LMFDB - Embedded newform 320.6.d.b.161.5

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 384x^{6} + 506x^{5} + 49869x^{4} + 29654x^{3} - 2235516x^{2} - 1528906x + 34180205 $ x^8 - 4x^7 - 384x^6 + 506x^5 + 49869x^4 + 29654x^3 - 2235516x^2 - 1528906*x + 34180205
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$5.36332 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.b.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.72664i q^{3} -25.0000i q^{5} -65.1432 q^{7} +148.393 q^{9} +O(q^{10})$	$q+9.72664i q^{3} -25.0000i q^{5} -65.1432 q^{7} +148.393 q^{9} -71.5642i q^{11} +733.370i q^{13} +243.166 q^{15} -716.077 q^{17} -2333.71i q^{19} -633.624i q^{21} +278.545 q^{23} -625.000 q^{25} +3806.93i q^{27} +2953.16i q^{29} +6047.10 q^{31} +696.079 q^{33} +1628.58i q^{35} +12195.5i q^{37} -7133.22 q^{39} +3629.90 q^{41} +230.070i q^{43} -3709.81i q^{45} -13998.1 q^{47} -12563.4 q^{49} -6965.02i q^{51} +22416.3i q^{53} -1789.10 q^{55} +22699.2 q^{57} +7285.64i q^{59} +19417.5i q^{61} -9666.77 q^{63} +18334.2 q^{65} +1390.88i q^{67} +2709.30i q^{69} -57315.7 q^{71} -35556.0 q^{73} -6079.15i q^{75} +4661.92i q^{77} -93011.5 q^{79} -969.268 q^{81} +1599.04i q^{83} +17901.9i q^{85} -28724.3 q^{87} +2029.74 q^{89} -47774.1i q^{91} +58817.9i q^{93} -58342.9 q^{95} -120706. q^{97} -10619.6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) $ 8 * q + 320 * q^7 - 1192 * q^9	$ 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) $ 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * 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
$3$	9.72664i	0.623964i	0.950088	+	0.311982i	$0.100993\pi$
$3$	9.72664i	0.623964i	−0.950088	+	0.311982i	$0.899007\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	−65.1432	−0.502486	−0.251243	−	0.967924i	$-0.580839\pi$
$7$	−65.1432	−0.502486	−0.251243	+	0.967924i	$0.580839\pi$
$8$	0	0
$9$	148.393	0.610669
$10$	0	0
$11$	− 71.5642i	− 0.178326i	−0.996017	−	0.0891628i	$-0.971581\pi$
$11$	− 71.5642i	− 0.178326i	0.996017	−	0.0891628i	$-0.0284192\pi$
$12$	0	0
$13$	733.370i	1.20355i	0.798665	+	0.601776i	$0.205540\pi$
$13$	733.370i	1.20355i	−0.798665	+	0.601776i	$0.794460\pi$
$14$	0	0
$15$	243.166	0.279045
$16$	0	0
$17$	−716.077	−0.600949	−0.300474	−	0.953790i	$-0.597145\pi$
$17$	−716.077	−0.600949	−0.300474	+	0.953790i	$0.597145\pi$
$18$	0	0
$19$	− 2333.71i	− 1.48308i	−0.670910	−	0.741538i	$-0.734096\pi$
$19$	− 2333.71i	− 1.48308i	0.670910	−	0.741538i	$-0.265904\pi$
$20$	0	0
$21$	− 633.624i	− 0.313533i
$22$	0	0
$23$	278.545	0.109793	0.0548966	−	0.998492i	$-0.482517\pi$
$23$	278.545	0.109793	0.0548966	+	0.998492i	$0.482517\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	3806.93i	1.00500i
$28$	0	0
$29$	2953.16i	0.652066i	0.945358	+	0.326033i	$0.105712\pi$
$29$	2953.16i	0.652066i	−0.945358	+	0.326033i	$0.894288\pi$
$30$	0	0
$31$	6047.10	1.13017	0.565084	−	0.825033i	$-0.308844\pi$
$31$	6047.10	1.13017	0.565084	+	0.825033i	$0.308844\pi$
$32$	0	0
$33$	696.079	0.111269
$34$	0	0
$35$	1628.58i	0.224719i
$36$	0	0
$37$	12195.5i	1.46452i	0.681024	+	0.732261i	$0.261535\pi$
$37$	12195.5i	1.46452i	−0.681024	+	0.732261i	$0.738465\pi$
$38$	0	0
$39$	−7133.22	−0.750973
$40$	0	0
$41$	3629.90	0.337236	0.168618	−	0.985681i	$-0.446069\pi$
$41$	3629.90	0.337236	0.168618	+	0.985681i	$0.446069\pi$
$42$	0	0
$43$	230.070i	0.0189753i	0.999955	+	0.00948766i	$0.00302006\pi$
$43$	230.070i	0.0189753i	−0.999955	+	0.00948766i	$0.996980\pi$
$44$	0	0
$45$	− 3709.81i	− 0.273099i
$46$	0	0
$47$	−13998.1	−0.924322	−0.462161	−	0.886796i	$-0.652926\pi$
$47$	−13998.1	−0.924322	−0.462161	+	0.886796i	$0.652926\pi$
$48$	0	0
$49$	−12563.4	−0.747508
$50$	0	0
$51$	− 6965.02i	− 0.374971i
$52$	0	0
$53$	22416.3i	1.09616i	0.836426	+	0.548080i	$0.184641\pi$
$53$	22416.3i	1.09616i	−0.836426	+	0.548080i	$0.815359\pi$
$54$	0	0
$55$	−1789.10	−0.0797497
$56$	0	0
$57$	22699.2	0.925387
$58$	0	0
$59$	7285.64i	0.272482i	0.990676	+	0.136241i	$0.0435021\pi$
$59$	7285.64i	0.272482i	−0.990676	+	0.136241i	$0.956498\pi$
$60$	0	0
$61$	19417.5i	0.668141i	0.942548	+	0.334070i	$0.108422\pi$
$61$	19417.5i	0.668141i	−0.942548	+	0.334070i	$0.891578\pi$
$62$	0	0
$63$	−9666.77	−0.306853
$64$	0	0
$65$	18334.2	0.538245
$66$	0	0
$67$	1390.88i	0.0378531i	0.999821	+	0.0189265i	$0.00602487\pi$
$67$	1390.88i	0.0378531i	−0.999821	+	0.0189265i	$0.993975\pi$
$68$	0	0
$69$	2709.30i	0.0685070i
$70$	0	0
$71$	−57315.7	−1.34936	−0.674679	−	0.738111i	$-0.735718\pi$
$71$	−57315.7	−1.34936	−0.674679	+	0.738111i	$0.735718\pi$
$72$	0	0
$73$	−35556.0	−0.780918	−0.390459	−	0.920620i	$-0.627684\pi$
$73$	−35556.0	−0.780918	−0.390459	+	0.920620i	$0.627684\pi$
$74$	0	0
$75$	− 6079.15i	− 0.124793i
$76$	0	0
$77$	4661.92i	0.0896062i
$78$	0	0
$79$	−93011.5	−1.67675	−0.838376	−	0.545092i	$-0.816495\pi$
