LMFDB - Embedded newform 3024.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.a (trivial)

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:	yes
Analytic conductor:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1512)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	3024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+2.64575 q^{5} +1.00000 q^{7} +0.645751 q^{11} +5.29150 q^{13} +1.00000 q^{19} -1.35425 q^{23} +2.00000 q^{25} +6.00000 q^{29} -2.29150 q^{31} +2.64575 q^{35} -2.29150 q^{37} +8.64575 q^{41} +7.29150 q^{43} -11.2915 q^{47} +1.00000 q^{49} +1.70850 q^{55} +3.29150 q^{59} -6.58301 q^{61} +14.0000 q^{65} +1.29150 q^{67} -13.2288 q^{71} +0.708497 q^{73} +0.645751 q^{77} +11.2915 q^{79} -10.0000 q^{83} -13.9373 q^{89} +5.29150 q^{91} +2.64575 q^{95} +18.5830 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^7	$ 2 q + 2 q^{7} - 4 q^{11} + 2 q^{19} - 8 q^{23} + 4 q^{25} + 12 q^{29} + 6 q^{31} + 6 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 14 q^{55} - 4 q^{59} + 8 q^{61} + 28 q^{65} - 8 q^{67} + 12 q^{73} - 4 q^{77} + 12 q^{79} - 20 q^{83} - 12 q^{89} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 4 * q^11 + 2 * q^19 - 8 * q^23 + 4 * q^25 + 12 * q^29 + 6 * q^31 + 6 * q^37 + 12 * q^41 + 4 * q^43 - 12 * q^47 + 2 * q^49 + 14 * q^55 - 4 * q^59 + 8 * q^61 + 28 * q^65 - 8 * q^67 + 12 * q^73 - 4 * q^77 + 12 * q^79 - 20 * q^83 - 12 * q^89 + 16 * q^97

Display 100 coefficients 10 coefficients

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
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.645751	0.194701	0.0973507	−	0.995250i	$-0.468963\pi$
$11$	0.645751	0.194701	0.0973507	+	0.995250i	$0.468963\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.35425	−0.282380	−0.141190	−	0.989982i	$-0.545093\pi$
$23$	−1.35425	−0.282380	−0.141190	+	0.989982i	$0.545093\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−2.29150	−0.411566	−0.205783	−	0.978598i	$-0.565974\pi$
$31$	−2.29150	−0.411566	−0.205783	+	0.978598i	$0.565974\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.64575	0.447214
$36$	0	0
$37$	−2.29150	−0.376721	−0.188360	−	0.982100i	$-0.560317\pi$
$37$	−2.29150	−0.376721	−0.188360	+	0.982100i	$0.560317\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.64575	1.35024	0.675120	−	0.737708i	$-0.264092\pi$
$41$	8.64575	1.35024	0.675120	+	0.737708i	$0.264092\pi$
$42$	0	0
$43$	7.29150	1.11194	0.555972	−	0.831201i	$-0.312346\pi$
$43$	7.29150	1.11194	0.555972	+	0.831201i	$0.312346\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.2915	−1.64703	−0.823517	−	0.567291i	$-0.807992\pi$
$47$	−11.2915	−1.64703	−0.823517	+	0.567291i	$0.807992\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	1.70850	0.230374
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.29150	0.428517	0.214259	−	0.976777i	$-0.431266\pi$
$59$	3.29150	0.428517	0.214259	+	0.976777i	$0.431266\pi$
$60$	0	0
$61$	−6.58301	−0.842867	−0.421434	−	0.906859i	$-0.638473\pi$
$61$	−6.58301	−0.842867	−0.421434	+	0.906859i	$0.638473\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	14.0000	1.73649
$66$	0	0
$67$	1.29150	0.157782	0.0788911	−	0.996883i	$-0.474862\pi$
$67$	1.29150	0.157782	0.0788911	+	0.996883i	$0.474862\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2288	−1.56996	−0.784982	−	0.619518i	$-0.787328\pi$
$71$	−13.2288	−1.56996	−0.784982	+	0.619518i	$0.787328\pi$
$72$	0	0
$73$	0.708497	0.0829233	0.0414617	−	0.999140i	$-0.486799\pi$
$73$	0.708497	0.0829233	0.0414617	+	0.999140i	$0.486799\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.645751	0.0735902
$78$	0	0
$79$	11.2915	1.27039	0.635197	−	0.772350i	$-0.280919\pi$
$79$	11.2915	1.27039	0.635197	+	0.772350i	$0.280919\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.0000	−1.09764	−0.548821	−	0.835940i	$-0.684923\pi$
$83$	−10.0000	−1.09764	−0.548821	+	0.835940i	$0.684923\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.9373	−1.47735	−0.738673	−	0.674064i	$-0.764547\pi$
$89$	−13.9373	−1.47735	−0.738673	+	0.674064i	$0.764547\pi$
$90$	0	0
$91$	5.29150	0.554700
