LMFDB - Embedded newform 4020.2.g.c.1609.30

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			1609.30
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.c.1609.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-0.471887 - 2.18571i) q^{5} +1.05400i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +(-0.471887 - 2.18571i) q^{5} +1.05400i q^{7} -1.00000 q^{9} -5.95761 q^{11} -5.75627i q^{13} +(2.18571 - 0.471887i) q^{15} +4.60912i q^{17} +4.92917 q^{19} -1.05400 q^{21} -5.41155i q^{23} +(-4.55464 + 2.06282i) q^{25} -1.00000i q^{27} -2.23932 q^{29} -3.73316 q^{31} -5.95761i q^{33} +(2.30373 - 0.497368i) q^{35} +10.8869i q^{37} +5.75627 q^{39} +12.0866 q^{41} -6.81718i q^{43} +(0.471887 + 2.18571i) q^{45} +8.17727i q^{47} +5.88909 q^{49} -4.60912 q^{51} +1.55304i q^{53} +(2.81132 + 13.0216i) q^{55} +4.92917i q^{57} -14.2977 q^{59} +2.79653 q^{61} -1.05400i q^{63} +(-12.5815 + 2.71631i) q^{65} +1.00000i q^{67} +5.41155 q^{69} -14.7028 q^{71} +7.57857i q^{73} +(-2.06282 - 4.55464i) q^{75} -6.27931i q^{77} +11.8213 q^{79} +1.00000 q^{81} +17.4407i q^{83} +(10.0742 - 2.17498i) q^{85} -2.23932i q^{87} +1.38538 q^{89} +6.06709 q^{91} -3.73316i q^{93} +(-2.32601 - 10.7737i) q^{95} +2.06742i q^{97} +5.95761 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.471887	−	2.18571i	−0.211034	−	0.977479i
$5$	−0.471887	−	2.18571i	−0.211034	−	0.977479i
$6$		0			0
$7$			1.05400i			0.398373i	0.979962	+	0.199187i	$0.0638300\pi$
$7$			1.05400i			0.398373i	−0.979962	+	0.199187i	$0.936170\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−5.95761			−1.79629			−0.898144	−	0.439701i	$-0.855084\pi$
$11$	−5.95761			−1.79629			−0.898144	+	0.439701i	$0.855084\pi$
$12$		0			0
$13$		−	5.75627i		−	1.59650i	−0.602324	−	0.798252i	$-0.705759\pi$
$13$		−	5.75627i		−	1.59650i	0.602324	−	0.798252i	$-0.294241\pi$
$14$		0			0
$15$	2.18571	−	0.471887i	0.564348	−	0.121841i
$16$		0			0
$17$			4.60912i			1.11788i	0.829210	+	0.558938i	$0.188791\pi$
$17$			4.60912i			1.11788i	−0.829210	+	0.558938i	$0.811209\pi$
$18$		0			0
$19$	4.92917			1.13083			0.565415	−	0.824807i	$-0.308716\pi$
$19$	4.92917			1.13083			0.565415	+	0.824807i	$0.308716\pi$
$20$		0			0
$21$	−1.05400			−0.230001
$22$		0			0
$23$		−	5.41155i		−	1.12839i	−0.825643	−	0.564193i	$-0.809187\pi$
$23$		−	5.41155i		−	1.12839i	0.825643	−	0.564193i	$-0.190813\pi$
$24$		0			0
$25$	−4.55464	+	2.06282i	−0.910929	+	0.412563i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−2.23932			−0.415831			−0.207916	−	0.978147i	$-0.566668\pi$
$29$	−2.23932			−0.415831			−0.207916	+	0.978147i	$0.566668\pi$
$30$		0			0
$31$	−3.73316			−0.670495			−0.335248	−	0.942130i	$-0.608820\pi$
$31$	−3.73316			−0.670495			−0.335248	+	0.942130i	$0.608820\pi$
$32$		0			0
$33$		−	5.95761i		−	1.03709i
$34$		0			0
$35$	2.30373	−	0.497368i	0.389401	−	0.0840705i
$36$		0			0
$37$			10.8869i			1.78979i	0.446274	+	0.894896i	$0.352751\pi$
$37$			10.8869i			1.78979i	−0.446274	+	0.894896i	$0.647249\pi$
$38$		0			0
$39$	5.75627			0.921742
$40$		0			0
$41$	12.0866			1.88761			0.943805	−	0.330502i	$-0.107218\pi$
$41$	12.0866			1.88761			0.943805	+	0.330502i	$0.107218\pi$
$42$		0			0
$43$		−	6.81718i		−	1.03961i	−0.854285	−	0.519805i	$-0.826005\pi$
$43$		−	6.81718i		−	1.03961i	0.854285	−	0.519805i	$-0.173995\pi$
$44$		0			0
$45$	0.471887	+	2.18571i	0.0703448	+	0.325826i
$46$		0			0
$47$			8.17727i			1.19278i	0.802696	+	0.596388i	$0.203398\pi$
$47$			8.17727i			1.19278i	−0.802696	+	0.596388i	$0.796602\pi$
$48$		0			0
$49$	5.88909			0.841299
$50$		0			0
$51$	−4.60912			−0.645406
$52$		0			0
$53$			1.55304i			0.213327i	0.994295	+	0.106663i	$0.0340167\pi$
$53$			1.55304i			0.213327i	−0.994295	+	0.106663i	$0.965983\pi$
$54$		0			0
$55$	2.81132	+	13.0216i	0.379079	+	1.75583i
$56$		0			0
$57$			4.92917i			0.652885i
$58$		0			0
$59$	−14.2977			−1.86140			−0.930699	−	0.365787i	$-0.880800\pi$
$59$	−14.2977			−1.86140			−0.930699	+	0.365787i	$0.880800\pi$
$60$		0			0
$61$	2.79653			0.358059			0.179030	−	0.983844i	$-0.442704\pi$
$61$	2.79653			0.358059			0.179030	+	0.983844i	$0.442704\pi$
$62$		0			0
$63$		−	1.05400i		−	0.132791i
$64$		0			0
$65$	−12.5815	+	2.71631i	−1.56055	+	0.336917i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	5.41155			0.651474
$70$		0			0
$71$	−14.7028			−1.74490			−0.872451	−	0.488701i	$-0.837471\pi$
$71$	−14.7028			−1.74490			−0.872451	+	0.488701i	$0.837471\pi$
$72$		0			0
$73$			7.57857i			0.887005i	0.896273	+	0.443502i	$0.146264\pi$
$73$			7.57857i			0.887005i	−0.896273	+	0.443502i	$0.853736\pi$
$74$		0			0
$75$	−2.06282	−	4.55464i	−0.238194	−	0.525925i
$76$		0			0
$77$		−	6.27931i		−	0.715593i
$78$		0			0
$79$	11.8213			1.33000			0.665002	−	0.746842i	$-0.268431\pi$
$79$	11.8213			1.33000			0.665002	+	0.746842i	$0.268431\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			17.4407i			1.91436i	0.289488	+	0.957182i	$0.406515\pi$
$83$			17.4407i			1.91436i	−0.289488	+	0.957182i	$0.593485\pi$
$84$		0			0
$85$	10.0742	−	2.17498i	1.09270	−	0.235910i
$86$		0			0
$87$		−	2.23932i		−	0.240080i
$88$		0			0
$89$	1.38538			0.146850			0.0734250	−	0.997301i	$-0.476607\pi$
$89$	1.38538			0.146850			0.0734250	+	0.997301i	$0.476607\pi$
$90$		0			0
$91$	6.06709			0.636004
$92$		0			0
$93$		−	3.73316i		−	0.387111i
$94$		0			0
