LMFDB - Embedded newform 4020.2.a.d.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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.95630$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -3.44512 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -3.44512 q^{7} +1.00000 q^{9} -0.102193 q^{11} +1.50361 q^{13} +1.00000 q^{15} -6.84187 q^{17} +4.86889 q^{19} -3.44512 q^{21} -6.49448 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.53976 q^{29} +4.02859 q^{31} -0.102193 q^{33} -3.44512 q^{35} -8.49448 q^{37} +1.50361 q^{39} -2.72023 q^{41} -6.71734 q^{43} +1.00000 q^{45} -1.84943 q^{47} +4.86889 q^{49} -6.84187 q^{51} -7.79252 q^{53} -0.102193 q^{55} +4.86889 q^{57} +10.9534 q^{59} +0.796590 q^{61} -3.44512 q^{63} +1.50361 q^{65} +1.00000 q^{67} -6.49448 q^{69} -6.59727 q^{71} -12.3791 q^{73} +1.00000 q^{75} +0.352068 q^{77} -8.29923 q^{79} +1.00000 q^{81} -0.0528416 q^{83} -6.84187 q^{85} +3.53976 q^{87} +5.48436 q^{89} -5.18014 q^{91} +4.02859 q^{93} +4.86889 q^{95} -0.284239 q^{97} -0.102193 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * q^95 - 11 * 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
$7$	−3.44512	−1.30213	−0.651067	−	0.759020i	$-0.725678\pi$
$7$	−3.44512	−1.30213	−0.651067	+	0.759020i	$0.725678\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.102193	−0.0308124	−0.0154062	−	0.999881i	$-0.504904\pi$
$11$	−0.102193	−0.0308124	−0.0154062	+	0.999881i	$0.504904\pi$
$12$	0	0
$13$	1.50361	0.417027	0.208514	−	0.978019i	$-0.433137\pi$
$13$	1.50361	0.417027	0.208514	+	0.978019i	$0.433137\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.84187	−1.65940	−0.829699	−	0.558211i	$-0.811488\pi$
$17$	−6.84187	−1.65940	−0.829699	+	0.558211i	$0.811488\pi$
$18$	0	0
$19$	4.86889	1.11700	0.558499	−	0.829505i	$-0.311377\pi$
$19$	4.86889	1.11700	0.558499	+	0.829505i	$0.311377\pi$
$20$	0	0
$21$	−3.44512	−0.751788
$22$	0	0
$23$	−6.49448	−1.35419	−0.677096	−	0.735895i	$-0.736762\pi$
$23$	−6.49448	−1.35419	−0.677096	+	0.735895i	$0.736762\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.53976	0.657317	0.328659	−	0.944449i	$-0.393403\pi$
$29$	3.53976	0.657317	0.328659	+	0.944449i	$0.393403\pi$
$30$	0	0
$31$	4.02859	0.723556	0.361778	−	0.932264i	$-0.382170\pi$
$31$	4.02859	0.723556	0.361778	+	0.932264i	$0.382170\pi$
$32$	0	0
$33$	−0.102193	−0.0177896
$34$	0	0
$35$	−3.44512	−0.582332
$36$	0	0
$37$	−8.49448	−1.39648	−0.698242	−	0.715862i	$-0.746034\pi$
$37$	−8.49448	−1.39648	−0.698242	+	0.715862i	$0.746034\pi$
$38$	0	0
$39$	1.50361	0.240771
$40$	0	0
$41$	−2.72023	−0.424828	−0.212414	−	0.977180i	$-0.568133\pi$
$41$	−2.72023	−0.424828	−0.212414	+	0.977180i	$0.568133\pi$
$42$	0	0
$43$	−6.71734	−1.02438	−0.512192	−	0.858871i	$-0.671167\pi$
$43$	−6.71734	−1.02438	−0.512192	+	0.858871i	$0.671167\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−1.84943	−0.269767	−0.134884	−	0.990861i	$-0.543066\pi$
$47$	−1.84943	−0.269767	−0.134884	+	0.990861i	$0.543066\pi$
$48$	0	0
$49$	4.86889	0.695555
$50$	0	0
$51$	−6.84187	−0.958054
$52$	0	0
$53$	−7.79252	−1.07039	−0.535193	−	0.844730i	$-0.679761\pi$
$53$	−7.79252	−1.07039	−0.535193	+	0.844730i	$0.679761\pi$
$54$	0	0
$55$	−0.102193	−0.0137797
$56$	0	0
$57$	4.86889	0.644900
$58$	0	0
$59$	10.9534	1.42601	0.713006	−	0.701158i	$-0.247333\pi$
$59$	10.9534	1.42601	0.713006	+	0.701158i	$0.247333\pi$
$60$	0	0
$61$	0.796590	0.101993	0.0509964	−	0.998699i	$-0.483760\pi$
$61$	0.796590	0.101993	0.0509964	+	0.998699i	$0.483760\pi$
$62$	0	0
$63$	−3.44512	−0.434045
$64$	0	0
$65$	1.50361	0.186500
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−6.49448	−0.781843
$70$	0	0
$71$	−6.59727	−0.782952	−0.391476	−	0.920188i	$-0.628035\pi$
$71$	−6.59727	−0.782952	−0.391476	+	0.920188i	$0.628035\pi$
$72$	0	0
$73$	−12.3791	−1.44886	−0.724431	−	0.689347i	$-0.757897\pi$
$73$	−12.3791	−1.44886	−0.724431	+	0.689347i	$0.757897\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0.352068	0.0401219
$78$	0	0
$79$	−8.29923	−0.933736	−0.466868	−	0.884327i	$-0.654618\pi$
$79$	−8.29923	−0.933736	−0.466868	+	0.884327i	$0.654618\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.0528416	−0.00580012	−0.00290006	−	0.999996i	$-0.500923\pi$
$83$	−0.0528416	−0.00580012	−0.00290006	+	0.999996i	$0.500923\pi$
$84$	0	0
$85$	−6.84187	−0.742105
$86$	0	0
$87$	3.53976	0.379502
$88$	0	0
$89$	5.48436	0.581341	0.290671	−	0.956823i	$-0.406122\pi$
$89$	5.48436	0.581341	0.290671	+	0.956823i	$0.406122\pi$
$90$	0	0
$91$	−5.18014	−0.543026
$92$	0	0
$93$	4.02859	0.417745
$94$	0	0
$95$	4.86889	0.499537
