26.6266			−1.49550			−0.747749	−	0.663981i	$-0.768865\pi$
$317$	−26.6266			−1.49550			−0.747749	+	0.663981i	$0.768865\pi$
$318$		0			0
$319$			0.973995i			0.0545333i
$320$		0			0
$321$	2.53705	−	7.17490i	0.141604	−	0.400464i
$322$		0			0
$323$	−12.9282			−0.719343
$324$		0			0
$325$	4.42958	−	7.27446i	0.245709	−	0.403514i
$326$		0			0
$327$	−7.76730	+	21.9663i	−0.429533	+	1.21474i
$328$		0			0
$329$	−11.0015			−0.606530
$330$		0			0
$331$	−20.5440			−1.12920			−0.564599	−	0.825365i	$-0.690969\pi$
$331$	−20.5440			−1.12920			−0.564599	+	0.825365i	$0.690969\pi$
$332$		0			0
$333$	−4.29302	−	3.46989i	−0.235256	−	0.190149i
$334$		0			0
$335$	−21.2550	−	5.96214i	−1.16129	−	0.325746i
$336$		0			0
$337$	31.4046			1.71072			0.855358	−	0.518037i	$-0.173337\pi$
$337$	31.4046			1.71072			0.855358	+	0.518037i	$0.173337\pi$
$338$		0			0
$339$	−9.97320	+	28.2047i	−0.541670	+	1.53187i
$340$		0			0
$341$	−0.571976			−0.0309742
$342$		0			0
$343$	−18.5319			−1.00063
$344$		0			0
$345$	17.6673	−	5.73291i	0.951176	−	0.308650i
$346$		0			0
$347$	8.55729			0.459380			0.229690	−	0.973264i	$-0.426229\pi$
$347$	8.55729			0.459380			0.229690	+	0.973264i	$0.426229\pi$
$348$		0			0
$349$	3.98001			0.213045			0.106522	−	0.994310i	$-0.466028\pi$
$349$	3.98001			0.213045			0.106522	+	0.994310i	$0.466028\pi$
$350$		0			0
$351$	7.54041	−	4.63510i	0.402478	−	0.247403i
$352$		0			0
$353$	−12.9378			−0.688610			−0.344305	−	0.938858i	$-0.611885\pi$
$353$	−12.9378			−0.688610			−0.344305	+	0.938858i	$0.611885\pi$
$354$		0			0
$355$	0.433082	−	1.54394i	0.0229856	−	0.0819436i
$356$		0			0
$357$	5.26989	−	14.9035i	0.278912	−	0.788777i
$358$		0			0
$359$	−34.1974			−1.80487			−0.902434	−	0.430829i	$-0.858221\pi$
$359$	−34.1974			−1.80487			−0.902434	+	0.430829i	$0.858221\pi$
$360$		0			0
$361$	13.6221			0.716951
$362$		0			0
$363$	−17.8927	−	6.32686i	−0.939121	−	0.332074i
$364$		0			0
$365$	7.40381	−	26.3946i	0.387533	−	1.38155i
$366$		0			0
$367$	−14.6984			−0.767250			−0.383625	−	0.923489i	$-0.625324\pi$
$367$	−14.6984			−0.767250			−0.383625	+	0.923489i	$0.625324\pi$
$368$		0			0
$369$	−14.7330	+	18.2280i	−0.766969	+	0.948912i
$370$		0			0
$371$			7.73628i			0.401648i
$372$		0			0
$373$	−15.5299			−0.804108			−0.402054	−	0.915616i	$-0.631704\pi$
$373$	−15.5299			−0.804108			−0.402054	+	0.915616i	$0.631704\pi$
$374$		0			0
$375$	7.72767	+	17.7562i	0.399055	+	0.916927i
$376$		0			0
$377$	8.01308			0.412694
$378$		0			0
$379$			24.4244i			1.25460i	0.778780	+	0.627298i	$0.215839\pi$
$379$			24.4244i			1.25460i	−0.778780	+	0.627298i	$0.784161\pi$
$380$		0			0
$381$	5.69578	−	16.1079i	0.291804	−	0.825234i
$382$		0			0
$383$			4.89011i			0.249873i	0.992165	+	0.124936i	$0.0398727\pi$
$383$			4.89011i			0.249873i	−0.992165	+	0.124936i	$0.960127\pi$
$384$		0			0
$385$	0.729779	+	0.204707i	0.0371930	+	0.0104328i
$386$		0			0
$387$	−8.12314	−	6.56562i	−0.412922	−	0.333749i
$388$		0			0
$389$	−7.54013			−0.382300			−0.191150	−	0.981561i	$-0.561222\pi$
$389$	−7.54013			−0.382300			−0.191150	+	0.981561i	$0.561222\pi$
$390$		0			0
$391$	−21.3860	−	16.0450i	−1.08154	−	0.811433i
$392$		0			0
$393$	3.47088	−	9.81583i	0.175083	−	0.495143i
$394$		0			0
$395$	−2.61303	+	9.31545i	−0.131476	+	0.468711i
$396$		0			0
$397$			2.34323i			0.117603i	0.998270	+	0.0588016i	$0.0187279\pi$
$397$			2.34323i			0.117603i	−0.998270	+	0.0588016i	$0.981272\pi$
$398$		0			0
$399$	2.19220	−	6.19964i	0.109747	−	0.310370i
$400$		0			0
$401$	−4.30594			−0.215028			−0.107514	−	0.994204i	$-0.534289\pi$
$401$	−4.30594			−0.215028			−0.107514	+	0.994204i	$0.534289\pi$
$402$		0			0
$403$			4.70566i			0.234406i
$404$		0			0
$405$	−1.25089	+	20.0857i	−0.0621570	+	0.998066i
$406$		0			0
$407$	−0.380969			−0.0188839
$408$		0			0
$409$	18.6197			0.920684			0.460342	−	0.887742i	$-0.347727\pi$
