LMFDB - Embedded newform 4020.2.g.c.1609.9

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			1609.9
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.28

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-0.599682 - 2.15415i) q^{5} +0.0778518i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-0.599682 - 2.15415i) q^{5} +0.0778518i q^{7} -1.00000 q^{9} -4.03380 q^{11} -0.176152i q^{13} +(-2.15415 + 0.599682i) q^{15} -7.72106i q^{17} -1.88421 q^{19} +0.0778518 q^{21} -2.47191i q^{23} +(-4.28076 + 2.58362i) q^{25} +1.00000i q^{27} +6.84971 q^{29} -7.45359 q^{31} +4.03380i q^{33} +(0.167705 - 0.0466864i) q^{35} -3.67413i q^{37} -0.176152 q^{39} -0.0931527 q^{41} +6.63231i q^{43} +(0.599682 + 2.15415i) q^{45} +8.43872i q^{47} +6.99394 q^{49} -7.72106 q^{51} -1.60464i q^{53} +(2.41900 + 8.68943i) q^{55} +1.88421i q^{57} -0.795026 q^{59} -5.94662 q^{61} -0.0778518i q^{63} +(-0.379458 + 0.105635i) q^{65} -1.00000i q^{67} -2.47191 q^{69} +10.2043 q^{71} -0.963933i q^{73} +(2.58362 + 4.28076i) q^{75} -0.314039i q^{77} -13.7522 q^{79} +1.00000 q^{81} +5.08460i q^{83} +(-16.6324 + 4.63019i) q^{85} -6.84971i q^{87} -12.5250 q^{89} +0.0137137 q^{91} +7.45359i q^{93} +(1.12993 + 4.05887i) q^{95} +13.8335i q^{97} +4.03380 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.599682	−	2.15415i	−0.268186	−	0.963367i
$5$	−0.599682	−	2.15415i	−0.268186	−	0.963367i
$6$		0			0
$7$			0.0778518i			0.0294252i	0.999892	+	0.0147126i	$0.00468334\pi$
$7$			0.0778518i			0.0294252i	−0.999892	+	0.0147126i	$0.995317\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.03380			−1.21624			−0.608119	−	0.793846i	$-0.708075\pi$
$11$	−4.03380			−1.21624			−0.608119	+	0.793846i	$0.708075\pi$
$12$		0			0
$13$		−	0.176152i		−	0.0488557i	−0.999702	−	0.0244278i	$-0.992224\pi$
$13$		−	0.176152i		−	0.0488557i	0.999702	−	0.0244278i	$-0.00777640\pi$
$14$		0			0
$15$	−2.15415	+	0.599682i	−0.556200	+	0.154837i
$16$		0			0
$17$		−	7.72106i		−	1.87263i	−0.351158	−	0.936316i	$-0.614212\pi$
$17$		−	7.72106i		−	1.87263i	0.351158	−	0.936316i	$-0.385788\pi$
$18$		0			0
$19$	−1.88421			−0.432267			−0.216133	−	0.976364i	$-0.569345\pi$
$19$	−1.88421			−0.432267			−0.216133	+	0.976364i	$0.569345\pi$
$20$		0			0
$21$	0.0778518			0.0169887
$22$		0			0
$23$		−	2.47191i		−	0.515429i	−0.966221	−	0.257715i	$-0.917031\pi$
$23$		−	2.47191i		−	0.515429i	0.966221	−	0.257715i	$-0.0829695\pi$
$24$		0			0
$25$	−4.28076	+	2.58362i	−0.856152	+	0.516723i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.84971			1.27196			0.635980	−	0.771706i	$-0.280596\pi$
$29$	6.84971			1.27196			0.635980	+	0.771706i	$0.280596\pi$
$30$		0			0
$31$	−7.45359			−1.33870			−0.669352	−	0.742945i	$-0.733428\pi$
$31$	−7.45359			−1.33870			−0.669352	+	0.742945i	$0.733428\pi$
$32$		0			0
$33$			4.03380i			0.702195i
$34$		0			0
$35$	0.167705	−	0.0466864i	0.0283473	−	0.00789144i
$36$		0			0
$37$		−	3.67413i		−	0.604024i	−0.953304	−	0.302012i	$-0.902342\pi$
$37$		−	3.67413i		−	0.604024i	0.953304	−	0.302012i	$-0.0976582\pi$
$38$		0			0
$39$	−0.176152			−0.0282068
$40$		0			0
$41$	−0.0931527			−0.0145480			−0.00727401	−	0.999974i	$-0.502315\pi$
$41$	−0.0931527			−0.0145480			−0.00727401	+	0.999974i	$0.502315\pi$
$42$		0			0
$43$			6.63231i			1.01142i	0.862704	+	0.505709i	$0.168769\pi$
$43$			6.63231i			1.01142i	−0.862704	+	0.505709i	$0.831231\pi$
$44$		0			0
$45$	0.599682	+	2.15415i	0.0893954	+	0.321122i
$46$		0			0
$47$			8.43872i			1.23091i	0.788170	+	0.615457i	$0.211029\pi$
$47$			8.43872i			1.23091i	−0.788170	+	0.615457i	$0.788971\pi$
$48$		0			0
$49$	6.99394			0.999134
$50$		0			0
$51$	−7.72106			−1.08117
$52$		0			0
$53$		−	1.60464i		−	0.220415i	−0.993909	−	0.110207i	$-0.964849\pi$
$53$		−	1.60464i		−	0.220415i	0.993909	−	0.110207i	$-0.0351515\pi$
$54$		0			0
$55$	2.41900	+	8.68943i	0.326178	+	1.17168i
$56$		0			0
$57$			1.88421i			0.249569i
$58$		0			0
$59$	−0.795026			−0.103504			−0.0517518	−	0.998660i	$-0.516480\pi$
$59$	−0.795026			−0.103504			−0.0517518	+	0.998660i	$0.516480\pi$
$60$		0			0
$61$	−5.94662			−0.761387			−0.380693	−	0.924701i	$-0.624315\pi$
$61$	−5.94662			−0.761387			−0.380693	+	0.924701i	$0.624315\pi$
$62$		0			0
$63$		−	0.0778518i		−	0.00980841i
$64$		0			0
$65$	−0.379458	+	0.105635i	−0.0470660	+	0.0131024i
$66$		0			0
$67$		−	1.00000i		−	0.122169i
$67$		−	1.00000i		−	0.122169i
$68$		0			0
$69$	−2.47191			−0.297583
$70$		0			0
$71$	10.2043			1.21103			0.605514	−	0.795834i	$-0.292967\pi$
$71$	10.2043			1.21103			0.605514	+	0.795834i	$0.292967\pi$
$72$		0			0
$73$		−	0.963933i		−	0.112820i	−0.998408	−	0.0564099i	$-0.982035\pi$
$73$		−	0.963933i		−	0.112820i	0.998408	−	0.0564099i	$-0.0179654\pi$
$74$		0			0
$75$	2.58362	+	4.28076i	0.298330	+	0.494300i
$76$		0			0
$77$		−	0.314039i		−	0.0357880i
$78$		0			0
$79$	−13.7522			−1.54725			−0.773623	−	0.633646i	$-0.781557\pi$
$79$	−13.7522			−1.54725			−0.773623	+	0.633646i	$0.781557\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.08460i			0.558108i	0.960275	+	0.279054i	$0.0900208\pi$
$83$			5.08460i			0.558108i	−0.960275	+	0.279054i	$0.909979\pi$
$84$		0			0
$85$	−16.6324	+	4.63019i	−1.80403	+	0.502214i
$86$		0			0
$87$		−	6.84971i		−	0.734366i
$88$		0			0
$89$	−12.5250			−1.32764			−0.663822	−	0.747891i	$-0.731067\pi$
$89$	−12.5250			−1.32764			−0.663822	+	0.747891i	$0.731067\pi$
