LMFDB - Embedded newform 4020.2.g.b.1609.5

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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1609.5
Character	$\chi$	$=$	4020.1609
Dual form			4020.2.g.b.1609.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +(-1.47707 + 1.67877i) q^{5} +3.23400i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +(-1.47707 + 1.67877i) q^{5} +3.23400i q^{7} -1.00000 q^{9} +1.20727 q^{11} -3.00762i q^{13} +(1.67877 + 1.47707i) q^{15} -1.30594i q^{17} +5.21168 q^{19} +3.23400 q^{21} +0.725616i q^{23} +(-0.636556 - 4.95931i) q^{25} +1.00000i q^{27} +6.60775 q^{29} -6.07405 q^{31} -1.20727i q^{33} +(-5.42915 - 4.77683i) q^{35} -1.93524i q^{37} -3.00762 q^{39} +6.64513 q^{41} +8.35230i q^{43} +(1.47707 - 1.67877i) q^{45} -11.4814i q^{47} -3.45877 q^{49} -1.30594 q^{51} +5.95836i q^{53} +(-1.78322 + 2.02673i) q^{55} -5.21168i q^{57} -0.552615 q^{59} -5.13959 q^{61} -3.23400i q^{63} +(5.04911 + 4.44245i) q^{65} +1.00000i q^{67} +0.725616 q^{69} +11.4976 q^{71} +9.22347i q^{73} +(-4.95931 + 0.636556i) q^{75} +3.90431i q^{77} -5.79731 q^{79} +1.00000 q^{81} +7.25526i q^{83} +(2.19238 + 1.92896i) q^{85} -6.60775i q^{87} +5.48396 q^{89} +9.72665 q^{91} +6.07405i q^{93} +(-7.69799 + 8.74923i) q^{95} +2.07762i q^{97} -1.20727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) $ 24 * q - 4 * q^5 - 24 * q^9	$ 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) $ 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * 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
$5$	−1.47707	+	1.67877i	−0.660564	+	0.750770i
$5$	−1.47707	+	1.67877i	−0.660564	+	0.750770i
$6$		0			0
$7$			3.23400i			1.22234i	0.791500	+	0.611169i	$0.209300\pi$
$7$			3.23400i			1.22234i	−0.791500	+	0.611169i	$0.790700\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.20727			0.364006			0.182003	−	0.983298i	$-0.441742\pi$
$11$	1.20727			0.364006			0.182003	+	0.983298i	$0.441742\pi$
$12$		0			0
$13$		−	3.00762i		−	0.834164i	−0.908869	−	0.417082i	$-0.863053\pi$
$13$		−	3.00762i		−	0.834164i	0.908869	−	0.417082i	$-0.136947\pi$
$14$		0			0
$15$	1.67877	+	1.47707i	0.433457	+	0.381377i
$16$		0			0
$17$		−	1.30594i		−	0.316737i	−0.987380	−	0.158369i	$-0.949377\pi$
$17$		−	1.30594i		−	0.316737i	0.987380	−	0.158369i	$-0.0506234\pi$
$18$		0			0
$19$	5.21168			1.19564			0.597821	−	0.801630i	$-0.296033\pi$
$19$	5.21168			1.19564			0.597821	+	0.801630i	$0.296033\pi$
$20$		0			0
$21$	3.23400			0.705717
$22$		0			0
$23$			0.725616i			0.151301i	0.997134	+	0.0756507i	$0.0241034\pi$
$23$			0.725616i			0.151301i	−0.997134	+	0.0756507i	$0.975897\pi$
$24$		0			0
$25$	−0.636556	−	4.95931i	−0.127311	−	0.991863i
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	6.60775			1.22703			0.613514	−	0.789684i	$-0.289755\pi$
$29$	6.60775			1.22703			0.613514	+	0.789684i	$0.289755\pi$
$30$		0			0
$31$	−6.07405			−1.09093			−0.545466	−	0.838133i	$-0.683647\pi$
$31$	−6.07405			−1.09093			−0.545466	+	0.838133i	$0.683647\pi$
$32$		0			0
$33$		−	1.20727i		−	0.210159i
$34$		0			0
$35$	−5.42915	−	4.77683i	−0.917695	−	0.807432i
$36$		0			0
$37$		−	1.93524i		−	0.318151i	−0.987266	−	0.159076i	$-0.949149\pi$
$37$		−	1.93524i		−	0.318151i	0.987266	−	0.159076i	$-0.0508514\pi$
$38$		0			0
$39$	−3.00762			−0.481605
$40$		0			0
$41$	6.64513			1.03779			0.518897	−	0.854837i	$-0.326343\pi$
$41$	6.64513			1.03779			0.518897	+	0.854837i	$0.326343\pi$
$42$		0			0
$43$			8.35230i			1.27371i	0.770982	+	0.636857i	$0.219766\pi$
$43$			8.35230i			1.27371i	−0.770982	+	0.636857i	$0.780234\pi$
$44$		0			0
$45$	1.47707	−	1.67877i	0.220188	−	0.250257i
$46$		0			0
$47$		−	11.4814i		−	1.67474i	−0.546635	−	0.837371i	$-0.684092\pi$
$47$		−	11.4814i		−	1.67474i	0.546635	−	0.837371i	$-0.315908\pi$
$48$		0			0
$49$	−3.45877			−0.494110
$50$		0			0
$51$	−1.30594			−0.182868
$52$		0			0
$53$			5.95836i			0.818444i	0.912435	+	0.409222i	$0.134200\pi$
$53$			5.95836i			0.818444i	−0.912435	+	0.409222i	$0.865800\pi$
$54$		0			0
$55$	−1.78322	+	2.02673i	−0.240449	+	0.273284i
$56$		0			0
$57$		−	5.21168i		−	0.690304i
$58$		0			0
$59$	−0.552615			−0.0719444			−0.0359722	−	0.999353i	$-0.511453\pi$
$59$	−0.552615			−0.0719444			−0.0359722	+	0.999353i	$0.511453\pi$
$60$		0			0
$61$	−5.13959			−0.658057			−0.329029	−	0.944320i	$-0.606721\pi$
$61$	−5.13959			−0.658057			−0.329029	+	0.944320i	$0.606721\pi$
$62$		0			0
$63$		−	3.23400i		−	0.407446i
$64$		0			0
$65$	5.04911	+	4.44245i	0.626265	+	0.551018i
$66$		0			0
$67$			1.00000i			0.122169i
$67$			1.00000i			0.122169i
$68$		0			0
$69$	0.725616			0.0873539
$70$		0			0
$71$	11.4976			1.36451			0.682255	−	0.731115i	$-0.261000\pi$
$71$	11.4976			1.36451			0.682255	+	0.731115i	$0.261000\pi$
$72$		0			0
$73$			9.22347i			1.07953i	0.841817	+	0.539763i	$0.181486\pi$
$73$			9.22347i			1.07953i	−0.841817	+	0.539763i	$0.818514\pi$
$74$		0			0
$75$	−4.95931	+	0.636556i	−0.572652	+	0.0735031i
$76$		0			0
$77$			3.90431i			0.444938i
$78$		0			0
$79$	−5.79731			−0.652249			−0.326124	−	0.945327i	$-0.605743\pi$
$79$	−5.79731			−0.652249			−0.326124	+	0.945327i	$0.605743\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			7.25526i			0.796368i	0.917306	+	0.398184i	$0.130359\pi$
$83$			7.25526i			0.796368i	−0.917306	+	0.398184i	$0.869641\pi$
$84$		0			0
$85$	2.19238	+	1.92896i	0.237797	+	0.209225i
$86$		0			0
$87$		−	6.60775i		−	0.708425i
$88$		0			0