$409$	18.6197			0.920684			0.460342	+	0.887742i	$0.347727\pi$
$410$		0			0
$411$	5.25019	−	14.8478i	0.258973	−	0.732387i
$412$		0			0
$413$			18.6076i			0.915618i
$414$		0			0
$415$	1.00689	−	3.58956i	0.0494262	−	0.176204i
$416$		0			0
$417$	16.4704	+	5.82394i	0.806558	+	0.285199i
$418$		0			0
$419$	35.9166			1.75464			0.877321	−	0.479905i	$-0.159329\pi$
$419$	35.9166			1.75464			0.877321	+	0.479905i	$0.159329\pi$
$420$		0			0
$421$			16.8166i			0.819592i	0.912177	+	0.409796i	$0.134400\pi$
$421$			16.8166i			0.819592i	−0.912177	+	0.409796i	$0.865600\pi$
$422$		0			0
$423$	−15.6789	−	12.6727i	−0.762336	−	0.616167i
$424$		0			0
$425$	14.4969	−	23.8076i	0.703205	−	1.15484i
$426$		0			0
$427$		−	14.2429i		−	0.689260i
$428$		0			0
$429$	0.203648	−	0.575925i	0.00983220	−	0.0278059i
$430$		0			0
$431$	−6.79697			−0.327399			−0.163699	−	0.986510i	$-0.552343\pi$
$431$	−6.79697			−0.327399			−0.163699	+	0.986510i	$0.552343\pi$
$432$		0			0
$433$	−30.0194			−1.44264			−0.721321	−	0.692601i	$-0.756465\pi$
$433$	−30.0194			−1.44264			−0.721321	+	0.692601i	$0.756465\pi$
$434$		0			0
$435$	−10.4873	+	14.8982i	−0.502825	+	0.714315i
$436$		0			0
$437$	−8.89626	−	6.67450i	−0.425566	−	0.319285i
$438$		0			0
$439$	9.50884			0.453832			0.226916	−	0.973914i	$-0.427136\pi$
$439$	9.50884			0.453832			0.226916	+	0.973914i	$0.427136\pi$
$440$		0			0
$441$	−10.0789	−	8.14641i	−0.479949	−	0.387924i
$442$		0			0
$443$	−15.6346			−0.742824			−0.371412	−	0.928468i	$-0.621126\pi$
$443$	−15.6346			−0.742824			−0.371412	+	0.928468i	$0.621126\pi$
$444$		0			0
$445$	31.6397	+	8.87509i	1.49986	+	0.420719i
$446$		0			0
$447$	8.47536	+	2.99690i	0.400871	+	0.141748i
$448$		0			0
$449$			2.72950i			0.128813i	0.997924	+	0.0644066i	$0.0205154\pi$
$449$			2.72950i			0.128813i	−0.997924	+	0.0644066i	$0.979485\pi$
$450$		0			0
$451$			1.61758i			0.0761687i
$452$		0			0
$453$	−5.36366	−	1.89659i	−0.252007	−	0.0891098i
$454$		0			0
$455$	1.68413	−	6.00391i	0.0789531	−	0.281468i
$456$		0			0
$457$	21.4599			1.00385			0.501926	−	0.864911i	$-0.332625\pi$
$457$	21.4599			1.00385			0.501926	+	0.864911i	$0.332625\pi$
$458$		0			0
$459$	24.6780	−	15.1696i	1.15187	−	0.708055i
$460$		0			0
$461$			24.8449i			1.15714i	0.815632	+	0.578571i	$0.196389\pi$
$461$			24.8449i			1.15714i	−0.815632	+	0.578571i	$0.803611\pi$
$462$		0			0
$463$		−	13.7756i		−	0.640206i	−0.947383	−	0.320103i	$-0.896283\pi$
$463$		−	13.7756i		−	0.640206i	0.947383	−	0.320103i	$-0.103717\pi$
$464$		0			0
$465$	−8.74895	−	6.15861i	−0.405723	−	0.285599i
$466$		0			0
$467$			3.89720i			0.180341i	0.995926	+	0.0901705i	$0.0287412\pi$
$467$			3.89720i			0.180341i	−0.995926	+	0.0901705i	$0.971259\pi$
$468$		0			0
$469$	−16.1623			−0.746307
$470$		0			0
$471$	−7.17428	−	2.53683i	−0.330573	−	0.116891i
$472$		0			0
$473$	−0.720858			−0.0331451
$474$		0			0
$475$	6.03052	−	9.90359i	0.276699	−	0.454408i
$476$		0			0
$477$	−8.91150	+	11.0255i	−0.408030	+	0.504824i
$478$		0			0
$479$	−11.9151			−0.544417			−0.272209	−	0.962238i	$-0.587754\pi$
$479$	−11.9151			−0.544417			−0.272209	+	0.962238i	$0.587754\pi$
$480$		0			0
$481$			3.13424i			0.142909i
$482$		0			0
$483$	11.3207	−	7.53482i	0.515109	−	0.342846i
$484$		0			0
$485$	−20.4793	−	5.74454i	−0.929916	−	0.260846i
$486$		0			0
$487$		−	30.0577i		−	1.36204i	−0.732264	−	0.681021i	$-0.761536\pi$
$487$		−	30.0577i		−	1.36204i	0.732264	−	0.681021i	$-0.238464\pi$
$488$		0			0
$489$	3.12560	−	8.83934i	0.141344	−	0.399729i
$490$		0			0
$491$			12.4656i			0.562563i	0.959625	+	0.281282i	$0.0907595\pi$
$491$			12.4656i			0.562563i	−0.959625	+	0.281282i	$0.909240\pi$
$492$		0			0
$493$	26.2249			1.18111
$494$		0			0
$495$	0.804256	+	1.13238i	0.0361486	+	0.0508968i
$496$		0			0
$497$		−	1.17401i		−	0.0526616i
$498$		0			0
