LMFDB - Embedded newform 464.6.a.k.1.2

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:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 $ x^7 - 3x^6 - 184x^5 + 584x^4 + 10145x^3 - 34491x^2 - 149754x + 524902
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.56883$ 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)

$f(q)$	$=$	$q-18.3274 q^{3} -84.3249 q^{5} -216.816 q^{7} +92.8952 q^{9} +O(q^{10})$	$q-18.3274 q^{3} -84.3249 q^{5} -216.816 q^{7} +92.8952 q^{9} -380.662 q^{11} +1058.05 q^{13} +1545.46 q^{15} +801.385 q^{17} +288.487 q^{19} +3973.69 q^{21} -334.571 q^{23} +3985.69 q^{25} +2751.04 q^{27} +841.000 q^{29} +3280.54 q^{31} +6976.57 q^{33} +18283.0 q^{35} -11158.5 q^{37} -19391.3 q^{39} -887.943 q^{41} -13247.9 q^{43} -7833.38 q^{45} +2338.13 q^{47} +30202.3 q^{49} -14687.3 q^{51} +1850.38 q^{53} +32099.3 q^{55} -5287.23 q^{57} -7102.05 q^{59} +9622.02 q^{61} -20141.2 q^{63} -89219.6 q^{65} -11714.6 q^{67} +6131.84 q^{69} +67084.1 q^{71} -2340.00 q^{73} -73047.5 q^{75} +82533.8 q^{77} -33928.9 q^{79} -72993.0 q^{81} +97350.5 q^{83} -67576.7 q^{85} -15413.4 q^{87} +101586. q^{89} -229401. q^{91} -60124.0 q^{93} -24326.7 q^{95} -84674.6 q^{97} -35361.7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9}+O(q^{10}) $ 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9	$ 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9} - 1106 q^{11} + 408 q^{13} + 614 q^{15} - 874 q^{17} - 4288 q^{19} - 4200 q^{21} + 4532 q^{23} + 5527 q^{25} - 5942 q^{27} + 5887 q^{29} - 7794 q^{31} + 34410 q^{33} - 7088 q^{35} + 5086 q^{37} - 33394 q^{39} + 19826 q^{41} - 19498 q^{43} + 7854 q^{45} - 14278 q^{47} + 38431 q^{49} - 23892 q^{51} - 58644 q^{53} + 25574 q^{55} - 88540 q^{57} - 12888 q^{59} + 102866 q^{61} + 88632 q^{63} - 149206 q^{65} - 102996 q^{67} - 107244 q^{69} + 51596 q^{71} - 17566 q^{73} - 39356 q^{75} - 94104 q^{77} - 212058 q^{79} - 128285 q^{81} + 122928 q^{83} - 109336 q^{85} - 21866 q^{87} - 66510 q^{89} - 194368 q^{91} - 474274 q^{93} + 131676 q^{95} - 118182 q^{97} - 300668 q^{99}+O(q^{100}) $ 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9 - 1106 * q^11 + 408 * q^13 + 614 * q^15 - 874 * q^17 - 4288 * q^19 - 4200 * q^21 + 4532 * q^23 + 5527 * q^25 - 5942 * q^27 + 5887 * q^29 - 7794 * q^31 + 34410 * q^33 - 7088 * q^35 + 5086 * q^37 - 33394 * q^39 + 19826 * q^41 - 19498 * q^43 + 7854 * q^45 - 14278 * q^47 + 38431 * q^49 - 23892 * q^51 - 58644 * q^53 + 25574 * q^55 - 88540 * q^57 - 12888 * q^59 + 102866 * q^61 + 88632 * q^63 - 149206 * q^65 - 102996 * q^67 - 107244 * q^69 + 51596 * q^71 - 17566 * q^73 - 39356 * q^75 - 94104 * q^77 - 212058 * q^79 - 128285 * q^81 + 122928 * q^83 - 109336 * q^85 - 21866 * q^87 - 66510 * q^89 - 194368 * q^91 - 474274 * q^93 + 131676 * q^95 - 118182 * q^97 - 300668 * 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
$3$	−18.3274	−1.17571	−0.587853	−	0.808968i	$-0.700027\pi$
$3$	−18.3274	−1.17571	−0.587853	+	0.808968i	$0.700027\pi$
$4$	0	0
$5$	−84.3249	−1.50845	−0.754225	−	0.656616i	$-0.771987\pi$
$5$	−84.3249	−1.50845	−0.754225	+	0.656616i	$0.771987\pi$
$6$	0	0
$7$	−216.816	−1.67242	−0.836212	−	0.548406i	$-0.815235\pi$
$7$	−216.816	−1.67242	−0.836212	+	0.548406i	$0.815235\pi$
$8$	0	0
$9$	92.8952	0.382285
$10$	0	0
$11$	−380.662	−0.948546	−0.474273	−	0.880378i	$-0.657289\pi$
$11$	−380.662	−0.948546	−0.474273	+	0.880378i	$0.657289\pi$
$12$	0	0
$13$	1058.05	1.73638	0.868192	−	0.496228i	$-0.165282\pi$
$13$	1058.05	1.73638	0.868192	+	0.496228i	$0.165282\pi$
$14$	0	0
$15$	1545.46	1.77349
$16$	0	0
$17$	801.385	0.672541	0.336271	−	0.941765i	$-0.390834\pi$
$17$	801.385	0.672541	0.336271	+	0.941765i	$0.390834\pi$
$18$	0	0
$19$	288.487	0.183334	0.0916669	−	0.995790i	$-0.470781\pi$
$19$	288.487	0.183334	0.0916669	+	0.995790i	$0.470781\pi$
$20$	0	0
$21$	3973.69	1.96628
$22$	0	0
$23$	−334.571	−0.131877	−0.0659385	−	0.997824i	$-0.521004\pi$
$23$	−334.571	−0.131877	−0.0659385	+	0.997824i	$0.521004\pi$
$24$	0	0
$25$	3985.69	1.27542
$26$	0	0
$27$	2751.04	0.726252
$28$	0	0
$29$	841.000	0.185695
$29$	841.000	0.185695
$30$	0	0
$31$	3280.54	0.613115	0.306557	−	0.951852i	$-0.400823\pi$
$31$	3280.54	0.613115	0.306557	+	0.951852i	$0.400823\pi$
$32$	0	0
$33$	6976.57	1.11521
$34$	0	0
$35$	18283.0	2.52277
$36$	0	0
$37$	−11158.5	−1.33999	−0.669994	−	0.742366i	$-0.733703\pi$
$37$	−11158.5	−1.33999	−0.669994	+	0.742366i	$0.733703\pi$
$38$	0	0
$39$	−19391.3	−2.04148
$40$	0	0
$41$	−887.943	−0.0824946	−0.0412473	−	0.999149i	$-0.513133\pi$
$41$	−887.943	−0.0824946	−0.0412473	+	0.999149i	$0.513133\pi$
$42$	0	0
$43$	−13247.9	−1.09264	−0.546318	−	0.837578i	$-0.683971\pi$
$43$	−13247.9	−1.09264	−0.546318	+	0.837578i	$0.683971\pi$
$44$	0	0
$45$	−7833.38	−0.576657
$46$	0	0
$47$	2338.13	0.154392	0.0771959	−	0.997016i	$-0.475403\pi$
$47$	2338.13	0.154392	0.0771959	+	0.997016i	$0.475403\pi$
$48$	0	0
$49$	30202.3	1.79700
$50$	0	0
$51$	−14687.3	−0.790711
$52$	0	0
$53$	1850.38	0.0904840	0.0452420	−	0.998976i	$-0.485594\pi$
