LMFDB - Embedded newform 3432.2.g.d.1585.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3432,2,Mod(1585,3432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

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

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

chi := DirichletCharacter("3432.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3432.g (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:	$27.4046579737$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 25x^{12} + 236x^{10} + 1040x^{8} + 2124x^{6} + 1676x^{4} + 340x^{2} + 16 $ x^14 + 25x^12 + 236x^10 + 1040x^8 + 2124x^6 + 1676x^4 + 340x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.10
Root			$1.08554i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.08554i q^{5} -2.42229i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.08554i q^{5} -2.42229i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(-2.23141 - 2.83210i) q^{13} +1.08554i q^{15} +1.16023 q^{17} +6.05565i q^{19} -2.42229i q^{21} +0.650519 q^{23} +3.82160 q^{25} +1.00000 q^{27} +0.993564 q^{29} -4.98238i q^{31} -1.00000i q^{33} +2.62949 q^{35} +5.84844i q^{37} +(-2.23141 - 2.83210i) q^{39} -9.99409i q^{41} +9.22816 q^{43} +1.08554i q^{45} -1.16722i q^{47} +1.13251 q^{49} +1.16023 q^{51} -8.43984 q^{53} +1.08554 q^{55} +6.05565i q^{57} -7.90211i q^{59} +2.12507 q^{61} -2.42229i q^{63} +(3.07436 - 2.42229i) q^{65} -11.7643i q^{67} +0.650519 q^{69} -13.7648i q^{71} -4.23833i q^{73} +3.82160 q^{75} -2.42229 q^{77} +14.2819 q^{79} +1.00000 q^{81} +0.514539i q^{83} +1.25948i q^{85} +0.993564 q^{87} +11.3065i q^{89} +(-6.86017 + 5.40513i) q^{91} -4.98238i q^{93} -6.57365 q^{95} -15.3992i q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) $ 14 * q + 14 * q^3 + 14 * q^9	$ 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) $ 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3432\mathbb{Z}\right)^\times$.

$n$	$937$	$1145$	$1717$	$2575$	$2641$
$\chi(n)$	$1$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			1.08554i			0.485469i	0.970093	+	0.242734i	$0.0780443\pi$
$5$			1.08554i			0.485469i	−0.970093	+	0.242734i	$0.921956\pi$
$6$		0			0
$7$		−	2.42229i		−	0.915539i	−0.889071	−	0.457770i	$-0.848648\pi$
$7$		−	2.42229i		−	0.915539i	0.889071	−	0.457770i	$-0.151352\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−2.23141	−	2.83210i	−0.618882	−	0.785484i
$13$	−2.23141	−	2.83210i	−0.618882	−	0.785484i
$14$		0			0
$15$			1.08554i			0.280285i
$16$		0			0
$17$	1.16023			0.281398			0.140699	−	0.990052i	$-0.455065\pi$
$17$	1.16023			0.281398			0.140699	+	0.990052i	$0.455065\pi$
$18$		0			0
$19$			6.05565i			1.38926i	0.719367	+	0.694630i	$0.244432\pi$
$19$			6.05565i			1.38926i	−0.719367	+	0.694630i	$0.755568\pi$
$20$		0			0
$21$		−	2.42229i		−	0.528587i
$22$		0			0
$23$	0.650519			0.135643			0.0678213	−	0.997697i	$-0.478395\pi$
$23$	0.650519			0.135643			0.0678213	+	0.997697i	$0.478395\pi$
$24$		0			0
$25$	3.82160			0.764320
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	0.993564			0.184500			0.0922501	−	0.995736i	$-0.470594\pi$
$29$	0.993564			0.184500			0.0922501	+	0.995736i	$0.470594\pi$
$30$		0			0
$31$		−	4.98238i		−	0.894863i	−0.894318	−	0.447431i	$-0.852339\pi$
$31$		−	4.98238i		−	0.894863i	0.894318	−	0.447431i	$-0.147661\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	2.62949			0.444466
$36$		0			0
$37$			5.84844i			0.961478i	0.876864	+	0.480739i	$0.159632\pi$
$37$			5.84844i			0.961478i	−0.876864	+	0.480739i	$0.840368\pi$
$38$		0			0
$39$	−2.23141	−	2.83210i	−0.357312	−	0.453499i
$40$		0			0
$41$		−	9.99409i		−	1.56081i	−0.625272	−	0.780407i	$-0.715012\pi$
$41$		−	9.99409i		−	1.56081i	0.625272	−	0.780407i	$-0.284988\pi$
$42$		0			0
$43$	9.22816			1.40728			0.703641	−	0.710556i	$-0.251557\pi$
$43$	9.22816			1.40728			0.703641	+	0.710556i	$0.251557\pi$
$44$		0			0
$45$			1.08554i			0.161823i
$46$		0			0
$47$		−	1.16722i		−	0.170257i	−0.996370	−	0.0851283i	$-0.972870\pi$
$47$		−	1.16722i		−	0.170257i	0.996370	−	0.0851283i	$-0.0271300\pi$
$48$		0			0
$49$	1.13251			0.161788
$50$		0			0
$51$	1.16023			0.162465
$52$		0			0
$53$	−8.43984			−1.15930			−0.579650	−	0.814865i	$-0.696811\pi$
$53$	−8.43984			−1.15930			−0.579650	+	0.814865i	$0.696811\pi$
$54$		0			0
$55$	1.08554			0.146374
$56$		0			0
$57$			6.05565i			0.802090i
$58$		0			0
$59$		−	7.90211i		−	1.02877i	−0.857560	−	0.514384i	$-0.828021\pi$
$59$		−	7.90211i		−	1.02877i	0.857560	−	0.514384i	$-0.171979\pi$
$60$		0			0
$61$	2.12507			0.272088			0.136044	−	0.990703i	$-0.456561\pi$
$61$	2.12507			0.272088			0.136044	+	0.990703i	$0.456561\pi$
$62$		0			0
$63$		−	2.42229i		−	0.305180i
$64$		0			0
$65$	3.07436	−	2.42229i	0.381328	−	0.300448i
$66$		0			0
$67$		−	11.7643i		−	1.43724i	−0.695404	−	0.718619i	$-0.744774\pi$
$67$		−	11.7643i		−	1.43724i	0.695404	−	0.718619i	$-0.255226\pi$
$68$		0			0
$69$	0.650519			0.0783132
$70$		0			0
$71$		−	13.7648i		−	1.63358i	−0.576935	−	0.816790i	$-0.695751\pi$
$71$		−	13.7648i		−	1.63358i	0.576935	−	0.816790i	$-0.304249\pi$
$72$		0			0
$73$		−	4.23833i		−	0.496060i	−0.968752	−	0.248030i	$-0.920217\pi$
$73$		−	4.23833i		−	0.496060i	0.968752	−	0.248030i	$-0.0797831\pi$
$74$		0			0
$75$	3.82160			0.441280
$76$		0			0
$77$	−2.42229			−0.276046
$78$		0			0
