LMFDB - Embedded newform 4020.2.g.c.1609.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1609,4020)]

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, 0, 1, 0]))

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

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

chi := DirichletCharacter("4020.1609");

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.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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.20
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.23543 - 0.0533841i) q^{5} +4.50874i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-2.23543 - 0.0533841i) q^{5} +4.50874i q^{7} -1.00000 q^{9} +0.971195 q^{11} -1.87337i q^{13} +(0.0533841 - 2.23543i) q^{15} +5.03013i q^{17} +1.77783 q^{19} -4.50874 q^{21} +5.44812i q^{23} +(4.99430 + 0.238673i) q^{25} -1.00000i q^{27} +3.85056 q^{29} +5.05491 q^{31} +0.971195i q^{33} +(0.240695 - 10.0790i) q^{35} +10.1759i q^{37} +1.87337 q^{39} +5.79630 q^{41} +8.65164i q^{43} +(2.23543 + 0.0533841i) q^{45} -1.27397i q^{47} -13.3288 q^{49} -5.03013 q^{51} -4.85441i q^{53} +(-2.17104 - 0.0518463i) q^{55} +1.77783i q^{57} +0.801017 q^{59} -9.97361 q^{61} -4.50874i q^{63} +(-0.100008 + 4.18779i) q^{65} +1.00000i q^{67} -5.44812 q^{69} -12.3349 q^{71} +9.73950i q^{73} +(-0.238673 + 4.99430i) q^{75} +4.37887i q^{77} +5.68322 q^{79} +1.00000 q^{81} -16.8058i q^{83} +(0.268529 - 11.2445i) q^{85} +3.85056i q^{87} -16.4007 q^{89} +8.44655 q^{91} +5.05491i q^{93} +(-3.97422 - 0.0949080i) q^{95} +11.3278i q^{97} -0.971195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) $ 38 * q - 2 * q^5 - 38 * q^9	$ 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) $ 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.23543	−	0.0533841i	−0.999715	−	0.0238741i
$5$	−2.23543	−	0.0533841i	−0.999715	−	0.0238741i
$6$		0			0
$7$			4.50874i			1.70414i	0.523424	+	0.852072i	$0.324654\pi$
$7$			4.50874i			1.70414i	−0.523424	+	0.852072i	$0.675346\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	0.971195			0.292826			0.146413	−	0.989224i	$-0.453227\pi$
$11$	0.971195			0.292826			0.146413	+	0.989224i	$0.453227\pi$
$12$		0			0
$13$		−	1.87337i		−	0.519580i	−0.965665	−	0.259790i	$-0.916347\pi$
$13$		−	1.87337i		−	0.519580i	0.965665	−	0.259790i	$-0.0836533\pi$
$14$		0			0
$15$	0.0533841	−	2.23543i	0.0137837	−	0.577186i
$16$		0			0
$17$			5.03013i			1.21999i	0.792407	+	0.609993i	$0.208828\pi$
$17$			5.03013i			1.21999i	−0.792407	+	0.609993i	$0.791172\pi$
$18$		0			0
$19$	1.77783			0.407863			0.203932	−	0.978985i	$-0.434628\pi$
$19$	1.77783			0.407863			0.203932	+	0.978985i	$0.434628\pi$
$20$		0			0
$21$	−4.50874			−0.983888
$22$		0			0
$23$			5.44812i			1.13601i	0.823024	+	0.568006i	$0.192285\pi$
$23$			5.44812i			1.13601i	−0.823024	+	0.568006i	$0.807715\pi$
$24$		0			0
$25$	4.99430	+	0.238673i	0.998860	+	0.0477345i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	3.85056			0.715031			0.357515	−	0.933907i	$-0.383624\pi$
$29$	3.85056			0.715031			0.357515	+	0.933907i	$0.383624\pi$
$30$		0			0
$31$	5.05491			0.907889			0.453945	−	0.891030i	$-0.350016\pi$
$31$	5.05491			0.907889			0.453945	+	0.891030i	$0.350016\pi$
$32$		0			0
$33$			0.971195i			0.169063i
$34$		0			0
$35$	0.240695	−	10.0790i	0.0406849	−	1.70366i
$36$		0			0
$37$			10.1759i			1.67291i	0.548036	+	0.836455i	$0.315376\pi$
$37$			10.1759i			1.67291i	−0.548036	+	0.836455i	$0.684624\pi$
$38$		0			0
$39$	1.87337			0.299980
$40$		0			0
$41$	5.79630			0.905230			0.452615	−	0.891706i	$-0.350491\pi$
$41$	5.79630			0.905230			0.452615	+	0.891706i	$0.350491\pi$
$42$		0			0
$43$			8.65164i			1.31936i	0.751546	+	0.659681i	$0.229309\pi$
$43$			8.65164i			1.31936i	−0.751546	+	0.659681i	$0.770691\pi$
$44$		0			0
$45$	2.23543	+	0.0533841i	0.333238	+	0.00795803i
$46$		0			0
$47$		−	1.27397i		−	0.185827i	−0.995674	−	0.0929137i	$-0.970382\pi$
$47$		−	1.27397i		−	0.185827i	0.995674	−	0.0929137i	$-0.0296181\pi$
$48$		0			0
$49$	−13.3288			−1.90411
$50$		0			0
$51$	−5.03013			−0.704359
$52$		0			0
$53$		−	4.85441i		−	0.666805i	−0.942785	−	0.333402i	$-0.891803\pi$
$53$		−	4.85441i		−	0.666805i	0.942785	−	0.333402i	$-0.108197\pi$
$54$		0			0
$55$	−2.17104	−	0.0518463i	−0.292743	−	0.00699096i
$56$		0			0
$57$			1.77783i			0.235480i
$58$		0			0
$59$	0.801017			0.104283			0.0521417	−	0.998640i	$-0.483395\pi$
$59$	0.801017			0.104283			0.0521417	+	0.998640i	$0.483395\pi$
$60$		0			0
$61$	−9.97361			−1.27699			−0.638495	−	0.769626i	$-0.720443\pi$
$61$	−9.97361			−1.27699			−0.638495	+	0.769626i	$0.720443\pi$
$62$		0			0
$63$		−	4.50874i		−	0.568048i
$64$		0			0
$65$	−0.100008	+	4.18779i	−0.0124045	+	0.519432i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	−5.44812			−0.655877
$70$		0			0
$71$	−12.3349			−1.46388			−0.731941	−	0.681368i	$-0.761385\pi$
$71$	−12.3349			−1.46388			−0.731941	+	0.681368i	$0.761385\pi$
$72$		0			0
$73$			9.73950i			1.13992i	0.821672	+	0.569961i	$0.193042\pi$
$73$			9.73950i			1.13992i	−0.821672	+	0.569961i	$0.806958\pi$
$74$		0			0
$75$	−0.238673	+	4.99430i	−0.0275596	+	0.576692i
$76$		0			0
$77$			4.37887i			0.499018i
$78$		0			0
$79$	5.68322			0.639413			0.319706	−	0.947517i	$-0.396416\pi$
$79$	5.68322			0.639413			0.319706	+	0.947517i	$0.396416\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	16.8058i		−	1.84467i	−0.386388	−	0.922336i	$-0.626277\pi$