$96$	0	0
$97$	−0.284239	−0.0288601	−0.0144301	−	0.999896i	$-0.504593\pi$
$97$	−0.284239	−0.0288601	−0.0144301	+	0.999896i	$0.504593\pi$
$98$	0	0
$99$	−0.102193	−0.0102708
$100$	0	0
$101$	−14.5240	−1.44520	−0.722598	−	0.691268i	$-0.757052\pi$
$101$	−14.5240	−1.44520	−0.722598	+	0.691268i	$0.757052\pi$
$102$	0	0
$103$	1.84536	0.181829	0.0909146	−	0.995859i	$-0.471021\pi$
$103$	1.84536	0.181829	0.0909146	+	0.995859i	$0.471021\pi$
$104$	0	0
$105$	−3.44512	−0.336210
$106$	0	0
$107$	−12.3803	−1.19684	−0.598422	−	0.801181i	$-0.704206\pi$
$107$	−12.3803	−1.19684	−0.598422	+	0.801181i	$0.704206\pi$
$108$	0	0
$109$	−5.81507	−0.556982	−0.278491	−	0.960439i	$-0.589834\pi$
$109$	−5.81507	−0.556982	−0.278491	+	0.960439i	$0.589834\pi$
$110$	0	0
$111$	−8.49448	−0.806260
$112$	0	0
$113$	2.42474	0.228100	0.114050	−	0.993475i	$-0.463618\pi$
$113$	2.42474	0.228100	0.114050	+	0.993475i	$0.463618\pi$
$114$	0	0
$115$	−6.49448	−0.605613
$116$	0	0
$117$	1.50361	0.139009
$118$	0	0
$119$	23.5711	2.16076
$120$	0	0
$121$	−10.9896	−0.999051
$122$	0	0
$123$	−2.72023	−0.245275
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.1238	1.34202	0.671010	−	0.741448i	$-0.265861\pi$
$127$	15.1238	1.34202	0.671010	+	0.741448i	$0.265861\pi$
$128$	0	0
$129$	−6.71734	−0.591429
$130$	0	0
$131$	7.80126	0.681599	0.340800	−	0.940136i	$-0.389302\pi$
$131$	7.80126	0.681599	0.340800	+	0.940136i	$0.389302\pi$
$132$	0	0
$133$	−16.7739	−1.45448
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−4.13387	−0.353181	−0.176590	−	0.984284i	$-0.556507\pi$
$137$	−4.13387	−0.353181	−0.176590	+	0.984284i	$0.556507\pi$
$138$	0	0
$139$	11.7311	0.995023	0.497511	−	0.867457i	$-0.334247\pi$
$139$	11.7311	0.995023	0.497511	+	0.867457i	$0.334247\pi$
$140$	0	0
$141$	−1.84943	−0.155750
$142$	0	0
$143$	−0.153659	−0.0128496
$144$	0	0
$145$	3.53976	0.293961
$146$	0	0
$147$	4.86889	0.401579
$148$	0	0
$149$	−17.2089	−1.40981	−0.704905	−	0.709301i	$-0.749011\pi$
$149$	−17.2089	−1.40981	−0.704905	+	0.709301i	$0.749011\pi$
$150$	0	0
$151$	−13.5456	−1.10233	−0.551163	−	0.834398i	$-0.685816\pi$
$151$	−13.5456	−1.10233	−0.551163	+	0.834398i	$0.685816\pi$
$152$	0	0
$153$	−6.84187	−0.553133
$154$	0	0
$155$	4.02859	0.323584
$156$	0	0
$157$	3.42278	0.273168	0.136584	−	0.990628i	$-0.456388\pi$
$157$	3.42278	0.273168	0.136584	+	0.990628i	$0.456388\pi$
$158$	0	0
$159$	−7.79252	−0.617987
$160$	0	0
$161$	22.3743	1.76334
$162$	0	0
$163$	−8.16442	−0.639487	−0.319743	−	0.947504i	$-0.603597\pi$
$163$	−8.16442	−0.639487	−0.319743	+	0.947504i	$0.603597\pi$
$164$	0	0
$165$	−0.102193	−0.00795573
$166$	0	0
$167$	6.99342	0.541167	0.270584	−	0.962697i	$-0.412783\pi$
$167$	6.99342	0.541167	0.270584	+	0.962697i	$0.412783\pi$
$168$	0	0
$169$	−10.7391	−0.826088
$170$	0	0
$171$	4.86889	0.372333
$172$	0	0
$173$	−4.15345	−0.315781	−0.157891	−	0.987457i	$-0.550469\pi$
$173$	−4.15345	−0.315781	−0.157891	+	0.987457i	$0.550469\pi$
$174$	0	0
$175$	−3.44512	−0.260427
$176$	0	0
$177$	10.9534	0.823309
$178$	0	0
$179$	−18.9018	−1.41279	−0.706394	−	0.707819i	$-0.749679\pi$
$179$	−18.9018	−1.41279	−0.706394	+	0.707819i	$0.749679\pi$
$180$	0	0
$181$	−1.63651	−0.121641	−0.0608205	−	0.998149i	$-0.519372\pi$
$181$	−1.63651	−0.121641	−0.0608205	+	0.998149i	$0.519372\pi$
$182$	0	0
$183$	0.796590	0.0588856
$184$	0	0
$185$	−8.49448	−0.624526
$186$	0	0
$187$	0.699193	0.0511301
$188$	0	0
$189$	−3.44512	−0.250596
$190$	0	0
$191$	−4.41247	−0.319275	−0.159637	−	0.987176i	$-0.551033\pi$
$191$	−4.41247	−0.319275	−0.159637	+	0.987176i	$0.551033\pi$
$192$	0	0
$193$	−2.83485	−0.204057	−0.102029	−	0.994781i	$-0.532533\pi$
$193$	−2.83485	−0.204057	−0.102029	+	0.994781i	$0.532533\pi$
$194$	0	0
$195$	1.50361	0.107676
$196$	0	0
$197$	−13.2785	−0.946052	−0.473026	−	0.881049i	$-0.656838\pi$
$197$	−13.2785	−0.946052	−0.473026	+	0.881049i	$0.656838\pi$
$198$	0	0
$199$	13.4658	0.954562	0.477281	−	0.878751i	$-0.341622\pi$
$199$	13.4658	0.954562	0.477281	+	0.878751i	$0.341622\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	−12.1949	−0.855916
$204$	0	0
$205$	−2.72023	−0.189989
$206$	0	0
$207$	−6.49448	−0.451397
$208$	0	0
$209$	−0.497567	−0.0344174
$210$	0	0
$211$	11.1568	0.768067	0.384034	−	0.923319i	$-0.374535\pi$
$211$	11.1568	0.768067	0.384034	+	0.923319i	$0.374535\pi$
$212$	0	0
$213$	−6.59727	−0.452038
$214$	0	0
$215$	−6.71734	−0.458119
$216$	0	0
$217$	−13.8790	−0.942168
$218$	0	0
$219$	−12.3791	−0.836501
$220$	0	0
$221$	−10.2875	−0.692014
$222$	0	0
$223$	−3.28911	−0.220255	−0.110128	−	0.993917i	$-0.535126\pi$