$79$	14.2819			1.60684			0.803421	−	0.595412i	$-0.203011\pi$
$79$	14.2819			1.60684			0.803421	+	0.595412i	$0.203011\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			0.514539i			0.0564780i	0.999601	+	0.0282390i	$0.00898994\pi$
$83$			0.514539i			0.0564780i	−0.999601	+	0.0282390i	$0.991010\pi$
$84$		0			0
$85$			1.25948i			0.136610i
$86$		0			0
$87$	0.993564			0.106521
$88$		0			0
$89$			11.3065i			1.19849i	0.800566	+	0.599245i	$0.204532\pi$
$89$			11.3065i			1.19849i	−0.800566	+	0.599245i	$0.795468\pi$
$90$		0			0
$91$	−6.86017	+	5.40513i	−0.719141	+	0.566611i
$92$		0			0
$93$		−	4.98238i		−	0.516649i
$94$		0			0
$95$	−6.57365			−0.674442
$96$		0			0
$97$		−	15.3992i		−	1.56355i	−0.623561	−	0.781775i	$-0.714315\pi$
$97$		−	15.3992i		−	1.56355i	0.623561	−	0.781775i	$-0.285685\pi$
$98$		0			0
$99$		−	1.00000i		−	0.100504i
$100$		0			0
$101$	−12.1396			−1.20793			−0.603966	−	0.797010i	$-0.706414\pi$
$101$	−12.1396			−1.20793			−0.603966	+	0.797010i	$0.706414\pi$
$102$		0			0
$103$	−14.6543			−1.44393			−0.721965	−	0.691930i	$-0.756761\pi$
$103$	−14.6543			−1.44393			−0.721965	+	0.691930i	$0.756761\pi$
$104$		0			0
$105$	2.62949			0.256612
$106$		0			0
$107$	9.73103			0.940734			0.470367	−	0.882471i	$-0.344122\pi$
$107$	9.73103			0.940734			0.470367	+	0.882471i	$0.344122\pi$
$108$		0			0
$109$		−	11.4991i		−	1.10141i	−0.834700	−	0.550705i	$-0.814359\pi$
$109$		−	11.4991i		−	1.10141i	0.834700	−	0.550705i	$-0.185641\pi$
$110$		0			0
$111$			5.84844i			0.555109i
$112$		0			0
$113$	−8.64375			−0.813136			−0.406568	−	0.913621i	$-0.633275\pi$
$113$	−8.64375			−0.813136			−0.406568	+	0.913621i	$0.633275\pi$
$114$		0			0
$115$			0.706165i			0.0658502i
$116$		0			0
$117$	−2.23141	−	2.83210i	−0.206294	−	0.261828i
$118$		0			0
$119$		−	2.81042i		−	0.257631i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	9.99409i		−	0.901136i
$124$		0			0
$125$			9.57621i			0.856522i
$126$		0			0
$127$	14.4234			1.27987			0.639933	−	0.768430i	$-0.278962\pi$
$127$	14.4234			1.27987			0.639933	+	0.768430i	$0.278962\pi$
$128$		0			0
$129$	9.22816			0.812494
$130$		0			0
$131$	5.79231			0.506077			0.253038	−	0.967456i	$-0.418570\pi$
$131$	5.79231			0.506077			0.253038	+	0.967456i	$0.418570\pi$
$132$		0			0
$133$	14.6685			1.27192
$134$		0			0
$135$			1.08554i			0.0934285i
$136$		0			0
$137$		−	9.45837i		−	0.808083i	−0.914741	−	0.404041i	$-0.867605\pi$
$137$		−	9.45837i		−	0.808083i	0.914741	−	0.404041i	$-0.132395\pi$
$138$		0			0
$139$	21.1832			1.79674			0.898368	−	0.439244i	$-0.144754\pi$
$139$	21.1832			1.79674			0.898368	+	0.439244i	$0.144754\pi$
$140$		0			0
$141$		−	1.16722i		−	0.0982977i
$142$		0			0
$143$	−2.83210	+	2.23141i	−0.236832	+	0.186600i
$144$		0			0
$145$			1.07855i			0.0895690i
$146$		0			0
$147$	1.13251			0.0934081
$148$		0			0
$149$			5.83285i			0.477846i	0.971039	+	0.238923i	$0.0767943\pi$
$149$			5.83285i			0.477846i	−0.971039	+	0.238923i	$0.923206\pi$
$150$		0			0
$151$		−	5.25702i		−	0.427811i	−0.976854	−	0.213905i	$-0.931382\pi$
$151$		−	5.25702i		−	0.427811i	0.976854	−	0.213905i	$-0.0686184\pi$
$152$		0			0
$153$	1.16023			0.0937993
$154$		0			0
$155$	5.40858			0.434428
$156$		0			0
$157$	−15.4388			−1.23215			−0.616075	−	0.787687i	$-0.711278\pi$
$157$	−15.4388			−1.23215			−0.616075	+	0.787687i	$0.711278\pi$
$158$		0			0
$159$	−8.43984			−0.669322
$160$		0			0
$161$		−	1.57574i		−	0.124186i
$162$		0			0
$163$			19.9209i			1.56033i	0.625576	+	0.780163i	$0.284864\pi$
$163$			19.9209i			1.56033i	−0.625576	+	0.780163i	$0.715136\pi$
$164$		0			0
$165$	1.08554			0.0845093
$166$		0			0
$167$			22.7588i			1.76113i	0.473928	+	0.880563i	$0.342836\pi$
$167$			22.7588i			1.76113i	−0.473928	+	0.880563i	$0.657164\pi$
$168$		0			0
$169$	−3.04160	+	12.6392i	−0.233969	+	0.972244i
$170$		0			0
$171$			6.05565i			0.463087i
$172$		0			0
$173$	12.7184			0.966963			0.483482	−	0.875354i	$-0.339372\pi$
$173$	12.7184			0.966963			0.483482	+	0.875354i	$0.339372\pi$
$174$		0			0
$175$		−	9.25702i		−	0.699765i
$176$		0			0
$177$		−	7.90211i		−	0.593959i
$178$		0			0
$179$	9.80928			0.733180			0.366590	−	0.930383i	$-0.380525\pi$
$179$	9.80928			0.733180			0.366590	+	0.930383i	$0.380525\pi$
$180$		0			0
$181$	20.0517			1.49043			0.745216	−	0.666823i	$-0.232346\pi$
$181$	20.0517			1.49043			0.745216	+	0.666823i	$0.232346\pi$
$182$		0			0
$183$	2.12507			0.157090
$184$		0			0
$185$	−6.34872			−0.466767
$186$		0			0
$187$		−	1.16023i		−	0.0848447i
$188$		0			0
$189$		−	2.42229i		−	0.176196i
$190$		0			0
$191$	−21.4452			−1.55172			−0.775859	−	0.630906i	$-0.782683\pi$
$191$	−21.4452			−1.55172			−0.775859	+	0.630906i	$0.782683\pi$
$192$		0			0
$193$		−	21.9974i		−	1.58341i	−0.610904	−	0.791705i	$-0.709194\pi$
$193$		−	21.9974i		−	1.58341i	0.610904	−	0.791705i	$-0.290806\pi$
$194$		0			0
$195$	3.07436	−	2.42229i	0.220160	−	0.173464i
$196$		0			0
$197$		−	13.2243i		−	0.942189i	−0.882083	−	0.471094i	$-0.843859\pi$
$197$		−	13.2243i		−	0.942189i	0.882083	−	0.471094i	$-0.156141\pi$
$198$		0			0
$199$	17.7891			1.26104			0.630518	−	0.776174i	$-0.282842\pi$
$199$	17.7891			1.26104			0.630518	+	0.776174i	$0.282842\pi$
$200$		0			0
$201$		−	11.7643i		−	0.829790i
$202$		0			0
$203$		−	2.40670i		−	0.168917i
$204$		0			0
$205$	10.8490			0.757726
$206$		0			0
$207$	0.650519			0.0452142
