LMFDB - Embedded newform 464.6.a.i.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,6,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("464.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	464.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:	$74.4180923932$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.275208$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.2828 q^{3} -97.7313 q^{5} -139.558 q^{7} +91.2623 q^{9} +O(q^{10})$	$q+18.2828 q^{3} -97.7313 q^{5} -139.558 q^{7} +91.2623 q^{9} -533.092 q^{11} -675.965 q^{13} -1786.81 q^{15} -268.994 q^{17} +2649.15 q^{19} -2551.52 q^{21} -794.438 q^{23} +6426.40 q^{25} -2774.20 q^{27} -841.000 q^{29} +4231.04 q^{31} -9746.44 q^{33} +13639.2 q^{35} -2689.54 q^{37} -12358.6 q^{39} +1395.36 q^{41} +23810.5 q^{43} -8919.18 q^{45} -11267.5 q^{47} +2669.53 q^{49} -4917.98 q^{51} -3396.67 q^{53} +52099.8 q^{55} +48433.9 q^{57} +2785.38 q^{59} +41551.7 q^{61} -12736.4 q^{63} +66062.9 q^{65} -8574.14 q^{67} -14524.6 q^{69} +6995.03 q^{71} -4994.73 q^{73} +117493. q^{75} +74397.5 q^{77} +23856.6 q^{79} -72896.9 q^{81} -43076.9 q^{83} +26289.2 q^{85} -15375.9 q^{87} +13806.4 q^{89} +94336.6 q^{91} +77355.4 q^{93} -258905. q^{95} -176400. q^{97} -48651.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) $ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9	$ 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) $ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 + 124 * q^11 - 460 * q^13 - 932 * q^15 + 184 * q^17 + 2392 * q^19 + 992 * q^21 + 1192 * q^23 + 1824 * q^25 - 2468 * q^27 - 3364 * q^29 + 19212 * q^31 - 10580 * q^33 + 22944 * q^35 - 10928 * q^37 + 8732 * q^39 - 1120 * q^41 + 21420 * q^43 - 8344 * q^45 - 23772 * q^47 + 10452 * q^49 - 12744 * q^51 + 8860 * q^53 + 52652 * q^55 + 48944 * q^57 + 10840 * q^59 + 49448 * q^61 - 27488 * q^63 + 97836 * q^65 + 7840 * q^67 + 58792 * q^69 + 48744 * q^71 - 74992 * q^73 + 90448 * q^75 + 128656 * q^77 + 106076 * q^79 - 59692 * q^81 - 62888 * q^83 + 23848 * q^85 - 23548 * q^87 + 107568 * q^89 + 268896 * q^91 + 221460 * q^93 - 147352 * q^95 - 49520 * q^97 - 166720 * 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
$3$	18.2828	1.17284	0.586422	−	0.810005i	$-0.300536\pi$
$3$	18.2828	1.17284	0.586422	+	0.810005i	$0.300536\pi$
$4$	0	0
$5$	−97.7313	−1.74827	−0.874135	−	0.485683i	$-0.838571\pi$
$5$	−97.7313	−1.74827	−0.874135	+	0.485683i	$0.838571\pi$
$6$	0	0
$7$	−139.558	−1.07649	−0.538246	−	0.842788i	$-0.680913\pi$
$7$	−139.558	−1.07649	−0.538246	+	0.842788i	$0.680913\pi$
$8$	0	0
$9$	91.2623	0.375565
$10$	0	0
$11$	−533.092	−1.32838	−0.664188	−	0.747566i	$-0.731222\pi$
$11$	−533.092	−1.32838	−0.664188	+	0.747566i	$0.731222\pi$
$12$	0	0
$13$	−675.965	−1.10934	−0.554672	−	0.832069i	$-0.687156\pi$
$13$	−675.965	−1.10934	−0.554672	+	0.832069i	$0.687156\pi$
$14$	0	0
$15$	−1786.81	−2.05045
$16$	0	0
$17$	−268.994	−0.225746	−0.112873	−	0.993609i	$-0.536005\pi$
$17$	−268.994	−0.225746	−0.112873	+	0.993609i	$0.536005\pi$
$18$	0	0
$19$	2649.15	1.68354	0.841768	−	0.539840i	$-0.181515\pi$
$19$	2649.15	1.68354	0.841768	+	0.539840i	$0.181515\pi$
$20$	0	0
$21$	−2551.52	−1.26256
$22$	0	0
$23$	−794.438	−0.313141	−0.156571	−	0.987667i	$-0.550044\pi$
$23$	−794.438	−0.313141	−0.156571	+	0.987667i	$0.550044\pi$
$24$	0	0
$25$	6426.40	2.05645
$26$	0	0
$27$	−2774.20	−0.732366
$28$	0	0
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	4231.04	0.790756	0.395378	−	0.918518i	$-0.370614\pi$
$31$	4231.04	0.790756	0.395378	+	0.918518i	$0.370614\pi$
$32$	0	0
$33$	−9746.44	−1.55798
$34$	0	0
$35$	13639.2	1.88200
$36$	0	0
$37$	−2689.54	−0.322979	−0.161489	−	0.986874i	$-0.551630\pi$
$37$	−2689.54	−0.322979	−0.161489	+	0.986874i	$0.551630\pi$
$38$	0	0
$39$	−12358.6	−1.30109
$40$	0	0
$41$	1395.36	0.129636	0.0648182	−	0.997897i	$-0.479353\pi$
$41$	1395.36	0.129636	0.0648182	+	0.997897i	$0.479353\pi$
$42$	0	0
$43$	23810.5	1.96380	0.981901	−	0.189395i	$-0.0606527\pi$
$43$	23810.5	1.96380	0.981901	+	0.189395i	$0.0606527\pi$
$44$	0	0
$45$	−8919.18	−0.656589
$46$	0	0
$47$	−11267.5	−0.744016	−0.372008	−	0.928229i	$-0.621331\pi$
$47$	−11267.5	−0.744016	−0.372008	+	0.928229i	$0.621331\pi$
$48$	0	0
$49$	2669.53	0.158834
$50$	0	0
$51$	−4917.98	−0.264766
$52$	0	0
$53$	−3396.67	−0.166097	−0.0830487	−	0.996545i	$-0.526466\pi$
$53$	−3396.67	−0.166097	−0.0830487	+	0.996545i	$0.526466\pi$
$54$	0	0
$55$	52099.8	2.32236
$56$	0	0
$57$	48433.9	1.97453
$58$	0	0
$59$	2785.38	0.104173	0.0520865	−	0.998643i	$-0.483413\pi$
$59$	2785.38	0.104173	0.0520865	+	0.998643i	$0.483413\pi$
$60$	0	0
$61$	41551.7	1.42976	0.714881	−	0.699246i	$-0.246481\pi$
$61$	41551.7	1.42976	0.714881	+	0.699246i	$0.246481\pi$
$62$	0	0
$63$	−12736.4	−0.404292
$64$	0	0
$65$	66062.9	1.93943
$66$	0	0
