LMFDB - Embedded newform 136.4.b.b.33.4

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 $ x^8 + 95x^6 + 756x^4 + 1780*x^2 + 1152
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.4
Root			$9.30031i$ of defining polynomial
Character	$\chi$	$=$	136.33
Dual form			136.4.b.b.33.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} +O(q^{10})$	$q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} -1.19452i q^{11} +58.3731 q^{13} +30.2245 q^{15} +(-54.8640 - 43.6227i) q^{17} +138.971 q^{19} -60.4096 q^{21} -180.911i q^{23} -7.98158 q^{25} -123.528i q^{27} -115.027i q^{29} +210.726i q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661i q^{37} -152.994i q^{39} +297.500i q^{41} +174.746 q^{43} +232.140i q^{45} -199.717 q^{47} -188.234 q^{49} +(-114.334 + 143.797i) q^{51} -706.217 q^{53} +13.7749 q^{55} -364.238i q^{57} -182.746 q^{59} -489.499i q^{61} -463.978i q^{63} +673.144i q^{65} -167.043 q^{67} -474.163 q^{69} +818.157i q^{71} +1090.15i q^{73} +20.9195i q^{75} -27.5319 q^{77} +110.196i q^{79} +219.760 q^{81} +118.672 q^{83} +(503.047 - 632.679i) q^{85} -301.483 q^{87} +400.332 q^{89} -1345.41i q^{91} +552.306 q^{93} +1602.58i q^{95} -924.543i q^{97} -24.0463i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 132 q^{9}+O(q^{10}) $ 8 * q - 132 * q^9	$ 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) $ 8 * q - 132 * q^9 + 44 * q^13 + 24 * q^15 + 28 * q^17 + 48 * q^19 + 308 * q^21 - 520 * q^25 + 812 * q^33 - 1064 * q^35 + 8 * q^43 + 312 * q^47 - 1124 * q^49 + 408 * q^51 + 472 * q^53 + 1416 * q^55 - 72 * q^59 - 624 * q^67 - 180 * q^69 + 1660 * q^77 + 3156 * q^81 + 2472 * q^83 - 2160 * q^85 - 6664 * q^87 + 68 * q^89 - 4036 * 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
$3$		−	2.62097i		−	0.504407i	−0.967674	−	0.252203i	$-0.918845\pi$
$3$		−	2.62097i		−	0.504407i	0.967674	−	0.252203i	$-0.0811552\pi$
$4$		0			0
$5$			11.5318i			1.03143i	0.856759	+	0.515716i	$0.172474\pi$
$5$			11.5318i			1.03143i	−0.856759	+	0.515716i	$0.827526\pi$
$6$		0			0
$7$		−	23.0485i		−	1.24450i	−0.782817	−	0.622251i	$-0.786218\pi$
$7$		−	23.0485i		−	1.24450i	0.782817	−	0.622251i	$-0.213782\pi$
$8$		0			0
$9$	20.1305			0.745574
$10$		0			0
$11$		−	1.19452i		−	0.0327419i	−0.999866	−	0.0163710i	$-0.994789\pi$
$11$		−	1.19452i		−	0.0327419i	0.999866	−	0.0163710i	$-0.00521127\pi$
$12$		0			0
$13$	58.3731			1.24537			0.622684	−	0.782474i	$-0.286042\pi$
$13$	58.3731			1.24537			0.622684	+	0.782474i	$0.286042\pi$
$14$		0			0
$15$	30.2245			0.520262
$16$		0			0
$17$	−54.8640	−	43.6227i	−0.782734	−	0.622357i
$17$	−54.8640	−	43.6227i	−0.782734	−	0.622357i
$18$		0			0
$19$	138.971			1.67800			0.839001	−	0.544130i	$-0.183140\pi$
$19$	138.971			1.67800			0.839001	+	0.544130i	$0.183140\pi$
$20$		0			0
$21$	−60.4096			−0.627736
$22$		0			0
$23$		−	180.911i		−	1.64011i	−0.572284	−	0.820055i	$-0.693943\pi$
$23$		−	180.911i		−	1.64011i	0.572284	−	0.820055i	$-0.306057\pi$
$24$		0			0
$25$	−7.98158			−0.0638527
$26$		0			0
$27$		−	123.528i		−	0.880479i
$28$		0			0
$29$		−	115.027i		−	0.736551i	−0.929717	−	0.368275i	$-0.879948\pi$
$29$		−	115.027i		−	0.736551i	0.929717	−	0.368275i	$-0.120052\pi$
$30$		0			0
$31$			210.726i			1.22088i	0.792061	+	0.610442i	$0.209008\pi$
$31$			210.726i			1.22088i	−0.792061	+	0.610442i	$0.790992\pi$
$32$		0			0
$33$	−3.13080			−0.0165152
$34$		0			0
$35$	265.790			1.28362
$36$		0			0
$37$			210.661i			0.936010i	0.883726	+	0.468005i	$0.155027\pi$
$37$			210.661i			0.936010i	−0.883726	+	0.468005i	$0.844973\pi$
$38$		0			0
$39$		−	152.994i		−	0.628172i
$40$		0			0
$41$			297.500i			1.13321i	0.823989	+	0.566605i	$0.191743\pi$
$41$			297.500i			1.13321i	−0.823989	+	0.566605i	$0.808257\pi$
$42$		0			0
$43$	174.746			0.619734			0.309867	−	0.950780i	$-0.399715\pi$
$43$	174.746			0.619734			0.309867	+	0.950780i	$0.399715\pi$
$44$		0			0
$45$			232.140i			0.769009i
$46$		0			0
$47$	−199.717			−0.619823			−0.309911	−	0.950765i	$-0.600299\pi$
$47$	−199.717			−0.619823			−0.309911	+	0.950765i	$0.600299\pi$
$48$		0			0
$49$	−188.234			−0.548787
$50$		0			0
$51$	−114.334	+	143.797i	−0.313921	+	0.394816i
$52$		0			0
$53$	−706.217			−1.83031			−0.915154	−	0.403105i	$-0.867931\pi$
$53$	−706.217			−1.83031			−0.915154	+	0.403105i	$0.867931\pi$
$54$		0			0
$55$	13.7749			0.0337711
$56$		0			0
$57$		−	364.238i		−	0.846396i
$58$		0			0
$59$	−182.746			−0.403247			−0.201623	−	0.979463i	$-0.564622\pi$
$59$	−182.746			−0.403247			−0.201623	+	0.979463i	$0.564622\pi$
$60$		0			0
$61$		−	489.499i		−	1.02744i	−0.857958	−	0.513720i	$-0.828267\pi$
$61$		−	489.499i		−	1.02744i	0.857958	−	0.513720i	$-0.171733\pi$
$62$		0			0
$63$		−	463.978i		−	0.927869i
$64$		0			0
$65$			673.144i			1.28451i
$66$		0			0
$67$	−167.043			−0.304590			−0.152295	−	0.988335i	$-0.548666\pi$
$67$	−167.043			−0.304590			−0.152295	+	0.988335i	$0.548666\pi$
$68$		0			0
$69$	−474.163			−0.827283
$70$		0			0
$71$			818.157i			1.36757i	0.729684	+	0.683784i	$0.239667\pi$
$71$			818.157i			1.36757i	−0.729684	+	0.683784i	$0.760333\pi$
$72$		0			0
$73$			1090.15i			1.74784i	0.486071	+	0.873919i	$0.338430\pi$
$73$			1090.15i			1.74784i	−0.486071	+	0.873919i	$0.661570\pi$
$74$		0			0
$75$			20.9195i			0.0322077i
$76$		0			0
$77$	−27.5319			−0.0407474
$78$		0			0
