LMFDB - Embedded newform 288.7.b.b.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [288,7,Mod(271,288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("288.271");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 288 = 2^{5} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	288.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:	$66.2555760825$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.3803625.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 6x^{2} - 16x + 256 $ x^4 - x^3 + 6x^2 - 16x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.1
Root			$-2.31174 - 3.26433i$ of defining polynomial
Character	$\chi$	$=$	288.271
Dual form			288.7.b.b.271.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-199.084i q^{5} +19.6656i q^{7} +O(q^{10})$	$q-199.084i q^{5} +19.6656i q^{7} -924.152 q^{11} +1550.92i q^{13} -5140.78 q^{17} +1696.10 q^{19} +19210.4i q^{23} -24009.6 q^{25} +16588.1i q^{29} -7550.64i q^{31} +3915.12 q^{35} -28960.5i q^{37} +52111.3 q^{41} -5896.43 q^{43} +64453.4i q^{47} +117262. q^{49} -197386. i q^{53} +183984. i q^{55} +142210. q^{59} -96476.4i q^{61} +308763. q^{65} +75260.9 q^{67} +556121. i q^{71} +285914. q^{73} -18174.0i q^{77} +342014. i q^{79} -929558. q^{83} +1.02345e6i q^{85} -434757. q^{89} -30499.7 q^{91} -337668. i q^{95} +643314. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 976 q^{11} - 4168 q^{17} + 1456 q^{19} - 23900 q^{25} - 49920 q^{35} + 117944 q^{41} - 197456 q^{43} + 2116 q^{49} + 542032 q^{59} + 205440 q^{65} + 790192 q^{67} + 443912 q^{73} - 3465008 q^{83} - 761224 q^{89} - 3398400 q^{91} - 926776 q^{97}+O(q^{100}) $ 4 * q + 976 * q^11 - 4168 * q^17 + 1456 * q^19 - 23900 * q^25 - 49920 * q^35 + 117944 * q^41 - 197456 * q^43 + 2116 * q^49 + 542032 * q^59 + 205440 * q^65 + 790192 * q^67 + 443912 * q^73 - 3465008 * q^83 - 761224 * q^89 - 3398400 * q^91 - 926776 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/288\mathbb{Z}\right)^\times$.

$n$	$37$	$65$	$127$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 199.084i	− 1.59268i	−0.604852	−	0.796338i	$-0.706768\pi$
$5$	− 199.084i	− 1.59268i	0.604852	−	0.796338i	$-0.293232\pi$
$6$	0	0
$7$	19.6656i	0.0573342i	0.999589	+	0.0286671i	$0.00912627\pi$
$7$	19.6656i	0.0573342i	−0.999589	+	0.0286671i	$0.990874\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−924.152	−0.694329	−0.347165	−	0.937804i	$-0.612856\pi$
$11$	−924.152	−0.694329	−0.347165	+	0.937804i	$0.612856\pi$
$12$	0	0
$13$	1550.92i	0.705925i	0.935638	+	0.352962i	$0.114826\pi$
$13$	1550.92i	0.705925i	−0.935638	+	0.352962i	$0.885174\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5140.78	−1.04636	−0.523181	−	0.852221i	$-0.675255\pi$
$17$	−5140.78	−1.04636	−0.523181	+	0.852221i	$0.675255\pi$
$18$	0	0
$19$	1696.10	0.247281	0.123641	−	0.992327i	$-0.460543\pi$
$19$	1696.10	0.247281	0.123641	+	0.992327i	$0.460543\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	19210.4i	1.57890i	0.613817	+	0.789448i	$0.289633\pi$
$23$	19210.4i	1.57890i	−0.613817	+	0.789448i	$0.710367\pi$
$24$	0	0
$25$	−24009.6	−1.53662
$26$	0	0
$27$	0	0
$28$	0	0
$29$	16588.1i	0.680147i	0.940399	+	0.340074i	$0.110452\pi$
$29$	16588.1i	0.680147i	−0.940399	+	0.340074i	$0.889548\pi$
$30$	0	0
$31$	− 7550.64i	− 0.253454i	−0.991938	−	0.126727i	$-0.959553\pi$
$31$	− 7550.64i	− 0.253454i	0.991938	−	0.126727i	$-0.0404472\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3915.12	0.0913148
$36$	0	0
$37$	− 28960.5i	− 0.571743i	−0.958268	−	0.285871i	$-0.907717\pi$
$37$	− 28960.5i	− 0.571743i	0.958268	−	0.285871i	$-0.0922830\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	52111.3	0.756101	0.378051	−	0.925785i	$-0.376594\pi$
$41$	52111.3	0.756101	0.378051	+	0.925785i	$0.376594\pi$
$42$	0	0
$43$	−5896.43	−0.0741625	−0.0370812	−	0.999312i	$-0.511806\pi$
$43$	−5896.43	−0.0741625	−0.0370812	+	0.999312i	$0.511806\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	64453.4i	0.620801i	0.950606	+	0.310400i	$0.100463\pi$
$47$	64453.4i	0.620801i	−0.950606	+	0.310400i	$0.899537\pi$
$48$	0	0
$49$	117262.	0.996713
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 197386.i	− 1.32583i	−0.748694	−	0.662916i	$-0.769319\pi$
$53$	− 197386.i	− 1.32583i	0.748694	−	0.662916i	$-0.230681\pi$
$54$	0	0
$55$	183984.i	1.10584i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	142210.	0.692425	0.346212	−	0.938156i	$-0.387468\pi$
$59$	142210.	0.692425	0.346212	+	0.938156i	$0.387468\pi$
$60$	0	0
$61$	− 96476.4i	− 0.425042i	−0.977157	−	0.212521i	$-0.931833\pi$
$61$	− 96476.4i	− 0.425042i	0.977157	−	0.212521i	$-0.0681673\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	308763.	1.12431
$66$	0	0
$67$	75260.9	0.250233	0.125117	−	0.992142i	$-0.460070\pi$
$67$	75260.9	0.250233	0.125117	+	0.992142i	$0.460070\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	556121.i	1.55380i	0.629625	+	0.776899i	$0.283208\pi$
