LMFDB - Embedded newform 192.3.h.c.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(161,192)]

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([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("192.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	192.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.12960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 $ x^8 - 3x^6 + 8x^4 - 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$1.40126 + 0.809017i$ of defining polynomial
Character	$\chi$	$=$	192.161
Dual form			192.3.h.c.161.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23607 - 2.00000i) q^{3} -7.74597 q^{5} -3.46410 q^{7} +(1.00000 - 8.94427i) q^{9} +O(q^{10})$	$q+(2.23607 - 2.00000i) q^{3} -7.74597 q^{5} -3.46410 q^{7} +(1.00000 - 8.94427i) q^{9} -13.4164 q^{11} -20.7846i q^{13} +(-17.3205 + 15.4919i) q^{15} -4.00000i q^{19} +(-7.74597 + 6.92820i) q^{21} +30.9839i q^{23} +35.0000 q^{25} +(-15.6525 - 22.0000i) q^{27} -7.74597 q^{29} +24.2487 q^{31} +(-30.0000 + 26.8328i) q^{33} +26.8328 q^{35} -34.6410i q^{37} +(-41.5692 - 46.4758i) q^{39} -53.6656i q^{41} +52.0000i q^{43} +(-7.74597 + 69.2820i) q^{45} -61.9677i q^{47} -37.0000 q^{49} +54.2218 q^{53} +103.923 q^{55} +(-8.00000 - 8.94427i) q^{57} +40.2492 q^{59} +6.92820i q^{61} +(-3.46410 + 30.9839i) q^{63} +160.997i q^{65} +28.0000i q^{67} +(61.9677 + 69.2820i) q^{69} -30.9839i q^{71} +74.0000 q^{73} +(78.2624 - 70.0000i) q^{75} +46.4758 q^{77} +51.9615 q^{79} +(-79.0000 - 17.8885i) q^{81} -120.748 q^{83} +(-17.3205 + 15.4919i) q^{87} -53.6656i q^{89} +72.0000i q^{91} +(54.2218 - 48.4974i) q^{93} +30.9839i q^{95} -62.0000 q^{97} +(-13.4164 + 120.000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 280 q^{25} - 240 q^{33} - 296 q^{49} - 64 q^{57} + 592 q^{73} - 632 q^{81} - 496 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 280 * q^25 - 240 * q^33 - 296 * q^49 - 64 * q^57 + 592 * q^73 - 632 * q^81 - 496 * 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
$3$	2.23607	−	2.00000i	0.745356	−	0.666667i
$3$	2.23607	−	2.00000i	0.745356	−	0.666667i
$4$		0			0
$5$	−7.74597			−1.54919			−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−7.74597			−1.54919			−0.774597	+	0.632456i	$0.782047\pi$
$6$		0			0
$7$	−3.46410			−0.494872			−0.247436	−	0.968904i	$-0.579588\pi$
$7$	−3.46410			−0.494872			−0.247436	+	0.968904i	$0.579588\pi$
$8$		0			0
$9$	1.00000	−	8.94427i	0.111111	−	0.993808i
$10$		0			0
$11$	−13.4164			−1.21967			−0.609837	−	0.792527i	$-0.708765\pi$
$11$	−13.4164			−1.21967			−0.609837	+	0.792527i	$0.708765\pi$
$12$		0			0
$13$		−	20.7846i		−	1.59882i	−0.600788	−	0.799408i	$-0.705147\pi$
$13$		−	20.7846i		−	1.59882i	0.600788	−	0.799408i	$-0.294853\pi$
$14$		0			0
$15$	−17.3205	+	15.4919i	−1.15470	+	1.03280i
$16$		0			0
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		−	4.00000i		−	0.210526i	−0.994444	−	0.105263i	$-0.966432\pi$
$19$		−	4.00000i		−	0.210526i	0.994444	−	0.105263i	$-0.0335685\pi$
$20$		0			0
$21$	−7.74597	+	6.92820i	−0.368856	+	0.329914i
$22$		0			0
$23$			30.9839i			1.34712i	0.739130	+	0.673562i	$0.235237\pi$
$23$			30.9839i			1.34712i	−0.739130	+	0.673562i	$0.764763\pi$
$24$		0			0
$25$	35.0000			1.40000
$26$		0			0
$27$	−15.6525	−	22.0000i	−0.579721	−	0.814815i
$28$		0			0
$29$	−7.74597			−0.267102			−0.133551	−	0.991042i	$-0.542638\pi$
$29$	−7.74597			−0.267102			−0.133551	+	0.991042i	$0.542638\pi$
$30$		0			0
$31$	24.2487			0.782216			0.391108	−	0.920345i	$-0.372092\pi$
$31$	24.2487			0.782216			0.391108	+	0.920345i	$0.372092\pi$
$32$		0			0
$33$	−30.0000	+	26.8328i	−0.909091	+	0.813116i
$34$		0			0
$35$	26.8328			0.766652
$36$		0			0
$37$		−	34.6410i		−	0.936244i	−0.883664	−	0.468122i	$-0.844931\pi$
$37$		−	34.6410i		−	0.936244i	0.883664	−	0.468122i	$-0.155069\pi$
$38$		0			0
$39$	−41.5692	−	46.4758i	−1.06588	−	1.19169i
$40$		0			0
$41$		−	53.6656i		−	1.30892i	−0.756098	−	0.654459i	$-0.772896\pi$
$41$		−	53.6656i		−	1.30892i	0.756098	−	0.654459i	$-0.227104\pi$
$42$		0			0
$43$			52.0000i			1.20930i	0.796490	+	0.604651i	$0.206687\pi$
$43$			52.0000i			1.20930i	−0.796490	+	0.604651i	$0.793313\pi$
$44$		0			0
$45$	−7.74597	+	69.2820i	−0.172133	+	1.53960i
$46$		0			0
$47$		−	61.9677i		−	1.31846i	−0.751940	−	0.659231i	$-0.770882\pi$
$47$		−	61.9677i		−	1.31846i	0.751940	−	0.659231i	$-0.229118\pi$
$48$		0			0
$49$	−37.0000			−0.755102
$50$		0			0
$51$		0			0
$52$		0			0
$53$	54.2218			1.02305			0.511526	−	0.859268i	$-0.329080\pi$
$53$	54.2218			1.02305			0.511526	+	0.859268i	$0.329080\pi$
$54$		0			0
$55$	103.923			1.88951
$56$		0			0
$57$	−8.00000	−	8.94427i	−0.140351	−	0.156917i
$58$		0			0
$59$	40.2492			0.682190			0.341095	−	0.940029i	$-0.389202\pi$
$59$	40.2492			0.682190			0.341095	+	0.940029i	$0.389202\pi$
$60$		0			0
$61$			6.92820i			0.113577i	0.998386	+	0.0567886i	$0.0180861\pi$
$61$			6.92820i			0.113577i	−0.998386	+	0.0567886i	$0.981914\pi$
$62$		0			0
$63$	−3.46410	+	30.9839i	−0.0549857	+	0.491807i
$64$		0			0
$65$			160.997i			2.47688i
$66$		0			0
$67$			28.0000i			0.417910i	0.977925	+	0.208955i	$0.0670063\pi$
$67$			28.0000i			0.417910i	−0.977925	+	0.208955i	$0.932994\pi$
$68$		0			0
$69$	61.9677	+	69.2820i	0.898083	+	1.00409i
$70$		0			0
$71$		−	30.9839i		−	0.436392i	−0.975905	−	0.218196i	$-0.929983\pi$
$71$		−	30.9839i		−	0.436392i	0.975905	−	0.218196i	$-0.0700173\pi$
$72$		0			0
$73$	74.0000			1.01370			0.506849	−	0.862035i	$-0.330810\pi$
$73$	74.0000			1.01370			0.506849	+	0.862035i	$0.330810\pi$
$74$		0			0
$75$	78.2624	−	70.0000i	1.04350	−	0.933333i
$76$		0			0
$77$	46.4758			0.603582
$78$		0			0
$79$	51.9615			0.657741			0.328870	−	0.944375i	$-0.393332\pi$
