LMFDB - Embedded newform 464.6.a.i.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:	$0$
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			1.2
Root			$-5.34807$ 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)

$f(q)$	$=$	$q-0.734546 q^{3} +41.6400 q^{5} +90.0205 q^{7} -242.460 q^{9} +O(q^{10})$	$q-0.734546 q^{3} +41.6400 q^{5} +90.0205 q^{7} -242.460 q^{9} +269.080 q^{11} -444.579 q^{13} -30.5865 q^{15} +485.649 q^{17} +1572.80 q^{19} -66.1242 q^{21} +398.704 q^{23} -1391.11 q^{25} +356.593 q^{27} -841.000 q^{29} +8469.77 q^{31} -197.652 q^{33} +3748.45 q^{35} +3339.33 q^{37} +326.564 q^{39} +18154.2 q^{41} -9996.71 q^{43} -10096.1 q^{45} -12568.0 q^{47} -8703.31 q^{49} -356.732 q^{51} -21343.1 q^{53} +11204.5 q^{55} -1155.29 q^{57} +30036.1 q^{59} +49792.6 q^{61} -21826.4 q^{63} -18512.3 q^{65} -47588.2 q^{67} -292.867 q^{69} +50164.6 q^{71} -44770.3 q^{73} +1021.83 q^{75} +24222.7 q^{77} +78464.6 q^{79} +58656.0 q^{81} -46721.8 q^{83} +20222.4 q^{85} +617.753 q^{87} -39465.7 q^{89} -40021.2 q^{91} -6221.43 q^{93} +65491.2 q^{95} +48336.3 q^{97} -65241.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.734546	−0.0471211	−0.0235606	−	0.999722i	$-0.507500\pi$
$3$	−0.734546	−0.0471211	−0.0235606	+	0.999722i	$0.507500\pi$
$4$	0	0
$5$	41.6400	0.744879	0.372440	−	0.928056i	$-0.378521\pi$
$5$	41.6400	0.744879	0.372440	+	0.928056i	$0.378521\pi$
$6$	0	0
$7$	90.0205	0.694379	0.347189	−	0.937795i	$-0.387136\pi$
$7$	90.0205	0.694379	0.347189	+	0.937795i	$0.387136\pi$
$8$	0	0
$9$	−242.460	−0.997780
$10$	0	0
$11$	269.080	0.670502	0.335251	−	0.942129i	$-0.391179\pi$
$11$	269.080	0.670502	0.335251	+	0.942129i	$0.391179\pi$
$12$	0	0
$13$	−444.579	−0.729610	−0.364805	−	0.931084i	$-0.618864\pi$
$13$	−444.579	−0.729610	−0.364805	+	0.931084i	$0.618864\pi$
$14$	0	0
$15$	−30.5865	−0.0350995
$16$	0	0
$17$	485.649	0.407569	0.203784	−	0.979016i	$-0.434676\pi$
$17$	485.649	0.407569	0.203784	+	0.979016i	$0.434676\pi$
$18$	0	0
$19$	1572.80	0.999513	0.499756	−	0.866166i	$-0.333423\pi$
$19$	1572.80	0.999513	0.499756	+	0.866166i	$0.333423\pi$
$20$	0	0
$21$	−66.1242	−0.0327199
$22$	0	0
$23$	398.704	0.157156	0.0785781	−	0.996908i	$-0.474962\pi$
$23$	398.704	0.157156	0.0785781	+	0.996908i	$0.474962\pi$
$24$	0	0
$25$	−1391.11	−0.445155
$26$	0	0
$27$	356.593	0.0941376
$28$	0	0
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	8469.77	1.58295	0.791475	−	0.611202i	$-0.209314\pi$
$31$	8469.77	1.58295	0.791475	+	0.611202i	$0.209314\pi$
$32$	0	0
$33$	−197.652	−0.0315948
$34$	0	0
$35$	3748.45	0.517228
$36$	0	0
$37$	3339.33	0.401010	0.200505	−	0.979693i	$-0.435742\pi$
$37$	3339.33	0.401010	0.200505	+	0.979693i	$0.435742\pi$
$38$	0	0
$39$	326.564	0.0343801
$40$	0	0
$41$	18154.2	1.68662	0.843311	−	0.537425i	$-0.180603\pi$
$41$	18154.2	1.68662	0.843311	+	0.537425i	$0.180603\pi$
$42$	0	0
$43$	−9996.71	−0.824491	−0.412246	−	0.911073i	$-0.635255\pi$
$43$	−9996.71	−0.824491	−0.412246	+	0.911073i	$0.635255\pi$
$44$	0	0
$45$	−10096.1	−0.743225
$46$	0	0
$47$	−12568.0	−0.829890	−0.414945	−	0.909846i	$-0.636199\pi$
$47$	−12568.0	−0.829890	−0.414945	+	0.909846i	$0.636199\pi$
$48$	0	0
$49$	−8703.31	−0.517838
$50$	0	0
$51$	−356.732	−0.0192051
$52$	0	0
$53$	−21343.1	−1.04368	−0.521841	−	0.853043i	$-0.674755\pi$
$53$	−21343.1	−1.04368	−0.521841	+	0.853043i	$0.674755\pi$
$54$	0	0
$55$	11204.5	0.499443
$56$	0	0
$57$	−1155.29	−0.0470982
$58$	0	0
$59$	30036.1	1.12334	0.561672	−	0.827360i	$-0.310158\pi$
$59$	30036.1	1.12334	0.561672	+	0.827360i	$0.310158\pi$
$60$	0	0
$61$	49792.6	1.71333	0.856663	−	0.515877i	$-0.172534\pi$
$61$	49792.6	1.71333	0.856663	+	0.515877i	$0.172534\pi$
$62$	0	0
$63$	−21826.4	−0.692837
$64$	0	0
$65$	−18512.3	−0.543471
$66$	0	0
$67$	−47588.2	−1.29513	−0.647563	−	0.762012i	$-0.724212\pi$
$67$	−47588.2	−1.29513	−0.647563	+	0.762012i	$0.724212\pi$
$68$	0	0
$69$	−292.867	−0.00740538
$70$	0	0
$71$	50164.6	1.18101	0.590503	−	0.807036i	$-0.298930\pi$
$71$	50164.6	1.18101	0.590503	+	0.807036i	$0.298930\pi$
$72$	0	0
$73$	−44770.3	−0.983294	−0.491647	−	0.870795i	$-0.663605\pi$
$73$	−44770.3	−0.983294	−0.491647	+	0.870795i	$0.663605\pi$
$74$	0	0
$75$	1021.83	0.0209762
$76$	0	0
$77$	24222.7	0.465582
$78$	0	0
$79$	78464.6	1.41451	0.707255	−	0.706959i	$-0.249933\pi$
$79$	78464.6	1.41451	0.707255	+	0.706959i	$0.249933\pi$
$80$	0	0
$81$	58656.0	0.993344
$82$	0	0
$83$	−46721.8	−0.744430	−0.372215	−	0.928146i	$-0.621402\pi$
$83$	−46721.8	−0.744430	−0.372215	+	0.928146i	$0.621402\pi$
$84$	0	0
$85$	20222.4	0.303589
$86$	0	0
$87$	617.753	0.00875017
$88$	0	0
