LMFDB - Embedded newform 6048.2.h.a.2591.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(2591,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("6048.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.5
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	6048.2591
Dual form			6048.2.h.a.2591.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.19980i q^{5} -1.00000i q^{7} +O(q^{10})$	$q+1.19980i q^{5} -1.00000i q^{7} +1.62863 q^{11} -2.09638 q^{13} -0.585786i q^{17} -5.89898i q^{19} -7.09273 q^{23} +3.56048 q^{25} +7.08516i q^{29} +6.72741i q^{31} +1.19980 q^{35} +2.17157 q^{37} +0.164525i q^{41} +5.75323i q^{43} +3.41421 q^{47} -1.00000 q^{49} +3.29858i q^{53} +1.95403i q^{55} -2.08641 q^{59} +4.92820 q^{61} -2.51523i q^{65} -12.4887i q^{67} +11.2497 q^{71} +3.97418 q^{73} -1.62863i q^{77} -0.702827i q^{79} +16.8831 q^{83} +0.702827 q^{85} +6.33485i q^{89} +2.09638i q^{91} +7.07760 q^{95} -6.58506 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 8 q^{11} - 8 q^{13} - 8 q^{23} - 8 q^{25} + 8 q^{35} + 40 q^{37} + 16 q^{47} - 8 q^{49} - 64 q^{59} - 16 q^{61} + 72 q^{71} + 24 q^{73} + 32 q^{83} + 8 q^{85} + 56 q^{95} + 48 q^{97}+O(q^{100}) $ 8 * q - 8 * q^11 - 8 * q^13 - 8 * q^23 - 8 * q^25 + 8 * q^35 + 40 * q^37 + 16 * q^47 - 8 * q^49 - 64 * q^59 - 16 * q^61 + 72 * q^71 + 24 * q^73 + 32 * q^83 + 8 * q^85 + 56 * q^95 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.19980i	0.536567i	0.963340	+	0.268284i	$0.0864564\pi$
$5$	1.19980i	0.536567i	−0.963340	+	0.268284i	$0.913544\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.62863	0.491049	0.245525	−	0.969390i	$-0.421040\pi$
$11$	1.62863	0.491049	0.245525	+	0.969390i	$0.421040\pi$
$12$	0	0
$13$	−2.09638	−0.581430	−0.290715	−	0.956810i	$-0.593893\pi$
$13$	−2.09638	−0.581430	−0.290715	+	0.956810i	$0.593893\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.585786i	− 0.142074i	−0.997474	−	0.0710370i	$-0.977369\pi$
$17$	− 0.585786i	− 0.142074i	0.997474	−	0.0710370i	$-0.0226309\pi$
$18$	0	0
$19$	− 5.89898i	− 1.35332i	−0.736296	−	0.676659i	$-0.763427\pi$
$19$	− 5.89898i	− 1.35332i	0.736296	−	0.676659i	$-0.236573\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.09273	−1.47894	−0.739468	−	0.673192i	$-0.764923\pi$
$23$	−7.09273	−1.47894	−0.739468	+	0.673192i	$0.764923\pi$
$24$	0	0
$25$	3.56048	0.712096
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.08516i	1.31568i	0.753157	+	0.657841i	$0.228530\pi$
$29$	7.08516i	1.31568i	−0.753157	+	0.657841i	$0.771470\pi$
$30$	0	0
$31$	6.72741i	1.20828i	0.796879	+	0.604139i	$0.206483\pi$
$31$	6.72741i	1.20828i	−0.796879	+	0.604139i	$0.793517\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.19980	0.202803
$36$	0	0
$37$	2.17157	0.357004	0.178502	−	0.983940i	$-0.442875\pi$
$37$	2.17157	0.357004	0.178502	+	0.983940i	$0.442875\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.164525i	0.0256944i	0.999917	+	0.0128472i	$0.00408951\pi$
$41$	0.164525i	0.0256944i	−0.999917	+	0.0128472i	$0.995910\pi$
$42$	0	0
$43$	5.75323i	0.877359i	0.898643	+	0.438680i	$0.144554\pi$
$43$	5.75323i	0.877359i	−0.898643	+	0.438680i	$0.855446\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.41421	0.498014	0.249007	−	0.968502i	$-0.419896\pi$
$47$	3.41421	0.498014	0.249007	+	0.968502i	$0.419896\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.29858i	0.453095i	0.974000	+	0.226547i	$0.0727438\pi$
$53$	3.29858i	0.453095i	−0.974000	+	0.226547i	$0.927256\pi$
$54$	0	0
$55$	1.95403i	0.263481i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.08641	−0.271627	−0.135814	−	0.990734i	$-0.543365\pi$
$59$	−2.08641	−0.271627	−0.135814	+	0.990734i	$0.543365\pi$
$60$	0	0
$61$	4.92820	0.630992	0.315496	−	0.948927i	$-0.397829\pi$
$61$	4.92820	0.630992	0.315496	+	0.948927i	$0.397829\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.51523i	− 0.311976i
$66$	0	0
$67$	− 12.4887i	− 1.52574i	−0.646555	−	0.762868i	$-0.723791\pi$
$67$	− 12.4887i	− 1.52574i	0.646555	−	0.762868i	$-0.276209\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	11.2497	1.33509	0.667546	−	0.744568i	$-0.267345\pi$
$71$	11.2497	1.33509	0.667546	+	0.744568i	$0.267345\pi$
$72$	0	0
$73$	3.97418	0.465142	0.232571	−	0.972579i	$-0.425286\pi$
$73$	3.97418	0.465142	0.232571	+	0.972579i	$0.425286\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.62863i	− 0.185599i
$78$	0	0
$79$	− 0.702827i	− 0.0790742i	−0.999218	−	0.0395371i	$-0.987412\pi$
$79$	− 0.702827i	− 0.0790742i	0.999218	−	0.0395371i	$-0.0125883\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.8831	1.85316	0.926582	−	0.376093i	$-0.122733\pi$