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.64575	0.271448
$96$	0	0
$97$	18.5830	1.88682	0.943409	−	0.331631i	$-0.107599\pi$
$97$	18.5830	1.88682	0.943409	+	0.331631i	$0.107599\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.00000	0.398015	0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000	0.398015	0.199007	+	0.979998i	$0.436228\pi$
$102$	0	0
$103$	4.29150	0.422854	0.211427	−	0.977394i	$-0.432189\pi$
$103$	4.29150	0.422854	0.211427	+	0.977394i	$0.432189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.70850	−0.261840	−0.130920	−	0.991393i	$-0.541793\pi$
$107$	−2.70850	−0.261840	−0.130920	+	0.991393i	$0.541793\pi$
$108$	0	0
$109$	14.8745	1.42472	0.712360	−	0.701815i	$-0.247626\pi$
$109$	14.8745	1.42472	0.712360	+	0.701815i	$0.247626\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.8745	−1.11706	−0.558530	−	0.829484i	$-0.688634\pi$
$113$	−11.8745	−1.11706	−0.558530	+	0.829484i	$0.688634\pi$
$114$	0	0
$115$	−3.58301	−0.334117
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.5830	−0.962091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	0	0
$127$	17.2915	1.53437	0.767186	−	0.641424i	$-0.221656\pi$
$127$	17.2915	1.53437	0.767186	+	0.641424i	$0.221656\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.70850	0.411383	0.205692	−	0.978617i	$-0.434056\pi$
$131$	4.70850	0.411383	0.205692	+	0.978617i	$0.434056\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.5830	1.07504	0.537519	−	0.843251i	$-0.319362\pi$
$137$	12.5830	1.07504	0.537519	+	0.843251i	$0.319362\pi$
$138$	0	0
$139$	−10.5830	−0.897639	−0.448819	−	0.893622i	$-0.648155\pi$
$139$	−10.5830	−0.897639	−0.448819	+	0.893622i	$0.648155\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.41699	0.285743
$144$	0	0
$145$	15.8745	1.31831
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.8745	1.30049	0.650245	−	0.759724i	$-0.274666\pi$
$149$	15.8745	1.30049	0.650245	+	0.759724i	$0.274666\pi$
$150$	0	0
$151$	−0.583005	−0.0474443	−0.0237221	−	0.999719i	$-0.507552\pi$
$151$	−0.583005	−0.0474443	−0.0237221	+	0.999719i	$0.507552\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.06275	−0.486971
$156$	0	0
$157$	9.29150	0.741543	0.370771	−	0.928724i	$-0.379093\pi$
$157$	9.29150	0.741543	0.370771	+	0.928724i	$0.379093\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.35425	−0.106730
$162$	0	0
$163$	−9.29150	−0.727767	−0.363883	−	0.931445i	$-0.618549\pi$
$163$	−9.29150	−0.727767	−0.363883	+	0.931445i	$0.618549\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.2915	−1.33806	−0.669028	−	0.743237i	$-0.733290\pi$
$167$	−17.2915	−1.33806	−0.669028	+	0.743237i	$0.733290\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.5203	−1.10395	−0.551977	−	0.833859i	$-0.686127\pi$
$173$	−14.5203	−1.10395	−0.551977	+	0.833859i	$0.686127\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.7085	0.800391	0.400195	−	0.916430i	$-0.368942\pi$
$179$	10.7085	0.800391	0.400195	+	0.916430i	$0.368942\pi$
$180$	0	0
$181$	3.41699	0.253983	0.126992	−	0.991904i	$-0.459468\pi$
$181$	3.41699	0.253983	0.126992	+	0.991904i	$0.459468\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−6.06275	−0.445742
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.6458	−1.92802	−0.964009	−	0.265868i	$-0.914341\pi$
$191$	−26.6458	−1.92802	−0.964009	+	0.265868i	$0.914341\pi$
$192$	0	0
$193$	−20.5830	−1.48160	−0.740799	−	0.671727i	$-0.765553\pi$
$193$	−20.5830	−1.48160	−0.740799	+	0.671727i	$0.765553\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.29150	−0.661992	−0.330996	−	0.943632i	$-0.607385\pi$
$197$	−9.29150	−0.661992	−0.330996	+	0.943632i	$0.607385\pi$
$198$	0	0
$199$	24.8745	1.76331	0.881654	−	0.471897i	$-0.156431\pi$
$199$	24.8745	1.76331	0.881654	+	0.471897i	$0.156431\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000	0.421117
$204$	0	0
$205$	22.8745	1.59762
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.645751	0.0446676
$210$	0	0
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	19.2915	1.31567
$216$	0	0
$217$	−2.29150	−0.155557
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	24.2915	1.62668	0.813340	−	0.581789i	$-0.197647\pi$
