LMFDB - Embedded newform 3432.2.g.d.1585.11

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			1585.11
Root			$1.90876i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.90876i q^{5} +4.43738i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.90876i q^{5} +4.43738i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(2.32474 + 2.75601i) q^{13} +1.90876i q^{15} -1.12570 q^{17} +6.14161i q^{19} +4.43738i q^{21} +4.17415 q^{23} +1.35663 q^{25} +1.00000 q^{27} +0.694710 q^{29} -0.657086i q^{31} +1.00000i q^{33} -8.46990 q^{35} -6.76566i q^{37} +(2.32474 + 2.75601i) q^{39} -5.69880i q^{41} -1.69155 q^{43} +1.90876i q^{45} -1.70843i q^{47} -12.6903 q^{49} -1.12570 q^{51} -3.66614 q^{53} -1.90876 q^{55} +6.14161i q^{57} -5.09533i q^{59} +0.988449 q^{61} +4.43738i q^{63} +(-5.26056 + 4.43738i) q^{65} +8.43652i q^{67} +4.17415 q^{69} -13.7136i q^{71} +11.6443i q^{73} +1.35663 q^{75} -4.43738 q^{77} -4.00469 q^{79} +1.00000 q^{81} +14.3846i q^{83} -2.14870i q^{85} +0.694710 q^{87} +1.93033i q^{89} +(-12.2294 + 10.3158i) q^{91} -0.657086i q^{93} -11.7229 q^{95} -9.00610i q^{97} +1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			1.90876i			0.853624i	0.904340	+	0.426812i	$0.140363\pi$
$5$			1.90876i			0.853624i	−0.904340	+	0.426812i	$0.859637\pi$
$6$		0			0
$7$			4.43738i			1.67717i	0.544769	+	0.838586i	$0.316617\pi$
$7$			4.43738i			1.67717i	−0.544769	+	0.838586i	$0.683383\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	2.32474	+	2.75601i	0.644768	+	0.764379i
$13$	2.32474	+	2.75601i	0.644768	+	0.764379i
$14$		0			0
$15$			1.90876i			0.492840i
$16$		0			0
$17$	−1.12570			−0.273023			−0.136511	−	0.990639i	$-0.543589\pi$
$17$	−1.12570			−0.273023			−0.136511	+	0.990639i	$0.543589\pi$
$18$		0			0
$19$			6.14161i			1.40898i	0.709712	+	0.704491i	$0.248825\pi$
$19$			6.14161i			1.40898i	−0.709712	+	0.704491i	$0.751175\pi$
$20$		0			0
$21$			4.43738i			0.968316i
$22$		0			0
$23$	4.17415			0.870371			0.435186	−	0.900341i	$-0.356683\pi$
$23$	4.17415			0.870371			0.435186	+	0.900341i	$0.356683\pi$
$24$		0			0
$25$	1.35663			0.271326
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	0.694710			0.129004			0.0645022	−	0.997918i	$-0.479454\pi$
$29$	0.694710			0.129004			0.0645022	+	0.997918i	$0.479454\pi$
$30$		0			0
$31$		−	0.657086i		−	0.118016i	−0.998258	−	0.0590081i	$-0.981206\pi$
$31$		−	0.657086i		−	0.118016i	0.998258	−	0.0590081i	$-0.0187938\pi$
$32$		0			0
$33$			1.00000i			0.174078i
$34$		0			0
$35$	−8.46990			−1.43167
$36$		0			0
$37$		−	6.76566i		−	1.11227i	−0.831093	−	0.556134i	$-0.812284\pi$
$37$		−	6.76566i		−	1.11227i	0.831093	−	0.556134i	$-0.187716\pi$
$38$		0			0
$39$	2.32474	+	2.75601i	0.372257	+	0.441314i
$40$		0			0
$41$		−	5.69880i		−	0.890003i	−0.895530	−	0.445002i	$-0.853203\pi$
$41$		−	5.69880i		−	0.890003i	0.895530	−	0.445002i	$-0.146797\pi$
$42$		0			0
$43$	−1.69155			−0.257959			−0.128979	−	0.991647i	$-0.541170\pi$
$43$	−1.69155			−0.257959			−0.128979	+	0.991647i	$0.541170\pi$
$44$		0			0
$45$			1.90876i			0.284541i
$46$		0			0
$47$		−	1.70843i		−	0.249200i	−0.992207	−	0.124600i	$-0.960235\pi$
$47$		−	1.70843i		−	0.249200i	0.992207	−	0.124600i	$-0.0397647\pi$
$48$		0			0
$49$	−12.6903			−1.81291
$50$		0			0
$51$	−1.12570			−0.157630
$52$		0			0
$53$	−3.66614			−0.503583			−0.251792	−	0.967781i	$-0.581020\pi$
$53$	−3.66614			−0.503583			−0.251792	+	0.967781i	$0.581020\pi$
$54$		0			0
$55$	−1.90876			−0.257377
$56$		0			0
$57$			6.14161i			0.813477i
$58$		0			0
$59$		−	5.09533i		−	0.663355i	−0.943393	−	0.331678i	$-0.892385\pi$
$59$		−	5.09533i		−	0.663355i	0.943393	−	0.331678i	$-0.107615\pi$
$60$		0			0
$61$	0.988449			0.126558			0.0632789	−	0.997996i	$-0.479844\pi$
$61$	0.988449			0.126558			0.0632789	+	0.997996i	$0.479844\pi$
$62$		0			0
$63$			4.43738i			0.559057i
$64$		0			0
$65$	−5.26056	+	4.43738i	−0.652492	+	0.550389i
$66$		0			0
$67$			8.43652i			1.03068i	0.856984	+	0.515342i	$0.172335\pi$
$67$			8.43652i			1.03068i	−0.856984	+	0.515342i	$0.827665\pi$
$68$		0			0
$69$	4.17415			0.502509
$70$		0			0
$71$		−	13.7136i		−	1.62750i	−0.581213	−	0.813751i	$-0.697422\pi$
$71$		−	13.7136i		−	1.62750i	0.581213	−	0.813751i	$-0.302578\pi$
$72$		0			0
$73$			11.6443i			1.36286i	0.731881	+	0.681432i	$0.238643\pi$
$73$			11.6443i			1.36286i	−0.731881	+	0.681432i	$0.761357\pi$
$74$		0			0
$75$	1.35663			0.156650
$76$		0			0
$77$	−4.43738			−0.505686
$78$		0			0
$79$	−4.00469			−0.450563			−0.225281	−	0.974294i	$-0.572330\pi$
$79$	−4.00469			−0.450563			−0.225281	+	0.974294i	$0.572330\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			14.3846i			1.57891i	0.613808	+	0.789455i	$0.289637\pi$
$83$			14.3846i			1.57891i	−0.613808	+	0.789455i	$0.710363\pi$
$84$		0			0
$85$		−	2.14870i		−	0.233059i
$86$		0			0
$87$	0.694710			0.0744808
$88$		0			0
$89$			1.93033i			0.204615i	0.994753	+	0.102307i	$0.0326225\pi$
$89$			1.93033i			0.204615i	−0.994753	+	0.102307i	$0.967378\pi$
$90$		0			0
$91$	−12.2294	+	10.3158i	−1.28199	+	1.08139i
$92$		0			0
$93$		−	0.657086i		−	0.0681367i
$94$		0			0
$95$	−11.7229			−1.20274
