LMFDB - Embedded newform 320.4.c.g.129.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(129,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.129"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,0,0,-4,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.3
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	320.129
Dual form			320.4.c.g.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.89898i q^{3} +(-8.79796 + 6.89898i) q^{5} -16.6969i q^{7} +18.5959 q^{9} +19.1918 q^{11} -61.7980i q^{13} +(-20.0000 - 25.5051i) q^{15} +30.3837i q^{17} +59.1918 q^{19} +48.4041 q^{21} +205.687i q^{23} +(29.8082 - 121.394i) q^{25} +132.182i q^{27} +8.38367 q^{29} +331.151 q^{31} +55.6367i q^{33} +(115.192 + 146.899i) q^{35} +266.565i q^{37} +179.151 q^{39} -320.788 q^{41} -83.1214i q^{43} +(-163.606 + 128.293i) q^{45} +276.434i q^{47} +64.2122 q^{49} -88.0816 q^{51} -390.888i q^{53} +(-168.849 + 132.404i) q^{55} +171.596i q^{57} +779.110 q^{59} +483.171 q^{61} -310.495i q^{63} +(426.343 + 543.696i) q^{65} +123.707i q^{67} -596.282 q^{69} +187.233 q^{71} -778.706i q^{73} +(351.918 + 86.4133i) q^{75} -320.445i q^{77} +446.384 q^{79} +118.898 q^{81} +1054.05i q^{83} +(-209.616 - 267.314i) q^{85} +24.3041i q^{87} +94.8490 q^{89} -1031.84 q^{91} +960.000i q^{93} +(-520.767 + 408.363i) q^{95} -252.041i q^{97} +356.890 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9} - 80 q^{11} - 80 q^{15} + 80 q^{19} + 272 q^{21} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} - 224 q^{39} - 1048 q^{41} - 772 q^{45} + 492 q^{49} - 1920 q^{51} - 1616 q^{55}+ \cdots + 3152 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 80 * q^11 - 80 * q^15 + 80 * q^19 + 272 * q^21 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 - 224 * q^39 - 1048 * q^41 - 772 * q^45 + 492 * q^49 - 1920 * q^51 - 1616 * q^55 + 1392 * q^59 + 1384 * q^61 + 608 * q^65 - 112 * q^69 + 1376 * q^71 - 160 * q^75 + 1472 * q^79 - 1484 * q^81 - 1152 * q^85 + 1320 * q^89 - 992 * q^91 - 1456 * q^95 + 3152 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			2.89898i			0.557909i	0.960304	+	0.278954i	$0.0899878\pi$
$3$			2.89898i			0.557909i	−0.960304	+	0.278954i	$0.910012\pi$
$4$		0			0
$5$	−8.79796	+	6.89898i	−0.786913	+	0.617063i
$5$	−8.79796	+	6.89898i	−0.786913	+	0.617063i
$6$		0			0
$7$		−	16.6969i		−	0.901550i	−0.892638	−	0.450775i	$-0.851148\pi$
$7$		−	16.6969i		−	0.901550i	0.892638	−	0.450775i	$-0.148852\pi$
$8$		0			0
$9$	18.5959			0.688738
$10$		0			0
$11$	19.1918			0.526051			0.263025	−	0.964789i	$-0.415280\pi$
$11$	19.1918			0.526051			0.263025	+	0.964789i	$0.415280\pi$
$12$		0			0
$13$		−	61.7980i		−	1.31844i	−0.751952	−	0.659218i	$-0.770887\pi$
$13$		−	61.7980i		−	1.31844i	0.751952	−	0.659218i	$-0.229113\pi$
$14$		0			0
$15$	−20.0000	−	25.5051i	−0.344265	−	0.439026i
$16$		0			0
$17$			30.3837i			0.433478i	0.976230	+	0.216739i	$0.0695420\pi$
$17$			30.3837i			0.433478i	−0.976230	+	0.216739i	$0.930458\pi$
$18$		0			0
$19$	59.1918			0.714713			0.357356	−	0.933968i	$-0.383678\pi$
$19$	59.1918			0.714713			0.357356	+	0.933968i	$0.383678\pi$
$20$		0			0
$21$	48.4041			0.502983
$22$		0			0
$23$			205.687i			1.86472i	0.361526	+	0.932362i	$0.382256\pi$
$23$			205.687i			1.86472i	−0.361526	+	0.932362i	$0.617744\pi$
$24$		0			0
$25$	29.8082	−	121.394i	0.238465	−	0.971151i
$26$		0			0
$27$			132.182i			0.942162i
$28$		0			0
$29$	8.38367			0.0536831			0.0268415	−	0.999640i	$-0.491455\pi$
$29$	8.38367			0.0536831			0.0268415	+	0.999640i	$0.491455\pi$
$30$		0			0
$31$	331.151			1.91860			0.959298	−	0.282396i	$-0.0911291\pi$
$31$	331.151			1.91860			0.959298	+	0.282396i	$0.0911291\pi$
$32$		0			0
$33$			55.6367i			0.293488i
$34$		0			0
$35$	115.192	+	146.899i	0.556314	+	0.709442i
$36$		0			0
$37$			266.565i			1.18441i	0.805788	+	0.592204i	$0.201742\pi$
$37$			266.565i			1.18441i	−0.805788	+	0.592204i	$0.798258\pi$
$38$		0			0
$39$	179.151			0.735567
$40$		0			0
$41$	−320.788			−1.22192			−0.610959	−	0.791662i	$-0.709216\pi$
$41$	−320.788			−1.22192			−0.610959	+	0.791662i	$0.709216\pi$
$42$		0			0
$43$		−	83.1214i		−	0.294788i	−0.989078	−	0.147394i	$-0.952911\pi$
$43$		−	83.1214i		−	0.294788i	0.989078	−	0.147394i	$-0.0470886\pi$
$44$		0			0
$45$	−163.606	+	128.293i	−0.541977	+	0.424995i
$46$		0			0
$47$			276.434i			0.857915i	0.903325	+	0.428957i	$0.141119\pi$
$47$			276.434i			0.857915i	−0.903325	+	0.428957i	$0.858881\pi$
$48$		0			0
$49$	64.2122			0.187208
$50$		0			0
$51$	−88.0816			−0.241841
$52$		0			0
$53$		−	390.888i		−	1.01307i	−0.862220	−	0.506534i	$-0.830927\pi$
$53$		−	390.888i		−	1.01307i	0.862220	−	0.506534i	$-0.169073\pi$
$54$		0			0
$55$	−168.849	+	132.404i	−0.413956	+	0.324607i
$56$		0			0
$57$			171.596i			0.398744i
$58$		0			0
$59$	779.110			1.71918			0.859589	−	0.510986i	$-0.170720\pi$
$59$	779.110			1.71918			0.859589	+	0.510986i	$0.170720\pi$
$60$		0			0
$61$	483.171			1.01416			0.507080	−	0.861899i	$-0.330725\pi$
$61$	483.171			1.01416			0.507080	+	0.861899i	$0.330725\pi$
$62$		0			0
$63$		−	310.495i		−	0.620931i
$64$		0			0
$65$	426.343	+	543.696i	0.813559	+	1.03750i
$66$		0			0
$67$			123.707i			0.225571i	0.993619	+	0.112785i	$0.0359772\pi$
$67$			123.707i			0.225571i	−0.993619	+	0.112785i	$0.964023\pi$
$68$		0			0
$69$	−596.282			−1.04035
$70$		0			0
$71$	187.233			0.312964			0.156482	−	0.987681i	$-0.449985\pi$
$71$	187.233			0.312964			0.156482	+	0.987681i	$0.449985\pi$
$72$		0			0
$73$		−	778.706i		−	1.24850i	−0.781224	−	0.624251i	$-0.785404\pi$
$73$		−	778.706i		−	1.24850i	0.781224	−	0.624251i	$-0.214596\pi$
$74$		0			0
$75$	351.918	+	86.4133i	0.541814	+	0.133042i
