LMFDB - Embedded newform 1224.4.c.e.577.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,4,Mod(577,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1224.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1224.c (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:	$72.2183378470$
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:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.6
Root			$-9.30031i$ of defining polynomial
Character	$\chi$	$=$	1224.577
Dual form			1224.4.c.e.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+11.5318i q^{5} +23.0485i q^{7} +O(q^{10})$	$q+11.5318i q^{5} +23.0485i q^{7} -1.19452i q^{11} +58.3731 q^{13} +(54.8640 - 43.6227i) q^{17} +138.971 q^{19} -180.911i q^{23} -7.98158 q^{25} -115.027i q^{29} -210.726i q^{31} -265.790 q^{35} -210.661i q^{37} +297.500i q^{41} +174.746 q^{43} +199.717 q^{47} -188.234 q^{49} +706.217 q^{53} +13.7749 q^{55} +182.746 q^{59} +489.499i q^{61} +673.144i q^{65} -167.043 q^{67} +818.157i q^{71} -1090.15i q^{73} +27.5319 q^{77} -110.196i q^{79} -118.672 q^{83} +(503.047 + 632.679i) q^{85} -400.332 q^{89} +1345.41i q^{91} +1602.58i q^{95} +924.543i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) $ 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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	$0.213782\pi$
$7$			23.0485i			1.24450i	−0.782817	+	0.622251i	$0.786218\pi$
$8$		0			0
$9$		0			0
$10$		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
$15$		0			0
$16$		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
$21$		0			0
$22$		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
$27$		0			0
$28$		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	$-0.790992\pi$
$31$		−	210.726i		−	1.22088i	0.792061	−	0.610442i	$-0.209008\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−265.790			−1.28362
$36$		0			0
$37$		−	210.661i		−	0.936010i	−0.883726	−	0.468005i	$-0.844973\pi$
$37$		−	210.661i		−	0.936010i	0.883726	−	0.468005i	$-0.155027\pi$
$38$		0			0
$39$		0			0
$40$		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
$45$		0			0
$46$		0			0
$47$	199.717			0.619823			0.309911	−	0.950765i	$-0.399701\pi$
$47$	199.717			0.619823			0.309911	+	0.950765i	$0.399701\pi$
$48$		0			0
$49$	−188.234			−0.548787
$50$		0			0
$51$		0			0
$52$		0			0
$53$	706.217			1.83031			0.915154	−	0.403105i	$-0.132069\pi$
$53$	706.217			1.83031			0.915154	+	0.403105i	$0.132069\pi$
$54$		0			0
$55$	13.7749			0.0337711
$56$		0			0
$57$		0			0
$58$		0			0
$59$	182.746			0.403247			0.201623	−	0.979463i	$-0.435378\pi$
$59$	182.746			0.403247			0.201623	+	0.979463i	$0.435378\pi$
$60$		0			0
$61$			489.499i			1.02744i	0.857958	+	0.513720i	$0.171733\pi$
$61$			489.499i			1.02744i	−0.857958	+	0.513720i	$0.828267\pi$
$62$		0			0
$63$		0			0
$64$		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
$69$		0			0
$70$		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	$-0.661570\pi$
$73$		−	1090.15i		−	1.74784i	0.486071	−	0.873919i	$-0.338430\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	27.5319			0.0407474
$78$		0			0
$79$		−	110.196i		−	0.156937i	−0.996917	−	0.0784685i	$-0.974997\pi$
$79$		−	110.196i		−	0.156937i	0.996917	−	0.0784685i	$-0.0250030\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−118.672			−0.156939			−0.0784696	−	0.996917i	$-0.525003\pi$
$83$	−118.672			−0.156939			−0.0784696	+	0.996917i	$0.525003\pi$
$84$		0			0