$53$	1850.38	0.0904840	0.0452420	+	0.998976i	$0.485594\pi$
$54$	0	0
$55$	32099.3	1.43083
$56$	0	0
$57$	−5287.23	−0.215547
$58$	0	0
$59$	−7102.05	−0.265616	−0.132808	−	0.991142i	$-0.542399\pi$
$59$	−7102.05	−0.265616	−0.132808	+	0.991142i	$0.542399\pi$
$60$	0	0
$61$	9622.02	0.331087	0.165543	−	0.986203i	$-0.447062\pi$
$61$	9622.02	0.331087	0.165543	+	0.986203i	$0.447062\pi$
$62$	0	0
$63$	−20141.2	−0.639343
$64$	0	0
$65$	−89219.6	−2.61925
$66$	0	0
$67$	−11714.6	−0.318816	−0.159408	−	0.987213i	$-0.550958\pi$
$67$	−11714.6	−0.318816	−0.159408	+	0.987213i	$0.550958\pi$
$68$	0	0
$69$	6131.84	0.155049
$70$	0	0
$71$	67084.1	1.57933	0.789667	−	0.613536i	$-0.210253\pi$
$71$	67084.1	1.57933	0.789667	+	0.613536i	$0.210253\pi$
$72$	0	0
$73$	−2340.00	−0.0513935	−0.0256967	−	0.999670i	$-0.508180\pi$
$73$	−2340.00	−0.0513935	−0.0256967	+	0.999670i	$0.508180\pi$
$74$	0	0
$75$	−73047.5	−1.49952
$76$	0	0
$77$	82533.8	1.58637
$78$	0	0
$79$	−33928.9	−0.611649	−0.305825	−	0.952088i	$-0.598932\pi$
$79$	−33928.9	−0.611649	−0.305825	+	0.952088i	$0.598932\pi$
$80$	0	0
$81$	−72993.0	−1.23614
$82$	0	0
$83$	97350.5	1.55111	0.775555	−	0.631279i	$-0.217470\pi$
$83$	97350.5	1.55111	0.775555	+	0.631279i	$0.217470\pi$
$84$	0	0
$85$	−67576.7	−1.01449
$86$	0	0
$87$	−15413.4	−0.218323
$88$	0	0
$89$	101586.	1.35943	0.679716	−	0.733475i	$-0.262103\pi$
$89$	101586.	1.35943	0.679716	+	0.733475i	$0.262103\pi$
$90$	0	0
$91$	−229401.	−2.90397
$92$	0	0
$93$	−60124.0	−0.720843
$94$	0	0
$95$	−24326.7	−0.276550
$96$	0	0
$97$	−84674.6	−0.913743	−0.456871	−	0.889533i	$-0.651030\pi$
$97$	−84674.6	−0.913743	−0.456871	+	0.889533i	$0.651030\pi$
$98$	0	0
$99$	−35361.7	−0.362615
$100$	0	0
$101$	59936.3	0.584637	0.292318	−	0.956321i	$-0.405573\pi$
$101$	59936.3	0.584637	0.292318	+	0.956321i	$0.405573\pi$
$102$	0	0
$103$	196075.	1.82108	0.910542	−	0.413417i	$-0.135665\pi$
$103$	196075.	1.82108	0.910542	+	0.413417i	$0.135665\pi$
$104$	0	0
$105$	−335081.	−2.96604
$106$	0	0
$107$	−82447.0	−0.696170	−0.348085	−	0.937463i	$-0.613168\pi$
$107$	−82447.0	−0.696170	−0.348085	+	0.937463i	$0.613168\pi$
$108$	0	0
$109$	−152195.	−1.22697	−0.613484	−	0.789707i	$-0.710233\pi$
$109$	−152195.	−1.22697	−0.613484	+	0.789707i	$0.710233\pi$
$110$	0	0
$111$	204507.	1.57543
$112$	0	0
$113$	168561.	1.24182	0.620912	−	0.783880i	$-0.286762\pi$
$113$	168561.	1.24182	0.620912	+	0.783880i	$0.286762\pi$
$114$	0	0
$115$	28212.7	0.198930
$116$	0	0
$117$	98287.3	0.663793
$118$	0	0
$119$	−173753.	−1.12477
$120$	0	0
$121$	−16147.1	−0.100261
$122$	0	0
$123$	16273.7	0.0969893
$124$	0	0
$125$	−72577.8	−0.415460
$126$	0	0
$127$	21587.4	0.118766	0.0593829	−	0.998235i	$-0.481087\pi$
$127$	21587.4	0.118766	0.0593829	+	0.998235i	$0.481087\pi$
$128$	0	0
$129$	242800.	1.28462
$130$	0	0
$131$	−137802.	−0.701581	−0.350790	−	0.936454i	$-0.614087\pi$
$131$	−137802.	−0.701581	−0.350790	+	0.936454i	$0.614087\pi$
$132$	0	0
$133$	−62548.7	−0.306612
$134$	0	0
$135$	−231981.	−1.09551
$136$	0	0
$137$	−422084.	−1.92131	−0.960654	−	0.277747i	$-0.910412\pi$
$137$	−422084.	−1.92131	−0.960654	+	0.277747i	$0.910412\pi$
$138$	0	0
$139$	−79419.8	−0.348652	−0.174326	−	0.984688i	$-0.555775\pi$
$139$	−79419.8	−0.348652	−0.174326	+	0.984688i	$0.555775\pi$
$140$	0	0
$141$	−42852.0	−0.181519
$142$	0	0
$143$	−402758.	−1.64704
$144$	0	0
$145$	−70917.3	−0.280112
$146$	0	0
$147$	−553530.	−2.11275
$148$	0	0
$149$	−144641.	−0.533736	−0.266868	−	0.963733i	$-0.585989\pi$
$149$	−144641.	−0.533736	−0.266868	+	0.963733i	$0.585989\pi$
$150$	0	0
$151$	446697.	1.59430	0.797152	−	0.603779i	$-0.206339\pi$
$151$	446697.	1.59430	0.797152	+	0.603779i	$0.206339\pi$
$152$	0	0
$153$	74444.8	0.257102
$154$	0	0
$155$	−276632.	−0.924853
$156$	0	0
$157$	−136908.	−0.443282	−0.221641	−	0.975128i	$-0.571141\pi$
$157$	−136908.	−0.443282	−0.221641	+	0.975128i	$0.571141\pi$
$158$	0	0
$159$	−33912.8	−0.106383
$160$	0	0
$161$	72540.5	0.220554
$162$	0	0
$163$	−5407.16	−0.0159404	−0.00797022	−	0.999968i	$-0.502537\pi$
$163$	−5407.16	−0.0159404	−0.00797022	+	0.999968i	$0.502537\pi$
$164$	0	0
$165$	−588299.	−1.68224
$166$	0	0
$167$	303826.	0.843013	0.421506	−	0.906825i	$-0.361502\pi$
$167$	303826.	0.843013	0.421506	+	0.906825i	$0.361502\pi$
$168$	0	0
$169$	748167.	2.01503
$170$	0	0
$171$	26799.1	0.0700857
$172$	0	0
$173$	−45069.8	−0.114491	−0.0572454	−	0.998360i	$-0.518232\pi$
$173$	−45069.8	−0.114491	−0.0572454	+	0.998360i	$0.518232\pi$
$174$	0	0
$175$	−864163.	−2.13305
$176$	0	0
$177$	130162.	0.312286
$178$	0	0
$179$	−348392.	−0.812710	−0.406355	−	0.913715i	$-0.633200\pi$
$179$	−348392.	−0.812710	−0.406355	+	0.913715i	$0.633200\pi$
$180$	0	0
$181$	−754876.	−1.71269	−0.856346	−	0.516403i	$-0.827271\pi$
$181$	−754876.	−1.71269	−0.856346	+	0.516403i	$0.827271\pi$
$182$	0	0
$183$	−176347.	−0.389261
$184$	0	0
$185$	940939.	2.02131
$186$	0	0
$187$	−305057.	−0.637936
$188$	0	0
$189$	−596469.	−1.21460
$190$	0	0
$191$	559504.	1.10974	0.554868	−	0.831938i	$-0.312769\pi$