$76$		0			0
$77$		−	320.445i		−	0.474261i
$78$		0			0
$79$	446.384			0.635723			0.317861	−	0.948137i	$-0.397035\pi$
$79$	446.384			0.635723			0.317861	+	0.948137i	$0.397035\pi$
$80$		0			0
$81$	118.898			0.163097
$82$		0			0
$83$			1054.05i			1.39394i	0.717100	+	0.696970i	$0.245469\pi$
$83$			1054.05i			1.39394i	−0.717100	+	0.696970i	$0.754531\pi$
$84$		0			0
$85$	−209.616	−	267.314i	−0.267483	−	0.341109i
$86$		0			0
$87$			24.3041i			0.0299503i
$88$		0			0
$89$	94.8490			0.112966			0.0564830	−	0.998404i	$-0.482011\pi$
$89$	94.8490			0.112966			0.0564830	+	0.998404i	$0.482011\pi$
$90$		0			0
$91$	−1031.84			−1.18864
$92$		0			0
$93$			960.000i			1.07040i
$94$		0			0
$95$	−520.767	+	408.363i	−0.562417	+	0.441023i
$96$		0			0
$97$		−	252.041i		−	0.263823i	−0.991261	−	0.131912i	$-0.957888\pi$
$97$		−	252.041i		−	0.263823i	0.991261	−	0.131912i	$-0.0421115\pi$
$98$		0			0
$99$	356.890			0.362311
$100$		0			0
$101$	−37.9184			−0.0373566			−0.0186783	−	0.999826i	$-0.505946\pi$
$101$	−37.9184			−0.0373566			−0.0186783	+	0.999826i	$0.505946\pi$
$102$		0			0
$103$			94.4133i			0.0903186i	0.998980	+	0.0451593i	$0.0143795\pi$
$103$			94.4133i			0.0903186i	−0.998980	+	0.0451593i	$0.985620\pi$
$104$		0			0
$105$	−425.857	+	333.939i	−0.395804	+	0.310372i
$106$		0			0
$107$		−	901.464i		−	0.814466i	−0.913324	−	0.407233i	$-0.866494\pi$
$107$		−	901.464i		−	0.814466i	0.913324	−	0.407233i	$-0.133506\pi$
$108$		0			0
$109$	1415.69			1.24403			0.622013	−	0.783007i	$-0.286315\pi$
$109$	1415.69			1.24403			0.622013	+	0.783007i	$0.286315\pi$
$110$		0			0
$111$	−772.767			−0.660791
$112$		0			0
$113$		−	293.576i		−	0.244401i	−0.992505	−	0.122200i	$-0.961005\pi$
$113$		−	293.576i		−	0.244401i	0.992505	−	0.122200i	$-0.0389950\pi$
$114$		0			0
$115$	−1419.03	−	1809.62i	−1.15065	−	1.46738i
$116$		0			0
$117$		−	1149.19i		−	0.908057i
$118$		0			0
$119$	507.314			0.390802
$120$		0			0
$121$	−962.673			−0.723271
$122$		0			0
$123$		−	929.957i		−	0.681719i
$124$		0			0
$125$	575.243	+	1273.66i	0.411610	+	0.911360i
$126$		0			0
$127$		−	774.717i		−	0.541300i	−0.962678	−	0.270650i	$-0.912761\pi$
$127$		−	774.717i		−	0.541300i	0.962678	−	0.270650i	$-0.0872385\pi$
$128$		0			0
$129$	240.967			0.164465
$130$		0			0
$131$	−334.343			−0.222990			−0.111495	−	0.993765i	$-0.535564\pi$
$131$	−334.343			−0.222990			−0.111495	+	0.993765i	$0.535564\pi$
$132$		0			0
$133$		−	988.322i		−	0.644349i
$134$		0			0
$135$	−911.918	−	1162.93i	−0.581374	−	0.741400i
$136$		0			0
$137$		−	323.514i		−	0.201750i	−0.994899	−	0.100875i	$-0.967836\pi$
$137$		−	323.514i		−	0.201750i	0.994899	−	0.100875i	$-0.0321641\pi$
$138$		0			0
$139$	−396.482			−0.241936			−0.120968	−	0.992656i	$-0.538600\pi$
$139$	−396.482			−0.241936			−0.120968	+	0.992656i	$0.538600\pi$
$140$		0			0
$141$	−801.376			−0.478638
$142$		0			0
$143$		−	1186.02i		−	0.693564i
$144$		0			0
$145$	−73.7592	+	57.8388i	−0.0422439	+	0.0331259i
$146$		0			0
$147$			186.150i			0.104445i
$148$		0			0
$149$	−1682.89			−0.925284			−0.462642	−	0.886545i	$-0.653099\pi$
$149$	−1682.89			−0.925284			−0.462642	+	0.886545i	$0.653099\pi$
$150$		0			0
$151$	−2924.52			−1.57612			−0.788060	−	0.615598i	$-0.788915\pi$
$151$	−2924.52			−1.57612			−0.788060	+	0.615598i	$0.788915\pi$
$152$		0			0
$153$			565.012i			0.298553i
$154$		0			0
$155$	−2913.45	+	2284.60i	−1.50977	+	1.18390i
$156$		0			0
$157$			2768.42i			1.40729i	0.710553	+	0.703644i	$0.248445\pi$
$157$			2768.42i			1.40729i	−0.710553	+	0.703644i	$0.751555\pi$
$158$		0			0
$159$	1133.18			0.565199
$160$		0			0
$161$	3434.34			1.68114
$162$		0			0
$163$		−	2816.33i		−	1.35333i	−0.736292	−	0.676663i	$-0.763425\pi$
$163$		−	2816.33i		−	1.35333i	0.736292	−	0.676663i	$-0.236575\pi$
$164$		0			0
$165$	−383.837	−	489.490i	−0.181101	−	0.230950i
$166$		0			0
$167$		−	1836.41i		−	0.850933i	−0.904974	−	0.425466i	$-0.860110\pi$
$167$		−	1836.41i		−	0.850933i	0.904974	−	0.425466i	$-0.139890\pi$
$168$		0			0
$169$	−1621.99			−0.738274
$170$		0			0
$171$	1100.73			0.492249
$172$		0			0
$173$		−	1224.22i		−	0.538011i	−0.963139	−	0.269006i	$-0.913305\pi$
$173$		−	1224.22i		−	0.538011i	0.963139	−	0.269006i	$-0.0866950\pi$
$174$		0			0
$175$	−2026.91	−	497.705i	−0.875541	−	0.214988i
$176$		0			0
$177$			2258.62i			0.959145i
$178$		0			0
$179$	2729.58			1.13977			0.569883	−	0.821726i	$-0.306989\pi$
$179$	2729.58			1.13977			0.569883	+	0.821726i	$0.306989\pi$
$180$		0			0
$181$	−2642.36			−1.08511			−0.542555	−	0.840020i	$-0.682543\pi$
$181$	−2642.36			−1.08511			−0.542555	+	0.840020i	$0.682543\pi$
$182$		0			0
$183$			1400.70i			0.565809i
$184$		0			0
$185$	−1839.03	−	2345.23i	−0.730854	−	0.932026i
$186$		0			0
$187$			583.118i			0.228031i
$188$		0			0
$189$	2207.03			0.849406
$190$		0			0
$191$	−2339.23			−0.886183			−0.443091	−	0.896476i	$-0.646118\pi$
$191$	−2339.23			−0.886183			−0.443091	+	0.896476i	$0.646118\pi$
$192$		0			0
$193$			4601.61i			1.71622i	0.513464	+	0.858111i	$0.328362\pi$
$193$			4601.61i			1.71622i	−0.513464	+	0.858111i	$0.671638\pi$
$194$		0			0
$195$	−1576.16	+	1235.96i	−0.578828	+	0.453892i
$196$		0			0
$197$		−	823.941i		−	0.297987i	−0.988838	−	0.148993i	$-0.952397\pi$
$197$		−	823.941i		−	0.297987i	0.988838	−	0.148993i	$-0.0476033\pi$
$198$		0			0
$199$	3329.70			1.18611			0.593055	−	0.805162i	$-0.297922\pi$
$199$	3329.70			1.18611			0.593055	+	0.805162i	$0.297922\pi$
$200$		0			0
$201$	−358.624			−0.125848
$202$		0			0
$203$		−	139.982i		−	0.0483980i
$204$		0			0
$205$	2822.28	−	2213.11i	0.961543	−	0.754001i
$206$		0			0
$207$			3824.93i			1.28431i
$208$		0			0
$209$	1136.00			0.375975
$210$		0			0