$83$	16.8831	1.85316	0.926582	+	0.376093i	$0.122733\pi$
$84$	0	0
$85$	0.702827	0.0762323
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.33485i	0.671493i	0.941952	+	0.335747i	$0.108989\pi$
$89$	6.33485i	0.671493i	−0.941952	+	0.335747i	$0.891011\pi$
$90$	0	0
$91$	2.09638i	0.219760i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	7.07760	0.726146
$96$	0	0
$97$	−6.58506	−0.668611	−0.334306	−	0.942465i	$-0.608502\pi$
$97$	−6.58506	−0.668611	−0.334306	+	0.942465i	$0.608502\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.34366i	− 0.929729i	−0.885382	−	0.464865i	$-0.846103\pi$
$101$	− 9.34366i	− 0.929729i	0.885382	−	0.464865i	$-0.153897\pi$
$102$	0	0
$103$	− 0.586302i	− 0.0577700i	−0.999583	−	0.0288850i	$-0.990804\pi$
$103$	− 0.586302i	− 0.0577700i	0.999583	−	0.0288850i	$-0.00919567\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.14162	0.303712	0.151856	−	0.988403i	$-0.451475\pi$
$107$	3.14162	0.303712	0.151856	+	0.988403i	$0.451475\pi$
$108$	0	0
$109$	11.6522	1.11608	0.558040	−	0.829814i	$-0.311554\pi$
$109$	11.6522	1.11608	0.558040	+	0.829814i	$0.311554\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.5546i	1.83954i	0.392457	+	0.919771i	$0.371625\pi$
$113$	19.5546i	1.83954i	−0.392457	+	0.919771i	$0.628375\pi$
$114$	0	0
$115$	− 8.50986i	− 0.793549i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.585786	−0.0536990
$120$	0	0
$121$	−8.34758	−0.758871
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.2709i	0.918654i
$126$	0	0
$127$	− 4.16693i	− 0.369755i	−0.982762	−	0.184878i	$-0.940811\pi$
$127$	− 4.16693i	− 0.369755i	0.982762	−	0.184878i	$-0.0591889\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.6410	1.19182	0.595910	−	0.803051i	$-0.296792\pi$
$131$	13.6410	1.19182	0.595910	+	0.803051i	$0.296792\pi$
$132$	0	0
$133$	−5.89898	−0.511506
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.15696i	0.696896i	0.937328	+	0.348448i	$0.113291\pi$
$137$	8.15696i	0.696896i	−0.937328	+	0.348448i	$0.886709\pi$
$138$	0	0
$139$	− 9.46410i	− 0.802735i	−0.915917	−	0.401367i	$-0.868535\pi$
$139$	− 9.46410i	− 0.802735i	0.915917	−	0.401367i	$-0.131465\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.41421	−0.285511
$144$	0	0
$145$	−8.50079	−0.705952
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.6848i	1.69457i	0.531140	+	0.847284i	$0.321764\pi$
$149$	20.6848i	1.69457i	−0.531140	+	0.847284i	$0.678236\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.07155	−0.648322
$156$	0	0
$157$	0.246769	0.0196943	0.00984717	−	0.999952i	$-0.496865\pi$
$157$	0.246769	0.0196943	0.00984717	+	0.999952i	$0.496865\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	7.09273i	0.558985i
$162$	0	0
$163$	7.89434i	0.618332i	0.951008	+	0.309166i	$0.100050\pi$
$163$	7.89434i	0.618332i	−0.951008	+	0.309166i	$0.899950\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.74150	0.134762	0.0673808	−	0.997727i	$-0.478536\pi$
$167$	1.74150	0.134762	0.0673808	+	0.997727i	$0.478536\pi$
$168$	0	0
$169$	−8.60521	−0.661939
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 10.2709i	− 0.780880i	−0.920628	−	0.390440i	$-0.872323\pi$
$173$	− 10.2709i	− 0.780880i	0.920628	−	0.390440i	$-0.127677\pi$
$174$	0	0
$175$	− 3.56048i	− 0.269147i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2.81257	0.210221	0.105111	−	0.994461i	$-0.466480\pi$
$179$	2.81257	0.210221	0.105111	+	0.994461i	$0.466480\pi$
$180$	0	0
$181$	−8.14110	−0.605124	−0.302562	−	0.953130i	$-0.597842\pi$
$181$	−8.14110	−0.605124	−0.302562	+	0.953130i	$0.597842\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.60545i	0.191557i
$186$	0	0
$187$	− 0.954027i	− 0.0697654i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.0482	1.16120	0.580602	−	0.814187i	$-0.302817\pi$
$191$	16.0482	1.16120	0.580602	+	0.814187i	$0.302817\pi$
$192$	0	0
$193$	−2.87904	−0.207238	−0.103619	−	0.994617i	$-0.533042\pi$
$193$	−2.87904	−0.207238	−0.103619	+	0.994617i	$0.533042\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.09778i	0.576943i	0.957488	+	0.288472i	$0.0931471\pi$
$197$	8.09778i	0.576943i	−0.957488	+	0.288472i	$0.906853\pi$
$198$	0	0
$199$	21.9024i	1.55262i	0.630352	+	0.776309i	$0.282910\pi$
$199$	21.9024i	1.55262i	−0.630352	+	0.776309i	$0.717090\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.08516	0.497281
$204$	0	0
$205$	−0.197397	−0.0137868
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 9.60723i	− 0.664546i
$210$	0	0
$211$	− 3.80725i	− 0.262102i	−0.991376	−	0.131051i	$-0.958165\pi$
