LMFDB - Embedded newform 192.4.c.b.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,Mod(191,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	192.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.3283667211$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + x^{2} + 4 $ x^4 + x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.1
Root			$0.866025 + 1.11803i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.4.c.b.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.46410 - 3.87298i) q^{3} +8.94427i q^{5} -7.74597i q^{7} +(-3.00000 + 26.8328i) q^{9} +O(q^{10})$	\(q+(-3.46410 - 3.87298i) q^{3} +8.94427i q^{5} -7.74597i q^{7} +(-3.00000 + 26.8328i) q^{9} +34.6410 q^{11} +10.0000 q^{13} +(34.6410 - 30.9839i) q^{15} -35.7771i q^{17} -69.7137i q^{19} +(-30.0000 + 26.8328i) q^{21} +96.9948 q^{23} +45.0000 q^{25} +(114.315 - 81.3327i) q^{27} -152.053i q^{29} -224.633i q^{31} +(-120.000 - 134.164i) q^{33} +69.2820 q^{35} +130.000 q^{37} +(-34.6410 - 38.7298i) q^{39} +125.220i q^{41} -224.633i q^{43} +(-240.000 - 26.8328i) q^{45} +193.990 q^{47} +283.000 q^{49} +(-138.564 + 123.935i) q^{51} +545.601i q^{53} +309.839i q^{55} +(-270.000 + 241.495i) q^{57} +173.205 q^{59} +442.000 q^{61} +(207.846 + 23.2379i) q^{63} +89.4427i q^{65} +735.867i q^{67} +(-336.000 - 375.659i) q^{69} -1039.23 q^{71} +410.000 q^{73} +(-155.885 - 174.284i) q^{75} -268.328i q^{77} +85.2056i q^{79} +(-711.000 - 160.997i) q^{81} -1254.00 q^{83} +320.000 q^{85} +(-588.897 + 526.726i) q^{87} -840.762i q^{89} -77.4597i q^{91} +(-870.000 + 778.152i) q^{93} +623.538 q^{95} +770.000 q^{97} +(-103.923 + 929.516i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 40 q^{13} - 120 q^{21} + 180 q^{25} - 480 q^{33} + 520 q^{37} - 960 q^{45} + 1132 q^{49} - 1080 q^{57} + 1768 q^{61} - 1344 q^{69} + 1640 q^{73} - 2844 q^{81} + 1280 q^{85} - 3480 q^{93} + 3080 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 40 * q^13 - 120 * q^21 + 180 * q^25 - 480 * q^33 + 520 * q^37 - 960 * q^45 + 1132 * q^49 - 1080 * q^57 + 1768 * q^61 - 1344 * q^69 + 1640 * q^73 - 2844 * q^81 + 1280 * q^85 - 3480 * q^93 + 3080 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−3.46410	−	3.87298i	−0.666667	−	0.745356i
$3$	−3.46410	−	3.87298i	−0.666667	−	0.745356i
$4$		0			0
$5$			8.94427i			0.800000i	0.916515	+	0.400000i	$0.130990\pi$
$5$			8.94427i			0.800000i	−0.916515	+	0.400000i	$0.869010\pi$
$6$		0			0
$7$		−	7.74597i		−	0.418243i	−0.977890	−	0.209121i	$-0.932940\pi$
$7$		−	7.74597i		−	0.418243i	0.977890	−	0.209121i	$-0.0670604\pi$
$8$		0			0
$9$	−3.00000	+	26.8328i	−0.111111	+	0.993808i
$10$		0			0
$11$	34.6410			0.949514			0.474757	−	0.880117i	$-0.342536\pi$
$11$	34.6410			0.949514			0.474757	+	0.880117i	$0.342536\pi$
$12$		0			0
$13$	10.0000			0.213346			0.106673	−	0.994294i	$-0.465980\pi$
$13$	10.0000			0.213346			0.106673	+	0.994294i	$0.465980\pi$
$14$		0			0
$15$	34.6410	−	30.9839i	0.596285	−	0.533333i
$16$		0			0
$17$		−	35.7771i		−	0.510425i	−0.966885	−	0.255212i	$-0.917855\pi$
$17$		−	35.7771i		−	0.510425i	0.966885	−	0.255212i	$-0.0821454\pi$
$18$		0			0
$19$		−	69.7137i		−	0.841759i	−0.907117	−	0.420879i	$-0.861722\pi$
$19$		−	69.7137i		−	0.841759i	0.907117	−	0.420879i	$-0.138278\pi$
$20$		0			0
$21$	−30.0000	+	26.8328i	−0.311740	+	0.278829i
$22$		0			0
$23$	96.9948			0.879340			0.439670	−	0.898159i	$-0.355095\pi$
$23$	96.9948			0.879340			0.439670	+	0.898159i	$0.355095\pi$
$24$		0			0
$25$	45.0000			0.360000
$26$		0			0
$27$	114.315	−	81.3327i	0.814815	−	0.579721i
$28$		0			0
$29$		−	152.053i		−	0.973637i	−0.873503	−	0.486818i	$-0.838157\pi$
$29$		−	152.053i		−	0.973637i	0.873503	−	0.486818i	$-0.161843\pi$
$30$		0			0
$31$		−	224.633i		−	1.30146i	−0.759309	−	0.650730i	$-0.774463\pi$
$31$		−	224.633i		−	1.30146i	0.759309	−	0.650730i	$-0.225537\pi$
$32$		0			0
$33$	−120.000	−	134.164i	−0.633010	−	0.707726i
$34$		0			0
$35$	69.2820			0.334594
$36$		0			0
$37$	130.000			0.577618			0.288809	−	0.957387i	$-0.406741\pi$
$37$	130.000			0.577618			0.288809	+	0.957387i	$0.406741\pi$
$38$		0			0
$39$	−34.6410	−	38.7298i	−0.142231	−	0.159019i
$40$		0			0
$41$			125.220i			0.476977i	0.971145	+	0.238488i	$0.0766519\pi$
$41$			125.220i			0.476977i	−0.971145	+	0.238488i	$0.923348\pi$
$42$		0			0
$43$		−	224.633i		−	0.796656i	−0.917243	−	0.398328i	$-0.869591\pi$
$43$		−	224.633i		−	0.796656i	0.917243	−	0.398328i	$-0.130409\pi$
$44$		0			0
$45$	−240.000	−	26.8328i	−0.795046	−	0.0888889i
$46$		0			0
$47$	193.990			0.602049			0.301025	−	0.953616i	$-0.402671\pi$
$47$	193.990			0.602049			0.301025	+	0.953616i	$0.402671\pi$
$48$		0			0
$49$	283.000			0.825073
$50$		0			0
$51$	−138.564	+	123.935i	−0.380448	+	0.340283i
$52$		0			0
$53$			545.601i			1.41404i	0.707195	+	0.707019i	$0.249960\pi$
$53$			545.601i			1.41404i	−0.707195	+	0.707019i	$0.750040\pi$
$54$		0			0
$55$			309.839i			0.759612i
$56$		0			0
$57$	−270.000	+	241.495i	−0.627410	+	0.561173i
$58$		0			0
$59$	173.205			0.382193			0.191096	−	0.981571i	$-0.438796\pi$
$59$	173.205			0.382193			0.191096	+	0.981571i	$0.438796\pi$
$60$		0			0
$61$	442.000			0.927743			0.463871	−	0.885903i	$-0.346460\pi$
$61$	442.000			0.927743			0.463871	+	0.885903i	$0.346460\pi$
$62$		0			0
$63$	207.846	+	23.2379i	0.415653	+	0.0464714i
$64$		0			0
$65$			89.4427i			0.170677i
$66$		0			0
$67$			735.867i			1.34180i	0.741549	+	0.670899i	$0.234092\pi$
$67$			735.867i			1.34180i	−0.741549	+	0.670899i	$0.765908\pi$
$68$		0			0
$69$	−336.000	−	375.659i	−0.586227	−	0.655421i
$70$		0			0
$71$	−1039.23			−1.73710			−0.868549	−	0.495603i	$-0.834947\pi$
$71$	−1039.23			−1.73710			−0.868549	+	0.495603i	$0.834947\pi$
$72$		0			0
$73$	410.000			0.657354			0.328677	−	0.944442i	$-0.393397\pi$
$73$	410.000			0.657354			0.328677	+	0.944442i	$0.393397\pi$
