LMFDB - Embedded newform 456.6.a.f.1.5

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			1.5
Root			$-15.6548$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} -125.529 q^{11} +351.083 q^{13} -419.452 q^{15} -885.933 q^{17} +361.000 q^{19} +527.793 q^{21} +936.962 q^{23} -952.906 q^{25} -729.000 q^{27} +3117.41 q^{29} -10558.4 q^{31} +1129.76 q^{33} -2733.13 q^{35} +10653.1 q^{37} -3159.75 q^{39} +11247.7 q^{41} +5845.74 q^{43} +3775.06 q^{45} -4849.81 q^{47} -13367.9 q^{49} +7973.40 q^{51} -36611.8 q^{53} -5850.36 q^{55} -3249.00 q^{57} -7920.92 q^{59} +49482.3 q^{61} -4750.14 q^{63} +16362.5 q^{65} +15138.3 q^{67} -8432.66 q^{69} -39065.7 q^{71} +36875.3 q^{73} +8576.15 q^{75} +7361.46 q^{77} +8875.46 q^{79} +6561.00 q^{81} -61884.5 q^{83} -41289.5 q^{85} -28056.7 q^{87} -25935.2 q^{89} -20588.8 q^{91} +95025.4 q^{93} +16824.7 q^{95} -88399.1 q^{97} -10167.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	46.6057	0.833709	0.416854	−	0.908973i	$-0.363133\pi$
$5$	46.6057	0.833709	0.416854	+	0.908973i	$0.363133\pi$
$6$	0	0
$7$	−58.6437	−0.452351	−0.226176	−	0.974087i	$-0.572622\pi$
$7$	−58.6437	−0.452351	−0.226176	+	0.974087i	$0.572622\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−125.529	−0.312796	−0.156398	−	0.987694i	$-0.549988\pi$
$11$	−125.529	−0.312796	−0.156398	+	0.987694i	$0.549988\pi$
$12$	0	0
$13$	351.083	0.576171	0.288085	−	0.957605i	$-0.406981\pi$
$13$	351.083	0.576171	0.288085	+	0.957605i	$0.406981\pi$
$14$	0	0
$15$	−419.452	−0.481342
$16$	0	0
$17$	−885.933	−0.743496	−0.371748	−	0.928334i	$-0.621241\pi$
$17$	−885.933	−0.743496	−0.371748	+	0.928334i	$0.621241\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	527.793	0.261165
$22$	0	0
$23$	936.962	0.369320	0.184660	−	0.982803i	$-0.440882\pi$
$23$	936.962	0.369320	0.184660	+	0.982803i	$0.440882\pi$
$24$	0	0
$25$	−952.906	−0.304930
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	3117.41	0.688333	0.344166	−	0.938909i	$-0.388162\pi$
$29$	3117.41	0.688333	0.344166	+	0.938909i	$0.388162\pi$
$30$	0	0
$31$	−10558.4	−1.97330	−0.986649	−	0.162859i	$-0.947929\pi$
$31$	−10558.4	−1.97330	−0.986649	+	0.162859i	$0.947929\pi$
$32$	0	0
$33$	1129.76	0.180593
$34$	0	0
$35$	−2733.13	−0.377129
$36$	0	0
$37$	10653.1	1.27929	0.639646	−	0.768670i	$-0.279081\pi$
$37$	10653.1	1.27929	0.639646	+	0.768670i	$0.279081\pi$
$38$	0	0
$39$	−3159.75	−0.332652
$40$	0	0
$41$	11247.7	1.04497	0.522486	−	0.852648i	$-0.325005\pi$
$41$	11247.7	1.04497	0.522486	+	0.852648i	$0.325005\pi$
$42$	0	0
$43$	5845.74	0.482134	0.241067	−	0.970508i	$-0.422503\pi$
$43$	5845.74	0.482134	0.241067	+	0.970508i	$0.422503\pi$
$44$	0	0
$45$	3775.06	0.277903
$46$	0	0
$47$	−4849.81	−0.320244	−0.160122	−	0.987097i	$-0.551189\pi$
$47$	−4849.81	−0.320244	−0.160122	+	0.987097i	$0.551189\pi$
$48$	0	0
$49$	−13367.9	−0.795378
$50$	0	0
$51$	7973.40	0.429257
$52$	0	0
$53$	−36611.8	−1.79033	−0.895163	−	0.445740i	$-0.852941\pi$
$53$	−36611.8	−1.79033	−0.895163	+	0.445740i	$0.852941\pi$
$54$	0	0
$55$	−5850.36	−0.260781
$56$	0	0
$57$	−3249.00	−0.132453
$58$	0	0
$59$	−7920.92	−0.296241	−0.148121	−	0.988969i	$-0.547322\pi$
$59$	−7920.92	−0.296241	−0.148121	+	0.988969i	$0.547322\pi$
$60$	0	0
$61$	49482.3	1.70265	0.851325	−	0.524639i	$-0.175800\pi$
$61$	49482.3	1.70265	0.851325	+	0.524639i	$0.175800\pi$
$62$	0	0
$63$	−4750.14	−0.150784
$64$	0	0
$65$	16362.5	0.480359
$66$	0	0
$67$	15138.3	0.411994	0.205997	−	0.978553i	$-0.433956\pi$
$67$	15138.3	0.411994	0.205997	+	0.978553i	$0.433956\pi$
$68$	0	0
$69$	−8432.66	−0.213227
$70$	0	0
$71$	−39065.7	−0.919708	−0.459854	−	0.887995i	$-0.652098\pi$
$71$	−39065.7	−0.919708	−0.459854	+	0.887995i	$0.652098\pi$
$72$	0	0
$73$	36875.3	0.809895	0.404947	−	0.914340i	$-0.367290\pi$
$73$	36875.3	0.809895	0.404947	+	0.914340i	$0.367290\pi$
$74$	0	0
$75$	8576.15	0.176051
$76$	0	0
$77$	7361.46	0.141494
$78$	0	0
$79$	8875.46	0.160001	0.0800006	−	0.996795i	$-0.474508\pi$
$79$	8875.46	0.160001	0.0800006	+	0.996795i	$0.474508\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−61884.5	−0.986021	−0.493011	−	0.870023i	$-0.664104\pi$
$83$	−61884.5	−0.986021	−0.493011	+	0.870023i	$0.664104\pi$
$84$	0	0
$85$	−41289.5	−0.619859
$86$	0	0
$87$	−28056.7	−0.397409
$88$	0	0
$89$	−25935.2	−0.347068	−0.173534	−	0.984828i	$-0.555519\pi$
$89$	−25935.2	−0.347068	−0.173534	+	0.984828i	$0.555519\pi$
$90$	0	0
$91$	−20588.8	−0.260632
$92$	0	0
$93$	95025.4	1.13928
$94$	0	0
$95$	16824.7	0.191266
$96$	0	0
$97$	−88399.1	−0.953934	−0.476967	−	0.878921i	$-0.658264\pi$