$223$	24.2915	1.62668	0.813340	+	0.581789i	$0.197647\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.2915	0.882188	0.441094	−	0.897461i	$-0.354591\pi$
$227$	13.2915	0.882188	0.441094	+	0.897461i	$0.354591\pi$
$228$	0	0
$229$	−26.5830	−1.75665	−0.878327	−	0.478060i	$-0.841340\pi$
$229$	−26.5830	−1.75665	−0.878327	+	0.478060i	$0.841340\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.2915	0.870755	0.435378	−	0.900248i	$-0.356615\pi$
$233$	13.2915	0.870755	0.435378	+	0.900248i	$0.356615\pi$
$234$	0	0
$235$	−29.8745	−1.94880
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.7085	0.692675	0.346338	−	0.938110i	$-0.387425\pi$
$239$	10.7085	0.692675	0.346338	+	0.938110i	$0.387425\pi$
$240$	0	0
$241$	12.7085	0.818626	0.409313	−	0.912394i	$-0.365768\pi$
$241$	12.7085	0.818626	0.409313	+	0.912394i	$0.365768\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.64575	0.169031
$246$	0	0
$247$	5.29150	0.336690
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.5830	1.17295	0.586474	−	0.809968i	$-0.300515\pi$
$251$	18.5830	1.17295	0.586474	+	0.809968i	$0.300515\pi$
$252$	0	0
$253$	−0.874508	−0.0549798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−31.2288	−1.94800	−0.973998	−	0.226556i	$-0.927253\pi$
$257$	−31.2288	−1.94800	−0.973998	+	0.226556i	$0.927253\pi$
$258$	0	0
$259$	−2.29150	−0.142387
$260$	0	0
$261$	0	0
$262$	0	0
$263$	17.2288	1.06237	0.531185	−	0.847256i	$-0.321747\pi$
$263$	17.2288	1.06237	0.531185	+	0.847256i	$0.321747\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−26.6458	−1.62462	−0.812310	−	0.583226i	$-0.801790\pi$
$269$	−26.6458	−1.62462	−0.812310	+	0.583226i	$0.801790\pi$
$270$	0	0
$271$	5.41699	0.329059	0.164529	−	0.986372i	$-0.447389\pi$
$271$	5.41699	0.329059	0.164529	+	0.986372i	$0.447389\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.29150	0.0778805
$276$	0	0
$277$	−14.2915	−0.858693	−0.429347	−	0.903140i	$-0.641256\pi$
$277$	−14.2915	−0.858693	−0.429347	+	0.903140i	$0.641256\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	21.8745	1.30492	0.652462	−	0.757822i	$-0.273736\pi$
$281$	21.8745	1.30492	0.652462	+	0.757822i	$0.273736\pi$
$282$	0	0
$283$	2.58301	0.153544	0.0767719	−	0.997049i	$-0.475539\pi$
$283$	2.58301	0.153544	0.0767719	+	0.997049i	$0.475539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.64575	0.510343
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.41699	0.316464	0.158232	−	0.987402i	$-0.449421\pi$
$293$	5.41699	0.316464	0.158232	+	0.987402i	$0.449421\pi$
$294$	0	0
$295$	8.70850	0.507028
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.16601	−0.414421
$300$	0	0
$301$	7.29150	0.420275
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−17.4170	−0.997294
$306$	0	0
$307$	18.1660	1.03679	0.518394	−	0.855142i	$-0.326530\pi$
$307$	18.1660	1.03679	0.518394	+	0.855142i	$0.326530\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.5830	1.16716	0.583578	−	0.812057i	$-0.301652\pi$
$311$	20.5830	1.16716	0.583578	+	0.812057i	$0.301652\pi$
$312$	0	0
$313$	13.2915	0.751280	0.375640	−	0.926766i	$-0.377423\pi$
$313$	13.2915	0.751280	0.375640	+	0.926766i	$0.377423\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.7085	1.16311	0.581553	−	0.813509i	$-0.302445\pi$
$317$	20.7085	1.16311	0.581553	+	0.813509i	$0.302445\pi$
$318$	0	0
$319$	3.87451	0.216931
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	10.5830	0.587040
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11.2915	−0.622521
$330$	0	0
$331$	10.5830	0.581695	0.290847	−	0.956769i	$-0.406063\pi$
$331$	10.5830	0.581695	0.290847	+	0.956769i	$0.406063\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.41699	0.186690
$336$	0	0
$337$	−6.41699	−0.349556	−0.174778	−	0.984608i	$-0.555921\pi$
$337$	−6.41699	−0.349556	−0.174778	+	0.984608i	$0.555921\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.47974	−0.0801325
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.35425	0.394797	0.197398	−	0.980323i	$-0.436751\pi$
$347$	7.35425	0.394797	0.197398	+	0.980323i	$0.436751\pi$
