LMFDB - Embedded newform 960.3.j.b.319.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(319,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("960.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.j (of order $2$, degree $1$, not 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:	$26.1581053786$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	960.319
Dual form			960.3.j.b.319.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +5.00000i q^{5} +10.3923 q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +5.00000i q^{5} +10.3923 q^{7} +3.00000 q^{9} +10.3923i q^{11} +18.0000i q^{13} +8.66025i q^{15} +10.0000i q^{17} -13.8564i q^{19} +18.0000 q^{21} -6.92820 q^{23} -25.0000 q^{25} +5.19615 q^{27} -36.0000 q^{29} +6.92820i q^{31} +18.0000i q^{33} +51.9615i q^{35} -54.0000i q^{37} +31.1769i q^{39} +18.0000 q^{41} +20.7846 q^{43} +15.0000i q^{45} +59.0000 q^{49} +17.3205i q^{51} -26.0000i q^{53} -51.9615 q^{55} -24.0000i q^{57} +31.1769i q^{59} +74.0000 q^{61} +31.1769 q^{63} -90.0000 q^{65} -41.5692 q^{67} -12.0000 q^{69} +103.923i q^{71} -36.0000i q^{73} -43.3013 q^{75} +108.000i q^{77} +90.0666i q^{79} +9.00000 q^{81} -90.0666 q^{83} -50.0000 q^{85} -62.3538 q^{87} +18.0000 q^{89} +187.061i q^{91} +12.0000i q^{93} +69.2820 q^{95} -72.0000i q^{97} +31.1769i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 72 q^{21} - 100 q^{25} - 144 q^{29} + 72 q^{41} + 236 q^{49} + 296 q^{61} - 360 q^{65} - 48 q^{69} + 36 q^{81} - 200 q^{85} + 72 q^{89}+O(q^{100}) $ 4 * q + 12 * q^9 + 72 * q^21 - 100 * q^25 - 144 * q^29 + 72 * q^41 + 236 * q^49 + 296 * q^61 - 360 * q^65 - 48 * q^69 + 36 * q^81 - 200 * q^85 + 72 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	5.00000i	1.00000i
$5$	5.00000i	1.00000i
$6$	0	0
$7$	10.3923	1.48461	0.742307	−	0.670059i	$-0.233731\pi$
$7$	10.3923	1.48461	0.742307	+	0.670059i	$0.233731\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	10.3923i	0.944755i	0.881396	+	0.472377i	$0.156604\pi$
$11$	10.3923i	0.944755i	−0.881396	+	0.472377i	$0.843396\pi$
$12$	0	0
$13$	18.0000i	1.38462i	0.721602	+	0.692308i	$0.243406\pi$
$13$	18.0000i	1.38462i	−0.721602	+	0.692308i	$0.756594\pi$
$14$	0	0
$15$	8.66025i	0.577350i
$16$	0	0
$17$	10.0000i	0.588235i	0.955769	+	0.294118i	$0.0950258\pi$
$17$	10.0000i	0.588235i	−0.955769	+	0.294118i	$0.904974\pi$
$18$	0	0
$19$	− 13.8564i	− 0.729285i	−0.931148	−	0.364642i	$-0.881191\pi$
$19$	− 13.8564i	− 0.729285i	0.931148	−	0.364642i	$-0.118809\pi$
$20$	0	0
$21$	18.0000	0.857143
$22$	0	0
$23$	−6.92820	−0.301226	−0.150613	−	0.988593i	$-0.548125\pi$
$23$	−6.92820	−0.301226	−0.150613	+	0.988593i	$0.548125\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	5.19615	0.192450
$28$	0	0
$29$	−36.0000	−1.24138	−0.620690	−	0.784056i	$-0.713147\pi$
$29$	−36.0000	−1.24138	−0.620690	+	0.784056i	$0.713147\pi$
$30$	0	0
$31$	6.92820i	0.223490i	0.993737	+	0.111745i	$0.0356441\pi$
$31$	6.92820i	0.223490i	−0.993737	+	0.111745i	$0.964356\pi$
$32$	0	0
$33$	18.0000i	0.545455i
$34$	0	0
$35$	51.9615i	1.48461i
$36$	0	0
$37$	− 54.0000i	− 1.45946i	−0.683736	−	0.729730i	$-0.739646\pi$
$37$	− 54.0000i	− 1.45946i	0.683736	−	0.729730i	$-0.260354\pi$
$38$	0	0
$39$	31.1769i	0.799408i
$40$	0	0
$41$	18.0000	0.439024	0.219512	−	0.975610i	$-0.429553\pi$
$41$	18.0000	0.439024	0.219512	+	0.975610i	$0.429553\pi$
$42$	0	0
$43$	20.7846	0.483363	0.241682	−	0.970356i	$-0.422301\pi$
$43$	20.7846	0.483363	0.241682	+	0.970356i	$0.422301\pi$
$44$	0	0
$45$	15.0000i	0.333333i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	59.0000	1.20408
$50$	0	0
$51$	17.3205i	0.339618i
$52$	0	0
$53$	− 26.0000i	− 0.490566i	−0.969452	−	0.245283i	$-0.921119\pi$
$53$	− 26.0000i	− 0.490566i	0.969452	−	0.245283i	$-0.0788809\pi$
$54$	0	0
$55$	−51.9615	−0.944755
$56$	0	0
$57$	− 24.0000i	− 0.421053i
$58$	0	0
$59$	31.1769i	0.528422i	0.964465	+	0.264211i	$0.0851116\pi$
$59$	31.1769i	0.528422i	−0.964465	+	0.264211i	$0.914888\pi$
$60$	0	0
$61$	74.0000	1.21311	0.606557	−	0.795040i	$-0.292550\pi$
$61$	74.0000	1.21311	0.606557	+	0.795040i	$0.292550\pi$
$62$	0	0
$63$	31.1769	0.494872
$64$	0	0
$65$	−90.0000	−1.38462
$66$	0	0
$67$	−41.5692	−0.620436	−0.310218	−	0.950665i	$-0.600402\pi$
$67$	−41.5692	−0.620436	−0.310218	+	0.950665i	$0.600402\pi$
$68$	0	0
$69$	−12.0000	−0.173913
$70$	0	0
$71$	103.923i	1.46370i	0.681463	+	0.731852i	$0.261344\pi$
$71$	103.923i	1.46370i	−0.681463	+	0.731852i	$0.738656\pi$
$72$	0	0
$73$	− 36.0000i	− 0.493151i	−0.969124	−	0.246575i	$-0.920695\pi$
$73$	− 36.0000i	− 0.493151i	0.969124	−	0.246575i	$-0.0793053\pi$
$74$	0	0
$75$	−43.3013	−0.577350
$76$	0	0
$77$	108.000i	1.40260i
$78$	0	0
$79$	90.0666i	1.14008i	0.821616	+	0.570042i	$0.193073\pi$