$208$		0			0
$209$	6.05565			0.418878
$210$		0			0
$211$	−21.5092			−1.48075			−0.740376	−	0.672193i	$-0.765352\pi$
$211$	−21.5092			−1.48075			−0.740376	+	0.672193i	$0.765352\pi$
$212$		0			0
$213$		−	13.7648i		−	0.943148i
$214$		0			0
$215$			10.0175i			0.683191i
$216$		0			0
$217$	−12.0688			−0.819282
$218$		0			0
$219$		−	4.23833i		−	0.286400i
$220$		0			0
$221$	−2.58896	−	3.28590i	−0.174152	−	0.221034i
$222$		0			0
$223$			3.79915i			0.254410i	0.991876	+	0.127205i	$0.0406005\pi$
$223$			3.79915i			0.254410i	−0.991876	+	0.127205i	$0.959399\pi$
$224$		0			0
$225$	3.82160			0.254773
$226$		0			0
$227$			25.2277i			1.67442i	0.546881	+	0.837210i	$0.315815\pi$
$227$			25.2277i			1.67442i	−0.546881	+	0.837210i	$0.684185\pi$
$228$		0			0
$229$		−	9.68915i		−	0.640277i	−0.947371	−	0.320138i	$-0.896271\pi$
$229$		−	9.68915i		−	0.640277i	0.947371	−	0.320138i	$-0.103729\pi$
$230$		0			0
$231$	−2.42229			−0.159375
$232$		0			0
$233$	22.3698			1.46549			0.732747	−	0.680501i	$-0.238238\pi$
$233$	22.3698			1.46549			0.732747	+	0.680501i	$0.238238\pi$
$234$		0			0
$235$	1.26707			0.0826543
$236$		0			0
$237$	14.2819			0.927710
$238$		0			0
$239$			9.53892i			0.617022i	0.951221	+	0.308511i	$0.0998306\pi$
$239$			9.53892i			0.617022i	−0.951221	+	0.308511i	$0.900169\pi$
$240$		0			0
$241$			10.9265i			0.703839i	0.936030	+	0.351919i	$0.114471\pi$
$241$			10.9265i			0.703839i	−0.936030	+	0.351919i	$0.885529\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			1.22939i			0.0785428i
$246$		0			0
$247$	17.1502	−	13.5126i	1.09124	−	0.859789i
$248$		0			0
$249$			0.514539i			0.0326076i
$250$		0			0
$251$	−17.6754			−1.11566			−0.557830	−	0.829955i	$-0.688366\pi$
$251$	−17.6754			−1.11566			−0.557830	+	0.829955i	$0.688366\pi$
$252$		0			0
$253$		−	0.650519i		−	0.0408978i
$254$		0			0
$255$			1.25948i			0.0788718i
$256$		0			0
$257$	24.2035			1.50978			0.754888	−	0.655854i	$-0.227691\pi$
$257$	24.2035			1.50978			0.754888	+	0.655854i	$0.227691\pi$
$258$		0			0
$259$	14.1666			0.880271
$260$		0			0
$261$	0.993564			0.0615000
$262$		0			0
$263$	5.83418			0.359751			0.179876	−	0.983689i	$-0.442430\pi$
$263$	5.83418			0.359751			0.179876	+	0.983689i	$0.442430\pi$
$264$		0			0
$265$		−	9.16179i		−	0.562804i
$266$		0			0
$267$			11.3065i			0.691948i
$268$		0			0
$269$	−23.8973			−1.45704			−0.728521	−	0.685024i	$-0.759792\pi$
$269$	−23.8973			−1.45704			−0.728521	+	0.685024i	$0.759792\pi$
$270$		0			0
$271$		−	21.8469i		−	1.32711i	−0.748129	−	0.663554i	$-0.769047\pi$
$271$		−	21.8469i		−	1.32711i	0.748129	−	0.663554i	$-0.230953\pi$
$272$		0			0
$273$	−6.86017	+	5.40513i	−0.415196	+	0.327133i
$274$		0			0
$275$		−	3.82160i		−	0.230451i
$276$		0			0
$277$	−21.8377			−1.31210			−0.656050	−	0.754717i	$-0.727774\pi$
$277$	−21.8377			−1.31210			−0.656050	+	0.754717i	$0.727774\pi$
$278$		0			0
$279$		−	4.98238i		−	0.298288i
$280$		0			0
$281$			12.3481i			0.736628i	0.929702	+	0.368314i	$0.120065\pi$
$281$			12.3481i			0.736628i	−0.929702	+	0.368314i	$0.879935\pi$
$282$		0			0
$283$	−1.50194			−0.0892811			−0.0446405	−	0.999003i	$-0.514214\pi$
$283$	−1.50194			−0.0892811			−0.0446405	+	0.999003i	$0.514214\pi$
$284$		0			0
$285$	−6.57365			−0.389390
$286$		0			0
$287$	−24.2086			−1.42899
$288$		0			0
$289$	−15.6539			−0.920815
$290$		0			0
$291$		−	15.3992i		−	0.902716i
$292$		0			0
$293$		−	4.84790i		−	0.283217i	−0.989923	−	0.141609i	$-0.954773\pi$
$293$		−	4.84790i		−	0.283217i	0.989923	−	0.141609i	$-0.0452275\pi$
$294$		0			0
$295$	8.57806			0.499434
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$		0			0
$299$	−1.45158	−	1.84233i	−0.0839468	−	0.106545i
$300$		0			0
$301$		−	22.3533i		−	1.28842i
$302$		0			0
$303$	−12.1396			−0.697400
$304$		0			0
$305$			2.30685i			0.132090i
$306$		0			0
$307$			31.1746i			1.77923i	0.456713	+	0.889614i	$0.349027\pi$
$307$			31.1746i			1.77923i	−0.456713	+	0.889614i	$0.650973\pi$
$308$		0			0
$309$	−14.6543			−0.833653
$310$		0			0
$311$	15.6659			0.888332			0.444166	−	0.895944i	$-0.353500\pi$
$311$	15.6659			0.888332			0.444166	+	0.895944i	$0.353500\pi$
$312$		0			0
$313$	−21.3093			−1.20447			−0.602237	−	0.798317i	$-0.705724\pi$
$313$	−21.3093			−1.20447			−0.602237	+	0.798317i	$0.705724\pi$
$314$		0			0
$315$	2.62949			0.148155
$316$		0			0
$317$			4.68183i			0.262958i	0.991319	+	0.131479i	$0.0419726\pi$
$317$			4.68183i			0.262958i	−0.991319	+	0.131479i	$0.958027\pi$
$318$		0			0
$319$		−	0.993564i		−	0.0556289i
$320$		0			0
$321$	9.73103			0.543133
$322$		0			0
$323$			7.02596i			0.390935i
$324$		0			0
$325$	−8.52757	−	10.8232i	−0.473024	−	0.600361i
$326$		0			0
$327$		−	11.4991i		−	0.635899i
$328$		0			0
$329$	−2.82735			−0.155877
$330$		0			0
$331$			24.1129i			1.32536i	0.748901	+	0.662682i	$0.230582\pi$
$331$			24.1129i			1.32536i	−0.748901	+	0.662682i	$0.769418\pi$
$332$		0			0
$333$			5.84844i			0.320493i
$334$		0			0
$335$	12.7706			0.697734
$336$		0			0
$337$	2.96334			0.161423			0.0807116	−	0.996737i	$-0.474281\pi$
$337$	2.96334			0.161423			0.0807116	+	0.996737i	$0.474281\pi$
$338$		0			0
$339$	−8.64375			−0.469464
$340$		0			0
$341$	−4.98238			−0.269811
$342$		0			0
$343$		−	19.6993i		−	1.06366i
$344$		0			0
$345$			0.706165i			0.0380186i
$346$		0			0
$347$	14.7305			0.790775			0.395387	−	0.918514i	$-0.370610\pi$