$211$	−1018.78			−0.332398			−0.166199	−	0.986092i	$-0.553149\pi$
$211$	−1018.78			−0.332398			−0.166199	+	0.986092i	$0.553149\pi$
$212$		0			0
$213$			542.784i			0.174605i
$214$		0			0
$215$	573.453	+	731.299i	0.181903	+	0.231973i
$216$		0			0
$217$		−	5529.21i		−	1.72971i
$218$		0			0
$219$	2257.45			0.696550
$220$		0			0
$221$	1877.65			0.571513
$222$		0			0
$223$		−	99.1581i		−	0.0297763i	−0.999889	−	0.0148882i	$-0.995261\pi$
$223$		−	99.1581i		−	0.0297763i	0.999889	−	0.0148882i	$-0.00473922\pi$
$224$		0			0
$225$	554.310	−	2257.43i	0.164240	−	0.668868i
$226$		0			0
$227$		−	1197.59i		−	0.350163i	−0.984554	−	0.175081i	$-0.943981\pi$
$227$		−	1197.59i		−	0.350163i	0.984554	−	0.175081i	$-0.0560188\pi$
$228$		0			0
$229$	−453.592			−0.130892			−0.0654458	−	0.997856i	$-0.520847\pi$
$229$	−453.592			−0.130892			−0.0654458	+	0.997856i	$0.520847\pi$
$230$		0			0
$231$	928.963			0.264594
$232$		0			0
$233$		−	3788.49i		−	1.06520i	−0.846367	−	0.532601i	$-0.821215\pi$
$233$		−	3788.49i		−	1.06520i	0.846367	−	0.532601i	$-0.178785\pi$
$234$		0			0
$235$	−1907.11	−	2432.05i	−0.529388	−	0.675105i
$236$		0			0
$237$			1294.06i			0.354675i
$238$		0			0
$239$	6000.47			1.62401			0.812005	−	0.583651i	$-0.198376\pi$
$239$	6000.47			1.62401			0.812005	+	0.583651i	$0.198376\pi$
$240$		0			0
$241$	1842.53			0.492480			0.246240	−	0.969209i	$-0.420805\pi$
$241$	1842.53			0.492480			0.246240	+	0.969209i	$0.420805\pi$
$242$		0			0
$243$			3913.59i			1.03316i
$244$		0			0
$245$	−564.937	+	442.999i	−0.147316	+	0.115519i
$246$		0			0
$247$		−	3657.93i		−	0.942303i
$248$		0			0
$249$	−3055.67			−0.777691
$250$		0			0
$251$	−1149.46			−0.289057			−0.144529	−	0.989501i	$-0.546167\pi$
$251$	−1149.46			−0.289057			−0.144529	+	0.989501i	$0.546167\pi$
$252$		0			0
$253$			3947.51i			0.980939i
$254$		0			0
$255$	774.939	−	607.673i	0.190308	−	0.149231i
$256$		0			0
$257$			5407.67i			1.31253i	0.754528	+	0.656267i	$0.227866\pi$
$257$			5407.67i			1.31253i	−0.754528	+	0.656267i	$0.772134\pi$
$258$		0			0
$259$	4450.82			1.06780
$260$		0			0
$261$	155.902			0.0369735
$262$		0			0
$263$			2067.34i			0.484705i	0.970188	+	0.242352i	$0.0779190\pi$
$263$			2067.34i			0.484705i	−0.970188	+	0.242352i	$0.922081\pi$
$264$		0			0
$265$	2696.73	+	3439.01i	0.625127	+	0.797196i
$266$		0			0
$267$			274.965i			0.0630247i
$268$		0			0
$269$	−592.388			−0.134270			−0.0671348	−	0.997744i	$-0.521386\pi$
$269$	−592.388			−0.134270			−0.0671348	+	0.997744i	$0.521386\pi$
$270$		0			0
$271$	−2583.27			−0.579049			−0.289524	−	0.957171i	$-0.593497\pi$
$271$	−2583.27			−0.579049			−0.289524	+	0.957171i	$0.593497\pi$
$272$		0			0
$273$		−	2991.27i		−	0.663151i
$274$		0			0
$275$	572.073	−	2329.77i	0.125445	−	0.510875i
$276$		0			0
$277$		−	4488.09i		−	0.973513i	−0.873538	−	0.486756i	$-0.838180\pi$
$277$		−	4488.09i		−	0.973513i	0.873538	−	0.486756i	$-0.161820\pi$
$278$		0			0
$279$	6158.06			1.32141
$280$		0			0
$281$	−6280.54			−1.33333			−0.666665	−	0.745357i	$-0.732279\pi$
$281$	−6280.54			−1.33333			−0.666665	+	0.745357i	$0.732279\pi$
$282$		0			0
$283$			5233.40i			1.09927i	0.835405	+	0.549635i	$0.185233\pi$
$283$			5233.40i			1.09927i	−0.835405	+	0.549635i	$0.814767\pi$
$284$		0			0
$285$	−1183.84	−	1509.69i	−0.246051	−	0.313777i
$286$		0			0
$287$			5356.17i			1.10162i
$288$		0			0
$289$	3989.83			0.812097
$290$		0			0
$291$	730.661			0.147189
$292$		0			0
$293$			2438.21i			0.486149i	0.970008	+	0.243074i	$0.0781559\pi$
$293$			2438.21i			0.486149i	−0.970008	+	0.243074i	$0.921844\pi$
$294$		0			0
$295$	−6854.58	+	5375.07i	−1.35284	+	1.06084i
$296$		0			0
$297$			2536.81i			0.495625i
$298$		0			0
$299$	12711.0			2.45852
$300$		0			0
$301$	−1387.87			−0.265766
$302$		0			0
$303$		−	109.925i		−	0.0208416i
$304$		0			0
$305$	−4250.92	+	3333.39i	−0.798056	+	0.625801i
$306$		0			0
$307$			7910.44i			1.47060i	0.677744	+	0.735298i	$0.262958\pi$
$307$			7910.44i			1.47060i	−0.677744	+	0.735298i	$0.737042\pi$
$308$		0			0
$309$	−273.702			−0.0503895
$310$		0			0
$311$	5419.71			0.988178			0.494089	−	0.869411i	$-0.335502\pi$
$311$	5419.71			0.988178			0.494089	+	0.869411i	$0.335502\pi$
$312$		0			0
$313$			5570.86i			1.00602i	0.864281	+	0.503009i	$0.167774\pi$
$313$			5570.86i			1.00602i	−0.864281	+	0.503009i	$0.832226\pi$
$314$		0			0
$315$	2142.10	+	2731.72i	0.383154	+	0.488619i
$316$		0			0
$317$		−	3724.09i		−	0.659829i	−0.944011	−	0.329915i	$-0.892980\pi$
$317$		−	3724.09i		−	0.659829i	0.944011	−	0.329915i	$-0.107020\pi$
$318$		0			0
$319$	160.898			0.0282400
$320$		0			0
$321$	2613.33			0.454398
$322$		0			0
$323$			1798.47i			0.309812i
$324$		0			0
$325$	−7501.89	−	1842.08i	−1.28040	−	0.314401i
$326$		0			0
$327$			4104.07i			0.694053i
$328$		0			0
$329$	4615.60			0.773453
$330$		0			0
$331$	−3223.96			−0.535362			−0.267681	−	0.963508i	$-0.586257\pi$
$331$	−3223.96			−0.535362			−0.267681	+	0.963508i	$0.586257\pi$
$332$		0			0
$333$			4957.03i			0.815746i
$334$		0			0
$335$	−853.453	−	1088.37i	−0.139191	−	0.177505i
$336$		0			0
$337$		−	9524.43i		−	1.53955i	−0.638314	−	0.769776i	$-0.720368\pi$
$337$		−	9524.43i		−	1.53955i	0.638314	−	0.769776i	$-0.279632\pi$
$338$		0			0
$339$	851.069			0.136353
$340$		0			0
$341$	6355.40			1.00928
$342$		0			0
$343$		−	6799.20i		−	1.07033i
$344$		0			0
$345$	5246.06	−	4113.73i	0.818662	−	0.641960i
$346$		0			0
$347$		−	6042.30i		−	0.934777i	−0.884052	−	0.467388i	$-0.845195\pi$
$347$		−	6042.30i		−	0.934777i	0.884052	−	0.467388i	$-0.154805\pi$
$348$		0			0
$349$	−1626.20			−0.249422			−0.124711	−	0.992193i	$-0.539800\pi$
$349$	−1626.20			−0.249422			−0.124711	+	0.992193i	$0.539800\pi$