$89$	−39465.7	−0.528135	−0.264068	−	0.964504i	$-0.585064\pi$
$89$	−39465.7	−0.528135	−0.264068	+	0.964504i	$0.585064\pi$
$90$	0	0
$91$	−40021.2	−0.506626
$92$	0	0
$93$	−6221.43	−0.0745904
$94$	0	0
$95$	65491.2	0.744516
$96$	0	0
$97$	48336.3	0.521608	0.260804	−	0.965392i	$-0.416012\pi$
$97$	48336.3	0.521608	0.260804	+	0.965392i	$0.416012\pi$
$98$	0	0
$99$	−65241.3	−0.669013
$100$	0	0
$101$	111417.	1.08680	0.543399	−	0.839475i	$-0.317137\pi$
$101$	111417.	1.08680	0.543399	+	0.839475i	$0.317137\pi$
$102$	0	0
$103$	1416.34	0.0131545	0.00657726	−	0.999978i	$-0.497906\pi$
$103$	1416.34	0.0131545	0.00657726	+	0.999978i	$0.497906\pi$
$104$	0	0
$105$	−2753.41	−0.0243724
$106$	0	0
$107$	162530.	1.37238	0.686188	−	0.727424i	$-0.259283\pi$
$107$	162530.	1.37238	0.686188	+	0.727424i	$0.259283\pi$
$108$	0	0
$109$	152425.	1.22882	0.614411	−	0.788986i	$-0.289394\pi$
$109$	152425.	1.22882	0.614411	+	0.788986i	$0.289394\pi$
$110$	0	0
$111$	−2452.89	−0.0188961
$112$	0	0
$113$	256365.	1.88870	0.944350	−	0.328942i	$-0.106692\pi$
$113$	256365.	1.88870	0.944350	+	0.328942i	$0.106692\pi$
$114$	0	0
$115$	16602.1	0.117062
$116$	0	0
$117$	107793.	0.727990
$118$	0	0
$119$	43718.4	0.283007
$120$	0	0
$121$	−88646.9	−0.550428
$122$	0	0
$123$	−13335.1	−0.0794756
$124$	0	0
$125$	−188051.	−1.07647
$126$	0	0
$127$	145336.	0.799585	0.399792	−	0.916606i	$-0.369082\pi$
$127$	145336.	0.799585	0.399792	+	0.916606i	$0.369082\pi$
$128$	0	0
$129$	7343.04	0.0388510
$130$	0	0
$131$	259373.	1.32052	0.660262	−	0.751035i	$-0.270445\pi$
$131$	259373.	1.32052	0.660262	+	0.751035i	$0.270445\pi$
$132$	0	0
$133$	141584.	0.694040
$134$	0	0
$135$	14848.5	0.0701212
$136$	0	0
$137$	94701.8	0.431079	0.215539	−	0.976495i	$-0.430849\pi$
$137$	94701.8	0.431079	0.215539	+	0.976495i	$0.430849\pi$
$138$	0	0
$139$	−237100.	−1.04086	−0.520432	−	0.853903i	$-0.674229\pi$
$139$	−237100.	−1.04086	−0.520432	+	0.853903i	$0.674229\pi$
$140$	0	0
$141$	9231.75	0.0391054
$142$	0	0
$143$	−119627.	−0.489205
$144$	0	0
$145$	−35019.2	−0.138321
$146$	0	0
$147$	6392.98	0.0244011
$148$	0	0
$149$	−81590.7	−0.301075	−0.150538	−	0.988604i	$-0.548100\pi$
$149$	−81590.7	−0.301075	−0.150538	+	0.988604i	$0.548100\pi$
$150$	0	0
$151$	199008.	0.710276	0.355138	−	0.934814i	$-0.384434\pi$
$151$	199008.	0.710276	0.355138	+	0.934814i	$0.384434\pi$
$152$	0	0
$153$	−117751.	−0.406664
$154$	0	0
$155$	352681.	1.17911
$156$	0	0
$157$	−321167.	−1.03988	−0.519938	−	0.854204i	$-0.674045\pi$
$157$	−321167.	−1.03988	−0.519938	+	0.854204i	$0.674045\pi$
$158$	0	0
$159$	15677.5	0.0491795
$160$	0	0
$161$	35891.6	0.109126
$162$	0	0
$163$	621023.	1.83079	0.915395	−	0.402557i	$-0.131879\pi$
$163$	621023.	1.83079	0.915395	+	0.402557i	$0.131879\pi$
$164$	0	0
$165$	−8230.21	−0.0235343
$166$	0	0
$167$	59437.0	0.164917	0.0824585	−	0.996594i	$-0.473723\pi$
$167$	59437.0	0.164917	0.0824585	+	0.996594i	$0.473723\pi$
$168$	0	0
$169$	−173642.	−0.467669
$170$	0	0
$171$	−381341.	−0.997293
$172$	0	0
$173$	175144.	0.444918	0.222459	−	0.974942i	$-0.428592\pi$
$173$	175144.	0.444918	0.222459	+	0.974942i	$0.428592\pi$
$174$	0	0
$175$	−125228.	−0.309106
$176$	0	0
$177$	−22062.9	−0.0529332
$178$	0	0
$179$	421857.	0.984086	0.492043	−	0.870571i	$-0.336250\pi$
$179$	421857.	0.984086	0.492043	+	0.870571i	$0.336250\pi$
$180$	0	0
$181$	313961.	0.712326	0.356163	−	0.934424i	$-0.384085\pi$
$181$	313961.	0.712326	0.356163	+	0.934424i	$0.384085\pi$
$182$	0	0
$183$	−36574.9	−0.0807338
$184$	0	0
$185$	139050.	0.298704
$186$	0	0
$187$	130679.	0.273275
$188$	0	0
$189$	32100.7	0.0653672
$190$	0	0
$191$	−85652.5	−0.169886	−0.0849428	−	0.996386i	$-0.527071\pi$
$191$	−85652.5	−0.169886	−0.0849428	+	0.996386i	$0.527071\pi$
$192$	0	0
$193$	356203.	0.688343	0.344171	−	0.938907i	$-0.388160\pi$
$193$	356203.	0.688343	0.344171	+	0.938907i	$0.388160\pi$
$194$	0	0
$195$	13598.1	0.0256090
$196$	0	0
$197$	2000.17	0.00367199	0.00183599	−	0.999998i	$-0.499416\pi$
$197$	2000.17	0.00367199	0.00183599	+	0.999998i	$0.499416\pi$
$198$	0	0
$199$	241348.	0.432027	0.216014	−	0.976390i	$-0.430694\pi$
$199$	241348.	0.432027	0.216014	+	0.976390i	$0.430694\pi$
$200$	0	0
$201$	34955.7	0.0610278
$202$	0	0
$203$	−75707.2	−0.128943
$204$	0	0
$205$	755942.	1.25633
$206$	0	0
$207$	−96670.0	−0.156807
$208$	0	0
$209$	423208.	0.670175
$210$	0	0
$211$	611130.	0.944991	0.472495	−	0.881333i	$-0.343353\pi$
$211$	611130.	0.944991	0.472495	+	0.881333i	$0.343353\pi$
$212$	0	0
$213$	−36848.2	−0.0556503
$214$	0	0
$215$	−416263.	−0.614146
$216$	0	0
$217$	762452.	1.09917
$218$	0	0
$219$	32885.9	0.0463339
$220$	0	0
$221$	−215910.	−0.297366