$96$		0			0
$97$		−	9.00610i		−	0.914431i	−0.889356	−	0.457215i	$-0.848847\pi$
$97$		−	9.00610i		−	0.914431i	0.889356	−	0.457215i	$-0.151153\pi$
$98$		0			0
$99$			1.00000i			0.100504i
$100$		0			0
$101$	1.60661			0.159863			0.0799317	−	0.996800i	$-0.474530\pi$
$101$	1.60661			0.159863			0.0799317	+	0.996800i	$0.474530\pi$
$102$		0			0
$103$	11.8688			1.16946			0.584732	−	0.811227i	$-0.301200\pi$
$103$	11.8688			1.16946			0.584732	+	0.811227i	$0.301200\pi$
$104$		0			0
$105$	−8.46990			−0.826577
$106$		0			0
$107$	2.72219			0.263164			0.131582	−	0.991305i	$-0.457994\pi$
$107$	2.72219			0.263164			0.131582	+	0.991305i	$0.457994\pi$
$108$		0			0
$109$		−	15.0452i		−	1.44107i	−0.693421	−	0.720533i	$-0.743897\pi$
$109$		−	15.0452i		−	1.44107i	0.693421	−	0.720533i	$-0.256103\pi$
$110$		0			0
$111$		−	6.76566i		−	0.642168i
$112$		0			0
$113$	−2.82525			−0.265777			−0.132888	−	0.991131i	$-0.542425\pi$
$113$	−2.82525			−0.265777			−0.132888	+	0.991131i	$0.542425\pi$
$114$		0			0
$115$			7.96746i			0.742970i
$116$		0			0
$117$	2.32474	+	2.75601i	0.214923	+	0.254793i
$118$		0			0
$119$		−	4.99517i		−	0.457906i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	5.69880i		−	0.513843i
$124$		0			0
$125$			12.1333i			1.08523i
$126$		0			0
$127$	0.155855			0.0138299			0.00691493	−	0.999976i	$-0.497799\pi$
$127$	0.155855			0.0138299			0.00691493	+	0.999976i	$0.497799\pi$
$128$		0			0
$129$	−1.69155			−0.148933
$130$		0			0
$131$	16.4978			1.44142			0.720712	−	0.693235i	$-0.243815\pi$
$131$	16.4978			1.44142			0.720712	+	0.693235i	$0.243815\pi$
$132$		0			0
$133$	−27.2527			−2.36311
$134$		0			0
$135$			1.90876i			0.164280i
$136$		0			0
$137$			1.76460i			0.150760i	0.997155	+	0.0753800i	$0.0240170\pi$
$137$			1.76460i			0.150760i	−0.997155	+	0.0753800i	$0.975983\pi$
$138$		0			0
$139$	−15.6726			−1.32933			−0.664666	−	0.747140i	$-0.731426\pi$
$139$	−15.6726			−1.32933			−0.664666	+	0.747140i	$0.731426\pi$
$140$		0			0
$141$		−	1.70843i		−	0.143875i
$142$		0			0
$143$	−2.75601	+	2.32474i	−0.230469	+	0.194405i
$144$		0			0
$145$			1.32604i			0.110121i
$146$		0			0
$147$	−12.6903			−1.04668
$148$		0			0
$149$		−	5.41098i		−	0.443285i	−0.975128	−	0.221642i	$-0.928858\pi$
$149$		−	5.41098i		−	0.443285i	0.975128	−	0.221642i	$-0.0711417\pi$
$150$		0			0
$151$			2.01988i			0.164376i	0.996617	+	0.0821879i	$0.0261908\pi$
$151$			2.01988i			0.164376i	−0.996617	+	0.0821879i	$0.973809\pi$
$152$		0			0
$153$	−1.12570			−0.0910076
$154$		0			0
$155$	1.25422			0.100741
$156$		0			0
$157$	−9.14472			−0.729828			−0.364914	−	0.931041i	$-0.618902\pi$
$157$	−9.14472			−0.729828			−0.364914	+	0.931041i	$0.618902\pi$
$158$		0			0
$159$	−3.66614			−0.290744
$160$		0			0
$161$			18.5223i			1.45976i
$162$		0			0
$163$		−	17.3336i		−	1.35767i	−0.734291	−	0.678835i	$-0.762485\pi$
$163$		−	17.3336i		−	1.35767i	0.734291	−	0.678835i	$-0.237515\pi$
$164$		0			0
$165$	−1.90876			−0.148597
$166$		0			0
$167$		−	21.4064i		−	1.65647i	−0.560378	−	0.828237i	$-0.689344\pi$
$167$		−	21.4064i		−	1.65647i	0.560378	−	0.828237i	$-0.310656\pi$
$168$		0			0
$169$	−2.19114	+	12.8140i	−0.168549	+	0.985693i
$170$		0			0
$171$			6.14161i			0.469661i
$172$		0			0
$173$	−6.89719			−0.524384			−0.262192	−	0.965016i	$-0.584445\pi$
$173$	−6.89719			−0.524384			−0.262192	+	0.965016i	$0.584445\pi$
$174$		0			0
$175$			6.01988i			0.455060i
$176$		0			0
$177$		−	5.09533i		−	0.382988i
$178$		0			0
$179$	5.85804			0.437850			0.218925	−	0.975742i	$-0.429745\pi$
$179$	5.85804			0.437850			0.218925	+	0.975742i	$0.429745\pi$
$180$		0			0
$181$	−6.12104			−0.454973			−0.227487	−	0.973781i	$-0.573051\pi$
$181$	−6.12104			−0.454973			−0.227487	+	0.973781i	$0.573051\pi$
$182$		0			0
$183$	0.988449			0.0730682
$184$		0			0
$185$	12.9140			0.949459
$186$		0			0
$187$		−	1.12570i		−	0.0823195i
$188$		0			0
$189$			4.43738i			0.322772i
$190$		0			0
$191$	7.81207			0.565261			0.282631	−	0.959229i	$-0.408793\pi$
$191$	7.81207			0.565261			0.282631	+	0.959229i	$0.408793\pi$
$192$		0			0
$193$			2.85984i			0.205856i	0.994689	+	0.102928i	$0.0328211\pi$
$193$			2.85984i			0.205856i	−0.994689	+	0.102928i	$0.967179\pi$
$194$		0			0
$195$	−5.26056	+	4.43738i	−0.376716	+	0.317767i
$196$		0			0
$197$			2.83452i			0.201951i	0.994889	+	0.100976i	$0.0321964\pi$
$197$			2.83452i			0.201951i	−0.994889	+	0.100976i	$0.967804\pi$
$198$		0			0
$199$	−7.13591			−0.505851			−0.252926	−	0.967486i	$-0.581393\pi$
$199$	−7.13591			−0.505851			−0.252926	+	0.967486i	$0.581393\pi$
$200$		0			0
$201$			8.43652i			0.595066i
$202$		0			0
$203$			3.08269i			0.216363i
$204$		0			0
$205$	10.8776			0.759728
$206$		0			0
$207$	4.17415			0.290124
$208$		0			0
$209$	−6.14161			−0.424824
$210$		0			0
$211$	3.79781			0.261452			0.130726	−	0.991419i	$-0.458269\pi$
$211$	3.79781			0.261452			0.130726	+	0.991419i	$0.458269\pi$
$212$		0			0
$213$		−	13.7136i		−	0.939639i
$214$		0			0
$215$		−	3.22876i		−	0.220200i
$216$		0			0
$217$	2.91574			0.197933
$218$		0			0
$219$			11.6443i			0.786850i
$220$		0			0
$221$	−2.61697	−	3.10244i	−0.176036	−	0.208693i