$71$	556121.i	1.55380i	−0.629625	+	0.776899i	$0.716792\pi$
$72$	0	0
$73$	285914.	0.734965	0.367483	−	0.930030i	$-0.380220\pi$
$73$	285914.	0.734965	0.367483	+	0.930030i	$0.380220\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 18174.0i	− 0.0398088i
$78$	0	0
$79$	342014.i	0.693686i	0.937923	+	0.346843i	$0.112746\pi$
$79$	342014.i	0.693686i	−0.937923	+	0.346843i	$0.887254\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−929558.	−1.62571	−0.812853	−	0.582469i	$-0.802087\pi$
$83$	−929558.	−1.62571	−0.812853	+	0.582469i	$0.802087\pi$
$84$	0	0
$85$	1.02345e6i	1.66652i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−434757.	−0.616704	−0.308352	−	0.951272i	$-0.599777\pi$
$89$	−434757.	−0.616704	−0.308352	+	0.951272i	$0.599777\pi$
$90$	0	0
$91$	−30499.7	−0.0404736
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 337668.i	− 0.393839i
$96$	0	0
$97$	643314.	0.704868	0.352434	−	0.935837i	$-0.385354\pi$
$97$	643314.	0.704868	0.352434	+	0.935837i	$0.385354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	858368.i	0.833123i	0.909107	+	0.416562i	$0.136765\pi$
$101$	858368.i	0.833123i	−0.909107	+	0.416562i	$0.863235\pi$
$102$	0	0
$103$	2.13382e6i	1.95275i	0.216089	+	0.976374i	$0.430670\pi$
$103$	2.13382e6i	1.95275i	−0.216089	+	0.976374i	$0.569330\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.81311e6	1.48004	0.740019	−	0.672585i	$-0.234816\pi$
$107$	1.81311e6	1.48004	0.740019	+	0.672585i	$0.234816\pi$
$108$	0	0
$109$	1.43023e6i	1.10440i	0.833711	+	0.552201i	$0.186212\pi$
$109$	1.43023e6i	1.10440i	−0.833711	+	0.552201i	$0.813788\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−628373.	−0.435494	−0.217747	−	0.976005i	$-0.569871\pi$
$113$	−628373.	−0.435494	−0.217747	+	0.976005i	$0.569871\pi$
$114$	0	0
$115$	3.82450e6	2.51467
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 101097.i	− 0.0599924i
$120$	0	0
$121$	−917503.	−0.517907
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.66925e6i	0.854656i
$126$	0	0
$127$	− 2.43195e6i	− 1.18725i	−0.804741	−	0.593626i	$-0.797696\pi$
$127$	− 2.43195e6i	− 1.18725i	0.804741	−	0.593626i	$-0.202304\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.66985e6	1.18761	0.593804	−	0.804610i	$-0.297626\pi$
$131$	2.66985e6	1.18761	0.593804	+	0.804610i	$0.297626\pi$
$132$	0	0
$133$	33354.9i	0.0141777i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.00205e6	1.55640	0.778199	−	0.628018i	$-0.216133\pi$
$137$	4.00205e6	1.55640	0.778199	+	0.628018i	$0.216133\pi$
$138$	0	0
$139$	−1.89422e6	−0.705319	−0.352660	−	0.935752i	$-0.614723\pi$
$139$	−1.89422e6	−0.705319	−0.352660	+	0.935752i	$0.614723\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.43328e6i	− 0.490144i
$144$	0	0
$145$	3.30243e6	1.08325
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.06394e6i	0.926235i	0.886297	+	0.463118i	$0.153269\pi$
$149$	3.06394e6i	0.926235i	−0.886297	+	0.463118i	$0.846731\pi$
$150$	0	0
$151$	2.24437e6i	0.651873i	0.945392	+	0.325936i	$0.105680\pi$
$151$	2.24437e6i	0.651873i	−0.945392	+	0.325936i	$0.894320\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.50322e6	−0.403670
$156$	0	0
$157$	4.89064e6i	1.26377i	0.775064	+	0.631883i	$0.217718\pi$
$157$	4.89064e6i	1.26377i	−0.775064	+	0.631883i	$0.782282\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−377785.	−0.0905248
$162$	0	0
$163$	−1.82987e6	−0.422529	−0.211265	−	0.977429i	$-0.567758\pi$
$163$	−1.82987e6	−0.422529	−0.211265	+	0.977429i	$0.567758\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.94279e6i	0.846554i	0.906000	+	0.423277i	$0.139120\pi$
$167$	3.94279e6i	0.846554i	−0.906000	+	0.423277i	$0.860880\pi$
$168$	0	0
$169$	2.42147e6	0.501670
$170$	0	0
$171$	0	0
$172$	0	0
$173$	205167.i	0.0396249i	0.999804	+	0.0198125i	$0.00630692\pi$
$173$	205167.i	0.0396249i	−0.999804	+	0.0198125i	$0.993693\pi$
$174$	0	0
$175$	− 472164.i	− 0.0881007i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.36007e6	−0.585853	−0.292927	−	0.956135i	$-0.594629\pi$
$179$	−3.36007e6	−0.585853	−0.292927	+	0.956135i	$0.594629\pi$
$180$	0	0
$181$	− 2.39145e6i	− 0.403298i	−0.979458	−	0.201649i	$-0.935370\pi$
$181$	− 2.39145e6i	− 0.403298i	0.979458	−	0.201649i	$-0.0646300\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.76558e6	−0.910601
$186$	0	0
$187$	4.75086e6	0.726520
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.02441e6i	0.290535i	0.989392	+	0.145268i	$0.0464043\pi$
$191$	2.02441e6i	0.290535i	−0.989392	+	0.145268i	$0.953596\pi$
$192$	0	0
$193$	−7.92510e6	−1.10238	−0.551192	−	0.834378i	$-0.685827\pi$
$193$	−7.92510e6	−1.10238	−0.551192	+	0.834378i	$0.685827\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.12787e6i	− 0.539917i	−0.962872	−	0.269959i	$-0.912990\pi$
$197$	− 4.12787e6i	− 0.539917i	0.962872	−	0.269959i	$-0.0870100\pi$
$198$	0	0
$199$	1.18196e7i	1.49983i	0.661532	+	0.749917i	$0.269907\pi$