$95$	−2.32601	−	10.7737i	−0.238644	−	1.10536i
$96$		0			0
$97$			2.06742i			0.209915i	0.994477	+	0.104957i	$0.0334706\pi$
$97$			2.06742i			0.209915i	−0.994477	+	0.104957i	$0.966529\pi$
$98$		0			0
$99$	5.95761			0.598763
$100$		0			0
$101$	4.77954			0.475582			0.237791	−	0.971316i	$-0.423577\pi$
$101$	4.77954			0.475582			0.237791	+	0.971316i	$0.423577\pi$
$102$		0			0
$103$			5.44398i			0.536411i	0.963362	+	0.268206i	$0.0864306\pi$
$103$			5.44398i			0.536411i	−0.963362	+	0.268206i	$0.913569\pi$
$104$		0			0
$105$	0.497368	+	2.30373i	0.0485381	+	0.224821i
$106$		0			0
$107$			3.33646i			0.322548i	0.986910	+	0.161274i	$0.0515603\pi$
$107$			3.33646i			0.322548i	−0.986910	+	0.161274i	$0.948440\pi$
$108$		0			0
$109$	−6.41914			−0.614842			−0.307421	−	0.951574i	$-0.599466\pi$
$109$	−6.41914			−0.614842			−0.307421	+	0.951574i	$0.599466\pi$
$110$		0			0
$111$	−10.8869			−1.03334
$112$		0			0
$113$			12.8095i			1.20501i	0.798114	+	0.602507i	$0.205831\pi$
$113$			12.8095i			1.20501i	−0.798114	+	0.602507i	$0.794169\pi$
$114$		0			0
$115$	−11.8281	+	2.55364i	−1.10297	+	0.238128i
$116$		0			0
$117$			5.75627i			0.532168i
$118$		0			0
$119$	−4.85799			−0.445332
$120$		0			0
$121$	24.4932			2.22665
$122$		0			0
$123$			12.0866i			1.08981i
$124$		0			0
$125$	6.65800	+	8.98171i	0.595509	+	0.803348i
$126$		0			0
$127$			7.82063i			0.693969i	0.937871	+	0.346984i	$0.112794\pi$
$127$			7.82063i			0.693969i	−0.937871	+	0.346984i	$0.887206\pi$
$128$		0			0
$129$	6.81718			0.600219
$130$		0			0
$131$	−11.0662			−0.966860			−0.483430	−	0.875383i	$-0.660609\pi$
$131$	−11.0662			−0.966860			−0.483430	+	0.875383i	$0.660609\pi$
$132$		0			0
$133$			5.19533i			0.450492i
$134$		0			0
$135$	−2.18571	+	0.471887i	−0.188116	+	0.0406136i
$136$		0			0
$137$			3.15865i			0.269861i	0.990855	+	0.134931i	$0.0430812\pi$
$137$			3.15865i			0.269861i	−0.990855	+	0.134931i	$0.956919\pi$
$138$		0			0
$139$	−12.8408			−1.08915			−0.544573	−	0.838714i	$-0.683308\pi$
$139$	−12.8408			−1.08915			−0.544573	+	0.838714i	$0.683308\pi$
$140$		0			0
$141$	−8.17727			−0.688650
$142$		0			0
$143$			34.2937i			2.86778i
$144$		0			0
$145$	1.05671	+	4.89450i	0.0877547	+	0.406466i
$146$		0			0
$147$			5.88909i			0.485724i
$148$		0			0
$149$	16.5610			1.35673			0.678365	−	0.734725i	$-0.262689\pi$
$149$	16.5610			1.35673			0.678365	+	0.734725i	$0.262689\pi$
$150$		0			0
$151$	−4.96949			−0.404411			−0.202206	−	0.979343i	$-0.564811\pi$
$151$	−4.96949			−0.404411			−0.202206	+	0.979343i	$0.564811\pi$
$152$		0			0
$153$		−	4.60912i		−	0.372625i
$154$		0			0
$155$	1.76163	+	8.15960i	0.141498	+	0.655395i
$156$		0			0
$157$		−	9.21604i		−	0.735520i	−0.929921	−	0.367760i	$-0.880125\pi$
$157$		−	9.21604i		−	0.735520i	0.929921	−	0.367760i	$-0.119875\pi$
$158$		0			0
$159$	−1.55304			−0.123164
$160$		0			0
$161$	5.70376			0.449519
$162$		0			0
$163$		−	1.88270i		−	0.147465i	−0.997278	−	0.0737324i	$-0.976509\pi$
$163$		−	1.88270i		−	0.147465i	0.997278	−	0.0737324i	$-0.0234911\pi$
$164$		0			0
$165$	−13.0216	+	2.81132i	−1.01373	+	0.218861i
$166$		0			0
$167$			6.39200i			0.494628i	0.968935	+	0.247314i	$0.0795479\pi$
$167$			6.39200i			0.494628i	−0.968935	+	0.247314i	$0.920452\pi$
$168$		0			0
$169$	−20.1347			−1.54882
$170$		0			0
$171$	−4.92917			−0.376943
$172$		0			0
$173$		−	14.0950i		−	1.07162i	−0.844339	−	0.535810i	$-0.820006\pi$
$173$		−	14.0950i		−	1.07162i	0.844339	−	0.535810i	$-0.179994\pi$
$174$		0			0
$175$	−2.17420	−	4.80058i	−0.164354	−	0.362890i
$176$		0			0
$177$		−	14.2977i		−	1.07468i
$178$		0			0
$179$	−6.33808			−0.473730			−0.236865	−	0.971543i	$-0.576120\pi$
$179$	−6.33808			−0.473730			−0.236865	+	0.971543i	$0.576120\pi$
$180$		0			0
$181$	10.5417			0.783558			0.391779	−	0.920059i	$-0.371860\pi$
$181$	10.5417			0.783558			0.391779	+	0.920059i	$0.371860\pi$
$182$		0			0
$183$			2.79653i			0.206726i
$184$		0			0
$185$	23.7956	−	5.13738i	1.74948	−	0.377708i
$186$		0			0
$187$		−	27.4593i		−	2.00803i
$188$		0			0
$189$	1.05400			0.0766670
$190$		0			0
$191$	17.3720			1.25699			0.628496	−	0.777813i	$-0.283671\pi$
$191$	17.3720			1.25699			0.628496	+	0.777813i	$0.283671\pi$
$192$		0			0
$193$		−	1.58895i		−	0.114375i	−0.998363	−	0.0571875i	$-0.981787\pi$
$193$		−	1.58895i		−	0.114375i	0.998363	−	0.0571875i	$-0.0182133\pi$
$194$		0			0
$195$	−2.71631	−	12.5815i	−0.194519	−	0.900983i
$196$		0			0
$197$			3.54762i			0.252757i	0.991982	+	0.126379i	$0.0403354\pi$
$197$			3.54762i			0.252757i	−0.991982	+	0.126379i	$0.959665\pi$
$198$		0			0
$199$	16.2568			1.15241			0.576206	−	0.817305i	$-0.304533\pi$
$199$	16.2568			1.15241			0.576206	+	0.817305i	$0.304533\pi$
$200$		0			0
$201$	−1.00000			−0.0705346
$202$		0			0
$203$		−	2.36024i		−	0.165656i
$204$		0			0
$205$	−5.70352	−	26.4178i	−0.398351	−	1.84510i
$206$		0			0
$207$			5.41155i			0.376129i
$208$		0			0
$209$	−29.3661			−2.03130
$210$		0			0
$211$	−21.6131			−1.48791			−0.743955	−	0.668230i	$-0.767052\pi$
$211$	−21.6131			−1.48791			−0.743955	+	0.668230i	$0.767052\pi$
$212$		0			0
$213$		−	14.7028i		−	1.00742i
$214$		0			0
$215$	−14.9004	+	3.21694i	−1.01620	+	0.219393i
$216$		0			0
$217$		−	3.93474i		−	0.267107i
$218$		0			0
$219$	−7.57857			−0.512112
$220$		0			0
$221$	26.5313			1.78469
$222$		0			0
