LMFDB - Embedded newform 96.4.c.a.95.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [96,4,Mod(95,96)] 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("96.95"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 96 = 2^{5} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	96.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$5.66418336055$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.26525057735983104.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 103x^{8} - 260x^{6} + 259x^{4} + 356x^{2} + 169 $ x^12 - 16x^10 + 103x^8 - 260x^6 + 259x^4 + 356*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{44}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			95.6
Root			$-0.248859 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	96.95
Dual form			96.4.c.a.95.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.497717 + 5.17226i) q^{3} -18.4532i q^{5} -17.1600i q^{7} +(-26.5046 - 5.14865i) q^{9} +48.1992 q^{11} +33.7471 q^{13} +(95.4449 + 9.18449i) q^{15} -84.4951i q^{17} +28.7135i q^{19} +(88.7562 + 8.54084i) q^{21} -96.4823 q^{23} -215.521 q^{25} +(39.8219 - 134.526i) q^{27} -141.202i q^{29} +51.8673i q^{31} +(-23.9896 + 249.299i) q^{33} -316.658 q^{35} +323.802 q^{37} +(-16.7965 + 174.549i) q^{39} +134.442i q^{41} +114.365i q^{43} +(-95.0091 + 489.094i) q^{45} -247.555 q^{47} +48.5332 q^{49} +(437.031 + 42.0547i) q^{51} +169.592i q^{53} -889.430i q^{55} +(-148.514 - 14.2912i) q^{57} -605.603 q^{59} -343.296 q^{61} +(-88.3509 + 454.819i) q^{63} -622.742i q^{65} +900.357i q^{67} +(48.0209 - 499.031i) q^{69} +331.924 q^{71} +777.525 q^{73} +(107.269 - 1114.73i) q^{75} -827.099i q^{77} +587.661i q^{79} +(675.983 + 272.925i) q^{81} +1463.98 q^{83} -1559.21 q^{85} +(730.332 + 70.2785i) q^{87} +78.6240i q^{89} -579.101i q^{91} +(-268.271 - 25.8152i) q^{93} +529.857 q^{95} -62.0627 q^{97} +(-1277.50 - 248.160i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9} + 72 q^{13} + 136 q^{21} - 132 q^{25} + 80 q^{33} - 24 q^{37} - 544 q^{45} - 540 q^{49} - 888 q^{57} + 456 q^{61} + 1312 q^{69} + 2424 q^{73} + 2924 q^{81} - 3072 q^{85} - 2360 q^{93} - 2952 q^{97}+O(q^{100}) $ 12 * q - 20 * q^9 + 72 * q^13 + 136 * q^21 - 132 * q^25 + 80 * q^33 - 24 * q^37 - 544 * q^45 - 540 * q^49 - 888 * q^57 + 456 * q^61 + 1312 * q^69 + 2424 * q^73 + 2924 * q^81 - 3072 * q^85 - 2360 * q^93 - 2952 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/96\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$65$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−0.497717	+	5.17226i	−0.0957857	+	0.995402i
$3$	−0.497717	+	5.17226i	−0.0957857	+	0.995402i
$4$		0			0
$5$		−	18.4532i		−	1.65051i	−0.564763	−	0.825253i	$-0.691032\pi$
$5$		−	18.4532i		−	1.65051i	0.564763	−	0.825253i	$-0.308968\pi$
$6$		0			0
$7$		−	17.1600i		−	0.926555i	−0.886213	−	0.463277i	$-0.846673\pi$
$7$		−	17.1600i		−	0.926555i	0.886213	−	0.463277i	$-0.153327\pi$
$8$		0			0
$9$	−26.5046	−	5.14865i	−0.981650	−	0.190691i
$10$		0			0
$11$	48.1992			1.32115			0.660573	−	0.750762i	$-0.270314\pi$
$11$	48.1992			1.32115			0.660573	+	0.750762i	$0.270314\pi$
$12$		0			0
$13$	33.7471			0.719981			0.359990	−	0.932956i	$-0.382780\pi$
$13$	33.7471			0.719981			0.359990	+	0.932956i	$0.382780\pi$
$14$		0			0
$15$	95.4449	+	9.18449i	1.64292	+	0.158095i
$16$		0			0
$17$		−	84.4951i		−	1.20547i	−0.797940	−	0.602737i	$-0.794077\pi$
$17$		−	84.4951i		−	1.20547i	0.797940	−	0.602737i	$-0.205923\pi$
$18$		0			0
$19$			28.7135i			0.346701i	0.984860	+	0.173351i	$0.0554594\pi$
$19$			28.7135i			0.346701i	−0.984860	+	0.173351i	$0.944541\pi$
$20$		0			0
$21$	88.7562	+	8.54084i	0.922294	+	0.0887507i
$22$		0			0
$23$	−96.4823			−0.874693			−0.437347	−	0.899293i	$-0.644082\pi$
$23$	−96.4823			−0.874693			−0.437347	+	0.899293i	$0.644082\pi$
$24$		0			0
$25$	−215.521			−1.72417
$26$		0			0
$27$	39.8219	−	134.526i	0.283842	−	0.958871i
$28$		0			0
$29$		−	141.202i		−	0.904155i	−0.891979	−	0.452078i	$-0.850683\pi$
$29$		−	141.202i		−	0.904155i	0.891979	−	0.452078i	$-0.149317\pi$
$30$		0			0
$31$			51.8673i			0.300505i	0.988648	+	0.150252i	$0.0480086\pi$
$31$			51.8673i			0.300505i	−0.988648	+	0.150252i	$0.951991\pi$
$32$		0			0
$33$	−23.9896	+	249.299i	−0.126547	+	1.31507i
$34$		0			0
$35$	−316.658			−1.52928
$36$		0			0
$37$	323.802			1.43872			0.719360	−	0.694637i	$-0.244435\pi$
$37$	323.802			1.43872			0.719360	+	0.694637i	$0.244435\pi$
$38$		0			0
$39$	−16.7965	+	174.549i	−0.0689639	+	0.716670i
$40$		0			0
$41$			134.442i			0.512105i	0.966663	+	0.256052i	$0.0824220\pi$
$41$			134.442i			0.512105i	−0.966663	+	0.256052i	$0.917578\pi$
$42$		0			0
$43$			114.365i			0.405594i	0.979221	+	0.202797i	$0.0650032\pi$
$43$			114.365i			0.405594i	−0.979221	+	0.202797i	$0.934997\pi$
$44$		0			0
$45$	−95.0091	+	489.094i	−0.314736	+	1.62022i
$46$		0			0
$47$	−247.555			−0.768288			−0.384144	−	0.923273i	$-0.625503\pi$
$47$	−247.555			−0.768288			−0.384144	+	0.923273i	$0.625503\pi$
$48$		0			0
$49$	48.5332			0.141496
$50$		0			0
$51$	437.031	+	42.0547i	1.19993	+	0.115467i
$52$		0			0
$53$			169.592i			0.439533i	0.975553	+	0.219766i	$0.0705295\pi$
$53$			169.592i			0.439533i	−0.975553	+	0.219766i	$0.929470\pi$
$54$		0			0
$55$		−	889.430i		−	2.18056i
$56$		0			0
$57$	−148.514	−	14.2912i	−0.345107	−	0.0332090i
$58$		0			0
$59$	−605.603			−1.33632			−0.668159	−	0.744019i	$-0.732917\pi$
$59$	−605.603			−1.33632			−0.668159	+	0.744019i	$0.732917\pi$
$60$		0			0
$61$	−343.296			−0.720566			−0.360283	−	0.932843i	$-0.617320\pi$
$61$	−343.296			−0.720566			−0.360283	+	0.932843i	$0.617320\pi$
$62$		0			0
$63$	−88.3509	+	454.819i	−0.176685	+	0.909553i
$64$		0			0
$65$		−	622.742i		−	1.18833i
$66$		0			0
$67$			900.357i			1.64173i	0.571120	+	0.820867i	$0.306509\pi$
$67$			900.357i			1.64173i	−0.571120	+	0.820867i	$0.693491\pi$
