LMFDB - Embedded newform 3584.2.b.j.1793.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3584,2,Mod(1793,3584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3584, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3584.1793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3584 = 2^{9} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3584.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:	$28.6183840844$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 2 x^{10} + 8 x^{9} + 2 x^{8} + 12 x^{7} + 40 x^{6} + 4 x^{5} + 49 x^{4} - 56 x^{3} + \cdots + 2 $ x^12 - 4x^11 + 2x^10 + 8x^9 + 2x^8 + 12x^7 + 40x^6 + 4x^5 + 49x^4 - 56x^3 + 34x^2 - 12*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1793.11
Root			$2.84980 - 1.18043i$ of defining polynomial
Character	$\chi$	$=$	3584.1793
Dual form			3584.2.b.j.1793.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36086i q^{3} +0.711757i q^{5} -1.00000 q^{7} -2.57364 q^{9} +O(q^{10})$	$q+2.36086i q^{3} +0.711757i q^{5} -1.00000 q^{7} -2.57364 q^{9} +2.98785i q^{11} +6.85246i q^{13} -1.68036 q^{15} +3.63038 q^{17} -1.35428i q^{19} -2.36086i q^{21} +1.07739 q^{23} +4.49340 q^{25} +1.00658i q^{27} +9.17376i q^{29} +8.57079 q^{31} -7.05389 q^{33} -0.711757i q^{35} +6.30788i q^{37} -16.1777 q^{39} +4.55764 q^{41} -8.12198i q^{43} -1.83181i q^{45} +0.349902 q^{47} +1.00000 q^{49} +8.57079i q^{51} -2.37638i q^{53} -2.12662 q^{55} +3.19726 q^{57} -4.30257i q^{59} +4.92519i q^{61} +2.57364 q^{63} -4.87728 q^{65} -2.16618i q^{67} +2.54356i q^{69} -3.44999 q^{71} -7.06704 q^{73} +10.6083i q^{75} -2.98785i q^{77} -13.7182 q^{79} -10.0973 q^{81} -14.7807i q^{83} +2.58394i q^{85} -21.6579 q^{87} -8.41460 q^{89} -6.85246i q^{91} +20.2344i q^{93} +0.963918 q^{95} +15.3981 q^{97} -7.68966i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{7} - 4 q^{9}+O(q^{10}) $ 12 * q - 12 * q^7 - 4 * q^9	$ 12 q - 12 q^{7} - 4 q^{9} - 16 q^{15} - 20 q^{25} + 16 q^{31} - 8 q^{33} - 8 q^{41} + 16 q^{47} + 12 q^{49} + 32 q^{55} + 24 q^{57} + 4 q^{63} + 16 q^{71} + 16 q^{73} - 48 q^{79} - 28 q^{81} - 16 q^{87} - 16 q^{89} + 32 q^{95}+O(q^{100}) $ 12 * q - 12 * q^7 - 4 * q^9 - 16 * q^15 - 20 * q^25 + 16 * q^31 - 8 * q^33 - 8 * q^41 + 16 * q^47 + 12 * q^49 + 32 * q^55 + 24 * q^57 + 4 * q^63 + 16 * q^71 + 16 * q^73 - 48 * q^79 - 28 * q^81 - 16 * q^87 - 16 * q^89 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3584\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	2.36086i	1.36304i	0.731799	+	0.681520i	$0.238681\pi$
$3$	2.36086i	1.36304i	−0.731799	+	0.681520i	$0.761319\pi$
$4$	0	0
$5$	0.711757i	0.318307i	0.987254	+	0.159154i	$0.0508765\pi$
$5$	0.711757i	0.318307i	−0.987254	+	0.159154i	$0.949123\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.57364	−0.857880
$10$	0	0
$11$	2.98785i	0.900872i	0.892809	+	0.450436i	$0.148731\pi$
$11$	2.98785i	0.900872i	−0.892809	+	0.450436i	$0.851269\pi$
$12$	0	0
$13$	6.85246i	1.90053i	0.311442	+	0.950265i	$0.399188\pi$
$13$	6.85246i	1.90053i	−0.311442	+	0.950265i	$0.600812\pi$
$14$	0	0
$15$	−1.68036	−0.433866
$16$	0	0
$17$	3.63038	0.880495	0.440248	−	0.897876i	$-0.354891\pi$
$17$	3.63038	0.880495	0.440248	+	0.897876i	$0.354891\pi$
$18$	0	0
$19$	− 1.35428i	− 0.310693i	−0.987860	−	0.155347i	$-0.950351\pi$
$19$	− 1.35428i	− 0.310693i	0.987860	−	0.155347i	$-0.0496494\pi$
$20$	0	0
$21$	− 2.36086i	− 0.515181i
$22$	0	0
$23$	1.07739	0.224651	0.112326	−	0.993671i	$-0.464170\pi$
$23$	1.07739	0.224651	0.112326	+	0.993671i	$0.464170\pi$
$24$	0	0
$25$	4.49340	0.898680
$26$	0	0
$27$	1.00658i	0.193716i
$28$	0	0
$29$	9.17376i	1.70352i	0.523929	+	0.851762i	$0.324466\pi$
$29$	9.17376i	1.70352i	−0.523929	+	0.851762i	$0.675534\pi$
$30$	0	0
$31$	8.57079	1.53936	0.769680	−	0.638430i	$-0.220416\pi$
$31$	8.57079	1.53936	0.769680	+	0.638430i	$0.220416\pi$
$32$	0	0
$33$	−7.05389	−1.22792
$34$	0	0
$35$	− 0.711757i	− 0.120309i
$36$	0	0
$37$	6.30788i	1.03701i	0.855075	+	0.518505i	$0.173511\pi$
$37$	6.30788i	1.03701i	−0.855075	+	0.518505i	$0.826489\pi$
$38$	0	0
$39$	−16.1777	−2.59050
$40$	0	0
$41$	4.55764	0.711784	0.355892	−	0.934527i	$-0.384177\pi$
$41$	4.55764	0.711784	0.355892	+	0.934527i	$0.384177\pi$
$42$	0	0
$43$	− 8.12198i	− 1.23859i	−0.785158	−	0.619295i	$-0.787418\pi$
$43$	− 8.12198i	− 1.23859i	0.785158	−	0.619295i	$-0.212582\pi$
$44$	0	0
$45$	− 1.83181i	− 0.273069i
$46$	0	0
$47$	0.349902	0.0510385	0.0255192	−	0.999674i	$-0.491876\pi$
$47$	0.349902	0.0510385	0.0255192	+	0.999674i	$0.491876\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	8.57079i	1.20015i
$52$	0	0
$53$	− 2.37638i	− 0.326421i	−0.986591	−	0.163211i	$-0.947815\pi$
$53$	− 2.37638i	− 0.326421i	0.986591	−	0.163211i	$-0.0521850\pi$
$54$	0	0
$55$	−2.12662	−0.286754
$56$	0	0
$57$	3.19726	0.423487
$58$	0	0
$59$	− 4.30257i	− 0.560148i	−0.959978	−	0.280074i	$-0.909641\pi$
$59$	− 4.30257i	− 0.560148i	0.959978	−	0.280074i	$-0.0903589\pi$
