LMFDB - Embedded newform 456.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,6,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("456.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	456.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:	$73.1350218347$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 $ x^6 - 2x^5 - 4725x^4 + 92430x^3 + 1610577x^2 - 16081740*x - 24661341
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-74.1084$ of defining polynomial
Character	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -86.8767 q^{5} +2.94760 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -86.8767 q^{5} +2.94760 q^{7} +81.0000 q^{9} +318.804 q^{11} -638.130 q^{13} +781.890 q^{15} +219.917 q^{17} +361.000 q^{19} -26.5284 q^{21} +3471.66 q^{23} +4422.56 q^{25} -729.000 q^{27} -7206.90 q^{29} +5620.43 q^{31} -2869.24 q^{33} -256.078 q^{35} +13965.2 q^{37} +5743.17 q^{39} +14877.4 q^{41} +1988.69 q^{43} -7037.01 q^{45} +14219.0 q^{47} -16798.3 q^{49} -1979.26 q^{51} -16286.1 q^{53} -27696.7 q^{55} -3249.00 q^{57} -26626.3 q^{59} -11066.0 q^{61} +238.756 q^{63} +55438.6 q^{65} -39426.5 q^{67} -31244.9 q^{69} +52126.0 q^{71} -73711.7 q^{73} -39803.1 q^{75} +939.708 q^{77} -8701.62 q^{79} +6561.00 q^{81} -53001.4 q^{83} -19105.7 q^{85} +64862.1 q^{87} -125964. q^{89} -1880.95 q^{91} -50583.9 q^{93} -31362.5 q^{95} -73604.0 q^{97} +25823.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	$ 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) $ 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * 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
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−86.8767	−1.55410	−0.777049	−	0.629440i	$-0.783284\pi$
$5$	−86.8767	−1.55410	−0.777049	+	0.629440i	$0.783284\pi$
$6$	0	0
$7$	2.94760	0.0227365	0.0113682	−	0.999935i	$-0.496381\pi$
$7$	2.94760	0.0227365	0.0113682	+	0.999935i	$0.496381\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	318.804	0.794406	0.397203	−	0.917731i	$-0.369981\pi$
$11$	318.804	0.794406	0.397203	+	0.917731i	$0.369981\pi$
$12$	0	0
$13$	−638.130	−1.04725	−0.523625	−	0.851949i	$-0.675421\pi$
$13$	−638.130	−1.04725	−0.523625	+	0.851949i	$0.675421\pi$
$14$	0	0
$15$	781.890	0.897259
$16$	0	0
$17$	219.917	0.184560	0.0922799	−	0.995733i	$-0.470585\pi$
$17$	219.917	0.184560	0.0922799	+	0.995733i	$0.470585\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−26.5284	−0.0131269
$22$	0	0
$23$	3471.66	1.36841	0.684207	−	0.729288i	$-0.260148\pi$
$23$	3471.66	1.36841	0.684207	+	0.729288i	$0.260148\pi$
$24$	0	0
$25$	4422.56	1.41522
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−7206.90	−1.59130	−0.795652	−	0.605753i	$-0.792872\pi$
$29$	−7206.90	−1.59130	−0.795652	+	0.605753i	$0.792872\pi$
$30$	0	0
$31$	5620.43	1.05043	0.525213	−	0.850971i	$-0.323986\pi$
$31$	5620.43	1.05043	0.525213	+	0.850971i	$0.323986\pi$
$32$	0	0
$33$	−2869.24	−0.458651
$34$	0	0
$35$	−256.078	−0.0353347
$36$	0	0
$37$	13965.2	1.67704	0.838518	−	0.544874i	$-0.183422\pi$
$37$	13965.2	1.67704	0.838518	+	0.544874i	$0.183422\pi$
$38$	0	0
$39$	5743.17	0.604630
$40$	0	0
$41$	14877.4	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$41$	14877.4	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$42$	0	0
$43$	1988.69	0.164020	0.0820098	−	0.996632i	$-0.473866\pi$
$43$	1988.69	0.164020	0.0820098	+	0.996632i	$0.473866\pi$
$44$	0	0
$45$	−7037.01	−0.518033
$46$	0	0
$47$	14219.0	0.938909	0.469455	−	0.882957i	$-0.344451\pi$
$47$	14219.0	0.938909	0.469455	+	0.882957i	$0.344451\pi$
$48$	0	0
$49$	−16798.3	−0.999483
$50$	0	0
$51$	−1979.26	−0.106556
$52$	0	0
$53$	−16286.1	−0.796395	−0.398197	−	0.917300i	$-0.630364\pi$
$53$	−16286.1	−0.796395	−0.398197	+	0.917300i	$0.630364\pi$
$54$	0	0
$55$	−27696.7	−1.23459
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−26626.3	−0.995819	−0.497910	−	0.867229i	$-0.665899\pi$
$59$	−26626.3	−0.995819	−0.497910	+	0.867229i	$0.665899\pi$
$60$	0	0
$61$	−11066.0	−0.380774	−0.190387	−	0.981709i	$-0.560974\pi$
$61$	−11066.0	−0.380774	−0.190387	+	0.981709i	$0.560974\pi$
$62$	0	0
$63$	238.756	0.00757883
$64$	0	0
$65$	55438.6	1.62753
$66$	0	0
$67$	−39426.5	−1.07300	−0.536502	−	0.843899i	$-0.680254\pi$
$67$	−39426.5	−1.07300	−0.536502	+	0.843899i	$0.680254\pi$
$68$	0	0
$69$	−31244.9	−0.790054
$70$	0	0
$71$	52126.0	1.22718	0.613590	−	0.789625i	$-0.289725\pi$
$71$	52126.0	1.22718	0.613590	+	0.789625i	$0.289725\pi$
$72$	0	0
$73$	−73711.7	−1.61893	−0.809467	−	0.587165i	$-0.800244\pi$
$73$	−73711.7	−1.61893	−0.809467	+	0.587165i	$0.800244\pi$
$74$	0	0
$75$	−39803.1	−0.817078
$76$	0	0
$77$	939.708	0.0180620
$78$	0	0
$79$	−8701.62	−0.156867	−0.0784336	−	0.996919i	$-0.524992\pi$
