LMFDB - Embedded newform 464.6.a.k.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:	$1$
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			1.4
Root			$-8.92709$ 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)

$f(q)$	$=$	$q-15.4219 q^{3} -58.0818 q^{5} +210.388 q^{7} -5.16616 q^{9} +O(q^{10})$	$q-15.4219 q^{3} -58.0818 q^{5} +210.388 q^{7} -5.16616 q^{9} -527.254 q^{11} -92.3017 q^{13} +895.729 q^{15} +1791.54 q^{17} -1639.87 q^{19} -3244.57 q^{21} +2765.87 q^{23} +248.491 q^{25} +3827.18 q^{27} +841.000 q^{29} -689.400 q^{31} +8131.24 q^{33} -12219.7 q^{35} -1274.51 q^{37} +1423.46 q^{39} +18048.8 q^{41} +6419.14 q^{43} +300.060 q^{45} -2066.40 q^{47} +27456.0 q^{49} -27628.8 q^{51} -28738.1 q^{53} +30623.8 q^{55} +25289.9 q^{57} +34729.5 q^{59} +35599.4 q^{61} -1086.90 q^{63} +5361.05 q^{65} +18011.5 q^{67} -42654.9 q^{69} -4174.40 q^{71} -55919.3 q^{73} -3832.19 q^{75} -110928. q^{77} -96202.6 q^{79} -57766.9 q^{81} -67821.3 q^{83} -104056. q^{85} -12969.8 q^{87} -78390.8 q^{89} -19419.1 q^{91} +10631.8 q^{93} +95246.6 q^{95} -10374.5 q^{97} +2723.88 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−15.4219	−0.989313	−0.494656	−	0.869089i	$-0.664706\pi$
$3$	−15.4219	−0.989313	−0.494656	+	0.869089i	$0.664706\pi$
$4$	0	0
$5$	−58.0818	−1.03900	−0.519499	−	0.854471i	$-0.673881\pi$
$5$	−58.0818	−1.03900	−0.519499	+	0.854471i	$0.673881\pi$
$6$	0	0
$7$	210.388	1.62284	0.811419	−	0.584464i	$-0.198695\pi$
$7$	210.388	1.62284	0.811419	+	0.584464i	$0.198695\pi$
$8$	0	0
$9$	−5.16616	−0.0212599
$10$	0	0
$11$	−527.254	−1.31383	−0.656914	−	0.753966i	$-0.728138\pi$
$11$	−527.254	−1.31383	−0.656914	+	0.753966i	$0.728138\pi$
$12$	0	0
$13$	−92.3017	−0.151479	−0.0757393	−	0.997128i	$-0.524132\pi$
$13$	−92.3017	−0.151479	−0.0757393	+	0.997128i	$0.524132\pi$
$14$	0	0
$15$	895.729	1.02789
$16$	0	0
$17$	1791.54	1.50350	0.751750	−	0.659448i	$-0.229210\pi$
$17$	1791.54	1.50350	0.751750	+	0.659448i	$0.229210\pi$
$18$	0	0
$19$	−1639.87	−1.04214	−0.521070	−	0.853514i	$-0.674467\pi$
$19$	−1639.87	−1.04214	−0.521070	+	0.853514i	$0.674467\pi$
$20$	0	0
$21$	−3244.57	−1.60550
$22$	0	0
$23$	2765.87	1.09021	0.545107	−	0.838366i	$-0.316489\pi$
$23$	2765.87	1.09021	0.545107	+	0.838366i	$0.316489\pi$
$24$	0	0
$25$	248.491	0.0795171
$26$	0	0
$27$	3827.18	1.01035
$28$	0	0
$29$	841.000	0.185695
$29$	841.000	0.185695
$30$	0	0
$31$	−689.400	−0.128845	−0.0644224	−	0.997923i	$-0.520521\pi$
$31$	−689.400	−0.128845	−0.0644224	+	0.997923i	$0.520521\pi$
$32$	0	0
$33$	8131.24	1.29979
$34$	0	0
$35$	−12219.7	−1.68613
$36$	0	0
$37$	−1274.51	−0.153052	−0.0765260	−	0.997068i	$-0.524383\pi$
$37$	−1274.51	−0.153052	−0.0765260	+	0.997068i	$0.524383\pi$
$38$	0	0
$39$	1423.46	0.149860
$40$	0	0
$41$	18048.8	1.67683	0.838414	−	0.545034i	$-0.183483\pi$
$41$	18048.8	1.67683	0.838414	+	0.545034i	$0.183483\pi$
$42$	0	0
$43$	6419.14	0.529427	0.264713	−	0.964327i	$-0.414723\pi$
$43$	6419.14	0.529427	0.264713	+	0.964327i	$0.414723\pi$
$44$	0	0
$45$	300.060	0.0220890
$46$	0	0
$47$	−2066.40	−0.136449	−0.0682244	−	0.997670i	$-0.521733\pi$
$47$	−2066.40	−0.136449	−0.0682244	+	0.997670i	$0.521733\pi$
$48$	0	0
$49$	27456.0	1.63361
$50$	0	0
$51$	−27628.8	−1.48743
$52$	0	0
$53$	−28738.1	−1.40530	−0.702650	−	0.711536i	$-0.748000\pi$
$53$	−28738.1	−1.40530	−0.702650	+	0.711536i	$0.748000\pi$
$54$	0	0
$55$	30623.8	1.36506
$56$	0	0
$57$	25289.9	1.03100
$58$	0	0
$59$	34729.5	1.29888	0.649439	−	0.760414i	$-0.275004\pi$
$59$	34729.5	1.29888	0.649439	+	0.760414i	$0.275004\pi$
$60$	0	0
$61$	35599.4	1.22495	0.612475	−	0.790490i	$-0.290174\pi$
$61$	35599.4	1.22495	0.612475	+	0.790490i	$0.290174\pi$
$62$	0	0
$63$	−1086.90	−0.0345014
$64$	0	0
$65$	5361.05	0.157386
$66$	0	0
$67$	18011.5	0.490189	0.245095	−	0.969499i	$-0.421181\pi$
$67$	18011.5	0.490189	0.245095	+	0.969499i	$0.421181\pi$
$68$	0	0
$69$	−42654.9	−1.07856
$70$	0	0
$71$	−4174.40	−0.0982762	−0.0491381	−	0.998792i	$-0.515647\pi$
$71$	−4174.40	−0.0982762	−0.0491381	+	0.998792i	$0.515647\pi$
$72$	0	0
$73$	−55919.3	−1.22816	−0.614080	−	0.789244i	$-0.710473\pi$
$73$	−55919.3	−1.22816	−0.614080	+	0.789244i	$0.710473\pi$
$74$	0	0
$75$	−3832.19	−0.0786673
$76$	0	0
$77$	−110928.	−2.13213
$78$	0	0
$79$	−96202.6	−1.73428	−0.867139	−	0.498066i	$-0.834044\pi$
$79$	−96202.6	−1.73428	−0.867139	+	0.498066i	$0.834044\pi$
$80$	0	0
$81$	−57766.9	−0.978288
$82$	0	0
$83$	−67821.3	−1.08061	−0.540307	−	0.841468i	$-0.681692\pi$
$83$	−67821.3	−1.08061	−0.540307	+	0.841468i	$0.681692\pi$
$84$	0	0
$85$	−104056.	−1.56213
$86$	0	0
$87$	−12969.8	−0.183711
$88$	0	0