$79$	51.9615			0.657741			0.328870	+	0.944375i	$0.393332\pi$
$80$		0			0
$81$	−79.0000	−	17.8885i	−0.975309	−	0.220846i
$82$		0			0
$83$	−120.748			−1.45479			−0.727396	−	0.686218i	$-0.759269\pi$
$83$	−120.748			−1.45479			−0.727396	+	0.686218i	$0.759269\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−17.3205	+	15.4919i	−0.199086	+	0.178068i
$88$		0			0
$89$		−	53.6656i		−	0.602985i	−0.953469	−	0.301492i	$-0.902515\pi$
$89$		−	53.6656i		−	0.602985i	0.953469	−	0.301492i	$-0.0974848\pi$
$90$		0			0
$91$			72.0000i			0.791209i
$92$		0			0
$93$	54.2218	−	48.4974i	0.583030	−	0.521478i
$94$		0			0
$95$			30.9839i			0.326146i
$96$		0			0
$97$	−62.0000			−0.639175			−0.319588	−	0.947557i	$-0.603544\pi$
$97$	−62.0000			−0.639175			−0.319588	+	0.947557i	$0.603544\pi$
$98$		0			0
$99$	−13.4164	+	120.000i	−0.135519	+	1.21212i
$100$		0			0
$101$	−131.681			−1.30378			−0.651888	−	0.758315i	$-0.726023\pi$
$101$	−131.681			−1.30378			−0.651888	+	0.758315i	$0.726023\pi$
$102$		0			0
$103$	−86.6025			−0.840801			−0.420401	−	0.907339i	$-0.638110\pi$
$103$	−86.6025			−0.840801			−0.420401	+	0.907339i	$0.638110\pi$
$104$		0			0
$105$	60.0000	−	53.6656i	0.571429	−	0.511101i
$106$		0			0
$107$	40.2492			0.376161			0.188080	−	0.982154i	$-0.439773\pi$
$107$	40.2492			0.376161			0.188080	+	0.982154i	$0.439773\pi$
$108$		0			0
$109$		−	76.2102i		−	0.699176i	−0.936903	−	0.349588i	$-0.886321\pi$
$109$		−	76.2102i		−	0.699176i	0.936903	−	0.349588i	$-0.113679\pi$
$110$		0			0
$111$	−69.2820	−	77.4597i	−0.624162	−	0.697835i
$112$		0			0
$113$		−	107.331i		−	0.949834i	−0.880031	−	0.474917i	$-0.842478\pi$
$113$		−	107.331i		−	0.949834i	0.880031	−	0.474917i	$-0.157522\pi$
$114$		0			0
$115$		−	240.000i		−	2.08696i
$116$		0			0
$117$	−185.903	−	20.7846i	−1.58892	−	0.177646i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	59.0000			0.487603
$122$		0			0
$123$	−107.331	−	120.000i	−0.872612	−	0.975610i
$124$		0			0
$125$	−77.4597			−0.619677
$126$		0			0
$127$	−86.6025			−0.681910			−0.340955	−	0.940080i	$-0.610750\pi$
$127$	−86.6025			−0.681910			−0.340955	+	0.940080i	$0.610750\pi$
$128$		0			0
$129$	104.000	+	116.276i	0.806202	+	0.901361i
$130$		0			0
$131$	147.580			1.12657			0.563284	−	0.826263i	$-0.309538\pi$
$131$	147.580			1.12657			0.563284	+	0.826263i	$0.309538\pi$
$132$		0			0
$133$			13.8564i			0.104184i
$134$		0			0
$135$	121.244	+	170.411i	0.898100	+	1.26231i
$136$		0			0
$137$		−	53.6656i		−	0.391720i	−0.980632	−	0.195860i	$-0.937250\pi$
$137$		−	53.6656i		−	0.391720i	0.980632	−	0.195860i	$-0.0627498\pi$
$138$		0			0
$139$			68.0000i			0.489209i	0.969623	+	0.244604i	$0.0786581\pi$
$139$			68.0000i			0.489209i	−0.969623	+	0.244604i	$0.921342\pi$
$140$		0			0
$141$	−123.935	−	138.564i	−0.878975	−	0.982724i
$142$		0			0
$143$			278.855i			1.95003i
$144$		0			0
$145$	60.0000			0.413793
$146$		0			0
$147$	−82.7345	+	74.0000i	−0.562820	+	0.503401i
$148$		0			0
$149$	116.190			0.779795			0.389898	−	0.920858i	$-0.372510\pi$
$149$	116.190			0.779795			0.389898	+	0.920858i	$0.372510\pi$
$150$		0			0
$151$	−252.879			−1.67470			−0.837349	−	0.546669i	$-0.815896\pi$
$151$	−252.879			−1.67470			−0.837349	+	0.546669i	$0.815896\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−187.830			−1.21180
$156$		0			0
$157$			117.779i			0.750188i	0.926987	+	0.375094i	$0.122390\pi$
$157$			117.779i			0.750188i	−0.926987	+	0.375094i	$0.877610\pi$
$158$		0			0
$159$	121.244	−	108.444i	0.762538	−	0.682035i
$160$		0			0
$161$		−	107.331i		−	0.666654i
$162$		0			0
$163$		−	52.0000i		−	0.319018i	−0.987196	−	0.159509i	$-0.949009\pi$
$163$		−	52.0000i		−	0.319018i	0.987196	−	0.159509i	$-0.0509912\pi$
$164$		0			0
$165$	232.379	−	207.846i	1.40836	−	1.25967i
$166$		0			0
$167$		−	30.9839i		−	0.185532i	−0.995688	−	0.0927661i	$-0.970429\pi$
$167$		−	30.9839i		−	0.185532i	0.995688	−	0.0927661i	$-0.0295709\pi$
$168$		0			0
$169$	−263.000			−1.55621
$170$		0			0
$171$	−35.7771	−	4.00000i	−0.209223	−	0.0233918i
$172$		0			0
$173$	−7.74597			−0.0447744			−0.0223872	−	0.999749i	$-0.507127\pi$
$173$	−7.74597			−0.0447744			−0.0223872	+	0.999749i	$0.507127\pi$
$174$		0			0
$175$	−121.244			−0.692820
$176$		0			0
$177$	90.0000	−	80.4984i	0.508475	−	0.454793i
$178$		0			0
$179$	−67.0820			−0.374760			−0.187380	−	0.982288i	$-0.560000\pi$
$179$	−67.0820			−0.374760			−0.187380	+	0.982288i	$0.560000\pi$
$180$		0			0
$181$		−	6.92820i		−	0.0382774i	−0.999817	−	0.0191387i	$-0.993908\pi$
$181$		−	6.92820i		−	0.0382774i	0.999817	−	0.0191387i	$-0.00609240\pi$
$182$		0			0
$183$	13.8564	+	15.4919i	0.0757181	+	0.0846554i
$184$		0			0
$185$			268.328i			1.45042i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	54.2218	+	76.2102i	0.286888	+	0.403229i
$190$		0			0
$191$		−	123.935i		−	0.648877i	−0.945907	−	0.324438i	$-0.894825\pi$
$191$		−	123.935i		−	0.648877i	0.945907	−	0.324438i	$-0.105175\pi$
$192$		0			0
$193$	−22.0000			−0.113990			−0.0569948	−	0.998374i	$-0.518152\pi$
$193$	−22.0000			−0.113990			−0.0569948	+	0.998374i	$0.518152\pi$
$194$		0			0
$195$	321.994	+	360.000i	1.65125	+	1.84615i
$196$		0			0
$197$	−69.7137			−0.353877			−0.176938	−	0.984222i	$-0.556619\pi$
$197$	−69.7137			−0.353877			−0.176938	+	0.984222i	$0.556619\pi$
$198$		0			0
$199$	301.377			1.51446			0.757228	−	0.653150i	$-0.226553\pi$
$199$	301.377			1.51446			0.757228	+	0.653150i	$0.226553\pi$
$200$		0			0
$201$	56.0000	+	62.6099i	0.278607	+	0.311492i
$202$		0			0
$203$	26.8328			0.132181
$204$		0			0
$205$			415.692i			2.02777i
$206$		0			0
$207$	277.128	+	30.9839i	1.33878	+	0.149681i
$208$		0			0
$209$			53.6656i			0.256773i
