LMFDB - Embedded newform 192.2.d.a.97.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,2,Mod(97,192)]

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, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("192.97");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	192.d (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:	$1.53312771881$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.3
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.97
Dual form			192.2.d.a.97.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -3.46410i q^{5} +3.46410 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -3.46410i q^{5} +3.46410 q^{7} -1.00000 q^{9} +3.46410 q^{15} +6.00000 q^{17} +4.00000i q^{19} +3.46410i q^{21} -6.92820 q^{23} -7.00000 q^{25} -1.00000i q^{27} -3.46410i q^{29} -3.46410 q^{31} -12.0000i q^{35} +6.92820i q^{37} -6.00000 q^{41} -4.00000i q^{43} +3.46410i q^{45} -6.92820 q^{47} +5.00000 q^{49} +6.00000i q^{51} +3.46410i q^{53} -4.00000 q^{57} +12.0000i q^{59} +6.92820i q^{61} -3.46410 q^{63} -4.00000i q^{67} -6.92820i q^{69} +6.92820 q^{71} +2.00000 q^{73} -7.00000i q^{75} +10.3923 q^{79} +1.00000 q^{81} -20.7846i q^{85} +3.46410 q^{87} +6.00000 q^{89} -3.46410i q^{93} +13.8564 q^{95} -2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 24 q^{17} - 28 q^{25} - 24 q^{41} + 20 q^{49} - 16 q^{57} + 8 q^{73} + 4 q^{81} + 24 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 24 * q^17 - 28 * q^25 - 24 * q^41 + 20 * q^49 - 16 * q^57 + 8 * q^73 + 4 * q^81 + 24 * q^89 - 8 * 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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	4.00000i	0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$	4.00000i	0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$	0	0
$21$	3.46410i	0.755929i
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 12.0000i	− 2.02837i
$36$	0	0
$37$	6.92820i	1.13899i	0.821995	+	0.569495i	$0.192861\pi$
$37$	6.92820i	1.13899i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	3.46410i	0.516398i
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	6.00000i	0.840168i
$52$	0	0
$53$	3.46410i	0.475831i	0.971286	+	0.237915i	$0.0764641\pi$
$53$	3.46410i	0.475831i	−0.971286	+	0.237915i	$0.923536\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	12.0000i	1.56227i	0.624364	+	0.781133i	$0.285358\pi$
$59$	12.0000i	1.56227i	−0.624364	+	0.781133i	$0.714642\pi$
$60$	0	0
$61$	6.92820i	0.887066i	0.896258	+	0.443533i	$0.146275\pi$
$61$	6.92820i	0.887066i	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	− 6.92820i	− 0.834058i
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	− 7.00000i	− 0.808290i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 20.7846i	− 2.25441i
$86$	0	0
$87$	3.46410	0.371391
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	− 3.46410i	− 0.359211i
$94$	0	0
$95$	13.8564	1.42164
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.46410i	0.344691i	0.985037	+	0.172345i	$0.0551346\pi$
$101$	3.46410i	0.344691i	−0.985037	+	0.172345i	$0.944865\pi$
$102$	0	0
$103$	−17.3205	−1.70664	−0.853320	−	0.521387i	$-0.825415\pi$
$103$	−17.3205	−1.70664	−0.853320	+	0.521387i	$0.825415\pi$
$104$	0	0
$105$	12.0000	1.17108
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 13.8564i	− 1.32720i	−0.748086	−	0.663602i	$-0.769027\pi$
$109$	− 13.8564i	− 1.32720i	0.748086	−	0.663602i	$-0.230973\pi$
$110$	0	0
$111$	−6.92820	−0.657596
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	24.0000i	2.23801i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	20.7846	1.90532
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	− 6.00000i	− 0.541002i
