LMFDB - Embedded newform 320.6.d.c.161.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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, 1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$51.3228223402$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + \cdots + 653157349 $ x^12 - 343x^10 - 696x^9 + 44406x^8 + 179640x^7 - 2401691x^6 - 15554592x^5 + 26901210x^4 + 434775816x^3 + 1271335685x^2 + 1475231592x + 653157349
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$-7.49748 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.c.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} +O(q^{10})$	$q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} -94.5683i q^{11} -887.073i q^{13} -126.669 q^{15} +1448.19 q^{17} -611.677i q^{19} -1268.38i q^{21} +814.118 q^{23} -625.000 q^{25} +2332.37i q^{27} +5291.31i q^{29} +3614.36 q^{31} +479.155 q^{33} -6258.32i q^{35} +9480.98i q^{37} +4494.59 q^{39} -9910.48 q^{41} +20213.1i q^{43} +5433.20i q^{45} -20765.1 q^{47} +45859.5 q^{49} +7337.66i q^{51} -13009.9i q^{53} +2364.21 q^{55} +3099.22 q^{57} -16632.8i q^{59} -715.975i q^{61} -54404.3 q^{63} +22176.8 q^{65} +21955.0i q^{67} +4124.94i q^{69} -38473.5 q^{71} -40857.0 q^{73} -3166.73i q^{75} +23673.5i q^{77} +84846.9 q^{79} +40993.1 q^{81} +116115. i q^{83} +36204.9i q^{85} -26809.8 q^{87} +121182. q^{89} +222064. i q^{91} +18313.1i q^{93} +15291.9 q^{95} +21598.7 q^{97} -20552.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 268 q^{7} - 428 q^{9}+O(q^{10}) $ 12 * q - 268 * q^7 - 428 * q^9	$ 12 q - 268 q^{7} - 428 q^{9} + 900 q^{15} + 2400 q^{17} - 8108 q^{23} - 7500 q^{25} - 7976 q^{31} - 20776 q^{33} - 40984 q^{39} - 56408 q^{41} - 21172 q^{47} - 5540 q^{49} - 18400 q^{55} + 39992 q^{57} - 179516 q^{63} + 44000 q^{65} - 367704 q^{71} + 58736 q^{73} - 26192 q^{79} + 411692 q^{81} - 183200 q^{87} + 87672 q^{89} - 121000 q^{95} - 172336 q^{97}+O(q^{100}) $ 12 * q - 268 * q^7 - 428 * q^9 + 900 * q^15 + 2400 * q^17 - 8108 * q^23 - 7500 * q^25 - 7976 * q^31 - 20776 * q^33 - 40984 * q^39 - 56408 * q^41 - 21172 * q^47 - 5540 * q^49 - 18400 * q^55 + 39992 * q^57 - 179516 * q^63 + 44000 * q^65 - 367704 * q^71 + 58736 * q^73 - 26192 * q^79 + 411692 * q^81 - 183200 * q^87 + 87672 * q^89 - 121000 * q^95 - 172336 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	5.06676i	0.325033i	0.986706	+	0.162516i	$0.0519611\pi$
$3$	5.06676i	0.325033i	−0.986706	+	0.162516i	$0.948039\pi$
$4$	0	0
$5$	25.0000i	0.447214i
$5$	25.0000i	0.447214i
$6$	0	0
$7$	−250.333	−1.93096	−0.965479	−	0.260481i	$-0.916119\pi$
$7$	−250.333	−1.93096	−0.965479	+	0.260481i	$0.916119\pi$
$8$	0	0
$9$	217.328	0.894354
$10$	0	0
$11$	− 94.5683i	− 0.235648i	−0.993034	−	0.117824i	$-0.962408\pi$
$11$	− 94.5683i	− 0.235648i	0.993034	−	0.117824i	$-0.0375919\pi$
$12$	0	0
$13$	− 887.073i	− 1.45580i	−0.685684	−	0.727899i	$-0.740497\pi$
$13$	− 887.073i	− 1.45580i	0.685684	−	0.727899i	$-0.259503\pi$
$14$	0	0
$15$	−126.669	−0.145359
$16$	0	0
$17$	1448.19	1.21536	0.607679	−	0.794182i	$-0.292101\pi$
$17$	1448.19	1.21536	0.607679	+	0.794182i	$0.292101\pi$
$18$	0	0
$19$	− 611.677i	− 0.388721i	−0.980930	−	0.194361i	$-0.937737\pi$
$19$	− 611.677i	− 0.388721i	0.980930	−	0.194361i	$-0.0622632\pi$
$20$	0	0
$21$	− 1268.38i	− 0.627625i
$22$	0	0
$23$	814.118	0.320899	0.160449	−	0.987044i	$-0.448706\pi$
$23$	814.118	0.320899	0.160449	+	0.987044i	$0.448706\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	2332.37i	0.615727i
$28$	0	0
$29$	5291.31i	1.16834i	0.811633	+	0.584168i	$0.198579\pi$
$29$	5291.31i	1.16834i	−0.811633	+	0.584168i	$0.801421\pi$
$30$	0	0
$31$	3614.36	0.675502	0.337751	−	0.941236i	$-0.390334\pi$
$31$	3614.36	0.675502	0.337751	+	0.941236i	$0.390334\pi$
$32$	0	0
$33$	479.155	0.0765934
$34$	0	0
$35$	− 6258.32i	− 0.863551i
$36$	0	0
$37$	9480.98i	1.13854i	0.822150	+	0.569271i	$0.192774\pi$
$37$	9480.98i	1.13854i	−0.822150	+	0.569271i	$0.807226\pi$
$38$	0	0
$39$	4494.59	0.473182
$40$	0	0
$41$	−9910.48	−0.920736	−0.460368	−	0.887728i	$-0.652282\pi$
$41$	−9910.48	−0.920736	−0.460368	+	0.887728i	$0.652282\pi$
$42$	0	0
$43$	20213.1i	1.66710i	0.552445	+	0.833549i	$0.313695\pi$
$43$	20213.1i	1.66710i	−0.552445	+	0.833549i	$0.686305\pi$
$44$	0	0
$45$	5433.20i	0.399967i
$46$	0	0
$47$	−20765.1	−1.37116	−0.685581	−	0.727996i	$-0.740452\pi$
$47$	−20765.1	−1.37116	−0.685581	+	0.727996i	$0.740452\pi$
$48$	0	0
$49$	45859.5	2.72860
$50$	0	0
$51$	7337.66i	0.395032i
$52$	0	0
$53$	− 13009.9i	− 0.636187i	−0.948059	−	0.318094i	$-0.896957\pi$
$53$	− 13009.9i	− 0.636187i	0.948059	−	0.318094i	$-0.103043\pi$
$54$	0	0
$55$	2364.21	0.105385
$56$	0	0
$57$	3099.22	0.126347
$58$	0	0
$59$	− 16632.8i	− 0.622065i	−0.950399	−	0.311032i	$-0.899325\pi$
$59$	− 16632.8i	− 0.622065i	0.950399	−	0.311032i	$-0.100675\pi$
$60$	0	0
$61$	− 715.975i	− 0.0246362i	−0.999924	−	0.0123181i	$-0.996079\pi$
$61$	− 715.975i	− 0.0246362i	0.999924	−	0.0123181i	$-0.00392107\pi$
$62$	0	0