$68$		0			0
$69$	48.0209	−	499.031i	0.0837831	−	0.870671i
$70$		0			0
$71$	331.924			0.554819			0.277409	−	0.960752i	$-0.410524\pi$
$71$	331.924			0.554819			0.277409	+	0.960752i	$0.410524\pi$
$72$		0			0
$73$	777.525			1.24661			0.623304	−	0.781979i	$-0.285790\pi$
$73$	777.525			1.24661			0.623304	+	0.781979i	$0.285790\pi$
$74$		0			0
$75$	107.269	−	1114.73i	0.165151	−	1.71624i
$76$		0			0
$77$		−	827.099i		−	1.22411i
$78$		0			0
$79$			587.661i			0.836924i	0.908234	+	0.418462i	$0.137431\pi$
$79$			587.661i			0.836924i	−0.908234	+	0.418462i	$0.862569\pi$
$80$		0			0
$81$	675.983	+	272.925i	0.927274	+	0.374383i
$82$		0			0
$83$	1463.98			1.93605			0.968027	−	0.250846i	$-0.0807087\pi$
$83$	1463.98			1.93605			0.968027	+	0.250846i	$0.0807087\pi$
$84$		0			0
$85$	−1559.21			−1.98964
$86$		0			0
$87$	730.332	+	70.2785i	0.899998	+	0.0866051i
$88$		0			0
$89$			78.6240i			0.0936419i	0.998903	+	0.0468210i	$0.0149090\pi$
$89$			78.6240i			0.0936419i	−0.998903	+	0.0468210i	$0.985091\pi$
$90$		0			0
$91$		−	579.101i		−	0.667102i
$92$		0			0
$93$	−268.271	−	25.8152i	−0.299123	−	0.0287840i
$94$		0			0
$95$	529.857			0.572233
$96$		0			0
$97$	−62.0627			−0.0649640			−0.0324820	−	0.999472i	$-0.510341\pi$
$97$	−62.0627			−0.0649640			−0.0324820	+	0.999472i	$0.510341\pi$
$98$		0			0
$99$	−1277.50	−	248.160i	−1.29690	−	0.251930i
$100$		0			0
$101$			789.157i			0.777466i	0.921351	+	0.388733i	$0.127087\pi$
$101$			789.157i			0.777466i	−0.921351	+	0.388733i	$0.872913\pi$
$102$		0			0
$103$		−	1329.46i		−	1.27180i	−0.771771	−	0.635901i	$-0.780629\pi$
$103$		−	1329.46i		−	1.27180i	0.771771	−	0.635901i	$-0.219371\pi$
$104$		0			0
$105$	157.606	−	1637.84i	0.146484	−	1.52225i
$106$		0			0
$107$	908.920			0.821202			0.410601	−	0.911815i	$-0.365319\pi$
$107$	908.920			0.821202			0.410601	+	0.911815i	$0.365319\pi$
$108$		0			0
$109$	1201.93			1.05619			0.528094	−	0.849186i	$-0.322907\pi$
$109$	1201.93			1.05619			0.528094	+	0.849186i	$0.322907\pi$
$110$		0			0
$111$	−161.162	+	1674.79i	−0.137809	+	1.43211i
$112$		0			0
$113$			547.390i			0.455700i	0.973696	+	0.227850i	$0.0731696\pi$
$113$			547.390i			0.455700i	−0.973696	+	0.227850i	$0.926830\pi$
$114$		0			0
$115$			1780.41i			1.44369i
$116$		0			0
$117$	−894.451	−	173.752i	−0.706769	−	0.137294i
$118$		0			0
$119$	−1449.94			−1.11694
$120$		0			0
$121$	992.160			0.745425
$122$		0			0
$123$	−695.369	−	66.9140i	−0.509750	−	0.0490523i
$124$		0			0
$125$			1670.41i			1.19525i
$126$		0			0
$127$		−	733.919i		−	0.512793i	−0.966572	−	0.256397i	$-0.917465\pi$
$127$		−	733.919i		−	0.512793i	0.966572	−	0.256397i	$-0.0825354\pi$
$128$		0			0
$129$	−591.528	−	56.9216i	−0.403730	−	0.0388502i
$130$		0			0
$131$	−1715.61			−1.14422			−0.572112	−	0.820176i	$-0.693876\pi$
$131$	−1715.61			−1.14422			−0.572112	+	0.820176i	$0.693876\pi$
$132$		0			0
$133$	492.725			0.321238
$134$		0			0
$135$	−2482.44	−	734.843i	−1.58262	−	0.468483i
$136$		0			0
$137$			2256.63i			1.40728i	0.710558	+	0.703639i	$0.248443\pi$
$137$			2256.63i			1.40728i	−0.710558	+	0.703639i	$0.751557\pi$
$138$		0			0
$139$		−	824.963i		−	0.503399i	−0.967805	−	0.251699i	$-0.919011\pi$
$139$		−	824.963i		−	0.503399i	0.967805	−	0.251699i	$-0.0809894\pi$
$140$		0			0
$141$	123.212	−	1280.42i	0.0735911	−	0.764756i
$142$		0			0
$143$	1626.58			0.951199
$144$		0			0
$145$	−2605.63			−1.49231
$146$		0			0
$147$	−24.1558	+	251.026i	−0.0135533	+	0.140846i
$148$		0			0
$149$			192.210i			0.105681i	0.998603	+	0.0528404i	$0.0168275\pi$
$149$			192.210i			0.105681i	−0.998603	+	0.0528404i	$0.983173\pi$
$150$		0			0
$151$		−	611.358i		−	0.329481i	−0.986337	−	0.164740i	$-0.947321\pi$
$151$		−	611.358i		−	0.329481i	0.986337	−	0.164740i	$-0.0526786\pi$
$152$		0			0
$153$	−435.035	+	2239.50i	−0.229873	+	1.18335i
$154$		0			0
$155$	957.119			0.495985
$156$		0			0
$157$	1387.68			0.705405			0.352703	−	0.935735i	$-0.385263\pi$
$157$	1387.68			0.705405			0.352703	+	0.935735i	$0.385263\pi$
$158$		0			0
$159$	−877.173	−	84.4088i	−0.437512	−	0.0421010i
$160$		0			0
$161$			1655.64i			0.810451i
$162$		0			0
$163$			2156.96i			1.03648i	0.855235	+	0.518240i	$0.173413\pi$
$163$			2156.96i			1.03648i	−0.855235	+	0.518240i	$0.826587\pi$
$164$		0			0
$165$	4600.36	+	442.685i	2.17053	+	0.208866i
$166$		0			0
$167$	2252.75			1.04385			0.521924	−	0.852992i	$-0.325214\pi$
$167$	2252.75			1.04385			0.521924	+	0.852992i	$0.325214\pi$
$168$		0			0
$169$	−1058.14			−0.481628
$170$		0			0
$171$	147.836	−	761.038i	0.0661127	−	0.340340i
$172$		0			0
$173$			868.408i			0.381640i	0.981625	+	0.190820i	$0.0611148\pi$
$173$			868.408i			0.381640i	−0.981625	+	0.190820i	$0.938885\pi$
$174$		0			0
$175$			3698.36i			1.59754i
$176$		0			0
$177$	301.419	−	3132.33i	0.128000	−	1.33017i
$178$		0			0
$179$	−201.084			−0.0839650			−0.0419825	−	0.999118i	$-0.513367\pi$
$179$	−201.084			−0.0839650			−0.0419825	+	0.999118i	$0.513367\pi$
$180$		0			0
$181$	−960.433			−0.394411			−0.197206	−	0.980362i	$-0.563187\pi$
$181$	−960.433			−0.394411			−0.197206	+	0.980362i	$0.563187\pi$
$182$		0			0
$183$	170.864	−	1775.62i	0.0690199	−	0.717253i
$184$		0			0
$185$		−	5975.19i		−	2.37462i
$186$		0			0
$187$		−	4072.59i		−	1.59261i
$188$		0			0
$189$	−2308.47	−	683.345i	−0.888447	−	0.262995i
$190$		0			0
$191$	−1113.13			−0.421694			−0.210847	−	0.977519i	$-0.567622\pi$
$191$	−1113.13			−0.421694			−0.210847	+	0.977519i	$0.567622\pi$
$192$		0			0
$193$	−812.451			−0.303013			−0.151507	−	0.988456i	$-0.548412\pi$
$193$	−812.451			−0.303013			−0.151507	+	0.988456i	$0.548412\pi$
$194$		0			0
$195$	3220.98	+	309.949i	1.18287	+	0.113825i
$196$		0			0
$197$		−	3762.51i		−	1.36075i	−0.732864	−	0.680375i	$-0.761817\pi$
$197$		−	3762.51i		−	1.36075i	0.732864	−	0.680375i	$-0.238183\pi$
$198$		0			0