$347$	14.7305			0.790775			0.395387	+	0.918514i	$0.370610\pi$
$348$		0			0
$349$		−	2.84195i		−	0.152126i	−0.997103	−	0.0760631i	$-0.975765\pi$
$349$		−	2.84195i		−	0.152126i	0.997103	−	0.0760631i	$-0.0242350\pi$
$350$		0			0
$351$	−2.23141	−	2.83210i	−0.119104	−	0.151166i
$352$		0			0
$353$			31.0912i			1.65482i	0.561599	+	0.827409i	$0.310186\pi$
$353$			31.0912i			1.65482i	−0.561599	+	0.827409i	$0.689814\pi$
$354$		0			0
$355$	14.9423			0.793052
$356$		0			0
$357$		−	2.81042i		−	0.148743i
$358$		0			0
$359$		−	7.97858i		−	0.421093i	−0.977584	−	0.210547i	$-0.932476\pi$
$359$		−	7.97858i		−	0.421093i	0.977584	−	0.210547i	$-0.0675244\pi$
$360$		0			0
$361$	−17.6708			−0.930045
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	4.60089			0.240821
$366$		0			0
$367$	32.8930			1.71700			0.858500	−	0.512813i	$-0.171397\pi$
$367$	32.8930			1.71700			0.858500	+	0.512813i	$0.171397\pi$
$368$		0			0
$369$		−	9.99409i		−	0.520271i
$370$		0			0
$371$			20.4437i			1.06139i
$372$		0			0
$373$	−11.6798			−0.604756			−0.302378	−	0.953188i	$-0.597781\pi$
$373$	−11.6798			−0.604756			−0.302378	+	0.953188i	$0.597781\pi$
$374$		0			0
$375$			9.57621i			0.494513i
$376$		0			0
$377$	−2.21705	−	2.81387i	−0.114184	−	0.144922i
$378$		0			0
$379$		−	28.2997i		−	1.45366i	−0.686819	−	0.726828i	$-0.740994\pi$
$379$		−	28.2997i		−	1.45366i	0.686819	−	0.726828i	$-0.259006\pi$
$380$		0			0
$381$	14.4234			0.738931
$382$		0			0
$383$			36.0137i			1.84021i	0.391670	+	0.920106i	$0.371898\pi$
$383$			36.0137i			1.84021i	−0.391670	+	0.920106i	$0.628102\pi$
$384$		0			0
$385$		−	2.62949i		−	0.134011i
$386$		0			0
$387$	9.22816			0.469094
$388$		0			0
$389$	−25.1309			−1.27419			−0.637093	−	0.770787i	$-0.719863\pi$
$389$	−25.1309			−1.27419			−0.637093	+	0.770787i	$0.719863\pi$
$390$		0			0
$391$	0.754754			0.0381695
$392$		0			0
$393$	5.79231			0.292184
$394$		0			0
$395$			15.5036i			0.780071i
$396$		0			0
$397$			6.42079i			0.322250i	0.986934	+	0.161125i	$0.0515123\pi$
$397$			6.42079i			0.322250i	−0.986934	+	0.161125i	$0.948488\pi$
$398$		0			0
$399$	14.6685			0.734345
$400$		0			0
$401$			1.67578i			0.0836846i	0.999124	+	0.0418423i	$0.0133227\pi$
$401$			1.67578i			0.0836846i	−0.999124	+	0.0418423i	$0.986677\pi$
$402$		0			0
$403$	−14.1106	+	11.1178i	−0.702900	+	0.553815i
$404$		0			0
$405$			1.08554i			0.0539410i
$406$		0			0
$407$	5.84844			0.289896
$408$		0			0
$409$		−	1.10830i		−	0.0548018i	−0.999625	−	0.0274009i	$-0.991277\pi$
$409$		−	1.10830i		−	0.0548018i	0.999625	−	0.0274009i	$-0.00872307\pi$
$410$		0			0
$411$		−	9.45837i		−	0.466547i
$412$		0			0
$413$	−19.1412			−0.941877
$414$		0			0
$415$	−0.558553			−0.0274183
$416$		0			0
$417$	21.1832			1.03735
$418$		0			0
$419$	−26.8317			−1.31081			−0.655407	−	0.755276i	$-0.727503\pi$
$419$	−26.8317			−1.31081			−0.655407	+	0.755276i	$0.727503\pi$
$420$		0			0
$421$		−	5.26460i		−	0.256581i	−0.991737	−	0.128291i	$-0.959051\pi$
$421$		−	5.26460i		−	0.256581i	0.991737	−	0.128291i	$-0.0409490\pi$
$422$		0			0
$423$		−	1.16722i		−	0.0567522i
$424$		0			0
$425$	4.43395			0.215078
$426$		0			0
$427$		−	5.14754i		−	0.249107i
$428$		0			0
$429$	−2.83210	+	2.23141i	−0.136735	+	0.107734i
$430$		0			0
$431$		−	30.0603i		−	1.44795i	−0.689824	−	0.723977i	$-0.742312\pi$
$431$		−	30.0603i		−	1.44795i	0.689824	−	0.723977i	$-0.257688\pi$
$432$		0			0
$433$	−5.51753			−0.265155			−0.132578	−	0.991173i	$-0.542325\pi$
$433$	−5.51753			−0.265155			−0.132578	+	0.991173i	$0.542325\pi$
$434$		0			0
$435$			1.07855i			0.0517127i
$436$		0			0
$437$			3.93931i			0.188443i
$438$		0			0
$439$	−2.78880			−0.133102			−0.0665511	−	0.997783i	$-0.521200\pi$
$439$	−2.78880			−0.133102			−0.0665511	+	0.997783i	$0.521200\pi$
$440$		0			0
$441$	1.13251			0.0539292
$442$		0			0
$443$	31.6470			1.50359			0.751797	−	0.659395i	$-0.229188\pi$
$443$	31.6470			1.50359			0.751797	+	0.659395i	$0.229188\pi$
$444$		0			0
$445$	−12.2737			−0.581829
$446$		0			0
$447$			5.83285i			0.275884i
$448$		0			0
$449$		−	10.6667i		−	0.503394i	−0.967806	−	0.251697i	$-0.919011\pi$
$449$		−	10.6667i		−	0.503394i	0.967806	−	0.251697i	$-0.0809887\pi$
$450$		0			0
$451$	−9.99409			−0.470603
$452$		0			0
$453$		−	5.25702i		−	0.246997i
$454$		0			0
$455$	−5.86749	−	7.44700i	−0.275072	−	0.349121i
$456$		0			0
$457$		−	8.15049i		−	0.381264i	−0.981662	−	0.190632i	$-0.938946\pi$
$457$		−	8.15049i		−	0.381264i	0.981662	−	0.190632i	$-0.0610537\pi$
$458$		0			0
$459$	1.16023			0.0541551
$460$		0			0
$461$			2.20887i			0.102877i	0.998676	+	0.0514387i	$0.0163807\pi$
$461$			2.20887i			0.102877i	−0.998676	+	0.0514387i	$0.983619\pi$
$462$		0			0
$463$			11.1983i			0.520430i	0.965551	+	0.260215i	$0.0837934\pi$
$463$			11.1983i			0.520430i	−0.965551	+	0.260215i	$0.916207\pi$
$464$		0			0
$465$	5.40858			0.250817
$466$		0			0
$467$	−13.4975			−0.624591			−0.312295	−	0.949985i	$-0.601098\pi$
$467$	−13.4975			−0.624591			−0.312295	+	0.949985i	$0.601098\pi$
$468$		0			0
$469$	−28.4966			−1.31585
$470$		0			0
$471$	−15.4388			−0.711383
$472$		0			0
$473$		−	9.22816i		−	0.424311i
$474$		0			0
$475$			23.1423i			1.06184i
$476$		0			0
$477$	−8.43984			−0.386434
$478$		0			0
$479$		−	13.7208i		−	0.626921i	−0.949601	−	0.313460i	$-0.898512\pi$