$223$	−3.28911	−0.220255	−0.110128	+	0.993917i	$0.535126\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−0.582490	−0.0386612	−0.0193306	−	0.999813i	$-0.506154\pi$
$227$	−0.582490	−0.0386612	−0.0193306	+	0.999813i	$0.506154\pi$
$228$	0	0
$229$	−0.0911463	−0.00602312	−0.00301156	−	0.999995i	$-0.500959\pi$
$229$	−0.0911463	−0.00602312	−0.00301156	+	0.999995i	$0.500959\pi$
$230$	0	0
$231$	0.352068	0.0231644
$232$	0	0
$233$	8.61482	0.564375	0.282188	−	0.959359i	$-0.408940\pi$
$233$	8.61482	0.564375	0.282188	+	0.959359i	$0.408940\pi$
$234$	0	0
$235$	−1.84943	−0.120644
$236$	0	0
$237$	−8.29923	−0.539093
$238$	0	0
$239$	15.2902	0.989038	0.494519	−	0.869167i	$-0.335344\pi$
$239$	15.2902	0.989038	0.494519	+	0.869167i	$0.335344\pi$
$240$	0	0
$241$	−8.08839	−0.521019	−0.260509	−	0.965471i	$-0.583890\pi$
$241$	−8.08839	−0.521019	−0.260509	+	0.965471i	$0.583890\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	4.86889	0.311062
$246$	0	0
$247$	7.32092	0.465819
$248$	0	0
$249$	−0.0528416	−0.00334870
$250$	0	0
$251$	−7.18887	−0.453758	−0.226879	−	0.973923i	$-0.572852\pi$
$251$	−7.18887	−0.453758	−0.226879	+	0.973923i	$0.572852\pi$
$252$	0	0
$253$	0.663692	0.0417259
$254$	0	0
$255$	−6.84187	−0.428455
$256$	0	0
$257$	14.2474	0.888726	0.444363	−	0.895847i	$-0.353430\pi$
$257$	14.2474	0.888726	0.444363	+	0.895847i	$0.353430\pi$
$258$	0	0
$259$	29.2645	1.81841
$260$	0	0
$261$	3.53976	0.219106
$262$	0	0
$263$	17.8631	1.10149	0.550743	−	0.834675i	$-0.314344\pi$
$263$	17.8631	1.10149	0.550743	+	0.834675i	$0.314344\pi$
$264$	0	0
$265$	−7.79252	−0.478691
$266$	0	0
$267$	5.48436	0.335638
$268$	0	0
$269$	−5.16791	−0.315093	−0.157546	−	0.987512i	$-0.550358\pi$
$269$	−5.16791	−0.315093	−0.157546	+	0.987512i	$0.550358\pi$
$270$	0	0
$271$	28.1095	1.70753	0.853765	−	0.520658i	$-0.174313\pi$
$271$	28.1095	1.70753	0.853765	+	0.520658i	$0.174313\pi$
$272$	0	0
$273$	−5.18014	−0.313516
$274$	0	0
$275$	−0.102193	−0.00616248
$276$	0	0
$277$	19.8573	1.19311	0.596554	−	0.802573i	$-0.296536\pi$
$277$	19.8573	1.19311	0.596554	+	0.802573i	$0.296536\pi$
$278$	0	0
$279$	4.02859	0.241185
$280$	0	0
$281$	−5.38973	−0.321524	−0.160762	−	0.986993i	$-0.551395\pi$
$281$	−5.38973	−0.321524	−0.160762	+	0.986993i	$0.551395\pi$
$282$	0	0
$283$	11.0292	0.655616	0.327808	−	0.944744i	$-0.393690\pi$
$283$	11.0292	0.655616	0.327808	+	0.944744i	$0.393690\pi$
$284$	0	0
$285$	4.86889	0.288408
$286$	0	0
$287$	9.37152	0.553183
$288$	0	0
$289$	29.8112	1.75360
$290$	0	0
$291$	−0.284239	−0.0166624
$292$	0	0
$293$	−27.1050	−1.58349	−0.791744	−	0.610853i	$-0.790827\pi$
$293$	−27.1050	−1.58349	−0.791744	+	0.610853i	$0.790827\pi$
$294$	0	0
$295$	10.9534	0.637732
$296$	0	0
$297$	−0.102193	−0.00592985
$298$	0	0
$299$	−9.76518	−0.564735
$300$	0	0
$301$	23.1421	1.33389
$302$	0	0
$303$	−14.5240	−0.834385
$304$	0	0
$305$	0.796590	0.0456126
$306$	0	0
$307$	24.5608	1.40176	0.700879	−	0.713280i	$-0.252791\pi$
$307$	24.5608	1.40176	0.700879	+	0.713280i	$0.252791\pi$
$308$	0	0
$309$	1.84536	0.104979
$310$	0	0
$311$	−20.5985	−1.16803	−0.584017	−	0.811742i	$-0.698520\pi$
$311$	−20.5985	−1.16803	−0.584017	+	0.811742i	$0.698520\pi$
$312$	0	0
$313$	29.7980	1.68429	0.842143	−	0.539255i	$-0.181294\pi$
$313$	29.7980	1.68429	0.842143	+	0.539255i	$0.181294\pi$
$314$	0	0
$315$	−3.44512	−0.194111
$316$	0	0
$317$	−0.174862	−0.00982122	−0.00491061	−	0.999988i	$-0.501563\pi$
$317$	−0.174862	−0.00982122	−0.00491061	+	0.999988i	$0.501563\pi$
$318$	0	0
$319$	−0.361740	−0.0202535
$320$	0	0
$321$	−12.3803	−0.690999
$322$	0	0
$323$	−33.3123	−1.85355
$324$	0	0
$325$	1.50361	0.0834055
$326$	0	0
$327$	−5.81507	−0.321574
$328$	0	0
$329$	6.37152	0.351273
$330$	0	0
$331$	2.36816	0.130166	0.0650829	−	0.997880i	$-0.479269\pi$
$331$	2.36816	0.130166	0.0650829	+	0.997880i	$0.479269\pi$
$332$	0	0
$333$	−8.49448	−0.465494
$334$	0	0
$335$	1.00000	0.0546358
$336$	0	0
$337$	2.08997	0.113848	0.0569238	−	0.998379i	$-0.481871\pi$
$337$	2.08997	0.113848	0.0569238	+	0.998379i	$0.481871\pi$
$338$	0	0
$339$	2.42474	0.131694
$340$	0	0
$341$	−0.411695	−0.0222945
$342$	0	0
$343$	7.34196	0.396428
$344$	0	0
$345$	−6.49448	−0.349651
$346$	0	0
$347$	−17.8864	−0.960194	−0.480097	−	0.877215i	$-0.659399\pi$
$347$	−17.8864	−0.960194	−0.480097	+	0.877215i	$0.659399\pi$
$348$	0	0
$349$	−34.5298	−1.84834	−0.924170	−	0.381982i	$-0.875242\pi$
$349$	−34.5298	−1.84834	−0.924170	+	0.381982i	$0.875242\pi$
$350$	0	0
$351$	1.50361	0.0802569
$352$	0	0
$353$	17.9949	0.957770	0.478885	−	0.877878i	$-0.341041\pi$
$353$	17.9949	0.957770	0.478885	+	0.877878i	$0.341041\pi$
$354$	0	0