$79$	−8701.62	−0.156867	−0.0784336	+	0.996919i	$0.524992\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−53001.4	−0.844485	−0.422242	−	0.906483i	$-0.638757\pi$
$83$	−53001.4	−0.844485	−0.422242	+	0.906483i	$0.638757\pi$
$84$	0	0
$85$	−19105.7	−0.286824
$86$	0	0
$87$	64862.1	0.918740
$88$	0	0
$89$	−125964.	−1.68567	−0.842833	−	0.538175i	$-0.819114\pi$
$89$	−125964.	−1.68567	−0.842833	+	0.538175i	$0.819114\pi$
$90$	0	0
$91$	−1880.95	−0.0238108
$92$	0	0
$93$	−50583.9	−0.606464
$94$	0	0
$95$	−31362.5	−0.356535
$96$	0	0
$97$	−73604.0	−0.794277	−0.397139	−	0.917759i	$-0.629997\pi$
$97$	−73604.0	−0.794277	−0.397139	+	0.917759i	$0.629997\pi$
$98$	0	0
$99$	25823.2	0.264802
$100$	0	0
$101$	127732.	1.24594	0.622968	−	0.782247i	$-0.285927\pi$
$101$	127732.	1.24594	0.622968	+	0.782247i	$0.285927\pi$
$102$	0	0
$103$	−127852.	−1.18745	−0.593723	−	0.804669i	$-0.702343\pi$
$103$	−127852.	−1.18745	−0.593723	+	0.804669i	$0.702343\pi$
$104$	0	0
$105$	2304.70	0.0204005
$106$	0	0
$107$	95172.5	0.803623	0.401811	−	0.915722i	$-0.368381\pi$
$107$	95172.5	0.803623	0.401811	+	0.915722i	$0.368381\pi$
$108$	0	0
$109$	235632.	1.89962	0.949812	−	0.312820i	$-0.101274\pi$
$109$	235632.	1.89962	0.949812	+	0.312820i	$0.101274\pi$
$110$	0	0
$111$	−125687.	−0.968237
$112$	0	0
$113$	256972.	1.89317	0.946584	−	0.322457i	$-0.104509\pi$
$113$	256972.	1.89317	0.946584	+	0.322457i	$0.104509\pi$
$114$	0	0
$115$	−301606.	−2.12665
$116$	0	0
$117$	−51688.5	−0.349083
$118$	0	0
$119$	648.229	0.00419624
$120$	0	0
$121$	−59414.7	−0.368919
$122$	0	0
$123$	−133897.	−0.798007
$124$	0	0
$125$	−112728.	−0.645293
$126$	0	0
$127$	−57230.7	−0.314862	−0.157431	−	0.987530i	$-0.550321\pi$
$127$	−57230.7	−0.314862	−0.157431	+	0.987530i	$0.550321\pi$
$128$	0	0
$129$	−17898.2	−0.0946967
$130$	0	0
$131$	5007.18	0.0254927	0.0127463	−	0.999919i	$-0.495943\pi$
$131$	5007.18	0.0254927	0.0127463	+	0.999919i	$0.495943\pi$
$132$	0	0
$133$	1064.08	0.00521611
$134$	0	0
$135$	63333.1	0.299086
$136$	0	0
$137$	−402860.	−1.83380	−0.916901	−	0.399115i	$-0.869317\pi$
$137$	−402860.	−1.83380	−0.916901	+	0.399115i	$0.869317\pi$
$138$	0	0
$139$	−395813.	−1.73761	−0.868807	−	0.495150i	$-0.835113\pi$
$139$	−395813.	−1.73761	−0.868807	+	0.495150i	$0.835113\pi$
$140$	0	0
$141$	−127971.	−0.542080
$142$	0	0
$143$	−203439.	−0.831942
$144$	0	0
$145$	626112.	2.47304
$146$	0	0
$147$	151185.	0.577052
$148$	0	0
$149$	−86000.9	−0.317349	−0.158675	−	0.987331i	$-0.550722\pi$
$149$	−86000.9	−0.317349	−0.158675	+	0.987331i	$0.550722\pi$
$150$	0	0
$151$	−257590.	−0.919362	−0.459681	−	0.888084i	$-0.652036\pi$
$151$	−257590.	−0.919362	−0.459681	+	0.888084i	$0.652036\pi$
$152$	0	0
$153$	17813.3	0.0615199
$154$	0	0
$155$	−488285.	−1.63246
$156$	0	0
$157$	80428.3	0.260411	0.130206	−	0.991487i	$-0.458436\pi$
$157$	80428.3	0.260411	0.130206	+	0.991487i	$0.458436\pi$
$158$	0	0
$159$	146575.	0.459799
$160$	0	0
$161$	10233.1	0.0311129
$162$	0	0
$163$	−482818.	−1.42336	−0.711680	−	0.702504i	$-0.752065\pi$
$163$	−482818.	−1.42336	−0.711680	+	0.702504i	$0.752065\pi$
$164$	0	0
$165$	249270.	0.712788
$166$	0	0
$167$	219499.	0.609035	0.304517	−	0.952507i	$-0.401505\pi$
$167$	219499.	0.609035	0.304517	+	0.952507i	$0.401505\pi$
$168$	0	0
$169$	35916.4	0.0967332
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	361299.	0.917807	0.458904	−	0.888486i	$-0.348242\pi$
$173$	361299.	0.917807	0.458904	+	0.888486i	$0.348242\pi$
$174$	0	0
$175$	13036.0	0.0321772
$176$	0	0
$177$	239637.	0.574937
$178$	0	0
$179$	−622262.	−1.45158	−0.725790	−	0.687916i	$-0.758526\pi$
$179$	−622262.	−1.45158	−0.725790	+	0.687916i	$0.758526\pi$
$180$	0	0
$181$	−104136.	−0.236267	−0.118134	−	0.992998i	$-0.537691\pi$
$181$	−104136.	−0.236267	−0.118134	+	0.992998i	$0.537691\pi$
$182$	0	0
$183$	99594.3	0.219840
$184$	0	0
$185$	−1.21325e6	−2.60628
$186$	0	0
$187$	70110.6	0.146615
$188$	0	0
$189$	−2148.80	−0.00437564
$190$	0	0
$191$	183238.	0.363439	0.181719	−	0.983350i	$-0.441834\pi$
$191$	183238.	0.363439	0.181719	+	0.983350i	$0.441834\pi$
$192$	0	0
$193$	−96854.3	−0.187165	−0.0935827	−	0.995612i	$-0.529832\pi$
$193$	−96854.3	−0.187165	−0.0935827	+	0.995612i	$0.529832\pi$
$194$	0	0
$195$	−498947.	−0.939655
$196$	0	0
$197$	−275049.	−0.504945	−0.252472	−	0.967604i	$-0.581244\pi$
$197$	−275049.	−0.504945	−0.252472	+	0.967604i	$0.581244\pi$
$198$	0	0
$199$	−256443.	−0.459047	−0.229524	−	0.973303i	$-0.573717\pi$
$199$	−256443.	−0.459047	−0.229524	+	0.973303i	$0.573717\pi$
$200$	0	0
$201$	354838.	0.619499
$202$	0	0
$203$	−21243.1	−0.0361807
$204$	0	0
$205$	−1.29250e6	−2.14806
$206$	0	0
$207$	281204.	0.456138
$208$	0	0
$209$	115088.	0.182249
$210$	0	0
$211$	150209.	0.232268	0.116134	−	0.993234i	$-0.462950\pi$
