LMFDB - Embedded newform 3432.2.g.d.1585.13

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			1585.13
Root			$2.68836i$ of defining polynomial
Character	$\chi$	$=$	3432.1585
Dual form			3432.2.g.d.1585.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.68836i q^{5} +0.456068i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.68836i q^{5} +0.456068i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(0.169646 - 3.60156i) q^{13} +2.68836i q^{15} -3.87365 q^{17} -8.52662i q^{19} +0.456068i q^{21} -8.60399 q^{23} -2.22727 q^{25} +1.00000 q^{27} -4.98687 q^{29} -4.00705i q^{31} -1.00000i q^{33} -1.22608 q^{35} -7.75661i q^{37} +(0.169646 - 3.60156i) q^{39} -2.73452i q^{41} +4.82176 q^{43} +2.68836i q^{45} -12.2212i q^{47} +6.79200 q^{49} -3.87365 q^{51} -8.50656 q^{53} +2.68836 q^{55} -8.52662i q^{57} +6.94071i q^{59} -6.82922 q^{61} +0.456068i q^{63} +(9.68228 + 0.456068i) q^{65} +7.85990i q^{67} -8.60399 q^{69} -1.05678i q^{71} +13.8065i q^{73} -2.22727 q^{75} +0.456068 q^{77} -10.2029 q^{79} +1.00000 q^{81} -11.2164i q^{83} -10.4138i q^{85} -4.98687 q^{87} +3.88488i q^{89} +(1.64256 + 0.0773700i) q^{91} -4.00705i q^{93} +22.9226 q^{95} +13.9248i q^{97} -1.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			2.68836i			1.20227i	0.799147	+	0.601135i	$0.205285\pi$
$5$			2.68836i			1.20227i	−0.799147	+	0.601135i	$0.794715\pi$
$6$		0			0
$7$			0.456068i			0.172378i	0.996279	+	0.0861888i	$0.0274688\pi$
$7$			0.456068i			0.172378i	−0.996279	+	0.0861888i	$0.972531\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	0.169646	−	3.60156i	0.0470512	−	0.998892i
$13$	0.169646	−	3.60156i	0.0470512	−	0.998892i
$14$		0			0
$15$			2.68836i			0.694131i
$16$		0			0
$17$	−3.87365			−0.939498			−0.469749	−	0.882800i	$-0.655656\pi$
$17$	−3.87365			−0.939498			−0.469749	+	0.882800i	$0.655656\pi$
$18$		0			0
$19$		−	8.52662i		−	1.95614i	−0.208275	−	0.978070i	$-0.566785\pi$
$19$		−	8.52662i		−	1.95614i	0.208275	−	0.978070i	$-0.433215\pi$
$20$		0			0
$21$			0.456068i			0.0995223i
$22$		0			0
$23$	−8.60399			−1.79406			−0.897028	−	0.441974i	$-0.854278\pi$
$23$	−8.60399			−1.79406			−0.897028	+	0.441974i	$0.854278\pi$
$24$		0			0
$25$	−2.22727			−0.445454
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−4.98687			−0.926038			−0.463019	−	0.886348i	$-0.653234\pi$
$29$	−4.98687			−0.926038			−0.463019	+	0.886348i	$0.653234\pi$
$30$		0			0
$31$		−	4.00705i		−	0.719688i	−0.933012	−	0.359844i	$-0.882830\pi$
$31$		−	4.00705i		−	0.719688i	0.933012	−	0.359844i	$-0.117170\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−1.22608			−0.207245
$36$		0			0
$37$		−	7.75661i		−	1.27518i	−0.770376	−	0.637590i	$-0.779932\pi$
$37$		−	7.75661i		−	1.27518i	0.770376	−	0.637590i	$-0.220068\pi$
$38$		0			0
$39$	0.169646	−	3.60156i	0.0271650	−	0.576711i
$40$		0			0
$41$		−	2.73452i		−	0.427060i	−0.976937	−	0.213530i	$-0.931504\pi$
$41$		−	2.73452i		−	0.427060i	0.976937	−	0.213530i	$-0.0684961\pi$
$42$		0			0
$43$	4.82176			0.735312			0.367656	−	0.929962i	$-0.380160\pi$
$43$	4.82176			0.735312			0.367656	+	0.929962i	$0.380160\pi$
$44$		0			0
$45$			2.68836i			0.400757i
$46$		0			0
$47$		−	12.2212i		−	1.78264i	−0.453371	−	0.891322i	$-0.649779\pi$
$47$		−	12.2212i		−	1.78264i	0.453371	−	0.891322i	$-0.350221\pi$
$48$		0			0
$49$	6.79200			0.970286
$50$		0			0
$51$	−3.87365			−0.542419
$52$		0			0
$53$	−8.50656			−1.16847			−0.584233	−	0.811586i	$-0.698605\pi$
$53$	−8.50656			−1.16847			−0.584233	+	0.811586i	$0.698605\pi$
$54$		0			0
$55$	2.68836			0.362498
$56$		0			0
$57$		−	8.52662i		−	1.12938i
$58$		0			0
$59$			6.94071i			0.903603i	0.892118	+	0.451802i	$0.149218\pi$
$59$			6.94071i			0.903603i	−0.892118	+	0.451802i	$0.850782\pi$
$60$		0			0
$61$	−6.82922			−0.874393			−0.437196	−	0.899366i	$-0.644029\pi$
$61$	−6.82922			−0.874393			−0.437196	+	0.899366i	$0.644029\pi$
$62$		0			0
$63$			0.456068i			0.0574592i
$64$		0			0
$65$	9.68228	+	0.456068i	1.20094	+	0.0565683i
$66$		0			0
$67$			7.85990i			0.960239i	0.877203	+	0.480120i	$0.159407\pi$
$67$			7.85990i			0.960239i	−0.877203	+	0.480120i	$0.840593\pi$
$68$		0			0
$69$	−8.60399			−1.03580
$70$		0			0
$71$		−	1.05678i		−	0.125416i	−0.998032	−	0.0627081i	$-0.980026\pi$
$71$		−	1.05678i		−	0.125416i	0.998032	−	0.0627081i	$-0.0199737\pi$
$72$		0			0
$73$			13.8065i			1.61593i	0.589230	+	0.807965i	$0.299431\pi$
$73$			13.8065i			1.61593i	−0.589230	+	0.807965i	$0.700569\pi$
$74$		0			0
$75$	−2.22727			−0.257183
$76$		0			0
$77$	0.456068			0.0519738
$78$		0			0
$79$	−10.2029			−1.14792			−0.573958	−	0.818885i	$-0.694593\pi$
$79$	−10.2029			−1.14792			−0.573958	+	0.818885i	$0.694593\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.2164i		−	1.23116i	−0.788073	−	0.615582i	$-0.788921\pi$
$83$		−	11.2164i		−	1.23116i	0.788073	−	0.615582i	$-0.211079\pi$
$84$		0			0
$85$		−	10.4138i		−	1.12953i
$86$		0			0
$87$	−4.98687			−0.534648
$88$		0			0
$89$			3.88488i			0.411797i	0.978573	+	0.205898i	$0.0660116\pi$
$89$			3.88488i			0.411797i	−0.978573	+	0.205898i	$0.933988\pi$
$90$		0			0
$91$	1.64256	+	0.0773700i	0.172187	+	0.00811058i
$92$		0			0
