LMFDB - Embedded newform 387.6.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,6,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("387.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	387.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:	$62.0685382676$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 $ x^8 - 4x^7 - 173x^6 + 462x^5 + 9118x^4 - 14192x^3 - 167688x^2 + 106368*x + 681984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-6.09504$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +O(q^{10})$	\(q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +513.028 q^{10} +631.897 q^{11} +28.5724 q^{13} +1809.15 q^{14} -972.709 q^{16} +1743.07 q^{17} -2027.92 q^{19} +2124.97 q^{20} +5115.23 q^{22} -2980.86 q^{23} +891.469 q^{25} +231.295 q^{26} +7493.51 q^{28} -766.139 q^{29} -8355.33 q^{31} -8270.38 q^{32} +14110.3 q^{34} +14163.7 q^{35} +14892.6 q^{37} -16416.1 q^{38} +784.783 q^{40} +5342.20 q^{41} -1849.00 q^{43} +21187.3 q^{44} -24130.2 q^{46} +6282.09 q^{47} +33140.1 q^{49} +7216.48 q^{50} +958.024 q^{52} +915.172 q^{53} +40046.8 q^{55} +2767.47 q^{56} -6201.92 q^{58} +14644.5 q^{59} -21324.9 q^{61} -67636.7 q^{62} -35822.4 q^{64} +1810.79 q^{65} -12868.9 q^{67} +58444.8 q^{68} +114656. q^{70} -56454.6 q^{71} -25591.3 q^{73} +120556. q^{74} -67995.4 q^{76} +141222. q^{77} +5795.13 q^{79} -61646.1 q^{80} +43245.4 q^{82} +7857.24 q^{83} +110468. q^{85} -14967.7 q^{86} +7824.80 q^{88} +7560.11 q^{89} +6385.60 q^{91} -99947.5 q^{92} +50853.8 q^{94} -128520. q^{95} +111712. q^{97} +268271. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	$ 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) $ 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

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 $

$2$	8.09504	1.43101	0.715507	−	0.698605i	$-0.246196\pi$
$2$	8.09504	1.43101	0.715507	+	0.698605i	$0.246196\pi$
$3$	0	0
$3$	0	0
$4$	33.5297	1.04780
$5$	63.3756	1.13370	0.566849	−	0.823822i	$-0.308162\pi$
$5$	63.3756	1.13370	0.566849	+	0.823822i	$0.308162\pi$
$6$	0	0
$7$	223.489	1.72389	0.861946	−	0.507000i	$-0.169245\pi$
$7$	223.489	1.72389	0.861946	+	0.507000i	$0.169245\pi$
$8$	12.3830	0.0684073
$9$	0	0
$10$	513.028	1.62234
$11$	631.897	1.57458	0.787289	−	0.616584i	$-0.211484\pi$
$11$	631.897	1.57458	0.787289	+	0.616584i	$0.211484\pi$
$12$	0	0
$13$	28.5724	0.0468909	0.0234454	−	0.999725i	$-0.492536\pi$
$13$	28.5724	0.0468909	0.0234454	+	0.999725i	$0.492536\pi$
$14$	1809.15	2.46692
$15$	0	0
$16$	−972.709	−0.949911
$17$	1743.07	1.46283	0.731414	−	0.681933i	$-0.238861\pi$
$17$	1743.07	1.46283	0.731414	+	0.681933i	$0.238861\pi$
$18$	0	0
$19$	−2027.92	−1.28874	−0.644371	−	0.764713i	$-0.722881\pi$
$19$	−2027.92	−1.28874	−0.644371	+	0.764713i	$0.722881\pi$
$20$	2124.97	1.18789
$21$	0	0
$22$	5115.23	2.25324
$23$	−2980.86	−1.17496	−0.587479	−	0.809239i	$-0.699880\pi$
$23$	−2980.86	−1.17496	−0.587479	+	0.809239i	$0.699880\pi$
$24$	0	0
$25$	891.469	0.285270
$26$	231.295	0.0671015
$27$	0	0
$28$	7493.51	1.80630
$29$	−766.139	−0.169166	−0.0845829	−	0.996416i	$-0.526956\pi$
$29$	−766.139	−0.169166	−0.0845829	+	0.996416i	$0.526956\pi$
$30$	0	0
$31$	−8355.33	−1.56156	−0.780781	−	0.624805i	$-0.785178\pi$
$31$	−8355.33	−1.56156	−0.780781	+	0.624805i	$0.785178\pi$
$32$	−8270.38	−1.42774
$33$	0	0
$34$	14110.3	2.09333
$35$	14163.7	1.95437
$36$	0	0
$37$	14892.6	1.78841	0.894204	−	0.447661i	$-0.147743\pi$
$37$	14892.6	1.78841	0.894204	+	0.447661i	$0.147743\pi$
$38$	−16416.1	−1.84421
$39$	0	0
$40$	784.783	0.0775532
$41$	5342.20	0.496319	0.248159	−	0.968719i	$-0.420174\pi$
$41$	5342.20	0.496319	0.248159	+	0.968719i	$0.420174\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	21187.3	1.64985
$45$	0	0
$46$	−24130.2	−1.68138
$47$	6282.09	0.414820	0.207410	−	0.978254i	$-0.433497\pi$
$47$	6282.09	0.414820	0.207410	+	0.978254i	$0.433497\pi$
$48$	0	0
$49$	33140.1	1.97181
$50$	7216.48	0.408226
$51$	0	0
$52$	958.024	0.0491324
$53$	915.172	0.0447521	0.0223760	−	0.999750i	$-0.492877\pi$
$53$	915.172	0.0447521	0.0223760	+	0.999750i	$0.492877\pi$
$54$	0	0
$55$	40046.8	1.78510
$56$	2767.47	0.117927
$57$	0	0
$58$	−6201.92	−0.242079
$59$	14644.5	0.547704	0.273852	−	0.961772i	$-0.411702\pi$
$59$	14644.5	0.547704	0.273852	+	0.961772i	$0.411702\pi$
$60$	0	0
$61$	−21324.9	−0.733775	−0.366887	−	0.930265i	$-0.619577\pi$
$61$	−21324.9	−0.733775	−0.366887	+	0.930265i	$0.619577\pi$
$62$	−67636.7	−2.23462
$63$	0	0
$64$	−35822.4	−1.09321
$65$	1810.79	0.0531601
$66$	0	0
$67$	−12868.9	−0.350232	−0.175116	−	0.984548i	$-0.556030\pi$
$67$	−12868.9	−0.350232	−0.175116	+	0.984548i	$0.556030\pi$
$68$	58444.8	1.53276
$69$	0	0
$70$	114656.	2.79674
$71$	−56454.6	−1.32909	−0.664544	−	0.747249i	$-0.731374\pi$
$71$	−56454.6	−1.32909	−0.664544	+	0.747249i	$0.731374\pi$
$72$	0	0
$73$	−25591.3	−0.562064	−0.281032	−	0.959698i	$-0.590677\pi$
$73$	−25591.3	−0.562064	−0.281032	+	0.959698i	$0.590677\pi$
$74$	120556.	2.55924
$75$	0	0
$76$	−67995.4	−1.35035
$77$	141222.	2.71440
$78$	0	0
$79$	5795.13	0.104471	0.0522355	−	0.998635i	$-0.483365\pi$
$79$	5795.13	0.104471	0.0522355	+	0.998635i	$0.483365\pi$
