LMFDB - Embedded newform 5808.2.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5808,2,Mod(1,5808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5808, 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("5808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5808 = 2^{4} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5808.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:	$46.3771134940$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.61803 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.76393 q^{13} +2.61803 q^{15} +1.61803 q^{17} -5.85410 q^{19} +3.00000 q^{21} -3.47214 q^{23} +1.85410 q^{25} -1.00000 q^{27} +4.47214 q^{29} -2.85410 q^{31} +7.85410 q^{35} +0.236068 q^{37} -1.76393 q^{39} -11.9443 q^{41} -6.23607 q^{43} -2.61803 q^{45} -1.61803 q^{47} +2.00000 q^{49} -1.61803 q^{51} -9.61803 q^{53} +5.85410 q^{57} -10.3262 q^{59} +7.85410 q^{61} -3.00000 q^{63} -4.61803 q^{65} +9.56231 q^{67} +3.47214 q^{69} +5.56231 q^{71} -3.23607 q^{73} -1.85410 q^{75} -9.47214 q^{79} +1.00000 q^{81} -0.708204 q^{83} -4.23607 q^{85} -4.47214 q^{87} +0.527864 q^{89} -5.29180 q^{91} +2.85410 q^{93} +15.3262 q^{95} -14.0344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 3 q^{5} - 6 q^{7} + 2 q^{9} + 8 q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + 6 q^{21} + 2 q^{23} - 3 q^{25} - 2 q^{27} + q^{31} + 9 q^{35} - 4 q^{37} - 8 q^{39} - 6 q^{41} - 8 q^{43} - 3 q^{45}+ \cdots + q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 3 * q^5 - 6 * q^7 + 2 * q^9 + 8 * q^13 + 3 * q^15 + q^17 - 5 * q^19 + 6 * q^21 + 2 * q^23 - 3 * q^25 - 2 * q^27 + q^31 + 9 * q^35 - 4 * q^37 - 8 * q^39 - 6 * q^41 - 8 * q^43 - 3 * q^45 - q^47 + 4 * q^49 - q^51 - 17 * q^53 + 5 * q^57 - 5 * q^59 + 9 * q^61 - 6 * q^63 - 7 * q^65 - q^67 - 2 * q^69 - 9 * q^71 - 2 * q^73 + 3 * q^75 - 10 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 + 10 * q^89 - 24 * q^91 - q^93 + 15 * q^95 + q^97

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.61803	−1.17082	−0.585410	−	0.810737i	$-0.699067\pi$
$5$	−2.61803	−1.17082	−0.585410	+	0.810737i	$0.699067\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	2.61803	0.675973
$16$	0	0
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	0	0
$21$	3.00000	0.654654
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	1.85410	0.370820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−2.85410	−0.512612	−0.256306	−	0.966596i	$-0.582505\pi$
$31$	−2.85410	−0.512612	−0.256306	+	0.966596i	$0.582505\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.85410	1.32759
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−1.76393	−0.282455
$40$	0	0
$41$	−11.9443	−1.86538	−0.932691	−	0.360677i	$-0.882546\pi$
$41$	−11.9443	−1.86538	−0.932691	+	0.360677i	$0.882546\pi$
$42$	0	0
$43$	−6.23607	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$43$	−6.23607	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$44$	0	0
$45$	−2.61803	−0.390273
$46$	0	0
$47$	−1.61803	−0.236015	−0.118007	−	0.993013i	$-0.537651\pi$
$47$	−1.61803	−0.236015	−0.118007	+	0.993013i	$0.537651\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	−1.61803	−0.226570
$52$	0	0
$53$	−9.61803	−1.32114	−0.660569	−	0.750765i	$-0.729685\pi$
$53$	−9.61803	−1.32114	−0.660569	+	0.750765i	$0.729685\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.85410	0.775395
$58$	0	0
$59$	−10.3262	−1.34436	−0.672181	−	0.740387i	$-0.734642\pi$
$59$	−10.3262	−1.34436	−0.672181	+	0.740387i	$0.734642\pi$
$60$	0	0
$61$	7.85410	1.00561	0.502807	−	0.864398i	$-0.332301\pi$
$61$	7.85410	1.00561	0.502807	+	0.864398i	$0.332301\pi$
$62$	0	0
$63$	−3.00000	−0.377964
$64$	0	0
$65$	−4.61803	−0.572797
$66$	0	0
$67$	9.56231	1.16822	0.584111	−	0.811674i	$-0.301443\pi$
$67$	9.56231	1.16822	0.584111	+	0.811674i	$0.301443\pi$
$68$	0	0
$69$	3.47214	0.417996
$70$	0	0
$71$	5.56231	0.660124	0.330062	−	0.943959i	$-0.392930\pi$
$71$	5.56231	0.660124	0.330062	+	0.943959i	$0.392930\pi$
$72$	0	0
$73$	−3.23607	−0.378753	−0.189377	−	0.981905i	$-0.560647\pi$
$73$	−3.23607	−0.378753	−0.189377	+	0.981905i	$0.560647\pi$
$74$	0	0
$75$	−1.85410	−0.214093
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.708204	−0.0777355	−0.0388677	−	0.999244i	$-0.512375\pi$
$83$	−0.708204	−0.0777355	−0.0388677	+	0.999244i	$0.512375\pi$
$84$	0	0
$85$	−4.23607	−0.459466
$86$	0	0
$87$	−4.47214	−0.479463
$88$	0	0
$89$	0.527864	0.0559535	0.0279767	−	0.999609i	$-0.491094\pi$
$89$	0.527864	0.0559535	0.0279767	+	0.999609i	$0.491094\pi$
$90$	0	0
$91$	−5.29180	−0.554731
$92$	0	0
$93$	2.85410	0.295957
$94$	0	0
$95$	15.3262	1.57244
$96$	0	0