$89$	5.48396			0.581299			0.290649	−	0.956830i	$-0.406129\pi$
$89$	5.48396			0.581299			0.290649	+	0.956830i	$0.406129\pi$
$90$		0			0
$91$	9.72665			1.01963
$92$		0			0
$93$			6.07405i			0.629850i
$94$		0			0
$95$	−7.69799	+	8.74923i	−0.789797	+	0.897652i
$96$		0			0
$97$			2.07762i			0.210950i	0.994422	+	0.105475i	$0.0336363\pi$
$97$			2.07762i			0.210950i	−0.994422	+	0.105475i	$0.966364\pi$
$98$		0			0
$99$	−1.20727			−0.121335
$100$		0			0
$101$	0.0202098			0.00201095			0.00100547	−	0.999999i	$-0.499680\pi$
$101$	0.0202098			0.00201095			0.00100547	+	0.999999i	$0.499680\pi$
$102$		0			0
$103$			17.1778i			1.69258i	0.532726	+	0.846288i	$0.321168\pi$
$103$			17.1778i			1.69258i	−0.532726	+	0.846288i	$0.678832\pi$
$104$		0			0
$105$	−4.77683	+	5.42915i	−0.466171	+	0.529831i
$106$		0			0
$107$		−	4.03449i		−	0.390029i	−0.980800	−	0.195014i	$-0.937525\pi$
$107$		−	4.03449i		−	0.390029i	0.980800	−	0.195014i	$-0.0624753\pi$
$108$		0			0
$109$	17.5493			1.68092			0.840458	−	0.541876i	$-0.182286\pi$
$109$	17.5493			1.68092			0.840458	+	0.541876i	$0.182286\pi$
$110$		0			0
$111$	−1.93524			−0.183685
$112$		0			0
$113$			11.6114i			1.09231i	0.837686	+	0.546153i	$0.183908\pi$
$113$			11.6114i			1.09231i	−0.837686	+	0.546153i	$0.816092\pi$
$114$		0			0
$115$	−1.21814	−	1.07178i	−0.113592	−	0.0999442i
$116$		0			0
$117$			3.00762i			0.278055i
$118$		0			0
$119$	4.22341			0.387160
$120$		0			0
$121$	−9.54250			−0.867500
$122$		0			0
$123$		−	6.64513i		−	0.599171i
$124$		0			0
$125$	9.26580	+	6.25660i	0.828758	+	0.559607i
$126$		0			0
$127$			16.4526i			1.45993i	0.683483	+	0.729966i	$0.260464\pi$
$127$			16.4526i			1.45993i	−0.683483	+	0.729966i	$0.739536\pi$
$128$		0			0
$129$	8.35230			0.735379
$130$		0			0
$131$	−8.90912			−0.778393			−0.389197	−	0.921155i	$-0.627247\pi$
$131$	−8.90912			−0.778393			−0.389197	+	0.921155i	$0.627247\pi$
$132$		0			0
$133$			16.8546i			1.46148i
$134$		0			0
$135$	−1.67877	−	1.47707i	−0.144486	−	0.127126i
$136$		0			0
$137$		−	3.75213i		−	0.320566i	−0.987071	−	0.160283i	$-0.948759\pi$
$137$		−	3.75213i		−	0.320566i	0.987071	−	0.160283i	$-0.0512407\pi$
$138$		0			0
$139$	19.5201			1.65568			0.827838	−	0.560968i	$-0.189571\pi$
$139$	19.5201			1.65568			0.827838	+	0.560968i	$0.189571\pi$
$140$		0			0
$141$	−11.4814			−0.966912
$142$		0			0
$143$		−	3.63101i		−	0.303640i
$144$		0			0
$145$	−9.76008	+	11.0929i	−0.810530	+	0.921216i
$146$		0			0
$147$			3.45877i			0.285274i
$148$		0			0
$149$	−7.12082			−0.583360			−0.291680	−	0.956516i	$-0.594214\pi$
$149$	−7.12082			−0.583360			−0.291680	+	0.956516i	$0.594214\pi$
$150$		0			0
$151$	−2.40258			−0.195519			−0.0977596	−	0.995210i	$-0.531168\pi$
$151$	−2.40258			−0.195519			−0.0977596	+	0.995210i	$0.531168\pi$
$152$		0			0
$153$			1.30594i			0.105579i
$154$		0			0
$155$	8.97177	−	10.1970i	0.720630	−	0.819039i
$156$		0			0
$157$			11.9264i			0.951829i	0.879492	+	0.475914i	$0.157883\pi$
$157$			11.9264i			0.951829i	−0.879492	+	0.475914i	$0.842117\pi$
$158$		0			0
$159$	5.95836			0.472529
$160$		0			0
$161$	−2.34664			−0.184941
$162$		0			0
$163$			3.26717i			0.255905i	0.991780	+	0.127952i	$0.0408405\pi$
$163$			3.26717i			0.255905i	−0.991780	+	0.127952i	$0.959160\pi$
$164$		0			0
$165$	2.02673	+	1.78322i	0.157781	+	0.138823i
$166$		0			0
$167$			3.07382i			0.237859i	0.992903	+	0.118930i	$0.0379463\pi$
$167$			3.07382i			0.237859i	−0.992903	+	0.118930i	$0.962054\pi$
$168$		0			0
$169$	3.95422			0.304171
$170$		0			0
$171$	−5.21168			−0.398547
$172$		0			0
$173$			15.7208i			1.19523i	0.801783	+	0.597615i	$0.203885\pi$
$173$			15.7208i			1.19523i	−0.801783	+	0.597615i	$0.796115\pi$
$174$		0			0
$175$	16.0384	−	2.05862i	1.21239	−	0.155617i
$176$		0			0
$177$			0.552615i			0.0415371i
$178$		0			0
$179$	2.10290			0.157178			0.0785892	−	0.996907i	$-0.474958\pi$
$179$	2.10290			0.157178			0.0785892	+	0.996907i	$0.474958\pi$
$180$		0			0
$181$	−22.3515			−1.66138			−0.830688	−	0.556738i	$-0.812053\pi$
$181$	−22.3515			−1.66138			−0.830688	+	0.556738i	$0.812053\pi$
$182$		0			0
$183$			5.13959i			0.379930i
$184$		0			0
$185$	3.24882	+	2.85847i	0.238858	+	0.210159i
$186$		0			0
$187$		−	1.57662i		−	0.115294i
$188$		0			0
$189$	−3.23400			−0.235239
$190$		0			0
$191$	18.4208			1.33288			0.666440	−	0.745558i	$-0.267817\pi$
$191$	18.4208			1.33288			0.666440	+	0.745558i	$0.267817\pi$
$192$		0			0
$193$		−	17.5629i		−	1.26420i	−0.774885	−	0.632102i	$-0.782192\pi$
$193$		−	17.5629i		−	1.26420i	0.774885	−	0.632102i	$-0.217808\pi$
$194$		0			0
$195$	4.44245	−	5.04911i	0.318131	−	0.361574i
$196$		0			0
$197$		−	11.2224i		−	0.799561i	−0.916611	−	0.399781i	$-0.869086\pi$
$197$		−	11.2224i		−	0.799561i	0.916611	−	0.399781i	$-0.130914\pi$
$198$		0			0
$199$	−16.5069			−1.17014			−0.585071	−	0.810982i	$-0.698934\pi$
$199$	−16.5069			−1.17014			−0.585071	+	0.810982i	$0.698934\pi$
$200$		0			0
$201$	1.00000			0.0705346
$202$		0			0
$203$			21.3695i			1.49984i
$204$		0			0
$205$	−9.81529	+	11.1557i	−0.685529	+	0.779145i
$206$		0			0
$207$		−	0.725616i		−	0.0504338i
$208$		0			0
$209$	6.29191			0.435220
$210$		0			0
$211$	−1.07243			−0.0738288			−0.0369144	−	0.999318i	$-0.511753\pi$
$211$	−1.07243			−0.0738288			−0.0369144	+	0.999318i	$0.511753\pi$
$212$		0			0
$213$		−	11.4976i		−	0.787800i
$214$		0			0
$215$	−14.0216	−	12.3369i	−0.956266	−	0.841369i
$216$		0			0
$217$		−	19.6435i		−	1.33349i
$218$		0			0
