LMFDB - Embedded newform 4620.2.a.v.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:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.85577$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.01121 q^{13} -1.00000 q^{15} +1.44389 q^{17} -2.55611 q^{19} +1.00000 q^{21} +4.55611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.45510 q^{29} +9.01121 q^{31} -1.00000 q^{33} -1.00000 q^{35} +12.1234 q^{37} -5.01121 q^{39} +3.89899 q^{41} -9.56732 q^{43} -1.00000 q^{45} -1.01121 q^{47} +1.00000 q^{49} +1.44389 q^{51} +12.5785 q^{53} +1.00000 q^{55} -2.55611 q^{57} +5.56732 q^{59} +11.4663 q^{61} +1.00000 q^{63} +5.01121 q^{65} +5.11222 q^{67} +4.55611 q^{69} -3.89899 q^{71} -6.02242 q^{73} +1.00000 q^{75} -1.00000 q^{77} +0.988791 q^{79} +1.00000 q^{81} +1.44389 q^{83} -1.44389 q^{85} -8.45510 q^{87} +2.45510 q^{89} -5.01121 q^{91} +9.01121 q^{93} +2.55611 q^{95} +5.54490 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{15} + 8 q^{17} - 4 q^{19} + 3 q^{21} + 10 q^{23} + 3 q^{25} + 3 q^{27} - 10 q^{29} + 8 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} + 4 q^{39} - 6 q^{43} - 3 q^{45} + 16 q^{47} + 3 q^{49} + 8 q^{51} - 4 q^{53} + 3 q^{55} - 4 q^{57} - 6 q^{59} + 3 q^{63} - 4 q^{65} + 8 q^{67} + 10 q^{69} + 20 q^{73} + 3 q^{75} - 3 q^{77} + 22 q^{79} + 3 q^{81} + 8 q^{83} - 8 q^{85} - 10 q^{87} - 8 q^{89} + 4 q^{91} + 8 q^{93} + 4 q^{95} + 32 q^{97} - 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 - 3 * q^15 + 8 * q^17 - 4 * q^19 + 3 * q^21 + 10 * q^23 + 3 * q^25 + 3 * q^27 - 10 * q^29 + 8 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 + 4 * q^39 - 6 * q^43 - 3 * q^45 + 16 * q^47 + 3 * q^49 + 8 * q^51 - 4 * q^53 + 3 * q^55 - 4 * q^57 - 6 * q^59 + 3 * q^63 - 4 * q^65 + 8 * q^67 + 10 * q^69 + 20 * q^73 + 3 * q^75 - 3 * q^77 + 22 * q^79 + 3 * q^81 + 8 * q^83 - 8 * q^85 - 10 * q^87 - 8 * q^89 + 4 * q^91 + 8 * q^93 + 4 * q^95 + 32 * q^97 - 3 * 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
$13$	−5.01121	−1.38986	−0.694930	−	0.719078i	$-0.744565\pi$
$13$	−5.01121	−1.38986	−0.694930	+	0.719078i	$0.744565\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.44389	0.350195	0.175098	−	0.984551i	$-0.443976\pi$
$17$	1.44389	0.350195	0.175098	+	0.984551i	$0.443976\pi$
$18$	0	0
$19$	−2.55611	−0.586411	−0.293206	−	0.956049i	$-0.594722\pi$
$19$	−2.55611	−0.586411	−0.293206	+	0.956049i	$0.594722\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	4.55611	0.950014	0.475007	−	0.879982i	$-0.342445\pi$
$23$	4.55611	0.950014	0.475007	+	0.879982i	$0.342445\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.45510	−1.57007	−0.785036	−	0.619450i	$-0.787356\pi$
$29$	−8.45510	−1.57007	−0.785036	+	0.619450i	$0.787356\pi$
$30$	0	0
$31$	9.01121	1.61846	0.809230	−	0.587491i	$-0.199884\pi$
$31$	9.01121	1.61846	0.809230	+	0.587491i	$0.199884\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	12.1234	1.99308	0.996539	−	0.0831209i	$-0.0264888\pi$
$37$	12.1234	1.99308	0.996539	+	0.0831209i	$0.0264888\pi$
$38$	0	0
$39$	−5.01121	−0.802436
$40$	0	0
$41$	3.89899	0.608920	0.304460	−	0.952525i	$-0.401524\pi$
$41$	3.89899	0.608920	0.304460	+	0.952525i	$0.401524\pi$
$42$	0	0
$43$	−9.56732	−1.45900	−0.729501	−	0.683980i	$-0.760248\pi$
$43$	−9.56732	−1.45900	−0.729501	+	0.683980i	$0.760248\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−1.01121	−0.147500	−0.0737500	−	0.997277i	$-0.523497\pi$
$47$	−1.01121	−0.147500	−0.0737500	+	0.997277i	$0.523497\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.44389	0.202185
$52$	0	0
$53$	12.5785	1.72779	0.863897	−	0.503669i	$-0.168017\pi$
$53$	12.5785	1.72779	0.863897	+	0.503669i	$0.168017\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.55611	−0.338565
$58$	0	0
$59$	5.56732	0.724803	0.362402	−	0.932022i	$-0.381957\pi$
$59$	5.56732	0.724803	0.362402	+	0.932022i	$0.381957\pi$
$60$	0	0
$61$	11.4663	1.46811	0.734055	−	0.679090i	$-0.237625\pi$
$61$	11.4663	1.46811	0.734055	+	0.679090i	$0.237625\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	5.01121	0.621564
$66$	0	0
$67$	5.11222	0.624557	0.312278	−	0.949991i	$-0.398908\pi$
$67$	5.11222	0.624557	0.312278	+	0.949991i	$0.398908\pi$
$68$	0	0
$69$	4.55611	0.548491
$70$	0	0
$71$	−3.89899	−0.462725	−0.231363	−	0.972868i	$-0.574318\pi$
$71$	−3.89899	−0.462725	−0.231363	+	0.972868i	$0.574318\pi$
$72$	0	0
$73$	−6.02242	−0.704871	−0.352435	−	0.935836i	$-0.614646\pi$
$73$	−6.02242	−0.704871	−0.352435	+	0.935836i	$0.614646\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	0.988791	0.111248	0.0556238	−	0.998452i	$-0.482285\pi$