$67$	−8574.14	−0.233348	−0.116674	−	0.993170i	$-0.537223\pi$
$67$	−8574.14	−0.233348	−0.116674	+	0.993170i	$0.537223\pi$
$68$	0	0
$69$	−14524.6	−0.367266
$70$	0	0
$71$	6995.03	0.164681	0.0823405	−	0.996604i	$-0.473760\pi$
$71$	6995.03	0.164681	0.0823405	+	0.996604i	$0.473760\pi$
$72$	0	0
$73$	−4994.73	−0.109699	−0.0548497	−	0.998495i	$-0.517468\pi$
$73$	−4994.73	−0.109699	−0.0548497	+	0.998495i	$0.517468\pi$
$74$	0	0
$75$	117493.	2.41190
$76$	0	0
$77$	74397.5	1.42998
$78$	0	0
$79$	23856.6	0.430071	0.215036	−	0.976606i	$-0.431013\pi$
$79$	23856.6	0.430071	0.215036	+	0.976606i	$0.431013\pi$
$80$	0	0
$81$	−72896.9	−1.23452
$82$	0	0
$83$	−43076.9	−0.686356	−0.343178	−	0.939270i	$-0.611503\pi$
$83$	−43076.9	−0.686356	−0.343178	+	0.939270i	$0.611503\pi$
$84$	0	0
$85$	26289.2	0.394666
$86$	0	0
$87$	−15375.9	−0.217792
$88$	0	0
$89$	13806.4	0.184759	0.0923793	−	0.995724i	$-0.470553\pi$
$89$	13806.4	0.184759	0.0923793	+	0.995724i	$0.470553\pi$
$90$	0	0
$91$	94336.6	1.19420
$92$	0	0
$93$	77355.4	0.927435
$94$	0	0
$95$	−258905.	−2.94327
$96$	0	0
$97$	−176400.	−1.90357	−0.951786	−	0.306762i	$-0.900755\pi$
$97$	−176400.	−1.90357	−0.951786	+	0.306762i	$0.900755\pi$
$98$	0	0
$99$	−48651.2	−0.498891
$100$	0	0
$101$	−112043.	−1.09290	−0.546452	−	0.837490i	$-0.684022\pi$
$101$	−112043.	−1.09290	−0.546452	+	0.837490i	$0.684022\pi$
$102$	0	0
$103$	−38255.6	−0.355305	−0.177653	−	0.984093i	$-0.556850\pi$
$103$	−38255.6	−0.355305	−0.177653	+	0.984093i	$0.556850\pi$
$104$	0	0
$105$	249364.	2.20729
$106$	0	0
$107$	19410.6	0.163900	0.0819499	−	0.996636i	$-0.473885\pi$
$107$	19410.6	0.163900	0.0819499	+	0.996636i	$0.473885\pi$
$108$	0	0
$109$	−51029.2	−0.411389	−0.205694	−	0.978616i	$-0.565945\pi$
$109$	−51029.2	−0.411389	−0.205694	+	0.978616i	$0.565945\pi$
$110$	0	0
$111$	−49172.5	−0.378804
$112$	0	0
$113$	45687.3	0.336588	0.168294	−	0.985737i	$-0.446174\pi$
$113$	45687.3	0.336588	0.168294	+	0.985737i	$0.446174\pi$
$114$	0	0
$115$	77641.5	0.547456
$116$	0	0
$117$	−61690.1	−0.416630
$118$	0	0
$119$	37540.4	0.243014
$120$	0	0
$121$	123136.	0.764581
$122$	0	0
$123$	25511.2	0.152043
$124$	0	0
$125$	−322650.	−1.84696
$126$	0	0
$127$	−267210.	−1.47009	−0.735044	−	0.678019i	$-0.762839\pi$
$127$	−267210.	−1.47009	−0.735044	+	0.678019i	$0.762839\pi$
$128$	0	0
$129$	435324.	2.30323
$130$	0	0
$131$	165523.	0.842713	0.421356	−	0.906895i	$-0.361554\pi$
$131$	165523.	0.842713	0.421356	+	0.906895i	$0.361554\pi$
$132$	0	0
$133$	−369711.	−1.81231
$134$	0	0
$135$	271126.	1.28037
$136$	0	0
$137$	91000.4	0.414230	0.207115	−	0.978317i	$-0.433593\pi$
$137$	91000.4	0.414230	0.207115	+	0.978317i	$0.433593\pi$
$138$	0	0
$139$	400431.	1.75788	0.878942	−	0.476928i	$-0.158250\pi$
$139$	400431.	1.75788	0.878942	+	0.476928i	$0.158250\pi$
$140$	0	0
$141$	−206002.	−0.872616
$142$	0	0
$143$	360352.	1.47362
$144$	0	0
$145$	82192.0	0.324646
$146$	0	0
$147$	48806.6	0.186288
$148$	0	0
$149$	−18312.8	−0.0675756	−0.0337878	−	0.999429i	$-0.510757\pi$
$149$	−18312.8	−0.0675756	−0.0337878	+	0.999429i	$0.510757\pi$
$150$	0	0
$151$	17899.8	0.0638859	0.0319430	−	0.999490i	$-0.489831\pi$
$151$	17899.8	0.0638859	0.0319430	+	0.999490i	$0.489831\pi$
$152$	0	0
$153$	−24549.0	−0.0847824
$154$	0	0
$155$	−413505.	−1.38246
$156$	0	0
$157$	−413598.	−1.33915	−0.669576	−	0.742744i	$-0.733524\pi$
$157$	−413598.	−1.33915	−0.669576	+	0.742744i	$0.733524\pi$
$158$	0	0
$159$	−62100.7	−0.194807
$160$	0	0
$161$	110870.	0.337094
$162$	0	0
$163$	−178276.	−0.525562	−0.262781	−	0.964856i	$-0.584640\pi$
$163$	−178276.	−0.525562	−0.262781	+	0.964856i	$0.584640\pi$
$164$	0	0
$165$	952532.	2.72377
$166$	0	0
$167$	452143.	1.25454	0.627269	−	0.778802i	$-0.284172\pi$
$167$	452143.	1.25454	0.627269	+	0.778802i	$0.284172\pi$
$168$	0	0
$169$	85635.9	0.230642
$170$	0	0
$171$	241767.	0.632277
$172$	0	0
$173$	754374.	1.91634	0.958168	−	0.286206i	$-0.0923943\pi$
$173$	754374.	1.91634	0.958168	+	0.286206i	$0.0923943\pi$
$174$	0	0
$175$	−896858.	−2.21375
$176$	0	0
$177$	50924.7	0.122179
$178$	0	0
$179$	352813.	0.823024	0.411512	−	0.911404i	$-0.365001\pi$
$179$	352813.	0.823024	0.411512	+	0.911404i	$0.365001\pi$
$180$	0	0
$181$	227087.	0.515224	0.257612	−	0.966248i	$-0.417064\pi$
$181$	227087.	0.515224	0.257612	+	0.966248i	$0.417064\pi$
$182$	0	0
$183$	759683.	1.67689
$184$	0	0
$185$	262852.	0.564654
$186$	0	0
$187$	143399.	0.299876
$188$	0	0
$189$	387162.	0.788385
$190$	0	0
$191$	−183415.	−0.363790	−0.181895	−	0.983318i	$-0.558223\pi$
$191$	−183415.	−0.363790	−0.181895	+	0.983318i	$0.558223\pi$
$192$	0	0
$193$	997005.	1.92665	0.963327	−	0.268329i	$-0.0864714\pi$
$193$	997005.	1.92665	0.963327	+	0.268329i	$0.0864714\pi$
$194$	0	0
$195$	1.20782e6	2.27465
$196$	0	0
$197$	−861288.	−1.58119	−0.790593	−	0.612342i	$-0.790227\pi$
$197$	−861288.	−1.58119	−0.790593	+	0.612342i	$0.790227\pi$