$199$	1.18196e7i	1.49983i	−0.661532	+	0.749917i	$0.730093\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−326215.	−0.0389957
$204$	0	0
$205$	− 1.03745e7i	− 1.20422i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.56746e6	−0.171695
$210$	0	0
$211$	−1.19422e7	−1.27127	−0.635635	−	0.771989i	$-0.719262\pi$
$211$	−1.19422e7	−1.27127	−0.635635	+	0.771989i	$0.719262\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.17389e6i	0.118117i
$216$	0	0
$217$	148488.	0.0145316
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 7.97292e6i	− 0.738653i
$222$	0	0
$223$	− 2.10926e7i	− 1.90202i	−0.309153	−	0.951012i	$-0.600045\pi$
$223$	− 2.10926e7i	− 1.90202i	0.309153	−	0.951012i	$-0.399955\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.73658e6	0.233954	0.116977	−	0.993135i	$-0.462680\pi$
$227$	2.73658e6	0.233954	0.116977	+	0.993135i	$0.462680\pi$
$228$	0	0
$229$	7.23777e6i	0.602696i	0.953514	+	0.301348i	$0.0974366\pi$
$229$	7.23777e6i	0.602696i	−0.953514	+	0.301348i	$0.902563\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.63958e7	1.29618	0.648088	−	0.761565i	$-0.275569\pi$
$233$	1.63958e7	1.29618	0.648088	+	0.761565i	$0.275569\pi$
$234$	0	0
$235$	1.28317e7	0.988734
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 1.73985e7i	− 1.27443i	−0.770685	−	0.637217i	$-0.780086\pi$
$239$	− 1.73985e7i	− 1.27443i	0.770685	−	0.637217i	$-0.219914\pi$
$240$	0	0
$241$	1.34047e7	0.957645	0.478822	−	0.877912i	$-0.341064\pi$
$241$	1.34047e7	0.957645	0.478822	+	0.877912i	$0.341064\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.33451e7i	− 1.58744i
$246$	0	0
$247$	2.63051e6i	0.174562i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.70047e7	−1.07534	−0.537672	−	0.843154i	$-0.680696\pi$
$251$	−1.70047e7	−1.07534	−0.537672	+	0.843154i	$0.680696\pi$
$252$	0	0
$253$	− 1.77534e7i	− 1.09627i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.94325e7	−1.14480	−0.572400	−	0.819974i	$-0.693988\pi$
$257$	−1.94325e7	−1.14480	−0.572400	+	0.819974i	$0.693988\pi$
$258$	0	0
$259$	569526.	0.0327804
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.33303e7i	0.732778i	0.930462	+	0.366389i	$0.119406\pi$
$263$	1.33303e7i	0.732778i	−0.930462	+	0.366389i	$0.880594\pi$
$264$	0	0
$265$	−3.92965e7	−2.11162
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.13509e6i	− 0.366558i	−0.983061	−	0.183279i	$-0.941329\pi$
$269$	− 7.13509e6i	− 0.366558i	0.983061	−	0.183279i	$-0.0586712\pi$
$270$	0	0
$271$	− 6.93561e6i	− 0.348479i	−0.984703	−	0.174240i	$-0.944253\pi$
$271$	− 6.93561e6i	− 0.348479i	0.984703	−	0.174240i	$-0.0557467\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.21886e7	1.06692
$276$	0	0
$277$	− 1.79332e7i	− 0.843757i	−0.906652	−	0.421879i	$-0.861371\pi$
$277$	− 1.79332e7i	− 0.843757i	0.906652	−	0.421879i	$-0.138629\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.29329e6	0.193496	0.0967479	−	0.995309i	$-0.469156\pi$
$281$	4.29329e6	0.193496	0.0967479	+	0.995309i	$0.469156\pi$
$282$	0	0
$283$	−3.39377e7	−1.49735	−0.748674	−	0.662938i	$-0.769309\pi$
$283$	−3.39377e7	−1.49735	−0.748674	+	0.662938i	$0.769309\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.02480e6i	0.0433505i
$288$	0	0
$289$	2.29005e6	0.0948751
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.15178e7i	1.25301i	0.779419	+	0.626504i	$0.215515\pi$
$293$	3.15178e7i	1.25301i	−0.779419	+	0.626504i	$0.784485\pi$
$294$	0	0
$295$	− 2.83117e7i	− 1.10281i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.97938e7	−1.11458
$300$	0	0
$301$	− 115957.i	− 0.00425204i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.92069e7	−0.676953
$306$	0	0
$307$	−3.45268e7	−1.19328	−0.596638	−	0.802510i	$-0.703497\pi$
$307$	−3.45268e7	−1.19328	−0.596638	+	0.802510i	$0.703497\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.55896e7i	− 0.518268i	−0.965841	−	0.259134i	$-0.916563\pi$
$311$	− 1.55896e7i	− 0.518268i	0.965841	−	0.259134i	$-0.0834371\pi$
$312$	0	0
$313$	2.13473e6	0.0696160	0.0348080	−	0.999394i	$-0.488918\pi$
$313$	2.13473e6	0.0696160	0.0348080	+	0.999394i	$0.488918\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.15440e7i	1.30416i	0.758151	+	0.652079i	$0.226103\pi$
$317$	4.15440e7i	1.30416i	−0.758151	+	0.652079i	$0.773897\pi$
$318$	0	0
$319$	− 1.53299e7i	− 0.472246i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−8.71930e6	−0.258746
$324$	0	0
$325$	− 3.72369e7i	− 1.08474i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.26752e6	−0.0355931
$330$	0	0
$331$	−1.78997e7	−0.493585	−0.246793	−	0.969068i	$-0.579377\pi$
$331$	−1.78997e7	−0.493585	−0.246793	+	0.969068i	$0.579377\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.49833e7i	− 0.398540i
$336$	0	0
$337$	1.86244e7	0.486624	0.243312	−	0.969948i	$-0.421766\pi$
$337$	1.86244e7	0.486624	0.243312	+	0.969948i	$0.421766\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.97794e6i	0.175980i
