LMFDB - Embedded newform 4020.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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, 0, 0]))

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

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

chi := DirichletCharacter("4020.1");

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.a (trivial)

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:	yes
Analytic conductor:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 21x^{5} - 12x^{4} + 93x^{3} + 18x^{2} - 120x + 48 $ x^7 - 21x^5 - 12x^4 + 93x^3 + 18x^2 - 120*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.819259$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -4.84231 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -4.84231 q^{7} +1.00000 q^{9} +2.99012 q^{11} +3.60594 q^{13} +1.00000 q^{15} +2.27070 q^{17} -7.97380 q^{19} -4.84231 q^{21} -5.80294 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.96063 q^{29} +4.58797 q^{31} +2.99012 q^{33} -4.84231 q^{35} +9.44146 q^{37} +3.60594 q^{39} -1.48060 q^{41} +10.9395 q^{43} +1.00000 q^{45} -0.843708 q^{47} +16.4480 q^{49} +2.27070 q^{51} +5.10704 q^{53} +2.99012 q^{55} -7.97380 q^{57} -11.3069 q^{59} +10.2161 q^{61} -4.84231 q^{63} +3.60594 q^{65} -1.00000 q^{67} -5.80294 q^{69} +1.79657 q^{71} +2.50789 q^{73} +1.00000 q^{75} -14.4791 q^{77} +5.85187 q^{79} +1.00000 q^{81} -0.213675 q^{83} +2.27070 q^{85} +2.96063 q^{87} -2.60419 q^{89} -17.4611 q^{91} +4.58797 q^{93} -7.97380 q^{95} -10.5722 q^{97} +2.99012 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	$ 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) $ 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.84231	−1.83022	−0.915111	−	0.403203i	$-0.867897\pi$
$7$	−4.84231	−1.83022	−0.915111	+	0.403203i	$0.867897\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.99012	0.901555	0.450777	−	0.892636i	$-0.351147\pi$
$11$	2.99012	0.901555	0.450777	+	0.892636i	$0.351147\pi$
$12$	0	0
$13$	3.60594	1.00011	0.500055	−	0.865994i	$-0.333313\pi$
$13$	3.60594	1.00011	0.500055	+	0.865994i	$0.333313\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.27070	0.550725	0.275362	−	0.961340i	$-0.411202\pi$
$17$	2.27070	0.550725	0.275362	+	0.961340i	$0.411202\pi$
$18$	0	0
$19$	−7.97380	−1.82932	−0.914658	−	0.404228i	$-0.867540\pi$
$19$	−7.97380	−1.82932	−0.914658	+	0.404228i	$0.867540\pi$
$20$	0	0
$21$	−4.84231	−1.05668
$22$	0	0
$23$	−5.80294	−1.21000	−0.604999	−	0.796226i	$-0.706826\pi$
$23$	−5.80294	−1.21000	−0.604999	+	0.796226i	$0.706826\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.96063	0.549776	0.274888	−	0.961476i	$-0.411359\pi$
$29$	2.96063	0.549776	0.274888	+	0.961476i	$0.411359\pi$
$30$	0	0
$31$	4.58797	0.824023	0.412012	−	0.911179i	$-0.364826\pi$
$31$	4.58797	0.824023	0.412012	+	0.911179i	$0.364826\pi$
$32$	0	0
$33$	2.99012	0.520513
$34$	0	0
$35$	−4.84231	−0.818500
$36$	0	0
$37$	9.44146	1.55217	0.776083	−	0.630630i	$-0.217204\pi$
$37$	9.44146	1.55217	0.776083	+	0.630630i	$0.217204\pi$
$38$	0	0
$39$	3.60594	0.577413
$40$	0	0
$41$	−1.48060	−0.231231	−0.115615	−	0.993294i	$-0.536884\pi$
$41$	−1.48060	−0.231231	−0.115615	+	0.993294i	$0.536884\pi$
$42$	0	0
$43$	10.9395	1.66825	0.834127	−	0.551572i	$-0.185972\pi$
$43$	10.9395	1.66825	0.834127	+	0.551572i	$0.185972\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.843708	−0.123067	−0.0615337	−	0.998105i	$-0.519599\pi$
$47$	−0.843708	−0.123067	−0.0615337	+	0.998105i	$0.519599\pi$
$48$	0	0
$49$	16.4480	2.34971
$50$	0	0
$51$	2.27070	0.317961
$52$	0	0
$53$	5.10704	0.701506	0.350753	−	0.936468i	$-0.385926\pi$
$53$	5.10704	0.701506	0.350753	+	0.936468i	$0.385926\pi$
$54$	0	0
$55$	2.99012	0.403187
$56$	0	0
$57$	−7.97380	−1.05616
$58$	0	0
$59$	−11.3069	−1.47204	−0.736018	−	0.676962i	$-0.763296\pi$
$59$	−11.3069	−1.47204	−0.736018	+	0.676962i	$0.763296\pi$
$60$	0	0
$61$	10.2161	1.30804	0.654018	−	0.756479i	$-0.273082\pi$
$61$	10.2161	1.30804	0.654018	+	0.756479i	$0.273082\pi$
$62$	0	0
$63$	−4.84231	−0.610074
$64$	0	0
$65$	3.60594	0.447262
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	0	0
$69$	−5.80294	−0.698592
$70$	0	0
$71$	1.79657	0.213213	0.106607	−	0.994301i	$-0.466001\pi$
$71$	1.79657	0.213213	0.106607	+	0.994301i	$0.466001\pi$
$72$	0	0
$73$	2.50789	0.293527	0.146763	−	0.989172i	$-0.453114\pi$
$73$	2.50789	0.293527	0.146763	+	0.989172i	$0.453114\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−14.4791	−1.65004
$78$	0	0
$79$	5.85187	0.658387	0.329193	−	0.944263i	$-0.393223\pi$
$79$	5.85187	0.658387	0.329193	+	0.944263i	$0.393223\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.213675	−0.0234539	−0.0117269	−	0.999931i	$-0.503733\pi$