$90$		0			0
$91$	0.0137137			0.00143759
$92$		0			0
$93$			7.45359i			0.772901i
$94$		0			0
$95$	1.12993	+	4.05887i	0.115928	+	0.416431i
$96$		0			0
$97$			13.8335i			1.40458i	0.711890	+	0.702291i	$0.247839\pi$
$97$			13.8335i			1.40458i	−0.711890	+	0.702291i	$0.752161\pi$
$98$		0			0
$99$	4.03380			0.405412
$100$		0			0
$101$	15.7346			1.56566			0.782828	−	0.622239i	$-0.213777\pi$
$101$	15.7346			1.56566			0.782828	+	0.622239i	$0.213777\pi$
$102$		0			0
$103$		−	13.3508i		−	1.31550i	−0.753237	−	0.657749i	$-0.771509\pi$
$103$		−	13.3508i		−	1.31550i	0.753237	−	0.657749i	$-0.228491\pi$
$104$		0			0
$105$	−0.0466864	−	0.167705i	−0.00455612	−	0.0163663i
$106$		0			0
$107$			4.89148i			0.472877i	0.971646	+	0.236438i	$0.0759801\pi$
$107$			4.89148i			0.472877i	−0.971646	+	0.236438i	$0.924020\pi$
$108$		0			0
$109$	−13.4979			−1.29286			−0.646430	−	0.762973i	$-0.723739\pi$
$109$	−13.4979			−1.29286			−0.646430	+	0.762973i	$0.723739\pi$
$110$		0			0
$111$	−3.67413			−0.348733
$112$		0			0
$113$			8.95878i			0.842771i	0.906882	+	0.421385i	$0.138456\pi$
$113$			8.95878i			0.842771i	−0.906882	+	0.421385i	$0.861544\pi$
$114$		0			0
$115$	−5.32488	+	1.48236i	−0.496548	+	0.138231i
$116$		0			0
$117$			0.176152i			0.0162852i
$118$		0			0
$119$	0.601099			0.0551026
$120$		0			0
$121$	5.27156			0.479233
$122$		0			0
$123$			0.0931527i			0.00839930i
$124$		0			0
$125$	8.13261	+	7.67207i	0.727403	+	0.686211i
$126$		0			0
$127$			8.12676i			0.721133i	0.932733	+	0.360567i	$0.117417\pi$
$127$			8.12676i			0.721133i	−0.932733	+	0.360567i	$0.882583\pi$
$128$		0			0
$129$	6.63231			0.583942
$130$		0			0
$131$	−4.45229			−0.388998			−0.194499	−	0.980903i	$-0.562308\pi$
$131$	−4.45229			−0.388998			−0.194499	+	0.980903i	$0.562308\pi$
$132$		0			0
$133$		−	0.146689i		−	0.0127195i
$134$		0			0
$135$	2.15415	−	0.599682i	0.185400	−	0.0516125i
$136$		0			0
$137$		−	2.00903i		−	0.171643i	−0.996311	−	0.0858217i	$-0.972648\pi$
$137$		−	2.00903i		−	0.171643i	0.996311	−	0.0858217i	$-0.0273515\pi$
$138$		0			0
$139$	−14.3601			−1.21801			−0.609003	−	0.793168i	$-0.708430\pi$
$139$	−14.3601			−1.21801			−0.609003	+	0.793168i	$0.708430\pi$
$140$		0			0
$141$	8.43872			0.710669
$142$		0			0
$143$			0.710561i			0.0594201i
$144$		0			0
$145$	−4.10765	−	14.7553i	−0.341122	−	1.22536i
$146$		0			0
$147$		−	6.99394i		−	0.576850i
$148$		0			0
$149$	−4.93926			−0.404640			−0.202320	−	0.979319i	$-0.564848\pi$
$149$	−4.93926			−0.404640			−0.202320	+	0.979319i	$0.564848\pi$
$150$		0			0
$151$	2.01310			0.163824			0.0819118	−	0.996640i	$-0.473897\pi$
$151$	2.01310			0.163824			0.0819118	+	0.996640i	$0.473897\pi$
$152$		0			0
$153$			7.72106i			0.624211i
$154$		0			0
$155$	4.46979	+	16.0562i	0.359022	+	1.28966i
$156$		0			0
$157$			9.56781i			0.763595i	0.924246	+	0.381797i	$0.124695\pi$
$157$			9.56781i			0.763595i	−0.924246	+	0.381797i	$0.875305\pi$
$158$		0			0
$159$	−1.60464			−0.127256
$160$		0			0
$161$	0.192443			0.0151666
$162$		0			0
$163$		−	14.5402i		−	1.13888i	−0.822033	−	0.569439i	$-0.807160\pi$
$163$		−	14.5402i		−	1.13888i	0.822033	−	0.569439i	$-0.192840\pi$
$164$		0			0
$165$	8.68943	−	2.41900i	0.676471	−	0.188319i
$166$		0			0
$167$			11.5213i			0.891542i	0.895147	+	0.445771i	$0.147070\pi$
$167$			11.5213i			0.891542i	−0.895147	+	0.445771i	$0.852930\pi$
$168$		0			0
$169$	12.9690			0.997613
$170$		0			0
$171$	1.88421			0.144089
$172$		0			0
$173$		−	4.32812i		−	0.329061i	−0.986372	−	0.164530i	$-0.947389\pi$
$173$		−	4.32812i		−	0.329061i	0.986372	−	0.164530i	$-0.0526109\pi$
$174$		0			0
$175$	−0.201139	−	0.333265i	−0.0152047	−	0.0251925i
$176$		0			0
$177$			0.795026i			0.0597578i
$178$		0			0
$179$	−0.803651			−0.0600677			−0.0300338	−	0.999549i	$-0.509562\pi$
$179$	−0.803651			−0.0600677			−0.0300338	+	0.999549i	$0.509562\pi$
$180$		0			0
$181$	1.58419			0.117752			0.0588760	−	0.998265i	$-0.481248\pi$
$181$	1.58419			0.117752			0.0588760	+	0.998265i	$0.481248\pi$
$182$		0			0
$183$			5.94662i			0.439587i
$184$		0			0
$185$	−7.91465	+	2.20331i	−0.581897	+	0.161991i
$186$		0			0
$187$			31.1452i			2.27757i
$188$		0			0
$189$	−0.0778518			−0.00566289
$190$		0			0
$191$	−7.17923			−0.519471			−0.259736	−	0.965680i	$-0.583635\pi$
$191$	−7.17923			−0.519471			−0.259736	+	0.965680i	$0.583635\pi$
$192$		0			0
$193$			10.4401i			0.751492i	0.926723	+	0.375746i	$0.122613\pi$
$193$			10.4401i			0.751492i	−0.926723	+	0.375746i	$0.877387\pi$
$194$		0			0
$195$	0.105635	+	0.379458i	0.00756468	+	0.0271735i
$196$		0			0
$197$		−	17.5284i		−	1.24884i	−0.781087	−	0.624422i	$-0.785335\pi$
$197$		−	17.5284i		−	1.24884i	0.781087	−	0.624422i	$-0.214665\pi$
$198$		0			0
$199$	4.68951			0.332430			0.166215	−	0.986090i	$-0.446845\pi$
$199$	4.68951			0.332430			0.166215	+	0.986090i	$0.446845\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$			0.533262i			0.0374277i
$204$		0			0
$205$	0.0558621	+	0.200665i	0.00390158	+	0.0140151i
$206$		0			0
$207$			2.47191i			0.171810i
$208$		0			0
$209$	7.60052			0.525739
$210$		0			0
$211$	24.4865			1.68572			0.842860	−	0.538133i	$-0.180870\pi$
$211$	24.4865			1.68572			0.842860	+	0.538133i	$0.180870\pi$
$212$		0			0
$213$		−	10.2043i		−	0.699188i
$214$		0			0
$215$	14.2870	−	3.97728i	0.974367	−	0.271248i
$216$		0			0
$217$		−	0.580276i		−	0.0393917i
$218$		0			0