$79$	0.988791	0.111248	0.0556238	+	0.998452i	$0.482285\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.44389	0.158488	0.0792438	−	0.996855i	$-0.474749\pi$
$83$	1.44389	0.158488	0.0792438	+	0.996855i	$0.474749\pi$
$84$	0	0
$85$	−1.44389	−0.156612
$86$	0	0
$87$	−8.45510	−0.906482
$88$	0	0
$89$	2.45510	0.260240	0.130120	−	0.991498i	$-0.458464\pi$
$89$	2.45510	0.260240	0.130120	+	0.991498i	$0.458464\pi$
$90$	0	0
$91$	−5.01121	−0.525317
$92$	0	0
$93$	9.01121	0.934419
$94$	0	0
$95$	2.55611	0.262251
$96$	0	0
$97$	5.54490	0.562999	0.281500	−	0.959561i	$-0.409168\pi$
$97$	5.54490	0.562999	0.281500	+	0.959561i	$0.409168\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−2.88778	−0.287345	−0.143673	−	0.989625i	$-0.545891\pi$
$101$	−2.88778	−0.287345	−0.143673	+	0.989625i	$0.545891\pi$
$102$	0	0
$103$	7.34288	0.723516	0.361758	−	0.932272i	$-0.382177\pi$
$103$	7.34288	0.723516	0.361758	+	0.932272i	$0.382177\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	9.89899	0.956972	0.478486	−	0.878095i	$-0.341186\pi$
$107$	9.89899	0.956972	0.478486	+	0.878095i	$0.341186\pi$
$108$	0	0
$109$	12.1234	1.16121	0.580607	−	0.814184i	$-0.302815\pi$
$109$	12.1234	1.16121	0.580607	+	0.814184i	$0.302815\pi$
$110$	0	0
$111$	12.1234	1.15070
$112$	0	0
$113$	15.4663	1.45495	0.727474	−	0.686135i	$-0.240694\pi$
$113$	15.4663	1.45495	0.727474	+	0.686135i	$0.240694\pi$
$114$	0	0
$115$	−4.55611	−0.424859
$116$	0	0
$117$	−5.01121	−0.463286
$118$	0	0
$119$	1.44389	0.132361
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.89899	0.351560
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	13.5673	1.20390	0.601952	−	0.798532i	$-0.294390\pi$
$127$	13.5673	1.20390	0.601952	+	0.798532i	$0.294390\pi$
$128$	0	0
$129$	−9.56732	−0.842355
$130$	0	0
$131$	−13.0112	−1.13679	−0.568397	−	0.822754i	$-0.692436\pi$
$131$	−13.0112	−1.13679	−0.568397	+	0.822754i	$0.692436\pi$
$132$	0	0
$133$	−2.55611	−0.221643
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−3.11222	−0.265895	−0.132947	−	0.991123i	$-0.542444\pi$
$137$	−3.11222	−0.265895	−0.132947	+	0.991123i	$0.542444\pi$
$138$	0	0
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−1.01121	−0.0851592
$142$	0	0
$143$	5.01121	0.419058
$144$	0	0
$145$	8.45510	0.702158
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−11.1346	−0.912185	−0.456092	−	0.889932i	$-0.650751\pi$
$149$	−11.1346	−0.912185	−0.456092	+	0.889932i	$0.650751\pi$
$150$	0	0
$151$	−4.22443	−0.343780	−0.171890	−	0.985116i	$-0.554987\pi$
$151$	−4.22443	−0.343780	−0.171890	+	0.985116i	$0.554987\pi$
$152$	0	0
$153$	1.44389	0.116732
$154$	0	0
$155$	−9.01121	−0.723798
$156$	0	0
$157$	−17.5897	−1.40381	−0.701907	−	0.712269i	$-0.747668\pi$
$157$	−17.5897	−1.40381	−0.701907	+	0.712269i	$0.747668\pi$
$158$	0	0
$159$	12.5785	0.997542
$160$	0	0
$161$	4.55611	0.359072
$162$	0	0
$163$	−19.0336	−1.49083	−0.745414	−	0.666601i	$-0.767748\pi$
$163$	−19.0336	−1.49083	−0.745414	+	0.666601i	$0.767748\pi$
$164$	0	0
$165$	1.00000	0.0778499
$166$	0	0
$167$	−22.0448	−1.70588	−0.852940	−	0.522008i	$-0.825183\pi$
$167$	−22.0448	−1.70588	−0.852940	+	0.522008i	$0.825183\pi$
$168$	0	0
$169$	12.1122	0.931709
$170$	0	0
$171$	−2.55611	−0.195470
$172$	0	0
$173$	−24.9326	−1.89559	−0.947796	−	0.318877i	$-0.896694\pi$
$173$	−24.9326	−1.89559	−0.947796	+	0.318877i	$0.896694\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	5.56732	0.418465
$178$	0	0
$179$	17.9214	1.33951	0.669755	−	0.742583i	$-0.266399\pi$
$179$	17.9214	1.33951	0.669755	+	0.742583i	$0.266399\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	11.4663	0.847614
$184$	0	0
$185$	−12.1234	−0.891332
$186$	0	0
$187$	−1.44389	−0.105588
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	19.1346	1.37734	0.688671	−	0.725074i	$-0.258195\pi$
$193$	19.1346	1.37734	0.688671	+	0.725074i	$0.258195\pi$
$194$	0	0
$195$	5.01121	0.358860
$196$	0	0
$197$	−17.1346	−1.22079	−0.610396	−	0.792096i	$-0.708990\pi$
$197$	−17.1346	−1.22079	−0.610396	+	0.792096i	$0.708990\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	5.11222	0.360588
$202$	0	0
$203$	−8.45510	−0.593432
$204$	0	0
$205$	−3.89899	−0.272317
$206$	0	0
$207$	4.55611	0.316671
$208$	0	0
$209$	2.55611	0.176810
$210$	0	0
$211$	22.2469	1.53154	0.765768	−	0.643117i	$-0.222359\pi$
$211$	22.2469	1.53154	0.765768	+	0.643117i	$0.222359\pi$
$212$	0	0
$213$	−3.89899	−0.267155
$214$	0	0
$215$	9.56732	0.652486
$216$	0	0
$217$	9.01121	0.611721