$83$	−0.213675	−0.0234539	−0.0117269	+	0.999931i	$0.503733\pi$
$84$	0	0
$85$	2.27070	0.246292
$86$	0	0
$87$	2.96063	0.317413
$88$	0	0
$89$	−2.60419	−0.276043	−0.138022	−	0.990429i	$-0.544074\pi$
$89$	−2.60419	−0.276043	−0.138022	+	0.990429i	$0.544074\pi$
$90$	0	0
$91$	−17.4611	−1.83042
$92$	0	0
$93$	4.58797	0.475750
$94$	0	0
$95$	−7.97380	−0.818095
$96$	0	0
$97$	−10.5722	−1.07345	−0.536723	−	0.843758i	$-0.680338\pi$
$97$	−10.5722	−1.07345	−0.536723	+	0.843758i	$0.680338\pi$
$98$	0	0
$99$	2.99012	0.300518
$100$	0	0
$101$	10.0779	1.00279	0.501396	−	0.865218i	$-0.332820\pi$
$101$	10.0779	1.00279	0.501396	+	0.865218i	$0.332820\pi$
$102$	0	0
$103$	4.02218	0.396317	0.198159	−	0.980170i	$-0.436504\pi$
$103$	4.02218	0.396317	0.198159	+	0.980170i	$0.436504\pi$
$104$	0	0
$105$	−4.84231	−0.472561
$106$	0	0
$107$	18.6804	1.80590	0.902950	−	0.429745i	$-0.141397\pi$
$107$	18.6804	1.80590	0.902950	+	0.429745i	$0.141397\pi$
$108$	0	0
$109$	−11.0634	−1.05969	−0.529843	−	0.848096i	$-0.677749\pi$
$109$	−11.0634	−1.05969	−0.529843	+	0.848096i	$0.677749\pi$
$110$	0	0
$111$	9.44146	0.896144
$112$	0	0
$113$	16.3320	1.53638	0.768191	−	0.640221i	$-0.221157\pi$
$113$	16.3320	1.53638	0.768191	+	0.640221i	$0.221157\pi$
$114$	0	0
$115$	−5.80294	−0.541127
$116$	0	0
$117$	3.60594	0.333370
$118$	0	0
$119$	−10.9954	−1.00795
$120$	0	0
$121$	−2.05919	−0.187199
$122$	0	0
$123$	−1.48060	−0.133501
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.14947	−0.368206	−0.184103	−	0.982907i	$-0.558938\pi$
$127$	−4.14947	−0.368206	−0.184103	+	0.982907i	$0.558938\pi$
$128$	0	0
$129$	10.9395	0.963167
$130$	0	0
$131$	19.9433	1.74245	0.871227	−	0.490880i	$-0.163324\pi$
$131$	19.9433	1.74245	0.871227	+	0.490880i	$0.163324\pi$
$132$	0	0
$133$	38.6116	3.34805
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−11.7577	−1.00453	−0.502263	−	0.864715i	$-0.667499\pi$
$137$	−11.7577	−1.00453	−0.502263	+	0.864715i	$0.667499\pi$
$138$	0	0
$139$	−8.12269	−0.688958	−0.344479	−	0.938794i	$-0.611944\pi$
$139$	−8.12269	−0.688958	−0.344479	+	0.938794i	$0.611944\pi$
$140$	0	0
$141$	−0.843708	−0.0710530
$142$	0	0
$143$	10.7822	0.901653
$144$	0	0
$145$	2.96063	0.245867
$146$	0	0
$147$	16.4480	1.35661
$148$	0	0
$149$	−8.87668	−0.727206	−0.363603	−	0.931554i	$-0.618454\pi$
$149$	−8.87668	−0.727206	−0.363603	+	0.931554i	$0.618454\pi$
$150$	0	0
$151$	−10.6821	−0.869299	−0.434650	−	0.900600i	$-0.643128\pi$
$151$	−10.6821	−0.869299	−0.434650	+	0.900600i	$0.643128\pi$
$152$	0	0
$153$	2.27070	0.183575
$154$	0	0
$155$	4.58797	0.368514
$156$	0	0
$157$	9.44956	0.754157	0.377078	−	0.926181i	$-0.376929\pi$
$157$	9.44956	0.754157	0.377078	+	0.926181i	$0.376929\pi$
$158$	0	0
$159$	5.10704	0.405015
$160$	0	0
$161$	28.0997	2.21456
$162$	0	0
$163$	−0.211368	−0.0165556	−0.00827781	−	0.999966i	$-0.502635\pi$
$163$	−0.211368	−0.0165556	−0.00827781	+	0.999966i	$0.502635\pi$
$164$	0	0
$165$	2.99012	0.232780
$166$	0	0
$167$	23.2644	1.80025	0.900126	−	0.435629i	$-0.143474\pi$
$167$	23.2644	1.80025	0.900126	+	0.435629i	$0.143474\pi$
$168$	0	0
$169$	0.00283565	0.000218127 0
$170$	0	0
$171$	−7.97380	−0.609772
$172$	0	0
$173$	14.8175	1.12656	0.563278	−	0.826267i	$-0.309540\pi$
$173$	14.8175	1.12656	0.563278	+	0.826267i	$0.309540\pi$
$174$	0	0
$175$	−4.84231	−0.366044
$176$	0	0
$177$	−11.3069	−0.849880
$178$	0	0
$179$	−4.45960	−0.333326	−0.166663	−	0.986014i	$-0.553299\pi$
$179$	−4.45960	−0.333326	−0.166663	+	0.986014i	$0.553299\pi$
$180$	0	0
$181$	8.72399	0.648449	0.324224	−	0.945980i	$-0.394897\pi$
$181$	8.72399	0.648449	0.324224	+	0.945980i	$0.394897\pi$
$182$	0	0
$183$	10.2161	0.755195
$184$	0	0
$185$	9.44146	0.694150
$186$	0	0
$187$	6.78965	0.496509
$188$	0	0
$189$	−4.84231	−0.352226
$190$	0	0
$191$	0.232896	0.0168518	0.00842589	−	0.999965i	$-0.497318\pi$
$191$	0.232896	0.0168518	0.00842589	+	0.999965i	$0.497318\pi$
$192$	0	0
$193$	16.7733	1.20737	0.603684	−	0.797224i	$-0.293699\pi$
$193$	16.7733	1.20737	0.603684	+	0.797224i	$0.293699\pi$
$194$	0	0
$195$	3.60594	0.258227
$196$	0	0
$197$	1.71308	0.122052	0.0610261	−	0.998136i	$-0.480563\pi$
$197$	1.71308	0.122052	0.0610261	+	0.998136i	$0.480563\pi$
$198$	0	0
$199$	−2.64605	−0.187573	−0.0937866	−	0.995592i	$-0.529897\pi$
$199$	−2.64605	−0.187573	−0.0937866	+	0.995592i	$0.529897\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−14.3363	−1.00621
$204$	0	0
$205$	−1.48060	−0.103410
$206$	0	0
$207$	−5.80294	−0.403333
$208$	0	0
$209$	−23.8426	−1.64923
$210$	0	0
$211$	−3.23271	−0.222549	−0.111274	−	0.993790i	$-0.535493\pi$