$219$	−0.963933			−0.0651365
$220$		0			0
$221$	−1.36008			−0.0914887
$222$		0			0
$223$		−	8.35896i		−	0.559758i	−0.960035	−	0.279879i	$-0.909706\pi$
$223$		−	8.35896i		−	0.559758i	0.960035	−	0.279879i	$-0.0902943\pi$
$224$		0			0
$225$	4.28076	−	2.58362i	0.285384	−	0.172241i
$226$		0			0
$227$			20.3926i			1.35350i	0.736212	+	0.676751i	$0.236613\pi$
$227$			20.3926i			1.35350i	−0.736212	+	0.676751i	$0.763387\pi$
$228$		0			0
$229$	6.28122			0.415074			0.207537	−	0.978227i	$-0.433455\pi$
$229$	6.28122			0.415074			0.207537	+	0.978227i	$0.433455\pi$
$230$		0			0
$231$	−0.314039			−0.0206622
$232$		0			0
$233$			17.2508i			1.13014i	0.825043	+	0.565070i	$0.191151\pi$
$233$			17.2508i			1.13014i	−0.825043	+	0.565070i	$0.808849\pi$
$234$		0			0
$235$	18.1783	−	5.06055i	1.18582	−	0.330114i
$236$		0			0
$237$			13.7522i			0.893303i
$238$		0			0
$239$	16.1232			1.04292			0.521461	−	0.853275i	$-0.325387\pi$
$239$	16.1232			1.04292			0.521461	+	0.853275i	$0.325387\pi$
$240$		0			0
$241$	10.5061			0.676757			0.338379	−	0.941010i	$-0.390122\pi$
$241$	10.5061			0.676757			0.338379	+	0.941010i	$0.390122\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	−4.19414	−	15.0660i	−0.267954	−	0.962533i
$246$		0			0
$247$			0.331906i			0.0211187i
$248$		0			0
$249$	5.08460			0.322224
$250$		0			0
$251$	−23.0076			−1.45223			−0.726114	−	0.687574i	$-0.758675\pi$
$251$	−23.0076			−1.45223			−0.726114	+	0.687574i	$0.758675\pi$
$252$		0			0
$253$			9.97121i			0.626884i
$254$		0			0
$255$	4.63019	+	16.6324i	0.289954	+	1.04156i
$256$		0			0
$257$		−	10.9038i		−	0.680163i	−0.940396	−	0.340082i	$-0.889545\pi$
$257$		−	10.9038i		−	0.680163i	0.940396	−	0.340082i	$-0.110455\pi$
$258$		0			0
$259$	0.286038			0.0177735
$260$		0			0
$261$	−6.84971			−0.423986
$262$		0			0
$263$			9.75034i			0.601232i	0.953745	+	0.300616i	$0.0971922\pi$
$263$			9.75034i			0.601232i	−0.953745	+	0.300616i	$0.902808\pi$
$264$		0			0
$265$	−3.45665	+	0.962276i	−0.212340	+	0.0591122i
$266$		0			0
$267$			12.5250i			0.766515i
$268$		0			0
$269$	−10.3564			−0.631442			−0.315721	−	0.948852i	$-0.602246\pi$
$269$	−10.3564			−0.631442			−0.315721	+	0.948852i	$0.602246\pi$
$270$		0			0
$271$	0.270042			0.0164039			0.00820194	−	0.999966i	$-0.497389\pi$
$271$	0.270042			0.0164039			0.00820194	+	0.999966i	$0.497389\pi$
$272$		0			0
$273$		−	0.0137137i		−	0.000829993i
$274$		0			0
$275$	17.2677	−	10.4218i	1.04128	−	0.628458i
$276$		0			0
$277$			25.9341i			1.55823i	0.626883	+	0.779113i	$0.284330\pi$
$277$			25.9341i			1.55823i	−0.626883	+	0.779113i	$0.715670\pi$
$278$		0			0
$279$	7.45359			0.446235
$280$		0			0
$281$	−10.2757			−0.612996			−0.306498	−	0.951871i	$-0.599157\pi$
$281$	−10.2757			−0.612996			−0.306498	+	0.951871i	$0.599157\pi$
$282$		0			0
$283$			0.341083i			0.0202753i	0.999949	+	0.0101377i	$0.00322697\pi$
$283$			0.341083i			0.0202753i	−0.999949	+	0.0101377i	$0.996773\pi$
$284$		0			0
$285$	4.05887	−	1.12993i	0.240427	−	0.0669310i
$286$		0			0
$287$		−	0.00725211i		−	0.000428079i
$288$		0			0
$289$	−42.6148			−2.50675
$290$		0			0
$291$	13.8335			0.810936
$292$		0			0
$293$		−	6.82592i		−	0.398775i	−0.979921	−	0.199387i	$-0.936105\pi$
$293$		−	6.82592i		−	0.398775i	0.979921	−	0.199387i	$-0.0638952\pi$
$294$		0			0
$295$	0.476763	+	1.71261i	0.0277582	+	0.0997119i
$296$		0			0
$297$		−	4.03380i		−	0.234065i
$298$		0			0
$299$	−0.435432			−0.0251817
$300$		0			0
$301$	−0.516337			−0.0297612
$302$		0			0
$303$		−	15.7346i		−	0.903932i
$304$		0			0
$305$	3.56609	+	12.8099i	0.204193	+	0.733495i
$306$		0			0
$307$			12.0892i			0.689968i	0.938609	+	0.344984i	$0.112116\pi$
$307$			12.0892i			0.689968i	−0.938609	+	0.344984i	$0.887884\pi$
$308$		0			0
$309$	−13.3508			−0.759503
$310$		0			0
$311$	−14.4102			−0.817128			−0.408564	−	0.912730i	$-0.633970\pi$
$311$	−14.4102			−0.817128			−0.408564	+	0.912730i	$0.633970\pi$
$312$		0			0
$313$			6.05120i			0.342034i	0.985268	+	0.171017i	$0.0547053\pi$
$313$			6.05120i			0.342034i	−0.985268	+	0.171017i	$0.945295\pi$
$314$		0			0
$315$	−0.167705	+	0.0466864i	−0.00944910	+	0.00263048i
$316$		0			0
$317$		−	4.43387i		−	0.249031i	−0.992218	−	0.124515i	$-0.960262\pi$
$317$		−	4.43387i		−	0.249031i	0.992218	−	0.124515i	$-0.0397376\pi$
$318$		0			0
$319$	−27.6304			−1.54700
$320$		0			0
$321$	4.89148			0.273016
$322$		0			0
$323$			14.5481i			0.809477i
$324$		0			0
$325$	0.455108	+	0.754063i	0.0252449	+	0.0418279i
$326$		0			0
$327$			13.4979i			0.746433i
$328$		0			0
$329$	−0.656970			−0.0362199
$330$		0			0
$331$	−11.5281			−0.633643			−0.316821	−	0.948485i	$-0.602616\pi$
$331$	−11.5281			−0.633643			−0.316821	+	0.948485i	$0.602616\pi$
$332$		0			0
$333$			3.67413i			0.201341i
$334$		0			0
$335$	−2.15415	+	0.599682i	−0.117694	+	0.0327642i
$336$		0			0
$337$		−	9.88171i		−	0.538291i	−0.963099	−	0.269146i	$-0.913259\pi$
$337$		−	9.88171i		−	0.538291i	0.963099	−	0.269146i	$-0.0867413\pi$
$338$		0			0
$339$	8.95878			0.486574
$340$		0			0
$341$	30.0663			1.62818
$342$		0			0
$343$			1.08945i			0.0588250i
$344$		0			0
$345$	1.48236	+	5.32488i	0.0798077	+	0.286682i
$346$		0			0
$347$		−	25.9320i		−	1.39210i	−0.717992	−	0.696051i	$-0.754939\pi$
$347$		−	25.9320i		−	1.39210i	0.717992	−	0.696051i	$-0.245061\pi$
$348$		0			0
$349$	−30.1666			−1.61478			−0.807390	−	0.590018i	$-0.799121\pi$