$63$	−54404.3	−1.72696
$64$	0	0
$65$	22176.8	0.651053
$66$	0	0
$67$	21955.0i	0.597513i	0.954329	+	0.298756i	$0.0965718\pi$
$67$	21955.0i	0.597513i	−0.954329	+	0.298756i	$0.903428\pi$
$68$	0	0
$69$	4124.94i	0.104303i
$70$	0	0
$71$	−38473.5	−0.905765	−0.452883	−	0.891570i	$-0.649604\pi$
$71$	−38473.5	−0.905765	−0.452883	+	0.891570i	$0.649604\pi$
$72$	0	0
$73$	−40857.0	−0.897344	−0.448672	−	0.893696i	$-0.648103\pi$
$73$	−40857.0	−0.897344	−0.448672	+	0.893696i	$0.648103\pi$
$74$	0	0
$75$	− 3166.73i	− 0.0650066i
$76$	0	0
$77$	23673.5i	0.455026i
$78$	0	0
$79$	84846.9	1.52957	0.764783	−	0.644288i	$-0.222846\pi$
$79$	84846.9	1.52957	0.764783	+	0.644288i	$0.222846\pi$
$80$	0	0
$81$	40993.1	0.694222
$82$	0	0
$83$	116115.i	1.85010i	0.379851	+	0.925048i	$0.375975\pi$
$83$	116115.i	1.85010i	−0.379851	+	0.925048i	$0.624025\pi$
$84$	0	0
$85$	36204.9i	0.543525i
$86$	0	0
$87$	−26809.8	−0.379748
$88$	0	0
$89$	121182.	1.62168	0.810838	−	0.585271i	$-0.199012\pi$
$89$	121182.	1.62168	0.810838	+	0.585271i	$0.199012\pi$
$90$	0	0
$91$	222064.i	2.81108i
$92$	0	0
$93$	18313.1i	0.219560i
$94$	0	0
$95$	15291.9	0.173841
$96$	0	0
$97$	21598.7	0.233077	0.116538	−	0.993186i	$-0.462820\pi$
$97$	21598.7	0.233077	0.116538	+	0.993186i	$0.462820\pi$
$98$	0	0
$99$	− 20552.3i	− 0.210753i
$100$	0	0
$101$	105401.i	1.02811i	0.857756	+	0.514057i	$0.171858\pi$
$101$	105401.i	1.02811i	−0.857756	+	0.514057i	$0.828142\pi$
$102$	0	0
$103$	−15362.8	−0.142684	−0.0713421	−	0.997452i	$-0.522728\pi$
$103$	−15362.8	−0.142684	−0.0713421	+	0.997452i	$0.522728\pi$
$104$	0	0
$105$	31709.4	0.280682
$106$	0	0
$107$	14774.1i	0.124750i	0.998053	+	0.0623750i	$0.0198675\pi$
$107$	14774.1i	0.124750i	−0.998053	+	0.0623750i	$0.980133\pi$
$108$	0	0
$109$	165494.i	1.33419i	0.744973	+	0.667094i	$0.232462\pi$
$109$	165494.i	1.33419i	−0.744973	+	0.667094i	$0.767538\pi$
$110$	0	0
$111$	−48037.9	−0.370064
$112$	0	0
$113$	72833.6	0.536582	0.268291	−	0.963338i	$-0.413541\pi$
$113$	72833.6	0.536582	0.268291	+	0.963338i	$0.413541\pi$
$114$	0	0
$115$	20352.9i	0.143510i
$116$	0	0
$117$	− 192786.i	− 1.30200i
$118$	0	0
$119$	−362531.	−2.34681
$120$	0	0
$121$	152108.	0.944470
$122$	0	0
$123$	− 50214.0i	− 0.299269i
$124$	0	0
$125$	− 15625.0i	− 0.0894427i
$126$	0	0
$127$	8287.17	0.0455929	0.0227964	−	0.999740i	$-0.492743\pi$
$127$	8287.17	0.0455929	0.0227964	+	0.999740i	$0.492743\pi$
$128$	0	0
$129$	−102415.	−0.541862
$130$	0	0
$131$	− 177213.i	− 0.902231i	−0.892466	−	0.451115i	$-0.851026\pi$
$131$	− 177213.i	− 0.902231i	0.892466	−	0.451115i	$-0.148974\pi$
$132$	0	0
$133$	153123.i	0.750605i
$134$	0	0
$135$	−58309.3	−0.275362
$136$	0	0
$137$	352108.	1.60278	0.801391	−	0.598141i	$-0.204094\pi$
$137$	352108.	1.60278	0.801391	+	0.598141i	$0.204094\pi$
$138$	0	0
$139$	145465.i	0.638589i	0.947656	+	0.319294i	$0.103446\pi$
$139$	145465.i	0.638589i	−0.947656	+	0.319294i	$0.896554\pi$
$140$	0	0
$141$	− 105212.i	− 0.445673i
$142$	0	0
$143$	−83889.0	−0.343056
$144$	0	0
$145$	−132283.	−0.522496
$146$	0	0
$147$	232359.i	0.886884i
$148$	0	0
$149$	− 162250.i	− 0.598712i	−0.954142	−	0.299356i	$-0.903228\pi$
$149$	− 162250.i	− 0.598712i	0.954142	−	0.299356i	$-0.0967718\pi$
$150$	0	0
$151$	427740.	1.52664	0.763321	−	0.646019i	$-0.223567\pi$
$151$	427740.	1.52664	0.763321	+	0.646019i	$0.223567\pi$
$152$	0	0
$153$	314733.	1.08696
$154$	0	0
$155$	90358.9i	0.302094i
$156$	0	0
$157$	278644.i	0.902194i	0.892475	+	0.451097i	$0.148967\pi$
$157$	278644.i	0.902194i	−0.892475	+	0.451097i	$0.851033\pi$
$158$	0	0
$159$	65918.2	0.206782
$160$	0	0
$161$	−203800.	−0.619642
$162$	0	0
$163$	− 276615.i	− 0.815467i	−0.913101	−	0.407733i	$-0.866319\pi$
$163$	− 276615.i	− 0.815467i	0.913101	−	0.407733i	$-0.133681\pi$
$164$	0	0
$165$	11978.9i	0.0342536i
$166$	0	0
$167$	−3008.18	−0.00834667	−0.00417334	−	0.999991i	$-0.501328\pi$
$167$	−3008.18	−0.00834667	−0.00417334	+	0.999991i	$0.501328\pi$
$168$	0	0
$169$	−415606.	−1.11935
$170$	0	0
$171$	− 132935.i	− 0.347654i
$172$	0	0
$173$	473850.i	1.20372i	0.798602	+	0.601860i	$0.205573\pi$
$173$	473850.i	1.20372i	−0.798602	+	0.601860i	$0.794427\pi$
$174$	0	0
$175$	156458.	0.386192
$176$	0	0
$177$	84274.5	0.202192
$178$	0	0
$179$	661760.i	1.54372i	0.635794	+	0.771859i	$0.280673\pi$
$179$	661760.i	1.54372i	−0.635794	+	0.771859i	$0.719327\pi$
$180$	0	0
$181$	390057.i	0.884977i	0.896774	+	0.442488i	$0.145904\pi$
$181$	390057.i	0.884977i	−0.896774	+	0.442488i	$0.854096\pi$
$182$	0	0
$183$	3627.67	0.00800757
$184$	0	0
$185$	−237025.	−0.509171
$186$	0	0
$187$	− 136953.i	− 0.286397i
$188$	0	0
$189$	− 583869.i	− 1.18894i
$190$	0	0
$191$	456202.	0.904845	0.452422	−	0.891804i	$-0.350560\pi$
$191$	456202.	0.904845	0.452422	+	0.891804i	$0.350560\pi$
$192$	0	0
$193$	−28265.7	−0.0546218	−0.0273109	−	0.999627i	$-0.508694\pi$
$193$	−28265.7	−0.0546218	−0.0273109	+	0.999627i	$0.508694\pi$
$194$	0	0
$195$	112365.i	0.211614i
$196$	0	0
$197$	− 746684.i	− 1.37079i	−0.728171	−	0.685396i	$-0.759629\pi$
