LMFDB - Embedded newform 1344.3.f.j.769.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,3,Mod(769,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1344.769"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,-48,0,0,0,0,0,0,0,0,0,0,0,-24,0,0,0,-64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2 x^{14} - 8 x^{13} - 57 x^{12} + 32 x^{11} + 466 x^{10} + 304 x^{9} + 3000 x^{8} + \cdots + 790321 $ x^16 + 2x^14 - 8x^13 - 57x^12 + 32x^11 + 466x^10 + 304x^9 + 3000x^8 + 1328x^7 - 11560x^6 + 22316x^5 - 23333x^4 - 87520x^3 - 135796x^2 - 88900x + 790321
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.10
Root			$2.79294 - 1.11885i$ of defining polynomial
Character	$\chi$	$=$	1344.769
Dual form			1344.3.f.j.769.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -5.15119i q^{5} +(2.53356 - 6.52542i) q^{7} -3.00000 q^{9} +14.9067 q^{11} +4.05181i q^{13} +8.92212 q^{15} +26.1871i q^{17} +10.3048i q^{19} +(11.3024 + 4.38825i) q^{21} -18.0230 q^{23} -1.53475 q^{25} -5.19615i q^{27} +42.4532 q^{29} +3.16847i q^{31} +25.8191i q^{33} +(-33.6137 - 13.0508i) q^{35} +34.4305 q^{37} -7.01793 q^{39} -41.4151i q^{41} -16.2894 q^{43} +15.4536i q^{45} +1.01371i q^{47} +(-36.1622 - 33.0651i) q^{49} -45.3574 q^{51} +84.3489 q^{53} -76.7870i q^{55} -17.8485 q^{57} -59.3790i q^{59} -86.6064i q^{61} +(-7.60067 + 19.5763i) q^{63} +20.8716 q^{65} +116.082 q^{67} -31.2167i q^{69} -97.7926 q^{71} +28.2973i q^{73} -2.65827i q^{75} +(37.7669 - 97.2722i) q^{77} +103.257 q^{79} +9.00000 q^{81} -34.8098i q^{83} +134.895 q^{85} +73.5311i q^{87} -64.9992i q^{89} +(26.4397 + 10.2655i) q^{91} -5.48795 q^{93} +53.0820 q^{95} +105.475i q^{97} -44.7200 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 48 q^{9} - 24 q^{21} - 64 q^{25} + 128 q^{29} - 48 q^{37} - 256 q^{49} + 160 q^{53} - 144 q^{57} - 288 q^{65} - 128 q^{77} + 144 q^{81} + 16 q^{85} - 96 q^{93}+O(q^{100}) $ 16 * q - 48 * q^9 - 24 * q^21 - 64 * q^25 + 128 * q^29 - 48 * q^37 - 256 * q^49 + 160 * q^53 - 144 * q^57 - 288 * q^65 - 128 * q^77 + 144 * q^81 + 16 * q^85 - 96 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$		0			0
$5$		−	5.15119i		−	1.03024i	−0.857119	−	0.515119i	$-0.827748\pi$
$5$		−	5.15119i		−	1.03024i	0.857119	−	0.515119i	$-0.172252\pi$
$6$		0			0
$7$	2.53356	−	6.52542i	0.361937	−	0.932203i
$7$	2.53356	−	6.52542i	0.361937	−	0.932203i
$8$		0			0
$9$	−3.00000			−0.333333
$10$		0			0
$11$	14.9067			1.35515			0.677575	−	0.735453i	$-0.263031\pi$
$11$	14.9067			1.35515			0.677575	+	0.735453i	$0.263031\pi$
$12$		0			0
$13$			4.05181i			0.311677i	0.987783	+	0.155839i	$0.0498080\pi$
$13$			4.05181i			0.311677i	−0.987783	+	0.155839i	$0.950192\pi$
$14$		0			0
$15$	8.92212			0.594808
$16$		0			0
$17$			26.1871i			1.54042i	0.637792	+	0.770209i	$0.279848\pi$
$17$			26.1871i			1.54042i	−0.637792	+	0.770209i	$0.720152\pi$
$18$		0			0
$19$			10.3048i			0.542359i	0.962529	+	0.271179i	$0.0874136\pi$
$19$			10.3048i			0.542359i	−0.962529	+	0.271179i	$0.912586\pi$
$20$		0			0
$21$	11.3024	+	4.38825i	0.538207	+	0.208964i
$22$		0			0
$23$	−18.0230			−0.783607			−0.391803	−	0.920049i	$-0.628149\pi$
$23$	−18.0230			−0.783607			−0.391803	+	0.920049i	$0.628149\pi$
$24$		0			0
$25$	−1.53475			−0.0613900
$26$		0			0
$27$		−	5.19615i		−	0.192450i
$28$		0			0
$29$	42.4532			1.46390			0.731951	−	0.681357i	$-0.238610\pi$
$29$	42.4532			1.46390			0.731951	+	0.681357i	$0.238610\pi$
$30$		0			0
$31$			3.16847i			0.102209i	0.998693	+	0.0511044i	$0.0162741\pi$
$31$			3.16847i			0.102209i	−0.998693	+	0.0511044i	$0.983726\pi$
$32$		0			0
$33$			25.8191i			0.782397i
$34$		0			0
$35$	−33.6137	−	13.0508i	−0.960390	−	0.372881i
$36$		0			0
$37$	34.4305			0.930553			0.465277	−	0.885165i	$-0.345955\pi$
$37$	34.4305			0.930553			0.465277	+	0.885165i	$0.345955\pi$
$38$		0			0
$39$	−7.01793			−0.179947
$40$		0			0
$41$		−	41.4151i		−	1.01012i	−0.863083	−	0.505062i	$-0.831470\pi$
$41$		−	41.4151i		−	1.01012i	0.863083	−	0.505062i	$-0.168530\pi$
$42$		0			0
$43$	−16.2894			−0.378823			−0.189412	−	0.981898i	$-0.560658\pi$
$43$	−16.2894			−0.378823			−0.189412	+	0.981898i	$0.560658\pi$
$44$		0			0
$45$			15.4536i			0.343413i
$46$		0			0
$47$			1.01371i			0.0215683i	0.999942	+	0.0107841i	$0.00343276\pi$
$47$			1.01371i			0.0215683i	−0.999942	+	0.0107841i	$0.996567\pi$
$48$		0			0
$49$	−36.1622	−	33.0651i	−0.738003	−	0.674797i
$50$		0			0
$51$	−45.3574			−0.889361
$52$		0			0
$53$	84.3489			1.59149			0.795744	−	0.605633i	$-0.207080\pi$
$53$	84.3489			1.59149			0.795744	+	0.605633i	$0.207080\pi$
$54$		0			0
$55$		−	76.7870i		−	1.39613i
$56$		0			0
$57$	−17.8485			−0.313131
$58$		0			0
$59$		−	59.3790i		−	1.00642i	−0.864163	−	0.503212i	$-0.832151\pi$
$59$		−	59.3790i		−	1.00642i	0.864163	−	0.503212i	$-0.167849\pi$
$60$		0			0
$61$		−	86.6064i		−	1.41978i	−0.704314	−	0.709889i	$-0.748745\pi$
$61$		−	86.6064i		−	1.41978i	0.704314	−	0.709889i	$-0.251255\pi$
$62$		0			0
$63$	−7.60067	+	19.5763i	−0.120646	+	0.310734i
$64$		0			0
$65$	20.8716			0.321102
$66$		0			0
$67$	116.082			1.73257			0.866286	−	0.499549i	$-0.166501\pi$
$67$	116.082			1.73257			0.866286	+	0.499549i	$0.166501\pi$
$68$		0			0
$69$		−	31.2167i		−	0.452416i
$70$		0			0
$71$	−97.7926			−1.37736			−0.688680	−	0.725065i	$-0.741810\pi$
$71$	−97.7926			−1.37736			−0.688680	+	0.725065i	$0.741810\pi$
$72$		0			0
$73$			28.2973i			0.387634i	0.981038	+	0.193817i	$0.0620868\pi$
$73$			28.2973i			0.387634i	−0.981038	+	0.193817i	$0.937913\pi$