$199$			1406.90i			0.501169i	0.968095	+	0.250585i	$0.0806228\pi$
$199$			1406.90i			0.501169i	−0.968095	+	0.250585i	$0.919377\pi$
$200$		0			0
$201$	−4656.88	−	448.123i	−1.63419	−	0.157255i
$202$		0			0
$203$	−2423.03			−0.837749
$204$		0			0
$205$	2480.89			0.845232
$206$		0			0
$207$	2557.22	+	496.753i	0.858643	+	0.166796i
$208$		0			0
$209$			1383.97i			0.458043i
$210$		0			0
$211$		−	3714.03i		−	1.21177i	−0.795551	−	0.605886i	$-0.792819\pi$
$211$		−	3714.03i		−	1.21177i	0.795551	−	0.605886i	$-0.207181\pi$
$212$		0			0
$213$	−165.204	+	1716.80i	−0.0531437	+	0.552267i
$214$		0			0
$215$	2110.41			0.669436
$216$		0			0
$217$	890.045			0.278434
$218$		0			0
$219$	−386.988	+	4021.56i	−0.119407	+	1.24088i
$220$		0			0
$221$		−	2851.46i		−	0.867919i
$222$		0			0
$223$		−	3020.76i		−	0.907107i	−0.891229	−	0.453553i	$-0.850156\pi$
$223$		−	3020.76i		−	0.907107i	0.891229	−	0.453553i	$-0.149844\pi$
$224$		0			0
$225$	5712.30	+	1109.64i	1.69253	+	0.328783i
$226$		0			0
$227$	5598.67			1.63699			0.818495	−	0.574514i	$-0.194809\pi$
$227$	5598.67			1.63699			0.818495	+	0.574514i	$0.194809\pi$
$228$		0			0
$229$	690.341			0.199210			0.0996048	−	0.995027i	$-0.468242\pi$
$229$	690.341			0.199210			0.0996048	+	0.995027i	$0.468242\pi$
$230$		0			0
$231$	4277.97	+	411.662i	1.21848	+	0.117253i
$232$		0			0
$233$			1398.79i			0.393295i	0.980474	+	0.196648i	$0.0630055\pi$
$233$			1398.79i			0.393295i	−0.980474	+	0.196648i	$0.936995\pi$
$234$		0			0
$235$			4568.18i			1.26807i
$236$		0			0
$237$	−3039.54	−	292.489i	−0.833076	−	0.0801654i
$238$		0			0
$239$	−6871.75			−1.85982			−0.929909	−	0.367789i	$-0.880115\pi$
$239$	−6871.75			−1.85982			−0.929909	+	0.367789i	$0.880115\pi$
$240$		0			0
$241$	2400.00			0.641484			0.320742	−	0.947167i	$-0.396068\pi$
$241$	2400.00			0.641484			0.320742	+	0.947167i	$0.396068\pi$
$242$		0			0
$243$	−1748.09	+	3360.52i	−0.461481	+	0.887150i
$244$		0			0
$245$		−	895.594i		−	0.233541i
$246$		0			0
$247$			968.996i			0.249618i
$248$		0			0
$249$	−728.647	+	7572.08i	−0.185446	+	1.92715i
$250$		0			0
$251$	196.846			0.0495011			0.0247506	−	0.999694i	$-0.492121\pi$
$251$	196.846			0.0495011			0.0247506	+	0.999694i	$0.492121\pi$
$252$		0			0
$253$	−4650.37			−1.15560
$254$		0			0
$255$	776.044	−	8064.62i	0.190579	−	1.98050i
$256$		0			0
$257$		−	5652.50i		−	1.37196i	−0.727621	−	0.685979i	$-0.759374\pi$
$257$		−	5652.50i		−	1.37196i	0.727621	−	0.685979i	$-0.240626\pi$
$258$		0			0
$259$		−	5556.45i		−	1.33305i
$260$		0			0
$261$	−726.998	+	3742.49i	−0.172414	+	0.887564i
$262$		0			0
$263$	−6241.47			−1.46337			−0.731684	−	0.681644i	$-0.761265\pi$
$263$	−6241.47			−1.46337			−0.731684	+	0.681644i	$0.761265\pi$
$264$		0			0
$265$	3129.52			0.725452
$266$		0			0
$267$	−406.664	−	39.1325i	−0.0932114	−	0.00896956i
$268$		0			0
$269$			7035.30i			1.59461i	0.603578	+	0.797304i	$0.293741\pi$
$269$			7035.30i			1.59461i	−0.603578	+	0.797304i	$0.706259\pi$
$270$		0			0
$271$			4877.01i			1.09320i	0.837394	+	0.546600i	$0.184078\pi$
$271$			4877.01i			1.09320i	−0.837394	+	0.546600i	$0.815922\pi$
$272$		0			0
$273$	2995.26	+	288.228i	0.664034	+	0.0638988i
$274$		0			0
$275$	−10388.0			−2.27788
$276$		0			0
$277$	−5872.49			−1.27380			−0.636902	−	0.770944i	$-0.719785\pi$
$277$	−5872.49			−1.27380			−0.636902	+	0.770944i	$0.719785\pi$
$278$		0			0
$279$	267.046	−	1374.72i	0.0573034	−	0.294990i
$280$		0			0
$281$		−	1669.01i		−	0.354324i	−0.984182	−	0.177162i	$-0.943308\pi$
$281$		−	1669.01i		−	0.354324i	0.984182	−	0.177162i	$-0.0566916\pi$
$282$		0			0
$283$			1247.63i			0.262063i	0.991378	+	0.131032i	$0.0418289\pi$
$283$			1247.63i			0.262063i	−0.991378	+	0.131032i	$0.958171\pi$
$284$		0			0
$285$	−263.719	+	2740.56i	−0.0548117	+	0.569602i
$286$		0			0
$287$	2307.03			0.474493
$288$		0			0
$289$	−2226.42			−0.453169
$290$		0			0
$291$	30.8897	−	321.004i	0.00622262	−	0.0646653i
$292$		0			0
$293$		−	1450.34i		−	0.289180i	−0.989492	−	0.144590i	$-0.953814\pi$
$293$		−	1450.34i		−	0.289180i	0.989492	−	0.144590i	$-0.0461863\pi$
$294$		0			0
$295$			11175.3i			2.20560i
$296$		0			0
$297$	1919.38	−	6484.04i	0.374996	−	1.26681i
$298$		0			0
$299$	−3255.99			−0.629762
$300$		0			0
$301$	1962.51			0.375805
$302$		0			0
$303$	−4081.72	−	392.777i	−0.773891	−	0.0744701i
$304$		0			0
$305$			6334.91i			1.18930i
$306$		0			0
$307$		−	4347.28i		−	0.808184i	−0.914718	−	0.404092i	$-0.867588\pi$
$307$		−	4347.28i		−	0.808184i	0.914718	−	0.404092i	$-0.132412\pi$
$308$		0			0
$309$	6876.31	+	661.695i	1.26595	+	0.121820i
$310$		0			0
$311$	−4944.21			−0.901480			−0.450740	−	0.892655i	$-0.648840\pi$
$311$	−4944.21			−0.901480			−0.450740	+	0.892655i	$0.648840\pi$
$312$		0			0
$313$	9110.42			1.64521			0.822606	−	0.568612i	$-0.192520\pi$
$313$	9110.42			1.64521			0.822606	+	0.568612i	$0.192520\pi$
$314$		0			0
$315$	8392.88	+	1630.36i	1.50122	+	0.291620i
$316$		0			0
$317$		−	3103.06i		−	0.549796i	−0.961473	−	0.274898i	$-0.911356\pi$
$317$		−	3103.06i		−	0.549796i	0.961473	−	0.274898i	$-0.0886440\pi$
$318$		0			0
$319$		−	6805.81i		−	1.19452i
$320$		0			0
$321$	−452.385	+	4701.17i	−0.0786594	+	0.817426i
$322$		0			0
$323$	2426.15			0.417940
$324$		0			0
$325$	−7273.21			−1.24137
$326$		0			0
$327$	−598.224	+	6216.72i	−0.101168	+	1.05133i
$328$		0			0
$329$			4248.05i			0.711861i
$330$		0			0
$331$		−	1431.07i		−	0.237639i	−0.992916	−	0.118819i	$-0.962089\pi$
$331$		−	1431.07i		−	0.237639i	0.992916	−	0.118819i	$-0.0379110\pi$
$332$		0			0
$333$	−8582.22	−	1667.14i	−1.41232	−	0.274351i
$334$		0			0
$335$	16614.5			2.70969
$336$		0			0
$337$	−6355.77			−1.02736			−0.513681	−	0.857981i	$-0.671718\pi$
$337$	−6355.77			−1.02736			−0.513681	+	0.857981i	$0.671718\pi$
$338$		0			0
$339$	−2831.24	−	272.445i	−0.453605	−	0.0436496i
$340$		0			0
$341$			2499.96i			0.397010i