$211$	− 3.80725i	− 0.262102i	0.991376	−	0.131051i	$-0.0418351\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.90273	−0.470762
$216$	0	0
$217$	6.72741	0.456686
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.22803i	0.0826062i
$222$	0	0
$223$	27.4088i	1.83543i	0.397237	+	0.917716i	$0.369969\pi$
$223$	27.4088i	1.83543i	−0.397237	+	0.917716i	$0.630031\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.79920	0.318534	0.159267	−	0.987236i	$-0.449087\pi$
$227$	4.79920	0.318534	0.159267	+	0.987236i	$0.449087\pi$
$228$	0	0
$229$	11.9907	0.792369	0.396184	−	0.918171i	$-0.370334\pi$
$229$	11.9907	0.792369	0.396184	+	0.918171i	$0.370334\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.898979i	0.0588941i	0.999566	+	0.0294471i	$0.00937464\pi$
$233$	0.898979i	0.0588941i	−0.999566	+	0.0294471i	$0.990625\pi$
$234$	0	0
$235$	4.09638i	0.267218i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.6702	−1.01362	−0.506811	−	0.862057i	$-0.669176\pi$
$239$	−15.6702	−1.01362	−0.506811	+	0.862057i	$0.669176\pi$
$240$	0	0
$241$	27.1443	1.74852	0.874259	−	0.485460i	$-0.161348\pi$
$241$	27.1443	1.74852	0.874259	+	0.485460i	$0.161348\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.19980i	− 0.0766525i
$246$	0	0
$247$	12.3665i	0.786860i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.8977	1.38217	0.691086	−	0.722772i	$-0.257132\pi$
$251$	21.8977	1.38217	0.691086	+	0.722772i	$0.257132\pi$
$252$	0	0
$253$	−11.5514	−0.726230
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.89245i	0.180426i	0.995923	+	0.0902129i	$0.0287548\pi$
$257$	2.89245i	0.180426i	−0.995923	+	0.0902129i	$0.971245\pi$
$258$	0	0
$259$	− 2.17157i	− 0.134935i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.5901	1.57795	0.788977	−	0.614423i	$-0.210611\pi$
$263$	25.5901	1.57795	0.788977	+	0.614423i	$0.210611\pi$
$264$	0	0
$265$	−3.95764	−0.243116
$266$	0	0
$267$	0	0
$268$	0	0
$269$	25.6692i	1.56508i	0.622600	+	0.782540i	$0.286076\pi$
$269$	25.6692i	1.56508i	−0.622600	+	0.782540i	$0.713924\pi$
$270$	0	0
$271$	6.05845i	0.368024i	0.982924	+	0.184012i	$0.0589086\pi$
$271$	6.05845i	0.368024i	−0.982924	+	0.184012i	$0.941091\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.79869	0.349674
$276$	0	0
$277$	−31.8166	−1.91168	−0.955839	−	0.293892i	$-0.905049\pi$
$277$	−31.8166	−1.91168	−0.955839	+	0.293892i	$0.905049\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.80295i	− 0.167210i	−0.996499	−	0.0836051i	$-0.973357\pi$
$281$	− 2.80295i	− 0.167210i	0.996499	−	0.0836051i	$-0.0266434\pi$
$282$	0	0
$283$	− 10.6367i	− 0.632287i	−0.948711	−	0.316143i	$-0.897612\pi$
$283$	− 10.6367i	− 0.632287i	0.948711	−	0.316143i	$-0.102388\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.164525	0.00971158
$288$	0	0
$289$	16.6569	0.979815
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.702148i	− 0.0410199i	−0.999790	−	0.0205100i	$-0.993471\pi$
$293$	− 0.702148i	− 0.0410199i	0.999790	−	0.0205100i	$-0.00652898\pi$
$294$	0	0
$295$	− 2.50327i	− 0.145746i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.8690	0.859898
$300$	0	0
$301$	5.75323	0.331611
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.91286i	0.338570i
$306$	0	0
$307$	− 21.4122i	− 1.22206i	−0.791607	−	0.611031i	$-0.790755\pi$
$307$	− 21.4122i	− 1.22206i	0.791607	−	0.611031i	$-0.209245\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.3424	0.813284	0.406642	−	0.913588i	$-0.366700\pi$
$311$	14.3424	0.813284	0.406642	+	0.913588i	$0.366700\pi$
$312$	0	0
$313$	17.8238	1.00746	0.503730	−	0.863861i	$-0.331961\pi$
$313$	17.8238	1.00746	0.503730	+	0.863861i	$0.331961\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 26.7113i	− 1.50026i	−0.661291	−	0.750129i	$-0.729991\pi$
$317$	− 26.7113i	− 1.50026i	0.661291	−	0.750129i	$-0.270009\pi$
$318$	0	0
$319$	11.5391i	0.646065i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.45554	−0.192272
$324$	0	0
$325$	−7.46410	−0.414034
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.41421i	− 0.188232i
$330$	0	0
$331$	− 3.01086i	− 0.165492i	−0.996571	−	0.0827459i	$-0.973631\pi$
$331$	− 3.01086i	− 0.165492i	0.996571	−	0.0827459i	$-0.0263690\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.9839	0.818660
$336$	0	0
$337$	−14.1210	−0.769217	−0.384609	−	0.923080i	$-0.625664\pi$
$337$	−14.1210	−0.769217	−0.384609	+	0.923080i	$0.625664\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.9564i	0.593324i
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.5835	−0.621837	−0.310919	−	0.950437i	$-0.600637\pi$
$347$	−11.5835	−0.621837	−0.310919	+	0.950437i	$0.600637\pi$