$74$		0			0
$75$	−155.885	−	174.284i	−0.240000	−	0.268328i
$76$		0			0
$77$		−	268.328i		−	0.397128i
$78$		0			0
$79$			85.2056i			0.121347i	0.998158	+	0.0606733i	$0.0193248\pi$
$79$			85.2056i			0.121347i	−0.998158	+	0.0606733i	$0.980675\pi$
$80$		0			0
$81$	−711.000	−	160.997i	−0.975309	−	0.220846i
$82$		0			0
$83$	−1254.00			−1.65837			−0.829186	−	0.558973i	$-0.811196\pi$
$83$	−1254.00			−1.65837			−0.829186	+	0.558973i	$0.811196\pi$
$84$		0			0
$85$	320.000			0.408340
$86$		0			0
$87$	−588.897	+	526.726i	−0.725706	+	0.649091i
$88$		0			0
$89$		−	840.762i		−	1.00135i	−0.865634	−	0.500677i	$-0.833084\pi$
$89$		−	840.762i		−	1.00135i	0.865634	−	0.500677i	$-0.166916\pi$
$90$		0			0
$91$		−	77.4597i		−	0.0892305i
$92$		0			0
$93$	−870.000	+	778.152i	−0.970052	+	0.867641i
$94$		0			0
$95$	623.538			0.673407
$96$		0			0
$97$	770.000			0.805996			0.402998	−	0.915201i	$-0.367968\pi$
$97$	770.000			0.805996			0.402998	+	0.915201i	$0.367968\pi$
$98$		0			0
$99$	−103.923	+	929.516i	−0.105502	+	0.943635i
$100$		0			0
$101$		−	1493.69i		−	1.47156i	−0.677218	−	0.735782i	$-0.736815\pi$
$101$		−	1493.69i		−	1.47156i	0.677218	−	0.735782i	$-0.263185\pi$
$102$		0			0
$103$			1355.54i			1.29675i	0.761319	+	0.648377i	$0.224552\pi$
$103$			1355.54i			1.29675i	−0.761319	+	0.648377i	$0.775448\pi$
$104$		0			0
$105$	−240.000	−	268.328i	−0.223063	−	0.249392i
$106$		0			0
$107$	644.323			0.582141			0.291070	−	0.956702i	$-0.405989\pi$
$107$	644.323			0.582141			0.291070	+	0.956702i	$0.405989\pi$
$108$		0			0
$109$	1066.00			0.936737			0.468368	−	0.883533i	$-0.344842\pi$
$109$	1066.00			0.936737			0.468368	+	0.883533i	$0.344842\pi$
$110$		0			0
$111$	−450.333	−	503.488i	−0.385079	−	0.430531i
$112$		0			0
$113$			1037.54i			0.863745i	0.901935	+	0.431872i	$0.142147\pi$
$113$			1037.54i			0.863745i	−0.901935	+	0.431872i	$0.857853\pi$
$114$		0			0
$115$			867.548i			0.703472i
$116$		0			0
$117$	−30.0000	+	268.328i	−0.0237051	+	0.212025i
$118$		0			0
$119$	−277.128			−0.213481
$120$		0			0
$121$	−131.000			−0.0984222
$122$		0			0
$123$	484.974	−	433.774i	0.355518	−	0.317985i
$124$		0			0
$125$			1520.53i			1.08800i
$126$		0			0
$127$		−	1835.79i		−	1.28268i	−0.767257	−	0.641340i	$-0.778379\pi$
$127$		−	1835.79i		−	1.28268i	0.767257	−	0.641340i	$-0.221621\pi$
$128$		0			0
$129$	−870.000	+	778.152i	−0.593792	+	0.531104i
$130$		0			0
$131$	−450.333			−0.300350			−0.150175	−	0.988659i	$-0.547984\pi$
$131$	−450.333			−0.300350			−0.150175	+	0.988659i	$0.547984\pi$
$132$		0			0
$133$	−540.000			−0.352060
$134$		0			0
$135$	727.461	+	1022.47i	0.463777	+	0.651852i
$136$		0			0
$137$		−	89.4427i		−	0.0557782i	−0.999611	−	0.0278891i	$-0.991121\pi$
$137$		−	89.4427i		−	0.0557782i	0.999611	−	0.0278891i	$-0.00887852\pi$
$138$		0			0
$139$		−	1959.73i		−	1.19584i	−0.801555	−	0.597921i	$-0.795994\pi$
$139$		−	1959.73i		−	1.19584i	0.801555	−	0.597921i	$-0.204006\pi$
$140$		0			0
$141$	−672.000	−	751.319i	−0.401366	−	0.448741i
$142$		0			0
$143$	346.410			0.202575
$144$		0			0
$145$	1360.00			0.778909
$146$		0			0
$147$	−980.341	−	1096.05i	−0.550049	−	0.614973i
$148$		0			0
$149$			1618.91i			0.890111i	0.895503	+	0.445055i	$0.146816\pi$
$149$			1618.91i			0.890111i	−0.895503	+	0.445055i	$0.853184\pi$
$150$		0			0
$151$		−	565.456i		−	0.304743i	−0.988323	−	0.152371i	$-0.951309\pi$
$151$		−	565.456i		−	0.304743i	0.988323	−	0.152371i	$-0.0486909\pi$
$152$		0			0
$153$	960.000	+	107.331i	0.507264	+	0.0567138i
$154$		0			0
$155$	2009.18			1.04117
$156$		0			0
$157$	730.000			0.371085			0.185542	−	0.982636i	$-0.440596\pi$
$157$	730.000			0.371085			0.185542	+	0.982636i	$0.440596\pi$
$158$		0			0
$159$	2113.10	−	1890.02i	1.05396	−	0.942692i
$160$		0			0
$161$		−	751.319i		−	0.367778i
$162$		0			0
$163$		−	255.617i		−	0.122831i	−0.998112	−	0.0614155i	$-0.980439\pi$
$163$		−	255.617i		−	0.122831i	0.998112	−	0.0614155i	$-0.0195615\pi$
$164$		0			0
$165$	1200.00	−	1073.31i	0.566181	−	0.506408i
$166$		0			0
$167$	13.8564			0.00642060			0.00321030	−	0.999995i	$-0.498978\pi$
$167$	13.8564			0.00642060			0.00321030	+	0.999995i	$0.498978\pi$
$168$		0			0
$169$	−2097.00			−0.954483
$170$		0			0
$171$	1870.61	+	209.141i	0.836547	+	0.0935288i
$172$		0			0
$173$		−	1118.03i		−	0.491344i	−0.969353	−	0.245672i	$-0.920991\pi$
$173$		−	1118.03i		−	0.491344i	0.969353	−	0.245672i	$-0.0790086\pi$
$174$		0			0
$175$		−	348.569i		−	0.150567i
$176$		0			0
$177$	−600.000	−	670.820i	−0.254795	−	0.284870i
$178$		0			0
$179$	1351.00			0.564125			0.282063	−	0.959396i	$-0.408981\pi$
$179$	1351.00			0.564125			0.282063	+	0.959396i	$0.408981\pi$
$180$		0			0
$181$	−1262.00			−0.518253			−0.259126	−	0.965843i	$-0.583435\pi$
$181$	−1262.00			−0.518253			−0.259126	+	0.965843i	$0.583435\pi$
$182$		0			0
$183$	−1531.13	−	1711.86i	−0.618495	−	0.691499i
$184$		0			0
$185$			1162.76i			0.462094i
$186$		0			0
$187$		−	1239.35i		−	0.484656i
$188$		0			0
$189$	−630.000	−	885.483i	−0.242464	−	0.340791i
$190$		0			0
$191$	2771.28			1.04986			0.524929	−	0.851146i	$-0.324092\pi$
$191$	2771.28			1.04986			0.524929	+	0.851146i	$0.324092\pi$
$192$		0			0
$193$	−190.000			−0.0708627			−0.0354313	−	0.999372i	$-0.511281\pi$
$193$	−190.000			−0.0708627			−0.0354313	+	0.999372i	$0.511281\pi$
$194$		0			0
$195$	346.410	−	309.839i	0.127215	−	0.113785i
$196$		0			0
$197$		−	2137.68i		−	0.773114i	−0.922266	−	0.386557i	$-0.873664\pi$
$197$		−	2137.68i		−	0.773114i	0.922266	−	0.386557i	$-0.126336\pi$
$198$		0			0
$199$		−	255.617i		−	0.0910563i	−0.998963	−	0.0455281i	$-0.985503\pi$
$199$		−	255.617i		−	0.0910563i	0.998963	−	0.0455281i	$-0.0144971\pi$
$200$		0			0
$201$	2850.00	−	2549.12i	1.00012	−	0.894532i
$202$		0			0
$203$	−1177.79			−0.407217
$204$		0			0
$205$	−1120.00			−0.381581