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	3.46410	0.307389	0.153695	−	0.988118i	$-0.450883\pi$
$127$	3.46410	0.307389	0.153695	+	0.988118i	$0.450883\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	13.8564i	1.20150i
$134$	0	0
$135$	−3.46410	−0.298142
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	− 20.0000i	− 1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	− 20.0000i	− 1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$	0	0
$141$	− 6.92820i	− 0.583460i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	5.00000i	0.412393i
$148$	0	0
$149$	10.3923i	0.851371i	0.904871	+	0.425685i	$0.139967\pi$
$149$	10.3923i	0.851371i	−0.904871	+	0.425685i	$0.860033\pi$
$150$	0	0
$151$	3.46410	0.281905	0.140952	−	0.990016i	$-0.454984\pi$
$151$	3.46410	0.281905	0.140952	+	0.990016i	$0.454984\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	12.0000i	0.963863i
$156$	0	0
$157$	− 6.92820i	− 0.552931i	−0.961024	−	0.276465i	$-0.910837\pi$
$157$	− 6.92820i	− 0.552931i	0.961024	−	0.276465i	$-0.0891631\pi$
$158$	0	0
$159$	−3.46410	−0.274721
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	− 4.00000i	− 0.305888i
$172$	0	0
$173$	17.3205i	1.31685i	0.752645	+	0.658427i	$0.228778\pi$
$173$	17.3205i	1.31685i	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	−24.2487	−1.83303
$176$	0	0
$177$	−12.0000	−0.901975
$178$	0	0
$179$	12.0000i	0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$	12.0000i	0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$	0	0
$181$	13.8564i	1.02994i	0.857209	+	0.514969i	$0.172197\pi$
$181$	13.8564i	1.02994i	−0.857209	+	0.514969i	$0.827803\pi$
$182$	0	0
$183$	−6.92820	−0.512148
$184$	0	0
$185$	24.0000	1.76452
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 3.46410i	− 0.251976i
$190$	0	0
$191$	−13.8564	−1.00261	−0.501307	−	0.865269i	$-0.667147\pi$
$191$	−13.8564	−1.00261	−0.501307	+	0.865269i	$0.667147\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 10.3923i	− 0.740421i	−0.928948	−	0.370211i	$-0.879286\pi$
$197$	− 10.3923i	− 0.740421i	0.928948	−	0.370211i	$-0.120714\pi$
$198$	0	0
$199$	10.3923	0.736691	0.368345	−	0.929689i	$-0.379924\pi$
$199$	10.3923	0.736691	0.368345	+	0.929689i	$0.379924\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	20.7846i	1.45166i
$206$	0	0
$207$	6.92820	0.481543
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	6.92820i	0.474713i
$214$	0	0
$215$	−13.8564	−0.944999
$216$	0	0
$217$	−12.0000	−0.814613
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3.46410	0.231973	0.115987	−	0.993251i	$-0.462997\pi$
$223$	3.46410	0.231973	0.115987	+	0.993251i	$0.462997\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	− 27.7128i	− 1.83131i	−0.401960	−	0.915657i	$-0.631671\pi$
$229$	− 27.7128i	− 1.83131i	0.401960	−	0.915657i	$-0.368329\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	24.0000i	1.56559i
$236$	0	0
$237$	10.3923i	0.675053i
$238$	0	0
$239$	27.7128	1.79259	0.896296	−	0.443455i	$-0.146248\pi$
$239$	27.7128	1.79259	0.896296	+	0.443455i	$0.146248\pi$
$240$	0	0
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 17.3205i	− 1.10657i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	20.7846	1.30158
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	24.0000i	1.49129i
$260$	0	0
$261$	3.46410i	0.214423i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	− 31.1769i	− 1.90089i	−0.310893	−	0.950445i	$-0.600628\pi$
$269$	− 31.1769i	− 1.90089i	0.310893	−	0.950445i	$-0.399372\pi$
$270$	0	0
$271$	3.46410	0.210429	0.105215	−	0.994450i	$-0.466447\pi$