$191$	559504.	1.10974	0.554868	+	0.831938i	$0.312769\pi$
$192$	0	0
$193$	915469.	1.76909	0.884546	−	0.466453i	$-0.154468\pi$
$193$	915469.	1.76909	0.884546	+	0.466453i	$0.154468\pi$
$194$	0	0
$195$	1.63517e6	3.07947
$196$	0	0
$197$	149660.	0.274751	0.137375	−	0.990519i	$-0.456133\pi$
$197$	149660.	0.274751	0.137375	+	0.990519i	$0.456133\pi$
$198$	0	0
$199$	−2043.12	−0.00365731	−0.00182865	−	0.999998i	$-0.500582\pi$
$199$	−2043.12	−0.00365731	−0.00182865	+	0.999998i	$0.500582\pi$
$200$	0	0
$201$	214698.	0.374834
$202$	0	0
$203$	−182342.	−0.310562
$204$	0	0
$205$	74875.7	0.124439
$206$	0	0
$207$	−31080.1	−0.0504146
$208$	0	0
$209$	−109816.	−0.173900
$210$	0	0
$211$	94629.7	0.146326	0.0731630	−	0.997320i	$-0.476691\pi$
$211$	94629.7	0.146326	0.0731630	+	0.997320i	$0.476691\pi$
$212$	0	0
$213$	−1.22948e6	−1.85683
$214$	0	0
$215$	1.11713e6	1.64819
$216$	0	0
$217$	−711275.	−1.02539
$218$	0	0
$219$	42886.1	0.0604236
$220$	0	0
$221$	847902.	1.16779
$222$	0	0
$223$	−33151.7	−0.0446420	−0.0223210	−	0.999751i	$-0.507106\pi$
$223$	−33151.7	−0.0446420	−0.0223210	+	0.999751i	$0.507106\pi$
$224$	0	0
$225$	370252.	0.487574
$226$	0	0
$227$	886146.	1.14141	0.570704	−	0.821156i	$-0.306671\pi$
$227$	886146.	1.14141	0.570704	+	0.821156i	$0.306671\pi$
$228$	0	0
$229$	−51324.5	−0.0646749	−0.0323375	−	0.999477i	$-0.510295\pi$
$229$	−51324.5	−0.0646749	−0.0323375	+	0.999477i	$0.510295\pi$
$230$	0	0
$231$	−1.51263e6	−1.86511
$232$	0	0
$233$	−854399.	−1.03103	−0.515515	−	0.856881i	$-0.672399\pi$
$233$	−854399.	−1.03103	−0.515515	+	0.856881i	$0.672399\pi$
$234$	0	0
$235$	−197163.	−0.232892
$236$	0	0
$237$	621831.	0.719120
$238$	0	0
$239$	−750885.	−0.850313	−0.425156	−	0.905120i	$-0.639781\pi$
$239$	−750885.	−0.850313	−0.425156	+	0.905120i	$0.639781\pi$
$240$	0	0
$241$	−71808.1	−0.0796400	−0.0398200	−	0.999207i	$-0.512678\pi$
$241$	−71808.1	−0.0796400	−0.0398200	+	0.999207i	$0.512678\pi$
$242$	0	0
$243$	669273.	0.727089
$244$	0	0
$245$	−2.54680e6	−2.71069
$246$	0	0
$247$	305233.	0.318338
$248$	0	0
$249$	−1.78419e6	−1.82365
$250$	0	0
$251$	−897843.	−0.899531	−0.449765	−	0.893147i	$-0.648492\pi$
$251$	−897843.	−0.899531	−0.449765	+	0.893147i	$0.648492\pi$
$252$	0	0
$253$	127359.	0.125091
$254$	0	0
$255$	1.23851e6	1.19275
$256$	0	0
$257$	848562.	0.801402	0.400701	−	0.916209i	$-0.368767\pi$
$257$	848562.	0.801402	0.400701	+	0.916209i	$0.368767\pi$
$258$	0	0
$259$	2.41934e6	2.24103
$260$	0	0
$261$	78124.9	0.0709885
$262$	0	0
$263$	592909.	0.528565	0.264283	−	0.964445i	$-0.414865\pi$
$263$	592909.	0.528565	0.264283	+	0.964445i	$0.414865\pi$
$264$	0	0
$265$	−156033.	−0.136491
$266$	0	0
$267$	−1.86181e6	−1.59829
$268$	0	0
$269$	833107.	0.701973	0.350986	−	0.936381i	$-0.385846\pi$
$269$	833107.	0.701973	0.350986	+	0.936381i	$0.385846\pi$
$270$	0	0
$271$	−728989.	−0.602973	−0.301487	−	0.953470i	$-0.597483\pi$
$271$	−728989.	−0.602973	−0.301487	+	0.953470i	$0.597483\pi$
$272$	0	0
$273$	4.20434e6	3.41422
$274$	0	0
$275$	−1.51720e6	−1.20980
$276$	0	0
$277$	1.44549e6	1.13192	0.565961	−	0.824432i	$-0.308505\pi$
$277$	1.44549e6	1.13192	0.565961	+	0.824432i	$0.308505\pi$
$278$	0	0
$279$	304747.	0.234384
$280$	0	0
$281$	1.71312e6	1.29426	0.647131	−	0.762379i	$-0.275969\pi$
$281$	1.71312e6	1.29426	0.647131	+	0.762379i	$0.275969\pi$
$282$	0	0
$283$	−2.14168e6	−1.58960	−0.794799	−	0.606872i	$-0.792424\pi$
$283$	−2.14168e6	−1.58960	−0.794799	+	0.606872i	$0.792424\pi$
$284$	0	0
$285$	445845.	0.325141
$286$	0	0
$287$	192520.	0.137966
$288$	0	0
$289$	−777639.	−0.547688
$290$	0	0
$291$	1.55187e6	1.07429
$292$	0	0
$293$	663344.	0.451408	0.225704	−	0.974196i	$-0.427532\pi$
$293$	663344.	0.451408	0.225704	+	0.974196i	$0.427532\pi$
$294$	0	0
$295$	598880.	0.400668
$296$	0	0
$297$	−1.04722e6	−0.688883
$298$	0	0
$299$	−353992.	−0.228989
$300$	0	0
$301$	2.87236e6	1.82735
$302$	0	0
$303$	−1.09848e6	−0.687361
$304$	0	0
$305$	−811376.	−0.499428
$306$	0	0
$307$	−615215.	−0.372547	−0.186274	−	0.982498i	$-0.559641\pi$
$307$	−615215.	−0.372547	−0.186274	+	0.982498i	$0.559641\pi$
$308$	0	0
$309$	−3.59356e6	−2.14106
$310$	0	0
$311$	1.17303e6	0.687712	0.343856	−	0.939022i	$-0.388267\pi$
$311$	1.17303e6	0.687712	0.343856	+	0.939022i	$0.388267\pi$
$312$	0	0
$313$	−2.58664e6	−1.49237	−0.746184	−	0.665740i	$-0.768116\pi$
$313$	−2.58664e6	−1.49237	−0.746184	+	0.665740i	$0.768116\pi$
$314$	0	0
$315$	1.69840e6	0.964416
$316$	0	0
$317$	−934239.	−0.522167	−0.261084	−	0.965316i	$-0.584080\pi$
$317$	−934239.	−0.522167	−0.261084	+	0.965316i	$0.584080\pi$
$318$	0	0
$319$	−320137.	−0.176141
$320$	0	0
$321$	1.51104e6	0.818491
$322$	0	0
$323$	231189.	0.123299
$324$	0	0
$325$	4.21704e6	2.21462
$326$	0	0
$327$	2.78934e6	1.44255
$328$	0	0
$329$	−506945.	−0.258209
$330$	0	0
$331$	3.77955e6	1.89614	0.948070	−	0.318061i	$-0.103032\pi$
$331$	3.77955e6	1.89614	0.948070	+	0.318061i	$0.103032\pi$
$332$	0	0
$333$	−1.03657e6	−0.512257
$334$	0	0
$335$	987831.	0.480918
$336$	0	0
$337$	2.10980e6	1.01197	0.505983	−	0.862544i	$-0.331130\pi$
