LMFDB - Embedded newform 4620.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1,4620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4620.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.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:	$36.8908857338$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
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			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	4620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{15} -6.70156 q^{17} -1.29844 q^{19} +1.00000 q^{21} +4.70156 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.1047 q^{29} +5.40312 q^{31} -1.00000 q^{33} +1.00000 q^{35} +7.40312 q^{37} -4.00000 q^{39} -5.40312 q^{41} +3.29844 q^{43} -1.00000 q^{45} -13.4031 q^{47} +1.00000 q^{49} +6.70156 q^{51} +12.1047 q^{53} -1.00000 q^{55} +1.29844 q^{57} +10.1047 q^{59} -6.70156 q^{61} -1.00000 q^{63} -4.00000 q^{65} -13.4031 q^{67} -4.70156 q^{69} +9.40312 q^{71} +8.00000 q^{73} -1.00000 q^{75} -1.00000 q^{77} -3.40312 q^{79} +1.00000 q^{81} -6.70156 q^{83} +6.70156 q^{85} +10.1047 q^{87} +8.10469 q^{89} -4.00000 q^{91} -5.40312 q^{93} +1.29844 q^{95} +10.7016 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 7 q^{17} - 9 q^{19} + 2 q^{21} + 3 q^{23} + 2 q^{25} - 2 q^{27} - q^{29} - 2 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} + 13 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} + 7 q^{51} + 5 q^{53} - 2 q^{55} + 9 q^{57} + q^{59} - 7 q^{61} - 2 q^{63} - 8 q^{65} - 14 q^{67} - 3 q^{69} + 6 q^{71} + 16 q^{73} - 2 q^{75} - 2 q^{77} + 6 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{85} + q^{87} - 3 q^{89} - 8 q^{91} + 2 q^{93} + 9 q^{95} + 15 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 8 * q^13 + 2 * q^15 - 7 * q^17 - 9 * q^19 + 2 * q^21 + 3 * q^23 + 2 * q^25 - 2 * q^27 - q^29 - 2 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 + 13 * q^43 - 2 * q^45 - 14 * q^47 + 2 * q^49 + 7 * q^51 + 5 * q^53 - 2 * q^55 + 9 * q^57 + q^59 - 7 * q^61 - 2 * q^63 - 8 * q^65 - 14 * q^67 - 3 * q^69 + 6 * q^71 + 16 * q^73 - 2 * q^75 - 2 * q^77 + 6 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^85 + q^87 - 3 * q^89 - 8 * q^91 + 2 * q^93 + 9 * q^95 + 15 * q^97 + 2 * 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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.70156	−1.62537	−0.812684	−	0.582705i	$-0.801994\pi$
$17$	−6.70156	−1.62537	−0.812684	+	0.582705i	$0.801994\pi$
$18$	0	0
$19$	−1.29844	−0.297882	−0.148941	−	0.988846i	$-0.547586\pi$
$19$	−1.29844	−0.297882	−0.148941	+	0.988846i	$0.547586\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	4.70156	0.980343	0.490172	−	0.871626i	$-0.336934\pi$
$23$	4.70156	0.980343	0.490172	+	0.871626i	$0.336934\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.1047	−1.87639	−0.938197	−	0.346103i	$-0.887505\pi$
$29$	−10.1047	−1.87639	−0.938197	+	0.346103i	$0.887505\pi$
$30$	0	0
$31$	5.40312	0.970430	0.485215	−	0.874395i	$-0.338741\pi$
$31$	5.40312	0.970430	0.485215	+	0.874395i	$0.338741\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	7.40312	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$37$	7.40312	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−5.40312	−0.843826	−0.421913	−	0.906636i	$-0.638641\pi$
$41$	−5.40312	−0.843826	−0.421913	+	0.906636i	$0.638641\pi$
$42$	0	0
$43$	3.29844	0.503007	0.251504	−	0.967856i	$-0.419075\pi$
$43$	3.29844	0.503007	0.251504	+	0.967856i	$0.419075\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−13.4031	−1.95505	−0.977523	−	0.210827i	$-0.932384\pi$
$47$	−13.4031	−1.95505	−0.977523	+	0.210827i	$0.932384\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.70156	0.938406
$52$	0	0
$53$	12.1047	1.66271	0.831353	−	0.555744i	$-0.187567\pi$
$53$	12.1047	1.66271	0.831353	+	0.555744i	$0.187567\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	1.29844	0.171982
$58$	0	0
$59$	10.1047	1.31552	0.657759	−	0.753228i	$-0.271505\pi$
$59$	10.1047	1.31552	0.657759	+	0.753228i	$0.271505\pi$
$60$	0	0
$61$	−6.70156	−0.858047	−0.429024	−	0.903293i	$-0.641142\pi$
$61$	−6.70156	−0.858047	−0.429024	+	0.903293i	$0.641142\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−13.4031	−1.63745	−0.818726	−	0.574184i	$-0.805319\pi$
$67$	−13.4031	−1.63745	−0.818726	+	0.574184i	$0.805319\pi$
$68$	0	0
$69$	−4.70156	−0.566002