$499$	6.51207			0.291520			0.145760	−	0.989320i	$-0.453437\pi$
$499$	6.51207			0.291520			0.145760	+	0.989320i	$0.453437\pi$
$500$		0			0
$501$	−4.09156	−	1.44678i	−0.182797	−	0.0646373i
$502$		0			0
$503$			5.18925i			0.231377i	0.993286	+	0.115689i	$0.0369074\pi$
$503$			5.18925i			0.231377i	−0.993286	+	0.115689i	$0.963093\pi$
$504$		0			0
$505$	−8.07785	+	28.7975i	−0.359460	+	1.28147i
$506$		0			0
$507$	16.4904	+	5.83104i	0.732366	+	0.258965i
$508$		0			0
$509$		−	0.697413i		−	0.0309123i	−0.999881	−	0.0154561i	$-0.995080\pi$
$509$		−	0.697413i		−	0.0309123i	0.999881	−	0.0154561i	$-0.00492004\pi$
$510$		0			0
$511$		−	20.0704i		−	0.887864i
$512$		0			0
$513$	10.2657	−	6.31032i	0.453240	−	0.278607i
$514$		0			0
$515$	−8.95160	−	2.51097i	−0.394455	−	0.110647i
$516$		0			0
$517$	−1.39137			−0.0611924
$518$		0			0
$519$	20.9112	+	7.39421i	0.917899	+	0.324570i
$520$		0			0
$521$	−10.0615			−0.440801			−0.220400	−	0.975410i	$-0.570736\pi$
$521$	−10.0615			−0.440801			−0.220400	+	0.975410i	$0.570736\pi$
$522$		0			0
$523$	−24.8331			−1.08588			−0.542939	−	0.839772i	$-0.682688\pi$
$523$	−24.8331			−1.08588			−0.542939	+	0.839772i	$0.682688\pi$
$524$		0			0
$525$	8.95857	+	10.9889i	0.390984	+	0.479596i
$526$		0			0
$527$			15.4005i			0.670856i
$528$		0			0
$529$	−6.43267	−	22.0821i	−0.279681	−	0.960093i
$530$		0			0
$531$	−21.4342	+	26.5189i	−0.930166	+	1.15082i
$532$		0			0
$533$	13.3078			0.576426
$534$		0			0
$535$	2.65349	−	9.45967i	0.114720	−	0.408977i
$536$		0			0
$537$	1.26617	−	3.58079i	0.0546392	−	0.154522i
$538$		0			0
$539$	−0.894417			−0.0385253
$540$		0			0
$541$	1.02582			0.0441036			0.0220518	−	0.999757i	$-0.492980\pi$
$541$	1.02582			0.0441036			0.0220518	+	0.999757i	$0.492980\pi$
$542$		0			0
$543$	10.5636	−	29.8744i	0.453328	−	1.28203i
$544$		0			0
$545$	−8.12377	+	28.9612i	−0.347984	+	1.24056i
$546$		0			0
$547$			15.7049i			0.671494i	0.941952	+	0.335747i	$0.108989\pi$
$547$			15.7049i			0.671494i	−0.941952	+	0.335747i	$0.891011\pi$
$548$		0			0
$549$	16.4065	−	20.2985i	0.700211	−	0.866318i
$550$		0			0
$551$	10.9092			0.464746
$552$		0			0
$553$			7.08347i			0.301220i
$554$		0			0
$555$	−5.82729	−	4.10198i	−0.247355	−	0.174120i
$556$		0			0
$557$		−	18.4785i		−	0.782958i	−0.920187	−	0.391479i	$-0.871964\pi$
$557$		−	18.4785i		−	0.782958i	0.920187	−	0.391479i	$-0.128036\pi$
$558$		0			0
$559$			5.93051i			0.250834i
$560$		0			0
$561$	0.666490	−	1.88487i	0.0281392	−	0.0795791i
$562$		0			0
$563$			7.85340i			0.330982i	0.986211	+	0.165491i	$0.0529208\pi$
$563$			7.85340i			0.330982i	−0.986211	+	0.165491i	$0.947079\pi$
$564$		0			0
$565$	−10.4309	+	37.1862i	−0.438832	+	1.56443i
$566$		0			0
$567$	3.08991	+	14.4064i	0.129764	+	0.605014i
$568$		0			0
$569$	−7.81276			−0.327528			−0.163764	−	0.986500i	$-0.552364\pi$
$569$	−7.81276			−0.327528			−0.163764	+	0.986500i	$0.552364\pi$
$570$		0			0
$571$			2.82295i			0.118137i	0.998254	+	0.0590684i	$0.0188130\pi$
$571$			2.82295i			0.118137i	−0.998254	+	0.0590684i	$0.981187\pi$
$572$		0			0
$573$	−34.0509	−	12.0404i	−1.42250	−	0.502996i
$574$		0			0
$575$	22.2670	−	8.89826i	0.928600	−	0.371083i
$576$		0			0
$577$		−	2.24759i		−	0.0935685i	−0.998905	−	0.0467842i	$-0.985103\pi$
$577$		−	2.24759i		−	0.0935685i	0.998905	−	0.0467842i	$-0.0148973\pi$
$578$		0			0
$579$	−7.44838	+	21.0644i	−0.309544	+	0.875406i
$580$		0			0
$581$		−	2.72950i		−	0.113239i
$582$		0			0
$583$			0.978418i			0.0405219i
$584$		0			0
$585$	9.31613	−	6.61663i	0.385175	−	0.273564i
$586$		0			0
$587$	28.2490			1.16596			0.582981	−	0.812486i	$-0.301886\pi$
$587$	28.2490			1.16596			0.582981	+	0.812486i	$0.301886\pi$
$588$		0			0
$589$			6.40638i			0.263970i
$590$		0			0
$591$	−17.3574	−	6.13760i	−0.713989	−	0.252467i
$592$		0			0
$593$	30.8749			1.26788			0.633940	−	0.773382i	$-0.281436\pi$
