LMFDB - Embedded newform 912.3.o.e.721.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [912,3,Mod(721,912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(912, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("912.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 912 = 2^{4} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	912.o (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:	$24.8502001097$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 80 x^{18} - 152 x^{17} + 4326 x^{16} - 10096 x^{15} + 70116 x^{14} - 93436 x^{13} + \cdots + 36100 $ x^20 - 4x^19 + 80x^18 - 152x^17 + 4326x^16 - 10096x^15 + 70116x^14 - 93436x^13 + 597327x^12 - 837564x^11 + 2838686x^10 - 2898276x^9 + 7652658x^8 - 6988600x^7 + 12843126x^6 - 6054124x^5 + 7434909x^4 - 621764x^3 + 3099194x^2 - 329080*x + 36100
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 456)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.6
Root			$-1.34238 + 2.32507i$ of defining polynomial
Character	$\chi$	$=$	912.721
Dual form			912.3.o.e.721.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -0.0118390 q^{5} -5.38136 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -0.0118390 q^{5} -5.38136 q^{7} -3.00000 q^{9} +20.1709 q^{11} +18.7435i q^{13} +0.0205058i q^{15} -2.61324 q^{17} +(-14.1212 - 12.7119i) q^{19} +9.32078i q^{21} -12.3236 q^{23} -24.9999 q^{25} +5.19615i q^{27} +4.07280i q^{29} +14.9617i q^{31} -34.9370i q^{33} +0.0637101 q^{35} +62.7522i q^{37} +32.4647 q^{39} +46.8535i q^{41} +72.6553 q^{43} +0.0355171 q^{45} +81.3241 q^{47} -20.0410 q^{49} +4.52627i q^{51} +32.5146i q^{53} -0.238804 q^{55} +(-22.0177 + 24.4586i) q^{57} +48.9963i q^{59} +81.6937 q^{61} +16.1441 q^{63} -0.221905i q^{65} -64.8329i q^{67} +21.3452i q^{69} +18.7083i q^{71} +117.506 q^{73} +43.3010i q^{75} -108.547 q^{77} +80.1054i q^{79} +9.00000 q^{81} -61.0368 q^{83} +0.0309383 q^{85} +7.05429 q^{87} -132.639i q^{89} -100.865i q^{91} +25.9144 q^{93} +(0.167181 + 0.150497i) q^{95} +91.8294i q^{97} -60.5127 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 16 q^{7} - 60 q^{9}+O(q^{10}) $ 20 * q + 16 * q^7 - 60 * q^9	$ 20 q + 16 q^{7} - 60 q^{9} - 16 q^{11} + 32 q^{17} - 40 q^{19} - 64 q^{23} + 68 q^{25} + 208 q^{35} - 48 q^{39} - 64 q^{43} - 48 q^{47} + 20 q^{49} + 336 q^{55} - 60 q^{57} + 184 q^{61} - 48 q^{63} + 104 q^{73} + 88 q^{77} + 180 q^{81} - 224 q^{83} - 136 q^{85} + 240 q^{87} + 120 q^{93} + 320 q^{95} + 48 q^{99}+O(q^{100}) $ 20 * q + 16 * q^7 - 60 * q^9 - 16 * q^11 + 32 * q^17 - 40 * q^19 - 64 * q^23 + 68 * q^25 + 208 * q^35 - 48 * q^39 - 64 * q^43 - 48 * q^47 + 20 * q^49 + 336 * q^55 - 60 * q^57 + 184 * q^61 - 48 * q^63 + 104 * q^73 + 88 * q^77 + 180 * q^81 - 224 * q^83 - 136 * q^85 + 240 * q^87 + 120 * q^93 + 320 * q^95 + 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$305$	$799$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		−	1.73205i		−	0.577350i
$3$		−	1.73205i		−	0.577350i
$4$		0			0
$5$	−0.0118390			−0.00236781			−0.00118390	−	0.999999i	$-0.500377\pi$
$5$	−0.0118390			−0.00236781			−0.00118390	+	0.999999i	$0.500377\pi$
$6$		0			0
$7$	−5.38136			−0.768765			−0.384383	−	0.923174i	$-0.625586\pi$
$7$	−5.38136			−0.768765			−0.384383	+	0.923174i	$0.625586\pi$
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	20.1709			1.83372			0.916859	−	0.399211i	$-0.130716\pi$
$11$	20.1709			1.83372			0.916859	+	0.399211i	$0.130716\pi$
$12$		0			0
$13$			18.7435i			1.44181i	0.693035	+	0.720904i	$0.256273\pi$
$13$			18.7435i			1.44181i	−0.693035	+	0.720904i	$0.743727\pi$
$14$		0			0
$15$			0.0205058i			0.00136705i
$16$		0			0
$17$	−2.61324			−0.153720			−0.0768600	−	0.997042i	$-0.524489\pi$
$17$	−2.61324			−0.153720			−0.0768600	+	0.997042i	$0.524489\pi$
$18$		0			0
$19$	−14.1212	−	12.7119i	−0.743219	−	0.669049i
$19$	−14.1212	−	12.7119i	−0.743219	−	0.669049i
$20$		0			0
$21$			9.32078i			0.443847i
$22$		0			0
$23$	−12.3236			−0.535810			−0.267905	−	0.963445i	$-0.586331\pi$
$23$	−12.3236			−0.535810			−0.267905	+	0.963445i	$0.586331\pi$
$24$		0			0
$25$	−24.9999			−0.999994
$26$		0			0
$27$			5.19615i			0.192450i
$28$		0			0
$29$			4.07280i			0.140441i	0.997531	+	0.0702206i	$0.0223703\pi$
$29$			4.07280i			0.140441i	−0.997531	+	0.0702206i	$0.977630\pi$
$30$		0			0
$31$			14.9617i			0.482635i	0.970446	+	0.241317i	$0.0775795\pi$
$31$			14.9617i			0.482635i	−0.970446	+	0.241317i	$0.922421\pi$
$32$		0			0
$33$		−	34.9370i		−	1.05870i
$34$		0			0
$35$	0.0637101			0.00182029
$36$		0			0
$37$			62.7522i			1.69601i	0.529992	+	0.848003i	$0.322195\pi$
$37$			62.7522i			1.69601i	−0.529992	+	0.848003i	$0.677805\pi$
$38$		0			0
$39$	32.4647			0.832428
$40$		0			0
$41$			46.8535i			1.14277i	0.820683	+	0.571384i	$0.193593\pi$
$41$			46.8535i			1.14277i	−0.820683	+	0.571384i	$0.806407\pi$
$42$		0			0
$43$	72.6553			1.68966			0.844829	−	0.535037i	$-0.179702\pi$
$43$	72.6553			1.68966			0.844829	+	0.535037i	$0.179702\pi$
$44$		0			0
$45$	0.0355171			0.000789269
$46$		0			0
$47$	81.3241			1.73030			0.865150	−	0.501512i	$-0.167223\pi$
$47$	81.3241			1.73030			0.865150	+	0.501512i	$0.167223\pi$
$48$		0			0
$49$	−20.0410			−0.409000
$50$		0			0
$51$			4.52627i			0.0887503i
$52$		0			0
$53$			32.5146i			0.613483i	0.951793	+	0.306741i	$0.0992387\pi$
$53$			32.5146i			0.613483i	−0.951793	+	0.306741i	$0.900761\pi$
$54$		0			0
$55$	−0.238804			−0.00434189
$56$		0			0
$57$	−22.0177	+	24.4586i	−0.386275	+	0.429097i
$58$		0			0
$59$			48.9963i			0.830446i	0.909720	+	0.415223i	$0.136296\pi$