$80$	−61646.1	−1.07691
$81$	0	0
$82$	43245.4	0.710240
$83$	7857.24	0.125192	0.0625958	−	0.998039i	$-0.480062\pi$
$83$	7857.24	0.125192	0.0625958	+	0.998039i	$0.480062\pi$
$84$	0	0
$85$	110468.	1.65841
$86$	−14967.7	−0.218228
$87$	0	0
$88$	7824.80	0.107713
$89$	7560.11	0.101170	0.0505852	−	0.998720i	$-0.483891\pi$
$89$	7560.11	0.101170	0.0505852	+	0.998720i	$0.483891\pi$
$90$	0	0
$91$	6385.60	0.0808348
$92$	−99947.5	−1.23113
$93$	0	0
$94$	50853.8	0.593613
$95$	−128520.	−1.46104
$96$	0	0
$97$	111712.	1.20551	0.602754	−	0.797927i	$-0.294070\pi$
$97$	111712.	1.20551	0.602754	+	0.797927i	$0.294070\pi$
$98$	268271.	2.82168
$99$	0	0
$100$	29890.7	0.298907
$101$	−10492.3	−0.102345	−0.0511724	−	0.998690i	$-0.516296\pi$
$101$	−10492.3	−0.102345	−0.0511724	+	0.998690i	$0.516296\pi$
$102$	0	0
$103$	−11286.6	−0.104826	−0.0524130	−	0.998625i	$-0.516691\pi$
$103$	−11286.6	−0.104826	−0.0524130	+	0.998625i	$0.516691\pi$
$104$	353.813	0.00320768
$105$	0	0
$106$	7408.36	0.0640409
$107$	−39045.4	−0.329693	−0.164847	−	0.986319i	$-0.552713\pi$
$107$	−39045.4	−0.329693	−0.164847	+	0.986319i	$0.552713\pi$
$108$	0	0
$109$	−14622.8	−0.117887	−0.0589433	−	0.998261i	$-0.518773\pi$
$109$	−14622.8	−0.117887	−0.0589433	+	0.998261i	$0.518773\pi$
$110$	324181.	2.55450
$111$	0	0
$112$	−217389.	−1.63755
$113$	859.121	0.00632934	0.00316467	−	0.999995i	$-0.498993\pi$
$113$	859.121	0.00632934	0.00316467	+	0.999995i	$0.498993\pi$
$114$	0	0
$115$	−188914.	−1.33205
$116$	−25688.4	−0.177252
$117$	0	0
$118$	118548.	0.783772
$119$	389557.	2.52176
$120$	0	0
$121$	238242.	1.47930
$122$	−172626.	−1.05004
$123$	0	0
$124$	−280152.	−1.63621
$125$	−141551.	−0.810288
$126$	0	0
$127$	193396.	1.06399	0.531997	−	0.846746i	$-0.321442\pi$
$127$	193396.	1.06399	0.531997	+	0.846746i	$0.321442\pi$
$128$	−25331.5	−0.136658
$129$	0	0
$130$	14658.4	0.0760728
$131$	269599.	1.37259	0.686293	−	0.727325i	$-0.259237\pi$
$131$	269599.	1.37259	0.686293	+	0.727325i	$0.259237\pi$
$132$	0	0
$133$	−453216.	−2.22165
$134$	−104175.	−0.501187
$135$	0	0
$136$	21584.6	0.100068
$137$	−148987.	−0.678183	−0.339091	−	0.940753i	$-0.610120\pi$
$137$	−148987.	−0.678183	−0.339091	+	0.940753i	$0.610120\pi$
$138$	0	0
$139$	−16355.6	−0.0718008	−0.0359004	−	0.999355i	$-0.511430\pi$
$139$	−16355.6	−0.0718008	−0.0359004	+	0.999355i	$0.511430\pi$
$140$	474906.	2.04780
$141$	0	0
$142$	−457003.	−1.90194
$143$	18054.8	0.0738333
$144$	0	0
$145$	−48554.5	−0.191783
$146$	−207163.	−0.804322
$147$	0	0
$148$	499345.	1.87390
$149$	150536.	0.555489	0.277744	−	0.960655i	$-0.410413\pi$
$149$	150536.	0.555489	0.277744	+	0.960655i	$0.410413\pi$
$150$	0	0
$151$	428141.	1.52807	0.764037	−	0.645172i	$-0.223214\pi$
$151$	428141.	1.52807	0.764037	+	0.645172i	$0.223214\pi$
$152$	−25111.8	−0.0881594
$153$	0	0
$154$	1.14320e6	3.88435
$155$	−529524.	−1.77034
$156$	0	0
$157$	−434190.	−1.40582	−0.702912	−	0.711277i	$-0.748117\pi$
$157$	−434190.	−1.40582	−0.702912	+	0.711277i	$0.748117\pi$
$158$	46911.8	0.149499
$159$	0	0
$160$	−524140.	−1.61863
$161$	−666189.	−2.02550
$162$	0	0
$163$	−571830.	−1.68577	−0.842884	−	0.538095i	$-0.819144\pi$
$163$	−571830.	−1.68577	−0.842884	+	0.538095i	$0.819144\pi$
$164$	179122.	0.520044
$165$	0	0
$166$	63604.7	0.179151
$167$	−605744.	−1.68073	−0.840365	−	0.542020i	$-0.817660\pi$
$167$	−605744.	−1.68073	−0.840365	+	0.542020i	$0.817660\pi$
$168$	0	0
$169$	−370477.	−0.997801
$170$	894246.	2.37320
$171$	0	0
$172$	−61996.4	−0.159789
$173$	85238.8	0.216532	0.108266	−	0.994122i	$-0.465470\pi$
$173$	85238.8	0.216532	0.108266	+	0.994122i	$0.465470\pi$
$174$	0	0
$175$	199233.	0.491775
$176$	−614652.	−1.49571
$177$	0	0
$178$	61199.4	0.144776
$179$	−77941.8	−0.181818	−0.0909092	−	0.995859i	$-0.528977\pi$
$179$	−77941.8	−0.181818	−0.0909092	+	0.995859i	$0.528977\pi$
$180$	0	0
$181$	−347661.	−0.788787	−0.394393	−	0.918942i	$-0.629045\pi$
$181$	−347661.	−0.788787	−0.394393	+	0.918942i	$0.629045\pi$
$182$	51691.7	0.115676
$183$	0	0
$184$	−36912.2	−0.0803757
$185$	943828.	2.02751
$186$	0	0
$187$	1.10144e6	2.30334
$188$	210637.	0.434650
$189$	0	0
$190$	−1.04038e6	−2.09078
$191$	192163.	0.381142	0.190571	−	0.981673i	$-0.438966\pi$
$191$	192163.	0.381142	0.190571	+	0.981673i	$0.438966\pi$
$192$	0	0
$193$	−271724.	−0.525091	−0.262546	−	0.964920i	$-0.584562\pi$
$193$	−271724.	−0.525091	−0.262546	+	0.964920i	$0.584562\pi$
$194$	904313.	1.72510
$195$	0	0
$196$	1.11118e6	2.06606
$197$	−76788.1	−0.140971	−0.0704853	−	0.997513i	$-0.522455\pi$
$197$	−76788.1	−0.140971	−0.0704853	+	0.997513i	$0.522455\pi$
$198$	0	0
$199$	−694776.	−1.24369	−0.621845	−	0.783140i	$-0.713617\pi$
$199$	−694776.	−1.24369	−0.621845	+	0.783140i	$0.713617\pi$
$200$	11039.1	0.0195146
$201$	0	0
$202$	−84935.3	−0.146457
$203$	−171223.	−0.291624
$204$	0	0
$205$	338565.	0.562675
$206$	−91365.3	−0.150008
$207$	0	0
$208$	−27792.6	−0.0445422
$209$	−1.28143e6	−2.02923
$210$	0	0
$211$	−433533.	−0.670372	−0.335186	−	0.942152i	$-0.608799\pi$
$211$	−433533.	−0.670372	−0.335186	+	0.942152i	$0.608799\pi$
$212$	30685.5	0.0468914
$213$	0	0
$214$	−316074.	−0.471796
$215$	−117182.	−0.172887