$60$	0	0
$61$	4.92519i	0.630606i	0.948991	+	0.315303i	$0.102106\pi$
$61$	4.92519i	0.630606i	−0.948991	+	0.315303i	$0.897894\pi$
$62$	0	0
$63$	2.57364	0.324248
$64$	0	0
$65$	−4.87728	−0.604953
$66$	0	0
$67$	− 2.16618i	− 0.264641i	−0.991207	−	0.132321i	$-0.957757\pi$
$67$	− 2.16618i	− 0.264641i	0.991207	−	0.132321i	$-0.0422429\pi$
$68$	0	0
$69$	2.54356i	0.306209i
$70$	0	0
$71$	−3.44999	−0.409439	−0.204719	−	0.978821i	$-0.565628\pi$
$71$	−3.44999	−0.409439	−0.204719	+	0.978821i	$0.565628\pi$
$72$	0	0
$73$	−7.06704	−0.827135	−0.413567	−	0.910474i	$-0.635717\pi$
$73$	−7.06704	−0.827135	−0.413567	+	0.910474i	$0.635717\pi$
$74$	0	0
$75$	10.6083i	1.22494i
$76$	0	0
$77$	− 2.98785i	− 0.340497i
$78$	0	0
$79$	−13.7182	−1.54342	−0.771712	−	0.635973i	$-0.780599\pi$
$79$	−13.7182	−1.54342	−0.771712	+	0.635973i	$0.780599\pi$
$80$	0	0
$81$	−10.0973	−1.12192
$82$	0	0
$83$	− 14.7807i	− 1.62240i	−0.584772	−	0.811198i	$-0.698816\pi$
$83$	− 14.7807i	− 1.62240i	0.584772	−	0.811198i	$-0.301184\pi$
$84$	0	0
$85$	2.58394i	0.280268i
$86$	0	0
$87$	−21.6579	−2.32197
$88$	0	0
$89$	−8.41460	−0.891946	−0.445973	−	0.895046i	$-0.647142\pi$
$89$	−8.41460	−0.891946	−0.445973	+	0.895046i	$0.647142\pi$
$90$	0	0
$91$	− 6.85246i	− 0.718333i
$92$	0	0
$93$	20.2344i	2.09821i
$94$	0	0
$95$	0.963918	0.0988959
$96$	0	0
$97$	15.3981	1.56344	0.781720	−	0.623630i	$-0.214343\pi$
$97$	15.3981	1.56344	0.781720	+	0.623630i	$0.214343\pi$
$98$	0	0
$99$	− 7.68966i	− 0.772839i
$100$	0	0
$101$	− 11.6052i	− 1.15476i	−0.816475	−	0.577381i	$-0.804075\pi$
$101$	− 11.6052i	− 1.15476i	0.816475	−	0.577381i	$-0.195925\pi$
$102$	0	0
$103$	−14.0581	−1.38519	−0.692594	−	0.721327i	$-0.743532\pi$
$103$	−14.0581	−1.38519	−0.692594	+	0.721327i	$0.743532\pi$
$104$	0	0
$105$	1.68036	0.163986
$106$	0	0
$107$	2.39821i	0.231844i	0.993258	+	0.115922i	$0.0369823\pi$
$107$	2.39821i	0.231844i	−0.993258	+	0.115922i	$0.963018\pi$
$108$	0	0
$109$	− 1.28723i	− 0.123294i	−0.998098	−	0.0616471i	$-0.980365\pi$
$109$	− 1.28723i	− 0.123294i	0.998098	−	0.0616471i	$-0.0196353\pi$
$110$	0	0
$111$	−14.8920	−1.41349
$112$	0	0
$113$	−4.85411	−0.456636	−0.228318	−	0.973587i	$-0.573323\pi$
$113$	−4.85411	−0.456636	−0.228318	+	0.973587i	$0.573323\pi$
$114$	0	0
$115$	0.766840i	0.0715082i
$116$	0	0
$117$	− 17.6358i	− 1.63043i
$118$	0	0
$119$	−3.63038	−0.332796
$120$	0	0
$121$	2.07274	0.188431
$122$	0	0
$123$	10.7599i	0.970190i
$124$	0	0
$125$	6.75699i	0.604364i
$126$	0	0
$127$	3.92442	0.348236	0.174118	−	0.984725i	$-0.444293\pi$
$127$	3.92442	0.348236	0.174118	+	0.984725i	$0.444293\pi$
$128$	0	0
$129$	19.1748	1.68825
$130$	0	0
$131$	13.4518i	1.17529i	0.809118	+	0.587646i	$0.199945\pi$
$131$	13.4518i	1.17529i	−0.809118	+	0.587646i	$0.800055\pi$
$132$	0	0
$133$	1.35428i	0.117431i
$134$	0	0
$135$	−0.716437	−0.0616611
$136$	0	0
$137$	18.6239	1.59115	0.795575	−	0.605855i	$-0.207169\pi$
$137$	18.6239	1.59115	0.795575	+	0.605855i	$0.207169\pi$
$138$	0	0
$139$	− 7.14960i	− 0.606421i	−0.952924	−	0.303211i	$-0.901941\pi$
$139$	− 7.14960i	− 0.606421i	0.952924	−	0.303211i	$-0.0980586\pi$
$140$	0	0
$141$	0.826069i	0.0695675i
$142$	0	0
$143$	−20.4741	−1.71213
$144$	0	0
$145$	−6.52948	−0.542244
$146$	0	0
$147$	2.36086i	0.194720i
$148$	0	0
$149$	− 0.489493i	− 0.0401008i	−0.999799	−	0.0200504i	$-0.993617\pi$
$149$	− 0.489493i	− 0.0401008i	0.999799	−	0.0200504i	$-0.00638267\pi$
$150$	0	0
$151$	22.4722	1.82876	0.914382	−	0.404853i	$-0.132677\pi$
$151$	22.4722	1.82876	0.914382	+	0.404853i	$0.132677\pi$
$152$	0	0
$153$	−9.34328	−0.755359
$154$	0	0
$155$	6.10032i	0.489989i
$156$	0	0
$157$	16.2478i	1.29672i	0.761335	+	0.648359i	$0.224544\pi$
$157$	16.2478i	1.29672i	−0.761335	+	0.648359i	$0.775456\pi$
$158$	0	0
$159$	5.61029	0.444925
$160$	0	0
$161$	−1.07739	−0.0849102
$162$	0	0
$163$	4.84263i	0.379304i	0.981851	+	0.189652i	$0.0607360\pi$
$163$	4.84263i	0.379304i	−0.981851	+	0.189652i	$0.939264\pi$
$164$	0	0
$165$	− 5.02065i	− 0.390857i
$166$	0	0
$167$	9.49149	0.734473	0.367237	−	0.930128i	$-0.380304\pi$
$167$	9.49149	0.734473	0.367237	+	0.930128i	$0.380304\pi$
$168$	0	0
$169$	−33.9562	−2.61202
$170$	0	0
$171$	3.48543i	0.266537i
$172$	0	0
$173$	− 7.89792i	− 0.600467i	−0.953866	−	0.300234i	$-0.902935\pi$
$173$	− 7.89792i	− 0.600467i	0.953866	−	0.300234i	$-0.0970647\pi$
$174$	0	0
$175$	−4.49340	−0.339669
$176$	0	0
$177$	10.1578	0.763504
$178$	0	0
$179$	22.5817i	1.68783i	0.536475	+	0.843916i	$0.319756\pi$
$179$	22.5817i	1.68783i	−0.536475	+	0.843916i	$0.680244\pi$
$180$	0	0
$181$	− 13.7043i	− 1.01863i	−0.860580	−	0.509315i	$-0.829899\pi$
$181$	− 13.7043i	− 1.01863i	0.860580	−	0.509315i	$-0.170101\pi$
$182$	0	0
$183$	−11.6277	−0.859541
$184$	0	0
$185$	−4.48968	−0.330088
$186$	0	0
$187$	10.8470i	0.793213i
$188$	0	0
$189$	− 1.00658i	− 0.0732176i
$190$	0	0
$191$	−19.1266	−1.38395	−0.691975	−	0.721921i	$-0.743259\pi$