$79$	−93011.5	−1.67675	−0.838376	+	0.545092i	$0.816495\pi$
$80$	0	0
$81$	−969.268	−0.0164146
$82$	0	0
$83$	1599.04i	0.0254779i	0.999919	+	0.0127389i	$0.00405504\pi$
$83$	1599.04i	0.0254779i	−0.999919	+	0.0127389i	$0.995945\pi$
$84$	0	0
$85$	17901.9i	0.268753i
$86$	0	0
$87$	−28724.3	−0.406866
$88$	0	0
$89$	2029.74	0.0271622	0.0135811	−	0.999908i	$-0.495677\pi$
$89$	2029.74	0.0271622	0.0135811	+	0.999908i	$0.495677\pi$
$90$	0	0
$91$	− 47774.1i	− 0.604768i
$92$	0	0
$93$	58817.9i	0.705184i
$94$	0	0
$95$	−58342.9	−0.663252
$96$	0	0
$97$	−120706.	−1.30257	−0.651284	−	0.758834i	$-0.725769\pi$
$97$	−120706.	−1.30257	−0.651284	+	0.758834i	$0.725769\pi$
$98$	0	0
$99$	− 10619.6i	− 0.108898i
$100$	0	0
$101$	173879.i	1.69607i	0.529942	+	0.848034i	$0.322214\pi$
$101$	173879.i	1.69607i	−0.529942	+	0.848034i	$0.677786\pi$
$102$	0	0
$103$	−111795.	−1.03832	−0.519160	−	0.854677i	$-0.673755\pi$
$103$	−111795.	−1.03832	−0.519160	+	0.854677i	$0.673755\pi$
$104$	0	0
$105$	−15840.6	−0.140216
$106$	0	0
$107$	110498.i	0.933030i	0.884513	+	0.466515i	$0.154491\pi$
$107$	110498.i	0.933030i	−0.884513	+	0.466515i	$0.845509\pi$
$108$	0	0
$109$	45521.0i	0.366983i	0.983021	+	0.183491i	$0.0587399\pi$
$109$	45521.0i	0.366983i	−0.983021	+	0.183491i	$0.941260\pi$
$110$	0	0
$111$	−118621.	−0.913809
$112$	0	0
$113$	181355.	1.33608	0.668041	−	0.744125i	$-0.267133\pi$
$113$	181355.	1.33608	0.668041	+	0.744125i	$0.267133\pi$
$114$	0	0
$115$	− 6963.62i	− 0.0491010i
$116$	0	0
$117$	108827.i	0.734971i
$118$	0	0
$119$	46647.6	0.301969
$120$	0	0
$121$	155930.	0.968200
$122$	0	0
$123$	35306.7i	0.210423i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	−99975.7	−0.550028	−0.275014	−	0.961440i	$-0.588683\pi$
$127$	−99975.7	−0.550028	−0.275014	+	0.961440i	$0.588683\pi$
$128$	0	0
$129$	−2237.81	−0.0118399
$130$	0	0
$131$	− 137164.i	− 0.698331i	−0.937061	−	0.349166i	$-0.886465\pi$
$131$	− 137164.i	− 0.698331i	0.937061	−	0.349166i	$-0.113535\pi$
$132$	0	0
$133$	152026.i	0.745226i
$134$	0	0
$135$	95173.3	0.449449
$136$	0	0
$137$	−181966.	−0.828303	−0.414151	−	0.910208i	$-0.635922\pi$
$137$	−181966.	−0.828303	−0.414151	+	0.910208i	$0.635922\pi$
$138$	0	0
$139$	134495.i	0.590431i	0.955431	+	0.295215i	$0.0953914\pi$
$139$	134495.i	0.590431i	−0.955431	+	0.295215i	$0.904609\pi$
$140$	0	0
$141$	− 136154.i	− 0.576744i
$142$	0	0
$143$	52483.0	0.214624
$144$	0	0
$145$	73828.9	0.291613
$146$	0	0
$147$	− 122199.i	− 0.466418i
$148$	0	0
$149$	348040.i	1.28429i	0.766583	+	0.642146i	$0.221956\pi$
$149$	348040.i	1.28429i	−0.766583	+	0.642146i	$0.778044\pi$
$150$	0	0
$151$	159935.	0.570821	0.285411	−	0.958405i	$-0.407870\pi$
$151$	159935.	0.570821	0.285411	+	0.958405i	$0.407870\pi$
$152$	0	0
$153$	−106261.	−0.366981
$154$	0	0
$155$	− 151177.i	− 0.505426i
$156$	0	0
$157$	212838.i	0.689127i	0.938763	+	0.344563i	$0.111973\pi$
$157$	212838.i	0.689127i	−0.938763	+	0.344563i	$0.888027\pi$
$158$	0	0
$159$	−218035.	−0.683964
$160$	0	0
$161$	−18145.3	−0.0551696
$162$	0	0
$163$	410171.i	1.20919i	0.796531	+	0.604597i	$0.206666\pi$
$163$	410171.i	1.20919i	−0.796531	+	0.604597i	$0.793334\pi$
$164$	0	0
$165$	− 17402.0i	− 0.0497609i
$166$	0	0
$167$	−35175.3	−0.0975994	−0.0487997	−	0.998809i	$-0.515540\pi$
$167$	−35175.3	−0.0975994	−0.0487997	+	0.998809i	$0.515540\pi$
$168$	0	0
$169$	−166538.	−0.448536
$170$	0	0
$171$	− 346306.i	− 0.905669i
$172$	0	0
$173$	− 325641.i	− 0.827224i	−0.910453	−	0.413612i	$-0.864267\pi$
$173$	− 325641.i	− 0.827224i	0.910453	−	0.413612i	$-0.135733\pi$
$174$	0	0
$175$	40714.5	0.100497
$176$	0	0
$177$	−70864.8	−0.170019
$178$	0	0
$179$	231118.i	0.539139i	0.962981	+	0.269569i	$0.0868813\pi$
$179$	231118.i	0.539139i	−0.962981	+	0.269569i	$0.913119\pi$
$180$	0	0
$181$	− 207075.i	− 0.469819i	−0.972017	−	0.234910i	$-0.924521\pi$
$181$	− 207075.i	− 0.469819i	0.972017	−	0.234910i	$-0.0754794\pi$
$182$	0	0
$183$	−188867.	−0.416896
$184$	0	0
$185$	304888.	0.654954
$186$	0	0
$187$	51245.5i	0.107165i
$188$	0	0
$189$	− 247996.i	− 0.504998i
$190$	0	0
$191$	190884.	0.378605	0.189302	−	0.981919i	$-0.439377\pi$
$191$	190884.	0.378605	0.189302	+	0.981919i	$0.439377\pi$
$192$	0	0
$193$	−4008.86	−0.00774690	−0.00387345	−	0.999992i	$-0.501233\pi$
$193$	−4008.86	−0.00774690	−0.00387345	+	0.999992i	$0.501233\pi$
$194$	0	0
$195$	178331.i	0.335845i
$196$	0	0
$197$	− 517405.i	− 0.949873i	−0.880020	−	0.474936i	$-0.842471\pi$
$197$	− 517405.i	− 0.949873i	0.880020	−	0.474936i	$-0.157529\pi$
$198$	0	0
$199$	984918.	1.76306	0.881531	−	0.472127i	$-0.156513\pi$
$199$	984918.	1.76306	0.881531	+	0.472127i	$0.156513\pi$
$200$	0	0
$201$	−13528.5	−0.0236190
$202$	0	0
$203$	− 192378.i	− 0.327654i
$204$	0	0
$205$	− 90747.4i	− 0.150817i
$206$	0	0
$207$	41334.0	0.0670473
$208$	0	0