$219$	9.22347			0.623264
$220$		0			0
$221$	−3.92777			−0.264211
$222$		0			0
$223$			20.8037i			1.39312i	0.717500	+	0.696559i	$0.245287\pi$
$223$			20.8037i			1.39312i	−0.717500	+	0.696559i	$0.754713\pi$
$224$		0			0
$225$	0.636556	+	4.95931i	0.0424371	+	0.330621i
$226$		0			0
$227$			27.6609i			1.83592i	0.396677	+	0.917958i	$0.370163\pi$
$227$			27.6609i			1.83592i	−0.396677	+	0.917958i	$0.629837\pi$
$228$		0			0
$229$	1.13274			0.0748536			0.0374268	−	0.999299i	$-0.488084\pi$
$229$	1.13274			0.0748536			0.0374268	+	0.999299i	$0.488084\pi$
$230$		0			0
$231$	3.90431			0.256885
$232$		0			0
$233$		−	23.6704i		−	1.55070i	−0.631533	−	0.775349i	$-0.717574\pi$
$233$		−	23.6704i		−	1.55070i	0.631533	−	0.775349i	$-0.282426\pi$
$234$		0			0
$235$	19.2747	+	16.9588i	1.25735	+	1.10627i
$236$		0			0
$237$			5.79731i			0.376576i
$238$		0			0
$239$	18.5005			1.19670			0.598349	−	0.801236i	$-0.295824\pi$
$239$	18.5005			1.19670			0.598349	+	0.801236i	$0.295824\pi$
$240$		0			0
$241$	−12.6062			−0.812035			−0.406017	−	0.913865i	$-0.633083\pi$
$241$	−12.6062			−0.812035			−0.406017	+	0.913865i	$0.633083\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	5.10883	−	5.80649i	0.326391	−	0.370963i
$246$		0			0
$247$		−	15.6748i		−	0.997361i
$248$		0			0
$249$	7.25526			0.459783
$250$		0			0
$251$	27.7683			1.75272			0.876360	−	0.481657i	$-0.159965\pi$
$251$	27.7683			1.75272			0.876360	+	0.481657i	$0.159965\pi$
$252$		0			0
$253$			0.876014i			0.0550745i
$254$		0			0
$255$	1.92896	−	2.19238i	0.120796	−	0.137292i
$256$		0			0
$257$		−	6.68359i		−	0.416911i	−0.978032	−	0.208455i	$-0.933156\pi$
$257$		−	6.68359i		−	0.416911i	0.978032	−	0.208455i	$-0.0668436\pi$
$258$		0			0
$259$	6.25856			0.388888
$260$		0			0
$261$	−6.60775			−0.409009
$262$		0			0
$263$			11.2651i			0.694636i	0.937747	+	0.347318i	$0.112908\pi$
$263$			11.2651i			0.694636i	−0.937747	+	0.347318i	$0.887092\pi$
$264$		0			0
$265$	−10.0027	−	8.80089i	−0.614463	−	0.540635i
$266$		0			0
$267$		−	5.48396i		−	0.335613i
$268$		0			0
$269$	−16.1349			−0.983764			−0.491882	−	0.870662i	$-0.663691\pi$
$269$	−16.1349			−0.983764			−0.491882	+	0.870662i	$0.663691\pi$
$270$		0			0
$271$	24.6627			1.49815			0.749076	−	0.662484i	$-0.230498\pi$
$271$	24.6627			1.49815			0.749076	+	0.662484i	$0.230498\pi$
$272$		0			0
$273$		−	9.72665i		−	0.588684i
$274$		0			0
$275$	−0.768495	−	5.98723i	−0.0463420	−	0.361044i
$276$		0			0
$277$			26.4497i			1.58921i	0.607126	+	0.794605i	$0.292322\pi$
$277$			26.4497i			1.58921i	−0.607126	+	0.794605i	$0.707678\pi$
$278$		0			0
$279$	6.07405			0.363644
$280$		0			0
$281$	23.6361			1.41001			0.705005	−	0.709202i	$-0.250945\pi$
$281$	23.6361			1.41001			0.705005	+	0.709202i	$0.250945\pi$
$282$		0			0
$283$		−	15.4054i		−	0.915757i	−0.889015	−	0.457878i	$-0.848610\pi$
$283$		−	15.4054i		−	0.915757i	0.889015	−	0.457878i	$-0.151390\pi$
$284$		0			0
$285$	8.74923	+	7.69799i	0.518259	+	0.455990i
$286$		0			0
$287$			21.4904i			1.26854i
$288$		0			0
$289$	15.2945			0.899678
$290$		0			0
$291$	2.07762			0.121792
$292$		0			0
$293$			3.15135i			0.184104i	0.995754	+	0.0920520i	$0.0293426\pi$
$293$			3.15135i			0.184104i	−0.995754	+	0.0920520i	$0.970657\pi$
$294$		0			0
$295$	0.816249	−	0.927715i	0.0475238	−	0.0540137i
$296$		0			0
$297$			1.20727i			0.0700529i
$298$		0			0
$299$	2.18238			0.126210
$300$		0			0
$301$	−27.0113			−1.55691
$302$		0			0
$303$		−	0.0202098i		−	0.00116102i
$304$		0			0
$305$	7.59151	−	8.62821i	0.434689	−	0.494050i
$306$		0			0
$307$		−	25.3961i		−	1.44943i	−0.689047	−	0.724717i	$-0.741971\pi$
$307$		−	25.3961i		−	1.44943i	0.689047	−	0.724717i	$-0.258029\pi$
$308$		0			0
$309$	17.1778			0.977209
$310$		0			0
$311$	22.7465			1.28983			0.644917	−	0.764253i	$-0.276892\pi$
$311$	22.7465			1.28983			0.644917	+	0.764253i	$0.276892\pi$
$312$		0			0
$313$		−	16.2629i		−	0.919233i	−0.888117	−	0.459617i	$-0.847987\pi$
$313$		−	16.2629i		−	0.919233i	0.888117	−	0.459617i	$-0.152013\pi$
$314$		0			0
$315$	5.42915	+	4.77683i	0.305898	+	0.269144i
$316$		0			0
$317$		−	3.56463i		−	0.200210i	−0.994977	−	0.100105i	$-0.968082\pi$
$317$		−	3.56463i		−	0.200210i	0.994977	−	0.100105i	$-0.0319178\pi$
$318$		0			0
$319$	7.97734			0.446645
$320$		0			0
$321$	−4.03449			−0.225183
$322$		0			0
$323$		−	6.80615i		−	0.378704i
$324$		0			0
$325$	−14.9157	+	1.91452i	−0.827376	+	0.106198i
$326$		0			0
$327$		−	17.5493i		−	0.970478i
$328$		0			0
$329$	37.1310			2.04710
$330$		0			0
$331$	−5.34386			−0.293725			−0.146863	−	0.989157i	$-0.546918\pi$
$331$	−5.34386			−0.293725			−0.146863	+	0.989157i	$0.546918\pi$
$332$		0			0
$333$			1.93524i			0.106050i
$334$		0			0
$335$	−1.67877	−	1.47707i	−0.0917212	−	0.0807007i
$336$		0			0
$337$			13.0537i			0.711081i	0.934661	+	0.355540i	$0.115703\pi$
$337$			13.0537i			0.711081i	−0.934661	+	0.355540i	$0.884297\pi$
$338$		0			0
$339$	11.6114			0.630643
$340$		0			0
$341$	−7.33302			−0.397105
$342$		0			0
$343$			11.4524i			0.618369i
$344$		0			0
$345$	−1.07178	+	1.21814i	−0.0577028	+	0.0655827i
$346$		0			0
$347$		−	10.5323i		−	0.565405i	−0.959208	−	0.282703i	$-0.908769\pi$
$347$		−	10.5323i		−	0.565405i	0.959208	−	0.282703i	$-0.0912309\pi$
$348$		0			0
$349$	−31.0843			−1.66391			−0.831953	−	0.554846i	$-0.812777\pi$
$349$	−31.0843			−1.66391			−0.831953	+	0.554846i	$0.812777\pi$
$350$		0			0
$351$	3.00762			0.160535
$352$		0			0