$97$	−14.0344	−1.42498	−0.712491	−	0.701681i	$-0.752433\pi$
$97$	−14.0344	−1.42498	−0.712491	+	0.701681i	$0.752433\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	0	0
$105$	−7.85410	−0.766482
$106$	0	0
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	0	0
$113$	0.708204	0.0666222	0.0333111	−	0.999445i	$-0.489395\pi$
$113$	0.708204	0.0666222	0.0333111	+	0.999445i	$0.489395\pi$
$114$	0	0
$115$	9.09017	0.847663
$116$	0	0
$117$	1.76393	0.163076
$118$	0	0
$119$	−4.85410	−0.444975
$120$	0	0
$121$	0	0
$122$	0	0
$123$	11.9443	1.07698
$124$	0	0
$125$	8.23607	0.736656
$126$	0	0
$127$	3.70820	0.329050	0.164525	−	0.986373i	$-0.447391\pi$
$127$	3.70820	0.329050	0.164525	+	0.986373i	$0.447391\pi$
$128$	0	0
$129$	6.23607	0.549055
$130$	0	0
$131$	7.14590	0.624340	0.312170	−	0.950026i	$-0.398944\pi$
$131$	7.14590	0.624340	0.312170	+	0.950026i	$0.398944\pi$
$132$	0	0
$133$	17.5623	1.52285
$134$	0	0
$135$	2.61803	0.225324
$136$	0	0
$137$	7.47214	0.638388	0.319194	−	0.947689i	$-0.396588\pi$
$137$	7.47214	0.638388	0.319194	+	0.947689i	$0.396588\pi$
$138$	0	0
$139$	0.854102	0.0724440	0.0362220	−	0.999344i	$-0.488468\pi$
$139$	0.854102	0.0724440	0.0362220	+	0.999344i	$0.488468\pi$
$140$	0	0
$141$	1.61803	0.136263
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−11.7082	−0.972313
$146$	0	0
$147$	−2.00000	−0.164957
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	1.61803	0.130810
$154$	0	0
$155$	7.47214	0.600176
$156$	0	0
$157$	−3.70820	−0.295947	−0.147973	−	0.988991i	$-0.547275\pi$
$157$	−3.70820	−0.295947	−0.147973	+	0.988991i	$0.547275\pi$
$158$	0	0
$159$	9.61803	0.762760
$160$	0	0
$161$	10.4164	0.820928
$162$	0	0
$163$	−18.2705	−1.43106	−0.715528	−	0.698584i	$-0.753814\pi$
$163$	−18.2705	−1.43106	−0.715528	+	0.698584i	$0.753814\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.0344	−0.776488	−0.388244	−	0.921557i	$-0.626918\pi$
$167$	−10.0344	−0.776488	−0.388244	+	0.921557i	$0.626918\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	−5.85410	−0.447674
$172$	0	0
$173$	15.3820	1.16947	0.584735	−	0.811225i	$-0.301199\pi$
$173$	15.3820	1.16947	0.584735	+	0.811225i	$0.301199\pi$
$174$	0	0
$175$	−5.56231	−0.420471
$176$	0	0
$177$	10.3262	0.776168
$178$	0	0
$179$	2.23607	0.167132	0.0835658	−	0.996502i	$-0.473369\pi$
$179$	2.23607	0.167132	0.0835658	+	0.996502i	$0.473369\pi$
$180$	0	0
$181$	−17.4721	−1.29869	−0.649347	−	0.760492i	$-0.724958\pi$
$181$	−17.4721	−1.29869	−0.649347	+	0.760492i	$0.724958\pi$
$182$	0	0
$183$	−7.85410	−0.580592
$184$	0	0
$185$	−0.618034	−0.0454388
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.00000	0.218218
$190$	0	0
$191$	7.47214	0.540665	0.270332	−	0.962767i	$-0.412866\pi$
$191$	7.47214	0.540665	0.270332	+	0.962767i	$0.412866\pi$
$192$	0	0
$193$	−18.5623	−1.33614	−0.668072	−	0.744097i	$-0.732880\pi$
$193$	−18.5623	−1.33614	−0.668072	+	0.744097i	$0.732880\pi$
$194$	0	0
$195$	4.61803	0.330704
$196$	0	0
$197$	24.3820	1.73714	0.868572	−	0.495564i	$-0.165039\pi$
$197$	24.3820	1.73714	0.868572	+	0.495564i	$0.165039\pi$
$198$	0	0
$199$	16.7082	1.18441	0.592207	−	0.805786i	$-0.298257\pi$
$199$	16.7082	1.18441	0.592207	+	0.805786i	$0.298257\pi$
$200$	0	0
$201$	−9.56231	−0.674473
$202$	0	0
$203$	−13.4164	−0.941647
$204$	0	0
$205$	31.2705	2.18403
$206$	0	0
$207$	−3.47214	−0.241330
$208$	0	0
$209$	0	0
$210$	0	0
$211$	22.2705	1.53317	0.766583	−	0.642146i	$-0.221956\pi$
$211$	22.2705	1.53317	0.766583	+	0.642146i	$0.221956\pi$
$212$	0	0
$213$	−5.56231	−0.381123
$214$	0	0
$215$	16.3262	1.11344
$216$	0	0
$217$	8.56231	0.581247
$218$	0	0
$219$	3.23607	0.218673
$220$	0	0
$221$	2.85410	0.191988
$222$	0	0
$223$	−0.708204	−0.0474248	−0.0237124	−	0.999719i	$-0.507549\pi$
$223$	−0.708204	−0.0474248	−0.0237124	+	0.999719i	$0.507549\pi$
$224$	0	0
$225$	1.85410	0.123607
$226$	0	0
$227$	24.8885	1.65191	0.825955	−	0.563736i	$-0.190636\pi$
$227$	24.8885	1.65191	0.825955	+	0.563736i	$0.190636\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.3262	1.59366	0.796832	−	0.604200i	$-0.206507\pi$
$233$	24.3262	1.59366	0.796832	+	0.604200i	$0.206507\pi$
$234$	0	0
$235$	4.23607	0.276331
$236$	0	0
$237$	9.47214	0.615281
$238$	0	0
$239$	−2.56231	−0.165742	−0.0828709	−	0.996560i	$-0.526409\pi$
$239$	−2.56231	−0.165742	−0.0828709	+	0.996560i	$0.526409\pi$
$240$	0	0
$241$	−23.1246	−1.48959	−0.744794	−	0.667295i	$-0.767452\pi$
$241$	−23.1246	−1.48959	−0.744794	+	0.667295i	$0.767452\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.23607	−0.334520
$246$	0	0
$247$	−10.3262	−0.657043
$248$	0	0
$249$	0.708204	0.0448806
$250$	0	0
$251$	7.79837	0.492229	0.246114	−	0.969241i	$-0.420846\pi$