$59$			48.9963i			0.830446i	−0.909720	+	0.415223i	$0.863704\pi$
$60$		0			0
$61$	81.6937			1.33924			0.669620	−	0.742704i	$-0.266457\pi$
$61$	81.6937			1.33924			0.669620	+	0.742704i	$0.266457\pi$
$62$		0			0
$63$	16.1441			0.256255
$64$		0			0
$65$		−	0.221905i		−	0.00341392i
$66$		0			0
$67$		−	64.8329i		−	0.967655i	−0.875164	−	0.483827i	$-0.839246\pi$
$67$		−	64.8329i		−	0.967655i	0.875164	−	0.483827i	$-0.160754\pi$
$68$		0			0
$69$			21.3452i			0.309350i
$70$		0			0
$71$			18.7083i			0.263498i	0.991283	+	0.131749i	$0.0420592\pi$
$71$			18.7083i			0.263498i	−0.991283	+	0.131749i	$0.957941\pi$
$72$		0			0
$73$	117.506			1.60966			0.804832	−	0.593502i	$-0.202255\pi$
$73$	117.506			1.60966			0.804832	+	0.593502i	$0.202255\pi$
$74$		0			0
$75$			43.3010i			0.577347i
$76$		0			0
$77$	−108.547			−1.40970
$78$		0			0
$79$			80.1054i			1.01399i	0.861948	+	0.506996i	$0.169244\pi$
$79$			80.1054i			1.01399i	−0.861948	+	0.506996i	$0.830756\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−61.0368			−0.735383			−0.367692	−	0.929948i	$-0.619852\pi$
$83$	−61.0368			−0.735383			−0.367692	+	0.929948i	$0.619852\pi$
$84$		0			0
$85$	0.0309383			0.000363979
$86$		0			0
$87$	7.05429			0.0810838
$88$		0			0
$89$		−	132.639i		−	1.49032i	−0.666884	−	0.745162i	$-0.732372\pi$
$89$		−	132.639i		−	1.49032i	0.666884	−	0.745162i	$-0.267628\pi$
$90$		0			0
$91$		−	100.865i		−	1.10841i
$92$		0			0
$93$	25.9144			0.278649
$94$		0			0
$95$	0.167181	+	0.150497i	0.00175980	+	0.00158418i
$96$		0			0
$97$			91.8294i			0.946695i	0.880876	+	0.473347i	$0.156954\pi$
$97$			91.8294i			0.946695i	−0.880876	+	0.473347i	$0.843046\pi$
$98$		0			0
$99$	−60.5127			−0.611239
$100$		0			0
$101$	−4.18062			−0.0413923			−0.0206961	−	0.999786i	$-0.506588\pi$
$101$	−4.18062			−0.0413923			−0.0206961	+	0.999786i	$0.506588\pi$
$102$		0			0
$103$		−	49.0424i		−	0.476140i	−0.971248	−	0.238070i	$-0.923485\pi$
$103$		−	49.0424i		−	0.476140i	0.971248	−	0.238070i	$-0.0765148\pi$
$104$		0			0
$105$		−	0.110349i		−	0.00105094i
$106$		0			0
$107$			123.082i			1.15030i	0.818049	+	0.575149i	$0.195056\pi$
$107$			123.082i			1.15030i	−0.818049	+	0.575149i	$0.804944\pi$
$108$		0			0
$109$		−	40.7625i		−	0.373968i	−0.982363	−	0.186984i	$-0.940129\pi$
$109$		−	40.7625i		−	0.373968i	0.982363	−	0.186984i	$-0.0598712\pi$
$110$		0			0
$111$	108.690			0.979189
$112$		0			0
$113$			188.610i			1.66911i	0.550921	+	0.834557i	$0.314277\pi$
$113$			188.610i			1.66911i	−0.550921	+	0.834557i	$0.685723\pi$
$114$		0			0
$115$	0.145900			0.00126870
$116$		0			0
$117$		−	56.2305i		−	0.480603i
$118$		0			0
$119$	14.0628			0.118175
$120$		0			0
$121$	285.865			2.36252
$122$		0			0
$123$	81.1526			0.659777
$124$		0			0
$125$	0.591950			0.00473560
$126$		0			0
$127$		−	189.176i		−	1.48958i	−0.667300	−	0.744789i	$-0.732550\pi$
$127$		−	189.176i		−	1.48958i	0.667300	−	0.744789i	$-0.267450\pi$
$128$		0			0
$129$		−	125.843i		−	0.975524i
$130$		0			0
$131$	−57.7649			−0.440953			−0.220477	−	0.975392i	$-0.570761\pi$
$131$	−57.7649			−0.440953			−0.220477	+	0.975392i	$0.570761\pi$
$132$		0			0
$133$	75.9910	+	68.4074i	0.571361	+	0.514341i
$134$		0			0
$135$		−	0.0615174i		−	0.000455685i
$136$		0			0
$137$	190.267			1.38881			0.694406	−	0.719584i	$-0.255667\pi$
$137$	190.267			1.38881			0.694406	+	0.719584i	$0.255667\pi$
$138$		0			0
$139$	−59.9097			−0.431005			−0.215502	−	0.976503i	$-0.569139\pi$
$139$	−59.9097			−0.431005			−0.215502	+	0.976503i	$0.569139\pi$
$140$		0			0
$141$		−	140.858i		−	0.998990i
$142$		0			0
$143$			378.073i			2.64387i
$144$		0			0
$145$		−	0.0482180i		−	0.000332538i
$146$		0			0
$147$			34.7120i			0.236136i
$148$		0			0
$149$	−126.063			−0.846063			−0.423032	−	0.906115i	$-0.639034\pi$
$149$	−126.063			−0.846063			−0.423032	+	0.906115i	$0.639034\pi$
$150$		0			0
$151$			255.446i			1.69170i	0.533422	+	0.845849i	$0.320906\pi$
$151$			255.446i			1.69170i	−0.533422	+	0.845849i	$0.679094\pi$
$152$		0			0
$153$	7.83972			0.0512400
$154$		0			0
$155$		−	0.177132i		−	0.00114279i
$156$		0			0
$157$	52.9750			0.337421			0.168710	−	0.985666i	$-0.446040\pi$
$157$	52.9750			0.337421			0.168710	+	0.985666i	$0.446040\pi$
$158$		0			0
$159$	56.3169			0.354195
$160$		0			0
$161$	66.3179			0.411912
$162$		0			0
$163$	−260.416			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$163$	−260.416			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$164$		0			0
$165$			0.413621i			0.00250679i
$166$		0			0
$167$			283.157i			1.69555i	0.530357	+	0.847774i	$0.322058\pi$
$167$			283.157i			1.69555i	−0.530357	+	0.847774i	$0.677942\pi$
$168$		0			0
$169$	−182.319			−1.07881
$170$		0			0
$171$	42.3635	+	38.1358i	0.247740	+	0.223016i
$172$		0			0
$173$			97.1696i			0.561674i	0.959755	+	0.280837i	$0.0906120\pi$
$173$			97.1696i			0.561674i	−0.959755	+	0.280837i	$0.909388\pi$
$174$		0			0
$175$	134.533			0.768761
$176$		0			0
$177$	84.8641			0.479458
$178$		0			0
$179$		−	309.155i		−	1.72712i	−0.504245	−	0.863560i	$-0.668229\pi$
$179$		−	309.155i		−	1.72712i	0.504245	−	0.863560i	$-0.331771\pi$
$180$		0			0
$181$		−	62.7290i		−	0.346569i	−0.984872	−	0.173284i	$-0.944562\pi$
$181$		−	62.7290i		−	0.346569i	0.984872	−	0.173284i	$-0.0554380\pi$
$182$		0			0
$183$		−	141.498i		−	0.773211i
$184$		0			0
$185$		−	0.742926i		−	0.00401581i
$186$		0			0
$187$	−52.7114			−0.281879
$188$		0			0
$189$		−	27.9624i		−	0.147949i
$190$		0			0
$191$	−238.858			−1.25057			−0.625283	−	0.780398i	$-0.715016\pi$
