LMFDB - Embedded newform 320.6.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.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:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.39180.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 36x - 24 $ x^3 - x^2 - 36*x - 24
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.11788$ of defining polynomial
Character	$\chi$	$=$	320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +O(q^{10})$	$q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +438.282 q^{11} +758.285 q^{13} +461.788 q^{15} -1534.86 q^{17} -75.8518 q^{19} +2251.66 q^{21} +3694.27 q^{23} +625.000 q^{25} -2674.73 q^{27} -6323.09 q^{29} +2691.35 q^{31} +8095.74 q^{33} +3047.47 q^{35} +7252.19 q^{37} +14006.7 q^{39} +4913.34 q^{41} -2533.81 q^{43} +2454.92 q^{45} +11380.6 q^{47} -1947.64 q^{49} -28351.1 q^{51} -29442.3 q^{53} +10957.1 q^{55} -1401.10 q^{57} +5681.08 q^{59} +48039.8 q^{61} +11970.1 q^{63} +18957.1 q^{65} -39570.6 q^{67} +68238.7 q^{69} +12622.3 q^{71} +57959.2 q^{73} +11544.7 q^{75} +53426.2 q^{77} -29504.0 q^{79} -73268.2 q^{81} -112209. q^{83} -38371.4 q^{85} -116797. q^{87} +66969.5 q^{89} +92434.2 q^{91} +49713.4 q^{93} -1896.30 q^{95} +131833. q^{97} +43037.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 10 q^{3} + 75 q^{5} - 6 q^{7} + 467 q^{9}+O(q^{10}) $ 3 * q - 10 * q^3 + 75 * q^5 - 6 * q^7 + 467 * q^9	$ 3 q - 10 q^{3} + 75 q^{5} - 6 q^{7} + 467 q^{9} + 396 q^{11} + 354 q^{13} - 250 q^{15} + 1158 q^{17} - 3192 q^{19} + 820 q^{21} + 6126 q^{23} + 1875 q^{25} - 13804 q^{27} - 426 q^{29} + 3276 q^{31} - 2408 q^{33} - 150 q^{35} + 11562 q^{37} + 21348 q^{39} + 12450 q^{41} - 26346 q^{43} + 11675 q^{45} + 36762 q^{47} - 3849 q^{49} - 71444 q^{51} + 21162 q^{53} + 9900 q^{55} + 69136 q^{57} - 35040 q^{59} + 24138 q^{61} + 80986 q^{63} + 8850 q^{65} + 9570 q^{67} + 27036 q^{69} + 88092 q^{71} + 66750 q^{73} - 6250 q^{75} + 136488 q^{77} - 92952 q^{79} + 151391 q^{81} - 30258 q^{83} + 28950 q^{85} - 26228 q^{87} + 172686 q^{89} + 106812 q^{91} + 318232 q^{93} - 79800 q^{95} + 170910 q^{97} + 351436 q^{99}+O(q^{100}) $ 3 * q - 10 * q^3 + 75 * q^5 - 6 * q^7 + 467 * q^9 + 396 * q^11 + 354 * q^13 - 250 * q^15 + 1158 * q^17 - 3192 * q^19 + 820 * q^21 + 6126 * q^23 + 1875 * q^25 - 13804 * q^27 - 426 * q^29 + 3276 * q^31 - 2408 * q^33 - 150 * q^35 + 11562 * q^37 + 21348 * q^39 + 12450 * q^41 - 26346 * q^43 + 11675 * q^45 + 36762 * q^47 - 3849 * q^49 - 71444 * q^51 + 21162 * q^53 + 9900 * q^55 + 69136 * q^57 - 35040 * q^59 + 24138 * q^61 + 80986 * q^63 + 8850 * q^65 + 9570 * q^67 + 27036 * q^69 + 88092 * q^71 + 66750 * q^73 - 6250 * q^75 + 136488 * q^77 - 92952 * q^79 + 151391 * q^81 - 30258 * q^83 + 28950 * q^85 - 26228 * q^87 + 172686 * q^89 + 106812 * q^91 + 318232 * q^93 - 79800 * q^95 + 170910 * q^97 + 351436 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	18.4715	1.18495	0.592474	−	0.805590i	$-0.298151\pi$
$3$	18.4715	1.18495	0.592474	+	0.805590i	$0.298151\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	121.899	0.940275	0.470138	−	0.882593i	$-0.344204\pi$
$7$	121.899	0.940275	0.470138	+	0.882593i	$0.344204\pi$
$8$	0	0
$9$	98.1968	0.404102
$10$	0	0
$11$	438.282	1.09213	0.546063	−	0.837744i	$-0.316126\pi$
$11$	438.282	1.09213	0.546063	+	0.837744i	$0.316126\pi$
$12$	0	0
$13$	758.285	1.24444	0.622220	−	0.782842i	$-0.286231\pi$
$13$	758.285	1.24444	0.622220	+	0.782842i	$0.286231\pi$
$14$	0	0
$15$	461.788	0.529925
$16$	0	0
$17$	−1534.86	−1.28809	−0.644044	−	0.764989i	$-0.722744\pi$
$17$	−1534.86	−1.28809	−0.644044	+	0.764989i	$0.722744\pi$
$18$	0	0
$19$	−75.8518	−0.0482039	−0.0241019	−	0.999710i	$-0.507673\pi$
$19$	−75.8518	−0.0482039	−0.0241019	+	0.999710i	$0.507673\pi$
$20$	0	0
$21$	2251.66	1.11418
$22$	0	0
$23$	3694.27	1.45616	0.728079	−	0.685493i	$-0.240413\pi$
$23$	3694.27	1.45616	0.728079	+	0.685493i	$0.240413\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−2674.73	−0.706108
$28$	0	0
$29$	−6323.09	−1.39616	−0.698079	−	0.716021i	$-0.745961\pi$
$29$	−6323.09	−1.39616	−0.698079	+	0.716021i	$0.745961\pi$
$30$	0	0
$31$	2691.35	0.502998	0.251499	−	0.967857i	$-0.419076\pi$
$31$	2691.35	0.502998	0.251499	+	0.967857i	$0.419076\pi$
$32$	0	0
$33$	8095.74	1.29411
$34$	0	0
$35$	3047.47	0.420504
$36$	0	0
$37$	7252.19	0.870893	0.435446	−	0.900215i	$-0.356591\pi$
$37$	7252.19	0.870893	0.435446	+	0.900215i	$0.356591\pi$
$38$	0	0
$39$	14006.7	1.47460
$40$	0	0
$41$	4913.34	0.456475	0.228238	−	0.973605i	$-0.426704\pi$
$41$	4913.34	0.456475	0.228238	+	0.973605i	$0.426704\pi$
$42$	0	0
$43$	−2533.81	−0.208979	−0.104490	−	0.994526i	$-0.533321\pi$
$43$	−2533.81	−0.208979	−0.104490	+	0.994526i	$0.533321\pi$
$44$	0	0
$45$	2454.92	0.180720
$46$	0	0
$47$	11380.6	0.751487	0.375743	−	0.926724i	$-0.377387\pi$
$47$	11380.6	0.751487	0.375743	+	0.926724i	$0.377387\pi$
$48$	0	0
$49$	−1947.64	−0.115882
$50$	0	0
$51$	−28351.1	−1.52632
$52$	0	0
$53$	−29442.3	−1.43973	−0.719866	−	0.694113i	$-0.755797\pi$