$251$	7.79837	0.492229	0.246114	+	0.969241i	$0.420846\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.23607	0.265273
$256$	0	0
$257$	−11.6738	−0.728189	−0.364095	−	0.931362i	$-0.618622\pi$
$257$	−11.6738	−0.728189	−0.364095	+	0.931362i	$0.618622\pi$
$258$	0	0
$259$	−0.708204	−0.0440057
$260$	0	0
$261$	4.47214	0.276818
$262$	0	0
$263$	16.3262	1.00672	0.503359	−	0.864077i	$-0.332097\pi$
$263$	16.3262	1.00672	0.503359	+	0.864077i	$0.332097\pi$
$264$	0	0
$265$	25.1803	1.54682
$266$	0	0
$267$	−0.527864	−0.0323048
$268$	0	0
$269$	15.5279	0.946751	0.473375	−	0.880861i	$-0.343035\pi$
$269$	15.5279	0.946751	0.473375	+	0.880861i	$0.343035\pi$
$270$	0	0
$271$	27.2705	1.65657	0.828283	−	0.560310i	$-0.189318\pi$
$271$	27.2705	1.65657	0.828283	+	0.560310i	$0.189318\pi$
$272$	0	0
$273$	5.29180	0.320274
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.5623	1.83631	0.918155	−	0.396220i	$-0.129678\pi$
$277$	30.5623	1.83631	0.918155	+	0.396220i	$0.129678\pi$
$278$	0	0
$279$	−2.85410	−0.170871
$280$	0	0
$281$	−0.763932	−0.0455724	−0.0227862	−	0.999740i	$-0.507254\pi$
$281$	−0.763932	−0.0455724	−0.0227862	+	0.999740i	$0.507254\pi$
$282$	0	0
$283$	−0.180340	−0.0107201	−0.00536005	−	0.999986i	$-0.501706\pi$
$283$	−0.180340	−0.0107201	−0.00536005	+	0.999986i	$0.501706\pi$
$284$	0	0
$285$	−15.3262	−0.907848
$286$	0	0
$287$	35.8328	2.11514
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	14.0344	0.822714
$292$	0	0
$293$	0.0557281	0.00325567	0.00162783	−	0.999999i	$-0.499482\pi$
$293$	0.0557281	0.00325567	0.00162783	+	0.999999i	$0.499482\pi$
$294$	0	0
$295$	27.0344	1.57401
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.12461	−0.354195
$300$	0	0
$301$	18.7082	1.07832
$302$	0	0
$303$	3.00000	0.172345
$304$	0	0
$305$	−20.5623	−1.17739
$306$	0	0
$307$	−0.562306	−0.0320925	−0.0160462	−	0.999871i	$-0.505108\pi$
$307$	−0.562306	−0.0320925	−0.0160462	+	0.999871i	$0.505108\pi$
$308$	0	0
$309$	−6.00000	−0.341328
$310$	0	0
$311$	−2.52786	−0.143342	−0.0716710	−	0.997428i	$-0.522833\pi$
$311$	−2.52786	−0.143342	−0.0716710	+	0.997428i	$0.522833\pi$
$312$	0	0
$313$	25.8328	1.46016	0.730079	−	0.683363i	$-0.239483\pi$
$313$	25.8328	1.46016	0.730079	+	0.683363i	$0.239483\pi$
$314$	0	0
$315$	7.85410	0.442529
$316$	0	0
$317$	−19.3607	−1.08740	−0.543702	−	0.839278i	$-0.682978\pi$
$317$	−19.3607	−1.08740	−0.543702	+	0.839278i	$0.682978\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.23607	−0.236434
$322$	0	0
$323$	−9.47214	−0.527044
$324$	0	0
$325$	3.27051	0.181415
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.85410	0.267615
$330$	0	0
$331$	−26.5967	−1.46189	−0.730945	−	0.682437i	$-0.760920\pi$
$331$	−26.5967	−1.46189	−0.730945	+	0.682437i	$0.760920\pi$
$332$	0	0
$333$	0.236068	0.0129364
$334$	0	0
$335$	−25.0344	−1.36778
$336$	0	0
$337$	−0.291796	−0.0158951	−0.00794757	−	0.999968i	$-0.502530\pi$
$337$	−0.291796	−0.0158951	−0.00794757	+	0.999968i	$0.502530\pi$
$338$	0	0
$339$	−0.708204	−0.0384644
$340$	0	0
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	−9.09017	−0.489398
$346$	0	0
$347$	20.9443	1.12435	0.562174	−	0.827019i	$-0.309965\pi$
$347$	20.9443	1.12435	0.562174	+	0.827019i	$0.309965\pi$
$348$	0	0
$349$	−10.1246	−0.541958	−0.270979	−	0.962585i	$-0.587347\pi$
$349$	−10.1246	−0.541958	−0.270979	+	0.962585i	$0.587347\pi$
$350$	0	0
$351$	−1.76393	−0.0941517
$352$	0	0
$353$	−10.4721	−0.557376	−0.278688	−	0.960382i	$-0.589899\pi$
$353$	−10.4721	−0.557376	−0.278688	+	0.960382i	$0.589899\pi$
$354$	0	0
$355$	−14.5623	−0.772887
$356$	0	0
$357$	4.85410	0.256906
$358$	0	0
$359$	12.7639	0.673655	0.336827	−	0.941566i	$-0.390646\pi$
$359$	12.7639	0.673655	0.336827	+	0.941566i	$0.390646\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.47214	0.443452
$366$	0	0
$367$	−5.56231	−0.290350	−0.145175	−	0.989406i	$-0.546375\pi$
$367$	−5.56231	−0.290350	−0.145175	+	0.989406i	$0.546375\pi$
$368$	0	0
$369$	−11.9443	−0.621794
$370$	0	0
$371$	28.8541	1.49803
$372$	0	0
$373$	−4.41641	−0.228673	−0.114336	−	0.993442i	$-0.536474\pi$
$373$	−4.41641	−0.228673	−0.114336	+	0.993442i	$0.536474\pi$
$374$	0	0
$375$	−8.23607	−0.425309
$376$	0	0
$377$	7.88854	0.406281
$378$	0	0
$379$	−1.58359	−0.0813437	−0.0406718	−	0.999173i	$-0.512950\pi$
$379$	−1.58359	−0.0813437	−0.0406718	+	0.999173i	$0.512950\pi$
$380$	0	0
$381$	−3.70820	−0.189977
$382$	0	0
$383$	−26.8885	−1.37394	−0.686970	−	0.726686i	$-0.741060\pi$
$383$	−26.8885	−1.37394	−0.686970	+	0.726686i	$0.741060\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.23607	−0.316997
$388$	0	0
$389$	24.2705	1.23056	0.615282	−	0.788307i	$-0.289042\pi$