$271$	3.46410	0.210429	0.105215	+	0.994450i	$0.466447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.8564i	0.832551i	0.909239	+	0.416275i	$0.136665\pi$
$277$	13.8564i	0.832551i	−0.909239	+	0.416275i	$0.863335\pi$
$278$	0	0
$279$	3.46410	0.207390
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	0	0
$285$	13.8564i	0.820783i
$286$	0	0
$287$	−20.7846	−1.22688
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	− 2.00000i	− 0.117242i
$292$	0	0
$293$	3.46410i	0.202375i	0.994867	+	0.101187i	$0.0322642\pi$
$293$	3.46410i	0.202375i	−0.994867	+	0.101187i	$0.967736\pi$
$294$	0	0
$295$	41.5692	2.42025
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 13.8564i	− 0.798670i
$302$	0	0
$303$	−3.46410	−0.199007
$304$	0	0
$305$	24.0000	1.37424
$306$	0	0
$307$	− 4.00000i	− 0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$	− 4.00000i	− 0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$	0	0
$309$	− 17.3205i	− 0.985329i
$310$	0	0
$311$	−13.8564	−0.785725	−0.392862	−	0.919597i	$-0.628515\pi$
$311$	−13.8564	−0.785725	−0.392862	+	0.919597i	$0.628515\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	0	0
$315$	12.0000i	0.676123i
$316$	0	0
$317$	− 31.1769i	− 1.75107i	−0.483155	−	0.875535i	$-0.660509\pi$
$317$	− 31.1769i	− 1.75107i	0.483155	−	0.875535i	$-0.339491\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.0000	−0.669775
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.8564	0.766261
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	− 20.0000i	− 1.09930i	−0.835395	−	0.549650i	$-0.814761\pi$
$331$	− 20.0000i	− 1.09930i	0.835395	−	0.549650i	$-0.185239\pi$
$332$	0	0
$333$	− 6.92820i	− 0.379663i
$334$	0	0
$335$	−13.8564	−0.757056
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	− 6.00000i	− 0.325875i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6.92820	−0.374088
$344$	0	0
$345$	−24.0000	−1.29212
$346$	0	0
$347$	− 24.0000i	− 1.28839i	−0.764862	−	0.644194i	$-0.777193\pi$
$347$	− 24.0000i	− 1.28839i	0.764862	−	0.644194i	$-0.222807\pi$
$348$	0	0
$349$	6.92820i	0.370858i	0.982658	+	0.185429i	$0.0593675\pi$
$349$	6.92820i	0.370858i	−0.982658	+	0.185429i	$0.940632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	− 24.0000i	− 1.27379i
$356$	0	0
$357$	20.7846i	1.10004i
$358$	0	0
$359$	6.92820	0.365657	0.182828	−	0.983145i	$-0.441475\pi$
$359$	6.92820	0.365657	0.182828	+	0.983145i	$0.441475\pi$
$360$	0	0
$361$	3.00000	0.157895
$362$	0	0
$363$	11.0000i	0.577350i
$364$	0	0
$365$	− 6.92820i	− 0.362639i
$366$	0	0
$367$	24.2487	1.26577	0.632886	−	0.774245i	$-0.281870\pi$
$367$	24.2487	1.26577	0.632886	+	0.774245i	$0.281870\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000i	0.623009i
$372$	0	0
$373$	20.7846i	1.07619i	0.842885	+	0.538093i	$0.180855\pi$
$373$	20.7846i	1.07619i	−0.842885	+	0.538093i	$0.819145\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	0	0
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	3.46410i	0.177471i
$382$	0	0
$383$	−13.8564	−0.708029	−0.354015	−	0.935240i	$-0.615184\pi$
$383$	−13.8564	−0.708029	−0.354015	+	0.935240i	$0.615184\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000i	0.203331i
$388$	0	0
$389$	24.2487i	1.22946i	0.788738	+	0.614729i	$0.210735\pi$
$389$	24.2487i	1.22946i	−0.788738	+	0.614729i	$0.789265\pi$
$390$	0	0
$391$	−41.5692	−2.10225
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	− 36.0000i	− 1.81136i
$396$	0	0
$397$	34.6410i	1.73858i	0.494300	+	0.869291i	$0.335424\pi$
$397$	34.6410i	1.73858i	−0.494300	+	0.869291i	$0.664576\pi$
$398$	0	0
$399$	−13.8564	−0.693688
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 3.46410i	− 0.172133i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	− 6.00000i	− 0.295958i