$353$		−	14.0479i		−	0.747692i	−0.927491	−	0.373846i	$-0.878039\pi$
$353$		−	14.0479i		−	0.747692i	0.927491	−	0.373846i	$-0.121961\pi$
$354$		0			0
$355$	−16.9826	+	19.3018i	−0.901345	+	1.02443i
$356$		0			0
$357$		−	4.22341i		−	0.223527i
$358$		0			0
$359$	−5.17155			−0.272944			−0.136472	−	0.990644i	$-0.543576\pi$
$359$	−5.17155			−0.272944			−0.136472	+	0.990644i	$0.543576\pi$
$360$		0			0
$361$	8.16161			0.429559
$362$		0			0
$363$			9.54250i			0.500851i
$364$		0			0
$365$	−15.4841	−	13.6237i	−0.810475	−	0.713095i
$366$		0			0
$367$		−	1.73630i		−	0.0906343i	−0.998973	−	0.0453171i	$-0.985570\pi$
$367$		−	1.73630i		−	0.0906343i	0.998973	−	0.0453171i	$-0.0144298\pi$
$368$		0			0
$369$	−6.64513			−0.345932
$370$		0			0
$371$	−19.2694			−1.00042
$372$		0			0
$373$		−	4.44734i		−	0.230275i	−0.993350	−	0.115137i	$-0.963269\pi$
$373$		−	4.44734i		−	0.230275i	0.993350	−	0.115137i	$-0.0367308\pi$
$374$		0			0
$375$	6.25660	−	9.26580i	0.323089	−	0.478484i
$376$		0			0
$377$		−	19.8736i		−	1.02354i
$378$		0			0
$379$	10.0256			0.514981			0.257491	−	0.966281i	$-0.417104\pi$
$379$	10.0256			0.514981			0.257491	+	0.966281i	$0.417104\pi$
$380$		0			0
$381$	16.4526			0.842892
$382$		0			0
$383$		−	18.6878i		−	0.954903i	−0.878658	−	0.477452i	$-0.841561\pi$
$383$		−	18.6878i		−	0.954903i	0.878658	−	0.477452i	$-0.158439\pi$
$384$		0			0
$385$	−6.55445	−	5.76693i	−0.334046	−	0.293910i
$386$		0			0
$387$		−	8.35230i		−	0.424571i
$388$		0			0
$389$	20.6954			1.04930			0.524649	−	0.851319i	$-0.324197\pi$
$389$	20.6954			1.04930			0.524649	+	0.851319i	$0.324197\pi$
$390$		0			0
$391$	0.947611			0.0479228
$392$		0			0
$393$			8.90912i			0.449405i
$394$		0			0
$395$	8.56301	−	9.73237i	0.430852	−	0.489689i
$396$		0			0
$397$			8.66735i			0.435002i	0.976060	+	0.217501i	$0.0697905\pi$
$397$			8.66735i			0.435002i	−0.976060	+	0.217501i	$0.930209\pi$
$398$		0			0
$399$	16.8546			0.843785
$400$		0			0
$401$	6.39608			0.319405			0.159703	−	0.987165i	$-0.448946\pi$
$401$	6.39608			0.319405			0.159703	+	0.987165i	$0.448946\pi$
$402$		0			0
$403$			18.2684i			0.910016i
$404$		0			0
$405$	−1.47707	+	1.67877i	−0.0733960	+	0.0834189i
$406$		0			0
$407$		−	2.33635i		−	0.115809i
$408$		0			0
$409$	−24.1003			−1.19168			−0.595841	−	0.803103i	$-0.703181\pi$
$409$	−24.1003			−1.19168			−0.595841	+	0.803103i	$0.703181\pi$
$410$		0			0
$411$	−3.75213			−0.185079
$412$		0			0
$413$		−	1.78716i		−	0.0879403i
$414$		0			0
$415$	−12.1799	−	10.7165i	−0.597889	−	0.526052i
$416$		0			0
$417$		−	19.5201i		−	0.955905i
$418$		0			0
$419$	−10.1154			−0.494167			−0.247084	−	0.968994i	$-0.579472\pi$
$419$	−10.1154			−0.494167			−0.247084	+	0.968994i	$0.579472\pi$
$420$		0			0
$421$	4.17960			0.203701			0.101851	−	0.994800i	$-0.467524\pi$
$421$	4.17960			0.203701			0.101851	+	0.994800i	$0.467524\pi$
$422$		0			0
$423$			11.4814i			0.558247i
$424$		0			0
$425$	−6.47657	+	0.831304i	−0.314160	+	0.0403242i
$426$		0			0
$427$		−	16.6215i		−	0.804368i
$428$		0			0
$429$	−3.63101			−0.175307
$430$		0			0
$431$	2.33388			0.112419			0.0562096	−	0.998419i	$-0.482098\pi$
$431$	2.33388			0.112419			0.0562096	+	0.998419i	$0.482098\pi$
$432$		0			0
$433$			12.3095i			0.591556i	0.955257	+	0.295778i	$0.0955788\pi$
$433$			12.3095i			0.591556i	−0.955257	+	0.295778i	$0.904421\pi$
$434$		0			0
$435$	11.0929	+	9.76008i	0.531864	+	0.467960i
$436$		0			0
$437$			3.78168i			0.180902i
$438$		0			0
$439$	22.5537			1.07643			0.538216	−	0.842807i	$-0.319099\pi$
$439$	22.5537			1.07643			0.538216	+	0.842807i	$0.319099\pi$
$440$		0			0
$441$	3.45877			0.164703
$442$		0			0
$443$		−	34.0947i		−	1.61989i	−0.586507	−	0.809944i	$-0.699497\pi$
$443$		−	34.0947i		−	1.61989i	0.586507	−	0.809944i	$-0.300503\pi$
$444$		0			0
$445$	−8.10017	+	9.20632i	−0.383985	+	0.436422i
$446$		0			0
$447$			7.12082i			0.336803i
$448$		0			0
$449$	−13.2088			−0.623362			−0.311681	−	0.950187i	$-0.600892\pi$
$449$	−13.2088			−0.623362			−0.311681	+	0.950187i	$0.600892\pi$
$450$		0			0
$451$	8.02246			0.377763
$452$		0			0
$453$			2.40258i			0.112883i
$454$		0			0
$455$	−14.3669	+	16.3288i	−0.673531	+	0.765508i
$456$		0			0
$457$			6.49253i			0.303708i	0.988403	+	0.151854i	$0.0485243\pi$
$457$			6.49253i			0.303708i	−0.988403	+	0.151854i	$0.951476\pi$
$458$		0			0
$459$	1.30594			0.0609561
$460$		0			0
$461$	30.4650			1.41890			0.709449	−	0.704757i	$-0.248944\pi$
$461$	30.4650			1.41890			0.709449	+	0.704757i	$0.248944\pi$
$462$		0			0
$463$			36.4308i			1.69308i	0.532323	+	0.846541i	$0.321319\pi$
$463$			36.4308i			1.69308i	−0.532323	+	0.846541i	$0.678681\pi$
$464$		0			0
$465$	−10.1970	−	8.97177i	−0.472872	−	0.416056i
$466$		0			0
$467$			22.3594i			1.03467i	0.855783	+	0.517335i	$0.173076\pi$
$467$			22.3594i			1.03467i	−0.855783	+	0.517335i	$0.826924\pi$
$468$		0			0
$469$	−3.23400			−0.149332
$470$		0			0
$471$	11.9264			0.549538
$472$		0			0
$473$			10.0835i			0.463639i
$474$		0			0
$475$	−3.31753	−	25.8464i	−0.152219	−	1.18591i
$476$		0			0
$477$		−	5.95836i		−	0.272815i
$478$		0			0
$479$	−24.6829			−1.12779			−0.563894	−	0.825847i	$-0.690697\pi$
$479$	−24.6829			−1.12779			−0.563894	+	0.825847i	$0.690697\pi$
$480$		0			0
$481$	−5.82046			−0.265390
$482$		0			0
$483$			2.34664i			0.106776i
$484$		0			0
$485$	−3.48785	−	3.06878i	−0.158375	−	0.139346i
$486$		0			0
$487$			1.05113i			0.0476310i	0.999716	+	0.0238155i	$0.00758143\pi$