$389$	24.2705	1.23056	0.615282	+	0.788307i	$0.289042\pi$
$390$	0	0
$391$	−5.61803	−0.284116
$392$	0	0
$393$	−7.14590	−0.360463
$394$	0	0
$395$	24.7984	1.24774
$396$	0	0
$397$	−38.7082	−1.94271	−0.971355	−	0.237635i	$-0.923628\pi$
$397$	−38.7082	−1.94271	−0.971355	+	0.237635i	$0.923628\pi$
$398$	0	0
$399$	−17.5623	−0.879215
$400$	0	0
$401$	−26.0902	−1.30288	−0.651440	−	0.758700i	$-0.725835\pi$
$401$	−26.0902	−1.30288	−0.651440	+	0.758700i	$0.725835\pi$
$402$	0	0
$403$	−5.03444	−0.250783
$404$	0	0
$405$	−2.61803	−0.130091
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11.0557	−0.546671	−0.273335	−	0.961919i	$-0.588127\pi$
$409$	−11.0557	−0.546671	−0.273335	+	0.961919i	$0.588127\pi$
$410$	0	0
$411$	−7.47214	−0.368573
$412$	0	0
$413$	30.9787	1.52436
$414$	0	0
$415$	1.85410	0.0910143
$416$	0	0
$417$	−0.854102	−0.0418256
$418$	0	0
$419$	−16.5066	−0.806399	−0.403200	−	0.915112i	$-0.632102\pi$
$419$	−16.5066	−0.806399	−0.403200	+	0.915112i	$0.632102\pi$
$420$	0	0
$421$	−37.2705	−1.81645	−0.908227	−	0.418478i	$-0.862564\pi$
$421$	−37.2705	−1.81645	−0.908227	+	0.418478i	$0.862564\pi$
$422$	0	0
$423$	−1.61803	−0.0786715
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	−23.5623	−1.14026
$428$	0	0
$429$	0	0
$430$	0	0
$431$	39.5066	1.90296	0.951482	−	0.307703i	$-0.0995603\pi$
$431$	39.5066	1.90296	0.951482	+	0.307703i	$0.0995603\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	11.7082	0.561365
$436$	0	0
$437$	20.3262	0.972336
$438$	0	0
$439$	−3.29180	−0.157109	−0.0785544	−	0.996910i	$-0.525030\pi$
$439$	−3.29180	−0.157109	−0.0785544	+	0.996910i	$0.525030\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	0	0
$443$	41.1246	1.95389	0.976945	−	0.213493i	$-0.0684840\pi$
$443$	41.1246	1.95389	0.976945	+	0.213493i	$0.0684840\pi$
$444$	0	0
$445$	−1.38197	−0.0655115
$446$	0	0
$447$	−15.0000	−0.709476
$448$	0	0
$449$	−24.4721	−1.15491	−0.577456	−	0.816422i	$-0.695954\pi$
$449$	−24.4721	−1.15491	−0.577456	+	0.816422i	$0.695954\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.00000	0.0939682
$454$	0	0
$455$	13.8541	0.649490
$456$	0	0
$457$	8.20163	0.383656	0.191828	−	0.981429i	$-0.438558\pi$
$457$	8.20163	0.383656	0.191828	+	0.981429i	$0.438558\pi$
$458$	0	0
$459$	−1.61803	−0.0755234
$460$	0	0
$461$	−21.0902	−0.982267	−0.491134	−	0.871084i	$-0.663417\pi$
$461$	−21.0902	−0.982267	−0.491134	+	0.871084i	$0.663417\pi$
$462$	0	0
$463$	15.7984	0.734213	0.367106	−	0.930179i	$-0.380349\pi$
$463$	15.7984	0.734213	0.367106	+	0.930179i	$0.380349\pi$
$464$	0	0
$465$	−7.47214	−0.346512
$466$	0	0
$467$	9.76393	0.451821	0.225910	−	0.974148i	$-0.427464\pi$
$467$	9.76393	0.451821	0.225910	+	0.974148i	$0.427464\pi$
$468$	0	0
$469$	−28.6869	−1.32464
$470$	0	0
$471$	3.70820	0.170865
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−10.8541	−0.498020
$476$	0	0
$477$	−9.61803	−0.440380
$478$	0	0
$479$	28.0902	1.28347	0.641736	−	0.766925i	$-0.278214\pi$
$479$	28.0902	1.28347	0.641736	+	0.766925i	$0.278214\pi$
$480$	0	0
$481$	0.416408	0.0189866
$482$	0	0
$483$	−10.4164	−0.473963
$484$	0	0
$485$	36.7426	1.66840
$486$	0	0
$487$	−16.8197	−0.762172	−0.381086	−	0.924540i	$-0.624450\pi$
$487$	−16.8197	−0.762172	−0.381086	+	0.924540i	$0.624450\pi$
$488$	0	0
$489$	18.2705	0.826221
$490$	0	0
$491$	−25.2148	−1.13793	−0.568964	−	0.822363i	$-0.692655\pi$
$491$	−25.2148	−1.13793	−0.568964	+	0.822363i	$0.692655\pi$
$492$	0	0
$493$	7.23607	0.325896
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−16.6869	−0.748511
$498$	0	0
$499$	17.5623	0.786197	0.393098	−	0.919496i	$-0.371403\pi$
$499$	17.5623	0.786197	0.393098	+	0.919496i	$0.371403\pi$
$500$	0	0
$501$	10.0344	0.448306
$502$	0	0
$503$	−28.0689	−1.25153	−0.625765	−	0.780012i	$-0.715213\pi$
$503$	−28.0689	−1.25153	−0.625765	+	0.780012i	$0.715213\pi$
$504$	0	0
$505$	7.85410	0.349503
$506$	0	0
$507$	9.88854	0.439166
$508$	0	0
$509$	23.6180	1.04685	0.523425	−	0.852071i	$-0.324654\pi$
$509$	23.6180	1.04685	0.523425	+	0.852071i	$0.324654\pi$
$510$	0	0
$511$	9.70820	0.429466
$512$	0	0
$513$	5.85410	0.258465
$514$	0	0
$515$	−15.7082	−0.692186
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−15.3820	−0.675193
$520$	0	0
$521$	−14.8328	−0.649837	−0.324919	−	0.945742i	$-0.605337\pi$
$521$	−14.8328	−0.649837	−0.324919	+	0.945742i	$0.605337\pi$
$522$	0	0
$523$	11.9787	0.523793	0.261896	−	0.965096i	$-0.415652\pi$
$523$	11.9787	0.523793	0.261896	+	0.965096i	$0.415652\pi$
$524$	0	0
$525$	5.56231	0.242759
$526$	0	0
$527$	−4.61803	−0.201165
$528$	0	0
$529$	−10.9443	−0.475838
$530$	0	0
$531$	−10.3262	−0.448121
$532$	0	0
$533$	−21.0689	−0.912595
$534$	0	0