$209$	−167010.	−0.264471
$210$	0	0
$211$	− 920941.i	− 1.42405i	−0.702154	−	0.712025i	$-0.747778\pi$
$211$	− 920941.i	− 1.42405i	0.702154	−	0.712025i	$-0.252222\pi$
$212$	0	0
$213$	− 557489.i	− 0.841951i
$214$	0	0
$215$	5751.75	0.00848602
$216$	0	0
$217$	−393927.	−0.567894
$218$	0	0
$219$	− 345840.i	− 0.487265i
$220$	0	0
$221$	− 525149.i	− 0.723273i
$222$	0	0
$223$	186992.	0.251803	0.125902	−	0.992043i	$-0.459818\pi$
$223$	186992.	0.251803	0.125902	+	0.992043i	$0.459818\pi$
$224$	0	0
$225$	−92745.3	−0.122134
$226$	0	0
$227$	796832.i	1.02637i	0.858279	+	0.513183i	$0.171534\pi$
$227$	796832.i	1.02637i	−0.858279	+	0.513183i	$0.828466\pi$
$228$	0	0
$229$	− 1.14511e6i	− 1.44298i	−0.692426	−	0.721489i	$-0.743458\pi$
$229$	− 1.14511e6i	− 1.44298i	0.692426	−	0.721489i	$-0.256542\pi$
$230$	0	0
$231$	−45344.8	−0.0559110
$232$	0	0
$233$	1.25471e6	1.51410	0.757048	−	0.653359i	$-0.226641\pi$
$233$	1.25471e6	1.51410	0.757048	+	0.653359i	$0.226641\pi$
$234$	0	0
$235$	349952.i	0.413369i
$236$	0	0
$237$	− 904689.i	− 1.04623i
$238$	0	0
$239$	−454297.	−0.514452	−0.257226	−	0.966351i	$-0.582809\pi$
$239$	−454297.	−0.514452	−0.257226	+	0.966351i	$0.582809\pi$
$240$	0	0
$241$	241513.	0.267853	0.133927	−	0.990991i	$-0.457241\pi$
$241$	241513.	0.267853	0.133927	+	0.990991i	$0.457241\pi$
$242$	0	0
$243$	915657.i	0.994757i
$244$	0	0
$245$	314084.i	0.334296i
$246$	0	0
$247$	1.71148e6	1.78496
$248$	0	0
$249$	−15553.2	−0.0158973
$250$	0	0
$251$	22467.7i	0.0225099i	0.999937	+	0.0112550i	$0.00358264\pi$
$251$	22467.7i	0.0225099i	−0.999937	+	0.0112550i	$0.996417\pi$
$252$	0	0
$253$	− 19933.8i	− 0.0195790i
$254$	0	0
$255$	−174126.	−0.167692
$256$	0	0
$257$	−391165.	−0.369426	−0.184713	−	0.982793i	$-0.559135\pi$
$257$	−391165.	−0.369426	−0.184713	+	0.982793i	$0.559135\pi$
$258$	0	0
$259$	− 794455.i	− 0.735902i
$260$	0	0
$261$	438227.i	0.398196i
$262$	0	0
$263$	2.20817e6	1.96854	0.984269	−	0.176677i	$-0.0565348\pi$
$263$	2.20817e6	1.96854	0.984269	+	0.176677i	$0.0565348\pi$
$264$	0	0
$265$	560407.	0.490217
$266$	0	0
$267$	19742.6i	0.0169483i
$268$	0	0
$269$	290107.i	0.244443i	0.992503	+	0.122221i	$0.0390018\pi$
$269$	290107.i	0.244443i	−0.992503	+	0.122221i	$0.960998\pi$
$270$	0	0
$271$	−252397.	−0.208766	−0.104383	−	0.994537i	$-0.533287\pi$
$271$	−252397.	−0.208766	−0.104383	+	0.994537i	$0.533287\pi$
$272$	0	0
$273$	464681.	0.377353
$274$	0	0
$275$	44727.6i	0.0356651i
$276$	0	0
$277$	− 698922.i	− 0.547305i	−0.961829	−	0.273653i	$-0.911768\pi$
$277$	− 698922.i	− 0.547305i	0.961829	−	0.273653i	$-0.0882318\pi$
$278$	0	0
$279$	897344.	0.690158
$280$	0	0
$281$	−2.49919e6	−1.88813	−0.944067	−	0.329754i	$-0.893034\pi$
$281$	−2.49919e6	−1.88813	−0.944067	+	0.329754i	$0.893034\pi$
$282$	0	0
$283$	1.39956e6i	1.03878i	0.854536	+	0.519392i	$0.173842\pi$
$283$	1.39956e6i	1.03878i	−0.854536	+	0.519392i	$0.826158\pi$
$284$	0	0
$285$	− 567480.i	− 0.413845i
$286$	0	0
$287$	−236463.	−0.169457
$288$	0	0
$289$	−907090.	−0.638860
$290$	0	0
$291$	− 1.17407e6i	− 0.812755i
$292$	0	0
$293$	− 407532.i	− 0.277327i	−0.990340	−	0.138664i	$-0.955719\pi$
$293$	− 407532.i	− 0.277327i	0.990340	−	0.138664i	$-0.0442807\pi$
$294$	0	0
$295$	182141.	0.121858
$296$	0	0
$297$	272440.	0.179217
$298$	0	0
$299$	204276.i	0.132142i
$300$	0	0
$301$	− 14987.5i	− 0.00953483i
$302$	0	0
$303$	−1.69126e6	−1.05828
$304$	0	0
$305$	485437.	0.298802
$306$	0	0
$307$	− 3.06612e6i	− 1.85671i	−0.371699	−	0.928353i	$-0.621225\pi$
$307$	− 3.06612e6i	− 1.85671i	0.371699	−	0.928353i	$-0.378775\pi$
$308$	0	0
$309$	− 1.08739e6i	− 0.647874i
$310$	0	0
$311$	3.37340e6	1.97773	0.988864	−	0.148822i	$-0.0475483\pi$
$311$	3.37340e6	1.97773	0.988864	+	0.148822i	$0.0475483\pi$
$312$	0	0
$313$	1.20582e6	0.695702	0.347851	−	0.937550i	$-0.386911\pi$
$313$	1.20582e6	0.695702	0.347851	+	0.937550i	$0.386911\pi$
$314$	0	0
$315$	241669.i	0.137229i
$316$	0	0
$317$	2.43849e6i	1.36293i	0.731851	+	0.681464i	$0.238657\pi$
$317$	2.43849e6i	1.36293i	−0.731851	+	0.681464i	$0.761343\pi$
$318$	0	0
$319$	211340.	0.116280
$320$	0	0
$321$	−1.07478e6	−0.582177
$322$	0	0
$323$	1.67112e6i	0.891254i
$324$	0	0
$325$	− 458356.i	− 0.240710i
$326$	0	0
$327$	−442766.	−0.228984
$328$	0	0
$329$	911879.	0.464459
$330$	0	0
$331$	− 465766.i	− 0.233667i	−0.993152	−	0.116834i	$-0.962726\pi$
$331$	− 465766.i	− 0.233667i	0.993152	−	0.116834i	$-0.0372744\pi$
$332$	0	0
$333$	1.80972e6i	0.894338i
$334$	0	0
$335$	34771.9	0.0169284
$336$	0	0
$337$	−119437.	−0.0572881	−0.0286441	−	0.999590i	$-0.509119\pi$
$337$	−119437.	−0.0572881	−0.0286441	+	0.999590i	$0.509119\pi$
$338$	0	0
$339$	1.76397e6i	0.833667i
$340$	0	0
$341$	− 432756.i	− 0.201538i
$342$	0	0
$343$	1.91328e6	0.878098
$344$	0	0
$345$	67732.6	0.0306373
$346$	0	0
$347$	− 2.44229e6i	− 1.08886i	−0.838805	−	0.544431i	$-0.816746\pi$