$222$	0	0
$223$	−702678.	−0.946225	−0.473112	−	0.881002i	$-0.656870\pi$
$223$	−702678.	−0.946225	−0.473112	+	0.881002i	$0.656870\pi$
$224$	0	0
$225$	337289.	0.444167
$226$	0	0
$227$	−933943.	−1.20297	−0.601487	−	0.798883i	$-0.705425\pi$
$227$	−933943.	−1.20297	−0.601487	+	0.798883i	$0.705425\pi$
$228$	0	0
$229$	−1.02337e6	−1.28956	−0.644782	−	0.764367i	$-0.723052\pi$
$229$	−1.02337e6	−1.28956	−0.644782	+	0.764367i	$0.723052\pi$
$230$	0	0
$231$	−17792.7	−0.0219387
$232$	0	0
$233$	−994807.	−1.20046	−0.600232	−	0.799826i	$-0.704925\pi$
$233$	−994807.	−1.20046	−0.600232	+	0.799826i	$0.704925\pi$
$234$	0	0
$235$	−523330.	−0.618168
$236$	0	0
$237$	−57635.8	−0.0666533
$238$	0	0
$239$	111950.	0.126773	0.0633867	−	0.997989i	$-0.479810\pi$
$239$	111950.	0.126773	0.0633867	+	0.997989i	$0.479810\pi$
$240$	0	0
$241$	1.31520e6	1.45865	0.729323	−	0.684169i	$-0.239835\pi$
$241$	1.31520e6	1.45865	0.729323	+	0.684169i	$0.239835\pi$
$242$	0	0
$243$	−129738.	−0.140945
$244$	0	0
$245$	−362406.	−0.385727
$246$	0	0
$247$	−699232.	−0.729255
$248$	0	0
$249$	34319.3	0.0350784
$250$	0	0
$251$	440669.	0.441498	0.220749	−	0.975331i	$-0.429150\pi$
$251$	440669.	0.441498	0.220749	+	0.975331i	$0.429150\pi$
$252$	0	0
$253$	107283.	0.105373
$254$	0	0
$255$	−14854.3	−0.0143055
$256$	0	0
$257$	−554939.	−0.524098	−0.262049	−	0.965055i	$-0.584398\pi$
$257$	−554939.	−0.524098	−0.262049	+	0.965055i	$0.584398\pi$
$258$	0	0
$259$	300608.	0.278453
$260$	0	0
$261$	203909.	0.185283
$262$	0	0
$263$	1.24351e6	1.10856	0.554279	−	0.832331i	$-0.312994\pi$
$263$	1.24351e6	1.10856	0.554279	+	0.832331i	$0.312994\pi$
$264$	0	0
$265$	−888728.	−0.777417
$266$	0	0
$267$	28989.4	0.0248863
$268$	0	0
$269$	598804.	0.504549	0.252275	−	0.967656i	$-0.418821\pi$
$269$	598804.	0.504549	0.252275	+	0.967656i	$0.418821\pi$
$270$	0	0
$271$	−654368.	−0.541251	−0.270626	−	0.962685i	$-0.587231\pi$
$271$	−654368.	−0.541251	−0.270626	+	0.962685i	$0.587231\pi$
$272$	0	0
$273$	29397.4	0.0238728
$274$	0	0
$275$	−374320.	−0.298477
$276$	0	0
$277$	−1.76929e6	−1.38548	−0.692739	−	0.721188i	$-0.743596\pi$
$277$	−1.76929e6	−1.38548	−0.692739	+	0.721188i	$0.743596\pi$
$278$	0	0
$279$	−2.05358e6	−1.57943
$280$	0	0
$281$	−495309.	−0.374206	−0.187103	−	0.982340i	$-0.559910\pi$
$281$	−495309.	−0.374206	−0.187103	+	0.982340i	$0.559910\pi$
$282$	0	0
$283$	785235.	0.582819	0.291409	−	0.956598i	$-0.405876\pi$
$283$	785235.	0.582819	0.291409	+	0.956598i	$0.405876\pi$
$284$	0	0
$285$	−48106.3	−0.0350824
$286$	0	0
$287$	1.63425e6	1.17115
$288$	0	0
$289$	−1.18400e6	−0.833888
$290$	0	0
$291$	−35505.2	−0.0245788
$292$	0	0
$293$	1.72437e6	1.17344	0.586719	−	0.809790i	$-0.300419\pi$
$293$	1.72437e6	1.17344	0.586719	+	0.809790i	$0.300419\pi$
$294$	0	0
$295$	1.25070e6	0.836755
$296$	0	0
$297$	95952.0	0.0631194
$298$	0	0
$299$	−177256.	−0.114663
$300$	0	0
$301$	−899909.	−0.572509
$302$	0	0
$303$	−81841.0	−0.0512111
$304$	0	0
$305$	2.07336e6	1.27622
$306$	0	0
$307$	3.12637e6	1.89319	0.946597	−	0.322419i	$-0.104496\pi$
$307$	3.12637e6	1.89319	0.946597	+	0.322419i	$0.104496\pi$
$308$	0	0
$309$	−1040.37	−0.000619856 0
$310$	0	0
$311$	−2.08430e6	−1.22196	−0.610982	−	0.791644i	$-0.709225\pi$
$311$	−2.08430e6	−1.22196	−0.610982	+	0.791644i	$0.709225\pi$
$312$	0	0
$313$	−2.27357e6	−1.31174	−0.655871	−	0.754873i	$-0.727698\pi$
$313$	−2.27357e6	−1.31174	−0.655871	+	0.754873i	$0.727698\pi$
$314$	0	0
$315$	−908852.	−0.516080
$316$	0	0
$317$	−891538.	−0.498301	−0.249151	−	0.968465i	$-0.580151\pi$
$317$	−891538.	−0.498301	−0.249151	+	0.968465i	$0.580151\pi$
$318$	0	0
$319$	−226296.	−0.124509
$320$	0	0
$321$	−119385.	−0.0646679
$322$	0	0
$323$	763827.	0.407370
$324$	0	0
$325$	618459.	0.324790
$326$	0	0
$327$	−111963.	−0.0579035
$328$	0	0
$329$	−1.13138e6	−0.576258
$330$	0	0
$331$	768207.	0.385397	0.192698	−	0.981258i	$-0.438276\pi$
$331$	768207.	0.385397	0.192698	+	0.981258i	$0.438276\pi$
$332$	0	0
$333$	−809656.	−0.400120
$334$	0	0
$335$	−1.98157e6	−0.964713
$336$	0	0
$337$	−740054.	−0.354968	−0.177484	−	0.984124i	$-0.556796\pi$
$337$	−740054.	−0.354968	−0.177484	+	0.984124i	$0.556796\pi$
$338$	0	0
$339$	−188312.	−0.0889977
$340$	0	0
$341$	2.27904e6	1.06137
$342$	0	0
$343$	−2.29645e6	−1.05395
$344$	0	0
$345$	−12195.0	−0.00551611
$346$	0	0
$347$	−2.65308e6	−1.18284	−0.591420	−	0.806363i	$-0.701433\pi$
$347$	−2.65308e6	−1.18284	−0.591420	+	0.806363i	$0.701433\pi$
$348$	0	0
$349$	1.02314e6	0.449646	0.224823	−	0.974400i	$-0.427820\pi$
$349$	1.02314e6	0.449646	0.224823	+	0.974400i	$0.427820\pi$
$350$	0	0
$351$	−158534.	−0.0686838
$352$	0	0