$223$			16.2068i			1.08529i	0.839963	+	0.542644i	$0.182577\pi$
$223$			16.2068i			1.08529i	−0.839963	+	0.542644i	$0.817423\pi$
$224$		0			0
$225$	4.55464	−	2.06282i	0.303643	−	0.137521i
$226$		0			0
$227$			27.5854i			1.83091i	0.402424	+	0.915453i	$0.368168\pi$
$227$			27.5854i			1.83091i	−0.402424	+	0.915453i	$0.631832\pi$
$228$		0			0
$229$	7.79803			0.515309			0.257654	−	0.966237i	$-0.417050\pi$
$229$	7.79803			0.515309			0.257654	+	0.966237i	$0.417050\pi$
$230$		0			0
$231$	6.27931			0.413148
$232$		0			0
$233$			0.283397i			0.0185660i	0.999957	+	0.00928299i	$0.00295491\pi$
$233$			0.283397i			0.0185660i	−0.999957	+	0.00928299i	$0.997045\pi$
$234$		0			0
$235$	17.8731	−	3.85875i	1.16591	−	0.251717i
$236$		0			0
$237$			11.8213i			0.767878i
$238$		0			0
$239$	−2.65421			−0.171687			−0.0858433	−	0.996309i	$-0.527358\pi$
$239$	−2.65421			−0.171687			−0.0858433	+	0.996309i	$0.527358\pi$
$240$		0			0
$241$	9.03842			0.582216			0.291108	−	0.956690i	$-0.405976\pi$
$241$	9.03842			0.582216			0.291108	+	0.956690i	$0.405976\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−2.77899	−	12.8718i	−0.177543	−	0.822351i
$246$		0			0
$247$		−	28.3737i		−	1.80537i
$248$		0			0
$249$	−17.4407			−1.10526
$250$		0			0
$251$	−6.51967			−0.411518			−0.205759	−	0.978603i	$-0.565966\pi$
$251$	−6.51967			−0.411518			−0.205759	+	0.978603i	$0.565966\pi$
$252$		0			0
$253$			32.2399i			2.02691i
$254$		0			0
$255$	2.17498	+	10.0742i	0.136203	+	0.630870i
$256$		0			0
$257$			20.0865i			1.25296i	0.779438	+	0.626479i	$0.215505\pi$
$257$			20.0865i			1.25296i	−0.779438	+	0.626479i	$0.784495\pi$
$258$		0			0
$259$	−11.4747			−0.713006
$260$		0			0
$261$	2.23932			0.138610
$262$		0			0
$263$		−	24.5886i		−	1.51620i	−0.652139	−	0.758099i	$-0.726128\pi$
$263$		−	24.5886i		−	1.51620i	0.652139	−	0.758099i	$-0.273872\pi$
$264$		0			0
$265$	3.39450	−	0.732861i	0.208522	−	0.0450193i
$266$		0			0
$267$			1.38538i			0.0847839i
$268$		0			0
$269$	−31.4202			−1.91572			−0.957861	−	0.287233i	$-0.907265\pi$
$269$	−31.4202			−1.91572			−0.957861	+	0.287233i	$0.907265\pi$
$270$		0			0
$271$	26.7654			1.62588			0.812942	−	0.582344i	$-0.197864\pi$
$271$	26.7654			1.62588			0.812942	+	0.582344i	$0.197864\pi$
$272$		0			0
$273$			6.06709i			0.367197i
$274$		0			0
$275$	27.1348	−	12.2895i	1.63629	−	0.741083i
$276$		0			0
$277$			14.2357i			0.855342i	0.903935	+	0.427671i	$0.140666\pi$
$277$			14.2357i			0.855342i	−0.903935	+	0.427671i	$0.859334\pi$
$278$		0			0
$279$	3.73316			0.223498
$280$		0			0
$281$	−23.1025			−1.37818			−0.689090	−	0.724676i	$-0.741989\pi$
$281$	−23.1025			−1.37818			−0.689090	+	0.724676i	$0.741989\pi$
$282$		0			0
$283$			0.425677i			0.0253039i	0.999920	+	0.0126519i	$0.00402734\pi$
$283$			0.425677i			0.0253039i	−0.999920	+	0.0126519i	$0.995973\pi$
$284$		0			0
$285$	10.7737	−	2.32601i	0.638181	−	0.137781i
$286$		0			0
$287$			12.7392i			0.751974i
$288$		0			0
$289$	−4.24396			−0.249645
$290$		0			0
$291$	−2.06742			−0.121194
$292$		0			0
$293$			8.67159i			0.506600i	0.967388	+	0.253300i	$0.0815160\pi$
$293$			8.67159i			0.506600i	−0.967388	+	0.253300i	$0.918484\pi$
$294$		0			0
$295$	6.74689	+	31.2505i	0.392819	+	1.81948i
$296$		0			0
$297$			5.95761i			0.345696i
$298$		0			0
$299$	−31.1504			−1.80147
$300$		0			0
$301$	7.18528			0.414153
$302$		0			0
$303$			4.77954i			0.274577i
$304$		0			0
$305$	−1.31965	−	6.11240i	−0.0755628	−	0.349995i
$306$		0			0
$307$		−	25.6728i		−	1.46522i	−0.680646	−	0.732612i	$-0.738301\pi$
$307$		−	25.6728i		−	1.46522i	0.680646	−	0.732612i	$-0.261699\pi$
$308$		0			0
$309$	−5.44398			−0.309697
$310$		0			0
$311$	−5.83109			−0.330651			−0.165325	−	0.986239i	$-0.552867\pi$
$311$	−5.83109			−0.330651			−0.165325	+	0.986239i	$0.552867\pi$
$312$		0			0
$313$			32.0059i			1.80908i	0.426389	+	0.904540i	$0.359786\pi$
$313$			32.0059i			1.80908i	−0.426389	+	0.904540i	$0.640214\pi$
$314$		0			0
$315$	−2.30373	+	0.497368i	−0.129800	+	0.0280235i
$316$		0			0
$317$			27.5282i			1.54614i	0.634323	+	0.773068i	$0.281279\pi$
$317$			27.5282i			1.54614i	−0.634323	+	0.773068i	$0.718721\pi$
$318$		0			0
$319$	13.3410			0.746953
$320$		0			0
$321$	−3.33646			−0.186223
$322$		0			0
$323$			22.7191i			1.26413i
$324$		0			0
$325$	11.8741	+	26.2178i	0.658659	+	1.45430i
$326$		0			0
$327$		−	6.41914i		−	0.354979i
$328$		0			0
$329$	−8.61881			−0.475170
$330$		0			0
$331$	−6.60882			−0.363254			−0.181627	−	0.983368i	$-0.558136\pi$
$331$	−6.60882			−0.363254			−0.181627	+	0.983368i	$0.558136\pi$
$332$		0			0
$333$		−	10.8869i		−	0.596598i
$334$		0			0
$335$	2.18571	−	0.471887i	0.119418	−	0.0257820i
$336$		0			0
$337$		−	10.8040i		−	0.588530i	−0.955724	−	0.294265i	$-0.904925\pi$
$337$		−	10.8040i		−	0.588530i	0.955724	−	0.294265i	$-0.0950749\pi$
$338$		0			0
$339$	−12.8095			−0.695715
$340$		0			0
$341$	22.2407			1.20440
$342$		0			0
$343$			13.5851i			0.733524i
$344$		0			0
$345$	−2.55364	−	11.8281i	−0.137483	−	0.636802i
$346$		0			0
$347$		−	23.1980i		−	1.24533i	−0.782488	−	0.622666i	$-0.786049\pi$
$347$		−	23.1980i		−	1.24533i	0.782488	−	0.622666i	$-0.213951\pi$
$348$		0			0
$349$	−1.29425			−0.0692795			−0.0346398	−	0.999400i	$-0.511028\pi$
$349$	−1.29425			−0.0692795			−0.0346398	+	0.999400i	$0.511028\pi$
$350$		0			0
$351$	−5.75627			−0.307247
$352$		0			0