$222$		0			0
$223$			20.5550i			1.37647i	0.725490	+	0.688233i	$0.241613\pi$
$223$			20.5550i			1.37647i	−0.725490	+	0.688233i	$0.758387\pi$
$224$		0			0
$225$	1.35663			0.0904420
$226$		0			0
$227$			14.6605i			0.973052i	0.873666	+	0.486526i	$0.161736\pi$
$227$			14.6605i			0.973052i	−0.873666	+	0.486526i	$0.838264\pi$
$228$		0			0
$229$		−	2.92954i		−	0.193589i	−0.995304	−	0.0967947i	$-0.969141\pi$
$229$		−	2.92954i		−	0.193589i	0.995304	−	0.0967947i	$-0.0308590\pi$
$230$		0			0
$231$	−4.43738			−0.291958
$232$		0			0
$233$	−15.8380			−1.03758			−0.518790	−	0.854901i	$-0.673618\pi$
$233$	−15.8380			−1.03758			−0.518790	+	0.854901i	$0.673618\pi$
$234$		0			0
$235$	3.26098			0.212723
$236$		0			0
$237$	−4.00469			−0.260133
$238$		0			0
$239$			6.10433i			0.394857i	0.980317	+	0.197428i	$0.0632590\pi$
$239$			6.10433i			0.394857i	−0.980317	+	0.197428i	$0.936741\pi$
$240$		0			0
$241$		−	7.03019i		−	0.452855i	−0.974028	−	0.226427i	$-0.927295\pi$
$241$		−	7.03019i		−	0.452855i	0.974028	−	0.226427i	$-0.0727046\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	24.2228i		−	1.54754i
$246$		0			0
$247$	−16.9263	+	14.2777i	−1.07700	+	0.908467i
$248$		0			0
$249$			14.3846i			0.911584i
$250$		0			0
$251$	18.2012			1.14885			0.574425	−	0.818557i	$-0.305226\pi$
$251$	18.2012			1.14885			0.574425	+	0.818557i	$0.305226\pi$
$252$		0			0
$253$			4.17415i			0.262427i
$254$		0			0
$255$		−	2.14870i		−	0.134557i
$256$		0			0
$257$	14.4077			0.898725			0.449362	−	0.893350i	$-0.351651\pi$
$257$	14.4077			0.898725			0.449362	+	0.893350i	$0.351651\pi$
$258$		0			0
$259$	30.0218			1.86546
$260$		0			0
$261$	0.694710			0.0430015
$262$		0			0
$263$	1.47052			0.0906761			0.0453380	−	0.998972i	$-0.485564\pi$
$263$	1.47052			0.0906761			0.0453380	+	0.998972i	$0.485564\pi$
$264$		0			0
$265$		−	6.99779i		−	0.429871i
$266$		0			0
$267$			1.93033i			0.118134i
$268$		0			0
$269$	−0.438313			−0.0267244			−0.0133622	−	0.999911i	$-0.504253\pi$
$269$	−0.438313			−0.0267244			−0.0133622	+	0.999911i	$0.504253\pi$
$270$		0			0
$271$			3.06975i			0.186474i	0.995644	+	0.0932371i	$0.0297215\pi$
$271$			3.06975i			0.186474i	−0.995644	+	0.0932371i	$0.970279\pi$
$272$		0			0
$273$	−12.2294	+	10.3158i	−0.740160	+	0.624339i
$274$		0			0
$275$			1.35663i			0.0818079i
$276$		0			0
$277$	31.0870			1.86784			0.933919	−	0.357484i	$-0.116365\pi$
$277$	31.0870			1.86784			0.933919	+	0.357484i	$0.116365\pi$
$278$		0			0
$279$		−	0.657086i		−	0.0393387i
$280$		0			0
$281$			3.94885i			0.235569i	0.993039	+	0.117784i	$0.0375791\pi$
$281$			3.94885i			0.235569i	−0.993039	+	0.117784i	$0.962421\pi$
$282$		0			0
$283$	5.32488			0.316531			0.158266	−	0.987397i	$-0.449410\pi$
$283$	5.32488			0.316531			0.158266	+	0.987397i	$0.449410\pi$
$284$		0			0
$285$	−11.7229			−0.694403
$286$		0			0
$287$	25.2877			1.49269
$288$		0			0
$289$	−15.7328			−0.925459
$290$		0			0
$291$		−	9.00610i		−	0.527947i
$292$		0			0
$293$		−	4.18008i		−	0.244203i	−0.992518	−	0.122101i	$-0.961037\pi$
$293$		−	4.18008i		−	0.244203i	0.992518	−	0.122101i	$-0.0389633\pi$
$294$		0			0
$295$	9.72577			0.566256
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	9.70383	+	11.5040i	0.561187	+	0.665293i
$300$		0			0
$301$		−	7.50605i		−	0.432641i
$302$		0			0
$303$	1.60661			0.0922972
$304$		0			0
$305$			1.88671i			0.108033i
$306$		0			0
$307$			19.4112i			1.10785i	0.832565	+	0.553927i	$0.186871\pi$
$307$			19.4112i			1.10785i	−0.832565	+	0.553927i	$0.813129\pi$
$308$		0			0
$309$	11.8688			0.675190
$310$		0			0
$311$	31.4499			1.78336			0.891679	−	0.452669i	$-0.149528\pi$
$311$	31.4499			1.78336			0.891679	+	0.452669i	$0.149528\pi$
$312$		0			0
$313$	−15.1078			−0.853945			−0.426972	−	0.904265i	$-0.640420\pi$
$313$	−15.1078			−0.853945			−0.426972	+	0.904265i	$0.640420\pi$
$314$		0			0
$315$	−8.46990			−0.477225
$316$		0			0
$317$		−	30.8482i		−	1.73261i	−0.499520	−	0.866303i	$-0.666490\pi$
$317$		−	30.8482i		−	1.73261i	0.499520	−	0.866303i	$-0.333510\pi$
$318$		0			0
$319$			0.694710i			0.0388963i
$320$		0			0
$321$	2.72219			0.151938
$322$		0			0
$323$		−	6.91363i		−	0.384684i
$324$		0			0
$325$	3.15382	+	3.73888i	0.174942	+	0.207396i
$326$		0			0
$327$		−	15.0452i		−	0.831999i
$328$		0			0
$329$	7.58094			0.417951
$330$		0			0
$331$			26.2094i			1.44060i	0.693665	+	0.720298i	$0.255995\pi$
$331$			26.2094i			1.44060i	−0.693665	+	0.720298i	$0.744005\pi$
$332$		0			0
$333$		−	6.76566i		−	0.370756i
$334$		0			0
$335$	−16.1033			−0.879817
$336$		0			0
$337$	0.144012			0.00784483			0.00392242	−	0.999992i	$-0.498751\pi$
$337$	0.144012			0.00784483			0.00392242	+	0.999992i	$0.498751\pi$
$338$		0			0
$339$	−2.82525			−0.153446
$340$		0			0
$341$	0.657086			0.0355832
$342$		0			0
$343$		−	25.2502i		−	1.36338i
$344$		0			0
$345$			7.96746i			0.428954i
$346$		0			0
$347$	7.38993			0.396712			0.198356	−	0.980130i	$-0.436440\pi$
$347$	7.38993			0.396712			0.198356	+	0.980130i	$0.436440\pi$
$348$		0			0
$349$		−	23.7656i		−	1.27214i	−0.771630	−	0.636072i	$-0.780558\pi$
$349$		−	23.7656i		−	1.27214i	0.771630	−	0.636072i	$-0.219442\pi$