$348$	0	0
$349$	−0.583005	−0.0312076	−0.0156038	−	0.999878i	$-0.504967\pi$
$349$	−0.583005	−0.0312076	−0.0156038	+	0.999878i	$0.504967\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.06275	−0.109789	−0.0548944	−	0.998492i	$-0.517482\pi$
$353$	−2.06275	−0.109789	−0.0548944	+	0.998492i	$0.517482\pi$
$354$	0	0
$355$	−35.0000	−1.85761
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.8745	0.837824	0.418912	−	0.908027i	$-0.362411\pi$
$359$	15.8745	0.837824	0.418912	+	0.908027i	$0.362411\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.87451	0.0981162
$366$	0	0
$367$	26.2915	1.37241	0.686203	−	0.727410i	$-0.259276\pi$
$367$	26.2915	1.37241	0.686203	+	0.727410i	$0.259276\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.8745	0.563061	0.281530	−	0.959552i	$-0.409158\pi$
$373$	10.8745	0.563061	0.281530	+	0.959552i	$0.409158\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.7490	1.63516
$378$	0	0
$379$	−17.8745	−0.918152	−0.459076	−	0.888397i	$-0.651819\pi$
$379$	−17.8745	−0.918152	−0.459076	+	0.888397i	$0.651819\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.70850	0.138398	0.0691989	−	0.997603i	$-0.477956\pi$
$383$	2.70850	0.138398	0.0691989	+	0.997603i	$0.477956\pi$
$384$	0	0
$385$	1.70850	0.0870731
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−28.0000	−1.41966	−0.709828	−	0.704375i	$-0.751227\pi$
$389$	−28.0000	−1.41966	−0.709828	+	0.704375i	$0.751227\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	29.8745	1.50315
$396$	0	0
$397$	10.0000	0.501886	0.250943	−	0.968002i	$-0.419259\pi$
$397$	10.0000	0.501886	0.250943	+	0.968002i	$0.419259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.1660	−0.957105	−0.478552	−	0.878059i	$-0.658838\pi$
$401$	−19.1660	−0.957105	−0.478552	+	0.878059i	$0.658838\pi$
$402$	0	0
$403$	−12.1255	−0.604014
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.47974	−0.0733480
$408$	0	0
$409$	13.8745	0.686050	0.343025	−	0.939326i	$-0.388548\pi$
$409$	13.8745	0.686050	0.343025	+	0.939326i	$0.388548\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.29150	0.161964
$414$	0	0
$415$	−26.4575	−1.29875
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.2915	−1.23557	−0.617785	−	0.786347i	$-0.711970\pi$
$419$	−25.2915	−1.23557	−0.617785	+	0.786347i	$0.711970\pi$
$420$	0	0
$421$	4.29150	0.209155	0.104578	−	0.994517i	$-0.466651\pi$
$421$	4.29150	0.209155	0.104578	+	0.994517i	$0.466651\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.58301	−0.318574
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.1033	1.20918	0.604591	−	0.796536i	$-0.293337\pi$
$431$	25.1033	1.20918	0.604591	+	0.796536i	$0.293337\pi$
$432$	0	0
$433$	−23.8745	−1.14734	−0.573668	−	0.819088i	$-0.694480\pi$
$433$	−23.8745	−1.14734	−0.573668	+	0.819088i	$0.694480\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.35425	−0.0647825
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.2288	−1.67377	−0.836884	−	0.547380i	$-0.815625\pi$
$443$	−35.2288	−1.67377	−0.836884	+	0.547380i	$0.815625\pi$
$444$	0	0
$445$	−36.8745	−1.74802
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.4575	−1.24861	−0.624304	−	0.781182i	$-0.714617\pi$
$449$	−26.4575	−1.24861	−0.624304	+	0.781182i	$0.714617\pi$
$450$	0	0
$451$	5.58301	0.262893
$452$	0	0
$453$	0	0
$454$	0	0
$455$	14.0000	0.656330
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−39.6863	−1.84837	−0.924187	−	0.381940i	$-0.875256\pi$
$461$	−39.6863	−1.84837	−0.924187	+	0.381940i	$0.875256\pi$
$462$	0	0
$463$	−33.7490	−1.56845	−0.784225	−	0.620477i	$-0.786939\pi$
$463$	−33.7490	−1.56845	−0.784225	+	0.620477i	$0.786939\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.87451	−0.271840	−0.135920	−	0.990720i	$-0.543399\pi$
$467$	−5.87451	−0.271840	−0.135920	+	0.990720i	$0.543399\pi$
$468$	0	0
$469$	1.29150	0.0596361
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.70850	0.216497
$474$	0	0
$475$	2.00000	0.0917663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−12.1255	−0.552875
$482$	0	0
$483$	0	0
$484$	0	0
$485$	49.1660	2.23251
$486$	0	0
$487$	−23.2915	−1.05544	−0.527719	−	0.849419i	$-0.676953\pi$