$216$	0	0
$217$	−1.86732e6	−2.69196
$218$	−118372.	−0.168698
$219$	0	0
$220$	1.34276e6	1.87043
$221$	49803.8	0.0685933
$222$	0	0
$223$	1.04204e6	1.40320	0.701602	−	0.712569i	$-0.252469\pi$
$223$	1.04204e6	1.40320	0.701602	+	0.712569i	$0.252469\pi$
$224$	−1.84834e6	−2.46128
$225$	0	0
$226$	6954.62	0.00905737
$227$	−924970.	−1.19141	−0.595707	−	0.803202i	$-0.703128\pi$
$227$	−924970.	−1.19141	−0.595707	+	0.803202i	$0.703128\pi$
$228$	0	0
$229$	−137140.	−0.172812	−0.0864062	−	0.996260i	$-0.527538\pi$
$229$	−137140.	−0.172812	−0.0864062	+	0.996260i	$0.527538\pi$
$230$	−1.52927e6	−1.90618
$231$	0	0
$232$	−9487.13	−0.0115722
$233$	−428602.	−0.517208	−0.258604	−	0.965983i	$-0.583262\pi$
$233$	−428602.	−0.517208	−0.258604	+	0.965983i	$0.583262\pi$
$234$	0	0
$235$	398131.	0.470280
$236$	491027.	0.573886
$237$	0	0
$238$	3.15348e6	3.60868
$239$	1.17134e6	1.32645	0.663223	−	0.748422i	$-0.269188\pi$
$239$	1.17134e6	1.32645	0.663223	+	0.748422i	$0.269188\pi$
$240$	0	0
$241$	119252.	0.132258	0.0661291	−	0.997811i	$-0.478935\pi$
$241$	119252.	0.132258	0.0661291	+	0.997811i	$0.478935\pi$
$242$	1.92858e6	2.11690
$243$	0	0
$244$	−715018.	−0.768852
$245$	2.10028e6	2.23543
$246$	0	0
$247$	−57942.4	−0.0604302
$248$	−103464.	−0.106822
$249$	0	0
$250$	−1.14586e6	−1.15953
$251$	141272.	0.141538	0.0707690	−	0.997493i	$-0.477455\pi$
$251$	141272.	0.141538	0.0707690	+	0.997493i	$0.477455\pi$
$252$	0	0
$253$	−1.88360e6	−1.85006
$254$	1.56555e6	1.52259
$255$	0	0
$256$	941257.	0.897652
$257$	196749.	0.185815	0.0929074	−	0.995675i	$-0.470384\pi$
$257$	196749.	0.185815	0.0929074	+	0.995675i	$0.470384\pi$
$258$	0	0
$259$	3.32833e6	3.08302
$260$	60715.4	0.0557013
$261$	0	0
$262$	2.18241e6	1.96419
$263$	1.96879e6	1.75513	0.877565	−	0.479457i	$-0.159166\pi$
$263$	1.96879e6	1.75513	0.877565	+	0.479457i	$0.159166\pi$
$264$	0	0
$265$	57999.6	0.0507353
$266$	−3.66880e6	−3.17922
$267$	0	0
$268$	−431492.	−0.366974
$269$	531007.	0.447424	0.223712	−	0.974655i	$-0.428182\pi$
$269$	531007.	0.447424	0.223712	+	0.974655i	$0.428182\pi$
$270$	0	0
$271$	−1.68769e6	−1.39595	−0.697975	−	0.716122i	$-0.745916\pi$
$271$	−1.68769e6	−1.39595	−0.697975	+	0.716122i	$0.745916\pi$
$272$	−1.69550e6	−1.38956
$273$	0	0
$274$	−1.20606e6	−0.970489
$275$	563316.	0.449180
$276$	0	0
$277$	13630.1	0.0106733	0.00533665	−	0.999986i	$-0.498301\pi$
$277$	13630.1	0.0106733	0.00533665	+	0.999986i	$0.498301\pi$
$278$	−132399.	−0.102748
$279$	0	0
$280$	175390.	0.133693
$281$	−1.12070e6	−0.846692	−0.423346	−	0.905968i	$-0.639145\pi$
$281$	−1.12070e6	−0.846692	−0.423346	+	0.905968i	$0.639145\pi$
$282$	0	0
$283$	−1.52947e6	−1.13521	−0.567605	−	0.823301i	$-0.692130\pi$
$283$	−1.52947e6	−1.13521	−0.567605	+	0.823301i	$0.692130\pi$
$284$	−1.89291e6	−1.39262
$285$	0	0
$286$	146154.	0.105657
$287$	1.19392e6	0.855600
$288$	0	0
$289$	1.61845e6	1.13987
$290$	−393051.	−0.274444
$291$	0	0
$292$	−858070.	−0.588932
$293$	1.58484e6	1.07849	0.539246	−	0.842148i	$-0.318709\pi$
$293$	1.58484e6	1.07849	0.539246	+	0.842148i	$0.318709\pi$
$294$	0	0
$295$	928107.	0.620930
$296$	184416.	0.122340
$297$	0	0
$298$	1.21860e6	0.794912
$299$	−85170.4	−0.0550948
$300$	0	0
$301$	−413230.	−0.262891
$302$	3.46582e6	2.18670
$303$	0	0
$304$	1.97257e6	1.22419
$305$	−1.35148e6	−0.831879
$306$	0	0
$307$	695751.	0.421316	0.210658	−	0.977560i	$-0.432439\pi$
$307$	695751.	0.421316	0.210658	+	0.977560i	$0.432439\pi$
$308$	4.73512e6	2.84416
$309$	0	0
$310$	−4.28652e6	−2.53338
$311$	988763.	0.579684	0.289842	−	0.957075i	$-0.406397\pi$
$311$	988763.	0.579684	0.289842	+	0.957075i	$0.406397\pi$
$312$	0	0
$313$	−3.07551e6	−1.77442	−0.887208	−	0.461369i	$-0.847358\pi$
$313$	−3.07551e6	−1.77442	−0.887208	+	0.461369i	$0.847358\pi$
$314$	−3.51479e6	−2.01176
$315$	0	0
$316$	194309.	0.109465
$317$	−904527.	−0.505561	−0.252780	−	0.967524i	$-0.581345\pi$
$317$	−904527.	−0.505561	−0.252780	+	0.967524i	$0.581345\pi$
$318$	0	0
$319$	−484120.	−0.266365
$320$	−2.27027e6	−1.23937
$321$	0	0
$322$	−5.39283e6	−2.89852
$323$	−3.53481e6	−1.88521
$324$	0	0
$325$	25471.4	0.0133766
$326$	−4.62899e6	−2.41236
$327$	0	0
$328$	66152.7	0.0339518
$329$	1.40398e6	0.715105
$330$	0	0
$331$	−2.09309e6	−1.05007	−0.525034	−	0.851081i	$-0.675947\pi$
$331$	−2.09309e6	−1.05007	−0.525034	+	0.851081i	$0.675947\pi$
$332$	263451.	0.131176
$333$	0	0
$334$	−4.90353e6	−2.40515
$335$	−815577.	−0.397057
$336$	0	0
$337$	−647484.	−0.310566	−0.155283	−	0.987870i	$-0.549629\pi$
$337$	−647484.	−0.310566	−0.155283	+	0.987870i	$0.549629\pi$
$338$	−2.99902e6	−1.42787
$339$	0	0
$340$	3.70397e6	1.73768
$341$	−5.27970e6	−2.45880
$342$	0	0
$343$	3.65027e6	1.67529
$344$	−22896.2	−0.0104320
$345$	0	0
$346$	690012.	0.309860
$347$	462018.	0.205985	0.102992	−	0.994682i	$-0.467158\pi$
$347$	462018.	0.205985	0.102992	+	0.994682i	$0.467158\pi$
$348$	0	0
$349$	2.85075e6	1.25284	0.626420	−	0.779486i	$-0.284520\pi$
$349$	2.85075e6	1.25284	0.626420	+	0.779486i	$0.284520\pi$
$350$	1.61280e6	0.703737
$351$	0	0
$352$	−5.22602e6	−2.24810
$353$	809444.	0.345740	0.172870	−	0.984945i	$-0.444696\pi$
$353$	809444.	0.345740	0.172870	+	0.984945i	$0.444696\pi$