$535$	−11.0902	−0.479470
$536$	0	0
$537$	−2.23607	−0.0964935
$538$	0	0
$539$	0	0
$540$	0	0
$541$	19.5623	0.841049	0.420525	−	0.907281i	$-0.361846\pi$
$541$	19.5623	0.841049	0.420525	+	0.907281i	$0.361846\pi$
$542$	0	0
$543$	17.4721	0.749801
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−21.6180	−0.924320	−0.462160	−	0.886796i	$-0.652925\pi$
$547$	−21.6180	−0.924320	−0.462160	+	0.886796i	$0.652925\pi$
$548$	0	0
$549$	7.85410	0.335205
$550$	0	0
$551$	−26.1803	−1.11532
$552$	0	0
$553$	28.4164	1.20839
$554$	0	0
$555$	0.618034	0.0262341
$556$	0	0
$557$	14.9098	0.631750	0.315875	−	0.948801i	$-0.397702\pi$
$557$	14.9098	0.631750	0.315875	+	0.948801i	$0.397702\pi$
$558$	0	0
$559$	−11.0000	−0.465250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.88854	0.374607	0.187304	−	0.982302i	$-0.440025\pi$
$563$	8.88854	0.374607	0.187304	+	0.982302i	$0.440025\pi$
$564$	0	0
$565$	−1.85410	−0.0780027
$566$	0	0
$567$	−3.00000	−0.125988
$568$	0	0
$569$	−24.0689	−1.00902	−0.504510	−	0.863406i	$-0.668327\pi$
$569$	−24.0689	−1.00902	−0.504510	+	0.863406i	$0.668327\pi$
$570$	0	0
$571$	−34.6869	−1.45160	−0.725801	−	0.687905i	$-0.758531\pi$
$571$	−34.6869	−1.45160	−0.725801	+	0.687905i	$0.758531\pi$
$572$	0	0
$573$	−7.47214	−0.312153
$574$	0	0
$575$	−6.43769	−0.268470
$576$	0	0
$577$	10.7639	0.448108	0.224054	−	0.974577i	$-0.428071\pi$
$577$	10.7639	0.448108	0.224054	+	0.974577i	$0.428071\pi$
$578$	0	0
$579$	18.5623	0.771423
$580$	0	0
$581$	2.12461	0.0881437
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−4.61803	−0.190932
$586$	0	0
$587$	38.3050	1.58101	0.790507	−	0.612453i	$-0.209817\pi$
$587$	38.3050	1.58101	0.790507	+	0.612453i	$0.209817\pi$
$588$	0	0
$589$	16.7082	0.688450
$590$	0	0
$591$	−24.3820	−1.00294
$592$	0	0
$593$	22.2148	0.912252	0.456126	−	0.889915i	$-0.349237\pi$
$593$	22.2148	0.912252	0.456126	+	0.889915i	$0.349237\pi$
$594$	0	0
$595$	12.7082	0.520986
$596$	0	0
$597$	−16.7082	−0.683821
$598$	0	0
$599$	8.29180	0.338794	0.169397	−	0.985548i	$-0.445818\pi$
$599$	8.29180	0.338794	0.169397	+	0.985548i	$0.445818\pi$
$600$	0	0
$601$	33.8328	1.38007	0.690035	−	0.723776i	$-0.257595\pi$
$601$	33.8328	1.38007	0.690035	+	0.723776i	$0.257595\pi$
$602$	0	0
$603$	9.56231	0.389407
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−13.3262	−0.540895	−0.270448	−	0.962735i	$-0.587172\pi$
$607$	−13.3262	−0.540895	−0.270448	+	0.962735i	$0.587172\pi$
$608$	0	0
$609$	13.4164	0.543660
$610$	0	0
$611$	−2.85410	−0.115465
$612$	0	0
$613$	34.6525	1.39960	0.699800	−	0.714339i	$-0.253272\pi$
$613$	34.6525	1.39960	0.699800	+	0.714339i	$0.253272\pi$
$614$	0	0
$615$	−31.2705	−1.26095
$616$	0	0
$617$	19.5836	0.788406	0.394203	−	0.919023i	$-0.371021\pi$
$617$	19.5836	0.788406	0.394203	+	0.919023i	$0.371021\pi$
$618$	0	0
$619$	8.81966	0.354492	0.177246	−	0.984167i	$-0.443281\pi$
$619$	8.81966	0.354492	0.177246	+	0.984167i	$0.443281\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	0	0
$623$	−1.58359	−0.0634453
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.381966	0.0152300
$630$	0	0
$631$	−46.2705	−1.84200	−0.921000	−	0.389563i	$-0.872626\pi$
$631$	−46.2705	−1.84200	−0.921000	+	0.389563i	$0.872626\pi$
$632$	0	0
$633$	−22.2705	−0.885173
$634$	0	0
$635$	−9.70820	−0.385258
$636$	0	0
$637$	3.52786	0.139779
$638$	0	0
$639$	5.56231	0.220041
$640$	0	0
$641$	35.7426	1.41175	0.705875	−	0.708337i	$-0.250554\pi$
$641$	35.7426	1.41175	0.705875	+	0.708337i	$0.250554\pi$
$642$	0	0
$643$	38.5623	1.52075	0.760374	−	0.649485i	$-0.225015\pi$
$643$	38.5623	1.52075	0.760374	+	0.649485i	$0.225015\pi$
$644$	0	0
$645$	−16.3262	−0.642845
$646$	0	0
$647$	−27.7984	−1.09287	−0.546433	−	0.837503i	$-0.684015\pi$
$647$	−27.7984	−1.09287	−0.546433	+	0.837503i	$0.684015\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−8.56231	−0.335583
$652$	0	0
$653$	51.0344	1.99713	0.998566	−	0.0535342i	$-0.0170486\pi$
$653$	51.0344	1.99713	0.998566	+	0.0535342i	$0.0170486\pi$
$654$	0	0
$655$	−18.7082	−0.730990
$656$	0	0
$657$	−3.23607	−0.126251
$658$	0	0
$659$	10.6525	0.414962	0.207481	−	0.978239i	$-0.433474\pi$
$659$	10.6525	0.414962	0.207481	+	0.978239i	$0.433474\pi$
$660$	0	0
$661$	−9.90983	−0.385448	−0.192724	−	0.981253i	$-0.561732\pi$
$661$	−9.90983	−0.385448	−0.192724	+	0.981253i	$0.561732\pi$
$662$	0	0
$663$	−2.85410	−0.110844
$664$	0	0
$665$	−45.9787	−1.78298
$666$	0	0
$667$	−15.5279	−0.601241
$668$	0	0
$669$	0.708204	0.0273807
$670$	0	0
$671$	0	0
$672$	0	0
$673$	12.4164	0.478617	0.239309	−	0.970944i	$-0.423079\pi$
$673$	12.4164	0.478617	0.239309	+	0.970944i	$0.423079\pi$
$674$	0	0
$675$	−1.85410	−0.0713644