$342$		0			0
$343$		−	6718.72i		−	1.05766i
$344$		0			0
$345$	−9208.74	−	886.140i	−1.43705	−	0.138285i
$346$		0			0
$347$	3904.41			0.604034			0.302017	−	0.953303i	$-0.402340\pi$
$347$	3904.41			0.604034			0.302017	+	0.953303i	$0.402340\pi$
$348$		0			0
$349$	11087.5			1.70058			0.850289	−	0.526317i	$-0.176427\pi$
$349$	11087.5			1.70058			0.850289	+	0.526317i	$0.176427\pi$
$350$		0			0
$351$	1343.87	−	4539.85i	0.204361	−	0.690369i
$352$		0			0
$353$		−	4998.69i		−	0.753692i	−0.926276	−	0.376846i	$-0.877009\pi$
$353$		−	4998.69i		−	0.753692i	0.926276	−	0.376846i	$-0.122991\pi$
$354$		0			0
$355$		−	6125.06i		−	0.915732i
$356$		0			0
$357$	721.659	−	7499.46i	0.106987	−	1.11180i
$358$		0			0
$359$	7166.46			1.05357			0.526784	−	0.849999i	$-0.323398\pi$
$359$	7166.46			1.05357			0.526784	+	0.849999i	$0.323398\pi$
$360$		0			0
$361$	6034.54			0.879798
$362$		0			0
$363$	−493.815	+	5131.71i	−0.0714010	+	0.741997i
$364$		0			0
$365$		−	14347.8i		−	2.05754i
$366$		0			0
$367$			3698.63i			0.526068i	0.964787	+	0.263034i	$0.0847231\pi$
$367$			3698.63i			0.526068i	−0.964787	+	0.263034i	$0.915277\pi$
$368$		0			0
$369$	692.194	−	3563.32i	0.0976536	−	0.502708i
$370$		0			0
$371$	2910.20			0.407251
$372$		0			0
$373$	2625.39			0.364444			0.182222	−	0.983257i	$-0.441671\pi$
$373$	2625.39			0.364444			0.182222	+	0.983257i	$0.441671\pi$
$374$		0			0
$375$	−8639.81	−	831.393i	−1.18975	−	0.114488i
$376$		0			0
$377$		−	4765.14i		−	0.650974i
$378$		0			0
$379$			11506.8i			1.55954i	0.626068	+	0.779769i	$0.284663\pi$
$379$			11506.8i			1.55954i	−0.626068	+	0.779769i	$0.715337\pi$
$380$		0			0
$381$	3796.02	+	365.284i	0.510436	+	0.0491183i
$382$		0			0
$383$	−11012.1			−1.46917			−0.734584	−	0.678517i	$-0.762623\pi$
$383$	−11012.1			−1.46917			−0.734584	+	0.678517i	$0.762623\pi$
$384$		0			0
$385$	−15262.7			−2.02041
$386$		0			0
$387$	588.827	−	3031.20i	0.0773430	−	0.398152i
$388$		0			0
$389$			14545.0i			1.89579i	0.318586	+	0.947894i	$0.396792\pi$
$389$			14545.0i			1.89579i	−0.318586	+	0.947894i	$0.603208\pi$
$390$		0			0
$391$			8152.28i			1.05442i
$392$		0			0
$393$	853.887	−	8873.56i	0.109600	−	1.13896i
$394$		0			0
$395$	10844.2			1.38135
$396$		0			0
$397$	14650.2			1.85207			0.926033	−	0.377443i	$-0.123197\pi$
$397$	14650.2			1.85207			0.926033	+	0.377443i	$0.123197\pi$
$398$		0			0
$399$	−245.237	+	2548.50i	−0.0307700	+	0.319761i
$400$		0			0
$401$		−	7703.28i		−	0.959311i	−0.877457	−	0.479655i	$-0.840762\pi$
$401$		−	7703.28i		−	0.959311i	0.877457	−	0.479655i	$-0.159238\pi$
$402$		0			0
$403$			1750.37i			0.216357i
$404$		0			0
$405$	5036.35	−	12474.1i	0.617921	−	1.53047i
$406$		0			0
$407$	15607.0			1.90076
$408$		0			0
$409$	−12422.4			−1.50182			−0.750912	−	0.660402i	$-0.770386\pi$
$409$	−12422.4			−1.50182			−0.750912	+	0.660402i	$0.770386\pi$
$410$		0			0
$411$	−11671.9	−	1123.16i	−1.40081	−	0.134797i
$412$		0			0
$413$			10392.2i			1.23817i
$414$		0			0
$415$		−	27015.1i		−	3.19547i
$416$		0			0
$417$	4266.92	+	410.598i	0.501084	+	0.0482184i
$418$		0			0
$419$	973.166			0.113466			0.0567330	−	0.998389i	$-0.481932\pi$
$419$	973.166			0.113466			0.0567330	+	0.998389i	$0.481932\pi$
$420$		0			0
$421$	−12537.3			−1.45138			−0.725688	−	0.688024i	$-0.758478\pi$
$421$	−12537.3			−1.45138			−0.725688	+	0.688024i	$0.758478\pi$
$422$		0			0
$423$	6561.33	+	1274.57i	0.754190	+	0.146505i
$424$		0			0
$425$			18210.5i			2.07845i
$426$		0			0
$427$			5890.97i			0.667644i
$428$		0			0
$429$	−809.577	+	8413.10i	−0.0911113	+	0.946825i
$430$		0			0
$431$	−10628.1			−1.18779			−0.593895	−	0.804543i	$-0.702410\pi$
$431$	−10628.1			−1.18779			−0.593895	+	0.804543i	$0.702410\pi$
$432$		0			0
$433$	132.773			0.0147360			0.00736799	−	0.999973i	$-0.497655\pi$
$433$	132.773			0.0147360			0.00736799	+	0.999973i	$0.497655\pi$
$434$		0			0
$435$	1296.87	−	13477.0i	0.142942	−	1.48545i
$436$		0			0
$437$		−	2770.34i		−	0.303257i
$438$		0			0
$439$			8526.83i			0.927024i	0.886091	+	0.463512i	$0.153411\pi$
$439$			8526.83i			0.927024i	−0.886091	+	0.463512i	$0.846589\pi$
$440$		0			0
$441$	−1286.35	−	249.880i	−0.138900	−	0.0269820i
$442$		0			0
$443$	−7107.83			−0.762309			−0.381155	−	0.924511i	$-0.624473\pi$
$443$	−7107.83			−0.762309			−0.381155	+	0.924511i	$0.624473\pi$
$444$		0			0
$445$	1450.87			0.154557
$446$		0			0
$447$	−994.160	−	95.6662i	−0.105195	−	0.0101227i
$448$		0			0
$449$			16844.5i			1.77047i	0.465140	+	0.885237i	$0.346004\pi$
$449$			16844.5i			1.77047i	−0.465140	+	0.885237i	$0.653996\pi$
$450$		0			0
$451$			6479.99i			0.676565i
$452$		0			0
$453$	3162.10	+	304.283i	0.327966	+	0.0315596i
$454$		0			0
$455$	−10686.3			−1.10106
$456$		0			0
$457$	1380.82			0.141339			0.0706695	−	0.997500i	$-0.477486\pi$
$457$	1380.82			0.141339			0.0706695	+	0.997500i	$0.477486\pi$
$458$		0			0
$459$	−11366.8	−	3364.76i	−1.15589	−	0.342164i
$460$		0			0
$461$		−	525.661i		−	0.0531073i	−0.999647	−	0.0265536i	$-0.991547\pi$
$461$		−	525.661i		−	0.0531073i	0.999647	−	0.0265536i	$-0.00845328\pi$
$462$		0			0
$463$			16412.0i			1.64737i	0.567051	+	0.823683i	$0.308084\pi$
$463$			16412.0i			1.64737i	−0.567051	+	0.823683i	$0.691916\pi$
$464$		0			0
$465$	−476.374	+	4950.47i	−0.0475083	+	0.493704i
$466$		0			0
$467$	15400.9			1.52606			0.763029	−	0.646365i	$-0.223711\pi$
$467$	15400.9			1.52606			0.763029	+	0.646365i	$0.223711\pi$
$468$		0			0
$469$	15450.2			1.52116
$470$		0			0
$471$	−690.670	+	7177.42i	−0.0675678	+	0.702162i
$472$		0			0
$473$			5512.32i			0.535849i
$474$		0			0
$475$		−	6188.37i		−	0.597773i
$476$		0			0
$477$	873.169	−	4494.96i	0.0838148	−	0.431467i
$478$		0			0
$479$	−649.281			−0.0619340			−0.0309670	−	0.999520i	$-0.509859\pi$
$479$	−649.281			−0.0619340			−0.0309670	+	0.999520i	$0.509859\pi$
$480$		0			0
$481$	10927.4			1.03585