$337$	2.10980e6	1.01197	0.505983	+	0.862544i	$0.331130\pi$
$338$	0	0
$339$	−3.08929e6	−1.46002
$340$	0	0
$341$	−1.24878e6	−0.581567
$342$	0	0
$343$	−2.90431e6	−1.33293
$344$	0	0
$345$	−517067.	−0.233883
$346$	0	0
$347$	−37808.7	−0.0168565	−0.00842827	−	0.999964i	$-0.502683\pi$
$347$	−37808.7	−0.0168565	−0.00842827	+	0.999964i	$0.502683\pi$
$348$	0	0
$349$	632608.	0.278017	0.139008	−	0.990291i	$-0.455608\pi$
$349$	632608.	0.278017	0.139008	+	0.990291i	$0.455608\pi$
$350$	0	0
$351$	2.91072e6	1.26105
$352$	0	0
$353$	−2.76871e6	−1.18261	−0.591304	−	0.806449i	$-0.701387\pi$
$353$	−2.76871e6	−1.18261	−0.591304	+	0.806449i	$0.701387\pi$
$354$	0	0
$355$	−5.65687e6	−2.38235
$356$	0	0
$357$	3.18445e6	1.32240
$358$	0	0
$359$	2.50490e6	1.02578	0.512889	−	0.858455i	$-0.328575\pi$
$359$	2.50490e6	1.02578	0.512889	+	0.858455i	$0.328575\pi$
$360$	0	0
$361$	−2.39287e6	−0.966389
$362$	0	0
$363$	295935.	0.117877
$364$	0	0
$365$	197320.	0.0775245
$366$	0	0
$367$	−3.09219e6	−1.19840	−0.599199	−	0.800600i	$-0.704514\pi$
$367$	−3.09219e6	−1.19840	−0.599199	+	0.800600i	$0.704514\pi$
$368$	0	0
$369$	−82485.6	−0.0315364
$370$	0	0
$371$	−401193.	−0.151328
$372$	0	0
$373$	2.44054e6	0.908266	0.454133	−	0.890934i	$-0.349949\pi$
$373$	2.44054e6	0.908266	0.454133	+	0.890934i	$0.349949\pi$
$374$	0	0
$375$	1.33017e6	0.488458
$376$	0	0
$377$	889816.	0.322439
$378$	0	0
$379$	−2.64306e6	−0.945168	−0.472584	−	0.881286i	$-0.656679\pi$
$379$	−2.64306e6	−0.945168	−0.472584	+	0.881286i	$0.656679\pi$
$380$	0	0
$381$	−395642.	−0.139634
$382$	0	0
$383$	460287.	0.160336	0.0801682	−	0.996781i	$-0.474454\pi$
$383$	460287.	0.160336	0.0801682	+	0.996781i	$0.474454\pi$
$384$	0	0
$385$	−6.95965e6	−2.39296
$386$	0	0
$387$	−1.23066e6	−0.417698
$388$	0	0
$389$	−2.72948e6	−0.914547	−0.457273	−	0.889326i	$-0.651174\pi$
$389$	−2.72948e6	−0.914547	−0.457273	+	0.889326i	$0.651174\pi$
$390$	0	0
$391$	−268120.	−0.0886927
$392$	0	0
$393$	2.52556e6	0.824853
$394$	0	0
$395$	2.86106e6	0.922643
$396$	0	0
$397$	530743.	0.169008	0.0845041	−	0.996423i	$-0.473069\pi$
$397$	530743.	0.169008	0.0845041	+	0.996423i	$0.473069\pi$
$398$	0	0
$399$	1.14636e6	0.360486
$400$	0	0
$401$	1.65453e6	0.513822	0.256911	−	0.966435i	$-0.417295\pi$
$401$	1.65453e6	0.513822	0.256911	+	0.966435i	$0.417295\pi$
$402$	0	0
$403$	3.47097e6	1.06460
$404$	0	0
$405$	6.15513e6	1.86466
$406$	0	0
$407$	4.24762e6	1.27104
$408$	0	0
$409$	−5.20484e6	−1.53851	−0.769253	−	0.638944i	$-0.779372\pi$
$409$	−5.20484e6	−1.53851	−0.769253	+	0.638944i	$0.779372\pi$
$410$	0	0
$411$	7.73571e6	2.25889
$412$	0	0
$413$	1.53984e6	0.444222
$414$	0	0
$415$	−8.20907e6	−2.33977
$416$	0	0
$417$	1.45556e6	0.409912
$418$	0	0
$419$	−191911.	−0.0534030	−0.0267015	−	0.999643i	$-0.508500\pi$
$419$	−191911.	−0.0534030	−0.0267015	+	0.999643i	$0.508500\pi$
$420$	0	0
$421$	5.00002e6	1.37488	0.687442	−	0.726239i	$-0.258734\pi$
$421$	5.00002e6	1.37488	0.687442	+	0.726239i	$0.258734\pi$
$422$	0	0
$423$	217201.	0.0590216
$424$	0	0
$425$	3.19407e6	0.857773
$426$	0	0
$427$	−2.08621e6	−0.553718
$428$	0	0
$429$	7.38153e6	1.93644
$430$	0	0
$431$	3.04923e6	0.790672	0.395336	−	0.918537i	$-0.370628\pi$
$431$	3.04923e6	0.790672	0.395336	+	0.918537i	$0.370628\pi$
$432$	0	0
$433$	3.79886e6	0.973719	0.486860	−	0.873480i	$-0.338142\pi$
$433$	3.79886e6	0.973719	0.486860	+	0.873480i	$0.338142\pi$
$434$	0	0
$435$	1.29973e6	0.329330
$436$	0	0
$437$	−96519.5	−0.0241775
$438$	0	0
$439$	−3.28829e6	−0.814345	−0.407173	−	0.913351i	$-0.633485\pi$
$439$	−3.28829e6	−0.814345	−0.407173	+	0.913351i	$0.633485\pi$
$440$	0	0
$441$	2.80565e6	0.686968
$442$	0	0
$443$	−97599.9	−0.0236287	−0.0118144	−	0.999930i	$-0.503761\pi$
$443$	−97599.9	−0.0236287	−0.0118144	+	0.999930i	$0.503761\pi$
$444$	0	0
$445$	−8.56621e6	−2.05064
$446$	0	0
$447$	2.65091e6	0.627517
$448$	0	0
$449$	4.55742e6	1.06685	0.533424	−	0.845848i	$-0.320905\pi$
$449$	4.55742e6	1.06685	0.533424	+	0.845848i	$0.320905\pi$
$450$	0	0
$451$	338006.	0.0782499
$452$	0	0
$453$	−8.18682e6	−1.87443
$454$	0	0
$455$	1.93443e7	4.38050
$456$	0	0
$457$	491699.	0.110131	0.0550654	−	0.998483i	$-0.482463\pi$
$457$	491699.	0.110131	0.0550654	+	0.998483i	$0.482463\pi$
$458$	0	0
$459$	2.20464e6	0.488434
$460$	0	0
$461$	2.55180e6	0.559234	0.279617	−	0.960112i	$-0.409792\pi$
$461$	2.55180e6	0.559234	0.279617	+	0.960112i	$0.409792\pi$
$462$	0	0
$463$	6.25797e6	1.35669	0.678345	−	0.734743i	$-0.262697\pi$
$463$	6.25797e6	1.35669	0.678345	+	0.734743i	$0.262697\pi$
$464$	0	0
$465$	5.06995e6	1.08736
$466$	0	0
$467$	−2.34350e6	−0.497247	−0.248624	−	0.968600i	$-0.579978\pi$
$467$	−2.34350e6	−0.497247	−0.248624	+	0.968600i	$0.579978\pi$
$468$	0	0
$469$	2.53991e6	0.533195
$470$	0	0
$471$	2.50918e6	0.521169
$472$	0	0
$473$	5.04297e6	1.03641
$474$	0	0
$475$	1.14982e6	0.233828
$476$	0	0
$477$	171892.	0.0345907
$478$	0	0
$479$	−4.98927e6	−0.993570	−0.496785	−	0.867874i	$-0.665486\pi$
$479$	−4.98927e6	−0.993570	−0.496785	+	0.867874i	$0.665486\pi$