$93$		−	4.00705i		−	0.415512i
$94$		0			0
$95$	22.9226			2.35181
$96$		0			0
$97$			13.9248i			1.41385i	0.707290	+	0.706923i	$0.249917\pi$
$97$			13.9248i			1.41385i	−0.707290	+	0.706923i	$0.750083\pi$
$98$		0			0
$99$		−	1.00000i		−	0.100504i
$100$		0			0
$101$	17.6109			1.75235			0.876176	−	0.481992i	$-0.160087\pi$
$101$	17.6109			1.75235			0.876176	+	0.481992i	$0.160087\pi$
$102$		0			0
$103$	11.8654			1.16913			0.584565	−	0.811347i	$-0.301265\pi$
$103$	11.8654			1.16913			0.584565	+	0.811347i	$0.301265\pi$
$104$		0			0
$105$	−1.22608			−0.119653
$106$		0			0
$107$	−8.31742			−0.804076			−0.402038	−	0.915623i	$-0.631698\pi$
$107$	−8.31742			−0.804076			−0.402038	+	0.915623i	$0.631698\pi$
$108$		0			0
$109$		−	19.4456i		−	1.86255i	−0.364311	−	0.931277i	$-0.618695\pi$
$109$		−	19.4456i		−	1.86255i	0.364311	−	0.931277i	$-0.381305\pi$
$110$		0			0
$111$		−	7.75661i		−	0.736225i
$112$		0			0
$113$	−6.65343			−0.625902			−0.312951	−	0.949769i	$-0.601318\pi$
$113$	−6.65343			−0.625902			−0.312951	+	0.949769i	$0.601318\pi$
$114$		0			0
$115$		−	23.1306i		−	2.15694i
$116$		0			0
$117$	0.169646	−	3.60156i	0.0156837	−	0.332964i
$118$		0			0
$119$		−	1.76665i		−	0.161948i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	2.73452i		−	0.246563i
$124$		0			0
$125$			7.45409i			0.666714i
$126$		0			0
$127$	20.3733			1.80784			0.903919	−	0.427704i	$-0.140678\pi$
$127$	20.3733			1.80784			0.903919	+	0.427704i	$0.140678\pi$
$128$		0			0
$129$	4.82176			0.424533
$130$		0			0
$131$	−3.14297			−0.274603			−0.137301	−	0.990529i	$-0.543843\pi$
$131$	−3.14297			−0.274603			−0.137301	+	0.990529i	$0.543843\pi$
$132$		0			0
$133$	3.88872			0.337195
$134$		0			0
$135$			2.68836i			0.231377i
$136$		0			0
$137$			14.6687i			1.25323i	0.779329	+	0.626615i	$0.215560\pi$
$137$			14.6687i			1.25323i	−0.779329	+	0.626615i	$0.784440\pi$
$138$		0			0
$139$	2.05859			0.174607			0.0873036	−	0.996182i	$-0.472175\pi$
$139$	2.05859			0.174607			0.0873036	+	0.996182i	$0.472175\pi$
$140$		0			0
$141$		−	12.2212i		−	1.02921i
$142$		0			0
$143$	−3.60156	−	0.169646i	−0.301177	−	0.0141865i
$144$		0			0
$145$		−	13.4065i		−	1.11335i
$146$		0			0
$147$	6.79200			0.560195
$148$		0			0
$149$		−	5.02619i		−	0.411762i	−0.978577	−	0.205881i	$-0.933994\pi$
$149$		−	5.02619i		−	0.411762i	0.978577	−	0.205881i	$-0.0660059\pi$
$150$		0			0
$151$			2.98421i			0.242852i	0.992601	+	0.121426i	$0.0387467\pi$
$151$			2.98421i			0.242852i	−0.992601	+	0.121426i	$0.961253\pi$
$152$		0			0
$153$	−3.87365			−0.313166
$154$		0			0
$155$	10.7724			0.865260
$156$		0			0
$157$	4.51439			0.360288			0.180144	−	0.983640i	$-0.442344\pi$
$157$	4.51439			0.360288			0.180144	+	0.983640i	$0.442344\pi$
$158$		0			0
$159$	−8.50656			−0.674614
$160$		0			0
$161$		−	3.92401i		−	0.309255i
$162$		0			0
$163$			7.42762i			0.581776i	0.956757	+	0.290888i	$0.0939507\pi$
$163$			7.42762i			0.581776i	−0.956757	+	0.290888i	$0.906049\pi$
$164$		0			0
$165$	2.68836			0.209288
$166$		0			0
$167$		−	22.8540i		−	1.76849i	−0.467021	−	0.884246i	$-0.654673\pi$
$167$		−	22.8540i		−	1.76849i	0.467021	−	0.884246i	$-0.345327\pi$
$168$		0			0
$169$	−12.9424	−	1.22198i	−0.995572	−	0.0939982i
$170$		0			0
$171$		−	8.52662i		−	0.652047i
$172$		0			0
$173$	−7.29517			−0.554641			−0.277321	−	0.960777i	$-0.589446\pi$
$173$	−7.29517			−0.554641			−0.277321	+	0.960777i	$0.589446\pi$
$174$		0			0
$175$		−	1.01579i		−	0.0767864i
$176$		0			0
$177$			6.94071i			0.521695i
$178$		0			0
$179$	1.90697			0.142534			0.0712669	−	0.997457i	$-0.477296\pi$
$179$	1.90697			0.142534			0.0712669	+	0.997457i	$0.477296\pi$
$180$		0			0
$181$	−15.1346			−1.12495			−0.562474	−	0.826815i	$-0.690150\pi$
$181$	−15.1346			−1.12495			−0.562474	+	0.826815i	$0.690150\pi$
$182$		0			0
$183$	−6.82922			−0.504831
$184$		0			0
$185$	20.8526			1.53311
$186$		0			0
$187$			3.87365i			0.283269i
$188$		0			0
$189$			0.456068i			0.0331741i
$190$		0			0
$191$	−4.28117			−0.309775			−0.154887	−	0.987932i	$-0.549501\pi$
$191$	−4.28117			−0.309775			−0.154887	+	0.987932i	$0.549501\pi$
$192$		0			0
$193$		−	10.3333i		−	0.743810i	−0.928271	−	0.371905i	$-0.878705\pi$
$193$		−	10.3333i		−	0.743810i	0.928271	−	0.371905i	$-0.121295\pi$
$194$		0			0
$195$	9.68228	+	0.456068i	0.693362	+	0.0326597i
$196$		0			0
$197$			0.399650i			0.0284739i	0.999899	+	0.0142370i	$0.00453192\pi$
$197$			0.399650i			0.0284739i	−0.999899	+	0.0142370i	$0.995468\pi$
$198$		0			0
$199$	0.761983			0.0540155			0.0270078	−	0.999635i	$-0.491402\pi$
$199$	0.761983			0.0540155			0.0270078	+	0.999635i	$0.491402\pi$
$200$		0			0
$201$			7.85990i			0.554394i
$202$		0			0
$203$		−	2.27435i		−	0.159628i
$204$		0			0
$205$	7.35136			0.513442
$206$		0			0
$207$	−8.60399			−0.598019
$208$		0			0
$209$	−8.52662			−0.589799
$210$		0			0
$211$	12.6351			0.869838			0.434919	−	0.900470i	$-0.356777\pi$
$211$	12.6351			0.869838			0.434919	+	0.900470i	$0.356777\pi$
$212$		0			0
$213$		−	1.05678i		−	0.0724090i
$214$		0			0
$215$			12.9626i			0.884044i
$216$		0			0
$217$	1.82749			0.124058
$218$		0			0
$219$			13.8065i			0.932958i
$220$		0			0