$354$	0	0
$355$	−3.57785e6	−1.50678
$356$	253488.	0.106007
$357$	0	0
$358$	−630942.	−0.260185
$359$	−3.46118e6	−1.41739	−0.708694	−	0.705516i	$-0.750715\pi$
$359$	−3.46118e6	−1.41739	−0.708694	+	0.705516i	$0.750715\pi$
$360$	0	0
$361$	1.63635e6	0.660856
$362$	−2.81433e6	−1.12877
$363$	0	0
$364$	214107.	0.0846990
$365$	−1.62187e6	−0.637210
$366$	0	0
$367$	2.13602e6	0.827827	0.413914	−	0.910316i	$-0.364162\pi$
$367$	2.13602e6	0.827827	0.413914	+	0.910316i	$0.364162\pi$
$368$	2.89951e6	1.11611
$369$	0	0
$370$	7.64033e6	2.90140
$371$	204531.	0.0771478
$372$	0	0
$373$	2.37193e6	0.882734	0.441367	−	0.897327i	$-0.354494\pi$
$373$	2.37193e6	0.882734	0.441367	+	0.897327i	$0.354494\pi$
$374$	8.91622e6	3.29611
$375$	0	0
$376$	77791.4	0.0283767
$377$	−21890.4	−0.00793233
$378$	0	0
$379$	−5.47562e6	−1.95810	−0.979051	−	0.203616i	$-0.934731\pi$
$379$	−5.47562e6	−1.95810	−0.979051	+	0.203616i	$0.934731\pi$
$380$	−4.30925e6	−1.53089
$381$	0	0
$382$	1.55557e6	0.545419
$383$	−3.16966e6	−1.10412	−0.552059	−	0.833805i	$-0.686158\pi$
$383$	−3.16966e6	−1.10412	−0.552059	+	0.833805i	$0.686158\pi$
$384$	0	0
$385$	8.95001e6	3.07731
$386$	−2.19962e6	−0.751413
$387$	0	0
$388$	3.74567e6	1.26314
$389$	311235.	0.104283	0.0521416	−	0.998640i	$-0.483395\pi$
$389$	311235.	0.104283	0.0521416	+	0.998640i	$0.483395\pi$
$390$	0	0
$391$	−5.19587e6	−1.71876
$392$	410376.	0.134886
$393$	0	0
$394$	−621603.	−0.201731
$395$	367270.	0.118438
$396$	0	0
$397$	−496314.	−0.158045	−0.0790224	−	0.996873i	$-0.525180\pi$
$397$	−496314.	−0.158045	−0.0790224	+	0.996873i	$0.525180\pi$
$398$	−5.62424e6	−1.77974
$399$	0	0
$400$	−867140.	−0.270981
$401$	5.63670e6	1.75051	0.875253	−	0.483665i	$-0.160695\pi$
$401$	5.63670e6	1.75051	0.875253	+	0.483665i	$0.160695\pi$
$402$	0	0
$403$	−238732.	−0.0732230
$404$	−351802.	−0.107237
$405$	0	0
$406$	−1.38606e6	−0.417318
$407$	9.41059e6	2.81599
$408$	0	0
$409$	−720515.	−0.212978	−0.106489	−	0.994314i	$-0.533961\pi$
$409$	−720515.	−0.212978	−0.106489	+	0.994314i	$0.533961\pi$
$410$	2.74070e6	0.805197
$411$	0	0
$412$	−378435.	−0.109837
$413$	3.27289e6	0.944182
$414$	0	0
$415$	497958.	0.141929
$416$	−236305.	−0.0669482
$417$	0	0
$418$	−1.03733e7	−2.90385
$419$	3.94151e6	1.09680	0.548400	−	0.836216i	$-0.315237\pi$
$419$	3.94151e6	1.09680	0.548400	+	0.836216i	$0.315237\pi$
$420$	0	0
$421$	4.39506e6	1.20854	0.604268	−	0.796781i	$-0.293466\pi$
$421$	4.39506e6	1.20854	0.604268	+	0.796781i	$0.293466\pi$
$422$	−3.50947e6	−0.959312
$423$	0	0
$424$	11332.6	0.00306137
$425$	1.55390e6	0.417301
$426$	0	0
$427$	−4.76588e6	−1.26495
$428$	−1.30918e6	−0.345454
$429$	0	0
$430$	−948589.	−0.247404
$431$	−1.24913e6	−0.323903	−0.161951	−	0.986799i	$-0.551779\pi$
$431$	−1.24913e6	−0.323903	−0.161951	+	0.986799i	$0.551779\pi$
$432$	0	0
$433$	971902.	0.249117	0.124558	−	0.992212i	$-0.460249\pi$
$433$	971902.	0.249117	0.124558	+	0.992212i	$0.460249\pi$
$434$	−1.51160e7	−3.85224
$435$	0	0
$436$	−490299.	−0.123522
$437$	6.04494e6	1.51422
$438$	0	0
$439$	−1.32453e6	−0.328021	−0.164010	−	0.986459i	$-0.552443\pi$
$439$	−1.32453e6	−0.328021	−0.164010	+	0.986459i	$0.552443\pi$
$440$	495902.	0.122114
$441$	0	0
$442$	403164.	0.0981581
$443$	−4.76446e6	−1.15346	−0.576732	−	0.816933i	$-0.695672\pi$
$443$	−4.76446e6	−1.15346	−0.576732	+	0.816933i	$0.695672\pi$
$444$	0	0
$445$	479127.	0.114697
$446$	8.43533e6	2.00801
$447$	0	0
$448$	−8.00589e6	−1.88458
$449$	5.40007e6	1.26411	0.632053	−	0.774926i	$-0.282213\pi$
$449$	5.40007e6	1.26411	0.632053	+	0.774926i	$0.282213\pi$
$450$	0	0
$451$	3.37572e6	0.781493
$452$	28806.1	0.00663190
$453$	0	0
$454$	−7.48767e6	−1.70493
$455$	404692.	0.0916422
$456$	0	0
$457$	2.12752e6	0.476523	0.238261	−	0.971201i	$-0.423423\pi$
$457$	2.12752e6	0.476523	0.238261	+	0.971201i	$0.423423\pi$
$458$	−1.11015e6	−0.247297
$459$	0	0
$460$	−6.33423e6	−1.39572
$461$	−2.80905e6	−0.615612	−0.307806	−	0.951449i	$-0.599595\pi$
$461$	−2.80905e6	−0.615612	−0.307806	+	0.951449i	$0.599595\pi$
$462$	0	0
$463$	−2.85342e6	−0.618605	−0.309302	−	0.950964i	$-0.600096\pi$
$463$	−2.85342e6	−0.618605	−0.309302	+	0.950964i	$0.600096\pi$
$464$	745230.	0.160692
$465$	0	0
$466$	−3.46955e6	−0.740132
$467$	559463.	0.118708	0.0593539	−	0.998237i	$-0.481096\pi$
$467$	559463.	0.118708	0.0593539	+	0.998237i	$0.481096\pi$
$468$	0	0
$469$	−2.87606e6	−0.603762
$470$	3.22289e6	0.672978
$471$	0	0
$472$	181344.	0.0374669
$473$	−1.16838e6	−0.240121
$474$	0	0
$475$	−1.80782e6	−0.367640
$476$	1.30617e7	2.64231
$477$	0	0
$478$	9.48208e6	1.89816
$479$	−5.46747e6	−1.08880	−0.544399	−	0.838826i	$-0.683242\pi$
$479$	−5.46747e6	−1.08880	−0.544399	+	0.838826i	$0.683242\pi$
$480$	0	0
$481$	425517.	0.0838600
$482$	965349.	0.189263
$483$	0	0
$484$	7.98819e6	1.55001
$485$	7.07981e6	1.36668
$486$	0	0
$487$	9.34759e6	1.78598	0.892991	−	0.450074i	$-0.148603\pi$
$487$	9.34759e6	1.78598	0.892991	+	0.450074i	$0.148603\pi$
$488$	−264067.	−0.0501955
$489$	0	0
$490$	1.70018e7	3.19893
$491$	1.04964e7	1.96488	0.982438	−	0.186589i	$-0.0597433\pi$
$491$	1.04964e7	1.96488	0.982438	+	0.186589i	$0.0597433\pi$
$492$	0	0
$493$	−1.33544e6	−0.247460