$480$	0	0
$481$	−1.18062e7	−2.32674
$482$	0	0
$483$	−1.32948e6	−0.259307
$484$	0	0
$485$	7.14018e6	1.37834
$486$	0	0
$487$	1.75440e6	0.335202	0.167601	−	0.985855i	$-0.446398\pi$
$487$	1.75440e6	0.335202	0.167601	+	0.985855i	$0.446398\pi$
$488$	0	0
$489$	99099.5	0.0187413
$490$	0	0
$491$	−2.28688e6	−0.428094	−0.214047	−	0.976823i	$-0.568665\pi$
$491$	−2.28688e6	−0.428094	−0.214047	+	0.976823i	$0.568665\pi$
$492$	0	0
$493$	673965.	0.124888
$494$	0	0
$495$	2.98187e6	0.546986
$496$	0	0
$497$	−1.45449e7	−2.64132
$498$	0	0
$499$	−6.03762e6	−1.08546	−0.542730	−	0.839907i	$-0.682609\pi$
$499$	−6.03762e6	−1.08546	−0.542730	+	0.839907i	$0.682609\pi$
$500$	0	0
$501$	−5.56836e6	−0.991135
$502$	0	0
$503$	1.11239e6	0.196036	0.0980182	−	0.995185i	$-0.468750\pi$
$503$	1.11239e6	0.196036	0.0980182	+	0.995185i	$0.468750\pi$
$504$	0	0
$505$	−5.05412e6	−0.881896
$506$	0	0
$507$	−1.37120e7	−2.36909
$508$	0	0
$509$	−3.45813e6	−0.591625	−0.295813	−	0.955246i	$-0.595590\pi$
$509$	−3.45813e6	−0.591625	−0.295813	+	0.955246i	$0.595590\pi$
$510$	0	0
$511$	507349.	0.0859517
$512$	0	0
$513$	793639.	0.133146
$514$	0	0
$515$	−1.65340e7	−2.74701
$516$	0	0
$517$	−890039.	−0.146448
$518$	0	0
$519$	826015.	0.134608
$520$	0	0
$521$	−9.29125e6	−1.49961	−0.749807	−	0.661656i	$-0.769854\pi$
$521$	−9.29125e6	−1.49961	−0.749807	+	0.661656i	$0.769854\pi$
$522$	0	0
$523$	−1.62992e6	−0.260563	−0.130282	−	0.991477i	$-0.541588\pi$
$523$	−1.62992e6	−0.260563	−0.130282	+	0.991477i	$0.541588\pi$
$524$	0	0
$525$	1.58379e7	2.50784
$526$	0	0
$527$	2.62898e6	0.412345
$528$	0	0
$529$	−6.32441e6	−0.982608
$530$	0	0
$531$	−659746.	−0.101541
$532$	0	0
$533$	−939484.	−0.143242
$534$	0	0
$535$	6.95233e6	1.05014
$536$	0	0
$537$	6.38514e6	0.955509
$538$	0	0
$539$	−1.14969e7	−1.70454
$540$	0	0
$541$	−1.04884e7	−1.54070	−0.770348	−	0.637624i	$-0.779917\pi$
$541$	−1.04884e7	−1.54070	−0.770348	+	0.637624i	$0.779917\pi$
$542$	0	0
$543$	1.38349e7	2.01362
$544$	0	0
$545$	1.28338e7	1.85082
$546$	0	0
$547$	−4.48107e6	−0.640345	−0.320172	−	0.947359i	$-0.603741\pi$
$547$	−4.48107e6	−0.640345	−0.320172	+	0.947359i	$0.603741\pi$
$548$	0	0
$549$	893839.	0.126569
$550$	0	0
$551$	242618.	0.0340442
$552$	0	0
$553$	7.35634e6	1.02294
$554$	0	0
$555$	−1.72450e7	−2.37646
$556$	0	0
$557$	−720322.	−0.0983759	−0.0491879	−	0.998790i	$-0.515663\pi$
$557$	−720322.	−0.0983759	−0.0491879	+	0.998790i	$0.515663\pi$
$558$	0	0
$559$	−1.40169e7	−1.89724
$560$	0	0
$561$	5.59092e6	0.750025
$562$	0	0
$563$	−1.26872e7	−1.68692	−0.843458	−	0.537196i	$-0.819484\pi$
$563$	−1.26872e7	−1.68692	−0.843458	+	0.537196i	$0.819484\pi$
$564$	0	0
$565$	−1.42139e7	−1.87323
$566$	0	0
$567$	1.58261e7	2.06736
$568$	0	0
$569$	−7.93605e6	−1.02760	−0.513799	−	0.857910i	$-0.671762\pi$
$569$	−7.93605e6	−1.02760	−0.513799	+	0.857910i	$0.671762\pi$
$570$	0	0
$571$	−9.14124e6	−1.17332	−0.586658	−	0.809835i	$-0.699557\pi$
$571$	−9.14124e6	−1.17332	−0.586658	+	0.809835i	$0.699557\pi$
$572$	0	0
$573$	−1.02543e7	−1.30472
$574$	0	0
$575$	−1.33350e6	−0.168199
$576$	0	0
$577$	2.52312e6	0.315499	0.157750	−	0.987479i	$-0.449576\pi$
$577$	2.52312e6	0.315499	0.157750	+	0.987479i	$0.449576\pi$
$578$	0	0
$579$	−1.67782e7	−2.07993
$580$	0	0
$581$	−2.11072e7	−2.59412
$582$	0	0
$583$	−704371.	−0.0858283
$584$	0	0
$585$	−8.28807e6	−1.00130
$586$	0	0
$587$	1.02895e7	1.23254	0.616268	−	0.787536i	$-0.288644\pi$
$587$	1.02895e7	1.23254	0.616268	+	0.787536i	$0.288644\pi$
$588$	0	0
$589$	946395.	0.112405
$590$	0	0
$591$	−2.74288e6	−0.323026
$592$	0	0
$593$	−573757.	−0.0670025	−0.0335013	−	0.999439i	$-0.510666\pi$
$593$	−573757.	−0.0670025	−0.0335013	+	0.999439i	$0.510666\pi$
$594$	0	0
$595$	1.46517e7	1.69667
$596$	0	0
$597$	37445.2	0.00429992
$598$	0	0
$599$	−1.61434e7	−1.83835	−0.919173	−	0.393853i	$-0.871142\pi$
$599$	−1.61434e7	−1.83835	−0.919173	+	0.393853i	$0.871142\pi$
$600$	0	0
$601$	3.97295e6	0.448670	0.224335	−	0.974512i	$-0.427979\pi$
$601$	3.97295e6	0.448670	0.224335	+	0.974512i	$0.427979\pi$
$602$	0	0
$603$	−1.08823e6	−0.121878
$604$	0	0
$605$	1.36160e6	0.151238
$606$	0	0
$607$	−5.65644e6	−0.623119	−0.311560	−	0.950227i	$-0.600851\pi$
$607$	−5.65644e6	−0.623119	−0.311560	+	0.950227i	$0.600851\pi$
$608$	0	0
$609$	3.34187e6	0.365129
$610$	0	0
$611$	2.47385e6	0.268084
$612$	0	0
$613$	7.65406e6	0.822699	0.411349	−	0.911478i	$-0.365058\pi$
$613$	7.65406e6	0.822699	0.411349	+	0.911478i	$0.365058\pi$
$614$	0	0
$615$	−1.37228e6	−0.146304
$616$	0	0
$617$	−1.06193e7	−1.12301	−0.561505	−	0.827473i	$-0.689778\pi$
$617$	−1.06193e7	−1.12301	−0.561505	+	0.827473i	$0.689778\pi$
$618$	0	0
$619$	4.45953e6	0.467802	0.233901	−	0.972260i	$-0.424851\pi$
$619$	4.45953e6	0.467802	0.233901	+	0.972260i	$0.424851\pi$
$620$	0	0
$621$	−920418.	−0.0957759
$622$	0	0
$623$	−2.20254e7	−2.27355
$624$	0	0
$625$	−6.33517e6	−0.648722
$626$	0	0
$627$	2.01265e6	0.204456
$628$	0	0
$629$	−8.94224e6	−0.901197
$630$	0	0
$631$	−1.51771e7	−1.51745	−0.758725	−	0.651411i	$-0.774177\pi$