$74$		0			0
$75$		−	2.65827i		−	0.0354436i
$76$		0			0
$77$	37.7669	−	97.2722i	0.490479	−	1.26327i
$78$		0			0
$79$	103.257			1.30705			0.653526	−	0.756904i	$-0.273289\pi$
$79$	103.257			1.30705			0.653526	+	0.756904i	$0.273289\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$		−	34.8098i		−	0.419396i	−0.977766	−	0.209698i	$-0.932752\pi$
$83$		−	34.8098i		−	0.419396i	0.977766	−	0.209698i	$-0.0672480\pi$
$84$		0			0
$85$	134.895			1.58700
$86$		0			0
$87$			73.5311i			0.845184i
$88$		0			0
$89$		−	64.9992i		−	0.730328i	−0.930943	−	0.365164i	$-0.881013\pi$
$89$		−	64.9992i		−	0.730328i	0.930943	−	0.365164i	$-0.118987\pi$
$90$		0			0
$91$	26.4397	+	10.2655i	0.290547	+	0.112808i
$92$		0			0
$93$	−5.48795			−0.0590103
$94$		0			0
$95$	53.0820			0.558758
$96$		0			0
$97$			105.475i			1.08737i	0.839289	+	0.543686i	$0.182972\pi$
$97$			105.475i			1.08737i	−0.839289	+	0.543686i	$0.817028\pi$
$98$		0			0
$99$	−44.7200			−0.451717
$100$		0			0
$101$			6.24742i			0.0618556i	0.999522	+	0.0309278i	$0.00984620\pi$
$101$			6.24742i			0.0618556i	−0.999522	+	0.0309278i	$0.990154\pi$
$102$		0			0
$103$		−	57.0493i		−	0.553876i	−0.960888	−	0.276938i	$-0.910680\pi$
$103$		−	57.0493i		−	0.553876i	0.960888	−	0.276938i	$-0.0893197\pi$
$104$		0			0
$105$	22.6047	−	58.2206i	0.215283	−	0.554482i
$106$		0			0
$107$	−133.600			−1.24860			−0.624299	−	0.781186i	$-0.714615\pi$
$107$	−133.600			−1.24860			−0.624299	+	0.781186i	$0.714615\pi$
$108$		0			0
$109$	37.9546			0.348207			0.174104	−	0.984727i	$-0.444297\pi$
$109$	37.9546			0.348207			0.174104	+	0.984727i	$0.444297\pi$
$110$		0			0
$111$			59.6353i			0.537255i
$112$		0			0
$113$	−1.35912			−0.0120276			−0.00601380	−	0.999982i	$-0.501914\pi$
$113$	−1.35912			−0.0120276			−0.00601380	+	0.999982i	$0.501914\pi$
$114$		0			0
$115$			92.8396i			0.807301i
$116$		0			0
$117$		−	12.1554i		−	0.103892i
$118$		0			0
$119$	170.882	+	66.3465i	1.43598	+	0.557534i
$120$		0			0
$121$	101.208			0.836433
$122$		0			0
$123$	71.7330			0.583195
$124$		0			0
$125$		−	120.874i		−	0.966992i
$126$		0			0
$127$	−139.975			−1.10217			−0.551083	−	0.834450i	$-0.685785\pi$
$127$	−139.975			−1.10217			−0.551083	+	0.834450i	$0.685785\pi$
$128$		0			0
$129$		−	28.2141i		−	0.218714i
$130$		0			0
$131$		−	24.0681i		−	0.183726i	−0.995772	−	0.0918629i	$-0.970718\pi$
$131$		−	24.0681i		−	0.183726i	0.995772	−	0.0918629i	$-0.0292822\pi$
$132$		0			0
$133$	67.2432	+	26.1078i	0.505588	+	0.196300i
$134$		0			0
$135$	−26.7664			−0.198269
$136$		0			0
$137$	185.628			1.35495			0.677475	−	0.735545i	$-0.263074\pi$
$137$	185.628			1.35495			0.677475	+	0.735545i	$0.263074\pi$
$138$		0			0
$139$			255.078i			1.83510i	0.397625	+	0.917548i	$0.369835\pi$
$139$			255.078i			1.83510i	−0.397625	+	0.917548i	$0.630165\pi$
$140$		0			0
$141$	−1.75579			−0.0124524
$142$		0			0
$143$			60.3989i			0.422370i
$144$		0			0
$145$		−	218.684i		−	1.50817i
$146$		0			0
$147$	57.2703	−	62.6347i	0.389594	−	0.426086i
$148$		0			0
$149$	−41.7178			−0.279985			−0.139993	−	0.990153i	$-0.544708\pi$
$149$	−41.7178			−0.279985			−0.139993	+	0.990153i	$0.544708\pi$
$150$		0			0
$151$	−109.711			−0.726565			−0.363282	−	0.931679i	$-0.618344\pi$
$151$	−109.711			−0.726565			−0.363282	+	0.931679i	$0.618344\pi$
$152$		0			0
$153$		−	78.5613i		−	0.513473i
$154$		0			0
$155$	16.3214			0.105299
$156$		0			0
$157$			22.5395i			0.143564i	0.997420	+	0.0717818i	$0.0228685\pi$
$157$			22.5395i			0.143564i	−0.997420	+	0.0717818i	$0.977131\pi$
$158$		0			0
$159$			146.097i			0.918846i
$160$		0			0
$161$	−45.6622	+	117.607i	−0.283616	+	0.730480i
$162$		0			0
$163$	272.331			1.67074			0.835371	−	0.549686i	$-0.185253\pi$
$163$	272.331			1.67074			0.835371	+	0.549686i	$0.185253\pi$
$164$		0			0
$165$	132.999			0.806055
$166$		0			0
$167$		−	310.691i		−	1.86043i	−0.367019	−	0.930214i	$-0.619621\pi$
$167$		−	310.691i		−	1.86043i	0.367019	−	0.930214i	$-0.380379\pi$
$168$		0			0
$169$	152.583			0.902857
$170$		0			0
$171$		−	30.9144i		−	0.180786i
$172$		0			0
$173$			321.021i			1.85561i	0.373064	+	0.927805i	$0.378307\pi$
$173$			321.021i			1.85561i	−0.373064	+	0.927805i	$0.621693\pi$
$174$		0			0
$175$	−3.88838	+	10.0149i	−0.0222193	+	0.0572280i
$176$		0			0
$177$	102.847			0.581059
$178$		0			0
$179$	−141.201			−0.788830			−0.394415	−	0.918932i	$-0.629053\pi$
$179$	−141.201			−0.788830			−0.394415	+	0.918932i	$0.629053\pi$
$180$		0			0
$181$			197.828i			1.09297i	0.837468	+	0.546486i	$0.184035\pi$
$181$			197.828i			1.09297i	−0.837468	+	0.546486i	$0.815965\pi$
$182$		0			0
$183$	150.007			0.819709
$184$		0			0
$185$		−	177.358i		−	0.958691i
$186$		0			0
$187$			390.362i			2.08750i
$188$		0			0
$189$	−33.9071	−	13.1648i	−0.179402	−	0.0696548i
$190$		0			0
$191$	−116.101			−0.607857			−0.303929	−	0.952695i	$-0.598298\pi$
$191$	−116.101			−0.607857			−0.303929	+	0.952695i	$0.598298\pi$
$192$		0			0
$193$	368.857			1.91118			0.955588	−	0.294705i	$-0.0952213\pi$
$193$	368.857			1.91118			0.955588	+	0.294705i	$0.0952213\pi$
$194$		0			0
$195$			36.1507i			0.185388i
$196$		0			0
$197$	65.8721			0.334376			0.167188	−	0.985925i	$-0.446531\pi$
$197$	65.8721			0.334376			0.167188	+	0.985925i	$0.446531\pi$
$198$		0			0
$199$			69.6424i			0.349962i	0.984572	+	0.174981i	$0.0559863\pi$
$199$			69.6424i			0.349962i	−0.984572	+	0.174981i	$0.944014\pi$
$200$		0			0
$201$			201.060i			1.00030i
$202$		0			0