$487$	−23.2915	−1.05544	−0.527719	+	0.849419i	$0.676953\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−29.8118	−1.34539	−0.672693	−	0.739922i	$-0.734863\pi$
$491$	−29.8118	−1.34539	−0.672693	+	0.739922i	$0.734863\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−13.2288	−0.593391
$498$	0	0
$499$	−35.2915	−1.57986	−0.789932	−	0.613194i	$-0.789884\pi$
$499$	−35.2915	−1.57986	−0.789932	+	0.613194i	$0.789884\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−20.5830	−0.917751	−0.458875	−	0.888501i	$-0.651748\pi$
$503$	−20.5830	−0.917751	−0.458875	+	0.888501i	$0.651748\pi$
$504$	0	0
$505$	10.5830	0.470938
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.4170	−0.949292	−0.474646	−	0.880177i	$-0.657424\pi$
$509$	−21.4170	−0.949292	−0.474646	+	0.880177i	$0.657424\pi$
$510$	0	0
$511$	0.708497	0.0313421
$512$	0	0
$513$	0	0
$514$	0	0
$515$	11.3542	0.500328
$516$	0	0
$517$	−7.29150	−0.320680
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.9373	−0.610602	−0.305301	−	0.952256i	$-0.598757\pi$
$521$	−13.9373	−0.610602	−0.305301	+	0.952256i	$0.598757\pi$
$522$	0	0
$523$	−6.41699	−0.280596	−0.140298	−	0.990109i	$-0.544806\pi$
$523$	−6.41699	−0.280596	−0.140298	+	0.990109i	$0.544806\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−21.1660	−0.920261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	45.7490	1.98161
$534$	0	0
$535$	−7.16601	−0.309814
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.645751	0.0278145
$540$	0	0
$541$	20.2915	0.872400	0.436200	−	0.899850i	$-0.356324\pi$
$541$	20.2915	0.872400	0.436200	+	0.899850i	$0.356324\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	39.3542	1.68575
$546$	0	0
$547$	−6.58301	−0.281469	−0.140734	−	0.990047i	$-0.544946\pi$
$547$	−6.58301	−0.281469	−0.140734	+	0.990047i	$0.544946\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	11.2915	0.480164
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.4575	−0.782070	−0.391035	−	0.920376i	$-0.627883\pi$
$557$	−18.4575	−0.782070	−0.391035	+	0.920376i	$0.627883\pi$
$558$	0	0
$559$	38.5830	1.63189
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−23.1660	−0.976331	−0.488165	−	0.872751i	$-0.662334\pi$
$563$	−23.1660	−0.976331	−0.488165	+	0.872751i	$0.662334\pi$
$564$	0	0
$565$	−31.4170	−1.32172
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.45751	−0.270713	−0.135357	−	0.990797i	$-0.543218\pi$
$569$	−6.45751	−0.270713	−0.135357	+	0.990797i	$0.543218\pi$
$570$	0	0
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.70850	−0.112952
$576$	0	0
$577$	−20.5830	−0.856882	−0.428441	−	0.903570i	$-0.640937\pi$
$577$	−20.5830	−0.856882	−0.428441	+	0.903570i	$0.640937\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.87451	−0.159918	−0.0799590	−	0.996798i	$-0.525479\pi$
$587$	−3.87451	−0.159918	−0.0799590	+	0.996798i	$0.525479\pi$
$588$	0	0
$589$	−2.29150	−0.0944197
$590$	0	0
$591$	0	0
$592$	0	0
$593$	24.6458	1.01208	0.506040	−	0.862510i	$-0.331109\pi$
$593$	24.6458	1.01208	0.506040	+	0.862510i	$0.331109\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.8118	−1.46323	−0.731614	−	0.681719i	$-0.761233\pi$
$599$	−35.8118	−1.46323	−0.731614	+	0.681719i	$0.761233\pi$
$600$	0	0
$601$	4.12549	0.168282	0.0841412	−	0.996454i	$-0.473185\pi$
$601$	4.12549	0.168282	0.0841412	+	0.996454i	$0.473185\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−28.0000	−1.13836
$606$	0	0
$607$	−13.4170	−0.544579	−0.272290	−	0.962215i	$-0.587781\pi$
$607$	−13.4170	−0.544579	−0.272290	+	0.962215i	$0.587781\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−59.7490	−2.41719
$612$	0	0
$613$	−29.7085	−1.19991	−0.599957	−	0.800032i	$-0.704816\pi$
$613$	−29.7085	−1.19991	−0.599957	+	0.800032i	$0.704816\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19.2915	−0.776647	−0.388323	−	0.921523i	$-0.626946\pi$
$617$	−19.2915	−0.776647	−0.388323	+	0.921523i	$0.626946\pi$
$618$	0	0
$619$	−0.416995	−0.0167604	−0.00838022	−	0.999965i	$-0.502668\pi$
$619$	−0.416995	−0.0167604	−0.00838022	+	0.999965i	$0.502668\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.9373	−0.558384