$348$	0	0
$349$	25.7887	1.38044	0.690218	−	0.723602i	$-0.257515\pi$
$349$	25.7887	1.38044	0.690218	+	0.723602i	$0.257515\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	28.2326i	1.50267i	0.659922	+	0.751335i	$0.270590\pi$
$353$	28.2326i	1.50267i	−0.659922	+	0.751335i	$0.729410\pi$
$354$	0	0
$355$	13.4974i	0.716367i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.4976	0.870711	0.435355	−	0.900259i	$-0.356623\pi$
$359$	16.4976	0.870711	0.435355	+	0.900259i	$0.356623\pi$
$360$	0	0
$361$	−15.7980	−0.831472
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.76822i	0.249580i
$366$	0	0
$367$	− 7.06818i	− 0.368956i	−0.982837	−	0.184478i	$-0.940941\pi$
$367$	− 7.06818i	− 0.368956i	0.982837	−	0.184478i	$-0.0590595\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.29858	0.171254
$372$	0	0
$373$	2.88369	0.149312	0.0746559	−	0.997209i	$-0.476214\pi$
$373$	2.88369	0.149312	0.0746559	+	0.997209i	$0.476214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14.8532i	− 0.764977i
$378$	0	0
$379$	− 29.4076i	− 1.51057i	−0.655398	−	0.755283i	$-0.727499\pi$
$379$	− 29.4076i	− 1.51057i	0.655398	−	0.755283i	$-0.272501\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.8977	0.710141	0.355071	−	0.934840i	$-0.384457\pi$
$383$	13.8977	0.710141	0.355071	+	0.934840i	$0.384457\pi$
$384$	0	0
$385$	1.95403	0.0995864
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.4263i	1.13706i	0.822662	+	0.568530i	$0.192488\pi$
$389$	22.4263i	1.13706i	−0.822662	+	0.568530i	$0.807512\pi$
$390$	0	0
$391$	4.15482i	0.210118i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.843253	0.0424286
$396$	0	0
$397$	−14.3028	−0.717839	−0.358920	−	0.933368i	$-0.616855\pi$
$397$	−14.3028	−0.717839	−0.358920	+	0.933368i	$0.616855\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.5964i	1.32816i	0.747660	+	0.664081i	$0.231177\pi$
$401$	26.5964i	1.32816i	−0.747660	+	0.664081i	$0.768823\pi$
$402$	0	0
$403$	− 14.1032i	− 0.702529i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.53668	0.175307
$408$	0	0
$409$	−16.2093	−0.801498	−0.400749	−	0.916188i	$-0.631250\pi$
$409$	−16.2093	−0.801498	−0.400749	+	0.916188i	$0.631250\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.08641i	0.102665i
$414$	0	0
$415$	20.2564i	0.994347i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.5437	−1.19904	−0.599520	−	0.800360i	$-0.704642\pi$
$419$	−24.5437	−1.19904	−0.599520	+	0.800360i	$0.704642\pi$
$420$	0	0
$421$	−0.373614	−0.0182088	−0.00910441	−	0.999959i	$-0.502898\pi$
$421$	−0.373614	−0.0182088	−0.00910441	+	0.999959i	$0.502898\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2.08568i	− 0.101170i
$426$	0	0
$427$	− 4.92820i	− 0.238492i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.7801	−0.856435	−0.428217	−	0.903676i	$-0.640858\pi$
$431$	−17.7801	−0.856435	−0.428217	+	0.903676i	$0.640858\pi$
$432$	0	0
$433$	10.7613	0.517154	0.258577	−	0.965991i	$-0.416746\pi$
$433$	10.7613	0.517154	0.258577	+	0.965991i	$0.416746\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.8399i	2.00147i
$438$	0	0
$439$	18.5758i	0.886573i	0.896380	+	0.443287i	$0.146188\pi$
$439$	18.5758i	0.886573i	−0.896380	+	0.443287i	$0.853812\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.3699	1.44292	0.721459	−	0.692457i	$-0.243472\pi$
$443$	30.3699	1.44292	0.721459	+	0.692457i	$0.243472\pi$
$444$	0	0
$445$	−7.60056	−0.360301
$446$	0	0
$447$	0	0
$448$	0	0
$449$	31.4382i	1.48366i	0.670587	+	0.741831i	$0.266042\pi$
$449$	31.4382i	1.48366i	−0.670587	+	0.741831i	$0.733958\pi$
$450$	0	0
$451$	0.267949i	0.0126172i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.51523	−0.117916
$456$	0	0
$457$	−30.3629	−1.42032	−0.710158	−	0.704043i	$-0.751376\pi$
$457$	−30.3629	−1.42032	−0.710158	+	0.704043i	$0.751376\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2.28745i	− 0.106537i	−0.998580	−	0.0532687i	$-0.983036\pi$
$461$	− 2.28745i	− 0.106537i	0.998580	−	0.0532687i	$-0.0169640\pi$
$462$	0	0
$463$	4.92140i	0.228717i	0.993440	+	0.114359i	$0.0364813\pi$
$463$	4.92140i	0.228717i	−0.993440	+	0.114359i	$0.963519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−17.1860	−0.795272	−0.397636	−	0.917543i	$-0.630169\pi$
$467$	−17.1860	−0.795272	−0.397636	+	0.917543i	$0.630169\pi$
$468$	0	0
$469$	−12.4887	−0.576674
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.36986i	0.430827i
$474$	0	0
$475$	− 21.0032i	− 0.963692i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.585057	0.0267320	0.0133660	−	0.999911i	$-0.495745\pi$