$53$	−29442.3	−1.43973	−0.719866	+	0.694113i	$0.755797\pi$
$54$	0	0
$55$	10957.1	0.488413
$56$	0	0
$57$	−1401.10	−0.0571191
$58$	0	0
$59$	5681.08	0.212472	0.106236	−	0.994341i	$-0.466120\pi$
$59$	5681.08	0.212472	0.106236	+	0.994341i	$0.466120\pi$
$60$	0	0
$61$	48039.8	1.65301	0.826507	−	0.562926i	$-0.190324\pi$
$61$	48039.8	1.65301	0.826507	+	0.562926i	$0.190324\pi$
$62$	0	0
$63$	11970.1	0.379967
$64$	0	0
$65$	18957.1	0.556531
$66$	0	0
$67$	−39570.6	−1.07693	−0.538463	−	0.842649i	$-0.680995\pi$
$67$	−39570.6	−1.07693	−0.538463	+	0.842649i	$0.680995\pi$
$68$	0	0
$69$	68238.7	1.72547
$70$	0	0
$71$	12622.3	0.297161	0.148581	−	0.988900i	$-0.452530\pi$
$71$	12622.3	0.297161	0.148581	+	0.988900i	$0.452530\pi$
$72$	0	0
$73$	57959.2	1.27296	0.636481	−	0.771292i	$-0.280389\pi$
$73$	57959.2	1.27296	0.636481	+	0.771292i	$0.280389\pi$
$74$	0	0
$75$	11544.7	0.236990
$76$	0	0
$77$	53426.2	1.02690
$78$	0	0
$79$	−29504.0	−0.531879	−0.265939	−	0.963990i	$-0.585682\pi$
$79$	−29504.0	−0.531879	−0.265939	+	0.963990i	$0.585682\pi$
$80$	0	0
$81$	−73268.2	−1.24080
$82$	0	0
$83$	−112209.	−1.78786	−0.893928	−	0.448211i	$-0.852061\pi$
$83$	−112209.	−1.78786	−0.893928	+	0.448211i	$0.852061\pi$
$84$	0	0
$85$	−38371.4	−0.576050
$86$	0	0
$87$	−116797.	−1.65437
$88$	0	0
$89$	66969.5	0.896194	0.448097	−	0.893985i	$-0.352102\pi$
$89$	66969.5	0.896194	0.448097	+	0.893985i	$0.352102\pi$
$90$	0	0
$91$	92434.2	1.17012
$92$	0	0
$93$	49713.4	0.596027
$94$	0	0
$95$	−1896.30	−0.0215574
$96$	0	0
$97$	131833.	1.42264	0.711322	−	0.702867i	$-0.248097\pi$
$97$	131833.	1.42264	0.711322	+	0.702867i	$0.248097\pi$
$98$	0	0
$99$	43037.9	0.441330
$100$	0	0
$101$	30688.2	0.299343	0.149671	−	0.988736i	$-0.452178\pi$
$101$	30688.2	0.299343	0.149671	+	0.988736i	$0.452178\pi$
$102$	0	0
$103$	−140744.	−1.30718	−0.653591	−	0.756848i	$-0.726738\pi$
$103$	−140744.	−1.30718	−0.653591	+	0.756848i	$0.726738\pi$
$104$	0	0
$105$	56291.5	0.498275
$106$	0	0
$107$	−14196.9	−0.119876	−0.0599381	−	0.998202i	$-0.519090\pi$
$107$	−14196.9	−0.119876	−0.0599381	+	0.998202i	$0.519090\pi$
$108$	0	0
$109$	−9873.09	−0.0795952	−0.0397976	−	0.999208i	$-0.512671\pi$
$109$	−9873.09	−0.0795952	−0.0397976	+	0.999208i	$0.512671\pi$
$110$	0	0
$111$	133959.	1.03196
$112$	0	0
$113$	245866.	1.81135	0.905675	−	0.423973i	$-0.139365\pi$
$113$	245866.	1.81135	0.905675	+	0.423973i	$0.139365\pi$
$114$	0	0
$115$	92356.7	0.651214
$116$	0	0
$117$	74461.2	0.502881
$118$	0	0
$119$	−187097.	−1.21116
$120$	0	0
$121$	31040.5	0.192737
$122$	0	0
$123$	90756.9	0.540900
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	88403.3	0.486361	0.243181	−	0.969981i	$-0.421809\pi$
$127$	88403.3	0.486361	0.243181	+	0.969981i	$0.421809\pi$
$128$	0	0
$129$	−46803.3	−0.247630
$130$	0	0
$131$	274297.	1.39651	0.698253	−	0.715851i	$-0.253961\pi$
$131$	274297.	1.39651	0.698253	+	0.715851i	$0.253961\pi$
$132$	0	0
$133$	−9246.26	−0.0453249
$134$	0	0
$135$	−66868.4	−0.315781
$136$	0	0
$137$	−309183.	−1.40739	−0.703694	−	0.710503i	$-0.748467\pi$
$137$	−309183.	−1.40739	−0.703694	+	0.710503i	$0.748467\pi$
$138$	0	0
$139$	381380.	1.67425	0.837125	−	0.547011i	$-0.184235\pi$
$139$	381380.	1.67425	0.837125	+	0.547011i	$0.184235\pi$
$140$	0	0
$141$	210217.	0.890473
$142$	0	0
$143$	332343.	1.35908
$144$	0	0
$145$	−158077.	−0.624381
$146$	0	0
$147$	−35975.8	−0.137315
$148$	0	0
$149$	101742.	0.375436	0.187718	−	0.982223i	$-0.439891\pi$
$149$	101742.	0.375436	0.187718	+	0.982223i	$0.439891\pi$
$150$	0	0
$151$	338843.	1.20936	0.604680	−	0.796469i	$-0.293301\pi$
$151$	338843.	1.20936	0.604680	+	0.796469i	$0.293301\pi$
$152$	0	0
$153$	−150718.	−0.520519
$154$	0	0
$155$	67283.9	0.224948
$156$	0	0
$157$	−101650.	−0.329123	−0.164561	−	0.986367i	$-0.552621\pi$
$157$	−101650.	−0.329123	−0.164561	+	0.986367i	$0.552621\pi$
$158$	0	0
$159$	−543843.	−1.70601
$160$	0	0
$161$	450327.	1.36919
$162$	0	0
$163$	−214136.	−0.631279	−0.315640	−	0.948879i	$-0.602219\pi$
$163$	−214136.	−0.631279	−0.315640	+	0.948879i	$0.602219\pi$
$164$	0	0
$165$	202393.	0.578744
$166$	0	0
$167$	−606370.	−1.68247	−0.841234	−	0.540672i	$-0.818170\pi$
$167$	−606370.	−1.68247	−0.841234	+	0.540672i	$0.818170\pi$
$168$	0	0
$169$	203703.	0.548633
$170$	0	0
$171$	−7448.41	−0.0194793
$172$	0	0
$173$	426958.	1.08460	0.542301	−	0.840184i	$-0.317553\pi$
$173$	426958.	1.08460	0.542301	+	0.840184i	$0.317553\pi$
$174$	0	0
$175$	76186.9	0.188055
$176$	0	0
$177$	104938.	0.251768
$178$	0	0
$179$	−682764.	−1.59271	−0.796357	−	0.604826i	$-0.793243\pi$
$179$	−682764.	−1.59271	−0.796357	+	0.604826i	$0.793243\pi$
$180$	0	0
$181$	−725666.	−1.64642	−0.823209	−	0.567738i	$-0.807819\pi$
$181$	−725666.	−1.64642	−0.823209	+	0.567738i	$0.807819\pi$
$182$	0	0
$183$	887368.	1.95874
$184$	0	0
$185$	181305.	0.389475
$186$	0	0
$187$	−672700.	−1.40675
$188$	0	0
$189$	−326047.	−0.663936
$190$	0	0
$191$	662000.	1.31303	0.656515	−	0.754313i	$-0.272030\pi$
$191$	662000.	1.31303	0.656515	+	0.754313i	$0.272030\pi$
$192$	0	0
$193$	−481557.	−0.930582	−0.465291	−	0.885158i	$-0.654050\pi$