$97$	−88399.1	−0.953934	−0.476967	+	0.878921i	$0.658264\pi$
$98$	0	0
$99$	−10167.8	−0.104265
$100$	0	0
$101$	−97916.9	−0.955112	−0.477556	−	0.878601i	$-0.658477\pi$
$101$	−97916.9	−0.955112	−0.477556	+	0.878601i	$0.658477\pi$
$102$	0	0
$103$	−158754.	−1.47445	−0.737225	−	0.675647i	$-0.763864\pi$
$103$	−158754.	−1.47445	−0.737225	+	0.675647i	$0.763864\pi$
$104$	0	0
$105$	24598.2	0.217736
$106$	0	0
$107$	−147837.	−1.24831	−0.624156	−	0.781300i	$-0.714557\pi$
$107$	−147837.	−1.24831	−0.624156	+	0.781300i	$0.714557\pi$
$108$	0	0
$109$	−81578.7	−0.657674	−0.328837	−	0.944387i	$-0.606657\pi$
$109$	−81578.7	−0.657674	−0.328837	+	0.944387i	$0.606657\pi$
$110$	0	0
$111$	−95877.5	−0.738600
$112$	0	0
$113$	29397.4	0.216577	0.108289	−	0.994119i	$-0.465463\pi$
$113$	29397.4	0.216577	0.108289	+	0.994119i	$0.465463\pi$
$114$	0	0
$115$	43667.8	0.307905
$116$	0	0
$117$	28437.7	0.192057
$118$	0	0
$119$	51954.3	0.336321
$120$	0	0
$121$	−145294.	−0.902159
$122$	0	0
$123$	−101229.	−0.603315
$124$	0	0
$125$	−190054.	−1.08793
$126$	0	0
$127$	−115145.	−0.633482	−0.316741	−	0.948512i	$-0.602589\pi$
$127$	−115145.	−0.633482	−0.316741	+	0.948512i	$0.602589\pi$
$128$	0	0
$129$	−52611.6	−0.278360
$130$	0	0
$131$	−60304.8	−0.307025	−0.153512	−	0.988147i	$-0.549059\pi$
$131$	−60304.8	−0.307025	−0.153512	+	0.988147i	$0.549059\pi$
$132$	0	0
$133$	−21170.4	−0.103777
$134$	0	0
$135$	−33975.6	−0.160447
$136$	0	0
$137$	−231240.	−1.05259	−0.526297	−	0.850301i	$-0.676420\pi$
$137$	−231240.	−1.05259	−0.526297	+	0.850301i	$0.676420\pi$
$138$	0	0
$139$	−50266.0	−0.220667	−0.110333	−	0.993895i	$-0.535192\pi$
$139$	−50266.0	−0.220667	−0.110333	+	0.993895i	$0.535192\pi$
$140$	0	0
$141$	43648.3	0.184893
$142$	0	0
$143$	−44071.0	−0.180224
$144$	0	0
$145$	145289.	0.573869
$146$	0	0
$147$	120311.	0.459212
$148$	0	0
$149$	353949.	1.30610	0.653048	−	0.757317i	$-0.273490\pi$
$149$	353949.	1.30610	0.653048	+	0.757317i	$0.273490\pi$
$150$	0	0
$151$	−337116.	−1.20320	−0.601599	−	0.798798i	$-0.705469\pi$
$151$	−337116.	−1.20320	−0.601599	+	0.798798i	$0.705469\pi$
$152$	0	0
$153$	−71760.6	−0.247832
$154$	0	0
$155$	−492081.	−1.64516
$156$	0	0
$157$	−494620.	−1.60148	−0.800741	−	0.599010i	$-0.795561\pi$
$157$	−494620.	−1.60148	−0.800741	+	0.599010i	$0.795561\pi$
$158$	0	0
$159$	329507.	1.03364
$160$	0	0
$161$	−54946.9	−0.167062
$162$	0	0
$163$	−117016.	−0.344966	−0.172483	−	0.985013i	$-0.555179\pi$
$163$	−117016.	−0.344966	−0.172483	+	0.985013i	$0.555179\pi$
$164$	0	0
$165$	52653.2	0.150562
$166$	0	0
$167$	−300735.	−0.834435	−0.417217	−	0.908807i	$-0.636995\pi$
$167$	−300735.	−0.834435	−0.417217	+	0.908807i	$0.636995\pi$
$168$	0	0
$169$	−248034.	−0.668027
$170$	0	0
$171$	29241.0	0.0764719
$172$	0	0
$173$	468620.	1.19043	0.595217	−	0.803565i	$-0.297066\pi$
$173$	468620.	1.19043	0.595217	+	0.803565i	$0.297066\pi$
$174$	0	0
$175$	55881.9	0.137936
$176$	0	0
$177$	71288.2	0.171035
$178$	0	0
$179$	695221.	1.62177	0.810887	−	0.585203i	$-0.198985\pi$
$179$	695221.	1.62177	0.810887	+	0.585203i	$0.198985\pi$
$180$	0	0
$181$	−606339.	−1.37568	−0.687842	−	0.725860i	$-0.741442\pi$
$181$	−606339.	−1.37568	−0.687842	+	0.725860i	$0.741442\pi$
$182$	0	0
$183$	−445341.	−0.983025
$184$	0	0
$185$	496493.	1.06656
$186$	0	0
$187$	111210.	0.232563
$188$	0	0
$189$	42751.2	0.0870551
$190$	0	0
$191$	84147.2	0.166900	0.0834500	−	0.996512i	$-0.473406\pi$
$191$	84147.2	0.166900	0.0834500	+	0.996512i	$0.473406\pi$
$192$	0	0
$193$	757920.	1.46464	0.732319	−	0.680962i	$-0.238438\pi$
$193$	757920.	1.46464	0.732319	+	0.680962i	$0.238438\pi$
$194$	0	0
$195$	−147262.	−0.277335
$196$	0	0
$197$	353050.	0.648143	0.324072	−	0.946033i	$-0.394948\pi$
$197$	353050.	0.648143	0.324072	+	0.946033i	$0.394948\pi$
$198$	0	0
$199$	90967.9	0.162838	0.0814189	−	0.996680i	$-0.474055\pi$
$199$	90967.9	0.162838	0.0814189	+	0.996680i	$0.474055\pi$
$200$	0	0
$201$	−136245.	−0.237865
$202$	0	0
$203$	−182816.	−0.311368
$204$	0	0
$205$	524208.	0.871203
$206$	0	0
$207$	75893.9	0.123107
$208$	0	0
$209$	−45315.9	−0.0717603
$210$	0	0
$211$	−612292.	−0.946788	−0.473394	−	0.880851i	$-0.656971\pi$
$211$	−612292.	−0.946788	−0.473394	+	0.880851i	$0.656971\pi$
$212$	0	0
$213$	351591.	0.530993
$214$	0	0
$215$	272445.	0.401960
$216$	0	0
$217$	619182.	0.892625
$218$	0	0
$219$	−331878.	−0.467593
$220$	0	0
$221$	−311036.	−0.428381
$222$	0	0
$223$	−904520.	−1.21802	−0.609012	−	0.793161i	$-0.708434\pi$
$223$	−904520.	−1.21802	−0.609012	+	0.793161i	$0.708434\pi$
$224$	0	0