$482$		0			0
$483$	−8563.40	−	824.040i	−0.806725	−	0.0776297i
$484$		0			0
$485$			1145.26i			0.107224i
$486$		0			0
$487$		−	14196.5i		−	1.32096i	−0.750845	−	0.660479i	$-0.770353\pi$
$487$		−	14196.5i		−	1.32096i	0.750845	−	0.660479i	$-0.229647\pi$
$488$		0			0
$489$	−11156.4	−	1073.56i	−1.03172	−	0.0992801i
$490$		0			0
$491$	−5333.43			−0.490212			−0.245106	−	0.969496i	$-0.578823\pi$
$491$	−5333.43			−0.490212			−0.245106	+	0.969496i	$0.578823\pi$
$492$		0			0
$493$	−11930.9			−1.08994
$494$		0			0
$495$	−4579.36	+	23573.9i	−0.415812	+	2.14055i
$496$		0			0
$497$		−	5695.82i		−	0.514070i
$498$		0			0
$499$			1129.82i			0.101358i	0.998715	+	0.0506789i	$0.0161385\pi$
$499$			1129.82i			0.101358i	−0.998715	+	0.0506789i	$0.983861\pi$
$500$		0			0
$501$	−1121.23	+	11651.8i	−0.0999858	+	1.03905i
$502$		0			0
$503$	318.246			0.0282105			0.0141053	−	0.999901i	$-0.495510\pi$
$503$	318.246			0.0282105			0.0141053	+	0.999901i	$0.495510\pi$
$504$		0			0
$505$	14562.5			1.28321
$506$		0			0
$507$	526.653	−	5472.96i	0.0461331	−	0.479413i
$508$		0			0
$509$		−	1454.04i		−	0.126619i	−0.997994	−	0.0633094i	$-0.979834\pi$
$509$		−	1454.04i		−	0.126619i	0.997994	−	0.0633094i	$-0.0201655\pi$
$510$		0			0
$511$		−	13342.4i		−	1.15505i
$512$		0			0
$513$	3862.71	+	1143.43i	0.332442	+	0.0984084i
$514$		0			0
$515$	−24532.8			−2.09912
$516$		0			0
$517$	−11931.9			−1.01502
$518$		0			0
$519$	−4491.63	−	432.221i	−0.379886	−	0.0365557i
$520$		0			0
$521$		−	17140.8i		−	1.44137i	−0.693263	−	0.720685i	$-0.743827\pi$
$521$		−	17140.8i		−	1.44137i	0.693263	−	0.720685i	$-0.256173\pi$
$522$		0			0
$523$		−	854.626i		−	0.0714535i	−0.999362	−	0.0357268i	$-0.988625\pi$
$523$		−	854.626i		−	0.0714535i	0.999362	−	0.0357268i	$-0.0113746\pi$
$524$		0			0
$525$	−19128.9	−	1840.74i	−1.59019	−	0.153021i
$526$		0			0
$527$	4382.53			0.362251
$528$		0			0
$529$	−2858.17			−0.234912
$530$		0			0
$531$	16051.2	+	3118.03i	1.31180	+	0.254823i
$532$		0			0
$533$			4537.02i			0.368706i
$534$		0			0
$535$		−	16772.5i		−	1.35540i
$536$		0			0
$537$	100.083	−	1040.06i	0.00804265	−	0.0835789i
$538$		0			0
$539$	2339.26			0.186937
$540$		0			0
$541$	−15044.4			−1.19558			−0.597792	−	0.801651i	$-0.703955\pi$
$541$	−15044.4			−1.19558			−0.597792	+	0.801651i	$0.703955\pi$
$542$		0			0
$543$	478.024	−	4967.61i	0.0377790	−	0.392598i
$544$		0			0
$545$		−	22179.6i		−	1.74324i
$546$		0			0
$547$			15376.9i			1.20195i	0.799266	+	0.600977i	$0.205222\pi$
$547$			15376.9i			1.20195i	−0.799266	+	0.600977i	$0.794778\pi$
$548$		0			0
$549$	9098.90	+	1767.51i	0.707344	+	0.137405i
$550$		0			0
$551$	4054.39			0.313472
$552$		0			0
$553$	10084.3			0.775456
$554$		0			0
$555$	30905.2	+	2973.95i	2.36370	+	0.227455i
$556$		0			0
$557$			8917.47i			0.678358i	0.940722	+	0.339179i	$0.110149\pi$
$557$			8917.47i			0.678358i	−0.940722	+	0.339179i	$0.889851\pi$
$558$		0			0
$559$			3859.50i			0.292020i
$560$		0			0
$561$	21064.5	+	2027.00i	1.58528	+	0.152549i
$562$		0			0
$563$	4528.00			0.338957			0.169478	−	0.985534i	$-0.445792\pi$
$563$	4528.00			0.338957			0.169478	+	0.985534i	$0.445792\pi$
$564$		0			0
$565$	10101.1			0.752136
$566$		0			0
$567$	4683.40	−	11599.9i	0.346886	−	0.859170i
$568$		0			0
$569$			2855.22i			0.210364i	0.994453	+	0.105182i	$0.0335425\pi$
$569$			2855.22i			0.210364i	−0.994453	+	0.105182i	$0.966457\pi$
$570$		0			0
$571$			6198.54i			0.454292i	0.973861	+	0.227146i	$0.0729395\pi$
$571$			6198.54i			0.454292i	−0.973861	+	0.227146i	$0.927061\pi$
$572$		0			0
$573$	554.026	−	5757.42i	0.0403922	−	0.419755i
$574$		0			0
$575$	20794.0			1.50812
$576$		0			0
$577$	1201.48			0.0866870			0.0433435	−	0.999060i	$-0.486199\pi$
$577$	1201.48			0.0866870			0.0433435	+	0.999060i	$0.486199\pi$
$578$		0			0
$579$	404.371	−	4202.21i	0.0290243	−	0.301620i
$580$		0			0
$581$		−	25121.9i		−	1.79386i
$582$		0			0
$583$			8174.19i			0.580687i
$584$		0			0
$585$	−3206.28	+	16505.5i	−0.226604	+	1.16653i
$586$		0			0
$587$	5963.60			0.419326			0.209663	−	0.977774i	$-0.432763\pi$
$587$	5963.60			0.419326			0.209663	+	0.977774i	$0.432763\pi$
$588$		0			0
$589$	−1489.29			−0.104185
$590$		0			0
$591$	19460.7	+	1872.67i	1.35449	+	0.130341i
$592$		0			0
$593$			14566.1i			1.00870i	0.863500	+	0.504348i	$0.168267\pi$
$593$			14566.1i			1.00870i	−0.863500	+	0.504348i	$0.831733\pi$
$594$		0			0
$595$			26756.0i			1.84351i
$596$		0			0
$597$	−7276.86	−	700.239i	−0.498865	−	0.0480048i
$598$		0			0
$599$	−21015.8			−1.43353			−0.716763	−	0.697317i	$-0.754377\pi$
$599$	−21015.8			−1.43353			−0.716763	+	0.697317i	$0.754377\pi$
$600$		0			0
$601$	−4904.60			−0.332883			−0.166442	−	0.986051i	$-0.553228\pi$
$601$	−4904.60			−0.332883			−0.166442	+	0.986051i	$0.553228\pi$
$602$		0			0
$603$	4635.62	−	23863.6i	0.313063	−	1.61161i
$604$		0			0
$605$		−	18308.6i		−	1.23033i
$606$		0			0
$607$			3014.03i			0.201541i	0.994910	+	0.100771i	$0.0321309\pi$
$607$			3014.03i			0.201541i	−0.994910	+	0.100771i	$0.967869\pi$
$608$		0			0
$609$	1205.98	−	12532.5i	0.0802444	−	0.833897i
$610$		0			0
$611$	−8354.24			−0.553153
$612$		0			0
$613$	−20267.4			−1.33539			−0.667695	−	0.744435i	$-0.732719\pi$
$613$	−20267.4			−1.33539			−0.667695	+	0.744435i	$0.732719\pi$
$614$		0			0
$615$	−1234.78	+	12831.8i	−0.0809612	+	0.841346i
$616$		0			0
$617$		−	20155.9i		−	1.31515i	−0.753389	−	0.657575i	$-0.771582\pi$
$617$		−	20155.9i		−	1.31515i	0.753389	−	0.657575i	$-0.228418\pi$
$618$		0			0
$619$		−	24145.1i		−	1.56781i	−0.620882	−	0.783904i	$-0.713225\pi$
$619$		−	24145.1i		−	1.56781i	0.620882	−	0.783904i	$-0.286775\pi$
$620$		0			0
$621$	−3842.11	+	12979.4i	−0.248275	+	0.838718i
$622$		0			0
$623$	1349.19			0.0867644
$624$		0			0
$625$	3884.31			0.248596
$626$		0			0
$627$	−7158.24	−	688.824i	−0.455937	−	0.0438740i
$628$		0			0
