LMFDB - Embedded newform 1100.6.b.a.749.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1100,6,Mod(749,1100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1100, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("1100.749");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1100 = 2^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1100.b (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:	$176.422201794$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 44)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			749.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1100.749
Dual form			1100.6.b.a.749.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +O(q^{10})$	$q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +121.000 q^{11} -380.000i q^{13} +1154.00i q^{17} +1824.00 q^{19} -350.000 q^{21} +3591.00i q^{23} +3059.00i q^{27} -8032.00 q^{29} -2945.00 q^{31} +847.000i q^{33} -6979.00i q^{37} +2660.00 q^{39} -520.000 q^{41} -2486.00i q^{43} +6920.00i q^{47} +14307.0 q^{49} -8078.00 q^{51} -13718.0i q^{53} +12768.0i q^{57} +31779.0 q^{59} +34156.0 q^{61} +9700.00i q^{63} +61503.0i q^{67} -25137.0 q^{69} -14971.0 q^{71} -36444.0i q^{73} +6050.00i q^{77} +28538.0 q^{79} +25729.0 q^{81} +77482.0i q^{83} -56224.0i q^{87} -36271.0 q^{89} +19000.0 q^{91} -20615.0i q^{93} +49799.0i q^{97} +23474.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 388 q^{9}+O(q^{10}) $ 2 * q + 388 * q^9	$ 2 q + 388 q^{9} + 242 q^{11} + 3648 q^{19} - 700 q^{21} - 16064 q^{29} - 5890 q^{31} + 5320 q^{39} - 1040 q^{41} + 28614 q^{49} - 16156 q^{51} + 63558 q^{59} + 68312 q^{61} - 50274 q^{69} - 29942 q^{71} + 57076 q^{79} + 51458 q^{81} - 72542 q^{89} + 38000 q^{91} + 46948 q^{99}+O(q^{100}) $ 2 * q + 388 * q^9 + 242 * q^11 + 3648 * q^19 - 700 * q^21 - 16064 * q^29 - 5890 * q^31 + 5320 * q^39 - 1040 * q^41 + 28614 * q^49 - 16156 * q^51 + 63558 * q^59 + 68312 * q^61 - 50274 * q^69 - 29942 * q^71 + 57076 * q^79 + 51458 * q^81 - 72542 * q^89 + 38000 * q^91 + 46948 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$551$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	7.00000i	0.449050i	0.974468	+	0.224525i	$0.0720831\pi$
$3$	7.00000i	0.449050i	−0.974468	+	0.224525i	$0.927917\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	50.0000i	0.385678i	0.981230	+	0.192839i	$0.0617695\pi$
$7$	50.0000i	0.385678i	−0.981230	+	0.192839i	$0.938230\pi$
$8$	0	0
$9$	194.000	0.798354
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	− 380.000i	− 0.623627i	−0.950143	−	0.311814i	$-0.899064\pi$
$13$	− 380.000i	− 0.623627i	0.950143	−	0.311814i	$-0.100936\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1154.00i	0.968464i	0.874940	+	0.484232i	$0.160901\pi$
$17$	1154.00i	0.968464i	−0.874940	+	0.484232i	$0.839099\pi$
$18$	0	0
$19$	1824.00	1.15915	0.579577	−	0.814918i	$-0.303218\pi$
$19$	1824.00	1.15915	0.579577	+	0.814918i	$0.303218\pi$
$20$	0	0
$21$	−350.000	−0.173189
$22$	0	0
$23$	3591.00i	1.41545i	0.706486	+	0.707727i	$0.250279\pi$
$23$	3591.00i	1.41545i	−0.706486	+	0.707727i	$0.749721\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3059.00i	0.807551i
$28$	0	0
$29$	−8032.00	−1.77349	−0.886745	−	0.462259i	$-0.847039\pi$
$29$	−8032.00	−1.77349	−0.886745	+	0.462259i	$0.847039\pi$
$30$	0	0
$31$	−2945.00	−0.550403	−0.275202	−	0.961387i	$-0.588745\pi$
$31$	−2945.00	−0.550403	−0.275202	+	0.961387i	$0.588745\pi$
$32$	0	0
$33$	847.000i	0.135394i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6979.00i	− 0.838087i	−0.907966	−	0.419043i	$-0.862366\pi$
$37$	− 6979.00i	− 0.838087i	0.907966	−	0.419043i	$-0.137634\pi$
$38$	0	0
$39$	2660.00	0.280040
$40$	0	0
$41$	−520.000	−0.0483107	−0.0241554	−	0.999708i	$-0.507690\pi$
$41$	−520.000	−0.0483107	−0.0241554	+	0.999708i	$0.507690\pi$
$42$	0	0
$43$	− 2486.00i	− 0.205036i	−0.994731	−	0.102518i	$-0.967310\pi$
$43$	− 2486.00i	− 0.205036i	0.994731	−	0.102518i	$-0.0326899\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6920.00i	0.456942i	0.973551	+	0.228471i	$0.0733727\pi$
$47$	6920.00i	0.456942i	−0.973551	+	0.228471i	$0.926627\pi$
$48$	0	0
$49$	14307.0	0.851252
$50$	0	0
$51$	−8078.00	−0.434889
$52$	0	0
$53$	− 13718.0i	− 0.670812i	−0.942074	−	0.335406i	$-0.891126\pi$
$53$	− 13718.0i	− 0.670812i	0.942074	−	0.335406i	$-0.108874\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12768.0i	0.520518i
$58$	0	0
$59$	31779.0	1.18853	0.594265	−	0.804269i	$-0.297443\pi$
$59$	31779.0	1.18853	0.594265	+	0.804269i	$0.297443\pi$
$60$	0	0
$61$	34156.0	1.17528	0.587641	−	0.809121i	$-0.300057\pi$
$61$	34156.0	1.17528	0.587641	+	0.809121i	$0.300057\pi$
$62$	0	0
$63$	9700.00i	0.307908i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	61503.0i	1.67382i	0.547339	+	0.836911i	$0.315641\pi$
$67$	61503.0i	1.67382i	−0.547339	+	0.836911i	$0.684359\pi$
$68$	0	0
$69$	−25137.0	−0.635610
$70$	0	0