$210$		0			0
$211$		−	404.000i		−	1.91469i	−0.288944	−	0.957346i	$-0.593304\pi$
$211$		−	404.000i		−	1.91469i	0.288944	−	0.957346i	$-0.406696\pi$
$212$		0			0
$213$	−61.9677	−	69.2820i	−0.290928	−	0.325268i
$214$		0			0
$215$		−	402.790i		−	1.87344i
$216$		0			0
$217$	−84.0000			−0.387097
$218$		0			0
$219$	165.469	−	148.000i	0.755566	−	0.675799i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−169.741			−0.761170			−0.380585	−	0.924746i	$-0.624277\pi$
$223$	−169.741			−0.761170			−0.380585	+	0.924746i	$0.624277\pi$
$224$		0			0
$225$	35.0000	−	313.050i	0.155556	−	1.39133i
$226$		0			0
$227$	308.577			1.35937			0.679686	−	0.733503i	$-0.262116\pi$
$227$	308.577			1.35937			0.679686	+	0.733503i	$0.262116\pi$
$228$		0			0
$229$			408.764i			1.78500i	0.451052	+	0.892498i	$0.351049\pi$
$229$			408.764i			1.78500i	−0.451052	+	0.892498i	$0.648951\pi$
$230$		0			0
$231$	103.923	−	92.9516i	0.449883	−	0.402388i
$232$		0			0
$233$			268.328i			1.15162i	0.817582	+	0.575811i	$0.195314\pi$
$233$			268.328i			1.15162i	−0.817582	+	0.575811i	$0.804686\pi$
$234$		0			0
$235$			480.000i			2.04255i
$236$		0			0
$237$	116.190	−	103.923i	0.490251	−	0.438494i
$238$		0			0
$239$		−	309.839i		−	1.29640i	−0.761472	−	0.648198i	$-0.775523\pi$
$239$		−	309.839i		−	1.29640i	0.761472	−	0.648198i	$-0.224477\pi$
$240$		0			0
$241$	74.0000			0.307054			0.153527	−	0.988144i	$-0.450937\pi$
$241$	74.0000			0.307054			0.153527	+	0.988144i	$0.450937\pi$
$242$		0			0
$243$	−212.426	+	118.000i	−0.874183	+	0.485597i
$244$		0			0
$245$	286.601			1.16980
$246$		0			0
$247$	−83.1384			−0.336593
$248$		0			0
$249$	−270.000	+	241.495i	−1.08434	+	0.969861i
$250$		0			0
$251$	−13.4164			−0.0534518			−0.0267259	−	0.999643i	$-0.508508\pi$
$251$	−13.4164			−0.0534518			−0.0267259	+	0.999643i	$0.508508\pi$
$252$		0			0
$253$		−	415.692i		−	1.64305i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	214.663i		−	0.835263i	−0.908617	−	0.417631i	$-0.862860\pi$
$257$		−	214.663i		−	0.835263i	0.908617	−	0.417631i	$-0.137140\pi$
$258$		0			0
$259$			120.000i			0.463320i
$260$		0			0
$261$	−7.74597	+	69.2820i	−0.0296780	+	0.265448i
$262$		0			0
$263$			464.758i			1.76714i	0.468298	+	0.883570i	$0.344867\pi$
$263$			464.758i			1.76714i	−0.468298	+	0.883570i	$0.655133\pi$
$264$		0			0
$265$	−420.000			−1.58491
$266$		0			0
$267$	−107.331	−	120.000i	−0.401990	−	0.449438i
$268$		0			0
$269$	302.093			1.12302			0.561511	−	0.827470i	$-0.310220\pi$
$269$	302.093			1.12302			0.561511	+	0.827470i	$0.310220\pi$
$270$		0			0
$271$	329.090			1.21435			0.607176	−	0.794567i	$-0.292302\pi$
$271$	329.090			1.21435			0.607176	+	0.794567i	$0.292302\pi$
$272$		0			0
$273$	144.000	+	160.997i	0.527473	+	0.589732i
$274$		0			0
$275$	−469.574			−1.70754
$276$		0			0
$277$			159.349i			0.575266i	0.957741	+	0.287633i	$0.0928683\pi$
$277$			159.349i			0.575266i	−0.957741	+	0.287633i	$0.907132\pi$
$278$		0			0
$279$	24.2487	−	216.887i	0.0869129	−	0.777373i
$280$		0			0
$281$		−	268.328i		−	0.954904i	−0.878658	−	0.477452i	$-0.841560\pi$
$281$		−	268.328i		−	0.954904i	0.878658	−	0.477452i	$-0.158440\pi$
$282$		0			0
$283$		−	172.000i		−	0.607774i	−0.952708	−	0.303887i	$-0.901715\pi$
$283$		−	172.000i		−	0.607774i	0.952708	−	0.303887i	$-0.0982845\pi$
$284$		0			0
$285$	61.9677	+	69.2820i	0.217431	+	0.243095i
$286$		0			0
$287$			185.903i			0.647746i
$288$		0			0
$289$	289.000			1.00000
$290$		0			0
$291$	−138.636	+	124.000i	−0.476413	+	0.426117i
$292$		0			0
$293$	−317.585			−1.08391			−0.541953	−	0.840409i	$-0.682315\pi$
$293$	−317.585			−1.08391			−0.541953	+	0.840409i	$0.682315\pi$
$294$		0			0
$295$	−311.769			−1.05684
$296$		0			0
$297$	210.000	+	295.161i	0.707071	+	0.993808i
$298$		0			0
$299$	643.988			2.15380
$300$		0			0
$301$		−	180.133i		−	0.598449i
$302$		0			0
$303$	−294.449	+	263.363i	−0.971778	+	0.869184i
$304$		0			0
$305$		−	53.6656i		−	0.175953i
$306$		0			0
$307$		−	212.000i		−	0.690554i	−0.938501	−	0.345277i	$-0.887785\pi$
$307$		−	212.000i		−	0.690554i	0.938501	−	0.345277i	$-0.112215\pi$
$308$		0			0
$309$	−193.649	+	173.205i	−0.626696	+	0.560534i
$310$		0			0
$311$			526.726i			1.69365i	0.531870	+	0.846826i	$0.321489\pi$
$311$			526.726i			1.69365i	−0.531870	+	0.846826i	$0.678511\pi$
$312$		0			0
$313$	202.000			0.645367			0.322684	−	0.946507i	$-0.395415\pi$
$313$	202.000			0.645367			0.322684	+	0.946507i	$0.395415\pi$
$314$		0			0
$315$	26.8328	−	240.000i	0.0851835	−	0.761905i
$316$		0			0
$317$	−69.7137			−0.219917			−0.109959	−	0.993936i	$-0.535072\pi$
$317$	−69.7137			−0.219917			−0.109959	+	0.993936i	$0.535072\pi$
$318$		0			0
$319$	103.923			0.325778
$320$		0			0
$321$	90.0000	−	80.4984i	0.280374	−	0.250774i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		−	727.461i		−	2.23834i
$326$		0			0
$327$	−152.420	−	170.411i	−0.466118	−	0.521135i
$328$		0			0
$329$			214.663i			0.652470i
$330$		0			0
$331$		−	28.0000i		−	0.0845921i	−0.999105	−	0.0422961i	$-0.986533\pi$
$331$		−	28.0000i		−	0.0845921i	0.999105	−	0.0422961i	$-0.0134673\pi$
$332$		0			0
$333$	−309.839	−	34.6410i	−0.930446	−	0.104027i
$334$		0			0
$335$		−	216.887i		−	0.647424i
$336$		0			0
$337$	298.000			0.884273			0.442136	−	0.896948i	$-0.354221\pi$
$337$	298.000			0.884273			0.442136	+	0.896948i	$0.354221\pi$
$338$		0			0
$339$	−214.663	−	240.000i	−0.633223	−	0.707965i
$340$		0			0
$341$	−325.331			−0.954049
$342$		0			0
$343$	297.913			0.868550
$344$		0			0
$345$	−480.000	−	536.656i	−1.39130	−	1.55553i
$346$		0			0
$347$	−442.741			−1.27591			−0.637956	−	0.770073i	$-0.720220\pi$
$347$	−442.741			−1.27591			−0.637956	+	0.770073i	$0.720220\pi$
$348$		0			0