$593$	30.8749			1.26788			0.633940	+	0.773382i	$0.281436\pi$
$594$		0			0
$595$	5.51175	−	19.6494i	0.225960	−	0.805545i
$596$		0			0
$597$	−10.8281	+	30.6223i	−0.443164	+	1.25329i
$598$		0			0
$599$		−	5.00576i		−	0.204530i	−0.994757	−	0.102265i	$-0.967391\pi$
$599$		−	5.00576i		−	0.204530i	0.994757	−	0.102265i	$-0.0326089\pi$
$600$		0			0
$601$	15.0456			0.613723			0.306861	−	0.951754i	$-0.400721\pi$
$601$	15.0456			0.613723			0.306861	+	0.951754i	$0.400721\pi$
$602$		0			0
$603$	−23.0341	−	18.6175i	−0.938019	−	0.758165i
$604$		0			0
$605$	−23.5904	−	6.61722i	−0.959085	−	0.269028i
$606$		0			0
$607$			41.7731i			1.69552i	0.530383	+	0.847758i	$0.322048\pi$
$607$			41.7731i			1.69552i	−0.530383	+	0.847758i	$0.677952\pi$
$608$		0			0
$609$	−4.44688	+	12.5760i	−0.180197	+	0.509605i
$610$		0			0
$611$			11.4468i			0.463089i
$612$		0			0
$613$	42.7894			1.72825			0.864124	−	0.503279i	$-0.167873\pi$
$613$	42.7894			1.72825			0.864124	+	0.503279i	$0.167873\pi$
$614$		0			0
$615$	−17.4168	+	24.7424i	−0.702315	+	0.997711i
$616$		0			0
$617$			46.8879i			1.88763i	0.330468	+	0.943817i	$0.392793\pi$
$617$			46.8879i			1.88763i	−0.330468	+	0.943817i	$0.607207\pi$
$618$		0			0
$619$			17.0556i			0.685523i	0.939422	+	0.342762i	$0.111362\pi$
$619$			17.0556i			0.685523i	−0.939422	+	0.342762i	$0.888638\pi$
$620$		0			0
$621$	24.8133	+	2.30200i	0.995724	+	0.0923762i
$622$		0			0
$623$	24.0588			0.963896
$624$		0			0
$625$	11.4754	+	22.2107i	0.459017	+	0.888428i
$626$		0			0
$627$	0.277250	−	0.784076i	0.0110723	−	0.0313130i
$628$		0			0
$629$			10.2576i			0.408997i
$630$		0			0
$631$		−	48.3233i		−	1.92372i	−0.273539	−	0.961861i	$-0.588194\pi$
$631$		−	48.3233i		−	1.92372i	0.273539	−	0.961861i	$-0.411806\pi$
$632$		0			0
$633$	−27.2490	−	9.63527i	−1.08305	−	0.382968i
$634$		0			0
$635$	5.95718	−	21.2373i	0.236404	−	0.842778i
$636$		0			0
$637$			7.35839i			0.291550i
$638$		0			0
$639$	1.35235	−	1.67316i	0.0534983	−	0.0661893i
$640$		0			0
$641$	9.06540			0.358062			0.179031	−	0.983843i	$-0.442704\pi$
$641$	9.06540			0.358062			0.179031	+	0.983843i	$0.442704\pi$
$642$		0			0
$643$	−22.5385			−0.888832			−0.444416	−	0.895820i	$-0.646589\pi$
$643$	−22.5385			−0.888832			−0.444416	+	0.895820i	$0.646589\pi$
$644$		0			0
$645$	−11.0262	−	7.76166i	−0.434158	−	0.305615i
$646$		0			0
$647$	−24.4503			−0.961242			−0.480621	−	0.876928i	$-0.659589\pi$
$647$	−24.4503			−0.961242			−0.480621	+	0.876928i	$0.659589\pi$
$648$		0			0
$649$			2.35332i			0.0923760i
$650$		0			0
$651$	−7.38522	−	2.61142i	−0.289450	−	0.102350i
$652$		0			0
$653$	18.8912			0.739271			0.369635	−	0.929177i	$-0.379483\pi$
$653$	18.8912			0.739271			0.369635	+	0.929177i	$0.379483\pi$
$654$		0			0
$655$	3.63018	−	12.9416i	0.141843	−	0.505669i
$656$		0			0
$657$	23.1193	−	28.6038i	0.901971	−	1.11594i
$658$		0			0
$659$	48.0612			1.87220			0.936100	−	0.351734i	$-0.114408\pi$
$659$	48.0612			1.87220			0.936100	+	0.351734i	$0.114408\pi$
$660$		0			0
$661$			15.1990i			0.591174i	0.955316	+	0.295587i	$0.0955153\pi$
$661$			15.1990i			0.591174i	−0.955316	+	0.295587i	$0.904485\pi$
$662$		0			0
$663$	−15.5068	−	5.48323i	−0.602235	−	0.212951i
$664$		0			0
$665$	2.29280	−	8.17384i	0.0889112	−	0.316968i
$666$		0			0
$667$	18.0461	+	13.5393i	0.698748	+	0.524243i
$668$		0			0
$669$	4.62303	−	13.0741i	0.178737	−	0.505475i
$670$		0			0
$671$		−	1.80131i		−	0.0695389i
$672$		0			0
$673$			11.9705i			0.461430i	0.973021	+	0.230715i	$0.0741065\pi$
$673$			11.9705i			0.461430i	−0.973021	+	0.230715i	$0.925894\pi$
$674$		0			0
$675$	0.109248	+	25.9805i	0.00420495	+	0.999991i
$676$		0			0
$677$		−	45.7040i		−	1.75655i	−0.478157	−	0.878274i	$-0.658695\pi$
$677$		−	45.7040i		−	1.75655i	0.478157	−	0.878274i	$-0.341305\pi$
$678$		0			0
$679$	−15.5725			−0.597616
$680$		0			0
$681$	1.55752	−	4.40473i	0.0596842	−	0.168790i
