LMFDB - Embedded newform 4020.2.f.a.401.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(401,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 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("4020.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.f (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.9
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.40436 - 1.01380i) q^{3} -1.00000 q^{5} -3.61166i q^{7} +(0.944435 + 2.84746i) q^{9} +O(q^{10})$	\(q+(-1.40436 - 1.01380i) q^{3} -1.00000 q^{5} -3.61166i q^{7} +(0.944435 + 2.84746i) q^{9} -1.04388 q^{11} -0.0250308i q^{13} +(1.40436 + 1.01380i) q^{15} -1.83039i q^{17} +3.45689 q^{19} +(-3.66149 + 5.07206i) q^{21} -6.66422i q^{23} +1.00000 q^{25} +(1.56042 - 4.95632i) q^{27} +5.90996i q^{29} -1.86192i q^{31} +(1.46599 + 1.05829i) q^{33} +3.61166i q^{35} +10.6139 q^{37} +(-0.0253761 + 0.0351522i) q^{39} +2.93548 q^{41} -11.4481i q^{43} +(-0.944435 - 2.84746i) q^{45} -4.63191i q^{47} -6.04409 q^{49} +(-1.85564 + 2.57052i) q^{51} -7.26440 q^{53} +1.04388 q^{55} +(-4.85470 - 3.50458i) q^{57} +3.59034i q^{59} -6.07828i q^{61} +(10.2841 - 3.41098i) q^{63} +0.0250308i q^{65} +(5.81460 - 5.76111i) q^{67} +(-6.75616 + 9.35894i) q^{69} +1.63263i q^{71} -9.89619 q^{73} +(-1.40436 - 1.01380i) q^{75} +3.77015i q^{77} +5.00529i q^{79} +(-7.21608 + 5.37849i) q^{81} +6.25585i q^{83} +1.83039i q^{85} +(5.99150 - 8.29970i) q^{87} -16.9978i q^{89} -0.0904028 q^{91} +(-1.88760 + 2.61480i) q^{93} -3.45689 q^{95} +11.7436i q^{97} +(-0.985881 - 2.97242i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$-1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.40436	−	1.01380i	−0.810806	−	0.585315i
$3$	−1.40436	−	1.01380i	−0.810806	−	0.585315i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	3.61166i		−	1.36508i	−0.730849	−	0.682540i	$-0.760876\pi$
$7$		−	3.61166i		−	1.36508i	0.730849	−	0.682540i	$-0.239124\pi$
$8$		0			0
$9$	0.944435	+	2.84746i	0.314812	+	0.949154i
$10$		0			0
$11$	−1.04388			−0.314743			−0.157371	−	0.987539i	$-0.550302\pi$
$11$	−1.04388			−0.314743			−0.157371	+	0.987539i	$0.550302\pi$
$12$		0			0
$13$		−	0.0250308i		−	0.00694230i	−0.999994	−	0.00347115i	$-0.998895\pi$
$13$		−	0.0250308i		−	0.00694230i	0.999994	−	0.00347115i	$-0.00110490\pi$
$14$		0			0
$15$	1.40436	+	1.01380i	0.362603	+	0.261761i
$16$		0			0
$17$		−	1.83039i		−	0.443935i	−0.975054	−	0.221967i	$-0.928752\pi$
$17$		−	1.83039i		−	0.443935i	0.975054	−	0.221967i	$-0.0712478\pi$
$18$		0			0
$19$	3.45689			0.793064			0.396532	−	0.918021i	$-0.370214\pi$
$19$	3.45689			0.793064			0.396532	+	0.918021i	$0.370214\pi$
$20$		0			0
$21$	−3.66149	+	5.07206i	−0.799002	+	1.10681i
$22$		0			0
$23$		−	6.66422i		−	1.38959i	−0.719210	−	0.694793i	$-0.755496\pi$
$23$		−	6.66422i		−	1.38959i	0.719210	−	0.694793i	$-0.244504\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	1.56042	−	4.95632i	0.300303	−	0.953844i
$28$		0			0
$29$			5.90996i			1.09745i	0.836002	+	0.548726i	$0.184887\pi$
$29$			5.90996i			1.09745i	−0.836002	+	0.548726i	$0.815113\pi$
$30$		0			0
$31$		−	1.86192i		−	0.334410i	−0.985922	−	0.167205i	$-0.946526\pi$
$31$		−	1.86192i		−	0.334410i	0.985922	−	0.167205i	$-0.0534742\pi$
$32$		0			0
$33$	1.46599	+	1.05829i	0.255195	+	0.184224i
$34$		0			0
$35$			3.61166i			0.610482i
$36$		0			0
$37$	10.6139			1.74491			0.872456	−	0.488693i	$-0.162526\pi$
$37$	10.6139			1.74491			0.872456	+	0.488693i	$0.162526\pi$
$38$		0			0
$39$	−0.0253761	+	0.0351522i	−0.00406343	+	0.00562885i
$40$		0			0
$41$	2.93548			0.458445			0.229223	−	0.973374i	$-0.426382\pi$
$41$	2.93548			0.458445			0.229223	+	0.973374i	$0.426382\pi$
$42$		0			0
$43$		−	11.4481i		−	1.74581i	−0.487886	−	0.872907i	$-0.662232\pi$
$43$		−	11.4481i		−	1.74581i	0.487886	−	0.872907i	$-0.337768\pi$
$44$		0			0
$45$	−0.944435	−	2.84746i	−0.140788	−	0.424475i
$46$		0			0
$47$		−	4.63191i		−	0.675633i	−0.941212	−	0.337817i	$-0.890312\pi$
$47$		−	4.63191i		−	0.675633i	0.941212	−	0.337817i	$-0.109688\pi$
$48$		0			0
$49$	−6.04409			−0.863441
$50$		0			0
$51$	−1.85564	+	2.57052i	−0.259842	+	0.359945i
$52$		0			0
$53$	−7.26440			−0.997842			−0.498921	−	0.866648i	$-0.666270\pi$
$53$	−7.26440			−0.997842			−0.498921	+	0.866648i	$0.666270\pi$
$54$		0			0
$55$	1.04388			0.140757
$56$		0			0
$57$	−4.85470	−	3.50458i	−0.643021	−	0.464193i
$58$		0			0
$59$			3.59034i			0.467423i	0.972306	+	0.233711i	$0.0750871\pi$
$59$			3.59034i			0.467423i	−0.972306	+	0.233711i	$0.924913\pi$
$60$		0			0
$61$		−	6.07828i		−	0.778244i	−0.921186	−	0.389122i	$-0.872778\pi$
$61$		−	6.07828i		−	0.778244i	0.921186	−	0.389122i	$-0.127222\pi$
$62$		0			0
$63$	10.2841	−	3.41098i	1.29567	−	0.429743i
$64$		0			0
$65$			0.0250308i			0.00310469i
$66$		0			0
$67$	5.81460	−	5.76111i	0.710367	−	0.703832i
$67$	5.81460	−	5.76111i	0.710367	−	0.703832i
$68$		0			0
$69$	−6.75616	+	9.35894i	−0.813346	+	1.12668i
$70$		0			0
$71$			1.63263i			0.193758i	0.995296	+	0.0968790i	$0.0308860\pi$
$71$			1.63263i			0.193758i	−0.995296	+	0.0968790i	$0.969114\pi$
$72$		0			0
$73$	−9.89619			−1.15826			−0.579131	−	0.815235i	$-0.696608\pi$
$73$	−9.89619			−1.15826			−0.579131	+	0.815235i	$0.696608\pi$
$74$		0			0
$75$	−1.40436	−	1.01380i	−0.162161	−	0.117063i
$76$		0			0
$77$			3.77015i			0.429649i
$78$		0			0
$79$			5.00529i			0.563140i	0.959541	+	0.281570i	$0.0908551\pi$
$79$			5.00529i			0.563140i	−0.959541	+	0.281570i	$0.909145\pi$
$80$		0			0
$81$	−7.21608	+	5.37849i	−0.801787	+	0.597610i
$82$		0			0
$83$			6.25585i			0.686668i	0.939213	+	0.343334i	$0.111556\pi$