$349$			173.205i			0.496290i	0.968723	+	0.248145i	$0.0798209\pi$
$349$			173.205i			0.496290i	−0.968723	+	0.248145i	$0.920179\pi$
$350$		0			0
$351$	−457.261	+	325.331i	−1.30274	+	0.926868i
$352$		0			0
$353$		−	643.988i		−	1.82433i	−0.409826	−	0.912164i	$-0.634411\pi$
$353$		−	643.988i		−	1.82433i	0.409826	−	0.912164i	$-0.365589\pi$
$354$		0			0
$355$			240.000i			0.676056i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	402.790i		−	1.12198i	−0.827823	−	0.560989i	$-0.810421\pi$
$359$		−	402.790i		−	1.12198i	0.827823	−	0.560989i	$-0.189579\pi$
$360$		0			0
$361$	345.000			0.955679
$362$		0			0
$363$	131.928	−	118.000i	0.363438	−	0.325069i
$364$		0			0
$365$	−573.202			−1.57042
$366$		0			0
$367$	−419.156			−1.14212			−0.571058	−	0.820910i	$-0.693467\pi$
$367$	−419.156			−1.14212			−0.571058	+	0.820910i	$0.693467\pi$
$368$		0			0
$369$	−480.000	−	53.6656i	−1.30081	−	0.145435i
$370$		0			0
$371$	−187.830			−0.506280
$372$		0			0
$373$		−	394.908i		−	1.05873i	−0.848393	−	0.529367i	$-0.822430\pi$
$373$		−	394.908i		−	1.05873i	0.848393	−	0.529367i	$-0.177570\pi$
$374$		0			0
$375$	−173.205	+	154.919i	−0.461880	+	0.413118i
$376$		0			0
$377$			160.997i			0.427047i
$378$		0			0
$379$			484.000i			1.27704i	0.769603	+	0.638522i	$0.220454\pi$
$379$			484.000i			1.27704i	−0.769603	+	0.638522i	$0.779546\pi$
$380$		0			0
$381$	−193.649	+	173.205i	−0.508266	+	0.454607i
$382$		0			0
$383$		−	123.935i		−	0.323591i	−0.986824	−	0.161796i	$-0.948271\pi$
$383$		−	123.935i		−	0.323591i	0.986824	−	0.161796i	$-0.0517285\pi$
$384$		0			0
$385$	−360.000			−0.935065
$386$		0			0
$387$	465.102	+	52.0000i	1.20181	+	0.134367i
$388$		0			0
$389$	426.028			1.09519			0.547594	−	0.836744i	$-0.315544\pi$
$389$	426.028			1.09519			0.547594	+	0.836744i	$0.315544\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	330.000	−	295.161i	0.839695	−	0.751046i
$394$		0			0
$395$	−402.492			−1.01897
$396$		0			0
$397$		−	464.190i		−	1.16924i	−0.811306	−	0.584622i	$-0.801243\pi$
$397$		−	464.190i		−	1.16924i	0.811306	−	0.584622i	$-0.198757\pi$
$398$		0			0
$399$	27.7128	+	30.9839i	0.0694557	+	0.0776538i
$400$		0			0
$401$			643.988i			1.60595i	0.596010	+	0.802977i	$0.296752\pi$
$401$			643.988i			1.60595i	−0.596010	+	0.802977i	$0.703248\pi$
$402$		0			0
$403$		−	504.000i		−	1.25062i
$404$		0			0
$405$	611.931	+	138.564i	1.51094	+	0.342133i
$406$		0			0
$407$			464.758i			1.14191i
$408$		0			0
$409$	386.000			0.943765			0.471883	−	0.881661i	$-0.343575\pi$
$409$	386.000			0.943765			0.471883	+	0.881661i	$0.343575\pi$
$410$		0			0
$411$	−107.331	−	120.000i	−0.261147	−	0.291971i
$412$		0			0
$413$	−139.427			−0.337597
$414$		0			0
$415$	935.307			2.25375
$416$		0			0
$417$	136.000	+	152.053i	0.326139	+	0.364635i
$418$		0			0
$419$	308.577			0.736462			0.368231	−	0.929734i	$-0.379964\pi$
$419$	308.577			0.736462			0.368231	+	0.929734i	$0.379964\pi$
$420$		0			0
$421$		−	34.6410i		−	0.0822827i	−0.999153	−	0.0411413i	$-0.986901\pi$
$421$		−	34.6410i		−	0.0822827i	0.999153	−	0.0411413i	$-0.0130994\pi$
$422$		0			0
$423$	−554.256	−	61.9677i	−1.31030	−	0.146496i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	24.0000i		−	0.0562061i
$428$		0			0
$429$	557.710	+	623.538i	1.30002	+	1.45347i
$430$		0			0
$431$			433.774i			1.00644i	0.864159	+	0.503218i	$0.167851\pi$
$431$			433.774i			1.00644i	−0.864159	+	0.503218i	$0.832149\pi$
$432$		0			0
$433$	322.000			0.743649			0.371824	−	0.928303i	$-0.378732\pi$
$433$	322.000			0.743649			0.371824	+	0.928303i	$0.378732\pi$
$434$		0			0
$435$	134.164	−	120.000i	0.308423	−	0.275862i
$436$		0			0
$437$	123.935			0.283605
$438$		0			0
$439$	633.931			1.44403			0.722017	−	0.691876i	$-0.243215\pi$
$439$	633.931			1.44403			0.722017	+	0.691876i	$0.243215\pi$
$440$		0			0
$441$	−37.0000	+	330.938i	−0.0839002	+	0.750426i
$442$		0			0
$443$	−13.4164			−0.0302853			−0.0151427	−	0.999885i	$-0.504820\pi$
$443$	−13.4164			−0.0302853			−0.0151427	+	0.999885i	$0.504820\pi$
$444$		0			0
$445$			415.692i			0.934140i
$446$		0			0
$447$	259.808	−	232.379i	0.581225	−	0.519864i
$448$		0			0
$449$			536.656i			1.19523i	0.801785	+	0.597613i	$0.203884\pi$
$449$			536.656i			1.19523i	−0.801785	+	0.597613i	$0.796116\pi$
$450$		0			0
$451$			720.000i			1.59645i
$452$		0			0
$453$	−565.456	+	505.759i	−1.24825	+	1.11647i
$454$		0			0
$455$		−	557.710i		−	1.22574i
$456$		0			0
$457$	−574.000			−1.25602			−0.628009	−	0.778206i	$-0.716130\pi$
$457$	−574.000			−1.25602			−0.628009	+	0.778206i	$0.716130\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	487.996			1.05856			0.529280	−	0.848447i	$-0.322462\pi$
$461$	487.996			1.05856			0.529280	+	0.848447i	$0.322462\pi$
$462$		0			0
$463$	−363.731			−0.785595			−0.392798	−	0.919625i	$-0.628493\pi$
$463$	−363.731			−0.785595			−0.392798	+	0.919625i	$0.628493\pi$
$464$		0			0
$465$	−420.000	+	375.659i	−0.903226	+	0.807870i
$466$		0			0
$467$	308.577			0.660765			0.330383	−	0.943847i	$-0.392822\pi$
$467$	308.577			0.660765			0.330383	+	0.943847i	$0.392822\pi$
$468$		0			0
$469$		−	96.9948i		−	0.206812i
$470$		0			0
$471$	235.559	+	263.363i	0.500125	+	0.559157i
$472$		0			0
$473$		−	697.653i		−	1.47495i
$474$		0			0
$475$		−	140.000i		−	0.294737i
$476$		0			0
$477$	54.2218	−	484.974i	0.113672	−	1.01672i
$478$		0			0
$479$		−	371.806i		−	0.776214i	−0.921614	−	0.388107i	$-0.873129\pi$
$479$		−	371.806i		−	0.776214i	0.921614	−	0.388107i	$-0.126871\pi$
$480$		0			0
$481$	−720.000			−1.49688
$482$		0			0
$483$	−214.663	−	240.000i	−0.444436	−	0.496894i
$484$		0			0
$485$	480.250			0.990206
$486$		0			0
$487$	384.515			0.789559			0.394780	−	0.918776i	$-0.370821\pi$
