LMFDB - Embedded newform 980.6.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,6,Mod(1,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, 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("980.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	980.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:	$157.176143417$
Analytic rank:	$0$
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} - x^{6} - 954x^{5} - 1014x^{4} + 229389x^{3} - 141213x^{2} - 15115156x + 53924500 $ x^7 - x^6 - 954x^5 - 1014x^4 + 229389x^3 - 141213x^2 - 15115156*x + 53924500
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.97812$ of defining polynomial
Character	$\chi$	$=$	980.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.978125 q^{3} -25.0000 q^{5} -242.043 q^{9} +O(q^{10})$	$q+0.978125 q^{3} -25.0000 q^{5} -242.043 q^{9} -221.105 q^{11} -1064.81 q^{13} -24.4531 q^{15} -1727.93 q^{17} -2239.73 q^{19} -3657.37 q^{23} +625.000 q^{25} -474.433 q^{27} +1946.84 q^{29} +88.0515 q^{31} -216.268 q^{33} -13412.0 q^{37} -1041.51 q^{39} +12293.0 q^{41} +3861.54 q^{43} +6051.08 q^{45} -1402.43 q^{47} -1690.13 q^{51} +18493.8 q^{53} +5527.62 q^{55} -2190.73 q^{57} +35406.3 q^{59} -38330.1 q^{61} +26620.1 q^{65} -31257.1 q^{67} -3577.36 q^{69} -24787.2 q^{71} -74336.5 q^{73} +611.328 q^{75} +67550.4 q^{79} +58352.5 q^{81} -16697.0 q^{83} +43198.2 q^{85} +1904.25 q^{87} -18688.8 q^{89} +86.1253 q^{93} +55993.2 q^{95} -48997.0 q^{97} +53517.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 27 q^{3} - 175 q^{5} + 312 q^{9}+O(q^{10}) $ 7 * q - 27 * q^3 - 175 * q^5 + 312 * q^9	$ 7 q - 27 q^{3} - 175 q^{5} + 312 q^{9} + 590 q^{11} - 854 q^{13} + 675 q^{15} - 1328 q^{17} - 2990 q^{19} + 4089 q^{23} + 4375 q^{25} - 4281 q^{27} - 3063 q^{29} - 5740 q^{31} + 1470 q^{33} + 12218 q^{37} - 1560 q^{39} - 8477 q^{41} - 9195 q^{43} - 7800 q^{45} - 28652 q^{47} + 41244 q^{51} + 24932 q^{53} - 14750 q^{55} - 62934 q^{57} - 38242 q^{59} - 61299 q^{61} + 21350 q^{65} + 24913 q^{67} - 86151 q^{69} - 36170 q^{71} - 94780 q^{73} - 16875 q^{75} - 594 q^{79} - 53265 q^{81} - 113585 q^{83} + 33200 q^{85} - 43875 q^{87} - 36823 q^{89} - 21828 q^{93} + 74750 q^{95} - 257950 q^{97} + 378432 q^{99}+O(q^{100}) $ 7 * q - 27 * q^3 - 175 * q^5 + 312 * q^9 + 590 * q^11 - 854 * q^13 + 675 * q^15 - 1328 * q^17 - 2990 * q^19 + 4089 * q^23 + 4375 * q^25 - 4281 * q^27 - 3063 * q^29 - 5740 * q^31 + 1470 * q^33 + 12218 * q^37 - 1560 * q^39 - 8477 * q^41 - 9195 * q^43 - 7800 * q^45 - 28652 * q^47 + 41244 * q^51 + 24932 * q^53 - 14750 * q^55 - 62934 * q^57 - 38242 * q^59 - 61299 * q^61 + 21350 * q^65 + 24913 * q^67 - 86151 * q^69 - 36170 * q^71 - 94780 * q^73 - 16875 * q^75 - 594 * q^79 - 53265 * q^81 - 113585 * q^83 + 33200 * q^85 - 43875 * q^87 - 36823 * q^89 - 21828 * q^93 + 74750 * q^95 - 257950 * q^97 + 378432 * 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
$3$	0.978125	0.0627467	0.0313734	−	0.999508i	$-0.490012\pi$
$3$	0.978125	0.0627467	0.0313734	+	0.999508i	$0.490012\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−242.043	−0.996063
$10$	0	0
$11$	−221.105	−0.550956	−0.275478	−	0.961307i	$-0.588836\pi$
$11$	−221.105	−0.550956	−0.275478	+	0.961307i	$0.588836\pi$
$12$	0	0
$13$	−1064.81	−1.74748	−0.873739	−	0.486395i	$-0.838312\pi$
$13$	−1064.81	−1.74748	−0.873739	+	0.486395i	$0.838312\pi$
$14$	0	0
$15$	−24.4531	−0.0280612
$16$	0	0
$17$	−1727.93	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$17$	−1727.93	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$18$	0	0
$19$	−2239.73	−1.42335	−0.711674	−	0.702510i	$-0.752063\pi$
$19$	−2239.73	−1.42335	−0.711674	+	0.702510i	$0.752063\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3657.37	−1.44161	−0.720807	−	0.693136i	$-0.756229\pi$
$23$	−3657.37	−1.44161	−0.720807	+	0.693136i	$0.756229\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−474.433	−0.125246
$28$	0	0
$29$	1946.84	0.429867	0.214934	−	0.976629i	$-0.431047\pi$
$29$	1946.84	0.429867	0.214934	+	0.976629i	$0.431047\pi$
$30$	0	0
$31$	88.0515	0.0164563	0.00822815	−	0.999966i	$-0.497381\pi$
$31$	88.0515	0.0164563	0.00822815	+	0.999966i	$0.497381\pi$
$32$	0	0
$33$	−216.268	−0.0345707
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13412.0	−1.61061	−0.805305	−	0.592860i	$-0.797999\pi$
$37$	−13412.0	−1.61061	−0.805305	+	0.592860i	$0.797999\pi$
$38$	0	0
$39$	−1041.51	−0.109649
$40$	0	0
$41$	12293.0	1.14208	0.571042	−	0.820921i	$-0.306539\pi$
$41$	12293.0	1.14208	0.571042	+	0.820921i	$0.306539\pi$
$42$	0	0
$43$	3861.54	0.318485	0.159242	−	0.987240i	$-0.449095\pi$
$43$	3861.54	0.318485	0.159242	+	0.987240i	$0.449095\pi$
$44$	0	0
$45$	6051.08	0.445453
$46$	0	0
$47$	−1402.43	−0.0926052	−0.0463026	−	0.998927i	$-0.514744\pi$
$47$	−1402.43	−0.0926052	−0.0463026	+	0.998927i	$0.514744\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1690.13	−0.0909902
$52$	0	0