$355$	−6.59727	−0.350147
$356$	0	0
$357$	23.5711	1.24752
$358$	0	0
$359$	−27.9871	−1.47710	−0.738552	−	0.674197i	$-0.764490\pi$
$359$	−27.9871	−1.47710	−0.738552	+	0.674197i	$0.764490\pi$
$360$	0	0
$361$	4.70605	0.247687
$362$	0	0
$363$	−10.9896	−0.576802
$364$	0	0
$365$	−12.3791	−0.647951
$366$	0	0
$367$	20.4131	1.06556	0.532778	−	0.846255i	$-0.321148\pi$
$367$	20.4131	1.06556	0.532778	+	0.846255i	$0.321148\pi$
$368$	0	0
$369$	−2.72023	−0.141609
$370$	0	0
$371$	26.8462	1.39379
$372$	0	0
$373$	0.0922842	0.00477829	0.00238915	−	0.999997i	$-0.499240\pi$
$373$	0.0922842	0.00477829	0.00238915	+	0.999997i	$0.499240\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	5.32243	0.274119
$378$	0	0
$379$	−14.9406	−0.767446	−0.383723	−	0.923448i	$-0.625358\pi$
$379$	−14.9406	−0.767446	−0.383723	+	0.923448i	$0.625358\pi$
$380$	0	0
$381$	15.1238	0.774816
$382$	0	0
$383$	−30.1996	−1.54313	−0.771563	−	0.636152i	$-0.780525\pi$
$383$	−30.1996	−1.54313	−0.771563	+	0.636152i	$0.780525\pi$
$384$	0	0
$385$	0.352068	0.0179431
$386$	0	0
$387$	−6.71734	−0.341462
$388$	0	0
$389$	15.4739	0.784559	0.392279	−	0.919846i	$-0.371687\pi$
$389$	15.4739	0.784559	0.392279	+	0.919846i	$0.371687\pi$
$390$	0	0
$391$	44.4344	2.24714
$392$	0	0
$393$	7.80126	0.393522
$394$	0	0
$395$	−8.29923	−0.417579
$396$	0	0
$397$	29.7452	1.49287	0.746434	−	0.665459i	$-0.231764\pi$
$397$	29.7452	1.49287	0.746434	+	0.665459i	$0.231764\pi$
$398$	0	0
$399$	−16.7739	−0.839746
$400$	0	0
$401$	−2.69866	−0.134765	−0.0673823	−	0.997727i	$-0.521465\pi$
$401$	−2.69866	−0.134765	−0.0673823	+	0.997727i	$0.521465\pi$
$402$	0	0
$403$	6.05744	0.301743
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.868078	0.0430290
$408$	0	0
$409$	−11.1955	−0.553580	−0.276790	−	0.960930i	$-0.589271\pi$
$409$	−11.1955	−0.553580	−0.276790	+	0.960930i	$0.589271\pi$
$410$	0	0
$411$	−4.13387	−0.203909
$412$	0	0
$413$	−37.7359	−1.85686
$414$	0	0
$415$	−0.0528416	−0.00259389
$416$	0	0
$417$	11.7311	0.574477
$418$	0	0
$419$	−4.27275	−0.208738	−0.104369	−	0.994539i	$-0.533282\pi$
$419$	−4.27275	−0.208738	−0.104369	+	0.994539i	$0.533282\pi$
$420$	0	0
$421$	2.92599	0.142604	0.0713020	−	0.997455i	$-0.477285\pi$
$421$	2.92599	0.142604	0.0713020	+	0.997455i	$0.477285\pi$
$422$	0	0
$423$	−1.84943	−0.0899224
$424$	0	0
$425$	−6.84187	−0.331880
$426$	0	0
$427$	−2.74435	−0.132808
$428$	0	0
$429$	−0.153659	−0.00741873
$430$	0	0
$431$	−20.8707	−1.00531	−0.502654	−	0.864488i	$-0.667643\pi$
$431$	−20.8707	−1.00531	−0.502654	+	0.864488i	$0.667643\pi$
$432$	0	0
$433$	15.6840	0.753726	0.376863	−	0.926269i	$-0.377003\pi$
$433$	15.6840	0.753726	0.376863	+	0.926269i	$0.377003\pi$
$434$	0	0
$435$	3.53976	0.169719
$436$	0	0
$437$	−31.6209	−1.51263
$438$	0	0
$439$	−7.41556	−0.353925	−0.176963	−	0.984218i	$-0.556627\pi$
$439$	−7.41556	−0.353925	−0.176963	+	0.984218i	$0.556627\pi$
$440$	0	0
$441$	4.86889	0.231852
$442$	0	0
$443$	11.8336	0.562231	0.281115	−	0.959674i	$-0.409296\pi$
$443$	11.8336	0.562231	0.281115	+	0.959674i	$0.409296\pi$
$444$	0	0
$445$	5.48436	0.259984
$446$	0	0
$447$	−17.2089	−0.813954
$448$	0	0
$449$	29.5338	1.39379	0.696893	−	0.717175i	$-0.254565\pi$
$449$	29.5338	1.39379	0.696893	+	0.717175i	$0.254565\pi$
$450$	0	0
$451$	0.277989	0.0130900
$452$	0	0
$453$	−13.5456	−0.636428
$454$	0	0
$455$	−5.18014	−0.242849
$456$	0	0
$457$	−39.0155	−1.82507	−0.912534	−	0.409000i	$-0.865877\pi$
$457$	−39.0155	−1.82507	−0.912534	+	0.409000i	$0.865877\pi$
$458$	0	0
$459$	−6.84187	−0.319351
$460$	0	0
$461$	−6.08708	−0.283504	−0.141752	−	0.989902i	$-0.545273\pi$
$461$	−6.08708	−0.283504	−0.141752	+	0.989902i	$0.545273\pi$
$462$	0	0
$463$	−4.85954	−0.225842	−0.112921	−	0.993604i	$-0.536021\pi$
$463$	−4.85954	−0.225842	−0.112921	+	0.993604i	$0.536021\pi$
$464$	0	0
$465$	4.02859	0.186821
$466$	0	0
$467$	−12.5083	−0.578816	−0.289408	−	0.957206i	$-0.593458\pi$
$467$	−12.5083	−0.578816	−0.289408	+	0.957206i	$0.593458\pi$
$468$	0	0
$469$	−3.44512	−0.159081
$470$	0	0
$471$	3.42278	0.157714
$472$	0	0
$473$	0.686467	0.0315638
$474$	0	0
$475$	4.86889	0.223400
$476$	0	0
$477$	−7.79252	−0.356795
$478$	0	0
$479$	−17.6997	−0.808722	−0.404361	−	0.914600i	$-0.632506\pi$
$479$	−17.6997	−0.808722	−0.404361	+	0.914600i	$0.632506\pi$
$480$	0	0
$481$	−12.7724	−0.582372
$482$	0	0
$483$	22.3743	1.01807
$484$	0	0
$485$	−0.284239	−0.0129066
$486$	0	0
$487$	0.845295	0.0383040	0.0191520	−	0.999817i	$-0.493903\pi$
$487$	0.845295	0.0383040	0.0191520	+	0.999817i	$0.493903\pi$
$488$	0	0
$489$	−8.16442	−0.369208
$490$	0	0
$491$	−3.02485	−0.136510	−0.0682549	−	0.997668i	$-0.521743\pi$