$198$	0	0
$199$	837743.	1.49961	0.749805	−	0.661659i	$-0.230147\pi$
$199$	837743.	1.49961	0.749805	+	0.661659i	$0.230147\pi$
$200$	0	0
$201$	−156760.	−0.273681
$202$	0	0
$203$	117369.	0.199899
$204$	0	0
$205$	−136371.	−0.226640
$206$	0	0
$207$	−72502.2	−0.117605
$208$	0	0
$209$	−1.41224e6	−2.23637
$210$	0	0
$211$	637594.	0.985912	0.492956	−	0.870054i	$-0.335916\pi$
$211$	637594.	0.985912	0.492956	+	0.870054i	$0.335916\pi$
$212$	0	0
$213$	127889.	0.193145
$214$	0	0
$215$	−2.32703e6	−3.43326
$216$	0	0
$217$	−590477.	−0.851243
$218$	0	0
$219$	−91317.8	−0.128660
$220$	0	0
$221$	181831.	0.250430
$222$	0	0
$223$	−45526.9	−0.0613064	−0.0306532	−	0.999530i	$-0.509759\pi$
$223$	−45526.9	−0.0613064	−0.0306532	+	0.999530i	$0.509759\pi$
$224$	0	0
$225$	586488.	0.772330
$226$	0	0
$227$	−1.33313e6	−1.71715	−0.858575	−	0.512688i	$-0.828650\pi$
$227$	−1.33313e6	−1.71715	−0.858575	+	0.512688i	$0.828650\pi$
$228$	0	0
$229$	830883.	1.04701	0.523506	−	0.852022i	$-0.324624\pi$
$229$	830883.	1.04701	0.523506	+	0.852022i	$0.324624\pi$
$230$	0	0
$231$	1.36020e6	1.67715
$232$	0	0
$233$	767627.	0.926319	0.463159	−	0.886275i	$-0.346716\pi$
$233$	767627.	0.926319	0.463159	+	0.886275i	$0.346716\pi$
$234$	0	0
$235$	1.10119e6	1.30074
$236$	0	0
$237$	436166.	0.504407
$238$	0	0
$239$	1.22858e6	1.39126	0.695631	−	0.718399i	$-0.255125\pi$
$239$	1.22858e6	1.39126	0.695631	+	0.718399i	$0.255125\pi$
$240$	0	0
$241$	304030.	0.337190	0.168595	−	0.985685i	$-0.446077\pi$
$241$	304030.	0.337190	0.168595	+	0.985685i	$0.446077\pi$
$242$	0	0
$243$	−658633.	−0.715530
$244$	0	0
$245$	−260897.	−0.277685
$246$	0	0
$247$	−1.79073e6	−1.86762
$248$	0	0
$249$	−787569.	−0.804989
$250$	0	0
$251$	−850953.	−0.852553	−0.426276	−	0.904593i	$-0.640175\pi$
$251$	−850953.	−0.852553	−0.426276	+	0.904593i	$0.640175\pi$
$252$	0	0
$253$	423509.	0.415969
$254$	0	0
$255$	480641.	0.462882
$256$	0	0
$257$	28252.8	0.0266826	0.0133413	−	0.999911i	$-0.495753\pi$
$257$	28252.8	0.0266826	0.0133413	+	0.999911i	$0.495753\pi$
$258$	0	0
$259$	375348.	0.347684
$260$	0	0
$261$	−76751.6	−0.0697406
$262$	0	0
$263$	−393978.	−0.351223	−0.175611	−	0.984460i	$-0.556190\pi$
$263$	−393978.	−0.351223	−0.175611	+	0.984460i	$0.556190\pi$
$264$	0	0
$265$	331960.	0.290383
$266$	0	0
$267$	252420.	0.216693
$268$	0	0
$269$	453727.	0.382309	0.191154	−	0.981560i	$-0.438777\pi$
$269$	453727.	0.382309	0.191154	+	0.981560i	$0.438777\pi$
$270$	0	0
$271$	1.54312e6	1.27637	0.638186	−	0.769882i	$-0.279685\pi$
$271$	1.54312e6	1.27637	0.638186	+	0.769882i	$0.279685\pi$
$272$	0	0
$273$	1.72474e6	1.40061
$274$	0	0
$275$	−3.42587e6	−2.73174
$276$	0	0
$277$	−1.13023e6	−0.885050	−0.442525	−	0.896756i	$-0.645917\pi$
$277$	−1.13023e6	−0.885050	−0.442525	+	0.896756i	$0.645917\pi$
$278$	0	0
$279$	386134.	0.296980
$280$	0	0
$281$	−1.17984e6	−0.891371	−0.445685	−	0.895190i	$-0.647040\pi$
$281$	−1.17984e6	−0.891371	−0.445685	+	0.895190i	$0.647040\pi$
$282$	0	0
$283$	−345340.	−0.256319	−0.128160	−	0.991754i	$-0.540907\pi$
$283$	−345340.	−0.256319	−0.128160	+	0.991754i	$0.540907\pi$
$284$	0	0
$285$	−4.73351e6	−3.45200
$286$	0	0
$287$	−194734.	−0.139553
$288$	0	0
$289$	−1.34750e6	−0.949039
$290$	0	0
$291$	−3.22509e6	−2.23260
$292$	0	0
$293$	2.40825e6	1.63883	0.819414	−	0.573202i	$-0.194299\pi$
$293$	2.40825e6	1.63883	0.819414	+	0.573202i	$0.194299\pi$
$294$	0	0
$295$	−272219.	−0.182123
$296$	0	0
$297$	1.47890e6	0.972856
$298$	0	0
$299$	537013.	0.347381
$300$	0	0
$301$	−3.32296e6	−2.11402
$302$	0	0
$303$	−2.04847e6	−1.28181
$304$	0	0
$305$	−4.06090e6	−2.49961
$306$	0	0
$307$	−595416.	−0.360558	−0.180279	−	0.983616i	$-0.557700\pi$
$307$	−595416.	−0.360558	−0.180279	+	0.983616i	$0.557700\pi$
$308$	0	0
$309$	−699420.	−0.416718
$310$	0	0
$311$	−999900.	−0.586213	−0.293107	−	0.956080i	$-0.594689\pi$
$311$	−999900.	−0.586213	−0.293107	+	0.956080i	$0.594689\pi$
$312$	0	0
$313$	−365270.	−0.210743	−0.105371	−	0.994433i	$-0.533603\pi$
$313$	−365270.	−0.210743	−0.105371	+	0.994433i	$0.533603\pi$
$314$	0	0
$315$	1.24475e6	0.706813
$316$	0	0
$317$	1.60635e6	0.897823	0.448912	−	0.893576i	$-0.351812\pi$
$317$	1.60635e6	0.897823	0.448912	+	0.893576i	$0.351812\pi$
$318$	0	0
$319$	448331.	0.246673
$320$	0	0
$321$	354880.	0.192229
$322$	0	0
$323$	−712606.	−0.380052
$324$	0	0
$325$	−4.34403e6	−2.28131
$326$	0	0
$327$	−932959.	−0.482495
$328$	0	0
$329$	1.57247e6	0.800928
$330$	0	0
$331$	535124.	0.268463	0.134232	−	0.990950i	$-0.457143\pi$
$331$	535124.	0.268463	0.134232	+	0.990950i	$0.457143\pi$
$332$	0	0
$333$	−245454.	−0.121299
$334$	0	0
$335$	837961.	0.407955
$336$	0	0
$337$	1.93303e6	0.927178	0.463589	−	0.886050i	$-0.346561\pi$
$337$	1.93303e6	0.927178	0.463589	+	0.886050i	$0.346561\pi$
$338$	0	0
$339$	835293.	0.394766
$340$	0	0
$341$	−2.25553e6	−1.05042
$342$	0	0
$343$	1.97300e6	0.905508