$225$	−77185.4	−0.101643
$226$	0	0
$227$	−10704.5	−0.0137881	−0.00689403	−	0.999976i	$-0.502194\pi$
$227$	−10704.5	−0.0137881	−0.00689403	+	0.999976i	$0.502194\pi$
$228$	0	0
$229$	−536171.	−0.675639	−0.337819	−	0.941211i	$-0.609689\pi$
$229$	−536171.	−0.675639	−0.337819	+	0.941211i	$0.609689\pi$
$230$	0	0
$231$	−66253.2	−0.0816915
$232$	0	0
$233$	124720.	0.150504	0.0752519	−	0.997165i	$-0.476024\pi$
$233$	124720.	0.150504	0.0752519	+	0.997165i	$0.476024\pi$
$234$	0	0
$235$	−226029.	−0.266990
$236$	0	0
$237$	−79879.1	−0.0923767
$238$	0	0
$239$	−580116.	−0.656931	−0.328466	−	0.944516i	$-0.606532\pi$
$239$	−580116.	−0.656931	−0.328466	+	0.944516i	$0.606532\pi$
$240$	0	0
$241$	−381173.	−0.422746	−0.211373	−	0.977406i	$-0.567793\pi$
$241$	−381173.	−0.422746	−0.211373	+	0.977406i	$0.567793\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−623022.	−0.663114
$246$	0	0
$247$	126741.	0.132183
$248$	0	0
$249$	556960.	0.569280
$250$	0	0
$251$	710860.	0.712197	0.356098	−	0.934448i	$-0.384107\pi$
$251$	710860.	0.712197	0.356098	+	0.934448i	$0.384107\pi$
$252$	0	0
$253$	−117616.	−0.115522
$254$	0	0
$255$	371606.	0.357876
$256$	0	0
$257$	143727.	0.135739	0.0678695	−	0.997694i	$-0.478380\pi$
$257$	143727.	0.135739	0.0678695	+	0.997694i	$0.478380\pi$
$258$	0	0
$259$	−624734.	−0.578690
$260$	0	0
$261$	252510.	0.229444
$262$	0	0
$263$	20933.3	0.0186616	0.00933079	−	0.999956i	$-0.497030\pi$
$263$	20933.3	0.0186616	0.00933079	+	0.999956i	$0.497030\pi$
$264$	0	0
$265$	−1.70632e6	−1.49261
$266$	0	0
$267$	233417.	0.200380
$268$	0	0
$269$	899788.	0.758158	0.379079	−	0.925364i	$-0.376241\pi$
$269$	899788.	0.758158	0.379079	+	0.925364i	$0.376241\pi$
$270$	0	0
$271$	−1.38128e6	−1.14251	−0.571255	−	0.820773i	$-0.693543\pi$
$271$	−1.38128e6	−1.14251	−0.571255	+	0.820773i	$0.693543\pi$
$272$	0	0
$273$	185299.	0.150476
$274$	0	0
$275$	119617.	0.0953809
$276$	0	0
$277$	1.93050e6	1.51172	0.755858	−	0.654735i	$-0.227220\pi$
$277$	1.93050e6	1.51172	0.755858	+	0.654735i	$0.227220\pi$
$278$	0	0
$279$	−855228.	−0.657766
$280$	0	0
$281$	1.99112e6	1.50429	0.752146	−	0.658996i	$-0.229019\pi$
$281$	1.99112e6	1.50429	0.752146	+	0.658996i	$0.229019\pi$
$282$	0	0
$283$	240309.	0.178363	0.0891813	−	0.996015i	$-0.471575\pi$
$283$	240309.	0.178363	0.0891813	+	0.996015i	$0.471575\pi$
$284$	0	0
$285$	−151422.	−0.110427
$286$	0	0
$287$	−659607.	−0.472695
$288$	0	0
$289$	−634980.	−0.447214
$290$	0	0
$291$	795592.	0.550754
$292$	0	0
$293$	−1.75213e6	−1.19234	−0.596168	−	0.802860i	$-0.703311\pi$
$293$	−1.75213e6	−1.19234	−0.596168	+	0.802860i	$0.703311\pi$
$294$	0	0
$295$	−369160.	−0.246979
$296$	0	0
$297$	91510.4	0.0601976
$298$	0	0
$299$	328951.	0.212791
$300$	0	0
$301$	−342815.	−0.218094
$302$	0	0
$303$	881252.	0.551434
$304$	0	0
$305$	2.30616e6	1.41951
$306$	0	0
$307$	2.56334e6	1.55224	0.776122	−	0.630583i	$-0.217184\pi$
$307$	2.56334e6	1.55224	0.776122	+	0.630583i	$0.217184\pi$
$308$	0	0
$309$	1.42878e6	0.851275
$310$	0	0
$311$	−1.22333e6	−0.717203	−0.358601	−	0.933491i	$-0.616746\pi$
$311$	−1.22333e6	−0.717203	−0.358601	+	0.933491i	$0.616746\pi$
$312$	0	0
$313$	2.36852e6	1.36652	0.683262	−	0.730174i	$-0.260561\pi$
$313$	2.36852e6	1.36652	0.683262	+	0.730174i	$0.260561\pi$
$314$	0	0
$315$	−221384.	−0.125710
$316$	0	0
$317$	2.41136e6	1.34776	0.673882	−	0.738839i	$-0.264626\pi$
$317$	2.41136e6	1.34776	0.673882	+	0.738839i	$0.264626\pi$
$318$	0	0
$319$	−391324.	−0.215308
$320$	0	0
$321$	1.33053e6	0.720713
$322$	0	0
$323$	−319822.	−0.170570
$324$	0	0
$325$	−334549.	−0.175692
$326$	0	0
$327$	734208.	0.379708
$328$	0	0
$329$	284411.	0.144863
$330$	0	0
$331$	−2.96614e6	−1.48806	−0.744032	−	0.668144i	$-0.767089\pi$
$331$	−2.96614e6	−1.48806	−0.744032	+	0.668144i	$0.767089\pi$
$332$	0	0
$333$	862897.	0.426431
$334$	0	0
$335$	705533.	0.343483
$336$	0	0
$337$	1.06940e6	0.512937	0.256469	−	0.966553i	$-0.417441\pi$
$337$	1.06940e6	0.512937	0.256469	+	0.966553i	$0.417441\pi$
$338$	0	0
$339$	−264577.	−0.125041
$340$	0	0
$341$	1.32538e6	0.617240
$342$	0	0
$343$	1.76957e6	0.812142
$344$	0	0
$345$	−393010.	−0.177769
$346$	0	0
$347$	−146634.	−0.0653748	−0.0326874	−	0.999466i	$-0.510407\pi$
$347$	−146634.	−0.0653748	−0.0326874	+	0.999466i	$0.510407\pi$
$348$	0	0
$349$	398157.	0.174981	0.0874905	−	0.996165i	$-0.472115\pi$
$349$	398157.	0.174981	0.0874905	+	0.996165i	$0.472115\pi$
$350$	0	0
$351$	−255939.	−0.110884
$352$	0	0
$353$	−3.57018e6	−1.52494	−0.762470	−	0.647023i	$-0.776014\pi$
$353$	−3.57018e6	−1.52494	−0.762470	+	0.647023i	$0.776014\pi$