$89$	−78390.8	−1.04904	−0.524518	−	0.851400i	$-0.675754\pi$
$89$	−78390.8	−1.04904	−0.524518	+	0.851400i	$0.675754\pi$
$90$	0	0
$91$	−19419.1	−0.245825
$92$	0	0
$93$	10631.8	0.127468
$94$	0	0
$95$	95246.6	1.08278
$96$	0	0
$97$	−10374.5	−0.111953	−0.0559765	−	0.998432i	$-0.517827\pi$
$97$	−10374.5	−0.111953	−0.0559765	+	0.998432i	$0.517827\pi$
$98$	0	0
$99$	2723.88	0.0279319
$100$	0	0
$101$	−15354.8	−0.149776	−0.0748880	−	0.997192i	$-0.523860\pi$
$101$	−15354.8	−0.149776	−0.0748880	+	0.997192i	$0.523860\pi$
$102$	0	0
$103$	−136830.	−1.27083	−0.635415	−	0.772171i	$-0.719171\pi$
$103$	−136830.	−1.27083	−0.635415	+	0.772171i	$0.719171\pi$
$104$	0	0
$105$	188450.	1.66811
$106$	0	0
$107$	−82551.8	−0.697055	−0.348527	−	0.937299i	$-0.613318\pi$
$107$	−82551.8	−0.697055	−0.348527	+	0.937299i	$0.613318\pi$
$108$	0	0
$109$	−31621.7	−0.254929	−0.127465	−	0.991843i	$-0.540684\pi$
$109$	−31621.7	−0.254929	−0.127465	+	0.991843i	$0.540684\pi$
$110$	0	0
$111$	19655.3	0.151416
$112$	0	0
$113$	26182.9	0.192895	0.0964476	−	0.995338i	$-0.469252\pi$
$113$	26182.9	0.192895	0.0964476	+	0.995338i	$0.469252\pi$
$114$	0	0
$115$	−160647.	−1.13273
$116$	0	0
$117$	476.846	0.00322042
$118$	0	0
$119$	376918.	2.43994
$120$	0	0
$121$	116946.	0.726142
$122$	0	0
$123$	−278346.	−1.65891
$124$	0	0
$125$	167073.	0.956380
$126$	0	0
$127$	−31565.7	−0.173663	−0.0868313	−	0.996223i	$-0.527674\pi$
$127$	−31565.7	−0.173663	−0.0868313	+	0.996223i	$0.527674\pi$
$128$	0	0
$129$	−98995.1	−0.523768
$130$	0	0
$131$	143630.	0.731250	0.365625	−	0.930762i	$-0.380855\pi$
$131$	143630.	0.731250	0.365625	+	0.930762i	$0.380855\pi$
$132$	0	0
$133$	−345009.	−1.69122
$134$	0	0
$135$	−222290.	−1.04975
$136$	0	0
$137$	−120976.	−0.550677	−0.275338	−	0.961347i	$-0.588790\pi$
$137$	−120976.	−0.550677	−0.275338	+	0.961347i	$0.588790\pi$
$138$	0	0
$139$	−70220.3	−0.308266	−0.154133	−	0.988050i	$-0.549258\pi$
$139$	−70220.3	−0.308266	−0.154133	+	0.988050i	$0.549258\pi$
$140$	0	0
$141$	31867.7	0.134990
$142$	0	0
$143$	48666.5	0.199017
$144$	0	0
$145$	−48846.8	−0.192937
$146$	0	0
$147$	−423423.	−1.61615
$148$	0	0
$149$	−249053.	−0.919024	−0.459512	−	0.888172i	$-0.651976\pi$
$149$	−249053.	−0.919024	−0.459512	+	0.888172i	$0.651976\pi$
$150$	0	0
$151$	−337447.	−1.20438	−0.602190	−	0.798353i	$-0.705705\pi$
$151$	−337447.	−1.20438	−0.602190	+	0.798353i	$0.705705\pi$
$152$	0	0
$153$	−9255.37	−0.0319643
$154$	0	0
$155$	40041.6	0.133870
$156$	0	0
$157$	−60595.1	−0.196195	−0.0980976	−	0.995177i	$-0.531276\pi$
$157$	−60595.1	−0.196195	−0.0980976	+	0.995177i	$0.531276\pi$
$158$	0	0
$159$	443195.	1.39028
$160$	0	0
$161$	581905.	1.76924
$162$	0	0
$163$	−316079.	−0.931808	−0.465904	−	0.884835i	$-0.654271\pi$
$163$	−316079.	−0.931808	−0.465904	+	0.884835i	$0.654271\pi$
$164$	0	0
$165$	−472277.	−1.35048
$166$	0	0
$167$	169743.	0.470978	0.235489	−	0.971877i	$-0.424331\pi$
$167$	169743.	0.470978	0.235489	+	0.971877i	$0.424331\pi$
$168$	0	0
$169$	−362773.	−0.977054
$170$	0	0
$171$	8471.84	0.0221558
$172$	0	0
$173$	−269463.	−0.684516	−0.342258	−	0.939606i	$-0.611192\pi$
$173$	−269463.	−0.684516	−0.342258	+	0.939606i	$0.611192\pi$
$174$	0	0
$175$	52279.4	0.129043
$176$	0	0
$177$	−535593.	−1.28500
$178$	0	0
$179$	637873.	1.48799	0.743997	−	0.668183i	$-0.232928\pi$
$179$	637873.	1.48799	0.743997	+	0.668183i	$0.232928\pi$
$180$	0	0
$181$	505742.	1.14745	0.573724	−	0.819049i	$-0.305498\pi$
$181$	505742.	1.14745	0.573724	+	0.819049i	$0.305498\pi$
$182$	0	0
$183$	−549009.	−1.21186
$184$	0	0
$185$	74025.8	0.159021
$186$	0	0
$187$	−944596.	−1.97534
$188$	0	0
$189$	805193.	1.63963
$190$	0	0
$191$	−310087.	−0.615035	−0.307517	−	0.951543i	$-0.599498\pi$
$191$	−310087.	−0.615035	−0.307517	+	0.951543i	$0.599498\pi$
$192$	0	0
$193$	−547378.	−1.05778	−0.528888	−	0.848692i	$-0.677391\pi$
$193$	−547378.	−1.05778	−0.528888	+	0.848692i	$0.677391\pi$
$194$	0	0
$195$	−82677.3	−0.155704
$196$	0	0
$197$	56434.3	0.103604	0.0518021	−	0.998657i	$-0.483503\pi$
$197$	56434.3	0.103604	0.0518021	+	0.998657i	$0.483503\pi$
$198$	0	0
$199$	−411565.	−0.736726	−0.368363	−	0.929682i	$-0.620082\pi$
$199$	−411565.	−0.736726	−0.368363	+	0.929682i	$0.620082\pi$
$200$	0	0
$201$	−277771.	−0.484951
$202$	0	0
$203$	176936.	0.301354
$204$	0	0
$205$	−1.04831e6	−1.74222
$206$	0	0
$207$	−14288.9	−0.0231779
$208$	0	0
$209$	864629.	1.36919
$210$	0	0
$211$	213777.	0.330563	0.165281	−	0.986246i	$-0.447147\pi$
$211$	213777.	0.330563	0.165281	+	0.986246i	$0.447147\pi$
$212$	0	0
$213$	64377.0	0.0972259
$214$	0	0
$215$	−372835.	−0.550073
$216$	0	0
$217$	−145041.	−0.209094
$218$	0	0
$219$	862380.	1.21503
$220$	0	0