$344$	0	0
$345$	1.41951e6	0.642081
$346$	0	0
$347$	2.83701e6	1.26485	0.632423	−	0.774623i	$-0.282060\pi$
$347$	2.83701e6	1.26485	0.632423	+	0.774623i	$0.282060\pi$
$348$	0	0
$349$	1.11395e6	0.489555	0.244778	−	0.969579i	$-0.421285\pi$
$349$	1.11395e6	0.489555	0.244778	+	0.969579i	$0.421285\pi$
$350$	0	0
$351$	1.87526e6	0.812445
$352$	0	0
$353$	−3.49053e6	−1.49092	−0.745461	−	0.666549i	$-0.767771\pi$
$353$	−3.49053e6	−1.49092	−0.745461	+	0.666549i	$0.767771\pi$
$354$	0	0
$355$	−683633.	−0.287907
$356$	0	0
$357$	686345.	0.285018
$358$	0	0
$359$	−469676.	−0.192337	−0.0961684	−	0.995365i	$-0.530659\pi$
$359$	−469676.	−0.192337	−0.0961684	+	0.995365i	$0.530659\pi$
$360$	0	0
$361$	4.54189e6	1.83429
$362$	0	0
$363$	2.25128e6	0.896735
$364$	0	0
$365$	488141.	0.191784
$366$	0	0
$367$	−2.70229e6	−1.04729	−0.523645	−	0.851936i	$-0.675428\pi$
$367$	−2.70229e6	−1.04729	−0.523645	+	0.851936i	$0.675428\pi$
$368$	0	0
$369$	127344.	0.0486869
$370$	0	0
$371$	474033.	0.178803
$372$	0	0
$373$	2.77666e6	1.03336	0.516678	−	0.856180i	$-0.327168\pi$
$373$	2.77666e6	1.03336	0.516678	+	0.856180i	$0.327168\pi$
$374$	0	0
$375$	−5.89897e6	−2.16620
$376$	0	0
$377$	568487.	0.206000
$378$	0	0
$379$	4.41837e6	1.58002	0.790012	−	0.613091i	$-0.210074\pi$
$379$	4.41837e6	1.58002	0.790012	+	0.613091i	$0.210074\pi$
$380$	0	0
$381$	−4.88536e6	−1.72419
$382$	0	0
$383$	−1.37796e6	−0.479999	−0.239999	−	0.970773i	$-0.577147\pi$
$383$	−1.37796e6	−0.479999	−0.239999	+	0.970773i	$0.577147\pi$
$384$	0	0
$385$	−7.27096e6	−2.50000
$386$	0	0
$387$	2.17300e6	0.737535
$388$	0	0
$389$	649334.	0.217568	0.108784	−	0.994065i	$-0.465304\pi$
$389$	649334.	0.217568	0.108784	+	0.994065i	$0.465304\pi$
$390$	0	0
$391$	213699.	0.0706906
$392$	0	0
$393$	3.02623e6	0.988371
$394$	0	0
$395$	−2.33154e6	−0.751881
$396$	0	0
$397$	−2.52909e6	−0.805356	−0.402678	−	0.915342i	$-0.631921\pi$
$397$	−2.52909e6	−0.805356	−0.402678	+	0.915342i	$0.631921\pi$
$398$	0	0
$399$	−6.75936e6	−2.12556
$400$	0	0
$401$	−3.64231e6	−1.13114	−0.565569	−	0.824701i	$-0.691344\pi$
$401$	−3.64231e6	−1.13114	−0.565569	+	0.824701i	$0.691344\pi$
$402$	0	0
$403$	−2.86003e6	−0.877220
$404$	0	0
$405$	7.12431e6	2.15827
$406$	0	0
$407$	1.43377e6	0.429037
$408$	0	0
$409$	2.68522e6	0.793729	0.396865	−	0.917877i	$-0.370098\pi$
$409$	2.68522e6	0.793729	0.396865	+	0.917877i	$0.370098\pi$
$410$	0	0
$411$	1.66375e6	0.485828
$412$	0	0
$413$	−388724.	−0.112141
$414$	0	0
$415$	4.20996e6	1.19994
$416$	0	0
$417$	7.32101e6	2.06173
$418$	0	0
$419$	4.33500e6	1.20630	0.603148	−	0.797629i	$-0.293913\pi$
$419$	4.33500e6	1.20630	0.603148	+	0.797629i	$0.293913\pi$
$420$	0	0
$421$	3.38962e6	0.932063	0.466032	−	0.884768i	$-0.345683\pi$
$421$	3.38962e6	0.932063	0.466032	+	0.884768i	$0.345683\pi$
$422$	0	0
$423$	−1.02830e6	−0.279426
$424$	0	0
$425$	−1.72867e6	−0.464236
$426$	0	0
$427$	−5.79888e6	−1.53913
$428$	0	0
$429$	6.58826e6	1.72833
$430$	0	0
$431$	−290831.	−0.0754132	−0.0377066	−	0.999289i	$-0.512005\pi$
$431$	−290831.	−0.0754132	−0.0377066	+	0.999289i	$0.512005\pi$
$432$	0	0
$433$	−2.64162e6	−0.677097	−0.338549	−	0.940949i	$-0.609936\pi$
$433$	−2.64162e6	−0.677097	−0.338549	+	0.940949i	$0.609936\pi$
$434$	0	0
$435$	1.50270e6	0.380759
$436$	0	0
$437$	−2.10458e6	−0.527185
$438$	0	0
$439$	−5.65486e6	−1.40043	−0.700214	−	0.713933i	$-0.746912\pi$
$439$	−5.65486e6	−1.40043	−0.700214	+	0.713933i	$0.746912\pi$
$440$	0	0
$441$	243627.	0.0596526
$442$	0	0
$443$	−1.31778e6	−0.319031	−0.159515	−	0.987195i	$-0.550993\pi$
$443$	−1.31778e6	−0.319031	−0.159515	+	0.987195i	$0.550993\pi$
$444$	0	0
$445$	−1.34932e6	−0.323008
$446$	0	0
$447$	−334810.	−0.0792556
$448$	0	0
$449$	−6.50617e6	−1.52303	−0.761517	−	0.648145i	$-0.775545\pi$
$449$	−6.50617e6	−1.52303	−0.761517	+	0.648145i	$0.775545\pi$
$450$	0	0
$451$	−743857.	−0.172206
$452$	0	0
$453$	327259.	0.0749283
$454$	0	0
$455$	−9.21963e6	−2.08778
$456$	0	0
$457$	−3.50691e6	−0.785477	−0.392739	−	0.919650i	$-0.628472\pi$
$457$	−3.50691e6	−0.785477	−0.392739	+	0.919650i	$0.628472\pi$
$458$	0	0
$459$	746244.	0.165329
$460$	0	0
$461$	−1.22247e6	−0.267908	−0.133954	−	0.990988i	$-0.542767\pi$
$461$	−1.22247e6	−0.267908	−0.133954	+	0.990988i	$0.542767\pi$
$462$	0	0
$463$	−382178.	−0.0828540	−0.0414270	−	0.999142i	$-0.513190\pi$
$463$	−382178.	−0.0828540	−0.0414270	+	0.999142i	$0.513190\pi$
$464$	0	0
$465$	−7.56004e6	−1.62141
$466$	0	0
$467$	1.81409e6	0.384916	0.192458	−	0.981305i	$-0.438354\pi$
$467$	1.81409e6	0.384916	0.192458	+	0.981305i	$0.438354\pi$
$468$	0	0
$469$	1.19659e6	0.251197
$470$	0	0
$471$	−7.56175e6	−1.57062
$472$	0	0
$473$	−1.26932e7	−2.60867
$474$	0	0
$475$	1.70245e7	3.46210
$476$	0	0
$477$	−309987.	−0.0623804
$478$	0	0
$479$	−4.87397e6	−0.970609	−0.485304	−	0.874345i	$-0.661291\pi$
$479$	−4.87397e6	−0.970609	−0.485304	+	0.874345i	$0.661291\pi$