$211$	150209.	0.232268	0.116134	+	0.993234i	$0.462950\pi$
$212$	0	0
$213$	−469134.	−0.708513
$214$	0	0
$215$	−172771.	−0.254902
$216$	0	0
$217$	16566.8	0.0238830
$218$	0	0
$219$	663405.	0.934692
$220$	0	0
$221$	−140336.	−0.193280
$222$	0	0
$223$	1.00599e6	1.35466	0.677330	−	0.735679i	$-0.263137\pi$
$223$	1.00599e6	1.35466	0.677330	+	0.735679i	$0.263137\pi$
$224$	0	0
$225$	358228.	0.471740
$226$	0	0
$227$	666788.	0.858862	0.429431	−	0.903100i	$-0.358714\pi$
$227$	666788.	0.858862	0.429431	+	0.903100i	$0.358714\pi$
$228$	0	0
$229$	161657.	0.203708	0.101854	−	0.994799i	$-0.467523\pi$
$229$	161657.	0.203708	0.101854	+	0.994799i	$0.467523\pi$
$230$	0	0
$231$	−8457.37	−0.0104281
$232$	0	0
$233$	932958.	1.12583	0.562914	−	0.826515i	$-0.309680\pi$
$233$	932958.	1.12583	0.562914	+	0.826515i	$0.309680\pi$
$234$	0	0
$235$	−1.23530e6	−1.45916
$236$	0	0
$237$	78314.5	0.0905673
$238$	0	0
$239$	1.01934e6	1.15431	0.577156	−	0.816634i	$-0.304162\pi$
$239$	1.01934e6	1.15431	0.577156	+	0.816634i	$0.304162\pi$
$240$	0	0
$241$	−21598.0	−0.0239537	−0.0119768	−	0.999928i	$-0.503812\pi$
$241$	−21598.0	−0.0239537	−0.0119768	+	0.999928i	$0.503812\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	1.45938e6	1.55329
$246$	0	0
$247$	−230365.	−0.240256
$248$	0	0
$249$	477012.	0.487564
$250$	0	0
$251$	−1.00782e6	−1.00972	−0.504858	−	0.863202i	$-0.668455\pi$
$251$	−1.00782e6	−1.00972	−0.504858	+	0.863202i	$0.668455\pi$
$252$	0	0
$253$	1.10678e6	1.08708
$254$	0	0
$255$	171951.	0.165598
$256$	0	0
$257$	112989.	0.106709	0.0533547	−	0.998576i	$-0.483009\pi$
$257$	112989.	0.106709	0.0533547	+	0.998576i	$0.483009\pi$
$258$	0	0
$259$	41163.8	0.0381299
$260$	0	0
$261$	−583759.	−0.530435
$262$	0	0
$263$	−289482.	−0.258067	−0.129033	−	0.991640i	$-0.541187\pi$
$263$	−289482.	−0.258067	−0.129033	+	0.991640i	$0.541187\pi$
$264$	0	0
$265$	1.41489e6	1.23768
$266$	0	0
$267$	1.13368e6	0.973220
$268$	0	0
$269$	−885103.	−0.745784	−0.372892	−	0.927875i	$-0.621634\pi$
$269$	−885103.	−0.745784	−0.372892	+	0.927875i	$0.621634\pi$
$270$	0	0
$271$	−1.75726e6	−1.45349	−0.726746	−	0.686906i	$-0.758968\pi$
$271$	−1.75726e6	−1.45349	−0.726746	+	0.686906i	$0.758968\pi$
$272$	0	0
$273$	16928.6	0.0137472
$274$	0	0
$275$	1.40993e6	1.12426
$276$	0	0
$277$	365208.	0.285984	0.142992	−	0.989724i	$-0.454328\pi$
$277$	365208.	0.285984	0.142992	+	0.989724i	$0.454328\pi$
$278$	0	0
$279$	455255.	0.350142
$280$	0	0
$281$	1.54749e6	1.16913	0.584566	−	0.811346i	$-0.301265\pi$
$281$	1.54749e6	1.16913	0.584566	+	0.811346i	$0.301265\pi$
$282$	0	0
$283$	389877.	0.289375	0.144688	−	0.989477i	$-0.453782\pi$
$283$	389877.	0.289375	0.144688	+	0.989477i	$0.453782\pi$
$284$	0	0
$285$	282262.	0.205845
$286$	0	0
$287$	43852.7	0.0314262
$288$	0	0
$289$	−1.37149e6	−0.965938
$290$	0	0
$291$	662436.	0.458576
$292$	0	0
$293$	289148.	0.196766	0.0983831	−	0.995149i	$-0.468633\pi$
$293$	289148.	0.196766	0.0983831	+	0.995149i	$0.468633\pi$
$294$	0	0
$295$	2.31320e6	1.54760
$296$	0	0
$297$	−232408.	−0.152884
$298$	0	0
$299$	−2.21537e6	−1.43307
$300$	0	0
$301$	5861.86	0.00372923
$302$	0	0
$303$	−1.14959e6	−0.719342
$304$	0	0
$305$	961380.	0.591760
$306$	0	0
$307$	−3.12009e6	−1.88939	−0.944695	−	0.327950i	$-0.893642\pi$
$307$	−3.12009e6	−1.88939	−0.944695	+	0.327950i	$0.893642\pi$
$308$	0	0
$309$	1.15067e6	0.685573
$310$	0	0
$311$	555696.	0.325789	0.162895	−	0.986643i	$-0.447917\pi$
$311$	555696.	0.325789	0.162895	+	0.986643i	$0.447917\pi$
$312$	0	0
$313$	1.68130e6	0.970028	0.485014	−	0.874506i	$-0.338814\pi$
$313$	1.68130e6	0.970028	0.485014	+	0.874506i	$0.338814\pi$
$314$	0	0
$315$	−20742.3	−0.0117782
$316$	0	0
$317$	−1.70075e6	−0.950586	−0.475293	−	0.879828i	$-0.657658\pi$
$317$	−1.70075e6	−0.950586	−0.475293	+	0.879828i	$0.657658\pi$
$318$	0	0
$319$	−2.29759e6	−1.26414
$320$	0	0
$321$	−856553.	−0.463972
$322$	0	0
$323$	79390.2	0.0423409
$324$	0	0
$325$	−2.82217e6	−1.48209
$326$	0	0
$327$	−2.12069e6	−1.09675
$328$	0	0
$329$	41911.9	0.0213475
$330$	0	0
$331$	268046.	0.134475	0.0672373	−	0.997737i	$-0.478582\pi$
$331$	268046.	0.134475	0.0672373	+	0.997737i	$0.478582\pi$
$332$	0	0
$333$	1.13118e6	0.559012
$334$	0	0
$335$	3.42524e6	1.66755
$336$	0	0
$337$	52952.3	0.0253986	0.0126993	−	0.999919i	$-0.495958\pi$
$337$	52952.3	0.0253986	0.0126993	+	0.999919i	$0.495958\pi$
$338$	0	0
$339$	−2.31275e6	−1.09302
$340$	0	0
$341$	1.79182e6	0.834465
$342$	0	0
$343$	−99055.1	−0.0454612
$344$	0	0
$345$	2.71446e6	1.22782
$346$	0	0
$347$	2.74946e6	1.22581	0.612905	−	0.790157i	$-0.290001\pi$
$347$	2.74946e6	1.22581	0.612905	+	0.790157i	$0.290001\pi$
$348$	0	0
$349$	1.23589e6	0.543147	0.271573	−	0.962418i	$-0.412456\pi$