$70$	0	0
$71$	9.40312	1.11595	0.557973	−	0.829859i	$-0.311579\pi$
$71$	9.40312	1.11595	0.557973	+	0.829859i	$0.311579\pi$
$72$	0	0
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−3.40312	−0.382881	−0.191441	−	0.981504i	$-0.561316\pi$
$79$	−3.40312	−0.382881	−0.191441	+	0.981504i	$0.561316\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.70156	−0.735592	−0.367796	−	0.929907i	$-0.619888\pi$
$83$	−6.70156	−0.735592	−0.367796	+	0.929907i	$0.619888\pi$
$84$	0	0
$85$	6.70156	0.726886
$86$	0	0
$87$	10.1047	1.08334
$88$	0	0
$89$	8.10469	0.859095	0.429548	−	0.903044i	$-0.358673\pi$
$89$	8.10469	0.859095	0.429548	+	0.903044i	$0.358673\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	−5.40312	−0.560278
$94$	0	0
$95$	1.29844	0.133217
$96$	0	0
$97$	10.7016	1.08658	0.543290	−	0.839545i	$-0.317179\pi$
$97$	10.7016	1.08658	0.543290	+	0.839545i	$0.317179\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−9.40312	−0.935646	−0.467823	−	0.883822i	$-0.654961\pi$
$101$	−9.40312	−0.935646	−0.467823	+	0.883822i	$0.654961\pi$
$102$	0	0
$103$	−14.1047	−1.38978	−0.694888	−	0.719118i	$-0.744546\pi$
$103$	−14.1047	−1.38978	−0.694888	+	0.719118i	$0.744546\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	0	0
$109$	−12.8062	−1.22662	−0.613308	−	0.789844i	$-0.710162\pi$
$109$	−12.8062	−1.22662	−0.613308	+	0.789844i	$0.710162\pi$
$110$	0	0
$111$	−7.40312	−0.702673
$112$	0	0
$113$	8.10469	0.762425	0.381212	−	0.924487i	$-0.375507\pi$
$113$	8.10469	0.762425	0.381212	+	0.924487i	$0.375507\pi$
$114$	0	0
$115$	−4.70156	−0.438423
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	6.70156	0.614331
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.40312	0.487183
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.701562	−0.0622536	−0.0311268	−	0.999515i	$-0.509910\pi$
$127$	−0.701562	−0.0622536	−0.0311268	+	0.999515i	$0.509910\pi$
$128$	0	0
$129$	−3.29844	−0.290411
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	1.29844	0.112589
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.80625	0.752369	0.376184	−	0.926545i	$-0.377236\pi$
$137$	8.80625	0.752369	0.376184	+	0.926545i	$0.377236\pi$
$138$	0	0
$139$	−0.596876	−0.0506263	−0.0253132	−	0.999680i	$-0.508058\pi$
$139$	−0.596876	−0.0506263	−0.0253132	+	0.999680i	$0.508058\pi$
$140$	0	0
$141$	13.4031	1.12875
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	10.1047	0.839149
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−21.4031	−1.75341	−0.876706	−	0.481026i	$-0.840264\pi$
$149$	−21.4031	−1.75341	−0.876706	+	0.481026i	$0.840264\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	0	0
$153$	−6.70156	−0.541789
$154$	0	0
$155$	−5.40312	−0.433989
$156$	0	0
$157$	6.70156	0.534843	0.267421	−	0.963580i	$-0.413828\pi$
$157$	6.70156	0.534843	0.267421	+	0.963580i	$0.413828\pi$
$158$	0	0
$159$	−12.1047	−0.959964
$160$	0	0
$161$	−4.70156	−0.370535
$162$	0	0
$163$	−6.80625	−0.533107	−0.266553	−	0.963820i	$-0.585885\pi$
$163$	−6.80625	−0.533107	−0.266553	+	0.963820i	$0.585885\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−0.596876	−0.0461876	−0.0230938	−	0.999733i	$-0.507352\pi$
$167$	−0.596876	−0.0461876	−0.0230938	+	0.999733i	$0.507352\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	−1.29844	−0.0992940
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−10.1047	−0.759515
$178$	0	0
$179$	−13.4031	−1.00180	−0.500898	−	0.865506i	$-0.666997\pi$
$179$	−13.4031	−1.00180	−0.500898	+	0.865506i	$0.666997\pi$
$180$	0	0
$181$	4.80625	0.357246	0.178623	−	0.983918i	$-0.442836\pi$
$181$	4.80625	0.357246	0.178623	+	0.983918i	$0.442836\pi$
$182$	0	0
$183$	6.70156	0.495394
$184$	0	0
$185$	−7.40312	−0.544289
$186$	0	0
$187$	−6.70156	−0.490067
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−18.8062	−1.36077	−0.680386	−	0.732854i	$-0.738188\pi$
$191$	−18.8062	−1.36077	−0.680386	+	0.732854i	$0.738188\pi$
$192$	0	0
$193$	16.2094	1.16678	0.583388	−	0.812194i	$-0.301727\pi$
$193$	16.2094	1.16678	0.583388	+	0.812194i	$0.301727\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	14.2094	1.01238	0.506188	−	0.862423i	$-0.331054\pi$
$197$	14.2094	1.01238	0.506188	+	0.862423i	$0.331054\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	13.4031	0.945383
$202$	0	0
$203$	10.1047	0.709210
$204$	0	0
$205$	5.40312	0.377371
$206$	0	0
$207$	4.70156	0.326781