$211$	−3.23271	−0.222549	−0.111274	+	0.993790i	$0.535493\pi$
$212$	0	0
$213$	1.79657	0.123099
$214$	0	0
$215$	10.9395	0.746066
$216$	0	0
$217$	−22.2164	−1.50814
$218$	0	0
$219$	2.50789	0.169468
$220$	0	0
$221$	8.18801	0.550785
$222$	0	0
$223$	15.1028	1.01136	0.505681	−	0.862721i	$-0.331241\pi$
$223$	15.1028	1.01136	0.505681	+	0.862721i	$0.331241\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	20.5149	1.36162	0.680812	−	0.732459i	$-0.261627\pi$
$227$	20.5149	1.36162	0.680812	+	0.732459i	$0.261627\pi$
$228$	0	0
$229$	5.13819	0.339541	0.169771	−	0.985484i	$-0.445697\pi$
$229$	5.13819	0.339541	0.169771	+	0.985484i	$0.445697\pi$
$230$	0	0
$231$	−14.4791	−0.952654
$232$	0	0
$233$	−10.4780	−0.686438	−0.343219	−	0.939255i	$-0.611517\pi$
$233$	−10.4780	−0.686438	−0.343219	+	0.939255i	$0.611517\pi$
$234$	0	0
$235$	−0.843708	−0.0550374
$236$	0	0
$237$	5.85187	0.380120
$238$	0	0
$239$	−6.36783	−0.411901	−0.205950	−	0.978562i	$-0.566029\pi$
$239$	−6.36783	−0.411901	−0.205950	+	0.978562i	$0.566029\pi$
$240$	0	0
$241$	5.78118	0.372398	0.186199	−	0.982512i	$-0.440383\pi$
$241$	5.78118	0.372398	0.186199	+	0.982512i	$0.440383\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.4480	1.05082
$246$	0	0
$247$	−28.7531	−1.82952
$248$	0	0
$249$	−0.213675	−0.0135411
$250$	0	0
$251$	18.7557	1.18385	0.591925	−	0.805993i	$-0.298368\pi$
$251$	18.7557	1.18385	0.591925	+	0.805993i	$0.298368\pi$
$252$	0	0
$253$	−17.3515	−1.09088
$254$	0	0
$255$	2.27070	0.142197
$256$	0	0
$257$	−24.2968	−1.51559	−0.757795	−	0.652493i	$-0.773723\pi$
$257$	−24.2968	−1.51559	−0.757795	+	0.652493i	$0.773723\pi$
$258$	0	0
$259$	−45.7185	−2.84081
$260$	0	0
$261$	2.96063	0.183259
$262$	0	0
$263$	8.04176	0.495876	0.247938	−	0.968776i	$-0.420247\pi$
$263$	8.04176	0.495876	0.247938	+	0.968776i	$0.420247\pi$
$264$	0	0
$265$	5.10704	0.313723
$266$	0	0
$267$	−2.60419	−0.159374
$268$	0	0
$269$	12.7458	0.777124	0.388562	−	0.921423i	$-0.372972\pi$
$269$	12.7458	0.777124	0.388562	+	0.921423i	$0.372972\pi$
$270$	0	0
$271$	2.10365	0.127788	0.0638939	−	0.997957i	$-0.479648\pi$
$271$	2.10365	0.127788	0.0638939	+	0.997957i	$0.479648\pi$
$272$	0	0
$273$	−17.4611	−1.05679
$274$	0	0
$275$	2.99012	0.180311
$276$	0	0
$277$	20.6658	1.24169	0.620843	−	0.783935i	$-0.286791\pi$
$277$	20.6658	1.24169	0.620843	+	0.783935i	$0.286791\pi$
$278$	0	0
$279$	4.58797	0.274674
$280$	0	0
$281$	−29.6454	−1.76849	−0.884247	−	0.467020i	$-0.845328\pi$
$281$	−29.6454	−1.76849	−0.884247	+	0.467020i	$0.845328\pi$
$282$	0	0
$283$	6.46126	0.384082	0.192041	−	0.981387i	$-0.438489\pi$
$283$	6.46126	0.384082	0.192041	+	0.981387i	$0.438489\pi$
$284$	0	0
$285$	−7.97380	−0.472327
$286$	0	0
$287$	7.16953	0.423204
$288$	0	0
$289$	−11.8439	−0.696702
$290$	0	0
$291$	−10.5722	−0.619754
$292$	0	0
$293$	−22.6857	−1.32531	−0.662655	−	0.748925i	$-0.730570\pi$
$293$	−22.6857	−1.32531	−0.662655	+	0.748925i	$0.730570\pi$
$294$	0	0
$295$	−11.3069	−0.658314
$296$	0	0
$297$	2.99012	0.173504
$298$	0	0
$299$	−20.9251	−1.21013
$300$	0	0
$301$	−52.9723	−3.05327
$302$	0	0
$303$	10.0779	0.578963
$304$	0	0
$305$	10.2161	0.584972
$306$	0	0
$307$	−0.646736	−0.0369112	−0.0184556	−	0.999830i	$-0.505875\pi$
$307$	−0.646736	−0.0369112	−0.0184556	+	0.999830i	$0.505875\pi$
$308$	0	0
$309$	4.02218	0.228814
$310$	0	0
$311$	−5.06710	−0.287329	−0.143665	−	0.989626i	$-0.545889\pi$
$311$	−5.06710	−0.287329	−0.143665	+	0.989626i	$0.545889\pi$
$312$	0	0
$313$	−5.41514	−0.306082	−0.153041	−	0.988220i	$-0.548907\pi$
$313$	−5.41514	−0.306082	−0.153041	+	0.988220i	$0.548907\pi$
$314$	0	0
$315$	−4.84231	−0.272833
$316$	0	0
$317$	0.998536	0.0560834	0.0280417	−	0.999607i	$-0.491073\pi$
$317$	0.998536	0.0560834	0.0280417	+	0.999607i	$0.491073\pi$
$318$	0	0
$319$	8.85265	0.495653
$320$	0	0
$321$	18.6804	1.04264
$322$	0	0
$323$	−18.1061	−1.00745
$324$	0	0
$325$	3.60594	0.200022
$326$	0	0
$327$	−11.0634	−0.611810
$328$	0	0
$329$	4.08549	0.225241
$330$	0	0
$331$	−19.4831	−1.07089	−0.535444	−	0.844570i	$-0.679856\pi$
$331$	−19.4831	−1.07089	−0.535444	+	0.844570i	$0.679856\pi$
$332$	0	0
$333$	9.44146	0.517389
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	−29.2212	−1.59178	−0.795890	−	0.605441i	$-0.792997\pi$
$337$	−29.2212	−1.59178	−0.795890	+	0.605441i	$0.792997\pi$
$338$	0	0
$339$	16.3320	0.887030
$340$	0	0
$341$	13.7186	0.742902
$342$	0	0
$343$	−45.7500	−2.47027
$344$	0	0
$345$	−5.80294	−0.312420
$346$	0	0
$347$	18.3741	0.986373	0.493186	−	0.869924i	$-0.335832\pi$
$347$	18.3741	0.986373	0.493186	+	0.869924i	$0.335832\pi$
$348$	0	0
$349$	15.7991	0.845705	0.422853	−	0.906198i	$-0.361029\pi$
$349$	15.7991	0.845705	0.422853	+	0.906198i	$0.361029\pi$