$191$	−238.858			−1.25057			−0.625283	+	0.780398i	$0.715016\pi$
$192$		0			0
$193$		−	32.4003i		−	0.167877i	−0.996471	−	0.0839385i	$-0.973250\pi$
$193$		−	32.4003i		−	0.167877i	0.996471	−	0.0839385i	$-0.0267499\pi$
$194$		0			0
$195$	−0.384351			−0.00197103
$196$		0			0
$197$	−186.657			−0.947495			−0.473748	−	0.880661i	$-0.657099\pi$
$197$	−186.657			−0.947495			−0.473748	+	0.880661i	$0.657099\pi$
$198$		0			0
$199$	−218.064			−1.09580			−0.547899	−	0.836545i	$-0.684572\pi$
$199$	−218.064			−1.09580			−0.547899	+	0.836545i	$0.684572\pi$
$200$		0			0
$201$	−112.294			−0.558676
$202$		0			0
$203$		−	21.9172i		−	0.107966i
$204$		0			0
$205$		−	0.554700i		−	0.00270585i
$206$		0			0
$207$	36.9709			0.178603
$208$		0			0
$209$	−284.836	−	256.411i	−1.36285	−	1.22685i
$210$		0			0
$211$		−	65.8204i		−	0.311945i	−0.987761	−	0.155973i	$-0.950149\pi$
$211$		−	65.8204i		−	0.311945i	0.987761	−	0.155973i	$-0.0498512\pi$
$212$		0			0
$213$	32.4038			0.152130
$214$		0			0
$215$	−0.860168			−0.00400078
$216$		0			0
$217$		−	80.5142i		−	0.371033i
$218$		0			0
$219$		−	203.526i		−	0.929340i
$220$		0			0
$221$		−	48.9813i		−	0.221635i
$222$		0			0
$223$		−	18.3832i		−	0.0824358i	−0.999150	−	0.0412179i	$-0.986876\pi$
$223$		−	18.3832i		−	0.0824358i	0.999150	−	0.0412179i	$-0.0131238\pi$
$224$		0			0
$225$	74.9996			0.333331
$226$		0			0
$227$		−	16.7659i		−	0.0738586i	−0.999318	−	0.0369293i	$-0.988242\pi$
$227$		−	16.7659i		−	0.0738586i	0.999318	−	0.0369293i	$-0.0117576\pi$
$228$		0			0
$229$	126.904			0.554165			0.277083	−	0.960846i	$-0.410632\pi$
$229$	126.904			0.554165			0.277083	+	0.960846i	$0.410632\pi$
$230$		0			0
$231$			188.009i			0.813890i
$232$		0			0
$233$	−286.202			−1.22834			−0.614168	−	0.789175i	$-0.710508\pi$
$233$	−286.202			−1.22834			−0.614168	+	0.789175i	$0.710508\pi$
$234$		0			0
$235$	−0.962799			−0.00409702
$236$		0			0
$237$	138.747			0.585429
$238$		0			0
$239$	−184.163			−0.770557			−0.385278	−	0.922800i	$-0.625895\pi$
$239$	−184.163			−0.770557			−0.385278	+	0.922800i	$0.625895\pi$
$240$		0			0
$241$			13.5417i			0.0561895i	0.999605	+	0.0280948i	$0.00894402\pi$
$241$			13.5417i			0.0561895i	−0.999605	+	0.0280948i	$0.991056\pi$
$242$		0			0
$243$		−	15.5885i		−	0.0641500i
$244$		0			0
$245$	0.237266			0.000968433
$246$		0			0
$247$	238.266	−	264.680i	0.964640	−	1.07158i
$248$		0			0
$249$			105.719i			0.424574i
$250$		0			0
$251$	−35.0962			−0.139825			−0.0699127	−	0.997553i	$-0.522272\pi$
$251$	−35.0962			−0.139825			−0.0699127	+	0.997553i	$0.522272\pi$
$252$		0			0
$253$	−248.579			−0.982525
$254$		0			0
$255$		−	0.0535866i		−	0.000210144i
$256$		0			0
$257$		−	157.231i		−	0.611794i	−0.952065	−	0.305897i	$-0.901044\pi$
$257$		−	157.231i		−	0.611794i	0.952065	−	0.305897i	$-0.0989564\pi$
$258$		0			0
$259$		−	337.692i		−	1.30383i
$260$		0			0
$261$		−	12.2184i		−	0.0468138i
$262$		0			0
$263$	268.630			1.02141			0.510704	−	0.859757i	$-0.329385\pi$
$263$	268.630			1.02141			0.510704	+	0.859757i	$0.329385\pi$
$264$		0			0
$265$		−	0.384941i		−	0.00145261i
$266$		0			0
$267$	−229.737			−0.860439
$268$		0			0
$269$			170.930i			0.635426i	0.948187	+	0.317713i	$0.102915\pi$
$269$			170.930i			0.635426i	−0.948187	+	0.317713i	$0.897085\pi$
$270$		0			0
$271$	372.857			1.37585			0.687927	−	0.725779i	$-0.258521\pi$
$271$	372.857			1.37585			0.687927	+	0.725779i	$0.258521\pi$
$272$		0			0
$273$	−174.704			−0.639942
$274$		0			0
$275$	−504.270			−1.83371
$276$		0			0
$277$	400.543			1.44600			0.723001	−	0.690847i	$-0.242762\pi$
$277$	400.543			1.44600			0.723001	+	0.690847i	$0.242762\pi$
$278$		0			0
$279$		−	44.8850i		−	0.160878i
$280$		0			0
$281$			149.073i			0.530510i	0.964178	+	0.265255i	$0.0854562\pi$
$281$			149.073i			0.530510i	−0.964178	+	0.265255i	$0.914544\pi$
$282$		0			0
$283$	−26.4609			−0.0935013			−0.0467506	−	0.998907i	$-0.514887\pi$
$283$	−26.4609			−0.0935013			−0.0467506	+	0.998907i	$0.514887\pi$
$284$		0			0
$285$	0.260668	−	0.289566i	0.000914626	−	0.00101602i
$286$		0			0
$287$		−	252.135i		−	0.878520i
$288$		0			0
$289$	−282.171			−0.976370
$290$		0			0
$291$	159.053			0.546574
$292$		0			0
$293$			538.517i			1.83794i	0.394326	+	0.918970i	$0.370978\pi$
$293$			538.517i			1.83794i	−0.394326	+	0.918970i	$0.629022\pi$
$294$		0			0
$295$		−	0.580069i		−	0.00196634i
$296$		0			0
$297$			104.811i			0.352899i
$298$		0			0
$299$		−	230.988i		−	0.772536i
$300$		0			0
$301$	−390.984			−1.29895
$302$		0			0
$303$			7.24104i			0.0238978i
$304$		0			0
$305$	−0.967174			−0.00317106
$306$		0			0
$307$			284.758i			0.927552i	0.885953	+	0.463776i	$0.153506\pi$
$307$			284.758i			0.927552i	−0.885953	+	0.463776i	$0.846494\pi$
$308$		0			0
$309$	−84.9440			−0.274900
$310$		0			0
$311$	−128.577			−0.413431			−0.206715	−	0.978401i	$-0.566277\pi$
$311$	−128.577			−0.413431			−0.206715	+	0.978401i	$0.566277\pi$
$312$		0			0
$313$	13.0647			0.0417402			0.0208701	−	0.999782i	$-0.493356\pi$
$313$	13.0647			0.0417402			0.0208701	+	0.999782i	$0.493356\pi$
$314$		0			0
$315$	−0.191130			−0.000606763
$316$		0			0
$317$		−	238.719i		−	0.753056i	−0.926405	−	0.376528i	$-0.877118\pi$
$317$		−	238.719i		−	0.753056i	0.926405	−	0.376528i	$-0.122882\pi$
$318$		0			0
$319$			82.1520i			0.257530i
$320$		0			0
$321$	213.184			0.664125
$322$		0			0
$323$	36.9020	+	33.2193i	0.114248	+	0.102846i
$324$		0			0
$325$		−	468.585i		−	1.44180i
$326$		0			0
$327$	−70.6027			−0.215910
$328$		0			0
$329$	−437.634			−1.33020
$330$		0			0