$193$	−481557.	−0.930582	−0.465291	+	0.885158i	$0.654050\pi$
$194$	0	0
$195$	350167.	0.659460
$196$	0	0
$197$	−569264.	−1.04508	−0.522538	−	0.852616i	$-0.675015\pi$
$197$	−569264.	−1.04508	−0.522538	+	0.852616i	$0.675015\pi$
$198$	0	0
$199$	−850151.	−1.52182	−0.760911	−	0.648857i	$-0.775247\pi$
$199$	−850151.	−1.52182	−0.760911	+	0.648857i	$0.775247\pi$
$200$	0	0
$201$	−730930.	−1.27610
$202$	0	0
$203$	−770778.	−1.31277
$204$	0	0
$205$	122834.	0.204142
$206$	0	0
$207$	362765.	0.588437
$208$	0	0
$209$	−33244.5	−0.0526447
$210$	0	0
$211$	−285552.	−0.441548	−0.220774	−	0.975325i	$-0.570858\pi$
$211$	−285552.	−0.441548	−0.220774	+	0.975325i	$0.570858\pi$
$212$	0	0
$213$	233153.	0.352120
$214$	0	0
$215$	−63345.3	−0.0934583
$216$	0	0
$217$	328073.	0.472957
$218$	0	0
$219$	1.07059e6	1.50839
$220$	0	0
$221$	−1.16386e6	−1.60295
$222$	0	0
$223$	279669.	0.376602	0.188301	−	0.982111i	$-0.439702\pi$
$223$	279669.	0.376602	0.188301	+	0.982111i	$0.439702\pi$
$224$	0	0
$225$	61373.0	0.0808204
$226$	0	0
$227$	−1.27295e6	−1.63964	−0.819819	−	0.572623i	$-0.805926\pi$
$227$	−1.27295e6	−1.63964	−0.819819	+	0.572623i	$0.805926\pi$
$228$	0	0
$229$	−584351.	−0.736352	−0.368176	−	0.929756i	$-0.620018\pi$
$229$	−584351.	−0.736352	−0.368176	+	0.929756i	$0.620018\pi$
$230$	0	0
$231$	986863.	1.21682
$232$	0	0
$233$	−658722.	−0.794900	−0.397450	−	0.917624i	$-0.630105\pi$
$233$	−658722.	−0.794900	−0.397450	+	0.917624i	$0.630105\pi$
$234$	0	0
$235$	284515.	0.336075
$236$	0	0
$237$	−544983.	−0.630248
$238$	0	0
$239$	−664166.	−0.752111	−0.376055	−	0.926597i	$-0.622720\pi$
$239$	−664166.	−0.752111	−0.376055	+	0.926597i	$0.622720\pi$
$240$	0	0
$241$	−192140.	−0.213095	−0.106548	−	0.994308i	$-0.533980\pi$
$241$	−192140.	−0.213095	−0.106548	+	0.994308i	$0.533980\pi$
$242$	0	0
$243$	−703414.	−0.764180
$244$	0	0
$245$	−48690.9	−0.0518242
$246$	0	0
$247$	−57517.3	−0.0599869
$248$	0	0
$249$	−2.07267e6	−2.11852
$250$	0	0
$251$	−1.20021e6	−1.20246	−0.601231	−	0.799075i	$-0.705323\pi$
$251$	−1.20021e6	−1.20246	−0.601231	+	0.799075i	$0.705323\pi$
$252$	0	0
$253$	1.61913e6	1.59031
$254$	0	0
$255$	−708778.	−0.682589
$256$	0	0
$257$	−494440.	−0.466961	−0.233481	−	0.972361i	$-0.575012\pi$
$257$	−494440.	−0.466961	−0.233481	+	0.972361i	$0.575012\pi$
$258$	0	0
$259$	884034.	0.818879
$260$	0	0
$261$	−620907.	−0.564190
$262$	0	0
$263$	535899.	0.477742	0.238871	−	0.971051i	$-0.423223\pi$
$263$	535899.	0.477742	0.238871	+	0.971051i	$0.423223\pi$
$264$	0	0
$265$	−736056.	−0.643867
$266$	0	0
$267$	1.23703e6	1.06194
$268$	0	0
$269$	886621.	0.747063	0.373532	−	0.927617i	$-0.378147\pi$
$269$	886621.	0.747063	0.373532	+	0.927617i	$0.378147\pi$
$270$	0	0
$271$	−1.31305e6	−1.08607	−0.543036	−	0.839710i	$-0.682725\pi$
$271$	−1.31305e6	−1.08607	−0.543036	+	0.839710i	$0.682725\pi$
$272$	0	0
$273$	1.70740e6	1.38653
$274$	0	0
$275$	273927.	0.218425
$276$	0	0
$277$	−175211.	−0.137203	−0.0686013	−	0.997644i	$-0.521854\pi$
$277$	−175211.	−0.137203	−0.0686013	+	0.997644i	$0.521854\pi$
$278$	0	0
$279$	264282.	0.203263
$280$	0	0
$281$	−1.78398e6	−1.34779	−0.673897	−	0.738826i	$-0.735381\pi$
$281$	−1.78398e6	−1.34779	−0.673897	+	0.738826i	$0.735381\pi$
$282$	0	0
$283$	−1.98478e6	−1.47315	−0.736573	−	0.676358i	$-0.763557\pi$
$283$	−1.98478e6	−1.47315	−0.736573	+	0.676358i	$0.763557\pi$
$284$	0	0
$285$	−35027.4	−0.0255444
$286$	0	0
$287$	598931.	0.429212
$288$	0	0
$289$	935925.	0.659168
$290$	0	0
$291$	2.43516e6	1.68576
$292$	0	0
$293$	−2.44146e6	−1.66143	−0.830714	−	0.556700i	$-0.812067\pi$
$293$	−2.44146e6	−1.66143	−0.830714	+	0.556700i	$0.812067\pi$
$294$	0	0
$295$	142027.	0.0950202
$296$	0	0
$297$	−1.17229e6	−0.771158
$298$	0	0
$299$	2.80131e6	1.81210
$300$	0	0
$301$	−308869.	−0.196498
$302$	0	0
$303$	566858.	0.354706
$304$	0	0
$305$	1.20100e6	0.739251
$306$	0	0
$307$	2.53943e6	1.53777	0.768883	−	0.639390i	$-0.220813\pi$
$307$	2.53943e6	1.53777	0.768883	+	0.639390i	$0.220813\pi$
$308$	0	0
$309$	−2.59975e6	−1.54894
$310$	0	0
$311$	1.86583e6	1.09388	0.546941	−	0.837171i	$-0.315792\pi$
$311$	1.86583e6	1.09388	0.546941	+	0.837171i	$0.315792\pi$
$312$	0	0
$313$	2.17459e6	1.25463	0.627316	−	0.778765i	$-0.284153\pi$
$313$	2.17459e6	1.25463	0.627316	+	0.778765i	$0.284153\pi$
$314$	0	0
$315$	299252.	0.169926
$316$	0	0
$317$	1.48421e6	0.829557	0.414779	−	0.909922i	$-0.363859\pi$
$317$	1.48421e6	0.829557	0.414779	+	0.909922i	$0.363859\pi$
$318$	0	0
$319$	−2.77130e6	−1.52478
$320$	0	0
$321$	−262238.	−0.142047
$322$	0	0
$323$	116422.	0.0620908
$324$	0	0
$325$	473928.	0.248888
$326$	0	0
$327$	−182371.	−0.0943162
$328$	0	0
$329$	1.38729e6	0.706604
$330$	0	0
$331$	−2.02880e6	−1.01782	−0.508908	−	0.860821i	$-0.669950\pi$
$331$	−2.02880e6	−1.01782	−0.508908	+	0.860821i	$0.669950\pi$
$332$	0	0
$333$	712141.	0.351930
$334$	0	0
$335$	−989266.	−0.481616
$336$	0	0
$337$	−1.34355e6	−0.644435	−0.322218	−	0.946666i	$-0.604428\pi$
$337$	−1.34355e6	−0.644435	−0.322218	+	0.946666i	$0.604428\pi$
$338$	0	0
$339$	4.54152e6	2.14636
$340$	0	0