$206$		0			0
$207$	−290.985	+	2602.64i	−0.0977045	+	0.873895i
$208$		0			0
$209$		−	2414.95i		−	0.799262i
$210$		0			0
$211$			549.964i			0.179436i	0.995967	+	0.0897181i	$0.0285966\pi$
$211$			549.964i			0.179436i	−0.995967	+	0.0897181i	$0.971403\pi$
$212$		0			0
$213$	3600.00	+	4024.92i	1.15807	+	1.29476i
$214$		0			0
$215$	2009.18			0.637325
$216$		0			0
$217$	−1740.00			−0.544327
$218$		0			0
$219$	−1420.28	−	1587.92i	−0.438236	−	0.489963i
$220$		0			0
$221$		−	357.771i		−	0.108897i
$222$		0			0
$223$		−	472.504i		−	0.141889i	−0.997480	−	0.0709444i	$-0.977399\pi$
$223$		−	472.504i		−	0.141889i	0.997480	−	0.0709444i	$-0.0226013\pi$
$224$		0			0
$225$	−135.000	+	1207.48i	−0.0400000	+	0.357771i
$226$		0			0
$227$	−505.759			−0.147878			−0.0739392	−	0.997263i	$-0.523557\pi$
$227$	−505.759			−0.147878			−0.0739392	+	0.997263i	$0.523557\pi$
$228$		0			0
$229$	−4094.00			−1.18139			−0.590697	−	0.806894i	$-0.701147\pi$
$229$	−4094.00			−1.18139			−0.590697	+	0.806894i	$0.701147\pi$
$230$		0			0
$231$	−1039.23	+	929.516i	−0.296001	+	0.264752i
$232$		0			0
$233$			5277.12i			1.48376i	0.670534	+	0.741879i	$0.266065\pi$
$233$			5277.12i			1.48376i	−0.670534	+	0.741879i	$0.733935\pi$
$234$		0			0
$235$			1735.10i			0.481639i
$236$		0			0
$237$	330.000	−	295.161i	0.0904464	−	0.0808977i
$238$		0			0
$239$	−5681.13			−1.53758			−0.768790	−	0.639502i	$-0.779141\pi$
$239$	−5681.13			−1.53758			−0.768790	+	0.639502i	$0.779141\pi$
$240$		0			0
$241$	−1198.00			−0.320207			−0.160104	−	0.987100i	$-0.551183\pi$
$241$	−1198.00			−0.320207			−0.160104	+	0.987100i	$0.551183\pi$
$242$		0			0
$243$	1839.44	+	3311.40i	0.485597	+	0.874183i
$244$		0			0
$245$			2531.23i			0.660058i
$246$		0			0
$247$		−	697.137i		−	0.179586i
$248$		0			0
$249$	4344.00	+	4856.74i	1.10558	+	1.23608i
$250$		0			0
$251$	−4260.84			−1.07148			−0.535741	−	0.844382i	$-0.679968\pi$
$251$	−4260.84			−1.07148			−0.535741	+	0.844382i	$0.679968\pi$
$252$		0			0
$253$	3360.00			0.834946
$254$		0			0
$255$	−1108.51	−	1239.35i	−0.272226	−	0.304358i
$256$		0			0
$257$		−	3148.38i		−	0.764166i	−0.924128	−	0.382083i	$-0.875207\pi$
$257$		−	3148.38i		−	0.764166i	0.924128	−	0.382083i	$-0.124793\pi$
$258$		0			0
$259$		−	1006.98i		−	0.241585i
$260$		0			0
$261$	4080.00	+	456.158i	0.967608	+	0.108182i
$262$		0			0
$263$	−4253.92			−0.997368			−0.498684	−	0.866784i	$-0.666183\pi$
$263$	−4253.92			−0.997368			−0.498684	+	0.866784i	$0.666183\pi$
$264$		0			0
$265$	−4880.00			−1.13123
$266$		0			0
$267$	−3256.26	+	2912.48i	−0.746366	+	0.667570i
$268$		0			0
$269$		−	44.7214i		−	0.0101365i	−0.999987	−	0.00506823i	$-0.998387\pi$
$269$		−	44.7214i		−	0.0101365i	0.999987	−	0.00506823i	$-0.00161328\pi$
$270$		0			0
$271$			8760.69i			1.96374i	0.189552	+	0.981871i	$0.439296\pi$
$271$			8760.69i			1.96374i	−0.189552	+	0.981871i	$0.560704\pi$
$272$		0			0
$273$	−300.000	+	268.328i	−0.0665085	+	0.0594870i
$274$		0			0
$275$	1558.85			0.341825
$276$		0			0
$277$	−6350.00			−1.37738			−0.688690	−	0.725055i	$-0.741814\pi$
$277$	−6350.00			−1.37738			−0.688690	+	0.725055i	$0.741814\pi$
$278$		0			0
$279$	6027.54	+	673.899i	1.29340	+	0.144607i
$280$		0			0
$281$		−	5563.34i		−	1.18107i	−0.807012	−	0.590535i	$-0.798917\pi$
$281$		−	5563.34i		−	1.18107i	0.807012	−	0.590535i	$-0.201083\pi$
$282$		0			0
$283$			6777.72i			1.42365i	0.702356	+	0.711826i	$0.252132\pi$
$283$			6777.72i			1.42365i	−0.702356	+	0.711826i	$0.747868\pi$
$284$		0			0
$285$	−2160.00	−	2414.95i	−0.448938	−	0.501928i
$286$		0			0
$287$	969.948			0.199492
$288$		0			0
$289$	3633.00			0.739467
$290$		0			0
$291$	−2667.36	−	2982.20i	−0.537331	−	0.600754i
$292$		0			0
$293$			652.932i			0.130187i	0.997879	+	0.0650933i	$0.0207345\pi$
$293$			652.932i			0.130187i	−0.997879	+	0.0650933i	$0.979265\pi$
$294$		0			0
$295$			1549.19i			0.305754i
$296$		0			0
$297$	3960.00	−	2817.45i	0.773678	−	0.550454i
$298$		0			0
$299$	969.948			0.187604
$300$		0			0
$301$	−1740.00			−0.333196
$302$		0			0
$303$	−5785.05	+	5174.31i	−1.09684	+	0.981043i
$304$		0			0
$305$			3953.37i			0.742194i
$306$		0			0
$307$		−	1556.94i		−	0.289444i	−0.989472	−	0.144722i	$-0.953771\pi$
$307$		−	1556.94i		−	0.289444i	0.989472	−	0.144722i	$-0.0462287\pi$
$308$		0			0
$309$	5250.00	−	4695.74i	0.966544	−	0.864503i
$310$		0			0
$311$	3256.26			0.593715			0.296857	−	0.954922i	$-0.404061\pi$
$311$	3256.26			0.593715			0.296857	+	0.954922i	$0.404061\pi$
$312$		0			0
$313$	−7030.00			−1.26952			−0.634759	−	0.772710i	$-0.718901\pi$
$313$	−7030.00			−1.26952			−0.634759	+	0.772710i	$0.718901\pi$
$314$		0			0
$315$	−207.846	+	1859.03i	−0.0371771	+	0.332522i
$316$		0			0
$317$			491.935i			0.0871603i	0.999050	+	0.0435802i	$0.0138764\pi$
$317$			491.935i			0.0871603i	−0.999050	+	0.0435802i	$0.986124\pi$
$318$		0			0
$319$		−	5267.26i		−	0.924482i
$320$		0			0
$321$	−2232.00	−	2495.45i	−0.388094	−	0.433902i
$322$		0			0
$323$	−2494.15			−0.429654
$324$		0			0
$325$	450.000			0.0768046
$326$		0			0
$327$	−3692.73	−	4128.60i	−0.624491	−	0.698202i
$328$		0			0
$329$		−	1502.64i		−	0.251803i
$330$		0			0
$331$			4237.04i			0.703592i	0.936077	+	0.351796i	$0.114429\pi$
$331$			4237.04i			0.703592i	−0.936077	+	0.351796i	$0.885571\pi$
$332$		0			0
$333$	−390.000	+	3488.27i	−0.0641798	+	0.574041i
$334$		0			0
$335$	−6581.79			−1.07344
$336$		0			0
$337$	1490.00			0.240847			0.120424	−	0.992723i	$-0.461575\pi$
$337$	1490.00			0.240847			0.120424	+	0.992723i	$0.461575\pi$
$338$		0			0
$339$	4018.36	−	3594.13i	0.643797	−	0.575830i
$340$		0			0
$341$		−	7781.52i		−	1.23576i
$342$		0			0
$343$		−	4848.98i		−	0.763324i
$344$		0			0
$345$	3360.00	−	3005.28i	0.524337	−	0.468981i
$346$		0			0
$347$	−1988.39			−0.307616			−0.153808	−	0.988101i	$-0.549154\pi$