$350$	0	0
$351$	3.60594	0.192471
$352$	0	0
$353$	−36.8675	−1.96226	−0.981129	−	0.193354i	$-0.938064\pi$
$353$	−36.8675	−1.96226	−0.981129	+	0.193354i	$0.938064\pi$
$354$	0	0
$355$	1.79657	0.0953519
$356$	0	0
$357$	−10.9954	−0.581939
$358$	0	0
$359$	−29.1788	−1.54000	−0.770000	−	0.638044i	$-0.779744\pi$
$359$	−29.1788	−1.54000	−0.770000	+	0.638044i	$0.779744\pi$
$360$	0	0
$361$	44.5815	2.34640
$362$	0	0
$363$	−2.05919	−0.108080
$364$	0	0
$365$	2.50789	0.131269
$366$	0	0
$367$	1.75370	0.0915422	0.0457711	−	0.998952i	$-0.485426\pi$
$367$	1.75370	0.0915422	0.0457711	+	0.998952i	$0.485426\pi$
$368$	0	0
$369$	−1.48060	−0.0770770
$370$	0	0
$371$	−24.7299	−1.28391
$372$	0	0
$373$	11.6526	0.603346	0.301673	−	0.953411i	$-0.402455\pi$
$373$	11.6526	0.603346	0.301673	+	0.953411i	$0.402455\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	10.6759	0.549836
$378$	0	0
$379$	26.6945	1.37121	0.685603	−	0.727976i	$-0.259539\pi$
$379$	26.6945	1.37121	0.685603	+	0.727976i	$0.259539\pi$
$380$	0	0
$381$	−4.14947	−0.212584
$382$	0	0
$383$	33.2707	1.70006	0.850028	−	0.526738i	$-0.176585\pi$
$383$	33.2707	1.70006	0.850028	+	0.526738i	$0.176585\pi$
$384$	0	0
$385$	−14.4791	−0.737922
$386$	0	0
$387$	10.9395	0.556085
$388$	0	0
$389$	11.0391	0.559704	0.279852	−	0.960043i	$-0.409715\pi$
$389$	11.0391	0.559704	0.279852	+	0.960043i	$0.409715\pi$
$390$	0	0
$391$	−13.1767	−0.666376
$392$	0	0
$393$	19.9433	1.00601
$394$	0	0
$395$	5.85187	0.294440
$396$	0	0
$397$	23.4420	1.17652	0.588259	−	0.808673i	$-0.299814\pi$
$397$	23.4420	1.17652	0.588259	+	0.808673i	$0.299814\pi$
$398$	0	0
$399$	38.6116	1.93300
$400$	0	0
$401$	−20.6954	−1.03348	−0.516739	−	0.856143i	$-0.672854\pi$
$401$	−20.6954	−1.03348	−0.516739	+	0.856143i	$0.672854\pi$
$402$	0	0
$403$	16.5440	0.824113
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	28.2311	1.39936
$408$	0	0
$409$	0.475379	0.0235060	0.0117530	−	0.999931i	$-0.496259\pi$
$409$	0.475379	0.0235060	0.0117530	+	0.999931i	$0.496259\pi$
$410$	0	0
$411$	−11.7577	−0.579963
$412$	0	0
$413$	54.7516	2.69415
$414$	0	0
$415$	−0.213675	−0.0104889
$416$	0	0
$417$	−8.12269	−0.397770
$418$	0	0
$419$	11.9860	0.585555	0.292777	−	0.956181i	$-0.405420\pi$
$419$	11.9860	0.585555	0.292777	+	0.956181i	$0.405420\pi$
$420$	0	0
$421$	−29.9447	−1.45942	−0.729708	−	0.683759i	$-0.760344\pi$
$421$	−29.9447	−1.45942	−0.729708	+	0.683759i	$0.760344\pi$
$422$	0	0
$423$	−0.843708	−0.0410225
$424$	0	0
$425$	2.27070	0.110145
$426$	0	0
$427$	−49.4695	−2.39400
$428$	0	0
$429$	10.7822	0.520570
$430$	0	0
$431$	29.8556	1.43809	0.719047	−	0.694961i	$-0.244578\pi$
$431$	29.8556	1.43809	0.719047	+	0.694961i	$0.244578\pi$
$432$	0	0
$433$	−8.05589	−0.387141	−0.193571	−	0.981086i	$-0.562007\pi$
$433$	−8.05589	−0.387141	−0.193571	+	0.981086i	$0.562007\pi$
$434$	0	0
$435$	2.96063	0.141952
$436$	0	0
$437$	46.2715	2.21347
$438$	0	0
$439$	−21.2178	−1.01267	−0.506336	−	0.862336i	$-0.669001\pi$
$439$	−21.2178	−1.01267	−0.506336	+	0.862336i	$0.669001\pi$
$440$	0	0
$441$	16.4480	0.783237
$442$	0	0
$443$	29.9788	1.42434	0.712169	−	0.702008i	$-0.247713\pi$
$443$	29.9788	1.42434	0.712169	+	0.702008i	$0.247713\pi$
$444$	0	0
$445$	−2.60419	−0.123450
$446$	0	0
$447$	−8.87668	−0.419853
$448$	0	0
$449$	32.5019	1.53386	0.766929	−	0.641732i	$-0.221784\pi$
$449$	32.5019	1.53386	0.766929	+	0.641732i	$0.221784\pi$
$450$	0	0
$451$	−4.42717	−0.208467
$452$	0	0
$453$	−10.6821	−0.501890
$454$	0	0
$455$	−17.4611	−0.818589
$456$	0	0
$457$	−36.5328	−1.70893	−0.854467	−	0.519506i	$-0.826116\pi$
$457$	−36.5328	−1.70893	−0.854467	+	0.519506i	$0.826116\pi$
$458$	0	0
$459$	2.27070	0.105987
$460$	0	0
$461$	−20.6772	−0.963034	−0.481517	−	0.876437i	$-0.659914\pi$
$461$	−20.6772	−0.963034	−0.481517	+	0.876437i	$0.659914\pi$
$462$	0	0
$463$	−10.7081	−0.497649	−0.248825	−	0.968549i	$-0.580044\pi$
$463$	−10.7081	−0.497649	−0.248825	+	0.968549i	$0.580044\pi$
$464$	0	0
$465$	4.58797	0.212762
$466$	0	0
$467$	−3.90238	−0.180581	−0.0902903	−	0.995915i	$-0.528779\pi$
$467$	−3.90238	−0.180581	−0.0902903	+	0.995915i	$0.528779\pi$
$468$	0	0
$469$	4.84231	0.223597
$470$	0	0
$471$	9.44956	0.435413
$472$	0	0
$473$	32.7103	1.50402
$474$	0	0
$475$	−7.97380	−0.365863
$476$	0	0
$477$	5.10704	0.233835
$478$	0	0
$479$	25.9061	1.18368	0.591841	−	0.806055i	$-0.298401\pi$
$479$	25.9061	1.18368	0.591841	+	0.806055i	$0.298401\pi$
$480$	0	0
$481$	34.0454	1.55234
$482$	0	0
$483$	28.0997	1.27858
$484$	0	0
$485$	−10.5722	−0.480060
$486$	0	0
$487$	−19.6816	−0.891857	−0.445928	−	0.895069i	$-0.647126\pi$
$487$	−19.6816	−0.891857	−0.445928	+	0.895069i	$0.647126\pi$