$342$	0	0
$343$	4.61968e6i	0.114480i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.96438e7	0.948827	0.474414	−	0.880302i	$-0.342660\pi$
$347$	3.96438e7	0.948827	0.474414	+	0.880302i	$0.342660\pi$
$348$	0	0
$349$	6.84414e7i	1.61006i	0.593232	+	0.805031i	$0.297852\pi$
$349$	6.84414e7i	1.61006i	−0.593232	+	0.805031i	$0.702148\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.01604e7	0.685666	0.342833	−	0.939396i	$-0.388613\pi$
$353$	3.01604e7	0.685666	0.342833	+	0.939396i	$0.388613\pi$
$354$	0	0
$355$	1.10715e8	2.47470
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.13950e7i	− 1.32694i	−0.748205	−	0.663468i	$-0.769084\pi$
$359$	− 6.13950e7i	− 1.32694i	0.748205	−	0.663468i	$-0.230916\pi$
$360$	0	0
$361$	−4.41691e7	−0.938852
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 5.69210e7i	− 1.17056i
$366$	0	0
$367$	4.17679e7i	0.844976i	0.906368	+	0.422488i	$0.138843\pi$
$367$	4.17679e7i	0.844976i	−0.906368	+	0.422488i	$0.861157\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.88172e6	0.0760155
$372$	0	0
$373$	5.61502e7i	1.08199i	0.841025	+	0.540997i	$0.181953\pi$
$373$	5.61502e7i	1.08199i	−0.841025	+	0.540997i	$0.818047\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.57268e7	−0.480133
$378$	0	0
$379$	1.96484e7	0.360919	0.180460	−	0.983582i	$-0.442242\pi$
$379$	1.96484e7	0.360919	0.180460	+	0.983582i	$0.442242\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.86630e7i	0.688176i	0.938937	+	0.344088i	$0.111812\pi$
$383$	3.86630e7i	0.688176i	−0.938937	+	0.344088i	$0.888188\pi$
$384$	0	0
$385$	−3.61817e6	−0.0634025
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 2.94915e6i	− 0.0501012i	−0.999686	−	0.0250506i	$-0.992025\pi$
$389$	− 2.94915e6i	− 0.0501012i	0.999686	−	0.0250506i	$-0.00797470\pi$
$390$	0	0
$391$	− 9.87566e7i	− 1.65210i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.80897e7	1.10482
$396$	0	0
$397$	− 6.88167e7i	− 1.09982i	−0.835223	−	0.549911i	$-0.814662\pi$
$397$	− 6.88167e7i	− 1.09982i	0.835223	−	0.549911i	$-0.185338\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.42281e7	−0.530823	−0.265412	−	0.964135i	$-0.585508\pi$
$401$	−3.42281e7	−0.530823	−0.265412	+	0.964135i	$0.585508\pi$
$402$	0	0
$403$	1.17104e7	0.178919
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.67639e7i	0.396978i
$408$	0	0
$409$	7.43891e7	1.08728	0.543638	−	0.839320i	$-0.317047\pi$
$409$	7.43891e7	1.08728	0.543638	+	0.839320i	$0.317047\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.79664e6i	0.0396996i
$414$	0	0
$415$	1.85061e8i	2.58922i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3.78200e7	−0.514138	−0.257069	−	0.966393i	$-0.582757\pi$
$419$	−3.78200e7	−0.514138	−0.257069	+	0.966393i	$0.582757\pi$
$420$	0	0
$421$	6.21731e7i	0.833213i	0.909087	+	0.416607i	$0.136781\pi$
$421$	6.21731e7i	0.833213i	−0.909087	+	0.416607i	$0.863219\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.23428e8	1.60786
$426$	0	0
$427$	1.89727e6	0.0243694
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.00561e8i	1.25603i	0.778202	+	0.628014i	$0.216132\pi$
$431$	1.00561e8i	1.25603i	−0.778202	+	0.628014i	$0.783868\pi$
$432$	0	0
$433$	1.07677e8	1.32636	0.663179	−	0.748461i	$-0.269207\pi$
$433$	1.07677e8	1.32636	0.663179	+	0.748461i	$0.269207\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.25829e7i	0.390432i
$438$	0	0
$439$	6.85796e7i	0.810590i	0.914186	+	0.405295i	$0.132831\pi$
$439$	6.85796e7i	0.810590i	−0.914186	+	0.405295i	$0.867169\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.84819e7	−1.01776	−0.508878	−	0.860839i	$-0.669939\pi$
$443$	−8.84819e7	−1.01776	−0.508878	+	0.860839i	$0.669939\pi$
$444$	0	0
$445$	8.65534e7i	0.982210i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.28752e7	−0.473661	−0.236830	−	0.971551i	$-0.576109\pi$
$449$	−4.28752e7	−0.473661	−0.236830	+	0.971551i	$0.576109\pi$
$450$	0	0
$451$	−4.81588e7	−0.524983
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.07203e6i	0.0644614i
$456$	0	0
$457$	−7.01681e7	−0.735176	−0.367588	−	0.929989i	$-0.619816\pi$
$457$	−7.01681e7	−0.735176	−0.367588	+	0.929989i	$0.619816\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.37805e7i	0.548937i	0.961596	+	0.274468i	$0.0885018\pi$
$461$	5.37805e7i	0.548937i	−0.961596	+	0.274468i	$0.911498\pi$
$462$	0	0
$463$	4.36059e7i	0.439342i	0.975574	+	0.219671i	$0.0704984\pi$
$463$	4.36059e7i	0.439342i	−0.975574	+	0.219671i	$0.929502\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.09469e7	−0.303855	−0.151928	−	0.988392i	$-0.548548\pi$
$467$	−3.09469e7	−0.303855	−0.151928	+	0.988392i	$0.548548\pi$
$468$	0	0
$469$	1.48005e6i	0.0143469i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.44920e6	0.0514932
$474$	0	0
$475$	−4.07228e7	−0.379977
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9.72219e7i	0.884622i	0.896862	+	0.442311i	$0.145841\pi$