$197$	− 746684.i	− 1.37079i	0.728171	−	0.685396i	$-0.240371\pi$
$198$	0	0
$199$	−271671.	−0.486307	−0.243153	−	0.969988i	$-0.578182\pi$
$199$	−271671.	−0.486307	−0.243153	+	0.969988i	$0.578182\pi$
$200$	0	0
$201$	−111241.	−0.194211
$202$	0	0
$203$	− 1.32459e6i	− 2.25601i
$204$	0	0
$205$	− 247762.i	− 0.411766i
$206$	0	0
$207$	176931.	0.286997
$208$	0	0
$209$	−57845.3	−0.0916014
$210$	0	0
$211$	20283.2i	0.0313640i	0.999877	+	0.0156820i	$0.00499194\pi$
$211$	20283.2i	0.0313640i	−0.999877	+	0.0156820i	$0.995008\pi$
$212$	0	0
$213$	− 194936.i	− 0.294404i
$214$	0	0
$215$	−505327.	−0.745549
$216$	0	0
$217$	−904792.	−1.30437
$218$	0	0
$219$	− 207013.i	− 0.291667i
$220$	0	0
$221$	− 1.28465e6i	− 1.76932i
$222$	0	0
$223$	−30380.0	−0.0409096	−0.0204548	−	0.999791i	$-0.506511\pi$
$223$	−30380.0	−0.0409096	−0.0204548	+	0.999791i	$0.506511\pi$
$224$	0	0
$225$	−135830.	−0.178871
$226$	0	0
$227$	− 1.45723e6i	− 1.87699i	−0.345286	−	0.938497i	$-0.612218\pi$
$227$	− 1.45723e6i	− 1.87699i	0.345286	−	0.938497i	$-0.387782\pi$
$228$	0	0
$229$	− 963464.i	− 1.21408i	−0.794672	−	0.607039i	$-0.792357\pi$
$229$	− 963464.i	− 1.21408i	0.794672	−	0.607039i	$-0.207643\pi$
$230$	0	0
$231$	−119948.	−0.147899
$232$	0	0
$233$	−1.31864e6	−1.59125	−0.795623	−	0.605792i	$-0.792856\pi$
$233$	−1.31864e6	−1.59125	−0.795623	+	0.605792i	$0.792856\pi$
$234$	0	0
$235$	− 519127.i	− 0.613202i
$236$	0	0
$237$	429899.i	0.497159i
$238$	0	0
$239$	1.17974e6	1.33595	0.667974	−	0.744184i	$-0.267162\pi$
$239$	1.17974e6	1.33595	0.667974	+	0.744184i	$0.267162\pi$
$240$	0	0
$241$	785926.	0.871644	0.435822	−	0.900033i	$-0.356458\pi$
$241$	785926.	0.871644	0.435822	+	0.900033i	$0.356458\pi$
$242$	0	0
$243$	774469.i	0.841372i
$244$	0	0
$245$	1.14649e6i	1.22027i
$246$	0	0
$247$	−542603.	−0.565900
$248$	0	0
$249$	−588329.	−0.601342
$250$	0	0
$251$	923722.i	0.925459i	0.886500	+	0.462729i	$0.153130\pi$
$251$	923722.i	0.925459i	−0.886500	+	0.462729i	$0.846870\pi$
$252$	0	0
$253$	− 76989.7i	− 0.0756191i
$254$	0	0
$255$	−183441.	−0.176664
$256$	0	0
$257$	−497776.	−0.470112	−0.235056	−	0.971982i	$-0.575527\pi$
$257$	−497776.	−0.470112	−0.235056	+	0.971982i	$0.575527\pi$
$258$	0	0
$259$	− 2.37340e6i	− 2.19848i
$260$	0	0
$261$	1.14995e6i	1.04491i
$262$	0	0
$263$	−745381.	−0.664491	−0.332245	−	0.943193i	$-0.607806\pi$
$263$	−745381.	−0.664491	−0.332245	+	0.943193i	$0.607806\pi$
$264$	0	0
$265$	325248.	0.284512
$266$	0	0
$267$	614002.i	0.527098i
$268$	0	0
$269$	1.80992e6i	1.52503i	0.646970	+	0.762516i	$0.276036\pi$
$269$	1.80992e6i	1.52503i	−0.646970	+	0.762516i	$0.723964\pi$
$270$	0	0
$271$	2.20319e6	1.82234	0.911169	−	0.412032i	$-0.135181\pi$
$271$	2.20319e6	1.82234	0.911169	+	0.412032i	$0.135181\pi$
$272$	0	0
$273$	−1.12514e6	−0.913695
$274$	0	0
$275$	59105.2i	0.0471296i
$276$	0	0
$277$	− 1.85372e6i	− 1.45159i	−0.687912	−	0.725795i	$-0.741472\pi$
$277$	− 1.85372e6i	− 1.45159i	0.687912	−	0.725795i	$-0.258528\pi$
$278$	0	0
$279$	785500.	0.604138
$280$	0	0
$281$	−1.06542e6	−0.804926	−0.402463	−	0.915436i	$-0.631846\pi$
$281$	−1.06542e6	−0.804926	−0.402463	+	0.915436i	$0.631846\pi$
$282$	0	0
$283$	− 38350.4i	− 0.0284645i	−0.999899	−	0.0142323i	$-0.995470\pi$
$283$	− 38350.4i	− 0.0284645i	0.999899	−	0.0142323i	$-0.00453042\pi$
$284$	0	0
$285$	77480.6i	0.0565042i
$286$	0	0
$287$	2.48092e6	1.77790
$288$	0	0
$289$	677410.	0.477097
$290$	0	0
$291$	109436.i	0.0757576i
$292$	0	0
$293$	463895.i	0.315683i	0.987464	+	0.157841i	$0.0504534\pi$
$293$	463895.i	0.315683i	−0.987464	+	0.157841i	$0.949547\pi$
$294$	0	0
$295$	415820.	0.278196
$296$	0	0
$297$	220568.	0.145095
$298$	0	0
$299$	− 722182.i	− 0.467164i
$300$	0	0
$301$	− 5.06000e6i	− 3.21910i
$302$	0	0
$303$	−534042.	−0.334171
$304$	0	0
$305$	17899.4	0.0110176
$306$	0	0
$307$	203618.i	0.123302i	0.998098	+	0.0616509i	$0.0196365\pi$
$307$	203618.i	0.123302i	−0.998098	+	0.0616509i	$0.980363\pi$
$308$	0	0
$309$	− 77839.4i	− 0.0463771i
$310$	0	0
$311$	−1.32625e6	−0.777541	−0.388770	−	0.921335i	$-0.627100\pi$
$311$	−1.32625e6	−0.777541	−0.388770	+	0.921335i	$0.627100\pi$
$312$	0	0
$313$	−1.62292e6	−0.936344	−0.468172	−	0.883637i	$-0.655087\pi$
$313$	−1.62292e6	−0.936344	−0.468172	+	0.883637i	$0.655087\pi$
$314$	0	0
$315$	− 1.36011e6i	− 0.772320i
$316$	0	0
$317$	920943.i	0.514736i	0.966313	+	0.257368i	$0.0828553\pi$
$317$	920943.i	0.514736i	−0.966313	+	0.257368i	$0.917145\pi$
$318$	0	0
$319$	500390.	0.275316
$320$	0	0
$321$	−74856.7	−0.0405479
$322$	0	0
$323$	− 885828.i	− 0.472436i
$324$	0	0
$325$	554421.i	0.291160i
$326$	0	0
$327$	−838521.	−0.433655
$328$	0	0
$329$	5.19818e6	2.64766
$330$	0	0
$331$	− 689680.i	− 0.346001i	−0.984922	−	0.173001i	$-0.944654\pi$
$331$	− 689680.i	− 0.346001i	0.984922	−	0.173001i	$-0.0553463\pi$
$332$	0	0
$333$	2.06048e6i	1.01826i
$334$	0	0
$335$	−548876.	−0.267216
$336$	0	0
$337$	1.23442e6	0.592089	0.296045	−	0.955174i	$-0.404332\pi$
$337$	1.23442e6	0.592089	0.296045	+	0.955174i	$0.404332\pi$
$338$	0	0
$339$	369031.i	0.174407i
