LMFDB - Embedded newform 192.3.g.c.127.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,3,Mod(127,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("192.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$5.23162107572$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 96)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.127
Dual form			192.3.g.c.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +8.92820 q^{5} -10.9282i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +8.92820 q^{5} -10.9282i q^{7} -3.00000 q^{9} -1.07180i q^{11} +3.85641 q^{13} +15.4641i q^{15} +7.85641 q^{17} +17.0718i q^{19} +18.9282 q^{21} -8.00000i q^{23} +54.7128 q^{25} -5.19615i q^{27} +3.07180 q^{29} +30.6410i q^{31} +1.85641 q^{33} -97.5692i q^{35} -45.7128 q^{37} +6.67949i q^{39} -35.8564 q^{41} +74.6410i q^{43} -26.7846 q^{45} -42.1436i q^{47} -70.4256 q^{49} +13.6077i q^{51} -12.9282 q^{53} -9.56922i q^{55} -29.5692 q^{57} +44.2102i q^{59} +14.0000 q^{61} +32.7846i q^{63} +34.4308 q^{65} -80.4974i q^{67} +13.8564 q^{69} -123.138i q^{71} -85.4256 q^{73} +94.7654i q^{75} -11.7128 q^{77} +55.2154i q^{79} +9.00000 q^{81} +49.0718i q^{83} +70.1436 q^{85} +5.32051i q^{87} -105.713 q^{89} -42.1436i q^{91} -53.0718 q^{93} +152.420i q^{95} +21.1384 q^{97} +3.21539i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 8 * q^5 - 12 * q^9	$ 4 q + 8 q^{5} - 12 q^{9} - 40 q^{13} - 24 q^{17} + 48 q^{21} + 108 q^{25} + 40 q^{29} - 48 q^{33} - 72 q^{37} - 88 q^{41} - 24 q^{45} - 60 q^{49} - 24 q^{53} + 48 q^{57} + 56 q^{61} + 304 q^{65} - 120 q^{73} + 64 q^{77} + 36 q^{81} + 336 q^{85} - 312 q^{89} - 240 q^{93} - 248 q^{97}+O(q^{100}) $ 4 * q + 8 * q^5 - 12 * q^9 - 40 * q^13 - 24 * q^17 + 48 * q^21 + 108 * q^25 + 40 * q^29 - 48 * q^33 - 72 * q^37 - 88 * q^41 - 24 * q^45 - 60 * q^49 - 24 * q^53 + 48 * q^57 + 56 * q^61 + 304 * q^65 - 120 * q^73 + 64 * q^77 + 36 * q^81 + 336 * q^85 - 312 * q^89 - 240 * q^93 - 248 * 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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	8.92820	1.78564	0.892820	−	0.450413i	$-0.148723\pi$
$5$	8.92820	1.78564	0.892820	+	0.450413i	$0.148723\pi$
$6$	0	0
$7$	− 10.9282i	− 1.56117i	−0.625048	−	0.780586i	$-0.714921\pi$
$7$	− 10.9282i	− 1.56117i	0.625048	−	0.780586i	$-0.285079\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 1.07180i	− 0.0974361i	−0.998813	−	0.0487180i	$-0.984486\pi$
$11$	− 1.07180i	− 0.0974361i	0.998813	−	0.0487180i	$-0.0155136\pi$
$12$	0	0
$13$	3.85641	0.296647	0.148323	−	0.988939i	$-0.452612\pi$
$13$	3.85641	0.296647	0.148323	+	0.988939i	$0.452612\pi$
$14$	0	0
$15$	15.4641i	1.03094i
$16$	0	0
$17$	7.85641	0.462142	0.231071	−	0.972937i	$-0.425777\pi$
$17$	7.85641	0.462142	0.231071	+	0.972937i	$0.425777\pi$
$18$	0	0
$19$	17.0718i	0.898516i	0.893402	+	0.449258i	$0.148312\pi$
$19$	17.0718i	0.898516i	−0.893402	+	0.449258i	$0.851688\pi$
$20$	0	0
$21$	18.9282	0.901343
$22$	0	0
$23$	− 8.00000i	− 0.347826i	−0.984761	−	0.173913i	$-0.944359\pi$
$23$	− 8.00000i	− 0.347826i	0.984761	−	0.173913i	$-0.0556412\pi$
$24$	0	0
$25$	54.7128	2.18851
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	3.07180	0.105924	0.0529620	−	0.998597i	$-0.483134\pi$
$29$	3.07180	0.105924	0.0529620	+	0.998597i	$0.483134\pi$
$30$	0	0
$31$	30.6410i	0.988420i	0.869343	+	0.494210i	$0.164543\pi$
$31$	30.6410i	0.988420i	−0.869343	+	0.494210i	$0.835457\pi$
$32$	0	0
$33$	1.85641	0.0562547
$34$	0	0
$35$	− 97.5692i	− 2.78769i
$36$	0	0
$37$	−45.7128	−1.23548	−0.617741	−	0.786382i	$-0.711952\pi$
$37$	−45.7128	−1.23548	−0.617741	+	0.786382i	$0.711952\pi$
$38$	0	0
$39$	6.67949i	0.171269i
$40$	0	0
$41$	−35.8564	−0.874546	−0.437273	−	0.899329i	$-0.644056\pi$
$41$	−35.8564	−0.874546	−0.437273	+	0.899329i	$0.644056\pi$
$42$	0	0
$43$	74.6410i	1.73584i	0.496706	+	0.867919i	$0.334543\pi$
$43$	74.6410i	1.73584i	−0.496706	+	0.867919i	$0.665457\pi$
$44$	0	0
$45$	−26.7846	−0.595214
$46$	0	0
$47$	− 42.1436i	− 0.896672i	−0.893865	−	0.448336i	$-0.852017\pi$
$47$	− 42.1436i	− 0.896672i	0.893865	−	0.448336i	$-0.147983\pi$
$48$	0	0
$49$	−70.4256	−1.43726
$50$	0	0
$51$	13.6077i	0.266818i
$52$	0	0
$53$	−12.9282	−0.243928	−0.121964	−	0.992535i	$-0.538919\pi$
$53$	−12.9282	−0.243928	−0.121964	+	0.992535i	$0.538919\pi$
$54$	0	0
$55$	− 9.56922i	− 0.173986i
$56$	0	0
$57$	−29.5692	−0.518758
$58$	0	0
$59$	44.2102i	0.749326i	0.927161	+	0.374663i	$0.122242\pi$
$59$	44.2102i	0.749326i	−0.927161	+	0.374663i	$0.877758\pi$
$60$	0	0
$61$	14.0000	0.229508	0.114754	−	0.993394i	$-0.463392\pi$
$61$	14.0000	0.229508	0.114754	+	0.993394i	$0.463392\pi$
$62$	0	0
$63$	32.7846i	0.520391i
$64$	0	0
$65$	34.4308	0.529704
$66$	0	0
$67$	− 80.4974i	− 1.20145i	−0.799454	−	0.600727i	$-0.794878\pi$
$67$	− 80.4974i	− 1.20145i	0.799454	−	0.600727i	$-0.205122\pi$
$68$	0	0
$69$	13.8564	0.200817
$70$	0	0
$71$	− 123.138i	− 1.73434i	−0.498009	−	0.867172i	$-0.665935\pi$
$71$	− 123.138i	− 1.73434i	0.498009	−	0.867172i	$-0.334065\pi$
