LMFDB - Embedded newform 960.4.f.g.769.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(769,960)]

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([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("960.769");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	960.f (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:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.4.f.g.769.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q+3.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -46.0000 q^{11} +34.0000i q^{13} +(15.0000 + 30.0000i) q^{15} -66.0000i q^{17} -104.000 q^{19} -30.0000 q^{21} -164.000i q^{23} +(75.0000 - 100.000i) q^{25} -27.0000i q^{27} +224.000 q^{29} +72.0000 q^{31} -138.000i q^{33} +(50.0000 + 100.000i) q^{35} -22.0000i q^{37} -102.000 q^{39} +194.000 q^{41} +108.000i q^{43} +(-90.0000 + 45.0000i) q^{45} -480.000i q^{47} +243.000 q^{49} +198.000 q^{51} -286.000i q^{53} +(-460.000 + 230.000i) q^{55} -312.000i q^{57} -426.000 q^{59} -698.000 q^{61} -90.0000i q^{63} +(170.000 + 340.000i) q^{65} -328.000i q^{67} +492.000 q^{69} -188.000 q^{71} -740.000i q^{73} +(300.000 + 225.000i) q^{75} -460.000i q^{77} +1168.00 q^{79} +81.0000 q^{81} +412.000i q^{83} +(-330.000 - 660.000i) q^{85} +672.000i q^{87} -1206.00 q^{89} -340.000 q^{91} +216.000i q^{93} +(-1040.00 + 520.000i) q^{95} +1384.00i q^{97} +414.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{5} - 18 q^{9}+O(q^{10}) $ 2 * q + 20 * q^5 - 18 * q^9	$ 2 q + 20 q^{5} - 18 q^{9} - 92 q^{11} + 30 q^{15} - 208 q^{19} - 60 q^{21} + 150 q^{25} + 448 q^{29} + 144 q^{31} + 100 q^{35} - 204 q^{39} + 388 q^{41} - 180 q^{45} + 486 q^{49} + 396 q^{51} - 920 q^{55} - 852 q^{59} - 1396 q^{61} + 340 q^{65} + 984 q^{69} - 376 q^{71} + 600 q^{75} + 2336 q^{79} + 162 q^{81} - 660 q^{85} - 2412 q^{89} - 680 q^{91} - 2080 q^{95} + 828 q^{99}+O(q^{100}) $ 2 * q + 20 * q^5 - 18 * q^9 - 92 * q^11 + 30 * q^15 - 208 * q^19 - 60 * q^21 + 150 * q^25 + 448 * q^29 + 144 * q^31 + 100 * q^35 - 204 * q^39 + 388 * q^41 - 180 * q^45 + 486 * q^49 + 396 * q^51 - 920 * q^55 - 852 * q^59 - 1396 * q^61 + 340 * q^65 + 984 * q^69 - 376 * q^71 + 600 * q^75 + 2336 * q^79 + 162 * q^81 - 660 * q^85 - 2412 * q^89 - 680 * q^91 - 2080 * q^95 + 828 * q^99

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
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	10.0000	−	5.00000i	0.894427	−	0.447214i
$5$	10.0000	−	5.00000i	0.894427	−	0.447214i
$6$		0			0
$7$			10.0000i			0.539949i	0.962867	+	0.269975i	$0.0870153\pi$
$7$			10.0000i			0.539949i	−0.962867	+	0.269975i	$0.912985\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−46.0000			−1.26087			−0.630433	−	0.776244i	$-0.717123\pi$
$11$	−46.0000			−1.26087			−0.630433	+	0.776244i	$0.717123\pi$
$12$		0			0
$13$			34.0000i			0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$			34.0000i			0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$		0			0
$15$	15.0000	+	30.0000i	0.258199	+	0.516398i
$16$		0			0
$17$		−	66.0000i		−	0.941609i	−0.882238	−	0.470804i	$-0.843964\pi$
$17$		−	66.0000i		−	0.941609i	0.882238	−	0.470804i	$-0.156036\pi$
$18$		0			0
$19$	−104.000			−1.25575			−0.627875	−	0.778314i	$-0.716075\pi$
$19$	−104.000			−1.25575			−0.627875	+	0.778314i	$0.716075\pi$
$20$		0			0
$21$	−30.0000			−0.311740
$22$		0			0
$23$		−	164.000i		−	1.48680i	−0.668848	−	0.743399i	$-0.733212\pi$
$23$		−	164.000i		−	1.48680i	0.668848	−	0.743399i	$-0.266788\pi$
$24$		0			0
$25$	75.0000	−	100.000i	0.600000	−	0.800000i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	224.000			1.43434			0.717168	−	0.696900i	$-0.245438\pi$
$29$	224.000			1.43434			0.717168	+	0.696900i	$0.245438\pi$
$30$		0			0
$31$	72.0000			0.417148			0.208574	−	0.978007i	$-0.433118\pi$
$31$	72.0000			0.417148			0.208574	+	0.978007i	$0.433118\pi$
$32$		0			0
$33$		−	138.000i		−	0.727961i
$34$		0			0
$35$	50.0000	+	100.000i	0.241473	+	0.482945i
$36$		0			0
$37$		−	22.0000i		−	0.0977507i	−0.998805	−	0.0488754i	$-0.984436\pi$
$37$		−	22.0000i		−	0.0977507i	0.998805	−	0.0488754i	$-0.0155637\pi$
$38$		0			0
$39$	−102.000			−0.418797
$40$		0			0
$41$	194.000			0.738969			0.369484	−	0.929237i	$-0.379534\pi$
$41$	194.000			0.738969			0.369484	+	0.929237i	$0.379534\pi$
$42$		0			0
$43$			108.000i			0.383020i	0.981491	+	0.191510i	$0.0613384\pi$
$43$			108.000i			0.383020i	−0.981491	+	0.191510i	$0.938662\pi$
$44$		0			0
$45$	−90.0000	+	45.0000i	−0.298142	+	0.149071i
$46$		0			0
$47$		−	480.000i		−	1.48969i	−0.667240	−	0.744843i	$-0.732525\pi$
