LMFDB - Embedded newform 192.3.h.c.161.2

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			161.2
Root			$-0.535233 - 0.309017i$ of defining polynomial
Character	$\chi$	$=$	192.161
Dual form			192.3.h.c.161.4

$q$-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	$-0.217953\pi$
$5$	7.74597			1.54919			0.774597	+	0.632456i	$0.217953\pi$
$6$		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	$-0.291235\pi$
$11$	13.4164			1.21967			0.609837	+	0.792527i	$0.291235\pi$
$12$		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	$-0.764763\pi$
$23$		−	30.9839i		−	1.34712i	0.739130	−	0.673562i	$-0.235237\pi$
$24$		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	$-0.457362\pi$
$29$	7.74597			0.267102			0.133551	+	0.991042i	$0.457362\pi$
$30$		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	$0.227104\pi$
$41$			53.6656i			1.30892i	−0.756098	+	0.654459i	$0.772896\pi$
$42$		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	$0.229118\pi$
$47$			61.9677i			1.31846i	−0.751940	+	0.659231i	$0.770882\pi$
$48$		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	$-0.670920\pi$
$53$	−54.2218			−1.02305			−0.511526	+	0.859268i	$0.670920\pi$
$54$		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	$-0.610798\pi$
$59$	−40.2492			−0.682190			−0.341095	+	0.940029i	$0.610798\pi$
$60$		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	$0.0700173\pi$
$71$			30.9839i			0.436392i	−0.975905	+	0.218196i	$0.929983\pi$
$72$		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	$-0.240731\pi$
$83$	120.748			1.45479			0.727396	+	0.686218i	$0.240731\pi$
$84$		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	$0.0974848\pi$
$89$			53.6656i			0.602985i	−0.953469	+	0.301492i	$0.902515\pi$
$90$		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	$-0.273977\pi$
$101$	131.681			1.30378			0.651888	+	0.758315i	$0.273977\pi$
$102$		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	$-0.560227\pi$
$107$	−40.2492			−0.376161			−0.188080	+	0.982154i	$0.560227\pi$
$108$		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	$0.157522\pi$
$113$			107.331i			0.949834i	−0.880031	+	0.474917i	$0.842478\pi$
$114$		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	$-0.690462\pi$
$131$	−147.580			−1.12657			−0.563284	+	0.826263i	$0.690462\pi$
$132$		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	$0.0627498\pi$
$137$			53.6656i			0.391720i	−0.980632	+	0.195860i	$0.937250\pi$
$138$		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	$-0.627490\pi$
$149$	−116.190			−0.779795			−0.389898	+	0.920858i	$0.627490\pi$
$150$		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	$0.0295709\pi$
$167$			30.9839i			0.185532i	−0.995688	+	0.0927661i	$0.970429\pi$
$168$		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	$-0.492873\pi$
$173$	7.74597			0.0447744			0.0223872	+	0.999749i	$0.492873\pi$
$174$		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	$-0.440000\pi$
$179$	67.0820			0.374760			0.187380	+	0.982288i	$0.440000\pi$
$180$		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	$0.105175\pi$
$191$			123.935i			0.648877i	−0.945907	+	0.324438i	$0.894825\pi$
$192$		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	$-0.443381\pi$
$197$	69.7137			0.353877			0.176938	+	0.984222i	$0.443381\pi$
$198$		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	$-0.737884\pi$
$227$	−308.577			−1.35937			−0.679686	+	0.733503i	$0.737884\pi$
$228$		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	$-0.804686\pi$
$233$		−	268.328i		−	1.15162i	0.817582	−	0.575811i	$-0.195314\pi$
$234$		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	$0.224477\pi$
$239$			309.839i			1.29640i	−0.761472	+	0.648198i	$0.775523\pi$
$240$		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	$-0.491492\pi$
$251$	13.4164			0.0534518			0.0267259	+	0.999643i	$0.491492\pi$
$252$		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	$0.137140\pi$
$257$			214.663i			0.835263i	−0.908617	+	0.417631i	$0.862860\pi$
$258$		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	$-0.655133\pi$
$263$		−	464.758i		−	1.76714i	0.468298	−	0.883570i	$-0.344867\pi$
$264$		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	$-0.689780\pi$
$269$	−302.093			−1.12302			−0.561511	+	0.827470i	$0.689780\pi$