$221$	−165362.	−0.227748
$222$	0	0
$223$	−704304.	−0.948415	−0.474207	−	0.880413i	$-0.657265\pi$
$223$	−704304.	−0.948415	−0.474207	+	0.880413i	$0.657265\pi$
$224$	0	0
$225$	−1283.74	−0.00169053
$226$	0	0
$227$	565745.	0.728712	0.364356	−	0.931260i	$-0.381289\pi$
$227$	565745.	0.728712	0.364356	+	0.931260i	$0.381289\pi$
$228$	0	0
$229$	525166.	0.661771	0.330886	−	0.943671i	$-0.392653\pi$
$229$	525166.	0.661771	0.330886	+	0.943671i	$0.392653\pi$
$230$	0	0
$231$	1.71071e6	2.10934
$232$	0	0
$233$	647478.	0.781332	0.390666	−	0.920533i	$-0.372245\pi$
$233$	647478.	0.781332	0.390666	+	0.920533i	$0.372245\pi$
$234$	0	0
$235$	120020.	0.141770
$236$	0	0
$237$	1.48362e6	1.71574
$238$	0	0
$239$	1.37154e6	1.55315	0.776577	−	0.630022i	$-0.216954\pi$
$239$	1.37154e6	1.55315	0.776577	+	0.630022i	$0.216954\pi$
$240$	0	0
$241$	125435.	0.139116	0.0695580	−	0.997578i	$-0.477841\pi$
$241$	125435.	0.139116	0.0695580	+	0.997578i	$0.477841\pi$
$242$	0	0
$243$	−39132.1	−0.0425126
$244$	0	0
$245$	−1.59469e6	−1.69731
$246$	0	0
$247$	151363.	0.157862
$248$	0	0
$249$	1.04593e6	1.06907
$250$	0	0
$251$	−1.07758e6	−1.07960	−0.539802	−	0.841792i	$-0.681501\pi$
$251$	−1.07758e6	−1.07960	−0.539802	+	0.841792i	$0.681501\pi$
$252$	0	0
$253$	−1.45832e6	−1.43235
$254$	0	0
$255$	1.60473e6	1.54544
$256$	0	0
$257$	−300653.	−0.283944	−0.141972	−	0.989871i	$-0.545344\pi$
$257$	−300653.	−0.283944	−0.141972	+	0.989871i	$0.545344\pi$
$258$	0	0
$259$	−268141.	−0.248379
$260$	0	0
$261$	−4344.74	−0.00394787
$262$	0	0
$263$	−1.14659e6	−1.02216	−0.511080	−	0.859533i	$-0.670755\pi$
$263$	−1.14659e6	−1.02216	−0.511080	+	0.859533i	$0.670755\pi$
$264$	0	0
$265$	1.66916e6	1.46010
$266$	0	0
$267$	1.20893e6	1.03782
$268$	0	0
$269$	−1.18627e6	−0.999547	−0.499774	−	0.866156i	$-0.666583\pi$
$269$	−1.18627e6	−0.999547	−0.499774	+	0.866156i	$0.666583\pi$
$270$	0	0
$271$	−1.28048e6	−1.05913	−0.529567	−	0.848268i	$-0.677646\pi$
$271$	−1.28048e6	−1.05913	−0.529567	+	0.848268i	$0.677646\pi$
$272$	0	0
$273$	299479.	0.243198
$274$	0	0
$275$	−131018.	−0.104472
$276$	0	0
$277$	1.77690e6	1.39144	0.695718	−	0.718315i	$-0.255086\pi$
$277$	1.77690e6	1.39144	0.695718	+	0.718315i	$0.255086\pi$
$278$	0	0
$279$	3561.55	0.00273923
$280$	0	0
$281$	−649378.	−0.490605	−0.245302	−	0.969447i	$-0.578887\pi$
$281$	−649378.	−0.490605	−0.245302	+	0.969447i	$0.578887\pi$
$282$	0	0
$283$	152296.	0.113038	0.0565188	−	0.998402i	$-0.482000\pi$
$283$	152296.	0.113038	0.0565188	+	0.998402i	$0.482000\pi$
$284$	0	0
$285$	−1.46888e6	−1.07121
$286$	0	0
$287$	3.79724e6	2.72122
$288$	0	0
$289$	1.78975e6	1.26051
$290$	0	0
$291$	159993.	0.110757
$292$	0	0
$293$	−799868.	−0.544314	−0.272157	−	0.962253i	$-0.587737\pi$
$293$	−799868.	−0.544314	−0.272157	+	0.962253i	$0.587737\pi$
$294$	0	0
$295$	−2.01715e6	−1.34953
$296$	0	0
$297$	−2.01790e6	−1.32742
$298$	0	0
$299$	−255294.	−0.165144
$300$	0	0
$301$	1.35051e6	0.859174
$302$	0	0
$303$	236800.	0.148175
$304$	0	0
$305$	−2.06768e6	−1.27272
$306$	0	0
$307$	−478184.	−0.289567	−0.144783	−	0.989463i	$-0.546249\pi$
$307$	−478184.	−0.289567	−0.144783	+	0.989463i	$0.546249\pi$
$308$	0	0
$309$	2.11017e6	1.25725
$310$	0	0
$311$	−2.63423e6	−1.54437	−0.772187	−	0.635395i	$-0.780837\pi$
$311$	−2.63423e6	−1.54437	−0.772187	+	0.635395i	$0.780837\pi$
$312$	0	0
$313$	−1.38599e6	−0.799651	−0.399826	−	0.916591i	$-0.630929\pi$
$313$	−1.38599e6	−0.799651	−0.399826	+	0.916591i	$0.630929\pi$
$314$	0	0
$315$	63128.9	0.0358469
$316$	0	0
$317$	2.58220e6	1.44325	0.721626	−	0.692283i	$-0.243395\pi$
$317$	2.58220e6	1.44325	0.721626	+	0.692283i	$0.243395\pi$
$318$	0	0
$319$	−443421.	−0.243972
$320$	0	0
$321$	1.27310e6	0.689605
$322$	0	0
$323$	−2.93789e6	−1.56686
$324$	0	0
$325$	−22936.1	−0.0120451
$326$	0	0
$327$	487666.	0.252205
$328$	0	0
$329$	−434745.	−0.221434
$330$	0	0
$331$	1.13450e6	0.569162	0.284581	−	0.958652i	$-0.408145\pi$
$331$	1.13450e6	0.569162	0.284581	+	0.958652i	$0.408145\pi$
$332$	0	0
$333$	6584.33	0.00325387
$334$	0	0
$335$	−1.04614e6	−0.509306
$336$	0	0
$337$	149357.	0.0716392	0.0358196	−	0.999358i	$-0.488596\pi$
$337$	149357.	0.0716392	0.0358196	+	0.999358i	$0.488596\pi$
$338$	0	0
$339$	−403789.	−0.190834
$340$	0	0
$341$	363489.	0.169280
$342$	0	0
$343$	2.24042e6	1.02824
$344$	0	0
$345$	2.47747e6	1.12063
$346$	0	0
$347$	2.53791e6	1.13150	0.565748	−	0.824578i	$-0.308587\pi$
$347$	2.53791e6	1.13150	0.565748	+	0.824578i	$0.308587\pi$
$348$	0	0
$349$	−1.77115e6	−0.778380	−0.389190	−	0.921157i	$-0.627245\pi$
$349$	−1.77115e6	−0.778380	−0.389190	+	0.921157i	$0.627245\pi$
$350$	0	0
$351$	−353256.	−0.153046
$352$	0	0