$350$		0			0
$351$	2.32474	+	2.75601i	0.124086	+	0.147105i
$352$		0			0
$353$			3.06913i			0.163353i	0.996659	+	0.0816766i	$0.0260275\pi$
$353$			3.06913i			0.163353i	−0.996659	+	0.0816766i	$0.973973\pi$
$354$		0			0
$355$	26.1759			1.38928
$356$		0			0
$357$		−	4.99517i		−	0.264372i
$358$		0			0
$359$			32.6780i			1.72468i	0.506331	+	0.862339i	$0.331001\pi$
$359$			32.6780i			1.72468i	−0.506331	+	0.862339i	$0.668999\pi$
$360$		0			0
$361$	−18.7194			−0.985233
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−22.2262			−1.16337
$366$		0			0
$367$	5.01623			0.261845			0.130923	−	0.991393i	$-0.458206\pi$
$367$	5.01623			0.261845			0.130923	+	0.991393i	$0.458206\pi$
$368$		0			0
$369$		−	5.69880i		−	0.296668i
$370$		0			0
$371$		−	16.2681i		−	0.844596i
$372$		0			0
$373$	−23.7005			−1.22716			−0.613582	−	0.789631i	$-0.710272\pi$
$373$	−23.7005			−1.22716			−0.613582	+	0.789631i	$0.710272\pi$
$374$		0			0
$375$			12.1333i			0.626560i
$376$		0			0
$377$	1.61502	+	1.91463i	0.0831779	+	0.0986083i
$378$		0			0
$379$			30.6857i			1.57622i	0.615534	+	0.788110i	$0.288940\pi$
$379$			30.6857i			1.57622i	−0.615534	+	0.788110i	$0.711060\pi$
$380$		0			0
$381$	0.155855			0.00798467
$382$		0			0
$383$		−	6.07050i		−	0.310188i	−0.987900	−	0.155094i	$-0.950432\pi$
$383$		−	6.07050i		−	0.310188i	0.987900	−	0.155094i	$-0.0495680\pi$
$384$		0			0
$385$		−	8.46990i		−	0.431666i
$386$		0			0
$387$	−1.69155			−0.0859863
$388$		0			0
$389$	17.4778			0.886162			0.443081	−	0.896482i	$-0.353885\pi$
$389$	17.4778			0.886162			0.443081	+	0.896482i	$0.353885\pi$
$390$		0			0
$391$	−4.69885			−0.237631
$392$		0			0
$393$	16.4978			0.832206
$394$		0			0
$395$		−	7.64400i		−	0.384611i
$396$		0			0
$397$			11.2585i			0.565049i	0.959260	+	0.282525i	$0.0911719\pi$
$397$			11.2585i			0.565049i	−0.959260	+	0.282525i	$0.908828\pi$
$398$		0			0
$399$	−27.2527			−1.36434
$400$		0			0
$401$		−	12.6100i		−	0.629712i	−0.949139	−	0.314856i	$-0.898044\pi$
$401$		−	12.6100i		−	0.629712i	0.949139	−	0.314856i	$-0.101956\pi$
$402$		0			0
$403$	1.81093	−	1.52756i	0.0902090	−	0.0760930i
$404$		0			0
$405$			1.90876i			0.0948471i
$406$		0			0
$407$	6.76566			0.335361
$408$		0			0
$409$			9.49544i			0.469519i	0.972053	+	0.234760i	$0.0754303\pi$
$409$			9.49544i			0.469519i	−0.972053	+	0.234760i	$0.924570\pi$
$410$		0			0
$411$			1.76460i			0.0870414i
$412$		0			0
$413$	22.6099			1.11256
$414$		0			0
$415$	−27.4567			−1.34780
$416$		0			0
$417$	−15.6726			−0.767490
$418$		0			0
$419$	−9.87821			−0.482582			−0.241291	−	0.970453i	$-0.577571\pi$
$419$	−9.87821			−0.482582			−0.241291	+	0.970453i	$0.577571\pi$
$420$		0			0
$421$			28.8638i			1.40674i	0.710825	+	0.703369i	$0.248322\pi$
$421$			28.8638i			1.40674i	−0.710825	+	0.703369i	$0.751678\pi$
$422$		0			0
$423$		−	1.70843i		−	0.0830666i
$424$		0			0
$425$	−1.52716			−0.0740782
$426$		0			0
$427$			4.38612i			0.212259i
$428$		0			0
$429$	−2.75601	+	2.32474i	−0.133061	+	0.112240i
$430$		0			0
$431$		−	20.7432i		−	0.999163i	−0.866267	−	0.499581i	$-0.833487\pi$
$431$		−	20.7432i		−	0.999163i	0.866267	−	0.499581i	$-0.166513\pi$
$432$		0			0
$433$	−15.5238			−0.746024			−0.373012	−	0.927826i	$-0.621675\pi$
$433$	−15.5238			−0.746024			−0.373012	+	0.927826i	$0.621675\pi$
$434$		0			0
$435$			1.32604i			0.0635786i
$436$		0			0
$437$			25.6360i			1.22634i
$438$		0			0
$439$	21.3342			1.01823			0.509114	−	0.860699i	$-0.329973\pi$
$439$	21.3342			1.01823			0.509114	+	0.860699i	$0.329973\pi$
$440$		0			0
$441$	−12.6903			−0.604302
$442$		0			0
$443$	−14.3952			−0.683938			−0.341969	−	0.939711i	$-0.611094\pi$
$443$	−14.3952			−0.683938			−0.341969	+	0.939711i	$0.611094\pi$
$444$		0			0
$445$	−3.68454			−0.174664
$446$		0			0
$447$		−	5.41098i		−	0.255930i
$448$		0			0
$449$		−	30.8496i		−	1.45588i	−0.685638	−	0.727942i	$-0.740477\pi$
$449$		−	30.8496i		−	1.45588i	0.685638	−	0.727942i	$-0.259523\pi$
$450$		0			0
$451$	5.69880			0.268346
$452$		0			0
$453$			2.01988i			0.0949024i
$454$		0			0
$455$	−19.6903	−	23.3431i	−0.923097	−	1.09434i
$456$		0			0
$457$		−	23.4296i		−	1.09599i	−0.836481	−	0.547996i	$-0.815391\pi$
$457$		−	23.4296i		−	1.09599i	0.836481	−	0.547996i	$-0.184609\pi$
$458$		0			0
$459$	−1.12570			−0.0525433
$460$		0			0
$461$		−	12.0390i		−	0.560714i	−0.959896	−	0.280357i	$-0.909547\pi$
$461$		−	12.0390i		−	0.560714i	0.959896	−	0.280357i	$-0.0904528\pi$
$462$		0			0
$463$			37.5611i			1.74561i	0.488067	+	0.872806i	$0.337702\pi$
$463$			37.5611i			1.74561i	−0.488067	+	0.872806i	$0.662298\pi$
$464$		0			0
$465$	1.25422			0.0581631
$466$		0			0
$467$	25.7326			1.19076			0.595381	−	0.803443i	$-0.297001\pi$
$467$	25.7326			1.19076			0.595381	+	0.803443i	$0.297001\pi$
$468$		0			0
$469$	−37.4360			−1.72864
$470$		0			0
$471$	−9.14472			−0.421367
$472$		0			0
$473$		−	1.69155i		−	0.0777775i
$474$		0			0
$475$			8.33190i			0.382294i
$476$		0			0
$477$	−3.66614			−0.167861
$478$		0			0
$479$		−	41.8842i		−	1.91374i	−0.290518	−	0.956870i	$-0.593827\pi$
$479$		−	41.8842i		−	1.91374i	0.290518	−	0.956870i	$-0.406173\pi$