$354$	0	0
$355$	−1.82069e6	−0.766768
$356$	0	0
$357$	−467589.	−0.194175
$358$	0	0
$359$	2.51582e6	1.03025	0.515125	−	0.857115i	$-0.327745\pi$
$359$	2.51582e6	1.03025	0.515125	+	0.857115i	$0.327745\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	1.30764e6	0.520862
$364$	0	0
$365$	1.71860e6	0.675216
$366$	0	0
$367$	−4.82115e6	−1.86847	−0.934233	−	0.356663i	$-0.883914\pi$
$367$	−4.82115e6	−1.86847	−0.934233	+	0.356663i	$0.883914\pi$
$368$	0	0
$369$	911065.	0.348324
$370$	0	0
$371$	2.14705e6	0.809856
$372$	0	0
$373$	−1.29172e6	−0.480723	−0.240362	−	0.970683i	$-0.577266\pi$
$373$	−1.29172e6	−0.480723	−0.240362	+	0.970683i	$0.577266\pi$
$374$	0	0
$375$	1.71048e6	0.628117
$376$	0	0
$377$	1.09447e6	0.396597
$378$	0	0
$379$	2.36358e6	0.845225	0.422613	−	0.906310i	$-0.361113\pi$
$379$	2.36358e6	0.845225	0.422613	+	0.906310i	$0.361113\pi$
$380$	0	0
$381$	1.03630e6	0.365741
$382$	0	0
$383$	3.86944e6	1.34788	0.673939	−	0.738787i	$-0.264601\pi$
$383$	3.86944e6	1.34788	0.673939	+	0.738787i	$0.264601\pi$
$384$	0	0
$385$	343086.	0.117965
$386$	0	0
$387$	473505.	0.160711
$388$	0	0
$389$	−3.20371e6	−1.07344	−0.536722	−	0.843759i	$-0.680338\pi$
$389$	−3.20371e6	−1.07344	−0.536722	+	0.843759i	$0.680338\pi$
$390$	0	0
$391$	−830085.	−0.274588
$392$	0	0
$393$	542743.	0.177261
$394$	0	0
$395$	413647.	0.133394
$396$	0	0
$397$	861302.	0.274270	0.137135	−	0.990552i	$-0.456211\pi$
$397$	861302.	0.274270	0.137135	+	0.990552i	$0.456211\pi$
$398$	0	0
$399$	190533.	0.0599154
$400$	0	0
$401$	5.75486e6	1.78720	0.893601	−	0.448862i	$-0.148171\pi$
$401$	5.75486e6	1.78720	0.893601	+	0.448862i	$0.148171\pi$
$402$	0	0
$403$	−3.70687e6	−1.13696
$404$	0	0
$405$	305780.	0.0926343
$406$	0	0
$407$	−1.33726e6	−0.400158
$408$	0	0
$409$	3.25999e6	0.963625	0.481813	−	0.876274i	$-0.339979\pi$
$409$	3.25999e6	0.963625	0.481813	+	0.876274i	$0.339979\pi$
$410$	0	0
$411$	2.08116e6	0.607715
$412$	0	0
$413$	464511.	0.134005
$414$	0	0
$415$	−2.88417e6	−0.822055
$416$	0	0
$417$	452394.	0.127402
$418$	0	0
$419$	−1.10038e6	−0.306203	−0.153101	−	0.988210i	$-0.548926\pi$
$419$	−1.10038e6	−0.306203	−0.153101	+	0.988210i	$0.548926\pi$
$420$	0	0
$421$	−4.45235e6	−1.22429	−0.612145	−	0.790746i	$-0.709693\pi$
$421$	−4.45235e6	−1.22429	−0.612145	+	0.790746i	$0.709693\pi$
$422$	0	0
$423$	−392835.	−0.106748
$424$	0	0
$425$	844211.	0.226714
$426$	0	0
$427$	−2.90182e6	−0.770196
$428$	0	0
$429$	396639.	0.104052
$430$	0	0
$431$	2.46438e6	0.639020	0.319510	−	0.947583i	$-0.396482\pi$
$431$	2.46438e6	0.639020	0.319510	+	0.947583i	$0.396482\pi$
$432$	0	0
$433$	6.18726e6	1.58591	0.792955	−	0.609280i	$-0.208542\pi$
$433$	6.18726e6	1.58591	0.792955	+	0.609280i	$0.208542\pi$
$434$	0	0
$435$	−1.30760e6	−0.331323
$436$	0	0
$437$	338243.	0.0847277
$438$	0	0
$439$	5.72940e6	1.41889	0.709443	−	0.704762i	$-0.248946\pi$
$439$	5.72940e6	1.41889	0.709443	+	0.704762i	$0.248946\pi$
$440$	0	0
$441$	−1.08280e6	−0.265126
$442$	0	0
$443$	−820524.	−0.198647	−0.0993234	−	0.995055i	$-0.531668\pi$
$443$	−820524.	−0.198647	−0.0993234	+	0.995055i	$0.531668\pi$
$444$	0	0
$445$	−1.20873e6	−0.289354
$446$	0	0
$447$	−3.18554e6	−0.754075
$448$	0	0
$449$	−1.12512e6	−0.263380	−0.131690	−	0.991291i	$-0.542040\pi$
$449$	−1.12512e6	−0.263380	−0.131690	+	0.991291i	$0.542040\pi$
$450$	0	0
$451$	−1.41191e6	−0.326863
$452$	0	0
$453$	3.03404e6	0.694667
$454$	0	0
$455$	−959555.	−0.217291
$456$	0	0
$457$	−1.66587e6	−0.373121	−0.186561	−	0.982443i	$-0.559734\pi$
$457$	−1.66587e6	−0.373121	−0.186561	+	0.982443i	$0.559734\pi$
$458$	0	0
$459$	645845.	0.143086
$460$	0	0
$461$	−3.75543e6	−0.823014	−0.411507	−	0.911407i	$-0.634997\pi$
$461$	−3.75543e6	−0.823014	−0.411507	+	0.911407i	$0.634997\pi$
$462$	0	0
$463$	−641032.	−0.138972	−0.0694860	−	0.997583i	$-0.522136\pi$
$463$	−641032.	−0.138972	−0.0694860	+	0.997583i	$0.522136\pi$
$464$	0	0
$465$	4.42873e6	0.949831
$466$	0	0
$467$	−2.90723e6	−0.616862	−0.308431	−	0.951247i	$-0.599804\pi$
$467$	−2.90723e6	−0.616862	−0.308431	+	0.951247i	$0.599804\pi$
$468$	0	0
$469$	−887767.	−0.186366
$470$	0	0
$471$	4.45158e6	0.924616
$472$	0	0
$473$	−733808.	−0.150810
$474$	0	0
$475$	−343999.	−0.0699557
$476$	0	0
$477$	−2.96556e6	−0.596775
$478$	0	0
$479$	8.55294e6	1.70324	0.851622	−	0.524157i	$-0.175619\pi$
$479$	8.55294e6	1.70324	0.851622	+	0.524157i	$0.175619\pi$
$480$	0	0
$481$	3.74010e6	0.737091
$482$	0	0
$483$	494522.	0.0964534
$484$	0	0
$485$	−4.11990e6	−0.795303
$486$	0	0
$487$	−6.51187e6	−1.24418	−0.622090	−	0.782945i	$-0.713716\pi$