$191$	−19.1266	−1.38395	−0.691975	+	0.721921i	$0.743259\pi$
$192$	0	0
$193$	−8.35271	−0.601241	−0.300620	−	0.953744i	$-0.597194\pi$
$193$	−8.35271	−0.601241	−0.300620	+	0.953744i	$0.597194\pi$
$194$	0	0
$195$	− 11.5146i	− 0.824575i
$196$	0	0
$197$	1.38203i	0.0984657i	0.998787	+	0.0492328i	$0.0156776\pi$
$197$	1.38203i	0.0984657i	−0.998787	+	0.0492328i	$0.984322\pi$
$198$	0	0
$199$	2.47653	0.175556	0.0877782	−	0.996140i	$-0.472023\pi$
$199$	2.47653	0.175556	0.0877782	+	0.996140i	$0.472023\pi$
$200$	0	0
$201$	5.11404	0.360717
$202$	0	0
$203$	− 9.17376i	− 0.643872i
$204$	0	0
$205$	3.24393i	0.226566i
$206$	0	0
$207$	−2.77281	−0.192724
$208$	0	0
$209$	4.04639	0.279894
$210$	0	0
$211$	− 9.71501i	− 0.668809i	−0.942430	−	0.334405i	$-0.891465\pi$
$211$	− 9.71501i	− 0.668809i	0.942430	−	0.334405i	$-0.108535\pi$
$212$	0	0
$213$	− 8.14493i	− 0.558082i
$214$	0	0
$215$	5.78087	0.394252
$216$	0	0
$217$	−8.57079	−0.581823
$218$	0	0
$219$	− 16.6843i	− 1.12742i
$220$	0	0
$221$	24.8770i	1.67341i
$222$	0	0
$223$	−2.70540	−0.181167	−0.0905835	−	0.995889i	$-0.528873\pi$
$223$	−2.70540	−0.181167	−0.0905835	+	0.995889i	$0.528873\pi$
$224$	0	0
$225$	−11.5644	−0.770960
$226$	0	0
$227$	− 16.4716i	− 1.09326i	−0.837375	−	0.546628i	$-0.815911\pi$
$227$	− 16.4716i	− 1.09326i	0.837375	−	0.546628i	$-0.184089\pi$
$228$	0	0
$229$	− 22.0757i	− 1.45880i	−0.684086	−	0.729401i	$-0.739799\pi$
$229$	− 22.0757i	− 1.45880i	0.684086	−	0.729401i	$-0.260201\pi$
$230$	0	0
$231$	7.05389	0.464112
$232$	0	0
$233$	18.2673	1.19673	0.598365	−	0.801224i	$-0.295817\pi$
$233$	18.2673	1.19673	0.598365	+	0.801224i	$0.295817\pi$
$234$	0	0
$235$	0.249045i	0.0162459i
$236$	0	0
$237$	− 32.3868i	− 2.10375i
$238$	0	0
$239$	−13.4441	−0.869627	−0.434814	−	0.900520i	$-0.643186\pi$
$239$	−13.4441	−0.869627	−0.434814	+	0.900520i	$0.643186\pi$
$240$	0	0
$241$	8.34893	0.537802	0.268901	−	0.963168i	$-0.413340\pi$
$241$	8.34893	0.537802	0.268901	+	0.963168i	$0.413340\pi$
$242$	0	0
$243$	− 20.8185i	− 1.33551i
$244$	0	0
$245$	0.711757i	0.0454725i
$246$	0	0
$247$	9.28014	0.590481
$248$	0	0
$249$	34.8952	2.21139
$250$	0	0
$251$	− 5.33731i	− 0.336888i	−0.985711	−	0.168444i	$-0.946126\pi$
$251$	− 5.33731i	− 0.336888i	0.985711	−	0.168444i	$-0.0538742\pi$
$252$	0	0
$253$	3.21908i	0.202382i
$254$	0	0
$255$	−6.10032	−0.382017
$256$	0	0
$257$	−25.4346	−1.58657	−0.793285	−	0.608851i	$-0.791631\pi$
$257$	−25.4346	−1.58657	−0.793285	+	0.608851i	$0.791631\pi$
$258$	0	0
$259$	− 6.30788i	− 0.391953i
$260$	0	0
$261$	− 23.6099i	− 1.46142i
$262$	0	0
$263$	−13.2187	−0.815102	−0.407551	−	0.913182i	$-0.633617\pi$
$263$	−13.2187	−0.815102	−0.407551	+	0.913182i	$0.633617\pi$
$264$	0	0
$265$	1.69141	0.103902
$266$	0	0
$267$	− 19.8657i	− 1.21576i
$268$	0	0
$269$	− 17.5765i	− 1.07166i	−0.844326	−	0.535830i	$-0.819999\pi$
$269$	− 17.5765i	− 1.07166i	0.844326	−	0.535830i	$-0.180001\pi$
$270$	0	0
$271$	−20.5468	−1.24813	−0.624064	−	0.781373i	$-0.714520\pi$
$271$	−20.5468	−1.24813	−0.624064	+	0.781373i	$0.714520\pi$
$272$	0	0
$273$	16.1777	0.979117
$274$	0	0
$275$	13.4256i	0.809596i
$276$	0	0
$277$	− 22.0715i	− 1.32615i	−0.748555	−	0.663073i	$-0.769252\pi$
$277$	− 22.0715i	− 1.32615i	0.748555	−	0.663073i	$-0.230748\pi$
$278$	0	0
$279$	−22.0581	−1.32059
$280$	0	0
$281$	17.0269	1.01574	0.507869	−	0.861434i	$-0.330433\pi$
$281$	17.0269	1.01574	0.507869	+	0.861434i	$0.330433\pi$
$282$	0	0
$283$	− 15.0782i	− 0.896304i	−0.893957	−	0.448152i	$-0.852082\pi$
$283$	− 15.0782i	− 0.896304i	0.893957	−	0.448152i	$-0.147918\pi$
$284$	0	0
$285$	2.27567i	0.134799i
$286$	0	0
$287$	−4.55764	−0.269029
$288$	0	0
$289$	−3.82037	−0.224728
$290$	0	0
$291$	36.3527i	2.13103i
$292$	0	0
$293$	− 7.28162i	− 0.425397i	−0.977118	−	0.212699i	$-0.931775\pi$
$293$	− 7.28162i	− 0.425397i	0.977118	−	0.212699i	$-0.0682252\pi$
$294$	0	0
$295$	3.06239	0.178299
$296$	0	0
$297$	−3.00750	−0.174513
$298$	0	0
$299$	7.38277i	0.426957i
$300$	0	0
$301$	8.12198i	0.468143i
$302$	0	0
$303$	27.3983	1.57399
$304$	0	0
$305$	−3.50554	−0.200726
$306$	0	0
$307$	− 8.84254i	− 0.504670i	−0.967640	−	0.252335i	$-0.918801\pi$
$307$	− 8.84254i	− 0.504670i	0.967640	−	0.252335i	$-0.0811986\pi$
$308$	0	0
$309$	− 33.1892i	− 1.88807i
$310$	0	0
$311$	31.9075	1.80931	0.904654	−	0.426147i	$-0.140129\pi$
$311$	31.9075	1.80931	0.904654	+	0.426147i	$0.140129\pi$
$312$	0	0
$313$	−9.06377	−0.512314	−0.256157	−	0.966635i	$-0.582456\pi$
$313$	−9.06377	−0.512314	−0.256157	+	0.966635i	$0.582456\pi$
$314$	0	0
$315$	1.83181i	0.103211i
$316$	0	0
$317$	− 29.5991i	− 1.66245i	−0.555936	−	0.831225i	$-0.687640\pi$
$317$	− 29.5991i	− 1.66245i	0.555936	−	0.831225i	$-0.312360\pi$
$318$	0	0
$319$	−27.4098	−1.53466
$320$	0	0
$321$	−5.66184	−0.316013
$322$	0	0
$323$	− 4.91654i	− 0.273564i
$324$	0	0
$325$	30.7909i	1.70797i
$326$	0	0
$327$	3.03896	0.168055
$328$	0	0
$329$	−0.349902	−0.0192907
$330$	0	0