$487$	384.515			0.789559			0.394780	+	0.918776i	$0.370821\pi$
$488$		0			0
$489$	−104.000	−	116.276i	−0.212679	−	0.237782i
$490$		0			0
$491$	254.912			0.519169			0.259584	−	0.965720i	$-0.416414\pi$
$491$	254.912			0.519169			0.259584	+	0.965720i	$0.416414\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	103.923	−	929.516i	0.209946	−	1.87781i
$496$		0			0
$497$			107.331i			0.215958i
$498$		0			0
$499$			844.000i			1.69138i	0.533672	+	0.845691i	$0.320812\pi$
$499$			844.000i			1.69138i	−0.533672	+	0.845691i	$0.679188\pi$
$500$		0			0
$501$	−61.9677	−	69.2820i	−0.123688	−	0.138287i
$502$		0			0
$503$		−	712.629i		−	1.41676i	−0.705833	−	0.708379i	$-0.749427\pi$
$503$		−	712.629i		−	1.41676i	0.705833	−	0.708379i	$-0.250573\pi$
$504$		0			0
$505$	1020.00			2.01980
$506$		0			0
$507$	−588.086	+	526.000i	−1.15993	+	1.03748i
$508$		0			0
$509$	983.738			1.93269			0.966344	−	0.257255i	$-0.0828180\pi$
$509$	983.738			1.93269			0.966344	+	0.257255i	$0.0828180\pi$
$510$		0			0
$511$	−256.344			−0.501651
$512$		0			0
$513$	−88.0000	+	62.6099i	−0.171540	+	0.122047i
$514$		0			0
$515$	670.820			1.30256
$516$		0			0
$517$			831.384i			1.60809i
$518$		0			0
$519$	−17.3205	+	15.4919i	−0.0333728	+	0.0298496i
$520$		0			0
$521$		−	912.316i		−	1.75109i	−0.483140	−	0.875543i	$-0.660504\pi$
$521$		−	912.316i		−	1.75109i	0.483140	−	0.875543i	$-0.339496\pi$
$522$		0			0
$523$			52.0000i			0.0994264i	0.998764	+	0.0497132i	$0.0158307\pi$
$523$			52.0000i			0.0994264i	−0.998764	+	0.0497132i	$0.984169\pi$
$524$		0			0
$525$	−271.109	+	242.487i	−0.516398	+	0.461880i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−431.000			−0.814745
$530$		0			0
$531$	40.2492	−	360.000i	0.0757989	−	0.677966i
$532$		0			0
$533$	−1115.42			−2.09272
$534$		0			0
$535$	−311.769			−0.582746
$536$		0			0
$537$	−150.000	+	134.164i	−0.279330	+	0.249840i
$538$		0			0
$539$	496.407			0.920978
$540$		0			0
$541$		−	491.902i		−	0.909247i	−0.890684	−	0.454623i	$-0.849774\pi$
$541$		−	491.902i		−	0.909247i	0.890684	−	0.454623i	$-0.150226\pi$
$542$		0			0
$543$	−13.8564	−	15.4919i	−0.0255182	−	0.0285303i
$544$		0			0
$545$			590.322i			1.08316i
$546$		0			0
$547$		−	724.000i		−	1.32358i	−0.749688	−	0.661792i	$-0.769796\pi$
$547$		−	724.000i		−	1.32358i	0.749688	−	0.661792i	$-0.230204\pi$
$548$		0			0
$549$	61.9677	+	6.92820i	0.112874	+	0.0126197i
$550$		0			0
$551$			30.9839i			0.0562321i
$552$		0			0
$553$	−180.000			−0.325497
$554$		0			0
$555$	536.656	+	600.000i	0.966948	+	1.08108i
$556$		0			0
$557$	−441.520			−0.792675			−0.396338	−	0.918105i	$-0.629719\pi$
$557$	−441.520			−0.792675			−0.396338	+	0.918105i	$0.629719\pi$
$558$		0			0
$559$	1080.80			1.93345
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−120.748			−0.214472			−0.107236	−	0.994234i	$-0.534200\pi$
$563$	−120.748			−0.214472			−0.107236	+	0.994234i	$0.534200\pi$
$564$		0			0
$565$			831.384i			1.47148i
$566$		0			0
$567$	273.664	+	61.9677i	0.482653	+	0.109291i
$568$		0			0
$569$			482.991i			0.848841i	0.905465	+	0.424421i	$0.139522\pi$
$569$			482.991i			0.848841i	−0.905465	+	0.424421i	$0.860478\pi$
$570$		0			0
$571$		−	1036.00i		−	1.81436i	−0.420742	−	0.907180i	$-0.638230\pi$
$571$		−	1036.00i		−	1.81436i	0.420742	−	0.907180i	$-0.361770\pi$
$572$		0			0
$573$	−247.871	−	277.128i	−0.432585	−	0.483644i
$574$		0			0
$575$			1084.44i			1.88597i
$576$		0			0
$577$	−182.000			−0.315425			−0.157712	−	0.987485i	$-0.550412\pi$
$577$	−182.000			−0.315425			−0.157712	+	0.987485i	$0.550412\pi$
$578$		0			0
$579$	−49.1935	+	44.0000i	−0.0849629	+	0.0759931i
$580$		0			0
$581$	418.282			0.719935
$582$		0			0
$583$	−727.461			−1.24779
$584$		0			0
$585$	1440.00	+	160.997i	2.46154	+	0.275208i
$586$		0			0
$587$	−1033.06			−1.75990			−0.879952	−	0.475063i	$-0.842425\pi$
$587$	−1033.06			−1.75990			−0.879952	+	0.475063i	$0.842425\pi$
$588$		0			0
$589$		−	96.9948i		−	0.164677i
$590$		0			0
$591$	−155.885	+	139.427i	−0.263764	+	0.235918i
$592$		0			0
$593$		−	107.331i		−	0.180997i	−0.995897	−	0.0904985i	$-0.971154\pi$
$593$		−	107.331i		−	0.180997i	0.995897	−	0.0904985i	$-0.0288460\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	673.899	−	602.754i	1.12881	−	1.00964i
$598$		0			0
$599$			30.9839i			0.0517260i	0.999665	+	0.0258630i	$0.00823336\pi$
$599$			30.9839i			0.0517260i	−0.999665	+	0.0258630i	$0.991767\pi$
$600$		0			0
$601$	554.000			0.921797			0.460899	−	0.887453i	$-0.347527\pi$
$601$	554.000			0.921797			0.460899	+	0.887453i	$0.347527\pi$
$602$		0			0
$603$	250.440	+	28.0000i	0.415323	+	0.0464345i
$604$		0			0
$605$	−457.012			−0.755392
$606$		0			0
$607$	−917.987			−1.51233			−0.756167	−	0.654379i	$-0.772930\pi$
$607$	−917.987			−1.51233			−0.756167	+	0.654379i	$0.772930\pi$
$608$		0			0
$609$	60.0000	−	53.6656i	0.0985222	−	0.0881209i
$610$		0			0
$611$	−1287.98			−2.10798
$612$		0			0
$613$		−	866.025i		−	1.41277i	−0.707830	−	0.706383i	$-0.750326\pi$
$613$		−	866.025i		−	1.41277i	0.707830	−	0.706383i	$-0.249674\pi$
$614$		0			0
$615$	831.384	+	929.516i	1.35184	+	1.51141i
$616$		0			0
$617$		−	375.659i		−	0.608848i	−0.952537	−	0.304424i	$-0.901536\pi$
$617$		−	375.659i		−	0.608848i	0.952537	−	0.304424i	$-0.0984640\pi$
$618$		0			0
$619$		−	124.000i		−	0.200323i	−0.994971	−	0.100162i	$-0.968064\pi$
$619$		−	124.000i		−	0.200323i	0.994971	−	0.100162i	$-0.0319360\pi$
$620$		0			0
$621$	681.645	−	484.974i	1.09766	−	0.780957i
$622$		0			0
$623$			185.903i			0.298400i
$624$		0			0
$625$	−275.000			−0.440000
$626$		0			0
$627$	107.331	+	120.000i	0.171182	+	0.191388i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−890.274			−1.41089			−0.705447	−	0.708763i	$-0.749254\pi$