$480$		0			0
$481$	18.6462	−	15.7284i	0.850194	−	0.717155i
$482$		0			0
$483$			18.5223i			0.842794i
$484$		0			0
$485$	17.1905			0.780580
$486$		0			0
$487$			26.6278i			1.20662i	0.797507	+	0.603310i	$0.206152\pi$
$487$			26.6278i			1.20662i	−0.797507	+	0.603310i	$0.793848\pi$
$488$		0			0
$489$		−	17.3336i		−	0.783852i
$490$		0			0
$491$	−3.21994			−0.145314			−0.0726570	−	0.997357i	$-0.523148\pi$
$491$	−3.21994			−0.145314			−0.0726570	+	0.997357i	$0.523148\pi$
$492$		0			0
$493$	−0.782037			−0.0352212
$494$		0			0
$495$	−1.90876			−0.0857924
$496$		0			0
$497$	60.8524			2.72960
$498$		0			0
$499$		−	22.5647i		−	1.01013i	−0.863080	−	0.505067i	$-0.831468\pi$
$499$		−	22.5647i		−	1.01013i	0.863080	−	0.505067i	$-0.168532\pi$
$500$		0			0
$501$		−	21.4064i		−	0.956366i
$502$		0			0
$503$	25.3810			1.13168			0.565841	−	0.824514i	$-0.308551\pi$
$503$	25.3810			1.13168			0.565841	+	0.824514i	$0.308551\pi$
$504$		0			0
$505$			3.06663i			0.136463i
$506$		0			0
$507$	−2.19114	+	12.8140i	−0.0973119	+	0.569090i
$508$		0			0
$509$		−	25.4287i		−	1.12711i	−0.826079	−	0.563554i	$-0.809434\pi$
$509$		−	25.4287i		−	1.12711i	0.826079	−	0.563554i	$-0.190566\pi$
$510$		0			0
$511$	−51.6703			−2.28576
$512$		0			0
$513$			6.14161i			0.271159i
$514$		0			0
$515$			22.6546i			0.998282i
$516$		0			0
$517$	1.70843			0.0751365
$518$		0			0
$519$	−6.89719			−0.302753
$520$		0			0
$521$	−22.4277			−0.982576			−0.491288	−	0.870997i	$-0.663474\pi$
$521$	−22.4277			−0.982576			−0.491288	+	0.870997i	$0.663474\pi$
$522$		0			0
$523$	17.3359			0.758045			0.379022	−	0.925387i	$-0.376260\pi$
$523$	17.3359			0.758045			0.379022	+	0.925387i	$0.376260\pi$
$524$		0			0
$525$			6.01988i			0.262729i
$526$		0			0
$527$			0.739683i			0.0322211i
$528$		0			0
$529$	−5.57645			−0.242454
$530$		0			0
$531$		−	5.09533i		−	0.221118i
$532$		0			0
$533$	15.7059	−	13.2482i	0.680299	−	0.573845i
$534$		0			0
$535$			5.19602i			0.224643i
$536$		0			0
$537$	5.85804			0.252793
$538$		0			0
$539$		−	12.6903i		−	0.546612i
$540$		0			0
$541$		−	41.3482i		−	1.77770i	−0.458201	−	0.888849i	$-0.651506\pi$
$541$		−	41.3482i		−	1.77770i	0.458201	−	0.888849i	$-0.348494\pi$
$542$		0			0
$543$	−6.12104			−0.262679
$544$		0			0
$545$	28.7176			1.23013
$546$		0			0
$547$	32.9040			1.40687			0.703436	−	0.710758i	$-0.251648\pi$
$547$	32.9040			1.40687			0.703436	+	0.710758i	$0.251648\pi$
$548$		0			0
$549$	0.988449			0.0421860
$550$		0			0
$551$			4.26664i			0.181765i
$552$		0			0
$553$		−	17.7703i		−	0.755671i
$554$		0			0
$555$	12.9140			0.548170
$556$		0			0
$557$			11.4579i			0.485487i	0.970091	+	0.242743i	$0.0780473\pi$
$557$			11.4579i			0.485487i	−0.970091	+	0.242743i	$0.921953\pi$
$558$		0			0
$559$	−3.93242	−	4.66192i	−0.166324	−	0.197178i
$560$		0			0
$561$		−	1.12570i		−	0.0475272i
$562$		0			0
$563$	−18.4249			−0.776515			−0.388258	−	0.921551i	$-0.626923\pi$
$563$	−18.4249			−0.776515			−0.388258	+	0.921551i	$0.626923\pi$
$564$		0			0
$565$		−	5.39272i		−	0.226873i
$566$		0			0
$567$			4.43738i			0.186352i
$568$		0			0
$569$	−13.2887			−0.557090			−0.278545	−	0.960423i	$-0.589852\pi$
$569$	−13.2887			−0.557090			−0.278545	+	0.960423i	$0.589852\pi$
$570$		0			0
$571$	36.1021			1.51082			0.755412	−	0.655250i	$-0.227437\pi$
$571$	36.1021			1.51082			0.755412	+	0.655250i	$0.227437\pi$
$572$		0			0
$573$	7.81207			0.326354
$574$		0			0
$575$	5.66278			0.236154
$576$		0			0
$577$		−	23.5919i		−	0.982146i	−0.871119	−	0.491073i	$-0.836605\pi$
$577$		−	23.5919i		−	0.982146i	0.871119	−	0.491073i	$-0.163395\pi$
$578$		0			0
$579$			2.85984i			0.118851i
$580$		0			0
$581$	−63.8297			−2.64810
$582$		0			0
$583$		−	3.66614i		−	0.151836i
$584$		0			0
$585$	−5.26056	+	4.43738i	−0.217497	+	0.183463i
$586$		0			0
$587$			30.0670i			1.24100i	0.784207	+	0.620500i	$0.213070\pi$
$587$			30.0670i			1.24100i	−0.784207	+	0.620500i	$0.786930\pi$
$588$		0			0
$589$	4.03557			0.166283
$590$		0			0
$591$			2.83452i			0.116597i
$592$		0			0
$593$			13.3708i			0.549072i	0.961577	+	0.274536i	$0.0885242\pi$
$593$			13.3708i			0.549072i	−0.961577	+	0.274536i	$0.911476\pi$
$594$		0			0
$595$	9.53458			0.390880
$596$		0			0
$597$	−7.13591			−0.292053
$598$		0			0
$599$	−38.0972			−1.55661			−0.778304	−	0.627887i	$-0.783920\pi$
$599$	−38.0972			−1.55661			−0.778304	+	0.627887i	$0.783920\pi$
$600$		0			0
$601$	17.8055			0.726302			0.363151	−	0.931730i	$-0.381701\pi$
$601$	17.8055			0.726302			0.363151	+	0.931730i	$0.381701\pi$
$602$		0			0
$603$			8.43652i			0.343562i
$604$		0			0
$605$		−	1.90876i		−	0.0776022i
$606$		0			0
$607$	−19.9035			−0.807858			−0.403929	−	0.914790i	$-0.632356\pi$
$607$	−19.9035			−0.807858			−0.403929	+	0.914790i	$0.632356\pi$
$608$		0			0
$609$			3.08269i			0.124917i
$610$		0			0
$611$	4.70843	−	3.97165i	0.190483	−	0.160676i
$612$		0			0
$613$			40.6689i			1.64260i	0.570497	+	0.821300i	$0.306751\pi$
$613$			40.6689i			1.64260i	−0.570497	+	0.821300i	$0.693249\pi$
$614$		0			0
$615$	10.8776			0.438629
$616$		0			0
$617$		−	35.9642i		−	1.44786i	−0.689872	−	0.723931i	$-0.742333\pi$