$353$		−	13.6477i		−	0.726393i	−0.931713	−	0.363197i	$-0.881685\pi$
$353$		−	13.6477i		−	0.726393i	0.931713	−	0.363197i	$-0.118315\pi$
$354$		0			0
$355$	6.93807	+	32.1361i	0.368235	+	1.70560i
$356$		0			0
$357$		−	4.85799i		−	0.257112i
$358$		0			0
$359$	24.2243			1.27851			0.639255	−	0.768995i	$-0.279243\pi$
$359$	24.2243			1.27851			0.639255	+	0.768995i	$0.279243\pi$
$360$		0			0
$361$	5.29673			0.278775
$362$		0			0
$363$			24.4932i			1.28556i
$364$		0			0
$365$	16.5645	−	3.57623i	0.867028	−	0.187189i
$366$		0			0
$367$			21.1096i			1.10191i	0.834535	+	0.550955i	$0.185736\pi$
$367$			21.1096i			1.10191i	−0.834535	+	0.550955i	$0.814264\pi$
$368$		0			0
$369$	−12.0866			−0.629203
$370$		0			0
$371$	−1.63690			−0.0849837
$372$		0			0
$373$		−	1.16636i		−	0.0603921i	−0.999544	−	0.0301960i	$-0.990387\pi$
$373$		−	1.16636i		−	0.0603921i	0.999544	−	0.0301960i	$-0.00961316\pi$
$374$		0			0
$375$	−8.98171	+	6.65800i	−0.463813	+	0.343817i
$376$		0			0
$377$			12.8901i			0.663876i
$378$		0			0
$379$	−23.3326			−1.19851			−0.599257	−	0.800556i	$-0.704537\pi$
$379$	−23.3326			−1.19851			−0.599257	+	0.800556i	$0.704537\pi$
$380$		0			0
$381$	−7.82063			−0.400663
$382$		0			0
$383$			21.1343i			1.07991i	0.841693	+	0.539956i	$0.181559\pi$
$383$			21.1343i			1.07991i	−0.841693	+	0.539956i	$0.818441\pi$
$384$		0			0
$385$	−13.7247	+	2.96313i	−0.699477	+	0.151015i
$386$		0			0
$387$			6.81718i			0.346536i
$388$		0			0
$389$	18.4327			0.934574			0.467287	−	0.884106i	$-0.345232\pi$
$389$	18.4327			0.934574			0.467287	+	0.884106i	$0.345232\pi$
$390$		0			0
$391$	24.9425			1.26139
$392$		0			0
$393$		−	11.0662i		−	0.558217i
$394$		0			0
$395$	−5.57834	−	25.8380i	−0.280677	−	1.30005i
$396$		0			0
$397$		−	5.41387i		−	0.271714i	−0.990728	−	0.135857i	$-0.956621\pi$
$397$		−	5.41387i		−	0.271714i	0.990728	−	0.135857i	$-0.0433788\pi$
$398$		0			0
$399$	−5.19533			−0.260092
$400$		0			0
$401$	−33.7402			−1.68491			−0.842453	−	0.538770i	$-0.818889\pi$
$401$	−33.7402			−1.68491			−0.842453	+	0.538770i	$0.818889\pi$
$402$		0			0
$403$			21.4891i			1.07045i
$404$		0			0
$405$	−0.471887	−	2.18571i	−0.0234483	−	0.108609i
$406$		0			0
$407$		−	64.8599i		−	3.21498i
$408$		0			0
$409$	−7.47591			−0.369660			−0.184830	−	0.982771i	$-0.559173\pi$
$409$	−7.47591			−0.369660			−0.184830	+	0.982771i	$0.559173\pi$
$410$		0			0
$411$	−3.15865			−0.155805
$412$		0			0
$413$		−	15.0697i		−	0.741531i
$414$		0			0
$415$	38.1202	−	8.23003i	1.87125	−	0.403997i
$416$		0			0
$417$		−	12.8408i		−	0.628819i
$418$		0			0
$419$	−10.5414			−0.514983			−0.257491	−	0.966281i	$-0.582896\pi$
$419$	−10.5414			−0.514983			−0.257491	+	0.966281i	$0.582896\pi$
$420$		0			0
$421$	−15.7034			−0.765338			−0.382669	−	0.923886i	$-0.624995\pi$
$421$	−15.7034			−0.765338			−0.382669	+	0.923886i	$0.624995\pi$
$422$		0			0
$423$		−	8.17727i		−	0.397592i
$424$		0			0
$425$	−9.50776	−	20.9929i	−0.461194	−	1.01830i
$426$		0			0
$427$			2.94754i			0.142641i
$428$		0			0
$429$	−34.2937			−1.65571
$430$		0			0
$431$	10.8294			0.521633			0.260816	−	0.965388i	$-0.416008\pi$
$431$	10.8294			0.521633			0.260816	+	0.965388i	$0.416008\pi$
$432$		0			0
$433$			31.8147i			1.52892i	0.644673	+	0.764459i	$0.276994\pi$
$433$			31.8147i			1.52892i	−0.644673	+	0.764459i	$0.723006\pi$
$434$		0			0
$435$	−4.89450	+	1.05671i	−0.234673	+	0.0506652i
$436$		0			0
$437$		−	26.6745i		−	1.27601i
$438$		0			0
$439$	10.0051			0.477517			0.238758	−	0.971079i	$-0.423260\pi$
$439$	10.0051			0.477517			0.238758	+	0.971079i	$0.423260\pi$
$440$		0			0
$441$	−5.88909			−0.280433
$442$		0			0
$443$		−	30.2793i		−	1.43861i	−0.694693	−	0.719306i	$-0.744460\pi$
$443$		−	30.2793i		−	1.43861i	0.694693	−	0.719306i	$-0.255540\pi$
$444$		0			0
$445$	−0.653743	−	3.02804i	−0.0309904	−	0.143543i
$446$		0			0
$447$			16.5610i			0.783309i
$448$		0			0
$449$	−28.9478			−1.36613			−0.683066	−	0.730357i	$-0.739354\pi$
$449$	−28.9478			−1.36613			−0.683066	+	0.730357i	$0.739354\pi$
$450$		0			0
$451$	−72.0073			−3.39069
$452$		0			0
$453$		−	4.96949i		−	0.233487i
$454$		0			0
$455$	−2.86299	−	13.2609i	−0.134219	−	0.621681i
$456$		0			0
$457$		−	13.2130i		−	0.618079i	−0.951049	−	0.309039i	$-0.899993\pi$
$457$		−	13.2130i		−	0.618079i	0.951049	−	0.309039i	$-0.100007\pi$
$458$		0			0
$459$	4.60912			0.215135
$460$		0			0
$461$	−12.4553			−0.580100			−0.290050	−	0.957011i	$-0.593672\pi$
$461$	−12.4553			−0.580100			−0.290050	+	0.957011i	$0.593672\pi$
$462$		0			0
$463$		−	7.23535i		−	0.336255i	−0.985765	−	0.168128i	$-0.946228\pi$
$463$		−	7.23535i		−	0.336255i	0.985765	−	0.168128i	$-0.0537721\pi$
$464$		0			0
$465$	−8.15960	+	1.76163i	−0.378392	+	0.0816937i
$466$		0			0
$467$			12.0723i			0.558641i	0.960198	+	0.279321i	$0.0901092\pi$
$467$			12.0723i			0.558641i	−0.960198	+	0.279321i	$0.909891\pi$
$468$		0			0
$469$	−1.05400			−0.0486690
$470$		0			0
$471$	9.21604			0.424653
$472$		0			0
$473$			40.6141i			1.86744i
$474$		0			0
$475$	−22.4506	+	10.1680i	−1.03011	+	0.466539i
$476$		0			0
$477$		−	1.55304i		−	0.0711090i
$478$		0			0
$479$	−21.3952			−0.977570			−0.488785	−	0.872404i	$-0.662560\pi$
$479$	−21.3952			−0.977570			−0.488785	+	0.872404i	$0.662560\pi$
$480$		0			0
$481$	62.6679			2.85741
$482$		0			0
$483$			5.70376i			0.259530i
$484$		0			0
