LMFDB - Embedded newform 6600.2.d.h.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1849,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6600.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.d (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:	$52.7012653340$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	6600.1849
Dual form			6600.2.d.h.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +4.00000i q^{17} -2.00000 q^{21} +4.00000i q^{23} +1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +1.00000i q^{33} -2.00000i q^{37} +4.00000 q^{39} -10.0000 q^{41} -10.0000i q^{43} -12.0000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} +12.0000 q^{59} +10.0000 q^{61} +2.00000i q^{63} -8.00000i q^{67} +4.00000 q^{69} -4.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -2.00000i q^{83} -6.00000i q^{87} +10.0000 q^{89} +8.00000 q^{91} +8.00000i q^{93} -2.00000i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} - 2 q^{11} - 4 q^{21} + 12 q^{29} - 16 q^{31} + 8 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} + 24 q^{59} + 20 q^{61} + 8 q^{69} - 8 q^{79} + 2 q^{81} + 20 q^{89} + 16 q^{91} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 - 2 * q^11 - 4 * q^21 + 12 * q^29 - 16 * q^31 + 8 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 + 24 * q^59 + 20 * q^61 + 8 * q^69 - 8 * q^79 + 2 * q^81 + 20 * q^89 + 16 * q^91 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times$.

$n$	$1201$	$2201$	$2377$	$3301$	$4951$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.00000i	− 0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$	− 2.00000i	− 0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	4.00000i	0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$	4.00000i	0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	1.00000i	0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 12.0000i	− 1.75038i	−0.483779	−	0.875190i	$-0.660736\pi$
$47$	− 12.0000i	− 1.75038i	0.483779	−	0.875190i	$-0.339264\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	2.00000i	0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000i	0.227921i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.00000i	− 0.219529i	−0.993958	−	0.109764i	$-0.964990\pi$
$83$	− 2.00000i	− 0.219529i	0.993958	−	0.109764i	$-0.0350096\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 6.00000i	− 0.643268i
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	8.00000i	0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.0000i	− 1.35343i	−0.736245	−	0.676716i	$-0.763403\pi$
$107$	− 14.0000i	− 1.35343i	0.736245	−	0.676716i	$-0.236597\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	8.00000	0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.0000i	0.901670i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	−10.0000	−0.880451
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.0000i	1.87959i	0.341743	+	0.939793i	$0.388983\pi$
$137$	22.0000i	1.87959i	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	−12.0000	−1.01058
$142$	0	0
$143$	− 4.00000i	− 0.334497i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000i	0.159617i	0.996810	+	0.0798087i	$0.0254309\pi$
$157$	2.00000i	0.159617i	−0.996810	+	0.0798087i	$0.974569\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 22.0000i	− 1.70241i	−0.524832	−	0.851206i	$-0.675872\pi$
$167$	− 22.0000i	− 1.70241i	0.524832	−	0.851206i	$-0.324128\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 16.0000i	− 1.21646i	−0.793762	−	0.608229i	$-0.791880\pi$
$173$	− 16.0000i	− 1.21646i	0.793762	−	0.608229i	$-0.208120\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	0	0
$183$	− 10.0000i	− 0.739221i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.00000i	− 0.292509i
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	− 20.0000i	− 1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$	− 20.0000i	− 1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	− 12.0000i	− 0.842235i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 4.00000i	− 0.278019i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	0	0
$219$	−4.00000	−0.270295
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	28.0000i	1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$	28.0000i	1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.00000i	0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$	6.00000i	0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$	0	0
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	24.0000i	1.57229i	0.618041	+	0.786146i	$0.287927\pi$
$233$	24.0000i	1.57229i	−0.618041	+	0.786146i	$0.712073\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	4.00000i	0.259828i
$238$	0	0
$239$	20.0000	1.29369	0.646846	−	0.762620i	$-0.276088\pi$
$239$	20.0000	1.29369	0.646846	+	0.762620i	$0.276088\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.00000	−0.126745
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 14.0000i	− 0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$	− 14.0000i	− 0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	− 6.00000i	− 0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$	− 6.00000i	− 0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 10.0000i	− 0.611990i