$350$		0			0
$351$	8168.55			1.24218
$352$		0			0
$353$			886.955i			0.133733i	0.997762	+	0.0668667i	$0.0213002\pi$
$353$			886.955i			0.133733i	−0.997762	+	0.0668667i	$0.978700\pi$
$354$		0			0
$355$	−1647.27	+	1291.71i	−0.246275	+	0.193119i
$356$		0			0
$357$			1470.69i			0.218032i
$358$		0			0
$359$	−1722.27			−0.253198			−0.126599	−	0.991954i	$-0.540406\pi$
$359$	−1722.27			−0.253198			−0.126599	+	0.991954i	$0.540406\pi$
$360$		0			0
$361$	−3355.33			−0.489186
$362$		0			0
$363$		−	2790.77i		−	0.403519i
$364$		0			0
$365$	5372.28	+	6851.02i	0.770405	+	0.982463i
$366$		0			0
$367$		−	9271.77i		−	1.31875i	−0.751813	−	0.659377i	$-0.770820\pi$
$367$		−	9271.77i		−	1.31875i	0.751813	−	0.659377i	$-0.229180\pi$
$368$		0			0
$369$	−5965.34			−0.841581
$370$		0			0
$371$	−6526.63			−0.913331
$372$		0			0
$373$		−	5697.15i		−	0.790850i	−0.918498	−	0.395425i	$-0.870597\pi$
$373$		−	5697.15i		−	0.790850i	0.918498	−	0.395425i	$-0.129403\pi$
$374$		0			0
$375$	−3692.33	+	1667.62i	−0.508456	+	0.229641i
$376$		0			0
$377$		−	518.094i		−	0.0707777i
$378$		0			0
$379$	−8526.24			−1.15558			−0.577789	−	0.816186i	$-0.696084\pi$
$379$	−8526.24			−1.15558			−0.577789	+	0.816186i	$0.696084\pi$
$380$		0			0
$381$	2245.89			0.301996
$382$		0			0
$383$			4069.23i			0.542893i	0.962453	+	0.271447i	$0.0875020\pi$
$383$			4069.23i			0.542893i	−0.962453	+	0.271447i	$0.912498\pi$
$384$		0			0
$385$	2210.74	+	2819.26i	0.292649	+	0.373202i
$386$		0			0
$387$		−	1545.72i		−	0.203032i
$388$		0			0
$389$	−2394.17			−0.312054			−0.156027	−	0.987753i	$-0.549869\pi$
$389$	−2394.17			−0.312054			−0.156027	+	0.987753i	$0.549869\pi$
$390$		0			0
$391$	−6249.52			−0.808316
$392$		0			0
$393$		−	969.253i		−	0.124408i
$394$		0			0
$395$	−3927.27	+	3079.59i	−0.500259	+	0.392281i
$396$		0			0
$397$			3497.79i			0.442190i	0.975252	+	0.221095i	$0.0709630\pi$
$397$			3497.79i			0.442190i	−0.975252	+	0.221095i	$0.929037\pi$
$398$		0			0
$399$	2865.13			0.359488
$400$		0			0
$401$	−8608.89			−1.07209			−0.536044	−	0.844190i	$-0.680082\pi$
$401$	−8608.89			−1.07209			−0.536044	+	0.844190i	$0.680082\pi$
$402$		0			0
$403$		−	20464.5i		−	2.52955i
$404$		0			0
$405$	−1046.06	+	820.275i	−0.128343	+	0.100641i
$406$		0			0
$407$			5115.88i			0.623058i
$408$		0			0
$409$	−1385.67			−0.167523			−0.0837615	−	0.996486i	$-0.526693\pi$
$409$	−1385.67			−0.167523			−0.0837615	+	0.996486i	$0.526693\pi$
$410$		0			0
$411$	937.861			0.112558
$412$		0			0
$413$		−	13008.8i		−	1.54992i
$414$		0			0
$415$	−7271.87	−	9273.49i	−0.860149	−	1.09691i
$416$		0			0
$417$		−	1149.39i		−	0.134978i
$418$		0			0
$419$	3738.23			0.435858			0.217929	−	0.975965i	$-0.430070\pi$
$419$	3738.23			0.435858			0.217929	+	0.975965i	$0.430070\pi$
$420$		0			0
$421$	8993.95			1.04118			0.520592	−	0.853806i	$-0.325711\pi$
$421$	8993.95			1.04118			0.520592	+	0.853806i	$0.325711\pi$
$422$		0			0
$423$			5140.54i			0.590878i
$424$		0			0
$425$	3688.39	+	905.681i	0.420972	+	0.103369i
$426$		0			0
$427$		−	8067.48i		−	0.914316i
$428$		0			0
$429$	3438.24			0.386946
$430$		0			0
$431$	462.547			0.0516940			0.0258470	−	0.999666i	$-0.491772\pi$
$431$	462.547			0.0516940			0.0258470	+	0.999666i	$0.491772\pi$
$432$		0			0
$433$		−	5231.82i		−	0.580659i	−0.956927	−	0.290329i	$-0.906235\pi$
$433$		−	5231.82i		−	0.580659i	0.956927	−	0.290329i	$-0.0937649\pi$
$434$		0			0
$435$	−167.673	−	213.826i	−0.0184812	−	0.0235683i
$436$		0			0
$437$			12175.0i			1.33274i
$438$		0			0
$439$	−8995.77			−0.978006			−0.489003	−	0.872282i	$-0.662639\pi$
$439$	−8995.77			−0.978006			−0.489003	+	0.872282i	$0.662639\pi$
$440$		0			0
$441$	1194.09			0.128937
$442$		0			0
$443$			5549.52i			0.595182i	0.954694	+	0.297591i	$0.0961831\pi$
$443$			5549.52i			0.595182i	−0.954694	+	0.297591i	$0.903817\pi$
$444$		0			0
$445$	−834.477	+	654.361i	−0.0888944	+	0.0697072i
$446$		0			0
$447$		−	4878.65i		−	0.516224i
$448$		0			0
$449$	5951.29			0.625521			0.312760	−	0.949832i	$-0.398746\pi$
$449$	5951.29			0.625521			0.312760	+	0.949832i	$0.398746\pi$
$450$		0			0
$451$	−6156.51			−0.642791
$452$		0			0
$453$		−	8478.13i		−	0.879332i
$454$		0			0
$455$	9078.06	−	7118.62i	0.935354	−	0.733464i
$456$		0			0
$457$			13912.7i			1.42409i	0.702132	+	0.712047i	$0.252232\pi$
$457$			13912.7i			1.42409i	−0.702132	+	0.712047i	$0.747768\pi$
$458$		0			0
$459$	−4016.16			−0.408406
$460$		0			0
$461$	−17467.6			−1.76475			−0.882374	−	0.470549i	$-0.844056\pi$
$461$	−17467.6			−1.76475			−0.882374	+	0.470549i	$0.844056\pi$
$462$		0			0
$463$			1575.36i			0.158127i	0.996870	+	0.0790637i	$0.0251931\pi$
$463$			1575.36i			0.158127i	−0.996870	+	0.0790637i	$0.974807\pi$
$464$		0			0
$465$	−6623.02	−	8446.04i	−0.660506	−	0.842313i
$466$		0			0
$467$			15618.3i			1.54760i	0.633430	+	0.773800i	$0.281646\pi$
$467$			15618.3i			1.54760i	−0.633430	+	0.773800i	$0.718354\pi$
$468$		0			0
$469$	2065.53			0.203363
$470$		0			0
$471$	−8025.60			−0.785138
$472$		0			0
$473$		−	1595.25i		−	0.155074i
$474$		0			0
$475$	1764.40	−	7185.53i	0.170434	−	0.694094i
$476$		0			0
$477$		−	7268.92i		−	0.697738i
$478$		0			0
$479$	−9527.90			−0.908854			−0.454427	−	0.890784i	$-0.650156\pi$
$479$	−9527.90			−0.908854			−0.454427	+	0.890784i	$0.650156\pi$
$480$		0			0
$481$	16473.2			1.56157
$482$		0			0
$483$			9956.08i			0.937924i
$484$		0			0
$485$	1738.82	+	2217.44i	0.162796	+	0.207606i
$486$		0			0
$487$		−	15729.6i		−	1.46361i	−0.681514	−	0.731805i	$-0.738678\pi$
$487$		−	15729.6i		−	1.46361i	0.681514	−	0.731805i	$-0.261322\pi$
$488$		0			0
$489$	8164.49			0.755033
$490$		0			0
$491$	−2566.49			−0.235894			−0.117947	−	0.993020i	$-0.537631\pi$