$631$	−1.51771e7	−1.51745	−0.758725	+	0.651411i	$0.774177\pi$
$632$	0	0
$633$	−1.73432e6	−0.172036
$634$	0	0
$635$	−1.82036e6	−0.179152
$636$	0	0
$637$	3.19554e7	3.12029
$638$	0	0
$639$	6.23180e6	0.603755
$640$	0	0
$641$	1.33769e7	1.28591	0.642957	−	0.765903i	$-0.277708\pi$
$641$	1.33769e7	1.28591	0.642957	+	0.765903i	$0.277708\pi$
$642$	0	0
$643$	5.68424e6	0.542182	0.271091	−	0.962554i	$-0.412616\pi$
$643$	5.68424e6	0.542182	0.271091	+	0.962554i	$0.412616\pi$
$644$	0	0
$645$	−2.04741e7	−1.93778
$646$	0	0
$647$	1.93095e7	1.81347	0.906735	−	0.421702i	$-0.138567\pi$
$647$	1.93095e7	1.81347	0.906735	+	0.421702i	$0.138567\pi$
$648$	0	0
$649$	2.70348e6	0.251949
$650$	0	0
$651$	1.30359e7	1.20556
$652$	0	0
$653$	151435.	0.0138977	0.00694884	−	0.999976i	$-0.497788\pi$
$653$	151435.	0.0138977	0.00694884	+	0.999976i	$0.497788\pi$
$654$	0	0
$655$	1.16201e7	1.05830
$656$	0	0
$657$	−217374.	−0.0196469
$658$	0	0
$659$	9.74503e6	0.874117	0.437058	−	0.899433i	$-0.356020\pi$
$659$	9.74503e6	0.874117	0.437058	+	0.899433i	$0.356020\pi$
$660$	0	0
$661$	−8.52857e6	−0.759229	−0.379614	−	0.925145i	$-0.623943\pi$
$661$	−8.52857e6	−0.759229	−0.379614	+	0.925145i	$0.623943\pi$
$662$	0	0
$663$	−1.55399e7	−1.37298
$664$	0	0
$665$	5.27441e6	0.462509
$666$	0	0
$667$	−281374.	−0.0244889
$668$	0	0
$669$	607586.	0.0524859
$670$	0	0
$671$	−3.66274e6	−0.314051
$672$	0	0
$673$	−1.53606e7	−1.30729	−0.653643	−	0.756803i	$-0.726760\pi$
$673$	−1.53606e7	−1.30729	−0.653643	+	0.756803i	$0.726760\pi$
$674$	0	0
$675$	1.09648e7	0.926277
$676$	0	0
$677$	−2.37603e6	−0.199241	−0.0996207	−	0.995025i	$-0.531763\pi$
$677$	−2.37603e6	−0.199241	−0.0996207	+	0.995025i	$0.531763\pi$
$678$	0	0
$679$	1.83588e7	1.52817
$680$	0	0
$681$	−1.62408e7	−1.34196
$682$	0	0
$683$	6.86181e6	0.562842	0.281421	−	0.959584i	$-0.409194\pi$
$683$	6.86181e6	0.562842	0.281421	+	0.959584i	$0.409194\pi$
$684$	0	0
$685$	3.55922e7	2.89820
$686$	0	0
$687$	940647.	0.0760387
$688$	0	0
$689$	1.95779e6	0.157115
$690$	0	0
$691$	−1.86887e7	−1.48896	−0.744480	−	0.667645i	$-0.767302\pi$
$691$	−1.86887e7	−1.48896	−0.744480	+	0.667645i	$0.767302\pi$
$692$	0	0
$693$	7.66699e6	0.606446
$694$	0	0
$695$	6.69706e6	0.525923
$696$	0	0
$697$	−711584.	−0.0554810
$698$	0	0
$699$	1.56589e7	1.21219
$700$	0	0
$701$	−5.85780e6	−0.450235	−0.225118	−	0.974332i	$-0.572277\pi$
$701$	−5.85780e6	−0.450235	−0.225118	+	0.974332i	$0.572277\pi$
$702$	0	0
$703$	−3.21908e6	−0.245665
$704$	0	0
$705$	3.61349e6	0.273813
$706$	0	0
$707$	−1.29952e7	−0.977761
$708$	0	0
$709$	−3.17834e6	−0.237457	−0.118728	−	0.992927i	$-0.537882\pi$
$709$	−3.17834e6	−0.237457	−0.118728	+	0.992927i	$0.537882\pi$
$710$	0	0
$711$	−3.15184e6	−0.233824
$712$	0	0
$713$	−1.09758e6	−0.0808557
$714$	0	0
$715$	3.39626e7	2.48448
$716$	0	0
$717$	1.37618e7	0.999718
$718$	0	0
$719$	−2.62515e7	−1.89379	−0.946894	−	0.321546i	$-0.895798\pi$
$719$	−2.62515e7	−1.89379	−0.946894	+	0.321546i	$0.895798\pi$
$720$	0	0
$721$	−4.25123e7	−3.04563
$722$	0	0
$723$	1.31606e6	0.0936332
$724$	0	0
$725$	3.35197e6	0.236840
$726$	0	0
$727$	1.53808e7	1.07930	0.539651	−	0.841889i	$-0.318556\pi$
$727$	1.53808e7	1.07930	0.539651	+	0.841889i	$0.318556\pi$
$728$	0	0
$729$	5.47123e6	0.381300
$730$	0	0
$731$	−1.06167e7	−0.734842
$732$	0	0
$733$	1.07403e7	0.738338	0.369169	−	0.929362i	$-0.379642\pi$
$733$	1.07403e7	0.738338	0.369169	+	0.929362i	$0.379642\pi$
$734$	0	0
$735$	4.66764e7	3.18698
$736$	0	0
$737$	4.45930e6	0.302411
$738$	0	0
$739$	−1.50225e7	−1.01189	−0.505943	−	0.862567i	$-0.668855\pi$
$739$	−1.50225e7	−1.01189	−0.505943	+	0.862567i	$0.668855\pi$
$740$	0	0
$741$	−5.59413e6	−0.374272
$742$	0	0
$743$	1.20298e7	0.799440	0.399720	−	0.916637i	$-0.369107\pi$
$743$	1.20298e7	0.799440	0.399720	+	0.916637i	$0.369107\pi$
$744$	0	0
$745$	1.21969e7	0.805115
$746$	0	0
$747$	9.04339e6	0.592966
$748$	0	0
$749$	1.78758e7	1.16429
$750$	0	0
$751$	−1.27308e7	−0.823674	−0.411837	−	0.911258i	$-0.635113\pi$
$751$	−1.27308e7	−0.823674	−0.411837	+	0.911258i	$0.635113\pi$
$752$	0	0
$753$	1.64552e7	1.05758
$754$	0	0
$755$	−3.76677e7	−2.40493
$756$	0	0
$757$	−2.61540e7	−1.65882	−0.829408	−	0.558643i	$-0.811322\pi$
$757$	−2.61540e7	−1.65882	−0.829408	+	0.558643i	$0.811322\pi$
$758$	0	0
$759$	−2.33416e6	−0.147071
$760$	0	0
$761$	1.26786e7	0.793612	0.396806	−	0.917903i	$-0.370119\pi$
$761$	1.26786e7	0.793612	0.396806	+	0.917903i	$0.370119\pi$
$762$	0	0
$763$	3.29983e7	2.05201
$764$	0	0
$765$	−6.27755e6	−0.387826
$766$	0	0
$767$	−7.51429e6	−0.461211
$768$	0	0
$769$	2.28743e7	1.39486	0.697431	−	0.716652i	$-0.254326\pi$
$769$	2.28743e7	1.39486	0.697431	+	0.716652i	$0.254326\pi$
$770$	0	0
$771$	−1.55520e7	−0.942213
$772$	0	0
$773$	2.65876e7	1.60041	0.800204	−	0.599728i	$-0.204725\pi$
$773$	2.65876e7	1.60041	0.800204	+	0.599728i	$0.204725\pi$
$774$	0	0
$775$	1.30752e7	0.781980
$776$	0	0
$777$	−4.43403e7	−2.63479
$778$	0	0
$779$	−256160.	−0.0151240
$780$	0	0
$781$	−2.55364e7	−1.49807
$782$	0	0
$783$	2.31362e6	0.134862
$784$	0	0
$785$	1.15448e7	0.668669