$617$		−	35.9642i		−	1.44786i	0.689872	−	0.723931i	$-0.257667\pi$
$618$		0			0
$619$			5.79740i			0.233017i	0.993190	+	0.116509i	$0.0371702\pi$
$619$			5.79740i			0.233017i	−0.993190	+	0.116509i	$0.962830\pi$
$620$		0			0
$621$	4.17415			0.167503
$622$		0			0
$623$	−8.56560			−0.343174
$624$		0			0
$625$	−16.3764			−0.655056
$626$		0			0
$627$	−6.14161			−0.245272
$628$		0			0
$629$			7.61612i			0.303675i
$630$		0			0
$631$		−	46.5298i		−	1.85232i	−0.377129	−	0.926161i	$-0.623089\pi$
$631$		−	46.5298i		−	1.85232i	0.377129	−	0.926161i	$-0.376911\pi$
$632$		0			0
$633$	3.79781			0.150949
$634$		0			0
$635$			0.297489i			0.0118055i
$636$		0			0
$637$	−29.5018	−	34.9746i	−1.16890	−	1.38575i
$638$		0			0
$639$		−	13.7136i		−	0.542501i
$640$		0			0
$641$	31.0865			1.22784			0.613921	−	0.789367i	$-0.289591\pi$
$641$	31.0865			1.22784			0.613921	+	0.789367i	$0.289591\pi$
$642$		0			0
$643$		−	19.7002i		−	0.776901i	−0.921470	−	0.388450i	$-0.873010\pi$
$643$		−	19.7002i		−	0.776901i	0.921470	−	0.388450i	$-0.126990\pi$
$644$		0			0
$645$		−	3.22876i		−	0.127132i
$646$		0			0
$647$	27.9379			1.09835			0.549176	−	0.835707i	$-0.314942\pi$
$647$	27.9379			1.09835			0.549176	+	0.835707i	$0.314942\pi$
$648$		0			0
$649$	5.09533			0.200009
$650$		0			0
$651$	2.91574			0.114277
$652$		0			0
$653$	−42.7610			−1.67337			−0.836683	−	0.547687i	$-0.815509\pi$
$653$	−42.7610			−1.67337			−0.836683	+	0.547687i	$0.815509\pi$
$654$		0			0
$655$			31.4904i			1.23043i
$656$		0			0
$657$			11.6443i			0.454288i
$658$		0			0
$659$	−29.5587			−1.15144			−0.575721	−	0.817646i	$-0.695279\pi$
$659$	−29.5587			−1.15144			−0.575721	+	0.817646i	$0.695279\pi$
$660$		0			0
$661$			8.49417i			0.330385i	0.986261	+	0.165192i	$0.0528245\pi$
$661$			8.49417i			0.330385i	−0.986261	+	0.165192i	$0.947176\pi$
$662$		0			0
$663$	−2.61697	−	3.10244i	−0.101635	−	0.120489i
$664$		0			0
$665$		−	52.0188i		−	2.01720i
$666$		0			0
$667$	2.89983			0.112282
$668$		0			0
$669$			20.5550i			0.794703i
$670$		0			0
$671$			0.988449i			0.0381586i
$672$		0			0
$673$	−3.92045			−0.151122			−0.0755611	−	0.997141i	$-0.524075\pi$
$673$	−3.92045			−0.151122			−0.0755611	+	0.997141i	$0.524075\pi$
$674$		0			0
$675$	1.35663			0.0522167
$676$		0			0
$677$	42.6141			1.63779			0.818896	−	0.573942i	$-0.194586\pi$
$677$	42.6141			1.63779			0.818896	+	0.573942i	$0.194586\pi$
$678$		0			0
$679$	39.9635			1.53366
$680$		0			0
$681$			14.6605i			0.561792i
$682$		0			0
$683$		−	28.1536i		−	1.07727i	−0.842539	−	0.538635i	$-0.818940\pi$
$683$		−	28.1536i		−	1.07727i	0.842539	−	0.538635i	$-0.181060\pi$
$684$		0			0
$685$	−3.36820			−0.128692
$686$		0			0
$687$		−	2.92954i		−	0.111769i
$688$		0			0
$689$	−8.52284	−	10.1039i	−0.324694	−	0.384928i
$690$		0			0
$691$		−	14.1382i		−	0.537844i	−0.963162	−	0.268922i	$-0.913333\pi$
$691$		−	14.1382i		−	0.537844i	0.963162	−	0.268922i	$-0.0866674\pi$
$692$		0			0
$693$	−4.43738			−0.168562
$694$		0			0
$695$		−	29.9152i		−	1.13475i
$696$		0			0
$697$			6.41515i			0.242991i
$698$		0			0
$699$	−15.8380			−0.599048
$700$		0			0
$701$	31.9918			1.20831			0.604157	−	0.796866i	$-0.293510\pi$
$701$	31.9918			1.20831			0.604157	+	0.796866i	$0.293510\pi$
$702$		0			0
$703$	41.5521			1.56717
$704$		0			0
$705$	3.26098			0.122816
$706$		0			0
$707$			7.12913i			0.268119i
$708$		0			0
$709$			26.5857i			0.998446i	0.866474	+	0.499223i	$0.166381\pi$
$709$			26.5857i			0.998446i	−0.866474	+	0.499223i	$0.833619\pi$
$710$		0			0
$711$	−4.00469			−0.150188
$712$		0			0
$713$		−	2.74278i		−	0.102718i
$714$		0			0
$715$	−4.43738	−	5.26056i	−0.165949	−	0.196734i
$716$		0			0
$717$			6.10433i			0.227971i
$718$		0			0
$719$	−47.4376			−1.76912			−0.884562	−	0.466424i	$-0.845542\pi$
$719$	−47.4376			−1.76912			−0.884562	+	0.466424i	$0.845542\pi$
$720$		0			0
$721$			52.6662i			1.96139i
$722$		0			0
$723$		−	7.03019i		−	0.261456i
$724$		0			0
$725$	0.942465			0.0350023
$726$		0			0
$727$	40.7675			1.51198			0.755991	−	0.654582i	$-0.227155\pi$
$727$	40.7675			1.51198			0.755991	+	0.654582i	$0.227155\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	1.90418			0.0704286
$732$		0			0
$733$		−	0.724390i		−	0.0267560i	−0.999911	−	0.0133780i	$-0.995742\pi$
$733$		−	0.724390i		−	0.0267560i	0.999911	−	0.0133780i	$-0.00425847\pi$
$734$		0			0
$735$		−	24.2228i		−	0.893472i
$736$		0			0
$737$	−8.43652			−0.310763
$738$		0			0
$739$		−	36.5601i		−	1.34488i	−0.740150	−	0.672442i	$-0.765246\pi$
$739$		−	36.5601i		−	1.34488i	0.740150	−	0.672442i	$-0.234754\pi$
$740$		0			0
$741$	−16.9263	+	14.2777i	−0.621804	+	0.524504i
$742$		0			0
$743$		−	8.72321i		−	0.320023i	−0.987115	−	0.160012i	$-0.948847\pi$
$743$		−	8.72321i		−	0.320023i	0.987115	−	0.160012i	$-0.0511532\pi$
$744$		0			0
$745$	10.3283			0.378398
$746$		0			0
$747$			14.3846i			0.526303i
$748$		0			0
$749$			12.0794i			0.441372i
$750$		0			0
$751$	5.91769			0.215939			0.107970	−	0.994154i	$-0.465565\pi$
$751$	5.91769			0.215939			0.107970	+	0.994154i	$0.465565\pi$
$752$		0			0
$753$	18.2012			0.663289
$754$		0			0
$755$	−3.85548			−0.140315
$756$		0			0