$479$	9.72219e7i	0.884622i	−0.896862	+	0.442311i	$0.854159\pi$
$480$	0	0
$481$	4.49153e7	0.403607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.28074e8i	− 1.12263i
$486$	0	0
$487$	9.02434e7i	0.781320i	0.920535	+	0.390660i	$0.127753\pi$
$487$	9.02434e7i	0.781320i	−0.920535	+	0.390660i	$0.872247\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.00966e8	−0.852962	−0.426481	−	0.904497i	$-0.640247\pi$
$491$	−1.00966e8	−0.852962	−0.426481	+	0.904497i	$0.640247\pi$
$492$	0	0
$493$	− 8.52758e7i	− 0.711681i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.09365e7	−0.0890857
$498$	0	0
$499$	1.47819e8	1.18968	0.594839	−	0.803845i	$-0.297216\pi$
$499$	1.47819e8	1.18968	0.594839	+	0.803845i	$0.297216\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.38774e7i	0.266199i	0.991103	+	0.133099i	$0.0424929\pi$
$503$	3.38774e7i	0.266199i	−0.991103	+	0.133099i	$0.957507\pi$
$504$	0	0
$505$	1.70888e8	1.32690
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.16056e7i	0.391331i	0.980671	+	0.195665i	$0.0626866\pi$
$509$	5.16056e7i	0.391331i	−0.980671	+	0.195665i	$0.937313\pi$
$510$	0	0
$511$	5.62268e6i	0.0421386i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.24810e8	3.11009
$516$	0	0
$517$	− 5.95647e7i	− 0.431040i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.06575e8	0.753602	0.376801	−	0.926294i	$-0.377024\pi$
$521$	1.06575e8	0.753602	0.376801	+	0.926294i	$0.377024\pi$
$522$	0	0
$523$	−1.10885e8	−0.775117	−0.387558	−	0.921845i	$-0.626681\pi$
$523$	−1.10885e8	−0.775117	−0.387558	+	0.921845i	$0.626681\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.88162e7i	0.265205i
$528$	0	0
$529$	−2.21005e8	−1.49292
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.08202e7i	0.533751i
$534$	0	0
$535$	− 3.60962e8i	− 2.35722i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.08368e8	−0.692047
$540$	0	0
$541$	8.74191e7i	0.552096i	0.961144	+	0.276048i	$0.0890248\pi$
$541$	8.74191e7i	0.552096i	−0.961144	+	0.276048i	$0.910975\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.84737e8	1.75896
$546$	0	0
$547$	1.11073e7	0.0678653	0.0339326	−	0.999424i	$-0.489197\pi$
$547$	1.11073e7	0.0678653	0.0339326	+	0.999424i	$0.489197\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.81351e7i	0.168188i
$552$	0	0
$553$	−6.72592e6	−0.0397719
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 4.54249e7i	− 0.262862i	−0.991325	−	0.131431i	$-0.958043\pi$
$557$	− 4.54249e7i	− 0.262862i	0.991325	−	0.131431i	$-0.0419573\pi$
$558$	0	0
$559$	− 9.14488e6i	− 0.0523531i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.57201e7	0.312239	0.156119	−	0.987738i	$-0.450102\pi$
$563$	5.57201e7	0.312239	0.156119	+	0.987738i	$0.450102\pi$
$564$	0	0
$565$	1.25099e8i	0.693600i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.43539e8	−1.86483	−0.932416	−	0.361387i	$-0.882303\pi$
$569$	−3.43539e8	−1.86483	−0.932416	+	0.361387i	$0.882303\pi$
$570$	0	0
$571$	2.25486e8	1.21119	0.605593	−	0.795775i	$-0.292936\pi$
$571$	2.25486e8	1.21119	0.605593	+	0.795775i	$0.292936\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 4.61236e8i	− 2.42616i
$576$	0	0
$577$	−8.79050e7	−0.457600	−0.228800	−	0.973473i	$-0.573480\pi$
$577$	−8.79050e7	−0.457600	−0.228800	+	0.973473i	$0.573480\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.82803e7i	− 0.0932085i
$582$	0	0
$583$	1.82415e8i	0.920564i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.94523e7	−0.195056	−0.0975278	−	0.995233i	$-0.531093\pi$
$587$	−3.94523e7	−0.195056	−0.0975278	+	0.995233i	$0.531093\pi$
$588$	0	0
$589$	− 1.28067e7i	− 0.0626744i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.30474e8	−1.10524	−0.552622	−	0.833432i	$-0.686373\pi$
$593$	−2.30474e8	−1.10524	−0.552622	+	0.833432i	$0.686373\pi$
$594$	0	0
$595$	−2.01268e7	−0.0955484
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.85523e8i	− 0.863212i	−0.902062	−	0.431606i	$-0.857947\pi$
$599$	− 1.85523e8i	− 0.863212i	0.902062	−	0.431606i	$-0.142053\pi$
$600$	0	0
$601$	−4.38545e7	−0.202018	−0.101009	−	0.994885i	$-0.532207\pi$
$601$	−4.38545e7	−0.202018	−0.101009	+	0.994885i	$0.532207\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.82661e8i	0.824858i
$606$	0	0
$607$	− 2.43622e8i	− 1.08931i	−0.838661	−	0.544654i	$-0.816661\pi$
$607$	− 2.43622e8i	− 1.08931i	0.838661	−	0.544654i	$-0.183339\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.99618e7	−0.438238
$612$	0	0
$613$	− 3.31194e8i	− 1.43781i	−0.695109	−	0.718905i	$-0.744644\pi$
$613$	− 3.31194e8i	− 1.43781i	0.695109	−	0.718905i	$-0.255356\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.49869e8	−0.638054	−0.319027	−	0.947746i	$-0.603356\pi$
$617$	−1.49869e8	−0.638054	−0.319027	+	0.947746i	$0.603356\pi$
$618$	0	0
$619$	−8.10701e7	−0.341813	−0.170907	−	0.985287i	$-0.554670\pi$
$619$	−8.10701e7	−0.341813	−0.170907	+	0.985287i	$0.554670\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 8.54977e6i	− 0.0353582i