$83$			6.25585i			0.686668i	−0.939213	+	0.343334i	$0.888444\pi$
$84$		0			0
$85$			1.83039i			0.198534i
$86$		0			0
$87$	5.99150	−	8.29970i	0.642356	−	0.889821i
$88$		0			0
$89$		−	16.9978i		−	1.80176i	−0.434069	−	0.900880i	$-0.642922\pi$
$89$		−	16.9978i		−	1.80176i	0.434069	−	0.900880i	$-0.357078\pi$
$90$		0			0
$91$	−0.0904028			−0.00947679
$92$		0			0
$93$	−1.88760	+	2.61480i	−0.195735	+	0.271142i
$94$		0			0
$95$	−3.45689			−0.354669
$96$		0			0
$97$			11.7436i			1.19238i	0.802842	+	0.596192i	$0.203320\pi$
$97$			11.7436i			1.19238i	−0.802842	+	0.596192i	$0.796680\pi$
$98$		0			0
$99$	−0.985881	−	2.97242i	−0.0990847	−	0.298739i
$100$		0			0
$101$	16.8963			1.68124			0.840620	−	0.541625i	$-0.182191\pi$
$101$	16.8963			1.68124			0.840620	+	0.541625i	$0.182191\pi$
$102$		0			0
$103$	2.00029			0.197094			0.0985471	−	0.995132i	$-0.468580\pi$
$103$	2.00029			0.197094			0.0985471	+	0.995132i	$0.468580\pi$
$104$		0			0
$105$	3.66149	−	5.07206i	0.357324	−	0.494982i
$106$		0			0
$107$			6.80047i			0.657427i	0.944430	+	0.328713i	$0.106615\pi$
$107$			6.80047i			0.657427i	−0.944430	+	0.328713i	$0.893385\pi$
$108$		0			0
$109$		−	10.9435i		−	1.04820i	−0.851658	−	0.524098i	$-0.824403\pi$
$109$		−	10.9435i		−	1.04820i	0.851658	−	0.524098i	$-0.175597\pi$
$110$		0			0
$111$	−14.9057	−	10.7603i	−1.41478	−	1.02132i
$112$		0			0
$113$	14.3089			1.34607			0.673034	−	0.739612i	$-0.264991\pi$
$113$	14.3089			1.34607			0.673034	+	0.739612i	$0.264991\pi$
$114$		0			0
$115$			6.66422i			0.621442i
$116$		0			0
$117$	0.0712743	−	0.0236400i	0.00658931	−	0.00218552i
$118$		0			0
$119$	−6.61074			−0.606006
$120$		0			0
$121$	−9.91031			−0.900937
$122$		0			0
$123$	−4.12246	−	2.97598i	−0.371710	−	0.268335i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−13.6288			−1.20936			−0.604679	−	0.796469i	$-0.706699\pi$
$127$	−13.6288			−1.20936			−0.604679	+	0.796469i	$0.706699\pi$
$128$		0			0
$129$	−11.6060	+	16.0772i	−1.02185	+	1.41552i
$130$		0			0
$131$		−	17.1393i		−	1.49747i	−0.662871	−	0.748734i	$-0.730662\pi$
$131$		−	17.1393i		−	1.49747i	0.662871	−	0.748734i	$-0.269338\pi$
$132$		0			0
$133$		−	12.4851i		−	1.08259i
$134$		0			0
$135$	−1.56042	+	4.95632i	−0.134300	+	0.426572i
$136$		0			0
$137$	5.29208			0.452133			0.226066	−	0.974112i	$-0.427413\pi$
$137$	5.29208			0.452133			0.226066	+	0.974112i	$0.427413\pi$
$138$		0			0
$139$			11.7980i			1.00069i	0.865825	+	0.500347i	$0.166794\pi$
$139$			11.7980i			1.00069i	−0.865825	+	0.500347i	$0.833206\pi$
$140$		0			0
$141$	−4.69581	+	6.50485i	−0.395459	+	0.547807i
$142$		0			0
$143$			0.0261293i			0.00218504i
$144$		0			0
$145$		−	5.90996i		−	0.490796i
$146$		0			0
$147$	8.48805	+	6.12747i	0.700083	+	0.505385i
$148$		0			0
$149$			10.5970i			0.868140i	0.900879	+	0.434070i	$0.142923\pi$
$149$			10.5970i			0.868140i	−0.900879	+	0.434070i	$0.857077\pi$
$150$		0			0
$151$	−10.4759			−0.852520			−0.426260	−	0.904601i	$-0.640169\pi$
$151$	−10.4759			−0.852520			−0.426260	+	0.904601i	$0.640169\pi$
$152$		0			0
$153$	5.21197	−	1.72868i	0.421362	−	0.139756i
$154$		0			0
$155$			1.86192i			0.149553i
$156$		0			0
$157$	−22.1666			−1.76908			−0.884542	−	0.466460i	$-0.845529\pi$
$157$	−22.1666			−1.76908			−0.884542	+	0.466460i	$0.845529\pi$
$158$		0			0
$159$	10.2018	+	7.36462i	0.809056	+	0.584052i
$160$		0			0
$161$	−24.0689			−1.89689
$162$		0			0
$163$	2.45032			0.191924			0.0959620	−	0.995385i	$-0.469407\pi$
$163$	2.45032			0.191924			0.0959620	+	0.995385i	$0.469407\pi$
$164$		0			0
$165$	−1.46599	−	1.05829i	−0.114127	−	0.0823874i
$166$		0			0
$167$		−	8.81054i		−	0.681780i	−0.940103	−	0.340890i	$-0.889272\pi$
$167$		−	8.81054i		−	0.681780i	0.940103	−	0.340890i	$-0.110728\pi$
$168$		0			0
$169$	12.9994			0.999952
$170$		0			0
$171$	3.26480	+	9.84335i	0.249666	+	0.752740i
$172$		0			0
$173$			17.2499i			1.31149i	0.754983	+	0.655744i	$0.227645\pi$
$173$			17.2499i			1.31149i	−0.754983	+	0.655744i	$0.772355\pi$
$174$		0			0
$175$		−	3.61166i		−	0.273016i
$176$		0			0
$177$	3.63988	−	5.04212i	0.273590	−	0.378989i
$178$		0			0
$179$	−15.8297			−1.18317			−0.591583	−	0.806244i	$-0.701497\pi$
$179$	−15.8297			−1.18317			−0.591583	+	0.806244i	$0.701497\pi$
$180$		0			0
$181$	−6.73071			−0.500289			−0.250145	−	0.968208i	$-0.580478\pi$
$181$	−6.73071			−0.500289			−0.250145	+	0.968208i	$0.580478\pi$
$182$		0			0
$183$	−6.16213	+	8.53607i	−0.455518	+	0.631005i
$184$		0			0
$185$	−10.6139			−0.780348
$186$		0			0
$187$			1.91071i			0.139725i
$188$		0			0
$189$	−17.9005	−	5.63571i	−1.30207	−	0.409938i
$190$		0			0
$191$	−17.4585			−1.26325			−0.631626	−	0.775273i	$-0.717612\pi$
$191$	−17.4585			−1.26325			−0.631626	+	0.775273i	$0.717612\pi$
$192$		0			0
$193$	−21.2061			−1.52645			−0.763225	−	0.646132i	$-0.776385\pi$
$193$	−21.2061			−1.52645			−0.763225	+	0.646132i	$0.776385\pi$
$194$		0			0
$195$	0.0253761	−	0.0351522i	0.00181722	−	0.00251730i
$196$		0			0
$197$	−22.5226			−1.60467			−0.802335	−	0.596874i	$-0.796409\pi$
$197$	−22.5226			−1.60467			−0.802335	+	0.596874i	$0.796409\pi$
$198$		0			0
$199$	−11.8880			−0.842717			−0.421359	−	0.906894i	$-0.638447\pi$
$199$	−11.8880			−0.842717			−0.421359	+	0.906894i	$0.638447\pi$
$200$		0			0
$201$	−14.0064	+	2.19584i	−0.987933	+	0.154882i
$202$		0			0
$203$	21.3448			1.49811
$204$		0			0
$205$	−2.93548			−0.205023
$206$		0			0
$207$	18.9761	−	6.29392i	1.31893	−	0.437458i
$208$		0			0
$209$	−3.60859			−0.249611
$210$		0			0
$211$	27.3285			1.88137			0.940685	−	0.339281i	$-0.110184\pi$
$211$	27.3285			1.88137			0.940685	+	0.339281i	$0.110184\pi$
$212$		0			0