$208$	0	0
$209$	−1.29844	−0.0898148
$210$	0	0
$211$	−3.19375	−0.219867	−0.109933	−	0.993939i	$-0.535064\pi$
$211$	−3.19375	−0.219867	−0.109933	+	0.993939i	$0.535064\pi$
$212$	0	0
$213$	−9.40312	−0.644291
$214$	0	0
$215$	−3.29844	−0.224952
$216$	0	0
$217$	−5.40312	−0.366788
$218$	0	0
$219$	−8.00000	−0.540590
$220$	0	0
$221$	−26.8062	−1.80318
$222$	0	0
$223$	−4.70156	−0.314840	−0.157420	−	0.987532i	$-0.550318\pi$
$223$	−4.70156	−0.314840	−0.157420	+	0.987532i	$0.550318\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−13.5078	−0.896545	−0.448272	−	0.893897i	$-0.647961\pi$
$227$	−13.5078	−0.896545	−0.448272	+	0.893897i	$0.647961\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	−11.4031	−0.747044	−0.373522	−	0.927621i	$-0.621850\pi$
$233$	−11.4031	−0.747044	−0.373522	+	0.927621i	$0.621850\pi$
$234$	0	0
$235$	13.4031	0.874323
$236$	0	0
$237$	3.40312	0.221057
$238$	0	0
$239$	−14.1047	−0.912356	−0.456178	−	0.889888i	$-0.650782\pi$
$239$	−14.1047	−0.912356	−0.456178	+	0.889888i	$0.650782\pi$
$240$	0	0
$241$	−16.8062	−1.08259	−0.541293	−	0.840834i	$-0.682065\pi$
$241$	−16.8062	−1.08259	−0.541293	+	0.840834i	$0.682065\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−5.19375	−0.330470
$248$	0	0
$249$	6.70156	0.424694
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	4.70156	0.295585
$254$	0	0
$255$	−6.70156	−0.419668
$256$	0	0
$257$	−19.4031	−1.21033	−0.605167	−	0.796099i	$-0.706894\pi$
$257$	−19.4031	−1.21033	−0.605167	+	0.796099i	$0.706894\pi$
$258$	0	0
$259$	−7.40312	−0.460008
$260$	0	0
$261$	−10.1047	−0.625464
$262$	0	0
$263$	−18.2094	−1.12284	−0.561419	−	0.827532i	$-0.689744\pi$
$263$	−18.2094	−1.12284	−0.561419	+	0.827532i	$0.689744\pi$
$264$	0	0
$265$	−12.1047	−0.743585
$266$	0	0
$267$	−8.10469	−0.495999
$268$	0	0
$269$	−22.7016	−1.38414	−0.692069	−	0.721831i	$-0.743301\pi$
$269$	−22.7016	−1.38414	−0.692069	+	0.721831i	$0.743301\pi$
$270$	0	0
$271$	−7.89531	−0.479606	−0.239803	−	0.970822i	$-0.577083\pi$
$271$	−7.89531	−0.479606	−0.239803	+	0.970822i	$0.577083\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	2.59688	0.156031	0.0780156	−	0.996952i	$-0.475142\pi$
$277$	2.59688	0.156031	0.0780156	+	0.996952i	$0.475142\pi$
$278$	0	0
$279$	5.40312	0.323477
$280$	0	0
$281$	16.2094	0.966970	0.483485	−	0.875353i	$-0.339371\pi$
$281$	16.2094	0.966970	0.483485	+	0.875353i	$0.339371\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	−1.29844	−0.0769128
$286$	0	0
$287$	5.40312	0.318936
$288$	0	0
$289$	27.9109	1.64182
$290$	0	0
$291$	−10.7016	−0.627337
$292$	0	0
$293$	−2.70156	−0.157827	−0.0789135	−	0.996881i	$-0.525145\pi$
$293$	−2.70156	−0.157827	−0.0789135	+	0.996881i	$0.525145\pi$
$294$	0	0
$295$	−10.1047	−0.588318
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	18.8062	1.08759
$300$	0	0
$301$	−3.29844	−0.190119
$302$	0	0
$303$	9.40312	0.540195
$304$	0	0
$305$	6.70156	0.383730
$306$	0	0
$307$	29.4031	1.67812	0.839062	−	0.544035i	$-0.183104\pi$
$307$	29.4031	1.67812	0.839062	+	0.544035i	$0.183104\pi$
$308$	0	0
$309$	14.1047	0.802388
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	0	0
$313$	−13.5078	−0.763507	−0.381753	−	0.924264i	$-0.624680\pi$
$313$	−13.5078	−0.763507	−0.381753	+	0.924264i	$0.624680\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	−10.1047	−0.565754
$320$	0	0
$321$	2.00000	0.111629
$322$	0	0
$323$	8.70156	0.484168
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	12.8062	0.708187
$328$	0	0
$329$	13.4031	0.738938
$330$	0	0
$331$	−15.2984	−0.840878	−0.420439	−	0.907321i	$-0.638124\pi$
$331$	−15.2984	−0.840878	−0.420439	+	0.907321i	$0.638124\pi$
$332$	0	0
$333$	7.40312	0.405689
$334$	0	0
$335$	13.4031	0.732291
$336$	0	0
$337$	−35.7172	−1.94564	−0.972819	−	0.231565i	$-0.925615\pi$
$337$	−35.7172	−1.94564	−0.972819	+	0.231565i	$0.925615\pi$
$338$	0	0
$339$	−8.10469	−0.440186
$340$	0	0
$341$	5.40312	0.292596
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.70156	0.253124
$346$	0	0
$347$	31.4031	1.68581	0.842904	−	0.538064i	$-0.180844\pi$
$347$	31.4031	1.68581	0.842904	+	0.538064i	$0.180844\pi$
$348$	0	0
$349$	20.1047	1.07618	0.538090	−	0.842888i	$-0.319146\pi$
$349$	20.1047	1.07618	0.538090	+	0.842888i	$0.319146\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	−9.40312	−0.499066
$356$	0	0
$357$	−6.70156	−0.354684
$358$	0	0
