LMFDB - Embedded newform 136.4.b.a.33.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(33,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("136.33");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.02425976078$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 81x^{4} + 222x^{2} + 64 $ x^6 + 81x^4 + 222x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.6
Root			$-1.58186i$ of defining polynomial
Character	$\chi$	$=$	136.33
Dual form			136.4.b.a.33.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.27144i q^{3} -6.42821i q^{5} +24.9151i q^{7} -41.4167 q^{9} +O(q^{10})$	$q+8.27144i q^{3} -6.42821i q^{5} +24.9151i q^{7} -41.4167 q^{9} +18.4869i q^{11} +10.7538 q^{13} +53.1705 q^{15} +(-67.0797 - 20.3301i) q^{17} -130.004 q^{19} -206.084 q^{21} -90.2427i q^{23} +83.6781 q^{25} -119.247i q^{27} -29.6470i q^{29} +132.901i q^{31} -152.913 q^{33} +160.159 q^{35} +412.040i q^{37} +88.9494i q^{39} +165.227i q^{41} +502.172 q^{43} +266.235i q^{45} -160.186 q^{47} -277.762 q^{49} +(168.159 - 554.846i) q^{51} +372.163 q^{53} +118.838 q^{55} -1075.32i q^{57} +453.800 q^{59} +254.958i q^{61} -1031.90i q^{63} -69.1276i q^{65} -383.360 q^{67} +746.437 q^{69} +818.663i q^{71} -640.109i q^{73} +692.139i q^{75} -460.602 q^{77} +380.550i q^{79} -131.905 q^{81} +984.831 q^{83} +(-130.686 + 431.202i) q^{85} +245.223 q^{87} +417.573 q^{89} +267.932i q^{91} -1099.29 q^{93} +835.693i q^{95} -1327.74i q^{97} -765.667i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 42 q^{9}+O(q^{10}) $ 6 * q - 42 * q^9	$ 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) $ 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$			8.27144i			1.59184i	0.605402	+	0.795920i	$0.293012\pi$
$3$			8.27144i			1.59184i	−0.605402	+	0.795920i	$0.706988\pi$
$4$		0			0
$5$		−	6.42821i		−	0.574956i	−0.957787	−	0.287478i	$-0.907183\pi$
$5$		−	6.42821i		−	0.574956i	0.957787	−	0.287478i	$-0.0928169\pi$
$6$		0			0
$7$			24.9151i			1.34529i	0.739966	+	0.672644i	$0.234842\pi$
$7$			24.9151i			1.34529i	−0.739966	+	0.672644i	$0.765158\pi$
$8$		0			0
$9$	−41.4167			−1.53395
$10$		0			0
$11$			18.4869i			0.506728i	0.967371	+	0.253364i	$0.0815370\pi$
$11$			18.4869i			0.506728i	−0.967371	+	0.253364i	$0.918463\pi$
$12$		0			0
$13$	10.7538			0.229428			0.114714	−	0.993399i	$-0.463405\pi$
$13$	10.7538			0.229428			0.114714	+	0.993399i	$0.463405\pi$
$14$		0			0
$15$	53.1705			0.915238
$16$		0			0
$17$	−67.0797	−	20.3301i	−0.957013	−	0.290046i
$17$	−67.0797	−	20.3301i	−0.957013	−	0.290046i
$18$		0			0
$19$	−130.004			−1.56974			−0.784868	−	0.619663i	$-0.787269\pi$
$19$	−130.004			−1.56974			−0.784868	+	0.619663i	$0.787269\pi$
$20$		0			0
$21$	−206.084			−2.14148
$22$		0			0
$23$		−	90.2427i		−	0.818126i	−0.912506	−	0.409063i	$-0.865856\pi$
$23$		−	90.2427i		−	0.818126i	0.912506	−	0.409063i	$-0.134144\pi$
$24$		0			0
$25$	83.6781			0.669425
$26$		0			0
$27$		−	119.247i		−	0.849969i
$28$		0			0
$29$		−	29.6470i		−	0.189838i	−0.995485	−	0.0949191i	$-0.969741\pi$
$29$		−	29.6470i		−	0.189838i	0.995485	−	0.0949191i	$-0.0302592\pi$
$30$		0			0
$31$			132.901i			0.769994i	0.922918	+	0.384997i	$0.125797\pi$
$31$			132.901i			0.769994i	−0.922918	+	0.384997i	$0.874203\pi$
$32$		0			0
$33$	−152.913			−0.806629
$34$		0			0
$35$	160.159			0.773482
$36$		0			0
$37$			412.040i			1.83078i	0.402567	+	0.915390i	$0.368118\pi$
$37$			412.040i			1.83078i	−0.402567	+	0.915390i	$0.631882\pi$
$38$		0			0
$39$			88.9494i			0.365213i
$40$		0			0
$41$			165.227i			0.629370i	0.949196	+	0.314685i	$0.101899\pi$
$41$			165.227i			0.629370i	−0.949196	+	0.314685i	$0.898101\pi$
$42$		0			0
$43$	502.172			1.78094			0.890470	−	0.455041i	$-0.150376\pi$
$43$	502.172			1.78094			0.890470	+	0.455041i	$0.150376\pi$
$44$		0			0
$45$			266.235i			0.881956i
$46$		0			0
$47$	−160.186			−0.497138			−0.248569	−	0.968614i	$-0.579960\pi$
$47$	−160.186			−0.497138			−0.248569	+	0.968614i	$0.579960\pi$
$48$		0			0
$49$	−277.762			−0.809801
$50$		0			0
$51$	168.159	−	554.846i	0.461706	−	1.52341i
$52$		0			0
$53$	372.163			0.964539			0.482270	−	0.876023i	$-0.339813\pi$
$53$	372.163			0.964539			0.482270	+	0.876023i	$0.339813\pi$
$54$		0			0
$55$	118.838			0.291346
$56$		0			0
$57$		−	1075.32i		−	2.49877i
$58$		0			0
$59$	453.800			1.00135			0.500676	−	0.865635i	$-0.333085\pi$
$59$	453.800			1.00135			0.500676	+	0.865635i	$0.333085\pi$
$60$		0			0
$61$			254.958i			0.535147i	0.963537	+	0.267574i	$0.0862219\pi$
$61$			254.958i			0.535147i	−0.963537	+	0.267574i	$0.913778\pi$
$62$		0			0
$63$		−	1031.90i		−	2.06361i
$64$		0			0
$65$		−	69.1276i		−	0.131911i
$66$		0			0
$67$	−383.360			−0.699029			−0.349514	−	0.936931i	$-0.613653\pi$
$67$	−383.360			−0.699029			−0.349514	+	0.936931i	$0.613653\pi$
$68$		0			0
$69$	746.437			1.30233
$70$		0			0
$71$			818.663i			1.36841i	0.729288	+	0.684207i	$0.239852\pi$
$71$			818.663i			1.36841i	−0.729288	+	0.684207i	$0.760148\pi$
$72$		0			0
$73$		−	640.109i		−	1.02629i	−0.858302	−	0.513144i	$-0.828481\pi$
$73$		−	640.109i		−	1.02629i	0.858302	−	0.513144i	$-0.171519\pi$
$74$		0			0
$75$			692.139i			1.06562i
$76$		0			0
$77$	−460.602			−0.681695
$78$		0			0
$79$			380.550i			0.541964i	0.962584	+	0.270982i	$0.0873484\pi$
$79$			380.550i			0.541964i	−0.962584	+	0.270982i	$0.912652\pi$
$80$		0			0