$480$	0	0
$481$	1.81804e6	0.358294
$482$	0	0
$483$	2.02703e6	0.395359
$484$	0	0
$485$	1.72398e7	3.32796
$486$	0	0
$487$	1.84954e6	0.353379	0.176690	−	0.984267i	$-0.443461\pi$
$487$	1.84954e6	0.353379	0.176690	+	0.984267i	$0.443461\pi$
$488$	0	0
$489$	−3.25939e6	−0.616403
$490$	0	0
$491$	5.47088e6	1.02413	0.512063	−	0.858948i	$-0.328881\pi$
$491$	5.47088e6	1.02413	0.512063	+	0.858948i	$0.328881\pi$
$492$	0	0
$493$	226224.	0.0419201
$494$	0	0
$495$	4.75475e6	0.872196
$496$	0	0
$497$	−976215.	−0.177278
$498$	0	0
$499$	−5.27760e6	−0.948823	−0.474411	−	0.880303i	$-0.657339\pi$
$499$	−5.27760e6	−0.948823	−0.474411	+	0.880303i	$0.657339\pi$
$500$	0	0
$501$	8.26645e6	1.47138
$502$	0	0
$503$	−6.07721e6	−1.07099	−0.535494	−	0.844539i	$-0.679875\pi$
$503$	−6.07721e6	−1.07099	−0.535494	+	0.844539i	$0.679875\pi$
$504$	0	0
$505$	1.09501e7	1.91069
$506$	0	0
$507$	1.56567e6	0.270508
$508$	0	0
$509$	4.28972e6	0.733896	0.366948	−	0.930242i	$-0.380403\pi$
$509$	4.28972e6	0.733896	0.366948	+	0.930242i	$0.380403\pi$
$510$	0	0
$511$	697056.	0.118091
$512$	0	0
$513$	−7.34926e6	−1.23296
$514$	0	0
$515$	3.73876e6	0.621169
$516$	0	0
$517$	6.00661e6	0.988333
$518$	0	0
$519$	1.37921e7	2.24756
$520$	0	0
$521$	3.12541e6	0.504444	0.252222	−	0.967669i	$-0.418839\pi$
$521$	3.12541e6	0.504444	0.252222	+	0.967669i	$0.418839\pi$
$522$	0	0
$523$	6.57035e6	1.05035	0.525176	−	0.850994i	$-0.324000\pi$
$523$	6.57035e6	1.05035	0.525176	+	0.850994i	$0.324000\pi$
$524$	0	0
$525$	−1.63971e7	−2.59639
$526$	0	0
$527$	−1.13813e6	−0.178510
$528$	0	0
$529$	−5.80521e6	−0.901942
$530$	0	0
$531$	254201.	0.0391237
$532$	0	0
$533$	−943216.	−0.143811
$534$	0	0
$535$	−1.89702e6	−0.286541
$536$	0	0
$537$	6.45043e6	0.965279
$538$	0	0
$539$	−1.42311e6	−0.210992
$540$	0	0
$541$	−6.72381e6	−0.987694	−0.493847	−	0.869549i	$-0.664410\pi$
$541$	−6.72381e6	−0.987694	−0.493847	+	0.869549i	$0.664410\pi$
$542$	0	0
$543$	4.15180e6	0.604278
$544$	0	0
$545$	4.98715e6	0.719219
$546$	0	0
$547$	−1.19150e6	−0.170264	−0.0851322	−	0.996370i	$-0.527131\pi$
$547$	−1.19150e6	−0.170264	−0.0851322	+	0.996370i	$0.527131\pi$
$548$	0	0
$549$	3.79210e6	0.536969
$550$	0	0
$551$	−2.22793e6	−0.312625
$552$	0	0
$553$	−3.32939e6	−0.462968
$554$	0	0
$555$	4.80569e6	0.662252
$556$	0	0
$557$	2.43064e6	0.331958	0.165979	−	0.986129i	$-0.446922\pi$
$557$	2.43064e6	0.331958	0.165979	+	0.986129i	$0.446922\pi$
$558$	0	0
$559$	−1.60951e7	−2.17853
$560$	0	0
$561$	2.62174e6	0.351708
$562$	0	0
$563$	1.05702e7	1.40544	0.702721	−	0.711466i	$-0.251968\pi$
$563$	1.05702e7	1.40544	0.702721	+	0.711466i	$0.251968\pi$
$564$	0	0
$565$	−4.46508e6	−0.588447
$566$	0	0
$567$	1.01734e7	1.32895
$568$	0	0
$569$	5.17877e6	0.670573	0.335286	−	0.942116i	$-0.391167\pi$
$569$	5.17877e6	0.670573	0.335286	+	0.942116i	$0.391167\pi$
$570$	0	0
$571$	3.92258e6	0.503479	0.251739	−	0.967795i	$-0.418997\pi$
$571$	3.92258e6	0.503479	0.251739	+	0.967795i	$0.418997\pi$
$572$	0	0
$573$	−3.35335e6	−0.426670
$574$	0	0
$575$	−5.10538e6	−0.643959
$576$	0	0
$577$	1.12133e7	1.40215	0.701074	−	0.713089i	$-0.252704\pi$
$577$	1.12133e7	1.40215	0.701074	+	0.713089i	$0.252704\pi$
$578$	0	0
$579$	1.82281e7	2.25967
$580$	0	0
$581$	6.01174e6	0.738857
$582$	0	0
$583$	1.81074e6	0.220640
$584$	0	0
$585$	6.02905e6	0.728382
$586$	0	0
$587$	−1.03337e7	−1.23783	−0.618915	−	0.785458i	$-0.712427\pi$
$587$	−1.03337e7	−1.23783	−0.618915	+	0.785458i	$0.712427\pi$
$588$	0	0
$589$	1.12086e7	1.33127
$590$	0	0
$591$	−1.57468e7	−1.85448
$592$	0	0
$593$	−1.58653e7	−1.85272	−0.926362	−	0.376634i	$-0.877081\pi$
$593$	−1.58653e7	−1.85272	−0.926362	+	0.376634i	$0.877081\pi$
$594$	0	0
$595$	−3.66887e6	−0.424855
$596$	0	0
$597$	1.53163e7	1.75881
$598$	0	0
$599$	9.66314e6	1.10040	0.550201	−	0.835032i	$-0.314551\pi$
$599$	9.66314e6	1.10040	0.550201	+	0.835032i	$0.314551\pi$
$600$	0	0
$601$	−1.07881e7	−1.21831	−0.609155	−	0.793051i	$-0.708491\pi$
$601$	−1.07881e7	−1.21831	−0.609155	+	0.793051i	$0.708491\pi$
$602$	0	0
$603$	−782495.	−0.0876372
$604$	0	0
$605$	−1.20343e7	−1.33669
$606$	0	0
$607$	−6.76786e6	−0.745555	−0.372778	−	0.927921i	$-0.621595\pi$
$607$	−6.76786e6	−0.745555	−0.372778	+	0.927921i	$0.621595\pi$
$608$	0	0
$609$	2.14583e6	0.234451
$610$	0	0
$611$	7.61643e6	0.825370
$612$	0	0
$613$	2.91439e6	0.313253	0.156627	−	0.987658i	$-0.449938\pi$
$613$	2.91439e6	0.313253	0.156627	+	0.987658i	$0.449938\pi$
$614$	0	0
$615$	−2.49324e6	−0.265813
$616$	0	0
$617$	−1.01418e7	−1.07251	−0.536254	−	0.844056i	$-0.680161\pi$
$617$	−1.01418e7	−1.07251	−0.536254	+	0.844056i	$0.680161\pi$
$618$	0	0
$619$	6.89280e6	0.723051	0.361526	−	0.932362i	$-0.382256\pi$
$619$	6.89280e6	0.723051	0.361526	+	0.932362i	$0.382256\pi$
$620$	0	0
$621$	2.20393e6	0.229334
$622$	0	0
$623$	−1.92680e6	−0.198891
$624$	0	0