$221$	−0.657148	+	13.9512i	−0.0442045	+	0.938458i
$222$		0			0
$223$		−	3.82102i		−	0.255875i	−0.991782	−	0.127937i	$-0.959164\pi$
$223$		−	3.82102i		−	0.255875i	0.991782	−	0.127937i	$-0.0408356\pi$
$224$		0			0
$225$	−2.22727			−0.148485
$226$		0			0
$227$			7.91248i			0.525170i	0.964909	+	0.262585i	$0.0845750\pi$
$227$			7.91248i			0.525170i	−0.964909	+	0.262585i	$0.915425\pi$
$228$		0			0
$229$		−	12.0503i		−	0.796307i	−0.917319	−	0.398154i	$-0.869651\pi$
$229$		−	12.0503i		−	0.796307i	0.917319	−	0.398154i	$-0.130349\pi$
$230$		0			0
$231$	0.456068			0.0300071
$232$		0			0
$233$	3.50989			0.229940			0.114970	−	0.993369i	$-0.463323\pi$
$233$	3.50989			0.229940			0.114970	+	0.993369i	$0.463323\pi$
$234$		0			0
$235$	32.8549			2.14322
$236$		0			0
$237$	−10.2029			−0.662750
$238$		0			0
$239$		−	6.31623i		−	0.408563i	−0.978912	−	0.204281i	$-0.934514\pi$
$239$		−	6.31623i		−	0.408563i	0.978912	−	0.204281i	$-0.0654858\pi$
$240$		0			0
$241$		−	27.0761i		−	1.74412i	−0.489396	−	0.872062i	$-0.662783\pi$
$241$		−	27.0761i		−	1.74412i	0.489396	−	0.872062i	$-0.337217\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$			18.2593i			1.16655i
$246$		0			0
$247$	−30.7091	−	1.44650i	−1.95397	−	0.0920388i
$248$		0			0
$249$		−	11.2164i		−	0.710813i
$250$		0			0
$251$	0.173701			0.0109639			0.00548195	−	0.999985i	$-0.498255\pi$
$251$	0.173701			0.0109639			0.00548195	+	0.999985i	$0.498255\pi$
$252$		0			0
$253$			8.60399i			0.540928i
$254$		0			0
$255$		−	10.4138i		−	0.652135i
$256$		0			0
$257$	−22.5272			−1.40521			−0.702604	−	0.711581i	$-0.747980\pi$
$257$	−22.5272			−1.40521			−0.702604	+	0.711581i	$0.747980\pi$
$258$		0			0
$259$	3.53755			0.219812
$260$		0			0
$261$	−4.98687			−0.308679
$262$		0			0
$263$	−9.63612			−0.594189			−0.297094	−	0.954848i	$-0.596018\pi$
$263$	−9.63612			−0.594189			−0.297094	+	0.954848i	$0.596018\pi$
$264$		0			0
$265$		−	22.8687i		−	1.40481i
$266$		0			0
$267$			3.88488i			0.237751i
$268$		0			0
$269$	−5.27311			−0.321507			−0.160753	−	0.986995i	$-0.551392\pi$
$269$	−5.27311			−0.321507			−0.160753	+	0.986995i	$0.551392\pi$
$270$		0			0
$271$		−	3.24141i		−	0.196901i	−0.995142	−	0.0984507i	$-0.968611\pi$
$271$		−	3.24141i		−	0.196901i	0.995142	−	0.0984507i	$-0.0313887\pi$
$272$		0			0
$273$	1.64256	+	0.0773700i	0.0994120	+	0.00468265i
$274$		0			0
$275$			2.22727i			0.134310i
$276$		0			0
$277$	−27.8030			−1.67052			−0.835259	−	0.549857i	$-0.814682\pi$
$277$	−27.8030			−1.67052			−0.835259	+	0.549857i	$0.814682\pi$
$278$		0			0
$279$		−	4.00705i		−	0.239896i
$280$		0			0
$281$		−	11.6043i		−	0.692252i	−0.938188	−	0.346126i	$-0.887497\pi$
$281$		−	11.6043i		−	0.692252i	0.938188	−	0.346126i	$-0.112503\pi$
$282$		0			0
$283$	26.0688			1.54963			0.774814	−	0.632190i	$-0.217844\pi$
$283$	26.0688			1.54963			0.774814	+	0.632190i	$0.217844\pi$
$284$		0			0
$285$	22.9226			1.35782
$286$		0			0
$287$	1.24713			0.0736156
$288$		0			0
$289$	−1.99484			−0.117343
$290$		0			0
$291$			13.9248i			0.816285i
$292$		0			0
$293$			24.0256i			1.40359i	0.712379	+	0.701795i	$0.247618\pi$
$293$			24.0256i			1.40359i	−0.712379	+	0.701795i	$0.752382\pi$
$294$		0			0
$295$	−18.6591			−1.08638
$296$		0			0
$297$		−	1.00000i		−	0.0580259i
$298$		0			0
$299$	−1.45963	+	30.9878i	−0.0844125	+	1.79207i
$300$		0			0
$301$			2.19905i			0.126751i
$302$		0			0
$303$	17.6109			1.01172
$304$		0			0
$305$		−	18.3594i		−	1.05126i
$306$		0			0
$307$			4.12304i			0.235314i	0.993054	+	0.117657i	$0.0375384\pi$
$307$			4.12304i			0.235314i	−0.993054	+	0.117657i	$0.962462\pi$
$308$		0			0
$309$	11.8654			0.674998
$310$		0			0
$311$	20.2962			1.15089			0.575445	−	0.817840i	$-0.304829\pi$
$311$	20.2962			1.15089			0.575445	+	0.817840i	$0.304829\pi$
$312$		0			0
$313$	0.295037			0.0166765			0.00833825	−	0.999965i	$-0.497346\pi$
$313$	0.295037			0.0166765			0.00833825	+	0.999965i	$0.497346\pi$
$314$		0			0
$315$	−1.22608			−0.0690815
$316$		0			0
$317$		−	6.67853i		−	0.375103i	−0.982255	−	0.187552i	$-0.939945\pi$
$317$		−	6.67853i		−	0.375103i	0.982255	−	0.187552i	$-0.0600552\pi$
$318$		0			0
$319$			4.98687i			0.279211i
$320$		0			0
$321$	−8.31742			−0.464233
$322$		0			0
$323$			33.0291i			1.83779i
$324$		0			0
$325$	−0.377847	+	8.02165i	−0.0209592	+	0.444961i
$326$		0			0
$327$		−	19.4456i		−	1.07535i
$328$		0			0
$329$	5.57370			0.307288
$330$		0			0
$331$			22.0794i			1.21359i	0.794857	+	0.606796i	$0.207546\pi$
$331$			22.0794i			1.21359i	−0.794857	+	0.606796i	$0.792454\pi$
$332$		0			0
$333$		−	7.75661i		−	0.425060i
$334$		0			0
$335$	−21.1302			−1.15447
$336$		0			0
$337$	−8.12224			−0.442447			−0.221223	−	0.975223i	$-0.571005\pi$
$337$	−8.12224			−0.442447			−0.221223	+	0.975223i	$0.571005\pi$
$338$		0			0
$339$	−6.65343			−0.361365
$340$		0			0
$341$	−4.00705			−0.216994
$342$		0			0
$343$			6.29009i			0.339633i
$344$		0			0
$345$		−	23.1306i		−	1.24531i
$346$		0			0
$347$	0.125877			0.00675743			0.00337872	−	0.999994i	$-0.498925\pi$
$347$	0.125877			0.00675743			0.00337872	+	0.999994i	$0.498925\pi$
$348$		0			0
$349$		−	31.7580i		−	1.69997i	−0.526811	−	0.849983i	$-0.676612\pi$