$488$	0	0
$489$	−0.211368	−0.00955839
$490$	0	0
$491$	−19.1474	−0.864110	−0.432055	−	0.901847i	$-0.642211\pi$
$491$	−19.1474	−0.864110	−0.432055	+	0.901847i	$0.642211\pi$
$492$	0	0
$493$	6.72270	0.302775
$494$	0	0
$495$	2.99012	0.134396
$496$	0	0
$497$	−8.69954	−0.390228
$498$	0	0
$499$	9.97016	0.446326	0.223163	−	0.974781i	$-0.428362\pi$
$499$	9.97016	0.446326	0.223163	+	0.974781i	$0.428362\pi$
$500$	0	0
$501$	23.2644	1.03938
$502$	0	0
$503$	−22.7211	−1.01308	−0.506542	−	0.862215i	$-0.669077\pi$
$503$	−22.7211	−1.01308	−0.506542	+	0.862215i	$0.669077\pi$
$504$	0	0
$505$	10.0779	0.448462
$506$	0	0
$507$	0.00283565	0.000125936 0
$508$	0	0
$509$	−35.0924	−1.55544	−0.777722	−	0.628609i	$-0.783625\pi$
$509$	−35.0924	−1.55544	−0.777722	+	0.628609i	$0.783625\pi$
$510$	0	0
$511$	−12.1440	−0.537219
$512$	0	0
$513$	−7.97380	−0.352052
$514$	0	0
$515$	4.02218	0.177239
$516$	0	0
$517$	−2.52279	−0.110952
$518$	0	0
$519$	14.8175	0.650418
$520$	0	0
$521$	−29.8820	−1.30915	−0.654577	−	0.755996i	$-0.727153\pi$
$521$	−29.8820	−1.30915	−0.654577	+	0.755996i	$0.727153\pi$
$522$	0	0
$523$	−30.8111	−1.34728	−0.673639	−	0.739061i	$-0.735270\pi$
$523$	−30.8111	−1.34728	−0.673639	+	0.739061i	$0.735270\pi$
$524$	0	0
$525$	−4.84231	−0.211336
$526$	0	0
$527$	10.4179	0.453810
$528$	0	0
$529$	10.6742	0.464094
$530$	0	0
$531$	−11.3069	−0.490678
$532$	0	0
$533$	−5.33896	−0.231256
$534$	0	0
$535$	18.6804	0.807623
$536$	0	0
$537$	−4.45960	−0.192446
$538$	0	0
$539$	49.1814	2.11839
$540$	0	0
$541$	19.7310	0.848304	0.424152	−	0.905591i	$-0.360572\pi$
$541$	19.7310	0.848304	0.424152	+	0.905591i	$0.360572\pi$
$542$	0	0
$543$	8.72399	0.374382
$544$	0	0
$545$	−11.0634	−0.473906
$546$	0	0
$547$	−22.7348	−0.972070	−0.486035	−	0.873939i	$-0.661557\pi$
$547$	−22.7348	−0.972070	−0.486035	+	0.873939i	$0.661557\pi$
$548$	0	0
$549$	10.2161	0.436012
$550$	0	0
$551$	−23.6075	−1.00571
$552$	0	0
$553$	−28.3366	−1.20499
$554$	0	0
$555$	9.44146	0.400768
$556$	0	0
$557$	−3.57813	−0.151610	−0.0758051	−	0.997123i	$-0.524153\pi$
$557$	−3.57813	−0.151610	−0.0758051	+	0.997123i	$0.524153\pi$
$558$	0	0
$559$	39.4471	1.66844
$560$	0	0
$561$	6.78965	0.286659
$562$	0	0
$563$	33.8007	1.42453	0.712264	−	0.701911i	$-0.247670\pi$
$563$	33.8007	1.42453	0.712264	+	0.701911i	$0.247670\pi$
$564$	0	0
$565$	16.3320	0.687091
$566$	0	0
$567$	−4.84231	−0.203358
$568$	0	0
$569$	13.1156	0.549836	0.274918	−	0.961468i	$-0.411349\pi$
$569$	13.1156	0.549836	0.274918	+	0.961468i	$0.411349\pi$
$570$	0	0
$571$	−1.02208	−0.0427726	−0.0213863	−	0.999771i	$-0.506808\pi$
$571$	−1.02208	−0.0427726	−0.0213863	+	0.999771i	$0.506808\pi$
$572$	0	0
$573$	0.232896	0.00972938
$574$	0	0
$575$	−5.80294	−0.242000
$576$	0	0
$577$	−26.8153	−1.11634	−0.558168	−	0.829728i	$-0.688495\pi$
$577$	−26.8153	−1.11634	−0.558168	+	0.829728i	$0.688495\pi$
$578$	0	0
$579$	16.7733	0.697075
$580$	0	0
$581$	1.03468	0.0429258
$582$	0	0
$583$	15.2707	0.632446
$584$	0	0
$585$	3.60594	0.149087
$586$	0	0
$587$	−26.1352	−1.07872	−0.539358	−	0.842077i	$-0.681333\pi$
$587$	−26.1352	−1.07872	−0.539358	+	0.842077i	$0.681333\pi$
$588$	0	0
$589$	−36.5835	−1.50740
$590$	0	0
$591$	1.71308	0.0704668
$592$	0	0
$593$	−5.21015	−0.213955	−0.106978	−	0.994261i	$-0.534117\pi$
$593$	−5.21015	−0.213955	−0.106978	+	0.994261i	$0.534117\pi$
$594$	0	0
$595$	−10.9954	−0.450768
$596$	0	0
$597$	−2.64605	−0.108295
$598$	0	0
$599$	−26.5596	−1.08519	−0.542597	−	0.839993i	$-0.682559\pi$
$599$	−26.5596	−1.08519	−0.542597	+	0.839993i	$0.682559\pi$
$600$	0	0
$601$	−39.1562	−1.59721	−0.798607	−	0.601853i	$-0.794429\pi$
$601$	−39.1562	−1.59721	−0.798607	+	0.601853i	$0.794429\pi$
$602$	0	0
$603$	−1.00000	−0.0407231
$604$	0	0
$605$	−2.05919	−0.0837181
$606$	0	0
$607$	−5.93066	−0.240718	−0.120359	−	0.992730i	$-0.538405\pi$
$607$	−5.93066	−0.240718	−0.120359	+	0.992730i	$0.538405\pi$
$608$	0	0
$609$	−14.3363	−0.580937
$610$	0	0
$611$	−3.04236	−0.123081
$612$	0	0
$613$	−10.5007	−0.424118	−0.212059	−	0.977257i	$-0.568017\pi$
$613$	−10.5007	−0.424118	−0.212059	+	0.977257i	$0.568017\pi$
$614$	0	0
$615$	−1.48060	−0.0597036
$616$	0	0
$617$	12.3657	0.497823	0.248911	−	0.968526i	$-0.419927\pi$
$617$	12.3657	0.497823	0.248911	+	0.968526i	$0.419927\pi$
$618$	0	0
$619$	−6.58166	−0.264539	−0.132270	−	0.991214i	$-0.542226\pi$
$619$	−6.58166	−0.264539	−0.132270	+	0.991214i	$0.542226\pi$
$620$	0	0
$621$	−5.80294	−0.232864
$622$	0	0
$623$	12.6103	0.505220
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−23.8426	−0.952182
$628$	0	0
$629$	21.4387	0.854817
$630$	0	0
$631$	−14.6455	−0.583029	−0.291515	−	0.956566i	$-0.594159\pi$