$47$		−	480.000i		−	1.48969i	0.667240	−	0.744843i	$-0.267475\pi$
$48$		0			0
$49$	243.000			0.708455
$50$		0			0
$51$	198.000			0.543638
$52$		0			0
$53$		−	286.000i		−	0.741229i	−0.928787	−	0.370614i	$-0.879147\pi$
$53$		−	286.000i		−	0.741229i	0.928787	−	0.370614i	$-0.120853\pi$
$54$		0			0
$55$	−460.000	+	230.000i	−1.12775	+	0.563876i
$56$		0			0
$57$		−	312.000i		−	0.725007i
$58$		0			0
$59$	−426.000			−0.940008			−0.470004	−	0.882664i	$-0.655748\pi$
$59$	−426.000			−0.940008			−0.470004	+	0.882664i	$0.655748\pi$
$60$		0			0
$61$	−698.000			−1.46508			−0.732539	−	0.680725i	$-0.761665\pi$
$61$	−698.000			−1.46508			−0.732539	+	0.680725i	$0.761665\pi$
$62$		0			0
$63$		−	90.0000i		−	0.179983i
$64$		0			0
$65$	170.000	+	340.000i	0.324399	+	0.648797i
$66$		0			0
$67$		−	328.000i		−	0.598083i	−0.954240	−	0.299042i	$-0.903333\pi$
$67$		−	328.000i		−	0.598083i	0.954240	−	0.299042i	$-0.0966669\pi$
$68$		0			0
$69$	492.000			0.858403
$70$		0			0
$71$	−188.000			−0.314246			−0.157123	−	0.987579i	$-0.550222\pi$
$71$	−188.000			−0.314246			−0.157123	+	0.987579i	$0.550222\pi$
$72$		0			0
$73$		−	740.000i		−	1.18644i	−0.805039	−	0.593222i	$-0.797856\pi$
$73$		−	740.000i		−	1.18644i	0.805039	−	0.593222i	$-0.202144\pi$
$74$		0			0
$75$	300.000	+	225.000i	0.461880	+	0.346410i
$76$		0			0
$77$		−	460.000i		−	0.680803i
$78$		0			0
$79$	1168.00			1.66342			0.831711	−	0.555209i	$-0.187362\pi$
$79$	1168.00			1.66342			0.831711	+	0.555209i	$0.187362\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			412.000i			0.544854i	0.962176	+	0.272427i	$0.0878263\pi$
$83$			412.000i			0.544854i	−0.962176	+	0.272427i	$0.912174\pi$
$84$		0			0
$85$	−330.000	−	660.000i	−0.421100	−	0.842201i
$86$		0			0
$87$			672.000i			0.828115i
$88$		0			0
$89$	−1206.00			−1.43636			−0.718178	−	0.695859i	$-0.755024\pi$
$89$	−1206.00			−1.43636			−0.718178	+	0.695859i	$0.755024\pi$
$90$		0			0
$91$	−340.000			−0.391667
$92$		0			0
$93$			216.000i			0.240840i
$94$		0			0
$95$	−1040.00	+	520.000i	−1.12318	+	0.561588i
$96$		0			0
$97$			1384.00i			1.44870i	0.689432	+	0.724350i	$0.257860\pi$
$97$			1384.00i			1.44870i	−0.689432	+	0.724350i	$0.742140\pi$
$98$		0			0
$99$	414.000			0.420289
$100$		0			0
$101$	1128.00			1.11129			0.555645	−	0.831420i	$-0.312472\pi$
$101$	1128.00			1.11129			0.555645	+	0.831420i	$0.312472\pi$
$102$		0			0
$103$			758.000i			0.725126i	0.931959	+	0.362563i	$0.118098\pi$
$103$			758.000i			0.725126i	−0.931959	+	0.362563i	$0.881902\pi$
$104$		0			0
$105$	−300.000	+	150.000i	−0.278829	+	0.139414i
$106$		0			0
$107$		−	1324.00i		−	1.19622i	−0.801413	−	0.598112i	$-0.795918\pi$
$107$		−	1324.00i		−	1.19622i	0.801413	−	0.598112i	$-0.204082\pi$
$108$		0			0
$109$	1602.00			1.40774			0.703871	−	0.710328i	$-0.251454\pi$
$109$	1602.00			1.40774			0.703871	+	0.710328i	$0.251454\pi$
$110$		0			0
$111$	66.0000			0.0564364
$112$		0			0
$113$		−	2074.00i		−	1.72660i	−0.504693	−	0.863299i	$-0.668394\pi$
$113$		−	2074.00i		−	1.72660i	0.504693	−	0.863299i	$-0.331606\pi$
$114$		0			0
$115$	−820.000	−	1640.00i	−0.664916	−	1.32983i
$116$		0			0
$117$		−	306.000i		−	0.241792i
$118$		0			0
$119$	660.000			0.508421
$120$		0			0
$121$	785.000			0.589782
$122$		0			0
$123$			582.000i			0.426644i
$124$		0			0
$125$	250.000	−	1375.00i	0.178885	−	0.983870i
$126$		0			0
$127$		−	534.000i		−	0.373109i	−0.982445	−	0.186554i	$-0.940268\pi$
$127$		−	534.000i		−	0.373109i	0.982445	−	0.186554i	$-0.0597321\pi$
$128$		0			0
$129$	−324.000			−0.221137
$130$		0			0
$131$	1806.00			1.20451			0.602256	−	0.798303i	$-0.294269\pi$
$131$	1806.00			1.20451			0.602256	+	0.798303i	$0.294269\pi$
$132$		0			0
$133$		−	1040.00i		−	0.678041i
$134$		0			0
$135$	−135.000	−	270.000i	−0.0860663	−	0.172133i
$136$		0			0
$137$		−	1822.00i		−	1.13623i	−0.822948	−	0.568117i	$-0.807672\pi$
$137$		−	1822.00i		−	1.13623i	0.822948	−	0.568117i	$-0.192328\pi$
$138$		0			0
$139$	−532.000			−0.324631			−0.162315	−	0.986739i	$-0.551896\pi$
$139$	−532.000			−0.324631			−0.162315	+	0.986739i	$0.551896\pi$
$140$		0			0
$141$	1440.00			0.860070
$142$		0			0
$143$		−	1564.00i		−	0.914603i
$144$		0			0
$145$	2240.00	−	1120.00i	1.28291	−	0.641455i
$146$		0			0
$147$			729.000i			0.409027i
$148$		0			0
$149$	−1284.00			−0.705969			−0.352984	−	0.935629i	$-0.614833\pi$
$149$	−1284.00			−0.705969			−0.352984	+	0.935629i	$0.614833\pi$
$150$		0			0
$151$	−184.000			−0.0991636			−0.0495818	−	0.998770i	$-0.515789\pi$