$53$	18493.8	0.904351	0.452176	−	0.891929i	$-0.350648\pi$
$53$	18493.8	0.904351	0.452176	+	0.891929i	$0.350648\pi$
$54$	0	0
$55$	5527.62	0.246395
$56$	0	0
$57$	−2190.73	−0.0893104
$58$	0	0
$59$	35406.3	1.32419	0.662095	−	0.749420i	$-0.269668\pi$
$59$	35406.3	1.32419	0.662095	+	0.749420i	$0.269668\pi$
$60$	0	0
$61$	−38330.1	−1.31891	−0.659455	−	0.751744i	$-0.729213\pi$
$61$	−38330.1	−1.31891	−0.659455	+	0.751744i	$0.729213\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26620.1	0.781496
$66$	0	0
$67$	−31257.1	−0.850671	−0.425336	−	0.905036i	$-0.639844\pi$
$67$	−31257.1	−0.850671	−0.425336	+	0.905036i	$0.639844\pi$
$68$	0	0
$69$	−3577.36	−0.0904565
$70$	0	0
$71$	−24787.2	−0.583555	−0.291778	−	0.956486i	$-0.594247\pi$
$71$	−24787.2	−0.583555	−0.291778	+	0.956486i	$0.594247\pi$
$72$	0	0
$73$	−74336.5	−1.63266	−0.816329	−	0.577587i	$-0.803994\pi$
$73$	−74336.5	−1.63266	−0.816329	+	0.577587i	$0.803994\pi$
$74$	0	0
$75$	611.328	0.0125493
$76$	0	0
$77$	0	0
$78$	0	0
$79$	67550.4	1.21776	0.608878	−	0.793264i	$-0.291620\pi$
$79$	67550.4	1.21776	0.608878	+	0.793264i	$0.291620\pi$
$80$	0	0
$81$	58352.5	0.988204
$82$	0	0
$83$	−16697.0	−0.266037	−0.133019	−	0.991114i	$-0.542467\pi$
$83$	−16697.0	−0.266037	−0.133019	+	0.991114i	$0.542467\pi$
$84$	0	0
$85$	43198.2	0.648513
$86$	0	0
$87$	1904.25	0.0269727
$88$	0	0
$89$	−18688.8	−0.250095	−0.125048	−	0.992151i	$-0.539908\pi$
$89$	−18688.8	−0.250095	−0.125048	+	0.992151i	$0.539908\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	86.1253	0.00103258
$94$	0	0
$95$	55993.2	0.636541
$96$	0	0
$97$	−48997.0	−0.528738	−0.264369	−	0.964422i	$-0.585164\pi$
$97$	−48997.0	−0.528738	−0.264369	+	0.964422i	$0.585164\pi$
$98$	0	0
$99$	53517.0	0.548787
$100$	0	0
$101$	−132421.	−1.29168	−0.645840	−	0.763473i	$-0.723493\pi$
$101$	−132421.	−1.29168	−0.645840	+	0.763473i	$0.723493\pi$
$102$	0	0
$103$	182740.	1.69723	0.848615	−	0.529010i	$-0.177437\pi$
$103$	182740.	1.69723	0.848615	+	0.529010i	$0.177437\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2911.90	−0.0245876	−0.0122938	−	0.999924i	$-0.503913\pi$
$107$	−2911.90	−0.0245876	−0.0122938	+	0.999924i	$0.503913\pi$
$108$	0	0
$109$	5811.83	0.0468540	0.0234270	−	0.999726i	$-0.492542\pi$
$109$	5811.83	0.0468540	0.0234270	+	0.999726i	$0.492542\pi$
$110$	0	0
$111$	−13118.6	−0.101061
$112$	0	0
$113$	−200359.	−1.47609	−0.738047	−	0.674750i	$-0.764252\pi$
$113$	−200359.	−1.47609	−0.738047	+	0.674750i	$0.764252\pi$
$114$	0	0
$115$	91434.2	0.644709
$116$	0	0
$117$	257729.	1.74060
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−112164.	−0.696448
$122$	0	0
$123$	12024.1	0.0716620
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	90848.6	0.499815	0.249907	−	0.968270i	$-0.419600\pi$
$127$	90848.6	0.499815	0.249907	+	0.968270i	$0.419600\pi$
$128$	0	0
$129$	3777.06	0.0199839
$130$	0	0
$131$	−23754.9	−0.120941	−0.0604706	−	0.998170i	$-0.519260\pi$
$131$	−23754.9	−0.120941	−0.0604706	+	0.998170i	$0.519260\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11860.8	0.0560119
$136$	0	0
$137$	−230944.	−1.05125	−0.525625	−	0.850716i	$-0.676169\pi$
$137$	−230944.	−1.05125	−0.525625	+	0.850716i	$0.676169\pi$
$138$	0	0
$139$	−312412.	−1.37148	−0.685742	−	0.727844i	$-0.740522\pi$
$139$	−312412.	−1.37148	−0.685742	+	0.727844i	$0.740522\pi$
$140$	0	0
$141$	−1371.75	−0.00581067
$142$	0	0
$143$	235434.	0.962783
$144$	0	0
$145$	−48670.9	−0.192242
$146$	0	0
$147$	0	0
$148$	0	0
$149$	113896.	0.420283	0.210141	−	0.977671i	$-0.432608\pi$
$149$	113896.	0.420283	0.210141	+	0.977671i	$0.432608\pi$
$150$	0	0
$151$	395350.	1.41104	0.705520	−	0.708690i	$-0.250713\pi$
$151$	395350.	1.41104	0.705520	+	0.708690i	$0.250713\pi$
$152$	0	0
$153$	418234.	1.44441
$154$	0	0
$155$	−2201.29	−0.00735948
$156$	0	0
$157$	−424588.	−1.37473	−0.687367	−	0.726310i	$-0.741234\pi$
$157$	−424588.	−1.37473	−0.687367	+	0.726310i	$0.741234\pi$
$158$	0	0
$159$	18089.3	0.0567451
$160$	0	0
$161$	0	0
$162$	0	0
$163$	222660.	0.656408	0.328204	−	0.944607i	$-0.393557\pi$
$163$	222660.	0.656408	0.328204	+	0.944607i	$0.393557\pi$
$164$	0	0
$165$	5406.70	0.0154605
$166$	0	0
$167$	32195.9	0.0893326	0.0446663	−	0.999002i	$-0.485778\pi$
$167$	32195.9	0.0893326	0.0446663	+	0.999002i	$0.485778\pi$
$168$	0	0
$169$	762517.	2.05368
$170$	0	0
$171$	542111.	1.41774
$172$	0	0
$173$	−75370.2	−0.191463	−0.0957314	−	0.995407i	$-0.530519\pi$
$173$	−75370.2	−0.191463	−0.0957314	+	0.995407i	$0.530519\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	34631.7	0.0830886
$178$	0	0
$179$	399136.	0.931084	0.465542	−	0.885026i	$-0.345859\pi$
$179$	399136.	0.931084	0.465542	+	0.885026i	$0.345859\pi$
$180$	0	0
$181$	−500148.	−1.13476	−0.567378	−	0.823458i	$-0.692042\pi$
$181$	−500148.	−1.13476	−0.567378	+	0.823458i	$0.692042\pi$
$182$	0	0
$183$	−37491.6	−0.0827573
$184$	0	0
$185$	335301.	0.720287
$186$	0	0
$187$	382054.	0.798951
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−933954.	−1.85243	−0.926215	−	0.376995i	$-0.876957\pi$