$757$	−29.2288			−1.06234			−0.531170	−	0.847265i	$-0.678247\pi$
$757$	−29.2288			−1.06234			−0.531170	+	0.847265i	$0.678247\pi$
$758$		0			0
$759$			4.17415i			0.151512i
$760$		0			0
$761$			25.9311i			0.940002i	0.882666	+	0.470001i	$0.155746\pi$
$761$			25.9311i			0.940002i	−0.882666	+	0.470001i	$0.844254\pi$
$762$		0			0
$763$	66.7611			2.41691
$764$		0			0
$765$		−	2.14870i		−	0.0776863i
$766$		0			0
$767$	14.0428	−	11.8453i	0.507055	−	0.427710i
$768$		0			0
$769$			15.2981i			0.551663i	0.961206	+	0.275831i	$0.0889531\pi$
$769$			15.2981i			0.551663i	−0.961206	+	0.275831i	$0.911047\pi$
$770$		0			0
$771$	14.4077			0.518879
$772$		0			0
$773$		−	5.38882i		−	0.193822i	−0.995293	−	0.0969112i	$-0.969104\pi$
$773$		−	5.38882i		−	0.193822i	0.995293	−	0.0969112i	$-0.0308963\pi$
$774$		0			0
$775$		−	0.891423i		−	0.0320209i
$776$		0			0
$777$	30.0218			1.07703
$778$		0			0
$779$	34.9998			1.25400
$780$		0			0
$781$	13.7136			0.490710
$782$		0			0
$783$	0.694710			0.0248269
$784$		0			0
$785$		−	17.4551i		−	0.622999i
$786$		0			0
$787$			45.4695i			1.62081i	0.585868	+	0.810406i	$0.300754\pi$
$787$			45.4695i			1.62081i	−0.585868	+	0.810406i	$0.699246\pi$
$788$		0			0
$789$	1.47052			0.0523519
$790$		0			0
$791$		−	12.5367i		−	0.445753i
$792$		0			0
$793$	2.29789	+	2.72417i	0.0816004	+	0.0967381i
$794$		0			0
$795$		−	6.99779i		−	0.248186i
$796$		0			0
$797$	−3.23457			−0.114574			−0.0572872	−	0.998358i	$-0.518245\pi$
$797$	−3.23457			−0.114574			−0.0572872	+	0.998358i	$0.518245\pi$
$798$		0			0
$799$			1.92318i			0.0680372i
$800$		0			0
$801$			1.93033i			0.0682048i
$802$		0			0
$803$	−11.6443			−0.410919
$804$		0			0
$805$	−35.3547			−1.24609
$806$		0			0
$807$	−0.438313			−0.0154293
$808$		0			0
$809$	10.7281			0.377178			0.188589	−	0.982056i	$-0.439609\pi$
$809$	10.7281			0.377178			0.188589	+	0.982056i	$0.439609\pi$
$810$		0			0
$811$		−	25.5349i		−	0.896652i	−0.893870	−	0.448326i	$-0.852020\pi$
$811$		−	25.5349i		−	0.896652i	0.893870	−	0.448326i	$-0.147980\pi$
$812$		0			0
$813$			3.06975i			0.107661i
$814$		0			0
$815$	33.0857			1.15894
$816$		0			0
$817$		−	10.3888i		−	0.363460i
$818$		0			0
$819$	−12.2294	+	10.3158i	−0.427331	+	0.360462i
$820$		0			0
$821$		−	35.1572i		−	1.22700i	−0.789696	−	0.613498i	$-0.789762\pi$
$821$		−	35.1572i		−	1.22700i	0.789696	−	0.613498i	$-0.210238\pi$
$822$		0			0
$823$	−34.7546			−1.21147			−0.605735	−	0.795666i	$-0.707121\pi$
$823$	−34.7546			−1.21147			−0.605735	+	0.795666i	$0.707121\pi$
$824$		0			0
$825$			1.35663i			0.0472318i
$826$		0			0
$827$			42.4611i			1.47652i	0.674518	+	0.738258i	$0.264351\pi$
$827$			42.4611i			1.47652i	−0.674518	+	0.738258i	$0.735649\pi$
$828$		0			0
$829$	29.4000			1.02110			0.510552	−	0.859847i	$-0.329441\pi$
$829$	29.4000			1.02110			0.510552	+	0.859847i	$0.329441\pi$
$830$		0			0
$831$	31.0870			1.07840
$832$		0			0
$833$	14.2855			0.494964
$834$		0			0
$835$	40.8596			1.41401
$836$		0			0
$837$		−	0.657086i		−	0.0227122i
$838$		0			0
$839$			40.3282i			1.39229i	0.717903	+	0.696143i	$0.245102\pi$
$839$			40.3282i			1.39229i	−0.717903	+	0.696143i	$0.754898\pi$
$840$		0			0
$841$	−28.5174			−0.983358
$842$		0			0
$843$			3.94885i			0.136006i
$844$		0			0
$845$	−24.4589	−	4.18236i	−0.841411	−	0.143878i
$846$		0			0
$847$		−	4.43738i		−	0.152470i
$848$		0			0
$849$	5.32488			0.182749
$850$		0			0
$851$		−	28.2409i		−	0.968086i
$852$		0			0
$853$			22.6196i			0.774482i	0.921979	+	0.387241i	$0.126572\pi$
$853$			22.6196i			0.774482i	−0.921979	+	0.387241i	$0.873428\pi$
$854$		0			0
$855$	−11.7229			−0.400914
$856$		0			0
$857$	51.4270			1.75671			0.878357	−	0.478005i	$-0.158640\pi$
$857$	51.4270			1.75671			0.878357	+	0.478005i	$0.158640\pi$
$858$		0			0
$859$	−15.1666			−0.517477			−0.258739	−	0.965947i	$-0.583307\pi$
$859$	−15.1666			−0.517477			−0.258739	+	0.965947i	$0.583307\pi$
$860$		0			0
$861$	25.2877			0.861804
$862$		0			0
$863$		−	50.6333i		−	1.72358i	−0.507266	−	0.861789i	$-0.669344\pi$
$863$		−	50.6333i		−	1.72358i	0.507266	−	0.861789i	$-0.330656\pi$
$864$		0			0
$865$		−	13.1651i		−	0.447626i
$866$		0			0
$867$	−15.7328			−0.534314
$868$		0			0
$869$		−	4.00469i		−	0.135850i
$870$		0			0
$871$	−23.2511	+	19.6127i	−0.787833	+	0.664552i
$872$		0			0
$873$		−	9.00610i		−	0.304810i
$874$		0			0
$875$	−53.8400			−1.82012
$876$		0			0
$877$			36.0276i			1.21657i	0.793720	+	0.608283i	$0.208141\pi$
$877$			36.0276i			1.21657i	−0.793720	+	0.608283i	$0.791859\pi$
$878$		0			0
$879$		−	4.18008i		−	0.140991i
$880$		0			0
$881$	−0.991025			−0.0333885			−0.0166942	−	0.999861i	$-0.505314\pi$
$881$	−0.991025			−0.0333885			−0.0166942	+	0.999861i	$0.505314\pi$
$882$		0			0
$883$	23.0855			0.776889			0.388444	−	0.921472i	$-0.373013\pi$
$883$	23.0855			0.776889			0.388444	+	0.921472i	$0.373013\pi$
$884$		0			0
$885$	9.72577			0.326928
$886$		0			0
$887$	29.0237			0.974522			0.487261	−	0.873256i	$-0.337996\pi$
$887$	29.0237			0.974522			0.487261	+	0.873256i	$0.337996\pi$
$888$		0			0
$889$			0.691586i			0.0231950i
$890$		0			0
$891$			1.00000i			0.0335013i
$892$		0			0