$487$	−6.51187e6	−1.24418	−0.622090	+	0.782945i	$0.713716\pi$
$488$	0	0
$489$	1.05314e6	0.199166
$490$	0	0
$491$	−1.47134e6	−0.275429	−0.137714	−	0.990472i	$-0.543976\pi$
$491$	−1.47134e6	−0.275429	−0.137714	+	0.990472i	$0.543976\pi$
$492$	0	0
$493$	−2.76181e6	−0.511772
$494$	0	0
$495$	−473879.	−0.0869269
$496$	0	0
$497$	2.29096e6	0.416031
$498$	0	0
$499$	−8.75236e6	−1.57352	−0.786762	−	0.617256i	$-0.788244\pi$
$499$	−8.75236e6	−1.57352	−0.786762	+	0.617256i	$0.788244\pi$
$500$	0	0
$501$	2.70661e6	0.481761
$502$	0	0
$503$	9.02461e6	1.59041	0.795204	−	0.606342i	$-0.207364\pi$
$503$	9.02461e6	1.59041	0.795204	+	0.606342i	$0.207364\pi$
$504$	0	0
$505$	−4.56349e6	−0.796285
$506$	0	0
$507$	2.23230e6	0.385686
$508$	0	0
$509$	890437.	0.152338	0.0761691	−	0.997095i	$-0.475731\pi$
$509$	890437.	0.152338	0.0761691	+	0.997095i	$0.475731\pi$
$510$	0	0
$511$	−2.16250e6	−0.366357
$512$	0	0
$513$	−263169.	−0.0441511
$514$	0	0
$515$	−7.39882e6	−1.22926
$516$	0	0
$517$	608791.	0.100171
$518$	0	0
$519$	−4.21758e6	−0.687298
$520$	0	0
$521$	−3.15267e6	−0.508844	−0.254422	−	0.967093i	$-0.581885\pi$
$521$	−3.15267e6	−0.508844	−0.254422	+	0.967093i	$0.581885\pi$
$522$	0	0
$523$	−4.92318e6	−0.787030	−0.393515	−	0.919318i	$-0.628741\pi$
$523$	−4.92318e6	−0.787030	−0.393515	+	0.919318i	$0.628741\pi$
$524$	0	0
$525$	−502937.	−0.0796371
$526$	0	0
$527$	9.35401e6	1.46714
$528$	0	0
$529$	−5.55845e6	−0.863603
$530$	0	0
$531$	−641594.	−0.0987470
$532$	0	0
$533$	3.94888e6	0.602083
$534$	0	0
$535$	−6.89004e6	−1.04073
$536$	0	0
$537$	−6.25699e6	−0.936331
$538$	0	0
$539$	1.67806e6	0.248791
$540$	0	0
$541$	−7.78529e6	−1.14362	−0.571810	−	0.820386i	$-0.693758\pi$
$541$	−7.78529e6	−1.14362	−0.571810	+	0.820386i	$0.693758\pi$
$542$	0	0
$543$	5.45705e6	0.794252
$544$	0	0
$545$	−3.80204e6	−0.548308
$546$	0	0
$547$	2.10139e6	0.300289	0.150144	−	0.988664i	$-0.452026\pi$
$547$	2.10139e6	0.300289	0.150144	+	0.988664i	$0.452026\pi$
$548$	0	0
$549$	4.00807e6	0.567550
$550$	0	0
$551$	1.12538e6	0.157914
$552$	0	0
$553$	−520489.	−0.0723768
$554$	0	0
$555$	−4.46844e6	−0.615777
$556$	0	0
$557$	−2.52411e6	−0.344722	−0.172361	−	0.985034i	$-0.555140\pi$
$557$	−2.52411e6	−0.344722	−0.172361	+	0.985034i	$0.555140\pi$
$558$	0	0
$559$	2.05234e6	0.277792
$560$	0	0
$561$	−1.00089e6	−0.134270
$562$	0	0
$563$	7.21244e6	0.958984	0.479492	−	0.877546i	$-0.340821\pi$
$563$	7.21244e6	0.958984	0.479492	+	0.877546i	$0.340821\pi$
$564$	0	0
$565$	1.37009e6	0.180562
$566$	0	0
$567$	−384761.	−0.0502613
$568$	0	0
$569$	−5.98706e6	−0.775234	−0.387617	−	0.921821i	$-0.626702\pi$
$569$	−5.98706e6	−0.775234	−0.387617	+	0.921821i	$0.626702\pi$
$570$	0	0
$571$	−3.41302e6	−0.438075	−0.219037	−	0.975716i	$-0.570292\pi$
$571$	−3.41302e6	−0.438075	−0.219037	+	0.975716i	$0.570292\pi$
$572$	0	0
$573$	−757325.	−0.0963598
$574$	0	0
$575$	−892837.	−0.112617
$576$	0	0
$577$	956178.	0.119564	0.0597818	−	0.998211i	$-0.480960\pi$
$577$	956178.	0.119564	0.0597818	+	0.998211i	$0.480960\pi$
$578$	0	0
$579$	−6.82128e6	−0.845609
$580$	0	0
$581$	3.62913e6	0.446028
$582$	0	0
$583$	4.59584e6	0.560007
$584$	0	0
$585$	1.32536e6	0.160120
$586$	0	0
$587$	5.95696e6	0.713559	0.356779	−	0.934189i	$-0.383875\pi$
$587$	5.95696e6	0.713559	0.356779	+	0.934189i	$0.383875\pi$
$588$	0	0
$589$	−3.81157e6	−0.452706
$590$	0	0
$591$	−3.17745e6	−0.374206
$592$	0	0
$593$	−1.17783e7	−1.37545	−0.687725	−	0.725971i	$-0.741391\pi$
$593$	−1.17783e7	−1.37545	−0.687725	+	0.725971i	$0.741391\pi$
$594$	0	0
$595$	2.42137e6	0.280394
$596$	0	0
$597$	−818711.	−0.0940145
$598$	0	0
$599$	131291.	0.0149509	0.00747544	−	0.999972i	$-0.497620\pi$
$599$	131291.	0.0149509	0.00747544	+	0.999972i	$0.497620\pi$
$600$	0	0
$601$	−5.73312e6	−0.647448	−0.323724	−	0.946152i	$-0.604935\pi$
$601$	−5.73312e6	−0.647448	−0.323724	+	0.946152i	$0.604935\pi$
$602$	0	0
$603$	1.22620e6	0.137331
$604$	0	0
$605$	−6.77151e6	−0.752137
$606$	0	0
$607$	4.70647e6	0.518470	0.259235	−	0.965814i	$-0.416530\pi$
$607$	4.70647e6	0.518470	0.259235	+	0.965814i	$0.416530\pi$
$608$	0	0
$609$	1.64534e6	0.179769
$610$	0	0
$611$	−1.70269e6	−0.184515
$612$	0	0
$613$	−5.56498e6	−0.598153	−0.299077	−	0.954229i	$-0.596679\pi$
$613$	−5.56498e6	−0.598153	−0.299077	+	0.954229i	$0.596679\pi$
$614$	0	0
$615$	−4.71787e6	−0.502989
$616$	0	0
$617$	−8.82742e6	−0.933514	−0.466757	−	0.884386i	$-0.654578\pi$
$617$	−8.82742e6	−0.933514	−0.466757	+	0.884386i	$0.654578\pi$
$618$	0	0
$619$	−1.31806e7	−1.38264	−0.691319	−	0.722550i	$-0.742970\pi$