$191$	−933954.	−1.85243	−0.926215	+	0.376995i	$0.876957\pi$
$192$	0	0
$193$	−417528.	−0.806850	−0.403425	−	0.915013i	$-0.632180\pi$
$193$	−417528.	−0.806850	−0.403425	+	0.915013i	$0.632180\pi$
$194$	0	0
$195$	26037.8	0.0490363
$196$	0	0
$197$	870857.	1.59875	0.799376	−	0.600831i	$-0.205163\pi$
$197$	870857.	1.59875	0.799376	+	0.600831i	$0.205163\pi$
$198$	0	0
$199$	−940950.	−1.68436	−0.842178	−	0.539200i	$-0.818727\pi$
$199$	−940950.	−1.68436	−0.842178	+	0.539200i	$0.818727\pi$
$200$	0	0
$201$	−30573.3	−0.0533768
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−307325.	−0.510755
$206$	0	0
$207$	885241.	1.43594
$208$	0	0
$209$	495215.	0.784202
$210$	0	0
$211$	529745.	0.819145	0.409573	−	0.912278i	$-0.365678\pi$
$211$	529745.	0.819145	0.409573	+	0.912278i	$0.365678\pi$
$212$	0	0
$213$	−24245.0	−0.0366162
$214$	0	0
$215$	−96538.4	−0.142431
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−72710.4	−0.102444
$220$	0	0
$221$	1.83991e6	2.53405
$222$	0	0
$223$	−137809.	−0.185573	−0.0927867	−	0.995686i	$-0.529577\pi$
$223$	−137809.	−0.185573	−0.0927867	+	0.995686i	$0.529577\pi$
$224$	0	0
$225$	−151277.	−0.199213
$226$	0	0
$227$	−1.24118e6	−1.59871	−0.799355	−	0.600859i	$-0.794825\pi$
$227$	−1.24118e6	−1.59871	−0.799355	+	0.600859i	$0.794825\pi$
$228$	0	0
$229$	635935.	0.801353	0.400677	−	0.916220i	$-0.368775\pi$
$229$	635935.	0.801353	0.400677	+	0.916220i	$0.368775\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	846201.	1.02114	0.510568	−	0.859837i	$-0.329435\pi$
$233$	846201.	1.02114	0.510568	+	0.859837i	$0.329435\pi$
$234$	0	0
$235$	35060.7	0.0414143
$236$	0	0
$237$	66072.7	0.0764102
$238$	0	0
$239$	−1.44232e6	−1.63330	−0.816651	−	0.577132i	$-0.804172\pi$
$239$	−1.44232e6	−1.63330	−0.816651	+	0.577132i	$0.804172\pi$
$240$	0	0
$241$	−288560.	−0.320032	−0.160016	−	0.987114i	$-0.551155\pi$
$241$	−288560.	−0.320032	−0.160016	+	0.987114i	$0.551155\pi$
$242$	0	0
$243$	172363.	0.187253
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.38487e6	2.48727
$248$	0	0
$249$	−16331.7	−0.0166930
$250$	0	0
$251$	−1.45550e6	−1.45824	−0.729121	−	0.684385i	$-0.760071\pi$
$251$	−1.45550e6	−1.45824	−0.729121	+	0.684385i	$0.760071\pi$
$252$	0	0
$253$	808662.	0.794265
$254$	0	0
$255$	42253.2	0.0406921
$256$	0	0
$257$	998610.	0.943111	0.471556	−	0.881836i	$-0.343693\pi$
$257$	998610.	0.943111	0.471556	+	0.881836i	$0.343693\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−471218.	−0.428175
$262$	0	0
$263$	1.52857e6	1.36269	0.681344	−	0.731963i	$-0.261396\pi$
$263$	1.52857e6	1.36269	0.681344	+	0.731963i	$0.261396\pi$
$264$	0	0
$265$	−462346.	−0.404438
$266$	0	0
$267$	−18279.9	−0.0156927
$268$	0	0
$269$	−404859.	−0.341133	−0.170566	−	0.985346i	$-0.554560\pi$
$269$	−404859.	−0.341133	−0.170566	+	0.985346i	$0.554560\pi$
$270$	0	0
$271$	−86061.5	−0.0711846	−0.0355923	−	0.999366i	$-0.511332\pi$
$271$	−86061.5	−0.0711846	−0.0355923	+	0.999366i	$0.511332\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−138191.	−0.110191
$276$	0	0
$277$	−1.46525e6	−1.14739	−0.573695	−	0.819069i	$-0.694491\pi$
$277$	−1.46525e6	−1.14739	−0.573695	+	0.819069i	$0.694491\pi$
$278$	0	0
$279$	−21312.3	−0.0163915
$280$	0	0
$281$	375945.	0.284026	0.142013	−	0.989865i	$-0.454642\pi$
$281$	375945.	0.284026	0.142013	+	0.989865i	$0.454642\pi$
$282$	0	0
$283$	269361.	0.199925	0.0999627	−	0.994991i	$-0.468128\pi$
$283$	269361.	0.199925	0.0999627	+	0.994991i	$0.468128\pi$
$284$	0	0
$285$	54768.3	0.0399408
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.56588e6	1.10284
$290$	0	0
$291$	−47925.2	−0.0331765
$292$	0	0
$293$	−2.17033e6	−1.47692	−0.738459	−	0.674298i	$-0.764446\pi$
$293$	−2.17033e6	−1.47692	−0.738459	+	0.674298i	$0.764446\pi$
$294$	0	0
$295$	−885157.	−0.592196
$296$	0	0
$297$	104899.	0.0690052
$298$	0	0
$299$	3.89438e6	2.51919
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−129525.	−0.0810487
$304$	0	0
$305$	958252.	0.589834
$306$	0	0
$307$	−1.01798e6	−0.616441	−0.308221	−	0.951315i	$-0.599734\pi$
$307$	−1.01798e6	−0.616441	−0.308221	+	0.951315i	$0.599734\pi$
$308$	0	0
$309$	178743.	0.106496
$310$	0	0
$311$	1.76210e6	1.03307	0.516536	−	0.856265i	$-0.327221\pi$
$311$	1.76210e6	1.03307	0.516536	+	0.856265i	$0.327221\pi$
$312$	0	0
$313$	1.93315e6	1.11533	0.557666	−	0.830065i	$-0.311697\pi$
$313$	1.93315e6	1.11533	0.557666	+	0.830065i	$0.311697\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.53458e6	−0.857714	−0.428857	−	0.903373i	$-0.641083\pi$
$317$	−1.53458e6	−0.857714	−0.428857	+	0.903373i	$0.641083\pi$
$318$	0	0
$319$	−430455.	−0.236838
$320$	0	0
$321$	−2848.20	−0.00154279
$322$	0	0
$323$	3.87009e6	2.06402
$324$	0	0
$325$	−665503.	−0.349496
$326$	0	0
$327$	5684.70	0.00293994
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.25484e6	0.629530	0.314765	−	0.949170i	$-0.398074\pi$
$331$	1.25484e6	0.629530	0.314765	+	0.949170i	$0.398074\pi$