$85$	503.047	+	632.679i	0.641919	+	0.807337i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−400.332			−0.476799			−0.238399	−	0.971167i	$-0.576623\pi$
$89$	−400.332			−0.476799			−0.238399	+	0.971167i	$0.576623\pi$
$90$		0			0
$91$			1345.41i			1.54986i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			1602.58i			1.73075i
$96$		0			0
$97$			924.543i			0.967764i	0.875133	+	0.483882i	$0.160774\pi$
$97$			924.543i			0.967764i	−0.875133	+	0.483882i	$0.839226\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−223.816			−0.220500			−0.110250	−	0.993904i	$-0.535165\pi$
$101$	−223.816			−0.220500			−0.110250	+	0.993904i	$0.535165\pi$
$102$		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
$105$		0			0
$106$		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	$-0.895572\pi$
$109$		−	733.360i		−	0.644433i	0.946666	−	0.322216i	$-0.104428\pi$
$110$		0			0
$111$		0			0
$112$		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
$117$		0			0
$118$		0			0
$119$	1005.44	+	1264.53i	0.774525	+	0.974115i
$120$		0			0
$121$	1329.57			0.998928
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		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
$135$		0			0
$136$		0			0
$137$	−18.9889			−0.0118418			−0.00592092	−	0.999982i	$-0.501885\pi$
$137$	−18.9889			−0.0118418			−0.00592092	+	0.999982i	$0.501885\pi$
$138$		0			0
$139$			276.969i			0.169009i	0.996423	+	0.0845043i	$0.0269307\pi$
$139$			276.969i			0.169009i	−0.996423	+	0.0845043i	$0.973069\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	69.7277i		−	0.0407757i
$144$		0			0
$145$	1326.46			0.759702
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−128.491			−0.0706468			−0.0353234	−	0.999376i	$-0.511246\pi$
$149$	−128.491			−0.0706468			−0.0353234	+	0.999376i	$0.511246\pi$
$150$		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
$153$		0			0
$154$		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
$159$		0			0
$160$		0			0
$161$	4169.73			2.04112
$162$		0			0
$163$		−	1574.26i		−	0.756478i	−0.925708	−	0.378239i	$-0.876530\pi$
$163$		−	1574.26i		−	0.756478i	0.925708	−	0.378239i	$-0.123470\pi$
$164$		0			0
$165$		0			0
$166$		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
$171$		0			0
$172$		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
$177$		0			0
$178$		0			0
$179$	−1719.87			−0.718153			−0.359077	−	0.933308i	$-0.616908\pi$
$179$	−1719.87			−0.718153			−0.359077	+	0.933308i	$0.616908\pi$
$180$		0			0
$181$		−	61.7101i		−	0.0253418i	−0.999920	−	0.0126709i	$-0.995967\pi$
$181$		−	61.7101i		−	0.0253418i	0.999920	−	0.0126709i	$-0.00403339\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2429.29			0.965431
$186$		0			0
$187$	−52.1082	−	65.5361i	−0.0203771	−	0.0256282i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2565.34			0.971839			0.485919	−	0.874004i	$-0.338485\pi$
$191$	2565.34			0.971839			0.485919	+	0.874004i	$0.338485\pi$
$192$		0			0
$193$		−	1991.65i		−	0.742810i	−0.928471	−	0.371405i	$-0.878876\pi$
$193$		−	1991.65i		−	0.742810i	0.928471	−	0.371405i	$-0.121124\pi$
$194$		0			0
$195$		0			0
$196$		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	$-0.873599\pi$
$199$		−	2171.38i		−	0.773492i	0.922186	−	0.386746i	$-0.126401\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2651.20			0.916639
$204$		0			0
$205$	−3430.69			−1.16883
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	166.003i		−	0.0549410i