$72$	0	0
$73$	−85.4256	−1.17021	−0.585107	−	0.810956i	$-0.698948\pi$
$73$	−85.4256	−1.17021	−0.585107	+	0.810956i	$0.698948\pi$
$74$	0	0
$75$	94.7654i	1.26354i
$76$	0	0
$77$	−11.7128	−0.152114
$78$	0	0
$79$	55.2154i	0.698929i	0.936950	+	0.349464i	$0.113636\pi$
$79$	55.2154i	0.698929i	−0.936950	+	0.349464i	$0.886364\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	49.0718i	0.591226i	0.955308	+	0.295613i	$0.0955240\pi$
$83$	49.0718i	0.591226i	−0.955308	+	0.295613i	$0.904476\pi$
$84$	0	0
$85$	70.1436	0.825219
$86$	0	0
$87$	5.32051i	0.0611553i
$88$	0	0
$89$	−105.713	−1.18778	−0.593892	−	0.804545i	$-0.702409\pi$
$89$	−105.713	−1.18778	−0.593892	+	0.804545i	$0.702409\pi$
$90$	0	0
$91$	− 42.1436i	− 0.463116i
$92$	0	0
$93$	−53.0718	−0.570664
$94$	0	0
$95$	152.420i	1.60443i
$96$	0	0
$97$	21.1384	0.217922	0.108961	−	0.994046i	$-0.465248\pi$
$97$	21.1384	0.217922	0.108961	+	0.994046i	$0.465248\pi$
$98$	0	0
$99$	3.21539i	0.0324787i
$100$	0	0
$101$	−172.928	−1.71216	−0.856080	−	0.516843i	$-0.827107\pi$
$101$	−172.928	−1.71216	−0.856080	+	0.516843i	$0.827107\pi$
$102$	0	0
$103$	− 16.7846i	− 0.162957i	−0.996675	−	0.0814787i	$-0.974036\pi$
$103$	− 16.7846i	− 0.162957i	0.996675	−	0.0814787i	$-0.0259643\pi$
$104$	0	0
$105$	168.995	1.60947
$106$	0	0
$107$	− 138.067i	− 1.29034i	−0.764038	−	0.645171i	$-0.776786\pi$
$107$	− 138.067i	− 1.29034i	0.764038	−	0.645171i	$-0.223214\pi$
$108$	0	0
$109$	−7.85641	−0.0720771	−0.0360386	−	0.999350i	$-0.511474\pi$
$109$	−7.85641	−0.0720771	−0.0360386	+	0.999350i	$0.511474\pi$
$110$	0	0
$111$	− 79.1769i	− 0.713306i
$112$	0	0
$113$	165.138	1.46140	0.730701	−	0.682698i	$-0.239193\pi$
$113$	165.138	1.46140	0.730701	+	0.682698i	$0.239193\pi$
$114$	0	0
$115$	− 71.4256i	− 0.621092i
$116$	0	0
$117$	−11.5692	−0.0988822
$118$	0	0
$119$	− 85.8564i	− 0.721482i
$120$	0	0
$121$	119.851	0.990506
$122$	0	0
$123$	− 62.1051i	− 0.504920i
$124$	0	0
$125$	265.282	2.12226
$126$	0	0
$127$	147.349i	1.16023i	0.814536	+	0.580113i	$0.196992\pi$
$127$	147.349i	1.16023i	−0.814536	+	0.580113i	$0.803008\pi$
$128$	0	0
$129$	−129.282	−1.00219
$130$	0	0
$131$	70.9282i	0.541437i	0.962659	+	0.270718i	$0.0872612\pi$
$131$	70.9282i	0.541437i	−0.962659	+	0.270718i	$0.912739\pi$
$132$	0	0
$133$	186.564	1.40274
$134$	0	0
$135$	− 46.3923i	− 0.343647i
$136$	0	0
$137$	206.133	1.50462	0.752311	−	0.658808i	$-0.228939\pi$
$137$	206.133	1.50462	0.752311	+	0.658808i	$0.228939\pi$
$138$	0	0
$139$	39.9230i	0.287216i	0.989635	+	0.143608i	$0.0458705\pi$
$139$	39.9230i	0.287216i	−0.989635	+	0.143608i	$0.954130\pi$
$140$	0	0
$141$	72.9948	0.517694
$142$	0	0
$143$	− 4.13328i	− 0.0289041i
$144$	0	0
$145$	27.4256	0.189142
$146$	0	0
$147$	− 121.981i	− 0.829801i
$148$	0	0
$149$	0.353829	0.00237469	0.00118735	−	0.999999i	$-0.499622\pi$
$149$	0.353829	0.00237469	0.00118735	+	0.999999i	$0.499622\pi$
$150$	0	0
$151$	104.210i	0.690134i	0.938578	+	0.345067i	$0.112144\pi$
$151$	104.210i	0.690134i	−0.938578	+	0.345067i	$0.887856\pi$
$152$	0	0
$153$	−23.5692	−0.154047
$154$	0	0
$155$	273.569i	1.76496i
$156$	0	0
$157$	−221.713	−1.41218	−0.706092	−	0.708120i	$-0.749543\pi$
$157$	−221.713	−1.41218	−0.706092	+	0.708120i	$0.749543\pi$
$158$	0	0
$159$	− 22.3923i	− 0.140832i
$160$	0	0
$161$	−87.4256	−0.543016
$162$	0	0
$163$	175.923i	1.07928i	0.841895	+	0.539641i	$0.181440\pi$
$163$	175.923i	1.07928i	−0.841895	+	0.539641i	$0.818560\pi$
$164$	0	0
$165$	16.5744	0.100451
$166$	0	0
$167$	237.856i	1.42429i	0.702032	+	0.712145i	$0.252276\pi$
$167$	237.856i	1.42429i	−0.702032	+	0.712145i	$0.747724\pi$
$168$	0	0
$169$	−154.128	−0.912001
$170$	0	0
$171$	− 51.2154i	− 0.299505i
$172$	0	0
$173$	71.7795	0.414910	0.207455	−	0.978245i	$-0.433482\pi$
$173$	71.7795	0.414910	0.207455	+	0.978245i	$0.433482\pi$
$174$	0	0
$175$	− 597.913i	− 3.41664i
$176$	0	0
$177$	−76.5744	−0.432624
$178$	0	0
$179$	− 5.93336i	− 0.0331473i	−0.999863	−	0.0165736i	$-0.994724\pi$
$179$	− 5.93336i	− 0.0331473i	0.999863	−	0.0165736i	$-0.00527579\pi$
$180$	0	0
$181$	150.995	0.834226	0.417113	−	0.908855i	$-0.363042\pi$
$181$	150.995	0.834226	0.417113	+	0.908855i	$0.363042\pi$
$182$	0	0
$183$	24.2487i	0.132507i
$184$	0	0
$185$	−408.133	−2.20613
$186$	0	0
$187$	− 8.42047i	− 0.0450293i
$188$	0	0
$189$	−56.7846	−0.300448
$190$	0	0
$191$	23.4256i	0.122647i	0.998118	+	0.0613236i	$0.0195322\pi$
$191$	23.4256i	0.122647i	−0.998118	+	0.0613236i	$0.980468\pi$
$192$	0	0
$193$	137.426	0.712050	0.356025	−	0.934476i	$-0.384132\pi$
$193$	137.426	0.712050	0.356025	+	0.934476i	$0.384132\pi$
$194$	0	0
$195$	59.6359i	0.305825i
$196$	0	0
$197$	−39.4923	−0.200468	−0.100234	−	0.994964i	$-0.531959\pi$
$197$	−39.4923	−0.200468	−0.100234	+	0.994964i	$0.531959\pi$
$198$	0	0
$199$	− 275.923i	− 1.38655i	−0.720674	−	0.693274i	$-0.756168\pi$
$199$	− 275.923i	− 1.38655i	0.720674	−	0.693274i	$-0.243832\pi$
$200$	0	0
$201$	139.426	0.693660
$202$	0	0
$203$	− 33.5692i	− 0.165366i