$81$	−131.905			−0.180940
$82$		0			0
$83$	984.831			1.30240			0.651201	−	0.758906i	$-0.274266\pi$
$83$	984.831			1.30240			0.651201	+	0.758906i	$0.274266\pi$
$84$		0			0
$85$	−130.686	+	431.202i	−0.166764	+	0.550241i
$86$		0			0
$87$	245.223			0.302192
$88$		0			0
$89$	417.573			0.497333			0.248667	−	0.968589i	$-0.420008\pi$
$89$	417.573			0.497333			0.248667	+	0.968589i	$0.420008\pi$
$90$		0			0
$91$			267.932i			0.308647i
$92$		0			0
$93$	−1099.29			−1.22571
$94$		0			0
$95$			835.693i			0.902529i
$96$		0			0
$97$		−	1327.74i		−	1.38981i	−0.719102	−	0.694904i	$-0.755447\pi$
$97$		−	1327.74i		−	1.38981i	0.719102	−	0.694904i	$-0.244553\pi$
$98$		0			0
$99$		−	765.667i		−	0.777297i
$100$		0			0
$101$	1615.11			1.59119			0.795593	−	0.605832i	$-0.207160\pi$
$101$	1615.11			1.59119			0.795593	+	0.605832i	$0.207160\pi$
$102$		0			0
$103$	−1785.31			−1.70788			−0.853940	−	0.520372i	$-0.825793\pi$
$103$	−1785.31			−1.70788			−0.853940	+	0.520372i	$0.825793\pi$
$104$		0			0
$105$			1324.75i			1.23126i
$106$		0			0
$107$			1387.87i			1.25393i	0.779047	+	0.626966i	$0.215703\pi$
$107$			1387.87i			1.25393i	−0.779047	+	0.626966i	$0.784297\pi$
$108$		0			0
$109$			865.306i			0.760379i	0.924909	+	0.380190i	$0.124141\pi$
$109$			865.306i			0.760379i	−0.924909	+	0.380190i	$0.875859\pi$
$110$		0			0
$111$	−3408.16			−2.91431
$112$		0			0
$113$			282.480i			0.235164i	0.993063	+	0.117582i	$0.0375142\pi$
$113$			282.480i			0.235164i	−0.993063	+	0.117582i	$0.962486\pi$
$114$		0			0
$115$	−580.099			−0.470387
$116$		0			0
$117$	−445.387			−0.351932
$118$		0			0
$119$	506.527	−	1671.30i	0.390195	−	1.28746i
$120$		0			0
$121$	989.235			0.743227
$122$		0			0
$123$	−1366.67			−1.00186
$124$		0			0
$125$		−	1341.43i		−	0.959847i
$126$		0			0
$127$	1101.83			0.769857			0.384929	−	0.922946i	$-0.374226\pi$
$127$	1101.83			0.769857			0.384929	+	0.922946i	$0.374226\pi$
$128$		0			0
$129$			4153.68i			2.83497i
$130$		0			0
$131$		−	2718.83i		−	1.81332i	−0.421861	−	0.906661i	$-0.638623\pi$
$131$		−	2718.83i		−	1.81332i	0.421861	−	0.906661i	$-0.361377\pi$
$132$		0			0
$133$		−	3239.06i		−	2.11175i
$134$		0			0
$135$	−766.546			−0.488695
$136$		0			0
$137$	−1446.52			−0.902076			−0.451038	−	0.892505i	$-0.648946\pi$
$137$	−1446.52			−0.902076			−0.451038	+	0.892505i	$0.648946\pi$
$138$		0			0
$139$			1978.07i			1.20703i	0.797350	+	0.603517i	$0.206235\pi$
$139$			1978.07i			1.20703i	−0.797350	+	0.603517i	$0.793765\pi$
$140$		0			0
$141$		−	1324.97i		−	0.791364i
$142$		0			0
$143$			198.804i			0.116258i
$144$		0			0
$145$	−190.577			−0.109149
$146$		0			0
$147$		−	2297.49i		−	1.28907i
$148$		0			0
$149$	−804.749			−0.442467			−0.221234	−	0.975221i	$-0.571008\pi$
$149$	−804.749			−0.442467			−0.221234	+	0.975221i	$0.571008\pi$
$150$		0			0
$151$	−3253.35			−1.75334			−0.876668	−	0.481096i	$-0.840239\pi$
$151$	−3253.35			−1.75334			−0.876668	+	0.481096i	$0.840239\pi$
$152$		0			0
$153$	2778.22	+	842.007i	1.46801	+	0.444917i
$154$		0			0
$155$	854.318			0.442713
$156$		0			0
$157$	−290.488			−0.147666			−0.0738328	−	0.997271i	$-0.523523\pi$
$157$	−290.488			−0.147666			−0.0738328	+	0.997271i	$0.523523\pi$
$158$		0			0
$159$			3078.33i			1.53539i
$160$		0			0
$161$	2248.40			1.10062
$162$		0			0
$163$			1346.81i			0.647181i	0.946197	+	0.323591i	$0.104890\pi$
$163$			1346.81i			0.647181i	−0.946197	+	0.323591i	$0.895110\pi$
$164$		0			0
$165$			982.958i			0.463777i
$166$		0			0
$167$		−	1082.94i		−	0.501798i	−0.968013	−	0.250899i	$-0.919274\pi$
$167$		−	1082.94i		−	0.501798i	0.968013	−	0.250899i	$-0.0807261\pi$
$168$		0			0
$169$	−2081.36			−0.947363
$170$		0			0
$171$	5384.34			2.40790
$172$		0			0
$173$		−	3355.97i		−	1.47485i	−0.675428	−	0.737426i	$-0.736041\pi$
$173$		−	3355.97i		−	1.47485i	0.675428	−	0.737426i	$-0.263959\pi$
$174$		0			0
$175$			2084.85i			0.900570i
$176$		0			0
$177$			3753.58i			1.59399i
$178$		0			0
$179$	3368.20			1.40643			0.703216	−	0.710976i	$-0.251747\pi$
$179$	3368.20			1.40643			0.703216	+	0.710976i	$0.251747\pi$
$180$		0			0
$181$			1744.07i			0.716219i	0.933680	+	0.358109i	$0.116578\pi$
$181$			1744.07i			0.716219i	−0.933680	+	0.358109i	$0.883422\pi$
$182$		0			0
$183$	−2108.87			−0.851869
$184$		0			0
$185$	2648.68			1.05262
$186$		0			0
$187$	375.841	−	1240.09i	0.146974	−	0.484945i
$188$		0			0
$189$	2971.06			1.14345
$190$		0			0
$191$	311.843			0.118137			0.0590684	−	0.998254i	$-0.481187\pi$
$191$	311.843			0.118137			0.0590684	+	0.998254i	$0.481187\pi$
$192$		0			0
$193$		−	2737.52i		−	1.02099i	−0.859881	−	0.510495i	$-0.829462\pi$
$193$		−	2737.52i		−	1.02099i	0.859881	−	0.510495i	$-0.170538\pi$
$194$		0			0
$195$	571.785			0.209982
$196$		0			0
$197$		−	916.615i		−	0.331503i	−0.986168	−	0.165752i	$-0.946995\pi$
$197$		−	916.615i		−	0.331503i	0.986168	−	0.165752i	$-0.0530050\pi$
$198$		0			0
$199$			1594.70i			0.568065i	0.958815	+	0.284033i	$0.0916724\pi$
$199$			1594.70i			0.568065i	−0.958815	+	0.284033i	$0.908328\pi$
$200$		0			0
$201$		−	3170.94i		−	1.11274i
$202$		0			0
$203$	738.657			0.255387
$204$		0			0
$205$	1062.12			0.361860
$206$		0			0
$207$			3737.56i			1.25497i
$208$		0			0
$209$		−	2403.37i		−	0.795429i
$210$		0			0
$211$		−	5223.03i		−	1.70412i	−0.523447	−	0.852058i	$-0.675354\pi$
$211$		−	5223.03i		−	1.70412i	0.523447	−	0.852058i	$-0.324646\pi$