$479$		−	13.7208i		−	0.626921i	0.949601	−	0.313460i	$-0.101488\pi$
$480$		0			0
$481$	16.5634	−	13.0503i	0.755225	−	0.595042i
$482$		0			0
$483$		−	1.57574i		−	0.0716989i
$484$		0			0
$485$	16.7164			0.759054
$486$		0			0
$487$			21.2306i			0.962050i	0.876707	+	0.481025i	$0.159735\pi$
$487$			21.2306i			0.962050i	−0.876707	+	0.481025i	$0.840265\pi$
$488$		0			0
$489$			19.9209i			0.900855i
$490$		0			0
$491$	−5.75786			−0.259849			−0.129924	−	0.991524i	$-0.541473\pi$
$491$	−5.75786			−0.259849			−0.129924	+	0.991524i	$0.541473\pi$
$492$		0			0
$493$	1.15277			0.0519180
$494$		0			0
$495$	1.08554			0.0487914
$496$		0			0
$497$	−33.3423			−1.49561
$498$		0			0
$499$			40.7899i			1.82601i	0.407951	+	0.913004i	$0.366243\pi$
$499$			40.7899i			1.82601i	−0.407951	+	0.913004i	$0.633757\pi$
$500$		0			0
$501$			22.7588i			1.01679i
$502$		0			0
$503$	−1.70990			−0.0762407			−0.0381204	−	0.999273i	$-0.512137\pi$
$503$	−1.70990			−0.0762407			−0.0381204	+	0.999273i	$0.512137\pi$
$504$		0			0
$505$		−	13.1780i		−	0.586413i
$506$		0			0
$507$	−3.04160	+	12.6392i	−0.135082	+	0.561325i
$508$		0			0
$509$		−	14.0612i		−	0.623251i	−0.950205	−	0.311625i	$-0.899127\pi$
$509$		−	14.0612i		−	0.623251i	0.950205	−	0.311625i	$-0.100873\pi$
$510$		0			0
$511$	−10.2665			−0.454162
$512$		0			0
$513$			6.05565i			0.267363i
$514$		0			0
$515$		−	15.9078i		−	0.700983i
$516$		0			0
$517$	−1.16722			−0.0513343
$518$		0			0
$519$	12.7184			0.558277
$520$		0			0
$521$	−26.8724			−1.17730			−0.588651	−	0.808387i	$-0.700341\pi$
$521$	−26.8724			−1.17730			−0.588651	+	0.808387i	$0.700341\pi$
$522$		0			0
$523$	39.3044			1.71866			0.859330	−	0.511422i	$-0.170881\pi$
$523$	39.3044			1.71866			0.859330	+	0.511422i	$0.170881\pi$
$524$		0			0
$525$		−	9.25702i		−	0.404010i
$526$		0			0
$527$		−	5.78073i		−	0.251813i
$528$		0			0
$529$	−22.5768			−0.981601
$530$		0			0
$531$		−	7.90211i		−	0.342922i
$532$		0			0
$533$	−28.3043	+	22.3009i	−1.22599	+	0.965961i
$534$		0			0
$535$			10.5634i			0.456697i
$536$		0			0
$537$	9.80928			0.423302
$538$		0			0
$539$		−	1.13251i		−	0.0487808i
$540$		0			0
$541$		−	34.5965i		−	1.48742i	−0.668503	−	0.743710i	$-0.733064\pi$
$541$		−	34.5965i		−	1.48742i	0.668503	−	0.743710i	$-0.266936\pi$
$542$		0			0
$543$	20.0517			0.860501
$544$		0			0
$545$	12.4827			0.534700
$546$		0			0
$547$	29.3928			1.25675			0.628373	−	0.777912i	$-0.283721\pi$
$547$	29.3928			1.25675			0.628373	+	0.777912i	$0.283721\pi$
$548$		0			0
$549$	2.12507			0.0906959
$550$		0			0
$551$			6.01667i			0.256319i
$552$		0			0
$553$		−	34.5950i		−	1.47113i
$554$		0			0
$555$	−6.34872			−0.269488
$556$		0			0
$557$			27.1152i			1.14891i	0.818538	+	0.574453i	$0.194785\pi$
$557$			27.1152i			1.14891i	−0.818538	+	0.574453i	$0.805215\pi$
$558$		0			0
$559$	−20.5918	−	26.1351i	−0.870942	−	1.10540i
$560$		0			0
$561$		−	1.16023i		−	0.0489851i
$562$		0			0
$563$	−26.5592			−1.11934			−0.559669	−	0.828716i	$-0.689072\pi$
$563$	−26.5592			−1.11934			−0.559669	+	0.828716i	$0.689072\pi$
$564$		0			0
$565$		−	9.38315i		−	0.394752i
$566$		0			0
$567$		−	2.42229i		−	0.101727i
$568$		0			0
$569$	−16.4283			−0.688709			−0.344355	−	0.938840i	$-0.611902\pi$
$569$	−16.4283			−0.688709			−0.344355	+	0.938840i	$0.611902\pi$
$570$		0			0
$571$	41.3599			1.73086			0.865430	−	0.501030i	$-0.167045\pi$
$571$	41.3599			1.73086			0.865430	+	0.501030i	$0.167045\pi$
$572$		0			0
$573$	−21.4452			−0.895885
$574$		0			0
$575$	2.48602			0.103674
$576$		0			0
$577$		−	10.5070i		−	0.437414i	−0.975791	−	0.218707i	$-0.929816\pi$
$577$		−	10.5070i		−	0.437414i	0.975791	−	0.218707i	$-0.0701839\pi$
$578$		0			0
$579$		−	21.9974i		−	0.914182i
$580$		0			0
$581$	1.24636			0.0517078
$582$		0			0
$583$			8.43984i			0.349542i
$584$		0			0
$585$	3.07436	−	2.42229i	0.127109	−	0.100149i
$586$		0			0
$587$			6.78015i			0.279847i	0.990162	+	0.139923i	$0.0446856\pi$
$587$			6.78015i			0.279847i	−0.990162	+	0.139923i	$0.955314\pi$
$588$		0			0
$589$	30.1716			1.24320
$590$		0			0
$591$		−	13.2243i		−	0.543973i
$592$		0			0
$593$			1.34608i			0.0552769i	0.999618	+	0.0276385i	$0.00879872\pi$
$593$			1.34608i			0.0552769i	−0.999618	+	0.0276385i	$0.991201\pi$
$594$		0			0
$595$	3.05083			0.125072
$596$		0			0
$597$	17.7891			0.728060
$598$		0			0
$599$	−25.5039			−1.04206			−0.521030	−	0.853539i	$-0.674452\pi$
$599$	−25.5039			−1.04206			−0.521030	+	0.853539i	$0.674452\pi$
$600$		0			0
$601$	8.57115			0.349624			0.174812	−	0.984602i	$-0.444068\pi$
$601$	8.57115			0.349624			0.174812	+	0.984602i	$0.444068\pi$
$602$		0			0
$603$		−	11.7643i		−	0.479080i
$604$		0			0
$605$		−	1.08554i		−	0.0441335i
$606$		0			0
$607$	−5.58231			−0.226579			−0.113289	−	0.993562i	$-0.536139\pi$
$607$	−5.58231			−0.226579			−0.113289	+	0.993562i	$0.536139\pi$
$608$		0			0
$609$		−	2.40670i		−	0.0975244i
$610$		0			0
$611$	−3.30569	+	2.60455i	−0.133734	+	0.105369i
$612$		0			0
$613$			32.7060i			1.32098i	0.750834	+	0.660491i	$0.229652\pi$
$613$			32.7060i			1.32098i	−0.750834	+	0.660491i	$0.770348\pi$
$614$		0			0
$615$	10.8490			0.437474
$616$		0			0
$617$			36.8938i			1.48529i	0.669685	+	0.742645i	$0.266429\pi$
$617$			36.8938i			1.48529i	−0.669685	+	0.742645i	$0.733571\pi$
$618$		0			0
$619$		−	2.16773i		−	0.0871283i	−0.999051	−	0.0435642i	$-0.986129\pi$