$347$	−1988.39			−0.307616			−0.153808	+	0.988101i	$0.549154\pi$
$348$		0			0
$349$	2074.00			0.318105			0.159053	−	0.987270i	$-0.449156\pi$
$349$	2074.00			0.318105			0.159053	+	0.987270i	$0.449156\pi$
$350$		0			0
$351$	1143.15	−	813.327i	0.173838	−	0.123681i
$352$		0			0
$353$			8658.06i			1.30544i	0.757597	+	0.652722i	$0.226373\pi$
$353$			8658.06i			1.30544i	−0.757597	+	0.652722i	$0.773627\pi$
$354$		0			0
$355$		−	9295.16i		−	1.38968i
$356$		0			0
$357$	960.000	+	1073.31i	0.142321	+	0.159120i
$358$		0			0
$359$	8106.00			1.19169			0.595847	−	0.803098i	$-0.296816\pi$
$359$	8106.00			1.19169			0.595847	+	0.803098i	$0.296816\pi$
$360$		0			0
$361$	1999.00			0.291442
$362$		0			0
$363$	453.797	+	507.361i	0.0656148	+	0.0733596i
$364$		0			0
$365$			3667.15i			0.525884i
$366$		0			0
$367$			7893.14i			1.12267i	0.827590	+	0.561333i	$0.189711\pi$
$367$			7893.14i			1.12267i	−0.827590	+	0.561333i	$0.810289\pi$
$368$		0			0
$369$	−3360.00	−	375.659i	−0.474023	−	0.0529974i
$370$		0			0
$371$	4226.20			0.591411
$372$		0			0
$373$	−4910.00			−0.681582			−0.340791	−	0.940139i	$-0.610695\pi$
$373$	−4910.00			−0.681582			−0.340791	+	0.940139i	$0.610695\pi$
$374$		0			0
$375$	5888.97	−	5267.26i	0.810947	−	0.725333i
$376$		0			0
$377$		−	1520.53i		−	0.207722i
$378$		0			0
$379$		−	3137.12i		−	0.425179i	−0.977142	−	0.212590i	$-0.931810\pi$
$379$		−	3137.12i		−	0.425179i	0.977142	−	0.212590i	$-0.0681897\pi$
$380$		0			0
$381$	−7110.00	+	6359.38i	−0.956053	+	0.855120i
$382$		0			0
$383$	6207.67			0.828191			0.414095	−	0.910233i	$-0.364098\pi$
$383$	6207.67			0.828191			0.414095	+	0.910233i	$0.364098\pi$
$384$		0			0
$385$	2400.00			0.317702
$386$		0			0
$387$	6027.54	+	673.899i	0.791723	+	0.0885174i
$388$		0			0
$389$			9454.10i			1.23224i	0.787652	+	0.616120i	$0.211297\pi$
$389$			9454.10i			1.23224i	−0.787652	+	0.616120i	$0.788703\pi$
$390$		0			0
$391$		−	3470.19i		−	0.448837i
$392$		0			0
$393$	1560.00	+	1744.13i	0.200233	+	0.223867i
$394$		0			0
$395$	−762.102			−0.0970773
$396$		0			0
$397$	10570.0			1.33625			0.668127	−	0.744047i	$-0.267096\pi$
$397$	10570.0			1.33625			0.668127	+	0.744047i	$0.267096\pi$
$398$		0			0
$399$	1870.61	+	2091.41i	0.234706	+	0.262410i
$400$		0			0
$401$			1681.52i			0.209405i	0.994504	+	0.104702i	$0.0333890\pi$
$401$			1681.52i			0.209405i	−0.994504	+	0.104702i	$0.966611\pi$
$402$		0			0
$403$		−	2246.33i		−	0.277662i
$404$		0			0
$405$	1440.00	−	6359.38i	0.176677	−	0.780247i
$406$		0			0
$407$	4503.33			0.548457
$408$		0			0
$409$	−3574.00			−0.432085			−0.216043	−	0.976384i	$-0.569315\pi$
$409$	−3574.00			−0.432085			−0.216043	+	0.976384i	$0.569315\pi$
$410$		0			0
$411$	−346.410	+	309.839i	−0.0415746	+	0.0371854i
$412$		0			0
$413$		−	1341.64i		−	0.159849i
$414$		0			0
$415$		−	11216.2i		−	1.32670i
$416$		0			0
$417$	−7590.00	+	6788.70i	−0.891328	+	0.797228i
$418$		0			0
$419$	15346.0			1.78926			0.894630	−	0.446808i	$-0.147439\pi$
$419$	15346.0			1.78926			0.894630	+	0.446808i	$0.147439\pi$
$420$		0			0
$421$	−3518.00			−0.407261			−0.203630	−	0.979048i	$-0.565274\pi$
$421$	−3518.00			−0.407261			−0.203630	+	0.979048i	$0.565274\pi$
$422$		0			0
$423$	−581.969	+	5205.29i	−0.0668943	+	0.598321i
$424$		0			0
$425$		−	1609.97i		−	0.183753i
$426$		0			0
$427$		−	3423.72i		−	0.388022i
$428$		0			0
$429$	−1200.00	−	1341.64i	−0.135050	−	0.150991i
$430$		0			0
$431$	−12886.5			−1.44018			−0.720091	−	0.693879i	$-0.755900\pi$
$431$	−12886.5			−1.44018			−0.720091	+	0.693879i	$0.755900\pi$
$432$		0			0
$433$	14450.0			1.60375			0.801874	−	0.597493i	$-0.203837\pi$
$433$	14450.0			1.60375			0.801874	+	0.597493i	$0.203837\pi$
$434$		0			0
$435$	−4711.18	−	5267.26i	−0.519273	−	0.580565i
$436$		0			0
$437$		−	6761.87i		−	0.740192i
$438$		0			0
$439$		−	15065.9i		−	1.63794i	−0.573835	−	0.818971i	$-0.694545\pi$
$439$		−	15065.9i		−	1.63794i	0.573835	−	0.818971i	$-0.305455\pi$
$440$		0			0
$441$	−849.000	+	7593.69i	−0.0916748	+	0.819964i
$442$		0			0
$443$	−3041.48			−0.326197			−0.163098	−	0.986610i	$-0.552149\pi$
$443$	−3041.48			−0.326197			−0.163098	+	0.986610i	$0.552149\pi$
$444$		0			0
$445$	7520.00			0.801084
$446$		0			0
$447$	6270.02	−	5608.08i	0.663450	−	0.593407i
$448$		0			0
$449$		−	14310.8i		−	1.50416i	−0.659069	−	0.752082i	$-0.729049\pi$
$449$		−	14310.8i		−	1.50416i	0.659069	−	0.752082i	$-0.270951\pi$
$450$		0			0
$451$			4337.74i			0.452896i
$452$		0			0
$453$	−2190.00	+	1958.80i	−0.227142	+	0.203162i
$454$		0			0
$455$	692.820			0.0713844
$456$		0			0
$457$	−3430.00			−0.351091			−0.175546	−	0.984471i	$-0.556169\pi$
$457$	−3430.00			−0.351091			−0.175546	+	0.984471i	$0.556169\pi$
$458$		0			0
$459$	−2909.85	−	4089.87i	−0.295904	−	0.415902i
$460$		0			0
$461$		−	3908.65i		−	0.394889i	−0.980314	−	0.197445i	$-0.936736\pi$
$461$		−	3908.65i		−	0.394889i	0.980314	−	0.197445i	$-0.0632642\pi$
$462$		0			0
$463$			18179.8i			1.82481i	0.409291	+	0.912404i	$0.365776\pi$
$463$			18179.8i			1.82481i	−0.409291	+	0.912404i	$0.634224\pi$
$464$		0			0
$465$	−6960.00	−	7781.52i	−0.694112	−	0.776041i
$466$		0			0
$467$	1849.83			0.183298			0.0916488	−	0.995791i	$-0.470786\pi$
$467$	1849.83			0.183298			0.0916488	+	0.995791i	$0.470786\pi$
$468$		0			0
$469$	5700.00			0.561197
$470$		0			0
$471$	−2528.79	−	2827.28i	−0.247390	−	0.276590i
$472$		0			0
$473$		−	7781.52i		−	0.756437i
$474$		0			0
$475$		−	3137.12i		−	0.303033i
$476$		0			0
$477$	−14640.0	−	1636.80i	−1.40528	−	0.157115i
$478$		0			0
$479$	15242.0			1.45392			0.726959	−	0.686681i	$-0.240933\pi$
$479$	15242.0			1.45392			0.726959	+	0.686681i	$0.240933\pi$
$480$		0			0
$481$	1300.00			0.123233
$482$		0			0
$483$	−2909.85	+	2602.64i	−0.274125	+	0.245185i
$484$		0			0
$485$			6887.09i			0.644797i
$486$		0			0