$151$	−184.000			−0.0991636			−0.0495818	+	0.998770i	$0.515789\pi$
$152$		0			0
$153$			594.000i			0.313870i
$154$		0			0
$155$	720.000	−	360.000i	0.373108	−	0.186554i
$156$		0			0
$157$		−	3746.00i		−	1.90423i	−0.305748	−	0.952113i	$-0.598906\pi$
$157$		−	3746.00i		−	1.90423i	0.305748	−	0.952113i	$-0.401094\pi$
$158$		0			0
$159$	858.000			0.427949
$160$		0			0
$161$	1640.00			0.802796
$162$		0			0
$163$			1504.00i			0.722714i	0.932428	+	0.361357i	$0.117686\pi$
$163$			1504.00i			0.722714i	−0.932428	+	0.361357i	$0.882314\pi$
$164$		0			0
$165$	−690.000	−	1380.00i	−0.325554	−	0.651108i
$166$		0			0
$167$			3012.00i			1.39566i	0.716262	+	0.697831i	$0.245851\pi$
$167$			3012.00i			1.39566i	−0.716262	+	0.697831i	$0.754149\pi$
$168$		0			0
$169$	1041.00			0.473828
$170$		0			0
$171$	936.000			0.418583
$172$		0			0
$173$			438.000i			0.192489i	0.995358	+	0.0962443i	$0.0306830\pi$
$173$			438.000i			0.192489i	−0.995358	+	0.0962443i	$0.969317\pi$
$174$		0			0
$175$	1000.00	+	750.000i	0.431959	+	0.323970i
$176$		0			0
$177$		−	1278.00i		−	0.542714i
$178$		0			0
$179$	1462.00			0.610475			0.305237	−	0.952276i	$-0.401264\pi$
$179$	1462.00			0.610475			0.305237	+	0.952276i	$0.401264\pi$
$180$		0			0
$181$	−586.000			−0.240647			−0.120323	−	0.992735i	$-0.538393\pi$
$181$	−586.000			−0.240647			−0.120323	+	0.992735i	$0.538393\pi$
$182$		0			0
$183$		−	2094.00i		−	0.845863i
$184$		0			0
$185$	−110.000	−	220.000i	−0.0437155	−	0.0874309i
$186$		0			0
$187$			3036.00i			1.18724i
$188$		0			0
$189$	270.000			0.103913
$190$		0			0
$191$	−60.0000			−0.0227301			−0.0113650	−	0.999935i	$-0.503618\pi$
$191$	−60.0000			−0.0227301			−0.0113650	+	0.999935i	$0.503618\pi$
$192$		0			0
$193$		−	4676.00i		−	1.74397i	−0.489534	−	0.871984i	$-0.662833\pi$
$193$		−	4676.00i		−	1.74397i	0.489534	−	0.871984i	$-0.337167\pi$
$194$		0			0
$195$	−1020.00	+	510.000i	−0.374583	+	0.187292i
$196$		0			0
$197$			2286.00i			0.826755i	0.910560	+	0.413378i	$0.135651\pi$
$197$			2286.00i			0.826755i	−0.910560	+	0.413378i	$0.864349\pi$
$198$		0			0
$199$	−3536.00			−1.25960			−0.629800	−	0.776757i	$-0.716863\pi$
$199$	−3536.00			−1.25960			−0.629800	+	0.776757i	$0.716863\pi$
$200$		0			0
$201$	984.000			0.345304
$202$		0			0
$203$			2240.00i			0.774469i
$204$		0			0
$205$	1940.00	−	970.000i	0.660954	−	0.330477i
$206$		0			0
$207$			1476.00i			0.495599i
$208$		0			0
$209$	4784.00			1.58333
$210$		0			0
$211$	−3500.00			−1.14194			−0.570971	−	0.820970i	$-0.693433\pi$
$211$	−3500.00			−1.14194			−0.570971	+	0.820970i	$0.693433\pi$
$212$		0			0
$213$		−	564.000i		−	0.181430i
$214$		0			0
$215$	540.000	+	1080.00i	0.171292	+	0.342583i
$216$		0			0
$217$			720.000i			0.225239i
$218$		0			0
$219$	2220.00			0.684994
$220$		0			0
$221$	2244.00			0.683022
$222$		0			0
$223$		−	5874.00i		−	1.76391i	−0.471333	−	0.881955i	$-0.656227\pi$
$223$		−	5874.00i		−	1.76391i	0.471333	−	0.881955i	$-0.343773\pi$
$224$		0			0
$225$	−675.000	+	900.000i	−0.200000	+	0.266667i
$226$		0			0
$227$			124.000i			0.0362563i	0.999836	+	0.0181281i	$0.00577068\pi$
$227$			124.000i			0.0362563i	−0.999836	+	0.0181281i	$0.994229\pi$
$228$		0			0
$229$	−1362.00			−0.393028			−0.196514	−	0.980501i	$-0.562962\pi$
$229$	−1362.00			−0.393028			−0.196514	+	0.980501i	$0.562962\pi$
$230$		0			0
$231$	1380.00			0.393062
$232$		0			0
$233$		−	3870.00i		−	1.08812i	−0.839046	−	0.544060i	$-0.816886\pi$
$233$		−	3870.00i		−	1.08812i	0.839046	−	0.544060i	$-0.183114\pi$
$234$		0			0
$235$	−2400.00	−	4800.00i	−0.666207	−	1.33241i
$236$		0			0
$237$			3504.00i			0.960377i
$238$		0			0
$239$	6116.00			1.65528			0.827638	−	0.561262i	$-0.189684\pi$
$239$	6116.00			1.65528			0.827638	+	0.561262i	$0.189684\pi$
$240$		0			0
$241$	−5962.00			−1.59355			−0.796776	−	0.604274i	$-0.793463\pi$
$241$	−5962.00			−1.59355			−0.796776	+	0.604274i	$0.793463\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	2430.00	−	1215.00i	0.633661	−	0.316831i
$246$		0			0
$247$		−	3536.00i		−	0.910892i
$248$		0			0
$249$	−1236.00			−0.314572
$250$		0			0
$251$	−1490.00			−0.374693			−0.187347	−	0.982294i	$-0.559989\pi$
$251$	−1490.00			−0.374693			−0.187347	+	0.982294i	$0.559989\pi$
$252$		0			0
$253$			7544.00i			1.87465i
$254$		0			0
$255$	1980.00	−	990.000i	0.486245	−	0.243122i
$256$		0			0
$257$			5394.00i			1.30922i	0.755969	+	0.654608i	$0.227166\pi$
$257$			5394.00i			1.30922i	−0.755969	+	0.654608i	$0.772834\pi$
$258$		0			0
$259$	220.000			0.0527804
$260$		0			0
$261$	−2016.00			−0.478112
$262$		0			0