$682$		0			0
$683$	35.7900			1.36947			0.684733	−	0.728794i	$-0.259919\pi$
$683$	35.7900			1.36947			0.684733	+	0.728794i	$0.259919\pi$
$684$		0			0
$685$	5.49114	−	19.5759i	0.209806	−	0.747957i
$686$		0			0
$687$	11.6047	−	32.8187i	0.442748	−	1.25211i
$688$		0			0
$689$	8.04947			0.306660
$690$		0			0
$691$	27.7248			1.05470			0.527351	−	0.849648i	$-0.323185\pi$
$691$	27.7248			1.05470			0.527351	+	0.849648i	$0.323185\pi$
$692$		0			0
$693$	0.790862	+	0.639223i	0.0300424	+	0.0242821i
$694$		0			0
$695$	21.7152	+	6.09122i	0.823704	+	0.231053i
$696$		0			0
$697$	43.5533			1.64970
$698$		0			0
$699$	39.3731	+	13.9224i	1.48923	+	0.526592i
$700$		0			0
$701$	38.4541			1.45239			0.726196	−	0.687488i	$-0.241286\pi$
$701$	38.4541			1.45239			0.726196	+	0.687488i	$0.241286\pi$
$702$		0			0
$703$			4.26701i			0.160933i
$704$		0			0
$705$	−21.2824	−	14.9812i	−0.801541	−	0.564226i
$706$		0			0
$707$			21.8977i			0.823546i
$708$		0			0
$709$		−	23.1467i		−	0.869293i	−0.900601	−	0.434646i	$-0.856873\pi$
$709$		−	23.1467i		−	0.869293i	0.900601	−	0.434646i	$-0.143127\pi$
$710$		0			0
$711$	−8.15952	+	10.0952i	−0.306006	+	0.378598i
$712$		0			0
$713$	−7.95090	+	10.5975i	−0.297764	+	0.396881i
$714$		0			0
$715$	0.212994	−	0.759323i	0.00796552	−	0.0283971i
$716$		0			0
$717$	11.9262	−	33.7277i	0.445391	−	1.25959i
$718$		0			0
$719$			19.7459i			0.736399i	0.929747	+	0.368200i	$0.120026\pi$
$719$			19.7459i			0.736399i	−0.929747	+	0.368200i	$0.879974\pi$
$720$		0			0
$721$	−6.80681			−0.253499
$722$		0			0
$723$	13.2150	−	37.3725i	0.491469	−	1.38990i
$724$		0			0
$725$	−12.2329	+	20.0895i	−0.454320	+	0.746105i
$726$		0			0
$727$	37.9626			1.40795			0.703977	−	0.710223i	$-0.251406\pi$
$727$	37.9626			1.40795			0.703977	+	0.710223i	$0.251406\pi$
$728$		0			0
$729$	−12.1913	+	24.0909i	−0.451529	+	0.892256i
$730$		0			0
$731$			19.4092i			0.717873i
$732$		0			0
$733$	50.4670			1.86404			0.932020	−	0.362407i	$-0.118045\pi$
$733$	50.4670			1.86404			0.932020	+	0.362407i	$0.118045\pi$
$734$		0			0
$735$	−13.6810	−	9.63042i	−0.504631	−	0.355223i
$736$		0			0
$737$	−2.04407			−0.0752944
$738$		0			0
$739$	41.4082			1.52322			0.761612	−	0.648033i	$-0.224408\pi$
$739$	41.4082			1.52322			0.761612	+	0.648033i	$0.224408\pi$
$740$		0			0
$741$	−6.45061	−	2.28094i	−0.236969	−	0.0837925i
$742$		0			0
$743$		−	10.8120i		−	0.396652i	−0.980136	−	0.198326i	$-0.936450\pi$
$743$		−	10.8120i		−	0.396652i	0.980136	−	0.198326i	$-0.0635505\pi$
$744$		0			0
$745$	11.1743	+	3.13444i	0.409393	+	0.114837i
$746$		0			0
$747$	3.14414	−	3.89000i	0.115038	−	0.142328i
$748$		0			0
$749$		−	7.19315i		−	0.262832i
$750$		0			0
$751$		−	15.8172i		−	0.577176i	−0.957453	−	0.288588i	$-0.906814\pi$
$751$		−	15.8172i		−	0.577176i	0.957453	−	0.288588i	$-0.0931858\pi$
$752$		0			0
$753$	−20.0205	−	7.07925i	−0.729586	−	0.257982i
$754$		0			0
$755$	−7.07166	−	1.98364i	−0.257364	−	0.0721919i
$756$		0			0
$757$	−41.0607			−1.49238			−0.746188	−	0.665735i	$-0.768118\pi$
$757$	−41.0607			−1.49238			−0.746188	+	0.665735i	$0.768118\pi$
$758$		0			0
$759$	1.43174	−	0.952939i	0.0519689	−	0.0345895i
$760$		0			0
$761$			27.2604i			0.988189i	0.869408	+	0.494095i	$0.164500\pi$
$761$			27.2604i			0.988189i	−0.869408	+	0.494095i	$0.835500\pi$
$762$		0			0
$763$			22.0221i			0.797255i
$764$		0			0
$765$	30.4895	−	21.6546i	1.10235	−	0.782925i
$766$		0			0
$767$	19.3608			0.699080
$768$		0			0
$769$		−	2.40554i		−	0.0867461i	−0.999059	−	0.0433730i	$-0.986190\pi$
$769$		−	2.40554i		−	0.0867461i	0.999059	−	0.0433730i	$-0.0138104\pi$
$770$		0			0
$771$	34.0649	+	12.0454i	1.22682	+	0.433803i
$772$		0			0
$773$			28.1266i			1.01164i	0.862638	+	0.505822i	$0.168811\pi$
$773$			28.1266i			1.01164i	−0.862638	+	0.505822i	$0.831189\pi$
$774$		0			0