$79$			110.196i			0.156937i	0.996917	+	0.0784685i	$0.0250030\pi$
$79$			110.196i			0.156937i	−0.996917	+	0.0784685i	$0.974997\pi$
$80$		0			0
$81$	219.760			0.301454
$82$		0			0
$83$	118.672			0.156939			0.0784696	−	0.996917i	$-0.474997\pi$
$83$	118.672			0.156939			0.0784696	+	0.996917i	$0.474997\pi$
$84$		0			0
$85$	503.047	−	632.679i	0.641919	−	0.807337i
$86$		0			0
$87$	−301.483			−0.371521
$88$		0			0
$89$	400.332			0.476799			0.238399	−	0.971167i	$-0.423377\pi$
$89$	400.332			0.476799			0.238399	+	0.971167i	$0.423377\pi$
$90$		0			0
$91$		−	1345.41i		−	1.54986i
$92$		0			0
$93$	552.306			0.615823
$94$		0			0
$95$			1602.58i			1.73075i
$96$		0			0
$97$		−	924.543i		−	0.967764i	−0.875133	−	0.483882i	$-0.839226\pi$
$97$		−	924.543i		−	0.967764i	0.875133	−	0.483882i	$-0.160774\pi$
$98$		0			0
$99$		−	24.0463i		−	0.0244115i
$100$		0			0
$101$	223.816			0.220500			0.110250	−	0.993904i	$-0.464835\pi$
$101$	223.816			0.220500			0.110250	+	0.993904i	$0.464835\pi$
$102$		0			0
$103$	−1280.06			−1.22454			−0.612271	−	0.790648i	$-0.709744\pi$
$103$	−1280.06			−1.22454			−0.612271	+	0.790648i	$0.709744\pi$
$104$		0			0
$105$		−	696.629i		−	0.647467i
$106$		0			0
$107$		−	1099.03i		−	0.992969i	−0.868046	−	0.496485i	$-0.834624\pi$
$107$		−	1099.03i		−	0.992969i	0.868046	−	0.496485i	$-0.165376\pi$
$108$		0			0
$109$			733.360i			0.644433i	0.946666	+	0.322216i	$0.104428\pi$
$109$			733.360i			0.644433i	−0.946666	+	0.322216i	$0.895572\pi$
$110$		0			0
$111$	552.136			0.472130
$112$		0			0
$113$		−	1637.39i		−	1.36312i	−0.731761	−	0.681561i	$-0.761301\pi$
$113$		−	1637.39i		−	1.36312i	0.731761	−	0.681561i	$-0.238699\pi$
$114$		0			0
$115$	2086.22			1.69166
$116$		0			0
$117$	1175.08			0.928513
$118$		0			0
$119$	−1005.44	+	1264.53i	−0.774525	+	0.974115i
$120$		0			0
$121$	1329.57			0.998928
$122$		0			0
$123$	779.739			0.571599
$124$		0			0
$125$			1349.43i			0.965573i
$126$		0			0
$127$	−1502.71			−1.04995			−0.524977	−	0.851117i	$-0.675926\pi$
$127$	−1502.71			−1.04995			−0.524977	+	0.851117i	$0.675926\pi$
$128$		0			0
$129$		−	458.006i		−	0.312598i
$130$		0			0
$131$		−	702.381i		−	0.468453i	−0.972182	−	0.234226i	$-0.924744\pi$
$131$		−	702.381i		−	0.468453i	0.972182	−	0.234226i	$-0.0752557\pi$
$132$		0			0
$133$		−	3203.07i		−	2.08828i
$134$		0			0
$135$	1424.49			0.908155
$136$		0			0
$137$	18.9889			0.0118418			0.00592092	−	0.999982i	$-0.498115\pi$
$137$	18.9889			0.0118418			0.00592092	+	0.999982i	$0.498115\pi$
$138$		0			0
$139$		−	276.969i		−	0.169009i	−0.996423	−	0.0845043i	$-0.973069\pi$
$139$		−	276.969i		−	0.169009i	0.996423	−	0.0845043i	$-0.0269307\pi$
$140$		0			0
$141$			523.452i			0.312643i
$142$		0			0
$143$		−	69.7277i		−	0.0407757i
$144$		0			0
$145$	1326.46			0.759702
$146$		0			0
$147$			493.357i			0.276812i
$148$		0			0
$149$	128.491			0.0706468			0.0353234	−	0.999376i	$-0.488754\pi$
$149$	128.491			0.0706468			0.0353234	+	0.999376i	$0.488754\pi$
$150$		0			0
$151$	−2097.44			−1.13038			−0.565190	−	0.824961i	$-0.691197\pi$
$151$	−2097.44			−1.13038			−0.565190	+	0.824961i	$0.691197\pi$
$152$		0			0
$153$	−1104.44	−	878.147i	−0.583586	−	0.464013i
$154$		0			0
$155$	−2430.04			−1.25926
$156$		0			0
$157$	2911.59			1.48007			0.740033	−	0.672571i	$-0.234810\pi$
$157$	2911.59			1.48007			0.740033	+	0.672571i	$0.234810\pi$
$158$		0			0
$159$			1850.98i			0.923220i
$160$		0			0
$161$	−4169.73			−2.04112
$162$		0			0
$163$			1574.26i			0.756478i	0.925708	+	0.378239i	$0.123470\pi$
$163$			1574.26i			0.756478i	−0.925708	+	0.378239i	$0.876530\pi$
$164$		0			0
$165$		−	36.1037i		−	0.0170344i
$166$		0			0
$167$		−	1290.10i		−	0.597788i	−0.954286	−	0.298894i	$-0.903382\pi$
$167$		−	1290.10i		−	0.597788i	0.954286	−	0.298894i	$-0.0966177\pi$
$168$		0			0
$169$	1210.41			0.550939
$170$		0			0
$171$	2797.55			1.25107
$172$		0			0
$173$			2370.19i			1.04163i	0.853669	+	0.520816i	$0.174372\pi$
$173$			2370.19i			1.04163i	−0.853669	+	0.520816i	$0.825628\pi$
$174$		0			0
$175$			183.964i			0.0794648i
$176$		0			0
$177$			478.974i			0.203400i
$178$		0			0
$179$	1719.87			0.718153			0.359077	−	0.933308i	$-0.383092\pi$
$179$	1719.87			0.718153			0.359077	+	0.933308i	$0.383092\pi$
$180$		0			0
$181$			61.7101i			0.0253418i	0.999920	+	0.0126709i	$0.00403339\pi$
$181$			61.7101i			0.0253418i	−0.999920	+	0.0126709i	$0.995967\pi$
$182$		0			0
$183$	−1282.96			−0.518248
$184$		0			0
$185$	−2429.29			−0.965431
$186$		0			0
$187$	−52.1082	+	65.5361i	−0.0203771	+	0.0256282i
$188$		0			0
$189$	−2847.13			−1.09576
$190$		0			0
$191$	−2565.34			−0.971839			−0.485919	−	0.874004i	$-0.661515\pi$
$191$	−2565.34			−0.971839			−0.485919	+	0.874004i	$0.661515\pi$
$192$		0			0
$193$			1991.65i			0.742810i	0.928471	+	0.371405i	$0.121124\pi$
$193$			1991.65i			0.742810i	−0.928471	+	0.371405i	$0.878876\pi$
$194$		0			0
$195$	1764.29			0.647917
$196$		0			0
$197$			3831.76i			1.38579i	0.721036	+	0.692897i	$0.243666\pi$
$197$			3831.76i			1.38579i	−0.721036	+	0.692897i	$0.756334\pi$
$198$		0			0
$199$			2171.38i			0.773492i	0.922186	+	0.386746i	$0.126401\pi$
$199$			2171.38i			0.773492i	−0.922186	+	0.386746i	$0.873599\pi$
$200$		0			0
$201$			437.815i			0.153637i
$202$		0			0
$203$	−2651.20			−0.916639
$204$		0			0
$205$	−3430.69			−1.16883
$206$		0			0
$207$		−	3641.83i		−	1.22282i
$208$		0			0