$349$		−	31.7580i		−	1.69997i	0.526811	−	0.849983i	$-0.323388\pi$
$350$		0			0
$351$	0.169646	−	3.60156i	0.00905501	−	0.192237i
$352$		0			0
$353$			4.38336i			0.233303i	0.993173	+	0.116651i	$0.0372161\pi$
$353$			4.38336i			0.233303i	−0.993173	+	0.116651i	$0.962784\pi$
$354$		0			0
$355$	2.84099			0.150784
$356$		0			0
$357$		−	1.76665i		−	0.0935010i
$358$		0			0
$359$			33.6499i			1.77598i	0.459868	+	0.887988i	$0.347897\pi$
$359$			33.6499i			1.77598i	−0.459868	+	0.887988i	$0.652103\pi$
$360$		0			0
$361$	−53.7032			−2.82649
$362$		0			0
$363$	−1.00000			−0.0524864
$364$		0			0
$365$	−37.1169			−1.94279
$366$		0			0
$367$	−36.1078			−1.88481			−0.942405	−	0.334473i	$-0.891442\pi$
$367$	−36.1078			−1.88481			−0.942405	+	0.334473i	$0.891442\pi$
$368$		0			0
$369$		−	2.73452i		−	0.142353i
$370$		0			0
$371$		−	3.87957i		−	0.201417i
$372$		0			0
$373$	−15.6337			−0.809481			−0.404741	−	0.914431i	$-0.632638\pi$
$373$	−15.6337			−0.809481			−0.404741	+	0.914431i	$0.632638\pi$
$374$		0			0
$375$			7.45409i			0.384927i
$376$		0			0
$377$	−0.846000	+	17.9605i	−0.0435712	+	0.925012i
$378$		0			0
$379$		−	38.2693i		−	1.96576i	−0.184244	−	0.982880i	$-0.558984\pi$
$379$		−	38.2693i		−	1.96576i	0.184244	−	0.982880i	$-0.441016\pi$
$380$		0			0
$381$	20.3733			1.04376
$382$		0			0
$383$		−	14.1076i		−	0.720863i	−0.932786	−	0.360431i	$-0.882630\pi$
$383$		−	14.1076i		−	0.720863i	0.932786	−	0.360431i	$-0.117370\pi$
$384$		0			0
$385$			1.22608i			0.0624866i
$386$		0			0
$387$	4.82176			0.245104
$388$		0			0
$389$	−4.79744			−0.243240			−0.121620	−	0.992577i	$-0.538809\pi$
$389$	−4.79744			−0.243240			−0.121620	+	0.992577i	$0.538809\pi$
$390$		0			0
$391$	33.3288			1.68551
$392$		0			0
$393$	−3.14297			−0.158542
$394$		0			0
$395$		−	27.4291i		−	1.38011i
$396$		0			0
$397$		−	1.45805i		−	0.0731774i	−0.999330	−	0.0365887i	$-0.988351\pi$
$397$		−	1.45805i		−	0.0731774i	0.999330	−	0.0365887i	$-0.0116491\pi$
$398$		0			0
$399$	3.88872			0.194680
$400$		0			0
$401$		−	14.8401i		−	0.741079i	−0.928817	−	0.370539i	$-0.879173\pi$
$401$		−	14.8401i		−	0.741079i	0.928817	−	0.370539i	$-0.120827\pi$
$402$		0			0
$403$	−14.4316	−	0.679779i	−0.718891	−	0.0338622i
$404$		0			0
$405$			2.68836i			0.133586i
$406$		0			0
$407$	−7.75661			−0.384481
$408$		0			0
$409$			28.6951i			1.41888i	0.704764	+	0.709442i	$0.251053\pi$
$409$			28.6951i			1.41888i	−0.704764	+	0.709442i	$0.748947\pi$
$410$		0			0
$411$			14.6687i			0.723553i
$412$		0			0
$413$	−3.16544			−0.155761
$414$		0			0
$415$	30.1538			1.48019
$416$		0			0
$417$	2.05859			0.100810
$418$		0			0
$419$	19.1434			0.935214			0.467607	−	0.883937i	$-0.345116\pi$
$419$	19.1434			0.935214			0.467607	+	0.883937i	$0.345116\pi$
$420$		0			0
$421$			36.1231i			1.76053i	0.474479	+	0.880267i	$0.342637\pi$
$421$			36.1231i			1.76053i	−0.474479	+	0.880267i	$0.657363\pi$
$422$		0			0
$423$		−	12.2212i		−	0.594215i
$424$		0			0
$425$	8.62767			0.418504
$426$		0			0
$427$		−	3.11459i		−	0.150726i
$428$		0			0
$429$	−3.60156	−	0.169646i	−0.173885	−	0.00819057i
$430$		0			0
$431$			26.9562i			1.29843i	0.760603	+	0.649217i	$0.224903\pi$
$431$			26.9562i			1.29843i	−0.760603	+	0.649217i	$0.775097\pi$
$432$		0			0
$433$	−10.7840			−0.518244			−0.259122	−	0.965845i	$-0.583433\pi$
$433$	−10.7840			−0.518244			−0.259122	+	0.965845i	$0.583433\pi$
$434$		0			0
$435$		−	13.4065i		−	0.642792i
$436$		0			0
$437$			73.3629i			3.50943i
$438$		0			0
$439$	20.0293			0.955946			0.477973	−	0.878374i	$-0.341372\pi$
$439$	20.0293			0.955946			0.477973	+	0.878374i	$0.341372\pi$
$440$		0			0
$441$	6.79200			0.323429
$442$		0			0
$443$	34.8709			1.65677			0.828384	−	0.560161i	$-0.189261\pi$
$443$	34.8709			1.65677			0.828384	+	0.560161i	$0.189261\pi$
$444$		0			0
$445$	−10.4440			−0.495091
$446$		0			0
$447$		−	5.02619i		−	0.237731i
$448$		0			0
$449$		−	28.1932i		−	1.33052i	−0.746612	−	0.665260i	$-0.768321\pi$
$449$		−	28.1932i		−	1.33052i	0.746612	−	0.665260i	$-0.231679\pi$
$450$		0			0
$451$	−2.73452			−0.128763
$452$		0			0
$453$			2.98421i			0.140211i
$454$		0			0
$455$	−0.207998	+	4.41578i	−0.00975111	+	0.207015i
$456$		0			0
$457$			18.3265i			0.857279i	0.903476	+	0.428640i	$0.141007\pi$
$457$			18.3265i			0.857279i	−0.903476	+	0.428640i	$0.858993\pi$
$458$		0			0
$459$	−3.87365			−0.180806
$460$		0			0
$461$		−	12.4136i		−	0.578159i	−0.957305	−	0.289079i	$-0.906651\pi$
$461$		−	12.4136i		−	0.578159i	0.957305	−	0.289079i	$-0.0933492\pi$
$462$		0			0
$463$		−	25.7458i		−	1.19651i	−0.801306	−	0.598254i	$-0.795861\pi$
$463$		−	25.7458i		−	1.19651i	0.801306	−	0.598254i	$-0.204139\pi$
$464$		0			0
$465$	10.7724			0.499558
$466$		0			0
$467$	0.292996			0.0135582			0.00677912	−	0.999977i	$-0.497842\pi$
$467$	0.292996			0.0135582			0.00677912	+	0.999977i	$0.497842\pi$
$468$		0			0
$469$	−3.58465			−0.165524
$470$		0			0
$471$	4.51439			0.208012
$472$		0			0
$473$		−	4.82176i		−	0.221705i
$474$		0			0
$475$			18.9911i			0.871371i
$476$		0			0
$477$	−8.50656			−0.389489
$478$		0			0
$479$		−	22.8562i		−	1.04433i	−0.852845	−	0.522164i	$-0.825125\pi$
$479$		−	22.8562i		−	1.04433i	0.852845	−	0.522164i	$-0.174875\pi$