$218$	0	0
$219$	−6.02242	−0.406957
$220$	0	0
$221$	−7.23564	−0.486722
$222$	0	0
$223$	10.6795	0.715155	0.357577	−	0.933884i	$-0.383603\pi$
$223$	10.6795	0.715155	0.357577	+	0.933884i	$0.383603\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.578527	0.0383982	0.0191991	−	0.999816i	$-0.493888\pi$
$227$	0.578527	0.0383982	0.0191991	+	0.999816i	$0.493888\pi$
$228$	0	0
$229$	5.89899	0.389816	0.194908	−	0.980821i	$-0.437559\pi$
$229$	5.89899	0.389816	0.194908	+	0.980821i	$0.437559\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	13.2356	0.867096	0.433548	−	0.901131i	$-0.357262\pi$
$233$	13.2356	0.867096	0.433548	+	0.901131i	$0.357262\pi$
$234$	0	0
$235$	1.01121	0.0659640
$236$	0	0
$237$	0.988791	0.0642289
$238$	0	0
$239$	2.67953	0.173325	0.0866623	−	0.996238i	$-0.472380\pi$
$239$	2.67953	0.173325	0.0866623	+	0.996238i	$0.472380\pi$
$240$	0	0
$241$	18.9102	1.21811	0.609057	−	0.793127i	$-0.291548\pi$
$241$	18.9102	1.21811	0.609057	+	0.793127i	$0.291548\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	12.8092	0.815029
$248$	0	0
$249$	1.44389	0.0915029
$250$	0	0
$251$	−16.0448	−1.01274	−0.506371	−	0.862316i	$-0.669013\pi$
$251$	−16.0448	−1.01274	−0.506371	+	0.862316i	$0.669013\pi$
$252$	0	0
$253$	−4.55611	−0.286440
$254$	0	0
$255$	−1.44389	−0.0904200
$256$	0	0
$257$	21.0336	1.31204	0.656021	−	0.754743i	$-0.272238\pi$
$257$	21.0336	1.31204	0.656021	+	0.754743i	$0.272238\pi$
$258$	0	0
$259$	12.1234	0.753313
$260$	0	0
$261$	−8.45510	−0.523358
$262$	0	0
$263$	−22.9102	−1.41270	−0.706352	−	0.707861i	$-0.749660\pi$
$263$	−22.9102	−1.41270	−0.706352	+	0.707861i	$0.749660\pi$
$264$	0	0
$265$	−12.5785	−0.772693
$266$	0	0
$267$	2.45510	0.150250
$268$	0	0
$269$	−26.4551	−1.61300	−0.806498	−	0.591237i	$-0.798640\pi$
$269$	−26.4551	−1.61300	−0.806498	+	0.591237i	$0.798640\pi$
$270$	0	0
$271$	15.6683	0.951783	0.475891	−	0.879504i	$-0.342125\pi$
$271$	15.6683	0.951783	0.475891	+	0.879504i	$0.342125\pi$
$272$	0	0
$273$	−5.01121	−0.303292
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	5.97758	0.359158	0.179579	−	0.983744i	$-0.442526\pi$
$277$	5.97758	0.359158	0.179579	+	0.983744i	$0.442526\pi$
$278$	0	0
$279$	9.01121	0.539487
$280$	0	0
$281$	26.0224	1.55237	0.776184	−	0.630507i	$-0.217153\pi$
$281$	26.0224	1.55237	0.776184	+	0.630507i	$0.217153\pi$
$282$	0	0
$283$	−2.98879	−0.177665	−0.0888326	−	0.996047i	$-0.528314\pi$
$283$	−2.98879	−0.177665	−0.0888326	+	0.996047i	$0.528314\pi$
$284$	0	0
$285$	2.55611	0.151411
$286$	0	0
$287$	3.89899	0.230150
$288$	0	0
$289$	−14.9152	−0.877363
$290$	0	0
$291$	5.54490	0.325048
$292$	0	0
$293$	4.33167	0.253059	0.126530	−	0.991963i	$-0.459616\pi$
$293$	4.33167	0.253059	0.126530	+	0.991963i	$0.459616\pi$
$294$	0	0
$295$	−5.56732	−0.324142
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−22.8316	−1.32039
$300$	0	0
$301$	−9.56732	−0.551451
$302$	0	0
$303$	−2.88778	−0.165899
$304$	0	0
$305$	−11.4663	−0.656559
$306$	0	0
$307$	12.9102	0.736824	0.368412	−	0.929663i	$-0.379902\pi$
$307$	12.9102	0.736824	0.368412	+	0.929663i	$0.379902\pi$
$308$	0	0
$309$	7.34288	0.417722
$310$	0	0
$311$	−7.79798	−0.442183	−0.221092	−	0.975253i	$-0.570962\pi$
$311$	−7.79798	−0.442183	−0.221092	+	0.975253i	$0.570962\pi$
$312$	0	0
$313$	13.7918	0.779556	0.389778	−	0.920909i	$-0.372552\pi$
$313$	13.7918	0.779556	0.389778	+	0.920909i	$0.372552\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	17.1346	0.962377	0.481188	−	0.876617i	$-0.340205\pi$
$317$	17.1346	0.962377	0.481188	+	0.876617i	$0.340205\pi$
$318$	0	0
$319$	8.45510	0.473395
$320$	0	0
$321$	9.89899	0.552508
$322$	0	0
$323$	−3.69074	−0.205358
$324$	0	0
$325$	−5.01121	−0.277972
$326$	0	0
$327$	12.1234	0.670427
$328$	0	0
$329$	−1.01121	−0.0557498
$330$	0	0
$331$	−8.55611	−0.470286	−0.235143	−	0.971961i	$-0.575556\pi$
$331$	−8.55611	−0.470286	−0.235143	+	0.971961i	$0.575556\pi$
$332$	0	0
$333$	12.1234	0.664360
$334$	0	0
$335$	−5.11222	−0.279310
$336$	0	0
$337$	21.3653	1.16384	0.581921	−	0.813245i	$-0.302301\pi$
$337$	21.3653	1.16384	0.581921	+	0.813245i	$0.302301\pi$
$338$	0	0
$339$	15.4663	0.840015
$340$	0	0
$341$	−9.01121	−0.487984
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−4.55611	−0.245293
$346$	0	0
$347$	22.9102	1.22988	0.614942	−	0.788572i	$-0.289179\pi$
$347$	22.9102	1.22988	0.614942	+	0.788572i	$0.289179\pi$
$348$	0	0
$349$	−3.42147	−0.183147	−0.0915736	−	0.995798i	$-0.529190\pi$
$349$	−3.42147	−0.183147	−0.0915736	+	0.995798i	$0.529190\pi$
$350$	0	0
$351$	−5.01121	−0.267479