$79$	90.0666i	1.14008i	−0.821616	+	0.570042i	$0.806927\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−90.0666	−1.08514	−0.542570	−	0.840011i	$-0.682549\pi$
$83$	−90.0666	−1.08514	−0.542570	+	0.840011i	$0.682549\pi$
$84$	0	0
$85$	−50.0000	−0.588235
$86$	0	0
$87$	−62.3538	−0.716711
$88$	0	0
$89$	18.0000	0.202247	0.101124	−	0.994874i	$-0.467756\pi$
$89$	18.0000	0.202247	0.101124	+	0.994874i	$0.467756\pi$
$90$	0	0
$91$	187.061i	2.05562i
$92$	0	0
$93$	12.0000i	0.129032i
$94$	0	0
$95$	69.2820	0.729285
$96$	0	0
$97$	− 72.0000i	− 0.742268i	−0.928579	−	0.371134i	$-0.878969\pi$
$97$	− 72.0000i	− 0.742268i	0.928579	−	0.371134i	$-0.121031\pi$
$98$	0	0
$99$	31.1769i	0.314918i
$100$	0	0
$101$	−36.0000	−0.356436	−0.178218	−	0.983991i	$-0.557033\pi$
$101$	−36.0000	−0.356436	−0.178218	+	0.983991i	$0.557033\pi$
$102$	0	0
$103$	−10.3923	−0.100896	−0.0504481	−	0.998727i	$-0.516065\pi$
$103$	−10.3923	−0.100896	−0.0504481	+	0.998727i	$0.516065\pi$
$104$	0	0
$105$	90.0000i	0.857143i
$106$	0	0
$107$	187.061	1.74824	0.874119	−	0.485712i	$-0.161439\pi$
$107$	187.061	1.74824	0.874119	+	0.485712i	$0.161439\pi$
$108$	0	0
$109$	−26.0000	−0.238532	−0.119266	−	0.992862i	$-0.538054\pi$
$109$	−26.0000	−0.238532	−0.119266	+	0.992862i	$0.538054\pi$
$110$	0	0
$111$	− 93.5307i	− 0.842619i
$112$	0	0
$113$	10.0000i	0.0884956i	0.999021	+	0.0442478i	$0.0140891\pi$
$113$	10.0000i	0.0884956i	−0.999021	+	0.0442478i	$0.985911\pi$
$114$	0	0
$115$	− 34.6410i	− 0.301226i
$116$	0	0
$117$	54.0000i	0.461538i
$118$	0	0
$119$	103.923i	0.873303i
$120$	0	0
$121$	13.0000	0.107438
$122$	0	0
$123$	31.1769	0.253471
$124$	0	0
$125$	− 125.000i	− 1.00000i
$126$	0	0
$127$	218.238	1.71841	0.859206	−	0.511629i	$-0.170958\pi$
$127$	218.238	1.71841	0.859206	+	0.511629i	$0.170958\pi$
$128$	0	0
$129$	36.0000	0.279070
$130$	0	0
$131$	− 135.100i	− 1.03130i	−0.856800	−	0.515649i	$-0.827551\pi$
$131$	− 135.100i	− 1.03130i	0.856800	−	0.515649i	$-0.172449\pi$
$132$	0	0
$133$	− 144.000i	− 1.08271i
$134$	0	0
$135$	25.9808i	0.192450i
$136$	0	0
$137$	110.000i	0.802920i	0.915877	+	0.401460i	$0.131497\pi$
$137$	110.000i	0.802920i	−0.915877	+	0.401460i	$0.868503\pi$
$138$	0	0
$139$	187.061i	1.34577i	0.739749	+	0.672883i	$0.234944\pi$
$139$	187.061i	1.34577i	−0.739749	+	0.672883i	$0.765056\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−187.061	−1.30812
$144$	0	0
$145$	− 180.000i	− 1.24138i
$146$	0	0
$147$	102.191	0.695177
$148$	0	0
$149$	288.000	1.93289	0.966443	−	0.256881i	$-0.0826950\pi$
$149$	288.000	1.93289	0.966443	+	0.256881i	$0.0826950\pi$
$150$	0	0
$151$	187.061i	1.23882i	0.785069	+	0.619409i	$0.212628\pi$
$151$	187.061i	1.23882i	−0.785069	+	0.619409i	$0.787372\pi$
$152$	0	0
$153$	30.0000i	0.196078i
$154$	0	0
$155$	−34.6410	−0.223490
$156$	0	0
$157$	− 234.000i	− 1.49045i	−0.666815	−	0.745223i	$-0.732343\pi$
$157$	− 234.000i	− 1.49045i	0.666815	−	0.745223i	$-0.267657\pi$
$158$	0	0
$159$	− 45.0333i	− 0.283228i
$160$	0	0
$161$	−72.0000	−0.447205
$162$	0	0
$163$	124.708	0.765078	0.382539	−	0.923939i	$-0.375050\pi$
$163$	124.708	0.765078	0.382539	+	0.923939i	$0.375050\pi$
$164$	0	0
$165$	−90.0000	−0.545455
$166$	0	0
$167$	131.636	0.788239	0.394119	−	0.919059i	$-0.371050\pi$
$167$	131.636	0.788239	0.394119	+	0.919059i	$0.371050\pi$
$168$	0	0
$169$	−155.000	−0.917160
$170$	0	0
$171$	− 41.5692i	− 0.243095i
$172$	0	0
$173$	146.000i	0.843931i	0.906612	+	0.421965i	$0.138660\pi$
$173$	146.000i	0.843931i	−0.906612	+	0.421965i	$0.861340\pi$
$174$	0	0
$175$	−259.808	−1.48461
$176$	0	0
$177$	54.0000i	0.305085i
$178$	0	0
$179$	72.7461i	0.406403i	0.979137	+	0.203201i	$0.0651346\pi$
$179$	72.7461i	0.406403i	−0.979137	+	0.203201i	$0.934865\pi$
$180$	0	0
$181$	−262.000	−1.44751	−0.723757	−	0.690055i	$-0.757586\pi$
$181$	−262.000	−1.44751	−0.723757	+	0.690055i	$0.757586\pi$
$182$	0	0
$183$	128.172	0.700392
$184$	0	0
$185$	270.000	1.45946
$186$	0	0
$187$	−103.923	−0.555738
$188$	0	0
$189$	54.0000	0.285714
$190$	0	0
$191$	− 187.061i	− 0.979380i	−0.871897	−	0.489690i	$-0.837110\pi$
$191$	− 187.061i	− 0.979380i	0.871897	−	0.489690i	$-0.162890\pi$
$192$	0	0
$193$	− 180.000i	− 0.932642i	−0.884615	−	0.466321i	$-0.845579\pi$
$193$	− 180.000i	− 0.932642i	0.884615	−	0.466321i	$-0.154421\pi$
$194$	0	0
$195$	−155.885	−0.799408
$196$	0	0
$197$	154.000i	0.781726i	0.920449	+	0.390863i	$0.127823\pi$
$197$	154.000i	0.781726i	−0.920449	+	0.390863i	$0.872177\pi$
$198$	0	0
$199$	− 187.061i	− 0.940007i	−0.882664	−	0.470004i	$-0.844253\pi$
$199$	− 187.061i	− 0.940007i	0.882664	−	0.470004i	$-0.155747\pi$
$200$	0	0
$201$	−72.0000	−0.358209
$202$	0	0
$203$	−374.123	−1.84297
$204$	0	0
$205$	90.0000i	0.439024i
$206$	0	0
$207$	−20.7846	−0.100409
$208$	0	0
$209$	144.000	0.688995
$210$	0	0
$211$	− 242.487i	− 1.14923i	−0.818425	−	0.574614i	$-0.805152\pi$
$211$	− 242.487i	− 1.14923i	0.818425	−	0.574614i	$-0.194848\pi$