$83$		−	16.8058i		−	1.84467i	0.386388	−	0.922336i	$-0.373723\pi$
$84$		0			0
$85$	0.268529	−	11.2445i	0.0291260	−	1.21964i
$86$		0			0
$87$			3.85056i			0.412823i
$88$		0			0
$89$	−16.4007			−1.73847			−0.869235	−	0.494399i	$-0.835388\pi$
$89$	−16.4007			−1.73847			−0.869235	+	0.494399i	$0.835388\pi$
$90$		0			0
$91$	8.44655			0.885439
$92$		0			0
$93$			5.05491i			0.524170i
$94$		0			0
$95$	−3.97422	−	0.0949080i	−0.407747	−	0.00973736i
$96$		0			0
$97$			11.3278i			1.15017i	0.818095	+	0.575083i	$0.195030\pi$
$97$			11.3278i			1.15017i	−0.818095	+	0.575083i	$0.804970\pi$
$98$		0			0
$99$	−0.971195			−0.0976087
$100$		0			0
$101$	8.75522			0.871177			0.435589	−	0.900146i	$-0.356540\pi$
$101$	8.75522			0.871177			0.435589	+	0.900146i	$0.356540\pi$
$102$		0			0
$103$		−	2.99346i		−	0.294954i	−0.989065	−	0.147477i	$-0.952885\pi$
$103$		−	2.99346i		−	0.294954i	0.989065	−	0.147477i	$-0.0471153\pi$
$104$		0			0
$105$	10.0790	+	0.240695i	0.983608	+	0.0234894i
$106$		0			0
$107$			1.83928i			0.177810i	0.996040	+	0.0889052i	$0.0283368\pi$
$107$			1.83928i			0.177810i	−0.996040	+	0.0889052i	$0.971663\pi$
$108$		0			0
$109$	11.2793			1.08036			0.540179	−	0.841550i	$-0.318356\pi$
$109$	11.2793			1.08036			0.540179	+	0.841550i	$0.318356\pi$
$110$		0			0
$111$	−10.1759			−0.965855
$112$		0			0
$113$		−	12.1469i		−	1.14269i	−0.820711	−	0.571344i	$-0.806422\pi$
$113$		−	12.1469i		−	1.14269i	0.820711	−	0.571344i	$-0.193578\pi$
$114$		0			0
$115$	0.290843	−	12.1789i	0.0271212	−	1.13569i
$116$		0			0
$117$			1.87337i			0.173193i
$118$		0			0
$119$	−22.6796			−2.07903
$120$		0			0
$121$	−10.0568			−0.914253
$122$		0			0
$123$			5.79630i			0.522635i
$124$		0			0
$125$	−11.1517	−	0.800152i	−0.997436	−	0.0715678i
$126$		0			0
$127$		−	21.5167i		−	1.90930i	−0.297730	−	0.954650i	$-0.596229\pi$
$127$		−	21.5167i		−	1.90930i	0.297730	−	0.954650i	$-0.403771\pi$
$128$		0			0
$129$	−8.65164			−0.761734
$130$		0			0
$131$	19.7230			1.72321			0.861603	−	0.507582i	$-0.169461\pi$
$131$	19.7230			1.72321			0.861603	+	0.507582i	$0.169461\pi$
$132$		0			0
$133$			8.01580i			0.695058i
$134$		0			0
$135$	−0.0533841	+	2.23543i	−0.00459457	+	0.192395i
$136$		0			0
$137$			17.0580i			1.45737i	0.684851	+	0.728683i	$0.259867\pi$
$137$			17.0580i			1.45737i	−0.684851	+	0.728683i	$0.740133\pi$
$138$		0			0
$139$	6.94806			0.589327			0.294663	−	0.955601i	$-0.404792\pi$
$139$	6.94806			0.589327			0.294663	+	0.955601i	$0.404792\pi$
$140$		0			0
$141$	1.27397			0.107287
$142$		0			0
$143$		−	1.81941i		−	0.152147i
$144$		0			0
$145$	−8.60766	−	0.205558i	−0.714827	−	0.0170707i
$146$		0			0
$147$		−	13.3288i		−	1.09934i
$148$		0			0
$149$	−17.4588			−1.43028			−0.715142	−	0.698979i	$-0.753638\pi$
$149$	−17.4588			−1.43028			−0.715142	+	0.698979i	$0.753638\pi$
$150$		0			0
$151$	−6.90694			−0.562079			−0.281040	−	0.959696i	$-0.590679\pi$
$151$	−6.90694			−0.562079			−0.281040	+	0.959696i	$0.590679\pi$
$152$		0			0
$153$		−	5.03013i		−	0.406662i
$154$		0			0
$155$	−11.2999	−	0.269852i	−0.907630	−	0.0216750i
$156$		0			0
$157$			9.22557i			0.736281i	0.929770	+	0.368141i	$0.120005\pi$
$157$			9.22557i			0.736281i	−0.929770	+	0.368141i	$0.879995\pi$
$158$		0			0
$159$	4.85441			0.384980
$160$		0			0
$161$	−24.5642			−1.93593
$162$		0			0
$163$		−	25.2659i		−	1.97897i	−0.144620	−	0.989487i	$-0.546196\pi$
$163$		−	25.2659i		−	1.97897i	0.144620	−	0.989487i	$-0.453804\pi$
$164$		0			0
$165$	0.0518463	−	2.17104i	0.00403623	−	0.169015i
$166$		0			0
$167$		−	0.666796i		−	0.0515982i	−0.999667	−	0.0257991i	$-0.991787\pi$
$167$		−	0.666796i		−	0.0515982i	0.999667	−	0.0257991i	$-0.00821302\pi$
$168$		0			0
$169$	9.49048			0.730037
$170$		0			0
$171$	−1.77783			−0.135954
$172$		0			0
$173$		−	16.8775i		−	1.28317i	−0.767052	−	0.641585i	$-0.778277\pi$
$173$		−	16.8775i		−	1.28317i	0.767052	−	0.641585i	$-0.221723\pi$
$174$		0			0
$175$	−1.07611	+	22.5180i	−0.0813466	+	1.70220i
$176$		0			0
$177$			0.801017i			0.0602081i
$178$		0			0
$179$	−25.1781			−1.88190			−0.940949	−	0.338549i	$-0.890064\pi$
$179$	−25.1781			−1.88190			−0.940949	+	0.338549i	$0.890064\pi$
$180$		0			0
$181$	4.21845			0.313555			0.156777	−	0.987634i	$-0.449890\pi$
$181$	4.21845			0.313555			0.156777	+	0.987634i	$0.449890\pi$
$182$		0			0
$183$		−	9.97361i		−	0.737271i
$184$		0			0
$185$	0.543232	−	22.7475i	0.0399392	−	1.67243i
$186$		0			0
$187$			4.88524i			0.357244i
$188$		0			0
$189$	4.50874			0.327963
$190$		0			0
$191$	−19.3167			−1.39771			−0.698854	−	0.715264i	$-0.746306\pi$
$191$	−19.3167			−1.39771			−0.698854	+	0.715264i	$0.746306\pi$
$192$		0			0
$193$			22.0344i			1.58607i	0.609174	+	0.793036i	$0.291501\pi$
$193$			22.0344i			1.58607i	−0.609174	+	0.793036i	$0.708499\pi$
$194$		0			0
$195$	−4.18779	−	0.100008i	−0.299894	−	0.00716174i
$196$		0			0
$197$		−	23.8537i		−	1.69951i	−0.527179	−	0.849754i	$-0.676750\pi$
$197$		−	23.8537i		−	1.69951i	0.527179	−	0.849754i	$-0.323250\pi$
$198$		0			0
$199$	−5.29391			−0.375276			−0.187638	−	0.982238i	$-0.560083\pi$
$199$	−5.29391			−0.375276			−0.187638	+	0.982238i	$0.560083\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			17.3612i			1.21852i
$204$		0			0
$205$	−12.9572	−	0.309430i	−0.904972	−	0.0216115i
$206$		0			0
$207$		−	5.44812i		−	0.378671i
$208$		0			0
$209$	1.72662			0.119433
$210$		0			0
$211$	4.36599			0.300567			0.150283	−	0.988643i	$-0.451981\pi$
$211$	4.36599			0.300567			0.150283	+	0.988643i	$0.451981\pi$