$213$	1.65516	−	2.29280i	0.113410	−	0.157100i
$214$		0			0
$215$			11.4481i			0.780752i
$216$		0			0
$217$	−6.72461			−0.456496
$218$		0			0
$219$	13.8978	+	10.0327i	0.939125	+	0.677948i
$220$		0			0
$221$	−0.0458161			−0.00308193
$222$		0			0
$223$	8.14688			0.545556			0.272778	−	0.962077i	$-0.412058\pi$
$223$	8.14688			0.545556			0.272778	+	0.962077i	$0.412058\pi$
$224$		0			0
$225$	0.944435	+	2.84746i	0.0629623	+	0.189831i
$226$		0			0
$227$		−	25.8364i		−	1.71482i	−0.514633	−	0.857410i	$-0.672072\pi$
$227$		−	25.8364i		−	1.71482i	0.514633	−	0.857410i	$-0.327928\pi$
$228$		0			0
$229$			16.0240i			1.05890i	0.848342	+	0.529448i	$0.177601\pi$
$229$			16.0240i			1.05890i	−0.848342	+	0.529448i	$0.822399\pi$
$230$		0			0
$231$	3.82217	−	5.29464i	0.251480	−	0.348362i
$232$		0			0
$233$	−12.0047			−0.786451			−0.393226	−	0.919442i	$-0.628641\pi$
$233$	−12.0047			−0.786451			−0.393226	+	0.919442i	$0.628641\pi$
$234$		0			0
$235$			4.63191i			0.302152i
$236$		0			0
$237$	5.07435	−	7.02922i	0.329614	−	0.456597i
$238$		0			0
$239$	7.36478			0.476388			0.238194	−	0.971218i	$-0.423445\pi$
$239$	7.36478			0.476388			0.238194	+	0.971218i	$0.423445\pi$
$240$		0			0
$241$	−3.66964			−0.236382			−0.118191	−	0.992991i	$-0.537710\pi$
$241$	−3.66964			−0.236382			−0.118191	+	0.992991i	$0.537710\pi$
$242$		0			0
$243$	15.5866	−	0.237677i	0.999884	−	0.0152470i
$244$		0			0
$245$	6.04409			0.386142
$246$		0			0
$247$		−	0.0865286i		−	0.00550569i
$248$		0			0
$249$	6.34215	−	8.78544i	0.401918	−	0.556755i
$250$		0			0
$251$	−27.8077			−1.75521			−0.877603	−	0.479388i	$-0.840859\pi$
$251$	−27.8077			−1.75521			−0.877603	+	0.479388i	$0.840859\pi$
$252$		0			0
$253$			6.95667i			0.437362i
$254$		0			0
$255$	1.85564	−	2.57052i	0.116205	−	0.160972i
$256$		0			0
$257$		−	11.2475i		−	0.701598i	−0.936451	−	0.350799i	$-0.885910\pi$
$257$		−	11.2475i		−	0.701598i	0.936451	−	0.350799i	$-0.114090\pi$
$258$		0			0
$259$		−	38.3337i		−	2.38194i
$260$		0			0
$261$	−16.8284	+	5.58158i	−1.04165	+	0.345491i
$262$		0			0
$263$			18.7112i			1.15378i	0.816822	+	0.576890i	$0.195734\pi$
$263$			18.7112i			1.15378i	−0.816822	+	0.576890i	$0.804266\pi$
$264$		0			0
$265$	7.26440			0.446248
$266$		0			0
$267$	−17.2323	+	23.8709i	−1.05460	+	1.46088i
$268$		0			0
$269$		−	3.89055i		−	0.237211i	−0.992941	−	0.118606i	$-0.962158\pi$
$269$		−	3.89055i		−	0.237211i	0.992941	−	0.118606i	$-0.0378424\pi$
$270$		0			0
$271$			2.89196i			0.175674i	0.996135	+	0.0878369i	$0.0279954\pi$
$271$			2.89196i			0.175674i	−0.996135	+	0.0878369i	$0.972005\pi$
$272$		0			0
$273$	0.126958	+	0.0916500i	0.00768383	+	0.00554691i
$274$		0			0
$275$	−1.04388			−0.0629486
$276$		0			0
$277$	−22.7677			−1.36798			−0.683989	−	0.729493i	$-0.739756\pi$
$277$	−22.7677			−1.36798			−0.683989	+	0.729493i	$0.739756\pi$
$278$		0			0
$279$	5.30174	−	1.75846i	0.317407	−	0.105276i
$280$		0			0
$281$	−17.9840			−1.07283			−0.536417	−	0.843953i	$-0.680223\pi$
$281$	−17.9840			−1.07283			−0.536417	+	0.843953i	$0.680223\pi$
$282$		0			0
$283$	25.4057			1.51021			0.755107	−	0.655601i	$-0.227585\pi$
$283$	25.4057			1.51021			0.755107	+	0.655601i	$0.227585\pi$
$284$		0			0
$285$	4.85470	+	3.50458i	0.287568	+	0.207593i
$286$		0			0
$287$		−	10.6020i		−	0.625814i
$288$		0			0
$289$	13.6497			0.802922
$290$		0			0
$291$	11.9056	−	16.4922i	0.697920	−	0.966791i
$292$		0			0
$293$		−	12.7453i		−	0.744591i	−0.928114	−	0.372295i	$-0.878571\pi$
$293$		−	12.7453i		−	0.744591i	0.928114	−	0.372295i	$-0.121429\pi$
$294$		0			0
$295$		−	3.59034i		−	0.209038i
$296$		0			0
$297$	−1.62890	+	5.17382i	−0.0945183	+	0.300215i
$298$		0			0
$299$	−0.166811			−0.00964692
$300$		0			0
$301$	−41.3465			−2.38318
$302$		0			0
$303$	−23.7284	−	17.1294i	−1.36316	−	0.984056i
$304$		0			0
$305$			6.07828i			0.348041i
$306$		0			0
$307$	−11.1643			−0.637180			−0.318590	−	0.947893i	$-0.603209\pi$
$307$	−11.1643			−0.637180			−0.318590	+	0.947893i	$0.603209\pi$
$308$		0			0
$309$	−2.80912	−	2.02788i	−0.159805	−	0.115362i
$310$		0			0
$311$	−23.1039			−1.31010			−0.655050	−	0.755586i	$-0.727352\pi$
$311$	−23.1039			−1.31010			−0.655050	+	0.755586i	$0.727352\pi$
$312$		0			0
$313$		−	10.0426i		−	0.567641i	−0.958877	−	0.283821i	$-0.908398\pi$
$313$		−	10.0426i		−	0.567641i	0.958877	−	0.283821i	$-0.0916020\pi$
$314$		0			0
$315$	−10.2841	+	3.41098i	−0.579441	+	0.192187i
$316$		0			0
$317$			8.75867i			0.491936i	0.969278	+	0.245968i	$0.0791058\pi$
$317$			8.75867i			0.491936i	−0.969278	+	0.245968i	$0.920894\pi$
$318$		0			0
$319$		−	6.16932i		−	0.345415i
$320$		0			0
$321$	6.89429	−	9.55029i	0.384802	−	0.533045i
$322$		0			0
$323$		−	6.32745i		−	0.352069i
$324$		0			0
$325$		−	0.0250308i		−	0.00138846i
$326$		0			0
$327$	−11.0945	+	15.3686i	−0.613525	+	0.849883i
$328$		0			0
$329$	−16.7289			−0.922293
$330$		0			0
$331$		−	0.262662i		−	0.0144372i	−0.999974	−	0.00721861i	$-0.997702\pi$
$331$		−	0.262662i		−	0.0144372i	0.999974	−	0.00721861i	$-0.00229778\pi$
$332$		0			0
$333$	10.0241	+	30.2226i	0.549319	+	1.65619i
$334$		0			0
$335$	−5.81460	+	5.76111i	−0.317686	+	0.314763i
$336$		0			0
$337$			3.41856i			0.186221i	0.995656	+	0.0931105i	$0.0296810\pi$
$337$			3.41856i			0.186221i	−0.995656	+	0.0931105i	$0.970319\pi$
$338$		0			0
$339$	−20.0948	−	14.5063i	−1.09140	−	0.787874i
$340$		0			0
$341$			1.94363i			0.105253i
$342$		0			0
$343$		−	3.45244i		−	0.186414i
$344$		0			0
$345$	6.75616	−	9.35894i	0.363739	−	0.503868i
$346$		0			0
$347$	6.86123			0.368330			0.184165	−	0.982895i	$-0.441042\pi$
$347$	6.86123			0.368330			0.184165	+	0.982895i	$0.441042\pi$