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	− 8.00000i	− 0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	20.0000i	1.20168i	0.799368	+	0.600842i	$0.205168\pi$
$277$	20.0000i	1.20168i	−0.799368	+	0.600842i	$0.794832\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	− 26.0000i	− 1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$	− 26.0000i	− 1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.0000i	1.18056i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.00000i	− 0.0580259i
$298$	0	0
$299$	−16.0000	−0.925304
$300$	0	0
$301$	−20.0000	−1.15278
$302$	0	0
$303$	− 6.00000i	− 0.344691i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.0000i	1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$	18.0000i	1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	0	0
$321$	−14.0000	−0.781404
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.00000i	0.331801i
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 4.00000i	− 0.217894i	−0.994048	−	0.108947i	$-0.965252\pi$
$337$	− 4.00000i	− 0.217894i	0.994048	−	0.108947i	$-0.0347479\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	− 20.0000i	− 1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.0000i	0.536828i	0.963304	+	0.268414i	$0.0864995\pi$
$347$	10.0000i	0.536828i	−0.963304	+	0.268414i	$0.913500\pi$
$348$	0	0
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	− 26.0000i	− 1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$	− 26.0000i	− 1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 8.00000i	− 0.423405i
$358$	0	0
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0000i	− 1.46159i	−0.682598	−	0.730794i	$-0.739150\pi$
$367$	− 28.0000i	− 1.46159i	0.682598	−	0.730794i	$-0.260850\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.0000i	0.508329i
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	−16.0000	−0.809155
$392$	0	0
$393$	16.0000i	0.807093i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.0000i	− 0.702640i	−0.936255	−	0.351320i	$-0.885733\pi$
$397$	− 14.0000i	− 0.702640i	0.936255	−	0.351320i	$-0.114267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	− 32.0000i	− 1.59403i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.00000i	0.0991363i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	22.0000	1.08518
$412$	0	0
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.0000i	0.979404i
$418$	0	0
$419$	20.0000	0.977064	0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000	0.977064	0.488532	+	0.872546i	$0.337533\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	0	0
$423$	12.0000i	0.583460i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 20.0000i	− 0.967868i
$428$	0	0
$429$	−4.00000	−0.193122
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.00000i	0.283790i
$448$	0	0
$449$	42.0000	1.98210	0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000	1.98210	0.991051	+	0.133482i	$0.0426157\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	0	0
$453$	4.00000i	0.187936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.0000i	0.748448i	0.927338	+	0.374224i	$0.122091\pi$
$457$	16.0000i	0.748448i	−0.927338	+	0.374224i	$0.877909\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.00000i	− 0.370196i	−0.982720	−	0.185098i	$-0.940740\pi$
$467$	− 8.00000i	− 0.370196i	0.982720	−	0.185098i	$-0.0592602\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	10.0000i	0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.00000i	0.274721i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	− 8.00000i	− 0.364013i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	0	0
$489$	4.00000	0.180886
$490$	0	0
$491$	28.0000	1.26362	0.631811	−	0.775122i	$-0.282312\pi$
$491$	28.0000	1.26362	0.631811	+	0.775122i	$0.282312\pi$
$492$	0	0
$493$	24.0000i	1.08091i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	−22.0000	−0.982888
$502$	0	0
$503$	− 2.00000i	− 0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.00000i	0.133235i
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−8.00000	−0.353899
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0000i	0.527759i
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	26.0000i	1.13690i	0.822718	+	0.568450i	$0.192457\pi$
$523$	26.0000i	1.13690i	−0.822718	+	0.568450i	$0.807543\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 32.0000i	− 1.39394i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	− 40.0000i	− 1.73259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 12.0000i	− 0.517838i
$538$	0	0
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	14.0000i	0.600798i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 2.00000i	− 0.0855138i	−0.999086	−	0.0427569i	$-0.986386\pi$
$547$	− 2.00000i	− 0.0855138i	0.999086	−	0.0427569i	$-0.0136141\pi$
$548$	0	0
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8.00000i	0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	10.0000i	0.421450i	0.977545	+	0.210725i	$0.0675824\pi$
$563$	10.0000i	0.421450i	−0.977545	+	0.210725i	$0.932418\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.00000i	− 0.0839921i
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	0	0
$579$	−20.0000	−0.831172