$71$	−14971.0	−0.352456	−0.176228	−	0.984349i	$-0.556390\pi$
$71$	−14971.0	−0.352456	−0.176228	+	0.984349i	$0.556390\pi$
$72$	0	0
$73$	− 36444.0i	− 0.800422i	−0.916423	−	0.400211i	$-0.868937\pi$
$73$	− 36444.0i	− 0.800422i	0.916423	−	0.400211i	$-0.131063\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6050.00i	0.116286i
$78$	0	0
$79$	28538.0	0.514465	0.257232	−	0.966350i	$-0.417189\pi$
$79$	28538.0	0.514465	0.257232	+	0.966350i	$0.417189\pi$
$80$	0	0
$81$	25729.0	0.435723
$82$	0	0
$83$	77482.0i	1.23454i	0.786751	+	0.617271i	$0.211762\pi$
$83$	77482.0i	1.23454i	−0.786751	+	0.617271i	$0.788238\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 56224.0i	− 0.796386i
$88$	0	0
$89$	−36271.0	−0.485383	−0.242691	−	0.970104i	$-0.578030\pi$
$89$	−36271.0	−0.485383	−0.242691	+	0.970104i	$0.578030\pi$
$90$	0	0
$91$	19000.0	0.240519
$92$	0	0
$93$	− 20615.0i	− 0.247159i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	49799.0i	0.537392i	0.963225	+	0.268696i	$0.0865927\pi$
$97$	49799.0i	0.537392i	−0.963225	+	0.268696i	$0.913407\pi$
$98$	0	0
$99$	23474.0	0.240713
$100$	0	0
$101$	−153406.	−1.49637	−0.748185	−	0.663490i	$-0.769074\pi$
$101$	−153406.	−1.49637	−0.748185	+	0.663490i	$0.769074\pi$
$102$	0	0
$103$	− 134720.i	− 1.25124i	−0.780130	−	0.625618i	$-0.784847\pi$
$103$	− 134720.i	− 1.25124i	0.780130	−	0.625618i	$-0.215153\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 169218.i	− 1.42885i	−0.699711	−	0.714426i	$-0.746688\pi$
$107$	− 169218.i	− 1.42885i	0.699711	−	0.714426i	$-0.253312\pi$
$108$	0	0
$109$	233206.	1.88007	0.940034	−	0.341081i	$-0.110793\pi$
$109$	233206.	1.88007	0.940034	+	0.341081i	$0.110793\pi$
$110$	0	0
$111$	48853.0	0.376343
$112$	0	0
$113$	94329.0i	0.694943i	0.937691	+	0.347471i	$0.112960\pi$
$113$	94329.0i	0.694943i	−0.937691	+	0.347471i	$0.887040\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 73720.0i	− 0.497875i
$118$	0	0
$119$	−57700.0	−0.373515
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	− 3640.00i	− 0.0216939i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	259480.i	1.42756i	0.700370	+	0.713780i	$0.253019\pi$
$127$	259480.i	1.42756i	−0.700370	+	0.713780i	$0.746981\pi$
$128$	0	0
$129$	17402.0	0.0920714
$130$	0	0
$131$	−85410.0	−0.434841	−0.217420	−	0.976078i	$-0.569764\pi$
$131$	−85410.0	−0.434841	−0.217420	+	0.976078i	$0.569764\pi$
$132$	0	0
$133$	91200.0i	0.447060i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	427703.i	1.94689i	0.228926	+	0.973444i	$0.426479\pi$
$137$	427703.i	1.94689i	−0.228926	+	0.973444i	$0.573521\pi$
$138$	0	0
$139$	−309690.	−1.35953	−0.679767	−	0.733428i	$-0.737919\pi$
$139$	−309690.	−1.35953	−0.679767	+	0.733428i	$0.737919\pi$
$140$	0	0
$141$	−48440.0	−0.205190
$142$	0	0
$143$	− 45980.0i	− 0.188031i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	100149.i	0.382255i
$148$	0	0
$149$	−449846.	−1.65996	−0.829981	−	0.557792i	$-0.811649\pi$
$149$	−449846.	−1.65996	−0.829981	+	0.557792i	$0.811649\pi$
$150$	0	0
$151$	405074.	1.44575	0.722873	−	0.690981i	$-0.242821\pi$
$151$	405074.	1.44575	0.722873	+	0.690981i	$0.242821\pi$
$152$	0	0
$153$	223876.i	0.773177i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 339321.i	− 1.09866i	−0.835607	−	0.549328i	$-0.814884\pi$
$157$	− 339321.i	− 1.09866i	0.835607	−	0.549328i	$-0.185116\pi$
$158$	0	0
$159$	96026.0	0.301228
$160$	0	0
$161$	−179550.	−0.545910
$162$	0	0
$163$	271396.i	0.800082i	0.916497	+	0.400041i	$0.131004\pi$
$163$	271396.i	0.800082i	−0.916497	+	0.400041i	$0.868996\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 72468.0i	− 0.201074i	−0.994933	−	0.100537i	$-0.967944\pi$
$167$	− 72468.0i	− 0.201074i	0.994933	−	0.100537i	$-0.0320560\pi$
$168$	0	0
$169$	226893.	0.611089
$170$	0	0
$171$	353856.	0.925414
$172$	0	0
$173$	479226.i	1.21738i	0.793409	+	0.608689i	$0.208304\pi$
$173$	479226.i	1.21738i	−0.793409	+	0.608689i	$0.791696\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	222453.i	0.533710i
$178$	0	0
$179$	40935.0	0.0954910	0.0477455	−	0.998860i	$-0.484796\pi$
$179$	40935.0	0.0954910	0.0477455	+	0.998860i	$0.484796\pi$
$180$	0	0
$181$	−90169.0	−0.204579	−0.102289	−	0.994755i	$-0.532617\pi$
$181$	−90169.0	−0.204579	−0.102289	+	0.994755i	$0.532617\pi$
$182$	0	0
$183$	239092.i	0.527761i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	139634.i	0.292003i
$188$	0	0
$189$	−152950.	−0.311455
$190$	0	0
$191$	−260375.	−0.516435	−0.258218	−	0.966087i	$-0.583135\pi$
$191$	−260375.	−0.516435	−0.258218	+	0.966087i	$0.583135\pi$
$192$	0	0
$193$	524324.i	1.01323i	0.862173	+	0.506613i	$0.169103\pi$
$193$	524324.i	1.01323i	−0.862173	+	0.506613i	$0.830897\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 759582.i	− 1.39447i	−0.716843	−	0.697235i	$-0.754413\pi$
$197$	− 759582.i	− 1.39447i	0.716843	−	0.697235i	$-0.245587\pi$
$198$	0	0
$199$	882736.	1.58015	0.790075	−	0.613011i	$-0.210042\pi$