$331$	15.4620i	0.849868i	0.905224	+	0.424934i	$0.139703\pi$
$331$	15.4620i	0.849868i	−0.905224	+	0.424934i	$0.860297\pi$
$332$	0	0
$333$	− 16.2342i	− 0.889630i
$334$	0	0
$335$	1.54179	0.0842373
$336$	0	0
$337$	2.21951	0.120904	0.0604522	−	0.998171i	$-0.480746\pi$
$337$	2.21951	0.120904	0.0604522	+	0.998171i	$0.480746\pi$
$338$	0	0
$339$	− 11.4599i	− 0.622414i
$340$	0	0
$341$	25.6083i	1.38677i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.81040	−0.0974685
$346$	0	0
$347$	13.5674i	0.728336i	0.931333	+	0.364168i	$0.118647\pi$
$347$	13.5674i	0.728336i	−0.931333	+	0.364168i	$0.881353\pi$
$348$	0	0
$349$	32.4963i	1.73949i	0.493504	+	0.869743i	$0.335716\pi$
$349$	32.4963i	1.73949i	−0.493504	+	0.869743i	$0.664284\pi$
$350$	0	0
$351$	−6.89752	−0.368162
$352$	0	0
$353$	12.1266	0.645435	0.322718	−	0.946495i	$-0.395404\pi$
$353$	12.1266	0.645435	0.322718	+	0.946495i	$0.395404\pi$
$354$	0	0
$355$	− 2.45556i	− 0.130327i
$356$	0	0
$357$	− 8.57079i	− 0.453614i
$358$	0	0
$359$	−14.1927	−0.749060	−0.374530	−	0.927215i	$-0.622196\pi$
$359$	−14.1927	−0.749060	−0.374530	+	0.927215i	$0.622196\pi$
$360$	0	0
$361$	17.1659	0.903470
$362$	0	0
$363$	4.89343i	0.256838i
$364$	0	0
$365$	− 5.03002i	− 0.263283i
$366$	0	0
$367$	25.2817	1.31969	0.659847	−	0.751400i	$-0.270621\pi$
$367$	25.2817	1.31969	0.659847	+	0.751400i	$0.270621\pi$
$368$	0	0
$369$	−11.7297	−0.610625
$370$	0	0
$371$	2.37638i	0.123376i
$372$	0	0
$373$	− 25.1188i	− 1.30060i	−0.759677	−	0.650300i	$-0.774643\pi$
$373$	− 25.1188i	− 1.30060i	0.759677	−	0.650300i	$-0.225357\pi$
$374$	0	0
$375$	−15.9523	−0.823773
$376$	0	0
$377$	−62.8628	−3.23760
$378$	0	0
$379$	19.4994i	1.00162i	0.865558	+	0.500809i	$0.166964\pi$
$379$	19.4994i	1.00162i	−0.865558	+	0.500809i	$0.833036\pi$
$380$	0	0
$381$	9.26498i	0.474659i
$382$	0	0
$383$	−33.8525	−1.72978	−0.864891	−	0.501960i	$-0.832613\pi$
$383$	−33.8525	−1.72978	−0.864891	+	0.501960i	$0.832613\pi$
$384$	0	0
$385$	2.12662	0.108383
$386$	0	0
$387$	20.9030i	1.06256i
$388$	0	0
$389$	30.2154i	1.53198i	0.642852	+	0.765990i	$0.277751\pi$
$389$	30.2154i	1.53198i	−0.642852	+	0.765990i	$0.722249\pi$
$390$	0	0
$391$	3.91133	0.197804
$392$	0	0
$393$	−31.7579	−1.60197
$394$	0	0
$395$	− 9.76405i	− 0.491283i
$396$	0	0
$397$	− 9.05698i	− 0.454557i	−0.973830	−	0.227278i	$-0.927017\pi$
$397$	− 9.05698i	− 0.454557i	0.973830	−	0.227278i	$-0.0729828\pi$
$398$	0	0
$399$	−3.19726	−0.160063
$400$	0	0
$401$	3.78707	0.189117	0.0945587	−	0.995519i	$-0.469856\pi$
$401$	3.78707	0.189117	0.0945587	+	0.995519i	$0.469856\pi$
$402$	0	0
$403$	58.7310i	2.92560i
$404$	0	0
$405$	− 7.18682i	− 0.357116i
$406$	0	0
$407$	−18.8470	−0.934212
$408$	0	0
$409$	23.7532	1.17452	0.587260	−	0.809398i	$-0.300207\pi$
$409$	23.7532	1.17452	0.587260	+	0.809398i	$0.300207\pi$
$410$	0	0
$411$	43.9684i	2.16880i
$412$	0	0
$413$	4.30257i	0.211716i
$414$	0	0
$415$	10.5203	0.516420
$416$	0	0
$417$	16.8792	0.826577
$418$	0	0
$419$	− 30.9658i	− 1.51278i	−0.654123	−	0.756388i	$-0.726962\pi$
$419$	− 30.9658i	− 1.51278i	0.654123	−	0.756388i	$-0.273038\pi$
$420$	0	0
$421$	15.9523i	0.777467i	0.921350	+	0.388733i	$0.127087\pi$
$421$	15.9523i	0.777467i	−0.921350	+	0.388733i	$0.872913\pi$
$422$	0	0
$423$	−0.900522	−0.0437849
$424$	0	0
$425$	16.3127	0.791284
$426$	0	0
$427$	− 4.92519i	− 0.238347i
$428$	0	0
$429$	− 48.3365i	− 2.33371i
$430$	0	0
$431$	9.50425	0.457804	0.228902	−	0.973449i	$-0.426486\pi$
$431$	9.50425	0.457804	0.228902	+	0.973449i	$0.426486\pi$
$432$	0	0
$433$	5.41658	0.260304	0.130152	−	0.991494i	$-0.458453\pi$
$433$	5.41658	0.260304	0.130152	+	0.991494i	$0.458453\pi$
$434$	0	0
$435$	− 15.4152i	− 0.739101i
$436$	0	0
$437$	− 1.45909i	− 0.0697976i
$438$	0	0
$439$	29.7082	1.41790	0.708948	−	0.705261i	$-0.249170\pi$
$439$	29.7082	1.41790	0.708948	+	0.705261i	$0.249170\pi$
$440$	0	0
$441$	−2.57364	−0.122554
$442$	0	0
$443$	− 17.0290i	− 0.809071i	−0.914522	−	0.404535i	$-0.867433\pi$
$443$	− 17.0290i	− 0.809071i	0.914522	−	0.404535i	$-0.132567\pi$
$444$	0	0
$445$	− 5.98915i	− 0.283913i
$446$	0	0
$447$	1.15562	0.0546590
$448$	0	0
$449$	41.6242	1.96437	0.982183	−	0.187929i	$-0.0601773\pi$
$449$	41.6242	1.96437	0.982183	+	0.187929i	$0.0601773\pi$
$450$	0	0
$451$	13.6176i	0.641226i
$452$	0	0
$453$	53.0537i	2.49268i
$454$	0	0
$455$	4.87728	0.228651
$456$	0	0
$457$	29.5834	1.38385	0.691925	−	0.721969i	$-0.256763\pi$
$457$	29.5834	1.38385	0.691925	+	0.721969i	$0.256763\pi$
$458$	0	0
$459$	3.65425i	0.170566i
$460$	0	0
$461$	− 0.815920i	− 0.0380012i	−0.999819	−	0.0190006i	$-0.993952\pi$
$461$	− 0.815920i	− 0.0380012i	0.999819	−	0.0190006i	$-0.00604844\pi$
$462$	0	0
$463$	22.9633	1.06719	0.533597	−	0.845739i	$-0.320840\pi$
$463$	22.9633	1.06719	0.533597	+	0.845739i	$0.320840\pi$
$464$	0	0
$465$	−14.4020	−0.667876
$466$	0	0
$467$	− 11.5170i	− 0.532945i	−0.963842	−	0.266473i	$-0.914142\pi$
$467$	− 11.5170i	− 0.532945i	0.963842	−	0.266473i	$-0.0858582\pi$