$359$	22.3141	1.17769	0.588845	−	0.808246i	$-0.299583\pi$
$359$	22.3141	1.17769	0.588845	+	0.808246i	$0.299583\pi$
$360$	0	0
$361$	−17.3141	−0.911266
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	−3.29844	−0.172177	−0.0860885	−	0.996287i	$-0.527437\pi$
$367$	−3.29844	−0.172177	−0.0860885	+	0.996287i	$0.527437\pi$
$368$	0	0
$369$	−5.40312	−0.281275
$370$	0	0
$371$	−12.1047	−0.628444
$372$	0	0
$373$	24.7016	1.27900	0.639499	−	0.768792i	$-0.279142\pi$
$373$	24.7016	1.27900	0.639499	+	0.768792i	$0.279142\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−40.4187	−2.08167
$378$	0	0
$379$	36.7016	1.88523	0.942616	−	0.333878i	$-0.108357\pi$
$379$	36.7016	1.88523	0.942616	+	0.333878i	$0.108357\pi$
$380$	0	0
$381$	0.701562	0.0359421
$382$	0	0
$383$	−33.6125	−1.71752	−0.858759	−	0.512379i	$-0.828764\pi$
$383$	−33.6125	−1.71752	−0.858759	+	0.512379i	$0.828764\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	3.29844	0.167669
$388$	0	0
$389$	38.2094	1.93729	0.968646	−	0.248445i	$-0.0799195\pi$
$389$	38.2094	1.93729	0.968646	+	0.248445i	$0.0799195\pi$
$390$	0	0
$391$	−31.5078	−1.59342
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	3.40312	0.171230
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	−1.29844	−0.0650032
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	21.6125	1.07660
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	7.40312	0.366959
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	−8.80625	−0.434380
$412$	0	0
$413$	−10.1047	−0.497219
$414$	0	0
$415$	6.70156	0.328967
$416$	0	0
$417$	0.596876	0.0292291
$418$	0	0
$419$	−7.29844	−0.356552	−0.178276	−	0.983981i	$-0.557052\pi$
$419$	−7.29844	−0.356552	−0.178276	+	0.983981i	$0.557052\pi$
$420$	0	0
$421$	−14.9109	−0.726714	−0.363357	−	0.931650i	$-0.618370\pi$
$421$	−14.9109	−0.726714	−0.363357	+	0.931650i	$0.618370\pi$
$422$	0	0
$423$	−13.4031	−0.651682
$424$	0	0
$425$	−6.70156	−0.325074
$426$	0	0
$427$	6.70156	0.324311
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−20.0000	−0.963366	−0.481683	−	0.876346i	$-0.659974\pi$
$431$	−20.0000	−0.963366	−0.481683	+	0.876346i	$0.659974\pi$
$432$	0	0
$433$	−4.80625	−0.230974	−0.115487	−	0.993309i	$-0.536843\pi$
$433$	−4.80625	−0.230974	−0.115487	+	0.993309i	$0.536843\pi$
$434$	0	0
$435$	−10.1047	−0.484483
$436$	0	0
$437$	−6.10469	−0.292027
$438$	0	0
$439$	29.5078	1.40833	0.704165	−	0.710036i	$-0.251321\pi$
$439$	29.5078	1.40833	0.704165	+	0.710036i	$0.251321\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	29.6125	1.40693	0.703466	−	0.710729i	$-0.251635\pi$
$443$	29.6125	1.40693	0.703466	+	0.710729i	$0.251635\pi$
$444$	0	0
$445$	−8.10469	−0.384199
$446$	0	0
$447$	21.4031	1.01233
$448$	0	0
$449$	−24.8062	−1.17068	−0.585340	−	0.810788i	$-0.699039\pi$
$449$	−24.8062	−1.17068	−0.585340	+	0.810788i	$0.699039\pi$
$450$	0	0
$451$	−5.40312	−0.254423
$452$	0	0
$453$	18.0000	0.845714
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	3.50781	0.164088	0.0820442	−	0.996629i	$-0.473855\pi$
$457$	3.50781	0.164088	0.0820442	+	0.996629i	$0.473855\pi$
$458$	0	0
$459$	6.70156	0.312802
$460$	0	0
$461$	27.0156	1.25824	0.629121	−	0.777307i	$-0.283415\pi$
$461$	27.0156	1.25824	0.629121	+	0.777307i	$0.283415\pi$
$462$	0	0
$463$	31.0156	1.44142	0.720709	−	0.693238i	$-0.243816\pi$
$463$	31.0156	1.44142	0.720709	+	0.693238i	$0.243816\pi$
$464$	0	0
$465$	5.40312	0.250564
$466$	0	0
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	13.4031	0.618899
$470$	0	0
$471$	−6.70156	−0.308792
$472$	0	0
$473$	3.29844	0.151662
$474$	0	0
$475$	−1.29844	−0.0595764
$476$	0	0
$477$	12.1047	0.554236
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	29.6125	1.35021
$482$	0	0
$483$	4.70156	0.213928
$484$	0	0
$485$	−10.7016	−0.485933
$486$	0	0
$487$	9.40312	0.426096	0.213048	−	0.977042i	$-0.431661\pi$
$487$	9.40312	0.426096	0.213048	+	0.977042i	$0.431661\pi$
$488$	0	0
$489$	6.80625	0.307789
$490$	0	0
$491$	−15.5078	−0.699858	−0.349929	−	0.936776i	$-0.613794\pi$
$491$	−15.5078	−0.699858	−0.349929	+	0.936776i	$0.613794\pi$
$492$	0	0
$493$	67.7172	3.04983
$494$	0	0
$495$	−1.00000	−0.0449467
$496$	0	0
$497$	−9.40312	−0.421788
$498$	0	0
$499$	−3.50781	−0.157031	−0.0785156	−	0.996913i	$-0.525018\pi$
$499$	−3.50781	−0.157031	−0.0785156	+	0.996913i	$0.525018\pi$
$500$	0	0
$501$	0.596876	0.0266664
$502$	0	0