$631$	−890.274			−1.41089			−0.705447	+	0.708763i	$0.749254\pi$
$632$		0			0
$633$	−808.000	−	903.371i	−1.27646	−	1.42713i
$634$		0			0
$635$	670.820			1.05641
$636$		0			0
$637$			769.031i			1.20727i
$638$		0			0
$639$	−277.128	−	30.9839i	−0.433690	−	0.0484881i
$640$		0			0
$641$			751.319i			1.17210i	0.810273	+	0.586052i	$0.199319\pi$
$641$			751.319i			1.17210i	−0.810273	+	0.586052i	$0.800681\pi$
$642$		0			0
$643$			524.000i			0.814930i	0.913221	+	0.407465i	$0.133587\pi$
$643$			524.000i			0.814930i	−0.913221	+	0.407465i	$0.866413\pi$
$644$		0			0
$645$	−805.581	−	900.666i	−1.24896	−	1.39638i
$646$		0			0
$647$			340.823i			0.526774i	0.964690	+	0.263387i	$0.0848395\pi$
$647$			340.823i			0.526774i	−0.964690	+	0.263387i	$0.915160\pi$
$648$		0			0
$649$	−540.000			−0.832049
$650$		0			0
$651$	−187.830	+	168.000i	−0.288525	+	0.258065i
$652$		0			0
$653$	−627.423			−0.960832			−0.480416	−	0.877041i	$-0.659514\pi$
$653$	−627.423			−0.960832			−0.480416	+	0.877041i	$0.659514\pi$
$654$		0			0
$655$	−1143.15			−1.74527
$656$		0			0
$657$	74.0000	−	661.876i	0.112633	−	1.00742i
$658$		0			0
$659$	−1140.39			−1.73049			−0.865246	−	0.501347i	$-0.832838\pi$
$659$	−1140.39			−1.73049			−0.865246	+	0.501347i	$0.832838\pi$
$660$		0			0
$661$		−	450.333i		−	0.681291i	−0.940192	−	0.340645i	$-0.889354\pi$
$661$		−	450.333i		−	0.681291i	0.940192	−	0.340645i	$-0.110646\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	107.331i		−	0.161400i
$666$		0			0
$667$		−	240.000i		−	0.359820i
$668$		0			0
$669$	−379.552	+	339.482i	−0.567343	+	0.507447i
$670$		0			0
$671$		−	92.9516i		−	0.138527i
$672$		0			0
$673$	226.000			0.335810			0.167905	−	0.985803i	$-0.446300\pi$
$673$	226.000			0.335810			0.167905	+	0.985803i	$0.446300\pi$
$674$		0			0
$675$	−547.837	−	770.000i	−0.811610	−	1.14074i
$676$		0			0
$677$	−7.74597			−0.0114416			−0.00572080	−	0.999984i	$-0.501821\pi$
$677$	−7.74597			−0.0114416			−0.00572080	+	0.999984i	$0.501821\pi$
$678$		0			0
$679$	214.774			0.316310
$680$		0			0
$681$	690.000	−	617.155i	1.01322	−	0.906248i
$682$		0			0
$683$	415.909			0.608944			0.304472	−	0.952521i	$-0.401520\pi$
$683$	415.909			0.608944			0.304472	+	0.952521i	$0.401520\pi$
$684$		0			0
$685$			415.692i			0.606850i
$686$		0			0
$687$	817.528	+	914.024i	1.19000	+	1.33046i
$688$		0			0
$689$		−	1126.98i		−	1.63567i
$690$		0			0
$691$			572.000i			0.827786i	0.910326	+	0.413893i	$0.135831\pi$
$691$			572.000i			0.827786i	−0.910326	+	0.413893i	$0.864169\pi$
$692$		0			0
$693$	46.4758	−	415.692i	0.0670646	−	0.599844i
$694$		0			0
$695$		−	526.726i		−	0.757879i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	536.656	+	600.000i	0.767749	+	0.858369i
$700$		0			0
$701$	−441.520			−0.629843			−0.314922	−	0.949118i	$-0.601978\pi$
$701$	−441.520			−0.629843			−0.314922	+	0.949118i	$0.601978\pi$
$702$		0			0
$703$	−138.564			−0.197104
$704$		0			0
$705$	960.000	+	1073.31i	1.36170	+	1.52243i
$706$		0			0
$707$	456.158			0.645202
$708$		0			0
$709$			353.338i			0.498362i	0.968457	+	0.249181i	$0.0801613\pi$
$709$			353.338i			0.498362i	−0.968457	+	0.249181i	$0.919839\pi$
$710$		0			0
$711$	51.9615	−	464.758i	0.0730823	−	0.653668i
$712$		0			0
$713$			751.319i			1.05374i
$714$		0			0
$715$		−	2160.00i		−	3.02098i
$716$		0			0
$717$	−619.677	−	692.820i	−0.864264	−	0.966277i
$718$		0			0
$719$		−	557.710i		−	0.775674i	−0.921728	−	0.387837i	$-0.873222\pi$
$719$		−	557.710i		−	0.775674i	0.921728	−	0.387837i	$-0.126778\pi$
$720$		0			0
$721$	300.000			0.416089
$722$		0			0
$723$	165.469	−	148.000i	0.228864	−	0.204703i
$724$		0			0
$725$	−271.109			−0.373943
$726$		0			0
$727$	−696.284			−0.957750			−0.478875	−	0.877883i	$-0.658955\pi$
$727$	−696.284			−0.957750			−0.478875	+	0.877883i	$0.658955\pi$
$728$		0			0
$729$	−239.000	+	688.709i	−0.327846	+	0.944731i
$730$		0			0
$731$		0			0
$732$		0			0
$733$			173.205i			0.236296i	0.992996	+	0.118148i	$0.0376957\pi$
$733$			173.205i			0.236296i	−0.992996	+	0.118148i	$0.962304\pi$
$734$		0			0
$735$	640.859	−	573.202i	0.871917	−	0.779866i
$736$		0			0
$737$		−	375.659i		−	0.509714i
$738$		0			0
$739$		−	548.000i		−	0.741543i	−0.928724	−	0.370771i	$-0.879093\pi$
$739$		−	548.000i		−	0.741543i	0.928724	−	0.370771i	$-0.120907\pi$
$740$		0			0
$741$	−185.903	+	166.277i	−0.250882	+	0.224395i
$742$		0			0
$743$			1456.24i			1.95995i	0.199124	+	0.979974i	$0.436190\pi$
$743$			1456.24i			1.95995i	−0.199124	+	0.979974i	$0.563810\pi$
$744$		0			0
$745$	−900.000			−1.20805
$746$		0			0
$747$	−120.748	+	1080.00i	−0.161643	+	1.44578i
$748$		0			0
$749$	−139.427			−0.186151
$750$		0			0
$751$	329.090			0.438202			0.219101	−	0.975702i	$-0.429688\pi$
$751$	329.090			0.438202			0.219101	+	0.975702i	$0.429688\pi$
$752$		0			0
$753$	−30.0000	+	26.8328i	−0.0398406	+	0.0356345i
$754$		0			0
$755$	1958.80			2.59443
$756$		0			0
$757$			325.626i			0.430153i	0.976597	+	0.215076i	$0.0690000\pi$
$757$			325.626i			0.430153i	−0.976597	+	0.215076i	$0.931000\pi$
$758$		0			0
$759$	−831.384	−	929.516i	−1.09537	−	1.22466i
$760$		0			0
$761$		−	160.997i		−	0.211560i	−0.994390	−	0.105780i	$-0.966266\pi$
$761$		−	160.997i		−	0.211560i	0.994390	−	0.105780i	$-0.0337339\pi$
$762$		0			0
$763$			264.000i			0.346003i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	836.564i		−	1.09070i
$768$		0			0
$769$	−214.000			−0.278283			−0.139142	−	0.990272i	$-0.544434\pi$
$769$	−214.000			−0.278283			−0.139142	+	0.990272i	$0.544434\pi$
$770$		0			0
$771$	−429.325	−	480.000i	−0.556842	−	0.622568i
$772$		0			0
$773$	611.931			0.791632			0.395816	−	0.918330i	$-0.370462\pi$
$773$	611.931			0.791632			0.395816	+	0.918330i	$0.370462\pi$