$331$		−	483.447i		−	1.46057i	−0.683144	−	0.730283i	$-0.739388\pi$
$331$		−	483.447i		−	1.46057i	0.683144	−	0.730283i	$-0.260612\pi$
$332$		0			0
$333$		−	188.257i		−	0.565335i
$334$		0			0
$335$			0.767558i			0.00229122i
$336$		0			0
$337$		−	238.148i		−	0.706670i	−0.935497	−	0.353335i	$-0.885048\pi$
$337$		−	238.148i		−	0.706670i	0.935497	−	0.353335i	$-0.114952\pi$
$338$		0			0
$339$	326.682			0.963664
$340$		0			0
$341$			301.791i			0.885016i
$342$		0			0
$343$	371.534			1.08319
$344$		0			0
$345$		−	0.252706i		−	0.000732482i
$346$		0			0
$347$	−283.931			−0.818244			−0.409122	−	0.912480i	$-0.634165\pi$
$347$	−283.931			−0.818244			−0.409122	+	0.912480i	$0.634165\pi$
$348$		0			0
$349$	−125.357			−0.359189			−0.179595	−	0.983741i	$-0.557479\pi$
$349$	−125.357			−0.359189			−0.179595	+	0.983741i	$0.557479\pi$
$350$		0			0
$351$	−97.3941			−0.277476
$352$		0			0
$353$	134.945			0.382281			0.191140	−	0.981563i	$-0.438781\pi$
$353$	134.945			0.382281			0.191140	+	0.981563i	$0.438781\pi$
$354$		0			0
$355$		−	0.221488i		−	0.000623911i
$356$		0			0
$357$		−	24.3575i		−	0.0682282i
$358$		0			0
$359$	434.019			1.20897			0.604484	−	0.796617i	$-0.293379\pi$
$359$	434.019			1.20897			0.604484	+	0.796617i	$0.293379\pi$
$360$		0			0
$361$	37.8139	+	359.014i	0.104748	+	0.994499i
$362$		0			0
$363$		−	495.133i		−	1.36400i
$364$		0			0
$365$	−1.39115			−0.00381137
$366$		0			0
$367$	190.842			0.520005			0.260002	−	0.965608i	$-0.416277\pi$
$367$	190.842			0.520005			0.260002	+	0.965608i	$0.416277\pi$
$368$		0			0
$369$		−	140.560i		−	0.380923i
$370$		0			0
$371$		−	174.973i		−	0.471624i
$372$		0			0
$373$		−	628.473i		−	1.68491i	−0.538763	−	0.842457i	$-0.681108\pi$
$373$		−	628.473i		−	1.68491i	0.538763	−	0.842457i	$-0.318892\pi$
$374$		0			0
$375$		−	1.02529i		−	0.00273410i
$376$		0			0
$377$	−76.3385			−0.202489
$378$		0			0
$379$		−	296.275i		−	0.781727i	−0.920449	−	0.390864i	$-0.872176\pi$
$379$		−	296.275i		−	0.781727i	0.920449	−	0.390864i	$-0.127824\pi$
$380$		0			0
$381$	−327.663			−0.860008
$382$		0			0
$383$		−	442.460i		−	1.15525i	−0.816303	−	0.577623i	$-0.803980\pi$
$383$		−	442.460i		−	1.15525i	0.816303	−	0.577623i	$-0.196020\pi$
$384$		0			0
$385$	1.28509			0.00333789
$386$		0			0
$387$	−217.966			−0.563219
$388$		0			0
$389$	−266.272			−0.684505			−0.342253	−	0.939608i	$-0.611190\pi$
$389$	−266.272			−0.684505			−0.342253	+	0.939608i	$0.611190\pi$
$390$		0			0
$391$	32.2046			0.0823648
$392$		0			0
$393$			100.052i			0.254585i
$394$		0			0
$395$		−	0.948371i		−	0.00240094i
$396$		0			0
$397$	−198.584			−0.500211			−0.250106	−	0.968219i	$-0.580465\pi$
$397$	−198.584			−0.500211			−0.250106	+	0.968219i	$0.580465\pi$
$398$		0			0
$399$	118.485	−	131.620i	0.296955	−	0.329875i
$400$		0			0
$401$			219.350i			0.547007i	0.961871	+	0.273503i	$0.0881825\pi$
$401$			219.350i			0.547007i	−0.961871	+	0.273503i	$0.911818\pi$
$402$		0			0
$403$	−280.434			−0.695867
$404$		0			0
$405$	−0.106551			−0.000263090
$406$		0			0
$407$			1265.77i			3.11000i
$408$		0			0
$409$		−	73.0707i		−	0.178657i	−0.996002	−	0.0893285i	$-0.971528\pi$
$409$		−	73.0707i		−	0.178657i	0.996002	−	0.0893285i	$-0.0284721\pi$
$410$		0			0
$411$		−	329.552i		−	0.801830i
$412$		0			0
$413$		−	263.667i		−	0.638418i
$414$		0			0
$415$	0.722617			0.00174125
$416$		0			0
$417$			103.767i			0.248841i
$418$		0			0
$419$	130.721			0.311983			0.155991	−	0.987758i	$-0.450143\pi$
$419$	130.721			0.311983			0.155991	+	0.987758i	$0.450143\pi$
$420$		0			0
$421$		−	205.731i		−	0.488673i	−0.969690	−	0.244337i	$-0.921430\pi$
$421$		−	205.731i		−	0.488673i	0.969690	−	0.244337i	$-0.0785702\pi$
$422$		0			0
$423$	−243.972			−0.576767
$424$		0			0
$425$	65.3307			0.153719
$426$		0			0
$427$	−439.623			−1.02956
$428$		0			0
$429$	654.842			1.52644
$430$		0			0
$431$		−	346.636i		−	0.804259i	−0.915583	−	0.402129i	$-0.868270\pi$
$431$		−	346.636i		−	0.804259i	0.915583	−	0.402129i	$-0.131730\pi$
$432$		0			0
$433$		−	44.3462i		−	0.102416i	−0.998688	−	0.0512081i	$-0.983693\pi$
$433$		−	44.3462i		−	0.102416i	0.998688	−	0.0512081i	$-0.0163072\pi$
$434$		0			0
$435$	−0.0835160			−0.000191991
$436$		0			0
$437$	174.024	+	156.657i	0.398224	+	0.358483i
$438$		0			0
$439$			227.421i			0.518043i	0.965872	+	0.259021i	$0.0834000\pi$
$439$			227.421i			0.518043i	−0.965872	+	0.259021i	$0.916600\pi$
$440$		0			0
$441$	60.1230			0.136333
$442$		0			0
$443$	390.532			0.881563			0.440782	−	0.897614i	$-0.354701\pi$
$443$	390.532			0.881563			0.440782	+	0.897614i	$0.354701\pi$
$444$		0			0
$445$			1.57032i			0.00352880i
$446$		0			0
$447$			218.348i			0.488475i
$448$		0			0
$449$		−	366.375i		−	0.815980i	−0.912986	−	0.407990i	$-0.866230\pi$
$449$		−	366.375i		−	0.815980i	0.912986	−	0.407990i	$-0.133770\pi$
$450$		0			0
$451$			945.077i			2.09551i
$452$		0			0
$453$	442.446			0.976702
$454$		0			0
$455$			1.19415i			0.00262451i
$456$		0			0
$457$	−152.655			−0.334038			−0.167019	−	0.985954i	$-0.553414\pi$
$457$	−152.655			−0.334038			−0.167019	+	0.985954i	$0.553414\pi$
$458$		0			0
$459$		−	13.5788i		−	0.0295834i
$460$		0			0
$461$	−295.846			−0.641749			−0.320874	−	0.947122i	$-0.603977\pi$
$461$	−295.846			−0.641749			−0.320874	+	0.947122i	$0.603977\pi$
$462$		0			0
$463$	596.448			1.28822			0.644112	−	0.764931i	$-0.277227\pi$
$463$	596.448			1.28822			0.644112	+	0.764931i	$0.277227\pi$
$464$		0			0
$465$	−0.306801			−0.000659788
$466$		0			0
$467$	−795.094			−1.70256			−0.851278	−	0.524715i	$-0.824172\pi$