$487$			1.05113i			0.0476310i	−0.999716	+	0.0238155i	$0.992419\pi$
$488$		0			0
$489$	3.26717			0.147747
$490$		0			0
$491$	−30.8746			−1.39335			−0.696677	−	0.717385i	$-0.745339\pi$
$491$	−30.8746			−1.39335			−0.696677	+	0.717385i	$0.745339\pi$
$492$		0			0
$493$		−	8.62933i		−	0.388645i
$494$		0			0
$495$	1.78322	−	2.02673i	0.0801496	−	0.0910948i
$496$		0			0
$497$			37.1831i			1.66789i
$498$		0			0
$499$	−15.8937			−0.711500			−0.355750	−	0.934581i	$-0.615775\pi$
$499$	−15.8937			−0.711500			−0.355750	+	0.934581i	$0.615775\pi$
$500$		0			0
$501$	3.07382			0.137328
$502$		0			0
$503$			6.00462i			0.267733i	0.990999	+	0.133866i	$0.0427393\pi$
$503$			6.00462i			0.267733i	−0.990999	+	0.133866i	$0.957261\pi$
$504$		0			0
$505$	−0.0298512	+	0.0339276i	−0.00132836	+	0.00150976i
$506$		0			0
$507$		−	3.95422i		−	0.175613i
$508$		0			0
$509$	−15.6354			−0.693029			−0.346514	−	0.938045i	$-0.612635\pi$
$509$	−15.6354			−0.693029			−0.346514	+	0.938045i	$0.612635\pi$
$510$		0			0
$511$	−29.8287			−1.31954
$512$		0			0
$513$			5.21168i			0.230101i
$514$		0			0
$515$	−28.8376	−	25.3727i	−1.27074	−	1.11805i
$516$		0			0
$517$		−	13.8612i		−	0.609615i
$518$		0			0
$519$	15.7208			0.690066
$520$		0			0
$521$	0.900031			0.0394311			0.0197155	−	0.999806i	$-0.493724\pi$
$521$	0.900031			0.0394311			0.0197155	+	0.999806i	$0.493724\pi$
$522$		0			0
$523$		−	32.1043i		−	1.40382i	−0.712264	−	0.701911i	$-0.752330\pi$
$523$		−	32.1043i		−	1.40382i	0.712264	−	0.701911i	$-0.247670\pi$
$524$		0			0
$525$	−2.05862	−	16.0384i	−0.0898457	−	0.699975i
$526$		0			0
$527$			7.93235i			0.345539i
$528$		0			0
$529$	22.4735			0.977108
$530$		0			0
$531$	0.552615			0.0239815
$532$		0			0
$533$		−	19.9860i		−	0.865691i
$534$		0			0
$535$	6.77299	+	5.95920i	0.292822	+	0.257639i
$536$		0			0
$537$		−	2.10290i		−	0.0907470i
$538$		0			0
$539$	−4.17567			−0.179859
$540$		0			0
$541$	−27.6021			−1.18671			−0.593354	−	0.804942i	$-0.702196\pi$
$541$	−27.6021			−1.18671			−0.593354	+	0.804942i	$0.702196\pi$
$542$		0			0
$543$			22.3515i			0.959196i
$544$		0			0
$545$	−25.9214	+	29.4613i	−1.11035	+	1.26198i
$546$		0			0
$547$			22.8312i			0.976192i	0.872790	+	0.488096i	$0.162308\pi$
$547$			22.8312i			0.976192i	−0.872790	+	0.488096i	$0.837692\pi$
$548$		0			0
$549$	5.13959			0.219352
$550$		0			0
$551$	34.4375			1.46709
$552$		0			0
$553$		−	18.7485i		−	0.797268i
$554$		0			0
$555$	2.85847	−	3.24882i	0.121335	−	0.137905i
$556$		0			0
$557$		−	12.1753i		−	0.515883i	−0.966161	−	0.257941i	$-0.916956\pi$
$557$		−	12.1753i		−	0.515883i	0.966161	−	0.257941i	$-0.0830441\pi$
$558$		0			0
$559$	25.1205			1.06249
$560$		0			0
$561$	−1.57662			−0.0665651
$562$		0			0
$563$			7.41435i			0.312478i	0.987719	+	0.156239i	$0.0499370\pi$
$563$			7.41435i			0.312478i	−0.987719	+	0.156239i	$0.950063\pi$
$564$		0			0
$565$	−19.4928	−	17.1507i	−0.820070	−	0.721537i
$566$		0			0
$567$			3.23400i			0.135815i
$568$		0			0
$569$	3.57390			0.149826			0.0749128	−	0.997190i	$-0.476132\pi$
$569$	3.57390			0.149826			0.0749128	+	0.997190i	$0.476132\pi$
$570$		0			0
$571$	−23.9459			−1.00211			−0.501053	−	0.865417i	$-0.667054\pi$
$571$	−23.9459			−1.00211			−0.501053	+	0.865417i	$0.667054\pi$
$572$		0			0
$573$		−	18.4208i		−	0.769539i
$574$		0			0
$575$	3.59856	−	0.461895i	0.150070	−	0.0192624i
$576$		0			0
$577$			12.8977i			0.536939i	0.963288	+	0.268470i	$0.0865179\pi$
$577$			12.8977i			0.536939i	−0.963288	+	0.268470i	$0.913482\pi$
$578$		0			0
$579$	−17.5629			−0.729889
$580$		0			0
$581$	−23.4635			−0.973431
$582$		0			0
$583$			7.19335i			0.297918i
$584$		0			0
$585$	−5.04911	−	4.44245i	−0.208755	−	0.183673i
$586$		0			0
$587$		−	10.8755i		−	0.448880i	−0.974488	−	0.224440i	$-0.927945\pi$
$587$		−	10.8755i		−	0.448880i	0.974488	−	0.224440i	$-0.0720553\pi$
$588$		0			0
$589$	−31.6560			−1.30436
$590$		0			0
$591$	−11.2224			−0.461627
$592$		0			0
$593$		−	22.6496i		−	0.930109i	−0.885282	−	0.465054i	$-0.846035\pi$
$593$		−	22.6496i		−	0.930109i	0.885282	−	0.465054i	$-0.153965\pi$
$594$		0			0
$595$	−6.23826	+	7.09015i	−0.255744	+	0.290668i
$596$		0			0
$597$			16.5069i			0.675582i
$598$		0			0
$599$	2.77235			0.113275			0.0566376	−	0.998395i	$-0.481962\pi$
$599$	2.77235			0.113275			0.0566376	+	0.998395i	$0.481962\pi$
$600$		0			0
$601$	−12.0030			−0.489612			−0.244806	−	0.969572i	$-0.578724\pi$
$601$	−12.0030			−0.489612			−0.244806	+	0.969572i	$0.578724\pi$
$602$		0			0
$603$		−	1.00000i		−	0.0407231i
$604$		0			0
$605$	14.0949	−	16.0197i	0.573039	−	0.651293i
$606$		0			0
$607$			38.4351i			1.56003i	0.625758	+	0.780017i	$0.284790\pi$
$607$			38.4351i			1.56003i	−0.625758	+	0.780017i	$0.715210\pi$
$608$		0			0
$609$	21.3695			0.865935
$610$		0			0
$611$	−34.5318			−1.39701
$612$		0			0
$613$		−	17.6710i		−	0.713727i	−0.934157	−	0.356863i	$-0.883846\pi$
$613$		−	17.6710i		−	0.713727i	0.934157	−	0.356863i	$-0.116154\pi$
$614$		0			0
$615$	11.1557	+	9.81529i	0.449840	+	0.395791i
$616$		0			0
$617$			24.8569i			1.00070i	0.865823	+	0.500351i	$0.166796\pi$
$617$			24.8569i			1.00070i	−0.865823	+	0.500351i	$0.833204\pi$
$618$		0			0
$619$	24.6909			0.992411			0.496205	−	0.868205i	$-0.334726\pi$
$619$	24.6909			0.992411			0.496205	+	0.868205i	$0.334726\pi$
$620$		0			0
$621$	−0.725616			−0.0291180
$622$		0			0
$623$			17.7351i			0.710543i
$624$		0			0
$625$	−24.1896	+	6.31376i	−0.967584	+	0.252550i
$626$		0			0