$199$	882736.	1.58015	0.790075	+	0.613011i	$0.210042\pi$
$200$	0	0
$201$	−430521.	−0.751630
$202$	0	0
$203$	− 401600.i	− 0.683996i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	696654.i	1.13003i
$208$	0	0
$209$	220704.	0.349498
$210$	0	0
$211$	−1.15285e6	−1.78266	−0.891328	−	0.453360i	$-0.850225\pi$
$211$	−1.15285e6	−1.78266	−0.891328	+	0.453360i	$0.850225\pi$
$212$	0	0
$213$	− 104797.i	− 0.158270i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 147250.i	− 0.212278i
$218$	0	0
$219$	255108.	0.359430
$220$	0	0
$221$	438520.	0.603961
$222$	0	0
$223$	− 65893.0i	− 0.0887314i	−0.999015	−	0.0443657i	$-0.985873\pi$
$223$	− 65893.0i	− 0.0887314i	0.999015	−	0.0443657i	$-0.0141267\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	314526.i	0.405128i	0.979269	+	0.202564i	$0.0649274\pi$
$227$	314526.i	0.405128i	−0.979269	+	0.202564i	$0.935073\pi$
$228$	0	0
$229$	−1.03846e6	−1.30859	−0.654293	−	0.756241i	$-0.727034\pi$
$229$	−1.03846e6	−1.30859	−0.654293	+	0.756241i	$0.727034\pi$
$230$	0	0
$231$	−42350.0	−0.0522184
$232$	0	0
$233$	509976.i	0.615403i	0.951483	+	0.307702i	$0.0995599\pi$
$233$	509976.i	0.615403i	−0.951483	+	0.307702i	$0.900440\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	199766.i	0.231021i
$238$	0	0
$239$	444494.	0.503351	0.251676	−	0.967812i	$-0.419018\pi$
$239$	444494.	0.503351	0.251676	+	0.967812i	$0.419018\pi$
$240$	0	0
$241$	−283464.	−0.314380	−0.157190	−	0.987568i	$-0.550244\pi$
$241$	−283464.	−0.314380	−0.157190	+	0.987568i	$0.550244\pi$
$242$	0	0
$243$	923440.i	1.00321i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 693120.i	− 0.722880i
$248$	0	0
$249$	−542374.	−0.554371
$250$	0	0
$251$	−773807.	−0.775262	−0.387631	−	0.921815i	$-0.626706\pi$
$251$	−773807.	−0.775262	−0.387631	+	0.921815i	$0.626706\pi$
$252$	0	0
$253$	434511.i	0.426775i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	387714.i	0.366167i	0.983097	+	0.183083i	$0.0586078\pi$
$257$	387714.i	0.366167i	−0.983097	+	0.183083i	$0.941392\pi$
$258$	0	0
$259$	348950.	0.323232
$260$	0	0
$261$	−1.55821e6	−1.41587
$262$	0	0
$263$	− 197602.i	− 0.176158i	−0.996113	−	0.0880789i	$-0.971927\pi$
$263$	− 197602.i	− 0.176158i	0.996113	−	0.0880789i	$-0.0280728\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 253897.i	− 0.217961i
$268$	0	0
$269$	262694.	0.221345	0.110672	−	0.993857i	$-0.464700\pi$
$269$	262694.	0.221345	0.110672	+	0.993857i	$0.464700\pi$
$270$	0	0
$271$	−159068.	−0.131571	−0.0657854	−	0.997834i	$-0.520955\pi$
$271$	−159068.	−0.131571	−0.0657854	+	0.997834i	$0.520955\pi$
$272$	0	0
$273$	133000.i	0.108005i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.29385e6i	− 1.01318i	−0.862188	−	0.506589i	$-0.830906\pi$
$277$	− 1.29385e6i	− 1.01318i	0.862188	−	0.506589i	$-0.169094\pi$
$278$	0	0
$279$	−571330.	−0.439417
$280$	0	0
$281$	−1.78114e6	−1.34565	−0.672824	−	0.739802i	$-0.734919\pi$
$281$	−1.78114e6	−1.34565	−0.672824	+	0.739802i	$0.734919\pi$
$282$	0	0
$283$	1.98279e6i	1.47167i	0.677161	+	0.735835i	$0.263210\pi$
$283$	1.98279e6i	1.47167i	−0.677161	+	0.735835i	$0.736790\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 26000.0i	− 0.0186324i
$288$	0	0
$289$	88141.0	0.0620774
$290$	0	0
$291$	−348593.	−0.241316
$292$	0	0
$293$	578360.i	0.393577i	0.980446	+	0.196788i	$0.0630512\pi$
$293$	578360.i	0.393577i	−0.980446	+	0.196788i	$0.936949\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	370139.i	0.243486i
$298$	0	0
$299$	1.36458e6	0.882716
$300$	0	0
$301$	124300.	0.0790779
$302$	0	0
$303$	− 1.07384e6i	− 0.671945i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.07602e6i	1.86270i	0.364120	+	0.931352i	$0.381370\pi$
$307$	3.07602e6i	1.86270i	−0.364120	+	0.931352i	$0.618630\pi$
$308$	0	0
$309$	943040.	0.561868
$310$	0	0
$311$	−3.13757e6	−1.83947	−0.919735	−	0.392540i	$-0.871597\pi$
$311$	−3.13757e6	−1.83947	−0.919735	+	0.392540i	$0.871597\pi$
$312$	0	0
$313$	2.61784e6i	1.51037i	0.655514	+	0.755183i	$0.272452\pi$
$313$	2.61784e6i	1.51037i	−0.655514	+	0.755183i	$0.727548\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.49220e6i	1.39294i	0.717584	+	0.696472i	$0.245248\pi$
$317$	2.49220e6i	1.39294i	−0.717584	+	0.696472i	$0.754752\pi$
$318$	0	0
$319$	−971872.	−0.534727
$320$	0	0
$321$	1.18453e6	0.641626
$322$	0	0
$323$	2.10490e6i	1.12260i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.63244e6i	0.844245i
$328$	0	0
$329$	−346000.	−0.176233
$330$	0	0
$331$	−2.70125e6	−1.35517	−0.677586	−	0.735443i	$-0.736974\pi$
$331$	−2.70125e6	−1.35517	−0.677586	+	0.735443i	$0.736974\pi$
$332$	0	0
$333$	− 1.35393e6i	− 0.669090i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.42610e6i	0.684031i	0.939694	+	0.342016i	$0.111110\pi$
$337$	1.42610e6i	0.684031i	−0.939694	+	0.342016i	$0.888890\pi$
$338$	0	0
$339$	−660303.	−0.312064
$340$	0	0
$341$	−356345.	−0.165953
$342$	0	0