$210$		0			0
$211$			2069.97i			0.675369i	0.941259	+	0.337685i	$0.109644\pi$
$211$			2069.97i			0.675369i	−0.941259	+	0.337685i	$0.890356\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			2015.13i			0.639214i
$216$		0			0
$217$	4856.91			1.51939
$218$		0			0
$219$		0			0
$220$		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
$225$		0			0
$226$		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
$231$		0			0
$232$		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
$237$		0			0
$238$		0			0
$239$	3432.31			0.928943			0.464472	−	0.885588i	$-0.346244\pi$
$239$	3432.31			0.928943			0.464472	+	0.885588i	$0.346244\pi$
$240$		0			0
$241$		−	1312.48i		−	0.350807i	−0.984497	−	0.175404i	$-0.943877\pi$
$241$		−	1312.48i		−	0.350807i	0.984497	−	0.175404i	$-0.0561230\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	2170.67i		−	0.566037i
$246$		0			0
$247$	8112.14			2.08973
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−6498.55			−1.63420			−0.817101	−	0.576494i	$-0.804420\pi$
$251$	−6498.55			−1.63420			−0.817101	+	0.576494i	$0.804420\pi$
$252$		0			0
$253$	−216.102			−0.0537004
$254$		0			0
$255$		0			0
$256$		0			0
$257$	4874.66			1.18316			0.591582	−	0.806245i	$-0.298504\pi$
$257$	4874.66			1.18316			0.591582	+	0.806245i	$0.298504\pi$
$258$		0			0
$259$	4855.41			1.16487
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1482.50			−0.347584			−0.173792	−	0.984782i	$-0.555602\pi$
$263$	−1482.50			−0.347584			−0.173792	+	0.984782i	$0.555602\pi$
$264$		0			0
$265$			8143.92i			1.88784i
$266$		0			0
$267$		0			0
$268$		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
$273$		0			0
$274$		0			0
$275$			9.53415i			0.00209066i
$276$		0			0
$277$			3698.34i			0.802208i	0.916033	+	0.401104i	$0.131373\pi$
$277$			3698.34i			0.802208i	−0.916033	+	0.401104i	$0.868627\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2190.08			−0.464945			−0.232472	−	0.972603i	$-0.574682\pi$
$281$	−2190.08			−0.464945			−0.232472	+	0.972603i	$0.574682\pi$
$282$		0			0
$283$		−	8159.66i		−	1.71393i	−0.515378	−	0.856963i	$-0.672348\pi$
$283$		−	8159.66i		−	1.71393i	0.515378	−	0.856963i	$-0.327652\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6856.92			−1.41028
$288$		0			0
$289$	1107.12	−	4786.63i	0.225345	−	0.974279i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4777.49			−0.952574			−0.476287	−	0.879290i	$-0.658018\pi$
$293$	−4777.49			−0.952574			−0.476287	+	0.879290i	$0.658018\pi$
$294$		0			0
$295$			2107.39i			0.415922i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	10560.3i		−	2.04254i
$300$		0			0
$301$			4027.65i			0.771261i
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		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	$0.283631\pi$
$313$			8613.47i			1.55547i	−0.628593	+	0.777735i	$0.716369\pi$
$314$		0			0
$315$		0			0
$316$		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
$321$		0			0
$322$		0			0
$323$	7624.48	−	6062.27i	1.31343	−	1.04432i
$324$		0			0
$325$	−465.909			−0.0795200
$326$		0			0
$327$		0			0
$328$		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
$333$		0			0
$334$		0			0
$335$		−	1926.30i		−	0.314164i
$336$		0			0
$337$		−	3664.37i		−	0.592318i	−0.955139	−	0.296159i	$-0.904294\pi$
$337$		−	3664.37i		−	0.592318i	0.955139	−	0.296159i	$-0.0957058\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−251.716			−0.0399741
$342$		0			0