$332$	0	0
$333$	3.24629e6	1.60427
$334$	0	0
$335$	781427.	0.380432
$336$	0	0
$337$	−345146.	−0.165549	−0.0827747	−	0.996568i	$-0.526378\pi$
$337$	−345146.	−0.165549	−0.0827747	+	0.996568i	$0.526378\pi$
$338$	0	0
$339$	−195977.	−0.0926200
$340$	0	0
$341$	−19468.6	−0.00906670
$342$	0	0
$343$	0	0
$344$	0	0
$345$	89434.0	0.0404534
$346$	0	0
$347$	2.24292e6	0.999975	0.499988	−	0.866033i	$-0.333338\pi$
$347$	2.24292e6	0.999975	0.499988	+	0.866033i	$0.333338\pi$
$348$	0	0
$349$	−326684.	−0.143570	−0.0717850	−	0.997420i	$-0.522870\pi$
$349$	−326684.	−0.143570	−0.0717850	+	0.997420i	$0.522870\pi$
$350$	0	0
$351$	505178.	0.218865
$352$	0	0
$353$	−2.57212e6	−1.09864	−0.549318	−	0.835613i	$-0.685112\pi$
$353$	−2.57212e6	−1.09864	−0.549318	+	0.835613i	$0.685112\pi$
$354$	0	0
$355$	619681.	0.260974
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4.77516e6	−1.95547	−0.977736	−	0.209839i	$-0.932706\pi$
$359$	−4.77516e6	−1.95547	−0.977736	+	0.209839i	$0.932706\pi$
$360$	0	0
$361$	2.54028e6	1.02592
$362$	0	0
$363$	−109710.	−0.0436998
$364$	0	0
$365$	1.85841e6	0.730147
$366$	0	0
$367$	−2.22919e6	−0.863936	−0.431968	−	0.901889i	$-0.642181\pi$
$367$	−2.22919e6	−0.863936	−0.431968	+	0.901889i	$0.642181\pi$
$368$	0	0
$369$	−2.97544e6	−1.13759
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−125608.	−0.0467460	−0.0233730	−	0.999727i	$-0.507441\pi$
$373$	−125608.	−0.0467460	−0.0233730	+	0.999727i	$0.507441\pi$
$374$	0	0
$375$	−15283.2	−0.00561224
$376$	0	0
$377$	−2.07300e6	−0.751183
$378$	0	0
$379$	3.05381e6	1.09205	0.546027	−	0.837767i	$-0.316140\pi$
$379$	3.05381e6	1.09205	0.546027	+	0.837767i	$0.316140\pi$
$380$	0	0
$381$	88861.2	0.0313617
$382$	0	0
$383$	1.15456e6	0.402180	0.201090	−	0.979573i	$-0.435552\pi$
$383$	1.15456e6	0.402180	0.201090	+	0.979573i	$0.435552\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−934659.	−0.317231
$388$	0	0
$389$	3.75994e6	1.25981	0.629907	−	0.776670i	$-0.283093\pi$
$389$	3.75994e6	1.25981	0.629907	+	0.776670i	$0.283093\pi$
$390$	0	0
$391$	6.31967e6	2.09051
$392$	0	0
$393$	−23235.2	−0.00758867
$394$	0	0
$395$	−1.68876e6	−0.544597
$396$	0	0
$397$	−630948.	−0.200917	−0.100459	−	0.994941i	$-0.532031\pi$
$397$	−630948.	−0.200917	−0.100459	+	0.994941i	$0.532031\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.90926e6	1.83515	0.917576	−	0.397561i	$-0.130143\pi$
$401$	5.90926e6	1.83515	0.917576	+	0.397561i	$0.130143\pi$
$402$	0	0
$403$	−93757.7	−0.0287570
$404$	0	0
$405$	−1.45881e6	−0.441938
$406$	0	0
$407$	2.96547e6	0.887375
$408$	0	0
$409$	632268.	0.186893	0.0934465	−	0.995624i	$-0.470212\pi$
$409$	632268.	0.186893	0.0934465	+	0.995624i	$0.470212\pi$
$410$	0	0
$411$	−225892.	−0.0659625
$412$	0	0
$413$	0	0
$414$	0	0
$415$	417424.	0.118975
$416$	0	0
$417$	−305578.	−0.0860561
$418$	0	0
$419$	−1.24571e6	−0.346643	−0.173321	−	0.984865i	$-0.555450\pi$
$419$	−1.24571e6	−0.346643	−0.173321	+	0.984865i	$0.555450\pi$
$420$	0	0
$421$	−145810.	−0.0400943	−0.0200471	−	0.999799i	$-0.506382\pi$
$421$	−145810.	−0.0400943	−0.0200471	+	0.999799i	$0.506382\pi$
$422$	0	0
$423$	339448.	0.0922406
$424$	0	0
$425$	−1.07996e6	−0.290024
$426$	0	0
$427$	0	0
$428$	0	0
$429$	230283.	0.0604115
$430$	0	0
$431$	−3.95635e6	−1.02589	−0.512945	−	0.858421i	$-0.671446\pi$
$431$	−3.95635e6	−1.02589	−0.512945	+	0.858421i	$0.671446\pi$
$432$	0	0
$433$	1.52292e6	0.390353	0.195177	−	0.980768i	$-0.437472\pi$
$433$	1.52292e6	0.390353	0.195177	+	0.980768i	$0.437472\pi$
$434$	0	0
$435$	−47606.2	−0.0120626
$436$	0	0
$437$	8.19151e6	2.05192
$438$	0	0
$439$	5.33251e6	1.32060	0.660299	−	0.751003i	$-0.270430\pi$
$439$	5.33251e6	1.32060	0.660299	+	0.751003i	$0.270430\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.91375e6	−1.18961	−0.594804	−	0.803871i	$-0.702770\pi$
$443$	−4.91375e6	−1.18961	−0.594804	+	0.803871i	$0.702770\pi$
$444$	0	0
$445$	467219.	0.111846
$446$	0	0
$447$	111404.	0.0263714
$448$	0	0
$449$	1.22232e6	0.286134	0.143067	−	0.989713i	$-0.454304\pi$
$449$	1.22232e6	0.286134	0.143067	+	0.989713i	$0.454304\pi$
$450$	0	0
$451$	−2.71804e6	−0.629238
$452$	0	0
$453$	386702.	0.0885381
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.22483e6	0.274338	0.137169	−	0.990548i	$-0.456200\pi$
$457$	1.22483e6	0.274338	0.137169	+	0.990548i	$0.456200\pi$
$458$	0	0
$459$	819786.	0.181622
$460$	0	0
$461$	−463384.	−0.101552	−0.0507760	−	0.998710i	$-0.516169\pi$
$461$	−463384.	−0.101552	−0.0507760	+	0.998710i	$0.516169\pi$
$462$	0	0
$463$	−3.09945e6	−0.671943	−0.335972	−	0.941872i	$-0.609065\pi$
$463$	−3.09945e6	−0.671943	−0.335972	+	0.941872i	$0.609065\pi$
$464$	0	0
$465$	−2153.13	−0.000461783 0
$466$	0	0
$467$	−1.26464e6	−0.268333	−0.134167	−	0.990959i	$-0.542836\pi$
$467$	−1.26464e6	−0.268333	−0.134167	+	0.990959i	$0.542836\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−415300.	−0.0862601
$472$	0	0
$473$	−853805.	−0.175471
$474$	0	0
$475$	−1.39983e6	−0.284670