$203$	107.558	−	277.025i	0.529840	−	1.36465i
$204$		0			0
$205$	−213.337			−1.04067
$206$		0			0
$207$	54.0689			0.261202
$208$		0			0
$209$			153.610i			0.734978i
$210$		0			0
$211$	−353.466			−1.67519			−0.837597	−	0.546289i	$-0.816040\pi$
$211$	−353.466			−1.67519			−0.837597	+	0.546289i	$0.816040\pi$
$212$		0			0
$213$		−	169.382i		−	0.795219i
$214$		0			0
$215$			83.9098i			0.390278i
$216$		0			0
$217$	20.6756	+	8.02751i	0.0952793	+	0.0369931i
$218$		0			0
$219$	−49.0123			−0.223801
$220$		0			0
$221$	−106.105			−0.480113
$222$		0			0
$223$		−	348.135i		−	1.56114i	−0.625065	−	0.780572i	$-0.714928\pi$
$223$		−	348.135i		−	1.56114i	0.625065	−	0.780572i	$-0.285072\pi$
$224$		0			0
$225$	4.60425			0.0204633
$226$		0			0
$227$		−	420.898i		−	1.85417i	−0.374845	−	0.927087i	$-0.622304\pi$
$227$		−	420.898i		−	1.85417i	0.374845	−	0.927087i	$-0.377696\pi$
$228$		0			0
$229$		−	367.551i		−	1.60503i	−0.596633	−	0.802514i	$-0.703495\pi$
$229$		−	367.551i		−	1.60503i	0.596633	−	0.802514i	$-0.296505\pi$
$230$		0			0
$231$	168.480	+	65.4142i	0.729352	+	0.283178i
$232$		0			0
$233$	234.452			1.00623			0.503115	−	0.864219i	$-0.332187\pi$
$233$	234.452			1.00623			0.503115	+	0.864219i	$0.332187\pi$
$234$		0			0
$235$	5.22180			0.0222204
$236$		0			0
$237$			178.847i			0.754627i
$238$		0			0
$239$	−53.3195			−0.223094			−0.111547	−	0.993759i	$-0.535581\pi$
$239$	−53.3195			−0.223094			−0.111547	+	0.993759i	$0.535581\pi$
$240$		0			0
$241$			31.2242i			0.129561i	0.997900	+	0.0647805i	$0.0206347\pi$
$241$			31.2242i			0.129561i	−0.997900	+	0.0647805i	$0.979365\pi$
$242$		0			0
$243$			15.5885i			0.0641500i
$244$		0			0
$245$	−170.324	+	186.278i	−0.695201	+	0.760319i
$246$		0			0
$247$	−41.7531			−0.169041
$248$		0			0
$249$	60.2924			0.242138
$250$		0			0
$251$		−	159.945i		−	0.637232i	−0.947884	−	0.318616i	$-0.896782\pi$
$251$		−	159.945i		−	0.637232i	0.947884	−	0.318616i	$-0.103218\pi$
$252$		0			0
$253$	−268.662			−1.06191
$254$		0			0
$255$			233.645i			0.916253i
$256$		0			0
$257$		−	33.1653i		−	0.129048i	−0.997916	−	0.0645239i	$-0.979447\pi$
$257$		−	33.1653i		−	0.129048i	0.997916	−	0.0645239i	$-0.0205529\pi$
$258$		0			0
$259$	87.2316	−	224.673i	0.336801	−	0.867464i
$260$		0			0
$261$	−127.360			−0.487967
$262$		0			0
$263$	53.6396			0.203953			0.101976	−	0.994787i	$-0.467483\pi$
$263$	53.6396			0.203953			0.101976	+	0.994787i	$0.467483\pi$
$264$		0			0
$265$		−	434.497i		−	1.63961i
$266$		0			0
$267$	112.582			0.421655
$268$		0			0
$269$			21.6712i			0.0805621i	0.999188	+	0.0402810i	$0.0128253\pi$
$269$			21.6712i			0.0805621i	−0.999188	+	0.0402810i	$0.987175\pi$
$270$		0			0
$271$			57.0119i			0.210376i	0.994452	+	0.105188i	$0.0335444\pi$
$271$			57.0119i			0.210376i	−0.994452	+	0.105188i	$0.966456\pi$
$272$		0			0
$273$	−17.7803	+	45.7950i	−0.0651295	+	0.167747i
$274$		0			0
$275$	−22.8780			−0.0831928
$276$		0			0
$277$	−35.3099			−0.127472			−0.0637362	−	0.997967i	$-0.520302\pi$
$277$	−35.3099			−0.127472			−0.0637362	+	0.997967i	$0.520302\pi$
$278$		0			0
$279$		−	9.50541i		−	0.0340696i
$280$		0			0
$281$	243.290			0.865799			0.432900	−	0.901442i	$-0.357490\pi$
$281$	243.290			0.865799			0.432900	+	0.901442i	$0.357490\pi$
$282$		0			0
$283$			78.1560i			0.276170i	0.990420	+	0.138085i	$0.0440947\pi$
$283$			78.1560i			0.276170i	−0.990420	+	0.138085i	$0.955905\pi$
$284$		0			0
$285$			91.9408i			0.322599i
$286$		0			0
$287$	−270.251	−	104.927i	−0.941640	−	0.365601i
$288$		0			0
$289$	−396.764			−1.37289
$290$		0			0
$291$	−182.688			−0.627795
$292$		0			0
$293$		−	351.888i		−	1.20098i	−0.799632	−	0.600491i	$-0.794972\pi$
$293$		−	351.888i		−	1.20098i	0.799632	−	0.600491i	$-0.205028\pi$
$294$		0			0
$295$	−305.873			−1.03686
$296$		0			0
$297$		−	77.4573i		−	0.260799i
$298$		0			0
$299$		−	73.0255i		−	0.244233i
$300$		0			0
$301$	−41.2701	+	106.295i	−0.137110	+	0.353140i
$302$		0			0
$303$	−10.8208			−0.0357124
$304$		0			0
$305$	−446.126			−1.46271
$306$		0			0
$307$			368.173i			1.19926i	0.800277	+	0.599631i	$0.204686\pi$
$307$			368.173i			1.19926i	−0.800277	+	0.599631i	$0.795314\pi$
$308$		0			0
$309$	98.8122			0.319781
$310$		0			0
$311$			93.4897i			0.300610i	0.988640	+	0.150305i	$0.0480256\pi$
$311$			93.4897i			0.300610i	−0.988640	+	0.150305i	$0.951974\pi$
$312$		0			0
$313$			333.448i			1.06533i	0.846326	+	0.532665i	$0.178810\pi$
$313$			333.448i			1.06533i	−0.846326	+	0.532665i	$0.821190\pi$
$314$		0			0
$315$	100.841	+	39.1525i	0.320130	+	0.124294i
$316$		0			0
$317$	−367.491			−1.15928			−0.579638	−	0.814874i	$-0.696806\pi$
$317$	−367.491			−1.15928			−0.579638	+	0.814874i	$0.696806\pi$
$318$		0			0
$319$	632.835			1.98381
$320$		0			0
$321$		−	231.402i		−	0.720878i
$322$		0			0
$323$	−269.853			−0.835459
$324$		0			0
$325$		−	6.21852i		−	0.0191339i
$326$		0			0
$327$			65.7393i			0.201038i
$328$		0			0
$329$	6.61487	+	2.56829i	0.0201060	+	0.00780635i
$330$		0			0
$331$	−118.816			−0.358961			−0.179481	−	0.983762i	$-0.557442\pi$
$331$	−118.816			−0.358961			−0.179481	+	0.983762i	$0.557442\pi$
$332$		0			0
$333$	−103.291			−0.310184
$334$		0			0
$335$		−	597.962i		−	1.78496i
$336$		0			0
$337$	−411.888			−1.22222			−0.611109	−	0.791546i	$-0.709276\pi$
$337$	−411.888			−1.22222			−0.611109	+	0.791546i	$0.709276\pi$
$338$		0			0
$339$		−	2.35406i		−	0.00694414i
$340$		0			0
$341$			47.2313i			0.138508i
$342$		0			0
$343$	−307.382	+	152.201i	−0.896158	+	0.443735i
$344$		0			0