$775$	−11.7975	−	7.18376i	−0.423779	−	0.258048i
$776$		0			0
$777$	−4.91898	−	1.73935i	−0.176467	−	0.0623990i
$778$		0			0
$779$	18.1175			0.649128
$780$		0			0
$781$		−	0.148479i		−	0.00531299i
$782$		0			0
$783$	−20.8240	+	12.8005i	−0.744188	+	0.457453i
$784$		0			0
$785$	−9.45885	−	2.65326i	−0.337601	−	0.0946988i
$786$		0			0
$787$	−36.6014			−1.30470			−0.652350	−	0.757918i	$-0.726217\pi$
$787$	−36.6014			−1.30470			−0.652350	+	0.757918i	$0.726217\pi$
$788$		0			0
$789$	−0.197541	+	0.558656i	−0.00703265	+	0.0198887i
$790$		0			0
$791$			28.2764i			1.00539i
$792$		0			0
$793$	−14.8194			−0.526254
$794$		0			0
$795$	−10.5349	+	14.9659i	−0.373634	+	0.530785i
$796$		0			0
$797$		−	34.7219i		−	1.22991i	−0.788562	−	0.614956i	$-0.789174\pi$
$797$		−	34.7219i		−	1.22991i	0.788562	−	0.614956i	$-0.210826\pi$
$798$		0			0
$799$			37.4627i			1.32534i
$800$		0			0
$801$	34.2879	+	27.7136i	1.21150	+	0.979211i
$802$		0			0
$803$		−	2.53834i		−	0.0895760i
$804$		0			0
$805$	13.9373	−	10.6757i	0.491225	−	0.376270i
$806$		0			0
$807$	−2.67382	+	7.56168i	−0.0941228	+	0.266184i
$808$		0			0
$809$		−	14.2714i		−	0.501756i	−0.968019	−	0.250878i	$-0.919281\pi$
$809$		−	14.2714i		−	0.501756i	0.968019	−	0.250878i	$-0.0807193\pi$
$810$		0			0
$811$	22.5273			0.791041			0.395520	−	0.918457i	$-0.370564\pi$
$811$	22.5273			0.791041			0.395520	+	0.918457i	$0.370564\pi$
$812$		0			0
$813$	−25.1148	−	8.88060i	−0.880813	−	0.311456i
$814$		0			0
$815$	3.26905	−	11.6541i	0.114510	−	0.408227i
$816$		0			0
$817$			8.07392i			0.282471i
$818$		0			0
$819$	5.25890	−	6.50644i	0.183761	−	0.227353i
$820$		0			0
$821$		−	38.1320i		−	1.33082i	−0.746480	−	0.665408i	$-0.768258\pi$
$821$		−	38.1320i		−	1.33082i	0.746480	−	0.665408i	$-0.231742\pi$
$822$		0			0
$823$			19.8857i			0.693170i	0.938018	+	0.346585i	$0.112659\pi$
$823$			19.8857i			0.693170i	−0.938018	+	0.346585i	$0.887341\pi$
$824$		0			0
$825$	1.13300	+	1.38978i	0.0394461	+	0.0483860i
$826$		0			0
$827$		−	14.8977i		−	0.518045i	−0.965871	−	0.259022i	$-0.916600\pi$
$827$		−	14.8977i		−	0.518045i	0.965871	−	0.259022i	$-0.0834003\pi$
$828$		0			0
$829$	−11.2770			−0.391665			−0.195833	−	0.980637i	$-0.562741\pi$
$829$	−11.2770			−0.391665			−0.195833	+	0.980637i	$0.562741\pi$
$830$		0			0
$831$	−7.53756	+	21.3166i	−0.261475	+	0.739464i
$832$		0			0
$833$			24.0822i			0.834400i
$834$		0			0
$835$	−5.39447	−	1.51318i	−0.186683	−	0.0523656i
$836$		0			0
$837$	−7.51707	−	12.2288i	−0.259828	−	0.422690i
$838$		0			0
$839$	11.2973			0.390026			0.195013	−	0.980801i	$-0.437525\pi$
$839$	11.2973			0.390026			0.195013	+	0.980801i	$0.437525\pi$
$840$		0			0
$841$	6.87069			0.236920
$842$		0			0
$843$	16.8007	+	5.94073i	0.578646	+	0.204610i
$844$		0			0
$845$	21.7417	+	6.09864i	0.747936	+	0.209800i
$846$		0			0
$847$	−17.9381			−0.616362
$848$		0			0
$849$	3.66143	+	1.29468i	0.125660	+	0.0444334i
$850$		0			0
$851$	−5.29575	+	7.05856i	−0.181536	+	0.241964i
$852$		0			0
$853$			35.1200i			1.20248i	0.799067	+	0.601242i	$0.205327\pi$
$853$			35.1200i			1.20248i	−0.799067	+	0.601242i	$0.794673\pi$
$854$		0			0
$855$	12.6832	−	9.00801i	0.433755	−	0.308067i
$856$		0			0
$857$	−0.883179			−0.0301688			−0.0150844	−	0.999886i	$-0.504802\pi$
$857$	−0.883179			−0.0301688			−0.0150844	+	0.999886i	$0.504802\pi$
$858$		0			0
$859$	4.79758			0.163691			0.0818457	−	0.996645i	$-0.473919\pi$
$859$	4.79758			0.163691			0.0818457	+	0.996645i	$0.473919\pi$
$860$		0			0
$861$	−7.38522	+	20.8858i	−0.251688	+	0.711785i
$862$		0			0
$863$	0.163247			0.00555699			0.00277850	−	0.999996i	$-0.499116\pi$
$863$	0.163247			0.00555699			0.00277850	+	0.999996i	$0.499116\pi$
$864$		0			0
$865$	27.5701	+	7.73356i	0.937412	+	0.262949i
$866$		0			0
$867$	−22.9897	−	8.12917i	−0.780770	−	0.276081i
$868$		0			0
$869$			0.895857i			0.0303899i
$870$		0			0