$412$	0	0
$413$	41.5692i	2.04549i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000	0.979404
$418$	0	0
$419$	24.0000i	1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$	24.0000i	1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$	0	0
$421$	− 13.8564i	− 0.675320i	−0.941268	−	0.337660i	$-0.890365\pi$
$421$	− 13.8564i	− 0.675320i	0.941268	−	0.337660i	$-0.109635\pi$
$422$	0	0
$423$	6.92820	0.336861
$424$	0	0
$425$	−42.0000	−2.03730
$426$	0	0
$427$	24.0000i	1.16144i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.92820	0.333720	0.166860	−	0.985981i	$-0.446637\pi$
$431$	6.92820	0.333720	0.166860	+	0.985981i	$0.446637\pi$
$432$	0	0
$433$	−2.00000	−0.0961139	−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000	−0.0961139	−0.0480569	+	0.998845i	$0.515303\pi$
$434$	0	0
$435$	− 12.0000i	− 0.575356i
$436$	0	0
$437$	− 27.7128i	− 1.32568i
$438$	0	0
$439$	−31.1769	−1.48799	−0.743996	−	0.668184i	$-0.767072\pi$
$439$	−31.1769	−1.48799	−0.743996	+	0.668184i	$0.767072\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	− 20.7846i	− 0.985285i
$446$	0	0
$447$	−10.3923	−0.491539
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.46410i	0.162758i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	− 6.00000i	− 0.280056i
$460$	0	0
$461$	31.1769i	1.45205i	0.687666	+	0.726027i	$0.258635\pi$
$461$	31.1769i	1.45205i	−0.687666	+	0.726027i	$0.741365\pi$
$462$	0	0
$463$	−10.3923	−0.482971	−0.241486	−	0.970404i	$-0.577635\pi$
$463$	−10.3923	−0.482971	−0.241486	+	0.970404i	$0.577635\pi$
$464$	0	0
$465$	−12.0000	−0.556487
$466$	0	0
$467$	− 24.0000i	− 1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$	− 24.0000i	− 1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$	0	0
$469$	− 13.8564i	− 0.639829i
$470$	0	0
$471$	6.92820	0.319235
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 28.0000i	− 1.28473i
$476$	0	0
$477$	− 3.46410i	− 0.158610i
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 24.0000i	− 1.09204i
$484$	0	0
$485$	6.92820i	0.314594i
$486$	0	0
$487$	−10.3923	−0.470920	−0.235460	−	0.971884i	$-0.575660\pi$
$487$	−10.3923	−0.470920	−0.235460	+	0.971884i	$0.575660\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	36.0000i	1.62466i	0.583200	+	0.812329i	$0.301800\pi$
$491$	36.0000i	1.62466i	−0.583200	+	0.812329i	$0.698200\pi$
$492$	0	0
$493$	− 20.7846i	− 0.936092i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	− 28.0000i	− 1.25345i	−0.779240	−	0.626726i	$-0.784395\pi$
$499$	− 28.0000i	− 1.25345i	0.779240	−	0.626726i	$-0.215605\pi$
$500$	0	0
$501$	− 13.8564i	− 0.619059i
$502$	0	0
$503$	34.6410	1.54457	0.772283	−	0.635278i	$-0.219115\pi$
$503$	34.6410	1.54457	0.772283	+	0.635278i	$0.219115\pi$
$504$	0	0
$505$	12.0000	0.533993
$506$	0	0
$507$	13.0000i	0.577350i
$508$	0	0
$509$	24.2487i	1.07481i	0.843326	+	0.537403i	$0.180594\pi$
$509$	24.2487i	1.07481i	−0.843326	+	0.537403i	$0.819406\pi$
$510$	0	0
$511$	6.92820	0.306486
$512$	0	0
$513$	4.00000	0.176604
$514$	0	0
$515$	60.0000i	2.64392i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−17.3205	−0.760286
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	− 24.2487i	− 1.05830i
$526$	0	0
$527$	−20.7846	−0.905392
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	− 12.0000i	− 0.520756i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	41.5692	1.79719
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 13.8564i	− 0.595733i	−0.954607	−	0.297867i	$-0.903725\pi$
$541$	− 13.8564i	− 0.595733i	0.954607	−	0.297867i	$-0.0962751\pi$
$542$	0	0
$543$	−13.8564	−0.594635
$544$	0	0
$545$	−48.0000	−2.05609