$480$		0			0
$481$	−27.9359	−	1.31588i	−1.27377	−	0.0599988i
$482$		0			0
$483$		−	3.92401i		−	0.178548i
$484$		0			0
$485$	−37.4348			−1.69983
$486$		0			0
$487$		−	0.990681i		−	0.0448920i	−0.999748	−	0.0224460i	$-0.992855\pi$
$487$		−	0.990681i		−	0.0448920i	0.999748	−	0.0224460i	$-0.00714539\pi$
$488$		0			0
$489$			7.42762i			0.335889i
$490$		0			0
$491$	30.6870			1.38489			0.692444	−	0.721472i	$-0.256534\pi$
$491$	30.6870			1.38489			0.692444	+	0.721472i	$0.256534\pi$
$492$		0			0
$493$	19.3174			0.870011
$494$		0			0
$495$	2.68836			0.120833
$496$		0			0
$497$	0.481962			0.0216189
$498$		0			0
$499$		−	30.0478i		−	1.34512i	−0.740041	−	0.672561i	$-0.765194\pi$
$499$		−	30.0478i		−	1.34512i	0.740041	−	0.672561i	$-0.234806\pi$
$500$		0			0
$501$		−	22.8540i		−	1.02104i
$502$		0			0
$503$	2.93922			0.131053			0.0655267	−	0.997851i	$-0.479127\pi$
$503$	2.93922			0.131053			0.0655267	+	0.997851i	$0.479127\pi$
$504$		0			0
$505$			47.3445i			2.10680i
$506$		0			0
$507$	−12.9424	−	1.22198i	−0.574794	−	0.0542699i
$508$		0			0
$509$		−	27.3029i		−	1.21018i	−0.796157	−	0.605090i	$-0.793137\pi$
$509$		−	27.3029i		−	1.21018i	0.796157	−	0.605090i	$-0.206863\pi$
$510$		0			0
$511$	−6.29672			−0.278550
$512$		0			0
$513$		−	8.52662i		−	0.376459i
$514$		0			0
$515$			31.8984i			1.40561i
$516$		0			0
$517$	−12.2212			−0.537487
$518$		0			0
$519$	−7.29517			−0.320222
$520$		0			0
$521$	11.4147			0.500086			0.250043	−	0.968235i	$-0.419555\pi$
$521$	11.4147			0.500086			0.250043	+	0.968235i	$0.419555\pi$
$522$		0			0
$523$	37.5911			1.64374			0.821872	−	0.569672i	$-0.192930\pi$
$523$	37.5911			1.64374			0.821872	+	0.569672i	$0.192930\pi$
$524$		0			0
$525$		−	1.01579i		−	0.0443326i
$526$		0			0
$527$			15.5219i			0.676146i
$528$		0			0
$529$	51.0286			2.21864
$530$		0			0
$531$			6.94071i			0.301201i
$532$		0			0
$533$	−9.84852	−	0.463899i	−0.426587	−	0.0200937i
$534$		0			0
$535$		−	22.3602i		−	0.966717i
$536$		0			0
$537$	1.90697			0.0822919
$538$		0			0
$539$		−	6.79200i		−	0.292552i
$540$		0			0
$541$			15.3730i			0.660936i	0.943817	+	0.330468i	$0.107207\pi$
$541$			15.3730i			0.660936i	−0.943817	+	0.330468i	$0.892793\pi$
$542$		0			0
$543$	−15.1346			−0.649489
$544$		0			0
$545$	52.2769			2.23929
$546$		0			0
$547$	−14.4302			−0.616992			−0.308496	−	0.951226i	$-0.599826\pi$
$547$	−14.4302			−0.616992			−0.308496	+	0.951226i	$0.599826\pi$
$548$		0			0
$549$	−6.82922			−0.291464
$550$		0			0
$551$			42.5211i			1.81146i
$552$		0			0
$553$		−	4.65322i		−	0.197875i
$554$		0			0
$555$	20.8526			0.885142
$556$		0			0
$557$			22.6891i			0.961370i	0.876893	+	0.480685i	$0.159612\pi$
$557$			22.6891i			0.961370i	−0.876893	+	0.480685i	$0.840388\pi$
$558$		0			0
$559$	0.817991	−	17.3659i	0.0345973	−	0.734498i
$560$		0			0
$561$			3.87365i			0.163546i
$562$		0			0
$563$	36.6350			1.54398			0.771990	−	0.635634i	$-0.219261\pi$
$563$	36.6350			1.54398			0.771990	+	0.635634i	$0.219261\pi$
$564$		0			0
$565$		−	17.8868i		−	0.752504i
$566$		0			0
$567$			0.456068i			0.0191531i
$568$		0			0
$569$	25.3383			1.06224			0.531118	−	0.847298i	$-0.321772\pi$
$569$	25.3383			1.06224			0.531118	+	0.847298i	$0.321772\pi$
$570$		0			0
$571$	−19.0980			−0.799228			−0.399614	−	0.916684i	$-0.630856\pi$
$571$	−19.0980			−0.799228			−0.399614	+	0.916684i	$0.630856\pi$
$572$		0			0
$573$	−4.28117			−0.178848
$574$		0			0
$575$	19.1634			0.799170
$576$		0			0
$577$		−	2.24163i		−	0.0933201i	−0.998911	−	0.0466601i	$-0.985142\pi$
$577$		−	2.24163i		−	0.0933201i	0.998911	−	0.0466601i	$-0.0148578\pi$
$578$		0			0
$579$		−	10.3333i		−	0.429439i
$580$		0			0
$581$	5.11546			0.212225
$582$		0			0
$583$			8.50656i			0.352306i
$584$		0			0
$585$	9.68228	+	0.456068i	0.400313	+	0.0188561i
$586$		0			0
$587$		−	40.9186i		−	1.68889i	−0.535642	−	0.844445i	$-0.679930\pi$
$587$		−	40.9186i		−	1.68889i	0.535642	−	0.844445i	$-0.320070\pi$
$588$		0			0
$589$	−34.1666			−1.40781
$590$		0			0
$591$			0.399650i			0.0164394i
$592$		0			0
$593$		−	9.45673i		−	0.388341i	−0.980968	−	0.194171i	$-0.937798\pi$
$593$		−	9.45673i		−	0.388341i	0.980968	−	0.194171i	$-0.0622016\pi$
$594$		0			0
$595$	4.74939			0.194706
$596$		0			0
$597$	0.761983			0.0311859
$598$		0			0
$599$	−30.8171			−1.25915			−0.629577	−	0.776938i	$-0.716772\pi$
$599$	−30.8171			−1.25915			−0.629577	+	0.776938i	$0.716772\pi$
$600$		0			0
$601$	−6.63070			−0.270472			−0.135236	−	0.990813i	$-0.543179\pi$
$601$	−6.63070			−0.270472			−0.135236	+	0.990813i	$0.543179\pi$
$602$		0			0
$603$			7.85990i			0.320080i
$604$		0			0
$605$		−	2.68836i		−	0.109297i
$606$		0			0
$607$	−4.74018			−0.192398			−0.0961991	−	0.995362i	$-0.530669\pi$
$607$	−4.74018			−0.192398			−0.0961991	+	0.995362i	$0.530669\pi$
$608$		0			0
$609$		−	2.27435i		−	0.0921614i
$610$		0			0
$611$	−44.0153	−	2.07327i	−1.78067	−	0.0838756i
$612$		0			0
$613$		−	15.1254i		−	0.610910i	−0.952207	−	0.305455i	$-0.901191\pi$
$613$		−	15.1254i		−	0.610910i	0.952207	−	0.305455i	$-0.0988085\pi$
$614$		0			0
$615$	7.35136			0.296436
$616$		0			0
$617$			15.9304i			0.641332i	0.947192	+	0.320666i	$0.103907\pi$