$204$	0	0
$205$	−320.133	−1.56163
$206$	0	0
$207$	24.0000i	0.115942i
$208$	0	0
$209$	18.2975	0.0875478
$210$	0	0
$211$	− 189.359i	− 0.897436i	−0.893673	−	0.448718i	$-0.851881\pi$
$211$	− 189.359i	− 0.897436i	0.893673	−	0.448718i	$-0.148119\pi$
$212$	0	0
$213$	213.282	1.00132
$214$	0	0
$215$	666.410i	3.09958i
$216$	0	0
$217$	334.851	1.54309
$218$	0	0
$219$	− 147.962i	− 0.675623i
$220$	0	0
$221$	30.2975	0.137093
$222$	0	0
$223$	144.210i	0.646683i	0.946282	+	0.323341i	$0.104806\pi$
$223$	144.210i	0.646683i	−0.946282	+	0.323341i	$0.895194\pi$
$224$	0	0
$225$	−164.138	−0.729504
$226$	0	0
$227$	225.072i	0.991506i	0.868464	+	0.495753i	$0.165108\pi$
$227$	225.072i	0.991506i	−0.868464	+	0.495753i	$0.834892\pi$
$228$	0	0
$229$	−233.846	−1.02116	−0.510581	−	0.859830i	$-0.670570\pi$
$229$	−233.846	−1.02116	−0.510581	+	0.859830i	$0.670570\pi$
$230$	0	0
$231$	− 20.2872i	− 0.0878233i
$232$	0	0
$233$	−218.862	−0.939320	−0.469660	−	0.882847i	$-0.655623\pi$
$233$	−218.862	−0.939320	−0.469660	+	0.882847i	$0.655623\pi$
$234$	0	0
$235$	− 376.267i	− 1.60113i
$236$	0	0
$237$	−95.6359	−0.403527
$238$	0	0
$239$	− 371.979i	− 1.55640i	−0.628017	−	0.778200i	$-0.716133\pi$
$239$	− 371.979i	− 1.55640i	0.628017	−	0.778200i	$-0.283867\pi$
$240$	0	0
$241$	−328.564	−1.36334	−0.681668	−	0.731661i	$-0.738745\pi$
$241$	−328.564	−1.36334	−0.681668	+	0.731661i	$0.738745\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−628.774	−2.56643
$246$	0	0
$247$	65.8358i	0.266542i
$248$	0	0
$249$	−84.9948	−0.341345
$250$	0	0
$251$	− 148.210i	− 0.590479i	−0.955423	−	0.295240i	$-0.904601\pi$
$251$	− 148.210i	− 0.590479i	0.955423	−	0.295240i	$-0.0953994\pi$
$252$	0	0
$253$	−8.57437	−0.0338908
$254$	0	0
$255$	121.492i	0.476440i
$256$	0	0
$257$	351.703	1.36849	0.684246	−	0.729251i	$-0.260131\pi$
$257$	351.703	1.36849	0.684246	+	0.729251i	$0.260131\pi$
$258$	0	0
$259$	499.559i	1.92880i
$260$	0	0
$261$	−9.21539	−0.0353080
$262$	0	0
$263$	− 242.144i	− 0.920698i	−0.887738	−	0.460349i	$-0.847724\pi$
$263$	− 242.144i	− 0.920698i	0.887738	−	0.460349i	$-0.152276\pi$
$264$	0	0
$265$	−115.426	−0.435568
$266$	0	0
$267$	− 183.100i	− 0.685768i
$268$	0	0
$269$	75.6462	0.281213	0.140606	−	0.990066i	$-0.455095\pi$
$269$	75.6462	0.281213	0.140606	+	0.990066i	$0.455095\pi$
$270$	0	0
$271$	150.795i	0.556439i	0.960518	+	0.278219i	$0.0897442\pi$
$271$	150.795i	0.556439i	−0.960518	+	0.278219i	$0.910256\pi$
$272$	0	0
$273$	72.9948	0.267380
$274$	0	0
$275$	− 58.6410i	− 0.213240i
$276$	0	0
$277$	−213.559	−0.770971	−0.385485	−	0.922714i	$-0.625966\pi$
$277$	−213.559	−0.770971	−0.385485	+	0.922714i	$0.625966\pi$
$278$	0	0
$279$	− 91.9230i	− 0.329473i
$280$	0	0
$281$	194.000	0.690391	0.345196	−	0.938531i	$-0.387813\pi$
$281$	194.000	0.690391	0.345196	+	0.938531i	$0.387813\pi$
$282$	0	0
$283$	− 179.790i	− 0.635300i	−0.948208	−	0.317650i	$-0.897106\pi$
$283$	− 179.790i	− 0.635300i	0.948208	−	0.317650i	$-0.102894\pi$
$284$	0	0
$285$	−264.000	−0.926316
$286$	0	0
$287$	391.846i	1.36532i
$288$	0	0
$289$	−227.277	−0.786425
$290$	0	0
$291$	36.6128i	0.125817i
$292$	0	0
$293$	517.902	1.76759	0.883793	−	0.467879i	$-0.154982\pi$
$293$	517.902	1.76759	0.883793	+	0.467879i	$0.154982\pi$
$294$	0	0
$295$	394.718i	1.33803i
$296$	0	0
$297$	−5.56922	−0.0187516
$298$	0	0
$299$	− 30.8513i	− 0.103181i
$300$	0	0
$301$	815.692	2.70994
$302$	0	0
$303$	− 299.520i	− 0.988516i
$304$	0	0
$305$	124.995	0.409819
$306$	0	0
$307$	407.769i	1.32824i	0.747627	+	0.664119i	$0.231193\pi$
$307$	407.769i	1.32824i	−0.747627	+	0.664119i	$0.768807\pi$
$308$	0	0
$309$	29.0718	0.0940835
$310$	0	0
$311$	− 368.995i	− 1.18648i	−0.805026	−	0.593239i	$-0.797849\pi$
$311$	− 368.995i	− 1.18648i	0.805026	−	0.593239i	$-0.202151\pi$
$312$	0	0
$313$	341.979	1.09259	0.546293	−	0.837594i	$-0.316039\pi$
$313$	341.979	1.09259	0.546293	+	0.837594i	$0.316039\pi$
$314$	0	0
$315$	292.708i	0.929231i
$316$	0	0
$317$	396.487	1.25075	0.625374	−	0.780325i	$-0.284946\pi$
$317$	396.487	1.25075	0.625374	+	0.780325i	$0.284946\pi$
$318$	0	0
$319$	− 3.29234i	− 0.0103208i
$320$	0	0
$321$	239.138	0.744980
$322$	0	0
$323$	134.123i	0.415241i
$324$	0	0
$325$	210.995	0.649215
$326$	0	0
$327$	− 13.6077i	− 0.0416137i
$328$	0	0
$329$	−460.554	−1.39986
$330$	0	0
$331$	267.061i	0.806832i	0.915017	+	0.403416i	$0.132177\pi$
$331$	267.061i	0.806832i	−0.915017	+	0.403416i	$0.867823\pi$
$332$	0	0
$333$	137.138	0.411827
$334$	0	0
$335$	− 718.697i	− 2.14537i
$336$	0	0
$337$	−149.426	−0.443399	−0.221700	−	0.975115i	$-0.571160\pi$
$337$	−149.426	−0.443399	−0.221700	+	0.975115i	$0.571160\pi$
$338$	0	0
$339$	286.028i	0.843741i
$340$	0	0
$341$	32.8409	0.0963077
$342$	0	0
$343$	234.144i	0.682634i
$344$	0	0
$345$	123.713	0.358588
$346$	0	0
$347$	− 529.913i	− 1.52713i	−0.645733	−	0.763563i	$-0.723448\pi$
$347$	− 529.913i	− 1.52713i	0.645733	−	0.763563i	$-0.276552\pi$
$348$	0	0
$349$	75.7025	0.216913	0.108456	−	0.994101i	$-0.465409\pi$