$353$	−2.17906e6	−0.930747	−0.465374	−	0.885114i	$-0.654080\pi$
$353$	−2.17906e6	−0.930747	−0.465374	+	0.885114i	$0.654080\pi$
$354$	0	0
$355$	2.08886e6	0.879706
$356$	0	0
$357$	−32113.2	−0.0133356
$358$	0	0
$359$	−4.09152e6	−1.67552	−0.837759	−	0.546041i	$-0.816134\pi$
$359$	−4.09152e6	−1.67552	−0.837759	+	0.546041i	$0.816134\pi$
$360$	0	0
$361$	−2413.03	−0.000974529 0
$362$	0	0
$363$	65115.2	0.0259368
$364$	0	0
$365$	−1.86424e6	−0.732435
$366$	0	0
$367$	113276.	0.0439009	0.0219504	−	0.999759i	$-0.493012\pi$
$367$	113276.	0.0439009	0.0219504	+	0.999759i	$0.493012\pi$
$368$	0	0
$369$	−4.40168e6	−1.68288
$370$	0	0
$371$	−1.92132e6	−0.724711
$372$	0	0
$373$	−1.74960e6	−0.651130	−0.325565	−	0.945520i	$-0.605554\pi$
$373$	−1.74960e6	−0.651130	−0.325565	+	0.945520i	$0.605554\pi$
$374$	0	0
$375$	138132.	0.0507243
$376$	0	0
$377$	373891.	0.135485
$378$	0	0
$379$	−2.62142e6	−0.937429	−0.468714	−	0.883350i	$-0.655283\pi$
$379$	−2.62142e6	−0.937429	−0.468714	+	0.883350i	$0.655283\pi$
$380$	0	0
$381$	−106756.	−0.0376773
$382$	0	0
$383$	−4.63046e6	−1.61297	−0.806486	−	0.591253i	$-0.798633\pi$
$383$	−4.63046e6	−1.61297	−0.806486	+	0.591253i	$0.798633\pi$
$384$	0	0
$385$	1.00863e6	0.346802
$386$	0	0
$387$	2.42381e6	0.822660
$388$	0	0
$389$	−3.41385e6	−1.14385	−0.571927	−	0.820305i	$-0.693804\pi$
$389$	−3.41385e6	−1.14385	−0.571927	+	0.820305i	$0.693804\pi$
$390$	0	0
$391$	193631.	0.0640519
$392$	0	0
$393$	−190521.	−0.0622246
$394$	0	0
$395$	3.26726e6	1.05364
$396$	0	0
$397$	−2.05992e6	−0.655954	−0.327977	−	0.944686i	$-0.606367\pi$
$397$	−2.05992e6	−0.655954	−0.327977	+	0.944686i	$0.606367\pi$
$398$	0	0
$399$	−104000.	−0.0327040
$400$	0	0
$401$	4.48615e6	1.39320	0.696599	−	0.717461i	$-0.254696\pi$
$401$	4.48615e6	1.39320	0.696599	+	0.717461i	$0.254696\pi$
$402$	0	0
$403$	−3.76548e6	−1.15494
$404$	0	0
$405$	2.44243e6	0.739921
$406$	0	0
$407$	898548.	0.268878
$408$	0	0
$409$	−5.12907e6	−1.51611	−0.758055	−	0.652191i	$-0.773850\pi$
$409$	−5.12907e6	−1.51611	−0.758055	+	0.652191i	$0.773850\pi$
$410$	0	0
$411$	−69562.8	−0.0203129
$412$	0	0
$413$	2.70386e6	0.780026
$414$	0	0
$415$	−1.94549e6	−0.554511
$416$	0	0
$417$	174161.	0.0490467
$418$	0	0
$419$	2.21170e6	0.615446	0.307723	−	0.951476i	$-0.400433\pi$
$419$	2.21170e6	0.615446	0.307723	+	0.951476i	$0.400433\pi$
$420$	0	0
$421$	751122.	0.206540	0.103270	−	0.994653i	$-0.467069\pi$
$421$	751122.	0.206540	0.103270	+	0.994653i	$0.467069\pi$
$422$	0	0
$423$	3.04724e6	0.828047
$424$	0	0
$425$	−675592.	−0.181431
$426$	0	0
$427$	4.48235e6	1.18970
$428$	0	0
$429$	87871.8	0.0230519
$430$	0	0
$431$	−2.98345e6	−0.773615	−0.386808	−	0.922160i	$-0.626422\pi$
$431$	−2.98345e6	−0.773615	−0.386808	+	0.922160i	$0.626422\pi$
$432$	0	0
$433$	−2.17337e6	−0.557075	−0.278537	−	0.960425i	$-0.589850\pi$
$433$	−2.17337e6	−0.557075	−0.278537	+	0.960425i	$0.589850\pi$
$434$	0	0
$435$	25723.2	0.00651782
$436$	0	0
$437$	627081.	0.157080
$438$	0	0
$439$	−24436.8	−0.00605178	−0.00302589	−	0.999995i	$-0.500963\pi$
$439$	−24436.8	−0.00605178	−0.00302589	+	0.999995i	$0.500963\pi$
$440$	0	0
$441$	2.11021e6	0.516689
$442$	0	0
$443$	6.19315e6	1.49935	0.749674	−	0.661808i	$-0.230211\pi$
$443$	6.19315e6	1.49935	0.749674	+	0.661808i	$0.230211\pi$
$444$	0	0
$445$	−1.64335e6	−0.393397
$446$	0	0
$447$	59932.1	0.0141870
$448$	0	0
$449$	7.56136e6	1.77004	0.885022	−	0.465548i	$-0.154143\pi$
$449$	7.56136e6	1.77004	0.885022	+	0.465548i	$0.154143\pi$
$450$	0	0
$451$	4.88494e6	1.13088
$452$	0	0
$453$	−146180.	−0.0334690
$454$	0	0
$455$	−1.66648e6	−0.377375
$456$	0	0
$457$	5.02930e6	1.12646	0.563232	−	0.826299i	$-0.309558\pi$
$457$	5.02930e6	1.12646	0.563232	+	0.826299i	$0.309558\pi$
$458$	0	0
$459$	173179.	0.0383675
$460$	0	0
$461$	3.58934e6	0.786615	0.393308	−	0.919407i	$-0.371331\pi$
$461$	3.58934e6	0.786615	0.393308	+	0.919407i	$0.371331\pi$
$462$	0	0
$463$	−8.86551e6	−1.92199	−0.960996	−	0.276563i	$-0.910804\pi$
$463$	−8.86551e6	−1.92199	−0.960996	+	0.276563i	$0.910804\pi$
$464$	0	0
$465$	−259060.	−0.0555608
$466$	0	0
$467$	−3.24975e6	−0.689537	−0.344769	−	0.938688i	$-0.612043\pi$
$467$	−3.24975e6	−0.689537	−0.344769	+	0.938688i	$0.612043\pi$
$468$	0	0
$469$	−4.28391e6	−0.899308
$470$	0	0
$471$	235912.	0.0490001
$472$	0	0
$473$	−2.68992e6	−0.552823
$474$	0	0
$475$	−2.18793e6	−0.444938
$476$	0	0
$477$	5.17486e6	1.04136
$478$	0	0
$479$	2.33334e6	0.464664	0.232332	−	0.972637i	$-0.425364\pi$
$479$	2.33334e6	0.464664	0.232332	+	0.972637i	$0.425364\pi$
$480$	0	0
$481$	−1.48460e6	−0.292581
$482$	0	0
$483$	−26364.0	−0.00514213
$484$	0	0