$349$	−30.1666			−1.61478			−0.807390	+	0.590018i	$0.799121\pi$
$350$		0			0
$351$	0.176152			0.00940228
$352$		0			0
$353$		−	17.6545i		−	0.939652i	−0.882759	−	0.469826i	$-0.844317\pi$
$353$		−	17.6545i		−	0.939652i	0.882759	−	0.469826i	$-0.155683\pi$
$354$		0			0
$355$	−6.11934	−	21.9817i	−0.324781	−	1.16667i
$356$		0			0
$357$		−	0.601099i		−	0.0318135i
$358$		0			0
$359$	6.56411			0.346440			0.173220	−	0.984883i	$-0.444583\pi$
$359$	6.56411			0.346440			0.173220	+	0.984883i	$0.444583\pi$
$360$		0			0
$361$	−15.4498			−0.813146
$362$		0			0
$363$		−	5.27156i		−	0.276685i
$364$		0			0
$365$	−2.07646	+	0.578054i	−0.108687	+	0.0302567i
$366$		0			0
$367$			17.8986i			0.934299i	0.884178	+	0.467149i	$0.154719\pi$
$367$			17.8986i			0.934299i	−0.884178	+	0.467149i	$0.845281\pi$
$368$		0			0
$369$	0.0931527			0.00484934
$370$		0			0
$371$	0.124924			0.00648575
$372$		0			0
$373$		−	27.6803i		−	1.43323i	−0.697469	−	0.716615i	$-0.745691\pi$
$373$		−	27.6803i		−	1.43323i	0.697469	−	0.716615i	$-0.254309\pi$
$374$		0			0
$375$	7.67207	−	8.13261i	0.396184	−	0.419966i
$376$		0			0
$377$		−	1.20659i		−	0.0621424i
$378$		0			0
$379$	−2.80412			−0.144038			−0.0720189	−	0.997403i	$-0.522944\pi$
$379$	−2.80412			−0.144038			−0.0720189	+	0.997403i	$0.522944\pi$
$380$		0			0
$381$	8.12676			0.416346
$382$		0			0
$383$			26.6124i			1.35983i	0.733290	+	0.679916i	$0.237984\pi$
$383$			26.6124i			1.35983i	−0.733290	+	0.679916i	$0.762016\pi$
$384$		0			0
$385$	−0.676488	+	0.188324i	−0.0344770	+	0.00959786i
$386$		0			0
$387$		−	6.63231i		−	0.337139i
$388$		0			0
$389$	−14.1640			−0.718144			−0.359072	−	0.933310i	$-0.616907\pi$
$389$	−14.1640			−0.718144			−0.359072	+	0.933310i	$0.616907\pi$
$390$		0			0
$391$	−19.0858			−0.965210
$392$		0			0
$393$			4.45229i			0.224588i
$394$		0			0
$395$	8.24697	+	29.6244i	0.414950	+	1.49057i
$396$		0			0
$397$		−	17.3011i		−	0.868316i	−0.900837	−	0.434158i	$-0.857046\pi$
$397$		−	17.3011i		−	0.868316i	0.900837	−	0.434158i	$-0.142954\pi$
$398$		0			0
$399$	−0.146689			−0.00734363
$400$		0			0
$401$	28.8970			1.44305			0.721524	−	0.692389i	$-0.243442\pi$
$401$	28.8970			1.44305			0.721524	+	0.692389i	$0.243442\pi$
$402$		0			0
$403$			1.31296i			0.0654033i
$404$		0			0
$405$	−0.599682	−	2.15415i	−0.0297985	−	0.107041i
$406$		0			0
$407$			14.8207i			0.734636i
$408$		0			0
$409$	−32.7055			−1.61718			−0.808592	−	0.588370i	$-0.799770\pi$
$409$	−32.7055			−1.61718			−0.808592	+	0.588370i	$0.799770\pi$
$410$		0			0
$411$	−2.00903			−0.0990983
$412$		0			0
$413$		−	0.0618942i		−	0.00304562i
$414$		0			0
$415$	10.9530	−	3.04915i	0.537663	−	0.149677i
$416$		0			0
$417$			14.3601i			0.703216i
$418$		0			0
$419$	−21.4604			−1.04841			−0.524204	−	0.851593i	$-0.675637\pi$
$419$	−21.4604			−1.04841			−0.524204	+	0.851593i	$0.675637\pi$
$420$		0			0
$421$	8.40929			0.409844			0.204922	−	0.978778i	$-0.434306\pi$
$421$	8.40929			0.409844			0.204922	+	0.978778i	$0.434306\pi$
$422$		0			0
$423$		−	8.43872i		−	0.410305i
$424$		0			0
$425$	19.9483	+	33.0520i	0.967633	+	1.60326i
$426$		0			0
$427$		−	0.462955i		−	0.0224040i
$428$		0			0
$429$	0.710561			0.0343062
$430$		0			0
$431$	−9.99239			−0.481317			−0.240658	−	0.970610i	$-0.577363\pi$
$431$	−9.99239			−0.481317			−0.240658	+	0.970610i	$0.577363\pi$
$432$		0			0
$433$			8.63842i			0.415136i	0.978221	+	0.207568i	$0.0665548\pi$
$433$			8.63842i			0.415136i	−0.978221	+	0.207568i	$0.933445\pi$
$434$		0			0
$435$	−14.7553	+	4.10765i	−0.707464	+	0.196947i
$436$		0			0
$437$			4.65759i			0.222803i
$438$		0			0
$439$	−9.55805			−0.456181			−0.228090	−	0.973640i	$-0.573248\pi$
$439$	−9.55805			−0.456181			−0.228090	+	0.973640i	$0.573248\pi$
$440$		0			0
$441$	−6.99394			−0.333045
$442$		0			0
$443$		−	10.9529i		−	0.520390i	−0.965556	−	0.260195i	$-0.916213\pi$
$443$		−	10.9529i		−	0.520390i	0.965556	−	0.260195i	$-0.0837868\pi$
$444$		0			0
$445$	7.51100	+	26.9807i	0.356056	+	1.27901i
$446$		0			0
$447$			4.93926i			0.233619i
$448$		0			0
$449$	13.1449			0.620346			0.310173	−	0.950680i	$-0.399613\pi$
$449$	13.1449			0.620346			0.310173	+	0.950680i	$0.399613\pi$
$450$		0			0
$451$	0.375760			0.0176938
$452$		0			0
$453$		−	2.01310i		−	0.0945836i
$454$		0			0
$455$	−0.00822388	−	0.0295415i	−0.000385542	−	0.00138493i
$456$		0			0
$457$		−	39.5239i		−	1.84885i	−0.381364	−	0.924425i	$-0.624546\pi$
$457$		−	39.5239i		−	1.84885i	0.381364	−	0.924425i	$-0.375454\pi$
$458$		0			0
$459$	7.72106			0.360388
$460$		0			0
$461$	5.13801			0.239301			0.119650	−	0.992816i	$-0.461823\pi$
$461$	5.13801			0.239301			0.119650	+	0.992816i	$0.461823\pi$
$462$		0			0
$463$			40.6052i			1.88708i	0.331255	+	0.943541i	$0.392528\pi$
$463$			40.6052i			1.88708i	−0.331255	+	0.943541i	$0.607472\pi$
$464$		0			0
$465$	16.0562	−	4.46979i	0.744588	−	0.207281i
$466$		0			0
$467$		−	38.8979i		−	1.79998i	−0.435910	−	0.899990i	$-0.643573\pi$
$467$		−	38.8979i		−	1.79998i	0.435910	−	0.899990i	$-0.356427\pi$
$468$		0			0
$469$	0.0778518			0.00359486
$470$		0			0
$471$	9.56781			0.440862
$472$		0			0
$473$		−	26.7534i		−	1.23012i
$474$		0			0
$475$	8.06584	−	4.86807i	0.370086	−	0.223362i
$476$		0			0
$477$			1.60464i			0.0734715i
$478$		0			0
$479$	21.2898			0.972756			0.486378	−	0.873748i	$-0.338318\pi$
$479$	21.2898			0.972756			0.486378	+	0.873748i	$0.338318\pi$