$349$	75.7025	0.216913	0.108456	+	0.994101i	$0.465409\pi$
$350$	0	0
$351$	− 20.0385i	− 0.0570897i
$352$	0	0
$353$	325.138	0.921072	0.460536	−	0.887641i	$-0.347657\pi$
$353$	325.138	0.921072	0.460536	+	0.887641i	$0.347657\pi$
$354$	0	0
$355$	− 1099.41i	− 3.09692i
$356$	0	0
$357$	148.708	0.416548
$358$	0	0
$359$	− 195.713i	− 0.545161i	−0.962133	−	0.272581i	$-0.912123\pi$
$359$	− 195.713i	− 0.545161i	0.962133	−	0.272581i	$-0.0878771\pi$
$360$	0	0
$361$	69.5538	0.192670
$362$	0	0
$363$	207.588i	0.571869i
$364$	0	0
$365$	−762.697	−2.08958
$366$	0	0
$367$	− 257.051i	− 0.700412i	−0.936673	−	0.350206i	$-0.886112\pi$
$367$	− 257.051i	− 0.700412i	0.936673	−	0.350206i	$-0.113888\pi$
$368$	0	0
$369$	107.569	0.291515
$370$	0	0
$371$	141.282i	0.380814i
$372$	0	0
$373$	339.128	0.909191	0.454595	−	0.890698i	$-0.349784\pi$
$373$	339.128	0.909191	0.454595	+	0.890698i	$0.349784\pi$
$374$	0	0
$375$	459.482i	1.22529i
$376$	0	0
$377$	11.8461	0.0314220
$378$	0	0
$379$	163.215i	0.430647i	0.976543	+	0.215324i	$0.0690807\pi$
$379$	163.215i	0.430647i	−0.976543	+	0.215324i	$0.930919\pi$
$380$	0	0
$381$	−255.215	−0.669857
$382$	0	0
$383$	76.5538i	0.199879i	0.994994	+	0.0999396i	$0.0318650\pi$
$383$	76.5538i	0.199879i	−0.994994	+	0.0999396i	$0.968135\pi$
$384$	0	0
$385$	−104.574	−0.271622
$386$	0	0
$387$	− 223.923i	− 0.578613i
$388$	0	0
$389$	−338.477	−0.870120	−0.435060	−	0.900401i	$-0.643273\pi$
$389$	−338.477	−0.870120	−0.435060	+	0.900401i	$0.643273\pi$
$390$	0	0
$391$	− 62.8513i	− 0.160745i
$392$	0	0
$393$	−122.851	−0.312599
$394$	0	0
$395$	492.974i	1.24804i
$396$	0	0
$397$	7.72312	0.0194537	0.00972685	−	0.999953i	$-0.496904\pi$
$397$	7.72312	0.0194537	0.00972685	+	0.999953i	$0.496904\pi$
$398$	0	0
$399$	323.138i	0.809871i
$400$	0	0
$401$	448.123	1.11751	0.558757	−	0.829332i	$-0.311279\pi$
$401$	448.123	1.11751	0.558757	+	0.829332i	$0.311279\pi$
$402$	0	0
$403$	118.164i	0.293211i
$404$	0	0
$405$	80.3538	0.198405
$406$	0	0
$407$	48.9948i	0.120380i
$408$	0	0
$409$	−317.692	−0.776754	−0.388377	−	0.921501i	$-0.626964\pi$
$409$	−317.692	−0.776754	−0.388377	+	0.921501i	$0.626964\pi$
$410$	0	0
$411$	357.033i	0.868694i
$412$	0	0
$413$	483.138	1.16983
$414$	0	0
$415$	438.123i	1.05572i
$416$	0	0
$417$	−69.1487	−0.165824
$418$	0	0
$419$	− 660.056i	− 1.57531i	−0.616114	−	0.787657i	$-0.711294\pi$
$419$	− 660.056i	− 1.57531i	0.616114	−	0.787657i	$-0.288706\pi$
$420$	0	0
$421$	42.4411	0.100810	0.0504051	−	0.998729i	$-0.483949\pi$
$421$	42.4411	0.100810	0.0504051	+	0.998729i	$0.483949\pi$
$422$	0	0
$423$	126.431i	0.298891i
$424$	0	0
$425$	429.846	1.01140
$426$	0	0
$427$	− 152.995i	− 0.358302i
$428$	0	0
$429$	7.15906	0.0166878
$430$	0	0
$431$	302.123i	0.700981i	0.936566	+	0.350491i	$0.113985\pi$
$431$	302.123i	0.700981i	−0.936566	+	0.350491i	$0.886015\pi$
$432$	0	0
$433$	376.277	0.869000	0.434500	−	0.900672i	$-0.356925\pi$
$433$	376.277	0.869000	0.434500	+	0.900672i	$0.356925\pi$
$434$	0	0
$435$	47.5026i	0.109201i
$436$	0	0
$437$	136.574	0.312527
$438$	0	0
$439$	394.928i	0.899609i	0.893127	+	0.449804i	$0.148506\pi$
$439$	394.928i	0.899609i	−0.893127	+	0.449804i	$0.851494\pi$
$440$	0	0
$441$	211.277	0.479086
$442$	0	0
$443$	− 666.487i	− 1.50449i	−0.658886	−	0.752243i	$-0.728972\pi$
$443$	− 666.487i	− 1.50449i	0.658886	−	0.752243i	$-0.271028\pi$
$444$	0	0
$445$	−943.825	−2.12096
$446$	0	0
$447$	0.612850i	0.00137103i
$448$	0	0
$449$	−793.825	−1.76799	−0.883993	−	0.467501i	$-0.845155\pi$
$449$	−793.825	−1.76799	−0.883993	+	0.467501i	$0.845155\pi$
$450$	0	0
$451$	38.4308i	0.0852124i
$452$	0	0
$453$	−180.497	−0.398449
$454$	0	0
$455$	− 376.267i	− 0.826959i
$456$	0	0
$457$	693.138	1.51671	0.758357	−	0.651839i	$-0.226002\pi$
$457$	693.138	1.51671	0.758357	+	0.651839i	$0.226002\pi$
$458$	0	0
$459$	− 40.8231i	− 0.0889392i
$460$	0	0
$461$	−99.0924	−0.214951	−0.107476	−	0.994208i	$-0.534277\pi$
$461$	−99.0924	−0.214951	−0.107476	+	0.994208i	$0.534277\pi$
$462$	0	0
$463$	− 308.077i	− 0.665393i	−0.943034	−	0.332696i	$-0.892042\pi$
$463$	− 308.077i	− 0.665393i	0.943034	−	0.332696i	$-0.107958\pi$
$464$	0	0
$465$	−473.836	−1.01900
$466$	0	0
$467$	− 295.195i	− 0.632109i	−0.948741	−	0.316054i	$-0.897642\pi$
$467$	− 295.195i	− 0.632109i	0.948741	−	0.316054i	$-0.102358\pi$
$468$	0	0
$469$	−879.692	−1.87568
$470$	0	0
$471$	− 384.018i	− 0.815325i
$472$	0	0
$473$	80.0000	0.169133
$474$	0	0
$475$	934.046i	1.96641i
$476$	0	0
$477$	38.7846	0.0813095
$478$	0	0
$479$	291.559i	0.608682i	0.952563	+	0.304341i	$0.0984363\pi$
$479$	291.559i	0.608682i	−0.952563	+	0.304341i	$0.901564\pi$
$480$	0	0
$481$	−176.287	−0.366501
$482$	0	0
$483$	− 151.426i	− 0.313511i
$484$	0	0
$485$	188.728	0.389130
$486$	0	0
$487$	− 729.779i	− 1.49852i	−0.662276	−	0.749260i	$-0.730409\pi$
$487$	− 729.779i	− 1.49852i	0.662276	−	0.749260i	$-0.269591\pi$
$488$	0	0
$489$	−304.708	−0.623124
$490$	0	0
$491$	258.487i	0.526450i	0.964734	+	0.263225i	$0.0847862\pi$
$491$	258.487i	0.526450i	−0.964734	+	0.263225i	$0.915214\pi$