$491$	−3.02485	−0.136510	−0.0682549	+	0.997668i	$0.521743\pi$
$492$	0	0
$493$	−24.2186	−1.09075
$494$	0	0
$495$	−0.102193	−0.00459324
$496$	0	0
$497$	22.7284	1.01951
$498$	0	0
$499$	6.10824	0.273442	0.136721	−	0.990610i	$-0.456344\pi$
$499$	6.10824	0.273442	0.136721	+	0.990610i	$0.456344\pi$
$500$	0	0
$501$	6.99342	0.312443
$502$	0	0
$503$	29.3757	1.30980	0.654898	−	0.755717i	$-0.272712\pi$
$503$	29.3757	1.30980	0.654898	+	0.755717i	$0.272712\pi$
$504$	0	0
$505$	−14.5240	−0.646311
$506$	0	0
$507$	−10.7391	−0.476942
$508$	0	0
$509$	13.5752	0.601709	0.300855	−	0.953670i	$-0.402728\pi$
$509$	13.5752	0.601709	0.300855	+	0.953670i	$0.402728\pi$
$510$	0	0
$511$	42.6475	1.88661
$512$	0	0
$513$	4.86889	0.214967
$514$	0	0
$515$	1.84536	0.0813165
$516$	0	0
$517$	0.188999	0.00831218
$518$	0	0
$519$	−4.15345	−0.182316
$520$	0	0
$521$	34.5900	1.51542	0.757708	−	0.652593i	$-0.226319\pi$
$521$	34.5900	1.51542	0.757708	+	0.652593i	$0.226319\pi$
$522$	0	0
$523$	−17.0201	−0.744237	−0.372119	−	0.928185i	$-0.621369\pi$
$523$	−17.0201	−0.744237	−0.372119	+	0.928185i	$0.621369\pi$
$524$	0	0
$525$	−3.44512	−0.150358
$526$	0	0
$527$	−27.5631	−1.20067
$528$	0	0
$529$	19.1782	0.833836
$530$	0	0
$531$	10.9534	0.475337
$532$	0	0
$533$	−4.09017	−0.177165
$534$	0	0
$535$	−12.3803	−0.535245
$536$	0	0
$537$	−18.9018	−0.815674
$538$	0	0
$539$	−0.497567	−0.0214317
$540$	0	0
$541$	−37.4214	−1.60887	−0.804436	−	0.594040i	$-0.797532\pi$
$541$	−37.4214	−1.60887	−0.804436	+	0.594040i	$0.797532\pi$
$542$	0	0
$543$	−1.63651	−0.0702294
$544$	0	0
$545$	−5.81507	−0.249090
$546$	0	0
$547$	34.9358	1.49375	0.746874	−	0.664966i	$-0.231554\pi$
$547$	34.9358	1.49375	0.746874	+	0.664966i	$0.231554\pi$
$548$	0	0
$549$	0.796590	0.0339976
$550$	0	0
$551$	17.2347	0.734223
$552$	0	0
$553$	28.5919	1.21585
$554$	0	0
$555$	−8.49448	−0.360570
$556$	0	0
$557$	−5.45768	−0.231249	−0.115625	−	0.993293i	$-0.536887\pi$
$557$	−5.45768	−0.231249	−0.115625	+	0.993293i	$0.536887\pi$
$558$	0	0
$559$	−10.1003	−0.427196
$560$	0	0
$561$	0.699193	0.0295200
$562$	0	0
$563$	12.1141	0.510548	0.255274	−	0.966869i	$-0.417834\pi$
$563$	12.1141	0.510548	0.255274	+	0.966869i	$0.417834\pi$
$564$	0	0
$565$	2.42474	0.102009
$566$	0	0
$567$	−3.44512	−0.144682
$568$	0	0
$569$	27.1461	1.13803	0.569013	−	0.822329i	$-0.307325\pi$
$569$	27.1461	1.13803	0.569013	+	0.822329i	$0.307325\pi$
$570$	0	0
$571$	−8.12408	−0.339982	−0.169991	−	0.985446i	$-0.554374\pi$
$571$	−8.12408	−0.339982	−0.169991	+	0.985446i	$0.554374\pi$
$572$	0	0
$573$	−4.41247	−0.184333
$574$	0	0
$575$	−6.49448	−0.270838
$576$	0	0
$577$	32.5321	1.35433	0.677166	−	0.735831i	$-0.263208\pi$
$577$	32.5321	1.35433	0.677166	+	0.735831i	$0.263208\pi$
$578$	0	0
$579$	−2.83485	−0.117812
$580$	0	0
$581$	0.182046	0.00755254
$582$	0	0
$583$	0.796343	0.0329812
$584$	0	0
$585$	1.50361	0.0621668
$586$	0	0
$587$	5.03476	0.207807	0.103903	−	0.994587i	$-0.466867\pi$
$587$	5.03476	0.207807	0.103903	+	0.994587i	$0.466867\pi$
$588$	0	0
$589$	19.6147	0.808212
$590$	0	0
$591$	−13.2785	−0.546203
$592$	0	0
$593$	−18.4273	−0.756716	−0.378358	−	0.925659i	$-0.623511\pi$
$593$	−18.4273	−0.756716	−0.378358	+	0.925659i	$0.623511\pi$
$594$	0	0
$595$	23.5711	0.966321
$596$	0	0
$597$	13.4658	0.551117
$598$	0	0
$599$	−47.1055	−1.92468	−0.962339	−	0.271853i	$-0.912363\pi$
$599$	−47.1055	−1.92468	−0.962339	+	0.271853i	$0.912363\pi$
$600$	0	0
$601$	−14.5631	−0.594040	−0.297020	−	0.954871i	$-0.595993\pi$
$601$	−14.5631	−0.594040	−0.297020	+	0.954871i	$0.595993\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	−10.9896	−0.446789
$606$	0	0
$607$	8.48825	0.344528	0.172264	−	0.985051i	$-0.444892\pi$
$607$	8.48825	0.344528	0.172264	+	0.985051i	$0.444892\pi$
$608$	0	0
$609$	−12.1949	−0.494163
$610$	0	0
$611$	−2.78083	−0.112500
$612$	0	0
$613$	15.1137	0.610436	0.305218	−	0.952282i	$-0.401271\pi$
$613$	15.1137	0.610436	0.305218	+	0.952282i	$0.401271\pi$
$614$	0	0
$615$	−2.72023	−0.109690
$616$	0	0
$617$	5.94054	0.239157	0.119578	−	0.992825i	$-0.461846\pi$
$617$	5.94054	0.239157	0.119578	+	0.992825i	$0.461846\pi$
$618$	0	0
$619$	−16.3925	−0.658870	−0.329435	−	0.944178i	$-0.606858\pi$
$619$	−16.3925	−0.658870	−0.329435	+	0.944178i	$0.606858\pi$
$620$	0	0
$621$	−6.49448	−0.260614
$622$	0	0
$623$	−18.8943	−0.756985
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.497567	−0.0198709
$628$	0	0
$629$	58.1181	2.31732
$630$	0	0
$631$	−23.1231	−0.920518	−0.460259	−	0.887785i	$-0.652243\pi$
$631$	−23.1231	−0.920518	−0.460259	+	0.887785i	$0.652243\pi$
$632$	0	0