$212$		0			0
$213$	−6771.52			−2.17830
$214$		0			0
$215$		−	3228.06i		−	1.02396i
$216$		0			0
$217$	−3311.25			−1.03586
$218$		0			0
$219$	5294.62			1.63369
$220$		0			0
$221$	−721.362	−	218.626i	−0.219566	−	0.0665447i
$222$		0			0
$223$	2116.60			0.635597			0.317799	−	0.948158i	$-0.397056\pi$
$223$	2116.60			0.635597			0.317799	+	0.948158i	$0.397056\pi$
$224$		0			0
$225$	−3465.68			−1.02687
$226$		0			0
$227$			3418.70i			0.999592i	0.866143	+	0.499796i	$0.166592\pi$
$227$			3418.70i			0.999592i	−0.866143	+	0.499796i	$0.833408\pi$
$228$		0			0
$229$	4713.81			1.36025			0.680125	−	0.733097i	$-0.261926\pi$
$229$	4713.81			1.36025			0.680125	+	0.733097i	$0.261926\pi$
$230$		0			0
$231$		−	3809.85i		−	1.08515i
$232$		0			0
$233$		−	348.123i		−	0.0978811i	−0.998802	−	0.0489405i	$-0.984416\pi$
$233$		−	348.123i		−	0.0978811i	0.998802	−	0.0489405i	$-0.0155845\pi$
$234$		0			0
$235$			1029.71i			0.285833i
$236$		0			0
$237$	−3147.69			−0.862720
$238$		0			0
$239$	6724.01			1.81983			0.909917	−	0.414791i	$-0.136145\pi$
$239$	6724.01			1.81983			0.909917	+	0.414791i	$0.136145\pi$
$240$		0			0
$241$		−	2305.58i		−	0.616247i	−0.951346	−	0.308124i	$-0.900299\pi$
$241$		−	2305.58i		−	0.616247i	0.951346	−	0.308124i	$-0.0997011\pi$
$242$		0			0
$243$		−	4310.72i		−	1.13800i
$244$		0			0
$245$			1785.51i			0.465601i
$246$		0			0
$247$	−1398.04			−0.360142
$248$		0			0
$249$			8145.98i			2.07321i
$250$		0			0
$251$	−2237.70			−0.562718			−0.281359	−	0.959603i	$-0.590785\pi$
$251$	−2237.70			−0.562718			−0.281359	+	0.959603i	$0.590785\pi$
$252$		0			0
$253$	1668.31			0.414567
$254$		0			0
$255$	−3566.66	−	1080.96i	−0.875895	−	0.265461i
$256$		0			0
$257$	−33.2292			−0.00806529			−0.00403264	−	0.999992i	$-0.501284\pi$
$257$	−33.2292			−0.00806529			−0.00403264	+	0.999992i	$0.501284\pi$
$258$		0			0
$259$	−10266.0			−2.46293
$260$		0			0
$261$			1227.88i			0.291203i
$262$		0			0
$263$	4735.59			1.11030			0.555150	−	0.831750i	$-0.312661\pi$
$263$	4735.59			1.11030			0.555150	+	0.831750i	$0.312661\pi$
$264$		0			0
$265$		−	2392.34i		−	0.554568i
$266$		0			0
$267$			3453.93i			0.791675i
$268$		0			0
$269$			8592.97i			1.94767i	0.227259	+	0.973834i	$0.427024\pi$
$269$			8592.97i			1.94767i	−0.227259	+	0.973834i	$0.572976\pi$
$270$		0			0
$271$	−4108.02			−0.920827			−0.460414	−	0.887705i	$-0.652299\pi$
$271$	−4108.02			−0.920827			−0.460414	+	0.887705i	$0.652299\pi$
$272$		0			0
$273$	−2216.18			−0.491317
$274$		0			0
$275$			1546.95i			0.339216i
$276$		0			0
$277$			6567.79i			1.42462i	0.701864	+	0.712311i	$0.252351\pi$
$277$			6567.79i			1.42462i	−0.701864	+	0.712311i	$0.747649\pi$
$278$		0			0
$279$		−	5504.35i		−	1.18113i
$280$		0			0
$281$	−5596.04			−1.18801			−0.594007	−	0.804460i	$-0.702455\pi$
$281$	−5596.04			−1.18801			−0.594007	+	0.804460i	$0.702455\pi$
$282$		0			0
$283$		−	693.528i		−	0.145675i	−0.997344	−	0.0728374i	$-0.976795\pi$
$283$		−	693.528i		−	0.145675i	0.997344	−	0.0728374i	$-0.0232054\pi$
$284$		0			0
$285$	−6912.39			−1.43668
$286$		0			0
$287$	−4116.65			−0.846684
$288$		0			0
$289$	4086.37	+	2727.48i	0.831747	+	0.555155i
$290$		0			0
$291$	10982.3			2.21235
$292$		0			0
$293$	1984.66			0.395716			0.197858	−	0.980231i	$-0.436601\pi$
$293$	1984.66			0.395716			0.197858	+	0.980231i	$0.436601\pi$
$294$		0			0
$295$		−	2917.12i		−	0.575733i
$296$		0			0
$297$	2204.51			0.430703
$298$		0			0
$299$		−	970.451i		−	0.187701i
$300$		0			0
$301$			12511.7i			2.39588i
$302$		0			0
$303$			13359.3i			2.53291i
$304$		0			0
$305$	1638.92			0.307686
$306$		0			0
$307$	−3966.28			−0.737353			−0.368677	−	0.929558i	$-0.620189\pi$
$307$	−3966.28			−0.737353			−0.368677	+	0.929558i	$0.620189\pi$
$308$		0			0
$309$		−	14767.1i		−	2.71867i
$310$		0			0
$311$		−	4101.94i		−	0.747909i	−0.927447	−	0.373954i	$-0.878002\pi$
$311$		−	4101.94i		−	0.747909i	0.927447	−	0.373954i	$-0.121998\pi$
$312$		0			0
$313$		−	4152.29i		−	0.749845i	−0.927056	−	0.374922i	$-0.877669\pi$
$313$		−	4152.29i		−	0.749845i	0.927056	−	0.374922i	$-0.122331\pi$
$314$		0			0
$315$	−6633.28			−1.18649
$316$		0			0
$317$		−	154.363i		−	0.0273498i	−0.999906	−	0.0136749i	$-0.995647\pi$
$317$		−	154.363i		−	0.0273498i	0.999906	−	0.0136749i	$-0.00435299\pi$
$318$		0			0
$319$	548.080			0.0961962
$320$		0			0
$321$	−11479.7			−1.99606
$322$		0			0
$323$	8720.63	+	2643.00i	1.50226	+	0.455295i
$324$		0			0
$325$	899.858			0.153585
$326$		0			0
$327$	−7157.33			−1.21040
$328$		0			0
$329$		−	3991.04i		−	0.668794i
$330$		0			0
$331$	−2181.69			−0.362286			−0.181143	−	0.983457i	$-0.557980\pi$
$331$	−2181.69			−0.362286			−0.181143	+	0.983457i	$0.557980\pi$
$332$		0			0
$333$		−	17065.3i		−	2.80833i
$334$		0			0
$335$			2464.32i			0.401911i
$336$		0			0
$337$		−	5187.30i		−	0.838488i	−0.907874	−	0.419244i	$-0.862295\pi$
$337$		−	5187.30i		−	0.838488i	0.907874	−	0.419244i	$-0.137705\pi$
$338$		0			0
$339$	−2336.52			−0.374343
$340$		0			0
$341$	−2456.93			−0.390177
$342$		0			0
$343$			1625.41i			0.255872i
$344$		0			0
$345$		−	4798.25i		−	0.748780i
$346$		0			0
$347$		−	6191.79i		−	0.957903i	−0.877841	−	0.478952i	$-0.841017\pi$
$347$		−	6191.79i		−	0.957903i	0.877841	−	0.478952i	$-0.158983\pi$
$348$		0			0
$349$	−3085.80			−0.473293			−0.236646	−	0.971596i	$-0.576048\pi$