$492$	0	0
$493$	24.1333	0.0489519
$494$	0	0
$495$	28.7077i	0.0579953i
$496$	0	0
$497$	−1345.68	−2.70761
$498$	0	0
$499$	495.195i	0.992374i	0.868216	+	0.496187i	$0.165267\pi$
$499$	495.195i	0.992374i	−0.868216	+	0.496187i	$0.834733\pi$
$500$	0	0
$501$	−411.979	−0.822314
$502$	0	0
$503$	− 698.831i	− 1.38933i	−0.719336	−	0.694663i	$-0.755554\pi$
$503$	− 698.831i	− 1.38933i	0.719336	−	0.694663i	$-0.244446\pi$
$504$	0	0
$505$	−1543.94	−3.05730
$506$	0	0
$507$	− 266.958i	− 0.526544i
$508$	0	0
$509$	−21.1948	−0.0416400	−0.0208200	−	0.999783i	$-0.506628\pi$
$509$	−21.1948	−0.0416400	−0.0208200	+	0.999783i	$0.506628\pi$
$510$	0	0
$511$	933.549i	1.82691i
$512$	0	0
$513$	88.7077	0.172919
$514$	0	0
$515$	− 149.856i	− 0.290983i
$516$	0	0
$517$	−45.1694	−0.0873682
$518$	0	0
$519$	124.326i	0.239548i
$520$	0	0
$521$	167.015	0.320567	0.160284	−	0.987071i	$-0.448759\pi$
$521$	167.015	0.320567	0.160284	+	0.987071i	$0.448759\pi$
$522$	0	0
$523$	450.908i	0.862156i	0.902315	+	0.431078i	$0.141867\pi$
$523$	450.908i	0.862156i	−0.902315	+	0.431078i	$0.858133\pi$
$524$	0	0
$525$	1035.62	1.97260
$526$	0	0
$527$	240.728i	0.456790i
$528$	0	0
$529$	465.000	0.879017
$530$	0	0
$531$	− 132.631i	− 0.249775i
$532$	0	0
$533$	−138.277	−0.259431
$534$	0	0
$535$	− 1232.69i	− 2.30409i
$536$	0	0
$537$	10.2769	0.0191376
$538$	0	0
$539$	75.4820i	0.140041i
$540$	0	0
$541$	79.5692	0.147078	0.0735390	−	0.997292i	$-0.476571\pi$
$541$	79.5692	0.147078	0.0735390	+	0.997292i	$0.476571\pi$
$542$	0	0
$543$	261.531i	0.481640i
$544$	0	0
$545$	−70.1436	−0.128704
$546$	0	0
$547$	349.933i	0.639732i	0.947463	+	0.319866i	$0.103638\pi$
$547$	349.933i	0.639732i	−0.947463	+	0.319866i	$0.896362\pi$
$548$	0	0
$549$	−42.0000	−0.0765027
$550$	0	0
$551$	52.4411i	0.0951744i
$552$	0	0
$553$	603.405	1.09115
$554$	0	0
$555$	− 706.908i	− 1.27371i
$556$	0	0
$557$	−874.210	−1.56950	−0.784749	−	0.619814i	$-0.787208\pi$
$557$	−874.210	−1.56950	−0.784749	+	0.619814i	$0.787208\pi$
$558$	0	0
$559$	287.846i	0.514930i
$560$	0	0
$561$	14.5847	0.0259977
$562$	0	0
$563$	104.497i	0.185608i	0.995684	+	0.0928041i	$0.0295830\pi$
$563$	104.497i	0.185608i	−0.995684	+	0.0928041i	$0.970417\pi$
$564$	0	0
$565$	1474.39	2.60954
$566$	0	0
$567$	− 98.3538i	− 0.173464i
$568$	0	0
$569$	136.697	0.240241	0.120121	−	0.992759i	$-0.461672\pi$
$569$	136.697	0.240241	0.120121	+	0.992759i	$0.461672\pi$
$570$	0	0
$571$	− 566.928i	− 0.992869i	−0.868074	−	0.496435i	$-0.834642\pi$
$571$	− 566.928i	− 0.992869i	0.868074	−	0.496435i	$-0.165358\pi$
$572$	0	0
$573$	−40.5744	−0.0708104
$574$	0	0
$575$	− 437.703i	− 0.761222i
$576$	0	0
$577$	778.574	1.34935	0.674675	−	0.738115i	$-0.264284\pi$
$577$	778.574	1.34935	0.674675	+	0.738115i	$0.264284\pi$
$578$	0	0
$579$	238.028i	0.411102i
$580$	0	0
$581$	536.267	0.923006
$582$	0	0
$583$	13.8564i	0.0237674i
$584$	0	0
$585$	−103.292	−0.176568
$586$	0	0
$587$	1053.89i	1.79539i	0.440621	+	0.897693i	$0.354758\pi$
$587$	1053.89i	1.79539i	−0.440621	+	0.897693i	$0.645242\pi$
$588$	0	0
$589$	−523.097	−0.888111
$590$	0	0
$591$	− 68.4026i	− 0.115740i
$592$	0	0
$593$	544.543	0.918286	0.459143	−	0.888362i	$-0.348157\pi$
$593$	544.543	0.918286	0.459143	+	0.888362i	$0.348157\pi$
$594$	0	0
$595$	− 766.543i	− 1.28831i
$596$	0	0
$597$	477.913	0.800524
$598$	0	0
$599$	948.246i	1.58305i	0.611138	+	0.791524i	$0.290712\pi$
$599$	948.246i	1.58305i	−0.611138	+	0.791524i	$0.709288\pi$
$600$	0	0
$601$	−542.000	−0.901830	−0.450915	−	0.892567i	$-0.648902\pi$
$601$	−542.000	−0.901830	−0.450915	+	0.892567i	$0.648902\pi$
$602$	0	0
$603$	241.492i	0.400485i
$604$	0	0
$605$	1070.06	1.76869
$606$	0	0
$607$	− 1096.06i	− 1.80569i	−0.429962	−	0.902847i	$-0.641473\pi$
$607$	− 1096.06i	− 1.80569i	0.429962	−	0.902847i	$-0.358527\pi$
$608$	0	0
$609$	58.1436	0.0954739
$610$	0	0
$611$	− 162.523i	− 0.265995i
$612$	0	0
$613$	−738.000	−1.20392	−0.601958	−	0.798528i	$-0.705612\pi$
$613$	−738.000	−1.20392	−0.601958	+	0.798528i	$0.705612\pi$
$614$	0	0
$615$	− 554.487i	− 0.901605i
$616$	0	0
$617$	−385.979	−0.625574	−0.312787	−	0.949823i	$-0.601263\pi$
$617$	−385.979	−0.625574	−0.312787	+	0.949823i	$0.601263\pi$
$618$	0	0
$619$	− 558.887i	− 0.902887i	−0.892300	−	0.451443i	$-0.850909\pi$
$619$	− 558.887i	− 0.902887i	0.892300	−	0.451443i	$-0.149091\pi$
$620$	0	0
$621$	−41.5692	−0.0669392
$622$	0	0
$623$	1155.25i	1.85434i
$624$	0	0
$625$	1000.67	1.60107
$626$	0	0
$627$	31.6922i	0.0505458i
$628$	0	0
$629$	−359.138	−0.570967
$630$	0	0
$631$	420.231i	0.665976i	0.942931	+	0.332988i	$0.108057\pi$
$631$	420.231i	0.665976i	−0.942931	+	0.332988i	$0.891943\pi$
$632$	0	0
$633$	327.979	0.518135
$634$	0	0
$635$	1315.56i	2.07175i
$636$	0	0
$637$	−271.590	−0.426358
$638$	0	0
$639$	369.415i	0.578115i
$640$	0	0
$641$	256.964	0.400880	0.200440	−	0.979706i	$-0.435763\pi$
$641$	256.964	0.400880	0.200440	+	0.979706i	$0.435763\pi$
$642$	0	0
$643$	− 467.215i	− 0.726618i	−0.931669	−	0.363309i	$-0.881647\pi$