$348$		0			0
$349$	29.6280			1.58595			0.792974	−	0.609255i	$-0.208531\pi$
$349$	29.6280			1.58595			0.792974	+	0.609255i	$0.208531\pi$
$350$		0			0
$351$	−0.124061	−	0.0390586i	−0.00662187	−	0.00208480i
$352$		0			0
$353$	−13.0326			−0.693653			−0.346826	−	0.937929i	$-0.612741\pi$
$353$	−13.0326			−0.693653			−0.346826	+	0.937929i	$0.612741\pi$
$354$		0			0
$355$		−	1.63263i		−	0.0866512i
$356$		0			0
$357$	9.28384	+	6.70195i	0.491353	+	0.354705i
$358$		0			0
$359$		−	17.2316i		−	0.909447i	−0.890633	−	0.454724i	$-0.849738\pi$
$359$		−	17.2316i		−	0.909447i	0.890633	−	0.454724i	$-0.150262\pi$
$360$		0			0
$361$	−7.04994			−0.371050
$362$		0			0
$363$	13.9176	+	10.0470i	0.730485	+	0.527332i
$364$		0			0
$365$	9.89619			0.517990
$366$		0			0
$367$			13.5515i			0.707381i	0.935362	+	0.353691i	$0.115073\pi$
$367$			13.5515i			0.707381i	−0.935362	+	0.353691i	$0.884927\pi$
$368$		0			0
$369$	2.77237	+	8.35867i	0.144324	+	0.435135i
$370$		0			0
$371$			26.2365i			1.36213i
$372$		0			0
$373$		−	34.2085i		−	1.77125i	−0.464404	−	0.885624i	$-0.653731\pi$
$373$		−	34.2085i		−	1.77125i	0.464404	−	0.885624i	$-0.346269\pi$
$374$		0			0
$375$	1.40436	+	1.01380i	0.0725207	+	0.0523522i
$376$		0			0
$377$	0.147931			0.00761885
$378$		0			0
$379$			7.84963i			0.403208i	0.979467	+	0.201604i	$0.0646154\pi$
$379$			7.84963i			0.403208i	−0.979467	+	0.201604i	$0.935385\pi$
$380$		0			0
$381$	19.1397	+	13.8168i	0.980555	+	0.707856i
$382$		0			0
$383$	−9.75234			−0.498321			−0.249161	−	0.968462i	$-0.580155\pi$
$383$	−9.75234			−0.498321			−0.249161	+	0.968462i	$0.580155\pi$
$384$		0			0
$385$		−	3.77015i		−	0.192145i
$386$		0			0
$387$	32.5980	−	10.8120i	1.65705	−	0.549603i
$388$		0			0
$389$			5.21031i			0.264173i	0.991238	+	0.132086i	$0.0421677\pi$
$389$			5.21031i			0.264173i	−0.991238	+	0.132086i	$0.957832\pi$
$390$		0			0
$391$	−12.1981			−0.616885
$392$		0			0
$393$	−17.3758	+	24.0697i	−0.876491	+	1.21416i
$394$		0			0
$395$		−	5.00529i		−	0.251844i
$396$		0			0
$397$	21.0907			1.05851			0.529255	−	0.848463i	$-0.322471\pi$
$397$	21.0907			1.05851			0.529255	+	0.848463i	$0.322471\pi$
$398$		0			0
$399$	−12.6573	+	17.5335i	−0.633659	+	0.877774i
$400$		0			0
$401$	−16.1506			−0.806520			−0.403260	−	0.915085i	$-0.632123\pi$
$401$	−16.1506			−0.806520			−0.403260	+	0.915085i	$0.632123\pi$
$402$		0			0
$403$	−0.0466053			−0.00232158
$404$		0			0
$405$	7.21608	−	5.37849i	0.358570	−	0.267259i
$406$		0			0
$407$	−11.0797			−0.549199
$408$		0			0
$409$			27.6503i			1.36722i	0.729848	+	0.683609i	$0.239591\pi$
$409$			27.6503i			1.36722i	−0.729848	+	0.683609i	$0.760409\pi$
$410$		0			0
$411$	−7.43196	−	5.36509i	−0.366592	−	0.264640i
$412$		0			0
$413$	12.9671			0.638069
$414$		0			0
$415$		−	6.25585i		−	0.307087i
$416$		0			0
$417$	11.9608	−	16.5686i	0.585722	−	0.811369i
$418$		0			0
$419$		−	6.79895i		−	0.332150i	−0.986113	−	0.166075i	$-0.946891\pi$
$419$		−	6.79895i		−	0.332150i	0.986113	−	0.166075i	$-0.0531095\pi$
$420$		0			0
$421$	−4.14838			−0.202180			−0.101090	−	0.994877i	$-0.532233\pi$
$421$	−4.14838			−0.202180			−0.101090	+	0.994877i	$0.532233\pi$
$422$		0			0
$423$	13.1892	−	4.37454i	0.641280	−	0.212697i
$424$		0			0
$425$		−	1.83039i		−	0.0887869i
$426$		0			0
$427$	−21.9527			−1.06236
$428$		0			0
$429$	0.0264897	−	0.0366948i	0.00127894	−	0.00177164i
$430$		0			0
$431$			19.7388i			0.950783i	0.879774	+	0.475392i	$0.157694\pi$
$431$			19.7388i			0.950783i	−0.879774	+	0.475392i	$0.842306\pi$
$432$		0			0
$433$		−	38.3034i		−	1.84074i	−0.391046	−	0.920371i	$-0.627887\pi$
$433$		−	38.3034i		−	1.84074i	0.391046	−	0.920371i	$-0.372113\pi$
$434$		0			0
$435$	−5.99150	+	8.29970i	−0.287270	+	0.397940i
$436$		0			0
$437$		−	23.0374i		−	1.10203i
$438$		0			0
$439$	35.9242			1.71457			0.857284	−	0.514844i	$-0.172150\pi$
$439$	35.9242			1.71457			0.857284	+	0.514844i	$0.172150\pi$
$440$		0			0
$441$	−5.70825	−	17.2103i	−0.271821	−	0.819538i
$442$		0			0
$443$	28.9987			1.37777			0.688884	−	0.724872i	$-0.258101\pi$
$443$	28.9987			1.37777			0.688884	+	0.724872i	$0.258101\pi$
$444$		0			0
$445$			16.9978i			0.805771i
$446$		0			0
$447$	10.7432	−	14.8820i	0.508136	−	0.703893i
$448$		0			0
$449$		−	21.3580i		−	1.00795i	−0.863720	−	0.503973i	$-0.831871\pi$
$449$		−	21.3580i		−	1.00795i	0.863720	−	0.503973i	$-0.168129\pi$
$450$		0			0
$451$	−3.06430			−0.144292
$452$		0			0
$453$	14.7120	+	10.6205i	0.691228	+	0.498993i
$454$		0			0
$455$	0.0904028			0.00423815
$456$		0			0
$457$	−7.78473			−0.364154			−0.182077	−	0.983284i	$-0.558282\pi$
$457$	−7.78473			−0.364154			−0.182077	+	0.983284i	$0.558282\pi$
$458$		0			0
$459$	−9.07199	−	2.85618i	−0.423444	−	0.133315i
$460$		0			0
$461$		−	27.5047i		−	1.28102i	−0.767950	−	0.640510i	$-0.778723\pi$
$461$		−	27.5047i		−	1.28102i	0.767950	−	0.640510i	$-0.221277\pi$
$462$		0			0
$463$			16.0762i			0.747123i	0.927605	+	0.373561i	$0.121864\pi$
$463$			16.0762i			0.747123i	−0.927605	+	0.373561i	$0.878136\pi$
$464$		0			0
$465$	1.88760	−	2.61480i	0.0875356	−	0.121258i
$466$		0			0
$467$		−	20.1445i		−	0.932176i	−0.884738	−	0.466088i	$-0.845663\pi$
$467$		−	20.1445i		−	0.932176i	0.884738	−	0.466088i	$-0.154337\pi$
$468$		0			0
$469$	−20.8072	−	21.0004i	−0.960786	−	0.969706i
$470$		0			0
$471$	31.1298	+	22.4724i	1.43438	+	1.03547i
$472$		0			0
$473$			11.9505i			0.549483i
$474$		0			0
$475$	3.45689			0.158613
$476$		0			0
$477$	−6.86075	−	20.6851i	−0.314132	−	0.947106i
$478$		0			0
$479$		−	0.897104i		−	0.0409897i	−0.999790	−	0.0204949i	$-0.993476\pi$