$343$	1.55570e6i	0.713987i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.86374e6i	− 1.27676i	−0.769721	−	0.638381i	$-0.779604\pi$
$347$	− 2.86374e6i	− 1.27676i	0.769721	−	0.638381i	$-0.220396\pi$
$348$	0	0
$349$	−296350.	−0.130239	−0.0651195	−	0.997877i	$-0.520743\pi$
$349$	−296350.	−0.130239	−0.0651195	+	0.997877i	$0.520743\pi$
$350$	0	0
$351$	1.16242e6	0.503611
$352$	0	0
$353$	− 2.12114e6i	− 0.906010i	−0.891508	−	0.453005i	$-0.850352\pi$
$353$	− 2.12114e6i	− 0.906010i	0.891508	−	0.453005i	$-0.149648\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 403900.i	− 0.167727i
$358$	0	0
$359$	3.47512e6	1.42310	0.711548	−	0.702638i	$-0.247994\pi$
$359$	3.47512e6	1.42310	0.711548	+	0.702638i	$0.247994\pi$
$360$	0	0
$361$	850877.	0.343636
$362$	0	0
$363$	102487.i	0.0408227i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.56190e6i	− 0.605322i	−0.953098	−	0.302661i	$-0.902125\pi$
$367$	− 1.56190e6i	− 0.605322i	0.953098	−	0.302661i	$-0.0978750\pi$
$368$	0	0
$369$	−100880.	−0.0385691
$370$	0	0
$371$	685900.	0.258718
$372$	0	0
$373$	1.93773e6i	0.721144i	0.932731	+	0.360572i	$0.117419\pi$
$373$	1.93773e6i	0.721144i	−0.932731	+	0.360572i	$0.882581\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.05216e6i	1.10600i
$378$	0	0
$379$	−3.07495e6	−1.09961	−0.549806	−	0.835292i	$-0.685298\pi$
$379$	−3.07495e6	−1.09961	−0.549806	+	0.835292i	$0.685298\pi$
$380$	0	0
$381$	−1.81636e6	−0.641046
$382$	0	0
$383$	4.31553e6i	1.50327i	0.659579	+	0.751635i	$0.270734\pi$
$383$	4.31553e6i	1.50327i	−0.659579	+	0.751635i	$0.729266\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 482284.i	− 0.163691i
$388$	0	0
$389$	−2.36251e6	−0.791590	−0.395795	−	0.918339i	$-0.629531\pi$
$389$	−2.36251e6	−0.791590	−0.395795	+	0.918339i	$0.629531\pi$
$390$	0	0
$391$	−4.14401e6	−1.37082
$392$	0	0
$393$	− 597870.i	− 0.195265i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.77598e6i	− 0.565539i	−0.959188	−	0.282769i	$-0.908747\pi$
$397$	− 1.77598e6i	− 0.565539i	0.959188	−	0.282769i	$-0.0912531\pi$
$398$	0	0
$399$	−638400.	−0.200752
$400$	0	0
$401$	1.56967e6	0.487468	0.243734	−	0.969842i	$-0.421628\pi$
$401$	1.56967e6	0.487468	0.243734	+	0.969842i	$0.421628\pi$
$402$	0	0
$403$	1.11910e6i	0.343247i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 844459.i	− 0.252693i
$408$	0	0
$409$	−1.29485e6	−0.382746	−0.191373	−	0.981517i	$-0.561294\pi$
$409$	−1.29485e6	−0.382746	−0.191373	+	0.981517i	$0.561294\pi$
$410$	0	0
$411$	−2.99392e6	−0.874250
$412$	0	0
$413$	1.58895e6i	0.458390i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.16783e6i	− 0.610499i
$418$	0	0
$419$	−272916.	−0.0759441	−0.0379720	−	0.999279i	$-0.512090\pi$
$419$	−272916.	−0.0759441	−0.0379720	+	0.999279i	$0.512090\pi$
$420$	0	0
$421$	−2.61801e6	−0.719890	−0.359945	−	0.932974i	$-0.617205\pi$
$421$	−2.61801e6	−0.719890	−0.359945	+	0.932974i	$0.617205\pi$
$422$	0	0
$423$	1.34248e6i	0.364802i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.70780e6i	0.453281i
$428$	0	0
$429$	321860.	0.0844352
$430$	0	0
$431$	2.81037e6	0.728735	0.364368	−	0.931255i	$-0.381285\pi$
$431$	2.81037e6	0.728735	0.364368	+	0.931255i	$0.381285\pi$
$432$	0	0
$433$	5.98509e6i	1.53409i	0.641593	+	0.767046i	$0.278274\pi$
$433$	5.98509e6i	1.53409i	−0.641593	+	0.767046i	$0.721726\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.54998e6i	1.64073i
$438$	0	0
$439$	7.50486e6	1.85858	0.929290	−	0.369352i	$-0.120420\pi$
$439$	7.50486e6	1.85858	0.929290	+	0.369352i	$0.120420\pi$
$440$	0	0
$441$	2.77556e6	0.679601
$442$	0	0
$443$	1.56806e6i	0.379624i	0.981820	+	0.189812i	$0.0607878\pi$
$443$	1.56806e6i	0.379624i	−0.981820	+	0.189812i	$0.939212\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.14892e6i	− 0.745406i
$448$	0	0
$449$	4.04044e6	0.945831	0.472915	−	0.881108i	$-0.343202\pi$
$449$	4.04044e6	0.945831	0.472915	+	0.881108i	$0.343202\pi$
$450$	0	0
$451$	−62920.0	−0.0145662
$452$	0	0
$453$	2.83552e6i	0.649213i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.21132e6i	− 0.495291i	−0.968851	−	0.247645i	$-0.920343\pi$
$457$	− 2.21132e6i	− 0.495291i	0.968851	−	0.247645i	$-0.0796568\pi$
$458$	0	0
$459$	−3.53009e6	−0.782084
$460$	0	0
$461$	−3.56735e6	−0.781795	−0.390898	−	0.920434i	$-0.627835\pi$
$461$	−3.56735e6	−0.781795	−0.390898	+	0.920434i	$0.627835\pi$
$462$	0	0
$463$	747757.i	0.162109i	0.996710	+	0.0810547i	$0.0258288\pi$
$463$	747757.i	0.162109i	−0.996710	+	0.0810547i	$0.974171\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.44511e6i	1.15535i	0.816266	+	0.577676i	$0.196040\pi$
$467$	5.44511e6i	1.15535i	−0.816266	+	0.577676i	$0.803960\pi$
$468$	0	0
$469$	−3.07515e6	−0.645556
$470$	0	0
$471$	2.37525e6	0.493352
$472$	0	0
$473$	− 300806.i	− 0.0618207i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.66129e6i	− 0.535546i
$478$	0	0
$479$	6.22046e6	1.23875	0.619375	−	0.785095i	$-0.287386\pi$
$479$	6.22046e6	1.23875	0.619375	+	0.785095i	$0.287386\pi$