$347$	− 2.44229e6i	− 1.08886i	0.838805	−	0.544431i	$-0.183254\pi$
$348$	0	0
$349$	908422.i	0.399231i	0.979874	+	0.199615i	$0.0639693\pi$
$349$	908422.i	0.399231i	−0.979874	+	0.199615i	$0.936031\pi$
$350$	0	0
$351$	−2.79189e6	−1.20957
$352$	0	0
$353$	4.33295e6	1.85075	0.925374	−	0.379055i	$-0.123751\pi$
$353$	4.33295e6	1.85075	0.925374	+	0.379055i	$0.123751\pi$
$354$	0	0
$355$	1.43289e6i	0.603452i
$356$	0	0
$357$	453724.i	0.188417i
$358$	0	0
$359$	−3.84357e6	−1.57398	−0.786988	−	0.616968i	$-0.788361\pi$
$359$	−3.84357e6	−1.57398	−0.786988	+	0.616968i	$0.788361\pi$
$360$	0	0
$361$	−2.97012e6	−1.19952
$362$	0	0
$363$	1.51667e6i	0.604122i
$364$	0	0
$365$	888899.i	0.349237i
$366$	0	0
$367$	−3.40048e6	−1.31788	−0.658938	−	0.752197i	$-0.728994\pi$
$367$	−3.40048e6	−1.31788	−0.658938	+	0.752197i	$0.728994\pi$
$368$	0	0
$369$	538649.	0.205940
$370$	0	0
$371$	− 1.46027e6i	− 0.550805i
$372$	0	0
$373$	728064.i	0.270955i	0.990780	+	0.135478i	$0.0432569\pi$
$373$	728064.i	0.270955i	−0.990780	+	0.135478i	$0.956743\pi$
$374$	0	0
$375$	−151979.	−0.0558090
$376$	0	0
$377$	−2.16576e6	−0.784795
$378$	0	0
$379$	− 1.03272e6i	− 0.369306i	−0.982804	−	0.184653i	$-0.940884\pi$
$379$	− 1.03272e6i	− 0.369306i	0.982804	−	0.184653i	$-0.0591161\pi$
$380$	0	0
$381$	− 972427.i	− 0.343198i
$382$	0	0
$383$	−4.21824e6	−1.46938	−0.734690	−	0.678403i	$-0.762672\pi$
$383$	−4.21824e6	−1.46938	−0.734690	+	0.678403i	$0.762672\pi$
$384$	0	0
$385$	116548.	0.0400731
$386$	0	0
$387$	34140.7i	0.0115876i
$388$	0	0
$389$	3.25817e6i	1.09169i	0.837886	+	0.545845i	$0.183791\pi$
$389$	3.25817e6i	1.09169i	−0.837886	+	0.545845i	$0.816209\pi$
$390$	0	0
$391$	−199460.	−0.0659801
$392$	0	0
$393$	1.33414e6	0.435734
$394$	0	0
$395$	2.32529e6i	0.749866i
$396$	0	0
$397$	174369.i	0.0555255i	0.999615	+	0.0277628i	$0.00883830\pi$
$397$	174369.i	0.0555255i	−0.999615	+	0.0277628i	$0.991162\pi$
$398$	0	0
$399$	−1.47870e6	−0.464994
$400$	0	0
$401$	−5.38282e6	−1.67166	−0.835831	−	0.548986i	$-0.815014\pi$
$401$	−5.38282e6	−1.67166	−0.835831	+	0.548986i	$0.815014\pi$
$402$	0	0
$403$	4.43476e6i	1.36021i
$404$	0	0
$405$	24231.7i	0.00734085i
$406$	0	0
$407$	872762.	0.261162
$408$	0	0
$409$	3.44650e6	1.01876	0.509378	−	0.860543i	$-0.329875\pi$
$409$	3.44650e6	1.01876	0.509378	+	0.860543i	$0.329875\pi$
$410$	0	0
$411$	− 1.76992e6i	− 0.516831i
$412$	0	0
$413$	− 474610.i	− 0.136918i
$414$	0	0
$415$	39975.9	0.0113941
$416$	0	0
$417$	−1.30818e6	−0.368407
$418$	0	0
$419$	− 4.73672e6i	− 1.31808i	−0.752107	−	0.659041i	$-0.770962\pi$
$419$	− 4.73672e6i	− 1.31808i	0.752107	−	0.659041i	$-0.229038\pi$
$420$	0	0
$421$	2.96214e6i	0.814516i	0.913313	+	0.407258i	$0.133515\pi$
$421$	2.96214e6i	0.814516i	−0.913313	+	0.407258i	$0.866485\pi$
$422$	0	0
$423$	−2.07721e6	−0.564455
$424$	0	0
$425$	447548.	0.120190
$426$	0	0
$427$	− 1.26492e6i	− 0.335731i
$428$	0	0
$429$	510483.i	0.133918i
$430$	0	0
$431$	−6.11636e6	−1.58599	−0.792995	−	0.609229i	$-0.791479\pi$
$431$	−6.11636e6	−1.58599	−0.792995	+	0.609229i	$0.791479\pi$
$432$	0	0
$433$	7.55385e6	1.93619	0.968097	−	0.250577i	$-0.0806204\pi$
$433$	7.55385e6	1.93619	0.968097	+	0.250577i	$0.0806204\pi$
$434$	0	0
$435$	718107.i	0.181956i
$436$	0	0
$437$	− 650044.i	− 0.162832i
$438$	0	0
$439$	4.84092e6	1.19886	0.599428	−	0.800429i	$-0.295395\pi$
$439$	4.84092e6	1.19886	0.599428	+	0.800429i	$0.295395\pi$
$440$	0	0
$441$	−1.86431e6	−0.456480
$442$	0	0
$443$	6.96723e6i	1.68675i	0.537325	+	0.843375i	$0.319435\pi$
$443$	6.96723e6i	1.68675i	−0.537325	+	0.843375i	$0.680565\pi$
$444$	0	0
$445$	− 50743.5i	− 0.0121473i
$446$	0	0
$447$	−3.38526e6	−0.801352
$448$	0	0
$449$	−3.85998e6	−0.903586	−0.451793	−	0.892123i	$-0.649215\pi$
$449$	−3.85998e6	−0.903586	−0.451793	+	0.892123i	$0.649215\pi$
$450$	0	0
$451$	− 259770.i	− 0.0601379i
$452$	0	0
$453$	1.55563e6i	0.356172i
$454$	0	0
$455$	−1.19435e6	−0.270460
$456$	0	0
$457$	1.52936e6	0.342547	0.171273	−	0.985224i	$-0.445212\pi$
$457$	1.52936e6	0.342547	0.171273	+	0.985224i	$0.445212\pi$
$458$	0	0
$459$	− 2.72606e6i	− 0.603953i
$460$	0	0
$461$	7.36871e6i	1.61488i	0.589952	+	0.807438i	$0.299146\pi$
$461$	7.36871e6i	1.61488i	−0.589952	+	0.807438i	$0.700854\pi$
$462$	0	0
$463$	−2.95746e6	−0.641161	−0.320580	−	0.947221i	$-0.603878\pi$
$463$	−2.95746e6	−0.641161	−0.320580	+	0.947221i	$0.603878\pi$
$464$	0	0
$465$	1.47045e6	0.315368
$466$	0	0
$467$	− 6.04555e6i	− 1.28275i	−0.767226	−	0.641377i	$-0.778363\pi$
$467$	− 6.04555e6i	− 1.28275i	0.767226	−	0.641377i	$-0.221637\pi$
$468$	0	0
$469$	− 90606.1i	− 0.0190207i
$470$	0	0
$471$	−2.07019e6	−0.429990
$472$	0	0
$473$	16464.8	0.00338379
$474$	0	0
$475$	1.45857e6i	0.296615i
$476$	0	0
$477$	3.32641e6i	0.669391i
$478$	0	0
$479$	−3.59287e6	−0.715489	−0.357745	−	0.933819i	$-0.616454\pi$