$786$	0	0
$787$	9.06273e6	0.521582	0.260791	−	0.965395i	$-0.416017\pi$
$787$	9.06273e6	0.521582	0.260791	+	0.965395i	$0.416017\pi$
$788$	0	0
$789$	−1.08665e7	−0.621437
$790$	0	0
$791$	−3.65467e7	−2.07686
$792$	0	0
$793$	1.01805e7	0.574894
$794$	0	0
$795$	2.85969e6	0.160473
$796$	0	0
$797$	−2.52271e7	−1.40676	−0.703382	−	0.710812i	$-0.748328\pi$
$797$	−2.52271e7	−1.40676	−0.703382	+	0.710812i	$0.748328\pi$
$798$	0	0
$799$	1.87374e6	0.103835
$800$	0	0
$801$	9.43683e6	0.519690
$802$	0	0
$803$	890748.	0.0487491
$804$	0	0
$805$	−6.11697e6	−0.332695
$806$	0	0
$807$	−1.52687e7	−0.825314
$808$	0	0
$809$	−8.69627e6	−0.467156	−0.233578	−	0.972338i	$-0.575043\pi$
$809$	−8.69627e6	−0.467156	−0.233578	+	0.972338i	$0.575043\pi$
$810$	0	0
$811$	−1.94112e7	−1.03633	−0.518166	−	0.855280i	$-0.673385\pi$
$811$	−1.94112e7	−1.03633	−0.518166	+	0.855280i	$0.673385\pi$
$812$	0	0
$813$	1.33605e7	0.708919
$814$	0	0
$815$	455959.	0.0240454
$816$	0	0
$817$	−3.82184e6	−0.200317
$818$	0	0
$819$	−2.13103e7	−1.11014
$820$	0	0
$821$	1.78189e7	0.922620	0.461310	−	0.887239i	$-0.347380\pi$
$821$	1.78189e7	0.922620	0.461310	+	0.887239i	$0.347380\pi$
$822$	0	0
$823$	1.73515e7	0.892969	0.446485	−	0.894791i	$-0.352676\pi$
$823$	1.73515e7	0.892969	0.446485	+	0.894791i	$0.352676\pi$
$824$	0	0
$825$	2.78065e7	1.42236
$826$	0	0
$827$	−8.83226e6	−0.449064	−0.224532	−	0.974467i	$-0.572085\pi$
$827$	−8.83226e6	−0.449064	−0.224532	+	0.974467i	$0.572085\pi$
$828$	0	0
$829$	−1.69640e7	−0.857316	−0.428658	−	0.903467i	$-0.641014\pi$
$829$	−1.69640e7	−0.857316	−0.428658	+	0.903467i	$0.641014\pi$
$830$	0	0
$831$	−2.64922e7	−1.33081
$832$	0	0
$833$	2.42036e7	1.20856
$834$	0	0
$835$	−2.56201e7	−1.27164
$836$	0	0
$837$	9.02490e6	0.445275
$838$	0	0
$839$	2.68046e7	1.31463	0.657316	−	0.753615i	$-0.271692\pi$
$839$	2.68046e7	1.31463	0.657316	+	0.753615i	$0.271692\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	0	0
$843$	−3.13971e7	−1.52167
$844$	0	0
$845$	−6.30891e7	−3.03957
$846$	0	0
$847$	3.50095e6	0.167678
$848$	0	0
$849$	3.92514e7	1.86890
$850$	0	0
$851$	3.73331e6	0.176714
$852$	0	0
$853$	2.65626e7	1.24997	0.624983	−	0.780638i	$-0.285106\pi$
$853$	2.65626e7	1.24997	0.624983	+	0.780638i	$0.285106\pi$
$854$	0	0
$855$	−2.25983e6	−0.105721
$856$	0	0
$857$	−1.81681e7	−0.845003	−0.422501	−	0.906362i	$-0.638848\pi$
$857$	−1.81681e7	−0.845003	−0.422501	+	0.906362i	$0.638848\pi$
$858$	0	0
$859$	−2.68029e7	−1.23936	−0.619681	−	0.784854i	$-0.712738\pi$
$859$	−2.68029e7	−1.23936	−0.619681	+	0.784854i	$0.712738\pi$
$860$	0	0
$861$	−3.52841e6	−0.162207
$862$	0	0
$863$	2.87179e7	1.31258	0.656291	−	0.754508i	$-0.272124\pi$
$863$	2.87179e7	1.31258	0.656291	+	0.754508i	$0.272124\pi$
$864$	0	0
$865$	3.80051e6	0.172704
$866$	0	0
$867$	1.42521e7	0.643921
$868$	0	0
$869$	1.29155e7	0.580178
$870$	0	0
$871$	−1.23946e7	−0.553587
$872$	0	0
$873$	−7.86586e6	−0.349310
$874$	0	0
$875$	1.57360e7	0.694825
$876$	0	0
$877$	139217.	0.00611213	0.00305606	−	0.999995i	$-0.499027\pi$
$877$	139217.	0.00611213	0.00305606	+	0.999995i	$0.499027\pi$
$878$	0	0
$879$	−1.21574e7	−0.530723
$880$	0	0
$881$	−8.68118e6	−0.376824	−0.188412	−	0.982090i	$-0.560334\pi$
$881$	−8.68118e6	−0.376824	−0.188412	+	0.982090i	$0.560334\pi$
$882$	0	0
$883$	2.65874e7	1.14756	0.573778	−	0.819011i	$-0.305477\pi$
$883$	2.65874e7	1.14756	0.573778	+	0.819011i	$0.305477\pi$
$884$	0	0
$885$	−1.09759e7	−0.471068
$886$	0	0
$887$	4.43687e7	1.89351	0.946754	−	0.321956i	$-0.104340\pi$
$887$	4.43687e7	1.89351	0.946754	+	0.321956i	$0.104340\pi$
$888$	0	0
$889$	−4.68050e6	−0.198627
$890$	0	0
$891$	2.77857e7	1.17254
$892$	0	0
$893$	674521.	0.0283052
$894$	0	0
$895$	2.93781e7	1.22593
$896$	0	0
$897$	6.48776e6	0.269224
$898$	0	0
$899$	2.75894e6	0.113853
$900$	0	0
$901$	1.48287e6	0.0608542
$902$	0	0
$903$	−5.26429e7	−2.14843
$904$	0	0
$905$	6.36548e7	2.58351
$906$	0	0
$907$	1.96923e7	0.794839	0.397420	−	0.917637i	$-0.369906\pi$
$907$	1.96923e7	0.794839	0.397420	+	0.917637i	$0.369906\pi$
$908$	0	0
$909$	5.56779e6	0.223498
$910$	0	0
$911$	−4.33698e6	−0.173138	−0.0865688	−	0.996246i	$-0.527590\pi$
$911$	−4.33698e6	−0.173138	−0.0865688	+	0.996246i	$0.527590\pi$
$912$	0	0
$913$	−3.70577e7	−1.47130
$914$	0	0
$915$	1.48704e7	0.587180
$916$	0	0
$917$	2.98777e7	1.17334
$918$	0	0
$919$	−2.13300e7	−0.833110	−0.416555	−	0.909110i	$-0.636763\pi$
$919$	−2.13300e7	−0.833110	−0.416555	+	0.909110i	$0.636763\pi$
$920$	0	0
$921$	1.12753e7	0.438006
$922$	0	0
$923$	7.09781e7	2.74233
$924$	0	0
$925$	−4.44743e7	−1.70905
$926$	0	0
$927$	1.82145e7	0.696172
$928$	0	0
$929$	4.41448e7	1.67819	0.839093	−	0.543988i	$-0.183086\pi$
$929$	4.41448e7	1.67819	0.839093	+	0.543988i	$0.183086\pi$
$930$	0	0
$931$	8.71296e6	0.329452
$932$	0	0
$933$	−2.14986e7	−0.808547
$934$	0	0
$935$	2.57239e7	0.962295
$936$	0	0
$937$	5.10338e7	1.89893	0.949466	−	0.313871i	$-0.101626\pi$
$937$	5.10338e7	1.89893	0.949466	+	0.313871i	$0.101626\pi$
$938$	0	0
$939$	4.74066e7	1.75459
$940$	0	0