$340$	0	0
$341$	− 341803.i	− 0.159181i
$342$	0	0
$343$	−7.27281e6	−3.33785
$344$	0	0
$345$	−103124.	−0.0466455
$346$	0	0
$347$	2.59951e6i	1.15896i	0.814987	+	0.579480i	$0.196744\pi$
$347$	2.59951e6i	1.15896i	−0.814987	+	0.579480i	$0.803256\pi$
$348$	0	0
$349$	277542.i	0.121974i	0.998139	+	0.0609868i	$0.0194247\pi$
$349$	277542.i	0.121974i	−0.998139	+	0.0609868i	$0.980575\pi$
$350$	0	0
$351$	2.06899e6	0.896375
$352$	0	0
$353$	−2.51736e6	−1.07525	−0.537624	−	0.843184i	$-0.680678\pi$
$353$	−2.51736e6	−1.07525	−0.537624	+	0.843184i	$0.680678\pi$
$354$	0	0
$355$	− 961837.i	− 0.405071i
$356$	0	0
$357$	− 1.83686e6i	− 0.762790i
$358$	0	0
$359$	−899617.	−0.368402	−0.184201	−	0.982889i	$-0.558970\pi$
$359$	−899617.	−0.368402	−0.184201	+	0.982889i	$0.558970\pi$
$360$	0	0
$361$	2.10195e6	0.848896
$362$	0	0
$363$	770694.i	0.306984i
$364$	0	0
$365$	− 1.02142e6i	− 0.401305i
$366$	0	0
$367$	2.07519e6	0.804253	0.402126	−	0.915584i	$-0.368271\pi$
$367$	2.07519e6	0.804253	0.402126	+	0.915584i	$0.368271\pi$
$368$	0	0
$369$	−2.15382e6	−0.823463
$370$	0	0
$371$	3.25681e6i	1.22845i
$372$	0	0
$373$	5.12372e6i	1.90684i	0.301651	+	0.953419i	$0.402462\pi$
$373$	5.12372e6i	1.90684i	−0.301651	+	0.953419i	$0.597538\pi$
$374$	0	0
$375$	79168.2	0.0290718
$376$	0	0
$377$	4.69378e6	1.70086
$378$	0	0
$379$	1.55330e6i	0.555465i	0.960659	+	0.277732i	$0.0895828\pi$
$379$	1.55330e6i	0.555465i	−0.960659	+	0.277732i	$0.910417\pi$
$380$	0	0
$381$	41989.1i	0.0148192i
$382$	0	0
$383$	1.17661e6	0.409861	0.204930	−	0.978777i	$-0.434303\pi$
$383$	1.17661e6	0.409861	0.204930	+	0.978777i	$0.434303\pi$
$384$	0	0
$385$	−591839.	−0.203494
$386$	0	0
$387$	4.39287e6i	1.49098i
$388$	0	0
$389$	1.56804e6i	0.525392i	0.964879	+	0.262696i	$0.0846116\pi$
$389$	1.56804e6i	0.525392i	−0.964879	+	0.262696i	$0.915388\pi$
$390$	0	0
$391$	1.17900e6	0.390007
$392$	0	0
$393$	897897.	0.293255
$394$	0	0
$395$	2.12117e6i	0.684043i
$396$	0	0
$397$	1.55070e6i	0.493799i	0.969041	+	0.246900i	$0.0794118\pi$
$397$	1.55070e6i	0.493799i	−0.969041	+	0.246900i	$0.920588\pi$
$398$	0	0
$399$	−775838.	−0.243971
$400$	0	0
$401$	1.78114e6	0.553144	0.276572	−	0.960993i	$-0.410802\pi$
$401$	1.78114e6	0.553144	0.276572	+	0.960993i	$0.410802\pi$
$402$	0	0
$403$	− 3.20620e6i	− 0.983394i
$404$	0	0
$405$	1.02483e6i	0.310465i
$406$	0	0
$407$	896600.	0.268295
$408$	0	0
$409$	−5.19593e6	−1.53587	−0.767936	−	0.640527i	$-0.778716\pi$
$409$	−5.19593e6	−1.53587	−0.767936	+	0.640527i	$0.778716\pi$
$410$	0	0
$411$	1.78405e6i	0.520957i
$412$	0	0
$413$	4.16374e6i	1.20118i
$414$	0	0
$415$	−2.90288e6	−0.827388
$416$	0	0
$417$	−737036.	−0.207562
$418$	0	0
$419$	6.45443e6i	1.79607i	0.439927	+	0.898033i	$0.355004\pi$
$419$	6.45443e6i	1.79607i	−0.439927	+	0.898033i	$0.644996\pi$
$420$	0	0
$421$	− 1.41145e6i	− 0.388114i	−0.980990	−	0.194057i	$-0.937835\pi$
$421$	− 1.41145e6i	− 0.388114i	0.980990	−	0.194057i	$-0.0621646\pi$
$422$	0	0
$423$	−4.51283e6	−1.22630
$424$	0	0
$425$	−905121.	−0.243072
$426$	0	0
$427$	179232.i	0.0475714i
$428$	0	0
$429$	− 425046.i	− 0.111504i
$430$	0	0
$431$	2.94825e6	0.764488	0.382244	−	0.924061i	$-0.375151\pi$
$431$	2.94825e6	0.764488	0.382244	+	0.924061i	$0.375151\pi$
$432$	0	0
$433$	4.63564e6	1.18820	0.594100	−	0.804391i	$-0.297508\pi$
$433$	4.63564e6	1.18820	0.594100	+	0.804391i	$0.297508\pi$
$434$	0	0
$435$	− 670245.i	− 0.169828i
$436$	0	0
$437$	− 497978.i	− 0.124740i
$438$	0	0
$439$	−2.15825e6	−0.534491	−0.267245	−	0.963629i	$-0.586113\pi$
$439$	−2.15825e6	−0.534491	−0.267245	+	0.963629i	$0.586113\pi$
$440$	0	0
$441$	9.96656e6	2.44033
$442$	0	0
$443$	− 4.27689e6i	− 1.03543i	−0.855554	−	0.517713i	$-0.826783\pi$
$443$	− 4.27689e6i	− 1.03543i	0.855554	−	0.517713i	$-0.173217\pi$
$444$	0	0
$445$	3.02956e6i	0.725236i
$446$	0	0
$447$	822080.	0.194601
$448$	0	0
$449$	1.31292e6	0.307342	0.153671	−	0.988122i	$-0.450890\pi$
$449$	1.31292e6	0.307342	0.153671	+	0.988122i	$0.450890\pi$
$450$	0	0
$451$	937217.i	0.216969i
$452$	0	0
$453$	2.16726e6i	0.496209i
$454$	0	0
$455$	−5.55159e6	−1.25716
$456$	0	0
$457$	−5.14752e6	−1.15294	−0.576471	−	0.817117i	$-0.695571\pi$
$457$	−5.14752e6	−1.15294	−0.576471	+	0.817117i	$0.695571\pi$
$458$	0	0
$459$	3.37773e6i	0.748330i
$460$	0	0
$461$	− 7.94846e6i	− 1.74193i	−0.491346	−	0.870965i	$-0.663495\pi$
$461$	− 7.94846e6i	− 1.74193i	0.491346	−	0.870965i	$-0.336505\pi$
$462$	0	0
$463$	374404.	0.0811687	0.0405843	−	0.999176i	$-0.487078\pi$
$463$	374404.	0.0811687	0.0405843	+	0.999176i	$0.487078\pi$
$464$	0	0
$465$	−457827.	−0.0981904
$466$	0	0
$467$	− 3.84308e6i	− 0.815432i	−0.913109	−	0.407716i	$-0.866325\pi$
$467$	− 3.84308e6i	− 0.815432i	0.913109	−	0.407716i	$-0.133675\pi$
$468$	0	0
$469$	− 5.49607e6i	− 1.15377i
$470$	0	0
$471$	−1.41182e6	−0.293243
$472$	0	0
$473$	1.91152e6	0.392848
$474$	0	0
$475$	382298.i	0.0777443i
$476$	0	0
$477$	− 2.82742e6i	− 0.568977i
$478$	0	0
$479$	−1.08933e6	−0.216930	−0.108465	−	0.994100i	$-0.534594\pi$