$491$	−2566.49			−0.235894			−0.117947	+	0.993020i	$0.537631\pi$
$492$		0			0
$493$			254.727i			0.0232704i
$494$		0			0
$495$	−3139.90	+	2462.18i	−0.285107	+	0.223569i
$496$		0			0
$497$		−	3126.21i		−	0.282152i
$498$		0			0
$499$	−13560.4			−1.21652			−0.608261	−	0.793737i	$-0.708133\pi$
$499$	−13560.4			−1.21652			−0.608261	+	0.793737i	$0.708133\pi$
$500$		0			0
$501$	5323.72			0.474743
$502$		0			0
$503$		−	5222.51i		−	0.462943i	−0.972842	−	0.231471i	$-0.925646\pi$
$503$		−	5222.51i		−	0.462943i	0.972842	−	0.231471i	$-0.0743540\pi$
$504$		0			0
$505$	333.604	−	261.598i	0.0293964	−	0.0230514i
$506$		0			0
$507$		−	4702.11i		−	0.411890i
$508$		0			0
$509$	−11875.9			−1.03416			−0.517082	−	0.855936i	$-0.672982\pi$
$509$	−11875.9			−1.03416			−0.517082	+	0.855936i	$0.672982\pi$
$510$		0			0
$511$	−13002.0			−1.12559
$512$		0			0
$513$			7824.07i			0.673375i
$514$		0			0
$515$	−651.355	−	830.644i	−0.0557323	−	0.0710729i
$516$		0			0
$517$			5305.27i			0.451307i
$518$		0			0
$519$	3549.00			0.300161
$520$		0			0
$521$	2456.04			0.206528			0.103264	−	0.994654i	$-0.467071\pi$
$521$	2456.04			0.206528			0.103264	+	0.994654i	$0.467071\pi$
$522$		0			0
$523$		−	634.460i		−	0.0530459i	−0.999648	−	0.0265229i	$-0.991556\pi$
$523$		−	634.460i		−	0.0530459i	0.999648	−	0.0265229i	$-0.00844351\pi$
$524$		0			0
$525$	1442.84	−	5875.96i	0.119944	−	0.488472i
$526$		0			0
$527$			10061.6i			0.831669i
$528$		0			0
$529$	−30140.0			−2.47720
$530$		0			0
$531$	14488.3			1.18406
$532$		0			0
$533$			19824.0i			1.61102i
$534$		0			0
$535$	6219.18	+	7931.05i	0.502577	+	0.640914i
$536$		0			0
$537$			7912.98i			0.635885i
$538$		0			0
$539$	1232.35			0.0984807
$540$		0			0
$541$	−18078.4			−1.43669			−0.718347	−	0.695685i	$-0.755101\pi$
$541$	−18078.4			−1.43669			−0.718347	+	0.695685i	$0.755101\pi$
$542$		0			0
$543$		−	7660.14i		−	0.605393i
$544$		0			0
$545$	−12455.2	+	9766.84i	−0.978941	+	0.767643i
$546$		0			0
$547$		−	24815.3i		−	1.93972i	−0.243661	−	0.969860i	$-0.578349\pi$
$547$		−	24815.3i		−	1.93972i	0.243661	−	0.969860i	$-0.421651\pi$
$548$		0			0
$549$	8985.02			0.698490
$550$		0			0
$551$	496.245			0.0383680
$552$		0			0
$553$		−	7453.24i		−	0.573136i
$554$		0			0
$555$	6798.78	−	5331.31i	0.519985	−	0.407750i
$556$		0			0
$557$		−	10073.7i		−	0.766310i	−0.923684	−	0.383155i	$-0.874838\pi$
$557$		−	10073.7i		−	0.766310i	0.923684	−	0.383155i	$-0.125162\pi$
$558$		0			0
$559$	−5136.73			−0.388660
$560$		0			0
$561$	−1690.45			−0.127221
$562$		0			0
$563$		−	7505.06i		−	0.561813i	−0.959735	−	0.280906i	$-0.909365\pi$
$563$		−	7505.06i		−	0.561813i	0.959735	−	0.280906i	$-0.0906350\pi$
$564$		0			0
$565$	2025.37	+	2582.87i	0.150811	+	0.192322i
$566$		0			0
$567$		−	1985.23i		−	0.147040i
$568$		0			0
$569$	15251.4			1.12368			0.561838	−	0.827247i	$-0.310094\pi$
$569$	15251.4			1.12368			0.561838	+	0.827247i	$0.310094\pi$
$570$		0			0
$571$	2683.78			0.196695			0.0983474	−	0.995152i	$-0.468644\pi$
$571$	2683.78			0.196695			0.0983474	+	0.995152i	$0.468644\pi$
$572$		0			0
$573$		−	6781.39i		−	0.494409i
$574$		0			0
$575$	24969.1	+	6131.14i	1.81093	+	0.444672i
$576$		0			0
$577$			7369.88i			0.531737i	0.964009	+	0.265868i	$0.0856586\pi$
$577$			7369.88i			0.531737i	−0.964009	+	0.265868i	$0.914341\pi$
$578$		0			0
$579$	−13340.0			−0.957496
$580$		0			0
$581$	17599.4			1.25671
$582$		0			0
$583$		−	7501.85i		−	0.532925i
$584$		0			0
$585$	7928.24	+	10110.5i	0.560329	+	0.714562i
$586$		0			0
$587$			20865.4i			1.46713i	0.679618	+	0.733567i	$0.262146\pi$
$587$			20865.4i			1.46713i	−0.679618	+	0.733567i	$0.737854\pi$
$588$		0			0
$589$	19601.4			1.37124
$590$		0			0
$591$	2388.59			0.166249
$592$		0			0
$593$		−	25894.0i		−	1.79316i	−0.442886	−	0.896578i	$-0.646046\pi$
$593$		−	25894.0i		−	1.79316i	0.442886	−	0.896578i	$-0.353954\pi$
$594$		0			0
$595$	−4463.33	+	3499.95i	−0.307527	+	0.241150i
$596$		0			0
$597$			9652.73i			0.661742i
$598$		0			0
$599$	−5632.42			−0.384198			−0.192099	−	0.981376i	$-0.561529\pi$
$599$	−5632.42			−0.384198			−0.192099	+	0.981376i	$0.561529\pi$
$600$		0			0
$601$	−13079.7			−0.887742			−0.443871	−	0.896091i	$-0.646395\pi$
$601$	−13079.7			−0.887742			−0.443871	+	0.896091i	$0.646395\pi$
$602$		0			0
$603$			2300.45i			0.155359i
$604$		0			0
$605$	8469.56	−	6641.46i	0.569151	−	0.446304i
$606$		0			0
$607$			18890.2i			1.26314i	0.775317	+	0.631572i	$0.217590\pi$
$607$			18890.2i			1.26314i	−0.775317	+	0.631572i	$0.782410\pi$
$608$		0			0
$609$	405.804			0.0270017
$610$		0			0
$611$	17083.0			1.13111
$612$		0			0
$613$			13631.5i			0.898155i	0.893493	+	0.449078i	$0.148247\pi$
$613$			13631.5i			0.898155i	−0.893493	+	0.449078i	$0.851753\pi$
$614$		0			0
$615$	6415.76	+	8181.72i	0.420664	+	0.536454i
$616$		0			0
$617$		−	14982.2i		−	0.977568i	−0.872405	−	0.488784i	$-0.837441\pi$
$617$		−	14982.2i		−	0.977568i	0.872405	−	0.488784i	$-0.162559\pi$
$618$		0			0
$619$	−25049.9			−1.62656			−0.813279	−	0.581873i	$-0.802320\pi$
$619$	−25049.9			−1.62656			−0.813279	+	0.581873i	$0.802320\pi$
$620$		0			0
$621$	−27188.0			−1.75687
$622$		0			0
$623$		−	1583.69i		−	0.101844i
$624$		0			0
$625$	−13847.9	−	7237.06i	−0.886269	−	0.463172i
$626$		0			0
$627$			3293.24i			0.209760i
$628$		0			0
$629$	−8099.23			−0.513414
$630$		0			0
$631$	18711.0			1.18046			0.590232	−	0.807233i	$-0.299036\pi$
$631$	18711.0			1.18046			0.590232	+	0.807233i	$0.299036\pi$
$632$		0			0
$633$		−	2953.43i		−	0.185448i
$634$		0			0
$635$	5344.76	+	6815.93i	0.334016	+	0.425956i
$636$		0			0
$637$		−	3968.19i		−	0.246821i
$638$		0			0
$639$	3481.76			0.215550