$467$	−795.094			−1.70256			−0.851278	+	0.524715i	$0.824172\pi$
$468$		0			0
$469$			348.889i			0.743899i
$470$		0			0
$471$		−	91.7555i		−	0.194810i
$472$		0			0
$473$	1465.52			3.09836
$474$		0			0
$475$	353.027	+	317.796i	0.743214	+	0.669045i
$476$		0			0
$477$		−	97.5438i		−	0.204494i
$478$		0			0
$479$	−258.081			−0.538791			−0.269396	−	0.963030i	$-0.586824\pi$
$479$	−258.081			−0.538791			−0.269396	+	0.963030i	$0.586824\pi$
$480$		0			0
$481$	−1176.20			−2.44531
$482$		0			0
$483$		−	114.866i		−	0.237818i
$484$		0			0
$485$		−	1.08717i		−	0.00224159i
$486$		0			0
$487$			608.037i			1.24854i	0.781210	+	0.624268i	$0.214603\pi$
$487$			608.037i			1.24854i	−0.781210	+	0.624268i	$0.785397\pi$
$488$		0			0
$489$			451.054i			0.922402i
$490$		0			0
$491$	−503.753			−1.02597			−0.512987	−	0.858396i	$-0.671461\pi$
$491$	−503.753			−1.02597			−0.512987	+	0.858396i	$0.671461\pi$
$492$		0			0
$493$		−	10.6432i		−	0.0215886i
$494$		0			0
$495$	0.716412			0.00144730
$496$		0			0
$497$		−	100.676i		−	0.202568i
$498$		0			0
$499$	485.555			0.973056			0.486528	−	0.873665i	$-0.338263\pi$
$499$	485.555			0.973056			0.486528	+	0.873665i	$0.338263\pi$
$500$		0			0
$501$	490.442			0.978925
$502$		0			0
$503$	−389.683			−0.774718			−0.387359	−	0.921929i	$-0.626613\pi$
$503$	−389.683			−0.774718			−0.387359	+	0.921929i	$0.626613\pi$
$504$		0			0
$505$	0.0494945			9.80089e−5
$506$		0			0
$507$			315.786i			0.622851i
$508$		0			0
$509$			361.385i			0.709990i	0.934868	+	0.354995i	$0.115517\pi$
$509$			361.385i			0.709990i	−0.934868	+	0.354995i	$0.884483\pi$
$510$		0			0
$511$	−632.339			−1.23745
$512$		0			0
$513$	66.0531	−	73.3757i	0.128758	−	0.143032i
$514$		0			0
$515$			0.580615i			0.00112741i
$516$		0			0
$517$	1640.38			3.17288
$518$		0			0
$519$	168.303			0.324283
$520$		0			0
$521$			798.564i			1.53275i	0.642392	+	0.766377i	$0.277942\pi$
$521$			798.564i			1.53275i	−0.642392	+	0.766377i	$0.722058\pi$
$522$		0			0
$523$		−	346.310i		−	0.662160i	−0.943603	−	0.331080i	$-0.892587\pi$
$523$		−	346.310i		−	0.662160i	0.943603	−	0.331080i	$-0.107413\pi$
$524$		0			0
$525$		−	233.018i		−	0.443844i
$526$		0			0
$527$		−	39.0985i		−	0.0741907i
$528$		0			0
$529$	−377.128			−0.712907
$530$		0			0
$531$		−	146.989i		−	0.276815i
$532$		0			0
$533$	−878.199			−1.64765
$534$		0			0
$535$		−	1.45717i		−	0.00272368i
$536$		0			0
$537$	−535.472			−0.997154
$538$		0			0
$539$	−404.245			−0.749990
$540$		0			0
$541$	−8.91567			−0.0164800			−0.00823999	−	0.999966i	$-0.502623\pi$
$541$	−8.91567			−0.0164800			−0.00823999	+	0.999966i	$0.502623\pi$
$542$		0			0
$543$	−108.650			−0.200092
$544$		0			0
$545$			0.482589i			0.000885484i
$546$		0			0
$547$		−	412.090i		−	0.753363i	−0.926343	−	0.376681i	$-0.877065\pi$
$547$		−	412.090i		−	0.753363i	0.926343	−	0.376681i	$-0.122935\pi$
$548$		0			0
$549$	−245.081			−0.446414
$550$		0			0
$551$	51.7731	−	57.5126i	0.0939620	−	0.104379i
$552$		0			0
$553$		−	431.076i		−	0.779522i
$554$		0			0
$555$	−1.28678			−0.00231853
$556$		0			0
$557$	164.634			0.295573			0.147786	−	0.989019i	$-0.452785\pi$
$557$	164.634			0.295573			0.147786	+	0.989019i	$0.452785\pi$
$558$		0			0
$559$			1361.81i			2.43616i
$560$		0			0
$561$			91.2989i			0.162743i
$562$		0			0
$563$		−	863.263i		−	1.53333i	−0.642050	−	0.766663i	$-0.721916\pi$
$563$		−	863.263i		−	1.53333i	0.642050	−	0.766663i	$-0.278084\pi$
$564$		0			0
$565$		−	2.23296i		−	0.00395214i
$566$		0			0
$567$	−48.4322			−0.0854184
$568$		0			0
$569$		−	635.576i		−	1.11700i	−0.829503	−	0.558502i	$-0.811376\pi$
$569$		−	635.576i		−	1.11700i	0.829503	−	0.558502i	$-0.188624\pi$
$570$		0			0
$571$	111.520			0.195306			0.0976530	−	0.995221i	$-0.468866\pi$
$571$	111.520			0.195306			0.0976530	+	0.995221i	$0.468866\pi$
$572$		0			0
$573$			413.714i			0.722014i
$574$		0			0
$575$	308.089			0.535807
$576$		0			0
$577$	658.831			1.14182			0.570911	−	0.821012i	$-0.306590\pi$
$577$	658.831			1.14182			0.570911	+	0.821012i	$0.306590\pi$
$578$		0			0
$579$	−56.1189			−0.0969238
$580$		0			0
$581$	328.461			0.565337
$582$		0			0
$583$			655.849i			1.12495i
$584$		0			0
$585$			0.665715i			0.00113797i
$586$		0			0
$587$	608.235			1.03618			0.518088	−	0.855327i	$-0.326644\pi$
$587$	608.235			1.03618			0.518088	+	0.855327i	$0.326644\pi$
$588$		0			0
$589$	190.192	−	211.276i	0.322906	−	0.358703i
$590$		0			0
$591$			323.299i			0.547037i
$592$		0			0
$593$	1086.74			1.83261			0.916305	−	0.400481i	$-0.131157\pi$
$593$	1086.74			1.83261			0.916305	+	0.400481i	$0.131157\pi$
$594$		0			0
$595$	−0.166490			−0.000279815
$596$		0			0
$597$			377.697i			0.632659i
$598$		0			0
$599$		−	420.739i		−	0.702402i	−0.936300	−	0.351201i	$-0.885773\pi$
$599$		−	420.739i		−	0.702402i	0.936300	−	0.351201i	$-0.114227\pi$
$600$		0			0
$601$		−	577.036i		−	0.960126i	−0.877234	−	0.480063i	$-0.840614\pi$
$601$		−	577.036i		−	0.960126i	0.877234	−	0.480063i	$-0.159386\pi$
$602$		0			0
$603$			194.499i			0.322552i
$604$		0			0
$605$	−3.38437			−0.00559400
$606$		0			0
$607$		−	150.995i		−	0.248756i	−0.992235	−	0.124378i	$-0.960306\pi$
$607$		−	150.995i		−	0.248756i	0.992235	−	0.124378i	$-0.0396935\pi$
$608$		0			0
$609$	−37.9617			−0.0623344
$610$		0			0
$611$			1524.30i			2.49476i
$612$		0			0
$613$	−451.848			−0.737109			−0.368554	−	0.929606i	$-0.620147\pi$
$613$	−451.848			−0.737109			−0.368554	+	0.929606i	$0.620147\pi$
$614$		0			0
$615$	−0.960769			−0.00156223
$616$		0			0
$617$	528.451			0.856485			0.428242	−	0.903664i	$-0.359133\pi$