$624$	0	0
$625$	−4.28287e7	−0.175426
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.48879e8i	0.598250i
$630$	0	0
$631$	2.70423e8i	1.07635i	0.842832	+	0.538177i	$0.180887\pi$
$631$	2.70423e8i	1.07635i	−0.842832	+	0.538177i	$0.819113\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−4.84163e8	−1.89091
$636$	0	0
$637$	1.81864e8i	0.703604i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7.98693e7	−0.303254	−0.151627	−	0.988438i	$-0.548451\pi$
$641$	−7.98693e7	−0.303254	−0.151627	+	0.988438i	$0.548451\pi$
$642$	0	0
$643$	−3.07378e8	−1.15622	−0.578110	−	0.815959i	$-0.696209\pi$
$643$	−3.07378e8	−1.15622	−0.578110	+	0.815959i	$0.696209\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.33867e7i	− 0.123271i	−0.998099	−	0.0616354i	$-0.980368\pi$
$647$	− 3.33867e7i	− 0.123271i	0.998099	−	0.0616354i	$-0.0196316\pi$
$648$	0	0
$649$	−1.31423e8	−0.480771
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3.92550e8i	− 1.40979i	−0.709309	−	0.704897i	$-0.750993\pi$
$653$	− 3.92550e8i	− 1.40979i	0.709309	−	0.704897i	$-0.249007\pi$
$654$	0	0
$655$	− 5.31525e8i	− 1.89147i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.88173e8	1.70576	0.852879	−	0.522109i	$-0.174855\pi$
$659$	4.88173e8	1.70576	0.852879	+	0.522109i	$0.174855\pi$
$660$	0	0
$661$	2.56127e8i	0.886853i	0.896311	+	0.443427i	$0.146237\pi$
$661$	2.56127e8i	0.886853i	−0.896311	+	0.443427i	$0.853763\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.64045e6	0.0225805
$666$	0	0
$667$	−3.18665e8	−1.07388
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.91589e7i	0.295119i
$672$	0	0
$673$	2.16653e8	0.710754	0.355377	−	0.934723i	$-0.384352\pi$
$673$	2.16653e8	0.710754	0.355377	+	0.934723i	$0.384352\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.50931e8i	0.486420i	0.969974	+	0.243210i	$0.0782005\pi$
$677$	1.50931e8i	0.486420i	−0.969974	+	0.243210i	$0.921800\pi$
$678$	0	0
$679$	1.26512e7i	0.0404130i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.69947e8	−1.16112	−0.580561	−	0.814217i	$-0.697167\pi$
$683$	−3.69947e8	−1.16112	−0.580561	+	0.814217i	$0.697167\pi$
$684$	0	0
$685$	− 7.96746e8i	− 2.47884i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.06129e8	0.935938
$690$	0	0
$691$	2.15167e7	0.0652141	0.0326070	−	0.999468i	$-0.489619\pi$
$691$	2.15167e7	0.0652141	0.0326070	+	0.999468i	$0.489619\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.77109e8i	1.12334i
$696$	0	0
$697$	−2.67893e8	−0.791156
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 2.95578e8i	− 0.858060i	−0.903290	−	0.429030i	$-0.858855\pi$
$701$	− 2.95578e8i	− 0.858060i	0.903290	−	0.429030i	$-0.141145\pi$
$702$	0	0
$703$	− 4.91200e7i	− 0.141381i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.68803e7	−0.0477665
$708$	0	0
$709$	− 3.43066e8i	− 0.962584i	−0.876560	−	0.481292i	$-0.840168\pi$
$709$	− 3.43066e8i	− 0.962584i	0.876560	−	0.481292i	$-0.159832\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.45051e8	0.400177
$714$	0	0
$715$	−2.85344e8	−0.780641
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 2.69203e8i	− 0.724257i	−0.932128	−	0.362129i	$-0.882050\pi$
$719$	− 2.69203e8i	− 0.724257i	0.932128	−	0.362129i	$-0.117950\pi$
$720$	0	0
$721$	−4.19629e7	−0.111959
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3.98274e8i	− 1.04513i
$726$	0	0
$727$	− 1.90694e8i	− 0.496287i	−0.968723	−	0.248144i	$-0.920180\pi$
$727$	− 1.90694e8i	− 0.496287i	0.968723	−	0.248144i	$-0.0798205\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.03123e7	0.0776008
$732$	0	0
$733$	4.68798e8i	1.19035i	0.803597	+	0.595174i	$0.202917\pi$
$733$	4.68798e8i	1.19035i	−0.803597	+	0.595174i	$0.797083\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−6.95525e7	−0.173744
$738$	0	0
$739$	6.73855e8	1.66968	0.834840	−	0.550493i	$-0.185560\pi$
$739$	6.73855e8	1.66968	0.834840	+	0.550493i	$0.185560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 7.42053e7i	− 0.180913i	−0.995900	−	0.0904563i	$-0.971167\pi$
$743$	− 7.42053e7i	− 0.180913i	0.995900	−	0.0904563i	$-0.0288325\pi$
$744$	0	0
$745$	6.09983e8	1.47519
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.56560e7i	0.0848568i
$750$	0	0
$751$	9.05182e7i	0.213706i	0.994275	+	0.106853i	$0.0340774\pi$
$751$	9.05182e7i	0.213706i	−0.994275	+	0.106853i	$0.965923\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.46819e8	1.03822
$756$	0	0
$757$	− 2.19809e8i	− 0.506709i	−0.967374	−	0.253354i	$-0.918466\pi$
$757$	− 2.19809e8i	− 0.506709i	0.967374	−	0.253354i	$-0.0815339\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.05220e7	−0.182709	−0.0913547	−	0.995818i	$-0.529120\pi$
$761$	−8.05220e7	−0.182709	−0.0913547	+	0.995818i	$0.529120\pi$
$762$	0	0
$763$	−2.81264e7	−0.0633200
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.20555e8i	0.488800i
$768$	0	0
$769$	−8.94416e8	−1.96680	−0.983400	−	0.181449i	$-0.941921\pi$