$479$		−	0.897104i		−	0.0409897i	0.999790	−	0.0204949i	$-0.00652418\pi$
$480$		0			0
$481$		−	0.265674i		−	0.0121137i
$482$		0			0
$483$	33.8013	+	24.4009i	1.53801	+	1.11028i
$484$		0			0
$485$		−	11.7436i		−	0.533250i
$486$		0			0
$487$		−	3.45512i		−	0.156567i	−0.996931	−	0.0782833i	$-0.975056\pi$
$487$		−	3.45512i		−	0.156567i	0.996931	−	0.0782833i	$-0.0249439\pi$
$488$		0			0
$489$	−3.44113	−	2.48413i	−0.155613	−	0.112336i
$490$		0			0
$491$		−	0.0174004i		−	0.000785270i	−1.00000		0.000392635i	$-0.999875\pi$
$491$		−	0.0174004i		−	0.000785270i	1.00000		0.000392635i	$-0.000124980\pi$
$492$		0			0
$493$	10.8175			0.487197
$494$		0			0
$495$	0.985881	+	2.97242i	0.0443120	+	0.133600i
$496$		0			0
$497$	5.89652			0.264495
$498$		0			0
$499$			6.26689i			0.280545i	0.990113	+	0.140272i	$0.0447978\pi$
$499$			6.26689i			0.280545i	−0.990113	+	0.140272i	$0.955202\pi$
$500$		0			0
$501$	−8.93209	+	12.3731i	−0.399056	+	0.552791i
$502$		0			0
$503$	−29.3157			−1.30712			−0.653562	−	0.756873i	$-0.726726\pi$
$503$	−29.3157			−1.30712			−0.653562	+	0.756873i	$0.726726\pi$
$504$		0			0
$505$	−16.8963			−0.751874
$506$		0			0
$507$	−18.2558	−	13.1787i	−0.810767	−	0.585287i
$508$		0			0
$509$			4.08486i			0.181058i	0.995894	+	0.0905291i	$0.0288558\pi$
$509$			4.08486i			0.181058i	−0.995894	+	0.0905291i	$0.971144\pi$
$510$		0			0
$511$			35.7417i			1.58112i
$512$		0			0
$513$	5.39420	−	17.1334i	0.238160	−	0.756459i
$514$		0			0
$515$	−2.00029			−0.0881432
$516$		0			0
$517$			4.83518i			0.212651i
$518$		0			0
$519$	17.4879	−	24.2250i	0.767634	−	1.06336i
$520$		0			0
$521$	0.476289			0.0208666			0.0104333	−	0.999946i	$-0.496679\pi$
$521$	0.476289			0.0208666			0.0104333	+	0.999946i	$0.496679\pi$
$522$		0			0
$523$	−6.33618			−0.277062			−0.138531	−	0.990358i	$-0.544238\pi$
$523$	−6.33618			−0.277062			−0.138531	+	0.990358i	$0.544238\pi$
$524$		0			0
$525$	−3.66149	+	5.07206i	−0.159800	+	0.221363i
$526$		0			0
$527$	−3.40803			−0.148456
$528$		0			0
$529$	−21.4118			−0.930949
$530$		0			0
$531$	−10.2234	+	3.39085i	−0.443656	+	0.147150i
$532$		0			0
$533$		−	0.0734775i		−	0.00318266i
$534$		0			0
$535$		−	6.80047i		−	0.294010i
$536$		0			0
$537$	22.2305	+	16.0481i	0.959318	+	0.692526i
$538$		0			0
$539$	6.30932			0.271762
$540$		0			0
$541$		−	9.21149i		−	0.396033i	−0.980199	−	0.198016i	$-0.936550\pi$
$541$		−	9.21149i		−	0.396033i	0.980199	−	0.198016i	$-0.0634499\pi$
$542$		0			0
$543$	9.45231	+	6.82356i	0.405638	+	0.292827i
$544$		0			0
$545$			10.9435i			0.468767i
$546$		0			0
$547$			41.8666i			1.79008i	0.445981	+	0.895042i	$0.352855\pi$
$547$			41.8666i			1.79008i	−0.445981	+	0.895042i	$0.647145\pi$
$548$		0			0
$549$	17.3077	−	5.74054i	0.738673	−	0.245000i
$550$		0			0
$551$			20.4301i			0.870350i
$552$		0			0
$553$	18.0774			0.768730
$554$		0			0
$555$	14.9057	+	10.7603i	0.632711	+	0.456750i
$556$		0			0
$557$			45.6502i			1.93426i	0.254282	+	0.967130i	$0.418161\pi$
$557$			45.6502i			1.93426i	−0.254282	+	0.967130i	$0.581839\pi$
$558$		0			0
$559$	−0.286555			−0.0121200
$560$		0			0
$561$	1.93707	−	2.68332i	0.0817833	−	0.113290i
$562$		0			0
$563$	25.5425			1.07649			0.538243	−	0.842790i	$-0.319088\pi$
$563$	25.5425			1.07649			0.538243	+	0.842790i	$0.319088\pi$
$564$		0			0
$565$	−14.3089			−0.601980
$566$		0			0
$567$	19.4253	+	26.0620i	0.815784	+	1.09450i
$568$		0			0
$569$		−	39.7915i		−	1.66815i	−0.551652	−	0.834074i	$-0.686002\pi$
$569$		−	39.7915i		−	1.66815i	0.551652	−	0.834074i	$-0.313998\pi$
$570$		0			0
$571$	−12.3059			−0.514984			−0.257492	−	0.966280i	$-0.582896\pi$
$571$	−12.3059			−0.514984			−0.257492	+	0.966280i	$0.582896\pi$
$572$		0			0
$573$	24.5180	+	17.6994i	1.02425	+	0.739401i
$574$		0			0
$575$		−	6.66422i		−	0.277917i
$576$		0			0
$577$			41.6717i			1.73482i	0.497597	+	0.867409i	$0.334216\pi$
$577$			41.6717i			1.73482i	−0.497597	+	0.867409i	$0.665784\pi$
$578$		0			0
$579$	29.7810	+	21.4987i	1.23765	+	0.893455i
$580$		0			0
$581$	22.5940			0.937357
$582$		0			0
$583$	7.58319			0.314064
$584$		0			0
$585$	−0.0712743	+	0.0236400i	−0.00294683	+	0.000977393i
$586$		0			0
$587$	7.72689			0.318923			0.159462	−	0.987204i	$-0.449024\pi$
$587$	7.72689			0.318923			0.159462	+	0.987204i	$0.449024\pi$
$588$		0			0
$589$		−	6.43643i		−	0.265209i
$590$		0			0
$591$	31.6298	+	22.8333i	1.30108	+	0.939238i
$592$		0			0
$593$	38.8814			1.59667			0.798334	−	0.602216i	$-0.205715\pi$
$593$	38.8814			1.59667			0.798334	+	0.602216i	$0.205715\pi$
$594$		0			0
$595$	6.61074			0.271014
$596$		0			0
$597$	16.6950	+	12.0520i	0.683280	+	0.493255i
$598$		0			0
$599$	−7.01344			−0.286561			−0.143281	−	0.989682i	$-0.545765\pi$
$599$	−7.01344			−0.286561			−0.143281	+	0.989682i	$0.545765\pi$
$600$		0			0
$601$	3.81810			0.155743			0.0778717	−	0.996963i	$-0.475188\pi$
$601$	3.81810			0.155743			0.0778717	+	0.996963i	$0.475188\pi$
$602$		0			0
$603$	21.8961	+	11.1159i	0.891677	+	0.452673i
$604$		0			0
$605$	9.91031			0.402911
$606$		0			0
$607$	−27.6472			−1.12217			−0.561083	−	0.827760i	$-0.689615\pi$
$607$	−27.6472			−1.12217			−0.561083	+	0.827760i	$0.689615\pi$
$608$		0			0
$609$	−29.9757	−	21.6393i	−1.21468	−	0.876867i
$610$		0			0
$611$	−0.115940			−0.00469045
$612$		0			0
$613$	17.3173			0.699438			0.349719	−	0.936855i	$-0.386277\pi$
$613$	17.3173			0.699438			0.349719	+	0.936855i	$0.386277\pi$
$614$		0			0
$615$	4.12246	+	2.97598i	0.166234	+	0.120003i
$616$		0			0
$617$			25.0843i			1.00985i	0.863162	+	0.504927i	$0.168481\pi$
$617$			25.0843i			1.00985i	−0.863162	+	0.504927i	$0.831519\pi$
$618$		0			0
$619$	−6.83455			−0.274704			−0.137352	−	0.990522i	$-0.543859\pi$