$349$	−3085.80			−0.473293			−0.236646	+	0.971596i	$0.576048\pi$
$350$		0			0
$351$		−	1282.36i		−	0.195007i
$352$		0			0
$353$	1222.09			0.184264			0.0921322	−	0.995747i	$-0.470632\pi$
$353$	1222.09			0.184264			0.0921322	+	0.995747i	$0.470632\pi$
$354$		0			0
$355$	5262.53			0.786778
$356$		0			0
$357$	13824.0	+	4189.71i	2.04943	+	0.621128i
$358$		0			0
$359$	−1420.57			−0.208844			−0.104422	−	0.994533i	$-0.533299\pi$
$359$	−1420.57			−0.208844			−0.104422	+	0.994533i	$0.533299\pi$
$360$		0			0
$361$	10042.1			1.46407
$362$		0			0
$363$			8182.40i			1.18310i
$364$		0			0
$365$	−4114.75			−0.590071
$366$		0			0
$367$		−	5050.42i		−	0.718337i	−0.933273	−	0.359168i	$-0.883060\pi$
$367$		−	5050.42i		−	0.718337i	0.933273	−	0.359168i	$-0.116940\pi$
$368$		0			0
$369$		−	6843.18i		−	0.965424i
$370$		0			0
$371$			9272.49i			1.29758i
$372$		0			0
$373$	10829.2			1.50326			0.751630	−	0.659585i	$-0.229268\pi$
$373$	10829.2			1.50326			0.751630	+	0.659585i	$0.229268\pi$
$374$		0			0
$375$	11095.5			1.52792
$376$		0			0
$377$		−	318.818i		−	0.0435542i
$378$		0			0
$379$			7763.78i			1.05224i	0.850411	+	0.526119i	$0.176353\pi$
$379$			7763.78i			1.05224i	−0.850411	+	0.526119i	$0.823647\pi$
$380$		0			0
$381$			9113.75i			1.22549i
$382$		0			0
$383$	−6636.56			−0.885411			−0.442706	−	0.896667i	$-0.645981\pi$
$383$	−6636.56			−0.885411			−0.442706	+	0.896667i	$0.645981\pi$
$384$		0			0
$385$			2960.85i			0.391945i
$386$		0			0
$387$	−20798.3			−2.73188
$388$		0			0
$389$	4500.32			0.586569			0.293284	−	0.956025i	$-0.405252\pi$
$389$	4500.32			0.586569			0.293284	+	0.956025i	$0.405252\pi$
$390$		0			0
$391$	−1834.64	+	6053.45i	−0.237294	+	0.782957i
$392$		0			0
$393$	22488.6			2.88652
$394$		0			0
$395$	2446.25			0.311606
$396$		0			0
$397$			1404.37i			0.177540i	0.996052	+	0.0887699i	$0.0282936\pi$
$397$			1404.37i			0.177540i	−0.996052	+	0.0887699i	$0.971706\pi$
$398$		0			0
$399$	26791.7			3.36156
$400$		0			0
$401$			7742.95i			0.964251i	0.876102	+	0.482125i	$0.160135\pi$
$401$			7742.95i			0.964251i	−0.876102	+	0.482125i	$0.839865\pi$
$402$		0			0
$403$			1429.20i			0.176658i
$404$		0			0
$405$			847.913i			0.104032i
$406$		0			0
$407$	−7617.33			−0.927707
$408$		0			0
$409$	6453.44			0.780201			0.390100	−	0.920772i	$-0.372440\pi$
$409$	6453.44			0.780201			0.390100	+	0.920772i	$0.372440\pi$
$410$		0			0
$411$		−	11964.8i		−	1.43596i
$412$		0			0
$413$			11306.5i			1.34711i
$414$		0			0
$415$		−	6330.70i		−	0.748824i
$416$		0			0
$417$	−16361.5			−1.92141
$418$		0			0
$419$		−	16893.6i		−	1.96970i	−0.173395	−	0.984852i	$-0.555474\pi$
$419$		−	16893.6i		−	1.96970i	0.173395	−	0.984852i	$-0.444526\pi$
$420$		0			0
$421$	3516.41			0.407077			0.203539	−	0.979067i	$-0.434756\pi$
$421$	3516.41			0.407077			0.203539	+	0.979067i	$0.434756\pi$
$422$		0			0
$423$	6634.37			0.762587
$424$		0			0
$425$	−5613.10	−	1701.19i	−0.640648	−	0.194164i
$426$		0			0
$427$	−6352.30			−0.719928
$428$		0			0
$429$	−1644.40			−0.185064
$430$		0			0
$431$			2497.94i			0.279168i	0.990210	+	0.139584i	$0.0445765\pi$
$431$			2497.94i			0.279168i	−0.990210	+	0.139584i	$0.955423\pi$
$432$		0			0
$433$	−8345.94			−0.926283			−0.463141	−	0.886284i	$-0.653278\pi$
$433$	−8345.94			−0.926283			−0.463141	+	0.886284i	$0.653278\pi$
$434$		0			0
$435$		−	1576.35i		−	0.173747i
$436$		0			0
$437$			11731.9i			1.28424i
$438$		0			0
$439$		−	4196.10i		−	0.456194i	−0.973638	−	0.228097i	$-0.926750\pi$
$439$		−	4196.10i		−	0.456194i	0.973638	−	0.228097i	$-0.0732503\pi$
$440$		0			0
$441$	11504.0			1.24220
$442$		0			0
$443$	−2830.64			−0.303584			−0.151792	−	0.988412i	$-0.548504\pi$
$443$	−2830.64			−0.303584			−0.151792	+	0.988412i	$0.548504\pi$
$444$		0			0
$445$		−	2684.25i		−	0.285945i
$446$		0			0
$447$		−	6656.44i		−	0.704337i
$448$		0			0
$449$		−	8540.06i		−	0.897617i	−0.893628	−	0.448809i	$-0.851848\pi$
$449$		−	8540.06i		−	0.897617i	0.893628	−	0.448809i	$-0.148152\pi$
$450$		0			0
$451$	−3054.54			−0.318919
$452$		0			0
$453$		−	26909.9i		−	2.79103i
$454$		0			0
$455$	1722.32			0.177459
$456$		0			0
$457$	13563.4			1.38834			0.694169	−	0.719812i	$-0.255772\pi$
$457$	13563.4			1.38834			0.694169	+	0.719812i	$0.255772\pi$
$458$		0			0
$459$	−2424.31	+	7999.07i	−0.246530	+	0.813431i
$460$		0			0
$461$	15908.0			1.60718			0.803590	−	0.595183i	$-0.202920\pi$
$461$	15908.0			1.60718			0.803590	+	0.595183i	$0.202920\pi$
$462$		0			0
$463$	−3390.90			−0.340364			−0.170182	−	0.985413i	$-0.554436\pi$
$463$	−3390.90			−0.340364			−0.170182	+	0.985413i	$0.554436\pi$
$464$		0			0
$465$			7066.44i			0.704728i
$466$		0			0
$467$	−2796.37			−0.277089			−0.138544	−	0.990356i	$-0.544242\pi$
$467$	−2796.37			−0.277089			−0.138544	+	0.990356i	$0.544242\pi$
$468$		0			0
$469$		−	9551.46i		−	0.940395i
$470$		0			0
$471$		−	2402.76i		−	0.235060i
$472$		0			0
$473$			9283.59i			0.902452i
$474$		0			0
$475$	−10878.5			−1.05082
$476$		0			0
$477$	−15413.8			−1.47956
$478$		0			0
$479$		−	13981.7i		−	1.33370i	−0.745193	−	0.666849i	$-0.767643\pi$
$479$		−	13981.7i		−	1.33370i	0.745193	−	0.666849i	$-0.232357\pi$
$480$		0			0
$481$			4430.99i			0.420033i
$482$		0			0
$483$			18597.5i			1.75200i
$484$		0			0
$485$	−8534.98			−0.799079
$486$		0			0
$487$			16670.8i			1.55119i	0.631234	+	0.775593i	$0.282549\pi$