$479$	0.585057	0.0267320	0.0133660	+	0.999911i	$0.495745\pi$
$480$	0	0
$481$	−4.55243	−0.207573
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.90076i	− 0.358755i
$486$	0	0
$487$	15.7318i	0.712878i	0.934319	+	0.356439i	$0.116009\pi$
$487$	15.7318i	0.712878i	−0.934319	+	0.356439i	$0.883991\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−27.7267	−1.25129	−0.625645	−	0.780108i	$-0.715164\pi$
$491$	−27.7267	−1.25129	−0.625645	+	0.780108i	$0.715164\pi$
$492$	0	0
$493$	4.15039	0.186924
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 11.2497i	− 0.504618i
$498$	0	0
$499$	− 1.41688i	− 0.0634285i	−0.999497	−	0.0317142i	$-0.989903\pi$
$499$	− 1.41688i	− 0.0634285i	0.999497	−	0.0317142i	$-0.0100966\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.69870	0.209505	0.104752	−	0.994498i	$-0.466595\pi$
$503$	4.69870	0.209505	0.104752	+	0.994498i	$0.466595\pi$
$504$	0	0
$505$	11.2105	0.498862
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.82645i	− 0.346901i	−0.984843	−	0.173451i	$-0.944508\pi$
$509$	− 7.82645i	− 0.346901i	0.984843	−	0.173451i	$-0.0554917\pi$
$510$	0	0
$511$	− 3.97418i	− 0.175807i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.703446	0.0309975
$516$	0	0
$517$	5.56048	0.244550
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.83496i	0.211823i	0.994376	+	0.105912i	$0.0337761\pi$
$521$	4.83496i	0.211823i	−0.994376	+	0.105912i	$0.966224\pi$
$522$	0	0
$523$	− 6.46750i	− 0.282804i	−0.989952	−	0.141402i	$-0.954839\pi$
$523$	− 6.46750i	− 0.282804i	0.989952	−	0.141402i	$-0.0451610\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.94082	0.171665
$528$	0	0
$529$	27.3068	1.18725
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 0.344906i	− 0.0149395i
$534$	0	0
$535$	3.76932i	0.162962i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.62863	−0.0701499
$540$	0	0
$541$	20.0864	0.863583	0.431791	−	0.901973i	$-0.357882\pi$
$541$	20.0864	0.863583	0.431791	+	0.901973i	$0.357882\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.9803i	0.598852i
$546$	0	0
$547$	− 19.2689i	− 0.823877i	−0.911212	−	0.411938i	$-0.864852\pi$
$547$	− 19.2689i	− 0.823877i	0.911212	−	0.411938i	$-0.135148\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	41.7952	1.78054
$552$	0	0
$553$	−0.702827	−0.0298872
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.90346i	− 0.123024i	−0.998106	−	0.0615118i	$-0.980408\pi$
$557$	− 2.90346i	− 0.123024i	0.998106	−	0.0615118i	$-0.0195922\pi$
$558$	0	0
$559$	− 12.0609i	− 0.510123i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.18743	0.387204	0.193602	−	0.981080i	$-0.437983\pi$
$563$	9.18743	0.387204	0.193602	+	0.981080i	$0.437983\pi$
$564$	0	0
$565$	−23.4616	−0.987037
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.31694i	0.0971314i	0.998820	+	0.0485657i	$0.0154650\pi$
$569$	2.31694i	0.0971314i	−0.998820	+	0.0485657i	$0.984535\pi$
$570$	0	0
$571$	− 18.6770i	− 0.781608i	−0.920474	−	0.390804i	$-0.872197\pi$
$571$	− 18.6770i	− 0.781608i	0.920474	−	0.390804i	$-0.127803\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−25.2535	−1.05314
$576$	0	0
$577$	39.2011	1.63196	0.815982	−	0.578077i	$-0.196197\pi$
$577$	39.2011	1.63196	0.815982	+	0.578077i	$0.196197\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 16.8831i	− 0.700430i
$582$	0	0
$583$	5.37216i	0.222492i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.9839	0.742275	0.371138	−	0.928578i	$-0.378968\pi$
$587$	17.9839	0.742275	0.371138	+	0.928578i	$0.378968\pi$
$588$	0	0
$589$	39.6848	1.63519
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 30.6480i	− 1.25856i	−0.777177	−	0.629282i	$-0.783349\pi$
$593$	− 30.6480i	− 1.25856i	0.777177	−	0.629282i	$-0.216651\pi$
$594$	0	0
$595$	− 0.702827i	− 0.0288131i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−23.8899	−0.976116	−0.488058	−	0.872811i	$-0.662295\pi$
$599$	−23.8899	−0.976116	−0.488058	+	0.872811i	$0.662295\pi$
$600$	0	0
$601$	−12.2161	−0.498305	−0.249152	−	0.968464i	$-0.580152\pi$
$601$	−12.2161	−0.498305	−0.249152	+	0.968464i	$0.580152\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 10.0154i	− 0.407185i
$606$	0	0
$607$	− 0.921404i	− 0.0373986i	−0.999825	−	0.0186993i	$-0.994047\pi$
$607$	− 0.921404i	− 0.0373986i	0.999825	−	0.0186993i	$-0.00595252\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.15748	−0.289561
$612$	0	0
$613$	−17.7716	−0.717788	−0.358894	−	0.933378i	$-0.616846\pi$
$613$	−17.7716	−0.717788	−0.358894	+	0.933378i	$0.616846\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.5702i	0.747609i	0.927507	+	0.373805i	$0.121947\pi$