$617$	528.451			0.856485			0.428242	+	0.903664i	$0.359133\pi$
$618$		0			0
$619$	63.6940			0.102898			0.0514491	−	0.998676i	$-0.483616\pi$
$619$	63.6940			0.102898			0.0514491	+	0.998676i	$0.483616\pi$
$620$		0			0
$621$		−	64.0355i		−	0.103117i
$622$		0			0
$623$			713.777i			1.14571i
$624$		0			0
$625$	624.989			0.999983
$626$		0			0
$627$	−444.117	+	493.351i	−0.708320	+	0.786844i
$628$		0			0
$629$		−	163.987i		−	0.260710i
$630$		0			0
$631$	−773.865			−1.22641			−0.613206	−	0.789923i	$-0.710120\pi$
$631$	−773.865			−1.22641			−0.613206	+	0.789923i	$0.710120\pi$
$632$		0			0
$633$	−114.004			−0.180102
$634$		0			0
$635$			2.23967i			0.00352703i
$636$		0			0
$637$		−	375.638i		−	0.589699i
$638$		0			0
$639$		−	56.1250i		−	0.0878325i
$640$		0			0
$641$			5.28345i			0.00824252i	0.999992	+	0.00412126i	$0.00131184\pi$
$641$			5.28345i			0.00824252i	−0.999992	+	0.00412126i	$0.998688\pi$
$642$		0			0
$643$	694.339			1.07984			0.539922	−	0.841715i	$-0.318454\pi$
$643$	694.339			1.07984			0.539922	+	0.841715i	$0.318454\pi$
$644$		0			0
$645$			1.48986i			0.00230985i
$646$		0			0
$647$	279.022			0.431256			0.215628	−	0.976476i	$-0.430820\pi$
$647$	279.022			0.431256			0.215628	+	0.976476i	$0.430820\pi$
$648$		0			0
$649$			988.300i			1.52280i
$650$		0			0
$651$	−139.455			−0.214216
$652$		0			0
$653$	421.633			0.645686			0.322843	−	0.946453i	$-0.395361\pi$
$653$	421.633			0.645686			0.322843	+	0.946453i	$0.395361\pi$
$654$		0			0
$655$	0.683880			0.00104409
$656$		0			0
$657$	−352.517			−0.536555
$658$		0			0
$659$			376.000i			0.570561i	0.958444	+	0.285280i	$0.0920867\pi$
$659$			376.000i			0.570561i	−0.958444	+	0.285280i	$0.907913\pi$
$660$		0			0
$661$			720.200i			1.08956i	0.838579	+	0.544781i	$0.183387\pi$
$661$			720.200i			1.08956i	−0.838579	+	0.544781i	$0.816613\pi$
$662$		0			0
$663$	−84.8381			−0.127961
$664$		0			0
$665$	−0.899660	−	0.809878i	−0.00135287	−	0.00121786i
$666$		0			0
$667$		−	50.1917i		−	0.0752499i
$668$		0			0
$669$	−31.8406			−0.0475944
$670$		0			0
$671$	1647.84			2.45579
$672$		0			0
$673$			868.848i			1.29101i	0.763757	+	0.645504i	$0.223353\pi$
$673$			868.848i			1.29101i	−0.763757	+	0.645504i	$0.776647\pi$
$674$		0			0
$675$		−	129.903i		−	0.192449i
$676$		0			0
$677$			977.114i			1.44330i	0.692258	+	0.721650i	$0.256616\pi$
$677$			977.114i			1.44330i	−0.692258	+	0.721650i	$0.743384\pi$
$678$		0			0
$679$		−	494.167i		−	0.727786i
$680$		0			0
$681$	−29.0394			−0.0426423
$682$		0			0
$683$			228.420i			0.334437i	0.985920	+	0.167218i	$0.0534785\pi$
$683$			228.420i			0.334437i	−0.985920	+	0.167218i	$0.946521\pi$
$684$		0			0
$685$	−2.25258			−0.00328844
$686$		0			0
$687$		−	219.804i		−	0.319948i
$688$		0			0
$689$	−609.437			−0.884525
$690$		0			0
$691$	−94.8120			−0.137210			−0.0686049	−	0.997644i	$-0.521855\pi$
$691$	−94.8120			−0.137210			−0.0686049	+	0.997644i	$0.521855\pi$
$692$		0			0
$693$	325.640			0.469900
$694$		0			0
$695$	0.709273			0.00102054
$696$		0			0
$697$		−	122.439i		−	0.175666i
$698$		0			0
$699$			495.717i			0.709180i
$700$		0			0
$701$	−828.374			−1.18170			−0.590852	−	0.806780i	$-0.701208\pi$
$701$	−828.374			−1.18170			−0.590852	+	0.806780i	$0.701208\pi$
$702$		0			0
$703$	797.701	−	886.134i	1.13471	−	1.26050i
$704$		0			0
$705$			1.66762i			0.00236541i
$706$		0			0
$707$	22.4974			0.0318209
$708$		0			0
$709$	−475.081			−0.670073			−0.335036	−	0.942205i	$-0.608749\pi$
$709$	−475.081			−0.670073			−0.335036	+	0.942205i	$0.608749\pi$
$710$		0			0
$711$		−	240.316i		−	0.337998i
$712$		0			0
$713$		−	184.382i		−	0.258601i
$714$		0			0
$715$		−	4.47602i		−	0.00626017i
$716$		0			0
$717$			318.980i			0.444881i
$718$		0			0
$719$	1015.22			1.41199			0.705997	−	0.708215i	$-0.250499\pi$
$719$	1015.22			1.41199			0.705997	+	0.708215i	$0.250499\pi$
$720$		0			0
$721$			263.915i			0.366040i
$722$		0			0
$723$	23.4549			0.0324410
$724$		0			0
$725$		−	101.819i		−	0.140440i
$726$		0			0
$727$	7.82363			0.0107615			0.00538076	−	0.999986i	$-0.498287\pi$
$727$	7.82363			0.0107615			0.00538076	+	0.999986i	$0.498287\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$	−189.866			−0.259734
$732$		0			0
$733$	168.524			0.229910			0.114955	−	0.993371i	$-0.463328\pi$
$733$	168.524			0.229910			0.114955	+	0.993371i	$0.463328\pi$
$734$		0			0
$735$		−	0.410957i		−	0.000559125i
$736$		0			0
$737$		−	1307.74i		−	1.77441i
$738$		0			0
$739$	−256.826			−0.347531			−0.173766	−	0.984787i	$-0.555594\pi$
$739$	−256.826			−0.347531			−0.173766	+	0.984787i	$0.555594\pi$
$740$		0			0
$741$	−458.439	−	412.689i	−0.618676	−	0.556935i
$742$		0			0
$743$		−	848.220i		−	1.14161i	−0.821084	−	0.570807i	$-0.806630\pi$
$743$		−	848.220i		−	1.14161i	0.821084	−	0.570807i	$-0.193370\pi$
$744$		0			0
$745$	1.49247			0.00200331
$746$		0			0
$747$	183.110			0.245128
$748$		0			0
$749$		−	662.348i		−	0.884310i
$750$		0			0
$751$			61.3237i			0.0816560i	0.999166	+	0.0408280i	$0.0129996\pi$
$751$			61.3237i			0.0816560i	−0.999166	+	0.0408280i	$0.987000\pi$
$752$		0			0
$753$			60.7884i			0.0807283i
$754$		0			0
$755$		−	3.02424i		−	0.00400561i
$756$		0			0
$757$	−275.977			−0.364566			−0.182283	−	0.983246i	$-0.558349\pi$
$757$	−275.977			−0.364566			−0.182283	+	0.983246i	$0.558349\pi$
$758$		0			0
$759$			430.551i			0.567261i
$760$		0			0
$761$	1142.17			1.50088			0.750441	−	0.660938i	$-0.229841\pi$
$761$	1142.17			1.50088			0.750441	+	0.660938i	$0.229841\pi$
$762$		0			0
$763$			219.358i			0.287494i