$480$		0			0
$481$	−0.647205			−0.0295100
$482$		0			0
$483$		−	0.192443i		−	0.00875646i
$484$		0			0
$485$	29.7995	−	8.29572i	1.35313	−	0.376689i
$486$		0			0
$487$			11.0189i			0.499312i	0.968335	+	0.249656i	$0.0803176\pi$
$487$			11.0189i			0.499312i	−0.968335	+	0.249656i	$0.919682\pi$
$488$		0			0
$489$	−14.5402			−0.657532
$490$		0			0
$491$	19.5214			0.880989			0.440495	−	0.897755i	$-0.354803\pi$
$491$	19.5214			0.880989			0.440495	+	0.897755i	$0.354803\pi$
$492$		0			0
$493$		−	52.8870i		−	2.38191i
$494$		0			0
$495$	−2.41900	−	8.68943i	−0.108726	−	0.390561i
$496$		0			0
$497$			0.794424i			0.0356348i
$498$		0			0
$499$	−15.7073			−0.703157			−0.351579	−	0.936158i	$-0.614355\pi$
$499$	−15.7073			−0.703157			−0.351579	+	0.936158i	$0.614355\pi$
$500$		0			0
$501$	11.5213			0.514732
$502$		0			0
$503$		−	11.1273i		−	0.496144i	−0.968742	−	0.248072i	$-0.920203\pi$
$503$		−	11.1273i		−	0.496144i	0.968742	−	0.248072i	$-0.0797969\pi$
$504$		0			0
$505$	−9.43579	−	33.8948i	−0.419887	−	1.50830i
$506$		0			0
$507$		−	12.9690i		−	0.575972i
$508$		0			0
$509$	−39.8436			−1.76604			−0.883019	−	0.469338i	$-0.844493\pi$
$509$	−39.8436			−1.76604			−0.883019	+	0.469338i	$0.844493\pi$
$510$		0			0
$511$	0.0750439			0.00331975
$512$		0			0
$513$		−	1.88421i		−	0.0831897i
$514$		0			0
$515$	−28.7598	+	8.00627i	−1.26731	+	0.352798i
$516$		0			0
$517$		−	34.0401i		−	1.49708i
$518$		0			0
$519$	−4.32812			−0.189983
$520$		0			0
$521$	5.17332			0.226647			0.113324	−	0.993558i	$-0.463850\pi$
$521$	5.17332			0.226647			0.113324	+	0.993558i	$0.463850\pi$
$522$		0			0
$523$		−	42.9653i		−	1.87874i	−0.342902	−	0.939371i	$-0.611410\pi$
$523$		−	42.9653i		−	1.87874i	0.342902	−	0.939371i	$-0.388590\pi$
$524$		0			0
$525$	−0.333265	+	0.201139i	−0.0145449	+	0.00877844i
$526$		0			0
$527$			57.5496i			2.50690i
$528$		0			0
$529$	16.8896			0.734332
$530$		0			0
$531$	0.795026			0.0345012
$532$		0			0
$533$			0.0164090i			0.000710753i
$534$		0			0
$535$	10.5370	−	2.93333i	0.455554	−	0.126819i
$536$		0			0
$537$			0.803651i			0.0346801i
$538$		0			0
$539$	−28.2122			−1.21518
$540$		0			0
$541$	−42.3274			−1.81980			−0.909899	−	0.414830i	$-0.863841\pi$
$541$	−42.3274			−1.81980			−0.909899	+	0.414830i	$0.863841\pi$
$542$		0			0
$543$		−	1.58419i		−	0.0679841i
$544$		0			0
$545$	8.09443	+	29.0765i	0.346727	+	1.24550i
$546$		0			0
$547$		−	0.119998i		−	0.00513075i	−0.999997	−	0.00256537i	$-0.999183\pi$
$547$		−	0.119998i		−	0.00513075i	0.999997	−	0.00256537i	$-0.000816585\pi$
$548$		0			0
$549$	5.94662			0.253796
$550$		0			0
$551$	−12.9063			−0.549825
$552$		0			0
$553$		−	1.07064i		−	0.0455281i
$554$		0			0
$555$	2.20331	+	7.91465i	0.0935254	+	0.335958i
$556$		0			0
$557$		−	1.87818i		−	0.0795812i	−0.999208	−	0.0397906i	$-0.987331\pi$
$557$		−	1.87818i		−	0.0795812i	0.999208	−	0.0397906i	$-0.0126691\pi$
$558$		0			0
$559$	1.16829			0.0494135
$560$		0			0
$561$	31.1452			1.31495
$562$		0			0
$563$			0.0755137i			0.00318252i	0.999999	+	0.00159126i	$0.000506515\pi$
$563$			0.0755137i			0.00318252i	−0.999999	+	0.00159126i	$0.999493\pi$
$564$		0			0
$565$	19.2986	−	5.37242i	0.811898	−	0.226019i
$566$		0			0
$567$			0.0778518i			0.00326947i
$568$		0			0
$569$	14.7697			0.619179			0.309589	−	0.950870i	$-0.399808\pi$
$569$	14.7697			0.619179			0.309589	+	0.950870i	$0.399808\pi$
$570$		0			0
$571$	7.45240			0.311873			0.155937	−	0.987767i	$-0.450160\pi$
$571$	7.45240			0.311873			0.155937	+	0.987767i	$0.450160\pi$
$572$		0			0
$573$			7.17923i			0.299917i
$574$		0			0
$575$	6.38648	+	10.5817i	0.266335	+	0.441286i
$576$		0			0
$577$			16.8741i			0.702477i	0.936286	+	0.351239i	$0.114239\pi$
$577$			16.8741i			0.702477i	−0.936286	+	0.351239i	$0.885761\pi$
$578$		0			0
$579$	10.4401			0.433874
$580$		0			0
$581$	−0.395846			−0.0164224
$582$		0			0
$583$			6.47281i			0.268076i
$584$		0			0
$585$	0.379458	−	0.105635i	0.0156887	−	0.00436747i
$586$		0			0
$587$		−	31.7975i		−	1.31242i	−0.754576	−	0.656212i	$-0.772158\pi$
$587$		−	31.7975i		−	1.31242i	0.754576	−	0.656212i	$-0.227842\pi$
$588$		0			0
$589$	14.0441			0.578677
$590$		0			0
$591$	−17.5284			−0.721020
$592$		0			0
$593$			11.0814i			0.455059i	0.973771	+	0.227530i	$0.0730649\pi$
$593$			11.0814i			0.455059i	−0.973771	+	0.227530i	$0.926935\pi$
$594$		0			0
$595$	−0.360468	−	1.29486i	−0.0147778	−	0.0530841i
$596$		0			0
$597$		−	4.68951i		−	0.191929i
$598$		0			0
$599$	−4.51312			−0.184401			−0.0922005	−	0.995740i	$-0.529390\pi$
$599$	−4.51312			−0.184401			−0.0922005	+	0.995740i	$0.529390\pi$
$600$		0			0
$601$	−1.03044			−0.0420325			−0.0210163	−	0.999779i	$-0.506690\pi$
$601$	−1.03044			−0.0420325			−0.0210163	+	0.999779i	$0.506690\pi$
$602$		0			0
$603$			1.00000i			0.0407231i
$604$		0			0
$605$	−3.16126	−	11.3558i	−0.128524	−	0.461677i
$606$		0			0
$607$		−	16.2834i		−	0.660924i	−0.943819	−	0.330462i	$-0.892795\pi$
$607$		−	16.2834i		−	0.660924i	0.943819	−	0.330462i	$-0.107205\pi$
$608$		0			0
$609$	0.533262			0.0216089
$610$		0			0
$611$	1.48650			0.0601372
$612$		0			0
$613$		−	24.9121i		−	1.00619i	−0.864231	−	0.503095i	$-0.832195\pi$
$613$		−	24.9121i		−	1.00619i	0.864231	−	0.503095i	$-0.167805\pi$
$614$		0			0
$615$	0.200665	−	0.0558621i	0.00809161	−	0.00225258i
$616$		0			0
$617$		−	1.55603i		−	0.0626433i	−0.999509	−	0.0313217i	$-0.990028\pi$