$263$			636.000i			0.149116i	0.997217	+	0.0745579i	$0.0237546\pi$
$263$			636.000i			0.149116i	−0.997217	+	0.0745579i	$0.976245\pi$
$264$		0			0
$265$	−1430.00	−	2860.00i	−0.331488	−	0.662975i
$266$		0			0
$267$		−	3618.00i		−	0.829281i
$268$		0			0
$269$	−3360.00			−0.761572			−0.380786	−	0.924663i	$-0.624347\pi$
$269$	−3360.00			−0.761572			−0.380786	+	0.924663i	$0.624347\pi$
$270$		0			0
$271$	−5768.00			−1.29292			−0.646459	−	0.762948i	$-0.723751\pi$
$271$	−5768.00			−1.29292			−0.646459	+	0.762948i	$0.723751\pi$
$272$		0			0
$273$		−	1020.00i		−	0.226129i
$274$		0			0
$275$	−3450.00	+	4600.00i	−0.756519	+	1.00869i
$276$		0			0
$277$		−	1398.00i		−	0.303241i	−0.988439	−	0.151620i	$-0.951551\pi$
$277$		−	1398.00i		−	0.303241i	0.988439	−	0.151620i	$-0.0484491\pi$
$278$		0			0
$279$	−648.000			−0.139049
$280$		0			0
$281$	−4194.00			−0.890367			−0.445183	−	0.895439i	$-0.646862\pi$
$281$	−4194.00			−0.890367			−0.445183	+	0.895439i	$0.646862\pi$
$282$		0			0
$283$		−	8256.00i		−	1.73416i	−0.498166	−	0.867082i	$-0.665993\pi$
$283$		−	8256.00i		−	1.73416i	0.498166	−	0.867082i	$-0.334007\pi$
$284$		0			0
$285$	−1560.00	−	3120.00i	−0.324233	−	0.648466i
$286$		0			0
$287$			1940.00i			0.399006i
$288$		0			0
$289$	557.000			0.113373
$290$		0			0
$291$	−4152.00			−0.836407
$292$		0			0
$293$		−	5534.00i		−	1.10341i	−0.834039	−	0.551706i	$-0.813977\pi$
$293$		−	5534.00i		−	1.10341i	0.834039	−	0.551706i	$-0.186023\pi$
$294$		0			0
$295$	−4260.00	+	2130.00i	−0.840769	+	0.420384i
$296$		0			0
$297$			1242.00i			0.242654i
$298$		0			0
$299$	5576.00			1.07849
$300$		0			0
$301$	−1080.00			−0.206811
$302$		0			0
$303$			3384.00i			0.641603i
$304$		0			0
$305$	−6980.00	+	3490.00i	−1.31041	+	0.655203i
$306$		0			0
$307$			484.000i			0.0899783i	0.998987	+	0.0449892i	$0.0143253\pi$
$307$			484.000i			0.0899783i	−0.998987	+	0.0449892i	$0.985675\pi$
$308$		0			0
$309$	−2274.00			−0.418652
$310$		0			0
$311$	2724.00			0.496668			0.248334	−	0.968674i	$-0.420117\pi$
$311$	2724.00			0.496668			0.248334	+	0.968674i	$0.420117\pi$
$312$		0			0
$313$			5308.00i			0.958549i	0.877665	+	0.479275i	$0.159100\pi$
$313$			5308.00i			0.958549i	−0.877665	+	0.479275i	$0.840900\pi$
$314$		0			0
$315$	−450.000	−	900.000i	−0.0804909	−	0.160982i
$316$		0			0
$317$			4218.00i			0.747339i	0.927562	+	0.373670i	$0.121901\pi$
$317$			4218.00i			0.747339i	−0.927562	+	0.373670i	$0.878099\pi$
$318$		0			0
$319$	−10304.0			−1.80851
$320$		0			0
$321$	3972.00			0.690640
$322$		0			0
$323$			6864.00i			1.18242i
$324$		0			0
$325$	3400.00	+	2550.00i	0.580302	+	0.435226i
$326$		0			0
$327$			4806.00i			0.812760i
$328$		0			0
$329$	4800.00			0.804354
$330$		0			0
$331$	−4640.00			−0.770506			−0.385253	−	0.922811i	$-0.625886\pi$
$331$	−4640.00			−0.770506			−0.385253	+	0.922811i	$0.625886\pi$
$332$		0			0
$333$			198.000i			0.0325836i
$334$		0			0
$335$	−1640.00	−	3280.00i	−0.267471	−	0.534942i
$336$		0			0
$337$			8156.00i			1.31835i	0.751987	+	0.659177i	$0.229095\pi$
$337$			8156.00i			1.31835i	−0.751987	+	0.659177i	$0.770905\pi$
$338$		0			0
$339$	6222.00			0.996851
$340$		0			0
$341$	−3312.00			−0.525967
$342$		0			0
$343$			5860.00i			0.922479i
$344$		0			0
$345$	4920.00	−	2460.00i	0.767779	−	0.383890i
$346$		0			0
$347$		−	4124.00i		−	0.638006i	−0.947754	−	0.319003i	$-0.896652\pi$
$347$		−	4124.00i		−	0.638006i	0.947754	−	0.319003i	$-0.103348\pi$
$348$		0			0
$349$	3650.00			0.559828			0.279914	−	0.960025i	$-0.409694\pi$
$349$	3650.00			0.559828			0.279914	+	0.960025i	$0.409694\pi$
$350$		0			0
$351$	918.000			0.139599
$352$		0			0
$353$			6834.00i			1.03042i	0.857065	+	0.515208i	$0.172285\pi$
$353$			6834.00i			1.03042i	−0.857065	+	0.515208i	$0.827715\pi$
$354$		0			0
$355$	−1880.00	+	940.000i	−0.281071	+	0.140535i
$356$		0			0
$357$			1980.00i			0.293537i
$358$		0			0
$359$	3904.00			0.573942			0.286971	−	0.957939i	$-0.407352\pi$
$359$	3904.00			0.573942			0.286971	+	0.957939i	$0.407352\pi$
$360$		0			0
$361$	3957.00			0.576906
$362$		0			0
$363$			2355.00i			0.340511i
$364$		0			0
$365$	−3700.00	−	7400.00i	−0.530594	−	1.06119i
$366$		0			0
$367$		−	13174.0i		−	1.87378i	−0.349624	−	0.936890i	$-0.613691\pi$
$367$		−	13174.0i		−	1.87378i	0.349624	−	0.936890i	$-0.386309\pi$
$368$		0			0
$369$	−1746.00			−0.246323
$370$		0			0
$371$	2860.00			0.400226
$372$		0			0
$373$			1090.00i			0.151308i	0.997134	+	0.0756542i	$0.0241045\pi$
$373$			1090.00i			0.151308i	−0.997134	+	0.0756542i	$0.975895\pi$
$374$		0			0