$352$	0	0
$353$	−3.97758	−0.211705	−0.105853	−	0.994382i	$-0.533757\pi$
$353$	−3.97758	−0.211705	−0.105853	+	0.994382i	$0.533757\pi$
$354$	0	0
$355$	3.89899	0.206937
$356$	0	0
$357$	1.44389	0.0764188
$358$	0	0
$359$	13.3653	0.705394	0.352697	−	0.935738i	$-0.385265\pi$
$359$	13.3653	0.705394	0.352697	+	0.935738i	$0.385265\pi$
$360$	0	0
$361$	−12.4663	−0.656122
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	6.02242	0.315228
$366$	0	0
$367$	−9.56732	−0.499410	−0.249705	−	0.968322i	$-0.580334\pi$
$367$	−9.56732	−0.499410	−0.249705	+	0.968322i	$0.580334\pi$
$368$	0	0
$369$	3.89899	0.202973
$370$	0	0
$371$	12.5785	0.653045
$372$	0	0
$373$	−7.54490	−0.390660	−0.195330	−	0.980738i	$-0.562578\pi$
$373$	−7.54490	−0.390660	−0.195330	+	0.980738i	$0.562578\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	42.3703	2.18218
$378$	0	0
$379$	−15.4887	−0.795603	−0.397801	−	0.917472i	$-0.630227\pi$
$379$	−15.4887	−0.795603	−0.397801	+	0.917472i	$0.630227\pi$
$380$	0	0
$381$	13.5673	0.695075
$382$	0	0
$383$	−8.10101	−0.413942	−0.206971	−	0.978347i	$-0.566361\pi$
$383$	−8.10101	−0.413942	−0.206971	+	0.978347i	$0.566361\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	0	0
$387$	−9.56732	−0.486334
$388$	0	0
$389$	−9.89899	−0.501899	−0.250949	−	0.968000i	$-0.580743\pi$
$389$	−9.89899	−0.501899	−0.250949	+	0.968000i	$0.580743\pi$
$390$	0	0
$391$	6.57853	0.332690
$392$	0	0
$393$	−13.0112	−0.656329
$394$	0	0
$395$	−0.988791	−0.0497515
$396$	0	0
$397$	−14.9102	−0.748322	−0.374161	−	0.927364i	$-0.622069\pi$
$397$	−14.9102	−0.748322	−0.374161	+	0.927364i	$0.622069\pi$
$398$	0	0
$399$	−2.55611	−0.127965
$400$	0	0
$401$	13.2356	0.660956	0.330478	−	0.943814i	$-0.392790\pi$
$401$	13.2356	0.660956	0.330478	+	0.943814i	$0.392790\pi$
$402$	0	0
$403$	−45.1571	−2.24943
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−12.1234	−0.600936
$408$	0	0
$409$	6.91020	0.341687	0.170844	−	0.985298i	$-0.445351\pi$
$409$	6.91020	0.341687	0.170844	+	0.985298i	$0.445351\pi$
$410$	0	0
$411$	−3.11222	−0.153514
$412$	0	0
$413$	5.56732	0.273950
$414$	0	0
$415$	−1.44389	−0.0708779
$416$	0	0
$417$	14.0000	0.685583
$418$	0	0
$419$	37.3653	1.82541	0.912707	−	0.408614i	$-0.133988\pi$
$419$	37.3653	1.82541	0.912707	+	0.408614i	$0.133988\pi$
$420$	0	0
$421$	5.24188	0.255473	0.127737	−	0.991808i	$-0.459229\pi$
$421$	5.24188	0.255473	0.127737	+	0.991808i	$0.459229\pi$
$422$	0	0
$423$	−1.01121	−0.0491667
$424$	0	0
$425$	1.44389	0.0700390
$426$	0	0
$427$	11.4663	0.554894
$428$	0	0
$429$	5.01121	0.241943
$430$	0	0
$431$	19.7980	0.953635	0.476818	−	0.879002i	$-0.341790\pi$
$431$	19.7980	0.953635	0.476818	+	0.879002i	$0.341790\pi$
$432$	0	0
$433$	−7.97758	−0.383378	−0.191689	−	0.981456i	$-0.561397\pi$
$433$	−7.97758	−0.383378	−0.191689	+	0.981456i	$0.561397\pi$
$434$	0	0
$435$	8.45510	0.405391
$436$	0	0
$437$	−11.6459	−0.557099
$438$	0	0
$439$	−16.5785	−0.791250	−0.395625	−	0.918412i	$-0.629472\pi$
$439$	−16.5785	−0.791250	−0.395625	+	0.918412i	$0.629472\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−14.8878	−0.707340	−0.353670	−	0.935370i	$-0.615066\pi$
$443$	−14.8878	−0.707340	−0.353670	+	0.935370i	$0.615066\pi$
$444$	0	0
$445$	−2.45510	−0.116383
$446$	0	0
$447$	−11.1346	−0.526650
$448$	0	0
$449$	−28.8316	−1.36065	−0.680324	−	0.732912i	$-0.738161\pi$
$449$	−28.8316	−1.36065	−0.680324	+	0.732912i	$0.738161\pi$
$450$	0	0
$451$	−3.89899	−0.183596
$452$	0	0
$453$	−4.22443	−0.198481
$454$	0	0
$455$	5.01121	0.234929
$456$	0	0
$457$	23.5449	1.10138	0.550692	−	0.834709i	$-0.314364\pi$
$457$	23.5449	1.10138	0.550692	+	0.834709i	$0.314364\pi$
$458$	0	0
$459$	1.44389	0.0673951
$460$	0	0
$461$	11.1346	0.518592	0.259296	−	0.965798i	$-0.416510\pi$
$461$	11.1346	0.518592	0.259296	+	0.965798i	$0.416510\pi$
$462$	0	0
$463$	−8.91020	−0.414092	−0.207046	−	0.978331i	$-0.566385\pi$
$463$	−8.91020	−0.414092	−0.207046	+	0.978331i	$0.566385\pi$
$464$	0	0
$465$	−9.01121	−0.417885
$466$	0	0
$467$	4.20202	0.194446	0.0972230	−	0.995263i	$-0.469004\pi$
$467$	4.20202	0.194446	0.0972230	+	0.995263i	$0.469004\pi$
$468$	0	0
$469$	5.11222	0.236060
$470$	0	0
$471$	−17.5897	−0.810492
$472$	0	0
$473$	9.56732	0.439906
$474$	0	0
$475$	−2.55611	−0.117282
$476$	0	0
$477$	12.5785	0.575931
$478$	0	0
$479$	−13.0112	−0.594497	−0.297249	−	0.954800i	$-0.596069\pi$
$479$	−13.0112	−0.594497	−0.297249	+	0.954800i	$0.596069\pi$
$480$	0	0
$481$	−60.7530	−2.77010
$482$	0	0
$483$	4.55611	0.207310
$484$	0	0
$485$	−5.54490	−0.251781
$486$	0	0