$487$			16670.8i			1.55119i	−0.631234	+	0.775593i	$0.717451\pi$
$488$		0			0
$489$	−11140.1			−1.03021
$490$		0			0
$491$	−8790.42			−0.807955			−0.403978	−	0.914769i	$-0.632373\pi$
$491$	−8790.42			−0.807955			−0.403978	+	0.914769i	$0.632373\pi$
$492$		0			0
$493$	−602.727	+	1988.71i	−0.0550617	+	0.181677i
$494$		0			0
$495$	−4921.86			−0.446912
$496$		0			0
$497$	−20397.1			−1.84091
$498$		0			0
$499$		−	7040.53i		−	0.631618i	−0.948823	−	0.315809i	$-0.897724\pi$
$499$		−	7040.53i		−	0.631618i	0.948823	−	0.315809i	$-0.102276\pi$
$500$		0			0
$501$	8957.45			0.798781
$502$		0			0
$503$			13853.9i			1.22806i	0.789283	+	0.614030i	$0.210453\pi$
$503$			13853.9i			1.22806i	−0.789283	+	0.614030i	$0.789547\pi$
$504$		0			0
$505$		−	10382.3i		−	0.914862i
$506$		0			0
$507$		−	17215.8i		−	1.50805i
$508$		0			0
$509$	−19381.2			−1.68774			−0.843869	−	0.536550i	$-0.819727\pi$
$509$	−19381.2			−1.68774			−0.843869	+	0.536550i	$0.819727\pi$
$510$		0			0
$511$	15948.4			1.38065
$512$		0			0
$513$			15502.6i			1.33423i
$514$		0			0
$515$			11476.3i			0.981956i
$516$		0			0
$517$		−	2961.34i		−	0.251914i
$518$		0			0
$519$	27758.7			2.34773
$520$		0			0
$521$			1224.16i			0.102939i	0.998675	+	0.0514697i	$0.0163906\pi$
$521$			1224.16i			0.102939i	−0.998675	+	0.0514697i	$0.983609\pi$
$522$		0			0
$523$	1614.81			0.135011			0.0675053	−	0.997719i	$-0.478496\pi$
$523$	1614.81			0.135011			0.0675053	+	0.997719i	$0.478496\pi$
$524$		0			0
$525$	−17244.7			−1.43356
$526$		0			0
$527$	2701.90	−	8914.99i	0.223333	−	0.736894i
$528$		0			0
$529$	4023.26			0.330670
$530$		0			0
$531$	−18794.9			−1.53603
$532$		0			0
$533$			1776.82i			0.144395i
$534$		0			0
$535$	8921.53			0.720956
$536$		0			0
$537$			27859.9i			2.23881i
$538$		0			0
$539$		−	5134.95i		−	0.410349i
$540$		0			0
$541$		−	13187.7i		−	1.04803i	−0.851709	−	0.524015i	$-0.824433\pi$
$541$		−	13187.7i		−	1.04803i	0.851709	−	0.524015i	$-0.175567\pi$
$542$		0			0
$543$	−14426.0			−1.14011
$544$		0			0
$545$	5562.37			0.437185
$546$		0			0
$547$			2702.54i			0.211247i	0.994406	+	0.105624i	$0.0336839\pi$
$547$			2702.54i			0.211247i	−0.994406	+	0.105624i	$0.966316\pi$
$548$		0			0
$549$		−	10559.5i		−	0.820891i
$550$		0			0
$551$			3854.23i			0.297996i
$552$		0			0
$553$	−9481.43			−0.729098
$554$		0			0
$555$			21908.4i			1.67560i
$556$		0			0
$557$	−11067.8			−0.841933			−0.420967	−	0.907076i	$-0.638309\pi$
$557$	−11067.8			−0.841933			−0.420967	+	0.907076i	$0.638309\pi$
$558$		0			0
$559$	5400.25			0.408598
$560$		0			0
$561$	10257.4	+	3108.74i	0.771955	+	0.233959i
$562$		0			0
$563$	−5796.49			−0.433912			−0.216956	−	0.976181i	$-0.569613\pi$
$563$	−5796.49			−0.433912			−0.216956	+	0.976181i	$0.569613\pi$
$564$		0			0
$565$	1815.84			0.135209
$566$		0			0
$567$		−	3286.43i		−	0.243416i
$568$		0			0
$569$	−8077.56			−0.595130			−0.297565	−	0.954702i	$-0.596174\pi$
$569$	−8077.56			−0.595130			−0.297565	+	0.954702i	$0.596174\pi$
$570$		0			0
$571$			11105.7i			0.813939i	0.913442	+	0.406970i	$0.133415\pi$
$571$			11105.7i			0.813939i	−0.913442	+	0.406970i	$0.866585\pi$
$572$		0			0
$573$			2579.39i			0.188055i
$574$		0			0
$575$		−	7551.34i		−	0.547674i
$576$		0			0
$577$	4687.93			0.338234			0.169117	−	0.985596i	$-0.445908\pi$
$577$	4687.93			0.338234			0.169117	+	0.985596i	$0.445908\pi$
$578$		0			0
$579$	22643.2			1.62525
$580$		0			0
$581$			24537.2i			1.75211i
$582$		0			0
$583$			6880.14i			0.488759i
$584$		0			0
$585$			2863.04i			0.202346i
$586$		0			0
$587$	14644.0			1.02968			0.514840	−	0.857286i	$-0.327851\pi$
$587$	14644.0			1.02968			0.514840	+	0.857286i	$0.327851\pi$
$588$		0			0
$589$		−	17277.7i		−	1.20869i
$590$		0			0
$591$	7581.73			0.527700
$592$		0			0
$593$	−18225.5			−1.26211			−0.631056	−	0.775737i	$-0.717378\pi$
$593$	−18225.5			−1.26211			−0.631056	+	0.775737i	$0.717378\pi$
$594$		0			0
$595$	−10743.4	−	3256.06i	−0.740232	−	0.224345i
$596$		0			0
$597$	−13190.4			−0.904269
$598$		0			0
$599$	−18264.8			−1.24588			−0.622939	−	0.782271i	$-0.714061\pi$
$599$	−18264.8			−1.24588			−0.622939	+	0.782271i	$0.714061\pi$
$600$		0			0
$601$		−	1302.27i		−	0.0883874i	−0.999023	−	0.0441937i	$-0.985928\pi$
$601$		−	1302.27i		−	0.0883874i	0.999023	−	0.0441937i	$-0.0140719\pi$
$602$		0			0
$603$	15877.5			1.07228
$604$		0			0
$605$		−	6359.01i		−	0.427323i
$606$		0			0
$607$		−	16244.2i		−	1.08621i	−0.839664	−	0.543106i	$-0.817248\pi$
$607$		−	16244.2i		−	1.08621i	0.839664	−	0.543106i	$-0.182752\pi$
$608$		0			0
$609$			6109.76i			0.406535i
$610$		0			0
$611$	−1722.61			−0.114058
$612$		0			0
$613$	1690.19			0.111364			0.0556820	−	0.998449i	$-0.482267\pi$
$613$	1690.19			0.111364			0.0556820	+	0.998449i	$0.482267\pi$
$614$		0			0
$615$			8785.23i			0.576024i
$616$		0			0
$617$		−	23802.4i		−	1.55308i	−0.630068	−	0.776540i	$-0.716973\pi$
$617$		−	23802.4i		−	1.55308i	0.630068	−	0.776540i	$-0.283027\pi$
$618$		0			0
$619$		−	82.5464i		−	0.00535997i	−0.999996	−	0.00267998i	$-0.999147\pi$
$619$		−	82.5464i		−	0.00535997i	0.999996	−	0.00267998i	$-0.000853066\pi$
$620$		0			0
$621$	−10761.2			−0.695381
$622$		0			0
$623$			10403.9i			0.669057i
$624$		0			0
$625$	1836.80			0.117555
$626$		0			0
$627$	19879.3			1.26619
$628$		0			0
$629$	8376.81	−	27639.5i	0.531010	−	1.75208i
$630$		0			0