$375$	4125.00	+	750.000i	0.568038	+	0.103280i
$376$		0			0
$377$			7616.00i			1.04043i
$378$		0			0
$379$	9220.00			1.24960			0.624802	−	0.780784i	$-0.285180\pi$
$379$	9220.00			1.24960			0.624802	+	0.780784i	$0.285180\pi$
$380$		0			0
$381$	1602.00			0.215415
$382$		0			0
$383$			3960.00i			0.528320i	0.964479	+	0.264160i	$0.0850947\pi$
$383$			3960.00i			0.528320i	−0.964479	+	0.264160i	$0.914905\pi$
$384$		0			0
$385$	−2300.00	−	4600.00i	−0.304465	−	0.608929i
$386$		0			0
$387$		−	972.000i		−	0.127673i
$388$		0			0
$389$	−1788.00			−0.233047			−0.116523	−	0.993188i	$-0.537175\pi$
$389$	−1788.00			−0.233047			−0.116523	+	0.993188i	$0.537175\pi$
$390$		0			0
$391$	−10824.0			−1.39998
$392$		0			0
$393$			5418.00i			0.695425i
$394$		0			0
$395$	11680.0	−	5840.00i	1.48781	−	0.743905i
$396$		0			0
$397$			9642.00i			1.21894i	0.792810	+	0.609469i	$0.208617\pi$
$397$			9642.00i			1.21894i	−0.792810	+	0.609469i	$0.791383\pi$
$398$		0			0
$399$	3120.00			0.391467
$400$		0			0
$401$	−410.000			−0.0510584			−0.0255292	−	0.999674i	$-0.508127\pi$
$401$	−410.000			−0.0510584			−0.0255292	+	0.999674i	$0.508127\pi$
$402$		0			0
$403$			2448.00i			0.302589i
$404$		0			0
$405$	810.000	−	405.000i	0.0993808	−	0.0496904i
$406$		0			0
$407$			1012.00i			0.123251i
$408$		0			0
$409$	−13766.0			−1.66427			−0.832133	−	0.554576i	$-0.812880\pi$
$409$	−13766.0			−1.66427			−0.832133	+	0.554576i	$0.812880\pi$
$410$		0			0
$411$	5466.00			0.656005
$412$		0			0
$413$		−	4260.00i		−	0.507557i
$414$		0			0
$415$	2060.00	+	4120.00i	0.243666	+	0.487332i
$416$		0			0
$417$		−	1596.00i		−	0.187426i
$418$		0			0
$419$	−16998.0			−1.98188			−0.990939	−	0.134315i	$-0.957117\pi$
$419$	−16998.0			−1.98188			−0.990939	+	0.134315i	$0.957117\pi$
$420$		0			0
$421$	2450.00			0.283624			0.141812	−	0.989894i	$-0.454707\pi$
$421$	2450.00			0.283624			0.141812	+	0.989894i	$0.454707\pi$
$422$		0			0
$423$			4320.00i			0.496562i
$424$		0			0
$425$	−6600.00	−	4950.00i	−0.753287	−	0.564965i
$426$		0			0
$427$		−	6980.00i		−	0.791068i
$428$		0			0
$429$	4692.00			0.528046
$430$		0			0
$431$	9248.00			1.03355			0.516776	−	0.856121i	$-0.327132\pi$
$431$	9248.00			1.03355			0.516776	+	0.856121i	$0.327132\pi$
$432$		0			0
$433$		−	5028.00i		−	0.558038i	−0.960286	−	0.279019i	$-0.909991\pi$
$433$		−	5028.00i		−	0.558038i	0.960286	−	0.279019i	$-0.0900092\pi$
$434$		0			0
$435$	3360.00	+	6720.00i	0.370344	+	0.740688i
$436$		0			0
$437$			17056.0i			1.86705i
$438$		0			0
$439$	3120.00			0.339202			0.169601	−	0.985513i	$-0.445752\pi$
$439$	3120.00			0.339202			0.169601	+	0.985513i	$0.445752\pi$
$440$		0			0
$441$	−2187.00			−0.236152
$442$		0			0
$443$		−	8220.00i		−	0.881589i	−0.897608	−	0.440795i	$-0.854697\pi$
$443$		−	8220.00i		−	0.881589i	0.897608	−	0.440795i	$-0.145303\pi$
$444$		0			0
$445$	−12060.0	+	6030.00i	−1.28472	+	0.642358i
$446$		0			0
$447$		−	3852.00i		−	0.407591i
$448$		0			0
$449$	5826.00			0.612352			0.306176	−	0.951975i	$-0.400950\pi$
$449$	5826.00			0.612352			0.306176	+	0.951975i	$0.400950\pi$
$450$		0			0
$451$	−8924.00			−0.931740
$452$		0			0
$453$		−	552.000i		−	0.0572521i
$454$		0			0
$455$	−3400.00	+	1700.00i	−0.350317	+	0.175159i
$456$		0			0
$457$			16896.0i			1.72946i	0.502240	+	0.864728i	$0.332509\pi$
$457$			16896.0i			1.72946i	−0.502240	+	0.864728i	$0.667491\pi$
$458$		0			0
$459$	−1782.00			−0.181213
$460$		0			0
$461$	996.000			0.100625			0.0503127	−	0.998734i	$-0.483978\pi$
$461$	996.000			0.100625			0.0503127	+	0.998734i	$0.483978\pi$
$462$		0			0
$463$		−	10046.0i		−	1.00837i	−0.863594	−	0.504187i	$-0.831792\pi$
$463$		−	10046.0i		−	1.00837i	0.863594	−	0.504187i	$-0.168208\pi$
$464$		0			0
$465$	1080.00	+	2160.00i	0.107707	+	0.215414i
$466$		0			0
$467$			8388.00i			0.831157i	0.909557	+	0.415579i	$0.136421\pi$
$467$			8388.00i			0.831157i	−0.909557	+	0.415579i	$0.863579\pi$
$468$		0			0
$469$	3280.00			0.322935
$470$		0			0
$471$	11238.0			1.09940
$472$		0			0
$473$		−	4968.00i		−	0.482936i
$474$		0			0
$475$	−7800.00	+	10400.0i	−0.753450	+	1.00460i
$476$		0			0
$477$			2574.00i			0.247076i
$478$		0			0
$479$	11396.0			1.08705			0.543525	−	0.839393i	$-0.317089\pi$
$479$	11396.0			1.08705			0.543525	+	0.839393i	$0.317089\pi$
$480$		0			0
$481$	748.000			0.0709062
$482$		0			0
$483$			4920.00i			0.463494i
$484$		0			0
$485$	6920.00	+	13840.0i	0.647878	+	1.29576i
$486$		0			0
$487$			6454.00i			0.600531i	0.953856	+	0.300266i	$0.0970753\pi$
$487$			6454.00i			0.600531i	−0.953856	+	0.300266i	$0.902925\pi$