$619$		−	2.16773i		−	0.0871283i	0.999051	−	0.0435642i	$-0.0138713\pi$
$620$		0			0
$621$	0.650519			0.0261044
$622$		0			0
$623$	27.3877			1.09726
$624$		0			0
$625$	8.71264			0.348505
$626$		0			0
$627$	6.05565			0.241839
$628$		0			0
$629$			6.78556i			0.270558i
$630$		0			0
$631$			34.1614i			1.35994i	0.733238	+	0.679972i	$0.238008\pi$
$631$			34.1614i			1.35994i	−0.733238	+	0.679972i	$0.761992\pi$
$632$		0			0
$633$	−21.5092			−0.854912
$634$		0			0
$635$			15.6572i			0.621335i
$636$		0			0
$637$	−2.52710	−	3.20739i	−0.100127	−	0.127081i
$638$		0			0
$639$		−	13.7648i		−	0.544527i
$640$		0			0
$641$	−28.3260			−1.11881			−0.559406	−	0.828894i	$-0.688971\pi$
$641$	−28.3260			−1.11881			−0.559406	+	0.828894i	$0.688971\pi$
$642$		0			0
$643$		−	21.6342i		−	0.853170i	−0.904447	−	0.426585i	$-0.859717\pi$
$643$		−	21.6342i		−	0.853170i	0.904447	−	0.426585i	$-0.140283\pi$
$644$		0			0
$645$			10.0175i			0.394440i
$646$		0			0
$647$	−22.7263			−0.893464			−0.446732	−	0.894668i	$-0.647412\pi$
$647$	−22.7263			−0.893464			−0.446732	+	0.894668i	$0.647412\pi$
$648$		0			0
$649$	−7.90211			−0.310185
$650$		0			0
$651$	−12.0688			−0.473013
$652$		0			0
$653$	47.5524			1.86087			0.930435	−	0.366458i	$-0.119430\pi$
$653$	47.5524			1.86087			0.930435	+	0.366458i	$0.119430\pi$
$654$		0			0
$655$			6.28780i			0.245685i
$656$		0			0
$657$		−	4.23833i		−	0.165353i
$658$		0			0
$659$	9.63533			0.375339			0.187670	−	0.982232i	$-0.439907\pi$
$659$	9.63533			0.375339			0.187670	+	0.982232i	$0.439907\pi$
$660$		0			0
$661$		−	9.80880i		−	0.381518i	−0.981637	−	0.190759i	$-0.938905\pi$
$661$		−	9.80880i		−	0.381518i	0.981637	−	0.190759i	$-0.0610949\pi$
$662$		0			0
$663$	−2.58896	−	3.28590i	−0.100547	−	0.127614i
$664$		0			0
$665$			15.9233i			0.617479i
$666$		0			0
$667$	0.646332			0.0250261
$668$		0			0
$669$			3.79915i			0.146883i
$670$		0			0
$671$		−	2.12507i		−	0.0820375i
$672$		0			0
$673$	8.77305			0.338176			0.169088	−	0.985601i	$-0.445918\pi$
$673$	8.77305			0.338176			0.169088	+	0.985601i	$0.445918\pi$
$674$		0			0
$675$	3.82160			0.147093
$676$		0			0
$677$	−21.9786			−0.844706			−0.422353	−	0.906431i	$-0.638796\pi$
$677$	−21.9786			−0.844706			−0.422353	+	0.906431i	$0.638796\pi$
$678$		0			0
$679$	−37.3013			−1.43149
$680$		0			0
$681$			25.2277i			0.966727i
$682$		0			0
$683$		−	11.2129i		−	0.429050i	−0.976719	−	0.214525i	$-0.931180\pi$
$683$		−	11.2129i		−	0.429050i	0.976719	−	0.214525i	$-0.0688203\pi$
$684$		0			0
$685$	10.2674			0.392299
$686$		0			0
$687$		−	9.68915i		−	0.369664i
$688$		0			0
$689$	18.8328	+	23.9025i	0.717471	+	0.910612i
$690$		0			0
$691$			21.4075i			0.814379i	0.913344	+	0.407189i	$0.133491\pi$
$691$			21.4075i			0.814379i	−0.913344	+	0.407189i	$0.866509\pi$
$692$		0			0
$693$	−2.42229			−0.0920152
$694$		0			0
$695$			22.9952i			0.872259i
$696$		0			0
$697$		−	11.5955i		−	0.439210i
$698$		0			0
$699$	22.3698			0.846104
$700$		0			0
$701$	−14.1535			−0.534572			−0.267286	−	0.963617i	$-0.586127\pi$
$701$	−14.1535			−0.534572			−0.267286	+	0.963617i	$0.586127\pi$
$702$		0			0
$703$	−35.4161			−1.33574
$704$		0			0
$705$	1.26707			0.0477205
$706$		0			0
$707$			29.4055i			1.10591i
$708$		0			0
$709$		−	24.8332i		−	0.932632i	−0.884618	−	0.466316i	$-0.845581\pi$
$709$		−	24.8332i		−	0.932632i	0.884618	−	0.466316i	$-0.154419\pi$
$710$		0			0
$711$	14.2819			0.535614
$712$		0			0
$713$		−	3.24113i		−	0.121381i
$714$		0			0
$715$	−2.42229	−	3.07436i	−0.0905885	−	0.114975i
$716$		0			0
$717$			9.53892i			0.356238i
$718$		0			0
$719$	−29.5706			−1.10280			−0.551398	−	0.834242i	$-0.685905\pi$
$719$	−29.5706			−1.10280			−0.551398	+	0.834242i	$0.685905\pi$
$720$		0			0
$721$			35.4969i			1.32197i
$722$		0			0
$723$			10.9265i			0.406361i
$724$		0			0
$725$	3.79700			0.141017
$726$		0			0
$727$	7.10950			0.263677			0.131838	−	0.991271i	$-0.457912\pi$
$727$	7.10950			0.263677			0.131838	+	0.991271i	$0.457912\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	10.7068			0.396006
$732$		0			0
$733$			0.996399i			0.0368029i	0.999831	+	0.0184014i	$0.00585769\pi$
$733$			0.996399i			0.0368029i	−0.999831	+	0.0184014i	$0.994142\pi$
$734$		0			0
$735$			1.22939i			0.0453467i
$736$		0			0
$737$	−11.7643			−0.433344
$738$		0			0
$739$		−	18.1830i		−	0.668873i	−0.942418	−	0.334437i	$-0.891454\pi$
$739$		−	18.1830i		−	0.668873i	0.942418	−	0.334437i	$-0.108546\pi$
$740$		0			0
$741$	17.1502	−	13.5126i	0.630028	−	0.496399i
$742$		0			0
$743$			31.3429i			1.14986i	0.818203	+	0.574930i	$0.194971\pi$
$743$			31.3429i			1.14986i	−0.818203	+	0.574930i	$0.805029\pi$
$744$		0			0
$745$	−6.33180			−0.231979
$746$		0			0
$747$			0.514539i			0.0188260i
$748$		0			0
$749$		−	23.5714i		−	0.861279i
$750$		0			0
$751$	41.8615			1.52755			0.763775	−	0.645483i	$-0.223344\pi$
$751$	41.8615			1.52755			0.763775	+	0.645483i	$0.223344\pi$
$752$		0			0
$753$	−17.6754			−0.644126
$754$		0			0
$755$	5.70672			0.207689
$756$		0			0
$757$	13.8847			0.504648			0.252324	−	0.967643i	$-0.418805\pi$
$757$	13.8847			0.504648			0.252324	+	0.967643i	$0.418805\pi$
$758$		0			0
$759$		−	0.650519i		−	0.0236123i
$760$		0			0
$761$			8.88745i			0.322170i	0.986941	+	0.161085i	$0.0514993\pi$
$761$			8.88745i			0.322170i	−0.986941	+	0.161085i	$0.948501\pi$
$762$		0			0
$763$	−27.8541			−1.00838