$209$		−	166.003i		−	0.0549410i
$210$		0			0
$211$		−	2069.97i		−	0.675369i	−0.941259	−	0.337685i	$-0.890356\pi$
$211$		−	2069.97i		−	0.675369i	0.941259	−	0.337685i	$-0.109644\pi$
$212$		0			0
$213$	2144.37			0.689811
$214$		0			0
$215$			2015.13i			0.639214i
$216$		0			0
$217$	4856.91			1.51939
$218$		0			0
$219$	2857.25			0.881622
$220$		0			0
$221$	−3202.58	−	2546.39i	−0.974791	−	0.775062i
$222$		0			0
$223$	−4297.20			−1.29041			−0.645206	−	0.764008i	$-0.723229\pi$
$223$	−4297.20			−1.29041			−0.645206	+	0.764008i	$0.723229\pi$
$224$		0			0
$225$	−160.673			−0.0476069
$226$		0			0
$227$		−	336.979i		−	0.0985291i	−0.998786	−	0.0492646i	$-0.984312\pi$
$227$		−	336.979i		−	0.0985291i	0.998786	−	0.0492646i	$-0.0156877\pi$
$228$		0			0
$229$	−3076.57			−0.887798			−0.443899	−	0.896077i	$-0.646405\pi$
$229$	−3076.57			−0.887798			−0.443899	+	0.896077i	$0.646405\pi$
$230$		0			0
$231$			72.1604i			0.0205533i
$232$		0			0
$233$			6537.86i			1.83824i	0.393980	+	0.919119i	$0.371098\pi$
$233$			6537.86i			1.83824i	−0.393980	+	0.919119i	$0.628902\pi$
$234$		0			0
$235$		−	2303.09i		−	0.639305i
$236$		0			0
$237$	288.821			0.0791601
$238$		0			0
$239$	−3432.31			−0.928943			−0.464472	−	0.885588i	$-0.653756\pi$
$239$	−3432.31			−0.928943			−0.464472	+	0.885588i	$0.653756\pi$
$240$		0			0
$241$			1312.48i			0.350807i	0.984497	+	0.175404i	$0.0561230\pi$
$241$			1312.48i			0.350807i	−0.984497	+	0.175404i	$0.943877\pi$
$242$		0			0
$243$		−	3911.24i		−	1.03253i
$244$		0			0
$245$		−	2170.67i		−	0.566037i
$246$		0			0
$247$	8112.14			2.08973
$248$		0			0
$249$		−	311.037i		−	0.0791612i
$250$		0			0
$251$	6498.55			1.63420			0.817101	−	0.576494i	$-0.195580\pi$
$251$	6498.55			1.63420			0.817101	+	0.576494i	$0.195580\pi$
$252$		0			0
$253$	−216.102			−0.0537004
$254$		0			0
$255$	−1658.24	−	1318.47i	−0.407226	−	0.323788i
$256$		0			0
$257$	−4874.66			−1.18316			−0.591582	−	0.806245i	$-0.701496\pi$
$257$	−4874.66			−1.18316			−0.591582	+	0.806245i	$0.701496\pi$
$258$		0			0
$259$	4855.41			1.16487
$260$		0			0
$261$		−	2315.55i		−	0.549153i
$262$		0			0
$263$	1482.50			0.347584			0.173792	−	0.984782i	$-0.444398\pi$
$263$	1482.50			0.347584			0.173792	+	0.984782i	$0.444398\pi$
$264$		0			0
$265$		−	8143.92i		−	1.88784i
$266$		0			0
$267$		−	1049.26i		−	0.240501i
$268$		0			0
$269$			5058.75i			1.14661i	0.819343	+	0.573304i	$0.194338\pi$
$269$			5058.75i			1.14661i	−0.819343	+	0.573304i	$0.805662\pi$
$270$		0			0
$271$	1133.70			0.254122			0.127061	−	0.991895i	$-0.459446\pi$
$271$	1133.70			0.254122			0.127061	+	0.991895i	$0.459446\pi$
$272$		0			0
$273$	−3526.29			−0.781761
$274$		0			0
$275$			9.53415i			0.00209066i
$276$		0			0
$277$		−	3698.34i		−	0.802208i	−0.916033	−	0.401104i	$-0.868627\pi$
$277$		−	3698.34i		−	0.802208i	0.916033	−	0.401104i	$-0.131373\pi$
$278$		0			0
$279$			4242.01i			0.910260i
$280$		0			0
$281$	2190.08			0.464945			0.232472	−	0.972603i	$-0.425318\pi$
$281$	2190.08			0.464945			0.232472	+	0.972603i	$0.425318\pi$
$282$		0			0
$283$			8159.66i			1.71393i	0.515378	+	0.856963i	$0.327652\pi$
$283$			8159.66i			1.71393i	−0.515378	+	0.856963i	$0.672348\pi$
$284$		0			0
$285$	4200.31			0.873000
$286$		0			0
$287$	6856.92			1.41028
$288$		0			0
$289$	1107.12	+	4786.63i	0.225345	+	0.974279i
$290$		0			0
$291$	−2423.20			−0.488147
$292$		0			0
$293$	4777.49			0.952574			0.476287	−	0.879290i	$-0.341982\pi$
$293$	4777.49			0.952574			0.476287	+	0.879290i	$0.341982\pi$
$294$		0			0
$295$		−	2107.39i		−	0.415922i
$296$		0			0
$297$	−147.556			−0.0288286
$298$		0			0
$299$		−	10560.3i		−	2.04254i
$300$		0			0
$301$		−	4027.65i		−	0.771261i
$302$		0			0
$303$		−	586.615i		−	0.111222i
$304$		0			0
$305$	5644.78			1.05974
$306$		0			0
$307$	531.103			0.0987351			0.0493675	−	0.998781i	$-0.484279\pi$
$307$	531.103			0.0987351			0.0493675	+	0.998781i	$0.484279\pi$
$308$		0			0
$309$			3355.00i			0.617667i
$310$		0			0
$311$			4767.26i			0.869217i	0.900619	+	0.434609i	$0.143113\pi$
$311$			4767.26i			0.869217i	−0.900619	+	0.434609i	$0.856887\pi$
$312$		0			0
$313$		−	8613.47i		−	1.55547i	−0.628593	−	0.777735i	$-0.716369\pi$
$313$		−	8613.47i		−	1.55547i	0.628593	−	0.777735i	$-0.283631\pi$
$314$		0			0
$315$	5350.48			0.957034
$316$		0			0
$317$		−	2453.20i		−	0.434654i	−0.976099	−	0.217327i	$-0.930266\pi$
$317$		−	2453.20i		−	0.434654i	0.976099	−	0.217327i	$-0.0697338\pi$
$318$		0			0
$319$	−137.402			−0.0241161
$320$		0			0
$321$	−2880.54			−0.500860
$322$		0			0
$323$	−7624.48	−	6062.27i	−1.31343	−	1.04432i
$324$		0			0
$325$	−465.909			−0.0795200
$326$		0			0
$327$	1922.12			0.325056
$328$		0			0
$329$			4603.17i			0.771371i
$330$		0			0
$331$	−11003.5			−1.82721			−0.913605	−	0.406604i	$-0.866713\pi$
$331$	−11003.5			−1.82721			−0.913605	+	0.406604i	$0.866713\pi$
$332$		0			0
$333$			4240.70i			0.697865i
$334$		0			0
$335$		−	1926.30i		−	0.314164i
$336$		0			0
$337$			3664.37i			0.592318i	0.955139	+	0.296159i	$0.0957058\pi$
$337$			3664.37i			0.592318i	−0.955139	+	0.296159i	$0.904294\pi$
$338$		0			0
$339$	−4291.56			−0.687568
$340$		0			0
$341$	251.716			0.0399741
$342$		0			0
$343$		−	3567.12i		−	0.561535i
$344$		0			0
$345$		−	5467.94i		−	0.853286i
$346$		0			0
$347$			3173.37i			0.490938i	0.969404	+	0.245469i	$0.0789419\pi$