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	47.0405	1.87265	0.936327	−	0.351130i	$-0.114202\pi$
$631$	47.0405	1.87265	0.936327	+	0.351130i	$0.114202\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	45.7490	1.81549
$636$	0	0
$637$	5.29150	0.209657
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.70850	−0.106979	−0.0534896	−	0.998568i	$-0.517034\pi$
$641$	−2.70850	−0.106979	−0.0534896	+	0.998568i	$0.517034\pi$
$642$	0	0
$643$	−7.00000	−0.276053	−0.138027	−	0.990429i	$-0.544076\pi$
$643$	−7.00000	−0.276053	−0.138027	+	0.990429i	$0.544076\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.2915	−0.915683	−0.457842	−	0.889034i	$-0.651377\pi$
$647$	−23.2915	−0.915683	−0.457842	+	0.889034i	$0.651377\pi$
$648$	0	0
$649$	2.12549	0.0834329
$650$	0	0
$651$	0	0
$652$	0	0
$653$	16.7085	0.653854	0.326927	−	0.945050i	$-0.393987\pi$
$653$	16.7085	0.653854	0.326927	+	0.945050i	$0.393987\pi$
$654$	0	0
$655$	12.4575	0.486755
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.8118	1.16130	0.580651	−	0.814153i	$-0.302798\pi$
$659$	29.8118	1.16130	0.580651	+	0.814153i	$0.302798\pi$
$660$	0	0
$661$	−19.8745	−0.773029	−0.386514	−	0.922283i	$-0.626321\pi$
$661$	−19.8745	−0.773029	−0.386514	+	0.922283i	$0.626321\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.64575	0.102598
$666$	0	0
$667$	−8.12549	−0.314620
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.25098	−0.164107
$672$	0	0
$673$	4.58301	0.176662	0.0883309	−	0.996091i	$-0.471847\pi$
$673$	4.58301	0.176662	0.0883309	+	0.996091i	$0.471847\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5.22876	−0.200957	−0.100479	−	0.994939i	$-0.532037\pi$
$677$	−5.22876	−0.200957	−0.100479	+	0.994939i	$0.532037\pi$
$678$	0	0
$679$	18.5830	0.713150
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−42.9778	−1.64450	−0.822249	−	0.569127i	$-0.807281\pi$
$683$	−42.9778	−1.64450	−0.822249	+	0.569127i	$0.807281\pi$
$684$	0	0
$685$	33.2915	1.27200
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−29.1660	−1.10953	−0.554764	−	0.832008i	$-0.687191\pi$
$691$	−29.1660	−1.10953	−0.554764	+	0.832008i	$0.687191\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.0000	−1.06210
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	23.4170	0.884448	0.442224	−	0.896905i	$-0.354190\pi$
$701$	23.4170	0.884448	0.442224	+	0.896905i	$0.354190\pi$
$702$	0	0
$703$	−2.29150	−0.0864257
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.00000	0.150435
$708$	0	0
$709$	−24.8745	−0.934182	−0.467091	−	0.884209i	$-0.654698\pi$
$709$	−24.8745	−0.934182	−0.467091	+	0.884209i	$0.654698\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.10326	0.116218
$714$	0	0
$715$	9.04052	0.338096
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.0405	1.15762	0.578808	−	0.815464i	$-0.303518\pi$
$719$	31.0405	1.15762	0.578808	+	0.815464i	$0.303518\pi$
$720$	0	0
$721$	4.29150	0.159824
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.0000	0.445669
$726$	0	0
$727$	1.41699	0.0525534	0.0262767	−	0.999655i	$-0.491635\pi$
$727$	1.41699	0.0525534	0.0262767	+	0.999655i	$0.491635\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−49.8745	−1.84216	−0.921078	−	0.389377i	$-0.872690\pi$
$733$	−49.8745	−1.84216	−0.921078	+	0.389377i	$0.872690\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.833990	0.0307204
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.93725	0.144444	0.0722219	−	0.997389i	$-0.476991\pi$
$743$	3.93725	0.144444	0.0722219	+	0.997389i	$0.476991\pi$
$744$	0	0
$745$	42.0000	1.53876
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.70850	−0.0989663
$750$	0	0
$751$	16.7085	0.609702	0.304851	−	0.952400i	$-0.401393\pi$
$751$	16.7085	0.609702	0.304851	+	0.952400i	$0.401393\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.54249	−0.0561368
$756$	0	0
$757$	28.5830	1.03887	0.519433	−	0.854511i	$-0.326143\pi$
$757$	28.5830	1.03887	0.519433	+	0.854511i	$0.326143\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	14.8745	0.538493
$764$	0	0
$765$	0	0
$766$	0	0
$767$	17.4170	0.628891
$768$	0	0