$353$	2.26199e6	0.966171	0.483085	−	0.875573i	$-0.339516\pi$
$353$	2.26199e6	0.966171	0.483085	+	0.875573i	$0.339516\pi$
$354$	0	0
$355$	242457.	0.102109
$356$	0	0
$357$	−5.81277e6	−2.41386
$358$	0	0
$359$	1.72972e6	0.708338	0.354169	−	0.935181i	$-0.384764\pi$
$359$	1.72972e6	0.708338	0.354169	+	0.935181i	$0.384764\pi$
$360$	0	0
$361$	213081.	0.0860550
$362$	0	0
$363$	−1.80352e6	−0.718381
$364$	0	0
$365$	3.24789e6	1.27606
$366$	0	0
$367$	−1.57623e6	−0.610877	−0.305438	−	0.952212i	$-0.598803\pi$
$367$	−1.57623e6	−0.610877	−0.305438	+	0.952212i	$0.598803\pi$
$368$	0	0
$369$	−93242.9	−0.0356492
$370$	0	0
$371$	−6.04615e6	−2.28057
$372$	0	0
$373$	−1.23586e6	−0.459935	−0.229968	−	0.973198i	$-0.573862\pi$
$373$	−1.23586e6	−0.459935	−0.229968	+	0.973198i	$0.573862\pi$
$374$	0	0
$375$	−2.57657e6	−0.946159
$376$	0	0
$377$	−77625.7	−0.0281289
$378$	0	0
$379$	−339945.	−0.121566	−0.0607829	−	0.998151i	$-0.519360\pi$
$379$	−339945.	−0.121566	−0.0607829	+	0.998151i	$0.519360\pi$
$380$	0	0
$381$	486802.	0.171807
$382$	0	0
$383$	3.13257e6	1.09120	0.545600	−	0.838046i	$-0.316302\pi$
$383$	3.13257e6	1.09120	0.545600	+	0.838046i	$0.316302\pi$
$384$	0	0
$385$	6.44288e6	2.21528
$386$	0	0
$387$	−33162.3	−0.0112556
$388$	0	0
$389$	3.26603e6	1.09433	0.547163	−	0.837026i	$-0.315708\pi$
$389$	3.26603e6	1.09433	0.547163	+	0.837026i	$0.315708\pi$
$390$	0	0
$391$	4.95516e6	1.63914
$392$	0	0
$393$	−2.21504e6	−0.723435
$394$	0	0
$395$	5.58761e6	1.80191
$396$	0	0
$397$	−1.47446e6	−0.469522	−0.234761	−	0.972053i	$-0.575431\pi$
$397$	−1.47446e6	−0.469522	−0.234761	+	0.972053i	$0.575431\pi$
$398$	0	0
$399$	5.32068e6	1.67315
$400$	0	0
$401$	2.37462e6	0.737450	0.368725	−	0.929539i	$-0.379794\pi$
$401$	2.37462e6	0.737450	0.368725	+	0.929539i	$0.379794\pi$
$402$	0	0
$403$	63632.8	0.0195172
$404$	0	0
$405$	3.35521e6	1.01644
$406$	0	0
$407$	671991.	0.201084
$408$	0	0
$409$	−3.82996e6	−1.13210	−0.566051	−	0.824370i	$-0.691530\pi$
$409$	−3.82996e6	−1.13210	−0.566051	+	0.824370i	$0.691530\pi$
$410$	0	0
$411$	1.86567e6	0.544792
$412$	0	0
$413$	7.30666e6	2.10787
$414$	0	0
$415$	3.93918e6	1.12276
$416$	0	0
$417$	1.08293e6	0.304972
$418$	0	0
$419$	−621328.	−0.172897	−0.0864483	−	0.996256i	$-0.527552\pi$
$419$	−621328.	−0.172897	−0.0864483	+	0.996256i	$0.527552\pi$
$420$	0	0
$421$	−4.99920e6	−1.37466	−0.687330	−	0.726345i	$-0.741217\pi$
$421$	−4.99920e6	−1.37466	−0.687330	+	0.726345i	$0.741217\pi$
$422$	0	0
$423$	10675.3	0.00290089
$424$	0	0
$425$	445181.	0.119554
$426$	0	0
$427$	7.48968e6	1.98790
$428$	0	0
$429$	−750527.	−0.196890
$430$	0	0
$431$	5.42664e6	1.40714	0.703570	−	0.710626i	$-0.251588\pi$
$431$	5.42664e6	1.40714	0.703570	+	0.710626i	$0.251588\pi$
$432$	0	0
$433$	1.21607e6	0.311702	0.155851	−	0.987781i	$-0.450188\pi$
$433$	1.21607e6	0.311702	0.155851	+	0.987781i	$0.450188\pi$
$434$	0	0
$435$	753308.	0.190875
$436$	0	0
$437$	−4.53567e6	−1.13616
$438$	0	0
$439$	2.34458e6	0.580635	0.290317	−	0.956930i	$-0.406239\pi$
$439$	2.34458e6	0.580635	0.290317	+	0.956930i	$0.406239\pi$
$440$	0	0
$441$	−141842.	−0.0347303
$442$	0	0
$443$	−4.17338e6	−1.01036	−0.505182	−	0.863013i	$-0.668575\pi$
$443$	−4.17338e6	−1.01036	−0.505182	+	0.863013i	$0.668575\pi$
$444$	0	0
$445$	4.55308e6	1.08995
$446$	0	0
$447$	3.84087e6	0.909202
$448$	0	0
$449$	−7.89264e6	−1.84759	−0.923797	−	0.382883i	$-0.874931\pi$
$449$	−7.89264e6	−1.84759	−0.923797	+	0.382883i	$0.874931\pi$
$450$	0	0
$451$	−9.51630e6	−2.20306
$452$	0	0
$453$	5.20406e6	1.19151
$454$	0	0
$455$	1.12790e6	0.255412
$456$	0	0
$457$	−3.36902e6	−0.754593	−0.377297	−	0.926092i	$-0.623146\pi$
$457$	−3.36902e6	−0.754593	−0.377297	+	0.926092i	$0.623146\pi$
$458$	0	0
$459$	6.85655e6	1.51906
$460$	0	0
$461$	−35186.9	−0.00771132	−0.00385566	−	0.999993i	$-0.501227\pi$
$461$	−35186.9	−0.00771132	−0.00385566	+	0.999993i	$0.501227\pi$
$462$	0	0
$463$	1.98494e6	0.430322	0.215161	−	0.976579i	$-0.430972\pi$
$463$	1.98494e6	0.430322	0.215161	+	0.976579i	$0.430972\pi$
$464$	0	0
$465$	−617516.	−0.132439
$466$	0	0
$467$	−5.84649e6	−1.24052	−0.620259	−	0.784397i	$-0.712972\pi$
$467$	−5.84649e6	−1.24052	−0.620259	+	0.784397i	$0.712972\pi$
$468$	0	0
$469$	3.78941e6	0.795498
$470$	0	0
$471$	934490.	0.194099
$472$	0	0
$473$	−3.38452e6	−0.695575
$474$	0	0
$475$	−407493.	−0.0828679
$476$	0	0
$477$	148466.	0.0298766
$478$	0	0
$479$	−3.76308e6	−0.749385	−0.374693	−	0.927149i	$-0.622252\pi$
$479$	−3.76308e6	−0.749385	−0.374693	+	0.927149i	$0.622252\pi$
$480$	0	0
$481$	117639.	0.0231841
$482$	0	0
$483$	−8.97406e6	−1.75033
$484$	0	0
$485$	602567.	0.116319
$486$	0	0
$487$	179343.	0.0342658	0.0171329	−	0.999853i	$-0.494546\pi$