$619$	−6.83455			−0.274704			−0.137352	+	0.990522i	$0.543859\pi$
$620$		0			0
$621$	−33.0300	−	10.3990i	−1.32545	−	0.417297i
$622$		0			0
$623$	−61.3901			−2.45954
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	5.06774	+	3.65837i	0.202386	+	0.146101i
$628$		0			0
$629$		−	19.4275i		−	0.774627i
$630$		0			0
$631$			1.82661i			0.0727160i	0.999339	+	0.0363580i	$0.0115757\pi$
$631$			1.82661i			0.0727160i	−0.999339	+	0.0363580i	$0.988424\pi$
$632$		0			0
$633$	−38.3789	−	27.7055i	−1.52543	−	1.10120i
$634$		0			0
$635$	13.6288			0.540842
$636$		0			0
$637$			0.151288i			0.00599426i
$638$		0			0
$639$	−4.64886	+	1.54192i	−0.183906	+	0.0609973i
$640$		0			0
$641$	−45.1654			−1.78393			−0.891964	−	0.452107i	$-0.850672\pi$
$641$	−45.1654			−1.78393			−0.891964	+	0.452107i	$0.850672\pi$
$642$		0			0
$643$	35.8424			1.41349			0.706743	−	0.707471i	$-0.250164\pi$
$643$	35.8424			1.41349			0.706743	+	0.707471i	$0.250164\pi$
$644$		0			0
$645$	11.6060	−	16.0772i	0.456986	−	0.633038i
$646$		0			0
$647$	−10.2163			−0.401643			−0.200822	−	0.979628i	$-0.564361\pi$
$647$	−10.2163			−0.401643			−0.200822	+	0.979628i	$0.564361\pi$
$648$		0			0
$649$		−	3.74790i		−	0.147118i
$650$		0			0
$651$	9.44375	+	6.81738i	0.370130	+	0.267194i
$652$		0			0
$653$	2.56494			0.100374			0.0501869	−	0.998740i	$-0.484018\pi$
$653$	2.56494			0.100374			0.0501869	+	0.998740i	$0.484018\pi$
$654$		0			0
$655$			17.1393i			0.669688i
$656$		0			0
$657$	−9.34631	−	28.1790i	−0.364634	−	1.09937i
$658$		0			0
$659$		−	30.3958i		−	1.18405i	−0.805919	−	0.592025i	$-0.798328\pi$
$659$		−	30.3958i		−	1.18405i	0.805919	−	0.592025i	$-0.201672\pi$
$660$		0			0
$661$		−	26.3946i		−	1.02663i	−0.858200	−	0.513315i	$-0.828417\pi$
$661$		−	26.3946i		−	1.02663i	0.858200	−	0.513315i	$-0.171583\pi$
$662$		0			0
$663$	0.0643422	+	0.0464482i	0.00249884	+	0.00180390i
$664$		0			0
$665$			12.4851i			0.484151i
$666$		0			0
$667$	39.3853			1.52501
$668$		0			0
$669$	−11.4411	−	8.25928i	−0.442340	−	0.319322i
$670$		0			0
$671$			6.34502i			0.244947i
$672$		0			0
$673$			8.38695i			0.323293i	0.986849	+	0.161647i	$0.0516805\pi$
$673$			8.38695i			0.323293i	−0.986849	+	0.161647i	$0.948320\pi$
$674$		0			0
$675$	1.56042	−	4.95632i	0.0600607	−	0.190769i
$676$		0			0
$677$	9.47437			0.364130			0.182065	−	0.983287i	$-0.441722\pi$
$677$	9.47437			0.364130			0.182065	+	0.983287i	$0.441722\pi$
$678$		0			0
$679$	42.4139			1.62770
$680$		0			0
$681$	−26.1928	+	36.2835i	−1.00371	+	1.39039i
$682$		0			0
$683$	34.4849			1.31953			0.659764	−	0.751473i	$-0.270656\pi$
$683$	34.4849			1.31953			0.659764	+	0.751473i	$0.270656\pi$
$684$		0			0
$685$	−5.29208			−0.202200
$686$		0			0
$687$	16.2451	−	22.5034i	0.619788	−	0.858559i
$688$		0			0
$689$			0.181834i			0.00692731i
$690$		0			0
$691$	31.8816			1.21283			0.606416	−	0.795148i	$-0.292607\pi$
$691$	31.8816			1.21283			0.606416	+	0.795148i	$0.292607\pi$
$692$		0			0
$693$	−10.7354	+	3.56067i	−0.407803	+	0.135258i
$694$		0			0
$695$		−	11.7980i		−	0.447524i
$696$		0			0
$697$		−	5.37307i		−	0.203520i
$698$		0			0
$699$	16.8588	+	12.1703i	0.637659	+	0.460322i
$700$		0			0
$701$	−33.1222			−1.25101			−0.625504	−	0.780221i	$-0.715107\pi$
$701$	−33.1222			−1.25101			−0.625504	+	0.780221i	$0.715107\pi$
$702$		0			0
$703$	36.6910			1.38383
$704$		0			0
$705$	4.69581	−	6.50485i	0.176854	−	0.244987i
$706$		0			0
$707$		−	61.0235i		−	2.29503i
$708$		0			0
$709$	26.1171			0.980849			0.490425	−	0.871484i	$-0.336842\pi$
$709$	26.1171			0.980849			0.490425	+	0.871484i	$0.336842\pi$
$710$		0			0
$711$	−14.2524	+	4.72718i	−0.534506	+	0.177283i
$712$		0			0
$713$	−12.4082			−0.464692
$714$		0			0
$715$		−	0.0261293i		−	0.000977179i
$716$		0			0
$717$	−10.3428	−	7.46639i	−0.386258	−	0.278837i
$718$		0			0
$719$			37.9764i			1.41628i	0.706072	+	0.708140i	$0.250465\pi$
$719$			37.9764i			1.41628i	−0.706072	+	0.708140i	$0.749535\pi$
$720$		0			0
$721$		−	7.22436i		−	0.269049i
$722$		0			0
$723$	5.15348	+	3.72026i	0.191660	+	0.138358i
$724$		0			0
$725$			5.90996i			0.219491i
$726$		0			0
$727$			35.3642i			1.31159i	0.754940	+	0.655793i	$0.227666\pi$
$727$			35.3642i			1.31159i	−0.754940	+	0.655793i	$0.772334\pi$
$728$		0			0
$729$	−22.1302	−	15.4679i	−0.819636	−	0.572885i
$730$		0			0
$731$	−20.9544			−0.775028
$732$		0			0
$733$		−	11.9426i		−	0.441112i	−0.975374	−	0.220556i	$-0.929213\pi$
$733$		−	11.9426i		−	0.441112i	0.975374	−	0.220556i	$-0.0707871\pi$
$734$		0			0
$735$	−8.48805	−	6.12747i	−0.313086	−	0.226015i
$736$		0			0
$737$	−6.06977	+	6.01393i	−0.223583	+	0.221526i
$738$		0			0
$739$		−	10.0688i		−	0.370388i	−0.982702	−	0.185194i	$-0.940709\pi$
$739$		−	10.0688i		−	0.370388i	0.982702	−	0.185194i	$-0.0592914\pi$
$740$		0			0
$741$	−0.0877224	+	0.121517i	−0.00322256	+	0.00446404i
$742$		0			0
$743$			11.8033i			0.433020i	0.976280	+	0.216510i	$0.0694675\pi$
$743$			11.8033i			0.433020i	−0.976280	+	0.216510i	$0.930533\pi$
$744$		0			0
$745$		−	10.5970i		−	0.388244i
$746$		0			0
$747$	−17.8133	+	5.90824i	−0.651754	+	0.216171i
$748$		0			0
$749$	24.5610			0.897439
$750$		0			0
$751$	8.15143			0.297450			0.148725	−	0.988879i	$-0.452483\pi$
$751$	8.15143			0.297450			0.148725	+	0.988879i	$0.452483\pi$
$752$		0			0
$753$	39.0519	+	28.1913i	1.42313	+	1.02735i
$754$		0			0
$755$	10.4759			0.381258
$756$		0			0
$757$		−	5.46805i		−	0.198740i	−0.995051	−	0.0993698i	$-0.968317\pi$
$757$		−	5.46805i		−	0.198740i	0.995051	−	0.0993698i	$-0.0316827\pi$
$758$		0			0
$759$	7.05265	−	9.76965i	0.255995	−	0.354616i
$760$		0			0