$341$	1.17957e6	0.549337
$342$	0	0
$343$	−2.28617e6	−1.04924
$344$	0	0
$345$	1.70597e6	0.771655
$346$	0	0
$347$	−1.52038e6	−0.677843	−0.338921	−	0.940815i	$-0.610062\pi$
$347$	−1.52038e6	−0.677843	−0.338921	+	0.940815i	$0.610062\pi$
$348$	0	0
$349$	1.80948e6	0.795224	0.397612	−	0.917554i	$-0.369839\pi$
$349$	1.80948e6	0.795224	0.397612	+	0.917554i	$0.369839\pi$
$350$	0	0
$351$	−2.02821e6	−0.878710
$352$	0	0
$353$	2.48843e6	1.06289	0.531446	−	0.847092i	$-0.321649\pi$
$353$	2.48843e6	1.06289	0.531446	+	0.847092i	$0.321649\pi$
$354$	0	0
$355$	315557.	0.132894
$356$	0	0
$357$	−3.45597e6	−1.43516
$358$	0	0
$359$	1.44154e6	0.590324	0.295162	−	0.955447i	$-0.404626\pi$
$359$	1.44154e6	0.590324	0.295162	+	0.955447i	$0.404626\pi$
$360$	0	0
$361$	−2.47035e6	−0.997676
$362$	0	0
$363$	573365.	0.228383
$364$	0	0
$365$	1.44898e6	0.569286
$366$	0	0
$367$	257353.	0.0997388	0.0498694	−	0.998756i	$-0.484119\pi$
$367$	257353.	0.0997388	0.0498694	+	0.998756i	$0.484119\pi$
$368$	0	0
$369$	482474.	0.184463
$370$	0	0
$371$	−3.58898e6	−1.35374
$372$	0	0
$373$	4.55163e6	1.69393	0.846964	−	0.531651i	$-0.178428\pi$
$373$	4.55163e6	1.69393	0.846964	+	0.531651i	$0.178428\pi$
$374$	0	0
$375$	288617.	0.105985
$376$	0	0
$377$	−4.79471e6	−1.73743
$378$	0	0
$379$	−2.96634e6	−1.06077	−0.530387	−	0.847756i	$-0.677953\pi$
$379$	−2.96634e6	−1.06077	−0.530387	+	0.847756i	$0.677953\pi$
$380$	0	0
$381$	1.63294e6	0.576313
$382$	0	0
$383$	3.57107e6	1.24395	0.621973	−	0.783039i	$-0.286331\pi$
$383$	3.57107e6	1.24395	0.621973	+	0.783039i	$0.286331\pi$
$384$	0	0
$385$	1.33565e6	0.459243
$386$	0	0
$387$	−248812.	−0.0844489
$388$	0	0
$389$	881685.	0.295420	0.147710	−	0.989031i	$-0.452810\pi$
$389$	881685.	0.295420	0.147710	+	0.989031i	$0.452810\pi$
$390$	0	0
$391$	−5.67017e6	−1.87566
$392$	0	0
$393$	5.06668e6	1.65479
$394$	0	0
$395$	−737599.	−0.237863
$396$	0	0
$397$	2.56117e6	0.815570	0.407785	−	0.913078i	$-0.366301\pi$
$397$	2.56117e6	0.815570	0.407785	+	0.913078i	$0.366301\pi$
$398$	0	0
$399$	−170792.	−0.0537077
$400$	0	0
$401$	−3.37596e6	−1.04842	−0.524211	−	0.851589i	$-0.675640\pi$
$401$	−3.37596e6	−1.04842	−0.524211	+	0.851589i	$0.675640\pi$
$402$	0	0
$403$	2.04081e6	0.625952
$404$	0	0
$405$	−1.83171e6	−0.554904
$406$	0	0
$407$	3.17851e6	0.951124
$408$	0	0
$409$	−2.56437e6	−0.758006	−0.379003	−	0.925395i	$-0.623733\pi$
$409$	−2.56437e6	−0.758006	−0.379003	+	0.925395i	$0.623733\pi$
$410$	0	0
$411$	−5.71107e6	−1.66768
$412$	0	0
$413$	692518.	0.199782
$414$	0	0
$415$	−2.80522e6	−0.799553
$416$	0	0
$417$	7.04466e6	1.98390
$418$	0	0
$419$	−6.64881e6	−1.85016	−0.925079	−	0.379775i	$-0.876001\pi$
$419$	−6.64881e6	−1.85016	−0.925079	+	0.379775i	$0.876001\pi$
$420$	0	0
$421$	−4.98840e6	−1.37169	−0.685846	−	0.727747i	$-0.740568\pi$
$421$	−4.98840e6	−1.37169	−0.685846	+	0.727747i	$0.740568\pi$
$422$	0	0
$423$	1.11754e6	0.303677
$424$	0	0
$425$	−959285.	−0.257617
$426$	0	0
$427$	5.85600e6	1.55429
$428$	0	0
$429$	6.13888e6	1.61045
$430$	0	0
$431$	802200.	0.208012	0.104006	−	0.994577i	$-0.466834\pi$
$431$	802200.	0.208012	0.104006	+	0.994577i	$0.466834\pi$
$432$	0	0
$433$	5.80978e6	1.48916	0.744578	−	0.667535i	$-0.232651\pi$
$433$	5.80978e6	1.48916	0.744578	+	0.667535i	$0.232651\pi$
$434$	0	0
$435$	−2.91993e6	−0.739859
$436$	0	0
$437$	−280217.	−0.0701925
$438$	0	0
$439$	585972.	0.145116	0.0725581	−	0.997364i	$-0.476884\pi$
$439$	585972.	0.145116	0.0725581	+	0.997364i	$0.476884\pi$
$440$	0	0
$441$	−191252.	−0.0468283
$442$	0	0
$443$	−4.41675e6	−1.06929	−0.534643	−	0.845078i	$-0.679554\pi$
$443$	−4.41675e6	−1.06929	−0.534643	+	0.845078i	$0.679554\pi$
$444$	0	0
$445$	1.67424e6	0.400790
$446$	0	0
$447$	1.87934e6	0.444872
$448$	0	0
$449$	2.28408e6	0.534681	0.267340	−	0.963602i	$-0.413855\pi$
$449$	2.28408e6	0.534681	0.267340	+	0.963602i	$0.413855\pi$
$450$	0	0
$451$	2.15343e6	0.498528
$452$	0	0
$453$	6.25893e6	1.43303
$454$	0	0
$455$	2.31086e6	0.523292
$456$	0	0
$457$	3.43086e6	0.768444	0.384222	−	0.923241i	$-0.374470\pi$
$457$	3.43086e6	0.768444	0.384222	+	0.923241i	$0.374470\pi$
$458$	0	0
$459$	4.10533e6	0.909529
$460$	0	0
$461$	−674699.	−0.147862	−0.0739311	−	0.997263i	$-0.523555\pi$
$461$	−674699.	−0.147862	−0.0739311	+	0.997263i	$0.523555\pi$
$462$	0	0
$463$	5.78627e6	1.25443	0.627215	−	0.778846i	$-0.284195\pi$
$463$	5.78627e6	1.25443	0.627215	+	0.778846i	$0.284195\pi$
$464$	0	0
$465$	1.24283e6	0.266551
$466$	0	0
$467$	−6.88097e6	−1.46002	−0.730008	−	0.683439i	$-0.760484\pi$
$467$	−6.88097e6	−1.46002	−0.730008	+	0.683439i	$0.760484\pi$
$468$	0	0
$469$	−4.82362e6	−1.01261
$470$	0	0
$471$	−1.87763e6	−0.389993
$472$	0	0
$473$	−1.11052e6	−0.228231
$474$	0	0
$475$	−47407.4	−0.00964078
$476$	0	0
$477$	−2.89114e6	−0.581798
$478$	0	0
$479$	4.94000e6	0.983757	0.491879	−	0.870664i	$-0.336310\pi$
$479$	4.94000e6	0.983757	0.491879	+	0.870664i	$0.336310\pi$
$480$	0	0
$481$	5.49923e6	1.08377
$482$	0	0
$483$	8.31823e6	1.62242
$484$	0	0
$485$	3.29583e6	0.636225
$486$	0	0
$487$	5.21802e6	0.996973	0.498487	−	0.866897i	$-0.333889\pi$