$479$	−3.59287e6	−0.715489	−0.357745	+	0.933819i	$0.616454\pi$
$480$	0	0
$481$	−8.94383e6	−1.76263
$482$	0	0
$483$	− 176493.i	− 0.0344238i
$484$	0	0
$485$	3.01765e6i	0.582526i
$486$	0	0
$487$	6.30036e6	1.20377	0.601884	−	0.798584i	$-0.294417\pi$
$487$	6.30036e6	1.20377	0.601884	+	0.798584i	$0.294417\pi$
$488$	0	0
$489$	−3.98959e6	−0.754494
$490$	0	0
$491$	− 8.99067e6i	− 1.68302i	−0.540245	−	0.841508i	$-0.681668\pi$
$491$	− 8.99067e6i	− 1.68302i	0.540245	−	0.841508i	$-0.318332\pi$
$492$	0	0
$493$	− 2.11469e6i	− 0.391858i
$494$	0	0
$495$	−265490.	−0.0487006
$496$	0	0
$497$	3.73373e6	0.678034
$498$	0	0
$499$	3.96986e6i	0.713713i	0.934159	+	0.356856i	$0.116151\pi$
$499$	3.96986e6i	0.713713i	−0.934159	+	0.356856i	$0.883849\pi$
$500$	0	0
$501$	− 342138.i	− 0.0608985i
$502$	0	0
$503$	4.44728e6	0.783745	0.391873	−	0.920020i	$-0.371827\pi$
$503$	4.44728e6	0.783745	0.391873	+	0.920020i	$0.371827\pi$
$504$	0	0
$505$	4.34697e6	0.758504
$506$	0	0
$507$	− 1.61986e6i	− 0.279870i
$508$	0	0
$509$	3.69162e6i	0.631572i	0.948830	+	0.315786i	$0.102268\pi$
$509$	3.69162e6i	0.631572i	−0.948830	+	0.315786i	$0.897732\pi$
$510$	0	0
$511$	2.31623e6	0.392400
$512$	0	0
$513$	8.88429e6	1.49049
$514$	0	0
$515$	2.79489e6i	0.464350i
$516$	0	0
$517$	1.00176e6i	0.164830i
$518$	0	0
$519$	3.16739e6	0.516158
$520$	0	0
$521$	−3.62317e6	−0.584783	−0.292391	−	0.956299i	$-0.594451\pi$
$521$	−3.62317e6	−0.584783	−0.292391	+	0.956299i	$0.594451\pi$
$522$	0	0
$523$	6.82738e6i	1.09144i	0.837968	+	0.545720i	$0.183744\pi$
$523$	6.82738e6i	1.09144i	−0.837968	+	0.545720i	$0.816256\pi$
$524$	0	0
$525$	396015.i	0.0627067i
$526$	0	0
$527$	−4.33019e6	−0.679173
$528$	0	0
$529$	−6.35876e6	−0.987945
$530$	0	0
$531$	1.08113e6i	0.166396i
$532$	0	0
$533$	2.66206e6i	0.405881i
$534$	0	0
$535$	2.76245e6	0.417264
$536$	0	0
$537$	−2.24800e6	−0.336403
$538$	0	0
$539$	899087.i	0.133300i
$540$	0	0
$541$	− 1.20284e7i	− 1.76692i	−0.468509	−	0.883459i	$-0.655209\pi$
$541$	− 1.20284e7i	− 1.76692i	0.468509	−	0.883459i	$-0.344791\pi$
$542$	0	0
$543$	2.01414e6	0.293150
$544$	0	0
$545$	1.13803e6	0.164120
$546$	0	0
$547$	742984.i	0.106172i	0.998590	+	0.0530861i	$0.0169058\pi$
$547$	742984.i	0.106172i	−0.998590	+	0.0530861i	$0.983094\pi$
$548$	0	0
$549$	2.88141e6i	0.408013i
$550$	0	0
$551$	6.89182e6	0.967064
$552$	0	0
$553$	6.05907e6	0.842545
$554$	0	0
$555$	2.96554e6i	0.408668i
$556$	0	0
$557$	− 9.35350e6i	− 1.27743i	−0.769444	−	0.638714i	$-0.779467\pi$
$557$	− 9.35350e6i	− 1.27743i	0.769444	−	0.638714i	$-0.220533\pi$
$558$	0	0
$559$	−168726.	−0.0228378
$560$	0	0
$561$	−498446.	−0.0668669
$562$	0	0
$563$	9.70970e6i	1.29103i	0.763749	+	0.645513i	$0.223356\pi$
$563$	9.70970e6i	1.29103i	−0.763749	+	0.645513i	$0.776644\pi$
$564$	0	0
$565$	− 4.53387e6i	− 0.597514i
$566$	0	0
$567$	63141.3	0.00824813
$568$	0	0
$569$	1.09705e7	1.42052	0.710258	−	0.703941i	$-0.248578\pi$
$569$	1.09705e7	1.42052	0.710258	+	0.703941i	$0.248578\pi$
$570$	0	0
$571$	− 9.87288e6i	− 1.26722i	−0.773651	−	0.633612i	$-0.781572\pi$
$571$	− 9.87288e6i	− 1.26722i	0.773651	−	0.633612i	$-0.218428\pi$
$572$	0	0
$573$	1.85666e6i	0.236236i
$574$	0	0
$575$	−174091.	−0.0219586
$576$	0	0
$577$	−9.27844e6	−1.16021	−0.580103	−	0.814543i	$-0.696988\pi$
$577$	−9.27844e6	−1.16021	−0.580103	+	0.814543i	$0.696988\pi$
$578$	0	0
$579$	− 38992.8i	− 0.00483379i
$580$	0	0
$581$	− 104166.i	− 0.0128023i
$582$	0	0
$583$	1.60420e6	0.195473
$584$	0	0
$585$	2.72067e6	0.328689
$586$	0	0
$587$	2.34170e6i	0.280502i	0.990116	+	0.140251i	$0.0447910\pi$
$587$	2.34170e6i	0.280502i	−0.990116	+	0.140251i	$0.955209\pi$
$588$	0	0
$589$	− 1.41122e7i	− 1.67613i
$590$	0	0
$591$	5.03261e6	0.592686
$592$	0	0
$593$	3.14474e6	0.367238	0.183619	−	0.982997i	$-0.441219\pi$
$593$	3.14474e6	0.367238	0.183619	+	0.982997i	$0.441219\pi$
$594$	0	0
$595$	− 1.16619e6i	− 0.135044i
$596$	0	0
$597$	9.57994e6i	1.10009i
$598$	0	0
$599$	3.65098e6	0.415759	0.207880	−	0.978154i	$-0.433344\pi$
$599$	3.65098e6	0.415759	0.207880	+	0.978154i	$0.433344\pi$
$600$	0	0
$601$	−1.23821e7	−1.39832	−0.699161	−	0.714965i	$-0.746443\pi$
$601$	−1.23821e7	−1.39832	−0.699161	+	0.714965i	$0.746443\pi$
$602$	0	0
$603$	206396.i	0.0231157i
$604$	0	0
$605$	− 3.89824e6i	− 0.432992i
$606$	0	0
$607$	1.06103e7	1.16884	0.584421	−	0.811451i	$-0.301322\pi$
$607$	1.06103e7	1.16884	0.584421	+	0.811451i	$0.301322\pi$
$608$	0	0
$609$	1.87119e6	0.204444
$610$	0	0
$611$	− 1.02658e7i	− 1.11247i
$612$	0	0
$613$	− 1.01958e7i	− 1.09590i	−0.836511	−	0.547950i	$-0.815408\pi$
$613$	− 1.01958e7i	− 1.09590i	0.836511	−	0.547950i	$-0.184592\pi$
$614$	0	0
$615$	882667.	0.0941042
$616$	0	0
$617$	−1.30889e7	−1.38417	−0.692086	−	0.721815i	$-0.743308\pi$
$617$	−1.30889e7	−1.38417	−0.692086	+	0.721815i	$0.743308\pi$
$618$	0	0