$503$	−21.2984	−0.949650	−0.474825	−	0.880080i	$-0.657489\pi$
$503$	−21.2984	−0.949650	−0.474825	+	0.880080i	$0.657489\pi$
$504$	0	0
$505$	9.40312	0.418434
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	−34.9109	−1.54740	−0.773700	−	0.633552i	$-0.781596\pi$
$509$	−34.9109	−1.54740	−0.773700	+	0.633552i	$0.781596\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	1.29844	0.0573274
$514$	0	0
$515$	14.1047	0.621527
$516$	0	0
$517$	−13.4031	−0.589469
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	5.50781	0.241302	0.120651	−	0.992695i	$-0.461502\pi$
$521$	5.50781	0.241302	0.120651	+	0.992695i	$0.461502\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−36.2094	−1.57731
$528$	0	0
$529$	−0.895314	−0.0389267
$530$	0	0
$531$	10.1047	0.438506
$532$	0	0
$533$	−21.6125	−0.936141
$534$	0	0
$535$	2.00000	0.0864675
$536$	0	0
$537$	13.4031	0.578388
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	−4.80625	−0.206256
$544$	0	0
$545$	12.8062	0.548559
$546$	0	0
$547$	−34.1047	−1.45821	−0.729106	−	0.684401i	$-0.760064\pi$
$547$	−34.1047	−1.45821	−0.729106	+	0.684401i	$0.760064\pi$
$548$	0	0
$549$	−6.70156	−0.286016
$550$	0	0
$551$	13.1203	0.558944
$552$	0	0
$553$	3.40312	0.144716
$554$	0	0
$555$	7.40312	0.314245
$556$	0	0
$557$	22.0000	0.932170	0.466085	−	0.884740i	$-0.345664\pi$
$557$	22.0000	0.932170	0.466085	+	0.884740i	$0.345664\pi$
$558$	0	0
$559$	13.1938	0.558036
$560$	0	0
$561$	6.70156	0.282940
$562$	0	0
$563$	−16.5969	−0.699475	−0.349737	−	0.936848i	$-0.613729\pi$
$563$	−16.5969	−0.699475	−0.349737	+	0.936848i	$0.613729\pi$
$564$	0	0
$565$	−8.10469	−0.340967
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−18.1047	−0.758988	−0.379494	−	0.925194i	$-0.623902\pi$
$569$	−18.1047	−0.758988	−0.379494	+	0.925194i	$0.623902\pi$
$570$	0	0
$571$	22.2094	0.929433	0.464717	−	0.885459i	$-0.346156\pi$
$571$	22.2094	0.929433	0.464717	+	0.885459i	$0.346156\pi$
$572$	0	0
$573$	18.8062	0.785642
$574$	0	0
$575$	4.70156	0.196069
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	−16.2094	−0.673639
$580$	0	0
$581$	6.70156	0.278028
$582$	0	0
$583$	12.1047	0.501325
$584$	0	0
$585$	−4.00000	−0.165380
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−7.01562	−0.289074
$590$	0	0
$591$	−14.2094	−0.584495
$592$	0	0
$593$	12.8062	0.525890	0.262945	−	0.964811i	$-0.415306\pi$
$593$	12.8062	0.525890	0.262945	+	0.964811i	$0.415306\pi$
$594$	0	0
$595$	−6.70156	−0.274737
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−41.8219	−1.70880	−0.854398	−	0.519620i	$-0.826074\pi$
$599$	−41.8219	−1.70880	−0.854398	+	0.519620i	$0.826074\pi$
$600$	0	0
$601$	−13.2984	−0.542455	−0.271227	−	0.962515i	$-0.587430\pi$
$601$	−13.2984	−0.542455	−0.271227	+	0.962515i	$0.587430\pi$
$602$	0	0
$603$	−13.4031	−0.545817
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	5.40312	0.219306	0.109653	−	0.993970i	$-0.465026\pi$
$607$	5.40312	0.219306	0.109653	+	0.993970i	$0.465026\pi$
$608$	0	0
$609$	−10.1047	−0.409463
$610$	0	0
$611$	−53.6125	−2.16893
$612$	0	0
$613$	−0.209373	−0.00845648	−0.00422824	−	0.999991i	$-0.501346\pi$
$613$	−0.209373	−0.00845648	−0.00422824	+	0.999991i	$0.501346\pi$
$614$	0	0
$615$	−5.40312	−0.217875
$616$	0	0
$617$	−11.1938	−0.450643	−0.225322	−	0.974284i	$-0.572343\pi$
$617$	−11.1938	−0.450643	−0.225322	+	0.974284i	$0.572343\pi$
$618$	0	0
$619$	−45.4031	−1.82491	−0.912453	−	0.409182i	$-0.865814\pi$
$619$	−45.4031	−1.82491	−0.912453	+	0.409182i	$0.865814\pi$
$620$	0	0
$621$	−4.70156	−0.188667
$622$	0	0
$623$	−8.10469	−0.324707
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.29844	0.0518546
$628$	0	0
$629$	−49.6125	−1.97818
$630$	0	0
$631$	1.89531	0.0754512	0.0377256	−	0.999288i	$-0.487989\pi$
$631$	1.89531	0.0754512	0.0377256	+	0.999288i	$0.487989\pi$
$632$	0	0
$633$	3.19375	0.126940
$634$	0	0
$635$	0.701562	0.0278406
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	9.40312	0.371982
$640$	0	0
$641$	31.6125	1.24862	0.624309	−	0.781177i	$-0.285381\pi$
$641$	31.6125	1.24862	0.624309	+	0.781177i	$0.285381\pi$
$642$	0	0
$643$	16.7016	0.658645	0.329323	−	0.944217i	$-0.393180\pi$
$643$	16.7016	0.658645	0.329323	+	0.944217i	$0.393180\pi$
$644$	0	0
$645$	3.29844	0.129876
$646$	0	0
$647$	−17.1938	−0.675956	−0.337978	−	0.941154i	$-0.609743\pi$
$647$	−17.1938	−0.675956	−0.337978	+	0.941154i	$0.609743\pi$
$648$	0	0
$649$	10.1047	0.396644