$349$	1.23589e6	0.543147	0.271573	+	0.962418i	$0.412456\pi$
$350$	0	0
$351$	465196.	0.201543
$352$	0	0
$353$	1.89335e6	0.808712	0.404356	−	0.914602i	$-0.367496\pi$
$353$	1.89335e6	0.808712	0.404356	+	0.914602i	$0.367496\pi$
$354$	0	0
$355$	−4.52854e6	−1.90716
$356$	0	0
$357$	−5834.06	−0.00242270
$358$	0	0
$359$	−2.28422e6	−0.935411	−0.467706	−	0.883884i	$-0.654919\pi$
$359$	−2.28422e6	−0.935411	−0.467706	+	0.883884i	$0.654919\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	534733.	0.212995
$364$	0	0
$365$	6.40383e6	2.51598
$366$	0	0
$367$	858143.	0.332579	0.166289	−	0.986077i	$-0.446821\pi$
$367$	858143.	0.332579	0.166289	+	0.986077i	$0.446821\pi$
$368$	0	0
$369$	1.20507e6	0.460730
$370$	0	0
$371$	−48005.0	−0.0181072
$372$	0	0
$373$	3.42136e6	1.27329	0.636644	−	0.771158i	$-0.280322\pi$
$373$	3.42136e6	1.27329	0.636644	+	0.771158i	$0.280322\pi$
$374$	0	0
$375$	1.01455e6	0.372560
$376$	0	0
$377$	4.59893e6	1.66649
$378$	0	0
$379$	−4.11049e6	−1.46993	−0.734963	−	0.678107i	$-0.762801\pi$
$379$	−4.11049e6	−1.46993	−0.734963	+	0.678107i	$0.762801\pi$
$380$	0	0
$381$	515076.	0.181785
$382$	0	0
$383$	287612.	0.100187	0.0500933	−	0.998745i	$-0.484048\pi$
$383$	287612.	0.100187	0.0500933	+	0.998745i	$0.484048\pi$
$384$	0	0
$385$	−81638.8	−0.0280701
$386$	0	0
$387$	161084.	0.0546732
$388$	0	0
$389$	−4.31602e6	−1.44614	−0.723068	−	0.690777i	$-0.757269\pi$
$389$	−4.31602e6	−1.44614	−0.723068	+	0.690777i	$0.757269\pi$
$390$	0	0
$391$	763478.	0.252554
$392$	0	0
$393$	−45064.6	−0.0147182
$394$	0	0
$395$	755968.	0.243787
$396$	0	0
$397$	−3.42142e6	−1.08951	−0.544754	−	0.838596i	$-0.683377\pi$
$397$	−3.42142e6	−1.08951	−0.544754	+	0.838596i	$0.683377\pi$
$398$	0	0
$399$	−9576.76	−0.00301152
$400$	0	0
$401$	4.54215e6	1.41059	0.705294	−	0.708914i	$-0.250815\pi$
$401$	4.54215e6	1.41059	0.705294	+	0.708914i	$0.250815\pi$
$402$	0	0
$403$	−3.58656e6	−1.10006
$404$	0	0
$405$	−569998.	−0.172678
$406$	0	0
$407$	4.45216e6	1.33225
$408$	0	0
$409$	−2.71102e6	−0.801354	−0.400677	−	0.916219i	$-0.631225\pi$
$409$	−2.71102e6	−0.801354	−0.400677	+	0.916219i	$0.631225\pi$
$410$	0	0
$411$	3.62574e6	1.05875
$412$	0	0
$413$	−78483.7	−0.0226414
$414$	0	0
$415$	4.60459e6	1.31241
$416$	0	0
$417$	3.56232e6	1.00321
$418$	0	0
$419$	−2.42371e6	−0.674442	−0.337221	−	0.941425i	$-0.609487\pi$
$419$	−2.42371e6	−0.674442	−0.337221	+	0.941425i	$0.609487\pi$
$420$	0	0
$421$	−3.06050e6	−0.841564	−0.420782	−	0.907162i	$-0.638244\pi$
$421$	−3.06050e6	−0.841564	−0.420782	+	0.907162i	$0.638244\pi$
$422$	0	0
$423$	1.15174e6	0.312970
$424$	0	0
$425$	972598.	0.261193
$426$	0	0
$427$	−32618.2	−0.00865747
$428$	0	0
$429$	1.83095e6	0.480322
$430$	0	0
$431$	−2.08114e6	−0.539645	−0.269823	−	0.962910i	$-0.586965\pi$
$431$	−2.08114e6	−0.539645	−0.269823	+	0.962910i	$0.586965\pi$
$432$	0	0
$433$	−4.21567e6	−1.08055	−0.540277	−	0.841487i	$-0.681681\pi$
$433$	−4.21567e6	−1.08055	−0.540277	+	0.841487i	$0.681681\pi$
$434$	0	0
$435$	−5.63500e6	−1.42781
$436$	0	0
$437$	1.25327e6	0.313936
$438$	0	0
$439$	2.24036e6	0.554826	0.277413	−	0.960751i	$-0.410523\pi$
$439$	2.24036e6	0.554826	0.277413	+	0.960751i	$0.410523\pi$
$440$	0	0
$441$	−1.36066e6	−0.333161
$442$	0	0
$443$	5.08323e6	1.23064	0.615319	−	0.788278i	$-0.289027\pi$
$443$	5.08323e6	1.23064	0.615319	+	0.788278i	$0.289027\pi$
$444$	0	0
$445$	1.09433e7	2.61969
$446$	0	0
$447$	774008.	0.183222
$448$	0	0
$449$	3.67478e6	0.860232	0.430116	−	0.902774i	$-0.358473\pi$
$449$	3.67478e6	0.860232	0.430116	+	0.902774i	$0.358473\pi$
$450$	0	0
$451$	4.74298e6	1.09802
$452$	0	0
$453$	2.31831e6	0.530794
$454$	0	0
$455$	163411.	0.0370043
$456$	0	0
$457$	−2.30561e6	−0.516412	−0.258206	−	0.966090i	$-0.583131\pi$
$457$	−2.30561e6	−0.516412	−0.258206	+	0.966090i	$0.583131\pi$
$458$	0	0
$459$	−160320.	−0.0355186
$460$	0	0
$461$	−4.00465e6	−0.877631	−0.438816	−	0.898577i	$-0.644602\pi$
$461$	−4.00465e6	−0.877631	−0.438816	+	0.898577i	$0.644602\pi$
$462$	0	0
$463$	−573291.	−0.124286	−0.0621430	−	0.998067i	$-0.519793\pi$
$463$	−573291.	−0.124286	−0.0621430	+	0.998067i	$0.519793\pi$
$464$	0	0
$465$	4.39456e6	0.942504
$466$	0	0
$467$	4.13715e6	0.877827	0.438914	−	0.898529i	$-0.355363\pi$
$467$	4.13715e6	0.877827	0.438914	+	0.898529i	$0.355363\pi$
$468$	0	0
$469$	−116214.	−0.0243963
$470$	0	0
$471$	−723855.	−0.150348
$472$	0	0
$473$	634003.	0.130298
$474$	0	0
$475$	1.59655e6	0.324674
$476$	0	0
$477$	−1.31918e6	−0.265465
$478$	0	0
$479$	−9.05846e6	−1.80391	−0.901956	−	0.431827i	$-0.857869\pi$
$479$	−9.05846e6	−1.80391	−0.901956	+	0.431827i	$0.857869\pi$
$480$	0	0
$481$	−8.91160e6	−1.75628
$482$	0	0
$483$	−92097.6	−0.0179631
$484$	0	0
$485$	6.39448e6	1.23439
$486$	0	0