$476$	0	0
$477$	−4.47631e6	−0.900791
$478$	0	0
$479$	2.53108e6	0.504042	0.252021	−	0.967722i	$-0.418905\pi$
$479$	2.53108e6	0.504042	0.252021	+	0.967722i	$0.418905\pi$
$480$	0	0
$481$	1.42812e7	2.81451
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.22493e6	0.236459
$486$	0	0
$487$	2.99555e6	0.572340	0.286170	−	0.958179i	$-0.407618\pi$
$487$	2.99555e6	0.572340	0.286170	+	0.958179i	$0.407618\pi$
$488$	0	0
$489$	217790.	0.0411875
$490$	0	0
$491$	−143990.	−0.0269543	−0.0134772	−	0.999909i	$-0.504290\pi$
$491$	−143990.	−0.0269543	−0.0134772	+	0.999909i	$0.504290\pi$
$492$	0	0
$493$	−3.36399e6	−0.623358
$494$	0	0
$495$	−1.33792e6	−0.245425
$496$	0	0
$497$	0	0
$498$	0	0
$499$	6.95831e6	1.25099	0.625493	−	0.780230i	$-0.284898\pi$
$499$	6.95831e6	1.25099	0.625493	+	0.780230i	$0.284898\pi$
$500$	0	0
$501$	31491.6	0.00560533
$502$	0	0
$503$	551360.	0.0971662	0.0485831	−	0.998819i	$-0.484529\pi$
$503$	551360.	0.0971662	0.0485831	+	0.998819i	$0.484529\pi$
$504$	0	0
$505$	3.31054e6	0.577657
$506$	0	0
$507$	745837.	0.128862
$508$	0	0
$509$	−8.97190e6	−1.53493	−0.767467	−	0.641088i	$-0.778483\pi$
$509$	−8.97190e6	−1.53493	−0.767467	+	0.641088i	$0.778483\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1.06260e6	0.178269
$514$	0	0
$515$	−4.56850e6	−0.759025
$516$	0	0
$517$	310083.	0.0510214
$518$	0	0
$519$	−73721.5	−0.0120137
$520$	0	0
$521$	−1.10741e7	−1.78737	−0.893684	−	0.448697i	$-0.851888\pi$
$521$	−1.10741e7	−1.78737	−0.893684	+	0.448697i	$0.851888\pi$
$522$	0	0
$523$	−4.06992e6	−0.650626	−0.325313	−	0.945606i	$-0.605470\pi$
$523$	−4.06992e6	−0.650626	−0.325313	+	0.945606i	$0.605470\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−152147.	−0.0238636
$528$	0	0
$529$	6.93999e6	1.07825
$530$	0	0
$531$	−8.56985e6	−1.31898
$532$	0	0
$533$	−1.30896e7	−1.99577
$534$	0	0
$535$	72797.5	0.0109959
$536$	0	0
$537$	390405.	0.0584224
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.27956e6	−0.334856	−0.167428	−	0.985884i	$-0.553546\pi$
$541$	−2.27956e6	−0.334856	−0.167428	+	0.985884i	$0.553546\pi$
$542$	0	0
$543$	−489207.	−0.0712022
$544$	0	0
$545$	−145296.	−0.0209538
$546$	0	0
$547$	140199.	0.0200344	0.0100172	−	0.999950i	$-0.496811\pi$
$547$	140199.	0.0200344	0.0100172	+	0.999950i	$0.496811\pi$
$548$	0	0
$549$	9.27754e6	1.31372
$550$	0	0
$551$	−4.36038e6	−0.611850
$552$	0	0
$553$	0	0
$554$	0	0
$555$	327966.	0.0451956
$556$	0	0
$557$	1.66449e6	0.227323	0.113661	−	0.993520i	$-0.463742\pi$
$557$	1.66449e6	0.227323	0.113661	+	0.993520i	$0.463742\pi$
$558$	0	0
$559$	−4.11178e6	−0.556546
$560$	0	0
$561$	373696.	0.0501316
$562$	0	0
$563$	−2.52071e6	−0.335160	−0.167580	−	0.985858i	$-0.553595\pi$
$563$	−2.52071e6	−0.335160	−0.167580	+	0.985858i	$0.553595\pi$
$564$	0	0
$565$	5.00899e6	0.660129
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.03168e6	−0.651527	−0.325763	−	0.945451i	$-0.605621\pi$
$569$	−5.03168e6	−0.651527	−0.325763	+	0.945451i	$0.605621\pi$
$570$	0	0
$571$	−1.25919e7	−1.61622	−0.808110	−	0.589032i	$-0.799509\pi$
$571$	−1.25919e7	−1.61622	−0.808110	+	0.589032i	$0.799509\pi$
$572$	0	0
$573$	−913523.	−0.116234
$574$	0	0
$575$	−2.28585e6	−0.288323
$576$	0	0
$577$	−6.05632e6	−0.757302	−0.378651	−	0.925540i	$-0.623612\pi$
$577$	−6.05632e6	−0.757302	−0.378651	+	0.925540i	$0.623612\pi$
$578$	0	0
$579$	−408395.	−0.0506272
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−4.08908e6	−0.498258
$584$	0	0
$585$	−6.44322e6	−0.778419
$586$	0	0
$587$	−1.17428e7	−1.40662	−0.703312	−	0.710881i	$-0.748296\pi$
$587$	−1.17428e7	−1.40662	−0.703312	+	0.710881i	$0.748296\pi$
$588$	0	0
$589$	−197211.	−0.0234231
$590$	0	0
$591$	851807.	0.100316
$592$	0	0
$593$	−2.19149e6	−0.255920	−0.127960	−	0.991779i	$-0.540843\pi$
$593$	−2.19149e6	−0.255920	−0.127960	+	0.991779i	$0.540843\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−920366.	−0.105688
$598$	0	0
$599$	−4.41915e6	−0.503236	−0.251618	−	0.967827i	$-0.580963\pi$
$599$	−4.41915e6	−0.503236	−0.251618	+	0.967827i	$0.580963\pi$
$600$	0	0
$601$	4.11214e6	0.464389	0.232194	−	0.972669i	$-0.425409\pi$
$601$	4.11214e6	0.464389	0.232194	+	0.972669i	$0.425409\pi$
$602$	0	0
$603$	7.56557e6	0.847322
$604$	0	0
$605$	2.80409e6	0.311461
$606$	0	0
$607$	−1.30343e7	−1.43587	−0.717935	−	0.696111i	$-0.754912\pi$
$607$	−1.30343e7	−1.43587	−0.717935	+	0.696111i	$0.754912\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.49331e6	0.161826
$612$	0	0
$613$	−7.04272e6	−0.756988	−0.378494	−	0.925604i	$-0.623558\pi$
$613$	−7.04272e6	−0.756988	−0.378494	+	0.925604i	$0.623558\pi$
$614$	0	0
$615$	−300602.	−0.0320482
$616$	0	0
$617$	5.80501e6	0.613889	0.306945	−	0.951727i	$-0.400693\pi$
$617$	5.80501e6	0.613889	0.306945	+	0.951727i	$0.400693\pi$
$618$	0	0
$619$	5.35648e6	0.561892	0.280946	−	0.959724i	$-0.409352\pi$
$619$	5.35648e6	0.561892	0.280946	+	0.959724i	$0.409352\pi$
$620$	0	0
$621$	1.73517e6	0.180557
$622$	0	0
$623$	0	0