$212$		0			0
$213$		−	12.3349i		−	0.845172i
$214$		0			0
$215$	0.461859	−	19.3401i	0.0314986	−	1.31899i
$216$		0			0
$217$			22.7913i			1.54717i
$218$		0			0
$219$	−9.73950			−0.658134
$220$		0			0
$221$	9.42330			0.633880
$222$		0			0
$223$			5.00754i			0.335329i	0.985844	+	0.167665i	$0.0536226\pi$
$223$			5.00754i			0.335329i	−0.985844	+	0.167665i	$0.946377\pi$
$224$		0			0
$225$	−4.99430	−	0.238673i	−0.332953	−	0.0159115i
$226$		0			0
$227$		−	18.3845i		−	1.22022i	−0.792316	−	0.610111i	$-0.791125\pi$
$227$		−	18.3845i		−	1.22022i	0.792316	−	0.610111i	$-0.208875\pi$
$228$		0			0
$229$	−24.6317			−1.62771			−0.813855	−	0.581068i	$-0.802635\pi$
$229$	−24.6317			−1.62771			−0.813855	+	0.581068i	$0.802635\pi$
$230$		0			0
$231$	−4.37887			−0.288108
$232$		0			0
$233$			8.36272i			0.547860i	0.961750	+	0.273930i	$0.0883237\pi$
$233$			8.36272i			0.547860i	−0.961750	+	0.273930i	$0.911676\pi$
$234$		0			0
$235$	−0.0680096	+	2.84787i	−0.00443646	+	0.185774i
$236$		0			0
$237$			5.68322i			0.369165i
$238$		0			0
$239$	−22.6245			−1.46346			−0.731730	−	0.681595i	$-0.761287\pi$
$239$	−22.6245			−1.46346			−0.731730	+	0.681595i	$0.761287\pi$
$240$		0			0
$241$	16.5883			1.06854			0.534272	−	0.845313i	$-0.320586\pi$
$241$	16.5883			1.06854			0.534272	+	0.845313i	$0.320586\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	29.7955	+	0.711543i	1.90357	+	0.0454588i
$246$		0			0
$247$		−	3.33054i		−	0.211917i
$248$		0			0
$249$	16.8058			1.06502
$250$		0			0
$251$	30.0957			1.89963			0.949813	−	0.312819i	$-0.101273\pi$
$251$	30.0957			1.89963			0.949813	+	0.312819i	$0.101273\pi$
$252$		0			0
$253$			5.29119i			0.332654i
$254$		0			0
$255$	11.2445	+	0.268529i	0.704158	+	0.0168159i
$256$		0			0
$257$		−	11.4033i		−	0.711319i	−0.934616	−	0.355659i	$-0.884256\pi$
$257$		−	11.4033i		−	0.711319i	0.934616	−	0.355659i	$-0.115744\pi$
$258$		0			0
$259$	−45.8806			−2.85088
$260$		0			0
$261$	−3.85056			−0.238344
$262$		0			0
$263$			20.9006i			1.28879i	0.764695	+	0.644393i	$0.222890\pi$
$263$			20.9006i			1.28879i	−0.764695	+	0.644393i	$0.777110\pi$
$264$		0			0
$265$	−0.259148	+	10.8517i	−0.0159194	+	0.666615i
$266$		0			0
$267$		−	16.4007i		−	1.00371i
$268$		0			0
$269$	−6.93394			−0.422770			−0.211385	−	0.977403i	$-0.567797\pi$
$269$	−6.93394			−0.422770			−0.211385	+	0.977403i	$0.567797\pi$
$270$		0			0
$271$	−2.74661			−0.166845			−0.0834223	−	0.996514i	$-0.526585\pi$
$271$	−2.74661			−0.166845			−0.0834223	+	0.996514i	$0.526585\pi$
$272$		0			0
$273$			8.44655i			0.511209i
$274$		0			0
$275$	4.85044	+	0.231798i	0.292492	+	0.0139779i
$276$		0			0
$277$			25.9747i			1.56067i	0.625362	+	0.780334i	$0.284951\pi$
$277$			25.9747i			1.56067i	−0.625362	+	0.780334i	$0.715049\pi$
$278$		0			0
$279$	−5.05491			−0.302630
$280$		0			0
$281$	26.2101			1.56356			0.781782	−	0.623552i	$-0.214311\pi$
$281$	26.2101			1.56356			0.781782	+	0.623552i	$0.214311\pi$
$282$		0			0
$283$			23.3253i			1.38654i	0.720677	+	0.693271i	$0.243831\pi$
$283$			23.3253i			1.38654i	−0.720677	+	0.693271i	$0.756169\pi$
$284$		0			0
$285$	0.0949080	−	3.97422i	0.00562186	−	0.235413i
$286$		0			0
$287$			26.1340i			1.54264i
$288$		0			0
$289$	−8.30221			−0.488365
$290$		0			0
$291$	−11.3278			−0.664049
$292$		0			0
$293$			17.1915i			1.00434i	0.864770	+	0.502168i	$0.167464\pi$
$293$			17.1915i			1.00434i	−0.864770	+	0.502168i	$0.832536\pi$
$294$		0			0
$295$	−1.79062	−	0.0427615i	−0.104254	−	0.00248967i
$296$		0			0
$297$		−	0.971195i		−	0.0563544i
$298$		0			0
$299$	10.2064			0.590249
$300$		0			0
$301$	−39.0080			−2.24838
$302$		0			0
$303$			8.75522i			0.502974i
$304$		0			0
$305$	22.2953	+	0.532432i	1.27663	+	0.0304870i
$306$		0			0
$307$			20.2599i			1.15629i	0.815934	+	0.578145i	$0.196223\pi$
$307$			20.2599i			1.15629i	−0.815934	+	0.578145i	$0.803777\pi$
$308$		0			0
$309$	2.99346			0.170292
$310$		0			0
$311$	−11.1037			−0.629632			−0.314816	−	0.949153i	$-0.601943\pi$
$311$	−11.1037			−0.629632			−0.314816	+	0.949153i	$0.601943\pi$
$312$		0			0
$313$		−	1.11630i		−	0.0630967i	−0.999502	−	0.0315484i	$-0.989956\pi$
$313$		−	1.11630i		−	0.0630967i	0.999502	−	0.0315484i	$-0.0100438\pi$
$314$		0			0
$315$	−0.240695	+	10.0790i	−0.0135616	+	0.567886i
$316$		0			0
$317$		−	2.89619i		−	0.162666i	−0.996687	−	0.0813331i	$-0.974082\pi$
$317$		−	2.89619i		−	0.162666i	0.996687	−	0.0813331i	$-0.0259178\pi$
$318$		0			0
$319$	3.73964			0.209380
$320$		0			0
$321$	−1.83928			−0.102659
$322$		0			0
$323$			8.94274i			0.497587i
$324$		0			0
$325$	0.447123	−	9.35618i	0.0248019	−	0.518988i
$326$		0			0
$327$			11.2793i			0.623745i
$328$		0			0
$329$	5.74399			0.316677
$330$		0			0
$331$	15.0647			0.828028			0.414014	−	0.910270i	$-0.364126\pi$
$331$	15.0647			0.828028			0.414014	+	0.910270i	$0.364126\pi$
$332$		0			0
$333$		−	10.1759i		−	0.557637i
$334$		0			0
$335$	0.0533841	−	2.23543i	0.00291668	−	0.122135i
$336$		0			0
$337$		−	17.1955i		−	0.936697i	−0.883544	−	0.468348i	$-0.844849\pi$
$337$		−	17.1955i		−	0.936697i	0.883544	−	0.468348i	$-0.155151\pi$
$338$		0			0
$339$	12.1469			0.659731
$340$		0			0
$341$	4.90930			0.265854
$342$		0			0
$343$		−	28.5348i		−	1.54073i
$344$		0			0
$345$	12.1789	+	0.290843i	0.655690	+	0.0156585i
$346$		0			0
$347$		−	4.96587i		−	0.266582i	−0.991077	−	0.133291i	$-0.957446\pi$
$347$		−	4.96587i		−	0.266582i	0.991077	−	0.133291i	$-0.0425544\pi$
$348$		0			0
$349$	−14.5284			−0.777689			−0.388844	−	0.921304i	$-0.627126\pi$