$343$			3567.12i			0.561535i
$344$		0			0
$345$		0			0
$346$		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
$351$		0			0
$352$		0			0
$353$	−1216.36			−0.183400			−0.0917002	−	0.995787i	$-0.529230\pi$
$353$	−1216.36			−0.183400			−0.0917002	+	0.995787i	$0.529230\pi$
$354$		0			0
$355$	−9434.80			−1.41055
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7448.93			1.09510			0.547548	−	0.836774i	$-0.315561\pi$
$359$	7448.93			1.09510			0.547548	+	0.836774i	$0.315561\pi$
$360$		0			0
$361$	12453.8			1.81569
$362$		0			0
$363$		0			0
$364$		0			0
$365$	12571.3			1.80278
$366$		0			0
$367$		−	2374.59i		−	0.337746i	−0.985638	−	0.168873i	$-0.945987\pi$
$367$		−	2374.59i		−	0.337746i	0.985638	−	0.168873i	$-0.0540128\pi$
$368$		0			0
$369$		0			0
$370$		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
$375$		0			0
$376$		0			0
$377$		−	6714.48i		−	0.917276i
$378$		0			0
$379$		−	14129.9i		−	1.91505i	−0.288358	−	0.957523i	$-0.593109\pi$
$379$		−	14129.9i		−	1.91505i	0.288358	−	0.957523i	$-0.406891\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13306.4			−1.77526			−0.887631	−	0.460556i	$-0.847650\pi$
$383$	−13306.4			−1.77526			−0.887631	+	0.460556i	$0.847650\pi$
$384$		0			0
$385$			317.491i			0.0420282i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8007.84			1.04374			0.521868	−	0.853026i	$-0.325235\pi$
$389$	8007.84			1.04374			0.521868	+	0.853026i	$0.325235\pi$
$390$		0			0
$391$	−7891.83	−	9925.50i	−1.02073	−	1.28377i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1270.76			0.161870
$396$		0			0
$397$			13185.2i			1.66687i	0.552619	+	0.833434i	$0.313628\pi$
$397$			13185.2i			1.66687i	−0.552619	+	0.833434i	$0.686372\pi$
$398$		0			0
$399$		0			0
$400$		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
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$			4212.03i			0.501842i
$414$		0			0
$415$		−	1368.50i		−	0.161872i
$416$		0			0
$417$		0			0
$418$		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
$423$		0			0
$424$		0			0
$425$	−437.902	+	348.178i	−0.0499796	+	0.0397391i
$426$		0			0
$427$	−11282.2			−1.27865
$428$		0			0
$429$		0			0
$430$		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
$435$		0			0
$436$		0			0
$437$		−	25141.3i		−	2.75211i
$438$		0			0
$439$			15069.5i			1.63834i	0.573553	+	0.819168i	$0.305565\pi$
$439$			15069.5i			1.63834i	−0.573553	+	0.819168i	$0.694435\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3026.24			0.324562			0.162281	−	0.986745i	$-0.448115\pi$
$443$	3026.24			0.324562			0.162281	+	0.986745i	$0.448115\pi$
$444$		0			0
$445$		−	4616.53i		−	0.491786i
$446$		0			0
$447$		0			0
$448$		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
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	−2451.93			−0.247718			−0.123859	−	0.992300i	$-0.539527\pi$
$461$	−2451.93			−0.247718			−0.123859	+	0.992300i	$0.539527\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$	−15195.6			−1.50572			−0.752859	−	0.658182i	$-0.771326\pi$
$467$	−15195.6			−1.50572			−0.752859	+	0.658182i	$0.771326\pi$
$468$		0			0
$469$		−	3850.09i		−	0.379063i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	208.738i		−	0.0202913i
$474$		0			0
$475$	−1109.21			−0.107145
$476$		0			0
$477$		0			0
$478$		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
$483$		0			0
$484$		0			0
$485$	−10661.6			−0.998183
$486$		0			0
$487$			17772.7i			1.65371i	0.562414	+	0.826856i	$0.309873\pi$