$769$	−8.94416e8	−1.96680	−0.983400	+	0.181449i	$0.941921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 7.83108e8i	− 1.69544i	−0.530442	−	0.847721i	$-0.677974\pi$
$773$	− 7.83108e8i	− 1.69544i	0.530442	−	0.847721i	$-0.322026\pi$
$774$	0	0
$775$	1.81288e8i	0.389461i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.83861e7	0.186970
$780$	0	0
$781$	− 5.13941e8i	− 1.07885i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.73650e8	2.01277
$786$	0	0
$787$	−4.40033e7	−0.0902737	−0.0451368	−	0.998981i	$-0.514372\pi$
$787$	−4.40033e7	−0.0902737	−0.0451368	+	0.998981i	$0.514372\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.23573e7i	− 0.0249687i
$792$	0	0
$793$	1.49627e8	0.300047
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.71867e8i	1.91969i	0.280526	+	0.959846i	$0.409491\pi$
$797$	9.71867e8i	1.91969i	−0.280526	+	0.959846i	$0.590509\pi$
$798$	0	0
$799$	− 3.31341e8i	− 0.649583i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.64228e8	−0.510308
$804$	0	0
$805$	7.52112e7i	0.144177i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−9.68480e8	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$809$	−9.68480e8	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$810$	0	0
$811$	−6.66747e8	−1.24997	−0.624984	−	0.780638i	$-0.714894\pi$
$811$	−6.66747e8	−1.24997	−0.624984	+	0.780638i	$0.714894\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.64298e8i	0.672952i
$816$	0	0
$817$	−1.00010e7	−0.0183390
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.01581e9i	1.83563i	0.397009	+	0.917815i	$0.370048\pi$
$821$	1.01581e9i	1.83563i	−0.397009	+	0.917815i	$0.629952\pi$
$822$	0	0
$823$	9.24417e8i	1.65832i	0.559011	+	0.829160i	$0.311181\pi$
$823$	9.24417e8i	1.65832i	−0.559011	+	0.829160i	$0.688819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.14562e8	−0.909748	−0.454874	−	0.890556i	$-0.650316\pi$
$827$	−5.14562e8	−0.909748	−0.454874	+	0.890556i	$0.650316\pi$
$828$	0	0
$829$	7.43047e8i	1.30423i	0.758122	+	0.652113i	$0.226117\pi$
$829$	7.43047e8i	1.30423i	−0.758122	+	0.652113i	$0.773883\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.02820e8	−1.04292
$834$	0	0
$835$	7.84949e8	1.34829
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9.70115e8i	1.64262i	0.570482	+	0.821310i	$0.306756\pi$
$839$	9.70115e8i	1.64262i	−0.570482	+	0.821310i	$0.693244\pi$
$840$	0	0
$841$	3.19658e8	0.537400
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.82077e8i	− 0.798998i
$846$	0	0
$847$	− 1.80433e7i	− 0.0296938i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.56344e8	0.902723
$852$	0	0
$853$	4.99318e8i	0.804507i	0.915528	+	0.402254i	$0.131773\pi$
$853$	4.99318e8i	0.804507i	−0.915528	+	0.402254i	$0.868227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.80356e8	1.55755	0.778774	−	0.627305i	$-0.215842\pi$
$857$	9.80356e8	1.55755	0.778774	+	0.627305i	$0.215842\pi$
$858$	0	0
$859$	−1.39198e8	−0.219610	−0.109805	−	0.993953i	$-0.535023\pi$
$859$	−1.39198e8	−0.219610	−0.109805	+	0.993953i	$0.535023\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.80087e7i	− 0.0746943i	−0.999302	−	0.0373471i	$-0.988109\pi$
$863$	− 4.80087e7i	− 0.0746943i	0.999302	−	0.0373471i	$-0.0118907\pi$
$864$	0	0
$865$	4.08455e7	0.0631097
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 3.16073e8i	− 0.481646i
$870$	0	0
$871$	1.16723e8i	0.176646i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.28269e7	−0.0490010
$876$	0	0
$877$	− 1.09245e8i	− 0.161959i	−0.996716	−	0.0809793i	$-0.974195\pi$
$877$	− 1.09245e8i	− 0.161959i	0.996716	−	0.0809793i	$-0.0258048\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.54386e8	−0.810746	−0.405373	−	0.914151i	$-0.632858\pi$
$881$	−5.54386e8	−0.810746	−0.405373	+	0.914151i	$0.632858\pi$
$882$	0	0
$883$	8.44750e8	1.22700	0.613502	−	0.789693i	$-0.289760\pi$
$883$	8.44750e8	1.22700	0.613502	+	0.789693i	$0.289760\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 7.33580e8i	− 1.05118i	−0.850738	−	0.525589i	$-0.823845\pi$
$887$	− 7.33580e8i	− 1.05118i	0.850738	−	0.525589i	$-0.176155\pi$
$888$	0	0
$889$	4.78257e7	0.0680701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.09320e8i	0.153512i
$894$	0	0
$895$	6.68937e8i	0.933075i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.25251e8	0.172386
$900$	0	0
$901$	1.01472e9i	1.38730i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.76101e8	−0.642323
$906$	0	0
$907$	−1.71358e8	−0.229658	−0.114829	−	0.993385i	$-0.536632\pi$
$907$	−1.71358e8	−0.229658	−0.114829	+	0.993385i	$0.536632\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 7.97986e8i	− 1.05546i	−0.849413	−	0.527728i	$-0.823044\pi$
$911$	− 7.97986e8i	− 1.05546i	0.849413	−	0.527728i	$-0.176956\pi$
$912$	0	0
$913$	8.59053e8	1.12878
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.25042e7i	0.0680905i
$918$	0	0
$919$	− 7.66471e7i	− 0.0987527i	−0.998780	−	0.0493764i	$-0.984277\pi$