$617$		−	1.55603i		−	0.0626433i	0.999509	−	0.0313217i	$-0.00997163\pi$
$618$		0			0
$619$	−34.3980			−1.38257			−0.691286	−	0.722582i	$-0.742955\pi$
$619$	−34.3980			−1.38257			−0.691286	+	0.722582i	$0.742955\pi$
$620$		0			0
$621$	2.47191			0.0991945
$622$		0			0
$623$		−	0.975091i		−	0.0390662i
$624$		0			0
$625$	11.6498	−	22.1197i	0.465994	−	0.884788i
$626$		0			0
$627$		−	7.60052i		−	0.303535i
$628$		0			0
$629$	−28.3682			−1.13111
$630$		0			0
$631$	−12.4889			−0.497176			−0.248588	−	0.968609i	$-0.579967\pi$
$631$	−12.4889			−0.497176			−0.248588	+	0.968609i	$0.579967\pi$
$632$		0			0
$633$		−	24.4865i		−	0.973251i
$634$		0			0
$635$	17.5063	−	4.87347i	0.694716	−	0.193398i
$636$		0			0
$637$		−	1.23199i		−	0.0488134i
$638$		0			0
$639$	−10.2043			−0.403676
$640$		0			0
$641$	41.0882			1.62289			0.811443	−	0.584431i	$-0.198682\pi$
$641$	41.0882			1.62289			0.811443	+	0.584431i	$0.198682\pi$
$642$		0			0
$643$			37.2853i			1.47039i	0.677857	+	0.735194i	$0.262909\pi$
$643$			37.2853i			1.47039i	−0.677857	+	0.735194i	$0.737091\pi$
$644$		0			0
$645$	−3.97728	−	14.2870i	−0.156605	−	0.562551i
$646$		0			0
$647$		−	4.45446i		−	0.175123i	−0.996159	−	0.0875614i	$-0.972093\pi$
$647$		−	4.45446i		−	0.175123i	0.996159	−	0.0875614i	$-0.0279074\pi$
$648$		0			0
$649$	3.20698			0.125885
$650$		0			0
$651$	−0.580276			−0.0227428
$652$		0			0
$653$		−	31.2109i		−	1.22138i	−0.791872	−	0.610688i	$-0.790893\pi$
$653$		−	31.2109i		−	1.22138i	0.791872	−	0.610688i	$-0.209107\pi$
$654$		0			0
$655$	2.66996	+	9.59091i	0.104324	+	0.374748i
$656$		0			0
$657$			0.963933i			0.0376066i
$658$		0			0
$659$	−13.1330			−0.511591			−0.255795	−	0.966731i	$-0.582337\pi$
$659$	−13.1330			−0.511591			−0.255795	+	0.966731i	$0.582337\pi$
$660$		0			0
$661$	−34.1253			−1.32732			−0.663660	−	0.748034i	$-0.730998\pi$
$661$	−34.1253			−1.32732			−0.663660	+	0.748034i	$0.730998\pi$
$662$		0			0
$663$			1.36008i			0.0528211i
$664$		0			0
$665$	−0.315991	+	0.0879668i	−0.0122536	+	0.00341121i
$666$		0			0
$667$		−	16.9319i		−	0.655605i
$668$		0			0
$669$	−8.35896			−0.323176
$670$		0			0
$671$	23.9875			0.926027
$672$		0			0
$673$		−	22.8699i		−	0.881571i	−0.897612	−	0.440786i	$-0.854700\pi$
$673$		−	22.8699i		−	0.881571i	0.897612	−	0.440786i	$-0.145300\pi$
$674$		0			0
$675$	−2.58362	−	4.28076i	−0.0994435	−	0.164767i
$676$		0			0
$677$		−	46.7344i		−	1.79615i	−0.439843	−	0.898075i	$-0.644966\pi$
$677$		−	46.7344i		−	1.79615i	0.439843	−	0.898075i	$-0.355034\pi$
$678$		0			0
$679$	−1.07697			−0.0413301
$680$		0			0
$681$	20.3926			0.781445
$682$		0			0
$683$			31.1832i			1.19319i	0.802542	+	0.596596i	$0.203481\pi$
$683$			31.1832i			1.19319i	−0.802542	+	0.596596i	$0.796519\pi$
$684$		0			0
$685$	−4.32777	+	1.20478i	−0.165356	+	0.0460324i
$686$		0			0
$687$		−	6.28122i		−	0.239643i
$688$		0			0
$689$	−0.282660			−0.0107685
$690$		0			0
$691$	5.21574			0.198416			0.0992080	−	0.995067i	$-0.468369\pi$
$691$	5.21574			0.198416			0.0992080	+	0.995067i	$0.468369\pi$
$692$		0			0
$693$			0.314039i			0.0119293i
$694$		0			0
$695$	8.61149	+	30.9338i	0.326652	+	1.17339i
$696$		0			0
$697$			0.719238i			0.0272431i
$698$		0			0
$699$	17.2508			0.652487
$700$		0			0
$701$	44.2520			1.67137			0.835687	−	0.549206i	$-0.185070\pi$
$701$	44.2520			1.67137			0.835687	+	0.549206i	$0.185070\pi$
$702$		0			0
$703$			6.92283i			0.261099i
$704$		0			0
$705$	−5.06055	−	18.1783i	−0.190592	−	0.684635i
$706$		0			0
$707$			1.22497i			0.0460698i
$708$		0			0
$709$	−9.48585			−0.356249			−0.178124	−	0.984008i	$-0.557003\pi$
$709$	−9.48585			−0.356249			−0.178124	+	0.984008i	$0.557003\pi$
$710$		0			0
$711$	13.7522			0.515749
$712$		0			0
$713$			18.4246i			0.690008i
$714$		0			0
$715$	1.53066	−	0.426111i	0.0572434	−	0.0159356i
$716$		0			0
$717$		−	16.1232i		−	0.602131i
$718$		0			0
$719$	−38.6200			−1.44028			−0.720141	−	0.693828i	$-0.755923\pi$
$719$	−38.6200			−1.44028			−0.720141	+	0.693828i	$0.755923\pi$
$720$		0			0
$721$	1.03939			0.0387088
$722$		0			0
$723$		−	10.5061i		−	0.390726i
$724$		0			0
$725$	−29.3220	+	17.6970i	−1.08899	+	0.657251i
$726$		0			0
$727$			43.5732i			1.61604i	0.589155	+	0.808020i	$0.299461\pi$
$727$			43.5732i			1.61604i	−0.589155	+	0.808020i	$0.700539\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	51.2085			1.89401
$732$		0			0
$733$			5.25802i			0.194209i	0.995274	+	0.0971047i	$0.0309582\pi$
$733$			5.25802i			0.194209i	−0.995274	+	0.0971047i	$0.969042\pi$
$734$		0			0
$735$	−15.0660	+	4.19414i	−0.555719	+	0.154703i
$736$		0			0
$737$			4.03380i			0.148587i
$738$		0			0
$739$	−26.6880			−0.981735			−0.490867	−	0.871234i	$-0.663320\pi$
$739$	−26.6880			−0.981735			−0.490867	+	0.871234i	$0.663320\pi$
$740$		0			0
$741$	0.331906			0.0121929
$742$		0			0
$743$			11.6192i			0.426268i	0.977023	+	0.213134i	$0.0683671\pi$
$743$			11.6192i			0.426268i	−0.977023	+	0.213134i	$0.931633\pi$
$744$		0			0
$745$	2.96199	+	10.6399i	0.108519	+	0.389817i
$746$		0			0
$747$		−	5.08460i		−	0.186036i
$748$		0			0
$749$	−0.380810			−0.0139145
$750$		0			0
$751$	13.4994			0.492598			0.246299	−	0.969194i	$-0.420785\pi$
$751$	13.4994			0.492598			0.246299	+	0.969194i	$0.420785\pi$
$752$		0			0
$753$			23.0076i			0.838444i
$754$		0			0
$755$	−1.20722	−	4.33653i	−0.0439352	−	0.157822i
$756$		0			0