$345$	−160.803			−0.466096
$346$		0			0
$347$	95.9406			0.276486			0.138243	−	0.990398i	$-0.455855\pi$
$347$	95.9406			0.276486			0.138243	+	0.990398i	$0.455855\pi$
$348$		0			0
$349$			71.7537i			0.205598i	0.994702	+	0.102799i	$0.0327798\pi$
$349$			71.7537i			0.205598i	−0.994702	+	0.102799i	$0.967220\pi$
$350$		0			0
$351$	21.0538			0.0599824
$352$		0			0
$353$		−	282.287i		−	0.799679i	−0.916585	−	0.399840i	$-0.869066\pi$
$353$		−	282.287i		−	0.799679i	0.916585	−	0.399840i	$-0.130934\pi$
$354$		0			0
$355$			503.748i			1.41901i
$356$		0			0
$357$	−114.916	+	295.976i	−0.321892	+	0.829064i
$358$		0			0
$359$	−199.095			−0.554583			−0.277291	−	0.960786i	$-0.589437\pi$
$359$	−199.095			−0.554583			−0.277291	+	0.960786i	$0.589437\pi$
$360$		0			0
$361$	254.811			0.705847
$362$		0			0
$363$			175.298i			0.482915i
$364$		0			0
$365$	145.765			0.399355
$366$		0			0
$367$			531.102i			1.44714i	0.690249	+	0.723572i	$0.257501\pi$
$367$			531.102i			1.44714i	−0.690249	+	0.723572i	$0.742499\pi$
$368$		0			0
$369$			124.245i			0.336708i
$370$		0			0
$371$	213.703	−	550.412i	0.576018	−	1.48359i
$372$		0			0
$373$	451.830			1.21134			0.605670	−	0.795716i	$-0.292905\pi$
$373$	451.830			1.21134			0.605670	+	0.795716i	$0.292905\pi$
$374$		0			0
$375$	209.360			0.558293
$376$		0			0
$377$			172.012i			0.456265i
$378$		0			0
$379$	499.730			1.31855			0.659274	−	0.751903i	$-0.270864\pi$
$379$	499.730			1.31855			0.659274	+	0.751903i	$0.270864\pi$
$380$		0			0
$381$		−	242.444i		−	0.636336i
$382$		0			0
$383$		−	391.943i		−	1.02335i	−0.859179	−	0.511674i	$-0.829025\pi$
$383$		−	391.943i		−	1.02335i	0.859179	−	0.511674i	$-0.170975\pi$
$384$		0			0
$385$	−501.067	−	194.544i	−1.30147	−	0.505310i
$386$		0			0
$387$	48.8682			0.126274
$388$		0			0
$389$	411.379			1.05753			0.528765	−	0.848769i	$-0.322655\pi$
$389$	411.379			1.05753			0.528765	+	0.848769i	$0.322655\pi$
$390$		0			0
$391$		−	471.969i		−	1.20708i
$392$		0			0
$393$	41.6871			0.106074
$394$		0			0
$395$		−	531.897i		−	1.34657i
$396$		0			0
$397$			314.331i			0.791765i	0.918301	+	0.395882i	$0.129561\pi$
$397$			314.331i			0.791765i	−0.918301	+	0.395882i	$0.870439\pi$
$398$		0			0
$399$	−45.2201	+	116.469i	−0.113334	+	0.291901i
$400$		0			0
$401$	−505.723			−1.26115			−0.630577	−	0.776127i	$-0.717182\pi$
$401$	−505.723			−1.26115			−0.630577	+	0.776127i	$0.717182\pi$
$402$		0			0
$403$	−12.8380			−0.0318562
$404$		0			0
$405$		−	46.3607i		−	0.114471i
$406$		0			0
$407$	513.243			1.26104
$408$		0			0
$409$			533.011i			1.30321i	0.758560	+	0.651603i	$0.225903\pi$
$409$			533.011i			1.30321i	−0.758560	+	0.651603i	$0.774097\pi$
$410$		0			0
$411$			321.518i			0.782281i
$412$		0			0
$413$	−387.473	−	150.440i	−0.938191	−	0.364262i
$414$		0			0
$415$	−179.312			−0.432077
$416$		0			0
$417$	−441.809			−1.05949
$418$		0			0
$419$			527.882i			1.25986i	0.776652	+	0.629930i	$0.216917\pi$
$419$			527.882i			1.25986i	−0.776652	+	0.629930i	$0.783083\pi$
$420$		0			0
$421$	−357.750			−0.849763			−0.424881	−	0.905249i	$-0.639684\pi$
$421$	−357.750			−0.849763			−0.424881	+	0.905249i	$0.639684\pi$
$422$		0			0
$423$		−	3.04112i		−	0.00718942i
$424$		0			0
$425$		−	40.1907i		−	0.0945663i
$426$		0			0
$427$	−565.143	−	219.422i	−1.32352	−	0.513870i
$428$		0			0
$429$	−104.614			−0.243855
$430$		0			0
$431$	−295.522			−0.685667			−0.342833	−	0.939396i	$-0.611387\pi$
$431$	−295.522			−0.685667			−0.342833	+	0.939396i	$0.611387\pi$
$432$		0			0
$433$			819.746i			1.89318i	0.322445	+	0.946588i	$0.395495\pi$
$433$			819.746i			1.89318i	−0.322445	+	0.946588i	$0.604505\pi$
$434$		0			0
$435$	378.772			0.870741
$436$		0			0
$437$		−	185.723i		−	0.424996i
$438$		0			0
$439$			580.965i			1.32338i	0.749776	+	0.661691i	$0.230161\pi$
$439$			580.965i			1.32338i	−0.749776	+	0.661691i	$0.769839\pi$
$440$		0			0
$441$	108.487	+	99.1952i	0.246001	+	0.224932i
$442$		0			0
$443$	−137.309			−0.309953			−0.154977	−	0.987918i	$-0.549530\pi$
$443$	−137.309			−0.309953			−0.154977	+	0.987918i	$0.549530\pi$
$444$		0			0
$445$	−334.823			−0.752412
$446$		0			0
$447$		−	72.2573i		−	0.161649i
$448$		0			0
$449$	−334.385			−0.744732			−0.372366	−	0.928086i	$-0.621453\pi$
$449$	−334.385			−0.744732			−0.372366	+	0.928086i	$0.621453\pi$
$450$		0			0
$451$		−	617.360i		−	1.36887i
$452$		0			0
$453$		−	190.026i		−	0.419482i
$454$		0			0
$455$	52.8795	−	136.196i	0.116219	−	0.299332i
$456$		0			0
$457$	−656.384			−1.43629			−0.718145	−	0.695894i	$-0.755008\pi$
$457$	−656.384			−1.43629			−0.718145	+	0.695894i	$0.755008\pi$
$458$		0			0
$459$	136.072			0.296454
$460$		0			0
$461$			134.660i			0.292105i	0.989277	+	0.146052i	$0.0466568\pi$
$461$			134.660i			0.292105i	−0.989277	+	0.146052i	$0.953343\pi$
$462$		0			0
$463$	471.861			1.01914			0.509569	−	0.860430i	$-0.329805\pi$
$463$	471.861			1.01914			0.509569	+	0.860430i	$0.329805\pi$
$464$		0			0
$465$			28.2695i			0.0607946i
$466$		0			0
$467$			529.473i			1.13377i	0.823795	+	0.566887i	$0.191853\pi$
$467$			529.473i			1.13377i	−0.823795	+	0.566887i	$0.808147\pi$
$468$		0			0
$469$	294.101	−	757.486i	0.627082	−	1.61511i
$470$		0			0
$471$	−39.0396			−0.0828865
$472$		0			0
$473$	−242.821			−0.513363
$474$		0			0
$475$		−	15.8153i		−	0.0332954i
$476$		0			0
$477$	−253.047			−0.530496
$478$		0			0
$479$			15.3586i			0.0320640i	0.999871	+	0.0160320i	$0.00510336\pi$
$479$			15.3586i			0.0320640i	−0.999871	+	0.0160320i	$0.994897\pi$
$480$		0			0