$487$			17772.7i			1.65371i	−0.562414	+	0.826856i	$0.690127\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4845.34			−0.445350			−0.222675	−	0.974893i	$-0.571479\pi$
$491$	−4845.34			−0.445350			−0.222675	+	0.974893i	$0.571479\pi$
$492$		0			0
$493$	−5017.79	−	6310.84i	−0.458397	−	0.576523i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18857.3			−1.70194
$498$		0			0
$499$			1060.23i			0.0951153i	0.998868	+	0.0475576i	$0.0151438\pi$
$499$			1060.23i			0.0951153i	−0.998868	+	0.0475576i	$0.984856\pi$
$500$		0			0
$501$		0			0
$502$		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
$507$		0			0
$508$		0			0
$509$	−1658.97			−0.144465			−0.0722323	−	0.997388i	$-0.523012\pi$
$509$	−1658.97			−0.144465			−0.0722323	+	0.997388i	$0.523012\pi$
$510$		0			0
$511$	25126.3			2.17519
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	14761.3i		−	1.26303i
$516$		0			0
$517$		−	238.565i		−	0.0202942i
$518$		0			0
$519$		0			0
$520$		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
$525$		0			0
$526$		0			0
$527$	−9192.42	−	11561.2i	−0.759826	−	0.955628i
$528$		0			0
$529$	−20561.8			−1.68996
$530$		0			0
$531$		0			0
$532$		0			0
$533$			17366.0i			1.41126i
$534$		0			0
$535$	12673.8			1.02418
$536$		0			0
$537$		0			0
$538$		0			0
$539$			224.849i			0.0179683i
$540$		0			0
$541$			10632.7i			0.844984i	0.906367	+	0.422492i	$0.138845\pi$
$541$			10632.7i			0.844984i	−0.906367	+	0.422492i	$0.861155\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8456.94			0.664689
$546$		0			0
$547$		−	17370.0i		−	1.35774i	−0.734257	−	0.678872i	$-0.762469\pi$
$547$		−	17370.0i		−	1.35774i	0.734257	−	0.678872i	$-0.237531\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	15985.4i		−	1.23593i
$552$		0			0
$553$	2539.86			0.195309
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1084.77			−0.0825190			−0.0412595	−	0.999148i	$-0.513137\pi$
$557$	−1084.77			−0.0825190			−0.0412595	+	0.999148i	$0.513137\pi$
$558$		0			0
$559$	10200.5			0.771797
$560$		0			0
$561$		0			0
$562$		0			0
$563$	5494.27			0.411289			0.205645	−	0.978627i	$-0.434071\pi$
$563$	5494.27			0.411289			0.205645	+	0.978627i	$0.434071\pi$
$564$		0			0
$565$	18882.0			1.40597
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11612.4			0.855569			0.427784	−	0.903881i	$-0.359294\pi$
$569$	11612.4			0.855569			0.427784	+	0.903881i	$0.359294\pi$
$570$		0			0
$571$		−	3413.97i		−	0.250210i	−0.992143	−	0.125105i	$-0.960073\pi$
$571$		−	3413.97i		−	0.250210i	0.992143	−	0.125105i	$-0.0399268\pi$
$572$		0			0
$573$		0			0
$574$		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
$579$		0			0
$580$		0			0
$581$		−	2735.22i		−	0.195311i
$582$		0			0
$583$		−	843.589i		−	0.0599278i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−25232.2			−1.77418			−0.887092	−	0.461593i	$-0.847278\pi$
$587$	−25232.2			−1.77418			−0.887092	+	0.461593i	$0.847278\pi$
$588$		0			0
$589$		−	29284.7i		−	2.04865i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14327.4			0.992171			0.496086	−	0.868274i	$-0.334770\pi$
$593$	14327.4			0.992171			0.496086	+	0.868274i	$0.334770\pi$
$594$		0			0
$595$	−14582.3	+	11594.5i	−1.00473	+	0.798870i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	2523.52			0.172134			0.0860671	−	0.996289i	$-0.472570\pi$
$599$	2523.52			0.172134			0.0860671	+	0.996289i	$0.472570\pi$
$600$		0			0
$601$			17286.6i			1.17327i	0.809853	+	0.586634i	$0.199547\pi$