$212$	0	0
$213$	180.000i	0.845070i
$214$	0	0
$215$	103.923i	0.483363i
$216$	0	0
$217$	72.0000i	0.331797i
$218$	0	0
$219$	− 62.3538i	− 0.284721i
$220$	0	0
$221$	−180.000	−0.814480
$222$	0	0
$223$	−93.5307	−0.419420	−0.209710	−	0.977764i	$-0.567252\pi$
$223$	−93.5307	−0.419420	−0.209710	+	0.977764i	$0.567252\pi$
$224$	0	0
$225$	−75.0000	−0.333333
$226$	0	0
$227$	214.774	0.946142	0.473071	−	0.881024i	$-0.343145\pi$
$227$	214.774	0.946142	0.473071	+	0.881024i	$0.343145\pi$
$228$	0	0
$229$	338.000	1.47598	0.737991	−	0.674810i	$-0.235775\pi$
$229$	338.000	1.47598	0.737991	+	0.674810i	$0.235775\pi$
$230$	0	0
$231$	187.061i	0.809790i
$232$	0	0
$233$	182.000i	0.781116i	0.920578	+	0.390558i	$0.127718\pi$
$233$	182.000i	0.781116i	−0.920578	+	0.390558i	$0.872282\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	156.000i	0.658228i
$238$	0	0
$239$	− 353.338i	− 1.47840i	−0.673484	−	0.739202i	$-0.735203\pi$
$239$	− 353.338i	− 1.47840i	0.673484	−	0.739202i	$-0.264797\pi$
$240$	0	0
$241$	−106.000	−0.439834	−0.219917	−	0.975519i	$-0.570579\pi$
$241$	−106.000	−0.439834	−0.219917	+	0.975519i	$0.570579\pi$
$242$	0	0
$243$	15.5885	0.0641500
$244$	0	0
$245$	295.000i	1.20408i
$246$	0	0
$247$	249.415	1.00978
$248$	0	0
$249$	−156.000	−0.626506
$250$	0	0
$251$	322.161i	1.28351i	0.766909	+	0.641756i	$0.221794\pi$
$251$	322.161i	1.28351i	−0.766909	+	0.641756i	$0.778206\pi$
$252$	0	0
$253$	− 72.0000i	− 0.284585i
$254$	0	0
$255$	−86.6025	−0.339618
$256$	0	0
$257$	14.0000i	0.0544747i	0.999629	+	0.0272374i	$0.00867099\pi$
$257$	14.0000i	0.0544747i	−0.999629	+	0.0272374i	$0.991329\pi$
$258$	0	0
$259$	− 561.184i	− 2.16674i
$260$	0	0
$261$	−108.000	−0.413793
$262$	0	0
$263$	−187.061	−0.711260	−0.355630	−	0.934627i	$-0.615734\pi$
$263$	−187.061	−0.711260	−0.355630	+	0.934627i	$0.615734\pi$
$264$	0	0
$265$	130.000	0.490566
$266$	0	0
$267$	31.1769	0.116767
$268$	0	0
$269$	−108.000	−0.401487	−0.200743	−	0.979644i	$-0.564336\pi$
$269$	−108.000	−0.401487	−0.200743	+	0.979644i	$0.564336\pi$
$270$	0	0
$271$	− 325.626i	− 1.20157i	−0.799411	−	0.600785i	$-0.794855\pi$
$271$	− 325.626i	− 1.20157i	0.799411	−	0.600785i	$-0.205145\pi$
$272$	0	0
$273$	324.000i	1.18681i
$274$	0	0
$275$	− 259.808i	− 0.944755i
$276$	0	0
$277$	− 270.000i	− 0.974729i	−0.873199	−	0.487365i	$-0.837958\pi$
$277$	− 270.000i	− 0.974729i	0.873199	−	0.487365i	$-0.162042\pi$
$278$	0	0
$279$	20.7846i	0.0744968i
$280$	0	0
$281$	−234.000	−0.832740	−0.416370	−	0.909195i	$-0.636698\pi$
$281$	−234.000	−0.832740	−0.416370	+	0.909195i	$0.636698\pi$
$282$	0	0
$283$	−83.1384	−0.293775	−0.146888	−	0.989153i	$-0.546926\pi$
$283$	−83.1384	−0.293775	−0.146888	+	0.989153i	$0.546926\pi$
$284$	0	0
$285$	120.000	0.421053
$286$	0	0
$287$	187.061	0.651782
$288$	0	0
$289$	189.000	0.653979
$290$	0	0
$291$	− 124.708i	− 0.428549i
$292$	0	0
$293$	− 58.0000i	− 0.197952i	−0.995090	−	0.0989761i	$-0.968443\pi$
$293$	− 58.0000i	− 0.197952i	0.995090	−	0.0989761i	$-0.0315567\pi$
$294$	0	0
$295$	−155.885	−0.528422
$296$	0	0
$297$	54.0000i	0.181818i
$298$	0	0
$299$	− 124.708i	− 0.417082i
$300$	0	0
$301$	216.000	0.717608
$302$	0	0
$303$	−62.3538	−0.205788
$304$	0	0
$305$	370.000i	1.21311i
$306$	0	0
$307$	270.200	0.880130	0.440065	−	0.897966i	$-0.354955\pi$
$307$	270.200	0.880130	0.440065	+	0.897966i	$0.354955\pi$
$308$	0	0
$309$	−18.0000	−0.0582524
$310$	0	0
$311$	− 270.200i	− 0.868810i	−0.900718	−	0.434405i	$-0.856959\pi$
$311$	− 270.200i	− 0.868810i	0.900718	−	0.434405i	$-0.143041\pi$
$312$	0	0
$313$	− 468.000i	− 1.49521i	−0.664145	−	0.747604i	$-0.731204\pi$
$313$	− 468.000i	− 1.49521i	0.664145	−	0.747604i	$-0.268796\pi$
$314$	0	0
$315$	155.885i	0.494872i
$316$	0	0
$317$	− 250.000i	− 0.788644i	−0.918972	−	0.394322i	$-0.870980\pi$
$317$	− 250.000i	− 0.788644i	0.918972	−	0.394322i	$-0.129020\pi$
$318$	0	0
$319$	− 374.123i	− 1.17280i
$320$	0	0
$321$	324.000	1.00935
$322$	0	0
$323$	138.564	0.428991
$324$	0	0
$325$	− 450.000i	− 1.38462i
$326$	0	0
$327$	−45.0333	−0.137717
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 374.123i	− 1.13028i	−0.824995	−	0.565140i	$-0.808822\pi$
$331$	− 374.123i	− 1.13028i	0.824995	−	0.565140i	$-0.191178\pi$
$332$	0	0
$333$	− 162.000i	− 0.486486i
$334$	0	0
$335$	− 207.846i	− 0.620436i
$336$	0	0
$337$	− 468.000i	− 1.38872i	−0.719626	−	0.694362i	$-0.755687\pi$
$337$	− 468.000i	− 1.38872i	0.719626	−	0.694362i	$-0.244313\pi$
$338$	0	0
$339$	17.3205i	0.0510929i
$340$	0	0
$341$	−72.0000	−0.211144
$342$	0	0
$343$	103.923	0.302983
$344$	0	0
$345$	− 60.0000i	− 0.173913i
$346$	0	0
$347$	561.184	1.61725	0.808623	−	0.588327i	$-0.200213\pi$
$347$	561.184	1.61725	0.808623	+	0.588327i	$0.200213\pi$
$348$	0	0
$349$	−434.000	−1.24355	−0.621777	−	0.783195i	$-0.713589\pi$
$349$	−434.000	−1.24355	−0.621777	+	0.783195i	$0.713589\pi$