$347$			3173.37i			0.490938i	−0.969404	+	0.245469i	$0.921058\pi$
$348$		0			0
$349$	−2185.47			−0.335202			−0.167601	−	0.985855i	$-0.553602\pi$
$349$	−2185.47			−0.335202			−0.167601	+	0.985855i	$0.553602\pi$
$350$		0			0
$351$		−	7210.70i		−	1.09652i
$352$		0			0
$353$	1216.36			0.183400			0.0917002	−	0.995787i	$-0.470770\pi$
$353$	1216.36			0.183400			0.0917002	+	0.995787i	$0.470770\pi$
$354$		0			0
$355$	−9434.80			−1.41055
$356$		0			0
$357$	3314.31	+	2635.23i	0.491350	+	0.390675i
$358$		0			0
$359$	−7448.93			−1.09510			−0.547548	−	0.836774i	$-0.684439\pi$
$359$	−7448.93			−1.09510			−0.547548	+	0.836774i	$0.684439\pi$
$360$		0			0
$361$	12453.8			1.81569
$362$		0			0
$363$		−	3484.78i		−	0.503866i
$364$		0			0
$365$	−12571.3			−1.80278
$366$		0			0
$367$			2374.59i			0.337746i	0.985638	+	0.168873i	$0.0540128\pi$
$367$			2374.59i			0.337746i	−0.985638	+	0.168873i	$0.945987\pi$
$368$		0			0
$369$			5988.81i			0.844892i
$370$		0			0
$371$			16277.2i			2.27782i
$372$		0			0
$373$	12231.7			1.69795			0.848974	−	0.528435i	$-0.177221\pi$
$373$	12231.7			1.69795			0.848974	+	0.528435i	$0.177221\pi$
$374$		0			0
$375$	3536.82			0.487041
$376$		0			0
$377$		−	6714.48i		−	0.917276i
$378$		0			0
$379$			14129.9i			1.91505i	0.288358	+	0.957523i	$0.406891\pi$
$379$			14129.9i			1.91505i	−0.288358	+	0.957523i	$0.593109\pi$
$380$		0			0
$381$			3938.57i			0.529604i
$382$		0			0
$383$	13306.4			1.77526			0.887631	−	0.460556i	$-0.152350\pi$
$383$	13306.4			1.77526			0.887631	+	0.460556i	$0.152350\pi$
$384$		0			0
$385$		−	317.491i		−	0.0420282i
$386$		0			0
$387$	3517.73			0.462058
$388$		0			0
$389$	−8007.84			−1.04374			−0.521868	−	0.853026i	$-0.674765\pi$
$389$	−8007.84			−1.04374			−0.521868	+	0.853026i	$0.674765\pi$
$390$		0			0
$391$	−7891.83	+	9925.50i	−1.02073	+	1.28377i
$392$		0			0
$393$	−1840.92			−0.236291
$394$		0			0
$395$	−1270.76			−0.161870
$396$		0			0
$397$		−	13185.2i		−	1.66687i	−0.552619	−	0.833434i	$-0.686372\pi$
$397$		−	13185.2i		−	1.66687i	0.552619	−	0.833434i	$-0.313628\pi$
$398$		0			0
$399$	−8395.15			−1.05334
$400$		0			0
$401$			3205.81i			0.399228i	0.979875	+	0.199614i	$0.0639688\pi$
$401$			3205.81i			0.399228i	−0.979875	+	0.199614i	$0.936031\pi$
$402$		0			0
$403$			12300.7i			1.52045i
$404$		0			0
$405$			2534.22i			0.310929i
$406$		0			0
$407$	251.638			0.0306468
$408$		0			0
$409$	−15693.3			−1.89727			−0.948636	−	0.316369i	$-0.897536\pi$
$409$	−15693.3			−1.89727			−0.948636	+	0.316369i	$0.897536\pi$
$410$		0			0
$411$		−	49.7695i		−	0.00597311i
$412$		0			0
$413$			4212.03i			0.501842i
$414$		0			0
$415$			1368.50i			0.161872i
$416$		0			0
$417$	−725.929			−0.0852491
$418$		0			0
$419$		−	6910.65i		−	0.805745i	−0.915256	−	0.402873i	$-0.868012\pi$
$419$		−	6910.65i		−	0.805745i	0.915256	−	0.402873i	$-0.131988\pi$
$420$		0			0
$421$	9225.05			1.06794			0.533968	−	0.845505i	$-0.320700\pi$
$421$	9225.05			1.06794			0.533968	+	0.845505i	$0.320700\pi$
$422$		0			0
$423$	−4020.40			−0.462124
$424$		0			0
$425$	437.902	+	348.178i	0.0499796	+	0.0397391i
$426$		0			0
$427$	−11282.2			−1.27865
$428$		0			0
$429$	−182.755			−0.0205675
$430$		0			0
$431$			8822.74i			0.986024i	0.870022	+	0.493012i	$0.164104\pi$
$431$			8822.74i			0.986024i	−0.870022	+	0.493012i	$0.835896\pi$
$432$		0			0
$433$	−3190.95			−0.354151			−0.177075	−	0.984197i	$-0.556664\pi$
$433$	−3190.95			−0.354151			−0.177075	+	0.984197i	$0.556664\pi$
$434$		0			0
$435$		−	3476.63i		−	0.383199i
$436$		0			0
$437$		−	25141.3i		−	2.75211i
$438$		0			0
$439$		−	15069.5i		−	1.63834i	−0.573553	−	0.819168i	$-0.694435\pi$
$439$		−	15069.5i		−	1.63834i	0.573553	−	0.819168i	$-0.305565\pi$
$440$		0			0
$441$	−3789.24			−0.409161
$442$		0			0
$443$	−3026.24			−0.324562			−0.162281	−	0.986745i	$-0.551885\pi$
$443$	−3026.24			−0.324562			−0.162281	+	0.986745i	$0.551885\pi$
$444$		0			0
$445$			4616.53i			0.491786i
$446$		0			0
$447$		−	336.771i		−	0.0356347i
$448$		0			0
$449$		−	4641.02i		−	0.487802i	−0.969800	−	0.243901i	$-0.921573\pi$
$449$		−	4641.02i		−	0.487802i	0.969800	−	0.243901i	$-0.0784272\pi$
$450$		0			0
$451$	355.369			0.0371035
$452$		0			0
$453$			5497.34i			0.570171i
$454$		0			0
$455$	15515.0			1.59858
$456$		0			0
$457$	17393.2			1.78035			0.890174	−	0.455620i	$-0.150582\pi$
$457$	17393.2			1.78035			0.890174	+	0.455620i	$0.150582\pi$
$458$		0			0
$459$	−5388.62	+	6777.23i	−0.547972	+	0.689181i
$460$		0			0
$461$	2451.93			0.247718			0.123859	−	0.992300i	$-0.460473\pi$
$461$	2451.93			0.247718			0.123859	+	0.992300i	$0.460473\pi$
$462$		0			0
$463$	−14586.3			−1.46411			−0.732054	−	0.681246i	$-0.761438\pi$
$463$	−14586.3			−1.46411			−0.732054	+	0.681246i	$0.761438\pi$
$464$		0			0
$465$			6369.07i			0.635179i
$466$		0			0
$467$	15195.6			1.50572			0.752859	−	0.658182i	$-0.228674\pi$
$467$	15195.6			1.50572			0.752859	+	0.658182i	$0.228674\pi$
$468$		0			0
$469$			3850.09i			0.379063i
$470$		0			0
$471$		−	7631.21i		−	0.746555i
$472$		0			0
$473$		−	208.738i		−	0.0202913i
$474$		0			0
$475$	−1109.21			−0.107145
$476$		0			0
$477$	−14216.5			−1.36463
$478$		0			0
$479$		−	4422.10i		−	0.421819i	−0.977506	−	0.210909i	$-0.932358\pi$
$479$		−	4422.10i		−	0.421819i	0.977506	−	0.210909i	$-0.0676425\pi$
$480$		0			0
$481$			12296.9i			1.16568i
$482$		0			0