$633$	11.1568	0.443444
$634$	0	0
$635$	15.1238	0.600170
$636$	0	0
$637$	7.32092	0.290065
$638$	0	0
$639$	−6.59727	−0.260984
$640$	0	0
$641$	−7.52758	−0.297321	−0.148661	−	0.988888i	$-0.547496\pi$
$641$	−7.52758	−0.297321	−0.148661	+	0.988888i	$0.547496\pi$
$642$	0	0
$643$	2.39679	0.0945203	0.0472601	−	0.998883i	$-0.484951\pi$
$643$	2.39679	0.0945203	0.0472601	+	0.998883i	$0.484951\pi$
$644$	0	0
$645$	−6.71734	−0.264495
$646$	0	0
$647$	−2.13014	−0.0837444	−0.0418722	−	0.999123i	$-0.513332\pi$
$647$	−2.13014	−0.0837444	−0.0418722	+	0.999123i	$0.513332\pi$
$648$	0	0
$649$	−1.11936	−0.0439389
$650$	0	0
$651$	−13.8790	−0.543961
$652$	0	0
$653$	−32.6084	−1.27606	−0.638032	−	0.770010i	$-0.720251\pi$
$653$	−32.6084	−1.27606	−0.638032	+	0.770010i	$0.720251\pi$
$654$	0	0
$655$	7.80126	0.304820
$656$	0	0
$657$	−12.3791	−0.482954
$658$	0	0
$659$	23.0047	0.896136	0.448068	−	0.893999i	$-0.352112\pi$
$659$	23.0047	0.896136	0.448068	+	0.893999i	$0.352112\pi$
$660$	0	0
$661$	16.8359	0.654839	0.327420	−	0.944879i	$-0.393821\pi$
$661$	16.8359	0.654839	0.327420	+	0.944879i	$0.393821\pi$
$662$	0	0
$663$	−10.2875	−0.399535
$664$	0	0
$665$	−16.7739	−0.650465
$666$	0	0
$667$	−22.9889	−0.890134
$668$	0	0
$669$	−3.28911	−0.127164
$670$	0	0
$671$	−0.0814061	−0.00314265
$672$	0	0
$673$	14.7569	0.568837	0.284418	−	0.958700i	$-0.408200\pi$
$673$	14.7569	0.568837	0.284418	+	0.958700i	$0.408200\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−47.2075	−1.81433	−0.907165	−	0.420775i	$-0.861758\pi$
$677$	−47.2075	−1.81433	−0.907165	+	0.420775i	$0.861758\pi$
$678$	0	0
$679$	0.979239	0.0375798
$680$	0	0
$681$	−0.582490	−0.0223210
$682$	0	0
$683$	−2.12953	−0.0814844	−0.0407422	−	0.999170i	$-0.512972\pi$
$683$	−2.12953	−0.0814844	−0.0407422	+	0.999170i	$0.512972\pi$
$684$	0	0
$685$	−4.13387	−0.157947
$686$	0	0
$687$	−0.0911463	−0.00347745
$688$	0	0
$689$	−11.7169	−0.446380
$690$	0	0
$691$	43.0569	1.63796	0.818980	−	0.573822i	$-0.194540\pi$
$691$	43.0569	1.63796	0.818980	+	0.573822i	$0.194540\pi$
$692$	0	0
$693$	0.352068	0.0133740
$694$	0	0
$695$	11.7311	0.444988
$696$	0	0
$697$	18.6115	0.704959
$698$	0	0
$699$	8.61482	0.325842
$700$	0	0
$701$	11.8575	0.447850	0.223925	−	0.974606i	$-0.428113\pi$
$701$	11.8575	0.447850	0.223925	+	0.974606i	$0.428113\pi$
$702$	0	0
$703$	−41.3586	−1.55987
$704$	0	0
$705$	−1.84943	−0.0696536
$706$	0	0
$707$	50.0371	1.88184
$708$	0	0
$709$	−8.25735	−0.310111	−0.155056	−	0.987906i	$-0.549556\pi$
$709$	−8.25735	−0.310111	−0.155056	+	0.987906i	$0.549556\pi$
$710$	0	0
$711$	−8.29923	−0.311245
$712$	0	0
$713$	−26.1636	−0.979834
$714$	0	0
$715$	−0.153659	−0.00574652
$716$	0	0
$717$	15.2902	0.571021
$718$	0	0
$719$	14.6745	0.547266	0.273633	−	0.961834i	$-0.411775\pi$
$719$	14.6745	0.547266	0.273633	+	0.961834i	$0.411775\pi$
$720$	0	0
$721$	−6.35751	−0.236766
$722$	0	0
$723$	−8.08839	−0.300810
$724$	0	0
$725$	3.53976	0.131463
$726$	0	0
$727$	32.2460	1.19594	0.597969	−	0.801519i	$-0.295975\pi$
$727$	32.2460	1.19594	0.597969	+	0.801519i	$0.295975\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	45.9592	1.69986
$732$	0	0
$733$	−24.6390	−0.910064	−0.455032	−	0.890475i	$-0.650372\pi$
$733$	−24.6390	−0.910064	−0.455032	+	0.890475i	$0.650372\pi$
$734$	0	0
$735$	4.86889	0.179592
$736$	0	0
$737$	−0.102193	−0.00376434
$738$	0	0
$739$	−46.5893	−1.71382	−0.856908	−	0.515469i	$-0.827618\pi$
$739$	−46.5893	−1.71382	−0.856908	+	0.515469i	$0.827618\pi$
$740$	0	0
$741$	7.32092	0.268941
$742$	0	0
$743$	50.0981	1.83792	0.918961	−	0.394349i	$-0.129030\pi$
$743$	50.0981	1.83792	0.918961	+	0.394349i	$0.129030\pi$
$744$	0	0
$745$	−17.2089	−0.630486
$746$	0	0
$747$	−0.0528416	−0.00193337
$748$	0	0
$749$	42.6515	1.55845
$750$	0	0
$751$	35.7406	1.30419	0.652097	−	0.758136i	$-0.273890\pi$
$751$	35.7406	1.30419	0.652097	+	0.758136i	$0.273890\pi$
$752$	0	0
$753$	−7.18887	−0.261977
$754$	0	0
$755$	−13.5456	−0.492975
$756$	0	0
$757$	−20.0721	−0.729534	−0.364767	−	0.931099i	$-0.618851\pi$
$757$	−20.0721	−0.729534	−0.364767	+	0.931099i	$0.618851\pi$
$758$	0	0
$759$	0.663692	0.0240905
$760$	0	0
$761$	−7.72859	−0.280161	−0.140081	−	0.990140i	$-0.544736\pi$
$761$	−7.72859	−0.280161	−0.140081	+	0.990140i	$0.544736\pi$
$762$	0	0
$763$	20.0336	0.725266
$764$	0	0
$765$	−6.84187	−0.247368
$766$	0	0
$767$	16.4697	0.594686
$768$	0	0
$769$	−5.90069	−0.212784	−0.106392	−	0.994324i	$-0.533930\pi$
$769$	−5.90069	−0.212784	−0.106392	+	0.994324i	$0.533930\pi$
$770$	0	0
$771$	14.2474	0.513106
$772$	0	0
$773$	−30.4425	−1.09494	−0.547470	−	0.836825i	$-0.684409\pi$