$640$		0			0
$641$	25792.2			1.58929			0.794643	−	0.607077i	$-0.207658\pi$
$641$	25792.2			1.58929			0.794643	+	0.607077i	$0.207658\pi$
$642$		0			0
$643$			20256.7i			1.24237i	0.783663	+	0.621186i	$0.213349\pi$
$643$			20256.7i			1.24237i	−0.783663	+	0.621186i	$0.786651\pi$
$644$		0			0
$645$	−2120.02	+	1662.43i	−0.129420	+	0.101485i
$646$		0			0
$647$			6655.43i			0.404408i	0.979343	+	0.202204i	$0.0648104\pi$
$647$			6655.43i			0.404408i	−0.979343	+	0.202204i	$0.935190\pi$
$648$		0			0
$649$	14952.6			0.904375
$650$		0			0
$651$	16029.1			0.965021
$652$		0			0
$653$		−	8490.91i		−	0.508844i	−0.967093	−	0.254422i	$-0.918115\pi$
$653$		−	8490.91i		−	0.508844i	0.967093	−	0.254422i	$-0.0818852\pi$
$654$		0			0
$655$	2941.53	−	2306.62i	0.175474	−	0.137599i
$656$		0			0
$657$		−	14480.8i		−	0.859890i
$658$		0			0
$659$	15543.8			0.918816			0.459408	−	0.888225i	$-0.348062\pi$
$659$	15543.8			0.918816			0.459408	+	0.888225i	$0.348062\pi$
$660$		0			0
$661$	13519.8			0.795553			0.397777	−	0.917482i	$-0.369782\pi$
$661$	13519.8			0.795553			0.397777	+	0.917482i	$0.369782\pi$
$662$		0			0
$663$			5443.27i			0.318852i
$664$		0			0
$665$	6818.42	+	8695.22i	0.397604	+	0.507047i
$666$		0			0
$667$			1724.41i			0.100104i
$668$		0			0
$669$	287.457			0.0166125
$670$		0			0
$671$	9272.95			0.533499
$672$		0			0
$673$			11565.3i			0.662421i	0.943557	+	0.331211i	$0.107457\pi$
$673$			11565.3i			0.662421i	−0.943557	+	0.331211i	$0.892543\pi$
$674$		0			0
$675$	16046.0	+	3940.09i	0.914981	+	0.224673i
$676$		0			0
$677$		−	28227.5i		−	1.60247i	−0.598350	−	0.801235i	$-0.704177\pi$
$677$		−	28227.5i		−	1.60247i	0.598350	−	0.801235i	$-0.295823\pi$
$678$		0			0
$679$	−4208.31			−0.237850
$680$		0			0
$681$	3471.79			0.195359
$682$		0			0
$683$			6425.99i			0.360005i	0.983666	+	0.180003i	$0.0576106\pi$
$683$			6425.99i			0.360005i	−0.983666	+	0.180003i	$0.942389\pi$
$684$		0			0
$685$	2231.92	+	2846.27i	0.124492	+	0.158759i
$686$		0			0
$687$		−	1314.95i		−	0.0730256i
$688$		0			0
$689$	−24156.1			−1.33566
$690$		0			0
$691$	19066.9			1.04969			0.524846	−	0.851197i	$-0.324123\pi$
$691$	19066.9			1.04969			0.524846	+	0.851197i	$0.324123\pi$
$692$		0			0
$693$		−	5958.97i		−	0.326641i
$694$		0			0
$695$	3488.23	−	2735.32i	0.190383	−	0.149290i
$696$		0			0
$697$		−	9746.71i		−	0.529674i
$698$		0			0
$699$	10982.7			0.594285
$700$		0			0
$701$	−4796.00			−0.258406			−0.129203	−	0.991618i	$-0.541242\pi$
$701$	−4796.00			−0.258406			−0.129203	+	0.991618i	$0.541242\pi$
$702$		0			0
$703$			15778.5i			0.846511i
$704$		0			0
$705$	7050.47	−	5528.67i	0.376647	−	0.295350i
$706$		0			0
$707$			633.121i			0.0336789i
$708$		0			0
$709$	−9805.33			−0.519389			−0.259695	−	0.965691i	$-0.583622\pi$
$709$	−9805.33			−0.519389			−0.259695	+	0.965691i	$0.583622\pi$
$710$		0			0
$711$	8300.91			0.437846
$712$		0			0
$713$			68113.4i			3.57765i
$714$		0			0
$715$	8182.30	+	10434.5i	0.427973	+	0.545775i
$716$		0			0
$717$			17395.2i			0.906049i
$718$		0			0
$719$	−27539.7			−1.42845			−0.714227	−	0.699915i	$-0.753221\pi$
$719$	−27539.7			−1.42845			−0.714227	+	0.699915i	$0.753221\pi$
$720$		0			0
$721$	1576.41			0.0814267
$722$		0			0
$723$			5341.45i			0.274759i
$724$		0			0
$725$	249.902	−	1017.73i	0.0128015	−	0.0521344i
$726$		0			0
$727$			16543.8i			0.843985i	0.906599	+	0.421993i	$0.138669\pi$
$727$			16543.8i			0.843985i	−0.906599	+	0.421993i	$0.861331\pi$
$728$		0			0
$729$	−8135.16			−0.413309
$730$		0			0
$731$	2525.53			0.127784
$732$		0			0
$733$			16718.6i			0.842450i	0.906956	+	0.421225i	$0.138400\pi$
$733$			16718.6i			0.842450i	−0.906956	+	0.421225i	$0.861600\pi$
$734$		0			0
$735$	−1284.24	−	1637.74i	−0.0644491	−	0.0821890i
$736$		0			0
$737$			2374.17i			0.118662i
$738$		0			0
$739$	−24022.6			−1.19579			−0.597893	−	0.801576i	$-0.703995\pi$
$739$	−24022.6			−1.19579			−0.597893	+	0.801576i	$0.703995\pi$
$740$		0			0
$741$	10604.3			0.525719
$742$		0			0
$743$			26201.8i			1.29374i	0.762600	+	0.646871i	$0.223923\pi$
$743$			26201.8i			1.29374i	−0.762600	+	0.646871i	$0.776077\pi$
$744$		0			0
$745$	14806.0	−	11610.2i	0.728119	−	0.570959i
$746$		0			0
$747$			19601.0i			0.960059i
$748$		0			0
$749$	−15051.7			−0.734282
$750$		0			0
$751$	24451.7			1.18809			0.594045	−	0.804432i	$-0.297530\pi$
$751$	24451.7			1.18809			0.594045	+	0.804432i	$0.297530\pi$
$752$		0			0
$753$		−	3332.26i		−	0.161268i
$754$		0			0
$755$	25729.8	−	20176.2i	1.24027	−	0.972567i
$756$		0			0
$757$		−	21403.8i		−	1.02765i	−0.857894	−	0.513827i	$-0.828227\pi$
$757$		−	21403.8i		−	1.02765i	0.857894	−	0.513827i	$-0.171773\pi$
$758$		0			0
$759$	−11443.7			−0.547275
$760$		0			0
$761$	−28935.9			−1.37835			−0.689176	−	0.724594i	$-0.742027\pi$
$761$	−28935.9			−1.37835			−0.689176	+	0.724594i	$0.742027\pi$
$762$		0			0
$763$		−	23637.8i		−	1.12155i
$764$		0			0
$765$	−3898.01	−	4970.95i	−0.184226	−	0.234935i
$766$		0			0
$767$		−	48147.4i		−	2.26663i
$768$		0			0
$769$	11479.3			0.538301			0.269151	−	0.963098i	$-0.413257\pi$
$769$	11479.3			0.538301			0.269151	+	0.963098i	$0.413257\pi$
$770$		0			0
$771$	−15676.7			−0.732275
$772$		0			0
$773$			2512.27i			0.116895i	0.998290	+	0.0584477i	$0.0186151\pi$
$773$			2512.27i			0.116895i	−0.998290	+	0.0584477i	$0.981385\pi$
$774$		0			0
$775$	9871.00	−	40199.7i	0.457519	−	1.86325i
$776$		0			0
$777$			12902.8i			0.595736i
$778$		0			0
$779$	−18988.0			−0.873320
$780$		0			0
$781$	3593.34			0.164635
$782$		0			0
$783$			1108.17i			0.0505781i
$784$		0			0
$785$	−19099.3	−	24356.5i	−0.868386	−	1.10741i
$786$		0			0
$787$		−	2650.18i		−	0.120036i	−0.998197	−	0.0600182i	$-0.980884\pi$