$619$	−1.31806e7	−1.38264	−0.691319	+	0.722550i	$0.742970\pi$
$620$	0	0
$621$	−683045.	−0.0710756
$622$	0	0
$623$	1.52094e6	0.156997
$624$	0	0
$625$	−5.87976e6	−0.602088
$626$	0	0
$627$	407843.	0.0414309
$628$	0	0
$629$	−9.43789e6	−0.951148
$630$	0	0
$631$	1.16412e7	1.16392	0.581959	−	0.813218i	$-0.302286\pi$
$631$	1.16412e7	1.16392	0.581959	+	0.813218i	$0.302286\pi$
$632$	0	0
$633$	5.51063e6	0.546628
$634$	0	0
$635$	−5.36640e6	−0.528139
$636$	0	0
$637$	−4.69325e6	−0.458274
$638$	0	0
$639$	−3.16432e6	−0.306569
$640$	0	0
$641$	−9.10465e6	−0.875222	−0.437611	−	0.899164i	$-0.644175\pi$
$641$	−9.10465e6	−0.875222	−0.437611	+	0.899164i	$0.644175\pi$
$642$	0	0
$643$	−1.08215e7	−1.03219	−0.516096	−	0.856531i	$-0.672615\pi$
$643$	−1.08215e7	−1.03219	−0.516096	+	0.856531i	$0.672615\pi$
$644$	0	0
$645$	−2.45200e6	−0.232071
$646$	0	0
$647$	1.42375e7	1.33713	0.668564	−	0.743655i	$-0.266909\pi$
$647$	1.42375e7	1.33713	0.668564	+	0.743655i	$0.266909\pi$
$648$	0	0
$649$	994302.	0.0926631
$650$	0	0
$651$	−5.57264e6	−0.515357
$652$	0	0
$653$	−919718.	−0.0844057	−0.0422029	−	0.999109i	$-0.513438\pi$
$653$	−919718.	−0.0844057	−0.0422029	+	0.999109i	$0.513438\pi$
$654$	0	0
$655$	−2.81055e6	−0.255969
$656$	0	0
$657$	2.98690e6	0.269965
$658$	0	0
$659$	−2.87830e6	−0.258180	−0.129090	−	0.991633i	$-0.541206\pi$
$659$	−2.87830e6	−0.258180	−0.129090	+	0.991633i	$0.541206\pi$
$660$	0	0
$661$	477439.	0.0425024	0.0212512	−	0.999774i	$-0.493235\pi$
$661$	477439.	0.0425024	0.0212512	+	0.999774i	$0.493235\pi$
$662$	0	0
$663$	2.79932e6	0.247326
$664$	0	0
$665$	−986660.	−0.0865194
$666$	0	0
$667$	2.92089e6	0.254215
$668$	0	0
$669$	8.14068e6	0.703227
$670$	0	0
$671$	−6.21145e6	−0.532582
$672$	0	0
$673$	−5.74481e6	−0.488920	−0.244460	−	0.969659i	$-0.578611\pi$
$673$	−5.74481e6	−0.488920	−0.244460	+	0.969659i	$0.578611\pi$
$674$	0	0
$675$	694669.	0.0586838
$676$	0	0
$677$	1.47442e7	1.23637	0.618187	−	0.786031i	$-0.287867\pi$
$677$	1.47442e7	1.23637	0.618187	+	0.786031i	$0.287867\pi$
$678$	0	0
$679$	5.18404e6	0.431513
$680$	0	0
$681$	96340.9	0.00796054
$682$	0	0
$683$	1.47287e7	1.20813	0.604065	−	0.796935i	$-0.293547\pi$
$683$	1.47287e7	1.20813	0.604065	+	0.796935i	$0.293547\pi$
$684$	0	0
$685$	−1.07771e7	−0.877556
$686$	0	0
$687$	4.82554e6	0.390080
$688$	0	0
$689$	−1.28538e7	−1.03153
$690$	0	0
$691$	1.51588e7	1.20773	0.603863	−	0.797088i	$-0.293627\pi$
$691$	1.51588e7	1.20773	0.603863	+	0.797088i	$0.293627\pi$
$692$	0	0
$693$	596279.	0.0471646
$694$	0	0
$695$	−2.34268e6	−0.183972
$696$	0	0
$697$	−9.96472e6	−0.776933
$698$	0	0
$699$	−1.12248e6	−0.0868934
$700$	0	0
$701$	−1.08937e7	−0.837295	−0.418647	−	0.908149i	$-0.637496\pi$
$701$	−1.08937e7	−0.837295	−0.418647	+	0.908149i	$0.637496\pi$
$702$	0	0
$703$	3.84575e6	0.293490
$704$	0	0
$705$	2.03426e6	0.154147
$706$	0	0
$707$	5.74221e6	0.432046
$708$	0	0
$709$	−1.52402e7	−1.13861	−0.569304	−	0.822127i	$-0.692787\pi$
$709$	−1.52402e7	−1.13861	−0.569304	+	0.822127i	$0.692787\pi$
$710$	0	0
$711$	718912.	0.0533337
$712$	0	0
$713$	−9.89280e6	−0.728778
$714$	0	0
$715$	−2.05396e6	−0.150254
$716$	0	0
$717$	5.22104e6	0.379279
$718$	0	0
$719$	1.55304e7	1.12037	0.560183	−	0.828369i	$-0.310731\pi$
$719$	1.55304e7	1.12037	0.560183	+	0.828369i	$0.310731\pi$
$720$	0	0
$721$	9.30989e6	0.666970
$722$	0	0
$723$	3.43055e6	0.244072
$724$	0	0
$725$	−2.97059e6	−0.209893
$726$	0	0
$727$	4.95054e6	0.347389	0.173695	−	0.984800i	$-0.444429\pi$
$727$	4.95054e6	0.347389	0.173695	+	0.984800i	$0.444429\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−5.17893e6	−0.358465
$732$	0	0
$733$	−7.63011e6	−0.524531	−0.262265	−	0.964996i	$-0.584470\pi$
$733$	−7.63011e6	−0.524531	−0.262265	+	0.964996i	$0.584470\pi$
$734$	0	0
$735$	5.60720e6	0.382849
$736$	0	0
$737$	−1.90029e6	−0.128870
$738$	0	0
$739$	−7.81572e6	−0.526451	−0.263226	−	0.964734i	$-0.584786\pi$
$739$	−7.81572e6	−0.526451	−0.263226	+	0.964734i	$0.584786\pi$
$740$	0	0
$741$	−1.14067e6	−0.0763157
$742$	0	0
$743$	8.78448e6	0.583773	0.291887	−	0.956453i	$-0.405717\pi$
$743$	8.78448e6	0.583773	0.291887	+	0.956453i	$0.405717\pi$
$744$	0	0
$745$	1.64960e7	1.08890
$746$	0	0
$747$	−5.01264e6	−0.328674
$748$	0	0
$749$	8.66969e6	0.564676
$750$	0	0
$751$	1.84385e7	1.19296	0.596479	−	0.802628i	$-0.296566\pi$
$751$	1.84385e7	1.19296	0.596479	+	0.802628i	$0.296566\pi$
$752$	0	0
$753$	−6.39774e6	−0.411187
$754$	0	0
$755$	−1.57115e7	−1.00312
$756$	0	0
$757$	2.01886e7	1.28046	0.640232	−	0.768181i	$-0.278838\pi$