$617$	18.5702i	0.747609i	−0.927507	+	0.373805i	$0.878053\pi$
$618$	0	0
$619$	28.9780i	1.16473i	0.812929	+	0.582363i	$0.197872\pi$
$619$	28.9780i	1.16473i	−0.812929	+	0.582363i	$0.802128\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.33485	0.253801
$624$	0	0
$625$	5.47939	0.219176
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.27208i	− 0.0507211i
$630$	0	0
$631$	32.3996i	1.28981i	0.764265	+	0.644903i	$0.223102\pi$
$631$	32.3996i	1.28981i	−0.764265	+	0.644903i	$0.776898\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.99948	0.198398
$636$	0	0
$637$	2.09638	0.0830615
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 41.1701i	− 1.62612i	−0.582179	−	0.813061i	$-0.697800\pi$
$641$	− 41.1701i	− 1.62612i	0.582179	−	0.813061i	$-0.302200\pi$
$642$	0	0
$643$	− 4.52433i	− 0.178422i	−0.996013	−	0.0892112i	$-0.971565\pi$
$643$	− 4.52433i	− 0.178422i	0.996013	−	0.0892112i	$-0.0284346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24.4377	−0.960746	−0.480373	−	0.877064i	$-0.659499\pi$
$647$	−24.4377	−0.960746	−0.480373	+	0.877064i	$0.659499\pi$
$648$	0	0
$649$	−3.39798	−0.133382
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.82564i	− 0.110576i	−0.998470	−	0.0552878i	$-0.982392\pi$
$653$	− 2.82564i	− 0.110576i	0.998470	−	0.0552878i	$-0.0176076\pi$
$654$	0	0
$655$	16.3665i	0.639491i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	34.8305	1.35680	0.678401	−	0.734692i	$-0.262673\pi$
$659$	34.8305	1.35680	0.678401	+	0.734692i	$0.262673\pi$
$660$	0	0
$661$	32.6230	1.26889	0.634444	−	0.772969i	$-0.281229\pi$
$661$	32.6230	1.26889	0.634444	+	0.772969i	$0.281229\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.07760i	− 0.274458i
$666$	0	0
$667$	− 50.2531i	− 1.94581i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.02620	0.309848
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 13.6947i	− 0.526330i	−0.964751	−	0.263165i	$-0.915234\pi$
$677$	− 13.6947i	− 0.526330i	0.964751	−	0.263165i	$-0.0847664\pi$
$678$	0	0
$679$	6.58506i	0.252711i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.7975	−1.21670	−0.608349	−	0.793670i	$-0.708168\pi$
$683$	−31.7975	−1.21670	−0.608349	+	0.793670i	$0.708168\pi$
$684$	0	0
$685$	−9.78673	−0.373932
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 6.91507i	− 0.263443i
$690$	0	0
$691$	− 29.8496i	− 1.13553i	−0.823190	−	0.567766i	$-0.807808\pi$
$691$	− 29.8496i	− 1.13553i	0.823190	−	0.567766i	$-0.192192\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3550	0.430721
$696$	0	0
$697$	0.0963763	0.00365051
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.0711i	1.09800i	0.835823	+	0.549000i	$0.184991\pi$
$701$	29.0711i	1.09800i	−0.835823	+	0.549000i	$0.815009\pi$
$702$	0	0
$703$	− 12.8101i	− 0.483141i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.34366	−0.351405
$708$	0	0
$709$	10.4368	0.391963	0.195981	−	0.980608i	$-0.437211\pi$
$709$	10.4368	0.391963	0.195981	+	0.980608i	$0.437211\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 47.7157i	− 1.78697i
$714$	0	0
$715$	− 4.09638i	− 0.153196i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	43.2268	1.61209	0.806044	−	0.591856i	$-0.201605\pi$
$719$	43.2268	1.61209	0.806044	+	0.591856i	$0.201605\pi$
$720$	0	0
$721$	−0.586302	−0.0218350
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25.2266i	0.936891i
$726$	0	0
$727$	− 5.31122i	− 0.196982i	−0.995138	−	0.0984911i	$-0.968598\pi$
$727$	− 5.31122i	− 0.196982i	0.995138	−	0.0984911i	$-0.0314016\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.37016	0.124650
$732$	0	0
$733$	16.1057	0.594876	0.297438	−	0.954741i	$-0.403868\pi$
$733$	16.1057	0.594876	0.297438	+	0.954741i	$0.403868\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 20.3394i	− 0.749211i
$738$	0	0
$739$	− 16.8673i	− 0.620472i	−0.950660	−	0.310236i	$-0.899592\pi$
$739$	− 16.8673i	− 0.620472i	0.950660	−	0.310236i	$-0.100408\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.2019	−1.65830	−0.829148	−	0.559029i	$-0.811174\pi$
$743$	−45.2019	−1.65830	−0.829148	+	0.559029i	$0.811174\pi$
$744$	0	0
$745$	−24.8177	−0.909249
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.14162i	− 0.114792i
$750$	0	0
$751$	0.479065i	0.0174813i	0.999962	+	0.00874067i	$0.00278228\pi$
$751$	0.479065i	0.0174813i	−0.999962	+	0.00874067i	$0.997218\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−3.71530	−0.135035	−0.0675175	−	0.997718i	$-0.521508\pi$
$757$	−3.71530	−0.135035	−0.0675175	+	0.997718i	$0.521508\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.2164i	0.805345i	0.915344	+	0.402673i	$0.131919\pi$