$349$	−14.5284			−0.777689			−0.388844	+	0.921304i	$0.627126\pi$
$350$		0			0
$351$	−1.87337			−0.0999932
$352$		0			0
$353$		−	11.2401i		−	0.598251i	−0.954214	−	0.299125i	$-0.903305\pi$
$353$		−	11.2401i		−	0.598251i	0.954214	−	0.299125i	$-0.0966948\pi$
$354$		0			0
$355$	27.5738	+	0.658486i	1.46346	+	0.0349488i
$356$		0			0
$357$		−	22.6796i		−	1.20033i
$358$		0			0
$359$	6.47309			0.341636			0.170818	−	0.985303i	$-0.445359\pi$
$359$	6.47309			0.341636			0.170818	+	0.985303i	$0.445359\pi$
$360$		0			0
$361$	−15.8393			−0.833648
$362$		0			0
$363$		−	10.0568i		−	0.527844i
$364$		0			0
$365$	0.519934	−	21.7720i	0.0272146	−	1.13960i
$366$		0			0
$367$			4.66944i			0.243743i	0.992546	+	0.121871i	$0.0388896\pi$
$367$			4.66944i			0.243743i	−0.992546	+	0.121871i	$0.961110\pi$
$368$		0			0
$369$	−5.79630			−0.301743
$370$		0			0
$371$	21.8873			1.13633
$372$		0			0
$373$			33.8473i			1.75255i	0.481815	+	0.876273i	$0.339978\pi$
$373$			33.8473i			1.75255i	−0.481815	+	0.876273i	$0.660022\pi$
$374$		0			0
$375$	0.800152	−	11.1517i	0.0413197	−	0.575870i
$376$		0			0
$377$		−	7.21353i		−	0.371516i
$378$		0			0
$379$	−26.6749			−1.37020			−0.685098	−	0.728450i	$-0.740241\pi$
$379$	−26.6749			−1.37020			−0.685098	+	0.728450i	$0.740241\pi$
$380$		0			0
$381$	21.5167			1.10234
$382$		0			0
$383$		−	4.72526i		−	0.241449i	−0.992686	−	0.120725i	$-0.961478\pi$
$383$		−	4.72526i		−	0.241449i	0.992686	−	0.120725i	$-0.0385218\pi$
$384$		0			0
$385$	0.233762	−	9.78865i	0.0119136	−	0.498876i
$386$		0			0
$387$		−	8.65164i		−	0.439787i
$388$		0			0
$389$	15.8613			0.804198			0.402099	−	0.915596i	$-0.368281\pi$
$389$	15.8613			0.804198			0.402099	+	0.915596i	$0.368281\pi$
$390$		0			0
$391$	−27.4048			−1.38592
$392$		0			0
$393$			19.7230i			0.994894i
$394$		0			0
$395$	−12.7044	−	0.303393i	−0.639230	−	0.0152654i
$396$		0			0
$397$		−	17.9367i		−	0.900218i	−0.892974	−	0.450109i	$-0.851385\pi$
$397$		−	17.9367i		−	0.900218i	0.892974	−	0.450109i	$-0.148615\pi$
$398$		0			0
$399$	−8.01580			−0.401292
$400$		0			0
$401$	7.83427			0.391225			0.195612	−	0.980681i	$-0.437331\pi$
$401$	7.83427			0.391225			0.195612	+	0.980681i	$0.437331\pi$
$402$		0			0
$403$		−	9.46973i		−	0.471721i
$404$		0			0
$405$	−2.23543	−	0.0533841i	−0.111079	−	0.00265268i
$406$		0			0
$407$			9.88279i			0.489872i
$408$		0			0
$409$	24.7241			1.22253			0.611265	−	0.791426i	$-0.290661\pi$
$409$	24.7241			1.22253			0.611265	+	0.791426i	$0.290661\pi$
$410$		0			0
$411$	−17.0580			−0.841411
$412$		0			0
$413$			3.61158i			0.177714i
$414$		0			0
$415$	−0.897160	+	37.5681i	−0.0440399	+	1.84415i
$416$		0			0
$417$			6.94806i			0.340248i
$418$		0			0
$419$	−13.4506			−0.657105			−0.328553	−	0.944486i	$-0.606561\pi$
$419$	−13.4506			−0.657105			−0.328553	+	0.944486i	$0.606561\pi$
$420$		0			0
$421$	−10.7666			−0.524732			−0.262366	−	0.964968i	$-0.584503\pi$
$421$	−10.7666			−0.524732			−0.262366	+	0.964968i	$0.584503\pi$
$422$		0			0
$423$			1.27397i			0.0619424i
$424$		0			0
$425$	−1.20055	+	25.1220i	−0.0582355	+	1.21859i
$426$		0			0
$427$		−	44.9685i		−	2.17618i
$428$		0			0
$429$	1.81941			0.0878419
$430$		0			0
$431$	−11.6621			−0.561746			−0.280873	−	0.959745i	$-0.590624\pi$
$431$	−11.6621			−0.561746			−0.280873	+	0.959745i	$0.590624\pi$
$432$		0			0
$433$			5.61780i			0.269974i	0.990847	+	0.134987i	$0.0430993\pi$
$433$			5.61780i			0.269974i	−0.990847	+	0.134987i	$0.956901\pi$
$434$		0			0
$435$	0.205558	−	8.60766i	0.00985578	−	0.412706i
$436$		0			0
$437$			9.68586i			0.463338i
$438$		0			0
$439$	4.31288			0.205842			0.102921	−	0.994690i	$-0.467181\pi$
$439$	4.31288			0.205842			0.102921	+	0.994690i	$0.467181\pi$
$440$		0			0
$441$	13.3288			0.634703
$442$		0			0
$443$		−	5.40726i		−	0.256907i	−0.991716	−	0.128453i	$-0.958999\pi$
$443$		−	5.40726i		−	0.256907i	0.991716	−	0.128453i	$-0.0410012\pi$
$444$		0			0
$445$	36.6626	+	0.875535i	1.73797	+	0.0415044i
$446$		0			0
$447$		−	17.4588i		−	0.825775i
$448$		0			0
$449$	−0.546480			−0.0257900			−0.0128950	−	0.999917i	$-0.504105\pi$
$449$	−0.546480			−0.0257900			−0.0128950	+	0.999917i	$0.504105\pi$
$450$		0			0
$451$	5.62933			0.265075
$452$		0			0
$453$		−	6.90694i		−	0.324517i
$454$		0			0
$455$	−18.8817	−	0.450911i	−0.885187	−	0.0211390i
$456$		0			0
$457$		−	16.7141i		−	0.781852i	−0.920422	−	0.390926i	$-0.872155\pi$
$457$		−	16.7141i		−	0.781852i	0.920422	−	0.390926i	$-0.127845\pi$
$458$		0			0
$459$	5.03013			0.234786
$460$		0			0
$461$	21.0447			0.980151			0.490075	−	0.871680i	$-0.336969\pi$
$461$	21.0447			0.980151			0.490075	+	0.871680i	$0.336969\pi$
$462$		0			0
$463$			30.1870i			1.40291i	0.712714	+	0.701455i	$0.247466\pi$
$463$			30.1870i			1.40291i	−0.712714	+	0.701455i	$0.752534\pi$
$464$		0			0
$465$	0.269852	−	11.2999i	0.0125141	−	0.524021i
$466$		0			0
$467$		−	8.97008i		−	0.415086i	−0.978226	−	0.207543i	$-0.933453\pi$
$467$		−	8.97008i		−	0.415086i	0.978226	−	0.207543i	$-0.0665467\pi$
$468$		0			0
$469$	−4.50874			−0.208194
$470$		0			0
$471$	−9.22557			−0.425092
$472$		0			0
$473$			8.40242i			0.386344i
$474$		0			0
$475$	8.87904	+	0.424321i	0.407398	+	0.0194692i
$476$		0			0
$477$			4.85441i			0.222268i
$478$		0			0
$479$	21.7966			0.995910			0.497955	−	0.867203i	$-0.334084\pi$
$479$	21.7966			0.995910			0.497955	+	0.867203i	$0.334084\pi$
$480$		0			0
$481$	19.0633			0.869210
$482$		0			0
$483$		−	24.5642i		−	1.11771i