$350$	0	0
$351$	93.5307i	0.266469i
$352$	0	0
$353$	158.000i	0.447592i	0.974636	+	0.223796i	$0.0718449\pi$
$353$	158.000i	0.447592i	−0.974636	+	0.223796i	$0.928155\pi$
$354$	0	0
$355$	−519.615	−1.46370
$356$	0	0
$357$	180.000i	0.504202i
$358$	0	0
$359$	457.261i	1.27371i	0.770984	+	0.636854i	$0.219765\pi$
$359$	457.261i	1.27371i	−0.770984	+	0.636854i	$0.780235\pi$
$360$	0	0
$361$	169.000	0.468144
$362$	0	0
$363$	22.5167	0.0620294
$364$	0	0
$365$	180.000	0.493151
$366$	0	0
$367$	218.238	0.594655	0.297328	−	0.954776i	$-0.403905\pi$
$367$	218.238	0.594655	0.297328	+	0.954776i	$0.403905\pi$
$368$	0	0
$369$	54.0000	0.146341
$370$	0	0
$371$	− 270.200i	− 0.728302i
$372$	0	0
$373$	− 270.000i	− 0.723861i	−0.932205	−	0.361930i	$-0.882118\pi$
$373$	− 270.000i	− 0.723861i	0.932205	−	0.361930i	$-0.117882\pi$
$374$	0	0
$375$	− 216.506i	− 0.577350i
$376$	0	0
$377$	− 648.000i	− 1.71883i
$378$	0	0
$379$	325.626i	0.859170i	0.903026	+	0.429585i	$0.141340\pi$
$379$	325.626i	0.859170i	−0.903026	+	0.429585i	$0.858660\pi$
$380$	0	0
$381$	378.000	0.992126
$382$	0	0
$383$	55.4256	0.144714	0.0723572	−	0.997379i	$-0.476948\pi$
$383$	55.4256	0.144714	0.0723572	+	0.997379i	$0.476948\pi$
$384$	0	0
$385$	−540.000	−1.40260
$386$	0	0
$387$	62.3538	0.161121
$388$	0	0
$389$	−288.000	−0.740360	−0.370180	−	0.928960i	$-0.620704\pi$
$389$	−288.000	−0.740360	−0.370180	+	0.928960i	$0.620704\pi$
$390$	0	0
$391$	− 69.2820i	− 0.177192i
$392$	0	0
$393$	− 234.000i	− 0.595420i
$394$	0	0
$395$	−450.333	−1.14008
$396$	0	0
$397$	306.000i	0.770781i	0.922754	+	0.385390i	$0.125933\pi$
$397$	306.000i	0.770781i	−0.922754	+	0.385390i	$0.874067\pi$
$398$	0	0
$399$	− 249.415i	− 0.625101i
$400$	0	0
$401$	−450.000	−1.12219	−0.561097	−	0.827750i	$-0.689621\pi$
$401$	−450.000	−1.12219	−0.561097	+	0.827750i	$0.689621\pi$
$402$	0	0
$403$	−124.708	−0.309448
$404$	0	0
$405$	45.0000i	0.111111i
$406$	0	0
$407$	561.184	1.37883
$408$	0	0
$409$	50.0000	0.122249	0.0611247	−	0.998130i	$-0.480531\pi$
$409$	50.0000	0.122249	0.0611247	+	0.998130i	$0.480531\pi$
$410$	0	0
$411$	190.526i	0.463566i
$412$	0	0
$413$	324.000i	0.784504i
$414$	0	0
$415$	− 450.333i	− 1.08514i
$416$	0	0
$417$	324.000i	0.776978i
$418$	0	0
$419$	− 737.854i	− 1.76099i	−0.474058	−	0.880494i	$-0.657211\pi$
$419$	− 737.854i	− 1.76099i	0.474058	−	0.880494i	$-0.342789\pi$
$420$	0	0
$421$	286.000	0.679335	0.339667	−	0.940546i	$-0.389685\pi$
$421$	286.000	0.679335	0.339667	+	0.940546i	$0.389685\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 250.000i	− 0.588235i
$426$	0	0
$427$	769.031	1.80101
$428$	0	0
$429$	−324.000	−0.755245
$430$	0	0
$431$	124.708i	0.289345i	0.989480	+	0.144672i	$0.0462128\pi$
$431$	124.708i	0.289345i	−0.989480	+	0.144672i	$0.953787\pi$
$432$	0	0
$433$	36.0000i	0.0831409i	0.999136	+	0.0415704i	$0.0132361\pi$
$433$	36.0000i	0.0831409i	−0.999136	+	0.0415704i	$0.986764\pi$
$434$	0	0
$435$	− 311.769i	− 0.716711i
$436$	0	0
$437$	96.0000i	0.219680i
$438$	0	0
$439$	782.887i	1.78334i	0.452684	+	0.891671i	$0.350466\pi$
$439$	782.887i	1.78334i	−0.452684	+	0.891671i	$0.649534\pi$
$440$	0	0
$441$	177.000	0.401361
$442$	0	0
$443$	−214.774	−0.484818	−0.242409	−	0.970174i	$-0.577938\pi$
$443$	−214.774	−0.484818	−0.242409	+	0.970174i	$0.577938\pi$
$444$	0	0
$445$	90.0000i	0.202247i
$446$	0	0
$447$	498.831	1.11595
$448$	0	0
$449$	−54.0000	−0.120267	−0.0601336	−	0.998190i	$-0.519153\pi$
$449$	−54.0000	−0.120267	−0.0601336	+	0.998190i	$0.519153\pi$
$450$	0	0
$451$	187.061i	0.414770i
$452$	0	0
$453$	324.000i	0.715232i
$454$	0	0
$455$	−935.307	−2.05562
$456$	0	0
$457$	288.000i	0.630197i	0.949059	+	0.315098i	$0.102038\pi$
$457$	288.000i	0.630197i	−0.949059	+	0.315098i	$0.897962\pi$
$458$	0	0
$459$	51.9615i	0.113206i
$460$	0	0
$461$	288.000	0.624729	0.312364	−	0.949962i	$-0.398879\pi$
$461$	288.000	0.624729	0.312364	+	0.949962i	$0.398879\pi$
$462$	0	0
$463$	−405.300	−0.875378	−0.437689	−	0.899126i	$-0.644203\pi$
$463$	−405.300	−0.875378	−0.437689	+	0.899126i	$0.644203\pi$
$464$	0	0
$465$	−60.0000	−0.129032
$466$	0	0
$467$	−575.041	−1.23135	−0.615675	−	0.788000i	$-0.711117\pi$
$467$	−575.041	−1.23135	−0.615675	+	0.788000i	$0.711117\pi$
$468$	0	0
$469$	−432.000	−0.921109
$470$	0	0
$471$	− 405.300i	− 0.860509i
$472$	0	0
$473$	216.000i	0.456660i
$474$	0	0
$475$	346.410i	0.729285i
$476$	0	0
$477$	− 78.0000i	− 0.163522i
$478$	0	0
$479$	− 145.492i	− 0.303742i	−0.988400	−	0.151871i	$-0.951470\pi$
$479$	− 145.492i	− 0.303742i	0.988400	−	0.151871i	$-0.0485298\pi$
$480$	0	0
$481$	972.000	2.02079
$482$	0	0
$483$	−124.708	−0.258194
$484$	0	0
$485$	360.000	0.742268
$486$	0	0
$487$	−259.808	−0.533486	−0.266743	−	0.963768i	$-0.585947\pi$
$487$	−259.808	−0.533486	−0.266743	+	0.963768i	$0.585947\pi$
$488$	0	0
$489$	216.000	0.441718
$490$	0	0