$764$		0			0
$765$			1.25948i			0.0455366i
$766$		0			0
$767$	−22.3796	+	17.6329i	−0.808080	+	0.636686i
$768$		0			0
$769$			27.5395i			0.993100i	0.868008	+	0.496550i	$0.165400\pi$
$769$			27.5395i			0.993100i	−0.868008	+	0.496550i	$0.834600\pi$
$770$		0			0
$771$	24.2035			0.871669
$772$		0			0
$773$			43.8230i			1.57620i	0.615544	+	0.788102i	$0.288936\pi$
$773$			43.8230i			1.57620i	−0.615544	+	0.788102i	$0.711064\pi$
$774$		0			0
$775$		−	19.0407i		−	0.683962i
$776$		0			0
$777$	14.1666			0.508225
$778$		0			0
$779$	60.5207			2.16838
$780$		0			0
$781$	−13.7648			−0.492543
$782$		0			0
$783$	0.993564			0.0355071
$784$		0			0
$785$		−	16.7595i		−	0.598171i
$786$		0			0
$787$			9.84788i			0.351039i	0.984476	+	0.175519i	$0.0561605\pi$
$787$			9.84788i			0.351039i	−0.984476	+	0.175519i	$0.943840\pi$
$788$		0			0
$789$	5.83418			0.207702
$790$		0			0
$791$			20.9377i			0.744458i
$792$		0			0
$793$	−4.74191	−	6.01842i	−0.168390	−	0.213720i
$794$		0			0
$795$		−	9.16179i		−	0.324935i
$796$		0			0
$797$	−45.8557			−1.62429			−0.812146	−	0.583454i	$-0.801701\pi$
$797$	−45.8557			−1.62429			−0.812146	+	0.583454i	$0.801701\pi$
$798$		0			0
$799$		−	1.35425i		−	0.0479099i
$800$		0			0
$801$			11.3065i			0.399496i
$802$		0			0
$803$	−4.23833			−0.149568
$804$		0			0
$805$	1.71054			0.0602885
$806$		0			0
$807$	−23.8973			−0.841223
$808$		0			0
$809$	41.9622			1.47531			0.737656	−	0.675176i	$-0.235933\pi$
$809$	41.9622			1.47531			0.737656	+	0.675176i	$0.235933\pi$
$810$		0			0
$811$			21.5435i			0.756495i	0.925705	+	0.378247i	$0.123473\pi$
$811$			21.5435i			0.756495i	−0.925705	+	0.378247i	$0.876527\pi$
$812$		0			0
$813$		−	21.8469i		−	0.766206i
$814$		0			0
$815$	−21.6250			−0.757490
$816$		0			0
$817$			55.8825i			1.95508i
$818$		0			0
$819$	−6.86017	+	5.40513i	−0.239714	+	0.188870i
$820$		0			0
$821$			39.9326i			1.39366i	0.717238	+	0.696828i	$0.245406\pi$
$821$			39.9326i			1.39366i	−0.717238	+	0.696828i	$0.754594\pi$
$822$		0			0
$823$	−8.75377			−0.305137			−0.152569	−	0.988293i	$-0.548755\pi$
$823$	−8.75377			−0.305137			−0.152569	+	0.988293i	$0.548755\pi$
$824$		0			0
$825$		−	3.82160i		−	0.133051i
$826$		0			0
$827$			20.2057i			0.702622i	0.936259	+	0.351311i	$0.114264\pi$
$827$			20.2057i			0.702622i	−0.936259	+	0.351311i	$0.885736\pi$
$828$		0			0
$829$	32.9767			1.14533			0.572663	−	0.819791i	$-0.305910\pi$
$829$	32.9767			1.14533			0.572663	+	0.819791i	$0.305910\pi$
$830$		0			0
$831$	−21.8377			−0.757542
$832$		0			0
$833$	1.31398			0.0455267
$834$		0			0
$835$	−24.7056			−0.854972
$836$		0			0
$837$		−	4.98238i		−	0.172216i
$838$		0			0
$839$		−	5.70002i		−	0.196787i	−0.995148	−	0.0983934i	$-0.968630\pi$
$839$		−	5.70002i		−	0.196787i	0.995148	−	0.0983934i	$-0.0313703\pi$
$840$		0			0
$841$	−28.0128			−0.965960
$842$		0			0
$843$			12.3481i			0.425292i
$844$		0			0
$845$	−13.7203	−	3.30178i	−0.471994	−	0.113585i
$846$		0			0
$847$			2.42229i			0.0832309i
$848$		0			0
$849$	−1.50194			−0.0515465
$850$		0			0
$851$			3.80452i			0.130417i
$852$		0			0
$853$			45.5258i			1.55877i	0.626543	+	0.779387i	$0.284469\pi$
$853$			45.5258i			1.55877i	−0.626543	+	0.779387i	$0.715531\pi$
$854$		0			0
$855$	−6.57365			−0.224814
$856$		0			0
$857$	40.1886			1.37282			0.686408	−	0.727217i	$-0.259187\pi$
$857$	40.1886			1.37282			0.686408	+	0.727217i	$0.259187\pi$
$858$		0			0
$859$	46.6485			1.59163			0.795814	−	0.605542i	$-0.207044\pi$
$859$	46.6485			1.59163			0.795814	+	0.605542i	$0.207044\pi$
$860$		0			0
$861$	−24.2086			−0.825026
$862$		0			0
$863$		−	29.0049i		−	0.987340i	−0.869649	−	0.493670i	$-0.835655\pi$
$863$		−	29.0049i		−	0.987340i	0.869649	−	0.493670i	$-0.164345\pi$
$864$		0			0
$865$			13.8064i			0.469430i
$866$		0			0
$867$	−15.6539			−0.531633
$868$		0			0
$869$		−	14.2819i		−	0.484481i
$870$		0			0
$871$	−33.3177	+	26.2510i	−1.12893	+	0.889482i
$872$		0			0
$873$		−	15.3992i		−	0.521183i
$874$		0			0
$875$	23.1964			0.784180
$876$		0			0
$877$		−	7.82712i		−	0.264303i	−0.991230	−	0.132151i	$-0.957811\pi$
$877$		−	7.82712i		−	0.264303i	0.991230	−	0.132151i	$-0.0421885\pi$
$878$		0			0
$879$		−	4.84790i		−	0.163516i
$880$		0			0
$881$	39.6171			1.33473			0.667367	−	0.744729i	$-0.267421\pi$
$881$	39.6171			1.33473			0.667367	+	0.744729i	$0.267421\pi$
$882$		0			0
$883$	8.23933			0.277276			0.138638	−	0.990343i	$-0.455728\pi$
$883$	8.23933			0.277276			0.138638	+	0.990343i	$0.455728\pi$
$884$		0			0
$885$	8.57806			0.288348
$886$		0			0
$887$	−35.6951			−1.19853			−0.599263	−	0.800552i	$-0.704540\pi$
$887$	−35.6951			−1.19853			−0.599263	+	0.800552i	$0.704540\pi$
$888$		0			0
$889$		−	34.9376i		−	1.17177i
$890$		0			0
$891$		−	1.00000i		−	0.0335013i
$892$		0			0
$893$	7.06827			0.236531
$894$		0			0
$895$			10.6484i			0.355936i
$896$		0			0
$897$	−1.45158	−	1.84233i	−0.0484667	−	0.0615138i
$898$		0			0
$899$		−	4.95032i		−	0.165102i
$900$		0			0
$901$	−9.79218			−0.326225
$902$		0			0
$903$		−	22.3533i		−	0.743870i
$904$		0			0
$905$			21.7670i			0.723558i
$906$		0			0
$907$	−10.8366			−0.359825			−0.179912	−	0.983683i	$-0.557581\pi$
$907$	−10.8366			−0.359825			−0.179912	+	0.983683i	$0.557581\pi$
$908$		0			0
$909$	−12.1396			−0.402644
$910$		0			0
$911$	−38.3842			−1.27172			−0.635862	−	0.771803i	$-0.719355\pi$