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	484382.	0.0492061
$628$	0	0
$629$	2.31751e7	2.33558
$630$	0	0
$631$	−8.17906e6	−0.817768	−0.408884	−	0.912586i	$-0.634082\pi$
$631$	−8.17906e6	−0.817768	−0.408884	+	0.912586i	$0.634082\pi$
$632$	0	0
$633$	518157.	0.0513987
$634$	0	0
$635$	−2.27121e6	−0.223524
$636$	0	0
$637$	0	0
$638$	0	0
$639$	5.99958e6	0.581258
$640$	0	0
$641$	133640.	0.0128467	0.00642334	−	0.999979i	$-0.497955\pi$
$641$	133640.	0.0128467	0.00642334	+	0.999979i	$0.497955\pi$
$642$	0	0
$643$	1.57783e7	1.50498	0.752491	−	0.658602i	$-0.228852\pi$
$643$	1.57783e7	1.50498	0.752491	+	0.658602i	$0.228852\pi$
$644$	0	0
$645$	−94426.6	−0.00893707
$646$	0	0
$647$	1.04053e7	0.977228	0.488614	−	0.872500i	$-0.337503\pi$
$647$	1.04053e7	0.977228	0.488614	+	0.872500i	$0.337503\pi$
$648$	0	0
$649$	−7.82850e6	−0.729570
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.01565e6	0.460304	0.230152	−	0.973155i	$-0.426078\pi$
$653$	5.01565e6	0.460304	0.230152	+	0.973155i	$0.426078\pi$
$654$	0	0
$655$	593872.	0.0540866
$656$	0	0
$657$	1.79927e7	1.62623
$658$	0	0
$659$	−1.87344e7	−1.68045	−0.840226	−	0.542236i	$-0.817578\pi$
$659$	−1.87344e7	−1.68045	−0.840226	+	0.542236i	$0.817578\pi$
$660$	0	0
$661$	7.35404e6	0.654670	0.327335	−	0.944908i	$-0.393850\pi$
$661$	7.35404e6	0.654670	0.327335	+	0.944908i	$0.393850\pi$
$662$	0	0
$663$	1.79966e6	0.159003
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.12029e6	−0.619702
$668$	0	0
$669$	−134794.	−0.0116441
$670$	0	0
$671$	8.47497e6	0.726661
$672$	0	0
$673$	2.42769e6	0.206612	0.103306	−	0.994650i	$-0.467058\pi$
$673$	2.42769e6	0.206612	0.103306	+	0.994650i	$0.467058\pi$
$674$	0	0
$675$	−296520.	−0.0250493
$676$	0	0
$677$	−1.67412e7	−1.40383	−0.701914	−	0.712262i	$-0.747671\pi$
$677$	−1.67412e7	−1.40383	−0.701914	+	0.712262i	$0.747671\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.21403e6	−0.100314
$682$	0	0
$683$	1.31240e7	1.07650	0.538251	−	0.842785i	$-0.319085\pi$
$683$	1.31240e7	1.07650	0.538251	+	0.842785i	$0.319085\pi$
$684$	0	0
$685$	5.77361e6	0.470133
$686$	0	0
$687$	622024.	0.0502823
$688$	0	0
$689$	−1.96923e7	−1.58033
$690$	0	0
$691$	−1.84927e7	−1.47335	−0.736675	−	0.676247i	$-0.763605\pi$
$691$	−1.84927e7	−1.47335	−0.736675	+	0.676247i	$0.763605\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.81030e6	0.613346
$696$	0	0
$697$	−2.12414e7	−1.65616
$698$	0	0
$699$	827690.	0.0640729
$700$	0	0
$701$	9.59372e6	0.737381	0.368691	−	0.929552i	$-0.379806\pi$
$701$	9.59372e6	0.737381	0.368691	+	0.929552i	$0.379806\pi$
$702$	0	0
$703$	3.00393e7	2.29246
$704$	0	0
$705$	34293.7	0.00259861
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−2.37424e7	−1.77381	−0.886907	−	0.461948i	$-0.847151\pi$
$709$	−2.37424e7	−1.77381	−0.886907	+	0.461948i	$0.847151\pi$
$710$	0	0
$711$	−1.63501e7	−1.21296
$712$	0	0
$713$	−322037.	−0.0237236
$714$	0	0
$715$	−5.88584e6	−0.430570
$716$	0	0
$717$	−1.41077e6	−0.102484
$718$	0	0
$719$	1.37265e7	0.990234	0.495117	−	0.868826i	$-0.335125\pi$
$719$	1.37265e7	0.990234	0.495117	+	0.868826i	$0.335125\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−282248.	−0.0200810
$724$	0	0
$725$	1.21677e6	0.0859734
$726$	0	0
$727$	−2.21306e7	−1.55295	−0.776474	−	0.630150i	$-0.782993\pi$
$727$	−2.21306e7	−1.55295	−0.776474	+	0.630150i	$0.782993\pi$
$728$	0	0
$729$	−1.40111e7	−0.976455
$730$	0	0
$731$	−6.67246e6	−0.461841
$732$	0	0
$733$	−1.19275e7	−0.819955	−0.409978	−	0.912096i	$-0.634464\pi$
$733$	−1.19275e7	−0.819955	−0.409978	+	0.912096i	$0.634464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.91110e6	0.468682
$738$	0	0
$739$	4.77278e6	0.321485	0.160742	−	0.986996i	$-0.448611\pi$
$739$	4.77278e6	0.321485	0.160742	+	0.986996i	$0.448611\pi$
$740$	0	0
$741$	2.33270e6	0.156068
$742$	0	0
$743$	−2.20892e6	−0.146794	−0.0733968	−	0.997303i	$-0.523384\pi$
$743$	−2.20892e6	−0.146794	−0.0733968	+	0.997303i	$0.523384\pi$
$744$	0	0
$745$	−2.84739e6	−0.187956
$746$	0	0
$747$	4.04139e6	0.264990
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.42635e7	−1.56983	−0.784917	−	0.619600i	$-0.787295\pi$
$751$	−2.42635e7	−1.56983	−0.784917	+	0.619600i	$0.787295\pi$
$752$	0	0
$753$	−1.42367e6	−0.0914999
$754$	0	0
$755$	−9.88375e6	−0.631036
$756$	0	0
$757$	4.49576e6	0.285143	0.142572	−	0.989784i	$-0.454463\pi$
$757$	4.49576e6	0.285143	0.142572	+	0.989784i	$0.454463\pi$
$758$	0	0
$759$	790972.	0.0498376
$760$	0	0
$761$	−1.15916e7	−0.725573	−0.362787	−	0.931872i	$-0.618175\pi$
$761$	−1.15916e7	−0.725573	−0.362787	+	0.931872i	$0.618175\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.04558e7	−0.645960
$766$	0	0
$767$	−3.77008e7	−2.31399
$768$	0	0
$769$	5.13304e6	0.313011	0.156505	−	0.987677i	$-0.449977\pi$
$769$	5.13304e6	0.313011	0.156505	+	0.987677i	$0.449977\pi$
$770$	0	0
$771$	976765.	0.0591771
$772$	0	0
$773$	1.16425e7	0.700806	0.350403	−	0.936599i	$-0.386045\pi$
$773$	1.16425e7	0.700806	0.350403	+	0.936599i	$0.386045\pi$