$631$	−14.6455	−0.583029	−0.291515	+	0.956566i	$0.594159\pi$
$632$	0	0
$633$	−3.23271	−0.128489
$634$	0	0
$635$	−4.14947	−0.164667
$636$	0	0
$637$	59.3105	2.34997
$638$	0	0
$639$	1.79657	0.0710711
$640$	0	0
$641$	−29.6272	−1.17020	−0.585101	−	0.810960i	$-0.698945\pi$
$641$	−29.6272	−1.17020	−0.585101	+	0.810960i	$0.698945\pi$
$642$	0	0
$643$	−16.4622	−0.649208	−0.324604	−	0.945850i	$-0.605231\pi$
$643$	−16.4622	−0.649208	−0.324604	+	0.945850i	$0.605231\pi$
$644$	0	0
$645$	10.9395	0.430741
$646$	0	0
$647$	13.2632	0.521431	0.260716	−	0.965416i	$-0.416041\pi$
$647$	13.2632	0.521431	0.260716	+	0.965416i	$0.416041\pi$
$648$	0	0
$649$	−33.8090	−1.32712
$650$	0	0
$651$	−22.2164	−0.870728
$652$	0	0
$653$	−11.1582	−0.436655	−0.218327	−	0.975876i	$-0.570060\pi$
$653$	−11.1582	−0.436655	−0.218327	+	0.975876i	$0.570060\pi$
$654$	0	0
$655$	19.9433	0.779250
$656$	0	0
$657$	2.50789	0.0978422
$658$	0	0
$659$	−12.6753	−0.493760	−0.246880	−	0.969046i	$-0.579405\pi$
$659$	−12.6753	−0.493760	−0.246880	+	0.969046i	$0.579405\pi$
$660$	0	0
$661$	29.7893	1.15867	0.579334	−	0.815090i	$-0.303313\pi$
$661$	29.7893	1.15867	0.579334	+	0.815090i	$0.303313\pi$
$662$	0	0
$663$	8.18801	0.317996
$664$	0	0
$665$	38.6116	1.49729
$666$	0	0
$667$	−17.1804	−0.665228
$668$	0	0
$669$	15.1028	0.583910
$670$	0	0
$671$	30.5473	1.17927
$672$	0	0
$673$	−10.6550	−0.410721	−0.205360	−	0.978686i	$-0.565837\pi$
$673$	−10.6550	−0.410721	−0.205360	+	0.978686i	$0.565837\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	23.7018	0.910934	0.455467	−	0.890253i	$-0.349472\pi$
$677$	23.7018	0.910934	0.455467	+	0.890253i	$0.349472\pi$
$678$	0	0
$679$	51.1940	1.96464
$680$	0	0
$681$	20.5149	0.786134
$682$	0	0
$683$	−10.7008	−0.409455	−0.204728	−	0.978819i	$-0.565631\pi$
$683$	−10.7008	−0.409455	−0.204728	+	0.978819i	$0.565631\pi$
$684$	0	0
$685$	−11.7577	−0.449238
$686$	0	0
$687$	5.13819	0.196034
$688$	0	0
$689$	18.4157	0.701583
$690$	0	0
$691$	46.1014	1.75378	0.876891	−	0.480690i	$-0.159614\pi$
$691$	46.1014	1.75378	0.876891	+	0.480690i	$0.159614\pi$
$692$	0	0
$693$	−14.4791	−0.550015
$694$	0	0
$695$	−8.12269	−0.308111
$696$	0	0
$697$	−3.36200	−0.127345
$698$	0	0
$699$	−10.4780	−0.396315
$700$	0	0
$701$	45.0067	1.69988	0.849940	−	0.526879i	$-0.176638\pi$
$701$	45.0067	1.69988	0.849940	+	0.526879i	$0.176638\pi$
$702$	0	0
$703$	−75.2844	−2.83940
$704$	0	0
$705$	−0.843708	−0.0317759
$706$	0	0
$707$	−48.8005	−1.83533
$708$	0	0
$709$	−31.4383	−1.18069	−0.590346	−	0.807150i	$-0.701009\pi$
$709$	−31.4383	−1.18069	−0.590346	+	0.807150i	$0.701009\pi$
$710$	0	0
$711$	5.85187	0.219462
$712$	0	0
$713$	−26.6237	−0.997066
$714$	0	0
$715$	10.7822	0.403231
$716$	0	0
$717$	−6.36783	−0.237811
$718$	0	0
$719$	−0.0337815	−0.00125984	−0.000629919	−	1.00000i	$-0.500201\pi$
$719$	−0.0337815	−0.00125984	−0.000629919		1.00000i	$0.500201\pi$
$720$	0	0
$721$	−19.4767	−0.725349
$722$	0	0
$723$	5.78118	0.215004
$724$	0	0
$725$	2.96063	0.109955
$726$	0	0
$727$	−51.6285	−1.91480	−0.957398	−	0.288772i	$-0.906753\pi$
$727$	−51.6285	−1.91480	−0.957398	+	0.288772i	$0.906753\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	24.8402	0.918749
$732$	0	0
$733$	−47.2560	−1.74544	−0.872720	−	0.488221i	$-0.837646\pi$
$733$	−47.2560	−1.74544	−0.872720	+	0.488221i	$0.837646\pi$
$734$	0	0
$735$	16.4480	0.606693
$736$	0	0
$737$	−2.99012	−0.110142
$738$	0	0
$739$	31.3711	1.15400	0.577002	−	0.816742i	$-0.304222\pi$
$739$	31.3711	1.15400	0.577002	+	0.816742i	$0.304222\pi$
$740$	0	0
$741$	−28.7531	−1.05627
$742$	0	0
$743$	−14.3245	−0.525516	−0.262758	−	0.964862i	$-0.584632\pi$
$743$	−14.3245	−0.525516	−0.262758	+	0.964862i	$0.584632\pi$
$744$	0	0
$745$	−8.87668	−0.325216
$746$	0	0
$747$	−0.213675	−0.00781796
$748$	0	0
$749$	−90.4562	−3.30520
$750$	0	0
$751$	−22.7173	−0.828964	−0.414482	−	0.910057i	$-0.636037\pi$
$751$	−22.7173	−0.828964	−0.414482	+	0.910057i	$0.636037\pi$
$752$	0	0
$753$	18.7557	0.683496
$754$	0	0
$755$	−10.6821	−0.388762
$756$	0	0
$757$	1.53605	0.0558286	0.0279143	−	0.999610i	$-0.491113\pi$
$757$	1.53605	0.0558286	0.0279143	+	0.999610i	$0.491113\pi$
$758$	0	0
$759$	−17.3515	−0.629819
$760$	0	0
$761$	−19.0667	−0.691168	−0.345584	−	0.938388i	$-0.612319\pi$
$761$	−19.0667	−0.691168	−0.345584	+	0.938388i	$0.612319\pi$
$762$	0	0
$763$	53.5726	1.93946
$764$	0	0
$765$	2.27070	0.0820972
$766$	0	0
$767$	−40.7721	−1.47220
$768$	0	0
$769$	−14.0583	−0.506956	−0.253478	−	0.967341i	$-0.581575\pi$
$769$	−14.0583	−0.506956	−0.253478	+	0.967341i	$0.581575\pi$
$770$	0	0
$771$	−24.2968	−0.875026
$772$	0	0
$773$	48.0621	1.72867	0.864337	−	0.502913i	$-0.167738\pi$