$485$	4.51878	−	0.975589i	0.205187	−	0.0442992i
$486$		0			0
$487$			38.9047i			1.76294i	0.472240	+	0.881470i	$0.343446\pi$
$487$			38.9047i			1.76294i	−0.472240	+	0.881470i	$0.656554\pi$
$488$		0			0
$489$	1.88270			0.0851388
$490$		0			0
$491$	22.6008			1.01996			0.509979	−	0.860187i	$-0.329653\pi$
$491$	22.6008			1.01996			0.509979	+	0.860187i	$0.329653\pi$
$492$		0			0
$493$		−	10.3213i		−	0.464848i
$494$		0			0
$495$	−2.81132	−	13.0216i	−0.126360	−	0.585278i
$496$		0			0
$497$		−	15.4967i		−	0.695123i
$498$		0			0
$499$	22.3659			1.00124			0.500618	−	0.865668i	$-0.333106\pi$
$499$	22.3659			1.00124			0.500618	+	0.865668i	$0.333106\pi$
$500$		0			0
$501$	−6.39200			−0.285573
$502$		0			0
$503$		−	19.1497i		−	0.853842i	−0.904289	−	0.426921i	$-0.859598\pi$
$503$		−	19.1497i		−	0.853842i	0.904289	−	0.426921i	$-0.140402\pi$
$504$		0			0
$505$	−2.25540	−	10.4467i	−0.100364	−	0.464871i
$506$		0			0
$507$		−	20.1347i		−	0.894213i
$508$		0			0
$509$	−27.5130			−1.21949			−0.609745	−	0.792598i	$-0.708728\pi$
$509$	−27.5130			−1.21949			−0.609745	+	0.792598i	$0.708728\pi$
$510$		0			0
$511$	−7.98779			−0.353359
$512$		0			0
$513$		−	4.92917i		−	0.217628i
$514$		0			0
$515$	11.8990	−	2.56894i	0.524330	−	0.113201i
$516$		0			0
$517$		−	48.7170i		−	2.14257i
$518$		0			0
$519$	14.0950			0.618700
$520$		0			0
$521$	−17.1764			−0.752513			−0.376257	−	0.926515i	$-0.622789\pi$
$521$	−17.1764			−0.752513			−0.376257	+	0.926515i	$0.622789\pi$
$522$		0			0
$523$		−	11.0736i		−	0.484216i	−0.970249	−	0.242108i	$-0.922161\pi$
$523$		−	11.0736i		−	0.484216i	0.970249	−	0.242108i	$-0.0778389\pi$
$524$		0			0
$525$	4.80058	−	2.17420i	0.209515	−	0.0948899i
$526$		0			0
$527$		−	17.2066i		−	0.749530i
$528$		0			0
$529$	−6.28487			−0.273255
$530$		0			0
$531$	14.2977			0.620466
$532$		0			0
$533$		−	69.5738i		−	3.01358i
$534$		0			0
$535$	7.29253	−	1.57443i	0.315284	−	0.0680687i
$536$		0			0
$537$		−	6.33808i		−	0.273508i
$538$		0			0
$539$	−35.0849			−1.51122
$540$		0			0
$541$	40.8480			1.75619			0.878097	−	0.478483i	$-0.158813\pi$
$541$	40.8480			1.75619			0.878097	+	0.478483i	$0.158813\pi$
$542$		0			0
$543$			10.5417i			0.452387i
$544$		0			0
$545$	3.02911	+	14.0304i	0.129753	+	0.600995i
$546$		0			0
$547$			0.246608i			0.0105442i	0.999986	+	0.00527210i	$0.00167817\pi$
$547$			0.246608i			0.0105442i	−0.999986	+	0.00527210i	$0.998322\pi$
$548$		0			0
$549$	−2.79653			−0.119353
$550$		0			0
$551$	−11.0380			−0.470234
$552$		0			0
$553$			12.4596i			0.529838i
$554$		0			0
$555$	5.13738	+	23.7956i	0.218070	+	1.01007i
$556$		0			0
$557$			34.2129i			1.44965i	0.688935	+	0.724823i	$0.258078\pi$
$557$			34.2129i			1.44965i	−0.688935	+	0.724823i	$0.741922\pi$
$558$		0			0
$559$	−39.2415			−1.65974
$560$		0			0
$561$	27.4593			1.15933
$562$		0			0
$563$		−	23.6249i		−	0.995670i	−0.867272	−	0.497835i	$-0.834129\pi$
$563$		−	23.6249i		−	0.995670i	0.867272	−	0.497835i	$-0.165871\pi$
$564$		0			0
$565$	27.9978	−	6.04463i	1.17788	−	0.254299i
$566$		0			0
$567$			1.05400i			0.0442637i
$568$		0			0
$569$	30.1495			1.26393			0.631966	−	0.774996i	$-0.282248\pi$
$569$	30.1495			1.26393			0.631966	+	0.774996i	$0.282248\pi$
$570$		0			0
$571$	−21.5928			−0.903630			−0.451815	−	0.892112i	$-0.649223\pi$
$571$	−21.5928			−0.903630			−0.451815	+	0.892112i	$0.649223\pi$
$572$		0			0
$573$			17.3720i			0.725725i
$574$		0			0
$575$	11.1630	+	24.6477i	0.465531	+	1.02788i
$576$		0			0
$577$			11.7363i			0.488587i	0.969701	+	0.244293i	$0.0785560\pi$
$577$			11.7363i			0.488587i	−0.969701	+	0.244293i	$0.921444\pi$
$578$		0			0
$579$	1.58895			0.0660344
$580$		0			0
$581$	−18.3824			−0.762631
$582$		0			0
$583$		−	9.25243i		−	0.383197i
$584$		0			0
$585$	12.5815	−	2.71631i	0.520183	−	0.112306i
$586$		0			0
$587$			20.3837i			0.841324i	0.907218	+	0.420662i	$0.138202\pi$
$587$			20.3837i			0.841324i	−0.907218	+	0.420662i	$0.861798\pi$
$588$		0			0
$589$	−18.4014			−0.758216
$590$		0			0
$591$	−3.54762			−0.145930
$592$		0			0
$593$		−	9.43935i		−	0.387627i	−0.981038	−	0.193814i	$-0.937914\pi$
$593$		−	9.43935i		−	0.387627i	0.981038	−	0.193814i	$-0.0620857\pi$
$594$		0			0
$595$	2.29243	+	10.6182i	0.0939803	+	0.435302i
$596$		0			0
$597$			16.2568i			0.665345i
$598$		0			0
$599$	−19.8634			−0.811597			−0.405799	−	0.913962i	$-0.633007\pi$
$599$	−19.8634			−0.811597			−0.405799	+	0.913962i	$0.633007\pi$
$600$		0			0
$601$	10.7313			0.437738			0.218869	−	0.975754i	$-0.429763\pi$
$601$	10.7313			0.437738			0.218869	+	0.975754i	$0.429763\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	−11.5580	−	53.5349i	−0.469900	−	2.17650i
$606$		0			0
$607$		−	11.0936i		−	0.450277i	−0.974327	−	0.225138i	$-0.927717\pi$
$607$		−	11.0936i		−	0.450277i	0.974327	−	0.225138i	$-0.0722834\pi$
$608$		0			0
$609$	2.36024			0.0956416
$610$		0			0
$611$	47.0706			1.90427
$612$		0			0
$613$		−	47.3529i		−	1.91257i	−0.292443	−	0.956283i	$-0.594468\pi$
$613$		−	47.3529i		−	1.91257i	0.292443	−	0.956283i	$-0.405532\pi$
$614$		0			0
$615$	26.4178	−	5.70352i	1.06527	−	0.229988i
$616$		0			0
$617$		−	10.2495i		−	0.412628i	−0.978486	−	0.206314i	$-0.933853\pi$
$617$		−	10.2495i		−	0.412628i	0.978486	−	0.206314i	$-0.0661469\pi$
$618$		0			0
$619$	17.3560			0.697598			0.348799	−	0.937197i	$-0.386590\pi$