$676$	0	0
$677$	13.5279	0.519918	0.259959	−	0.965620i	$-0.416291\pi$
$677$	13.5279	0.519918	0.259959	+	0.965620i	$0.416291\pi$
$678$	0	0
$679$	42.1033	1.61578
$680$	0	0
$681$	−24.8885	−0.953731
$682$	0	0
$683$	3.11146	0.119057	0.0595283	−	0.998227i	$-0.481040\pi$
$683$	3.11146	0.119057	0.0595283	+	0.998227i	$0.481040\pi$
$684$	0	0
$685$	−19.5623	−0.747437
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−16.9656	−0.646336
$690$	0	0
$691$	26.2918	1.00019	0.500094	−	0.865971i	$-0.333299\pi$
$691$	26.2918	1.00019	0.500094	+	0.865971i	$0.333299\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.23607	−0.0848189
$696$	0	0
$697$	−19.3262	−0.732033
$698$	0	0
$699$	−24.3262	−0.920103
$700$	0	0
$701$	−10.6869	−0.403639	−0.201820	−	0.979423i	$-0.564685\pi$
$701$	−10.6869	−0.403639	−0.201820	+	0.979423i	$0.564685\pi$
$702$	0	0
$703$	−1.38197	−0.0521218
$704$	0	0
$705$	−4.23607	−0.159540
$706$	0	0
$707$	9.00000	0.338480
$708$	0	0
$709$	48.7426	1.83057	0.915284	−	0.402809i	$-0.131966\pi$
$709$	48.7426	1.83057	0.915284	+	0.402809i	$0.131966\pi$
$710$	0	0
$711$	−9.47214	−0.355233
$712$	0	0
$713$	9.90983	0.371126
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.56231	0.0956911
$718$	0	0
$719$	1.58359	0.0590580	0.0295290	−	0.999564i	$-0.490599\pi$
$719$	1.58359	0.0590580	0.0295290	+	0.999564i	$0.490599\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	23.1246	0.860014
$724$	0	0
$725$	8.29180	0.307950
$726$	0	0
$727$	−38.8541	−1.44102	−0.720509	−	0.693445i	$-0.756092\pi$
$727$	−38.8541	−1.44102	−0.720509	+	0.693445i	$0.756092\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0902	−0.373198
$732$	0	0
$733$	−37.7082	−1.39278	−0.696392	−	0.717661i	$-0.745213\pi$
$733$	−37.7082	−1.39278	−0.696392	+	0.717661i	$0.745213\pi$
$734$	0	0
$735$	5.23607	0.193135
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	0	0
$741$	10.3262	0.379344
$742$	0	0
$743$	−35.1803	−1.29064	−0.645321	−	0.763912i	$-0.723276\pi$
$743$	−35.1803	−1.29064	−0.645321	+	0.763912i	$0.723276\pi$
$744$	0	0
$745$	−39.2705	−1.43876
$746$	0	0
$747$	−0.708204	−0.0259118
$748$	0	0
$749$	−12.7082	−0.464348
$750$	0	0
$751$	−11.9230	−0.435076	−0.217538	−	0.976052i	$-0.569803\pi$
$751$	−11.9230	−0.435076	−0.217538	+	0.976052i	$0.569803\pi$
$752$	0	0
$753$	−7.79837	−0.284189
$754$	0	0
$755$	5.23607	0.190560
$756$	0	0
$757$	1.94427	0.0706658	0.0353329	−	0.999376i	$-0.488751\pi$
$757$	1.94427	0.0706658	0.0353329	+	0.999376i	$0.488751\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.8885	−1.11971	−0.559854	−	0.828591i	$-0.689143\pi$
$761$	−30.8885	−1.11971	−0.559854	+	0.828591i	$0.689143\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.23607	−0.153155
$766$	0	0
$767$	−18.2148	−0.657698
$768$	0	0
$769$	12.6869	0.457502	0.228751	−	0.973485i	$-0.426536\pi$
$769$	12.6869	0.457502	0.228751	+	0.973485i	$0.426536\pi$
$770$	0	0
$771$	11.6738	0.420420
$772$	0	0
$773$	−31.2492	−1.12396	−0.561978	−	0.827152i	$-0.689960\pi$
$773$	−31.2492	−1.12396	−0.561978	+	0.827152i	$0.689960\pi$
$774$	0	0
$775$	−5.29180	−0.190087
$776$	0	0
$777$	0.708204	0.0254067
$778$	0	0
$779$	69.9230	2.50525
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.47214	−0.159821
$784$	0	0
$785$	9.70820	0.346501
$786$	0	0
$787$	10.2918	0.366863	0.183431	−	0.983033i	$-0.441279\pi$
$787$	10.2918	0.366863	0.183431	+	0.983033i	$0.441279\pi$
$788$	0	0
$789$	−16.3262	−0.581229
$790$	0	0
$791$	−2.12461	−0.0755425
$792$	0	0
$793$	13.8541	0.491974
$794$	0	0
$795$	−25.1803	−0.893055
$796$	0	0
$797$	10.7639	0.381278	0.190639	−	0.981660i	$-0.438944\pi$
$797$	10.7639	0.381278	0.190639	+	0.981660i	$0.438944\pi$
$798$	0	0
$799$	−2.61803	−0.0926194
$800$	0	0
$801$	0.527864	0.0186512
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−27.2705	−0.961159
$806$	0	0
$807$	−15.5279	−0.546607
$808$	0	0
$809$	9.67376	0.340111	0.170056	−	0.985434i	$-0.445605\pi$
$809$	9.67376	0.340111	0.170056	+	0.985434i	$0.445605\pi$
$810$	0	0
$811$	−3.25735	−0.114381	−0.0571906	−	0.998363i	$-0.518214\pi$
$811$	−3.25735	−0.114381	−0.0571906	+	0.998363i	$0.518214\pi$
$812$	0	0
$813$	−27.2705	−0.956419
$814$	0	0
$815$	47.8328	1.67551
$816$	0	0
$817$	36.5066	1.27720
$818$	0	0
$819$	−5.29180	−0.184910
$820$	0	0
$821$	40.4164	1.41054	0.705271	−	0.708938i	$-0.250825\pi$
$821$	40.4164	1.41054	0.705271	+	0.708938i	$0.250825\pi$
$822$	0	0
$823$	35.4721	1.23648	0.618240	−	0.785989i	$-0.287846\pi$
$823$	35.4721	1.23648	0.618240	+	0.785989i	$0.287846\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.1246	−1.84732	−0.923662	−	0.383208i	$-0.874819\pi$
$827$	−53.1246	−1.84732	−0.923662	+	0.383208i	$0.874819\pi$