$769$	−18.7085	−0.674646	−0.337323	−	0.941389i	$-0.609521\pi$
$769$	−18.7085	−0.674646	−0.337323	+	0.941389i	$0.609521\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.81176	−0.280970	−0.140485	−	0.990083i	$-0.544866\pi$
$773$	−7.81176	−0.280970	−0.140485	+	0.990083i	$0.544866\pi$
$774$	0	0
$775$	−4.58301	−0.164626
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.64575	0.309766
$780$	0	0
$781$	−8.54249	−0.305674
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.5830	0.877405
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11.8745	−0.422209
$792$	0	0
$793$	−34.8340	−1.23699
$794$	0	0
$795$	0	0
$796$	0	0
$797$	43.9373	1.55634	0.778169	−	0.628055i	$-0.216149\pi$
$797$	43.9373	1.55634	0.778169	+	0.628055i	$0.216149\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0.457513	0.0161453
$804$	0	0
$805$	−3.58301	−0.126284
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.5830	1.00492	0.502462	−	0.864599i	$-0.332427\pi$
$809$	28.5830	1.00492	0.502462	+	0.864599i	$0.332427\pi$
$810$	0	0
$811$	−1.58301	−0.0555868	−0.0277934	−	0.999614i	$-0.508848\pi$
$811$	−1.58301	−0.0555868	−0.0277934	+	0.999614i	$0.508848\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.5830	−0.861105
$816$	0	0
$817$	7.29150	0.255097
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.0405	1.36252	0.681262	−	0.732040i	$-0.261432\pi$
$821$	39.0405	1.36252	0.681262	+	0.732040i	$0.261432\pi$
$822$	0	0
$823$	−31.1660	−1.08638	−0.543189	−	0.839610i	$-0.682783\pi$
$823$	−31.1660	−1.08638	−0.543189	+	0.839610i	$0.682783\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.9373	−1.04102	−0.520510	−	0.853855i	$-0.674258\pi$
$827$	−29.9373	−1.04102	−0.520510	+	0.853855i	$0.674258\pi$
$828$	0	0
$829$	−1.87451	−0.0651043	−0.0325522	−	0.999470i	$-0.510364\pi$
$829$	−1.87451	−0.0651043	−0.0325522	+	0.999470i	$0.510364\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−45.7490	−1.58321
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39.6863	1.36525
$846$	0	0
$847$	−10.5830	−0.363636
$848$	0	0
$849$	0	0
$850$	0	0
$851$	3.10326	0.106379
$852$	0	0
$853$	15.7490	0.539236	0.269618	−	0.962967i	$-0.413103\pi$
$853$	15.7490	0.539236	0.269618	+	0.962967i	$0.413103\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.2288	0.383567	0.191784	−	0.981437i	$-0.438573\pi$
$857$	11.2288	0.383567	0.191784	+	0.981437i	$0.438573\pi$
$858$	0	0
$859$	−48.1660	−1.64340	−0.821702	−	0.569918i	$-0.806975\pi$
$859$	−48.1660	−1.64340	−0.821702	+	0.569918i	$0.806975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−53.2915	−1.81406	−0.907032	−	0.421062i	$-0.861658\pi$
$863$	−53.2915	−1.81406	−0.907032	+	0.421062i	$0.861658\pi$
$864$	0	0
$865$	−38.4170	−1.30622
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.29150	0.247347
$870$	0	0
$871$	6.83399	0.231561
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.93725	−0.268328
$876$	0	0
$877$	4.83399	0.163232	0.0816161	−	0.996664i	$-0.473992\pi$
$877$	4.83399	0.163232	0.0816161	+	0.996664i	$0.473992\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	13.9373	0.469558	0.234779	−	0.972049i	$-0.424563\pi$
$881$	13.9373	0.469558	0.234779	+	0.972049i	$0.424563\pi$
$882$	0	0
$883$	−11.7490	−0.395386	−0.197693	−	0.980264i	$-0.563345\pi$
$883$	−11.7490	−0.395386	−0.197693	+	0.980264i	$0.563345\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32.5830	1.09403	0.547015	−	0.837123i	$-0.315764\pi$
$887$	32.5830	1.09403	0.547015	+	0.837123i	$0.315764\pi$
$888$	0	0
$889$	17.2915	0.579938
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.2915	−0.377856
$894$	0	0
$895$	28.3320	0.947035
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.7490	−0.458555
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.04052	0.300517
$906$	0	0
$907$	−45.7490	−1.51907	−0.759536	−	0.650466i	$-0.774574\pi$
$907$	−45.7490	−1.51907	−0.759536	+	0.650466i	$0.774574\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.12549	−0.269210	−0.134605	−	0.990899i	$-0.542976\pi$
$911$	−8.12549	−0.269210	−0.134605	+	0.990899i	$0.542976\pi$
$912$	0	0