$494$	−469046.	−0.0864766
$495$	0	0
$496$	8.12730e6	1.48335
$497$	−1.26170e7	−2.29120
$498$	0	0
$499$	5.74842e6	1.03347	0.516734	−	0.856146i	$-0.327148\pi$
$499$	5.74842e6	1.03347	0.516734	+	0.856146i	$0.327148\pi$
$500$	−4.74618e6	−0.849022
$501$	0	0
$502$	1.14361e6	0.202543
$503$	−3.10341e6	−0.546915	−0.273457	−	0.961884i	$-0.588167\pi$
$503$	−3.10341e6	−0.546915	−0.273457	+	0.961884i	$0.588167\pi$
$504$	0	0
$505$	−664953.	−0.116028
$506$	−1.52478e7	−2.64747
$507$	0	0
$508$	6.48453e6	1.11486
$509$	7.92714e6	1.35620	0.678098	−	0.734972i	$-0.262805\pi$
$509$	7.92714e6	1.35620	0.678098	+	0.734972i	$0.262805\pi$
$510$	0	0
$511$	−5.71937e6	−0.968938
$512$	8.43012e6	1.42121
$513$	0	0
$514$	1.59269e6	0.265904
$515$	−715293.	−0.118841
$516$	0	0
$517$	3.96963e6	0.653166
$518$	2.69429e7	4.41185
$519$	0	0
$520$	22423.1	0.00363654
$521$	4.37136e6	0.705541	0.352771	−	0.935710i	$-0.385240\pi$
$521$	4.37136e6	0.705541	0.352771	+	0.935710i	$0.385240\pi$
$522$	0	0
$523$	1.11680e6	0.178534	0.0892671	−	0.996008i	$-0.471548\pi$
$523$	1.11680e6	0.178534	0.0892671	+	0.996008i	$0.471548\pi$
$524$	9.03957e6	1.43820
$525$	0	0
$526$	1.59374e7	2.51162
$527$	−1.45640e7	−2.28430
$528$	0	0
$529$	2.44920e6	0.380527
$530$	469509.	0.0726030
$531$	0	0
$532$	−1.51962e7	−2.32786
$533$	152640.	0.0232728
$534$	0	0
$535$	−2.47453e6	−0.373773
$536$	−159357.	−0.0239584
$537$	0	0
$538$	4.29852e6	0.640271
$539$	2.09411e7	3.10476
$540$	0	0
$541$	−6212.12	−0.000912528 0	−0.000456264	−	1.00000i	$-0.500145\pi$
$541$	−6212.12	−0.000912528 0	−0.000456264		1.00000i	$0.500145\pi$
$542$	−1.36619e7	−1.99763
$543$	0	0
$544$	−1.44159e7	−2.08855
$545$	−926730.	−0.133648
$546$	0	0
$547$	−9.82839e6	−1.40448	−0.702238	−	0.711942i	$-0.747816\pi$
$547$	−9.82839e6	−1.40448	−0.702238	+	0.711942i	$0.747816\pi$
$548$	−4.99549e6	−0.710602
$549$	0	0
$550$	4.56007e6	0.642783
$551$	1.55367e6	0.218011
$552$	0	0
$553$	1.29515e6	0.180097
$554$	110336.	0.0152737
$555$	0	0
$556$	−548398.	−0.0752332
$557$	6.27045e6	0.856369	0.428184	−	0.903691i	$-0.359153\pi$
$557$	6.27045e6	0.856369	0.428184	+	0.903691i	$0.359153\pi$
$558$	0	0
$559$	−52830.4	−0.00715079
$560$	−1.37772e7	−1.85648
$561$	0	0
$562$	−9.07215e6	−1.21163
$563$	6.71129e6	0.892350	0.446175	−	0.894946i	$-0.352786\pi$
$563$	6.71129e6	0.892350	0.446175	+	0.894946i	$0.352786\pi$
$564$	0	0
$565$	54447.3	0.00717555
$566$	−1.23812e7	−1.62450
$567$	0	0
$568$	−699080.	−0.0909193
$569$	7.01697e6	0.908592	0.454296	−	0.890851i	$-0.349891\pi$
$569$	7.01697e6	0.908592	0.454296	+	0.890851i	$0.349891\pi$
$570$	0	0
$571$	1.61627e6	0.207455	0.103727	−	0.994606i	$-0.466923\pi$
$571$	1.61627e6	0.207455	0.103727	+	0.994606i	$0.466923\pi$
$572$	605372.	0.0773628
$573$	0	0
$574$	9.66484e6	1.22438
$575$	−2.65735e6	−0.335181
$576$	0	0
$577$	1.08712e7	1.35937	0.679686	−	0.733503i	$-0.262116\pi$
$577$	1.08712e7	1.35937	0.679686	+	0.733503i	$0.262116\pi$
$578$	1.31014e7	1.63117
$579$	0	0
$580$	−1.62802e6	−0.200951
$581$	1.75600e6	0.215817
$582$	0	0
$583$	578294.	0.0704656
$584$	−316898.	−0.0384493
$585$	0	0
$586$	1.28294e7	1.54334
$587$	−8.65320e6	−1.03653	−0.518264	−	0.855221i	$-0.673422\pi$
$587$	−8.65320e6	−1.03653	−0.518264	+	0.855221i	$0.673422\pi$
$588$	0	0
$589$	1.69439e7	2.01245
$590$	7.51306e6	0.888560
$591$	0	0
$592$	−1.44862e7	−1.69883
$593$	−5.13574e6	−0.599745	−0.299872	−	0.953979i	$-0.596944\pi$
$593$	−5.13574e6	−0.599745	−0.299872	+	0.953979i	$0.596944\pi$
$594$	0	0
$595$	2.46884e7	2.85891
$596$	5.04743e6	0.582043
$597$	0	0
$598$	−689458.	−0.0788415
$599$	−7.56853e6	−0.861875	−0.430937	−	0.902382i	$-0.641817\pi$
$599$	−7.56853e6	−0.861875	−0.430937	+	0.902382i	$0.641817\pi$
$600$	0	0
$601$	2.56891e6	0.290110	0.145055	−	0.989424i	$-0.453664\pi$
$601$	2.56891e6	0.290110	0.145055	+	0.989424i	$0.453664\pi$
$602$	−3.34512e6	−0.376201
$603$	0	0
$604$	1.43554e7	1.60112
$605$	1.50987e7	1.67707
$606$	0	0
$607$	2.62172e6	0.288812	0.144406	−	0.989519i	$-0.453873\pi$
$607$	2.62172e6	0.288812	0.144406	+	0.989519i	$0.453873\pi$
$608$	1.67716e7	1.83999
$609$	0	0
$610$	−1.09403e7	−1.19043
$611$	179494.	0.0194513
$612$	0	0
$613$	537167.	0.0577376	0.0288688	−	0.999583i	$-0.490810\pi$
$613$	537167.	0.0577376	0.0288688	+	0.999583i	$0.490810\pi$
$614$	5.63213e6	0.602909
$615$	0	0
$616$	1.74875e6	0.185685
$617$	−1.15522e7	−1.22166	−0.610830	−	0.791762i	$-0.709164\pi$
$617$	−1.15522e7	−1.22166	−0.610830	+	0.791762i	$0.709164\pi$
$618$	0	0
$619$	2.20752e6	0.231568	0.115784	−	0.993274i	$-0.463062\pi$
$619$	2.20752e6	0.231568	0.115784	+	0.993274i	$0.463062\pi$
$620$	−1.77548e7	−1.85497
$621$	0	0
$622$	8.00408e6	0.829536
$623$	1.68960e6	0.174407
$624$	0	0
$625$	−1.17567e7	−1.20389
$626$	−2.48963e7	−2.53922
$627$	0	0
$628$	−1.45583e7	−1.47303
$629$	2.59589e7	2.61613
$630$	0	0
$631$	−1.43943e7	−1.43919	−0.719594	−	0.694395i	$-0.755672\pi$
$631$	−1.43943e7	−1.43919	−0.719594	+	0.694395i	$0.755672\pi$
$632$	71761.3	0.00714657
$633$	0	0
$634$	−7.32219e6	−0.723465
$635$	1.22566e7	1.20625
$636$	0	0
$637$	946893.	0.0924597
$638$	−3.91897e6	−0.381172
$639$	0	0
$640$	−1.60540e6	−0.154929