$631$	10391.9			0.655616			0.327808	−	0.944744i	$-0.393690\pi$
$631$	10391.9			0.655616			0.327808	+	0.944744i	$0.393690\pi$
$632$		0			0
$633$	43202.0			2.71268
$634$		0			0
$635$		−	7082.81i		−	0.442634i
$636$		0			0
$637$	−2987.00			−0.185791
$638$		0			0
$639$		−	33906.3i		−	2.09908i
$640$		0			0
$641$		−	24781.2i		−	1.52699i	−0.645816	−	0.763493i	$-0.723483\pi$
$641$		−	24781.2i		−	1.52699i	0.645816	−	0.763493i	$-0.276517\pi$
$642$		0			0
$643$		−	16855.6i		−	1.03378i	−0.856052	−	0.516890i	$-0.827090\pi$
$643$		−	16855.6i		−	1.03378i	0.856052	−	0.516890i	$-0.172910\pi$
$644$		0			0
$645$	26700.7			1.62999
$646$		0			0
$647$	1895.84			0.115198			0.0575991	−	0.998340i	$-0.481655\pi$
$647$	1895.84			0.115198			0.0575991	+	0.998340i	$0.481655\pi$
$648$		0			0
$649$			8389.35i			0.507413i
$650$		0			0
$651$		−	27388.8i		−	1.64893i
$652$		0			0
$653$		−	5103.25i		−	0.305828i	−0.988239	−	0.152914i	$-0.951134\pi$
$653$		−	5103.25i		−	0.305828i	0.988239	−	0.152914i	$-0.0488658\pi$
$654$		0			0
$655$	−17477.2			−1.04258
$656$		0			0
$657$			26511.2i			1.57428i
$658$		0			0
$659$	−6060.64			−0.358253			−0.179127	−	0.983826i	$-0.557327\pi$
$659$	−6060.64			−0.358253			−0.179127	+	0.983826i	$0.557327\pi$
$660$		0			0
$661$	295.173			0.0173690			0.00868449	−	0.999962i	$-0.497236\pi$
$661$	295.173			0.0173690			0.00868449	+	0.999962i	$0.497236\pi$
$662$		0			0
$663$	1808.35	−	5966.70i	0.105928	−	0.349513i
$664$		0			0
$665$	−20821.4			−1.21416
$666$		0			0
$667$	−2675.42			−0.155311
$668$		0			0
$669$			17507.3i			1.01177i
$670$		0			0
$671$	−4713.37			−0.271174
$672$		0			0
$673$			25350.4i			1.45199i	0.687702	+	0.725993i	$0.258620\pi$
$673$			25350.4i			1.45199i	−0.687702	+	0.725993i	$0.741380\pi$
$674$		0			0
$675$		−	9978.39i		−	0.568990i
$676$		0			0
$677$			9954.70i			0.565126i	0.959249	+	0.282563i	$0.0911846\pi$
$677$			9954.70i			0.565126i	−0.959249	+	0.282563i	$0.908815\pi$
$678$		0			0
$679$	33080.7			1.86969
$680$		0			0
$681$	−28277.6			−1.59119
$682$		0			0
$683$		−	3129.30i		−	0.175314i	−0.996151	−	0.0876569i	$-0.972062\pi$
$683$		−	3129.30i		−	0.175314i	0.996151	−	0.0876569i	$-0.0279379\pi$
$684$		0			0
$685$			9298.52i			0.518654i
$686$		0			0
$687$			38990.0i			2.16530i
$688$		0			0
$689$	4002.17			0.221293
$690$		0			0
$691$			23129.7i			1.27337i	0.771126	+	0.636683i	$0.219694\pi$
$691$			23129.7i			1.27337i	−0.771126	+	0.636683i	$0.780306\pi$
$692$		0			0
$693$	19076.7			1.04569
$694$		0			0
$695$	12715.5			0.693992
$696$		0			0
$697$	3359.09	−	11083.4i	0.182546	−	0.602315i
$698$		0			0
$699$	2879.48			0.155811
$700$		0			0
$701$	36272.1			1.95432			0.977160	−	0.212506i	$-0.0681625\pi$
$701$	36272.1			1.95432			0.977160	+	0.212506i	$0.0681625\pi$
$702$		0			0
$703$		−	53566.8i		−	2.87384i
$704$		0			0
$705$	−8517.16			−0.455000
$706$		0			0
$707$			40240.7i			2.14060i
$708$		0			0
$709$			27866.8i			1.47610i	0.674743	+	0.738052i	$0.264254\pi$
$709$			27866.8i			1.47610i	−0.674743	+	0.738052i	$0.735746\pi$
$710$		0			0
$711$		−	15761.1i		−	0.831348i
$712$		0			0
$713$	11993.4			0.629952
$714$		0			0
$715$	1277.95			0.0668431
$716$		0			0
$717$			55617.3i			2.89688i
$718$		0			0
$719$			8631.42i			0.447702i	0.974623	+	0.223851i	$0.0718629\pi$
$719$			8631.42i			0.447702i	−0.974623	+	0.223851i	$0.928137\pi$
$720$		0			0
$721$		−	44481.1i		−	2.29759i
$722$		0			0
$723$	19070.5			0.980967
$724$		0			0
$725$		−	2480.80i		−	0.127082i
$726$		0			0
$727$	−3101.76			−0.158237			−0.0791183	−	0.996865i	$-0.525210\pi$
$727$	−3101.76			−0.158237			−0.0791183	+	0.996865i	$0.525210\pi$
$728$		0			0
$729$	32094.5			1.63057
$730$		0			0
$731$	−33685.5	−	10209.2i	−1.70438	−	0.516554i
$732$		0			0
$733$	26358.4			1.32820			0.664101	−	0.747643i	$-0.268815\pi$
$733$	26358.4			1.32820			0.664101	+	0.747643i	$0.268815\pi$
$734$		0			0
$735$	−14768.8			−0.741161
$736$		0			0
$737$		−	7087.14i		−	0.354217i
$738$		0			0
$739$	−2119.43			−0.105500			−0.0527500	−	0.998608i	$-0.516799\pi$
$739$	−2119.43			−0.105500			−0.0527500	+	0.998608i	$0.516799\pi$
$740$		0			0
$741$		−	11563.8i		−	0.573288i
$742$		0			0
$743$			29090.4i			1.43637i	0.695852	+	0.718185i	$0.255027\pi$
$743$			29090.4i			1.43637i	−0.695852	+	0.718185i	$0.744973\pi$
$744$		0			0
$745$			5173.09i			0.254399i
$746$		0			0
$747$	−40788.5			−1.99782
$748$		0			0
$749$	−34579.0			−1.68690
$750$		0			0
$751$			7393.28i			0.359234i	0.983737	+	0.179617i	$0.0574858\pi$
$751$			7393.28i			0.359234i	−0.983737	+	0.179617i	$0.942514\pi$
$752$		0			0
$753$		−	18509.0i		−	0.895757i
$754$		0			0
$755$			20913.2i			1.00809i
$756$		0			0
$757$	3370.78			0.161840			0.0809201	−	0.996721i	$-0.474214\pi$
$757$	3370.78			0.161840			0.0809201	+	0.996721i	$0.474214\pi$
$758$		0			0
$759$			13799.3i			0.659924i
$760$		0			0
$761$	14659.3			0.698293			0.349146	−	0.937068i	$-0.386472\pi$
$761$	14659.3			0.698293			0.349146	+	0.937068i	$0.386472\pi$
$762$		0			0
$763$	−21559.2			−1.02293
$764$		0			0
$765$	5412.60	−	17859.0i	0.255808	−	0.844044i
$766$		0			0
$767$	4880.07			0.229738
$768$		0			0
$769$	2764.69			0.129646			0.0648228	−	0.997897i	$-0.479352\pi$
$769$	2764.69			0.129646			0.0648228	+	0.997897i	$0.479352\pi$
$770$		0			0
$771$		−	274.853i		−	0.0128386i
$772$		0			0
$773$	−17274.6			−0.803781			−0.401891	−	0.915688i	$-0.631647\pi$