$761$	22.2164i	0.805345i	−0.915344	+	0.402673i	$0.868081\pi$
$762$	0	0
$763$	− 11.6522i	− 0.421838i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.37390	0.157932
$768$	0	0
$769$	−40.1576	−1.44812	−0.724061	−	0.689736i	$-0.757727\pi$
$769$	−40.1576	−1.44812	−0.724061	+	0.689736i	$0.757727\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.25750i	− 0.153132i	−0.997065	−	0.0765658i	$-0.975604\pi$
$773$	− 4.25750i	− 0.153132i	0.997065	−	0.0765658i	$-0.0243955\pi$
$774$	0	0
$775$	23.9528i	0.860409i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.970528	0.0347728
$780$	0	0
$781$	18.3215	0.655596
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.296074i	0.0105673i
$786$	0	0
$787$	5.70850i	0.203486i	0.994811	+	0.101743i	$0.0324420\pi$
$787$	5.70850i	0.203486i	−0.994811	+	0.101743i	$0.967558\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	19.5546	0.695281
$792$	0	0
$793$	−10.3314	−0.366878
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.02499i	− 0.0363071i	−0.999835	−	0.0181535i	$-0.994221\pi$
$797$	− 1.02499i	− 0.0363071i	0.999835	−	0.0181535i	$-0.00577877\pi$
$798$	0	0
$799$	− 2.00000i	− 0.0707549i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.47245	0.228408
$804$	0	0
$805$	−8.50986	−0.299933
$806$	0	0
$807$	0	0
$808$	0	0
$809$	15.4989i	0.544911i	0.962168	+	0.272455i	$0.0878358\pi$
$809$	15.4989i	0.544911i	−0.962168	+	0.272455i	$0.912164\pi$
$810$	0	0
$811$	32.4913i	1.14092i	0.821324	+	0.570462i	$0.193236\pi$
$811$	32.4913i	1.14092i	−0.821324	+	0.570462i	$0.806764\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.47163	−0.331777
$816$	0	0
$817$	33.9382	1.18735
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 40.6559i	− 1.41890i	−0.704755	−	0.709451i	$-0.748943\pi$
$821$	− 40.6559i	− 1.41890i	0.704755	−	0.709451i	$-0.251057\pi$
$822$	0	0
$823$	3.07180i	0.107076i	0.998566	+	0.0535381i	$0.0170498\pi$
$823$	3.07180i	0.107076i	−0.998566	+	0.0535381i	$0.982950\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.7665	−0.965536	−0.482768	−	0.875748i	$-0.660369\pi$
$827$	−27.7665	−0.965536	−0.482768	+	0.875748i	$0.660369\pi$
$828$	0	0
$829$	−55.3673	−1.92299	−0.961493	−	0.274830i	$-0.911378\pi$
$829$	−55.3673	−1.92299	−0.961493	+	0.274830i	$0.911378\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0.585786i	0.0202963i
$834$	0	0
$835$	2.08946i	0.0723086i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.38698	−0.185979	−0.0929896	−	0.995667i	$-0.529642\pi$
$839$	−5.38698	−0.185979	−0.0929896	+	0.995667i	$0.529642\pi$
$840$	0	0
$841$	−21.1996	−0.731019
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 10.3245i	− 0.355175i
$846$	0	0
$847$	8.34758i	0.286826i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−15.4024	−0.527987
$852$	0	0
$853$	−15.9597	−0.546450	−0.273225	−	0.961950i	$-0.588090\pi$
$853$	−15.9597	−0.546450	−0.273225	+	0.961950i	$0.588090\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.7050i	− 1.15134i	−0.817682	−	0.575671i	$-0.804741\pi$
$857$	− 33.7050i	− 1.15134i	0.817682	−	0.575671i	$-0.195259\pi$
$858$	0	0
$859$	− 20.3739i	− 0.695150i	−0.937652	−	0.347575i	$-0.887005\pi$
$859$	− 20.3739i	− 0.695150i	0.937652	−	0.347575i	$-0.112995\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.986419	−0.0335781	−0.0167890	−	0.999859i	$-0.505344\pi$
$863$	−0.986419	−0.0335781	−0.0167890	+	0.999859i	$0.505344\pi$
$864$	0	0
$865$	12.3230	0.418994
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1.14464i	− 0.0388293i
$870$	0	0
$871$	26.1810i	0.887109i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.2709	0.347219
$876$	0	0
$877$	54.0958	1.82669	0.913343	−	0.407191i	$-0.133492\pi$
$877$	54.0958	1.82669	0.913343	+	0.407191i	$0.133492\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.7649i	1.00281i	0.865214	+	0.501403i	$0.167183\pi$
$881$	29.7649i	1.00281i	−0.865214	+	0.501403i	$0.832817\pi$
$882$	0	0
$883$	1.48582i	0.0500019i	0.999687	+	0.0250010i	$0.00795888\pi$
$883$	1.48582i	0.0500019i	−0.999687	+	0.0250010i	$0.992041\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.8854	−0.466226	−0.233113	−	0.972450i	$-0.574891\pi$
$887$	−13.8854	−0.466226	−0.233113	+	0.972450i	$0.574891\pi$
$888$	0	0
$889$	−4.16693	−0.139754
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 20.1404i	− 0.673972i
$894$	0	0
$895$	3.37452i	0.112798i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−47.6648	−1.58971
$900$	0	0
$901$	1.93226	0.0643731
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 9.76770i	− 0.324689i