$627$		−	6.29191i		−	0.251274i
$628$		0			0
$629$	−2.52731			−0.100770
$630$		0			0
$631$	26.0132			1.03557			0.517784	−	0.855511i	$-0.326757\pi$
$631$	26.0132			1.03557			0.517784	+	0.855511i	$0.326757\pi$
$632$		0			0
$633$			1.07243i			0.0426251i
$634$		0			0
$635$	−27.6202	−	24.3016i	−1.09607	−	0.964378i
$636$		0			0
$637$			10.4027i			0.412168i
$638$		0			0
$639$	−11.4976			−0.454836
$640$		0			0
$641$	21.0892			0.832973			0.416486	−	0.909142i	$-0.363261\pi$
$641$	21.0892			0.832973			0.416486	+	0.909142i	$0.363261\pi$
$642$		0			0
$643$			4.02367i			0.158678i	0.996848	+	0.0793391i	$0.0252810\pi$
$643$			4.02367i			0.158678i	−0.996848	+	0.0793391i	$0.974719\pi$
$644$		0			0
$645$	−12.3369	+	14.0216i	−0.485765	+	0.552100i
$646$		0			0
$647$			39.6116i			1.55729i	0.627463	+	0.778646i	$0.284093\pi$
$647$			39.6116i			1.55729i	−0.627463	+	0.778646i	$0.715907\pi$
$648$		0			0
$649$	−0.667156			−0.0261881
$650$		0			0
$651$	−19.6435			−0.769889
$652$		0			0
$653$			37.2351i			1.45712i	0.684980	+	0.728562i	$0.259811\pi$
$653$			37.2351i			1.45712i	−0.684980	+	0.728562i	$0.740189\pi$
$654$		0			0
$655$	13.1593	−	14.9564i	0.514178	−	0.584394i
$656$		0			0
$657$		−	9.22347i		−	0.359842i
$658$		0			0
$659$	44.0553			1.71615			0.858074	−	0.513525i	$-0.171661\pi$
$659$	44.0553			1.71615			0.858074	+	0.513525i	$0.171661\pi$
$660$		0			0
$661$	−29.0006			−1.12799			−0.563996	−	0.825778i	$-0.690737\pi$
$661$	−29.0006			−1.12799			−0.563996	+	0.825778i	$0.690737\pi$
$662$		0			0
$663$			3.92777i			0.152542i
$664$		0			0
$665$	−28.2950	−	24.8953i	−1.09723	−	0.965399i
$666$		0			0
$667$			4.79469i			0.185651i
$668$		0			0
$669$	20.8037			0.804317
$670$		0			0
$671$	−6.20488			−0.239537
$672$		0			0
$673$		−	30.6887i		−	1.18296i	−0.806319	−	0.591481i	$-0.798543\pi$
$673$		−	30.6887i		−	1.18296i	0.806319	−	0.591481i	$-0.201457\pi$
$674$		0			0
$675$	4.95931	−	0.636556i	0.190884	−	0.0245010i
$676$		0			0
$677$			13.4091i			0.515353i	0.966231	+	0.257676i	$0.0829569\pi$
$677$			13.4091i			0.515353i	−0.966231	+	0.257676i	$0.917043\pi$
$678$		0			0
$679$	−6.71902			−0.257852
$680$		0			0
$681$	27.6609			1.05997
$682$		0			0
$683$		−	24.9058i		−	0.952994i	−0.879176	−	0.476497i	$-0.841906\pi$
$683$		−	24.9058i		−	0.952994i	0.879176	−	0.476497i	$-0.158094\pi$
$684$		0			0
$685$	6.29898	+	5.54215i	0.240672	+	0.211754i
$686$		0			0
$687$		−	1.13274i		−	0.0432167i
$688$		0			0
$689$	17.9205			0.682717
$690$		0			0
$691$	35.4854			1.34993			0.674964	−	0.737851i	$-0.264159\pi$
$691$	35.4854			1.34993			0.674964	+	0.737851i	$0.264159\pi$
$692$		0			0
$693$		−	3.90431i		−	0.148313i
$694$		0			0
$695$	−28.8325	+	32.7699i	−1.09368	+	1.24303i
$696$		0			0
$697$		−	8.67814i		−	0.328708i
$698$		0			0
$699$	−23.6704			−0.895296
$700$		0			0
$701$	−35.9639			−1.35834			−0.679169	−	0.733982i	$-0.737660\pi$
$701$	−35.9639			−1.35834			−0.679169	+	0.733982i	$0.737660\pi$
$702$		0			0
$703$		−	10.0858i		−	0.380395i
$704$		0			0
$705$	16.9588	−	19.2747i	0.638707	−	0.725929i
$706$		0			0
$707$			0.0653585i			0.00245806i
$708$		0			0
$709$	−9.79286			−0.367778			−0.183889	−	0.982947i	$-0.558869\pi$
$709$	−9.79286			−0.367778			−0.183889	+	0.982947i	$0.558869\pi$
$710$		0			0
$711$	5.79731			0.217416
$712$		0			0
$713$		−	4.40743i		−	0.165059i
$714$		0			0
$715$	6.09564	+	5.36324i	0.227964	+	0.200574i
$716$		0			0
$717$		−	18.5005i		−	0.690914i
$718$		0			0
$719$	22.7630			0.848917			0.424459	−	0.905447i	$-0.360464\pi$
$719$	22.7630			0.848917			0.424459	+	0.905447i	$0.360464\pi$
$720$		0			0
$721$	−55.5529			−2.06890
$722$		0			0
$723$			12.6062i			0.468828i
$724$		0			0
$725$	−4.20620	−	32.7699i	−0.156214	−	1.21704i
$726$		0			0
$727$		−	3.30626i		−	0.122623i	−0.998119	−	0.0613113i	$-0.980472\pi$
$727$		−	3.30626i		−	0.122623i	0.998119	−	0.0613113i	$-0.0195282\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	10.9076			0.403432
$732$		0			0
$733$			13.1087i			0.484183i	0.970253	+	0.242091i	$0.0778334\pi$
$733$			13.1087i			0.484183i	−0.970253	+	0.242091i	$0.922167\pi$
$734$		0			0
$735$	−5.80649	−	5.10883i	−0.214175	−	0.188442i
$736$		0			0
$737$			1.20727i			0.0444704i
$738$		0			0
$739$	34.0763			1.25352			0.626758	−	0.779214i	$-0.284381\pi$
$739$	34.0763			1.25352			0.626758	+	0.779214i	$0.284381\pi$
$740$		0			0
$741$	−15.6748			−0.575827
$742$		0			0
$743$		−	23.8779i		−	0.875995i	−0.898976	−	0.437998i	$-0.855688\pi$
$743$		−	23.8779i		−	0.875995i	0.898976	−	0.437998i	$-0.144312\pi$
$744$		0			0
$745$	10.5179	−	11.9542i	0.385347	−	0.437969i
$746$		0			0
$747$		−	7.25526i		−	0.265456i
$748$		0			0
$749$	13.0475			0.476747
$750$		0			0
$751$	−49.3932			−1.80238			−0.901191	−	0.433422i	$-0.857306\pi$
$751$	−49.3932			−1.80238			−0.901191	+	0.433422i	$0.857306\pi$
$752$		0			0
$753$		−	27.7683i		−	1.01193i
$754$		0			0
$755$	3.54877	−	4.03338i	0.129153	−	0.146790i
$756$		0			0
$757$		−	16.7364i		−	0.608296i	−0.952625	−	0.304148i	$-0.901628\pi$
$757$		−	16.7364i		−	0.608296i	0.952625	−	0.304148i	$-0.0983718\pi$
$758$		0			0
$759$	0.876014			0.0317973
$760$		0			0
$761$	−1.53218			−0.0555413			−0.0277707	−	0.999614i	$-0.508841\pi$
$761$	−1.53218			−0.0555413			−0.0277707	+	0.999614i	$0.508841\pi$
$762$		0			0
$763$			56.7544i			2.05465i
$764$		0			0
$765$	−2.19238	−	1.92896i	−0.0792656	−	0.0697417i
$766$		0			0
$767$			1.66206i			0.0600134i
$768$		0			0