$487$	−8.05132e6	−1.53831	−0.769157	−	0.639060i	$-0.779323\pi$
$487$	−8.05132e6	−1.53831	−0.769157	+	0.639060i	$0.779323\pi$
$488$	0	0
$489$	4.34536e6	0.821777
$490$	0	0
$491$	5.43572e6	1.01754	0.508772	−	0.860901i	$-0.330100\pi$
$491$	5.43572e6	1.01754	0.508772	+	0.860901i	$0.330100\pi$
$492$	0	0
$493$	−1.58492e6	−0.293691
$494$	0	0
$495$	−2.24343e6	−0.411528
$496$	0	0
$497$	153647.	0.0279018
$498$	0	0
$499$	9.75490e6	1.75377	0.876883	−	0.480704i	$-0.159619\pi$
$499$	9.75490e6	1.75377	0.876883	+	0.480704i	$0.159619\pi$
$500$	0	0
$501$	−1.97549e6	−0.351626
$502$	0	0
$503$	61342.4	0.0108104	0.00540519	−	0.999985i	$-0.498279\pi$
$503$	61342.4	0.0108104	0.00540519	+	0.999985i	$0.498279\pi$
$504$	0	0
$505$	−1.10969e7	−1.93631
$506$	0	0
$507$	−323247.	−0.0558490
$508$	0	0
$509$	−9.96677e6	−1.70514	−0.852570	−	0.522613i	$-0.824957\pi$
$509$	−9.96677e6	−1.70514	−0.852570	+	0.522613i	$0.824957\pi$
$510$	0	0
$511$	−217273.	−0.0368089
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	1.11074e7	1.84541
$516$	0	0
$517$	4.53307e6	0.745876
$518$	0	0
$519$	−3.25169e6	−0.529896
$520$	0	0
$521$	−9.39975e6	−1.51713	−0.758563	−	0.651599i	$-0.774098\pi$
$521$	−9.39975e6	−1.51713	−0.758563	+	0.651599i	$0.774098\pi$
$522$	0	0
$523$	−6.85510e6	−1.09587	−0.547936	−	0.836520i	$-0.684586\pi$
$523$	−6.85510e6	−1.09587	−0.547936	+	0.836520i	$0.684586\pi$
$524$	0	0
$525$	−117324.	−0.0185775
$526$	0	0
$527$	1.23603e6	0.193866
$528$	0	0
$529$	5.61608e6	0.872557
$530$	0	0
$531$	−2.15673e6	−0.331940
$532$	0	0
$533$	−9.49371e6	−1.44750
$534$	0	0
$535$	−8.26828e6	−1.24891
$536$	0	0
$537$	5.60036e6	0.838070
$538$	0	0
$539$	−5.35538e6	−0.793996
$540$	0	0
$541$	−5.56585e6	−0.817595	−0.408797	−	0.912625i	$-0.634052\pi$
$541$	−5.56585e6	−0.817595	−0.408797	+	0.912625i	$0.634052\pi$
$542$	0	0
$543$	937221.	0.136409
$544$	0	0
$545$	−2.04709e7	−2.95220
$546$	0	0
$547$	−1.05049e7	−1.50115	−0.750573	−	0.660787i	$-0.770223\pi$
$547$	−1.05049e7	−1.50115	−0.750573	+	0.660787i	$0.770223\pi$
$548$	0	0
$549$	−896349.	−0.126925
$550$	0	0
$551$	−2.60169e6	−0.365070
$552$	0	0
$553$	−25648.9	−0.00356661
$554$	0	0
$555$	1.09192e7	1.50474
$556$	0	0
$557$	−8.51679e6	−1.16316	−0.581578	−	0.813490i	$-0.697565\pi$
$557$	−8.51679e6	−1.16316	−0.581578	+	0.813490i	$0.697565\pi$
$558$	0	0
$559$	−1.26904e6	−0.171769
$560$	0	0
$561$	−630996.	−0.0846485
$562$	0	0
$563$	−8.65897e6	−1.15132	−0.575659	−	0.817690i	$-0.695254\pi$
$563$	−8.65897e6	−1.15132	−0.575659	+	0.817690i	$0.695254\pi$
$564$	0	0
$565$	−2.23249e7	−2.94217
$566$	0	0
$567$	19339.2	0.00252628
$568$	0	0
$569$	−5.60596e6	−0.725887	−0.362944	−	0.931811i	$-0.618228\pi$
$569$	−5.60596e6	−0.725887	−0.362944	+	0.931811i	$0.618228\pi$
$570$	0	0
$571$	7.13232e6	0.915463	0.457731	−	0.889090i	$-0.348662\pi$
$571$	7.13232e6	0.915463	0.457731	+	0.889090i	$0.348662\pi$
$572$	0	0
$573$	−1.64914e6	−0.209831
$574$	0	0
$575$	1.53536e7	1.93661
$576$	0	0
$577$	−3.42233e6	−0.427940	−0.213970	−	0.976840i	$-0.568639\pi$
$577$	−3.42233e6	−0.427940	−0.213970	+	0.976840i	$0.568639\pi$
$578$	0	0
$579$	871689.	0.108060
$580$	0	0
$581$	−156227.	−0.0192006
$582$	0	0
$583$	−5.19209e6	−0.632661
$584$	0	0
$585$	4.49053e6	0.542510
$586$	0	0
$587$	−1.09057e7	−1.30634	−0.653171	−	0.757210i	$-0.726562\pi$
$587$	−1.09057e7	−1.30634	−0.653171	+	0.757210i	$0.726562\pi$
$588$	0	0
$589$	2.02898e6	0.240984
$590$	0	0
$591$	2.47544e6	0.291530
$592$	0	0
$593$	−8.51651e6	−0.994546	−0.497273	−	0.867594i	$-0.665665\pi$
$593$	−8.51651e6	−0.994546	−0.497273	+	0.867594i	$0.665665\pi$
$594$	0	0
$595$	−56316.0	−0.00652137
$596$	0	0
$597$	2.30798e6	0.265031
$598$	0	0
$599$	−30851.7	−0.00351327	−0.00175664	−	0.999998i	$-0.500559\pi$
$599$	−30851.7	−0.00351327	−0.00175664	+	0.999998i	$0.500559\pi$
$600$	0	0
$601$	8.18341e6	0.924162	0.462081	−	0.886838i	$-0.347103\pi$
$601$	8.18341e6	0.924162	0.462081	+	0.886838i	$0.347103\pi$
$602$	0	0
$603$	−3.19355e6	−0.357668
$604$	0	0
$605$	5.16176e6	0.573336
$606$	0	0
$607$	4.22459e6	0.465385	0.232693	−	0.972550i	$-0.425246\pi$
$607$	4.22459e6	0.465385	0.232693	+	0.972550i	$0.425246\pi$
$608$	0	0
$609$	191188.	0.0208889
$610$	0	0
$611$	−9.07355e6	−0.983273
$612$	0	0
$613$	1.66199e6	0.178640	0.0893199	−	0.996003i	$-0.471531\pi$
$613$	1.66199e6	0.178640	0.0893199	+	0.996003i	$0.471531\pi$
$614$	0	0
$615$	1.16325e7	1.24018
$616$	0	0
$617$	6.20540e6	0.656231	0.328115	−	0.944638i	$-0.393587\pi$
$617$	6.20540e6	0.656231	0.328115	+	0.944638i	$0.393587\pi$
$618$	0	0
$619$	1.59871e7	1.67704	0.838519	−	0.544872i	$-0.183422\pi$
$619$	1.59871e7	1.67704	0.838519	+	0.544872i	$0.183422\pi$
$620$	0	0
$621$	−2.53084e6	−0.263351
$622$	0	0
$623$	−371292.	−0.0383261
$624$	0	0
$625$	−4.02706e6	−0.412371
$626$	0	0