$787$		−	2650.18i		−	0.120036i	0.998197	−	0.0600182i	$-0.0191159\pi$
$788$		0			0
$789$	−5993.16			−0.270421
$790$		0			0
$791$	−4901.81			−0.220339
$792$		0			0
$793$		−	29859.0i		−	1.33711i
$794$		0			0
$795$	−9969.63	+	7817.75i	−0.444763	+	0.348764i
$796$		0			0
$797$			24516.7i			1.08962i	0.838560	+	0.544810i	$0.183398\pi$
$797$			24516.7i			1.08962i	−0.838560	+	0.544810i	$0.816602\pi$
$798$		0			0
$799$	−8399.07			−0.371887
$800$		0			0
$801$	1763.80			0.0778039
$802$		0			0
$803$		−	14944.8i		−	0.656775i
$804$		0			0
$805$	−30215.2	+	23693.4i	−1.32291	+	1.03737i
$806$		0			0
$807$		−	1717.32i		−	0.0749102i
$808$		0			0
$809$	29725.5			1.29183			0.645917	−	0.763408i	$-0.276475\pi$
$809$	29725.5			1.29183			0.645917	+	0.763408i	$0.276475\pi$
$810$		0			0
$811$	2050.38			0.0887773			0.0443887	−	0.999014i	$-0.485866\pi$
$811$	2050.38			0.0887773			0.0443887	+	0.999014i	$0.485866\pi$
$812$		0			0
$813$		−	7488.83i		−	0.323056i
$814$		0			0
$815$	19429.8	+	24778.0i	0.835089	+	1.06495i
$816$		0			0
$817$		−	4920.11i		−	0.210689i
$818$		0			0
$819$	−19188.0			−0.818658
$820$		0			0
$821$	33863.4			1.43952			0.719758	−	0.694225i	$-0.244253\pi$
$821$	33863.4			1.43952			0.719758	+	0.694225i	$0.244253\pi$
$822$		0			0
$823$		−	1216.52i		−	0.0515251i	−0.999668	−	0.0257625i	$-0.991799\pi$
$823$		−	1216.52i		−	0.0515251i	0.999668	−	0.0257625i	$-0.00820138\pi$
$824$		0			0
$825$	6753.96	+	1658.43i	0.285021	+	0.0699868i
$826$		0			0
$827$		−	25654.4i		−	1.07871i	−0.842079	−	0.539355i	$-0.818668\pi$
$827$		−	25654.4i		−	1.07871i	0.842079	−	0.539355i	$-0.181332\pi$
$828$		0			0
$829$	24544.6			1.02831			0.514154	−	0.857698i	$-0.328106\pi$
$829$	24544.6			1.02831			0.514154	+	0.857698i	$0.328106\pi$
$830$		0			0
$831$	13010.9			0.543131
$832$		0			0
$833$			1951.00i			0.0811504i
$834$		0			0
$835$	12669.4	+	16156.7i	0.525080	+	0.669610i
$836$		0			0
$837$			43772.1i			1.80763i
$838$		0			0
$839$	−16548.6			−0.680954			−0.340477	−	0.940253i	$-0.610589\pi$
$839$	−16548.6			−0.680954			−0.340477	+	0.940253i	$0.610589\pi$
$840$		0			0
$841$	−24318.7			−0.997118
$842$		0			0
$843$		−	18207.2i		−	0.743877i
$844$		0			0
$845$	14270.2	−	11190.1i	0.580958	−	0.455562i
$846$		0			0
$847$			16073.7i			0.652065i
$848$		0			0
$849$	−15171.5			−0.613292
$850$		0			0
$851$	−54828.9			−2.20859
$852$		0			0
$853$		−	24363.5i		−	0.977950i	−0.872298	−	0.488975i	$-0.837371\pi$
$853$		−	24363.5i		−	0.977950i	0.872298	−	0.488975i	$-0.162629\pi$
$854$		0			0
$855$	−9684.15	+	7593.89i	−0.387358	+	0.303749i
$856$		0			0
$857$		−	575.718i		−	0.0229477i	−0.999934	−	0.0114738i	$-0.996348\pi$
$857$		−	575.718i		−	0.0229477i	0.999934	−	0.0114738i	$-0.00365232\pi$
$858$		0			0
$859$	1531.31			0.0608236			0.0304118	−	0.999537i	$-0.490318\pi$
$859$	1531.31			0.0608236			0.0304118	+	0.999537i	$0.490318\pi$
$860$		0			0
$861$	−15527.4			−0.614604
$862$		0			0
$863$		−	7706.51i		−	0.303978i	−0.988382	−	0.151989i	$-0.951432\pi$
$863$		−	7706.51i		−	0.303978i	0.988382	−	0.151989i	$-0.0485678\pi$
$864$		0			0
$865$	8445.89	+	10770.7i	0.331987	+	0.423368i
$866$		0			0
$867$			11566.4i			0.453076i
$868$		0			0
$869$	8566.92			0.334422
$870$		0			0
$871$	7644.85			0.297400
$872$		0			0
$873$		−	4686.93i		−	0.181705i
$874$		0			0
$875$	21266.3	−	9604.79i	0.821637	−	0.371087i
$876$		0			0
$877$		−	43618.9i		−	1.67948i	−0.542988	−	0.839741i	$-0.682707\pi$
$877$		−	43618.9i		−	1.67948i	0.542988	−	0.839741i	$-0.317293\pi$
$878$		0			0
$879$	−7068.31			−0.271227
$880$		0			0
$881$	−13416.4			−0.513066			−0.256533	−	0.966536i	$-0.582580\pi$
$881$	−13416.4			−0.513066			−0.256533	+	0.966536i	$0.582580\pi$
$882$		0			0
$883$			29538.2i			1.12575i	0.826542	+	0.562875i	$0.190305\pi$
$883$			29538.2i			1.12575i	−0.826542	+	0.562875i	$0.809695\pi$
$884$		0			0
$885$	−15582.2	−	19871.3i	−0.591853	−	0.754764i
$886$		0			0
$887$		−	1950.88i		−	0.0738490i	−0.999318	−	0.0369245i	$-0.988244\pi$
$887$		−	1950.88i		−	0.0738490i	0.999318	−	0.0369245i	$-0.0117561\pi$
$888$		0			0
$889$	−12935.4			−0.488009
$890$		0			0
$891$	2281.87			0.0857974
$892$		0			0
$893$			16362.6i			0.613162i
$894$		0			0
$895$	−24014.7	+	18831.3i	−0.896897	+	0.703308i
$896$		0			0
$897$			36849.0i			1.37163i
$898$		0			0
$899$	2776.26			0.102996
$900$		0			0
$901$	11876.6			0.439142
$902$		0			0
$903$		−	4023.42i		−	0.148273i
$904$		0			0
$905$	23247.4	−	18229.6i	0.853888	−	0.669582i
$906$		0			0
$907$		−	3507.55i		−	0.128408i	−0.997937	−	0.0642042i	$-0.979549\pi$
$907$		−	3507.55i		−	0.128408i	0.997937	−	0.0642042i	$-0.0204509\pi$
$908$		0			0
$909$	−705.127			−0.0257289
$910$		0			0
$911$	−33841.4			−1.23075			−0.615377	−	0.788233i	$-0.710996\pi$
$911$	−33841.4			−1.23075			−0.615377	+	0.788233i	$0.710996\pi$
$912$		0			0
$913$			20229.2i			0.733283i
$914$		0			0
$915$	−9663.43	−	12323.3i	−0.349140	−	0.445243i
$916$		0			0
$917$			5582.50i			0.201036i
$918$		0			0
$919$	−25440.7			−0.913178			−0.456589	−	0.889678i	$-0.650929\pi$
$919$	−25440.7			−0.913178			−0.456589	+	0.889678i	$0.650929\pi$
$920$		0			0
$921$	−22932.2			−0.820458
$922$		0			0
$923$		−	11570.6i		−	0.412623i
$924$		0			0
$925$	32359.4	+	7945.82i	1.15024	+	0.282440i
$926$		0			0
$927$			1755.70i			0.0622058i
$928$		0			0
$929$	26416.7			0.932941			0.466471	−	0.884537i	$-0.345525\pi$
$929$	26416.7			0.932941			0.466471	+	0.884537i	$0.345525\pi$
$930$		0			0
$931$	3800.84			0.133800
$932$		0			0
$933$			15711.6i			0.551313i
$934$		0			0
$935$	−4022.92	−	5130.25i	−0.140710	−	0.179441i
$936$		0			0
$937$		−	1936.70i		−	0.0675231i	−0.999430	−	0.0337616i	$-0.989251\pi$