$774$	0	0
$775$	55032.2	0.00329126
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.75330e7	−1.62558
$780$	0	0
$781$	5.48058e6	0.321513
$782$	0	0
$783$	−923642.	−0.0538393
$784$	0	0
$785$	1.06147e7	0.614800
$786$	0	0
$787$	−1.56107e6	−0.0898433	−0.0449217	−	0.998991i	$-0.514304\pi$
$787$	−1.56107e6	−0.0898433	−0.0449217	+	0.998991i	$0.514304\pi$
$788$	0	0
$789$	1.49513e6	0.0855042
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.08141e7	2.30477
$794$	0	0
$795$	−452232.	−0.0253772
$796$	0	0
$797$	−2.19483e7	−1.22393	−0.611963	−	0.790886i	$-0.709620\pi$
$797$	−2.19483e7	−1.22393	−0.611963	+	0.790886i	$0.709620\pi$
$798$	0	0
$799$	2.42329e6	0.134289
$800$	0	0
$801$	4.52349e6	0.249111
$802$	0	0
$803$	1.64362e7	0.899522
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−396003.	−0.0214050
$808$	0	0
$809$	−3.47284e7	−1.86558	−0.932789	−	0.360422i	$-0.882633\pi$
$809$	−3.47284e7	−1.86558	−0.932789	+	0.360422i	$0.882633\pi$
$810$	0	0
$811$	−1.07921e7	−0.576174	−0.288087	−	0.957604i	$-0.593019\pi$
$811$	−1.07921e7	−0.576174	−0.288087	+	0.957604i	$0.593019\pi$
$812$	0	0
$813$	−84178.9	−0.00446660
$814$	0	0
$815$	−5.56651e6	−0.293555
$816$	0	0
$817$	−8.64879e6	−0.453315
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.17198e7	1.64238	0.821188	−	0.570658i	$-0.193312\pi$
$821$	3.17198e7	1.64238	0.821188	+	0.570658i	$0.193312\pi$
$822$	0	0
$823$	−3.30028e7	−1.69845	−0.849223	−	0.528034i	$-0.822929\pi$
$823$	−3.30028e7	−1.69845	−0.849223	+	0.528034i	$0.822929\pi$
$824$	0	0
$825$	−135168.	−0.00691413
$826$	0	0
$827$	−2.33424e7	−1.18681	−0.593407	−	0.804903i	$-0.702217\pi$
$827$	−2.33424e7	−1.18681	−0.593407	+	0.804903i	$0.702217\pi$
$828$	0	0
$829$	−264764.	−0.0133805	−0.00669026	−	0.999978i	$-0.502130\pi$
$829$	−264764.	−0.0133805	−0.00669026	+	0.999978i	$0.502130\pi$
$830$	0	0
$831$	−1.43319e6	−0.0719950
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−804899.	−0.0399508
$836$	0	0
$837$	−41774.5	−0.00206109
$838$	0	0
$839$	6.62082e6	0.324719	0.162359	−	0.986732i	$-0.448090\pi$
$839$	6.62082e6	0.324719	0.162359	+	0.986732i	$0.448090\pi$
$840$	0	0
$841$	−1.67210e7	−0.815214
$842$	0	0
$843$	367721.	0.0178217
$844$	0	0
$845$	−1.90629e7	−0.918434
$846$	0	0
$847$	0	0
$848$	0	0
$849$	263468.	0.0125447
$850$	0	0
$851$	4.90528e7	2.32188
$852$	0	0
$853$	1.08548e7	0.510800	0.255400	−	0.966836i	$-0.417793\pi$
$853$	1.08548e7	0.510800	0.255400	+	0.966836i	$0.417793\pi$
$854$	0	0
$855$	−1.35528e7	−0.634035
$856$	0	0
$857$	−9.49231e6	−0.441489	−0.220744	−	0.975332i	$-0.570849\pi$
$857$	−9.49231e6	−0.441489	−0.220744	+	0.975332i	$0.570849\pi$
$858$	0	0
$859$	3.16670e7	1.46428	0.732140	−	0.681155i	$-0.238522\pi$
$859$	3.16670e7	1.46428	0.732140	+	0.681155i	$0.238522\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.37049e7	−1.54052	−0.770258	−	0.637733i	$-0.779872\pi$
$863$	−3.37049e7	−1.54052	−0.770258	+	0.637733i	$0.779872\pi$
$864$	0	0
$865$	1.88426e6	0.0856248
$866$	0	0
$867$	1.53163e6	0.0691999
$868$	0	0
$869$	−1.49357e7	−0.670930
$870$	0	0
$871$	3.32827e7	1.48653
$872$	0	0
$873$	1.18594e7	0.526656
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.48904e7	1.53182	0.765908	−	0.642950i	$-0.222290\pi$
$877$	3.48904e7	1.53182	0.765908	+	0.642950i	$0.222290\pi$
$878$	0	0
$879$	−2.12285e6	−0.0926718
$880$	0	0
$881$	2.52144e7	1.09448	0.547241	−	0.836975i	$-0.315678\pi$
$881$	2.52144e7	1.09448	0.547241	+	0.836975i	$0.315678\pi$
$882$	0	0
$883$	−1.69692e7	−0.732419	−0.366210	−	0.930532i	$-0.619345\pi$
$883$	−1.69692e7	−0.732419	−0.366210	+	0.930532i	$0.619345\pi$
$884$	0	0
$885$	−865794.	−0.0371583
$886$	0	0
$887$	637699.	0.0272149	0.0136074	−	0.999907i	$-0.495668\pi$
$887$	637699.	0.0272149	0.0136074	+	0.999907i	$0.495668\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.29020e7	−0.544457
$892$	0	0
$893$	3.14105e6	0.131809
$894$	0	0
$895$	−9.97841e6	−0.416393
$896$	0	0
$897$	3.80919e6	0.158071
$898$	0	0
$899$	171422.	0.00707402
$900$	0	0
$901$	−3.19560e7	−1.31142
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.25037e7	0.507478
$906$	0	0
$907$	−3.82732e7	−1.54481	−0.772407	−	0.635128i	$-0.780947\pi$
$907$	−3.82732e7	−1.54481	−0.772407	+	0.635128i	$0.780947\pi$
$908$	0	0
$909$	3.20517e7	1.28659
$910$	0	0
$911$	−1.18499e7	−0.473063	−0.236531	−	0.971624i	$-0.576011\pi$
$911$	−1.18499e7	−0.473063	−0.236531	+	0.971624i	$0.576011\pi$
$912$	0	0
$913$	3.69178e6	0.146575
$914$	0	0
$915$	937290.	0.0370102
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.29719e7	1.67840	0.839201	−	0.543821i	$-0.183023\pi$
$919$	4.29719e7	1.67840	0.839201	+	0.543821i	$0.183023\pi$
$920$	0	0
$921$	−995708.	−0.0386797
$922$	0	0
$923$	2.63936e7	1.01975
$924$	0	0
$925$	−8.38253e6	−0.322122
$926$	0	0
$927$	−4.42310e7	−1.69055
$928$	0	0
$929$	−9.20387e6	−0.349890	−0.174945	−	0.984578i	$-0.555975\pi$
$929$	−9.20387e6	−0.349890	−0.174945	+	0.984578i	$0.555975\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1.72356e6	0.0648219