$625$	1.14505e7	1.17253
$626$	0	0
$627$	−2.58198e7	−2.62291
$628$	0	0
$629$	723472.	0.0729113
$630$	0	0
$631$	7.52515e6	0.752388	0.376194	−	0.926541i	$-0.377233\pi$
$631$	7.52515e6	0.752388	0.376194	+	0.926541i	$0.377233\pi$
$632$	0	0
$633$	1.16570e7	1.15632
$634$	0	0
$635$	2.61148e7	2.57011
$636$	0	0
$637$	−1.80451e6	−0.176202
$638$	0	0
$639$	638382.	0.0618484
$640$	0	0
$641$	−278549.	−0.0267767	−0.0133883	−	0.999910i	$-0.504262\pi$
$641$	−278549.	−0.0267767	−0.0133883	+	0.999910i	$0.504262\pi$
$642$	0	0
$643$	3.63623e6	0.346836	0.173418	−	0.984848i	$-0.444519\pi$
$643$	3.63623e6	0.346836	0.173418	+	0.984848i	$0.444519\pi$
$644$	0	0
$645$	−4.25448e7	−4.02668
$646$	0	0
$647$	1.08234e7	1.01649	0.508243	−	0.861214i	$-0.330295\pi$
$647$	1.08234e7	1.01649	0.508243	+	0.861214i	$0.330295\pi$
$648$	0	0
$649$	−1.48487e6	−0.138381
$650$	0	0
$651$	−1.07956e7	−0.998376
$652$	0	0
$653$	8.86232e6	0.813326	0.406663	−	0.913578i	$-0.366692\pi$
$653$	8.86232e6	0.813326	0.406663	+	0.913578i	$0.366692\pi$
$654$	0	0
$655$	−1.61768e7	−1.47329
$656$	0	0
$657$	−455830.	−0.0411993
$658$	0	0
$659$	4.94619e6	0.443667	0.221833	−	0.975085i	$-0.428796\pi$
$659$	4.94619e6	0.443667	0.221833	+	0.975085i	$0.428796\pi$
$660$	0	0
$661$	−6.34440e6	−0.564790	−0.282395	−	0.959298i	$-0.591129\pi$
$661$	−6.34440e6	−0.564790	−0.282395	+	0.959298i	$0.591129\pi$
$662$	0	0
$663$	3.32438e6	0.293716
$664$	0	0
$665$	3.61323e7	3.16841
$666$	0	0
$667$	668122.	0.0581489
$668$	0	0
$669$	−832361.	−0.0719029
$670$	0	0
$671$	−2.21509e7	−1.89926
$672$	0	0
$673$	1.00239e7	0.853095	0.426548	−	0.904465i	$-0.359730\pi$
$673$	1.00239e7	0.853095	0.426548	+	0.904465i	$0.359730\pi$
$674$	0	0
$675$	−1.78281e7	−1.50607
$676$	0	0
$677$	3.44231e6	0.288654	0.144327	−	0.989530i	$-0.453898\pi$
$677$	3.44231e6	0.288654	0.144327	+	0.989530i	$0.453898\pi$
$678$	0	0
$679$	2.46181e7	2.04918
$680$	0	0
$681$	−2.43734e7	−2.01395
$682$	0	0
$683$	−2.51352e6	−0.206172	−0.103086	−	0.994672i	$-0.532872\pi$
$683$	−2.51352e6	−0.206172	−0.103086	+	0.994672i	$0.532872\pi$
$684$	0	0
$685$	−8.89359e6	−0.724187
$686$	0	0
$687$	1.51909e7	1.22798
$688$	0	0
$689$	2.29603e6	0.184259
$690$	0	0
$691$	2.01696e7	1.60695	0.803474	−	0.595340i	$-0.202983\pi$
$691$	2.01696e7	1.60695	0.803474	+	0.595340i	$0.202983\pi$
$692$	0	0
$693$	6.78968e6	0.537052
$694$	0	0
$695$	−3.91346e7	−3.07326
$696$	0	0
$697$	−375345.	−0.0292650
$698$	0	0
$699$	1.40344e7	1.08643
$700$	0	0
$701$	1.67322e6	0.128605	0.0643025	−	0.997930i	$-0.479518\pi$
$701$	1.67322e6	0.128605	0.0643025	+	0.997930i	$0.479518\pi$
$702$	0	0
$703$	−7.12499e6	−0.543746
$704$	0	0
$705$	2.01328e7	1.52557
$706$	0	0
$707$	1.56366e7	1.17650
$708$	0	0
$709$	6.28913e6	0.469867	0.234933	−	0.972011i	$-0.424513\pi$
$709$	6.28913e6	0.469867	0.234933	+	0.972011i	$0.424513\pi$
$710$	0	0
$711$	2.17721e6	0.161520
$712$	0	0
$713$	−3.36130e6	−0.247619
$714$	0	0
$715$	−3.52177e7	−2.57629
$716$	0	0
$717$	2.24620e7	1.63173
$718$	0	0
$719$	−2.87519e6	−0.207417	−0.103709	−	0.994608i	$-0.533071\pi$
$719$	−2.87519e6	−0.207417	−0.103709	+	0.994608i	$0.533071\pi$
$720$	0	0
$721$	5.33888e6	0.382483
$722$	0	0
$723$	5.55854e6	0.395471
$724$	0	0
$725$	−5.40461e6	−0.381873
$726$	0	0
$727$	−2.76368e6	−0.193933	−0.0969665	−	0.995288i	$-0.530914\pi$
$727$	−2.76368e6	−0.193933	−0.0969665	+	0.995288i	$0.530914\pi$
$728$	0	0
$729$	5.67227e6	0.395310
$730$	0	0
$731$	−6.40490e6	−0.443321
$732$	0	0
$733$	−1.06643e7	−0.733115	−0.366557	−	0.930395i	$-0.619464\pi$
$733$	−1.06643e7	−0.733115	−0.366557	+	0.930395i	$0.619464\pi$
$734$	0	0
$735$	−4.76993e6	−0.325682
$736$	0	0
$737$	4.57081e6	0.309973
$738$	0	0
$739$	3.98603e6	0.268491	0.134245	−	0.990948i	$-0.457139\pi$
$739$	3.98603e6	0.268491	0.134245	+	0.990948i	$0.457139\pi$
$740$	0	0
$741$	−3.27397e7	−2.19043
$742$	0	0
$743$	2.19748e7	1.46034	0.730170	−	0.683266i	$-0.239441\pi$
$743$	2.19748e7	1.46034	0.730170	+	0.683266i	$0.239441\pi$
$744$	0	0
$745$	1.78974e6	0.118140
$746$	0	0
$747$	−3.93130e6	−0.257771
$748$	0	0
$749$	−2.70891e6	−0.176437
$750$	0	0
$751$	1.84068e7	1.19091	0.595456	−	0.803388i	$-0.296972\pi$
$751$	1.84068e7	1.19091	0.595456	+	0.803388i	$0.296972\pi$
$752$	0	0
$753$	−1.55578e7	−0.999912
$754$	0	0
$755$	−1.74937e6	−0.111690
$756$	0	0
$757$	2.78199e7	1.76447	0.882237	−	0.470805i	$-0.156037\pi$
$757$	2.78199e7	1.76447	0.882237	+	0.470805i	$0.156037\pi$
$758$	0	0
$759$	7.74295e6	0.487867
$760$	0	0
$761$	1.96481e6	0.122987	0.0614935	−	0.998107i	$-0.480414\pi$
$761$	1.96481e6	0.122987	0.0614935	+	0.998107i	$0.480414\pi$
$762$	0	0
$763$	7.12155e6	0.442857
$764$	0	0
$765$	2.39921e6	0.148223
$766$	0	0
$767$	−1.88282e6	−0.115564
$768$	0	0
$769$	−1.16064e7	−0.707754	−0.353877	−	0.935292i	$-0.615137\pi$
$769$	−1.16064e7	−0.707754	−0.353877	+	0.935292i	$0.615137\pi$
$770$	0	0