$483$			10928.8i			1.02956i
$484$		0			0
$485$	10661.6			0.998183
$486$		0			0
$487$		−	17772.7i		−	1.65371i	−0.562414	−	0.826856i	$-0.690127\pi$
$487$		−	17772.7i		−	1.65371i	0.562414	−	0.826856i	$-0.309873\pi$
$488$		0			0
$489$	4126.11			0.381573
$490$		0			0
$491$	4845.34			0.445350			0.222675	−	0.974893i	$-0.428521\pi$
$491$	4845.34			0.445350			0.222675	+	0.974893i	$0.428521\pi$
$492$		0			0
$493$	−5017.79	+	6310.84i	−0.458397	+	0.576523i
$494$		0			0
$495$	277.296			0.0251788
$496$		0			0
$497$	18857.3			1.70194
$498$		0			0
$499$		−	1060.23i		−	0.0951153i	−0.998868	−	0.0475576i	$-0.984856\pi$
$499$		−	1060.23i		−	0.0951153i	0.998868	−	0.0475576i	$-0.0151438\pi$
$500$		0			0
$501$	−3381.31			−0.301528
$502$		0			0
$503$			2803.19i			0.248485i	0.992252	+	0.124243i	$0.0396501\pi$
$503$			2803.19i			0.248485i	−0.992252	+	0.124243i	$0.960350\pi$
$504$		0			0
$505$			2580.99i			0.227431i
$506$		0			0
$507$		−	3172.46i		−	0.277898i
$508$		0			0
$509$	1658.97			0.144465			0.0722323	−	0.997388i	$-0.476988\pi$
$509$	1658.97			0.144465			0.0722323	+	0.997388i	$0.476988\pi$
$510$		0			0
$511$	25126.3			2.17519
$512$		0			0
$513$		−	17166.7i		−	1.47745i
$514$		0			0
$515$		−	14761.3i		−	1.26303i
$516$		0			0
$517$			238.565i			0.0202942i
$518$		0			0
$519$	6212.21			0.525406
$520$		0			0
$521$		−	12553.6i		−	1.05563i	−0.849359	−	0.527815i	$-0.823011\pi$
$521$		−	12553.6i		−	1.05563i	0.849359	−	0.527815i	$-0.176989\pi$
$522$		0			0
$523$	−12031.6			−1.00593			−0.502967	−	0.864306i	$-0.667758\pi$
$523$	−12031.6			−1.00593			−0.502967	+	0.864306i	$0.667758\pi$
$524$		0			0
$525$	482.164			0.0400826
$526$		0			0
$527$	9192.42	−	11561.2i	0.759826	−	0.955628i
$528$		0			0
$529$	−20561.8			−1.68996
$530$		0			0
$531$	−3678.78			−0.300650
$532$		0			0
$533$			17366.0i			1.41126i
$534$		0			0
$535$	12673.8			1.02418
$536$		0			0
$537$		−	4507.75i		−	0.362241i
$538$		0			0
$539$			224.849i			0.0179683i
$540$		0			0
$541$		−	10632.7i		−	0.844984i	−0.906367	−	0.422492i	$-0.861155\pi$
$541$		−	10632.7i		−	0.844984i	0.906367	−	0.422492i	$-0.138845\pi$
$542$		0			0
$543$	161.741			0.0127826
$544$		0			0
$545$	−8456.94			−0.664689
$546$		0			0
$547$			17370.0i			1.35774i	0.734257	+	0.678872i	$0.237531\pi$
$547$			17370.0i			1.35774i	−0.734257	+	0.678872i	$0.762469\pi$
$548$		0			0
$549$		−	9853.85i		−	0.766033i
$550$		0			0
$551$		−	15985.4i		−	1.23593i
$552$		0			0
$553$	2539.86			0.195309
$554$		0			0
$555$			6367.10i			0.486970i
$556$		0			0
$557$	1084.77			0.0825190			0.0412595	−	0.999148i	$-0.486863\pi$
$557$	1084.77			0.0825190			0.0412595	+	0.999148i	$0.486863\pi$
$558$		0			0
$559$	10200.5			0.771797
$560$		0			0
$561$	171.768	+	136.574i	0.0129270	+	0.0102784i
$562$		0			0
$563$	−5494.27			−0.411289			−0.205645	−	0.978627i	$-0.565929\pi$
$563$	−5494.27			−0.411289			−0.205645	+	0.978627i	$0.565929\pi$
$564$		0			0
$565$	18882.0			1.40597
$566$		0			0
$567$		−	5065.14i		−	0.375160i
$568$		0			0
$569$	−11612.4			−0.855569			−0.427784	−	0.903881i	$-0.640706\pi$
$569$	−11612.4			−0.855569			−0.427784	+	0.903881i	$0.640706\pi$
$570$		0			0
$571$			3413.97i			0.250210i	0.992143	+	0.125105i	$0.0399268\pi$
$571$			3413.97i			0.250210i	−0.992143	+	0.125105i	$0.960073\pi$
$572$		0			0
$573$			6723.68i			0.490202i
$574$		0			0
$575$			1443.96i			0.104725i
$576$		0			0
$577$	1355.70			0.0978141			0.0489070	−	0.998803i	$-0.484426\pi$
$577$	1355.70			0.0978141			0.0489070	+	0.998803i	$0.484426\pi$
$578$		0			0
$579$	5220.07			0.374678
$580$		0			0
$581$		−	2735.22i		−	0.195311i
$582$		0			0
$583$			843.589i			0.0599278i
$584$		0			0
$585$			13550.7i			0.957698i
$586$		0			0
$587$	25232.2			1.77418			0.887092	−	0.461593i	$-0.152722\pi$
$587$	25232.2			1.77418			0.887092	+	0.461593i	$0.152722\pi$
$588$		0			0
$589$			29284.7i			2.04865i
$590$		0			0
$591$	10042.9			0.699004
$592$		0			0
$593$	−14327.4			−0.992171			−0.496086	−	0.868274i	$-0.665230\pi$
$593$	−14327.4			−0.992171			−0.496086	+	0.868274i	$0.665230\pi$
$594$		0			0
$595$	−14582.3	−	11594.5i	−1.00473	−	0.798870i
$596$		0			0
$597$	5691.13			0.390155
$598$		0			0
$599$	−2523.52			−0.172134			−0.0860671	−	0.996289i	$-0.527430\pi$
$599$	−2523.52			−0.172134			−0.0860671	+	0.996289i	$0.527430\pi$
$600$		0			0
$601$		−	17286.6i		−	1.17327i	−0.809853	−	0.586634i	$-0.800453\pi$
$601$		−	17286.6i		−	1.17327i	0.809853	−	0.586634i	$-0.199547\pi$
$602$		0			0
$603$	−3362.66			−0.227095
$604$		0			0
$605$			15332.3i			1.03033i
$606$		0			0
$607$			6413.65i			0.428867i	0.976739	+	0.214433i	$0.0687905\pi$
$607$			6413.65i			0.428867i	−0.976739	+	0.214433i	$0.931210\pi$
$608$		0			0
$609$			6948.73i			0.462359i
$610$		0			0
$611$	−11658.1			−0.771907
$612$		0			0
$613$	485.788			0.0320078			0.0160039	−	0.999872i	$-0.494906\pi$
$613$	485.788			0.0320078			0.0160039	+	0.999872i	$0.494906\pi$
$614$		0			0
$615$			8991.76i			0.589566i
$616$		0			0
$617$			6830.32i			0.445670i	0.974856	+	0.222835i	$0.0715311\pi$
$617$			6830.32i			0.445670i	−0.974856	+	0.222835i	$0.928469\pi$
$618$		0			0
$619$		−	10667.8i		−	0.692689i	−0.938107	−	0.346344i	$-0.887423\pi$
$619$		−	10667.8i		−	0.692689i	0.938107	−	0.346344i	$-0.112577\pi$
$620$		0			0
$621$	−22347.5			−1.44408
$622$		0			0
$623$		−	9227.05i		−	0.593377i
$624$		0			0