$487$	−36.9550	−1.67459	−0.837296	−	0.546749i	$-0.815865\pi$
$487$	−36.9550	−1.67459	−0.837296	+	0.546749i	$0.815865\pi$
$488$	0	0
$489$	−19.0336	−0.860730
$490$	0	0
$491$	5.11845	0.230992	0.115496	−	0.993308i	$-0.463154\pi$
$491$	5.11845	0.230992	0.115496	+	0.993308i	$0.463154\pi$
$492$	0	0
$493$	−12.2082	−0.549832
$494$	0	0
$495$	1.00000	0.0449467
$496$	0	0
$497$	−3.89899	−0.174894
$498$	0	0
$499$	29.9152	1.33919	0.669594	−	0.742728i	$-0.266468\pi$
$499$	29.9152	1.33919	0.669594	+	0.742728i	$0.266468\pi$
$500$	0	0
$501$	−22.0448	−0.984891
$502$	0	0
$503$	−22.5561	−1.00573	−0.502864	−	0.864366i	$-0.667720\pi$
$503$	−22.5561	−1.00573	−0.502864	+	0.864366i	$0.667720\pi$
$504$	0	0
$505$	2.88778	0.128505
$506$	0	0
$507$	12.1122	0.537922
$508$	0	0
$509$	−22.7020	−1.00625	−0.503123	−	0.864215i	$-0.667816\pi$
$509$	−22.7020	−1.00625	−0.503123	+	0.864215i	$0.667816\pi$
$510$	0	0
$511$	−6.02242	−0.266416
$512$	0	0
$513$	−2.55611	−0.112855
$514$	0	0
$515$	−7.34288	−0.323566
$516$	0	0
$517$	1.01121	0.0444729
$518$	0	0
$519$	−24.9326	−1.09442
$520$	0	0
$521$	−29.7918	−1.30520	−0.652600	−	0.757702i	$-0.726322\pi$
$521$	−29.7918	−1.30520	−0.652600	+	0.757702i	$0.726322\pi$
$522$	0	0
$523$	−6.02242	−0.263342	−0.131671	−	0.991293i	$-0.542034\pi$
$523$	−6.02242	−0.263342	−0.131671	+	0.991293i	$0.542034\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	13.0112	0.566777
$528$	0	0
$529$	−2.24188	−0.0974729
$530$	0	0
$531$	5.56732	0.241601
$532$	0	0
$533$	−19.5387	−0.846314
$534$	0	0
$535$	−9.89899	−0.427971
$536$	0	0
$537$	17.9214	0.773366
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−28.3703	−1.21973	−0.609867	−	0.792504i	$-0.708777\pi$
$541$	−28.3703	−1.21973	−0.609867	+	0.792504i	$0.708777\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−12.1234	−0.519311
$546$	0	0
$547$	44.9488	1.92187	0.960936	−	0.276771i	$-0.0892642\pi$
$547$	44.9488	1.92187	0.960936	+	0.276771i	$0.0892642\pi$
$548$	0	0
$549$	11.4663	0.489370
$550$	0	0
$551$	21.6122	0.920709
$552$	0	0
$553$	0.988791	0.0420477
$554$	0	0
$555$	−12.1234	−0.514611
$556$	0	0
$557$	−25.7980	−1.09310	−0.546548	−	0.837428i	$-0.684058\pi$
$557$	−25.7980	−1.09310	−0.546548	+	0.837428i	$0.684058\pi$
$558$	0	0
$559$	47.9438	2.02781
$560$	0	0
$561$	−1.44389	−0.0609612
$562$	0	0
$563$	−10.2020	−0.429964	−0.214982	−	0.976618i	$-0.568969\pi$
$563$	−10.2020	−0.429964	−0.214982	+	0.976618i	$0.568969\pi$
$564$	0	0
$565$	−15.4663	−0.650673
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	4.70195	0.197116	0.0985581	−	0.995131i	$-0.468577\pi$
$569$	4.70195	0.197116	0.0985581	+	0.995131i	$0.468577\pi$
$570$	0	0
$571$	37.2805	1.56014	0.780070	−	0.625693i	$-0.215184\pi$
$571$	37.2805	1.56014	0.780070	+	0.625693i	$0.215184\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.55611	0.190003
$576$	0	0
$577$	18.0448	0.751216	0.375608	−	0.926779i	$-0.377434\pi$
$577$	18.0448	0.751216	0.375608	+	0.926779i	$0.377434\pi$
$578$	0	0
$579$	19.1346	0.795208
$580$	0	0
$581$	1.44389	0.0599027
$582$	0	0
$583$	−12.5785	−0.520949
$584$	0	0
$585$	5.01121	0.207188
$586$	0	0
$587$	−14.1683	−0.584787	−0.292393	−	0.956298i	$-0.594452\pi$
$587$	−14.1683	−0.584787	−0.292393	+	0.956298i	$0.594452\pi$
$588$	0	0
$589$	−23.0336	−0.949084
$590$	0	0
$591$	−17.1346	−0.704825
$592$	0	0
$593$	−41.1346	−1.68920	−0.844599	−	0.535400i	$-0.820161\pi$
$593$	−41.1346	−1.68920	−0.844599	+	0.535400i	$0.820161\pi$
$594$	0	0
$595$	−1.44389	−0.0591938
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	16.9102	0.690932	0.345466	−	0.938431i	$-0.387721\pi$
$599$	16.9102	0.690932	0.345466	+	0.938431i	$0.387721\pi$
$600$	0	0
$601$	12.7805	0.521329	0.260665	−	0.965429i	$-0.416058\pi$
$601$	12.7805	0.521329	0.260665	+	0.965429i	$0.416058\pi$
$602$	0	0
$603$	5.11222	0.208186
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	46.7306	1.89674	0.948368	−	0.317171i	$-0.102733\pi$
$607$	46.7306	1.89674	0.948368	+	0.317171i	$0.102733\pi$
$608$	0	0
$609$	−8.45510	−0.342618
$610$	0	0
$611$	5.06738	0.205004
$612$	0	0
$613$	−25.1122	−1.01427	−0.507136	−	0.861866i	$-0.669296\pi$
$613$	−25.1122	−1.01427	−0.507136	+	0.861866i	$0.669296\pi$
$614$	0	0
$615$	−3.89899	−0.157223
$616$	0	0
$617$	−32.7306	−1.31768	−0.658842	−	0.752281i	$-0.728954\pi$
$617$	−32.7306	−1.31768	−0.658842	+	0.752281i	$0.728954\pi$
$618$	0	0
$619$	14.9326	0.600193	0.300096	−	0.953909i	$-0.402981\pi$
$619$	14.9326	0.600193	0.300096	+	0.953909i	$0.402981\pi$
$620$	0	0
$621$	4.55611	0.182830
$622$	0	0
$623$	2.45510	0.0983615