$774$		0			0
$775$	848.705			1.09510
$776$		0			0
$777$	240.000	+	268.328i	0.308880	+	0.345339i
$778$		0			0
$779$	−214.663			−0.275562
$780$		0			0
$781$			415.692i			0.532256i
$782$		0			0
$783$	121.244	+	170.411i	0.154845	+	0.217639i
$784$		0			0
$785$		−	912.316i		−	1.16219i
$786$		0			0
$787$			1052.00i			1.33672i	0.743837	+	0.668361i	$0.233004\pi$
$787$			1052.00i			1.33672i	−0.743837	+	0.668361i	$0.766996\pi$
$788$		0			0
$789$	929.516	+	1039.23i	1.17809	+	1.31715i
$790$		0			0
$791$			371.806i			0.470046i
$792$		0			0
$793$	144.000			0.181589
$794$		0			0
$795$	−939.149	+	840.000i	−1.18132	+	1.05660i
$796$		0			0
$797$	1293.58			1.62306			0.811529	−	0.584313i	$-0.198636\pi$
$797$	1293.58			1.62306			0.811529	+	0.584313i	$0.198636\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−480.000	−	53.6656i	−0.599251	−	0.0669983i
$802$		0			0
$803$	−992.814			−1.23638
$804$		0			0
$805$			831.384i			1.03278i
$806$		0			0
$807$	675.500	−	604.185i	0.837051	−	0.748681i
$808$		0			0
$809$			375.659i			0.464350i	0.972674	+	0.232175i	$0.0745843\pi$
$809$			375.659i			0.464350i	−0.972674	+	0.232175i	$0.925416\pi$
$810$		0			0
$811$			916.000i			1.12947i	0.825272	+	0.564735i	$0.191022\pi$
$811$			916.000i			1.12947i	−0.825272	+	0.564735i	$0.808978\pi$
$812$		0			0
$813$	735.867	−	658.179i	0.905125	−	0.809569i
$814$		0			0
$815$			402.790i			0.494221i
$816$		0			0
$817$	208.000			0.254590
$818$		0			0
$819$	643.988	+	72.0000i	0.786310	+	0.0879121i
$820$		0			0
$821$	364.060			0.443435			0.221718	−	0.975111i	$-0.428834\pi$
$821$	364.060			0.443435			0.221718	+	0.975111i	$0.428834\pi$
$822$		0			0
$823$	384.515			0.467212			0.233606	−	0.972331i	$-0.424947\pi$
$823$	384.515			0.467212			0.233606	+	0.972331i	$0.424947\pi$
$824$		0			0
$825$	−1050.00	+	939.149i	−1.27273	+	1.13836i
$826$		0			0
$827$	−389.076			−0.470467			−0.235233	−	0.971939i	$-0.575585\pi$
$827$	−389.076			−0.470467			−0.235233	+	0.971939i	$0.575585\pi$
$828$		0			0
$829$		−	935.307i		−	1.12824i	−0.825694	−	0.564118i	$-0.809216\pi$
$829$		−	935.307i		−	1.12824i	0.825694	−	0.564118i	$-0.190784\pi$
$830$		0			0
$831$	318.697	+	356.314i	0.383511	+	0.428778i
$832$		0			0
$833$		0			0
$834$		0			0
$835$			240.000i			0.287425i
$836$		0			0
$837$	−379.552	−	533.472i	−0.453468	−	0.637362i
$838$		0			0
$839$		−	526.726i		−	0.627802i	−0.949456	−	0.313901i	$-0.898364\pi$
$839$		−	526.726i		−	0.627802i	0.949456	−	0.313901i	$-0.101636\pi$
$840$		0			0
$841$	−781.000			−0.928656
$842$		0			0
$843$	−536.656	−	600.000i	−0.636603	−	0.711744i
$844$		0			0
$845$	2037.19			2.41087
$846$		0			0
$847$	−204.382			−0.241301
$848$		0			0
$849$	−344.000	−	384.604i	−0.405183	−	0.453008i
$850$		0			0
$851$	1073.31			1.26124
$852$		0			0
$853$			325.626i			0.381742i	0.981615	+	0.190871i	$0.0611311\pi$
$853$			325.626i			0.381742i	−0.981615	+	0.190871i	$0.938869\pi$
$854$		0			0
$855$	277.128	+	30.9839i	0.324126	+	0.0362384i
$856$		0			0
$857$			1556.30i			1.81599i	0.418981	+	0.907995i	$0.362387\pi$
$857$			1556.30i			1.81599i	−0.418981	+	0.907995i	$0.637613\pi$
$858$		0			0
$859$		−	188.000i		−	0.218859i	−0.993995	−	0.109430i	$-0.965098\pi$
$859$		−	188.000i		−	0.218859i	0.993995	−	0.109430i	$-0.0349024\pi$
$860$		0			0
$861$	371.806	+	415.692i	0.431831	+	0.482802i
$862$		0			0
$863$		−	1239.35i		−	1.43610i	−0.695991	−	0.718050i	$-0.745035\pi$
$863$		−	1239.35i		−	1.43610i	0.695991	−	0.718050i	$-0.254965\pi$
$864$		0			0
$865$	60.0000			0.0693642
$866$		0			0
$867$	646.224	−	578.000i	0.745356	−	0.666667i
$868$		0			0
$869$	−697.137			−0.802229
$870$		0			0
$871$	581.969			0.668162
$872$		0			0
$873$	−62.0000	+	554.545i	−0.0710195	+	0.635217i
$874$		0			0
$875$	268.328			0.306661
$876$		0			0
$877$			1087.73i			1.24028i	0.784490	+	0.620141i	$0.212925\pi$
$877$			1087.73i			1.24028i	−0.784490	+	0.620141i	$0.787075\pi$
$878$		0			0
$879$	−710.141	+	635.169i	−0.807896	+	0.722604i
$880$		0			0
$881$			751.319i			0.852802i	0.904534	+	0.426401i	$0.140219\pi$
$881$			751.319i			0.852802i	−0.904534	+	0.426401i	$0.859781\pi$
$882$		0			0
$883$			668.000i			0.756512i	0.925701	+	0.378256i	$0.123476\pi$
$883$			668.000i			0.756512i	−0.925701	+	0.378256i	$0.876524\pi$
$884$		0			0
$885$	−697.137	+	623.538i	−0.787725	+	0.704563i
$886$		0			0
$887$		−	464.758i		−	0.523966i	−0.965072	−	0.261983i	$-0.915624\pi$
$887$		−	464.758i		−	0.523966i	0.965072	−	0.261983i	$-0.0843765\pi$
$888$		0			0
$889$	300.000			0.337458
$890$		0			0
$891$	1059.90	+	240.000i	1.18956	+	0.269360i
$892$		0			0
$893$	−247.871			−0.277571
$894$		0			0
$895$	519.615			0.580576
$896$		0			0
$897$	1440.00	−	1287.98i	1.60535	−	1.43587i
$898$		0			0
$899$	−187.830			−0.208932
$900$		0			0
$901$		0			0
$902$		0			0
$903$	−360.267	−	402.790i	−0.398966	−	0.446058i
$904$		0			0
$905$			53.6656i			0.0592990i
$906$		0			0
$907$		−	1196.00i		−	1.31863i	−0.751866	−	0.659316i	$-0.770846\pi$
$907$		−	1196.00i		−	1.31863i	0.751866	−	0.659316i	$-0.229154\pi$
$908$		0			0
$909$	−131.681	+	1177.79i	−0.144864	+	1.29570i
$910$		0			0
$911$		−	433.774i		−	0.476152i	−0.971247	−	0.238076i	$-0.923483\pi$
$911$		−	433.774i		−	0.476152i	0.971247	−	0.238076i	$-0.0765167\pi$
$912$		0			0
$913$	1620.00			1.77437
$914$		0			0
$915$	−107.331	−	120.000i	−0.117302	−	0.131148i
$916$		0			0
$917$	−511.234			−0.557507
$918$		0			0
$919$	1049.62			1.14214			0.571068	−	0.820903i	$-0.306529\pi$
$919$	1049.62			1.14214			0.571068	+	0.820903i	$0.306529\pi$
$920$		0			0
$921$	−424.000	−	474.046i	−0.460369	−	0.514708i
$922$		0			0
$923$	−643.988			−0.697711
$924$		0			0
$925$		−	1212.44i		−	1.31074i
$926$		0			0