$619$	17.3560			0.697598			0.348799	+	0.937197i	$0.386590\pi$
$620$		0			0
$621$	−5.41155			−0.217158
$622$		0			0
$623$			1.46019i			0.0585011i
$624$		0			0
$625$	16.4896	−	18.7908i	0.659583	−	0.751632i
$626$		0			0
$627$		−	29.3661i		−	1.17277i
$628$		0			0
$629$	−50.1789			−2.00077
$630$		0			0
$631$	21.3285			0.849074			0.424537	−	0.905411i	$-0.360437\pi$
$631$	21.3285			0.849074			0.424537	+	0.905411i	$0.360437\pi$
$632$		0			0
$633$		−	21.6131i		−	0.859045i
$634$		0			0
$635$	17.0936	−	3.69046i	0.678340	−	0.146451i
$636$		0			0
$637$		−	33.8992i		−	1.34314i
$638$		0			0
$639$	14.7028			0.581634
$640$		0			0
$641$	1.63011			0.0643855			0.0321928	−	0.999482i	$-0.489751\pi$
$641$	1.63011			0.0643855			0.0321928	+	0.999482i	$0.489751\pi$
$642$		0			0
$643$			4.46775i			0.176191i	0.996112	+	0.0880955i	$0.0280781\pi$
$643$			4.46775i			0.176191i	−0.996112	+	0.0880955i	$0.971922\pi$
$644$		0			0
$645$	−3.21694	−	14.9004i	−0.126667	−	0.586701i
$646$		0			0
$647$			24.0164i			0.944184i	0.881549	+	0.472092i	$0.156501\pi$
$647$			24.0164i			0.944184i	−0.881549	+	0.472092i	$0.843499\pi$
$648$		0			0
$649$	85.1800			3.34361
$650$		0			0
$651$	3.93474			0.154215
$652$		0			0
$653$			7.55753i			0.295749i	0.989006	+	0.147875i	$0.0472432\pi$
$653$			7.55753i			0.295749i	−0.989006	+	0.147875i	$0.952757\pi$
$654$		0			0
$655$	5.22201	+	24.1875i	0.204041	+	0.945085i
$656$		0			0
$657$		−	7.57857i		−	0.295668i
$658$		0			0
$659$	28.1442			1.09634			0.548172	−	0.836366i	$-0.315324\pi$
$659$	28.1442			1.09634			0.548172	+	0.836366i	$0.315324\pi$
$660$		0			0
$661$	−10.5584			−0.410675			−0.205338	−	0.978691i	$-0.565829\pi$
$661$	−10.5584			−0.410675			−0.205338	+	0.978691i	$0.565829\pi$
$662$		0			0
$663$			26.5313i			1.03039i
$664$		0			0
$665$	11.3555	−	2.45161i	0.440347	−	0.0950694i
$666$		0			0
$667$			12.1182i			0.469218i
$668$		0			0
$669$	−16.2068			−0.626591
$670$		0			0
$671$	−16.6607			−0.643178
$672$		0			0
$673$		−	33.5266i		−	1.29235i	−0.763188	−	0.646177i	$-0.776367\pi$
$673$		−	33.5266i		−	1.29235i	0.763188	−	0.646177i	$-0.223633\pi$
$674$		0			0
$675$	2.06282	+	4.55464i	0.0793978	+	0.175308i
$676$		0			0
$677$		−	3.27727i		−	0.125956i	−0.998015	−	0.0629778i	$-0.979940\pi$
$677$		−	3.27727i		−	0.125956i	0.998015	−	0.0629778i	$-0.0200597\pi$
$678$		0			0
$679$	−2.17905			−0.0836244
$680$		0			0
$681$	−27.5854			−1.05707
$682$		0			0
$683$		−	31.7957i		−	1.21663i	−0.793697	−	0.608314i	$-0.791846\pi$
$683$		−	31.7957i		−	1.21663i	0.793697	−	0.608314i	$-0.208154\pi$
$684$		0			0
$685$	6.90388	−	1.49053i	0.263784	−	0.0569500i
$686$		0			0
$687$			7.79803i			0.297514i
$688$		0			0
$689$	8.93974			0.340577
$690$		0			0
$691$	13.4097			0.510130			0.255065	−	0.966924i	$-0.417903\pi$
$691$	13.4097			0.510130			0.255065	+	0.966924i	$0.417903\pi$
$692$		0			0
$693$			6.27931i			0.238531i
$694$		0			0
$695$	6.05943	+	28.0663i	0.229847	+	1.06462i
$696$		0			0
$697$			55.7086i			2.11011i
$698$		0			0
$699$	−0.283397			−0.0107191
$700$		0			0
$701$	2.29925			0.0868415			0.0434207	−	0.999057i	$-0.486174\pi$
$701$	2.29925			0.0868415			0.0434207	+	0.999057i	$0.486174\pi$
$702$		0			0
$703$			53.6633i			2.02395i
$704$		0			0
$705$	3.85875	+	17.8731i	0.145329	+	0.673141i
$706$		0			0
$707$			5.03762i			0.189459i
$708$		0			0
$709$	30.4751			1.14452			0.572258	−	0.820074i	$-0.306068\pi$
$709$	30.4751			1.14452			0.572258	+	0.820074i	$0.306068\pi$
$710$		0			0
$711$	−11.8213			−0.443335
$712$		0			0
$713$			20.2022i			0.756578i
$714$		0			0
$715$	74.9560	−	16.1827i	2.80319	−	0.605200i
$716$		0			0
$717$		−	2.65421i		−	0.0991233i
$718$		0			0
$719$	−14.2577			−0.531723			−0.265861	−	0.964011i	$-0.585656\pi$
$719$	−14.2577			−0.531723			−0.265861	+	0.964011i	$0.585656\pi$
$720$		0			0
$721$	−5.73794			−0.213692
$722$		0			0
$723$			9.03842i			0.336143i
$724$		0			0
$725$	10.1993	−	4.61931i	0.378793	−	0.171557i
$726$		0			0
$727$			16.4092i			0.608584i	0.952579	+	0.304292i	$0.0984199\pi$
$727$			16.4092i			0.608584i	−0.952579	+	0.304292i	$0.901580\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	31.4212			1.16215
$732$		0			0
$733$		−	2.72078i		−	0.100494i	−0.998737	−	0.0502472i	$-0.983999\pi$
$733$		−	2.72078i		−	0.100494i	0.998737	−	0.0502472i	$-0.0160009\pi$
$734$		0			0
$735$	12.8718	−	2.77899i	0.474785	−	0.102504i
$736$		0			0
$737$		−	5.95761i		−	0.219452i
$738$		0			0
$739$	−20.0392			−0.737153			−0.368577	−	0.929597i	$-0.620155\pi$
$739$	−20.0392			−0.737153			−0.368577	+	0.929597i	$0.620155\pi$
$740$		0			0
$741$	28.3737			1.04233
$742$		0			0
$743$		−	2.34914i		−	0.0861817i	−0.999071	−	0.0430909i	$-0.986280\pi$
$743$		−	2.34914i		−	0.0861817i	0.999071	−	0.0430909i	$-0.0137205\pi$
$744$		0			0
$745$	−7.81493	−	36.1975i	−0.286317	−	1.32617i
$746$		0			0
$747$		−	17.4407i		−	0.638121i
$748$		0			0
$749$	−3.51662			−0.128494
$750$		0			0
$751$	−38.0028			−1.38674			−0.693371	−	0.720581i	$-0.743875\pi$
$751$	−38.0028			−1.38674			−0.693371	+	0.720581i	$0.743875\pi$
$752$		0			0
$753$		−	6.51967i		−	0.237590i
$754$		0			0
$755$	2.34504	+	10.8619i	0.0853447	+	0.395304i
$756$		0			0
$757$		−	9.99948i		−	0.363437i	−0.983351	−	0.181719i	$-0.941834\pi$
$757$		−	9.99948i		−	0.363437i	0.983351	−	0.181719i	$-0.0581660\pi$
$758$		0			0