$629$		−	27359.7i		−	1.73434i
$630$		0			0
$631$		−	23860.5i		−	1.50534i	−0.658397	−	0.752671i	$-0.728765\pi$
$631$		−	23860.5i		−	1.50534i	0.658397	−	0.752671i	$-0.271235\pi$
$632$		0			0
$633$	19209.9	+	1848.53i	1.20620	+	0.116071i
$634$		0			0
$635$	−13543.2			−0.846369
$636$		0			0
$637$	1637.85			0.101875
$638$		0			0
$639$	−8797.49	−	1708.96i	−0.544638	−	0.105799i
$640$		0			0
$641$			2767.39i			0.170523i	0.996359	+	0.0852616i	$0.0271726\pi$
$641$			2767.39i			0.170523i	−0.996359	+	0.0852616i	$0.972827\pi$
$642$		0			0
$643$		−	19313.5i		−	1.18453i	−0.805744	−	0.592263i	$-0.798235\pi$
$643$		−	19313.5i		−	1.18453i	0.805744	−	0.592263i	$-0.201765\pi$
$644$		0			0
$645$	−1050.39	+	10915.6i	−0.0641224	+	0.666358i
$646$		0			0
$647$	−4268.41			−0.259364			−0.129682	−	0.991556i	$-0.541396\pi$
$647$	−4268.41			−0.259364			−0.129682	+	0.991556i	$0.541396\pi$
$648$		0			0
$649$	−29189.5			−1.76547
$650$		0			0
$651$	−442.990	+	4603.54i	−0.0266700	+	0.277154i
$652$		0			0
$653$			13550.4i			0.812048i	0.913862	+	0.406024i	$0.133085\pi$
$653$			13550.4i			0.812048i	−0.913862	+	0.406024i	$0.866915\pi$
$654$		0			0
$655$			31658.5i			1.88855i
$656$		0			0
$657$	−20608.0	−	4003.20i	−1.22373	−	0.237717i
$658$		0			0
$659$	6645.62			0.392833			0.196416	−	0.980521i	$-0.437070\pi$
$659$	6645.62			0.392833			0.196416	+	0.980521i	$0.437070\pi$
$660$		0			0
$661$	2133.53			0.125544			0.0627719	−	0.998028i	$-0.480006\pi$
$661$	2133.53			0.125544			0.0627719	+	0.998028i	$0.480006\pi$
$662$		0			0
$663$	14748.5	+	1419.22i	0.863928	+	0.0831342i
$664$		0			0
$665$		−	9092.36i		−	0.530205i
$666$		0			0
$667$			13623.5i			0.790858i
$668$		0			0
$669$	15624.1	+	1503.48i	0.902936	+	0.0868879i
$670$		0			0
$671$	−16546.6			−0.951972
$672$		0			0
$673$	18933.9			1.08447			0.542235	−	0.840227i	$-0.317578\pi$
$673$	18933.9			1.08447			0.542235	+	0.840227i	$0.317578\pi$
$674$		0			0
$675$	−8582.48	+	28993.2i	−0.489392	+	1.65326i
$676$		0			0
$677$			22893.2i			1.29964i	0.760088	+	0.649820i	$0.225156\pi$
$677$			22893.2i			1.29964i	−0.760088	+	0.649820i	$0.774844\pi$
$678$		0			0
$679$			1065.00i			0.0601927i
$680$		0			0
$681$	−2786.55	+	28957.8i	−0.156800	+	1.62946i
$682$		0			0
$683$	−11506.4			−0.644625			−0.322313	−	0.946633i	$-0.604460\pi$
$683$	−11506.4			−0.644625			−0.322313	+	0.946633i	$0.604460\pi$
$684$		0			0
$685$	41642.1			2.32272
$686$		0			0
$687$	−343.594	+	3570.62i	−0.0190814	+	0.198294i
$688$		0			0
$689$			5723.23i			0.316455i
$690$		0			0
$691$		−	16969.7i		−	0.934235i	−0.884195	−	0.467117i	$-0.845293\pi$
$691$		−	16969.7i		−	0.934235i	0.884195	−	0.467117i	$-0.154707\pi$
$692$		0			0
$693$	−4258.44	+	21921.9i	−0.233427	+	1.20165i
$694$		0			0
$695$	−15223.2			−0.830863
$696$		0			0
$697$	11359.7			0.617329
$698$		0			0
$699$	−7234.91	−	696.202i	−0.391487	−	0.0376721i
$700$		0			0
$701$			18959.8i			1.02154i	0.859717	+	0.510771i	$0.170640\pi$
$701$			18959.8i			1.02154i	−0.859717	+	0.510771i	$0.829360\pi$
$702$		0			0
$703$			9297.48i			0.498807i
$704$		0			0
$705$	−23627.8	−	2273.66i	−1.26223	−	0.121463i
$706$		0			0
$707$	13542.0			0.720364
$708$		0			0
$709$	29888.7			1.58321			0.791604	−	0.611034i	$-0.209246\pi$
$709$	29888.7			1.58321			0.791604	+	0.611034i	$0.209246\pi$
$710$		0			0
$711$	3025.66	−	15575.7i	0.159594	−	0.821567i
$712$		0			0
$713$		−	5004.27i		−	0.262849i
$714$		0			0
$715$		−	30015.6i		−	1.56996i
$716$		0			0
$717$	3420.19	−	35542.5i	0.178144	−	1.85127i
$718$		0			0
$719$	−17686.8			−0.917393			−0.458696	−	0.888593i	$-0.651683\pi$
$719$	−17686.8			−0.917393			−0.458696	+	0.888593i	$0.651683\pi$
$720$		0			0
$721$	−22813.6			−1.17839
$722$		0			0
$723$	−1194.52	+	12413.4i	−0.0614450	+	0.638535i
$724$		0			0
$725$			30432.0i			1.55892i
$726$		0			0
$727$		−	14109.5i		−	0.719796i	−0.932992	−	0.359898i	$-0.882811\pi$
$727$		−	14109.5i		−	0.719796i	0.932992	−	0.359898i	$-0.117189\pi$
$728$		0			0
$729$	−16511.4	−	10714.2i	−0.838868	−	0.544336i
$730$		0			0
$731$	9663.32			0.488934
$732$		0			0
$733$	−23217.2			−1.16992			−0.584958	−	0.811064i	$-0.698889\pi$
$733$	−23217.2			−1.16992			−0.584958	+	0.811064i	$0.698889\pi$
$734$		0			0
$735$	4632.25	+	445.753i	0.232467	+	0.0223698i
$736$		0			0
$737$			43396.5i			2.16897i
$738$		0			0
$739$			30548.3i			1.52062i	0.649560	+	0.760311i	$0.274953\pi$
$739$			30548.3i			1.52062i	−0.649560	+	0.760311i	$0.725047\pi$
$740$		0			0
$741$	−5011.90	−	482.286i	−0.248471	−	0.0239099i
$742$		0			0
$743$	25859.5			1.27684			0.638420	−	0.769688i	$-0.279588\pi$
$743$	25859.5			1.27684			0.638420	+	0.769688i	$0.279588\pi$
$744$		0			0
$745$	3546.89			0.174427
$746$		0			0
$747$	−38802.1	−	7537.51i	−1.90053	−	0.369187i
$748$		0			0
$749$		−	15597.1i		−	0.760888i
$750$		0			0
$751$			16289.9i			0.791516i	0.918355	+	0.395758i	$0.129518\pi$
$751$			16289.9i			0.791516i	−0.918355	+	0.395758i	$0.870482\pi$
$752$		0			0
$753$	−97.9734	+	1018.14i	−0.00474150	+	0.0492735i
$754$		0			0
$755$	−11281.5			−0.543810
$756$		0			0
$757$	−25624.7			−1.23031			−0.615155	−	0.788406i	$-0.710906\pi$
$757$	−25624.7			−1.23031			−0.615155	+	0.788406i	$0.710906\pi$
$758$		0			0
$759$	2314.57	−	24052.9i	0.110690	−	1.15028i
$760$		0			0
$761$			28456.4i			1.35551i	0.735287	+	0.677756i	$0.237047\pi$
$761$			28456.4i			1.35551i	−0.735287	+	0.677756i	$0.762953\pi$
$762$		0			0
$763$		−	20625.2i		−	0.978616i
$764$		0			0
$765$	41326.1	+	8027.80i	1.95313	+	0.379406i
$766$		0			0
$767$	−20437.3			−0.962123
$768$		0			0
$769$	−26858.7			−1.25949			−0.629747	−	0.776800i	$-0.716842\pi$
$769$	−26858.7			−1.25949			−0.629747	+	0.776800i	$0.716842\pi$
$770$		0			0
$771$	29236.2	+	2813.35i	1.36565	+	0.131414i
$772$		0			0
$773$		−	33227.3i		−	1.54606i	−0.634369	−	0.773030i	$-0.718740\pi$