$487$			9783.16i			0.910302i	0.890414	+	0.455151i	$0.150415\pi$
$487$			9783.16i			0.910302i	−0.890414	+	0.455151i	$0.849585\pi$
$488$		0			0
$489$	−990.000	+	885.483i	−0.0915529	+	0.0818874i
$490$		0			0
$491$	−6893.56			−0.633609			−0.316805	−	0.948491i	$-0.602610\pi$
$491$	−6893.56			−0.633609			−0.316805	+	0.948491i	$0.602610\pi$
$492$		0			0
$493$	−5440.00			−0.496968
$494$		0			0
$495$	−8313.84	−	929.516i	−0.754908	−	0.0844013i
$496$		0			0
$497$			8049.84i			0.726529i
$498$		0			0
$499$		−	1309.07i		−	0.117439i	−0.998275	−	0.0587194i	$-0.981298\pi$
$499$		−	1309.07i		−	0.117439i	0.998275	−	0.0587194i	$-0.0187017\pi$
$500$		0			0
$501$	−48.0000	−	53.6656i	−0.00428040	−	0.00478564i
$502$		0			0
$503$	−7939.72			−0.703806			−0.351903	−	0.936036i	$-0.614465\pi$
$503$	−7939.72			−0.703806			−0.351903	+	0.936036i	$0.614465\pi$
$504$		0			0
$505$	13360.0			1.17725
$506$		0			0
$507$	7264.22	+	8121.65i	0.636322	+	0.711430i
$508$		0			0
$509$		−	14534.4i		−	1.26567i	−0.774285	−	0.632837i	$-0.781890\pi$
$509$		−	14534.4i		−	1.26567i	0.774285	−	0.632837i	$-0.218110\pi$
$510$		0			0
$511$		−	3175.85i		−	0.274934i
$512$		0			0
$513$	−5670.00	−	7969.35i	−0.487986	−	0.685878i
$514$		0			0
$515$	−12124.4			−1.03740
$516$		0			0
$517$	6720.00			0.571654
$518$		0			0
$519$	−4330.13	+	3872.98i	−0.366226	+	0.327563i
$520$		0			0
$521$			9355.71i			0.786720i	0.919385	+	0.393360i	$0.128687\pi$
$521$			9355.71i			0.786720i	−0.919385	+	0.393360i	$0.871313\pi$
$522$		0			0
$523$			10062.0i			0.841264i	0.907231	+	0.420632i	$0.138192\pi$
$523$			10062.0i			0.841264i	−0.907231	+	0.420632i	$0.861808\pi$
$524$		0			0
$525$	−1350.00	+	1207.48i	−0.112226	+	0.100378i
$526$		0			0
$527$	−8036.72			−0.664298
$528$		0			0
$529$	−2759.00			−0.226761
$530$		0			0
$531$	−519.615	+	4647.58i	−0.0424659	+	0.379826i
$532$		0			0
$533$			1252.20i			0.101761i
$534$		0			0
$535$			5763.00i			0.465712i
$536$		0			0
$537$	−4680.00	−	5232.40i	−0.376084	−	0.420474i
$538$		0			0
$539$	9803.41			0.783419
$540$		0			0
$541$	23962.0			1.90426			0.952132	−	0.305687i	$-0.0988862\pi$
$541$	23962.0			1.90426			0.952132	+	0.305687i	$0.0988862\pi$
$542$		0			0
$543$	4371.70	+	4887.70i	0.345502	+	0.386283i
$544$		0			0
$545$			9534.59i			0.749389i
$546$		0			0
$547$			15112.4i			1.18128i	0.806936	+	0.590639i	$0.201124\pi$
$547$			15112.4i			1.18128i	−0.806936	+	0.590639i	$0.798876\pi$
$548$		0			0
$549$	−1326.00	+	11860.1i	−0.103083	+	0.921998i
$550$		0			0
$551$	−10600.2			−0.819567
$552$		0			0
$553$	660.000			0.0507524
$554$		0			0
$555$	4503.33	−	4027.90i	0.344425	−	0.308063i
$556$		0			0
$557$			16055.0i			1.22131i	0.791896	+	0.610656i	$0.209094\pi$
$557$			16055.0i			1.22131i	−0.791896	+	0.610656i	$0.790906\pi$
$558$		0			0
$559$		−	2246.33i		−	0.169964i
$560$		0			0
$561$	−4800.00	+	4293.25i	−0.361241	+	0.323104i
$562$		0			0
$563$	−25142.4			−1.88211			−0.941055	−	0.338254i	$-0.890164\pi$
$563$	−25142.4			−1.88211			−0.941055	+	0.338254i	$0.890164\pi$
$564$		0			0
$565$	−9280.00			−0.690996
$566$		0			0
$567$	−1247.08	+	5507.38i	−0.0923674	+	0.407916i
$568$		0			0
$569$			23416.1i			1.72523i	0.505864	+	0.862613i	$0.331174\pi$
$569$			23416.1i			1.72523i	−0.505864	+	0.862613i	$0.668826\pi$
$570$		0			0
$571$			4918.69i			0.360492i	0.983622	+	0.180246i	$0.0576893\pi$
$571$			4918.69i			0.360492i	−0.983622	+	0.180246i	$0.942311\pi$
$572$		0			0
$573$	−9600.00	−	10733.1i	−0.699905	−	0.782518i
$574$		0			0
$575$	4364.77			0.316562
$576$		0			0
$577$	19490.0			1.40620			0.703102	−	0.711089i	$-0.251798\pi$
$577$	19490.0			1.40620			0.703102	+	0.711089i	$0.251798\pi$
$578$		0			0
$579$	658.179	+	735.867i	0.0472418	+	0.0528179i
$580$		0			0
$581$			9713.48i			0.693602i
$582$		0			0
$583$			18900.2i			1.34265i
$584$		0			0
$585$	−2400.00	−	268.328i	−0.169620	−	0.0189641i
$586$		0			0
$587$	1364.86			0.0959687			0.0479844	−	0.998848i	$-0.484720\pi$
$587$	1364.86			0.0959687			0.0479844	+	0.998848i	$0.484720\pi$
$588$		0			0
$589$	−15660.0			−1.09552
$590$		0			0
$591$	−8279.20	+	7405.14i	−0.576245	+	0.515409i
$592$		0			0
$593$		−	25795.3i		−	1.78632i	−0.449743	−	0.893158i	$-0.648485\pi$
$593$		−	25795.3i		−	1.78632i	0.449743	−	0.893158i	$-0.351515\pi$
$594$		0			0
$595$		−	2478.71i		−	0.170785i
$596$		0			0
$597$	−990.000	+	885.483i	−0.0678694	+	0.0607042i
$598$		0			0
$599$	2424.87			0.165405			0.0827025	−	0.996574i	$-0.473645\pi$
$599$	2424.87			0.165405			0.0827025	+	0.996574i	$0.473645\pi$
$600$		0			0
$601$	−8758.00			−0.594420			−0.297210	−	0.954812i	$-0.596056\pi$
$601$	−8758.00			−0.594420			−0.297210	+	0.954812i	$0.596056\pi$
$602$		0			0
$603$	−19745.4	−	2207.60i	−1.33349	−	0.149089i
$604$		0			0
$605$		−	1171.70i		−	0.0787378i
$606$		0			0
$607$		−	19558.6i		−	1.30784i	−0.756565	−	0.653919i	$-0.773124\pi$
$607$		−	19558.6i		−	1.30784i	0.756565	−	0.653919i	$-0.226876\pi$
$608$		0			0
$609$	4080.00	+	4561.58i	0.271478	+	0.303521i
$610$		0			0
$611$	1939.90			0.128445
$612$		0			0
$613$	16450.0			1.08386			0.541932	−	0.840422i	$-0.317693\pi$
$613$	16450.0			1.08386			0.541932	+	0.840422i	$0.317693\pi$
$614$		0			0
$615$	3879.79	+	4337.74i	0.254388	+	0.284414i
$616$		0			0
$617$		−	8461.28i		−	0.552088i	−0.961145	−	0.276044i	$-0.910976\pi$
$617$		−	8461.28i		−	0.552088i	0.961145	−	0.276044i	$-0.0890236\pi$
$618$		0			0
$619$		−	19930.4i		−	1.29413i	−0.762433	−	0.647067i	$-0.775995\pi$
$619$		−	19930.4i		−	1.29413i	0.762433	−	0.647067i	$-0.224005\pi$
$620$		0			0
$621$	11088.0	−	7888.85i	0.716499	−	0.509772i
$622$		0			0
$623$	−6512.51			−0.418809
$624$		0			0
$625$	−7975.00			−0.510400
$626$		0			0
$627$	−9353.07	+	8365.64i	−0.595735	+	0.532842i
$628$		0			0
$629$		−	4651.02i		−	0.294830i
$630$		0			0
$631$			12199.9i			0.769683i	0.922983	+	0.384842i	$0.125744\pi$