$650$	0	0
$651$	5.40312	0.211765
$652$	0	0
$653$	38.7016	1.51451	0.757255	−	0.653120i	$-0.226540\pi$
$653$	38.7016	1.51451	0.757255	+	0.653120i	$0.226540\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	8.00000	0.312110
$658$	0	0
$659$	−22.3141	−0.869232	−0.434616	−	0.900616i	$-0.643116\pi$
$659$	−22.3141	−0.869232	−0.434616	+	0.900616i	$0.643116\pi$
$660$	0	0
$661$	−7.19375	−0.279805	−0.139902	−	0.990165i	$-0.544679\pi$
$661$	−7.19375	−0.279805	−0.139902	+	0.990165i	$0.544679\pi$
$662$	0	0
$663$	26.8062	1.04107
$664$	0	0
$665$	−1.29844	−0.0503513
$666$	0	0
$667$	−47.5078	−1.83951
$668$	0	0
$669$	4.70156	0.181773
$670$	0	0
$671$	−6.70156	−0.258711
$672$	0	0
$673$	−27.5078	−1.06035	−0.530174	−	0.847889i	$-0.677873\pi$
$673$	−27.5078	−1.06035	−0.530174	+	0.847889i	$0.677873\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	33.2984	1.27976	0.639881	−	0.768474i	$-0.278983\pi$
$677$	33.2984	1.27976	0.639881	+	0.768474i	$0.278983\pi$
$678$	0	0
$679$	−10.7016	−0.410688
$680$	0	0
$681$	13.5078	0.517620
$682$	0	0
$683$	−32.4187	−1.24047	−0.620234	−	0.784417i	$-0.712963\pi$
$683$	−32.4187	−1.24047	−0.620234	+	0.784417i	$0.712963\pi$
$684$	0	0
$685$	−8.80625	−0.336469
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	48.4187	1.84461
$690$	0	0
$691$	6.59688	0.250957	0.125478	−	0.992096i	$-0.459953\pi$
$691$	6.59688	0.250957	0.125478	+	0.992096i	$0.459953\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	0.596876	0.0226408
$696$	0	0
$697$	36.2094	1.37153
$698$	0	0
$699$	11.4031	0.431306
$700$	0	0
$701$	38.1047	1.43919	0.719597	−	0.694392i	$-0.244327\pi$
$701$	38.1047	1.43919	0.719597	+	0.694392i	$0.244327\pi$
$702$	0	0
$703$	−9.61250	−0.362542
$704$	0	0
$705$	−13.4031	−0.504791
$706$	0	0
$707$	9.40312	0.353641
$708$	0	0
$709$	−21.7172	−0.815606	−0.407803	−	0.913070i	$-0.633705\pi$
$709$	−21.7172	−0.815606	−0.407803	+	0.913070i	$0.633705\pi$
$710$	0	0
$711$	−3.40312	−0.127627
$712$	0	0
$713$	25.4031	0.951354
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	14.1047	0.526749
$718$	0	0
$719$	27.2984	1.01806	0.509030	−	0.860749i	$-0.330004\pi$
$719$	27.2984	1.01806	0.509030	+	0.860749i	$0.330004\pi$
$720$	0	0
$721$	14.1047	0.525286
$722$	0	0
$723$	16.8062	0.625031
$724$	0	0
$725$	−10.1047	−0.375279
$726$	0	0
$727$	3.29844	0.122332	0.0611661	−	0.998128i	$-0.480518\pi$
$727$	3.29844	0.122332	0.0611661	+	0.998128i	$0.480518\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−22.1047	−0.817571
$732$	0	0
$733$	−1.40312	−0.0518256	−0.0259128	−	0.999664i	$-0.508249\pi$
$733$	−1.40312	−0.0518256	−0.0259128	+	0.999664i	$0.508249\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−13.4031	−0.493710
$738$	0	0
$739$	20.8062	0.765370	0.382685	−	0.923879i	$-0.374999\pi$
$739$	20.8062	0.765370	0.382685	+	0.923879i	$0.374999\pi$
$740$	0	0
$741$	5.19375	0.190797
$742$	0	0
$743$	40.8062	1.49704	0.748518	−	0.663114i	$-0.230766\pi$
$743$	40.8062	1.49704	0.748518	+	0.663114i	$0.230766\pi$
$744$	0	0
$745$	21.4031	0.784150
$746$	0	0
$747$	−6.70156	−0.245197
$748$	0	0
$749$	2.00000	0.0730784
$750$	0	0
$751$	1.89531	0.0691610	0.0345805	−	0.999402i	$-0.488990\pi$
$751$	1.89531	0.0691610	0.0345805	+	0.999402i	$0.488990\pi$
$752$	0	0
$753$	4.00000	0.145768
$754$	0	0
$755$	18.0000	0.655087
$756$	0	0
$757$	50.4187	1.83250	0.916250	−	0.400606i	$-0.131201\pi$
$757$	50.4187	1.83250	0.916250	+	0.400606i	$0.131201\pi$
$758$	0	0
$759$	−4.70156	−0.170656
$760$	0	0
$761$	−45.8219	−1.66104	−0.830521	−	0.556988i	$-0.811957\pi$
$761$	−45.8219	−1.66104	−0.830521	+	0.556988i	$0.811957\pi$
$762$	0	0
$763$	12.8062	0.463617
$764$	0	0
$765$	6.70156	0.242295
$766$	0	0
$767$	40.4187	1.45944
$768$	0	0
$769$	−1.50781	−0.0543730	−0.0271865	−	0.999630i	$-0.508655\pi$
$769$	−1.50781	−0.0543730	−0.0271865	+	0.999630i	$0.508655\pi$
$770$	0	0
$771$	19.4031	0.698786
$772$	0	0
$773$	52.8062	1.89931	0.949654	−	0.313299i	$-0.101434\pi$
$773$	52.8062	1.89931	0.949654	+	0.313299i	$0.101434\pi$
$774$	0	0
$775$	5.40312	0.194086
$776$	0	0
$777$	7.40312	0.265586
$778$	0	0
$779$	7.01562	0.251361
$780$	0	0
$781$	9.40312	0.336470
$782$	0	0
$783$	10.1047	0.361112
$784$	0	0
$785$	−6.70156	−0.239189
$786$	0	0
$787$	29.6125	1.05557	0.527786	−	0.849378i	$-0.323022\pi$
$787$	29.6125	1.05557	0.527786	+	0.849378i	$0.323022\pi$
$788$	0	0
$789$	18.2094	0.648271