$773$		−	33227.3i		−	1.54606i	0.634369	−	0.773030i	$-0.281260\pi$
$774$		0			0
$775$		−	11178.5i		−	0.518121i
$776$		0			0
$777$	28739.4	+	2765.54i	1.32692	+	0.127688i
$778$		0			0
$779$	−3860.30			−0.177547
$780$		0			0
$781$	15998.5			0.732996
$782$		0			0
$783$	−18995.3	−	5622.92i	−0.866968	−	0.256637i
$784$		0			0
$785$		−	25607.1i		−	1.16428i
$786$		0			0
$787$			13493.5i			0.611171i	0.952165	+	0.305586i	$0.0988522\pi$
$787$			13493.5i			0.611171i	−0.952165	+	0.305586i	$0.901148\pi$
$788$		0			0
$789$	3106.49	−	32282.5i	0.140170	−	1.45664i
$790$		0			0
$791$	9393.23			0.422231
$792$		0			0
$793$	−11585.2			−0.518794
$794$		0			0
$795$	−1557.61	+	16186.7i	−0.0694879	+	0.722116i
$796$		0			0
$797$			4559.97i			0.202663i	0.994853	+	0.101331i	$0.0323103\pi$
$797$			4559.97i			0.202663i	−0.994853	+	0.101331i	$0.967690\pi$
$798$		0			0
$799$			20917.2i			0.926152i
$800$		0			0
$801$	404.807	−	2083.90i	0.0178566	−	0.0919236i
$802$		0			0
$803$	37476.1			1.64695
$804$		0			0
$805$	30551.9			1.33765
$806$		0			0
$807$	−36388.4	−	3501.59i	−1.58728	−	0.152741i
$808$		0			0
$809$		−	26274.6i		−	1.14186i	−0.820998	−	0.570931i	$-0.806582\pi$
$809$		−	26274.6i		−	1.14186i	0.820998	−	0.570931i	$-0.193418\pi$
$810$		0			0
$811$			33570.4i			1.45354i	0.686883	+	0.726768i	$0.258979\pi$
$811$			33570.4i			1.45354i	−0.686883	+	0.726768i	$0.741021\pi$
$812$		0			0
$813$	−25225.2	−	2427.37i	−1.08817	−	0.104713i
$814$		0			0
$815$	39802.9			1.71072
$816$		0			0
$817$	−3283.83			−0.140620
$818$		0			0
$819$	−2981.58	+	15348.8i	−0.127210	+	0.654860i
$820$		0			0
$821$		−	15122.2i		−	0.642835i	−0.946938	−	0.321417i	$-0.895841\pi$
$821$		−	15122.2i		−	0.642835i	0.946938	−	0.321417i	$-0.104159\pi$
$822$		0			0
$823$			26434.7i			1.11963i	0.828617	+	0.559816i	$0.189128\pi$
$823$			26434.7i			1.11963i	−0.828617	+	0.559816i	$0.810872\pi$
$824$		0			0
$825$	5170.26	−	53729.2i	0.218188	−	2.26741i
$826$		0			0
$827$	9808.95			0.412443			0.206222	−	0.978505i	$-0.433883\pi$
$827$	9808.95			0.412443			0.206222	+	0.978505i	$0.433883\pi$
$828$		0			0
$829$	−23250.9			−0.974110			−0.487055	−	0.873371i	$-0.661929\pi$
$829$	−23250.9			−0.974110			−0.487055	+	0.873371i	$0.661929\pi$
$830$		0			0
$831$	2922.84	−	30374.1i	0.122012	−	1.26795i
$832$		0			0
$833$		−	4100.82i		−	0.170570i
$834$		0			0
$835$		−	41570.4i		−	1.72288i
$836$		0			0
$837$	6977.49	+	2065.45i	0.288145	+	0.0852958i
$838$		0			0
$839$	43358.6			1.78416			0.892078	−	0.451882i	$-0.149247\pi$
$839$	43358.6			1.78416			0.892078	+	0.451882i	$0.149247\pi$
$840$		0			0
$841$	4451.08			0.182504
$842$		0			0
$843$	8632.57	+	830.697i	0.352695	+	0.0339392i
$844$		0			0
$845$			19526.0i			0.794930i
$846$		0			0
$847$		−	17025.5i		−	0.690677i
$848$		0			0
$849$	−6453.07	−	620.967i	−0.260858	−	0.0251019i
$850$		0			0
$851$	−31241.1			−1.25844
$852$		0			0
$853$	−6641.65			−0.266595			−0.133298	−	0.991076i	$-0.542557\pi$
$853$	−6641.65			−0.266595			−0.133298	+	0.991076i	$0.542557\pi$
$854$		0			0
$855$	−14043.6	−	2728.04i	−0.561733	−	0.109119i
$856$		0			0
$857$			39138.2i			1.56002i	0.625767	+	0.780010i	$0.284786\pi$
$857$			39138.2i			1.56002i	−0.625767	+	0.780010i	$0.715214\pi$
$858$		0			0
$859$		−	40377.3i		−	1.60379i	−0.597465	−	0.801895i	$-0.703825\pi$
$859$		−	40377.3i		−	1.60379i	0.597465	−	0.801895i	$-0.296175\pi$
$860$		0			0
$861$	−1148.25	+	11932.5i	−0.0454497	+	0.472311i
$862$		0			0
$863$	39324.1			1.55111			0.775554	−	0.631281i	$-0.217471\pi$
$863$	39324.1			1.55111			0.775554	+	0.631281i	$0.217471\pi$
$864$		0			0
$865$	16024.9			0.629900
$866$		0			0
$867$	1108.13	−	11515.6i	0.0434072	−	0.451086i
$868$		0			0
$869$			28324.8i			1.10570i
$870$		0			0
$871$			30384.4i			1.18202i
$872$		0			0
$873$	1644.94	+	319.539i	0.0637719	+	0.0123880i
$874$		0			0
$875$	28664.3			1.10746
$876$		0			0
$877$	1540.57			0.0593175			0.0296588	−	0.999560i	$-0.490558\pi$
$877$	1540.57			0.0593175			0.0296588	+	0.999560i	$0.490558\pi$
$878$		0			0
$879$	7501.53	+	721.859i	0.287850	+	0.0276993i
$880$		0			0
$881$		−	37069.1i		−	1.41758i	−0.705419	−	0.708790i	$-0.749241\pi$
$881$		−	37069.1i		−	1.41758i	0.705419	−	0.708790i	$-0.250759\pi$
$882$		0			0
$883$		−	18496.2i		−	0.704922i	−0.935826	−	0.352461i	$-0.885345\pi$
$883$		−	18496.2i		−	0.704922i	0.935826	−	0.352461i	$-0.114655\pi$
$884$		0			0
$885$	−57801.7	−	5562.15i	−2.19546	−	0.211265i
$886$		0			0
$887$	24375.0			0.922698			0.461349	−	0.887219i	$-0.347366\pi$
$887$	24375.0			0.922698			0.461349	+	0.887219i	$0.347366\pi$
$888$		0			0
$889$	−12594.1			−0.475131
$890$		0			0
$891$	32581.8	+	13154.8i	1.22506	+	0.494614i
$892$		0			0
$893$		−	7108.16i		−	0.266367i
$894$		0			0
$895$			3710.65i			0.138585i
$896$		0			0
$897$	1620.56	−	16840.8i	0.0603222	−	0.626867i
$898$		0			0
$899$	7323.75			0.271703
$900$		0			0
$901$	14329.7			0.529846
$902$		0			0
$903$	−976.777	+	10150.6i	−0.0359968	+	0.374078i
$904$		0			0
$905$			17723.1i			0.650978i
$906$		0			0
$907$		−	21352.8i		−	0.781708i	−0.920453	−	0.390854i	$-0.872180\pi$
$907$		−	21352.8i		−	0.781708i	0.920453	−	0.390854i	$-0.127820\pi$
$908$		0			0
$909$	4063.09	−	20916.2i	0.148255	−	0.763199i
$910$		0			0
$911$	−39237.3			−1.42699			−0.713496	−	0.700659i	$-0.752889\pi$
$911$	−39237.3			−1.42699			−0.713496	+	0.700659i	$0.752889\pi$
$912$		0			0
$913$	70562.5			2.55781
$914$		0			0
$915$	−32765.8	−	3153.00i	−1.18383	−	0.113918i
$916$		0			0
$917$			29439.9i			1.06019i
$918$		0			0
$919$			11761.3i			0.422166i	0.977468	+	0.211083i	$0.0676990\pi$
$919$			11761.3i			0.422166i	−0.977468	+	0.211083i	$0.932301\pi$
$920$		0			0
$921$	22485.3	+	2163.72i	0.804468	+	0.0774125i
$922$		0			0
$923$	11201.5			0.399459
$924$		0			0
$925$	−69786.2			−2.48060
$926$		0			0
$927$	−6844.92	+	35236.7i	−0.242521	+	1.24846i
$928$		0			0