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	− 6.92820i	− 0.295689i
$550$	0	0
$551$	13.8564	0.590303
$552$	0	0
$553$	36.0000	1.53088
$554$	0	0
$555$	24.0000i	1.01874i
$556$	0	0
$557$	− 10.3923i	− 0.440336i	−0.975462	−	0.220168i	$-0.929339\pi$
$557$	− 10.3923i	− 0.440336i	0.975462	−	0.220168i	$-0.0706606\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	20.7846i	0.874415i
$566$	0	0
$567$	3.46410	0.145479
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	4.00000i	0.167395i	0.996491	+	0.0836974i	$0.0266729\pi$
$571$	4.00000i	0.167395i	−0.996491	+	0.0836974i	$0.973327\pi$
$572$	0	0
$573$	− 13.8564i	− 0.578860i
$574$	0	0
$575$	48.4974	2.02248
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	0	0
$579$	2.00000i	0.0831172i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12.0000i	− 0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$	− 12.0000i	− 0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	10.3923	0.427482
$592$	0	0
$593$	−6.00000	−0.246390	−0.123195	−	0.992382i	$-0.539314\pi$
$593$	−6.00000	−0.246390	−0.123195	+	0.992382i	$0.539314\pi$
$594$	0	0
$595$	− 72.0000i	− 2.95171i
$596$	0	0
$597$	10.3923i	0.425329i
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	0	0
$605$	− 38.1051i	− 1.54919i
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	0	0
$609$	12.0000	0.486265
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 34.6410i	− 1.39914i	−0.714565	−	0.699569i	$-0.753375\pi$
$613$	− 34.6410i	− 1.39914i	0.714565	−	0.699569i	$-0.246625\pi$
$614$	0	0
$615$	−20.7846	−0.838116
$616$	0	0
$617$	−18.0000	−0.724653	−0.362326	−	0.932051i	$-0.618017\pi$
$617$	−18.0000	−0.724653	−0.362326	+	0.932051i	$0.618017\pi$
$618$	0	0
$619$	4.00000i	0.160774i	0.996764	+	0.0803868i	$0.0256155\pi$
$619$	4.00000i	0.160774i	−0.996764	+	0.0803868i	$0.974384\pi$
$620$	0	0
$621$	6.92820i	0.278019i
$622$	0	0
$623$	20.7846	0.832718
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	41.5692i	1.65747i
$630$	0	0
$631$	17.3205	0.689519	0.344759	−	0.938691i	$-0.387961\pi$
$631$	17.3205	0.689519	0.344759	+	0.938691i	$0.387961\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.92820	−0.274075
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	− 13.8564i	− 0.545595i
$646$	0	0
$647$	−34.6410	−1.36188	−0.680939	−	0.732340i	$-0.738428\pi$
$647$	−34.6410	−1.36188	−0.680939	+	0.732340i	$0.738428\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	− 12.0000i	− 0.470317i
$652$	0	0
$653$	− 10.3923i	− 0.406682i	−0.979108	−	0.203341i	$-0.934820\pi$
$653$	− 10.3923i	− 0.406682i	0.979108	−	0.203341i	$-0.0651801\pi$
$654$	0	0
$655$	−41.5692	−1.62424
$656$	0	0
$657$	−2.00000	−0.0780274
$658$	0	0
$659$	− 12.0000i	− 0.467454i	−0.972302	−	0.233727i	$-0.924908\pi$
$659$	− 12.0000i	− 0.467454i	0.972302	−	0.233727i	$-0.0750921\pi$
$660$	0	0
$661$	6.92820i	0.269476i	0.990881	+	0.134738i	$0.0430193\pi$
$661$	6.92820i	0.269476i	−0.990881	+	0.134738i	$0.956981\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	48.0000	1.86136
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	3.46410i	0.133930i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	0	0
$675$	7.00000i	0.269430i
$676$	0	0
$677$	− 3.46410i	− 0.133136i	−0.997782	−	0.0665681i	$-0.978795\pi$
$677$	− 3.46410i	− 0.133136i	0.997782	−	0.0665681i	$-0.0212050\pi$
$678$	0	0
$679$	−6.92820	−0.265880
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	20.7846i	0.794139i
$686$	0	0
$687$	27.7128	1.05731
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.0000i	1.06517i	0.846376	+	0.532585i	$0.178779\pi$
$691$	28.0000i	1.06517i	−0.846376	+	0.532585i	$0.821221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−69.2820	−2.62802