$771$	516541.	0.0312946
$772$	0	0
$773$	2.10057e7	1.26441	0.632207	−	0.774800i	$-0.282149\pi$
$773$	2.10057e7	1.26441	0.632207	+	0.774800i	$0.282149\pi$
$774$	0	0
$775$	2.71904e7	1.62615
$776$	0	0
$777$	6.86243e6	0.407779
$778$	0	0
$779$	3.69652e6	0.218248
$780$	0	0
$781$	−3.72900e6	−0.218758
$782$	0	0
$783$	2.33310e6	0.135997
$784$	0	0
$785$	4.04215e7	2.34120
$786$	0	0
$787$	8.11927e6	0.467283	0.233642	−	0.972323i	$-0.424936\pi$
$787$	8.11927e6	0.467283	0.233642	+	0.972323i	$0.424936\pi$
$788$	0	0
$789$	−7.20304e6	−0.411930
$790$	0	0
$791$	−6.37604e6	−0.362335
$792$	0	0
$793$	−2.80875e7	−1.58610
$794$	0	0
$795$	6.06918e6	0.340575
$796$	0	0
$797$	−1.42204e7	−0.792985	−0.396493	−	0.918038i	$-0.629773\pi$
$797$	−1.42204e7	−0.792985	−0.396493	+	0.918038i	$0.629773\pi$
$798$	0	0
$799$	3.03089e6	0.167959
$800$	0	0
$801$	1.26000e6	0.0693889
$802$	0	0
$803$	2.66265e6	0.145722
$804$	0	0
$805$	−1.08355e7	−0.589332
$806$	0	0
$807$	8.29542e6	0.448389
$808$	0	0
$809$	2.40784e7	1.29347	0.646736	−	0.762714i	$-0.276134\pi$
$809$	2.40784e7	1.29347	0.646736	+	0.762714i	$0.276134\pi$
$810$	0	0
$811$	−2.71158e7	−1.44767	−0.723836	−	0.689972i	$-0.757623\pi$
$811$	−2.71158e7	−1.44767	−0.723836	+	0.689972i	$0.757623\pi$
$812$	0	0
$813$	2.82127e7	1.49699
$814$	0	0
$815$	1.74231e7	0.918824
$816$	0	0
$817$	6.30776e7	3.30613
$818$	0	0
$819$	8.60937e6	0.448499
$820$	0	0
$821$	−1.57023e7	−0.813025	−0.406513	−	0.913645i	$-0.633255\pi$
$821$	−1.57023e7	−0.813025	−0.406513	+	0.913645i	$0.633255\pi$
$822$	0	0
$823$	1.52411e7	0.784364	0.392182	−	0.919888i	$-0.371720\pi$
$823$	1.52411e7	0.784364	0.392182	+	0.919888i	$0.371720\pi$
$824$	0	0
$825$	−6.26346e7	−3.20390
$826$	0	0
$827$	2.27417e7	1.15627	0.578136	−	0.815940i	$-0.303780\pi$
$827$	2.27417e7	1.15627	0.578136	+	0.815940i	$0.303780\pi$
$828$	0	0
$829$	1.10029e7	0.556060	0.278030	−	0.960572i	$-0.410319\pi$
$829$	1.10029e7	0.556060	0.278030	+	0.960572i	$0.410319\pi$
$830$	0	0
$831$	−2.06638e7	−1.03803
$832$	0	0
$833$	−718089.	−0.0358563
$834$	0	0
$835$	−4.41885e7	−2.19327
$836$	0	0
$837$	−1.17377e7	−0.579123
$838$	0	0
$839$	2.33372e7	1.14458	0.572288	−	0.820053i	$-0.306056\pi$
$839$	2.33372e7	1.14458	0.572288	+	0.820053i	$0.306056\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	0	0
$843$	−2.15709e7	−1.04544
$844$	0	0
$845$	−8.36931e6	−0.403225
$846$	0	0
$847$	−1.71847e7	−0.823065
$848$	0	0
$849$	−6.31380e6	−0.300622
$850$	0	0
$851$	2.13667e6	0.101138
$852$	0	0
$853$	1.63119e7	0.767595	0.383797	−	0.923417i	$-0.374616\pi$
$853$	1.63119e7	0.767595	0.383797	+	0.923417i	$0.374616\pi$
$854$	0	0
$855$	−2.36282e7	−1.10539
$856$	0	0
$857$	−1.11103e7	−0.516743	−0.258372	−	0.966046i	$-0.583186\pi$
$857$	−1.11103e7	−0.516743	−0.258372	+	0.966046i	$0.583186\pi$
$858$	0	0
$859$	−2.94971e7	−1.36394	−0.681971	−	0.731379i	$-0.738877\pi$
$859$	−2.94971e7	−1.36394	−0.681971	+	0.731379i	$0.738877\pi$
$860$	0	0
$861$	−3.56030e6	−0.163674
$862$	0	0
$863$	−2.33489e7	−1.06718	−0.533592	−	0.845742i	$-0.679158\pi$
$863$	−2.33489e7	−1.06718	−0.533592	+	0.845742i	$0.679158\pi$
$864$	0	0
$865$	−7.37260e7	−3.35027
$866$	0	0
$867$	−2.46361e7	−1.11307
$868$	0	0
$869$	−1.27178e7	−0.571296
$870$	0	0
$871$	5.79582e6	0.258863
$872$	0	0
$873$	−1.60987e7	−0.714915
$874$	0	0
$875$	4.50286e7	1.98824
$876$	0	0
$877$	−3.29425e7	−1.44630	−0.723149	−	0.690692i	$-0.757306\pi$
$877$	−3.29425e7	−1.44630	−0.723149	+	0.690692i	$0.757306\pi$
$878$	0	0
$879$	4.40297e7	1.92209
$880$	0	0
$881$	3.22782e7	1.40110	0.700550	−	0.713603i	$-0.252938\pi$
$881$	3.22782e7	1.40110	0.700550	+	0.713603i	$0.252938\pi$
$882$	0	0
$883$	1.31542e7	0.567758	0.283879	−	0.958860i	$-0.408379\pi$
$883$	1.31542e7	0.567758	0.283879	+	0.958860i	$0.408379\pi$
$884$	0	0
$885$	−4.97694e6	−0.213602
$886$	0	0
$887$	5.84285e6	0.249354	0.124677	−	0.992197i	$-0.460211\pi$
$887$	5.84285e6	0.249354	0.124677	+	0.992197i	$0.460211\pi$
$888$	0	0
$889$	3.72914e7	1.58254
$890$	0	0
$891$	3.88608e7	1.63990
$892$	0	0
$893$	−2.98492e7	−1.25258
$894$	0	0
$895$	−3.44809e7	−1.43887
$896$	0	0
$897$	9.81811e6	0.407424
$898$	0	0
$899$	−3.55830e6	−0.146840
$900$	0	0
$901$	913684.	0.0374959
$902$	0	0
$903$	−6.07531e7	−2.47941
$904$	0	0
$905$	−2.21935e7	−0.900751
$906$	0	0
$907$	2.27296e7	0.917431	0.458716	−	0.888583i	$-0.348310\pi$
$907$	2.27296e7	0.917431	0.458716	+	0.888583i	$0.348310\pi$
$908$	0	0
$909$	−1.02253e7	−0.410457
$910$	0	0
$911$	−2.85916e7	−1.14141	−0.570706	−	0.821154i	$-0.693330\pi$
$911$	−2.85916e7	−1.14141	−0.570706	+	0.821154i	$0.693330\pi$
$912$	0	0
$913$	2.29640e7	0.911738
$914$	0	0
$915$	−7.42448e7	−2.93166
$916$	0	0
$917$	−2.31001e7	−0.907173
$918$	0	0
$919$	−4.07727e7	−1.59250	−0.796252	−	0.604966i	$-0.793187\pi$
$919$	−4.07727e7	−1.59250	−0.796252	+	0.604966i	$0.793187\pi$
$920$	0	0
$921$	−1.08859e7	−0.422878