$757$		−	51.7616i		−	1.88131i	−0.339371	−	0.940653i	$-0.610214\pi$
$757$		−	51.7616i		−	1.88131i	0.339371	−	0.940653i	$-0.389786\pi$
$758$		0			0
$759$	9.97121			0.361932
$760$		0			0
$761$	−51.3327			−1.86081			−0.930404	−	0.366537i	$-0.880543\pi$
$761$	−51.3327			−1.86081			−0.930404	+	0.366537i	$0.880543\pi$
$762$		0			0
$763$		−	1.05083i		−	0.0380427i
$764$		0			0
$765$	16.6324	−	4.63019i	0.601344	−	0.167405i
$766$		0			0
$767$			0.140045i			0.00505674i
$768$		0			0
$769$	−40.1613			−1.44825			−0.724127	−	0.689667i	$-0.757757\pi$
$769$	−40.1613			−1.44825			−0.724127	+	0.689667i	$0.757757\pi$
$770$		0			0
$771$	−10.9038			−0.392692
$772$		0			0
$773$			35.3015i			1.26971i	0.772633	+	0.634853i	$0.218939\pi$
$773$			35.3015i			1.26971i	−0.772633	+	0.634853i	$0.781061\pi$
$774$		0			0
$775$	31.9071	−	19.2572i	1.14614	−	0.691740i
$776$		0			0
$777$		−	0.286038i		−	0.0102616i
$778$		0			0
$779$	0.175519			0.00628862
$780$		0			0
$781$	−41.1622			−1.47290
$782$		0			0
$783$			6.84971i			0.244789i
$784$		0			0
$785$	20.6105	−	5.73765i	0.735622	−	0.204786i
$786$		0			0
$787$		−	37.9520i		−	1.35284i	−0.736515	−	0.676421i	$-0.763530\pi$
$787$		−	37.9520i		−	1.35284i	0.736515	−	0.676421i	$-0.236470\pi$
$788$		0			0
$789$	9.75034			0.347121
$790$		0			0
$791$	−0.697457			−0.0247987
$792$		0			0
$793$			1.04751i			0.0371981i
$794$		0			0
$795$	0.962276	+	3.45665i	0.0341284	+	0.122595i
$796$		0			0
$797$		−	16.2069i		−	0.574077i	−0.957919	−	0.287039i	$-0.907329\pi$
$797$		−	16.2069i		−	0.574077i	0.957919	−	0.287039i	$-0.0926708\pi$
$798$		0			0
$799$	65.1559			2.30505
$800$		0			0
$801$	12.5250			0.442548
$802$		0			0
$803$			3.88831i			0.137216i
$804$		0			0
$805$	−0.115405	−	0.414552i	−0.00406748	−	0.0146110i
$806$		0			0
$807$			10.3564i			0.364563i
$808$		0			0
$809$	−13.9421			−0.490178			−0.245089	−	0.969501i	$-0.578817\pi$
$809$	−13.9421			−0.490178			−0.245089	+	0.969501i	$0.578817\pi$
$810$		0			0
$811$	−15.9924			−0.561570			−0.280785	−	0.959771i	$-0.590595\pi$
$811$	−15.9924			−0.561570			−0.280785	+	0.959771i	$0.590595\pi$
$812$		0			0
$813$		−	0.270042i		−	0.00947078i
$814$		0			0
$815$	−31.3219	+	8.71952i	−1.09716	+	0.305432i
$816$		0			0
$817$		−	12.4966i		−	0.437202i
$818$		0			0
$819$	−0.0137137			−0.000479196
$820$		0			0
$821$	−5.26133			−0.183622			−0.0918108	−	0.995776i	$-0.529266\pi$
$821$	−5.26133			−0.183622			−0.0918108	+	0.995776i	$0.529266\pi$
$822$		0			0
$823$			9.88084i			0.344425i	0.985060	+	0.172212i	$0.0550915\pi$
$823$			9.88084i			0.344425i	−0.985060	+	0.172212i	$0.944909\pi$
$824$		0			0
$825$	−10.4218	−	17.2677i	−0.362841	−	0.601186i
$826$		0			0
$827$		−	48.1730i		−	1.67514i	−0.546332	−	0.837569i	$-0.683976\pi$
$827$		−	48.1730i		−	1.67514i	0.546332	−	0.837569i	$-0.316024\pi$
$828$		0			0
$829$	−19.5330			−0.678408			−0.339204	−	0.940713i	$-0.610158\pi$
$829$	−19.5330			−0.678408			−0.339204	+	0.940713i	$0.610158\pi$
$830$		0			0
$831$	25.9341			0.899643
$832$		0			0
$833$		−	54.0006i		−	1.87101i
$834$		0			0
$835$	24.8186	−	6.90910i	0.858882	−	0.239099i
$836$		0			0
$837$		−	7.45359i		−	0.257634i
$838$		0			0
$839$	47.9169			1.65428			0.827138	−	0.561999i	$-0.189967\pi$
$839$	47.9169			1.65428			0.827138	+	0.561999i	$0.189967\pi$
$840$		0			0
$841$	17.9185			0.617880
$842$		0			0
$843$			10.2757i			0.353914i
$844$		0			0
$845$	−7.77726	−	27.9372i	−0.267546	−	0.961068i
$846$		0			0
$847$			0.410400i			0.0141015i
$848$		0			0
$849$	0.341083			0.0117060
$850$		0			0
$851$	−9.08214			−0.311332
$852$		0			0
$853$		−	11.1888i		−	0.383097i	−0.981483	−	0.191549i	$-0.938649\pi$
$853$		−	11.1888i		−	0.383097i	0.981483	−	0.191549i	$-0.0613509\pi$
$854$		0			0
$855$	−1.12993	−	4.05887i	−0.0386426	−	0.138810i
$856$		0			0
$857$			7.49345i			0.255972i	0.991776	+	0.127986i	$0.0408512\pi$
$857$			7.49345i			0.255972i	−0.991776	+	0.127986i	$0.959149\pi$
$858$		0			0
$859$	−34.7007			−1.18397			−0.591987	−	0.805948i	$-0.701656\pi$
$859$	−34.7007			−1.18397			−0.591987	+	0.805948i	$0.701656\pi$
$860$		0			0
$861$	−0.00725211			−0.000247151
$862$		0			0
$863$			24.6939i			0.840590i	0.907388	+	0.420295i	$0.138073\pi$
$863$			24.6939i			0.840590i	−0.907388	+	0.420295i	$0.861927\pi$
$864$		0			0
$865$	−9.32344	+	2.59550i	−0.317006	+	0.0882496i
$866$		0			0
$867$			42.6148i			1.44727i
$868$		0			0
$869$	55.4737			1.88182
$870$		0			0
$871$	−0.176152			−0.00596867
$872$		0			0
$873$		−	13.8335i		−	0.468194i
$874$		0			0
$875$	−0.597285	+	0.633138i	−0.0201919	+	0.0214040i
$876$		0			0
$877$			24.2554i			0.819045i	0.912300	+	0.409523i	$0.134305\pi$
$877$			24.2554i			0.819045i	−0.912300	+	0.409523i	$0.865695\pi$
$878$		0			0
$879$	−6.82592			−0.230233
$880$		0			0
$881$	53.7899			1.81223			0.906113	−	0.423035i	$-0.139035\pi$
$881$	53.7899			1.81223			0.906113	+	0.423035i	$0.139035\pi$
$882$		0			0
$883$			7.02218i			0.236315i	0.992995	+	0.118158i	$0.0376988\pi$
$883$			7.02218i			0.236315i	−0.992995	+	0.118158i	$0.962301\pi$
$884$		0			0
$885$	1.71261	−	0.476763i	0.0575687	−	0.0160262i
$886$		0			0
$887$			26.7478i			0.898105i	0.893505	+	0.449052i	$0.148238\pi$
$887$			26.7478i			0.898105i	−0.893505	+	0.449052i	$0.851762\pi$
$888$		0			0
$889$	−0.632683			−0.0212195
$890$		0			0
$891$	−4.03380			−0.135137
$892$		0			0
$893$		−	15.9003i		−	0.532083i
$894$		0			0