$696$	0	0
$697$	−36.0000	−1.36360
$698$	0	0
$699$	− 18.0000i	− 0.680823i
$700$	0	0
$701$	10.3923i	0.392512i	0.980553	+	0.196256i	$0.0628784\pi$
$701$	10.3923i	0.392512i	−0.980553	+	0.196256i	$0.937122\pi$
$702$	0	0
$703$	−27.7128	−1.04521
$704$	0	0
$705$	−24.0000	−0.903892
$706$	0	0
$707$	12.0000i	0.451306i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	−10.3923	−0.389742
$712$	0	0
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	0	0
$717$	27.7128i	1.03495i
$718$	0	0
$719$	−20.7846	−0.775135	−0.387568	−	0.921841i	$-0.626685\pi$
$719$	−20.7846	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$720$	0	0
$721$	−60.0000	−2.23452
$722$	0	0
$723$	− 22.0000i	− 0.818189i
$724$	0	0
$725$	24.2487i	0.900575i
$726$	0	0
$727$	−10.3923	−0.385429	−0.192715	−	0.981255i	$-0.561729\pi$
$727$	−10.3923	−0.385429	−0.192715	+	0.981255i	$0.561729\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 24.0000i	− 0.887672i
$732$	0	0
$733$	27.7128i	1.02360i	0.859106	+	0.511798i	$0.171020\pi$
$733$	27.7128i	1.02360i	−0.859106	+	0.511798i	$0.828980\pi$
$734$	0	0
$735$	17.3205	0.638877
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 28.0000i	− 1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$	− 28.0000i	− 1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	27.7128	1.01668	0.508342	−	0.861155i	$-0.330258\pi$
$743$	27.7128	1.01668	0.508342	+	0.861155i	$0.330258\pi$
$744$	0	0
$745$	36.0000	1.31894
$746$	0	0
$747$	0	0
$748$	0	0
$749$	41.5692i	1.51891i
$750$	0	0
$751$	24.2487	0.884848	0.442424	−	0.896806i	$-0.354119\pi$
$751$	24.2487	0.884848	0.442424	+	0.896806i	$0.354119\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 12.0000i	− 0.436725i
$756$	0	0
$757$	− 27.7128i	− 1.00724i	−0.863925	−	0.503620i	$-0.832001\pi$
$757$	− 27.7128i	− 1.00724i	0.863925	−	0.503620i	$-0.167999\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	− 48.0000i	− 1.73772i
$764$	0	0
$765$	20.7846i	0.751469i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	18.0000i	0.648254i
$772$	0	0
$773$	− 38.1051i	− 1.37055i	−0.728286	−	0.685273i	$-0.759683\pi$
$773$	− 38.1051i	− 1.37055i	0.728286	−	0.685273i	$-0.240317\pi$
$774$	0	0
$775$	24.2487	0.871039
$776$	0	0
$777$	−24.0000	−0.860995
$778$	0	0
$779$	− 24.0000i	− 0.859889i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−3.46410	−0.123797
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.7846	−0.739016
$792$	0	0
$793$	0	0
$794$	0	0
$795$	12.0000i	0.425596i
$796$	0	0
$797$	− 24.2487i	− 0.858933i	−0.903083	−	0.429467i	$-0.858702\pi$
$797$	− 24.2487i	− 0.858933i	0.903083	−	0.429467i	$-0.141298\pi$
$798$	0	0
$799$	−41.5692	−1.47061
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	83.1384i	2.93024i
$806$	0	0
$807$	31.1769	1.09748
$808$	0	0
$809$	−54.0000	−1.89854	−0.949269	−	0.314464i	$-0.898175\pi$
$809$	−54.0000	−1.89854	−0.949269	+	0.314464i	$0.898175\pi$
$810$	0	0
$811$	20.0000i	0.702295i	0.936320	+	0.351147i	$0.114208\pi$
$811$	20.0000i	0.702295i	−0.936320	+	0.351147i	$0.885792\pi$
$812$	0	0
$813$	3.46410i	0.121491i
$814$	0	0
$815$	−69.2820	−2.42684
$816$	0	0
$817$	16.0000	0.559769
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 24.2487i	− 0.846286i	−0.906063	−	0.423143i	$-0.860927\pi$
$821$	− 24.2487i	− 0.846286i	0.906063	−	0.423143i	$-0.139073\pi$
$822$	0	0
$823$	31.1769	1.08676	0.543379	−	0.839487i	$-0.317144\pi$
$823$	31.1769	1.08676	0.543379	+	0.839487i	$0.317144\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	41.5692i	1.44376i	0.692019	+	0.721879i	$0.256721\pi$
$829$	41.5692i	1.44376i	−0.692019	+	0.721879i	$0.743279\pi$