$481$			139.506i			0.290032i
$482$		0			0
$483$	−203.702	−	79.0892i	−0.421743	−	0.163746i
$484$		0			0
$485$	543.322			1.12025
$486$		0			0
$487$	−288.676			−0.592763			−0.296382	−	0.955070i	$-0.595780\pi$
$487$	−288.676			−0.592763			−0.296382	+	0.955070i	$0.595780\pi$
$488$		0			0
$489$			471.691i			0.964604i
$490$		0			0
$491$	−444.046			−0.904371			−0.452185	−	0.891924i	$-0.649355\pi$
$491$	−444.046			−0.904371			−0.452185	+	0.891924i	$0.649355\pi$
$492$		0			0
$493$			1111.73i			2.25502i
$494$		0			0
$495$			230.361i			0.465376i
$496$		0			0
$497$	−247.763	+	638.138i	−0.498517	+	1.28398i
$498$		0			0
$499$	466.334			0.934537			0.467269	−	0.884115i	$-0.345238\pi$
$499$	466.334			0.934537			0.467269	+	0.884115i	$0.345238\pi$
$500$		0			0
$501$	538.133			1.07412
$502$		0			0
$503$		−	935.154i		−	1.85915i	−0.368628	−	0.929577i	$-0.620172\pi$
$503$		−	935.154i		−	1.85915i	0.368628	−	0.929577i	$-0.379828\pi$
$504$		0			0
$505$	32.1816			0.0637260
$506$		0			0
$507$			264.281i			0.521265i
$508$		0			0
$509$			393.830i			0.773732i	0.922136	+	0.386866i	$0.126442\pi$
$509$			393.830i			0.773732i	−0.922136	+	0.386866i	$0.873558\pi$
$510$		0			0
$511$	184.652	+	71.6928i	0.361353	+	0.140299i
$512$		0			0
$513$	53.5454			0.104377
$514$		0			0
$515$	−293.872			−0.570624
$516$		0			0
$517$			15.1110i			0.0292282i
$518$		0			0
$519$	−556.024			−1.07134
$520$		0			0
$521$			383.993i			0.737030i	0.929622	+	0.368515i	$0.120134\pi$
$521$			383.993i			0.737030i	−0.929622	+	0.368515i	$0.879866\pi$
$522$		0			0
$523$			53.3845i			0.102074i	0.998697	+	0.0510368i	$0.0162526\pi$
$523$			53.3845i			0.102074i	−0.998697	+	0.0510368i	$0.983747\pi$
$524$		0			0
$525$	−17.3463	−	6.73487i	−0.0330406	−	0.0128283i
$526$		0			0
$527$	−82.9731			−0.157444
$528$		0			0
$529$	−204.173			−0.385961
$530$		0			0
$531$			178.137i			0.335475i
$532$		0			0
$533$	167.806			0.314833
$534$		0			0
$535$			688.198i			1.28635i
$536$		0			0
$537$		−	244.567i		−	0.455431i
$538$		0			0
$539$	−539.057	−	492.889i	−1.00011	−	0.914452i
$540$		0			0
$541$	−33.8848			−0.0626336			−0.0313168	−	0.999510i	$-0.509970\pi$
$541$	−33.8848			−0.0626336			−0.0313168	+	0.999510i	$0.509970\pi$
$542$		0			0
$543$	−342.648			−0.631028
$544$		0			0
$545$		−	195.511i		−	0.358736i
$546$		0			0
$547$	344.630			0.630037			0.315019	−	0.949086i	$-0.397989\pi$
$547$	344.630			0.630037			0.315019	+	0.949086i	$0.397989\pi$
$548$		0			0
$549$			259.819i			0.473259i
$550$		0			0
$551$			437.472i			0.793960i
$552$		0			0
$553$	261.608	−	673.796i	0.473070	−	1.21844i
$554$		0			0
$555$	307.193			0.553501
$556$		0			0
$557$	−129.194			−0.231947			−0.115973	−	0.993252i	$-0.536999\pi$
$557$	−129.194			−0.231947			−0.115973	+	0.993252i	$0.536999\pi$
$558$		0			0
$559$		−	66.0015i		−	0.118071i
$560$		0			0
$561$	−676.127			−1.20522
$562$		0			0
$563$			783.083i			1.39091i	0.718569	+	0.695456i	$0.244797\pi$
$563$			783.083i			1.39091i	−0.718569	+	0.695456i	$0.755203\pi$
$564$		0			0
$565$			7.00108i			0.0123913i
$566$		0			0
$567$	22.8020	−	58.7288i	0.0402152	−	0.103578i
$568$		0			0
$569$	−420.659			−0.739296			−0.369648	−	0.929172i	$-0.620522\pi$
$569$	−420.659			−0.739296			−0.369648	+	0.929172i	$0.620522\pi$
$570$		0			0
$571$	66.0186			0.115619			0.0578097	−	0.998328i	$-0.481588\pi$
$571$	66.0186			0.115619			0.0578097	+	0.998328i	$0.481588\pi$
$572$		0			0
$573$		−	201.092i		−	0.350947i
$574$		0			0
$575$	27.6608			0.0481057
$576$		0			0
$577$			476.719i			0.826203i	0.910685	+	0.413101i	$0.135555\pi$
$577$			476.719i			0.826203i	−0.910685	+	0.413101i	$0.864445\pi$
$578$		0			0
$579$			638.879i			1.10342i
$580$		0			0
$581$	−227.149	−	88.1927i	−0.390962	−	0.151795i
$582$		0			0
$583$	1257.36			2.15671
$584$		0			0
$585$	−62.6149			−0.107034
$586$		0			0
$587$		−	782.519i		−	1.33308i	−0.745469	−	0.666541i	$-0.767774\pi$
$587$		−	782.519i		−	1.33308i	0.745469	−	0.666541i	$-0.232226\pi$
$588$		0			0
$589$	−32.6505			−0.0554338
$590$		0			0
$591$			114.094i			0.193052i
$592$		0			0
$593$		−	114.539i		−	0.193152i	−0.995326	−	0.0965762i	$-0.969211\pi$
$593$		−	114.539i		−	0.193152i	0.995326	−	0.0965762i	$-0.0307892\pi$
$594$		0			0
$595$	341.764	−	880.245i	0.574393	−	1.47940i
$596$		0			0
$597$	−120.624			−0.202050
$598$		0			0
$599$	−1073.21			−1.79167			−0.895837	−	0.444383i	$-0.853423\pi$
$599$	−1073.21			−1.79167			−0.895837	+	0.444383i	$0.853423\pi$
$600$		0			0
$601$		−	573.754i		−	0.954666i	−0.878723	−	0.477333i	$-0.841604\pi$
$601$		−	573.754i		−	0.954666i	0.878723	−	0.477333i	$-0.158396\pi$
$602$		0			0
$603$	−348.247			−0.577524
$604$		0			0
$605$		−	521.344i		−	0.861725i
$606$		0			0
$607$			1163.67i			1.91708i	0.284959	+	0.958540i	$0.408020\pi$
$607$			1163.67i			1.91708i	−0.284959	+	0.958540i	$0.591980\pi$
$608$		0			0
$609$	479.821	+	186.295i	0.787883	+	0.305903i
$610$		0			0
$611$	−4.10735			−0.00672234
$612$		0			0
$613$	−466.983			−0.761799			−0.380900	−	0.924616i	$-0.624386\pi$
$613$	−466.983			−0.761799			−0.380900	+	0.924616i	$0.624386\pi$
$614$		0			0
$615$		−	369.510i		−	0.600830i
$616$		0			0
$617$	543.328			0.880596			0.440298	−	0.897852i	$-0.354873\pi$
$617$	543.328			0.880596			0.440298	+	0.897852i	$0.354873\pi$
$618$		0			0
$619$		−	132.036i		−	0.213305i	−0.994296	−	0.106653i	$-0.965987\pi$
$619$		−	132.036i		−	0.213305i	0.994296	−	0.106653i	$-0.0340132\pi$
$620$		0			0
$621$			93.6500i			0.150805i
$622$		0			0