$761$		−	15.0981i		−	0.547306i	−0.961829	−	0.273653i	$-0.911768\pi$
$761$		−	15.0981i		−	0.547306i	0.961829	−	0.273653i	$-0.0882319\pi$
$762$		0			0
$763$	−39.5241			−1.43087
$764$		0			0
$765$	−5.21197	+	1.72868i	−0.188439	+	0.0625007i
$766$		0			0
$767$	0.0898692			0.00324499
$768$		0			0
$769$		−	13.2743i		−	0.478683i	−0.970935	−	0.239342i	$-0.923068\pi$
$769$		−	13.2743i		−	0.478683i	0.970935	−	0.239342i	$-0.0769315\pi$
$770$		0			0
$771$	−11.4026	+	15.7955i	−0.410656	+	0.568860i
$772$		0			0
$773$		−	10.6097i		−	0.381604i	−0.981629	−	0.190802i	$-0.938891\pi$
$773$		−	10.6097i		−	0.381604i	0.981629	−	0.190802i	$-0.0611088\pi$
$774$		0			0
$775$		−	1.86192i		−	0.0668820i
$776$		0			0
$777$	−38.8626	+	53.8342i	−1.39419	+	1.93129i
$778$		0			0
$779$	10.1476			0.363576
$780$		0			0
$781$		−	1.70428i		−	0.0609839i
$782$		0			0
$783$	29.2917	+	9.22204i	1.04680	+	0.329569i
$784$		0			0
$785$	22.1666			0.791159
$786$		0			0
$787$		−	23.9019i		−	0.852010i	−0.904721	−	0.426005i	$-0.859921\pi$
$787$		−	23.9019i		−	0.852010i	0.904721	−	0.426005i	$-0.140079\pi$
$788$		0			0
$789$	18.9693	−	26.2771i	0.675325	−	0.935491i
$790$		0			0
$791$		−	51.6788i		−	1.83749i
$792$		0			0
$793$	−0.152144			−0.00540280
$794$		0			0
$795$	−10.2018	−	7.36462i	−0.361821	−	0.261196i
$796$		0			0
$797$			5.15056i			0.182442i	0.995831	+	0.0912210i	$0.0290770\pi$
$797$			5.15056i			0.182442i	−0.995831	+	0.0912210i	$0.970923\pi$
$798$		0			0
$799$	−8.47820			−0.299937
$800$		0			0
$801$	48.4005	−	16.0533i	1.71015	−	0.567215i
$802$		0			0
$803$	10.3305			0.364555
$804$		0			0
$805$	24.0689			0.848317
$806$		0			0
$807$	−3.94423	+	5.46372i	−0.138843	+	0.192332i
$808$		0			0
$809$	45.8969			1.61365			0.806825	−	0.590791i	$-0.201184\pi$
$809$	45.8969			1.61365			0.806825	+	0.590791i	$0.201184\pi$
$810$		0			0
$811$		−	9.00698i		−	0.316278i	−0.987417	−	0.158139i	$-0.949451\pi$
$811$		−	9.00698i		−	0.316278i	0.987417	−	0.158139i	$-0.0505494\pi$
$812$		0			0
$813$	2.93185	−	4.06134i	0.102825	−	0.142437i
$814$		0			0
$815$	−2.45032			−0.0858310
$816$		0			0
$817$		−	39.5747i		−	1.38454i
$818$		0			0
$819$	−0.0853796	−	0.257418i	−0.00298340	−	0.00899493i
$820$		0			0
$821$		−	27.7219i		−	0.967501i	−0.875206	−	0.483750i	$-0.839274\pi$
$821$		−	27.7219i		−	0.967501i	0.875206	−	0.483750i	$-0.160726\pi$
$822$		0			0
$823$	1.92740			0.0671851			0.0335926	−	0.999436i	$-0.489305\pi$
$823$	1.92740			0.0671851			0.0335926	+	0.999436i	$0.489305\pi$
$824$		0			0
$825$	1.46599	+	1.05829i	0.0510391	+	0.0368448i
$826$		0			0
$827$		−	11.8711i		−	0.412799i	−0.978468	−	0.206399i	$-0.933825\pi$
$827$		−	11.8711i		−	0.412799i	0.978468	−	0.206399i	$-0.0661746\pi$
$828$		0			0
$829$	−2.12041			−0.0736450			−0.0368225	−	0.999322i	$-0.511724\pi$
$829$	−2.12041			−0.0736450			−0.0368225	+	0.999322i	$0.511724\pi$
$830$		0			0
$831$	31.9739	+	23.0818i	1.10916	+	0.800698i
$832$		0			0
$833$			11.0630i			0.383311i
$834$		0			0
$835$			8.81054i			0.304901i
$836$		0			0
$837$	−9.22825	−	2.90538i	−0.318975	−	0.100425i
$838$		0			0
$839$			1.29970i			0.0448707i	0.999748	+	0.0224353i	$0.00714199\pi$
$839$			1.29970i			0.0448707i	−0.999748	+	0.0224353i	$0.992858\pi$
$840$		0			0
$841$	−5.92768			−0.204403
$842$		0			0
$843$	25.2559	+	18.2321i	0.869860	+	0.627947i
$844$		0			0
$845$	−12.9994			−0.447192
$846$		0			0
$847$			35.7927i			1.22985i
$848$		0			0
$849$	−35.6787	−	25.7562i	−1.22449	−	0.883952i
$850$		0			0
$851$		−	70.7333i		−	2.42471i
$852$		0			0
$853$	−36.3155			−1.24342			−0.621709	−	0.783248i	$-0.713561\pi$
$853$	−36.3155			−1.24342			−0.621709	+	0.783248i	$0.713561\pi$
$854$		0			0
$855$	−3.26480	−	9.84335i	−0.111654	−	0.336636i
$856$		0			0
$857$	20.8515			0.712274			0.356137	−	0.934434i	$-0.384094\pi$
$857$	20.8515			0.712274			0.356137	+	0.934434i	$0.384094\pi$
$858$		0			0
$859$	22.1920			0.757180			0.378590	−	0.925564i	$-0.376409\pi$
$859$	22.1920			0.757180			0.378590	+	0.925564i	$0.376409\pi$
$860$		0			0
$861$	−10.7482	+	14.8889i	−0.366298	+	0.507413i
$862$		0			0
$863$		−	0.735115i		−	0.0250236i	−0.999922	−	0.0125118i	$-0.996017\pi$
$863$		−	0.735115i		−	0.0250236i	0.999922	−	0.0125118i	$-0.00398273\pi$
$864$		0			0
$865$		−	17.2499i		−	0.586515i
$866$		0			0
$867$	−19.1690	−	13.8380i	−0.651014	−	0.469963i
$868$		0			0
$869$		−	5.22495i		−	0.177244i
$870$		0			0
$871$	−0.144205	−	0.145544i	−0.00488621	−	0.00493158i
$872$		0			0
$873$	−33.4395	+	11.0911i	−1.13176	+	0.375376i
$874$		0			0
$875$			3.61166i			0.122096i
$876$		0			0
$877$	40.3920			1.36394			0.681971	−	0.731380i	$-0.261123\pi$
$877$	40.3920			1.36394			0.681971	+	0.731380i	$0.261123\pi$
$878$		0			0
$879$	−12.9212	+	17.8990i	−0.435821	+	0.603718i
$880$		0			0
$881$			5.74180i			0.193446i	0.995311	+	0.0967232i	$0.0308361\pi$
$881$			5.74180i			0.193446i	−0.995311	+	0.0967232i	$0.969164\pi$
$882$		0			0
$883$			1.87869i			0.0632231i	0.999500	+	0.0316115i	$0.0100639\pi$
$883$			1.87869i			0.0632231i	−0.999500	+	0.0316115i	$0.989936\pi$
$884$		0			0
$885$	−3.63988	+	5.04212i	−0.122353	+	0.169489i
$886$		0			0
$887$		−	50.3507i		−	1.69061i	−0.534282	−	0.845306i	$-0.679418\pi$
$887$		−	50.3507i		−	1.69061i	0.534282	−	0.845306i	$-0.320582\pi$
$888$		0			0
$889$			49.2225i			1.65087i
$890$		0			0
$891$	7.53275	−	5.61452i	0.252357	−	0.188093i
$892$		0			0
$893$		−	16.0120i		−	0.535820i
$894$		0			0
$895$	15.8297			0.529128
$896$		0			0
$897$	0.234262	+	0.169112i	0.00782178	+	0.00564649i
$898$		0			0
$899$	11.0039			0.366999
$900$		0			0
$901$			13.2967i			0.442977i
$902$		0			0