$601$			17286.6i			1.17327i	−0.809853	+	0.586634i	$0.800453\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			15332.3i			1.03033i
$606$		0			0
$607$		−	6413.65i		−	0.428867i	−0.976739	−	0.214433i	$-0.931210\pi$
$607$		−	6413.65i		−	0.428867i	0.976739	−	0.214433i	$-0.0687905\pi$
$608$		0			0
$609$		0			0
$610$		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
$615$		0			0
$616$		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	$0.112577\pi$
$619$			10667.8i			0.692689i	−0.938107	+	0.346344i	$0.887423\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	9227.05i		−	0.593377i
$624$		0			0
$625$	−16559.0			−1.05978
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$		−	17328.9i		−	1.08296i
$636$		0			0
$637$	−10987.8			−0.683442
$638$		0			0
$639$		0			0
$640$		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	$0.0553543\pi$
$643$			5642.30i			0.346051i	−0.984917	+	0.173025i	$0.944646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7790.69			−0.473391			−0.236695	−	0.971584i	$-0.576064\pi$
$647$	−7790.69			−0.473391			−0.236695	+	0.971584i	$0.576064\pi$
$648$		0			0
$649$		−	218.294i		−	0.0132031i
$650$		0			0
$651$		0			0
$652$		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
$657$		0			0
$658$		0			0
$659$	14303.3			0.845492			0.422746	−	0.906248i	$-0.361066\pi$
$659$	14303.3			0.845492			0.422746	+	0.906248i	$0.361066\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	−36937.0			−2.15392
$666$		0			0
$667$	−20809.6			−1.20802
$668$		0			0
$669$		0			0
$670$		0			0
$671$	584.715			0.0336404
$672$		0			0
$673$			19026.5i			1.08977i	0.838509	+	0.544887i	$0.183428\pi$
$673$			19026.5i			1.08977i	−0.838509	+	0.544887i	$0.816572\pi$
$674$		0			0
$675$		0			0
$676$		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
$681$		0			0
$682$		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
$687$		0			0
$688$		0			0
$689$	41224.0			2.27941
$690$		0			0
$691$		−	35037.4i		−	1.92892i	−0.264221	−	0.964462i	$-0.585115\pi$
$691$		−	35037.4i		−	1.92892i	0.264221	−	0.964462i	$-0.414885\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3193.94			−0.174321
$696$		0			0
$697$	12977.7	+	16322.0i	0.705261	+	0.887002i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3539.51			0.190707			0.0953535	−	0.995443i	$-0.469602\pi$
$701$	3539.51			0.190707			0.0953535	+	0.995443i	$0.469602\pi$
$702$		0			0
$703$		−	29275.6i		−	1.57063i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	5158.62i		−	0.274413i
$708$		0			0
$709$		−	7264.39i		−	0.384795i	−0.981317	−	0.192398i	$-0.938374\pi$
$709$		−	7264.39i		−	0.384795i	0.981317	−	0.192398i	$-0.0616263\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−38122.6			−2.00239
$714$		0			0
$715$	804.084			0.0420574
$716$		0			0
$717$		0			0
$718$		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
$723$		0			0
$724$		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
$729$		0			0
$730$		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
$735$		0			0
$736$		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
$741$		0			0
$742$		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
$747$		0			0
$748$		0			0
$749$	25331.1			1.23575
$750$		0			0
$751$			16018.0i			0.778302i	0.921174	+	0.389151i	$0.127232\pi$
$751$			16018.0i			0.778302i	−0.921174	+	0.389151i	$0.872768\pi$
$752$		0			0