$631$			12199.9i			0.769683i	−0.922983	+	0.384842i	$0.874256\pi$
$632$		0			0
$633$	2130.00	−	1905.13i	0.133744	−	0.119624i
$634$		0			0
$635$	16419.8			1.02614
$636$		0			0
$637$	2830.00			0.176026
$638$		0			0
$639$	3117.69	−	27885.5i	0.193011	−	1.72634i
$640$		0			0
$641$		−	7012.31i		−	0.432090i	−0.976383	−	0.216045i	$-0.930684\pi$
$641$		−	7012.31i		−	0.432090i	0.976383	−	0.216045i	$-0.0693158\pi$
$642$		0			0
$643$			15979.9i			0.980073i	0.871702	+	0.490036i	$0.163017\pi$
$643$			15979.9i			0.980073i	−0.871702	+	0.490036i	$0.836983\pi$
$644$		0			0
$645$	−6960.00	−	7781.52i	−0.424883	−	0.475034i
$646$		0			0
$647$	−17999.5			−1.09371			−0.546856	−	0.837226i	$-0.684176\pi$
$647$	−17999.5			−1.09371			−0.546856	+	0.837226i	$0.684176\pi$
$648$		0			0
$649$	6000.00			0.362898
$650$		0			0
$651$	6027.54	+	6738.99i	0.362884	+	0.405717i
$652$		0			0
$653$		−	5196.62i		−	0.311423i	−0.987803	−	0.155712i	$-0.950233\pi$
$653$		−	5196.62i		−	0.311423i	0.987803	−	0.155712i	$-0.0497671\pi$
$654$		0			0
$655$		−	4027.90i		−	0.240280i
$656$		0			0
$657$	−1230.00	+	11001.5i	−0.0730394	+	0.653284i
$658$		0			0
$659$	6062.18			0.358344			0.179172	−	0.983818i	$-0.442658\pi$
$659$	6062.18			0.358344			0.179172	+	0.983818i	$0.442658\pi$
$660$		0			0
$661$	−9422.00			−0.554423			−0.277211	−	0.960809i	$-0.589410\pi$
$661$	−9422.00			−0.554423			−0.277211	+	0.960809i	$0.589410\pi$
$662$		0			0
$663$	−1385.64	+	1239.35i	−0.0811672	+	0.0725981i
$664$		0			0
$665$		−	4829.91i		−	0.281648i
$666$		0			0
$667$		−	14748.3i		−	0.856158i
$668$		0			0
$669$	−1830.00	+	1636.80i	−0.105758	+	0.0945925i
$670$		0			0
$671$	15311.3			0.880905
$672$		0			0
$673$	−17470.0			−1.00062			−0.500311	−	0.865846i	$-0.666781\pi$
$673$	−17470.0			−1.00062			−0.500311	+	0.865846i	$0.666781\pi$
$674$		0			0
$675$	5144.19	−	3659.97i	0.293333	−	0.208700i
$676$		0			0
$677$		−	20813.3i		−	1.18157i	−0.806830	−	0.590784i	$-0.798819\pi$
$677$		−	20813.3i		−	1.18157i	0.806830	−	0.590784i	$-0.201181\pi$
$678$		0			0
$679$		−	5964.39i		−	0.337102i
$680$		0			0
$681$	1752.00	+	1958.80i	0.0985856	+	0.110222i
$682$		0			0
$683$	12616.3			0.706805			0.353402	−	0.935471i	$-0.385025\pi$
$683$	12616.3			0.706805			0.353402	+	0.935471i	$0.385025\pi$
$684$		0			0
$685$	800.000			0.0446225
$686$		0			0
$687$	14182.0	+	15856.0i	0.787596	+	0.880559i
$688$		0			0
$689$			5456.01i			0.301680i
$690$		0			0
$691$			3028.67i			0.166738i	0.996519	+	0.0833691i	$0.0265681\pi$
$691$			3028.67i			0.166738i	−0.996519	+	0.0833691i	$0.973432\pi$
$692$		0			0
$693$	7200.00	+	804.984i	0.394669	+	0.0441253i
$694$		0			0
$695$	17528.4			0.956674
$696$		0			0
$697$	4480.00			0.243461
$698$		0			0
$699$	20438.2	−	18280.5i	1.10593	−	0.989172i
$700$		0			0
$701$			17664.9i			0.951777i	0.879506	+	0.475888i	$0.157873\pi$
$701$			17664.9i			0.951777i	−0.879506	+	0.475888i	$0.842127\pi$
$702$		0			0
$703$		−	9062.78i		−	0.486215i
$704$		0			0
$705$	6720.00	−	6010.55i	0.358993	−	0.321093i
$706$		0			0
$707$	−11570.1			−0.615472
$708$		0			0
$709$	−14174.0			−0.750798			−0.375399	−	0.926863i	$-0.622494\pi$
$709$	−14174.0			−0.750798			−0.375399	+	0.926863i	$0.622494\pi$
$710$		0			0
$711$	−2286.31	−	255.617i	−0.120595	−	0.0134830i
$712$		0			0
$713$		−	21788.2i		−	1.14443i
$714$		0			0
$715$			3098.39i			0.162060i
$716$		0			0
$717$	19680.0	+	22002.9i	1.02505	+	1.14604i
$718$		0			0
$719$	32839.7			1.70336			0.851678	−	0.524065i	$-0.175585\pi$
$719$	32839.7			1.70336			0.851678	+	0.524065i	$0.175585\pi$
$720$		0			0
$721$	10500.0			0.542358
$722$		0			0
$723$	4149.99	+	4639.83i	0.213472	+	0.238668i
$724$		0			0
$725$		−	6842.37i		−	0.350509i
$726$		0			0
$727$		−	8001.58i		−	0.408201i	−0.978950	−	0.204101i	$-0.934573\pi$
$727$		−	8001.58i		−	0.408201i	0.978950	−	0.204101i	$-0.0654270\pi$
$728$		0			0
$729$	6453.00	−	18595.1i	0.327846	−	0.944731i
$730$		0			0
$731$	−8036.72			−0.406633
$732$		0			0
$733$	−11750.0			−0.592082			−0.296041	−	0.955175i	$-0.595667\pi$
$733$	−11750.0			−0.592082			−0.296041	+	0.955175i	$0.595667\pi$
$734$		0			0
$735$	9803.41	−	8768.43i	0.491978	−	0.440039i
$736$		0			0
$737$			25491.2i			1.27406i
$738$		0			0
$739$		−	19961.4i		−	0.993627i	−0.867857	−	0.496814i	$-0.834503\pi$
$739$		−	19961.4i		−	0.993627i	0.867857	−	0.496814i	$-0.165497\pi$
$740$		0			0
$741$	−2700.00	+	2414.95i	−0.133856	+	0.119724i
$742$		0			0
$743$	−25592.8			−1.26367			−0.631836	−	0.775102i	$-0.717698\pi$
$743$	−25592.8			−1.26367			−0.631836	+	0.775102i	$0.717698\pi$
$744$		0			0
$745$	−14480.0			−0.712089
$746$		0			0
$747$	3762.01	−	33648.5i	0.184264	−	1.64810i
$748$		0			0
$749$		−	4990.90i		−	0.243476i
$750$		0			0
$751$		−	5244.02i		−	0.254803i	−0.991851	−	0.127401i	$-0.959336\pi$
$751$		−	5244.02i		−	0.254803i	0.991851	−	0.127401i	$-0.0406637\pi$
$752$		0			0
$753$	14760.0	+	16502.2i	0.714322	+	0.798636i
$754$		0			0
$755$	5057.59			0.243794
$756$		0			0
$757$	14290.0			0.686102			0.343051	−	0.939317i	$-0.388540\pi$
$757$	14290.0			0.686102			0.343051	+	0.939317i	$0.388540\pi$
$758$		0			0
$759$	−11639.4	−	13013.2i	−0.556631	−	0.622332i
$760$		0			0
$761$			16976.2i			0.808657i	0.914614	+	0.404328i	$0.132495\pi$
$761$			16976.2i			0.808657i	−0.914614	+	0.404328i	$0.867505\pi$
$762$		0			0
$763$		−	8257.20i		−	0.391783i
$764$		0			0
$765$	−960.000	+	8586.50i	−0.0453711	+	0.405811i
$766$		0			0
$767$	1732.05			0.0815394
$768$		0			0
$769$	−29566.0			−1.38645			−0.693223	−	0.720723i	$-0.743810\pi$
$769$	−29566.0			−1.38645			−0.693223	+	0.720723i	$0.743810\pi$
$770$		0			0
$771$	−12193.6	+	10906.3i	−0.569576	+	0.509444i
$772$		0			0
$773$		−	21457.3i		−	0.998403i	−0.866486	−	0.499202i	$-0.833627\pi$
$773$		−	21457.3i		−	0.998403i	0.866486	−	0.499202i	$-0.166373\pi$