$480$	0	0
$481$	−2.65202e6	−0.522654
$482$	0	0
$483$	− 1.25685e6i	− 0.245141i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.34398e6i	0.638913i	0.947601	+	0.319457i	$0.103500\pi$
$487$	3.34398e6i	0.638913i	−0.947601	+	0.319457i	$0.896500\pi$
$488$	0	0
$489$	−1.89977e6	−0.359277
$490$	0	0
$491$	5.58646e6	1.04576	0.522881	−	0.852406i	$-0.324857\pi$
$491$	5.58646e6	1.04576	0.522881	+	0.852406i	$0.324857\pi$
$492$	0	0
$493$	− 9.26893e6i	− 1.71756i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 748550.i	− 0.135935i
$498$	0	0
$499$	8.29348e6	1.49103	0.745514	−	0.666490i	$-0.232204\pi$
$499$	8.29348e6	1.49103	0.745514	+	0.666490i	$0.232204\pi$
$500$	0	0
$501$	507276.	0.0902922
$502$	0	0
$503$	5.29951e6i	0.933933i	0.884275	+	0.466967i	$0.154653\pi$
$503$	5.29951e6i	0.933933i	−0.884275	+	0.466967i	$0.845347\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.58825e6i	0.274410i
$508$	0	0
$509$	−24415.0	−0.00417698	−0.00208849	−	0.999998i	$-0.500665\pi$
$509$	−24415.0	−0.00417698	−0.00208849	+	0.999998i	$0.500665\pi$
$510$	0	0
$511$	1.82220e6	0.308705
$512$	0	0
$513$	5.57962e6i	0.936076i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	837320.i	0.137773i
$518$	0	0
$519$	−3.35458e6	−0.546663
$520$	0	0
$521$	4.76275e6	0.768712	0.384356	−	0.923185i	$-0.374424\pi$
$521$	4.76275e6	0.768712	0.384356	+	0.923185i	$0.374424\pi$
$522$	0	0
$523$	735248.i	0.117538i	0.998272	+	0.0587692i	$0.0187176\pi$
$523$	735248.i	0.117538i	−0.998272	+	0.0587692i	$0.981282\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3.39853e6i	− 0.533046i
$528$	0	0
$529$	−6.45894e6	−1.00351
$530$	0	0
$531$	6.16513e6	0.948868
$532$	0	0
$533$	197600.i	0.0301279i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	286545.i	0.0428802i
$538$	0	0
$539$	1.73115e6	0.256662
$540$	0	0
$541$	−3.19649e6	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$541$	−3.19649e6	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$542$	0	0
$543$	− 631183.i	− 0.0918662i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 8.85902e6i	− 1.26595i	−0.774171	−	0.632976i	$-0.781833\pi$
$547$	− 8.85902e6i	− 1.26595i	0.774171	−	0.632976i	$-0.218167\pi$
$548$	0	0
$549$	6.62626e6	0.938292
$550$	0	0
$551$	−1.46504e7	−2.05575
$552$	0	0
$553$	1.42690e6i	0.198418i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.74512e6i	0.238335i	0.992874	+	0.119167i	$0.0380225\pi$
$557$	1.74512e6i	0.238335i	−0.992874	+	0.119167i	$0.961977\pi$
$558$	0	0
$559$	−944680.	−0.127866
$560$	0	0
$561$	−977438.	−0.131124
$562$	0	0
$563$	− 1.32333e7i	− 1.75953i	−0.475410	−	0.879764i	$-0.657700\pi$
$563$	− 1.32333e7i	− 1.75953i	0.475410	−	0.879764i	$-0.342300\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.28645e6i	0.168049i
$568$	0	0
$569$	−1.04156e7	−1.34867	−0.674335	−	0.738426i	$-0.735570\pi$
$569$	−1.04156e7	−1.34867	−0.674335	+	0.738426i	$0.735570\pi$
$570$	0	0
$571$	−2.48163e6	−0.318527	−0.159264	−	0.987236i	$-0.550912\pi$
$571$	−2.48163e6	−0.318527	−0.159264	+	0.987236i	$0.550912\pi$
$572$	0	0
$573$	− 1.82262e6i	− 0.231905i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.31244e7i	1.64112i	0.571557	+	0.820562i	$0.306339\pi$
$577$	1.31244e7i	1.64112i	−0.571557	+	0.820562i	$0.693661\pi$
$578$	0	0
$579$	−3.67027e6	−0.454989
$580$	0	0
$581$	−3.87410e6	−0.476135
$582$	0	0
$583$	− 1.65988e6i	− 0.202258i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.86010e6i	0.582170i	0.956697	+	0.291085i	$0.0940163\pi$
$587$	4.86010e6i	0.582170i	−0.956697	+	0.291085i	$0.905984\pi$
$588$	0	0
$589$	−5.37168e6	−0.638002
$590$	0	0
$591$	5.31707e6	0.626187
$592$	0	0
$593$	− 1.58559e6i	− 0.185163i	−0.995705	−	0.0925814i	$-0.970488\pi$
$593$	− 1.58559e6i	− 0.185163i	0.995705	−	0.0925814i	$-0.0295118\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.17915e6i	0.709566i
$598$	0	0
$599$	−9.04294e6	−1.02978	−0.514888	−	0.857258i	$-0.672166\pi$
$599$	−9.04294e6	−1.02978	−0.514888	+	0.857258i	$0.672166\pi$
$600$	0	0
$601$	729186.	0.0823478	0.0411739	−	0.999152i	$-0.486890\pi$
$601$	729186.	0.0823478	0.0411739	+	0.999152i	$0.486890\pi$
$602$	0	0
$603$	1.19316e7i	1.33630i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.91130e6i	0.430873i	0.976518	+	0.215437i	$0.0691175\pi$
$607$	3.91130e6i	0.430873i	−0.976518	+	0.215437i	$0.930883\pi$
$608$	0	0
$609$	2.81120e6	0.307149
$610$	0	0
$611$	2.62960e6	0.284962
$612$	0	0
$613$	− 5.52184e6i	− 0.593516i	−0.954953	−	0.296758i	$-0.904094\pi$
$613$	− 5.52184e6i	− 0.593516i	0.954953	−	0.296758i	$-0.0959055\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.88539e6i	0.516638i	0.966060	+	0.258319i	$0.0831686\pi$
$617$	4.88539e6i	0.516638i	−0.966060	+	0.258319i	$0.916831\pi$
$618$	0	0
$619$	4.11150e6	0.431295	0.215647	−	0.976471i	$-0.430814\pi$
$619$	4.11150e6	0.431295	0.215647	+	0.976471i	$0.430814\pi$
$620$	0	0
$621$	−1.09849e7	−1.14305
$622$	0	0