$643$	− 467.215i	− 0.726618i	0.931669	−	0.363309i	$-0.118353\pi$
$644$	0	0
$645$	−1154.26	−1.78954
$646$	0	0
$647$	971.138i	1.50099i	0.660878	+	0.750493i	$0.270184\pi$
$647$	971.138i	1.50099i	−0.660878	+	0.750493i	$0.729816\pi$
$648$	0	0
$649$	47.3844	0.0730114
$650$	0	0
$651$	579.979i	0.890905i
$652$	0	0
$653$	−439.380	−0.672863	−0.336432	−	0.941708i	$-0.609220\pi$
$653$	−439.380	−0.672863	−0.336432	+	0.941708i	$0.609220\pi$
$654$	0	0
$655$	633.261i	0.966811i
$656$	0	0
$657$	256.277	0.390071
$658$	0	0
$659$	− 1092.48i	− 1.65778i	−0.559412	−	0.828890i	$-0.688973\pi$
$659$	− 1092.48i	− 1.65778i	0.559412	−	0.828890i	$-0.311027\pi$
$660$	0	0
$661$	−122.267	−0.184972	−0.0924861	−	0.995714i	$-0.529481\pi$
$661$	−122.267	−0.184972	−0.0924861	+	0.995714i	$0.529481\pi$
$662$	0	0
$663$	52.4768i	0.0791505i
$664$	0	0
$665$	1665.68	2.50478
$666$	0	0
$667$	− 24.5744i	− 0.0368431i
$668$	0	0
$669$	−249.779	−0.373362
$670$	0	0
$671$	− 15.0052i	− 0.0223624i
$672$	0	0
$673$	−796.851	−1.18403	−0.592014	−	0.805927i	$-0.701667\pi$
$673$	−796.851	−1.18403	−0.592014	+	0.805927i	$0.701667\pi$
$674$	0	0
$675$	− 284.296i	− 0.421179i
$676$	0	0
$677$	1185.19	1.75066	0.875328	−	0.483529i	$-0.160645\pi$
$677$	1185.19	1.75066	0.875328	+	0.483529i	$0.160645\pi$
$678$	0	0
$679$	− 231.005i	− 0.340214i
$680$	0	0
$681$	−389.836	−0.572446
$682$	0	0
$683$	− 850.446i	− 1.24516i	−0.782555	−	0.622581i	$-0.786084\pi$
$683$	− 850.446i	− 1.24516i	0.782555	−	0.622581i	$-0.213916\pi$
$684$	0	0
$685$	1840.40	2.68672
$686$	0	0
$687$	− 405.033i	− 0.589568i
$688$	0	0
$689$	−49.8564	−0.0723605
$690$	0	0
$691$	− 1161.49i	− 1.68089i	−0.541900	−	0.840443i	$-0.682295\pi$
$691$	− 1161.49i	− 1.68089i	0.541900	−	0.840443i	$-0.317705\pi$
$692$	0	0
$693$	35.1384	0.0507048
$694$	0	0
$695$	356.441i	0.512865i
$696$	0	0
$697$	−281.703	−0.404164
$698$	0	0
$699$	− 379.079i	− 0.542317i
$700$	0	0
$701$	−177.215	−0.252804	−0.126402	−	0.991979i	$-0.540343\pi$
$701$	−177.215	−0.252804	−0.126402	+	0.991979i	$0.540343\pi$
$702$	0	0
$703$	− 780.400i	− 1.11010i
$704$	0	0
$705$	651.713	0.924415
$706$	0	0
$707$	1889.79i	2.67298i
$708$	0	0
$709$	−329.846	−0.465227	−0.232614	−	0.972569i	$-0.574728\pi$
$709$	−329.846	−0.465227	−0.232614	+	0.972569i	$0.574728\pi$
$710$	0	0
$711$	− 165.646i	− 0.232976i
$712$	0	0
$713$	245.128	0.343798
$714$	0	0
$715$	− 36.9028i	− 0.0516123i
$716$	0	0
$717$	644.287	0.898587
$718$	0	0
$719$	− 1343.27i	− 1.86825i	−0.356946	−	0.934125i	$-0.616182\pi$
$719$	− 1343.27i	− 1.86825i	0.356946	−	0.934125i	$-0.383818\pi$
$720$	0	0
$721$	−183.426	−0.254404
$722$	0	0
$723$	− 569.090i	− 0.787123i
$724$	0	0
$725$	168.067	0.231816
$726$	0	0
$727$	− 6.64102i	− 0.00913482i	−0.999990	−	0.00456741i	$-0.998546\pi$
$727$	− 6.64102i	− 0.00913482i	0.999990	−	0.00456741i	$-0.00145386\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	586.410i	0.802203i
$732$	0	0
$733$	1396.39	1.90503	0.952517	−	0.304486i	$-0.0984848\pi$
$733$	1396.39	1.90503	0.952517	+	0.304486i	$0.0984848\pi$
$734$	0	0
$735$	− 1089.07i	− 1.48173i
$736$	0	0
$737$	−86.2769	−0.117065
$738$	0	0
$739$	276.939i	0.374748i	0.982289	+	0.187374i	$0.0599976\pi$
$739$	276.939i	0.374748i	−0.982289	+	0.187374i	$0.940002\pi$
$740$	0	0
$741$	−114.031	−0.153888
$742$	0	0
$743$	− 424.728i	− 0.571640i	−0.958283	−	0.285820i	$-0.907734\pi$
$743$	− 424.728i	− 0.571640i	0.958283	−	0.285820i	$-0.0922659\pi$
$744$	0	0
$745$	3.15906	0.00424035
$746$	0	0
$747$	− 147.215i	− 0.197075i
$748$	0	0
$749$	−1508.82	−2.01445
$750$	0	0
$751$	1008.32i	1.34264i	0.741167	+	0.671320i	$0.234272\pi$
$751$	1008.32i	1.34264i	−0.741167	+	0.671320i	$0.765728\pi$
$752$	0	0
$753$	256.708	0.340913
$754$	0	0
$755$	930.410i	1.23233i
$756$	0	0
$757$	2.70766	0.00357683	0.00178841	−	0.999998i	$-0.499431\pi$
$757$	2.70766	0.00357683	0.00178841	+	0.999998i	$0.499431\pi$
$758$	0	0
$759$	− 14.8513i	− 0.0195669i
$760$	0	0
$761$	656.431	0.862590	0.431295	−	0.902211i	$-0.358057\pi$
$761$	656.431	0.862590	0.431295	+	0.902211i	$0.358057\pi$
$762$	0	0
$763$	85.8564i	0.112525i
$764$	0	0
$765$	−210.431	−0.275073
$766$	0	0
$767$	170.493i	0.222285i
$768$	0	0
$769$	−533.959	−0.694355	−0.347177	−	0.937799i	$-0.612860\pi$
$769$	−533.959	−0.694355	−0.347177	+	0.937799i	$0.612860\pi$
$770$	0	0
$771$	609.167i	0.790099i
$772$	0	0
$773$	−350.918	−0.453969	−0.226984	−	0.973898i	$-0.572887\pi$
$773$	−350.918	−0.453969	−0.226984	+	0.973898i	$0.572887\pi$
$774$	0	0
$775$	1676.46i	2.16317i
$776$	0	0
$777$	−865.261	−1.11359
$778$	0	0
$779$	− 612.133i	− 0.785794i
$780$	0	0
$781$	−131.979	−0.168988
$782$	0	0
$783$	− 15.9615i	− 0.0203851i
$784$	0	0
$785$	−1979.50	−2.52165
$786$	0	0
$787$	391.041i	0.496875i	0.968648	+	0.248438i	$0.0799171\pi$
$787$	391.041i	0.496875i	−0.968648	+	0.248438i	$0.920083\pi$
$788$	0	0
$789$	419.405	0.531565
$790$	0	0
$791$	− 1804.67i	− 2.28150i
$792$	0	0
$793$	53.9897	0.0680828
$794$	0	0
$795$	− 199.923i	− 0.251476i
$796$	0	0