$641$	−4.90224e6	−0.471248	−0.235624	−	0.971844i	$-0.575713\pi$
$641$	−4.90224e6	−0.471248	−0.235624	+	0.971844i	$0.575713\pi$
$642$	0	0
$643$	−1.80253e7	−1.71931	−0.859656	−	0.510874i	$-0.829322\pi$
$643$	−1.80253e7	−1.71931	−0.859656	+	0.510874i	$0.829322\pi$
$644$	−2.23371e7	−2.12233
$645$	0	0
$646$	−2.86144e7	−2.69776
$647$	1.24242e7	1.16683	0.583415	−	0.812174i	$-0.301716\pi$
$647$	1.24242e7	1.16683	0.583415	+	0.812174i	$0.301716\pi$
$648$	0	0
$649$	9.25383e6	0.862402
$650$	206192.	0.0191421
$651$	0	0
$652$	−1.91733e7	−1.76635
$653$	1.28553e7	1.17977	0.589887	−	0.807486i	$-0.299172\pi$
$653$	1.28553e7	1.17977	0.589887	+	0.807486i	$0.299172\pi$
$654$	0	0
$655$	1.70860e7	1.55610
$656$	−5.19641e6	−0.471459
$657$	0	0
$658$	1.13652e7	1.02333
$659$	5.18218e6	0.464835	0.232417	−	0.972616i	$-0.425336\pi$
$659$	5.18218e6	0.464835	0.232417	+	0.972616i	$0.425336\pi$
$660$	0	0
$661$	6.47130e6	0.576087	0.288043	−	0.957617i	$-0.406995\pi$
$661$	6.47130e6	0.576087	0.288043	+	0.957617i	$0.406995\pi$
$662$	−1.69436e7	−1.50266
$663$	0	0
$664$	97296.6	0.00856402
$665$	−2.87229e7	−2.51868
$666$	0	0
$667$	2.28375e6	0.198763
$668$	−2.03104e7	−1.76108
$669$	0	0
$670$	−6.60213e6	−0.568194
$671$	−1.34751e7	−1.15539
$672$	0	0
$673$	6.21171e6	0.528657	0.264328	−	0.964433i	$-0.414850\pi$
$673$	6.21171e6	0.528657	0.264328	+	0.964433i	$0.414850\pi$
$674$	−5.24141e6	−0.444425
$675$	0	0
$676$	−1.24220e7	−1.04550
$677$	−1.85078e7	−1.55197	−0.775984	−	0.630752i	$-0.782747\pi$
$677$	−1.85078e7	−1.55197	−0.775984	+	0.630752i	$0.782747\pi$
$678$	0	0
$679$	2.49663e7	2.07817
$680$	1.36793e6	0.113447
$681$	0	0
$682$	−4.27394e7	−3.51858
$683$	4.91590e6	0.403228	0.201614	−	0.979465i	$-0.435381\pi$
$683$	4.91590e6	0.403228	0.201614	+	0.979465i	$0.435381\pi$
$684$	0	0
$685$	−9.44214e6	−0.768854
$686$	2.95491e7	2.39736
$687$	0	0
$688$	1.79854e6	0.144860
$689$	26148.7	0.00209846
$690$	0	0
$691$	−1.07645e7	−0.857630	−0.428815	−	0.903392i	$-0.641069\pi$
$691$	−1.07645e7	−0.857630	−0.428815	+	0.903392i	$0.641069\pi$
$692$	2.85803e6	0.226883
$693$	0	0
$694$	3.74005e6	0.294767
$695$	−1.03655e6	−0.0814004
$696$	0	0
$697$	9.31185e6	0.726029
$698$	2.30769e7	1.79283
$699$	0	0
$700$	6.68023e6	0.515284
$701$	−9.02819e6	−0.693914	−0.346957	−	0.937881i	$-0.612785\pi$
$701$	−9.02819e6	−0.693914	−0.346957	+	0.937881i	$0.612785\pi$
$702$	0	0
$703$	−3.02010e7	−2.30480
$704$	−2.26360e7	−1.72135
$705$	0	0
$706$	6.55248e6	0.494760
$707$	−2.34490e6	−0.176431
$708$	0	0
$709$	4.91059e6	0.366875	0.183438	−	0.983031i	$-0.441277\pi$
$709$	4.91059e6	0.366875	0.183438	+	0.983031i	$0.441277\pi$
$710$	−2.89628e7	−2.15623
$711$	0	0
$712$	93617.2	0.00692079
$713$	2.49061e7	1.83477
$714$	0	0
$715$	1.14423e6	0.0837047
$716$	−2.61336e6	−0.190510
$717$	0	0
$718$	−2.80184e7	−2.02830
$719$	−1.44751e7	−1.04424	−0.522119	−	0.852873i	$-0.674858\pi$
$719$	−1.44751e7	−1.04424	−0.522119	+	0.852873i	$0.674858\pi$
$720$	0	0
$721$	−2.52242e6	−0.180709
$722$	1.32463e7	0.945695
$723$	0	0
$724$	−1.16570e7	−0.826493
$725$	−682989.	−0.0482579
$726$	0	0
$727$	1.55189e7	1.08899	0.544495	−	0.838764i	$-0.316721\pi$
$727$	1.55189e7	1.08899	0.544495	+	0.838764i	$0.316721\pi$
$728$	79073.2	0.00552969
$729$	0	0
$730$	−1.31291e7	−0.911857
$731$	−3.22294e6	−0.223079
$732$	0	0
$733$	1.03870e6	0.0714050	0.0357025	−	0.999362i	$-0.488633\pi$
$733$	1.03870e6	0.0714050	0.0357025	+	0.999362i	$0.488633\pi$
$734$	1.72912e7	1.18463
$735$	0	0
$736$	2.46529e7	1.67754
$737$	−8.13184e6	−0.551467
$738$	0	0
$739$	1.33918e6	0.0902046	0.0451023	−	0.998982i	$-0.485639\pi$
$739$	1.33918e6	0.0902046	0.0451023	+	0.998982i	$0.485639\pi$
$740$	3.16463e7	2.12443
$741$	0	0
$742$	1.65568e6	0.110400
$743$	−1.56518e7	−1.04014	−0.520070	−	0.854124i	$-0.674094\pi$
$743$	−1.56518e7	−1.04014	−0.520070	+	0.854124i	$0.674094\pi$
$744$	0	0
$745$	9.54032e6	0.629756
$746$	1.92009e7	1.26321
$747$	0	0
$748$	3.69310e7	2.41345
$749$	−8.72620e6	−0.568356
$750$	0	0
$751$	1.92274e7	1.24400	0.622002	−	0.783016i	$-0.286320\pi$
$751$	1.92274e7	1.24400	0.622002	+	0.783016i	$0.286320\pi$
$752$	−6.11065e6	−0.394042
$753$	0	0
$754$	−177204.	−0.0113513
$755$	2.71337e7	1.73237
$756$	0	0
$757$	−2.73627e7	−1.73548	−0.867739	−	0.497021i	$-0.834427\pi$
$757$	−2.73627e7	−1.73548	−0.867739	+	0.497021i	$0.834427\pi$
$758$	−4.43254e7	−2.80207
$759$	0	0
$760$	−1.59147e6	−0.0999461
$761$	−5.86301e6	−0.366994	−0.183497	−	0.983020i	$-0.558742\pi$
$761$	−5.86301e6	−0.366994	−0.183497	+	0.983020i	$0.558742\pi$
$762$	0	0
$763$	−3.26803e6	−0.203224
$764$	6.44317e6	0.399361
$765$	0	0
$766$	−2.56585e7	−1.58001
$767$	418430.	0.0256823
$768$	0	0
$769$	−1.36162e7	−0.830308	−0.415154	−	0.909751i	$-0.636272\pi$
$769$	−1.36162e7	−0.830308	−0.415154	+	0.909751i	$0.636272\pi$
$770$	7.24507e7	4.40368
$771$	0	0
$772$	−9.11083e6	−0.550192
$773$	9.44787e6	0.568703	0.284351	−	0.958720i	$-0.408222\pi$
$773$	9.44787e6	0.568703	0.284351	+	0.958720i	$0.408222\pi$
$774$	0	0
$775$	−7.44851e6	−0.445467
$776$	1.38333e6	0.0824655
$777$	0	0
$778$	2.51946e6	0.149231
$779$	−1.08335e7	−0.639627
$780$	0	0
$781$	−3.56735e7	−2.09275
$782$	−4.20608e7	−2.45958