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	6.00000i	0.248495i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 16.0000i	− 0.660391i	−0.943913	−	0.330195i	$-0.892885\pi$
$587$	− 16.0000i	− 0.660391i	0.943913	−	0.330195i	$-0.107115\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 44.0000i	− 1.80686i	−0.428732	−	0.903432i	$-0.641040\pi$
$593$	− 44.0000i	− 1.80686i	0.428732	−	0.903432i	$-0.358960\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 8.00000i	− 0.327418i
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−34.0000	−1.38689	−0.693444	−	0.720510i	$-0.743908\pi$
$601$	−34.0000	−1.38689	−0.693444	+	0.720510i	$0.743908\pi$
$602$	0	0
$603$	8.00000i	0.325785i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 14.0000i	− 0.568242i	−0.958788	−	0.284121i	$-0.908298\pi$
$607$	− 14.0000i	− 0.568242i	0.958788	−	0.284121i	$-0.0917018\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	48.0000	1.94187
$612$	0	0
$613$	16.0000i	0.646234i	0.946359	+	0.323117i	$0.104731\pi$
$613$	16.0000i	0.646234i	−0.946359	+	0.323117i	$0.895269\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	− 20.0000i	− 0.801283i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.00000	0.318981
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	8.00000i	0.317971i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	0	0
$640$	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
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 40.0000i	− 1.57256i	−0.617869	−	0.786281i	$-0.712004\pi$
$647$	− 40.0000i	− 1.57256i	0.617869	−	0.786281i	$-0.287996\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	42.0000i	1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$	42.0000i	1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4.00000i	0.156055i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	0	0
$663$	16.0000i	0.621389i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	24.0000i	0.929284i
$668$	0	0
$669$	28.0000	1.08254
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	44.0000i	1.69608i	0.529936	+	0.848038i	$0.322216\pi$
$673$	44.0000i	1.69608i	−0.529936	+	0.848038i	$0.677784\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	12.0000i	0.461197i	0.973049	+	0.230599i	$0.0740685\pi$
$677$	12.0000i	0.461197i	−0.973049	+	0.230599i	$0.925932\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	− 8.00000i	− 0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$	− 8.00000i	− 0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	22.0000i	0.839352i
$688$	0	0
$689$	24.0000	0.914327
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	− 2.00000i	− 0.0759737i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 40.0000i	− 1.51511i
$698$	0	0
$699$	24.0000	0.907763
$700$	0	0
$701$	−22.0000	−0.830929	−0.415464	−	0.909610i	$-0.636381\pi$
$701$	−22.0000	−0.830929	−0.415464	+	0.909610i	$0.636381\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 12.0000i	− 0.451306i
$708$	0	0
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 20.0000i	− 0.746914i
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	0	0
$723$	2.00000i	0.0743808i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	− 8.00000i	− 0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$	− 8.00000i	− 0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 6.00000i	− 0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$	− 6.00000i	− 0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.00000i	0.0731762i
$748$	0	0
$749$	−28.0000	−1.02310
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.0000i	0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$	18.0000i	0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	12.0000i	0.434429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48.0000i	1.73318i
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	− 30.0000i	− 1.07903i	−0.841978	−	0.539513i	$-0.818609\pi$
$773$	− 30.0000i	− 1.07903i	0.841978	−	0.539513i	$-0.181391\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000i	0.143499i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	6.00000i	0.214423i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 38.0000i	− 1.35455i	−0.735728	−	0.677277i	$-0.763160\pi$
$787$	− 38.0000i	− 1.35455i	0.735728	−	0.677277i	$-0.236840\pi$
$788$	0	0
$789$	−6.00000	−0.213606
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	40.0000i	1.42044i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 22.0000i	− 0.779280i	−0.920967	−	0.389640i	$-0.872599\pi$
$797$	− 22.0000i	− 0.779280i	0.920967	−	0.389640i	$-0.127401\pi$
$798$	0	0
$799$	48.0000	1.69812
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	4.00000i	0.141157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.0000i	0.492823i
$808$	0	0
$809$	−14.0000	−0.492214	−0.246107	−	0.969243i	$-0.579151\pi$
$809$	−14.0000	−0.492214	−0.246107	+	0.969243i	$0.579151\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	− 20.0000i	− 0.701431i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−8.00000	−0.279543
$820$	0	0
$821$	50.0000	1.74501	0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000	1.74501	0.872506	+	0.488603i	$0.162493\pi$
$822$	0	0
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.0000i	0.904109i	0.891990	+	0.452054i	$0.149309\pi$
$827$	26.0000i	0.904109i	−0.891990	+	0.452054i	$0.850691\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	0	0