$911$	−38.3842			−1.27172			−0.635862	+	0.771803i	$0.719355\pi$
$912$		0			0
$913$	0.514539			0.0170287
$914$		0			0
$915$			2.30685i			0.0762622i
$916$		0			0
$917$		−	14.0307i		−	0.463333i
$918$		0			0
$919$	−5.34536			−0.176327			−0.0881636	−	0.996106i	$-0.528100\pi$
$919$	−5.34536			−0.176327			−0.0881636	+	0.996106i	$0.528100\pi$
$920$		0			0
$921$			31.1746i			1.02724i
$922$		0			0
$923$	−38.9833	+	30.7149i	−1.28315	+	1.01099i
$924$		0			0
$925$			22.3504i			0.734877i
$926$		0			0
$927$	−14.6543			−0.481310
$928$		0			0
$929$		−	9.57045i		−	0.313996i	−0.987599	−	0.156998i	$-0.949818\pi$
$929$		−	9.57045i		−	0.313996i	0.987599	−	0.156998i	$-0.0501817\pi$
$930$		0			0
$931$			6.85810i			0.224765i
$932$		0			0
$933$	15.6659			0.512879
$934$		0			0
$935$	1.25948			0.0411894
$936$		0			0
$937$	−39.7627			−1.29899			−0.649495	−	0.760365i	$-0.725020\pi$
$937$	−39.7627			−1.29899			−0.649495	+	0.760365i	$0.725020\pi$
$938$		0			0
$939$	−21.3093			−0.695403
$940$		0			0
$941$			22.2887i			0.726589i	0.931674	+	0.363295i	$0.118348\pi$
$941$			22.2887i			0.726589i	−0.931674	+	0.363295i	$0.881652\pi$
$942$		0			0
$943$		−	6.50134i		−	0.211713i
$944$		0			0
$945$	2.62949			0.0855375
$946$		0			0
$947$			2.88312i			0.0936888i	0.998902	+	0.0468444i	$0.0149165\pi$
$947$			2.88312i			0.0936888i	−0.998902	+	0.0468444i	$0.985084\pi$
$948$		0			0
$949$	−12.0034	+	9.45747i	−0.389647	+	0.307003i
$950$		0			0
$951$			4.68183i			0.151819i
$952$		0			0
$953$	−53.4726			−1.73215			−0.866074	−	0.499915i	$-0.833364\pi$
$953$	−53.4726			−1.73215			−0.866074	+	0.499915i	$0.833364\pi$
$954$		0			0
$955$		−	23.2796i		−	0.753311i
$956$		0			0
$957$		−	0.993564i		−	0.0321174i
$958$		0			0
$959$	−22.9109			−0.739832
$960$		0			0
$961$	6.17584			0.199221
$962$		0			0
$963$	9.73103			0.313578
$964$		0			0
$965$	23.8791			0.768696
$966$		0			0
$967$			20.3761i			0.655252i	0.944808	+	0.327626i	$0.106249\pi$
$967$			20.3761i			0.655252i	−0.944808	+	0.327626i	$0.893751\pi$
$968$		0			0
$969$			7.02596i			0.225706i
$970$		0			0
$971$	−58.3840			−1.87363			−0.936815	−	0.349824i	$-0.886241\pi$
$971$	−58.3840			−1.87363			−0.936815	+	0.349824i	$0.886241\pi$
$972$		0			0
$973$		−	51.3118i		−	1.64498i
$974$		0			0
$975$	−8.52757	−	10.8232i	−0.273101	−	0.346619i
$976$		0			0
$977$		−	38.8217i		−	1.24202i	−0.783804	−	0.621008i	$-0.786723\pi$
$977$		−	38.8217i		−	1.24202i	0.783804	−	0.621008i	$-0.213277\pi$
$978$		0			0
$979$	11.3065			0.361358
$980$		0			0
$981$		−	11.4991i		−	0.367137i
$982$		0			0
$983$		−	51.3898i		−	1.63908i	−0.573022	−	0.819540i	$-0.694229\pi$
$983$		−	51.3898i		−	1.63908i	0.573022	−	0.819540i	$-0.305771\pi$
$984$		0			0
$985$	14.3555			0.457403
$986$		0			0
$987$	−2.82735			−0.0899954
$988$		0			0
$989$	6.00309			0.190887
$990$		0			0
$991$	41.0166			1.30293			0.651467	−	0.758677i	$-0.274154\pi$
$991$	41.0166			1.30293			0.651467	+	0.758677i	$0.274154\pi$
$992$		0			0
$993$			24.1129i			0.765199i
$994$		0			0
$995$			19.3108i			0.612194i
$996$		0			0
$997$	17.7135			0.560994			0.280497	−	0.959855i	$-0.409501\pi$
$997$	17.7135			0.560994			0.280497	+	0.959855i	$0.409501\pi$
$998$		0			0
$999$			5.84844i			0.185036i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.10	yes	14
13.12	even	2	inner	3432.2.g.d.1585.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.5	✓	14	13.12	even	2	inner
3432.2.g.d.1585.10	yes	14	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:	\( N \)	\(=\)	\( 3432 = 2^{3} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3432.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.08554i q^{5} -2.42229i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.08554i q^{5} -2.42229i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(-2.23141 - 2.83210i) q^{13} +1.08554i q^{15} +1.16023 q^{17} +6.05565i q^{19} -2.42229i q^{21} +0.650519 q^{23} +3.82160 q^{25} +1.00000 q^{27} +0.993564 q^{29} -4.98238i q^{31} -1.00000i q^{33} +2.62949 q^{35} +5.84844i q^{37} +(-2.23141 - 2.83210i) q^{39} -9.99409i q^{41} +9.22816 q^{43} +1.08554i q^{45} -1.16722i q^{47} +1.13251 q^{49} +1.16023 q^{51} -8.43984 q^{53} +1.08554 q^{55} +6.05565i q^{57} -7.90211i q^{59} +2.12507 q^{61} -2.42229i q^{63} +(3.07436 - 2.42229i) q^{65} -11.7643i q^{67} +0.650519 q^{69} -13.7648i q^{71} -4.23833i q^{73} +3.82160 q^{75} -2.42229 q^{77} +14.2819 q^{79} +1.00000 q^{81} +0.514539i q^{83} +1.25948i q^{85} +0.993564 q^{87} +11.3065i q^{89} +(-6.86017 + 5.40513i) q^{91} -4.98238i q^{93} -6.57365 q^{95} -15.3992i q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 14 * q + 14 * q^3 + 14 * q^9	\( 14 q + 14 q^{3} + 14 q^{9} - 4 q^{13} - 4 q^{17} + 2 q^{23} + 20 q^{25} + 14 q^{27} - 10 q^{29} - 14 q^{35} - 4 q^{39} + 22 q^{43} + 8 q^{49} - 4 q^{51} - 20 q^{53} + 2 q^{55} - 2 q^{61} - 20 q^{65} + 2 q^{69} + 20 q^{75} + 2 q^{77} - 40 q^{79} + 14 q^{81} - 10 q^{87} - 44 q^{91} + 4 q^{95}+O(q^{100}) \) 14 * q + 14 * q^3 + 14 * q^9 - 4 * q^13 - 4 * q^17 + 2 * q^23 + 20 * q^25 + 14 * q^27 - 10 * q^29 - 14 * q^35 - 4 * q^39 + 22 * q^43 + 8 * q^49 - 4 * q^51 - 20 * q^53 + 2 * q^55 - 2 * q^61 - 20 * q^65 + 2 * q^69 + 20 * q^75 + 2 * q^77 - 40 * q^79 + 14 * q^81 - 10 * q^87 - 44 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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