$619$	− 8.92428e6i	− 0.936152i	−0.883688	−	0.468076i	$-0.844947\pi$
$619$	− 8.92428e6i	− 0.936152i	0.883688	−	0.468076i	$-0.155053\pi$
$620$	0	0
$621$	1.06040e6i	0.110342i
$622$	0	0
$623$	−132224.	−0.0136486
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	− 1.62445e6i	− 0.165020i
$628$	0	0
$629$	− 8.73294e6i	− 0.880103i
$630$	0	0
$631$	6.87718e6	0.687602	0.343801	−	0.939043i	$-0.388285\pi$
$631$	6.87718e6	0.687602	0.343801	+	0.939043i	$0.388285\pi$
$632$	0	0
$633$	8.95765e6	0.888556
$634$	0	0
$635$	2.49939e6i	0.245980i
$636$	0	0
$637$	− 9.21359e6i	− 0.899664i
$638$	0	0
$639$	−8.50522e6	−0.824011
$640$	0	0
$641$	1.39914e7	1.34498	0.672489	−	0.740107i	$-0.265225\pi$
$641$	1.39914e7	1.34498	0.672489	+	0.740107i	$0.265225\pi$
$642$	0	0
$643$	6.77228e6i	0.645962i	0.946405	+	0.322981i	$0.104685\pi$
$643$	6.77228e6i	0.645962i	−0.946405	+	0.322981i	$0.895315\pi$
$644$	0	0
$645$	55945.2i	0.00529497i
$646$	0	0
$647$	−1.16544e7	−1.09454	−0.547268	−	0.836958i	$-0.684332\pi$
$647$	−1.16544e7	−1.09454	−0.547268	+	0.836958i	$0.684332\pi$
$648$	0	0
$649$	521391.	0.0485905
$650$	0	0
$651$	− 3.83159e6i	− 0.354345i
$652$	0	0
$653$	− 6.39818e6i	− 0.587183i	−0.955931	−	0.293591i	$-0.905149\pi$
$653$	− 6.39818e6i	− 0.587183i	0.955931	−	0.293591i	$-0.0948505\pi$
$654$	0	0
$655$	−3.42910e6	−0.312303
$656$	0	0
$657$	−5.27624e6	−0.476882
$658$	0	0
$659$	2.45934e6i	0.220600i	0.993898	+	0.110300i	$0.0351812\pi$
$659$	2.45934e6i	0.220600i	−0.993898	+	0.110300i	$0.964819\pi$
$660$	0	0
$661$	− 1.99968e7i	− 1.78015i	−0.455816	−	0.890074i	$-0.650652\pi$
$661$	− 1.99968e7i	− 1.78015i	0.455816	−	0.890074i	$-0.349348\pi$
$662$	0	0
$663$	5.10794e6	0.451296
$664$	0	0
$665$	3.80064e6	0.333275
$666$	0	0
$667$	822587.i	0.0715924i
$668$	0	0
$669$	1.81881e6i	0.157116i
$670$	0	0
$671$	1.38959e6	0.119147
$672$	0	0
$673$	8.57475e6	0.729766	0.364883	−	0.931053i	$-0.381109\pi$
$673$	8.57475e6	0.729766	0.364883	+	0.931053i	$0.381109\pi$
$674$	0	0
$675$	− 2.37933e6i	− 0.201000i
$676$	0	0
$677$	− 5.45187e6i	− 0.457166i	−0.973524	−	0.228583i	$-0.926591\pi$
$677$	− 5.45187e6i	− 0.457166i	0.973524	−	0.228583i	$-0.0734092\pi$
$678$	0	0
$679$	7.86319e6	0.654522
$680$	0	0
$681$	−7.75050e6	−0.640416
$682$	0	0
$683$	1.77200e7i	1.45349i	0.686909	+	0.726743i	$0.258967\pi$
$683$	1.77200e7i	1.45349i	−0.686909	+	0.726743i	$0.741033\pi$
$684$	0	0
$685$	4.54915e6i	0.370428i
$686$	0	0
$687$	1.11381e7	0.900366
$688$	0	0
$689$	−1.64394e7	−1.31928
$690$	0	0
$691$	− 1.58363e7i	− 1.26171i	−0.775902	−	0.630853i	$-0.782705\pi$
$691$	− 1.58363e7i	− 1.26171i	0.775902	−	0.630853i	$-0.217295\pi$
$692$	0	0
$693$	691794.i	0.0547197i
$694$	0	0
$695$	3.36237e6	0.264049
$696$	0	0
$697$	−2.59929e6	−0.202662
$698$	0	0
$699$	1.22041e7i	0.944742i
$700$	0	0
$701$	1.17168e7i	0.900565i	0.892886	+	0.450283i	$0.148677\pi$
$701$	1.17168e7i	0.900565i	−0.892886	+	0.450283i	$0.851323\pi$
$702$	0	0
$703$	2.84609e7	2.17200
$704$	0	0
$705$	−3.40385e6	−0.257928
$706$	0	0
$707$	− 1.13270e7i	− 0.852250i
$708$	0	0
$709$	− 5.36977e6i	− 0.401181i	−0.979675	−	0.200591i	$-0.935714\pi$
$709$	− 5.36977e6i	− 0.401181i	0.979675	−	0.200591i	$-0.0642861\pi$
$710$	0	0
$711$	−1.38022e7	−1.02394
$712$	0	0
$713$	1.68439e6	0.124085
$714$	0	0
$715$	− 1.31207e6i	− 0.0959828i
$716$	0	0
$717$	− 4.41878e6i	− 0.321000i
$718$	0	0
$719$	7.02495e6	0.506782	0.253391	−	0.967364i	$-0.418454\pi$
$719$	7.02495e6	0.506782	0.253391	+	0.967364i	$0.418454\pi$
$720$	0	0
$721$	7.28271e6	0.521741
$722$	0	0
$723$	2.34911e6i	0.167131i
$724$	0	0
$725$	− 1.84572e6i	− 0.130413i
$726$	0	0
$727$	1.57023e7	1.10187	0.550933	−	0.834550i	$-0.314272\pi$
$727$	1.57023e7	1.10187	0.550933	+	0.834550i	$0.314272\pi$
$728$	0	0
$729$	−9.14180e6	−0.637107
$730$	0	0
$731$	− 164748.i	− 0.0114032i
$732$	0	0
$733$	− 1.01796e6i	− 0.0699794i	−0.999388	−	0.0349897i	$-0.988860\pi$
$733$	− 1.01796e6i	− 0.0699794i	0.999388	−	0.0349897i	$-0.0111398\pi$
$734$	0	0
$735$	−3.05498e6	−0.208588
$736$	0	0
$737$	99536.9	0.00675018
$738$	0	0
$739$	1.30624e7i	0.879857i	0.898033	+	0.439929i	$0.144996\pi$
$739$	1.30624e7i	0.879857i	−0.898033	+	0.439929i	$0.855004\pi$
$740$	0	0
$741$	1.66469e7i	1.11375i
$742$	0	0
$743$	−2.27890e7	−1.51444	−0.757222	−	0.653157i	$-0.773444\pi$
$743$	−2.27890e7	−1.51444	−0.757222	+	0.653157i	$0.773444\pi$
$744$	0	0
$745$	8.70100e6	0.574352
$746$	0	0
$747$	237285.i	0.0155585i
$748$	0	0
$749$	− 7.19821e6i	− 0.468835i
$750$	0	0
$751$	6.67237e6	0.431699	0.215849	−	0.976427i	$-0.430748\pi$
$751$	6.67237e6	0.431699	0.215849	+	0.976427i	$0.430748\pi$
$752$	0	0
$753$	−218535.	−0.0140454
$754$	0	0
$755$	− 3.99837e6i	− 0.255279i
$756$	0	0
$757$	− 1.93030e7i	− 1.22429i	−0.790745	−	0.612146i	$-0.790307\pi$
$757$	− 1.93030e7i	− 1.22429i	0.790745	−	0.612146i	$-0.209693\pi$