$790$	0	0
$791$	−8.10469	−0.288169
$792$	0	0
$793$	−26.8062	−0.951918
$794$	0	0
$795$	12.1047	0.429309
$796$	0	0
$797$	11.6125	0.411336	0.205668	−	0.978622i	$-0.434063\pi$
$797$	11.6125	0.411336	0.205668	+	0.978622i	$0.434063\pi$
$798$	0	0
$799$	89.8219	3.17767
$800$	0	0
$801$	8.10469	0.286365
$802$	0	0
$803$	8.00000	0.282314
$804$	0	0
$805$	4.70156	0.165708
$806$	0	0
$807$	22.7016	0.799133
$808$	0	0
$809$	25.4031	0.893126	0.446563	−	0.894752i	$-0.352648\pi$
$809$	25.4031	0.893126	0.446563	+	0.894752i	$0.352648\pi$
$810$	0	0
$811$	23.4031	0.821795	0.410897	−	0.911682i	$-0.365215\pi$
$811$	23.4031	0.821795	0.410897	+	0.911682i	$0.365215\pi$
$812$	0	0
$813$	7.89531	0.276901
$814$	0	0
$815$	6.80625	0.238412
$816$	0	0
$817$	−4.28282	−0.149837
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	31.7172	1.10694	0.553469	−	0.832870i	$-0.313304\pi$
$821$	31.7172	1.10694	0.553469	+	0.832870i	$0.313304\pi$
$822$	0	0
$823$	−10.8062	−0.376682	−0.188341	−	0.982104i	$-0.560311\pi$
$823$	−10.8062	−0.376682	−0.188341	+	0.982104i	$0.560311\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	−5.79063	−0.201360	−0.100680	−	0.994919i	$-0.532102\pi$
$827$	−5.79063	−0.201360	−0.100680	+	0.994919i	$0.532102\pi$
$828$	0	0
$829$	−4.80625	−0.166928	−0.0834640	−	0.996511i	$-0.526598\pi$
$829$	−4.80625	−0.166928	−0.0834640	+	0.996511i	$0.526598\pi$
$830$	0	0
$831$	−2.59688	−0.0900846
$832$	0	0
$833$	−6.70156	−0.232195
$834$	0	0
$835$	0.596876	0.0206557
$836$	0	0
$837$	−5.40312	−0.186759
$838$	0	0
$839$	−28.7016	−0.990888	−0.495444	−	0.868640i	$-0.664995\pi$
$839$	−28.7016	−0.990888	−0.495444	+	0.868640i	$0.664995\pi$
$840$	0	0
$841$	73.1047	2.52085
$842$	0	0
$843$	−16.2094	−0.558280
$844$	0	0
$845$	−3.00000	−0.103203
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	16.0000	0.549119
$850$	0	0
$851$	34.8062	1.19314
$852$	0	0
$853$	29.4031	1.00674	0.503372	−	0.864070i	$-0.332093\pi$
$853$	29.4031	1.00674	0.503372	+	0.864070i	$0.332093\pi$
$854$	0	0
$855$	1.29844	0.0444056
$856$	0	0
$857$	−50.4187	−1.72227	−0.861136	−	0.508375i	$-0.830246\pi$
$857$	−50.4187	−1.72227	−0.861136	+	0.508375i	$0.830246\pi$
$858$	0	0
$859$	−43.0156	−1.46767	−0.733837	−	0.679326i	$-0.762272\pi$
$859$	−43.0156	−1.46767	−0.733837	+	0.679326i	$0.762272\pi$
$860$	0	0
$861$	−5.40312	−0.184138
$862$	0	0
$863$	19.5078	0.664054	0.332027	−	0.943270i	$-0.392268\pi$
$863$	19.5078	0.664054	0.332027	+	0.943270i	$0.392268\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	−27.9109	−0.947905
$868$	0	0
$869$	−3.40312	−0.115443
$870$	0	0
$871$	−53.6125	−1.81659
$872$	0	0
$873$	10.7016	0.362193
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−38.3141	−1.29377	−0.646887	−	0.762586i	$-0.723929\pi$
$877$	−38.3141	−1.29377	−0.646887	+	0.762586i	$0.723929\pi$
$878$	0	0
$879$	2.70156	0.0911214
$880$	0	0
$881$	−57.5078	−1.93749	−0.968744	−	0.248064i	$-0.920206\pi$
$881$	−57.5078	−1.93749	−0.968744	+	0.248064i	$0.920206\pi$
$882$	0	0
$883$	−6.80625	−0.229048	−0.114524	−	0.993420i	$-0.536534\pi$
$883$	−6.80625	−0.229048	−0.114524	+	0.993420i	$0.536534\pi$
$884$	0	0
$885$	10.1047	0.339665
$886$	0	0
$887$	16.1047	0.540742	0.270371	−	0.962756i	$-0.412854\pi$
$887$	16.1047	0.540742	0.270371	+	0.962756i	$0.412854\pi$
$888$	0	0
$889$	0.701562	0.0235296
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	17.4031	0.582373
$894$	0	0
$895$	13.4031	0.448017
$896$	0	0
$897$	−18.8062	−0.627922
$898$	0	0
$899$	−54.5969	−1.82091
$900$	0	0
$901$	−81.1203	−2.70251
$902$	0	0
$903$	3.29844	0.109765
$904$	0	0
$905$	−4.80625	−0.159765
$906$	0	0
$907$	−49.6125	−1.64736	−0.823678	−	0.567058i	$-0.808082\pi$
$907$	−49.6125	−1.64736	−0.823678	+	0.567058i	$0.808082\pi$
$908$	0	0
$909$	−9.40312	−0.311882
$910$	0	0
$911$	42.8062	1.41823	0.709117	−	0.705091i	$-0.249094\pi$
$911$	42.8062	1.41823	0.709117	+	0.705091i	$0.249094\pi$
$912$	0	0
$913$	−6.70156	−0.221789
$914$	0	0
$915$	−6.70156	−0.221547
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	16.8062	0.554387	0.277193	−	0.960814i	$-0.410596\pi$
$919$	16.8062	0.554387	0.277193	+	0.960814i	$0.410596\pi$
$920$	0	0
$921$	−29.4031	−0.968866
$922$	0	0
$923$	37.6125	1.23803
$924$	0	0
$925$	7.40312	0.243413
$926$	0	0
$927$	−14.1047	−0.463259
$928$	0	0
$929$	−2.00000	−0.0656179	−0.0328089	−	0.999462i	$-0.510445\pi$