$487$	179343.	0.0342658	0.0171329	+	0.999853i	$0.494546\pi$
$488$	0	0
$489$	4.87452e6	0.921850
$490$	0	0
$491$	−1.00202e7	−1.87574	−0.937869	−	0.346990i	$-0.887204\pi$
$491$	−1.00202e7	−1.87574	−0.937869	+	0.346990i	$0.887204\pi$
$492$	0	0
$493$	1.50668e6	0.279193
$494$	0	0
$495$	−158208.	−0.0290212
$496$	0	0
$497$	−878243.	−0.159486
$498$	0	0
$499$	−8.59738e6	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$499$	−8.59738e6	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$500$	0	0
$501$	−2.61775e6	−0.465944
$502$	0	0
$503$	9.46324e6	1.66771	0.833854	−	0.551985i	$-0.186130\pi$
$503$	9.46324e6	1.66771	0.833854	+	0.551985i	$0.186130\pi$
$504$	0	0
$505$	891837.	0.155617
$506$	0	0
$507$	5.59464e6	0.966612
$508$	0	0
$509$	−742258.	−0.126987	−0.0634937	−	0.997982i	$-0.520224\pi$
$509$	−742258.	−0.126987	−0.0634937	+	0.997982i	$0.520224\pi$
$510$	0	0
$511$	−1.17647e7	−1.99310
$512$	0	0
$513$	−6.27609e6	−1.05292
$514$	0	0
$515$	7.94732e6	1.32039
$516$	0	0
$517$	1.08952e6	0.179270
$518$	0	0
$519$	4.15562e6	0.677200
$520$	0	0
$521$	−3.00978e6	−0.485781	−0.242890	−	0.970054i	$-0.578096\pi$
$521$	−3.00978e6	−0.485781	−0.242890	+	0.970054i	$0.578096\pi$
$522$	0	0
$523$	−8.78961e6	−1.40513	−0.702563	−	0.711621i	$-0.747961\pi$
$523$	−8.78961e6	−1.40513	−0.702563	+	0.711621i	$0.747961\pi$
$524$	0	0
$525$	−806246.	−0.127664
$526$	0	0
$527$	−1.23509e6	−0.193718
$528$	0	0
$529$	1.21369e6	0.188568
$530$	0	0
$531$	−179418.	−0.0276140
$532$	0	0
$533$	−1.66593e6	−0.254004
$534$	0	0
$535$	4.79475e6	0.724239
$536$	0	0
$537$	−9.83718e6	−1.47209
$538$	0	0
$539$	−1.44763e7	−2.14627
$540$	0	0
$541$	8.42234e6	1.23720	0.618599	−	0.785707i	$-0.287700\pi$
$541$	8.42234e6	1.23720	0.618599	+	0.785707i	$0.287700\pi$
$542$	0	0
$543$	−7.79949e6	−1.13518
$544$	0	0
$545$	1.83665e6	0.264871
$546$	0	0
$547$	2.55152e6	0.364612	0.182306	−	0.983242i	$-0.441644\pi$
$547$	2.55152e6	0.364612	0.182306	+	0.983242i	$0.441644\pi$
$548$	0	0
$549$	−183912.	−0.0260423
$550$	0	0
$551$	−1.37913e6	−0.193520
$552$	0	0
$553$	−2.02398e7	−2.81445
$554$	0	0
$555$	−1.14162e6	−0.157321
$556$	0	0
$557$	4.38436e6	0.598782	0.299391	−	0.954131i	$-0.403217\pi$
$557$	4.38436e6	0.598782	0.299391	+	0.954131i	$0.403217\pi$
$558$	0	0
$559$	−592498.	−0.0801968
$560$	0	0
$561$	1.45674e7	1.95423
$562$	0	0
$563$	−9.56112e6	−1.27127	−0.635635	−	0.771990i	$-0.719262\pi$
$563$	−9.56112e6	−1.27127	−0.635635	+	0.771990i	$0.719262\pi$
$564$	0	0
$565$	−1.52075e6	−0.200418
$566$	0	0
$567$	−1.21535e7	−1.58760
$568$	0	0
$569$	4.02277e6	0.520889	0.260444	−	0.965489i	$-0.416131\pi$
$569$	4.02277e6	0.520889	0.260444	+	0.965489i	$0.416131\pi$
$570$	0	0
$571$	−8.31378e6	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$571$	−8.31378e6	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$572$	0	0
$573$	4.78211e6	0.608462
$574$	0	0
$575$	687293.	0.0866907
$576$	0	0
$577$	−1.03727e7	−1.29703	−0.648517	−	0.761200i	$-0.724611\pi$
$577$	−1.03727e7	−1.29703	−0.648517	+	0.761200i	$0.724611\pi$
$578$	0	0
$579$	8.44158e6	1.04647
$580$	0	0
$581$	−1.42688e7	−1.75366
$582$	0	0
$583$	1.51523e7	1.84632
$584$	0	0
$585$	−27696.0	−0.00334601
$586$	0	0
$587$	9.76113e6	1.16924	0.584622	−	0.811306i	$-0.301243\pi$
$587$	9.76113e6	1.16924	0.584622	+	0.811306i	$0.301243\pi$
$588$	0	0
$589$	1.13053e6	0.134274
$590$	0	0
$591$	−870322.	−0.102497
$592$	0	0
$593$	−6.43984e6	−0.752036	−0.376018	−	0.926612i	$-0.622707\pi$
$593$	−6.43984e6	−0.752036	−0.376018	+	0.926612i	$0.622707\pi$
$594$	0	0
$595$	−2.18920e7	−2.53509
$596$	0	0
$597$	6.34710e6	0.728853
$598$	0	0
$599$	1.74577e7	1.98802	0.994009	−	0.109301i	$-0.0348612\pi$
$599$	1.74577e7	1.98802	0.994009	+	0.109301i	$0.0348612\pi$
$600$	0	0
$601$	1.70010e7	1.91994	0.959969	−	0.280106i	$-0.0903695\pi$
$601$	1.70010e7	1.91994	0.959969	+	0.280106i	$0.0903695\pi$
$602$	0	0
$603$	−93050.5	−0.0104214
$604$	0	0
$605$	−6.79242e6	−0.754460
$606$	0	0
$607$	9.77787e6	1.07714	0.538570	−	0.842581i	$-0.318965\pi$
$607$	9.77787e6	1.07714	0.538570	+	0.842581i	$0.318965\pi$
$608$	0	0
$609$	−2.72868e6	−0.298133
$610$	0	0
$611$	190732.	0.0206691
$612$	0	0
$613$	−1.43306e7	−1.54033	−0.770165	−	0.637844i	$-0.779826\pi$
$613$	−1.43306e7	−1.54033	−0.770165	+	0.637844i	$0.779826\pi$
$614$	0	0
$615$	1.61668e7	1.72360
$616$	0	0
$617$	−1.05568e6	−0.111640	−0.0558202	−	0.998441i	$-0.517777\pi$
$617$	−1.05568e6	−0.111640	−0.0558202	+	0.998441i	$0.517777\pi$
$618$	0	0
$619$	3.57533e6	0.375051	0.187525	−	0.982260i	$-0.439953\pi$
$619$	3.57533e6	0.375051	0.187525	+	0.982260i	$0.439953\pi$
$620$	0	0
$621$	1.05855e7	1.10149