$773$	48.0621	1.72867	0.864337	+	0.502913i	$0.167738\pi$
$774$	0	0
$775$	4.58797	0.164805
$776$	0	0
$777$	−45.7185	−1.64014
$778$	0	0
$779$	11.8060	0.422995
$780$	0	0
$781$	5.37195	0.192223
$782$	0	0
$783$	2.96063	0.105804
$784$	0	0
$785$	9.44956	0.337269
$786$	0	0
$787$	22.8497	0.814505	0.407253	−	0.913316i	$-0.366487\pi$
$787$	22.8497	0.814505	0.407253	+	0.913316i	$0.366487\pi$
$788$	0	0
$789$	8.04176	0.286294
$790$	0	0
$791$	−79.0844	−2.81192
$792$	0	0
$793$	36.8387	1.30818
$794$	0	0
$795$	5.10704	0.181128
$796$	0	0
$797$	39.6882	1.40583	0.702914	−	0.711275i	$-0.251882\pi$
$797$	39.6882	1.40583	0.702914	+	0.711275i	$0.251882\pi$
$798$	0	0
$799$	−1.91580	−0.0677763
$800$	0	0
$801$	−2.60419	−0.0920144
$802$	0	0
$803$	7.49890	0.264630
$804$	0	0
$805$	28.0997	0.990383
$806$	0	0
$807$	12.7458	0.448673
$808$	0	0
$809$	−16.0296	−0.563572	−0.281786	−	0.959477i	$-0.590927\pi$
$809$	−16.0296	−0.563572	−0.281786	+	0.959477i	$0.590927\pi$
$810$	0	0
$811$	−12.5429	−0.440440	−0.220220	−	0.975450i	$-0.570678\pi$
$811$	−12.5429	−0.440440	−0.220220	+	0.975450i	$0.570678\pi$
$812$	0	0
$813$	2.10365	0.0737784
$814$	0	0
$815$	−0.211368	−0.00740389
$816$	0	0
$817$	−87.2292	−3.05176
$818$	0	0
$819$	−17.4611	−0.610140
$820$	0	0
$821$	18.2626	0.637370	0.318685	−	0.947861i	$-0.396759\pi$
$821$	18.2626	0.637370	0.318685	+	0.947861i	$0.396759\pi$
$822$	0	0
$823$	17.6214	0.614242	0.307121	−	0.951670i	$-0.400634\pi$
$823$	17.6214	0.614242	0.307121	+	0.951670i	$0.400634\pi$
$824$	0	0
$825$	2.99012	0.104103
$826$	0	0
$827$	20.0581	0.697487	0.348743	−	0.937218i	$-0.386608\pi$
$827$	20.0581	0.697487	0.348743	+	0.937218i	$0.386608\pi$
$828$	0	0
$829$	21.7833	0.756564	0.378282	−	0.925690i	$-0.376515\pi$
$829$	21.7833	0.756564	0.378282	+	0.925690i	$0.376515\pi$
$830$	0	0
$831$	20.6658	0.716887
$832$	0	0
$833$	37.3484	1.29404
$834$	0	0
$835$	23.2644	0.805097
$836$	0	0
$837$	4.58797	0.158583
$838$	0	0
$839$	−31.2430	−1.07863	−0.539313	−	0.842105i	$-0.681316\pi$
$839$	−31.2430	−1.07863	−0.539313	+	0.842105i	$0.681316\pi$
$840$	0	0
$841$	−20.2346	−0.697746
$842$	0	0
$843$	−29.6454	−1.02104
$844$	0	0
$845$	0.00283565	9.75493e−5 0
$846$	0	0
$847$	9.97125	0.342616
$848$	0	0
$849$	6.46126	0.221750
$850$	0	0
$851$	−54.7883	−1.87812
$852$	0	0
$853$	54.0138	1.84940	0.924699	−	0.380698i	$-0.124316\pi$
$853$	54.0138	1.84940	0.924699	+	0.380698i	$0.124316\pi$
$854$	0	0
$855$	−7.97380	−0.272698
$856$	0	0
$857$	−1.54790	−0.0528754	−0.0264377	−	0.999650i	$-0.508416\pi$
$857$	−1.54790	−0.0528754	−0.0264377	+	0.999650i	$0.508416\pi$
$858$	0	0
$859$	24.5244	0.836761	0.418381	−	0.908272i	$-0.362598\pi$
$859$	24.5244	0.836761	0.418381	+	0.908272i	$0.362598\pi$
$860$	0	0
$861$	7.16953	0.244337
$862$	0	0
$863$	−4.70124	−0.160032	−0.0800160	−	0.996794i	$-0.525497\pi$
$863$	−4.70124	−0.160032	−0.0800160	+	0.996794i	$0.525497\pi$
$864$	0	0
$865$	14.8175	0.503811
$866$	0	0
$867$	−11.8439	−0.402241
$868$	0	0
$869$	17.4978	0.593572
$870$	0	0
$871$	−3.60594	−0.122183
$872$	0	0
$873$	−10.5722	−0.357815
$874$	0	0
$875$	−4.84231	−0.163700
$876$	0	0
$877$	−12.0193	−0.405862	−0.202931	−	0.979193i	$-0.565047\pi$
$877$	−12.0193	−0.405862	−0.202931	+	0.979193i	$0.565047\pi$
$878$	0	0
$879$	−22.6857	−0.765168
$880$	0	0
$881$	−31.4205	−1.05858	−0.529292	−	0.848440i	$-0.677542\pi$
$881$	−31.4205	−1.05858	−0.529292	+	0.848440i	$0.677542\pi$
$882$	0	0
$883$	−58.9950	−1.98534	−0.992670	−	0.120854i	$-0.961437\pi$
$883$	−58.9950	−1.98534	−0.992670	+	0.120854i	$0.961437\pi$
$884$	0	0
$885$	−11.3069	−0.380078
$886$	0	0
$887$	−0.783128	−0.0262949	−0.0131474	−	0.999914i	$-0.504185\pi$
$887$	−0.783128	−0.0262949	−0.0131474	+	0.999914i	$0.504185\pi$
$888$	0	0
$889$	20.0930	0.673898
$890$	0	0
$891$	2.99012	0.100173
$892$	0	0
$893$	6.72756	0.225129
$894$	0	0
$895$	−4.45960	−0.149068
$896$	0	0
$897$	−20.9251	−0.698669
$898$	0	0
$899$	13.5833	0.453028
$900$	0	0
$901$	11.5965	0.386337
$902$	0	0
$903$	−52.9723	−1.76281
$904$	0	0
$905$	8.72399	0.289995
$906$	0	0
$907$	−39.1048	−1.29845	−0.649227	−	0.760595i	$-0.724907\pi$
$907$	−39.1048	−1.29845	−0.649227	+	0.760595i	$0.724907\pi$
$908$	0	0
$909$	10.0779	0.334264
$910$	0	0
$911$	−49.0296	−1.62442	−0.812212	−	0.583363i	$-0.801737\pi$
$911$	−49.0296	−1.62442	−0.812212	+	0.583363i	$0.801737\pi$
$912$	0	0
$913$	−0.638914	−0.0211450
$914$	0	0
$915$	10.2161	0.337734
$916$	0	0
$917$	−96.5717	−3.18908
$918$	0	0
$919$	−26.1951	−0.864096	−0.432048	−	0.901851i	$-0.642209\pi$
$919$	−26.1951	−0.864096	−0.432048	+	0.901851i	$0.642209\pi$
$920$	0	0
$921$	−0.646736	−0.0213107
$922$	0	0
$923$	6.47832	0.213237
$924$	0	0
$925$	9.44146	0.310433
$926$	0	0