$468$	0	0
$469$	2.16618i	0.100025i
$470$	0	0
$471$	−38.3588	−1.76748
$472$	0	0
$473$	24.2673	1.11581
$474$	0	0
$475$	− 6.08532i	− 0.279214i
$476$	0	0
$477$	6.11595i	0.280030i
$478$	0	0
$479$	−14.9202	−0.681719	−0.340860	−	0.940114i	$-0.610718\pi$
$479$	−14.9202	−0.681719	−0.340860	+	0.940114i	$0.610718\pi$
$480$	0	0
$481$	−43.2245	−1.97087
$482$	0	0
$483$	− 2.54356i	− 0.115736i
$484$	0	0
$485$	10.9597i	0.497654i
$486$	0	0
$487$	36.1278	1.63711	0.818554	−	0.574429i	$-0.194776\pi$
$487$	36.1278	1.63711	0.818554	+	0.574429i	$0.194776\pi$
$488$	0	0
$489$	−11.4327	−0.517007
$490$	0	0
$491$	33.6926i	1.52052i	0.649616	+	0.760262i	$0.274930\pi$
$491$	33.6926i	1.52052i	−0.649616	+	0.760262i	$0.725070\pi$
$492$	0	0
$493$	33.3042i	1.49994i
$494$	0	0
$495$	5.47316	0.246000
$496$	0	0
$497$	3.44999	0.154753
$498$	0	0
$499$	− 16.1230i	− 0.721766i	−0.932611	−	0.360883i	$-0.882475\pi$
$499$	− 16.1230i	− 0.721766i	0.932611	−	0.360883i	$-0.117525\pi$
$500$	0	0
$501$	22.4080i	1.00112i
$502$	0	0
$503$	11.3607	0.506549	0.253275	−	0.967394i	$-0.418492\pi$
$503$	11.3607	0.506549	0.253275	+	0.967394i	$0.418492\pi$
$504$	0	0
$505$	8.26010	0.367569
$506$	0	0
$507$	− 80.1657i	− 3.56028i
$508$	0	0
$509$	36.1459i	1.60214i	0.598572	+	0.801069i	$0.295735\pi$
$509$	36.1459i	1.60214i	−0.598572	+	0.801069i	$0.704265\pi$
$510$	0	0
$511$	7.06704	0.312628
$512$	0	0
$513$	1.36319	0.0601861
$514$	0	0
$515$	− 10.0060i	− 0.440916i
$516$	0	0
$517$	1.04546i	0.0459791i
$518$	0	0
$519$	18.6458	0.818461
$520$	0	0
$521$	−36.6860	−1.60724	−0.803622	−	0.595140i	$-0.797097\pi$
$521$	−36.6860	−1.60724	−0.803622	+	0.595140i	$0.797097\pi$
$522$	0	0
$523$	32.5558i	1.42357i	0.702399	+	0.711783i	$0.252112\pi$
$523$	32.5558i	1.42357i	−0.702399	+	0.711783i	$0.747888\pi$
$524$	0	0
$525$	− 10.6083i	− 0.462983i
$526$	0	0
$527$	31.1152	1.35540
$528$	0	0
$529$	−21.8392	−0.949532
$530$	0	0
$531$	11.0733i	0.480539i
$532$	0	0
$533$	31.2310i	1.35277i
$534$	0	0
$535$	−1.70695	−0.0737977
$536$	0	0
$537$	−53.3120	−2.30058
$538$	0	0
$539$	2.98785i	0.128696i
$540$	0	0
$541$	33.9654i	1.46029i	0.683293	+	0.730144i	$0.260547\pi$
$541$	33.9654i	1.46029i	−0.683293	+	0.730144i	$0.739453\pi$
$542$	0	0
$543$	32.3538	1.38843
$544$	0	0
$545$	0.916195	0.0392455
$546$	0	0
$547$	− 2.41390i	− 0.103211i	−0.998668	−	0.0516055i	$-0.983566\pi$
$547$	− 2.41390i	− 0.103211i	0.998668	−	0.0516055i	$-0.0164338\pi$
$548$	0	0
$549$	− 12.6757i	− 0.540984i
$550$	0	0
$551$	12.4238	0.529273
$552$	0	0
$553$	13.7182	0.583359
$554$	0	0
$555$	− 10.5995i	− 0.449923i
$556$	0	0
$557$	11.0466i	0.468060i	0.972229	+	0.234030i	$0.0751914\pi$
$557$	11.0466i	0.468060i	−0.972229	+	0.234030i	$0.924809\pi$
$558$	0	0
$559$	55.6555	2.35398
$560$	0	0
$561$	−25.6083	−1.08118
$562$	0	0
$563$	− 12.5465i	− 0.528773i	−0.964417	−	0.264387i	$-0.914830\pi$
$563$	− 12.5465i	− 0.528773i	0.964417	−	0.264387i	$-0.0851695\pi$
$564$	0	0
$565$	− 3.45495i	− 0.145351i
$566$	0	0
$567$	10.0973	0.424047
$568$	0	0
$569$	27.7545	1.16353	0.581764	−	0.813358i	$-0.302363\pi$
$569$	27.7545	1.16353	0.581764	+	0.813358i	$0.302363\pi$
$570$	0	0
$571$	4.50930i	0.188708i	0.995539	+	0.0943541i	$0.0300786\pi$
$571$	4.50930i	0.188708i	−0.995539	+	0.0943541i	$0.969921\pi$
$572$	0	0
$573$	− 45.1551i	− 1.88638i
$574$	0	0
$575$	4.84115	0.201890
$576$	0	0
$577$	−2.11497	−0.0880475	−0.0440238	−	0.999030i	$-0.514018\pi$
$577$	−2.11497	−0.0880475	−0.0440238	+	0.999030i	$0.514018\pi$
$578$	0	0
$579$	− 19.7195i	− 0.819516i
$580$	0	0
$581$	14.7807i	0.613208i
$582$	0	0
$583$	7.10028	0.294063
$584$	0	0
$585$	12.5524	0.518977
$586$	0	0
$587$	− 44.9688i	− 1.85606i	−0.372505	−	0.928030i	$-0.621501\pi$
$587$	− 44.9688i	− 1.85606i	0.372505	−	0.928030i	$-0.378499\pi$
$588$	0	0
$589$	− 11.6072i	− 0.478268i
$590$	0	0
$591$	−3.26278	−0.134213
$592$	0	0
$593$	−24.5463	−1.00800	−0.503998	−	0.863705i	$-0.668138\pi$
$593$	−24.5463	−1.00800	−0.503998	+	0.863705i	$0.668138\pi$
$594$	0	0
$595$	− 2.58394i	− 0.105931i
$596$	0	0
$597$	5.84672i	0.239290i
$598$	0	0
$599$	−30.0946	−1.22963	−0.614817	−	0.788670i	$-0.710770\pi$
$599$	−30.0946	−1.22963	−0.614817	+	0.788670i	$0.710770\pi$
$600$	0	0
$601$	−1.02003	−0.0416078	−0.0208039	−	0.999784i	$-0.506623\pi$
$601$	−1.02003	−0.0416078	−0.0208039	+	0.999784i	$0.506623\pi$
$602$	0	0
$603$	5.57497i	0.227030i
$604$	0	0
$605$	1.47528i	0.0599788i
$606$	0	0
$607$	19.1251	0.776263	0.388132	−	0.921604i	$-0.373121\pi$
$607$	19.1251	0.776263	0.388132	+	0.921604i	$0.373121\pi$
$608$	0	0
$609$	21.6579	0.877623
$610$	0	0
$611$	2.39769i	0.0970002i
$612$	0	0
$613$	− 25.0418i	− 1.01143i	−0.862701	−	0.505715i	$-0.831229\pi$
$613$	− 25.0418i	− 1.01143i	0.862701	−	0.505715i	$-0.168771\pi$
$614$	0	0
$615$	−7.65845	−0.308819
$616$	0	0
$617$	−43.1687	−1.73790	−0.868952	−	0.494896i	$-0.835206\pi$
$617$	−43.1687	−1.73790	−0.868952	+	0.494896i	$0.835206\pi$