$833$	12.0000i	0.415775i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 8.00000i	− 0.276520i
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	10.0000i	0.344418i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	0	0
$849$	−26.0000	−0.892318
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	56.0000i	1.91740i	0.284413	+	0.958702i	$0.408201\pi$
$853$	56.0000i	1.91740i	−0.284413	+	0.958702i	$0.591799\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 28.0000i	− 0.956462i	−0.878234	−	0.478231i	$-0.841278\pi$
$857$	− 28.0000i	− 0.956462i	0.878234	−	0.478231i	$-0.158722\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	20.0000	0.681598
$862$	0	0
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	32.0000	1.08428
$872$	0	0
$873$	2.00000i	0.0676897i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	8.00000i	0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$	8.00000i	0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.0000i	1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$	42.0000i	1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$	0	0
$889$	4.00000	0.134156
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	16.0000i	0.534224i
$898$	0	0
$899$	−48.0000	−1.60089
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	20.0000i	0.665558i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.0000i	0.531271i	0.964073	+	0.265636i	$0.0855818\pi$
$907$	16.0000i	0.531271i	−0.964073	+	0.265636i	$0.914418\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	2.00000i	0.0661903i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	32.0000i	1.05673i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	18.0000	0.593120
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	8.00000i	0.262754i
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 16.0000i	− 0.523816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.0000i	0.522697i	0.965244	+	0.261349i	$0.0841672\pi$
$937$	16.0000i	0.522697i	−0.965244	+	0.261349i	$0.915833\pi$
$938$	0	0
$939$	−26.0000	−0.848478
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	− 40.0000i	− 1.30258i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	8.00000i	0.259145i	0.991570	+	0.129573i	$0.0413606\pi$
$953$	8.00000i	0.259145i	−0.991570	+	0.129573i	$0.958639\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.00000i	0.193952i
$958$	0	0
$959$	44.0000	1.42083
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	14.0000i	0.451144i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 6.00000i	− 0.192947i	−0.995336	−	0.0964735i	$-0.969244\pi$
$967$	− 6.00000i	− 0.192947i	0.995336	−	0.0964735i	$-0.0307563\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	36.0000	1.15529	0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000	1.15529	0.577647	+	0.816286i	$0.303971\pi$
$972$	0	0
$973$	40.0000i	1.28234i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	−10.0000	−0.319601
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	− 16.0000i	− 0.510321i	−0.966899	−	0.255160i	$-0.917872\pi$
$983$	− 16.0000i	− 0.510321i	0.966899	−	0.255160i	$-0.0821283\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	24.0000i	0.763928i
$988$	0	0
$989$	40.0000	1.27193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 16.0000i	− 0.506725i	−0.967371	−	0.253363i	$-0.918463\pi$
$997$	− 16.0000i	− 0.506725i	0.967371	−	0.253363i	$-0.0815366\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.d.h.1849.1		2
5.2	odd	4		6600.2.a.m.1.1		1
5.3	odd	4		1320.2.a.i.1.1	✓	1
5.4	even	2	inner	6600.2.d.h.1849.2		2
15.8	even	4		3960.2.a.m.1.1		1
20.3	even	4		2640.2.a.d.1.1		1
60.23	odd	4		7920.2.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.i.1.1	✓	1	5.3	odd	4
2640.2.a.d.1.1		1	20.3	even	4
3960.2.a.m.1.1		1	15.8	even	4
6600.2.a.m.1.1		1	5.2	odd	4
6600.2.d.h.1849.1		2	1.1	even	1	trivial
6600.2.d.h.1849.2		2	5.4	even	2	inner
7920.2.a.bd.1.1		1	60.23	odd	4

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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:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.00000i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.00000i q^{13} +4.00000i q^{17} -2.00000 q^{21} +4.00000i q^{23} +1.00000i q^{27} +6.00000 q^{29} -8.00000 q^{31} +1.00000i q^{33} -2.00000i q^{37} +4.00000 q^{39} -10.0000 q^{41} -10.0000i q^{43} -12.0000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -6.00000i q^{53} +12.0000 q^{59} +10.0000 q^{61} +2.00000i q^{63} -8.00000i q^{67} +4.00000 q^{69} -4.00000i q^{73} +2.00000i q^{77} -4.00000 q^{79} +1.00000 q^{81} -2.00000i q^{83} -6.00000i q^{87} +10.0000 q^{89} +8.00000 q^{91} +8.00000i q^{93} -2.00000i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} - 2 q^{11} - 4 q^{21} + 12 q^{29} - 16 q^{31} + 8 q^{39} - 20 q^{41} + 6 q^{49} + 8 q^{51} + 24 q^{59} + 20 q^{61} + 8 q^{69} - 8 q^{79} + 2 q^{81} + 20 q^{89} + 16 q^{91} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 - 2 * q^11 - 4 * q^21 + 12 * q^29 - 16 * q^31 + 8 * q^39 - 20 * q^41 + 6 * q^49 + 8 * q^51 + 24 * q^59 + 20 * q^61 + 8 * q^69 - 8 * q^79 + 2 * q^81 + 20 * q^89 + 16 * q^91 + 2 * q^99

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