$774$		0			0
$775$		−	10108.5i		−	0.468526i
$776$		0			0
$777$	−3900.00	+	3488.27i	−0.180067	+	0.161056i
$778$		0			0
$779$	8729.54			0.401499
$780$		0			0
$781$	−36000.0			−1.64940
$782$		0			0
$783$	−12366.8	−	17381.9i	−0.564438	−	0.793334i
$784$		0			0
$785$			6529.32i			0.296868i
$786$		0			0
$787$			3896.22i			0.176474i	0.996099	+	0.0882372i	$0.0281233\pi$
$787$			3896.22i			0.176474i	−0.996099	+	0.0882372i	$0.971877\pi$
$788$		0			0
$789$	14736.0	+	16475.3i	0.664912	+	0.743394i
$790$		0			0
$791$	8036.72			0.361255
$792$		0			0
$793$	4420.00			0.197930
$794$		0			0
$795$	16904.8	+	18900.2i	0.754154	+	0.843169i
$796$		0			0
$797$			30759.4i			1.36707i	0.729919	+	0.683533i	$0.239558\pi$
$797$			30759.4i			1.36707i	−0.729919	+	0.683533i	$0.760442\pi$
$798$		0			0
$799$		−	6940.39i		−	0.307301i
$800$		0			0
$801$	22560.0	+	2522.28i	0.995154	+	0.111262i
$802$		0			0
$803$	14202.8			0.624168
$804$		0			0
$805$	6720.00			0.294222
$806$		0			0
$807$	−173.205	+	154.919i	−0.00755528	+	0.00675764i
$808$		0			0
$809$			10429.0i			0.453232i	0.973984	+	0.226616i	$0.0727663\pi$
$809$			10429.0i			0.453232i	−0.973984	+	0.226616i	$0.927234\pi$
$810$		0			0
$811$		−	8156.50i		−	0.353161i	−0.984286	−	0.176580i	$-0.943496\pi$
$811$		−	8156.50i		−	0.353161i	0.984286	−	0.176580i	$-0.0565036\pi$
$812$		0			0
$813$	33930.0	−	30347.9i	1.46369	−	1.30916i
$814$		0			0
$815$	2286.31			0.0982648
$816$		0			0
$817$	−15660.0			−0.670592
$818$		0			0
$819$	2078.46	+	232.379i	0.0886780	+	0.00991450i
$820$		0			0
$821$			35750.3i			1.51972i	0.650085	+	0.759861i	$0.274733\pi$
$821$			35750.3i			1.51972i	−0.650085	+	0.759861i	$0.725267\pi$
$822$		0			0
$823$			20875.4i			0.884168i	0.896974	+	0.442084i	$0.145761\pi$
$823$			20875.4i			0.884168i	−0.896974	+	0.442084i	$0.854239\pi$
$824$		0			0
$825$	−5400.00	−	6037.38i	−0.227883	−	0.254781i
$826$		0			0
$827$	−12907.2			−0.542719			−0.271360	−	0.962478i	$-0.587473\pi$
$827$	−12907.2			−0.542719			−0.271360	+	0.962478i	$0.587473\pi$
$828$		0			0
$829$	14074.0			0.589638			0.294819	−	0.955553i	$-0.404741\pi$
$829$	14074.0			0.589638			0.294819	+	0.955553i	$0.404741\pi$
$830$		0			0
$831$	21997.0	+	24593.4i	0.918254	+	1.02664i
$832$		0			0
$833$		−	10124.9i		−	0.421138i
$834$		0			0
$835$			123.935i			0.00513648i
$836$		0			0
$837$	−18270.0	−	25679.0i	−0.754485	−	1.06045i
$838$		0			0
$839$	16697.0			0.687060			0.343530	−	0.939142i	$-0.388377\pi$
$839$	16697.0			0.687060			0.343530	+	0.939142i	$0.388377\pi$
$840$		0			0
$841$	1269.00			0.0520317
$842$		0			0
$843$	−21546.7	+	19272.0i	−0.880318	+	0.787380i
$844$		0			0
$845$		−	18756.1i		−	0.763587i
$846$		0			0
$847$			1014.72i			0.0411644i
$848$		0			0
$849$	26250.0	−	23478.7i	1.06113	−	0.949102i
$850$		0			0
$851$	12609.3			0.507923
$852$		0			0
$853$	−23630.0			−0.948506			−0.474253	−	0.880389i	$-0.657282\pi$
$853$	−23630.0			−0.948506			−0.474253	+	0.880389i	$0.657282\pi$
$854$		0			0
$855$	−1870.61	+	16731.3i	−0.0748230	+	0.669237i
$856$		0			0
$857$		−	31322.8i		−	1.24850i	−0.781223	−	0.624252i	$-0.785404\pi$
$857$		−	31322.8i		−	1.24850i	0.781223	−	0.624252i	$-0.214596\pi$
$858$		0			0
$859$		−	13671.6i		−	0.543038i	−0.962433	−	0.271519i	$-0.912474\pi$
$859$		−	13671.6i		−	0.543038i	0.962433	−	0.271519i	$-0.0875261\pi$
$860$		0			0
$861$	−3360.00	−	3756.59i	−0.132995	−	0.148693i
$862$		0			0
$863$	−25107.8			−0.990359			−0.495179	−	0.868791i	$-0.664898\pi$
$863$	−25107.8			−0.990359			−0.495179	+	0.868791i	$0.664898\pi$
$864$		0			0
$865$	10000.0			0.393075
$866$		0			0
$867$	−12585.1	−	14070.5i	−0.492978	−	0.551166i
$868$		0			0
$869$			2951.61i			0.115220i
$870$		0			0
$871$			7358.67i			0.286267i
$872$		0			0
$873$	−2310.00	+	20661.3i	−0.0895552	+	0.801006i
$874$		0			0
$875$	11777.9			0.455048
$876$		0			0
$877$	−41750.0			−1.60752			−0.803761	−	0.594952i	$-0.797171\pi$
$877$	−41750.0			−1.60752			−0.803761	+	0.594952i	$0.797171\pi$
$878$		0			0
$879$	2528.79	−	2261.82i	0.0970354	−	0.0867911i
$880$		0			0
$881$		−	2397.06i		−	0.0916676i	−0.998949	−	0.0458338i	$-0.985406\pi$
$881$		−	2397.06i		−	0.0916676i	0.998949	−	0.0458338i	$-0.0145945\pi$
$882$		0			0
$883$			43431.6i			1.65526i	0.561277	+	0.827628i	$0.310310\pi$
$883$			43431.6i			1.65526i	−0.561277	+	0.827628i	$0.689690\pi$
$884$		0			0
$885$	6000.00	−	5366.56i	0.227896	−	0.203836i
$886$		0			0
$887$	−6387.80			−0.241805			−0.120903	−	0.992664i	$-0.538579\pi$
$887$	−6387.80			−0.241805			−0.120903	+	0.992664i	$0.538579\pi$
$888$		0			0
$889$	−14220.0			−0.536472
$890$		0			0
$891$	−24629.8	−	5577.10i	−0.926070	−	0.209697i
$892$		0			0
$893$		−	13523.7i		−	0.506780i
$894$		0			0
$895$			12083.7i			0.451300i
$896$		0			0
$897$	−3360.00	−	3756.59i	−0.125069	−	0.139832i
$898$		0			0
$899$	−34156.0			−1.26715
$900$		0			0
$901$	19520.0			0.721760
$902$		0			0
$903$	6027.54	+	6738.99i	0.222131	+	0.248349i
$904$		0			0
$905$		−	11287.7i		−	0.414602i
$906$		0			0
$907$			37327.8i			1.36654i	0.730167	+	0.683269i	$0.239442\pi$
$907$			37327.8i			1.36654i	−0.730167	+	0.683269i	$0.760558\pi$
$908$		0			0
$909$	40080.0	+	4481.08i	1.46245	+	0.163507i
$910$		0			0
$911$	−50853.0			−1.84944			−0.924718	−	0.380654i	$-0.875699\pi$
$911$	−50853.0			−1.84944			−0.924718	+	0.380654i	$0.875699\pi$
$912$		0			0
$913$	−43440.0			−1.57465
$914$		0			0
$915$	15311.3	−	13694.9i	0.553199	−	0.494796i
$916$		0			0
$917$			3488.27i			0.125619i
$918$		0			0
$919$		−	14198.4i		−	0.509642i	−0.966988	−	0.254821i	$-0.917984\pi$
$919$		−	14198.4i		−	0.509642i	0.966988	−	0.254821i	$-0.0820165\pi$
$920$		0			0
$921$	−6030.00	+	5393.40i	−0.215739	+	0.192962i
$922$		0			0
$923$	−10392.3			−0.370603
$924$		0			0
$925$	5850.00			0.207943
$926$		0			0
$927$	−36373.1	−	4066.63i	−1.28873	−	0.144084i