$627$	−1.03580e6	−0.105222
$628$	0	0
$629$	3.07119e6	0.309513
$630$	0	0
$631$	−1.12004e7	−1.11985	−0.559923	−	0.828544i	$-0.689169\pi$
$631$	−1.12004e7	−1.11985	−0.559923	+	0.828544i	$0.689169\pi$
$632$	0	0
$633$	−1.35188e6	−0.134100
$634$	0	0
$635$	4.97201e6	0.489326
$636$	0	0
$637$	1.07195e7	1.04671
$638$	0	0
$639$	4.22221e6	0.409060
$640$	0	0
$641$	9.74956e6	0.937217	0.468608	−	0.883406i	$-0.344756\pi$
$641$	9.74956e6	0.937217	0.468608	+	0.883406i	$0.344756\pi$
$642$	0	0
$643$	−6.53602e6	−0.623427	−0.311714	−	0.950176i	$-0.600903\pi$
$643$	−6.53602e6	−0.623427	−0.311714	+	0.950176i	$0.600903\pi$
$644$	0	0
$645$	1.55494e6	0.147168
$646$	0	0
$647$	8.79430e6	0.825925	0.412962	−	0.910748i	$-0.364494\pi$
$647$	8.79430e6	0.825925	0.412962	+	0.910748i	$0.364494\pi$
$648$	0	0
$649$	−8.48858e6	−0.791085
$650$	0	0
$651$	−149101.	−0.0137889
$652$	0	0
$653$	−1.15593e7	−1.06084	−0.530418	−	0.847736i	$-0.677965\pi$
$653$	−1.15593e7	−1.06084	−0.530418	+	0.847736i	$0.677965\pi$
$654$	0	0
$655$	−435008.	−0.0396181
$656$	0	0
$657$	−5.97065e6	−0.539645
$658$	0	0
$659$	1.52916e7	1.37163	0.685817	−	0.727774i	$-0.259445\pi$
$659$	1.52916e7	1.37163	0.685817	+	0.727774i	$0.259445\pi$
$660$	0	0
$661$	−5.89824e6	−0.525072	−0.262536	−	0.964922i	$-0.584559\pi$
$661$	−5.89824e6	−0.525072	−0.262536	+	0.964922i	$0.584559\pi$
$662$	0	0
$663$	1.26302e6	0.111590
$664$	0	0
$665$	−92444.1	−0.00810635
$666$	0	0
$667$	−2.50199e7	−2.17756
$668$	0	0
$669$	−9.05389e6	−0.782114
$670$	0	0
$671$	−3.52790e6	−0.302489
$672$	0	0
$673$	−1.20024e7	−1.02148	−0.510740	−	0.859735i	$-0.670628\pi$
$673$	−1.20024e7	−1.02148	−0.510740	+	0.859735i	$0.670628\pi$
$674$	0	0
$675$	−3.22405e6	−0.272359
$676$	0	0
$677$	−1.65605e7	−1.38868	−0.694338	−	0.719649i	$-0.744303\pi$
$677$	−1.65605e7	−1.38868	−0.694338	+	0.719649i	$0.744303\pi$
$678$	0	0
$679$	−216955.	−0.0180591
$680$	0	0
$681$	−6.00109e6	−0.495864
$682$	0	0
$683$	401168.	0.0329060	0.0164530	−	0.999865i	$-0.494763\pi$
$683$	401168.	0.0329060	0.0164530	+	0.999865i	$0.494763\pi$
$684$	0	0
$685$	3.49991e7	2.84991
$686$	0	0
$687$	−1.45492e6	−0.117611
$688$	0	0
$689$	1.03927e7	0.834025
$690$	0	0
$691$	1.65388e6	0.131767	0.0658837	−	0.997827i	$-0.479013\pi$
$691$	1.65388e6	0.131767	0.0658837	+	0.997827i	$0.479013\pi$
$692$	0	0
$693$	76116.4	0.00602067
$694$	0	0
$695$	3.43870e7	2.70042
$696$	0	0
$697$	3.27180e6	0.255097
$698$	0	0
$699$	−8.39662e6	−0.649998
$700$	0	0
$701$	−2.31470e6	−0.177910	−0.0889549	−	0.996036i	$-0.528353\pi$
$701$	−2.31470e6	−0.177910	−0.0889549	+	0.996036i	$0.528353\pi$
$702$	0	0
$703$	5.04143e6	0.384739
$704$	0	0
$705$	1.11177e7	0.842445
$706$	0	0
$707$	376503.	0.0283282
$708$	0	0
$709$	167819.	0.0125379	0.00626897	−	0.999980i	$-0.498005\pi$
$709$	167819.	0.0125379	0.00626897	+	0.999980i	$0.498005\pi$
$710$	0	0
$711$	−704831.	−0.0522891
$712$	0	0
$713$	1.95122e7	1.43742
$714$	0	0
$715$	1.76741e7	1.29292
$716$	0	0
$717$	−9.17404e6	−0.666443
$718$	0	0
$719$	5.38526e6	0.388494	0.194247	−	0.980953i	$-0.437774\pi$
$719$	5.38526e6	0.388494	0.194247	+	0.980953i	$0.437774\pi$
$720$	0	0
$721$	−376856.	−0.0269984
$722$	0	0
$723$	194382.	0.0138297
$724$	0	0
$725$	−3.18730e7	−2.25205
$726$	0	0
$727$	−2.38223e7	−1.67166	−0.835828	−	0.548991i	$-0.815012\pi$
$727$	−2.38223e7	−1.67166	−0.835828	+	0.548991i	$0.815012\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	437347.	0.0302714
$732$	0	0
$733$	3.43335e6	0.236025	0.118013	−	0.993012i	$-0.462348\pi$
$733$	3.43335e6	0.236025	0.118013	+	0.993012i	$0.462348\pi$
$734$	0	0
$735$	−1.31344e7	−0.896795
$736$	0	0
$737$	−1.25693e7	−0.852401
$738$	0	0
$739$	−1.72898e7	−1.16460	−0.582302	−	0.812973i	$-0.697848\pi$
$739$	−1.72898e7	−1.16460	−0.582302	+	0.812973i	$0.697848\pi$
$740$	0	0
$741$	2.07328e6	0.138712
$742$	0	0
$743$	6.69285e6	0.444773	0.222387	−	0.974959i	$-0.428615\pi$
$743$	6.69285e6	0.444773	0.222387	+	0.974959i	$0.428615\pi$
$744$	0	0
$745$	7.47147e6	0.493192
$746$	0	0
$747$	−4.29311e6	−0.281495
$748$	0	0
$749$	280531.	0.0182716
$750$	0	0
$751$	−1.81435e7	−1.17387	−0.586937	−	0.809633i	$-0.699666\pi$
$751$	−1.81435e7	−1.17387	−0.586937	+	0.809633i	$0.699666\pi$
$752$	0	0
$753$	9.07039e6	0.582960
$754$	0	0
$755$	2.23786e7	1.42878
$756$	0	0
$757$	2.79893e6	0.177522	0.0887610	−	0.996053i	$-0.471709\pi$
$757$	2.79893e6	0.177522	0.0887610	+	0.996053i	$0.471709\pi$
$758$	0	0
$759$	−9.96102e6	−0.627624
$760$	0	0
$761$	−6.44239e6	−0.403261	−0.201630	−	0.979462i	$-0.564624\pi$
$761$	−6.44239e6	−0.403261	−0.201630	+	0.979462i	$0.564624\pi$
$762$	0	0
$763$	694549.	0.0431908
$764$	0	0
$765$	−1.54756e6	−0.0956080
$766$	0	0
$767$	1.69910e7	1.04287
$768$	0	0
$769$	−1.03180e7	−0.629188	−0.314594	−	0.949226i	$-0.601868\pi$