$769$	13.6503			0.492243			0.246121	−	0.969239i	$-0.420844\pi$
$769$	13.6503			0.492243			0.246121	+	0.969239i	$0.420844\pi$
$770$		0			0
$771$	−6.68359			−0.240703
$772$		0			0
$773$		−	45.2072i		−	1.62599i	−0.582271	−	0.812995i	$-0.697836\pi$
$773$		−	45.2072i		−	1.62599i	0.582271	−	0.812995i	$-0.302164\pi$
$774$		0			0
$775$	3.86647	+	30.1231i	0.138888	+	1.08205i
$776$		0			0
$777$		−	6.25856i		−	0.224525i
$778$		0			0
$779$	34.6323			1.24083
$780$		0			0
$781$	13.8807			0.496689
$782$		0			0
$783$			6.60775i			0.236142i
$784$		0			0
$785$	−20.0217	−	17.6160i	−0.714604	−	0.628743i
$786$		0			0
$787$		−	38.8290i		−	1.38410i	−0.721848	−	0.692052i	$-0.756707\pi$
$787$		−	38.8290i		−	1.38410i	0.721848	−	0.692052i	$-0.243293\pi$
$788$		0			0
$789$	11.2651			0.401048
$790$		0			0
$791$	−37.5512			−1.33517
$792$		0			0
$793$			15.4579i			0.548928i
$794$		0			0
$795$	−8.80089	+	10.0027i	−0.312136	+	0.354761i
$796$		0			0
$797$			1.36571i			0.0483761i	0.999707	+	0.0241880i	$0.00770004\pi$
$797$			1.36571i			0.0483761i	−0.999707	+	0.0241880i	$0.992300\pi$
$798$		0			0
$799$	−14.9941			−0.530453
$800$		0			0
$801$	−5.48396			−0.193766
$802$		0			0
$803$			11.1352i			0.392953i
$804$		0			0
$805$	3.46614	−	3.93948i	0.122166	−	0.138848i
$806$		0			0
$807$			16.1349i			0.567977i
$808$		0			0
$809$	2.38023			0.0836843			0.0418422	−	0.999124i	$-0.486677\pi$
$809$	2.38023			0.0836843			0.0418422	+	0.999124i	$0.486677\pi$
$810$		0			0
$811$	−11.8158			−0.414907			−0.207454	−	0.978245i	$-0.566518\pi$
$811$	−11.8158			−0.414907			−0.207454	+	0.978245i	$0.566518\pi$
$812$		0			0
$813$		−	24.6627i		−	0.864958i
$814$		0			0
$815$	−5.48484	−	4.82583i	−0.192126	−	0.169041i
$816$		0			0
$817$			43.5295i			1.52290i
$818$		0			0
$819$	−9.72665			−0.339877
$820$		0			0
$821$	−39.9197			−1.39321			−0.696603	−	0.717457i	$-0.745306\pi$
$821$	−39.9197			−1.39321			−0.696603	+	0.717457i	$0.745306\pi$
$822$		0			0
$823$			19.6220i			0.683978i	0.939704	+	0.341989i	$0.111101\pi$
$823$			19.6220i			0.683978i	−0.939704	+	0.341989i	$0.888899\pi$
$824$		0			0
$825$	−5.98723	+	0.768495i	−0.208449	+	0.0267556i
$826$		0			0
$827$			23.3852i			0.813184i	0.913610	+	0.406592i	$0.133283\pi$
$827$			23.3852i			0.813184i	−0.913610	+	0.406592i	$0.866717\pi$
$828$		0			0
$829$	19.9841			0.694078			0.347039	−	0.937851i	$-0.387187\pi$
$829$	19.9841			0.694078			0.347039	+	0.937851i	$0.387187\pi$
$830$		0			0
$831$	26.4497			0.917531
$832$		0			0
$833$			4.51695i			0.156503i
$834$		0			0
$835$	−5.16025	−	4.54023i	−0.178578	−	0.157121i
$836$		0			0
$837$		−	6.07405i		−	0.209950i
$838$		0			0
$839$	−17.6716			−0.610090			−0.305045	−	0.952338i	$-0.598672\pi$
$839$	−17.6716			−0.610090			−0.305045	+	0.952338i	$0.598672\pi$
$840$		0			0
$841$	14.6623			0.505598
$842$		0			0
$843$		−	23.6361i		−	0.814070i
$844$		0			0
$845$	−5.84064	+	6.63823i	−0.200924	+	0.228362i
$846$		0			0
$847$		−	30.8605i		−	1.06038i
$848$		0			0
$849$	−15.4054			−0.528712
$850$		0			0
$851$	1.40424			0.0481367
$852$		0			0
$853$			27.3921i			0.937888i	0.883228	+	0.468944i	$0.155365\pi$
$853$			27.3921i			0.937888i	−0.883228	+	0.468944i	$0.844635\pi$
$854$		0			0
$855$	7.69799	−	8.74923i	0.263266	−	0.299217i
$856$		0			0
$857$		−	50.9789i		−	1.74140i	−0.491811	−	0.870702i	$-0.663665\pi$
$857$		−	50.9789i		−	1.74140i	0.491811	−	0.870702i	$-0.336335\pi$
$858$		0			0
$859$	−37.1183			−1.26646			−0.633230	−	0.773963i	$-0.718271\pi$
$859$	−37.1183			−1.26646			−0.633230	+	0.773963i	$0.718271\pi$
$860$		0			0
$861$	21.4904			0.732389
$862$		0			0
$863$		−	6.13629i		−	0.208882i	−0.994531	−	0.104441i	$-0.966695\pi$
$863$		−	6.13629i		−	0.208882i	0.994531	−	0.104441i	$-0.0333053\pi$
$864$		0			0
$865$	−26.3916	−	23.2206i	−0.897343	−	0.789526i
$866$		0			0
$867$		−	15.2945i		−	0.519429i
$868$		0			0
$869$	−6.99892			−0.237422
$870$		0			0
$871$	3.00762			0.101909
$872$		0			0
$873$		−	2.07762i		−	0.0703167i
$874$		0			0
$875$	−20.2339	+	29.9656i	−0.684029	+	1.01302i
$876$		0			0
$877$			48.1883i			1.62720i	0.581423	+	0.813601i	$0.302496\pi$
$877$			48.1883i			1.62720i	−0.581423	+	0.813601i	$0.697504\pi$
$878$		0			0
$879$	3.15135			0.106292
$880$		0			0
$881$	−33.4487			−1.12691			−0.563457	−	0.826145i	$-0.690529\pi$
$881$	−33.4487			−1.12691			−0.563457	+	0.826145i	$0.690529\pi$
$882$		0			0
$883$			32.2616i			1.08569i	0.839833	+	0.542844i	$0.182653\pi$
$883$			32.2616i			1.08569i	−0.839833	+	0.542844i	$0.817347\pi$
$884$		0			0
$885$	−0.927715	−	0.816249i	−0.0311848	−	0.0274379i
$886$		0			0
$887$		−	35.4643i		−	1.19077i	−0.803439	−	0.595387i	$-0.796999\pi$
$887$		−	35.4643i		−	1.19077i	0.803439	−	0.595387i	$-0.203001\pi$
$888$		0			0
$889$	−53.2078			−1.78453
$890$		0			0
$891$	1.20727			0.0404451
$892$		0			0
$893$		−	59.8376i		−	2.00239i
$894$		0			0
$895$	−3.10613	+	3.53030i	−0.103826	+	0.118005i
$896$		0			0
$897$		−	2.18238i		−	0.0728674i
$898$		0			0
$899$	−40.1358			−1.33860
$900$		0			0
$901$	7.78127			0.259232
$902$		0			0
$903$			27.0113i			0.898881i
$904$		0			0
$905$	33.0147	−	37.5231i	1.09745	−	1.24731i
$906$		0			0
$907$		−	37.9813i		−	1.26115i	−0.776129	−	0.630574i	$-0.782819\pi$
$907$		−	37.9813i		−	1.26115i	0.776129	−	0.630574i	$-0.217181\pi$
$908$		0			0
$909$	−0.0202098			−0.000670316
$910$		0			0
$911$	20.6087			0.682798			0.341399	−	0.939918i	$-0.389099\pi$
$911$	20.6087			0.682798			0.341399	+	0.939918i	$0.389099\pi$