$623$	− 1.81355e6i	− 0.187202i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.54493e6i	0.156942i
$628$	0	0
$629$	8.05377e6	0.811657
$630$	0	0
$631$	8.24910e6	0.824771	0.412385	−	0.911009i	$-0.364696\pi$
$631$	8.24910e6	0.824771	0.412385	+	0.911009i	$0.364696\pi$
$632$	0	0
$633$	− 8.06996e6i	− 0.800502i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.43666e6i	− 0.530864i
$638$	0	0
$639$	−2.90437e6	−0.281385
$640$	0	0
$641$	−4.29330e6	−0.412711	−0.206355	−	0.978477i	$-0.566160\pi$
$641$	−4.29330e6	−0.412711	−0.206355	+	0.978477i	$0.566160\pi$
$642$	0	0
$643$	− 1.63045e7i	− 1.55518i	−0.628774	−	0.777588i	$-0.716443\pi$
$643$	− 1.63045e7i	− 1.55518i	0.628774	−	0.777588i	$-0.283557\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 4.42624e6i	− 0.415695i	−0.978161	−	0.207847i	$-0.933354\pi$
$647$	− 4.42624e6i	− 0.415695i	0.978161	−	0.207847i	$-0.0666457\pi$
$648$	0	0
$649$	3.84526e6	0.358355
$650$	0	0
$651$	1.03075e6	0.0953237
$652$	0	0
$653$	− 6.27529e6i	− 0.575905i	−0.957645	−	0.287952i	$-0.907025\pi$
$653$	− 6.27529e6i	− 0.575905i	0.957645	−	0.287952i	$-0.0929745\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 7.07014e6i	− 0.639020i
$658$	0	0
$659$	1.09748e7	0.984422	0.492211	−	0.870476i	$-0.336189\pi$
$659$	1.09748e7	0.984422	0.492211	+	0.870476i	$0.336189\pi$
$660$	0	0
$661$	2.02025e7	1.79846	0.899229	−	0.437478i	$-0.144128\pi$
$661$	2.02025e7	1.79846	0.899229	+	0.437478i	$0.144128\pi$
$662$	0	0
$663$	3.06964e6i	0.271209i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 2.88429e7i	− 2.51029i
$668$	0	0
$669$	461251.	0.0398448
$670$	0	0
$671$	4.13288e6	0.354361
$672$	0	0
$673$	1.14233e7i	0.972200i	0.873903	+	0.486100i	$0.161581\pi$
$673$	1.14233e7i	0.972200i	−0.873903	+	0.486100i	$0.838419\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.43918e6i	0.204537i	0.994757	+	0.102268i	$0.0326101\pi$
$677$	2.43918e6i	0.204537i	−0.994757	+	0.102268i	$0.967390\pi$
$678$	0	0
$679$	−2.48995e6	−0.207260
$680$	0	0
$681$	−2.20168e6	−0.181923
$682$	0	0
$683$	1.01384e6i	0.0831606i	0.999135	+	0.0415803i	$0.0132392\pi$
$683$	1.01384e6i	0.0831606i	−0.999135	+	0.0415803i	$0.986761\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 7.26924e6i	− 0.587621i
$688$	0	0
$689$	−5.21284e6	−0.418337
$690$	0	0
$691$	−8.03186e6	−0.639913	−0.319957	−	0.947432i	$-0.603668\pi$
$691$	−8.03186e6	−0.639913	−0.319957	+	0.947432i	$0.603668\pi$
$692$	0	0
$693$	1.17370e6i	0.0928376i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 600080.i	− 0.0467872i
$698$	0	0
$699$	−3.56983e6	−0.276347
$700$	0	0
$701$	−259806.	−0.0199689	−0.00998445	−	0.999950i	$-0.503178\pi$
$701$	−259806.	−0.0199689	−0.00998445	+	0.999950i	$0.503178\pi$
$702$	0	0
$703$	− 1.27297e7i	− 0.971471i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.67030e6i	− 0.577117i
$708$	0	0
$709$	1.92848e7	1.44079	0.720393	−	0.693566i	$-0.243961\pi$
$709$	1.92848e7	1.44079	0.720393	+	0.693566i	$0.243961\pi$
$710$	0	0
$711$	5.53637e6	0.410725
$712$	0	0
$713$	− 1.05755e7i	− 0.779071i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.11146e6i	0.226030i
$718$	0	0
$719$	−926119.	−0.0668105	−0.0334052	−	0.999442i	$-0.510635\pi$
$719$	−926119.	−0.0668105	−0.0334052	+	0.999442i	$0.510635\pi$
$720$	0	0
$721$	6.73600e6	0.482574
$722$	0	0
$723$	− 1.98425e6i	− 0.141173i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 2.02599e7i	− 1.42168i	−0.703354	−	0.710840i	$-0.748315\pi$
$727$	− 2.02599e7i	− 1.42168i	0.703354	−	0.710840i	$-0.251685\pi$
$728$	0	0
$729$	−211933.	−0.0147700
$730$	0	0
$731$	2.86884e6	0.198570
$732$	0	0
$733$	1.10982e7i	0.762944i	0.924380	+	0.381472i	$0.124583\pi$
$733$	1.10982e7i	0.762944i	−0.924380	+	0.381472i	$0.875417\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.44186e6i	0.504676i
$738$	0	0
$739$	−624962.	−0.0420962	−0.0210481	−	0.999778i	$-0.506700\pi$
$739$	−624962.	−0.0420962	−0.0210481	+	0.999778i	$0.506700\pi$
$740$	0	0
$741$	4.85184e6	0.324609
$742$	0	0
$743$	46436.0i	0.00308591i	0.999999	+	0.00154295i	$0.000491137\pi$
$743$	46436.0i	0.00308591i	−0.999999	+	0.00154295i	$0.999509\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.50315e7i	0.985601i
$748$	0	0
$749$	8.46090e6	0.551077
$750$	0	0
$751$	−6.12144e6	−0.396053	−0.198027	−	0.980197i	$-0.563453\pi$
$751$	−6.12144e6	−0.396053	−0.198027	+	0.980197i	$0.563453\pi$
$752$	0	0
$753$	− 5.41665e6i	− 0.348131i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.26458e6i	0.207056i	0.994627	+	0.103528i	$0.0330131\pi$
$757$	3.26458e6i	0.207056i	−0.994627	+	0.103528i	$0.966987\pi$
$758$	0	0
$759$	−3.04158e6	−0.191644
$760$	0	0
$761$	1.60311e7	1.00346	0.501732	−	0.865023i	$-0.332696\pi$
$761$	1.60311e7	1.00346	0.501732	+	0.865023i	$0.332696\pi$
$762$	0	0
$763$	1.16603e7i	0.725101i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.20760e7i	− 0.741200i
$768$	0	0
$769$	−2.64617e7	−1.61362	−0.806811	−	0.590810i	$-0.798808\pi$