$871$			16.8166i			0.569809i
$872$		0			0
$873$	−22.1934	−	17.9381i	−0.751132	−	0.607111i
$874$		0			0
$875$	12.4813	+	13.3880i	0.421945	+	0.452596i
$876$		0			0
$877$			20.7964i			0.702244i	0.936330	+	0.351122i	$0.114200\pi$
$877$			20.7964i			0.702244i	−0.936330	+	0.351122i	$0.885800\pi$
$878$		0			0
$879$	−3.43856	+	9.72442i	−0.115980	+	0.327996i
$880$		0			0
$881$	−6.45300			−0.217407			−0.108704	−	0.994074i	$-0.534670\pi$
$881$	−6.45300			−0.217407			−0.108704	+	0.994074i	$0.534670\pi$
$882$		0			0
$883$		−	37.3247i		−	1.25608i	−0.778182	−	0.628039i	$-0.783858\pi$
$883$		−	37.3247i		−	1.25608i	0.778182	−	0.628039i	$-0.216142\pi$
$884$		0			0
$885$	−25.3388	+	35.9964i	−0.851756	+	1.21001i
$886$		0			0
$887$	9.48401			0.318442			0.159221	−	0.987243i	$-0.449102\pi$
$887$	9.48401			0.318442			0.159221	+	0.987243i	$0.449102\pi$
$888$		0			0
$889$		−	16.1489i		−	0.541616i
$890$		0			0
$891$	0.390785	+	1.82200i	0.0130918	+	0.0610394i
$892$		0			0
$893$			15.5839i			0.521496i
$894$		0			0
$895$	1.32428	−	4.72105i	0.0442658	−	0.157807i
$896$		0			0
$897$	−7.83985	−	11.7790i	−0.261765	−	0.393288i
$898$		0			0
$899$		−	12.9954i		−	0.433420i
$900$		0			0
$901$	26.3440			0.877645
$902$		0			0
$903$	−9.30755	−	3.29116i	−0.309736	−	0.109523i
$904$		0			0
$905$	11.0484	−	39.3876i	0.367262	−	1.30929i
$906$		0			0
$907$	50.2287			1.66782			0.833908	−	0.551904i	$-0.186099\pi$
$907$	50.2287			1.66782			0.833908	+	0.551904i	$0.186099\pi$
$908$		0			0
$909$	−25.2241	+	31.2079i	−0.836631	+	1.03510i
$910$		0			0
$911$	−21.9604			−0.727581			−0.363790	−	0.931481i	$-0.618518\pi$
$911$	−21.9604			−0.727581			−0.363790	+	0.931481i	$0.618518\pi$
$912$		0			0
$913$		−	0.345204i		−	0.0114246i
$914$		0			0
$915$	19.3952	−	27.5529i	0.641185	−	0.910870i
$916$		0			0
$917$		−	9.84078i		−	0.324971i
$918$		0			0
$919$		−	7.80540i		−	0.257476i	−0.991679	−	0.128738i	$-0.958907\pi$
$919$		−	7.80540i		−	0.257476i	0.991679	−	0.128738i	$-0.0410927\pi$
$920$		0			0
$921$	14.7333	−	41.6665i	0.485479	−	1.37296i
$922$		0			0
$923$	−1.22154			−0.0402074
$924$		0			0
$925$	−7.85781	−	4.78479i	−0.258363	−	0.157323i
$926$		0			0
$927$	−9.70085	−	7.84083i	−0.318618	−	0.257527i
$928$		0			0
$929$			7.68969i			0.252291i	0.992012	+	0.126145i	$0.0402606\pi$
$929$			7.68969i			0.252291i	−0.992012	+	0.126145i	$0.959739\pi$
$930$		0			0
$931$			10.0179i			0.328322i
$932$		0			0
$933$	18.8186	−	53.2200i	0.616095	−	1.74235i
$934$		0			0
$935$	0.697078	−	2.48508i	0.0227969	−	0.0812709i
$936$		0			0
$937$	−37.6380			−1.22958			−0.614789	−	0.788691i	$-0.710759\pi$
$937$	−37.6380			−1.22958			−0.614789	+	0.788691i	$0.710759\pi$
$938$		0			0
$939$	34.5122	+	12.2036i	1.12626	+	0.398248i
$940$		0			0
$941$	1.54187			0.0502637			0.0251318	−	0.999684i	$-0.491999\pi$
$941$	1.54187			0.0502637			0.0251318	+	0.999684i	$0.491999\pi$
$942$		0			0
$943$	29.9703	+	22.4855i	0.975968	+	0.732229i
$944$		0			0
$945$	5.21434	+	18.2930i	0.169623	+	0.595070i
$946$		0			0
$947$	−9.95144			−0.323378			−0.161689	−	0.986842i	$-0.551694\pi$
$947$	−9.95144			−0.323378			−0.161689	+	0.986842i	$0.551694\pi$
$948$		0			0
$949$	−20.8829			−0.677889
$950$		0			0
$951$	−43.4804	−	15.3747i	−1.40995	−	0.498559i
$952$		0			0
$953$			27.8854i			0.903295i	0.892196	+	0.451648i	$0.149164\pi$
$953$			27.8854i			0.903295i	−0.892196	+	0.451648i	$0.850836\pi$
$954$		0			0
$955$	−44.8940	−	12.5930i	−1.45274	−	0.407500i
$956$		0			0
$957$	−0.562403	+	1.59050i	−0.0181799	+	0.0514137i
$958$		0			0
$959$		−	14.8855i		−	0.480679i
$960$		0			0
$961$	−23.3685			−0.753822
$962$		0			0
$963$	8.28585	−	10.2514i	0.267008	−	0.330348i
$964$		0			0
$965$	−7.79022	+	27.7721i	−0.250776	+	0.894016i
$966$		0			0
$967$		−	54.8270i		−	1.76312i	−0.472076	−	0.881558i	$-0.656495\pi$
$967$		−	54.8270i		−	1.76312i	0.472076	−	0.881558i	$-0.343505\pi$