$485$	2.01272e6	0.388535
$486$	0	0
$487$	−1.35787e6	−0.259440	−0.129720	−	0.991551i	$-0.541408\pi$
$487$	−1.35787e6	−0.259440	−0.129720	+	0.991551i	$0.541408\pi$
$488$	0	0
$489$	−456170.	−0.0862689
$490$	0	0
$491$	−4.42097e6	−0.827587	−0.413794	−	0.910371i	$-0.635797\pi$
$491$	−4.42097e6	−0.827587	−0.413794	+	0.910371i	$0.635797\pi$
$492$	0	0
$493$	−408431.	−0.0756836
$494$	0	0
$495$	−2.71665e6	−0.498334
$496$	0	0
$497$	4.51585e6	0.820065
$498$	0	0
$499$	9.61078e6	1.72785	0.863927	−	0.503617i	$-0.167998\pi$
$499$	9.61078e6	1.72785	0.863927	+	0.503617i	$0.167998\pi$
$500$	0	0
$501$	−43659.2	−0.00777108
$502$	0	0
$503$	6.10366e6	1.07565	0.537824	−	0.843057i	$-0.319246\pi$
$503$	6.10366e6	1.07565	0.537824	+	0.843057i	$0.319246\pi$
$504$	0	0
$505$	4.63941e6	0.809532
$506$	0	0
$507$	127548.	0.0220371
$508$	0	0
$509$	−1.01562e7	−1.73755	−0.868775	−	0.495208i	$-0.835092\pi$
$509$	−1.01562e7	−1.73755	−0.868775	+	0.495208i	$0.835092\pi$
$510$	0	0
$511$	−4.03025e6	−0.682778
$512$	0	0
$513$	560848.	0.0940918
$514$	0	0
$515$	58976.4	0.00979852
$516$	0	0
$517$	−3.38179e6	−0.556443
$518$	0	0
$519$	−128651.	−0.0209650
$520$	0	0
$521$	2.42580e6	0.391526	0.195763	−	0.980651i	$-0.437282\pi$
$521$	2.42580e6	0.391526	0.195763	+	0.980651i	$0.437282\pi$
$522$	0	0
$523$	328610.	0.0525322	0.0262661	−	0.999655i	$-0.491638\pi$
$523$	328610.	0.0525322	0.0262661	+	0.999655i	$0.491638\pi$
$524$	0	0
$525$	91986.0	0.0145654
$526$	0	0
$527$	4.11334e6	0.645160
$528$	0	0
$529$	−6.27738e6	−0.975302
$530$	0	0
$531$	−7.28255e6	−1.12085
$532$	0	0
$533$	−8.07099e6	−1.23058
$534$	0	0
$535$	6.76773e6	1.02225
$536$	0	0
$537$	−309874.	−0.0463713
$538$	0	0
$539$	−2.34189e6	−0.347212
$540$	0	0
$541$	5.52172e6	0.811113	0.405557	−	0.914070i	$-0.367078\pi$
$541$	5.52172e6	0.811113	0.405557	+	0.914070i	$0.367078\pi$
$542$	0	0
$543$	−230618.	−0.0335656
$544$	0	0
$545$	6.34697e6	0.915324
$546$	0	0
$547$	−2.61929e6	−0.374296	−0.187148	−	0.982332i	$-0.559924\pi$
$547$	−2.61929e6	−0.374296	−0.187148	+	0.982332i	$0.559924\pi$
$548$	0	0
$549$	−1.20727e7	−1.70952
$550$	0	0
$551$	−1.32272e6	−0.185605
$552$	0	0
$553$	7.06342e6	0.982205
$554$	0	0
$555$	−102138.	−0.0140753
$556$	0	0
$557$	4.95349e6	0.676509	0.338254	−	0.941055i	$-0.390164\pi$
$557$	4.95349e6	0.676509	0.338254	+	0.941055i	$0.390164\pi$
$558$	0	0
$559$	4.44433e6	0.601557
$560$	0	0
$561$	−95989.4	−0.0128770
$562$	0	0
$563$	2.28738e6	0.304135	0.152068	−	0.988370i	$-0.451407\pi$
$563$	2.28738e6	0.304135	0.152068	+	0.988370i	$0.451407\pi$
$564$	0	0
$565$	1.06751e7	1.40685
$566$	0	0
$567$	5.28024e6	0.689757
$568$	0	0
$569$	−3.25672e6	−0.421696	−0.210848	−	0.977519i	$-0.567622\pi$
$569$	−3.25672e6	−0.421696	−0.210848	+	0.977519i	$0.567622\pi$
$570$	0	0
$571$	9.92284e6	1.27364	0.636819	−	0.771013i	$-0.280250\pi$
$571$	9.92284e6	1.27364	0.636819	+	0.771013i	$0.280250\pi$
$572$	0	0
$573$	62915.7	0.00800520
$574$	0	0
$575$	−554642.	−0.0699589
$576$	0	0
$577$	1.23237e6	0.154100	0.0770498	−	0.997027i	$-0.475450\pi$
$577$	1.23237e6	0.154100	0.0770498	+	0.997027i	$0.475450\pi$
$578$	0	0
$579$	−261648.	−0.0324355
$580$	0	0
$581$	−4.20592e6	−0.516916
$582$	0	0
$583$	−5.74301e6	−0.699791
$584$	0	0
$585$	4.48850e6	0.542265
$586$	0	0
$587$	1.02407e7	1.22669	0.613345	−	0.789815i	$-0.289823\pi$
$587$	1.02407e7	1.22669	0.613345	+	0.789815i	$0.289823\pi$
$588$	0	0
$589$	1.33212e7	1.58218
$590$	0	0
$591$	−1469.22	−0.000173028 0
$592$	0	0
$593$	−1.53106e7	−1.78795	−0.893977	−	0.448114i	$-0.852096\pi$
$593$	−1.53106e7	−1.78795	−0.893977	+	0.448114i	$0.852096\pi$
$594$	0	0
$595$	1.82043e6	0.210806
$596$	0	0
$597$	−177281.	−0.0203576
$598$	0	0
$599$	1.37662e6	0.156764	0.0783819	−	0.996923i	$-0.475025\pi$
$599$	1.37662e6	0.156764	0.0783819	+	0.996923i	$0.475025\pi$
$600$	0	0
$601$	−1.39459e7	−1.57492	−0.787462	−	0.616363i	$-0.788606\pi$
$601$	−1.39459e7	−1.57492	−0.787462	+	0.616363i	$0.788606\pi$
$602$	0	0
$603$	1.15383e7	1.29225
$604$	0	0
$605$	−3.69126e6	−0.410002
$606$	0	0
$607$	2.10503e6	0.231892	0.115946	−	0.993255i	$-0.463010\pi$
$607$	2.10503e6	0.231892	0.115946	+	0.993255i	$0.463010\pi$
$608$	0	0
$609$	55610.4	0.00607593
$610$	0	0
$611$	5.58746e6	0.605496
$612$	0	0
$613$	1.75643e7	1.88790	0.943950	−	0.330088i	$-0.107078\pi$
$613$	1.75643e7	1.88790	0.943950	+	0.330088i	$0.107078\pi$
$614$	0	0
$615$	−555274.	−0.0591997
$616$	0	0
$617$	−6.68190e6	−0.706622	−0.353311	−	0.935506i	$-0.614944\pi$
$617$	−6.68190e6	−0.706622	−0.353311	+	0.935506i	$0.614944\pi$
$618$	0	0
$619$	6.69302e6	0.702095	0.351047	−	0.936358i	$-0.385826\pi$
$619$	6.69302e6	0.702095	0.351047	+	0.936358i	$0.385826\pi$