$830$	0	0
$831$	−13.8564	−0.480673
$832$	0	0
$833$	30.0000	1.03944
$834$	0	0
$835$	48.0000i	1.66111i
$836$	0	0
$837$	3.46410i	0.119737i
$838$	0	0
$839$	−6.92820	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$839$	−6.92820	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	30.0000i	1.03325i
$844$	0	0
$845$	− 45.0333i	− 1.54919i
$846$	0	0
$847$	38.1051	1.30931
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	− 48.0000i	− 1.64542i
$852$	0	0
$853$	− 48.4974i	− 1.66052i	−0.557376	−	0.830260i	$-0.688192\pi$
$853$	− 48.4974i	− 1.66052i	0.557376	−	0.830260i	$-0.311808\pi$
$854$	0	0
$855$	−13.8564	−0.473879
$856$	0	0
$857$	42.0000	1.43469	0.717346	−	0.696717i	$-0.245357\pi$
$857$	42.0000	1.43469	0.717346	+	0.696717i	$0.245357\pi$
$858$	0	0
$859$	− 4.00000i	− 0.136478i	−0.997669	−	0.0682391i	$-0.978262\pi$
$859$	− 4.00000i	− 0.136478i	0.997669	−	0.0682391i	$-0.0217381\pi$
$860$	0	0
$861$	− 20.7846i	− 0.708338i
$862$	0	0
$863$	−13.8564	−0.471678	−0.235839	−	0.971792i	$-0.575784\pi$
$863$	−13.8564	−0.471678	−0.235839	+	0.971792i	$0.575784\pi$
$864$	0	0
$865$	60.0000	2.04006
$866$	0	0
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	24.0000i	0.811348i
$876$	0	0
$877$	6.92820i	0.233949i	0.993135	+	0.116974i	$0.0373195\pi$
$877$	6.92820i	0.233949i	−0.993135	+	0.116974i	$0.962680\pi$
$878$	0	0
$879$	−3.46410	−0.116841
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	− 20.0000i	− 0.673054i	−0.941674	−	0.336527i	$-0.890748\pi$
$883$	− 20.0000i	− 0.673054i	0.941674	−	0.336527i	$-0.109252\pi$
$884$	0	0
$885$	41.5692i	1.39733i
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 27.7128i	− 0.927374i
$894$	0	0
$895$	41.5692	1.38951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	12.0000i	0.400222i
$900$	0	0
$901$	20.7846i	0.692436i
$902$	0	0
$903$	13.8564	0.461112
$904$	0	0
$905$	48.0000	1.59557
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	0	0
$909$	− 3.46410i	− 0.114897i
$910$	0	0
$911$	55.4256	1.83633	0.918166	−	0.396195i	$-0.129670\pi$
$911$	55.4256	1.83633	0.918166	+	0.396195i	$0.129670\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	24.0000i	0.793416i
$916$	0	0
$917$	− 41.5692i	− 1.37274i
$918$	0	0
$919$	−51.9615	−1.71405	−0.857026	−	0.515273i	$-0.827691\pi$
$919$	−51.9615	−1.71405	−0.857026	+	0.515273i	$0.827691\pi$
$920$	0	0
$921$	4.00000	0.131804
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 48.4974i	− 1.59459i
$926$	0	0
$927$	17.3205	0.568880
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	20.0000i	0.655474i
$932$	0	0
$933$	− 13.8564i	− 0.453638i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	− 26.0000i	− 0.848478i
$940$	0	0
$941$	− 24.2487i	− 0.790485i	−0.918577	−	0.395243i	$-0.870660\pi$
$941$	− 24.2487i	− 0.790485i	0.918577	−	0.395243i	$-0.129340\pi$
$942$	0	0
$943$	41.5692	1.35368
$944$	0	0
$945$	−12.0000	−0.390360
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	31.1769	1.01098
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	48.0000i	1.55324i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.7846	−0.671170
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	− 12.0000i	− 0.386695i
$964$	0	0
$965$	− 6.92820i	− 0.223027i
$966$	0	0
$967$	38.1051	1.22538	0.612689	−	0.790324i	$-0.290088\pi$
$967$	38.1051	1.22538	0.612689	+	0.790324i	$0.290088\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	48.0000i	1.54039i	0.637806	+	0.770197i	$0.279842\pi$
$971$	48.0000i	1.54039i	−0.637806	+	0.770197i	$0.720158\pi$
$972$	0	0
$973$	− 69.2820i	− 2.22108i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	13.8564i	0.442401i