$922$	0	0
$923$	−4.72840e6	−0.182688
$924$	0	0
$925$	−1.72841e7	−0.664189
$926$	0	0
$927$	−3.49129e6	−0.133440
$928$	0	0
$929$	−4.12838e7	−1.56943	−0.784713	−	0.619860i	$-0.787190\pi$
$929$	−4.12838e7	−1.56943	−0.784713	+	0.619860i	$0.787190\pi$
$930$	0	0
$931$	7.07198e6	0.267403
$932$	0	0
$933$	−1.82810e7	−0.687537
$934$	0	0
$935$	−1.40146e7	−0.524264
$936$	0	0
$937$	4.54057e7	1.68951	0.844755	−	0.535153i	$-0.179746\pi$
$937$	4.54057e7	1.68951	0.844755	+	0.535153i	$0.179746\pi$
$938$	0	0
$939$	−6.67817e6	−0.247169
$940$	0	0
$941$	4.70900e7	1.73362	0.866812	−	0.498635i	$-0.166165\pi$
$941$	4.70900e7	1.73362	0.866812	+	0.498635i	$0.166165\pi$
$942$	0	0
$943$	−1.10853e6	−0.0405946
$944$	0	0
$945$	−3.78379e7	−1.37831
$946$	0	0
$947$	−2.49040e6	−0.0902390	−0.0451195	−	0.998982i	$-0.514367\pi$
$947$	−2.49040e6	−0.0902390	−0.0451195	+	0.998982i	$0.514367\pi$
$948$	0	0
$949$	3.37626e6	0.121694
$950$	0	0
$951$	2.93686e7	1.05301
$952$	0	0
$953$	−1.43265e7	−0.510985	−0.255493	−	0.966811i	$-0.582238\pi$
$953$	−1.43265e7	−0.510985	−0.255493	+	0.966811i	$0.582238\pi$
$954$	0	0
$955$	1.79254e7	0.636004
$956$	0	0
$957$	8.19676e6	0.289309
$958$	0	0
$959$	−1.26999e7	−0.445916
$960$	0	0
$961$	−1.07275e7	−0.374704
$962$	0	0
$963$	1.77145e6	0.0615550
$964$	0	0
$965$	−9.74385e7	−3.36831
$966$	0	0
$967$	1.92616e7	0.662407	0.331204	−	0.943559i	$-0.392545\pi$
$967$	1.92616e7	0.662407	0.331204	+	0.943559i	$0.392545\pi$
$968$	0	0
$969$	−1.30285e7	−0.445742
$970$	0	0
$971$	3.31255e7	1.12749	0.563747	−	0.825948i	$-0.309359\pi$
$971$	3.31255e7	1.12749	0.563747	+	0.825948i	$0.309359\pi$
$972$	0	0
$973$	−5.58834e7	−1.89235
$974$	0	0
$975$	−7.94211e7	−2.67562
$976$	0	0
$977$	−5.00959e7	−1.67906	−0.839529	−	0.543314i	$-0.817169\pi$
$977$	−5.00959e7	−1.67906	−0.839529	+	0.543314i	$0.817169\pi$
$978$	0	0
$979$	−7.36008e6	−0.245429
$980$	0	0
$981$	−4.65704e6	−0.154503
$982$	0	0
$983$	6.69976e6	0.221144	0.110572	−	0.993868i	$-0.464732\pi$
$983$	6.69976e6	0.221144	0.110572	+	0.993868i	$0.464732\pi$
$984$	0	0
$985$	8.41748e7	2.76434
$986$	0	0
$987$	2.87493e7	0.939364
$988$	0	0
$989$	−1.89160e7	−0.614948
$990$	0	0
$991$	3.54819e7	1.14768	0.573842	−	0.818966i	$-0.305452\pi$
$991$	3.54819e7	1.14768	0.573842	+	0.818966i	$0.305452\pi$
$992$	0	0
$993$	9.78359e6	0.314866
$994$	0	0
$995$	−8.18737e7	−2.62172
$996$	0	0
$997$	−2.99768e7	−0.955097	−0.477549	−	0.878605i	$-0.658475\pi$
$997$	−2.99768e7	−0.955097	−0.477549	+	0.878605i	$0.658475\pi$
$998$	0	0
$999$	7.46132e6	0.236538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.a.i.1.4		4
4.3	odd	2		29.6.a.a.1.4	✓	4
12.11	even	2		261.6.a.a.1.1		4
20.19	odd	2		725.6.a.a.1.1		4
116.115	odd	2		841.6.a.a.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.4	✓	4	4.3	odd	2
261.6.a.a.1.1		4	12.11	even	2
464.6.a.i.1.4		4	1.1	even	1	trivial
725.6.a.a.1.1		4	20.19	odd	2
841.6.a.a.1.1		4	116.115	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q+18.2828 q^{3} -97.7313 q^{5} -139.558 q^{7} +91.2623 q^{9} +O(q^{10})\)	\(q+18.2828 q^{3} -97.7313 q^{5} -139.558 q^{7} +91.2623 q^{9} -533.092 q^{11} -675.965 q^{13} -1786.81 q^{15} -268.994 q^{17} +2649.15 q^{19} -2551.52 q^{21} -794.438 q^{23} +6426.40 q^{25} -2774.20 q^{27} -841.000 q^{29} +4231.04 q^{31} -9746.44 q^{33} +13639.2 q^{35} -2689.54 q^{37} -12358.6 q^{39} +1395.36 q^{41} +23810.5 q^{43} -8919.18 q^{45} -11267.5 q^{47} +2669.53 q^{49} -4917.98 q^{51} -3396.67 q^{53} +52099.8 q^{55} +48433.9 q^{57} +2785.38 q^{59} +41551.7 q^{61} -12736.4 q^{63} +66062.9 q^{65} -8574.14 q^{67} -14524.6 q^{69} +6995.03 q^{71} -4994.73 q^{73} +117493. q^{75} +74397.5 q^{77} +23856.6 q^{79} -72896.9 q^{81} -43076.9 q^{83} +26289.2 q^{85} -15375.9 q^{87} +13806.4 q^{89} +94336.6 q^{91} +77355.4 q^{93} -258905. q^{95} -176400. q^{97} -48651.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) \) 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9	\( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) \) 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 + 124 * q^11 - 460 * q^13 - 932 * q^15 + 184 * q^17 + 2392 * q^19 + 992 * q^21 + 1192 * q^23 + 1824 * q^25 - 2468 * q^27 - 3364 * q^29 + 19212 * q^31 - 10580 * q^33 + 22944 * q^35 - 10928 * q^37 + 8732 * q^39 - 1120 * q^41 + 21420 * q^43 - 8344 * q^45 - 23772 * q^47 + 10452 * q^49 - 12744 * q^51 + 8860 * q^53 + 52652 * q^55 + 48944 * q^57 + 10840 * q^59 + 49448 * q^61 - 27488 * q^63 + 97836 * q^65 + 7840 * q^67 + 58792 * q^69 + 48744 * q^71 - 74992 * q^73 + 90448 * q^75 + 128656 * q^77 + 106076 * q^79 - 59692 * q^81 - 62888 * q^83 + 23848 * q^85 - 23548 * q^87 + 107568 * q^89 + 268896 * q^91 + 221460 * q^93 - 147352 * q^95 - 49520 * q^97 - 166720 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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