$479$	−1.08933e6	−0.216930	−0.108465	+	0.994100i	$0.534594\pi$
$480$	0	0
$481$	8.41033e6	1.65749
$482$	0	0
$483$	− 1.03261e6i	− 0.201404i
$484$	0	0
$485$	539968.i	0.104235i
$486$	0	0
$487$	2.02078e6	0.386098	0.193049	−	0.981189i	$-0.438162\pi$
$487$	2.02078e6	0.386098	0.193049	+	0.981189i	$0.438162\pi$
$488$	0	0
$489$	1.40154e6	0.265054
$490$	0	0
$491$	9.05818e6i	1.69565i	0.530273	+	0.847827i	$0.322089\pi$
$491$	9.05818e6i	1.69565i	−0.530273	+	0.847827i	$0.677911\pi$
$492$	0	0
$493$	7.66284e6i	1.41995i
$494$	0	0
$495$	513808.	0.0942514
$496$	0	0
$497$	9.63118e6	1.74899
$498$	0	0
$499$	2.83503e6i	0.509690i	0.966982	+	0.254845i	$0.0820245\pi$
$499$	2.83503e6i	0.509690i	−0.966982	+	0.254845i	$0.917976\pi$
$500$	0	0
$501$	− 15241.8i	− 0.00271294i
$502$	0	0
$503$	−7.09022e6	−1.24951	−0.624755	−	0.780821i	$-0.714801\pi$
$503$	−7.09022e6	−1.24951	−0.624755	+	0.780821i	$0.714801\pi$
$504$	0	0
$505$	−2.63503e6	−0.459787
$506$	0	0
$507$	− 2.10578e6i	− 0.363825i
$508$	0	0
$509$	− 3.09922e6i	− 0.530222i	−0.964218	−	0.265111i	$-0.914591\pi$
$509$	− 3.09922e6i	− 0.530222i	0.964218	−	0.265111i	$-0.0854086\pi$
$510$	0	0
$511$	1.02278e7	1.73273
$512$	0	0
$513$	1.42666e6	0.239346
$514$	0	0
$515$	− 384069.i	− 0.0638103i
$516$	0	0
$517$	1.96372e6i	0.323112i
$518$	0	0
$519$	−2.40088e6	−0.391248
$520$	0	0
$521$	−6.38166e6	−1.03000	−0.515002	−	0.857189i	$-0.672209\pi$
$521$	−6.38166e6	−1.03000	−0.515002	+	0.857189i	$0.672209\pi$
$522$	0	0
$523$	− 4.15218e6i	− 0.663776i	−0.943319	−	0.331888i	$-0.892314\pi$
$523$	− 4.15218e6i	− 0.663776i	0.943319	−	0.331888i	$-0.107686\pi$
$524$	0	0
$525$	792736.i	0.125525i
$526$	0	0
$527$	5.23429e6	0.820977
$528$	0	0
$529$	−5.77355e6	−0.897024
$530$	0	0
$531$	− 3.61477e6i	− 0.556346i
$532$	0	0
$533$	8.79132e6i	1.34041i
$534$	0	0
$535$	−369352.	−0.0557899
$536$	0	0
$537$	−3.35298e6	−0.501759
$538$	0	0
$539$	− 4.33686e6i	− 0.642989i
$540$	0	0
$541$	6.36079e6i	0.934367i	0.884160	+	0.467184i	$0.154731\pi$
$541$	6.36079e6i	0.934367i	−0.884160	+	0.467184i	$0.845269\pi$
$542$	0	0
$543$	−1.97633e6	−0.287647
$544$	0	0
$545$	−4.13736e6	−0.596667
$546$	0	0
$547$	9.40603e6i	1.34412i	0.740497	+	0.672060i	$0.234590\pi$
$547$	9.40603e6i	1.34412i	−0.740497	+	0.672060i	$0.765410\pi$
$548$	0	0
$549$	− 155601.i	− 0.0220334i
$550$	0	0
$551$	3.23657e6	0.454157
$552$	0	0
$553$	−2.12400e7	−2.95353
$554$	0	0
$555$	− 1.20095e6i	− 0.165497i
$556$	0	0
$557$	3.24324e6i	0.442936i	0.975168	+	0.221468i	$0.0710848\pi$
$557$	3.24324e6i	0.442936i	−0.975168	+	0.221468i	$0.928915\pi$
$558$	0	0
$559$	1.79305e7	2.42696
$560$	0	0
$561$	693910.	0.0930884
$562$	0	0
$563$	− 8.80932e6i	− 1.17131i	−0.810561	−	0.585654i	$-0.800838\pi$
$563$	− 8.80932e6i	− 1.17131i	0.810561	−	0.585654i	$-0.199162\pi$
$564$	0	0
$565$	1.82084e6i	0.239967i
$566$	0	0
$567$	−1.02619e7	−1.34051
$568$	0	0
$569$	9.59867e6	1.24288	0.621441	−	0.783461i	$-0.286547\pi$
$569$	9.59867e6	1.24288	0.621441	+	0.783461i	$0.286547\pi$
$570$	0	0
$571$	− 1.28215e7i	− 1.64569i	−0.568262	−	0.822847i	$-0.692384\pi$
$571$	− 1.28215e7i	− 1.64569i	0.568262	−	0.822847i	$-0.307616\pi$
$572$	0	0
$573$	2.31147e6i	0.294104i
$574$	0	0
$575$	−508824.	−0.0641797
$576$	0	0
$577$	−5.39655e6	−0.674803	−0.337401	−	0.941361i	$-0.609548\pi$
$577$	−5.39655e6	−0.674803	−0.337401	+	0.941361i	$0.609548\pi$
$578$	0	0
$579$	− 143216.i	− 0.0177539i
$580$	0	0
$581$	− 2.90675e7i	− 3.57246i
$582$	0	0
$583$	−1.23033e6	−0.149916
$584$	0	0
$585$	4.81965e6	0.582271
$586$	0	0
$587$	− 2.42415e6i	− 0.290379i	−0.989404	−	0.145189i	$-0.953621\pi$
$587$	− 2.42415e6i	− 0.290379i	0.989404	−	0.145189i	$-0.0463792\pi$
$588$	0	0
$589$	− 2.21082e6i	− 0.262582i
$590$	0	0
$591$	3.78327e6	0.445552
$592$	0	0
$593$	−9.84341e6	−1.14950	−0.574750	−	0.818329i	$-0.694901\pi$
$593$	−9.84341e6	−1.14950	−0.574750	+	0.818329i	$0.694901\pi$
$594$	0	0
$595$	− 9.06327e6i	− 1.04952i
$596$	0	0
$597$	− 1.37649e6i	− 0.158066i
$598$	0	0
$599$	1.63365e6	0.186034	0.0930169	−	0.995665i	$-0.470349\pi$
$599$	1.63365e6	0.186034	0.0930169	+	0.995665i	$0.470349\pi$
$600$	0	0
$601$	−8.99906e6	−1.01627	−0.508137	−	0.861276i	$-0.669666\pi$
$601$	−8.99906e6	−1.01627	−0.508137	+	0.861276i	$0.669666\pi$
$602$	0	0
$603$	4.77144e6i	0.534388i
$604$	0	0
$605$	3.80270e6i	0.422380i
$606$	0	0
$607$	6.70178e6	0.738276	0.369138	−	0.929375i	$-0.379653\pi$
$607$	6.70178e6	0.738276	0.369138	+	0.929375i	$0.379653\pi$
$608$	0	0
$609$	6.71137e6	0.733277
$610$	0	0
$611$	1.84201e7i	1.99613i
$612$	0	0
$613$	− 5.70585e6i	− 0.613294i	−0.951823	−	0.306647i	$-0.900793\pi$
$613$	− 5.70585e6i	− 0.613294i	0.951823	−	0.306647i	$-0.0992072\pi$
$614$	0	0
$615$	1.25535e6	0.133837
$616$	0	0
$617$	1.55652e7	1.64605	0.823023	−	0.568008i	$-0.192286\pi$
$617$	1.55652e7	1.64605	0.823023	+	0.568008i	$0.192286\pi$
$618$	0	0
$619$	− 1.26590e7i	− 1.32792i	−0.747766	−	0.663962i	$-0.768874\pi$
$619$	− 1.26590e7i	− 1.32792i	0.747766	−	0.663962i	$-0.231126\pi$
$620$	0	0
$621$	1.89883e6i	0.197586i
$622$	0	0