$797$	−138.210	−0.173413	−0.0867065	−	0.996234i	$-0.527634\pi$
$797$	−138.210	−0.173413	−0.0867065	+	0.996234i	$0.527634\pi$
$798$	0	0
$799$	− 331.097i	− 0.414389i
$800$	0	0
$801$	317.138	0.395928
$802$	0	0
$803$	91.5589i	0.114021i
$804$	0	0
$805$	−780.554	−0.969632
$806$	0	0
$807$	131.023i	0.162358i
$808$	0	0
$809$	−919.528	−1.13662	−0.568311	−	0.822814i	$-0.692403\pi$
$809$	−919.528	−1.13662	−0.568311	+	0.822814i	$0.692403\pi$
$810$	0	0
$811$	− 181.359i	− 0.223624i	−0.993729	−	0.111812i	$-0.964335\pi$
$811$	− 181.359i	− 0.223624i	0.993729	−	0.111812i	$-0.0356654\pi$
$812$	0	0
$813$	−261.184	−0.321260
$814$	0	0
$815$	1570.68i	1.92721i
$816$	0	0
$817$	−1274.26	−1.55968
$818$	0	0
$819$	126.431i	0.154372i
$820$	0	0
$821$	839.359	1.02236	0.511181	−	0.859473i	$-0.329208\pi$
$821$	839.359	1.02236	0.511181	+	0.859473i	$0.329208\pi$
$822$	0	0
$823$	26.1999i	0.0318347i	0.999873	+	0.0159173i	$0.00506686\pi$
$823$	26.1999i	0.0318347i	−0.999873	+	0.0159173i	$0.994933\pi$
$824$	0	0
$825$	101.569	0.123114
$826$	0	0
$827$	7.38991i	0.00893581i	0.999990	+	0.00446790i	$0.00142218\pi$
$827$	7.38991i	0.00893581i	−0.999990	+	0.00446790i	$0.998578\pi$
$828$	0	0
$829$	−340.410	−0.410627	−0.205314	−	0.978696i	$-0.565821\pi$
$829$	−340.410	−0.410627	−0.205314	+	0.978696i	$0.565821\pi$
$830$	0	0
$831$	− 369.895i	− 0.445120i
$832$	0	0
$833$	−553.292	−0.664216
$834$	0	0
$835$	2123.63i	2.54327i
$836$	0	0
$837$	159.215	0.190221
$838$	0	0
$839$	734.810i	0.875816i	0.899020	+	0.437908i	$0.144281\pi$
$839$	734.810i	0.875816i	−0.899020	+	0.437908i	$0.855719\pi$
$840$	0	0
$841$	−831.564	−0.988780
$842$	0	0
$843$	336.018i	0.398598i
$844$	0	0
$845$	−1376.09	−1.62851
$846$	0	0
$847$	− 1309.76i	− 1.54635i
$848$	0	0
$849$	311.405	0.366790
$850$	0	0
$851$	365.703i	0.429733i
$852$	0	0
$853$	−778.800	−0.913013	−0.456506	−	0.889720i	$-0.650899\pi$
$853$	−778.800	−0.913013	−0.456506	+	0.889720i	$0.650899\pi$
$854$	0	0
$855$	− 457.261i	− 0.534809i
$856$	0	0
$857$	−1129.52	−1.31799	−0.658995	−	0.752147i	$-0.729018\pi$
$857$	−1129.52	−1.31799	−0.658995	+	0.752147i	$0.729018\pi$
$858$	0	0
$859$	− 1373.85i	− 1.59936i	−0.600426	−	0.799680i	$-0.705002\pi$
$859$	− 1373.85i	− 1.59936i	0.600426	−	0.799680i	$-0.294998\pi$
$860$	0	0
$861$	−678.697	−0.788266
$862$	0	0
$863$	− 539.405i	− 0.625035i	−0.949912	−	0.312517i	$-0.898828\pi$
$863$	− 539.405i	− 0.625035i	0.949912	−	0.312517i	$-0.101172\pi$
$864$	0	0
$865$	640.862	0.740880
$866$	0	0
$867$	− 393.655i	− 0.454043i
$868$	0	0
$869$	59.1797	0.0681009
$870$	0	0
$871$	− 310.431i	− 0.356407i
$872$	0	0
$873$	−63.4153	−0.0726407
$874$	0	0
$875$	− 2899.06i	− 3.31321i
$876$	0	0
$877$	668.236	0.761956	0.380978	−	0.924584i	$-0.375587\pi$
$877$	668.236	0.761956	0.380978	+	0.924584i	$0.375587\pi$
$878$	0	0
$879$	897.033i	1.02052i
$880$	0	0
$881$	1170.53	1.32864	0.664321	−	0.747448i	$-0.268721\pi$
$881$	1170.53	1.32864	0.664321	+	0.747448i	$0.268721\pi$
$882$	0	0
$883$	− 32.9179i	− 0.0372796i	−0.999826	−	0.0186398i	$-0.994066\pi$
$883$	− 32.9179i	− 0.0372796i	0.999826	−	0.0186398i	$-0.00593358\pi$
$884$	0	0
$885$	−683.672	−0.772510
$886$	0	0
$887$	655.846i	0.739398i	0.929152	+	0.369699i	$0.120539\pi$
$887$	655.846i	0.739398i	−0.929152	+	0.369699i	$0.879461\pi$
$888$	0	0
$889$	1610.26	1.81131
$890$	0	0
$891$	− 9.64617i	− 0.0108262i
$892$	0	0
$893$	719.467	0.805674
$894$	0	0
$895$	− 52.9742i	− 0.0591891i
$896$	0	0
$897$	53.4359	0.0595718
$898$	0	0
$899$	94.1230i	0.104697i
$900$	0	0
$901$	−101.569	−0.112729
$902$	0	0
$903$	1412.82i	1.56459i
$904$	0	0
$905$	1348.11	1.48963
$906$	0	0
$907$	378.949i	0.417805i	0.977937	+	0.208902i	$0.0669891\pi$
$907$	378.949i	0.417805i	−0.977937	+	0.208902i	$0.933011\pi$
$908$	0	0
$909$	518.785	0.570720
$910$	0	0
$911$	571.938i	0.627814i	0.949454	+	0.313907i	$0.101638\pi$
$911$	571.938i	0.627814i	−0.949454	+	0.313907i	$0.898362\pi$
$912$	0	0
$913$	52.5950	0.0576068
$914$	0	0
$915$	216.497i	0.236609i
$916$	0	0
$917$	775.118	0.845276
$918$	0	0
$919$	− 97.8207i	− 0.106443i	−0.998583	−	0.0532213i	$-0.983051\pi$
$919$	− 97.8207i	− 0.106443i	0.998583	−	0.0532213i	$-0.0169489\pi$
$920$	0	0
$921$	−706.277	−0.766859
$922$	0	0
$923$	− 474.872i	− 0.514487i
$924$	0	0
$925$	−2501.08	−2.70387
$926$	0	0
$927$	50.3538i	0.0543191i
$928$	0	0
$929$	1228.06	1.32192	0.660959	−	0.750422i	$-0.270150\pi$
$929$	1228.06	1.32192	0.660959	+	0.750422i	$0.270150\pi$
$930$	0	0
$931$	− 1202.29i	− 1.29140i
$932$	0	0
$933$	639.118	0.685014
$934$	0	0
$935$	− 75.1797i	− 0.0804061i
$936$	0	0
$937$	494.554	0.527806	0.263903	−	0.964549i	$-0.414990\pi$
$937$	494.554	0.527806	0.263903	+	0.964549i	$0.414990\pi$
$938$	0	0
$939$	592.326i	0.630805i
$940$	0	0
$941$	−1154.48	−1.22686	−0.613431	−	0.789748i	$-0.710211\pi$
$941$	−1154.48	−1.22686	−0.613431	+	0.789748i	$0.710211\pi$
$942$	0	0
$943$	286.851i	0.304190i
$944$	0	0
$945$	−506.985	−0.536492
$946$	0	0
$947$	1085.43i	1.14618i	0.819493	+	0.573089i	$0.194255\pi$