$913$	−6.45751	−0.213712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.70850	0.155488
$918$	0	0
$919$	11.7490	0.387564	0.193782	−	0.981045i	$-0.437925\pi$
$919$	11.7490	0.387564	0.193782	+	0.981045i	$0.437925\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−70.0000	−2.30408
$924$	0	0
$925$	−4.58301	−0.150688
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33.4170	−1.09638	−0.548188	−	0.836355i	$-0.684682\pi$
$929$	−33.4170	−1.09638	−0.548188	+	0.836355i	$0.684682\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	47.7490	1.55989	0.779946	−	0.625847i	$-0.215246\pi$
$937$	47.7490	1.55989	0.779946	+	0.625847i	$0.215246\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.47974	0.309031	0.154515	−	0.987990i	$-0.450618\pi$
$941$	9.47974	0.309031	0.154515	+	0.987990i	$0.450618\pi$
$942$	0	0
$943$	−11.7085	−0.381281
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.77124	−0.285027	−0.142514	−	0.989793i	$-0.545518\pi$
$947$	−8.77124	−0.285027	−0.142514	+	0.989793i	$0.545518\pi$
$948$	0	0
$949$	3.74902	0.121698
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−7.16601	−0.232130	−0.116065	−	0.993242i	$-0.537028\pi$
$953$	−7.16601	−0.232130	−0.116065	+	0.993242i	$0.537028\pi$
$954$	0	0
$955$	−70.4980	−2.28126
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.5830	0.406326
$960$	0	0
$961$	−25.7490	−0.830613
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−54.4575	−1.75305
$966$	0	0
$967$	47.1660	1.51676	0.758378	−	0.651815i	$-0.225992\pi$
$967$	47.1660	1.51676	0.758378	+	0.651815i	$0.225992\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.16601	0.229968	0.114984	−	0.993367i	$-0.463318\pi$
$971$	7.16601	0.229968	0.114984	+	0.993367i	$0.463318\pi$
$972$	0	0
$973$	−10.5830	−0.339276
$974$	0	0
$975$	0	0
$976$	0	0
$977$	22.5830	0.722494	0.361247	−	0.932470i	$-0.382351\pi$
$977$	22.5830	0.722494	0.361247	+	0.932470i	$0.382351\pi$
$978$	0	0
$979$	−9.00000	−0.287641
$980$	0	0
$981$	0	0
$982$	0	0
$983$	52.4575	1.67313	0.836567	−	0.547864i	$-0.184559\pi$
$983$	52.4575	1.67313	0.836567	+	0.547864i	$0.184559\pi$
$984$	0	0
$985$	−24.5830	−0.783280
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.87451	−0.313991
$990$	0	0
$991$	−10.7085	−0.340167	−0.170083	−	0.985430i	$-0.554404\pi$
$991$	−10.7085	−0.340167	−0.170083	+	0.985430i	$0.554404\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	65.8118	2.08637
$996$	0	0
$997$	−38.0000	−1.20347	−0.601736	−	0.798695i	$-0.705524\pi$
$997$	−38.0000	−1.20347	−0.601736	+	0.798695i	$0.705524\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.a.bh.1.2		2
3.2	odd	2		3024.2.a.bk.1.1		2
4.3	odd	2		1512.2.a.p.1.2	yes	2
12.11	even	2		1512.2.a.o.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.a.o.1.1	✓	2	12.11	even	2
1512.2.a.p.1.2	yes	2	4.3	odd	2
3024.2.a.bh.1.2		2	1.1	even	1	trivial
3024.2.a.bk.1.1		2	3.2	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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+2.64575 q^{5} +1.00000 q^{7} +0.645751 q^{11} +5.29150 q^{13} +1.00000 q^{19} -1.35425 q^{23} +2.00000 q^{25} +6.00000 q^{29} -2.29150 q^{31} +2.64575 q^{35} -2.29150 q^{37} +8.64575 q^{41} +7.29150 q^{43} -11.2915 q^{47} +1.00000 q^{49} +1.70850 q^{55} +3.29150 q^{59} -6.58301 q^{61} +14.0000 q^{65} +1.29150 q^{67} -13.2288 q^{71} +0.708497 q^{73} +0.645751 q^{77} +11.2915 q^{79} -10.0000 q^{83} -13.9373 q^{89} +5.29150 q^{91} +2.64575 q^{95} +18.5830 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^7	\( 2 q + 2 q^{7} - 4 q^{11} + 2 q^{19} - 8 q^{23} + 4 q^{25} + 12 q^{29} + 6 q^{31} + 6 q^{37} + 12 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 14 q^{55} - 4 q^{59} + 8 q^{61} + 28 q^{65} - 8 q^{67} + 12 q^{73} - 4 q^{77} + 12 q^{79} - 20 q^{83} - 12 q^{89} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 4 * q^11 + 2 * q^19 - 8 * q^23 + 4 * q^25 + 12 * q^29 + 6 * q^31 + 6 * q^37 + 12 * q^41 + 4 * q^43 - 12 * q^47 + 2 * q^49 + 14 * q^55 - 4 * q^59 + 8 * q^61 + 28 * q^65 - 8 * q^67 + 12 * q^73 - 4 * q^77 + 12 * q^79 - 20 * q^83 - 12 * q^89 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more