$769$	−1.03180e7	−0.629188	−0.314594	+	0.949226i	$0.601868\pi$
$770$	0	0
$771$	−1.01690e6	−0.0616087
$772$	0	0
$773$	2.30416e6	0.138696	0.0693480	−	0.997593i	$-0.477908\pi$
$773$	2.30416e6	0.138696	0.0693480	+	0.997593i	$0.477908\pi$
$774$	0	0
$775$	2.48567e7	1.48658
$776$	0	0
$777$	−370474.	−0.0220143
$778$	0	0
$779$	5.37074e6	0.317096
$780$	0	0
$781$	1.66180e7	0.974880
$782$	0	0
$783$	5.25383e6	0.306247
$784$	0	0
$785$	−6.98735e6	−0.404705
$786$	0	0
$787$	1.05381e7	0.606495	0.303247	−	0.952912i	$-0.401929\pi$
$787$	1.05381e7	0.606495	0.303247	+	0.952912i	$0.401929\pi$
$788$	0	0
$789$	2.60534e6	0.148995
$790$	0	0
$791$	757450.	0.0430440
$792$	0	0
$793$	7.06156e6	0.398766
$794$	0	0
$795$	−1.27340e7	−0.714572
$796$	0	0
$797$	1.28930e6	0.0718966	0.0359483	−	0.999354i	$-0.488555\pi$
$797$	1.28930e6	0.0718966	0.0359483	+	0.999354i	$0.488555\pi$
$798$	0	0
$799$	3.12700e6	0.173285
$800$	0	0
$801$	−1.02031e7	−0.561889
$802$	0	0
$803$	−2.34996e7	−1.28609
$804$	0	0
$805$	−889015.	−0.0483526
$806$	0	0
$807$	7.96593e6	0.430579
$808$	0	0
$809$	8.06076e6	0.433017	0.216508	−	0.976281i	$-0.430533\pi$
$809$	8.06076e6	0.433017	0.216508	+	0.976281i	$0.430533\pi$
$810$	0	0
$811$	−3.63148e7	−1.93879	−0.969397	−	0.245499i	$-0.921048\pi$
$811$	−3.63148e7	−1.93879	−0.969397	+	0.245499i	$0.921048\pi$
$812$	0	0
$813$	1.58153e7	0.839174
$814$	0	0
$815$	4.19457e7	2.21204
$816$	0	0
$817$	717916.	0.0376287
$818$	0	0
$819$	−152357.	−0.00793694
$820$	0	0
$821$	−2.27674e7	−1.17884	−0.589420	−	0.807827i	$-0.700644\pi$
$821$	−2.27674e7	−1.17884	−0.589420	+	0.807827i	$0.700644\pi$
$822$	0	0
$823$	2.46863e7	1.27045	0.635224	−	0.772328i	$-0.280908\pi$
$823$	2.46863e7	1.27045	0.635224	+	0.772328i	$0.280908\pi$
$824$	0	0
$825$	−1.26894e7	−0.649092
$826$	0	0
$827$	−2.14922e6	−0.109274	−0.0546369	−	0.998506i	$-0.517400\pi$
$827$	−2.14922e6	−0.109274	−0.0546369	+	0.998506i	$0.517400\pi$
$828$	0	0
$829$	2.31358e7	1.16922	0.584612	−	0.811313i	$-0.301247\pi$
$829$	2.31358e7	1.16922	0.584612	+	0.811313i	$0.301247\pi$
$830$	0	0
$831$	−3.28687e6	−0.165113
$832$	0	0
$833$	−3.69424e6	−0.184464
$834$	0	0
$835$	−1.90694e7	−0.946500
$836$	0	0
$837$	−4.09729e6	−0.202155
$838$	0	0
$839$	−2.41020e7	−1.18208	−0.591042	−	0.806641i	$-0.701283\pi$
$839$	−2.41020e7	−1.18208	−0.591042	+	0.806641i	$0.701283\pi$
$840$	0	0
$841$	3.14282e7	1.53225
$842$	0	0
$843$	−1.39275e7	−0.674998
$844$	0	0
$845$	−3.12030e6	−0.150333
$846$	0	0
$847$	−175131.	−0.00838792
$848$	0	0
$849$	−3.50889e6	−0.167071
$850$	0	0
$851$	4.84824e7	2.29488
$852$	0	0
$853$	−2.00928e7	−0.945515	−0.472758	−	0.881192i	$-0.656741\pi$
$853$	−2.00928e7	−0.945515	−0.472758	+	0.881192i	$0.656741\pi$
$854$	0	0
$855$	−2.54036e6	−0.118845
$856$	0	0
$857$	−2.38204e7	−1.10789	−0.553944	−	0.832554i	$-0.686878\pi$
$857$	−2.38204e7	−1.10789	−0.553944	+	0.832554i	$0.686878\pi$
$858$	0	0
$859$	5.90779e6	0.273176	0.136588	−	0.990628i	$-0.456386\pi$
$859$	5.90779e6	0.273176	0.136588	+	0.990628i	$0.456386\pi$
$860$	0	0
$861$	−394674.	−0.0181439
$862$	0	0
$863$	3.98849e7	1.82298	0.911490	−	0.411322i	$-0.134933\pi$
$863$	3.98849e7	1.82298	0.911490	+	0.411322i	$0.134933\pi$
$864$	0	0
$865$	−3.13885e7	−1.42636
$866$	0	0
$867$	1.23434e7	0.557684
$868$	0	0
$869$	−2.77411e6	−0.124616
$870$	0	0
$871$	2.51592e7	1.12370
$872$	0	0
$873$	−5.96193e6	−0.264759
$874$	0	0
$875$	−332278.	−0.0146717
$876$	0	0
$877$	2.37640e7	1.04333	0.521664	−	0.853151i	$-0.325311\pi$
$877$	2.37640e7	1.04333	0.521664	+	0.853151i	$0.325311\pi$
$878$	0	0
$879$	−2.60233e6	−0.113603
$880$	0	0
$881$	−3.96169e7	−1.71965	−0.859827	−	0.510586i	$-0.829429\pi$
$881$	−3.96169e7	−1.71965	−0.859827	+	0.510586i	$0.829429\pi$
$882$	0	0
$883$	8.43589e6	0.364107	0.182054	−	0.983289i	$-0.441726\pi$
$883$	8.43589e6	0.364107	0.182054	+	0.983289i	$0.441726\pi$
$884$	0	0
$885$	−2.08188e7	−0.893508
$886$	0	0
$887$	6.39703e6	0.273004	0.136502	−	0.990640i	$-0.456414\pi$
$887$	6.39703e6	0.273004	0.136502	+	0.990640i	$0.456414\pi$
$888$	0	0
$889$	−168693.	−0.00715885
$890$	0	0
$891$	2.09168e6	0.0882674
$892$	0	0
$893$	5.13305e6	0.215401
$894$	0	0
$895$	5.40601e7	2.25590
$896$	0	0
$897$	1.99383e7	0.827385
$898$	0	0
$899$	−4.05059e7	−1.67155
$900$	0	0
$901$	−3.58160e6	−0.146982
$902$	0	0
$903$	−52756.7	−0.00215307
$904$	0	0
$905$	9.04697e6	0.367182
$906$	0	0
$907$	2.81799e7	1.13742	0.568711	−	0.822538i	$-0.307442\pi$
$907$	2.81799e7	1.13742	0.568711	+	0.822538i	$0.307442\pi$
$908$	0	0
$909$	1.03463e7	0.415312
$910$	0	0
$911$	2.13420e7	0.852000	0.426000	−	0.904723i	$-0.359922\pi$
$911$	2.13420e7	0.852000	0.426000	+	0.904723i	$0.359922\pi$
$912$	0	0
$913$	−1.68971e7	−0.670864
$914$	0	0