$927$	4.02218	0.132106
$928$	0	0
$929$	−54.4215	−1.78551	−0.892756	−	0.450541i	$-0.851231\pi$
$929$	−54.4215	−1.78551	−0.892756	+	0.450541i	$0.851231\pi$
$930$	0	0
$931$	−131.153	−4.29836
$932$	0	0
$933$	−5.06710	−0.165890
$934$	0	0
$935$	6.78965	0.222045
$936$	0	0
$937$	19.5397	0.638333	0.319167	−	0.947699i	$-0.396597\pi$
$937$	19.5397	0.638333	0.319167	+	0.947699i	$0.396597\pi$
$938$	0	0
$939$	−5.41514	−0.176716
$940$	0	0
$941$	3.88074	0.126508	0.0632542	−	0.997997i	$-0.479852\pi$
$941$	3.88074	0.126508	0.0632542	+	0.997997i	$0.479852\pi$
$942$	0	0
$943$	8.59184	0.279789
$944$	0	0
$945$	−4.84231	−0.157520
$946$	0	0
$947$	−2.02797	−0.0659003	−0.0329501	−	0.999457i	$-0.510490\pi$
$947$	−2.02797	−0.0659003	−0.0329501	+	0.999457i	$0.510490\pi$
$948$	0	0
$949$	9.04332	0.293559
$950$	0	0
$951$	0.998536	0.0323797
$952$	0	0
$953$	51.5700	1.67052	0.835258	−	0.549859i	$-0.185318\pi$
$953$	51.5700	1.67052	0.835258	+	0.549859i	$0.185318\pi$
$954$	0	0
$955$	0.232896	0.00753635
$956$	0	0
$957$	8.85265	0.286165
$958$	0	0
$959$	56.9343	1.83850
$960$	0	0
$961$	−9.95056	−0.320986
$962$	0	0
$963$	18.6804	0.601967
$964$	0	0
$965$	16.7733	0.539952
$966$	0	0
$967$	36.6930	1.17997	0.589983	−	0.807416i	$-0.299134\pi$
$967$	36.6930	1.17997	0.589983	+	0.807416i	$0.299134\pi$
$968$	0	0
$969$	−18.1061	−0.581651
$970$	0	0
$971$	13.8299	0.443822	0.221911	−	0.975067i	$-0.428771\pi$
$971$	13.8299	0.443822	0.221911	+	0.975067i	$0.428771\pi$
$972$	0	0
$973$	39.3326	1.26094
$974$	0	0
$975$	3.60594	0.115483
$976$	0	0
$977$	−17.8856	−0.572212	−0.286106	−	0.958198i	$-0.592361\pi$
$977$	−17.8856	−0.572212	−0.286106	+	0.958198i	$0.592361\pi$
$978$	0	0
$979$	−7.78683	−0.248868
$980$	0	0
$981$	−11.0634	−0.353228
$982$	0	0
$983$	−48.4955	−1.54677	−0.773383	−	0.633939i	$-0.781437\pi$
$983$	−48.4955	−1.54677	−0.773383	+	0.633939i	$0.781437\pi$
$984$	0	0
$985$	1.71308	0.0545834
$986$	0	0
$987$	4.08549	0.130043
$988$	0	0
$989$	−63.4812	−2.01858
$990$	0	0
$991$	13.0942	0.415950	0.207975	−	0.978134i	$-0.433313\pi$
$991$	13.0942	0.415950	0.207975	+	0.978134i	$0.433313\pi$
$992$	0	0
$993$	−19.4831	−0.618278
$994$	0	0
$995$	−2.64605	−0.0838853
$996$	0	0
$997$	−2.67368	−0.0846764	−0.0423382	−	0.999103i	$-0.513481\pi$
$997$	−2.67368	−0.0846764	−0.0423382	+	0.999103i	$0.513481\pi$
$998$	0	0
$999$	9.44146	0.298715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.i.1.1	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.i.1.1	✓	7	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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.84231 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.84231 q^{7} +1.00000 q^{9} +2.99012 q^{11} +3.60594 q^{13} +1.00000 q^{15} +2.27070 q^{17} -7.97380 q^{19} -4.84231 q^{21} -5.80294 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.96063 q^{29} +4.58797 q^{31} +2.99012 q^{33} -4.84231 q^{35} +9.44146 q^{37} +3.60594 q^{39} -1.48060 q^{41} +10.9395 q^{43} +1.00000 q^{45} -0.843708 q^{47} +16.4480 q^{49} +2.27070 q^{51} +5.10704 q^{53} +2.99012 q^{55} -7.97380 q^{57} -11.3069 q^{59} +10.2161 q^{61} -4.84231 q^{63} +3.60594 q^{65} -1.00000 q^{67} -5.80294 q^{69} +1.79657 q^{71} +2.50789 q^{73} +1.00000 q^{75} -14.4791 q^{77} +5.85187 q^{79} +1.00000 q^{81} -0.213675 q^{83} +2.27070 q^{85} +2.96063 q^{87} -2.60419 q^{89} -17.4611 q^{91} +4.58797 q^{93} -7.97380 q^{95} -10.5722 q^{97} +2.99012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9	\( 7 q + 7 q^{3} + 7 q^{5} + q^{7} + 7 q^{9} + 5 q^{11} + 9 q^{13} + 7 q^{15} + 11 q^{17} + 2 q^{19} + q^{21} + 7 q^{23} + 7 q^{25} + 7 q^{27} + 8 q^{29} + 9 q^{31} + 5 q^{33} + q^{35} + 7 q^{37} + 9 q^{39} + 9 q^{41} + 5 q^{43} + 7 q^{45} + 2 q^{47} + 12 q^{49} + 11 q^{51} + 15 q^{53} + 5 q^{55} + 2 q^{57} + 11 q^{61} + q^{63} + 9 q^{65} - 7 q^{67} + 7 q^{69} + 18 q^{71} + 21 q^{73} + 7 q^{75} + 5 q^{77} + 5 q^{79} + 7 q^{81} + 6 q^{83} + 11 q^{85} + 8 q^{87} + 7 q^{89} - 9 q^{91} + 9 q^{93} + 2 q^{95} - 9 q^{97} + 5 q^{99}+O(q^{100}) \) 7 * q + 7 * q^3 + 7 * q^5 + q^7 + 7 * q^9 + 5 * q^11 + 9 * q^13 + 7 * q^15 + 11 * q^17 + 2 * q^19 + q^21 + 7 * q^23 + 7 * q^25 + 7 * q^27 + 8 * q^29 + 9 * q^31 + 5 * q^33 + q^35 + 7 * q^37 + 9 * q^39 + 9 * q^41 + 5 * q^43 + 7 * q^45 + 2 * q^47 + 12 * q^49 + 11 * q^51 + 15 * q^53 + 5 * q^55 + 2 * q^57 + 11 * q^61 + q^63 + 9 * q^65 - 7 * q^67 + 7 * q^69 + 18 * q^71 + 21 * q^73 + 7 * q^75 + 5 * q^77 + 5 * q^79 + 7 * q^81 + 6 * q^83 + 11 * q^85 + 8 * q^87 + 7 * q^89 - 9 * q^91 + 9 * q^93 + 2 * q^95 - 9 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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