$623$	−424.147	−	164.679i	−0.680814	−	0.264333i
$624$		0			0
$625$	−661.013			−1.05762
$626$		0			0
$627$	−266.061			−0.424340
$628$		0			0
$629$			901.634i			1.43344i
$630$		0			0
$631$	790.993			1.25355			0.626777	−	0.779198i	$-0.284374\pi$
$631$	790.993			1.25355			0.626777	+	0.779198i	$0.284374\pi$
$632$		0			0
$633$		−	612.221i		−	0.967174i
$634$		0			0
$635$			721.039i			1.13549i
$636$		0			0
$637$	133.973	−	146.522i	0.210319	−	0.230019i
$638$		0			0
$639$	293.378			0.459120
$640$		0			0
$641$	−461.410			−0.719829			−0.359915	−	0.932985i	$-0.617194\pi$
$641$	−461.410			−0.719829			−0.359915	+	0.932985i	$0.617194\pi$
$642$		0			0
$643$		−	308.047i		−	0.479078i	−0.970887	−	0.239539i	$-0.923004\pi$
$643$		−	308.047i		−	0.479078i	0.970887	−	0.239539i	$-0.0769962\pi$
$644$		0			0
$645$	−145.336			−0.225327
$646$		0			0
$647$			911.607i			1.40898i	0.709716	+	0.704488i	$0.248823\pi$
$647$			911.607i			1.40898i	−0.709716	+	0.704488i	$0.751177\pi$
$648$		0			0
$649$		−	885.143i		−	1.36386i
$650$		0			0
$651$	−13.9040	+	35.8112i	−0.0213580	+	0.0550095i
$652$		0			0
$653$	−761.550			−1.16623			−0.583116	−	0.812389i	$-0.698167\pi$
$653$	−761.550			−1.16623			−0.583116	+	0.812389i	$0.698167\pi$
$654$		0			0
$655$	−123.979			−0.189281
$656$		0			0
$657$		−	84.8918i		−	0.129211i
$658$		0			0
$659$	1169.93			1.77531			0.887655	−	0.460509i	$-0.152333\pi$
$659$	1169.93			1.77531			0.887655	+	0.460509i	$0.152333\pi$
$660$		0			0
$661$			51.1358i			0.0773613i	0.999252	+	0.0386806i	$0.0123155\pi$
$661$			51.1358i			0.0773613i	−0.999252	+	0.0386806i	$0.987685\pi$
$662$		0			0
$663$		−	183.779i		−	0.277194i
$664$		0			0
$665$	134.486	−	346.383i	0.202235	−	0.520876i
$666$		0			0
$667$	−765.132			−1.14712
$668$		0			0
$669$	602.988			0.901327
$670$		0			0
$671$		−	1291.01i		−	1.92401i
$672$		0			0
$673$	596.694			0.886619			0.443309	−	0.896369i	$-0.353804\pi$
$673$	596.694			0.886619			0.443309	+	0.896369i	$0.353804\pi$
$674$		0			0
$675$			7.97480i			0.0118145i
$676$		0			0
$677$		−	889.654i		−	1.31411i	−0.753842	−	0.657056i	$-0.771801\pi$
$677$		−	889.654i		−	1.31411i	0.753842	−	0.657056i	$-0.228199\pi$
$678$		0			0
$679$	688.269	+	267.227i	1.01365	+	0.393560i
$680$		0			0
$681$	729.016			1.07051
$682$		0			0
$683$	468.595			0.686083			0.343042	−	0.939320i	$-0.388543\pi$
$683$	468.595			0.686083			0.343042	+	0.939320i	$0.388543\pi$
$684$		0			0
$685$		−	956.206i		−	1.39592i
$686$		0			0
$687$	636.618			0.926663
$688$		0			0
$689$			341.765i			0.496031i
$690$		0			0
$691$			154.796i			0.224018i	0.993707	+	0.112009i	$0.0357286\pi$
$691$			154.796i			0.224018i	−0.993707	+	0.112009i	$0.964271\pi$
$692$		0			0
$693$	−113.301	+	291.817i	−0.163493	+	0.421092i
$694$		0			0
$695$	1313.96			1.89059
$696$		0			0
$697$	1084.54			1.55601
$698$		0			0
$699$			406.082i			0.580947i
$700$		0			0
$701$	−509.180			−0.726362			−0.363181	−	0.931719i	$-0.618309\pi$
$701$	−509.180			−0.726362			−0.363181	+	0.931719i	$0.618309\pi$
$702$		0			0
$703$			354.800i			0.504693i
$704$		0			0
$705$			9.04443i			0.0128290i
$706$		0			0
$707$	40.7670	+	15.8282i	0.0576620	+	0.0223878i
$708$		0			0
$709$	1058.71			1.49325			0.746623	−	0.665248i	$-0.231674\pi$
$709$	1058.71			1.49325			0.746623	+	0.665248i	$0.231674\pi$
$710$		0			0
$711$	−309.771			−0.435684
$712$		0			0
$713$		−	57.1052i		−	0.0800915i
$714$		0			0
$715$	311.126			0.435141
$716$		0			0
$717$		−	92.3521i		−	0.128804i
$718$		0			0
$719$			8.06477i			0.0112167i	0.999984	+	0.00560833i	$0.00178519\pi$
$719$			8.06477i			0.0112167i	−0.999984	+	0.00560833i	$0.998215\pi$
$720$		0			0
$721$	−372.270	−	144.538i	−0.516325	−	0.200468i
$722$		0			0
$723$	−54.0819			−0.0748020
$724$		0			0
$725$	−65.1551			−0.0898690
$726$		0			0
$727$		−	848.459i		−	1.16707i	−0.812088	−	0.583534i	$-0.801669\pi$
$727$		−	848.459i		−	1.16707i	0.812088	−	0.583534i	$-0.198331\pi$
$728$		0			0
$729$	−27.0000			−0.0370370
$730$		0			0
$731$		−	426.572i		−	0.583546i
$732$		0			0
$733$		−	1117.22i		−	1.52418i	−0.647471	−	0.762090i	$-0.724173\pi$
$733$		−	1117.22i		−	1.52418i	0.647471	−	0.762090i	$-0.275827\pi$
$734$		0			0
$735$	−322.643	−	295.010i	−0.438970	−	0.401375i
$736$		0			0
$737$	1730.40			2.34790
$738$		0			0
$739$	−733.946			−0.993161			−0.496580	−	0.867991i	$-0.665411\pi$
$739$	−733.946			−0.993161			−0.496580	+	0.867991i	$0.665411\pi$
$740$		0			0
$741$		−	72.3185i		−	0.0975958i
$742$		0			0
$743$	−1392.52			−1.87419			−0.937096	−	0.349071i	$-0.886497\pi$
$743$	−1392.52			−1.87419			−0.937096	+	0.349071i	$0.886497\pi$
$744$		0			0
$745$			214.896i			0.288451i
$746$		0			0
$747$			104.429i			0.139799i
$748$		0			0
$749$	−338.483	+	871.795i	−0.451913	+	1.16395i
$750$		0			0
$751$	−1196.00			−1.59255			−0.796275	−	0.604935i	$-0.793199\pi$
$751$	−1196.00			−1.59255			−0.796275	+	0.604935i	$0.793199\pi$
$752$		0			0
$753$	277.033			0.367906
$754$		0			0
$755$			565.144i			0.748535i
$756$		0			0
$757$	−326.138			−0.430830			−0.215415	−	0.976523i	$-0.569110\pi$
$757$	−326.138			−0.430830			−0.215415	+	0.976523i	$0.569110\pi$
$758$		0			0
$759$		−	465.336i		−	0.613091i
$760$		0			0
$761$			507.940i			0.667463i	0.942668	+	0.333732i	$0.108308\pi$
$761$			507.940i			0.667463i	−0.942668	+	0.333732i	$0.891692\pi$
$762$		0			0
$763$	96.1602	−	247.670i	0.126029	−	0.324600i
$764$		0			0
$765$	−404.684			−0.528999
$766$		0			0
$767$	240.592			0.313680
$768$		0			0
$769$		−	54.2112i		−	0.0704957i	−0.999379	−	0.0352478i	$-0.988778\pi$