$618$	0	0
$619$	12.7180i	0.511180i	0.966785	+	0.255590i	$0.0822697\pi$
$619$	12.7180i	0.511180i	−0.966785	+	0.255590i	$0.917730\pi$
$620$	0	0
$621$	1.08448i	0.0435185i
$622$	0	0
$623$	8.41460	0.337124
$624$	0	0
$625$	17.6577	0.706307
$626$	0	0
$627$	9.55294i	0.381508i
$628$	0	0
$629$	22.9000i	0.913082i
$630$	0	0
$631$	33.4744	1.33260	0.666298	−	0.745685i	$-0.267878\pi$
$631$	33.4744	1.33260	0.666298	+	0.745685i	$0.267878\pi$
$632$	0	0
$633$	22.9357	0.911614
$634$	0	0
$635$	2.79323i	0.110846i
$636$	0	0
$637$	6.85246i	0.271504i
$638$	0	0
$639$	8.87904	0.351249
$640$	0	0
$641$	−10.3148	−0.407408	−0.203704	−	0.979033i	$-0.565298\pi$
$641$	−10.3148	−0.407408	−0.203704	+	0.979033i	$0.565298\pi$
$642$	0	0
$643$	20.7404i	0.817920i	0.912552	+	0.408960i	$0.134108\pi$
$643$	20.7404i	0.817920i	−0.912552	+	0.408960i	$0.865892\pi$
$644$	0	0
$645$	13.6478i	0.537382i
$646$	0	0
$647$	−15.0656	−0.592291	−0.296146	−	0.955143i	$-0.595701\pi$
$647$	−15.0656	−0.592291	−0.296146	+	0.955143i	$0.595701\pi$
$648$	0	0
$649$	12.8555	0.504621
$650$	0	0
$651$	− 20.2344i	− 0.793049i
$652$	0	0
$653$	21.9568i	0.859236i	0.903011	+	0.429618i	$0.141352\pi$
$653$	21.9568i	0.859236i	−0.903011	+	0.429618i	$0.858648\pi$
$654$	0	0
$655$	−9.57444	−0.374104
$656$	0	0
$657$	18.1880	0.709582
$658$	0	0
$659$	− 29.8657i	− 1.16340i	−0.813403	−	0.581700i	$-0.802388\pi$
$659$	− 29.8657i	− 1.16340i	0.813403	−	0.581700i	$-0.197612\pi$
$660$	0	0
$661$	14.5152i	0.564575i	0.959330	+	0.282287i	$0.0910932\pi$
$661$	14.5152i	0.564575i	−0.959330	+	0.282287i	$0.908907\pi$
$662$	0	0
$663$	−58.7310	−2.28092
$664$	0	0
$665$	−0.963918	−0.0373791
$666$	0	0
$667$	9.88371i	0.382699i
$668$	0	0
$669$	− 6.38706i	− 0.246938i
$670$	0	0
$671$	−14.7157	−0.568095
$672$	0	0
$673$	16.1998	0.624456	0.312228	−	0.950007i	$-0.398925\pi$
$673$	16.1998	0.624456	0.312228	+	0.950007i	$0.398925\pi$
$674$	0	0
$675$	4.52295i	0.174088i
$676$	0	0
$677$	− 10.7247i	− 0.412185i	−0.978533	−	0.206092i	$-0.933925\pi$
$677$	− 10.7247i	− 0.412185i	0.978533	−	0.206092i	$-0.0660748\pi$
$678$	0	0
$679$	−15.3981	−0.590925
$680$	0	0
$681$	38.8870	1.49015
$682$	0	0
$683$	− 22.7354i	− 0.869946i	−0.900443	−	0.434973i	$-0.856758\pi$
$683$	− 22.7354i	− 0.869946i	0.900443	−	0.434973i	$-0.143242\pi$
$684$	0	0
$685$	13.2557i	0.506475i
$686$	0	0
$687$	52.1175	1.98841
$688$	0	0
$689$	16.2841	0.620373
$690$	0	0
$691$	− 23.6241i	− 0.898705i	−0.893355	−	0.449352i	$-0.851655\pi$
$691$	− 23.6241i	− 0.898705i	0.893355	−	0.449352i	$-0.148345\pi$
$692$	0	0
$693$	7.68966i	0.292106i
$694$	0	0
$695$	5.08878	0.193028
$696$	0	0
$697$	16.5459	0.626722
$698$	0	0
$699$	43.1264i	1.63119i
$700$	0	0
$701$	− 1.12363i	− 0.0424387i	−0.999775	−	0.0212194i	$-0.993245\pi$
$701$	− 1.12363i	− 0.0424387i	0.999775	−	0.0212194i	$-0.00675484\pi$
$702$	0	0
$703$	8.54264	0.322192
$704$	0	0
$705$	−0.587960	−0.0221439
$706$	0	0
$707$	11.6052i	0.436459i
$708$	0	0
$709$	16.1324i	0.605865i	0.953012	+	0.302933i	$0.0979657\pi$
$709$	16.1324i	0.605865i	−0.953012	+	0.302933i	$0.902034\pi$
$710$	0	0
$711$	35.3058	1.32407
$712$	0	0
$713$	9.23409	0.345819
$714$	0	0
$715$	− 14.5726i	− 0.544985i
$716$	0	0
$717$	− 31.7396i	− 1.18534i
$718$	0	0
$719$	46.6686	1.74044	0.870222	−	0.492660i	$-0.163975\pi$
$719$	46.6686	1.74044	0.870222	+	0.492660i	$0.163975\pi$
$720$	0	0
$721$	14.0581	0.523552
$722$	0	0
$723$	19.7106i	0.733046i
$724$	0	0
$725$	41.2214i	1.53092i
$726$	0	0
$727$	10.4515	0.387625	0.193812	−	0.981039i	$-0.437915\pi$
$727$	10.4515	0.387625	0.193812	+	0.981039i	$0.437915\pi$
$728$	0	0
$729$	18.8577	0.698432
$730$	0	0
$731$	− 29.4858i	− 1.09057i
$732$	0	0
$733$	− 15.5723i	− 0.575176i	−0.957754	−	0.287588i	$-0.907147\pi$
$733$	− 15.5723i	− 0.575176i	0.957754	−	0.287588i	$-0.0928534\pi$
$734$	0	0
$735$	−1.68036	−0.0619808
$736$	0	0
$737$	6.47223	0.238408
$738$	0	0
$739$	44.0571i	1.62067i	0.585969	+	0.810334i	$0.300714\pi$
$739$	44.0571i	1.62067i	−0.585969	+	0.810334i	$0.699286\pi$
$740$	0	0
$741$	21.9091i	0.804850i
$742$	0	0
$743$	−27.0389	−0.991960	−0.495980	−	0.868334i	$-0.665191\pi$
$743$	−27.0389	−0.991960	−0.495980	+	0.868334i	$0.665191\pi$
$744$	0	0
$745$	0.348400	0.0127644
$746$	0	0
$747$	38.0403i	1.39182i
$748$	0	0
$749$	− 2.39821i	− 0.0876288i
$750$	0	0
$751$	−0.0575811	−0.00210116	−0.00105058	−	0.999999i	$-0.500334\pi$
$751$	−0.0575811	−0.00210116	−0.00105058	+	0.999999i	$0.500334\pi$
$752$	0	0
$753$	12.6006	0.459192
$754$	0	0
$755$	15.9948i	0.582109i
$756$	0	0
$757$	18.1685i	0.660345i	0.943921	+	0.330172i	$0.107107\pi$
$757$	18.1685i	0.660345i	−0.943921	+	0.330172i	$0.892893\pi$
$758$	0	0
$759$	−7.59979	−0.275855
$760$	0	0
$761$	−21.8713	−0.792835	−0.396418	−	0.918070i	$-0.629747\pi$
$761$	−21.8713	−0.792835	−0.396418	+	0.918070i	$0.629747\pi$
$762$	0	0
$763$	1.28723i	0.0466008i
$764$	0	0
$765$	− 6.65014i	− 0.240436i
$766$	0	0
$767$	29.4832	1.06458