$625$	−16559.0			−1.05978
$626$		0			0
$627$	−435.090			−0.0277126
$628$		0			0
$629$	9189.58	−	11557.7i	0.582532	−	0.732647i
$630$		0			0
$631$	−15251.2			−0.962188			−0.481094	−	0.876669i	$-0.659760\pi$
$631$	−15251.2			−0.962188			−0.481094	+	0.876669i	$0.659760\pi$
$632$		0			0
$633$	−5425.35			−0.340661
$634$		0			0
$635$		−	17328.9i		−	1.08296i
$636$		0			0
$637$	−10987.8			−0.683442
$638$		0			0
$639$			16469.9i			1.01962i
$640$		0			0
$641$			12000.9i			0.739480i	0.929135	+	0.369740i	$0.120553\pi$
$641$			12000.9i			0.739480i	−0.929135	+	0.369740i	$0.879447\pi$
$642$		0			0
$643$		−	5642.30i		−	0.346051i	−0.984917	−	0.173025i	$-0.944646\pi$
$643$		−	5642.30i		−	0.346051i	0.984917	−	0.173025i	$-0.0553543\pi$
$644$		0			0
$645$	5281.62			0.322424
$646$		0			0
$647$	7790.69			0.473391			0.236695	−	0.971584i	$-0.423936\pi$
$647$	7790.69			0.473391			0.236695	+	0.971584i	$0.423936\pi$
$648$		0			0
$649$			218.294i			0.0132031i
$650$		0			0
$651$		−	12729.8i		−	0.766393i
$652$		0			0
$653$		−	6755.02i		−	0.404815i	−0.979301	−	0.202408i	$-0.935123\pi$
$653$		−	6755.02i		−	0.404815i	0.979301	−	0.202408i	$-0.0648766\pi$
$654$		0			0
$655$	8099.69			0.483177
$656$		0			0
$657$			21945.2i			1.30314i
$658$		0			0
$659$	−14303.3			−0.845492			−0.422746	−	0.906248i	$-0.638934\pi$
$659$	−14303.3			−0.845492			−0.422746	+	0.906248i	$0.638934\pi$
$660$		0			0
$661$	−26136.0			−1.53793			−0.768966	−	0.639289i	$-0.779229\pi$
$661$	−26136.0			−1.53793			−0.768966	+	0.639289i	$0.779229\pi$
$662$		0			0
$663$	−6674.03	+	8393.88i	−0.390947	+	0.491691i
$664$		0			0
$665$	36937.0			2.15392
$666$		0			0
$667$	−20809.6			−1.20802
$668$		0			0
$669$			11262.9i			0.650893i
$670$		0			0
$671$	−584.715			−0.0336404
$672$		0			0
$673$		−	19026.5i		−	1.08977i	−0.838509	−	0.544887i	$-0.816572\pi$
$673$		−	19026.5i		−	1.08977i	0.838509	−	0.544887i	$-0.183428\pi$
$674$		0			0
$675$			985.948i			0.0562210i
$676$		0			0
$677$		−	25683.6i		−	1.45805i	−0.684486	−	0.729026i	$-0.739974\pi$
$677$		−	25683.6i		−	1.45805i	0.684486	−	0.729026i	$-0.260026\pi$
$678$		0			0
$679$	−21309.3			−1.20439
$680$		0			0
$681$	−883.214			−0.0496988
$682$		0			0
$683$			28196.4i			1.57966i	0.613327	+	0.789829i	$0.289831\pi$
$683$			28196.4i			1.57966i	−0.613327	+	0.789829i	$0.710169\pi$
$684$		0			0
$685$			218.976i			0.0122141i
$686$		0			0
$687$			8063.62i			0.447811i
$688$		0			0
$689$	−41224.0			−2.27941
$690$		0			0
$691$			35037.4i			1.92892i	0.264221	+	0.964462i	$0.414885\pi$
$691$			35037.4i			1.92892i	−0.264221	+	0.964462i	$0.585115\pi$
$692$		0			0
$693$	−554.230			−0.0303802
$694$		0			0
$695$	3193.94			0.174321
$696$		0			0
$697$	12977.7	−	16322.0i	0.705261	−	0.887002i
$698$		0			0
$699$	17135.6			0.927220
$700$		0			0
$701$	−3539.51			−0.190707			−0.0953535	−	0.995443i	$-0.530398\pi$
$701$	−3539.51			−0.190707			−0.0953535	+	0.995443i	$0.530398\pi$
$702$		0			0
$703$			29275.6i			1.57063i
$704$		0			0
$705$	−6036.33			−0.322470
$706$		0			0
$707$		−	5158.62i		−	0.274413i
$708$		0			0
$709$			7264.39i			0.384795i	0.981317	+	0.192398i	$0.0616263\pi$
$709$			7264.39i			0.384795i	−0.981317	+	0.192398i	$0.938374\pi$
$710$		0			0
$711$			2218.30i			0.117008i
$712$		0			0
$713$	38122.6			2.00239
$714$		0			0
$715$	804.084			0.0420574
$716$		0			0
$717$			8995.99i			0.468565i
$718$		0			0
$719$		−	9704.92i		−	0.503383i	−0.967807	−	0.251692i	$-0.919013\pi$
$719$		−	9704.92i		−	0.503383i	0.967807	−	0.251692i	$-0.0809869\pi$
$720$		0			0
$721$			29503.4i			1.52395i
$722$		0			0
$723$	3439.99			0.176949
$724$		0			0
$725$			918.097i			0.0470307i
$726$		0			0
$727$	10937.3			0.557967			0.278983	−	0.960296i	$-0.410003\pi$
$727$	10937.3			0.557967			0.278983	+	0.960296i	$0.410003\pi$
$728$		0			0
$729$	−4317.73			−0.219364
$730$		0			0
$731$	−9587.29	−	7622.91i	−0.485087	−	0.385696i
$732$		0			0
$733$	14196.3			0.715351			0.357675	−	0.933846i	$-0.383569\pi$
$733$	14196.3			0.715351			0.357675	+	0.933846i	$0.383569\pi$
$734$		0			0
$735$	−5689.27			−0.285513
$736$		0			0
$737$			199.536i			0.00997287i
$738$		0			0
$739$	8477.05			0.421966			0.210983	−	0.977490i	$-0.432333\pi$
$739$	8477.05			0.421966			0.210983	+	0.977490i	$0.432333\pi$
$740$		0			0
$741$		−	21261.7i		−	1.05407i
$742$		0			0
$743$		−	5251.86i		−	0.259316i	−0.991559	−	0.129658i	$-0.958612\pi$
$743$		−	5251.86i		−	0.259316i	0.991559	−	0.129658i	$-0.0413880\pi$
$744$		0			0
$745$			1481.72i			0.0728674i
$746$		0			0
$747$	2388.93			0.117010
$748$		0			0
$749$	−25331.1			−1.23575
$750$		0			0
$751$		−	16018.0i		−	0.778302i	−0.921174	−	0.389151i	$-0.872768\pi$
$751$		−	16018.0i		−	0.778302i	0.921174	−	0.389151i	$-0.127232\pi$
$752$		0			0
$753$		−	17032.5i		−	0.824303i
$754$		0			0
$755$		−	24187.2i		−	1.16591i
$756$		0			0
$757$	−9344.23			−0.448642			−0.224321	−	0.974515i	$-0.572016\pi$
$757$	−9344.23			−0.448642			−0.224321	+	0.974515i	$0.572016\pi$
$758$		0			0
$759$			566.397i			0.0270868i
$760$		0			0
$761$	35043.1			1.66927			0.834633	−	0.550807i	$-0.185680\pi$
$761$	35043.1			1.66927			0.834633	+	0.550807i	$0.185680\pi$
$762$		0			0
$763$	16902.9			0.801998
$764$		0			0
$765$	10126.6	−	12736.1i	0.478598	−	0.601929i
$766$		0			0
$767$	−10667.5			−0.502190
$768$		0			0
$769$	−6719.33			−0.315091			−0.157546	−	0.987512i	$-0.550358\pi$