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−35043.1			−1.66927			−0.834633	−	0.550807i	$-0.814320\pi$
$761$	−35043.1			−1.66927			−0.834633	+	0.550807i	$0.814320\pi$
$762$		0			0
$763$	16902.9			0.801998
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$	−18716.9			−0.870892			−0.435446	−	0.900215i	$-0.643409\pi$
$773$	−18716.9			−0.870892			−0.435446	+	0.900215i	$0.643409\pi$
$774$		0			0
$775$			1681.92i			0.0779568i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			41343.7i			1.90153i
$780$		0			0
$781$	977.304			0.0447768
$782$		0			0
$783$		0			0
$784$		0			0
$785$			33575.8i			1.52659i
$786$		0			0
$787$			15916.1i			0.720898i	0.932779	+	0.360449i	$0.117377\pi$
$787$			15916.1i			0.720898i	−0.932779	+	0.360449i	$0.882623\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	37739.4			1.69641
$792$		0			0
$793$			28573.5i			1.27954i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−39802.4			−1.76898			−0.884489	−	0.466562i	$-0.845492\pi$
$797$	−39802.4			−1.76898			−0.884489	+	0.466562i	$0.845492\pi$
$798$		0			0
$799$	10957.3	−	8712.18i	0.485156	−	0.385751i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1302.20			−0.0572276
$804$		0			0
$805$			48084.3i			2.10528i
$806$		0			0
$807$		0			0
$808$		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	$0.112121\pi$
$811$			15936.1i			0.690002i	−0.938602	+	0.345001i	$0.887879\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18154.0			0.780256
$816$		0			0
$817$	24284.6			1.03992
$818$		0			0
$819$		0			0
$820$		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	$0.0129155\pi$
$823$			1915.45i			0.0811281i	−0.999177	+	0.0405640i	$0.987085\pi$
$824$		0			0
$825$		0			0
$826$		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
$831$		0			0
$832$		0			0
$833$	−10327.3	+	8211.28i	−0.429554	+	0.341541i
$834$		0			0
$835$	14877.1			0.616578
$836$		0			0
$837$		0			0
$838$		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
$843$		0			0
$844$		0			0
$845$			13958.2i			0.568257i
$846$		0			0
$847$			30644.7i			1.24317i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−38110.8			−1.53516
$852$		0			0
$853$			26801.3i			1.07580i	0.843008	+	0.537900i	$0.180782\pi$
$853$			26801.3i			1.07580i	−0.843008	+	0.537900i	$0.819218\pi$
$854$		0			0
$855$		0			0
$856$		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
$861$		0			0
$862$		0			0
$863$	43090.0			1.69965			0.849827	−	0.527062i	$-0.176706\pi$
$863$	43090.0			1.69965			0.849827	+	0.527062i	$0.176706\pi$
$864$		0			0
$865$	−27332.5			−1.07437
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−131.631			−0.00513842
$870$		0			0
$871$	−9750.81			−0.379327
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−31102.3			−1.20166
$876$		0			0
$877$		−	38515.6i		−	1.48299i	−0.670960	−	0.741493i	$-0.734118\pi$
$877$		−	38515.6i		−	1.48299i	0.670960	−	0.741493i	$-0.265882\pi$
$878$		0			0
$879$		0			0
$880$		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
$885$		0			0
$886$		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
$891$		0			0
$892$		0			0
$893$	27754.7			1.04006
$894$		0			0
$895$		−	19833.2i		−	0.740726i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24239.1			−0.899244
$900$		0			0
$901$	38745.9	−	30807.1i	1.43264	−	1.13910i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	711.626			0.0261384
$906$		0			0
$907$		−	2865.79i		−	0.104914i	−0.998623	−	0.0524570i	$-0.983295\pi$