$903$	58.0653	+	41.9170i	1.93229	+	1.39491i
$904$		0			0
$905$	6.73071			0.223736
$906$		0			0
$907$	21.6579			0.719139			0.359570	−	0.933118i	$-0.382923\pi$
$907$	21.6579			0.719139			0.359570	+	0.933118i	$0.382923\pi$
$908$		0			0
$909$	15.9574	+	48.1115i	0.529274	+	1.59576i
$910$		0			0
$911$		−	44.7859i		−	1.48382i	−0.670498	−	0.741912i	$-0.733919\pi$
$911$		−	44.7859i		−	1.48382i	0.670498	−	0.741912i	$-0.266081\pi$
$912$		0			0
$913$		−	6.53038i		−	0.216124i
$914$		0			0
$915$	6.16213	−	8.53607i	0.203714	−	0.282194i
$916$		0			0
$917$	−61.9013			−2.04416
$918$		0			0
$919$		−	4.85602i		−	0.160185i	−0.996787	−	0.0800926i	$-0.974478\pi$
$919$		−	4.85602i		−	0.160185i	0.996787	−	0.0800926i	$-0.0255216\pi$
$920$		0			0
$921$	15.6787	+	11.3183i	0.516629	+	0.372951i
$922$		0			0
$923$	0.0408661			0.00134513
$924$		0			0
$925$	10.6139			0.348982
$926$		0			0
$927$	1.88914	+	5.69574i	0.0620476	+	0.187073i
$928$		0			0
$929$	0.681168			0.0223484			0.0111742	−	0.999938i	$-0.496443\pi$
$929$	0.681168			0.0223484			0.0111742	+	0.999938i	$0.496443\pi$
$930$		0			0
$931$	−20.8937			−0.684764
$932$		0			0
$933$	32.4460	+	23.4226i	1.06224	+	0.766821i
$934$		0			0
$935$		−	1.91071i		−	0.0624870i
$936$		0			0
$937$		−	41.3093i		−	1.34952i	−0.738039	−	0.674758i	$-0.764248\pi$
$937$		−	41.3093i		−	1.34952i	0.738039	−	0.674758i	$-0.235752\pi$
$938$		0			0
$939$	−10.1811	+	14.1034i	−0.332249	+	0.460247i
$940$		0			0
$941$	5.75144			0.187492			0.0937458	−	0.995596i	$-0.470116\pi$
$941$	5.75144			0.187492			0.0937458	+	0.995596i	$0.470116\pi$
$942$		0			0
$943$		−	19.5627i		−	0.637049i
$944$		0			0
$945$	17.9005	+	5.63571i	0.582304	+	0.183330i
$946$		0			0
$947$		−	21.8426i		−	0.709789i	−0.934906	−	0.354894i	$-0.884517\pi$
$947$		−	21.8426i		−	0.709789i	0.934906	−	0.354894i	$-0.115483\pi$
$948$		0			0
$949$			0.247710i			0.00804100i
$950$		0			0
$951$	8.87951	−	12.3003i	0.287938	−	0.398865i
$952$		0			0
$953$			35.9173i			1.16347i	0.813377	+	0.581737i	$0.197627\pi$
$953$			35.9173i			1.16347i	−0.813377	+	0.581737i	$0.802373\pi$
$954$		0			0
$955$	17.4585			0.564944
$956$		0			0
$957$	−6.25443	+	8.66392i	−0.202177	+	0.280065i
$958$		0			0
$959$		−	19.1132i		−	0.617197i
$960$		0			0
$961$	27.5333			0.888170
$962$		0			0
$963$	−19.3641	+	6.42261i	−0.623999	+	0.206966i
$964$		0			0
$965$	21.2061			0.682649
$966$		0			0
$967$	−1.68822			−0.0542894			−0.0271447	−	0.999632i	$-0.508641\pi$
$967$	−1.68822			−0.0542894			−0.0271447	+	0.999632i	$0.508641\pi$
$968$		0			0
$969$	−6.41474	+	8.88599i	−0.206071	+	0.285459i
$970$		0			0
$971$		−	51.9817i		−	1.66817i	−0.551634	−	0.834086i	$-0.685996\pi$
$971$		−	51.9817i		−	1.66817i	0.551634	−	0.834086i	$-0.314004\pi$
$972$		0			0
$973$	42.6104			1.36603
$974$		0			0
$975$	−0.0253761	+	0.0351522i	−0.000812687	+	0.00112577i
$976$		0			0
$977$		−	27.0923i		−	0.866760i	−0.901211	−	0.433380i	$-0.857321\pi$
$977$		−	27.0923i		−	0.866760i	0.901211	−	0.433380i	$-0.142679\pi$
$978$		0			0
$979$			17.7437i			0.567091i
$980$		0			0
$981$	31.1612	−	10.3354i	0.994899	−	0.329984i
$982$		0			0
$983$	44.9841			1.43477			0.717384	−	0.696678i	$-0.245339\pi$
$983$	44.9841			1.43477			0.717384	+	0.696678i	$0.245339\pi$
$984$		0			0
$985$	22.5226			0.717630
$986$		0			0
$987$	23.4933	+	16.9597i	0.747800	+	0.539832i
$988$		0			0
$989$	−76.2925			−2.42596
$990$		0			0
$991$			34.0448i			1.08147i	0.841193	+	0.540734i	$0.181854\pi$
$991$			34.0448i			1.08147i	−0.841193	+	0.540734i	$0.818146\pi$
$992$		0			0
$993$	−0.266286	+	0.368872i	−0.00845033	+	0.0117058i
$994$		0			0
$995$	11.8880			0.376875
$996$		0			0
$997$	33.6165			1.06464			0.532322	−	0.846542i	$-0.321319\pi$
$997$	33.6165			1.06464			0.532322	+	0.846542i	$0.321319\pi$
$998$		0			0
$999$	16.5621	−	52.6058i	0.524003	−	1.66437i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.9	✓	46
3.2	odd	2		4020.2.f.b.401.37	yes	46
67.66	odd	2		4020.2.f.b.401.38	yes	46
201.200	even	2	inner	4020.2.f.a.401.10	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.9	✓	46	1.1	even	1	trivial
4020.2.f.a.401.10	yes	46	201.200	even	2	inner
4020.2.f.b.401.37	yes	46	3.2	odd	2
4020.2.f.b.401.38	yes	46	67.66	odd	2

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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.40436 - 1.01380i) q^{3} -1.00000 q^{5} -3.61166i q^{7} +(0.944435 + 2.84746i) q^{9} +O(q^{10})\)	\(q+(-1.40436 - 1.01380i) q^{3} -1.00000 q^{5} -3.61166i q^{7} +(0.944435 + 2.84746i) q^{9} -1.04388 q^{11} -0.0250308i q^{13} +(1.40436 + 1.01380i) q^{15} -1.83039i q^{17} +3.45689 q^{19} +(-3.66149 + 5.07206i) q^{21} -6.66422i q^{23} +1.00000 q^{25} +(1.56042 - 4.95632i) q^{27} +5.90996i q^{29} -1.86192i q^{31} +(1.46599 + 1.05829i) q^{33} +3.61166i q^{35} +10.6139 q^{37} +(-0.0253761 + 0.0351522i) q^{39} +2.93548 q^{41} -11.4481i q^{43} +(-0.944435 - 2.84746i) q^{45} -4.63191i q^{47} -6.04409 q^{49} +(-1.85564 + 2.57052i) q^{51} -7.26440 q^{53} +1.04388 q^{55} +(-4.85470 - 3.50458i) q^{57} +3.59034i q^{59} -6.07828i q^{61} +(10.2841 - 3.41098i) q^{63} +0.0250308i q^{65} +(5.81460 - 5.76111i) q^{67} +(-6.75616 + 9.35894i) q^{69} +1.63263i q^{71} -9.89619 q^{73} +(-1.40436 - 1.01380i) q^{75} +3.77015i q^{77} +5.00529i q^{79} +(-7.21608 + 5.37849i) q^{81} +6.25585i q^{83} +1.83039i q^{85} +(5.99150 - 8.29970i) q^{87} -16.9978i q^{89} -0.0904028 q^{91} +(-1.88760 + 2.61480i) q^{93} -3.45689 q^{95} +11.7436i q^{97} +(-0.985881 - 2.97242i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q - 46 * q^5 - 4 * q^9	\( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more