$769$	−6719.33			−0.315091			−0.157546	+	0.987512i	$0.550358\pi$
$770$		0			0
$771$			12776.4i			0.596796i
$772$		0			0
$773$	18716.9			0.870892			0.435446	−	0.900215i	$-0.356591\pi$
$773$	18716.9			0.870892			0.435446	+	0.900215i	$0.356591\pi$
$774$		0			0
$775$		−	1681.92i		−	0.0779568i
$776$		0			0
$777$		−	12725.9i		−	0.587567i
$778$		0			0
$779$			41343.7i			1.90153i
$780$		0			0
$781$	977.304			0.0447768
$782$		0			0
$783$	−14209.0			−0.648518
$784$		0			0
$785$			33575.8i			1.52659i
$786$		0			0
$787$		−	15916.1i		−	0.720898i	−0.932779	−	0.360449i	$-0.882623\pi$
$787$		−	15916.1i		−	0.720898i	0.932779	−	0.360449i	$-0.117377\pi$
$788$		0			0
$789$		−	3885.59i		−	0.175324i
$790$		0			0
$791$	−37739.4			−1.69641
$792$		0			0
$793$		−	28573.5i		−	1.27954i
$794$		0			0
$795$	−21345.0			−0.952239
$796$		0			0
$797$	39802.4			1.76898			0.884489	−	0.466562i	$-0.154508\pi$
$797$	39802.4			1.76898			0.884489	+	0.466562i	$0.154508\pi$
$798$		0			0
$799$	10957.3	+	8712.18i	0.485156	+	0.385751i
$800$		0			0
$801$	8058.88			0.355489
$802$		0			0
$803$	1302.20			0.0572276
$804$		0			0
$805$		−	48084.3i		−	2.10528i
$806$		0			0
$807$	13258.9			0.578357
$808$		0			0
$809$			12250.3i			0.532384i	0.963920	+	0.266192i	$0.0857655\pi$
$809$			12250.3i			0.532384i	−0.963920	+	0.266192i	$0.914234\pi$
$810$		0			0
$811$		−	15936.1i		−	0.690002i	−0.938602	−	0.345001i	$-0.887879\pi$
$811$		−	15936.1i		−	0.690002i	0.938602	−	0.345001i	$-0.112121\pi$
$812$		0			0
$813$		−	2971.39i		−	0.128181i
$814$		0			0
$815$	−18154.0			−0.780256
$816$		0			0
$817$	24284.6			1.03992
$818$		0			0
$819$		−	27083.8i		−	1.15554i
$820$		0			0
$821$		−	10389.6i		−	0.441656i	−0.975313	−	0.220828i	$-0.929124\pi$
$821$		−	10389.6i		−	0.441656i	0.975313	−	0.220828i	$-0.0708759\pi$
$822$		0			0
$823$		−	1915.45i		−	0.0811281i	−0.999177	−	0.0405640i	$-0.987085\pi$
$823$		−	1915.45i		−	0.0811281i	0.999177	−	0.0405640i	$-0.0129155\pi$
$824$		0			0
$825$	24.9888			0.00105454
$826$		0			0
$827$		−	1406.72i		−	0.0591492i	−0.999563	−	0.0295746i	$-0.990585\pi$
$827$		−	1406.72i		−	0.0591492i	0.999563	−	0.0295746i	$-0.00941527\pi$
$828$		0			0
$829$	−22131.8			−0.927226			−0.463613	−	0.886038i	$-0.653447\pi$
$829$	−22131.8			−0.927226			−0.463613	+	0.886038i	$0.653447\pi$
$830$		0			0
$831$	−9693.25			−0.404639
$832$		0			0
$833$	10327.3	+	8211.28i	0.429554	+	0.341541i
$834$		0			0
$835$	14877.1			0.616578
$836$		0			0
$837$	26030.5			1.07496
$838$		0			0
$839$		−	5716.82i		−	0.235240i	−0.993059	−	0.117620i	$-0.962473\pi$
$839$		−	5716.82i		−	0.235240i	0.993059	−	0.117620i	$-0.0375265\pi$
$840$		0			0
$841$	11157.8			0.457493
$842$		0			0
$843$		−	5740.16i		−	0.234521i
$844$		0			0
$845$			13958.2i			0.568257i
$846$		0			0
$847$		−	30644.7i		−	1.24317i
$848$		0			0
$849$	21386.3			0.864516
$850$		0			0
$851$	38110.8			1.53516
$852$		0			0
$853$		−	26801.3i		−	1.07580i	−0.843008	−	0.537900i	$-0.819218\pi$
$853$		−	26801.3i		−	1.07580i	0.843008	−	0.537900i	$-0.180782\pi$
$854$		0			0
$855$			32260.6i			1.29040i
$856$		0			0
$857$			6701.95i			0.267135i	0.991040	+	0.133567i	$0.0426432\pi$
$857$			6701.95i			0.267135i	−0.991040	+	0.133567i	$0.957357\pi$
$858$		0			0
$859$	33950.5			1.34852			0.674259	−	0.738495i	$-0.264463\pi$
$859$	33950.5			1.34852			0.674259	+	0.738495i	$0.264463\pi$
$860$		0			0
$861$		−	17971.8i		−	0.711357i
$862$		0			0
$863$	−43090.0			−1.69965			−0.849827	−	0.527062i	$-0.823294\pi$
$863$	−43090.0			−1.69965			−0.849827	+	0.527062i	$0.823294\pi$
$864$		0			0
$865$	−27332.5			−1.07437
$866$		0			0
$867$	12545.6	−	2901.73i	0.491433	−	0.113665i
$868$		0			0
$869$	131.631			0.00513842
$870$		0			0
$871$	−9750.81			−0.379327
$872$		0			0
$873$		−	18611.5i		−	0.721540i
$874$		0			0
$875$	31102.3			1.20166
$876$		0			0
$877$			38515.6i			1.48299i	0.670960	+	0.741493i	$0.265882\pi$
$877$			38515.6i			1.48299i	−0.670960	+	0.741493i	$0.734118\pi$
$878$		0			0
$879$		−	12521.7i		−	0.480485i
$880$		0			0
$881$		−	32178.8i		−	1.23057i	−0.788305	−	0.615284i	$-0.789041\pi$
$881$		−	32178.8i		−	1.23057i	0.788305	−	0.615284i	$-0.210959\pi$
$882$		0			0
$883$	−33578.7			−1.27974			−0.639871	−	0.768482i	$-0.721012\pi$
$883$	−33578.7			−1.27974			−0.639871	+	0.768482i	$0.721012\pi$
$884$		0			0
$885$	−5523.41			−0.209794
$886$		0			0
$887$			17336.8i			0.656273i	0.944630	+	0.328137i	$0.106421\pi$
$887$			17336.8i			0.656273i	−0.944630	+	0.328137i	$0.893579\pi$
$888$		0			0
$889$			34635.3i			1.30667i
$890$		0			0
$891$		−	262.507i		−	0.00987018i
$892$		0			0
$893$	−27754.7			−1.04006
$894$		0			0
$895$			19833.2i			0.740726i
$896$		0			0
$897$	−27678.4			−1.03027
$898$		0			0
$899$	24239.1			0.899244
$900$		0			0
$901$	38745.9	+	30807.1i	1.43264	+	1.13910i
$902$		0			0
$903$	−10556.4			−0.389029
$904$		0			0
$905$	−711.626			−0.0261384
$906$		0			0
$907$			2865.79i			0.104914i	0.998623	+	0.0524570i	$0.0167052\pi$
$907$			2865.79i			0.104914i	−0.998623	+	0.0524570i	$0.983295\pi$
$908$		0			0
$909$	4505.52			0.164399
$910$		0			0
$911$			23972.0i			0.871819i	0.899990	+	0.435910i	$0.143573\pi$
$911$			23972.0i			0.871819i	−0.899990	+	0.435910i	$0.856427\pi$
$912$		0			0
$913$		−	141.756i		−	0.00513849i
$914$		0			0
$915$		−	14794.8i		−	0.534538i
$916$		0			0