$759$	−32.2399			−1.17024
$760$		0			0
$761$	15.8632			0.575042			0.287521	−	0.957774i	$-0.407169\pi$
$761$	15.8632			0.575042			0.287521	+	0.957774i	$0.407169\pi$
$762$		0			0
$763$		−	6.76575i		−	0.244937i
$764$		0			0
$765$	−10.0742	+	2.17498i	−0.364233	+	0.0786367i
$766$		0			0
$767$			82.3013i			2.97173i
$768$		0			0
$769$	35.0656			1.26450			0.632248	−	0.774766i	$-0.282132\pi$
$769$	35.0656			1.26450			0.632248	+	0.774766i	$0.282132\pi$
$770$		0			0
$771$	−20.0865			−0.723396
$772$		0			0
$773$		−	17.5430i		−	0.630980i	−0.948929	−	0.315490i	$-0.897831\pi$
$773$		−	17.5430i		−	0.630980i	0.948929	−	0.315490i	$-0.102169\pi$
$774$		0			0
$775$	17.0032	−	7.70082i	0.610774	−	0.276622i
$776$		0			0
$777$		−	11.4747i		−	0.411654i
$778$		0			0
$779$	59.5769			2.13457
$780$		0			0
$781$	87.5937			3.13435
$782$		0			0
$783$			2.23932i			0.0800268i
$784$		0			0
$785$	−20.1436	+	4.34893i	−0.718955	+	0.155220i
$786$		0			0
$787$			19.5528i			0.696981i	0.937312	+	0.348491i	$0.113306\pi$
$787$			19.5528i			0.696981i	−0.937312	+	0.348491i	$0.886694\pi$
$788$		0			0
$789$	24.5886			0.875378
$790$		0			0
$791$	−13.5011			−0.480045
$792$		0			0
$793$		−	16.0976i		−	0.571643i
$794$		0			0
$795$	0.732861	+	3.39450i	0.0259919	+	0.120390i
$796$		0			0
$797$		−	8.89801i		−	0.315184i	−0.987504	−	0.157592i	$-0.949627\pi$
$797$		−	8.89801i		−	0.315184i	0.987504	−	0.157592i	$-0.0503730\pi$
$798$		0			0
$799$	−37.6900			−1.33338
$800$		0			0
$801$	−1.38538			−0.0489500
$802$		0			0
$803$		−	45.1502i		−	1.59332i
$804$		0			0
$805$	−2.69153	−	12.4667i	−0.0948640	−	0.439395i
$806$		0			0
$807$		−	31.4202i		−	1.10604i
$808$		0			0
$809$	30.1130			1.05872			0.529358	−	0.848399i	$-0.322433\pi$
$809$	30.1130			1.05872			0.529358	+	0.848399i	$0.322433\pi$
$810$		0			0
$811$	−4.13516			−0.145205			−0.0726026	−	0.997361i	$-0.523130\pi$
$811$	−4.13516			−0.145205			−0.0726026	+	0.997361i	$0.523130\pi$
$812$		0			0
$813$			26.7654i			0.938705i
$814$		0			0
$815$	−4.11504	+	0.888424i	−0.144144	+	0.0311201i
$816$		0			0
$817$		−	33.6030i		−	1.17562i
$818$		0			0
$819$	−6.06709			−0.212001
$820$		0			0
$821$	6.81459			0.237831			0.118915	−	0.992904i	$-0.462058\pi$
$821$	6.81459			0.237831			0.118915	+	0.992904i	$0.462058\pi$
$822$		0			0
$823$			7.85021i			0.273641i	0.990596	+	0.136821i	$0.0436884\pi$
$823$			7.85021i			0.273641i	−0.990596	+	0.136821i	$0.956312\pi$
$824$		0			0
$825$	12.2895	+	27.1348i	0.427864	+	0.944713i
$826$		0			0
$827$		−	38.7621i		−	1.34789i	−0.738781	−	0.673946i	$-0.764598\pi$
$827$		−	38.7621i		−	1.34789i	0.738781	−	0.673946i	$-0.235402\pi$
$828$		0			0
$829$	7.95665			0.276346			0.138173	−	0.990408i	$-0.455877\pi$
$829$	7.95665			0.276346			0.138173	+	0.990408i	$0.455877\pi$
$830$		0			0
$831$	−14.2357			−0.493832
$832$		0			0
$833$			27.1435i			0.940467i
$834$		0			0
$835$	13.9710	−	3.01630i	0.483488	−	0.104383i
$836$		0			0
$837$			3.73316i			0.129037i
$838$		0			0
$839$	0.479751			0.0165628			0.00828142	−	0.999966i	$-0.497364\pi$
$839$	0.479751			0.0165628			0.00828142	+	0.999966i	$0.497364\pi$
$840$		0			0
$841$	−23.9854			−0.827084
$842$		0			0
$843$		−	23.1025i		−	0.795692i
$844$		0			0
$845$	9.50131	+	44.0086i	0.326855	+	1.51394i
$846$		0			0
$847$			25.8157i			0.887039i
$848$		0			0
$849$	−0.425677			−0.0146092
$850$		0			0
$851$	58.9149			2.01958
$852$		0			0
$853$		−	10.1671i		−	0.348114i	−0.984736	−	0.174057i	$-0.944312\pi$
$853$		−	10.1671i		−	0.348114i	0.984736	−	0.174057i	$-0.0556877\pi$
$854$		0			0
$855$	2.32601	+	10.7737i	0.0795480	+	0.368454i
$856$		0			0
$857$		−	16.1488i		−	0.551632i	−0.961210	−	0.275816i	$-0.911052\pi$
$857$		−	16.1488i		−	0.551632i	0.961210	−	0.275816i	$-0.0889481\pi$
$858$		0			0
$859$	−33.1353			−1.13056			−0.565281	−	0.824898i	$-0.691232\pi$
$859$	−33.1353			−1.13056			−0.565281	+	0.824898i	$0.691232\pi$
$860$		0			0
$861$	−12.7392			−0.434152
$862$		0			0
$863$		−	37.1515i		−	1.26465i	−0.774702	−	0.632326i	$-0.782100\pi$
$863$		−	37.1515i		−	1.26465i	0.774702	−	0.632326i	$-0.217900\pi$
$864$		0			0
$865$	−30.8075	+	6.65123i	−1.04749	+	0.226149i
$866$		0			0
$867$		−	4.24396i		−	0.144133i
$868$		0			0
$869$	−70.4269			−2.38907
$870$		0			0
$871$	5.75627			0.195044
$872$		0			0
$873$		−	2.06742i		−	0.0699716i
$874$		0			0
$875$	−9.46669	+	7.01751i	−0.320033	+	0.237235i
$876$		0			0
$877$		−	11.4414i		−	0.386349i	−0.981164	−	0.193174i	$-0.938122\pi$
$877$		−	11.4414i		−	0.386349i	0.981164	−	0.193174i	$-0.0618783\pi$
$878$		0			0
$879$	−8.67159			−0.292486
$880$		0			0
$881$	−13.0790			−0.440641			−0.220321	−	0.975427i	$-0.570710\pi$
$881$	−13.0790			−0.440641			−0.220321	+	0.975427i	$0.570710\pi$
$882$		0			0
$883$		−	5.10061i		−	0.171649i	−0.996310	−	0.0858247i	$-0.972648\pi$
$883$		−	5.10061i		−	0.171649i	0.996310	−	0.0858247i	$-0.0273525\pi$
$884$		0			0
$885$	−31.2505	+	6.74689i	−1.05048	+	0.226794i
$886$		0			0
$887$		−	5.71010i		−	0.191726i	−0.995395	−	0.0958632i	$-0.969439\pi$
$887$		−	5.71010i		−	0.191726i	0.995395	−	0.0958632i	$-0.0305611\pi$
$888$		0			0
$889$	−8.24292			−0.276459
$890$		0			0
$891$	−5.95761			−0.199588
$892$		0			0
$893$			40.3071i			1.34883i
$894$		0			0
$895$	2.99086	+	13.8532i	0.0999734	+	0.463061i
$896$		0			0
$897$		−	31.1504i		−	1.04008i
$898$		0			0
$899$	8.35974			0.278813