$620$	0	0
$621$	142175.	0.0147943
$622$	0	0
$623$	−3.55272e6	−0.366726
$624$	0	0
$625$	−3.48322e6	−0.356682
$626$	0	0
$627$	−310866.	−0.0315794
$628$	0	0
$629$	1.62175e6	0.163439
$630$	0	0
$631$	−7.79549e6	−0.779417	−0.389709	−	0.920938i	$-0.627424\pi$
$631$	−7.79549e6	−0.779417	−0.389709	+	0.920938i	$0.627424\pi$
$632$	0	0
$633$	−448903.	−0.0445290
$634$	0	0
$635$	6.05180e6	0.595594
$636$	0	0
$637$	3.86931e6	0.377820
$638$	0	0
$639$	−1.21629e7	−1.17838
$640$	0	0
$641$	6.27067e6	0.602794	0.301397	−	0.953499i	$-0.402547\pi$
$641$	6.27067e6	0.602794	0.301397	+	0.953499i	$0.402547\pi$
$642$	0	0
$643$	1.59445e7	1.52084	0.760420	−	0.649432i	$-0.224993\pi$
$643$	1.59445e7	1.52084	0.760420	+	0.649432i	$0.224993\pi$
$644$	0	0
$645$	305764.	0.0289393
$646$	0	0
$647$	−1.59290e7	−1.49598	−0.747992	−	0.663707i	$-0.768982\pi$
$647$	−1.59290e7	−1.49598	−0.747992	+	0.663707i	$0.768982\pi$
$648$	0	0
$649$	8.08210e6	0.753204
$650$	0	0
$651$	−560056.	−0.0517940
$652$	0	0
$653$	2.65262e6	0.243440	0.121720	−	0.992564i	$-0.461159\pi$
$653$	2.65262e6	0.243440	0.121720	+	0.992564i	$0.461159\pi$
$654$	0	0
$655$	1.08003e7	0.983631
$656$	0	0
$657$	1.08550e7	0.981111
$658$	0	0
$659$	−1.35066e7	−1.21152	−0.605761	−	0.795647i	$-0.707131\pi$
$659$	−1.35066e7	−1.21152	−0.605761	+	0.795647i	$0.707131\pi$
$660$	0	0
$661$	1.17111e7	1.04254	0.521272	−	0.853391i	$-0.325458\pi$
$661$	1.17111e7	1.04254	0.521272	+	0.853391i	$0.325458\pi$
$662$	0	0
$663$	158596.	0.0140122
$664$	0	0
$665$	5.89555e6	0.516976
$666$	0	0
$667$	−335310.	−0.0291832
$668$	0	0
$669$	516149.	0.0445872
$670$	0	0
$671$	1.33982e7	1.14879
$672$	0	0
$673$	1.81529e7	1.54492	0.772462	−	0.635061i	$-0.219025\pi$
$673$	1.81529e7	1.54492	0.772462	+	0.635061i	$0.219025\pi$
$674$	0	0
$675$	−496060.	−0.0419059
$676$	0	0
$677$	−3.81675e6	−0.320053	−0.160027	−	0.987113i	$-0.551158\pi$
$677$	−3.81675e6	−0.320053	−0.160027	+	0.987113i	$0.551158\pi$
$678$	0	0
$679$	4.35126e6	0.362193
$680$	0	0
$681$	686024.	0.0566855
$682$	0	0
$683$	2.60743e6	0.213876	0.106938	−	0.994266i	$-0.465895\pi$
$683$	2.60743e6	0.213876	0.106938	+	0.994266i	$0.465895\pi$
$684$	0	0
$685$	3.94338e6	0.321102
$686$	0	0
$687$	751710.	0.0607657
$688$	0	0
$689$	9.48871e6	0.761481
$690$	0	0
$691$	−81457.6	−0.00648988	−0.00324494	−	0.999995i	$-0.501033\pi$
$691$	−81457.6	−0.00648988	−0.00324494	+	0.999995i	$0.501033\pi$
$692$	0	0
$693$	−5.87305e6	−0.464548
$694$	0	0
$695$	−9.87283e6	−0.775318
$696$	0	0
$697$	8.81659e6	0.687414
$698$	0	0
$699$	730731.	0.0565672
$700$	0	0
$701$	−2.32967e7	−1.79060	−0.895300	−	0.445464i	$-0.853039\pi$
$701$	−2.32967e7	−1.79060	−0.895300	+	0.445464i	$0.853039\pi$
$702$	0	0
$703$	5.25209e6	0.400815
$704$	0	0
$705$	384410.	0.0291288
$706$	0	0
$707$	1.00298e7	0.754649
$708$	0	0
$709$	7.52252e6	0.562015	0.281008	−	0.959706i	$-0.409331\pi$
$709$	7.52252e6	0.562015	0.281008	+	0.959706i	$0.409331\pi$
$710$	0	0
$711$	−1.90246e7	−1.41137
$712$	0	0
$713$	3.37693e6	0.248770
$714$	0	0
$715$	−4.98129e6	−0.364398
$716$	0	0
$717$	−82232.2	−0.00597371
$718$	0	0
$719$	−1.30064e7	−0.938289	−0.469144	−	0.883121i	$-0.655438\pi$
$719$	−1.30064e7	−0.938289	−0.469144	+	0.883121i	$0.655438\pi$
$720$	0	0
$721$	127500.	0.00913421
$722$	0	0
$723$	−966076.	−0.0687331
$724$	0	0
$725$	1.16992e6	0.0826632
$726$	0	0
$727$	−1.98038e7	−1.38967	−0.694835	−	0.719169i	$-0.744523\pi$
$727$	−1.98038e7	−1.38967	−0.694835	+	0.719169i	$0.744523\pi$
$728$	0	0
$729$	−1.41581e7	−0.986702
$730$	0	0
$731$	−4.85490e6	−0.336037
$732$	0	0
$733$	1.81414e7	1.24713	0.623565	−	0.781772i	$-0.285684\pi$
$733$	1.81414e7	1.24713	0.623565	+	0.781772i	$0.285684\pi$
$734$	0	0
$735$	266204.	0.0181759
$736$	0	0
$737$	−1.28050e7	−0.868385
$738$	0	0
$739$	2.22326e7	1.49754	0.748771	−	0.662829i	$-0.230645\pi$
$739$	2.22326e7	1.49754	0.748771	+	0.662829i	$0.230645\pi$
$740$	0	0
$741$	513618.	0.0343633
$742$	0	0
$743$	1.40248e6	0.0932021	0.0466011	−	0.998914i	$-0.485161\pi$
$743$	1.40248e6	0.0932021	0.0466011	+	0.998914i	$0.485161\pi$
$744$	0	0
$745$	−3.39744e6	−0.224264
$746$	0	0
$747$	1.13282e7	0.742777
$748$	0	0
$749$	1.46310e7	0.952948
$750$	0	0
$751$	−1.92135e7	−1.24310	−0.621552	−	0.783373i	$-0.713497\pi$
$751$	−1.92135e7	−1.24310	−0.621552	+	0.783373i	$0.713497\pi$
$752$	0	0
$753$	−323692.	−0.0208039
$754$	0	0
$755$	8.28668e6	0.529070
$756$	0	0
$757$	−1.23078e7	−0.780621	−0.390310	−	0.920683i	$-0.627632\pi$
$757$	−1.23078e7	−0.780621	−0.390310	+	0.920683i	$0.627632\pi$
$758$	0	0
$759$	−78804.6	−0.00496532
$760$	0	0
$761$	4.65071e6	0.291110	0.145555	−	0.989350i	$-0.453503\pi$