$928$		0			0
$929$		−	10232.2i		−	0.361366i	−0.983541	−	0.180683i	$-0.942169\pi$
$929$		−	10232.2i		−	0.361366i	0.983541	−	0.180683i	$-0.0578308\pi$
$930$		0			0
$931$		−	19729.0i		−	0.694512i
$932$		0			0
$933$	−11280.0	−	12611.4i	−0.395810	−	0.442529i
$934$		0			0
$935$	11085.1			0.387724
$936$		0			0
$937$	37850.0			1.31964			0.659822	−	0.751422i	$-0.270632\pi$
$937$	37850.0			1.31964			0.659822	+	0.751422i	$0.270632\pi$
$938$		0			0
$939$	24352.6	+	27227.1i	0.846345	+	0.946243i
$940$		0			0
$941$		−	13997.8i		−	0.484925i	−0.970161	−	0.242463i	$-0.922045\pi$
$941$		−	13997.8i		−	0.484925i	0.970161	−	0.242463i	$-0.0779552\pi$
$942$		0			0
$943$			12145.7i			0.419425i
$944$		0			0
$945$	7920.00	−	5634.89i	0.272632	−	0.193971i
$946$		0			0
$947$	−7738.80			−0.265552			−0.132776	−	0.991146i	$-0.542389\pi$
$947$	−7738.80			−0.265552			−0.132776	+	0.991146i	$0.542389\pi$
$948$		0			0
$949$	4100.00			0.140244
$950$		0			0
$951$	1905.26	−	1704.11i	0.0649655	−	0.0581069i
$952$		0			0
$953$			44882.4i			1.52558i	0.646644	+	0.762792i	$0.276172\pi$
$953$			44882.4i			1.52558i	−0.646644	+	0.762792i	$0.723828\pi$
$954$		0			0
$955$			24787.1i			0.839886i
$956$		0			0
$957$	−20400.0	+	18246.3i	−0.689068	+	0.616321i
$958$		0			0
$959$	−692.820			−0.0233288
$960$		0			0
$961$	−20669.0			−0.693800
$962$		0			0
$963$	−1932.97	+	17289.0i	−0.0646823	+	0.578536i
$964$		0			0
$965$		−	1699.41i		−	0.0566902i
$966$		0			0
$967$			15856.0i			0.527295i	0.964619	+	0.263648i	$0.0849256\pi$
$967$			15856.0i			0.527295i	−0.964619	+	0.263648i	$0.915074\pi$
$968$		0			0
$969$	8640.00	+	9659.81i	0.286436	+	0.320246i
$970$		0			0
$971$	−40426.1			−1.33608			−0.668040	−	0.744125i	$-0.732867\pi$
$971$	−40426.1			−1.33608			−0.668040	+	0.744125i	$0.732867\pi$
$972$		0			0
$973$	−15180.0			−0.500153
$974$		0			0
$975$	−1558.85	−	1742.84i	−0.0512031	−	0.0572468i
$976$		0			0
$977$			6404.10i			0.209709i	0.994488	+	0.104854i	$0.0334376\pi$
$977$			6404.10i			0.209709i	−0.994488	+	0.104854i	$0.966562\pi$
$978$		0			0
$979$		−	29124.8i		−	0.950801i
$980$		0			0
$981$	−3198.00	+	28603.8i	−0.104082	+	0.930936i
$982$		0			0
$983$	−25953.0			−0.842089			−0.421045	−	0.907040i	$-0.638336\pi$
$983$	−25953.0			−0.842089			−0.421045	+	0.907040i	$0.638336\pi$
$984$		0			0
$985$	19120.0			0.618491
$986$		0			0
$987$	−5819.69	+	5205.29i	−0.187683	+	0.167868i
$988$		0			0
$989$		−	21788.2i		−	0.700532i
$990$		0			0
$991$			48358.1i			1.55010i	0.631903	+	0.775048i	$0.282274\pi$
$991$			48358.1i			1.55010i	−0.631903	+	0.775048i	$0.717726\pi$
$992$		0			0
$993$	16410.0	−	14677.6i	0.524427	−	0.469061i
$994$		0			0
$995$	2286.31			0.0728450
$996$		0			0
$997$	18370.0			0.583534			0.291767	−	0.956489i	$-0.405757\pi$
$997$	18370.0			0.583534			0.291767	+	0.956489i	$0.405757\pi$
$998$		0			0
$999$	14861.0	−	10573.2i	0.470652	−	0.334858i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.4.c.b.191.1		4
3.2	odd	2	inner	192.4.c.b.191.3		4
4.3	odd	2	inner	192.4.c.b.191.4		4
8.3	odd	2		12.4.b.a.11.4	yes	4
8.5	even	2		12.4.b.a.11.2	yes	4
12.11	even	2	inner	192.4.c.b.191.2		4
16.3	odd	4		768.4.f.c.383.4		8
16.5	even	4		768.4.f.c.383.3		8
16.11	odd	4		768.4.f.c.383.5		8
16.13	even	4		768.4.f.c.383.6		8
24.5	odd	2		12.4.b.a.11.3	yes	4
24.11	even	2		12.4.b.a.11.1	✓	4
48.5	odd	4		768.4.f.c.383.2		8
48.11	even	4		768.4.f.c.383.8		8
48.29	odd	4		768.4.f.c.383.7		8
48.35	even	4		768.4.f.c.383.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.4.b.a.11.1	✓	4	24.11	even	2
12.4.b.a.11.2	yes	4	8.5	even	2
12.4.b.a.11.3	yes	4	24.5	odd	2
12.4.b.a.11.4	yes	4	8.3	odd	2
192.4.c.b.191.1		4	1.1	even	1	trivial
192.4.c.b.191.2		4	12.11	even	2	inner
192.4.c.b.191.3		4	3.2	odd	2	inner
192.4.c.b.191.4		4	4.3	odd	2	inner
768.4.f.c.383.1		8	48.35	even	4
768.4.f.c.383.2		8	48.5	odd	4
768.4.f.c.383.3		8	16.5	even	4
768.4.f.c.383.4		8	16.3	odd	4
768.4.f.c.383.5		8	16.11	odd	4
768.4.f.c.383.6		8	16.13	even	4
768.4.f.c.383.7		8	48.29	odd	4
768.4.f.c.383.8		8	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.46410 - 3.87298i) q^{3} +8.94427i q^{5} -7.74597i q^{7} +(-3.00000 + 26.8328i) q^{9} +O(q^{10})\)	\(q+(-3.46410 - 3.87298i) q^{3} +8.94427i q^{5} -7.74597i q^{7} +(-3.00000 + 26.8328i) q^{9} +34.6410 q^{11} +10.0000 q^{13} +(34.6410 - 30.9839i) q^{15} -35.7771i q^{17} -69.7137i q^{19} +(-30.0000 + 26.8328i) q^{21} +96.9948 q^{23} +45.0000 q^{25} +(114.315 - 81.3327i) q^{27} -152.053i q^{29} -224.633i q^{31} +(-120.000 - 134.164i) q^{33} +69.2820 q^{35} +130.000 q^{37} +(-34.6410 - 38.7298i) q^{39} +125.220i q^{41} -224.633i q^{43} +(-240.000 - 26.8328i) q^{45} +193.990 q^{47} +283.000 q^{49} +(-138.564 + 123.935i) q^{51} +545.601i q^{53} +309.839i q^{55} +(-270.000 + 241.495i) q^{57} +173.205 q^{59} +442.000 q^{61} +(207.846 + 23.2379i) q^{63} +89.4427i q^{65} +735.867i q^{67} +(-336.000 - 375.659i) q^{69} -1039.23 q^{71} +410.000 q^{73} +(-155.885 - 174.284i) q^{75} -268.328i q^{77} +85.2056i q^{79} +(-711.000 - 160.997i) q^{81} -1254.00 q^{83} +320.000 q^{85} +(-588.897 + 526.726i) q^{87} -840.762i q^{89} -77.4597i q^{91} +(-870.000 + 778.152i) q^{93} +623.538 q^{95} +770.000 q^{97} +(-103.923 + 929.516i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 40 q^{13} - 120 q^{21} + 180 q^{25} - 480 q^{33} + 520 q^{37} - 960 q^{45} + 1132 q^{49} - 1080 q^{57} + 1768 q^{61} - 1344 q^{69} + 1640 q^{73} - 2844 q^{81} + 1280 q^{85} - 3480 q^{93} + 3080 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 40 * q^13 - 120 * q^21 + 180 * q^25 - 480 * q^33 + 520 * q^37 - 960 * q^45 + 1132 * q^49 - 1080 * q^57 + 1768 * q^61 - 1344 * q^69 + 1640 * q^73 - 2844 * q^81 + 1280 * q^85 - 3480 * q^93 + 3080 * q^97

Introduction

L-functions

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