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.55611	0.102081
$628$	0	0
$629$	17.5049	0.697967
$630$	0	0
$631$	−15.4887	−0.616597	−0.308298	−	0.951290i	$-0.599759\pi$
$631$	−15.4887	−0.616597	−0.308298	+	0.951290i	$0.599759\pi$
$632$	0	0
$633$	22.2469	0.884233
$634$	0	0
$635$	−13.5673	−0.538403
$636$	0	0
$637$	−5.01121	−0.198551
$638$	0	0
$639$	−3.89899	−0.154242
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	28.4551	1.12216	0.561080	−	0.827762i	$-0.310386\pi$
$643$	28.4551	1.12216	0.561080	+	0.827762i	$0.310386\pi$
$644$	0	0
$645$	9.56732	0.376713
$646$	0	0
$647$	14.0224	0.551278	0.275639	−	0.961261i	$-0.411111\pi$
$647$	14.0224	0.551278	0.275639	+	0.961261i	$0.411111\pi$
$648$	0	0
$649$	−5.56732	−0.218536
$650$	0	0
$651$	9.01121	0.353177
$652$	0	0
$653$	−30.8030	−1.20541	−0.602706	−	0.797963i	$-0.705911\pi$
$653$	−30.8030	−1.20541	−0.602706	+	0.797963i	$0.705911\pi$
$654$	0	0
$655$	13.0112	0.508390
$656$	0	0
$657$	−6.02242	−0.234957
$658$	0	0
$659$	35.6346	1.38813	0.694063	−	0.719914i	$-0.255819\pi$
$659$	35.6346	1.38813	0.694063	+	0.719914i	$0.255819\pi$
$660$	0	0
$661$	−0.325441	−0.0126582	−0.00632910	−	0.999980i	$-0.502015\pi$
$661$	−0.325441	−0.0126582	−0.00632910	+	0.999980i	$0.502015\pi$
$662$	0	0
$663$	−7.23564	−0.281009
$664$	0	0
$665$	2.55611	0.0991216
$666$	0	0
$667$	−38.5224	−1.49159
$668$	0	0
$669$	10.6795	0.412895
$670$	0	0
$671$	−11.4663	−0.442652
$672$	0	0
$673$	1.56732	0.0604157	0.0302078	−	0.999544i	$-0.490383\pi$
$673$	1.56732	0.0604157	0.0302078	+	0.999544i	$0.490383\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−4.33167	−0.166480	−0.0832399	−	0.996530i	$-0.526527\pi$
$677$	−4.33167	−0.166480	−0.0832399	+	0.996530i	$0.526527\pi$
$678$	0	0
$679$	5.54490	0.212794
$680$	0	0
$681$	0.578527	0.0221692
$682$	0	0
$683$	−5.77557	−0.220996	−0.110498	−	0.993876i	$-0.535245\pi$
$683$	−5.77557	−0.220996	−0.110498	+	0.993876i	$0.535245\pi$
$684$	0	0
$685$	3.11222	0.118912
$686$	0	0
$687$	5.89899	0.225061
$688$	0	0
$689$	−63.0336	−2.40139
$690$	0	0
$691$	50.9326	1.93757	0.968784	−	0.247906i	$-0.0797424\pi$
$691$	50.9326	1.93757	0.968784	+	0.247906i	$0.0797424\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	5.62972	0.213241
$698$	0	0
$699$	13.2356	0.500618
$700$	0	0
$701$	−31.5897	−1.19313	−0.596564	−	0.802566i	$-0.703468\pi$
$701$	−31.5897	−1.19313	−0.596564	+	0.802566i	$0.703468\pi$
$702$	0	0
$703$	−30.9888	−1.16876
$704$	0	0
$705$	1.01121	0.0380843
$706$	0	0
$707$	−2.88778	−0.108606
$708$	0	0
$709$	48.6234	1.82609	0.913044	−	0.407860i	$-0.133725\pi$
$709$	48.6234	1.82609	0.913044	+	0.407860i	$0.133725\pi$
$710$	0	0
$711$	0.988791	0.0370826
$712$	0	0
$713$	41.0560	1.53756
$714$	0	0
$715$	−5.01121	−0.187409
$716$	0	0
$717$	2.67953	0.100069
$718$	0	0
$719$	5.11845	0.190886	0.0954430	−	0.995435i	$-0.469573\pi$
$719$	5.11845	0.190886	0.0954430	+	0.995435i	$0.469573\pi$
$720$	0	0
$721$	7.34288	0.273463
$722$	0	0
$723$	18.9102	0.703278
$724$	0	0
$725$	−8.45510	−0.314015
$726$	0	0
$727$	−30.5224	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$727$	−30.5224	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.8142	−0.510936
$732$	0	0
$733$	12.9102	0.476849	0.238425	−	0.971161i	$-0.423369\pi$
$733$	12.9102	0.476849	0.238425	+	0.971161i	$0.423369\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−5.11222	−0.188311
$738$	0	0
$739$	9.79798	0.360425	0.180212	−	0.983628i	$-0.442322\pi$
$739$	9.79798	0.360425	0.180212	+	0.983628i	$0.442322\pi$
$740$	0	0
$741$	12.8092	0.470557
$742$	0	0
$743$	−1.79798	−0.0659617	−0.0329808	−	0.999456i	$-0.510500\pi$
$743$	−1.79798	−0.0659617	−0.0329808	+	0.999456i	$0.510500\pi$
$744$	0	0
$745$	11.1346	0.407941
$746$	0	0
$747$	1.44389	0.0528292
$748$	0	0
$749$	9.89899	0.361701
$750$	0	0
$751$	31.4887	1.14904	0.574520	−	0.818491i	$-0.305189\pi$
$751$	31.4887	1.14904	0.574520	+	0.818491i	$0.305189\pi$
$752$	0	0
$753$	−16.0448	−0.584706
$754$	0	0
$755$	4.22443	0.153743
$756$	0	0
$757$	32.0673	1.16550	0.582752	−	0.812650i	$-0.301976\pi$
$757$	32.0673	1.16550	0.582752	+	0.812650i	$0.301976\pi$
$758$	0	0
$759$	−4.55611	−0.165376
$760$	0	0
$761$	−33.1122	−1.20032	−0.600158	−	0.799881i	$-0.704896\pi$
$761$	−33.1122	−1.20032	−0.600158	+	0.799881i	$0.704896\pi$
$762$	0	0
$763$	12.1234	0.438897
$764$	0	0
$765$	−1.44389	−0.0522040
$766$	0	0
$767$	−27.8990	−1.00737
$768$	0	0
$769$	−25.6907	−0.926432	−0.463216	−	0.886245i	$-0.653305\pi$
$769$	−25.6907	−0.926432	−0.463216	+	0.886245i	$0.653305\pi$
$770$	0	0