$487$	5.21802e6	0.996973	0.498487	+	0.866897i	$0.333889\pi$
$488$	0	0
$489$	−3.95542e6	−0.748033
$490$	0	0
$491$	4.20805e6	0.787730	0.393865	−	0.919168i	$-0.371138\pi$
$491$	4.20805e6	0.787730	0.393865	+	0.919168i	$0.371138\pi$
$492$	0	0
$493$	9.70503e6	1.79837
$494$	0	0
$495$	1.07595e6	0.197369
$496$	0	0
$497$	1.53864e6	0.279413
$498$	0	0
$499$	−7.74013e6	−1.39154	−0.695771	−	0.718263i	$-0.744937\pi$
$499$	−7.74013e6	−1.39154	−0.695771	+	0.718263i	$0.744937\pi$
$500$	0	0
$501$	−1.12006e7	−1.99364
$502$	0	0
$503$	−6.20893e6	−1.09420	−0.547100	−	0.837067i	$-0.684268\pi$
$503$	−6.20893e6	−1.09420	−0.547100	+	0.837067i	$0.684268\pi$
$504$	0	0
$505$	767206.	0.133870
$506$	0	0
$507$	3.76271e6	0.650101
$508$	0	0
$509$	−8.43892e6	−1.44375	−0.721876	−	0.692022i	$-0.756720\pi$
$509$	−8.43892e6	−1.44375	−0.721876	+	0.692022i	$0.756720\pi$
$510$	0	0
$511$	7.06517e6	1.19693
$512$	0	0
$513$	202883.	0.0340372
$514$	0	0
$515$	−3.51859e6	−0.584590
$516$	0	0
$517$	4.98793e6	0.820717
$518$	0	0
$519$	7.88657e6	1.28520
$520$	0	0
$521$	−6.61888e6	−1.06829	−0.534146	−	0.845392i	$-0.679367\pi$
$521$	−6.61888e6	−1.06829	−0.534146	+	0.845392i	$0.679367\pi$
$522$	0	0
$523$	5.48657e6	0.877095	0.438548	−	0.898708i	$-0.355493\pi$
$523$	5.48657e6	0.877095	0.438548	+	0.898708i	$0.355493\pi$
$524$	0	0
$525$	1.40729e6	0.222835
$526$	0	0
$527$	−4.13084e6	−0.647906
$528$	0	0
$529$	7.21127e6	1.12040
$530$	0	0
$531$	557864.	0.0858602
$532$	0	0
$533$	3.72571e6	0.568057
$534$	0	0
$535$	−354922.	−0.0536103
$536$	0	0
$537$	−1.26117e7	−1.88728
$538$	0	0
$539$	−853614.	−0.126558
$540$	0	0
$541$	427673.	0.0628230	0.0314115	−	0.999507i	$-0.490000\pi$
$541$	427673.	0.0628230	0.0314115	+	0.999507i	$0.490000\pi$
$542$	0	0
$543$	−1.34041e7	−1.95092
$544$	0	0
$545$	−246827.	−0.0355960
$546$	0	0
$547$	8.94393e6	1.27809	0.639043	−	0.769171i	$-0.279331\pi$
$547$	8.94393e6	1.27809	0.639043	+	0.769171i	$0.279331\pi$
$548$	0	0
$549$	4.71736e6	0.667987
$550$	0	0
$551$	479618.	0.0673002
$552$	0	0
$553$	−3.59650e6	−0.500112
$554$	0	0
$555$	3.34897e6	0.461508
$556$	0	0
$557$	−7.93887e6	−1.08423	−0.542114	−	0.840305i	$-0.682376\pi$
$557$	−7.93887e6	−1.08423	−0.542114	+	0.840305i	$0.682376\pi$
$558$	0	0
$559$	−1.92135e6	−0.260062
$560$	0	0
$561$	−1.24258e7	−1.66693
$562$	0	0
$563$	1.61543e6	0.214792	0.107396	−	0.994216i	$-0.465749\pi$
$563$	1.61543e6	0.214792	0.107396	+	0.994216i	$0.465749\pi$
$564$	0	0
$565$	6.14665e6	0.810060
$566$	0	0
$567$	−8.93132e6	−1.16670
$568$	0	0
$569$	1.97353e6	0.255543	0.127771	−	0.991804i	$-0.459218\pi$
$569$	1.97353e6	0.255543	0.127771	+	0.991804i	$0.459218\pi$
$570$	0	0
$571$	7.18140e6	0.921762	0.460881	−	0.887462i	$-0.347533\pi$
$571$	7.18140e6	0.921762	0.460881	+	0.887462i	$0.347533\pi$
$572$	0	0
$573$	1.22281e7	1.55587
$574$	0	0
$575$	2.30892e6	0.291232
$576$	0	0
$577$	−9.61498e6	−1.20229	−0.601145	−	0.799140i	$-0.705288\pi$
$577$	−9.61498e6	−1.20229	−0.601145	+	0.799140i	$0.705288\pi$
$578$	0	0
$579$	−8.89509e6	−1.10269
$580$	0	0
$581$	−1.36782e7	−1.68108
$582$	0	0
$583$	−1.29040e7	−1.57237
$584$	0	0
$585$	1.86153e6	0.224895
$586$	0	0
$587$	−2.66522e6	−0.319255	−0.159627	−	0.987177i	$-0.551029\pi$
$587$	−2.66522e6	−0.319255	−0.159627	+	0.987177i	$0.551029\pi$
$588$	0	0
$589$	−204144.	−0.0242465
$590$	0	0
$591$	−1.05152e7	−1.23836
$592$	0	0
$593$	5.83235e6	0.681093	0.340547	−	0.940228i	$-0.389388\pi$
$593$	5.83235e6	0.681093	0.340547	+	0.940228i	$0.389388\pi$
$594$	0	0
$595$	−4.67743e6	−0.541646
$596$	0	0
$597$	−1.57036e7	−1.80328
$598$	0	0
$599$	6.39890e6	0.728683	0.364341	−	0.931265i	$-0.381294\pi$
$599$	6.39890e6	0.728683	0.364341	+	0.931265i	$0.381294\pi$
$600$	0	0
$601$	1.00505e7	1.13502	0.567509	−	0.823367i	$-0.307907\pi$
$601$	1.00505e7	1.13502	0.567509	+	0.823367i	$0.307907\pi$
$602$	0	0
$603$	−3.88571e6	−0.435188
$604$	0	0
$605$	776012.	0.0861946
$606$	0	0
$607$	1.83636e6	0.202295	0.101148	−	0.994871i	$-0.467749\pi$
$607$	1.83636e6	0.202295	0.101148	+	0.994871i	$0.467749\pi$
$608$	0	0
$609$	−1.42374e7	−1.55557
$610$	0	0
$611$	8.62976e6	0.935181
$612$	0	0
$613$	−1.33155e7	−1.43122	−0.715611	−	0.698499i	$-0.753852\pi$
$613$	−1.33155e7	−1.43122	−0.715611	+	0.698499i	$0.753852\pi$
$614$	0	0
$615$	2.26892e6	0.241898
$616$	0	0
$617$	3.09296e6	0.327085	0.163543	−	0.986536i	$-0.447708\pi$
$617$	3.09296e6	0.327085	0.163543	+	0.986536i	$0.447708\pi$
$618$	0	0
$619$	9.64802e6	1.01207	0.506036	−	0.862512i	$-0.331110\pi$
$619$	9.64802e6	1.01207	0.506036	+	0.862512i	$0.331110\pi$
$620$	0	0
$621$	−9.88118e6	−1.02821
$622$	0	0
$623$	8.16352e6	0.842669
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−614077.	−0.0623812
$628$	0	0
$629$	−1.11311e7	−1.12179
$630$	0	0
$631$	−1.72048e7	−1.72019	−0.860096	−	0.510132i	$-0.829597\pi$
$631$	−1.72048e7	−1.72019	−0.860096	+	0.510132i	$0.829597\pi$
$632$	0	0
$633$	−5.27457e6	−0.523212
$634$	0	0
$635$	2.21008e6	0.217507
$636$	0	0
$637$	−1.47686e6	−0.144209
$638$	0	0
$639$	1.23947e6	0.120083
$640$	0	0
$641$	1.86292e6	0.179081	0.0895403	−	0.995983i	$-0.471460\pi$