$622$	0	0
$623$	−1.64925e7	−1.70242
$624$	0	0
$625$	−1.04804e7	−1.07319
$626$	0	0
$627$	−1.33342e7	−1.35456
$628$	0	0
$629$	−2.28333e6	−0.230114
$630$	0	0
$631$	−1.66113e7	−1.66085	−0.830423	−	0.557134i	$-0.811901\pi$
$631$	−1.66113e7	−1.66085	−0.830423	+	0.557134i	$0.811901\pi$
$632$	0	0
$633$	−3.29683e6	−0.327030
$634$	0	0
$635$	1.83339e6	0.180435
$636$	0	0
$637$	−2.53424e6	−0.247456
$638$	0	0
$639$	21565.6	0.00208934
$640$	0	0
$641$	−1.77657e7	−1.70780	−0.853902	−	0.520434i	$-0.825770\pi$
$641$	−1.77657e7	−1.70780	−0.853902	+	0.520434i	$0.825770\pi$
$642$	0	0
$643$	5.29461e6	0.505018	0.252509	−	0.967595i	$-0.418744\pi$
$643$	5.29461e6	0.505018	0.252509	+	0.967595i	$0.418744\pi$
$644$	0	0
$645$	5.74981e6	0.544194
$646$	0	0
$647$	1.47774e7	1.38783	0.693916	−	0.720056i	$-0.255884\pi$
$647$	1.47774e7	1.38783	0.693916	+	0.720056i	$0.255884\pi$
$648$	0	0
$649$	−1.83113e7	−1.70650
$650$	0	0
$651$	2.23681e6	0.206860
$652$	0	0
$653$	−1.36335e7	−1.25119	−0.625595	−	0.780148i	$-0.715144\pi$
$653$	−1.36335e7	−1.25119	−0.625595	+	0.780148i	$0.715144\pi$
$654$	0	0
$655$	−8.34227e6	−0.759768
$656$	0	0
$657$	288888.	0.0261106
$658$	0	0
$659$	−1.60545e7	−1.44007	−0.720033	−	0.693940i	$-0.755873\pi$
$659$	−1.60545e7	−1.44007	−0.720033	+	0.693940i	$0.755873\pi$
$660$	0	0
$661$	−2.01227e7	−1.79136	−0.895678	−	0.444704i	$-0.853309\pi$
$661$	−2.01227e7	−1.79136	−0.895678	+	0.444704i	$0.853309\pi$
$662$	0	0
$663$	2.55019e6	0.225314
$664$	0	0
$665$	2.00387e7	1.75718
$666$	0	0
$667$	2.32610e6	0.202448
$668$	0	0
$669$	1.08617e7	0.938279
$670$	0	0
$671$	−1.87699e7	−1.60937
$672$	0	0
$673$	2.01674e7	1.71637	0.858187	−	0.513338i	$-0.171591\pi$
$673$	2.01674e7	1.71637	0.858187	+	0.513338i	$0.171591\pi$
$674$	0	0
$675$	951020.	0.0803397
$676$	0	0
$677$	3.35832e6	0.281612	0.140806	−	0.990037i	$-0.455031\pi$
$677$	3.35832e6	0.281612	0.140806	+	0.990037i	$0.455031\pi$
$678$	0	0
$679$	−2.18266e6	−0.181682
$680$	0	0
$681$	−8.72484e6	−0.720924
$682$	0	0
$683$	−6.40198e6	−0.525125	−0.262563	−	0.964915i	$-0.584568\pi$
$683$	−6.40198e6	−0.525125	−0.262563	+	0.964915i	$0.584568\pi$
$684$	0	0
$685$	7.02648e6	0.572152
$686$	0	0
$687$	−8.09904e6	−0.654699
$688$	0	0
$689$	2.65258e6	0.212873
$690$	0	0
$691$	2.19756e7	1.75084	0.875418	−	0.483367i	$-0.160586\pi$
$691$	2.19756e7	1.75084	0.875418	+	0.483367i	$0.160586\pi$
$692$	0	0
$693$	573071.	0.0453289
$694$	0	0
$695$	4.07852e6	0.320288
$696$	0	0
$697$	3.23351e7	2.52111
$698$	0	0
$699$	−9.98532e6	−0.772982
$700$	0	0
$701$	7.90411e6	0.607516	0.303758	−	0.952749i	$-0.401759\pi$
$701$	7.90411e6	0.607516	0.303758	+	0.952749i	$0.401759\pi$
$702$	0	0
$703$	2.09003e6	0.159502
$704$	0	0
$705$	−1.85093e6	−0.140255
$706$	0	0
$707$	−3.23047e6	−0.243062
$708$	0	0
$709$	1.88248e7	1.40642	0.703211	−	0.710981i	$-0.251749\pi$
$709$	1.88248e7	1.40642	0.703211	+	0.710981i	$0.251749\pi$
$710$	0	0
$711$	496998.	0.0368706
$712$	0	0
$713$	−1.90679e6	−0.140469
$714$	0	0
$715$	−2.82663e6	−0.206778
$716$	0	0
$717$	−2.11517e7	−1.53656
$718$	0	0
$719$	−9.95692e6	−0.718295	−0.359148	−	0.933281i	$-0.616933\pi$
$719$	−9.95692e6	−0.718295	−0.359148	+	0.933281i	$0.616933\pi$
$720$	0	0
$721$	−2.87873e7	−2.06235
$722$	0	0
$723$	−1.93445e6	−0.137629
$724$	0	0
$725$	208981.	0.0147660
$726$	0	0
$727$	−7.47960e6	−0.524859	−0.262429	−	0.964951i	$-0.584524\pi$
$727$	−7.47960e6	−0.524859	−0.262429	+	0.964951i	$0.584524\pi$
$728$	0	0
$729$	1.46409e7	1.02035
$730$	0	0
$731$	1.15001e7	0.795993
$732$	0	0
$733$	−1.87960e7	−1.29213	−0.646065	−	0.763282i	$-0.723587\pi$
$733$	−1.87960e7	−1.29213	−0.646065	+	0.763282i	$0.723587\pi$
$734$	0	0
$735$	2.45931e7	1.67917
$736$	0	0
$737$	−9.49666e6	−0.644024
$738$	0	0
$739$	1.54929e7	1.04357	0.521784	−	0.853078i	$-0.325267\pi$
$739$	1.54929e7	1.04357	0.521784	+	0.853078i	$0.325267\pi$
$740$	0	0
$741$	−2.33430e6	−0.156175
$742$	0	0
$743$	1.97660e7	1.31355	0.656776	−	0.754086i	$-0.271920\pi$
$743$	1.97660e7	1.31355	0.656776	+	0.754086i	$0.271920\pi$
$744$	0	0
$745$	1.44655e7	0.954864
$746$	0	0
$747$	350376.	0.0229738
$748$	0	0
$749$	−1.73679e7	−1.13121
$750$	0	0
$751$	1.72461e6	0.111581	0.0557905	−	0.998442i	$-0.482232\pi$
$751$	1.72461e6	0.111581	0.0557905	+	0.998442i	$0.482232\pi$
$752$	0	0
$753$	1.66183e7	1.06807
$754$	0	0
$755$	1.95995e7	1.25135
$756$	0	0
$757$	−2.52697e7	−1.60273	−0.801364	−	0.598177i	$-0.795892\pi$
$757$	−2.52697e7	−1.60273	−0.801364	+	0.598177i	$0.795892\pi$
$758$	0	0
$759$	2.24899e7	1.41705
$760$	0	0
$761$	1.08483e7	0.679050	0.339525	−	0.940597i	$-0.389734\pi$
$761$	1.08483e7	0.679050	0.339525	+	0.940597i	$0.389734\pi$