$768$	0	0
$769$	−41.3141	−1.48982	−0.744912	−	0.667163i	$-0.767508\pi$
$769$	−41.3141	−1.48982	−0.744912	+	0.667163i	$0.767508\pi$
$770$	0	0
$771$	− 60.0475i	− 2.16256i
$772$	0	0
$773$	30.2428i	1.08776i	0.839164	+	0.543879i	$0.183045\pi$
$773$	30.2428i	1.08776i	−0.839164	+	0.543879i	$0.816955\pi$
$774$	0	0
$775$	38.5120	1.38339
$776$	0	0
$777$	14.8920	0.534248
$778$	0	0
$779$	− 6.17232i	− 0.221146i
$780$	0	0
$781$	− 10.3081i	− 0.368852i
$782$	0	0
$783$	−9.23409	−0.329999
$784$	0	0
$785$	−11.5645	−0.412755
$786$	0	0
$787$	− 16.3754i	− 0.583721i	−0.956461	−	0.291861i	$-0.905726\pi$
$787$	− 16.3754i	− 0.583721i	0.956461	−	0.291861i	$-0.0942744\pi$
$788$	0	0
$789$	− 31.2075i	− 1.11102i
$790$	0	0
$791$	4.85411	0.172592
$792$	0	0
$793$	−33.7497	−1.19849
$794$	0	0
$795$	3.99316i	0.141623i
$796$	0	0
$797$	− 17.9392i	− 0.635440i	−0.948185	−	0.317720i	$-0.897083\pi$
$797$	− 17.9392i	− 0.635440i	0.948185	−	0.317720i	$-0.102917\pi$
$798$	0	0
$799$	1.27028	0.0449391
$800$	0	0
$801$	21.6561	0.765182
$802$	0	0
$803$	− 21.1153i	− 0.745142i
$804$	0	0
$805$	− 0.766840i	− 0.0270275i
$806$	0	0
$807$	41.4956	1.46072
$808$	0	0
$809$	−33.9919	−1.19509	−0.597545	−	0.801835i	$-0.703857\pi$
$809$	−33.9919	−1.19509	−0.597545	+	0.801835i	$0.703857\pi$
$810$	0	0
$811$	− 17.5862i	− 0.617534i	−0.951138	−	0.308767i	$-0.900084\pi$
$811$	− 17.5862i	− 0.617534i	0.951138	−	0.308767i	$-0.0999163\pi$
$812$	0	0
$813$	− 48.5080i	− 1.70125i
$814$	0	0
$815$	−3.44677	−0.120735
$816$	0	0
$817$	−10.9994	−0.384821
$818$	0	0
$819$	17.6358i	0.616243i
$820$	0	0
$821$	36.7270i	1.28178i	0.767632	+	0.640891i	$0.221435\pi$
$821$	36.7270i	1.28178i	−0.767632	+	0.640891i	$0.778565\pi$
$822$	0	0
$823$	37.5706	1.30963	0.654814	−	0.755790i	$-0.272747\pi$
$823$	37.5706	1.30963	0.654814	+	0.755790i	$0.272747\pi$
$824$	0	0
$825$	−31.6960	−1.10351
$826$	0	0
$827$	32.4315i	1.12775i	0.825859	+	0.563877i	$0.190691\pi$
$827$	32.4315i	1.12775i	−0.825859	+	0.563877i	$0.809309\pi$
$828$	0	0
$829$	− 20.3050i	− 0.705222i	−0.935770	−	0.352611i	$-0.885294\pi$
$829$	− 20.3050i	− 0.705222i	0.935770	−	0.352611i	$-0.114706\pi$
$830$	0	0
$831$	52.1075	1.80759
$832$	0	0
$833$	3.63038	0.125785
$834$	0	0
$835$	6.75563i	0.233788i
$836$	0	0
$837$	8.62716i	0.298198i
$838$	0	0
$839$	−24.6124	−0.849713	−0.424857	−	0.905261i	$-0.639675\pi$
$839$	−24.6124	−0.849713	−0.424857	+	0.905261i	$0.639675\pi$
$840$	0	0
$841$	−55.1578	−1.90199
$842$	0	0
$843$	40.1980i	1.38449i
$844$	0	0
$845$	− 24.1686i	− 0.831424i
$846$	0	0
$847$	−2.07274	−0.0712200
$848$	0	0
$849$	35.5974	1.22170
$850$	0	0
$851$	6.79605i	0.232966i
$852$	0	0
$853$	32.0447i	1.09719i	0.836088	+	0.548595i	$0.184837\pi$
$853$	32.0447i	1.09719i	−0.836088	+	0.548595i	$0.815163\pi$
$854$	0	0
$855$	−2.48078	−0.0848408
$856$	0	0
$857$	29.0599	0.992668	0.496334	−	0.868132i	$-0.334679\pi$
$857$	29.0599	0.992668	0.496334	+	0.868132i	$0.334679\pi$
$858$	0	0
$859$	− 0.261668i	− 0.00892799i	−0.999990	−	0.00446399i	$-0.998579\pi$
$859$	− 0.261668i	− 0.00892799i	0.999990	−	0.00446399i	$-0.00142094\pi$
$860$	0	0
$861$	− 10.7599i	− 0.366697i
$862$	0	0
$863$	−17.9690	−0.611673	−0.305836	−	0.952084i	$-0.598936\pi$
$863$	−17.9690	−0.611673	−0.305836	+	0.952084i	$0.598936\pi$
$864$	0	0
$865$	5.62140	0.191133
$866$	0	0
$867$	− 9.01935i	− 0.306313i
$868$	0	0
$869$	− 40.9881i	− 1.39043i
$870$	0	0
$871$	14.8437	0.502959
$872$	0	0
$873$	−39.6291	−1.34124
$874$	0	0
$875$	− 6.75699i	− 0.228428i
$876$	0	0
$877$	− 4.35104i	− 0.146924i	−0.997298	−	0.0734621i	$-0.976595\pi$
$877$	− 4.35104i	− 0.146924i	0.997298	−	0.0734621i	$-0.0234048\pi$
$878$	0	0
$879$	17.1909	0.579833
$880$	0	0
$881$	11.2303	0.378357	0.189178	−	0.981943i	$-0.439418\pi$
$881$	11.2303	0.378357	0.189178	+	0.981943i	$0.439418\pi$
$882$	0	0
$883$	39.7974i	1.33929i	0.742681	+	0.669645i	$0.233554\pi$
$883$	39.7974i	1.33929i	−0.742681	+	0.669645i	$0.766446\pi$
$884$	0	0
$885$	7.22985i	0.243029i
$886$	0	0
$887$	−11.8390	−0.397515	−0.198757	−	0.980049i	$-0.563691\pi$
$887$	−11.8390	−0.397515	−0.198757	+	0.980049i	$0.563691\pi$
$888$	0	0
$889$	−3.92442	−0.131621
$890$	0	0
$891$	− 30.1692i	− 1.01071i
$892$	0	0
$893$	− 0.473865i	− 0.0158573i
$894$	0	0
$895$	−16.0726	−0.537249
$896$	0	0
$897$	−17.4297	−0.581959
$898$	0	0
$899$	78.6264i	2.62234i
$900$	0	0
$901$	− 8.62716i	− 0.287412i
$902$	0	0
$903$	−19.1748	−0.638098
$904$	0	0
$905$	9.75410	0.324237
$906$	0	0
$907$	− 2.14412i	− 0.0711943i	−0.999366	−	0.0355972i	$-0.988667\pi$
$907$	− 2.14412i	− 0.0711943i	0.999366	−	0.0355972i	$-0.0113333\pi$
$908$	0	0
$909$	29.8677i	0.990648i
$910$	0	0
$911$	−54.1211	−1.79311	−0.896557	−	0.442929i	$-0.853939\pi$
$911$	−54.1211	−1.79311	−0.896557	+	0.442929i	$0.853939\pi$
$912$	0	0
$913$	44.1626	1.46157
$914$	0	0
$915$	− 8.27607i	− 0.273598i
$916$	0	0
$917$	− 13.4518i	− 0.444219i
$918$	0	0
$919$	47.0290	1.55134	0.775672	−	0.631136i	$-0.217411\pi$