$900$		0			0
$901$	−7.15816			−0.238473
$902$		0			0
$903$			7.18528i			0.239111i
$904$		0			0
$905$	−4.97449	−	23.0411i	−0.165358	−	0.765911i
$906$		0			0
$907$			43.7074i			1.45128i	0.688074	+	0.725640i	$0.258456\pi$
$907$			43.7074i			1.45128i	−0.688074	+	0.725640i	$0.741544\pi$
$908$		0			0
$909$	−4.77954			−0.158527
$910$		0			0
$911$	3.63409			0.120403			0.0602014	−	0.998186i	$-0.480826\pi$
$911$	3.63409			0.120403			0.0602014	+	0.998186i	$0.480826\pi$
$912$		0			0
$913$		−	103.905i		−	3.43875i
$914$		0			0
$915$	6.11240	−	1.31965i	0.202070	−	0.0436262i
$916$		0			0
$917$		−	11.6638i		−	0.385171i
$918$		0			0
$919$	−4.09079			−0.134943			−0.0674714	−	0.997721i	$-0.521493\pi$
$919$	−4.09079			−0.134943			−0.0674714	+	0.997721i	$0.521493\pi$
$920$		0			0
$921$	25.6728			0.845948
$922$		0			0
$923$			84.6334i			2.78574i
$924$		0			0
$925$	−22.4576	−	49.5859i	−0.738403	−	1.63037i
$926$		0			0
$927$		−	5.44398i		−	0.178804i
$928$		0			0
$929$	15.3348			0.503119			0.251559	−	0.967842i	$-0.419057\pi$
$929$	15.3348			0.503119			0.251559	+	0.967842i	$0.419057\pi$
$930$		0			0
$931$	29.0283			0.951365
$932$		0			0
$933$		−	5.83109i		−	0.190901i
$934$		0			0
$935$	−60.0181	+	12.9577i	−1.96280	+	0.423763i
$936$		0			0
$937$		−	21.3325i		−	0.696903i	−0.937327	−	0.348451i	$-0.886708\pi$
$937$		−	21.3325i		−	0.696903i	0.937327	−	0.348451i	$-0.113292\pi$
$938$		0			0
$939$	−32.0059			−1.04447
$940$		0			0
$941$	21.1809			0.690477			0.345239	−	0.938515i	$-0.387798\pi$
$941$	21.1809			0.690477			0.345239	+	0.938515i	$0.387798\pi$
$942$		0			0
$943$		−	65.4073i		−	2.12995i
$944$		0			0
$945$	−0.497368	−	2.30373i	−0.0161794	−	0.0749403i
$946$		0			0
$947$			8.23161i			0.267491i	0.991016	+	0.133746i	$0.0427005\pi$
$947$			8.23161i			0.267491i	−0.991016	+	0.133746i	$0.957299\pi$
$948$		0			0
$949$	43.6243			1.41611
$950$		0			0
$951$	−27.5282			−0.892662
$952$		0			0
$953$		−	44.8858i		−	1.45399i	−0.686641	−	0.726996i	$-0.740916\pi$
$953$		−	44.8858i		−	1.45399i	0.686641	−	0.726996i	$-0.259084\pi$
$954$		0			0
$955$	−8.19762	−	37.9701i	−0.265269	−	1.22868i
$956$		0			0
$957$			13.3410i			0.431254i
$958$		0			0
$959$	−3.32920			−0.107506
$960$		0			0
$961$	−17.0635			−0.550436
$962$		0			0
$963$		−	3.33646i		−	0.107516i
$964$		0			0
$965$	−3.47298	+	0.749804i	−0.111799	+	0.0241371i
$966$		0			0
$967$			47.6937i			1.53373i	0.641810	+	0.766863i	$0.278184\pi$
$967$			47.6937i			1.53373i	−0.641810	+	0.766863i	$0.721816\pi$
$968$		0			0
$969$	−22.7191			−0.729844
$970$		0			0
$971$	−12.0702			−0.387350			−0.193675	−	0.981066i	$-0.562041\pi$
$971$	−12.0702			−0.387350			−0.193675	+	0.981066i	$0.562041\pi$
$972$		0			0
$973$		−	13.5342i		−	0.433887i
$974$		0			0
$975$	−26.2178	+	11.8741i	−0.839641	+	0.380277i
$976$		0			0
$977$		−	20.5820i		−	0.658477i	−0.944247	−	0.329238i	$-0.893208\pi$
$977$		−	20.5820i		−	0.658477i	0.944247	−	0.329238i	$-0.106792\pi$
$978$		0			0
$979$	−8.25356			−0.263785
$980$		0			0
$981$	6.41914			0.204947
$982$		0			0
$983$		−	26.3186i		−	0.839434i	−0.907655	−	0.419717i	$-0.862129\pi$
$983$		−	26.3186i		−	0.839434i	0.907655	−	0.419717i	$-0.137871\pi$
$984$		0			0
$985$	7.75406	−	1.67408i	0.247065	−	0.0533405i
$986$		0			0
$987$		−	8.61881i		−	0.274340i
$988$		0			0
$989$	−36.8915			−1.17308
$990$		0			0
$991$	−36.1491			−1.14832			−0.574158	−	0.818745i	$-0.694670\pi$
$991$	−36.1491			−1.14832			−0.574158	+	0.818745i	$0.694670\pi$
$992$		0			0
$993$		−	6.60882i		−	0.209725i
$994$		0			0
$995$	−7.67136	−	35.5326i	−0.243199	−	1.12646i
$996$		0			0
$997$		−	7.32045i		−	0.231841i	−0.993258	−	0.115920i	$-0.963018\pi$
$997$		−	7.32045i		−	0.231841i	0.993258	−	0.115920i	$-0.0369818\pi$
$998$		0			0
$999$	10.8869			0.344446

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.c.1609.30	yes	38
5.4	even	2	inner	4020.2.g.c.1609.11	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.c.1609.11	✓	38	5.4	even	2	inner
4020.2.g.c.1609.30	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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-0.471887 - 2.18571i) q^{5} +1.05400i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +(-0.471887 - 2.18571i) q^{5} +1.05400i q^{7} -1.00000 q^{9} -5.95761 q^{11} -5.75627i q^{13} +(2.18571 - 0.471887i) q^{15} +4.60912i q^{17} +4.92917 q^{19} -1.05400 q^{21} -5.41155i q^{23} +(-4.55464 + 2.06282i) q^{25} -1.00000i q^{27} -2.23932 q^{29} -3.73316 q^{31} -5.95761i q^{33} +(2.30373 - 0.497368i) q^{35} +10.8869i q^{37} +5.75627 q^{39} +12.0866 q^{41} -6.81718i q^{43} +(0.471887 + 2.18571i) q^{45} +8.17727i q^{47} +5.88909 q^{49} -4.60912 q^{51} +1.55304i q^{53} +(2.81132 + 13.0216i) q^{55} +4.92917i q^{57} -14.2977 q^{59} +2.79653 q^{61} -1.05400i q^{63} +(-12.5815 + 2.71631i) q^{65} +1.00000i q^{67} +5.41155 q^{69} -14.7028 q^{71} +7.57857i q^{73} +(-2.06282 - 4.55464i) q^{75} -6.27931i q^{77} +11.8213 q^{79} +1.00000 q^{81} +17.4407i q^{83} +(10.0742 - 2.17498i) q^{85} -2.23932i q^{87} +1.38538 q^{89} +6.06709 q^{91} -3.73316i q^{93} +(-2.32601 - 10.7737i) q^{95} +2.06742i q^{97} +5.95761 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q - 2 q^{5} - 38 q^{9}+O(q^{10}) \) 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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