$484$		0			0
$485$	0.604725	−	25.3226i	0.0274591	−	1.14984i
$486$		0			0
$487$		−	15.5188i		−	0.703222i	−0.936146	−	0.351611i	$-0.885634\pi$
$487$		−	15.5188i		−	0.703222i	0.936146	−	0.351611i	$-0.114366\pi$
$488$		0			0
$489$	25.2659			1.14256
$490$		0			0
$491$	−13.9191			−0.628160			−0.314080	−	0.949397i	$-0.601696\pi$
$491$	−13.9191			−0.628160			−0.314080	+	0.949397i	$0.601696\pi$
$492$		0			0
$493$			19.3688i			0.872327i
$494$		0			0
$495$	2.17104	+	0.0518463i	0.0975809	+	0.00233032i
$496$		0			0
$497$		−	55.6148i		−	2.49467i
$498$		0			0
$499$	10.0156			0.448359			0.224179	−	0.974548i	$-0.428030\pi$
$499$	10.0156			0.448359			0.224179	+	0.974548i	$0.428030\pi$
$500$		0			0
$501$	0.666796			0.0297902
$502$		0			0
$503$			1.95986i			0.0873860i	0.999045	+	0.0436930i	$0.0139123\pi$
$503$			1.95986i			0.0873860i	−0.999045	+	0.0436930i	$0.986088\pi$
$504$		0			0
$505$	−19.5717	−	0.467389i	−0.870929	−	0.0207986i
$506$		0			0
$507$			9.49048i			0.421487i
$508$		0			0
$509$	−14.3393			−0.635579			−0.317790	−	0.948161i	$-0.602941\pi$
$509$	−14.3393			−0.635579			−0.317790	+	0.948161i	$0.602941\pi$
$510$		0			0
$511$	−43.9129			−1.94259
$512$		0			0
$513$		−	1.77783i		−	0.0784933i
$514$		0			0
$515$	−0.159803	+	6.69167i	−0.00704176	+	0.294870i
$516$		0			0
$517$		−	1.23727i		−	0.0544151i
$518$		0			0
$519$	16.8775			0.740839
$520$		0			0
$521$	−22.8190			−0.999717			−0.499858	−	0.866107i	$-0.666615\pi$
$521$	−22.8190			−0.999717			−0.499858	+	0.866107i	$0.666615\pi$
$522$		0			0
$523$			2.43411i			0.106436i	0.998583	+	0.0532182i	$0.0169479\pi$
$523$			2.43411i			0.106436i	−0.998583	+	0.0532182i	$0.983052\pi$
$524$		0			0
$525$	−22.5180	−	1.07611i	−0.982767	−	0.0469655i
$526$		0			0
$527$			25.4269i			1.10761i
$528$		0			0
$529$	−6.68206			−0.290524
$530$		0			0
$531$	−0.801017			−0.0347612
$532$		0			0
$533$		−	10.8586i		−	0.470339i
$534$		0			0
$535$	0.0981885	−	4.11159i	0.00424506	−	0.177760i
$536$		0			0
$537$		−	25.1781i		−	1.08651i
$538$		0			0
$539$	−12.9448			−0.557573
$540$		0			0
$541$	11.0605			0.475528			0.237764	−	0.971323i	$-0.423585\pi$
$541$	11.0605			0.475528			0.237764	+	0.971323i	$0.423585\pi$
$542$		0			0
$543$			4.21845i			0.181031i
$544$		0			0
$545$	−25.2140	−	0.602133i	−1.08005	−	0.0257926i
$546$		0			0
$547$			6.45735i			0.276096i	0.990426	+	0.138048i	$0.0440829\pi$
$547$			6.45735i			0.276096i	−0.990426	+	0.138048i	$0.955917\pi$
$548$		0			0
$549$	9.97361			0.425663
$550$		0			0
$551$	6.84566			0.291635
$552$		0			0
$553$			25.6242i			1.08965i
$554$		0			0
$555$	22.7475	+	0.543232i	0.965580	+	0.0230589i
$556$		0			0
$557$		−	6.22074i		−	0.263581i	−0.991278	−	0.131791i	$-0.957927\pi$
$557$		−	6.22074i		−	0.263581i	0.991278	−	0.131791i	$-0.0420727\pi$
$558$		0			0
$559$	16.2077			0.685514
$560$		0			0
$561$	−4.88524			−0.206255
$562$		0			0
$563$			0.338641i			0.0142720i	0.999975	+	0.00713601i	$0.00227148\pi$
$563$			0.338641i			0.0142720i	−0.999975	+	0.00713601i	$0.997729\pi$
$564$		0			0
$565$	−0.648453	+	27.1536i	−0.0272806	+	1.14236i
$566$		0			0
$567$			4.50874i			0.189349i
$568$		0			0
$569$	19.4703			0.816238			0.408119	−	0.912929i	$-0.366185\pi$
$569$	19.4703			0.816238			0.408119	+	0.912929i	$0.366185\pi$
$570$		0			0
$571$	42.0804			1.76101			0.880506	−	0.474036i	$-0.157203\pi$
$571$	42.0804			1.76101			0.880506	+	0.474036i	$0.157203\pi$
$572$		0			0
$573$		−	19.3167i		−	0.806967i
$574$		0			0
$575$	−1.30032	+	27.2096i	−0.0542270	+	1.13472i
$576$		0			0
$577$		−	39.7519i		−	1.65489i	−0.561545	−	0.827446i	$-0.689793\pi$
$577$		−	39.7519i		−	1.65489i	0.561545	−	0.827446i	$-0.310207\pi$
$578$		0			0
$579$	−22.0344			−0.915720
$580$		0			0
$581$	75.7729			3.14359
$582$		0			0
$583$		−	4.71458i		−	0.195258i
$584$		0			0
$585$	0.100008	−	4.18779i	0.00413483	−	0.173144i
$586$		0			0
$587$			35.0378i			1.44616i	0.690762	+	0.723082i	$0.257275\pi$
$587$			35.0378i			1.44616i	−0.690762	+	0.723082i	$0.742725\pi$
$588$		0			0
$589$	8.98680			0.370295
$590$		0			0
$591$	23.8537			0.981212
$592$		0			0
$593$			6.07946i			0.249653i	0.992179	+	0.124827i	$0.0398375\pi$
$593$			6.07946i			0.249653i	−0.992179	+	0.124827i	$0.960163\pi$
$594$		0			0
$595$	50.6986	+	1.21073i	2.07844	+	0.0496350i
$596$		0			0
$597$		−	5.29391i		−	0.216665i
$598$		0			0
$599$	29.3951			1.20105			0.600526	−	0.799605i	$-0.294958\pi$
$599$	29.3951			1.20105			0.600526	+	0.799605i	$0.294958\pi$
$600$		0			0
$601$	36.7322			1.49834			0.749168	−	0.662380i	$-0.230453\pi$
$601$	36.7322			1.49834			0.749168	+	0.662380i	$0.230453\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	22.4812	+	0.536872i	0.913992	+	0.0218269i
$606$		0			0
$607$			19.9776i			0.810868i	0.914124	+	0.405434i	$0.132880\pi$
$607$			19.9776i			0.810868i	−0.914124	+	0.405434i	$0.867120\pi$
$608$		0			0
$609$	−17.3612			−0.703511
$610$		0			0
$611$	−2.38662			−0.0965521
$612$		0			0
$613$		−	42.8707i		−	1.73153i	−0.500449	−	0.865766i	$-0.666832\pi$
$613$		−	42.8707i		−	1.73153i	0.500449	−	0.865766i	$-0.333168\pi$
$614$		0			0
$615$	0.309430	−	12.9572i	0.0124774	−	0.522486i
$616$		0			0
$617$		−	22.0503i		−	0.887713i	−0.896098	−	0.443856i	$-0.853610\pi$
$617$		−	22.0503i		−	0.887713i	0.896098	−	0.443856i	$-0.146390\pi$
$618$		0			0
$619$	−25.2362			−1.01433			−0.507164	−	0.861850i	$-0.669306\pi$
$619$	−25.2362			−1.01433			−0.507164	+	0.861850i	$0.669306\pi$
$620$		0			0
$621$	5.44812			0.218626