$906$	0	0
$907$	4.42580i	0.146956i	0.997297	+	0.0734782i	$0.0234099\pi$
$907$	4.42580i	0.146956i	−0.997297	+	0.0734782i	$0.976590\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.6657	−0.982870	−0.491435	−	0.870914i	$-0.663527\pi$
$911$	−29.6657	−0.982870	−0.491435	+	0.870914i	$0.663527\pi$
$912$	0	0
$913$	27.4963	0.909995
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 13.6410i	− 0.450465i
$918$	0	0
$919$	− 42.4838i	− 1.40141i	−0.713450	−	0.700706i	$-0.752868\pi$
$919$	− 42.4838i	− 1.40141i	0.713450	−	0.700706i	$-0.247132\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−23.5836	−0.776263
$924$	0	0
$925$	7.73184	0.254221
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 31.3523i	− 1.02864i	−0.857600	−	0.514318i	$-0.828045\pi$
$929$	− 31.3523i	− 1.02864i	0.857600	−	0.514318i	$-0.171955\pi$
$930$	0	0
$931$	5.89898i	0.193331i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.14464	0.0374338
$936$	0	0
$937$	17.0557	0.557186	0.278593	−	0.960409i	$-0.410132\pi$
$937$	17.0557	0.557186	0.278593	+	0.960409i	$0.410132\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	45.1949i	1.47331i	0.676268	+	0.736656i	$0.263596\pi$
$941$	45.1949i	1.47331i	−0.676268	+	0.736656i	$0.736404\pi$
$942$	0	0
$943$	− 1.16693i	− 0.0380004i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.6571	−1.02872	−0.514359	−	0.857575i	$-0.671970\pi$
$947$	−31.6571	−1.02872	−0.514359	+	0.857575i	$0.671970\pi$
$948$	0	0
$949$	−8.33137	−0.270448
$950$	0	0
$951$	0	0
$952$	0	0
$953$	39.1230i	1.26732i	0.773612	+	0.633660i	$0.218448\pi$
$953$	39.1230i	1.26732i	−0.773612	+	0.633660i	$0.781552\pi$
$954$	0	0
$955$	19.2546i	0.623064i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.15696	0.263402
$960$	0	0
$961$	−14.2580	−0.459935
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3.45428i	− 0.111197i
$966$	0	0
$967$	50.8761i	1.63607i	0.575171	+	0.818033i	$0.304935\pi$
$967$	50.8761i	1.63607i	−0.575171	+	0.818033i	$0.695065\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	45.9782	1.47551	0.737755	−	0.675068i	$-0.235886\pi$
$971$	45.9782	1.47551	0.737755	+	0.675068i	$0.235886\pi$
$972$	0	0
$973$	−9.46410	−0.303405
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.1822i	− 0.485722i	−0.970061	−	0.242861i	$-0.921914\pi$
$977$	− 15.1822i	− 0.485722i	0.970061	−	0.242861i	$-0.0780859\pi$
$978$	0	0
$979$	10.3171i	0.329736i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−42.6625	−1.36072	−0.680361	−	0.732877i	$-0.738177\pi$
$983$	−42.6625	−1.36072	−0.680361	+	0.732877i	$0.738177\pi$
$984$	0	0
$985$	−9.71573	−0.309569
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 40.8061i	− 1.29756i
$990$	0	0
$991$	− 38.0072i	− 1.20734i	−0.797234	−	0.603670i	$-0.793704\pi$
$991$	− 38.0072i	− 1.20734i	0.797234	−	0.603670i	$-0.206296\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−26.2785	−0.833084
$996$	0	0
$997$	−24.0560	−0.761860	−0.380930	−	0.924604i	$-0.624396\pi$
$997$	−24.0560	−0.761860	−0.380930	+	0.924604i	$0.624396\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.h.a.2591.5	yes	8
3.2	odd	2		6048.2.h.g.2591.4	yes	8
4.3	odd	2		6048.2.h.g.2591.5	yes	8
12.11	even	2	inner	6048.2.h.a.2591.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6048.2.h.a.2591.4	✓	8	12.11	even	2	inner
6048.2.h.a.2591.5	yes	8	1.1	even	1	trivial
6048.2.h.g.2591.4	yes	8	3.2	odd	2
6048.2.h.g.2591.5	yes	8	4.3	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.19980i q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q+1.19980i q^{5} -1.00000i q^{7} +1.62863 q^{11} -2.09638 q^{13} -0.585786i q^{17} -5.89898i q^{19} -7.09273 q^{23} +3.56048 q^{25} +7.08516i q^{29} +6.72741i q^{31} +1.19980 q^{35} +2.17157 q^{37} +0.164525i q^{41} +5.75323i q^{43} +3.41421 q^{47} -1.00000 q^{49} +3.29858i q^{53} +1.95403i q^{55} -2.08641 q^{59} +4.92820 q^{61} -2.51523i q^{65} -12.4887i q^{67} +11.2497 q^{71} +3.97418 q^{73} -1.62863i q^{77} -0.702827i q^{79} +16.8831 q^{83} +0.702827 q^{85} +6.33485i q^{89} +2.09638i q^{91} +7.07760 q^{95} -6.58506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 8 q^{11} - 8 q^{13} - 8 q^{23} - 8 q^{25} + 8 q^{35} + 40 q^{37} + 16 q^{47} - 8 q^{49} - 64 q^{59} - 16 q^{61} + 72 q^{71} + 24 q^{73} + 32 q^{83} + 8 q^{85} + 56 q^{95} + 48 q^{97}+O(q^{100}) \) 8 * q - 8 * q^11 - 8 * q^13 - 8 * q^23 - 8 * q^25 + 8 * q^35 + 40 * q^37 + 16 * q^47 - 8 * q^49 - 64 * q^59 - 16 * q^61 + 72 * q^71 + 24 * q^73 + 32 * q^83 + 8 * q^85 + 56 * q^95 + 48 * q^97

Introduction

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