$641$	1.86292e6	0.179081	0.0895403	+	0.995983i	$0.471460\pi$
$642$	0	0
$643$	1.39819e7	1.33364	0.666821	−	0.745218i	$-0.267655\pi$
$643$	1.39819e7	1.33364	0.666821	+	0.745218i	$0.267655\pi$
$644$	0	0
$645$	−1.17008e6	−0.110743
$646$	0	0
$647$	−1.28310e7	−1.20504	−0.602520	−	0.798104i	$-0.705837\pi$
$647$	−1.28310e7	−1.20504	−0.602520	+	0.798104i	$0.705837\pi$
$648$	0	0
$649$	2.48992e6	0.232045
$650$	0	0
$651$	6.06001e6	0.560430
$652$	0	0
$653$	−423926.	−0.0389051	−0.0194526	−	0.999811i	$-0.506192\pi$
$653$	−423926.	−0.0389051	−0.0194526	+	0.999811i	$0.506192\pi$
$654$	0	0
$655$	6.85743e6	0.624537
$656$	0	0
$657$	5.69141e6	0.514406
$658$	0	0
$659$	−1.64781e7	−1.47807	−0.739035	−	0.673668i	$-0.764718\pi$
$659$	−1.64781e7	−1.47807	−0.739035	+	0.673668i	$0.764718\pi$
$660$	0	0
$661$	1.71556e7	1.52723	0.763613	−	0.645674i	$-0.223423\pi$
$661$	1.71556e7	1.52723	0.763613	+	0.645674i	$0.223423\pi$
$662$	0	0
$663$	−2.14982e7	−1.89941
$664$	0	0
$665$	−231157.	−0.0202699
$666$	0	0
$667$	−2.33592e7	−2.03303
$668$	0	0
$669$	5.16592e6	0.446254
$670$	0	0
$671$	2.10550e7	1.80530
$672$	0	0
$673$	2.24172e7	1.90785	0.953923	−	0.300053i	$-0.0970043\pi$
$673$	2.24172e7	1.90785	0.953923	+	0.300053i	$0.0970043\pi$
$674$	0	0
$675$	−1.67171e6	−0.141222
$676$	0	0
$677$	5.63329e6	0.472379	0.236190	−	0.971707i	$-0.424101\pi$
$677$	5.63329e6	0.472379	0.236190	+	0.971707i	$0.424101\pi$
$678$	0	0
$679$	1.60704e7	1.33768
$680$	0	0
$681$	−2.35134e7	−1.94289
$682$	0	0
$683$	1.27876e6	0.104891	0.0524453	−	0.998624i	$-0.483298\pi$
$683$	1.27876e6	0.104891	0.0524453	+	0.998624i	$0.483298\pi$
$684$	0	0
$685$	−7.72957e6	−0.629403
$686$	0	0
$687$	−1.07939e7	−0.872539
$688$	0	0
$689$	−2.23256e7	−1.79166
$690$	0	0
$691$	712345.	0.0567539	0.0283769	−	0.999597i	$-0.490966\pi$
$691$	712345.	0.0567539	0.0283769	+	0.999597i	$0.490966\pi$
$692$	0	0
$693$	5.24628e6	0.414972
$694$	0	0
$695$	9.53449e6	0.748748
$696$	0	0
$697$	−7.54127e6	−0.587980
$698$	0	0
$699$	−1.21676e7	−0.941915
$700$	0	0
$701$	−7.89586e6	−0.606882	−0.303441	−	0.952850i	$-0.598136\pi$
$701$	−7.89586e6	−0.606882	−0.303441	+	0.952850i	$0.598136\pi$
$702$	0	0
$703$	−550091.	−0.0419804
$704$	0	0
$705$	5.25543e6	0.398231
$706$	0	0
$707$	3.74087e6	0.281464
$708$	0	0
$709$	812363.	0.0606924	0.0303462	−	0.999539i	$-0.490339\pi$
$709$	812363.	0.0606924	0.0303462	+	0.999539i	$0.490339\pi$
$710$	0	0
$711$	−2.89719e6	−0.214933
$712$	0	0
$713$	9.94258e6	0.732446
$714$	0	0
$715$	8.30858e6	0.607801
$716$	0	0
$717$	−1.22681e7	−0.891212
$718$	0	0
$719$	−1.46975e7	−1.06028	−0.530142	−	0.847909i	$-0.677861\pi$
$719$	−1.46975e7	−1.06028	−0.530142	+	0.847909i	$0.677861\pi$
$720$	0	0
$721$	−1.71565e7	−1.22911
$722$	0	0
$723$	−3.54911e6	−0.252507
$724$	0	0
$725$	−3.95193e6	−0.279231
$726$	0	0
$727$	946478.	0.0664163	0.0332081	−	0.999448i	$-0.489428\pi$
$727$	946478.	0.0664163	0.0332081	+	0.999448i	$0.489428\pi$
$728$	0	0
$729$	4.81105e6	0.335290
$730$	0	0
$731$	3.88903e6	0.269183
$732$	0	0
$733$	−2.33370e6	−0.160430	−0.0802150	−	0.996778i	$-0.525561\pi$
$733$	−2.33370e6	−0.160430	−0.0802150	+	0.996778i	$0.525561\pi$
$734$	0	0
$735$	−899394.	−0.0614090
$736$	0	0
$737$	−1.73431e7	−1.17614
$738$	0	0
$739$	−1.17842e7	−0.793757	−0.396878	−	0.917871i	$-0.629906\pi$
$739$	−1.17842e7	−0.793757	−0.396878	+	0.917871i	$0.629906\pi$
$740$	0	0
$741$	−1.06243e6	−0.0710813
$742$	0	0
$743$	1.02537e7	0.681413	0.340706	−	0.940170i	$-0.389334\pi$
$743$	1.02537e7	0.681413	0.340706	+	0.940170i	$0.389334\pi$
$744$	0	0
$745$	2.54356e6	0.167900
$746$	0	0
$747$	−1.10186e7	−0.722476
$748$	0	0
$749$	−1.73058e6	−0.112717
$750$	0	0
$751$	2.02684e7	1.31135	0.655676	−	0.755043i	$-0.272384\pi$
$751$	2.02684e7	1.31135	0.655676	+	0.755043i	$0.272384\pi$
$752$	0	0
$753$	−2.21696e7	−1.42486
$754$	0	0
$755$	8.47106e6	0.540842
$756$	0	0
$757$	1.54478e7	0.979776	0.489888	−	0.871785i	$-0.337038\pi$
$757$	1.54478e7	0.979776	0.489888	+	0.871785i	$0.337038\pi$
$758$	0	0
$759$	2.99078e7	1.88443
$760$	0	0
$761$	−1.68238e7	−1.05308	−0.526540	−	0.850150i	$-0.676511\pi$
$761$	−1.68238e7	−1.05308	−0.526540	+	0.850150i	$0.676511\pi$
$762$	0	0
$763$	−1.20352e6	−0.0748414
$764$	0	0
$765$	−3.76795e6	−0.232783
$766$	0	0
$767$	4.30788e6	0.264408
$768$	0	0
$769$	2.25563e7	1.37547	0.687737	−	0.725960i	$-0.258604\pi$
$769$	2.25563e7	1.37547	0.687737	+	0.725960i	$0.258604\pi$
$770$	0	0
$771$	−9.13305e6	−0.553325
$772$	0	0
$773$	5.87038e6	0.353360	0.176680	−	0.984268i	$-0.443464\pi$
$773$	5.87038e6	0.353360	0.176680	+	0.984268i	$0.443464\pi$
$774$	0	0
$775$	1.68210e6	0.100600
$776$	0	0
$777$	1.63294e7	0.970329
$778$	0	0
$779$	−372686.	−0.0220039
$780$	0	0
$781$	5.53212e6	0.324537
$782$	0	0
$783$	1.69126e7	0.985838
$784$	0	0
$785$	−2.54125e6	−0.147188
$786$	0	0
$787$	2.45154e6	0.141092	0.0705459	−	0.997509i	$-0.477526\pi$
$787$	2.45154e6	0.141092	0.0705459	+	0.997509i	$0.477526\pi$
$788$	0	0
$789$	9.89886e6	0.566099
$790$	0	0
$791$	2.99708e7	1.70317
$792$	0	0
$793$	3.64279e7	2.05708
$794$	0	0