$769$	−2.64617e7	−1.61362	−0.806811	+	0.590810i	$0.798808\pi$
$770$	0	0
$771$	−2.71400e6	−0.164427
$772$	0	0
$773$	− 2.63836e7i	− 1.58813i	−0.607836	−	0.794063i	$-0.707962\pi$
$773$	− 2.63836e7i	− 1.58813i	0.607836	−	0.794063i	$-0.292038\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.44265e6i	0.145147i
$778$	0	0
$779$	−948480.	−0.0559995
$780$	0	0
$781$	−1.81149e6	−0.106269
$782$	0	0
$783$	− 2.45699e7i	− 1.43218i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.68115e6i	0.326964i	0.986546	+	0.163482i	$0.0522725\pi$
$787$	5.68115e6i	0.326964i	−0.986546	+	0.163482i	$0.947727\pi$
$788$	0	0
$789$	1.38321e6	0.0791037
$790$	0	0
$791$	−4.71645e6	−0.268024
$792$	0	0
$793$	− 1.29793e7i	− 0.732939i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 9.99383e6i	− 0.557296i	−0.960393	−	0.278648i	$-0.910114\pi$
$797$	− 9.99383e6i	− 0.557296i	0.960393	−	0.278648i	$-0.0898863\pi$
$798$	0	0
$799$	−7.98568e6	−0.442532
$800$	0	0
$801$	−7.03657e6	−0.387507
$802$	0	0
$803$	− 4.40972e6i	− 0.241336i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.83886e6i	0.0993950i
$808$	0	0
$809$	−2.32455e7	−1.24873	−0.624364	−	0.781134i	$-0.714642\pi$
$809$	−2.32455e7	−1.24873	−0.624364	+	0.781134i	$0.714642\pi$
$810$	0	0
$811$	1.27367e7	0.679991	0.339995	−	0.940427i	$-0.389574\pi$
$811$	1.27367e7	0.679991	0.339995	+	0.940427i	$0.389574\pi$
$812$	0	0
$813$	− 1.11348e6i	− 0.0590819i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.53446e6i	− 0.237668i
$818$	0	0
$819$	3.68600e6	0.192020
$820$	0	0
$821$	−7.85748e6	−0.406842	−0.203421	−	0.979091i	$-0.565206\pi$
$821$	−7.85748e6	−0.406842	−0.203421	+	0.979091i	$0.565206\pi$
$822$	0	0
$823$	− 1.09499e7i	− 0.563524i	−0.959484	−	0.281762i	$-0.909081\pi$
$823$	− 1.09499e7i	− 0.563524i	0.959484	−	0.281762i	$-0.0909188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.20638e7i	− 1.12180i	−0.827883	−	0.560901i	$-0.810455\pi$
$827$	− 2.20638e7i	− 1.12180i	0.827883	−	0.560901i	$-0.189545\pi$
$828$	0	0
$829$	−7.05255e6	−0.356418	−0.178209	−	0.983993i	$-0.557030\pi$
$829$	−7.05255e6	−0.356418	−0.178209	+	0.983993i	$0.557030\pi$
$830$	0	0
$831$	9.05698e6	0.454968
$832$	0	0
$833$	1.65103e7i	0.824407i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 9.00876e6i	− 0.444479i
$838$	0	0
$839$	2.26195e7	1.10937	0.554686	−	0.832060i	$-0.312838\pi$
$839$	2.26195e7	1.10937	0.554686	+	0.832060i	$0.312838\pi$
$840$	0	0
$841$	4.40019e7	2.14527
$842$	0	0
$843$	− 1.24680e7i	− 0.604264i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	732050.i	0.0350616i
$848$	0	0
$849$	−1.38795e7	−0.660853
$850$	0	0
$851$	2.50616e7	1.18627
$852$	0	0
$853$	9.46645e6i	0.445466i	0.974880	+	0.222733i	$0.0714978\pi$
$853$	9.46645e6i	0.445466i	−0.974880	+	0.222733i	$0.928502\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 941480.i	− 0.0437884i	−0.999760	−	0.0218942i	$-0.993030\pi$
$857$	− 941480.i	− 0.0437884i	0.999760	−	0.0218942i	$-0.00696970\pi$
$858$	0	0
$859$	806423.	0.0372889	0.0186445	−	0.999826i	$-0.494065\pi$
$859$	806423.	0.0372889	0.0186445	+	0.999826i	$0.494065\pi$
$860$	0	0
$861$	182000.	0.00836688
$862$	0	0
$863$	1.19485e7i	0.546119i	0.961997	+	0.273059i	$0.0880356\pi$
$863$	1.19485e7i	0.546119i	−0.961997	+	0.273059i	$0.911964\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	616987.i	0.0278759i
$868$	0	0
$869$	3.45310e6	0.155117
$870$	0	0
$871$	2.33711e7	1.04384
$872$	0	0
$873$	9.66101e6i	0.429029i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 7.84853e6i	− 0.344580i	−0.985046	−	0.172290i	$-0.944883\pi$
$877$	− 7.84853e6i	− 0.344580i	0.985046	−	0.172290i	$-0.0551165\pi$
$878$	0	0
$879$	−4.04852e6	−0.176736
$880$	0	0
$881$	−1.73933e7	−0.754991	−0.377496	−	0.926011i	$-0.623215\pi$
$881$	−1.73933e7	−0.754991	−0.377496	+	0.926011i	$0.623215\pi$
$882$	0	0
$883$	− 4.31619e7i	− 1.86294i	−0.363818	−	0.931470i	$-0.618527\pi$
$883$	− 4.31619e7i	− 1.86294i	0.363818	−	0.931470i	$-0.381473\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.25652e6i	0.395038i	0.980299	+	0.197519i	$0.0632884\pi$
$887$	9.25652e6i	0.395038i	−0.980299	+	0.197519i	$0.936712\pi$
$888$	0	0
$889$	−1.29740e7	−0.550579
$890$	0	0
$891$	3.11321e6	0.131375
$892$	0	0
$893$	1.26221e7i	0.529666i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	9.55206e6i	0.396384i
$898$	0	0
$899$	2.36542e7	0.976135
$900$	0	0
$901$	1.58306e7	0.649658
$902$	0	0
$903$	870100.i	0.0355099i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 4.47293e7i	− 1.80540i	−0.430270	−	0.902700i	$-0.641582\pi$
$907$	− 4.47293e7i	− 1.80540i	0.430270	−	0.902700i	$-0.358418\pi$
$908$	0	0
$909$	−2.97608e7	−1.19463
$910$	0	0
$911$	2.57577e7	1.02828	0.514139	−	0.857707i	$-0.328112\pi$
$911$	2.57577e7	1.02828	0.514139	+	0.857707i	$0.328112\pi$
$912$	0	0
$913$	9.37532e6i	0.372228i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.27050e6i	− 0.167709i