$771$	21.0336	0.757508
$772$	0	0
$773$	−37.7980	−1.35950	−0.679750	−	0.733444i	$-0.737912\pi$
$773$	−37.7980	−1.35950	−0.679750	+	0.733444i	$0.737912\pi$
$774$	0	0
$775$	9.01121	0.323692
$776$	0	0
$777$	12.1234	0.434925
$778$	0	0
$779$	−9.96625	−0.357078
$780$	0	0
$781$	3.89899	0.139517
$782$	0	0
$783$	−8.45510	−0.302161
$784$	0	0
$785$	17.5897	0.627805
$786$	0	0
$787$	−34.2244	−1.21997	−0.609985	−	0.792413i	$-0.708824\pi$
$787$	−34.2244	−1.21997	−0.609985	+	0.792413i	$0.708824\pi$
$788$	0	0
$789$	−22.9102	−0.815625
$790$	0	0
$791$	15.4663	0.549919
$792$	0	0
$793$	−57.4601	−2.04047
$794$	0	0
$795$	−12.5785	−0.446114
$796$	0	0
$797$	−4.42645	−0.156793	−0.0783964	−	0.996922i	$-0.524980\pi$
$797$	−4.42645	−0.156793	−0.0783964	+	0.996922i	$0.524980\pi$
$798$	0	0
$799$	−1.46008	−0.0516538
$800$	0	0
$801$	2.45510	0.0867467
$802$	0	0
$803$	6.02242	0.212527
$804$	0	0
$805$	−4.55611	−0.160582
$806$	0	0
$807$	−26.4551	−0.931264
$808$	0	0
$809$	−35.5835	−1.25105	−0.625525	−	0.780204i	$-0.715115\pi$
$809$	−35.5835	−1.25105	−0.625525	+	0.780204i	$0.715115\pi$
$810$	0	0
$811$	−23.1571	−0.813154	−0.406577	−	0.913616i	$-0.633278\pi$
$811$	−23.1571	−0.813154	−0.406577	+	0.913616i	$0.633278\pi$
$812$	0	0
$813$	15.6683	0.549512
$814$	0	0
$815$	19.0336	0.666719
$816$	0	0
$817$	24.4551	0.855576
$818$	0	0
$819$	−5.01121	−0.175106
$820$	0	0
$821$	−6.43268	−0.224502	−0.112251	−	0.993680i	$-0.535806\pi$
$821$	−6.43268	−0.224502	−0.112251	+	0.993680i	$0.535806\pi$
$822$	0	0
$823$	−17.4601	−0.608620	−0.304310	−	0.952573i	$-0.598426\pi$
$823$	−17.4601	−0.608620	−0.304310	+	0.952573i	$0.598426\pi$
$824$	0	0
$825$	−1.00000	−0.0348155
$826$	0	0
$827$	17.1346	0.595830	0.297915	−	0.954592i	$-0.403709\pi$
$827$	17.1346	0.595830	0.297915	+	0.954592i	$0.403709\pi$
$828$	0	0
$829$	−12.0224	−0.417556	−0.208778	−	0.977963i	$-0.566949\pi$
$829$	−12.0224	−0.417556	−0.208778	+	0.977963i	$0.566949\pi$
$830$	0	0
$831$	5.97758	0.207360
$832$	0	0
$833$	1.44389	0.0500279
$834$	0	0
$835$	22.0448	0.762893
$836$	0	0
$837$	9.01121	0.311473
$838$	0	0
$839$	42.2755	1.45951	0.729756	−	0.683707i	$-0.239634\pi$
$839$	42.2755	1.45951	0.729756	+	0.683707i	$0.239634\pi$
$840$	0	0
$841$	42.4887	1.46513
$842$	0	0
$843$	26.0224	0.896260
$844$	0	0
$845$	−12.1122	−0.416673
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−2.98879	−0.102575
$850$	0	0
$851$	55.2356	1.89345
$852$	0	0
$853$	−32.3479	−1.10757	−0.553785	−	0.832660i	$-0.686817\pi$
$853$	−32.3479	−1.10757	−0.553785	+	0.832660i	$0.686817\pi$
$854$	0	0
$855$	2.55611	0.0874171
$856$	0	0
$857$	−55.8652	−1.90832	−0.954160	−	0.299297i	$-0.903248\pi$
$857$	−55.8652	−1.90832	−0.954160	+	0.299297i	$0.903248\pi$
$858$	0	0
$859$	−49.1009	−1.67530	−0.837650	−	0.546207i	$-0.816071\pi$
$859$	−49.1009	−1.67530	−0.837650	+	0.546207i	$0.816071\pi$
$860$	0	0
$861$	3.89899	0.132877
$862$	0	0
$863$	21.9152	0.746001	0.373001	−	0.927831i	$-0.378329\pi$
$863$	21.9152	0.746001	0.373001	+	0.927831i	$0.378329\pi$
$864$	0	0
$865$	24.9326	0.847735
$866$	0	0
$867$	−14.9152	−0.506546
$868$	0	0
$869$	−0.988791	−0.0335424
$870$	0	0
$871$	−25.6184	−0.868046
$872$	0	0
$873$	5.54490	0.187666
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	32.9488	1.11260	0.556301	−	0.830981i	$-0.312220\pi$
$877$	32.9488	1.11260	0.556301	+	0.830981i	$0.312220\pi$
$878$	0	0
$879$	4.33167	0.146104
$880$	0	0
$881$	28.9264	0.974555	0.487277	−	0.873247i	$-0.337990\pi$
$881$	28.9264	0.974555	0.487277	+	0.873247i	$0.337990\pi$
$882$	0	0
$883$	−17.5736	−0.591397	−0.295699	−	0.955281i	$-0.595552\pi$
$883$	−17.5736	−0.591397	−0.295699	+	0.955281i	$0.595552\pi$
$884$	0	0
$885$	−5.56732	−0.187143
$886$	0	0
$887$	42.8030	1.43718	0.718591	−	0.695433i	$-0.244787\pi$
$887$	42.8030	1.43718	0.718591	+	0.695433i	$0.244787\pi$
$888$	0	0
$889$	13.5673	0.455033
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	2.58476	0.0864957
$894$	0	0
$895$	−17.9214	−0.599047
$896$	0	0
$897$	−22.8316	−0.762325
$898$	0	0
$899$	−76.1907	−2.54110
$900$	0	0
$901$	18.1620	0.605065
$902$	0	0
$903$	−9.56732	−0.318380
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−14.9888	−0.497695	−0.248847	−	0.968543i	$-0.580052\pi$
$907$	−14.9888	−0.497695	−0.248847	+	0.968543i	$0.580052\pi$
$908$	0	0
$909$	−2.88778	−0.0957817
$910$	0	0
$911$	50.0224	1.65732	0.828658	−	0.559755i	$-0.189105\pi$
$911$	50.0224	1.65732	0.828658	+	0.559755i	$0.189105\pi$
$912$	0	0
$913$	−1.44389	−0.0477858
$914$	0	0
$915$	−11.4663	−0.379065
$916$	0	0
$917$	−13.0112	−0.429668