$623$	−3.03359e7	−3.13139
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 293088.i	− 0.0297735i
$628$	0	0
$629$	1.37303e7i	1.38374i
$630$	0	0
$631$	1.23076e7	1.23055	0.615274	−	0.788313i	$-0.289045\pi$
$631$	1.23076e7	1.23055	0.615274	+	0.788313i	$0.289045\pi$
$632$	0	0
$633$	−102770.	−0.0101943
$634$	0	0
$635$	207179.i	0.0203897i
$636$	0	0
$637$	− 4.06808e7i	− 3.97229i
$638$	0	0
$639$	−8.36136e6	−0.810075
$640$	0	0
$641$	−1.25160e7	−1.20315	−0.601576	−	0.798816i	$-0.705460\pi$
$641$	−1.25160e7	−1.20315	−0.601576	+	0.798816i	$0.705460\pi$
$642$	0	0
$643$	− 5.85603e6i	− 0.558568i	−0.960209	−	0.279284i	$-0.909903\pi$
$643$	− 5.85603e6i	− 0.558568i	0.960209	−	0.279284i	$-0.0900970\pi$
$644$	0	0
$645$	− 2.56037e6i	− 0.242328i
$646$	0	0
$647$	1.02686e7	0.964387	0.482194	−	0.876065i	$-0.339840\pi$
$647$	1.02686e7	0.964387	0.482194	+	0.876065i	$0.339840\pi$
$648$	0	0
$649$	−1.57294e6	−0.146588
$650$	0	0
$651$	− 4.58437e6i	− 0.423962i
$652$	0	0
$653$	− 6277.16i	0 0.000576076i	−1.00000		0.000288038i	$-0.999908\pi$
$653$	− 6277.16i	0 0.000576076i	1.00000		0.000288038i	$-9.16854e-5\pi$
$654$	0	0
$655$	4.43033e6	0.403490
$656$	0	0
$657$	−8.87936e6	−0.802543
$658$	0	0
$659$	− 1.90209e7i	− 1.70615i	−0.521784	−	0.853077i	$-0.674733\pi$
$659$	− 1.90209e7i	− 1.70615i	0.521784	−	0.853077i	$-0.325267\pi$
$660$	0	0
$661$	− 9.84622e6i	− 0.876528i	−0.898846	−	0.438264i	$-0.855594\pi$
$661$	− 9.84622e6i	− 0.876528i	0.898846	−	0.438264i	$-0.144406\pi$
$662$	0	0
$663$	6.50904e6	0.575086
$664$	0	0
$665$	−3.82807e6	−0.335681
$666$	0	0
$667$	4.30775e6i	0.374918i
$668$	0	0
$669$	− 153928.i	− 0.0132970i
$670$	0	0
$671$	−67708.5	−0.00580546
$672$	0	0
$673$	−1.82184e7	−1.55050	−0.775252	−	0.631652i	$-0.782377\pi$
$673$	−1.82184e7	−1.55050	−0.775252	+	0.631652i	$0.782377\pi$
$674$	0	0
$675$	− 1.45773e6i	− 0.123145i
$676$	0	0
$677$	8.10909e6i	0.679987i	0.940428	+	0.339993i	$0.110425\pi$
$677$	8.10909e6i	0.679987i	−0.940428	+	0.339993i	$0.889575\pi$
$678$	0	0
$679$	−5.40687e6	−0.450061
$680$	0	0
$681$	7.38343e6	0.610085
$682$	0	0
$683$	1.11760e7i	0.916717i	0.888767	+	0.458358i	$0.151562\pi$
$683$	1.11760e7i	0.916717i	−0.888767	+	0.458358i	$0.848438\pi$
$684$	0	0
$685$	8.80269e6i	0.716785i
$686$	0	0
$687$	4.88164e6	0.394616
$688$	0	0
$689$	−1.15408e7	−0.926161
$690$	0	0
$691$	− 1.40751e6i	− 0.112139i	−0.998427	−	0.0560695i	$-0.982143\pi$
$691$	− 1.40751e6i	− 0.112139i	0.998427	−	0.0560695i	$-0.0178568\pi$
$692$	0	0
$693$	5.14492e6i	0.406954i
$694$	0	0
$695$	−3.63662e6	−0.285586
$696$	0	0
$697$	−1.43523e7	−1.11902
$698$	0	0
$699$	− 6.68125e6i	− 0.517208i
$700$	0	0
$701$	1.40860e7i	1.08266i	0.840809	+	0.541331i	$0.182079\pi$
$701$	1.40860e7i	1.08266i	−0.840809	+	0.541331i	$0.817921\pi$
$702$	0	0
$703$	5.79930e6	0.442576
$704$	0	0
$705$	2.63029e6	0.199311
$706$	0	0
$707$	− 2.63854e7i	− 1.98525i
$708$	0	0
$709$	− 1.38968e6i	− 0.103824i	−0.998652	−	0.0519122i	$-0.983468\pi$
$709$	− 1.38968e6i	− 0.103824i	0.998652	−	0.0519122i	$-0.0165316\pi$
$710$	0	0
$711$	1.84396e7	1.36797
$712$	0	0
$713$	2.94251e6	0.216768
$714$	0	0
$715$	− 2.09722e6i	− 0.153419i
$716$	0	0
$717$	5.97744e6i	0.434227i
$718$	0	0
$719$	1.91426e7	1.38096	0.690478	−	0.723354i	$-0.257400\pi$
$719$	1.91426e7	1.38096	0.690478	+	0.723354i	$0.257400\pi$
$720$	0	0
$721$	3.84580e6	0.275517
$722$	0	0
$723$	3.98210e6i	0.283313i
$724$	0	0
$725$	− 3.30707e6i	− 0.233667i
$726$	0	0
$727$	1.38873e7	0.974497	0.487248	−	0.873263i	$-0.338001\pi$
$727$	1.38873e7	0.974497	0.487248	+	0.873263i	$0.338001\pi$
$728$	0	0
$729$	6.03727e6	0.420748
$730$	0	0
$731$	2.92725e7i	2.02612i
$732$	0	0
$733$	− 9.41321e6i	− 0.647110i	−0.946209	−	0.323555i	$-0.895122\pi$
$733$	− 9.41321e6i	− 0.647110i	0.946209	−	0.323555i	$-0.104878\pi$
$734$	0	0
$735$	−5.80899e6	−0.396627
$736$	0	0
$737$	2.07625e6	0.140803
$738$	0	0
$739$	− 118264.i	− 0.00796605i	−0.999992	−	0.00398302i	$-0.998732\pi$
$739$	− 118264.i	− 0.00796605i	0.999992	−	0.00398302i	$-0.00126784\pi$
$740$	0	0
$741$	− 2.74924e6i	− 0.183936i
$742$	0	0
$743$	1.62763e6	0.108164	0.0540820	−	0.998537i	$-0.482777\pi$
$743$	1.62763e6	0.108164	0.0540820	+	0.998537i	$0.482777\pi$
$744$	0	0
$745$	4.05624e6	0.267752
$746$	0	0
$747$	2.52351e7i	1.65464i
$748$	0	0
$749$	− 3.69843e6i	− 0.240887i
$750$	0	0
$751$	−9.02408e6	−0.583853	−0.291926	−	0.956441i	$-0.594296\pi$
$751$	−9.02408e6	−0.583853	−0.291926	+	0.956441i	$0.594296\pi$
$752$	0	0
$753$	−4.68028e6	−0.300805
$754$	0	0
$755$	1.06935e7i	0.682735i
$756$	0	0
$757$	645135.i	0.0409177i	0.999791	+	0.0204588i	$0.00651271\pi$
$757$	645135.i	0.0409177i	−0.999791	+	0.0204588i	$0.993487\pi$
$758$	0	0
$759$	390089.	0.0245787
$760$	0	0
$761$	4.61414e6	0.288821	0.144411	−	0.989518i	$-0.453871\pi$
$761$	4.61414e6	0.288821	0.144411	+	0.989518i	$0.453871\pi$
$762$	0	0
$763$	− 4.14287e7i	− 2.57626i
$764$	0	0
$765$	7.86833e6i	0.486104i
$766$	0	0
$767$	−1.47545e7	−0.905601
$768$	0	0
$769$	59875.3	0.00365117	0.00182559	−	0.999998i	$-0.499419\pi$