$491$	72.7461i	0.148159i	0.997252	+	0.0740796i	$0.0236019\pi$
$491$	72.7461i	0.148159i	−0.997252	+	0.0740796i	$0.976398\pi$
$492$	0	0
$493$	− 360.000i	− 0.730223i
$494$	0	0
$495$	−155.885	−0.314918
$496$	0	0
$497$	1080.00i	2.17304i
$498$	0	0
$499$	443.405i	0.888587i	0.895881	+	0.444294i	$0.146545\pi$
$499$	443.405i	0.888587i	−0.895881	+	0.444294i	$0.853455\pi$
$500$	0	0
$501$	228.000	0.455090
$502$	0	0
$503$	−110.851	−0.220380	−0.110190	−	0.993911i	$-0.535146\pi$
$503$	−110.851	−0.220380	−0.110190	+	0.993911i	$0.535146\pi$
$504$	0	0
$505$	− 180.000i	− 0.356436i
$506$	0	0
$507$	−268.468	−0.529522
$508$	0	0
$509$	−252.000	−0.495088	−0.247544	−	0.968877i	$-0.579624\pi$
$509$	−252.000	−0.495088	−0.247544	+	0.968877i	$0.579624\pi$
$510$	0	0
$511$	− 374.123i	− 0.732139i
$512$	0	0
$513$	− 72.0000i	− 0.140351i
$514$	0	0
$515$	− 51.9615i	− 0.100896i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	252.879i	0.487244i
$520$	0	0
$521$	54.0000	0.103647	0.0518234	−	0.998656i	$-0.483497\pi$
$521$	54.0000	0.103647	0.0518234	+	0.998656i	$0.483497\pi$
$522$	0	0
$523$	−623.538	−1.19223	−0.596117	−	0.802898i	$-0.703291\pi$
$523$	−623.538	−1.19223	−0.596117	+	0.802898i	$0.703291\pi$
$524$	0	0
$525$	−450.000	−0.857143
$526$	0	0
$527$	−69.2820	−0.131465
$528$	0	0
$529$	−481.000	−0.909263
$530$	0	0
$531$	93.5307i	0.176141i
$532$	0	0
$533$	324.000i	0.607880i
$534$	0	0
$535$	935.307i	1.74824i
$536$	0	0
$537$	126.000i	0.234637i
$538$	0	0
$539$	613.146i	1.13756i
$540$	0	0
$541$	650.000	1.20148	0.600739	−	0.799445i	$-0.294873\pi$
$541$	650.000	1.20148	0.600739	+	0.799445i	$0.294873\pi$
$542$	0	0
$543$	−453.797	−0.835722
$544$	0	0
$545$	− 130.000i	− 0.238532i
$546$	0	0
$547$	685.892	1.25392	0.626958	−	0.779053i	$-0.284300\pi$
$547$	685.892	1.25392	0.626958	+	0.779053i	$0.284300\pi$
$548$	0	0
$549$	222.000	0.404372
$550$	0	0
$551$	498.831i	0.905319i
$552$	0	0
$553$	936.000i	1.69259i
$554$	0	0
$555$	467.654	0.842619
$556$	0	0
$557$	− 574.000i	− 1.03052i	−0.857034	−	0.515260i	$-0.827695\pi$
$557$	− 574.000i	− 1.03052i	0.857034	−	0.515260i	$-0.172305\pi$
$558$	0	0
$559$	374.123i	0.669272i
$560$	0	0
$561$	−180.000	−0.320856
$562$	0	0
$563$	561.184	0.996775	0.498388	−	0.866954i	$-0.333926\pi$
$563$	561.184	0.996775	0.498388	+	0.866954i	$0.333926\pi$
$564$	0	0
$565$	−50.0000	−0.0884956
$566$	0	0
$567$	93.5307	0.164957
$568$	0	0
$569$	−198.000	−0.347979	−0.173989	−	0.984748i	$-0.555666\pi$
$569$	−198.000	−0.347979	−0.173989	+	0.984748i	$0.555666\pi$
$570$	0	0
$571$	180.133i	0.315470i	0.987481	+	0.157735i	$0.0504192\pi$
$571$	180.133i	0.315470i	−0.987481	+	0.157735i	$0.949581\pi$
$572$	0	0
$573$	− 324.000i	− 0.565445i
$574$	0	0
$575$	173.205	0.301226
$576$	0	0
$577$	− 504.000i	− 0.873484i	−0.899587	−	0.436742i	$-0.856132\pi$
$577$	− 504.000i	− 0.873484i	0.899587	−	0.436742i	$-0.143868\pi$
$578$	0	0
$579$	− 311.769i	− 0.538461i
$580$	0	0
$581$	−936.000	−1.61102
$582$	0	0
$583$	270.200	0.463465
$584$	0	0
$585$	−270.000	−0.461538
$586$	0	0
$587$	408.764	0.696361	0.348181	−	0.937427i	$-0.386800\pi$
$587$	408.764	0.696361	0.348181	+	0.937427i	$0.386800\pi$
$588$	0	0
$589$	96.0000	0.162988
$590$	0	0
$591$	266.736i	0.451330i
$592$	0	0
$593$	− 998.000i	− 1.68297i	−0.540282	−	0.841484i	$-0.681682\pi$
$593$	− 998.000i	− 1.68297i	0.540282	−	0.841484i	$-0.318318\pi$
$594$	0	0
$595$	−519.615	−0.873303
$596$	0	0
$597$	− 324.000i	− 0.542714i
$598$	0	0
$599$	− 540.400i	− 0.902170i	−0.892481	−	0.451085i	$-0.851037\pi$
$599$	− 540.400i	− 0.902170i	0.892481	−	0.451085i	$-0.148963\pi$
$600$	0	0
$601$	−614.000	−1.02163	−0.510815	−	0.859690i	$-0.670656\pi$
$601$	−614.000	−1.02163	−0.510815	+	0.859690i	$0.670656\pi$
$602$	0	0
$603$	−124.708	−0.206812
$604$	0	0
$605$	65.0000i	0.107438i
$606$	0	0
$607$	654.715	1.07861	0.539304	−	0.842111i	$-0.318687\pi$
$607$	654.715	1.07861	0.539304	+	0.842111i	$0.318687\pi$
$608$	0	0
$609$	−648.000	−1.06404
$610$	0	0
$611$	0	0
$612$	0	0
$613$	414.000i	0.675367i	0.941260	+	0.337684i	$0.109643\pi$
$613$	414.000i	0.675367i	−0.941260	+	0.337684i	$0.890357\pi$
$614$	0	0
$615$	155.885i	0.253471i
$616$	0	0
$617$	58.0000i	0.0940032i	0.998895	+	0.0470016i	$0.0149666\pi$
$617$	58.0000i	0.0940032i	−0.998895	+	0.0470016i	$0.985033\pi$
$618$	0	0
$619$	− 187.061i	− 0.302199i	−0.988519	−	0.151100i	$-0.951719\pi$
$619$	− 187.061i	− 0.302199i	0.988519	−	0.151100i	$-0.0482815\pi$
$620$	0	0
$621$	−36.0000	−0.0579710
$622$	0	0
$623$	187.061	0.300259
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	249.415	0.397792
$628$	0	0
$629$	540.000	0.858506
$630$	0	0
$631$	824.456i	1.30659i	0.757105	+	0.653293i	$0.226613\pi$
$631$	824.456i	1.30659i	−0.757105	+	0.653293i	$0.773387\pi$
$632$	0	0
$633$	− 420.000i	− 0.663507i
$634$	0	0
$635$	1091.19i	1.71841i
$636$	0	0
$637$	1062.00i	1.66719i