$893$	10.4925			0.351118
$894$		0			0
$895$			11.1816i			0.373759i
$896$		0			0
$897$	9.70383	+	11.5040i	0.324002	+	0.384107i
$898$		0			0
$899$		−	0.456485i		−	0.0152246i
$900$		0			0
$901$	4.12698			0.137490
$902$		0			0
$903$		−	7.50605i		−	0.249786i
$904$		0			0
$905$		−	11.6836i		−	0.388376i
$906$		0			0
$907$	15.9209			0.528645			0.264323	−	0.964434i	$-0.414852\pi$
$907$	15.9209			0.528645			0.264323	+	0.964434i	$0.414852\pi$
$908$		0			0
$909$	1.60661			0.0532878
$910$		0			0
$911$	0.302832			0.0100333			0.00501663	−	0.999987i	$-0.498403\pi$
$911$	0.302832			0.0100333			0.00501663	+	0.999987i	$0.498403\pi$
$912$		0			0
$913$	−14.3846			−0.476059
$914$		0			0
$915$			1.88671i			0.0623728i
$916$		0			0
$917$			73.2072i			2.41751i
$918$		0			0
$919$	3.21040			0.105901			0.0529507	−	0.998597i	$-0.483137\pi$
$919$	3.21040			0.105901			0.0529507	+	0.998597i	$0.483137\pi$
$920$		0			0
$921$			19.4112i			0.639620i
$922$		0			0
$923$	37.7947	−	31.8805i	1.24403	−	1.04936i
$924$		0			0
$925$		−	9.17850i		−	0.301787i
$926$		0			0
$927$	11.8688			0.389821
$928$		0			0
$929$		−	29.5350i		−	0.969011i	−0.874788	−	0.484506i	$-0.839000\pi$
$929$		−	29.5350i		−	0.969011i	0.874788	−	0.484506i	$-0.161000\pi$
$930$		0			0
$931$		−	77.9392i		−	2.55435i
$932$		0			0
$933$	31.4499			1.02962
$934$		0			0
$935$	2.14870			0.0702699
$936$		0			0
$937$	−57.0853			−1.86489			−0.932447	−	0.361307i	$-0.882331\pi$
$937$	−57.0853			−1.86489			−0.932447	+	0.361307i	$0.882331\pi$
$938$		0			0
$939$	−15.1078			−0.493025
$940$		0			0
$941$			32.9059i			1.07270i	0.843995	+	0.536351i	$0.180198\pi$
$941$			32.9059i			1.07270i	−0.843995	+	0.536351i	$0.819802\pi$
$942$		0			0
$943$		−	23.7877i		−	0.774633i
$944$		0			0
$945$	−8.46990			−0.275526
$946$		0			0
$947$		−	10.9223i		−	0.354927i	−0.984127	−	0.177463i	$-0.943211\pi$
$947$		−	10.9223i		−	0.354927i	0.984127	−	0.177463i	$-0.0567891\pi$
$948$		0			0
$949$	−32.0918	+	27.0701i	−1.04174	+	0.878731i
$950$		0			0
$951$		−	30.8482i		−	1.00032i
$952$		0			0
$953$	38.0788			1.23349			0.616746	−	0.787162i	$-0.288450\pi$
$953$	38.0788			1.23349			0.616746	+	0.787162i	$0.288450\pi$
$954$		0			0
$955$			14.9114i			0.482521i
$956$		0			0
$957$			0.694710i			0.0224568i
$958$		0			0
$959$	−7.83021			−0.252851
$960$		0			0
$961$	30.5682			0.986072
$962$		0			0
$963$	2.72219			0.0877215
$964$		0			0
$965$	−5.45876			−0.175724
$966$		0			0
$967$		−	36.0872i		−	1.16049i	−0.814443	−	0.580243i	$-0.802958\pi$
$967$		−	36.0872i		−	1.16049i	0.814443	−	0.580243i	$-0.197042\pi$
$968$		0			0
$969$		−	6.91363i		−	0.222098i
$970$		0			0
$971$	32.9675			1.05798			0.528989	−	0.848629i	$-0.322571\pi$
$971$	32.9675			1.05798			0.528989	+	0.848629i	$0.322571\pi$
$972$		0			0
$973$		−	69.5453i		−	2.22952i
$974$		0			0
$975$	3.15382	+	3.73888i	0.101003	+	0.119740i
$976$		0			0
$977$		−	16.9309i		−	0.541667i	−0.962626	−	0.270833i	$-0.912701\pi$
$977$		−	16.9309i		−	0.541667i	0.962626	−	0.270833i	$-0.0872993\pi$
$978$		0			0
$979$	−1.93033			−0.0616936
$980$		0			0
$981$		−	15.0452i		−	0.480355i
$982$		0			0
$983$		−	52.1737i		−	1.66408i	−0.554714	−	0.832041i	$-0.687172\pi$
$983$		−	52.1737i		−	1.66408i	0.554714	−	0.832041i	$-0.312828\pi$
$984$		0			0
$985$	−5.41043			−0.172391
$986$		0			0
$987$	7.58094			0.241304
$988$		0			0
$989$	−7.06078			−0.224520
$990$		0			0
$991$	2.15407			0.0684264			0.0342132	−	0.999415i	$-0.489107\pi$
$991$	2.15407			0.0684264			0.0342132	+	0.999415i	$0.489107\pi$
$992$		0			0
$993$			26.2094i			0.831729i
$994$		0			0
$995$		−	13.6207i		−	0.431807i
$996$		0			0
$997$	61.4739			1.94690			0.973448	−	0.228907i	$-0.0735153\pi$
$997$	61.4739			1.94690			0.973448	+	0.228907i	$0.0735153\pi$
$998$		0			0
$999$		−	6.76566i		−	0.214056i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.11	yes	14
13.12	even	2	inner	3432.2.g.d.1585.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.4	✓	14	13.12	even	2	inner
3432.2.g.d.1585.11	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.90876i q^{5} +4.43738i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.90876i q^{5} +4.43738i q^{7} +1.00000 q^{9} +1.00000i q^{11} +(2.32474 + 2.75601i) q^{13} +1.90876i q^{15} -1.12570 q^{17} +6.14161i q^{19} +4.43738i q^{21} +4.17415 q^{23} +1.35663 q^{25} +1.00000 q^{27} +0.694710 q^{29} -0.657086i q^{31} +1.00000i q^{33} -8.46990 q^{35} -6.76566i q^{37} +(2.32474 + 2.75601i) q^{39} -5.69880i q^{41} -1.69155 q^{43} +1.90876i q^{45} -1.70843i q^{47} -12.6903 q^{49} -1.12570 q^{51} -3.66614 q^{53} -1.90876 q^{55} +6.14161i q^{57} -5.09533i q^{59} +0.988449 q^{61} +4.43738i q^{63} +(-5.26056 + 4.43738i) q^{65} +8.43652i q^{67} +4.17415 q^{69} -13.7136i q^{71} +11.6443i q^{73} +1.35663 q^{75} -4.43738 q^{77} -4.00469 q^{79} +1.00000 q^{81} +14.3846i q^{83} -2.14870i q^{85} +0.694710 q^{87} +1.93033i q^{89} +(-12.2294 + 10.3158i) q^{91} -0.657086i q^{93} -11.7229 q^{95} -9.00610i q^{97} +1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 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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