$769$	59875.3	0.00365117	0.00182559	+	0.999998i	$0.499419\pi$
$770$	0	0
$771$	− 2.52211e6i	− 0.152802i
$772$	0	0
$773$	1.78468e7i	1.07426i	0.843499	+	0.537132i	$0.180492\pi$
$773$	1.78468e7i	1.07426i	−0.843499	+	0.537132i	$0.819508\pi$
$774$	0	0
$775$	−2.25897e6	−0.135100
$776$	0	0
$777$	1.20255e7	0.714577
$778$	0	0
$779$	6.06202e6i	0.357910i
$780$	0	0
$781$	3.63837e6i	0.213442i
$782$	0	0
$783$	−1.23413e7	−0.719377
$784$	0	0
$785$	−6.96609e6	−0.403473
$786$	0	0
$787$	− 1.87666e7i	− 1.08006i	−0.841645	−	0.540031i	$-0.818412\pi$
$787$	− 1.87666e7i	− 1.08006i	0.841645	−	0.540031i	$-0.181588\pi$
$788$	0	0
$789$	− 3.77667e6i	− 0.215981i
$790$	0	0
$791$	−1.82326e7	−1.03612
$792$	0	0
$793$	−635122.	−0.0358653
$794$	0	0
$795$	1.64796e6i	0.0924757i
$796$	0	0
$797$	− 3.95963e6i	− 0.220805i	−0.993887	−	0.110403i	$-0.964786\pi$
$797$	− 3.95963e6i	− 0.220805i	0.993887	−	0.110403i	$-0.0352140\pi$
$798$	0	0
$799$	−3.00719e7	−1.66645
$800$	0	0
$801$	2.63363e7	1.45035
$802$	0	0
$803$	3.86377e6i	0.211457i
$804$	0	0
$805$	− 5.09501e6i	− 0.277112i
$806$	0	0
$807$	−9.17044e6	−0.495685
$808$	0	0
$809$	−8.74535e6	−0.469792	−0.234896	−	0.972020i	$-0.575475\pi$
$809$	−8.74535e6	−0.469792	−0.234896	+	0.972020i	$0.575475\pi$
$810$	0	0
$811$	− 2.74240e7i	− 1.46413i	−0.681236	−	0.732064i	$-0.738557\pi$
$811$	− 2.74240e7i	− 1.46413i	0.681236	−	0.732064i	$-0.261443\pi$
$812$	0	0
$813$	1.11631e7i	0.592320i
$814$	0	0
$815$	6.91537e6	0.364688
$816$	0	0
$817$	1.23639e7	0.648037
$818$	0	0
$819$	4.82606e7i	2.51410i
$820$	0	0
$821$	1.48659e7i	0.769719i	0.922975	+	0.384859i	$0.125750\pi$
$821$	1.48659e7i	0.769719i	−0.922975	+	0.384859i	$0.874250\pi$
$822$	0	0
$823$	−9.27591e6	−0.477372	−0.238686	−	0.971097i	$-0.576717\pi$
$823$	−9.27591e6	−0.477372	−0.238686	+	0.971097i	$0.576717\pi$
$824$	0	0
$825$	−299472.	−0.0153187
$826$	0	0
$827$	6.99433e6i	0.355617i	0.984065	+	0.177808i	$0.0569007\pi$
$827$	6.99433e6i	0.355617i	−0.984065	+	0.177808i	$0.943099\pi$
$828$	0	0
$829$	− 3.29484e7i	− 1.66513i	−0.553929	−	0.832564i	$-0.686872\pi$
$829$	− 3.29484e7i	− 1.66513i	0.553929	−	0.832564i	$-0.313128\pi$
$830$	0	0
$831$	9.39234e6	0.471814
$832$	0	0
$833$	6.64135e7	3.31623
$834$	0	0
$835$	− 75204.6i	− 0.00373274i
$836$	0	0
$837$	8.43002e6i	0.415925i
$838$	0	0
$839$	6.29215e6	0.308599	0.154299	−	0.988024i	$-0.450688\pi$
$839$	6.29215e6	0.308599	0.154299	+	0.988024i	$0.450688\pi$
$840$	0	0
$841$	−7.48678e6	−0.365010
$842$	0	0
$843$	− 5.39824e6i	− 0.261627i
$844$	0	0
$845$	− 1.03902e7i	− 0.500588i
$846$	0	0
$847$	−3.80776e7	−1.82373
$848$	0	0
$849$	194312.	0.00925191
$850$	0	0
$851$	7.71864e6i	0.365356i
$852$	0	0
$853$	− 2.28452e6i	− 0.107504i	−0.998554	−	0.0537518i	$-0.982882\pi$
$853$	− 2.28452e6i	− 0.107504i	0.998554	−	0.0537518i	$-0.0171180\pi$
$854$	0	0
$855$	3.32336e6	0.155476
$856$	0	0
$857$	1.33739e6	0.0622023	0.0311012	−	0.999516i	$-0.490099\pi$
$857$	1.33739e6	0.0622023	0.0311012	+	0.999516i	$0.490099\pi$
$858$	0	0
$859$	3.12847e7i	1.44660i	0.690533	+	0.723301i	$0.257376\pi$
$859$	3.12847e7i	1.44660i	−0.690533	+	0.723301i	$0.742624\pi$
$860$	0	0
$861$	1.25702e7i	0.577877i
$862$	0	0
$863$	3.83963e7	1.75494	0.877469	−	0.479633i	$-0.159230\pi$
$863$	3.83963e7	1.75494	0.877469	+	0.479633i	$0.159230\pi$
$864$	0	0
$865$	−1.18462e7	−0.538320
$866$	0	0
$867$	3.43228e6i	0.155072i
$868$	0	0
$869$	− 8.02382e6i	− 0.360439i
$870$	0	0
$871$	1.94757e7	0.869858
$872$	0	0
$873$	4.69401e6	0.208453
$874$	0	0
$875$	3.91145e6i	0.172710i
$876$	0	0
$877$	1.62074e7i	0.711565i	0.934569	+	0.355783i	$0.115786\pi$
$877$	1.62074e7i	0.711565i	−0.934569	+	0.355783i	$0.884214\pi$
$878$	0	0
$879$	−2.35045e6	−0.102607
$880$	0	0
$881$	−3.64879e6	−0.158383	−0.0791916	−	0.996859i	$-0.525234\pi$
$881$	−3.64879e6	−0.158383	−0.0791916	+	0.996859i	$0.525234\pi$
$882$	0	0
$883$	3.21031e7i	1.38562i	0.721119	+	0.692811i	$0.243628\pi$
$883$	3.21031e7i	1.38562i	−0.721119	+	0.692811i	$0.756372\pi$
$884$	0	0
$885$	2.10686e6i	0.0904228i
$886$	0	0
$887$	1.29249e7	0.551594	0.275797	−	0.961216i	$-0.411058\pi$
$887$	1.29249e7	0.551594	0.275797	+	0.961216i	$0.411058\pi$
$888$	0	0
$889$	−2.07455e6	−0.0880379
$890$	0	0
$891$	− 3.87665e6i	− 0.163592i
$892$	0	0
$893$	1.27015e7i	0.533000i
$894$	0	0
$895$	−1.65440e7	−0.690372
$896$	0	0
$897$	3.65913e6	0.151844
$898$	0	0
$899$	1.91247e7i	0.789214i
$900$	0	0
$901$	− 1.88409e7i	− 0.773196i
$902$	0	0
$903$	2.56378e7	1.04631
$904$	0	0
$905$	−9.75143e6	−0.395774
$906$	0	0
$907$	− 541993.i	− 0.0218764i	−0.999940	−	0.0109382i	$-0.996518\pi$
$907$	− 541993.i	− 0.0218764i	0.999940	−	0.0109382i	$-0.00348181\pi$
$908$	0	0
$909$	2.29066e7i	0.919498i
$910$	0	0
$911$	−3.25168e7	−1.29811	−0.649055	−	0.760741i	$-0.724836\pi$
$911$	−3.25168e7	−1.29811	−0.649055	+	0.760741i	$0.724836\pi$
$912$	0	0
$913$	1.09808e7	0.435971
$914$	0	0
$915$	90691.8i	0.00358109i
$916$	0	0
$917$	4.43623e7i	1.74217i
$918$	0	0
$919$	3.14025e7	1.22652	0.613261	−	0.789880i	$-0.289857\pi$