$488$		0			0
$489$	−4512.00			−0.417259
$490$		0			0
$491$	18638.0			1.71308			0.856539	−	0.516083i	$-0.172610\pi$
$491$	18638.0			1.71308			0.856539	+	0.516083i	$0.172610\pi$
$492$		0			0
$493$		−	14784.0i		−	1.35058i
$494$		0			0
$495$	4140.00	−	2070.00i	0.375917	−	0.187959i
$496$		0			0
$497$		−	1880.00i		−	0.169677i
$498$		0			0
$499$	−17768.0			−1.59400			−0.796999	−	0.603981i	$-0.793580\pi$
$499$	−17768.0			−1.59400			−0.796999	+	0.603981i	$0.793580\pi$
$500$		0			0
$501$	−9036.00			−0.805786
$502$		0			0
$503$			7952.00i			0.704895i	0.935832	+	0.352447i	$0.114650\pi$
$503$			7952.00i			0.704895i	−0.935832	+	0.352447i	$0.885350\pi$
$504$		0			0
$505$	11280.0	−	5640.00i	0.993967	−	0.496984i
$506$		0			0
$507$			3123.00i			0.273565i
$508$		0			0
$509$	−12896.0			−1.12300			−0.561498	−	0.827478i	$-0.689775\pi$
$509$	−12896.0			−1.12300			−0.561498	+	0.827478i	$0.689775\pi$
$510$		0			0
$511$	7400.00			0.640620
$512$		0			0
$513$			2808.00i			0.241669i
$514$		0			0
$515$	3790.00	+	7580.00i	0.324286	+	0.648572i
$516$		0			0
$517$			22080.0i			1.87829i
$518$		0			0
$519$	−1314.00			−0.111133
$520$		0			0
$521$	−2714.00			−0.228220			−0.114110	−	0.993468i	$-0.536402\pi$
$521$	−2714.00			−0.228220			−0.114110	+	0.993468i	$0.536402\pi$
$522$		0			0
$523$		−	13792.0i		−	1.15312i	−0.817055	−	0.576560i	$-0.804395\pi$
$523$		−	13792.0i		−	1.15312i	0.817055	−	0.576560i	$-0.195605\pi$
$524$		0			0
$525$	−2250.00	+	3000.00i	−0.187044	+	0.249392i
$526$		0			0
$527$		−	4752.00i		−	0.392790i
$528$		0			0
$529$	−14729.0			−1.21057
$530$		0			0
$531$	3834.00			0.313336
$532$		0			0
$533$			6596.00i			0.536031i
$534$		0			0
$535$	−6620.00	−	13240.0i	−0.534967	−	1.06993i
$536$		0			0
$537$			4386.00i			0.352458i
$538$		0			0
$539$	−11178.0			−0.893266
$540$		0			0
$541$	−6802.00			−0.540556			−0.270278	−	0.962782i	$-0.587116\pi$
$541$	−6802.00			−0.540556			−0.270278	+	0.962782i	$0.587116\pi$
$542$		0			0
$543$		−	1758.00i		−	0.138937i
$544$		0			0
$545$	16020.0	−	8010.00i	1.25912	−	0.629561i
$546$		0			0
$547$			18188.0i			1.42169i	0.703350	+	0.710843i	$0.251687\pi$
$547$			18188.0i			1.42169i	−0.703350	+	0.710843i	$0.748313\pi$
$548$		0			0
$549$	6282.00			0.488359
$550$		0			0
$551$	−23296.0			−1.80117
$552$		0			0
$553$			11680.0i			0.898163i
$554$		0			0
$555$	660.000	−	330.000i	0.0504783	−	0.0252391i
$556$		0			0
$557$			21462.0i			1.63263i	0.577608	+	0.816314i	$0.303986\pi$
$557$			21462.0i			1.63263i	−0.577608	+	0.816314i	$0.696014\pi$
$558$		0			0
$559$	−3672.00			−0.277834
$560$		0			0
$561$	−9108.00			−0.685455
$562$		0			0
$563$		−	17244.0i		−	1.29085i	−0.763824	−	0.645424i	$-0.776681\pi$
$563$		−	17244.0i		−	1.29085i	0.763824	−	0.645424i	$-0.223319\pi$
$564$		0			0
$565$	−10370.0	−	20740.0i	−0.772158	−	1.54432i
$566$		0			0
$567$			810.000i			0.0599944i
$568$		0			0
$569$	−8790.00			−0.647620			−0.323810	−	0.946122i	$-0.604964\pi$
$569$	−8790.00			−0.647620			−0.323810	+	0.946122i	$0.604964\pi$
$570$		0			0
$571$	−5984.00			−0.438568			−0.219284	−	0.975661i	$-0.570372\pi$
$571$	−5984.00			−0.438568			−0.219284	+	0.975661i	$0.570372\pi$
$572$		0			0
$573$		−	180.000i		−	0.0131232i
$574$		0			0
$575$	−16400.0	−	12300.0i	−1.18944	−	0.892079i
$576$		0			0
$577$			9344.00i			0.674170i	0.941474	+	0.337085i	$0.109441\pi$
$577$			9344.00i			0.674170i	−0.941474	+	0.337085i	$0.890559\pi$
$578$		0			0
$579$	14028.0			1.00688
$580$		0			0
$581$	−4120.00			−0.294193
$582$		0			0
$583$			13156.0i			0.934590i
$584$		0			0
$585$	−1530.00	−	3060.00i	−0.108133	−	0.216266i
$586$		0			0
$587$			6932.00i			0.487418i	0.969848	+	0.243709i	$0.0783642\pi$
$587$			6932.00i			0.487418i	−0.969848	+	0.243709i	$0.921636\pi$
$588$		0			0
$589$	−7488.00			−0.523833
$590$		0			0
$591$	−6858.00			−0.477327
$592$		0			0
$593$			9382.00i			0.649701i	0.945765	+	0.324850i	$0.105314\pi$
$593$			9382.00i			0.649701i	−0.945765	+	0.324850i	$0.894686\pi$
$594$		0			0
$595$	6600.00	−	3300.00i	0.454746	−	0.227373i
$596$		0			0
$597$		−	10608.0i		−	0.727230i
$598$		0			0
$599$	4096.00			0.279396			0.139698	−	0.990194i	$-0.455387\pi$
$599$	4096.00			0.279396			0.139698	+	0.990194i	$0.455387\pi$
$600$		0			0
$601$	22962.0			1.55847			0.779234	−	0.626733i	$-0.215608\pi$
$601$	22962.0			1.55847			0.779234	+	0.626733i	$0.215608\pi$
$602$		0			0
$603$			2952.00i			0.199361i
$604$		0			0
$605$	7850.00	−	3925.00i	0.527517	−	0.263759i
$606$		0			0
$607$		−	3490.00i		−	0.233369i	−0.993169	−	0.116684i	$-0.962773\pi$