$773$	−17274.6			−0.803781			−0.401891	+	0.915688i	$0.631647\pi$
$774$		0			0
$775$			11120.9i			0.515453i
$776$		0			0
$777$		−	84914.7i		−	3.92059i
$778$		0			0
$779$		−	21480.2i		−	0.987944i
$780$		0			0
$781$	−15134.5			−0.693413
$782$		0			0
$783$	−3535.32			−0.161356
$784$		0			0
$785$			1867.32i			0.0849013i
$786$		0			0
$787$			9612.72i			0.435396i	0.976016	+	0.217698i	$0.0698547\pi$
$787$			9612.72i			0.435396i	−0.976016	+	0.217698i	$0.930145\pi$
$788$		0			0
$789$			39170.2i			1.76742i
$790$		0			0
$791$	−7038.02			−0.316363
$792$		0			0
$793$			2741.76i			0.122778i
$794$		0			0
$795$	19788.1			0.882783
$796$		0			0
$797$	−11594.4			−0.515301			−0.257650	−	0.966238i	$-0.582948\pi$
$797$	−11594.4			−0.515301			−0.257650	+	0.966238i	$0.582948\pi$
$798$		0			0
$799$	10745.2	+	3256.60i	0.475768	+	0.144193i
$800$		0			0
$801$	−17294.5			−0.762886
$802$		0			0
$803$	11833.6			0.520049
$804$		0			0
$805$		−	14453.2i		−	0.632806i
$806$		0			0
$807$	−71076.3			−3.10038
$808$		0			0
$809$		−	9646.09i		−	0.419207i	−0.977786	−	0.209603i	$-0.932783\pi$
$809$		−	9646.09i		−	0.419207i	0.977786	−	0.209603i	$-0.0672173\pi$
$810$		0			0
$811$			18032.2i			0.780759i	0.920654	+	0.390379i	$0.127656\pi$
$811$			18032.2i			0.780759i	−0.920654	+	0.390379i	$0.872344\pi$
$812$		0			0
$813$		−	33979.2i		−	1.46581i
$814$		0			0
$815$	8657.59			0.372101
$816$		0			0
$817$	−65284.3			−2.79561
$818$		0			0
$819$		−	11096.9i		−	0.473450i
$820$		0			0
$821$			33425.9i			1.42092i	0.703740	+	0.710458i	$0.251512\pi$
$821$			33425.9i			1.42092i	−0.703740	+	0.710458i	$0.748488\pi$
$822$		0			0
$823$			19119.2i			0.809785i	0.914364	+	0.404893i	$0.132691\pi$
$823$			19119.2i			0.809785i	−0.914364	+	0.404893i	$0.867309\pi$
$824$		0			0
$825$	−12795.5			−0.539978
$826$		0			0
$827$		−	16467.1i		−	0.692403i	−0.938160	−	0.346201i	$-0.887471\pi$
$827$		−	16467.1i		−	0.692403i	0.938160	−	0.346201i	$-0.112529\pi$
$828$		0			0
$829$	17600.4			0.737380			0.368690	−	0.929552i	$-0.379806\pi$
$829$	17600.4			0.737380			0.368690	+	0.929552i	$0.379806\pi$
$830$		0			0
$831$	−54325.1			−2.26777
$832$		0			0
$833$	18632.2	+	5646.93i	0.774990	+	0.234879i
$834$		0			0
$835$	−6961.35			−0.288512
$836$		0			0
$837$	15848.1			0.654471
$838$		0			0
$839$			21102.5i			0.868343i	0.900830	+	0.434171i	$0.142959\pi$
$839$			21102.5i			0.868343i	−0.900830	+	0.434171i	$0.857041\pi$
$840$		0			0
$841$	23510.1			0.963961
$842$		0			0
$843$		−	46287.3i		−	1.89113i
$844$		0			0
$845$			13379.4i			0.544692i
$846$		0			0
$847$			24646.9i			0.999855i
$848$		0			0
$849$	5736.48			0.231891
$850$		0			0
$851$	37183.6			1.49781
$852$		0			0
$853$		−	13921.0i		−	0.558788i	−0.960177	−	0.279394i	$-0.909866\pi$
$853$		−	13921.0i		−	0.558788i	0.960177	−	0.279394i	$-0.0901335\pi$
$854$		0			0
$855$		−	34611.7i		−	1.38444i
$856$		0			0
$857$		−	48524.7i		−	1.93416i	−0.254476	−	0.967079i	$-0.581903\pi$
$857$		−	48524.7i		−	1.93416i	0.254476	−	0.967079i	$-0.418097\pi$
$858$		0			0
$859$	37057.5			1.47193			0.735964	−	0.677021i	$-0.236729\pi$
$859$	37057.5			1.47193			0.735964	+	0.677021i	$0.236729\pi$
$860$		0			0
$861$		−	34050.7i		−	1.34779i
$862$		0			0
$863$	−20999.5			−0.828310			−0.414155	−	0.910206i	$-0.635923\pi$
$863$	−20999.5			−0.828310			−0.414155	+	0.910206i	$0.635923\pi$
$864$		0			0
$865$	−21572.9			−0.847976
$866$		0			0
$867$	−22560.2	+	33800.2i	−0.883718	+	1.32401i
$868$		0			0
$869$	−7035.18			−0.274628
$870$		0			0
$871$	−4122.58			−0.160377
$872$		0			0
$873$			54990.6i			2.13190i
$874$		0			0
$875$	33421.8			1.29127
$876$		0			0
$877$			27765.1i			1.06905i	0.845151	+	0.534527i	$0.179510\pi$
$877$			27765.1i			1.06905i	−0.845151	+	0.534527i	$0.820490\pi$
$878$		0			0
$879$			16416.0i			0.629917i
$880$		0			0
$881$			45723.9i			1.74855i	0.485428	+	0.874277i	$0.338664\pi$
$881$			45723.9i			1.74855i	−0.485428	+	0.874277i	$0.661336\pi$
$882$		0			0
$883$	−22660.0			−0.863611			−0.431806	−	0.901967i	$-0.642123\pi$
$883$	−22660.0			−0.863611			−0.431806	+	0.901967i	$0.642123\pi$
$884$		0			0
$885$	24128.8			0.916475
$886$		0			0
$887$			6963.94i			0.263615i	0.991275	+	0.131807i	$0.0420780\pi$
$887$			6963.94i			0.263615i	−0.991275	+	0.131807i	$0.957922\pi$
$888$		0			0
$889$			27452.3i			1.03568i
$890$		0			0
$891$		−	2438.51i		−	0.0916872i
$892$		0			0
$893$	20824.8			0.780375
$894$		0			0
$895$		−	21651.5i		−	0.808637i
$896$		0			0
$897$	8027.03			0.298790
$898$		0			0
$899$	3940.13			0.146174
$900$		0			0
$901$	−24964.6	−	7566.13i	−0.923076	−	0.279761i
$902$		0			0
$903$	−103489.			−3.81386
$904$		0			0
$905$	11211.2			0.411795
$906$		0			0
$907$		−	20777.0i		−	0.760626i	−0.924858	−	0.380313i	$-0.875816\pi$
$907$		−	20777.0i		−	0.760626i	0.924858	−	0.380313i	$-0.124184\pi$
$908$		0			0
$909$	−66892.7			−2.44081
$910$		0			0
$911$			41231.6i			1.49952i	0.661710	+	0.749760i	$0.269831\pi$
$911$			41231.6i			1.49952i	−0.661710	+	0.749760i	$0.730169\pi$
$912$		0			0
$913$			18206.5i			0.659963i
$914$		0			0
$915$			13556.2i			0.489787i
$916$		0			0
$917$	67739.8			2.43944
$918$		0			0
$919$	−12152.3			−0.436200			−0.218100	−	0.975926i	$-0.569986\pi$
$919$	−12152.3			−0.436200			−0.218100	+	0.975926i	$0.569986\pi$
$920$		0			0
$921$		−	32806.8i		−	1.17375i
$922$		0			0