$761$	4.65071e6	0.291110	0.145555	+	0.989350i	$0.453503\pi$
$762$	0	0
$763$	1.37214e7	0.853268
$764$	0	0
$765$	−4.90314e6	−0.302915
$766$	0	0
$767$	−1.33534e7	−0.819603
$768$	0	0
$769$	1.54937e6	0.0944800	0.0472400	−	0.998884i	$-0.484957\pi$
$769$	1.54937e6	0.0944800	0.0472400	+	0.998884i	$0.484957\pi$
$770$	0	0
$771$	407628.	0.0246961
$772$	0	0
$773$	−1.71208e7	−1.03056	−0.515282	−	0.857021i	$-0.672313\pi$
$773$	−1.71208e7	−1.03056	−0.515282	+	0.857021i	$0.672313\pi$
$774$	0	0
$775$	−1.17824e7	−0.704658
$776$	0	0
$777$	−220811.	−0.0131210
$778$	0	0
$779$	2.85529e7	1.68580
$780$	0	0
$781$	1.34983e7	0.791866
$782$	0	0
$783$	−299895.	−0.0174809
$784$	0	0
$785$	−1.33734e7	−0.774581
$786$	0	0
$787$	681275.	0.0392090	0.0196045	−	0.999808i	$-0.493759\pi$
$787$	681275.	0.0392090	0.0196045	+	0.999808i	$0.493759\pi$
$788$	0	0
$789$	−913412.	−0.0522365
$790$	0	0
$791$	2.30781e7	1.31147
$792$	0	0
$793$	−2.21367e7	−1.25006
$794$	0	0
$795$	652811.	0.0366328
$796$	0	0
$797$	530445.	0.0295798	0.0147899	−	0.999891i	$-0.495292\pi$
$797$	530445.	0.0295798	0.0147899	+	0.999891i	$0.495292\pi$
$798$	0	0
$799$	−6.10363e6	−0.338237
$800$	0	0
$801$	9.56888e6	0.526962
$802$	0	0
$803$	−1.20468e7	−0.659300
$804$	0	0
$805$	1.49452e6	0.0812856
$806$	0	0
$807$	−439849.	−0.0237749
$808$	0	0
$809$	−2.65631e7	−1.42694	−0.713472	−	0.700684i	$-0.752878\pi$
$809$	−2.65631e7	−1.42694	−0.713472	+	0.700684i	$0.752878\pi$
$810$	0	0
$811$	−1.11660e7	−0.596137	−0.298069	−	0.954544i	$-0.596342\pi$
$811$	−1.11660e7	−0.596137	−0.298069	+	0.954544i	$0.596342\pi$
$812$	0	0
$813$	480663.	0.0255044
$814$	0	0
$815$	2.58594e7	1.36372
$816$	0	0
$817$	−1.57228e7	−0.824089
$818$	0	0
$819$	9.70357e6	0.505501
$820$	0	0
$821$	−3.42932e7	−1.77562	−0.887811	−	0.460209i	$-0.847774\pi$
$821$	−3.42932e7	−1.77562	−0.887811	+	0.460209i	$0.847774\pi$
$822$	0	0
$823$	−1.93467e7	−0.995651	−0.497825	−	0.867277i	$-0.665868\pi$
$823$	−1.93467e7	−0.995651	−0.497825	+	0.867277i	$0.665868\pi$
$824$	0	0
$825$	274955.	0.0140646
$826$	0	0
$827$	−2.67747e7	−1.36132	−0.680662	−	0.732598i	$-0.738308\pi$
$827$	−2.67747e7	−1.36132	−0.680662	+	0.732598i	$0.738308\pi$
$828$	0	0
$829$	−2.44308e7	−1.23467	−0.617335	−	0.786700i	$-0.711788\pi$
$829$	−2.44308e7	−1.23467	−0.617335	+	0.786700i	$0.711788\pi$
$830$	0	0
$831$	1.29962e6	0.0652853
$832$	0	0
$833$	−4.22676e6	−0.211055
$834$	0	0
$835$	2.47496e6	0.122843
$836$	0	0
$837$	3.02026e6	0.149015
$838$	0	0
$839$	−9.17559e6	−0.450017	−0.225009	−	0.974357i	$-0.572241\pi$
$839$	−9.17559e6	−0.450017	−0.225009	+	0.974357i	$0.572241\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	0	0
$843$	363827.	0.0176330
$844$	0	0
$845$	−7.23046e6	−0.348357
$846$	0	0
$847$	−7.98004e6	−0.382205
$848$	0	0
$849$	−576791.	−0.0274631
$850$	0	0
$851$	1.33141e6	0.0630212
$852$	0	0
$853$	1.34195e7	0.631487	0.315744	−	0.948845i	$-0.397746\pi$
$853$	1.34195e7	0.631487	0.315744	+	0.948845i	$0.397746\pi$
$854$	0	0
$855$	−1.58790e7	−0.742863
$856$	0	0
$857$	4.82712e6	0.224510	0.112255	−	0.993679i	$-0.464193\pi$
$857$	4.82712e6	0.224510	0.112255	+	0.993679i	$0.464193\pi$
$858$	0	0
$859$	−2.23698e7	−1.03438	−0.517188	−	0.855872i	$-0.673021\pi$
$859$	−2.23698e7	−1.03438	−0.517188	+	0.855872i	$0.673021\pi$
$860$	0	0
$861$	−1.20043e6	−0.0551861
$862$	0	0
$863$	2.28440e6	0.104411	0.0522054	−	0.998636i	$-0.483375\pi$
$863$	2.28440e6	0.104411	0.0522054	+	0.998636i	$0.483375\pi$
$864$	0	0
$865$	7.29299e6	0.331410
$866$	0	0
$867$	869703.	0.0392937
$868$	0	0
$869$	2.11132e7	0.948431
$870$	0	0
$871$	2.11567e7	0.944938
$872$	0	0
$873$	−1.17196e7	−0.520450
$874$	0	0
$875$	−1.69284e7	−0.747475
$876$	0	0
$877$	2.19674e7	0.964452	0.482226	−	0.876047i	$-0.339828\pi$
$877$	2.19674e7	0.964452	0.482226	+	0.876047i	$0.339828\pi$
$878$	0	0
$879$	−1.26663e6	−0.0552938
$880$	0	0
$881$	2.81070e6	0.122004	0.0610021	−	0.998138i	$-0.480570\pi$
$881$	2.81070e6	0.122004	0.0610021	+	0.998138i	$0.480570\pi$
$882$	0	0
$883$	3.24911e7	1.40237	0.701186	−	0.712978i	$-0.252654\pi$
$883$	3.24911e7	1.40237	0.701186	+	0.712978i	$0.252654\pi$
$884$	0	0
$885$	−918697.	−0.0394289
$886$	0	0
$887$	−3.68463e7	−1.57248	−0.786239	−	0.617923i	$-0.787974\pi$
$887$	−3.68463e7	−1.57248	−0.786239	+	0.617923i	$0.787974\pi$
$888$	0	0
$889$	1.30832e7	0.555215
$890$	0	0
$891$	1.57831e7	0.666039
$892$	0	0
$893$	−1.97669e7	−0.829486
$894$	0	0
$895$	1.75661e7	0.733025
$896$	0	0
$897$	130202.	0.00540304
$898$	0	0
$899$	−7.12307e6	−0.293946
$900$	0	0
$901$	−1.03653e7	−0.425372
$902$	0	0
$903$	661024.	0.0269773
$904$	0	0
$905$	1.30733e7	0.530597