$758$	0	0
$759$	193889.	0.0122166
$760$	0	0
$761$	171195.	0.0107159	0.00535795	−	0.999986i	$-0.498295\pi$
$761$	171195.	0.0107159	0.00535795	+	0.999986i	$0.498295\pi$
$762$	0	0
$763$	− 2.96538e6i	− 0.184404i
$764$	0	0
$765$	2.65651e6i	0.164119i
$766$	0	0
$767$	−5.34307e6	−0.327946
$768$	0	0
$769$	5.91465e6	0.360673	0.180336	−	0.983605i	$-0.442281\pi$
$769$	5.91465e6	0.360673	0.180336	+	0.983605i	$0.442281\pi$
$770$	0	0
$771$	− 3.80472e6i	− 0.230508i
$772$	0	0
$773$	2.30974e7i	1.39032i	0.718856	+	0.695160i	$0.244666\pi$
$773$	2.30974e7i	1.39032i	−0.718856	+	0.695160i	$0.755334\pi$
$774$	0	0
$775$	−3.77944e6	−0.226034
$776$	0	0
$777$	7.72738e6	0.459177
$778$	0	0
$779$	− 8.47114e6i	− 0.500148i
$780$	0	0
$781$	4.10175e6i	0.240625i
$782$	0	0
$783$	−1.12425e7	−0.655326
$784$	0	0
$785$	5.32094e6	0.308187
$786$	0	0
$787$	1.06371e7i	0.612192i	0.952001	+	0.306096i	$0.0990229\pi$
$787$	1.06371e7i	0.612192i	−0.952001	+	0.306096i	$0.900977\pi$
$788$	0	0
$789$	2.14781e7i	1.22830i
$790$	0	0
$791$	−1.18140e7	−0.671362
$792$	0	0
$793$	−1.42402e7	−0.804142
$794$	0	0
$795$	5.45087e6i	0.305878i
$796$	0	0
$797$	3.28380e7i	1.83118i	0.402115	+	0.915589i	$0.368275\pi$
$797$	3.28380e7i	1.83118i	−0.402115	+	0.915589i	$0.631725\pi$
$798$	0	0
$799$	1.00237e7	0.555470
$800$	0	0
$801$	301198.	0.0165871
$802$	0	0
$803$	2.54453e6i	0.139258i
$804$	0	0
$805$	453633.i	0.0246726i
$806$	0	0
$807$	−2.82177e6	−0.152524
$808$	0	0
$809$	1.93286e7	1.03831	0.519157	−	0.854679i	$-0.326246\pi$
$809$	1.93286e7	1.03831	0.519157	+	0.854679i	$0.326246\pi$
$810$	0	0
$811$	− 95836.0i	− 0.00511654i	−0.999997	−	0.00255827i	$-0.999186\pi$
$811$	− 95836.0i	− 0.00511654i	0.999997	−	0.00255827i	$-0.000814324\pi$
$812$	0	0
$813$	− 2.45497e6i	− 0.130263i
$814$	0	0
$815$	1.02543e7	0.540768
$816$	0	0
$817$	536918.	0.0281419
$818$	0	0
$819$	− 7.08931e6i	− 0.369313i
$820$	0	0
$821$	− 3.19867e6i	− 0.165620i	−0.996565	−	0.0828098i	$-0.973611\pi$
$821$	− 3.19867e6i	− 0.165620i	0.996565	−	0.0828098i	$-0.0263894\pi$
$822$	0	0
$823$	−4.78006e6	−0.245999	−0.123000	−	0.992407i	$-0.539251\pi$
$823$	−4.78006e6	−0.245999	−0.123000	+	0.992407i	$0.539251\pi$
$824$	0	0
$825$	−435049.	−0.0222538
$826$	0	0
$827$	2.64456e7i	1.34459i	0.740283	+	0.672296i	$0.234692\pi$
$827$	2.64456e7i	1.34459i	−0.740283	+	0.672296i	$0.765308\pi$
$828$	0	0
$829$	1.72636e7i	0.872460i	0.899835	+	0.436230i	$0.143687\pi$
$829$	1.72636e7i	0.872460i	−0.899835	+	0.436230i	$0.856313\pi$
$830$	0	0
$831$	6.79816e6	0.341499
$832$	0	0
$833$	8.99634e6	0.449214
$834$	0	0
$835$	879384.i	0.0436478i
$836$	0	0
$837$	2.30209e7i	1.13582i
$838$	0	0
$839$	−2.59843e7	−1.27440	−0.637200	−	0.770699i	$-0.719907\pi$
$839$	−2.59843e7	−1.27440	−0.637200	+	0.770699i	$0.719907\pi$
$840$	0	0
$841$	1.17900e7	0.574810
$842$	0	0
$843$	− 2.43087e7i	− 1.17813i
$844$	0	0
$845$	4.16346e6i	0.200591i
$846$	0	0
$847$	−1.01578e7	−0.486507
$848$	0	0
$849$	−1.36130e7	−0.648164
$850$	0	0
$851$	3.39700e6i	0.160795i
$852$	0	0
$853$	− 3.60072e7i	− 1.69440i	−0.531271	−	0.847202i	$-0.678285\pi$
$853$	− 3.60072e7i	− 1.69440i	0.531271	−	0.847202i	$-0.321715\pi$
$854$	0	0
$855$	−8.65764e6	−0.405027
$856$	0	0
$857$	−1.44304e7	−0.671158	−0.335579	−	0.942012i	$-0.608932\pi$
$857$	−1.44304e7	−0.671158	−0.335579	+	0.942012i	$0.608932\pi$
$858$	0	0
$859$	− 2.88104e7i	− 1.33219i	−0.745867	−	0.666095i	$-0.767965\pi$
$859$	− 2.88104e7i	− 1.33219i	0.745867	−	0.666095i	$-0.232035\pi$
$860$	0	0
$861$	− 2.29999e6i	− 0.105735i
$862$	0	0
$863$	−1.48508e7	−0.678772	−0.339386	−	0.940647i	$-0.610219\pi$
$863$	−1.48508e7	−0.678772	−0.339386	+	0.940647i	$0.610219\pi$
$864$	0	0
$865$	−8.14101e6	−0.369946
$866$	0	0
$867$	− 8.82294e6i	− 0.398626i
$868$	0	0
$869$	6.65629e6i	0.299008i
$870$	0	0
$871$	−1.02003e6	−0.0455581
$872$	0	0
$873$	−1.79119e7	−0.795437
$874$	0	0
$875$	− 1.01786e6i	− 0.0449437i
$876$	0	0
$877$	− 8.37357e6i	− 0.367630i	−0.982961	−	0.183815i	$-0.941155\pi$
$877$	− 8.37357e6i	− 0.367630i	0.982961	−	0.183815i	$-0.0588448\pi$
$878$	0	0
$879$	3.96391e6	0.173042
$880$	0	0
$881$	−2.64394e7	−1.14766	−0.573829	−	0.818975i	$-0.694543\pi$
$881$	−2.64394e7	−1.14766	−0.573829	+	0.818975i	$0.694543\pi$
$882$	0	0
$883$	595075.i	0.0256844i	0.999918	+	0.0128422i	$0.00408792\pi$
$883$	595075.i	0.0256844i	−0.999918	+	0.0128422i	$0.995912\pi$
$884$	0	0
$885$	1.77162e6i	0.0760348i
$886$	0	0
$887$	−4.09777e7	−1.74879	−0.874396	−	0.485213i	$-0.838742\pi$
$887$	−4.09777e7	−1.74879	−0.874396	+	0.485213i	$0.838742\pi$
$888$	0	0
$889$	6.51274e6	0.276382
$890$	0	0
$891$	69364.9i	0.00292715i
$892$	0	0
$893$	3.26675e7i	1.37084i
$894$	0	0
$895$	5.77794e6	0.241110
$896$	0	0
$897$	−1.98692e6	−0.0824517
$898$	0	0
$899$	1.78580e7i	0.736944i
$900$	0	0
$901$	− 1.60518e7i	− 0.658736i
$902$	0	0