$762$	0	0
$763$	−6.65282e6	−0.413709
$764$	0	0
$765$	537568.	0.0332109
$766$	0	0
$767$	−3.20559e6	−0.196752
$768$	0	0
$769$	3.05501e7	1.86293	0.931465	−	0.363830i	$-0.118531\pi$
$769$	3.05501e7	1.86293	0.931465	+	0.363830i	$0.118531\pi$
$770$	0	0
$771$	4.63663e6	0.280910
$772$	0	0
$773$	2.59485e6	0.156194	0.0780968	−	0.996946i	$-0.475116\pi$
$773$	2.59485e6	0.156194	0.0780968	+	0.996946i	$0.475116\pi$
$774$	0	0
$775$	−171310.	−0.0102454
$776$	0	0
$777$	4.13524e6	0.245724
$778$	0	0
$779$	−2.95977e7	−1.74749
$780$	0	0
$781$	2.20097e6	0.129118
$782$	0	0
$783$	3.21866e6	0.187616
$784$	0	0
$785$	3.51947e6	0.203847
$786$	0	0
$787$	5.86903e6	0.337776	0.168888	−	0.985635i	$-0.445982\pi$
$787$	5.86903e6	0.337776	0.168888	+	0.985635i	$0.445982\pi$
$788$	0	0
$789$	1.76826e7	1.01124
$790$	0	0
$791$	5.50856e6	0.313038
$792$	0	0
$793$	−3.28589e6	−0.185554
$794$	0	0
$795$	−2.57416e7	−1.44450
$796$	0	0
$797$	−5.23510e6	−0.291930	−0.145965	−	0.989290i	$-0.546629\pi$
$797$	−5.23510e6	−0.291930	−0.145965	+	0.989290i	$0.546629\pi$
$798$	0	0
$799$	−3.70203e6	−0.205151
$800$	0	0
$801$	404980.	0.0223024
$802$	0	0
$803$	2.94837e7	1.61359
$804$	0	0
$805$	−3.37981e7	−1.83824
$806$	0	0
$807$	1.82945e7	0.988865
$808$	0	0
$809$	1.68943e7	0.907548	0.453774	−	0.891117i	$-0.350077\pi$
$809$	1.68943e7	0.907548	0.453774	+	0.891117i	$0.350077\pi$
$810$	0	0
$811$	−332001.	−0.0177251	−0.00886253	−	0.999961i	$-0.502821\pi$
$811$	−332001.	−0.0177251	−0.00886253	+	0.999961i	$0.502821\pi$
$812$	0	0
$813$	1.97474e7	1.04782
$814$	0	0
$815$	1.83584e7	0.968147
$816$	0	0
$817$	−1.05266e7	−0.551736
$818$	0	0
$819$	100322.	0.00522623
$820$	0	0
$821$	−1.19256e6	−0.0617479	−0.0308739	−	0.999523i	$-0.509829\pi$
$821$	−1.19256e6	−0.0617479	−0.0308739	+	0.999523i	$0.509829\pi$
$822$	0	0
$823$	−1.34488e7	−0.692123	−0.346061	−	0.938212i	$-0.612481\pi$
$823$	−1.34488e7	−0.692123	−0.346061	+	0.938212i	$0.612481\pi$
$824$	0	0
$825$	2.02054e6	0.103355
$826$	0	0
$827$	1.99983e6	0.101678	0.0508392	−	0.998707i	$-0.483810\pi$
$827$	1.99983e6	0.101678	0.0508392	+	0.998707i	$0.483810\pi$
$828$	0	0
$829$	−1.19093e7	−0.601866	−0.300933	−	0.953645i	$-0.597298\pi$
$829$	−1.19093e7	−0.601866	−0.300933	+	0.953645i	$0.597298\pi$
$830$	0	0
$831$	−2.74031e7	−1.37657
$832$	0	0
$833$	4.91885e7	2.45613
$834$	0	0
$835$	−9.85896e6	−0.489345
$836$	0	0
$837$	−2.63846e6	−0.130178
$838$	0	0
$839$	−1.45360e7	−0.712919	−0.356459	−	0.934311i	$-0.616016\pi$
$839$	−1.45360e7	−0.712919	−0.356459	+	0.934311i	$0.616016\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	0	0
$843$	1.00146e7	0.485362
$844$	0	0
$845$	2.10705e7	1.01516
$846$	0	0
$847$	2.46040e7	1.17841
$848$	0	0
$849$	−2.34869e6	−0.111830
$850$	0	0
$851$	−3.52513e6	−0.166860
$852$	0	0
$853$	−6.18563e6	−0.291079	−0.145540	−	0.989352i	$-0.546492\pi$
$853$	−6.18563e6	−0.291079	−0.145540	+	0.989352i	$0.546492\pi$
$854$	0	0
$855$	−492060.	−0.0230198
$856$	0	0
$857$	−2.85410e7	−1.32745	−0.663724	−	0.747978i	$-0.731025\pi$
$857$	−2.85410e7	−1.32745	−0.663724	+	0.747978i	$0.731025\pi$
$858$	0	0
$859$	−3.50491e7	−1.62067	−0.810333	−	0.585970i	$-0.800714\pi$
$859$	−3.50491e7	−1.62067	−0.810333	+	0.585970i	$0.800714\pi$
$860$	0	0
$861$	−5.85606e7	−2.69214
$862$	0	0
$863$	−1.75590e7	−0.802550	−0.401275	−	0.915958i	$-0.631433\pi$
$863$	−1.75590e7	−0.802550	−0.401275	+	0.915958i	$0.631433\pi$
$864$	0	0
$865$	1.56509e7	0.711210
$866$	0	0
$867$	−2.76013e7	−1.24704
$868$	0	0
$869$	5.07232e7	2.27854
$870$	0	0
$871$	−1.66250e6	−0.0742532
$872$	0	0
$873$	53596.1	0.00238011
$874$	0	0
$875$	3.51501e7	1.55205
$876$	0	0
$877$	−9.57523e6	−0.420388	−0.210194	−	0.977660i	$-0.567410\pi$
$877$	−9.57523e6	−0.420388	−0.210194	+	0.977660i	$0.567410\pi$
$878$	0	0
$879$	1.23355e7	0.538497
$880$	0	0
$881$	−2.60927e6	−0.113261	−0.0566303	−	0.998395i	$-0.518036\pi$
$881$	−2.60927e6	−0.113261	−0.0566303	+	0.998395i	$0.518036\pi$
$882$	0	0
$883$	−2.20388e7	−0.951231	−0.475616	−	0.879653i	$-0.657775\pi$
$883$	−2.20388e7	−0.951231	−0.475616	+	0.879653i	$0.657775\pi$
$884$	0	0
$885$	3.11082e7	1.33511
$886$	0	0
$887$	−1.32296e7	−0.564596	−0.282298	−	0.959327i	$-0.591097\pi$
$887$	−1.32296e7	−0.564596	−0.282298	+	0.959327i	$0.591097\pi$
$888$	0	0
$889$	−6.64104e6	−0.281826
$890$	0	0
$891$	3.04579e7	1.28530
$892$	0	0
$893$	3.38863e6	0.142199
$894$	0	0
$895$	−3.70488e7	−1.54602
$896$	0	0
$897$	3.93712e6	0.163379
$898$	0	0
$899$	−579785.	−0.0239259
$900$	0	0
$901$	−5.14854e7	−2.11287
$902$	0	0
$903$	−2.08274e7	−0.849992
$904$	0	0
$905$	−2.93744e7	−1.19220
$906$	0	0
$907$	−2.28630e7	−0.922814	−0.461407	−	0.887189i	$-0.652655\pi$