$923$			8803.73i			0.313953i
$924$		0			0
$925$			34478.7i			1.22557i
$926$		0			0
$927$	73941.6			2.61981
$928$		0			0
$929$			36648.3i			1.29428i	0.762369	+	0.647142i	$0.224036\pi$
$929$			36648.3i			1.29428i	−0.762369	+	0.647142i	$0.775964\pi$
$930$		0			0
$931$	36110.2			1.27117
$932$		0			0
$933$	33928.9			1.19055
$934$		0			0
$935$	−7971.59	−	2415.98i	−0.278822	−	0.0845038i
$936$		0			0
$937$	−3637.93			−0.126837			−0.0634183	−	0.997987i	$-0.520200\pi$
$937$	−3637.93			−0.126837			−0.0634183	+	0.997987i	$0.520200\pi$
$938$		0			0
$939$	34345.4			1.19363
$940$		0			0
$941$		−	760.820i		−	0.0263571i	−0.999913	−	0.0131786i	$-0.995805\pi$
$941$		−	760.820i		−	0.0263571i	0.999913	−	0.0131786i	$-0.00419499\pi$
$942$		0			0
$943$	14910.6			0.514904
$944$		0			0
$945$		−	19098.6i		−	0.657436i
$946$		0			0
$947$		−	29101.6i		−	0.998599i	−0.866429	−	0.499300i	$-0.833591\pi$
$947$		−	29101.6i		−	0.998599i	0.866429	−	0.499300i	$-0.166409\pi$
$948$		0			0
$949$		−	6883.60i		−	0.235460i
$950$		0			0
$951$	1276.80			0.0435365
$952$		0			0
$953$	50520.6			1.71723			0.858616	−	0.512620i	$-0.171325\pi$
$953$	50520.6			1.71723			0.858616	+	0.512620i	$0.171325\pi$
$954$		0			0
$955$		−	2004.59i		−	0.0679235i
$956$		0			0
$957$			4533.41i			0.153129i
$958$		0			0
$959$		−	36040.1i		−	1.21355i
$960$		0			0
$961$	12128.2			0.407110
$962$		0			0
$963$		−	57481.1i		−	1.92347i
$964$		0			0
$965$	−17597.3			−0.587025
$966$		0			0
$967$	53192.1			1.76892			0.884458	−	0.466621i	$-0.154529\pi$
$967$	53192.1			1.76892			0.884458	+	0.466621i	$0.154529\pi$
$968$		0			0
$969$	−21861.4	+	72132.2i	−0.724757	+	2.39135i
$970$		0			0
$971$	−16152.1			−0.533828			−0.266914	−	0.963720i	$-0.586004\pi$
$971$	−16152.1			−0.533828			−0.266914	+	0.963720i	$0.586004\pi$
$972$		0			0
$973$	−49283.8			−1.62381
$974$		0			0
$975$			7443.12i			0.244483i
$976$		0			0
$977$	−3758.89			−0.123088			−0.0615442	−	0.998104i	$-0.519603\pi$
$977$	−3758.89			−0.123088			−0.0615442	+	0.998104i	$0.519603\pi$
$978$		0			0
$979$			7719.63i			0.252013i
$980$		0			0
$981$		−	35838.2i		−	1.16639i
$982$		0			0
$983$		−	43642.6i		−	1.41606i	−0.706184	−	0.708029i	$-0.749585\pi$
$983$		−	43642.6i		−	1.41606i	0.706184	−	0.708029i	$-0.250415\pi$
$984$		0			0
$985$	−5892.19			−0.190600
$986$		0			0
$987$	33011.7			1.06461
$988$		0			0
$989$		−	45317.3i		−	1.45703i
$990$		0			0
$991$		−	2046.19i		−	0.0655896i	−0.999462	−	0.0327948i	$-0.989559\pi$
$991$		−	2046.19i		−	0.0655896i	0.999462	−	0.0327948i	$-0.0104408\pi$
$992$		0			0
$993$		−	18045.8i		−	0.576702i
$994$		0			0
$995$	10251.0			0.326613
$996$		0			0
$997$		−	33294.8i		−	1.05763i	−0.848737	−	0.528815i	$-0.822636\pi$
$997$		−	33294.8i		−	1.05763i	0.848737	−	0.528815i	$-0.177364\pi$
$998$		0			0
$999$	49134.6			1.55611

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.b.a.33.6	yes	6
3.2	odd	2		1224.4.c.c.577.5		6
4.3	odd	2		272.4.b.e.33.1		6
17.4	even	4		2312.4.a.h.1.6		6
17.13	even	4		2312.4.a.h.1.1		6
17.16	even	2	inner	136.4.b.a.33.1	✓	6
51.50	odd	2		1224.4.c.c.577.2		6
68.67	odd	2		272.4.b.e.33.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.a.33.1	✓	6	17.16	even	2	inner
136.4.b.a.33.6	yes	6	1.1	even	1	trivial
272.4.b.e.33.1		6	4.3	odd	2
272.4.b.e.33.6		6	68.67	odd	2
1224.4.c.c.577.2		6	51.50	odd	2
1224.4.c.c.577.5		6	3.2	odd	2
2312.4.a.h.1.1		6	17.13	even	4
2312.4.a.h.1.6		6	17.4	even	4

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.27144i q^{3} -6.42821i q^{5} +24.9151i q^{7} -41.4167 q^{9} +O(q^{10})\)	\(q+8.27144i q^{3} -6.42821i q^{5} +24.9151i q^{7} -41.4167 q^{9} +18.4869i q^{11} +10.7538 q^{13} +53.1705 q^{15} +(-67.0797 - 20.3301i) q^{17} -130.004 q^{19} -206.084 q^{21} -90.2427i q^{23} +83.6781 q^{25} -119.247i q^{27} -29.6470i q^{29} +132.901i q^{31} -152.913 q^{33} +160.159 q^{35} +412.040i q^{37} +88.9494i q^{39} +165.227i q^{41} +502.172 q^{43} +266.235i q^{45} -160.186 q^{47} -277.762 q^{49} +(168.159 - 554.846i) q^{51} +372.163 q^{53} +118.838 q^{55} -1075.32i q^{57} +453.800 q^{59} +254.958i q^{61} -1031.90i q^{63} -69.1276i q^{65} -383.360 q^{67} +746.437 q^{69} +818.663i q^{71} -640.109i q^{73} +692.139i q^{75} -460.602 q^{77} +380.550i q^{79} -131.905 q^{81} +984.831 q^{83} +(-130.686 + 431.202i) q^{85} +245.223 q^{87} +417.573 q^{89} +267.932i q^{91} -1099.29 q^{93} +835.693i q^{95} -1327.74i q^{97} -765.667i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 42 q^{9}+O(q^{10}) \) 6 * q - 42 * q^9	\( 6 q - 42 q^{9} - 72 q^{13} - 24 q^{15} - 126 q^{17} - 24 q^{19} - 204 q^{21} - 114 q^{25} - 228 q^{33} + 408 q^{35} + 192 q^{43} - 72 q^{47} - 18 q^{49} + 456 q^{51} + 924 q^{53} - 456 q^{55} + 1680 q^{59} - 312 q^{67} + 492 q^{69} - 1668 q^{77} - 1614 q^{81} + 1296 q^{83} + 1344 q^{85} + 456 q^{87} + 24 q^{89} + 828 q^{93}+O(q^{100}) \) 6 * q - 42 * q^9 - 72 * q^13 - 24 * q^15 - 126 * q^17 - 24 * q^19 - 204 * q^21 - 114 * q^25 - 228 * q^33 + 408 * q^35 + 192 * q^43 - 72 * q^47 - 18 * q^49 + 456 * q^51 + 924 * q^53 - 456 * q^55 + 1680 * q^59 - 312 * q^67 + 492 * q^69 - 1668 * q^77 - 1614 * q^81 + 1296 * q^83 + 1344 * q^85 + 456 * q^87 + 24 * q^89 + 828 * q^93

Introduction

L-functions

Modular forms

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Database

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