$918$	0	0
$919$	3.63488e7	1.41972	0.709858	−	0.704344i	$-0.248759\pi$
$919$	3.63488e7	1.41972	0.709858	+	0.704344i	$0.248759\pi$
$920$	0	0
$921$	−2.15322e7	−0.836447
$922$	0	0
$923$	5.68898e6i	0.219801i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 2.61357e7i	− 0.998929i
$928$	0	0
$929$	3.96617e6	0.150776	0.0753880	−	0.997154i	$-0.475980\pi$
$929$	3.96617e6	0.150776	0.0753880	+	0.997154i	$0.475980\pi$
$930$	0	0
$931$	2.60960e7	0.986732
$932$	0	0
$933$	− 2.19630e7i	− 0.826014i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.50528e7i	1.30429i	0.758094	+	0.652145i	$0.226131\pi$
$937$	3.50528e7i	1.30429i	−0.758094	+	0.652145i	$0.773869\pi$
$938$	0	0
$939$	−1.83249e7	−0.678230
$940$	0	0
$941$	−1.40738e7	−0.518130	−0.259065	−	0.965860i	$-0.583414\pi$
$941$	−1.40738e7	−0.518130	−0.259065	+	0.965860i	$0.583414\pi$
$942$	0	0
$943$	− 1.86732e6i	− 0.0683816i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.73759e7i	1.35431i	0.735842	+	0.677153i	$0.236786\pi$
$947$	3.73759e7i	1.35431i	−0.735842	+	0.677153i	$0.763214\pi$
$948$	0	0
$949$	−1.38487e7	−0.499165
$950$	0	0
$951$	−1.74454e7	−0.625502
$952$	0	0
$953$	− 3.18424e7i	− 1.13572i	−0.823124	−	0.567862i	$-0.807771\pi$
$953$	− 3.18424e7i	− 1.13572i	0.823124	−	0.567862i	$-0.192229\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 6.80310e6i	− 0.240119i
$958$	0	0
$959$	−2.13852e7	−0.750872
$960$	0	0
$961$	−1.99561e7	−0.697056
$962$	0	0
$963$	− 3.28283e7i	− 1.14073i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.16276e7i	1.08768i	0.839190	+	0.543838i	$0.183029\pi$
$967$	3.16276e7i	1.08768i	−0.839190	+	0.543838i	$0.816971\pi$
$968$	0	0
$969$	−1.47343e7	−0.504103
$970$	0	0
$971$	−2.73412e7	−0.930614	−0.465307	−	0.885149i	$-0.654056\pi$
$971$	−2.73412e7	−0.930614	−0.465307	+	0.885149i	$0.654056\pi$
$972$	0	0
$973$	− 1.54845e7i	− 0.524343i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.81630e6i	0.194944i	0.995238	+	0.0974721i	$0.0310757\pi$
$977$	5.81630e6i	0.194944i	−0.995238	+	0.0974721i	$0.968924\pi$
$978$	0	0
$979$	−4.38879e6	−0.146348
$980$	0	0
$981$	4.52420e7	1.50096
$982$	0	0
$983$	− 3.81817e7i	− 1.26029i	−0.776476	−	0.630146i	$-0.782995\pi$
$983$	− 3.81817e7i	− 1.26029i	0.776476	−	0.630146i	$-0.217005\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.42200e6i	− 0.0791373i
$988$	0	0
$989$	8.92723e6	0.290219
$990$	0	0
$991$	5.44564e6	0.176143	0.0880714	−	0.996114i	$-0.471930\pi$
$991$	5.44564e6	0.176143	0.0880714	+	0.996114i	$0.471930\pi$
$992$	0	0
$993$	− 1.89087e7i	− 0.608541i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.77967e6i	0.120425i	0.998186	+	0.0602125i	$0.0191778\pi$
$997$	3.77967e6i	0.120425i	−0.998186	+	0.0602125i	$0.980822\pi$
$998$	0	0
$999$	2.13488e7	0.676798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1100.6.b.a.749.2		2
5.2	odd	4		44.6.a.a.1.1	✓	1
5.3	odd	4		1100.6.a.a.1.1		1
5.4	even	2	inner	1100.6.b.a.749.1		2
15.2	even	4		396.6.a.e.1.1		1
20.7	even	4		176.6.a.a.1.1		1
40.27	even	4		704.6.a.g.1.1		1
40.37	odd	4		704.6.a.d.1.1		1
55.32	even	4		484.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.6.a.a.1.1	✓	1	5.2	odd	4
176.6.a.a.1.1		1	20.7	even	4
396.6.a.e.1.1		1	15.2	even	4
484.6.a.b.1.1		1	55.32	even	4
704.6.a.d.1.1		1	40.37	odd	4
704.6.a.g.1.1		1	40.27	even	4
1100.6.a.a.1.1		1	5.3	odd	4
1100.6.b.a.749.1		2	5.4	even	2	inner
1100.6.b.a.749.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1100.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +O(q^{10})\)	\(q+7.00000i q^{3} +50.0000i q^{7} +194.000 q^{9} +121.000 q^{11} -380.000i q^{13} +1154.00i q^{17} +1824.00 q^{19} -350.000 q^{21} +3591.00i q^{23} +3059.00i q^{27} -8032.00 q^{29} -2945.00 q^{31} +847.000i q^{33} -6979.00i q^{37} +2660.00 q^{39} -520.000 q^{41} -2486.00i q^{43} +6920.00i q^{47} +14307.0 q^{49} -8078.00 q^{51} -13718.0i q^{53} +12768.0i q^{57} +31779.0 q^{59} +34156.0 q^{61} +9700.00i q^{63} +61503.0i q^{67} -25137.0 q^{69} -14971.0 q^{71} -36444.0i q^{73} +6050.00i q^{77} +28538.0 q^{79} +25729.0 q^{81} +77482.0i q^{83} -56224.0i q^{87} -36271.0 q^{89} +19000.0 q^{91} -20615.0i q^{93} +49799.0i q^{97} +23474.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 388 q^{9}+O(q^{10}) \) 2 * q + 388 * q^9	\( 2 q + 388 q^{9} + 242 q^{11} + 3648 q^{19} - 700 q^{21} - 16064 q^{29} - 5890 q^{31} + 5320 q^{39} - 1040 q^{41} + 28614 q^{49} - 16156 q^{51} + 63558 q^{59} + 68312 q^{61} - 50274 q^{69} - 29942 q^{71} + 57076 q^{79} + 51458 q^{81} - 72542 q^{89} + 38000 q^{91} + 46948 q^{99}+O(q^{100}) \) 2 * q + 388 * q^9 + 242 * q^11 + 3648 * q^19 - 700 * q^21 - 16064 * q^29 - 5890 * q^31 + 5320 * q^39 - 1040 * q^41 + 28614 * q^49 - 16156 * q^51 + 63558 * q^59 + 68312 * q^61 - 50274 * q^69 - 29942 * q^71 + 57076 * q^79 + 51458 * q^81 - 72542 * q^89 + 38000 * q^91 + 46948 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more