$968$		0			0
$969$	−21.1113	−	7.46498i	−0.678193	−	0.239810i
$970$		0			0
$971$	38.1975			1.22581			0.612907	−	0.790155i	$-0.290000\pi$
$971$	38.1975			1.22581			0.612907	+	0.790155i	$0.290000\pi$
$972$		0			0
$973$	16.5123			0.529358
$974$		0			0
$975$	11.4338	−	9.32124i	0.366174	−	0.298519i
$976$		0			0
$977$		−	9.12990i		−	0.292091i	−0.989278	−	0.146046i	$-0.953345\pi$
$977$		−	9.12990i		−	0.292091i	0.989278	−	0.146046i	$-0.0466546\pi$
$978$		0			0
$979$	3.04275			0.0972468
$980$		0			0
$981$	−25.3675	+	31.3853i	−0.809922	+	1.00205i
$982$		0			0
$983$			41.2738i			1.31643i	0.752830	+	0.658216i	$0.228688\pi$
$983$			41.2738i			1.31643i	−0.752830	+	0.658216i	$0.771312\pi$
$984$		0			0
$985$	−22.8847	−	6.41928i	−0.729167	−	0.204535i
$986$		0			0
$987$	−17.9650	−	6.35245i	−0.571833	−	0.202201i
$988$		0			0
$989$	−10.0205	+	13.3560i	−0.318632	+	0.424696i
$990$		0			0
$991$	−38.9675			−1.23784			−0.618922	−	0.785453i	$-0.712430\pi$
$991$	−38.9675			−1.23784			−0.618922	+	0.785453i	$0.712430\pi$
$992$		0			0
$993$	−33.5477	−	11.8625i	−1.06460	−	0.376445i
$994$		0			0
$995$	−11.3250	+	40.3737i	−0.359027	+	1.27993i
$996$		0			0
$997$			3.06107i			0.0969451i	0.998825	+	0.0484726i	$0.0154353\pi$
$997$			3.06107i			0.0969451i	−0.998825	+	0.0484726i	$0.984565\pi$
$998$		0			0
$999$	−5.00680	−	8.14509i	−0.158408	−	0.257699i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.2.n.a.689.44	yes	48
3.2	odd	2	inner	1380.2.n.a.689.7	yes	48
5.4	even	2	inner	1380.2.n.a.689.6	yes	48
15.14	odd	2	inner	1380.2.n.a.689.41	yes	48
23.22	odd	2	inner	1380.2.n.a.689.43	yes	48
69.68	even	2	inner	1380.2.n.a.689.8	yes	48
115.114	odd	2	inner	1380.2.n.a.689.5	✓	48
345.344	even	2	inner	1380.2.n.a.689.42	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.n.a.689.5	✓	48	115.114	odd	2	inner
1380.2.n.a.689.6	yes	48	5.4	even	2	inner
1380.2.n.a.689.7	yes	48	3.2	odd	2	inner
1380.2.n.a.689.8	yes	48	69.68	even	2	inner
1380.2.n.a.689.41	yes	48	15.14	odd	2	inner
1380.2.n.a.689.42	yes	48	345.344	even	2	inner
1380.2.n.a.689.43	yes	48	23.22	odd	2	inner
1380.2.n.a.689.44	yes	48	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.n (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.63297 + 0.577419i) q^{3} +(2.15297 + 0.603919i) q^{5} +1.63712 q^{7} +(2.33317 + 1.88581i) q^{9} +O(q^{10})\)	\(q+(1.63297 + 0.577419i) q^{3} +(2.15297 + 0.603919i) q^{5} +1.63712 q^{7} +(2.33317 + 1.88581i) q^{9} +0.207049 q^{11} -1.70340i q^{13} +(3.16702 + 2.22935i) q^{15} -5.57481i q^{17} -2.31904i q^{19} +(2.67337 + 0.945305i) q^{21} +(2.87814 - 3.83619i) q^{23} +(4.27056 + 2.60044i) q^{25} +(2.72110 + 4.42670i) q^{27} +4.70418i q^{29} -2.76252 q^{31} +(0.338104 + 0.119554i) q^{33} +(3.52467 + 0.988689i) q^{35} -1.83999 q^{37} +(0.983573 - 2.78159i) q^{39} +7.81253i q^{41} -3.48158 q^{43} +(3.88438 + 5.46915i) q^{45} -6.72000 q^{47} -4.31984 q^{49} +(3.21900 - 9.10348i) q^{51} +4.72554i q^{53} +(0.445770 + 0.125041i) q^{55} +(1.33906 - 3.78691i) q^{57} +11.3660i q^{59} -8.69994i q^{61} +(3.81969 + 3.08731i) q^{63} +(1.02871 - 3.66736i) q^{65} -9.87241 q^{67} +(6.91499 - 4.60248i) q^{69} -0.717119i q^{71} -12.2596i q^{73} +(5.47215 + 6.71234i) q^{75} +0.338964 q^{77} +4.32679i q^{79} +(1.88740 + 8.79987i) q^{81} -1.66726i q^{83} +(3.36673 - 12.0024i) q^{85} +(-2.71628 + 7.68178i) q^{87} +14.6958 q^{89} -2.78866i q^{91} +(-4.51110 - 1.59513i) q^{93} +(1.40051 - 4.99282i) q^{95} -9.51210 q^{97} +(0.483081 + 0.390456i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 16 q^{31} + 24 q^{39} + 112 q^{49} + 8 q^{55} - 16 q^{69} + 20 q^{75} - 8 q^{81} + 32 q^{85}+O(q^{100}) \) 48 * q + 16 * q^31 + 24 * q^39 + 112 * q^49 + 8 * q^55 - 16 * q^69 + 20 * q^75 - 8 * q^81 + 32 * q^85

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