$764$		0			0
$765$	−0.0928148			−0.000121326
$766$		0			0
$767$	−918.362			−1.19734
$768$		0			0
$769$	−71.7054			−0.0932450			−0.0466225	−	0.998913i	$-0.514846\pi$
$769$	−71.7054			−0.0932450			−0.0466225	+	0.998913i	$0.514846\pi$
$770$		0			0
$771$	−272.332			−0.353220
$772$		0			0
$773$			744.760i			0.963467i	0.876318	+	0.481734i	$0.159993\pi$
$773$			744.760i			0.963467i	−0.876318	+	0.481734i	$0.840007\pi$
$774$		0			0
$775$		−	374.040i		−	0.482632i
$776$		0			0
$777$	−584.900			−0.752767
$778$		0			0
$779$	595.598	−	661.625i	0.764567	−	0.849326i
$780$		0			0
$781$			377.364i			0.483180i
$782$		0			0
$783$	−21.1629			−0.0270279
$784$		0			0
$785$	−0.627173			−0.000798947
$786$		0			0
$787$			426.395i			0.541797i	0.962608	+	0.270899i	$0.0873209\pi$
$787$			426.395i			0.541797i	−0.962608	+	0.270899i	$0.912679\pi$
$788$		0			0
$789$		−	465.281i		−	0.589710i
$790$		0			0
$791$		−	1014.98i		−	1.28316i
$792$		0			0
$793$			1531.23i			1.93093i
$794$		0			0
$795$	−0.666738			−0.000838664
$796$		0			0
$797$		−	490.997i		−	0.616056i	−0.951377	−	0.308028i	$-0.900331\pi$
$797$		−	490.997i		−	0.616056i	0.951377	−	0.308028i	$-0.0996691\pi$
$798$		0			0
$799$	−212.520			−0.265982
$800$		0			0
$801$			397.916i			0.496774i
$802$		0			0
$803$	2370.19			2.95167
$804$		0			0
$805$	−0.785140			−0.000975329
$806$		0			0
$807$	296.059			0.366864
$808$		0			0
$809$	1223.89			1.51284			0.756420	−	0.654087i	$-0.226947\pi$
$809$	1223.89			1.51284			0.756420	+	0.654087i	$0.226947\pi$
$810$		0			0
$811$		−	590.727i		−	0.728393i	−0.931322	−	0.364197i	$-0.881344\pi$
$811$		−	590.727i		−	0.728393i	0.931322	−	0.364197i	$-0.118656\pi$
$812$		0			0
$813$		−	645.807i		−	0.794350i
$814$		0			0
$815$	3.08308			0.00378292
$816$		0			0
$817$	−1025.98	−	923.588i	−1.25578	−	1.13046i
$818$		0			0
$819$			302.596i			0.369471i
$820$		0			0
$821$	188.273			0.229321			0.114661	−	0.993405i	$-0.463422\pi$
$821$	188.273			0.229321			0.114661	+	0.993405i	$0.463422\pi$
$822$		0			0
$823$	−1419.13			−1.72434			−0.862168	−	0.506623i	$-0.830894\pi$
$823$	−1419.13			−1.72434			−0.862168	+	0.506623i	$0.830894\pi$
$824$		0			0
$825$			873.421i			1.05869i
$826$		0			0
$827$		−	1005.56i		−	1.21591i	−0.793972	−	0.607954i	$-0.791991\pi$
$827$		−	1005.56i		−	1.21591i	0.793972	−	0.607954i	$-0.208009\pi$
$828$		0			0
$829$		−	280.910i		−	0.338854i	−0.985543	−	0.169427i	$-0.945808\pi$
$829$		−	280.910i		−	0.338854i	0.985543	−	0.169427i	$-0.0541917\pi$
$830$		0			0
$831$		−	693.760i		−	0.834850i
$832$		0			0
$833$	52.3719			0.0628715
$834$		0			0
$835$		−	3.35230i		−	0.00401473i
$836$		0			0
$837$	−77.7432			−0.0928831
$838$		0			0
$839$		−	351.194i		−	0.418586i	−0.977853	−	0.209293i	$-0.932884\pi$
$839$		−	351.194i		−	0.418586i	0.977853	−	0.209293i	$-0.0671163\pi$
$840$		0			0
$841$	824.412			0.980276
$842$		0			0
$843$	258.203			0.306290
$844$		0			0
$845$	2.15848			0.00255441
$846$		0			0
$847$	−1538.34			−1.81623
$848$		0			0
$849$			45.8316i			0.0539830i
$850$		0			0
$851$		−	773.336i		−	0.908737i
$852$		0			0
$853$	−341.736			−0.400628			−0.200314	−	0.979732i	$-0.564196\pi$
$853$	−341.736			−0.400628			−0.200314	+	0.979732i	$0.564196\pi$
$854$		0			0
$855$	−0.501542	−	0.451491i	−0.000586599	−	0.000528059i
$856$		0			0
$857$		−	738.972i		−	0.862278i	−0.902286	−	0.431139i	$-0.858112\pi$
$857$		−	738.972i		−	0.862278i	0.902286	−	0.431139i	$-0.141888\pi$
$858$		0			0
$859$	375.173			0.436756			0.218378	−	0.975864i	$-0.429923\pi$
$859$	375.173			0.436756			0.218378	+	0.975864i	$0.429923\pi$
$860$		0			0
$861$	−436.711			−0.507214
$862$		0			0
$863$			195.029i			0.225989i	0.993596	+	0.112995i	$0.0360443\pi$
$863$			195.029i			0.225989i	−0.993596	+	0.112995i	$0.963956\pi$
$864$		0			0
$865$		−	1.15039i		−	0.00132994i
$866$		0			0
$867$			488.734i			0.563708i
$868$		0			0
$869$			1615.80i			1.85938i
$870$		0			0
$871$	1215.19			1.39517
$872$		0			0
$873$		−	275.488i		−	0.315565i
$874$		0			0
$875$	−3.18549			−0.00364057
$876$		0			0
$877$			1744.13i			1.98874i	0.105952	+	0.994371i	$0.466211\pi$
$877$			1744.13i			1.98874i	−0.105952	+	0.994371i	$0.533789\pi$
$878$		0			0
$879$	932.738			1.06114
$880$		0			0
$881$	−387.876			−0.440267			−0.220134	−	0.975470i	$-0.570649\pi$
$881$	−387.876			−0.440267			−0.220134	+	0.975470i	$0.570649\pi$
$882$		0			0
$883$	−348.266			−0.394413			−0.197206	−	0.980362i	$-0.563187\pi$
$883$	−348.266			−0.394413			−0.197206	+	0.980362i	$0.563187\pi$
$884$		0			0
$885$	−1.00471			−0.00113526
$886$		0			0
$887$		−	1499.32i		−	1.69032i	−0.534510	−	0.845162i	$-0.679504\pi$
$887$		−	1499.32i		−	1.69032i	0.534510	−	0.845162i	$-0.320496\pi$
$888$		0			0
$889$			1018.03i			1.14514i
$890$		0			0
$891$	181.538			0.203746
$892$		0			0
$893$	−1148.39	−	1033.79i	−1.28599	−	1.15766i
$894$		0			0
$895$			3.66009i			0.00408949i
$896$		0			0
$897$	−400.083			−0.446024
$898$		0			0
$899$	−60.9359			−0.0677818
$900$		0			0
$901$		−	84.9685i		−	0.0943046i
$902$		0			0
$903$			677.204i			0.749949i
$904$		0			0
$905$			0.742650i			0.000820608i
$906$		0			0
$907$			1270.98i			1.40130i	0.713505	+	0.700650i	$0.247107\pi$
$907$			1270.98i			1.40130i	−0.713505	+	0.700650i	$0.752893\pi$
$908$		0			0
$909$	12.5419			0.0137974
$910$		0			0
$911$			1691.31i			1.85654i	0.371910	+	0.928269i	$0.378703\pi$
$911$			1691.31i			1.85654i	−0.371910	+	0.928269i	$0.621297\pi$
$912$		0			0
$913$	−1231.17			−1.34849
$914$		0			0
$915$			1.67520i			0.00183081i
$916$		0			0
$917$	310.854			0.338990