$622$		0			0
$623$		−	73.9465i		−	2.96260i
$624$		0			0
$625$	24.8861	+	2.38401i	0.995443	+	0.0953603i
$626$		0			0
$627$			1.72662i			0.0689547i
$628$		0			0
$629$	−51.1862			−2.04093
$630$		0			0
$631$	−45.3673			−1.80604			−0.903021	−	0.429596i	$-0.858656\pi$
$631$	−45.3673			−1.80604			−0.903021	+	0.429596i	$0.858656\pi$
$632$		0			0
$633$			4.36599i			0.173532i
$634$		0			0
$635$	−1.14865	+	48.0991i	−0.0455828	+	1.90876i
$636$		0			0
$637$			24.9697i			0.989337i
$638$		0			0
$639$	12.3349			0.487960
$640$		0			0
$641$	2.22317			0.0878099			0.0439050	−	0.999036i	$-0.486020\pi$
$641$	2.22317			0.0878099			0.0439050	+	0.999036i	$0.486020\pi$
$642$		0			0
$643$			11.2031i			0.441806i	0.975296	+	0.220903i	$0.0709005\pi$
$643$			11.2031i			0.441806i	−0.975296	+	0.220903i	$0.929100\pi$
$644$		0			0
$645$	19.3401	+	0.461859i	0.761517	+	0.0181857i
$646$		0			0
$647$			10.0190i			0.393889i	0.980415	+	0.196944i	$0.0631019\pi$
$647$			10.0190i			0.393889i	−0.980415	+	0.196944i	$0.936898\pi$
$648$		0			0
$649$	0.777943			0.0305369
$650$		0			0
$651$	−22.7913			−0.893262
$652$		0			0
$653$			36.4424i			1.42610i	0.701112	+	0.713051i	$0.252687\pi$
$653$			36.4424i			1.42610i	−0.701112	+	0.713051i	$0.747313\pi$
$654$		0			0
$655$	−44.0894	−	1.05289i	−1.72272	−	0.0411400i
$656$		0			0
$657$		−	9.73950i		−	0.379974i
$658$		0			0
$659$	8.19584			0.319264			0.159632	−	0.987177i	$-0.448969\pi$
$659$	8.19584			0.319264			0.159632	+	0.987177i	$0.448969\pi$
$660$		0			0
$661$	20.5546			0.799480			0.399740	−	0.916629i	$-0.369100\pi$
$661$	20.5546			0.799480			0.399740	+	0.916629i	$0.369100\pi$
$662$		0			0
$663$			9.42330i			0.365971i
$664$		0			0
$665$	0.427916	−	17.9188i	0.0165939	−	0.694860i
$666$		0			0
$667$			20.9783i			0.812284i
$668$		0			0
$669$	−5.00754			−0.193603
$670$		0			0
$671$	−9.68632			−0.373936
$672$		0			0
$673$		−	0.667917i		−	0.0257463i	−0.999917	−	0.0128731i	$-0.995902\pi$
$673$		−	0.667917i		−	0.0257463i	0.999917	−	0.0128731i	$-0.00409776\pi$
$674$		0			0
$675$	0.238673	−	4.99430i	0.00918652	−	0.192231i
$676$		0			0
$677$		−	11.6730i		−	0.448628i	−0.974517	−	0.224314i	$-0.927986\pi$
$677$		−	11.6730i		−	0.448628i	0.974517	−	0.224314i	$-0.0720141\pi$
$678$		0			0
$679$	−51.0742			−1.96005
$680$		0			0
$681$	18.3845			0.704496
$682$		0			0
$683$			4.30766i			0.164828i	0.996598	+	0.0824141i	$0.0262630\pi$
$683$			4.30766i			0.164828i	−0.996598	+	0.0824141i	$0.973737\pi$
$684$		0			0
$685$	0.910627	−	38.1321i	0.0347933	−	1.45695i
$686$		0			0
$687$		−	24.6317i		−	0.939759i
$688$		0			0
$689$	−9.09412			−0.346458
$690$		0			0
$691$	3.59363			0.136708			0.0683541	−	0.997661i	$-0.478225\pi$
$691$	3.59363			0.136708			0.0683541	+	0.997661i	$0.478225\pi$
$692$		0			0
$693$		−	4.37887i		−	0.166339i
$694$		0			0
$695$	−15.5319	−	0.370916i	−0.589159	−	0.0140696i
$696$		0			0
$697$			29.1561i			1.10437i
$698$		0			0
$699$	−8.36272			−0.316307
$700$		0			0
$701$	3.93948			0.148792			0.0743961	−	0.997229i	$-0.476297\pi$
$701$	3.93948			0.148792			0.0743961	+	0.997229i	$0.476297\pi$
$702$		0			0
$703$			18.0911i			0.682318i
$704$		0			0
$705$	−2.84787	−	0.0680096i	−0.107257	−	0.00256139i
$706$		0			0
$707$			39.4750i			1.48461i
$708$		0			0
$709$	12.7293			0.478057			0.239029	−	0.971013i	$-0.423171\pi$
$709$	12.7293			0.478057			0.239029	+	0.971013i	$0.423171\pi$
$710$		0			0
$711$	−5.68322			−0.213138
$712$		0			0
$713$			27.5398i			1.03137i
$714$		0			0
$715$	−0.0971274	+	4.06716i	−0.00363236	+	0.152103i
$716$		0			0
$717$		−	22.6245i		−	0.844929i
$718$		0			0
$719$	−42.9125			−1.60037			−0.800184	−	0.599755i	$-0.795265\pi$
$719$	−42.9125			−1.60037			−0.800184	+	0.599755i	$0.795265\pi$
$720$		0			0
$721$	13.4967			0.502645
$722$		0			0
$723$			16.5883i			0.616924i
$724$		0			0
$725$	19.2308	+	0.919023i	0.714216	+	0.0341317i
$726$		0			0
$727$			21.4563i			0.795769i	0.917435	+	0.397885i	$0.130256\pi$
$727$			21.4563i			0.795769i	−0.917435	+	0.397885i	$0.869744\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−43.5189			−1.60960
$732$		0			0
$733$			28.7221i			1.06087i	0.847724	+	0.530437i	$0.177972\pi$
$733$			28.7221i			1.06087i	−0.847724	+	0.530437i	$0.822028\pi$
$734$		0			0
$735$	−0.711543	+	29.7955i	−0.0262457	+	1.09902i
$736$		0			0
$737$			0.971195i			0.0357744i
$738$		0			0
$739$	−9.08106			−0.334052			−0.167026	−	0.985952i	$-0.553416\pi$
$739$	−9.08106			−0.334052			−0.167026	+	0.985952i	$0.553416\pi$
$740$		0			0
$741$	3.33054			0.122351
$742$		0			0
$743$		−	31.5678i		−	1.15811i	−0.815288	−	0.579055i	$-0.803421\pi$
$743$		−	31.5678i		−	1.15811i	0.815288	−	0.579055i	$-0.196579\pi$
$744$		0			0
$745$	39.0280	+	0.932024i	1.42988	+	0.0341467i
$746$		0			0
$747$			16.8058i			0.614891i
$748$		0			0
$749$	−8.29286			−0.303014
$750$		0			0
$751$	0.509836			0.0186042			0.00930208	−	0.999957i	$-0.497039\pi$
$751$	0.509836			0.0186042			0.00930208	+	0.999957i	$0.497039\pi$
$752$		0			0
$753$			30.0957i			1.09675i
$754$		0			0
$755$	15.4400	+	0.368721i	0.561919	+	0.0134191i
$756$		0			0
$757$			28.3209i			1.02934i	0.857388	+	0.514671i	$0.172086\pi$
$757$			28.3209i			1.02934i	−0.857388	+	0.514671i	$0.827914\pi$
$758$		0			0
$759$	−5.29119			−0.192058
$760$		0			0
$761$	49.7335			1.80284			0.901419	−	0.432947i	$-0.142526\pi$
$761$	49.7335			1.80284			0.901419	+	0.432947i	$0.142526\pi$
$762$		0			0
$763$			50.8553i			1.84109i
$764$		0			0