$828$	0	0
$829$	17.6869	0.614292	0.307146	−	0.951662i	$-0.400626\pi$
$829$	17.6869	0.614292	0.307146	+	0.951662i	$0.400626\pi$
$830$	0	0
$831$	−30.5623	−1.06019
$832$	0	0
$833$	3.23607	0.112123
$834$	0	0
$835$	26.2705	0.909128
$836$	0	0
$837$	2.85410	0.0986522
$838$	0	0
$839$	36.7082	1.26731	0.633654	−	0.773617i	$-0.281554\pi$
$839$	36.7082	1.26731	0.633654	+	0.773617i	$0.281554\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	0	0
$843$	0.763932	0.0263112
$844$	0	0
$845$	25.8885	0.890593
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.180340	0.00618925
$850$	0	0
$851$	−0.819660	−0.0280976
$852$	0	0
$853$	−9.94427	−0.340485	−0.170243	−	0.985402i	$-0.554455\pi$
$853$	−9.94427	−0.340485	−0.170243	+	0.985402i	$0.554455\pi$
$854$	0	0
$855$	15.3262	0.524146
$856$	0	0
$857$	47.7214	1.63013	0.815065	−	0.579369i	$-0.196701\pi$
$857$	47.7214	1.63013	0.815065	+	0.579369i	$0.196701\pi$
$858$	0	0
$859$	−7.11146	−0.242640	−0.121320	−	0.992613i	$-0.538713\pi$
$859$	−7.11146	−0.242640	−0.121320	+	0.992613i	$0.538713\pi$
$860$	0	0
$861$	−35.8328	−1.22118
$862$	0	0
$863$	−11.8885	−0.404691	−0.202345	−	0.979314i	$-0.564856\pi$
$863$	−11.8885	−0.404691	−0.202345	+	0.979314i	$0.564856\pi$
$864$	0	0
$865$	−40.2705	−1.36924
$866$	0	0
$867$	14.3820	0.488437
$868$	0	0
$869$	0	0
$870$	0	0
$871$	16.8673	0.571525
$872$	0	0
$873$	−14.0344	−0.474994
$874$	0	0
$875$	−24.7082	−0.835290
$876$	0	0
$877$	6.41641	0.216667	0.108333	−	0.994115i	$-0.465449\pi$
$877$	6.41641	0.216667	0.108333	+	0.994115i	$0.465449\pi$
$878$	0	0
$879$	−0.0557281	−0.00187966
$880$	0	0
$881$	13.9098	0.468634	0.234317	−	0.972160i	$-0.424715\pi$
$881$	13.9098	0.468634	0.234317	+	0.972160i	$0.424715\pi$
$882$	0	0
$883$	−10.5836	−0.356166	−0.178083	−	0.984015i	$-0.556990\pi$
$883$	−10.5836	−0.356166	−0.178083	+	0.984015i	$0.556990\pi$
$884$	0	0
$885$	−27.0344	−0.908753
$886$	0	0
$887$	−3.00000	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$888$	0	0
$889$	−11.1246	−0.373108
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.47214	0.316973
$894$	0	0
$895$	−5.85410	−0.195681
$896$	0	0
$897$	6.12461	0.204495
$898$	0	0
$899$	−12.7639	−0.425701
$900$	0	0
$901$	−15.5623	−0.518456
$902$	0	0
$903$	−18.7082	−0.622570
$904$	0	0
$905$	45.7426	1.52054
$906$	0	0
$907$	42.9787	1.42708	0.713542	−	0.700612i	$-0.247090\pi$
$907$	42.9787	1.42708	0.713542	+	0.700612i	$0.247090\pi$
$908$	0	0
$909$	−3.00000	−0.0995037
$910$	0	0
$911$	−18.0557	−0.598213	−0.299106	−	0.954220i	$-0.596689\pi$
$911$	−18.0557	−0.598213	−0.299106	+	0.954220i	$0.596689\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	20.5623	0.679769
$916$	0	0
$917$	−21.4377	−0.707935
$918$	0	0
$919$	−46.9574	−1.54898	−0.774491	−	0.632585i	$-0.781994\pi$
$919$	−46.9574	−1.54898	−0.774491	+	0.632585i	$0.781994\pi$
$920$	0	0
$921$	0.562306	0.0185286
$922$	0	0
$923$	9.81153	0.322950
$924$	0	0
$925$	0.437694	0.0143913
$926$	0	0
$927$	6.00000	0.197066
$928$	0	0
$929$	−32.8885	−1.07904	−0.539519	−	0.841973i	$-0.681394\pi$
$929$	−32.8885	−1.07904	−0.539519	+	0.841973i	$0.681394\pi$
$930$	0	0
$931$	−11.7082	−0.383721
$932$	0	0
$933$	2.52786	0.0827586
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.5967	−0.542192	−0.271096	−	0.962552i	$-0.587386\pi$
$937$	−16.5967	−0.542192	−0.271096	+	0.962552i	$0.587386\pi$
$938$	0	0
$939$	−25.8328	−0.843022
$940$	0	0
$941$	−44.6312	−1.45494	−0.727468	−	0.686142i	$-0.759303\pi$
$941$	−44.6312	−1.45494	−0.727468	+	0.686142i	$0.759303\pi$
$942$	0	0
$943$	41.4721	1.35052
$944$	0	0
$945$	−7.85410	−0.255494
$946$	0	0
$947$	−18.3262	−0.595523	−0.297761	−	0.954640i	$-0.596240\pi$
$947$	−18.3262	−0.595523	−0.297761	+	0.954640i	$0.596240\pi$
$948$	0	0
$949$	−5.70820	−0.185296
$950$	0	0
$951$	19.3607	0.627813
$952$	0	0
$953$	37.8197	1.22510	0.612549	−	0.790432i	$-0.290144\pi$
$953$	37.8197	1.22510	0.612549	+	0.790432i	$0.290144\pi$
$954$	0	0
$955$	−19.5623	−0.633021
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.4164	−0.723864
$960$	0	0
$961$	−22.8541	−0.737229
$962$	0	0
$963$	4.23607	0.136505
$964$	0	0
$965$	48.5967	1.56438
$966$	0	0
$967$	34.6869	1.11546	0.557728	−	0.830024i	$-0.311673\pi$
$967$	34.6869	1.11546	0.557728	+	0.830024i	$0.311673\pi$
$968$	0	0
$969$	9.47214	0.304289
$970$	0	0
$971$	−37.7771	−1.21232	−0.606162	−	0.795341i	$-0.707292\pi$
$971$	−37.7771	−1.21232	−0.606162	+	0.795341i	$0.707292\pi$
$972$	0	0
$973$	−2.56231	−0.0821438
$974$	0	0
$975$	−3.27051	−0.104740
$976$	0	0
$977$	−4.63932	−0.148425	−0.0742125	−	0.997242i	$-0.523644\pi$