$795$	−1.35961e7	−0.762949
$796$	0	0
$797$	2.18622e7	1.21913	0.609564	−	0.792737i	$-0.291345\pi$
$797$	2.18622e7	1.21913	0.609564	+	0.792737i	$0.291345\pi$
$798$	0	0
$799$	−1.74676e7	−0.967980
$800$	0	0
$801$	6.57619e6	0.362154
$802$	0	0
$803$	2.54025e7	1.39023
$804$	0	0
$805$	1.12582e7	0.612320
$806$	0	0
$807$	1.63772e7	0.885231
$808$	0	0
$809$	2.66048e7	1.42919	0.714593	−	0.699541i	$-0.246612\pi$
$809$	2.66048e7	1.42919	0.714593	+	0.699541i	$0.246612\pi$
$810$	0	0
$811$	1.44067e7	0.769152	0.384576	−	0.923093i	$-0.374348\pi$
$811$	1.44067e7	0.769152	0.384576	+	0.923093i	$0.374348\pi$
$812$	0	0
$813$	−2.42540e7	−1.28694
$814$	0	0
$815$	−5.35341e6	−0.282317
$816$	0	0
$817$	192194.	0.0100736
$818$	0	0
$819$	9.07674e6	0.472847
$820$	0	0
$821$	−1.88597e6	−0.0976508	−0.0488254	−	0.998807i	$-0.515548\pi$
$821$	−1.88597e6	−0.0976508	−0.0488254	+	0.998807i	$0.515548\pi$
$822$	0	0
$823$	1.52832e7	0.786530	0.393265	−	0.919425i	$-0.371346\pi$
$823$	1.52832e7	0.786530	0.393265	+	0.919425i	$0.371346\pi$
$824$	0	0
$825$	5.05984e6	0.258822
$826$	0	0
$827$	2.78935e6	0.141821	0.0709103	−	0.997483i	$-0.477410\pi$
$827$	2.78935e6	0.141821	0.0709103	+	0.997483i	$0.477410\pi$
$828$	0	0
$829$	1.87077e6	0.0945440	0.0472720	−	0.998882i	$-0.484947\pi$
$829$	1.87077e6	0.0945440	0.0472720	+	0.998882i	$0.484947\pi$
$830$	0	0
$831$	−3.23641e6	−0.162578
$832$	0	0
$833$	2.98934e6	0.149267
$834$	0	0
$835$	−1.51593e7	−0.752422
$836$	0	0
$837$	−7.19866e6	−0.355171
$838$	0	0
$839$	−2.01293e6	−0.0987244	−0.0493622	−	0.998781i	$-0.515719\pi$
$839$	−2.01293e6	−0.0987244	−0.0493622	+	0.998781i	$0.515719\pi$
$840$	0	0
$841$	1.94703e7	0.949255
$842$	0	0
$843$	−3.29527e7	−1.59706
$844$	0	0
$845$	5.09259e6	0.245356
$846$	0	0
$847$	3.78381e6	0.181226
$848$	0	0
$849$	−3.66618e7	−1.74560
$850$	0	0
$851$	2.67915e7	1.26816
$852$	0	0
$853$	−1.09443e7	−0.515008	−0.257504	−	0.966277i	$-0.582900\pi$
$853$	−1.09443e7	−0.515008	−0.257504	+	0.966277i	$0.582900\pi$
$854$	0	0
$855$	−186210.	−0.00871140
$856$	0	0
$857$	−2.17692e7	−1.01249	−0.506245	−	0.862390i	$-0.668967\pi$
$857$	−2.17692e7	−1.01249	−0.506245	+	0.862390i	$0.668967\pi$
$858$	0	0
$859$	6.86741e6	0.317549	0.158774	−	0.987315i	$-0.449246\pi$
$859$	6.86741e6	0.317549	0.158774	+	0.987315i	$0.449246\pi$
$860$	0	0
$861$	1.10632e7	0.508595
$862$	0	0
$863$	−1.75635e7	−0.802755	−0.401378	−	0.915913i	$-0.631468\pi$
$863$	−1.75635e7	−0.802755	−0.401378	+	0.915913i	$0.631468\pi$
$864$	0	0
$865$	1.06740e7	0.485049
$866$	0	0
$867$	1.72880e7	0.781080
$868$	0	0
$869$	−1.29311e7	−0.580878
$870$	0	0
$871$	−3.00058e7	−1.34017
$872$	0	0
$873$	1.29456e7	0.574893
$874$	0	0
$875$	1.90467e6	0.0841008
$876$	0	0
$877$	2.95517e6	0.129743	0.0648715	−	0.997894i	$-0.479336\pi$
$877$	2.95517e6	0.129743	0.0648715	+	0.997894i	$0.479336\pi$
$878$	0	0
$879$	−4.50975e7	−1.96870
$880$	0	0
$881$	3.53773e7	1.53562	0.767811	−	0.640676i	$-0.221346\pi$
$881$	3.53773e7	1.53562	0.767811	+	0.640676i	$0.221346\pi$
$882$	0	0
$883$	−2.26106e7	−0.975912	−0.487956	−	0.872868i	$-0.662257\pi$
$883$	−2.26106e7	−0.975912	−0.487956	+	0.872868i	$0.662257\pi$
$884$	0	0
$885$	2.62345e6	0.112594
$886$	0	0
$887$	−8.57410e6	−0.365914	−0.182957	−	0.983121i	$-0.558567\pi$
$887$	−8.57410e6	−0.365914	−0.182957	+	0.983121i	$0.558567\pi$
$888$	0	0
$889$	1.07763e7	0.457314
$890$	0	0
$891$	−3.21122e7	−1.35511
$892$	0	0
$893$	−863241.	−0.0362246
$894$	0	0
$895$	−1.70691e7	−0.712284
$896$	0	0
$897$	5.17444e7	2.14725
$898$	0	0
$899$	−1.70177e7	−0.702265
$900$	0	0
$901$	4.51896e7	1.85450
$902$	0	0
$903$	−5.70528e6	−0.232840
$904$	0	0
$905$	−1.81416e7	−0.736301
$906$	0	0
$907$	−1.76363e7	−0.711852	−0.355926	−	0.934514i	$-0.615834\pi$
$907$	−1.76363e7	−0.711852	−0.355926	+	0.934514i	$0.615834\pi$
$908$	0	0
$909$	3.01349e6	0.120965
$910$	0	0
$911$	−1.48438e7	−0.592585	−0.296292	−	0.955097i	$-0.595750\pi$
$911$	−1.48438e7	−0.592585	−0.296292	+	0.955097i	$0.595750\pi$
$912$	0	0
$913$	−4.91792e7	−1.95256
$914$	0	0
$915$	2.21842e7	0.875973
$916$	0	0
$917$	3.34365e7	1.31310
$918$	0	0
$919$	−448602.	−0.0175215	−0.00876077	−	0.999962i	$-0.502789\pi$
$919$	−448602.	−0.0175215	−0.00876077	+	0.999962i	$0.502789\pi$
$920$	0	0
$921$	4.69071e7	1.82217
$922$	0	0
$923$	9.57129e6	0.369799
$924$	0	0
$925$	4.53262e6	0.174179
$926$	0	0
$927$	−1.38206e7	−0.528235
$928$	0	0
$929$	2.11976e6	0.0805836	0.0402918	−	0.999188i	$-0.487171\pi$
$929$	2.11976e6	0.0805836	0.0402918	+	0.999188i	$0.487171\pi$
$930$	0	0
$931$	147732.	0.00558598
$932$	0	0
$933$	3.44647e7	1.29619
$934$	0	0
$935$	−1.68175e7	−0.629119
$936$	0	0
$937$	5.54945e6	0.206491	0.103245	−	0.994656i	$-0.467077\pi$
$937$	5.54945e6	0.206491	0.103245	+	0.994656i	$0.467077\pi$
$938$	0	0
$939$	4.01680e7	1.48667
$940$	0	0
$941$	−6.00433e6	−0.221050	−0.110525	−	0.993873i	$-0.535253\pi$
$941$	−6.00433e6	−0.221050	−0.110525	+	0.993873i	$0.535253\pi$
$942$	0	0
$943$	1.81512e7	0.664701
$944$	0	0
$945$	−8.15119e6	−0.296921
$946$	0	0
$947$	1.19597e7	0.433357	0.216679	−	0.976243i	$-0.430478\pi$