$757$	2.01886e7	1.28046	0.640232	+	0.768181i	$0.278838\pi$
$758$	0	0
$759$	1.05854e6	0.0666965
$760$	0	0
$761$	5.59816e6	0.350416	0.175208	−	0.984531i	$-0.443940\pi$
$761$	5.59816e6	0.350416	0.175208	+	0.984531i	$0.443940\pi$
$762$	0	0
$763$	4.78407e6	0.297500
$764$	0	0
$765$	−3.34445e6	−0.206620
$766$	0	0
$767$	−2.78090e6	−0.170686
$768$	0	0
$769$	−3.22353e6	−0.196569	−0.0982846	−	0.995158i	$-0.531336\pi$
$769$	−3.22353e6	−0.196569	−0.0982846	+	0.995158i	$0.531336\pi$
$770$	0	0
$771$	−1.29354e6	−0.0783689
$772$	0	0
$773$	−3.00281e6	−0.180750	−0.0903752	−	0.995908i	$-0.528807\pi$
$773$	−3.00281e6	−0.180750	−0.0903752	+	0.995908i	$0.528807\pi$
$774$	0	0
$775$	1.00611e7	0.601718
$776$	0	0
$777$	5.62261e6	0.334107
$778$	0	0
$779$	4.06043e6	0.239733
$780$	0	0
$781$	4.90387e6	0.287681
$782$	0	0
$783$	−2.27259e6	−0.132470
$784$	0	0
$785$	−2.30521e7	−1.33517
$786$	0	0
$787$	9.05219e6	0.520975	0.260487	−	0.965477i	$-0.416117\pi$
$787$	9.05219e6	0.520975	0.260487	+	0.965477i	$0.416117\pi$
$788$	0	0
$789$	−188400.	−0.0107743
$790$	0	0
$791$	−1.72397e6	−0.0979690
$792$	0	0
$793$	1.73724e7	0.981017
$794$	0	0
$795$	1.53569e7	0.861759
$796$	0	0
$797$	77865.6	0.00434210	0.00217105	−	0.999998i	$-0.499309\pi$
$797$	77865.6	0.00434210	0.00217105	+	0.999998i	$0.499309\pi$
$798$	0	0
$799$	4.29661e6	0.238100
$800$	0	0
$801$	−2.10075e6	−0.115689
$802$	0	0
$803$	−4.62891e6	−0.253332
$804$	0	0
$805$	−2.56084e6	−0.139281
$806$	0	0
$807$	−8.09809e6	−0.437722
$808$	0	0
$809$	2.45959e6	0.132127	0.0660634	−	0.997815i	$-0.478956\pi$
$809$	2.45959e6	0.132127	0.0660634	+	0.997815i	$0.478956\pi$
$810$	0	0
$811$	1.57822e7	0.842587	0.421294	−	0.906924i	$-0.361576\pi$
$811$	1.57822e7	0.842587	0.421294	+	0.906924i	$0.361576\pi$
$812$	0	0
$813$	1.24316e7	0.659628
$814$	0	0
$815$	−5.45362e6	−0.287601
$816$	0	0
$817$	2.11031e6	0.110609
$818$	0	0
$819$	−1.66769e6	−0.0868772
$820$	0	0
$821$	1.10300e7	0.571106	0.285553	−	0.958363i	$-0.407823\pi$
$821$	1.10300e7	0.571106	0.285553	+	0.958363i	$0.407823\pi$
$822$	0	0
$823$	−1.90977e7	−0.982839	−0.491420	−	0.870923i	$-0.663522\pi$
$823$	−1.90977e7	−0.982839	−0.491420	+	0.870923i	$0.663522\pi$
$824$	0	0
$825$	−1.07655e6	−0.0550682
$826$	0	0
$827$	2.63862e7	1.34157	0.670784	−	0.741652i	$-0.265958\pi$
$827$	2.63862e7	1.34157	0.670784	+	0.741652i	$0.265958\pi$
$828$	0	0
$829$	1.35815e7	0.686374	0.343187	−	0.939267i	$-0.388494\pi$
$829$	1.35815e7	0.686374	0.343187	+	0.939267i	$0.388494\pi$
$830$	0	0
$831$	−1.73745e7	−0.872790
$832$	0	0
$833$	1.18431e7	0.591360
$834$	0	0
$835$	−1.40160e7	−0.695675
$836$	0	0
$837$	7.69706e6	0.379762
$838$	0	0
$839$	4.73494e6	0.232225	0.116113	−	0.993236i	$-0.462957\pi$
$839$	4.73494e6	0.232225	0.116113	+	0.993236i	$0.462957\pi$
$840$	0	0
$841$	−1.07929e7	−0.526198
$842$	0	0
$843$	−1.79201e7	−0.868503
$844$	0	0
$845$	−1.15598e7	−0.556940
$846$	0	0
$847$	8.52055e6	0.408093
$848$	0	0
$849$	−2.16278e6	−0.102978
$850$	0	0
$851$	9.98150e6	0.472468
$852$	0	0
$853$	7.94247e6	0.373751	0.186876	−	0.982384i	$-0.440164\pi$
$853$	7.94247e6	0.373751	0.186876	+	0.982384i	$0.440164\pi$
$854$	0	0
$855$	1.36280e6	0.0637553
$856$	0	0
$857$	−1.50658e7	−0.700713	−0.350356	−	0.936616i	$-0.613940\pi$
$857$	−1.50658e7	−0.700713	−0.350356	+	0.936616i	$0.613940\pi$
$858$	0	0
$859$	−2.28464e7	−1.05641	−0.528207	−	0.849116i	$-0.677135\pi$
$859$	−2.28464e7	−1.05641	−0.528207	+	0.849116i	$0.677135\pi$
$860$	0	0
$861$	5.93647e6	0.272910
$862$	0	0
$863$	7.03849e6	0.321701	0.160851	−	0.986979i	$-0.448576\pi$
$863$	7.03849e6	0.321701	0.160851	+	0.986979i	$0.448576\pi$
$864$	0	0
$865$	2.18404e7	0.992475
$866$	0	0
$867$	5.71482e6	0.258199
$868$	0	0
$869$	−1.11412e6	−0.0500477
$870$	0	0
$871$	5.31481e6	0.237379
$872$	0	0
$873$	−7.16032e6	−0.317978
$874$	0	0
$875$	1.11455e7	0.492127
$876$	0	0
$877$	2.02855e7	0.890608	0.445304	−	0.895379i	$-0.353096\pi$
$877$	2.02855e7	0.890608	0.445304	+	0.895379i	$0.353096\pi$
$878$	0	0
$879$	1.57692e7	0.688395
$880$	0	0
$881$	3.70861e7	1.60980	0.804898	−	0.593413i	$-0.202220\pi$
$881$	3.70861e7	1.60980	0.804898	+	0.593413i	$0.202220\pi$
$882$	0	0
$883$	1.26049e7	0.544048	0.272024	−	0.962290i	$-0.412307\pi$
$883$	1.26049e7	0.544048	0.272024	+	0.962290i	$0.412307\pi$
$884$	0	0
$885$	3.32244e6	0.142593
$886$	0	0
$887$	3.69844e7	1.57837	0.789187	−	0.614153i	$-0.210502\pi$
$887$	3.69844e7	1.57837	0.789187	+	0.614153i	$0.210502\pi$
$888$	0	0
$889$	6.75250e6	0.286556
$890$	0	0
$891$	−823594.	−0.0347551
$892$	0	0
$893$	−1.75078e6	−0.0734689
$894$	0	0