$918$		0			0
$919$	−803.862			−0.874713			−0.437357	−	0.899288i	$-0.644085\pi$
$919$	−803.862			−0.874713			−0.437357	+	0.899288i	$0.644085\pi$
$920$		0			0
$921$	493.216			0.535522
$922$		0			0
$923$	−350.660			−0.379913
$924$		0			0
$925$		−	1568.80i		−	1.69600i
$926$		0			0
$927$			147.127i			0.158713i
$928$		0			0
$929$	−818.827			−0.881407			−0.440703	−	0.897653i	$-0.645271\pi$
$929$	−818.827			−0.881407			−0.440703	+	0.897653i	$0.645271\pi$
$930$		0			0
$931$	283.002	+	254.760i	0.303976	+	0.273641i
$932$		0			0
$933$			222.702i			0.238694i
$934$		0			0
$935$	0.624052			0.000667436
$936$		0			0
$937$	−298.683			−0.318765			−0.159383	−	0.987217i	$-0.550950\pi$
$937$	−298.683			−0.318765			−0.159383	+	0.987217i	$0.550950\pi$
$938$		0			0
$939$		−	22.6287i		−	0.0240987i
$940$		0			0
$941$			416.099i			0.442188i	0.975253	+	0.221094i	$0.0709627\pi$
$941$			416.099i			0.442188i	−0.975253	+	0.221094i	$0.929037\pi$
$942$		0			0
$943$		−	577.405i		−	0.612307i
$944$		0			0
$945$			0.331047i			0.000350315i
$946$		0			0
$947$	700.253			0.739444			0.369722	−	0.929142i	$-0.379453\pi$
$947$	700.253			0.739444			0.369722	+	0.929142i	$0.379453\pi$
$948$		0			0
$949$			2202.47i			2.32083i
$950$		0			0
$951$	−413.473			−0.434777
$952$		0			0
$953$		−	264.222i		−	0.277253i	−0.990345	−	0.138627i	$-0.955731\pi$
$953$		−	264.222i		−	0.277253i	0.990345	−	0.138627i	$-0.0442688\pi$
$954$		0			0
$955$	2.82785			0.00296110
$956$		0			0
$957$	142.291			0.148685
$958$		0			0
$959$	−1023.90			−1.06767
$960$		0			0
$961$	737.148			0.767064
$962$		0			0
$963$		−	369.246i		−	0.383433i
$964$		0			0
$965$			0.383588i			0.000397500i
$966$		0			0
$967$	275.678			0.285086			0.142543	−	0.989789i	$-0.454472\pi$
$967$	275.678			0.285086			0.142543	+	0.989789i	$0.454472\pi$
$968$		0			0
$969$	57.5376	−	63.9161i	0.0593783	−	0.0659609i
$970$		0			0
$971$			1535.48i			1.58133i	0.612246	+	0.790667i	$0.290266\pi$
$971$			1535.48i			1.58133i	−0.612246	+	0.790667i	$0.709734\pi$
$972$		0			0
$973$	322.395			0.331342
$974$		0			0
$975$	−811.613			−0.832424
$976$		0			0
$977$		−	1030.08i		−	1.05432i	−0.849765	−	0.527162i	$-0.823256\pi$
$977$		−	1030.08i		−	1.05432i	0.849765	−	0.527162i	$-0.176744\pi$
$978$		0			0
$979$		−	2675.44i		−	2.73283i
$980$		0			0
$981$			122.287i			0.124656i
$982$		0			0
$983$			718.441i			0.730866i	0.930838	+	0.365433i	$0.119079\pi$
$983$			718.441i			0.730866i	−0.930838	+	0.365433i	$0.880921\pi$
$984$		0			0
$985$	2.20983			0.00224349
$986$		0			0
$987$			758.005i			0.767989i
$988$		0			0
$989$	−895.377			−0.905336
$990$		0			0
$991$		−	183.179i		−	0.184842i	−0.995720	−	0.0924211i	$-0.970539\pi$
$991$		−	183.179i		−	0.184842i	0.995720	−	0.0924211i	$-0.0294606\pi$
$992$		0			0
$993$	−837.356			−0.843258
$994$		0			0
$995$	2.58166			0.00259464
$996$		0			0
$997$	1980.18			1.98613			0.993067	−	0.117549i	$-0.0375037\pi$
$997$	1980.18			1.98613			0.993067	+	0.117549i	$0.0375037\pi$
$998$		0			0
$999$	−326.070			−0.326396

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	912.3.o.e.721.6		20
3.2	odd	2		2736.3.o.r.721.10		20
4.3	odd	2		456.3.o.a.265.16	yes	20
12.11	even	2		1368.3.o.c.721.10		20
19.18	odd	2	inner	912.3.o.e.721.16		20
57.56	even	2		2736.3.o.r.721.9		20
76.75	even	2		456.3.o.a.265.6	✓	20
228.227	odd	2		1368.3.o.c.721.9		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.3.o.a.265.6	✓	20	76.75	even	2
456.3.o.a.265.16	yes	20	4.3	odd	2
912.3.o.e.721.6		20	1.1	even	1	trivial
912.3.o.e.721.16		20	19.18	odd	2	inner
1368.3.o.c.721.9		20	228.227	odd	2
1368.3.o.c.721.10		20	12.11	even	2
2736.3.o.r.721.9		20	57.56	even	2
2736.3.o.r.721.10		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	912.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -0.0118390 q^{5} -5.38136 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -0.0118390 q^{5} -5.38136 q^{7} -3.00000 q^{9} +20.1709 q^{11} +18.7435i q^{13} +0.0205058i q^{15} -2.61324 q^{17} +(-14.1212 - 12.7119i) q^{19} +9.32078i q^{21} -12.3236 q^{23} -24.9999 q^{25} +5.19615i q^{27} +4.07280i q^{29} +14.9617i q^{31} -34.9370i q^{33} +0.0637101 q^{35} +62.7522i q^{37} +32.4647 q^{39} +46.8535i q^{41} +72.6553 q^{43} +0.0355171 q^{45} +81.3241 q^{47} -20.0410 q^{49} +4.52627i q^{51} +32.5146i q^{53} -0.238804 q^{55} +(-22.0177 + 24.4586i) q^{57} +48.9963i q^{59} +81.6937 q^{61} +16.1441 q^{63} -0.221905i q^{65} -64.8329i q^{67} +21.3452i q^{69} +18.7083i q^{71} +117.506 q^{73} +43.3010i q^{75} -108.547 q^{77} +80.1054i q^{79} +9.00000 q^{81} -61.0368 q^{83} +0.0309383 q^{85} +7.05429 q^{87} -132.639i q^{89} -100.865i q^{91} +25.9144 q^{93} +(0.167181 + 0.150497i) q^{95} +91.8294i q^{97} -60.5127 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 16 q^{7} - 60 q^{9}+O(q^{10}) \) 20 * q + 16 * q^7 - 60 * q^9	\( 20 q + 16 q^{7} - 60 q^{9} - 16 q^{11} + 32 q^{17} - 40 q^{19} - 64 q^{23} + 68 q^{25} + 208 q^{35} - 48 q^{39} - 64 q^{43} - 48 q^{47} + 20 q^{49} + 336 q^{55} - 60 q^{57} + 184 q^{61} - 48 q^{63} + 104 q^{73} + 88 q^{77} + 180 q^{81} - 224 q^{83} - 136 q^{85} + 240 q^{87} + 120 q^{93} + 320 q^{95} + 48 q^{99}+O(q^{100}) \) 20 * q + 16 * q^7 - 60 * q^9 - 16 * q^11 + 32 * q^17 - 40 * q^19 - 64 * q^23 + 68 * q^25 + 208 * q^35 - 48 * q^39 - 64 * q^43 - 48 * q^47 + 20 * q^49 + 336 * q^55 - 60 * q^57 + 184 * q^61 - 48 * q^63 + 104 * q^73 + 88 * q^77 + 180 * q^81 - 224 * q^83 - 136 * q^85 + 240 * q^87 + 120 * q^93 + 320 * q^95 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more