$929$	−2.00000	−0.0656179	−0.0328089	+	0.999462i	$0.510445\pi$
$930$	0	0
$931$	−1.29844	−0.0425546
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	6.70156	0.219165
$936$	0	0
$937$	28.2094	0.921560	0.460780	−	0.887514i	$-0.347570\pi$
$937$	28.2094	0.921560	0.460780	+	0.887514i	$0.347570\pi$
$938$	0	0
$939$	13.5078	0.440811
$940$	0	0
$941$	8.00000	0.260793	0.130396	−	0.991462i	$-0.458375\pi$
$941$	8.00000	0.260793	0.130396	+	0.991462i	$0.458375\pi$
$942$	0	0
$943$	−25.4031	−0.827240
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−8.91093	−0.289566	−0.144783	−	0.989463i	$-0.546248\pi$
$947$	−8.91093	−0.289566	−0.144783	+	0.989463i	$0.546248\pi$
$948$	0	0
$949$	32.0000	1.03876
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	58.0000	1.87880	0.939402	−	0.342817i	$-0.111381\pi$
$953$	58.0000	1.87880	0.939402	+	0.342817i	$0.111381\pi$
$954$	0	0
$955$	18.8062	0.608556
$956$	0	0
$957$	10.1047	0.326638
$958$	0	0
$959$	−8.80625	−0.284369
$960$	0	0
$961$	−1.80625	−0.0582661
$962$	0	0
$963$	−2.00000	−0.0644491
$964$	0	0
$965$	−16.2094	−0.521798
$966$	0	0
$967$	18.1047	0.582207	0.291104	−	0.956691i	$-0.405978\pi$
$967$	18.1047	0.582207	0.291104	+	0.956691i	$0.405978\pi$
$968$	0	0
$969$	−8.70156	−0.279534
$970$	0	0
$971$	23.2984	0.747682	0.373841	−	0.927493i	$-0.378040\pi$
$971$	23.2984	0.747682	0.373841	+	0.927493i	$0.378040\pi$
$972$	0	0
$973$	0.596876	0.0191350
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−33.5078	−1.07201	−0.536005	−	0.844215i	$-0.680067\pi$
$977$	−33.5078	−1.07201	−0.536005	+	0.844215i	$0.680067\pi$
$978$	0	0
$979$	8.10469	0.259027
$980$	0	0
$981$	−12.8062	−0.408872
$982$	0	0
$983$	8.20937	0.261838	0.130919	−	0.991393i	$-0.458207\pi$
$983$	8.20937	0.261838	0.130919	+	0.991393i	$0.458207\pi$
$984$	0	0
$985$	−14.2094	−0.452748
$986$	0	0
$987$	−13.4031	−0.426626
$988$	0	0
$989$	15.5078	0.493120
$990$	0	0
$991$	34.1047	1.08337	0.541686	−	0.840581i	$-0.317786\pi$
$991$	34.1047	1.08337	0.541686	+	0.840581i	$0.317786\pi$
$992$	0	0
$993$	15.2984	0.485481
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	2.59688	0.0822439	0.0411219	−	0.999154i	$-0.486907\pi$
$997$	2.59688	0.0822439	0.0411219	+	0.999154i	$0.486907\pi$
$998$	0	0
$999$	−7.40312	−0.234224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.p.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.p.1.1	✓	2	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:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +4.00000 q^{13} +1.00000 q^{15} -6.70156 q^{17} -1.29844 q^{19} +1.00000 q^{21} +4.70156 q^{23} +1.00000 q^{25} -1.00000 q^{27} -10.1047 q^{29} +5.40312 q^{31} -1.00000 q^{33} +1.00000 q^{35} +7.40312 q^{37} -4.00000 q^{39} -5.40312 q^{41} +3.29844 q^{43} -1.00000 q^{45} -13.4031 q^{47} +1.00000 q^{49} +6.70156 q^{51} +12.1047 q^{53} -1.00000 q^{55} +1.29844 q^{57} +10.1047 q^{59} -6.70156 q^{61} -1.00000 q^{63} -4.00000 q^{65} -13.4031 q^{67} -4.70156 q^{69} +9.40312 q^{71} +8.00000 q^{73} -1.00000 q^{75} -1.00000 q^{77} -3.40312 q^{79} +1.00000 q^{81} -6.70156 q^{83} +6.70156 q^{85} +10.1047 q^{87} +8.10469 q^{89} -4.00000 q^{91} -5.40312 q^{93} +1.29844 q^{95} +10.7016 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 8 q^{13} + 2 q^{15} - 7 q^{17} - 9 q^{19} + 2 q^{21} + 3 q^{23} + 2 q^{25} - 2 q^{27} - q^{29} - 2 q^{31} - 2 q^{33} + 2 q^{35} + 2 q^{37} - 8 q^{39} + 2 q^{41} + 13 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} + 7 q^{51} + 5 q^{53} - 2 q^{55} + 9 q^{57} + q^{59} - 7 q^{61} - 2 q^{63} - 8 q^{65} - 14 q^{67} - 3 q^{69} + 6 q^{71} + 16 q^{73} - 2 q^{75} - 2 q^{77} + 6 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{85} + q^{87} - 3 q^{89} - 8 q^{91} + 2 q^{93} + 9 q^{95} + 15 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 + 8 * q^13 + 2 * q^15 - 7 * q^17 - 9 * q^19 + 2 * q^21 + 3 * q^23 + 2 * q^25 - 2 * q^27 - q^29 - 2 * q^31 - 2 * q^33 + 2 * q^35 + 2 * q^37 - 8 * q^39 + 2 * q^41 + 13 * q^43 - 2 * q^45 - 14 * q^47 + 2 * q^49 + 7 * q^51 + 5 * q^53 - 2 * q^55 + 9 * q^57 + q^59 - 7 * q^61 - 2 * q^63 - 8 * q^65 - 14 * q^67 - 3 * q^69 + 6 * q^71 + 16 * q^73 - 2 * q^75 - 2 * q^77 + 6 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^85 + q^87 - 3 * q^89 - 8 * q^91 + 2 * q^93 + 9 * q^95 + 15 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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