$929$		−	1473.03i		−	0.0520222i	−0.999662	−	0.0260111i	$-0.991719\pi$
$929$		−	1473.03i		−	0.0520222i	0.999662	−	0.0260111i	$-0.00828053\pi$
$930$		0			0
$931$			1393.56i			0.0490570i
$932$		0			0
$933$	2460.82	−	25572.7i	0.0863489	−	0.897335i
$934$		0			0
$935$	−75152.5			−2.62861
$936$		0			0
$937$	18439.9			0.642907			0.321453	−	0.946925i	$-0.395829\pi$
$937$	18439.9			0.642907			0.321453	+	0.946925i	$0.395829\pi$
$938$		0			0
$939$	−4534.41	+	47121.4i	−0.157588	+	1.63765i
$940$		0			0
$941$		−	37937.5i		−	1.31427i	−0.753773	−	0.657135i	$-0.771768\pi$
$941$		−	37937.5i		−	1.31427i	0.753773	−	0.657135i	$-0.228232\pi$
$942$		0			0
$943$		−	12971.3i		−	0.447935i
$944$		0			0
$945$	−12609.9	+	42598.7i	−0.434075	+	1.46639i
$946$		0			0
$947$	−36960.6			−1.26828			−0.634138	−	0.773220i	$-0.718645\pi$
$947$	−36960.6			−1.26828			−0.634138	+	0.773220i	$0.718645\pi$
$948$		0			0
$949$	26239.2			0.897534
$950$		0			0
$951$	16049.8	+	1544.45i	0.547268	+	0.0526626i
$952$		0			0
$953$			2649.69i			0.0900650i	0.998986	+	0.0450325i	$0.0143391\pi$
$953$			2649.69i			0.0900650i	−0.998986	+	0.0450325i	$0.985661\pi$
$954$		0			0
$955$			20540.9i			0.696008i
$956$		0			0
$957$	35201.4	+	3387.37i	1.18903	+	0.114418i
$958$		0			0
$959$	38723.9			1.30392
$960$		0			0
$961$	27100.8			0.909697
$962$		0			0
$963$	−24090.5	−	4679.71i	−0.806133	−	0.156595i
$964$		0			0
$965$			14992.3i			0.500125i
$966$		0			0
$967$		−	4333.01i		−	0.144095i	−0.997401	−	0.0720476i	$-0.977047\pi$
$967$		−	4333.01i		−	0.144095i	0.997401	−	0.0720476i	$-0.0229534\pi$
$968$		0			0
$969$	−1207.54	+	12548.7i	−0.0400327	+	0.416018i
$970$		0			0
$971$	−35460.3			−1.17196			−0.585981	−	0.810325i	$-0.699291\pi$
$971$	−35460.3			−1.17196			−0.585981	+	0.810325i	$0.699291\pi$
$972$		0			0
$973$	−14156.4			−0.466427
$974$		0			0
$975$	3620.00	−	37619.0i	0.118906	−	1.23566i
$976$		0			0
$977$			52275.3i			1.71181i	0.517135	+	0.855904i	$0.326998\pi$
$977$			52275.3i			1.71181i	−0.517135	+	0.855904i	$0.673002\pi$
$978$		0			0
$979$			3789.61i			0.123715i
$980$		0			0
$981$	−31856.7	−	6188.34i	−1.03681	−	0.201405i
$982$		0			0
$983$	−53985.9			−1.75166			−0.875831	−	0.482618i	$-0.839686\pi$
$983$	−53985.9			−1.75166			−0.875831	+	0.482618i	$0.839686\pi$
$984$		0			0
$985$	−69430.5			−2.24593
$986$		0			0
$987$	−21972.0	−	2114.33i	−0.708588	−	0.0681861i
$988$		0			0
$989$		−	11034.2i		−	0.354771i
$990$		0			0
$991$		−	44415.5i		−	1.42372i	−0.702322	−	0.711860i	$-0.747853\pi$
$991$		−	44415.5i		−	1.42372i	0.702322	−	0.711860i	$-0.252147\pi$
$992$		0			0
$993$	7401.84	+	712.266i	0.236546	+	0.0227624i
$994$		0			0
$995$	25961.9			0.827183
$996$		0			0
$997$	−21929.9			−0.696615			−0.348308	−	0.937380i	$-0.613244\pi$
$997$	−21929.9			−0.696615			−0.348308	+	0.937380i	$0.613244\pi$
$998$		0			0
$999$	12894.4	−	43559.7i	0.408369	−	1.37955i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	96.4.c.a.95.6	yes	12
3.2	odd	2	inner	96.4.c.a.95.8	yes	12
4.3	odd	2	inner	96.4.c.a.95.7	yes	12
8.3	odd	2		192.4.c.d.191.6		12
8.5	even	2		192.4.c.d.191.7		12
12.11	even	2	inner	96.4.c.a.95.5	✓	12
16.3	odd	4		768.4.f.i.383.11		12
16.5	even	4		768.4.f.i.383.12		12
16.11	odd	4		768.4.f.d.383.2		12
16.13	even	4		768.4.f.d.383.1		12
24.5	odd	2		192.4.c.d.191.5		12
24.11	even	2		192.4.c.d.191.8		12
48.5	odd	4		768.4.f.i.383.9		12
48.11	even	4		768.4.f.d.383.3		12
48.29	odd	4		768.4.f.d.383.4		12
48.35	even	4		768.4.f.i.383.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.4.c.a.95.5	✓	12	12.11	even	2	inner
96.4.c.a.95.6	yes	12	1.1	even	1	trivial
96.4.c.a.95.7	yes	12	4.3	odd	2	inner
96.4.c.a.95.8	yes	12	3.2	odd	2	inner
192.4.c.d.191.5		12	24.5	odd	2
192.4.c.d.191.6		12	8.3	odd	2
192.4.c.d.191.7		12	8.5	even	2
192.4.c.d.191.8		12	24.11	even	2
768.4.f.d.383.1		12	16.13	even	4
768.4.f.d.383.2		12	16.11	odd	4
768.4.f.d.383.3		12	48.11	even	4
768.4.f.d.383.4		12	48.29	odd	4
768.4.f.i.383.9		12	48.5	odd	4
768.4.f.i.383.10		12	48.35	even	4
768.4.f.i.383.11		12	16.3	odd	4
768.4.f.i.383.12		12	16.5	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:	\( N \)	\(=\)	\( 96 = 2^{5} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	96.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.497717 + 5.17226i) q^{3} -18.4532i q^{5} -17.1600i q^{7} +(-26.5046 - 5.14865i) q^{9} +48.1992 q^{11} +33.7471 q^{13} +(95.4449 + 9.18449i) q^{15} -84.4951i q^{17} +28.7135i q^{19} +(88.7562 + 8.54084i) q^{21} -96.4823 q^{23} -215.521 q^{25} +(39.8219 - 134.526i) q^{27} -141.202i q^{29} +51.8673i q^{31} +(-23.9896 + 249.299i) q^{33} -316.658 q^{35} +323.802 q^{37} +(-16.7965 + 174.549i) q^{39} +134.442i q^{41} +114.365i q^{43} +(-95.0091 + 489.094i) q^{45} -247.555 q^{47} +48.5332 q^{49} +(437.031 + 42.0547i) q^{51} +169.592i q^{53} -889.430i q^{55} +(-148.514 - 14.2912i) q^{57} -605.603 q^{59} -343.296 q^{61} +(-88.3509 + 454.819i) q^{63} -622.742i q^{65} +900.357i q^{67} +(48.0209 - 499.031i) q^{69} +331.924 q^{71} +777.525 q^{73} +(107.269 - 1114.73i) q^{75} -827.099i q^{77} +587.661i q^{79} +(675.983 + 272.925i) q^{81} +1463.98 q^{83} -1559.21 q^{85} +(730.332 + 70.2785i) q^{87} +78.6240i q^{89} -579.101i q^{91} +(-268.271 - 25.8152i) q^{93} +529.857 q^{95} -62.0627 q^{97} +(-1277.50 - 248.160i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9} + 72 q^{13} + 136 q^{21} - 132 q^{25} + 80 q^{33} - 24 q^{37} - 544 q^{45} - 540 q^{49} - 888 q^{57} + 456 q^{61} + 1312 q^{69} + 2424 q^{73} + 2924 q^{81} - 3072 q^{85} - 2360 q^{93} - 2952 q^{97}+O(q^{100}) \) 12 * q - 20 * q^9 + 72 * q^13 + 136 * q^21 - 132 * q^25 + 80 * q^33 - 24 * q^37 - 544 * q^45 - 540 * q^49 - 888 * q^57 + 456 * q^61 + 1312 * q^69 + 2424 * q^73 + 2924 * q^81 - 3072 * q^85 - 2360 * q^93 - 2952 * q^97

Introduction

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