$947$	1085.43i	1.14618i	−0.819493	+	0.573089i	$0.805745\pi$
$948$	0	0
$949$	−329.436	−0.347140
$950$	0	0
$951$	686.736i	0.722120i
$952$	0	0
$953$	53.8667	0.0565233	0.0282617	−	0.999601i	$-0.491003\pi$
$953$	53.8667	0.0565233	0.0282617	+	0.999601i	$0.491003\pi$
$954$	0	0
$955$	209.149i	0.219004i
$956$	0	0
$957$	5.70250	0.00595873
$958$	0	0
$959$	− 2252.67i	− 2.34897i
$960$	0	0
$961$	22.1281	0.0230261
$962$	0	0
$963$	414.200i	0.430114i
$964$	0	0
$965$	1226.96	1.27147
$966$	0	0
$967$	172.918i	0.178819i	0.995995	+	0.0894095i	$0.0284980\pi$
$967$	172.918i	0.178819i	−0.995995	+	0.0894095i	$0.971502\pi$
$968$	0	0
$969$	−232.308	−0.239740
$970$	0	0
$971$	984.877i	1.01429i	0.861860	+	0.507146i	$0.169299\pi$
$971$	984.877i	1.01429i	−0.861860	+	0.507146i	$0.830701\pi$
$972$	0	0
$973$	436.287	0.448394
$974$	0	0
$975$	365.454i	0.374824i
$976$	0	0
$977$	−675.856	−0.691767	−0.345884	−	0.938277i	$-0.612421\pi$
$977$	−675.856	−0.691767	−0.345884	+	0.938277i	$0.612421\pi$
$978$	0	0
$979$	113.303i	0.115733i
$980$	0	0
$981$	23.5692	0.0240257
$982$	0	0
$983$	564.441i	0.574203i	0.957900	+	0.287101i	$0.0926916\pi$
$983$	564.441i	0.574203i	−0.957900	+	0.287101i	$0.907308\pi$
$984$	0	0
$985$	−352.595	−0.357964
$986$	0	0
$987$	− 797.703i	− 0.808209i
$988$	0	0
$989$	597.128	0.603770
$990$	0	0
$991$	− 1392.44i	− 1.40508i	−0.711644	−	0.702541i	$-0.752049\pi$
$991$	− 1392.44i	− 1.40508i	0.711644	−	0.702541i	$-0.247951\pi$
$992$	0	0
$993$	−462.564	−0.465825
$994$	0	0
$995$	− 2463.50i	− 2.47588i
$996$	0	0
$997$	−123.723	−0.124095	−0.0620477	−	0.998073i	$-0.519763\pi$
$997$	−123.723	−0.124095	−0.0620477	+	0.998073i	$0.519763\pi$
$998$	0	0
$999$	237.531i	0.237769i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.3.g.c.127.4		4
3.2	odd	2		576.3.g.j.127.1		4
4.3	odd	2	inner	192.3.g.c.127.2		4
8.3	odd	2		96.3.g.a.31.3	yes	4
8.5	even	2		96.3.g.a.31.1	✓	4
12.11	even	2		576.3.g.j.127.2		4
16.3	odd	4		768.3.b.d.127.3		4
16.5	even	4		768.3.b.d.127.4		4
16.11	odd	4		768.3.b.a.127.2		4
16.13	even	4		768.3.b.a.127.1		4
24.5	odd	2		288.3.g.d.127.3		4
24.11	even	2		288.3.g.d.127.4		4
40.3	even	4		2400.3.j.b.799.1		4
40.13	odd	4		2400.3.j.a.799.4		4
40.19	odd	2		2400.3.e.a.1951.1		4
40.27	even	4		2400.3.j.a.799.3		4
40.29	even	2		2400.3.e.a.1951.4		4
40.37	odd	4		2400.3.j.b.799.2		4
48.5	odd	4		2304.3.b.k.127.1		4
48.11	even	4		2304.3.b.o.127.1		4
48.29	odd	4		2304.3.b.o.127.4		4
48.35	even	4		2304.3.b.k.127.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.3.g.a.31.1	✓	4	8.5	even	2
96.3.g.a.31.3	yes	4	8.3	odd	2
192.3.g.c.127.2		4	4.3	odd	2	inner
192.3.g.c.127.4		4	1.1	even	1	trivial
288.3.g.d.127.3		4	24.5	odd	2
288.3.g.d.127.4		4	24.11	even	2
576.3.g.j.127.1		4	3.2	odd	2
576.3.g.j.127.2		4	12.11	even	2
768.3.b.a.127.1		4	16.13	even	4
768.3.b.a.127.2		4	16.11	odd	4
768.3.b.d.127.3		4	16.3	odd	4
768.3.b.d.127.4		4	16.5	even	4
2304.3.b.k.127.1		4	48.5	odd	4
2304.3.b.k.127.4		4	48.35	even	4
2304.3.b.o.127.1		4	48.11	even	4
2304.3.b.o.127.4		4	48.29	odd	4
2400.3.e.a.1951.1		4	40.19	odd	2
2400.3.e.a.1951.4		4	40.29	even	2
2400.3.j.a.799.3		4	40.27	even	4
2400.3.j.a.799.4		4	40.13	odd	4
2400.3.j.b.799.1		4	40.3	even	4
2400.3.j.b.799.2		4	40.37	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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +8.92820 q^{5} -10.9282i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +8.92820 q^{5} -10.9282i q^{7} -3.00000 q^{9} -1.07180i q^{11} +3.85641 q^{13} +15.4641i q^{15} +7.85641 q^{17} +17.0718i q^{19} +18.9282 q^{21} -8.00000i q^{23} +54.7128 q^{25} -5.19615i q^{27} +3.07180 q^{29} +30.6410i q^{31} +1.85641 q^{33} -97.5692i q^{35} -45.7128 q^{37} +6.67949i q^{39} -35.8564 q^{41} +74.6410i q^{43} -26.7846 q^{45} -42.1436i q^{47} -70.4256 q^{49} +13.6077i q^{51} -12.9282 q^{53} -9.56922i q^{55} -29.5692 q^{57} +44.2102i q^{59} +14.0000 q^{61} +32.7846i q^{63} +34.4308 q^{65} -80.4974i q^{67} +13.8564 q^{69} -123.138i q^{71} -85.4256 q^{73} +94.7654i q^{75} -11.7128 q^{77} +55.2154i q^{79} +9.00000 q^{81} +49.0718i q^{83} +70.1436 q^{85} +5.32051i q^{87} -105.713 q^{89} -42.1436i q^{91} -53.0718 q^{93} +152.420i q^{95} +21.1384 q^{97} +3.21539i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 8 * q^5 - 12 * q^9	\( 4 q + 8 q^{5} - 12 q^{9} - 40 q^{13} - 24 q^{17} + 48 q^{21} + 108 q^{25} + 40 q^{29} - 48 q^{33} - 72 q^{37} - 88 q^{41} - 24 q^{45} - 60 q^{49} - 24 q^{53} + 48 q^{57} + 56 q^{61} + 304 q^{65} - 120 q^{73} + 64 q^{77} + 36 q^{81} + 336 q^{85} - 312 q^{89} - 240 q^{93} - 248 q^{97}+O(q^{100}) \) 4 * q + 8 * q^5 - 12 * q^9 - 40 * q^13 - 24 * q^17 + 48 * q^21 + 108 * q^25 + 40 * q^29 - 48 * q^33 - 72 * q^37 - 88 * q^41 - 24 * q^45 - 60 * q^49 - 24 * q^53 + 48 * q^57 + 56 * q^61 + 304 * q^65 - 120 * q^73 + 64 * q^77 + 36 * q^81 + 336 * q^85 - 312 * q^89 - 240 * q^93 - 248 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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