$907$		−	2865.79i		−	0.104914i	0.998623	−	0.0524570i	$-0.0167052\pi$
$908$		0			0
$909$		0			0
$910$		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
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$			47758.3i			1.70313i
$924$		0			0
$925$			1681.40i			0.0597668i
$926$		0			0
$927$		0			0
$928$		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
$933$		0			0
$934$		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
$939$		0			0
$940$		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
$945$		0			0
$946$		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
$951$		0			0
$952$		0			0
$953$	39914.2			1.35671			0.678357	−	0.734733i	$-0.262693\pi$
$953$	39914.2			1.35671			0.678357	+	0.734733i	$0.262693\pi$
$954$		0			0
$955$			29582.8i			1.00239i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	437.666i		−	0.0147372i
$960$		0			0
$961$	−14614.3			−0.490560
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	−29860.5			−0.986890			−0.493445	−	0.869777i	$-0.664263\pi$
$971$	−29860.5			−0.986890			−0.493445	+	0.869777i	$0.664263\pi$
$972$		0			0
$973$	−6383.73			−0.210332
$974$		0			0
$975$		0			0
$976$		0			0
$977$	35041.0			1.14745			0.573726	−	0.819048i	$-0.305498\pi$
$977$	35041.0			1.14745			0.573726	+	0.819048i	$0.305498\pi$
$978$		0			0
$979$			478.204i			0.0156113i
$980$		0			0
$981$		0			0
$982$		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
$987$		0			0
$988$		0			0
$989$		−	31613.5i		−	1.01643i
$990$		0			0
$991$			15918.2i			0.510251i	0.966908	+	0.255126i	$0.0821168\pi$
$991$			15918.2i			0.510251i	−0.966908	+	0.255126i	$0.917883\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	25039.8			0.797805
$996$		0			0
$997$			15294.7i			0.485845i	0.970046	+	0.242923i	$0.0781061\pi$
$997$			15294.7i			0.485845i	−0.970046	+	0.242923i	$0.921894\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.b.b.33.4	✓	8	51.50	odd	2
136.4.b.b.33.5	yes	8	3.2	odd	2
272.4.b.f.33.4		8	12.11	even	2
272.4.b.f.33.5		8	204.203	even	2
1224.4.c.e.577.3		8	17.16	even	2	inner
1224.4.c.e.577.6		8	1.1	even	1	trivial
2312.4.a.k.1.4		8	51.47	odd	4
2312.4.a.k.1.5		8	51.38	odd	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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+11.5318i q^{5} +23.0485i q^{7} +O(q^{10})\)	\(q+11.5318i q^{5} +23.0485i q^{7} -1.19452i q^{11} +58.3731 q^{13} +(54.8640 - 43.6227i) q^{17} +138.971 q^{19} -180.911i q^{23} -7.98158 q^{25} -115.027i q^{29} -210.726i q^{31} -265.790 q^{35} -210.661i q^{37} +297.500i q^{41} +174.746 q^{43} +199.717 q^{47} -188.234 q^{49} +706.217 q^{53} +13.7749 q^{55} +182.746 q^{59} +489.499i q^{61} +673.144i q^{65} -167.043 q^{67} +818.157i q^{71} -1090.15i q^{73} +27.5319 q^{77} -110.196i q^{79} -118.672 q^{83} +(503.047 + 632.679i) q^{85} -400.332 q^{89} +1345.41i q^{91} +1602.58i q^{95} +924.543i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 44 q^{13} - 28 q^{17} + 48 q^{19} - 520 q^{25} + 1064 q^{35} + 8 q^{43} - 312 q^{47} - 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 1660 q^{77} - 2472 q^{83} - 2160 q^{85} - 68 q^{89}+O(q^{100}) \) 8 * q + 44 * q^13 - 28 * q^17 + 48 * q^19 - 520 * q^25 + 1064 * q^35 + 8 * q^43 - 312 * q^47 - 1124 * q^49 - 472 * q^53 + 1416 * q^55 + 72 * q^59 - 624 * q^67 - 1660 * q^77 - 2472 * q^83 - 2160 * q^85 - 68 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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