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−36.0000	−1.14706
$986$	0	0
$987$	− 24.0000i	− 0.763928i
$988$	0	0
$989$	27.7128i	0.881216i
$990$	0	0
$991$	24.2487	0.770286	0.385143	−	0.922857i	$-0.374152\pi$
$991$	24.2487	0.770286	0.385143	+	0.922857i	$0.374152\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	0	0
$995$	− 36.0000i	− 1.14128i
$996$	0	0
$997$	20.7846i	0.658255i	0.944285	+	0.329128i	$0.106755\pi$
$997$	20.7846i	0.658255i	−0.944285	+	0.329128i	$0.893245\pi$
$998$	0	0
$999$	6.92820	0.219199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.2.d.a.97.3	yes	4
3.2	odd	2		576.2.d.b.289.4		4
4.3	odd	2	inner	192.2.d.a.97.1	✓	4
5.2	odd	4		4800.2.d.o.1249.4		4
5.3	odd	4		4800.2.d.j.1249.1		4
5.4	even	2		4800.2.k.j.2401.1		4
8.3	odd	2	inner	192.2.d.a.97.4	yes	4
8.5	even	2	inner	192.2.d.a.97.2	yes	4
12.11	even	2		576.2.d.b.289.3		4
16.3	odd	4		768.2.a.k.1.1		2
16.5	even	4		768.2.a.k.1.2		2
16.11	odd	4		768.2.a.j.1.2		2
16.13	even	4		768.2.a.j.1.1		2
20.3	even	4		4800.2.d.o.1249.3		4
20.7	even	4		4800.2.d.j.1249.2		4
20.19	odd	2		4800.2.k.j.2401.4		4
24.5	odd	2		576.2.d.b.289.2		4
24.11	even	2		576.2.d.b.289.1		4
40.3	even	4		4800.2.d.j.1249.3		4
40.13	odd	4		4800.2.d.o.1249.1		4
40.19	odd	2		4800.2.k.j.2401.2		4
40.27	even	4		4800.2.d.o.1249.2		4
40.29	even	2		4800.2.k.j.2401.3		4
40.37	odd	4		4800.2.d.j.1249.4		4
48.5	odd	4		2304.2.a.u.1.1		2
48.11	even	4		2304.2.a.s.1.1		2
48.29	odd	4		2304.2.a.s.1.2		2
48.35	even	4		2304.2.a.u.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.2.d.a.97.1	✓	4	4.3	odd	2	inner
192.2.d.a.97.2	yes	4	8.5	even	2	inner
192.2.d.a.97.3	yes	4	1.1	even	1	trivial
192.2.d.a.97.4	yes	4	8.3	odd	2	inner
576.2.d.b.289.1		4	24.11	even	2
576.2.d.b.289.2		4	24.5	odd	2
576.2.d.b.289.3		4	12.11	even	2
576.2.d.b.289.4		4	3.2	odd	2
768.2.a.j.1.1		2	16.13	even	4
768.2.a.j.1.2		2	16.11	odd	4
768.2.a.k.1.1		2	16.3	odd	4
768.2.a.k.1.2		2	16.5	even	4
2304.2.a.s.1.1		2	48.11	even	4
2304.2.a.s.1.2		2	48.29	odd	4
2304.2.a.u.1.1		2	48.5	odd	4
2304.2.a.u.1.2		2	48.35	even	4
4800.2.d.j.1249.1		4	5.3	odd	4
4800.2.d.j.1249.2		4	20.7	even	4
4800.2.d.j.1249.3		4	40.3	even	4
4800.2.d.j.1249.4		4	40.37	odd	4
4800.2.d.o.1249.1		4	40.13	odd	4
4800.2.d.o.1249.2		4	40.27	even	4
4800.2.d.o.1249.3		4	20.3	even	4
4800.2.d.o.1249.4		4	5.2	odd	4
4800.2.k.j.2401.1		4	5.4	even	2
4800.2.k.j.2401.2		4	40.19	odd	2
4800.2.k.j.2401.3		4	40.29	even	2
4800.2.k.j.2401.4		4	20.19	odd	2

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	192.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -3.46410i q^{5} +3.46410 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -3.46410i q^{5} +3.46410 q^{7} -1.00000 q^{9} +3.46410 q^{15} +6.00000 q^{17} +4.00000i q^{19} +3.46410i q^{21} -6.92820 q^{23} -7.00000 q^{25} -1.00000i q^{27} -3.46410i q^{29} -3.46410 q^{31} -12.0000i q^{35} +6.92820i q^{37} -6.00000 q^{41} -4.00000i q^{43} +3.46410i q^{45} -6.92820 q^{47} +5.00000 q^{49} +6.00000i q^{51} +3.46410i q^{53} -4.00000 q^{57} +12.0000i q^{59} +6.92820i q^{61} -3.46410 q^{63} -4.00000i q^{67} -6.92820i q^{69} +6.92820 q^{71} +2.00000 q^{73} -7.00000i q^{75} +10.3923 q^{79} +1.00000 q^{81} -20.7846i q^{85} +3.46410 q^{87} +6.00000 q^{89} -3.46410i q^{93} +13.8564 q^{95} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 24 q^{17} - 28 q^{25} - 24 q^{41} + 20 q^{49} - 16 q^{57} + 8 q^{73} + 4 q^{81} + 24 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 24 * q^17 - 28 * q^25 - 24 * q^41 + 20 * q^49 - 16 * q^57 + 8 * q^73 + 4 * q^81 + 24 * q^89 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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