$607$		−	3490.00i		−	0.233369i	0.993169	−	0.116684i	$-0.0372266\pi$
$608$		0			0
$609$	−6720.00			−0.447140
$610$		0			0
$611$	16320.0			1.08058
$612$		0			0
$613$		−	6386.00i		−	0.420764i	−0.977619	−	0.210382i	$-0.932529\pi$
$613$		−	6386.00i		−	0.420764i	0.977619	−	0.210382i	$-0.0674707\pi$
$614$		0			0
$615$	2910.00	+	5820.00i	0.190801	+	0.381602i
$616$		0			0
$617$			19534.0i			1.27457i	0.770629	+	0.637285i	$0.219942\pi$
$617$			19534.0i			1.27457i	−0.770629	+	0.637285i	$0.780058\pi$
$618$		0			0
$619$	−8764.00			−0.569071			−0.284535	−	0.958666i	$-0.591839\pi$
$619$	−8764.00			−0.569071			−0.284535	+	0.958666i	$0.591839\pi$
$620$		0			0
$621$	−4428.00			−0.286134
$622$		0			0
$623$		−	12060.0i		−	0.775560i
$624$		0			0
$625$	−4375.00	−	15000.0i	−0.280000	−	0.960000i
$626$		0			0
$627$			14352.0i			0.914137i
$628$		0			0
$629$	−1452.00			−0.0920430
$630$		0			0
$631$	−7856.00			−0.495630			−0.247815	−	0.968807i	$-0.579712\pi$
$631$	−7856.00			−0.495630			−0.247815	+	0.968807i	$0.579712\pi$
$632$		0			0
$633$		−	10500.0i		−	0.659301i
$634$		0			0
$635$	−2670.00	−	5340.00i	−0.166859	−	0.333719i
$636$		0			0
$637$			8262.00i			0.513897i
$638$		0			0
$639$	1692.00			0.104749
$640$		0			0
$641$	−22974.0			−1.41563			−0.707815	−	0.706398i	$-0.750319\pi$
$641$	−22974.0			−1.41563			−0.707815	+	0.706398i	$0.750319\pi$
$642$		0			0
$643$		−	6216.00i		−	0.381237i	−0.981664	−	0.190618i	$-0.938951\pi$
$643$		−	6216.00i		−	0.381237i	0.981664	−	0.190618i	$-0.0610493\pi$
$644$		0			0
$645$	−3240.00	+	1620.00i	−0.197791	+	0.0988953i
$646$		0			0
$647$			13384.0i			0.813260i	0.913593	+	0.406630i	$0.133296\pi$
$647$			13384.0i			0.813260i	−0.913593	+	0.406630i	$0.866704\pi$
$648$		0			0
$649$	19596.0			1.18522
$650$		0			0
$651$	−2160.00			−0.130042
$652$		0			0
$653$		−	12882.0i		−	0.771993i	−0.922500	−	0.385997i	$-0.873858\pi$
$653$		−	12882.0i		−	0.771993i	0.922500	−	0.385997i	$-0.126142\pi$
$654$		0			0
$655$	18060.0	−	9030.00i	1.07735	−	0.538674i
$656$		0			0
$657$			6660.00i			0.395482i
$658$		0			0
$659$	2082.00			0.123070			0.0615351	−	0.998105i	$-0.480400\pi$
$659$	2082.00			0.123070			0.0615351	+	0.998105i	$0.480400\pi$
$660$		0			0
$661$	9430.00			0.554893			0.277447	−	0.960741i	$-0.410512\pi$
$661$	9430.00			0.554893			0.277447	+	0.960741i	$0.410512\pi$
$662$		0			0
$663$			6732.00i			0.394343i
$664$		0			0
$665$	−5200.00	−	10400.0i	−0.303229	−	0.606458i
$666$		0			0
$667$		−	36736.0i		−	2.13257i
$668$		0			0
$669$	17622.0			1.01839
$670$		0			0
$671$	32108.0			1.84727
$672$		0			0
$673$			3268.00i			0.187180i	0.995611	+	0.0935900i	$0.0298343\pi$
$673$			3268.00i			0.187180i	−0.995611	+	0.0935900i	$0.970166\pi$
$674$		0			0
$675$	−2700.00	−	2025.00i	−0.153960	−	0.115470i
$676$		0			0
$677$		−	15606.0i		−	0.885949i	−0.896534	−	0.442974i	$-0.853923\pi$
$677$		−	15606.0i		−	0.885949i	0.896534	−	0.442974i	$-0.146077\pi$
$678$		0			0
$679$	−13840.0			−0.782225
$680$		0			0
$681$	−372.000			−0.0209326
$682$		0			0
$683$			428.000i			0.0239780i	0.999928	+	0.0119890i	$0.00381631\pi$
$683$			428.000i			0.0239780i	−0.999928	+	0.0119890i	$0.996184\pi$
$684$		0			0
$685$	−9110.00	−	18220.0i	−0.508139	−	1.01628i
$686$		0			0
$687$		−	4086.00i		−	0.226915i
$688$		0			0
$689$	9724.00			0.537670
$690$		0			0
$691$	−6384.00			−0.351460			−0.175730	−	0.984438i	$-0.556229\pi$
$691$	−6384.00			−0.351460			−0.175730	+	0.984438i	$0.556229\pi$
$692$		0			0
$693$			4140.00i			0.226934i
$694$		0			0
$695$	−5320.00	+	2660.00i	−0.290358	+	0.145179i
$696$		0			0
$697$		−	12804.0i		−	0.695819i
$698$		0			0
$699$	11610.0			0.628227
$700$		0			0
$701$	12224.0			0.658622			0.329311	−	0.944221i	$-0.393184\pi$
$701$	12224.0			0.658622			0.329311	+	0.944221i	$0.393184\pi$
$702$		0			0
$703$			2288.00i			0.122750i
$704$		0			0
$705$	14400.0	−	7200.00i	0.769270	−	0.384635i
$706$		0			0
$707$			11280.0i			0.600040i
$708$		0			0
$709$	19510.0			1.03345			0.516723	−	0.856153i	$-0.327152\pi$
$709$	19510.0			1.03345			0.516723	+	0.856153i	$0.327152\pi$
$710$		0			0
$711$	−10512.0			−0.554474
$712$		0			0
$713$		−	11808.0i		−	0.620215i
$714$		0			0
$715$	−7820.00	−	15640.0i	−0.409023	−	0.818046i
$716$		0			0
$717$			18348.0i			0.955674i
$718$		0			0
$719$	−3368.00			−0.174694			−0.0873472	−	0.996178i	$-0.527839\pi$
$719$	−3368.00			−0.174694			−0.0873472	+	0.996178i	$0.527839\pi$
$720$		0			0
$721$	−7580.00			−0.391531
$722$		0			0
$723$		−	17886.0i		−	0.920038i
$724$		0			0
$725$	16800.0	−	22400.0i	0.860602	−	1.14747i