$918$	0	0
$919$	−19.8204	−0.653815	−0.326907	−	0.945056i	$-0.606007\pi$
$919$	−19.8204	−0.653815	−0.326907	+	0.945056i	$0.606007\pi$
$920$	0	0
$921$	12.9102	0.425406
$922$	0	0
$923$	19.5387	0.643123
$924$	0	0
$925$	12.1234	0.398616
$926$	0	0
$927$	7.34288	0.241172
$928$	0	0
$929$	17.1346	0.562169	0.281085	−	0.959683i	$-0.409306\pi$
$929$	17.1346	0.562169	0.281085	+	0.959683i	$0.409306\pi$
$930$	0	0
$931$	−2.55611	−0.0837731
$932$	0	0
$933$	−7.79798	−0.255295
$934$	0	0
$935$	1.44389	0.0472203
$936$	0	0
$937$	−32.1907	−1.05162	−0.525812	−	0.850601i	$-0.676238\pi$
$937$	−32.1907	−1.05162	−0.525812	+	0.850601i	$0.676238\pi$
$938$	0	0
$939$	13.7918	0.450077
$940$	0	0
$941$	41.5049	1.35302	0.676511	−	0.736433i	$-0.263491\pi$
$941$	41.5049	1.35302	0.676511	+	0.736433i	$0.263491\pi$
$942$	0	0
$943$	17.7642	0.578483
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−33.9152	−1.10210	−0.551048	−	0.834474i	$-0.685772\pi$
$947$	−33.9152	−1.10210	−0.551048	+	0.834474i	$0.685772\pi$
$948$	0	0
$949$	30.1796	0.979671
$950$	0	0
$951$	17.1346	0.555629
$952$	0	0
$953$	12.2244	0.395988	0.197994	−	0.980203i	$-0.436557\pi$
$953$	12.2244	0.395988	0.197994	+	0.980203i	$0.436557\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.45510	0.273315
$958$	0	0
$959$	−3.11222	−0.100499
$960$	0	0
$961$	50.2019	1.61942
$962$	0	0
$963$	9.89899	0.318991
$964$	0	0
$965$	−19.1346	−0.615966
$966$	0	0
$967$	−43.8366	−1.40969	−0.704845	−	0.709362i	$-0.748983\pi$
$967$	−43.8366	−1.40969	−0.704845	+	0.709362i	$0.748983\pi$
$968$	0	0
$969$	−3.69074	−0.118564
$970$	0	0
$971$	−23.7918	−0.763514	−0.381757	−	0.924263i	$-0.624681\pi$
$971$	−23.7918	−0.763514	−0.381757	+	0.924263i	$0.624681\pi$
$972$	0	0
$973$	14.0000	0.448819
$974$	0	0
$975$	−5.01121	−0.160487
$976$	0	0
$977$	40.6234	1.29966	0.649828	−	0.760081i	$-0.274841\pi$
$977$	40.6234	1.29966	0.649828	+	0.760081i	$0.274841\pi$
$978$	0	0
$979$	−2.45510	−0.0784654
$980$	0	0
$981$	12.1234	0.387071
$982$	0	0
$983$	−10.6858	−0.340823	−0.170412	−	0.985373i	$-0.554510\pi$
$983$	−10.6858	−0.340823	−0.170412	+	0.985373i	$0.554510\pi$
$984$	0	0
$985$	17.1346	0.545955
$986$	0	0
$987$	−1.01121	−0.0321871
$988$	0	0
$989$	−43.5897	−1.38607
$990$	0	0
$991$	24.8478	0.789316	0.394658	−	0.918828i	$-0.370863\pi$
$991$	24.8478	0.789316	0.394658	+	0.918828i	$0.370863\pi$
$992$	0	0
$993$	−8.55611	−0.271520
$994$	0	0
$995$	−8.00000	−0.253617
$996$	0	0
$997$	−19.3367	−0.612398	−0.306199	−	0.951968i	$-0.599057\pi$
$997$	−19.3367	−0.612398	−0.306199	+	0.951968i	$0.599057\pi$
$998$	0	0
$999$	12.1234	0.383568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.a.v.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.a.v.1.1	✓	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -5.01121 q^{13} -1.00000 q^{15} +1.44389 q^{17} -2.55611 q^{19} +1.00000 q^{21} +4.55611 q^{23} +1.00000 q^{25} +1.00000 q^{27} -8.45510 q^{29} +9.01121 q^{31} -1.00000 q^{33} -1.00000 q^{35} +12.1234 q^{37} -5.01121 q^{39} +3.89899 q^{41} -9.56732 q^{43} -1.00000 q^{45} -1.01121 q^{47} +1.00000 q^{49} +1.44389 q^{51} +12.5785 q^{53} +1.00000 q^{55} -2.55611 q^{57} +5.56732 q^{59} +11.4663 q^{61} +1.00000 q^{63} +5.01121 q^{65} +5.11222 q^{67} +4.55611 q^{69} -3.89899 q^{71} -6.02242 q^{73} +1.00000 q^{75} -1.00000 q^{77} +0.988791 q^{79} +1.00000 q^{81} +1.44389 q^{83} -1.44389 q^{85} -8.45510 q^{87} +2.45510 q^{89} -5.01121 q^{91} +9.01121 q^{93} +2.55611 q^{95} +5.54490 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} - 3 q^{15} + 8 q^{17} - 4 q^{19} + 3 q^{21} + 10 q^{23} + 3 q^{25} + 3 q^{27} - 10 q^{29} + 8 q^{31} - 3 q^{33} - 3 q^{35} + 10 q^{37} + 4 q^{39} - 6 q^{43} - 3 q^{45} + 16 q^{47} + 3 q^{49} + 8 q^{51} - 4 q^{53} + 3 q^{55} - 4 q^{57} - 6 q^{59} + 3 q^{63} - 4 q^{65} + 8 q^{67} + 10 q^{69} + 20 q^{73} + 3 q^{75} - 3 q^{77} + 22 q^{79} + 3 q^{81} + 8 q^{83} - 8 q^{85} - 10 q^{87} - 8 q^{89} + 4 q^{91} + 8 q^{93} + 4 q^{95} + 32 q^{97} - 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 - 3 * q^15 + 8 * q^17 - 4 * q^19 + 3 * q^21 + 10 * q^23 + 3 * q^25 + 3 * q^27 - 10 * q^29 + 8 * q^31 - 3 * q^33 - 3 * q^35 + 10 * q^37 + 4 * q^39 - 6 * q^43 - 3 * q^45 + 16 * q^47 + 3 * q^49 + 8 * q^51 - 4 * q^53 + 3 * q^55 - 4 * q^57 - 6 * q^59 + 3 * q^63 - 4 * q^65 + 8 * q^67 + 10 * q^69 + 20 * q^73 + 3 * q^75 - 3 * q^77 + 22 * q^79 + 3 * q^81 + 8 * q^83 - 8 * q^85 - 10 * q^87 - 8 * q^89 + 4 * q^91 + 8 * q^93 + 4 * q^95 + 32 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more