$638$	0	0
$639$	311.769i	0.487902i
$640$	0	0
$641$	810.000	1.26365	0.631825	−	0.775111i	$-0.282306\pi$
$641$	810.000	1.26365	0.631825	+	0.775111i	$0.282306\pi$
$642$	0	0
$643$	−415.692	−0.646489	−0.323244	−	0.946316i	$-0.604774\pi$
$643$	−415.692	−0.646489	−0.323244	+	0.946316i	$0.604774\pi$
$644$	0	0
$645$	180.000i	0.279070i
$646$	0	0
$647$	983.805	1.52056	0.760282	−	0.649593i	$-0.225061\pi$
$647$	983.805	1.52056	0.760282	+	0.649593i	$0.225061\pi$
$648$	0	0
$649$	−324.000	−0.499230
$650$	0	0
$651$	124.708i	0.191563i
$652$	0	0
$653$	− 950.000i	− 1.45482i	−0.686201	−	0.727412i	$-0.740723\pi$
$653$	− 950.000i	− 1.45482i	0.686201	−	0.727412i	$-0.259277\pi$
$654$	0	0
$655$	675.500	1.03130
$656$	0	0
$657$	− 108.000i	− 0.164384i
$658$	0	0
$659$	1132.76i	1.71891i	0.511212	+	0.859455i	$0.329197\pi$
$659$	1132.76i	1.71891i	−0.511212	+	0.859455i	$0.670803\pi$
$660$	0	0
$661$	242.000	0.366112	0.183056	−	0.983102i	$-0.441401\pi$
$661$	242.000	0.366112	0.183056	+	0.983102i	$0.441401\pi$
$662$	0	0
$663$	−311.769	−0.470240
$664$	0	0
$665$	720.000	1.08271
$666$	0	0
$667$	249.415	0.373936
$668$	0	0
$669$	−162.000	−0.242152
$670$	0	0
$671$	769.031i	1.14610i
$672$	0	0
$673$	− 324.000i	− 0.481426i	−0.970596	−	0.240713i	$-0.922619\pi$
$673$	− 324.000i	− 0.481426i	0.970596	−	0.240713i	$-0.0773813\pi$
$674$	0	0
$675$	−129.904	−0.192450
$676$	0	0
$677$	806.000i	1.19055i	0.803523	+	0.595273i	$0.202956\pi$
$677$	806.000i	1.19055i	−0.803523	+	0.595273i	$0.797044\pi$
$678$	0	0
$679$	− 748.246i	− 1.10198i
$680$	0	0
$681$	372.000	0.546256
$682$	0	0
$683$	575.041	0.841934	0.420967	−	0.907076i	$-0.361691\pi$
$683$	575.041	0.841934	0.420967	+	0.907076i	$0.361691\pi$
$684$	0	0
$685$	−550.000	−0.802920
$686$	0	0
$687$	585.433	0.852159
$688$	0	0
$689$	468.000	0.679245
$690$	0	0
$691$	775.959i	1.12295i	0.827494	+	0.561475i	$0.189766\pi$
$691$	775.959i	1.12295i	−0.827494	+	0.561475i	$0.810234\pi$
$692$	0	0
$693$	324.000i	0.467532i
$694$	0	0
$695$	−935.307	−1.34577
$696$	0	0
$697$	180.000i	0.258250i
$698$	0	0
$699$	315.233i	0.450977i
$700$	0	0
$701$	−756.000	−1.07846	−0.539230	−	0.842159i	$-0.681285\pi$
$701$	−756.000	−1.07846	−0.539230	+	0.842159i	$0.681285\pi$
$702$	0	0
$703$	−748.246	−1.06436
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−374.123	−0.529170
$708$	0	0
$709$	−310.000	−0.437236	−0.218618	−	0.975811i	$-0.570155\pi$
$709$	−310.000	−0.437236	−0.218618	+	0.975811i	$0.570155\pi$
$710$	0	0
$711$	270.200i	0.380028i
$712$	0	0
$713$	− 48.0000i	− 0.0673212i
$714$	0	0
$715$	− 935.307i	− 1.30812i
$716$	0	0
$717$	− 612.000i	− 0.853556i
$718$	0	0
$719$	83.1384i	0.115631i	0.998327	+	0.0578153i	$0.0184135\pi$
$719$	83.1384i	0.115631i	−0.998327	+	0.0578153i	$0.981587\pi$
$720$	0	0
$721$	−108.000	−0.149792
$722$	0	0
$723$	−183.597	−0.253938
$724$	0	0
$725$	900.000	1.24138
$726$	0	0
$727$	−1091.19	−1.50095	−0.750476	−	0.660898i	$-0.770176\pi$
$727$	−1091.19	−1.50095	−0.750476	+	0.660898i	$0.770176\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	207.846i	0.284331i
$732$	0	0
$733$	1206.00i	1.64529i	0.568553	+	0.822647i	$0.307503\pi$
$733$	1206.00i	1.64529i	−0.568553	+	0.822647i	$0.692497\pi$
$734$	0	0
$735$	510.955i	0.695177i
$736$	0	0
$737$	− 432.000i	− 0.586160i
$738$	0	0
$739$	− 484.974i	− 0.656257i	−0.944633	−	0.328129i	$-0.893582\pi$
$739$	− 484.974i	− 0.656257i	0.944633	−	0.328129i	$-0.106418\pi$
$740$	0	0
$741$	432.000	0.582996
$742$	0	0
$743$	1122.37	1.51059	0.755295	−	0.655385i	$-0.227493\pi$
$743$	1122.37	1.51059	0.755295	+	0.655385i	$0.227493\pi$
$744$	0	0
$745$	1440.00i	1.93289i
$746$	0	0
$747$	−270.200	−0.361713
$748$	0	0
$749$	1944.00	2.59546
$750$	0	0
$751$	242.487i	0.322886i	0.986882	+	0.161443i	$0.0516147\pi$
$751$	242.487i	0.322886i	−0.986882	+	0.161443i	$0.948385\pi$
$752$	0	0
$753$	558.000i	0.741036i
$754$	0	0
$755$	−935.307	−1.23882
$756$	0	0
$757$	− 846.000i	− 1.11757i	−0.829313	−	0.558785i	$-0.811268\pi$
$757$	− 846.000i	− 1.11757i	0.829313	−	0.558785i	$-0.188732\pi$
$758$	0	0
$759$	− 124.708i	− 0.164305i
$760$	0	0
$761$	−1458.00	−1.91590	−0.957950	−	0.286935i	$-0.907364\pi$
$761$	−1458.00	−1.91590	−0.957950	+	0.286935i	$0.907364\pi$
$762$	0	0
$763$	−270.200	−0.354128
$764$	0	0
$765$	−150.000	−0.196078
$766$	0	0
$767$	−561.184	−0.731662
$768$	0	0
$769$	−1282.00	−1.66710	−0.833550	−	0.552444i	$-0.813695\pi$
$769$	−1282.00	−1.66710	−0.833550	+	0.552444i	$0.813695\pi$
$770$	0	0
$771$	24.2487i	0.0314510i
$772$	0	0
$773$	− 422.000i	− 0.545925i	−0.962025	−	0.272962i	$-0.911997\pi$
$773$	− 422.000i	− 0.545925i	0.962025	−	0.272962i	$-0.0880035\pi$
$774$	0	0
$775$	− 173.205i	− 0.223490i
$776$	0	0
$777$	− 972.000i	− 1.25097i
$778$	0	0
$779$	− 249.415i	− 0.320174i
$780$	0	0
$781$	−1080.00	−1.38284
$782$	0	0
$783$	−187.061	−0.238904