$912$		0			0
$913$			8.75906i			0.289882i
$914$		0			0
$915$	−8.62821	−	7.59151i	−0.285240	−	0.250968i
$916$		0			0
$917$		−	28.8121i		−	0.951459i
$918$		0			0
$919$	9.55243			0.315106			0.157553	−	0.987511i	$-0.449640\pi$
$919$	9.55243			0.315106			0.157553	+	0.987511i	$0.449640\pi$
$920$		0			0
$921$	−25.3961			−0.836831
$922$		0			0
$923$		−	34.5803i		−	1.13822i
$924$		0			0
$925$	−9.59745	+	1.23189i	−0.315562	+	0.0405042i
$926$		0			0
$927$		−	17.1778i		−	0.564192i
$928$		0			0
$929$	45.2822			1.48566			0.742831	−	0.669479i	$-0.233483\pi$
$929$	45.2822			1.48566			0.742831	+	0.669479i	$0.233483\pi$
$930$		0			0
$931$	−18.0260			−0.590778
$932$		0			0
$933$		−	22.7465i		−	0.744686i
$934$		0			0
$935$	2.64679	+	2.32878i	0.0865593	+	0.0761591i
$936$		0			0
$937$			1.34952i			0.0440869i	0.999757	+	0.0220435i	$0.00701722\pi$
$937$			1.34952i			0.0440869i	−0.999757	+	0.0220435i	$0.992983\pi$
$938$		0			0
$939$	−16.2629			−0.530720
$940$		0			0
$941$	−12.4715			−0.406559			−0.203279	−	0.979121i	$-0.565160\pi$
$941$	−12.4715			−0.406559			−0.203279	+	0.979121i	$0.565160\pi$
$942$		0			0
$943$			4.82181i			0.157020i
$944$		0			0
$945$	4.77683	−	5.42915i	0.155390	−	0.176610i
$946$		0			0
$947$			25.0060i			0.812587i	0.913743	+	0.406294i	$0.133179\pi$
$947$			25.0060i			0.812587i	−0.913743	+	0.406294i	$0.866821\pi$
$948$		0			0
$949$	27.7407			0.900501
$950$		0			0
$951$	−3.56463			−0.115591
$952$		0			0
$953$		−	9.23382i		−	0.299113i	−0.988753	−	0.149556i	$-0.952215\pi$
$953$		−	9.23382i		−	0.299113i	0.988753	−	0.149556i	$-0.0477845\pi$
$954$		0			0
$955$	−27.2087	+	30.9243i	−0.880453	+	1.00069i
$956$		0			0
$957$		−	7.97734i		−	0.257871i
$958$		0			0
$959$	12.1344			0.391840
$960$		0			0
$961$	5.89409			0.190132
$962$		0			0
$963$			4.03449i			0.130010i
$964$		0			0
$965$	29.4841	+	25.9415i	0.949127	+	0.835088i
$966$		0			0
$967$			20.0547i			0.644916i	0.946584	+	0.322458i	$0.104509\pi$
$967$			20.0547i			0.644916i	−0.946584	+	0.322458i	$0.895491\pi$
$968$		0			0
$969$	−6.80615			−0.218645
$970$		0			0
$971$	53.7702			1.72557			0.862784	−	0.505573i	$-0.168719\pi$
$971$	53.7702			1.72557			0.862784	+	0.505573i	$0.168719\pi$
$972$		0			0
$973$			63.1281i			2.02379i
$974$		0			0
$975$	1.91452	+	14.9157i	0.0613137	+	0.477686i
$976$		0			0
$977$		−	12.0924i		−	0.386869i	−0.981113	−	0.193435i	$-0.938037\pi$
$977$		−	12.0924i		−	0.386869i	0.981113	−	0.193435i	$-0.0619628\pi$
$978$		0			0
$979$	6.62062			0.211596
$980$		0			0
$981$	−17.5493			−0.560306
$982$		0			0
$983$			44.9619i			1.43406i	0.697042	+	0.717030i	$0.254499\pi$
$983$			44.9619i			1.43406i	−0.697042	+	0.717030i	$0.745501\pi$
$984$		0			0
$985$	18.8398	+	16.5762i	0.600287	+	0.528161i
$986$		0			0
$987$		−	37.1310i		−	1.18189i
$988$		0			0
$989$	−6.06056			−0.192715
$990$		0			0
$991$	2.90251			0.0922012			0.0461006	−	0.998937i	$-0.485321\pi$
$991$	2.90251			0.0922012			0.0461006	+	0.998937i	$0.485321\pi$
$992$		0			0
$993$			5.34386i			0.169582i
$994$		0			0
$995$	24.3818	−	27.7113i	0.772954	−	0.878508i
$996$		0			0
$997$		−	40.6112i		−	1.28617i	−0.765795	−	0.643085i	$-0.777654\pi$
$997$		−	40.6112i		−	1.28617i	0.765795	−	0.643085i	$-0.222346\pi$
$998$		0			0
$999$	1.93524			0.0612282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.g.b.1609.5	✓	24
5.4	even	2	inner	4020.2.g.b.1609.17	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.g.b.1609.5	✓	24	1.1	even	1	trivial
4020.2.g.b.1609.17	yes	24	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +(-1.47707 + 1.67877i) q^{5} +3.23400i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +(-1.47707 + 1.67877i) q^{5} +3.23400i q^{7} -1.00000 q^{9} +1.20727 q^{11} -3.00762i q^{13} +(1.67877 + 1.47707i) q^{15} -1.30594i q^{17} +5.21168 q^{19} +3.23400 q^{21} +0.725616i q^{23} +(-0.636556 - 4.95931i) q^{25} +1.00000i q^{27} +6.60775 q^{29} -6.07405 q^{31} -1.20727i q^{33} +(-5.42915 - 4.77683i) q^{35} -1.93524i q^{37} -3.00762 q^{39} +6.64513 q^{41} +8.35230i q^{43} +(1.47707 - 1.67877i) q^{45} -11.4814i q^{47} -3.45877 q^{49} -1.30594 q^{51} +5.95836i q^{53} +(-1.78322 + 2.02673i) q^{55} -5.21168i q^{57} -0.552615 q^{59} -5.13959 q^{61} -3.23400i q^{63} +(5.04911 + 4.44245i) q^{65} +1.00000i q^{67} +0.725616 q^{69} +11.4976 q^{71} +9.22347i q^{73} +(-4.95931 + 0.636556i) q^{75} +3.90431i q^{77} -5.79731 q^{79} +1.00000 q^{81} +7.25526i q^{83} +(2.19238 + 1.92896i) q^{85} -6.60775i q^{87} +5.48396 q^{89} +9.72665 q^{91} +6.07405i q^{93} +(-7.69799 + 8.74923i) q^{95} +2.07762i q^{97} -1.20727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 4 q^{5} - 24 q^{9}+O(q^{10}) \) 24 * q - 4 * q^5 - 24 * q^9	\( 24 q - 4 q^{5} - 24 q^{9} - 8 q^{11} - 2 q^{15} + 16 q^{19} - 20 q^{21} + 10 q^{25} + 36 q^{29} - 2 q^{35} + 4 q^{39} - 24 q^{41} + 4 q^{45} - 4 q^{51} - 4 q^{55} + 24 q^{59} - 4 q^{61} - 20 q^{65} - 4 q^{69} + 20 q^{71} - 12 q^{75} - 28 q^{79} + 24 q^{81} - 16 q^{85} + 48 q^{89} - 20 q^{91} - 4 q^{95} + 8 q^{99}+O(q^{100}) \) 24 * q - 4 * q^5 - 24 * q^9 - 8 * q^11 - 2 * q^15 + 16 * q^19 - 20 * q^21 + 10 * q^25 + 36 * q^29 - 2 * q^35 + 4 * q^39 - 24 * q^41 + 4 * q^45 - 4 * q^51 - 4 * q^55 + 24 * q^59 - 4 * q^61 - 20 * q^65 - 4 * q^69 + 20 * q^71 - 12 * q^75 - 28 * q^79 + 24 * q^81 - 16 * q^85 + 48 * q^89 - 20 * q^91 - 4 * q^95 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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