$895$	3.24013e7	1.35209
$896$	0	0
$897$	−2.96056e6	−0.122855
$898$	0	0
$899$	−3.29147e7	−1.35829
$900$	0	0
$901$	3.24356e7	1.33110
$902$	0	0
$903$	3.08534e6	0.125917
$904$	0	0
$905$	−2.82589e7	−1.14692
$906$	0	0
$907$	−1.95026e6	−0.0787180	−0.0393590	−	0.999225i	$-0.512532\pi$
$907$	−1.95026e6	−0.0787180	−0.0393590	+	0.999225i	$0.512532\pi$
$908$	0	0
$909$	−7.93127e6	−0.318371
$910$	0	0
$911$	−1.72654e7	−0.689258	−0.344629	−	0.938739i	$-0.611995\pi$
$911$	−1.72654e7	−0.689258	−0.344629	+	0.938739i	$0.611995\pi$
$912$	0	0
$913$	7.76828e6	0.308424
$914$	0	0
$915$	−2.07554e7	−0.819556
$916$	0	0
$917$	3.53649e6	0.138883
$918$	0	0
$919$	−1.36353e7	−0.532570	−0.266285	−	0.963894i	$-0.585796\pi$
$919$	−1.36353e7	−0.532570	−0.266285	+	0.963894i	$0.585796\pi$
$920$	0	0
$921$	−2.30700e7	−0.896188
$922$	0	0
$923$	−1.37153e7	−0.529909
$924$	0	0
$925$	−1.01514e7	−0.390094
$926$	0	0
$927$	−1.28590e7	−0.491484
$928$	0	0
$929$	−1.31495e7	−0.499885	−0.249943	−	0.968261i	$-0.580412\pi$
$929$	−1.31495e7	−0.499885	−0.249943	+	0.968261i	$0.580412\pi$
$930$	0	0
$931$	−4.82582e6	−0.182472
$932$	0	0
$933$	1.10100e7	0.414077
$934$	0	0
$935$	5.18302e6	0.193889
$936$	0	0
$937$	−1.19341e7	−0.444058	−0.222029	−	0.975040i	$-0.571268\pi$
$937$	−1.19341e7	−0.444058	−0.222029	+	0.975040i	$0.571268\pi$
$938$	0	0
$939$	−2.13167e7	−0.788963
$940$	0	0
$941$	−4.51597e7	−1.66256	−0.831279	−	0.555856i	$-0.812391\pi$
$941$	−4.51597e7	−1.66256	−0.831279	+	0.555856i	$0.812391\pi$
$942$	0	0
$943$	1.05387e7	0.385929
$944$	0	0
$945$	1.99245e6	0.0725786
$946$	0	0
$947$	−4.17275e7	−1.51198	−0.755992	−	0.654581i	$-0.772845\pi$
$947$	−4.17275e7	−1.51198	−0.755992	+	0.654581i	$0.772845\pi$
$948$	0	0
$949$	1.29463e7	0.466638
$950$	0	0
$951$	−2.17022e7	−0.778132
$952$	0	0
$953$	−4.35052e7	−1.55170	−0.775851	−	0.630916i	$-0.782679\pi$
$953$	−4.35052e7	−1.55170	−0.775851	+	0.630916i	$0.782679\pi$
$954$	0	0
$955$	3.92174e6	0.139146
$956$	0	0
$957$	3.52192e6	0.124308
$958$	0	0
$959$	1.35607e7	0.476142
$960$	0	0
$961$	8.28501e7	2.89391
$962$	0	0
$963$	−1.19748e7	−0.416104
$964$	0	0
$965$	3.53234e7	1.22108
$966$	0	0
$967$	−164114.	−0.00564391	−0.00282195	−	0.999996i	$-0.500898\pi$
$967$	−164114.	−0.00564391	−0.00282195	+	0.999996i	$0.500898\pi$
$968$	0	0
$969$	2.87840e6	0.0984784
$970$	0	0
$971$	3.59222e7	1.22268	0.611342	−	0.791366i	$-0.290630\pi$
$971$	3.59222e7	1.22268	0.611342	+	0.791366i	$0.290630\pi$
$972$	0	0
$973$	2.94778e6	0.0998190
$974$	0	0
$975$	3.01094e6	0.101436
$976$	0	0
$977$	1.59785e7	0.535550	0.267775	−	0.963481i	$-0.413712\pi$
$977$	1.59785e7	0.535550	0.267775	+	0.963481i	$0.413712\pi$
$978$	0	0
$979$	3.25561e6	0.108562
$980$	0	0
$981$	−6.60788e6	−0.219225
$982$	0	0
$983$	5.38598e7	1.77779	0.888896	−	0.458109i	$-0.151473\pi$
$983$	5.38598e7	1.77779	0.888896	+	0.458109i	$0.151473\pi$
$984$	0	0
$985$	1.64542e7	0.540363
$986$	0	0
$987$	−2.55970e6	−0.0836365
$988$	0	0
$989$	5.47723e6	0.178062
$990$	0	0
$991$	−7.75820e6	−0.250944	−0.125472	−	0.992097i	$-0.540045\pi$
$991$	−7.75820e6	−0.250944	−0.125472	+	0.992097i	$0.540045\pi$
$992$	0	0
$993$	2.66953e7	0.859134
$994$	0	0
$995$	4.23962e6	0.135759
$996$	0	0
$997$	967884.	0.0308380	0.0154190	−	0.999881i	$-0.495092\pi$
$997$	967884.	0.0308380	0.0154190	+	0.999881i	$0.495092\pi$
$998$	0	0
$999$	−7.76607e6	−0.246200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.6.a.f.1.5	✓	6
4.3	odd	2		912.6.a.y.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.6.a.f.1.5	✓	6	1.1	even	1	trivial
912.6.a.y.1.5		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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} -125.529 q^{11} +351.083 q^{13} -419.452 q^{15} -885.933 q^{17} +361.000 q^{19} +527.793 q^{21} +936.962 q^{23} -952.906 q^{25} -729.000 q^{27} +3117.41 q^{29} -10558.4 q^{31} +1129.76 q^{33} -2733.13 q^{35} +10653.1 q^{37} -3159.75 q^{39} +11247.7 q^{41} +5845.74 q^{43} +3775.06 q^{45} -4849.81 q^{47} -13367.9 q^{49} +7973.40 q^{51} -36611.8 q^{53} -5850.36 q^{55} -3249.00 q^{57} -7920.92 q^{59} +49482.3 q^{61} -4750.14 q^{63} +16362.5 q^{65} +15138.3 q^{67} -8432.66 q^{69} -39065.7 q^{71} +36875.3 q^{73} +8576.15 q^{75} +7361.46 q^{77} +8875.46 q^{79} +6561.00 q^{81} -61884.5 q^{83} -41289.5 q^{85} -28056.7 q^{87} -25935.2 q^{89} -20588.8 q^{91} +95025.4 q^{93} +16824.7 q^{95} -88399.1 q^{97} -10167.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) \) 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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