$934$	0	0
$935$	−9.55134e6	−0.357302
$936$	0	0
$937$	−8.46665e6	−0.315038	−0.157519	−	0.987516i	$-0.550350\pi$
$937$	−8.46665e6	−0.315038	−0.157519	+	0.987516i	$0.550350\pi$
$938$	0	0
$939$	1.89086e6	0.0699834
$940$	0	0
$941$	1.60703e7	0.591628	0.295814	−	0.955245i	$-0.404409\pi$
$941$	1.60703e7	0.591628	0.295814	+	0.955245i	$0.404409\pi$
$942$	0	0
$943$	−4.49600e7	−1.64644
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.00021e7	−0.362424	−0.181212	−	0.983444i	$-0.558002\pi$
$947$	−1.00021e7	−0.362424	−0.181212	+	0.983444i	$0.558002\pi$
$948$	0	0
$949$	7.91540e7	2.85303
$950$	0	0
$951$	−1.50101e6	−0.0538187
$952$	0	0
$953$	−8.32957e6	−0.297092	−0.148546	−	0.988906i	$-0.547459\pi$
$953$	−8.32957e6	−0.297092	−0.148546	+	0.988906i	$0.547459\pi$
$954$	0	0
$955$	2.33488e7	0.828432
$956$	0	0
$957$	−421038.	−0.0148608
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.86214e7	−0.999729
$962$	0	0
$963$	704806.	0.0244908
$964$	0	0
$965$	1.04382e7	0.360834
$966$	0	0
$967$	1.10680e7	0.380631	0.190316	−	0.981723i	$-0.439049\pi$
$967$	1.10680e7	0.380631	0.190316	+	0.981723i	$0.439049\pi$
$968$	0	0
$969$	3.78543e6	0.129511
$970$	0	0
$971$	2.86163e7	0.974014	0.487007	−	0.873398i	$-0.338089\pi$
$971$	2.86163e7	0.974014	0.487007	+	0.873398i	$0.338089\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−650945.	−0.0219297
$976$	0	0
$977$	−1.64863e7	−0.552569	−0.276284	−	0.961076i	$-0.589103\pi$
$977$	−1.64863e7	−0.552569	−0.276284	+	0.961076i	$0.589103\pi$
$978$	0	0
$979$	4.13218e6	0.137791
$980$	0	0
$981$	−1.40672e6	−0.0466696
$982$	0	0
$983$	−3.41756e7	−1.12806	−0.564031	−	0.825754i	$-0.690750\pi$
$983$	−3.41756e7	−1.12806	−0.564031	+	0.825754i	$0.690750\pi$
$984$	0	0
$985$	−2.17714e7	−0.714984
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.41231e7	−0.459132
$990$	0	0
$991$	2.23058e6	0.0721495	0.0360747	−	0.999349i	$-0.488515\pi$
$991$	2.23058e6	0.0721495	0.0360747	+	0.999349i	$0.488515\pi$
$992$	0	0
$993$	1.22739e6	0.0395010
$994$	0	0
$995$	2.35237e7	0.753267
$996$	0	0
$997$	−2.25235e7	−0.717625	−0.358813	−	0.933410i	$-0.616818\pi$
$997$	−2.25235e7	−0.717625	−0.358813	+	0.933410i	$0.616818\pi$
$998$	0	0
$999$	6.36311e6	0.201723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.6.a.n.1.4		7
7.2	even	3		140.6.i.d.81.4	✓	14
7.4	even	3		140.6.i.d.121.4	yes	14
7.6	odd	2		980.6.a.o.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.6.i.d.81.4	✓	14	7.2	even	3
140.6.i.d.121.4	yes	14	7.4	even	3
980.6.a.n.1.4		7	1.1	even	1	trivial
980.6.a.o.1.4		7	7.6	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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	980.a (trivial)

\(f(q)\)	\(=\)	\(q+0.978125 q^{3} -25.0000 q^{5} -242.043 q^{9} +O(q^{10})\)	\(q+0.978125 q^{3} -25.0000 q^{5} -242.043 q^{9} -221.105 q^{11} -1064.81 q^{13} -24.4531 q^{15} -1727.93 q^{17} -2239.73 q^{19} -3657.37 q^{23} +625.000 q^{25} -474.433 q^{27} +1946.84 q^{29} +88.0515 q^{31} -216.268 q^{33} -13412.0 q^{37} -1041.51 q^{39} +12293.0 q^{41} +3861.54 q^{43} +6051.08 q^{45} -1402.43 q^{47} -1690.13 q^{51} +18493.8 q^{53} +5527.62 q^{55} -2190.73 q^{57} +35406.3 q^{59} -38330.1 q^{61} +26620.1 q^{65} -31257.1 q^{67} -3577.36 q^{69} -24787.2 q^{71} -74336.5 q^{73} +611.328 q^{75} +67550.4 q^{79} +58352.5 q^{81} -16697.0 q^{83} +43198.2 q^{85} +1904.25 q^{87} -18688.8 q^{89} +86.1253 q^{93} +55993.2 q^{95} -48997.0 q^{97} +53517.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 27 q^{3} - 175 q^{5} + 312 q^{9}+O(q^{10}) \) 7 * q - 27 * q^3 - 175 * q^5 + 312 * q^9	\( 7 q - 27 q^{3} - 175 q^{5} + 312 q^{9} + 590 q^{11} - 854 q^{13} + 675 q^{15} - 1328 q^{17} - 2990 q^{19} + 4089 q^{23} + 4375 q^{25} - 4281 q^{27} - 3063 q^{29} - 5740 q^{31} + 1470 q^{33} + 12218 q^{37} - 1560 q^{39} - 8477 q^{41} - 9195 q^{43} - 7800 q^{45} - 28652 q^{47} + 41244 q^{51} + 24932 q^{53} - 14750 q^{55} - 62934 q^{57} - 38242 q^{59} - 61299 q^{61} + 21350 q^{65} + 24913 q^{67} - 86151 q^{69} - 36170 q^{71} - 94780 q^{73} - 16875 q^{75} - 594 q^{79} - 53265 q^{81} - 113585 q^{83} + 33200 q^{85} - 43875 q^{87} - 36823 q^{89} - 21828 q^{93} + 74750 q^{95} - 257950 q^{97} + 378432 q^{99}+O(q^{100}) \) 7 * q - 27 * q^3 - 175 * q^5 + 312 * q^9 + 590 * q^11 - 854 * q^13 + 675 * q^15 - 1328 * q^17 - 2990 * q^19 + 4089 * q^23 + 4375 * q^25 - 4281 * q^27 - 3063 * q^29 - 5740 * q^31 + 1470 * q^33 + 12218 * q^37 - 1560 * q^39 - 8477 * q^41 - 9195 * q^43 - 7800 * q^45 - 28652 * q^47 + 41244 * q^51 + 24932 * q^53 - 14750 * q^55 - 62934 * q^57 - 38242 * q^59 - 61299 * q^61 + 21350 * q^65 + 24913 * q^67 - 86151 * q^69 - 36170 * q^71 - 94780 * q^73 - 16875 * q^75 - 594 * q^79 - 53265 * q^81 - 113585 * q^83 + 33200 * q^85 - 43875 * q^87 - 36823 * q^89 - 21828 * q^93 + 74750 * q^95 - 257950 * q^97 + 378432 * q^99

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