$903$	145778.	0.00594939
$904$	0	0
$905$	−5.17687e6	−0.210110
$906$	0	0
$907$	1.30125e7i	0.525222i	0.964902	+	0.262611i	$0.0845836\pi$
$907$	1.30125e7i	0.525222i	−0.964902	+	0.262611i	$0.915416\pi$
$908$	0	0
$909$	2.58023e7i	1.03574i
$910$	0	0
$911$	−2.09280e7	−0.835472	−0.417736	−	0.908568i	$-0.637176\pi$
$911$	−2.09280e7	−0.835472	−0.417736	+	0.908568i	$0.637176\pi$
$912$	0	0
$913$	114434.	0.00454336
$914$	0	0
$915$	4.72167e6i	0.186441i
$916$	0	0
$917$	8.93530e6i	0.350902i
$918$	0	0
$919$	−2.58899e7	−1.01121	−0.505605	−	0.862765i	$-0.668731\pi$
$919$	−2.58899e7	−1.01121	−0.505605	+	0.862765i	$0.668731\pi$
$920$	0	0
$921$	2.98230e7	1.15852
$922$	0	0
$923$	− 4.20336e7i	− 1.62402i
$924$	0	0
$925$	− 7.62220e6i	− 0.292905i
$926$	0	0
$927$	−1.65896e7	−0.634069
$928$	0	0
$929$	−6.83846e6	−0.259967	−0.129984	−	0.991516i	$-0.541493\pi$
$929$	−6.83846e6	−0.259967	−0.129984	+	0.991516i	$0.541493\pi$
$930$	0	0
$931$	2.93193e7i	1.10861i
$932$	0	0
$933$	3.28118e7i	1.23403i
$934$	0	0
$935$	1.28114e6	0.0479255
$936$	0	0
$937$	2.70839e7	1.00777	0.503886	−	0.863770i	$-0.331903\pi$
$937$	2.70839e7	1.00777	0.503886	+	0.863770i	$0.331903\pi$
$938$	0	0
$939$	1.17286e7i	0.434093i
$940$	0	0
$941$	1.67445e7i	0.616452i	0.951313	+	0.308226i	$0.0997353\pi$
$941$	1.67445e7i	0.616452i	−0.951313	+	0.308226i	$0.900265\pi$
$942$	0	0
$943$	1.01109e6	0.0370263
$944$	0	0
$945$	−6.19990e6	−0.225842
$946$	0	0
$947$	− 1.30092e7i	− 0.471386i	−0.971828	−	0.235693i	$-0.924264\pi$
$947$	− 1.30092e7i	− 0.471386i	0.971828	−	0.235693i	$-0.0757360\pi$
$948$	0	0
$949$	− 2.60757e7i	− 0.939875i
$950$	0	0
$951$	−2.37183e7	−0.850419
$952$	0	0
$953$	−1.37164e7	−0.489224	−0.244612	−	0.969621i	$-0.578661\pi$
$953$	−1.37164e7	−0.489224	−0.244612	+	0.969621i	$0.578661\pi$
$954$	0	0
$955$	− 4.77210e6i	− 0.169317i
$956$	0	0
$957$	2.05563e6i	0.0725546i
$958$	0	0
$959$	1.18539e7	0.416211
$960$	0	0
$961$	7.93826e6	0.277279
$962$	0	0
$963$	1.63971e7i	0.569772i
$964$	0	0
$965$	100222.i	0.00346452i
$966$	0	0
$967$	−7.73160e6	−0.265891	−0.132945	−	0.991123i	$-0.542443\pi$
$967$	−7.73160e6	−0.265891	−0.132945	+	0.991123i	$0.542443\pi$
$968$	0	0
$969$	−1.62544e7	−0.556110
$970$	0	0
$971$	4.66660e7i	1.58837i	0.607674	+	0.794187i	$0.292103\pi$
$971$	4.66660e7i	1.58837i	−0.607674	+	0.794187i	$0.707897\pi$
$972$	0	0
$973$	− 8.76143e6i	− 0.296683i
$974$	0	0
$975$	4.45826e6	0.150195
$976$	0	0
$977$	3.88117e7	1.30085	0.650424	−	0.759572i	$-0.274591\pi$
$977$	3.88117e7	1.30085	0.650424	+	0.759572i	$0.274591\pi$
$978$	0	0
$979$	− 145257.i	− 0.00484373i
$980$	0	0
$981$	6.75498e6i	0.224105i
$982$	0	0
$983$	−4.79223e7	−1.58181	−0.790904	−	0.611940i	$-0.790389\pi$
$983$	−4.79223e7	−1.58181	−0.790904	+	0.611940i	$0.790389\pi$
$984$	0	0
$985$	−1.29351e7	−0.424796
$986$	0	0
$987$	8.86951e6i	0.289806i
$988$	0	0
$989$	64084.8i	0.00208336i
$990$	0	0
$991$	2.44268e7	0.790101	0.395050	−	0.918659i	$-0.370727\pi$
$991$	2.44268e7	0.790101	0.395050	+	0.918659i	$0.370727\pi$
$992$	0	0
$993$	4.53034e6	0.145800
$994$	0	0
$995$	− 2.46230e7i	− 0.788465i
$996$	0	0
$997$	1.25248e7i	0.399057i	0.979892	+	0.199528i	$0.0639410\pi$
$997$	1.25248e7i	0.399057i	−0.979892	+	0.199528i	$0.936059\pi$
$998$	0	0
$999$	−4.64275e7	−1.47184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.b.161.5	yes	8
4.3	odd	2		320.6.d.a.161.4	✓	8
8.3	odd	2		320.6.d.a.161.5	yes	8
8.5	even	2	inner	320.6.d.b.161.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.a.161.4	✓	8	4.3	odd	2
320.6.d.a.161.5	yes	8	8.3	odd	2
320.6.d.b.161.4	yes	8	8.5	even	2	inner
320.6.d.b.161.5	yes	8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+9.72664i q^{3} -25.0000i q^{5} -65.1432 q^{7} +148.393 q^{9} +O(q^{10})\)	\(q+9.72664i q^{3} -25.0000i q^{5} -65.1432 q^{7} +148.393 q^{9} -71.5642i q^{11} +733.370i q^{13} +243.166 q^{15} -716.077 q^{17} -2333.71i q^{19} -633.624i q^{21} +278.545 q^{23} -625.000 q^{25} +3806.93i q^{27} +2953.16i q^{29} +6047.10 q^{31} +696.079 q^{33} +1628.58i q^{35} +12195.5i q^{37} -7133.22 q^{39} +3629.90 q^{41} +230.070i q^{43} -3709.81i q^{45} -13998.1 q^{47} -12563.4 q^{49} -6965.02i q^{51} +22416.3i q^{53} -1789.10 q^{55} +22699.2 q^{57} +7285.64i q^{59} +19417.5i q^{61} -9666.77 q^{63} +18334.2 q^{65} +1390.88i q^{67} +2709.30i q^{69} -57315.7 q^{71} -35556.0 q^{73} -6079.15i q^{75} +4661.92i q^{77} -93011.5 q^{79} -969.268 q^{81} +1599.04i q^{83} +17901.9i q^{85} -28724.3 q^{87} +2029.74 q^{89} -47774.1i q^{91} +58817.9i q^{93} -58342.9 q^{95} -120706. q^{97} -10619.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) \) 8 * q + 320 * q^7 - 1192 * q^9	\( 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) \) 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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