$783$	0	0
$784$	−3.22357e7	−1.87304
$785$	−2.75171e7	−1.59378
$786$	0	0
$787$	2.97678e7	1.71320	0.856602	−	0.515977i	$-0.172571\pi$
$787$	2.97678e7	1.71320	0.856602	+	0.515977i	$0.172571\pi$
$788$	−2.57468e6	−0.147709
$789$	0	0
$790$	2.97307e6	0.169487
$791$	192004.	0.0109111
$792$	0	0
$793$	−609304.	−0.0344073
$794$	−4.01768e6	−0.226164
$795$	0	0
$796$	−2.32956e7	−1.30314
$797$	1.80924e7	1.00891	0.504453	−	0.863439i	$-0.331694\pi$
$797$	1.80924e7	1.00891	0.504453	+	0.863439i	$0.331694\pi$
$798$	0	0
$799$	1.09501e7	0.606810
$800$	−7.37279e6	−0.407293
$801$	0	0
$802$	4.56293e7	2.50500
$803$	−1.61711e7	−0.885013
$804$	0	0
$805$	−4.22201e7	−2.29631
$806$	−1.93254e6	−0.104783
$807$	0	0
$808$	−129926.	−0.00700113
$809$	1.57182e7	0.844366	0.422183	−	0.906511i	$-0.361264\pi$
$809$	1.57182e7	0.844366	0.422183	+	0.906511i	$0.361264\pi$
$810$	0	0
$811$	−1.57562e6	−0.0841202	−0.0420601	−	0.999115i	$-0.513392\pi$
$811$	−1.57562e6	−0.0841202	−0.0420601	+	0.999115i	$0.513392\pi$
$812$	−5.74106e6	−0.305564
$813$	0	0
$814$	7.61791e7	4.02972
$815$	−3.62401e7	−1.91115
$816$	0	0
$817$	3.74962e6	0.196531
$818$	−5.83260e6	−0.304775
$819$	0	0
$820$	1.13520e7	0.589573
$821$	−2.61911e7	−1.35611	−0.678056	−	0.735010i	$-0.737177\pi$
$821$	−2.61911e7	−1.35611	−0.678056	+	0.735010i	$0.737177\pi$
$822$	0	0
$823$	−3.66910e7	−1.88825	−0.944126	−	0.329585i	$-0.893091\pi$
$823$	−3.66910e7	−1.88825	−0.944126	+	0.329585i	$0.893091\pi$
$824$	−139762.	−0.00717086
$825$	0	0
$826$	2.64942e7	1.35114
$827$	−1.38402e7	−0.703685	−0.351843	−	0.936059i	$-0.614445\pi$
$827$	−1.38402e7	−0.703685	−0.351843	+	0.936059i	$0.614445\pi$
$828$	0	0
$829$	−2.83219e7	−1.43132	−0.715660	−	0.698449i	$-0.753874\pi$
$829$	−2.83219e7	−1.43132	−0.715660	+	0.698449i	$0.753874\pi$
$830$	4.03099e6	0.203103
$831$	0	0
$832$	−1.02353e6	−0.0512617
$833$	5.77657e7	2.88441
$834$	0	0
$835$	−3.83894e7	−1.90544
$836$	−4.29661e7	−2.12623
$837$	0	0
$838$	3.19067e7	1.56954
$839$	−2.80231e7	−1.37439	−0.687197	−	0.726471i	$-0.741159\pi$
$839$	−2.80231e7	−1.37439	−0.687197	+	0.726471i	$0.741159\pi$
$840$	0	0
$841$	−1.99242e7	−0.971383
$842$	3.55782e7	1.72943
$843$	0	0
$844$	−1.45362e7	−0.702418
$845$	−2.34792e7	−1.13120
$846$	0	0
$847$	5.32444e7	2.55015
$848$	−890197.	−0.0425105
$849$	0	0
$850$	1.25789e7	0.597164
$851$	−4.43928e7	−2.10130
$852$	0	0
$853$	−2.38300e7	−1.12138	−0.560689	−	0.828027i	$-0.689463\pi$
$853$	−2.38300e7	−1.12138	−0.560689	+	0.828027i	$0.689463\pi$
$854$	−3.85800e7	−1.81016
$855$	0	0
$856$	−483501.	−0.0225534
$857$	3.83129e7	1.78194	0.890971	−	0.454061i	$-0.150025\pi$
$857$	3.83129e7	1.78194	0.890971	+	0.454061i	$0.150025\pi$
$858$	0	0
$859$	2.68650e7	1.24224	0.621118	−	0.783717i	$-0.286679\pi$
$859$	2.68650e7	1.24224	0.621118	+	0.783717i	$0.286679\pi$
$860$	−3.92906e6	−0.181152
$861$	0	0
$862$	−1.01118e7	−0.463510
$863$	1.63558e7	0.747558	0.373779	−	0.927518i	$-0.378062\pi$
$863$	1.63558e7	0.747558	0.373779	+	0.927518i	$0.378062\pi$
$864$	0	0
$865$	5.40206e6	0.245482
$866$	7.86758e6	0.356489
$867$	0	0
$868$	−6.26107e7	−2.82065
$869$	3.66192e6	0.164498
$870$	0	0
$871$	−367696.	−0.0164227
$872$	−181075.	−0.00806431
$873$	0	0
$874$	4.89341e7	2.16687
$875$	−3.16351e7	−1.39685
$876$	0	0
$877$	−2.30604e7	−1.01243	−0.506217	−	0.862406i	$-0.668957\pi$
$877$	−2.30604e7	−1.01243	−0.506217	+	0.862406i	$0.668957\pi$
$878$	−1.07221e7	−0.469402
$879$	0	0
$880$	−3.89539e7	−1.69568
$881$	3.15095e7	1.36774	0.683868	−	0.729606i	$-0.260296\pi$
$881$	3.15095e7	1.36774	0.683868	+	0.729606i	$0.260296\pi$
$882$	0	0
$883$	2.86348e7	1.23593	0.617963	−	0.786208i	$-0.287958\pi$
$883$	2.86348e7	1.23593	0.617963	+	0.786208i	$0.287958\pi$
$884$	1.66991e6	0.0718723
$885$	0	0
$886$	−3.85685e7	−1.65063
$887$	2.45670e7	1.04844	0.524220	−	0.851583i	$-0.324357\pi$
$887$	2.45670e7	1.04844	0.524220	+	0.851583i	$0.324357\pi$
$888$	0	0
$889$	4.32219e7	1.83421
$890$	3.87855e6	0.164132
$891$	0	0
$892$	3.49392e7	1.47028
$893$	−1.27396e7	−0.534596
$894$	0	0
$895$	−4.93961e6	−0.206127
$896$	−5.66130e6	−0.235584
$897$	0	0
$898$	4.37138e7	1.80895
$899$	6.40134e6	0.264163
$900$	0	0
$901$	1.59521e6	0.0654646
$902$	2.73266e7	1.11833
$903$	0	0
$904$	10638.5	0.000432973 0
$905$	−2.20332e7	−0.894245
$906$	0	0
$907$	1.90832e7	0.770254	0.385127	−	0.922864i	$-0.374158\pi$
$907$	1.90832e7	0.770254	0.385127	+	0.922864i	$0.374158\pi$
$908$	−3.10140e7	−1.24837
$909$	0	0
$910$	3.27600e6	0.131141
$911$	3.04616e7	1.21606	0.608032	−	0.793912i	$-0.291959\pi$
$911$	3.04616e7	1.21606	0.608032	+	0.793912i	$0.291959\pi$
$912$	0	0
$913$	4.96497e6	0.197124
$914$	1.72224e7	0.681911
$915$	0	0
$916$	−4.59826e6	−0.181073
$917$	6.02522e7	2.36619
$918$	0	0
$919$	2.73866e7	1.06967	0.534834	−	0.844957i	$-0.320374\pi$
$919$	2.73866e7	1.06967	0.534834	+	0.844957i	$0.320374\pi$
$920$	−2.33933e6	−0.0911218
$921$	0	0
$922$	−2.27394e7	−0.880950
$923$	−1.61304e6	−0.0623221
$924$	0	0
$925$	1.32763e7	0.510179
$926$	−2.30986e7	−0.885232
$927$	0	0
$928$	6.33626e6	0.241525
$929$	2.65451e7	1.00913	0.504563	−	0.863375i	$-0.331654\pi$
$929$	2.65451e7	1.00913	0.504563	+	0.863375i	$0.331654\pi$