$617$			15.9304i			0.641332i	−0.947192	+	0.320666i	$0.896093\pi$
$618$		0			0
$619$			15.4833i			0.622326i	0.950357	+	0.311163i	$0.100718\pi$
$619$			15.4833i			0.622326i	−0.950357	+	0.311163i	$0.899282\pi$
$620$		0			0
$621$	−8.60399			−0.345266
$622$		0			0
$623$	−1.77177			−0.0709846
$624$		0			0
$625$	−31.1756			−1.24702
$626$		0			0
$627$	−8.52662			−0.340520
$628$		0			0
$629$			30.0464i			1.19803i
$630$		0			0
$631$		−	27.0874i		−	1.07833i	−0.842199	−	0.539166i	$-0.818739\pi$
$631$		−	27.0874i		−	1.07833i	0.842199	−	0.539166i	$-0.181261\pi$
$632$		0			0
$633$	12.6351			0.502201
$634$		0			0
$635$			54.7707i			2.17351i
$636$		0			0
$637$	1.15223	−	24.4618i	0.0456532	−	0.969211i
$638$		0			0
$639$		−	1.05678i		−	0.0418054i
$640$		0			0
$641$	−18.8894			−0.746085			−0.373042	−	0.927814i	$-0.621685\pi$
$641$	−18.8894			−0.746085			−0.373042	+	0.927814i	$0.621685\pi$
$642$		0			0
$643$			10.7751i			0.424930i	0.977169	+	0.212465i	$0.0681492\pi$
$643$			10.7751i			0.424930i	−0.977169	+	0.212465i	$0.931851\pi$
$644$		0			0
$645$			12.9626i			0.510403i
$646$		0			0
$647$	−21.0573			−0.827850			−0.413925	−	0.910311i	$-0.635842\pi$
$647$	−21.0573			−0.827850			−0.413925	+	0.910311i	$0.635842\pi$
$648$		0			0
$649$	6.94071			0.272447
$650$		0			0
$651$	1.82749			0.0716250
$652$		0			0
$653$	33.7581			1.32106			0.660529	−	0.750801i	$-0.270332\pi$
$653$	33.7581			1.32106			0.660529	+	0.750801i	$0.270332\pi$
$654$		0			0
$655$		−	8.44943i		−	0.330147i
$656$		0			0
$657$			13.8065i			0.538644i
$658$		0			0
$659$	39.3947			1.53460			0.767300	−	0.641288i	$-0.221600\pi$
$659$	39.3947			1.53460			0.767300	+	0.641288i	$0.221600\pi$
$660$		0			0
$661$		−	0.340365i		−	0.0132387i	−0.999978	−	0.00661934i	$-0.997893\pi$
$661$		−	0.340365i		−	0.0132387i	0.999978	−	0.00661934i	$-0.00210702\pi$
$662$		0			0
$663$	−0.657148	+	13.9512i	−0.0255215	+	0.541819i
$664$		0			0
$665$			10.4543i			0.405399i
$666$		0			0
$667$	42.9069			1.66136
$668$		0			0
$669$		−	3.82102i		−	0.147729i
$670$		0			0
$671$			6.82922i			0.263639i
$672$		0			0
$673$	−36.9501			−1.42432			−0.712160	−	0.702017i	$-0.752283\pi$
$673$	−36.9501			−1.42432			−0.712160	+	0.702017i	$0.752283\pi$
$674$		0			0
$675$	−2.22727			−0.0857277
$676$		0			0
$677$	47.3511			1.81985			0.909925	−	0.414772i	$-0.136139\pi$
$677$	47.3511			1.81985			0.909925	+	0.414772i	$0.136139\pi$
$678$		0			0
$679$	−6.35065			−0.243715
$680$		0			0
$681$			7.91248i			0.303207i
$682$		0			0
$683$			7.01764i			0.268522i	0.990946	+	0.134261i	$0.0428661\pi$
$683$			7.01764i			0.268522i	−0.990946	+	0.134261i	$0.957134\pi$
$684$		0			0
$685$	−39.4347			−1.50672
$686$		0			0
$687$		−	12.0503i		−	0.459748i
$688$		0			0
$689$	−1.44310	+	30.6369i	−0.0549778	+	1.16717i
$690$		0			0
$691$			13.8821i			0.528101i	0.964509	+	0.264050i	$0.0850585\pi$
$691$			13.8821i			0.528101i	−0.964509	+	0.264050i	$0.914941\pi$
$692$		0			0
$693$	0.456068			0.0173246
$694$		0			0
$695$			5.53423i			0.209925i
$696$		0			0
$697$			10.5926i			0.401222i
$698$		0			0
$699$	3.50989			0.132756
$700$		0			0
$701$	16.8397			0.636027			0.318013	−	0.948086i	$-0.396984\pi$
$701$	16.8397			0.636027			0.318013	+	0.948086i	$0.396984\pi$
$702$		0			0
$703$	−66.1377			−2.49443
$704$		0			0
$705$	32.8549			1.23739
$706$		0			0
$707$			8.03178i			0.302066i
$708$		0			0
$709$			40.3379i			1.51492i	0.652880	+	0.757461i	$0.273561\pi$
$709$			40.3379i			1.51492i	−0.652880	+	0.757461i	$0.726439\pi$
$710$		0			0
$711$	−10.2029			−0.382639
$712$		0			0
$713$			34.4767i			1.29116i
$714$		0			0
$715$	0.456068	−	9.68228i	0.0170560	−	0.362097i
$716$		0			0
$717$		−	6.31623i		−	0.235884i
$718$		0			0
$719$	−46.8723			−1.74804			−0.874021	−	0.485887i	$-0.838496\pi$
$719$	−46.8723			−1.74804			−0.874021	+	0.485887i	$0.838496\pi$
$720$		0			0
$721$			5.41142i			0.201532i
$722$		0			0
$723$		−	27.0761i		−	1.00697i
$724$		0			0
$725$	11.1071			0.412508
$726$		0			0
$727$	45.3770			1.68294			0.841470	−	0.540303i	$-0.181690\pi$
$727$	45.3770			1.68294			0.841470	+	0.540303i	$0.181690\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−18.6778			−0.690824
$732$		0			0
$733$		−	2.43013i		−	0.0897588i	−0.998992	−	0.0448794i	$-0.985710\pi$
$733$		−	2.43013i		−	0.0897588i	0.998992	−	0.0448794i	$-0.0142904\pi$
$734$		0			0
$735$			18.2593i			0.673506i
$736$		0			0
$737$	7.85990			0.289523
$738$		0			0
$739$		−	9.02856i		−	0.332121i	−0.986116	−	0.166061i	$-0.946895\pi$
$739$		−	9.02856i		−	0.332121i	0.986116	−	0.166061i	$-0.0531047\pi$
$740$		0			0
$741$	−30.7091	−	1.44650i	−1.12813	−	0.0531386i
$742$		0			0
$743$		−	25.1440i		−	0.922444i	−0.887285	−	0.461222i	$-0.847411\pi$
$743$		−	25.1440i		−	0.922444i	0.887285	−	0.461222i	$-0.152589\pi$
$744$		0			0
$745$	13.5122			0.495049
$746$		0			0
$747$		−	11.2164i		−	0.410388i
$748$		0			0
$749$		−	3.79331i		−	0.138605i
$750$		0			0
$751$	38.9733			1.42215			0.711077	−	0.703114i	$-0.248208\pi$
$751$	38.9733			1.42215			0.711077	+	0.703114i	$0.248208\pi$
$752$		0			0
$753$	0.173701			0.00633001
$754$		0			0
$755$	−8.02263			−0.291973
$756$		0			0