$765$	−0.268529	+	11.2445i	−0.00970868	+	0.406546i
$766$		0			0
$767$		−	1.50060i		−	0.0541836i
$768$		0			0
$769$	−35.9613			−1.29680			−0.648398	−	0.761301i	$-0.724561\pi$
$769$	−35.9613			−1.29680			−0.648398	+	0.761301i	$0.724561\pi$
$770$		0			0
$771$	11.4033			0.410680
$772$		0			0
$773$			17.5692i			0.631919i	0.948773	+	0.315960i	$0.102326\pi$
$773$			17.5692i			0.631919i	−0.948773	+	0.315960i	$0.897674\pi$
$774$		0			0
$775$	25.2458	+	1.20647i	0.906854	+	0.0433377i
$776$		0			0
$777$		−	45.8806i		−	1.64596i
$778$		0			0
$779$	10.3049			0.369210
$780$		0			0
$781$	−11.9796			−0.428663
$782$		0			0
$783$		−	3.85056i		−	0.137608i
$784$		0			0
$785$	0.492499	−	20.6231i	0.0175780	−	0.736071i
$786$		0			0
$787$			0.129874i			0.00462949i	0.999997	+	0.00231475i	$0.000736808\pi$
$787$			0.129874i			0.00462949i	−0.999997	+	0.00231475i	$0.999263\pi$
$788$		0			0
$789$	−20.9006			−0.744080
$790$		0			0
$791$	54.7674			1.94731
$792$		0			0
$793$			18.6843i			0.663498i
$794$		0			0
$795$	−10.8517	−	0.259148i	−0.384870	−	0.00919104i
$796$		0			0
$797$		−	3.40948i		−	0.120770i	−0.998175	−	0.0603850i	$-0.980767\pi$
$797$		−	3.40948i		−	0.120770i	0.998175	−	0.0603850i	$-0.0192328\pi$
$798$		0			0
$799$	6.40822			0.226707
$800$		0			0
$801$	16.4007			0.579490
$802$		0			0
$803$			9.45895i			0.333799i
$804$		0			0
$805$	54.9115	+	1.31134i	1.93538	+	0.0462185i
$806$		0			0
$807$		−	6.93394i		−	0.244086i
$808$		0			0
$809$	−38.2513			−1.34484			−0.672422	−	0.740168i	$-0.734746\pi$
$809$	−38.2513			−1.34484			−0.672422	+	0.740168i	$0.734746\pi$
$810$		0			0
$811$	38.2871			1.34444			0.672221	−	0.740351i	$-0.265341\pi$
$811$	38.2871			1.34444			0.672221	+	0.740351i	$0.265341\pi$
$812$		0			0
$813$		−	2.74661i		−	0.0963278i
$814$		0			0
$815$	−1.34879	+	56.4801i	−0.0472462	+	1.97841i
$816$		0			0
$817$			15.3812i			0.538119i
$818$		0			0
$819$	−8.44655			−0.295146
$820$		0			0
$821$	30.6119			1.06836			0.534182	−	0.845370i	$-0.320620\pi$
$821$	30.6119			1.06836			0.534182	+	0.845370i	$0.320620\pi$
$822$		0			0
$823$		−	38.0704i		−	1.32705i	−0.748153	−	0.663526i	$-0.769059\pi$
$823$		−	38.0704i		−	1.32705i	0.748153	−	0.663526i	$-0.230941\pi$
$824$		0			0
$825$	−0.231798	+	4.85044i	−0.00807016	+	0.168871i
$826$		0			0
$827$		−	10.4687i		−	0.364033i	−0.983295	−	0.182017i	$-0.941738\pi$
$827$		−	10.4687i		−	0.364033i	0.983295	−	0.182017i	$-0.0582625\pi$
$828$		0			0
$829$	20.1155			0.698641			0.349321	−	0.937003i	$-0.386412\pi$
$829$	20.1155			0.698641			0.349321	+	0.937003i	$0.386412\pi$
$830$		0			0
$831$	−25.9747			−0.901053
$832$		0			0
$833$		−	67.0454i		−	2.32299i
$834$		0			0
$835$	−0.0355963	+	1.49058i	−0.00123186	+	0.0515835i
$836$		0			0
$837$		−	5.05491i		−	0.174723i
$838$		0			0
$839$	37.6037			1.29822			0.649111	−	0.760694i	$-0.275141\pi$
$839$	37.6037			1.29822			0.649111	+	0.760694i	$0.275141\pi$
$840$		0			0
$841$	−14.1732			−0.488731
$842$		0			0
$843$			26.2101i			0.902724i
$844$		0			0
$845$	−21.2153	−	0.506640i	−0.729829	−	0.0174290i
$846$		0			0
$847$		−	45.3434i		−	1.55802i
$848$		0			0
$849$	−23.3253			−0.800521
$850$		0			0
$851$	−55.4396			−1.90045
$852$		0			0
$853$		−	21.2936i		−	0.729080i	−0.931188	−	0.364540i	$-0.881226\pi$
$853$		−	21.2936i		−	0.729080i	0.931188	−	0.364540i	$-0.118774\pi$
$854$		0			0
$855$	3.97422	+	0.0949080i	0.135916	+	0.00324579i
$856$		0			0
$857$			39.2788i			1.34174i	0.741575	+	0.670870i	$0.234079\pi$
$857$			39.2788i			1.34174i	−0.741575	+	0.670870i	$0.765921\pi$
$858$		0			0
$859$	29.4314			1.00419			0.502094	−	0.864813i	$-0.332563\pi$
$859$	29.4314			1.00419			0.502094	+	0.864813i	$0.332563\pi$
$860$		0			0
$861$	−26.1340			−0.890645
$862$		0			0
$863$		−	23.1452i		−	0.787870i	−0.919138	−	0.393935i	$-0.871113\pi$
$863$		−	23.1452i		−	0.787870i	0.919138	−	0.393935i	$-0.128887\pi$
$864$		0			0
$865$	−0.900988	+	37.7284i	−0.0306345	+	1.28280i
$866$		0			0
$867$		−	8.30221i		−	0.281958i
$868$		0			0
$869$	5.51952			0.187237
$870$		0			0
$871$	1.87337			0.0634768
$872$		0			0
$873$		−	11.3278i		−	0.383389i
$874$		0			0
$875$	3.60768	−	50.2800i	0.121962	−	1.69977i
$876$		0			0
$877$			35.5897i			1.20178i	0.799332	+	0.600889i	$0.205187\pi$
$877$			35.5897i			1.20178i	−0.799332	+	0.600889i	$0.794813\pi$
$878$		0			0
$879$	−17.1915			−0.579854
$880$		0			0
$881$	41.5078			1.39843			0.699216	−	0.714911i	$-0.253533\pi$
$881$	41.5078			1.39843			0.699216	+	0.714911i	$0.253533\pi$
$882$		0			0
$883$		−	11.7409i		−	0.395113i	−0.980291	−	0.197557i	$-0.936699\pi$
$883$		−	11.7409i		−	0.395113i	0.980291	−	0.197557i	$-0.0633006\pi$
$884$		0			0
$885$	0.0427615	−	1.79062i	0.00143741	−	0.0601909i
$886$		0			0
$887$		−	6.61930i		−	0.222254i	−0.993806	−	0.111127i	$-0.964554\pi$
$887$		−	6.61930i		−	0.222254i	0.993806	−	0.111127i	$-0.0354461\pi$
$888$		0			0
$889$	97.0134			3.25372
$890$		0			0
$891$	0.971195			0.0325362
$892$		0			0
$893$		−	2.26490i		−	0.0757921i
$894$		0			0
$895$	56.2838	+	1.34411i	1.88136	+	0.0449286i
$896$		0			0
$897$			10.2064i			0.340781i
$898$		0			0
$899$	19.4642			0.649169
$900$		0			0
$901$	24.4183			0.813492
$902$		0			0
$903$		−	39.0080i		−	1.29810i
$904$		0			0
$905$	−9.43004	−	0.225198i	−0.313465	−	0.00748583i
$906$		0			0
$907$		−	48.2030i		−	1.60055i	−0.599631	−	0.800276i	$-0.704686\pi$
$907$		−	48.2030i		−	1.60055i	0.599631	−	0.800276i	$-0.295314\pi$
$908$		0			0
$909$	−8.75522			−0.290392
$910$		0			0