$784$	0	0
$785$	1170.00	1.49045
$786$	0	0
$787$	1205.51	1.53178	0.765888	−	0.642974i	$-0.222300\pi$
$787$	1205.51	1.53178	0.765888	+	0.642974i	$0.222300\pi$
$788$	0	0
$789$	−324.000	−0.410646
$790$	0	0
$791$	103.923i	0.131382i
$792$	0	0
$793$	1332.00i	1.67970i
$794$	0	0
$795$	225.167	0.283228
$796$	0	0
$797$	94.0000i	0.117942i	0.998260	+	0.0589711i	$0.0187820\pi$
$797$	94.0000i	0.117942i	−0.998260	+	0.0589711i	$0.981218\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	54.0000	0.0674157
$802$	0	0
$803$	374.123	0.465907
$804$	0	0
$805$	− 360.000i	− 0.447205i
$806$	0	0
$807$	−187.061	−0.231799
$808$	0	0
$809$	270.000	0.333745	0.166873	−	0.985978i	$-0.446633\pi$
$809$	270.000	0.333745	0.166873	+	0.985978i	$0.446633\pi$
$810$	0	0
$811$	− 187.061i	− 0.230655i	−0.993328	−	0.115328i	$-0.963208\pi$
$811$	− 187.061i	− 0.230655i	0.993328	−	0.115328i	$-0.0367918\pi$
$812$	0	0
$813$	− 564.000i	− 0.693727i
$814$	0	0
$815$	623.538i	0.765078i
$816$	0	0
$817$	− 288.000i	− 0.352509i
$818$	0	0
$819$	561.184i	0.685207i
$820$	0	0
$821$	−1188.00	−1.44702	−0.723508	−	0.690316i	$-0.757471\pi$
$821$	−1188.00	−1.44702	−0.723508	+	0.690316i	$0.757471\pi$
$822$	0	0
$823$	384.515	0.467212	0.233606	−	0.972331i	$-0.424947\pi$
$823$	384.515	0.467212	0.233606	+	0.972331i	$0.424947\pi$
$824$	0	0
$825$	− 450.000i	− 0.545455i
$826$	0	0
$827$	450.333	0.544538	0.272269	−	0.962221i	$-0.412226\pi$
$827$	450.333	0.544538	0.272269	+	0.962221i	$0.412226\pi$
$828$	0	0
$829$	718.000	0.866104	0.433052	−	0.901369i	$-0.357437\pi$
$829$	718.000	0.866104	0.433052	+	0.901369i	$0.357437\pi$
$830$	0	0
$831$	− 467.654i	− 0.562760i
$832$	0	0
$833$	590.000i	0.708283i
$834$	0	0
$835$	658.179i	0.788239i
$836$	0	0
$837$	36.0000i	0.0430108i
$838$	0	0
$839$	914.523i	1.09002i	0.838431	+	0.545008i	$0.183473\pi$
$839$	914.523i	1.09002i	−0.838431	+	0.545008i	$0.816527\pi$
$840$	0	0
$841$	455.000	0.541023
$842$	0	0
$843$	−405.300	−0.480783
$844$	0	0
$845$	− 775.000i	− 0.917160i
$846$	0	0
$847$	135.100	0.159504
$848$	0	0
$849$	−144.000	−0.169611
$850$	0	0
$851$	374.123i	0.439627i
$852$	0	0
$853$	− 666.000i	− 0.780774i	−0.920651	−	0.390387i	$-0.872341\pi$
$853$	− 666.000i	− 0.780774i	0.920651	−	0.390387i	$-0.127659\pi$
$854$	0	0
$855$	207.846	0.243095
$856$	0	0
$857$	− 182.000i	− 0.212369i	−0.994346	−	0.106184i	$-0.966137\pi$
$857$	− 182.000i	− 0.212369i	0.994346	−	0.106184i	$-0.0338634\pi$
$858$	0	0
$859$	990.733i	1.15336i	0.816971	+	0.576678i	$0.195651\pi$
$859$	990.733i	1.15336i	−0.816971	+	0.576678i	$0.804349\pi$
$860$	0	0
$861$	324.000	0.376307
$862$	0	0
$863$	−1170.87	−1.35674	−0.678370	−	0.734721i	$-0.737313\pi$
$863$	−1170.87	−1.35674	−0.678370	+	0.734721i	$0.737313\pi$
$864$	0	0
$865$	−730.000	−0.843931
$866$	0	0
$867$	327.358	0.377575
$868$	0	0
$869$	−936.000	−1.07710
$870$	0	0
$871$	− 748.246i	− 0.859065i
$872$	0	0
$873$	− 216.000i	− 0.247423i
$874$	0	0
$875$	− 1299.04i	− 1.48461i
$876$	0	0
$877$	774.000i	0.882554i	0.897371	+	0.441277i	$0.145474\pi$
$877$	774.000i	0.882554i	−0.897371	+	0.441277i	$0.854526\pi$
$878$	0	0
$879$	− 100.459i	− 0.114288i
$880$	0	0
$881$	−1602.00	−1.81839	−0.909194	−	0.416373i	$-0.863301\pi$
$881$	−1602.00	−1.81839	−0.909194	+	0.416373i	$0.863301\pi$
$882$	0	0
$883$	415.692	0.470773	0.235386	−	0.971902i	$-0.424364\pi$
$883$	415.692	0.470773	0.235386	+	0.971902i	$0.424364\pi$
$884$	0	0
$885$	−270.000	−0.305085
$886$	0	0
$887$	−367.195	−0.413974	−0.206987	−	0.978344i	$-0.566366\pi$
$887$	−367.195	−0.413974	−0.206987	+	0.978344i	$0.566366\pi$
$888$	0	0
$889$	2268.00	2.55118
$890$	0	0
$891$	93.5307i	0.104973i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−363.731	−0.406403
$896$	0	0
$897$	− 216.000i	− 0.240803i
$898$	0	0
$899$	− 249.415i	− 0.277436i
$900$	0	0
$901$	260.000	0.288568
$902$	0	0
$903$	374.123	0.414311
$904$	0	0
$905$	− 1310.00i	− 1.44751i
$906$	0	0
$907$	−1434.14	−1.58119	−0.790594	−	0.612340i	$-0.790228\pi$
$907$	−1434.14	−1.58119	−0.790594	+	0.612340i	$0.790228\pi$
$908$	0	0
$909$	−108.000	−0.118812
$910$	0	0
$911$	− 1080.80i	− 1.18639i	−0.805059	−	0.593194i	$-0.797867\pi$
$911$	− 1080.80i	− 1.18639i	0.805059	−	0.593194i	$-0.202133\pi$
$912$	0	0
$913$	− 936.000i	− 1.02519i
$914$	0	0
$915$	640.859i	0.700392i
$916$	0	0
$917$	− 1404.00i	− 1.53108i
$918$	0	0
$919$	− 187.061i	− 0.203549i	−0.994807	−	0.101774i	$-0.967548\pi$
$919$	− 187.061i	− 0.203549i	0.994807	−	0.101774i	$-0.0324520\pi$
$920$	0	0
$921$	468.000	0.508143
$922$	0	0
$923$	−1870.61	−2.02667
$924$	0	0
$925$	1350.00i	1.45946i
$926$	0	0
$927$	−31.1769	−0.0336321
$928$	0	0
$929$	−54.0000	−0.0581270	−0.0290635	−	0.999578i	$-0.509253\pi$
$929$	−54.0000	−0.0581270	−0.0290635	+	0.999578i	$0.509253\pi$
$930$	0	0
$931$	− 817.528i	− 0.878118i
$932$	0	0
$933$	− 468.000i	− 0.501608i
$934$	0	0