$927$	−86.6025	+	774.597i	−0.0934224	+	0.835595i
$928$		0			0
$929$			107.331i			0.115534i	0.998330	+	0.0577671i	$0.0183981\pi$
$929$			107.331i			0.115534i	−0.998330	+	0.0577671i	$0.981602\pi$
$930$		0			0
$931$			148.000i			0.158969i
$932$		0			0
$933$	1053.45	+	1177.79i	1.12910	+	1.26237i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1502.00			−1.60299			−0.801494	−	0.598003i	$-0.795961\pi$
$937$	−1502.00			−1.60299			−0.801494	+	0.598003i	$0.795961\pi$
$938$		0			0
$939$	451.686	−	404.000i	0.481028	−	0.430245i
$940$		0			0
$941$	−1866.78			−1.98382			−0.991912	−	0.126929i	$-0.959488\pi$
$941$	−1866.78			−1.98382			−0.991912	+	0.126929i	$0.959488\pi$
$942$		0			0
$943$	1662.77			1.76328
$944$		0			0
$945$	−420.000	−	590.322i	−0.444444	−	0.624679i
$946$		0			0
$947$	1220.89			1.28922			0.644611	−	0.764511i	$-0.277019\pi$
$947$	1220.89			1.28922			0.644611	+	0.764511i	$0.277019\pi$
$948$		0			0
$949$		−	1538.06i		−	1.62072i
$950$		0			0
$951$	−155.885	+	139.427i	−0.163916	+	0.146611i
$952$		0			0
$953$			53.6656i			0.0563123i	0.999604	+	0.0281562i	$0.00896357\pi$
$953$			53.6656i			0.0563123i	−0.999604	+	0.0281562i	$0.991036\pi$
$954$		0			0
$955$			960.000i			1.00524i
$956$		0			0
$957$	232.379	−	207.846i	0.242820	−	0.217185i
$958$		0			0
$959$			185.903i			0.193851i
$960$		0			0
$961$	−373.000			−0.388137
$962$		0			0
$963$	40.2492	−	360.000i	0.0417957	−	0.373832i
$964$		0			0
$965$	170.411			0.176592
$966$		0			0
$967$	−1887.94			−1.95236			−0.976182	−	0.216955i	$-0.930388\pi$
$967$	−1887.94			−1.95236			−0.976182	+	0.216955i	$0.930388\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−872.067			−0.898112			−0.449056	−	0.893504i	$-0.648240\pi$
$971$	−872.067			−0.898112			−0.449056	+	0.893504i	$0.648240\pi$
$972$		0			0
$973$		−	235.559i		−	0.242095i
$974$		0			0
$975$	−1454.92	−	1626.65i	−1.49223	−	1.66836i
$976$		0			0
$977$		−	429.325i		−	0.439432i	−0.975564	−	0.219716i	$-0.929487\pi$
$977$		−	429.325i		−	0.439432i	0.975564	−	0.219716i	$-0.0705131\pi$
$978$		0			0
$979$			720.000i			0.735444i
$980$		0			0
$981$	−681.645	−	76.2102i	−0.694847	−	0.0776863i
$982$		0			0
$983$			650.661i			0.661914i	0.943646	+	0.330957i	$0.107371\pi$
$983$			650.661i			0.661914i	−0.943646	+	0.330957i	$0.892629\pi$
$984$		0			0
$985$	540.000			0.548223
$986$		0			0
$987$	429.325	+	480.000i	0.434980	+	0.486322i
$988$		0			0
$989$	−1611.16			−1.62908
$990$		0			0
$991$	−668.572			−0.674643			−0.337322	−	0.941389i	$-0.609521\pi$
$991$	−668.572			−0.674643			−0.337322	+	0.941389i	$0.609521\pi$
$992$		0			0
$993$	−56.0000	−	62.6099i	−0.0563948	−	0.0630513i
$994$		0			0
$995$	−2334.45			−2.34619
$996$		0			0
$997$		−	1143.15i		−	1.14659i	−0.819348	−	0.573297i	$-0.805664\pi$
$997$		−	1143.15i		−	1.14659i	0.819348	−	0.573297i	$-0.194336\pi$
$998$		0			0
$999$	−762.102	+	542.218i	−0.762865	+	0.542760i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.h.c.161.5	yes	8
3.2	odd	2	inner	192.3.h.c.161.2	yes	8
4.3	odd	2	inner	192.3.h.c.161.3	yes	8
8.3	odd	2	inner	192.3.h.c.161.6	yes	8
8.5	even	2	inner	192.3.h.c.161.4	yes	8
12.11	even	2	inner	192.3.h.c.161.8	yes	8
16.3	odd	4		768.3.e.g.257.2		4
16.5	even	4		768.3.e.g.257.1		4
16.11	odd	4		768.3.e.n.257.3		4
16.13	even	4		768.3.e.n.257.4		4
24.5	odd	2	inner	192.3.h.c.161.7	yes	8
24.11	even	2	inner	192.3.h.c.161.1	✓	8
48.5	odd	4		768.3.e.g.257.4		4
48.11	even	4		768.3.e.n.257.2		4
48.29	odd	4		768.3.e.n.257.1		4
48.35	even	4		768.3.e.g.257.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.3.h.c.161.1	✓	8	24.11	even	2	inner
192.3.h.c.161.2	yes	8	3.2	odd	2	inner
192.3.h.c.161.3	yes	8	4.3	odd	2	inner
192.3.h.c.161.4	yes	8	8.5	even	2	inner
192.3.h.c.161.5	yes	8	1.1	even	1	trivial
192.3.h.c.161.6	yes	8	8.3	odd	2	inner
192.3.h.c.161.7	yes	8	24.5	odd	2	inner
192.3.h.c.161.8	yes	8	12.11	even	2	inner
768.3.e.g.257.1		4	16.5	even	4
768.3.e.g.257.2		4	16.3	odd	4
768.3.e.g.257.3		4	48.35	even	4
768.3.e.g.257.4		4	48.5	odd	4
768.3.e.n.257.1		4	48.29	odd	4
768.3.e.n.257.2		4	48.11	even	4
768.3.e.n.257.3		4	16.11	odd	4
768.3.e.n.257.4		4	16.13	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	192.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.23607 - 2.00000i) q^{3} -7.74597 q^{5} -3.46410 q^{7} +(1.00000 - 8.94427i) q^{9} +O(q^{10})\)	\(q+(2.23607 - 2.00000i) q^{3} -7.74597 q^{5} -3.46410 q^{7} +(1.00000 - 8.94427i) q^{9} -13.4164 q^{11} -20.7846i q^{13} +(-17.3205 + 15.4919i) q^{15} -4.00000i q^{19} +(-7.74597 + 6.92820i) q^{21} +30.9839i q^{23} +35.0000 q^{25} +(-15.6525 - 22.0000i) q^{27} -7.74597 q^{29} +24.2487 q^{31} +(-30.0000 + 26.8328i) q^{33} +26.8328 q^{35} -34.6410i q^{37} +(-41.5692 - 46.4758i) q^{39} -53.6656i q^{41} +52.0000i q^{43} +(-7.74597 + 69.2820i) q^{45} -61.9677i q^{47} -37.0000 q^{49} +54.2218 q^{53} +103.923 q^{55} +(-8.00000 - 8.94427i) q^{57} +40.2492 q^{59} +6.92820i q^{61} +(-3.46410 + 30.9839i) q^{63} +160.997i q^{65} +28.0000i q^{67} +(61.9677 + 69.2820i) q^{69} -30.9839i q^{71} +74.0000 q^{73} +(78.2624 - 70.0000i) q^{75} +46.4758 q^{77} +51.9615 q^{79} +(-79.0000 - 17.8885i) q^{81} -120.748 q^{83} +(-17.3205 + 15.4919i) q^{87} -53.6656i q^{89} +72.0000i q^{91} +(54.2218 - 48.4974i) q^{93} +30.9839i q^{95} -62.0000 q^{97} +(-13.4164 + 120.000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 280 q^{25} - 240 q^{33} - 296 q^{49} - 64 q^{57} + 592 q^{73} - 632 q^{81} - 496 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 280 * q^25 - 240 * q^33 - 296 * q^49 - 64 * q^57 + 592 * q^73 - 632 * q^81 - 496 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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