$757$	32.7993			1.19211			0.596055	−	0.802944i	$-0.296734\pi$
$757$	32.7993			1.19211			0.596055	+	0.802944i	$0.296734\pi$
$758$		0			0
$759$			8.60399i			0.312305i
$760$		0			0
$761$		−	22.6992i		−	0.822845i	−0.911445	−	0.411422i	$-0.865032\pi$
$761$		−	22.6992i		−	0.822845i	0.911445	−	0.411422i	$-0.134968\pi$
$762$		0			0
$763$	8.86854			0.321063
$764$		0			0
$765$		−	10.4138i		−	0.376510i
$766$		0			0
$767$	24.9974	+	1.17746i	0.902602	+	0.0425156i
$768$		0			0
$769$		−	0.569358i		−	0.0205316i	−0.999947	−	0.0102658i	$-0.996732\pi$
$769$		−	0.569358i		−	0.0205316i	0.999947	−	0.0102658i	$-0.00326776\pi$
$770$		0			0
$771$	−22.5272			−0.811297
$772$		0			0
$773$		−	18.8373i		−	0.677532i	−0.940871	−	0.338766i	$-0.889990\pi$
$773$		−	18.8373i		−	0.677532i	0.940871	−	0.338766i	$-0.110010\pi$
$774$		0			0
$775$			8.92480i			0.320588i
$776$		0			0
$777$	3.53755			0.126909
$778$		0			0
$779$	−23.3162			−0.835389
$780$		0			0
$781$	−1.05678			−0.0378144
$782$		0			0
$783$	−4.98687			−0.178216
$784$		0			0
$785$			12.1363i			0.433163i
$786$		0			0
$787$			38.8165i			1.38366i	0.722060	+	0.691830i	$0.243195\pi$
$787$			38.8165i			1.38366i	−0.722060	+	0.691830i	$0.756805\pi$
$788$		0			0
$789$	−9.63612			−0.343055
$790$		0			0
$791$		−	3.03442i		−	0.107892i
$792$		0			0
$793$	−1.15855	+	24.5958i	−0.0411413	+	0.873424i
$794$		0			0
$795$		−	22.8687i		−	0.811069i
$796$		0			0
$797$	20.2925			0.718797			0.359399	−	0.933184i	$-0.382982\pi$
$797$	20.2925			0.718797			0.359399	+	0.933184i	$0.382982\pi$
$798$		0			0
$799$			47.3406i			1.67479i
$800$		0			0
$801$			3.88488i			0.137266i
$802$		0			0
$803$	13.8065			0.487221
$804$		0			0
$805$	10.5491			0.371808
$806$		0			0
$807$	−5.27311			−0.185622
$808$		0			0
$809$	−21.3274			−0.749830			−0.374915	−	0.927059i	$-0.622328\pi$
$809$	−21.3274			−0.749830			−0.374915	+	0.927059i	$0.622328\pi$
$810$		0			0
$811$			17.5592i			0.616586i	0.951291	+	0.308293i	$0.0997578\pi$
$811$			17.5592i			0.616586i	−0.951291	+	0.308293i	$0.900242\pi$
$812$		0			0
$813$		−	3.24141i		−	0.113681i
$814$		0			0
$815$	−19.9681			−0.699452
$816$		0			0
$817$		−	41.1133i		−	1.43837i
$818$		0			0
$819$	1.64256	+	0.0773700i	0.0573956	+	0.00270353i
$820$		0			0
$821$			31.3260i			1.09328i	0.837366	+	0.546642i	$0.184094\pi$
$821$			31.3260i			1.09328i	−0.837366	+	0.546642i	$0.815906\pi$
$822$		0			0
$823$	7.61168			0.265327			0.132663	−	0.991161i	$-0.457647\pi$
$823$	7.61168			0.265327			0.132663	+	0.991161i	$0.457647\pi$
$824$		0			0
$825$			2.22727i			0.0775437i
$826$		0			0
$827$		−	4.35959i		−	0.151598i	−0.997123	−	0.0757988i	$-0.975849\pi$
$827$		−	4.35959i		−	0.151598i	0.997123	−	0.0757988i	$-0.0241507\pi$
$828$		0			0
$829$	17.4826			0.607196			0.303598	−	0.952800i	$-0.401812\pi$
$829$	17.4826			0.607196			0.303598	+	0.952800i	$0.401812\pi$
$830$		0			0
$831$	−27.8030			−0.964474
$832$		0			0
$833$	−26.3098			−0.911582
$834$		0			0
$835$	61.4397			2.12621
$836$		0			0
$837$		−	4.00705i		−	0.138504i
$838$		0			0
$839$			11.1193i			0.383880i	0.981407	+	0.191940i	$0.0614778\pi$
$839$			11.1193i			0.383880i	−0.981407	+	0.191940i	$0.938522\pi$
$840$		0			0
$841$	−4.13117			−0.142454
$842$		0			0
$843$		−	11.6043i		−	0.399672i
$844$		0			0
$845$	3.28511	−	34.7939i	0.113011	−	1.19695i
$846$		0			0
$847$		−	0.456068i		−	0.0156707i
$848$		0			0
$849$	26.0688			0.894678
$850$		0			0
$851$			66.7378i			2.28774i
$852$		0			0
$853$		−	26.2068i		−	0.897304i	−0.893707	−	0.448652i	$-0.851904\pi$
$853$		−	26.2068i		−	0.897304i	0.893707	−	0.448652i	$-0.148096\pi$
$854$		0			0
$855$	22.9226			0.783937
$856$		0			0
$857$	9.50250			0.324599			0.162300	−	0.986742i	$-0.448109\pi$
$857$	9.50250			0.324599			0.162300	+	0.986742i	$0.448109\pi$
$858$		0			0
$859$	−41.1007			−1.40234			−0.701169	−	0.712995i	$-0.747338\pi$
$859$	−41.1007			−1.40234			−0.701169	+	0.712995i	$0.747338\pi$
$860$		0			0
$861$	1.24713			0.0425020
$862$		0			0
$863$			13.4967i			0.459432i	0.973258	+	0.229716i	$0.0737797\pi$
$863$			13.4967i			0.459432i	−0.973258	+	0.229716i	$0.926220\pi$
$864$		0			0
$865$		−	19.6120i		−	0.666829i
$866$		0			0
$867$	−1.99484			−0.0677482
$868$		0			0
$869$			10.2029i			0.346110i
$870$		0			0
$871$	28.3079	+	1.33340i	0.959176	+	0.0451804i
$872$		0			0
$873$			13.9248i			0.471282i
$874$		0			0
$875$	−3.39957			−0.114927
$876$		0			0
$877$			14.5544i			0.491466i	0.969338	+	0.245733i	$0.0790286\pi$
$877$			14.5544i			0.491466i	−0.969338	+	0.245733i	$0.920971\pi$
$878$		0			0
$879$			24.0256i			0.810363i
$880$		0			0
$881$	35.3315			1.19035			0.595175	−	0.803596i	$-0.297083\pi$
$881$	35.3315			1.19035			0.595175	+	0.803596i	$0.297083\pi$
$882$		0			0
$883$	4.31806			0.145314			0.0726572	−	0.997357i	$-0.476852\pi$
$883$	4.31806			0.145314			0.0726572	+	0.997357i	$0.476852\pi$
$884$		0			0
$885$	−18.6591			−0.627219
$886$		0			0
$887$	32.0857			1.07733			0.538666	−	0.842519i	$-0.318929\pi$
$887$	32.0857			1.07733			0.538666	+	0.842519i	$0.318929\pi$
$888$		0			0
$889$			9.29161i			0.311631i
$890$		0			0
$891$		−	1.00000i		−	0.0335013i
$892$		0			0
$893$	−104.205			−3.48710