$773$	−30.4425	−1.09494	−0.547470	+	0.836825i	$0.684409\pi$
$774$	0	0
$775$	4.02859	0.144711
$776$	0	0
$777$	29.2645	1.04986
$778$	0	0
$779$	−13.2445	−0.474533
$780$	0	0
$781$	0.674197	0.0241247
$782$	0	0
$783$	3.53976	0.126501
$784$	0	0
$785$	3.42278	0.122164
$786$	0	0
$787$	36.1832	1.28979	0.644896	−	0.764270i	$-0.276901\pi$
$787$	36.1832	1.28979	0.644896	+	0.764270i	$0.276901\pi$
$788$	0	0
$789$	17.8631	0.635944
$790$	0	0
$791$	−8.35352	−0.297017
$792$	0	0
$793$	1.19776	0.0425338
$794$	0	0
$795$	−7.79252	−0.276372
$796$	0	0
$797$	22.8037	0.807750	0.403875	−	0.914814i	$-0.367663\pi$
$797$	22.8037	0.807750	0.403875	+	0.914814i	$0.367663\pi$
$798$	0	0
$799$	12.6536	0.447651
$800$	0	0
$801$	5.48436	0.193780
$802$	0	0
$803$	1.26506	0.0446429
$804$	0	0
$805$	22.3743	0.788590
$806$	0	0
$807$	−5.16791	−0.181919
$808$	0	0
$809$	37.4671	1.31727	0.658637	−	0.752461i	$-0.271133\pi$
$809$	37.4671	1.31727	0.658637	+	0.752461i	$0.271133\pi$
$810$	0	0
$811$	31.1110	1.09246	0.546228	−	0.837636i	$-0.316063\pi$
$811$	31.1110	1.09246	0.546228	+	0.837636i	$0.316063\pi$
$812$	0	0
$813$	28.1095	0.985843
$814$	0	0
$815$	−8.16442	−0.285987
$816$	0	0
$817$	−32.7060	−1.14424
$818$	0	0
$819$	−5.18014	−0.181009
$820$	0	0
$821$	−19.9473	−0.696165	−0.348082	−	0.937464i	$-0.613167\pi$
$821$	−19.9473	−0.696165	−0.348082	+	0.937464i	$0.613167\pi$
$822$	0	0
$823$	12.6234	0.440022	0.220011	−	0.975497i	$-0.429391\pi$
$823$	12.6234	0.440022	0.220011	+	0.975497i	$0.429391\pi$
$824$	0	0
$825$	−0.102193	−0.00355791
$826$	0	0
$827$	43.5312	1.51373	0.756864	−	0.653572i	$-0.226730\pi$
$827$	43.5312	1.51373	0.756864	+	0.653572i	$0.226730\pi$
$828$	0	0
$829$	52.7891	1.83344	0.916721	−	0.399529i	$-0.130826\pi$
$829$	52.7891	1.83344	0.916721	+	0.399529i	$0.130826\pi$
$830$	0	0
$831$	19.8573	0.688841
$832$	0	0
$833$	−33.3123	−1.15420
$834$	0	0
$835$	6.99342	0.242017
$836$	0	0
$837$	4.02859	0.139248
$838$	0	0
$839$	21.2287	0.732898	0.366449	−	0.930438i	$-0.380574\pi$
$839$	21.2287	0.732898	0.366449	+	0.930438i	$0.380574\pi$
$840$	0	0
$841$	−16.4701	−0.567934
$842$	0	0
$843$	−5.38973	−0.185632
$844$	0	0
$845$	−10.7391	−0.369438
$846$	0	0
$847$	37.8604	1.30090
$848$	0	0
$849$	11.0292	0.378520
$850$	0	0
$851$	55.1672	1.89111
$852$	0	0
$853$	35.6325	1.22003	0.610017	−	0.792389i	$-0.291163\pi$
$853$	35.6325	1.22003	0.610017	+	0.792389i	$0.291163\pi$
$854$	0	0
$855$	4.86889	0.166512
$856$	0	0
$857$	13.0360	0.445301	0.222650	−	0.974898i	$-0.428529\pi$
$857$	13.0360	0.445301	0.222650	+	0.974898i	$0.428529\pi$
$858$	0	0
$859$	6.98082	0.238182	0.119091	−	0.992883i	$-0.462002\pi$
$859$	6.98082	0.238182	0.119091	+	0.992883i	$0.462002\pi$
$860$	0	0
$861$	9.37152	0.319381
$862$	0	0
$863$	−9.47708	−0.322604	−0.161302	−	0.986905i	$-0.551569\pi$
$863$	−9.47708	−0.322604	−0.161302	+	0.986905i	$0.551569\pi$
$864$	0	0
$865$	−4.15345	−0.141222
$866$	0	0
$867$	29.8112	1.01244
$868$	0	0
$869$	0.848125	0.0287707
$870$	0	0
$871$	1.50361	0.0509480
$872$	0	0
$873$	−0.284239	−0.00962004
$874$	0	0
$875$	−3.44512	−0.116466
$876$	0	0
$877$	36.7906	1.24233	0.621164	−	0.783680i	$-0.286660\pi$
$877$	36.7906	1.24233	0.621164	+	0.783680i	$0.286660\pi$
$878$	0	0
$879$	−27.1050	−0.914227
$880$	0	0
$881$	−54.3062	−1.82962	−0.914811	−	0.403882i	$-0.867661\pi$
$881$	−54.3062	−1.82962	−0.914811	+	0.403882i	$0.867661\pi$
$882$	0	0
$883$	−18.5094	−0.622891	−0.311446	−	0.950264i	$-0.600813\pi$
$883$	−18.5094	−0.622891	−0.311446	+	0.950264i	$0.600813\pi$
$884$	0	0
$885$	10.9534	0.368195
$886$	0	0
$887$	−24.6772	−0.828579	−0.414290	−	0.910145i	$-0.635970\pi$
$887$	−24.6772	−0.828579	−0.414290	+	0.910145i	$0.635970\pi$
$888$	0	0
$889$	−52.1034	−1.74749
$890$	0	0
$891$	−0.102193	−0.00342360
$892$	0	0
$893$	−9.00467	−0.301330
$894$	0	0
$895$	−18.9018	−0.631818
$896$	0	0
$897$	−9.76518	−0.326050
$898$	0	0
$899$	14.2602	0.475606
$900$	0	0
$901$	53.3155	1.77620
$902$	0	0
$903$	23.1421	0.770120
$904$	0	0
$905$	−1.63651	−0.0543995
$906$	0	0
$907$	47.7308	1.58487	0.792437	−	0.609953i	$-0.208812\pi$
$907$	47.7308	1.58487	0.792437	+	0.609953i	$0.208812\pi$
$908$	0	0
$909$	−14.5240	−0.481732
$910$	0	0
$911$	−2.41520	−0.0800192	−0.0400096	−	0.999199i	$-0.512739\pi$
$911$	−2.41520	−0.0800192	−0.0400096	+	0.999199i	$0.512739\pi$
$912$	0	0
$913$	0.00540005	0.000178716 0
$914$	0	0
$915$	0.796590	0.0263345
$916$	0	0
$917$	−26.8763	−0.887534
$918$	0	0
$919$	−46.6947	−1.54032	−0.770158	−	0.637853i	$-0.779823\pi$
$919$	−46.6947	−1.54032	−0.770158	+	0.637853i	$0.779823\pi$
$920$	0	0
$921$	24.5608	0.809306
$922$	0	0
$923$	−9.91975	−0.326512