$937$		−	1936.70i		−	0.0675231i	0.999430	−	0.0337616i	$-0.0107487\pi$
$938$		0			0
$939$	−16149.8			−0.561266
$940$		0			0
$941$	−23459.1			−0.812694			−0.406347	−	0.913719i	$-0.633198\pi$
$941$	−23459.1			−0.812694			−0.406347	+	0.913719i	$0.633198\pi$
$942$		0			0
$943$		−	65981.8i		−	2.27854i
$944$		0			0
$945$	−19417.3	+	15226.2i	−0.668409	+	0.524137i
$946$		0			0
$947$		−	23606.8i		−	0.810049i	−0.914306	−	0.405025i	$-0.867263\pi$
$947$		−	23606.8i		−	0.810049i	0.914306	−	0.405025i	$-0.132737\pi$
$948$		0			0
$949$	−48122.4			−1.64607
$950$		0			0
$951$	10796.1			0.368125
$952$		0			0
$953$		−	30164.2i		−	1.02530i	−0.858596	−	0.512652i	$-0.828663\pi$
$953$		−	30164.2i		−	1.02530i	0.858596	−	0.512652i	$-0.171337\pi$
$954$		0			0
$955$	20580.5	−	16138.3i	0.697349	−	0.546831i
$956$		0			0
$957$			466.440i			0.0157553i
$958$		0			0
$959$	−5401.70			−0.181887
$960$		0			0
$961$	79870.0			2.68101
$962$		0			0
$963$		−	16763.6i		−	0.560953i
$964$		0			0
$965$	−31746.4	−	40484.8i	−1.05902	−	1.35052i
$966$		0			0
$967$			9034.04i			0.300429i	0.988653	+	0.150215i	$0.0479965\pi$
$967$			9034.04i			0.300429i	−0.988653	+	0.150215i	$0.952004\pi$
$968$		0			0
$969$	−5213.71			−0.172847
$970$		0			0
$971$	−36159.0			−1.19506			−0.597528	−	0.801848i	$-0.703850\pi$
$971$	−36159.0			−1.19506			−0.597528	+	0.801848i	$0.703850\pi$
$972$		0			0
$973$			6620.03i			0.218118i
$974$		0			0
$975$	5340.16	−	21747.8i	0.175407	−	0.714347i
$976$		0			0
$977$		−	38584.0i		−	1.26347i	−0.775184	−	0.631736i	$-0.782343\pi$
$977$		−	38584.0i		−	1.26347i	0.775184	−	0.631736i	$-0.217657\pi$
$978$		0			0
$979$	1820.33			0.0594258
$980$		0			0
$981$	26326.1			0.856808
$982$		0			0
$983$		−	53046.3i		−	1.72117i	−0.509303	−	0.860587i	$-0.670097\pi$
$983$		−	53046.3i		−	1.72117i	0.509303	−	0.860587i	$-0.329903\pi$
$984$		0			0
$985$	5684.35	+	7249.00i	0.183877	+	0.234490i
$986$		0			0
$987$			13380.5i			0.431516i
$988$		0			0
$989$	17097.0			0.549699
$990$		0			0
$991$	−16425.7			−0.526517			−0.263259	−	0.964725i	$-0.584797\pi$
$991$	−16425.7			−0.526517			−0.263259	+	0.964725i	$0.584797\pi$
$992$		0			0
$993$		−	9346.19i		−	0.298683i
$994$		0			0
$995$	−29294.5	+	22971.5i	−0.933366	+	0.731906i
$996$		0			0
$997$		−	20024.6i		−	0.636092i	−0.948075	−	0.318046i	$-0.896973\pi$
$997$		−	20024.6i		−	0.636092i	0.948075	−	0.318046i	$-0.103027\pi$
$998$		0			0
$999$	−35235.0			−1.11590

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.c.g.129.3		4
4.3	odd	2		320.4.c.h.129.2		4
5.2	odd	4		1600.4.a.ce.1.2		2
5.3	odd	4		1600.4.a.cl.1.1		2
5.4	even	2	inner	320.4.c.g.129.2		4
8.3	odd	2		80.4.c.c.49.3		4
8.5	even	2		40.4.c.a.9.2	✓	4
20.3	even	4		1600.4.a.cf.1.2		2
20.7	even	4		1600.4.a.cm.1.1		2
20.19	odd	2		320.4.c.h.129.3		4
24.5	odd	2		360.4.f.e.289.2		4
24.11	even	2		720.4.f.m.289.2		4
40.3	even	4		400.4.a.x.1.1		2
40.13	odd	4		200.4.a.k.1.2		2
40.19	odd	2		80.4.c.c.49.2		4
40.27	even	4		400.4.a.v.1.2		2
40.29	even	2		40.4.c.a.9.3	yes	4
40.37	odd	4		200.4.a.l.1.1		2
120.29	odd	2		360.4.f.e.289.1		4
120.53	even	4		1800.4.a.bk.1.1		2
120.59	even	2		720.4.f.m.289.1		4
120.77	even	4		1800.4.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.4.c.a.9.2	✓	4	8.5	even	2
40.4.c.a.9.3	yes	4	40.29	even	2
80.4.c.c.49.2		4	40.19	odd	2
80.4.c.c.49.3		4	8.3	odd	2
200.4.a.k.1.2		2	40.13	odd	4
200.4.a.l.1.1		2	40.37	odd	4
320.4.c.g.129.2		4	5.4	even	2	inner
320.4.c.g.129.3		4	1.1	even	1	trivial
320.4.c.h.129.2		4	4.3	odd	2
320.4.c.h.129.3		4	20.19	odd	2
360.4.f.e.289.1		4	120.29	odd	2
360.4.f.e.289.2		4	24.5	odd	2
400.4.a.v.1.2		2	40.27	even	4
400.4.a.x.1.1		2	40.3	even	4
720.4.f.m.289.1		4	120.59	even	2
720.4.f.m.289.2		4	24.11	even	2
1600.4.a.ce.1.2		2	5.2	odd	4
1600.4.a.cf.1.2		2	20.3	even	4
1600.4.a.cl.1.1		2	5.3	odd	4
1600.4.a.cm.1.1		2	20.7	even	4
1800.4.a.bk.1.1		2	120.53	even	4
1800.4.a.bp.1.2		2	120.77	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.89898i q^{3} +(-8.79796 + 6.89898i) q^{5} -16.6969i q^{7} +18.5959 q^{9} +19.1918 q^{11} -61.7980i q^{13} +(-20.0000 - 25.5051i) q^{15} +30.3837i q^{17} +59.1918 q^{19} +48.4041 q^{21} +205.687i q^{23} +(29.8082 - 121.394i) q^{25} +132.182i q^{27} +8.38367 q^{29} +331.151 q^{31} +55.6367i q^{33} +(115.192 + 146.899i) q^{35} +266.565i q^{37} +179.151 q^{39} -320.788 q^{41} -83.1214i q^{43} +(-163.606 + 128.293i) q^{45} +276.434i q^{47} +64.2122 q^{49} -88.0816 q^{51} -390.888i q^{53} +(-168.849 + 132.404i) q^{55} +171.596i q^{57} +779.110 q^{59} +483.171 q^{61} -310.495i q^{63} +(426.343 + 543.696i) q^{65} +123.707i q^{67} -596.282 q^{69} +187.233 q^{71} -778.706i q^{73} +(351.918 + 86.4133i) q^{75} -320.445i q^{77} +446.384 q^{79} +118.898 q^{81} +1054.05i q^{83} +(-209.616 - 267.314i) q^{85} +24.3041i q^{87} +94.8490 q^{89} -1031.84 q^{91} +960.000i q^{93} +(-520.767 + 408.363i) q^{95} -252.041i q^{97} +356.890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9} - 80 q^{11} - 80 q^{15} + 80 q^{19} + 272 q^{21} + 276 q^{25} - 280 q^{29} + 384 q^{31} + 304 q^{35} - 224 q^{39} - 1048 q^{41} - 772 q^{45} + 492 q^{49} - 1920 q^{51} - 1616 q^{55}+ \cdots + 3152 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 80 * q^11 - 80 * q^15 + 80 * q^19 + 272 * q^21 + 276 * q^25 - 280 * q^29 + 384 * q^31 + 304 * q^35 - 224 * q^39 - 1048 * q^41 - 772 * q^45 + 492 * q^49 - 1920 * q^51 - 1616 * q^55 + 1392 * q^59 + 1384 * q^61 + 608 * q^65 - 112 * q^69 + 1376 * q^71 - 160 * q^75 + 1472 * q^79 - 1484 * q^81 - 1152 * q^85 + 1320 * q^89 - 992 * q^91 - 1456 * q^95 + 3152 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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