$894$		0			0
$895$			5.12663i			0.171364i
$896$		0			0
$897$	−1.45963	+	30.9878i	−0.0487356	+	1.03465i
$898$		0			0
$899$			19.9826i			0.666459i
$900$		0			0
$901$	32.9514			1.09777
$902$		0			0
$903$			2.19905i			0.0731799i
$904$		0			0
$905$		−	40.6873i		−	1.35249i
$906$		0			0
$907$	33.8292			1.12328			0.561641	−	0.827381i	$-0.310170\pi$
$907$	33.8292			1.12328			0.561641	+	0.827381i	$0.310170\pi$
$908$		0			0
$909$	17.6109			0.584117
$910$		0			0
$911$	−30.3039			−1.00401			−0.502007	−	0.864864i	$-0.667405\pi$
$911$	−30.3039			−1.00401			−0.502007	+	0.864864i	$0.667405\pi$
$912$		0			0
$913$	−11.2164			−0.371210
$914$		0			0
$915$		−	18.3594i		−	0.606943i
$916$		0			0
$917$		−	1.43341i		−	0.0473353i
$918$		0			0
$919$	53.5528			1.76655			0.883273	−	0.468860i	$-0.155335\pi$
$919$	53.5528			1.76655			0.883273	+	0.468860i	$0.155335\pi$
$920$		0			0
$921$			4.12304i			0.135859i
$922$		0			0
$923$	−3.80604	−	0.179277i	−0.125277	−	0.00590098i
$924$		0			0
$925$			17.2761i			0.568034i
$926$		0			0
$927$	11.8654			0.389710
$928$		0			0
$929$			6.74707i			0.221364i	0.993856	+	0.110682i	$0.0353035\pi$
$929$			6.74707i			0.221364i	−0.993856	+	0.110682i	$0.964696\pi$
$930$		0			0
$931$		−	57.9128i		−	1.89802i
$932$		0			0
$933$	20.2962			0.664467
$934$		0			0
$935$	−10.4138			−0.340566
$936$		0			0
$937$	−22.4089			−0.732068			−0.366034	−	0.930601i	$-0.619285\pi$
$937$	−22.4089			−0.732068			−0.366034	+	0.930601i	$0.619285\pi$
$938$		0			0
$939$	0.295037			0.00962818
$940$		0			0
$941$			25.6145i			0.835009i	0.908675	+	0.417505i	$0.137095\pi$
$941$			25.6145i			0.835009i	−0.908675	+	0.417505i	$0.862905\pi$
$942$		0			0
$943$			23.5278i			0.766169i
$944$		0			0
$945$	−1.22608			−0.0398842
$946$		0			0
$947$		−	27.2251i		−	0.884696i	−0.896844	−	0.442348i	$-0.854146\pi$
$947$		−	27.2251i		−	0.884696i	0.896844	−	0.442348i	$-0.145854\pi$
$948$		0			0
$949$	49.7250	+	2.34222i	1.61414	+	0.0760315i
$950$		0			0
$951$		−	6.67853i		−	0.216566i
$952$		0			0
$953$	−19.1817			−0.621356			−0.310678	−	0.950515i	$-0.600556\pi$
$953$	−19.1817			−0.621356			−0.310678	+	0.950515i	$0.600556\pi$
$954$		0			0
$955$		−	11.5093i		−	0.372433i
$956$		0			0
$957$			4.98687i			0.161202i
$958$		0			0
$959$	−6.68992			−0.216029
$960$		0			0
$961$	14.9435			0.482049
$962$		0			0
$963$	−8.31742			−0.268025
$964$		0			0
$965$	27.7797			0.894261
$966$		0			0
$967$			8.61230i			0.276953i	0.990366	+	0.138477i	$0.0442205\pi$
$967$			8.61230i			0.276953i	−0.990366	+	0.138477i	$0.955779\pi$
$968$		0			0
$969$			33.0291i			1.06105i
$970$		0			0
$971$	−59.0691			−1.89562			−0.947809	−	0.318838i	$-0.896707\pi$
$971$	−59.0691			−1.89562			−0.947809	+	0.318838i	$0.896707\pi$
$972$		0			0
$973$			0.938857i			0.0300984i
$974$		0			0
$975$	−0.377847	+	8.02165i	−0.0121008	+	0.256898i
$976$		0			0
$977$			33.2422i			1.06351i	0.846898	+	0.531755i	$0.178467\pi$
$977$			33.2422i			1.06351i	−0.846898	+	0.531755i	$0.821533\pi$
$978$		0			0
$979$	3.88488			0.124161
$980$		0			0
$981$		−	19.4456i		−	0.620852i
$982$		0			0
$983$			15.4292i			0.492114i	0.969255	+	0.246057i	$0.0791350\pi$
$983$			15.4292i			0.492114i	−0.969255	+	0.246057i	$0.920865\pi$
$984$		0			0
$985$	−1.07440			−0.0342333
$986$		0			0
$987$	5.57370			0.177413
$988$		0			0
$989$	−41.4864			−1.31919
$990$		0			0
$991$	−60.9746			−1.93692			−0.968460	−	0.249168i	$-0.919843\pi$
$991$	−60.9746			−1.93692			−0.968460	+	0.249168i	$0.919843\pi$
$992$		0			0
$993$			22.0794i			0.700668i
$994$		0			0
$995$			2.04848i			0.0649413i
$996$		0			0
$997$	−36.7306			−1.16327			−0.581635	−	0.813450i	$-0.697587\pi$
$997$	−36.7306			−1.16327			−0.581635	+	0.813450i	$0.697587\pi$
$998$		0			0
$999$		−	7.75661i		−	0.245408i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3432.2.g.d.1585.13	yes	14
13.12	even	2	inner	3432.2.g.d.1585.2	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.g.d.1585.2	✓	14	13.12	even	2	inner
3432.2.g.d.1585.13	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.68836i q^{5} +0.456068i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.68836i q^{5} +0.456068i q^{7} +1.00000 q^{9} -1.00000i q^{11} +(0.169646 - 3.60156i) q^{13} +2.68836i q^{15} -3.87365 q^{17} -8.52662i q^{19} +0.456068i q^{21} -8.60399 q^{23} -2.22727 q^{25} +1.00000 q^{27} -4.98687 q^{29} -4.00705i q^{31} -1.00000i q^{33} -1.22608 q^{35} -7.75661i q^{37} +(0.169646 - 3.60156i) q^{39} -2.73452i q^{41} +4.82176 q^{43} +2.68836i q^{45} -12.2212i q^{47} +6.79200 q^{49} -3.87365 q^{51} -8.50656 q^{53} +2.68836 q^{55} -8.52662i q^{57} +6.94071i q^{59} -6.82922 q^{61} +0.456068i q^{63} +(9.68228 + 0.456068i) q^{65} +7.85990i q^{67} -8.60399 q^{69} -1.05678i q^{71} +13.8065i q^{73} -2.22727 q^{75} +0.456068 q^{77} -10.2029 q^{79} +1.00000 q^{81} -11.2164i q^{83} -10.4138i q^{85} -4.98687 q^{87} +3.88488i q^{89} +(1.64256 + 0.0773700i) q^{91} -4.00705i q^{93} +22.9226 q^{95} +13.9248i q^{97} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 14 q^{3} + 14 q^{9}+O(q^{10}) \) 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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