$935$	− 519.615i	− 0.555738i
$936$	0	0
$937$	936.000i	0.998933i	0.866333	+	0.499466i	$0.166471\pi$
$937$	936.000i	0.998933i	−0.866333	+	0.499466i	$0.833529\pi$
$938$	0	0
$939$	− 810.600i	− 0.863259i
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	−124.708	−0.132246
$944$	0	0
$945$	270.000i	0.285714i
$946$	0	0
$947$	−1032.30	−1.09008	−0.545038	−	0.838411i	$-0.683485\pi$
$947$	−1032.30	−1.09008	−0.545038	+	0.838411i	$0.683485\pi$
$948$	0	0
$949$	648.000	0.682824
$950$	0	0
$951$	− 433.013i	− 0.455324i
$952$	0	0
$953$	− 1550.00i	− 1.62644i	−0.581954	−	0.813221i	$-0.697712\pi$
$953$	− 1550.00i	− 1.62644i	0.581954	−	0.813221i	$-0.302288\pi$
$954$	0	0
$955$	935.307	0.979380
$956$	0	0
$957$	− 648.000i	− 0.677116i
$958$	0	0
$959$	1143.15i	1.19203i
$960$	0	0
$961$	913.000	0.950052
$962$	0	0
$963$	561.184	0.582746
$964$	0	0
$965$	900.000	0.932642
$966$	0	0
$967$	1215.90	1.25739	0.628697	−	0.777650i	$-0.283589\pi$
$967$	1215.90	1.25739	0.628697	+	0.777650i	$0.283589\pi$
$968$	0	0
$969$	240.000	0.247678
$970$	0	0
$971$	− 1839.44i	− 1.89437i	−0.320681	−	0.947187i	$-0.603912\pi$
$971$	− 1839.44i	− 1.89437i	0.320681	−	0.947187i	$-0.396088\pi$
$972$	0	0
$973$	1944.00i	1.99794i
$974$	0	0
$975$	− 779.423i	− 0.799408i
$976$	0	0
$977$	− 206.000i	− 0.210850i	−0.994427	−	0.105425i	$-0.966380\pi$
$977$	− 206.000i	− 0.210850i	0.994427	−	0.105425i	$-0.0336202\pi$
$978$	0	0
$979$	187.061i	0.191074i
$980$	0	0
$981$	−78.0000	−0.0795107
$982$	0	0
$983$	720.533	0.732994	0.366497	−	0.930419i	$-0.380557\pi$
$983$	720.533	0.732994	0.366497	+	0.930419i	$0.380557\pi$
$984$	0	0
$985$	−770.000	−0.781726
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−144.000	−0.145602
$990$	0	0
$991$	1323.29i	1.33530i	0.744473	+	0.667652i	$0.232701\pi$
$991$	1323.29i	1.33530i	−0.744473	+	0.667652i	$0.767299\pi$
$992$	0	0
$993$	− 648.000i	− 0.652568i
$994$	0	0
$995$	935.307	0.940007
$996$	0	0
$997$	198.000i	0.198596i	0.995058	+	0.0992979i	$0.0316597\pi$
$997$	198.000i	0.198596i	−0.995058	+	0.0992979i	$0.968340\pi$
$998$	0	0
$999$	− 280.592i	− 0.280873i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.3.j.b.319.4		4
4.3	odd	2	inner	960.3.j.b.319.2		4
5.4	even	2	inner	960.3.j.b.319.1		4
8.3	odd	2		60.3.f.a.19.4	yes	4
8.5	even	2		60.3.f.a.19.2	yes	4
20.19	odd	2	inner	960.3.j.b.319.3		4
24.5	odd	2		180.3.f.e.19.3		4
24.11	even	2		180.3.f.e.19.1		4
40.3	even	4		300.3.c.c.151.1		2
40.13	odd	4		300.3.c.c.151.2		2
40.19	odd	2		60.3.f.a.19.1	✓	4
40.27	even	4		300.3.c.a.151.2		2
40.29	even	2		60.3.f.a.19.3	yes	4
40.37	odd	4		300.3.c.a.151.1		2
120.29	odd	2		180.3.f.e.19.2		4
120.53	even	4		900.3.c.f.451.1		2
120.59	even	2		180.3.f.e.19.4		4
120.77	even	4		900.3.c.j.451.2		2
120.83	odd	4		900.3.c.f.451.2		2
120.107	odd	4		900.3.c.j.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.f.a.19.1	✓	4	40.19	odd	2
60.3.f.a.19.2	yes	4	8.5	even	2
60.3.f.a.19.3	yes	4	40.29	even	2
60.3.f.a.19.4	yes	4	8.3	odd	2
180.3.f.e.19.1		4	24.11	even	2
180.3.f.e.19.2		4	120.29	odd	2
180.3.f.e.19.3		4	24.5	odd	2
180.3.f.e.19.4		4	120.59	even	2
300.3.c.a.151.1		2	40.37	odd	4
300.3.c.a.151.2		2	40.27	even	4
300.3.c.c.151.1		2	40.3	even	4
300.3.c.c.151.2		2	40.13	odd	4
900.3.c.f.451.1		2	120.53	even	4
900.3.c.f.451.2		2	120.83	odd	4
900.3.c.j.451.1		2	120.107	odd	4
900.3.c.j.451.2		2	120.77	even	4
960.3.j.b.319.1		4	5.4	even	2	inner
960.3.j.b.319.2		4	4.3	odd	2	inner
960.3.j.b.319.3		4	20.19	odd	2	inner
960.3.j.b.319.4		4	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +5.00000i q^{5} +10.3923 q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +5.00000i q^{5} +10.3923 q^{7} +3.00000 q^{9} +10.3923i q^{11} +18.0000i q^{13} +8.66025i q^{15} +10.0000i q^{17} -13.8564i q^{19} +18.0000 q^{21} -6.92820 q^{23} -25.0000 q^{25} +5.19615 q^{27} -36.0000 q^{29} +6.92820i q^{31} +18.0000i q^{33} +51.9615i q^{35} -54.0000i q^{37} +31.1769i q^{39} +18.0000 q^{41} +20.7846 q^{43} +15.0000i q^{45} +59.0000 q^{49} +17.3205i q^{51} -26.0000i q^{53} -51.9615 q^{55} -24.0000i q^{57} +31.1769i q^{59} +74.0000 q^{61} +31.1769 q^{63} -90.0000 q^{65} -41.5692 q^{67} -12.0000 q^{69} +103.923i q^{71} -36.0000i q^{73} -43.3013 q^{75} +108.000i q^{77} +90.0666i q^{79} +9.00000 q^{81} -90.0666 q^{83} -50.0000 q^{85} -62.3538 q^{87} +18.0000 q^{89} +187.061i q^{91} +12.0000i q^{93} +69.2820 q^{95} -72.0000i q^{97} +31.1769i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 72 q^{21} - 100 q^{25} - 144 q^{29} + 72 q^{41} + 236 q^{49} + 296 q^{61} - 360 q^{65} - 48 q^{69} + 36 q^{81} - 200 q^{85} + 72 q^{89}+O(q^{100}) \) 4 * q + 12 * q^9 + 72 * q^21 - 100 * q^25 - 144 * q^29 + 72 * q^41 + 236 * q^49 + 296 * q^61 - 360 * q^65 - 48 * q^69 + 36 * q^81 - 200 * q^85 + 72 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more