$924$	0	0
$925$	−8.49448	−0.279297
$926$	0	0
$927$	1.84536	0.0606097
$928$	0	0
$929$	6.95289	0.228117	0.114058	−	0.993474i	$-0.463615\pi$
$929$	6.95289	0.228117	0.114058	+	0.993474i	$0.463615\pi$
$930$	0	0
$931$	23.7060	0.776934
$932$	0	0
$933$	−20.5985	−0.674364
$934$	0	0
$935$	0.699193	0.0228661
$936$	0	0
$937$	−58.0892	−1.89769	−0.948845	−	0.315741i	$-0.897747\pi$
$937$	−58.0892	−1.89769	−0.948845	+	0.315741i	$0.897747\pi$
$938$	0	0
$939$	29.7980	0.972422
$940$	0	0
$941$	−21.9560	−0.715745	−0.357872	−	0.933770i	$-0.616498\pi$
$941$	−21.9560	−0.715745	−0.357872	+	0.933770i	$0.616498\pi$
$942$	0	0
$943$	17.6665	0.575299
$944$	0	0
$945$	−3.44512	−0.112070
$946$	0	0
$947$	24.2497	0.788011	0.394005	−	0.919108i	$-0.371089\pi$
$947$	24.2497	0.788011	0.394005	+	0.919108i	$0.371089\pi$
$948$	0	0
$949$	−18.6133	−0.604215
$950$	0	0
$951$	−0.174862	−0.00567028
$952$	0	0
$953$	−59.4174	−1.92472	−0.962359	−	0.271783i	$-0.912387\pi$
$953$	−59.4174	−1.92472	−0.962359	+	0.271783i	$0.912387\pi$
$954$	0	0
$955$	−4.41247	−0.142784
$956$	0	0
$957$	−0.361740	−0.0116934
$958$	0	0
$959$	14.2417	0.459889
$960$	0	0
$961$	−14.7705	−0.476466
$962$	0	0
$963$	−12.3803	−0.398948
$964$	0	0
$965$	−2.83485	−0.0912571
$966$	0	0
$967$	36.0448	1.15912	0.579561	−	0.814929i	$-0.303224\pi$
$967$	36.0448	1.15912	0.579561	+	0.814929i	$0.303224\pi$
$968$	0	0
$969$	−33.3123	−1.07015
$970$	0	0
$971$	25.2763	0.811155	0.405578	−	0.914061i	$-0.367070\pi$
$971$	25.2763	0.811155	0.405578	+	0.914061i	$0.367070\pi$
$972$	0	0
$973$	−40.4153	−1.29565
$974$	0	0
$975$	1.50361	0.0481542
$976$	0	0
$977$	28.3632	0.907418	0.453709	−	0.891150i	$-0.350101\pi$
$977$	28.3632	0.907418	0.453709	+	0.891150i	$0.350101\pi$
$978$	0	0
$979$	−0.560465	−0.0179125
$980$	0	0
$981$	−5.81507	−0.185661
$982$	0	0
$983$	51.9588	1.65723	0.828614	−	0.559821i	$-0.189130\pi$
$983$	51.9588	1.65723	0.828614	+	0.559821i	$0.189130\pi$
$984$	0	0
$985$	−13.2785	−0.423087
$986$	0	0
$987$	6.37152	0.202808
$988$	0	0
$989$	43.6256	1.38721
$990$	0	0
$991$	27.1770	0.863305	0.431653	−	0.902040i	$-0.357931\pi$
$991$	27.1770	0.863305	0.431653	+	0.902040i	$0.357931\pi$
$992$	0	0
$993$	2.36816	0.0751512
$994$	0	0
$995$	13.4658	0.426893
$996$	0	0
$997$	24.7593	0.784136	0.392068	−	0.919936i	$-0.371760\pi$
$997$	24.7593	0.784136	0.392068	+	0.919936i	$0.371760\pi$
$998$	0	0
$999$	−8.49448	−0.268753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.d.1.1	✓	4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.44512 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.44512 q^{7} +1.00000 q^{9} -0.102193 q^{11} +1.50361 q^{13} +1.00000 q^{15} -6.84187 q^{17} +4.86889 q^{19} -3.44512 q^{21} -6.49448 q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.53976 q^{29} +4.02859 q^{31} -0.102193 q^{33} -3.44512 q^{35} -8.49448 q^{37} +1.50361 q^{39} -2.72023 q^{41} -6.71734 q^{43} +1.00000 q^{45} -1.84943 q^{47} +4.86889 q^{49} -6.84187 q^{51} -7.79252 q^{53} -0.102193 q^{55} +4.86889 q^{57} +10.9534 q^{59} +0.796590 q^{61} -3.44512 q^{63} +1.50361 q^{65} +1.00000 q^{67} -6.49448 q^{69} -6.59727 q^{71} -12.3791 q^{73} +1.00000 q^{75} +0.352068 q^{77} -8.29923 q^{79} +1.00000 q^{81} -0.0528416 q^{83} -6.84187 q^{85} +3.53976 q^{87} +5.48436 q^{89} -5.18014 q^{91} +4.02859 q^{93} +4.86889 q^{95} -0.284239 q^{97} -0.102193 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{5} - 3 q^{7} + 4 q^{9} - 5 q^{11} - 9 q^{13} + 4 q^{15} - 8 q^{17} - 7 q^{19} - 3 q^{21} - 12 q^{23} + 4 q^{25} + 4 q^{27} - 9 q^{29} - 9 q^{31} - 5 q^{33} - 3 q^{35} - 20 q^{37} - 9 q^{39} - 11 q^{41} + q^{43} + 4 q^{45} - 5 q^{47} - 7 q^{49} - 8 q^{51} - 15 q^{53} - 5 q^{55} - 7 q^{57} + 7 q^{59} - 7 q^{61} - 3 q^{63} - 9 q^{65} + 4 q^{67} - 12 q^{69} - 26 q^{73} + 4 q^{75} - 15 q^{77} - 9 q^{79} + 4 q^{81} - 8 q^{83} - 8 q^{85} - 9 q^{87} + 10 q^{89} + 3 q^{91} - 9 q^{93} - 7 q^{95} - 11 q^{97} - 5 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^5 - 3 * q^7 + 4 * q^9 - 5 * q^11 - 9 * q^13 + 4 * q^15 - 8 * q^17 - 7 * q^19 - 3 * q^21 - 12 * q^23 + 4 * q^25 + 4 * q^27 - 9 * q^29 - 9 * q^31 - 5 * q^33 - 3 * q^35 - 20 * q^37 - 9 * q^39 - 11 * q^41 + q^43 + 4 * q^45 - 5 * q^47 - 7 * q^49 - 8 * q^51 - 15 * q^53 - 5 * q^55 - 7 * q^57 + 7 * q^59 - 7 * q^61 - 3 * q^63 - 9 * q^65 + 4 * q^67 - 12 * q^69 - 26 * q^73 + 4 * q^75 - 15 * q^77 - 9 * q^79 + 4 * q^81 - 8 * q^83 - 8 * q^85 - 9 * q^87 + 10 * q^89 + 3 * q^91 - 9 * q^93 - 7 * q^95 - 11 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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