$919$	47.0290	1.55134	0.775672	+	0.631136i	$0.217411\pi$
$920$	0	0
$921$	20.8760	0.687886
$922$	0	0
$923$	− 23.6409i	− 0.778151i
$924$	0	0
$925$	28.3439i	0.931940i
$926$	0	0
$927$	36.1805	1.18833
$928$	0	0
$929$	−53.9999	−1.77168	−0.885839	−	0.463993i	$-0.846416\pi$
$929$	−53.9999	−1.77168	−0.885839	+	0.463993i	$0.846416\pi$
$930$	0	0
$931$	− 1.35428i	− 0.0443847i
$932$	0	0
$933$	75.3290i	2.46616i
$934$	0	0
$935$	−7.72045	−0.252486
$936$	0	0
$937$	10.4089	0.340044	0.170022	−	0.985440i	$-0.445616\pi$
$937$	10.4089	0.340044	0.170022	+	0.985440i	$0.445616\pi$
$938$	0	0
$939$	− 21.3982i	− 0.698305i
$940$	0	0
$941$	− 32.6610i	− 1.06472i	−0.846518	−	0.532359i	$-0.821306\pi$
$941$	− 32.6610i	− 1.06472i	0.846518	−	0.532359i	$-0.178694\pi$
$942$	0	0
$943$	4.91036	0.159903
$944$	0	0
$945$	0.716437	0.0233057
$946$	0	0
$947$	25.1457i	0.817126i	0.912730	+	0.408563i	$0.133970\pi$
$947$	25.1457i	0.817126i	−0.912730	+	0.408563i	$0.866030\pi$
$948$	0	0
$949$	− 48.4266i	− 1.57199i
$950$	0	0
$951$	69.8792	2.26599
$952$	0	0
$953$	43.5674	1.41129	0.705643	−	0.708567i	$-0.250658\pi$
$953$	43.5674	1.41129	0.705643	+	0.708567i	$0.250658\pi$
$954$	0	0
$955$	− 13.6135i	− 0.440522i
$956$	0	0
$957$	− 64.7107i	− 2.09180i
$958$	0	0
$959$	−18.6239	−0.601398
$960$	0	0
$961$	42.4585	1.36963
$962$	0	0
$963$	− 6.17214i	− 0.198894i
$964$	0	0
$965$	− 5.94510i	− 0.191379i
$966$	0	0
$967$	−8.97147	−0.288503	−0.144252	−	0.989541i	$-0.546077\pi$
$967$	−8.97147	−0.288503	−0.144252	+	0.989541i	$0.546077\pi$
$968$	0	0
$969$	11.6072	0.372879
$970$	0	0
$971$	53.5414i	1.71822i	0.511787	+	0.859112i	$0.328984\pi$
$971$	53.5414i	1.71822i	−0.511787	+	0.859112i	$0.671016\pi$
$972$	0	0
$973$	7.14960i	0.229206i
$974$	0	0
$975$	−72.6928	−2.32803
$976$	0	0
$977$	37.3412	1.19465	0.597326	−	0.801998i	$-0.296230\pi$
$977$	37.3412	1.19465	0.597326	+	0.801998i	$0.296230\pi$
$978$	0	0
$979$	− 25.1416i	− 0.803529i
$980$	0	0
$981$	3.31286i	0.105772i
$982$	0	0
$983$	36.2225	1.15532	0.577659	−	0.816278i	$-0.303966\pi$
$983$	36.2225	1.15532	0.577659	+	0.816278i	$0.303966\pi$
$984$	0	0
$985$	−0.983671	−0.0313424
$986$	0	0
$987$	− 0.826069i	− 0.0262941i
$988$	0	0
$989$	− 8.75054i	− 0.278251i
$990$	0	0
$991$	−11.4960	−0.365182	−0.182591	−	0.983189i	$-0.558448\pi$
$991$	−11.4960	−0.365182	−0.182591	+	0.983189i	$0.558448\pi$
$992$	0	0
$993$	−36.5035	−1.15840
$994$	0	0
$995$	1.76268i	0.0558809i
$996$	0	0
$997$	− 20.5111i	− 0.649594i	−0.945784	−	0.324797i	$-0.894704\pi$
$997$	− 20.5111i	− 0.649594i	0.945784	−	0.324797i	$-0.105296\pi$
$998$	0	0
$999$	−6.34937	−0.200885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3584.2.b.j.1793.11		12
4.3	odd	2		3584.2.b.l.1793.2		12
8.3	odd	2		3584.2.b.l.1793.11		12
8.5	even	2	inner	3584.2.b.j.1793.2		12
16.3	odd	4		3584.2.a.g.1.6	✓	6
16.5	even	4		3584.2.a.j.1.6	yes	6
16.11	odd	4		3584.2.a.i.1.1	yes	6
16.13	even	4		3584.2.a.h.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3584.2.a.g.1.6	✓	6	16.3	odd	4
3584.2.a.h.1.1	yes	6	16.13	even	4
3584.2.a.i.1.1	yes	6	16.11	odd	4
3584.2.a.j.1.6	yes	6	16.5	even	4
3584.2.b.j.1793.2		12	8.5	even	2	inner
3584.2.b.j.1793.11		12	1.1	even	1	trivial
3584.2.b.l.1793.2		12	4.3	odd	2
3584.2.b.l.1793.11		12	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3584 = 2^{9} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.36086i q^{3} +0.711757i q^{5} -1.00000 q^{7} -2.57364 q^{9} +O(q^{10})\)	\(q+2.36086i q^{3} +0.711757i q^{5} -1.00000 q^{7} -2.57364 q^{9} +2.98785i q^{11} +6.85246i q^{13} -1.68036 q^{15} +3.63038 q^{17} -1.35428i q^{19} -2.36086i q^{21} +1.07739 q^{23} +4.49340 q^{25} +1.00658i q^{27} +9.17376i q^{29} +8.57079 q^{31} -7.05389 q^{33} -0.711757i q^{35} +6.30788i q^{37} -16.1777 q^{39} +4.55764 q^{41} -8.12198i q^{43} -1.83181i q^{45} +0.349902 q^{47} +1.00000 q^{49} +8.57079i q^{51} -2.37638i q^{53} -2.12662 q^{55} +3.19726 q^{57} -4.30257i q^{59} +4.92519i q^{61} +2.57364 q^{63} -4.87728 q^{65} -2.16618i q^{67} +2.54356i q^{69} -3.44999 q^{71} -7.06704 q^{73} +10.6083i q^{75} -2.98785i q^{77} -13.7182 q^{79} -10.0973 q^{81} -14.7807i q^{83} +2.58394i q^{85} -21.6579 q^{87} -8.41460 q^{89} -6.85246i q^{91} +20.2344i q^{93} +0.963918 q^{95} +15.3981 q^{97} -7.68966i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{7} - 4 q^{9}+O(q^{10}) \) 12 * q - 12 * q^7 - 4 * q^9	\( 12 q - 12 q^{7} - 4 q^{9} - 16 q^{15} - 20 q^{25} + 16 q^{31} - 8 q^{33} - 8 q^{41} + 16 q^{47} + 12 q^{49} + 32 q^{55} + 24 q^{57} + 4 q^{63} + 16 q^{71} + 16 q^{73} - 48 q^{79} - 28 q^{81} - 16 q^{87} - 16 q^{89} + 32 q^{95}+O(q^{100}) \) 12 * q - 12 * q^7 - 4 * q^9 - 16 * q^15 - 20 * q^25 + 16 * q^31 - 8 * q^33 - 8 * q^41 + 16 * q^47 + 12 * q^49 + 32 * q^55 + 24 * q^57 + 4 * q^63 + 16 * q^71 + 16 * q^73 - 48 * q^79 - 28 * q^81 - 16 * q^87 - 16 * q^89 + 32 * q^95

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