$911$	15.0787			0.499580			0.249790	−	0.968300i	$-0.419638\pi$
$911$	15.0787			0.499580			0.249790	+	0.968300i	$0.419638\pi$
$912$		0			0
$913$		−	16.3217i		−	0.540169i
$914$		0			0
$915$	−0.532432	+	22.2953i	−0.0176017	+	0.737060i
$916$		0			0
$917$			88.9260i			2.93659i
$918$		0			0
$919$	−27.4880			−0.906747			−0.453374	−	0.891321i	$-0.649780\pi$
$919$	−27.4880			−0.906747			−0.453374	+	0.891321i	$0.649780\pi$
$920$		0			0
$921$	−20.2599			−0.667585
$922$		0			0
$923$			23.1078i			0.760603i
$924$		0			0
$925$	−2.42871	+	50.8216i	−0.0798556	+	1.67100i
$926$		0			0
$927$			2.99346i			0.0983181i
$928$		0			0
$929$	17.8975			0.587200			0.293600	−	0.955928i	$-0.405147\pi$
$929$	17.8975			0.587200			0.293600	+	0.955928i	$0.405147\pi$
$930$		0			0
$931$	−23.6963			−0.776616
$932$		0			0
$933$		−	11.1037i		−	0.363518i
$934$		0			0
$935$	0.260794	−	10.9206i	0.00852887	−	0.357142i
$936$		0			0
$937$			33.2302i			1.08558i	0.839868	+	0.542791i	$0.182632\pi$
$937$			33.2302i			1.08558i	−0.839868	+	0.542791i	$0.817368\pi$
$938$		0			0
$939$	1.11630			0.0364289
$940$		0			0
$941$	2.94439			0.0959843			0.0479921	−	0.998848i	$-0.484718\pi$
$941$	2.94439			0.0959843			0.0479921	+	0.998848i	$0.484718\pi$
$942$		0			0
$943$			31.5789i			1.02835i
$944$		0			0
$945$	−10.0790	−	0.240695i	−0.327869	−	0.00782981i
$946$		0			0
$947$			4.65790i			0.151362i	0.997132	+	0.0756808i	$0.0241130\pi$
$947$			4.65790i			0.151362i	−0.997132	+	0.0756808i	$0.975887\pi$
$948$		0			0
$949$	18.2457			0.592280
$950$		0			0
$951$	2.89619			0.0939154
$952$		0			0
$953$		−	46.5089i		−	1.50657i	−0.657694	−	0.753285i	$-0.728468\pi$
$953$		−	46.5089i		−	1.50657i	0.657694	−	0.753285i	$-0.271532\pi$
$954$		0			0
$955$	43.1812	+	1.03120i	1.39731	+	0.0333690i
$956$		0			0
$957$			3.73964i			0.120885i
$958$		0			0
$959$	−76.9103			−2.48356
$960$		0			0
$961$	−5.44785			−0.175737
$962$		0			0
$963$		−	1.83928i		−	0.0592701i
$964$		0			0
$965$	1.17629	−	49.2564i	0.0378660	−	1.58562i
$966$		0			0
$967$		−	45.2301i		−	1.45450i	−0.686371	−	0.727251i	$-0.740797\pi$
$967$		−	45.2301i		−	1.45450i	0.686371	−	0.727251i	$-0.259203\pi$
$968$		0			0
$969$	−8.94274			−0.287282
$970$		0			0
$971$	2.66050			0.0853796			0.0426898	−	0.999088i	$-0.486407\pi$
$971$	2.66050			0.0853796			0.0426898	+	0.999088i	$0.486407\pi$
$972$		0			0
$973$			31.3270i			1.00430i
$974$		0			0
$975$	9.35618	+	0.447123i	0.299638	+	0.0143194i
$976$		0			0
$977$			18.8779i			0.603959i	0.953314	+	0.301980i	$0.0976474\pi$
$977$			18.8779i			0.603959i	−0.953314	+	0.301980i	$0.902353\pi$
$978$		0			0
$979$	−15.9283			−0.509070
$980$		0			0
$981$	−11.2793			−0.360120
$982$		0			0
$983$			22.8392i			0.728458i	0.931309	+	0.364229i	$0.118667\pi$
$983$			22.8392i			0.728458i	−0.931309	+	0.364229i	$0.881333\pi$
$984$		0			0
$985$	−1.27341	+	53.3234i	−0.0405742	+	1.69902i
$986$		0			0
$987$			5.74399i			0.182833i
$988$		0			0
$989$	−47.1352			−1.49881
$990$		0			0
$991$	−12.1334			−0.385431			−0.192716	−	0.981255i	$-0.561729\pi$
$991$	−12.1334			−0.385431			−0.192716	+	0.981255i	$0.561729\pi$
$992$		0			0
$993$			15.0647i			0.478062i
$994$		0			0
$995$	11.8342	+	0.282611i	0.375169	+	0.00895936i
$996$		0			0
$997$			24.9901i			0.791443i	0.918371	+	0.395722i	$0.129505\pi$
$997$			24.9901i			0.791443i	−0.918371	+	0.395722i	$0.870495\pi$
$998$		0			0
$999$	10.1759			0.321952

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.20	yes	38
5.4	even	2	inner	4020.2.g.c.1609.1	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.1	✓	38	5.4	even	2	inner
4020.2.g.c.1609.20	yes	38	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.23543 - 0.0533841i) q^{5} +4.50874i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-2.23543 - 0.0533841i) q^{5} +4.50874i q^{7} -1.00000 q^{9} +0.971195 q^{11} -1.87337i q^{13} +(0.0533841 - 2.23543i) q^{15} +5.03013i q^{17} +1.77783 q^{19} -4.50874 q^{21} +5.44812i q^{23} +(4.99430 + 0.238673i) q^{25} -1.00000i q^{27} +3.85056 q^{29} +5.05491 q^{31} +0.971195i q^{33} +(0.240695 - 10.0790i) q^{35} +10.1759i q^{37} +1.87337 q^{39} +5.79630 q^{41} +8.65164i q^{43} +(2.23543 + 0.0533841i) q^{45} -1.27397i q^{47} -13.3288 q^{49} -5.03013 q^{51} -4.85441i q^{53} +(-2.17104 - 0.0518463i) q^{55} +1.77783i q^{57} +0.801017 q^{59} -9.97361 q^{61} -4.50874i q^{63} +(-0.100008 + 4.18779i) q^{65} +1.00000i q^{67} -5.44812 q^{69} -12.3349 q^{71} +9.73950i q^{73} +(-0.238673 + 4.99430i) q^{75} +4.37887i q^{77} +5.68322 q^{79} +1.00000 q^{81} -16.8058i q^{83} +(0.268529 - 11.2445i) q^{85} +3.85056i q^{87} -16.4007 q^{89} +8.44655 q^{91} +5.05491i q^{93} +(-3.97422 - 0.0949080i) q^{95} +11.3278i q^{97} -0.971195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 38 * q - 2 * q^5 - 38 * q^9	\( 38 q - 2 q^{5} - 38 q^{9} + 24 q^{11} + 2 q^{15} - 16 q^{19} + 12 q^{21} + 4 q^{25} - 56 q^{29} - 4 q^{31} - 2 q^{35} + 60 q^{41} + 2 q^{45} - 70 q^{49} + 12 q^{55} - 52 q^{59} + 48 q^{61} + 16 q^{65} - 12 q^{69} + 12 q^{75} - 24 q^{79} + 38 q^{81} + 16 q^{85} - 76 q^{89} + 44 q^{91} + 36 q^{95} - 24 q^{99}+O(q^{100}) \) 38 * q - 2 * q^5 - 38 * q^9 + 24 * q^11 + 2 * q^15 - 16 * q^19 + 12 * q^21 + 4 * q^25 - 56 * q^29 - 4 * q^31 - 2 * q^35 + 60 * q^41 + 2 * q^45 - 70 * q^49 + 12 * q^55 - 52 * q^59 + 48 * q^61 + 16 * q^65 - 12 * q^69 + 12 * q^75 - 24 * q^79 + 38 * q^81 + 16 * q^85 - 76 * q^89 + 44 * q^91 + 36 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more