$769$		−	54.2112i		−	0.0704957i	0.999379	−	0.0352478i	$-0.0112221\pi$
$770$		0			0
$771$	57.4439			0.0745057
$772$		0			0
$773$			487.877i			0.631147i	0.948901	+	0.315574i	$0.102197\pi$
$773$			487.877i			0.631147i	−0.948901	+	0.315574i	$0.897803\pi$
$774$		0			0
$775$		−	4.86282i		−	0.00627460i
$776$		0			0
$777$	389.145	+	151.090i	0.500831	+	0.194452i
$778$		0			0
$779$	426.774			0.547849
$780$		0			0
$781$	−1457.76			−1.86653
$782$		0			0
$783$		−	220.593i		−	0.281728i
$784$		0			0
$785$	116.105			0.147905
$786$		0			0
$787$		−	316.008i		−	0.401535i	−0.979639	−	0.200768i	$-0.935656\pi$
$787$		−	316.008i		−	0.401535i	0.979639	−	0.200768i	$-0.0643437\pi$
$788$		0			0
$789$			92.9064i			0.117752i
$790$		0			0
$791$	−3.44340	+	8.86882i	−0.00435323	+	0.0112122i
$792$		0			0
$793$	350.913			0.442513
$794$		0			0
$795$	752.571			0.946630
$796$		0			0
$797$			868.063i			1.08916i	0.838708	+	0.544582i	$0.183312\pi$
$797$			868.063i			1.08916i	−0.838708	+	0.544582i	$0.816688\pi$
$798$		0			0
$799$	−26.5461			−0.0332241
$800$		0			0
$801$			194.998i			0.243443i
$802$		0			0
$803$			421.818i			0.525302i
$804$		0			0
$805$	605.818	+	235.215i	0.752568	+	0.292192i
$806$		0			0
$807$	−37.5356			−0.0465125
$808$		0			0
$809$	−484.513			−0.598904			−0.299452	−	0.954111i	$-0.596804\pi$
$809$	−484.513			−0.598904			−0.299452	+	0.954111i	$0.596804\pi$
$810$		0			0
$811$		−	1234.55i		−	1.52226i	−0.648599	−	0.761130i	$-0.724645\pi$
$811$		−	1234.55i		−	1.52226i	0.648599	−	0.761130i	$-0.275355\pi$
$812$		0			0
$813$	−98.7475			−0.121461
$814$		0			0
$815$		−	1402.83i		−	1.72126i
$816$		0			0
$817$		−	167.859i		−	0.205458i
$818$		0			0
$819$	−79.3192	−	30.7965i	−0.0968488	−	0.0376025i
$820$		0			0
$821$	189.106			0.230336			0.115168	−	0.993346i	$-0.463259\pi$
$821$	189.106			0.230336			0.115168	+	0.993346i	$0.463259\pi$
$822$		0			0
$823$	−1606.53			−1.95204			−0.976018	−	0.217688i	$-0.930148\pi$
$823$	−1606.53			−1.95204			−0.976018	+	0.217688i	$0.930148\pi$
$824$		0			0
$825$		−	39.6259i		−	0.0480314i
$826$		0			0
$827$	315.159			0.381088			0.190544	−	0.981679i	$-0.438975\pi$
$827$	315.159			0.381088			0.190544	+	0.981679i	$0.438975\pi$
$828$		0			0
$829$		−	7.08792i		−	0.00854997i	−0.999991	−	0.00427498i	$-0.998639\pi$
$829$		−	7.08792i		−	0.00854997i	0.999991	−	0.00427498i	$-0.00136077\pi$
$830$		0			0
$831$		−	61.1585i		−	0.0735963i
$832$		0			0
$833$	865.878	−	946.982i	1.03947	−	1.13683i
$834$		0			0
$835$	−1600.43			−1.91668
$836$		0			0
$837$	16.4639			0.0196701
$838$		0			0
$839$			375.971i			0.448118i	0.974576	+	0.224059i	$0.0719308\pi$
$839$			375.971i			0.448118i	−0.974576	+	0.224059i	$0.928069\pi$
$840$		0			0
$841$	961.272			1.14301
$842$		0			0
$843$			421.390i			0.499869i
$844$		0			0
$845$		−	785.983i		−	0.930158i
$846$		0			0
$847$	256.417	−	660.427i	0.302736	−	0.779725i
$848$		0			0
$849$	−135.370			−0.159447
$850$		0			0
$851$	−620.539			−0.729188
$852$		0			0
$853$			402.432i			0.471785i	0.971779	+	0.235892i	$0.0758013\pi$
$853$			402.432i			0.471785i	−0.971779	+	0.235892i	$0.924199\pi$
$854$		0			0
$855$	−159.246			−0.186253
$856$		0			0
$857$		−	1625.19i		−	1.89637i	−0.317726	−	0.948183i	$-0.602919\pi$
$857$		−	1625.19i		−	1.89637i	0.317726	−	0.948183i	$-0.397081\pi$
$858$		0			0
$859$			779.473i			0.907419i	0.891150	+	0.453710i	$0.149900\pi$
$859$			779.473i			0.907419i	−0.891150	+	0.453710i	$0.850100\pi$
$860$		0			0
$861$	181.740	−	468.088i	0.211080	−	0.543656i
$862$		0			0
$863$	−904.421			−1.04800			−0.523998	−	0.851719i	$-0.675560\pi$
$863$	−904.421			−1.04800			−0.523998	+	0.851719i	$0.675560\pi$
$864$		0			0
$865$	1653.64			1.91172
$866$		0			0
$867$		−	687.216i		−	0.792637i
$868$		0			0
$869$	1539.22			1.77125
$870$		0			0
$871$			470.343i			0.540004i
$872$		0			0
$873$		−	316.425i		−	0.362457i
$874$		0			0
$875$	−788.753	−	306.241i	−0.901432	−	0.349990i
$876$		0			0
$877$	430.365			0.490724			0.245362	−	0.969432i	$-0.421093\pi$
$877$	430.365			0.490724			0.245362	+	0.969432i	$0.421093\pi$
$878$		0			0
$879$	609.487			0.693387
$880$		0			0
$881$		−	868.272i		−	0.985553i	−0.870156	−	0.492777i	$-0.835982\pi$
$881$		−	868.272i		−	0.985553i	0.870156	−	0.492777i	$-0.164018\pi$
$882$		0			0
$883$	−1245.18			−1.41017			−0.705084	−	0.709124i	$-0.749091\pi$
$883$	−1245.18			−1.41017			−0.705084	+	0.709124i	$0.749091\pi$
$884$		0			0
$885$		−	529.787i		−	0.598629i
$886$		0			0
$887$			898.252i			1.01269i	0.862332	+	0.506343i	$0.169003\pi$
$887$			898.252i			1.01269i	−0.862332	+	0.506343i	$0.830997\pi$
$888$		0			0
$889$	−354.635	+	913.397i	−0.398915	+	1.02744i
$890$		0			0
$891$	134.160			0.150572
$892$		0			0
$893$	−10.4461			−0.0116977
$894$		0			0
$895$			727.351i			0.812683i
$896$		0			0
$897$	126.484			0.141008
$898$		0			0
$899$			134.512i			0.149624i
$900$		0			0
$901$			2208.85i			2.45156i
$902$		0			0
$903$	−184.109	−	71.4820i	−0.203886	−	0.0791606i
$904$		0			0
$905$	1019.05			1.12602
$906$		0			0
$907$	−71.4730			−0.0788015			−0.0394007	−	0.999223i	$-0.512545\pi$
$907$	−71.4730			−0.0788015			−0.0394007	+	0.999223i	$0.512545\pi$
$908$		0			0
$909$		−	18.7423i		−	0.0206185i
$910$		0			0
$911$	773.022			0.848542			0.424271	−	0.905535i	$-0.360530\pi$
$911$	773.022			0.848542			0.424271	+	0.905535i	$0.360530\pi$
$912$		0			0
$913$		−	518.898i		−	0.568344i
$914$		0			0
$915$		−	772.713i		−	0.844495i
$916$		0			0