$270$		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	$0.158440\pi$
$281$			268.328i			0.954904i	−0.878658	+	0.477452i	$0.841560\pi$
$282$		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	$-0.317685\pi$
$293$	317.585			1.08391			0.541953	+	0.840409i	$0.317685\pi$
$294$		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	$-0.678511\pi$
$311$		−	526.726i		−	1.69365i	0.531870	−	0.846826i	$-0.321489\pi$
$312$		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	$-0.464928\pi$
$317$	69.7137			0.219917			0.109959	+	0.993936i	$0.464928\pi$
$318$		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	$-0.279780\pi$
$347$	442.741			1.27591			0.637956	+	0.770073i	$0.279780\pi$
$348$		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	$0.365589\pi$
$353$			643.988i			1.82433i	−0.409826	+	0.912164i	$0.634411\pi$
$354$		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	$0.189579\pi$
$359$			402.790i			1.12198i	−0.827823	+	0.560989i	$0.810421\pi$
$360$		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	$0.0517285\pi$
$383$			123.935i			0.323591i	−0.986824	+	0.161796i	$0.948271\pi$
$384$		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	$-0.684456\pi$
$389$	−426.028			−1.09519			−0.547594	+	0.836744i	$0.684456\pi$
$390$		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	$-0.703248\pi$
$401$		−	643.988i		−	1.60595i	0.596010	−	0.802977i	$-0.296752\pi$
$402$		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	$-0.620036\pi$
$419$	−308.577			−0.736462			−0.368231	+	0.929734i	$0.620036\pi$
$420$		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	$-0.832149\pi$
$431$		−	433.774i		−	1.00644i	0.864159	−	0.503218i	$-0.167851\pi$
$432$		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	$-0.495180\pi$
$443$	13.4164			0.0302853			0.0151427	+	0.999885i	$0.495180\pi$
$444$		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	$-0.796116\pi$
$449$		−	536.656i		−	1.19523i	0.801785	−	0.597613i	$-0.203884\pi$
$450$		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	$-0.677538\pi$
$461$	−487.996			−1.05856			−0.529280	+	0.848447i	$0.677538\pi$
$462$		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	$-0.607178\pi$
$467$	−308.577			−0.660765			−0.330383	+	0.943847i	$0.607178\pi$
$468$		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	$0.126871\pi$
$479$			371.806i			0.776214i	−0.921614	+	0.388107i	$0.873129\pi$
$480$		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	$-0.583586\pi$
$491$	−254.912			−0.519169			−0.259584	+	0.965720i	$0.583586\pi$
$492$		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	$0.250573\pi$
$503$			712.629i			1.41676i	−0.705833	+	0.708379i	$0.749427\pi$
$504$		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	$-0.917182\pi$
$509$	−983.738			−1.93269			−0.966344	+	0.257255i	$0.917182\pi$
$510$		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	$0.339496\pi$
$521$			912.316i			1.75109i	−0.483140	+	0.875543i	$0.660504\pi$
$522$		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	$-0.370281\pi$
$557$	441.520			0.792675			0.396338	+	0.918105i	$0.370281\pi$
$558$		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	$-0.465800\pi$
$563$	120.748			0.214472			0.107236	+	0.994234i	$0.465800\pi$
$564$		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	$-0.860478\pi$
$569$		−	482.991i		−	0.848841i	0.905465	−	0.424421i	$-0.139522\pi$
$570$		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	$-0.157575\pi$
$587$	1033.06			1.75990			0.879952	+	0.475063i	$0.157575\pi$
$588$		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	$0.0288460\pi$
$593$			107.331i			0.180997i	−0.995897	+	0.0904985i	$0.971154\pi$
$594$		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	$-0.991767\pi$
$599$		−	30.9839i		−	0.0517260i	0.999665	−	0.0258630i	$-0.00823336\pi$
$600$		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	$0.0984640\pi$
$617$			375.659i			0.608848i	−0.952537	+	0.304424i	$0.901536\pi$
$618$		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	$-0.800681\pi$
$641$		−	751.319i		−	1.17210i	0.810273	−	0.586052i	$-0.199319\pi$
$642$		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	$-0.915160\pi$
$647$		−	340.823i		−	0.526774i	0.964690	−	0.263387i	$-0.0848395\pi$
$648$		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	$-0.340486\pi$