$919$	− 7.66471e7i	− 0.0987527i	0.998780	−	0.0493764i	$-0.0157234\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.62498e8	−1.09686
$924$	0	0
$925$	6.95331e8i	0.878549i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	5.09008e8	0.634860	0.317430	−	0.948282i	$-0.397180\pi$
$929$	5.09008e8	0.634860	0.317430	+	0.948282i	$0.397180\pi$
$930$	0	0
$931$	1.98889e8	0.246469
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 9.45823e8i	− 1.15711i
$936$	0	0
$937$	8.59374e8	1.04463	0.522316	−	0.852752i	$-0.325068\pi$
$937$	8.59374e8	1.04463	0.522316	+	0.852752i	$0.325068\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.07856e8i	0.129442i	0.997903	+	0.0647208i	$0.0206157\pi$
$941$	1.07856e8i	0.129442i	−0.997903	+	0.0647208i	$0.979384\pi$
$942$	0	0
$943$	1.00108e9i	1.19381i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.41217e9	1.66279	0.831396	−	0.555680i	$-0.187542\pi$
$947$	1.41217e9	1.66279	0.831396	+	0.555680i	$0.187542\pi$
$948$	0	0
$949$	4.43429e8i	0.518830i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.38550e9	1.60077	0.800385	−	0.599486i	$-0.204628\pi$
$953$	1.38550e9	1.60077	0.800385	+	0.599486i	$0.204628\pi$
$954$	0	0
$955$	4.03029e8	0.462729
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7.87028e7i	0.0892348i
$960$	0	0
$961$	8.30491e8	0.935761
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.57777e9i	1.75574i
$966$	0	0
$967$	− 1.22400e9i	− 1.35364i	−0.736148	−	0.676821i	$-0.763357\pi$
$967$	− 1.22400e9i	− 1.35364i	0.736148	−	0.676821i	$-0.236643\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8.21087e8	−0.896874	−0.448437	−	0.893814i	$-0.648019\pi$
$971$	−8.21087e8	−0.896874	−0.448437	+	0.893814i	$0.648019\pi$
$972$	0	0
$973$	− 3.72510e7i	− 0.0404389i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.89473e8	0.417632	0.208816	−	0.977955i	$-0.433039\pi$
$977$	3.89473e8	0.417632	0.208816	+	0.977955i	$0.433039\pi$
$978$	0	0
$979$	4.01782e8	0.428196
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.66072e9i	1.74838i	0.485584	+	0.874190i	$0.338607\pi$
$983$	1.66072e9i	1.74838i	−0.485584	+	0.874190i	$0.661393\pi$
$984$	0	0
$985$	−8.21794e8	−0.859913
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.13273e8i	− 0.117095i
$990$	0	0
$991$	− 3.03432e8i	− 0.311775i	−0.987775	−	0.155887i	$-0.950176\pi$
$991$	− 3.03432e8i	− 0.311775i	0.987775	−	0.155887i	$-0.0498237\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.35310e9	2.38875
$996$	0	0
$997$	− 1.20614e9i	− 1.21706i	−0.793532	−	0.608528i	$-0.791760\pi$
$997$	− 1.20614e9i	− 1.21706i	0.793532	−	0.608528i	$-0.208240\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	288.7.b.b.271.1		4
3.2	odd	2		32.7.d.b.15.4		4
4.3	odd	2		72.7.b.b.19.3		4
8.3	odd	2	inner	288.7.b.b.271.4		4
8.5	even	2		72.7.b.b.19.4		4
12.11	even	2		8.7.d.b.3.2	yes	4
24.5	odd	2		8.7.d.b.3.1	✓	4
24.11	even	2		32.7.d.b.15.3		4
48.5	odd	4		256.7.c.l.255.1		8
48.11	even	4		256.7.c.l.255.7		8
48.29	odd	4		256.7.c.l.255.8		8
48.35	even	4		256.7.c.l.255.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.7.d.b.3.1	✓	4	24.5	odd	2
8.7.d.b.3.2	yes	4	12.11	even	2
32.7.d.b.15.3		4	24.11	even	2
32.7.d.b.15.4		4	3.2	odd	2
72.7.b.b.19.3		4	4.3	odd	2
72.7.b.b.19.4		4	8.5	even	2
256.7.c.l.255.1		8	48.5	odd	4
256.7.c.l.255.2		8	48.35	even	4
256.7.c.l.255.7		8	48.11	even	4
256.7.c.l.255.8		8	48.29	odd	4
288.7.b.b.271.1		4	1.1	even	1	trivial
288.7.b.b.271.4		4	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 288 = 2^{5} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	288.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-199.084i q^{5} +19.6656i q^{7} +O(q^{10})\)	\(q-199.084i q^{5} +19.6656i q^{7} -924.152 q^{11} +1550.92i q^{13} -5140.78 q^{17} +1696.10 q^{19} +19210.4i q^{23} -24009.6 q^{25} +16588.1i q^{29} -7550.64i q^{31} +3915.12 q^{35} -28960.5i q^{37} +52111.3 q^{41} -5896.43 q^{43} +64453.4i q^{47} +117262. q^{49} -197386. i q^{53} +183984. i q^{55} +142210. q^{59} -96476.4i q^{61} +308763. q^{65} +75260.9 q^{67} +556121. i q^{71} +285914. q^{73} -18174.0i q^{77} +342014. i q^{79} -929558. q^{83} +1.02345e6i q^{85} -434757. q^{89} -30499.7 q^{91} -337668. i q^{95} +643314. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 976 q^{11} - 4168 q^{17} + 1456 q^{19} - 23900 q^{25} - 49920 q^{35} + 117944 q^{41} - 197456 q^{43} + 2116 q^{49} + 542032 q^{59} + 205440 q^{65} + 790192 q^{67} + 443912 q^{73} - 3465008 q^{83} - 761224 q^{89} - 3398400 q^{91} - 926776 q^{97}+O(q^{100}) \) 4 * q + 976 * q^11 - 4168 * q^17 + 1456 * q^19 - 23900 * q^25 - 49920 * q^35 + 117944 * q^41 - 197456 * q^43 + 2116 * q^49 + 542032 * q^59 + 205440 * q^65 + 790192 * q^67 + 443912 * q^73 - 3465008 * q^83 - 761224 * q^89 - 3398400 * q^91 - 926776 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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