$977$	−4.63932	−0.148425	−0.0742125	+	0.997242i	$0.523644\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.3050	0.870893	0.435446	−	0.900215i	$-0.356591\pi$
$983$	27.3050	0.870893	0.435446	+	0.900215i	$0.356591\pi$
$984$	0	0
$985$	−63.8328	−2.03388
$986$	0	0
$987$	−4.85410	−0.154508
$988$	0	0
$989$	21.6525	0.688509
$990$	0	0
$991$	−21.2705	−0.675680	−0.337840	−	0.941204i	$-0.609696\pi$
$991$	−21.2705	−0.675680	−0.337840	+	0.941204i	$0.609696\pi$
$992$	0	0
$993$	26.5967	0.844022
$994$	0	0
$995$	−43.7426	−1.38674
$996$	0	0
$997$	−12.9787	−0.411040	−0.205520	−	0.978653i	$-0.565889\pi$
$997$	−12.9787	−0.411040	−0.205520	+	0.978653i	$0.565889\pi$
$998$	0	0
$999$	−0.236068	−0.00746886

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5808.2.a.bl.1.1		2
4.3	odd	2		363.2.a.h.1.1		2
11.3	even	5		528.2.y.f.97.1		4
11.4	even	5		528.2.y.f.49.1		4
11.10	odd	2		5808.2.a.bm.1.1		2
12.11	even	2		1089.2.a.m.1.2		2
20.19	odd	2		9075.2.a.x.1.2		2
44.3	odd	10		33.2.e.a.31.1	yes	4
44.7	even	10		363.2.e.j.148.1		4
44.15	odd	10		33.2.e.a.16.1	✓	4
44.19	even	10		363.2.e.j.130.1		4
44.27	odd	10		363.2.e.h.124.1		4
44.31	odd	10		363.2.e.h.202.1		4
44.35	even	10		363.2.e.c.202.1		4
44.39	even	10		363.2.e.c.124.1		4
44.43	even	2		363.2.a.e.1.2		2
132.47	even	10		99.2.f.b.64.1		4
132.59	even	10		99.2.f.b.82.1		4
132.131	odd	2		1089.2.a.s.1.1		2
220.3	even	20		825.2.bx.b.724.2		8
220.47	even	20		825.2.bx.b.724.1		8
220.59	odd	10		825.2.n.f.676.1		4
220.103	even	20		825.2.bx.b.49.1		8
220.147	even	20		825.2.bx.b.49.2		8
220.179	odd	10		825.2.n.f.526.1		4
220.219	even	2		9075.2.a.bv.1.1		2
396.47	even	30		891.2.n.a.757.1		8
396.59	even	30		891.2.n.a.379.1		8
396.103	odd	30		891.2.n.d.379.1		8
396.191	even	30		891.2.n.a.676.1		8
396.223	odd	30		891.2.n.d.757.1		8
396.311	even	30		891.2.n.a.460.1		8
396.355	odd	30		891.2.n.d.460.1		8
396.367	odd	30		891.2.n.d.676.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.a.16.1	✓	4	44.15	odd	10
33.2.e.a.31.1	yes	4	44.3	odd	10
99.2.f.b.64.1		4	132.47	even	10
99.2.f.b.82.1		4	132.59	even	10
363.2.a.e.1.2		2	44.43	even	2
363.2.a.h.1.1		2	4.3	odd	2
363.2.e.c.124.1		4	44.39	even	10
363.2.e.c.202.1		4	44.35	even	10
363.2.e.h.124.1		4	44.27	odd	10
363.2.e.h.202.1		4	44.31	odd	10
363.2.e.j.130.1		4	44.19	even	10
363.2.e.j.148.1		4	44.7	even	10
528.2.y.f.49.1		4	11.4	even	5
528.2.y.f.97.1		4	11.3	even	5
825.2.n.f.526.1		4	220.179	odd	10
825.2.n.f.676.1		4	220.59	odd	10
825.2.bx.b.49.1		8	220.103	even	20
825.2.bx.b.49.2		8	220.147	even	20
825.2.bx.b.724.1		8	220.47	even	20
825.2.bx.b.724.2		8	220.3	even	20
891.2.n.a.379.1		8	396.59	even	30
891.2.n.a.460.1		8	396.311	even	30
891.2.n.a.676.1		8	396.191	even	30
891.2.n.a.757.1		8	396.47	even	30
891.2.n.d.379.1		8	396.103	odd	30
891.2.n.d.460.1		8	396.355	odd	30
891.2.n.d.676.1		8	396.367	odd	30
891.2.n.d.757.1		8	396.223	odd	30
1089.2.a.m.1.2		2	12.11	even	2
1089.2.a.s.1.1		2	132.131	odd	2
5808.2.a.bl.1.1		2	1.1	even	1	trivial
5808.2.a.bm.1.1		2	11.10	odd	2
9075.2.a.x.1.2		2	20.19	odd	2
9075.2.a.bv.1.1		2	220.219	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5808.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.61803 q^{5} -3.00000 q^{7} +1.00000 q^{9} +1.76393 q^{13} +2.61803 q^{15} +1.61803 q^{17} -5.85410 q^{19} +3.00000 q^{21} -3.47214 q^{23} +1.85410 q^{25} -1.00000 q^{27} +4.47214 q^{29} -2.85410 q^{31} +7.85410 q^{35} +0.236068 q^{37} -1.76393 q^{39} -11.9443 q^{41} -6.23607 q^{43} -2.61803 q^{45} -1.61803 q^{47} +2.00000 q^{49} -1.61803 q^{51} -9.61803 q^{53} +5.85410 q^{57} -10.3262 q^{59} +7.85410 q^{61} -3.00000 q^{63} -4.61803 q^{65} +9.56231 q^{67} +3.47214 q^{69} +5.56231 q^{71} -3.23607 q^{73} -1.85410 q^{75} -9.47214 q^{79} +1.00000 q^{81} -0.708204 q^{83} -4.23607 q^{85} -4.47214 q^{87} +0.527864 q^{89} -5.29180 q^{91} +2.85410 q^{93} +15.3262 q^{95} -14.0344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 3 q^{5} - 6 q^{7} + 2 q^{9} + 8 q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + 6 q^{21} + 2 q^{23} - 3 q^{25} - 2 q^{27} + q^{31} + 9 q^{35} - 4 q^{37} - 8 q^{39} - 6 q^{41} - 8 q^{43} - 3 q^{45}+ \cdots + q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 3 * q^5 - 6 * q^7 + 2 * q^9 + 8 * q^13 + 3 * q^15 + q^17 - 5 * q^19 + 6 * q^21 + 2 * q^23 - 3 * q^25 - 2 * q^27 + q^31 + 9 * q^35 - 4 * q^37 - 8 * q^39 - 6 * q^41 - 8 * q^43 - 3 * q^45 - q^47 + 4 * q^49 - q^51 - 17 * q^53 + 5 * q^57 - 5 * q^59 + 9 * q^61 - 6 * q^63 - 7 * q^65 - q^67 - 2 * q^69 - 9 * q^71 - 2 * q^73 + 3 * q^75 - 10 * q^79 + 2 * q^81 + 12 * q^83 - 4 * q^85 + 10 * q^89 - 24 * q^91 - q^93 + 15 * q^95 + q^97

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