$917$	−16188.8			−0.582991
$918$		0			0
$919$	49371.5			1.77216			0.886081	−	0.463530i	$-0.153418\pi$
$919$	49371.5			1.77216			0.886081	+	0.463530i	$0.153418\pi$
$920$		0			0
$921$		−	1392.01i		−	0.0498026i
$922$		0			0
$923$			47758.3i			1.70313i
$924$		0			0
$925$		−	1681.40i		−	0.0597668i
$926$		0			0
$927$	−25768.2			−0.912986
$928$		0			0
$929$		−	47434.7i		−	1.67522i	−0.546266	−	0.837612i	$-0.683951\pi$
$929$		−	47434.7i		−	1.67522i	0.546266	−	0.837612i	$-0.316049\pi$
$930$		0			0
$931$	−26159.0			−0.920866
$932$		0			0
$933$	12494.9			0.438439
$934$		0			0
$935$	−755.747	−	600.899i	−0.0264338	−	0.0210176i
$936$		0			0
$937$	37330.2			1.30152			0.650760	−	0.759283i	$-0.274450\pi$
$937$	37330.2			1.30152			0.650760	+	0.759283i	$0.274450\pi$
$938$		0			0
$939$	−22575.7			−0.784589
$940$		0			0
$941$			32070.6i			1.11102i	0.831509	+	0.555511i	$0.187477\pi$
$941$			32070.6i			1.11102i	−0.831509	+	0.555511i	$0.812523\pi$
$942$		0			0
$943$	53820.9			1.85859
$944$		0			0
$945$		−	32832.5i		−	1.13020i
$946$		0			0
$947$		−	23171.6i		−	0.795118i	−0.917577	−	0.397559i	$-0.869857\pi$
$947$		−	23171.6i		−	0.795118i	0.917577	−	0.397559i	$-0.130143\pi$
$948$		0			0
$949$			63635.3i			2.17670i
$950$		0			0
$951$	−6429.77			−0.219242
$952$		0			0
$953$	−39914.2			−1.35671			−0.678357	−	0.734733i	$-0.737307\pi$
$953$	−39914.2			−1.35671			−0.678357	+	0.734733i	$0.737307\pi$
$954$		0			0
$955$		−	29582.8i		−	1.00239i
$956$		0			0
$957$			360.127i			0.0121643i
$958$		0			0
$959$		−	437.666i		−	0.0147372i
$960$		0			0
$961$	−14614.3			−0.490560
$962$		0			0
$963$		−	22124.1i		−	0.740332i
$964$		0			0
$965$	−22967.3			−0.766158
$966$		0			0
$967$	−40591.6			−1.34988			−0.674941	−	0.737871i	$-0.735831\pi$
$967$	−40591.6			−1.34988			−0.674941	+	0.737871i	$0.735831\pi$
$968$		0			0
$969$	−15889.1	+	19983.6i	−0.526760	+	0.662503i
$970$		0			0
$971$	29860.5			0.986890			0.493445	−	0.869777i	$-0.335737\pi$
$971$	29860.5			0.986890			0.493445	+	0.869777i	$0.335737\pi$
$972$		0			0
$973$	−6383.73			−0.210332
$974$		0			0
$975$			1221.14i			0.0401104i
$976$		0			0
$977$	−35041.0			−1.14745			−0.573726	−	0.819048i	$-0.694502\pi$
$977$	−35041.0			−1.14745			−0.573726	+	0.819048i	$0.694502\pi$
$978$		0			0
$979$		−	478.204i		−	0.0156113i
$980$		0			0
$981$			14762.9i			0.480472i
$982$		0			0
$983$			54091.0i			1.75507i	0.479513	+	0.877535i	$0.340813\pi$
$983$			54091.0i			1.75507i	−0.479513	+	0.877535i	$0.659187\pi$
$984$		0			0
$985$	−44187.0			−1.42935
$986$		0			0
$987$	12064.8			0.389085
$988$		0			0
$989$		−	31613.5i		−	1.01643i
$990$		0			0
$991$		−	15918.2i		−	0.510251i	−0.966908	−	0.255126i	$-0.917883\pi$
$991$		−	15918.2i		−	0.510251i	0.966908	−	0.255126i	$-0.0821168\pi$
$992$		0			0
$993$			28839.9i			0.921657i
$994$		0			0
$995$	−25039.8			−0.797805
$996$		0			0
$997$		−	15294.7i		−	0.485845i	−0.970046	−	0.242923i	$-0.921894\pi$
$997$		−	15294.7i		−	0.485845i	0.970046	−	0.242923i	$-0.0781061\pi$
$998$		0			0
$999$	26022.4			0.824138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.4.b.b.33.4	✓	8
3.2	odd	2		1224.4.c.e.577.3		8
4.3	odd	2		272.4.b.f.33.5		8
17.4	even	4		2312.4.a.k.1.4		8
17.13	even	4		2312.4.a.k.1.5		8
17.16	even	2	inner	136.4.b.b.33.5	yes	8
51.50	odd	2		1224.4.c.e.577.6		8
68.67	odd	2		272.4.b.f.33.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.4	✓	8	1.1	even	1	trivial
136.4.b.b.33.5	yes	8	17.16	even	2	inner
272.4.b.f.33.4		8	68.67	odd	2
272.4.b.f.33.5		8	4.3	odd	2
1224.4.c.e.577.3		8	3.2	odd	2
1224.4.c.e.577.6		8	51.50	odd	2
2312.4.a.k.1.4		8	17.4	even	4
2312.4.a.k.1.5		8	17.13	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)

\(f(q)\)	\(=\)	\(q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} +O(q^{10})\)	\(q-2.62097i q^{3} +11.5318i q^{5} -23.0485i q^{7} +20.1305 q^{9} -1.19452i q^{11} +58.3731 q^{13} +30.2245 q^{15} +(-54.8640 - 43.6227i) q^{17} +138.971 q^{19} -60.4096 q^{21} -180.911i q^{23} -7.98158 q^{25} -123.528i q^{27} -115.027i q^{29} +210.726i q^{31} -3.13080 q^{33} +265.790 q^{35} +210.661i q^{37} -152.994i q^{39} +297.500i q^{41} +174.746 q^{43} +232.140i q^{45} -199.717 q^{47} -188.234 q^{49} +(-114.334 + 143.797i) q^{51} -706.217 q^{53} +13.7749 q^{55} -364.238i q^{57} -182.746 q^{59} -489.499i q^{61} -463.978i q^{63} +673.144i q^{65} -167.043 q^{67} -474.163 q^{69} +818.157i q^{71} +1090.15i q^{73} +20.9195i q^{75} -27.5319 q^{77} +110.196i q^{79} +219.760 q^{81} +118.672 q^{83} +(503.047 - 632.679i) q^{85} -301.483 q^{87} +400.332 q^{89} -1345.41i q^{91} +552.306 q^{93} +1602.58i q^{95} -924.543i q^{97} -24.0463i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 132 q^{9}+O(q^{10}) \) 8 * q - 132 * q^9	\( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) \) 8 * q - 132 * q^9 + 44 * q^13 + 24 * q^15 + 28 * q^17 + 48 * q^19 + 308 * q^21 - 520 * q^25 + 812 * q^33 - 1064 * q^35 + 8 * q^43 + 312 * q^47 - 1124 * q^49 + 408 * q^51 + 472 * q^53 + 1416 * q^55 - 72 * q^59 - 624 * q^67 - 180 * q^69 + 1660 * q^77 + 3156 * q^81 + 2472 * q^83 - 2160 * q^85 - 6664 * q^87 + 68 * q^89 - 4036 * q^93

Introduction

L-functions

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Varieties

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Database

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