$941$	4.69231e7	1.72748	0.863739	−	0.503939i	$-0.168116\pi$
$941$	4.69231e7	1.72748	0.863739	+	0.503939i	$0.168116\pi$
$942$	0	0
$943$	297080.	0.0108791
$944$	0	0
$945$	5.02972e7	1.83217
$946$	0	0
$947$	−1.22405e7	−0.443533	−0.221766	−	0.975100i	$-0.571182\pi$
$947$	−1.22405e7	−0.443533	−0.221766	+	0.975100i	$0.571182\pi$
$948$	0	0
$949$	−2.47582e6	−0.0892388
$950$	0	0
$951$	1.71222e7	0.613915
$952$	0	0
$953$	−4.02315e7	−1.43494	−0.717470	−	0.696590i	$-0.754700\pi$
$953$	−4.02315e7	−1.43494	−0.717470	+	0.696590i	$0.754700\pi$
$954$	0	0
$955$	−4.71801e7	−1.67398
$956$	0	0
$957$	5.86730e6	0.207090
$958$	0	0
$959$	9.15146e7	3.21324
$960$	0	0
$961$	−1.78672e7	−0.624090
$962$	0	0
$963$	−7.65893e6	−0.266135
$964$	0	0
$965$	−7.71969e7	−2.66859
$966$	0	0
$967$	−1.72628e7	−0.593668	−0.296834	−	0.954929i	$-0.595931\pi$
$967$	−1.72628e7	−0.593668	−0.296834	+	0.954929i	$0.595931\pi$
$968$	0	0
$969$	−4.23711e6	−0.144964
$970$	0	0
$971$	−1.32907e7	−0.452378	−0.226189	−	0.974083i	$-0.572627\pi$
$971$	−1.32907e7	−0.452378	−0.226189	+	0.974083i	$0.572627\pi$
$972$	0	0
$973$	1.72195e7	0.583093
$974$	0	0
$975$	−7.72876e7	−2.60374
$976$	0	0
$977$	−4.16592e7	−1.39629	−0.698144	−	0.715957i	$-0.745990\pi$
$977$	−4.16592e7	−1.39629	−0.698144	+	0.715957i	$0.745990\pi$
$978$	0	0
$979$	−3.86699e7	−1.28948
$980$	0	0
$981$	−1.41382e7	−0.469051
$982$	0	0
$983$	−2.91834e6	−0.0963278	−0.0481639	−	0.998839i	$-0.515337\pi$
$983$	−2.91834e6	−0.0963278	−0.0481639	+	0.998839i	$0.515337\pi$
$984$	0	0
$985$	−1.26200e7	−0.414448
$986$	0	0
$987$	9.29100e6	0.303577
$988$	0	0
$989$	4.43236e6	0.144094
$990$	0	0
$991$	5.12994e7	1.65931	0.829657	−	0.558274i	$-0.188536\pi$
$991$	5.12994e7	1.65931	0.829657	+	0.558274i	$0.188536\pi$
$992$	0	0
$993$	−6.92696e7	−2.22930
$994$	0	0
$995$	172286.	0.00551686
$996$	0	0
$997$	3.15013e7	1.00367	0.501834	−	0.864964i	$-0.332659\pi$
$997$	3.15013e7	1.00367	0.501834	+	0.864964i	$0.332659\pi$
$998$	0	0
$999$	−3.06974e7	−0.973169

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.a.k.1.2		7
4.3	odd	2		29.6.a.b.1.1	✓	7
12.11	even	2		261.6.a.e.1.7		7
20.19	odd	2		725.6.a.b.1.7		7
116.115	odd	2		841.6.a.b.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.b.1.1	✓	7	4.3	odd	2
261.6.a.e.1.7		7	12.11	even	2
464.6.a.k.1.2		7	1.1	even	1	trivial
725.6.a.b.1.7		7	20.19	odd	2
841.6.a.b.1.7		7	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)

\(f(q)\)	\(=\)	\(q-18.3274 q^{3} -84.3249 q^{5} -216.816 q^{7} +92.8952 q^{9} +O(q^{10})\)	\(q-18.3274 q^{3} -84.3249 q^{5} -216.816 q^{7} +92.8952 q^{9} -380.662 q^{11} +1058.05 q^{13} +1545.46 q^{15} +801.385 q^{17} +288.487 q^{19} +3973.69 q^{21} -334.571 q^{23} +3985.69 q^{25} +2751.04 q^{27} +841.000 q^{29} +3280.54 q^{31} +6976.57 q^{33} +18283.0 q^{35} -11158.5 q^{37} -19391.3 q^{39} -887.943 q^{41} -13247.9 q^{43} -7833.38 q^{45} +2338.13 q^{47} +30202.3 q^{49} -14687.3 q^{51} +1850.38 q^{53} +32099.3 q^{55} -5287.23 q^{57} -7102.05 q^{59} +9622.02 q^{61} -20141.2 q^{63} -89219.6 q^{65} -11714.6 q^{67} +6131.84 q^{69} +67084.1 q^{71} -2340.00 q^{73} -73047.5 q^{75} +82533.8 q^{77} -33928.9 q^{79} -72993.0 q^{81} +97350.5 q^{83} -67576.7 q^{85} -15413.4 q^{87} +101586. q^{89} -229401. q^{91} -60124.0 q^{93} -24326.7 q^{95} -84674.6 q^{97} -35361.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9}+O(q^{10}) \) 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9	\( 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9} - 1106 q^{11} + 408 q^{13} + 614 q^{15} - 874 q^{17} - 4288 q^{19} - 4200 q^{21} + 4532 q^{23} + 5527 q^{25} - 5942 q^{27} + 5887 q^{29} - 7794 q^{31} + 34410 q^{33} - 7088 q^{35} + 5086 q^{37} - 33394 q^{39} + 19826 q^{41} - 19498 q^{43} + 7854 q^{45} - 14278 q^{47} + 38431 q^{49} - 23892 q^{51} - 58644 q^{53} + 25574 q^{55} - 88540 q^{57} - 12888 q^{59} + 102866 q^{61} + 88632 q^{63} - 149206 q^{65} - 102996 q^{67} - 107244 q^{69} + 51596 q^{71} - 17566 q^{73} - 39356 q^{75} - 94104 q^{77} - 212058 q^{79} - 128285 q^{81} + 122928 q^{83} - 109336 q^{85} - 21866 q^{87} - 66510 q^{89} - 194368 q^{91} - 474274 q^{93} + 131676 q^{95} - 118182 q^{97} - 300668 q^{99}+O(q^{100}) \) 7 * q - 26 * q^3 + 32 * q^5 - 184 * q^7 + 1005 * q^9 - 1106 * q^11 + 408 * q^13 + 614 * q^15 - 874 * q^17 - 4288 * q^19 - 4200 * q^21 + 4532 * q^23 + 5527 * q^25 - 5942 * q^27 + 5887 * q^29 - 7794 * q^31 + 34410 * q^33 - 7088 * q^35 + 5086 * q^37 - 33394 * q^39 + 19826 * q^41 - 19498 * q^43 + 7854 * q^45 - 14278 * q^47 + 38431 * q^49 - 23892 * q^51 - 58644 * q^53 + 25574 * q^55 - 88540 * q^57 - 12888 * q^59 + 102866 * q^61 + 88632 * q^63 - 149206 * q^65 - 102996 * q^67 - 107244 * q^69 + 51596 * q^71 - 17566 * q^73 - 39356 * q^75 - 94104 * q^77 - 212058 * q^79 - 128285 * q^81 + 122928 * q^83 - 109336 * q^85 - 21866 * q^87 - 66510 * q^89 - 194368 * q^91 - 474274 * q^93 + 131676 * q^95 - 118182 * q^97 - 300668 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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