$906$	0	0
$907$	−6.77134e6	−0.273311	−0.136655	−	0.990619i	$-0.543635\pi$
$907$	−6.77134e6	−0.273311	−0.136655	+	0.990619i	$0.543635\pi$
$908$	0	0
$909$	−2.70142e7	−1.08438
$910$	0	0
$911$	−1.38061e7	−0.551156	−0.275578	−	0.961279i	$-0.588869\pi$
$911$	−1.38061e7	−0.551156	−0.275578	+	0.961279i	$0.588869\pi$
$912$	0	0
$913$	−1.25719e7	−0.499142
$914$	0	0
$915$	−1.52298e6	−0.0601369
$916$	0	0
$917$	2.33489e7	0.916943
$918$	0	0
$919$	3.45509e7	1.34949	0.674746	−	0.738050i	$-0.264253\pi$
$919$	3.45509e7	1.34949	0.674746	+	0.738050i	$0.264253\pi$
$920$	0	0
$921$	−2.29647e6	−0.0892094
$922$	0	0
$923$	−2.23022e7	−0.861673
$924$	0	0
$925$	−4.64538e6	−0.178512
$926$	0	0
$927$	−343407.	−0.0131253
$928$	0	0
$929$	−2.47472e7	−0.940776	−0.470388	−	0.882460i	$-0.655886\pi$
$929$	−2.47472e7	−0.940776	−0.470388	+	0.882460i	$0.655886\pi$
$930$	0	0
$931$	−1.36885e7	−0.517586
$932$	0	0
$933$	1.53101e6	0.0575803
$934$	0	0
$935$	5.44146e6	0.203557
$936$	0	0
$937$	1.40114e6	0.0521352	0.0260676	−	0.999660i	$-0.491701\pi$
$937$	1.40114e6	0.0521352	0.0260676	+	0.999660i	$0.491701\pi$
$938$	0	0
$939$	1.67004e6	0.0618107
$940$	0	0
$941$	−2.79452e7	−1.02881	−0.514403	−	0.857548i	$-0.671987\pi$
$941$	−2.79452e7	−1.02881	−0.514403	+	0.857548i	$0.671987\pi$
$942$	0	0
$943$	7.23817e6	0.265063
$944$	0	0
$945$	1.33667e6	0.0486906
$946$	0	0
$947$	−3.03936e7	−1.10131	−0.550653	−	0.834734i	$-0.685621\pi$
$947$	−3.03936e7	−1.10131	−0.550653	+	0.834734i	$0.685621\pi$
$948$	0	0
$949$	1.99040e7	0.717421
$950$	0	0
$951$	654876.	0.0234805
$952$	0	0
$953$	3.75892e7	1.34070	0.670349	−	0.742046i	$-0.266144\pi$
$953$	3.75892e7	1.34070	0.670349	+	0.742046i	$0.266144\pi$
$954$	0	0
$955$	−3.56657e6	−0.126544
$956$	0	0
$957$	166225.	0.00586701
$958$	0	0
$959$	8.52510e6	0.299332
$960$	0	0
$961$	4.31078e7	1.50573
$962$	0	0
$963$	−3.94070e7	−1.36933
$964$	0	0
$965$	1.48323e7	0.512732
$966$	0	0
$967$	−4.14600e7	−1.42582	−0.712908	−	0.701258i	$-0.752622\pi$
$967$	−4.14600e7	−1.42582	−0.712908	+	0.701258i	$0.752622\pi$
$968$	0	0
$969$	−561066.	−0.0191957
$970$	0	0
$971$	5.71685e6	0.194585	0.0972924	−	0.995256i	$-0.468982\pi$
$971$	5.71685e6	0.194585	0.0972924	+	0.995256i	$0.468982\pi$
$972$	0	0
$973$	−2.13438e7	−0.722754
$974$	0	0
$975$	−454286.	−0.0153045
$976$	0	0
$977$	2.35225e6	0.0788400	0.0394200	−	0.999223i	$-0.487449\pi$
$977$	2.35225e6	0.0788400	0.0394200	+	0.999223i	$0.487449\pi$
$978$	0	0
$979$	−1.06194e7	−0.354115
$980$	0	0
$981$	−3.69570e7	−1.22609
$982$	0	0
$983$	2.67280e7	0.882232	0.441116	−	0.897450i	$-0.354583\pi$
$983$	2.67280e7	0.882232	0.441116	+	0.897450i	$0.354583\pi$
$984$	0	0
$985$	83287.1	0.00273519
$986$	0	0
$987$	831047.	0.0271539
$988$	0	0
$989$	−3.98573e6	−0.129574
$990$	0	0
$991$	1.24011e7	0.401122	0.200561	−	0.979681i	$-0.435724\pi$
$991$	1.24011e7	0.401122	0.200561	+	0.979681i	$0.435724\pi$
$992$	0	0
$993$	−564283.	−0.0181603
$994$	0	0
$995$	1.00497e7	0.321808
$996$	0	0
$997$	1.41176e7	0.449803	0.224902	−	0.974381i	$-0.427794\pi$
$997$	1.41176e7	0.449803	0.224902	+	0.974381i	$0.427794\pi$
$998$	0	0
$999$	1.19078e6	0.0377502

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.a.i.1.2		4
4.3	odd	2		29.6.a.a.1.1	✓	4
12.11	even	2		261.6.a.a.1.4		4
20.19	odd	2		725.6.a.a.1.4		4
116.115	odd	2		841.6.a.a.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.1	✓	4	4.3	odd	2
261.6.a.a.1.4		4	12.11	even	2
464.6.a.i.1.2		4	1.1	even	1	trivial
725.6.a.a.1.4		4	20.19	odd	2
841.6.a.a.1.4		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)

\(f(q)\)	\(=\)	\(q-0.734546 q^{3} +41.6400 q^{5} +90.0205 q^{7} -242.460 q^{9} +O(q^{10})\)	\(q-0.734546 q^{3} +41.6400 q^{5} +90.0205 q^{7} -242.460 q^{9} +269.080 q^{11} -444.579 q^{13} -30.5865 q^{15} +485.649 q^{17} +1572.80 q^{19} -66.1242 q^{21} +398.704 q^{23} -1391.11 q^{25} +356.593 q^{27} -841.000 q^{29} +8469.77 q^{31} -197.652 q^{33} +3748.45 q^{35} +3339.33 q^{37} +326.564 q^{39} +18154.2 q^{41} -9996.71 q^{43} -10096.1 q^{45} -12568.0 q^{47} -8703.31 q^{49} -356.732 q^{51} -21343.1 q^{53} +11204.5 q^{55} -1155.29 q^{57} +30036.1 q^{59} +49792.6 q^{61} -21826.4 q^{63} -18512.3 q^{65} -47588.2 q^{67} -292.867 q^{69} +50164.6 q^{71} -44770.3 q^{73} +1021.83 q^{75} +24222.7 q^{77} +78464.6 q^{79} +58656.0 q^{81} -46721.8 q^{83} +20222.4 q^{85} +617.753 q^{87} -39465.7 q^{89} -40021.2 q^{91} -6221.43 q^{93} +65491.2 q^{95} +48336.3 q^{97} -65241.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) \) 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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