$930$	0	0
$931$	−6.72054e7	−2.54115
$932$	−1.43709e7	−0.541932
$933$	0	0
$934$	4.52888e6	0.169873
$935$	6.98046e7	2.61129
$936$	0	0
$937$	−2.34660e7	−0.873151	−0.436576	−	0.899668i	$-0.643809\pi$
$937$	−2.34660e7	−0.873151	−0.436576	+	0.899668i	$0.643809\pi$
$938$	−2.32818e7	−0.863993
$939$	0	0
$940$	1.33492e7	0.492761
$941$	2.18303e7	0.803686	0.401843	−	0.915709i	$-0.368370\pi$
$941$	2.18303e7	0.803686	0.401843	+	0.915709i	$0.368370\pi$
$942$	0	0
$943$	−1.59244e7	−0.583154
$944$	−1.42449e7	−0.520270
$945$	0	0
$946$	−9.45806e6	−0.343617
$947$	−1.87197e7	−0.678305	−0.339152	−	0.940731i	$-0.610140\pi$
$947$	−1.87197e7	−0.678305	−0.339152	+	0.940731i	$0.610140\pi$
$948$	0	0
$949$	−731206.	−0.0263557
$950$	−1.46344e7	−0.526098
$951$	0	0
$952$	4.82390e6	0.172507
$953$	−5.37413e7	−1.91680	−0.958399	−	0.285432i	$-0.907863\pi$
$953$	−5.37413e7	−1.91680	−0.958399	+	0.285432i	$0.907863\pi$
$954$	0	0
$955$	1.21785e7	0.432099
$956$	3.92748e7	1.38986
$957$	0	0
$958$	−4.42594e7	−1.55809
$959$	−3.32969e7	−1.16911
$960$	0	0
$961$	4.11823e7	1.43847
$962$	3.44458e6	0.120005
$963$	0	0
$964$	3.99848e6	0.138581
$965$	−1.72207e7	−0.595295
$966$	0	0
$967$	1.78136e7	0.612611	0.306306	−	0.951933i	$-0.400907\pi$
$967$	1.78136e7	0.612611	0.306306	+	0.951933i	$0.400907\pi$
$968$	2.95016e6	0.101195
$969$	0	0
$970$	5.73114e7	1.95574
$971$	−3.06231e7	−1.04232	−0.521160	−	0.853459i	$-0.674500\pi$
$971$	−3.06231e7	−1.04232	−0.521160	+	0.853459i	$0.674500\pi$
$972$	0	0
$973$	−3.65529e6	−0.123777
$974$	7.56691e7	2.55577
$975$	0	0
$976$	2.07429e7	0.697021
$977$	1.93935e7	0.650011	0.325005	−	0.945712i	$-0.394634\pi$
$977$	1.93935e7	0.650011	0.325005	+	0.945712i	$0.394634\pi$
$978$	0	0
$979$	4.77721e6	0.159301
$980$	7.04217e7	2.34229
$981$	0	0
$982$	8.49685e7	2.81177
$983$	−3.41089e7	−1.12586	−0.562929	−	0.826505i	$-0.690325\pi$
$983$	−3.41089e7	−1.12586	−0.562929	+	0.826505i	$0.690325\pi$
$984$	0	0
$985$	−4.86650e6	−0.159818
$986$	−1.08104e7	−0.354120
$987$	0	0
$988$	−1.94279e6	−0.0633190
$989$	5.51162e6	0.179179
$990$	0	0
$991$	−1.15321e7	−0.373013	−0.186507	−	0.982454i	$-0.559717\pi$
$991$	−1.15321e7	−0.373013	−0.186507	+	0.982454i	$0.559717\pi$
$992$	6.91017e7	2.22951
$993$	0	0
$994$	−1.02135e8	−3.27875
$995$	−4.40319e7	−1.40997
$996$	0	0
$997$	3.45745e7	1.10159	0.550793	−	0.834642i	$-0.314325\pi$
$997$	3.45745e7	1.10159	0.550793	+	0.834642i	$0.314325\pi$
$998$	4.65337e7	1.47891
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.6.a.c.1.7		8
3.2	odd	2		43.6.a.a.1.2	✓	8
12.11	even	2		688.6.a.e.1.2		8
15.14	odd	2		1075.6.a.a.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.a.1.2	✓	8	3.2	odd	2
387.6.a.c.1.7		8	1.1	even	1	trivial
688.6.a.e.1.2		8	12.11	even	2
1075.6.a.a.1.7		8	15.14	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +O(q^{10})\)	\(q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +513.028 q^{10} +631.897 q^{11} +28.5724 q^{13} +1809.15 q^{14} -972.709 q^{16} +1743.07 q^{17} -2027.92 q^{19} +2124.97 q^{20} +5115.23 q^{22} -2980.86 q^{23} +891.469 q^{25} +231.295 q^{26} +7493.51 q^{28} -766.139 q^{29} -8355.33 q^{31} -8270.38 q^{32} +14110.3 q^{34} +14163.7 q^{35} +14892.6 q^{37} -16416.1 q^{38} +784.783 q^{40} +5342.20 q^{41} -1849.00 q^{43} +21187.3 q^{44} -24130.2 q^{46} +6282.09 q^{47} +33140.1 q^{49} +7216.48 q^{50} +958.024 q^{52} +915.172 q^{53} +40046.8 q^{55} +2767.47 q^{56} -6201.92 q^{58} +14644.5 q^{59} -21324.9 q^{61} -67636.7 q^{62} -35822.4 q^{64} +1810.79 q^{65} -12868.9 q^{67} +58444.8 q^{68} +114656. q^{70} -56454.6 q^{71} -25591.3 q^{73} +120556. q^{74} -67995.4 q^{76} +141222. q^{77} +5795.13 q^{79} -61646.1 q^{80} +43245.4 q^{82} +7857.24 q^{83} +110468. q^{85} -14967.7 q^{86} +7824.80 q^{88} +7560.11 q^{89} +6385.60 q^{91} -99947.5 q^{92} +50853.8 q^{94} -128520. q^{95} +111712. q^{97} +268271. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8	\( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) \) 8 * q + 12 * q^2 + 122 * q^4 + 212 * q^5 - 136 * q^7 + 666 * q^8 - 617 * q^10 + 532 * q^11 - 2492 * q^13 + 4240 * q^14 + 1882 * q^16 + 2534 * q^17 - 1678 * q^19 + 2607 * q^20 + 11502 * q^22 + 2488 * q^23 + 4378 * q^25 - 4586 * q^26 + 18640 * q^28 + 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 30007 * q^34 - 5640 * q^35 - 3772 * q^37 + 6559 * q^38 + 14869 * q^40 + 10698 * q^41 - 14792 * q^43 + 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 7188 * q^49 - 26877 * q^50 - 60736 * q^52 + 62352 * q^53 - 49552 * q^55 + 144528 * q^56 + 52951 * q^58 + 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 153858 * q^64 + 5000 * q^65 + 27784 * q^67 - 40507 * q^68 + 185910 * q^70 + 9504 * q^71 + 14260 * q^73 + 15239 * q^74 + 1279 * q^76 + 218140 * q^77 + 160248 * q^79 + 1291 * q^80 - 47781 * q^82 + 77176 * q^83 + 141096 * q^85 - 22188 * q^86 + 129544 * q^88 + 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 248737 * q^94 - 135884 * q^95 + 144742 * q^97 + 292244 * q^98

Introduction

L-functions

Modular forms

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Database

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