$915$	−8.65242e6	−0.341653
$916$	0	0
$917$	14759.2	0.000579614 0
$918$	0	0
$919$	2.45601e6	0.0959272	0.0479636	−	0.998849i	$-0.484727\pi$
$919$	2.45601e6	0.0959272	0.0479636	+	0.998849i	$0.484727\pi$
$920$	0	0
$921$	2.80808e7	1.09084
$922$	0	0
$923$	−3.32631e7	−1.28517
$924$	0	0
$925$	6.17619e7	2.37338
$926$	0	0
$927$	−1.03560e7	−0.395816
$928$	0	0
$929$	−1.58818e7	−0.603755	−0.301877	−	0.953347i	$-0.597613\pi$
$929$	−1.58818e7	−0.603755	−0.301877	+	0.953347i	$0.597613\pi$
$930$	0	0
$931$	−6.06419e6	−0.229297
$932$	0	0
$933$	−5.00127e6	−0.188094
$934$	0	0
$935$	−6.09098e6	−0.227855
$936$	0	0
$937$	−2.86186e7	−1.06488	−0.532439	−	0.846469i	$-0.678724\pi$
$937$	−2.86186e7	−1.06488	−0.532439	+	0.846469i	$0.678724\pi$
$938$	0	0
$939$	−1.51317e7	−0.560046
$940$	0	0
$941$	5.22102e7	1.92212	0.961061	−	0.276337i	$-0.0891205\pi$
$941$	5.22102e7	1.92212	0.961061	+	0.276337i	$0.0891205\pi$
$942$	0	0
$943$	5.16493e7	1.89141
$944$	0	0
$945$	186681.	0.00680018
$946$	0	0
$947$	−3.01070e7	−1.09092	−0.545459	−	0.838138i	$-0.683645\pi$
$947$	−3.01070e7	−1.09092	−0.545459	+	0.838138i	$0.683645\pi$
$948$	0	0
$949$	4.70376e7	1.69543
$950$	0	0
$951$	1.53067e7	0.548821
$952$	0	0
$953$	4.11241e7	1.46678	0.733388	−	0.679810i	$-0.237938\pi$
$953$	4.11241e7	1.46678	0.733388	+	0.679810i	$0.237938\pi$
$954$	0	0
$955$	−1.59191e7	−0.564819
$956$	0	0
$957$	2.06783e7	0.729853
$958$	0	0
$959$	−1.18747e6	−0.0416942
$960$	0	0
$961$	2.96010e6	0.103395
$962$	0	0
$963$	7.70898e6	0.267874
$964$	0	0
$965$	8.41438e6	0.290873
$966$	0	0
$967$	8.91988e6	0.306756	0.153378	−	0.988168i	$-0.450985\pi$
$967$	8.91988e6	0.306756	0.153378	+	0.988168i	$0.450985\pi$
$968$	0	0
$969$	−714511.	−0.0244455
$970$	0	0
$971$	−7.68387e6	−0.261536	−0.130768	−	0.991413i	$-0.541744\pi$
$971$	−7.68387e6	−0.261536	−0.130768	+	0.991413i	$0.541744\pi$
$972$	0	0
$973$	−1.16670e6	−0.0395073
$974$	0	0
$975$	2.53995e7	0.855685
$976$	0	0
$977$	−4.77434e7	−1.60021	−0.800105	−	0.599860i	$-0.795223\pi$
$977$	−4.77434e7	−1.60021	−0.800105	+	0.599860i	$0.795223\pi$
$978$	0	0
$979$	−4.01579e7	−1.33910
$980$	0	0
$981$	1.90862e7	0.633208
$982$	0	0
$983$	2.36245e7	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$983$	2.36245e7	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$984$	0	0
$985$	2.38953e7	0.784734
$986$	0	0
$987$	−377207.	−0.0123250
$988$	0	0
$989$	6.90405e6	0.224447
$990$	0	0
$991$	1.16867e6	0.0378014	0.0189007	−	0.999821i	$-0.493983\pi$
$991$	1.16867e6	0.0378014	0.0189007	+	0.999821i	$0.493983\pi$
$992$	0	0
$993$	−2.41242e6	−0.0776389
$994$	0	0
$995$	2.22789e7	0.713405
$996$	0	0
$997$	7.06390e6	0.225064	0.112532	−	0.993648i	$-0.464104\pi$
$997$	7.06390e6	0.225064	0.112532	+	0.993648i	$0.464104\pi$
$998$	0	0
$999$	−1.01806e7	−0.322746

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.6.a.f.1.1	✓	6
4.3	odd	2		912.6.a.y.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.6.a.f.1.1	✓	6	1.1	even	1	trivial
912.6.a.y.1.1		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -86.8767 q^{5} +2.94760 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -86.8767 q^{5} +2.94760 q^{7} +81.0000 q^{9} +318.804 q^{11} -638.130 q^{13} +781.890 q^{15} +219.917 q^{17} +361.000 q^{19} -26.5284 q^{21} +3471.66 q^{23} +4422.56 q^{25} -729.000 q^{27} -7206.90 q^{29} +5620.43 q^{31} -2869.24 q^{33} -256.078 q^{35} +13965.2 q^{37} +5743.17 q^{39} +14877.4 q^{41} +1988.69 q^{43} -7037.01 q^{45} +14219.0 q^{47} -16798.3 q^{49} -1979.26 q^{51} -16286.1 q^{53} -27696.7 q^{55} -3249.00 q^{57} -26626.3 q^{59} -11066.0 q^{61} +238.756 q^{63} +55438.6 q^{65} -39426.5 q^{67} -31244.9 q^{69} +52126.0 q^{71} -73711.7 q^{73} -39803.1 q^{75} +939.708 q^{77} -8701.62 q^{79} +6561.00 q^{81} -53001.4 q^{83} -19105.7 q^{85} +64862.1 q^{87} -125964. q^{89} -1880.95 q^{91} -50583.9 q^{93} -31362.5 q^{95} -73604.0 q^{97} +25823.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) \) 6 * q - 54 * q^3 - 65 * q^5 - 149 * q^7 + 486 * q^9 - 203 * q^11 - 298 * q^13 + 585 * q^15 + 1319 * q^17 + 2166 * q^19 + 1341 * q^21 + 1234 * q^23 + 4589 * q^25 - 4374 * q^27 + 7356 * q^29 + 1632 * q^31 + 1827 * q^33 + 4383 * q^35 + 14204 * q^37 + 2682 * q^39 + 14734 * q^41 - 4693 * q^43 - 5265 * q^45 - 10955 * q^47 + 38561 * q^49 - 11871 * q^51 + 47500 * q^53 + 769 * q^55 - 19494 * q^57 - 61744 * q^59 - 81581 * q^61 - 12069 * q^63 - 59686 * q^65 - 45756 * q^67 - 11106 * q^69 - 10416 * q^71 - 54615 * q^73 - 41301 * q^75 - 29515 * q^77 - 145594 * q^79 + 39366 * q^81 - 160548 * q^83 - 53947 * q^85 - 66204 * q^87 - 97728 * q^89 - 418294 * q^91 - 14688 * q^93 - 23465 * q^95 - 760 * q^97 - 16443 * q^99

Introduction

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