$947$	1.19597e7	0.433357	0.216679	+	0.976243i	$0.430478\pi$
$948$	0	0
$949$	4.39496e7	1.58413
$950$	0	0
$951$	2.74156e7	0.982982
$952$	0	0
$953$	−2.84885e7	−1.01610	−0.508051	−	0.861327i	$-0.669634\pi$
$953$	−2.84885e7	−1.01610	−0.508051	+	0.861327i	$0.669634\pi$
$954$	0	0
$955$	1.65500e7	0.587205
$956$	0	0
$957$	−5.11901e7	−1.80678
$958$	0	0
$959$	−3.76891e7	−1.32333
$960$	0	0
$961$	−2.13858e7	−0.746993
$962$	0	0
$963$	−1.39409e6	−0.0484422
$964$	0	0
$965$	−1.20389e7	−0.416169
$966$	0	0
$967$	2.44173e6	0.0839714	0.0419857	−	0.999118i	$-0.486632\pi$
$967$	2.44173e6	0.0839714	0.0419857	+	0.999118i	$0.486632\pi$
$968$	0	0
$969$	2.15048e6	0.0735744
$970$	0	0
$971$	−2.63888e7	−0.898198	−0.449099	−	0.893482i	$-0.648255\pi$
$971$	−2.63888e7	−0.898198	−0.449099	+	0.893482i	$0.648255\pi$
$972$	0	0
$973$	4.64898e7	1.57426
$974$	0	0
$975$	8.75417e6	0.294920
$976$	0	0
$977$	−6.67324e6	−0.223666	−0.111833	−	0.993727i	$-0.535672\pi$
$977$	−6.67324e6	−0.223666	−0.111833	+	0.993727i	$0.535672\pi$
$978$	0	0
$979$	2.93516e7	0.978756
$980$	0	0
$981$	−969506.	−0.0321646
$982$	0	0
$983$	−1.21646e7	−0.401528	−0.200764	−	0.979640i	$-0.564342\pi$
$983$	−1.21646e7	−0.401528	−0.200764	+	0.979640i	$0.564342\pi$
$984$	0	0
$985$	−1.42316e7	−0.467372
$986$	0	0
$987$	2.56253e7	0.837289
$988$	0	0
$989$	−9.36057e6	−0.304307
$990$	0	0
$991$	−1.16204e7	−0.375869	−0.187935	−	0.982182i	$-0.560179\pi$
$991$	−1.16204e7	−0.375869	−0.187935	+	0.982182i	$0.560179\pi$
$992$	0	0
$993$	−3.74750e7	−1.20606
$994$	0	0
$995$	−2.12538e7	−0.680579
$996$	0	0
$997$	4.61789e7	1.47132	0.735658	−	0.677354i	$-0.236873\pi$
$997$	4.61789e7	1.47132	0.735658	+	0.677354i	$0.236873\pi$
$998$	0	0
$999$	−1.93977e7	−0.614944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.a.x.1.3		3
4.3	odd	2		320.6.a.y.1.1		3
8.3	odd	2		160.6.a.f.1.3	✓	3
8.5	even	2		160.6.a.g.1.1	yes	3
40.3	even	4		800.6.c.k.449.5		6
40.13	odd	4		800.6.c.j.449.2		6
40.19	odd	2		800.6.a.o.1.1		3
40.27	even	4		800.6.c.k.449.2		6
40.29	even	2		800.6.a.n.1.3		3
40.37	odd	4		800.6.c.j.449.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.6.a.f.1.3	✓	3	8.3	odd	2
160.6.a.g.1.1	yes	3	8.5	even	2
320.6.a.x.1.3		3	1.1	even	1	trivial
320.6.a.y.1.1		3	4.3	odd	2
800.6.a.n.1.3		3	40.29	even	2
800.6.a.o.1.1		3	40.19	odd	2
800.6.c.j.449.2		6	40.13	odd	4
800.6.c.j.449.5		6	40.37	odd	4
800.6.c.k.449.2		6	40.27	even	4
800.6.c.k.449.5		6	40.3	even	4

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

\(f(q)\)	\(=\)	\(q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +O(q^{10})\)	\(q+18.4715 q^{3} +25.0000 q^{5} +121.899 q^{7} +98.1968 q^{9} +438.282 q^{11} +758.285 q^{13} +461.788 q^{15} -1534.86 q^{17} -75.8518 q^{19} +2251.66 q^{21} +3694.27 q^{23} +625.000 q^{25} -2674.73 q^{27} -6323.09 q^{29} +2691.35 q^{31} +8095.74 q^{33} +3047.47 q^{35} +7252.19 q^{37} +14006.7 q^{39} +4913.34 q^{41} -2533.81 q^{43} +2454.92 q^{45} +11380.6 q^{47} -1947.64 q^{49} -28351.1 q^{51} -29442.3 q^{53} +10957.1 q^{55} -1401.10 q^{57} +5681.08 q^{59} +48039.8 q^{61} +11970.1 q^{63} +18957.1 q^{65} -39570.6 q^{67} +68238.7 q^{69} +12622.3 q^{71} +57959.2 q^{73} +11544.7 q^{75} +53426.2 q^{77} -29504.0 q^{79} -73268.2 q^{81} -112209. q^{83} -38371.4 q^{85} -116797. q^{87} +66969.5 q^{89} +92434.2 q^{91} +49713.4 q^{93} -1896.30 q^{95} +131833. q^{97} +43037.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 10 q^{3} + 75 q^{5} - 6 q^{7} + 467 q^{9}+O(q^{10}) \) 3 * q - 10 * q^3 + 75 * q^5 - 6 * q^7 + 467 * q^9	\( 3 q - 10 q^{3} + 75 q^{5} - 6 q^{7} + 467 q^{9} + 396 q^{11} + 354 q^{13} - 250 q^{15} + 1158 q^{17} - 3192 q^{19} + 820 q^{21} + 6126 q^{23} + 1875 q^{25} - 13804 q^{27} - 426 q^{29} + 3276 q^{31} - 2408 q^{33} - 150 q^{35} + 11562 q^{37} + 21348 q^{39} + 12450 q^{41} - 26346 q^{43} + 11675 q^{45} + 36762 q^{47} - 3849 q^{49} - 71444 q^{51} + 21162 q^{53} + 9900 q^{55} + 69136 q^{57} - 35040 q^{59} + 24138 q^{61} + 80986 q^{63} + 8850 q^{65} + 9570 q^{67} + 27036 q^{69} + 88092 q^{71} + 66750 q^{73} - 6250 q^{75} + 136488 q^{77} - 92952 q^{79} + 151391 q^{81} - 30258 q^{83} + 28950 q^{85} - 26228 q^{87} + 172686 q^{89} + 106812 q^{91} + 318232 q^{93} - 79800 q^{95} + 170910 q^{97} + 351436 q^{99}+O(q^{100}) \) 3 * q - 10 * q^3 + 75 * q^5 - 6 * q^7 + 467 * q^9 + 396 * q^11 + 354 * q^13 - 250 * q^15 + 1158 * q^17 - 3192 * q^19 + 820 * q^21 + 6126 * q^23 + 1875 * q^25 - 13804 * q^27 - 426 * q^29 + 3276 * q^31 - 2408 * q^33 - 150 * q^35 + 11562 * q^37 + 21348 * q^39 + 12450 * q^41 - 26346 * q^43 + 11675 * q^45 + 36762 * q^47 - 3849 * q^49 - 71444 * q^51 + 21162 * q^53 + 9900 * q^55 + 69136 * q^57 - 35040 * q^59 + 24138 * q^61 + 80986 * q^63 + 8850 * q^65 + 9570 * q^67 + 27036 * q^69 + 88092 * q^71 + 66750 * q^73 - 6250 * q^75 + 136488 * q^77 - 92952 * q^79 + 151391 * q^81 - 30258 * q^83 + 28950 * q^85 - 26228 * q^87 + 172686 * q^89 + 106812 * q^91 + 318232 * q^93 - 79800 * q^95 + 170910 * q^97 + 351436 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more