$895$	0.481935	+	1.73119i	0.0161093	+	0.0578672i
$896$		0			0
$897$			0.435432i			0.0145386i
$898$		0			0
$899$	−51.0549			−1.70278
$900$		0			0
$901$	−12.3895			−0.412756
$902$		0			0
$903$			0.516337i			0.0171826i
$904$		0			0
$905$	−0.950011	−	3.41259i	−0.0315794	−	0.113438i
$906$		0			0
$907$		−	6.44857i		−	0.214121i	−0.994252	−	0.107061i	$-0.965856\pi$
$907$		−	6.44857i		−	0.214121i	0.994252	−	0.107061i	$-0.0341439\pi$
$908$		0			0
$909$	−15.7346			−0.521885
$910$		0			0
$911$	−3.80158			−0.125952			−0.0629760	−	0.998015i	$-0.520059\pi$
$911$	−3.80158			−0.125952			−0.0629760	+	0.998015i	$0.520059\pi$
$912$		0			0
$913$		−	20.5103i		−	0.678791i
$914$		0			0
$915$	12.8099	−	3.56609i	0.423484	−	0.117891i
$916$		0			0
$917$		−	0.346619i		−	0.0114464i
$918$		0			0
$919$	35.3736			1.16687			0.583434	−	0.812161i	$-0.301709\pi$
$919$	35.3736			1.16687			0.583434	+	0.812161i	$0.301709\pi$
$920$		0			0
$921$	12.0892			0.398353
$922$		0			0
$923$		−	1.79751i		−	0.0591656i
$924$		0			0
$925$	9.49255	+	15.7281i	0.312113	+	0.517136i
$926$		0			0
$927$			13.3508i			0.438499i
$928$		0			0
$929$	3.95053			0.129613			0.0648063	−	0.997898i	$-0.479357\pi$
$929$	3.95053			0.129613			0.0648063	+	0.997898i	$0.479357\pi$
$930$		0			0
$931$	−13.1780			−0.431892
$932$		0			0
$933$			14.4102i			0.471769i
$934$		0			0
$935$	67.0916	−	18.6773i	2.19413	−	0.610812i
$936$		0			0
$937$		−	21.3331i		−	0.696921i	−0.937324	−	0.348460i	$-0.886705\pi$
$937$		−	21.3331i		−	0.696921i	0.937324	−	0.348460i	$-0.113295\pi$
$938$		0			0
$939$	6.05120			0.197474
$940$		0			0
$941$	21.7230			0.708149			0.354075	−	0.935217i	$-0.384796\pi$
$941$	21.7230			0.708149			0.354075	+	0.935217i	$0.384796\pi$
$942$		0			0
$943$			0.230265i			0.00749848i
$944$		0			0
$945$	0.0466864	+	0.167705i	0.00151871	+	0.00545544i
$946$		0			0
$947$		−	9.11251i		−	0.296117i	−0.988979	−	0.148058i	$-0.952698\pi$
$947$		−	9.11251i		−	0.296117i	0.988979	−	0.148058i	$-0.0473024\pi$
$948$		0			0
$949$	−0.169798			−0.00551189
$950$		0			0
$951$	−4.43387			−0.143778
$952$		0			0
$953$		−	49.0738i		−	1.58966i	−0.606834	−	0.794829i	$-0.707561\pi$
$953$		−	49.0738i		−	1.58966i	0.606834	−	0.794829i	$-0.292439\pi$
$954$		0			0
$955$	4.30526	+	15.4652i	0.139315	+	0.500441i
$956$		0			0
$957$			27.6304i			0.893163i
$958$		0			0
$959$	0.156407			0.00505064
$960$		0			0
$961$	24.5560			0.792130
$962$		0			0
$963$		−	4.89148i		−	0.157626i
$964$		0			0
$965$	22.4895	−	6.26072i	0.723962	−	0.201540i
$966$		0			0
$967$			4.86983i			0.156603i	0.996930	+	0.0783016i	$0.0249497\pi$
$967$			4.86983i			0.156603i	−0.996930	+	0.0783016i	$0.975050\pi$
$968$		0			0
$969$	14.5481			0.467352
$970$		0			0
$971$	−48.7128			−1.56327			−0.781633	−	0.623738i	$-0.785613\pi$
$971$	−48.7128			−1.56327			−0.781633	+	0.623738i	$0.785613\pi$
$972$		0			0
$973$		−	1.11796i		−	0.0358401i
$974$		0			0
$975$	0.754063	−	0.455108i	0.0241494	−	0.0145751i
$976$		0			0
$977$		−	49.0264i		−	1.56849i	−0.620450	−	0.784246i	$-0.713050\pi$
$977$		−	49.0264i		−	1.56849i	0.620450	−	0.784246i	$-0.286950\pi$
$978$		0			0
$979$	50.5232			1.61473
$980$		0			0
$981$	13.4979			0.430953
$982$		0			0
$983$		−	10.6233i		−	0.338830i	−0.985545	−	0.169415i	$-0.945812\pi$
$983$		−	10.6233i		−	0.338830i	0.985545	−	0.169415i	$-0.0541878\pi$
$984$		0			0
$985$	−37.7588	+	10.5114i	−1.20310	+	0.334923i
$986$		0			0
$987$			0.656970i			0.0209116i
$988$		0			0
$989$	16.3945			0.521315
$990$		0			0
$991$	13.8210			0.439038			0.219519	−	0.975608i	$-0.429551\pi$
$991$	13.8210			0.439038			0.219519	+	0.975608i	$0.429551\pi$
$992$		0			0
$993$			11.5281i			0.365834i
$994$		0			0
$995$	−2.81221	−	10.1019i	−0.0891532	−	0.320252i
$996$		0			0
$997$			11.9375i			0.378064i	0.981971	+	0.189032i	$0.0605349\pi$
$997$			11.9375i			0.378064i	−0.981971	+	0.189032i	$0.939465\pi$
$998$		0			0
$999$	3.67413			0.116244

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.9	✓	38
5.4	even	2	inner	4020.2.g.c.1609.28	yes	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.9	✓	38	1.1	even	1	trivial
4020.2.g.c.1609.28	yes	38	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-0.599682 - 2.15415i) q^{5} +0.0778518i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-0.599682 - 2.15415i) q^{5} +0.0778518i q^{7} -1.00000 q^{9} -4.03380 q^{11} -0.176152i q^{13} +(-2.15415 + 0.599682i) q^{15} -7.72106i q^{17} -1.88421 q^{19} +0.0778518 q^{21} -2.47191i q^{23} +(-4.28076 + 2.58362i) q^{25} +1.00000i q^{27} +6.84971 q^{29} -7.45359 q^{31} +4.03380i q^{33} +(0.167705 - 0.0466864i) q^{35} -3.67413i q^{37} -0.176152 q^{39} -0.0931527 q^{41} +6.63231i q^{43} +(0.599682 + 2.15415i) q^{45} +8.43872i q^{47} +6.99394 q^{49} -7.72106 q^{51} -1.60464i q^{53} +(2.41900 + 8.68943i) q^{55} +1.88421i q^{57} -0.795026 q^{59} -5.94662 q^{61} -0.0778518i q^{63} +(-0.379458 + 0.105635i) q^{65} -1.00000i q^{67} -2.47191 q^{69} +10.2043 q^{71} -0.963933i q^{73} +(2.58362 + 4.28076i) q^{75} -0.314039i q^{77} -13.7522 q^{79} +1.00000 q^{81} +5.08460i q^{83} +(-16.6324 + 4.63019i) q^{85} -6.84971i q^{87} -12.5250 q^{89} +0.0137137 q^{91} +7.45359i q^{93} +(1.12993 + 4.05887i) q^{95} +13.8335i q^{97} +4.03380 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 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

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