$907$	−2.28630e7	−0.922814	−0.461407	+	0.887189i	$0.652655\pi$
$908$	0	0
$909$	79325.6	0.00318422
$910$	0	0
$911$	1.86891e7	0.746091	0.373046	−	0.927813i	$-0.378313\pi$
$911$	1.86891e7	0.746091	0.373046	+	0.927813i	$0.378313\pi$
$912$	0	0
$913$	3.57590e7	1.41974
$914$	0	0
$915$	3.18874e7	1.25912
$916$	0	0
$917$	3.02179e7	1.18670
$918$	0	0
$919$	6.34791e6	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$919$	6.34791e6	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$920$	0	0
$921$	7.37448e6	0.286472
$922$	0	0
$923$	385304.	0.0148867
$924$	0	0
$925$	−316704.	−0.0121703
$926$	0	0
$927$	706885.	0.0270178
$928$	0	0
$929$	2.80133e7	1.06494	0.532469	−	0.846449i	$-0.321264\pi$
$929$	2.80133e7	1.06494	0.532469	+	0.846449i	$0.321264\pi$
$930$	0	0
$931$	−4.50243e7	−1.70244
$932$	0	0
$933$	4.06247e7	1.52787
$934$	0	0
$935$	5.48638e7	2.05237
$936$	0	0
$937$	8.32451e6	0.309749	0.154874	−	0.987934i	$-0.450503\pi$
$937$	8.32451e6	0.309749	0.154874	+	0.987934i	$0.450503\pi$
$938$	0	0
$939$	2.13746e7	0.791105
$940$	0	0
$941$	5.98374e6	0.220292	0.110146	−	0.993915i	$-0.464868\pi$
$941$	5.98374e6	0.220292	0.110146	+	0.993915i	$0.464868\pi$
$942$	0	0
$943$	4.99206e7	1.82810
$944$	0	0
$945$	−4.67670e7	−1.70357
$946$	0	0
$947$	−1.30352e7	−0.472326	−0.236163	−	0.971713i	$-0.575890\pi$
$947$	−1.30352e7	−0.472326	−0.236163	+	0.971713i	$0.575890\pi$
$948$	0	0
$949$	5.16145e6	0.186040
$950$	0	0
$951$	−3.98224e7	−1.42783
$952$	0	0
$953$	−3.21255e7	−1.14582	−0.572911	−	0.819617i	$-0.694186\pi$
$953$	−3.21255e7	−1.14582	−0.572911	+	0.819617i	$0.694186\pi$
$954$	0	0
$955$	1.80104e7	0.639020
$956$	0	0
$957$	6.83837e6	0.241364
$958$	0	0
$959$	−2.54518e7	−0.893660
$960$	0	0
$961$	−2.81539e7	−0.983399
$962$	0	0
$963$	426476.	0.0148193
$964$	0	0
$965$	3.17927e7	1.09903
$966$	0	0
$967$	4.86640e7	1.67356	0.836781	−	0.547538i	$-0.184435\pi$
$967$	4.86640e7	1.67356	0.836781	+	0.547538i	$0.184435\pi$
$968$	0	0
$969$	4.53078e7	1.55011
$970$	0	0
$971$	−448444.	−0.0152637	−0.00763185	−	0.999971i	$-0.502429\pi$
$971$	−448444.	−0.0152637	−0.00763185	+	0.999971i	$0.502429\pi$
$972$	0	0
$973$	−1.47735e7	−0.500266
$974$	0	0
$975$	353718.	0.0119164
$976$	0	0
$977$	2.42795e7	0.813773	0.406887	−	0.913479i	$-0.366614\pi$
$977$	2.42795e7	0.813773	0.406887	+	0.913479i	$0.366614\pi$
$978$	0	0
$979$	4.13319e7	1.37825
$980$	0	0
$981$	163363.	0.00541977
$982$	0	0
$983$	1.75735e7	0.580062	0.290031	−	0.957017i	$-0.406334\pi$
$983$	1.75735e7	0.580062	0.290031	+	0.957017i	$0.406334\pi$
$984$	0	0
$985$	−3.27780e6	−0.107645
$986$	0	0
$987$	6.70458e6	0.219068
$988$	0	0
$989$	1.77545e7	0.577189
$990$	0	0
$991$	−2.20320e7	−0.712640	−0.356320	−	0.934364i	$-0.615969\pi$
$991$	−2.20320e7	−0.712640	−0.356320	+	0.934364i	$0.615969\pi$
$992$	0	0
$993$	−1.74962e7	−0.563080
$994$	0	0
$995$	2.39044e7	0.765457
$996$	0	0
$997$	1.20049e7	0.382490	0.191245	−	0.981542i	$-0.438748\pi$
$997$	1.20049e7	0.382490	0.191245	+	0.981542i	$0.438748\pi$
$998$	0	0
$999$	−4.87779e6	−0.154635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.a.k.1.4		7
4.3	odd	2		29.6.a.b.1.6	✓	7
12.11	even	2		261.6.a.e.1.2		7
20.19	odd	2		725.6.a.b.1.2		7
116.115	odd	2		841.6.a.b.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.b.1.6	✓	7	4.3	odd	2
261.6.a.e.1.2		7	12.11	even	2
464.6.a.k.1.4		7	1.1	even	1	trivial
725.6.a.b.1.2		7	20.19	odd	2
841.6.a.b.1.2		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)

\(f(q)\)	\(=\)	\(q-15.4219 q^{3} -58.0818 q^{5} +210.388 q^{7} -5.16616 q^{9} +O(q^{10})\)	\(q-15.4219 q^{3} -58.0818 q^{5} +210.388 q^{7} -5.16616 q^{9} -527.254 q^{11} -92.3017 q^{13} +895.729 q^{15} +1791.54 q^{17} -1639.87 q^{19} -3244.57 q^{21} +2765.87 q^{23} +248.491 q^{25} +3827.18 q^{27} +841.000 q^{29} -689.400 q^{31} +8131.24 q^{33} -12219.7 q^{35} -1274.51 q^{37} +1423.46 q^{39} +18048.8 q^{41} +6419.14 q^{43} +300.060 q^{45} -2066.40 q^{47} +27456.0 q^{49} -27628.8 q^{51} -28738.1 q^{53} +30623.8 q^{55} +25289.9 q^{57} +34729.5 q^{59} +35599.4 q^{61} -1086.90 q^{63} +5361.05 q^{65} +18011.5 q^{67} -42654.9 q^{69} -4174.40 q^{71} -55919.3 q^{73} -3832.19 q^{75} -110928. q^{77} -96202.6 q^{79} -57766.9 q^{81} -67821.3 q^{83} -104056. q^{85} -12969.8 q^{87} -78390.8 q^{89} -19419.1 q^{91} +10631.8 q^{93} +95246.6 q^{95} -10374.5 q^{97} +2723.88 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 26 q^{3} + 32 q^{5} - 184 q^{7} + 1005 q^{9}+O(q^{10}) \) 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

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Database

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