$917$	−157.054	−	60.9779i	−0.171270	−	0.0664972i
$918$		0			0
$919$	−586.713			−0.638426			−0.319213	−	0.947683i	$-0.603418\pi$
$919$	−586.713			−0.638426			−0.319213	+	0.947683i	$0.603418\pi$
$920$		0			0
$921$	−637.695			−0.692394
$922$		0			0
$923$		−	396.237i		−	0.429292i
$924$		0			0
$925$	−52.8422			−0.0571267
$926$		0			0
$927$			171.148i			0.184625i
$928$		0			0
$929$			1564.85i			1.68445i	0.539126	+	0.842225i	$0.318755\pi$
$929$			1564.85i			1.68445i	−0.539126	+	0.842225i	$0.681245\pi$
$930$		0			0
$931$	340.729	−	372.644i	0.365982	−	0.400262i
$932$		0			0
$933$	−161.929			−0.173557
$934$		0			0
$935$	2010.83			2.15062
$936$		0			0
$937$			128.110i			0.136723i	0.997661	+	0.0683615i	$0.0217771\pi$
$937$			128.110i			0.136723i	−0.997661	+	0.0683615i	$0.978223\pi$
$938$		0			0
$939$	−577.550			−0.615069
$940$		0			0
$941$		−	361.450i		−	0.384113i	−0.981384	−	0.192056i	$-0.938484\pi$
$941$		−	361.450i		−	0.384113i	0.981384	−	0.192056i	$-0.0615157\pi$
$942$		0			0
$943$			746.422i			0.791539i
$944$		0			0
$945$	−67.8141	+	174.662i	−0.0717610	+	0.184827i
$946$		0			0
$947$	1247.00			1.31679			0.658393	−	0.752674i	$-0.271236\pi$
$947$	1247.00			1.31679			0.658393	+	0.752674i	$0.271236\pi$
$948$		0			0
$949$	−114.655			−0.120817
$950$		0			0
$951$		−	636.512i		−	0.669308i
$952$		0			0
$953$	1200.01			1.25919			0.629594	−	0.776924i	$-0.283221\pi$
$953$	1200.01			1.25919			0.629594	+	0.776924i	$0.283221\pi$
$954$		0			0
$955$			598.057i			0.626238i
$956$		0			0
$957$			1096.10i			1.14535i
$958$		0			0
$959$	470.300	−	1211.30i	0.490407	−	1.26309i
$960$		0			0
$961$	950.961			0.989553
$962$		0			0
$963$	400.800			0.416199
$964$		0			0
$965$		−	1900.05i		−	1.96897i
$966$		0			0
$967$	1277.50			1.32110			0.660548	−	0.750784i	$-0.270324\pi$
$967$	1277.50			1.32110			0.660548	+	0.750784i	$0.270324\pi$
$968$		0			0
$969$		−	467.399i		−	0.482352i
$970$		0			0
$971$			602.816i			0.620819i	0.950603	+	0.310410i	$0.100466\pi$
$971$			602.816i			0.620819i	−0.950603	+	0.310410i	$0.899534\pi$
$972$		0			0
$973$	1664.49	+	646.256i	1.71068	+	0.664189i
$974$		0			0
$975$	10.7708			0.0110470
$976$		0			0
$977$	−1217.87			−1.24654			−0.623271	−	0.782006i	$-0.714197\pi$
$977$	−1217.87			−1.24654			−0.623271	+	0.782006i	$0.714197\pi$
$978$		0			0
$979$		−	968.921i		−	0.989705i
$980$		0			0
$981$	−113.864			−0.116069
$982$		0			0
$983$			1467.03i			1.49240i	0.665722	+	0.746199i	$0.268123\pi$
$983$			1467.03i			1.49240i	−0.665722	+	0.746199i	$0.731877\pi$
$984$		0			0
$985$		−	339.320i		−	0.344487i
$986$		0			0
$987$	−4.44841	+	11.4573i	−0.00450700	+	0.0116082i
$988$		0			0
$989$	293.583			0.296848
$990$		0			0
$991$	1112.89			1.12299			0.561497	−	0.827479i	$-0.310225\pi$
$991$	1112.89			1.12299			0.561497	+	0.827479i	$0.310225\pi$
$992$		0			0
$993$		−	205.796i		−	0.207246i
$994$		0			0
$995$	358.741			0.360544
$996$		0			0
$997$		−	462.397i		−	0.463788i	−0.972741	−	0.231894i	$-0.925508\pi$
$997$		−	462.397i		−	0.463788i	0.972741	−	0.231894i	$-0.0744923\pi$
$998$		0			0
$999$		−	178.906i		−	0.179085i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.f.j.769.10		16
4.3	odd	2	inner	1344.3.f.j.769.2		16
7.6	odd	2	inner	1344.3.f.j.769.7		16
8.3	odd	2		672.3.f.b.97.15	yes	16
8.5	even	2		672.3.f.b.97.7	yes	16
24.5	odd	2		2016.3.f.g.1441.4		16
24.11	even	2		2016.3.f.g.1441.3		16
28.27	even	2	inner	1344.3.f.j.769.15		16
56.13	odd	2		672.3.f.b.97.10	yes	16
56.27	even	2		672.3.f.b.97.2	✓	16
168.83	odd	2		2016.3.f.g.1441.13		16
168.125	even	2		2016.3.f.g.1441.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.3.f.b.97.2	✓	16	56.27	even	2
672.3.f.b.97.7	yes	16	8.5	even	2
672.3.f.b.97.10	yes	16	56.13	odd	2
672.3.f.b.97.15	yes	16	8.3	odd	2
1344.3.f.j.769.2		16	4.3	odd	2	inner
1344.3.f.j.769.7		16	7.6	odd	2	inner
1344.3.f.j.769.10		16	1.1	even	1	trivial
1344.3.f.j.769.15		16	28.27	even	2	inner
2016.3.f.g.1441.3		16	24.11	even	2
2016.3.f.g.1441.4		16	24.5	odd	2
2016.3.f.g.1441.13		16	168.83	odd	2
2016.3.f.g.1441.14		16	168.125	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -5.15119i q^{5} +(2.53356 - 6.52542i) q^{7} -3.00000 q^{9} +14.9067 q^{11} +4.05181i q^{13} +8.92212 q^{15} +26.1871i q^{17} +10.3048i q^{19} +(11.3024 + 4.38825i) q^{21} -18.0230 q^{23} -1.53475 q^{25} -5.19615i q^{27} +42.4532 q^{29} +3.16847i q^{31} +25.8191i q^{33} +(-33.6137 - 13.0508i) q^{35} +34.4305 q^{37} -7.01793 q^{39} -41.4151i q^{41} -16.2894 q^{43} +15.4536i q^{45} +1.01371i q^{47} +(-36.1622 - 33.0651i) q^{49} -45.3574 q^{51} +84.3489 q^{53} -76.7870i q^{55} -17.8485 q^{57} -59.3790i q^{59} -86.6064i q^{61} +(-7.60067 + 19.5763i) q^{63} +20.8716 q^{65} +116.082 q^{67} -31.2167i q^{69} -97.7926 q^{71} +28.2973i q^{73} -2.65827i q^{75} +(37.7669 - 97.2722i) q^{77} +103.257 q^{79} +9.00000 q^{81} -34.8098i q^{83} +134.895 q^{85} +73.5311i q^{87} -64.9992i q^{89} +(26.4397 + 10.2655i) q^{91} -5.48795 q^{93} +53.0820 q^{95} +105.475i q^{97} -44.7200 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 48 q^{9} - 24 q^{21} - 64 q^{25} + 128 q^{29} - 48 q^{37} - 256 q^{49} + 160 q^{53} - 144 q^{57} - 288 q^{65} - 128 q^{77} + 144 q^{81} + 16 q^{85} - 96 q^{93}+O(q^{100}) \) 16 * q - 48 * q^9 - 24 * q^21 - 64 * q^25 + 128 * q^29 - 48 * q^37 - 256 * q^49 + 160 * q^53 - 144 * q^57 - 288 * q^65 - 128 * q^77 + 144 * q^81 + 16 * q^85 - 96 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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