$919$	3.14025e7	1.22652	0.613261	+	0.789880i	$0.289857\pi$
$920$	0	0
$921$	−1.03168e6	−0.0400772
$922$	0	0
$923$	3.41288e7i	1.31861i
$924$	0	0
$925$	− 5.92561e6i	− 0.227708i
$926$	0	0
$927$	−3.33876e6	−0.127610
$928$	0	0
$929$	1.06286e7	0.404052	0.202026	−	0.979380i	$-0.435247\pi$
$929$	1.06286e7	0.404052	0.202026	+	0.979380i	$0.435247\pi$
$930$	0	0
$931$	− 2.80513e7i	− 1.06066i
$932$	0	0
$933$	− 6.71978e6i	− 0.252726i
$934$	0	0
$935$	3.42383e6	0.128081
$936$	0	0
$937$	−1.41673e7	−0.527154	−0.263577	−	0.964638i	$-0.584902\pi$
$937$	−1.41673e7	−0.527154	−0.263577	+	0.964638i	$0.584902\pi$
$938$	0	0
$939$	− 8.22293e6i	− 0.304343i
$940$	0	0
$941$	4.78895e6i	0.176306i	0.996107	+	0.0881528i	$0.0280964\pi$
$941$	4.78895e6i	0.176306i	−0.996107	+	0.0881528i	$0.971904\pi$
$942$	0	0
$943$	−8.06830e6	−0.295463
$944$	0	0
$945$	1.45967e7	0.531712
$946$	0	0
$947$	4.35818e7i	1.57917i	0.613639	+	0.789587i	$0.289705\pi$
$947$	4.35818e7i	1.57917i	−0.613639	+	0.789587i	$0.710295\pi$
$948$	0	0
$949$	3.62431e7i	1.30635i
$950$	0	0
$951$	−4.66620e6	−0.167306
$952$	0	0
$953$	−639034.	−0.0227925	−0.0113962	−	0.999935i	$-0.503628\pi$
$953$	−639034.	−0.0227925	−0.0113962	+	0.999935i	$0.503628\pi$
$954$	0	0
$955$	1.14051e7i	0.404659i
$956$	0	0
$957$	2.53536e6i	0.0894868i
$958$	0	0
$959$	−8.81442e7	−3.09490
$960$	0	0
$961$	−1.55656e7	−0.543697
$962$	0	0
$963$	3.21082e6i	0.111571i
$964$	0	0
$965$	− 706642.i	− 0.0244276i
$966$	0	0
$967$	1.89639e6	0.0652172	0.0326086	−	0.999468i	$-0.489619\pi$
$967$	1.89639e6	0.0652172	0.0326086	+	0.999468i	$0.489619\pi$
$968$	0	0
$969$	4.48828e6	0.153557
$970$	0	0
$971$	2.30710e7i	0.785268i	0.919695	+	0.392634i	$0.128436\pi$
$971$	2.30710e7i	0.785268i	−0.919695	+	0.392634i	$0.871564\pi$
$972$	0	0
$973$	− 3.64147e7i	− 1.23309i
$974$	0	0
$975$	−2.80912e6	−0.0946365
$976$	0	0
$977$	4.36091e7	1.46164	0.730820	−	0.682571i	$-0.239138\pi$
$977$	4.36091e7	1.46164	0.730820	+	0.682571i	$0.239138\pi$
$978$	0	0
$979$	− 1.14600e7i	− 0.382145i
$980$	0	0
$981$	3.59666e7i	1.19324i
$982$	0	0
$983$	−5.07856e6	−0.167632	−0.0838160	−	0.996481i	$-0.526711\pi$
$983$	−5.07856e6	−0.167632	−0.0838160	+	0.996481i	$0.526711\pi$
$984$	0	0
$985$	1.86671e7	0.613036
$986$	0	0
$987$	2.63379e7i	0.860575i
$988$	0	0
$989$	1.64558e7i	0.534970i
$990$	0	0
$991$	−1.28580e7	−0.415901	−0.207950	−	0.978139i	$-0.566679\pi$
$991$	−1.28580e7	−0.415901	−0.207950	+	0.978139i	$0.566679\pi$
$992$	0	0
$993$	3.49445e6	0.112462
$994$	0	0
$995$	− 6.79177e6i	− 0.217483i
$996$	0	0
$997$	− 3.44037e7i	− 1.09614i	−0.836431	−	0.548072i	$-0.815362\pi$
$997$	− 3.44037e7i	− 1.09614i	0.836431	−	0.548072i	$-0.184638\pi$
$998$	0	0
$999$	−2.21132e7	−0.701031

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.c.161.8	yes	12
4.3	odd	2		320.6.d.d.161.5	yes	12
8.3	odd	2		320.6.d.d.161.8	yes	12
8.5	even	2	inner	320.6.d.c.161.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.c.161.5	✓	12	8.5	even	2	inner
320.6.d.c.161.8	yes	12	1.1	even	1	trivial
320.6.d.d.161.5	yes	12	4.3	odd	2
320.6.d.d.161.8	yes	12	8.3	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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} +O(q^{10})\)	\(q+5.06676i q^{3} +25.0000i q^{5} -250.333 q^{7} +217.328 q^{9} -94.5683i q^{11} -887.073i q^{13} -126.669 q^{15} +1448.19 q^{17} -611.677i q^{19} -1268.38i q^{21} +814.118 q^{23} -625.000 q^{25} +2332.37i q^{27} +5291.31i q^{29} +3614.36 q^{31} +479.155 q^{33} -6258.32i q^{35} +9480.98i q^{37} +4494.59 q^{39} -9910.48 q^{41} +20213.1i q^{43} +5433.20i q^{45} -20765.1 q^{47} +45859.5 q^{49} +7337.66i q^{51} -13009.9i q^{53} +2364.21 q^{55} +3099.22 q^{57} -16632.8i q^{59} -715.975i q^{61} -54404.3 q^{63} +22176.8 q^{65} +21955.0i q^{67} +4124.94i q^{69} -38473.5 q^{71} -40857.0 q^{73} -3166.73i q^{75} +23673.5i q^{77} +84846.9 q^{79} +40993.1 q^{81} +116115. i q^{83} +36204.9i q^{85} -26809.8 q^{87} +121182. q^{89} +222064. i q^{91} +18313.1i q^{93} +15291.9 q^{95} +21598.7 q^{97} -20552.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 268 q^{7} - 428 q^{9}+O(q^{10}) \) 12 * q - 268 * q^7 - 428 * q^9	\( 12 q - 268 q^{7} - 428 q^{9} + 900 q^{15} + 2400 q^{17} - 8108 q^{23} - 7500 q^{25} - 7976 q^{31} - 20776 q^{33} - 40984 q^{39} - 56408 q^{41} - 21172 q^{47} - 5540 q^{49} - 18400 q^{55} + 39992 q^{57} - 179516 q^{63} + 44000 q^{65} - 367704 q^{71} + 58736 q^{73} - 26192 q^{79} + 411692 q^{81} - 183200 q^{87} + 87672 q^{89} - 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) 12 * q - 268 * q^7 - 428 * q^9 + 900 * q^15 + 2400 * q^17 - 8108 * q^23 - 7500 * q^25 - 7976 * q^31 - 20776 * q^33 - 40984 * q^39 - 56408 * q^41 - 21172 * q^47 - 5540 * q^49 - 18400 * q^55 + 39992 * q^57 - 179516 * q^63 + 44000 * q^65 - 367704 * q^71 + 58736 * q^73 - 26192 * q^79 + 411692 * q^81 - 183200 * q^87 + 87672 * q^89 - 121000 * q^95 - 172336 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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