$653$	627.423			0.960832			0.480416	+	0.877041i	$0.340486\pi$
$654$		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	$-0.167162\pi$
$659$	1140.39			1.73049			0.865246	+	0.501347i	$0.167162\pi$
$660$		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	$-0.498179\pi$
$677$	7.74597			0.0114416			0.00572080	+	0.999984i	$0.498179\pi$
$678$		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	$-0.598480\pi$
$683$	−415.909			−0.608944			−0.304472	+	0.952521i	$0.598480\pi$
$684$		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	$-0.398022\pi$
$701$	441.520			0.629843			0.314922	+	0.949118i	$0.398022\pi$
$702$		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	$0.126778\pi$
$719$			557.710i			0.775674i	−0.921728	+	0.387837i	$0.873222\pi$
$720$		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	$-0.563810\pi$
$743$		−	1456.24i		−	1.95995i	0.199124	−	0.979974i	$-0.436190\pi$
$744$		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	$0.0337339\pi$
$761$			160.997i			0.211560i	−0.994390	+	0.105780i	$0.966266\pi$
$762$		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	$-0.629538\pi$
$773$	−611.931			−0.791632			−0.395816	+	0.918330i	$0.629538\pi$
$774$		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	$-0.801364\pi$
$797$	−1293.58			−1.62306			−0.811529	+	0.584313i	$0.801364\pi$
$798$		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	$-0.925416\pi$
$809$		−	375.659i		−	0.464350i	0.972674	−	0.232175i	$-0.0745843\pi$
$810$		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	$-0.571166\pi$
$821$	−364.060			−0.443435			−0.221718	+	0.975111i	$0.571166\pi$
$822$		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	$-0.424415\pi$
$827$	389.076			0.470467			0.235233	+	0.971939i	$0.424415\pi$
$828$		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	$0.101636\pi$
$839$			526.726i			0.627802i	−0.949456	+	0.313901i	$0.898364\pi$
$840$		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	$-0.637613\pi$
$857$		−	1556.30i		−	1.81599i	0.418981	−	0.907995i	$-0.362387\pi$
$858$		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	$0.254965\pi$
$863$			1239.35i			1.43610i	−0.695991	+	0.718050i	$0.745035\pi$
$864$		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	$-0.859781\pi$
$881$		−	751.319i		−	0.852802i	0.904534	−	0.426401i	$-0.140219\pi$
$882$		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	$0.0843765\pi$
$887$			464.758i			0.523966i	−0.965072	+	0.261983i	$0.915624\pi$
$888$		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	$0.0765167\pi$
$911$			433.774i			0.476152i	−0.971247	+	0.238076i	$0.923483\pi$
$912$		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	$-0.981602\pi$
$929$		−	107.331i		−	0.115534i	0.998330	−	0.0577671i	$-0.0183981\pi$
$930$		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	$-0.0405121\pi$
$941$	1866.78			1.98382			0.991912	+	0.126929i	$0.0405121\pi$
$942$		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	$-0.722981\pi$
$947$	−1220.89			−1.28922			−0.644611	+	0.764511i	$0.722981\pi$
$948$		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	$-0.991036\pi$
$953$		−	53.6656i		−	0.0563123i	0.999604	−	0.0281562i	$-0.00896357\pi$
$954$		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	$-0.351760\pi$
$971$	872.067			0.898112			0.449056	+	0.893504i	$0.351760\pi$
$972$		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	$0.0705131\pi$
$977$			429.325i			0.439432i	−0.975564	+	0.219716i	$0.929487\pi$
$978$		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	$-0.892629\pi$
$983$		−	650.661i		−	0.661914i	0.943646	−	0.330957i	$-0.107371\pi$
$984$		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.161.2	yes	8
3.2	odd	2	inner	192.3.h.c.161.5	yes	8
4.3	odd	2	inner	192.3.h.c.161.8	yes	8
8.3	odd	2	inner	192.3.h.c.161.1	✓	8
8.5	even	2	inner	192.3.h.c.161.7	yes	8
12.11	even	2	inner	192.3.h.c.161.3	yes	8
16.3	odd	4		768.3.e.g.257.3		4
16.5	even	4		768.3.e.g.257.4		4
16.11	odd	4		768.3.e.n.257.2		4
16.13	even	4		768.3.e.n.257.1		4
24.5	odd	2	inner	192.3.h.c.161.4	yes	8
24.11	even	2	inner	192.3.h.c.161.6	yes	8
48.5	odd	4		768.3.e.g.257.1		4
48.11	even	4		768.3.e.n.257.3		4
48.29	odd	4		768.3.e.n.257.4		4
48.35	even	4		768.3.e.g.257.2		4

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