$726$		0			0
$727$			22134.0i			1.12917i	0.825376	+	0.564584i	$0.190963\pi$
$727$			22134.0i			1.12917i	−0.825376	+	0.564584i	$0.809037\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	7128.00			0.360655
$732$		0			0
$733$		−	32298.0i		−	1.62750i	−0.581219	−	0.813748i	$-0.697424\pi$
$733$		−	32298.0i		−	1.62750i	0.581219	−	0.813748i	$-0.302576\pi$
$734$		0			0
$735$	3645.00	+	7290.00i	0.182922	+	0.365844i
$736$		0			0
$737$			15088.0i			0.754103i
$738$		0			0
$739$	−25104.0			−1.24962			−0.624808	−	0.780779i	$-0.714823\pi$
$739$	−25104.0			−1.24962			−0.624808	+	0.780779i	$0.714823\pi$
$740$		0			0
$741$	10608.0			0.525904
$742$		0			0
$743$			27696.0i			1.36752i	0.729707	+	0.683760i	$0.239657\pi$
$743$			27696.0i			1.36752i	−0.729707	+	0.683760i	$0.760343\pi$
$744$		0			0
$745$	−12840.0	+	6420.00i	−0.631438	+	0.315719i
$746$		0			0
$747$		−	3708.00i		−	0.181618i
$748$		0			0
$749$	13240.0			0.645900
$750$		0			0
$751$	2176.00			0.105730			0.0528651	−	0.998602i	$-0.483165\pi$
$751$	2176.00			0.105730			0.0528651	+	0.998602i	$0.483165\pi$
$752$		0			0
$753$		−	4470.00i		−	0.216329i
$754$		0			0
$755$	−1840.00	+	920.000i	−0.0886946	+	0.0443473i
$756$		0			0
$757$			25514.0i			1.22500i	0.790472	+	0.612498i	$0.209835\pi$
$757$			25514.0i			1.22500i	−0.790472	+	0.612498i	$0.790165\pi$
$758$		0			0
$759$	−22632.0			−1.08233
$760$		0			0
$761$	18238.0			0.868761			0.434380	−	0.900730i	$-0.356967\pi$
$761$	18238.0			0.868761			0.434380	+	0.900730i	$0.356967\pi$
$762$		0			0
$763$			16020.0i			0.760109i
$764$		0			0
$765$	2970.00	+	5940.00i	0.140367	+	0.280734i
$766$		0			0
$767$		−	14484.0i		−	0.681860i
$768$		0			0
$769$	14462.0			0.678170			0.339085	−	0.940756i	$-0.389882\pi$
$769$	14462.0			0.678170			0.339085	+	0.940756i	$0.389882\pi$
$770$		0			0
$771$	−16182.0			−0.755876
$772$		0			0
$773$		−	34034.0i		−	1.58359i	−0.610785	−	0.791797i	$-0.709146\pi$
$773$		−	34034.0i		−	1.58359i	0.610785	−	0.791797i	$-0.290854\pi$
$774$		0			0
$775$	5400.00	−	7200.00i	0.250289	−	0.333718i
$776$		0			0
$777$			660.000i			0.0304728i
$778$		0			0
$779$	−20176.0			−0.927959
$780$		0			0
$781$	8648.00			0.396222
$782$		0			0
$783$		−	6048.00i		−	0.276038i
$784$		0			0
$785$	−18730.0	−	37460.0i	−0.851595	−	1.70319i
$786$		0			0
$787$		−	22064.0i		−	0.999360i	−0.866210	−	0.499680i	$-0.833451\pi$
$787$		−	22064.0i		−	0.999360i	0.866210	−	0.499680i	$-0.166549\pi$
$788$		0			0
$789$	−1908.00			−0.0860920
$790$		0			0
$791$	20740.0

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	960.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q+3.00000i q^{3} +(10.0000 - 5.00000i) q^{5} +10.0000i q^{7} -9.00000 q^{9} -46.0000 q^{11} +34.0000i q^{13} +(15.0000 + 30.0000i) q^{15} -66.0000i q^{17} -104.000 q^{19} -30.0000 q^{21} -164.000i q^{23} +(75.0000 - 100.000i) q^{25} -27.0000i q^{27} +224.000 q^{29} +72.0000 q^{31} -138.000i q^{33} +(50.0000 + 100.000i) q^{35} -22.0000i q^{37} -102.000 q^{39} +194.000 q^{41} +108.000i q^{43} +(-90.0000 + 45.0000i) q^{45} -480.000i q^{47} +243.000 q^{49} +198.000 q^{51} -286.000i q^{53} +(-460.000 + 230.000i) q^{55} -312.000i q^{57} -426.000 q^{59} -698.000 q^{61} -90.0000i q^{63} +(170.000 + 340.000i) q^{65} -328.000i q^{67} +492.000 q^{69} -188.000 q^{71} -740.000i q^{73} +(300.000 + 225.000i) q^{75} -460.000i q^{77} +1168.00 q^{79} +81.0000 q^{81} +412.000i q^{83} +(-330.000 - 660.000i) q^{85} +672.000i q^{87} -1206.00 q^{89} -340.000 q^{91} +216.000i q^{93} +(-1040.00 + 520.000i) q^{95} +1384.00i q^{97} +414.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{5} - 18 q^{9}+O(q^{10}) \) 2 * q + 20 * q^5 - 18 * q^9	\( 2 q + 20 q^{5} - 18 q^{9} - 92 q^{11} + 30 q^{15} - 208 q^{19} - 60 q^{21} + 150 q^{25} + 448 q^{29} + 144 q^{31} + 100 q^{35} - 204 q^{39} + 388 q^{41} - 180 q^{45} + 486 q^{49} + 396 q^{51} - 920 q^{55} - 852 q^{59} - 1396 q^{61} + 340 q^{65} + 984 q^{69} - 376 q^{71} + 600 q^{75} + 2336 q^{79} + 162 q^{81} - 660 q^{85} - 2412 q^{89} - 680 q^{91} - 2080 q^{95} + 828 q^{99}+O(q^{100}) \) 2 * q + 20 * q^5 - 18 * q^9 - 92 * q^11 + 30 * q^15 - 208 * q^19 - 60 * q^21 + 150 * q^25 + 448 * q^29 + 144 * q^31 + 100 * q^35 - 204 * q^39 + 388 * q^41 - 180 * q^45 + 486 * q^49 + 396 * q^51 - 920 * q^55 - 852 * q^59 - 1396 * q^61 + 340 * q^65 + 984 * q^69 - 376 * q^71 + 600 * q^75 + 2336 * q^79 + 162 * q^81 - 660 * q^85 - 2412 * q^89 - 680 * q^91 - 2080 * q^95 + 828 * q^99

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